Properties

Label 966.2.i.d
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} - q^{6} + ( -3 + \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} - q^{6} + ( -3 + \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{10} + ( -1 + \zeta_{6} ) q^{11} + \zeta_{6} q^{12} -6 q^{13} + ( 1 + 2 \zeta_{6} ) q^{14} + 3 q^{15} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} -3 q^{20} + ( -2 + 3 \zeta_{6} ) q^{21} + q^{22} + \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( -4 + 4 \zeta_{6} ) q^{25} + 6 \zeta_{6} q^{26} - q^{27} + ( 2 - 3 \zeta_{6} ) q^{28} -7 q^{29} -3 \zeta_{6} q^{30} + ( -7 + 7 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + \zeta_{6} q^{33} -2 q^{34} + ( -3 - 6 \zeta_{6} ) q^{35} + q^{36} + 2 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + ( -6 + 6 \zeta_{6} ) q^{39} + 3 \zeta_{6} q^{40} -10 q^{41} + ( 3 - \zeta_{6} ) q^{42} + 2 q^{43} -\zeta_{6} q^{44} + ( 3 - 3 \zeta_{6} ) q^{45} + ( 1 - \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} - q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + 4 q^{50} -2 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{52} + ( -11 + 11 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} -3 q^{55} + ( -3 + \zeta_{6} ) q^{56} + 2 q^{57} + 7 \zeta_{6} q^{58} + ( -15 + 15 \zeta_{6} ) q^{59} + ( -3 + 3 \zeta_{6} ) q^{60} + 2 \zeta_{6} q^{61} + 7 q^{62} + ( 1 + 2 \zeta_{6} ) q^{63} + q^{64} -18 \zeta_{6} q^{65} + ( 1 - \zeta_{6} ) q^{66} + ( -2 + 2 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + q^{69} + ( -6 + 9 \zeta_{6} ) q^{70} -\zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{75} -2 q^{76} + ( 2 - 3 \zeta_{6} ) q^{77} + 6 q^{78} + 11 \zeta_{6} q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 10 \zeta_{6} q^{82} + 13 q^{83} + ( -1 - 2 \zeta_{6} ) q^{84} + 6 q^{85} -2 \zeta_{6} q^{86} + ( -7 + 7 \zeta_{6} ) q^{87} + ( -1 + \zeta_{6} ) q^{88} -8 \zeta_{6} q^{89} -3 q^{90} + ( 18 - 6 \zeta_{6} ) q^{91} - q^{92} + 7 \zeta_{6} q^{93} + ( -6 + 6 \zeta_{6} ) q^{94} + ( -6 + 6 \zeta_{6} ) q^{95} + \zeta_{6} q^{96} + q^{97} + ( -5 - 3 \zeta_{6} ) q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} + 3q^{5} - 2q^{6} - 5q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} + 3q^{5} - 2q^{6} - 5q^{7} + 2q^{8} - q^{9} + 3q^{10} - q^{11} + q^{12} - 12q^{13} + 4q^{14} + 6q^{15} - q^{16} + 2q^{17} - q^{18} + 2q^{19} - 6q^{20} - q^{21} + 2q^{22} + q^{23} + q^{24} - 4q^{25} + 6q^{26} - 2q^{27} + q^{28} - 14q^{29} - 3q^{30} - 7q^{31} - q^{32} + q^{33} - 4q^{34} - 12q^{35} + 2q^{36} + 2q^{37} + 2q^{38} - 6q^{39} + 3q^{40} - 20q^{41} + 5q^{42} + 4q^{43} - q^{44} + 3q^{45} + q^{46} - 6q^{47} - 2q^{48} + 11q^{49} + 8q^{50} - 2q^{51} + 6q^{52} - 11q^{53} + q^{54} - 6q^{55} - 5q^{56} + 4q^{57} + 7q^{58} - 15q^{59} - 3q^{60} + 2q^{61} + 14q^{62} + 4q^{63} + 2q^{64} - 18q^{65} + q^{66} - 2q^{67} + 2q^{68} + 2q^{69} - 3q^{70} - q^{72} + 10q^{73} + 2q^{74} + 4q^{75} - 4q^{76} + q^{77} + 12q^{78} + 11q^{79} + 3q^{80} - q^{81} + 10q^{82} + 26q^{83} - 4q^{84} + 12q^{85} - 2q^{86} - 7q^{87} - q^{88} - 8q^{89} - 6q^{90} + 30q^{91} - 2q^{92} + 7q^{93} - 6q^{94} - 6q^{95} + q^{96} + 2q^{97} - 13q^{98} + 2q^{99} + O(q^{100}) \)

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 1.50000 + 2.59808i −1.00000 −2.50000 + 0.866025i 1.00000 −0.500000 0.866025i 1.50000 2.59808i
415.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.50000 2.59808i −1.00000 −2.50000 0.866025i 1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
\(n\): 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.d 2
7.c even 3 1 inner 966.2.i.d 2
7.c even 3 1 6762.2.a.x 1
7.d odd 6 1 6762.2.a.bm 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.d 2 1.a even 1 1 trivial
966.2.i.d 2 7.c even 3 1 inner
6762.2.a.x 1 7.c even 3 1
6762.2.a.bm 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( ( 7 + T )^{2} \)
$31$ \( 49 + 7 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 121 + 11 T + T^{2} \)
$59$ \( 225 + 15 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 121 - 11 T + T^{2} \)
$83$ \( ( -13 + T )^{2} \)
$89$ \( 64 + 8 T + T^{2} \)
$97$ \( ( -1 + T )^{2} \)
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