# Properties

 Label 69.2.e.c Level $69$ Weight $2$ Character orbit 69.e Analytic conductor $0.551$ 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] = [69,2,Mod(4,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(22))

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

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

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

chi := DirichletCharacter("69.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 69.e (of order $$11$$, degree $$10$$, 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: $$0.550967773947$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + 4165 x^{12} - 7040 x^{11} + 10649 x^{10} - 13519 x^{9} + 15111 x^{8} - 12101 x^{7} + \cdots + 529$$ x^20 - 7*x^19 + 24*x^18 - 70*x^17 + 209*x^16 - 527*x^15 + 1115*x^14 - 2187*x^13 + 4165*x^12 - 7040*x^11 + 10649*x^10 - 13519*x^9 + 15111*x^8 - 12101*x^7 + 8244*x^6 - 2837*x^5 + 1859*x^4 - 1730*x^3 + 1957*x^2 - 1656*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

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + 4165 x^{12} - 7040 x^{11} + 10649 x^{10} - 13519 x^{9} + 15111 x^{8} - 12101 x^{7} + \cdots + 529$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 44\!\cdots\!58 \nu^{19} + \cdots + 45\!\cdots\!20 ) / 24\!\cdots\!19$$ (44129229939565069392132843620424558*v^19 - 185728658963314769969938260816354131*v^18 + 344183167862028214811341538962220588*v^17 - 1080585998655109119600845200373005528*v^16 + 3590541613033472256059607911614737655*v^15 - 6018912826998824899930514546275418553*v^14 + 9720305626917136064774756365703612411*v^13 - 20773536187803823879823011954431024341*v^12 + 39187574336451032756944440248298951190*v^11 - 36132389005628476175051141224458095391*v^10 + 50933335514531249607785469440681394141*v^9 - 9195721952109525480776746941470112533*v^8 + 49016930055050580722939716994047241254*v^7 + 136034097515107792543185685088052206401*v^6 + 96892469823289542512882128828274671432*v^5 + 232145779038748840739715159326733138080*v^4 + 84351390301369344362855643656488502276*v^3 + 491755303909348262087547274545912945144*v^2 + 17440195884177906484704214889508870437*v + 45070732273283703518137469135864552820) / 247769057231947568873315158172319850019 $$\beta_{3}$$ $$=$$ $$( - 75\!\cdots\!64 \nu^{19} + \cdots + 26\!\cdots\!36 ) / 24\!\cdots\!19$$ (-75778449230406417576637040082199664*v^19 + 460491619320833625802481972982458153*v^18 - 1532313697080502716503216523833143883*v^17 + 4812316267872747747762674400871936859*v^16 - 14278535148765187844897125728960256505*v^15 + 34754460806533553415617755475909056055*v^14 - 76573896536743693477930087759099724695*v^13 + 153598289510873814040881970209054845445*v^12 - 289402787336897066225100520989724736199*v^11 + 486987371922927033748700283274752702746*v^10 - 773332362848448009085579410151110532602*v^9 + 973987776402668758068190164177590763928*v^8 - 1179893078154280187491703891157600183447*v^7 + 906443438416630153187162808656220686124*v^6 - 823444210355578171633821918924393601025*v^5 - 3654939204868939216357278413113789059*v^4 - 293070843450195871994779914965114198610*v^3 - 53246526139003461903420388227002206483*v^2 - 350103968065764333236953706497399029833*v + 267263930287969394034192040010044688836) / 247769057231947568873315158172319850019 $$\beta_{4}$$ $$=$$ $$( - 12\!\cdots\!75 \nu^{19} + \cdots + 23\!\cdots\!82 ) / 24\!\cdots\!19$$ (-123175950613640715774991644526617775*v^19 + 714918350687533450599846707927968804*v^18 - 2008460097114445737848453853056713532*v^17 + 5632467444335627246896156405053994967*v^16 - 17237191351151966669723494041688323513*v^15 + 39483785755697916307453364271069769759*v^14 - 75737089690024982880771517043437484005*v^13 + 144610668361837481261288853430769332880*v^12 - 274537389768909612345564077863330792929*v^11 + 418998834111897174349037182273219724001*v^10 - 587387337600870647631467165963049487069*v^9 + 617819863561717182861579682954188254684*v^8 - 670041909013784697257385225738809782759*v^7 + 266908901798484889555861033978505384720*v^6 - 357340404377294942605196036677877609126*v^5 - 2315151843717880362880687366119248954*v^4 - 568098871704795832135937094009247430484*v^3 + 68920707107550934315699760075661989569*v^2 - 118148737053203458431509458171287620868*v + 23344362638029921708438274275204591182) / 247769057231947568873315158172319850019 $$\beta_{5}$$ $$=$$ $$( - 14\!