# Properties

 Label 483.2.i.h Level $483$ Weight $2$ Character orbit 483.i Analytic conductor $3.857$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 483.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 3 x^{19} + 22 x^{18} - 43 x^{17} + 245 x^{16} - 416 x^{15} + 1707 x^{14} - 2021 x^{13} + 7135 x^{12} - 6640 x^{11} + 20315 x^{10} - 10565 x^{9} + 29358 x^{8} - 9009 x^{7} + \cdots + 4$$ x^20 - 3*x^19 + 22*x^18 - 43*x^17 + 245*x^16 - 416*x^15 + 1707*x^14 - 2021*x^13 + 7135*x^12 - 6640*x^11 + 20315*x^10 - 10565*x^9 + 29358*x^8 - 9009*x^7 + 30661*x^6 - 4026*x^5 + 15786*x^4 - 84*x^3 + 5800*x^2 + 152*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_1) q^{2} - \beta_{9} q^{3} + ( - \beta_{17} + \beta_{9} - \beta_{8} - \beta_1) q^{4} + (\beta_{18} + \beta_{9} - \beta_{6} + 1) q^{5} + \beta_{4} q^{6} + ( - \beta_{16} + \beta_{14}) q^{7} + (\beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{9} - 1) q^{9}+O(q^{10})$$ q + (b4 + b1) * q^2 - b9 * q^3 + (-b17 + b9 - b8 - b1) * q^4 + (b18 + b9 - b6 + 1) * q^5 + b4 * q^6 + (-b16 + b14) * q^7 + (b14 + b12 + b11 + b10 - b7 + b5 - 3*b4 + b3) * q^8 + (-b9 - 1) * q^9 $$q + (\beta_{4} + \beta_1) q^{2} - \beta_{9} q^{3} + ( - \beta_{17} + \beta_{9} - \beta_{8} - \beta_1) q^{4} + (\beta_{18} + \beta_{9} - \beta_{6} + 1) q^{5} + \beta_{4} q^{6} + ( - \beta_{16} + \beta_{14}) q^{7} + (\beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{9} - 1) q^{9} + (\beta_{19} - \beta_{17} + \beta_{16} + \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5}) q^{10} + (\beta_{19} + \beta_{18} + \beta_{16} + \beta_{15} - \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_1) q^{11} + ( - \beta_{17} + \beta_{9} - \beta_{8} + \beta_{2} - \beta_1 + 1) q^{12} + ( - \beta_{14} - \beta_{12} + \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{19} - \beta_{18} + \beta_{17} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + \cdots + 1) q^{14}+ \cdots + (\beta_{7} + 1) q^{99}+O(q^{100})$$ q + (b4 + b1) * q^2 - b9 * q^3 + (-b17 + b9 - b8 - b1) * q^4 + (b18 + b9 - b6 + 1) * q^5 + b4 * q^6 + (-b16 + b14) * q^7 + (b14 + b12 + b11 + b10 - b7 + b5 - 3*b4 + b3) * q^8 + (-b9 - 1) * q^9 + (b19 - b17 + b16 + b15 - 2*b14 - b13 - b12 + b9 + b7 + b6 + b5) * q^10 + (b19 + b18 + b16 + b15 - b14 - b11 + b10 + b9 - b1) * q^11 + (-b17 + b9 - b8 + b2 - b1 + 1) * q^12 + (-b14 - b12 + b4 - b3 + b2) * q^13 + (-b19 - b18 + b17 + b13 + b12 + b11 - b10 + b8 - b7 - b6 + b1 + 1) * q^14 + (-b6 + 1) * q^15 + (-b19 + 2*b17 + b15 + b14 - b11 - 3*b9 + b8 - b3 - b2 + b1 - 3) * q^16 + (b17 + b14 + b13 + b12 - b11 + b10 - b9 - b8 - b7 - b6 - b5) * q^17 - b1 * q^18 + (b18 - b17 - b16 - b13 - 2*b12 - b11 - b8 + b7 + b4 - b3 + b2 - b1) * q^19 + (-b16 + b15 + 3*b14 + 3*b12 - b7 - b5 - 3*b4 - 2*b2 - 2) * q^20 - b16 * q^21 + (-b16 + b15 - b14 - b12 - b11 - b10 + 2*b7 + 2*b6 + b4 - b3 + b2) * q^22 + (-b9 - 1) * q^23 + (b18 - b13 + b10 + b5 - b4 + b1) * q^24 + (2*b19 - b17 + b16 + b15 - 3*b14 - b13 - 2*b12 - b10 + 2*b9 + b8 + 2*b7 + 2*b6 + b5 + b4) * q^25 + (b16 + b15 - b14 - b13 - b11 + b7 + b6 + b4 - b3) * q^26 - q^27 + (-b18 + b17 + b16 + b14 - b13 + b10 - b7 + 2) * q^28 + (b6 + b3 + b2 + 1) * q^29 + (b19 - b17 + b16 + b15 - b14 - b13 + b9 + b7 + b6 + b4 + 1) * q^30 + (-b19 - b18 - b17 - b15 + 2*b14 + b10 - b9 - 2*b8 - b7 - b6 - b4) * q^31 + (-3*b19 - 2*b18 + 2*b17 - 2*b16 - b15 + 2*b14 + 2*b13 + 2*b12 + 2*b11 - 3*b10 - b7 - b6 - 2*b5 + b4 - b1) * q^32 + (b19 + b18 + b16 + b15 - b14 - b11 + b10 + b9 + b7 - b1 + 1) * q^33 + (-2*b14 - 2*b12 + 2*b7 + 2*b4 + b3 + b2) * q^34 + (b19 + b18 + 2*b17 + b16 - b14 - b12 + b10 + b9 + b8 + b7 + b5 + b4 - b2 + 1) * q^35 + (b2 + 1) * q^36 + (2*b19 - b18 - b17 - b14 - b13 - b12 + b11 + b8 + b7 + 2*b6 + b4 + b3 - b2 + b1) * q^37 + (-b18 - 2*b17 - b16 + b15 - 2*b14 + b12 + b11 - 2*b10 + b9 + b7 + b6 + b4 + b1) * q^38 + (b16 + b15 - b14 + b8) * q^39 + (-b19 + b18 + 4*b17 - b15 + 2*b14 + 4*b13 + 3*b12 + b11 + b10 - 3*b9 + b8 - 3*b7 - 4*b6 - 4*b4 + b3 - b2 - 3) * q^40 + (b16 - b15 + b5 - 3*b4 + 2*b3 - b2 - 3) * q^41 + (-b18 + b16 + b12 + b11 - b9 + b8 - b7 + b5 - 2*b4 + b3 - b2) * q^42 + (-b5 + b4 - 2*b3 + 3) * q^43 + (-b19 - b18 + 2*b17 + b16 - b15 + b14 + 2*b13 + 4*b12 + 3*b11 - b10 - b9 + 2*b8 - 3*b7 - 2*b6 - 2*b4 + 2*b3 - 2*b2 + 3*b1 - 1) * q^44 + (-b18 - b9) * q^45 - b1 * q^46 + (-2*b19 - b18 + 2*b14 + 2*b12 + 2*b11 - 2*b10 - b9 + b8 - 2*b7 - b6 - b4 + b3 - b2 + 2*b1 - 1) * q^47 + (-b16 + b15 - b11 - b10 + b7 + b6 + b4 - b3 - b2 - 3) * q^48 + (b18 - b17 + b16 - b15 + 2*b10 + b9 - b6 + b5 - b4 + b3 - b1 + 2) * q^49 + (-b16 + b15 + 3*b14 + 3*b12 - 2*b7 + 2*b6 - b5 - 2*b4 - b3 - 2) * q^50 + (b17 + b13 - b11 + b10 - b9 - b8 + b4 - b3 + b2 - 1) * q^51 + (-b19 + b18 - b17 - b16 + b14 + 2*b13 + 2*b12 - b10 + 3*b9 - b8 - b7 - b6 - 2*b5 + b4) * q^52 + (-2*b19 - b18 + b17 - b15 + 2*b14 - b13 + b11 + b10 - b7 - b6 + b5 - 2*b4 - b1) * q^53 + (-b4 - b1) * q^54 + (-b14 - b12 - b11 - b10 - b7 + b6 - b5 - 2*b3 + b2 - 3) * q^55 + (b18 - b17 - b16 + b15 + b13 + b12 - 3*b11 + 2*b9 - b8 + 2*b7 - 3*b5 + 6*b4 + b1 + 4) * q^56 + (-b16 + b15 - b14 - b12 - b11 - b10 + b7 - b5 + 2*b4 - b3 + b2) * q^57 + (b18 - b17 - b16 - b15 - b14 - b13 - 2*b12 - b11 + b7 + 5*b4 + 4*b1) * q^58 + (-b18 - 2*b17 + b13 - b10 - 2*b9 - b5 + b4) * q^59 + (b18 - b17 - 2*b16 - b15 + 2*b14 + b13 + 2*b12 - b11 + b10 + 2*b9 - 2*b8 - b7 - b6 - b5 + b1) * q^60 + (-b19 - b18 - b16 - b15 + b14 + b11 - b10 - 3*b9 - b7 - 2*b4 - b1 - 3) * q^61 + (-b16 + b15 + 2*b7 - b5 - b4 - b2 - 3) * q^62 - b14 * q^63 + (3*b16 - 3*b15 + b14 + b12 + 2*b11 + 2*b10 - 4*b7 - 4*b6 + b5 - 5*b4 + 5*b3 + 6) * q^64 + (-2*b19 - 2*b18 - 3*b17 - 2*b16 - b15 + 2*b14 - b13 - b12 + b11 - 2*b10 - b9 - 2*b8 - b7 + b6 - b4 + 2*b2 - 2*b1 - 1) * q^65 + (-b16 - b14 + b11 - 2*b10 + b8 + b7 + b6 + b4 + 2*b1) * q^66 + (-2*b19 - b18 + b17 - b16 - b15 + 2*b14 + b13 + b12 + b11 - b10 + b8 - b7 - b6 - b5 + 2*b1) * q^67 + (3*b19 - b18 - 4*b17 + 2*b16 - 4*b14 - 2*b13 + b11 + b9 + 2*b7 + 3*b6 + 2*b4 + 2*b3 + 1) * q^68 - q^69 + (-2*b18 + 2*b15 - b14 + b11 - 3*b10 + 3*b9 + b8 + b7 + 2*b6 - 2*b5 + 2*b4 + b2 - 2*b1 + 5) * q^70 + (-3*b14 - 3*b12 + 2*b7 + 3*b6 + 2*b5 + 2*b4 + 2*b2 + 2) * q^71 + (b18 - b14 - b13 - b12 - b11 + b7 + 2*b4 - b3 + b1) * q^72 + (-b19 - b17 - b16 - 2*b15 + b14 - 2*b12 + 2*b11 - 3*b10 - b9 + b7 + b6 + b4) * q^73 + (b19 + 3*b18 - b17 - b16 + b14 + b13 + 2*b12 - 2*b11 + 2*b10 + 2*b9 - b7 - b6 - b5 - 2*b1) * q^74 + (2*b19 - b17 + b16 + b15 - b14 - b13 + b11 + 2*b9 + b8 + b7 + b6 + b3 - b2 + 2) * q^75 + (b16 - b15 + 4*b14 + 4*b12 + 2*b11 + 2*b10 - 6*b7 - 2*b6 + 2*b5 - 10*b4 + 2*b3 - 2*b2 - 4) * q^76 + (b19 + b18 + b16 - b10 + 3*b9 + b8 - b7 - b5 + b3 - 2*b2 - b1 + 2) * q^77 + (-b11 - b10 + b7 + b6 - b5 + 2*b4 - b3) * q^78 + (3*b19 + b18 + b16 + b15 - 2*b14 - b12 - b11 + 2*b10 - b9 + 2*b8 + 2*b7 + b6 + 2*b4 + 2*b3 - 2*b2 + 2*b1 - 1) * q^79 + (-3*b18 + 5*b17 + 3*b16 + b15 - b14 - b13 - 2*b12 + b10 - 6*b9 + 3*b8 + b5 - b4 - b1) * q^80 + b9 * q^81 + (2*b19 - b18 - b17 - b16 + b15 - 2*b13 - 3*b12 - 5*b9 + b8 + 2*b7 + 3*b6 - 2*b4 - b2 - 3*b1 - 5) * q^82 + (-2*b14 - 2*b12 + b11 + b10 + b7 - b6 + b5 - 2*b4 - b3 + b2 - 1) * q^83 + (2*b17 + b15 + b14 - 2*b11 + b10 - 2*b9 + b6 - b5 + b4 - 2*b3 - b1) * q^84 + (b16 - b15 - 2*b14 - 2*b12 - 3*b6 - b5 + 3*b4 - b3 - b2 + 5) * q^85 + (-b19 + 2*b18 + 2*b17 - b15 + 2*b13 + b12 - b11 + b10 - b8 - b7 - 3*b6 + 2*b4 - b3 + b2 + 2*b1) * q^86 + (b18 + 2*b17 - b9 + b8 + b1) * q^87 + (2*b19 + 2*b18 + 2*b17 + 3*b16 - b14 - 2*b13 - 4*b12 - b11 + 2*b10 + b8 + b7 + b6 + 2*b5 - b4 - 2*b1) * q^88 + (-b19 - 2*b16 - 2*b15 - 2*b12 - b11 - 2*b8 + 4*b4 - 2*b3 + 2*b2 + 2*b1) * q^89 + (b14 + b12 - b5 + b4 + 1) * q^90 + (b18 + b17 + b15 - 2*b14 - b13 - b12 - 2*b11 - 2*b10 + 2*b9 + b8 + 2*b7 + b6 - 2*b3 - 2*b1 - 2) * q^91 + (b2 + 1) * q^92 + (-b19 - b18 - b17 - b16 + b14 - b12 - b11 - b9 - 2*b8 + b6 + 2*b4 - b3 + 2*b2 - 1) * q^93 + (b18 + b16 - b15 + 3*b14 + b13 - 2*b11 + 3*b10 + 3*b9 - 3*b8 - 2*b7 - 2*b6 - b5 - b4 - 3*b1) * q^94 + (b19 - 2*b17 - b16 - 2*b15 + b14 - 3*b13 - 2*b12 + 2*b10 + 4*b9 - 2*b8 + b7 + b6 + 3*b5 - 2*b4 - b1) * q^95 + (-3*b19 - 2*b18 + 2*b17 - b16 - 2*b15 + 3*b14 + 2*b13 + 3*b12 + 3*b11 - 2*b10 - 4*b7 - 2*b6 - 5*b4 - b1) * q^96 + (b16 - b15 - 3*b14 - 3*b12 - b11 - b10 + 4*b7 + 2*b6 + b4 + b3 + b2) * q^97 + (2*b19 - 2*b17 + 2*b15 - 4*b14 - 3*b13 - 4*b12 - b11 - 2*b10 + b9 + b8 + 5*b7 + 3*b6 + 2*b5 + 3*b4 - 3*b3 + b1 + 2) * q^98 + (b7 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 3 q^{2} + 10 q^{3} - 15 q^{4} + 5 q^{5} - 6 q^{6} + 18 q^{8} - 10 q^{9}+O(q^{10})$$ 20 * q - 3 * q^2 + 10 * q^3 - 15 * q^4 + 5 * q^5 - 6 * q^6 + 18 * q^8 - 10 * q^9 $$20 q - 3 q^{2} + 10 q^{3} - 15 q^{4} + 5 q^{5} - 6 q^{6} + 18 q^{8} - 10 q^{9} - 11 q^{10} - 8 q^{11} + 15 q^{12} + 11 q^{14} + 10 q^{15} - 37 q^{16} + 11 q^{17} - 3 q^{18} - q^{19} - 30 q^{20} + 12 q^{22} - 10 q^{23} + 9 q^{24} - 21 q^{25} - q^{26} - 20 q^{27} + 44 q^{28} + 44 q^{29} + 11 q^{30} + 3 q^{31} - 11 q^{32} + 8 q^{33} - 6 q^{34} - 9 q^{35} + 30 q^{36} + 3 q^{37} - 16 q^{38} - 39 q^{40} - 52 q^{41} + 7 q^{42} + 54 q^{43} - 16 q^{44} + 5 q^{45} - 3 q^{46} - 11 q^{47} - 74 q^{48} + 22 q^{49} + 4 q^{50} - 11 q^{51} - 29 q^{52} - 5 q^{53} + 3 q^{54} - 36 q^{55} + 43 q^{56} - 2 q^{57} - 16 q^{58} + 10 q^{59} - 15 q^{60} - 22 q^{61} - 64 q^{62} + 138 q^{64} + 11 q^{65} + 6 q^{66} + 2 q^{67} + 21 q^{68} - 20 q^{69} + 84 q^{70} + 54 q^{71} - 9 q^{72} + 8 q^{73} - 14 q^{74} + 21 q^{75} - 44 q^{76} + 8 q^{77} - 2 q^{78} - 21 q^{79} + 53 q^{80} - 10 q^{81} - 36 q^{82} - 24 q^{83} + 25 q^{84} + 46 q^{85} - 18 q^{86} + 22 q^{87} + 10 q^{88} - 6 q^{89} + 22 q^{90} - 62 q^{91} + 30 q^{92} - 3 q^{93} - 35 q^{94} - 44 q^{95} + 11 q^{96} + 12 q^{97} + 2 q^{98} + 16 q^{99}+O(q^{100})$$ 20 * q - 3 * q^2 + 10 * q^3 - 15 * q^4 + 5 * q^5 - 6 * q^6 + 18 * q^8 - 10 * q^9 - 11 * q^10 - 8 * q^11 + 15 * q^12 + 11 * q^14 + 10 * q^15 - 37 * q^16 + 11 * q^17 - 3 * q^18 - q^19 - 30 * q^20 + 12 * q^22 - 10 * q^23 + 9 * q^24 - 21 * q^25 - q^26 - 20 * q^27 + 44 * q^28 + 44 * q^29 + 11 * q^30 + 3 * q^31 - 11 * q^32 + 8 * q^33 - 6 * q^34 - 9 * q^35 + 30 * q^36 + 3 * q^37 - 16 * q^38 - 39 * q^40 - 52 * q^41 + 7 * q^42 + 54 * q^43 - 16 * q^44 + 5 * q^45 - 3 * q^46 - 11 * q^47 - 74 * q^48 + 22 * q^49 + 4 * q^50 - 11 * q^51 - 29 * q^52 - 5 * q^53 + 3 * q^54 - 36 * q^55 + 43 * q^56 - 2 * q^57 - 16 * q^58 + 10 * q^59 - 15 * q^60 - 22 * q^61 - 64 * q^62 + 138 * q^64 + 11 * q^65 + 6 * q^66 + 2 * q^67 + 21 * q^68 - 20 * q^69 + 84 * q^70 + 54 * q^71 - 9 * q^72 + 8 * q^73 - 14 * q^74 + 21 * q^75 - 44 * q^76 + 8 * q^77 - 2 * q^78 - 21 * q^79 + 53 * q^80 - 10 * q^81 - 36 * q^82 - 24 * q^83 + 25 * q^84 + 46 * q^85 - 18 * q^86 + 22 * q^87 + 10 * q^88 - 6 * q^89 + 22 * q^90 - 62 * q^91 + 30 * q^92 - 3 * q^93 - 35 * q^94 - 44 * q^95 + 11 * q^96 + 12 * q^97 + 2 * q^98 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 3 x^{19} + 22 x^{18} - 43 x^{17} + 245 x^{16} - 416 x^{15} + 1707 x^{14} - 2021 x^{13} + 7135 x^{12} - 6640 x^{11} + 20315 x^{10} - 10565 x^{9} + 29358 x^{8} - 9009 x^{7} + \cdots + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 22\!