# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.y (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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-15 \nu^{15} + 62 \nu^{14} - 94 \nu^{13} - 26 \nu^{12} + 342 \nu^{11} - 530 \nu^{10} - 2 \nu^{9} + 1110 \nu^{8} - 1415 \nu^{7} - 424 \nu^{6} + 3188 \nu^{5} - 3360 \nu^{4} - 240 \nu^{3} + 4480 \nu^{2} - 5056 \nu + 1792$$$$)/256$$ $$\beta_{2}$$ $$=$$ $$($$$$-13 \nu^{15} + 42 \nu^{14} - 36 \nu^{13} - 70 \nu^{12} + 238 \nu^{11} - 202 \nu^{10} - 250 \nu^{9} + 734 \nu^{8} - 409 \nu^{7} - 892 \nu^{6} + 1834 \nu^{5} - 908 \nu^{4} - 1232 \nu^{3} + 2624 \nu^{2} - 1760 \nu + 64$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$27 \nu^{15} - 88 \nu^{14} + 78 \nu^{13} + 142 \nu^{12} - 498 \nu^{11} + 438 \nu^{10} + 502 \nu^{9} - 1538 \nu^{8} + 911 \nu^{7} + 1822 \nu^{6} - 3856 \nu^{5} + 2064 \nu^{4} + 2544 \nu^{3} - 5408 \nu^{2} + 3584 \nu - 256$$$$)/256$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{15} - 43 \nu^{14} + 66 \nu^{13} + 16 \nu^{12} - 232 \nu^{11} + 368 \nu^{10} - 24 \nu^{9} - 752 \nu^{8} + 1001 \nu^{7} + 207 \nu^{6} - 2162 \nu^{5} + 2520 \nu^{4} - 80 \nu^{3} - 3056 \nu^{2} + 3680 \nu - 1664$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{15} - 21 \nu^{14} + 40 \nu^{13} - 8 \nu^{12} - 114 \nu^{11} + 226 \nu^{10} - 78 \nu^{9} - 382 \nu^{8} + 646 \nu^{7} - 53 \nu^{6} - 1158 \nu^{5} + 1594 \nu^{4} - 308 \nu^{3} - 1640 \nu^{2} + 2272 \nu - 1120$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-17 \nu^{15} + 98 \nu^{14} - 192 \nu^{13} + 50 \nu^{12} + 526 \nu^{11} - 1098 \nu^{10} + 454 \nu^{9} + 1758 \nu^{8} - 3173 \nu^{7} + 492 \nu^{6} + 5462 \nu^{5} - 7876 \nu^{4} + 1904 \nu^{3} + 7744 \nu^{2} - 11360 \nu + 5696$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$81 \nu^{15} - 348 \nu^{14} + 538 \nu^{13} + 114 \nu^{12} - 1918 \nu^{11} + 3034 \nu^{10} - 134 \nu^{9} - 6254 \nu^{8} + 8165 \nu^{7} + 2070 \nu^{6} - 17864 \nu^{5} + 19808 \nu^{4} + 432 \nu^{3} - 25312 \nu^{2} + 29184 \nu - 12032$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$87 \nu^{15} - 378 \nu^{14} + 582 \nu^{13} + 130 \nu^{12} - 2094 \nu^{11} + 3290 \nu^{10} - 86 \nu^{9} - 6830 \nu^{8} + 8807 \nu^{7} + 2428 \nu^{6} - 19540 \nu^{5} + 21248 \nu^{4} + 912 \nu^{3} - 27584 \nu^{2} + 31296 \nu - 12544$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$-50 \nu^{15} + 213 \nu^{14} - 322 \nu^{13} - 82 \nu^{12} + 1170 \nu^{11} - 1806 \nu^{10} + 10 \nu^{9} + 3786 \nu^{8} - 4820 \nu^{7} - 1407 \nu^{6} + 10772 \nu^{5} - 11660 \nu^{4} - 448 \nu^{3} + 15088 \nu^{2} - 17216 \nu + 7104$$$$)/128$$ $$\beta_{10}$$ $$=$$ $$($$$$-81 \nu^{15} + 