\cdots\!14 \nu^{19} + \cdots + 30\!\cdots\!65 ) / 24\!\cdots\!19$$ (-146688990501587513378202173314086614*v^19 + 1107426101600578582418913035541153504*v^18 - 4012315843323532472205065302785743372*v^17 + 11731145698133296531210793733143323557*v^16 - 34855286980450561108124999989186253726*v^15 + 90155390760930112868230415828824200450*v^14 - 193890204152062098337201761160362942171*v^13 + 381250984161312850876216118488493608109*v^12 - 729238107530221505895675050299755980571*v^11 + 1257490536382267661435384602763465831497*v^10 - 1917616541891526533031099276926681769647*v^9 + 2498909377151688876717886971198193717444*v^8 - 2821872760612467861469430099124067023840*v^7 + 2429169773523188992894935037515130364883*v^6 - 1597663481818455304766895169549931018099*v^5 + 681087951339902634556097636384556357000*v^4 - 291536748805863761529880006583926686473*v^3 + 290321570929084733390462782766035956049*v^2 - 424535101926573754590443619533836717299*v + 307178639386885276189718988576926758365) / 247769057231947568873315158172319850019 $$\beta_{6}$$ $$=$$ $$( 16\!\cdots\!92 \nu^{19} + \cdots - 41\!\cdots\!10 ) / 24\!\cdots\!19$$ (163990736626458594904774824597389892*v^19 - 1140880467355818566618878386451331781*v^18 + 4052625403633862396790575808798356533*v^17 - 12376536077420658039718642528514120531*v^16 + 36926615238824454189270704207819224524*v^15 - 93347440912094831564758042048891133653*v^14 + 204877018192385987857254825782563426022*v^13 - 412452651193601187063125576112612958562*v^12 + 783543381572551814850196802998695508148*v^11 - 1341641801921208172896420812160974548862*v^10 + 2114612245974388675470511629685006212942*v^9 - 2792617989005838475016427611857092885155*v^8 + 3245768873363117937058639785901495835344*v^7 - 2811501314518163783142458013052340736658*v^6 + 2160850592896136163721839930656687792540*v^5 - 777382692946263753982666895065119592590*v^4 + 426436569759908214148739129942310842048*v^3 - 322081123290876306439012292580551575795*v^2 + 424847292916448491301607442790248520334*v - 417755886485009033292983480719295431410) / 247769057231947568873315158172319850019 $$\beta_{7}$$ $$=$$ $$( 18\!\cdots\!74 \nu^{19} + \cdots - 80\!\cdots\!94 ) / 24\!\cdots\!19$$ (186008788582778465279739934793279874*v^19 - 1237210387360572289223529808535911498*v^18 + 3822462585575383956558360684601544456*v^17 - 10427246773059433374396849083933784258*v^16 + 31742841583668119289596599843713434282*v^15 - 77560880759433663616872696207184234762*v^14 + 151266301206160640506385378972379354412*v^13 - 287191914171483771974145139665322476313*v^12 + 551896547338843143152065136598370635974*v^11 - 886693151771650227272251000942254682466*v^10 + 1230831790618737796898577314766708834000*v^9 - 1429068898299412718002647563349064708708*v^8 + 1448874077362920265299543349341754456638*v^7 - 882897050768440776417606554537394991958*v^6 + 419432135535227881038419332393518606944*v^5 - 216198106753019903015995433981657470504*v^4 + 62002343686895669608150049917883692488*v^3 - 249502617365520623993005943336531439384*v^2 - 163239204928199163361208347050450734985*v - 80899648254244879676902564114333850894) / 247769057231947568873315158172319850019 $$\beta_{8}$$ $$=$$ $$( 25\!\cdots\!75 \nu^{19} + \cdots - 32\!\cdots\!90 ) / 24\!\cdots\!19$$ (253988479167783285976982314414037475*v^19 - 1809803702978039084423838150431927865*v^18 + 6302362689732621189786628403737565412*v^17 - 18258642735170328563853685671338498500*v^16 + 53821283152684160540027500924967848894*v^15 - 136131480472383856719930295581695529232*v^14 + 288069678624324562406846003132389943675*v^13 - 556343289627969997775616950907423252734*v^12 + 1046714419196361958525903664283047397189*v^11 - 1769032190180510499122824403664161014462*v^10 + 2647486047553598974313461774436386599684*v^9 - 3242415955373487937535477343153729190140*v^8 + 3450511530573250926018367510882699209177*v^7 - 2502157963452656582963004893253135313971*v^6 + 1352349220634328151308564833753839876204*v^5 + 30106614293495600815323116475322491917*v^4 + 146940976711903166669242754117890164521*v^3 - 364618125035724563325878868218798332328*v^2 + 498814834960022197089896253029542193779*v - 320343375617840997031015155847975793090) / 247769057231947568873315158172319850019 $$\beta_{9}$$ $$=$$ $$( 28\!