# Properties

 Label 230.2.g.b Level $230$ Weight $2$ Character orbit 230.g Analytic conductor $1.837$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.g (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} - 5910 x^{11} + 14651 x^{10} - 22501 x^{9} + 26003 x^{8} - 19607 x^{7} + \cdots + 529$$ x^20 - 5*x^19 + 12*x^18 + 16*x^17 - 49*x^16 + 59*x^15 - 197*x^14 + 42*x^13 + 1625*x^12 - 5910*x^11 + 14651*x^10 - 22501*x^9 + 26003*x^8 - 19607*x^7 + 13040*x^6 + 348*x^5 + 7193*x^4 + 10771*x^3 + 8781*x^2 + 3105*x + 529 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} - 5910 x^{11} + 14651 x^{10} - 22501 x^{9} + 26003 x^{8} - 19607 x^{7} + \cdots + 529$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 25\!\cdots\!20 \nu^{19} + \cdots + 57\!\cdots\!44 ) / 14\!\cdots\!87$$ (255528629899781294702654313316170404120*v^19 - 410065944618450785823791000938804115721*v^18 - 2045645195370603517782253699682784016717*v^17 + 18700026437074353967806730698312963396740*v^16 - 10100647106470328934588880284628304589826*v^15 - 32874605548804069769781723574766053706270*v^14 + 35860499624853403870730225286669603856475*v^13 - 226364459343756073862592953227587281586869*v^12 + 657862496724753733551235025714026116364721*v^11 - 253883448813212434204194256899857548498357*v^10 - 2518013112170390490193506200136566382621953*v^9 + 11990364229975521405450639018988032975556466*v^8 - 27007821815923048275387059034539645322859467*v^7 + 43737296124529339274974693135429145385804401*v^6 - 50913487129694830485159016227496209407638511*v^5 + 51151910357941535274913602092411660382726981*v^4 - 37788887720258171885457800239681042901471107*v^3 + 24468466196889426756571437367101920213282078*v^2 + 3995305575111579509743542587368443364808612*v + 576596384662590640840290327098545945888844) / 14101225796484310218514733144554606230186187 $$\beta_{3}$$ $$=$$ $$( - 68\!\cdots\!51 \nu^{19} + \cdots - 28\!\cdots\!27 ) / 14\!\cdots\!87$$ (-683711706220360357728869142279446030651*v^19 - 7515695955299456518531760708164809853716*v^18 + 54371436216124698240133980795507547417021*v^17 - 184016117701359584376047038342543631711519*v^16 - 34978847337389456927194009470630063932959*v^15 + 593925873283190331164125245494862516528175*v^14 - 936002093998199943625269212372932896240855*v^13 + 2706224873653860245317885341300806241904644*v^12 - 3274263846347939519738348015822023998750848*v^11 - 12969028133335738055725515055484609968222671*v^10 + 67394893208217107313092045925015835528161190*v^9 - 195148871823597025536938331897965777481939414*v^8 + 357998503905020636943657220568952301656136344*v^7 - 483111368540189198855984189348495980926931449*v^6 + 461735688925344546288322804779075251008898884*v^5 - 356346834891480775843375612809044370609802022*v^4 + 137892034011603810527497871341931218462980454*v^3 - 113925911564217741556141155877827230401784365*v^2 - 69880007139624976089742242363709863918284220*v - 28663337597692015312050660097057632995793027) / 14101225796484310218514733144554606230186187 $$\beta_{4}$$ $$=$$ $$( 10\!\cdots\!36 \nu^{19} + \cdots - 61\!\cdots\!32 ) / 14\!\cdots\!87$$ (1089974262122099510095066780904623716236*v^19 - 5705399940510278845177988217839288985300*v^18 + 13489757090083644906964592371794288710553*v^17 + 19485233389324195679303322194156763476493*v^16 - 72108765281057229962465002962639525492304*v^15 + 74409128571674200030197820358001103847750*v^14 - 181850324089249533718946432263444818392222*v^13 + 9918419384274775553262579511324592225437*v^12 + 1997572635292167777767076472197600820470369*v^11 - 7099610385866361838213079700860352279319481*v^10 + 16223096363164092356607017663933499615071993*v^9 - 22007497759838970586455591436998371856404283*v^8 + 16352236507985432155551382484874897517728242*v^7 + 5636696458495043180953084661342688118620215*v^6 - 29524031746457161663335022312432852126086961*v^5 + 51292798172913321114672099467251018460888639*v^4 - 43311725490497273498799786737364701991841433*v^3 + 49529000497575305708691764536804744949049063*v^2 - 14897402201195270958426655963978419361013762*v - 610935491222460530898360232659586725895832) / 14101225796484310218514733144554606230186187 $$\beta_{5}$$ $$=$$ $$( 64\!\cdots\!88 \nu^{19} + \cdots - 88\!\cdots\!56 ) / 14\!