# Properties

 Label 273.2.j.c Level $273$ Weight $2$ Character orbit 273.j Analytic conductor $2.180$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(100,273)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("273.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} + 18 x^{18} - 4 x^{17} + 211 x^{16} - 59 x^{15} + 1458 x^{14} - 526 x^{13} + 7324 x^{12} - 2645 x^{11} + 23428 x^{10} - 8506 x^{9} + 54235 x^{8} - 18801 x^{7} + 74141 x^{6} + \cdots + 1369$$ x^20 + 18*x^18 - 4*x^17 + 211*x^16 - 59*x^15 + 1458*x^14 - 526*x^13 + 7324*x^12 - 2645*x^11 + 23428*x^10 - 8506*x^9 + 54235*x^8 - 18801*x^7 + 74141*x^6 - 25533*x^5 + 68867*x^4 - 16327*x^3 + 16588*x^2 + 2997*x + 1369 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 18 x^{18} - 4 x^{17} + 211 x^{16} - 59 x^{15} + 1458 x^{14} - 526 x^{13} + 7324 x^{12} - 2645 x^{11} + 23428 x^{10} - 8506 x^{9} + 54235 x^{8} - 18801 x^{7} + 74141 x^{6} + \cdots + 1369$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 40\!\cdots\!97 \nu^{19} + \cdots - 54\!\cdots\!82 ) / 13\!\cdots\!02$$ (-4085487136543173326688997*v^19 + 2948586670918701808551791*v^18 - 71989800983906175327149653*v^17 + 64392367693258135889221094*v^16 - 847126094504028583409134957*v^15 + 775116859497906120872527767*v^14 - 5804068931644273128789469255*v^13 + 5431359145409041149804181685*v^12 - 29186840814893966044352762673*v^11 + 25779796624540228220353572936*v^10 - 91498734861179839043604586484*v^9 + 71968215133683284893975622780*v^8 - 208089548964465772315175897620*v^7 + 148022831628311368404020457341*v^6 - 268061444077698210708445476719*v^5 + 106832788127261578410932469239*v^4 - 235294624201578712546452422958*v^3 - 3734581071136679412423316370*v^2 - 4053938625265300732448395495*v - 541048332998388422025224026482) / 138515715760567765339564646802 $$\beta_{3}$$ $$=$$ $$( - 30\!\cdots\!58 \nu^{19} + \cdots + 31\!\cdots\!39 ) / 66\!\cdots\!26$$ (-300133994374942407958277244158*v^19 + 1371391336152384902356738569499*v^18 - 6153372527727959839784962018196*v^17 + 26547883599804116440388906209725*v^16 - 81977332045861207285091886808268*v^15 + 320844822250980836735900849124641*v^14 - 672485662073818096289969998302433*v^13 + 2319813098230545135177055987966447*v^12 - 3970015679584422038303839535352523*v^11 + 11943361514385690862839544910032821*v^10 - 15926811006029639049421851830493578*v^9 + 39739199709410044884437241045620313*v^8 - 44573378608605928791145945360996892*v^7 + 93226723298245350986656262736003598*v^6 - 86473198222264444011573080268539148*v^5 + 133061047576959130423042494261455679*v^4 - 108703474688815797770677787428453507*v^3 + 116276216222505912564463304237197753*v^2 - 75811065291850596623486187751281406*v + 31118642540475146064694445912463539) / 6657480846600168505515495619244526 $$\beta_{4}$$ $$=$$ $$( 95\!\cdots\!54 \nu^{19} + \cdots + 32\!\cdots\!33 ) / 13\!\cdots\!02$$ (9506596087569495495277254*v^19 + 4085487136543173326688997*v^18 + 168170142905332217106438781*v^17 + 33963416633628193346040637*v^16 + 1941499406783905413614279500*v^15 + 286236925337428349187776971*v^14 + 13085500236178418311241708565*v^13 + 803599389582718498273633651*v^12 + 64194950599949943857606426611*v^11 + 4041894163272650459344425843*v^10 + 196940736515037912243001933776*v^9 + 10635628540313710360776263960*v^8 + 443622023675648303292386247910*v^7 + 29356035922071687508468245166*v^6 + 556805708900178597111330431473*v^5 + 25329526173786282227531350337*v^4 + 547857964635386867862326181979*v^3 + 80080429879831559595060696900*v^2 + 22914281211171705348517758920*v + 32545207099711078731794325733) / 138515715760567765339564646802 $$\beta_{5}$$ $$=$$ $$( 69\!\cdots\!90 \nu^{19} + \cdots + 76\!\cdots\!03 ) / 59\!