# Properties

 Label 370.2.o.a Level $370$ Weight $2$ Character orbit 370.o Analytic conductor $2.954$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.o (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 3 x^{17} + 21 x^{16} - 20 x^{15} + 180 x^{14} - 126 x^{13} + 1002 x^{12} - 270 x^{11} + 3294 x^{10} - 1172 x^{9} + 6585 x^{8} - 1464 x^{7} + 8092 x^{6} - 2088 x^{5} + 5598 x^{4} + 180 x^{3} + 972 x^{2} - 162 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 3 x^{17} + 21 x^{16} - 20 x^{15} + 180 x^{14} - 126 x^{13} + 1002 x^{12} - 270 x^{11} + 3294 x^{10} - 1172 x^{9} + 6585 x^{8} - 1464 x^{7} + 8092 x^{6} - 2088 x^{5} + 5598 x^{4} + 180 x^{3} + 972 x^{2} - 162 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$818926776113450 \nu^{17} + 1168948310054943 \nu^{16} - 6575180516792982 \nu^{15} + 88274181421948508 \nu^{14} - 161936164282093929 \nu^{13} + 599355555270778623 \nu^{12} - 1719553300676054571 \nu^{11} + 3440460419591776794 \nu^{10} - 9417896954572503888 \nu^{9} + 6461070778398536891 \nu^{8} - 35425848059464854696 \nu^{7} + 16907622448111329699 \nu^{6} - 55782175847628730981 \nu^{5} + 17769089814196128606 \nu^{4} - 64375578410400988971 \nu^{3} + 17007792309413422245 \nu^{2} - 19030051147159693224 \nu - 7567962043410516870$$$$)/ 4380528236019758289$$ $$\beta_{3}$$ $$=$$ $$($$$$499294071546270 \nu^{17} - 2721639144721310 \nu^{16} + 13676689434132363 \nu^{15} - 32498254335735984 \nu^{14} + 102395689358813416 \nu^{13} - 246315143875649301 \nu^{12} + 595394964910153635 \nu^{11} - 1022503944873859389 \nu^{10} + 1744788526995328527 \nu^{9} - 2988952374660893847 \nu^{8} + 5014438636486947670 \nu^{7} - 2678540448022587378 \nu^{6} + 6751844527958597640 \nu^{5} - 2642108318207903348 \nu^{4} + 6370349104699899546 \nu^{3} + 2969751211689398130 \nu^{2} + 228906530597159046 \nu - 100569647840624811$$$$)/ 1460176078673252763$$ $$\beta_{4}$$ $$=$$ $$($$$$1158444016982230 \nu^{17} - 3358564818141241 \nu^{16} + 21976365938138304 \nu^{15} - 16634814653196692 \nu^{14} + 171200020882457357 \nu^{13} - 125932385573767422 \nu^{12} + 843827733246411840 \nu^{11} - 253551695574624171 \nu^{10} + 2218036824200902107 \nu^{9} - 1931107066075216232 \nu^{8} + 2493837154338654137 \nu^{7} - 2092968422676952368 \nu^{6} + 2604418207020529492 \nu^{5} - 2178380492339021125 \nu^{4} - 367753186422064248 \nu^{3} + 2549371046378354559 \nu^{2} + 909344666754639756 \nu - 114866908525424781$$$$)/ 1460176078673252763$$ $$\beta_{5}$$ $$=$$ $$($$$$1204279094480923 \nu^{17} - 5954783198188891 \nu^{16} + 33531632441868808 \nu^{15} - 74545020497725025 \nu^{14} + 282294091959178700 \nu^{13} - 552031408995578435 \nu^{12} + 1686336103651727064 \nu^{11} - 2466405053472983976 \nu^{10} + 5577079971178863147 \nu^{9} - 7642190055074617019 \nu^{8} + 14234181376387420895 \nu^{7} - 13639751901756577757 \nu^{6} + 17684528634454187932 \nu^{5} - 16715967465828789160 \nu^{4} + 16356863104185447883 \nu^{3} - 9343774333584952497 \nu^{2} + 315081883606798107 \nu - 1484206343635846122$$$$)/ 1460176078673252763$$ $$\beta_{6}$$ $$=$$ $$($$$$3724801771874993 \nu^{17} - 9676523100986169 \nu^{16} + 70055919775210923 \nu^{15} - 33465967135102771 \nu^{14} + 