Properties

Label 959.1.t.a
Level $959$
Weight $1$
Character orbit 959.t
Analytic conductor $0.479$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 959 = 7 \cdot 137 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 959.t (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.478603347115\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{17}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{17} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{34}^{2} - \zeta_{34}^{7} ) q^{2} + ( \zeta_{34}^{4} - \zeta_{34}^{9} + \zeta_{34}^{14} ) q^{4} -\zeta_{34}^{7} q^{7} + ( \zeta_{34}^{4} + \zeta_{34}^{6} - \zeta_{34}^{11} + \zeta_{34}^{16} ) q^{8} + \zeta_{34}^{6} q^{9} +O(q^{10})\) \( q + ( \zeta_{34}^{2} - \zeta_{34}^{7} ) q^{2} + ( \zeta_{34}^{4} - \zeta_{34}^{9} + \zeta_{34}^{14} ) q^{4} -\zeta_{34}^{7} q^{7} + ( \zeta_{34}^{4} + \zeta_{34}^{6} - \zeta_{34}^{11} + \zeta_{34}^{16} ) q^{8} + \zeta_{34}^{6} q^{9} + ( -\zeta_{34}^{5} - \zeta_{34}^{13} ) q^{11} + ( -\zeta_{34}^{9} + \zeta_{34}^{14} ) q^{14} + ( -\zeta_{34} + \zeta_{34}^{6} + \zeta_{34}^{8} - \zeta_{34}^{11} - \zeta_{34}^{13} ) q^{16} + ( \zeta_{34}^{8} - \zeta_{34}^{13} ) q^{18} + ( -\zeta_{34}^{3} - \zeta_{34}^{7} + \zeta_{34}^{12} - \zeta_{34}^{15} ) q^{22} + ( 1 + \zeta_{34}^{2} ) q^{23} + \zeta_{34}^{8} q^{25} + ( \zeta_{34}^{4} - \zeta_{34}^{11} + \zeta_{34}^{16} ) q^{28} + ( -\zeta_{34}^{5} + \zeta_{34}^{14} ) q^{29} + ( -\zeta_{34} - \zeta_{34}^{3} + \zeta_{34}^{8} + \zeta_{34}^{10} - \zeta_{34}^{13} - \zeta_{34}^{15} ) q^{32} + ( -\zeta_{34}^{3} + \zeta_{34}^{10} - \zeta_{34}^{15} ) q^{36} + ( \zeta_{34}^{8} - \zeta_{34}^{9} ) q^{37} + ( -\zeta_{34} - \zeta_{34}^{3} ) q^{43} + ( 1 + \zeta_{34}^{2} - \zeta_{34}^{5} - \zeta_{34}^{9} + \zeta_{34}^{10} + \zeta_{34}^{14} ) q^{44} + ( \zeta_{34}^{2} + \zeta_{34}^{4} - \zeta_{34}^{7} - \zeta_{34}^{9} ) q^{46} + \zeta_{34}^{14} q^{49} + ( \zeta_{34}^{10} - \zeta_{34}^{15} ) q^{50} + ( -\zeta_{34}^{5} + \zeta_{34}^{16} ) q^{53} + ( -\zeta_{34} + \zeta_{34}^{6} - \zeta_{34}^{11} - \zeta_{34}^{13} ) q^{56} + ( \zeta_{34}^{4} - \zeta_{34}^{7} + \zeta_{34}^{12} + \zeta_{34}^{16} ) q^{58} -\zeta_{34}^{13} q^{63} + ( 1 - \zeta_{34}^{3} - \zeta_{34}^{5} + \zeta_{34}^{8} + \zeta_{34}^{10} + \zeta_{34}^{12} - \zeta_{34}^{15} ) q^{64} + ( -\zeta_{34} - \zeta_{34}^{13} ) q^{67} + ( 1 + \zeta_{34}^{16} ) q^{71} + ( 1 - \zeta_{34}^{5} + \zeta_{34}^{10} + \zeta_{34}^{12} ) q^{72} + ( \zeta_{34}^{10} - \zeta_{34}^{11} - \zeta_{34}^{15} + \zeta_{34}^{16} ) q^{74} + ( -\zeta_{34}^{3} + \zeta_{34}^{12} ) q^{77} + ( \zeta_{34}^{12} + \zeta_{34}^{16} ) q^{79} + \zeta_{34}^{12} q^{81} + ( -\zeta_{34}^{3} - \zeta_{34}^{5} + \zeta_{34}^{8} + \zeta_{34}^{10} ) q^{86} + ( 1 + \zeta_{34}^{2} + \zeta_{34}^{4} - \zeta_{34}^{7} - \zeta_{34}^{9} - \zeta_{34}^{11} + \zeta_{34}^{12} + \zeta_{34}^{16} ) q^{88} + ( \zeta_{34}^{4} + \zeta_{34}^{6} - \zeta_{34}^{9} - \zeta_{34}^{11} + \zeta_{34}^{14} + \zeta_{34}^{16} ) q^{92} + ( \zeta_{34}^{4} + \zeta_{34}^{16} ) q^{98} + ( \zeta_{34}^{2} - \zeta_{34}^{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{2} - 3q^{4} - q^{7} - 4q^{8} - q^{9} + O(q^{10}) \) \( 16q - 2q^{2} - 3q^{4} - q^{7} - 4q^{8} - q^{9} - 2q^{11} - 2q^{14} - 5q^{16} - 2q^{18} - 4q^{22} + 15q^{23} - q^{25} - 3q^{28} - 2q^{29} - 6q^{32} - 3q^{36} - 2q^{37} - 2q^{43} + 11q^{44} - 4q^{46} - q^{49} - 2q^{50} - 2q^{53} - 4q^{56} - 4q^{58} - q^{63} + 10q^{64} - 2q^{67} + 15q^{71} + 13q^{72} - 4q^{74} - 2q^{77} - 2q^{79} - q^{81} - 4q^{86} + 9q^{88} - 6q^{92} - 2q^{98} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/959\mathbb{Z}\right)^\times\).

