# Properties

 Label 637.2.k.h Level $637$ Weight $2$ Character orbit 637.k Analytic conductor $5.086$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.58891012706304.1 Defining polynomial: $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 91) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{10} - 2 \nu^{9} - 7 \nu^{8} + 4 \nu^{7} + 17 \nu^{6} - 24 \nu^{5} - 14 \nu^{4} + 40 \nu^{3} + 36 \nu^{2} - 40 \nu$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} - 9 \nu^{7} + 24 \nu^{6} + 4 \nu^{5} - 44 \nu^{4} + 8 \nu^{3} + 72 \nu^{2} - 40 \nu - 48$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{11} + \nu^{10} - 6 \nu^{9} - 5 \nu^{8} + 16 \nu^{7} - \nu^{6} - 30 \nu^{5} + 6 \nu^{4} + 52 \nu^{3} - 4 \nu^{2} - 32 \nu - 16$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} - 3 \nu^{9} - 10 \nu^{8} + 9 \nu^{7} + 26 \nu^{6} - 42 \nu^{5} - 12 \nu^{4} + 60 \nu^{3} + 8 \nu^{2} - 96 \nu + 48$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} - 6 \nu^{10} - 9 \nu^{9} + 20 \nu^{8} + 31 \nu^{7} - 56 \nu^{6} - 38 \nu^{5} + 136 \nu^{4} + 28 \nu^{3} - 232 \nu^{2} - 32 \nu + 192$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} - 5 \nu^{9} + 13 \nu^{8} + 13 \nu^{7} - 35 \nu^{6} - 12 \nu^{5} + 70 \nu^{4} - 8 \nu^{3} - 108 \nu^{2} + 16 \nu + 80$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 20 \nu^{8} - 27 \nu^{7} - 32 \nu^{6} + 46 \nu^{5} + 64 \nu^{4} - 124 \nu^{3} - 136 \nu^{2} + 128 \nu + 224$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} + 3 \nu^{10} - 14 \nu^{9} - 7 \nu^{8} + 36 \nu^{7} + 5 \nu^{6} - 82 \nu^{5} + 34 \nu^{4} + 124 \nu^{3} - 60 \nu^{2} - 128 \nu + 64$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$7 \nu^{11} - 8 \nu^{10} - 19 \nu^{9} + 26 \nu^{8} + 41 \nu^{7} - 90 \nu^{6} - 18 \nu^{5} + 156 \nu^{4} + 4 \nu^{3} - 192 \nu^{2} + 80 \nu + 32$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{11} - 12 \nu^{10} + 13 \nu^{9} + 46 \nu^{8} - 31 \nu^{7} - 102 \nu^{6} + 126 \nu^{5} + 116 \nu^{4} - 252 \nu^{3} - 176 \nu^{2} + 304 \nu + 96$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-7 \nu^{10} + 8 \nu^{9} + 19 \nu^{8} - 26 \nu^{7} - 41 \nu^{6} + 90 \nu^{5} + 18 \nu^{4} - 140 \nu^{3} - 4 \nu^{2} + 176 \nu - 80$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{9} - 3 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 4 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 4 \beta_{3} - 3 \beta_{2} - 2$$ $$\nu^{7}$$ $$=$$ $$\beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_{1} + 1$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - \beta_{5} - 7 \beta_{4} - 8 \beta_{3} - \beta_{2} + 3 \beta_{1} - 3$$ $$\nu^{9}$$ $$=$$ $$-6 \beta_{11} + 6 \beta_{10} - \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + 5 \beta_{1} - 8$$ $$\nu^{10}$$ $$=$$ $$-6 \beta_{11} + 14 \beta_{10} + 5 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} - 13 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + \beta_{1} - 11$$ $$\nu^{11}$$ $$=$$ $$-29 \beta_{11} - 11 \beta_{10} + 24 \beta_{9} - 20 \beta_{8} + 5 \beta_{7} + 6 \beta_{6} - 10 \beta_{5} - 19 \beta_{4} - 18 \beta_{3} + 5 \beta_{2} - 12 \beta_{1} - 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$-1 - \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1
 −1.30089 − 0.554694i 1.40744 − 0.138282i −1.08105 − 0.911778i 1.34408 − 0.439820i 0.759479 + 1.19298i −1.12906 + 0.851598i −1.12906 − 0.851598i 0.759479 − 1.19298i 1.34408 + 0.439820i −1.08105 + 0.911778i 1.40744 + 0.138282i −1.30089 + 0.554694i
2.10939i −1.13082 1.95864i −2.44952 −3.11923 + 1.80089i −4.13154 + 2.38535i 0 0.948212i −1.05753 + 1.83169i 3.79878 + 6.57967i
459.2 1.27656i 0.583963 + 1.01145i 0.370384 1.57173 0.907437i 1.29118 0.745466i 0 3.02595i 0.817975 1.41677i −1.15840 2.00641i
459.3 0.823556i −1.33015 2.30388i 1.32176 2.73845 1.58105i −1.89737 + 1.09545i 0 2.73565i −2.03858 + 3.53092i −1.30208 2.25527i
459.4 0.120360i 0.291146 + 0.504280i 1.98551 −1.46199 + 0.844083i −0.0606950 + 0.0350423i 0 0.479696i 1.33047 2.30444i −0.101594 0.175965i
459.5 1.38595i 1.41289 + 2.44719i 0.0791355 0.449430 0.259479i −3.39169 + 1.95819i 0 2.88158i −2.49250 + 4.31714i 0.359625 + 0.622889i
459.6 2.70320i 0.172975 + 0.299601i −5.30727 2.82162 1.62906i −0.809880 + 0.467584i 0 8.94020i 1.44016 2.49443i 4.40367 + 7.62739i
569.1 2.70320i 0.172975 0.299601i −5.30727 2.82162 + 1.62906i −0.809880 0.467584i 0 8.94020i 1.44016 + 2.49443i 4.40367 7.62739i
569.2 1.38595i 1.41289 2.44719i 0.0791355 0.449430 + 0.259479i −3.39169 1.95819i 0 2.88158i −2.49250 4.31714i 0.359625 0.622889i
569.3 0.120360i 0.291146 0.504280i 1.98551 −1.46199 0.844083i −0.0606950 0.0350423i 0 0.479696i 1.33047 + 2.30444i −0.101594 + 0.175965i
569.4 0.823556i −1.33015 + 2.30388i 1.32176 2.73845 + 1.58105i −1.89737 1.09545i 0 2.73565i −2.03858 3.53092i −1.30208 + 2.25527i
569.5 1.27656i 0.583963 1.01145i 0.370384 1.57173 + 0.907437i 1.29118 + 0.745466i 0 3.02595i 0.817975 + 1.41677i −1.15840 + 2.00641i
569.6 2.10939i −1.13082 + 1.95864i −2.44952 −3.11923 1.80089i −4.13154 2.38535i 0 0.948212i −1.05753 1.83169i 3.79878 6.57967i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 569.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.k.h 12