572969555930290788 \nu^{13} - 162137955179808870 \nu^{12} + 2993305943791795083 \nu^{11} + 780488416324212795 \nu^{10} + 9201985201934648775 \nu^{9} + 868897904348493785 \nu^{8} + 15560962543814147364 \nu^{7} + 9590206115435853258 \nu^{6} + 22105474593944681222 \nu^{5} + 12478147484200807536 \nu^{4} + 12925115364332500770 \nu^{3} + 19781511633037197378 \nu^{2} + 12529760957330687586 \nu + 83301704747728272$$$$)/ 4380528236019758289$$ $$\beta_{7}$$ $$=$$ $$($$$$-4254329945386103 \nu^{17} + 9287657785211619 \nu^{16} - 79265234398684440 \nu^{15} + 19157501093307148 \nu^{14} - 715874946209908464 \nu^{13} + 22445510471276907 \nu^{12} - 3885041448555572940 \nu^{11} - 1382814114484987710 \nu^{10} - 13253107753377950769 \nu^{9} - 1668035776610193605 \nu^{8} - 22221441492141839559 \nu^{7} - 1253172422970707619 \nu^{6} - 28147132650033488372 \nu^{5} + 1069786304904594588 \nu^{4} - 17280597557254341219 \nu^{3} + 337480169096694204 \nu^{2} - 11783321846050355793 \nu - 2038832549111370582$$$$)/ 4380528236019758289$$ $$\beta_{8}$$ $$=$$ $$($$$$-1267556030104 \nu^{17} + 4354177580338 \nu^{16} - 28169998789216 \nu^{15} + 36316701473586 \nu^{14} - 236227920479190 \nu^{13} + 251178641320566 \nu^{12} - 1318392048421013 \nu^{11} + 835148629257425 \nu^{10} - 4208175506255429 \nu^{9} + 3029343409835692 \nu^{8} - 8721597516478930 \nu^{7} + 4627567306298074 \nu^{6} - 10434330425144378 \nu^{5} + 5742682106965238 \nu^{4} - 8224605492983498 \nu^{3} + 1370625148511829 \nu^{2} - 319405327219224 \nu + 286512494647932$$$$)/ 1226008462362093$$ $$\beta_{9}$$ $$=$$ $$($$$$-1608269163526979 \nu^{17} + 7464499626095967 \nu^{16} - 39300697503789709 \nu^{15} + 86603805094546990 \nu^{14} - 308217910543906914 \nu^{13} + 748493056049228333 \nu^{12} - 1584067116408603843 \nu^{11} + 3776592036866820048 \nu^{10} - 4041188146070796150 \nu^{9} + 15091405846123164385 \nu^{8} - 5857738869525808794 \nu^{7} + 32409341070005974469 \nu^{6} - 5999284505867822948 \nu^{5} + 42124635771071240727 \nu^{4} - 3254508721402921765 \nu^{3} + 28116932768154919890 \nu^{2} - 797235383878929522 \nu + 3083195759469350319$$$$)/ 1460176078673252763$$ $$\beta_{10}$$ $$=$$ $$($$$$-5016628441071265 \nu^{17} + 7945986134670438 \nu^{16} - 92620020357043155 \nu^{15} - 19157539538789038 \nu^{14} - 942367029256716237 \nu^{13} - 413922723022866399 \nu^{12} - 5558637828600121638 \nu^{11} - 4003654291201377279 \nu^{10} - 21898221323156152389 \nu^{9} - 11090112882698566576 \nu^{8} - 44501743613370800418 \nu^{7} - 11817850596196080816 \nu^{6} - 59210188596546087127 \nu^{5} - 345018565987338078 \nu^{4} - 43248007998969414801 \nu^{3} + 14216167338948015102 \nu^{2} - 18663065766954133506 \nu + 11869047272825256069$$$$)/ 4380528236019758289$$ $$\beta_{11}$$ $$=$$ $$($$$$1747149617299206 \nu^{17} - 3310706614803512 \nu^{16} + 29950034928556371 \nu^{15} + 5931712550586714 \nu^{14} + 261909460360795762 \nu^{13} + 98622791126670672 \nu^{12} + 1357360533669232632 \nu^{11} + 1157768336905185450 \nu^{10} + 4386324441667719132 \nu^{9} + 2327502314864056293 \nu^{8} + 5755355352829325728 \nu^{7} + 4121486045802849084 \nu^{6} + 5671881240457294854 \nu^{5} + 3310673230920648118 \nu^{4} - 260674456248688923 \nu^{3} + 3979724713820519661 \nu^{2} + 69542365907931030 \nu + 460163997512224074$$$$)/ 1460176078673252763$$ $$\beta_{12}$$ $$=$$ $$($$$$-1842651843199852 \nu^{17} + 5520132313048115 \nu^{16} - 36782749173974202 \nu^{15} + 33025600616876576 \nu^{14} - 297896553732168721 \nu^{13} + 233333307106192917 \nu^{12} - 1545854733052981131 \nu^{11} + 587680689084843396 \nu^{10} - 4485986429347646736 \nu^{9} + 3315051164569649093 \nu^{8} - 6862248670949558815 \nu^{7} + 5260961753891971638 \nu^{6} - 7449588188594024296 \nu^{5} + 6973200415073337581 \nu^{4} - 3190195465891619091 \nu^{3} + 2114273815460739081 \nu^{2} - 1060206599732777550 \nu - 198116254493579574$$$$)/ 1460176078673252763$$ $$\beta_{13}$$ $$=$$ $$($$$$-5620847765081348 \nu^{17} + 24439772893259580 \nu^{16} - 131639613552674358 \nu^{15} + 243888831419198332 \nu^{14} - 973779413938527384 \nu^{13} + 1887522185584370097 \nu^{12} - 4981305944148475623 \nu^{11} + 7852861286980767207 \nu^{10} - 11770104427488580716 \nu^{9} + 28303729782398665213 \nu^{8} - 17855621264659680327 \nu^{7} + 42778468938890827506 \nu^{6} - 3821132360267647616 \nu^{5} + 49399342351848381789 \nu^{4} + 14648305680789040665 \nu^{3} + 6007150563448188213 \nu^{2} + 26248727870423819802 \nu + 1705343908649137686$$$$)/ 4380528236019758289$$ $$\beta_{14}$$ $$=$$ $$($$$$31834721627548 \nu^{17} - 84096160611708 \nu^{16} + 629341555955466 \nu^{15} - 383164443448016 \nu^{14} + 5403399579696366 \nu^{13} - 1885123640758338 \nu^{12} + 29637783298918002 \nu^{11} + 3270153596351157 \nu^{10} + 97347235377826287 \nu^{9} + 563285808812605 \nu^{8} + 182367551228882352 \nu^{7} + 31888345185580098 \nu^{6} + 215958461653435750 \nu^{5} + 27438075067979178 \nu^{4} + 126526632708326562 \nu^{3} + 79751699329810122 \nu^{2} + 18607723085370195 \nu + 8751499202569077$$$$)/ 11034076161258837$$ $$\beta_{15}$$ $$=$$ $$($$$$-5681037006323754 \nu^{17} + 18790260636270468 \nu^{16} - 122612483747602346 \nu^{15} + 143570775055031451 \nu^{14} - 1016654948587689006 \nu^{13} + 977720123157588766 \nu^{12} - 5593776289209730836 \nu^{11} + 2891240525376646212 \nu^{10} - 17555567561925260226 \nu^{9} + 11044499813079158820 \nu^{8} - 35082126371777863797 \nu^{7} + 14072393530087301584 \nu^{6} - 41849465409368968284 \nu^{5} + 17533886509661293206 \nu^{4} - 28491771930479726774 \nu^{3} - 1283261117386964643 \nu^{2} - 1542243256326169227 \nu + 989870360932379178$$$$)/ 1460176078673252763$$ $$\beta_{16}$$ $$=$$ $$($$$$2336734652230229 \nu^{17} - 7113341176639761 \nu^{16} + 49550512020649467 \nu^{15} - 49579987811698819 \nu^{14} + 426571673025729261 \nu^{13} - 319539102874662600 \nu^{12} + 2385072605568062112 \nu^{11} - 784848787400866644 \nu^{10} + 7887089071484164314 \nu^{9} - 3281545288952069491 \nu^{8} + 15951750840218955021 \nu^{7} - 4848305052775317546 \nu^{6} + 19834941650594150651 \nu^{5} - 6812758408013485269 \nu^{4} + 13994669934541536201 \nu^{3} - 1417664721326570901 \nu^{2} + 2393014574569657977 \nu - 350091564194958651$$$$)/ 257678131530574017$$ $$\beta_{17}$$ $$=$$ $$($$$$44829263570167675 \nu^{17} - 122366721773643879 \nu^{16} + 910012790294620134 \nu^{15} - 657616327326221213 \nu^{14} + 7920989180632748889 \nu^{13} - 