# Properties

 Label 370.2.l.c Level $370$ Weight $2$ Character orbit 370.l Analytic conductor $2.954$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.116304318664704.2 Defining polynomial: $$x^{12} - 2 x^{11} + x^{10} + 6 x^{9} - 9 x^{8} - 2 x^{7} + 18 x^{6} - 4 x^{5} - 36 x^{4} + 48 x^{3} + 16 x^{2} - 64 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + x^{10} + 6 x^{9} - 9 x^{8} - 2 x^{7} + 18 x^{6} - 4 x^{5} - 36 x^{4} + 48 x^{3} + 16 x^{2} - 64 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} + 2 \nu^{10} + \nu^{9} - 6 \nu^{8} + 7 \nu^{7} + 10 \nu^{6} - 14 \nu^{5} - 12 \nu^{4} + 44 \nu^{3} + 32 \nu^{2} - 48 \nu + 64$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{10} + \nu^{9} - 6 \nu^{8} + 3 \nu^{7} + 10 \nu^{6} - 12 \nu^{5} - 12 \nu^{4} + 32 \nu^{3} - 8 \nu^{2} - 24 \nu + 32$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} - 3 \nu^{9} + 3 \nu^{8} + 7 \nu^{7} - 11 \nu^{6} - 10 \nu^{5} + 24 \nu^{4} + 8 \nu^{3} - 32 \nu^{2} + 32 \nu + 40$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 10 \nu^{10} + 5 \nu^{9} + 14 \nu^{8} - 37 \nu^{7} - 10 \nu^{6} + 78 \nu^{5} - 20 \nu^{4} - 156 \nu^{3} + 144 \nu^{2} + 32 \nu - 256$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} + \nu^{9} - 14 \nu^{8} + 19 \nu^{7} + 18 \nu^{6} - 44 \nu^{5} - 16 \nu^{4} + 96 \nu^{3} - 40 \nu^{2} - 72 \nu + 160$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{11} + 5 \nu^{10} + \nu^{9} - 15 \nu^{8} + 19 \nu^{7} + 19 \nu^{6} - 48 \nu^{5} - 18 \nu^{4} + 104 \nu^{3} - 44 \nu^{2} - 96 \nu + 176$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{11} + 4 \nu^{10} + 19 \nu^{9} - 44 \nu^{8} - 3 \nu^{7} + 88 \nu^{6} - 38 \nu^{5} - 160 \nu^{4} + 156 \nu^{3} + 72 \nu^{2} - 368 \nu + 128$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{11} - 9 \nu^{10} + 3 \nu^{9} + 19 \nu^{8} - 39 \nu^{7} - 15 \nu^{6} + 84 \nu^{5} - 6 \nu^{4} - 168 \nu^{3} + 116 \nu^{2} + 96 \nu - 288$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$-7 \nu^{11} + 18 \nu^{10} - 3 \nu^{9} - 46 \nu^{8} + 51 \nu^{7} + 66 \nu^{6} - 130 \nu^{5} - 68 \nu^{4} + 308 \nu^{3} - 176 \nu^{2} - 224 \nu + 288$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-9 \nu^{11} + 8 \nu^{10} + 19 \nu^{9} - 56 \nu^{8} + 13 \nu^{7} + 100 \nu^{6} - 70 \nu^{5} - 168 \nu^{4} + 236 \nu^{3} + 24 \nu^{2} - 432 \nu + 256$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-17 \nu^{11} + 14 \nu^{10} + 27 \nu^{9} - 82 \nu^{8} + 5 \nu^{7} + 158 \nu^{6} - 78 \nu^{5} - 284 \nu^{4} + 316 \nu^{3} + 48 \nu^{2} - 544 \nu + 160$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} - \beta_{9} - 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + 3 \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{11} + 3 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - 5 \beta_{6} + \beta_{5} - \beta_{3} + 5 \beta_{2}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{10} - \beta_{8} - 3 \beta_{7} - 7 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} + 3 \beta_{1} + 7$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{11} + \beta_{9} + \beta_{5} - 5 \beta_{4} - 9 \beta_{3} - \beta_{1} + 3$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 9 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} + 3 \beta_{5} - 9 \beta_{3} - 11 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-11 \beta_{10} + 9 \beta_{8} + 15 \beta_{7} + 