Jacobi sums of pairs of Dirichlet characters modulo a prime

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Numbers

$p$

$m$

$n$

$J(\chi_p(m,\cdot),\chi_p(n,\cdot))$

-1 + i * -2

comment: $\chi$ of order $4$, $\psi$ of order $4$; $J(\chi,\psi)=-1 - 2i$
equals: $-1-2i$

1 + i * 2

comment: $\chi$ of order $4$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 2i$
equals: $1+2i$

-1 + i * 2

comment: $\chi$ of order $4$, $\psi$ of order $4$; $J(\chi,\psi)=-1 + 2i$
equals: $-1+2i$

1 + i * -2

comment: $\chi$ of order $4$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 2i$
equals: $1-2i$

0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $3$, $\psi$ of order $3$; $J(\chi,\psi)=-1 - 3\zeta_{3}$

2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=1 + 2\zeta_{6}$

-2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=-2 - \zeta_{6}$

comment: $\chi$ of order $3$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 2\zeta_{6}$

2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $\chi$ of order $6$, $\psi$ of order $6$; $J(\chi,\psi)=3 - \zeta_{6}$

-2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $\chi$ of order $6$, $\psi$ of order $3$; $J(\chi,\psi)=-3 + \zeta_{6}$

-2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $6$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{6}$

0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $3$, $\psi$ of order $3$; $J(\chi,\psi)=2 + 3\zeta_{3}$

2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=3 - 2\zeta_{6}$

comment: $\chi$ of order $3$, $\psi$ of order $2$; $J(\chi,\psi)=3 - 2\zeta_{6}$

2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $\chi$ of order $6$, $\psi$ of order $6$; $J(\chi,\psi)=2 + \zeta_{6}$

-2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $6$, $\psi$ of order $2$; $J(\chi,\psi)=-3 + 2\zeta_{6}$

-3.309016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569 + i * 0.2245139882897926862209725758987633937896062411605417596972393170844037782395104132982152887546712902

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-4 + 2\zeta_{10} - 2\zeta_{10}^{2} + \zeta_{10}^{3}$

3.309016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569 + i * -0.2245139882897926862209725758987633937896062411605417596972393170844037782395104132982152887546712902

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=4 - 2\zeta_{10} + 2\zeta_{10}^{2} - \zeta_{10}^{3}$

0.8090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * -3.216440812888141159297051378878455805025142098859854898716950096963334206178422049366667194020504191

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=\zeta_{10} - 2\zeta_{10}^{2} - 2\zeta_{10}^{3}$

1.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * 3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=-1 + 2\zeta_{10} + 2\zeta_{10}^{2}$

-0.8090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * 3.216440812888141159297051378878455805025142098859854898716950096963334206178422049366667194020504191

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-\zeta_{10} + 2\zeta_{10}^{2} + 2\zeta_{10}^{3}$

0.3090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * -3.302197525465046088791263151935673217796308384698334186736395567459267066734601331752809010421776726

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=2 - 4\zeta_{10} + 2\zeta_{10}^{2} - 3\zeta_{10}^{3}$

-0.3090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * 3.302197525465046088791263151935673217796308384698334186736395567459267066734601331752809010421776726

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=-2 + 4\zeta_{10} - 2\zeta_{10}^{2} + 3\zeta_{10}^{3}$

10:

-1.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * -3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435

comment: $\chi$ of order $10$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 2\zeta_{10} - 2\zeta_{10}^{2}$

3.045084971874737120511467085914095294300772949514407155338621556763151157047256124268018010473477844 + i * -1.314327780297834015064172712119691518213744830608354453822338808103027866008035045929081572078881427

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=-2 - 2\zeta_{5} - 3\zeta_{5}^{2} - 4\zeta_{5}^{3}$

0.8090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * 3.216440812888141159297051378878455805025142098859854898716950096963334206178422049366667194020504191

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=1 - \zeta_{10} + 3\zeta_{10}^{2} + \zeta_{10}^{3}$

-3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * -0.7265425280053608858954667574806187496160923929652084627500663273457493918456830884205775222161400914

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=-3 + 2\zeta_{10} - 4\zeta_{10}^{2} + 2\zeta_{10}^{3}$

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=1 - \zeta_{10} + 3\zeta_{10}^{2} + \zeta_{10}^{3}$

-2.545084971874737120511467085914095294300772949514407155338621556763151157047256124268018010473477844 + i * -2.126627020880099830453851242657527680601003509412042204591850605944710118409897416735800910696294054

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=1 - \zeta_{5} + \zeta_{5}^{2} + 3\zeta_{5}^{3}$

10:

comment: $\chi$ of order $5$, $\psi$ of order $2$; $J(\chi,\psi)=-3 + 2\zeta_{10} - 4\zeta_{10}^{2} + 2\zeta_{10}^{3}$

3.045084971874737120511467085914095294300772949514407155338621556763151157047256124268018010473477844 + i * 1.314327780297834015064172712119691518213744830608354453822338808103027866008035045929081572078881427

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=2\zeta_{5} - 2\zeta_{5}^{2} - \zeta_{5}^{3}$

-2.545084971874737120511467085914095294300772949514407155338621556763151157047256124268018010473477844 + i * 2.126627020880099830453851242657527680601003509412042204591850605944710118409897416735800910696294054

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=2 + \zeta_{5} + 4\zeta_{5}^{2} + 2\zeta_{5}^{3}$

3.309016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569 + i * 0.2245139882897926862209725758987633937896062411605417596972393170844037782395104132982152887546712902

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=2 + 2\zeta_{10} - \zeta_{10}^{2}$

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=\zeta_{10} - 2\zeta_{10}^{2} - 2\zeta_{10}^{3}$

-3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * 0.7265425280053608858954667574806187496160923929652084627500663273457493918456830884205775222161400914

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=-1 - 2\zeta_{10} + 2\zeta_{10}^{3}$

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=2\zeta_{5} - 2\zeta_{5}^{2} - \zeta_{5}^{3}$

10:

comment: $\chi$ of order $5$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{10} + 2\zeta_{10}^{3}$

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=2 + \zeta_{5} + 4\zeta_{5}^{2} + 2\zeta_{5}^{3}$

-0.3090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * -3.302197525465046088791263151935673217796308384698334186736395567459267066734601331752809010421776726

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=2 - 4\zeta_{10} + \zeta_{10}^{2} - 2\zeta_{10}^{3}$

2.190983005625052575897706582817180941139845410097118568932275688647369768590548775146396397905304431 + i * -2.489898284882780273401584621397837055409049705894646435966883769617584814332738960946089671804364099

comment: $\chi$ of order $5$, $\psi$ of order $10$; $J(\chi,\psi)=3 - \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3}$

10:

comment: $\chi$ of order $5$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{10} + 2\zeta_{10}^{2}$

-3.309016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569 + i * -0.2245139882897926862209725758987633937896062411605417596972393170844037782395104132982152887546712902

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-2 - 2\zeta_{10} + \zeta_{10}^{2}$

0.3090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * 3.302197525465046088791263151935673217796308384698334186736395567459267066734601331752809010421776726

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-2 + 4\zeta_{10} - \zeta_{10}^{2} + 2\zeta_{10}^{3}$

-0.8090169943749474241022934171828190588601545899028814310677243113526302314094512248536036020946955687 + i * -3.216440812888141159297051378878455805025142098859854898716950096963334206178422049366667194020504191

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-1 + \zeta_{10} - 3\zeta_{10}^{2} - \zeta_{10}^{3}$

1.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * -3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=1 - 2\zeta_{10} + 2\zeta_{10}^{2} - 4\zeta_{10}^{3}$

10:

-1.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * 3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435

comment: $\chi$ of order $10$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{10} - 2\zeta_{10}^{2} + 4\zeta_{10}^{3}$

-2.190983005625052575897706582817180941139845410097118568932275688647369768590548775146396397905304431 + i * -2.489898284882780273401584621397837055409049705894646435966883769617584814332738960946089671804364099

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-2 - \zeta_{10} - 2\zeta_{10}^{3}$

2.190983005625052575897706582817180941139845410097118568932275688647369768590548775146396397905304431 + i * 2.489898284882780273401584621397837055409049705894646435966883769617584814332738960946089671804364099

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=2 + \zeta_{10} + 2\zeta_{10}^{3}$

10:

3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * 0.7265425280053608858954667574806187496160923929652084627500663273457493918456830884205775222161400914

comment: $\chi$ of order $10$, $\psi$ of order $2$; $J(\chi,\psi)=3 - 2\zeta_{10} + 4\zeta_{10}^{2} - 2\zeta_{10}^{3}$

-2.190983005625052575897706582817180941139845410097118568932275688647369768590548775146396397905304431 + i * 2.489898284882780273401584621397837055409049705894646435966883769617584814332738960946089671804364099

comment: $\chi$ of order $10$, $\psi$ of order $10$; $J(\chi,\psi)=-3 + \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3}$

comment: $\chi$ of order $10$, $\psi$ of order $5$; $J(\chi,\psi)=-2 + 4\zeta_{10} - 2\zeta_{10}^{2} + 3\zeta_{10}^{3}$

10:

3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275 + i * -0.7265425280053608858954667574806187496160923929652084627500663273457493918456830884205775222161400914

comment: $\chi$ of order $10$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 2\zeta_{10} - 2\zeta_{10}^{3}$

comment: $\chi$ of order $5$, $\psi$ of order $5$; $J(\chi,\psi)=1 - \zeta_{5} + \zeta_{5}^{2} + 3\zeta_{5}^{3}$

10:

comment: $\chi$ of order $5$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 2\zeta_{10} + 2\zeta_{10}^{2} - 4\zeta_{10}^{3}$

-0.2320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485757 + i * -3.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=3 - 2\zeta_{12} - 3\zeta_{12}^{2}$