\cdots\!38 \nu^{19} + \cdots - 25\!\cdots\!46 ) / 24\!\cdots\!19$$ (283180632785492837531281967221909338*v^19 - 1867477061235734662529996394897141774*v^18 + 6129730009831354102905136029502596508*v^17 - 17910702771014942789208138221491594322*v^16 + 53624766532151101730865366384186121223*v^15 - 132131161251042844224688963242012744320*v^14 + 276384669954561626595749200325468364969*v^13 - 541311477888818396746059997818438193190*v^12 + 1025185171139321887083197253879323169276*v^11 - 1701111412947018346863219947733304025080*v^10 + 2564144098796138573757596682469918604984*v^9 - 3163907699798990812822970593541834709893*v^8 + 3503159664439463147060098223729237672087*v^7 - 2572682249192683784099619584418828935132*v^6 + 1903244391237928735196277954546942733707*v^5 - 327189702210258892280418717214922009663*v^4 + 503447488753387011926455756592183193382*v^3 - 307207594743749096352657365815376447730*v^2 + 537486442187375127021624501253350106274*v - 250147335734356225466229497286733790446) / 247769057231947568873315158172319850019 $$\beta_{10}$$ $$=$$ $$( - 32\!\cdots\!70 \nu^{19} + \cdots + 21\!\cdots\!72 ) / 24\!\cdots\!19$$ (-329388148289868140407481852375921470*v^19 + 2061015356971214960213953291224985842*v^18 - 6255752916217573397219882923348429594*v^17 + 17716766384188596041401882112815597483*v^16 - 53852977010647541425164864017380285360*v^15 + 128805825853824508331475514503739475471*v^14 - 256464487465115616813783580973739514559*v^13 + 495581245674601198158776233737371444500*v^12 - 944030469481885492474792273619166799600*v^11 + 1506663259807631579757270013685452081863*v^10 - 2168487782869164401725904231317184510580*v^9 + 2525329724348453473261530216449397211618*v^8 - 2720571169372014360075763996746599657942*v^7 + 1601174748179591278483618603853444963493*v^6 - 1144649856981521250090311529055127185472*v^5 + 54295540830144867606570191835636048631*v^4 - 471837279315864364663715431195250655576*v^3 + 81335028047431771756735483979400694569*v^2 - 221124814413289888262950937341876224955*v + 217739822933360827553877556497216583072) / 247769057231947568873315158172319850019 $$\beta_{11}$$ $$=$$ $$( - 35\!\cdots\!92 \nu^{19} + \cdots + 38\!\cdots\!17 ) / 24\!\cdots\!19$$ (-352021761605070307458976599675278592*v^19 + 2030787980687168414682354309714074904*v^18 - 5577361592247507859921202337016421401*v^17 + 15500986553550086077065940429160306291*v^16 - 47934213098020878888259046930132360549*v^15 + 108672055331685244312799451496209254252*v^14 - 204388044681504225772310956808629695647*v^13 + 391671964786790620314240209753982652487*v^12 - 745583905562499432400063513938901433399*v^11 + 1115027744842053421993188391903849594009*v^10 - 1537696600783247469472107875427684013685*v^9 + 1597167995600812959443731064689578953417*v^8 - 1665151099242372301577757184786179941429*v^7 + 542179082076253840312549113645279438971*v^6 - 795107494898588715685999638626654440140*v^5 - 33646457975515755187982243498347948444*v^4 - 924385071068171079945708135848131045952*v^3 + 197294699137554410764337578067918862618*v^2 - 255941250062798673763572073553280368714*v + 38765431170815992938754312453347885417) / 247769057231947568873315158172319850019 $$\beta_{12}$$ $$=$$ $$( 40\!\cdots\!23 \nu^{19} + \cdots - 23\!\cdots\!35 ) / 24\!\cdots\!19$$ (405807242324735876654611797876769623*v^19 - 2539036680489373521531421212744321216*v^18 + 7707323327853626592789939379816201473*v^17 - 21817226237502607091415612535800881965*v^16 + 66100410996971649230286423060221103294*v^15 - 157632333467088985997345766430132336500*v^14 + 313796593469239346369739901829180355492*v^13 - 604718769943499408524311956752893298737*v^12 + 1144282801641889339320986921552361669865*v^11 - 1822045557552755651785541455103087889832*v^10 + 2624698856540090151832957437784541540928*v^9 - 3016526223867189671350661881725570564724*v^8 + 3165828068795436236934702783587171421607*v^7 - 1805464100303572769415083233602696425421*v^6 + 1303511838727261268095340452126902534485*v^5 - 103817284929669772874639142136522453364*v^4 + 727929513746256704418191472823993137339*v^3 - 575382055592278775170980239926706825513*v^2 + 385465888813562374334258442045546636020*v - 236062564195337513727459141809601241635) / 247769057231947568873315158172319850019 $$\beta_{13}$$ $$=$$ $$( - 41\!