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

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

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 \(S_{6}^{\mathrm{new}}(\Gamma_0(966))\). Copy content Toggle raw display

Hecke characteristic polynomials

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