# Properties

 Label 370.2.o.b 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: $$x^{18} - 3 x^{17} + 18 x^{16} - 25 x^{15} + 132 x^{14} - 135 x^{13} + 666 x^{12} - 297 x^{11} + 1845 x^{10} - 382 x^{9} + 3810 x^{8} - 63 x^{7} + 3325 x^{6} - 1329 x^{5} + 1332 x^{4} - 192 x^{3} + 117 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 3 x^{17} + 18 x^{16} - 25 x^{15} + 132 x^{14} - 135 x^{13} + 666 x^{12} - 297 x^{11} + 1845 x^{10} - 382 x^{9} + 3810 x^{8} - 63 x^{7} + 3325 x^{6} - 1329 x^{5} + 1332 x^{4} - 192 x^{3} + 117 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$467011630737 \nu^{17} + 7985628307079 \nu^{16} - 20288175144489 \nu^{15} + 158233129097535 \nu^{14} - 180198535062678 \nu^{13} + 1176501229728729 \nu^{12} - 1003199343912756 \nu^{11} + 6093996383348280 \nu^{10} - 2144939147208948 \nu^{9} + 16849968308636067 \nu^{8} - 2363608345702478 \nu^{7} + 34848378812526090 \nu^{6} - 214003037263167 \nu^{5} + 28577726301336609 \nu^{4} - 12308248046456505 \nu^{3} + 12364918060114449 \nu^{2} - 828245540915145 \nu + 1086225957699273$$$$)/ 947673457589169$$ $$\beta_{3}$$ $$=$$ $$($$$$-1286483210027 \nu^{17} + 14299357227244 \nu^{16} - 49682295365806 \nu^{15} + 207475771916758 \nu^{14} - 351354719981117 \nu^{13} + 1466126504711854 \nu^{12} - 1693474035059939 \nu^{11} + 6902922464553826 \nu^{10} - 2584072669612421 \nu^{9} + 19291752438371271 \nu^{8} - 787898339306631 \nu^{7} + 39643831895724317 \nu^{6} + 12565644693953733 \nu^{5} + 38464552842327453 \nu^{4} - 1461697716769071 \nu^{3} + 5948453104470423 \nu^{2} + 1298656915658397 \nu + 734083447796847$$$$)/ 947673457589169$$ $$\beta_{4}$$ $$=$$ $$($$$$4104600398475 \nu^{17} - 27243735217429 \nu^{16} + 106486954036400 \nu^{15} - 338955758102881 \nu^{14} + 708857919905234 \nu^{13} - 2293194019700023 \nu^{12} + 3243674232670454 \nu^{11} - 10011579132233233 \nu^{10} + 4432213352185154 \nu^{9} - 27875816530580689 \nu^{8} - 1648646059467 \nu^{7} - 58120735114955745 \nu^{6} - 30120769249294290 \nu^{5} - 65459963102298117 \nu^{4} - 12784748682299277 \nu^{3} - 9546047555721093 \nu^{2} - 4115805002670663 \nu - 1530279201448305$$$$)/ 947673457589169$$ $$\beta_{5}$$ $$=$$ $$($$$$9386663199290 \nu^{17} - 28694384497755 \nu^{16} + 169908419865960 \nu^{15} - 241844070319962 \nu^{14} + 1239547799878224 \nu^{13} - 1314229089983598 \nu^{12} + 6232698837677169 \nu^{11} - 3006575605918713 \nu^{10} + 17028366751577601 \nu^{9} - 4142922658810448 \nu^{8} + 34877800545262521 \nu^{7} - 1766816709463692 \nu^{6} + 29198384758586082 \nu^{5} - 12930307538598189 \nu^{4} + 12454584293215953 \nu^{3} - 882885901711374 \nu^{2} + 138552500110104 \nu - 4203104676633$$$$)/ 947673457589169$$ $$\beta_{6}$$ $$=$$ $$($$$$-14929934022004 \nu^{17} + 32604146863850 \nu^{16} - 236340748141006 \nu^{15} + 167050667306534 \nu^{14} - 1739072965905898 \nu^{13} + 510010367286104 \nu^{12} - 8792512813886158 \nu^{11} - 3140774383001221 \nu^{10} - 26307859178363239 \nu^{9} - 15640176164249217 \nu^{8} - 57862145289851820 \nu^{7} - 43768565574223665 \nu^{6} - 60004949172724842 \nu^{5} - 