# Properties

 Label 280.2.b.d Level $280$ Weight $2$ Character orbit 280.b Analytic conductor $2.236$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.8272021826830336.1 Defining polynomial: $$x^{12} + 3 x^{10} - 4 x^{9} + 4 x^{8} - 12 x^{7} + 10 x^{6} - 24 x^{5} + 16 x^{4} - 32 x^{3} + 48 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 3 x^{10} - 4 x^{9} + 4 x^{8} - 12 x^{7} + 10 x^{6} - 24 x^{5} + 16 x^{4} - 32 x^{3} + 48 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{11} - 5 \nu^{10} + 7 \nu^{9} - 17 \nu^{8} + 43 \nu^{7} - 30 \nu^{6} + 92 \nu^{5} - 62 \nu^{4} + 170 \nu^{3} - 140 \nu^{2} + 64 \nu - 384$$$$)/136$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{11} + 116 \nu^{10} - 57 \nu^{9} + 136 \nu^{8} - 440 \nu^{7} + 356 \nu^{6} - 822 \nu^{5} + 704 \nu^{4} - 1496 \nu^{3} + 1344 \nu^{2} - 560 \nu + 3360$$$$)/544$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} + 3 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} - 12 \nu^{6} + 10 \nu^{5} - 24 \nu^{4} + 16 \nu^{3} + 48 \nu$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} - 3 \nu^{9} + 4 \nu^{8} - 4 \nu^{7} + 12 \nu^{6} - 10 \nu^{5} + 24 \nu^{4} - 16 \nu^{3} + 32 \nu^{2} - 48 \nu$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$23 \nu^{11} + 66 \nu^{10} + 13 \nu^{9} + 34 \nu^{8} - 180 \nu^{7} + 124 \nu^{6} - 242 \nu^{5} - 52 \nu^{4} - 816 \nu^{3} + 80 \nu^{2} + 80 \nu + 2240$$$$)/544$$ $$\beta_{6}$$ $$=$$ $$($$$$-15 \nu^{11} - 29 \nu^{10} - 7 \nu^{9} + 17 \nu^{8} + 42 \nu^{7} - 4 \nu^{6} + 78 \nu^{5} + 62 \nu^{4} + 340 \nu^{3} - 200 \nu^{2} - 336 \nu - 976$$$$)/272$$ $$\beta_{7}$$ $$=$$ $$($$$$29 \nu^{11} - 38 \nu^{10} + 43 \nu^{9} - 102 \nu^{8} + 184 \nu^{7} - 228 \nu^{6} + 298 \nu^{5} - 580 \nu^{4} + 680 \nu^{3} - 384 \nu^{2} + 976 \nu - 960$$$$)/544$$ $$\beta_{8}$$ $$=$$ $$($$$$-19 \nu^{11} - 22 \nu^{10} + 7 \nu^{9} + 34 \nu^{8} + 60 \nu^{7} + 4 \nu^{6} + 58 \nu^{5} + 108 \nu^{4} + 272 \nu^{3} - 208 \nu^{2} - 752 \nu - 928$$$$)/272$$ $$\beta_{9}$$ $$=$$ $$($$$$-19 \nu^{11} - 22 \nu^{10} + 7 \nu^{9} + 34 \nu^{8} + 60 \nu^{7} + 4 \nu^{6} + 58 \nu^{5} + 108 \nu^{4} + 272 \nu^{3} - 208 \nu^{2} - 208 \nu - 928$$$$)/272$$ $$\beta_{10}$$ $$=$$ $$($$$$45 \nu^{11} - 32 \nu^{10} + 55 \nu^{9} - 204 \nu^{8} + 180 \nu^{7} - 260 \nu^{6} + 242 \nu^{5} - 696 \nu^{4} + 272 \nu^{3} - 80 \nu^{2} + 1552 \nu + 480$$$$)/544$$ $$\beta_{11}$$ $$=$$ $$($$$$93 \nu^{11} + 20 \nu^{10} + 91 \nu^{9} - 272 \nu^{8} + 32 \nu^{7} - 356 \nu^{6} + 210 \nu^{5} - 704 \nu^{4} - 136 \nu^{3} + 16 \nu^{2} + 2192 \nu + 2080$$$$)/544$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} - \beta_{8}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{2} + \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} + 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{11} - 2 \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 1$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + 3 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + \beta_{2} + 3 