# Properties

 Label 637.2.q.g 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: 12.0.2346760387617129.1 Defining polynomial: $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800$$$$)/224$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256$$$$)/224$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288$$$$)/224$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544$$$$)/224$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464$$$$)/112$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664$$$$)/224$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88$$$$)/28$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48$$$$)/28$$ $$\beta_{10}$$ $$=$$ $$($$$$-15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464$$$$)/112$$ $$\beta_{11}$$ $$=$$ $$($$$$-17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056$$$$)/112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{9}$$ $$=$$ $$2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta_{1} + 4$$ $$\nu^{10}$$ $$=$$ $$5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} + 1$$ $$\nu^{11}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta_{1} - 5$$

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1 + \beta_{4}$$ $$1$$

## 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
 1.21245 + 0.727987i 0.874681 − 1.11128i −1.18541 + 0.771231i 0.655911 + 1.25291i −1.38488 − 0.286553i 1.32725 − 0.488273i 1.21245 − 0.727987i 0.874681 + 1.11128i −1.18541 − 0.771231i 0.655911 − 1.25291i −1.38488 + 0.286553i 1.32725 + 0.488273i
−1.99469 + 1.15163i 0.736680 + 1.27597i 1.65252 2.86225i 0.847292i −2.93889 1.69677i 0 3.00585i 0.414604 0.718115i 0.975769 + 1.69008i
491.2 −1.16500 + 0.672613i −1.02505 1.77544i −0.0951832 + 0.164862i 3.56778i 2.38837 + 1.37893i 0 2.94654i −0.601462 + 1.04176i 2.39973 + 4.15646i
491.3 −0.433001 + 0.249993i −0.424801 0.735776i −0.875007 + 1.51556i 1.04248i 0.367878 + 0.212395i 0 1.87496i 1.13909 1.97296i −0.260612 0.451393i
491.4 0.156598 0.0904119i 0.913006 + 1.58137i −0.983651 + 1.70373i 2.68664i 0.285950 + 0.165093i 0 0.717383i −0.167162 + 0.289532i 0.242904 + 0.420723i
491.5 1.19430 0.689527i −1.44060 2.49520i −0.0491037 + 0.0850501i 0.805948i −3.44101 1.98667i 0 2.89354i −2.65067 + 4.59109i 0.555723 + 0.962541i
491.6 2.24179 1.29430i −0.259233 0.449005i 2.35043 4.07106i 1.61205i −1.16229 0.671051i 0 6.99143i 1.36560 2.36528i 2.08648 + 3.61389i
589.1 −1.99469 1.15163i 0.736680 1.27597i 1.65252 + 2.86225i 0.847292i −2.93889 + 1.69677i 0 3.00585i 0.414604 + 0.718115i 0.975769 1.69008i
589.2 −1.16500 0.672613i −1.02505 + 1.77544i −0.0951832 0.164862i 3.56778i 2.38837 1.37893i 0 2.94654i −0.601462 1.04176i 2.39973 4.15646i
589.3 −0.433001 0.249993i −0.424801 + 0.735776i −0.875007 1.51556i 1.04248i 0.367878 0.212395i 0 1.87496i 1.13909 + 1.97296i −0.260612 + 0.451393i
589.4 0.156598 + 0.0904119i 0.913006 1.58137i −0.983651 1.70373i 2.68664i 0.285950 0.165093i 0 0.717383i −0.167162 0.289532i 0.242904 0.420723i
589.5 1.19430 + 0.689527i −1.44060 + 2.49520i −0.0491037 0.0850501i 0.805948i −3.44101 + 1.98667i 0 2.89354i −2.65067 4.59109i 0.555723 0.962541i
589.6 2.24179 + 1.29430i −0.259233 + 0.449005i 2.35043 + 4.07106i 1.61205i −1.16229 + 0.671051i 0 6.99143i 1.36560 + 2.36528i 2.08648 3.61389i
 $$n$$: 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.g 12
7.b odd 2 1 637.2.q.i 12
7.c even 3 1 91.2.k.b 12
7.c even 3 1 91.2.u.b yes 12
7.d odd 6 1 637.2.k.i 12
7.d odd 6 1 637.2.u.g 12
13.e even 6 1 inner 637.2.q.g 12
13.f odd 12 2 8281.2.a.cp 12
21.h odd 6 1 819.2.bm.f 12
21.h odd 6 1 819.2.do.e 12
91.k even 6 1 91.2.u.b yes 12
91.l odd 6 1 637.2.u.g 12
91.p odd 6 1 637.2.k.i 12
91.t odd 6 1 637.2.q.i 12
91.u even 6 1 91.2.k.b 12
91.x odd 12 2 1183.2.e.j 24
91.bc even 12 2 8281.2.a.co 12
91.bd odd 12 2 1183.2.e.j 24
273.x odd 6 1 819.2.bm.f 12
273.bp odd 6 1 819.2.do.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 7.c even 3 1
91.2.k.b 12 91.u even 6 1
91.2.u.b yes 12 7.c even 3 1
91.2.u.b yes 12 91.k even 6 1
637.2.k.i 12 7.d odd 6 1
637.2.k.i 12 91.p odd 6 1
637.2.q.g 12 1.a even 1 1 trivial
637.2.q.g 12 13.e even 6 1 inner
637.2.q.i 12 7.b odd 2 1
637.2.q.i 12 91.t odd 6 1
637.2.u.g 12 7.d odd 6 1
637.2.u.g 12 91.l odd 6 1
819.2.bm.f 12 21.h odd 6 1
819.2.bm.f 12 273.x odd 6 1
819.2.do.e 12 21.h odd 6 1
819.2.do.e 12 273.bp odd 6 1
1183.2.e.j 24 91.x odd 12 2
1183.2.e.j 24 91.bd odd 12 2
8281.2.a.co 12 91.bc even 12 2
8281.2.a.cp 12 13.f odd 12 2

