Properties

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

Newform invariants

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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} - 2q^{13} + q^{14} + 2q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} - 2q^{20} - q^{21} - 4q^{22} - q^{23} - q^{24} - q^{25} - 2q^{26} - q^{27} + q^{28} - 2q^{29} + 2q^{30} - 8q^{31} + q^{32} + 4q^{33} - 6q^{34} - 2q^{35} + q^{36} + 6q^{37} + 4q^{38} + 2q^{39} - 2q^{40} - 6q^{41} - q^{42} - 4q^{43} - 4q^{44} - 2q^{45} - q^{46} - 8q^{47} - q^{48} + q^{49} - q^{50} + 6q^{51} - 2q^{52} + 6q^{53} - q^{54} + 8q^{55} + q^{56} - 4q^{57} - 2q^{58} + 4q^{59} + 2q^{60} - 10q^{61} - 8q^{62} + q^{63} + q^{64} + 4q^{65} + 4q^{66} + 4q^{67} - 6q^{68} + q^{69} - 2q^{70} - 8q^{71} + q^{72} - 6q^{73} + 6q^{74} + q^{75} + 4q^{76} - 4q^{77} + 2q^{78} - 2q^{80} + q^{81} - 6q^{82} - 12q^{83} - q^{84} + 12q^{85} - 4q^{86} + 2q^{87} - 4q^{88} + 2q^{89} - 2q^{90} - 2q^{91} - q^{92} + 8q^{93} - 8q^{94} - 8q^{95} - q^{96} + 10q^{97} + q^{98} - 4q^{99} + O(q^{100}) \)

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 −2.00000 −1.00000 1.00000 1.00000 1.00000 −2.00000
\(n\): 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.g 1
3.b odd 2 1 2898.2.a.i 1
4.b odd 2 1 7728.2.a.o 1
7.b odd 2 1 6762.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.g 1 1.a even 1 1 trivial
2898.2.a.i 1 3.b odd 2 1
6762.2.a.bl 1 7.b odd 2 1
7728.2.a.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5} + 2 \)
\( T_{11} + 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 2 + T \)
$7$ \( -1 + T \)
$11$ \( 4 + T \)
$13$ \( 2 + T \)
$17$ \( 6 + T \)
$19$ \( -4 + T \)
$23$ \( 1 + T \)
$29$ \( 2 + T \)
$31$ \( 8 + T \)
$37$ \( -6 + T \)
$41$ \( 6 + T \)
$43$ \( 4 + T \)
$47$ \( 8 + T \)
$53$ \( -6 + T \)
$59$ \( -4 + T \)
$61$ \( 10 + T \)
$67$ \( -4 + T \)
$71$ \( 8 + T \)
$73$ \( 6 + T \)
$79$ \( T \)
$83$ \( 12 + T \)
$89$ \( -2 + T \)
$97$ \( -10 + T \)
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