408 \nu^{14} - 718 \nu^{13} + 38 \nu^{12} + 2222 \nu^{11} - 4042 \nu^{10} + 950 \nu^{9} + 7358 \nu^{8} - 11253 \nu^{7} - 466 \nu^{6} + 21852 \nu^{5} - 27464 \nu^{4} + 3120 \nu^{3} + 30688 \nu^{2} - 39616 \nu + 18048$$$$)/256$$ $$\beta_{11}$$ $$=$$ $$($$$$105 \nu^{15} - 484 \nu^{14} + 790 \nu^{13} + 82 \nu^{12} - 2662 \nu^{11} + 4466 \nu^{10} - 558 \nu^{9} - 8758 \nu^{8} + 12197 \nu^{7} + 2006 \nu^{6} - 25404 \nu^{5} + 29624 \nu^{4} - 976 \nu^{3} - 35872 \nu^{2} + 43328 \nu - 18560$$$$)/256$$ $$\beta_{12}$$ $$=$$ $$($$$$117 \nu^{15} - 612 \nu^{14} + 1114 \nu^{13} - 134 \nu^{12} - 3318 \nu^{11} + 6306 \nu^{10} - 1854 \nu^{9} - 11014 \nu^{8} + 17785 \nu^{7} - 442 \nu^{6} - 33152 \nu^{5} + 43712 \nu^{4} - 7024 \nu^{3} - 47008 \nu^{2} + 63232 \nu - 29952$$$$)/256$$ $$\beta_{13}$$ $$=$$ $$($$$$-103 \nu^{15} + 474 \nu^{14} - 774 \nu^{13} - 66 \nu^{12} + 2590 \nu^{11} - 4378 \nu^{10} + 598 \nu^{9} + 8494 \nu^{8} - 11991 \nu^{7} - 1756 \nu^{6} + 24788 \nu^{5} - 29248 \nu^{4} + 1408 \nu^{3} + 35072 \nu^{2} - 42752 \nu + 18432$$$$)/128$$ $$\beta_{14}$$ $$=$$ $$($$$$-223 \nu^{15} + 1030 \nu^{14} - 1694 \nu^{13} - 122 \nu^{12} + 5638 \nu^{11} - 9602 \nu^{10} + 1422 \nu^{9} + 18534 \nu^{8} - 26407 \nu^{7} - 3584 \nu^{6} + 54212 \nu^{5} - 64304 \nu^{4} + 3376 \nu^{3} + 76992 \nu^{2} - 93888 \nu + 40448$$$$)/256$$ $$\beta_{15}$$ $$=$$ $$($$$$127 \nu^{15} - 602 \nu^{14} + 1016 \nu^{13} + 26 \nu^{12} - 3290 \nu^{11} + 5742 \nu^{10} - 1058 \nu^{9} - 10826 \nu^{8} + 15875 \nu^{7} + 1520 \nu^{6} - 31850 \nu^{5} + 38796 \nu^{4} - 3120 \nu^{3} - 45120 \nu^{2} + 56544 \nu - 25152$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_{1} + 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} - \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + 7 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 3 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + 8 \beta_{11} + 11 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} - 6 \beta_{7} + 3 \beta_{6} - \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - \beta_{12} + 5 \beta_{11} + 5 \beta_{10} - 6 \beta_{9} - 9 \beta_{8} - 11 \beta_{7} + 4 \beta_{6} + 7 \beta_{5} - 5 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} - 9 \beta_{1} - 2$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$9 \beta_{15} + 3 \beta_{14} + \beta_{13} + 4 \beta_{12} - 6 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} - 10 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 11 \beta_{3} + 14 \beta_{2} - 12 \beta_{1} + 1$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-7 \beta_{15} - \beta_{14} - 3 \beta_{13} + 11 \beta_{12} - 3 