Properties

Label 966.2.q.b
Level $966$
Weight $2$
Character orbit 966.q
Analytic conductor $7.714$
Analytic rank $0$
Dimension $10$
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.q (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \(x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{22}^{4} q^{2} + ( -1 + \zeta_{22} - \zeta_{22}^{2} + \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{6} + \zeta_{22}^{7} - \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{3} + \zeta_{22}^{8} q^{4} + ( -\zeta_{22}^{6} + \zeta_{22}^{9} ) q^{5} -\zeta_{22}^{3} q^{6} + \zeta_{22}^{5} q^{7} -\zeta_{22} q^{8} -\zeta_{22}^{9} q^{9} +O(q^{10})\) \( q + \zeta_{22}^{4} q^{2} + ( -1 + \zeta_{22} - \zeta_{22}^{2} + \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{6} + \zeta_{22}^{7} - \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{3} + \zeta_{22}^{8} q^{4} + ( -\zeta_{22}^{6} + \zeta_{22}^{9} ) q^{5} -\zeta_{22}^{3} q^{6} + \zeta_{22}^{5} q^{7} -\zeta_{22} q^{8} -\zeta_{22}^{9} q^{9} + ( 1 - \zeta_{22} - \zeta_{22}^{3} + \zeta_{22}^{4} - \zeta_{22}^{5} + \zeta_{22}^{6} - \zeta_{22}^{7} + \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{10} + ( 1 + \zeta_{22} - \zeta_{22}^{2} - \zeta_{22}^{3} - \zeta_{22}^{5} - 2 \zeta_{22}^{7} - \zeta_{22}^{9} ) q^{11} -\zeta_{22}^{7} q^{12} + ( -1 + \zeta_{22} - \zeta_{22}^{4} + \zeta_{22}^{5} - 3 \zeta_{22}^{6} + \zeta_{22}^{7} - \zeta_{22}^{8} ) q^{13} + \zeta_{22}^{9} q^{14} + ( \zeta_{22}^{5} - \zeta_{22}^{8} ) q^{15} -\zeta_{22}^{5} q^{16} + ( -1 + 2 \zeta_{22} + \zeta_{22}^{2} + \zeta_{22}^{3} + \zeta_{22}^{4} + 2 \zeta_{22}^{5} - \zeta_{22}^{6} ) q^{17} + \zeta_{22}^{2} q^{18} + ( 2 - 3 \zeta_{22} + \zeta_{22}^{2} - \zeta_{22}^{3} + 3 \zeta_{22}^{4} - 2 \zeta_{22}^{5} + \zeta_{22}^{7} + \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{19} + ( \zeta_{22}^{3} - \zeta_{22}^{6} ) q^{20} -\zeta_{22}^{4} q^{21} + ( 2 + \zeta_{22}^{2} + \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{6} - \zeta_{22}^{7} - \zeta_{22}^{9} ) q^{22} + ( 2 + \zeta_{22} + \zeta_{22}^{2} - \zeta_{22}^{3} - 2 \zeta_{22}^{4} + \zeta_{22}^{5} - 2 \zeta_{22}^{6} - 2 \zeta_{22}^{7} + \zeta_{22}^{8} ) q^{23} + q^{24} + ( -\zeta_{22} - 3 \zeta_{22}^{4} - \zeta_{22}^{7} ) q^{25} + ( 2 - 2 \zeta_{22} + 3 \zeta_{22}^{2} - 3 \zeta_{22}^{3} + 2 \zeta_{22}^{4} - 2 \zeta_{22}^{5} + 3 \zeta_{22}^{6} - 3 \zeta_{22}^{7} + 2 \zeta_{22}^{8} - 2 \zeta_{22}^{9} ) q^{26} + \zeta_{22}^{8} q^{27} -\zeta_{22}^{2} q^{28} + ( -3 + 2 \zeta_{22} + 5 \zeta_{22}^{3} + 2 \zeta_{22}^{5} - 3 \zeta_{22}^{6} + \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{29} + ( \zeta_{22} + \zeta_{22}^{9} ) q^{30} + ( 4 \zeta_{22} + 5 \zeta_{22}^{5} - 2 \zeta_{22}^{6} + 2 \zeta_{22}^{7} - 5 \zeta_{22}^{8} ) q^{31} -\zeta_{22}^{9} q^{32} + ( -2 + 2 \zeta_{22} + \zeta_{22}^{3} + \zeta_{22}^{5} + \zeta_{22}^{6} + \zeta_{22}^{7} + \zeta_{22}^{9} ) q^{33} + ( 1 - \zeta_{22} + \zeta_{22}^{2} - \zeta_{22}^{3} + \zeta_{22}^{5} + 2 \zeta_{22}^{6} + 2 \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{34} + ( 1 - \zeta_{22}^{3} ) q^{35} + \zeta_{22}^{6} q^{36} + ( 2 - 5 \zeta_{22}^{3} + 5 \zeta_{22}^{4} - 2 \zeta_{22}^{7} - 3 \zeta_{22}^{9} ) q^{37} + ( -1 - \zeta_{22} - \zeta_{22}^{2} + 2 \zeta_{22}^{4} - 3 \zeta_{22}^{5} + \zeta_{22}^{6} - \zeta_{22}^{7} + 3 \zeta_{22}^{8} - 2 \zeta_{22}^{9} ) q^{38} + ( -\zeta_{22} + \zeta_{22}^{2} + 2 \zeta_{22}^{5} + \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{39} + ( 1 - \zeta_{22} + \zeta_{22}^{2} - \zeta_{22}^{3} + \zeta_{22}^{4} - \zeta_{22}^{5} + \zeta_{22}^{6} + \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{40} + ( \zeta_{22} + \zeta_{22}^{3} - 2 \zeta_{22}^{6} + 2 \zeta_{22}^{9} ) q^{41} -\zeta_{22}^{8} q^{42} + ( -5 \zeta_{22} + \zeta_{22}^{4} - \zeta_{22}^{5} + 5 \zeta_{22}^{8} ) q^{43} + ( 2 - \zeta_{22} + 2 \zeta_{22}^{2} - \zeta_{22}^{3} + 3 \zeta_{22}^{4} - \zeta_{22}^{5} + 2 \zeta_{22}^{6} - \zeta_{22}^{7} + 2 \zeta_{22}^{8} ) q^{44} + ( -\zeta_{22}^{4} + \zeta_{22}^{7} ) q^{45} + ( 4 - 3 \zeta_{22} + 2 \zeta_{22}^{2} - 2 \zeta_{22}^{3} + 4 \zeta_{22}^{4} - \zeta_{22}^{5} + 3 \zeta_{22}^{6} - 3 \zeta_{22}^{7} - \zeta_{22}^{9} ) q^{46} + ( -2 + 5 \zeta_{22}^{2} + 2 \zeta_{22}^{3} + 3 \zeta_{22}^{4} - 3 \zeta_{22}^{7} - 2 \zeta_{22}^{8} - 5 \zeta_{22}^{9} ) q^{47} + \zeta_{22}^{4} q^{48} + ( -1 + \zeta_{22} - \zeta_{22}^{2} + \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{6} + \zeta_{22}^{7} - \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{49} + ( 1 - \zeta_{22}^{5} - 3 \zeta_{22}^{8} ) q^{50} + ( -1 - 2 \zeta_{22} - 2 \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{6} - \zeta_{22}^{7} + \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{51} + ( \zeta_{22} - \zeta_{22}^{2} + 3 \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{52} + ( -5 \zeta_{22} + 4 \zeta_{22}^{2} - 2 \zeta_{22}^{3} + 8 \zeta_{22}^{4} - \zeta_{22}^{5} + 8 \zeta_{22}^{6} - 2 \zeta_{22}^{7} + 4 \zeta_{22}^{8} - 5 \zeta_{22}^{9} ) q^{53} -\zeta_{22} q^{54} + ( -1 + 2 \zeta_{22} - 3 \zeta_{22}^{2} + 2 \zeta_{22}^{3} - 2 \zeta_{22}^{4} + 3 \zeta_{22}^{5} - 2 \zeta_{22}^{6} + \zeta_{22}^{7} + 3 \zeta_{22}^{9} ) q^{55} -\zeta_{22}^{6} q^{56} + ( 1 + \zeta_{22} - \zeta_{22}^{2} - \zeta_{22}^{3} + 2 \zeta_{22}^{5} - 3 \zeta_{22}^{6} + \zeta_{22}^{7} - 3 \zeta_{22}^{8} + 2 \zeta_{22}^{9} ) q^{57} + ( 3 - 4 \zeta_{22} + 4 \zeta_{22}^{2} - 3 \zeta_{22}^{3} - \zeta_{22}^{5} + 3 \zeta_{22}^{6} + 2 \zeta_{22}^{7} + 3 \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{58} + ( -1 + \zeta_{22} + 4 \zeta_{22}^{3} - \zeta_{22}^{4} + 5 \zeta_{22}^{5} - 4 \zeta_{22}^{6} + 5 \zeta_{22}^{7} - \zeta_{22}^{8} + 4 \zeta_{22}^{9} ) q^{59} + ( -\zeta_{22}^{2} + \zeta_{22}^{5} ) q^{60} + ( 4 - \zeta_{22} + 4 \zeta_{22}^{2} + \zeta_{22}^{4} + 6 \zeta_{22}^{6} - 6 \zeta_{22}^{7} - \zeta_{22}^{9} ) q^{61} + ( 3 \zeta_{22} + 2 \zeta_{22}^{2} - 2 \zeta_{22}^{3} + 2 \zeta_{22}^{4} + 2 \zeta_{22}^{5} + 2 \zeta_{22}^{6} - 2 \zeta_{22}^{7} + 2 \zeta_{22}^{8} + 3 \zeta_{22}^{9} ) q^{62} + \zeta_{22}^{3} q^{63} + \zeta_{22}^{2} q^{64} + ( -1 - \zeta_{22} + \zeta_{22}^{4} + \zeta_{22}^{5} + \zeta_{22}^{7} - 2 \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{65} + ( -2 + \zeta_{22} - 2 \zeta_{22}^{2} + \zeta_{22}^{3} - 3 \zeta_{22}^{4} + 3 \zeta_{22}^{5} - \zeta_{22}^{6} + 2 \zeta_{22}^{7} - \zeta_{22}^{8} + 2 \zeta_{22}^{9} ) q^{66} + ( -2 - 4 \zeta_{22} + \zeta_{22}^{2} - 2 \zeta_{22}^{4} + \zeta_{22}^{6} - 4 \zeta_{22}^{7} - 2 \zeta_{22}^{8} ) q^{67} + ( -2 - 3 \zeta_{22}^{2} + 2 \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{6} + \zeta_{22}^{7} - 2 \zeta_{22}^{8} + 3 \zeta_{22}^{9} ) q^{68} + ( -3 + \zeta_{22} - \zeta_{22}^{2} + 4 \zeta_{22}^{3} - 3 \zeta_{22}^{4} + 4 \zeta_{22}^{5} + \zeta_{22}^{7} - 2 \zeta_{22}^{8} + 2 \zeta_{22}^{9} ) q^{69} + ( \zeta_{22}^{4} - \zeta_{22}^{7} ) q^{70} + ( -1 + 6 \zeta_{22} - 8 \zeta_{22}^{2} + \zeta_{22}^{3} - 6 \zeta_{22}^{4} + \zeta_{22}^{5} - 8 \zeta_{22}^{6} + 6 \zeta_{22}^{7} - \zeta_{22}^{8} ) q^{71} + ( -1 + \zeta_{22} - \zeta_{22}^{2} + \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{5} - \zeta_{22}^{6} + \zeta_{22}^{7} - \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{72} + ( -1 - 2 \zeta_{22}^{2} + 2 \zeta_{22}^{3} + \zeta_{22}^{5} - 2 \zeta_{22}^{7} - 7 \zeta_{22}^{8} - 2 \zeta_{22}^{9} ) q^{73} + ( 2 + 3 \zeta_{22}^{2} + 2 \zeta_{22}^{4} - 5 \zeta_{22}^{7} + 5 \zeta_{22}^{8} ) q^{74} + ( 1 + 3 \zeta_{22}^{3} + \zeta_{22}^{6} ) q^{75} + ( -2 \zeta_{22} + \zeta_{22}^{2} + \zeta_{22}^{3} - 2 \zeta_{22}^{4} - 2 \zeta_{22}^{6} + \zeta_{22}^{7} + \zeta_{22}^{8} - 2 \zeta_{22}^{9} ) q^{76} + ( 1 + \zeta_{22} + \zeta_{22}^{2} + \zeta_{22}^{4} + 2 \zeta_{22}^{6} - 2 \zeta_{22}^{7} - \zeta_{22}^{9} ) q^{77} + ( -\zeta_{22} + \zeta_{22}^{2} - \zeta_{22}^{5} + \zeta_{22}^{6} + 2 \zeta_{22}^{9} ) q^{78} + ( 5 - 5 \zeta_{22} - 3 \zeta_{22}^{3} - 2 \zeta_{22}^{4} - \zeta_{22}^{5} - 6 \zeta_{22}^{6} - \zeta_{22}^{7} - 2 \zeta_{22}^{8} - 3 \zeta_{22}^{9} ) q^{79} + ( -1 + \zeta_{22}^{3} ) q^{80} -\zeta_{22}^{7} q^{81} + ( 2 - 2 \zeta_{22} - 2 \zeta_{22}^{3} + 2 \zeta_{22}^{4} - \zeta_{22}^{5} + 2 \zeta_{22}^{6} - \zeta_{22}^{7} + 2 \zeta_{22}^{8} - 2 \zeta_{22}^{9} ) q^{82} + ( -1 + 7 \zeta_{22} + \zeta_{22}^{2} + \zeta_{22}^{3} - \zeta_{22}^{4} - \zeta_{22}^{5} - 7 \zeta_{22}^{6} + \zeta_{22}^{7} ) q^{83} + \zeta_{22} q^{84} + ( -\zeta_{22} - 2 \zeta_{22}^{2} - \zeta_{22}^{3} + \zeta_{22}^{5} - \zeta_{22}^{7} - 2 \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{85} + ( -5 \zeta_{22} - 5 \zeta_{22}^{5} + \zeta_{22}^{8} - \zeta_{22}^{9} ) q^{86} + ( 1 - 3 \zeta_{22} - 2 \zeta_{22}^{2} - 3 \zeta_{22}^{3} + \zeta_{22}^{4} + 3 \zeta_{22}^{6} - 4 \zeta_{22}^{7} + 4 \zeta_{22}^{8} - 3 \zeta_{22}^{9} ) q^{87} + ( -1 - 2 \zeta_{22}^{2} + 2 \zeta_{22}^{3} + \zeta_{22}^{5} + \zeta_{22}^{7} + \zeta_{22}^{8} + \zeta_{22}^{9} ) q^{88} + ( -5 + 9 \zeta_{22} - 4 \zeta_{22}^{2} + 7 \zeta_{22}^{3} - 5 \zeta_{22}^{4} + 5 \zeta_{22}^{5} - 7 \zeta_{22}^{6} + 4 \zeta_{22}^{7} - 9 \zeta_{22}^{8} + 5 \zeta_{22}^{9} ) q^{89} + ( -1 - \zeta_{22}^{8} ) q^{90} + ( 2 + \zeta_{22}^{3} - \zeta_{22}^{4} + \zeta_{22}^{7} - \zeta_{22}^{8} ) q^{91} + ( 3 \zeta_{22} - 2 \zeta_{22}^{2} + 3 \zeta_{22}^{3} + \zeta_{22}^{4} - \zeta_{22}^{6} + \zeta_{22}^{7} + \zeta_{22}^{8} + 2 \zeta_{22}^{9} ) q^{92} + ( -4 - 5 \zeta_{22}^{4} + 2 \zeta_{22}^{5} - 2 \zeta_{22}^{6} + 5 \zeta_{22}^{7} ) q^{93} + ( 3 + 2 \zeta_{22} + 5 \zeta_{22}^{2} - 2 \zeta_{22}^{4} + 5 \zeta_{22}^{6} + 2 \zeta_{22}^{7} + 3 \zeta_{22}^{8} ) q^{94} + ( 3 - 5 \zeta_{22} + 4 \zeta_{22}^{2} - 3 \zeta_{22}^{3} + 7 \zeta_{22}^{4} - 7 \zeta_{22}^{5} + 3 \zeta_{22}^{6} - 4 \zeta_{22}^{7} + 5 \zeta_{22}^{8} - 3 \zeta_{22}^{9} ) q^{95} + \zeta_{22}^{8} q^{96} + ( -1 + \zeta_{22} + \zeta_{22}^{2} + \zeta_{22}^{3} - \zeta_{22}^{4} + 5 \zeta_{22}^{7} - 5 \zeta_{22}^{8} ) q^{97} -\zeta_{22}^{3} q^{98} + ( -2 \zeta_{22} + \zeta_{22}^{2} - 2 \zeta_{22}^{3} + \zeta_{22}^{4} - 3 \zeta_{22}^{5} + \zeta_{22}^{6} - 2 \zeta_{22}^{7} + \zeta_{22}^{8} - 2 \zeta_{22}^{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + q^{7} - q^{8} - q^{9} + O(q^{10}) \) \( 10q - q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + q^{7} - q^{8} - q^{9} + 2q^{10} + 7q^{11} - q^{12} - 2q^{13} + q^{14} + 2q^{15} - q^{16} - 6q^{17} - q^{18} + 11q^{19} + 2q^{20} + q^{21} + 18q^{22} + 21q^{23} + 10q^{24} + q^{25} - 2q^{26} - q^{27} + q^{28} - 20q^{29} + 2q^{30} + 18q^{31} - q^{32} - 15q^{33} + 5q^{34} + 9q^{35} - q^{36} + 5q^{37} - 22q^{38} - 2q^{39} + 2q^{40} + 6q^{41} + q^{42} - 12q^{43} + 7q^{44} + 2q^{45} + 21q^{46} - 32q^{47} - q^{48} - q^{49} + 12q^{50} - 17q^{51} + 9q^{52} - 39q^{53} - q^{54} + 8q^{55} + q^{56} + 22q^{57} + 13q^{58} + 15q^{59} + 2q^{60} + 21q^{61} - 4q^{62} + q^{63} - q^{64} - 7q^{65} - 4q^{66} - 26q^{67} - 6q^{68} - 12q^{69} - 2q^{70} + 27q^{71} - q^{72} - 2q^{73} + 5q^{74} + 12q^{75} + 4q^{77} - 2q^{78} + 47q^{79} - 9q^{80} - q^{81} + 6q^{82} + 5q^{83} + q^{84} + q^{85} - 12q^{86} - 9q^{87} - 4q^{88} + 5q^{89} - 9q^{90} + 24q^{91} + 10q^{92} - 26q^{93} + 23q^{94} - 11q^{95} - q^{96} + 2q^{97} - q^{98} - 15q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 + 0.909632i