-2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $3$; $J(\chi,\psi)=-1 - 3\zeta_{12}^{2}$

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151 + i * 1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $12$, $\psi$ of order $6$; $J(\chi,\psi)=4\zeta_{12} - \zeta_{12}^{3}$

-3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151 + i * -1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $12$, $\psi$ of order $4$; $J(\chi,\psi)=-4\zeta_{12} + \zeta_{12}^{3}$

-3 + i * 2

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=-3 + 2\zeta_{12}^{3}$

-2 + i * -3

comment: $\chi$ of order $12$, $\psi$ of order $4$; $J(\chi,\psi)=-2 - 3\zeta_{12}^{3}$

2 + i * 3

comment: $\chi$ of order $12$, $\psi$ of order $3$; $J(\chi,\psi)=2 + 3\zeta_{12}^{3}$

10:

0.2320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485757 + i * 3.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $6$; $J(\chi,\psi)=-3 + 2\zeta_{12} + 3\zeta_{12}^{2}$

11:

2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=1 + 3\zeta_{12}^{2}$

12:

3 + i * -2

comment: $\chi$ of order $12$, $\psi$ of order $2$; $J(\chi,\psi)=3 - 2\zeta_{12}^{3}$

comment: $\chi$ of order $3$, $\psi$ of order $3$; $J(\chi,\psi)=-4 - 3\zeta_{3}$

-1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=-3 + 4\zeta_{6}$

2 + i * -3

comment: $\chi$ of order $3$, $\psi$ of order $4$; $J(\chi,\psi)=2 - 3\zeta_{12}^{3}$

2 + i * 3

comment: $\chi$ of order $3$, $\psi$ of order $12$; $J(\chi,\psi)=2 + 3\zeta_{12}^{3}$

2 + i * -3

comment: $\chi$ of order $3$, $\psi$ of order $12$; $J(\chi,\psi)=2 - 3\zeta_{12}^{3}$

2 + i * 3

comment: $\chi$ of order $3$, $\psi$ of order $4$; $J(\chi,\psi)=2 + 3\zeta_{12}^{3}$

10:

3.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=3 + \zeta_{6}$

11:

comment: $\chi$ of order $3$, $\psi$ of order $12$; $J(\chi,\psi)=-1 - 3\zeta_{12}^{2}$

12:

comment: $\chi$ of order $3$, $\psi$ of order $2$; $J(\chi,\psi)=-3 + 4\zeta_{6}$

3.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -0.8660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878

comment: $\chi$ of order $6$, $\psi$ of order $6$; $J(\chi,\psi)=4 - \zeta_{6}$

comment: $\chi$ of order $6$, $\psi$ of order $4$; $J(\chi,\psi)=4\zeta_{12} - \zeta_{12}^{3}$

-3.232050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576 + i * -1.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $6$, $\psi$ of order $12$; $J(\chi,\psi)=-2\zeta_{12} - 3\zeta_{12}^{2} + 2\zeta_{12}^{3}$

0.2320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485757 + i * -3.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $6$, $\psi$ of order $12$; $J(\chi,\psi)=2\zeta_{12} - 3\zeta_{12}^{2} - 2\zeta_{12}^{3}$

comment: $\chi$ of order $6$, $\psi$ of order $4$; $J(\chi,\psi)=-4\zeta_{12} + \zeta_{12}^{3}$

comment: $\chi$ of order $6$, $\psi$ of order $3$; $J(\chi,\psi)=4 - \zeta_{6}$

11:

comment: $\chi$ of order $6$, $\psi$ of order $12$; $J(\chi,\psi)=-4\zeta_{12} + \zeta_{12}^{3}$

12:

comment: $\chi$ of order $6$, $\psi$ of order $2$; $J(\chi,\psi)=-3 + 4\zeta_{6}$

3 + i * 2

comment: $\chi$ of order $4$, $\psi$ of order $4$; $J(\chi,\psi)=3 + 2i$

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151 + i * -1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $4$, $\psi$ of order $12$; $J(\chi,\psi)=4\zeta_{12} - 3\zeta_{12}^{3}$

-2 + i * 3

comment: $\chi$ of order $4$, $\psi$ of order $12$; $J(\chi,\psi)=-2 + 3\zeta_{12}^{3}$

2 + i * -3

comment: $\chi$ of order $4$, $\psi$ of order $3$; $J(\chi,\psi)=2 - 3\zeta_{12}^{3}$

10:

-3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151 + i * 1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $4$, $\psi$ of order $6$; $J(\chi,\psi)=-4\zeta_{12} + 3\zeta_{12}^{3}$

11:

-2 + i * 3

comment: $\chi$ of order $4$, $\psi$ of order $12$; $J(\chi,\psi)=-2 + 3\zeta_{12}^{3}$

12:

-3 + i * -2

comment: $\chi$ of order $4$, $\psi$ of order $2$; $J(\chi,\psi)=-3 - 2i$

3.232050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576 + i * 1.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=2\zeta_{12} + 3\zeta_{12}^{2} - 2\zeta_{12}^{3}$

2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=4 - 3\zeta_{12}^{2}$

-2 + i * -3

comment: $\chi$ of order $12$, $\psi$ of order $4$; $J(\chi,\psi)=-2 - 3\zeta_{12}^{3}$

-2.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $3$; $J(\chi,\psi)=-4 + 3\zeta_{12}^{2}$

10:

comment: $\chi$ of order $12$, $\psi$ of order $6$; $J(\chi,\psi)=-4\zeta_{12} + 3\zeta_{12}^{3}$

12:

3 + i * -2

comment: $\chi$ of order $12$, $\psi$ of order $2$; $J(\chi,\psi)=3 - 2\zeta_{12}^{3}$

-0.2320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485757 + i * 3.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=-2\zeta_{12} + 3\zeta_{12}^{2} + 2\zeta_{12}^{3}$

comment: $\chi$ of order $12$, $\psi$ of order $4$; $J(\chi,\psi)=-4\zeta_{12} + 3\zeta_{12}^{3}$

comment: $\chi$ of order $12$, $\psi$ of order $3$; $J(\chi,\psi)=-4 + 3\zeta_{12}^{2}$

10:

comment: $\chi$ of order $12$, $\psi$ of order $6$; $J(\chi,\psi)=4\zeta_{12} - 3\zeta_{12}^{3}$

11:

-3 + i * -2

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=-3 - 2\zeta_{12}^{3}$

12:

3 + i * 2

comment: $\chi$ of order $12$, $\psi$ of order $2$; $J(\chi,\psi)=3 + 2\zeta_{12}^{3}$

3 + i * -2

comment: $\chi$ of order $4$, $\psi$ of order $4$; $J(\chi,\psi)=3 - 2i$

2 + i * 3

comment: $\chi$ of order $4$, $\psi$ of order $3$; $J(\chi,\psi)=2 + 3\zeta_{12}^{3}$

10:

comment: $\chi$ of order $4$, $\psi$ of order $6$; $J(\chi,\psi)=4\zeta_{12} - 3\zeta_{12}^{3}$

11:

comment: $\chi$ of order $4$, $\psi$ of order $12$; $J(\chi,\psi)=4\zeta_{12} - \zeta_{12}^{3}$

12:

-3 + i * 2

comment: $\chi$ of order $4$, $\psi$ of order $2$; $J(\chi,\psi)=-3 + 2i$

comment: $\chi$ of order $3$, $\psi$ of order $3$; $J(\chi,\psi)=-1 + 3\zeta_{3}$

10:

-1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=1 - 4\zeta_{6}$

11:

2 + i * -3

comment: $\chi$ of order $3$, $\psi$ of order $12$; $J(\chi,\psi)=2 - 3\zeta_{12}^{3}$

12:

comment: $\chi$ of order $3$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 4\zeta_{6}$

10:

comment: $\chi$ of order $6$, $\psi$ of order $6$; $J(\chi,\psi)=3 + \zeta_{6}$

11:

-3.232050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576 + i * 1.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $6$, $\psi$ of order $12$; $J(\chi,\psi)=-3 - 2\zeta_{12} + 3\zeta_{12}^{2}$

12:

comment: $\chi$ of order $6$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 4\zeta_{6}$

11:

3.232050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576 + i * -1.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $12$, $\psi$ of order $12$; $J(\chi,\psi)=3 + 2\zeta_{12} - 3\zeta_{12}^{2}$

12:

3 + i * 2

comment: $\chi$ of order $12$, $\psi$ of order $2$; $J(\chi,\psi)=3 + 2\zeta_{12}^{3}$

-3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=-3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384 + i * -1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{4} - 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

1 + i * 4

comment: $\chi$ of order $8$, $\psi$ of order $4$; $J(\chi,\psi)=1 + 4\zeta_{8}^{2}$

-2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * -3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} - 3\zeta_{16}^{4} + 2\zeta_{16}^{6}$

0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748 + i * 4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-\zeta_{16} + 2\zeta_{16}^{2} + 3\zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{7}$

2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671 + i * -3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=\zeta_{16} + 2\zeta_{16}^{2} - 3\zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{7}$

3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

10:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-\zeta_{16} + 2\zeta_{16}^{2} + 3\zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{7}$

11:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=\zeta_{16} + 2\zeta_{16}^{2} - 3\zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{7}$

12:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} - 3\zeta_{16}^{4} + 2\zeta_{16}^{6}$

13:

comment: $\chi$ of order $8$, $\psi$ of order $4$; $J(\chi,\psi)=-3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

14:

-3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384 + i * -1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{4} - 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

15:

1 + i * 4

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=1 + 4\zeta_{8}^{2}$

16:

comment: $\chi$ of order $8$, $\psi$ of order $2$; $J(\chi,\psi)=3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

-3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384 + i * 1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{4} + 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841 + i * -4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 - \zeta_{16} - 3\zeta_{16}^{3} + \zeta_{16}^{5} - 2\zeta_{16}^{6} - \zeta_{16}^{7}$

