Properties

 Label 731.2.a.e Level 731 Weight 2 Character orbit 731.a Self dual Yes Analytic conductor 5.837 Analytic rank 0 Dimension 19 CM No Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.a (trivial)

Newform invariants

 Self dual: Yes Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$19$$ Coefficient field: $$\mathbb{Q}[x]/(x^{19} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{18}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{16} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{14} ) q^{5} -\beta_{17} q^{6} + \beta_{10} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{16} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{14} ) q^{5} -\beta_{17} q^{6} + \beta_{10} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{5} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} + \beta_{17} + \beta_{18} ) q^{10} + ( \beta_{1} + \beta_{8} ) q^{11} + ( -1 - \beta_{3} - \beta_{7} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{12} + ( 1 + \beta_{9} ) q^{13} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} + \beta_{12} ) q^{14} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{13} - \beta_{14} + \beta_{16} - \beta_{18} ) q^{15} + ( 2 - \beta_{1} - \beta_{4} - 2 \beta_{6} + \beta_{9} - \beta_{11} + \beta_{15} ) q^{16} + q^{17} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{10} ) q^{18} + ( 1 - \beta_{3} + \beta_{5} - \beta_{15} ) q^{19} + ( 3 + \beta_{7} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{20} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{17} - \beta_{18} ) q^{21} + ( 1 + 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{22} + ( -1 + \beta_{1} - \beta_{3} - \beta_{11} - \beta_{12} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{24} + ( 2 - \beta_{3} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{17} + \beta_{18} ) q^{25} + ( -2 + 2 \beta_{1} + \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{18} ) q^{26} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{18} ) q^{27} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{28} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{29} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{16} - 2 \beta_{17} ) q^{30} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{15} ) q^{31} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} ) q^{32} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{33} + \beta_{1} q^{34} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} ) q^{35} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} ) q^{36} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{11} + 3 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} ) q^{37} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{38} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{18} ) q^{39} + ( -2 + 2 \beta_{1} + \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{40} + ( 3 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{17} - \beta_{18} ) q^{41} + ( -1 + \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{12} + 3 \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{42} - q^{43} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{44} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{45} + ( 