# 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 $$15$$, 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{15}^{6} q^{2} + ( \zeta_{15} + \zeta_{15}^{4} + \zeta_{15}^{7} ) q^{3} + ( -\zeta_{15}^{2} - \zeta_{15}^{7} ) q^{4} + ( -1 + \zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{3} + 2 \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{5} + ( -\zeta_{15} - \zeta_{15}^{4} ) q^{6} + ( 1 - \zeta_{15} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} + \zeta_{15}^{3} q^{8} + ( 1 - 2 \zeta_{15} + \zeta_{15}^{3} - \zeta_{15}^{4} + 2 \zeta_{15}^{5} - \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{9} +O(q^{10})$$ $$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 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.a 8
3.b odd 2 1 558.2.ba.f 8
4.b odd 2 1 496.2.bg.a 8
31.g even 15 1 inner 62.2.g.a 8
31.g even 15 1 1922.2.a.q 4
31.h odd 30 1 1922.2.a.o 4
93.o odd 30 1 558.2.ba.f 8
124.n odd 30 1 496.2.bg.a 8

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

## Hecke kernels

This newform subspace 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 $$S_{2}^{\mathrm{new}}(62, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$3$ $$1 - 3 T + 8 T^{2} - 5 T^{3} + 28 T^{5} + 24 T^{6} - 110 T^{7} + 409 T^{8} - 330 T^{9} + 216 T^{10} + 756 T^{11} - 1215 T^{13} + 5832 T^{14} - 6561 T^{15} + 6561 T^{16}$$
$5$ $$1 - T - 10 T^{2} + 21 T^{3} + 44 T^{4} - 134 T^{5} + 55 T^{6} + 384 T^{7} - 959 T^{8} + 1920 T^{9} + 1375 T^{10} - 16750 T^{11} + 27500 T^{12} + 65625 T^{13} - 156250 T^{14} - 78125 T^{15} + 390625 T^{16}$$
$7$ $$1 - 2 T + 17 T^{2} - 29 T^{3} + 178 T^{4} - 281 T^{5} + 1444 T^{6} - 1692 T^{7} + 10263 T^{8} - 11844 T^{9} + 70756 T^{10} - 96383 T^{11} + 427378 T^{12} - 487403 T^{13} + 2000033 T^{14} - 1647086 T^{15} + 5764801 T^{16}$$
$11$ $$1 + 13 T + 111 T^{2} + 702 T^{3} + 3806 T^{4} + 17889 T^{5} + 76094 T^{6} + 291046 T^{7} + 1016667 T^{8} + 3201506 T^{9} + 9207374 T^{10} + 23810259 T^{11} + 55723646 T^{12} + 113057802 T^{13} + 196643271 T^{14} + 253333223 T^{15} + 214358881 T^{16}$$
$13$ $$1 - 27 T^{2} - 50 T^{3} + 480 T^{4} + 1000 T^{5} - 4157 T^{6} - 9000 T^{7} + 47779 T^{8} - 117000 T^{9} - 702533 T^{10} + 2197000 T^{11} + 13709280 T^{12} - 18564650 T^{13} - 130323843 T^{14} + 815730721 T^{16}$$
$17$ $$1 + 2 T - 43 T^{2} - 51 T^{3} + 328 T^{4} - 759 T^{5} + 12554 T^{6} + 17852 T^{7} - 360647 T^{8} + 303484 T^{9} + 3628106 T^{10} - 3728967 T^{11} + 27394888 T^{12} - 72412707 T^{13} - 1037915467 T^{14} + 820677346 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 3 T - 26 T^{2} + 9 T^{3} + 762 T^{4} - 582 T^{5} - 13702 T^{6} - 3486 T^{7} + 152081 T^{8} - 66234 T^{9} - 4946422 T^{10} - 3991938 T^{11} + 99304602 T^{12} + 22284891 T^{13} - 1223192906 T^{14} + 2681615217 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 3 T + 7 T^{2} - 204 T^{3} + 1161 T^{4} - 3168 T^{5} + 22910 T^{6} - 131367 T^{7} + 625349 T^{8} - 3021441 T^{9} + 12119390 T^{10} - 38545056 T^{11} + 324895401 T^{12} - 1313013972 T^{13} + 1036251223 T^{14} - 10214476341 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 11 T + 14 T^{2} + 74 T^{3} + 506 T^{4} - 1933 T^{5} + 11316 T^{6} - 73148 T^{7} + 5907 T^{8} - 2121292 T^{9} + 9516756 T^{10} - 47143937 T^{11} + 357884186 T^{12} + 1517825026 T^{13} + 8327526494 T^{14} - 189748639399 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 11 T + 60 T^{2} + 11 T^{3} - 991 T^{4} + 341 T^{5} + 57660 T^{6} - 327701 T^{7} + 923521 T^{8}$$
