# Properties

 Label 370.2.m.d Level $370$ Weight $2$ Character orbit 370.m Analytic conductor $2.954$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 37 x^{14} + 559 x^{12} + 4431 x^{10} + 19684 x^{8} + 48248 x^{6} + 58656 x^{4} + 25392 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 37 x^{14} + 559 x^{12} + 4431 x^{10} + 19684 x^{8} + 48248 x^{6} + 58656 x^{4} + 25392 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{15} + 31 \nu^{13} + 381 \nu^{11} + 2361 \nu^{9} + 7734 \nu^{7} + 12700 \nu^{5} + 8872 \nu^{3} + 2272 \nu - 128$$$$)/256$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} + 2 \nu^{14} - 31 \nu^{13} + 54 \nu^{12} - 381 \nu^{11} + 546 \nu^{10} - 2361 \nu^{9} + 2538 \nu^{8} - 7734 \nu^{7} + 5252 \nu^{6} - 12636 \nu^{5} + 3304 \nu^{4} - 8168 \nu^{3} - 1232 \nu^{2} - 736 \nu - 384$$$$)/256$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} - 2 \nu^{14} - 31 \nu^{13} - 54 \nu^{12} - 381 \nu^{11} - 546 \nu^{10} - 2361 \nu^{9} - 2538 \nu^{8} - 7734 \nu^{7} - 5252 \nu^{6} - 12636 \nu^{5} - 3304 \nu^{4} - 8168 \nu^{3} + 1232 \nu^{2} - 736 \nu + 384$$$$)/256$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 162 \nu^{12} - 273 \nu^{11} + 1670 \nu^{10} - 1237 \nu^{9} + 8286 \nu^{8} - 2018 \nu^{7} + 20652 \nu^{6} + 2188 \nu^{5} + 24280 \nu^{4} + 9576 \nu^{3} + 10640 \nu^{2} + 5632 \nu$$$$)/256$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} - 6 \nu^{14} - 27 \nu^{13} - 162 \nu^{12} - 273 \nu^{11} - 1670 \nu^{10} - 1237 \nu^{9} - 8286 \nu^{8} - 2018 \nu^{7} - 20652 \nu^{6} + 2188 \nu^{5} - 24280 \nu^{4} + 9576 \nu^{3} - 10640 \nu^{2} + 5632 \nu$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{15} + 6 \nu^{14} + 27 \nu^{13} + 170 \nu^{12} + 289 \nu^{11} + 1870 \nu^{10} + 1637 \nu^{9} + 10102 \nu^{8} + 5682 \nu^{7} + 27876 \nu^{6} + 12708 \nu^{5} + 36184 \nu^{4} + 15864 \nu^{3} + 16240 \nu^{2} + 6528 \nu + 768$$$$)/256$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 170 \nu^{12} - 289 \nu^{11} + 1870 \nu^{10} - 1637 \nu^{9} + 10102 \nu^{8} - 5682 \nu^{7} + 27876 \nu^{6} - 12708 \nu^{5} + 36184 \nu^{4} - 15864 \nu^{3} + 16240 \nu^{2} - 6528 \nu + 768$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 170 \nu^{12} - 381 \nu^{11} + 1870 \nu^{10} - 2361 \nu^{9} + 10102 \nu^{8} - 7766 \nu^{7} + 27876 \nu^{6} - 13084 \nu^{5} + 36120 \nu^{4} - 9800 \nu^{3} + 15536 \nu^{2} - 1952 \nu - 384$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 170 \nu^{12} + 381 \nu^{11} + 1870 \nu^{10} + 2361 \nu^{9} + 10102 \nu^{8} + 7766 \nu^{7} + 27876 \nu^{6} + 13084 \nu^{5} + 36120 \nu^{4} + 9800 \nu^{3} + 15536 \nu^{2} + 1952 \nu - 384$$$$)/256$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{14} + 89 \nu^{12} + 1035 \nu^{10} + 5975 \nu^{8} + 17774 \nu^{6} + 24892 \nu^{4} + 11560 \nu^{2} + 64 \nu + 128$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{14} - 89 \nu^{12} - 1035 \nu^{10} - 5975 \nu^{8} - 17774 \nu^{6} - 24892 \nu^{4} - 11560 \nu^{2} + 64 \nu - 128$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$3 \nu^{15} + 6 \nu^{14} + 81 \nu^{13} + 170 \nu^{12} + 835 \nu^{11} + 1870 \nu^{10} + 4143 \nu^{9} + 10102 \nu^{8} + 10358 \nu^{7} + 27876 \nu^{6} + 12588 \nu^{5} + 36184 \nu^{4} + 7016 \nu^{3} + 16112 \nu^{2} + 1664 \nu + 256$$$$)/256$$ $$\beta_{13}$$ $$=$$ $$($$$$-3 \nu^{14} - 85 \nu^{12} - 935 \nu^{10} - 5051 \nu^{8} - 13938 \nu^{6} - 18092 \nu^{4} - 8056 \nu^{2} - 128$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$3 \nu^{14} + 87 \nu^{12} + 985 \nu^{10} + 5513 \nu^{8} + 8 \nu^{7} + 15872 \nu^{6} + 112 \nu^{5} + 21684 \nu^{4} + 440 \nu^{3} + 10336 \nu^{2} + 496 \nu + 320$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$-3 \nu^{14} - 87 \nu^{12} - 985 \nu^{10} - 5513 \nu^{8} + 8 \nu^{7} - 15872 \nu^{6} + 112 \nu^{5} - 21684 \nu^{4} + 440 \nu^{3} - 10336 \nu^{2} + 496 \nu - 320$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{11} + \beta_{10}$$ $$\nu^{2}$$ $$=$$ $$\beta_{13} + \beta_{7} + \beta_{6} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} - 6 \beta_{11} - 6 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-11 \beta_{13} - 2 \beta_{9} - 2 \beta_{8} - 9 \beta_{7} - 9 \beta_{6} + 26$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{15} - 11 \beta_{14} + 42 \beta_{11} + 42 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + 13 \beta_{3} + 13 \beta_{2} + 4 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{15} + 2 \beta_{14} + 101 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} + 75 \beta_{7} + 75 \beta_{6} - 192$$ $$\nu^{7}$$ $$=$$ $$103 \beta_{15} + 103 \beta_{14} - 320 \beta_{11} - 320 \beta_{10} - 99 \beta_{9} + 99 \beta_{8} - 127 \beta_{3} - 127 \beta_{2} - 56 \beta_{1} - 28$$ $$\nu^{8}$$ $$=$$ $$28 \beta_{15} - 28 \beta_{14} - 871 \beta_{13} - 32 \beta_{11} + 32 \beta_{10} - 230 \beta_{9} - 230 \beta_{8} - 621 \beta_{7} - 621 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 1510$$ $$\nu^{9}$$ $$=$$ $$-919 \beta_{15} - 919 \beta_{14} + 4 \beta_{13} + 8 \beta_{12} + 2546 \beta_{11} + 2546 \beta_{10} + 839 \beta_{9} - 839 \beta_{8} + 4 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 1133 \beta_{3} + 1133 \beta_{2} + 588 \beta_{1} + 294$$ $$\nu^{10}$$ $$=$$ $$-282 \beta_{15} + 282 \beta_{14} + 7329 \beta_{13} + 366 \beta_{11} - 366 \beta_{10} + 2056 \beta_{9} + 2056 \beta_{8} + 5159 \beta_{7} + 5159 \beta_{6} + 80 \beta_{5} - 80 \beta_{4} + 12 \beta_{3} - 12 \beta_{2} - 12252$$ $$\nu^{11}$$ $$=$$ $$8039 \beta_{15} + 8039 \beta_{14} - 100 \beta_{13} - 200 \beta_{12} - 20692 \beta_{11} - 20692 \beta_{10} - 6955 \beta_{9} + 6955 \beta_{8} - 108 \beta_{7} + 108 \beta_{6} - 192 \beta_{5} - 192 \beta_{4} - 9755 \beta_{3} - 9755 \beta_{2} - 5600 \beta_{1} - 2800$$ $$\nu^{12}$$ $$=$$ $$2500 \beta_{15} - 2500 \beta_{14} - 61043 \beta_{13} - 3692 \beta_{11} + 3692 \beta_{10} - 17886 \beta_{9} - 17886 \beta_{8} - 43025 \beta_{7} - 43025 \beta_{6} - 1076 \beta_{5} + 1076 \beta_{4} - 300 \beta_{3} + 300 \beta_{2} + 100922$$ $$\nu^{13}$$ $$=$$ $$-69671 \beta_{15} - 69671 \beta_{14} + 1576 \beta_{13} + 3152 \beta_{12} + 169910 \beta_{11} + 169910 \beta_{10} + 57167 \beta_{9} - 57167 \beta_{8} + 1792 \beta_{7} - 1792 \beta_{6} + 2968 \beta_{5} + 2968 \beta_{4} + 82721 \beta_{3} + 82721 \beta_{2} + 51172 \beta_{1} + 25586$$ $$\nu^{14}$$ $$=$$ $$-20794 \beta_{15} + 20794 \beta_{14} + 506205 \beta_{13} + 35122 \beta_{11} - 35122 \beta_{10} + 153784 \beta_{9} + 153784 \beta_{8} + 359851 \beta_{7} + 359851 \beta_{6} + 12288 \beta_{5} - 12288 \beta_{4} + 4760 \beta_{3} - 4760 \beta_{2} - 837320$$ $$\nu^{15}$$ $$=$$ $$600927 \beta_{15} + 600927 \beta_{14} - 20200 \beta_{13} - 40400 \beta_{12} - 1402224 \beta_{11} - 1402224 \beta_{10} - 468363 \beta_{9} + 468363 \beta_{8} - 23848 \beta_{7} + 23848 \beta_{6} - 37744 \beta_{5} - 37744 \beta_{4} - 696719 \beta_{3} - 696719 \beta_{2} - 458440 \beta_{1} - 229220$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
159.1
 2.90925i 2.88937i 1.76701i 0.926756i 0.101618i − 1.93403i − 2.06794i − 2.85998i − 2.90925i − 2.88937i − 1.76701i − 0.926756i − 0.101618i 1.93403i 2.06794i 2.85998i
0.500000 + 0.866025i −2.51948 1.45462i −0.500000 + 0.866025i −2.23418 0.0919631i 2.90925i 0.191824 + 0.110750i −1.00000 2.73187 + 4.73173i −1.03745 1.98083i
159.2 0.500000 + 0.866025i −2.50227 1.44468i −0.500000 + 0.866025i 1.41509 1.73134i 2.88937i −0.668346 0.385870i −1.00000 2.67422 + 4.63189i 2.20693 + 0.359832i
159.3 0.500000 + 0.866025i −1.53027 0.883503i −0.500000 + 0.866025i 0.209495 + 2.22623i 1.76701i −3.45647 1.99560i −1.00000 0.0611550 + 0.105924i −1.82323 + 1.29454i
159.4 0.500000 + 0.866025i −0.802594 0.463378i −0.500000 + 0.866025i −0.214614 + 2.22574i 0.926756i 3.13584 + 1.81048i −1.00000 −1.07056 1.85427i −2.03486 + 0.927011i