-1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714 + i * -3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-3\zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} - 2\zeta_{16}^{6} - \zeta_{16}^{7}$

4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957 + i * 0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 + 3\zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7}$

1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714 + i * 3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=3\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + 2\zeta_{16}^{6} + \zeta_{16}^{7}$

0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748 + i * -4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=-\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{4} - 3\zeta_{16}^{5} - 2\zeta_{16}^{6} + \zeta_{16}^{7}$

10:

-0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841 + i * 4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 + \zeta_{16} + 3\zeta_{16}^{3} - \zeta_{16}^{5} + 2\zeta_{16}^{6} + \zeta_{16}^{7}$

11:

-1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * -3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=1 - 2\zeta_{16}^{2} - 2\zeta_{16}^{3} - 2\zeta_{16}^{5} + 2\zeta_{16}^{6}$

12:

-0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748 + i * 4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=\zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{4} + 3\zeta_{16}^{5} + 2\zeta_{16}^{6} - \zeta_{16}^{7}$

13:

-4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957 + i * -0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 - 3\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7}$

14:

-2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * 3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} + 3\zeta_{16}^{4} + 2\zeta_{16}^{6}$

15:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * -3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=2\zeta_{16}^{2} - 3\zeta_{16}^{4} - 2\zeta_{16}^{6}$

16:

1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * 3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{16}^{2} + 2\zeta_{16}^{3} + 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

-1 + i * -4

comment: $\chi$ of order $4$, $\psi$ of order $4$; $J(\chi,\psi)=-1 - 4i$

-4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957 + i * 0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 - \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + 2\zeta_{16}^{6} + 3\zeta_{16}^{7}$

3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761 + i * 1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 + \zeta_{16} + 3\zeta_{16}^{3} - \zeta_{16}^{5} - 2\zeta_{16}^{6} + \zeta_{16}^{7}$

comment: $\chi$ of order $4$, $\psi$ of order $8$; $J(\chi,\psi)=-3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

-3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $4$, $\psi$ of order $8$; $J(\chi,\psi)=-3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

10:

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 - \zeta_{16} - 3\zeta_{16}^{3} + \zeta_{16}^{5} - 2\zeta_{16}^{6} - \zeta_{16}^{7}$

11:

3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812 + i * 2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 + \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + 2\zeta_{16}^{6} - 3\zeta_{16}^{7}$

12:

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 + \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + 2\zeta_{16}^{6} - 3\zeta_{16}^{7}$

14:

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 + \zeta_{16} + 3\zeta_{16}^{3} - \zeta_{16}^{5} - 2\zeta_{16}^{6} + \zeta_{16}^{7}$

15:

1 + i * 4

comment: $\chi$ of order $4$, $\psi$ of order $8$; $J(\chi,\psi)=1 + 4\zeta_{8}^{2}$

16:

-1 + i * -4

comment: $\chi$ of order $4$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 4i$

1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189 + i * -3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} + 2\zeta_{16}^{3} - \zeta_{16}^{4} - 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957 + i * -0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 + \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} - 2\zeta_{16}^{6} - 3\zeta_{16}^{7}$

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=3\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} + 2\zeta_{16}^{6} + \zeta_{16}^{7}$

2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671 + i * 3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{4} + 3\zeta_{16}^{5} - 2\zeta_{16}^{6} - \zeta_{16}^{7}$

10:

3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * 1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=1 + 2\zeta_{16} + 2\zeta_{16}^{2} - 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

11:

-3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761 + i * 1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 + \zeta_{16} - 2\zeta_{16}^{2} - \zeta_{16}^{3} + 3\zeta_{16}^{5} + \zeta_{16}^{7}$

12:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * 3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16}^{2} + 3\zeta_{16}^{4} - 2\zeta_{16}^{6}$

13:

3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761 + i * -1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 - \zeta_{16} + 2\zeta_{16}^{2} + \zeta_{16}^{3} - 3\zeta_{16}^{5} - \zeta_{16}^{7}$

14:

-2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671 + i * -3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-\zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{4} - 3\zeta_{16}^{5} + 2\zeta_{16}^{6} + \zeta_{16}^{7}$

15:

-1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189 + i * 3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=2\zeta_{16}^{2} - 2\zeta_{16}^{3} + \zeta_{16}^{4} + 2\zeta_{16}^{5} + 2\zeta_{16}^{6}$

16:

-3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * -1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{16} - 2\zeta_{16}^{2} + 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{4} - 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

-1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714 + i * 3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=\zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + 3\zeta_{16}^{7}$

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=2\zeta_{16}^{2} + 3\zeta_{16}^{4} - 2\zeta_{16}^{6}$

3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384 + i * 1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=2\zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{4} + 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

10:

-0.3318213620807010800022423134769192384475277486209171838140855466847545313800817058947352353115194748 + i * -4.109731692418242073314421481796851225859338378831518017637070663112872508466516473709747273729019957

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=\zeta_{16} - 2\zeta_{16}^{2} - 3\zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{7}$

11:

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} - 3\zeta_{16}^{4} + 2\zeta_{16}^{6}$

12:

-0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841 + i * -4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 - \zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{3} - 3\zeta_{16}^{5} - \zeta_{16}^{7}$

13:

0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841 + i * 4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 + \zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{3} + 3\zeta_{16}^{5} + \zeta_{16}^{7}$

14:

-1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * 3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=1 - 2\zeta_{16}^{2} + 2\zeta_{16}^{3} + 2\zeta_{16}^{5} + 2\zeta_{16}^{6}$

15:

1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714 + i * -3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=-\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - 3\zeta_{16}^{7}$

16:

1.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * -3.695518130045147024512732757587153147289666503454569944460390925122140030004409434859359739401378384

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{16}^{2} - 2\zeta_{16}^{3} - 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189 + i * 3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16}^{2} + 2\zeta_{16}^{3} + \zeta_{16}^{4} - 2\zeta_{16}^{5} + 2\zeta_{16}^{6}$

-1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189 + i * -3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=-2\zeta_{16}^{2} - 2\zeta_{16}^{3} - \zeta_{16}^{4} + 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=-2\zeta_{16}^{2} + 3\zeta_{16}^{4} + 2\zeta_{16}^{6}$

10:

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16}^{2} - 3\zeta_{16}^{4} - 2\zeta_{16}^{6}$

11:

-2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671 + i * 3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-\zeta_{16} - 2\zeta_{16}^{2} + 3\zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{7}$

12:

3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * -1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=1 - 2\zeta_{16} + 2\zeta_{16}^{2} - 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

13:

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 - 3\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7}$

14:

-3.944947291833454135715528660331293545615050125319456238911882280500916614302465807798442675696775761 + i * -1.198912367379658006911597622644676228597850501321590981921116995825429604460277063053319906057614714

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 - \zeta_{16} - 3\zeta_{16}^{3} + \zeta_{16}^{5} + 2\zeta_{16}^{6} - \zeta_{16}^{7}$

15:

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=-\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - 3\zeta_{16}^{7}$

16:

-3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145 + i * 1.530733729460359086913839936121595467045378249942508165735202542510184135840358768948055141369134189

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{16} - 2\zeta_{16}^{2} + 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=-3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

1 + i * -4

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=1 - 4\zeta_{8}^{2}$

10:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} + 2\zeta_{16}^{3} - \zeta_{16}^{4} - 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

11:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16}^{2} + 3\zeta_{16}^{4} - 2\zeta_{16}^{6}$

12:

-4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859 + i * 0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-3\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + 2\zeta_{16}^{6} - \zeta_{16}^{7}$

13:

1 + i * -4

comment: $\chi$ of order $8$, $\psi$ of order $4$; $J(\chi,\psi)=1 - 4\zeta_{8}^{2}$

14:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-3\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + 2\zeta_{16}^{6} - \zeta_{16}^{7}$

16:

comment: $\chi$ of order $8$, $\psi$ of order $2$; $J(\chi,\psi)=3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=-3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

10:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} + 3\zeta_{16}^{4} + 2\zeta_{16}^{6}$

11:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{4} + 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

12:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=-\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{4} - 3\zeta_{16}^{5} - 2\zeta_{16}^{6} + \zeta_{16}^{7}$

13:

1 + i * -4

comment: $\chi$ of order $8$, $\psi$ of order $4$; $J(\chi,\psi)=1 - 4\zeta_{8}^{2}$

14:

comment: $\chi$ of order $8$, $\psi$ of order $16$; $J(\chi,\psi)=\zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{4} + 3\zeta_{16}^{5} - 2\zeta_{16}^{6} - \zeta_{16}^{7}$

15:

3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

16:

comment: $\chi$ of order $8$, $\psi$ of order $2$; $J(\chi,\psi)=3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

10:

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16}^{2} - 2\zeta_{16}^{3} + \zeta_{16}^{4} + 2\zeta_{16}^{5} + 2\zeta_{16}^{6}$

11:

4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859 + i * 0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-\zeta_{16} + 2\zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{4} - \zeta_{16}^{5} - 3\zeta_{16}^{7}$

13:

3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812 + i * -2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 + 3\zeta_{16} - 2\zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7}$

14:

-3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812 + i * 2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 - 3\zeta_{16} + 2\zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7}$

15:

-4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859 + i * -0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=\zeta_{16} - 2\zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + 3\zeta_{16}^{7}$

16:

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{16} - 2\zeta_{16}^{2} + 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

11:

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16} - 2\zeta_{16}^{2} + \zeta_{16}^{4} - 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

12:

-3.281304567672051975711044033377455068719994628077621871283711187131407551542302396008972205073736812 + i * -2.496605762665489017601135134942476918691816002132978962539273929296710425544132371806039833343763671

