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Holomorphic periods of Fermat curves

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Numbers
$N$
$a$
$b$ 
$P_N(a,b)$
3
1
1:
2.649958125428174935970534249472658053857802807303836295190357862753179502592161870408230490000075381 + i * -4.589862111171578624739582516926473075718486841482788488185435057728567085398122829406515561697626034
4
1
1:
0 + i * -3.708149354602743836867700694390520092435197647043533811171856090112043553676239956714543723300743795
comment: The quartic period equals $-2iK(1/2)$ by (7).
4
1
2:
2.622057554292119810464839589891119413682754951431623162816821703800790587070414250230295532961429093 + i * -2.622057554292119810464839589891119413682754951431623162816821703800790587070414250230295532961429093
4
2
1:
2.622057554292119810464839589891119413682754951431623162816821703800790587070414250230295532961429093 + i * -2.622057554292119810464839589891119413682754951431623162816821703800790587070414250230295532961429093
5
1
1:
-0.8115251019424348769524244140505528069561943898033505751527630613279129210324859320323843698356495427 + i * -2.497617446252701077877710989564784783937838063912122789757208669203690926068838876259737017809545101
5
1
2:
0.9450001622708105969152988710723230177232842209175940073734786132070106289965632127034770177513306507 + i * -2.908411442048816803638794608237424753972745267785101528312545167602168695084736001055580432335921469
5
1
3:
2.124600299609017820473379033402123520489938091999343413007036668950217054929994497371530962347228248 + i * -1.543612472678882958663457390361921988791065707421576126967735234741676898625676380818793523119489358
5
2
1:
0.9450001622708105969152988710723230177232842209175940073734786132070106289965632127034770177513306507 + i * -2.908411442048816803638794608237424753972745267785101528312545167602168695084736001055580432335921469
5
2
2:
2.474042544199148156274199840729958453526272820446682216525480362135851903372306375742347465735774237 + i * -1.797497124455263896685978118748000989724517911945075330253729212506765345946169081568069810460647060
5
3
1:
2.124600299609017820473379033402123520489938091999343413007036668950217054929994497371530962347228248 + i * -1.543612472678882958663457390361921988791065707421576126967735234741676898625676380818793523119489358
6
1
1:
-0.9638106483299990655228488941631365297935682809517607085309735535318584336949171985077862349232018153 + i * -1.669369011783458076842999543527547066238520313990466327614882448719715494563532490478238369168887550
6
1
2:
0 + i * -2.428650647887581611819941689780931248555034844874909274416629418805405687361769174454672727088835444
6
1
3:
1.214325323943790805909970844890465624277517422437454637208314709402702843680884587227336363544417722 + i * -2.103273157988181391762528618575441203194533308135979144272990980614427148972704944106278399727537177
6
1
4:
1.669369011783458076842999543527547066238520313990466327614882448719715494563532490478238369168887550 + i * -0.9638106483299990655228488941631365297935682809517607085309735535318584336949171985077862349232018153
6
2
1:
0 + i * -2.428650647887581611819941689780931248555034844874909274416629418805405687361769174454672727088835444
6
2
2:
1.324979062714087467985267124736329026928901403651918147595178931376589751296080935204115245000037690 + i * -2.294931055585789312369791258463236537859243420741394244092717528864283542699061414703257780848813017
6
2
3:
2.103273157988181391762528618575441203194533308135979144272990980614427148972704944106278399727537177 + i * -1.214325323943790805909970844890465624277517422437454637208314709402702843680884587227336363544417722
6
3
1:
1.214325323943790805909970844890465624277517422437454637208314709402702843680884587227336363544417722 + i * -2.103273157988181391762528618575441203194533308135979144272990980614427148972704944106278399727537177
6
3
2:
2.103273157988181391762528618575441203194533308135979144272990980614427148972704944106278399727537177 + i * -1.214325323943790805909970844890465624277517422437454637208314709402702843680884587227336363544417722
6
4
1:
1.669369011783458076842999543527547066238520313990466327614882448719715494563532490478238369168887550 + i * -0.9638106483299990655228488941631365297935682809517607085309735535318584336949171985077862349232018153