0.959493 + 0.281733i
−0.415415 0.909632i
0.654861 + 0.755750i
0.959493 0.281733i
0.142315 0.989821i
−0.841254 + 0.540641i
0.654861 0.755750i
−0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.544078 + 1.19136i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −1.25667 0.368991i
127.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 1.61435 0.474017i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 1.10181 1.27155i
169.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.186393 0.215109i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 0.239446 0.153882i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.698939 0.449181i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.118239 0.822373i
463.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.186393 + 0.215109i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 0.239446 + 0.153882i
547.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.273100 + 1.89945i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0.797176 1.74557i
673.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.698939 + 0.449181i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.118239 + 0.822373i
715.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 1.61435 + 0.474017i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 1.10181 + 1.27155i
841.1 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.544078 1.19136i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −1.25667 + 0.368991i
883.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.273100 1.89945i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.797176 + 1.74557i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.b 10
23.c even 11 1 inner 966.2.q.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.b 10 1.a even 1 1 trivial
966.2.q.b 10 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 1 + 5 T + 14 T^{2} + 4 T^{3} - 2 T^{4} + T^{5} + 5 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$11$ \( 4489 - 10854 T + 12219 T^{2} - 9587 T^{3} + 5240 T^{4} - 1385 T^{5} + 300 T^{6} - 101 T^{7} + 38 T^{8} - 7 T^{9} + T^{10} \)
$13$ \( 1849 + 4945 T + 7978 T^{2} + 1789 T^{3} + 1340 T^{4} - 155 T^{5} + 16 T^{6} - 47 T^{7} - 7 T^{8} + 2 T^{9} + T^{10} \)
$17$ \( 7921 - 1780 T + 17120 T^{2} + 3781 T^{3} + 2557 T^{4} + 2507 T^{5} + 790 T^{6} + 51 T^{7} + 3 T^{8} + 6 T^{9} + T^{10} \)