3557026050437209335 \nu^{12} + 44087812469757882759 \nu^{11} - 727543682080048044 \nu^{10} + 147561837470326393524 \nu^{9} - 15028897244007323873 \nu^{8} + 287440370602493057961 \nu^{7} + 185463703219146162 \nu^{6} + 344932178321087000308 \nu^{5} - 14513725851626591784 \nu^{4} + 218272535576307398955 \nu^{3} + 54521885568296338344 \nu^{2} + 27536678314126707582 \nu + 5792611081801999815$$$$)/ 4380528236019758289$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{17} + \beta_{16} + \beta_{15} + 3 \beta_{14} + \beta_{12} + \beta_{10} - \beta_{6} - \beta_{3} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{17} + 3 \beta_{16} + \beta_{15} - 3 \beta_{13} - 2 \beta_{12} + 3 \beta_{11} - \beta_{10} + 3 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{17} - 12 \beta_{15} - 19 \beta_{14} - 14 \beta_{13} - 25 \beta_{12} + 8 \beta_{11} - 15 \beta_{10} + 11 \beta_{9} + 15 \beta_{8} - 10 \beta_{7} - 2 \beta_{6} - 12 \beta_{5} + 3 \beta_{4} + 8 \beta_{3} - 15 \beta_{2} - 31 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$34 \beta_{17} - 48 \beta_{16} - 29 \beta_{15} - 37 \beta_{14} - 50 \beta_{12} - 36 \beta_{11} - 34 \beta_{10} - 17 \beta_{9} + 23 \beta_{8} + 2 \beta_{7} + 29 \beta_{6} - 2 \beta_{5} - \beta_{4} + 30 \beta_{3} - 17 \beta_{2} - 74 \beta_{1} + 37$$ $$\nu^{6}$$ $$=$$ $$208 \beta_{17} - 188 \beta_{16} + 42 \beta_{15} + 188 \beta_{13} + 136 \beta_{12} - 204 \beta_{11} + 67 \beta_{10} - 208 \beta_{9} - 174 \beta_{8} + 161 \beta_{7} + 161 \beta_{6} + 119 \beta_{5} - 72 \beta_{4} + 48 \beta_{3} + 141 \beta_{2} + 188 \beta_{1} + 183$$ $$\nu^{7}$$ $$=$$ $$256 \beta_{17} + 541 \beta_{15} + 542 \beta_{14} + 682 \beta_{13} + 1162 \beta_{12} - 180 \beta_{11} + 754 \beta_{10} - 498 \beta_{9} - 754 \beta_{8} + 403 \beta_{7} + 138 \beta_{6} + 541 \beta_{5} - 300 \beta_{4} - 180 \beta_{3} + 754 \beta_{2} + 1621 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1921 \beta_{17} + 2585 \beta_{16} + 1557 \beta_{15} + 2201 \beta_{14} + 2718 \beta_{12} + 2097 \beta_{11} + 1921 \beta_{10} + 982 \beta_{9} - 466 \beta_{8} - 694 \beta_{7} - 1557 \beta_{6} + 694 \beta_{5} + 77 \beta_{4} - 1526 \beta_{3} + 982 \beta_{2} + 3369 \beta_{1} - 2201$$ $$\nu^{9}$$ $$=$$ $$-10769 \beta_{17} + 9577 \beta_{16} - 2560 \beta_{15} - 9577 \beta_{13} - 6057 \beta_{12} + 10482 \beta_{11} - 3700 \beta_{10} + 10769 \beta_{9} + 8821 \beta_{8} - 8175 \beta_{7} - 8175 \beta_{6} - 5615 \beta_{5} + 4712 \beta_{4} - 3233 \beta_{3} - 7069 \beta_{2} - 9577 \beta_{1} - 7606$$ $$\nu^{10}$$ $$=$$ $$-14002 \beta_{17} - 31733 \beta_{15} - 29136 \beta_{14} - 36009 \beta_{13} - 61561 \beta_{12} + 8965 \beta_{11} - 40721 \beta_{10} + 26719 \beta_{9} + 40721 \beta_{8} - 21253 \beta_{7} - 10480 \beta_{6} - 31733 \beta_{5} + 16587 \beta_{4} + 8965 \beta_{3} - 40721 \beta_{2} - 82993 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$99580 \beta_{17} - 134422 \beta_{16} - 78532 \beta_{15} - 106227 \beta_{14} - 145315 \beta_{12} - 116085 \beta_{11} - 99580 \beta_{10} - 52596 \beta_{9} + 24099 \beta_{8} + 39378 \beta_{7} + 78532 \beta_{6} - 39378 \beta_{5} - 4299 \beta_{4} + 77527 \beta_{3} - 52596 \beta_{2} - 176275 \beta_{1} + 106227$$ $$\nu^{12}$$ $$=$$ $$572097 \beta_{17} - 504309 \beta_{16} + 152081 \beta_{15} + 504309 \beta_{13} + 317054 \beta_{12} - 555348 \beta_{11} + 197911 \beta_{10} - 572097 \beta_{9} - 486674 \beta_{8} + 447319 \beta_{7} + 447319 \beta_{6} + 295238 \beta_{5} - 255043 \beta_{4} + 170506 \beta_{3} + 374186 \beta_{2} + 504309 \beta_{1} + 400331$$ $$\nu^{13}$$ $$=$$ $$742603 \beta_{17} + 1673031 \beta_{15} + 1486432 \beta_{14} + 1887683 \beta_{13} + 3225607 \beta_{12} - 437651 \beta_{11} + 2142726 \beta_{10} - 1400123 \beta_{9} - 2142726 \beta_{8} + 1100926 \beta_{7} + 572105 \beta_{6} + 1673031 \beta_{5} - 900273 \beta_{4} - 437651 \beta_{3} + 2142726 \beta_{2} + 4351972 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-5252245 \beta_{17} + 7077729 \beta_{16} + 4130201 \beta_{15} + 5578129 \beta_{14} + 7667879 \beta_{12} + 6166062 \beta_{11} + 5252245 \beta_{10} + 2787956 \beta_{9} - 1181480 \beta_{8} - 2167697 \beta_{7} - 4130201 \beta_{6} + 2167697 \beta_{5} + 236998 \beta_{4} - 4052775 \beta_{3} + 2787956 \beta_{2} + 9221681 \beta_{1} - 5578129$$ $$\nu^{15}$$ $$=$$ $$-30133351 \beta_{17} + 26518247 \beta_{16} - 8144025 \beta_{15} - 26518247 \beta_{13} - 16547347 \beta_{12} + 29248305 \beta_{11} - 10455835 \beta_{10} + 30133351 \beta_{9} + 25706037 \beta_{8} - 23597117 \beta_{7} - 23597117 \beta_{6} - 15453092 \beta_{5} + 13586004 \beta_{4} - 9085854 \beta_{3} - 19677516 \beta_{2} - 26518247 \beta_{1} - 20841006$$ $$\nu^{16}$$ $$=$$ $$-39219205 \beta_{17} - 88590124 \beta_{15} - 78134348 \beta_{14} - 99413071 \beta_{13} - 169853563 \beta_{12} + 22801536 \beta_{11} - 112999075 \beta_{10} + 73779870 \beta_{9} + 112999075 \beta_{8} - 57938839 \beta_{7} - 30651285 \beta_{6} - 88590124 \beta_{5} + 47638956 \beta_{4} + 22801536 \beta_{3} - 112999075 \beta_{2} - 228888829 \beta_{1}$$ $$\nu^{17}$$ $$=$$ $$276527785 \beta_{17} - 372601478 \beta_{16} - 217043820 \beta_{15} - 292555769 \beta_{14} - 404337114 \beta_{12} - 325839690 \beta_{11} - 276527785 \beta_{10} - 147052027 \beta_{9} + 61716679 \beta_{8} + 115034959 \beta_{7} + 217043820 \beta_{6} - 115034959 \beta_{5} - 12501653 \beta_{4} + 213047798 \beta_{3} - 147052027 \beta_{2} - 485296491 \beta_{1} + 292555769$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 −0.978711 − 1.69518i 0.571872 + 0.990512i 0.140794 + 0.243862i −1.05539 − 1.82798i 1.87438 + 3.24652i 0.620701 + 1.07509i −0.231777 − 0.401450i −0.649209 − 1.12446i 1.20734 + 2.09117i −1.05539 + 1.82798i 1.87438 − 3.24652i 0.620701 − 1.07509i −0.231777 + 0.401450i −0.649209 + 1.12446i 1.20734 − 2.09117i −0.978711 + 1.69518i 0.571872 − 0.990512i 0.140794 − 0.243862i
0.766044 + 0.642788i −1.90337 + 1.59711i 0.173648 + 0.984808i −0.939693 0.342020i −2.48467 −4.17420 1.51928i −0.500000 + 0.866025i 0.551085 3.12536i −0.500000 0.866025i
71.2 0.766044 + 0.642788i −0.646156 + 0.542189i 0.173648 + 0.984808i −0.939693 0.342020i −0.843496 0.282141 + 0.102691i −0.500000 + 0.866025i −0.397396 + 2.25375i −0.500000 0.866025i
71.3 0.766044 + 0.642788i 1.78348 1.49651i 0.173648 + 0.984808i −0.939693 0.342020i 2.32816 2.45236 + 0.892588i −0.500000 + 0.866025i 0.420289 2.38358i −0.500000 0.866025i
