Properties

Label 966.6.a.a
Level $966$
Weight $6$
Character orbit 966.a
Self dual yes
Analytic conductor $154.931$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(154.930769939\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 4q^{2} - 9q^{3} + 16q^{4} - 54q^{5} + 36q^{6} + 49q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} - 9q^{3} + 16q^{4} - 54q^{5} + 36q^{6} + 49q^{7} - 64q^{8} + 81q^{9} + 216q^{10} + 564q^{11} - 144q^{12} + 476q^{13} - 196q^{14} + 486q^{15} + 256q^{16} - 1308q^{17} - 324q^{18} + 14q^{19} - 864q^{20} - 441q^{21} - 2256q^{22} + 529q^{23} + 576q^{24} - 209q^{25} - 1904q^{26} - 729q^{27} + 784q^{28} - 6702q^{29} - 1944q^{30} - 10258q^{31} - 1024q^{32} - 5076q^{33} + 5232q^{34} - 2646q^{35} + 1296q^{36} + 14078q^{37} - 56q^{38} - 4284q^{39} + 3456q^{40} - 2826q^{41} + 1764q^{42} + 14216q^{43} + 9024q^{44} - 4374q^{45} - 2116q^{46} - 9990q^{47} - 2304q^{48} + 2401q^{49} + 836q^{50} + 11772q^{51} + 7616q^{52} + 16698q^{53} + 2916q^{54} - 30456q^{55} - 3136q^{56} - 126q^{57} + 26808q^{58} + 19044q^{59} + 7776q^{60} + 5678q^{61} + 41032q^{62} + 3969q^{63} + 4096q^{64} - 25704q^{65} + 20304q^{66} - 39736q^{67} - 20928q^{68} - 4761q^{69} + 10584q^{70} + 60108q^{71} - 5184q^{72} - 7714q^{73} - 56312q^{74} + 1881q^{75} + 224q^{76} + 27636q^{77} + 17136q^{78} + 106472q^{79} - 13824q^{80} + 6561q^{81} + 11304q^{82} - 7662q^{83} - 7056q^{84} + 70632q^{85} - 56864q^{86} + 60318q^{87} - 36096q^{88} - 76536q^{89} + 17496q^{90} + 23324q^{91} + 8464q^{92} + 92322q^{93} + 39960q^{94} - 756q^{95} + 9216q^{96} - 19048q^{97} - 9604q^{98} + 45684q^{99} + O(q^{100}) \)

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
−4.00000 −9.00000 16.0000 −54.0000 36.0000 49.0000 −64.0000 81.0000 216.000
\(n\): 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 966.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 54 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(966))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( 9 + T \)
$5$ \( 54 + T \)
$7$ \( -49 + T \)
$11$ \( -564 + T \)
$13$ \( -476 + T \)
$17$ \( 1308 + T \)
$19$ \( -14 + T \)
$23$ \( -529 + T \)
$29$ \( 6702 + T \)
$31$ \( 10258 + T \)
$37$ \( -14078 + T \)
$41$ \( 2826 + T \)
$43$ \( -14216 + T \)
$47$ \( 9990 + T \)
$53$ \( -16698 + T \)
$59$ \( -19044 + T \)
$61$ \( -5678 + T \)
$67$ \( 39736 + T \)
$71$ \( -60108 + T \)
$73$ \( 7714 + T \)
$79$ \( -106472 + T \)
$83$ \( 7662 + T \)
$89$ \( 76536 + T \)
$97$ \( 19048 + T \)
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