# Properties

 Label 322.2.i.a Level $322$ Weight $2$ Character orbit 322.i Analytic conductor $2.571$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.i (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$1$$ $$\zeta_{22}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.959493 − 0.281733i 0.142315 − 0.989821i −0.841254 − 0.540641i 0.142315 + 0.989821i −0.415415 − 0.909632i −0.415415 + 0.909632i −0.841254 + 0.540641i 0.959493 + 0.281733i 0.654861 + 0.755750i 0.654861 − 0.755750i
−0.415415 + 0.909632i −2.57385 0.755750i −0.654861 0.755750i 0.0741615 0.0476607i 1.75667 2.02730i 0.142315 0.989821i 0.959493 0.281733i 3.52977 + 2.26844i 0.0125459 + 0.0872586i
71.1 −0.841254 0.540641i 0.130785 + 0.909632i 0.415415 + 0.909632i 2.30075 + 0.675560i 0.381761 0.835939i 0.654861 0.755750i 0.142315 0.989821i 2.06815 0.607265i −1.57028 1.81219i
85.1 0.654861 0.755750i 1.54019 0.989821i −0.142315 0.989821i −0.570276 1.24873i 0.260554 1.81219i 0.959493 0.281733i −0.841254 0.540641i 0.146201 0.320135i −1.31718 0.386758i
127.1 −0.841254 + 0.540641i 0.130785 0.909632i 0.415415 0.909632i 2.30075 0.675560i 0.381761 + 0.835939i 0.654861 + 0.755750i 0.142315 + 0.989821i 2.06815 + 0.607265i −1.57028 + 1.81219i
141.1 0.142315 + 0.989821i −0.128663 + 0.281733i −0.959493 + 0.281733i 1.01255 1.16854i −0.297176 0.0872586i −0.841254 + 0.540641i −0.415415 0.909632i 1.90176 + 2.19475i 1.30075 + 0.835939i
169.1 0.142315 0.989821i −0.128663 0.281733i −0.959493 0.281733i 1.01255 + 1.16854i −0.297176 + 0.0872586i −0.841254 0.540641i −0.415415 + 0.909632i 1.90176 2.19475i 1.30075 0.835939i
197.1 0.654861 + 0.755750i 1.54019 + 0.989821i −0.142315 + 0.989821i −0.570276 + 1.24873i 0.260554 + 1.81219i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.146201 + 0.320135i −1.31718 + 0.386758i
211.1 −0.415415 0.909632i −2.57385 + 0.755750i −0.654861 + 0.755750i 0.0741615 + 0.0476607i 1.75667 + 2.02730i 0.142315 + 0.989821i 0.959493 + 0.281733i 3.52977 2.26844i 0.0125459 0.0872586i
225.1 0.959493 + 0.281733i −0.468468 + 0.540641i 0.841254 + 0.540641i −0.317178 + 2.20602i −0.601808 + 0.386758i −0.415415 0.909632i 0.654861 + 0.755750i 0.354114 + 2.46292i −0.925839 + 2.02730i
239.1 0.959493 0.281733i −0.468468 0.540641i 0.841254 0.540641i −0.317178 2.20602i −0.601808 0.386758i −0.415415 + 0.909632i 0.654861 0.755750i 0.354114 2.46292i −0.925839 2.02730i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.i.a 10
23.c even 11 1 inner 322.2.i.a 10
23.c even 11 1 7406.2.a.bg 5
23.d odd 22 1 7406.2.a.bh 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.a 10 1.a even 1 1 trivial
322.2.i.a 10 23.c even 11 1 inner
7406.2.a.bg 5 23.c even 11 1
7406.2.a.bh 5 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}$$
$3$ $$1 + 4 T + 16 T^{2} + 20 T^{3} + 25 T^{4} + 12 T^{5} + 15 T^{6} - 6 T^{7} - 2 T^{8} + 3 T^{9} + T^{10}$$
$5$ $$1 - 20 T + 147 T^{2} - 135 T^{3} + 126 T^{4} - 100 T^{5} + 75 T^{6} - 37 T^{7} + 14 T^{8} - 5 T^{9} + T^{10}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}$$
