# Properties

 Label 280.2.h.a Level $280$ Weight $2$ Character orbit 280.h Analytic conductor $2.236$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{15} - 2 x^{12} + 6 x^{11} - 12 x^{9} + 8 x^{8} - 24 x^{7} + 48 x^{5} - 32 x^{4} - 128 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} - 2 x^{12} + 6 x^{11} - 12 x^{9} + 8 x^{8} - 24 x^{7} + 48 x^{5} - 32 x^{4} - 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} + 2 \nu^{8} - 2 \nu^{7} - 2 \nu^{6} - 8 \nu^{2} - 32 \nu - 16$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 2 \nu^{13} + 4 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + 4 \nu^{9} - 4 \nu^{8} + 16 \nu^{7} - 8 \nu^{6} - 48 \nu^{5} - 16 \nu^{4} + 64 \nu^{3} + 64 \nu^{2} + 128 \nu$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 2 \nu^{13} + 4 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + 4 \nu^{9} - 4 \nu^{8} + 16 \nu^{7} - 8 \nu^{6} - 48 \nu^{5} - 16 \nu^{4} - 64 \nu^{3} - 64 \nu^{2} + 128 \nu$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + 32 \nu^{3} + 128$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{14} + \nu^{13} + 2 \nu^{12} + 4 \nu^{11} - 2 \nu^{10} - 6 \nu^{9} + 4 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} - 8 \nu^{5} - 48 \nu^{4} - 16 \nu^{3} - 64 \nu^{2} - 64 \nu + 128$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{14} + \nu^{13} + 2 \nu^{12} + 4 \nu^{11} - 2 \nu^{10} - 6 \nu^{9} + 4 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} - 8 \nu^{5} - 48 \nu^{4} - 80 \nu^{3} - 64 \nu + 128$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 2 \nu^{13} + 4 \nu^{12} + 2 \nu^{11} - 2 \nu^{10} - 4 \nu^{9} - 12 \nu^{8} + 16 \nu^{7} + 56 \nu^{6} + 16 \nu^{5} + 48 \nu^{4} - 64 \nu^{3} - 128 \nu^{2} + 128 \nu + 128$$$$)/128$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} + 2 \nu^{13} + 2 \nu^{11} - 2 \nu^{10} - 12 \nu^{9} + 12 \nu^{8} + 16 \nu^{7} + 8 \nu^{6} + 48 \nu^{5} - 48 \nu^{4} - 64 \nu^{3} + 64 \nu^{2} + 128$$$$)/128$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{13} + 2 \nu^{12} - 6 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + 8 \nu^{5} + 16 \nu^{3} - 64 \nu^{2} + 128 \nu + 192$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} - 4 \nu^{13} - 4 \nu^{12} + 2 \nu^{11} + 2 \nu^{10} - 8 \nu^{9} + 20 \nu^{8} - 8 \nu^{7} - 8 \nu^{6} + 32 \nu^{5} + 48 \nu^{4} + 96 \nu^{3} + 128 \nu^{2} - 256 \nu$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{13} - 2 \nu^{12} + 4 \nu^{11} + 2 \nu^{10} + 6 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} + 40 \nu^{5} + 32 \nu^{4} + 48 \nu^{3} - 64 \nu^{2} - 128 \nu - 64$$$$)/64$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} + 6 \nu^{11} - 10 \nu^{10} - 8 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 32 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} - 32 \nu^{2} + 320$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{15} - \nu^{13} - 4 \nu^{12} - 6 \nu^{11} + 4 \nu^{10} + 6 \nu^{9} + 4 \nu^{8} + 4 \nu^{7} - 32 \nu^{6} - 40 \nu^{5} + 64 \nu^{4} + 48 \nu^{3} + 96 \nu^{2} - 256$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$2 \nu^{15} - \nu^{14} - 3 \nu^{13} - 2 \nu^{12} - 4 \nu^{11} + 6 \nu^{10} + 18 \nu^{9} - 12 \nu^{8} + 