Properties

Label 690.2.m.a
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.415415 + 0.909632i
0.959493 0.281733i
−0.841254 + 0.540641i
0.959493 + 0.281733i
0.654861 + 0.755750i
0.142315 0.989821i
0.654861 0.755750i
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 0.909632i
−0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 0.654861 + 0.755750i −0.959493 + 0.281733i 2.31329 + 1.48666i 0.415415 0.909632i −0.654861 + 0.755750i −0.841254 + 0.540641i
121.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.841254 + 0.540641i −0.654861 + 0.755750i 0.0867074 0.603063i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.142315 + 0.989821i
151.1 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i −0.142315 0.989821i −1.88745 0.554206i 0.841254 0.540641i 0.415415 + 0.909632i 0.959493 0.281733i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.841254 0.540641i −0.654861 0.755750i 0.0867074 + 0.603063i −0.959493 0.281733i 0.841254 0.540641i 0.142315 0.989821i
271.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.142315 0.989821i 0.841254 0.540641i −1.24302 2.72183i −0.654861 0.755750i −0.142315 0.989821i −0.415415 + 0.909632i
301.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i 0.415415 0.909632i 0.730471 0.843008i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.654861 + 0.755750i
331.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.142315 + 0.989821i 0.841254 + 0.540641i −1.24302 + 2.72183i −0.654861 + 0.755750i −0.142315 + 0.989821i −0.415415 0.909632i
361.1 −0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.415415 0.909632i −0.142315 + 0.989821i −1.88745 + 0.554206i 0.841254 + 0.540641i 0.415415 0.909632i 0.959493 + 0.281733i
541.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 0.959493 0.281733i 0.415415 + 0.909632i 0.730471 + 0.843008i −0.142315 0.989821i −0.959493 0.281733i 0.654861 0.755750i
601.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 0.654861 0.755750i −0.959493 0.281733i 2.31329 1.48666i 0.415415 + 0.909632i −0.654861 0.755750i −0.841254 0.540641i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.1
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 690.2.m.a 10
23.c even 11 1 inner 690.2.m.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.a 10 1.a even 1 1 trivial
690.2.m.a 10 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 11 T_{7}^{7} + 22 T_{7}^{6} + 143 T_{7}^{5} - 121 T_{7}^{3} + 363 T_{7}^{2} - 121 T_{7} + 121 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$7$ \( 121 - 121 T + 363 T^{2} - 121 T^{3} + 143 T^{5} + 22 T^{6} - 11 T^{7} + T^{10} \)
$11$ \( 1 - 5 T + 58 T^{2} - 147 T^{3} + 64 T^{4} + 307 T^{5} + 412 T^{6} + 151 T^{7} + 15 T^{8} + 2 T^{9} + T^{10} \)
$13$ \( 958441 - 1001517 T + 213444 T^{2} + 15367 T^{3} + 19481 T^{4} + 3410 T^{5} + 1320 T^{6} + 33 T^{7} + 33 T^{8} + T^{10} \)