7
1
1:
-0.9132195654801113941719955335476096297880349752768462807723225073639574464934708421293526337995145110 + i * -1.145141114689628161221964998879307603175673486307674234932213891615775284747350963278812316235856044
7
1
2:
-0.4302023263625280779259206566003346717738740911037072294368401729761833129620542736251188405640024313 + i * -1.884839544753678461159594054251854001315606333308219546225885442680334268480592542721196750834897407
7
1
3:
0.4672078636563557579104824676864249017993928602108287722468856572564605063604779234309282704658885278 + i * -2.046971397121873434491896712652548153666693646460857365108399784534303707692166170645083196811426069
7
1
4:
1.205400132266196000096751773106655808048269006373061394471263321679367716167167742089338082309142604 + i * -1.511523956875187003450666649467337024393799170236362263666423142555157406333176072154289977435157173
7
1
5:
1.319640506715739404416328303115410917888091574349470278722644919232486949625801688561681841088504398 + i * -0.6355053739846061476633096461410704102243344366171262737468210941737469788287458780510394357559139555
7
2
1:
-0.4302023263625280779259206566003346717738740911037072294368401729761833129620542736251188405640024313 + i * -1.884839544753678461159594054251854001315606333308219546225885442680334268480592542721196750834897407
7
2
2:
0.4945461452570726574811702750134045332869531851336730299755500568828580058359053731318552829393294464 + i * -2.166748234877094307695729392436657020649959659605328184369151723626773533472977124955679435941262911
7
2
3:
1.385687506492612809948788477801846998826286144958754224417067497994061428261324059355652408577119207 + i * -1.737597173536466875400620883674848484064684749154134884157189107693496110291361720319293495145855875
7
2
4:
1.741853658712073695537369536021228001799465748670428956311871836762501655468321158721647535765699523 + i * -0.8388325117127659855610349067783167407109946655657961846846644638111820015928211655953664790969383936
7
3
1:
0.4672078636563557579104824676864249017993928602108287722468856572564605063604779234309282704658885278 + i * -2.046971397121873434491896712652548153666693646460857365108399784534303707692166170645083196811426069
7
3
2:
1.385687506492612809948788477801846998826286144958754224417067497994061428261324059355652408577119207 + i * -1.737597173536466875400620883674848484064684749154134884157189107693496110291361720319293495145855875
7
3
3:
1.891686020319391915463315769321818963616870621198549806676620590759609113294618837912466674805053245 + i * -0.9109879741388420471721564647774505722941736041312594033531412567343123995550614910948897213416794972
7
4
1:
1.205400132266196000096751773106655808048269006373061394471263321679367716167167742089338082309142604 + i * -1.511523956875187003450666649467337024393799170236362263666423142555157406333176072154289977435157173
7
4
2:
1.741853658712073695537369536021228001799465748670428956311871836762501655468321158721647535765699523 + i * -0.8388325117127659855610349067783167407109946655657961846846644638111820015928211655953664790969383936
7
5
1:
1.319640506715739404416328303115410917888091574349470278722644919232486949625801688561681841088504398 + i * -0.6355053739846061476633096461410704102243344366171262737468210941737469788287458780510394357559139555
8
1
1:
-0.8105834854075622288747990246553301715699347722948067017986566749470698788711553293581123905498935000 + i * -0.8105834854075622288747990246553301715699347722948067017986566749470698788711553293581123905498935000
8
1
2:
-0.5966359033844678255721678001414464601304898715821599988592224179416766306584838820750028876685186340 + i * -1.440406489749505826139045574879483830249157377191959727760450438947146043394580736866649587690942715
8
1
3:
0 + i * -1.781150396113309967409093856902172935886260196372572110435783354449873007938974507898371953431638204
8
1
4:
0.6816167471430807107389846003312843615156247321771401375279160622965399620020296241890748771790180390 + i * -1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067
8
1
5:
1.102439398996582772779449375495543449441588471125200985829441485358670625813406629240903162016925097 + i * -1.102439398996582772779449375495543449441588471125200985829441485358670625813406629240903162016925097
