Properties

Label 62.2.g.b
Level 62
Weight 2
Character orbit 62.g
Analytic conductor 0.495
Analytic rank 0
Dimension 8
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 62 = 2 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 62.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.495072492532\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.669131 0.743145i
0.669131 + 0.743145i
−0.104528 + 0.994522i
0.913545 0.406737i
−0.978148 + 0.207912i
−0.104528 0.994522i
−0.978148 0.207912i
0.913545 + 0.406737i
−0.309017 0.951057i 2.56082 0.544320i −0.809017 + 0.587785i −1.47815 + 2.56023i −1.30902 2.26728i −3.64728 1.62387i 0.809017 + 0.587785i 3.52090 1.56760i 2.89169 + 0.614648i
9.1 −0.309017 + 0.951057i 2.56082 + 0.544320i −0.809017 0.587785i −1.47815 2.56023i −1.30902 + 2.26728i −3.64728 + 1.62387i 0.809017 0.587785i 3.52090 + 1.56760i 2.89169 0.614648i
19.1 0.809017 + 0.587785i −0.348943 0.155360i 0.309017 + 0.951057i 0.413545 0.716282i −0.190983 0.330792i −0.981926 + 1.09054i −0.309017 + 0.951057i −1.90977 2.12101i 0.755585 0.336408i
41.1 0.809017 + 0.587785i 0.0399263 + 0.379874i 0.309017 + 0.951057i −0.604528 1.04707i −0.190983 + 0.330792i −3.01807 0.641511i −0.309017 + 0.951057i 2.79173 0.593401i 0.126381 1.20243i
45.1 −0.309017 + 0.951057i −1.75181 + 1.94558i −0.809017 0.587785i 0.169131 0.292943i −1.30902 2.26728i −0.352722 + 3.35592i 0.809017 0.587785i −0.402863 3.83299i 0.226341 + 0.251377i
49.1 0.809017 0.587785i −0.348943 + 0.155360i 0.309017 0.951057i 0.413545 + 0.716282i −0.190983 + 0.330792i −0.981926 1.09054i −0.309017 0.951057i −1.90977 + 2.12101i 0.755585 + 0.336408i
51.1 −0.309017 0.951057i −1.75181 1.94558i −0.809017 + 0.587785i 0.169131 + 0.292943i −1.30902 + 2.26728i −0.352722 3.35592i 0.809017 + 0.587785i −0.402863 + 3.83299i 0.226341 0.251377i
59.1 0.809017 0.587785i 0.0399263 0.379874i 0.309017 0.951057i −0.604528 + 1.04707i −0.190983 0.330792i −3.01807 + 0.641511i −0.309017 0.951057i 2.79173 + 0.593401i 0.126381 + 1.20243i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.2.g.b 8
3.b odd 2 1 558.2.ba.c 8
4.b odd 2 1 496.2.bg.b 8
31.g even 15 1 inner 62.2.g.b 8
31.g even 15 1 1922.2.a.h 4
31.h odd 30 1 1922.2.a.m 4
93.o odd 30 1 558.2.ba.c 8
124.n odd 30 1 496.2.bg.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.2.g.b 8 1.a even 1 1 trivial
62.2.g.b 8 31.g even 15 1 inner
496.2.bg.b 8 4.b odd 2 1
496.2.bg.b 8 124.n odd 30 1
558.2.ba.c 8 3.b odd 2 1
558.2.ba.c 8 93.o odd 30 1
1922.2.a.h 4 31.g even 15 1
1922.2.a.m 4 31.h odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(62, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( 1 - T - 2 T^{2} + 7 T^{3} - 12 T^{4} + 2 T^{5} + 16 T^{6} - 44 T^{7} + 73 T^{8} - 132 T^{9} + 144 T^{10} + 54 T^{11} - 972 T^{12} + 1701 T^{13} - 1458 T^{14} - 2187 T^{15} + 6561 T^{16} \)