4 \nu^{7} - 24 \nu^{6} - 40 \nu^{5} + 96 \nu^{4} + 48 \nu^{3} - 32 \nu^{2} - 320$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} + 2 \beta_{14} + 2 \beta_{12} - 2 \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 4 \beta_{2} + 2 \beta_{1} - 2$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 5 \beta_{4} + \beta_{3} - 4 \beta_{2} - 2$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} + 8 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} + \beta_{3} + 6$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$2 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 12 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 4 \beta_{1} + 2$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$4 \beta_{15} - 6 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} + 3 \beta_{7} - 9 \beta_{6} + 16 \beta_{5} + 3 \beta_{4} + \beta_{3} + 18$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$4 \beta_{15} - 6 \beta_{14} - 10 \beta_{13} - 6 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} - \beta_{7} + 11 \beta_{6} + 8 \beta_{5} - 17 \beta_{4} - 3 \beta_{3} + 12 \beta_{1} + 26$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$4 \beta_{15} + 2 \beta_{14} + 22 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 12 \beta_{10} + 8 \beta_{9} + 14 \beta_{8} - \beta_{7} - 13 \beta_{6} - 24 \beta_{5} - \beta_{4} + 21 \beta_{3} + 24 \beta_{2} + 12 \beta_{1} + 10$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-4 \beta_{15} - 22 \beta_{14} - 26 \beta_{13} - 6 \beta_{12} + 14 \beta_{11} + 20 \beta_{10} - 16 \beta_{9} - 10 \beta_{8} + 15 \beta_{7} - 13 \beta_{6} + 16 \beta_{5} - \beta_{4} + 13 \beta_{3} + 8 \beta_{2} - 4 \beta_{1} - 30$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-44 \beta_{15} + 42 \beta_{14} - 2 \beta_{13} + 42 \beta_{12} - 42 \beta_{11} - 12 \beta_{10} - 24 \beta_{9} + 22 \beta_{8} + 7 \beta_{7} + 3 \beta_{6} - 24 \beta_{5} - 9 \beta_{4} + 29 \beta_{3} - 8 \beta_{2} - 12 \beta_{1} + 34$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-12 \beta_{15} - 22 \beta_{14} - 18 \beta_{13} + 26 \beta_{12} + 54 \beta_{11} - 44 \beta_{10} - 56 \beta_{9} - 10 \beta_{8} + 7 \beta_{7} + 35 \beta_{6} - 8 \beta_{5} + 55 \beta_{4} - 19 \beta_{3} - 72 \beta_{2} + 60 \beta_{1} + 18$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$36 \beta_{15} + 10 \beta_{14} - 50 \beta_{13} + 58 \beta_{12} + 22 \beta_{11} + 36 \beta_{10} + 40 \beta_{9} - 58 \beta_{8} + 79 \beta_{7} + 107 \beta_{6} - 24 \beta_{5} - 33 \beta_{4} + 5 \beta_{3} - 120 \beta_{2} + 60 \beta_{1} - 30$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.38133 + 0.303194i −1.38133 − 0.303194i −1.24098 + 0.678208i −1.24098 − 0.678208i −0.470943 + 1.33350i −0.470943 − 1.33350i −0.275585 + 1.38710i −0.275585 − 1.38710i 0.244064 + 1.39299i 0.244064 − 1.39299i 1.07046 + 0.924187i 1.07046 − 0.924187i 1.14218 + 0.833926i 1.14218 − 0.833926i 1.41214 + 0.0765298i 1.41214 − 0.0765298i
−1.38133 0.303194i 1.34113i 1.81615 + 0.837621i −1.00000 0.406623 1.85255i 1.28003 2.31550i −2.25474 1.70768i 1.20136 1.38133 + 0.303194i
251.2 −1.38133 + 0.303194i 1.34113i 1.81615 0.837621i −1.00000 0.406623 + 1.85255i 1.28003 + 2.31550i −2.25474 + 1.70768i 1.20136 1.38133 0.303194i