\cdots\!68 \nu^{19} + \cdots + 46\!\cdots\!53 ) / 24\!\cdots\!19$$ (-411606470573460921651942450845399968*v^19 + 2551857145724358311156115303541878306*v^18 - 7817539936791847159432665529064613390*v^17 + 22556700023924691118416088635829568166*v^16 - 68308985965664736583854090113872995829*v^15 + 163063632981566364285408807578145497776*v^14 - 330135388835584419310440318188881488849*v^13 + 643718863679043418839014559025150215457*v^12 - 1218759704263863540521564074033719422220*v^11 + 1953679083355279395954882580332448975120*v^10 - 2876534045329153774914265145367212177369*v^9 + 3396020092813453798086705761661777656812*v^8 - 3694455652487114513820972158275441704830*v^7 + 2260278731037436252834391600933585354826*v^6 - 1792108995228020559614994960916032372699*v^5 + 23077700035387384636249203993272523744*v^4 - 710880887965918985744390824285962491881*v^3 + 240241914776223029794145008767291289064*v^2 - 724178834864831251916115892325047042807*v + 460495500856361397992665761258106122053) / 247769057231947568873315158172319850019 $$\beta_{14}$$ $$=$$ $$( - 43\!\cdots\!69 \nu^{19} + \cdots + 61\!\cdots\!51 ) / 24\!\cdots\!19$$ (-433111936391964446633437862196129169*v^19 + 3181672615409035568458754729208715059*v^18 - 11417790188041180467947826158636224205*v^17 + 33399440883327639952251217178279769432*v^16 - 98302201953673744528988131627524267123*v^15 + 251879926303142283082999698273223182400*v^14 - 541289861795936387013443641356389551139*v^13 + 1053692239522409538572772630760380483625*v^12 - 1992490852991684671094718623779022609646*v^11 + 3419709835633195485491671197703486121244*v^10 - 5201370924393127271440560498811804501397*v^9 + 6574127286653434146085504869638328192175*v^8 - 7260509323606896793880957062779627252138*v^7 + 5896656408675681488750551201298911486090*v^6 - 3818453667325766573635434858943249645323*v^5 + 1210300895062592399694702912666656842345*v^4 - 1148826598605221619086531883785152953414*v^3 + 1245348324026654859527691342506167921695*v^2 - 1170380400673913899181254877970390650279*v + 617264174328601228737326885548093797451) / 247769057231947568873315158172319850019 $$\beta_{15}$$ $$=$$ $$( - 44\!\cdots\!15 \nu^{19} + \cdots + 35\!\cdots\!20 ) / 24\!\cdots\!19$$ (-446243032505363920089714823836675315*v^19 + 2717893985212811563973391968979957582*v^18 - 8170796099639360560621734559335886344*v^17 + 23529688947521847813490098288751070577*v^16 - 71447567556118452207334785646064258870*v^15 + 169069667133355136656993289101706787711*v^14 - 339928647776391784902686262147760639725*v^13 + 662136918619991546866466417901628558413*v^12 - 1253883460441341318649350284526859388238*v^11 + 1997268147195872658110605438257832547735*v^10 - 2929996495596864733249831703933667539603*v^9 + 3408060699899924683859897265663472042557*v^8 - 3726652240321364525125018821270430120241*v^7 + 2234158867551972560070936299660436565208*v^6 - 1873363459670647387804525774106854871439*v^5 - 37520355509543826800819496902254665830*v^4 - 725748512497801754572140715375856957221*v^3 + 44070932488022877337015172413455157611*v^2 - 297915559020718416444591670321666765942*v + 353512573015320277334309306227987685620) / 247769057231947568873315158172319850019 $$\beta_{16}$$ $$=$$ $$( - 47\!\cdots\!74 \nu^{19} + \cdots + 24\!\cdots\!70 ) / 24\!\cdots\!19$$ (-472868309516741447006104909804789774*v^19 + 3026897533831697291511452401411619080*v^18 - 9481362367166060065616521440417812802*v^17 + 26971051656340547187522207656832687672*v^16 - 80918773917984019635067787927709468444*v^15 + 195576832583171640841351921082938089675*v^14 - 395117003860123869187118011190327853690*v^13 + 757778322958551918006602237417606870769*v^12 - 1428185031248409730034366951518511215520*v^11 + 2303807727858537899839781311146396839684*v^10 - 3334463215096761322304791236777902278246*v^9 + 3828562577560689048317935593181034349722*v^8 - 3981605325308489192886280698518343565021*v^7 + 2219019749022625103160777289818523383087*v^6 - 1325644094463332705018709292011857961724*v^5 - 561716997138933250039958325430754144869*v^4 - 551872485181363457703930310112182180203*v^3 + 314614686710575691394105737370103115638*v^2 - 618195686980513915438289942672597139988*v + 245583478372348709220485229383381759470) / 247769057231947568873315158172319850019 $$\beta_{17}$$ $$=$$ $$( - 50\!\cdots\!84 \nu^{19} + \cdots + 48\!