159.5 0.500000 + 0.866025i −0.0880035 0.0508088i −0.500000 + 0.866025i −2.11174 0.735216i 0.101618i −1.94895 1.12523i −1.00000 −1.49484 2.58913i −0.419156 2.19643i
159.6 0.500000 + 0.866025i 1.67492 + 0.967016i −0.500000 + 0.866025i 1.59185 1.57036i 1.93403i −0.358580 0.207027i −1.00000 0.370239 + 0.641273i 2.15589 + 0.593399i
159.7 0.500000 + 0.866025i 1.79089 + 1.03397i −0.500000 + 0.866025i 1.99857 + 1.00285i 2.06794i 1.25580 + 0.725037i −1.00000 0.638185 + 1.10537i 0.130790 + 2.23224i
159.8 0.500000 + 0.866025i 2.47681 + 1.42999i −0.500000 + 0.866025i −0.654464 + 2.13815i 2.85998i −4.15112 2.39665i −1.00000 2.58973 + 4.48555i −2.17892 + 0.502291i
249.1 0.500000 0.866025i −2.51948 + 1.45462i −0.500000 0.866025i −2.23418 + 0.0919631i 2.90925i 0.191824 0.110750i −1.00000 2.73187 4.73173i −1.03745 + 1.98083i
249.2 0.500000 0.866025i −2.50227 + 1.44468i −0.500000 0.866025i 1.41509 + 1.73134i 2.88937i −0.668346 + 0.385870i −1.00000 2.67422 4.63189i 2.20693 0.359832i
249.3 0.500000 0.866025i −1.53027 + 0.883503i −0.500000 0.866025i 0.209495 2.22623i 1.76701i −3.45647 + 1.99560i −1.00000 0.0611550 0.105924i −1.82323 1.29454i
249.4 0.500000 0.866025i −0.802594 + 0.463378i −0.500000 0.866025i −0.214614 2.22574i 0.926756i 3.13584 1.81048i −1.00000 −1.07056 + 1.85427i −2.03486 0.927011i
249.5 0.500000 0.866025i −0.0880035 + 0.0508088i −0.500000 0.866025i −2.11174 + 0.735216i 0.101618i −1.94895 + 1.12523i −1.00000 −1.49484 + 2.58913i −0.419156 + 2.19643i
249.6 0.500000 0.866025i 1.67492 0.967016i −0.500000 0.866025i 1.59185 + 1.57036i 1.93403i −0.358580 + 0.207027i −1.00000 0.370239 0.641273i 2.15589 0.593399i
249.7 0.500000 0.866025i 1.79089 1.03397i −0.500000 0.866025i 1.99857 1.00285i 2.06794i 1.25580 0.725037i −1.00000 0.638185 1.10537i 0.130790 2.23224i
249.8 0.500000 0.866025i 2.47681 1.42999i −0.500000 0.866025i −0.654464 2.13815i 2.85998i −4.15112 + 2.39665i −1.00000 2.58973 4.48555i −2.17892 0.502291i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 249.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.m.d yes 16
5.b even 2 1 370.2.m.c 16
37.e even 6 1 370.2.m.c 16
185.l even 6 1 inner 370.2.m.d yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.m.c 16 5.b even 2 1
370.2.m.c 16 37.e even 6 1
370.2.m.d yes 16 1.a even 1 1 trivial
370.2.m.d yes 16 185.l even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{8}$$
$3$ $$256 + 4800 T + 32304 T^{2} + 43200 T^{3} + 1776 T^{4} - 30096 T^{5} - 4132 T^{6} + 13308 T^{7} + 3208 T^{8} - 3318 T^{9} - 771 T^{10} + 561 T^{11} + 154 T^{12} - 51 T^{13} - 14 T^{14} + 3 T^{15} + T^{16}$$