\cdots\!01 \nu^{19} + \cdots - 20\!\cdots\!14 ) / 69\!\cdots\!38$$ (-2213201457183645337254101*v^19 - 72273515421915791353480230*v^18 + 187843348068867281776440945*v^17 - 1594508673564660159222272911*v^16 + 2702114895961082421844363400*v^15 - 17439344230046598654017193895*v^14 + 27088708473123312284969881789*v^13 - 119993201418817554758303406100*v^12 + 125394894201007064484994501200*v^11 - 481197907522635478395232759785*v^10 + 399666252882203229880542002975*v^9 - 1333433981037634169446698023720*v^8 + 527446607381174890390845734785*v^7 - 1671597695031203157391016193865*v^6 + 375647816135984156579116872210*v^5 - 2017431287596442205117175096217*v^4 + 101337786109969689566869793460*v^3 - 1050637386278119481166986396338*v^2 - 27549356344608130143878078690*v - 2077218414231396583211507347914) / 692323187363073553855733195538 $$\beta_{3}$$ $$=$$ $$( 22\!\cdots\!60 \nu^{19} + \cdots - 13\!\cdots\!22 ) / 36\!\cdots\!14$$ (2210988050055098216401380760*v^19 - 25698888055210344006251868528*v^18 + 104517880269381602053471469388*v^17 - 492683051148489581717386541089*v^16 + 1273119146972405495692043982194*v^15 - 5143221214434175768499725759708*v^14 + 10560684306705372889072572257512*v^13 - 32302603075073871856906184416117*v^12 + 44345501150248093716650460759300*v^11 - 120405308020888090342132961297604*v^10 + 127380551189497685791241752783352*v^9 - 300275599054522325059704471274895*v^8 + 138353104708822885980934759803412*v^7 - 320050831801058777606292712083790*v^6 + 92100566239291496667953706679764*v^5 - 421601832703848840574751340839498*v^4 + 83652837775332729943951914454278*v^3 - 217438483346490820575307883072080*v^2 - 5701496676322008526942895944388*v - 130750257555532932513190288510122) / 36693128930242898354353859363514 $$\beta_{4}$$ $$=$$ $$( - 62\!\cdots\!25 \nu^{19} + \cdots + 18\!\cdots\!90 ) / 69\!\cdots\!38$$ (-62213035543980411076940325*v^19 + 188852308089124878568075076*v^18 - 1296413266545653252339206920*v^17 + 2487317180322290394531993030*v^16 - 13647685034710540554628106714*v^15 + 23178507890334768586162811800*v^14 - 88758307443527963054319940880*v^13 + 98643836361261098501526515036*v^12 - 323896807187482678275665812775*v^11 + 287699661811022865065889256800*v^10 - 782659909553326572632809942590*v^9 + 257614467639949813147332530650*v^8 - 493016316462542738950116037630*v^7 + 33030629834544633001309653140*v^6 - 235916187782780226639051110960*v^5 - 125178135035919021583355123760*v^4 + 1035336308499167435856595125767*v^3 - 96111891124275335036406806160*v^2 - 2521407240040456935000684200*v + 18092974941923107660183149290) / 692323187363073553855733195538 $$\beta_{5}$$ $$=$$ $$( - 18\!\cdots\!73 \nu^{19} + \cdots - 89\!\cdots\!04 ) / 36\!\cdots\!14$$ (-18737857859988518657293389273*v^19 + 43015353136851625740627478907*v^18 - 349884007034168909777684330022*v^17 + 456088339773085505148341425254*v^16 - 3559853809606135470642794158277*v^15 + 3795943526086822597668972728026*v^14 - 21476149413769790275924146282206*v^13 + 8198799269497588146583899648086*v^12 - 73708366448064762959555425190645*v^11 + 517825278630851583954077198958*v^10 - 159602195157262183351985963535036*v^9 - 158432209004916928111809504893900*v^8 - 52874178114228375519723820365928*v^7 - 293510296569450136413420900264955*v^6 - 3574244857247743024453069087076*v^5 - 392316606593562070881619547065158*v^4 + 169474939526094388982793897633530*v^3 - 213486311054201382807788336115972*v^2 - 5598302002687177816652994965924*v - 89663458273712954002902986933404) / 36693128930242898354353859363514 $$\beta_{6}$$ $$=$$ $$( 20\!\cdots\!92 \nu^{19} + \cdots + 13\!\cdots\!44 ) / 36\!\cdots\!14$$ (20166016333219382972648435592*v^19 - 24163513814374738538561287870*v^18 + 320749549341974267412151082199*v^17 - 5653714093171285615556839902*v^16 + 3013005276627953593325732016610*v^15 + 1532535380422876592072464473943*v^14 + 15123499383961574786191501457143*v^13 + 31480533491369750263186192178615*v^12 + 39565259815972009915702287289465*v^11 + 182001129647270287993585589165169*v^10 + 39732711492383003467366957502931*v^9 + 720653801393401328725759785656545*v^8 - 152589750851261163068591863312879*v^7 + 1274636630528543252950277336251385*v^6 - 151214756610902429598820965261758*v^5 + 1301239064837991853698982418218023*v^4 - 133048476058791637833911084656459*v^3 + 691164855678887249237355758154042*v^2 + 18123939032242841951678861948038*v + 139115902664874132108643343216144) / 36693128930242898354353859363514 $$\beta_{7}$$ $$=$$ $$( - 51\!\cdots\!36 \nu^{19} + \cdots - 12\!\cdots\!86 ) / 36\!