\cdots\!71 ) / 24\!\cdots\!19$$ (-505224820960244601198850737259063684*v^19 + 3460795297491305790815318120731246124*v^18 - 11664904083725036802969935721235070263*v^17 + 33833423770136619367416335084301313997*v^16 - 100779671312818373902797129686272373097*v^15 + 251974945497283716986897212806566304963*v^14 - 528571214564139176921100816567946951605*v^13 + 1028352786903311249343956474626472552213*v^12 - 1950663089788544949952331350474945398415*v^11 + 3267379952223224926214808669314083599161*v^10 - 4893151746482717724417861217797016468170*v^9 + 6056801991713098754521683706854171411394*v^8 - 6660464493127587410647643326544120564996*v^7 + 4933832480285639731615588880414329456637*v^6 - 3258629985579626339096162669307500324772*v^5 + 609878606708635761967317622679570070483*v^4 - 942867881369963652845020798977713177615*v^3 + 580968096811027288079231860493065974710*v^2 - 1041971500758202146449571281042989836071*v + 486548335444400726348343114403610430871) / 247769057231947568873315158172319850019 $$\beta_{18}$$ $$=$$ $$( - 57\!\cdots\!30 \nu^{19} + \cdots + 12\!\cdots\!30 ) / 24\!\cdots\!19$$ (-572098190074954816207962312560720130*v^19 + 3755240763873382559810661985895250269*v^18 - 11960546114498357504727147916447397597*v^17 + 33663622703339193984235830429095774883*v^16 - 100510178412037266897739121993746333676*v^15 + 245872177221077814398590574368827895898*v^14 - 496425246306149505922417666349214474943*v^13 + 943210243433446075358977660418159809061*v^12 - 1783615786290759592647251064819216318186*v^11 + 2904048542371690027904698498961632506068*v^10 - 4170988225224008075635036763894271342811*v^9 + 4794187826920105644488946401657090490324*v^8 - 4998967941515682941799743772483134977759*v^7 + 2895637514616486799095830116512153841304*v^6 - 1784963578503173613004529805910825897462*v^5 - 133560656677674152872440461002731347777*v^4 - 1451106625874016969974783898821492431667*v^3 + 733937459476881388325210602084801270552*v^2 - 1263435568471587751099091026759810039886*v + 125056881036252614450193807528974087330) / 247769057231947568873315158172319850019 $$\beta_{19}$$ $$=$$ $$( - 60\!\cdots\!10 \nu^{19} + \cdots + 50\!\cdots\!81 ) / 24\!\cdots\!19$$ (-605564037084765589850690275705058210*v^19 + 3984959780425575842977849615521369995*v^18 - 12723733187056335071992728466489469175*v^17 + 36087119906200970099761690895616509288*v^16 - 108304241015545679714940581951018667390*v^15 + 265310964390987305311286274371597827776*v^14 - 539072420877129775963589361829444374918*v^13 + 1036298870480057782596613629834572361595*v^12 - 1965830924830078683952508047404144191916*v^11 + 3216456401880387794022955876680562401211*v^10 - 4679619240735158267197176342319003863828*v^9 + 5539134169795347034878020062820295341306*v^8 - 5908262209014404890698303413025405421170*v^7 + 3877418882189497476764835515424210190033*v^6 - 2490111958274150939766085739659364569269*v^5 + 365635952575151827097843478421410265566*v^4 - 1155850159234074832347756339011025704307*v^3 + 900684807444741303772451422851860538779*v^2 - 820470695539161696011922001336000584642*v + 503999210452349619702846843538034201981) / 247769057231947568873315158172319850019
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$- \beta_{19} + \beta_{17} + \beta_{15} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1$$ -b19 + b17 + b15 + b11 + b9 + b7 + b6 + b5 - b4 + 2*b2 + b1 $$\nu^{3}$$ $$=$$ $$2 \beta_{17} + \beta_{15} - \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 5 \beta_{4} + 2 \beta _1 + 1$$ 2*b17 + b15 - b13 - b12 + 2*b11 - b10 + 2*b9 + b8 + 2*b7 + b6 + b5 - 5*b4 + 2*b1 + 1 $$\nu^{4}$$ $$=$$ $$4 \beta_{19} - \beta_{18} + 2 \beta_{17} - \beta_{16} + \beta_{14} + 5 \beta_{13} - \beta_{12} + 2 \beta_{11} - 7 \beta_{10} + 8 \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} + 8 \beta _1 - 5$$ 4*b19 - b18 + 2*b17 - b16 + b14 + 5*b13 - b12 + 2*b11 - 7*b10 + 8*b9 + 2*b8 + b7 + 2*b6 + 2*b5 - 8*b4 + b3 - 5*b2 + 8*b1 - 5 $$\nu^{5}$$ $$=$$ $$- 7 \beta_{19} + 6 \beta_{17} + 8 \beta_{16} + 7 \beta_{15} + 8 \beta_{14} + 8 \beta_{13} + 7 \beta_{11} - 9 \beta_{10} + 29 \beta_{9} + 10 \beta_{8} + 2 \beta_{6} + 8 \beta_{5} - 17 \beta_{4} - \beta_{2} + 10 \beta _1 - 6$$ -7*b19 + 6*b17 + 8*b16 + 7*b15 + 8*b14 + 8*b13 + 7*b11 - 9*b10 + 29*b9 + 10*b8 + 2*b6 + 8*b5 - 17*b4 - b2 + 10*b1 - 6 $$\nu^{6}$$ $$=$$ $$15 \beta_{19} - 3 \beta_{18} + 13 \beta_{17} + 29 \beta_{16} + 10 \beta_{15} + 14 \beta_{14} + 7 \beta_{13} + 3 \beta_{11} - 36 \beta_{10} + 50 \beta_{9} + 48 \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} - 48 \beta_{4} - 9 \beta_{3} - 26 \beta_{2} + 14 \beta _1 - 1$$ 15*b19 - 3*b18 + 13*b17 + 29*b16 + 10*b15 + 14*b14 + 7*b13 + 3*b11 - 36*b10 + 50*b9 + 48*b8 + 2*b7 - 9*b6 + 3*b5 - 48*b4 - 9*b3 - 26*b2 + 14*b1 - 1 $$\nu^{7}$$ $$=$$ $$21 \beta_{19} - 56 \beta_{18} + 56 \beta_{17} + 61 \beta_{16} - 7 \beta_{15} + 81 \beta_{14} + 102 \beta_{13} + 4 \beta_{12} + 14 \beta_{11} - 100 \beta_{10} + 125 \beta_{9} + 125 \beta_{8} - 23 \beta_{7} + 76 \beta_{6} - 81 \beta_{4} - 4 \beta_{3} + \cdots - 42$$ 21*b19 - 56*b18 + 56*b17 + 61*b16 - 7*b15 + 81*b14 + 102*b13 + 4*b12 + 14*b11 - 100*b10 + 125*b9 + 125*b8 - 23*b7 + 76*b6 - 81*b4 - 4*b3 - 79*b2 + 56*b1 - 42 $$\nu^{8}$$ $$=$$ $$- 116 \beta_{19} - 92 \beta_{18} + 19 \beta_{17} + 225 \beta_{16} + 291 \beta_{14} + 262 \beta_{13} + 106 \beta_{12} + 2 \beta_{11} - 28 \beta_{10} + 291 \beta_{9} + 319 \beta_{8} - 133 \beta_{7} + 121 \beta_{6} - 43 \beta_{5} - 92 \beta_{4} + \cdots - 73$$ -116*b19 - 92*b18 + 19*b17 + 225*b16 + 291*b14 + 262*b13 + 106*b12 + 2*b11 - 28*b10 + 291*b9 + 319*b8 - 133*b7 + 121*b6 - 43*b5 - 92*b4 - 43*b3 - 90*b2 - 73 $$\nu^{9}$$ $$=$$ $$145 \beta_{19} - 227 \beta_{18} + 13 \beta_{17} + 455 \beta_{16} + 34 \beta_{15} + 509 \beta_{14} + 419 \beta_{13} + 442 \beta_{12} - 228 \beta_{11} + 227 \beta_{9} + 795 \beta_{8} - 192 \beta_{7} + 82 \beta_{6} - 442 \beta_{5} - 380 \beta_{3} + \cdots + 34$$ 145*b19 - 227*b18 + 13*b17 + 455*b16 + 34*b15 + 509*b14 + 419*b13 + 442*b12 - 228*b11 + 227*b9 + 795*b8 - 192*b7 + 82*b6 - 442*b5 - 380*b3 - 214*b2 - 380*b1 + 34 $$\nu^{10}$$ $$=$$ $$297 \beta_{19} - 1211 \beta_{18} + 537 \beta_{17} + 674 \beta_{16} - 483 \beta_{15} + 1485 \beta_{14} + 1457 \beta_{13} + 1131 \beta_{12} - 246 \beta_{11} + 274 \beta_{10} + 1485 \beta_{8} - 571 \beta_{7} + 1259 \beta_{6} + \cdots - 1131 \beta_1$$ 297*b19 - 1211*b18 + 537*b17 + 674*b16 - 483*b15 + 1485*b14 + 1457*b13 + 1131*b12 - 246*b11 + 274*b10 + 1485*b8 - 571*b7 + 1259*b6 - 1545*b5 + 710*b4 - 710*b3 - 483*b2 - 1131*b1 $$\nu^{11}$$ $$=$$ $$- 122 \beta_{19} - 1870 \beta_{18} - 1240 \beta_{17} + 1500 \beta_{16} - 1500 \beta_{15} + 3426 \beta_{14} + 3110 \beta_{13} + 4398 \beta_{12} - 1434 \beta_{11} + 3426 \beta_{10} - 1833 \beta_{9} + 1870 \beta_{8} + \cdots - 118$$ -122*b19 - 1870*b18 - 1240*b17 + 1500*b16 - 1500*b15 + 3426*b14 + 3110*b13 + 4398*b12 - 1434*b11 + 3426*b10 - 1833*b9 + 1870*b8 - 2807*b7 + 619*b6 - 5046*b5 + 4398*b4 - 1833*b3 - 5046*b1 - 118 $$\nu^{12}$$ $$=$$ $$4262 \beta_{19} - 2330 \beta_{18} - 4262 \beta_{17} - 1786 \beta_{15} + 2330 \beta_{14} + 1786 \beta_{13} + 12936 \beta_{12} - 6790 \beta_{11} + 9960 \beta_{10} - 12936 \beta_{9} - 5511 \beta_{7} - 4460 \beta_{6} + \cdots + 3102$$ 4262*b19 - 2330*b18 - 4262*b17 - 1786*b15 + 2330*b14 + 1786*b13 + 12936*b12 - 6790*b11 + 9960*b10 - 12936*b9 - 5511*b7 - 4460*b6 - 16482*b5 + 16482*b4 - 7630*b3 + 3102*b2 - 19165*b1 + 3102 $$\nu^{13}$$ $$=$$ $$10883 \beta_{19} - 8909 \beta_{18} - 5526 \beta_{17} - 9390 \beta_{16} - 10883 \beta_{15} + 26744 \beta_{12} - 13802 \beta_{11} + 26744 \beta_{10} - 44123 \beta_{9} - 12738 \beta_{8} - 13802 \beta_{7} + \cdots + 9390$$ 10883*b19 - 8909*b18 - 5526*b17 - 9390*b16 - 10883*b15 + 26744*b12 - 13802*b11 + 26744*b10 - 44123*b9 - 12738*b8 - 13802*b7 - 5526*b6 - 44123*b5 + 49873*b4 - 12738*b3 + 9330*b2 - 49873*b1 + 9390 $$\nu^{14}$$ $$=$$ $$18468 \beta_{19} - 49417 \beta_{17} - 35000 \beta_{16} - 30770 \beta_{15} - 18468 \beta_{14} - 15115 \beta_{13} + 67948 \beta_{12} - 45605 \beta_{11} + 88470 \beta_{10} - 134683 \beta_{9} - 67948 \beta_{8} + \cdots + 15115$$ 18468*b19 - 49417*b17 - 35000*b16 - 30770*b15 - 18468*b14 - 15115*b13 + 67948*b12 - 45605*b11 + 88470*b10 - 134683*b9 - 67948*b8 - 49417*b7 - 49238*b6 - 106938*b5 + 152122*b4 - 18468*b3 + 35000*b2 - 134683*b1 + 15115 $$\nu^{15}$$ $$=$$ $$67441 \beta_{19} + 53292 \beta_{18} - 140549 \beta_{17} - 138979 \beta_{16} - 35723 \beta_{15} - 141583 \beta_{14} - 134867 \beta_{13} + 141583 \beta_{12} - 140549 \beta_{11} + 222452 \beta_{10} + \cdots + 74142$$ 67441*b19 + 53292*b18 - 