Properties

Label 966.2.g.c
Level $966$
Weight $2$
Character orbit 966.g
Analytic conductor $7.714$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \(x^{4} - 3 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + \beta_{3} q^{7} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + \beta_{3} q^{7} - q^{8} - q^{9} -2 \beta_{3} q^{11} -\beta_{1} q^{12} + 2 \beta_{1} q^{13} -\beta_{3} q^{14} + q^{16} -2 \beta_{2} q^{17} + q^{18} -2 \beta_{2} q^{19} + \beta_{2} q^{21} + 2 \beta_{3} q^{22} + ( 4 - \beta_{3} ) q^{23} + \beta_{1} q^{24} -5 q^{25} -2 \beta_{1} q^{26} + \beta_{1} q^{27} + \beta_{3} q^{28} + 6 q^{29} -10 \beta_{1} q^{31} - q^{32} -2 \beta_{2} q^{33} + 2 \beta_{2} q^{34} - q^{36} + 2 \beta_{2} q^{38} + 2 q^{39} -12 \beta_{1} q^{41} -\beta_{2} q^{42} -2 \beta_{3} q^{43} -2 \beta_{3} q^{44} + ( -4 + \beta_{3} ) q^{46} + 6 \beta_{1} q^{47} -\beta_{1} q^{48} -7 q^{49} + 5 q^{50} + 2 \beta_{3} q^{51} + 2 \beta_{1} q^{52} -\beta_{1} q^{54} -\beta_{3} q^{56} + 2 \beta_{3} q^{57} -6 q^{58} -4 \beta_{2} q^{61} + 10 \beta_{1} q^{62} -\beta_{3} q^{63} + q^{64} + 2 \beta_{2} q^{66} + 2 \beta_{3} q^{67} -2 \beta_{2} q^{68} + ( -4 \beta_{1} - \beta_{2} ) q^{69} -8 q^{71} + q^{72} -4 \beta_{1} q^{73} + 5 \beta_{1} q^{75} -2 \beta_{2} q^{76} + 14 q^{77} -2 q^{78} + 2 \beta_{3} q^{79} + q^{81} + 12 \beta_{1} q^{82} -6 \beta_{2} q^{83} + \beta_{2} q^{84} + 2 \beta_{3} q^{86} -6 \beta_{1} q^{87} + 2 \beta_{3} q^{88} -2 \beta_{2} q^{89} -2 \beta_{2} q^{91} + ( 4 - \beta_{3} ) q^{92} -10 q^{93} -6 \beta_{1} q^{94} + \beta_{1} q^{96} -2 \beta_{2} q^{97} + 7 q^{98} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + 4q^{16} + 4q^{18} + 16q^{23} - 20q^{25} + 24q^{29} - 4q^{32} - 4q^{36} + 8q^{39} - 16q^{46} - 28q^{49} + 20q^{50} - 24q^{58} + 4q^{64} - 32q^{71} + 4q^{72} + 56q^{77} - 8q^{78} + 4q^{81} + 16q^{92} - 40q^{93} + 28q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 3 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 5 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
643.1
−1.32288 + 0.500000i
1.32288 + 0.500000i
1.32288 0.500000i
−1.32288 0.500000i
−1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
643.2 −1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
643.3 −1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
643.4 −1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.g.c 4
3.b odd 2 1 2898.2.g.c 4
7.b odd 2 1 inner 966.2.g.c 4
21.c even 2 1 2898.2.g.c 4
23.b odd 2 1 inner 966.2.g.c 4
69.c even 2 1 2898.2.g.c 4
161.c even 2 1 inner 966.2.g.c 4
483.c odd 2 1 2898.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.c 4 1.a even 1 1 trivial
966.2.g.c 4 7.b odd 2 1 inner
966.2.g.c 4 23.b odd 2 1 inner
966.2.g.c 4 161.c even 2 1 inner
2898.2.g.c 4 3.b odd 2 1
2898.2.g.c 4 21.c even 2 1
2898.2.g.c 4 69.c even 2 1
2898.2.g.c 4 483.c odd 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 7 + T^{2} )^{2} \)
$11$ \( ( 28 + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( -28 + T^{2} )^{2} \)
$19$ \( ( -28 + T^{2} )^{2} \)
$23$ \( ( 23 - 8 T + T^{2} )^{2} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 100 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 144 + T^{2} )^{2} \)
$43$ \( ( 28 + T^{2} )^{2} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -112 + T^{2} )^{2} \)
$67$ \( ( 28 + T^{2} )^{2} \)
$71$ \( ( 8 + T )^{4} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 28 + T^{2} )^{2} \)
$83$ \( ( -252 + T^{2} )^{2} \)
$89$ \( ( -28 + T^{2} )^{2} \)
$97$ \( ( -28 + T^{2} )^{2} \)
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