Properties

Label 1096.1.u.a
Level $1096$
Weight $1$
Character orbit 1096.u
Analytic conductor $0.547$
Analytic rank $0$
Dimension $16$
Projective image $D_{34}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1096 = 2^{3} \cdot 137 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1096.u (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.546975253846\)
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.

\(f(q)\) \(=\) \( q + \zeta_{34} q^{2} + ( \zeta_{34}^{9} - \zeta_{34}^{15} ) q^{3} + \zeta_{34}^{2} q^{4} + ( \zeta_{34}^{10} - \zeta_{34}^{16} ) q^{6} + \zeta_{34}^{3} q^{8} + ( -\zeta_{34} + \zeta_{34}^{7} - \zeta_{34}^{13} ) q^{9} +O(q^{10})\) \( q + \zeta_{34} q^{2} + ( \zeta_{34}^{9} - \zeta_{34}^{15} ) q^{3} + \zeta_{34}^{2} q^{4} + ( \zeta_{34}^{10} - \zeta_{34}^{16} ) q^{6} + \zeta_{34}^{3} q^{8} + ( -\zeta_{34} + \zeta_{34}^{7} - \zeta_{34}^{13} ) q^{9} + ( -\zeta_{34}^{6} + \zeta_{34}^{15} ) q^{11} + ( 1 + \zeta_{34}^{11} ) q^{12} + \zeta_{34}^{4} q^{16} + ( -\zeta_{34} + \zeta_{34}^{10} ) q^{17} + ( -\zeta_{34}^{2} + \zeta_{34}^{8} - \zeta_{34}^{14} ) q^{18} + ( -\zeta_{34}^{5} - \zeta_{34}^{11} ) q^{19} + ( -\zeta_{34}^{7} + \zeta_{34}^{16} ) q^{22} + ( \zeta_{34} + \zeta_{34}^{12} ) q^{24} + \zeta_{34}^{15} q^{25} + ( \zeta_{34}^{5} - \zeta_{34}^{10} - \zeta_{34}^{11} + \zeta_{34}^{16} ) q^{27} + \zeta_{34}^{5} q^{32} + ( -\zeta_{34}^{4} - \zeta_{34}^{7} + \zeta_{34}^{13} - \zeta_{34}^{15} ) q^{33} + ( -\zeta_{34}^{2} + \zeta_{34}^{11} ) q^{34} + ( -\zeta_{34}^{3} + \zeta_{34}^{9} - \zeta_{34}^{15} ) q^{36} + ( -\zeta_{34}^{6} - \zeta_{34}^{12} ) q^{38} + ( \zeta_{34}^{5} + \zeta_{34}^{12} ) q^{41} + ( \zeta_{34}^{4} - \zeta_{34}^{12} ) q^{43} + ( -1 - \zeta_{34}^{8} ) q^{44} + ( \zeta_{34}^{2} + \zeta_{34}^{13} ) q^{48} -\zeta_{34}^{5} q^{49} + \zeta_{34}^{16} q^{50} + ( -\zeta_{34}^{2} + \zeta_{34}^{8} - \zeta_{34}^{10} + \zeta_{34}^{16} ) q^{51} + ( -1 + \zeta_{34}^{6} - \zeta_{34}^{11} - \zeta_{34}^{12} ) q^{54} + ( -\zeta_{34}^{9} - \zeta_{34}^{14} ) q^{57} + ( \zeta_{34}^{6} + \zeta_{34}^{8} ) q^{59} + \zeta_{34}^{6} q^{64} + ( -\zeta_{34}^{5} - \zeta_{34}^{8} + \zeta_{34}^{14} - \zeta_{34}^{16} ) q^{66} + ( -\zeta_{34}^{8} + \zeta_{34}^{14} ) q^{67} + ( -\zeta_{34}^{3} + \zeta_{34}^{12} ) q^{68} + ( -\zeta_{34}^{4} + \zeta_{34}^{10} - \zeta_{34}^{16} ) q^{72} + ( \zeta_{34} + \zeta_{34}^{11} ) q^{73} + ( -\zeta_{34}^{7} + \zeta_{34}^{13} ) q^{75} + ( -\zeta_{34}^{7} - \zeta_{34}^{13} ) q^{76} + ( \zeta_{34}^{2} + \zeta_{34}^{3} - \zeta_{34}^{8} - \zeta_{34}^{9} + \zeta_{34}^{14} ) q^{81} + ( \zeta_{34}^{6} + \zeta_{34}^{13} ) q^{82} + ( -\zeta_{34}^{9} - \zeta_{34}^{14} ) q^{83} + ( \zeta_{34}^{5} - \zeta_{34}^{13} ) q^{86} + ( -\zeta_{34} - \zeta_{34}^{9} ) q^{88} + ( \zeta_{34}^{7} + \zeta_{34}^{8} ) q^{89} + ( \zeta_{34}^{3} + \zeta_{34}^{14} ) q^{96} + ( 