\cdots\!87$$ (6476514582945195858656628750998724817188*v^19 - 38687092091201713322372497466921394207551*v^18 + 111517364566217824023671376451260372023138*v^17 + 16076846287841508950946650016784100314032*v^16 - 386544856665726987808062887910663372641827*v^15 + 712998418514475131818711491778424701935828*v^14 - 1751497648546915203549767311469003133556029*v^13 + 1722195047701118037322924022379471671912043*v^12 + 9681708840001116589702303829554304152935206*v^11 - 48311052759391872698723658595773595692251056*v^10 + 135684398743592120995495185846777760609907314*v^9 - 252374329159592617648928644870704893168119933*v^8 + 349805859315143263117062281430669630322851408*v^7 - 359665139487091793426842167912681318631903823*v^6 + 295843137223870893981556343476036638579321795*v^5 - 154852187331187761486932670911601545031916205*v^4 + 93400055757043017022463592287576584799713461*v^3 + 26319005670857127872201514849478997648013333*v^2 + 4332975788016591397875261281677805620571183*v - 8833190003122577263656870121846560228283556) / 14101225796484310218514733144554606230186187 $$\beta_{6}$$ $$=$$ $$( 87\!\cdots\!15 \nu^{19} + \cdots + 23\!\cdots\!08 ) / 14\!\cdots\!87$$ (8747732413834047309201477943789463603215*v^19 - 51328602075384238825906557635201253635536*v^18 + 147033335776553980640951298734486994011778*v^17 + 26377813899777646111536319238164661482255*v^16 - 490308205569854506041126850555514787140587*v^15 + 927317067710029391953644180736717731760913*v^14 - 2398371887118165610479224185111077838552009*v^13 + 2220294220021207556665343494537415044535054*v^12 + 12934142271602553920459926727527943892616989*v^11 - 63474010248239463693622396043456116179881842*v^10 + 179541306359628038595940521005548867747546333*v^9 - 335571248287979877240319435872028792221618922*v^8 + 471373974380930328382460886769827497664534425*v^7 - 493251318284699366523764695323251088252314975*v^6 + 419764769109708616050858888604026850940442703*v^5 - 233809071217892289350762262500677227806634017*v^4 + 149380725014266169603363128970030184519234178*v^3 + 28730746479361724535132345775748768099483784*v^2 + 13047290258404863365491221899412099380758127*v + 2328523002888232391255165484176693209728908) / 14101225796484310218514733144554606230186187 $$\beta_{7}$$ $$=$$ $$( 16\!\cdots\!91 \nu^{19} + \cdots + 24\!\cdots\!54 ) / 14\!\cdots\!87$$ (16536306228823750469564300872492896090691*v^19 - 84790869234325897867664011448996187055991*v^18 + 208154721811081404277418554456812970218363*v^17 + 244695937436087844565900700353077996645544*v^16 - 859986836676896851994304579587997487027257*v^15 + 1078228342747432660151229815006837603114695*v^14 - 3315132479291106335425159251849312610673506*v^13 + 1011207777763368006969814944968169950938136*v^12 + 26951585664829314478911940836988387979007524*v^11 - 101470820349941220049155375517792121281432753*v^10 + 253506035775708135929899590123782878521811551*v^9 - 395889152668169512292039953493206450467391794*v^8 + 458679344004392751189388126254954589652877650*v^7 - 345527745421769814972418422989672681499320609*v^6 + 211632443854979487195582679481528964424757879*v^5 + 26503862940068459940016799602495847939013801*v^4 + 75927792695013080016060671228178123477008145*v^3 + 190244597491972680733202374016228627537462308*v^2 + 105731295205226232427814527252339487725722021*v + 24563085414072492567969196132598850579363354) / 14101225796484310218514733144554606230186187 $$\beta_{8}$$ $$=$$ $$( 16\!\cdots\!64 \nu^{19} + \cdots + 56\!\cdots\!03 ) / 14\!\cdots\!87$$ (16697901707226043976667051270031304779364*v^19 - 77012993953185024024678627599157799079632*v^18 + 161687728395510814397632117773454263144817*v^17 + 378683791881834527650344196771761248492962*v^16 - 802120337366234645905738862214749833874804*v^15 + 598631344060609606815293137021183609340649*v^14 - 2576488217809055531584697608417742339598880*v^13 - 1050185776843421356529751158127688332822741*v^12 + 28856285321943439499406882336180341938378543*v^11 - 89002890249704803312399969176330707093106034*v^10 + 196329905153176897603425309561455050630210908*v^9 - 240035087570701094523490134780196628230562050*v^8 + 181821208933406203876344689303919125009682159*v^7 + 22410100541562218866551407179165837513861460*v^6 - 141924501224864179971103819351473104308997263*v^5 + 301654007017985557285436477318007532642540467*v^4 - 34744180351110827162766571126266369753950953*v^3 + 273253155045574736695144401517083768578243105*v^2 + 172943280562009020031314892051623884915608617*v + 56179960588953457945426455475125006960496403) / 14101225796484310218514733144554606230186187 $$\beta_{9}$$ $$=$$ $$( - 76\!\cdots\!98 \nu^{19} + \cdots - 94\!\cdots\!93 ) / 61\!