Properties

Label 1.94.a.a
Level $1$
Weight $94$
Character orbit 1.a
Self dual yes
Analytic conductor $54.773$
Analytic rank $1$
Dimension $7$
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] = [1,94,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, 94, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 94);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 94 \)
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: \(54.7725430605\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{88}\cdot 3^{34}\cdot 5^{10}\cdot 7^{6}\cdot 13^{2}\cdot 19\cdot 23\cdot 31^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 6247918101970) q^{2} + (\beta_{2} - 31923367 \beta_1 - 52\!\cdots\!50) q^{3}+ \cdots + (77220 \beta_{6} - 74272308 \beta_{5} + \cdots + 51\!\cdots\!05) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 6247918101970) q^{2} + (\beta_{2} - 31923367 \beta_1 - 52\!\cdots\!50) q^{3}+ \cdots + ( - 13\!\cdots\!40 \beta_{6} + \cdots + 43\!\cdots\!06) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 43735426713792 q^{2} - 36\!\cdots\!84 q^{3}+ \cdots + 36\!\cdots\!11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 43735426713792 q^{2} - 36\!\cdots\!84 q^{3}+ \cdots + 30\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} + \cdots - 13\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 576\nu - 82 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 72\!\cdots\!11 \nu^{6} + \cdots - 37\!\cdots\!40 ) / 15\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44\!\cdots\!29 \nu^{6} + \cdots - 25\!\cdots\!28 ) / 15\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!13 \nu^{6} + \cdots - 13\!\cdots\!40 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47\!\cdots\!31 \nu^{6} + \cdots + 16\!\cdots\!40 ) / 38\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!85 \nu^{6} + \cdots - 29\!\cdots\!36 ) / 76\!\cdots\!28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 82 ) / 576 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 61939\beta_{2} - 4721797423044\beta _1 + 15212166611438802125090784970 ) / 331776 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 27 \beta_{6} - 864 \beta_{5} + 37744952 \beta_{4} - 1456222316363 \beta_{3} + \cdots - 44\!\cdots\!75 ) / 11943936 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4463654783831 \beta_{6} + \cdots + 34\!\cdots\!71 ) / 107495424 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40\!\cdots\!93 \beta_{6} + \cdots - 15\!\cdots\!33 ) / 17915904 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 13\!\cdots\!09 \beta_{6} + \cdots + 26\!\cdots\!21 ) / 107495424 \) 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
−3.01170e11
−2.45391e11
−9.86534e10
−1.98834e10
1.30998e11
2.59428e11
2.74671e11
−1.67226e14 1.22637e22 1.80609e28 −5.69707e32 −2.05080e36 3.48367e39 −1.36413e42 −8.52575e43 9.52696e46
1.2 −1.35097e14 −1.21702e22 8.34770e27 3.52234e32 1.64416e36 −7.46408e38 2.10186e41 −8.75419e43 −4.75858e46
1.3 −5.05764e13 2.16277e22 −7.34554e27 1.20105e32 −1.09385e36 −1.85159e39 8.72396e41 2.32102e44 −6.07451e45
1.4 −5.20495e12 −1.59625e22 −9.87643e27 −3.34454e32 8.30840e34 −2.46426e38 1.02954e41 1.91468e43 1.74082e45
1.5 8.17029e13 3.94920e21 −3.22816e27 2.56302e32 3.22661e35 2.42990e39 −1.07290e42 −2.20059e44 2.09406e46
1.6 1.55678e14 1.43559e22 1.43322e28 −3.85879e32 2.23491e36 −1.93687e39 6.89447e41 −2.95620e43 −6.00730e46
1.7 1.64459e14 −2.77426e22 1.71431e28 3.18897e32 −4.56250e36 −2.05341e39 1.19061e42 5.33996e44 5.24454e46
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 1.94.a.a 7
3.b odd 2 1 9.94.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.94.a.a 7 1.a even 1 1 trivial
9.94.a.b 7 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{94}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 43735426713792 T^{6} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$3$ \( T^{7} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 97\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 68\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 99\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 60\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 17\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 53\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 21\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 59\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 40\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 47\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 86\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 27\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 50\!\cdots\!72 \) Copy content Toggle raw display
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