Properties

Label 322.2.i.b
Level $322$
Weight $2$
Character orbit 322.i
Analytic conductor $2.571$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.959493 0.281733i
0.142315 0.989821i
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 0.909632i
−0.415415 + 0.909632i
−0.841254 + 0.540641i
0.959493 + 0.281733i
0.654861 + 0.755750i
0.654861 0.755750i
−0.415415 + 0.909632i 2.30075 + 0.675560i −0.654861 0.755750i 1.74412 1.12088i −1.57028 + 1.81219i 0.142315 0.989821i 0.959493 0.281733i 2.31329 + 1.48666i 0.295053 + 2.05214i
71.1 −0.841254 0.540641i −0.317178 2.20602i 0.415415 + 0.909632i 1.95204 + 0.573170i −0.925839 + 2.02730i 0.654861 0.755750i 0.142315 0.989821i −1.88745 + 0.554206i −1.33228 1.53753i
85.1 0.654861 0.755750i 0.0741615 0.0476607i −0.142315 0.989821i 1.57773 + 3.45475i 0.0125459 0.0872586i 0.959493 0.281733i −0.841254 0.540641i −1.24302 + 2.72183i 3.64412 + 1.07001i
127.1 −0.841254 + 0.540641i −0.317178 + 2.20602i 0.415415 0.909632i 1.95204 0.573170i −0.925839 2.02730i 0.654861 + 0.755750i 0.142315 + 0.989821i −1.88745 0.554206i −1.33228 + 1.53753i
141.1 0.142315 + 0.989821i −0.570276 + 1.24873i −0.959493 + 0.281733i −1.59792 + 1.84410i −1.31718 0.386758i −0.841254 + 0.540641i −0.415415 0.909632i 0.730471 + 0.843008i −2.05274 1.31921i
169.1 0.142315 0.989821i −0.570276 1.24873i −0.959493 0.281733i −1.59792 1.84410i −1.31718 + 0.386758i −0.841254 0.540641i −0.415415 + 0.909632i 0.730471 0.843008i −2.05274 + 1.31921i
197.1 0.654861 + 0.755750i 0.0741615 + 0.0476607i −0.142315 + 0.989821i 1.57773 3.45475i 0.0125459 + 0.0872586i 0.959493 + 0.281733i −0.841254 + 0.540641i −1.24302 2.72183i 3.64412 1.07001i
211.1 −0.415415 0.909632i 2.30075 0.675560i −0.654861 + 0.755750i 1.74412 + 1.12088i −1.57028 1.81219i 0.142315 + 0.989821i 0.959493 + 0.281733i 2.31329 1.48666i 0.295053 2.05214i
225.1 0.959493 + 0.281733i 1.01255 1.16854i 0.841254 + 0.540641i 0.324031 2.25368i 1.30075 0.835939i −0.415415 0.909632i 0.654861 + 0.755750i 0.0867074 + 0.603063i 0.945841 2.07110i
239.1 0.959493 0.281733i 1.01255 + 1.16854i 0.841254 0.540641i 0.324031 + 2.25368i 1.30075 + 0.835939i −0.415415 + 0.909632i 0.654861 0.755750i 0.0867074 0.603063i 0.945841 + 2.07110i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.1
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 322.2.i.b 10
23.c even 11 1 inner 322.2.i.b 10
23.c even 11 1 7406.2.a.bf 5
23.d odd 22 1 7406.2.a.be 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.b 10 1.a even 1 1 trivial
322.2.i.b 10 23.c even 11 1 inner
7406.2.a.be 5 23.d odd 22 1
7406.2.a.bf 5 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$3$ \( 1 - 20 T + 147 T^{2} - 135 T^{3} + 126 T^{4} - 100 T^{5} + 75 T^{6} - 37 T^{7} + 14 T^{8} - 5 T^{9} + T^{10} \)
$5$ \( 7921 - 12371 T + 9300 T^{2} - 4820 T^{3} + 2115 T^{4} - 846 T^{5} + 345 T^{6} - 138 T^{7} + 42 T^{8} - 8 T^{9} + T^{10} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$11$ \( 121 + 484 T + 2299 T^{2} + 1210 T^{3} + 121 T^{4} + 385 T^{5} + 22 T^{6} - 110 T^{7} + 55 T^{8} - 11 T^{9} + T^{10} \)
