# 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: $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{8} - \beta_{4}) q^{2} + ( - \beta_{9} - \beta_{2}) q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} - \beta_{5} - \beta_{2} - 1) q^{4} + ( - \beta_{11} - \beta_{8} - \beta_{5} - \beta_{3}) q^{5} + (\beta_{6} + \beta_1 - 1) q^{6} + ( - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{8}+ \cdots + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b8 - b4) * q^2 + (-b9 - b2) * q^3 + (b11 - b10 - b8 - b5 - b2 - 1) * q^4 + (-b11 - b8 - b5 - b3) * q^5 + (b6 + b1 - 1) * q^6 + (-2*b9 - 2*b8 - b7 - 2*b6 - b5 + b4 + b3 - b2 + b1 - 1) * q^8 + (b11 + b10 - b9 + b8 - b7 + b5 + b4 + b3 + 2*b1) * q^9 $$q + (\beta_{8} - \beta_{4}) q^{2} + ( - \beta_{9} - \beta_{2}) q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} - \beta_{5} - \beta_{2} - 1) q^{4} + ( - \beta_{11} - \beta_{8} - \beta_{5} - \beta_{3}) q^{5} + (\beta_{6} + \beta_1 - 1) q^{6} + ( - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{8}+ \cdots + ( - 4 \beta_{11} - 4 \beta_{10} - 5 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + \cdots - 3) q^{99}+O(q^{100})$$ q + (b8 - b4) * q^2 + (-b9 - b2) * q^3 + (b11 - b10 - b8 - b5 - b2 - 1) * q^4 + (-b11 - b8 - b5 - b3) * q^5 + (b6 + b1 - 1) * q^6 + (-2*b9 - 2*b8 - b7 - 2*b6 - b5 + b4 + b3 - b2 + b1 - 1) * q^8 + (b11 + b10 - b9 + b8 - b7 + b5 + b4 + b3 + 2*b1) * q^9 + (-2*b11 - b10 + 2*b9 + b8 + 3*b6 + b5 - 2*b4 - b3 + 2*b2 + 3) * q^10 + (b10 + b8 + b5 + b4 + b3 + b1) * q^11 + (-b11 - b10 - 2*b8 + b6 + 1) * q^12 + (-b11 - 2*b10 + b9 - b8 + b7 + b6 - b5 + b2 - b1 + 1) * q^13 + (-2*b11 - b8 + b6 + 2*b5 + 2) * q^15 + (3*b8 + b7 + b5 + b3 + 3*b2 + b1 + 2) * q^16 + (b11 - b10 - b4 + b3 + b2 + 1) * q^17 + (-b11 + 2*b9 + 2*b8 + b7 + b6 + b5 - 2*b4 + 4*b2 + 2) * q^18 + (-b11 + b10 + b7 + b5 - b3) * q^19 + (-b10 + b9 - b8 + b6 - b5 - 4*b4 - b3 - b2 - 2*b1 - 1) * q^20 + (2*b9 + 2*b8 + b6 - b4 + 2*b2 + 1) * q^22 + (2*b11 - 2*b10 - b7 - b5 + b4 + b3 - b2 - b1 + 2) * q^23 + (2*b10 - b9 + 2*b8 + 2*b5 + 2*b3 + b2 + 2*b1) * q^24 + (b10 - 3*b9 - 2*b6 - b4 + b3) * q^25 + (-b11 + b9 + b8 + 4*b6 + 2*b5 - b4 - b3 - b1 + 4) * q^26 + (b7 - b4 + b2 + b1 + 1) * q^27 + (-3*b11 - b10 + b9 - 3*b8 + b7 - 3*b5 - b4 - b3 - 2*b1) * q^29 + (2*b11 + 3*b10 - b9 + 5*b8 - 3*b6 + 2*b5 - b4 + 3*b3) * q^30 + (b11 - b10 + 2*b7 - b6 - b5 + b3 - 2) * q^31 + (-3*b11 - 3*b10 + 4*b9 + b8 + b7 + 6*b6 - 5*b4 - 3*b3 + 2*b2 - b1 + 3) * q^32 + (b11 + b10 + b9 + b6 - b5 - b4 - b3 + 2*b2 + 2) * q^33 + (b11 + b10 - 2*b9 + b7 - 4*b6 + b4 + b3 - b2 - b1 - 2) * q^34 + (2*b11 + 2*b9 + 4*b8 + b6 + 2*b5 - 2*b4) * q^36 + (b11 + b10 + 2*b8 - 3*b7 + 4*b6 + 2*b5 - b3 + 3*b1 + 2) * q^37 + (-b11 - b10 + 3*b9 + b6 - b5 - b4 - b3) * q^38 + (b10 + b9 + 2*b8 - b7 + b5 + 2*b4 + 2*b3 - b2 - b1 + 2) * q^39 + (2*b11 - b10 - 5*b9 - 7*b8 - 2*b7 - 8*b6 - 3*b5 + 4*b4 + 3*b3 - 5*b2 + b1 - 8) * q^40 + (-3*b11 + 2*b10 - b8 - b7 + 2*b6 + 3*b5 - 2*b3 + 4) * q^41 + (2*b11 + 2*b10 - 2*b9 - 2*b8 - b6 + 3*b4 - 2*b2 - 1) * q^43 + (b11 + b9 + b8 + b6 + b5 + b3 - b2 - 1) * q^44 + (3*b11 + 3*b10 - 2*b9 + 2*b8 - b7 - 2*b6 - b5 + 3*b4 + 4*b3 - b2 + b1 - 1) * q^45 + (2*b11 + 2*b10 - 6*b9 - b7 + b5 + b4 + b3 - 3*b2 + b1) * q^46 + (-2*b11 + b9 - 2*b8 + 3*b6 - 2*b5 - 2*b3 - b2 - 3*b1 - 3) * q^47 + (-2*b11 - 2*b10 + 4*b9 + 2*b7 + b6 - 2*b4 + 4*b2 - b1 + 1) * q^48 + (-b11 + b10 + b9 + 5*b8 + 3*b7 - b6 + b5 - b4 - b3 + 2*b2 - 2) * q^50 + (-b10 + b8 + 2*b7 - 4*b6 - b5 - b4 + b3 - b1 - 4) * q^51 + (2*b11 - 3*b10 - b9 - 3*b8 + b7 - b6 - 3*b5 - 2*b4 + 2*b3 - 3*b2 - 2*b1 - 3) * q^52 + (-2*b11 + 2*b9 - 2*b8 - 3*b6 - 2*b5 + 2*b4) * q^53 + (-2*b11 - 2*b10 + 4*b9 + b7 - b5 - 2*b4 - b3 + 2*b2 - b1) * q^54 + (2*b11 + 4*b10 + 4*b8 - b7 - b6 + 2*b5 + 4*b4 + 4*b3 + 2*b1) * q^55 + (2*b9 - b8 + b7 + 2*b6 + b5 + 2*b4 - b3 + b2 - b1 + 1) * q^57 + (-b11 + 2*b8 - b7 + b6 + b5 + 2) * q^58 + (4*b9 + 2*b8 + 2*b6 + 3*b5 - 3*b3 + 2*b2 + 1) * q^59 + (b11 - 2*b9 + b7 - 4*b6 - b5 + 2*b4 - 4*b2 - 8) * q^60 + (b11 - 2*b10 + 4*b9 + b8 + b7 - 2*b6 + b5 - 2*b3 - 2*b1) * q^61 + (-2*b11 - 2*b10 + 3*b9 - 4*b8 + 2*b6 - 2*b5 + 3*b4 - 2*b3) * q^62 + (4*b11 - 4*b10 - 2*b8 - 2*b4 + 4*b3 - 3) * q^64 + (-b11 + 2*b10 - 3*b9 + 2*b8 + b7 - 6*b6 + 2*b5 + b4 - b1 - 4) * q^65 + (-2*b11 - 2*b10 + b9 - 2*b8 + b7 - 3*b6 - 2*b5 - 3*b4 - 2*b3 - 2*b1) * q^66 + (3*b11 - 4*b10 + b9 - b8 - b6 - b5 + 2*b4 - b3 - b2 + 1) * q^67 + (b11 - b10 + 2*b8 + b7 - b5 + b4 + b2 + b1 + 1) * q^68 + (2*b11 - 6*b9 + 2*b8 - 4*b7 - 2*b5 + 2*b3 - 6*b2 + 2*b1) * q^69 + (-2*b11 - 2*b8 + 2*b6 - 2*b5 - 2*b4 - 2*b3 - 2) * q^71 + (4*b11 - 2*b10 - 5*b8 - 4*b6 - 4*b5 + 2*b3 - 8) * q^72 + (b11 + 2*b9 + 3*b8 + b7 - 2*b6 - b5 - 2*b4 + 4*b2 - 4) * q^73 + (b11 - b10 - 2*b8 + b5 - 5*b4 + 2*b3 + 3*b2) * q^74 + (2*b8 + 2*b7 + b5 + b4 + b3 + b2 + 2*b1 - 7) * q^75 + (-b9 - 4*b8 - b7 - b6 + b4 - 2*b2 - 2) * q^76 + (5*b11 + 3*b10 - 5*b9 + 5*b8 - 2*b7 - b6 + 2*b3 - 3*b2 + b1 - 2) * q^78 + (b11 + b10 - b9 - 2*b8 + 2*b7 + 4*b6 + 2*b4 - b2 - b1 + 4) * q^79 + (-3*b11 + 6*b10 - 5*b9 + 3*b8 - 5*b6 + 3*b5 + 4*b4 + 3*b3 + 5*b2 + b1 + 5) * q^80 + (b10 - 3*b8 - 2*b7 + b5 + 2*b4 - b3 + b1) * q^81 + (2*b11 + 3*b10 - b9 + 7*b8 - 4*b6 + 2*b5 - 5*b4 + 3*b3) * q^82 + (-2*b11 - 2*b10 + 4*b9 + 2*b8 + 2*b7 - 2*b6 - b5 - 4*b4 - b3 + 2*b2 - 2*b1 - 1) * q^83 + (-b11 - 3*b10 + b9 - 4*b8 + 4*b6 - 4*b5 - b4 - 4*b3 - b2 - b1 - 4) * q^85 + (-3*b11 + 2*b10 - b9 - b8 - 3*b6 - b5 + 2*b4 - b3 + b2 + 2*b1 + 3) * q^86 + (-2*b11 + 2*b10 - b7 + 2*b5 + 4*b4 - 4*b2 - b1 + 1) * q^87 + (4*b11 + b10 - b8 - 2*b7 - 4*b6 - 3*b5 + 3*b4 + 3*b3 + b1 - 4) * q^88 + (2*b11 + 2*b10 - 8*b9 - 4*b8 - b7 + b5 + 4*b4 + b3 - 4*b2 + b1) * q^89 + (-b11 + b10 + 4*b8 + b7 - b5 - 2*b3 + 4*b2 + b1 + 1) * q^90 + (2*b11 - 2*b10 + 2*b8 + b7 - b5 + b4 + b3 + b2 + b1 + 2) * q^92 + (-3*b11 - 3*b10 + 6*b9 - 5*b8 + 2*b7 + 4*b6 - b5 + 2*b4 - 2*b3 + 3*b2 - 2*b1 + 2) * q^93 + (b11 + b10 - 4*b9 - 6*b8 - 2*b7 + 4*b4 - 4*b2 + b1) * q^94 + (-2*b11 + 2*b10 + b8 + b7 + 3*b5 + b3 + b2 + b1 + 5) * q^95 + (b11 + 3*b10 + 4*b8 - b6 + 4*b5 + 4*b3 + 1) * q^96 + (3*b11 - b10 - b9 + 2*b8 - b6 + 2*b5 + 4*b4 + 2*b3 + b2 + 1) * q^97 + (-4*b11 - 4*b10 - 5*b8 + 2*b7 - 6*b6 - 2*b5 - 3*b4 - 2*b3 - 2*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{4} + 6 q^{5} - 18 q^{6} - 4 q^{9}+O(q^{10})$$ 12 * q - 8 * q^4 + 6 * q^5 - 18 * q^6 - 4 * q^9 $$12 q - 8 q^{4} + 6 q^{5} - 18 q^{6} - 4 q^{9} + 12 q^{10} - 6 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{15} + 16 q^{16} + 8 q^{17} + 12 q^{18} - 12 q^{20} + 6 q^{22} + 24 q^{23} - 12 q^{24} + 10 q^{25} + 18 q^{26} + 12 q^{27} + 8 q^{29} + 8 q^{30} - 18 q^{31} + 30 q^{33} - 10 q^{36} - 2 q^{38} + 14 q^{39} - 46 q^{40} + 30 q^{41} + 2 q^{43} - 24 q^{44} - 42 q^{47} - 2 q^{48} - 18 q^{50} - 26 q^{51} - 28 q^{52} + 22 q^{53} - 6 q^{55} + 12 q^{58} - 66 q^{60} + 14 q^{61} - 4 q^{62} - 52 q^{64} - 18 q^{65} + 26 q^{66} + 24 q^{67} + 16 q^{68} + 4 q^{69} - 24 q^{71} - 60 q^{72} - 30 q^{73} - 12 q^{74} - 92 q^{75} - 18 q^{76} - 10 q^{78} + 28 q^{79} + 72 q^{80} + 2 q^{81} + 14 q^{82} - 48 q^{85} + 60 q^{86} + 4 q^{87} - 