251.3 −1.24098 0.678208i 1.61069i 1.08007 + 1.68329i −1.00000 −1.09238 + 1.99883i −2.13463 + 1.56312i −0.198727 2.82144i 0.405694 1.24098 + 0.678208i
251.4 −1.24098 + 0.678208i 1.61069i 1.08007 1.68329i −1.00000 −1.09238 1.99883i −2.13463 1.56312i −0.198727 + 2.82144i 0.405694 1.24098 0.678208i
251.5 −0.470943 1.33350i 0.528177i −1.55642 + 1.25600i −1.00000 0.704322 0.248741i −2.17448 + 1.50719i 2.40786 + 1.48398i 2.72103 0.470943 + 1.33350i
251.6 −0.470943 + 1.33350i 0.528177i −1.55642 1.25600i −1.00000 0.704322 + 0.248741i −2.17448 1.50719i 2.40786 1.48398i 2.72103 0.470943 1.33350i
251.7 −0.275585 1.38710i 3.19977i −1.84811 + 0.764529i −1.00000 4.43840 0.881807i 2.59303 + 0.525543i 1.56979 + 2.35282i −7.23851 0.275585 + 1.38710i
251.8 −0.275585 + 1.38710i 3.19977i −1.84811 0.764529i −1.00000 4.43840 + 0.881807i 2.59303 0.525543i 1.56979 2.35282i −7.23851 0.275585 1.38710i
251.9 0.244064 1.39299i 1.68420i −1.88087 0.679959i −1.00000 −2.34608 0.411052i −0.695780 2.55262i −1.40623 + 2.45408i 0.163484 −0.244064 + 1.39299i
251.10 0.244064 + 1.39299i 1.68420i −1.88087 + 0.679959i −1.00000 −2.34608 + 0.411052i −0.695780 + 2.55262i −1.40623 2.45408i 0.163484 −0.244064 1.39299i
251.11 1.07046 0.924187i 2.99734i 0.291758 1.97860i −1.00000 −2.77010 3.20852i −0.183359 + 2.63939i −1.51629 2.38765i −5.98405 −1.07046 + 0.924187i
251.12 1.07046 + 0.924187i 2.99734i 0.291758 + 1.97860i −1.00000 −2.77010 + 3.20852i −0.183359 2.63939i −1.51629 + 2.38765i −5.98405 −1.07046 0.924187i
251.13 1.14218 0.833926i 0.586834i 0.609136 1.90498i −1.00000 0.489376 + 0.670268i 2.52442 0.792014i −0.892871 2.68380i 2.65563 −1.14218 + 0.833926i
251.14 1.14218 + 0.833926i 0.586834i 0.609136 + 1.90498i −1.00000 0.489376 0.670268i 2.52442 + 0.792014i −0.892871 + 2.68380i 2.65563 −1.14218 0.833926i
251.15 1.41214 0.0765298i 2.21915i 1.98829 0.216142i −1.00000 0.169831 + 3.13376i −1.20923 + 2.35325i 2.79120 0.457386i −1.92464 −1.41214 + 0.0765298i
251.16 1.41214 + 0.0765298i 2.21915i 1.98829 + 0.216142i −1.00000 0.169831 3.13376i −1.20923 2.35325i 2.79120 + 0.457386i −1.92464 −1.41214 0.0765298i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.h.a 16
4.b odd 2 1 1120.2.h.a 16
7.b odd 2 1 280.2.h.b yes 16
8.b even 2 1 1120.2.h.b 16
8.d odd 2 1 280.2.h.b yes 16
28.d even 2 1 1120.2.h.b 16
56.e even 2 1 inner 280.2.h.a 16
56.h odd 2 1 1120.2.h.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.h.a 16 1.a even 1 1 trivial
280.2.h.a 16 56.e even 2 1 inner
280.2.h.b yes 16 7.b odd 2 1
280.2.h.b yes 16 8.d odd 2 1
1120.2.h.a 16 4.b odd 2 1
1120.2.h.a 16 56.h odd 2 1
1120.2.h.b 16 8.b even 2 1
1120.2.h.b 16 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{8} - 62 T_{13}^{6} - 12 T_{13}^{5} + 1213 T_{13}^{4} + 228 T_{13}^{3} - 7792 T_{13}^{2} + 1232 T_{13} + 10032$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 128 T - 32 T^{4} + 48 T^{5} - 24 T^{7} + 8 T^{8} - 12 T^{9} + 6 T^{11} - 2 T^{12} - T^{15} + T^{16}$$
$3$ $$576 + 4720 T^{2} + 13024 T^{4} + 14676 T^{6} + 8217 T^{8} + 2468 T^{10} + 398 T^{12} + 32 T^{14} + T^{16}$$
$5$ $$( 1 + T )^{16}$$
