Properties

Label 115.2.g.a
Level $115$
Weight $2$
Character orbit 115.g
Analytic conductor $0.918$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
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: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0.959493 0.281733i
−0.841254 0.540641i
−0.415415 0.909632i
−0.415415 + 0.909632i
−0.841254 + 0.540641i
0.654861 + 0.755750i
0.142315 0.989821i
0.142315 + 0.989821i
0.959493 + 0.281733i
0.654861 0.755750i
0.925839 2.02730i 0.959493 + 0.281733i −1.94306 2.24241i −0.841254 + 0.540641i 1.45949 1.68434i −0.0125459 + 0.0872586i −2.06815 + 0.607265i −1.68251 1.08128i 0.317178 + 2.20602i
16.1 1.57028 1.81219i −0.841254 + 0.540641i −0.533654 3.71165i −0.415415 0.909632i −0.341254 + 2.37347i 1.31718 0.386758i −3.52977 2.26844i −0.830830 + 1.81926i −2.30075 0.675560i
26.1 −0.0125459 0.0872586i −0.415415 + 0.909632i 1.91153 0.561276i 0.654861 0.755750i 0.0845850 + 0.0248364i −1.30075 + 0.835939i −0.146201 0.320135i 1.30972 + 1.51150i −0.0741615 0.0476607i
31.1 −0.0125459 + 0.0872586i −0.415415 0.909632i 1.91153 + 0.561276i 0.654861 + 0.755750i 0.0845850 0.0248364i −1.30075 0.835939i −0.146201 + 0.320135i 1.30972 1.51150i −0.0741615 + 0.0476607i
36.1 1.57028 + 1.81219i −0.841254 0.540641i −0.533654 + 3.71165i −0.415415 + 0.909632i −0.341254 2.37347i 1.31718 + 0.386758i −3.52977 + 2.26844i −0.830830 1.81926i −2.30075 + 0.675560i
41.1 1.31718 + 0.386758i 0.654861 0.755750i −0.0971309 0.0624222i 0.142315 0.989821i 1.15486 0.742184i 0.925839 + 2.02730i −1.90176 2.19475i 0.284630 + 1.97964i 0.570276 1.24873i
71.1 −1.30075 0.835939i 0.142315 + 0.989821i 0.162317 + 0.355426i 0.959493 + 0.281733i 0.642315 1.40647i 1.57028 1.81219i −0.354114 + 2.46292i 1.91899 0.563465i −1.01255 1.16854i
81.1 −1.30075 + 0.835939i 0.142315 0.989821i 0.162317 0.355426i 0.959493 0.281733i 0.642315 + 1.40647i 1.57028 + 1.81219i −0.354114 2.46292i 1.91899 + 0.563465i −1.01255 + 1.16854i
96.1 0.925839 + 2.02730i 0.959493 0.281733i −1.94306 + 2.24241i −0.841254 0.540641i 1.45949 + 1.68434i −0.0125459 0.0872586i −2.06815 0.607265i −1.68251 + 1.08128i 0.317178 2.20602i
101.1 1.31718 0.386758i 0.654861 + 0.755750i −0.0971309 + 0.0624222i 0.142315 + 0.989821i 1.15486 + 0.742184i 0.925839 2.02730i −1.90176 + 2.19475i 0.284630 1.97964i 0.570276 + 1.24873i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.1
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 115.2.g.a 10
5.b even 2 1 575.2.k.a 10
5.c odd 4 2 575.2.p.a 20
23.c even 11 1 inner 115.2.g.a 10
23.c even 11 1 2645.2.a.n 5
23.d odd 22 1 2645.2.a.o 5
115.j even 22 1 575.2.k.a 10
115.k odd 44 2 575.2.p.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.a 10 1.a even 1 1 trivial
115.2.g.a 10 23.c even 11 1 inner
575.2.k.a 10 5.b even 2 1
575.2.k.a 10 115.j even 22 1
575.2.p.a 20 5.c odd 4 2
575.2.p.a 20 115.k odd 44 2
2645.2.a.n 5 23.c even 11 1
2645.2.a.o 5 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(115, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 125 T^{2} - 157 T^{3} + 38 T^{4} + 21 T^{5} - 2 T^{6} - 15 T^{7} + 14 T^{8} - 5 T^{9} + T^{10} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$5$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$7$ \( 1 + 2 T + 125 T^{2} - 157 T^{3} + 38 T^{4} + 21 T^{5} - 2 T^{6} - 15 T^{7} + 14 T^{8} - 5 T^{9} + T^{10} \)
