Properties

Label 966.4.a.b
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.9958450655\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} - 3q^{3} + 4q^{4} - 9q^{5} - 6q^{6} + 7q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} - 3q^{3} + 4q^{4} - 9q^{5} - 6q^{6} + 7q^{7} + 8q^{8} + 9q^{9} - 18q^{10} - 12q^{11} - 12q^{12} + 11q^{13} + 14q^{14} + 27q^{15} + 16q^{16} + 96q^{17} + 18q^{18} - 40q^{19} - 36q^{20} - 21q^{21} - 24q^{22} - 23q^{23} - 24q^{24} - 44q^{25} + 22q^{26} - 27q^{27} + 28q^{28} - 231q^{29} + 54q^{30} - 94q^{31} + 32q^{32} + 36q^{33} + 192q^{34} - 63q^{35} + 36q^{36} + 47q^{37} - 80q^{38} - 33q^{39} - 72q^{40} - 27q^{41} - 42q^{42} + 479q^{43} - 48q^{44} - 81q^{45} - 46q^{46} + 423q^{47} - 48q^{48} + 49q^{49} - 88q^{50} - 288q^{51} + 44q^{52} + 516q^{53} - 54q^{54} + 108q^{55} + 56q^{56} + 120q^{57} - 462q^{58} + 882q^{59} + 108q^{60} + 842q^{61} - 188q^{62} + 63q^{63} + 64q^{64} - 99q^{65} + 72q^{66} - 844q^{67} + 384q^{68} + 69q^{69} - 126q^{70} + 654q^{71} + 72q^{72} - 496q^{73} + 94q^{74} + 132q^{75} - 160q^{76} - 84q^{77} - 66q^{78} + 260q^{79} - 144q^{80} + 81q^{81} - 54q^{82} - 156q^{83} - 84q^{84} - 864q^{85} + 958q^{86} + 693q^{87} - 96q^{88} - 414q^{89} - 162q^{90} + 77q^{91} - 92q^{92} + 282q^{93} + 846q^{94} + 360q^{95} - 96q^{96} + 1343q^{97} + 98q^{98} - 108q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 −3.00000 4.00000 −9.00000 −6.00000 7.00000 8.00000 9.00000 −18.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 966.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.b 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( 9 + T \)
$7$ \( -7 + T \)
$11$ \( 12 + T \)
$13$ \( -11 + T \)
$17$ \( -96 + T \)
$19$ \( 40 + T \)
$23$ \( 23 + T \)
$29$ \( 231 + T \)
$31$ \( 94 + T \)
$37$ \( -47 + T \)
$41$ \( 27 + T \)
$43$ \( -479 + T \)
$47$ \( -423 + T \)
$53$ \( -516 + T \)
$59$ \( -882 + T \)
$61$ \( -842 + T \)
$67$ \( 844 + T \)
$71$ \( -654 + T \)
$73$ \( 496 + T \)
$79$ \( -260 + T \)
$83$ \( 156 + T \)
$89$ \( 414 + T \)
$97$ \( -1343 + T \)
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