4 - \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{13} - 3 \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{46} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{47} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{17} ) q^{48} + ( 2 - 2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{12} - 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{49} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + 3 \beta_{11} - \beta_{13} - 2 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} ) q^{50} + \beta_{16} q^{51} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{52} + ( 1 + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{16} ) q^{53} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} ) q^{54} + ( 1 + \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{55} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{56} + ( 1 - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} ) q^{57} + ( -5 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{17} ) q^{58} + ( -1 - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - 2 \beta_{16} + \beta_{18} ) q^{59} + ( 1 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 4 \beta_{6} - \beta_{8} + \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} - 2 \beta_{17} - 2 \beta_{18} ) q^{60} + ( 6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{18} ) q^{61} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - 4 \beta_{14} + \beta_{16} - 2 \beta_{18} ) q^{62} + ( 2 - 3 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{63} + ( -3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + 3 \beta_{10} + \beta_{11} - 3 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{64} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{65} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{66} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{14} - \beta_{17} - \beta_{18} ) q^{67} + ( 1 + \beta_{2} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{69} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 3 \beta_{17} - \beta_{18} ) q^{70} + ( 2 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + \beta_{17} + \beta_{18} ) q^{71} + ( -\beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{16} + \beta_{18} ) q^{72} + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{17} - \beta_{18} ) q^{73} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{74} + ( 1 - 5 \beta_{1} + 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{75} + ( 4 - 4 \beta_{1} + \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{76} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{77} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{17} + \beta_{18} ) q^{78} + ( -1 + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{79} + ( 3 - 4 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{13} - 4 \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{80} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{81} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{82} + ( -3 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} - 3 \beta_{17} ) q^{83} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - 4 \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} ) q^{84} + ( 1 - \beta_{14} ) q^{85} -\beta_{1} q^{86} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} + 3 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{87} + ( 2 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} + 3 \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{88} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{89} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - 4 \beta_{16} + \beta_{17} - \beta_{18} ) q^{90} + ( 3 - 2 \beta_{1} - 2 \beta_{4} - \beta_{8} + \beta_{14} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{91} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{7} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} + \beta_{17} - \beta_{18} ) q^{93} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 4 \beta_{14} + \beta_{16} + 2 \beta_{18} ) q^{94} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{95} + ( -7 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{18} ) q^{96} + ( 3 + \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{11} - \beta_{15} - 2 \beta_{18} ) q^{97} + ( -7 + 3 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} + 3 \beta_{16} + \beta_{17} ) q^{98} + ( 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{10} + 3 \beta_{11} - \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + 3 \beta_{17} + \beta_{18} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$19q + 2q^{2} + 5q^{3} + 26q^{4} + 11q^{5} + 3q^{6} + 7q^{7} - 6q^{8} + 28q^{9} + O(q^{10})$$ $$19q + 2q^{2} + 5q^{3} + 26q^{4} + 11q^{5} + 3q^{6} + 7q^{7} - 6q^{8} + 28q^{9} - 2q^{10} + 4q^{11} + 9q^{12} + 14q^{13} + 5q^{14} - 7q^{15} + 32q^{16} + 19q^{17} + 12q^{18} + 12q^{19} + 23q^{20} + 16q^{21} + 36q^{22} - q^{23} - 13q^{24} + 30q^{25} - 21q^{26} + 8q^{27} + 5q^{28} + 41q^{29} - 26q^{30} - 8q^{31} - 20q^{32} - 14q^{33} + 2q^{34} + 3q^{35} - 5q^{36} + 50q^{37} - 29q^{38} + 17q^{39} - 15q^{40} + 6q^{41} - q^{42} - 19q^{43} + 16q^{44} + 24q^{45} + 38q^{46} - 21q^{47} - 2q^{48} + 46q^{49} - 36q^{50} + 5q^{51} + 39q^{52} - 9q^{53} + 53q^{54} + 10q^{55} - 12q^{56} - 5q^{57} - 45q^{58} - 4q^{59} - 7q^{60} + 68q^{61} - 25q^{62} + 61q^{63} - 14q^{64} + 22q^{65} - 17q^{66} + 26q^{68} - 9q^{69} - 37q^{70} + 23q^{71} - 4q^{72} - q^{73} - 30q^{74} - 25q^{75} + 47q^{76} - 19q^{77} + 12q^{78} + 16q^{79} + 28q^{80} - 21q^{81} - 13q^{82} - 32q^{83} - 47q^{84} + 11q^{85} - 2q^{86} - 8q^{87} + 108q^{88} + 11q^{89} + 5q^{90} + 52q^{91} - 23q^{92} - 23q^{93} + 47q^{94} - 25q^{95} - 103q^{96} + 36q^{97} - 100q^{98} - 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + 7816 x^{11} - 19517 x^{10} - 13527 x^{9} + 40173 x^{8} + 8942 x^{7} - 41911 x^{6} + 1140 x^{5} + 18520 x^{4} - 1520 x^{3} - 2640 x^{2} - 64 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-14507 \nu^{18} + 35963 \nu^{17} + 407182 \nu^{16} - 1107200 \nu^{15} - 4547261 \nu^{14} + 13926787 \nu^{13} + 25263537 \nu^{12} - 91831214 \nu^{11} - 69013557 \nu^{10} + 337721009 \nu^{9} + 62942576 \nu^{8} - 677587536 \nu^{7} + 79630505 \nu^{6} + 666659837 \nu^{5} - 