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comment-normalisation: The eigenform is normalised so that $a_1=1$. The Petersson inner product is $\int_{\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}} f(z)\overline{g(z)}- y^k\,dx\,dy/y^2$, with no division by the volume of the fundamental domain.+ y^k\,dx\,dy/y^2$, with no division by the volume of the fundamental domain. The+ $k=12$ entry is the Petersson norm of the modular discriminant HREF{Q-expansion_of_the_modular_discriminant}[$\Delta$],+ whose coefficients are the Ramanujan tau function. comment-ordering: The index $i$ counts the eigenforms of weight $k$ in increasing order of the embedded $T_2$-eigenvalue $a_2$; this is not necessarily the order
whose roots are the $a_2$ values in that weight. Formulas:- formula-symmetric-square: In Sage's arithmetic normalisation CITE{SageModform},+ formula-symmetric-square: Let $\alpha_p$ and $\beta_p$ be the roots of $X^2-a_pX+p^{k-1}$.+ With $L(\mathrm{Sym}^2 f,s)=\prod_p((1-\alpha_p^2p^{-s})(1-p^{k-1-s})(1-\beta_p^2p^{-s}))^{-1}$, $\langle f,f\rangle=(k-1)!L(\mathrm{Sym}^2 f,k)/(2^{2k-1}\pi^{k+1})$. Programs:
a2 = mfcoefs(F, 2)[3]; - nf = nfinit(component(a2, 1));-- A2 = nfeltembed(nf, a2);+ A2 = if(type(a2) == "t_POLMOD", nfeltembed(nfinit(component(a2, 1)), a2), [a2]); P = mfpetersson(mfsymbol(mf, F)); - vecsort(vector(#A2, j, [real(A2[j]), P[j,j]]), 1)[i][2]'+ vecsort(vector(#A2, j, [real(A2[j]), if (#A2 == 1, P, P[j,j])]), 1)[i][2]' Similar tables: - table: HREF{T323}[Special values of the $L$-functions of level one cusp forms]
rigour details: The generator evaluates Sage's `petersson_norm` CITE{SageModform} at $80$ and $120$ decimal working digits, passing explicit bit precisions to Sage's- symmetric-square $L$-function computation, and stores only $30$ digits. It sorts- the embeddings of each Hecke orbit by the embedded value of $a_2$. An independent- check compares every stored value with PARI/GP's `mfsymbol` and `mfpetersson`- CITE{PARIModular}, after sorting PARI's embeddings by the same $a_2$ values. Neither- computation gives a certified error bound, so the table does not claim proven- digits.+ symmetric-square $L$-function computation. It sorts the embeddings of each Hecke+ orbit by the embedded value of $a_2$. An independent check compares every stored+ value with PARI/GP's `mfsymbol` and `mfpetersson` CITE{PARIModular}, after sorting+ PARI's embeddings by the same $a_2$ values, with tolerance $10^{-40}\max(1,|x|)$.+ The table stores $30$ digits because that is comfortably inside the Sage-PARI+ comparison tolerance. Neither computation gives a certified error bound, so the+ table does not claim proven digits. Display properties: number-header: $\langle f_{k,i},f_{k,i}\rangle$
- - i Numbers:-- params:- k: '12'- i: '1'- number: '0.00000103536205680432092234781681223'- comment: $a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\chi_{12,2}$].-- params:- k: '16'- i: '1'- number: '0.00000216906134759063332432422387084'- comment: $a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\chi_{16,2}$].-- params:- k: '18'- i: '1'- number: '0.00000459473619763924661011587324802'- comment: $a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\chi_{18,2}$].-- params:- k: '20'- i: '1'- number: '0.00000826554153165970316423006276026'- comment: $a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\chi_{20,2}$].-- params:- k: '22'- i: '1'- number: '0.0000200998183274306452315831330634'- comment: $a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\chi_{22,2}$].-- params:- k: '24'- i: '1'- number: '0.000107836545077234026820943374035'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\chi_{24,2}$]- in increasing order, with $a_2\approx -4016.351171716245$.-- params:- k: '24'- i: '2'- number: '0.000128992800758160472019099313271'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\chi_{24,2}$]- in increasing order, with $a_2\approx 5096.351171716245$.-- params:- k: '26'- i: '1'- number: '0.000205334716815372379243519562956'- comment: $a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\chi_{26,2}$].-- params:- k: '28'- i: '1'- number: '0.00236428002656855536683619961491'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\chi_{28,2}$]- in increasing order, with $a_2\approx -18713.59859471915$.-- params:- k: '28'- i: '2'- number: '0.00131542730871571614006541980065'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\chi_{28,2}$]- in increasing order, with $a_2\approx 10433.59859471915$.-- params:- k: '30'- i: '1'- number: '0.00675086417656698477174246013613'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\chi_{30,2}$]- in increasing order, with $a_2\approx -17433.90502875288$.-- params:- k: '30'- i: '2'- number: '0.0100997800473221939384580046955'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\chi_{30,2}$]- in increasing order, with $a_2\approx 26073.90502875288$.-- params:- k: '32'- i: '1'- number: '0.0320925558714875814394415038967'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\chi_{32,2}$]- in increasing order, with $a_2\approx -31347.87172677239$.-- params:- k: '32'- i: '2'- number: '0.0729994924629061249311411160214'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\chi_{32,2}$]- in increasing order, with $a_2\approx 71307.87172677239$.-- params:- k: '34'- i: '1'- number: '0.623070453157424716952166922143'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\chi_{34,2}$]- in increasing order, with $a_2\approx -171359.4371321172$.-- params:- k: '34'- i: '2'- number: '0.180637193843065066573942997006'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\chi_{34,2}$]- in increasing order, with $a_2\approx 49679.43713211717$.-- params:- k: '36'- i: '1'- number: '2.67742333549761657636239939753'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\chi_{36,2}$]- in increasing order, with $a_2\approx -165109.1678324939$.