Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} - q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} - q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + 2 q^{13} + ( -3 + 2 \zeta_{6} ) q^{14} - q^{15} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} + q^{20} + ( -2 - \zeta_{6} ) q^{21} -3 q^{22} -\zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( 4 - 4 \zeta_{6} ) q^{25} -2 \zeta_{6} q^{26} - q^{27} + ( 2 + \zeta_{6} ) q^{28} -7 q^{29} + \zeta_{6} q^{30} + ( -7 + 7 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -3 \zeta_{6} q^{33} -2 q^{34} + ( -3 + 2 \zeta_{6} ) q^{35} + q^{36} + 2 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + ( 2 - 2 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} -2 q^{41} + ( -1 + 3 \zeta_{6} ) q^{42} -6 q^{43} + 3 \zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{45} + ( -1 + \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} - q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} -4 q^{50} -2 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} -3 q^{55} + ( 1 - 3 \zeta_{6} ) q^{56} + 2 q^{57} + 7 \zeta_{6} q^{58} + ( 9 - 9 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} -6 \zeta_{6} q^{61} + 7 q^{62} + ( -3 + 2 \zeta_{6} ) q^{63} + q^{64} -2 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{66} + ( -10 + 10 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} - q^{69} + ( 2 + \zeta_{6} ) q^{70} -8 q^{71} -\zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} -4 \zeta_{6} q^{75} -2 q^{76} + ( -6 - 3 \zeta_{6} ) q^{77} -2 q^{78} + 15 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 2 \zeta_{6} q^{82} + q^{83} + ( 3 - 2 \zeta_{6} ) q^{84} -2 q^{85} + 6 \zeta_{6} q^{86} + ( -7 + 7 \zeta_{6} ) q^{87} + ( 3 - 3 \zeta_{6} ) q^{88} + q^{90} + ( 2 - 6 \zeta_{6} ) q^{91} + q^{92} + 7 \zeta_{6} q^{93} + ( -6 + 6 \zeta_{6} ) q^{94} + ( 2 - 2 \zeta_{6} ) q^{95} + \zeta_{6} q^{96} + 5 q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} - q^{5} - 2q^{6} - q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} - q^{5} - 2q^{6} - q^{7} + 2q^{8} - q^{9} - q^{10} + 3q^{11} + q^{12} + 4q^{13} - 4q^{14} - 2q^{15} - q^{16} + 2q^{17} - q^{18} + 2q^{19} + 2q^{20} - 5q^{21} - 6q^{22} - q^{23} + q^{24} + 4q^{25} - 2q^{26} - 2q^{27} + 5q^{28} - 14q^{29} + q^{30} - 7q^{31} - q^{32} - 3q^{33} - 4q^{34} - 4q^{35} + 2q^{36} + 2q^{37} + 2q^{38} + 2q^{39} - q^{40} - 4q^{41} + q^{42} - 12q^{43} + 3q^{44} - q^{45} - q^{46} - 6q^{47} - 2q^{48} - 13q^{49} - 8q^{50} - 2q^{51} - 2q^{52} + 9q^{53} + q^{54} - 6q^{55} - q^{56} + 4q^{57} + 7q^{58} + 9q^{59} + q^{60} - 6q^{61} + 14q^{62} - 4q^{63} + 2q^{64} - 2q^{65} - 3q^{66} - 10q^{67} + 2q^{68} - 2q^{69} + 5q^{70} - 16q^{71} - q^{72} + 10q^{73} + 2q^{74} - 4q^{75} - 4q^{76} - 15q^{77} - 4q^{78} + 15q^{79} - q^{80} - q^{81} + 2q^{82} + 2q^{83} + 4q^{84} - 4q^{85} + 6q^{86} - 7q^{87} + 3q^{88} + 2q^{90} - 2q^{91} + 2q^{92} + 7q^{93} - 6q^{94} + 2q^{95} + q^{96} + 10q^{97} + 11q^{98} - 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −0.500000 2.59808i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
415.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 −0.500000 + 2.59808i 1.00000 −0.500000 + 0.866025i −0.500000 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.i.c 2
7.c even 3 1 inner 966.2.i.c 2
7.c even 3 1 6762.2.a.bb 1
7.d odd 6 1 6762.2.a.bh 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.c 2 1.a even 1 1 trivial
966.2.i.c 2 7.c even 3 1 inner
6762.2.a.bb 1 7.c even 3 1
6762.2.a.bh 1 7.d odd 6 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 1 + T + T^{2} \)
$29$ \( ( 7 + T )^{2} \)
$31$ \( 49 + 7 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 81 - 9 T + T^{2} \)
$59$ \( 81 - 9 T + T^{2} \)
$61$ \( 36 + 6 T + T^{2} \)
$67$ \( 100 + 10 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 225 - 15 T + T^{2} \)
$83$ \( ( -1 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -5 + T )^{2} \)
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