\cdots\!66$$ (6905285611257289835653862890*v^19 - 8704509733109891846887882359*v^18 + 119717171362891862549077508377*v^17 - 180911357890774773273854317013*v^16 + 1413985969910790012325989263903*v^15 - 2124310771437863705198405800610*v^14 + 9692490867745127666643417399500*v^13 - 14718547408603054413942371662676*v^12 + 49188995418896324074745959195092*v^11 - 68255083814261259751423798054546*v^10 + 155223655426875839155891688089833*v^9 - 190199029453549464511799689740259*v^8 + 355004914948736606316975639667295*v^7 - 343249866809578308629773307357472*v^6 + 462931980944888790145921131781233*v^5 - 286082773563193033532835097647413*v^4 + 425011978776888589922216066006378*v^3 - 23286403199775164886189082413337*v^2 + 1040616193353465426806646546015*v + 76498439551155901605372825973303) / 59977304924325842392031492065266 $$\beta_{6}$$ $$=$$ $$( - 22\!\cdots\!28 \nu^{19} + \cdots + 69\!\cdots\!02 ) / 17\!\cdots\!98$$ (-22059565808638195111123708928*v^19 + 11384290415918989293568645273*v^18 - 409002972042913302267281368259*v^17 + 322382846246591900724621455712*v^16 - 4850842095428855703737495302835*v^15 + 4267936839669219355854820123001*v^14 - 34404684966066569774905181195101*v^13 + 34661762238567164155615375239799*v^12 - 174627552403471720341439771630759*v^11 + 185429221828389633696249134384313*v^10 - 572969530730428504376747894693240*v^9 + 659851727456421235501118013184470*v^8 - 1289676220270047542003975257645181*v^7 + 1635437266101970074637306641856777*v^6 - 1762655089830846690969078378496851*v^5 + 2685397279235594541360400347441666*v^4 - 1300802053762476375461092954576432*v^3 + 2485045796758903408408523246806288*v^2 - 460703310925526697682439300187295*v + 697109471565452723846450865976802) / 179931914772977527176094476195798 $$\beta_{7}$$ $$=$$ $$( - 87\!\cdots\!09 \nu^{19} + \cdots + 33\!\cdots\!41 ) / 51\!\cdots\!74$$ (-879600191884083208967414209*v^19 + 351744055240071333325258398*v^18 - 15681640429861400348325962873*v^17 + 9740696055033624868807891733*v^16 - 184338994072097313938320894530*v^15 + 123731889372165409632805779831*v^14 - 1271866313529508469754542168795*v^13 + 946833209669629245432803090839*v^12 - 6412458627944464838041217221629*v^11 + 4701755679731548010450248367412*v^10 - 20457723211419213352692836332261*v^9 + 14768686483222414528467896811466*v^8 - 47311598150841645554998987858595*v^7 + 32951378083611635633614645716079*v^6 - 64128264497361160758239731798327*v^5 + 43060642928682904667684212962898*v^4 - 59638233946051065909540254368734*v^3 + 34631977024400740663717040523566*v^2 - 11627832077419404565334221113592*v + 3336748112877763038183708627341) / 5125081483141007317563891931674 $$\beta_{8}$$ $$=$$ $$( 76\!\cdots\!29 \nu^{19} + \cdots + 39\!\cdots\!22 ) / 31\!\cdots\!06$$ (76928540504194365699792961129*v^19 - 38185185301840964082466607032*v^18 + 1387051403995825637601406358919*v^17 - 925188925555534473871396350365*v^16 + 16470861038462149920915230808112*v^15 - 11391515290637933883849482903572*v^14 + 115483059750272710211614687247338*v^13 - 82566308075111392091968624505486*v^12 + 593041204676302258474852776666840*v^11 - 395422648016227477077032809945425*v^10 + 1958689413598781412311416498295752*v^9 - 1143796078428707053884894387348873*v^8 + 4748028320841348040725199420195220*v^7 - 2416871137970853714564377948372994*v^6 + 7212248291147255902630496617640679*v^5 - 2792272658701624453569179841756060*v^4 + 7671470824449844209712451840114816*v^3 - 2276905601557612803506801898935177*v^2 + 3072478929710746354692840844883131*v + 399903162476216056702950743151422) / 317022897457150881215023600916406 $$\beta_{9}$$ $$=$$ $$( - 54\!\cdots\!61 \nu^{19} + \cdots + 41\!\cdots\!48 ) / 22\!