163996951 \nu^{4} - 247910982 \nu^{3} + 52696508 \nu^{2} + 15739768 \nu - 1655744$$$$)/1001648$$ $$\beta_{5}$$ $$=$$ $$($$$$-19398 \nu^{18} + 88717 \nu^{17} + 621578 \nu^{16} - 2632114 \nu^{15} - 7988484 \nu^{14} + 31920325 \nu^{13} + 52311436 \nu^{12} - 202906417 \nu^{11} - 181206471 \nu^{10} + 719163718 \nu^{9} + 300452097 \nu^{8} - 1391970549 \nu^{7} - 132742247 \nu^{6} + 1330230316 \nu^{5} - 140251151 \nu^{4} - 495302392 \nu^{3} + 81782228 \nu^{2} + 43925008 \nu - 2849472$$$$)/1001648$$ $$\beta_{6}$$ $$=$$ $$($$$$22069 \nu^{18} - 55896 \nu^{17} - 639438 \nu^{16} + 1743250 \nu^{15} + 7321881 \nu^{14} - 22176524 \nu^{13} - 40995911 \nu^{12} + 147507547 \nu^{11} + 106449974 \nu^{10} - 545283805 \nu^{9} - 49648401 \nu^{8} + 1094742551 \nu^{7} - 304438164 \nu^{6} - 1071066367 \nu^{5} + 504053710 \nu^{4} + 390497808 \nu^{3} - 193398836 \nu^{2} - 24616080 \nu + 8403120$$$$)/1001648$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{18} + 2 \nu^{17} + 30 \nu^{16} - 62 \nu^{15} - 365 \nu^{14} + 786 \nu^{13} + 2295 \nu^{12} - 5233 \nu^{11} - 7816 \nu^{10} + 19517 \nu^{9} + 13527 \nu^{8} - 40173 \nu^{7} - 8942 \nu^{6} + 41911 \nu^{5} - 1140 \nu^{4} - 18520 \nu^{3} + 1520 \nu^{2} + 2608 \nu + 64$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$46849 \nu^{18} + 114064 \nu^{17} - 1249518 \nu^{16} - 3525550 \nu^{15} + 13856033 \nu^{14} + 44592496 \nu^{13} - 85069127 \nu^{12} - 295404013 \nu^{11} + 328498046 \nu^{10} + 1089310335 \nu^{9} - 849007097 \nu^{8} - 2193115813 \nu^{7} + 1416278580 \nu^{6} + 2194168945 \nu^{5} - 1268174794 \nu^{4} - 874664780 \nu^{3} + 413760296 \nu^{2} + 83986416 \nu - 16602464$$$$)/1001648$$ $$\beta_{9}$$ $$=$$ $$($$$$-46276 \nu^{18} + 139217 \nu^{17} + 1428222 \nu^{16} - 4360874 \nu^{15} - 17616370 \nu^{14} + 55719265 \nu^{13} + 109411734 \nu^{12} - 372082139 \nu^{11} - 347631687 \nu^{10} + 1380032872 \nu^{9} + 463467287 \nu^{8} - 2779140163 \nu^{7} + 84220369 \nu^{6} + 2732732762 \nu^{5} - 641766275 \nu^{4} - 1015008288 \nu^{3} + 286268432 \nu^{2} + 74770784 \nu - 9733264$$$$)/1001648$$ $$\beta_{10}$$ $$=$$ $$($$$$27406 \nu^{18} - 6558 \nu^{17} - 821917 \nu^{16} + 308350 \nu^{15} + 10057360 \nu^{14} - 5100256 \nu^{13} - 64222291 \nu^{12} + 40741750 \nu^{11} + 226382933 \nu^{10} - 171693679 \nu^{9} - 425102108 \nu^{8} + 377548841 \nu^{7} + 364806745 \nu^{6} - 390853955 \nu^{5} - 79898616 \nu^{4} + 149343515 \nu^{3} - 18185716 \nu^{2} - 13633972 \nu + 1217600$$$$)/500824$$ $$\beta_{11}$$ $$=$$ $$($$$$76183 \nu^{18} + 21343 \nu^{17} - 2186962 \nu^{16} - 589856 \nu^{15} + 25862933 \nu^{14} + 6731423 \nu^{13} - 163094641 \nu^{12} - 40556760 \nu^{11} + 593377623 \nu^{10} + 137369087 \nu^{9} - 1259490850 \nu^{8} - 260638810 \nu^{7} + 1501254445 \nu^{6} + 265807551 \nu^{5} - 899855619 \nu^{4} - 126736682 \nu^{3} + 204586816 \nu^{2} + 15724232 \nu - 6465728$$$$)/1001648$$ $$\beta_{12}$$ $$=$$ $$($$$$84137 \nu^{18} + 15063 \nu^{17} - 2382182 \nu^{16} - 513220 \nu^{15} + 27801095 \nu^{14} + 7221111 \nu^{13} - 173510935 \nu^{12} - 52989234 \nu^{11} + 628966787 \nu^{10} + 214385853 \nu^{9} - 1346792180 \nu^{8} - 470097492 \nu^{7} + 1652849933 \nu^{6} + 