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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} + 6q^{11} + q^{12} + 2q^{13} - q^{14} + q^{16} - 6q^{17} - q^{18} + 2q^{19} + q^{21} - 6q^{22} + q^{23} - q^{24} - 5q^{25} - 2q^{26} + q^{27} + q^{28} - 6q^{29} + 8q^{31} - q^{32} + 6q^{33} + 6q^{34} + q^{36} + 8q^{37} - 2q^{38} + 2q^{39} + 6q^{41} - q^{42} + 2q^{43} + 6q^{44} - q^{46} + q^{48} + q^{49} + 5q^{50} - 6q^{51} + 2q^{52} - 12q^{53} - q^{54} - q^{56} + 2q^{57} + 6q^{58} + 8q^{61} - 8q^{62} + q^{63} + q^{64} - 6q^{66} - 10q^{67} - 6q^{68} + q^{69} - q^{72} + 14q^{73} - 8q^{74} - 5q^{75} + 2q^{76} + 6q^{77} - 2q^{78} + 8q^{79} + q^{81} - 6q^{82} + 6q^{83} + q^{84} - 2q^{86} - 6q^{87} - 6q^{88} + 6q^{89} + 2q^{91} + q^{92} + 8q^{93} - q^{96} - 10q^{97} - q^{98} + 6q^{99} + O(q^{100}) \)

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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5} \)
\( T_{11} - 6 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

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