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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Newspace parameters

Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1156.l (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.576919154604\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
Defining polynomial: \(x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
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

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

\(f(q)\) \(=\) \( q -\zeta_{34}^{10} q^{2} -\zeta_{34}^{3} q^{4} + ( -1 + \zeta_{34}^{8} ) q^{5} + \zeta_{34}^{13} q^{8} + \zeta_{34} q^{9} +O(q^{10})\) \( q -\zeta_{34}^{10} q^{2} -\zeta_{34}^{3} q^{4} + ( -1 + \zeta_{34}^{8} ) q^{5} + \zeta_{34}^{13} q^{8} + \zeta_{34} q^{9} + ( \zeta_{34} + \zeta_{34}^{10} ) q^{10} + ( -\zeta_{34}^{12} - \zeta_{34}^{14} ) q^{13} + \zeta_{34}^{6} q^{16} + \zeta_{34}^{6} q^{17} -\zeta_{34}^{11} q^{18} + ( \zeta_{34}^{3} - \zeta_{34}^{11} ) q^{20} + ( 1 - \zeta_{34}^{8} + \zeta_{34}^{16} ) q^{25} + ( -\zeta_{34}^{5} - \zeta_{34}^{7} ) q^{26} + ( -\zeta_{34}^{2} + \zeta_{34}^{4} ) q^{29} -\zeta_{34}^{16} q^{32} -\zeta_{34}^{16} q^{34} -\zeta_{34}^{4} q^{36} + ( -\zeta_{34}^{12} - \zeta_{34}^{15} ) q^{37} + ( -\zeta_{34}^{4} - \zeta_{34}^{13} ) q^{40} + ( \zeta_{34}^{5} - \zeta_{34}^{13} ) q^{41} + ( -\zeta_{34} + \zeta_{34}^{9} ) q^{45} -\zeta_{34}^{2} q^{49} + ( -\zeta_{34} + \zeta_{34}^{9} - \zeta_{34}^{10} ) q^{50} + ( -1 + \zeta_{34}^{15} ) q^{52} + ( -\zeta_{34}^{3} + \zeta_{34}^{16} ) q^{53} + ( \zeta_{34}^{12} - \zeta_{34}^{14} ) q^{58} + ( \zeta_{34}^{10} + \zeta_{34}^{11} ) q^{61} -\zeta_{34}^{9} q^{64} + ( \zeta_{34}^{3} + \zeta_{34}^{5} + \zeta_{34}^{12} + \zeta_{34}^{14} ) q^{65} -\zeta_{34}^{9} q^{68} + \zeta_{34}^{14} q^{72} + ( -\zeta_{34}^{5} + \zeta_{34}^{7} ) q^{73} + ( -\zeta_{34}^{5} - \zeta_{34}^{8} ) q^{74} + ( -\zeta_{34}^{6} + \zeta_{34}^{14} ) q^{80} + \zeta_{34}^{2} q^{81} + ( -\zeta_{34}^{6} - \zeta_{34}^{15} ) q^{82} + ( -\zeta_{34}^{6} + \zeta_{34}^{14} ) q^{85} + ( \zeta_{34}^{11} - \zeta_{34}^{14} ) q^{89} + ( \zeta_{34}^{2} + \zeta_{34}^{11} ) q^{90} + ( \zeta_{34}^{4} + \zeta_{34}^{15} ) q^{97} + \zeta_{34}^{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + q^{2} - q^{4} - 17q^{5} + q^{8} + q^{9} + O(q^{10}) \) \( 16q + q^{2} - q^{4} - 17q^{5} + q^{8} + q^{9} + 2q^{13} - q^{16} - q^{17} - q^{18} + 16q^{25} - 2q^{26} + q^{32} + q^{34} + q^{36} + q^{49} + q^{50} - 15q^{52} - 2q^{53} - q^{64} - q^{68} - q^{72} - q^{81} + 2q^{89} - q^{98} + O(q^{100}) \)

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.

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 990-1000
Embeddings: e.g. 1-3 or 1087.1
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

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 17 + 136 T + 680 T^{2} + 2380 T^{3} + 6188 T^{4} + 12376 T^{5} + 19448 T^{6} + 24310 T^{7} + 24310 T^{8} + 19448 T^{9} + 12376 T^{10} + 6188 T^{11} + 2380 T^{12} + 680 T^{13} + 136 T^{14} + 17 T^{15} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 + 8 T + 13 T^{2} - 15 T^{3} + 118 T^{4} - 59 T^{5} + 72 T^{6} - 2 T^{7} + T^{8} - 60 T^{9} + 30 T^{10} - 15 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$17$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( 17 + 119 T + 442 T^{2} + 935 T^{3} + 1122 T^{4} + 714 T^{5} + 204 T^{6} + 17 T^{7} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( 17 + 68 T + 34 T^{2} + 119 T^{4} - 221 T^{5} + 85 T^{8} + 17 T^{9} + 17 T^{12} + T^{16} \)
$41$ \( 17 + 17 T + 85 T^{2} - 102 T^{3} + 17 T^{4} + 255 T^{5} - 238 T^{7} + 51 T^{9} + T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( 1 - 8 T + 30 T^{2} - 2 T^{3} - T^{4} + 59 T^{5} + 140 T^{6} + 70 T^{7} + 35 T^{8} - 25 T^{9} - 4 T^{10} - 2 T^{11} - T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( 17 + 34 T + 17 T^{2} - 221 T^{3} + 85 T^{5} + 68 T^{6} + 119 T^{8} + 17 T^{11} + T^{16} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( 17 + 51 T - 85 T^{3} + 238 T^{4} + 17 T^{6} + 255 T^{7} + 102 T^{10} + 17 T^{13} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( 1 - 9 T + 64 T^{2} - 253 T^{3} + 594 T^{4} - 858 T^{5} + 786 T^{6} - 495 T^{7} + 256 T^{8} - 128 T^{9} + 64 T^{10} - 32 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$97$ \( 17 - 51 T + 85 T^{3} + 238 T^{4} + 17 T^{6} - 255 T^{7} + 102 T^{10} - 17 T^{13} + T^{16} \)
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