# Properties

 Label 240.2.bc.e Level $240$ Weight $2$ Character orbit 240.bc Analytic conductor $1.916$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.bc (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 26 x^{12} + 78 x^{11} - 66 x^{10} - 74 x^{9} + 233 x^{8} - 148 x^{7} - 264 x^{6} + 624 x^{5} - 416 x^{4} - 320 x^{3} + 896 x^{2} - 768 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 14 x^{14} - 10 x^{13} - 26 x^{12} + 78 x^{11} - 66 x^{10} - 74 x^{9} + 233 x^{8} - 148 x^{7} - 264 x^{6} + 624 x^{5} - 416 x^{4} - 320 x^{3} + 896 x^{2} - 768 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-15 \nu^{15} + 62 \nu^{14} - 94 \nu^{13} - 26 \nu^{12} + 342 \nu^{11} - 530 \nu^{10} - 2 \nu^{9} + 1110 \nu^{8} - 1415 \nu^{7} - 424 \nu^{6} + 3188 \nu^{5} - 3360 \nu^{4} - 240 \nu^{3} + 4480 \nu^{2} - 5056 \nu + 1792$$$$)/256$$ $$\beta_{2}$$ $$=$$ $$($$$$-8 \nu^{15} + 37 \nu^{14} - 64 \nu^{13} + 2 \nu^{12} + 202 \nu^{11} - 366 \nu^{10} + 90 \nu^{9} + 666 \nu^{8} - 1030 \nu^{7} - 31 \nu^{6} + 1994 \nu^{5} - 2544 \nu^{4} + 328 \nu^{3} + 2848 \nu^{2} - 3712 \nu + 1728$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{15} - 115 \nu^{14} + 222 \nu^{13} - 52 \nu^{12} - 628 \nu^{11} + 1268 \nu^{10} - 468 \nu^{9} - 2108 \nu^{8} + 3643 \nu^{7} - 357 \nu^{6} - 6458 \nu^{5} + 8944 \nu^{4} - 1808 \nu^{3} - 9200 \nu^{2} + 12832 \nu - 6272$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$57 \nu^{15} - 268 \nu^{14} + 450 \nu^{13} + 18 \nu^{12} - 1470 \nu^{11} + 2522 \nu^{10} - 390 \nu^{9} - 4846 \nu^{8} + 6941 \nu^{7} + 902 \nu^{6} - 14160 \nu^{5} + 16896 \nu^{4} - 976 \nu^{3} - 19808 \nu^{2} + 24320 \nu - 10752$$$$)/256$$ $$\beta_{5}$$ $$=$$ $$($$$$87 \nu^{15} - 378 \nu^{14} + 582 \nu^{13} + 130 \nu^{12} - 2094 \nu^{11} + 3290 \nu^{10} - 86 \nu^{9} - 6830 \nu^{8} + 8807 \nu^{7} + 2428 \nu^{6} - 19540 \nu^{5} + 21248 \nu^{4} + 912 \nu^{3} - 27584 \nu^{2} + 31296 \nu - 12544$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$-53 \nu^{15} + 224 \nu^{14} - 334 \nu^{13} - 98 \nu^{12} + 1238 \nu^{11} - 1874 \nu^{10} - 50 \nu^{9} + 4022 \nu^{8} - 4969 \nu^{7} - 1698 \nu^{6} + 11380 \nu^{5} - 11976 \nu^{4} - 960 \nu^{3} + 16000 \nu^{2} - 17664 \nu + 6912$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$-29 \nu^{15} + 127 \nu^{14} - 200 \nu^{13} - 32 \nu^{12} + 696 \nu^{11} - 1128 \nu^{10} + 96 \nu^{9} + 2264 \nu^{8} - 3063 \nu^{7} - 619 \nu^{6} + 6536 \nu^{5} - 7472 \nu^{4} + 120 \nu^{3} + 9232 \nu^{2} - 10976 \nu + 4672$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$119 \nu^{15} - 570 \nu^{14} + 966 \nu^{13} + 2 \nu^{12} - 3086 \nu^{11} + 5466 \nu^{10} - 1110 \nu^{9} - 10158 \nu^{8} + 15175 \nu^{7} + 1084 \nu^{6} - 30036 \nu^{5} + 37248 \nu^{4} - 3728 \nu^{3} - 42560 \nu^{2} + 54208 \nu - 24576$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$36 \nu^{15} - 159 \nu^{14} + 250 \nu^{13} + 42 \nu^{12} - 874 \nu^{11} + 1414 \nu^{10} - 98 \nu^{9} - 2866 \nu^{8} + 3826 \nu^{7} + 857 \nu^{6} - 8260 \nu^{5} + 9276 \nu^{4} + 64 \nu^{3} - 11664 \nu^{2} + 13632 \nu - 5568$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$-35 \nu^{15} + 161 \nu^{14} - 263 \nu^{13} - 24 \nu^{12} + 882 \nu^{11} - 1486 \nu^{10} + 202 \nu^{9} + 2890 \nu^{8} - 4071 \nu^{7} - 603 \nu^{6} + 8421 \nu^{5} - 9926 \nu^{4} + 472 \nu^{3} + 11872 \nu^{2} - 14512 \nu + 6304$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$71 \nu^{15} - 326 \nu^{14} + 532 \nu^{13} + 50 \nu^{12} - 1786 \nu^{11} + 3006 \nu^{10} - 402 \nu^{9} - 5850 \nu^{8} + 8235 \nu^{7} + 1244 \nu^{6} - 17030 \nu^{5} + 20100 \nu^{4} - 992 \nu^{3} - 24160 \nu^{2} + 29472 \nu - 12864$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$-41 \nu^{15} + 190 \nu^{14} - 310 \nu^{13} - 30 \nu^{12} + 1042 \nu^{11} - 1742 \nu^{10} + 210 \nu^{9} + 3418 \nu^{8} - 4753 \nu^{7} - 788 \nu^{6} + 9920 \nu^{5} - 11544 \nu^{4} + 408 \nu^{3} + 13936 \nu^{2} - 16864 \nu + 7232$$$$)/64$$ $$\beta_{13}$$ $$=$$ $$($$$$-81 \nu^{15} + 378 \nu^{14} - 628 \nu^{13} - 38 \nu^{12} + 2070 \nu^{11} - 3538 \nu^{10} + 542 \nu^{9} + 6806 \nu^{8} - 9709 \nu^{7} - 1268 \nu^{6} + 19858 \nu^{5} - 23636 \nu^{4} + 1344 \nu^{3} + 28000 \nu^{2} - 34400 \nu + 15040$$$$)/128$$ $$\beta_{14}$$ $$=$$ $$($$$$-115 \nu^{15} + 532 \nu^{14} - 882 \nu^{13} - 54 \nu^{12} + 2914 \nu^{11} - 4982 \nu^{10} + 778 \nu^{9} + 9570 \nu^{8} - 13719 \nu^{7} - 1738 \nu^{6} + 28004 \nu^{5} - 33512 \nu^{4} + 2048 \nu^{3} + 39680 \nu^{2} - 48896 \nu + 21504$$$$)/128$$ $$\beta_{15}$$ $$=$$ $$($$$$307 \nu^{15} - 1386 \nu^{14} + 2222 \nu^{13} + 282 \nu^{12} - 7606 \nu^{11} + 12562 \nu^{10} - 1310 \nu^{9} - 24918 \nu^{8} + 34259 \nu^{7} + 6308 \nu^{6} - 72228 \nu^{5} + 83264 \nu^{4} - 1808 \nu^{3} - 102208 \nu^{2} + 121792 \nu - 51456$$$$)/256$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{13} + \beta_{12} - 3 \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 3 \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 6 \beta_{2} - \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} - 2 \beta_{14} + \beta_{13} + 3 \beta_{12} - 3 \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + 4 \beta_{2} + 3 \beta_{1} + 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} - \beta_{13} + 6 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} + 3 \beta_{3} + 12 \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{15} + 3 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - 11 \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} + 6$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$5 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} - \beta_{9} - 4 \beta_{8} - 3 \beta_{6} - 13 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 6 \beta_{2} - 15 \beta_{1} - 12$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-3 \beta_{15} - 6 \beta_{14} + 11 \beta_{13} + \beta_{12} - 6 \beta_{11} - 3 \beta_{10} - 13 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} - 9 \beta_{6} + 17 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 15 \beta_{1} + 10$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$\beta_{15} + 5 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} + 18 \beta_{11} + \beta_{10} + 9 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} + 25 \beta_{6} + 3 \beta_{5} + 25 \beta_{4} - 5 \beta_{3} + 16 \beta_{2} - 11 \beta_{1} - 10$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-17 \beta_{15} + 16 \beta_{14} - 7 \beta_{13} + 33 \beta_{12} + 24 \beta_{11} - 19 \beta_{10} + 9 \beta_{9} + 16 \beta_{8} - \beta_{6} + 31 \beta_{5} + 25 \beta_{4} + 9 \beta_{3} + 26 \beta_{2} - 9 \beta_{1} + 16$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$3 \beta_{15} + 17 \beta_{14} - 19 \beta_{13} + 10 \beta_{12} + 22 \beta_{11} + 9 \beta_{10} + 29 \beta_{9} + 16 \beta_{7} + 11 \beta_{6} - 29 \beta_{5} + 19 \beta_{4} - 19 \beta_{3} - 18 \beta_{2} - 13 \beta_{1} - 42$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$19 \beta_{15} + 14 \beta_{14} - 27 \beta_{13} + 15 \beta_{12} + 24 \beta_{11} - 23 \beta_{10} - 11 \beta_{9} - 24 \beta_{8} + 24 \beta_{7} + 37 \beta_{6} - 11 \beta_{5} + 23 \beta_{4} - 45 \beta_{3} - 36 \beta_{2} - 25 \beta_{1} - 20$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$11 \beta_{15} - 53 \beta_{14} - 29 \beta_{13} - 50 \beta_{12} - 4 \beta_{11} + 81 \beta_{10} - 25 \beta_{9} - 34 \beta_{8} + 14 \beta_{7} + 51 \beta_{6} - 17 \beta_{5} + 5 \beta_{4} - 81 \beta_{3} - 60 \beta_{2} - 53 \beta_{1} - 24$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-23 \beta_{15} - 17 \beta_{13} + 23 \beta_{12} + 98 \beta_{11} - 7 \beta_{10} + 11 \beta_{9} - 100 \beta_{8} - 20 \beta_{7} + 121 \beta_{6} + 75 \beta_{5} + 125 \beta_{4} - 47 \beta_{3} - 62 \beta_{2} - 23 \beta_{1} + 54$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-71 \beta_{15} + 23 \beta_{14} - 73 \beta_{13} + 114 \beta_{12} + 44 \beta_{11} + 49 \beta_{10} + 83 \beta_{9} + 4 \beta_{8} - 88 \beta_{7} - 55 \beta_{6} - 41 \beta_{5} + 95 \beta_{4} + 57 \beta_{3} + 6 \beta_{2} - 171 \beta_{1} - 36$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$\beta_{5}$$ $$1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.885279 − 1.10285i 1.28040 + 0.600471i 0.424183 + 1.34910i −1.40988 + 0.110627i −1.20803 − 0.735291i 1.38194 + 0.300388i 1.40838 − 0.128355i 0.237728 − 1.39409i 0.885279 + 1.10285i 1.28040 − 0.600471i 0.424183 − 1.34910i −1.40988 − 0.110627i −1.20803 + 0.735291i 1.38194 − 0.300388i 1.40838 + 0.128355i 0.237728 + 1.39409i
−1.41211 + 0.0770377i 1.00000i 1.98813 0.217572i −0.658594 2.13688i −0.0770377 1.41211i −3.54781 + 3.54781i −2.79070 + 0.460397i −1.00000 1.09463 + 2.96678i