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 9 \beta_{8} - 17 \beta_{7} + 10 \beta_{6} + 19 \beta_{5} + 5 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} - 13 \beta_{1} - 6$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-23 \beta_{15} + 17 \beta_{14} - 25 \beta_{13} + 22 \beta_{12} + 2 \beta_{11} + 17 \beta_{10} + 7 \beta_{9} + 15 \beta_{8} + 46 \beta_{7} - 9 \beta_{6} - 23 \beta_{5} - 9 \beta_{4} - 9 \beta_{3} + 24 \beta_{2} - 16 \beta_{1} - 7$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-27 \beta_{15} - 3 \beta_{14} - 29 \beta_{13} + 3 \beta_{12} + 35 \beta_{11} - 5 \beta_{10} + 10 \beta_{9} - 29 \beta_{8} - 11 \beta_{7} + 16 \beta_{6} + 5 \beta_{5} + 19 \beta_{4} - 25 \beta_{3} - 19 \beta_{2} - 23 \beta_{1} - 52$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-19 \beta_{15} - 19 \beta_{14} + 35 \beta_{13} + 40 \beta_{12} + 8 \beta_{11} - 11 \beta_{10} + 5 \beta_{9} + 13 \beta_{8} + 6 \beta_{7} + 5 \beta_{6} - 43 \beta_{5} + 45 \beta_{4} + 39 \beta_{3} + 26 \beta_{2} - 30 \beta_{1} - 1$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$27 \beta_{15} - 11 \beta_{14} + 59 \beta_{13} + 47 \beta_{12} - 37 \beta_{11} - 51 \beta_{10} + 26 \beta_{9} + 17 \beta_{8} - 53 \beta_{7} + 90 \beta_{6} + 13 \beta_{5} + 81 \beta_{4} + 57 \beta_{3} - 25 \beta_{2} - 79 \beta_{1} - 16$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$49 \beta_{15} + 23 \beta_{14} + 89 \beta_{13} + 38 \beta_{12} + 40 \beta_{11} + 119 \beta_{10} - 49 \beta_{9} + 175 \beta_{8} - 102 \beta_{7} + 7 \beta_{6} + 43 \beta_{5} + 47 \beta_{4} - 49 \beta_{3} + 12 \beta_{2} + 26 \beta_{1} + 203$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$15 \beta_{15} + 71 \beta_{14} - 87 \beta_{13} - 77 \beta_{12} + 33 \beta_{11} + 113 \beta_{10} - 38 \beta_{9} - 45 \beta_{8} + 153 \beta_{7} - 52 \beta_{6} + 27 \beta_{5} - 57 \beta_{4} - 205 \beta_{3} + 31 \beta_{2} - 133 \beta_{1} - 2$$$$)/2$$

## Character values

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

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$\beta_{8}$$ $$1$$ $$\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 −1.40988 + 0.110627i 1.38194 + 0.300388i 1.40838 − 0.128355i 0.885279 − 1.10285i 0.237728 − 1.39409i 1.28040 + 0.600471i −1.20803 − 0.735291i 0.424183 + 1.34910i −1.40988 − 0.110627i 1.38194 − 0.300388i 1.40838 + 0.128355i 0.885279 + 1.10285i 0.237728 + 1.39409i 1.28040 − 0.600471i −1.20803 + 0.735291i 0.424183 − 1.34910i
−1.32675 + 0.489639i 1.00000 1.52051 1.29925i 2.06823 + 0.849960i −1.32675 + 0.489639i −2.08016 + 2.08016i −1.38116 + 2.46828i 1.00000 −3.16019 0.114996i