$11$ \( 7921 - 4539 T + 2447 T^{2} + 482 T^{3} - 602 T^{4} + 353 T^{5} - 7 T^{6} - 50 T^{7} + 31 T^{8} - 8 T^{9} + T^{10} \)
$13$ \( 121 + 363 T + 484 T^{2} + 363 T^{3} + 121 T^{4} - 22 T^{5} + 11 T^{7} + 11 T^{8} + T^{10} \)
$17$ \( 466489 + 751983 T + 642951 T^{2} + 365487 T^{3} + 150844 T^{4} + 46938 T^{5} + 11166 T^{6} + 2014 T^{7} + 265 T^{8} + 23 T^{9} + T^{10} \)
$19$ \( 11881 + 20928 T + 17867 T^{2} + 8037 T^{3} + 4068 T^{4} + 890 T^{5} + 665 T^{6} + 85 T^{7} + 48 T^{8} + 13 T^{9} + T^{10} \)
$23$ \( 6436343 + 5876661 T + 2956581 T^{2} + 1011977 T^{3} + 270733 T^{4} + 60543 T^{5} + 11771 T^{6} + 1913 T^{7} + 243 T^{8} + 21 T^{9} + T^{10} \)
$29$ \( 127893481 + 13491637 T - 1971142 T^{2} + 348308 T^{3} + 41578 T^{4} + 4599 T^{5} + 4185 T^{6} - 338 T^{7} + 70 T^{8} - 2 T^{9} + T^{10} \)
$31$ \( 7921 + 445 T + 8616 T^{2} - 6068 T^{3} + 3815 T^{4} - 3112 T^{5} + 27 T^{6} + 344 T^{7} + 158 T^{8} + 20 T^{9} + T^{10} \)
$37$ \( 11485321 - 26898493 T + 22157281 T^{2} + 401273 T^{3} + 265219 T^{4} + 99221 T^{5} - 3185 T^{6} + 245 T^{7} + 169 T^{8} - 13 T^{9} + T^{10} \)
$41$ \( 193293409 + 46825304 T + 4364782 T^{2} + 1304768 T^{3} + 382255 T^{4} + 37676 T^{5} + 4684 T^{6} + 1082 T^{7} + 91 T^{8} + 5 T^{9} + T^{10} \)
$43$ \( 4280761 - 2031758 T + 115828 T^{2} - 380000 T^{3} + 347593 T^{4} - 113312 T^{5} + 23642 T^{6} - 2946 T^{7} + 247 T^{8} - 15 T^{9} + T^{10} \)
$47$ \( ( -9811 - 4754 T - 82 T^{2} + 189 T^{3} + 26 T^{4} + T^{5} )^{2} \)
$53$ \( 1408969 + 8541652 T + 18111768 T^{2} - 1311582 T^{3} + 118398 T^{4} + 114522 T^{5} + 2693 T^{6} + 609 T^{7} + 179 T^{8} - 6 T^{9} + T^{10} \)
$59$ \( 437688241 + 275424965 T + 40431593 T^{2} - 1827862 T^{3} + 295674 T^{4} - 2080 T^{5} + 14672 T^{6} - 587 T^{7} + 201 T^{8} - 5 T^{9} + T^{10} \)
$61$ \( 436921 - 564494 T + 157327 T^{2} + 161258 T^{3} + 71481 T^{4} + 15951 T^{5} + 1126 T^{6} + 192 T^{7} + 75 T^{8} + 3 T^{9} + T^{10} \)
$67$ \( 25593481 - 25628894 T + 17666322 T^{2} - 8328841 T^{3} + 2427808 T^{4} - 430156 T^{5} + 47745 T^{6} - 3765 T^{7} + 290 T^{8} - 20 T^{9} + T^{10} \)
$71$ \( 1437601 + 1411223 T + 931216 T^{2} + 486783 T^{3} + 275759 T^{4} + 113036 T^{5} + 27962 T^{6} + 3333 T^{7} + 275 T^{8} + 22 T^{9} + T^{10} \)
$73$ \( 78623689 - 19400996 T + 14577157 T^{2} - 8353971 T^{3} + 2921326 T^{4} - 648020 T^{5} + 99727 T^{6} - 11087 T^{7} + 870 T^{8} - 43 T^{9} + T^{10} \)
$79$ \( 246772681 - 230812337 T + 113149199 T^{2} - 35175355 T^{3} + 7864278 T^{4} - 1264064 T^{5} + 167896 T^{6} - 17008 T^{7} + 1193 T^{8} - 51 T^{9} + T^{10} \)
$83$ \( 10029889 - 494052 T + 2128361 T^{2} - 1225689 T^{3} + 379692 T^{4} - 44078 T^{5} + 3485 T^{6} + 29 T^{7} + 36 T^{8} + 17 T^{9} + T^{10} \)
$89$ \( 43256929 - 81350913 T + 144725243 T^{2} - 71841315 T^{3} + 18714199 T^{4} - 3006321 T^{5} + 336506 T^{6} - 26969 T^{7} + 1544 T^{8} - 57 T^{9} + T^{10} \)
$97$ \( 62047129 - 33760822 T + 36209530 T^{2} - 21752381 T^{3} + 7342415 T^{4} - 1552902 T^{5} + 217122 T^{6} - 20565 T^{7} + 1318 T^{8} - 52 T^{9} + T^{10} \)
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