8
1
6:
1.059078361973931594252519173502856912226988925314159659694485326084552110473461239486734631848267393 + i * -0.4386846211454844499451512580852525825690678911394952076326072831231819032681138703680181821335275161
8
2
1:
-0.5966359033844678255721678001414464601304898715821599988592224179416766306584838820750028876685186340 + i * -1.440406489749505826139045574879483830249157377191959727760450438947146043394580736866649587690942715
8
2
2:
0 + i * -1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897
8
2
3:
0.7488814915597080220988352157940547473980284082268274336635463046038670068707875549273764069908974548 + i * -1.807959853533639616351354389296911659625017333540987093358031630688419117344248794414111038839164848
8
2
4:
1.311028777146059905232419794945559706841377475715811581408410851900395293535207125115147766480714547 + i * -1.311028777146059905232419794945559706841377475715811581408410851900395293535207125115147766480714547
8
2
5:
1.440406489749505826139045574879483830249157377191959727760450438947146043394580736866649587690942715 + i * -0.5966359033844678255721678001414464601304898715821599988592224179416766306584838820750028876685186340
8
3
1:
0 + i * -1.781150396113309967409093856902172935886260196372572110435783354449873007938974507898371953431638204
8
3
2:
0.7488814915597080220988352157940547473980284082268274336635463046038670068707875549273764069908974548 + i * -1.807959853533639616351354389296911659625017333540987093358031630688419117344248794414111038839164848
8
3
3:
1.383752564657076372114632099814734879041749385508285296451133301794609624050569315436431723927055695 + i * -1.383752564657076372114632099814734879041749385508285296451133301794609624050569315436431723927055695
8
3
4:
1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067 + i * -0.6816167471430807107389846003312843615156247321771401375279160622965399620020296241890748771790180390
8
4
1:
0.6816167471430807107389846003312843615156247321771401375279160622965399620020296241890748771790180390 + i * -1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067
8
4
2:
1.311028777146059905232419794945559706841377475715811581408410851900395293535207125115147766480714547 + i * -1.311028777146059905232419794945559706841377475715811581408410851900395293535207125115147766480714547
8
4
3:
1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067 + i * -0.6816167471430807107389846003312843615156247321771401375279160622965399620020296241890748771790180390
8
5
1:
1.102439398996582772779449375495543449441588471125200985829441485358670625813406629240903162016925097 + i * -1.102439398996582772779449375495543449441588471125200985829441485358670625813406629240903162016925097
8
5
2:
1.440406489749505826139045574879483830249157377191959727760450438947146043394580736866649587690942715 + i * -0.5966359033844678255721678001414464601304898715821599988592224179416766306584838820750028876685186340
8
6
1:
1.059078361973931594252519173502856912226988925314159659694485326084552110473461239486734631848267393 + i * -0.4386846211454844499451512580852525825690678911394952076326072831231819032681138703680181821335275161
9
1
1:
-0.7044461015437884188511423613050374093834137105023625793279381169882011872169154616187178795295879504 + i * -0.5911004639896656057682389773196482107280126865869131838211376320648323847704991807434168971523551234
9
1
2:
-0.6382645991542058011133952902760957364969223814656434075545217349345998784288886885616256079734893291 + i * -1.105506714407667912983762824471593540048141305827820152855710586669870677522547673056737290746685736
9
1
3:
-0.2618938372938204752371455587699012431190486278693726471029130912060811297356853563511042330802331974 + i * -1.485273758114362541667457699663158645172269378432080228400922187989692840051716470137897634539833303
9
1
4:
0.2758272885084240542914728160296354394285518965533199535814240496354165353949453926972820814794801470 + i * -1.564294286672274112465499473809138039229919274135263791889571746380393699376251292039717962772870561
9
1
5:
0.7540932499624373445302549599566192278949700056939809030883838094365820242870789154821737149939738779 + i * -1.306127822579678849386512573394055718955474664357722951675847511840323069873774172749746604185003901
9
1
6:
