Properties

Label 1.114.a.a
Level $1$
Weight $114$
Character orbit 1.a
Self dual yes
Analytic conductor $80.863$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,114,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, 114, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 114);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 114 \)
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: \(80.8627478904\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} + \cdots - 66\!\cdots\!92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{144}\cdot 3^{48}\cdot 5^{19}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 19^{3}\cdot 23 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 552636039763781) q^{2} + (\beta_{2} - 53786200 \beta_1 - 11\!\cdots\!03) q^{3}+ \cdots + ( - \beta_{8} + 93 \beta_{7} - 209767 \beta_{6} + \cdots + 61\!\cdots\!67) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 552636039763781) q^{2} + (\beta_{2} - 53786200 \beta_1 - 11\!\cdots\!03) q^{3}+ \cdots + ( - 74\!\cdots\!48 \beta_{8} + \cdots + 35\!\cdots\!87) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 49\!\cdots\!32 q^{2}+ \cdots + 55\!\cdots\!77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 49\!\cdots\!32 q^{2}+ \cdots + 32\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} + \cdots - 66\!\cdots\!92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 48\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83\!\cdots\!59 \nu^{8} + \cdots - 21\!\cdots\!40 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!57 \nu^{8} + \cdots - 73\!\cdots\!48 ) / 45\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!33 \nu^{8} + \cdots - 49\!\cdots\!88 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 61\!\cdots\!93 \nu^{8} + \cdots - 12\!\cdots\!48 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!91 \nu^{8} + \cdots + 12\!\cdots\!76 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!73 \nu^{8} + \cdots - 85\!\cdots\!28 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!73 \nu^{8} + \cdots + 44\!\cdots\!28 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 5 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 233291\beta_{2} - 8626466785316074\beta _1 + 16733190786889893764973572264708832 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 54 \beta_{6} + 1827020 \beta_{5} - 49231023484 \beta_{4} + \cdots - 14\!\cdots\!25 ) / 110592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 832514701824 \beta_{8} + \cdots + 26\!\cdots\!65 ) / 331776 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 50\!\cdots\!20 \beta_{8} + \cdots - 33\!\cdots\!57 ) / 995328 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 45\!\cdots\!40 \beta_{8} + \cdots + 33\!\cdots\!81 ) / 331776 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 40\!\cdots\!20 \beta_{8} + \cdots - 23\!\cdots\!35 ) / 331776 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 35\!\cdots\!88 \beta_{8} + \cdots + 13\!\cdots\!55 ) / 995328 \) 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
−3.94089e15
−3.29839e15
−2.30101e15
−1.42245e15
5.24199e14
9.66339e14
2.64772e15
3.15688e15
3.66758e15
−1.88610e17 4.27041e26 2.51891e34 −5.60099e39 −8.05442e43 −6.14078e47 −2.79228e51 −6.39314e53 1.05640e57
1.2 −1.57770e17 −1.34800e27 1.45067e34 3.80102e39 2.12674e44 5.08173e47 −6.50350e50 9.95431e53 −5.99686e56
1.3 −1.09896e17 1.02220e27 1.69249e33 3.22293e39 −1.12336e44 −9.90814e46 9.55225e50 2.23221e53 −3.54186e56
1.4 −6.77248e16 −4.98640e26 −5.79795e33 −2.60538e39 3.37703e43 3.08065e47 1.09596e51 −5.73036e53 1.76449e56
1.5 2.57142e16 −9.84309e26 −9.72337e33 1.11425e39 −2.53107e43 −8.63240e47 −5.17060e50 1.47185e53 2.86521e55
1.6 4.69369e16 1.07652e27 −8.18152e33 −1.53406e39 5.05287e43 2.72254e47 −8.71436e50 3.37224e53 −7.20041e55
1.7 1.27643e17 −2.19803e26 5.90823e33 5.13288e39 −2.80563e43 6.06875e47 −5.71378e50 −7.73365e53 6.55179e56
1.8 1.52083e17 −1.34317e27 1.27446e34 −5.29218e39 −2.04274e44 7.47667e47 3.58920e50 9.82432e53 −8.04850e56
1.9 1.76597e17 8.20335e26 2.08018e34 −1.35461e39 1.44868e44 −1.04223e48 1.83964e51 −1.48729e53 −2.39220e56
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 1.114.a.a 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.114.a.a 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace is the entire newspace \(S_{114}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + \cdots - 91\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 10\!\cdots\!72 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 38\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 72\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 75\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 76\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 73\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 28\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 89\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 29\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 40\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 71\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 76\!\cdots\!92 \) Copy content Toggle raw display
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