Properties

Label 483.2.q.a
Level $483$
Weight $2$
Character orbit 483.q
Analytic conductor $3.857$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.q (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.654861 + 0.755750i
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 + 0.909632i
0.959493 0.281733i
0.959493 + 0.281733i
0.142315 0.989821i
−0.841254 + 0.540641i
0.654861 0.755750i
−0.415415 0.909632i
−1.52977 0.449181i −0.654861 + 0.755750i 0.455922 + 0.293003i 0.142315 0.989821i 1.34125 0.861971i 0.415415 + 0.909632i 1.52231 + 1.75684i −0.142315 0.989821i −0.662317 + 1.45027i
85.1 1.64589 1.89945i 0.841254 0.540641i −0.614354 4.27292i −0.415415 0.909632i 0.357685 2.48775i −0.959493 + 0.281733i −4.89867 3.14818i 0.415415 0.909632i −2.41153 0.708089i
127.1 1.85380 1.19136i −0.142315 + 0.989821i 1.18639 2.59784i 0.959493 0.281733i 0.915415 + 2.00448i −0.654861 0.755750i −0.268423 1.86692i −0.959493 0.281733i 1.44306 1.66538i
169.1 −0.0681534 + 0.474017i 0.415415 + 0.909632i 1.69894 + 0.498853i 0.654861 + 0.755750i −0.459493 + 0.134919i 0.841254 + 0.540641i −0.750131 + 1.64256i −0.654861 + 0.755750i −0.402869 + 0.258908i
190.1 0.0982369 0.215109i −0.959493 0.281733i 1.27310 + 1.46924i −0.841254 + 0.540641i −0.154861 + 0.178719i −0.142315 + 0.989821i 0.894911 0.262769i 0.841254 + 0.540641i 0.0336545 + 0.234072i
211.1 0.0982369 + 0.215109i −0.959493 + 0.281733i 1.27310 1.46924i −0.841254 0.540641i −0.154861 0.178719i −0.142315 0.989821i 0.894911 + 0.262769i 0.841254 0.540641i 0.0336545 0.234072i
232.1 1.85380 + 1.19136i −0.142315 0.989821i 1.18639 + 2.59784i 0.959493 + 0.281733i 0.915415 2.00448i −0.654861 + 0.755750i −0.268423 + 1.86692i −0.959493 + 0.281733i 1.44306 + 1.66538i
358.1 1.64589 + 1.89945i 0.841254 + 0.540641i −0.614354 + 4.27292i −0.415415 + 0.909632i 0.357685 + 2.48775i −0.959493 0.281733i −4.89867 + 3.14818i 0.415415 + 0.909632i −2.41153 + 0.708089i
400.1 −1.52977 + 0.449181i −0.654861 0.755750i 0.455922 0.293003i 0.142315 + 0.989821i 1.34125 + 0.861971i 0.415415 0.909632i 1.52231 1.75684i −0.142315 + 0.989821i −0.662317 1.45027i
463.1 −0.0681534 0.474017i 0.415415 0.909632i 1.69894 0.498853i 0.654861 0.755750i −0.459493 0.134919i 0.841254 0.540641i −0.750131 1.64256i −0.654861 0.755750i −0.402869 0.258908i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 463.1
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 483.2.q.a 10
23.c even 11 1 inner 483.2.q.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.q.a 10 1.a even 1 1 trivial
483.2.q.a 10 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 20 T^{2} - 5 T^{3} + 70 T^{4} - T^{5} - 30 T^{6} + 13 T^{7} + 5 T^{8} - 4 T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$11$ \( 1 - 10 T + 67 T^{2} + 23 T^{3} + 496 T^{4} + 298 T^{5} + 111 T^{6} + T^{7} - 10 T^{8} + T^{9} + T^{10} \)