$7$ $$5764801 + 470596 T^{2} - 605052 T^{3} + 9604 T^{4} - 69972 T^{5} + 34300 T^{6} - 6328 T^{7} + 5814 T^{8} - 904 T^{9} + 700 T^{10} - 204 T^{11} + 4 T^{12} - 36 T^{13} + 4 T^{14} + T^{16}$$
$11$ $$( 2400 + 1960 T - 3932 T^{2} + 138 T^{3} + 761 T^{4} - 52 T^{5} - 50 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$13$ $$( 10032 + 1232 T - 7792 T^{2} + 228 T^{3} + 1213 T^{4} - 12 T^{5} - 62 T^{6} + T^{8} )^{2}$$
$17$ $$2359296 + 27167680 T^{2} + 34001456 T^{4} + 13596164 T^{6} + 2255465 T^{8} + 178476 T^{10} + 7086 T^{12} + 136 T^{14} + T^{16}$$
$19$ $$1327104 + 26099712 T^{2} + 81127424 T^{4} + 49375232 T^{6} + 6949696 T^{8} + 423168 T^{10} + 12736 T^{12} + 184 T^{14} + T^{16}$$
$23$ $$7573504 + 585268224 T^{2} + 289346560 T^{4} + 57190400 T^{6} + 5807360 T^{8} + 325872 T^{10} + 10096 T^{12} + 160 T^{14} + T^{16}$$
$29$ $$65536 + 2408448 T^{2} + 15058432 T^{4} + 9795008 T^{6} + 2200529 T^{8} + 217548 T^{10} + 9718 T^{12} + 172 T^{14} + T^{16}$$
$31$ $$( 98304 + 2816 T - 33920 T^{2} - 3040 T^{3} + 3496 T^{4} + 440 T^{5} - 92 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$37$ $$4063297536 + 30390333440 T^{2} + 7832304640 T^{4} + 825741568 T^{6} + 45832768 T^{8} + 1448768 T^{10} + 26192 T^{12} + 252 T^{14} + T^{16}$$
$41$ $$6423183360000 + 1819708211200 T^{2} + 201056874496 T^{4} + 11370832896 T^{6} + 364052032 T^{8} + 6823040 T^{10} + 73760 T^{12} + 424 T^{14} + T^{16}$$
$43$ $$( -69504 - 91360 T - 17664 T^{2} + 14912 T^{3} + 4232 T^{4} - 332 T^{5} - 128 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$47$ $$( -10368 - 5820 T + 36188 T^{2} + 28292 T^{3} + 5461 T^{4} - 424 T^{5} - 166 T^{6} + T^{8} )^{2}$$
$53$ $$262144 + 27312128 T^{2} + 158445568 T^{4} + 251149312 T^{6} + 69431040 T^{8} + 3445312 T^{10} + 58128 T^{12} + 404 T^{14} + T^{16}$$
$59$ $$1090584576 + 2931687424 T^{2} + 2362048512 T^{4} + 734793728 T^{6} + 82300416 T^{8} + 3287040 T^{10} + 55552 T^{12} + 400 T^{14} + T^{16}$$
$61$ $$( 10368 + 51712 T + 26432 T^{2} - 85216 T^{3} + 15176 T^{4} + 1256 T^{5} - 268 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$67$ $$( 165632 + 197184 T - 55616 T^{2} - 42240 T^{3} + 7712 T^{4} + 1204 T^{5} - 180 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$71$ $$2621440000 + 5845811200 T^{2} + 4596301824 T^{4} + 1494106112 T^{6} + 180376576 T^{8} + 6017280 T^{10} + 82752 T^{12} + 492 T^{14} + T^{16}$$
$73$ $$28179280429056 + 12310919774208 T^{2} + 1248988233728 T^{4} + 57276760064 T^{6} + 1406521344 T^{8} + 19483584 T^{10} + 151344 T^{12} + 612 T^{14} + T^{16}$$
$79$ $$39806206534656 + 12666930453504 T^{2} + 1373008608512 T^{4} + 61858049132 T^{6} + 1444670401 T^{8} + 19139020 T^{10} + 145478 T^{12} + 592 T^{14} + T^{16}$$
$83$ $$88794464256 + 165451399168 T^{2} + 51045990400 T^{4} + 6096578560 T^{6} + 337953344 T^{8} + 8985456 T^{10} + 109632 T^{12} + 564 T^{14} + T^{16}$$
$89$ $$347892350976 + 245081571328 T^{2} + 61230546944 T^{4} + 6628880384 T^{6} + 331338304 T^{8} + 8073856 T^{10} + 95808 T^{12} + 520 T^{14} + T^{16}$$
$97$ $$17227945230336 + 27670537460160 T^{2} + 4314469148336 T^{4} + 212781536324 T^{6} + 4708474873 T^{8} + 52726332 T^{10} + 307550 T^{12} + 888 T^{14} + T^{16}$$
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