513559697 \nu^{5} - 1053949383 \nu^{4} - 224852106 \nu^{3} + 270227080 \nu^{2} + 21049176 \nu - 12742608$$$$)/1001648$$ $$\beta_{13}$$ $$=$$ $$($$$$134774 \nu^{18} + 35561 \nu^{17} - 3930504 \nu^{16} - 732498 \nu^{15} + 47051744 \nu^{14} + 5132785 \nu^{13} - 297715422 \nu^{12} - 10186117 \nu^{11} + 1067473299 \nu^{10} - 35345492 \nu^{9} - 2158905863 \nu^{8} + 174884357 \nu^{7} + 2311814895 \nu^{6} - 194920570 \nu^{5} - 1139251251 \nu^{4} + 8823282 \nu^{3} + 193838436 \nu^{2} + 26201424 \nu - 2665232$$$$)/1001648$$ $$\beta_{14}$$ $$=$$ $$($$$$-303511 \nu^{18} + 415338 \nu^{17} + 8739918 \nu^{16} - 12671162 \nu^{15} - 101762115 \nu^{14} + 157656938 \nu^{13} + 609330069 \nu^{12} - 1026759715 \nu^{11} - 1954126856 \nu^{10} + 3723986055 \nu^{9} + 3068433189 \nu^{8} - 7352057555 \nu^{7} - 1424975210 \nu^{6} + 7078671629 \nu^{5} - 1148618448 \nu^{4} - 2541110364 \nu^{3} + 783180016 \nu^{2} + 177094112 \nu - 37885984$$$$)/2003296$$ $$\beta_{15}$$ $$=$$ $$($$$$152090 \nu^{18} - 193703 \nu^{17} - 4486878 \nu^{16} + 6150318 \nu^{15} + 53575804 \nu^{14} - 79414103 \nu^{13} - 329234660 \nu^{12} + 534709259 \nu^{11} + 1084895701 \nu^{10} - 1995510386 \nu^{9} - 1759312363 \nu^{8} + 4030398919 \nu^{7} + 887788253 \nu^{6} - 3942398108 \nu^{5} + 587022773 \nu^{4} + 1421356240 \nu^{3} - 421792668 \nu^{2} - 91537296 \nu + 20421328$$$$)/1001648$$ $$\beta_{16}$$ $$=$$ $$($$$$-305305 \nu^{18} + 224774 \nu^{17} + 8944762 \nu^{16} - 7408198 \nu^{15} - 106577725 \nu^{14} + 98696774 \nu^{13} + 659713075 \nu^{12} - 682712685 \nu^{11} - 2235618960 \nu^{10} + 2609190145 \nu^{9} + 3953271755 \nu^{8} - 5381877125 \nu^{7} - 2957972206 \nu^{6} + 5358915547 \nu^{5} + 162879536 \nu^{4} - 1963553948 \nu^{3} + 372234608 \nu^{2} + 130925968 \nu - 20921664$$$$)/2003296$$ $$\beta_{17}$$ $$=$$ $$($$$$96459 \nu^{18} + 53597 \nu^{17} - 2880178 \nu^{16} - 1214650 \nu^{15} + 35318239 \nu^{14} + 10240475 \nu^{13} - 228737095 \nu^{12} - 37661230 \nu^{11} + 837361885 \nu^{10} + 44147245 \nu^{9} - 1720785160 \nu^{8} + 56983724 \nu^{7} + 1859180577 \nu^{6} - 127731809 \nu^{5} - 922673663 \nu^{4} + 22957248 \nu^{3} + 168769808 \nu^{2} + 10115296 \nu - 4884880$$$$)/500824$$ $$\beta_{18}$$ $$=$$ $$($$$$420357 \nu^{18} - 380718 \nu^{17} - 12473386 \nu^{16} + 12686510 \nu^{15} + 150115353 \nu^{14} - 170545198 \nu^{13} - 933210671 \nu^{12} + 1186965433 \nu^{11} + 3136981480 \nu^{10} - 4546760309 \nu^{9} - 5324951247 \nu^{8} + 9351656601 \nu^{7} + 3303758878 \nu^{6} - 9214484999 \nu^{5} + 869105056 \nu^{4} + 3290672804 \nu^{3} - 947271184 \nu^{2} - 212098560 \nu + 52644032$$$$)/2003296$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{11} + \beta_{9} - 2 \beta_{6} - \beta_{4} + 6 \beta_{2} - \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$-\beta_{17} - \beta_{16} + \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} + 8 \beta_{3} - \beta_{2} + 29 \beta_{1} - 2$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{18} + \beta_{17} - \beta_{16} + 11 \beta_{15} - \beta_{14} - 3 \beta_{13} - 9 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + 2 \beta_{7} - 22 \beta_{6} + \beta_{5} - 11 \beta_{4} + \beta_{3} + 