\cdots\!69$$ (-769371535886807415898597577509623037698*v^19 + 4043925502351004022252394345300132242647*v^18 - 10257825320669159719676662031503102516232*v^17 - 9791592359961488718024879550214478041342*v^16 + 40683897201610450183343548841590059452399*v^15 - 56643028302091917383693901116276844910571*v^14 + 165179392279918365016073504241536939675739*v^13 - 69227027605335702320980223738303834383016*v^12 - 1241808263869936723305052047121722261801417*v^11 + 4874361340443562452315334474643744817579082*v^10 - 12513229441821131425880141513186276997554733*v^9 + 20367600221720125710905610762940896776551995*v^8 - 24590831297892265410847899855488217876885605*v^7 + 19976064985309219775165780273828837839038816*v^6 - 13048444172218640662679492730281400652407879*v^5 + 843775304518894854268712074714495753719292*v^4 - 4392032758627480541859741567334125045257830*v^3 - 7630342556389315320751670403675230862185028*v^2 - 4607990941231312597548017957592660897569865*v - 946456592727141154685840012732485527169593) / 613096773760187400804988397589330705660269 $$\beta_{10}$$ $$=$$ $$( - 21\!\cdots\!27 \nu^{19} + \cdots - 39\!\cdots\!82 ) / 14\!\cdots\!87$$ (-21873890644790335368591682094765884389727*v^19 + 111099096538971443010113106124099381325932*v^18 - 270996218937047637305494370195531335080156*v^17 - 329104033264265202114102349746687690968205*v^16 + 1098017905004229069707705459249424325698722*v^15 - 1367780115582579081121261326534776449547660*v^14 + 4427388024728001695090479269893454529219571*v^13 - 1272312851080890695617314286544472487604287*v^12 - 35496505027516390065291569747742985480575428*v^11 + 131967595884455422273585896903312099726893217*v^10 - 330590272636364135564494497344066372088698234*v^9 + 518043740716420026955822155876408110076524672*v^8 - 608948157553671478454124051937979736609937318*v^7 + 477498884873362845592892337810393323199371137*v^6 - 321903225020597943510510050636389918787788733*v^5 + 16823561877542925579441480931023518858087050*v^4 - 153573523106294298075007374390768209737547984*v^3 - 223360047759643716125225035115220214526993327*v^2 - 165743594427561308049979553992118595077976868*v - 39729165249764770550346595559753547479300682) / 14101225796484310218514733144554606230186187 $$\beta_{11}$$ $$=$$ $$( 22\!\cdots\!84 \nu^{19} + \cdots - 32\!\cdots\!98 ) / 14\!\cdots\!87$$ (22315072069352036275731225375839158346984*v^19 - 124190669593342636449796597232019284752905*v^18 + 333649083792539766403124552382408975922910*v^17 + 192887323174586445818713308119134495123303*v^16 - 1267039767240736863804775861090444104695000*v^15 + 1988862847498207369219214673565291902049504*v^14 - 5245461272330918182356709260264740867397975*v^13 + 3534165768200183712709546924491751862285459*v^12 + 35174505313945765457219761698807712370717288*v^11 - 152447730832141306920027963429064915272392281*v^10 + 405966819073834521486984516230994782868801443*v^9 - 701269028091899134610169673260844778472049781*v^8 + 899384080864409135392553044665613495082352690*v^7 - 817198582029018682382463095627492156823324146*v^6 + 598459795550571884905989148767397045313523319*v^5 - 204061943452705573211915848792317666732492524*v^4 + 191232083573495132599382541120708429070266799*v^3 + 150084429704491305804491818412078071184944182*v^2 + 84033198592638875273710486667241941843448734*v - 3267510901729584255396732734812457302887298) / 14101225796484310218514733144554606230186187 $$\beta_{12}$$ $$=$$ $$( - 22\!\cdots\!71 \nu^{19} + \cdots - 58\!\cdots\!01 ) / 14\!\cdots\!87$$ (-22723689447299579085238474191326170657771*v^19 + 117189285033456819960566109543276415036705*v^18 - 290592304699077562689496873841191552683755*v^17 - 323297075499936086717575684384512806368469*v^16 + 1184927411501310750830722299510273217275781*v^15 - 1554154792060874089908292653854971969901321*v^14 + 4652185040059003122354353424221157795545216*v^13 - 1500812324070008816745740238458199130700013*v^12 - 36972543315670805904321824417035942819049046*v^11 + 140687621806141886343867187624221924807218645*v^10 - 354214033275370059707740737807481655952799247*v^9 + 559453262027633949833426855925928738914292295*v^8 - 654007899654260032483215198423772666549925696*v^7 + 493181586521668037202758029592033376461753363*v^6 - 296333885670754645940623688653025495862283604*v^5 - 42311450258491943077580438348632915887235236*v^4 - 101333018089881289266516558014439982117693318*v^3 - 254132965792762171471149908074124313960802683*v^2 - 142026771572912347981857631580881552496387976*v - 58707105872182896830887227450865342929784301) / 14101225796484310218514733144554606230186187 $$\beta_{13}$$ $$=$$ $$( 28\!