$5$ \( 1 + 3 T - 10 T^{2} - 27 T^{3} + 94 T^{4} + 162 T^{5} - 595 T^{6} - 258 T^{7} + 3661 T^{8} - 1290 T^{9} - 14875 T^{10} + 20250 T^{11} + 58750 T^{12} - 84375 T^{13} - 156250 T^{14} + 234375 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 16 T + 127 T^{2} + 665 T^{3} + 2660 T^{4} + 9169 T^{5} + 29786 T^{6} + 92120 T^{7} + 259039 T^{8} + 644840 T^{9} + 1459514 T^{10} + 3144967 T^{11} + 6386660 T^{12} + 11176655 T^{13} + 14941423 T^{14} + 13176688 T^{15} + 5764801 T^{16} \)
$11$ \( 1 - 3 T + T^{2} - 12 T^{3} + 36 T^{4} + 261 T^{5} - 586 T^{6} + 3414 T^{7} - 22153 T^{8} + 37554 T^{9} - 70906 T^{10} + 347391 T^{11} + 527076 T^{12} - 1932612 T^{13} + 1771561 T^{14} - 58461513 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 27 T^{2} + 50 T^{3} + 480 T^{4} - 1000 T^{5} - 4157 T^{6} + 9000 T^{7} + 47779 T^{8} + 117000 T^{9} - 702533 T^{10} - 2197000 T^{11} + 13709280 T^{12} + 18564650 T^{13} - 130323843 T^{14} + 815730721 T^{16} \)
$17$ \( 1 + 2 T - 3 T^{2} + 9 T^{3} + 128 T^{4} + 981 T^{5} + 1394 T^{6} + 2132 T^{7} + 112113 T^{8} + 36244 T^{9} + 402866 T^{10} + 4819653 T^{11} + 10690688 T^{12} + 12778713 T^{13} - 72412707 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 3 T + 64 T^{2} - 279 T^{3} + 2202 T^{4} - 11568 T^{5} + 53798 T^{6} - 308064 T^{7} + 1089161 T^{8} - 5853216 T^{9} + 19421078 T^{10} - 79344912 T^{11} + 286966842 T^{12} - 690831621 T^{13} + 3010936384 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 15 T + 79 T^{2} - 60 T^{3} - 633 T^{4} - 3810 T^{5} + 36242 T^{6} + 75525 T^{7} - 1303075 T^{8} + 1737075 T^{9} + 19172018 T^{10} - 46356270 T^{11} - 177139353 T^{12} - 386180580 T^{13} + 11694835231 T^{14} - 51072381705 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 13 T + 30 T^{2} + 300 T^{3} - 1390 T^{4} - 7929 T^{5} + 57662 T^{6} + 293690 T^{7} - 3872925 T^{8} + 8517010 T^{9} + 48493742 T^{10} - 193380381 T^{11} - 983120590 T^{12} + 6153344700 T^{13} + 17844699630 T^{14} - 224248392017 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 11 T + 120 T^{2} + 799 T^{3} + 5429 T^{4} + 24769 T^{5} + 115320 T^{6} + 327701 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 8 T - 78 T^{2} + 554 T^{3} + 5603 T^{4} - 26439 T^{5} - 255041 T^{6} + 327007 T^{7} + 11441583 T^{8} + 12099259 T^{9} - 349151129 T^{10} - 1339214667 T^{11} + 10500924083 T^{12} + 38416552178 T^{13} - 200126659902 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 13 T + 6 T^{2} + 183 T^{3} + 2231 T^{4} + 2181 T^{5} - 186826 T^{6} + 468209 T^{7} + 1750392 T^{8} + 19196569 T^{9} - 314054506 T^{10} + 150316701 T^{11} + 6304272791 T^{12} + 21201684783 T^{13} + 28500625446 T^{14} - 2531805560453 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 73 T^{2} + 420 T^{3} + 4260 T^{4} + 29865 T^{5} + 245708 T^{6} + 2033505 T^{7} + 9196889 T^{8} + 87440715 T^{9} + 454314092 T^{10} + 2374476555 T^{11} + 14564092260 T^{12} + 61743546060 T^{13} + 461459502577 T^{14} + 11688200277601 T^{16} \)
