# Properties

 Label 62.2.g.a 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.

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

## 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 0.604528 0.128496i −0.809017 + 0.587785i 0.139886 0.242290i 0.309017 + 0.535233i −0.309017 0.137583i −0.809017 0.587785i −2.39169 + 1.06485i 0.273659 + 0.0581680i
9.1 0.309017 0.951057i 0.604528 + 0.128496i −0.809017 0.587785i 0.139886 + 0.242290i 0.309017 0.535233i −0.309017 + 0.137583i −0.809017 + 0.587785i −2.39169 1.06485i 0.273659 0.0581680i
19.1 −0.809017 0.587785i 1.47815 + 0.658114i 0.309017 + 0.951057i −0.204489 + 0.354185i −0.809017 1.40126i 0.809017 0.898504i 0.309017 0.951057i −0.255585 0.283856i 0.373619 0.166346i
41.1 −0.809017 0.587785i −0.169131 1.60917i 0.309017 + 0.951057i −1.22256 2.11754i −0.809017 + 1.40126i 0.809017 + 0.171962i 0.309017 0.951057i 0.373619 0.0794152i −0.255585 + 2.43173i
45.1 0.309017 0.951057i −0.413545 + 0.459289i −0.809017 0.587785i 1.78716 3.09546i 0.309017 + 0.535233i −0.309017 + 2.94010i −0.809017 + 0.587785i 0.273659 + 2.60369i −2.39169 2.65624i
49.1 −0.809017 + 0.587785i 1.47815 0.658114i 0.309017 0.951057i −0.204489 0.354185i −0.809017 + 1.40126i 0.809017 + 0.898504i 0.309017 + 0.951057i −0.255585 + 0.283856i 0.373619 + 0.166346i
51.1 0.309017 + 0.951057i −0.413545 0.459289i −0.809017 + 0.587785i 1.78716 + 3.09546i 0.309017 0.535233i −0.309017 2.94010i −0.809017 0.587785i 0.273659 2.60369i −2.39169 + 2.65624i
59.1 −0.809017 + 0.587785i −0.169131 + 1.60917i 0.309017 0.951057i −1.22256 + 2.11754i −0.809017 1.40126i 0.809017 0.171962i 0.309017 + 0.951057i 0.373619 + 0.0794152i −0.255585 2.43173i
 $$n$$: 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 $$T_{3}^{8} - 3 T_{3}^{7} + 5 T_{3}^{6} - 8 T_{3}^{5} + 9 T_{3}^{4} - 2 T_{3}^{3} - 2 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(62, [\chi])$$.