# Properties

 Label 637.2.q.h Level $637$ Weight $2$ Character orbit 637.q 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.q (of order $$6$$, degree $$2$$, 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.

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

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 $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 −1.12906 − 0.851598i 0.759479 − 1.19298i 1.34408 + 0.439820i −1.08105 + 0.911778i 1.40744 + 0.138282i −1.30089 + 0.554694i −1.12906 + 0.851598i 0.759479 + 1.19298i 1.34408 − 0.439820i −1.08105 − 0.911778i 1.40744 − 0.138282i −1.30089 − 0.554694i
−2.34104 + 1.35160i −0.172975 0.299601i 2.65363 4.59623i 3.25812i 0.809880 + 0.467584i 0 8.94020i 1.44016 2.49443i −4.40367 7.62739i
491.2 −1.20027 + 0.692976i −1.41289 2.44719i −0.0395678 + 0.0685334i 0.518957i 3.39169 + 1.95819i 0 2.88158i −2.49250 + 4.31714i −0.359625 0.622889i
491.3 −0.104235 + 0.0601799i −0.291146 0.504280i −0.992757 + 1.71951i 1.68817i 0.0606950 + 0.0350423i 0 0.479696i 1.33047 2.30444i 0.101594 + 0.175965i
491.4 0.713220 0.411778i 1.33015 + 2.30388i −0.660878 + 1.14467i 3.16209i 1.89737 + 1.09545i 0 2.73565i −2.03858 + 3.53092i 1.30208 + 2.25527i
491.5 1.10554 0.638282i −0.583963 1.01145i −0.185192 + 0.320762i 1.81487i −1.29118 0.745466i 0 3.02595i 0.817975 1.41677i 1.15840 + 2.00641i
491.6 1.82678 1.05469i 1.13082 + 1.95864i 1.22476 2.12135i 3.60178i 4.13154 + 2.38535i 0 0.948212i −1.05753 + 1.83169i −3.79878 6.57967i
589.1 −2.34104 1.35160i −0.172975 + 0.299601i 2.65363 + 4.59623i 3.25812i 0.809880 0.467584i 0 8.94020i 1.44016 + 2.49443i −4.40367 + 7.62739i
589.2 −1.20027 0.692976i −1.41289 + 2.44719i −0.0395678 0.0685334i 0.518957i 3.39169 1.95819i 0 2.88158i −2.49250 4.31714i −0.359625 + 0.622889i
589.3 −0.104235 0.0601799i −0.291146 + 0.504280i −0.992757 1.71951i 1.68817i 0.0606950 0.0350423i 0 0.479696i 1.33047 + 2.30444i 0.101594 0.175965i
589.4 0.713220 + 0.411778i 1.33015 2.30388i −0.660878 1.14467i 3.16209i 1.89737 1.09545i 0 2.73565i −2.03858 3.53092i 1.30208 2.25527i
589.5 1.10554 + 0.638282i −0.583963 + 1.01145i −0.185192 0.320762i 1.81487i −1.29118 + 0.745466i 0 3.02595i 0.817975 + 1.41677i 1.15840 2.00641i
589.6 1.82678 + 1.05469i 1.13082 1.95864i 1.22476 + 2.12135i 3.60178i 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 589.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.q.h 12
7.b odd 2 1 91.2.q.a 12
7.c even 3 1 637.2.k.g 12
7.c even 3 1 637.2.u.i 12
7.d odd 6 1 637.2.k.h 12
7.d odd 6 1 637.2.u.h 12
13.e even 6 1 inner 637.2.q.h 12
13.f odd 12 1 8281.2.a.by 6
13.f odd 12 1 8281.2.a.ch 6
21.c even 2 1 819.2.ct.a 12
28.d even 2 1 1456.2.cc.c 12
91.k even 6 1 637.2.u.i 12
91.l odd 6 1 637.2.u.h 12
91.n odd 6 1 1183.2.c.i 12
91.p odd 6 1 637.2.k.h 12
91.t odd 6 1 91.2.q.a 12
91.t odd 6 1 1183.2.c.i 12
91.u even 6 1 637.2.k.g 12
91.bc even 12 1 1183.2.a.m 6
91.bc even 12 1 1183.2.a.p 6
273.u even 6 1 819.2.ct.a 12
364.bc even 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.b odd 2 1
91.2.q.a 12 91.t odd 6 1
637.2.k.g 12 7.c even 3 1
637.2.k.g 12 91.u even 6 1
637.2.k.h 12 7.d odd 6 1
637.2.k.h 12 91.p odd 6 1
637.2.q.h 12 1.a even 1 1 trivial
637.2.q.h 12 13.e even 6 1 inner
637.2.u.h 12 7.d odd 6 1
637.2.u.h 12 91.l odd 6 1
637.2.u.i 12 7.c even 3 1
637.2.u.i 12 91.k even 6 1
819.2.ct.a 12 21.c even 2 1
819.2.ct.a 12 273.u even 6 1
1183.2.a.m 6 91.bc even 12 1
1183.2.a.p 6 91.bc even 12 1
1183.2.c.i 12 91.n odd 6 1
1183.2.c.i 12 91.t odd 6 1
1456.2.cc.c 12 28.d even 2 1
1456.2.cc.c 12 364.bc even 6 1
8281.2.a.by 6 13.f odd 12 1
8281.2.a.ch 6 13.f odd 12 1