18252076413067977 \nu^{4} - 8757964279213893 \nu^{3} - 4596043249292238 \nu^{2} - 1530279201448305 \nu - 36941403586275$$$$)/ 947673457589169$$ $$\beta_{7}$$ $$=$$ $$($$$$-18337730018439 \nu^{17} - 31292997456654 \nu^{16} - 118353124680056 \nu^{15} - 958004874913547 \nu^{14} - 1090615262838797 \nu^{13} - 7841873831279933 \nu^{12} - 6531955559730773 \nu^{11} - 46444554101576459 \nu^{10} - 37956330109318568 \nu^{9} - 142124563458383873 \nu^{8} - 117336074417295878 \nu^{7} - 319175285405624424 \nu^{6} - 219067215001880949 \nu^{5} - 275132314454322585 \nu^{4} - 37141584626266959 \nu^{3} - 53458401099252276 \nu^{2} - 16248709438677036 \nu - 5741236777511763$$$$)/ 947673457589169$$ $$\beta_{8}$$ $$=$$ $$($$$$26105354366996 \nu^{17} - 63238568869762 \nu^{16} + 423969596505896 \nu^{15} - 381333125877343 \nu^{14} + 3063723006584738 \nu^{13} - 1560445387916167 \nu^{12} + 15334609027455074 \nu^{11} + 2031082540412975 \nu^{10} + 43675230745520513 \nu^{9} + 16223020221568818 \nu^{8} + 93780870549494967 \nu^{7} + 50595698500465794 \nu^{6} + 85426159566591603 \nu^{5} + 4910375220472239 \nu^{4} + 13142287530355716 \nu^{3} + 5641722247626054 \nu^{2} + 1091488115192319 \nu + 345236246921304$$$$)/ 947673457589169$$ $$\beta_{9}$$ $$=$$ $$($$$$31986364509206 \nu^{17} - 91923387999016 \nu^{16} + 560235379575988 \nu^{15} - 717818795015503 \nu^{14} + 4060815680685065 \nu^{13} - 3713312318060329 \nu^{12} + 20303037641375396 \nu^{11} - 6471368802252214 \nu^{10} + 55522394222138021 \nu^{9} - 4448183141004396 \nu^{8} + 113671935032633766 \nu^{7} + 11946136116344506 \nu^{6} + 92650095300789093 \nu^{5} - 35555126719393029 \nu^{4} + 23532588363210324 \nu^{3} - 4266379383670638 \nu^{2} + 1026744091475571 \nu - 522277892930232$$$$)/ 947673457589169$$ $$\beta_{10}$$ $$=$$ $$($$$$-38359582991256 \nu^{17} + 141184103340764 \nu^{16} - 753711062712370 \nu^{15} + 1382959171287296 \nu^{14} - 5444798080723135 \nu^{13} + 8242266710404298 \nu^{12} - 27107927660092663 \nu^{11} + 26727405175858106 \nu^{10} - 68742348078454345 \nu^{9} + 58328591448180305 \nu^{8} - 129926990975116542 \nu^{7} + 96197524277944095 \nu^{6} - 76949914945460406 \nu^{5} + 136406045361970827 \nu^{4} - 46184589323880753 \nu^{3} + 20507327464676868 \nu^{2} + 1153651037649102 \nu + 1091488115192319$$$$)/ 947673457589169$$ $$\beta_{11}$$ $$=$$ $$($$$$43205244541727 \nu^{17} - 102223896048158 \nu^{16} + 700156475654080 \nu^{15} - 606479221671473 \nu^{14} + 5114283381713863 \nu^{13} - 2417630286567290 \nu^{12} + 25748120972162161 \nu^{11} + 4196125433622094 \nu^{10} + 74841836255345464 \nu^{9} + 29754900000193220 \nu^{8} + 161543249487177417 \nu^{7} + 91846838482672797 \nu^{6} + 154609304705983752 \nu^{5} + 15440744876682687 \nu^{4} + 23995208107725150 \nu^{3} + 6308578295113203 \nu^{2} + 4748282754537690 \nu - 207168800466078$$$$)/ 947673457589169$$ $$\beta_{12}$$ $$=$$ $$($$$$-58030876992248 \nu^{17} + 142106266467538 \nu^{16} - 952632397861448 \nu^{15} + 890536545230212 \nu^{14} - 6942256967961233 \nu^{13} + 3773352713268415 \nu^{12} - 34935251758776839 \nu^{11} - 3067867174677740 \nu^{10} - 100595599248445346 \nu^{9} - 33354599211099285 \nu^{8} - 216649458199460484 \nu^{7} - 110015989782122142 \nu^{6} - 204898802115569106 \nu^{5} - 15527059778091501 \nu^{4} - 41742001434281307 \nu^{3} - 12390659980698708 \nu^{2} - 2523233224422378 \nu - 1026744091475571$$$$)/ 947673457589169$$ $$\beta_{13}$$ $$=$$ $$($$$$59935273415844 \nu^{17} - 162040208301516 \nu^{16} + 1046593157200005 \nu^{15} - 1239117699341179 \nu^{14} + 7831896426734648 \nu^{13} - 6204271734960604 \nu^{12} + 40111892119961741 \nu^{11} - 8286392505447271 \nu^{10} + 118122926298266579 \nu^{9} + 6255636797747057 \nu^{8} + 255527698711368809 \nu^{7} + 62324159954199482 \nu^{6} + 266993801956578345 \nu^{5} - 12418421825717859 \nu^{4} + 106546815974665644 \nu^{3} - 12730041007282206 \nu^{2} + 12520519668190170 \nu - 54728443433070$$$$)/ 947673457589169$$ $$\beta_{14}$$ $$=$$ $$($$$$-1086451530723 \nu^{17} + 3619950724209 \nu^{16} - 20271176257636 \nu^{15} + 32616183042599 \nu^{14} - 146051003273701 \nu^{13} + 186265436063468 \nu^{12} - 725759892762523 \nu^{11} + 521010684894404 \nu^{10} - 1878167341158724 \nu^{9} + 1007632244640773 \nu^{8} - 3630767110174270 \nu^{7} + 1378105011657021 \nu^{6} - 2306955299614101 \nu^{5} + 2754867338283537 \nu^{4} - 842254955235942 \nu^{3} + 195918028216722 \nu^{2} + 47754510619860 \nu + 7416666919458$$$$)/ 12981828186153$$ $$\beta_{15}$$ $$=$$ $$($$$$90945361447588 \nu^{17} - 266358564289231 \nu^{16} + 1595034014270366 \nu^{15} - 2096293527119680 \nu^{14} + 11456653991062028 \nu^{13} - 10982542015727782 \nu^{12} + 56886368391164660 \nu^{11} - 20600500303944889 \nu^{10} + 151807188139139543 \nu^{9} - 20967078031398935 \nu^{8} + 304661895761477523 \nu^{7} + 13281434748853485 \nu^{6} + 219849080693715819 \nu^{5} - 126851113956033651 \nu^{4} + 38454064317268581 \nu^{3} - 3798337484608731 \nu^{2} - 7086379227931701 \nu + 627260144849775$$$$)/ 947673457589169$$ $$\beta_{16}$$ $$=$$ $$($$$$123305861608837 \nu^{17} - 339448293361062 \nu^{16} + 2103997077941108 \nu^{15} - 2480047842347340 \nu^{14} + 15134450162561409 \nu^{13} - 12348838612287261 \nu^{12} + 75279305856451767 \nu^{11} - 15509008597603083 \nu^{10} + 204866941050703278 \nu^{9} + 3975338912085122 \nu^{8} + 419149801440595713 \nu^{7} + 83086841103048088 \nu^{6} + 323338808238491463 \nu^{5} - 130113913119154452 \nu^{4} + 42761379534961740 \nu^{3} - 8429153801682627 \nu^{2} + 1641102954087024 \nu - 4360253579320527$$$$)/ 947673457589169$$ $$\beta_{17}$$ $$=$$ $$($$$$-153621363022881 \nu^{17} + 442400867850384 \nu^{16} - 2690072800467970 \nu^{15} + 3461283024183768 \nu^{14} - 19496773708261137 \nu^{13} + 18021853966873653 \nu^{12} - 97486410097108869 \nu^{11} + 32191540454597769 \nu^{10} - 266197581002689578 \nu^{9} + 26352445182246558 \nu^{8} - 544171307488865919 \nu^{7} - 47790022541484575 \nu^{6} - 435991069423046835 \nu^{5} + 183780060259854618 \nu^{4} - 110672281044251580 \nu^{3} + 12225009009679767 \nu^{2} - 2144815543427040 \nu + 4564935534281661$$$$)/ 