\cdots\!65 \nu^{19} + \cdots + 30\!\cdots\!95 ) / 14\!\cdots\!87$$ (28655829761855603460481536402724378294665*v^19 - 152929258400319699449343275673988309388846*v^18 + 393163869833839493488342004296049292581836*v^17 + 339275693884478729310359047635164245252963*v^16 - 1556965425124294870835836275919520639910634*v^15 + 2211531591667697242425122449354688041455767*v^14 - 6259702648052365870061621984400254697331911*v^13 + 3046027756252274313065325763628105948718908*v^12 + 46156650467760771107510637382048540892270656*v^11 - 185456562107727962849271040833732589238042570*v^10 + 479044442953872058443550937425932484415531507*v^9 - 789414348955430435324805626293666362810310892*v^8 + 963641701344194353209198297237835972015310059*v^7 - 798970717067029265810793352819415342822336077*v^6 + 521128199668302903687416257208616501109724321*v^5 - 46228261754998232299955197986731363778319979*v^4 + 135081920991343867972808548031480810855714700*v^3 + 296728419327850674746985061559737022256796425*v^2 + 134920730515552617831168581306937707052071860*v + 30160879958235933823805063274773415495415295) / 14101225796484310218514733144554606230186187 $$\beta_{14}$$ $$=$$ $$( - 29\!\cdots\!15 \nu^{19} + \cdots - 61\!\cdots\!61 ) / 14\!\cdots\!87$$ (-29172331874072462435043417722259372253615*v^19 + 146520591771491872633796181609421011945915*v^18 - 349170806371069393656358560269393536554973*v^17 - 482790289948575015641589854915813070466559*v^16 + 1505336891244577456619388047997070186879161*v^15 - 1720701107935415999199574794795325575759403*v^14 + 5525259202865127777141470013867255808254640*v^13 - 943996833099952397110608948808599791967703*v^12 - 48325322062945283849365806325999444609856242*v^11 + 174167432123784450029353060466066836573109055*v^10 - 424677567020075623909020520022005933849012380*v^9 + 636087929231009609446669733696919531723840413*v^8 - 694290651793983221262703389653614358565247683*v^7 + 454209441234005308153040507826187767966417585*v^6 - 224722741132134411420781888389154741732235929*v^5 - 146071972989336574996255603995174904823167424*v^4 - 112267337452606718372798967701994118491659611*v^3 - 338831200527084780206297080640110932723759833*v^2 - 221615902435964933740124667274850155086200721*v - 61634652886615800646076394144178882047885461) / 14101225796484310218514733144554606230186187 $$\beta_{15}$$ $$=$$ $$( 40\!\cdots\!38 \nu^{19} + \cdots + 34\!\cdots\!17 ) / 14\!\cdots\!87$$ (40913136099619595177301457578571434180638*v^19 - 223779285468785851240631445054381385593038*v^18 + 594079252671684575889760312112805149685435*v^17 + 386523204736672026801763479127760810385491*v^16 - 2213216433425112835374272612773283214565111*v^15 + 3429358167987133661371546874530988903357283*v^14 - 9556504410757143157787079144794063919022407*v^13 + 6121715495289043385805628019927431953718853*v^12 + 64053441114383902547115315738342166789745662*v^11 - 272293501886605448603423627923409292586095810*v^10 + 723967806354081683808607240042931034014761964*v^9 - 1248019941273591484411092021728536321670622824*v^8 + 1617738834389240482345807457736485116052480965*v^7 - 1506113120485974694461081840177329595667531399*v^6 + 1167568890025050255504706463301319231794427659*v^5 - 448109040954917531146597475369258862249536518*v^4 + 422461487378593922687890922349124998452021149*v^3 + 304392056568020942986649382211968728556336065*v^2 + 173147375844075744715776195033078722358470374*v + 34746952256129854699237875135418588328720217) / 14101225796484310218514733144554606230186187 $$\beta_{16}$$ $$=$$ $$( 17\!\cdots\!17 \nu^{19} + \cdots + 94\!\cdots\!20 ) / 61\!\cdots\!69$$ (1789142897404803695058298700817552981417*v^19 - 9715086022910825891190091081597387944783*v^18 + 25513640271208648362951978755110768019651*v^17 + 18368461037807699401256117181577745186440*v^16 - 97459594332796869775881515890274574130775*v^15 + 146243328148493868191783172189825685356002*v^14 - 409104179090838245310178745177334782249720*v^13 + 240323393970920120208522049675874164895253*v^12 + 2838130180677470302148755165090219760419609*v^11 - 11815642787532326561099597368953460381975887*v^10 + 31087093930321341388614468740321713548319549*v^9 - 52770733776326619368386920580282036632418650*v^8 + 66890682982937236193506551880299726952338246*v^7 - 59670556087308251459855962482417979183528724*v^6 + 43306488367467859958725995332489728716716496*v^5 - 12425822443921768976799204782396892214874763*v^4 + 13713080165551647832823054629695154349051773*v^3 + 14878825389319660057613193739171738117584677*v^2 + 8080121225722265925555250488203701867637649*v + 947297755210602875607999508445841109729920) / 613096773760187400804988397589330705660269 $$\beta_{17}$$ $$=$$ $$( - 56\!