$13$ \( 529 - 4623 T + 16553 T^{2} - 23908 T^{3} + 18176 T^{4} - 8790 T^{5} + 3094 T^{6} - 735 T^{7} + 137 T^{8} - 15 T^{9} + T^{10} \)
$17$ \( 978121 + 1756464 T + 1455237 T^{2} + 758259 T^{3} + 283215 T^{4} + 79166 T^{5} + 16868 T^{6} + 2692 T^{7} + 312 T^{8} + 24 T^{9} + T^{10} \)
$19$ \( 1 + 13 T + 103 T^{2} - 212 T^{3} + 225 T^{4} - 177 T^{5} + 97 T^{6} - 37 T^{7} + 14 T^{8} - 5 T^{9} + T^{10} \)
$23$ \( 6436343 - 5876661 T + 2555070 T^{2} - 808312 T^{3} + 224434 T^{4} - 52491 T^{5} + 9758 T^{6} - 1528 T^{7} + 210 T^{8} - 21 T^{9} + T^{10} \)
$29$ \( 123904 + 61952 T + 61952 T^{2} + 46464 T^{3} + 23232 T^{4} + 10912 T^{5} + 4400 T^{6} + 1232 T^{7} + 220 T^{8} + 22 T^{9} + T^{10} \)
$31$ \( 2105401 + 1857280 T + 1352004 T^{2} + 537165 T^{3} + 174738 T^{4} + 46584 T^{5} + 7253 T^{6} + 457 T^{7} + 20 T^{8} + 8 T^{9} + T^{10} \)
$37$ \( 2105401 - 4350098 T + 9592586 T^{2} - 3935221 T^{3} + 707617 T^{4} - 27556 T^{5} - 4450 T^{6} + 965 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} \)
$41$ \( 14645929 - 8052008 T + 1361490 T^{2} + 297984 T^{3} - 86049 T^{4} + 10264 T^{5} + 2236 T^{6} - 178 T^{7} + 115 T^{8} - 7 T^{9} + T^{10} \)
$43$ \( 6848689 + 607144 T + 1989362 T^{2} + 170114 T^{3} + 156190 T^{4} + 39104 T^{5} + 3928 T^{6} - 551 T^{7} - 87 T^{8} - T^{9} + T^{10} \)
$47$ \( ( 991 + 1189 T + 62 T^{2} - 102 T^{3} + 3 T^{4} + T^{5} )^{2} \)
$53$ \( 737881 + 1257576 T + 1201685 T^{2} + 656195 T^{3} + 196186 T^{4} + 20428 T^{5} + 837 T^{6} + 749 T^{7} + 122 T^{8} + T^{9} + T^{10} \)
$59$ \( 229734649 - 48532714 T - 8037160 T^{2} + 200190 T^{3} + 447030 T^{4} - 38312 T^{5} + 25609 T^{6} - 857 T^{7} + 177 T^{8} - 10 T^{9} + T^{10} \)
$61$ \( 212521 - 87129 T + 10608 T^{2} - 18527 T^{3} + 7023 T^{4} - 463 T^{5} + 581 T^{6} - 136 T^{7} + 25 T^{8} - 5 T^{9} + T^{10} \)
$67$ \( 174266401 - 101027253 T + 286106555 T^{2} - 63133747 T^{3} + 9452403 T^{4} - 468159 T^{5} - 19005 T^{6} + 1329 T^{7} + 247 T^{8} - 29 T^{9} + T^{10} \)
$71$ \( 33860761 - 27971933 T + 17986529 T^{2} - 5529821 T^{3} + 1171764 T^{4} - 172689 T^{5} + 17424 T^{6} - 979 T^{7} + 154 T^{8} - 22 T^{9} + T^{10} \)
$73$ \( 543169 + 826914 T + 2072004 T^{2} + 1807014 T^{3} + 826551 T^{4} + 235928 T^{5} + 45474 T^{6} + 6094 T^{7} + 561 T^{8} + 33 T^{9} + T^{10} \)
$79$ \( 867832681 - 181732571 T + 101363541 T^{2} - 16137729 T^{3} + 3881146 T^{4} + 480424 T^{5} - 43056 T^{6} + 656 T^{7} + 333 T^{8} - 27 T^{9} + T^{10} \)
$83$ \( 3695059369 - 2067730592 T + 543086813 T^{2} - 89553803 T^{3} + 11362200 T^{4} - 1219494 T^{5} + 107397 T^{6} - 7573 T^{7} + 460 T^{8} - 25 T^{9} + T^{10} \)
$89$ \( 14645929 - 34443 T + 8596251 T^{2} + 8137803 T^{3} + 3396376 T^{4} + 810932 T^{5} + 122494 T^{6} + 12016 T^{7} + 806 T^{8} + 38 T^{9} + T^{10} \)
$97$ \( 5993391889 + 421458148 T + 62311657 T^{2} + 13148719 T^{3} + 1204270 T^{4} + 48016 T^{5} + 4093 T^{6} + 661 T^{7} + 34 T^{8} + T^{9} + T^{10} \)
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