43.2 −1.15757 0.812425i 1.00000i 0.679932 + 1.88088i −1.69674 + 1.45639i 0.812425 1.15757i 1.12791 1.12791i 0.741001 2.72964i −1.00000 3.14730 0.307392i
43.3 −0.733173 1.20932i 1.00000i −0.924916 + 1.77328i −0.609492 2.15140i 1.20932 0.733173i 0.566689 0.566689i 2.82259 0.181602i −1.00000 −2.15487 + 2.31442i
43.4 −0.489639 + 1.32675i 1.00000i −1.52051 1.29925i −0.849960 + 2.06823i −1.32675 0.489639i −2.08016 + 2.08016i 2.46828 1.38116i −1.00000 −2.32784 2.14037i
43.5 0.873858 1.11192i 1.00000i −0.472743 1.94333i 1.54804 1.61356i 1.11192 + 0.873858i 0.143894 0.143894i −2.57394 1.17254i −1.00000 −0.441393 3.13132i
43.6 1.24128 + 0.677660i 1.00000i 1.08155 + 1.68233i −2.21420 + 0.311968i −0.677660 + 1.24128i −1.96597 + 1.96597i 0.202464 + 2.82117i −1.00000 −2.95985 1.11323i
43.7 1.32980 + 0.481284i 1.00000i 1.53673 + 1.28002i 0.539352 2.17005i −0.481284 + 1.32980i 3.00806 3.00806i 1.42749 + 2.44178i −1.00000 1.76164 2.62614i
43.8 1.34756 0.429059i 1.00000i 1.63182 1.15636i −0.0583995 + 2.23531i 0.429059 + 1.34756i 0.747384 0.747384i 1.70282 2.25841i −1.00000 0.880380 + 3.03726i
67.1 −1.41211 0.0770377i 1.00000i 1.98813 + 0.217572i −0.658594 + 2.13688i −0.0770377 + 1.41211i −3.54781 3.54781i −2.79070 0.460397i −1.00000 1.09463 2.96678i
67.2 −1.15757 + 0.812425i 1.00000i 0.679932 1.88088i −1.69674 1.45639i 0.812425 + 1.15757i 1.12791 + 1.12791i 0.741001 + 2.72964i −1.00000 3.14730 + 0.307392i
67.3 −0.733173 + 1.20932i 1.00000i −0.924916 1.77328i −0.609492 + 2.15140i 1.20932 + 0.733173i 0.566689 + 0.566689i 2.82259 + 0.181602i −1.00000 −2.15487 2.31442i
67.4 −0.489639 1.32675i 1.00000i −1.52051 + 1.29925i −0.849960 2.06823i −1.32675 + 0.489639i −2.08016 2.08016i 2.46828 + 1.38116i −1.00000 −2.32784 + 2.14037i
67.5 0.873858 + 1.11192i 1.00000i −0.472743 + 1.94333i 1.54804 + 1.61356i 1.11192 0.873858i 0.143894 + 0.143894i −2.57394 + 1.17254i −1.00000 −0.441393 + 3.13132i
67.6 1.24128 0.677660i 1.00000i 1.08155 1.68233i −2.21420 0.311968i −0.677660 1.24128i −1.96597 1.96597i 0.202464 2.82117i −1.00000 −2.95985 + 1.11323i
67.7 1.32980 0.481284i 1.00000i 1.53673 1.28002i 0.539352 + 2.17005i −0.481284 1.32980i 3.00806 + 3.00806i 1.42749 2.44178i −1.00000 1.76164 + 2.62614i
67.8 1.34756 + 0.429059i 1.00000i 1.63182 + 1.15636i −0.0583995 2.23531i 0.429059 1.34756i 0.747384 + 0.747384i 1.70282 + 2.25841i −1.00000 0.880380 3.03726i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.bc.e yes 16
3.b odd 2 1 720.2.bd.f 16
4.b odd 2 1 960.2.bc.e 16
5.c odd 4 1 240.2.y.e 16
8.b even 2 1 1920.2.bc.i 16
8.d odd 2 1 1920.2.bc.j 16
15.e even 4 1 720.2.z.f 16
16.e even 4 1 960.2.y.e 16
16.e even 4 1 1920.2.y.j 16
16.f odd 4 1 240.2.y.e 16
16.f odd 4 1 1920.2.y.i 16
20.e even 4 1 960.2.y.e 16
40.i odd 4 1 1920.2.y.i 16
40.k even 4 1 1920.2.y.j 16
48.k even 4 1 720.2.z.f 16
80.i odd 4 1 1920.2.bc.j 16