## 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

$p$ $F_p(T)$
$2$ $$1 - 6 T + 72 T^{3} + 130 T^{4} - 36 T^{5} - 91 T^{6} + 18 T^{7} + 52 T^{8} - 8 T^{10} + T^{12}$$
$3$ $$49 + 133 T + 333 T^{2} + 300 T^{3} + 355 T^{4} + 147 T^{5} + 233 T^{6} + 75 T^{7} + 69 T^{8} + 17 T^{9} + 14 T^{10} + 3 T^{11} + T^{12}$$
$5$ $$121 + 539 T^{2} + 878 T^{4} + 636 T^{6} + 201 T^{8} + 25 T^{10} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$85849 + 83505 T - 20684 T^{2} - 46455 T^{3} + 9240 T^{4} + 29736 T^{5} + 13287 T^{6} + 1143 T^{7} - 604 T^{8} - 84 T^{9} + 41 T^{10} + 12 T^{11} + T^{12}$$
$13$ $$4826809 + 742586 T - 514098 T^{2} - 37349 T^{3} + 57629 T^{4} - 819 T^{5} - 6395 T^{6} - 63 T^{7} + 341 T^{8} - 17 T^{9} - 18 T^{10} + 2 T^{11} + T^{12}$$
$17$ $$361 + 2774 T + 20252 T^{2} + 15700 T^{3} + 30220 T^{4} - 39443 T^{5} + 36348 T^{6} - 16958 T^{7} + 5794 T^{8} - 1236 T^{9} + 193 T^{10} - 17 T^{11} + T^{12}$$
$19$ $$1 + 48 T + 915 T^{2} + 7056 T^{3} + 25855 T^{4} + 39258 T^{5} + 27731 T^{6} + 6990 T^{7} + 22 T^{8} - 234 T^{9} + T^{10} + 9 T^{11} + T^{12}$$
$23$ $$628849 - 512278 T + 564021 T^{2} - 291264 T^{3} + 241189 T^{4} - 114894 T^{5} + 57479 T^{6} - 14706 T^{7} + 3462 T^{8} - 368 T^{9} + 59 T^{10} - 3 T^{11} + T^{12}$$
$29$ $$16072081 + 20205360 T + 19636658 T^{2} + 8770940 T^{3} + 3370218 T^{4} + 597669 T^{5} + 162746 T^{6} + 18504 T^{7} + 6148 T^{8} + 294 T^{9} + 87 T^{10} + T^{11} + T^{12}$$
$31$ $$241274089 + 82568820 T^{2} + 10607464 T^{4} + 639617 T^{6} + 18424 T^{8} + 232 T^{10} + T^{12}$$
$37$ $$123201 + 151632 T - 92583 T^{2} - 190512 T^{3} + 109917 T^{4} + 292410 T^{5} + 164889 T^{6} + 23868 T^{7} - 1638 T^{8} - 540 T^{9} + 39 T^{10} + 15 T^{11} + T^{12}$$
$41$ $$389707081 - 591933885 T + 198922270 T^{2} + 153073425 T^{3} + 9911704 T^{4} - 6405744 T^{5} - 349015 T^{6} + 188553 T^{7} + 21580 T^{8} - 1026 T^{9} - 159 T^{10} + 6 T^{11} + T^{12}$$