## Hecke kernels

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

 $$T_{2}^{12} - \cdots$$ $$T_{3}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 12 T + 36 T^{2} - 144 T^{3} + 112 T^{4} + 72 T^{5} - 82 T^{6} - 36 T^{7} + 52 T^{8} - 8 T^{10} + T^{12}$$
$3$ $$16 + 80 T + 300 T^{2} + 516 T^{3} + 709 T^{4} + 390 T^{5} + 287 T^{6} + 42 T^{7} + 96 T^{8} + 4 T^{9} + 11 T^{10} + T^{12}$$
$5$ $$3481 + 16148 T^{2} + 13040 T^{4} + 4146 T^{6} + 600 T^{8} + 40 T^{10} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$256 + 768 T - 80 T^{2} - 2544 T^{3} + 2025 T^{4} + 1494 T^{5} - 771 T^{6} - 522 T^{7} + 248 T^{8} + 114 T^{9} - 7 T^{10} - 6 T^{11} + T^{12}$$
$13$ $$4826809 + 1485172 T + 599781 T^{2} + 70304 T^{3} - 23998 T^{4} - 12012 T^{5} - 6587 T^{6} - 924 T^{7} - 142 T^{8} + 32 T^{9} + 21 T^{10} + 4 T^{11} + T^{12}$$
$17$ $$241081 + 109984 T + 132173 T^{2} + 21512 T^{3} + 31018 T^{4} + 2576 T^{5} + 5229 T^{6} - 148 T^{7} + 514 T^{8} - 36 T^{9} + 37 T^{10} - 4 T^{11} + T^{12}$$
$19$ $$55696 + 138768 T + 137196 T^{2} + 54684 T^{3} - 4499 T^{4} - 9486 T^{5} + 299 T^{6} + 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12}$$
$23$ $$38539264 + 9138176 T + 12198912 T^{2} + 5170176 T^{3} + 3630592 T^{4} + 1115904 T^{5} + 332096 T^{6} + 52416 T^{7} + 9312 T^{8} + 976 T^{9} + 164 T^{10} + 12 T^{11} + T^{12}$$
$29$ $$10042561 - 1064784 T + 4587524 T^{2} - 3112876 T^{3} + 2323356 T^{4} - 854112 T^{5} + 261878 T^{6} - 47832 T^{7} + 7876 T^{8} - 780 T^{9} + 108 T^{10} - 8 T^{11} + T^{12}$$
$31$ $$913936 + 1285560 T^{2} + 568225 T^{4} + 96896 T^{6} + 5854 T^{8} + 136 T^{10} + T^{12}$$
$37$ $$1755945216 - 633588480 T - 260200512 T^{2} + 121383360 T^{3} + 65323584 T^{4} + 8644320 T^{5} - 396036 T^{6} - 159840 T^{7} + 8109 T^{8} + 5670 T^{9} + 723 T^{10} + 42 T^{11} + T^{12}$$