\cdots\!42$$ (-547032892879214620350249147761*v^19 + 519035587247795853558443240459*v^18 - 10083199051821045769543385227201*v^17 + 11414554751613308422288814197776*v^16 - 122348425158084174468378036586158*v^15 + 140502377566984528040814712521412*v^14 - 886787509369351797071740745434444*v^13 + 1034388272145588058265759106144284*v^12 - 4713762858399919160082457680351292*v^11 + 5213186203916746026577518419615797*v^10 - 16399385849756527608088938758473141*v^9 + 16723467622644562549839486026744829*v^8 - 41482740246878481269743699563265421*v^7 + 38237790580296755340119187293923767*v^6 - 66160950939776705056606355614636631*v^5 + 52044721190489281147690452269887453*v^4 - 70989919562992066718448032595181971*v^3 + 44122561469480118425811710714121747*v^2 - 27302764799446949145615669852967362*v + 4182204103110874071297385847626848) / 2219160282200056168505165206414842 $$\beta_{10}$$ $$=$$ $$( 25\!\cdots\!27 \nu^{19} + \cdots + 49\!\cdots\!78 ) / 85\!\cdots\!38$$ (2592049278636087802657452227*v^19 - 660823901855692754332108674*v^18 + 46025780075609179116409549487*v^17 - 24634582481399738567098646190*v^16 + 537962157850176888307584496807*v^15 - 326748068612455370728726305585*v^14 + 3692246215347241728671550066957*v^13 - 2681156313251144971337417132923*v^12 + 18512554076498152147852594534837*v^11 - 13585884359044387449861562464470*v^10 + 58786074808962877919634751957201*v^9 - 44957598056872240854663862971214*v^8 + 135367928967498852239233487044924*v^7 - 97038888715487812292067041678316*v^6 + 181459129725459673940572068041447*v^5 - 127071818495169742977683956650873*v^4 + 169762115362412024437743322934801*v^3 - 65427083417999056397580191370166*v^2 + 32463056619900070843169307975053*v + 4904613101529405434832308787078) / 8568186417760834627433070295038 $$\beta_{11}$$ $$=$$ $$( - 10\!\cdots\!62 \nu^{19} + \cdots - 95\!\cdots\!36 ) / 31\!\cdots\!06$$ (-101821382051282373258671320462*v^19 - 73101611845366533817908434270*v^18 - 1852595082836908045721076066747*v^17 - 938583595257175127278138560594*v^16 - 21598404211828921261432769216251*v^15 - 9781372300994797067491508537861*v^14 - 149023170146987408049036059706653*v^13 - 56357334455999473786148494284293*v^12 - 743794169437154400721899818070155*v^11 - 278133207208836159741267032708863*v^10 - 2381566229158775041086438489527451*v^9 - 870430745096515881789068262931750*v^8 - 5562982321692739636853342377781223*v^7 - 1917349306696282402134285987322912*v^6 - 7701363420589751993110442011126113*v^5 - 2264254782915633046246063874308508*v^4 - 7070661971480829655082981461295971*v^3 - 1985436304259972786884689054741820*v^2 - 1485009820604195292527129026572033*v - 954636370439460057642453047153636) / 317022897457150881215023600916406 $$\beta_{12}$$ $$=$$ $$( - 81\!\cdots\!78 \nu^{19} + \cdots + 33\!\cdots\!27 ) / 22\!\cdots\!42$$ (-811347505638254673016242624978*v^19 - 119965180563481247720775643043*v^18 - 14996805006720435737873426277344*v^17 + 728933639287800660202845293091*v^16 - 177133376756155825899168281364319*v^15 + 18644798449409268253868538198082*v^14 - 1246607413112704486289128458061110*v^13 + 213053415623996835572168454063792*v^12 - 6314531343789698308882705308303288*v^11 + 1100503461168874083136273858269168*v^10 - 20677975530269450195061250508661791*v^9 + 3410838479643168671219531666900686*v^8 - 48204591625362680749063509544147081*v^7 + 8048557738538132872158760349398162*v^6 - 69002254441313896963403206636275922*v^5 + 12201844829134768961909988890829499*v^4 - 62991863936276004364949006290251674*v^3 + 11368317417355663790559997120818214*v^2 - 22651855290345595347313524174590961*v + 33167858081967647089695029280927) / 2219160282200056168505165206414842 $$\beta_{13}$$ $$=$$ $$( 79\!\cdots\!49 \nu^{19} + \cdots - 49\!\cdots\!74 ) / 17\!