\cdots\!14$$ (-51552071574013772618745005036*v^19 + 156790006262972014055184606447*v^18 - 1078926765888776299974767257242*v^17 + 2073652364466845887698407498219*v^16 - 11368205772812903719944866624524*v^15 + 19254540035206727024363003837438*v^14 - 73957584474390831580266312325694*v^13 + 81691142280010637850130900201943*v^12 - 269343125076239502199630285504530*v^11 + 231810708276379774979934785333874*v^10 - 644209976671998561931473954601096*v^9 + 182574554022567954999096629072455*v^8 - 399459084767881915144078366691438*v^7 - 85463586358976782785746921393173*v^6 - 187229987701075232919440575235616*v^5 - 163642137964115670161832077696576*v^4 + 314254056321907924354673462839332*v^3 - 111843163196428121719674554561620*v^2 - 2933735913090620891211292487552*v - 12552730951844104709584916300286) / 36693128930242898354353859363514 $$\beta_{8}$$ $$=$$ $$( - 14\!\cdots\!44 \nu^{19} + \cdots + 92\!\cdots\!32 ) / 36\!\cdots\!14$$ (-141818203699631164412929489044*v^19 + 432578457280165036320094917998*v^18 - 3134662782336161466489252768462*v^17 + 6226678203464539889670468704469*v^16 - 34883476322017825187554489306106*v^15 + 60307077285517458196443049593466*v^14 - 243208948254653293332233751205778*v^13 + 294632011037539224437409221141210*v^12 - 1013371213184294563087498017361814*v^11 + 971727116503994892183281067169194*v^10 - 2880005077769354259841528226083980*v^9 + 1582255714785901521411908183964442*v^8 - 4118314264723798866257243056428910*v^7 + 1401412983226271914361160524764544*v^6 - 4329553503905479412291080631684870*v^5 + 830603796879309882691062678884883*v^4 - 2224446757913046160765921111126204*v^3 + 76495788094608557036522185784532*v^2 - 817165367569152243292989829127468*v + 92947381995830568455091678032) / 36693128930242898354353859363514 $$\beta_{9}$$ $$=$$ $$( 90\!\cdots\!45 \nu^{19} + \cdots - 45\!\cdots\!36 ) / 13\!\cdots\!76$$ (9046487470961553830091574645*v^19 - 27015036341796700668120843285*v^18 + 198645019744975934504878492038*v^17 - 386406134718255508189259295895*v^16 + 2211414796024936107583371801965*v^15 - 3736043417850585312208838838892*v^14 + 15395997097150702850793992295415*v^13 - 18105434563926244364506432475785*v^12 + 64349400432588164380700332062003*v^11 - 59420883192809752075256724017250*v^10 + 183203993648961920328178560399575*v^9 - 94010820311602163069651866239245*v^8 + 265071550237209397717533783366610*v^7 - 80513772992967552977394763901545*v^6 + 277308291087483112718435150884065*v^5 - 35949326182525655266670577298850*v^4 + 143058207486670926804992307593490*v^3 - 2830577564559105393440882521714*v^2 + 52661851113825562884603946553320*v - 4537464659910011667545676636) / 1384646374726147107711466391076 $$\beta_{10}$$ $$=$$ $$( 74\!\cdots\!85 \nu^{19} + \cdots + 40\!\cdots\!04 ) / 73\!\cdots\!28$$ (743549806326306037299807476785*v^19 - 2244347089827480051120851081047*v^18 + 16422843491828392024104544076572*v^17 - 32345664962140078827643674936527*v^16 + 183269992198833972226318111716299*v^15 - 313616742667039008118563854575794*v^14 + 1280363641428900967274100131728077*v^13 - 1534487613175479514462666595622029*v^12 + 5368888339888671971400663352462019*v^11 - 5067133441187310157221373920519668*v^10 + 15329487830624883741675966182136809*v^9 - 8193852741310185902379534437143013*v^8 + 22308895574473230429339371677735082*v^7 - 6991098209568113955488249697237077*v^6 + 23205467409385592412646167872255789*v^5 - 2780107804200635033825425411754242*v^4 + 12091256066589073010418468208680656*v^3 + 342408076410390610048060472394698*v^2 + 4318385494502782504260529355965840*v + 400823423878412256571792415489104) / 73386257860485796708707718727028 $$\beta_{11}$$ $$=$$ $$( - 77\!\cdots\!09 \nu^{19} + \cdots + 28\!\cdots\!96 ) / 73\!\cdots\!28$$ (-772571159447585664564201845209*v^19 + 2433251837340193757560377714871*v^18 - 17328028696855863576525829553722*v^17 + 35649813506551642686711144536639*v^16 - 193719782117281550552023332543377*v^15 + 347621721658378688101756600837428*v^14 - 1360447118347432884455497539298011*v^13 + 1736761464448823352502028991597167*v^12 - 5694331515092250617713414421606189*v^11 + 5818921260736396894331557294247330*v^10 - 16236196394367117607145123881317935*v^9 + 9990671903188582429781670732412463*v^8 - 23222308780848818579541886156559852*v^7 + 9084307116159317665581235784931683*v^6 - 23795585449795088718846371838197249*v^5 + 5224372087349683355220219389807260*v^4 - 11863870093225785481624557883366160*v^3 + 913876165334712864244956527684482*v^2 - 4285444401041013866983833669793996*v + 288502434248044438370578233077396) / 73386257860485796708707718727028 $$\beta_{12}$$ $$=$$ $$( 10\!\cdots\!99 \nu^{19} + \cdots - 47\!\cdots\!12 ) / 73\!\cdots\!28$$ (1057625224828485092013384556499*v^19 - 3244875807615690833062867601985*v^18 + 23522298813263490571150366896518*v^17 - 47146691513758951479193983072953*v^16 + 262989549636250316840116543860953*v^15 - 458651581064160468483188972305516*v^14 + 1843815247910785822515306421239653*v^13 - 2269618041593166006569445710089625*v^12 + 7748412389450419469783885429707577*v^11 - 7568860587187297355880363069999062*v^10 + 22198521147395020898344549302581837*v^9 - 12703340574822968871560316036311777*v^8 + 32455266742493995592999975077882994*v^7 - 11522238474937900431207705492090797*v^6 + 33985024833234042849914747880970295*v^5 - 6050990944128842376878409419306628*v^4 + 17732691882787055608600205858406904*v^3 - 670396637927994778632056331267382*v^2 + 6330976639022414299790503375986568*v - 47285135823024251830041450808212) / 73386257860485796708707718727028 $$\beta_{13}$$ $$=$$ $$( - 55\!