Properties

Label 966.4.a.a
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

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{5} - 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{5} - 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + 12q^{10} + 48q^{11} + 12q^{12} + 38q^{13} - 14q^{14} - 18q^{15} + 16q^{16} + 114q^{17} - 18q^{18} + 56q^{19} - 24q^{20} + 21q^{21} - 96q^{22} - 23q^{23} - 24q^{24} - 89q^{25} - 76q^{26} + 27q^{27} + 28q^{28} - 162q^{29} + 36q^{30} - 16q^{31} - 32q^{32} + 144q^{33} - 228q^{34} - 42q^{35} + 36q^{36} - 46q^{37} - 112q^{38} + 114q^{39} + 48q^{40} - 342q^{41} - 42q^{42} + 248q^{43} + 192q^{44} - 54q^{45} + 46q^{46} - 24q^{47} + 48q^{48} + 49q^{49} + 178q^{50} + 342q^{51} + 152q^{52} + 426q^{53} - 54q^{54} - 288q^{55} - 56q^{56} + 168q^{57} + 324q^{58} - 852q^{59} - 72q^{60} + 338q^{61} + 32q^{62} + 63q^{63} + 64q^{64} - 228q^{65} - 288q^{66} + 488q^{67} + 456q^{68} - 69q^{69} + 84q^{70} + 336q^{71} - 72q^{72} + 362q^{73} + 92q^{74} - 267q^{75} + 224q^{76} + 336q^{77} - 228q^{78} + 1184q^{79} - 96q^{80} + 81q^{81} + 684q^{82} - 336q^{83} + 84q^{84} - 684q^{85} - 496q^{86} - 486q^{87} - 384q^{88} - 78q^{89} + 108q^{90} + 266q^{91} - 92q^{92} - 48q^{93} + 48q^{94} - 336q^{95} - 96q^{96} + 746q^{97} - 98q^{98} + 432q^{99} + O(q^{100}) \)

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 −6.00000 −6.00000 7.00000 −8.00000 9.00000 12.0000
\(n\): 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.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( 6 + T \)
$7$ \( -7 + T \)
$11$ \( -48 + T \)
$13$ \( -38 + T \)
$17$ \( -114 + T \)
$19$ \( -56 + T \)
$23$ \( 23 + T \)
$29$ \( 162 + T \)
$31$ \( 16 + T \)
$37$ \( 46 + T \)
$41$ \( 342 + T \)
$43$ \( -248 + T \)
$47$ \( 24 + T \)
$53$ \( -426 + T \)
$59$ \( 852 + T \)
$61$ \( -338 + T \)
$67$ \( -488 + T \)
$71$ \( -336 + T \)
$73$ \( -362 + T \)
$79$ \( -1184 + T \)
$83$ \( 336 + T \)
$89$ \( 78 + T \)
$97$ \( -746 + T \)
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