163.2 −0.677660 1.24128i 1.00000 −1.08155 + 1.68233i 0.311968 + 2.21420i −0.677660 1.24128i −1.96597 + 1.96597i 2.82117 + 0.202464i 1.00000 2.53703 1.88771i
163.3 −0.481284 1.32980i 1.00000 −1.53673 + 1.28002i −2.17005 0.539352i −0.481284 1.32980i 3.00806 3.00806i 2.44178 + 1.42749i 1.00000 0.327178 + 3.14531i
163.4 −0.0770377 + 1.41211i 1.00000 −1.98813 0.217572i −2.13688 + 0.658594i −0.0770377 + 1.41211i −3.54781 + 3.54781i 0.460397 2.79070i 1.00000 −0.765389 3.06825i
163.5 0.429059 1.34756i 1.00000 −1.63182 1.15636i 2.23531 + 0.0583995i 0.429059 1.34756i 0.747384 0.747384i −2.25841 + 1.70282i 1.00000 1.03777 2.98714i
163.6 0.812425 + 1.15757i 1.00000 −0.679932 + 1.88088i 1.45639 + 1.69674i 0.812425 + 1.15757i 1.12791 1.12791i −2.72964 + 0.741001i 1.00000 −0.780893 + 3.06434i
163.7 1.11192 0.873858i 1.00000 0.472743 1.94333i −1.61356 1.54804i 1.11192 0.873858i 0.143894 0.143894i −1.17254 2.57394i 1.00000 −3.14692 0.311271i
163.8 1.20932 + 0.733173i 1.00000 0.924916 + 1.77328i −2.15140 + 0.609492i 1.20932 + 0.733173i 0.566689 0.566689i −0.181602 + 2.82259i 1.00000 −3.04860 0.840275i
187.1 −1.32675 0.489639i 1.00000 1.52051 + 1.29925i 2.06823 0.849960i −1.32675 0.489639i −2.08016 2.08016i −1.38116 2.46828i 1.00000 −3.16019 + 0.114996i
187.2 −0.677660 + 1.24128i 1.00000 −1.08155 1.68233i 0.311968 2.21420i −0.677660 + 1.24128i −1.96597 1.96597i 2.82117 0.202464i 1.00000 2.53703 + 1.88771i
187.3 −0.481284 + 1.32980i 1.00000 −1.53673 1.28002i −2.17005 + 0.539352i −0.481284 + 1.32980i 3.00806 + 3.00806i 2.44178 1.42749i 1.00000 0.327178 3.14531i
187.4 −0.0770377 1.41211i 1.00000 −1.98813 + 0.217572i −2.13688 0.658594i −0.0770377 1.41211i −3.54781 3.54781i 0.460397 + 2.79070i 1.00000 −0.765389 + 3.06825i
187.5 0.429059 + 1.34756i 1.00000 −1.63182 + 1.15636i 2.23531 0.0583995i 0.429059 + 1.34756i 0.747384 + 0.747384i −2.25841 1.70282i 1.00000 1.03777 + 2.98714i
187.6 0.812425 1.15757i 1.00000 −0.679932 1.88088i 1.45639 1.69674i 0.812425 1.15757i 1.12791 + 1.12791i −2.72964 0.741001i 1.00000 −0.780893 3.06434i
187.7 1.11192 + 0.873858i 1.00000 0.472743 + 1.94333i −1.61356 + 1.54804i 1.11192 + 0.873858i 0.143894 + 0.143894i −1.17254 + 2.57394i 1.00000 −3.14692 + 0.311271i
187.8 1.20932 0.733173i 1.00000 0.924916 1.77328i −2.15140 0.609492i 1.20932 0.733173i 0.566689 + 0.566689i −0.181602 2.82259i 1.00000 −3.04860 + 0.840275i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 187.8 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.s 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.y.e 16
3.b odd 2 1 720.2.z.f 16
4.b odd 2 1 960.2.y.e 16
5.c odd 4 1 240.2.bc.e yes 16
8.b even 2 1 1920.2.y.i 16
8.d odd 2 1 1920.2.y.j 16