5 \beta_{6} - 21 \beta_{4} - 21 \beta_{2} + 7 \beta_{1} - 5$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$3 \beta_{11} - 11 \beta_{9} + 3 \beta_{5} - 13 \beta_{4} - \beta_{3} - 3 \beta_{1} - 39$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-13 \beta_{11} + 13 \beta_{10} + 15 \beta_{9} + 25 \beta_{8} + 15 \beta_{7} - 19 \beta_{6} + 31 \beta_{5} + 25 \beta_{3} - 5 \beta_{2}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$19 \beta_{10} + 25 \beta_{8} - 29 \beta_{7} + 39 \beta_{6} - 5 \beta_{4} - 5 \beta_{2} - 3 \beta_{1} - 39$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.41362 + 0.0408194i 1.33544 − 0.465413i 0.578188 + 1.29062i 0.742163 − 1.20382i −1.07078 + 0.923815i 0.828615 + 1.14604i −1.41362 − 0.0408194i 1.33544 + 0.465413i 0.578188 − 1.29062i 0.742163 + 1.20382i −1.07078 − 0.923815i 0.828615 − 1.14604i
−0.866025 + 0.500000i −0.671462 + 1.16301i 0.500000 0.866025i −0.866025 0.500000i 1.34292i −0.461662 + 0.799622i 1.00000i 0.598279 + 1.03625i 1.00000
11.2 −0.866025 + 0.500000i 0.264658 0.458402i 0.500000 0.866025i −0.866025 0.500000i 0.529317i −0.146963 + 0.254547i 1.00000i 1.35991 + 2.35544i 1.00000
11.3 −0.866025 + 0.500000i 1.40680 2.43665i 0.500000 0.866025i −0.866025 0.500000i 2.81361i 1.97465 3.42019i 1.00000i −2.45819 4.25771i 1.00000
11.4 0.866025 0.500000i −0.671462 + 1.16301i 0.500000 0.866025i 0.866025 + 0.500000i 1.34292i −1.45444 + 2.51917i 1.00000i 0.598279 + 1.03625i 1.00000
11.5 0.866025 0.500000i 0.264658 0.458402i 0.500000 0.866025i 0.866025 + 0.500000i 0.529317i 1.80085 3.11916i 1.00000i 1.35991 + 2.35544i 1.00000
11.6 0.866025 0.500000i 1.40680 2.43665i 0.500000 0.866025i 0.866025 + 0.500000i 2.81361i −0.712432 + 1.23397i 1.00000i −2.45819 4.25771i 1.00000
101.1 −0.866025 0.500000i −0.671462 1.16301i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.34292i −0.461662 0.799622i 1.00000i 0.598279 1.03625i 1.00000
101.2 −0.866025 0.500000i 0.264658 + 0.458402i 0.500000 + 0.866025i −0.866025 + 0.500000i 0.529317i −0.146963 0.254547i 1.00000i 1.35991 2.35544i 1.00000
101.3 −0.866025 0.500000i 1.40680 + 2.43665i 0.500000 + 0.866025i −0.866025 + 0.500000i 2.81361i 1.97465 + 3.42019i 1.00000i −2.45819 + 4.25771i 1.00000
101.4 0.866025 + 0.500000i −0.671462 1.16301i 0.500000 + 0.866025i 0.866025 0.500000i 1.34292i −1.45444 2.51917i 1.00000i 0.598279 1.03625i 1.00000
101.5 0.866025 + 0.500000i 0.264658 + 0.458402i 0.500000 + 0.866025i 0.866025 0.500000i 0.529317i 1.80085 + 3.11916i 1.00000i 1.35991 2.35544i 1.00000
101.6 0.866025 + 0.500000i 1.40680 + 2.43665i 0.500000 + 0.866025i 0.866025 0.500000i 2.81361i −0.712432 1.23397i 1.00000i −2.45819 + 4.25771i 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.6 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.e 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.l.c 12
37.e even 6 1 inner 370.2.l.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.l.c 12 1.a even 1 1 trivial
370.2.l.c 12 37.e even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}^{6} - 2 T_{3}^{5} + 7 T_{3}^{4} + 2 T_{3}^{3} + 13 T_{3}^{2} - 6 T_{3} + 4$$ $$T_{7}^{12} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$( 4 - 6 T + 13 T^{2} + 2 T^{3} + 7 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$5$ $$( 1 - T^{2} + T^{4} )^{3}$$