81.1 −0.939693 + 0.342020i −1.84510 0.671562i 0.766044 0.642788i 0.173648 + 0.984808i 1.96352 0.0255713 + 0.145022i −0.500000 + 0.866025i 0.655269 + 0.549836i −0.500000 0.866025i
81.2 −0.939693 + 0.342020i 0.528795 + 0.192466i 0.766044 0.642788i 0.173648 + 0.984808i −0.562732 −0.553248 3.13762i −0.500000 + 0.866025i −2.05555 1.72481i −0.500000 0.866025i
81.3 −0.939693 + 0.342020i 2.25600 + 0.821116i 0.766044 0.642788i 0.173648 + 0.984808i −2.40078 0.201324 + 1.14177i −0.500000 + 0.866025i 2.11716 + 1.77651i −0.500000 0.866025i
181.1 0.173648 0.984808i −0.568073 3.22170i −0.939693 0.342020i 0.766044 0.642788i −3.27140 2.86114 2.40078i −0.500000 + 0.866025i −7.23759 + 2.63427i −0.500000 0.866025i
181.2 0.173648 0.984808i −0.0317483 0.180053i −0.939693 0.342020i 0.766044 0.642788i −0.182831 1.13470 0.952128i −0.500000 + 0.866025i 2.78767 1.01463i −0.500000 0.866025i
181.3 0.173648 0.984808i 0.426173 + 2.41695i −0.939693 0.342020i 0.766044 0.642788i 2.45423 −3.72980 + 3.12967i −0.500000 + 0.866025i −2.84094 + 1.03402i −0.500000 0.866025i
201.1 −0.939693 0.342020i −1.84510 + 0.671562i 0.766044 + 0.642788i 0.173648 0.984808i 1.96352 0.0255713 0.145022i −0.500000 0.866025i 0.655269 0.549836i −0.500000 + 0.866025i
201.2 −0.939693 0.342020i 0.528795 0.192466i 0.766044 + 0.642788i 0.173648 0.984808i −0.562732 −0.553248 + 3.13762i −0.500000 0.866025i −2.05555 + 1.72481i −0.500000 + 0.866025i
201.3 −0.939693 0.342020i 2.25600 0.821116i 0.766044 + 0.642788i 0.173648 0.984808i −2.40078 0.201324 1.14177i −0.500000 0.866025i 2.11716 1.77651i −0.500000 + 0.866025i
231.1 0.173648 + 0.984808i −0.568073 + 3.22170i −0.939693 + 0.342020i 0.766044 + 0.642788i −3.27140 2.86114 + 2.40078i −0.500000 0.866025i −7.23759 2.63427i −0.500000 + 0.866025i
231.2 0.173648 + 0.984808i −0.0317483 + 0.180053i −0.939693 + 0.342020i 0.766044 + 0.642788i −0.182831 1.13470 + 0.952128i −0.500000 0.866025i 2.78767 + 1.01463i −0.500000 + 0.866025i
231.3 0.173648 + 0.984808i 0.426173 2.41695i −0.939693 + 0.342020i 0.766044 + 0.642788i 2.45423 −3.72980 3.12967i −0.500000 0.866025i −2.84094 1.03402i −0.500000 + 0.866025i
271.1 0.766044 0.642788i −1.90337 1.59711i 0.173648 0.984808i −0.939693 + 0.342020i −2.48467 −4.17420 + 1.51928i −0.500000 0.866025i 0.551085 + 3.12536i −0.500000 + 0.866025i
271.2 0.766044 0.642788i −0.646156 0.542189i 0.173648 0.984808i −0.939693 + 0.342020i −0.843496 0.282141 0.102691i −0.500000 0.866025i −0.397396 2.25375i −0.500000 + 0.866025i
271.3 0.766044 0.642788i 1.78348 + 1.49651i 0.173648 0.984808i −0.939693 + 0.342020i 2.32816 2.45236 0.892588i −0.500000 0.866025i 0.420289 + 2.38358i −0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.o.a 18
37.f even 9 1 inner 370.2.o.a 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.o.a 18 1.a even 1 1 trivial
370.2.o.a 18 37.f even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{3} + T^{6} )^{3}$$
$3$ $$361 + 171 T + 9174 T^{2} - 16219 T^{3} - 16890 T^{4} + 15603 T^{5} + 51104 T^{6} + 1905 T^{7} - 6651 T^{8} + 1412 T^{9} + 783 T^{10} - 240 T^{11} + 254 T^{12} + 21 T^{13} - 12 T^{14} - 10 T^{15} + 6 T^{16} + T^{18}$$
$5$ $$( 1 + T^{3} + T^{6} )^{3}$$
