Properties

Label 966.2.g.b
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: \(\Q(i, \sqrt{14})\)
Defining polynomial: \(x^{4} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{2} q^{3} + q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} -\beta_{2} q^{6} + \beta_{1} q^{7} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{2} q^{3} + q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} -\beta_{2} q^{6} + \beta_{1} q^{7} - q^{8} - q^{9} + ( \beta_{1} - \beta_{3} ) q^{10} + \beta_{2} q^{12} -2 \beta_{2} q^{13} -\beta_{1} q^{14} + ( -\beta_{1} - \beta_{3} ) q^{15} + q^{16} + ( -\beta_{1} + \beta_{3} ) q^{17} + q^{18} + ( -\beta_{1} + \beta_{3} ) q^{19} + ( -\beta_{1} + \beta_{3} ) q^{20} + \beta_{3} q^{21} + ( -3 + \beta_{1} + \beta_{3} ) q^{23} -\beta_{2} q^{24} + 9 q^{25} + 2 \beta_{2} q^{26} -\beta_{2} q^{27} + \beta_{1} q^{28} + 6 q^{29} + ( \beta_{1} + \beta_{3} ) q^{30} -4 \beta_{2} q^{31} - q^{32} + ( \beta_{1} - \beta_{3} ) q^{34} + ( -7 - 7 \beta_{2} ) q^{35} - q^{36} + ( \beta_{1} - \beta_{3} ) q^{38} + 2 q^{39} + ( \beta_{1} - \beta_{3} ) q^{40} -2 \beta_{2} q^{41} -\beta_{3} q^{42} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{43} + ( \beta_{1} - \beta_{3} ) q^{45} + ( 3 - \beta_{1} - \beta_{3} ) q^{46} -6 \beta_{2} q^{47} + \beta_{2} q^{48} + 7 \beta_{2} q^{49} -9 q^{50} + ( -\beta_{1} - \beta_{3} ) q^{51} -2 \beta_{2} q^{52} + ( -\beta_{1} - \beta_{3} ) q^{53} + \beta_{2} q^{54} -\beta_{1} q^{56} + ( -\beta_{1} - \beta_{3} ) q^{57} -6 q^{58} -14 \beta_{2} q^{59} + ( -\beta_{1} - \beta_{3} ) q^{60} + 4 \beta_{2} q^{62} -\beta_{1} q^{63} + q^{64} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{65} + ( -\beta_{1} - \beta_{3} ) q^{67} + ( -\beta_{1} + \beta_{3} ) q^{68} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{69} + ( 7 + 7 \beta_{2} ) q^{70} + 6 q^{71} + q^{72} + 4 \beta_{2} q^{73} + 9 \beta_{2} q^{75} + ( -\beta_{1} + \beta_{3} ) q^{76} -2 q^{78} + ( -\beta_{1} - \beta_{3} ) q^{79} + ( -\beta_{1} + \beta_{3} ) q^{80} + q^{81} + 2 \beta_{2} q^{82} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{83} + \beta_{3} q^{84} + 14 q^{85} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{86} + 6 \beta_{2} q^{87} + ( -\beta_{1} + \beta_{3} ) q^{89} + ( -\beta_{1} + \beta_{3} ) q^{90} -2 \beta_{3} q^{91} + ( -3 + \beta_{1} + \beta_{3} ) q^{92} + 4 q^{93} + 6 \beta_{2} q^{94} + 14 q^{95} -\beta_{2} q^{96} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{97} -7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + 4q^{16} + 4q^{18} - 12q^{23} + 36q^{25} + 24q^{29} - 4q^{32} - 28q^{35} - 4q^{36} + 8q^{39} + 12q^{46} - 36q^{50} - 24q^{58} + 4q^{64} + 28q^{70} + 24q^{71} + 4q^{72} - 8q^{78} + 4q^{81} + 56q^{85} - 12q^{92} + 16q^{93} + 56q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

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
1.87083 1.87083i
−1.87083 + 1.87083i
1.87083 + 1.87083i
−1.87083 1.87083i
−1.00000 1.00000i 1.00000 −3.74166 1.00000i 1.87083 1.87083i −1.00000 −1.00000 3.74166
643.2 −1.00000 1.00000i 1.00000 3.74166 1.00000i −1.87083 + 1.87083i −1.00000 −1.00000 −3.74166
643.3 −1.00000 1.00000i 1.00000 −3.74166 1.00000i 1.87083 + 1.87083i −1.00000 −1.00000 3.74166
643.4 −1.00000 1.00000i 1.00000 3.74166 1.00000i −1.87083 1.87083i −1.00000 −1.00000 −3.74166
\(n\): 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.b 4
3.b odd 2 1 2898.2.g.h 4
7.b odd 2 1 inner 966.2.g.b 4
21.c even 2 1 2898.2.g.h 4
23.b odd 2 1 inner 966.2.g.b 4
69.c even 2 1 2898.2.g.h 4
161.c even 2 1 inner 966.2.g.b 4
483.c odd 2 1 2898.2.g.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.b 4 1.a even 1 1 trivial
966.2.g.b 4 7.b odd 2 1 inner
966.2.g.b 4 23.b odd 2 1 inner
966.2.g.b 4 161.c even 2 1 inner
2898.2.g.h 4 3.b odd 2 1
2898.2.g.h 4 21.c even 2 1
2898.2.g.h 4 69.c even 2 1
2898.2.g.h 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{2} - 14 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( -14 + T^{2} )^{2} \)
$7$ \( 49 + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( -14 + T^{2} )^{2} \)
$19$ \( ( -14 + T^{2} )^{2} \)
$23$ \( ( 23 + 6 T + T^{2} )^{2} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 16 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 4 + T^{2} )^{2} \)
$43$ \( ( 126 + T^{2} )^{2} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( ( 14 + T^{2} )^{2} \)
$59$ \( ( 196 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 14 + T^{2} )^{2} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 14 + T^{2} )^{2} \)
$83$ \( ( -56 + T^{2} )^{2} \)
$89$ \( ( -14 + T^{2} )^{2} \)
$97$ \( ( -56 + T^{2} )^{2} \)
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