14 q^{88} + 24 q^{90} + 24 q^{92} + 4 q^{94} + 44 q^{95} - 6 q^{96} + 6 q^{97}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{10} - 2\nu^{9} - 7\nu^{8} + 4\nu^{7} + 17\nu^{6} - 24\nu^{5} - 14\nu^{4} + 40\nu^{3} + 36\nu^{2} - 40\nu ) / 16$$ (3*v^10 - 2*v^9 - 7*v^8 + 4*v^7 + 17*v^6 - 24*v^5 - 14*v^4 + 40*v^3 + 36*v^2 - 40*v) / 16 $$\beta_{2}$$ $$=$$ $$( - \nu^{11} + 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} - 9 \nu^{7} + 24 \nu^{6} + 4 \nu^{5} - 44 \nu^{4} + 8 \nu^{3} + 72 \nu^{2} - 40 \nu - 48 ) / 16$$ (-v^11 + 2*v^10 + 3*v^9 - 8*v^8 - 9*v^7 + 24*v^6 + 4*v^5 - 44*v^4 + 8*v^3 + 72*v^2 - 40*v - 48) / 16 $$\beta_{3}$$ $$=$$ $$( 2 \nu^{11} + \nu^{10} - 6 \nu^{9} - 5 \nu^{8} + 16 \nu^{7} - \nu^{6} - 30 \nu^{5} + 6 \nu^{4} + 52 \nu^{3} - 4 \nu^{2} - 32 \nu - 16 ) / 16$$ (2*v^11 + v^10 - 6*v^9 - 5*v^8 + 16*v^7 - v^6 - 30*v^5 + 6*v^4 + 52*v^3 - 4*v^2 - 32*v - 16) / 16 $$\beta_{4}$$ $$=$$ $$( - \nu^{11} + 4 \nu^{10} - 3 \nu^{9} - 10 \nu^{8} + 9 \nu^{7} + 26 \nu^{6} - 42 \nu^{5} - 12 \nu^{4} + 60 \nu^{3} + 8 \nu^{2} - 96 \nu + 48 ) / 16$$ (-v^11 + 4*v^10 - 3*v^9 - 10*v^8 + 9*v^7 + 26*v^6 - 42*v^5 - 12*v^4 + 60*v^3 + 8*v^2 - 96*v + 48) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{11} - 6 \nu^{10} - 9 \nu^{9} + 20 \nu^{8} + 31 \nu^{7} - 56 \nu^{6} - 38 \nu^{5} + 136 \nu^{4} + 28 \nu^{3} - 232 \nu^{2} - 32 \nu + 192 ) / 32$$ (v^11 - 6*v^10 - 9*v^9 + 20*v^8 + 31*v^7 - 56*v^6 - 38*v^5 + 136*v^4 + 28*v^3 - 232*v^2 - 32*v + 192) / 32 $$\beta_{6}$$ $$=$$ $$( \nu^{11} - 3 \nu^{10} - 5 \nu^{9} + 13 \nu^{8} + 13 \nu^{7} - 35 \nu^{6} - 12 \nu^{5} + 70 \nu^{4} - 8 \nu^{3} - 108 \nu^{2} + 16 \nu + 80 ) / 16$$ (v^11 - 3*v^10 - 5*v^9 + 13*v^8 + 13*v^7 - 35*v^6 - 12*v^5 + 70*v^4 - 8*v^3 - 108*v^2 + 16*v + 80) / 16 $$\beta_{7}$$ $$=$$ $$( - 5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 20 \nu^{8} - 27 \nu^{7} - 32 \nu^{6} + 46 \nu^{5} + 64 \nu^{4} - 124 \nu^{3} - 136 \nu^{2} + 128 \nu + 224 ) / 32$$ (-5*v^11 - 2*v^10 + 13*v^9 + 20*v^8 - 27*v^7 - 32*v^6 + 46*v^5 + 64*v^4 - 124*v^3 - 136*v^2 + 128*v + 224) / 32 $$\beta_{8}$$ $$=$$ $$( 2 \nu^{11} + 3 \nu^{10} - 14 \nu^{9} - 7 \nu^{8} + 36 \nu^{7} + 5 \nu^{6} - 82 \nu^{5} + 34 \nu^{4} + 124 \nu^{3} - 60 \nu^{2} - 128 \nu + 64 ) / 16$$ (2*v^11 + 3*v^10 - 14*v^9 - 7*v^8 + 36*v^7 + 5*v^6 - 82*v^5 + 34*v^4 + 124*v^3 - 60*v^2 - 128*v + 64) / 16 $$\beta_{9}$$ $$=$$ $$( 7 \nu^{11} - 8 \nu^{10} - 19 \nu^{9} + 26 \nu^{8} + 41 \nu^{7} - 90 \nu^{6} - 18 \nu^{5} + 156 \nu^{4} + 4 \nu^{3} - 192 \nu^{2} + 80 \nu + 32 ) / 32$$ (7*v^11 - 8*v^10 - 19*v^9 + 26*v^8 + 41*v^7 - 90*v^6 - 18*v^5 + 156*v^4 + 4*v^3 - 192*v^2 + 80*v + 32) / 32 $$\beta_{10}$$ $$=$$ $$( - \nu^{11} - 12 \nu^{10} + 13 \nu^{9} + 46 \nu^{8} - 31 \nu^{7} - 102 \nu^{6} + 126 \nu^{5} + 116 \nu^{4} - 252 \nu^{3} - 176 \nu^{2} + 304 \nu + 96 ) / 32$$ (-v^11 - 12*v^10 + 13*v^9 + 46*v^8 - 31*v^7 - 102*v^6 + 126*v^5 + 116*v^4 - 252*v^3 - 176*v^2 + 304*v + 96) / 32 $$\beta_{11}$$ $$=$$ $$( - 7 \nu^{10} + 8 \nu^{9} + 19 \nu^{8} - 26 \nu^{7} - 41 \nu^{6} + 90 \nu^{5} + 18 \nu^{4} - 140 \nu^{3} - 4 \nu^{2} + 176 \nu - 80 ) / 16$$ (-7*v^10 + 8*v^9 + 19*v^8 - 26*v^7 - 41*v^6 + 90*v^5 + 18*v^4 - 140*v^3 - 4*v^2 + 176*v - 80) / 16
 $$\nu$$ $$=$$ $$( \beta_{11} - \beta_{10} - \beta_{9} - 3\beta_{2} + \beta_1 ) / 4$$ (b11 - b10 - b9 - 3*b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta _1 + 2 ) / 2$$ (b11 + b10 - b9 - b7 + b5 + b4 + b3 + 2*b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} - 3\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (2*b8 + b7 - 2*b6 + b5 - 3*b4 - b3 + b2 + b1) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta _1 - 1$$ b9 - b8 - b6 + b5 + b4 + b1 - 1 $$\nu^{5}$$ $$=$$ $$2\beta_{10} + \beta_{9} + 2\beta_{8} - 4\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1$$ 2*b10 + b9 + 2*b8 - 4*b6 + b5 - b3 + b2 + 1 $$\nu^{6}$$ $$=$$ $$-2\beta_{11} - \beta_{10} + 2\beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - 4\beta_{3} - 3\beta_{2} - 2$$ -2*b11 - b10 + 2*b9 - 2*b8 - b7 - b6 - 4*b3 - 3*b2 - 2 $$\nu^{7}$$ $$=$$ $$\beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta _1 + 1$$ b11 + 7*b10 - 3*b9 + 2*b8 - 5*b7 - 5*b6 + 3*b5 + 4*b4 + b3 - 3*b2 + 3*b1 + 1 $$\nu^{8}$$ $$=$$ $$- 4 \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - \beta_{5} - 7 \beta_{4} - 8 \beta_{3} - \beta_{2} + 3 \beta _1 - 3$$ -4*b11 + 3*b10 - b9 + 3*b8 - 3*b7 - b5 - 7*b4 - 8*b3 - b2 + 3*b1 - 3 $$\nu^{9}$$ $$=$$ $$- 6 \beta_{11} + 6 \beta_{10} - \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + 5 \beta _1 - 8$$ -6*b11 + 6*b10 - b9 - 4*b8 - 4*b7 - 3*b6 + 5*b5 - 3*b4 + b3 + 5*b1 - 8 $$\nu^{10}$$ $$=$$ $$- 6 \beta_{11} + 14 \beta_{10} + 5 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} - 