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Properties

Label 1160.4.a.c
Level $1160$
Weight $4$
Character orbit 1160.a
Self dual yes
Analytic conductor $68.442$
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] = [1160,4,Mod(1,1160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1160.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1160.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: \(68.4422156067\)
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: \( x^{7} - x^{6} - 121x^{5} + 38x^{4} + 4100x^{3} + 836x^{2} - 34400x - 6912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 5 q^{5} + ( - \beta_{5} + 9) q^{7} + (\beta_{6} - \beta_{5} + \beta_1 + 7) q^{9} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{11} + (\beta_{5} - \beta_{3} + 2 \beta_{2} + \cdots - 20) q^{13}+ \cdots + ( - 31 \beta_{6} + 37 \beta_{5} + \cdots - 727) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{3} - 35 q^{5} + 65 q^{7} + 54 q^{9} - 14 q^{11} - 137 q^{13} - 5 q^{15} - 181 q^{17} + 14 q^{19} - 84 q^{21} + 279 q^{23} + 175 q^{25} + 196 q^{27} - 203 q^{29} - 55 q^{31} - 656 q^{33} - 325 q^{35}+ \cdots - 5216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 121x^{5} + 38x^{4} + 4100x^{3} + 836x^{2} - 34400x - 6912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 182\nu^{5} - 716\nu^{4} + 13747\nu^{3} + 34126\nu^{2} - 140258\nu - 446310 ) / 26514 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -41\nu^{6} - 407\nu^{5} + 4825\nu^{4} + 44266\nu^{3} - 122864\nu^{2} - 998696\nu + 34668 ) / 53028 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 73\nu^{6} - 425\nu^{5} - 5573\nu^{4} + 28678\nu^{3} + 56656\nu^{2} - 288920\nu + 546156 ) / 53028 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{6} - 107\nu^{5} - 2563\nu^{4} + 9466\nu^{3} + 68780\nu^{2} - 160496\nu - 182412 ) / 17676 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\nu^{6} - 107\nu^{5} - 2563\nu^{4} + 9466\nu^{3} + 86456\nu^{2} - 178172\nu - 783396 ) / 17676 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{6} + 3\beta_{3} - 6\beta_{2} + 55\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 76\beta_{6} - 88\beta_{5} + 21\beta_{4} + 9\beta_{3} - 24\beta_{2} + 124\beta _1 + 1834 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 339\beta_{6} - 90\beta_{5} + 9\beta_{4} + 222\beta_{3} - 624\beta_{2} + 3529\beta _1 + 3354 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5821\beta_{6} - 6466\beta_{5} + 2382\beta_{4} + 801\beta_{3} - 3108\beta_{2} + 11587\beta _1 + 113884 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−7.91868
−5.82299
−3.80888
−0.200913
3.38396
6.60950
8.75801
0 −7.91868 0 −5.00000 0 30.1700 0 35.7055 0
1.2 0 −5.82299 0 −5.00000 0 15.6996 0 6.90719 0
1.3 0 −3.80888 0 −5.00000 0 −20.4315 0 −12.4924 0
1.4 0 −0.200913 0 −5.00000 0 17.3430 0 −26.9596 0
1.5 0 3.38396 0 −5.00000 0 4.48136 0 −15.5488 0
1.6 0 6.60950 0 −5.00000 0 −0.688237 0 16.6854 0
1.7 0 8.75801 0 −5.00000 0 18.4257 0 49.7027 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(29\) \( +1 \)

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 1160.4.a.c ✓ 7
4.b odd 2 1 2320.4.a.r 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.4.a.c ✓ 7 1.a even 1 1 trivial
2320.4.a.r 7 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - T_{3}^{6} - 121T_{3}^{5} + 38T_{3}^{4} + 4100T_{3}^{3} + 836T_{3}^{2} - 34400T_{3} - 6912 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1160))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - T^{6} + \cdots - 6912 \) Copy content Toggle raw display
$5$ \( (T + 5)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 65 T^{6} + \cdots - 9538128 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 4188627648 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 18394305376 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 2985572880 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 7768105053120 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 196426245750768 \) Copy content Toggle raw display
$29$ \( (T + 29)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 4923354983088 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 99\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 71\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 62\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 76\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 73\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 119827736937120 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 30\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 12\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 30\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
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