\cdots\!79 \nu^{19} + \cdots - 13\!\cdots\!22 ) / 14\!\cdots\!87$$ (-56837741895859895981977729887560044637079*v^19 + 281412490926045977784849930230594089864987*v^18 - 661716886196020019763909606195365027710763*v^17 - 978681025991690210977060283477422926708810*v^16 + 2837345646587309861982100983995760917154550*v^15 - 3162871370094926975565702175787372966446356*v^14 + 10672618165827860172236799768724814822727710*v^13 - 1281902122268742254727291729636530758450629*v^12 - 93997278188033684288819448644157438381568142*v^11 + 332557207283642377714399004571086327996239961*v^10 - 806285540883241859924960114091385246663998138*v^9 + 1194129360072479021635986786577593633598344653*v^8 - 1298066221492429173540772094475172079242490975*v^7 + 837284220766102412002833869741542953530578939*v^6 - 416046843472643213022700297066447825529261395*v^5 - 315099680382193675877284390906362658401225545*v^4 - 207000763825589537058177447220183089695361366*v^3 - 713188839621533760375195817130553246390914659*v^2 - 472607435397105781278239550640576491270109159*v - 132043499403884808473727246995608557861326022) / 14101225796484310218514733144554606230186187 $$\beta_{18}$$ $$=$$ $$( - 72\!\cdots\!74 \nu^{19} + \cdots - 38\!\cdots\!34 ) / 14\!\cdots\!87$$ (-72656757924966472395861756755461538472174*v^19 + 396794838038150763476259434693638985175918*v^18 - 1049892623096345124277599696105528934534349*v^17 - 706727365291444669616857275459523341739330*v^16 + 3964945370144445618646407478512061019636231*v^15 - 6087327824019610560390998931578102460491931*v^14 + 16848769748339348214603180694287979914898085*v^13 - 10349142104176530986356544195263087399506485*v^12 - 114637342250346968537830296080375330832535275*v^11 + 483305543018569519041892564392698894230360354*v^10 - 1279748370659339364955041708536420181233315960*v^9 + 2190453063768646872079402360581290470140136014*v^8 - 2802623038436800049224853047667703082101937918*v^7 + 2540969058715135610246188560658799501091280747*v^6 - 1879773354864261776573405961972384784617373768*v^5 + 597663131664822432184685258837927701513615134*v^4 - 613097620083465940557902478215738598141050418*v^3 - 583327950292746040874628908692860544488507258*v^2 - 331098661991388112641837049623469903363975247*v - 38774114061701155818828730370421032720617134) / 14101225796484310218514733144554606230186187 $$\beta_{19}$$ $$=$$ $$( - 14\!\cdots\!30 \nu^{19} + \cdots - 16\!\cdots\!18 ) / 14\!\cdots\!87$$ (-149541693966899828206102875482725321217030*v^19 + 795970026859118670783706564677887382703664*v^18 - 2042445237419603301439701922854221450735894*v^17 - 1783893559232673159254844069041052368865182*v^16 + 8037927055166529805931483603760138968433263*v^15 - 11339691599972417859754016392110545457043405*v^14 + 32593345287018255414849383516853956476695496*v^13 - 16048251041227017376549429147287817323556252*v^12 - 239913016015833120851238288937096720298095987*v^11 + 962885161344286984531374416298755925203087428*v^10 - 2487322500209337719959822054214233990692709205*v^9 + 4105753256362206871081763461223403021969539143*v^8 - 5051272759061300938029491706446027624135657787*v^7 + 4279172248368804886564339860435636835824738434*v^6 - 2965669553789372331967805274201233682545730716*v^5 + 563019783445096387198243636042628199928997977*v^4 - 997135112667328439314290261599595734182626529*v^3 - 1398870771000610015420354770258151718994721399*v^2 - 774342268439531127495830312099237506561928412*v - 163489267414847069538698054005666427513686918) / 14101225796484310218514733144554606230186187
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 2$$ b18 - b17 + b16 + b15 - b13 + b12 + b11 - b9 - b8 - b7 - 2*b6 - b5 + 3*b4 + b3 - b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$3 \beta_{19} + \beta_{18} - 3 \beta_{17} + 6 \beta_{16} + 3 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} + 5 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 2$$ 3*b19 + b18 - 3*b17 + 6*b16 + 3*b15 - 3*b14 + 5*b13 + 5*b10 + 2*b9 - 3*b8 - 3*b7 + b6 - 2*b5 - b4 - 2*b3 - 5*b2 - 2 $$\nu^{4}$$ $$=$$ $$18 \beta_{19} - 2 \beta_{18} + 25 \beta_{16} + 12 \beta_{15} - 27 \beta_{14} + 18 \beta_{13} - 15 \beta_{12} - 4 \beta_{11} + 15 \beta_{9} - 4 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 13 \beta_{5} - 4 \beta_{3} - 5 \beta _1 - 19$$ 18*b19 - 2*b18 + 25*b16 + 