Petersson norms of level one cusp forms

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Numbers

$k$

$i$

$\langle f_{k,i},f_{k,i}\rangle$

0.00000103536205680432092234781681223

comment: $a_2=-24$, the root of $\chi_{12,2}$.

0.00000216906134759063332432422387084

comment: $a_2=216$, the root of $\chi_{16,2}$.

0.00000459473619763924661011587324802

comment: $a_2=-528$, the root of $\chi_{18,2}$.

0.00000826554153165970316423006276026

comment: $a_2=456$, the root of $\chi_{20,2}$.

0.0000200998183274306452315831330634

comment: $a_2=-288$, the root of $\chi_{22,2}$.

0.000107836545077234026820943374035

comment: $a_2$ is the first root of $\chi_{24,2}$ in increasing order, with $a_2\approx -4016.351171716245$.

0.000128992800758160472019099313271

comment: $a_2$ is the second root of $\chi_{24,2}$ in increasing order, with $a_2\approx 5096.351171716245$.

0.000205334716815372379243519562956

comment: $a_2=-48$, the root of $\chi_{26,2}$.

0.00236428002656855536683619961491

comment: $a_2$ is the first root of $\chi_{28,2}$ in increasing order, with $a_2\approx -18713.59859471915$.

0.00131542730871571614006541980065

comment: $a_2$ is the second root of $\chi_{28,2}$ in increasing order, with $a_2\approx 10433.59859471915$.

0.00675086417656698477174246013613

comment: $a_2$ is the first root of $\chi_{30,2}$ in increasing order, with $a_2\approx -17433.90502875288$.

0.0100997800473221939384580046955

comment: $a_2$ is the second root of $\chi_{30,2}$ in increasing order, with $a_2\approx 26073.90502875288$.

0.0320925558714875814394415038967

comment: $a_2$ is the first root of $\chi_{32,2}$ in increasing order, with $a_2\approx -31347.87172677239$.

0.0729994924629061249311411160214

comment: $a_2$ is the second root of $\chi_{32,2}$ in increasing order, with $a_2\approx 71307.87172677239$.

0.623070453157424716952166922143

comment: $a_2$ is the first root of $\chi_{34,2}$ in increasing order, with $a_2\approx -171359.4371321172$.

0.180637193843065066573942997006

comment: $a_2$ is the second root of $\chi_{34,2}$ in increasing order, with $a_2\approx 49679.43713211717$.

2.67742333549761657636239939753

comment: $a_2$ is the first root of $\chi_{36,2}$ in increasing order, with $a_2\approx -165109.1678324939$.

2.42584430383642443883211696130

comment: $a_2$ is the second root of $\chi_{36,2}$ in increasing order, with $a_2\approx -26808.00761087159$.

4.55969950662225591954959544292

comment: $a_2$ is the third root of $\chi_{36,2}$ in increasing order, with $a_2\approx 331573.1754433655$.

21.2742918225116448063549453159

comment: $a_2$ is the first root of $\chi_{38,2}$ in increasing order, with $a_2\approx -480411.7539742225$.

12.6655885373653394405955792016

comment: $a_2$ is the second root of $\chi_{38,2}$ in increasing order, with $a_2\approx 286011.7539742225$.

171.581094673690839287635257177

comment: $a_2$ is the first root of $\chi_{40,2}$ in increasing order, with $a_2\approx -799151.7631007462$.

190.225804606637761285194728054

comment: $a_2$ is the second root of $\chi_{40,2}$ in increasing order, with $a_2\approx 241487.7444577613$.

326.590585884552929043398261462

comment: $a_2$ is the third root of $\chi_{40,2}$ in increasing order, with $a_2\approx 1106520.018642985$.

Definition

For $f_{k,i}(q)=\sum_{n\geq1}a_nq^n$, the $i$th normalised Hecke eigenform in $S_k(\mathrm{SL}_2(\mathbb{Z}))$ [1] ordered by increasing embedded $a_2$, this table gives its Petersson norm $\langle f_{k,i},f_{k,i}\rangle$ [2].

Parameters

$k$

— weight ($k$ is even and $\dim S_k(\mathrm{SL}_2(\mathbb{Z}))>0$)

$i$

— position by increasing $a_2$ ($1\leq i\leq\dim S_k(\mathrm{SL}_2(\mathbb{Z}))$, with positions counted after sorting the eigenforms by increasing embedded $a_2$)

Formulas

(1)

Let $\alpha_p$ and $\beta_p$ be the roots of $X^2-a_pX+p^{k-1}$. With $L(\mathrm{Sym}^2 f,s)=\prod_p((1-\alpha_p^2p^{-s})(1-p^{k-1-s})(1-\beta_p^2p^{-s}))^{-1}$, $\langle f,f\rangle=(k-1)!L(\mathrm{Sym}^2 f,k)/(2^{2k-1}\pi^{k+1})$.

Comments

(2)

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. The $k=12$ entry is the Petersson norm of the modular discriminant $\Delta$, whose coefficients are the Ramanujan tau function.

(3)

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 Sage or PARI lists them in. Each entry gives its own $a_2$ and links to $\chi_{k,2}$ in the Hecke polynomial table, whose roots are the $a_2$ values in that weight.

Programs

(P1)

Sage

import numberdb.sage
import sage.symbolic.expression
import sage.rings.polynomial.laurent_polynomial_ring
from sage.modular.modform.constructor import Newforms
from sage.rings.complex_mpfr import ComplexField
k = 24; i = 1; bits = 330
C = ComplexField(bits)
forms = []
for f in Newforms(1, k, names='a'):
    a2 = f[2]
    for embedding, phi in enumerate(a2.parent().embeddings(C)):
        forms.append((phi(a2).real(), embedding, f))
forms.sort(key=lambda row: row[0])
forms[i - 1][2].petersson_norm(embedding=forms[i - 1][1], prec=bits)

(P2)

PARI/GP

default(realbitprecision, 330);
k = 24; i = 1;
mf = mfinit([1,k], 0);
F = mfeigenbasis(mf)[1];
a2 = mfcoefs(F, 2)[3];
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]), if (#A2 == 1, P, P[j,j])]), 1)[i][2]

Links

[1]

Wikipedia: cusp form

[2]

Wikipedia: Petersson inner product

[3]

SageMath reference: elements of modular forms spaces

[4]

PARI/GP reference: modular forms

Similar tables

Special values of the $L$-functions of level one cusp forms — stores critical values of the same eigenforms in the same ordering

Zeros of the $L$-functions of level one cusp forms — stores critical-line zero ordinates of the same eigenforms in the same ordering

Normalised Hecke eigenvalues of level one cusp forms — stores Deligne-normalised Hecke eigenvalues of the same eigenforms in the same ordering

Hecke polynomials of level one cusp forms — stores the characteristic polynomials whose $T_2$ roots order the eigenforms used here

$q$-expansion of the modular discriminant $\Delta$ — stores the coefficients of the weight $12$ eigenform whose Petersson norm is the $k=12$ row here

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds every even weight $12\leq k\leq40$ with $\dim S_k(\mathrm{SL}_2(\mathbb{Z}))>0$ and every eigenform in the increasing-$a_2$ ordering)

How they were obtained:

The generator evaluates Sage's petersson_norm [3] at $80$ and $120$ decimal working digits, passing explicit bit precisions to Sage's 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 [4], 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.