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-1 - \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - 2\zeta_{16}^{6} + 3\zeta_{16}^{7}$

13:

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 - \zeta_{16} + 2\zeta_{16}^{2} + \zeta_{16}^{3} - 3\zeta_{16}^{5} - \zeta_{16}^{7}$

15:

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=\zeta_{16} - 2\zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{4} + \zeta_{16}^{5} + 3\zeta_{16}^{7}$

16:

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{16}^{2} + 2\zeta_{16}^{3} + 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

12:

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=-2\zeta_{16}^{2} - 2\zeta_{16}^{3} - \zeta_{16}^{4} + 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

13:

comment: $\chi$ of order $16$, $\psi$ of order $4$; $J(\chi,\psi)=1 + \zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{3} + 3\zeta_{16}^{5} + \zeta_{16}^{7}$

14:

4.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859 + i * -0.8834798329127359618878487880881026115242936254344399074414771954805483426217482699023323929585073841

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=3\zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - 2\zeta_{16}^{6} + \zeta_{16}^{7}$

15:

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=2\zeta_{16}^{2} + 2\zeta_{16}^{3} + \zeta_{16}^{4} - 2\zeta_{16}^{5} + 2\zeta_{16}^{6}$

16:

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{16} - 2\zeta_{16}^{2} + 2\zeta_{16}^{6} + 2\zeta_{16}^{7}$

13:

-1 + i * 4

comment: $\chi$ of order $4$, $\psi$ of order $4$; $J(\chi,\psi)=-1 + 4i$

14:

comment: $\chi$ of order $4$, $\psi$ of order $16$; $J(\chi,\psi)=1 + 3\zeta_{16} - 2\zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7}$

15:

comment: $\chi$ of order $4$, $\psi$ of order $8$; $J(\chi,\psi)=-3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

16:

-1 + i * 4

comment: $\chi$ of order $4$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 4i$

14:

comment: $\chi$ of order $16$, $\psi$ of order $16$; $J(\chi,\psi)=2\zeta_{16} + 2\zeta_{16}^{2} - \zeta_{16}^{4} + 2\zeta_{16}^{6} - 2\zeta_{16}^{7}$

15:

comment: $\chi$ of order $16$, $\psi$ of order $8$; $J(\chi,\psi)=2\zeta_{16}^{2} - 3\zeta_{16}^{4} - 2\zeta_{16}^{6}$

16:

comment: $\chi$ of order $16$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{16}^{2} - 2\zeta_{16}^{3} - 2\zeta_{16}^{5} - 2\zeta_{16}^{6}$

15:

comment: $\chi$ of order $8$, $\psi$ of order $8$; $J(\chi,\psi)=-3 - 2\zeta_{8} - 2\zeta_{8}^{3}$

16:

comment: $\chi$ of order $8$, $\psi$ of order $2$; $J(\chi,\psi)=3 + 2\zeta_{8} + 2\zeta_{8}^{3}$

-1.460637153786699430609259157632675857937335165816672794799010797170418677412933665214295729310146789 + i * 4.106889224824293913377536092919008166793170391826443046436591213574978870058558713872780601232091503

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - \zeta_{18} + 4\zeta_{18}^{3} - \zeta_{18}^{4} + 2\zeta_{18}^{5}$

4.286988976119769081757542530863361061936959488632603407562769419038352855273918929792779116152747215 + i * -0.7884957315214673739262591363736050818907688529131776758709009677944221081289603572243599136605854200

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=4 - \zeta_{18} + 2\zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

2.233955556881021964797607349444583326064167542919604754145954715357844611143125276471770723319451507 + i * -3.742918990558091552026488210142174122902698608072704932240978224861904907958069075048395143828945117

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=4 - 2\zeta_{18} + \zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{5}$

1.460637153786699430609259157632675857937335165816672794799010797170418677412933665214295729310146789 + i * -4.106889224824293913377536092919008166793170391826443046436591213574978870058558713872780601232091503

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{18} - 4\zeta_{18}^{3} + \zeta_{18}^{4} - 2\zeta_{18}^{5}$

3.166374217691585849865761085512824001809375757161532673743342744586795277265697847692270895828643350 + i * 2.995675936001422317865784893225824769630536605982166722975196997765799490992305277241927367127816441

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=3\zeta_{18} + 2\zeta_{18}^{4}$

4.298133329356934105607177951666250021807497371241185737562135853926466166570624170584687830041645480 + i * 0.7252929636161356058537550488217748901447735858509607235352167360677462185673517585462725016365078385

comment: $\chi$ of order $18$, $\psi$ of order $3$; $J(\chi,\psi)=2 + 4\zeta_{18} - \zeta_{18}^{2} - 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

-3.166374217691585849865761085512824001809375757161532673743342744586795277265697847692270895828643350 + i * -2.995675936001422317865784893225824769630536605982166722975196997765799490992305277241927367127816441

comment: $\chi$ of order $18$, $\psi$ of order $6$; $J(\chi,\psi)=-3\zeta_{18} - 2\zeta_{18}^{4}$

3.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270 + i * -2.207180204479954943939525756852219687739767753068989047104296029971377382863344920017567453467231021

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-1 + 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 4\zeta_{18}^{5}$

11:

-0.8190778623577251521623278319741944098086244027933938992708599883226636329876683768492376695138442024 + i * 4.281251155374500970511071713040351635445656374393530303625151507108025163474222556662903822563005144

comment: $\chi$ of order $18$, $\psi$ of order $3$; $J(\chi,\psi)=2 - \zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{4} + 4\zeta_{18}^{5}$

12:

-2.233955556881021964797607349444583326064167542919604754145954715357844611143125276471770723319451507 + i * 3.742918990558091552026488210142174122902698608072704932240978224861904907958069075048395143828945117

comment: $\chi$ of order $18$, $\psi$ of order $6$; $J(\chi,\psi)=-4 + 2\zeta_{18} - \zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{5}$

13:

0.8190778623577251521623278319741944098086244027933938992708599883226636329876683768492376695138442024 + i * -4.281251155374500970511071713040351635445656374393530303625151507108025163474222556662903822563005144

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + \zeta_{18} + 2\zeta_{18}^{2} - 2\zeta_{18}^{4} - 4\zeta_{18}^{5}$

14:

0.3545703153092051966893887947951203861917512743906754879653813898094021891513463585722789436435553716 + i * -4.344453923279832738583575800592181827191651641455747255960835738834701053035831155340991234587082726

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=4 - \zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - \zeta_{18}^{4}$

15:

-4.298133329356934105607177951666250021807497371241185737562135853926466166570624170584687830041645480 + i * -0.7252929636161356058537550488217748901447735858509607235352167360677462185673517585462725016365078385

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 4\zeta_{18} + \zeta_{18}^{2} + 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

16:

-4.286988976119769081757542530863361061936959488632603407562769419038352855273918929792779116152747215 + i * 0.7884957315214673739262591363736050818907688529131776758709009677944221081289603572243599136605854200

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-4 + \zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

17:

-0.3545703153092051966893887947951203861917512743906754879653813898094021891513463585722789436435553716 + i * 4.344453923279832738583575800592181827191651641455747255960835738834701053035831155340991234587082726

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-4 + \zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + \zeta_{18}^{4}$

18:

-3.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270 + i * 2.207180204479954943939525756852219687739767753068989047104296029971377382863344920017567453467231021

comment: $\chi$ of order $18$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + 4\zeta_{18}^{5}$

-2.124485151690003756755461324896935225807121694057116350325894475794400344431609435163171216884245906 + i * 3.806121758463423320098992297694004314648693875134921884576662456588580797519677673726482555853022699

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - \zeta_{18} - 2\zeta_{18}^{2} + 4\zeta_{18}^{3} + 2\zeta_{18}^{4}$

1.011144353237165023849635420802888959870537882608582329999366434888113311296705240791908713888898265 + i * -4.239998478409702830745515893583095906827049523570890837020031766409708472311025997245011685603548401

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=2\zeta_{18} - 5\zeta_{18}^{4}$

-4.358440708571025721553068674341518551871289236976721104471849191152244955574734711634941940203697413 + i * 0.06320276790533176807250408755183019174599526706221695233568423172667588956160859867808741202407758143

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-4 - 2\zeta_{18} + 2\zeta_{18}^{3} + \zeta_{18}^{4} - 2\zeta_{18}^{5}$

-0.6945927106677213954068665070772591840015027087362775489449655125282632885560589416860664526295982960 + i * 4.303201246315034598818019981134926098573044790633107789355715998136384361872634595923099097627625982

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-1 - 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{5}$

-0.8190778623577251521623278319741944098086244027933938992708599883226636329876683768492376695138442024 + i * -4.281251155374500970511071713040351635445656374393530303625151507108025163474222556662903822563005144

comment: $\chi$ of order $18$, $\psi$ of order $3$; $J(\chi,\psi)=2 - 2\zeta_{18} - \zeta_{18}^{2} - 2\zeta_{18}^{4} - \zeta_{18}^{5}$

-1.011144353237165023849635420802888959870537882608582329999366434888113311296705240791908713888898265 + i * 4.239998478409702830745515893583095906827049523570890837020031766409708472311025997245011685603548401

comment: $\chi$ of order $18$, $\psi$ of order $6$; $J(\chi,\psi)=-2\zeta_{18} + 5\zeta_{18}^{4}$

3.939692620785908384054109277324731469936208134264464633090286662774221210995889458949745889837948067 + i * 1.865160061154286210895426142169960106976684385554828418639247532286662619161267320453377271128705473

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=4 + \zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + \zeta_{18}^{5}$

10:

0.8190778623577251521623278319741944098086244027933938992708599883226636329876683768492376695138442024 + i * 4.281251155374500970511071713040351635445656374393530303625151507108025163474222556662903822563005144

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} + \zeta_{18}^{2} + 2\zeta_{18}^{4} + \zeta_{18}^{5}$