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} + \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 8 \beta_{6} - 3 \beta_{5} - \beta_{4} - 4 \beta_{3} + 3$$ $$\nu^{8}$$ $$=$$ $$-6 \beta_{10} + 2 \beta_{9} - \beta_{8} + 5 \beta_{7} - 5 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 5 \beta_{1} + 4$$ $$\nu^{9}$$ $$=$$ $$3 \beta_{11} + 4 \beta_{10} + 9 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} + 7 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 2 \beta_{1} - 5$$ $$\nu^{10}$$ $$=$$ $$-2 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 3 \beta_{8} + 5 \beta_{7} - 11 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 4 \beta_{3} + 5 \beta_{2} + 11 \beta_{1} + 2$$ $$\nu^{11}$$ $$=$$ $$9 \beta_{11} - 16 \beta_{10} + \beta_{9} + 4 \beta_{8} + 23 \beta_{7} - 10 \beta_{6} - \beta_{5} + 9 \beta_{4} - 12 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} + 1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
141.1
 −0.258252 − 1.39043i −0.258252 + 1.39043i 0.832593 + 1.14315i 0.832593 − 1.14315i −1.11909 − 0.864661i −1.11909 + 0.864661i 1.39608 + 0.225774i 1.39608 − 0.225774i −0.722588 + 1.21568i −0.722588 − 1.21568i −0.128739 + 1.40834i −0.128739 − 1.40834i
−1.39043 0.258252i 0.861041i 1.86661 + 0.718165i 1.00000i 0.222366 1.19722i 1.00000 −2.40993 1.48062i 2.25861 0.258252 1.39043i
141.2 −1.39043 + 0.258252i 0.861041i 1.86661 0.718165i 1.00000i 0.222366 + 1.19722i 1.00000 −2.40993 + 1.48062i 2.25861 0.258252 + 1.39043i
141.3 −1.14315 0.832593i 3.42822i 0.613577 + 1.90356i 1.00000i −2.85431 + 3.91897i 1.00000 0.883477 2.68691i −8.75270 −0.832593 + 1.14315i
141.4 −1.14315 + 0.832593i 3.42822i 0.613577 1.90356i 1.00000i −2.85431 3.91897i 1.00000 0.883477 + 2.68691i −8.75270 −0.832593 1.14315i
141.5 −0.864661 1.11909i 0.903031i −0.504724 + 1.93527i 1.00000i −1.01057 + 0.780815i 1.00000 2.60215 1.10852i 2.18454 1.11909 0.864661i
141.6 −0.864661 + 1.11909i 0.903031i −0.504724 1.93527i 1.00000i −1.01057 0.780815i 1.00000 2.60215 + 1.10852i 2.18454 1.11909 + 0.864661i
141.7 −0.225774 1.39608i 2.07981i −1.89805 + 0.630396i 1.00000i 2.90357 0.469568i 1.00000 1.30861 + 2.50750i −1.32561 −1.39608 + 0.225774i
141.8 −0.225774 + 1.39608i 2.07981i −1.89805 0.630396i 1.00000i 2.90357 + 0.469568i 1.00000 1.30861 2.50750i −1.32561 −1.39608 0.225774i
141.9 1.21568 0.722588i 1.52755i 0.955734 1.75686i 1.00000i 1.10379 + 1.85701i 1.00000 −0.107626 2.82638i 0.666582 0.722588 + 1.21568i
141.10 1.21568 + 0.722588i 1.52755i 0.955734 + 1.75686i 1.00000i 1.10379 1.85701i 1.00000 −0.107626 + 2.82638i 0.666582 0.722588 1.21568i
141.11 1.40834 0.128739i 2.83397i 1.96685 0.362616i 1.00000i −0.364842 3.99120i 1.00000 2.72332 0.763897i −5.03141 0.128739 + 1.40834i
141.12 1.40834 + 0.128739i 2.83397i 1.96685 + 0.362616i 1.00000i −0.364842 + 3.99120i 1.00000 2.72332 + 0.763897i −5.03141 0.128739 1.40834i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 141.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.b.d 12
4.b odd 2 1 1120.2.b.d 12
8.b even 2 1 inner 280.2.b.d 12
8.d odd 2 1 1120.2.b.d 12