140549*b17 - 138979*b16 - 35723*b15 - 141583*b14 - 134867*b13 + 141583*b12 - 140549*b11 + 222452*b10 - 410885*b9 - 245496*b8 - 89015*b7 - 194875*b6 - 245496*b5 + 410885*b4 - 53292*b3 + 134867*b2 - 364035*b1 + 74142 $$\nu^{16}$$ $$=$$ $$145408 \beta_{19} + 140670 \beta_{18} - 272160 \beta_{17} - 452326 \beta_{16} - 134161 \beta_{15} - 468982 \beta_{14} - 452326 \beta_{13} + 140670 \beta_{12} - 274831 \beta_{11} + 492028 \beta_{10} + \cdots + 203059$$ 145408*b19 + 140670*b18 - 272160*b17 - 452326*b16 - 134161*b15 - 468982*b14 - 452326*b13 + 140670*b12 - 274831*b11 + 492028*b10 - 1075437*b9 - 752433*b8 - 140670*b7 - 406593*b6 - 468982*b5 + 961010*b4 + 323574*b2 - 752433*b1 + 203059 $$\nu^{17}$$ $$=$$ $$160116 \beta_{19} + 618272 \beta_{18} - 948695 \beta_{17} - 1216107 \beta_{16} - 330423 \beta_{15} - 1376223 \beta_{14} - 1270704 \beta_{13} - 618272 \beta_{11} + 1177528 \beta_{10} + \cdots + 304386$$ 160116*b19 + 618272*b18 - 948695*b17 - 1216107*b16 - 330423*b15 - 1376223*b14 - 1270704*b13 - 618272*b11 + 1177528*b10 - 2553751*b9 - 2190475*b8 - 313886*b7 - 1181574*b6 - 618272*b5 + 2190475*b4 + 330983*b3 + 812921*b2 - 1376223*b1 + 304386 $$\nu^{18}$$ $$=$$ $$168627 \beta_{19} + 2217072 \beta_{18} - 2217072 \beta_{17} - 3248010 \beta_{16} - 95985 \beta_{15} - 4262288 \beta_{14} - 4093661 \beta_{13} - 1045800 \beta_{12} - 1263141 \beta_{11} + \cdots + 953931$$ 168627*b19 + 2217072*b18 - 2217072*b17 - 3248010*b16 - 95985*b15 - 4262288*b14 - 4093661*b13 - 1045800*b12 - 1263141*b11 + 2291948*b10 - 5794581*b9 - 5794581*b8 + 105691*b7 - 3231350*b6 + 4262288*b4 + 1045800*b3 + 2322763*b2 - 2217072*b1 + 953931 $$\nu^{19}$$ $$=$$ $$- 241892 \beta_{19} + 5471549 \beta_{18} - 3300861 \beta_{17} - 8086529 \beta_{16} - 11100538 \beta_{14} - 10816524 \beta_{13} - 6475244 \beta_{12} - 894624 \beta_{11} + \cdots + 2170688$$ -241892*b19 + 5471549*b18 - 3300861*b17 - 8086529*b16 - 11100538*b14 - 10816524*b13 - 6475244*b12 - 894624*b11 + 2731858*b10 - 11100538*b9 - 13832396*b8 + 2614980*b7 - 5755563*b6 + 4354157*b5 + 5471549*b4 + 4354157*b3 + 4576925*b2 + 2170688

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$47$$ $$\chi(n)$$ $$-\beta_{16}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.545935 + 0.160301i 2.55199 + 0.749331i −0.591994 + 0.380451i 1.76796 − 1.13620i −0.591994 − 0.380451i 1.76796 + 1.13620i −0.201819 + 1.40368i 0.124087 − 0.863041i −0.932269 + 2.04138i 0.480233 − 1.05156i −0.932269 − 2.04138i 0.480233 + 1.05156i 0.545935 − 0.160301i 2.55199 − 0.749331i 1.05639 − 1.21914i −1.30051 + 1.50087i −0.201819 − 1.40368i 0.124087 + 0.863041i 1.05639 + 1.21914i −1.30051 − 1.50087i
−1.10489 2.41937i −0.959493 + 0.281733i −3.32285 + 3.83478i −2.44470 1.57111i 1.74175 + 2.01009i −0.326826 2.27313i 7.84516 + 2.30355i 0.841254 0.540641i −1.09998 + 7.65055i
4.2 −0.236364 0.517565i −0.959493 + 0.281733i 1.09772 1.26683i 2.27341 + 1.46103i 0.372604 + 0.430008i −0.481879 3.35154i −2.00700 0.589308i 0.841254 0.540641i 0.218827 1.52197i
13.1 −1.37624 1.58827i 0.841254 + 0.540641i −0.343924 + 2.39205i 1.43308 3.13801i −0.299086 2.08019i −1.51768 0.445631i 0.736612 0.473392i 0.415415 + 0.909632i −6.95628 + 2.04255i
13.2 0.460828 + 0.531824i 0.841254 + 0.540641i 0.214155 1.48948i −0.700489 + 1.53386i 0.100148 + 0.696541i −3.11747 0.915371i 2.07482 1.33341i 0.415415 + 0.909632i −1.13855 + 0.334308i
16.1 −1.37624 + 1.58827i 0.841254 0.540641i −0.343924 2.39205i 1.43308 + 3.13801i −0.299086 + 2.08019i −1.51768 + 0.445631i 0.736612 + 0.473392i 0.415415 0.909632i −6.95628 2.04255i
16.2 0.460828 0.531824i 0.841254 0.540641i 0.214155 + 1.48948i −0.700489 1.53386i 0.100148 0.696541i −3.11747 + 0.915371i 2.07482 + 1.33341i 0.415415 0.909632i −1.13855 0.334308i
25.1 −0.733503 0.471394i −0.142315 0.989821i −0.515016 1.12773i −1.20424 0.353596i −0.362207 + 0.793123i 1.95301 2.25389i −0.402011 + 2.79605i −0.959493 + 0.281733i 0.716629 + 0.827034i
25.2 1.19300 + 0.766692i −0.142315 0.989821i 0.00459227 + 0.0100557i 0.815022 + 0.239312i 0.589107 1.28996i −3.31578 + 3.82661i 0.401407 2.79185i −0.959493 + 0.281733i 0.788839 + 0.910368i
31.1 −0.164520 + 1.14426i 0.415415 + 0.909632i 0.636711 + 0.186955i −1.13298 1.30753i −1.10920 + 0.325692i −0.589836 0.379064i −1.27914 + 2.80093i −0.654861 + 0.755750i 1.68256 1.08131i