947673457589169$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{17} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{5} + 3 \beta_{2} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} + \beta_{11} + \beta_{9} + \beta_{8} + 5 \beta_{5} + \beta_{4} - \beta_{3} - 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{14} - 6 \beta_{13} + \beta_{12} - 8 \beta_{10} + \beta_{9} + \beta_{8} - 7 \beta_{7} + \beta_{6} - \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 16 \beta_{2} - 9 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{14} - 9 \beta_{13} + 12 \beta_{12} - 5 \beta_{11} - 6 \beta_{10} - \beta_{9} - 10 \beta_{8} - 11 \beta_{7} + 9 \beta_{6} - 30 \beta_{5} - 6 \beta_{4} - \beta_{3} - 21 \beta_{2} - 30 \beta_{1} + 21$$ $$\nu^{6}$$ $$=$$ $$-39 \beta_{17} + 11 \beta_{16} + 10 \beta_{15} - 10 \beta_{14} + 10 \beta_{13} + 48 \beta_{12} - 58 \beta_{11} + 58 \beta_{10} - 24 \beta_{9} - 24 \beta_{8} + 2 \beta_{6} - 48 \beta_{5} - 58 \beta_{4} + 16 \beta_{3} + 10 \beta_{1} + 100$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{17} + 8 \beta_{16} - 8 \beta_{15} - 24 \beta_{14} + 82 \beta_{13} - 53 \beta_{12} - 29 \beta_{11} + 101 \beta_{10} - 84 \beta_{9} - 9 \beta_{8} + 98 \beta_{7} - 69 \beta_{6} + 24 \beta_{5} - 72 \beta_{4} + 93 \beta_{3} + 176 \beta_{2} + 211 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$280 \beta_{17} - 84 \beta_{16} - 93 \beta_{15} - 9 \beta_{14} + 196 \beta_{13} - 426 \beta_{12} + 425 \beta_{11} - 8 \beta_{10} + 46 \beta_{9} + 172 \beta_{8} + 364 \beta_{7} - 163 \beta_{6} + 430 \beta_{5} - 8 \beta_{4} + 46 \beta_{3} + 672 \beta_{2} + 430 \beta_{1} - 672$$ $$\nu^{9}$$ $$=$$ $$593 \beta_{17} - 218 \beta_{16} - 172 \beta_{15} + 172 \beta_{14} - 172 \beta_{13} - 573 \beta_{12} + 852 \beta_{11} - 745 \beta_{10} + 728 \beta_{9} + 728 \beta_{8} - 125 \beta_{6} + 1108 \beta_{5} + 852 \beta_{4} - 681 \beta_{3} - 172 \beta_{1} - 1391$$ $$\nu^{10}$$ $$=$$ $$-47 \beta_{17} - 47 \beta_{16} + 47 \beta_{15} + 728 \beta_{14} - 2008 \beta_{13} + 562 \beta_{12} - 166 \beta_{11} - 3027 \beta_{10} + 1598 \beta_{9} + 200 \beta_{8} - 2689 \beta_{7} + 1070 \beta_{6} - 728 \beta_{5} + 3193 \beta_{4} - 1798 \beta_{3} - 4702 \beta_{2} - 3820 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-4890 \beta_{17} + 1598 \beta_{16} + 1798 \beta_{15} + 200 \beta_{14} - 3292 \beta_{13} + 7077 \beta_{12} - 6795 \beta_{11} - 82 \beta_{10} - 81 \beta_{9} - 5472 \beta_{8} - 6488 \beta_{7} + 5272 \beta_{6} - 9274 \beta_{5} - 82 \beta_{4} - 81 \beta_{3} - 10758 \beta_{2} - 9274 \beta_{1} + 10758$$ $$\nu^{12}$$ $$=$$ $$-14546 \beta_{17} + 5553 \beta_{16} + 5472 \beta_{15} - 5472 \beta_{14} + 5472 \beta_{13} + 19002 \beta_{12} - 22256 \beta_{11} + 24474 \beta_{10} - 14228 \beta_{9} - 14228 \beta_{8} + 5109 \beta_{6} - 17452 \beta_{5} - 22256 \beta_{4} + 13865 \beta_{3} + 5472 \beta_{1} + 33757$$ $$\nu^{13}$$ $$=$$ $$363 \beta_{17} + 363 \beta_{16} - 363 \beta_{15} - 14228 \beta_{14} + 37152 \beta_{13} - 10163 \beta_{12} + 4065 \beta_{11} + 54220 \beta_{10} - 43811 \beta_{9} + 1774 \beta_{8} + 51017 \beta_{7} - 27809 \beta_{6} + 14228 \beta_{5} - 58285 \beta_{4} + 42037 \beta_{3} + 82400 \beta_{2} + 79983 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$107792 \beta_{17} - 43811 \beta_{16} - 42037 \beta_{15} + 1774 \beta_{14} + 63981 \beta_{13} - 163773 \beta_{12} + 189982 \beta_{11} - 24435 \beta_{10} - 5476 \beta_{9} + 115961 \beta_{8} + 151603 \beta_{7} - 117735 \beta_{6} + 169305 \beta_{5} - 24435 \beta_{4} - 5476 \beta_{3} + 246687 \beta_{2} + 169305 \beta_{1} - 246687$$ $$\nu^{15}$$ $$=$$ $$287040 \beta_{17} - 110485 \beta_{16} - 115961 \beta_{15} + 115961 \beta_{14} - 115961 \beta_{13} - 367359 \beta_{12} + 422162 \beta_{11} - 483320 \beta_{10} + 318069 \beta_{9} + 318069 \beta_{8} - 147646 \beta_{6} + 367560 \beta_{5} + 422162 \beta_{4} - 349754 \beta_{3} - 115961 \beta_{1} - 628343$$ $$\nu^{16}$$ $$=$$ $$31685 \beta_{17} + 31685 \beta_{16} - 31685 \beta_{15} + 318069 \beta_{14} - 801590 \beta_{13} + 76024 \beta_{12} - 242045 \beta_{11} - 1242967 \beta_{10} + 949035 \beta_{9} - 98319 \beta_{8} - 1151344 \beta_{7} + 532647 \beta_{6} - 318069 \beta_{5} + 1485012 \beta_{4} - 850716 \beta_{3} - 1825870 \beta_{2} - 1601012 \beta_{1}$$ $$\nu^{17}$$ $$=$$ $$-2133659 \beta_{17} + 949035 \beta_{16} + 850716 \beta_{15} - 98319 \beta_{14} - 1184624 \beta_{13} + 3167268 \beta_{12} - 3927739 \beta_{11} + 662152 \beta_{10} + 372049 \beta_{9} - 2783801 \beta_{8} - 3082694 \beta_{7} + 2882120 \beta_{6} - 3494015 \beta_{5} + 662152 \beta_{4} + 372049 \beta_{3} - 4783620 \beta_{2} - 3494015 \beta_{1} + 4783620$$

## Character values

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

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\beta_{10} + \beta_{11}$$ $$1$$

## 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
 1.28031 − 2.21756i −0.132544 + 0.229572i −0.647764 + 1.12196i −0.707815 + 1.22597i 0.247857 − 0.429301i 0.959958 − 1.66270i 1.39238 − 2.41167i 0.206805 − 0.358197i −1.09918 + 1.90384i −0.707815 − 1.22597i 0.247857 + 0.429301i 0.959958 + 1.66270i 1.39238 + 2.41167i 0.206805 + 0.358197i −1.09918 − 1.90384i 1.28031 + 2.21756i −0.132544 − 0.229572i −0.647764 − 1.12196i
−0.766044 0.642788i −1.96154 + 1.64593i 0.173648 + 0.984808i 0.939693 + 0.342020i 2.56061 1.15931 + 0.421953i 0.500000 0.866025i 0.617622 3.50271i −0.500000 0.866025i
71.2 −0.766044 0.642788i 0.203068 0.170395i 0.173648 + 0.984808i 0.939693 + 0.342020i −0.265087 1.72587 + 0.628164i 0.500000 0.866025i −0.508742 + 2.88522i −0.500000 0.866025i
71.3 −0.766044 0.642788i 0.992431 0.832749i 0.173648 + 0.984808i 0.939693 + 0.342020i −1.29553 −3.09818 1.12765i 0.500000 0.866025i −0.229495 + 1.30153i −0.500000 0.866025i
81.1 0.939693 0.342020i −1.33026 0.484174i 0.766044 0.642788i −0.173648 0.984808i −1.41563 0.162212 + 0.919948i 0.500000 0.866025i −0.762973 0.640210i −0.500000 0.866025i
81.2 0.939693 0.342020i 0.465818 + 0.169544i 0.766044 0.642788i −0.173648 0.984808i 0.495714 −0.827523 4.69312i 0.500000 0.866025i −2.10989 1.77041i −0.500000 0.866025i