$5$ $$390625 + 46875 T^{2} + 50000 T^{3} - 6875 T^{4} + 16000 T^{5} + 3550 T^{6} - 200 T^{7} + 1630 T^{8} - 40 T^{9} + 142 T^{10} + 128 T^{11} - 11 T^{12} + 16 T^{13} + 3 T^{14} + T^{16}$$
$7$ $$256 - 384 T - 3392 T^{2} + 5376 T^{3} + 51328 T^{4} + 77664 T^{5} + 20304 T^{6} - 38544 T^{7} - 9488 T^{8} + 17376 T^{9} + 10292 T^{10} + 756 T^{11} - 600 T^{12} - 72 T^{13} + 42 T^{14} + 12 T^{15} + T^{16}$$
$11$ $$( -11840 - 15360 T - 1688 T^{2} + 3384 T^{3} + 798 T^{4} - 198 T^{5} - 56 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$13$ $$1849600 + 15558400 T + 110517120 T^{2} + 155272960 T^{3} + 156844464 T^{4} + 86183424 T^{5} + 38008152 T^{6} + 11361740 T^{7} + 3133569 T^{8} + 645564 T^{9} + 147624 T^{10} + 23904 T^{11} + 4583 T^{12} + 488 T^{13} + 84 T^{14} + 6 T^{15} + T^{16}$$
$17$ $$226576 - 105672 T + 879428 T^{2} - 1612032 T^{3} + 3807140 T^{4} - 3990228 T^{5} + 3345596 T^{6} - 1689552 T^{7} + 711469 T^{8} - 179220 T^{9} + 50374 T^{10} - 8016 T^{11} + 2735 T^{12} - 204 T^{13} + 58 T^{14} + T^{16}$$
$19$ $$1435500544 + 4073717760 T + 4482760704 T^{2} + 1785692160 T^{3} - 57965568 T^{4} - 212138496 T^{5} - 13862144 T^{6} + 19838976 T^{7} + 3487552 T^{8} - 741216 T^{9} - 170160 T^{10} + 21888 T^{11} + 5980 T^{12} - 282 T^{13} - 91 T^{14} + 3 T^{15} + T^{16}$$
$23$ $$( -9224 + 19864 T + 43960 T^{2} + 15584 T^{3} - 606 T^{4} - 860 T^{5} - 56 T^{6} + 11 T^{7} + T^{8} )^{2}$$
$29$ $$14500864 + 256263936 T^{2} + 198324352 T^{4} + 54563244 T^{6} + 7007273 T^{8} + 446220 T^{10} + 13646 T^{12} + 192 T^{14} + T^{16}$$
$31$ $$16384 + 474112 T^{2} + 1391616 T^{4} + 1378800 T^{6} + 548664 T^{8} + 86971 T^{10} + 5087 T^{12} + 121 T^{14} + T^{16}$$
$37$ $$3512479453921 - 1518910034128 T + 159075037358 T^{2} + 19000244218 T^{3} - 1690493222 T^{4} - 640355226 T^{5} + 1730416 T^{6} + 34110522 T^{7} - 7424265 T^{8} + 921906 T^{9} + 1264 T^{10} - 12642 T^{11} - 902 T^{12} + 274 T^{13} + 62 T^{14} - 16 T^{15} + T^{16}$$
$41$ $$23754764740996 - 11245921700338 T + 5235862332607 T^{2} - 1257368534669 T^{3} + 346531080914 T^{4} - 61457739217 T^{5} + 13883750897 T^{6} - 1890879656 T^{7} + 326675395 T^{8} - 30100061 T^{9} + 4593461 T^{10} - 322528 T^{11} + 45395 T^{12} - 1943 T^{13} + 262 T^{14} - 7 T^{15} + T^{16}$$
$43$ $$( 177208 + 262724 T + 137352 T^{2} + 24832 T^{3} - 2200 T^{4} - 1199 T^{5} - 69 T^{6} + 11 T^{7} + T^{8} )^{2}$$
$47$ $$35452370944 + 45429778432 T^{2} + 12943589952 T^{4} + 1582370592 T^{6} + 97315044 T^{8} + 3104440 T^{10} + 49616 T^{12} + 367 T^{14} + T^{16}$$