\cdots\!40 \nu^{19} + \cdots + 11\!\cdots\!72 ) / 36\!\cdots\!14$$ (-552533071735041510070348137940*v^19 + 1698637994076486173197178145624*v^18 - 12262268014715240085293012246521*v^17 + 24614157418095182003659661227541*v^16 - 136778198157620007026056213196014*v^15 + 239233509001313912935434228326011*v^14 - 956271951891639050330782799550977*v^13 + 1180207275232436321422505565039466*v^12 - 3997801004297506220457790850298688*v^11 + 3930407961278306560116366380703323*v^10 - 11382035726820085476209926770766889*v^9 + 6569067117072807196853803639234320*v^8 - 16329464835913017031776651157026858*v^7 + 6025928348418313546526177144981977*v^6 - 16832743625921472428575577524067500*v^5 + 3340055241347910208550291233553107*v^4 - 8422889299470635200537578214001598*v^3 + 607771425328864917810194903314822*v^2 - 2926107040355326303334915467188906*v + 110497578928214909756328385433872) / 36693128930242898354353859363514 $$\beta_{14}$$ $$=$$ $$( - 11\!\cdots\!71 \nu^{19} + \cdots - 21\!\cdots\!28 ) / 73\!\cdots\!28$$ (-1144816347608999634123785902271*v^19 + 3475046555067769251162470666689*v^18 - 25246173530260871716465881747224*v^17 + 49908626468831027021404157529209*v^16 - 280829284233916307095933611112545*v^15 + 483048690374557347181316546137722*v^14 - 1955674673850004482307043886404679*v^13 + 2352127809071051746739122358211147*v^12 - 8143261164736798636665129035788827*v^11 + 7733942298739263612125296863682040*v^10 - 23096522901949823478510396068501315*v^9 + 12380236072097429521854721689267127*v^8 - 32877547665658567006309025311237528*v^7 + 10509733379706993101269624548937215*v^6 - 34121469222325607445006602265017763*v^5 + 4814397087449732624057843559839850*v^4 - 17142992015063485085841375017626266*v^3 - 19039364622792830931131189729654*v^2 - 6349056491124455825208200372989292*v - 212893765141721933615456582698628) / 73386257860485796708707718727028 $$\beta_{15}$$ $$=$$ $$( - 71\!\cdots\!56 \nu^{19} + \cdots - 83\!\cdots\!36 ) / 36\!\cdots\!14$$ (-714510889437795874221495514456*v^19 + 2145209242843280081569137185656*v^18 - 15741624473958951166937646679568*v^17 + 30800318375406970120843847140123*v^16 - 175460831275899196122077598951424*v^15 + 298100452001858920704013266843749*v^14 - 1223864119280972047660169450098418*v^13 + 1450831422992759172158627748511361*v^12 - 5123360112750606152813876750274423*v^11 + 4768297948829627328169257818179085*v^10 - 14607742776055729953868397272847406*v^9 + 7603885046088057273013504311468503*v^8 - 21186538344804289968240489868258173*v^7 + 6413621175882948056333988423131183*v^6 - 22074097091390421472937650019778582*v^5 + 2786730131277803674655078793050368*v^4 - 11326157282238495333477299534487063*v^3 - 50343101924424457070991338365278*v^2 - 4061150439884899074098954226934528*v - 83174023864456525544052045649436) / 36693128930242898354353859363514 $$\beta_{16}$$ $$=$$ $$( - 73\!\cdots\!06 \nu^{19} + \cdots - 22\!\cdots\!76 ) / 36\!\cdots\!14$$ (-736062553610786264565068935306*v^19 + 2243445044281651537083007967191*v^18 - 16282234263920024740652737171757*v^17 + 32380711382575211971313023824322*v^16 - 181486123525976025532896967744039*v^15 + 314184070225833804846797617585136*v^14 - 1267001492946133503904161722139899*v^13 + 1541355345396083002680311805706385*v^12 - 5294244077316824924036310300919488*v^11 + 5107339601335414736511466885375706*v^10 - 15063214812831939379930559889551649*v^9 + 8384201661340536080179544690378123*v^8 - 21627872898014546065282050865061364*v^7 + 7477313649741584438374915642300307*v^6 - 22351836551169182357662365071937144*v^5 + 3818606528557603418576726412247369*v^4 - 11323889921761051929835169697843639*v^3 + 478704959699654069039747478241596*v^2 - 4047278303491946781972533765399446*v - 22716485702877342995746762114976) / 36693128930242898354353859363514 $$\beta_{17}$$ $$=$$ $$( 17\!\cdots\!37 \nu^{19} + \cdots - 93\!\cdots\!88 ) / 73\!\cdots\!28$$ (1721793315927687921404670411937*v^19 - 5168208685540728552754872822691*v^18 + 37873795099018796451122488511136*v^17 - 74060949746529559558310726384809*v^16 + 421668329390972366217580597645047*v^15 - 716493628497662920491417297123630*v^14 + 2937252838054419411042919084852175*v^13 - 3480747397089745963635721366978835*v^12 + 12271588953935414011541758249709305*v^11 - 11442381638862139725042275925618348*v^10 + 34931809768529167394230785008015735*v^9 - 18253575861106536192859813090483159*v^8 + 50438914357546296507983787316036020*v^7 - 15781705228007692286811536226424433*v^6 + 52790943959231188067410736642390335*v^5 - 7590998173265421826525167708485918*v^4 + 27205890311534426470619707327724078*v^3 - 641034714039109739925137531806390*v^2 + 9931258572603554726734048365164648*v - 933729971988000686346568638988) / 73386257860485796708707718727028 $$\beta_{18}$$ $$=$$ $$( - 18\!\cdots\!67 \nu^{19} + \cdots + 14\!\cdots\!12 ) / 73\!