Properties

Label 483.2.q.b
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.

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

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
2.30075 + 0.675560i −0.654861 + 0.755750i 3.15455 + 2.02730i −0.195368 + 1.35881i −2.01722 + 1.29639i 0.415415 + 0.909632i 2.74769 + 3.17101i −0.142315 0.989821i −1.36745 + 2.99430i
85.1 1.01255 1.16854i 0.841254 0.540641i −0.0556075 0.386758i 0.996114 + 2.18119i 0.220047 1.53046i −0.959493 + 0.281733i 2.09325 + 1.34525i 0.415415 0.909632i 3.55742 + 1.04455i
127.1 0.0741615 0.0476607i −0.142315 + 0.989821i −0.827602 + 1.81219i −1.48357 + 0.435615i 0.0366213 + 0.0801894i −0.654861 0.755750i 0.0500861 + 0.348356i −0.959493 0.281733i −0.0892619 + 0.103014i
169.1 −0.317178 + 2.20602i 0.415415 + 0.909632i −2.84695 0.835939i 0.0577299 + 0.0666238i −2.13843 + 0.627899i 0.841254 + 0.540641i 0.895412 1.96068i −0.654861 + 0.755750i −0.165284 + 0.106222i
190.1 −0.570276 + 1.24873i −0.959493 0.281733i 0.0756100 + 0.0872586i −1.87491 + 1.20493i 0.898983 1.03748i −0.142315 + 0.989821i −2.78644 + 0.818172i 0.841254 + 0.540641i −0.435418 3.02840i
211.1 −0.570276 1.24873i −0.959493 + 0.281733i 0.0756100 0.0872586i −1.87491 1.20493i 0.898983 + 1.03748i −0.142315 0.989821i −2.78644 0.818172i 0.841254 0.540641i −0.435418 + 3.02840i
232.1 0.0741615 + 0.0476607i −0.142315 0.989821i −0.827602 1.81219i −1.48357 0.435615i 0.0366213 0.0801894i −0.654861 + 0.755750i 0.0500861 0.348356i −0.959493 + 0.281733i −0.0892619 0.103014i
358.1 1.01255 + 1.16854i 0.841254 + 0.540641i −0.0556075 + 0.386758i 0.996114 2.18119i 0.220047 + 1.53046i −0.959493 0.281733i 2.09325 1.34525i 0.415415 + 0.909632i 3.55742 1.04455i
400.1 2.30075 0.675560i −0.654861 0.755750i 3.15455 2.02730i −0.195368 1.35881i −2.01722 1.29639i 0.415415 0.909632i 2.74769 3.17101i −0.142315 + 0.989821i −1.36745 2.99430i
463.1 −0.317178 2.20602i 0.415415 0.909632i −2.84695 + 0.835939i 0.0577299 0.0666238i −2.13843 0.627899i 0.841254 0.540641i 0.895412 + 1.96068i −0.654861 0.755750i −0.165284 0.106222i
\(n\): 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.b 10
23.c even 11 1 inner 483.2.q.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.q.b 10 1.a even 1 1 trivial
483.2.q.b 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