$53$ $$802816 - 21547008 T + 167222016 T^{2} + 685656576 T^{3} + 402965120 T^{4} - 1418504832 T^{5} + 470242512 T^{6} + 614103792 T^{7} + 148140344 T^{8} - 397692 T^{9} - 2694681 T^{10} - 43677 T^{11} + 38200 T^{12} + 675 T^{13} - 222 T^{14} - 3 T^{15} + T^{16}$$
$59$ $$159386189824 - 54499958784 T - 70785638400 T^{2} + 26328250368 T^{3} + 33011164416 T^{4} - 28263109632 T^{5} + 8803598336 T^{6} - 1033139760 T^{7} - 61118156 T^{8} + 22852908 T^{9} + 334368 T^{10} - 411972 T^{11} + 16180 T^{12} + 2910 T^{13} - 119 T^{14} - 15 T^{15} + T^{16}$$
$61$ $$899128754176 + 1580477005824 T + 712427715776 T^{2} - 375497964384 T^{3} - 18813834368 T^{4} + 21949041672 T^{5} + 459130896 T^{6} - 849480150 T^{7} + 18517765 T^{8} + 18904626 T^{9} - 646868 T^{10} - 295806 T^{11} + 17373 T^{12} + 2304 T^{13} - 144 T^{14} - 12 T^{15} + T^{16}$$
$67$ $$1893990400 + 116647526400 T + 2398480547840 T^{2} + 232523120640 T^{3} - 232338328576 T^{4} - 23556708864 T^{5} + 18546956288 T^{6} - 450238080 T^{7} - 501703296 T^{8} + 26247648 T^{9} + 10827792 T^{10} - 1321632 T^{11} + 3288 T^{12} + 5856 T^{13} - 52 T^{14} - 24 T^{15} + T^{16}$$
$71$ $$451477504 + 2659909632 T + 13650264064 T^{2} + 13041332224 T^{3} + 12639290368 T^{4} + 409718784 T^{5} + 1706118208 T^{6} + 144080160 T^{7} + 140061424 T^{8} + 656912 T^{9} + 3757984 T^{10} + 198232 T^{11} + 56012 T^{12} + 1096 T^{13} + 268 T^{14} + 4 T^{15} + T^{16}$$
$73$ $$10676686950400 + 9373196180480 T^{2} + 1377356153344 T^{4} + 71152240512 T^{6} + 1752551248 T^{8} + 23233516 T^{10} + 170184 T^{12} + 648 T^{14} + T^{16}$$
$79$ $$5065884565504 - 100257497088 T - 705894678528 T^{2} + 13983252480 T^{3} + 69162463232 T^{4} - 627176448 T^{5} - 3174048000 T^{6} + 20331264 T^{7} + 103724096 T^{8} - 424896 T^{9} - 2031888 T^{10} + 5088 T^{11} + 28576 T^{12} - 204 T^{14} + T^{16}$$
$83$ $$1363677184 + 32926777344 T + 275151003648 T^{2} + 244810874880 T^{3} + 55236196608 T^{4} - 19256559360 T^{5} - 3573728512 T^{6} + 1090003008 T^{7} + 279080080 T^{8} - 1886832 T^{9} - 4210944 T^{10} - 54528 T^{11} + 48868 T^{12} + 1560 T^{13} - 248 T^{14} - 6 T^{15} + T^{16}$$
$89$ $$5611687210000 - 16239520170000 T + 17900141082400 T^{2} - 6468085234800 T^{3} + 489043348856 T^{4} + 207232731012 T^{5} - 26599576529 T^{6} - 5400321249 T^{7} + 864443211 T^{8} + 75868470 T^{9} - 9426321 T^{10} - 662343 T^{11} + 78537 T^{12} + 3150 T^{13} - 323 T^{14} - 9 T^{15} + T^{16}$$
$97$ $$( 3678976 - 1843200 T - 596944 T^{2} + 501530 T^{3} - 102459 T^{4} + 7016 T^{5} + 138 T^{6} - 34 T^{7} + T^{8} )^{2}$$