\cdots\!98$$ (79974693311878674703640697149*v^19 + 13138839787370564488601661635*v^18 + 1438850555740982062458979078160*v^17 - 121160742218425489032481707987*v^16 + 16767119761634346661079598505079*v^15 - 2607018145755562210278700490285*v^14 + 115131180607893452464444544052139*v^13 - 30397496001346965294340493986699*v^12 + 572122942140396394028725245012775*v^11 - 164653778286039728687437829100396*v^10 + 1803754651441828823097315894995258*v^9 - 608292732892936936449960787987095*v^8 + 4078497411264945318359090066278175*v^7 - 1490027827156975678844184270869149*v^6 + 5375837228591700429922273111733424*v^5 - 2580714750078383904713071941742710*v^4 + 4770707009381107269392946697269550*v^3 - 2047690647161828431914062644517347*v^2 + 1476380044502233373600269716680920*v - 492993699877845225448670891064974) / 179931914772977527176094476195798 $$\beta_{14}$$ $$=$$ $$( 96\!\cdots\!47 \nu^{19} + \cdots + 53\!\cdots\!65 ) / 17\!\cdots\!98$$ (96819344902331263752639532847*v^19 - 36533770266434602873655340226*v^18 + 1692468395960390495557615498748*v^17 - 1038900563964696036247502732412*v^16 + 19725786405028462002656362856711*v^15 - 13090056078075075826777266841520*v^14 + 133669765967236327140238343479876*v^13 - 99815416730223525730094168189362*v^12 + 665964050906290548992564847538168*v^11 - 487402577433659750652445331140959*v^10 + 2069094372205210597488842165859680*v^9 - 1502579578852427417313528005999226*v^8 + 4732236146131896612002957056412861*v^7 - 3229188369133569554909210689419637*v^6 + 6157622112715051110144040655149386*v^5 - 3912434351393369072068114251110661*v^4 + 5920068097547458588645664266359253*v^3 - 2270782672098573370476154723632826*v^2 + 557810311606027021326458426778760*v + 530495014785748571929566404024965) / 179931914772977527176094476195798 $$\beta_{15}$$ $$=$$ $$( - 11\!\cdots\!66 \nu^{19} + \cdots + 17\!\cdots\!44 ) / 17\!\cdots\!98$$ (-115048696625446065214454465266*v^19 + 49930169430911949141898714106*v^18 - 2039053873531239807910274694313*v^17 + 1380653697218536641062518377714*v^16 - 23882867519601343002311216009290*v^15 + 17556183053897235165030578758450*v^14 - 163810622541150952673167990402340*v^13 + 135432290766910247670721912739306*v^12 - 821140788900012057666136351712498*v^11 + 678308458310155344073325206091640*v^10 - 2593718939981868180567860500979818*v^9 + 2179950719114701902556730628406521*v^8 - 5929408513623506200477998442515586*v^7 + 4909883649604400844120896212641956*v^6 - 7917006262000091551828222614465675*v^5 + 6657664373970494879849002313292186*v^4 - 7312369541394705678448896887870996*v^3 + 4838595711329907781960136594515673*v^2 - 1885362099641700168688105299274280*v + 173879805204720109882108632675844) / 179931914772977527176094476195798 $$\beta_{16}$$ $$=$$ $$( 17\!\cdots\!18 \nu^{19} + \cdots + 35\!\cdots\!66 ) / 25\!\cdots\!37$$ (1759200383768166417934828418*v^19 - 703488110480142666650516796*v^18 + 31363280859722800696651925746*v^17 - 19481392110067249737615783466*v^16 + 368677988144194627876641789060*v^15 - 247463778744330819265611559662*v^14 + 2543732627059016939509084337590*v^13 - 1893666419339258490865606181678*v^12 + 12824917255888929676082434443258*v^11 - 9403511359463096020900496734824*v^10 + 40915446422838426705385672664522*v^9 - 29537372966444829056935793622932*v^8 + 94623196301683291109997975717190*v^7 - 65902756167223271267229291432158*v^6 + 128256528994722321516479463596654*v^5 - 86121285857365809335368425925796*v^4 + 119276467892102131819080508737468*v^3 - 66701413307230977668652135081295*v^2 + 23255664154838809130668442227184*v + 3576666740526488558760366608666) / 2562540741570503658781945965837 $$\beta_{17}$$ $$=$$ $$( 16\!\cdots\!22 \nu^{19} + \cdots + 53\!\cdots\!99 ) / 22\!