# 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

## Newspace parameters

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

## 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: $$1$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
31.g Even 1 yes

## Hecke kernels

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