81.3 0.939693 0.342020i 1.80413 + 0.656650i 0.766044 0.642788i −0.173648 0.984808i 1.91992 0.523752 + 2.97035i 0.500000 0.866025i 0.525568 + 0.441004i −0.500000 0.866025i
181.1 −0.173648 + 0.984808i −0.483568 2.74245i −0.939693 0.342020i −0.766044 + 0.642788i 2.78476 −2.74168 + 2.30055i 0.500000 0.866025i −4.46812 + 1.62626i −0.500000 0.866025i
181.2 −0.173648 + 0.984808i −0.0718226 0.407326i −0.939693 0.342020i −0.766044 + 0.642788i 0.413610 1.40064 1.17528i 0.500000 0.866025i 2.65832 0.967550i −0.500000 0.866025i
181.3 −0.173648 + 0.984808i 0.381743 + 2.16497i −0.939693 0.342020i −0.766044 + 0.642788i −2.19837 −2.80439 + 2.35316i 0.500000 0.866025i −1.72229 + 0.626862i −0.500000 0.866025i
201.1 0.939693 + 0.342020i −1.33026 + 0.484174i 0.766044 + 0.642788i −0.173648 + 0.984808i −1.41563 0.162212 0.919948i 0.500000 + 0.866025i −0.762973 + 0.640210i −0.500000 + 0.866025i
201.2 0.939693 + 0.342020i 0.465818 0.169544i 0.766044 + 0.642788i −0.173648 + 0.984808i 0.495714 −0.827523 + 4.69312i 0.500000 + 0.866025i −2.10989 + 1.77041i −0.500000 + 0.866025i
201.3 0.939693 + 0.342020i 1.80413 0.656650i 0.766044 + 0.642788i −0.173648 + 0.984808i 1.91992 0.523752 2.97035i 0.500000 + 0.866025i 0.525568 0.441004i −0.500000 + 0.866025i
231.1 −0.173648 0.984808i −0.483568 + 2.74245i −0.939693 + 0.342020i −0.766044 0.642788i 2.78476 −2.74168 2.30055i 0.500000 + 0.866025i −4.46812 1.62626i −0.500000 + 0.866025i
231.2 −0.173648 0.984808i −0.0718226 + 0.407326i −0.939693 + 0.342020i −0.766044 0.642788i 0.413610 1.40064 + 1.17528i 0.500000 + 0.866025i 2.65832 + 0.967550i −0.500000 + 0.866025i
231.3 −0.173648 0.984808i 0.381743 2.16497i −0.939693 + 0.342020i −0.766044 0.642788i −2.19837 −2.80439 2.35316i 0.500000 + 0.866025i −1.72229 0.626862i −0.500000 + 0.866025i
271.1 −0.766044 + 0.642788i −1.96154 1.64593i 0.173648 0.984808i 0.939693 0.342020i 2.56061 1.15931 0.421953i 0.500000 + 0.866025i 0.617622 + 3.50271i −0.500000 + 0.866025i
271.2 −0.766044 + 0.642788i 0.203068 + 0.170395i 0.173648 0.984808i 0.939693 0.342020i −0.265087 1.72587 0.628164i 0.500000 + 0.866025i −0.508742 2.88522i −0.500000 + 0.866025i
271.3 −0.766044 + 0.642788i 0.992431 + 0.832749i 0.173648 0.984808i 0.939693 0.342020i −1.29553 −3.09818 + 1.12765i 0.500000 + 0.866025i −0.229495 1.30153i −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.b 18
37.f even 9 1 inner 370.2.o.b 18

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

$p$ $F_p(T)$
$2$ $$( 1 - T^{3} + T^{6} )^{3}$$
$3$ $$9 - 81 T + 360 T^{2} - 957 T^{3} + 2142 T^{4} - 3759 T^{5} + 3028 T^{6} + 1047 T^{7} - 1971 T^{8} + 362 T^{9} + 675 T^{10} - 306 T^{11} + 168 T^{12} - 63 T^{13} + 6 T^{14} - 10 T^{15} + 6 T^{16} + T^{18}$$
$5$ $$( 1 - T^{3} + T^{6} )^{3}$$
$7$ $$5774409 - 13754772 T + 18358164 T^{2} - 16349256 T^{3} + 7512210 T^{4} + 127098 T^{5} - 1113875 T^{6} - 13401 T^{7} + 187497 T^{8} - 3080 T^{9} - 6444 T^{10} - 1755 T^{11} - 1023 T^{12} - 63 T^{13} + 414 T^{14} + 196 T^{15} + 54 T^{16} + 9 T^{17} + T^{18}$$