\cdots\!42$$ (1686658510188675797689348187322*v^19 - 103190391697568836589356040596*v^18 + 30747373632935788536610003440016*v^17 - 8481915521695374654811004848836*v^16 + 363122793789656617209181994299800*v^15 - 120725570991746153888218773946909*v^14 + 2542333408179121209705748046590437*v^13 - 1036161739181998753486000177393007*v^12 + 12910547648560679288653495760188011*v^11 - 5252737286386413425111213884950263*v^10 + 42105228864217754332077886907028721*v^9 - 16934804998751859991868454159659457*v^8 + 98683681794708528028308383984701413*v^7 - 37738780061852986639038213859764727*v^6 + 138221106727360520241360746084271352*v^5 - 51273350773318583073480584300169314*v^4 + 127145978581831908427754519993007532*v^3 - 34486284259171854565973654535156721*v^2 + 30122856852506216649132371205625705*v + 5383522455165565036271733610858799) / 2219160282200056168505165206414842 $$\beta_{18}$$ $$=$$ $$( - 17\!\cdots\!04 \nu^{19} + \cdots - 43\!\cdots\!26 ) / 22\!\cdots\!42$$ (-1719118167780304038462420340704*v^19 - 513618288757343180281210438627*v^18 - 30754824652996742687170144972568*v^17 - 2153803386285737978957438111916*v^16 - 357003786002852251562654871409361*v^15 - 3186979705758439077407323004119*v^14 - 2433349996389391293580607412013645*v^13 + 198169718585141852182648539900149*v^12 - 12020282287313518222143761042060889*v^11 + 1093882144731424140859966833361143*v^10 - 37473063640521001529457375779529102*v^9 + 4176061120900396325278360352335609*v^8 - 84419605494219927958411562011813869*v^7 + 10213197170234755874175292102644985*v^6 - 109119549393686683662607279405126348*v^5 + 18326133580024093197043474646299372*v^4 - 94661561842689218647202901056098078*v^3 + 10149084621939645881515370251707393*v^2 - 16445804578374024653829603156301341*v - 4311806491182162040374557605989326) / 2219160282200056168505165206414842 $$\beta_{19}$$ $$=$$ $$( 14\!\cdots\!24 \nu^{19} + \cdots - 55\!\cdots\!06 ) / 17\!\cdots\!98$$ (145325051963474911389414146224*v^19 - 42081747011923910253210122375*v^18 + 2630251435731729180571983055783*v^17 - 1345705424515536726892269781377*v^16 + 31068256910262393098872149362431*v^15 - 17646610739331526917293322185605*v^14 + 217087838816891391822336106423763*v^13 - 140139535317884650512095653688609*v^12 + 1104254923813722910620595015217507*v^11 - 710449888262703659894656395765423*v^10 + 3603791201747180826034705420524430*v^9 - 2305940925222579256293389596864134*v^8 + 8500691893352716489040694290590264*v^7 - 5272367002415732960272623326427365*v^6 + 12155833781258047530985537654144269*v^5 - 7286142243257359708692038277015453*v^4 + 11747992316164129740457928900888084*v^3 - 5726660849761505580395631915937244*v^2 + 3696028165143525643865708873305820*v - 5568334016711834991228921863806) / 179931914772977527176094476195798
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{16} + 4\beta_{7} - 4$$ b16 + 4*b7 - 4 $$\nu^{3}$$ $$=$$ $$-\beta_{15} - \beta_{14} - \beta_{13} + 5\beta_{4}$$ -b15 - b14 - b13 + 5*b4 $$\nu^{4}$$ $$=$$ $$\beta_{18} - 7\beta_{16} + \beta_{14} + \beta_{13} - \beta_{11} - 23\beta_{7} - \beta_{6} - 7\beta_{2}$$ b18 - 7*b16 + b14 + b13 - b11 - 23*b7 - b6 - 7*b2 $$\nu^{5}$$ $$=$$ $$2 \beta_{18} - \beta_{17} + 2 \beta_{16} + 9 \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{8} + \beta_{7} + 9 \beta_{6} - 29 \beta_{4} - 29 \beta _1 - 1$$ 2*b18 - b17 + 2*b16 + 9*b13 - b12 - b11 - 2*b8 + b7 + 9*b6 - 29*b4 - 29*b1 - 1 $$\nu^{6}$$ $$=$$ $$- \beta_{19} - 3 \beta_{17} + \beta_{16} - 10 \beta_{15} - 10 \beta_{13} - 13 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} - 14 \beta_{8} - \beta_{7} + 10 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 45 \beta_{2} + 149$$ -b19 - 3*b17 + b16 - 10*b15 - 10*b13 - 13*b12 - 2*b11 - 2*b9 - 14*b8 - b7 + 10*b6 + b5 + 2*b4 - 2*b3 + 45*b2 + 149 $$\nu^{7}$$ $$=$$ $$- 3 \beta_{19} - 15 \beta_{18} - 26 \beta_{16} + 67 \beta_{15} + 68 \beta_{14} + \beta_{13} - 12 \beta_{12} + 27 \beta_{11} - 16 \beta_{9} + 12 \beta_{8} - 17 \beta_{7} - 68 \beta_{6} + 3 \beta_{3} - 26 \beta_{2} + 181 \beta_1$$ -3*b19 - 15*b18 - 26*b16 + 67*b15 + 68*b14 + b13 - 12*b12 + 27*b11 - 16*b9 + 12*b8 - 17*b7 - 68*b6 + 3*b3 - 26*b2 + 181*b1 $$\nu^{8}$$ $$=$$ $$30 \beta_{19} - 106 \beta_{18} + 42 \beta_{17} + 289 \beta_{16} + 79 \beta_{15} - 79 \beta_{14} + 137 \beta_{12} + 122 \beta_{11} - 19 \beta_{10} + 15 \beta_{9} + 121 \beta_{8} + 997 \beta_{7} - 30 \beta_{4} + 15 \beta_{3} + 15 \beta_{2} + \cdots - 1012$$ 30*b19 - 106*b18 + 42*b17 + 289*b16 + 79*b15 - 79*b14 + 137*b12 + 122*b11 - 19*b10 + 15*b9 + 121*b8 + 997*b7 - 30*b4 + 15*b3 + 15*b2 - 30*b1 - 1012 $$\nu^{9}$$ $$=$$ $$- 50 \beta_{19} - 60 \beta_{18} + 79 \beta_{17} + 50 \beta_{16} - 490 \beta_{15} - 474 \beta_{14} - 490 \beta_{13} + 163 \beta_{12} - 160 \beta_{11} + 129 \beta_{9} + 53 \beta_{8} - 50 \beta_{7} + 16 \beta_{6} + 3 \beta_{5} + 1175 \beta_{4} + \cdots + 294$$ -50*b19 - 60*b18 + 79*b17 + 50*b16 - 490*b15 - 474*b14 - 490*b13 + 163*b12 - 160*b11 + 129*b9 + 53*b8 - 50*b7 + 16*b6 + 3*b5 + 1175*b4 - 100*b3 + 200*b2 + 294 $$\nu^{10}$$ $$=$$ $$- 163 \beta_{19} + 853 \beta_{18} - 2039 \beta_{16} + 3 \beta_{15} + 587 \beta_{14} + 584 \beta_{13} - 176 \beta_{12} - 677 \beta_{11} + 229 \beta_{10} + 94 \beta_{9} + 176 \beta_{8} - 6699 \beta_{7} - 587 \beta_{6} - 229 \beta_{5} + \cdots + 308 \beta_1$$ -163*b19 + 853*b18 - 2039*b16 + 3*b15 + 587*b14 + 584*b13 - 176*b12 - 677*b11 + 229*b10 + 94*b9 + 176*b8 - 6699*b7 - 587*b6 - 229*b5 + 163*b3 - 2039*b2 + 308*b1 $$\nu^{11}$$ $$=$$ $$1136 \beta_{19} + 1690 \beta_{18} - 567 \beta_{17} + 1575 \beta_{16} + 179 \beta_{15} - 179 \beta_{14} + 3300 \beta_{13} - 199 \beta_{12} - 767 \beta_{11} - 69 \beta_{10} + 568 \beta_{9} - 1122 \beta_{8} + 2436 \beta_{7} + \cdots - 3004$$ 1136*b19 + 1690*b18 - 567*b17 + 1575*b16 + 179*b15 - 179*b14 + 3300*b13 - 199*b12 - 767*b11 - 69*b10 + 568*b9 - 1122*b8 + 2436*b7 + 3300*b6 - 7803*b4 + 568*b3 + 568*b2 - 7803*b1 - 3004 $$\nu^{12}$$ $$=$$ $$- 1560 \beta_{19} - 110 \beta_{18} - 3680 \beta_{17} + 1560 \beta_{16} - 4287 \beta_{15} - 64 \beta_{14} - 4287 \beta_{13} - 7885 \beta_{12} - 3230 \beta_{11} - 2120 \beta_{9} - 9555 \beta_{8} - 1560 \beta_{7} + \cdots + 49354$$ -1560*b19 - 110*b18 - 3680*b17 + 1560*b16 - 4287*b15 - 64*b14 - 4287*b13 - 7885*b12 - 3230*b11 - 2120*b9 - 9555*b8 - 1560*b7 + 4223*b6 + 2271*b5 + 2720*b4 - 3120*b3 + 12324*b2 + 49354 $$\nu^{13}$$ $$=$$ $$- 5501 \beta_{19} - 10744 \beta_{18} - 17337 \beta_{16} + 22872 \beta_{15} + 24606 \beta_{14} + 1734 \beta_{13} - 7453 \beta_{12} + 18197 \beta_{11} + 1000 \beta_{10} - 14808 \beta_{9} + 7453 \beta_{8} + \cdots + 52573 \beta_1$$ -5501*b19 - 10744*b18 - 17337*b16 + 22872*b15 + 24606*b14 + 1734*b13 - 7453*b12 + 18197*b11 + 1000*b10 - 14808*b9 + 7453*b8 - 16482*b7 - 24606*b6 - 1000*b5 + 5501*b3 - 17337*b2 + 52573*b1 $$\nu^{14}$$ $$=$$ $$27908 \beta_{19} - 46312 \beta_{18} + 30151 \beta_{17} + 81854 \beta_{16} + 30325 \beta_{15} - 30325 \beta_{14} + 874 \beta_{13} + 74954 \beta_{12} + 61000 \beta_{11} - 20309 \beta_{10} + 13954 \beta_{9} + \cdots - 349412$$ 27908*b19 - 46312*b18 + 30151*b17 + 81854*b16 + 30325*b15 - 30325*b14 + 874*b13 + 74954*b12 + 61000*b11 - 20309*b10 + 13954*b9 + 60266*b8 + 335458*b7 + 874*b6 - 22302*b4 + 13954*b3 + 13954*b2 - 22302*b1 - 349412 $$\nu^{15}$$ $$=$$ $$- 48951 \beta_{19} - 10016 \beta_{18} + 24025 \beta_{17} + 48951 \beta_{16} - 174053 \beta_{15} - 158491 \beta_{14} - 174053 \beta_{13} + 43533 \beta_{12} - 107918 \beta_{11} + 72976 \beta_{9} + \cdots + 236300$$ -48951*b19 - 10016*b18 + 24025*b17 + 48951*b16 - 174053*b15 - 158491*b14 - 174053*b13 + 43533*b12 - 107918*b11 + 72976*b9 - 15434*b8 - 48951*b7 + 15562*b6 + 11711*b5 + 357821*b4 - 97902*b3 + 86733*b2 + 236300 $$\nu^{16}$$ $$=$$ $$- 119629 \beta_{19} + 337294 \beta_{18} - 668559 \beta_{16} + 9811 \beta_{15} + 227269 \beta_{14} + 217458 \beta_{13} - 123480 \beta_{12} - 213814 \beta_{11} + 170878 \beta_{10} + \cdots + 175656 \beta_1$$ -119629*b19 + 337294*b18 - 668559*b16 + 9811*b15 + 227269*b14 + 217458*b13 - 123480*b12 - 213814*b11 + 170878*b10 - 1616*b9 + 123480*b8 - 2248024*b7 - 227269*b6 - 170878*b5 + 119629*b3 - 668559*b2 + 175656*b1 $$\nu^{17}$$ $$=$$ $$827974 \beta_{19} + 660926 \beta_{18} - 141583 \beta_{17} + 626550 \beta_{16} + 133291 \beta_{15} - 133291 \beta_{14} + 1099831 \beta_{13} + 213628 \beta_{12} - 200359 \beta_{11} + \cdots - 1955655$$ 827974*b19 + 660926*b18 - 141583*b17 + 626550*b16 + 133291*b15 - 133291*b14 + 1099831*b13 + 213628*b12 - 200359*b11 - 121245*b10 + 413987*b9 - 246939*b8 + 1541668*b7 + 1099831*b6 - 2454193*b4 + 413987*b3 + 413987*b2 - 2454193*b1 - 1955655 $$\nu^{18}$$ $$=$$ $$- 995799 \beta_{19} - 12046 \beta_{18} - 1828222 \beta_{17} + 995799 \beta_{16} - 1659313 \beta_{15} - 98915 \beta_{14} - 1659313 \beta_{13} - 3370902 \beta_{12} - 2003644 \beta_{11} + \cdots + 17778807$$ -995799*b19 - 12046*b18 - 1828222*b17 + 995799*b16 - 1659313*b15 - 98915*b14 - 1659313*b13 - 3370902*b12 - 2003644*b11 - 832423*b9 - 4378747*b8 - 995799*b7 + 1560398*b6 + 1383544*b5 + 1352440*b4 - 1991598*b3 + 3712233*b2 + 17778807 $$\nu^{19}$$ $$=$$ $$- 3387188 \beta_{19} - 4486483 \beta_{18} - 7877160 \beta_{16} + 7647734 \beta_{15} + 8754494 \beta_{14} + 1106760 \beta_{13} - 3564042 \beta_{12} + 8050525 \beta_{11} + \cdots + 16936871 \beta_1$$ -3387188*b19 - 4486483*b18 - 7877160*b16 + 7647734*b15 + 8754494*b14 + 1106760*b13 - 3564042*b12 + 8050525*b11 + 1159175*b10 - 7533580*b9 + 3564042*b8 - 9012635*b7 - 8754494*b6 - 1159175*b5 + 3387188*b3 - 7877160*b2 + 16936871*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{7}$$ $$-\beta_{7}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 1.31285 − 2.27393i 1.14017 − 1.97483i 0.904928 − 1.56738i 0.613260 − 1.06220i 0.328258 − 0.568560i −0.130586 + 0.226181i −0.707433 + 1.22531i −0.828334 + 1.43472i −1.27537 + 2.20901i −1.35774 + 2.35168i 1.31285 + 2.27393i 1.14017 + 1.97483i 0.904928 + 1.56738i 0.613260 + 1.06220i 0.328258 + 0.568560i −0.130586 − 0.226181i −0.707433 − 1.22531i −0.828334 − 1.43472i −1.27537 − 2.20901i −1.35774 − 2.35168i
−1.31285 2.27393i 1.00000 −2.44716 + 4.23861i 0.734607 1.27238i −1.31285 2.27393i 2.40805 1.09603i 7.59966 1.00000 −3.85772
100.2 −1.14017 1.97483i 1.00000 −1.59997 + 2.77124i −1.46862 + 2.54373i −1.14017 1.97483i −2.34076 1.23322i 2.73629 1.00000 6.69792
100.3 −0.904928 1.56738i 1.00000 −0.637789 + 1.10468i 1.98776 3.44291i −0.904928 1.56738i −2.60384 + 0.469078i −1.31110 1.00000 −7.19513
100.4 −0.613260 1.06220i 1.00000 0.247823 0.429243i −2.10660 + 3.64874i −0.613260 1.06220i 2.23241 + 1.41998i −3.06096 1.00000 5.16759
100.5 −0.328258 0.568560i 1.00000 0.784493 1.35878i −0.0109774 + 0.0190133i −0.328258 0.568560i −1.07162 2.41901i −2.34310 1.00000 0.0144136
100.6 0.130586 + 0.226181i 1.00000 0.965895 1.67298i 0.708533 1.22721i 0.130586 + 0.226181i −0.675578 + 2.55804i 1.02687 1.00000 0.370097
100.7 0.707433 + 1.22531i 1.00000 −0.000924008 0.00160043i −1.42962 + 2.47618i 0.707433 + 1.22531i −1.85598 + 1.88556i 2.82712 1.00000 −4.04546
100.8 0.828334 + 1.43472i 1.00000 −0.372274 + 0.644798i 1.05011 1.81885i 0.828334 + 1.43472i −1.14598 2.38469i 2.07987 1.00000 3.47937
100.9 1.27537 + 2.20901i 1.00000 −2.25315 + 3.90257i −1.40932 + 2.44101i 1.27537 + 2.20901i −0.0337632 2.64554i −6.39292 1.00000 −7.18961