80.j even 4 1 inner 240.2.bc.e yes 16
80.s even 4 1 1920.2.bc.i 16
80.t odd 4 1 960.2.bc.e 16
240.bd odd 4 1 720.2.bd.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.e 16 5.c odd 4 1
240.2.y.e 16 16.f odd 4 1
240.2.bc.e yes 16 1.a even 1 1 trivial
240.2.bc.e yes 16 80.j even 4 1 inner
720.2.z.f 16 15.e even 4 1
720.2.z.f 16 48.k even 4 1
720.2.bd.f 16 3.b odd 2 1
720.2.bd.f 16 240.bd odd 4 1
960.2.y.e 16 16.e even 4 1
960.2.y.e 16 20.e even 4 1
960.2.bc.e 16 4.b odd 2 1
960.2.bc.e 16 80.t odd 4 1
1920.2.y.i 16 16.f odd 4 1
1920.2.y.i 16 40.i odd 4 1
1920.2.y.j 16 16.e even 4 1
1920.2.y.j 16 40.k even 4 1
1920.2.bc.i 16 8.b even 2 1
1920.2.bc.i 16 80.s even 4 1
1920.2.bc.j 16 8.d odd 2 1
1920.2.bc.j 16 80.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{16} + \cdots$$ $$T_{11}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 256 T - 128 T^{2} + 128 T^{3} + 128 T^{4} - 32 T^{5} - 112 T^{6} + 16 T^{7} + 52 T^{8} + 8 T^{9} - 28 T^{10} - 4 T^{11} + 8 T^{12} + 4 T^{13} - 2 T^{14} - 2 T^{15} + T^{16}$$
$3$ $$( 1 + T^{2} )^{8}$$
$5$ $$390625 + 625000 T + 750000 T^{2} + 650000 T^{3} + 470000 T^{4} + 292000 T^{5} + 160800 T^{6} + 81000 T^{7} + 37486 T^{8} + 16200 T^{9} + 6432 T^{10} + 2336 T^{11} + 752 T^{12} + 208 T^{13} + 48 T^{14} + 8 T^{15} + T^{16}$$
$7$ $$2304 - 23040 T + 115200 T^{2} - 251904 T^{3} + 314752 T^{4} - 202368 T^{5} + 56832 T^{6} + 7360 T^{7} + 7856 T^{8} - 7648 T^{9} + 2432 T^{10} + 1264 T^{11} + 392 T^{12} - 32 T^{13} + 8 T^{14} + 4 T^{15} + T^{16}$$
$11$ $$952576 + 31232 T + 512 T^{2} - 36096 T^{3} + 1229440 T^{4} - 4608 T^{5} - 128 T^{6} + 3264 T^{7} + 335152 T^{8} + 4864 T^{9} + 32 T^{10} - 400 T^{11} + 1288 T^{12} - 8 T^{13} + T^{16}$$
$13$ $$( 2896 - 2688 T - 2336 T^{2} + 1696 T^{3} + 660 T^{4} - 184 T^{5} - 52 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$17$ $$9339136 + 11637248 T + 7250432 T^{2} + 1576192 T^{3} + 1092736 T^{4} + 1090048 T^{5} + 642944 T^{6} + 154688 T^{7} + 33072 T^{8} + 16704 T^{9} + 9632 T^{10} + 2384 T^{11} + 344 T^{12} + 56 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$19$ $$5308416 - 28311552 T + 75497472 T^{2} + 31162368 T^{3} + 12132352 T^{4} - 20144128 T^{5} + 26353664 T^{6} + 4669440 T^{7} + 365312 T^{8} + 8704 T^{9} + 159232 T^{10} + 37376 T^{11} + 4384 T^{12} + 32 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$23$ $$330366976 - 632815616 T + 606076928 T^{2} - 281018368 T^{3} + 78913536 T^{4} - 20127744 T^{5} + 13303808 T^{6} - 6053888 T^{7} + 1573632 T^{8} - 164352 T^{9} + 12800 T^{10} - 5376 T^{11} + 2144 T^{12} - 160 T^{13} + T^{16}$$
$29$ $$45373696 + 15088640 T + 2508800 T^{2} - 57830144 T^{3} + 205079424 T^{4} - 54760320 T^{5} + 7303808 T^{6} - 317056 T^{7} + 2256816 T^{8} - 658272 T^{9} + 93152 T^{10} + 8640 T^{11} + 968 T^{12} - 344 T^{13} + 72 T^{14} + 12 T^{15} + T^{16}$$
