Properties

Label 966.2.f.a
Level $966$
Weight $2$
Character orbit 966.f
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{2} + ( -1 + 2 \zeta_{12}^{2} ) q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{6} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} -3 q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{2} + ( -1 + 2 \zeta_{12}^{2} ) q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{6} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} -3 q^{9} + ( -1 + 2 \zeta_{12}^{2} ) q^{10} + ( 1 - 2 \zeta_{12}^{2} ) q^{12} + ( -1 + 2 \zeta_{12}^{2} ) q^{13} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{14} -3 \zeta_{12}^{3} q^{15} + q^{16} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + 3 \zeta_{12}^{3} q^{18} + ( -4 + 8 \zeta_{12}^{2} ) q^{19} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{20} + ( 1 + 4 \zeta_{12}^{2} ) q^{21} + \zeta_{12}^{3} q^{23} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{24} -2 q^{25} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{26} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( -3 + 2 \zeta_{12}^{2} ) q^{28} -9 \zeta_{12}^{3} q^{29} -3 q^{30} + ( -6 + 12 \zeta_{12}^{2} ) q^{31} -\zeta_{12}^{3} q^{32} + ( -2 + 4 \zeta_{12}^{2} ) q^{34} + ( -4 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{35} + 3 q^{36} -5 q^{37} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{38} -3 q^{39} + ( 1 - 2 \zeta_{12}^{2} ) q^{40} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{41} + ( 4 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{42} + q^{43} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{45} + q^{46} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{47} + ( -1 + 2 \zeta_{12}^{2} ) q^{48} + ( 5 - 8 \zeta_{12}^{2} ) q^{49} + 2 \zeta_{12}^{3} q^{50} -6 \zeta_{12}^{3} q^{51} + ( 1 - 2 \zeta_{12}^{2} ) q^{52} + 12 \zeta_{12}^{3} q^{53} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{54} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{56} -12 q^{57} -9 q^{58} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} + 3 \zeta_{12}^{3} q^{60} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{62} + ( -9 + 6 \zeta_{12}^{2} ) q^{63} - q^{64} -3 \zeta_{12}^{3} q^{65} -4 q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{69} + ( 1 + 4 \zeta_{12}^{2} ) q^{70} + 12 \zeta_{12}^{3} q^{71} -3 \zeta_{12}^{3} q^{72} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + 5 \zeta_{12}^{3} q^{74} + ( 2 - 4 \zeta_{12}^{2} ) q^{75} + ( 4 - 8 \zeta_{12}^{2} ) q^{76} + 3 \zeta_{12}^{3} q^{78} -10 q^{79} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{80} + 9 q^{81} + ( -7 + 14 \zeta_{12}^{2} ) q^{82} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} + ( -1 - 4 \zeta_{12}^{2} ) q^{84} + 6 q^{85} -\zeta_{12}^{3} q^{86} + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{87} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{89} + ( 3 - 6 \zeta_{12}^{2} ) q^{90} + ( 1 + 4 \zeta_{12}^{2} ) q^{91} -\zeta_{12}^{3} q^{92} -18 q^{93} + ( -5 + 10 \zeta_{12}^{2} ) q^{94} -12 \zeta_{12}^{3} q^{95} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{96} + ( 7 - 14 \zeta_{12}^{2} ) q^{97} + ( -8 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 8q^{7} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 8q^{7} - 12q^{9} + 4q^{16} + 12q^{21} - 8q^{25} - 8q^{28} - 12q^{30} + 12q^{36} - 20q^{37} - 12q^{39} + 4q^{43} + 4q^{46} + 4q^{49} - 48q^{57} - 36q^{58} - 24q^{63} - 4q^{64} - 16q^{67} + 12q^{70} - 40q^{79} + 36q^{81} - 12q^{84} + 24q^{85} + 12q^{91} - 72q^{93} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
1.00000i 1.73205i −1.00000 1.73205 −1.73205 2.00000 + 1.73205i 1.00000i −3.00000 1.73205i
461.2 1.00000i 1.73205i −1.00000 −1.73205 1.73205 2.00000 1.73205i 1.00000i −3.00000 1.73205i
461.3 1.00000i 1.73205i −1.00000 −1.73205 1.73205 2.00000 + 1.73205i 1.00000i −3.00000 1.73205i
461.4 1.00000i 1.73205i −1.00000 1.73205 −1.73205 2.00000 1.73205i 1.00000i −3.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.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.f.a 4
3.b odd 2 1 inner 966.2.f.a 4
7.b odd 2 1 inner 966.2.f.a 4
21.c even 2 1 inner 966.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.a 4 1.a even 1 1 trivial
966.2.f.a 4 3.b odd 2 1 inner
966.2.f.a 4 7.b odd 2 1 inner
966.2.f.a 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -3 + T^{2} )^{2} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 48 + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 81 + T^{2} )^{2} \)
$31$ \( ( 108 + T^{2} )^{2} \)
$37$ \( ( 5 + T )^{4} \)
$41$ \( ( -147 + T^{2} )^{2} \)
$43$ \( ( -1 + T )^{4} \)
$47$ \( ( -75 + T^{2} )^{2} \)
$53$ \( ( 144 + T^{2} )^{2} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 4 + T )^{4} \)
$71$ \( ( 144 + T^{2} )^{2} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( ( -12 + T^{2} )^{2} \)
$89$ \( ( -12 + T^{2} )^{2} \)
$97$ \( ( 147 + T^{2} )^{2} \)
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