$7$ $$256 + 1280 T + 4928 T^{2} + 6976 T^{3} + 7792 T^{4} + 3952 T^{5} + 1992 T^{6} + 232 T^{7} + 256 T^{8} + 12 T^{9} + 22 T^{10} - 2 T^{11} + T^{12}$$
$11$ $$( 4 - 20 T - 66 T^{2} - 30 T^{3} + 11 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$13$ $$32761 + 38010 T - 323 T^{2} - 17430 T^{3} - 866 T^{4} + 4890 T^{5} + 425 T^{6} - 870 T^{7} - 2 T^{8} + 90 T^{9} - 3 T^{10} - 6 T^{11} + T^{12}$$
$17$ $$1024 - 23040 T + 153472 T^{2} + 434880 T^{3} + 484240 T^{4} + 296976 T^{5} + 107432 T^{6} + 21384 T^{7} + 1384 T^{8} - 252 T^{9} - 30 T^{10} + 6 T^{11} + T^{12}$$
$19$ $$55830784 - 42321408 T + 1787008 T^{2} + 6751488 T^{3} - 326288 T^{4} - 1124256 T^{5} + 362120 T^{6} - 29184 T^{7} - 4055 T^{8} + 594 T^{9} + 75 T^{10} - 18 T^{11} + T^{12}$$
$23$ $$80496784 + 35093824 T^{2} + 5528236 T^{4} + 390800 T^{6} + 12685 T^{8} + 186 T^{10} + T^{12}$$
$29$ $$2166784 + 4016128 T^{2} + 2178816 T^{4} + 311040 T^{6} + 13040 T^{8} + 200 T^{10} + T^{12}$$
$31$ $$99042304 + 39556480 T^{2} + 6040585 T^{4} + 438344 T^{6} + 15142 T^{8} + 216 T^{10} + T^{12}$$
$37$ $$2565726409 + 1802942882 T + 534135885 T^{2} + 84387898 T^{3} + 7838894 T^{4} + 690642 T^{5} + 104809 T^{6} + 18666 T^{7} + 5726 T^{8} + 1666 T^{9} + 285 T^{10} + 26 T^{11} + T^{12}$$
$41$ $$3139984 - 645008 T + 5117132 T^{2} + 2725052 T^{3} + 7493713 T^{4} + 1457792 T^{5} + 616506 T^{6} - 43372 T^{7} + 18151 T^{8} - 408 T^{9} + 154 T^{10} - 4 T^{11} + T^{12}$$
$43$ $$14348944 + 39296008 T^{2} + 14667049 T^{4} + 972812 T^{6} + 25198 T^{8} + 276 T^{10} + T^{12}$$
$47$ $$( 436 - 1924 T - 4258 T^{2} - 1434 T^{3} - 105 T^{4} + 10 T^{5} + T^{6} )^{2}$$
$53$ $$16384 - 14336 T + 27008 T^{2} - 4240 T^{3} + 16321 T^{4} - 994 T^{5} + 7266 T^{6} + 1640 T^{7} + 919 T^{8} + 72 T^{9} + 34 T^{10} + 2 T^{11} + T^{12}$$
$59$ $$61895468944 + 14321232432 T - 2314306664 T^{2} - 791044488 T^{3} + 96258496 T^{4} + 22162404 T^{5} - 1790782 T^{6} - 365166 T^{7} + 32299 T^{8} + 2772 T^{9} - 183 T^{10} - 12 T^{11} + T^{12}$$
$61$ $$11546791936 - 4085047296 T - 510295040 T^{2} + 350963712 T^{3} + 40161792 T^{4} - 39962112 T^{5} + 7703616 T^{6} - 533376 T^{7} - 12256 T^{8} + 2976 T^{9} + 68 T^{10} - 24 T^{11} + T^{12}$$
$67$ $$2279489536 - 1708089344 T + 906373120 T^{2} - 289078272 T^{3} + 75917056 T^{4} - 14798336 T^{5} + 2761920 T^{6} - 425024 T^{7} + 60832 T^{8} - 6416 T^{9} + 548 T^{10} - 28 T^{11} + T^{12}$$
$71$ $$1020972826624 + 461597671424 T + 141214785536 T^{2} + 27114016768 T^{3} + 4141256704 T^{4} + 477446656 T^{5} + 48726912 T^{6} + 4139968 T^{7} + 331120 T^{8} + 21120 T^{9} + 1156 T^{10} + 40 T^{11} + T^{12}$$
$73$ $$( -7328 - 5360 T + 13108 T^{2} - 580 T^{3} - 254 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$79$ $$31564496896 + 17942642688 T + 1122852864 T^{2} - 1294313472 T^{3} + 11274496 T^{4} + 41258496 T^{5} + 1343552 T^{6} - 566784 T^{7} - 5888 T^{8} + 4512 T^{9} + 4 T^{10} - 24 T^{11} + T^{12}$$
$83$ $$304986689536 + 83836878848 T + 29310460928 T^{2} + 3791613952 T^{3} + 1012426240 T^{4} + 134689792 T^{5} + 21009216 T^{6} + 1652992 T^{7} + 143200 T^{8} + 6336 T^{9} + 484 T^{10} + 16 T^{11} + T^{12}$$
$89$ $$1867622656 - 771664896 T - 150769856 T^{2} + 106207488 T^{3} + 18392640 T^{4} - 6825888 T^{5} - 905052 T^{6} + 248208 T^{7} + 54341 T^{8} + 1446 T^{9} - 229 T^{10} - 6 T^{11} + T^{12}$$
$97$ $$3957919744 + 10598152192 T^{2} + 731004672 T^{4} + 15755520 T^{6} + 148208 T^{8} + 632 T^{10} + T^{12}$$