\cdots\!28$$ (-1826099606542181416800788357067*v^19 + 5600683297057648575956947676239*v^18 - 40564137005084070652207920964014*v^17 + 81269359760713368612610697325651*v^16 - 453099499328381942118052908494023*v^15 + 790220659650085468521047862835340*v^14 - 3172557031619414735877974110086325*v^13 + 3903926536935039969096099803253115*v^12 - 13303971395060451547938422377817305*v^11 + 13004540386030028230383309849616938*v^10 - 38001139388533057558629013136198265*v^9 + 21738955235266153101625784846634311*v^8 - 55093639480873488157714778520361766*v^7 + 19680488465623660394529639086438707*v^6 - 57103605500905301225264121971303755*v^5 + 10246027808506957672176652211247888*v^4 - 29277626165172875869440196817146742*v^3 + 1288359980379916849032139629967590*v^2 - 10526592248903126532076325751194632*v + 1417444266705166224272738809012) / 73386257860485796708707718727028 $$\beta_{19}$$ $$=$$ $$( 18\!\cdots\!75 \nu^{19} + \cdots - 11\!\cdots\!52 ) / 73\!\cdots\!28$$ (1833981508035291645456074210975*v^19 - 5516705114195740125534601004957*v^18 + 40391894722295708225441224273822*v^17 - 79237444392604844982176185950999*v^16 + 450118751920076775798883350947615*v^15 - 767705811009130326827628156780186*v^14 + 3139066576793927867957333440281575*v^13 - 3744583191604188609572462800524499*v^12 + 13137732669741883737226084854162837*v^11 - 12373289502896088311683132421340920*v^10 + 37466060565530754537307744161576151*v^9 - 20052254972029576785164512111629879*v^8 + 54357407904080407571991254321340050*v^7 - 17999535457709512909699353706048315*v^6 + 56941254038601058489275791311112181*v^5 - 9226225712566575585948160363852092*v^4 + 29437750334778600206601123879961342*v^3 - 1374663186639443313724633547987006*v^2 + 10919788645650447787124763885862392*v - 114627832208351724265196570066352) / 73386257860485796708707718727028
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{17} - 3\beta_{9} + \beta_{8} - \beta_{2} + \beta _1 - 3$$ b17 - 3*b9 + b8 - b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{7} - \beta_{5} + 7\beta_{4} - \beta_{3}$$ -b14 - b12 - b11 - b10 + b7 - b5 + 7*b4 - b3 $$\nu^{4}$$ $$=$$ $$\beta_{19} - 8 \beta_{17} - \beta_{16} - \beta_{14} - \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - 7 \beta_1$$ b19 - 8*b17 - b16 - b14 - b10 + 17*b9 - 7*b8 + b7 + b6 + b4 - 7*b1 $$\nu^{5}$$ $$=$$ $$- 3 \beta_{19} - 10 \beta_{18} + 2 \beta_{17} - \beta_{16} - 2 \beta_{15} + 11 \beta_{14} + 10 \beta_{13} + 11 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} - 12 \beta_{7} - 2 \beta_{6} - 41 \beta_{4} + 8 \beta_{3} - 29 \beta_1$$ -3*b19 - 10*b18 + 2*b17 - b16 - 2*b15 + 11*b14 + 10*b13 + 11*b12 + 11*b11 - 2*b10 - 12*b7 - 2*b6 - 41*b4 + 8*b3 - 29*b1 $$\nu^{6}$$ $$=$$ $$13 \beta_{16} - 13 \beta_{15} + \beta_{14} + \beta_{12} + 12 \beta_{11} + 12 \beta_{10} - 14 \beta_{7} - 14 \beta_{6} + \beta_{5} - 15 \beta_{4} + 15 \beta_{3} + 46 \beta_{2} + 112$$ 13*b16 - 13*b15 + b14 + b12 + 12*b11 + 12*b10 - 14*b7 - 14*b6 + b5 - 15*b4 + 15*b3 + 46*b2 + 112 $$\nu^{7}$$ $$=$$ $$40 \beta_{19} + 83 \beta_{18} - 28 \beta_{17} + 28 \beta_{16} + 12 \beta_{15} - 26 \beta_{14} - 87 \beta_{13} - 30 \beta_{12} - 26 \beta_{11} + 99 \beta_{10} - 2 \beta_{8} + 14 \beta_{7} + 14 \beta_{6} + 87 \beta_{5} - 73 \beta_{4} + 179 \beta_1$$ 40*b19 + 83*b18 - 28*b17 + 28*b16 + 12*b15 - 26*b14 - 87*b13 - 30*b12 - 26*b11 + 99*b10 - 2*b8 + 14*b7 + 14*b6 + 87*b5 - 73*b4 + 179*b1 $$\nu^{8}$$ $$=$$ $$- 65 \beta_{19} + 444 \beta_{17} + 2 \beta_{16} + 131 \beta_{15} + 81 \beta_{14} - 20 \beta_{13} - 46 \beta_{12} - 141 \beta_{11} + 28 \beta_{10} - 779 \beta_{9} + 311 \beta_{8} + 48 \beta_{7} + 48 \beta_{6} + 40 \beta_{4} - 153 \beta_{3} + \cdots - 779$$ -65*b19 + 444*b17 + 2*b16 + 131*b15 + 81*b14 - 20*b13 - 46*b12 - 141*b11 + 28*b10 - 779*b9 + 311*b8 + 48*b7 + 48*b6 + 40*b4 - 153*b3 - 311*b2 + 303*b1 - 779 $$\nu^{9}$$ $$=$$ $$- 177 \beta_{16} + 177 \beta_{15} - 539 \beta_{14} - 539 \beta_{12} - 581 \beta_{11} - 581 \beta_{10} + 831 \beta_{7} + 179 \beta_{6} - 724 \beta_{5} + 2738 \beta_{4} - 474 \beta_{3} + 30 \beta_{2} + 4$$ -177*b16 + 177*b15 - 539*b14 - 539*b12 - 581*b11 - 581*b10 + 831*b7 + 179*b6 - 724*b5 + 2738*b4 - 474*b3 + 30*b2 + 4 $$\nu^{10}$$ $$=$$ $$433 \beta_{19} + 4 \beta_{18} - 3278 \beta_{17} - 1195 \beta_{16} - 44 \beta_{15} - 673 \beta_{14} + 261 \beta_{13} + 434 \beta_{12} + 284 \beta_{11} - 1262 \beta_{10} + 5582 \beta_{9} - 2176 \beta_{8} + 717 \beta_{7} + \cdots - 2026 \beta_1$$ 433*b19 + 4*b18 - 3278*b17 - 1195*b16 - 44*b15 - 673*b14 + 261*b13 + 434*b12 + 284*b11 - 1262*b10 + 5582*b9 - 2176*b8 + 717*b7 + 717*b6 - 261*b5 + 978*b4 - 2026*b1 $$\nu^{11}$$ $$=$$ $$- 3452 \beta_{19} - 5001 \beta_{18} + 2464 \beta_{17} - 1026 \beta_{16} - 2722 \beta_{15} + 6345 \beta_{14} + 5889 \beta_{13} + 7015 \beta_{12} + 6741 \beta_{11} - 2152 \beta_{10} - 56 \beta_{9} + 302 \beta_{8} + \cdots - 56$$ -3452*b19 - 5001*b18 + 2464*b17 - 1026*b16 - 2722*b15 + 6345*b14 + 5889*b13 + 7015*b12 + 6741*b11 - 2152*b10 - 56*b9 + 302*b8 - 8041*b7 - 3040*b6 - 16038*b4 + 3727*b3 - 302*b2 - 7695*b1 - 56 $$\nu^{12}$$ $$=$$ $$9737 \beta_{16} - 9737 \beta_{15} + 332 \beta_{14} + 332 \beta_{12} + 8135 \beta_{11} + 8135 \beta_{10} - 10683 \beta_{7} - 10763 \beta_{6} + 2842 \beta_{5} - 11681 \beta_{4} + 11423 \beta_{3} + 