100.10 1.35774 + 2.35168i 1.00000 −2.68694 + 4.65391i 1.94413 3.36734i 1.35774 + 2.35168i 0.587055 + 2.57980i −9.16172 1.00000 10.5585
172.1 −1.31285 + 2.27393i 1.00000 −2.44716 4.23861i 0.734607 + 1.27238i −1.31285 + 2.27393i 2.40805 + 1.09603i 7.59966 1.00000 −3.85772
172.2 −1.14017 + 1.97483i 1.00000 −1.59997 2.77124i −1.46862 2.54373i −1.14017 + 1.97483i −2.34076 + 1.23322i 2.73629 1.00000 6.69792
172.3 −0.904928 + 1.56738i 1.00000 −0.637789 1.10468i 1.98776 + 3.44291i −0.904928 + 1.56738i −2.60384 0.469078i −1.31110 1.00000 −7.19513
172.4 −0.613260 + 1.06220i 1.00000 0.247823 + 0.429243i −2.10660 3.64874i −0.613260 + 1.06220i 2.23241 1.41998i −3.06096 1.00000 5.16759
172.5 −0.328258 + 0.568560i 1.00000 0.784493 + 1.35878i −0.0109774 0.0190133i −0.328258 + 0.568560i −1.07162 + 2.41901i −2.34310 1.00000 0.0144136
172.6 0.130586 0.226181i 1.00000 0.965895 + 1.67298i 0.708533 + 1.22721i 0.130586 0.226181i −0.675578 2.55804i 1.02687 1.00000 0.370097
172.7 0.707433 1.22531i 1.00000 −0.000924008 0.00160043i −1.42962 2.47618i 0.707433 1.22531i −1.85598 1.88556i 2.82712 1.00000 −4.04546
172.8 0.828334 1.43472i 1.00000 −0.372274 0.644798i 1.05011 + 1.81885i 0.828334 1.43472i −1.14598 + 2.38469i 2.07987 1.00000 3.47937
172.9 1.27537 2.20901i 1.00000 −2.25315 3.90257i −1.40932 2.44101i 1.27537 2.20901i −0.0337632 + 2.64554i −6.39292 1.00000 −7.18961
172.10 1.35774 2.35168i 1.00000 −2.68694 4.65391i 1.94413 + 3.36734i 1.35774 2.35168i 0.587055 2.57980i −9.16172 1.00000 10.5585
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 100.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.j.c 20
3.b odd 2 1 819.2.n.f 20
7.c even 3 1 273.2.l.c yes 20
13.c even 3 1 273.2.l.c yes 20
21.h odd 6 1 819.2.s.f 20
39.i odd 6 1 819.2.s.f 20
91.g even 3 1 inner 273.2.j.c 20
273.bm odd 6 1 819.2.n.f 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.c 20 1.a even 1 1 trivial
273.2.j.c 20 91.g even 3 1 inner
273.2.l.c yes 20 7.c even 3 1
273.2.l.c yes 20 13.c even 3 1
819.2.n.f 20 3.b odd 2 1
819.2.n.f 20 273.bm odd 6 1
819.2.s.f 20 21.h odd 6 1
819.2.s.f 20 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 18 T_{2}^{18} + 4 T_{2}^{17} + 211 T_{2}^{16} + 59 T_{2}^{15} + 1458 T_{2}^{14} + 526 T_{2}^{13} + 7324 T_{2}^{12} + 2645 T_{2}^{11} + 23428 T_{2}^{10} + 8506 T_{2}^{9} + 54235 T_{2}^{8} + 18801 T_{2}^{7} + 74141 T_{2}^{6} + \cdots + 1369$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 18 T^{18} + 4 T^{17} + \cdots + 1369$$
$3$ $$(T - 1)^{20}$$
$5$ $$T^{20} + 41 T^{18} - 6 T^{17} + \cdots + 21904$$
$7$ $$T^{20} + 9 T^{19} + 51 T^{18} + \cdots + 282475249$$
$11$ $$(T^{10} - 8 T^{9} - 17 T^{8} + 251 T^{7} + \cdots - 64)^{2}$$
$13$ $$T^{20} + 5 T^{19} + \cdots + 137858491849$$
$17$ $$T^{20} + 102 T^{18} + \cdots + 353590416$$
$19$ $$(T^{10} + 7 T^{9} - 65 T^{8} - 325 T^{7} + \cdots + 65216)^{2}$$
$23$ $$T^{20} + 14 T^{19} + \cdots + 24437192976$$
$29$ $$T^{20} + 9 T^{19} + 190 T^{18} + \cdots + 1882384$$
$31$ $$T^{20} + 9 T^{19} + \cdots + 64085935104$$
$37$ $$T^{20} - 18 T^{19} + \cdots + 300571569$$
$41$ $$T^{20} + T^{19} + 205 T^{18} + \cdots + 5541909136$$
$43$ $$T^{20} + 11 T^{19} + \cdots + 29\!\cdots\!89$$
$47$ $$T^{20} - 13 T^{19} + \cdots + 6617497104$$
$53$ $$T^{20} + 6 T^{19} + \cdots + 406192078224$$
$59$ $$T^{20} + 15 T^{19} + \cdots + 9269028962064$$
$61$ $$(T^{10} - 308 T^{8} + 195 T^{7} + \cdots - 163441132)^{2}$$
$67$ $$(T^{10} - 22 T^{9} - 11 T^{8} + \cdots - 289024)^{2}$$
$71$ $$T^{20} + 11 T^{19} + \cdots + 1342437380496$$
$73$ $$T^{20} + 398 T^{18} + \cdots + 21167631081$$
$79$ $$T^{20} + 36 T^{19} + \cdots + 22\!\cdots\!76$$
$83$ $$(T^{10} - 20 T^{9} - 378 T^{8} + \cdots + 548326464)^{2}$$
$89$ $$T^{20} - 2 T^{19} + \cdots + 204256994704$$
$97$ $$T^{20} - 21 T^{19} + \cdots + 479579795289$$