11:

2.520944533000791046555149880307944388001127031552208161708724134396197466417044206264549839472198722 + i * -3.555958191758365364657316664218576745300882788542569580089934771040278944906870798116631320926497306

comment: $\chi$ of order $18$, $\psi$ of order $3$; $J(\chi,\psi)=2 + 2\zeta_{18} - 2\zeta_{18}^{2} - \zeta_{18}^{4} - 2\zeta_{18}^{5}$

12:

-3.939692620785908384054109277324731469936208134264464633090286662774221210995889458949745889837948067 + i * -1.865160061154286210895426142169960106976684385554828418639247532286662619161267320453377271128705473

comment: $\chi$ of order $18$, $\psi$ of order $6$; $J(\chi,\psi)=-4 - \zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - \zeta_{18}^{5}$

14:

4.358440708571025721553068674341518551871289236976721104471849191152244955574734711634941940203697413 + i * -0.06320276790533176807250408755183019174599526706221695233568423172667588956160859867808741202407758143

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=4 + 2\zeta_{18} - 2\zeta_{18}^{3} - \zeta_{18}^{4} + 2\zeta_{18}^{5}$

15:

-2.520944533000791046555149880307944388001127031552208161708724134396197466417044206264549839472198722 + i * 3.555958191758365364657316664218576745300882788542569580089934771040278944906870798116631320926497306

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 2\zeta_{18} + 2\zeta_{18}^{2} + \zeta_{18}^{4} + 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-4 + \zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

17:

2.124485151690003756755461324896935225807121694057116350325894475794400344431609435163171216884245906 + i * -3.806121758463423320098992297694004314648693875134921884576662456588580797519677673726482555853022699

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{18} + 2\zeta_{18}^{2} - 4\zeta_{18}^{3} - 2\zeta_{18}^{4}$

18:

0.6945927106677213954068665070772591840015027087362775489449655125282632885560589416860664526295982960 + i * -4.303201246315034598818019981134926098573044790633107789355715998136384361872634595923099097627625982

comment: $\chi$ of order $18$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{5}$

2.520944533000791046555149880307944388001127031552208161708724134396197466417044206264549839472198722 + i * 3.555958191758365364657316664218576745300882788542569580089934771040278944906870798116631320926497306

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{9} + 4\zeta_{9}^{2} - \zeta_{9}^{4} + 2\zeta_{9}^{5}$

-4.340022395358516198717477712282138797809751434345602060979584122718861099404712583113787508986042924 + i * 0.4052229112310005011166037022340897725090786345763775808007327294113906532636861982422775943626031290

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=\zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{3} + 2\zeta_{9}^{4} + 2\zeta_{9}^{5}$

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=2\zeta_{18} - 5\zeta_{18}^{4}$

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{4} - 4\zeta_{9}^{5}$

10:

1.460637153786699430609259157632675857937335165816672794799010797170418677412933665214295729310146789 + i * 4.106889224824293913377536092919008166793170391826443046436591213574978870058558713872780601232091503

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + \zeta_{18}^{2} + 4\zeta_{18}^{3} + 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

11:

4.117211191714659257769505783640444431616121774034579636832995842249129799558292547433925499555489683 + i * -1.431283341208006700248902546280868514798328737118859466195878222465534944369918996934848141378179772

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=-2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{3} - 2\zeta_{9}^{4} - 2\zeta_{9}^{5}$

12:

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=4 - 2\zeta_{18} + \zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{5}$

13:

-4.358440708571025721553068674341518551871289236976721104471849191152244955574734711634941940203697413 + i * -0.06320276790533176807250408755183019174599526706221695233568423172667588956160859867808741202407758143

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + \zeta_{18}^{5}$

14:

-3.064177772475912140809570602221666695743329828321580983416181138568621555427498894112917106722193974 + i * 3.100131380871552225703844800234910689995138723867023141916000970190792071717110435337906916111350766

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-1 - 4\zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

15:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + \zeta_{18}^{5}$

16:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=-2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{3} - 2\zeta_{9}^{4} - 2\zeta_{9}^{5}$

17:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=\zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{3} + 2\zeta_{9}^{4} + 2\zeta_{9}^{5}$

18:

comment: $\chi$ of order $9$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 4\zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 4\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{4} - 2\zeta_{9}^{5}$

4.117211191714659257769505783640444431616121774034579636832995842249129799558292547433925499555489683 + i * 1.431283341208006700248902546280868514798328737118859466195878222465534944369918996934848141378179772

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=\zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{3} - \zeta_{9}^{4} - 4\zeta_{9}^{5}$

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=\zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{3} - \zeta_{9}^{4} - 4\zeta_{9}^{5}$

2.233955556881021964797607349444583326064167542919604754145954715357844611143125276471770723319451507 + i * 3.742918990558091552026488210142174122902698608072704932240978224861904907958069075048395143828945117

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=2 + \zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + \zeta_{18}^{4} + 2\zeta_{18}^{5}$

-4.340022395358516198717477712282138797809751434345602060979584122718861099404712583113787508986042924 + i * -0.4052229112310005011166037022340897725090786345763775808007327294113906532636861982422775943626031290

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=-2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{3} + 4\zeta_{9}^{4} + \zeta_{9}^{5}$

10:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=2 + \zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + \zeta_{18}^{4} + 2\zeta_{18}^{5}$

11:

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=-2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{3} + 4\zeta_{9}^{4} + \zeta_{9}^{5}$

12:

1.011144353237165023849635420802888959870537882608582329999366434888113311296705240791908713888898265 + i * 4.239998478409702830745515893583095906827049523570890837020031766409708472311025997245011685603548401

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=2\zeta_{18}^{2} + 3\zeta_{18}^{5}$

13:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=2\zeta_{18}^{2} + 3\zeta_{18}^{5}$

14:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-4 - 2\zeta_{18} + 2\zeta_{18}^{3} + \zeta_{18}^{4} - 2\zeta_{18}^{5}$

15:

-3.064177772475912140809570602221666695743329828321580983416181138568621555427498894112917106722193974 + i * -3.100131380871552225703844800234910689995138723867023141916000970190792071717110435337906916111350766

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=1 - 4\zeta_{18} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 4\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{4} - 2\zeta_{9}^{5}$

17:

4.298133329356934105607177951666250021807497371241185737562135853926466166570624170584687830041645480 + i * -0.7252929636161356058537550488217748901447735858509607235352167360677462185673517585462725016365078385

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{4} + \zeta_{9}^{5}$

18:

comment: $\chi$ of order $9$, $\psi$ of order $2$; $J(\chi,\psi)=1 - 4\zeta_{18} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 2\zeta_{9} - 2\zeta_{9}^{2} - \zeta_{9}^{4} + 2\zeta_{9}^{5}$

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=\zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{3} - \zeta_{9}^{4} - 4\zeta_{9}^{5}$

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=3\zeta_{18} + 2\zeta_{18}^{4}$

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 2\zeta_{9} - 2\zeta_{9}^{2} - \zeta_{9}^{4} + 2\zeta_{9}^{5}$

10:

-4.286988976119769081757542530863361061936959488632603407562769419038352855273918929792779116152747215 + i * -0.7884957315214673739262591363736050818907688529131776758709009677944221081289603572243599136605854200

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 2\zeta_{18} + \zeta_{18}^{2} - 2\zeta_{18}^{3} + \zeta_{18}^{5}$

11:

-2.777188796356143059052028071358305633806370339688977575853411719530268700153579964320137990569446758 + i * -3.359646170267624679216832776002658813521008389735904837170809986478873453092794916066312824530962826

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=-2 - 4\zeta_{9} + 2\zeta_{9}^{2} - 2\zeta_{9}^{3} - 2\zeta_{9}^{4} + \zeta_{9}^{5}$

12:

2.826351822333069651148283373230685203999624322815930612763758621867934177860985264578483386842600426 + i * -3.318393493302826539451276956545403084902401538913265370565690245780556761929598356648420687571506083

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=4 + 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{4} + \zeta_{18}^{5}$

13:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 2\zeta_{18} + \zeta_{18}^{2} - 2\zeta_{18}^{3} + \zeta_{18}^{5}$

14:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=4 + 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{4} + \zeta_{18}^{5}$

15:

-3.585122305476703187364720482529611083744456859873789145124905272964819021844543100377466946194392696 + i * 2.479293862125546527688149658422221720214967255900918837321588206548038433874563834887613963458377252

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - 2\zeta_{18} - 2\zeta_{18}^{2} + 4\zeta_{18}^{3} + \zeta_{18}^{5}$

17:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=-2 - 4\zeta_{9} + 2\zeta_{9}^{2} - 2\zeta_{9}^{3} - 2\zeta_{9}^{4} + \zeta_{9}^{5}$

18:

comment: $\chi$ of order $9$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{5}$

3.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $3$, $\psi$ of order $3$; $J(\chi,\psi)=2 - 3\zeta_{3}$

-4.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=-3 - 2\zeta_{6}$

-2.777188796356143059052028071358305633806370339688977575853411719530268700153579964320137990569446758 + i * 3.359646170267624679216832776002658813521008389735904837170809986478873453092794916066312824530962826

comment: $\chi$ of order $3$, $\psi$ of order $9$; $J(\chi,\psi)=-2\zeta_{9} + 4\zeta_{9}^{2} + 2\zeta_{9}^{3} - \zeta_{9}^{4} + 2\zeta_{9}^{5}$

10:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 - 2\zeta_{18} - \zeta_{18}^{2} - 2\zeta_{18}^{4} - \zeta_{18}^{5}$

12:

0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -4.330127018922193233818615853764680917357013134525951570139517448629832542272000092702865466893121439