12*b15 - 27*b14 + 18*b13 - 15*b12 - 4*b11 + 15*b9 - 4*b8 - 15*b7 - 4*b6 + 13*b5 - 4*b3 - 5*b1 - 19 $$\nu^{5}$$ $$=$$ $$29 \beta_{19} - 79 \beta_{18} + 52 \beta_{17} - 70 \beta_{16} - 27 \beta_{15} - 32 \beta_{14} + 15 \beta_{13} - 62 \beta_{12} + 20 \beta_{11} - 80 \beta_{10} + 79 \beta_{9} + 80 \beta_{8} + 42 \beta_{6} + 39 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 52 \beta_{2} + \cdots - 62$$ 29*b19 - 79*b18 + 52*b17 - 70*b16 - 27*b15 - 32*b14 + 15*b13 - 62*b12 + 20*b11 - 80*b10 + 79*b9 + 80*b8 + 42*b6 + 39*b5 - 5*b4 - 5*b3 + 52*b2 - 79*b1 - 62 $$\nu^{6}$$ $$=$$ $$- 56 \beta_{19} - 239 \beta_{18} + 282 \beta_{17} - 369 \beta_{16} - 239 \beta_{15} + 37 \beta_{14} - 210 \beta_{13} - 166 \beta_{12} - 19 \beta_{11} - 356 \beta_{10} + 56 \beta_{9} + 359 \beta_{8} + 221 \beta_{7} + 123 \beta_{6} + \cdots - 89$$ -56*b19 - 239*b18 + 282*b17 - 369*b16 - 239*b15 + 37*b14 - 210*b13 - 166*b12 - 19*b11 - 356*b10 + 56*b9 + 359*b8 + 221*b7 + 123*b6 + 226*b5 + 60*b3 + 222*b2 - 178*b1 - 89 $$\nu^{7}$$ $$=$$ $$- 1081 \beta_{19} - 361 \beta_{18} + 535 \beta_{17} - 1933 \beta_{16} - 1081 \beta_{15} + 1400 \beta_{14} - 1420 \beta_{13} + 361 \beta_{12} + 14 \beta_{11} - 854 \beta_{10} - 535 \beta_{9} + 640 \beta_{8} + 1038 \beta_{7} + \cdots + 852$$ -1081*b19 - 361*b18 + 535*b17 - 1933*b16 - 1081*b15 + 1400*b14 - 1420*b13 + 361*b12 + 14*b11 - 854*b10 - 535*b9 + 640*b8 + 1038*b7 - 45*b5 + 105*b4 + 339*b3 + 503*b2 - 45*b1 + 852 $$\nu^{8}$$ $$=$$ $$- 3798 \beta_{19} + 2438 \beta_{18} - 1058 \beta_{17} - 1773 \beta_{16} - 1058 \beta_{15} + 4464 \beta_{14} - 3798 \beta_{13} + 3798 \beta_{12} - 810 \beta_{11} + 1868 \beta_{10} - 4325 \beta_{9} - 1922 \beta_{8} + \cdots + 4464$$ -3798*b19 + 2438*b18 - 1058*b17 - 1773*b16 - 1058*b15 + 4464*b14 - 3798*b13 + 3798*b12 - 810*b11 + 1868*b10 - 4325*b9 - 1922*b8 + 2438*b7 - 1876*b6 - 2797*b5 + 413*b4 + 1221*b3 - 1001*b2 + 3267*b1 + 4464 $$\nu^{9}$$ $$=$$ $$- 5135 \beta_{19} + 16473 \beta_{18} - 15916 \beta_{17} + 15193 \beta_{16} + 8789 \beta_{15} + 6870 \beta_{14} + 420 \beta_{13} + 15916 \beta_{12} - 1210 \beta_{11} + 21605 \beta_{10} - 12724 \beta_{9} + \cdots + 14706$$ -5135*b19 + 16473*b18 - 15916*b17 + 15193*b16 + 8789*b15 + 6870*b14 + 420*b13 + 15916*b12 - 1210*b11 + 21605*b10 - 12724*b9 - 21471*b8 - 5135*b7 - 10069*b6 - 15916*b5 + 1735*b4 - 12724*b2 + 14481*b1 + 14706 $$\nu^{10}$$ $$=$$ $$36734 \beta_{19} + 50703 \beta_{18} - 59591 \beta_{17} + 118069 \beta_{16} + 66327 \beta_{15} - 47053 \beta_{14} + 67366 \beta_{13} + 17683 \beta_{12} - 4813 \beta_{11} + 87212 \beta_{10} - 77404 \beta_{8} + \cdots - 9938$$ 36734*b19 + 50703*b18 - 59591*b17 + 118069*b16 + 66327*b15 - 47053*b14 + 67366*b13 + 17683*b12 - 4813*b11 + 87212*b10 - 77404*b8 - 59591*b7 - 19274*b6 - 33307*b5 - 4813*b4 - 17813*b3 - 50703*b2 + 33020*b1 - 9938 $$\nu^{11}$$ $$=$$ $$256857 \beta_{19} - 76135 \beta_{17} + 344581 \beta_{16} + 196934 \beta_{15} - 318507 \beta_{14} + 318507 \beta_{13} - 138918 \beta_{12} + 22073 \beta_{11} + 101338 \beta_{10} + 196934 \beta_{9} + \cdots - 232313$$ 256857*b19 - 76135*b17 + 344581*b16 + 196934*b15 - 318507*b14 + 318507*b13 - 138918*b12 + 22073*b11 + 101338*b10 + 196934*b9 - 76135*b8 - 240976*b7 + 37850*b6 + 59923*b5 - 25203*b4 - 87724*b3 - 59923*b2 - 76135*b1 - 232313 $$\nu^{12}$$ $$=$$ $$778490 \beta_{19} - 778490 \beta_{18} + 560354 \beta_{17} - 972564 \beta_{14} + 654831 \beta_{13} - 1019617 \beta_{12} + 123659 \beta_{11} - 777936 \beta_{10} + 1019617 \beta_{9} + \cdots - 1153146$$ 778490*b19 - 778490*b18 + 560354*b17 - 972564*b14 + 654831*b13 - 1019617*b12 + 123659*b11 - 777936*b10 + 1019617*b9 + 830113*b8 - 282060*b7 + 560908*b6 + 842414*b5 - 133529*b4 - 194074*b3 + 459263*b2 - 863156*b1 - 1153146 $$\nu^{13}$$ $$=$$ $$- 3979257 \beta_{18} + 3979257 \beta_{17} - 5344858 \beta_{16} - 3042920 \beta_{15} - 1672508 \beta_{13} - 3042920 \beta_{12} + 397588 \beta_{11} - 5651765 \beta_{10} + 2164279 \beta_{9} + \cdots - 2111735$$ -3979257*b18 + 3979257*b17 - 5344858*b16 - 3042920*b15 - 1672508*b13 - 3042920*b12 + 397588*b11 - 5651765*b10 + 2164279*b9 + 5344858*b8 + 2164279*b7 + 2111735*b6 + 3414324*b5 - 117728*b4 + 397588*b3 + 3317425*b2 - 3317425*b1 - 2111735 $$\nu^{14}$$ $$=$$ $$- 11985926 \beta_{19} - 8599271 \beta_{18} + 11985926 \beta_{17} - 27688350 \beta_{16} - 15700939 \beta_{15} + 14954050 \beta_{14} - 18419890 \beta_{13} - 16918620 \beta_{10} + \cdots + 6433964$$ -11985926*b19 - 8599271*b18 + 11985926*b17 - 27688350*b16 - 15700939*b15 + 