Properties

Label 23.2.c.a
Level 23
Weight 2
Character orbit 23.c
Analytic conductor 0.184
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 23.c (of order \(11\) and degree \(10\))

Newform invariants

Self dual: No
Analytic conductor: \(0.183655924649\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-\zeta_{22}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 0.909632i
−0.415415 + 0.909632i
0.142315 0.989821i
0.959493 0.281733i
−0.841254 + 0.540641i
0.654861 + 0.755750i
0.654861 0.755750i
0.959493 + 0.281733i
−2.11435 1.35881i −0.226900 1.57812i 1.79329 + 3.92676i 1.41899 + 0.416652i −1.66463 + 3.64502i −0.804632 + 0.928595i 0.828708 5.76379i 0.439490 0.129046i −2.43409 2.80909i
3.1 −0.313607 2.18119i −1.04408 + 2.28621i −2.74024 + 0.804606i 0.809721 0.934468i 5.31408 + 1.56036i −1.99611 + 1.28282i 0.783524 + 1.71568i −2.17208 2.50672i −2.29218 1.47310i
4.1 0.198939 + 0.435615i −2.11435 + 0.620830i 1.15954 1.33818i −2.18251 1.40261i −0.691070 0.797537i 0.483568 + 3.36329i 1.73259 + 0.508735i 1.56130 1.00339i 0.176814 1.22977i
6.1 0.198939 0.435615i −2.11435 0.620830i 1.15954 + 1.33818i −2.18251 + 1.40261i −0.691070 + 0.797537i 0.483568 3.36329i 1.73259 0.508735i 1.56130 + 1.00339i 0.176814 + 1.22977i
8.1 −0.313607 + 2.18119i −1.04408 2.28621i −2.74024 0.804606i 0.809721 + 0.934468i 5.31408 1.56036i −1.99611 1.28282i 0.783524 1.71568i −2.17208 + 2.50672i −2.29218 + 1.47310i
9.1 −0.226900 + 0.0666238i −0.313607 0.361922i −1.63546 + 1.05105i −0.215370 1.49793i 0.0952700 + 0.0612263i −1.05773 + 2.31611i 0.610783 0.704881i 0.394306 2.74246i 0.148666 + 0.325532i
12.1 −2.11435 + 1.35881i −0.226900 + 1.57812i 1.79329 3.92676i 1.41899 0.416652i −1.66463 3.64502i −0.804632 0.928595i 0.828708 + 5.76379i 0.439490 + 0.129046i −2.43409 + 2.80909i
13.1 −1.04408 1.20493i 0.198939 + 0.127850i −0.0771283 + 0.536439i −1.33083 + 2.91411i −0.0536570 0.373193i 0.874908 + 0.256896i −1.95561 + 1.25679i −1.22301 2.67803i 4.90079 1.43900i
16.1 −1.04408 + 1.20493i 0.198939 0.127850i −0.0771283 0.536439i −1.33083 2.91411i −0.0536570 + 0.373193i 0.874908 0.256896i −1.95561 1.25679i −1.22301 + 2.67803i 4.90079 + 1.43900i
18.1 −0.226900 0.0666238i −0.313607 + 0.361922i −1.63546 1.05105i −0.215370 + 1.49793i 0.0952700 0.0612263i −1.05773 2.31611i 0.610783 + 0.704881i 0.394306 + 2.74246i 0.148666 0.325532i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
23.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(23, [\chi])\).