$47$ \( 1 - 9 T - 82 T^{2} + 897 T^{3} + 846 T^{4} - 48114 T^{5} + 392170 T^{6} + 968706 T^{7} - 29959051 T^{8} + 45529182 T^{9} + 866303530 T^{10} - 4995339822 T^{11} + 4128210126 T^{12} + 205722471279 T^{13} - 883895656978 T^{14} - 4559608084167 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 51 T + 1333 T^{2} + 23358 T^{3} + 305973 T^{4} + 3190833 T^{5} + 27951071 T^{6} + 218219904 T^{7} + 1611432353 T^{8} + 11565654912 T^{9} + 78514558439 T^{10} + 475041644541 T^{11} + 2414274143013 T^{12} + 9768210325494 T^{13} + 29545093384957 T^{14} + 59910268131687 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 18 T + 434 T^{2} + 6189 T^{3} + 85872 T^{4} + 974688 T^{5} + 9899353 T^{6} + 90561729 T^{7} + 724594301 T^{8} + 5343142011 T^{9} + 34459647793 T^{10} + 200180446752 T^{11} + 1040542023792 T^{12} + 4424666486511 T^{13} + 18306351600194 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \)
$61$ \( ( 1 + 5 T + 27 T^{2} + 305 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( 1 + 18 T - 28 T^{2} - 954 T^{3} + 20113 T^{4} + 152739 T^{5} - 1389301 T^{6} + 1098423 T^{7} + 186163333 T^{8} + 73594341 T^{9} - 6236572189 T^{10} + 45938239857 T^{11} + 405299496673 T^{12} - 1288019352078 T^{13} - 2532834700732 T^{14} + 109092808895814 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 7 T + 66 T^{2} + 1092 T^{3} - 8389 T^{4} + 122679 T^{5} + 285629 T^{6} - 2628694 T^{7} + 99216147 T^{8} - 186637274 T^{9} + 1439855789 T^{10} + 43908163569 T^{11} - 213178591909 T^{12} + 1970218451292 T^{13} + 8454618738786 T^{14} - 63665841108737 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 13 T + 88 T^{2} + 380 T^{3} - 655 T^{4} + 2272 T^{5} + 334724 T^{6} + 4833155 T^{7} + 47712949 T^{8} + 352820315 T^{9} + 1783744196 T^{10} + 883846624 T^{11} - 18600847855 T^{12} + 787767205340 T^{13} + 13317411913432 T^{14} + 143616180748261 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 9 T - 146 T^{2} + 1086 T^{3} + 13821 T^{4} - 24237 T^{5} - 1649569 T^{6} - 1358472 T^{7} + 173053847 T^{8} - 107319288 T^{9} - 10294960129 T^{10} - 11949786243 T^{11} + 538329069501 T^{12} + 3341683249314 T^{13} - 35490768506066 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 6 T + 128 T^{2} - 483 T^{3} + 7158 T^{4} + 32232 T^{5} + 54781 T^{6} + 6441891 T^{7} - 233647 T^{8} + 534676953 T^{9} + 377386309 T^{10} + 18429838584 T^{11} + 339706661718 T^{12} - 1902556630569 T^{13} + 41848367791232 T^{14} - 162816305937762 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 28 T + 435 T^{2} - 5355 T^{3} + 60455 T^{4} - 536409 T^{5} + 3669257 T^{6} - 27140740 T^{7} + 255213705 T^{8} - 2415525860 T^{9} + 29064184697 T^{10} - 378151716321 T^{11} + 3793082179655 T^{12} - 29902638349395 T^{13} + 216186861568035 T^{14} - 1238477377074812 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - T + 88 T^{2} + 1093 T^{3} + 17636 T^{4} + 161624 T^{5} + 1223990 T^{6} + 26988604 T^{7} + 155964619 T^{8} + 2617894588 T^{9} + 11516521910 T^{10} + 147509860952 T^{11} + 1561302399716 T^{12} + 9385962900901 T^{13} + 73301536433752 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \)
show more
show less