$11$ $$81 - 1215 T + 48600 T^{2} + 405999 T^{3} + 11756907 T^{4} - 9135126 T^{5} + 9705195 T^{6} - 3929256 T^{7} + 3016152 T^{8} - 1182864 T^{9} + 606177 T^{10} - 188943 T^{11} + 72238 T^{12} - 20193 T^{13} + 5604 T^{14} - 1054 T^{15} + 165 T^{16} - 15 T^{17} + T^{18}$$
$13$ $$811801 + 2784090 T + 851046 T^{2} - 4354806 T^{3} + 1378428 T^{4} + 1445376 T^{5} - 794379 T^{6} - 78927 T^{7} + 208491 T^{8} - 81070 T^{9} + 33654 T^{10} - 4395 T^{11} + 3489 T^{12} - 1245 T^{13} + 660 T^{14} - 102 T^{15} + 48 T^{16} - 3 T^{17} + T^{18}$$
$17$ $$172423161 + 2781697302 T + 12047307957 T^{2} - 6767851617 T^{3} + 3216741336 T^{4} - 743005899 T^{5} + 4975119 T^{6} + 44595171 T^{7} - 8189604 T^{8} - 129459 T^{9} + 802710 T^{10} - 296538 T^{11} + 69037 T^{12} - 9018 T^{13} + 1542 T^{14} - 83 T^{15} + 27 T^{16} - 6 T^{17} + T^{18}$$
$19$ $$8051293441 + 19912300764 T + 22552192695 T^{2} + 15194020335 T^{3} + 6424198680 T^{4} + 1166073837 T^{5} - 307349022 T^{6} - 189768873 T^{7} + 20364780 T^{8} + 9763310 T^{9} + 291030 T^{10} + 179742 T^{11} + 68343 T^{12} + 1821 T^{13} + 1563 T^{14} + 393 T^{15} + 60 T^{16} + 3 T^{17} + T^{18}$$
$23$ $$729 - 8748 T + 75087 T^{2} - 329508 T^{3} + 1096416 T^{4} - 1662363 T^{5} + 2613627 T^{6} - 2093445 T^{7} + 3941136 T^{8} - 2686716 T^{9} + 1546371 T^{10} - 497097 T^{11} + 146970 T^{12} - 29349 T^{13} + 7020 T^{14} - 1155 T^{15} + 180 T^{16} - 15 T^{17} + T^{18}$$
$29$ $$375017837769 + 445338849366 T + 425935608948 T^{2} + 208517354244 T^{3} + 95790149478 T^{4} + 27500450211 T^{5} + 10015621326 T^{6} + 2348343738 T^{7} + 685803510 T^{8} + 125867736 T^{9} + 30125763 T^{10} + 4645026 T^{11} + 920034 T^{12} + 109026 T^{13} + 16821 T^{14} + 1500 T^{15} + 198 T^{16} + 12 T^{17} + T^{18}$$
$31$ $$( -6497261 + 10168164 T - 5195916 T^{2} + 944337 T^{3} + 34101 T^{4} - 31329 T^{5} + 2802 T^{6} + 117 T^{7} - 27 T^{8} + T^{9} )^{2}$$
$37$ $$129961739795077 + 42149753447052 T + 2563160682591 T^{2} - 1344440638316 T^{3} - 183900173964 T^{4} + 25143743976 T^{5} + 9754349516 T^{6} + 676201122 T^{7} - 90133887 T^{8} - 31673777 T^{9} - 2436051 T^{10} + 493938 T^{11} + 192572 T^{12} + 13416 T^{13} - 2652 T^{14} - 524 T^{15} + 27 T^{16} + 12 T^{17} + T^{18}$$
$41$ $$15742469961 - 102300647274 T + 296961351039 T^{2} - 261283758615 T^{3} + 79007457753 T^{4} + 46277972676 T^{5} + 16283877444 T^{6} + 5714527239 T^{7} + 1125116730 T^{8} + 149484879 T^{9} + 25057512 T^{10} - 743769 T^{11} - 20448 T^{12} + 12501 T^{13} + 279 T^{14} + 699 T^{15} + 9 T^{16} - 15 T^{17} + T^{18}$$
$43$ $$( -31280563 + 11739282 T + 13617924 T^{2} + 2821553 T^{3} + 4539 T^{4} - 56511 T^{5} - 5039 T^{6} + 105 T^{7} + 30 T^{8} + T^{9} )^{2}$$
$47$ $$4886429409 + 53931912075 T + 641091535353 T^{2} - 505856984112 T^{3} + 424439578476 T^{4} - 135013738599 T^{5} + 65101061205 T^{6} - 14906461509 T^{7} + 6956482176 T^{8} - 1049058216 T^{9} + 233521038 T^{10} - 20822616 T^{11} + 3965149 T^{12} - 282264 T^{13} + 43944 T^{14} - 1730 T^{15} + 249 T^{16} - 6 T^{17} + T^{18}$$
$53$ $$29987502561 + 128325502098 T + 221807277207 T^{2} + 189233974413 T^{3} + 79015111119 T^{4} + 11888851194 T^{5} + 3837427209 T^{6} + 4378677048 T^{7} + 1467112995 T^{8} + 280984371 T^{9} + 119952576 T^{10} + 11003844 T^{11} + 629638 T^{12} - 147396 T^{13} + 16362 T^{14} - 2855 T^{15} + 246 T^{16} - 12 T^{17} + T^{18}$$