13 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + \beta _1 - 11$$ -6*b11 + 14*b10 + 5*b9 + 11*b8 + 2*b7 - 13*b6 - 3*b5 - 5*b4 - 4*b3 + 10*b2 + b1 - 11 $$\nu^{11}$$ $$=$$ $$- 29 \beta_{11} - 11 \beta_{10} + 24 \beta_{9} - 20 \beta_{8} + 5 \beta_{7} + 6 \beta_{6} - 10 \beta_{5} - 19 \beta_{4} - 18 \beta_{3} + 5 \beta_{2} - 12 \beta _1 - 7$$ -29*b11 - 11*b10 + 24*b9 - 20*b8 + 5*b7 + 6*b6 - 10*b5 - 19*b4 - 18*b3 + 5*b2 - 12*b1 - 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 206T_{2}^{6} + 208T_{2}^{4} + 72T_{2}^{2} + 1$$ T2^12 + 16*T2^10 + 88*T2^8 + 206*T2^6 + 208*T2^4 + 72*T2^2 + 1 $$T_{3}^{12} + 11 T_{3}^{10} - 4 T_{3}^{9} + 96 T_{3}^{8} - 42 T_{3}^{7} + 287 T_{3}^{6} - 390 T_{3}^{5} + 709 T_{3}^{4} - 516 T_{3}^{3} + 300 T_{3}^{2} - 80 T_{3} + 16$$ T3^12 + 11*T3^10 - 4*T3^9 + 96*T3^8 - 42*T3^7 + 287*T3^6 - 390*T3^5 + 709*T3^4 - 516*T3^3 + 300*T3^2 - 80*T3 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 16 T^{10} + 88 T^{8} + 206 T^{6} + \cdots + 1$$
$3$ $$T^{12} + 11 T^{10} - 4 T^{9} + 96 T^{8} + \cdots + 16$$
$5$ $$T^{12} - 6 T^{11} - 2 T^{10} + 84 T^{9} + \cdots + 3481$$
$7$ $$T^{12}$$
$11$ $$T^{12} + 6 T^{11} - 7 T^{10} - 114 T^{9} + \cdots + 256$$
$13$ $$T^{12} - 4 T^{11} + 21 T^{10} + \cdots + 4826809$$
$17$ $$(T^{6} - 4 T^{5} - 21 T^{4} + 60 T^{3} + \cdots - 491)^{2}$$
$19$ $$T^{12} - 29 T^{10} + 748 T^{8} + \cdots + 55696$$
$23$ $$(T^{6} - 12 T^{5} - 20 T^{4} + 608 T^{3} + \cdots + 6208)^{2}$$
$29$ $$T^{12} - 8 T^{11} + 108 T^{10} + \cdots + 10042561$$
$31$ $$T^{12} + 18 T^{11} + 94 T^{10} + \cdots + 913936$$
$37$ $$T^{12} + 318 T^{10} + \cdots + 1755945216$$
$41$ $$T^{12} - 30 T^{11} + \cdots + 884705536$$
$43$ $$T^{12} - 2 T^{11} + 113 T^{10} + \cdots + 2408704$$
$47$ $$T^{12} + 42 T^{11} + 746 T^{10} + \cdots + 9461776$$
$53$ $$T^{12} - 22 T^{11} + 393 T^{10} + \cdots + 5470921$$
$59$ $$T^{12} + 328 T^{10} + \cdots + 4571923456$$
$61$ $$T^{12} - 14 T^{11} + 283 T^{10} + \cdots + 5607424$$
$67$ $$T^{12} - 24 T^{11} + \cdots + 613651984$$
$71$ $$T^{12} + 24 T^{11} + 212 T^{10} + \cdots + 46895104$$
$73$ $$T^{12} + 30 T^{11} + \cdots + 1386221824$$
$79$ $$T^{12} - 28 T^{11} + 572 T^{10} + \cdots + 262144$$
$83$ $$T^{12} + 304 T^{10} + \cdots + 141324544$$
$89$ $$T^{12} + 658 T^{10} + \cdots + 1834580224$$
$97$ $$T^{12} - 6 T^{11} - 173 T^{10} + \cdots + 53465344$$