$13$ \( 1 - 9 T + 125 T^{2} + 52 T^{3} + 38 T^{4} + 54 T^{5} + 31 T^{6} + 18 T^{7} + 14 T^{8} + 6 T^{9} + T^{10} \)
$17$ \( 4489 + 402 T + 5668 T^{2} - 7539 T^{3} + 4365 T^{4} - 672 T^{5} - 116 T^{6} + 85 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$19$ \( 1 - 7 T + 82 T^{2} + 75 T^{3} - 19 T^{4} - 109 T^{5} - 29 T^{6} + 38 T^{7} + 31 T^{8} + 3 T^{9} + T^{10} \)
$23$ \( 6436343 + 5876661 T + 2153559 T^{2} + 267145 T^{3} - 62215 T^{4} - 27325 T^{5} - 2705 T^{6} + 505 T^{7} + 177 T^{8} + 21 T^{9} + T^{10} \)
$29$ \( 124609 + 180383 T + 139934 T^{2} + 55345 T^{3} + 9964 T^{4} - 2375 T^{5} - 248 T^{6} - 85 T^{7} + 15 T^{8} - 2 T^{9} + T^{10} \)
$31$ \( 11881 - 28340 T + 53058 T^{2} - 38432 T^{3} + 10739 T^{4} - 793 T^{5} - 18 T^{6} + 6 T^{7} + 20 T^{8} - 3 T^{9} + T^{10} \)
$37$ \( 7921 - 20381 T + 21696 T^{2} - 19704 T^{3} + 94286 T^{4} - 3345 T^{5} + 4673 T^{6} + 414 T^{7} - 30 T^{8} + 6 T^{9} + T^{10} \)
$41$ \( 73599241 + 24432992 T + 17476207 T^{2} + 2665928 T^{3} + 384957 T^{4} - 80739 T^{5} - 16796 T^{6} - 362 T^{7} + 221 T^{8} + 23 T^{9} + T^{10} \)
$43$ \( 4489 - 12462 T + 31516 T^{2} - 39918 T^{3} + 40701 T^{4} - 8964 T^{5} + 397 T^{6} + 254 T^{7} - 32 T^{8} + T^{9} + T^{10} \)
$47$ \( ( 4643 - 2782 T + 293 T^{2} + 85 T^{3} - 19 T^{4} + T^{5} )^{2} \)
$53$ \( 49266361 + 60054564 T + 53130576 T^{2} + 23632238 T^{3} + 6616285 T^{4} + 1239910 T^{5} + 165900 T^{6} + 16240 T^{7} + 1125 T^{8} + 49 T^{9} + T^{10} \)
$59$ \( 17161 + 56854 T + 287455 T^{2} + 42423 T^{3} + 213794 T^{4} - 64736 T^{5} + 1501 T^{6} + 2131 T^{7} + 4 T^{8} - 9 T^{9} + T^{10} \)
$61$ \( 89510521 + 10321951 T + 27361690 T^{2} + 7609170 T^{3} + 791774 T^{4} - 31901 T^{5} + 8435 T^{6} - 48 T^{7} + 14 T^{8} - 16 T^{9} + T^{10} \)
$67$ \( 529 - 4485 T + 16003 T^{2} - 31257 T^{3} + 36359 T^{4} - 23737 T^{5} + 10618 T^{6} - 2191 T^{7} + 214 T^{8} - 7 T^{9} + T^{10} \)
$71$ \( 310499641 - 73232876 T + 60922008 T^{2} + 9839261 T^{3} + 1984306 T^{4} - 78880 T^{5} + 17037 T^{6} - 51 T^{7} - 52 T^{8} + 16 T^{9} + T^{10} \)
$73$ \( 802985569 + 491080210 T + 94748195 T^{2} + 5675355 T^{3} + 198702 T^{4} - 64976 T^{5} + 23567 T^{6} + 2137 T^{7} + 268 T^{8} + 9 T^{9} + T^{10} \)
$79$ \( 189090001 + 105250154 T + 165179469 T^{2} - 60906975 T^{3} + 8268595 T^{4} - 557512 T^{5} + 43998 T^{6} - 3758 T^{7} + 102 T^{8} - 6 T^{9} + T^{10} \)
$83$ \( 79798489 - 80513129 T + 29738857 T^{2} - 5095286 T^{3} + 555514 T^{4} - 17150 T^{5} + 9244 T^{6} + 1249 T^{7} + 229 T^{8} + 19 T^{9} + T^{10} \)
$89$ \( 30724849 - 44549091 T + 24008781 T^{2} - 6369801 T^{3} + 1383275 T^{4} - 302787 T^{5} + 55334 T^{6} - 6099 T^{7} + 504 T^{8} - 25 T^{9} + T^{10} \)
$97$ \( 436921 + 1237392 T + 1652644 T^{2} + 1194729 T^{3} + 526091 T^{4} + 123617 T^{5} + 30226 T^{6} - 1951 T^{7} + 25 T^{8} - 16 T^{9} + T^{10} \)
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