$7$ $$2601 - 26622 T + 228258 T^{2} - 1166400 T^{3} + 3031380 T^{4} - 3919680 T^{5} + 3617871 T^{6} - 2171529 T^{7} + 781767 T^{8} - 85116 T^{9} - 9906 T^{10} - 345 T^{11} - 1319 T^{12} + 495 T^{13} + 324 T^{14} - 52 T^{15} - 12 T^{16} + 3 T^{17} + T^{18}$$
$11$ $$651249 + 7676991 T + 97583436 T^{2} - 80529999 T^{3} + 91984347 T^{4} - 50764890 T^{5} + 39718771 T^{6} - 19260630 T^{7} + 10302240 T^{8} - 3375034 T^{9} + 1167255 T^{10} - 246375 T^{11} + 70110 T^{12} - 11097 T^{13} + 2928 T^{14} - 256 T^{15} + 63 T^{16} - 3 T^{17} + T^{18}$$
$13$ $$289 - 4080 T + 16452 T^{2} - 6716 T^{3} + 62130 T^{4} - 8514 T^{5} + 44165 T^{6} + 12219 T^{7} + 29097 T^{8} + 57328 T^{9} + 34464 T^{10} + 11079 T^{11} + 4799 T^{12} - 3225 T^{13} + 252 T^{14} + 88 T^{15} + 12 T^{16} - 9 T^{17} + T^{18}$$
$17$ $$9 + 270 T + 1359 T^{2} - 17025 T^{3} + 76932 T^{4} + 26529 T^{5} + 146029 T^{6} + 87195 T^{7} + 168414 T^{8} - 133397 T^{9} + 70758 T^{10} - 60246 T^{11} + 24447 T^{12} - 1098 T^{13} - 498 T^{14} + 67 T^{15} + 21 T^{16} + 6 T^{17} + T^{18}$$
$19$ $$122556706561 + 185483416230 T + 132469680807 T^{2} + 1201293973 T^{3} + 12588258780 T^{4} - 7090933473 T^{5} + 812650736 T^{6} + 37028673 T^{7} + 31147974 T^{8} - 8028260 T^{9} + 551664 T^{10} - 309690 T^{11} + 61067 T^{12} + 7155 T^{13} - 1593 T^{14} - 35 T^{15} + 36 T^{16} - 9 T^{17} + T^{18}$$
$23$ $$153909489969 + 176216014836 T + 212546839275 T^{2} + 57233105736 T^{3} + 38304553374 T^{4} - 468371373 T^{5} + 4721886661 T^{6} - 586391727 T^{7} + 453098892 T^{8} - 95645518 T^{9} + 31671459 T^{10} - 5614587 T^{11} + 1222044 T^{12} - 185571 T^{13} + 30444 T^{14} - 3439 T^{15} + 354 T^{16} - 21 T^{17} + T^{18}$$
$29$ $$8485569 + 59197986 T + 658287414 T^{2} - 2232218280 T^{3} + 5369047362 T^{4} - 6201958083 T^{5} + 5221886782 T^{6} - 2553847764 T^{7} + 960606456 T^{8} - 234973336 T^{9} + 49735071 T^{10} - 7195734 T^{11} + 1276956 T^{12} - 142902 T^{13} + 21543 T^{14} - 1240 T^{15} + 168 T^{16} - 6 T^{17} + T^{18}$$
$31$ $$( 3187 - 3654 T - 14106 T^{2} + 19341 T^{3} + 2637 T^{4} - 4767 T^{5} - 1260 T^{6} - 27 T^{7} + 15 T^{8} + T^{9} )^{2}$$
$37$ $$129961739795077 - 21074876723526 T + 4841525733783 T^{2} - 184732301448 T^{3} - 70314772398 T^{4} + 40155773586 T^{5} - 2966847516 T^{6} + 458456196 T^{7} + 112390719 T^{8} - 20967095 T^{9} + 3037587 T^{10} + 334884 T^{11} - 58572 T^{12} + 21426 T^{13} - 1014 T^{14} - 72 T^{15} + 51 T^{16} - 6 T^{17} + T^{18}$$
$41$ $$5887606220721 - 22196661183126 T + 23960622705831 T^{2} - 761270042931 T^{3} + 355246462089 T^{4} - 221320778556 T^{5} + 49929853048 T^{6} - 8778979989 T^{7} + 2336004978 T^{8} - 370339201 T^{9} + 44493984 T^{10} - 7831953 T^{11} + 1130136 T^{12} - 112419 T^{13} + 18255 T^{14} - 2077 T^{15} + 153 T^{16} - 15 T^{17} + T^{18}$$
$43$ $$( 1578531 + 578628 T - 299448 T^{2} - 89613 T^{3} + 20355 T^{4} + 4971 T^{5} - 587 T^{6} - 117 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$47$ $$197880489 - 39178437777 T + 7753799233989 T^{2} - 634734708408 T^{3} + 1039424816664 T^{4} - 163700710101 T^{5} + 101649865095 T^{6} - 15619366215 T^{7} + 5421524184 T^{8} - 817541028 T^{9} + 206268210 T^{10} - 27073890 T^{11} + 4597143 T^{12} - 492948 T^{13} + 65556 T^{14} - 5700 T^{15} + 507 T^{16} - 24 T^{17} + T^{18}$$
