Properties

Label 966.2.q.a
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.

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

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 −1.27098 2.78305i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 2.93560 + 0.861971i
127.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −3.15843 + 0.927399i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i −2.15565 + 2.48775i
169.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −2.42796 2.80202i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 3.11903 2.00448i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.114669 0.0736930i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.0193985 0.134919i
463.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −2.42796 + 2.80202i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 3.11903 + 2.00448i
547.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.0279611 + 0.194474i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0.0816181 0.178719i
673.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.114669 + 0.0736930i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.0193985 + 0.134919i
715.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −3.15843 0.927399i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i −2.15565 2.48775i
841.1 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −1.27098 + 2.78305i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i 2.93560 0.861971i
883.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.0279611 0.194474i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.0816181 + 0.178719i
\(n\): 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.a 10
23.c even 11 1 inner 966.2.q.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.a 10 1.a even 1 1 trivial
966.2.q.a 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

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 1 + 15 T + 115 T^{2} + 526 T^{3} + 1950 T^{4} + 2003 T^{5} + 1137 T^{6} + 412 T^{7} + 97 T^{8} + 14 T^{9} + T^{10} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$11$ \( 529 + 4623 T + 11823 T^{2} + 4900 T^{3} + 3216 T^{4} + 342 T^{5} + 311 T^{6} + 64 T^{7} + 16 T^{8} + 4 T^{9} + T^{10} \)
$13$ \( 69169 + 70484 T + 52816 T^{2} + 21155 T^{3} + 5481 T^{4} - 1992 T^{5} + 114 T^{6} - 203 T^{7} + 86 T^{8} - 14 T^{9} + T^{10} \)
$17$ \( 124609 - 253454 T + 657545 T^{2} - 121115 T^{3} + 9112 T^{4} - 14378 T^{5} + 5371 T^{6} - 911 T^{7} + 126 T^{8} - 15 T^{9} + T^{10} \)
$19$ \( 121 + 3146 T + 22869 T^{2} + 10527 T^{3} + 7986 T^{4} + 6072 T^{5} + 3003 T^{6} + 803 T^{7} + 132 T^{8} + 11 T^{9} + T^{10} \)
$23$ \( 6436343 + 279841 T - 255507 T^{2} - 115851 T^{3} + 12673 T^{4} + 4379 T^{5} + 551 T^{6} - 219 T^{7} - 21 T^{8} + T^{9} + T^{10} \)
$29$ \( 6285049 + 3547405 T + 371497 T^{2} - 247062 T^{3} + 181470 T^{4} - 18864 T^{5} + 8584 T^{6} - 1821 T^{7} + 207 T^{8} - 19 T^{9} + T^{10} \)
$31$ \( 109561 - 401172 T + 866661 T^{2} + 229034 T^{3} - 14592 T^{4} - 17346 T^{5} + 53 T^{6} - 73 T^{7} + 91 T^{8} + 5 T^{9} + T^{10} \)
$37$ \( 80982001 + 25098211 T + 4554256 T^{2} + 1458354 T^{3} + 286955 T^{4} + 30238 T^{5} + 5747 T^{6} + 690 T^{7} + 26 T^{8} + 2 T^{9} + T^{10} \)
$41$ \( 1400331241 - 289152067 T + 39251908 T^{2} - 7330462 T^{3} + 1333660 T^{4} - 109493 T^{5} - 1425 T^{6} + 124 T^{7} + 92 T^{8} - 2 T^{9} + T^{10} \)
$43$ \( 73599241 - 15159093 T - 2934751 T^{2} + 5343318 T^{3} + 122895 T^{4} - 214391 T^{5} + 50153 T^{6} - 5989 T^{7} + 504 T^{8} - 25 T^{9} + T^{10} \)
$47$ \( ( 109 + 356 T + 75 T^{2} - 82 T^{3} + 2 T^{4} + T^{5} )^{2} \)
$53$ \( 212521 - 297806 T + 253691 T^{2} - 146811 T^{3} + 57402 T^{4} - 15599 T^{5} + 2522 T^{6} - 119 T^{7} + 82 T^{8} - 15 T^{9} + T^{10} \)
$59$ \( 21557449 - 12146088 T + 5148906 T^{2} + 1698848 T^{3} + 207399 T^{4} + 114322 T^{5} + 33746 T^{6} + 4198 T^{7} + 289 T^{8} + 17 T^{9} + T^{10} \)
$61$ \( 60668521 + 25438874 T + 6734575 T^{2} + 1671471 T^{3} + 398806 T^{4} + 30524 T^{5} + 5501 T^{6} + 1099 T^{7} + 12 T^{8} - T^{9} + T^{10} \)
$67$ \( 29300569 + 7161399 T + 16148702 T^{2} + 411746 T^{3} - 99293 T^{4} - 49004 T^{5} + 4689 T^{6} - 112 T^{7} - 28 T^{8} + 4 T^{9} + T^{10} \)
$71$ \( 460059601 + 9244519 T + 24902739 T^{2} + 4682483 T^{3} + 761458 T^{4} + 68399 T^{5} + 21096 T^{6} - 1503 T^{7} - 96 T^{8} - 6 T^{9} + T^{10} \)
$73$ \( 127396369 - 56446287 T + 23683302 T^{2} - 5039333 T^{3} + 1370086 T^{4} - 51611 T^{5} - 14681 T^{6} + 1501 T^{7} + 317 T^{8} + 25 T^{9} + T^{10} \)
$79$ \( 978121 + 787244 T + 572632 T^{2} + 106372 T^{3} + 54189 T^{4} - 14994 T^{5} + 961 T^{6} + 72 T^{7} - 24 T^{8} - 3 T^{9} + T^{10} \)
$83$ \( 3659569 - 13452216 T + 15919264 T^{2} - 5695234 T^{3} + 1224997 T^{4} - 164779 T^{5} + 23576 T^{6} - 3060 T^{7} + 280 T^{8} - 7 T^{9} + T^{10} \)
$89$ \( 23203489 + 13530953 T + 10827019 T^{2} + 5235353 T^{3} + 4388761 T^{4} + 1084049 T^{5} + 132829 T^{6} + 11369 T^{7} + 823 T^{8} + 41 T^{9} + T^{10} \)
$97$ \( 18809299609 + 222863875 T + 204514198 T^{2} + 56356239 T^{3} + 3427021 T^{4} - 125036 T^{5} - 5398 T^{6} + 2693 T^{7} + 225 T^{8} + 4 T^{9} + T^{10} \)
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