$17$ \( 39601 - 117808 T + 150484 T^{2} - 110949 T^{3} + 54939 T^{4} - 19581 T^{5} + 5234 T^{6} - 1049 T^{7} + 157 T^{8} - 16 T^{9} + T^{10} \)
$19$ \( 17161 - 14803 T + 15926 T^{2} - 11105 T^{3} + 7242 T^{4} - 3277 T^{5} + 1466 T^{6} - 563 T^{7} + 137 T^{8} - 18 T^{9} + T^{10} \)
$23$ \( 6436343 + 279841 T - 255507 T^{2} - 69299 T^{3} + 12167 T^{4} + 3279 T^{5} + 529 T^{6} - 131 T^{7} - 21 T^{8} + T^{9} + T^{10} \)
$29$ \( 64009 + 203159 T + 274912 T^{2} + 149072 T^{3} + 30250 T^{4} - 15103 T^{5} + 7513 T^{6} - 1540 T^{7} + 220 T^{8} - 22 T^{9} + T^{10} \)
$31$ \( 978121 - 569664 T + 4281293 T^{2} + 15563 T^{3} + 155741 T^{4} - 51666 T^{5} + 4338 T^{6} + 390 T^{7} - 46 T^{8} - 8 T^{9} + T^{10} \)
$37$ \( 351649 - 305395 T + 228958 T^{2} - 53226 T^{3} + 5593 T^{4} + 1772 T^{5} + 361 T^{6} + 290 T^{7} + 102 T^{8} + 16 T^{9} + T^{10} \)
$41$ \( 248787529 + 133518445 T + 58586465 T^{2} + 15152779 T^{3} + 2140301 T^{4} + 23761 T^{5} - 13294 T^{6} - 2417 T^{7} + 4 T^{8} + 9 T^{9} + T^{10} \)
$43$ \( 77070841 + 35967563 T + 5314169 T^{2} - 426114 T^{3} + 386868 T^{4} - 35507 T^{5} + 2711 T^{6} - 844 T^{7} + 37 T^{8} - 2 T^{9} + T^{10} \)
$47$ \( ( -4643 - 1514 T + 317 T^{2} + 182 T^{3} + 24 T^{4} + T^{5} )^{2} \)
$53$ \( 978121 + 2486346 T + 2591856 T^{2} + 1014886 T^{3} + 206974 T^{4} - 17588 T^{5} - 1535 T^{6} - 1097 T^{7} + 125 T^{8} - 2 T^{9} + T^{10} \)
$59$ \( 4695889 + 10535954 T + 12025101 T^{2} + 7230839 T^{3} + 2492842 T^{4} + 506847 T^{5} + 63767 T^{6} + 5863 T^{7} + 462 T^{8} + 22 T^{9} + T^{10} \)
$61$ \( 148035889 + 130685747 T + 15323014 T^{2} - 6838245 T^{3} + 1439634 T^{4} - 200705 T^{5} + 38417 T^{6} - 2065 T^{7} + 235 T^{8} - 13 T^{9} + T^{10} \)
$67$ \( 212521 - 179790 T + 112720 T^{2} + 418719 T^{3} + 243824 T^{4} + 56200 T^{5} + 6517 T^{6} + 751 T^{7} + 136 T^{8} - 2 T^{9} + T^{10} \)
$71$ \( 1113757129 + 798415652 T + 291205025 T^{2} + 70242899 T^{3} + 12627902 T^{4} + 1799216 T^{5} + 203501 T^{6} + 17513 T^{7} + 1090 T^{8} + 45 T^{9} + T^{10} \)
$73$ \( 2042041 + 312951 T + 296825 T^{2} - 192334 T^{3} + 34266 T^{4} - 4762 T^{5} + 10924 T^{6} - 2463 T^{7} + 265 T^{8} - 21 T^{9} + T^{10} \)
$79$ \( 11335647961 - 5629761313 T + 1709437059 T^{2} - 345692523 T^{3} + 51721571 T^{4} - 6110478 T^{5} + 574640 T^{6} - 41118 T^{7} + 2079 T^{8} - 66 T^{9} + T^{10} \)
$83$ \( 2327773009 - 751640013 T + 317932954 T^{2} - 111427211 T^{3} + 15637923 T^{4} - 608213 T^{5} + 8495 T^{6} + 2290 T^{7} + 159 T^{8} - 15 T^{9} + T^{10} \)
$89$ \( 44742721 - 56267868 T + 34530021 T^{2} - 12982118 T^{3} + 3260712 T^{4} - 577158 T^{5} + 74099 T^{6} - 6899 T^{7} + 511 T^{8} - 29 T^{9} + T^{10} \)
$97$ \( 392951329 + 256172629 T + 88450300 T^{2} + 17651923 T^{3} + 2760785 T^{4} + 235709 T^{5} + 41083 T^{6} + 1412 T^{7} + 179 T^{8} + 27 T^{9} + T^{10} \)
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