$p$ $F_p(T)$
$2$ \( 1 - 20 T + 147 T^{2} - 135 T^{3} + 126 T^{4} - 100 T^{5} + 75 T^{6} - 37 T^{7} + 14 T^{8} - 5 T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 1 - 13 T + 103 T^{2} + 212 T^{3} + 225 T^{4} + 177 T^{5} + 97 T^{6} + 37 T^{7} + 14 T^{8} + 5 T^{9} + T^{10} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$11$ \( 1 - 38 T + 542 T^{2} - 1456 T^{3} + 1758 T^{4} - 1244 T^{5} + 632 T^{6} - 223 T^{7} + 59 T^{8} - 9 T^{9} + T^{10} \)
$13$ \( 529 + 1840 T + 3298 T^{2} + 3976 T^{3} + 3491 T^{4} + 2300 T^{5} + 1147 T^{6} + 416 T^{7} + 104 T^{8} + 15 T^{9} + T^{10} \)
$17$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$19$ \( 1849 + 5375 T + 10851 T^{2} + 14885 T^{3} + 13676 T^{4} + 8392 T^{5} + 3546 T^{6} + 1018 T^{7} + 185 T^{8} + 19 T^{9} + T^{10} \)
$23$ \( 6436343 + 5876661 T + 2956581 T^{2} + 1011977 T^{3} + 270733 T^{4} + 60543 T^{5} + 11771 T^{6} + 1913 T^{7} + 243 T^{8} + 21 T^{9} + T^{10} \)
$29$ \( 121 - 242 T + 7986 T^{2} - 34969 T^{3} + 56144 T^{4} + 6512 T^{5} + 517 T^{6} - 649 T^{7} + 22 T^{8} + T^{10} \)
$31$ \( 1024 - 512 T + 256 T^{2} - 128 T^{3} + 64 T^{4} - 32 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} \)
$37$ \( 17161 + 10349 T + 166687 T^{2} + 251177 T^{3} + 170351 T^{4} + 68025 T^{5} + 17730 T^{6} + 3095 T^{7} + 366 T^{8} + 27 T^{9} + T^{10} \)
$41$ \( 58081 + 132309 T + 2663 T^{2} - 299133 T^{3} + 211663 T^{4} - 55362 T^{5} + 20516 T^{6} - 934 T^{7} + 155 T^{8} - 10 T^{9} + T^{10} \)
$43$ \( 1849 + 8170 T + 12032 T^{2} - 4262 T^{3} + 4514 T^{4} - 1002 T^{5} + 1246 T^{6} - 207 T^{7} + 71 T^{8} - 15 T^{9} + T^{10} \)
$47$ \( ( 1 + 4 T - 9 T^{2} - 5 T^{3} + 9 T^{4} + T^{5} )^{2} \)
$53$ \( 6285049 + 10993195 T + 3543820 T^{2} - 1478721 T^{3} + 450101 T^{4} - 87647 T^{5} + 15305 T^{6} - 1722 T^{7} + 251 T^{8} - 19 T^{9} + T^{10} \)
$59$ \( 549011761 - 42902161 T + 34946090 T^{2} + 170045 T^{3} + 11295 T^{4} + 8975 T^{5} + 8651 T^{6} + 382 T^{7} + 15 T^{8} + 13 T^{9} + T^{10} \)
$61$ \( 389825536 - 8845312 T - 30119168 T^{2} - 5040128 T^{3} + 863680 T^{4} + 319648 T^{5} + 60448 T^{6} + 5392 T^{7} + 412 T^{8} + 18 T^{9} + T^{10} \)
$67$ \( 529 - 2093 T + 9733 T^{2} - 20245 T^{3} + 29011 T^{4} - 9186 T^{5} - 162 T^{6} - 108 T^{7} + 115 T^{8} + 18 T^{9} + T^{10} \)
$71$ \( 12538681 - 10173293 T + 14414668 T^{2} - 6689906 T^{3} + 1612792 T^{4} - 273725 T^{5} + 48079 T^{6} - 7418 T^{7} + 718 T^{8} - 38 T^{9} + T^{10} \)
$73$ \( 51739249 - 74109479 T + 43270914 T^{2} - 13485437 T^{3} + 2618322 T^{4} - 350109 T^{5} + 37866 T^{6} - 3327 T^{7} + 273 T^{8} - 14 T^{9} + T^{10} \)
$79$ \( 9765625 + 25390625 T + 40234375 T^{2} - 16562500 T^{3} + 3515625 T^{4} - 553125 T^{5} + 60625 T^{6} - 4625 T^{7} + 350 T^{8} - 25 T^{9} + T^{10} \)
$83$ \( 69169 + 70484 T + 119014 T^{2} + 138448 T^{3} + 70700 T^{4} + 17984 T^{5} + 4338 T^{6} + 1029 T^{7} + 185 T^{8} + 19 T^{9} + T^{10} \)
$89$ \( 570875449 + 278902989 T + 584187773 T^{2} + 116895589 T^{3} + 9060397 T^{4} - 212917 T^{5} - 67839 T^{6} - 4151 T^{7} + 113 T^{8} + 17 T^{9} + T^{10} \)
$97$ \( 2220577129 + 987556711 T + 253613188 T^{2} + 53103823 T^{3} + 9201758 T^{4} + 1333949 T^{5} + 166197 T^{6} + 16251 T^{7} + 1147 T^{8} + 49 T^{9} + T^{10} \)
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