Properties

Label 966.2.q.c
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.

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

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 0.817178 + 1.78937i 0.142315 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i 1.88745 + 0.554206i
127.1 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 1.07028 0.314261i −0.415415 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −0.730471 + 0.843008i
169.1 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −1.80075 2.07817i 0.959493 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −2.31329 + 1.48666i
211.1 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.512546 0.329393i 0.654861 + 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.0867074 + 0.603063i
463.1 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −1.80075 + 2.07817i 0.959493 + 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −2.31329 1.48666i
547.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.425839 2.96177i −0.841254 + 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 1.24302 2.72183i
673.1 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.512546 + 0.329393i 0.654861 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.0867074 0.603063i
715.1 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 1.07028 + 0.314261i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i −0.730471 0.843008i
841.1 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.817178 1.78937i 0.142315 + 0.989821i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i 1.88745 0.554206i
883.1 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.425839 + 2.96177i −0.841254 0.540641i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i 1.24302 + 2.72183i
\(n\): 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.c 10
23.c even 11 1 inner 966.2.q.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.c 10 1.a even 1 1 trivial
966.2.q.c 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

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 121 + 121 T - 121 T^{2} - 242 T^{3} + 242 T^{4} - 99 T^{5} + 55 T^{6} + 11 T^{8} + T^{10} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$11$ \( 1 - 9 T + 125 T^{2} + 52 T^{3} + 38 T^{4} + 54 T^{5} + 31 T^{6} + 18 T^{7} + 14 T^{8} + 6 T^{9} + T^{10} \)
$13$ \( 978121 + 189888 T + 247668 T^{2} + 10941 T^{3} + 3441 T^{4} - 848 T^{5} + 346 T^{6} - 113 T^{7} - 18 T^{8} + 2 T^{9} + T^{10} \)
$17$ \( 139129 - 253640 T + 284057 T^{2} + 6559 T^{3} + 21444 T^{4} + 2586 T^{5} + 273 T^{6} - 7 T^{7} - 8 T^{8} + 5 T^{9} + T^{10} \)
$19$ \( 8814961 - 10795284 T + 6528635 T^{2} - 2556459 T^{3} + 791008 T^{4} - 192402 T^{5} + 35909 T^{6} - 4975 T^{7} + 488 T^{8} - 31 T^{9} + T^{10} \)
$23$ \( 6436343 + 279841 T - 255507 T^{2} - 115851 T^{3} + 12673 T^{4} + 4379 T^{5} + 551 T^{6} - 219 T^{7} - 21 T^{8} + T^{9} + T^{10} \)
$29$ \( 529 - 1817 T + 3689 T^{2} + 1290 T^{3} - 138 T^{4} - 10 T^{5} + 20 T^{6} + 37 T^{7} + 25 T^{8} + 5 T^{9} + T^{10} \)
$31$ \( 69172489 - 7352228 T + 4302985 T^{2} + 195340 T^{3} - 72058 T^{4} + 17082 T^{5} + 1797 T^{6} - 71 T^{7} + 9 T^{8} - 3 T^{9} + T^{10} \)
$37$ \( 4489 - 6901 T + 16120 T^{2} - 10734 T^{3} + 23015 T^{4} + 8348 T^{5} - 931 T^{6} - 60 T^{7} + 86 T^{8} - 14 T^{9} + T^{10} \)
$41$ \( 5442889 + 7113317 T + 5351174 T^{2} + 2554428 T^{3} + 836148 T^{4} + 190783 T^{5} + 31337 T^{6} + 3890 T^{7} + 388 T^{8} + 28 T^{9} + T^{10} \)
$43$ \( 4489 - 4757 T + 340541 T^{2} + 513278 T^{3} + 230085 T^{4} + 18809 T^{5} - 523 T^{6} + 305 T^{7} + 114 T^{8} - 9 T^{9} + T^{10} \)
$47$ \( ( -4609 - 2530 T - 77 T^{2} + 132 T^{3} + 22 T^{4} + T^{5} )^{2} \)
$53$ \( 18983449 + 15092648 T + 7435429 T^{2} + 2059991 T^{3} + 341254 T^{4} + 20987 T^{5} - 1682 T^{6} - 595 T^{7} - 32 T^{8} - T^{9} + T^{10} \)
$59$ \( 16834609 + 17872668 T + 8589306 T^{2} + 2365308 T^{3} + 681351 T^{4} + 123442 T^{5} + 8206 T^{6} + 550 T^{7} + 165 T^{8} + 11 T^{9} + T^{10} \)
$61$ \( 125372809 - 6606230 T + 14634911 T^{2} + 2551441 T^{3} + 503594 T^{4} + 256728 T^{5} + 69733 T^{6} + 9741 T^{7} + 812 T^{8} + 41 T^{9} + T^{10} \)
$67$ \( 4866876169 + 1996826349 T + 356317842 T^{2} + 42045290 T^{3} + 3384767 T^{4} + 132298 T^{5} + 18327 T^{6} - 582 T^{7} + 144 T^{8} + 12 T^{9} + T^{10} \)
$71$ \( 543169 - 1159301 T - 214533 T^{2} + 909799 T^{3} + 820622 T^{4} + 294899 T^{5} + 60060 T^{6} + 5797 T^{7} + 528 T^{8} + 22 T^{9} + T^{10} \)
$73$ \( 1 - 29 T + 522 T^{2} + 8347 T^{3} + 24874 T^{4} - 14079 T^{5} + 7451 T^{6} - 1447 T^{7} + 207 T^{8} - 19 T^{9} + T^{10} \)
$79$ \( 1068832249 - 319475996 T + 36147930 T^{2} - 24682590 T^{3} + 5991945 T^{4} - 76176 T^{5} + 25469 T^{6} + 1722 T^{7} + 86 T^{8} - 3 T^{9} + T^{10} \)
$83$ \( 1033043881 + 142063220 T + 36553708 T^{2} + 9074986 T^{3} + 1818789 T^{4} + 348435 T^{5} + 55636 T^{6} + 7212 T^{7} + 696 T^{8} + 39 T^{9} + T^{10} \)
$89$ \( 30198445729 + 13191238293 T + 2319994439 T^{2} + 236714607 T^{3} + 20101779 T^{4} + 1466761 T^{5} + 105455 T^{6} + 6431 T^{7} + 399 T^{8} + 17 T^{9} + T^{10} \)
$97$ \( 580376281 - 776958841 T + 624960172 T^{2} + 129435395 T^{3} + 4680875 T^{4} - 18534 T^{5} + 12948 T^{6} - 571 T^{7} + 375 T^{8} + 12 T^{9} + T^{10} \)
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