# Properties

 Label 350.2.h.a Level 350 Weight 2 Character orbit 350.h Analytic conductor 2.795 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\Q(\zeta_{15})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\zeta_{15}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 −0.978148 + 0.207912i 0.669131 + 0.743145i −0.104528 + 0.994522i 0.913545 − 0.406737i −0.104528 − 0.994522i 0.913545 + 0.406737i −0.978148 − 0.207912i 0.669131 − 0.743145i
0.309017 0.951057i −0.169131 0.122881i −0.809017 0.587785i 1.11803 1.93649i −0.169131 + 0.122881i 1.00000 −0.809017 + 0.587785i −0.913545 2.81160i −1.49622 1.66172i
71.2 0.309017 0.951057i 1.47815 + 1.07394i −0.809017 0.587785i 1.11803 + 1.93649i 1.47815 1.07394i 1.00000 −0.809017 + 0.587785i 0.104528 + 0.321706i 2.18720 0.464905i
141.1 −0.809017 0.587785i −0.413545 1.27276i 0.309017 + 0.951057i −1.11803 + 1.93649i −0.413545 + 1.27276i 1.00000 0.309017 0.951057i 0.978148 0.710666i 2.04275 0.909491i
141.2 −0.809017 0.587785i 0.604528 + 1.86055i 0.309017 + 0.951057i −1.11803 1.93649i 0.604528 1.86055i 1.00000 0.309017 0.951057i −0.669131 + 0.486152i −0.233733 + 2.22382i
211.1 −0.809017 + 0.587785i −0.413545 + 1.27276i 0.309017 0.951057i −1.11803 1.93649i −0.413545 1.27276i 1.00000 0.309017 + 0.951057i 0.978148 + 0.710666i 2.04275 + 0.909491i
211.2 −0.809017 + 0.587785i 0.604528 1.86055i 0.309017 0.951057i −1.11803 + 1.93649i 0.604528 + 1.86055i 1.00000 0.309017 + 0.951057i −0.669131 0.486152i −0.233733 2.22382i
281.1 0.309017 + 0.951057i −0.169131 + 0.122881i −0.809017 + 0.587785i 1.11803 + 1.93649i −0.169131 0.122881i 1.00000 −0.809017 0.587785i −0.913545 + 2.81160i −1.49622 + 1.66172i
281.2 0.309017 + 0.951057i 1.47815 1.07394i −0.809017 + 0.587785i 1.11803 1.93649i 1.47815 + 1.07394i 1.00000 −0.809017 0.587785i 0.104528 0.321706i 2.18720 + 0.464905i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 281.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.h.a 8
25.d even 5 1 inner 350.2.h.a 8
25.d even 5 1 8750.2.a.j 4
25.e even 10 1 8750.2.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.a 8 1.a even 1 1 trivial
350.2.h.a 8 25.d even 5 1 inner
8750.2.a.e 4 25.e even 10 1
8750.2.a.j 4 25.d even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$3$ $$1 - 3 T + 2 T^{2} + T^{3} + 6 T^{4} - 8 T^{5} - 30 T^{6} + 88 T^{7} - 131 T^{8} + 264 T^{9} - 270 T^{10} - 216 T^{11} + 486 T^{12} + 243 T^{13} + 1458 T^{14} - 6561 T^{15} + 6561 T^{16}$$
$5$ $$( 1 + 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - T )^{8}$$
$11$ $$1 + T - 30 T^{2} + 15 T^{3} + 260 T^{4} - 732 T^{5} + 3428 T^{6} + 5530 T^{7} - 83595 T^{8} + 60830 T^{9} + 414788 T^{10} - 974292 T^{11} + 3806660 T^{12} + 2415765 T^{13} - 53146830 T^{14} + 19487171 T^{15} + 214358881 T^{16}$$
$13$ $$1 - 11 T + 31 T^{2} + 71 T^{3} - 547 T^{4} + 1564 T^{5} - 4156 T^{6} - 10012 T^{7} + 113083 T^{8} - 130156 T^{9} - 702364 T^{10} + 3436108 T^{11} - 15622867 T^{12} + 26361803 T^{13} + 149631079 T^{14} - 690233687 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 14 T + 113 T^{2} - 798 T^{3} + 5296 T^{4} - 29904 T^{5} + 149375 T^{6} - 707504 T^{7} + 3101359 T^{8} - 12027568 T^{9} + 43169375 T^{10} - 146918352 T^{11} + 442327216 T^{12} - 1133045886 T^{13} + 2727545297 T^{14} - 5744741422 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 6 T + 24 T^{2} + 106 T^{3} + 681 T^{4} + 2398 T^{5} + 8776 T^{6} + 57768 T^{7} + 306817 T^{8} + 1097592 T^{9} + 3168136 T^{10} + 16447882 T^{11} + 88748601 T^{12} + 262466494 T^{13} + 1129101144 T^{14} + 5363230434 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 15 T + 64 T^{2} - 15 T^{3} + 192 T^{4} - 2310 T^{5} - 10768 T^{6} - 66000 T^{7} + 1144505 T^{8} - 1518000 T^{9} - 5696272 T^{10} - 28105770 T^{11} + 53729472 T^{12} - 96545145 T^{13} + 9474296896 T^{14} - 51072381705 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 8 T + 35 T^{2} + 120 T^{3} + 1480 T^{4} + 24 T^{5} - 47443 T^{6} - 312460 T^{7} - 647105 T^{8} - 9061340 T^{9} - 39899563 T^{10} + 585336 T^{11} + 1046775880 T^{12} + 2461337880 T^{13} + 20818816235 T^{14} + 137999010472 T^{15} + 500246412961 T^{16}$$