36 \beta_{2} - 13 \beta_{1} + 96$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{18} - 13 \beta_{17} - 12 \beta_{16} - \beta_{15} + 13 \beta_{14} + 12 \beta_{13} + \beta_{12} - 13 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 13 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 13 \beta_{4} + 55 \beta_{3} - 14 \beta_{2} + 178 \beta_{1} - 25$$ $$\nu^{8}$$ $$=$$ $$-30 \beta_{18} + 15 \beta_{17} - 15 \beta_{16} + 94 \beta_{15} - 17 \beta_{14} - 43 \beta_{13} - 2 \beta_{12} - 66 \beta_{11} + 45 \beta_{10} + 68 \beta_{9} - \beta_{8} + 30 \beta_{7} - 192 \beta_{6} + 13 \beta_{5} - 96 \beta_{4} + 14 \beta_{3} + 223 \beta_{2} - 125 \beta_{1} + 607$$ $$\nu^{9}$$ $$=$$ $$33 \beta_{18} - 123 \beta_{17} - 110 \beta_{16} - 19 \beta_{15} + 124 \beta_{14} + 108 \beta_{13} + 18 \beta_{12} - 124 \beta_{11} + 101 \beta_{10} - 126 \beta_{9} + 122 \beta_{8} - 15 \beta_{7} - 14 \beta_{6} + 17 \beta_{5} + 126 \beta_{4} + 368 \beta_{3} - 140 \beta_{2} + 1131 \beta_{1} - 231$$ $$\nu^{10}$$ $$=$$ $$-320 \beta_{18} + 163 \beta_{17} - 158 \beta_{16} + 737 \beta_{15} - 199 \beta_{14} - 442 \beta_{13} - 36 \beta_{12} - 458 \beta_{11} + 474 \beta_{10} + 497 \beta_{9} - 21 \beta_{8} + 319 \beta_{7} - 1547 \beta_{6} + 119 \beta_{5} - 781 \beta_{4} + 144 \beta_{3} + 1427 \beta_{2} - 1072 \beta_{1} + 3967$$ $$\nu^{11}$$ $$=$$ $$374 \beta_{18} - 1036 \beta_{17} - 911 \beta_{16} - 239 \beta_{15} + 1054 \beta_{14} + 889 \beta_{13} + 216 \beta_{12} - 1059 \beta_{11} + 733 \beta_{10} - 1096 \beta_{9} + 1016 \beta_{8} - 167 \beta_{7} - 123 \beta_{6} + 195 \beta_{5} + 1106 \beta_{4} + 2455 \beta_{3} - 1235 \beta_{2} + 7385 \beta_{1} - 1912$$ $$\nu^{12}$$ $$=$$ $$-2976 \beta_{18} + 1555 \beta_{17} - 1450 \beta_{16} + 5561 \beta_{15} - 1983 \beta_{14} - 4000 \beta_{13} - 430 \beta_{12} - 3109 \beta_{11} + 4334 \beta_{10} + 3628 \beta_{9} - 290 \beta_{8} + 2961 \beta_{7} - 12003 \beta_{6} + 950 \beta_{5} - 6170 \beta_{4} + 1317 \beta_{3} + 9406 \beta_{2} - 8692 \beta_{1} + 26575$$ $$\nu^{13}$$ $$=$$ $$3640 \beta_{18} - 8269 \beta_{17} - 7143 \beta_{16} - 2511 \beta_{15} + 8481 \beta_{14} + 7089 \beta_{13} + 2179 \beta_{12} - 8584 \beta_{11} + 4875 \beta_{10} - 9047 \beta_{9} + 8008 \beta_{8} - 1668 \beta_{7} - 812 \beta_{6} + 1893 \beta_{5} + 9283 \beta_{4} + 16423 \beta_{3} - 10253 \beta_{2} + 49363 \beta_{1} - 15060$$ $$\nu^{14}$$ $$=$$ $$-25750 \beta_{18} + 13820 \beta_{17} - 12388 \beta_{16} + 41181 \beta_{15} - 18074 \beta_{14} - 33948 \beta_{13} - 4313 \beta_{12} - 20841 \beta_{11} + 36830 \beta_{10} + 26660 \beta_{9} - 3298 \beta_{8} + 25608 \beta_{7} - 91224 \beta_{6} + 7085 \beta_{5} - 48012 \beta_{4} + 11318 \beta_{3} + 63617 \beta_{2} - 68331 \beta_{1} + 181555$$ $$\nu^{15}$$ $$=$$ $$32737 \beta_{18} - 64208 \beta_{17} - 54118 \beta_{16} - 23874 \beta_{15} + 66307 \beta_{14} + 55884 \beta_{13} + 20000 \beta_{12} - 67574 \beta_{11} + 30289 \beta_{10} - 72509 \beta_{9} + 61368 \beta_{8} - 15718 \beta_{7} - 3735 \beta_{6} + 16798 \beta_{5} + 75989 \beta_{4} + 110314 \beta_{3} - 82304 \beta_{2} + 336706 \beta_{1} - 115867$$ $$\nu^{16}$$ $$=$$ $$-213323 \beta_{18} + 117466 \beta_{17} - 