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Properties

Label 1156.1.l.a
Level $1156$
Weight $1$
Character orbit 1156.l
Analytic conductor $0.577$
Analytic rank $0$
Dimension $16$
Projective image $D_{34}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1156,1,Mod(67,1156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1156.67"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([17, 23])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1156.l (of order \(34\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.576919154604\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{34}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{34} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{34}^{10} q^{2} - \zeta_{34}^{3} q^{4} + (\zeta_{34}^{8} - 1) q^{5} + \zeta_{34}^{13} q^{8} + \zeta_{34} q^{9} + (\zeta_{34}^{10} + \zeta_{34}) q^{10} + ( - \zeta_{34}^{14} - \zeta_{34}^{12}) q^{13} + \cdots + \zeta_{34}^{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - q^{4} - 17 q^{5} + q^{8} + q^{9} + 2 q^{13} - q^{16} - q^{17} - q^{18} + 16 q^{25} - 2 q^{26} + q^{32} + q^{34} + q^{36} + q^{49} + q^{50} - 15 q^{52} - 2 q^{53} - q^{64} - q^{68}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(\zeta_{34}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.445738 − 0.895163i
0.602635 − 0.798017i
0.982973 + 0.183750i
0.273663 + 0.961826i
−0.739009 + 0.673696i
−0.932472 − 0.361242i
−0.0922684 − 0.995734i
0.850217 − 0.526432i
0.850217 + 0.526432i
−0.0922684 + 0.995734i
−0.932472 + 0.361242i
−0.739009 − 0.673696i
0.273663 − 0.961826i
0.982973 − 0.183750i
0.602635 + 0.798017i
−0.445738 + 0.895163i
−0.0922684 + 0.995734i 0 −0.982973 − 0.183750i −1.85022 + 0.526432i 0 0 0.273663 − 0.961826i −0.445738 − 0.895163i −0.353470 − 1.89090i
135.1 0.982973 + 0.183750i 0 0.932472 + 0.361242i −0.554262 − 0.895163i 0 0 0.850217 + 0.526432i 0.602635 − 0.798017i −0.380338 − 0.981767i
203.1 0.273663 − 0.961826i 0 −0.850217 − 0.526432i −0.907732 + 0.995734i 0 0 −0.739009 + 0.673696i 0.982973 + 0.183750i 0.709310 + 1.14558i
271.1 −0.932472 − 0.361242i 0 0.739009 + 0.673696i −1.60263 − 0.798017i 0 0 −0.445738 − 0.895163i 0.273663 + 0.961826i 1.20614 + 1.32307i
339.1 −0.445738 + 0.895163i 0 −0.602635 − 0.798017i −0.0675278 + 0.361242i 0 0 0.982973 − 0.183750i −0.739009 + 0.673696i −0.293271 − 0.221468i
407.1 0.850217 + 0.526432i 0 0.445738 + 0.895163i −1.98297 + 0.183750i 0 0 −0.0922684 + 0.995734i −0.932472 − 0.361242i −1.78269 − 0.887674i
475.1 0.602635 − 0.798017i 0 −0.273663 − 0.961826i −0.260991 − 0.673696i 0 0 −0.932472 − 0.361242i −0.0922684 − 0.995734i −0.694903 − 0.197717i
543.1 −0.739009 − 0.673696i 0 0.0922684 + 0.995734i −1.27366 + 0.961826i 0 0 0.602635 − 0.798017i 0.850217 − 0.526432i 1.58923 + 0.147263i
611.1 −0.739009 + 0.673696i 0 0.0922684 − 0.995734i −1.27366 − 0.961826i 0 0 0.602635 + 0.798017i 0.850217 + 0.526432i 1.58923 − 0.147263i
679.1 0.602635 + 0.798017i 0 −0.273663 + 0.961826i −0.260991 + 0.673696i 0 0 −0.932472 + 0.361242i −0.0922684 + 0.995734i −0.694903 + 0.197717i
747.1 0.850217 − 0.526432i 0 0.445738 − 0.895163i −1.98297 − 0.183750i 0 0 −0.0922684 − 0.995734i −0.932472 + 0.361242i −1.78269 + 0.887674i
815.1 −0.445738 − 0.895163i 0 −0.602635 + 0.798017i −0.0675278 − 0.361242i 0 0 0.982973 + 0.183750i −0.739009 − 0.673696i −0.293271 + 0.221468i
883.1 −0.932472 + 0.361242i 0 0.739009 − 0.673696i −1.60263 + 0.798017i 0 0 −0.445738 + 0.895163i 0.273663 − 0.961826i 1.20614 − 1.32307i
951.1 0.273663 + 0.961826i 0 −0.850217 + 0.526432i −0.907732 − 0.995734i 0 0 −0.739009 − 0.673696i 0.982973 − 0.183750i 0.709310 − 1.14558i
1019.1 0.982973 − 0.183750i 0 0.932472 − 0.361242i −0.554262 + 0.895163i 0 0 0.850217 − 0.526432i 0.602635 + 0.798017i −0.380338 + 0.981767i
1087.1 −0.0922684 − 0.995734i 0 −0.982973 + 0.183750i −1.85022 − 0.526432i 0 0 0.273663 + 0.961826i −0.445738 + 0.895163i −0.353470 + 1.89090i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 67.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
289.g even 34 1 inner
1156.l odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.l.a ✓ 16
4.b odd 2 1 CM 1156.1.l.a ✓ 16
289.g even 34 1 inner 1156.1.l.a ✓ 16
1156.l odd 34 1 inner 1156.1.l.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1156.1.l.a ✓ 16 1.a even 1 1 trivial
1156.1.l.a ✓ 16 4.b odd 2 1 CM
1156.1.l.a ✓ 16 289.g even 34 1 inner
1156.1.l.a ✓ 16 1156.l odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1156, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 17 T^{15} + \cdots + 17 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} + T^{15} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 17 T^{7} + \cdots + 17 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 17 T^{12} + \cdots + 17 \) Copy content Toggle raw display
$41$ \( T^{16} + 51 T^{9} + \cdots + 17 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} + 17 T^{11} + \cdots + 17 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + 17 T^{13} + \cdots + 17 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{16} - 17 T^{13} + \cdots + 17 \) Copy content Toggle raw display
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