7.b odd 2 1 637.2.k.g 12
7.c even 3 1 91.2.q.a 12
7.c even 3 1 637.2.u.h 12
7.d odd 6 1 637.2.q.h 12
7.d odd 6 1 637.2.u.i 12
13.e even 6 1 637.2.u.h 12
21.h odd 6 1 819.2.ct.a 12
28.g odd 6 1 1456.2.cc.c 12
91.h even 3 1 1183.2.c.i 12
91.k even 6 1 inner 637.2.k.h 12
91.k even 6 1 1183.2.c.i 12
91.l odd 6 1 637.2.k.g 12
91.p odd 6 1 637.2.q.h 12
91.t odd 6 1 637.2.u.i 12
91.u even 6 1 91.2.q.a 12
91.x odd 12 1 1183.2.a.m 6
91.x odd 12 1 1183.2.a.p 6
91.ba even 12 1 8281.2.a.by 6
91.ba even 12 1 8281.2.a.ch 6
273.x odd 6 1 819.2.ct.a 12
364.s odd 6 1 1456.2.cc.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 7.c even 3 1
91.2.q.a 12 91.u even 6 1
637.2.k.g 12 7.b odd 2 1
637.2.k.g 12 91.l odd 6 1
637.2.k.h 12 1.a even 1 1 trivial
637.2.k.h 12 91.k even 6 1 inner
637.2.q.h 12 7.d odd 6 1
637.2.q.h 12 91.p odd 6 1
637.2.u.h 12 7.c even 3 1
637.2.u.h 12 13.e even 6 1
637.2.u.i 12 7.d odd 6 1
637.2.u.i 12 91.t odd 6 1
819.2.ct.a 12 21.h odd 6 1
819.2.ct.a 12 273.x odd 6 1
1183.2.a.m 6 91.x odd 12 1
1183.2.a.p 6 91.x odd 12 1
1183.2.c.i 12 91.h even 3 1
1183.2.c.i 12 91.k even 6 1
1456.2.cc.c 12 28.g odd 6 1
1456.2.cc.c 12 364.s odd 6 1
8281.2.a.by 6 91.ba even 12 1
8281.2.a.ch 6 91.ba even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{12} + 16 T_{2}^{10} + 88 T_{2}^{8} + 206 T_{2}^{6} + 208 T_{2}^{4} + 72 T_{2}^{2} + 1$$ $$T_{3}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 72 T^{2} + 208 T^{4} + 206 T^{6} + 88 T^{8} + 16 T^{10} + T^{12}$$
$3$ $$16 - 80 T + 300 T^{2} - 516 T^{3} + 709 T^{4} - 390 T^{5} + 287 T^{6} - 42 T^{7} + 96 T^{8} - 4 T^{9} + 11 T^{10} + T^{12}$$
$5$ $$3481 - 13452 T + 17918 T^{2} - 2280 T^{3} - 6370 T^{4} + 1314 T^{5} + 2358 T^{6} - 1116 T^{7} - 6 T^{8} + 84 T^{9} - 2 T^{10} - 6 T^{11} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$256 - 768 T - 80 T^{2} + 2544 T^{3} + 2025 T^{4} - 1494 T^{5} - 771 T^{6} + 522 T^{7} + 248 T^{8} - 114 T^{9} - 7 T^{10} + 6 T^{11} + T^{12}$$
$13$ $$4826809 - 1485172 T + 599781 T^{2} - 70304 T^{3} - 23998 T^{4} + 12012 T^{5} - 6587 T^{6} + 924 T^{7} - 142 T^{8} - 32 T^{9} + 21 T^{10} - 4 T^{11} + T^{12}$$
$17$ $$( -491 - 224 T + 167 T^{2} + 60 T^{3} - 21 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$19$ $$55696 + 138768 T + 137196 T^{2} + 54684 T^{3} - 4499 T^{4} - 9486 T^{5} + 299 T^{6} + 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12}$$
$23$ $$( 6208 - 1472 T - 1616 T^{2} + 608 T^{3} - 20 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$29$ $$10042561 - 1064784 T + 4587524 T^{2} - 3112876 T^{3} + 2323356 T^{4} - 854112 T^{5} + 261878 T^{6} - 47832 T^{7} + 7876 T^{8} - 780 T^{9} + 108 T^{10} - 8 T^{11} + T^{12}$$