1 - \zeta_{34}^{4} ) q^{97} -\zeta_{34}^{6} q^{98} + ( -\zeta_{34}^{2} - \zeta_{34}^{5} + \zeta_{34}^{7} + \zeta_{34}^{11} - \zeta_{34}^{13} - \zeta_{34}^{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + q^{2} - q^{4} + q^{8} - q^{9} + O(q^{10}) \) \( 16q + q^{2} - q^{4} + q^{8} - q^{9} + 2q^{11} + 17q^{12} - q^{16} - 2q^{17} + q^{18} - 2q^{19} - 2q^{22} + q^{25} + q^{32} + 2q^{34} - q^{36} + 2q^{38} - 15q^{44} - q^{49} - q^{50} - 17q^{54} - 2q^{59} - q^{64} - 2q^{68} + q^{72} + 2q^{73} - 2q^{76} - q^{81} - 2q^{88} + 17q^{97} + q^{98} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1096\mathbb{Z}\right)^\times\).

\(n\) \(549\) \(823\) \(825\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{34}^{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.932472 0.361242i
−0.932472 + 0.361242i
0.850217 + 0.526432i
−0.445738 0.895163i
0.602635 + 0.798017i
0.982973 0.183750i
0.273663 + 0.961826i
0.273663 0.961826i
−0.739009 0.673696i
−0.739009 + 0.673696i
−0.0922684 + 0.995734i
0.850217 0.526432i
0.602635 0.798017i
−0.445738 + 0.895163i
0.982973 + 0.183750i
−0.0922684 0.995734i
−0.932472 0.361242i 1.72198 0.489946i 0.739009 + 0.673696i 0 −1.78269 0.165190i 0 −0.445738 0.895163i 1.87496 1.16092i 0
155.1 −0.932472 + 0.361242i 1.72198 + 0.489946i 0.739009 0.673696i 0 −1.78269 + 0.165190i 0 −0.445738 + 0.895163i 1.87496 + 1.16092i 0
323.1 0.850217 + 0.526432i 0.719401 1.85699i 0.445738 + 0.895163i 0 1.58923 1.20013i 0 −0.0922684 + 0.995734i −2.19186 1.99815i 0
339.1 −0.445738 0.895163i 0.247582 0.271585i −0.602635 + 0.798017i 0 −0.353470 0.100571i 0 0.982973 + 0.183750i 0.0798070 + 0.861255i 0
355.1 0.602635 + 0.798017i −0.719401 0.0666624i −0.273663 + 0.961826i 0 −0.380338 0.614268i 0 −0.932472 + 0.361242i −0.469879 0.0878355i 0
395.1 0.982973 0.183750i 0.840204 0.634493i 0.932472 0.361242i 0 0.709310 0.778076i 0 0.850217 0.526432i 0.0296988 0.104380i 0
475.1 0.273663 + 0.961826i −0.247582 1.32445i −0.850217 + 0.526432i 0 1.20614 0.600584i 0 −0.739009 0.673696i −0.760397 + 0.294579i 0
563.1 0.273663 0.961826i −0.247582 + 1.32445i −0.850217 0.526432i 0 1.20614 + 0.600584i 0 −0.739009 + 0.673696i −0.760397 0.294579i 0
611.1 −0.739009 0.673696i −0.840204 1.35698i 0.0922684 + 0.995734i 0 −0.293271 + 1.56886i 0 0.602635 0.798017i −0.689703 + 1.38511i 0
635.1 −0.739009 + 0.673696i −0.840204 + 1.35698i 0.0922684 0.995734i 0 −0.293271 1.56886i 0 0.602635 + 0.798017i −0.689703 1.38511i 0
651.1 −0.0922684 + 0.995734i −1.72198 + 0.857445i −0.982973 0.183750i 0 −0.694903 1.79375i 0 0.273663 0.961826i 1.62738 2.15499i 0