0.9778780988432826439357796143179158599613870130529557360037970795371251748312505453313712587834751922 + i * -0.8205371520757382342210587679286371763329802887192139334445439876252119001922486870776100037598588567
9
1
7:
0.8641310687757134015697116319594012704605563583807761113845222164147571620833448392090223484773396251 + i * -0.3145179875390002313111635790893804566946772397835652237088184129054985527463820482002572272496228760
9
2
1:
-0.6382645991542058011133952902760957364969223814656434075545217349345998784288886885616256079734893291 + i * -1.105506714407667912983762824471593540048141305827820152855710586669870677522547673056737290746685736
9
2
2:
-0.2698355049415498924868013999582755137012802167109167954734847173487652070964792840446168904332579842 + i * -1.530313193462376526144888340450569009706685203943110381995440196406028926481002266396964636753039959
9
2
3:
0.2952667293690957829128058868556182910557376842772146553359667600398388023895999022562253332019836871 + i * -1.674540834208935327019806921351442079769295827049147485022738503473332563302490010289128454732178368
9
2
4:
0.8501866628725541057521936288660111692455745526666921238463023271535165028961695297714368321219613547 + i * -1.472566496012696164787320345503191932858019415477720252340747331182460253946094812644258158584181982
9
2
5:
1.190372348814133460753264599916984932646592238842498689627426828679553264584884429447525585550961623 + i * -0.9988409988535718985030883590245863667046484520787206932635872068799925911979051559444277173729799354
9
2
6:
1.199545067868165865975342043147704616465522919425621493275167012237660416154469894240549354072655507 + i * -0.4365986993648439410033666283072670938625568759597681981740629389786644416360951115623000963219361134
9
3
1:
-0.2618938372938204752371455587699012431190486278693726471029130912060811297356853563511042330802331974 + i * -1.485273758114362541667457699663158645172269378432080228400922187989692840051716470137897634539833303
9
3
2:
0.2952667293690957829128058868556182910557376842772146553359667600398388023895999022562253332019836871 + i * -1.674540834208935327019806921351442079769295827049147485022738503473332563302490010289128454732178368
9
3
3:
0.8833193751427249786568447498242193512859342691012787650634526209177265008640539568027434966666917936 + i * -1.529954037057192874913194172308824358572828947160929496061811685909522361799374276468838520565875345
9
3
4:
1.302561537414776057264315479104062257854695571198214442182728483980411776166076277390438401325907962 + i * -1.092978905630449408273210665664676594833265281055726586316095887347304362787459978403912839812831715
9
3
5:
1.417231724748331714743137175043160189380198971595401777830447844584912237520227510184717622331245409 + i * -0.5158301628661443170343051171474222821316758787047861684432189274185883206400719245065323100363276568
9
4
1:
0.2758272885084240542914728160296354394285518965533199535814240496354165353949453926972820814794801470 + i * -1.564294286672274112465499473809138039229919274135263791889571746380393699376251292039717962772870561
9
4
2:
0.8501866628725541057521936288660111692455745526666921238463023271535165028961695297714368321219613547 + i * -1.472566496012696164787320345503191932858019415477720252340747331182460253946094812644258158584181982
9
4
3:
1.302561537414776057264315479104062257854695571198214442182728483980411776166076277390438401325907962 + i * -1.092978905630449408273210665664676594833265281055726586316095887347304362787459978403912839812831715
9
4
4:
1.492632235507258637697560256548004270483779355168155662644761725269956890489009533458119989508797438 + i * -0.5432737044308622607820278958368871667341375689840871931721695203642505777055099284061846748180999209
9
5
1:
0.7540932499624373445302549599566192278949700056939809030883838094365820242870789154821737149939738779 + i * -1.306127822579678849386512573394055718955474664357722951675847511840323069873774172749746604185003901
9
5
2:
1.190372348814133460753264599916984932646592238842498689627426828679553264584884429447525585550961623 + i * -0.9988409988535718985030883590245863667046484520787206932635872068799925911979051559444277173729799354
9
5
3:
1.417231724748331714743137175043160189380198971595401777830447844584912237520227510184717622331245409 + i * -0.5158301628661443170343051171474222821316758787047861684432189274185883206400719245065323100363276568