$31$ $$913936 + 1640496 T + 829548 T^{2} - 272844 T^{3} - 246191 T^{4} + 61974 T^{5} + 72686 T^{6} + 5256 T^{7} - 2633 T^{8} - 252 T^{9} + 94 T^{10} + 18 T^{11} + T^{12}$$
$37$ $$1755945216 + 749015424 T^{2} + 62699184 T^{4} + 2220264 T^{6} + 38457 T^{8} + 318 T^{10} + T^{12}$$
$41$ $$884705536 - 421175040 T - 8238656 T^{2} + 35739840 T^{3} - 872384 T^{4} - 2742240 T^{5} + 452252 T^{6} + 35760 T^{7} - 11651 T^{8} - 450 T^{9} + 315 T^{10} - 30 T^{11} + T^{12}$$
$43$ $$2408704 + 5860352 T + 10500784 T^{2} + 8862336 T^{3} + 5690569 T^{4} + 1037954 T^{5} + 282645 T^{6} + 3650 T^{7} + 9640 T^{8} + 38 T^{9} + 113 T^{10} - 2 T^{11} + T^{12}$$
$47$ $$9461776 + 44331312 T + 80071996 T^{2} + 50773476 T^{3} + 13560489 T^{4} + 615366 T^{5} - 342366 T^{6} - 7392 T^{7} + 26555 T^{8} + 6636 T^{9} + 746 T^{10} + 42 T^{11} + T^{12}$$
$53$ $$5470921 + 17079378 T + 45476537 T^{2} + 27758206 T^{3} + 16566858 T^{4} - 966678 T^{5} + 629801 T^{6} - 91134 T^{7} + 27034 T^{8} - 3402 T^{9} + 393 T^{10} - 22 T^{11} + T^{12}$$
$59$ $$4571923456 + 977387904 T^{2} + 71782249 T^{4} + 2425880 T^{6} + 40774 T^{8} + 328 T^{10} + T^{12}$$
$61$ $$5607424 - 3788800 T + 7030784 T^{2} - 3685376 T^{3} + 6036160 T^{4} - 2984960 T^{5} + 1823136 T^{6} - 177656 T^{7} + 29281 T^{8} - 1614 T^{9} + 283 T^{10} - 14 T^{11} + T^{12}$$
$67$ $$613651984 + 1248211536 T + 790406444 T^{2} - 113725716 T^{3} - 37845559 T^{4} + 4849356 T^{5} + 1364502 T^{6} - 185988 T^{7} - 11949 T^{8} + 2352 T^{9} + 94 T^{10} - 24 T^{11} + T^{12}$$
$71$ $$46895104 + 26296320 T - 3083264 T^{2} - 4485120 T^{3} + 367104 T^{4} + 620544 T^{5} + 82752 T^{6} - 17280 T^{7} - 3808 T^{8} + 480 T^{9} + 212 T^{10} + 24 T^{11} + T^{12}$$
$73$ $$1386221824 + 922162176 T + 50046272 T^{2} - 102737664 T^{3} - 2173696 T^{4} + 8937792 T^{5} + 1706964 T^{6} + 16440 T^{7} - 19803 T^{8} - 510 T^{9} + 283 T^{10} + 30 T^{11} + T^{12}$$
$79$ $$262144 + 851968 T + 1957888 T^{2} + 2439168 T^{3} + 2298112 T^{4} + 1024000 T^{5} + 325056 T^{6} - 131072 T^{7} + 41152 T^{8} - 5552 T^{9} + 572 T^{10} - 28 T^{11} + T^{12}$$
$83$ $$141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12}$$
$89$ $$1834580224 + 1726865504 T^{2} + 256467377 T^{4} + 12139518 T^{6} + 146499 T^{8} + 658 T^{10} + T^{12}$$
$97$ $$53465344 + 190755456 T + 211235504 T^{2} - 55750056 T^{3} - 14421439 T^{4} + 4289964 T^{5} + 938031 T^{6} - 349092 T^{7} + 28032 T^{8} + 1110 T^{9} - 173 T^{10} - 6 T^{11} + T^{12}$$