-- params:- k: '36'- i: '2'- number: '2.42584430383642443883211696130'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\chi_{36,2}$]- in increasing order, with $a_2\approx -26808.00761087159$.-- params:- k: '36'- i: '3'- number: '4.55969950662225591954959544292'- comment: $a_2$ is the third root of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\chi_{36,2}$]- in increasing order, with $a_2\approx 331573.1754433655$.-- params:- k: '38'- i: '1'- number: '21.2742918225116448063549453159'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\chi_{38,2}$]- in increasing order, with $a_2\approx -480411.7539742225$.-- params:- k: '38'- i: '2'- number: '12.6655885373653394405955792016'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\chi_{38,2}$]- in increasing order, with $a_2\approx 286011.7539742225$.-- params:- k: '40'- i: '1'- number: '171.581094673690839287635257177'- comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\chi_{40,2}$]- in increasing order, with $a_2\approx -799151.7631007462$.-- params:- k: '40'- i: '2'- number: '190.225804606637761285194728054'- comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\chi_{40,2}$]- in increasing order, with $a_2\approx 241487.7444577613$.-- params:- k: '40'- i: '3'- number: '326.590585884552929043398261462'- comment: $a_2$ is the third root of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\chi_{40,2}$]- in increasing order, with $a_2\approx 1106520.018642985$.+ '12':+ '1':+ number: '0.00000103536205680432092234781681223'+ comment: $a_2=-24$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#12,2}[$\chi_{12,2}$].+ '16':+ '1':+ number: '0.00000216906134759063332432422387084'+ comment: $a_2=216$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#16,2}[$\chi_{16,2}$].+ '18':+ '1':+ number: '0.00000459473619763924661011587324802'+ comment: $a_2=-528$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#18,2}[$\chi_{18,2}$].+ '20':+ '1':+ number: '0.00000826554153165970316423006276026'+ comment: $a_2=456$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#20,2}[$\chi_{20,2}$].+ '22':+ '1':+ number: '0.0000200998183274306452315831330634'+ comment: $a_2=-288$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#22,2}[$\chi_{22,2}$].+ '24':+ '1':+ number: '0.000107836545077234026820943374035'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\chi_{24,2}$]+ in increasing order, with $a_2\approx -4016.351171716245$.+ '2':+ number: '0.000128992800758160472019099313271'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#24,2}[$\chi_{24,2}$]+ in increasing order, with $a_2\approx 5096.351171716245$.+ '26':+ '1':+ number: '0.000205334716815372379243519562956'+ comment: $a_2=-48$, the root of HREF{Hecke_polynomials_of_level_one_cusp_forms#26,2}[$\chi_{26,2}$].+ '28':+ '1':+ number: '0.00236428002656855536683619961491'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\chi_{28,2}$]+ in increasing order, with $a_2\approx -18713.59859471915$.+ '2':+ number: '0.00131542730871571614006541980065'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#28,2}[$\chi_{28,2}$]+ in increasing order, with $a_2\approx 10433.59859471915$.+ '30':+ '1':+ number: '0.00675086417656698477174246013613'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\chi_{30,2}$]+ in increasing order, with $a_2\approx -17433.90502875288$.+ '2':+ number: '0.0100997800473221939384580046955'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#30,2}[$\chi_{30,2}$]+ in increasing order, with $a_2\approx 26073.90502875288$.+ '32':+ '1':+ number: '0.0320925558714875814394415038967'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\chi_{32,2}$]+ in increasing order, with $a_2\approx -31347.87172677239$.+ '2':+ number: '0.0729994924629061249311411160214'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#32,2}[$\chi_{32,2}$]+ in increasing order, with $a_2\approx 71307.87172677239$.+ '34':+ '1':+ number: '0.623070453157424716952166922143'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\chi_{34,2}$]+ in increasing order, with $a_2\approx -171359.4371321172$.+ '2':+ number: '0.180637193843065066573942997006'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#34,2}[$\chi_{34,2}$]+ in increasing order, with $a_2\approx 49679.43713211717$.+ '36':+ '1':+ number: '2.67742333549761657636239939753'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\chi_{36,2}$]+ in increasing order, with $a_2\approx -165109.1678324939$.+ '2':+ number: '2.42584430383642443883211696130'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\chi_{36,2}$]+ in increasing order, with $a_2\approx -26808.00761087159$.+ '3':+ number: '4.55969950662225591954959544292'+ comment: $a_2$ is the third root of HREF{Hecke_polynomials_of_level_one_cusp_forms#36,2}[$\chi_{36,2}$]+ in increasing order, with $a_2\approx 331573.1754433655$.+ '38':+ '1':+ number: '21.2742918225116448063549453159'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\chi_{38,2}$]+ in increasing order, with $a_2\approx -480411.7539742225$.+ '2':+ number: '12.6655885373653394405955792016'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#38,2}[$\chi_{38,2}$]+ in increasing order, with $a_2\approx 286011.7539742225$.+ '40':+ '1':+ number: '171.581094673690839287635257177'+ comment: $a_2$ is the first root of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\chi_{40,2}$]+ in increasing order, with $a_2\approx -799151.7631007462$.+ '2':+ number: '190.225804606637761285194728054'+ comment: $a_2$ is the second root of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\chi_{40,2}$]+ in increasing order, with $a_2\approx 241487.7444577613$.+ '3':+ number: '326.590585884552929043398261462'+ comment: $a_2$ is the third root of HREF{Hecke_polynomials_of_level_one_cusp_forms#40,2}[$\chi_{40,2}$]+ in increasing order, with $a_2\approx 1106520.018642985$.
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