# Properties

 Label 315.2.i.d Level 315 Weight 2 Character orbit 315.i Analytic conductor 2.515 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{15})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{15}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 −0.104528 − 0.994522i 0.669131 + 0.743145i 0.913545 − 0.406737i −0.978148 − 0.207912i −0.104528 + 0.994522i 0.669131 − 0.743145i 0.913545 + 0.406737i −0.978148 + 0.207912i
−0.978148 + 1.69420i 1.72256 + 0.181049i −0.913545 1.58231i 0.500000 + 0.866025i −1.99165 + 2.74128i 0.500000 0.866025i −0.338261 2.93444 + 0.623735i −1.95630
106.2 −0.104528 + 0.181049i −1.28716 1.15897i 0.978148 + 1.69420i 0.500000 + 0.866025i 0.344375 0.111894i 0.500000 0.866025i −0.827091 0.313585 + 2.98357i −0.209057
106.3 0.669131 1.15897i 0.704489 1.58231i 0.104528 + 0.181049i 0.500000 + 0.866025i −1.36245 1.87525i 0.500000 0.866025i 2.95630 −2.00739 2.22943i 1.33826
106.4 0.913545 1.58231i 0.360114 + 1.69420i −0.669131 1.15897i 0.500000 + 0.866025i 3.00973 + 0.977920i 0.500000 0.866025i 1.20906 −2.74064 + 1.22021i 1.82709
211.1 −0.978148 1.69420i 1.72256 0.181049i −0.913545 + 1.58231i 0.500000 0.866025i −1.99165 2.74128i 0.500000 + 0.866025i −0.338261 2.93444 0.623735i −1.95630
211.2 −0.104528 0.181049i −1.28716 + 1.15897i 0.978148 1.69420i 0.500000 0.866025i 0.344375 + 0.111894i 0.500000 + 0.866025i −0.827091 0.313585 2.98357i −0.209057
211.3 0.669131 + 1.15897i 0.704489 + 1.58231i 0.104528 0.181049i 0.500000 0.866025i −1.36245 + 1.87525i 0.500000 + 0.866025i 2.95630 −2.00739 + 2.22943i 1.33826
211.4 0.913545 + 1.58231i 0.360114 1.69420i −0.669131 + 1.15897i 0.500000 0.866025i 3.00973 0.977920i 0.500000 + 0.866025i 1.20906 −2.74064 1.22021i 1.82709
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.i.d 8
3.b odd 2 1 945.2.i.c 8
9.c even 3 1 inner 315.2.i.d 8
9.c even 3 1 2835.2.a.l 4
9.d odd 6 1 945.2.i.c 8
9.d odd 6 1 2835.2.a.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.d 8 1.a even 1 1 trivial
315.2.i.d 8 9.c even 3 1 inner
945.2.i.c 8 3.b odd 2 1
945.2.i.c 8 9.d odd 6 1
2835.2.a.l 4 9.c even 3 1
2835.2.a.q 4 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 50 T^{5} + 52 T^{6} - 40 T^{7} + 16 T^{8} )( 1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8} )$$
$3$ $$1 - 3 T + 6 T^{2} - 9 T^{3} + 9 T^{4} - 27 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8}$$
$5$ $$( 1 - T + T^{2} )^{4}$$
$7$ $$( 1 - T + T^{2} )^{4}$$
$11$ $$1 + 7 T - 9 T^{2} - 72 T^{3} + 656 T^{4} + 2001 T^{5} - 7276 T^{6} + 844 T^{7} + 151647 T^{8} + 9284 T^{9} - 880396 T^{10} + 2663331 T^{11} + 9604496 T^{12} - 11595672 T^{13} - 15944049 T^{14} + 136410197 T^{15} + 214358881 T^{16}$$
$13$ $$1 - 8 T - 2 T^{2} + 50 T^{3} + 725 T^{4} - 1727 T^{5} - 11281 T^{6} + 7925 T^{7} + 170479 T^{8} + 103025 T^{9} - 1906489 T^{10} - 3794219 T^{11} + 20706725 T^{12} + 18564650 T^{13} - 9653618 T^{14} - 501988136 T^{15} + 815730721 T^{16}$$
$17$ $$( 1 - T + 64 T^{2} - 47 T^{3} + 1599 T^{4} - 799 T^{5} + 18496 T^{6} - 4913 T^{7} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 3 T + 45 T^{2} - 168 T^{3} + 989 T^{4} - 3192 T^{5} + 16245 T^{6} - 20577 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 - 8 T + 18 T^{2} - 240 T^{3} + 1445 T^{4} - 1872 T^{5} + 33674 T^{6} - 166760 T^{7} + 89244 T^{8} - 3835480 T^{9} + 17813546 T^{10} - 22776624 T^{11} + 404370245 T^{12} - 1544722320 T^{13} + 2664646002 T^{14} - 27238603576 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + T - 61 T^{2} + 156 T^{3} + 1796 T^{4} - 8467 T^{5} - 22034 T^{6} + 147078 T^{7} + 433657 T^{8} + 