699.1 0.850217 0.526432i 0.719401 + 1.85699i 0.445738 0.895163i 0 1.58923 + 1.20013i 0 −0.0922684 0.995734i −2.19186 + 1.99815i 0
707.1 0.602635 0.798017i −0.719401 + 0.0666624i −0.273663 0.961826i 0 −0.380338 + 0.614268i 0 −0.932472 0.361242i −0.469879 + 0.0878355i 0
763.1 −0.445738 + 0.895163i 0.247582 + 0.271585i −0.602635 0.798017i 0 −0.353470 + 0.100571i 0 0.982973 0.183750i 0.0798070 0.861255i 0
899.1 0.982973 + 0.183750i 0.840204 + 0.634493i 0.932472 + 0.361242i 0 0.709310 + 0.778076i 0 0.850217 + 0.526432i 0.0296988 + 0.104380i 0
963.1 −0.0922684 0.995734i −1.72198 0.857445i −0.982973 + 0.183750i 0 −0.694903 + 1.79375i 0 0.273663 + 0.961826i 1.62738 + 2.15499i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 963.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
137.f even 34 1 inner
1096.u odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1096.1.u.a 16
8.d odd 2 1 CM 1096.1.u.a 16
137.f even 34 1 inner 1096.1.u.a 16
1096.u odd 34 1 inner 1096.1.u.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1096.1.u.a 16 1.a even 1 1 trivial
1096.1.u.a 16 8.d odd 2 1 CM
1096.1.u.a 16 137.f even 34 1 inner
1096.1.u.a 16 1096.u odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1096, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \)
$3$ \( 17 - 34 T + 17 T^{2} + 221 T^{3} - 85 T^{5} + 68 T^{6} + 119 T^{8} - 17 T^{11} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( T^{16} \)
$11$ \( 1 + 8 T + 30 T^{2} + 2 T^{3} - T^{4} - 59 T^{5} + 140 T^{6} - 70 T^{7} + 35 T^{8} + 25 T^{9} - 4 T^{10} + 2 T^{11} - T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$13$ \( T^{16} \)
$17$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$19$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( 17 + 204 T^{2} + 714 T^{4} + 1122 T^{6} + 935 T^{8} + 442 T^{10} + 119 T^{12} + 17 T^{14} + T^{16} \)
$43$ \( 17 + 17 T + 85 T^{2} - 102 T^{3} + 17 T^{4} + 255 T^{5} - 238 T^{7} + 51 T^{9} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$61$ \( T^{16} \)
$67$ \( 17 + 17 T + 85 T^{2} - 102 T^{3} + 17 T^{4} + 255 T^{5} - 238 T^{7} + 51 T^{9} + T^{16} \)
$71$ \( T^{16} \)
$73$ \( 1 + 8 T + 30 T^{2} + 2 T^{3} - T^{4} - 59 T^{5} + 140 T^{6} - 70 T^{7} + 35 T^{8} + 25 T^{9} - 4 T^{10} + 2 T^{11} - T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( 17 + 68 T + 34 T^{2} + 119 T^{4} - 221 T^{5} + 85 T^{8} + 17 T^{9} + 17 T^{12} + T^{16} \)
$89$ \( 17 + 17 T + 85 T^{2} - 102 T^{3} + 17 T^{4} + 255 T^{5} - 238 T^{7} + 51 T^{9} + T^{16} \)
$97$ \( 17 - 136 T + 680 T^{2} - 2380 T^{3} + 6188 T^{4} - 12376 T^{5} + 19448 T^{6} - 24310 T^{7} + 24310 T^{8} - 19448 T^{9} + 12376 T^{10} - 6188 T^{11} + 2380 T^{12} - 680 T^{13} + 136 T^{14} - 17 T^{15} + T^{16} \)
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