\(n\) \(414\) \(549\)
\(\chi(n)\) \(\zeta_{34}^{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
−0.932472 0.361242i
−0.739009 + 0.673696i
0.273663 + 0.961826i
0.982973 + 0.183750i
−0.739009 0.673696i
0.982973 0.183750i
0.602635 0.798017i
0.602635 + 0.798017i
0.850217 + 0.526432i
−0.0922684 + 0.995734i
−0.932472 + 0.361242i
−0.0922684 0.995734i
0.273663 0.961826i
−0.445738 + 0.895163i
0.850217 0.526432i
−0.445738 0.895163i
−0.111208 + 1.20013i 0 −0.444966 0.0831786i 0 0 −0.850217 + 0.526432i −0.180529 + 0.634493i −0.602635 + 0.798017i 0
153.1 0.538007 0.100571i 0 −0.653136 + 0.253026i 0 0 0.445738 + 0.895163i −0.791290 + 0.489946i −0.273663 + 0.961826i 0
209.1 0.0822551 + 0.165190i 0 0.582113 0.770842i 0 0 0.932472 0.361242i 0.356612 + 0.0666624i 0.0922684 + 0.995734i 0
461.1 0.658809 0.600584i 0 −0.0189399 + 0.204394i 0 0 −0.273663 0.961826i 0.647513 + 0.857445i 0.445738 + 0.895163i 0
608.1 0.538007 + 0.100571i 0 −0.653136 0.253026i 0 0 0.445738 0.895163i −0.791290 0.489946i −0.273663 0.961826i 0
622.1 0.658809 + 0.600584i 0 −0.0189399 0.204394i 0 0 −0.273663 + 0.961826i 0.647513 0.857445i 0.445738 0.895163i 0
636.1 −1.25664 0.778076i 0 0.527993 + 1.06035i 0 0 −0.982973 + 0.183750i 0.0251661 0.271585i 0.739009 + 0.673696i 0
671.1 −1.25664 + 0.778076i 0 0.527993 1.06035i 0 0 −0.982973 0.183750i 0.0251661 + 0.271585i 0.739009 0.673696i 0
741.1 1.18475 + 1.56886i 0 −0.784029 + 2.75558i 0 0 0.739009 + 0.673696i −3.41880 + 1.32445i −0.982973 0.183750i 0
804.1 −1.58561 + 0.614268i 0 1.39782 1.27428i 0 0 −0.602635 + 0.798017i −0.675694 + 1.35698i −0.850217 0.526432i 0
818.1 −0.111208 1.20013i 0 −0.444966 + 0.0831786i 0 0 −0.850217 0.526432i −0.180529 0.634493i −0.602635 0.798017i 0
860.1 −1.58561 0.614268i 0 1.39782 + 1.27428i 0 0 −0.602635 0.798017i −0.675694 1.35698i −0.850217 + 0.526432i 0
881.1 0.0822551 0.165190i 0 0.582113 + 0.770842i 0 0 0.932472 + 0.361242i 0.356612 0.0666624i 0.0922684 0.995734i 0
895.1 −0.510366 1.79375i 0 −2.10685 + 1.30451i 0 0 0.0922684 0.995734i 2.03702 + 1.85699i 0.932472 0.361242i 0
937.1 1.18475 1.56886i 0 −0.784029 2.75558i 0 0 0.739009 0.673696i −3.41880 1.32445i −0.982973 + 0.183750i 0
944.1 −0.510366 + 1.79375i 0 −2.10685 1.30451i 0 0 0.0922684 + 0.995734i 2.03702 1.85699i 0.932472 + 0.361242i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 944.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
137.e even 17 1 inner
959.t odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 959.1.t.a 16
7.b odd 2 1 CM 959.1.t.a 16
137.e even 17 1 inner 959.1.t.a 16
959.t odd 34 1 inner 959.1.t.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
959.1.t.a 16 1.a even 1 1 trivial
959.1.t.a 16 7.b odd 2 1 CM
959.1.t.a 16 137.e even 17 1 inner
959.1.t.a 16 959.t odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(959, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
$11$ \( 1 + 9 T + 13 T^{2} - 36 T^{3} + 33 T^{4} + 25 T^{5} + 140 T^{6} + 70 T^{7} + 154 T^{8} + 77 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$13$ \( T^{16} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( 1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16} \)
$29$ \( 1 - 8 T + 13 T^{2} + 15 T^{3} + 118 T^{4} + 59 T^{5} + 72 T^{6} + 2 T^{7} + T^{8} + 60 T^{9} + 30 T^{10} + 15 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( ( 1 - 4 T - 10 T^{2} + 10 T^{3} + 15 T^{4} - 6 T^{5} - 7 T^{6} + T^{7} + T^{8} )^{2} \)
$41$ \( T^{16} \)
$43$ \( 1 + 9 T + 13 T^{2} - 36 T^{3} + 33 T^{4} + 25 T^{5} + 140 T^{6} + 70 T^{7} + 154 T^{8} + 77 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( T^{16} \)
$67$ \( 1 - 8 T + 13 T^{2} + 15 T^{3} + 118 T^{4} + 59 T^{5} + 72 T^{6} + 2 T^{7} + T^{8} + 60 T^{9} + 30 T^{10} + 15 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$71$ \( 1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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