$19$ \( 64009 - 164197 T + 160083 T^{2} - 39325 T^{3} + 7865 T^{4} - 8613 T^{5} + 3586 T^{6} - 737 T^{7} + 132 T^{8} - 11 T^{9} + T^{10} \)
$23$ \( 6436343 - 5876661 T + 2555070 T^{2} - 645380 T^{3} + 108560 T^{4} - 17863 T^{5} + 4720 T^{6} - 1220 T^{7} + 210 T^{8} - 21 T^{9} + T^{10} \)
$29$ \( 62742241 + 26788822 T + 6384326 T^{2} + 1344523 T^{3} + 300650 T^{4} + 73008 T^{5} + 14305 T^{6} + 1917 T^{7} + 224 T^{8} + 20 T^{9} + T^{10} \)
$31$ \( 122921569 - 8936122 T + 1915989 T^{2} - 1673249 T^{3} + 391571 T^{4} - 56528 T^{5} + 11674 T^{6} - 1982 T^{7} + 214 T^{8} - 18 T^{9} + T^{10} \)
$37$ \( 108722329 - 26057073 T + 5128248 T^{2} - 1536857 T^{3} + 333954 T^{4} - 31329 T^{5} + 5465 T^{6} - 785 T^{7} + 25 T^{8} - 5 T^{9} + T^{10} \)
$41$ \( 1 - 13 T + 81 T^{2} - 283 T^{3} + 588 T^{4} - 670 T^{5} + 526 T^{6} - 194 T^{7} + 36 T^{8} - 6 T^{9} + T^{10} \)
$43$ \( 326041 - 842225 T + 1200453 T^{2} - 346433 T^{3} - 1110 T^{4} + 9835 T^{5} + 8086 T^{6} + 1673 T^{7} + 144 T^{8} + 12 T^{9} + T^{10} \)
$47$ \( ( -7963 - 8949 T - 2047 T^{2} - 67 T^{3} + 16 T^{4} + T^{5} )^{2} \)
$53$ \( 1256206249 + 1104581095 T + 456584132 T^{2} + 113344437 T^{3} + 19096170 T^{4} + 2332747 T^{5} + 219437 T^{6} + 16441 T^{7} + 949 T^{8} + 39 T^{9} + T^{10} \)
$59$ \( 1067089 + 1605282 T + 650736 T^{2} - 372168 T^{3} + 208509 T^{4} - 13091 T^{5} + 9474 T^{6} - 1516 T^{7} + 126 T^{8} - 15 T^{9} + T^{10} \)
$61$ \( 73599241 + 47742135 T + 16438148 T^{2} + 1025597 T^{3} - 68386 T^{4} - 3453 T^{5} + 4159 T^{6} - 2001 T^{7} + 287 T^{8} - 21 T^{9} + T^{10} \)
$67$ \( 591361 + 2924507 T + 8087077 T^{2} + 5205117 T^{3} + 1717434 T^{4} + 345874 T^{5} + 49316 T^{6} + 5168 T^{7} + 412 T^{8} + 26 T^{9} + T^{10} \)
$71$ \( 3335178001 - 2555539501 T + 833765849 T^{2} - 128656597 T^{3} + 14786117 T^{4} - 1591393 T^{5} + 149246 T^{6} - 10729 T^{7} + 586 T^{8} - 27 T^{9} + T^{10} \)
$73$ \( 4932841 + 2112171 T + 20361245 T^{2} - 3213269 T^{3} + 253757 T^{4} - 95470 T^{5} + 14514 T^{6} - 212 T^{7} - 73 T^{8} + 2 T^{9} + T^{10} \)
$79$ \( 214369 - 3551673 T + 21498075 T^{2} - 14044259 T^{3} + 6750054 T^{4} - 1593472 T^{5} + 234920 T^{6} - 20080 T^{7} + 1153 T^{8} - 47 T^{9} + T^{10} \)
$83$ \( 6285049 - 16992446 T + 17821654 T^{2} - 9763427 T^{3} + 3444072 T^{4} - 332553 T^{5} + 18940 T^{6} - 1269 T^{7} + 201 T^{8} - 5 T^{9} + T^{10} \)
$89$ \( 392951329 - 306760925 T + 101971720 T^{2} - 18247727 T^{3} + 2146402 T^{4} - 226073 T^{5} + 20953 T^{6} - 1511 T^{7} + 157 T^{8} - 5 T^{9} + T^{10} \)
$97$ \( 5139289 - 4676821 T + 3874236 T^{2} - 1816349 T^{3} + 455431 T^{4} - 30920 T^{5} - 1898 T^{6} + 993 T^{7} - 29 T^{8} - 2 T^{9} + T^{10} \)
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