9
6
1:
0.9778780988432826439357796143179158599613870130529557360037970795371251748312505453313712587834751922 + i * -0.8205371520757382342210587679286371763329802887192139334445439876252119001922486870776100037598588567
9
6
2:
1.199545067868165865975342043147704616465522919425621493275167012237660416154469894240549354072655507 + i * -0.4365986993648439410033666283072670938625568759597681981740629389786644416360951115623000963219361134
9
7
1:
0.8641310687757134015697116319594012704605563583807761113845222164147571620833448392090223484773396251 + i * -0.3145179875390002313111635790893804566946772397835652237088184129054985527463820482002572272496228760
10
1
1:
-0.6092158640091930165890739372354169010229686834690334088566263731967632899405851637546251449076163123 + i * -0.4426212339382092462452331925927358995956207131176518839057730369990264043369536744297683823447990178
10
1
2:
-0.6234675948990016726340841487639054863382252015491106226666321569549972278874460854499978031732607120 + i * -0.8581295256186315609094140285541578924295006569630219652062843652712136886789717835618185053908828497
10
1
3:
-0.3964872995725607950066968070951680943770239171059816389901388980420803980074968729599135961796728705 + i * -1.220262434593543244049739963588566390565297581975071452434414061579020729793972146527549723332235775
10
1
4:
0 + i * -1.399610521650914804276802149678014201115012823609235730054300129943115165190203721975101753933976565
10
1
5:
0.4325034366961179700071934203179892001416571345207196802134643287815628451174510404736990404072120774 + i * -1.331108706891361647005337909775863557412620375623182495019110786880367053531928625141618793967807287
10
1
6:
0.7541636597735052033298027414265226808504206501818197263906195633602377361193499040674348479660009905 + i * -1.038017226388625816851844952414266319029619285013956392536085058467283356228282137458396098627923313
10
1
7:
0.8581295256186315609094140285541578924295006569630219652062843652712136886789717835618185053908828497 + i * -0.6234675948990016726340841487639054863382252015491106226666321569549972278874460854499978031732607120
10
1
8:
0.7161762006544410353168690297804910192964002907085621120957885565913237106372367474509436605755539807 + i * -0.2326997535658779500197784017382575322566862708010984691678687519026315289818848674233048956622774851
10
2
1:
-0.6234675948990016726340841487639054863382252015491106226666321569549972278874460854499978031732607120 + i * -0.8581295256186315609094140285541578924295006569630219652062843652712136886789717835618185053908828497
10
2
2:
-0.4057625509712174384762122070252764034780971949016752875763815306639564605162429660161921849178247713 + i * -1.248808723126350538938855494782392391968919031956061394878604334601845463034419438129868508904772551
10
2
3:
0 + i * -1.473848743229964292590147508171499378439199953600288516152987869698913570705609168999019416674137184
10
2
4:
0.4725000811354052984576494355361615088616421104587970036867393066035053144982816063517385088756653253 + i * -1.454205721024408401819397304118712376986372633892550764156272583801084347542368000527790216167960734
10
2
5:
0.8663065553803690095249338682400775418576998047212202167960175600591669756824796966479438827275911954 + i * -1.192368680411199352509095186551751264003230280191736428095432293150993081690985329739715152359473049
10
2
6:
1.062300149804508910236689516701061760244969045999671706503518334475108527464997248685765481173614124 + i * -0.7718062363394414793317286951809609943955328537107880634838676173708384493128381904093967615597446792
10
2
7:
1.008791759430735271028063336173674376694827257432721614299851661449238301222606865506259063776468555 + i * -0.3277763120364936198642461462855154545115800088698902025393192045353860261581865502520532193209485039
10
3
1:
-0.3964872995725607950066968070951680943770239171059816389901388980420803980074968729599135961796728705 + i * -1.220262434593543244049739963588566390565297581975071452434414061579020729793972146527549723332235775
10
3
2:
0 + i * -1.473848743229964292590147508171499378439199953600288516152987869698913570705609168999019416674137184
10
3
3:
0.4861887689936565164795702197198346721485978064674346698542960094146188973888619403678904627784746182 + i * -1.496335170291278954259541756131790124364199018663175425499254123666813597621700570647818925444216370
10
3
4:
0.9247859938018722891102574013645725161107157069700995059286146180847105344722378143178267131725773252 + i * -1.272858722173863769664469930401350513589616951918121868663920847938529780531326312748189748931215620
10
3
5:
1.192368680411199352509095186551751264003230280191736428095432293150993081690985329739715152359473049 + i * -0.8663065553803690095249338682400775418576998047212202167960175600591669756824796966479438827275911954
10
3
6:
1.220262434593543244049739963588566390565297581975071452434414061579020729793972146527549723332235775 + i * -0.3964872995725607950066968070951680943770239171059816389901388980420803980074968729599135961796728705
10
4
1:
0 + i * -1.399610521650914804276802149678014201115012823609235730054300129943115165190203721975101753933976565
10
4
2:
0.4725000811354052984576494355361615088616421104587970036867393066035053144982816063517385088756653253 + i * -1.454205721024408401819397304118712376986372633892550764156272583801084347542368000527790216167960734
10
4
3:
0.9247859938018722891102574013645725161107157069700995059286146180847105344722378143178267131725773252 + i * -1.272858722173863769664469930401350513589616951918121868663920847938529780531326312748189748931215620
10
4
4:
1.237021272099574078137099920364979226763136410223341108262740181067925951686153187871173732867887118 + i * -0.8987485622276319483429890593740004948622589559725376651268646062533826729730845407840349052303235298
10
4
5:
1.331108706891361647005337909775863557412620375623182495019110786880367053531928625141618793967807287 + i * -0.4325034366961179700071934203179892001416571345207196802134643287815628451174510404736990404072120774
10
5
1:
0.4325034366961179700071934203179892001416571345207196802134643287815628451174510404736990404072120774 + i * -1.331108706891361647005337909775863557412620375623182495019110786880367053531928625141618793967807287
10
5
2:
0.8663065553803690095249338682400775418576998047212202167960175600591669756824796966479438827275911954 + i * -1.192368680411199352509095186551751264003230280191736428095432293150993081690985329739715152359473049
10
5
3:
1.192368680411199352509095186551751264003230280191736428095432293150993081690985329739715152359473049 + i * -0.8663065553803690095249338682400775418576998047212202167960175600591669756824796966479438827275911954
10
5
4:
1.331108706891361647005337909775863557412620375623182495019110786880367053531928625141618793967807287 + i * -0.4325034366961179700071934203179892001416571345207196802134643287815628451174510404736990404072120774
10
6
1:
0.7541636597735052033298027414265226808504206501818197263906195633602377361193499040674348479660009905 + i * -1.038017226388625816851844952414266319029619285013956392536085058467283356228282137458396098627923313
10
6
2:
1.062300149804508910236689516701061760244969045999671706503518334475108527464997248685765481173614124 + i * -0.7718062363394414793317286951809609943955328537107880634838676173708384493128381904093967615597446792
10
6
3:
1.220262434593543244049739963588566390565297581975071452434414061579020729793972146527549723332235775 + i * -0.3964872995725607950066968070951680943770239171059816389901388980420803980074968729599135961796728705
10
7
1:
0.8581295256186315609094140285541578924295006569630219652062843652712136886789717835618185053908828497 + i * -0.6234675948990016726340841487639054863382252015491106226666321569549972278874460854499978031732607120
10
7
2:
1.008791759430735271028063336173674376694827257432721614299851661449238301222606865506259063776468555 + i * -0.3277763120364936198642461462855154545115800088698902025393192045353860261581865502520532193209485039
10
8
1:
0.7161762006544410353168690297804910192964002907085621120957885565913237106372367474509436605755539807 + i * -0.2326997535658779500197784017382575322566862708010984691678687519026315289818848674233048956622774851
Definition
The marked holomorphic periods are $P_N(a,b)=\int_{\kappa_N}\omega_N^{a,b}$ on $C_N:X^N+Y^N=Z^N$, where $\omega_N^{a,b}=x^{a-1}y^{b-N}\,dx$, $x=X/Z$, and $y=Y/Z$ [1]. The oriented integral cycle $\kappa_N$ is fixed by (8).