$11$ $$121 - 726 T + 484 T^{2} + 2057 T^{3} + 3146 T^{4} + 2321 T^{5} + 1188 T^{6} + 363 T^{7} + 77 T^{8} + 11 T^{9} + T^{10}$$
$13$ $$529 - 1495 T + 2410 T^{2} - 2089 T^{3} + 1816 T^{4} - 1561 T^{5} + 892 T^{6} - 307 T^{7} + 67 T^{8} - 10 T^{9} + T^{10}$$
$17$ $$192721 + 150138 T + 54286 T^{2} + 5510 T^{3} + 6007 T^{4} - 2278 T^{5} + 650 T^{6} - 144 T^{7} + T^{8} - T^{9} + T^{10}$$
$19$ $$1 - 4 T + 5 T^{2} + 13 T^{3} - 30 T^{4} - T^{5} + 70 T^{6} - 5 T^{7} + 20 T^{8} - 3 T^{9} + T^{10}$$
$23$ $$6436343 - 3358092 T + 1216700 T^{2} - 372945 T^{3} + 97428 T^{4} - 22595 T^{5} + 4236 T^{6} - 705 T^{7} + 100 T^{8} - 12 T^{9} + T^{10}$$
$29$ $$1849 + 258 T + 1730 T^{2} - 469 T^{3} + 647 T^{4} - 197 T^{5} - 6 T^{6} + 47 T^{7} - 7 T^{8} - 2 T^{9} + T^{10}$$
$31$ $$157609 + 143317 T + 55785 T^{2} + 14369 T^{3} + 2722 T^{4} + 386 T^{5} + 196 T^{6} + 48 T^{7} + 25 T^{8} + 5 T^{9} + T^{10}$$
$37$ $$279841 - 377177 T + 840052 T^{2} + 73301 T^{3} + 62254 T^{4} - 22243 T^{5} + 4915 T^{6} - 1489 T^{7} + 289 T^{8} - 27 T^{9} + T^{10}$$
$41$ $$4012009 + 773158 T + 747550 T^{2} + 147907 T^{3} + 34772 T^{4} + 26808 T^{5} + 6909 T^{6} + 727 T^{7} + 78 T^{8} + 12 T^{9} + T^{10}$$
$43$ $$734449 - 1029257 T + 1026480 T^{2} - 828930 T^{3} + 508161 T^{4} - 184548 T^{5} + 45945 T^{6} - 7286 T^{7} + 696 T^{8} - 38 T^{9} + T^{10}$$
$47$ $$( 857 + 322 T - 129 T^{2} - 59 T^{3} - T^{4} + T^{5} )^{2}$$
$53$ $$19562929 + 9332530 T - 289956 T^{2} - 385943 T^{3} + 103031 T^{4} + 45002 T^{5} + 16080 T^{6} + 1687 T^{7} + 234 T^{8} + 16 T^{9} + T^{10}$$
$59$ $$529 - 2323 T + 3865 T^{2} - 1493 T^{3} + 335 T^{4} - 417 T^{5} + 75 T^{6} + 15 T^{7} + 25 T^{8} + 5 T^{9} + T^{10}$$
$61$ $$6285049 + 9148043 T + 5945949 T^{2} + 2379503 T^{3} + 690936 T^{4} + 155376 T^{5} + 28020 T^{6} + 3870 T^{7} + 390 T^{8} + 26 T^{9} + T^{10}$$
$67$ $$69169 - 76007 T + 135617 T^{2} - 80955 T^{3} + 73902 T^{4} - 27202 T^{5} + 3886 T^{6} + 284 T^{7} - 72 T^{8} + 4 T^{9} + T^{10}$$
$71$ $$292170649 - 17469046 T + 5214573 T^{2} - 82897 T^{3} + 180896 T^{4} - 749 T^{5} + 6348 T^{6} - 1849 T^{7} + 78 T^{8} - T^{9} + T^{10}$$
$73$ $$57957769 - 21727502 T - 925504 T^{2} - 328948 T^{3} + 349611 T^{4} + 44296 T^{5} + 4867 T^{6} - 256 T^{7} + 158 T^{8} - 9 T^{9} + T^{10}$$
$79$ $$978121 + 2663377 T + 7787421 T^{2} + 4712518 T^{3} + 2288456 T^{4} + 719531 T^{5} + 136299 T^{6} + 15968 T^{7} + 1169 T^{8} + 50 T^{9} + T^{10}$$
$83$ $$53333809 - 42393915 T + 77366056 T^{2} + 22630691 T^{3} + 5447798 T^{4} + 800117 T^{5} + 61779 T^{6} + 1245 T^{7} + 203 T^{8} + 29 T^{9} + T^{10}$$
$89$ $$11361641281 + 7816211439 T + 1676566021 T^{2} + 104093258 T^{3} + 9459322 T^{4} + 731336 T^{5} + 59480 T^{6} + 1857 T^{7} + 379 T^{8} - 7 T^{9} + T^{10}$$
$97$ $$3892886449 - 3045963867 T + 955454248 T^{2} - 114299076 T^{3} + 6953332 T^{4} - 775655 T^{5} + 82677 T^{6} - 4610 T^{7} + 452 T^{8} - 12 T^{9} + T^{10}$$