4265262 T^{9} - 18530594 T^{10} - 206501663 T^{11} + 1270276676 T^{12} + 3199739244 T^{13} - 36284222581 T^{14} + 17249876309 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 94 T^{2} - 90 T^{3} + 4885 T^{4} + 5625 T^{5} - 188701 T^{6} - 84465 T^{7} + 6132709 T^{8} - 2618415 T^{9} - 181341661 T^{10} + 167574375 T^{11} + 4511400085 T^{12} - 2576623590 T^{13} - 83425346014 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 + 21 T + 289 T^{2} + 2592 T^{3} + 18369 T^{4} + 95904 T^{5} + 395641 T^{6} + 1063713 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 + 20 T + 126 T^{2} + 390 T^{3} + 5225 T^{4} + 43635 T^{5} + 118739 T^{6} + 1639535 T^{7} + 19690659 T^{8} + 67220935 T^{9} + 199600259 T^{10} + 3007367835 T^{11} + 14764601225 T^{12} + 45183918390 T^{13} + 598513134366 T^{14} + 3895085477620 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 - 7 T - 77 T^{2} + 160 T^{3} + 5570 T^{4} + 5177 T^{5} - 258466 T^{6} - 11150 T^{7} + 7688479 T^{8} - 479450 T^{9} - 477903634 T^{10} + 411607739 T^{11} + 19042721570 T^{12} + 23521350880 T^{13} - 486744954773 T^{14} - 1902730277749 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 2 T - 58 T^{2} - 714 T^{3} + 2873 T^{4} + 38399 T^{5} + 291889 T^{6} - 1740327 T^{7} - 14193947 T^{8} - 81795369 T^{9} + 644782801 T^{10} + 3986699377 T^{11} + 14019323513 T^{12} - 163752334998 T^{13} - 625194489082 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$
$53$ $$( 1 + 8 T + 46 T^{2} - 401 T^{3} - 3531 T^{4} - 21253 T^{5} + 129214 T^{6} + 1191016 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 + 19 T + 59 T^{2} - 876 T^{3} - 244 T^{4} + 106457 T^{5} + 689896 T^{6} + 657732 T^{7} - 7735313 T^{8} + 38806188 T^{9} + 2401527976 T^{10} + 21864032203 T^{11} - 2956636084 T^{12} - 626273685924 T^{13} + 2488651484819 T^{14} + 47284378211561 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 12 T - 49 T^{2} + 852 T^{3} + 1876 T^{4} - 9576 T^{5} - 372571 T^{6} + 33096 T^{7} + 27681967 T^{8} + 2018856 T^{9} - 1386336691 T^{10} - 2173570056 T^{11} + 25974797716 T^{12} + 719596048452 T^{13} - 2524498343689 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16}$$
$67$ $$( 1 - 11 T - 42 T^{2} - 319 T^{3} + 13973 T^{4} - 21373 T^{5} - 188538 T^{6} - 3308393 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 13 T + 248 T^{2} - 1951 T^{3} + 23005 T^{4} - 138521 T^{5} + 1250168 T^{6} - 4652843 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 4 T + 78 T^{2} + 545 T^{3} + 761 T^{4} + 39785 T^{5} + 415662 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 - 24 T + 279 T^{2} - 2304 T^{3} + 10516 T^{4} + 43728 T^{5} - 930339 T^{6} + 9097068 T^{7} - 88845273 T^{8} + 718668372 T^{9} - 5806245699 T^{10} + 21559609392 T^{11} + 409599051796 T^{12} - 7089537943296 T^{13} + 67821400090359 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 19 T + 8 T^{2} + 1653 T^{3} - 472 T^{4} - 87362 T^{5} - 374159 T^{6} - 2991816 T^{7} + 141658213 T^{8} - 248320728 T^{9} - 2577581351 T^{10} - 49952455894 T^{11} - 22400327512 T^{12} + 6511234182879 T^{13} + 2615522986952 T^{14} - 515584968802913 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 15 T + 241 T^{2} - 1380 T^{3} + 17331 T^{4} - 122820 T^{5} + 1908961 T^{6} - 10574535 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 12 T - 33 T^{2} + 1956 T^{3} - 21452 T^{4} + 120264 T^{5} + 108549 T^{6} - 18621312 T^{7} + 276817383 T^{8} - 1806267264 T^{9} + 1021337541 T^{10} + 109761705672 T^{11} - 1899130136012 T^{12} + 16796837542692 T^{13} - 27488076162657 T^{14} - 969579413737356 T^{15} + 7837433594376961 T^{16}$$