Parameters
$N$
—   curve degree ($N\geq3$)
$a$
—   first character index ($a\geq1$)
$b$
—   second character index ($b\geq1$ and $a+b<N$)
Formulas
(1)
$P_N(a,b)=\dfrac{(1-\zeta_N^a)(1-\zeta_N^b)}{N}B(a/N,b/N)$, where $B$ is Euler's beta function [1].
(2)
$B(a/N,b/N)=\Gamma(a/N)\Gamma(b/N)/\Gamma((a+b)/N)$, with the Gamma function in its usual Euler normalization.
(3)
$\displaystyle\int_{(\alpha^r\beta^s)_*\kappa_N}\omega_N^{a,b}=\zeta_N^{ar+bs}P_N(a,b)$ for integers $r,s$ [1].
(4)
$P_N(a,b)=P_N(b,a)$.
(5)
$P_{kN}(ka,kb)=P_N(a,b)/k$ for every positive integer $k$.
(6)
$P_N(a,b)=-\dfrac{4}{N}\exp(\pi i(a+b)/N)\sin(\pi a/N)\sin(\pi b/N)B(a/N,b/N)$. In particular, $\operatorname{Im}P_N(a,b)<0$, and $\operatorname{Re}P_N(a,b)=0$ exactly when $2(a+b)=N$.
(7)
$P_4(1,1)=-2iK(1/2)$, where $K(1/2)$ uses the elliptic parameter $m=1/2$, rather than the modulus. The beta duplication identity gives $B(1/4,1/4)=4K(1/2)$.
Comments
(8)
Put $\zeta_N=\exp(2\pi i/N)$, $\alpha(x,y)=(\zeta_N x,y)$ and $\beta(x,y)=(x,\zeta_N y)$. Let $\delta_N(t)=(t^{1/N},(1-t)^{1/N})$, $0\leq t\leq1$, using the nonnegative real roots and traversing from $(0,1)$ to $(1,0)$. The oriented chain is $\kappa_N=\delta_N-\beta_*\delta_N+(\alpha\beta)_*\delta_N-\alpha_*\delta_N$, which is an integral cycle [1].
(9)
The forms $\omega_N^{a,b}$ for $a,b\geq1$ and $a+b<N$ form a basis of holomorphic differentials, with $g_N=(N-1)(N-2)/2$ elements [1]. Both ordered pairs $(a,b)$ and $(b,a)$ are retained: equal scalar integrals do not identify their basis vectors.
(10)
These differentials are $\omega_0^{a,b}$ in [1] and $\omega_N^{a,b}$ in [2]; they are not divided by their positive open-path beta integrals. The values retain the complex phase of the marked cycle. They are neither a period matrix in a symplectic basis nor a period covolume.