$31$ $$1 + 7 T - 4 T^{2} - 2 T^{3} + 526 T^{4} - 2249 T^{5} + 16384 T^{6} + 151404 T^{7} + 78027 T^{8} + 4693524 T^{9} + 15745024 T^{10} - 66999959 T^{11} + 485772046 T^{12} - 57258302 T^{13} - 3550014724 T^{14} + 192588298777 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 14 T + 8 T^{2} + 1162 T^{3} - 7679 T^{4} - 16574 T^{5} + 334840 T^{6} - 260964 T^{7} - 8129271 T^{8} - 9655668 T^{9} + 458395960 T^{10} - 839522822 T^{11} - 14391682319 T^{12} + 80577678034 T^{13} + 20525811272 T^{14} - 1329046279862 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 3 T - 19 T^{2} + 18 T^{3} - 459 T^{4} - 19074 T^{5} + 109024 T^{6} + 227799 T^{7} - 525013 T^{8} + 9339759 T^{9} + 183269344 T^{10} - 1314599154 T^{11} - 1297024299 T^{12} + 2085411618 T^{13} - 90251980579 T^{14} - 584262821643 T^{15} + 7984925229121 T^{16}$$
$43$ $$( 1 + 3 T + 111 T^{2} + 234 T^{3} + 6089 T^{4} + 10062 T^{5} + 205239 T^{6} + 238521 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 2 T - 16 T^{2} - 162 T^{3} - 1727 T^{4} - 26682 T^{5} + 136976 T^{6} + 626824 T^{7} + 4715593 T^{8} + 29460728 T^{9} + 302579984 T^{10} - 2770205286 T^{11} - 8427209087 T^{12} - 37153891134 T^{13} - 172467445264 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 4 T - 19 T^{2} + 634 T^{3} - 412 T^{4} - 2174 T^{5} + 75189 T^{6} + 261992 T^{7} + 4266723 T^{8} + 13885576 T^{9} + 211205901 T^{10} - 323658598 T^{11} - 3250878172 T^{12} + 265135942562 T^{13} - 421122861451 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 11 T - 76 T^{2} + 894 T^{3} + 3886 T^{4} - 58383 T^{5} + 315956 T^{6} + 623272 T^{7} - 26942453 T^{8} + 36773048 T^{9} + 1099842836 T^{10} - 11990642157 T^{11} + 47088064846 T^{12} + 639142323306 T^{13} - 3205720556716 T^{14} - 27375166333009 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 3 T^{2} + 650 T^{3} + 3438 T^{4} + 16250 T^{5} + 181801 T^{6} + 1901250 T^{7} + 15718855 T^{8} + 115976250 T^{9} + 676481521 T^{10} + 3688441250 T^{11} + 47602001358 T^{12} + 548987595650 T^{13} + 154561123083 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 14 T + 13 T^{2} + 482 T^{3} + 2336 T^{4} - 29384 T^{5} - 340345 T^{6} + 3464096 T^{7} - 12764441 T^{8} + 232094432 T^{9} - 1527808705 T^{10} - 8837619992 T^{11} + 47073018656 T^{12} + 650760301574 T^{13} + 1175958968197 T^{14} - 84849962474522 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 23 T + 251 T^{2} - 2812 T^{3} + 33341 T^{4} - 344824 T^{5} + 3544134 T^{6} - 31567271 T^{7} + 253196517 T^{8} - 2241276241 T^{9} + 17865979494 T^{10} - 123416302664 T^{11} + 847250856221 T^{12} - 5073492935012 T^{13} + 32153171264171 T^{14} - 209187763642993 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 5 T + 14 T^{2} - 640 T^{3} + 1222 T^{4} - 60055 T^{5} - 28808 T^{6} - 934650 T^{7} + 60949575 T^{8} - 68229450 T^{9} - 153517832 T^{10} - 23362415935 T^{11} + 34702650502 T^{12} - 1326765819520 T^{13} + 2118679168046 T^{14} + 55236992595485 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 2 T - 175 T^{2} + 940 T^{3} + 7090 T^{4} - 127586 T^{5} + 1192447 T^{6} + 5744430 T^{7} - 171664185 T^{8} + 453809970 T^{9} + 7442061727 T^{10} - 62904873854 T^{11} + 276156074290 T^{12} + 2892433015060 T^{13} - 42540304716175 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 27 T + 322 T^{2} + 2361 T^{3} + 19056 T^{4} + 273252 T^{5} + 2808440 T^{6} + 13313418 T^{7} + 42193589 T^{8} + 1105013694 T^{9} + 19347343160 T^{10} + 156241941324 T^{11} + 904365764976 T^{12} + 9300074958123 T^{13} + 105274800224818 T^{14} + 732673376719929 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 + 15 T - 43 T^{2} - 135 T^{3} + 11073 T^{4} - 6060 T^{5} + 887044 T^{6} + 4526550 T^{7} - 143552425 T^{8} + 402862950 T^{9} + 7026275524 T^{10} - 4272112140 T^{11} + 694744834593 T^{12} - 753848025615 T^{13} - 21370195511323 T^{14} + 663470023432935 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 - 40 T + 1006 T^{2} - 19760 T^{3} + 332807 T^{4} - 4819960 T^{5} + 61842548 T^{6} - 712007600 T^{7} + 7409982325 T^{8} - 69064737200 T^{9} + 581876534132 T^{10} - 4399047353080 T^{11} + 29463164421767 T^{12} - 169685843478320 T^{13} + 837969836958574 T^{14} - 3231931379124520 T^{15} + 7837433594376961 T^{16}$$