Properties

Label 1.96.a.a
Level $1$
Weight $96$
Character orbit 1.a
Self dual yes
Analytic conductor $57.154$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,96,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 96, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 96);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 96 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.1535908815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{38}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 729457392285) q^{2} + ( - \beta_{2} + 20242501 \beta_1 - 11\!\cdots\!60) q^{3}+ \cdots + ( - 192 \beta_{7} + 121524 \beta_{6} + \cdots + 11\!\cdots\!17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 729457392285) q^{2} + ( - \beta_{2} + 20242501 \beta_1 - 11\!\cdots\!60) q^{3}+ \cdots + ( - 79\!\cdots\!36 \beta_{7} + \cdots + 38\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3}+ \cdots + 92\!\cdots\!36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + \cdots + 12\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 24\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 76\!\cdots\!51 \nu^{7} + \cdots - 69\!\cdots\!36 ) / 58\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\!\cdots\!49 \nu^{7} + \cdots - 43\!\cdots\!12 ) / 64\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!97 \nu^{7} + \cdots - 31\!\cdots\!68 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 60\!\cdots\!17 \nu^{7} + \cdots + 10\!\cdots\!52 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!57 \nu^{7} + \cdots + 49\!\cdots\!92 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!87 \nu^{7} + \cdots - 15\!\cdots\!72 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 3 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 62091\beta_{2} - 26761022262152\beta _1 + 65733123937758428360847400200 ) / 576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 295 \beta_{6} + 160002 \beta_{5} - 2004356999 \beta_{4} + 51851570017067 \beta_{3} + \cdots - 17\!\cdots\!04 ) / 13824 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3926523411123 \beta_{7} + \cdots + 92\!\cdots\!72 ) / 41472 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 26\!\cdots\!33 \beta_{7} + \cdots - 24\!\cdots\!48 ) / 124416 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24\!\cdots\!87 \beta_{7} + \cdots + 20\!\cdots\!00 ) / 41472 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22\!\cdots\!91 \beta_{7} + \cdots + 40\!\cdots\!88 ) / 41472 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48943e13
−1.46162e13
−6.56924e12
−1.67307e12
4.79736e12
6.86315e12
9.80521e12
1.62871e13
−3.58194e14 −7.12911e22 8.86885e28 −1.64420e33 2.55360e37 −1.35904e40 −1.75781e43 2.96152e45 5.88943e47
1.2 −3.51519e14 6.56090e22 8.39514e28 2.79946e33 −2.30628e37 5.53054e39 −1.55854e43 2.18364e45 −9.84062e47
1.3 −1.58391e14 9.80327e21 −1.45263e28 −7.31691e32 −1.55275e36 −1.37055e39 8.57536e42 −2.02479e45 1.15893e47
1.4 −4.08831e13 −7.46060e22 −3.79427e28 2.52173e33 3.05013e36 2.11876e40 3.17076e42 3.44515e45 −1.03096e47
1.5 1.14407e14 8.75958e22 −2.65251e28 −1.07741e33 1.00216e37 1.86215e40 −7.56680e42 5.55214e45 −1.23264e47
1.6 1.63986e14 6.46739e21 −1.27226e28 1.20020e33 1.06056e36 −2.14893e40 −8.58249e42 −2.07907e45 1.96817e47
1.7 2.34596e14 −5.48368e22 1.54210e28 −2.52565e33 −1.28645e37 4.80687e39 −5.67559e42 8.86182e44 −5.92506e47
1.8 3.90162e14 2.16924e22 1.12612e29 1.39914e33 8.46353e36 1.75361e40 2.84812e43 −1.65034e45 5.45891e47
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.96.a.a 8
3.b odd 2 1 9.96.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.96.a.a 8 1.a even 1 1 trivial
9.96.a.c 8 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{96}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5835659138280 T^{7} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 73\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 36\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 54\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 69\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 23\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
show more
show less