$59$ $$4307434855364601 + 8781889053197124 T + 7793977142916243 T^{2} + 547069832126559 T^{3} + 222037215926058 T^{4} + 19871833300650 T^{5} + 232787564244 T^{6} - 19153528299 T^{7} - 14786666496 T^{8} - 2045972649 T^{9} + 279368730 T^{10} + 70500882 T^{11} + 8319391 T^{12} + 652998 T^{13} + 25458 T^{14} + 241 T^{15} + 204 T^{16} + 27 T^{17} + T^{18}$$
$61$ $$821069424267481 - 156414703452120 T - 22729993114014 T^{2} + 40788619250687 T^{3} + 252031362804 T^{4} - 25456607136 T^{5} + 1193021075315 T^{6} - 160141035897 T^{7} + 10790540271 T^{8} + 57834656 T^{9} - 61696080 T^{10} - 863460 T^{11} + 2295416 T^{12} - 111771 T^{13} - 4851 T^{14} + 1223 T^{15} + 18 T^{16} + T^{18}$$
$67$ $$4747623409 - 112684517424 T + 1135688290890 T^{2} + 1203591419891 T^{3} + 6478413894366 T^{4} + 4536647330898 T^{5} + 4812750320231 T^{6} + 1735956327333 T^{7} + 250651718937 T^{8} + 7085546684 T^{9} - 2093975154 T^{10} - 237839718 T^{11} - 1334986 T^{12} + 1902687 T^{13} + 254721 T^{14} + 22121 T^{15} + 1380 T^{16} + 54 T^{17} + T^{18}$$
$71$ $$81305839834120521 - 45919784594496348 T + 16430599712833218 T^{2} - 4243230567911691 T^{3} + 779514466965354 T^{4} - 112980906957420 T^{5} + 15338685137337 T^{6} - 1978465305033 T^{7} + 239690959407 T^{8} - 31574682738 T^{9} + 4183446690 T^{10} - 412379262 T^{11} + 26191998 T^{12} - 1336257 T^{13} + 107811 T^{14} - 10017 T^{15} + 702 T^{16} - 36 T^{17} + T^{18}$$
$73$ $$( -563149 + 3780636 T + 236091 T^{2} - 1127098 T^{3} + 2142 T^{4} + 35826 T^{5} - 242 T^{6} - 357 T^{7} + 3 T^{8} + T^{9} )^{2}$$
$79$ $$21673019174041 - 28453656167970 T + 20674211319996 T^{2} - 10685579843028 T^{3} + 4503196914459 T^{4} - 1447711965708 T^{5} + 328803940347 T^{6} - 49535409480 T^{7} + 3518899296 T^{8} + 474024854 T^{9} - 112101171 T^{10} + 4685505 T^{11} + 1585221 T^{12} - 201045 T^{13} + 19875 T^{14} + 411 T^{15} - 69 T^{16} - 12 T^{17} + T^{18}$$
$83$ $$244054772361 - 222926567769 T + 910804075803 T^{2} - 1389076332687 T^{3} + 890527915476 T^{4} - 267874108884 T^{5} + 42999349851 T^{6} - 2645204931 T^{7} + 496129698 T^{8} + 326112075 T^{9} + 149873247 T^{10} + 17109954 T^{11} + 2422395 T^{12} + 227421 T^{13} + 1872 T^{14} - 3102 T^{15} - 135 T^{16} + 15 T^{17} + T^{18}$$
$89$ $$167859825849 + 854966734632 T + 1113971228091 T^{2} - 139015479960 T^{3} + 1003534764966 T^{4} - 901193672826 T^{5} + 607655700837 T^{6} - 355646849688 T^{7} + 167555936619 T^{8} - 55151434482 T^{9} + 12589312176 T^{10} - 2069670225 T^{11} + 254025736 T^{12} - 23773167 T^{13} + 1699653 T^{14} - 90907 T^{15} + 3456 T^{16} - 84 T^{17} + T^{18}$$
$97$ $$133252011369 - 749821406589 T + 3368908228581 T^{2} - 5158348251990 T^{3} + 6544795679388 T^{4} + 416821002999 T^{5} + 651656302024 T^{6} + 2142031863 T^{7} + 36987318747 T^{8} + 1598255230 T^{9} + 894899088 T^{10} + 47709396 T^{11} + 15965523 T^{12} + 926784 T^{13} + 126501 T^{14} + 3517 T^{15} + 426 T^{16} + 9 T^{17} + T^{18}$$