Properties

Label 966.2.l.a
Level $966$
Weight $2$
Character orbit 966.l
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.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{6} + ( 2 - 3 \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{6} + ( 2 - 3 \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} + ( -1 - \zeta_{12}^{2} ) q^{10} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{12} + ( -1 + 2 \zeta_{12}^{2} ) q^{13} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{14} -3 q^{15} -\zeta_{12}^{2} q^{16} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{17} -3 \zeta_{12} q^{18} + ( 4 - 2 \zeta_{12}^{2} ) q^{19} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{20} + ( -4 \zeta_{12} - \zeta_{12}^{3} ) q^{21} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{23} + ( -1 - \zeta_{12}^{2} ) q^{24} + ( 2 - 2 \zeta_{12}^{2} ) q^{25} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -1 - 2 \zeta_{12}^{2} ) q^{28} + 6 \zeta_{12}^{3} q^{29} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{30} + ( 2 + 2 \zeta_{12}^{2} ) q^{31} -\zeta_{12} q^{32} + ( -3 + 6 \zeta_{12}^{2} ) q^{34} + ( -5 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{35} -3 q^{36} -4 \zeta_{12}^{2} q^{37} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{38} + 3 \zeta_{12} q^{39} + ( -2 + \zeta_{12}^{2} ) q^{40} + ( -4 - \zeta_{12}^{2} ) q^{42} + 4 q^{43} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{45} + ( 1 - \zeta_{12}^{2} ) q^{46} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{47} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{48} + ( -5 - 3 \zeta_{12}^{2} ) q^{49} -2 \zeta_{12}^{3} q^{50} + 9 \zeta_{12}^{2} q^{51} + ( 1 + \zeta_{12}^{2} ) q^{52} -9 \zeta_{12} q^{53} + ( -6 + 3 \zeta_{12}^{2} ) q^{54} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{56} -6 \zeta_{12}^{3} q^{57} + 6 \zeta_{12}^{2} q^{58} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{59} + ( -3 + 3 \zeta_{12}^{2} ) q^{60} + ( 16 - 8 \zeta_{12}^{2} ) q^{61} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{62} + ( -9 + 3 \zeta_{12}^{2} ) q^{63} - q^{64} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{65} + ( -5 + 5 \zeta_{12}^{2} ) q^{67} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{68} + ( 1 - 2 \zeta_{12}^{2} ) q^{69} + ( -5 + 4 \zeta_{12}^{2} ) q^{70} -15 \zeta_{12}^{3} q^{71} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{72} + ( 7 + 7 \zeta_{12}^{2} ) q^{73} -4 \zeta_{12} q^{74} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{75} + ( 2 - 4 \zeta_{12}^{2} ) q^{76} + 3 q^{78} -8 \zeta_{12}^{2} q^{79} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{80} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{83} + ( -5 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{84} + 9 q^{85} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{86} + ( 6 + 6 \zeta_{12}^{2} ) q^{87} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -3 + 6 \zeta_{12}^{2} ) q^{90} + ( 4 + \zeta_{12}^{2} ) q^{91} -\zeta_{12}^{3} q^{92} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{93} + ( 3 + 3 \zeta_{12}^{2} ) q^{94} -6 \zeta_{12} q^{95} + ( -2 + \zeta_{12}^{2} ) q^{96} + ( -6 + 12 \zeta_{12}^{2} ) q^{97} + ( -8 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{7} - 6q^{9} - 6q^{10} - 12q^{15} - 2q^{16} + 12q^{19} - 6q^{24} + 4q^{25} - 8q^{28} + 12q^{31} - 12q^{36} - 8q^{37} - 6q^{40} - 18q^{42} + 16q^{43} + 2q^{46} - 26q^{49} + 18q^{51} + 6q^{52} - 18q^{54} + 12q^{58} - 6q^{60} + 48q^{61} - 30q^{63} - 4q^{64} - 10q^{67} - 12q^{70} + 42q^{73} + 12q^{78} - 16q^{79} - 18q^{81} + 36q^{85} + 36q^{87} + 18q^{91} + 18q^{94} - 6q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 0.866025 1.50000i 1.73205i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
47.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i −0.866025 + 1.50000i 1.73205i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
185.1 −0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i 0.866025 + 1.50000i 1.73205i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
185.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i −0.866025 1.50000i 1.73205i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
\(n\): 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.l.a 4
3.b odd 2 1 inner 966.2.l.a 4
7.d odd 6 1 inner 966.2.l.a 4
21.g even 6 1 inner 966.2.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.l.a 4 1.a even 1 1 trivial
966.2.l.a 4 3.b odd 2 1 inner
966.2.l.a 4 7.d odd 6 1 inner
966.2.l.a 4 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( 9 + 3 T^{2} + T^{4} \)
$7$ \( ( 7 - T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( 729 + 27 T^{2} + T^{4} \)
$19$ \( ( 12 - 6 T + T^{2} )^{2} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( 12 - 6 T + T^{2} )^{2} \)
$37$ \( ( 16 + 4 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -4 + T )^{4} \)
$47$ \( 729 + 27 T^{2} + T^{4} \)
$53$ \( 6561 - 81 T^{2} + T^{4} \)
$59$ \( 11664 + 108 T^{2} + T^{4} \)
$61$ \( ( 192 - 24 T + T^{2} )^{2} \)
$67$ \( ( 25 + 5 T + T^{2} )^{2} \)
$71$ \( ( 225 + T^{2} )^{2} \)
$73$ \( ( 147 - 21 T + T^{2} )^{2} \)
$79$ \( ( 64 + 8 T + T^{2} )^{2} \)
$83$ \( ( -48 + T^{2} )^{2} \)
$89$ \( 2304 + 48 T^{2} + T^{4} \)
$97$ \( ( 108 + T^{2} )^{2} \)
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