$41$ $$884705536 + 421175040 T - 8238656 T^{2} - 35739840 T^{3} - 872384 T^{4} + 2742240 T^{5} + 452252 T^{6} - 35760 T^{7} - 11651 T^{8} + 450 T^{9} + 315 T^{10} + 30 T^{11} + T^{12}$$
$43$ $$2408704 + 5860352 T + 10500784 T^{2} + 8862336 T^{3} + 5690569 T^{4} + 1037954 T^{5} + 282645 T^{6} + 3650 T^{7} + 9640 T^{8} + 38 T^{9} + 113 T^{10} - 2 T^{11} + T^{12}$$
$47$ $$9461776 + 47561752 T^{2} + 10113609 T^{4} + 722232 T^{6} + 21782 T^{8} + 272 T^{10} + T^{12}$$
$53$ $$( -2339 + 7302 T - 3353 T^{2} - 700 T^{3} + 91 T^{4} + 22 T^{5} + T^{6} )^{2}$$
$59$ $$4571923456 + 3919023360 T + 1190986848 T^{2} + 61031880 T^{3} - 34227911 T^{4} - 3548430 T^{5} + 1035110 T^{6} + 185580 T^{7} - 2237 T^{8} - 1980 T^{9} - 2 T^{10} + 18 T^{11} + T^{12}$$
$61$ $$5607424 + 3788800 T + 7030784 T^{2} + 3685376 T^{3} + 6036160 T^{4} + 2984960 T^{5} + 1823136 T^{6} + 177656 T^{7} + 29281 T^{8} + 1614 T^{9} + 283 T^{10} + 14 T^{11} + T^{12}$$
$67$ $$613651984 - 1248211536 T + 790406444 T^{2} + 113725716 T^{3} - 37845559 T^{4} - 4849356 T^{5} + 1364502 T^{6} + 185988 T^{7} - 11949 T^{8} - 2352 T^{9} + 94 T^{10} + 24 T^{11} + T^{12}$$
$71$ $$46895104 + 26296320 T - 3083264 T^{2} - 4485120 T^{3} + 367104 T^{4} + 620544 T^{5} + 82752 T^{6} - 17280 T^{7} - 3808 T^{8} + 480 T^{9} + 212 T^{10} + 24 T^{11} + T^{12}$$
$73$ $$1386221824 + 513361280 T^{2} + 55965104 T^{4} + 2238456 T^{6} + 40473 T^{8} + 334 T^{10} + T^{12}$$
$79$ $$( -512 + 1664 T - 1584 T^{2} + 192 T^{3} + 212 T^{4} + 28 T^{5} + T^{6} )^{2}$$
$83$ $$141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12}$$
$89$ $$1834580224 - 1271596416 T - 422744080 T^{2} + 496650552 T^{3} + 291555473 T^{4} + 41341974 T^{5} - 2811261 T^{6} - 792798 T^{7} + 66288 T^{8} + 3660 T^{9} - 257 T^{10} - 12 T^{11} + T^{12}$$
$97$ $$53465344 - 190755456 T + 211235504 T^{2} + 55750056 T^{3} - 14421439 T^{4} - 4289964 T^{5} + 938031 T^{6} + 349092 T^{7} + 28032 T^{8} - 1110 T^{9} - 173 T^{10} + 6 T^{11} + T^{12}$$