14954050*b14 - 18419890*b13 - 16918620*b10 - 4413673*b9 + 14954050*b8 + 15700939*b7 + 2687957*b6 + 4413673*b5 + 1027056*b4 + 4932694*b3 + 9944941*b2 - 3715013*b1 + 6433964 $$\nu^{15}$$ $$=$$ $$- 61492387 \beta_{19} + 17642007 \beta_{18} - 58907654 \beta_{16} - 33525393 \beta_{15} + 76549661 \beta_{14} - 69535225 \beta_{13} + 46978173 \beta_{12} - 6084356 \beta_{11} + \cdots + 66536352$$ -61492387*b19 + 17642007*b18 - 58907654*b16 - 33525393*b15 + 76549661*b14 - 69535225*b13 + 46978173*b12 - 6084356*b11 - 56720847*b9 - 6084356*b8 + 46978173*b7 - 19834963*b6 - 29336166*b5 + 8042838*b4 + 19558179*b3 + 32156221*b1 + 66536352 $$\nu^{16}$$ $$=$$ $$- 132340428 \beta_{19} + 221906105 \beta_{18} - 184749586 \beta_{17} + 120271015 \beta_{16} + 68506823 \beta_{15} + 165278556 \beta_{14} - 72491788 \beta_{13} + \cdots + 242001342$$ -132340428*b19 + 221906105*b18 - 184749586*b17 + 120271015*b16 + 68506823*b15 + 165278556*b14 - 72491788*b13 + 242001342*b12 - 30705338*b11 + 261063925*b10 - 221906105*b9 - 261063925*b8 - 140998611*b6 - 220942488*b5 + 23474981*b4 + 23474981*b3 - 153399282*b2 + 221906105*b1 + 242001342 $$\nu^{17}$$ $$=$$ $$271144644 \beta_{19} + 873784229 \beta_{18} - 949293604 \beta_{17} + 1539365880 \beta_{16} + 873784229 \beta_{15} - 338975821 \beta_{14} + 698859189 \beta_{13} + \cdots + 217095417$$ 271144644*b19 + 873784229*b18 - 949293604*b17 + 1539365880*b16 + 873784229*b15 - 338975821*b14 + 698859189*b13 + 518087766*b12 - 67831177*b11 + 1343730611*b10 - 271144644*b9 - 1250285953*b8 - 724971372*b7 - 424471848*b6 - 678148960*b5 - 178167737*b3 - 789232410*b2 + 649461997*b1 + 217095417 $$\nu^{18}$$ $$=$$ $$3426020339 \beta_{19} + 1059630164 \beta_{18} - 2039347103 \beta_{17} + 6028541248 \beta_{16} + 3426020339 \beta_{15} - 4266560602 \beta_{14} + 4710719625 \beta_{13} + \cdots - 2602520909$$ 3426020339*b19 + 1059630164*b18 - 2039347103*b17 + 6028541248*b16 + 3426020339*b15 - 4266560602*b14 + 4710719625*b13 - 1059630164*b12 + 140416675*b11 + 2879887366*b10 + 2039347103*b9 - 2403439748*b8 - 3731718622*b7 - 97039871*b5 - 364092645*b4 - 1284699286*b3 - 1692371519*b2 - 97039871*b1 - 2602520909 $$\nu^{19}$$ $$=$$ $$13471243579 \beta_{19} - 7998279224 \beta_{18} + 4175877645 \beta_{17} + 7359415616 \beta_{16} + 4175877645 \beta_{15} - 16796005904 \beta_{14} + 13471243579 \beta_{13} + \cdots - 16796005904$$ 13471243579*b19 - 7998279224*b18 + 4175877645*b17 + 7359415616*b16 + 4175877645*b15 - 16796005904*b14 + 13471243579*b13 - 13471243579*b12 + 1716522174*b11 - 5892399819*b10 + 14648648527*b9 + 6927879328*b8 - 7998279224*b7 + 6543364251*b6 + 10008173667*b5 - 1886451261*b4 - 3850103282*b3 + 3463069912*b2 - 10472770882*b1 - 16796005904

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$1$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.588453 − 1.28853i −0.905632 + 1.98306i 2.57205 + 2.96831i −0.271308 − 0.313106i −0.138474 + 0.963109i 0.212636 − 1.47891i −0.138474 − 0.963109i 0.212636 + 1.47891i 2.57205 − 2.96831i −0.271308 + 0.313106i −2.49760 + 0.733362i 1.92732 − 0.565914i −0.319826 − 0.205540i 1.33237 + 0.856263i 0.588453 + 1.28853i −0.905632 − 1.98306i −0.319826 + 0.205540i 1.33237 − 0.856263i −2.49760 − 0.733362i 1.92732 + 0.565914i
0.142315 0.989821i −1.10832 2.42687i −0.959493 0.281733i −0.654861 0.755750i −2.55990 + 0.751655i −0.986597 0.634047i −0.415415 + 0.909632i −2.69677 + 3.11224i −0.841254 + 0.540641i
31.2 0.142315 0.989821i 1.40549 + 3.07760i −0.959493 0.281733i −0.654861 0.755750i 3.24630 0.953198i 2.97617 + 1.91267i −0.415415 + 0.909632i −5.53162 + 6.38383i −0.841254 + 0.540641i
41.1 0.959493 + 0.281733i −0.880272 + 1.01589i 0.841254 + 0.540641i −0.142315 + 0.989821i −1.13082 + 0.726736i 0.394126 + 0.863015i 0.654861 + 0.755750i 0.169795 + 1.18095i −0.415415 + 0.909632i
41.2 0.959493 + 0.281733i 1.48208 1.71041i 0.841254 + 0.540641i −0.142315 + 0.989821i 1.90392 1.22358i −0.119262 0.261148i 0.654861 + 0.755750i −0.302001 2.10046i −0.415415 + 0.909632i
71.1 −0.841254 0.540641i −0.420808 2.92679i 0.415415 + 0.909632i −0.959493 0.281733i −1.22833 + 2.68968i 2.73530 3.15671i 0.142315 0.989821i −5.51052 + 1.61803i 0.654861 + 0.755750i
71.2 −0.841254 0.540641i 0.0390478 + 0.271583i 0.415415 + 0.909632i −0.959493 0.281733i 0.113980 0.249581i −0.365070 + 0.421313i 0.142315 0.989821i 2.80625 0.823988i 0.654861 + 0.755750i
81.1 −0.841254 + 0.540641i −0.420808 + 2.92679i 0.415415 0.909632i −0.959493 + 0.281733i −1.22833 2.68968i 2.73530 + 3.15671i 0.142315 + 0.989821i −5.51052 1.61803i 0.654861 0.755750i