$37$ $$1 - 6 T - 78 T^{2} + 330 T^{3} + 4225 T^{4} - 7809 T^{5} - 196479 T^{6} + 132135 T^{7} + 7291959 T^{8} + 4888995 T^{9} - 268979751 T^{10} - 395549277 T^{11} + 7918330225 T^{12} + 22883505810 T^{13} - 200126659902 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 + 19 T + 266 T^{2} + 2343 T^{3} + 19987 T^{4} + 136629 T^{5} + 992162 T^{6} + 5775793 T^{7} + 39291616 T^{8} + 236807513 T^{9} + 1667824322 T^{10} + 9416607309 T^{11} + 56478485107 T^{12} + 271451078943 T^{13} + 1263527728106 T^{14} + 3700331203739 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 26 T + 283 T^{2} + 1288 T^{3} - 1732 T^{4} - 36097 T^{5} + 136466 T^{6} + 4437349 T^{7} + 39368833 T^{8} + 190806007 T^{9} + 252325634 T^{10} - 2869964179 T^{11} - 5921363332 T^{12} + 189346874584 T^{13} + 1788945742867 T^{14} + 7067283888782 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - T - 22 T^{2} - 137 T^{3} + 2366 T^{4} + 4924 T^{5} - 33810 T^{6} - 202576 T^{7} + 5800929 T^{8} - 9521072 T^{9} - 74686290 T^{10} + 511224452 T^{11} + 11545325246 T^{12} - 31420265959 T^{13} - 237142737238 T^{14} - 506623120463 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 17 T + 153 T^{2} - 1080 T^{3} + 5585 T^{4} - 1443 T^{5} - 181831 T^{6} + 2916400 T^{7} - 28141191 T^{8} + 154569200 T^{9} - 510763279 T^{10} - 214829511 T^{11} + 44068336385 T^{12} - 451651132440 T^{13} + 3391147252737 T^{14} - 19970089377229 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 8 T + 44 T^{2} - 949 T^{3} + 10412 T^{4} - 60868 T^{5} + 591603 T^{6} - 5355299 T^{7} + 38621781 T^{8} - 315962641 T^{9} + 2059370043 T^{10} - 12501008972 T^{11} + 126165962732 T^{12} - 678463159751 T^{13} + 1855943480204 T^{14} - 19909211878552 T^{15} + 146830437604321 T^{16}$$
$61$ $$( 1 - 24 T + 385 T^{2} - 4356 T^{3} + 39249 T^{4} - 265716 T^{5} + 1432585 T^{6} - 5447544 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 12 T - 138 T^{2} + 1146 T^{3} + 23563 T^{4} - 99741 T^{5} - 2275281 T^{6} + 1600923 T^{7} + 194106273 T^{8} + 107261841 T^{9} - 10213736409 T^{10} - 29998402383 T^{11} + 474820864123 T^{12} + 1547243372622 T^{13} - 12483256739322 T^{14} - 72728539263876 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 9 T + 146 T^{2} - 1158 T^{3} + 10347 T^{4} - 88539 T^{5} + 812197 T^{6} - 7118208 T^{7} + 60753011 T^{8} - 505392768 T^{9} + 4094285077 T^{10} - 31689082029 T^{11} + 262934663307 T^{12} - 2089297588458 T^{13} + 18702641452466 T^{14} - 81856081425519 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 33 T + 768 T^{2} + 12580 T^{3} + 176025 T^{4} + 2064592 T^{5} + 22155244 T^{6} + 211797255 T^{7} + 1902972349 T^{8} + 15461199615 T^{9} + 118065295276 T^{10} + 803161386064 T^{11} + 4998800372025 T^{12} + 26079240639940 T^{13} + 116224685789952 T^{14} + 364564151130201 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 19 T + 394 T^{2} + 5504 T^{3} + 71021 T^{4} + 825307 T^{5} + 8678231 T^{6} + 88869362 T^{7} + 794929807 T^{8} + 7020679598 T^{9} + 54160839671 T^{10} + 406908537973 T^{11} + 2766273702701 T^{12} + 16936118420096 T^{13} + 95776457475274 T^{14} + 364874270737021 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 40 T + 698 T^{2} - 6665 T^{3} + 27170 T^{4} + 192160 T^{5} - 3936477 T^{6} + 31340435 T^{7} - 222267711 T^{8} + 2601256105 T^{9} - 27118390053 T^{10} + 109874589920 T^{11} + 1289442581570 T^{12} - 26253705885595 T^{13} + 228204380611562 T^{14} - 1085442039585080 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 - 8 T - 145 T^{2} + 1605 T^{3} + 12835 T^{4} - 42189 T^{5} - 2038663 T^{6} - 1693160 T^{7} + 286639345 T^{8} - 150691240 T^{9} - 16148249623 T^{10} - 29741937141 T^{11} + 805296663235 T^{12} + 8962415415645 T^{13} - 72062287189345 T^{14} - 353850679164232 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 + 23 T + 244 T^{2} + 3433 T^{3} + 48488 T^{4} + 513248 T^{5} + 6282506 T^{6} + 62989744 T^{7} + 526428043 T^{8} + 6110005168 T^{9} + 59112098954 T^{10} + 468427591904 T^{11} + 4292607777128 T^{12} + 29480339102281 T^{13} + 203245169202676 T^{14} + 1858360542996599 T^{15} + 7837433594376961 T^{16}$$