$43$ $$418898089 - 158496448 T + 88152595 T^{2} - 22738656 T^{3} + 9218116 T^{4} - 2107681 T^{5} + 554133 T^{6} - 78022 T^{7} + 12754 T^{8} - 1093 T^{9} + 170 T^{10} - 11 T^{11} + T^{12}$$
$47$ $$121 + 1243 T^{2} + 2634 T^{4} + 1968 T^{6} + 509 T^{8} + 41 T^{10} + T^{12}$$
$53$ $$( -17 - 264 T - 611 T^{2} + 404 T^{3} - 38 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$59$ $$35582408689 + 40868659881 T + 15977236899 T^{2} + 379583064 T^{3} - 342075677 T^{4} - 14175459 T^{5} + 6359213 T^{6} + 586863 T^{7} - 24383 T^{8} - 4185 T^{9} + 88 T^{10} + 27 T^{11} + T^{12}$$
$61$ $$3157729 - 8574025 T + 21325925 T^{2} - 6565616 T^{3} + 3051325 T^{4} - 343235 T^{5} + 187245 T^{6} - 20375 T^{7} + 6295 T^{8} - 333 T^{9} + 100 T^{10} - 5 T^{11} + T^{12}$$
$67$ $$5708255809 - 3991238331 T + 441704945 T^{2} + 341579382 T^{3} - 11086297 T^{4} - 13240623 T^{5} + 357402 T^{6} + 351759 T^{7} + 15543 T^{8} - 2730 T^{9} - 107 T^{10} + 15 T^{11} + T^{12}$$
$71$ $$639230089 - 907078191 T + 408851926 T^{2} + 28665723 T^{3} - 35115036 T^{4} - 1753566 T^{5} + 2943903 T^{6} - 193635 T^{7} - 25312 T^{8} + 2190 T^{9} + 227 T^{10} - 30 T^{11} + T^{12}$$
$73$ $$484396081 + 886799840 T^{2} + 87039596 T^{4} + 3205221 T^{6} + 53832 T^{8} + 400 T^{10} + T^{12}$$
$79$ $$( 255121 + 36737 T - 20673 T^{2} + 504 T^{3} + 344 T^{4} - 35 T^{5} + T^{6} )^{2}$$
$83$ $$402363481 + 194879694 T^{2} + 33361345 T^{4} + 2359793 T^{6} + 59836 T^{8} + 463 T^{10} + T^{12}$$
$89$ $$145033849 + 341491308 T + 348395894 T^{2} + 189247944 T^{3} + 54744071 T^{4} + 6344400 T^{5} - 401292 T^{6} - 117660 T^{7} + 21411 T^{8} + 8112 T^{9} + 937 T^{10} + 48 T^{11} + T^{12}$$
$97$ $$1681 - 8364 T + 8255 T^{2} + 27948 T^{3} - 34684 T^{4} - 122601 T^{5} + 291057 T^{6} - 159822 T^{7} + 33072 T^{8} - 537 T^{9} - 176 T^{10} + 3 T^{11} + T^{12}$$