81.2 −0.841254 + 0.540641i 0.0390478 0.271583i 0.415415 0.909632i −0.959493 + 0.281733i 0.113980 + 0.249581i −0.365070 0.421313i 0.142315 + 0.989821i 2.80625 + 0.823988i 0.654861 0.755750i
101.1 0.959493 0.281733i −0.880272 1.01589i 0.841254 0.540641i −0.142315 0.989821i −1.13082 0.726736i 0.394126 0.863015i 0.654861 0.755750i 0.169795 1.18095i −0.415415 0.909632i
101.2 0.959493 0.281733i 1.48208 + 1.71041i 0.841254 0.540641i −0.142315 0.989821i 1.90392 + 1.22358i −0.119262 + 0.261148i 0.654861 0.755750i −0.302001 + 2.10046i −0.415415 0.909632i
121.1 −0.415415 + 0.909632i −1.50807 0.442808i −0.654861 0.755750i 0.841254 0.540641i 1.02927 1.18784i −0.440465 + 3.06350i 0.959493 0.281733i −0.445574 0.286353i 0.142315 + 0.989821i
121.2 −0.415415 + 0.909632i −0.248602 0.0729960i −0.654861 0.755750i 0.841254 0.540641i 0.169672 0.195812i 0.663794 4.61679i 0.959493 0.281733i −2.46729 1.58563i 0.142315 + 0.989821i
131.1 0.654861 0.755750i −1.71555 + 1.10252i −0.142315 0.989821i 0.415415 + 0.909632i −0.290219 + 2.01852i 3.10381 0.911362i −0.841254 0.540641i 0.481321 1.05395i 0.959493 + 0.281733i
131.2 0.654861 0.755750i 1.45499 0.935068i −0.142315 0.989821i 0.415415 + 0.909632i 0.246141 1.71195i 1.53819 0.451652i −0.841254 0.540641i −0.00358844 + 0.00785758i 0.959493 + 0.281733i
141.1 0.142315 + 0.989821i −1.10832 + 2.42687i −0.959493 + 0.281733i −0.654861 + 0.755750i −2.55990 0.751655i −0.986597 + 0.634047i −0.415415 0.909632i −2.69677 3.11224i −0.841254 0.540641i
141.2 0.142315 + 0.989821i 1.40549 3.07760i −0.959493 + 0.281733i −0.654861 + 0.755750i 3.24630 + 0.953198i 2.97617 1.91267i −0.415415 0.909632i −5.53162 6.38383i −0.841254 0.540641i
151.1 0.654861 + 0.755750i −1.71555 1.10252i −0.142315 + 0.989821i 0.415415 0.909632i −0.290219 2.01852i 3.10381 + 0.911362i −0.841254 + 0.540641i 0.481321 + 1.05395i 0.959493 0.281733i
151.2 0.654861 + 0.755750i 1.45499 + 0.935068i −0.142315 + 0.989821i 0.415415 0.909632i 0.246141 + 1.71195i 1.53819 + 0.451652i −0.841254 + 0.540641i −0.00358844 0.00785758i 0.959493 0.281733i
211.1 −0.415415 0.909632i −1.50807 + 0.442808i −0.654861 + 0.755750i 0.841254 + 0.540641i 1.02927 + 1.18784i −0.440465 3.06350i 0.959493 + 0.281733i −0.445574 + 0.286353i 0.142315 0.989821i
211.2 −0.415415 0.909632i −0.248602 + 0.0729960i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.169672 + 0.195812i 0.663794 + 4.61679i 0.959493 + 0.281733i −2.46729 + 1.58563i 0.142315 0.989821i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.g.b 20
23.c even 11 1 inner 230.2.g.b 20
23.c even 11 1 5290.2.a.bj 10
23.d odd 22 1 5290.2.a.bi 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.b 20 1.a even 1 1 trivial
230.2.g.b 20 23.c even 11 1 inner
5290.2.a.bi 10 23.d odd 22 1
5290.2.a.bj 10 23.c even 11 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} + 3 T_{3}^{19} + 21 T_{3}^{18} + 66 T_{3}^{17} + 252 T_{3}^{16} + 701 T_{3}^{15} + 1695 T_{3}^{14} + 3432 T_{3}^{13} + 7173 T_{3}^{12} + 15634 T_{3}^{11} + 34605 T_{3}^{10} + 46692 T_{3}^{9} + 31164 T_{3}^{8} + \cdots + 1024$$ acting on $$S_{2}^{\mathrm{new}}(230, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2}$$
$3$ $$T^{20} + 3 T^{19} + 21 T^{18} + \cdots + 1024$$
$5$ $$(T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2}$$
$7$ $$T^{20} - 19 T^{19} + 197 T^{18} + \cdots + 38809$$
$11$ $$T^{20} - 7 T^{19} + 40 T^{18} + \cdots + 2374681$$
$13$ $$T^{20} - 8 T^{19} + \cdots + 13416820561$$
$17$ $$T^{20} - 8 T^{19} + 25 T^{18} + \cdots + 605553664$$
$19$ $$T^{20} + T^{19} + 8 T^{18} + \cdots + 392951329$$
$23$ $$T^{20} + 9 T^{19} + \cdots + 41426511213649$$
$29$ $$T^{20} - 20 T^{19} + \cdots + 3969466491904$$
$31$ $$T^{20} + 17 T^{19} + 158 T^{18} + \cdots + 1024$$
$37$ $$T^{20} - 6 T^{19} + \cdots + 7650363233041$$
$41$ $$T^{20} + 2 T^{19} + \cdots + 1804635361$$
$43$ $$T^{20} + 18 T^{19} + \cdots + 23760372736$$
$47$ $$(T^{10} - 21 T^{9} - 141 T^{8} + \cdots + 130098649)^{2}$$
$53$ $$T^{20} - 19 T^{19} + 78 T^{18} + \cdots + 7447441$$
$59$ $$T^{20} - 25 T^{19} + \cdots + 27292338778849$$
$61$ $$T^{20} + 49 T^{19} + \cdots + 22\!\cdots\!04$$
$67$ $$T^{20} + \cdots + 246200702534656$$
$71$ $$T^{20} - 7 T^{19} + \cdots + 95950331339776$$
$73$ $$T^{20} + \cdots + 859836921496576$$
$79$ $$T^{20} + 33 T^{19} + \cdots + 90\!\cdots\!76$$
$83$ $$T^{20} + 41 T^{19} + \cdots + 160435495936$$
$89$ $$T^{20} - 55 T^{19} + \cdots + 20\!\cdots\!09$$
$97$ $$T^{20} + \cdots + 286047133533184$$