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.

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

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 \(S_{2}^{\mathrm{new}}(23, [\chi])\).