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=3 - 5\zeta_{6}$

13:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 - 2\zeta_{18} + 2\zeta_{18}^{2} + 4\zeta_{18}^{4} - \zeta_{18}^{5}$

14:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 - 2\zeta_{18} + 2\zeta_{18}^{2} + 4\zeta_{18}^{4} - \zeta_{18}^{5}$

15:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 + 4\zeta_{18} - \zeta_{18}^{2} - 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $3$, $\psi$ of order $9$; $J(\chi,\psi)=-2\zeta_{9} + 4\zeta_{9}^{2} + 2\zeta_{9}^{3} - \zeta_{9}^{4} + 2\zeta_{9}^{5}$

17:

comment: $\chi$ of order $3$, $\psi$ of order $9$; $J(\chi,\psi)=\zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{3} + 2\zeta_{9}^{4} + 2\zeta_{9}^{5}$

18:

comment: $\chi$ of order $3$, $\psi$ of order $2$; $J(\chi,\psi)=-3 - 2\zeta_{6}$

-0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -4.330127018922193233818615853764680917357013134525951570139517448629832542272000092702865466893121439

comment: $\chi$ of order $6$, $\psi$ of order $6$; $J(\chi,\psi)=2 - 5\zeta_{6}$

-4.177518570928750873715396506315712961679913639770115003742709179474908588562403088484179609717541615 + i * 1.244322542408280512879731000357271137196512917588724114044834768643908981318720720003084318475731960

comment: $\chi$ of order $6$, $\psi$ of order $9$; $J(\chi,\psi)=-5\zeta_{18} + 3\zeta_{18}^{4}$

10:

-2.233955556881021964797607349444583326064167542919604754145954715357844611143125276471770723319451507 + i * -3.742918990558091552026488210142174122902698608072704932240978224861904907958069075048395143828945117

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - \zeta_{18} + 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - \zeta_{18}^{4} - 2\zeta_{18}^{5}$

11:

0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 4.330127018922193233818615853764680917357013134525951570139517448629832542272000092702865466893121439

comment: $\chi$ of order $6$, $\psi$ of order $3$; $J(\chi,\psi)=-2 + 5\zeta_{6}$

13:

-3.939692620785908384054109277324731469936208134264464633090286662774221210995889458949745889837948067 + i * 1.865160061154286210895426142169960106976684385554828418639247532286662619161267320453377271128705473

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=-2 - \zeta_{18} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

14:

4.177518570928750873715396506315712961679913639770115003742709179474908588562403088484179609717541615 + i * -1.244322542408280512879731000357271137196512917588724114044834768643908981318720720003084318475731960

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=5\zeta_{18} - 3\zeta_{18}^{4}$

15:

-2.826351822333069651148283373230685203999624322815930612763758621867934177860985264578483386842600426 + i * -3.318393493302826539451276956545403084902401538913265370565690245780556761929598356648420687571506083

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - \zeta_{18}^{4}$

16:

2.826351822333069651148283373230685203999624322815930612763758621867934177860985264578483386842600426 + i * 3.318393493302826539451276956545403084902401538913265370565690245780556761929598356648420687571506083

comment: $\chi$ of order $6$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + \zeta_{18}^{4}$

17:

3.939692620785908384054109277324731469936208134264464633090286662774221210995889458949745889837948067 + i * -1.865160061154286210895426142169960106976684385554828418639247532286662619161267320453377271128705473

comment: $\chi$ of order $6$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{18} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

18:

4.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $6$, $\psi$ of order $2$; $J(\chi,\psi)=3 + 2\zeta_{6}$

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{9} - 2\zeta_{9}^{2} + 2\zeta_{9}^{4} - 4\zeta_{9}^{5}$

10:

-0.3545703153092051966893887947951203861917512743906754879653813898094021891513463585722789436435553716 + i * -4.344453923279832738583575800592181827191651641455747255960835738834701053035831155340991234587082726

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} + \zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

11:

comment: $\chi$ of order $9$, $\psi$ of order $3$; $J(\chi,\psi)=-2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{3} + 4\zeta_{9}^{4} + \zeta_{9}^{5}$

12:

comment: $\chi$ of order $9$, $\psi$ of order $6$; $J(\chi,\psi)=4 + \zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + \zeta_{18}^{5}$

13:

2.124485151690003756755461324896935225807121694057116350325894475794400344431609435163171216884245906 + i * 3.806121758463423320098992297694004314648693875134921884576662456588580797519677673726482555853022699

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} + \zeta_{18}^{2} + 4\zeta_{18}^{3} - 2\zeta_{18}^{4} + \zeta_{18}^{5}$

14:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-5\zeta_{18} + 3\zeta_{18}^{4}$

15:

comment: $\chi$ of order $9$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} + \zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=-2\zeta_{9} + 4\zeta_{9}^{2} + 2\zeta_{9}^{3} - \zeta_{9}^{4} + 2\zeta_{9}^{5}$

18:

comment: $\chi$ of order $9$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 4\zeta_{18}^{5}$

10:

-1.460637153786699430609259157632675857937335165816672794799010797170418677412933665214295729310146789 + i * -4.106889224824293913377536092919008166793170391826443046436591213574978870058558713872780601232091503

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=2 - \zeta_{18}^{2} - 4\zeta_{18}^{3} - 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

11:

comment: $\chi$ of order $18$, $\psi$ of order $3$; $J(\chi,\psi)=2 - \zeta_{18} + 4\zeta_{18}^{2} - \zeta_{18}^{4} - 2\zeta_{18}^{5}$

12:

-3.166374217691585849865761085512824001809375757161532673743342744586795277265697847692270895828643350 + i * 2.995675936001422317865784893225824769630536605982166722975196997765799490992305277241927367127816441

comment: $\chi$ of order $18$, $\psi$ of order $6$; $J(\chi,\psi)=-3\zeta_{18}^{2} + 5\zeta_{18}^{5}$

13:

4.286988976119769081757542530863361061936959488632603407562769419038352855273918929792779116152747215 + i * 0.7884957315214673739262591363736050818907688529131776758709009677944221081289603572243599136605854200

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=2 + 2\zeta_{18} - \zeta_{18}^{2} + 2\zeta_{18}^{3} - \zeta_{18}^{5}$

14:

-4.298133329356934105607177951666250021807497371241185737562135853926466166570624170584687830041645480 + i * 0.7252929636161356058537550488217748901447735858509607235352167360677462185673517585462725016365078385

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + \zeta_{18} - 4\zeta_{18}^{2} + \zeta_{18}^{4} + 2\zeta_{18}^{5}$

15:

0.3545703153092051966893887947951203861917512743906754879653813898094021891513463585722789436435553716 + i * 4.344453923279832738583575800592181827191651641455747255960835738834701053035831155340991234587082726

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=2 - 2\zeta_{18} - \zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

16:

3.166374217691585849865761085512824001809375757161532673743342744586795277265697847692270895828643350 + i * -2.995675936001422317865784893225824769630536605982166722975196997765799490992305277241927367127816441

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=3\zeta_{18}^{2} - 5\zeta_{18}^{5}$

17:

3.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270 + i * 2.207180204479954943939525756852219687739767753068989047104296029971377382863344920017567453467231021

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=1 + 2\zeta_{18} + 2\zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4}$

18:

-3.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270 + i * -2.207180204479954943939525756852219687739767753068989047104296029971377382863344920017567453467231021

comment: $\chi$ of order $18$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4}$

11:

3.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 2.598076211353315940291169512258808550414207880715570942083710469177899525363200055621719280135872864

comment: $\chi$ of order $3$, $\psi$ of order $3$; $J(\chi,\psi)=5 + 3\zeta_{3}$

12:

-4.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $3$, $\psi$ of order $6$; $J(\chi,\psi)=-5 + 2\zeta_{6}$

13:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 - \zeta_{18} - 2\zeta_{18}^{2} + 2\zeta_{18}^{4} + 4\zeta_{18}^{5}$

14:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 - \zeta_{18} + 4\zeta_{18}^{2} - \zeta_{18}^{4} - 2\zeta_{18}^{5}$

15:

comment: $\chi$ of order $3$, $\psi$ of order $18$; $J(\chi,\psi)=2 + 2\zeta_{18} - 2\zeta_{18}^{2} - \zeta_{18}^{4} - 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $3$, $\psi$ of order $9$; $J(\chi,\psi)=-2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{3} - 2\zeta_{9}^{4} - 2\zeta_{9}^{5}$

17:

comment: $\chi$ of order $3$, $\psi$ of order $9$; $J(\chi,\psi)=-2 - 4\zeta_{9} + 2\zeta_{9}^{2} - 2\zeta_{9}^{3} - 2\zeta_{9}^{4} + \zeta_{9}^{5}$

18:

comment: $\chi$ of order $3$, $\psi$ of order $2$; $J(\chi,\psi)=-5 + 2\zeta_{6}$

12:

-0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 4.330127018922193233818615853764680917357013134525951570139517448629832542272000092702865466893121439

comment: $\chi$ of order $6$, $\psi$ of order $6$; $J(\chi,\psi)=-3 + 5\zeta_{6}$

13:

-1.011144353237165023849635420802888959870537882608582329999366434888113311296705240791908713888898265 + i * -4.239998478409702830745515893583095906827049523570890837020031766409708472311025997245011685603548401

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=-2\zeta_{18}^{2} - 3\zeta_{18}^{5}$

14:

-2.826351822333069651148283373230685203999624322815930612763758621867934177860985264578483386842600426 + i * 3.318393493302826539451276956545403084902401538913265370565690245780556761929598356648420687571506083

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=-4 - 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + 2\zeta_{18}^{4} - \zeta_{18}^{5}$

15:

4.177518570928750873715396506315712961679913639770115003742709179474908588562403088484179609717541615 + i * 1.244322542408280512879731000357271137196512917588724114044834768643908981318720720003084318475731960

comment: $\chi$ of order $6$, $\psi$ of order $18$; $J(\chi,\psi)=5\zeta_{18}^{2} - 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $6$, $\psi$ of order $9$; $J(\chi,\psi)=3\zeta_{18}^{2} - 5\zeta_{18}^{5}$

17:

-4.177518570928750873715396506315712961679913639770115003742709179474908588562403088484179609717541615 + i * -1.244322542408280512879731000357271137196512917588724114044834768643908981318720720003084318475731960

comment: $\chi$ of order $6$, $\psi$ of order $9$; $J(\chi,\psi)=-5\zeta_{18}^{2} + 2\zeta_{18}^{5}$

18:

4.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: $\chi$ of order $6$, $\psi$ of order $2$; $J(\chi,\psi)=5 - 2\zeta_{6}$

13:

-2.124485151690003756755461324896935225807121694057116350325894475794400344431609435163171216884245906 + i * -3.806121758463423320098992297694004314648693875134921884576662456588580797519677673726482555853022699

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=2 - 2\zeta_{18} - \zeta_{18}^{2} - 4\zeta_{18}^{3} + 2\zeta_{18}^{4} - \zeta_{18}^{5}$

14:

-2.520944533000791046555149880307944388001127031552208161708724134396197466417044206264549839472198722 + i * -3.555958191758365364657316664218576745300882788542569580089934771040278944906870798116631320926497306

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} - 2\zeta_{18}^{2} - 4\zeta_{18}^{4} + \zeta_{18}^{5}$

15:

4.358440708571025721553068674341518551871289236976721104471849191152244955574734711634941940203697413 + i * 0.06320276790533176807250408755183019174599526706221695233568423172667588956160859867808741202407758143

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=2 + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - \zeta_{18}^{5}$

16:

-0.6945927106677213954068665070772591840015027087362775489449655125282632885560589416860664526295982960 + i * -4.303201246315034598818019981134926098573044790633107789355715998136384361872634595923099097627625982

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=1 + 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - 4\zeta_{18}^{4} + 2\zeta_{18}^{5}$

17:

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=2 + \zeta_{18} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

18:

0.6945927106677213954068665070772591840015027087362775489449655125282632885560589416860664526295982960 + i * 4.303201246315034598818019981134926098573044790633107789355715998136384361872634595923099097627625982

comment: $\chi$ of order $18$, $\psi$ of order $2$; $J(\chi,\psi)=-1 - 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + 4\zeta_{18}^{4} - 2\zeta_{18}^{5}$

14:

3.585122305476703187364720482529611083744456859873789145124905272964819021844543100377466946194392696 + i * 2.479293862125546527688149658422221720214967255900918837321588206548038433874563834887613963458377252

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=-2 + 2\zeta_{18} + 2\zeta_{18}^{2} + 4\zeta_{18}^{3} - \zeta_{18}^{4} - 2\zeta_{18}^{5}$

16:

-3.585122305476703187364720482529611083744456859873789145124905272964819021844543100377466946194392696 + i * -2.479293862125546527688149658422221720214967255900918837321588206548038433874563834887613963458377252

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 2\zeta_{18} - 2\zeta_{18}^{2} - 4\zeta_{18}^{3} + \zeta_{18}^{4} + 2\zeta_{18}^{5}$

17:

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-4 + \zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + \zeta_{18}^{4}$

18:

3.064177772475912140809570602221666695743329828321580983416181138568621555427498894112917106722193974 + i * -3.100131380871552225703844800234910689995138723867023141916000970190792071717110435337906916111350766

comment: $\chi$ of order $18$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 4\zeta_{18}^{2} - 2\zeta_{18}^{3} - 2\zeta_{18}^{4} - 2\zeta_{18}^{5}$

15:

3.585122305476703187364720482529611083744456859873789145124905272964819021844543100377466946194392696 + i * -2.479293862125546527688149658422221720214967255900918837321588206548038433874563834887613963458377252

comment: $\chi$ of order $18$, $\psi$ of order $18$; $J(\chi,\psi)=2 + 2\zeta_{18} + 2\zeta_{18}^{2} - 4\zeta_{18}^{3} - \zeta_{18}^{5}$

16:

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=2 - 2\zeta_{18} + 2\zeta_{18}^{2} + 2\zeta_{18}^{3} + \zeta_{18}^{4}$

17:

comment: $\chi$ of order $18$, $\psi$ of order $9$; $J(\chi,\psi)=-5\zeta_{18}^{2} + 2\zeta_{18}^{5}$

18:

3.064177772475912140809570602221666695743329828321580983416181138568621555427498894112917106722193974 + i * 3.100131380871552225703844800234910689995138723867023141916000970190792071717110435337906916111350766

comment: $\chi$ of order $18$, $\psi$ of order $2$; $J(\chi,\psi)=-1 + 4\zeta_{18} + 2\zeta_{18}^{3} - 2\zeta_{18}^{4} + 2\zeta_{18}^{5}$

16:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + 2\zeta_{9} + 2\zeta_{9}^{2} + 4\zeta_{9}^{4} + \zeta_{9}^{5}$

17:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + 2\zeta_{9} + 2\zeta_{9}^{2} + 4\zeta_{9}^{4} + \zeta_{9}^{5}$

18:

comment: $\chi$ of order $9$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 2\zeta_{18} - 2\zeta_{18}^{2} - 2\zeta_{18}^{3} - 4\zeta_{18}^{4} + 2\zeta_{18}^{5}$

17:

comment: $\chi$ of order $9$, $\psi$ of order $9$; $J(\chi,\psi)=2 + 2\zeta_{9} - \zeta_{9}^{2} - 2\zeta_{9}^{4} + \zeta_{9}^{5}$

18:

comment: $\chi$ of order $9$, $\psi$ of order $2$; $J(\chi,\psi)=1 + 2\zeta_{18} + 2\zeta_{18}^{2} - 2\zeta_{18}^{3} + 2\zeta_{18}^{4}$

Definition

The Jacobi sum $J(\chi,\psi)=\sum_{a=0}^{p-1}\chi(a)\,\psi(1-a)$ [4] of two nontrivial Dirichlet characters $\chi=\chi_p(m,\cdot)$ and $\psi=\chi_p(n,\cdot)$ modulo a prime $p$ with $\chi\psi$ nontrivial, indexed by the Conrey labels [7] $(p,m)$ and $(p,n)$, $m\leq n$.

Parameters

$p$

— prime modulus ($p$ prime, $p\geq 5$)

$m$

— Conrey index of $\chi$ ($2\leq m\leq p-1$)

$n$

— Conrey index of $\psi$ ($m\leq n\leq p-1$, $mn\not\equiv 1\pmod p$)

Formulas

(1)

$J(\chi,\psi)=J(\psi,\chi)$, and $\overline{J(\chi,\psi)}=J(\bar\chi,\bar\psi)$.

(2)

$J(\chi,\psi)=\dfrac{\tau(\chi)\,\tau(\psi)}{\tau(\chi\psi)}$ for $\chi$, $\psi$ and $\chi\psi$ nontrivial, where $\tau(\chi)=\sum_{a=1}^{p}\chi(a)\,e^{2\pi i a/p}$ is the Gauss sum [1].

(3)

$J(\chi,\psi)\,\overline{J(\chi,\psi)}=p$ for $\chi$, $\psi$ and $\chi\psi$ nontrivial, so $|J(\chi,\psi)|=\sqrt{p}$ [1].

(4)

For nontrivial $\chi$, $J(\chi,\bar\chi)=-\chi(-1)$; and with the trivial character $\varepsilon$ taking the value $\varepsilon(0)=1$, $J(\varepsilon,\chi)=0$ for nontrivial $\chi$ and $J(\varepsilon,\varepsilon)=p$ [1].

(5)

For $\gcd(c,d)=1$ and $\sigma_c$ the automorphism of $\mathbb{Q}(\zeta_d)$ with $\sigma_c(\zeta_d)=\zeta_d^c$: $\sigma_c(J(\chi,\psi))=J(\chi^c,\psi^c)$, where $\chi_p(n,\cdot)^c=\chi_p(n^c,\cdot)$.

(6)

For $k$ and $l$ dividing $p-1$, the number of solutions $(x,y)\in\mathbb{F}_p^2$ of $x^k+y^l=1$ is $\sum_{\chi^k=\varepsilon,\ \psi^l=\varepsilon}J(\chi,\psi)$, the sum over the characters of order dividing $k$ and of order dividing $l$, with $\varepsilon(0)=1$ as in (4) [1].

(7)

For $p\equiv 1\pmod 3$ and $\chi$ of order $3$: $J(\chi,\chi)=a+b\zeta_3$ with $a\equiv -1\pmod 3$, $b\equiv 0\pmod 3$ and $a^2-ab+b^2=p$; so $4p=L^2+27M^2$ with $L=2a-b\equiv 1\pmod 3$ and $M=b/3$, and $x^3+y^3=1$ has $p-2+L$ solutions over $\mathbb{F}_p$ [1].

(8)

For $p\equiv 1\pmod 4$ and $\chi$ of order $4$: $-\chi(-1)\,J(\chi,\chi)=a+bi\equiv 1\pmod{2+2i}$, that is $b$ even and $a\equiv 1-b\pmod 4$, with $a^2+b^2=p$ [1].

(9)

For the quadratic character $\chi=\left(\frac{\cdot}{p}\right)=\chi_p(p-1,\cdot)$ and every nontrivial $\psi\neq\chi$: $J(\chi,\psi)=\psi(4)\,J(\psi,\psi)$, so for $\psi=\chi_p(m,\cdot)$ with $2\leq m\leq p-2$ the entry at $(p,m,p-1)$ is $\psi(4)$ times the entry at $(p,m,m)$ [2].