15.e even 4 1 720.2.bd.f 16
16.e even 4 1 960.2.bc.e 16
16.e even 4 1 1920.2.bc.j 16
16.f odd 4 1 240.2.bc.e yes 16
16.f odd 4 1 1920.2.bc.i 16
20.e even 4 1 960.2.bc.e 16
40.i odd 4 1 1920.2.bc.i 16
40.k even 4 1 1920.2.bc.j 16
48.k even 4 1 720.2.bd.f 16
80.i odd 4 1 960.2.y.e 16
80.j even 4 1 1920.2.y.i 16
80.s even 4 1 inner 240.2.y.e 16
80.t odd 4 1 1920.2.y.j 16
240.z odd 4 1 720.2.z.f 16

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

$p$ $F_p(T)$
$2$ $$256 - 256 T + 384 T^{2} - 256 T^{3} + 256 T^{4} - 96 T^{5} + 80 T^{6} + 20 T^{8} + 20 T^{10} - 12 T^{11} + 16 T^{12} - 8 T^{13} + 6 T^{14} - 2 T^{15} + T^{16}$$
$3$ $$( -1 + T )^{16}$$
$5$ $$390625 + 312500 T - 125000 T^{2} - 137500 T^{3} + 20000 T^{4} + 23500 T^{5} - 5800 T^{6} - 1700 T^{7} + 1486 T^{8} - 340 T^{9} - 232 T^{10} + 188 T^{11} + 32 T^{12} - 44 T^{13} - 8 T^{14} + 4 T^{15} + T^{16}$$
$7$ $$2304 - 23040 T + 115200 T^{2} - 251904 T^{3} + 314752 T^{4} - 202368 T^{5} + 56832 T^{6} + 7360 T^{7} + 7856 T^{8} - 7648 T^{9} + 2432 T^{10} + 1264 T^{11} + 392 T^{12} - 32 T^{13} + 8 T^{14} + 4 T^{15} + T^{16}$$
$11$ $$952576 + 31232 T + 512 T^{2} - 36096 T^{3} + 1229440 T^{4} - 4608 T^{5} - 128 T^{6} + 3264 T^{7} + 335152 T^{8} + 4864 T^{9} + 32 T^{10} - 400 T^{11} + 1288 T^{12} - 8 T^{13} + T^{16}$$
$13$ $$8386816 + 20755456 T^{2} + 18397312 T^{4} + 7250304 T^{6} + 1329968 T^{8} + 120736 T^{10} + 5496 T^{12} + 120 T^{14} + T^{16}$$
$17$ $$9339136 + 11637248 T + 7250432 T^{2} + 1576192 T^{3} + 1092736 T^{4} + 1090048 T^{5} + 642944 T^{6} + 154688 T^{7} + 33072 T^{8} + 16704 T^{9} + 9632 T^{10} + 2384 T^{11} + 344 T^{12} + 56 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$19$ $$5308416 + 28311552 T + 75497472 T^{2} - 31162368 T^{3} + 12132352 T^{4} + 20144128 T^{5} + 26353664 T^{6} - 4669440 T^{7} + 365312 T^{8} - 8704 T^{9} + 159232 T^{10} - 37376 T^{11} + 4384 T^{12} - 32 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$23$ $$330366976 - 632815616 T + 606076928 T^{2} - 281018368 T^{3} + 78913536 T^{4} - 20127744 T^{5} + 13303808 T^{6} - 6053888 T^{7} + 1573632 T^{8} - 164352 T^{9} + 12800 T^{10} - 5376 T^{11} + 2144 T^{12} - 160 T^{13} + T^{16}$$
$29$ $$45373696 - 15088640 T + 2508800 T^{2} + 57830144 T^{3} + 205079424 T^{4} + 54760320 T^{5} + 7303808 T^{6} + 317056 T^{7} + 2256816 T^{8} + 658272 T^{9} + 93152 T^{10} - 8640 T^{11} + 968 T^{12} + 344 T^{13} + 72 T^{14} - 12 T^{15} + T^{16}$$
$31$ $$1655277803776 + 715784229888 T^{2} + 109950837504 T^{4} + 7591641856 T^{6} + 275577056 T^{8} + 5639616 T^{10} + 65392 T^{12} + 400 T^{14} + T^{16}$$
