Properties

Label 966.2.a.f
Level $966$
Weight $2$
Character orbit 966.a
Self dual yes
Analytic conductor $7.714$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} + 6 q^{11} + q^{12} + 2 q^{13} - q^{14} + q^{16} - 6 q^{17} - q^{18} + 2 q^{19} + q^{21} - 6 q^{22} + q^{23} - q^{24} - 5 q^{25} - 2 q^{26} + q^{27} + q^{28} - 6 q^{29} + 8 q^{31} - q^{32} + 6 q^{33} + 6 q^{34} + q^{36} + 8 q^{37} - 2 q^{38} + 2 q^{39} + 6 q^{41} - q^{42} + 2 q^{43} + 6 q^{44} - q^{46} + q^{48} + q^{49} + 5 q^{50} - 6 q^{51} + 2 q^{52} - 12 q^{53} - q^{54} - q^{56} + 2 q^{57} + 6 q^{58} + 8 q^{61} - 8 q^{62} + q^{63} + q^{64} - 6 q^{66} - 10 q^{67} - 6 q^{68} + q^{69} - q^{72} + 14 q^{73} - 8 q^{74} - 5 q^{75} + 2 q^{76} + 6 q^{77} - 2 q^{78} + 8 q^{79} + q^{81} - 6 q^{82} + 6 q^{83} + q^{84} - 2 q^{86} - 6 q^{87} - 6 q^{88} + 6 q^{89} + 2 q^{91} + q^{92} + 8 q^{93} - q^{96} - 10 q^{97} - q^{98} + 6 q^{99}+O(q^{100}) \) 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
−1.00000 1.00000 1.00000 0 −1.00000 1.00000 −1.00000 1.00000 0
\(n\): 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 966.2.a.f 1
3.b odd 2 1 2898.2.a.o 1
4.b odd 2 1 7728.2.a.c 1
7.b odd 2 1 6762.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.f 1 1.a even 1 1 trivial
2898.2.a.o 1 3.b odd 2 1
6762.2.a.h 1 7.b odd 2 1
7728.2.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 10 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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