(10)

For $\chi$ of order $d\geq 3$: $\tau(\chi)^d=\chi(-1)\,p\prod_{j=1}^{d-2}J(\chi,\chi^j)$, the $J(\chi,\chi^j)$ being the entries at $(p,m,m^j)$, the indices reduced modulo $p$ and ordered [1].

Comments

(11)

The Conrey label $(p,n)$ [7] names the character $\chi_p(n,\cdot)$ with $\chi_p(g^a,g^b)=e^{2\pi i ab/(p-1)}$, where $g$ is the least positive integer generating $(\mathbb{Z}/p^k\mathbb{Z})^\times$ for every $k\geq 1$, which for $p<40487$ is the least primitive root modulo $p$, and $\chi_p(n,m)=0$ when $p\mid m$. So $\chi_p(n,m)=\chi_p(m,n)$, the trivial character is $\chi_p(1,\cdot)$, $\chi_p(m,\cdot)\chi_p(n,\cdot)=\chi_p(mn,\cdot)$, $\overline{\chi_p(n,\cdot)}=\chi_p(n',\cdot)$ with $nn'\equiv 1\pmod p$, the quadratic character $\left(\frac{\cdot}{p}\right)$ is $\chi_p(p-1,\cdot)$, and the order of $\chi_p(m,\cdot)$ is the multiplicative order of $m$ modulo $p$. The Gauss sums $\tau(\chi)$, the zeros of $L(s,\chi)$ and the generalized Bernoulli numbers on this site, and the characters in the LMFDB [6], are indexed the same way; the LMFDB's page for $\chi_p(m,\cdot)$ computes its Jacobi sums exactly.

(12)

Modulo $2$ there is no nontrivial character, and modulo $3$ the only one is the Legendre symbol, whose square is trivial, so the first prime with an admissible pair is $5$. The product $\chi\psi=\chi_p(mn,\cdot)$ is trivial exactly when $mn\equiv 1\pmod p$, that is when $\psi=\bar\chi$; those pairs are left out, and their sums are $J(\chi,\bar\chi)=-\chi(-1)$ by (4). Since $J(\chi,\psi)=J(\psi,\chi)$ by (1), each pair is listed once, with $m\leq n$, which leaves $(p-1)(p-3)/2$ pairs for each $p$.

(13)

The characters listed are nontrivial, so $\chi(0)=\psi(0)=0$, the terms $a=0$ and $a=1$ vanish, and the sum runs over $2\leq a\leq p-1$. What the trivial character $\varepsilon$ is worth at $0$ is a convention that changes $J(\varepsilon,\chi)$: with $\varepsilon(0)=1$, as in [1], $J(\varepsilon,\chi)=0$ and $J(\varepsilon,\varepsilon)=p$, which is what the point count (6) needs; with $\varepsilon(0)=0$, as in Sage, $J(\varepsilon,\chi)=-1$.

(14)

$J(\chi,\psi)$ is an algebraic integer in $\mathbb{Z}[\zeta_d]$, $d$ the least common multiple of the orders of $\chi$ and $\psi$ and $\zeta_d=e^{2\pi i/d}$ a root of unity, because every term $\chi(a)\psi(1-a)$ is a $d$-th root of unity; its absolute value is $\sqrt{p}$ by (3). Each entry's comment gives the orders of $\chi$ and $\psi$ and the exact value, written in $\zeta_d$ with the constant term first and with $i$ for $\zeta_4$. The Galois conjugates of an entry are entries: $\sigma_c(J(\chi,\psi))$ of (5) is the entry at $(p,m^c,n^c)$, the indices reduced modulo $p$ and ordered.

(15)

No entry is real or purely imaginary: $J\bar J=p$ with $p$ prime makes $J$ irrational, and a purely imaginary $J$ would put $\sqrt{-p}$ into $\mathbb{Q}(\zeta_{p-1})$, in which $p$ is unramified. Every entry at $p=5$ and $p=7$ has degree $2$ over $\mathbb{Q}$, and an entry of degree $2$ is a root of $x^2-tx+p$ with $t=J+\bar J$ an integer; the four entries at $p=5$, $1\pm 2i$ and $-1\pm 2i$, are the roots of $x^2-2x+5$ and of $x^2+2x+5$ in the table of algebraic numbers of degree 2.

(16)

For $p\equiv 1\pmod 3$ and $\chi$ of order $3$, the entry $J(\chi,\chi)=a+b\zeta_3$ writes $p=a^2-ab+b^2$ as a norm from $\mathbb{Z}[\zeta_3]$, with $a\equiv -1$ and $b\equiv 0\pmod 3$ by (7), and $L=2a-b$ is the integer with $4p=L^2+27M^2$, $L\equiv 1\pmod 3$, for which $x^3+y^3=1$ has $p-2+L$ solutions over $\mathbb{F}_p$. For $p\equiv 1\pmod 4$ and $\chi$ of order $4$, $J(\chi,\chi)=a+bi$ writes $p=a^2+b^2$, with the sign fixed by (8). Weil [3] wrote the zeta function of every diagonal hypersurface over a finite field in terms of Jacobi sums, which is how they enter the point counts of Fermat curves (6).

Programs

(P1)

Sage

G = DirichletGroup(7)
chi = [c for c in G if c.conrey_number() == 2][0]        # chi_7(2, .)
chi.jacobi_sum(chi)                                      # -3*zeta6 + 2, exact: -1 - 3 zeta_3
CBF = ComplexBallField(400)
sum(CBF(chi(a)) * CBF(chi(1 - a)) for a in range(2, 7))   # 0.5 - 2.598...i

(P2)

PARI/GP

p = 7; G = znstar(p, 1); chi1 = znconreychar(G, 2); chi2 = znconreychar(G, 2);   \\ chi_7(2, .) twice; chi2 rather than psi, which is PARI's digamma
z = exp(2*Pi*I/(p-1)); sum(a = 2, p-1, z^((chareval(G, chi1, a) + chareval(G, chi2, 1-a)) * (p-1)))   \\ 0.5 - 2.598...*I

References

[1]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, second edition, Graduate Texts in Mathematics 84, Springer, 1990, Chapter 8.

[2]

B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts 21, Wiley, 1998.

[3]

A. Weil, Numbers of solutions of equations in finite fields, Bulletin of the American Mathematical Society 55 (1949), 497-508.

Links

[4]

Wikipedia: Jacobi sum

[5]

LMFDB knowl: Jacobi sum

[6]

LMFDB: Dirichlet characters

[7]

LMFDB knowl: Conrey label

[8]

OEIS A001479: $x$ with $p=x^2+3y^2$ for the primes $p\equiv 1\pmod 3$

[9]

OEIS A394567: product of the three cubic Gauss periods, $((L+3)p-1)/27$

[10]

OEIS A002972: the odd $x$ with $p=x^2+y^2$ for the primes $p\equiv 1\pmod 4$

Similar tables

Gauss sums of primitive Dirichlet characters — $J(\chi,\psi)=\tau(\chi)\tau(\psi)/\tau(\chi\psi)$, the same characters under the same Conrey labels

Zeros of Dirichlet $L$-functions — zeros of $L(s,\chi)$ for the same characters under the same Conrey labels

Generalized Bernoulli numbers — $B_{k,\chi}$ under the same labels

Roots of unity — the values $\chi(a)\psi(1-a)$ that are summed, and the $\zeta_d$ the exact values are written in

Algebraic numbers of degree 2 — holds the four entries at $p=5$, the roots $\pm 1\pm 2i$ of $x^2-2x+5$ and $x^2+2x+5$

Data properties

Entries are of type: complex number

How well the digits are known: proven

Table is complete: no (every admissible pair for the primes $p\leq 19$ is here, 372 entries)

How they were obtained:

Each value is a complex ball at 397 bits: the sum of the $p-2$ terms $e^{2\pi i r}$, $r$ the exact rational with $\chi(a)\psi(1-a)=e^{2\pi i r}$, each term arb's exponential of a ball containing $r$.

The characters are built from Conrey's definition (discrete logarithms to the least primitive root), not from a library, and compared with Sage's on every value of every character modulo every prime in range. The generator requires every value to overlap an enclosure of Sage's exact Jacobi sum of the characters Sage numbers $(p,m)$ and $(p,n)$; that exact value to satisfy $J\bar J=p$, to lie in $\mathbb{Z}[\zeta_d]$, to equal $\tau(\chi)\tau(\psi)/\tau(\chi\psi)$ in Sage's exact Gauss sums and to equal the sum with $\chi$ and $\psi$ exchanged; and the congruences of (7) and (8) and the identity (9) to hold, before returning it. A hundred digits are written; the widest ball, at $p=19$, supports $117$. Outside the generator: five Jacobi sums computed by the LMFDB's calculator, four of them entries here and one at $p=23$, agree exactly with Sage's; the number of solutions of $x^k+y^l=1$ over $\mathbb{F}_p$ for every $k$ and $l$ dividing $p-1$, found by enumeration, agrees with the sum of (6) in all $138$ cases and disagrees for every $p$ when the trivial character is given the value $0$ at $0$; the cubic entries $J(\chi,\chi)$ reproduce OEIS A001479 and A001480 [8] and, through $L$, OEIS A394567 [9], and the quartic entries reproduce OEIS A002972 and A002973 [10]; the conjugate, Galois, Legendre-row and Gauss-sum power identities hold exactly on every entry, and $\tau(\chi)\tau(\psi)/\tau(\chi\psi)$ from the stored digits of the Gauss-sum table encloses every entry; both programs were run and give the values their comments state; and every comment's exact value, read back from the stored table, encloses the stored digits. Two controls that must fail did: the conjugate of each LMFDB value is refused, and the quotient $\tau(\chi_7(2,\cdot))^2/\tau(\chi_7(3,\cdot))$ does not enclose $J(\chi_7(2,\cdot),\chi_7(2,\cdot))$.