Properties

Label 966.2.g.d
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{2} q^{5} -\beta_{1} q^{6} + \beta_{2} q^{7} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} + \beta_{2} q^{5} -\beta_{1} q^{6} + \beta_{2} q^{7} - q^{8} - q^{9} -\beta_{2} q^{10} + \beta_{1} q^{12} + 5 \beta_{1} q^{13} -\beta_{2} q^{14} + \beta_{3} q^{15} + q^{16} + 2 \beta_{2} q^{17} + q^{18} -2 \beta_{2} q^{19} + \beta_{2} q^{20} + \beta_{3} q^{21} + ( 4 + \beta_{3} ) q^{23} -\beta_{1} q^{24} + 2 q^{25} -5 \beta_{1} q^{26} -\beta_{1} q^{27} + \beta_{2} q^{28} - q^{29} -\beta_{3} q^{30} -4 \beta_{1} q^{31} - q^{32} -2 \beta_{2} q^{34} + 7 q^{35} - q^{36} + 3 \beta_{3} q^{37} + 2 \beta_{2} q^{38} -5 q^{39} -\beta_{2} q^{40} -9 \beta_{1} q^{41} -\beta_{3} q^{42} -\beta_{3} q^{43} -\beta_{2} q^{45} + ( -4 - \beta_{3} ) q^{46} -13 \beta_{1} q^{47} + \beta_{1} q^{48} + 7 q^{49} -2 q^{50} + 2 \beta_{3} q^{51} + 5 \beta_{1} q^{52} + 2 \beta_{3} q^{53} + \beta_{1} q^{54} -\beta_{2} q^{56} -2 \beta_{3} q^{57} + q^{58} + 14 \beta_{1} q^{59} + \beta_{3} q^{60} -4 \beta_{2} q^{61} + 4 \beta_{1} q^{62} -\beta_{2} q^{63} + q^{64} + 5 \beta_{3} q^{65} -2 \beta_{3} q^{67} + 2 \beta_{2} q^{68} + ( 4 \beta_{1} - \beta_{2} ) q^{69} -7 q^{70} + 6 q^{71} + q^{72} + 4 \beta_{1} q^{73} -3 \beta_{3} q^{74} + 2 \beta_{1} q^{75} -2 \beta_{2} q^{76} + 5 q^{78} + 2 \beta_{3} q^{79} + \beta_{2} q^{80} + q^{81} + 9 \beta_{1} q^{82} + 6 \beta_{2} q^{83} + \beta_{3} q^{84} + 14 q^{85} + \beta_{3} q^{86} -\beta_{1} q^{87} + \beta_{2} q^{90} + 5 \beta_{3} q^{91} + ( 4 + \beta_{3} ) q^{92} + 4 q^{93} + 13 \beta_{1} q^{94} -14 q^{95} -\beta_{1} q^{96} -\beta_{2} q^{97} -7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + 4q^{16} + 4q^{18} + 16q^{23} + 8q^{25} - 4q^{29} - 4q^{32} + 28q^{35} - 4q^{36} - 20q^{39} - 16q^{46} + 28q^{49} - 8q^{50} + 4q^{58} + 4q^{64} - 28q^{70} + 24q^{71} + 4q^{72} + 20q^{78} + 4q^{81} + 56q^{85} + 16q^{92} + 16q^{93} - 56q^{95} - 28q^{98} + O(q^{100}) \)

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 −2.64575 1.00000i −2.64575 −1.00000 −1.00000 2.64575
643.2 −1.00000 1.00000i 1.00000 2.64575 1.00000i 2.64575 −1.00000 −1.00000 −2.64575
643.3 −1.00000 1.00000i 1.00000 −2.64575 1.00000i −2.64575 −1.00000 −1.00000 2.64575
643.4 −1.00000 1.00000i 1.00000 2.64575 1.00000i 2.64575 −1.00000 −1.00000 −2.64575
\(n\): 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.d 4
3.b odd 2 1 2898.2.g.d 4
7.b odd 2 1 inner 966.2.g.d 4
21.c even 2 1 2898.2.g.d 4
23.b odd 2 1 inner 966.2.g.d 4
69.c even 2 1 2898.2.g.d 4
161.c even 2 1 inner 966.2.g.d 4
483.c odd 2 1 2898.2.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.d 4 1.a even 1 1 trivial
966.2.g.d 4 7.b odd 2 1 inner
966.2.g.d 4 23.b odd 2 1 inner
966.2.g.d 4 161.c even 2 1 inner
2898.2.g.d 4 3.b odd 2 1
2898.2.g.d 4 21.c even 2 1
2898.2.g.d 4 69.c even 2 1
2898.2.g.d 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{2} - 7 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( -7 + T^{2} )^{2} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 25 + T^{2} )^{2} \)
$17$ \( ( -28 + T^{2} )^{2} \)
$19$ \( ( -28 + T^{2} )^{2} \)
$23$ \( ( 23 - 8 T + T^{2} )^{2} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( ( 16 + T^{2} )^{2} \)
$37$ \( ( 63 + T^{2} )^{2} \)
$41$ \( ( 81 + T^{2} )^{2} \)
$43$ \( ( 7 + T^{2} )^{2} \)
$47$ \( ( 169 + T^{2} )^{2} \)
$53$ \( ( 28 + T^{2} )^{2} \)
$59$ \( ( 196 + T^{2} )^{2} \)
$61$ \( ( -112 + T^{2} )^{2} \)
$67$ \( ( 28 + T^{2} )^{2} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 28 + T^{2} )^{2} \)
$83$ \( ( -252 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -7 + T^{2} )^{2} \)
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