$37$ $$793886976 + 1520590848 T^{2} + 825907840 T^{4} + 194837376 T^{6} + 21750192 T^{8} + 1105568 T^{10} + 25464 T^{12} + 264 T^{14} + T^{16}$$
$41$ $$177209344 + 5163712512 T^{2} + 3275882496 T^{4} + 662142976 T^{6} + 51522560 T^{8} + 1887744 T^{10} + 34624 T^{12} + 304 T^{14} + T^{16}$$
$43$ $$37555339264 + 186591346688 T^{2} + 47801065472 T^{4} + 4693061632 T^{6} + 224705280 T^{8} + 5581568 T^{10} + 70816 T^{12} + 432 T^{14} + T^{16}$$
$47$ $$65536 - 1966080 T + 29491200 T^{2} - 208142336 T^{3} + 705069056 T^{4} + 436641792 T^{5} + 149700608 T^{6} - 53747712 T^{7} + 32079360 T^{8} + 10061312 T^{9} + 1450496 T^{10} + 58368 T^{11} + 14976 T^{12} + 3808 T^{13} + 512 T^{14} + 32 T^{15} + T^{16}$$
$53$ $$( 189328 + 119456 T - 35808 T^{2} - 19712 T^{3} + 3316 T^{4} + 792 T^{5} - 112 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$59$ $$8785580546304 - 3405193191936 T + 659907482112 T^{2} - 5539065600 T^{3} + 76917010048 T^{4} - 30963593984 T^{5} + 6225456512 T^{6} - 22966592 T^{7} - 6863056 T^{8} - 6636224 T^{9} + 3162272 T^{10} + 159344 T^{11} + 2312 T^{12} + 88 T^{13} + 288 T^{14} + 24 T^{15} + T^{16}$$
$61$ $$55857327698176 + 14044002376704 T + 1765515921408 T^{2} - 3702308508160 T^{3} + 1391650424064 T^{4} - 221883207424 T^{5} + 22923310592 T^{6} - 2889433984 T^{7} + 699602272 T^{8} - 111801536 T^{9} + 10869120 T^{10} - 663648 T^{11} + 63696 T^{12} - 8400 T^{13} + 800 T^{14} - 40 T^{15} + T^{16}$$
$67$ $$330366976 + 758972416 T^{2} + 531005440 T^{4} + 139972608 T^{6} + 15904256 T^{8} + 843520 T^{10} + 21120 T^{12} + 240 T^{14} + T^{16}$$
$71$ $$( 3900672 + 74496 T - 834176 T^{2} - 20416 T^{3} + 42640 T^{4} + 112 T^{5} - 392 T^{6} + T^{8} )^{2}$$
$73$ $$2179567984896 + 4135347053568 T + 3923046991872 T^{2} + 2106688255488 T^{3} + 689259012352 T^{4} + 121945550592 T^{5} + 8882210304 T^{6} - 423075200 T^{7} + 148347488 T^{8} + 40257728 T^{9} + 2951552 T^{10} - 649952 T^{11} + 70928 T^{12} + 208 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$79$ $$( -1507376 + 352960 T + 368096 T^{2} + 9824 T^{3} - 18952 T^{4} - 2512 T^{5} + 56 T^{6} + 24 T^{7} + T^{8} )^{2}$$
$83$ $$( -5888 + 256 T + 122368 T^{2} + 86336 T^{3} + 11168 T^{4} - 1264 T^{5} - 224 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$89$ $$( -3899136 + 2790144 T - 305024 T^{2} - 143552 T^{3} + 25168 T^{4} + 1424 T^{5} - 344 T^{6} + T^{8} )^{2}$$
$97$ $$16602430353664 - 1147409612800 T + 39649280000 T^{2} - 1260709630976 T^{3} + 2748398726400 T^{4} - 706044478464 T^{5} + 90097928192 T^{6} + 1208939264 T^{7} + 1104328032 T^{8} - 208141824 T^{9} + 17785856 T^{10} - 554688 T^{11} + 85328 T^{12} - 13376 T^{13} + 1152 T^{14} - 48 T^{15} + T^{16}$$