16.e even 4 1 8960.2.a.ca 6
16.e even 4 1 8960.2.a.cf 6
16.f odd 4 1 8960.2.a.cd 6
16.f odd 4 1 8960.2.a.cg 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.d 12 1.a even 1 1 trivial
280.2.b.d 12 8.b even 2 1 inner
1120.2.b.d 12 4.b odd 2 1
1120.2.b.d 12 8.d odd 2 1
8960.2.a.ca 6 16.e even 4 1
8960.2.a.cd 6 16.f odd 4 1
8960.2.a.cf 6 16.e even 4 1
8960.2.a.cg 6 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$:

 $$T_{3}^{12} + 28 T_{3}^{10} + 278 T_{3}^{8} + 1212 T_{3}^{6} + 2385 T_{3}^{4} + 1984 T_{3}^{2} + 576$$ $$T_{13}^{12} + 76 T_{13}^{10} + 1958 T_{13}^{8} + 19860 T_{13}^{6} + 65713 T_{13}^{4} + 77896 T_{13}^{2} + 26896$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 64 T - 16 T^{2} - 64 T^{3} - 32 T^{4} + 16 T^{5} + 26 T^{6} + 8 T^{7} - 8 T^{8} - 8 T^{9} - T^{10} + 2 T^{11} + T^{12}$$
$3$ $$576 + 1984 T^{2} + 2385 T^{4} + 1212 T^{6} + 278 T^{8} + 28 T^{10} + T^{12}$$
$5$ $$( 1 + T^{2} )^{6}$$
$7$ $$( -1 + T )^{12}$$
$11$ $$138384 + 222040 T^{2} + 119185 T^{4} + 24740 T^{6} + 2086 T^{8} + 76 T^{10} + T^{12}$$
$13$ $$26896 + 77896 T^{2} + 65713 T^{4} + 19860 T^{6} + 1958 T^{8} + 76 T^{10} + T^{12}$$
$17$ $$( -396 - 860 T + 729 T^{2} + 44 T^{3} - 74 T^{4} + T^{6} )^{2}$$
$19$ $$4096 + 16384 T^{2} + 16960 T^{4} + 6208 T^{6} + 928 T^{8} + 56 T^{10} + T^{12}$$
$23$ $$( -768 - 1984 T + 232 T^{2} + 352 T^{3} - 68 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$29$ $$123904 + 267008 T^{2} + 185025 T^{4} + 47612 T^{6} + 4038 T^{8} + 124 T^{10} + T^{12}$$
$31$ $$( 1152 - 256 T - 504 T^{2} + 176 T^{3} + 20 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$37$ $$435306496 + 196720640 T^{2} + 24890944 T^{4} + 1273408 T^{6} + 28384 T^{8} + 280 T^{10} + T^{12}$$
$41$ $$( 7200 + 6880 T + 632 T^{2} - 480 T^{3} - 64 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$43$ $$6718464 + 79069696 T^{2} + 59151936 T^{4} + 3380032 T^{6} + 59696 T^{8} + 416 T^{10} + T^{12}$$
$47$ $$( -224128 + 47536 T + 11865 T^{2} - 1256 T^{3} - 194 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$53$ $$6379536384 + 1375731712 T^{2} + 101519360 T^{4} + 3366912 T^{6} + 53376 T^{8} + 384 T^{10} + T^{12}$$
$59$ $$17314349056 + 2316173312 T^{2} + 123826176 T^{4} + 3362816 T^{6} + 48576 T^{8} + 352 T^{10} + T^{12}$$
$61$ $$70829056 + 29191680 T^{2} + 4629568 T^{4} + 353856 T^{6} + 13168 T^{8} + 208 T^{10} + T^{12}$$
$67$ $$18939904 + 9413558272 T^{2} + 485136384 T^{4} + 9919488 T^{6} + 100496 T^{8} + 504 T^{10} + T^{12}$$
$71$ $$( 8704 - 16512 T - 13424 T^{2} - 2784 T^{3} - 104 T^{4} + 16 T^{5} + T^{6} )^{2}$$
$73$ $$( 8192 + 8192 T - 256 T^{2} - 1472 T^{3} - 204 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$79$ $$( 90784 + 48664 T + 4953 T^{2} - 1176 T^{3} - 214 T^{4} + T^{6} )^{2}$$
$83$ $$46022656 + 69009408 T^{2} + 11815168 T^{4} + 743168 T^{6} + 20384 T^{8} + 240 T^{10} + T^{12}$$
$89$ $$( -51808 + 89952 T - 18312 T^{2} - 4864 T^{3} - 80 T^{4} + 24 T^{5} + T^{6} )^{2}$$
$97$ $$( 38068 - 1420 T - 9439 T^{2} + 2172 T^{3} - 66 T^{4} - 16 T^{5} + T^{6} )^{2}$$