$31$ $$1655277803776 + 715784229888 T^{2} + 109950837504 T^{4} + 7591641856 T^{6} + 275577056 T^{8} + 5639616 T^{10} + 65392 T^{12} + 400 T^{14} + T^{16}$$
$37$ $$( 28176 - 35712 T - 4352 T^{2} + 11232 T^{3} + 84 T^{4} - 904 T^{5} - 60 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$41$ $$177209344 + 5163712512 T^{2} + 3275882496 T^{4} + 662142976 T^{6} + 51522560 T^{8} + 1887744 T^{10} + 34624 T^{12} + 304 T^{14} + T^{16}$$
$43$ $$( 193792 + 389888 T - 89216 T^{2} - 48000 T^{3} + 6224 T^{4} + 1568 T^{5} - 144 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$47$ $$65536 + 1966080 T + 29491200 T^{2} + 208142336 T^{3} + 705069056 T^{4} - 436641792 T^{5} + 149700608 T^{6} + 53747712 T^{7} + 32079360 T^{8} - 10061312 T^{9} + 1450496 T^{10} - 58368 T^{11} + 14976 T^{12} - 3808 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$53$ $$35845091584 + 27828649984 T^{2} + 7247269504 T^{4} + 857669376 T^{6} + 52530608 T^{8} + 1757056 T^{10} + 31848 T^{12} + 288 T^{14} + T^{16}$$
$59$ $$8785580546304 + 3405193191936 T + 659907482112 T^{2} + 5539065600 T^{3} + 76917010048 T^{4} + 30963593984 T^{5} + 6225456512 T^{6} + 22966592 T^{7} - 6863056 T^{8} + 6636224 T^{9} + 3162272 T^{10} - 159344 T^{11} + 2312 T^{12} - 88 T^{13} + 288 T^{14} - 24 T^{15} + T^{16}$$
$61$ $$55857327698176 + 14044002376704 T + 1765515921408 T^{2} - 3702308508160 T^{3} + 1391650424064 T^{4} - 221883207424 T^{5} + 22923310592 T^{6} - 2889433984 T^{7} + 699602272 T^{8} - 111801536 T^{9} + 10869120 T^{10} - 663648 T^{11} + 63696 T^{12} - 8400 T^{13} + 800 T^{14} - 40 T^{15} + T^{16}$$
$67$ $$( -18176 + 6144 T + 19840 T^{2} - 12032 T^{3} + 288 T^{4} + 800 T^{5} - 88 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$71$ $$( 3900672 + 74496 T - 834176 T^{2} - 20416 T^{3} + 42640 T^{4} + 112 T^{5} - 392 T^{6} + T^{8} )^{2}$$
$73$ $$2179567984896 - 4135347053568 T + 3923046991872 T^{2} - 2106688255488 T^{3} + 689259012352 T^{4} - 121945550592 T^{5} + 8882210304 T^{6} + 423075200 T^{7} + 148347488 T^{8} - 40257728 T^{9} + 2951552 T^{10} + 649952 T^{11} + 70928 T^{12} - 208 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$79$ $$( -1507376 - 352960 T + 368096 T^{2} - 9824 T^{3} - 18952 T^{4} + 2512 T^{5} + 56 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$83$ $$34668544 + 1441071104 T^{2} + 14798209024 T^{4} + 4717408256 T^{6} + 288146944 T^{8} + 7046912 T^{10} + 82624 T^{12} + 464 T^{14} + T^{16}$$
$89$ $$( -3899136 - 2790144 T - 305024 T^{2} + 143552 T^{3} + 25168 T^{4} - 1424 T^{5} - 344 T^{6} + T^{8} )^{2}$$
$97$ $$16602430353664 - 1147409612800 T + 39649280000 T^{2} - 1260709630976 T^{3} + 2748398726400 T^{4} - 706044478464 T^{5} + 90097928192 T^{6} + 1208939264 T^{7} + 1104328032 T^{8} - 208141824 T^{9} + 17785856 T^{10} - 554688 T^{11} + 85328 T^{12} - 13376 T^{13} + 1152 T^{14} - 48 T^{15} + T^{16}$$
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