15653 \beta_{2} + \cdots + 40791$$ 9737*b16 - 9737*b15 + 332*b14 + 332*b12 + 8135*b11 + 8135*b10 - 10683*b7 - 10763*b6 + 2842*b5 - 11681*b4 + 11423*b3 + 15653*b2 + 40791 $$\nu^{13}$$ $$=$$ $$28749 \beta_{19} + 37910 \beta_{18} - 20356 \beta_{17} + 23956 \beta_{16} + 8829 \beta_{15} - 20000 \beta_{14} - 47244 \beta_{13} - 26298 \beta_{12} - 17578 \beta_{11} + 53651 \beta_{10} + 386 \beta_{9} + \cdots + 52927 \beta_1$$ 28749*b19 + 37910*b18 - 20356*b17 + 23956*b16 + 8829*b15 - 20000*b14 - 47244*b13 - 26298*b12 - 17578*b11 + 53651*b10 + 386*b9 - 2538*b8 + 11171*b7 + 11171*b6 + 47244*b5 - 36073*b4 + 52927*b1 $$\nu^{14}$$ $$=$$ $$- 16645 \beta_{19} - 1004 \beta_{18} + 179652 \beta_{17} + 6992 \beta_{16} + 86941 \beta_{15} + 30354 \beta_{14} - 28061 \beta_{13} - 42603 \beta_{12} - 87774 \beta_{11} + 21534 \beta_{10} + \cdots - 302152$$ -16645*b19 - 1004*b18 + 179652*b17 + 6992*b16 + 86941*b15 + 30354*b14 - 28061*b13 - 42603*b12 - 87774*b11 + 21534*b10 - 302152*b9 + 114906*b8 + 49595*b7 + 50599*b6 + 30763*b4 - 92807*b3 - 114906*b2 + 96074*b1 - 302152 $$\nu^{15}$$ $$=$$ $$- 129544 \beta_{16} + 129544 \beta_{15} - 218511 \beta_{14} - 218511 \beta_{12} - 282629 \beta_{11} - 282629 \beta_{10} + 422291 \beta_{7} + 136536 \beta_{6} - 375607 \beta_{5} + 1190219 \beta_{4} + \cdots + 252$$ -129544*b16 + 129544*b15 - 218511*b14 - 218511*b12 - 282629*b11 - 282629*b10 + 422291*b7 + 136536*b6 - 375607*b5 + 1190219*b4 - 232425*b3 + 19022*b2 + 252 $$\nu^{16}$$ $$=$$ $$95613 \beta_{19} + 10008 \beta_{18} - 1336036 \beta_{17} - 715923 \beta_{16} - 71110 \beta_{15} - 200731 \beta_{14} + 260972 \beta_{13} + 372972 \beta_{12} + 176228 \beta_{11} - 709041 \beta_{10} + \cdots - 682063 \beta_1$$ 95613*b19 + 10008*b18 - 1336036*b17 - 715923*b16 - 71110*b15 - 200731*b14 + 260972*b13 + 372972*b12 + 176228*b11 - 709041*b10 + 2259523*b9 - 855587*b8 + 271841*b7 + 271841*b6 - 260972*b5 + 532813*b4 - 682063*b1 $$\nu^{17}$$ $$=$$ $$- 1851193 \beta_{19} - 2149200 \beta_{18} + 1263312 \beta_{17} - 613931 \beta_{16} - 1695944 \beta_{15} + 2978621 \beta_{14} + 2967752 \beta_{13} + 3446703 \beta_{12} + 3302323 \beta_{11} + \cdots + 36976$$ -1851193*b19 - 2149200*b18 + 1263312*b17 - 613931*b16 - 1695944*b15 + 2978621*b14 + 2967752*b13 + 3446703*b12 + 3302323*b11 - 1092882*b10 + 36976*b9 + 129670*b8 - 4060634*b7 - 1911434*b6 - 6877705*b4 + 1834110*b3 - 129670*b2 - 2687401*b1 + 36976 $$\nu^{18}$$ $$=$$ $$5142647 \beta_{16} - 5142647 \beta_{15} - 145569 \beta_{14} - 145569 \beta_{12} + 4254666 \beta_{11} + 4254666 \beta_{10} - 5671250 \beta_{7} - 5756578 \beta_{6} + 2333197 \beta_{5} + \cdots + 17011682$$ 5142647*b16 - 5142647*b15 - 145569*b14 - 145569*b12 + 4254666*b11 + 4254666*b10 - 5671250*b7 - 5756578*b6 + 2333197*b5 - 6504849*b4 + 5867259*b3 + 6433300*b2 + 17011682 $$\nu^{19}$$ $$=$$ $$14578856 \beta_{19} + 16161297 \beta_{18} - 9686600 \beta_{17} + 13906594 \beta_{16} + 5014166 \beta_{15} - 11112322 \beta_{14} - 23345353 \beta_{13} - 14990584 \beta_{12} + \cdots + 19667087 \beta_1$$ 14578856*b19 + 16161297*b18 - 9686600*b17 + 13906594*b16 + 5014166*b15 - 11112322*b14 - 23345353*b13 - 14990584*b12 - 8480700*b11 + 25727897*b10 - 706004*b9 - 796794*b8 + 6098156*b7 + 6098156*b6 + 23345353*b5 - 17247197*b4 + 19667087*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$346$$ $$442$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{9}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 1.39981 − 2.42454i 1.25760 − 2.17823i 1.13751 − 1.97023i 0.599601 − 1.03854i 0.432430 − 0.748990i −0.0131282 + 0.0227388i −0.384545 + 0.666052i −0.526755 + 0.912367i −1.03253 + 1.78839i −1.37000 + 2.37290i 1.39981 + 2.42454i 1.25760 + 2.17823i 1.13751 + 1.97023i 0.599601 + 1.03854i 0.432430 + 0.748990i −0.0131282 − 0.0227388i −0.384545 − 0.666052i −0.526755 − 0.912367i −1.03253 − 1.78839i −1.37000 − 2.37290i
−1.39981 2.42454i 0.500000 0.866025i −2.91893 + 5.05574i 0.973538 + 1.68622i −2.79962 −2.64333 + 0.113245i 10.7446 −0.500000 0.866025i 2.72553 4.72076i
277.2 −1.25760 2.17823i 0.500000 0.866025i −2.16313 + 3.74664i −2.15868 3.73894i −2.51520 1.42694 2.22797i 5.85100 −0.500000 0.866025i −5.42951 + 9.40419i
277.3 −1.13751 1.97023i 0.500000 0.866025i −1.58786 + 2.75026i 0.619347 + 1.07274i −2.27502 1.84753 + 1.89384i 2.67481 −0.500000 0.866025i 1.40903 2.44051i
277.4 −0.599601 1.03854i 0.500000 0.866025i 0.280958 0.486633i 0.286327 + 0.495932i −1.19920 −1.87398 1.86767i −3.07225 −0.500000 0.866025i 0.343363 0.594723i
277.5 −0.432430 0.748990i 0.500000 0.866025i 0.626009 1.08428i 1.24779 + 2.16123i −0.864859 −0.0271380 + 2.64561i −2.81254 −0.500000 0.866025i 1.07916 1.86916i
277.6 0.0131282 + 0.0227388i 0.500000 0.866025i 0.999655 1.73145i −1.38914 2.40606i 0.0262565 −2.58821 + 0.548771i 0.105008 −0.500000 0.866025i 0.0364740 0.0631748i
277.7 0.384545 + 0.666052i 0.500000 0.866025i 0.704250 1.21980i 2.01355 + 3.48756i 0.769091 2.63574 0.229917i 2.62145 −0.500000 0.866025i −1.54860 + 2.68225i
277.8 0.526755 + 0.912367i 0.500000 0.866025i 0.445058 0.770863i −1.08806 1.88457i 1.05351 1.36439 + 2.26681i 3.04477 −0.500000 0.866025i 1.14628 1.98542i
277.9 1.03253 + 1.78839i 0.500000 0.866025i −1.13223 + 1.96108i 0.304427 + 0.527282i 2.06506 2.05425 1.66735i −0.546135 −0.500000 0.866025i −0.628658 + 1.08887i