101288 \beta_{16} + 302197 \beta_{15} - 155829 \beta_{14} - 277452 \beta_{13} - 39403 \beta_{12} - 138414 \beta_{11} + 299765 \beta_{10} + 197382 \beta_{9} - 33451 \beta_{8} + 212210 \beta_{7} - 684839 \beta_{6} + 50856 \beta_{5} - 370182 \beta_{4} + 93468 \beta_{3} + 439660 \beta_{2} - 527518 \beta_{1} + 1260042$$ $$\nu^{17}$$ $$=$$ $$281000 \beta_{18} - 491243 \beta_{17} - 400716 \beta_{16} - 213278 \beta_{15} + 510343 \beta_{14} + 438610 \beta_{13} + 173230 \beta_{12} - 522216 \beta_{11} + 175442 \beta_{10} - 570527 \beta_{9} + 463223 \beta_{8} - 142365 \beta_{7} - 1891 \beta_{6} + 141154 \beta_{5} + 611863 \beta_{4} + 744198 \beta_{3} - 647404 \beta_{2} + 2336751 \beta_{1} - 882149$$ $$\nu^{18}$$ $$=$$ $$-1717747 \beta_{18} + 968471 \beta_{17} - 803761 \beta_{16} + 2209007 \beta_{15} - 1294572 \beta_{14} - 2212803 \beta_{13} - 340414 \beta_{12} - 911937 \beta_{11} + 2373614 \beta_{10} + 1470612 \beta_{9} - 314775 \beta_{8} + 1709532 \beta_{7} - 5101973 \beta_{6} + 356621 \beta_{5} - 2836072 \beta_{4} + 750317 \beta_{3} + 3092290 \beta_{2} - 4027899 \beta_{1} + 8855947$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.74066 −2.48303 −2.48141 −2.21579 −1.86082 −1.32959 −0.753901 −0.276381 −0.248120 0.148448 0.614204 1.03821 1.32390 1.49769 2.18694 2.20570 2.30539 2.40179 2.66743
−2.74066 1.82797 5.51123 3.35226 −5.00984 3.40276 −9.62308 0.341469 −9.18741
1.2 −2.48303 1.10293 4.16544 −3.73374 −2.73861 −2.89905 −5.37685 −1.78354 9.27098
1.3 −2.48141 −0.945917 4.15740 3.33601 2.34721 −4.81122 −5.35339 −2.10524 −8.27801
1.4 −2.21579 −2.91752 2.90974 −0.513978 6.46463 5.13398 −2.01580 5.51193 1.13887
1.5 −1.86082 2.21039 1.46266 1.15323 −4.11315 2.59124 0.999890 1.88583 −2.14595
1.6 −1.32959 −1.38064 −0.232191 −0.595657 1.83568 −3.72326 2.96790 −1.09384 0.791980
1.7 −0.753901 2.87509 −1.43163 3.13521 −2.16753 0.322864 2.58711 5.26614 −2.36364
1.8 −0.276381 2.26539 −1.92361 −3.42185 −0.626111 1.42090 1.08441 2.13201 0.945732
1.9 −0.248120 −2.62183 −1.93844 3.83673 0.650531 2.64030 0.977206 3.87401 −0.951972
1.10 0.148448 0.359282 −1.97796 2.35987 0.0533348 −1.55967 −0.590522 −2.87092 0.350319
1.11 0.614204 −2.75618 −1.62275 −1.26314 −1.69286 −3.83612 −2.22511 4.59652 −0.775829
1.12 1.03821 −1.17568 −0.922121 −2.88687 −1.22060 3.15156 −3.03377 −1.61778 −2.99717
1.13 1.32390 3.32287 −0.247283 −0.817922 4.39916 5.14151 −2.97518 8.04147 −1.08285
1.14 1.49769 1.88586 0.243066 2.23917 2.82443 1.15776 −2.63134 0.556465 3.35357
1.15 2.18694 −2.61834 2.78271 4.32234 −5.72616 −1.52237 1.71175 3.85571 9.45271
1.16 2.20570 2.69039 2.86513 0.920247 5.93420 −2.62293 1.90823 4.23819 2.02979
1.17 2.30539 −0.0652591 3.31484 2.38814 −0.150448 2.55322 3.03122 −2.99574 5.50561
1.18 2.40179 2.16240 3.76858 −1.65227 5.19362 −1.91106 4.24776 1.67596 −3.96840
1.19 2.66743 −1.22120 5.11519 −1.15779 −3.25748 2.36958 8.30956 −1.50866 −3.08834
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.19 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$-1$$
$$43$$ $$1$$

Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{19} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(731))$$.