(11)
The translates $(\alpha^r\beta^s)_*\kappa_N$ generate integral homology as an overcomplete orbit [2]. The attachments record their relations and the character orbits, including $d=N/\gcd(N,a,b)$ and the holomorphic embeddings of $\mathbb Q(\zeta_d)$. The Jacobian carries its canonical principal polarization; no polarization-dependent scalar is tabulated.
(12)
The beta-value table holds the real analytic ingredients of (1), without the cycle factor. The genus-two real-period table instead holds real period covolumes.
(13)
Jacobi sums describe the finite-field character factors of the same Fermat cohomology [2]. This connection does not assert equality between complex periods and finite-field sums.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField

def fermat_period(N, a, b, digits=100):
    assert N >= 3 and a >= 1 and b >= 1 and a+b < N
    bits = numberdb.bits(digits, losing=96)
    R, C = RealBallField(bits), ComplexBallField(bits)
    z = (2*C.pi()*C.gen(0)/N).exp()
    B = R(QQ(a)/N).gamma()*R(QQ(b)/N).gamma()/R(QQ(a+b)/N).gamma()
    P = (1-z**a)*(1-z**b)*B/N
    return C(0, P.imag()) if 2*(a+b) == N else P

fermat_period(3, 1, 1)
References
[1]
N. Otsubo, On the Abel-Jacobi maps of Fermat Jacobians, Section 4.1, equation (4.1). (arXiv)
[2]
N. Otsubo, On special values of Jacobi-sum Hecke L-functions, 2014 preprint, Sections 2.1, 2.2 and 5.1, equation (5.2). (arXiv)
Links
Similar tables
Values of the Beta function $B(a,b)$ at pairs of rational numbers —   gives the positive beta factors before multiplication by the marked cycle factor
Values of the Gamma function at rational numbers —   gives the Gamma factors in the analytic evaluation of each period
Complete elliptic integral of the first kind $K(m)$ —   contains $K(1/2)$, with $P_4(1,1)=-2iK(1/2)$
Jacobi sums of pairs of Dirichlet characters modulo a prime —   gives the character sums underlying the finite-field realization of Fermat cohomology
Real periods of genus 2 curves over $\mathbb{Q}$ —   stores real period covolumes, whereas this table records individual marked complex integrals
Zeta polynomials of Fermat curves —   records local zeta numerators of the same curves $X^N+Y^N=Z^N$. Its Jacobi-sum product uses all first-cohomology character pairs, whereas these periods use the holomorphic pairs $a+b<N$; this relates two cohomological realizations and does not identify their numerical values
Data properties
Entries are of type: complex number
How well the digits are known: proven
Table is complete: no (it holds every ordered holomorphic character pair $a,b\geq1$, $a+b<N$, for every degree $3\leq N\leq10$. This initial segment includes the elliptic cubic and higher-genus prime and composite degrees, retaining both primitive and imprimitive characters. The noncanonical cutoff $N=10$ keeps each complete holomorphic basis inspectable, with genus at most 36; the attached argument-selection.md explains the coverage choice)
How they were obtained:

The attached generator evaluates the Gamma quotient and complex cycle factor in Arb ball arithmetic at 429 bits, then records 100 significant digits per nonzero component. The smallest measured relative accuracy exceeds 421 bits. The real component is declared exactly zero when $2(a+b)=N$, as follows from (6). All other components retain certified bounds containing the exact values.

more

All 120 values were compared with a separately bounded Euler integral split at $1/2$: each positive binomial series uses 450 terms at 500 bits, with its omitted tail bounded by twice the next term. This check uses no beta or Gamma evaluation. Its controls give $B(1,1)=1$, $B(1/2,1/2)=\pi$ and $K(0)=\pi/2$; a perturbation by $10^{-90}$ is rejected in every row. Symmetry, phase, complex conjugation, all 8514 translated cycle integrals and all 11 imprimitive reductions were checked across the range. The quartic value was independently compared with the arithmetic-geometric mean evaluation of $K(1/2)$ [3]. The attached checks also test these identities after reading the stored values and compare the quartic with the stored elliptic-integral entry.



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