277.10 1.37000 + 2.37290i 0.500000 0.866025i −2.75378 + 4.76968i 1.69091 + 2.92874i 2.73999 −2.19620 1.47537i −9.61066 −0.500000 0.866025i −4.63307 + 8.02471i
415.1 −1.39981 + 2.42454i 0.500000 + 0.866025i −2.91893 5.05574i 0.973538 1.68622i −2.79962 −2.64333 0.113245i 10.7446 −0.500000 + 0.866025i 2.72553 + 4.72076i
415.2 −1.25760 + 2.17823i 0.500000 + 0.866025i −2.16313 3.74664i −2.15868 + 3.73894i −2.51520 1.42694 + 2.22797i 5.85100 −0.500000 + 0.866025i −5.42951 9.40419i
415.3 −1.13751 + 1.97023i 0.500000 + 0.866025i −1.58786 2.75026i 0.619347 1.07274i −2.27502 1.84753 1.89384i 2.67481 −0.500000 + 0.866025i 1.40903 + 2.44051i
415.4 −0.599601 + 1.03854i 0.500000 + 0.866025i 0.280958 + 0.486633i 0.286327 0.495932i −1.19920 −1.87398 + 1.86767i −3.07225 −0.500000 + 0.866025i 0.343363 + 0.594723i
415.5 −0.432430 + 0.748990i 0.500000 + 0.866025i 0.626009 + 1.08428i 1.24779 2.16123i −0.864859 −0.0271380 2.64561i −2.81254 −0.500000 + 0.866025i 1.07916 + 1.86916i
415.6 0.0131282 0.0227388i 0.500000 + 0.866025i 0.999655 + 1.73145i −1.38914 + 2.40606i 0.0262565 −2.58821 0.548771i 0.105008 −0.500000 + 0.866025i 0.0364740 + 0.0631748i
415.7 0.384545 0.666052i 0.500000 + 0.866025i 0.704250 + 1.21980i 2.01355 3.48756i 0.769091 2.63574 + 0.229917i 2.62145 −0.500000 + 0.866025i −1.54860 2.68225i
415.8 0.526755 0.912367i 0.500000 + 0.866025i 0.445058 + 0.770863i −1.08806 + 1.88457i 1.05351 1.36439 2.26681i 3.04477 −0.500000 + 0.866025i 1.14628 + 1.98542i
415.9 1.03253 1.78839i 0.500000 + 0.866025i −1.13223 1.96108i 0.304427 0.527282i 2.06506 2.05425 + 1.66735i −0.546135 −0.500000 + 0.866025i −0.628658 1.08887i
415.10 1.37000 2.37290i 0.500000 + 0.866025i −2.75378 4.76968i 1.69091 2.92874i 2.73999 −2.19620 + 1.47537i −9.61066 −0.500000 + 0.866025i −4.63307 8.02471i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.h 20
7.c even 3 1 inner 483.2.i.h 20
7.c even 3 1 3381.2.a.bi 10
7.d odd 6 1 3381.2.a.bj 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.h 20 1.a even 1 1 trivial
483.2.i.h 20 7.c even 3 1 inner
3381.2.a.bi 10 7.c even 3 1
3381.2.a.bj 10 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$:

 $$T_{2}^{20} + 3 T_{2}^{19} + 22 T_{2}^{18} + 43 T_{2}^{17} + 245 T_{2}^{16} + 416 T_{2}^{15} + 1707 T_{2}^{14} + 2021 T_{2}^{13} + 7135 T_{2}^{12} + 6640 T_{2}^{11} + 20315 T_{2}^{10} + 10565 T_{2}^{9} + 29358 T_{2}^{8} + 9009 T_{2}^{7} + \cdots + 4$$ T2^20 + 3*T2^19 + 22*T2^18 + 43*T2^17 + 245*T2^16 + 416*T2^15 + 1707*T2^14 + 2021*T2^13 + 7135*T2^12 + 6640*T2^11 + 20315*T2^10 + 10565*T2^9 + 29358*T2^8 + 9009*T2^7 + 30661*T2^6 + 4026*T2^5 + 15786*T2^4 + 84*T2^3 + 5800*T2^2 - 152*T2 + 4 $$T_{5}^{20} - 5 T_{5}^{19} + 48 T_{5}^{18} - 187 T_{5}^{17} + 1263 T_{5}^{16} - 4454 T_{5}^{15} + 20191 T_{5}^{14} - 56713 T_{5}^{13} + 197781 T_{5}^{12} - 485572 T_{5}^{11} + 1332915 T_{5}^{10} - 2655713 T_{5}^{9} + \cdots + 556516$$ T5^20 - 5*T5^19 + 48*T5^18 - 187*T5^17 + 1263*T5^16 - 4454*T5^15 + 20191*T5^14 - 56713*T5^13 + 197781*T5^12 - 485572*T5^11 + 1332915*T5^10 - 2655713*T5^9 + 5625834*T5^8 - 9378557*T5^7 + 15668047*T5^6 - 19545514*T5^5 + 20588124*T5^4 - 14636660*T5^3 + 7945980*T5^2 - 2561764*T5 + 556516 $$T_{11}^{20} + 8 T_{11}^{19} + 88 T_{11}^{18} + 320 T_{11}^{17} + 2504 T_{11}^{16} + 6744 T_{11}^{15} + 49792 T_{11}^{14} + 87344 T_{11}^{13} + 539456 T_{11}^{12} + 847040 T_{11}^{11} + 4167744 T_{11}^{10} + 5391104 T_{11}^{9} + \cdots + 31719424$$ T11^20 + 8*T11^19 + 88*T11^18 + 320*T11^17 + 2504*T11^16 + 6744*T11^15 + 49792*T11^14 + 87344*T11^13 + 539456*T11^12 + 847040*T11^11 + 4167744*T11^10 + 5391104*T11^9 + 20048768*T11^8 + 23564288*T11^7 + 67148032*T11^6 + 57151488*T11^5 + 107479040*T11^4 + 51494912*T11^3 + 101711872*T11^2 + 46137344*T11 + 31719424

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 3 T^{19} + 22 T^{18} + 43 T^{17} + \cdots + 4$$
$3$ $$(T^{2} - T + 1)^{10}$$
$5$ $$T^{20} - 5 T^{19} + 48 T^{18} + \cdots + 556516$$
$7$ $$T^{20} - 11 T^{18} + \cdots + 282475249$$
$11$ $$T^{20} + 8 T^{19} + 88 T^{18} + \cdots + 31719424$$
$13$ $$(T^{10} - 57 T^{8} + 32 T^{7} + 1150 T^{6} + \cdots + 30667)^{2}$$
$17$ $$T^{20} - 11 T^{19} + \cdots + 53029799524$$
$19$ $$T^{20} + T^{19} + 104 T^{18} + \cdots + 13176284944$$
$23$ $$(T^{2} + T + 1)^{10}$$
$29$ $$(T^{10} - 22 T^{9} + 106 T^{8} + \cdots + 198784)^{2}$$
$31$ $$T^{20} - 3 T^{19} + \cdots + 3370484748544$$
$37$ $$T^{20} + \cdots + 345393601613824$$
$41$ $$(T^{10} + 26 T^{9} + 102 T^{8} + \cdots - 30634016)^{2}$$
$43$ $$(T^{10} - 27 T^{9} + 151 T^{8} + \cdots + 59884)^{2}$$
$47$ $$T^{20} + 11 T^{19} + 252 T^{18} + \cdots + 5161984$$
$53$ $$T^{20} + 5 T^{19} + \cdots + 7340588259904$$
$59$ $$T^{20} - 10 T^{19} + \cdots + 119638508544$$
$61$ $$T^{20} + 22 T^{19} + \cdots + 155728101376$$
$67$ $$T^{20} - 2 T^{19} + \cdots + 3442498027609$$
$71$ $$(T^{10} - 27 T^{9} - 73 T^{8} + \cdots + 5367896)^{2}$$
$73$ $$T^{20} - 8 T^{19} + \cdots + 60216094569409$$
$79$ $$T^{20} + 21 T^{19} + \cdots + 21\!\cdots\!64$$
$83$ $$(T^{10} + 12 T^{9} - 468 T^{8} + \cdots - 947876608)^{2}$$
$89$ $$T^{20} + 6 T^{19} + \cdots + 175748762176$$
$97$ $$(T^{10} - 6 T^{9} - 684 T^{8} + \cdots - 21133952)^{2}$$