$53$ $$53445579489 + 106812094392 T + 77950139805 T^{2} - 12468785031 T^{3} + 37374433803 T^{4} + 88657911936 T^{5} + 67113571095 T^{6} + 31150603944 T^{7} + 10727230383 T^{8} + 2595932253 T^{9} + 471484134 T^{10} + 58697946 T^{11} + 3874902 T^{12} - 19008 T^{13} - 24192 T^{14} - 1965 T^{15} + 12 T^{16} + 12 T^{17} + T^{18}$$
$59$ $$165096801 + 25672302 T + 1212518061 T^{2} - 506185071 T^{3} + 1675919088 T^{4} + 332462346 T^{5} + 195000280 T^{6} + 1187016369 T^{7} + 1142049072 T^{8} + 523455047 T^{9} + 139299084 T^{10} + 22185828 T^{11} + 1907781 T^{12} + 18504 T^{13} - 15288 T^{14} - 1735 T^{15} - 6 T^{16} + 15 T^{17} + T^{18}$$
$61$ $$5817438882546769 - 8840809417940430 T + 6623354818511154 T^{2} - 3045424282347051 T^{3} + 974554531758852 T^{4} - 231134691014730 T^{5} + 42671186371953 T^{6} - 6268535582973 T^{7} + 774724990383 T^{8} - 90036949084 T^{9} + 10789478652 T^{10} - 1309157982 T^{11} + 146443230 T^{12} - 13793223 T^{13} + 1034163 T^{14} - 59769 T^{15} + 2550 T^{16} - 72 T^{17} + T^{18}$$
$67$ $$6156844727401 - 2688988688898 T + 194256176742 T^{2} - 24618670803 T^{3} + 138946116480 T^{4} - 57829486824 T^{5} + 12912545493 T^{6} - 3426330939 T^{7} + 638482563 T^{8} - 6200440 T^{9} - 5576658 T^{10} + 54126 T^{11} - 142758 T^{12} - 12819 T^{13} + 11763 T^{14} - 1143 T^{15} + 192 T^{16} - 18 T^{17} + T^{18}$$
$71$ $$458961767632449 + 553190012359182 T + 95197897131678 T^{2} - 68210472303189 T^{3} + 14927580626256 T^{4} + 1166322295584 T^{5} - 62957055149 T^{6} + 14042654625 T^{7} + 1751098221 T^{8} - 526195898 T^{9} + 65267460 T^{10} + 1966410 T^{11} + 674190 T^{12} - 81531 T^{13} + 9423 T^{14} - 1763 T^{15} + 282 T^{16} + T^{18}$$
$73$ $$( 31500201 + 31351752 T + 11220597 T^{2} + 1376844 T^{3} - 126408 T^{4} - 47478 T^{5} - 2876 T^{6} + 237 T^{7} + 33 T^{8} + T^{9} )^{2}$$
$79$ $$68548580271649 - 18038112207504 T + 9819945785964 T^{2} - 4868479475624 T^{3} + 496872084603 T^{4} + 14588333910 T^{5} + 35188159241 T^{6} - 2937393414 T^{7} - 371090460 T^{8} - 191254826 T^{9} + 16521639 T^{10} + 3208173 T^{11} + 493427 T^{12} - 45705 T^{13} - 7083 T^{14} - 857 T^{15} + 63 T^{16} + 12 T^{17} + T^{18}$$
$83$ $$21019490361 + 259704030357 T + 546068809521 T^{2} - 2900336555493 T^{3} + 2879068240548 T^{4} - 563825630034 T^{5} + 267974337763 T^{6} + 2387972235 T^{7} - 52941726 T^{8} - 1122496379 T^{9} + 130778937 T^{10} - 9329202 T^{11} + 1773423 T^{12} - 22095 T^{13} + 38700 T^{14} - 4004 T^{15} + 327 T^{16} - 9 T^{17} + T^{18}$$
$89$ $$1197340549606881 - 417366012154428 T - 88084622558205 T^{2} + 14110342320768 T^{3} + 15578905898466 T^{4} - 1499626468842 T^{5} + 116756199121 T^{6} + 16025523312 T^{7} - 10681614405 T^{8} + 225933014 T^{9} + 191542968 T^{10} - 12495213 T^{11} + 6142128 T^{12} + 30861 T^{13} + 33045 T^{14} - 307 T^{15} + 96 T^{16} + T^{18}$$
$97$ $$50074347200329 - 18876778005831 T + 18859260078201 T^{2} - 5048464284218 T^{3} + 4424856820134 T^{4} - 1175970906027 T^{5} + 422115360380 T^{6} - 59258274999 T^{7} + 13784496765 T^{8} - 1123957778 T^{9} + 295372932 T^{10} - 14599848 T^{11} + 4129451 T^{12} - 73410 T^{13} + 41427 T^{14} - 173 T^{15} + 258 T^{16} + 3 T^{17} + T^{18}$$