31.2 0.319381 2.22134i 0.415415 + 0.909632i −2.91338 0.855446i −1.82263 2.10342i 2.15328 0.632261i 2.85304 + 1.83354i −0.966182 + 2.11564i −0.654861 + 0.755750i −5.25454 + 3.37689i
49.1 −0.164520 1.14426i 0.415415 0.909632i 0.636711 0.186955i −1.13298 + 1.30753i −1.10920 0.325692i −0.589836 + 0.379064i −1.27914 2.80093i −0.654861 0.755750i 1.68256 + 1.08131i
49.2 0.319381 + 2.22134i 0.415415 0.909632i −2.91338 + 0.855446i −1.82263 + 2.10342i 2.15328 + 0.632261i 2.85304 1.83354i −0.966182 2.11564i −0.654861 0.755750i −5.25454 3.37689i
52.1 −1.10489 + 2.41937i −0.959493 0.281733i −3.32285 3.83478i −2.44470 + 1.57111i 1.74175 2.01009i −0.326826 + 2.27313i 7.84516 2.30355i 0.841254 + 0.540641i −1.09998 7.65055i
52.2 −0.236364 + 0.517565i −0.959493 0.281733i 1.09772 + 1.26683i 2.27341 1.46103i 0.372604 0.430008i −0.481879 + 3.35154i −2.00700 + 0.589308i 0.841254 + 0.540641i 0.218827 + 1.52197i
55.1 −1.90549 + 0.559503i −0.654861 0.755750i 1.63535 1.05098i −0.291446 2.02705i 1.67068 + 1.07368i 1.34249 2.93963i 0.0729013 0.0841325i −0.142315 + 0.989821i 1.68949 + 3.69946i
55.2 1.54781 0.454477i −0.654861 0.755750i 0.506650 0.325604i 0.0749695 + 0.521424i −1.35707 0.872135i 0.200934 0.439985i −1.47656 + 1.70404i −0.142315 + 0.989821i 0.353014 + 0.772992i
58.1 −0.733503 + 0.471394i −0.142315 + 0.989821i −0.515016 + 1.12773i −1.20424 + 0.353596i −0.362207 0.793123i 1.95301 + 2.25389i −0.402011 2.79605i −0.959493 0.281733i 0.716629 0.827034i
58.2 1.19300 0.766692i −0.142315 + 0.989821i 0.00459227 0.0100557i 0.815022 0.239312i 0.589107 + 1.28996i −3.31578 3.82661i 0.401407 + 2.79185i −0.959493 0.281733i 0.788839 0.910368i
64.1 −1.90549 0.559503i −0.654861 + 0.755750i 1.63535 + 1.05098i −0.291446 + 2.02705i 1.67068 1.07368i 1.34249 + 2.93963i 0.0729013 + 0.0841325i −0.142315 0.989821i 1.68949 3.69946i
64.2 1.54781 + 0.454477i −0.654861 + 0.755750i 0.506650 + 0.325604i 0.0749695 0.521424i −1.35707 + 0.872135i 0.200934 + 0.439985i −1.47656 1.70404i −0.142315 0.989821i 0.353014 0.772992i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.2 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 69.2.e.c 20
3.b odd 2 1 207.2.i.d 20
23.c even 11 1 inner 69.2.e.c 20
23.c even 11 1 1587.2.a.u 10
23.d odd 22 1 1587.2.a.t 10
69.g even 22 1 4761.2.a.bu 10
69.h odd 22 1 207.2.i.d 20
69.h odd 22 1 4761.2.a.bt 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.c 20 1.a even 1 1 trivial
69.2.e.c 20 23.c even 11 1 inner
207.2.i.d 20 3.b odd 2 1
207.2.i.d 20 69.h odd 22 1
1587.2.a.t 10 23.d odd 22 1
1587.2.a.u 10 23.c even 11 1
4761.2.a.bt 10 69.h odd 22 1
4761.2.a.bu 10 69.g even 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 4 T_{2}^{19} + 13 T_{2}^{18} + 18 T_{2}^{17} + 11 T_{2}^{16} - 32 T_{2}^{15} - 117 T_{2}^{14} - 130 T_{2}^{13} + 238 T_{2}^{12} + 891 T_{2}^{11} + 584 T_{2}^{10} - 1463 T_{2}^{9} - 3 T_{2}^{8} - 518 T_{2}^{7} + 3360 T_{2}^{6} + \cdots + 529$$ acting on $$S_{2}^{\mathrm{new}}(69, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 4 T^{19} + 13 T^{18} + 18 T^{17} + \cdots + 529$$
$3$ $$(T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2}$$
$5$ $$T^{20} + 6 T^{19} + 25 T^{18} + \cdots + 64009$$
$7$ $$T^{20} + 6 T^{19} + 30 T^{18} + \cdots + 5031049$$
$11$ $$T^{20} + 16 T^{19} + 164 T^{18} + \cdots + 212521$$
$13$ $$T^{20} - 14 T^{19} + 137 T^{18} + \cdots + 27867841$$
$17$ $$T^{20} - 11 T^{19} + 93 T^{18} + \cdots + 21743569$$
$19$ $$T^{20} + 11 T^{19} + \cdots + 417211354561$$
$23$ $$T^{20} + 67 T^{18} + \cdots + 41426511213649$$
$29$ $$T^{20} - 12 T^{19} + \cdots + 3983644752649$$
$31$ $$T^{20} + \cdots + 811602438117769$$
$37$ $$T^{20} + 18 T^{19} + \cdots + 185232969769$$
$41$ $$T^{20} + 92 T^{18} + \cdots + 664350515929$$
$43$ $$T^{20} + \cdots + 131462437009849$$
$47$ $$(T^{10} - 9 T^{9} - 247 T^{8} + \cdots + 87731039)^{2}$$
$53$ $$T^{20} + 20 T^{19} + \cdots + 828516832441$$
$59$ $$T^{20} - 40 T^{19} + \cdots + 57652331881$$
$61$ $$T^{20} + \cdots + 559921052347801$$
$67$ $$T^{20} + 47 T^{19} + 964 T^{18} + \cdots + 2647129$$
$71$ $$T^{20} + 47 T^{19} + \cdots + 18\!\cdots\!41$$
$73$ $$T^{20} - 39 T^{19} + 621 T^{18} + \cdots + 11485321$$
$79$ $$T^{20} + 2 T^{19} + \cdots + 39\!\cdots\!81$$
$83$ $$T^{20} + 52 T^{19} + \cdots + 79\!\cdots\!61$$
$89$ $$T^{20} - 36 T^{19} + \cdots + 586753729$$
$97$ $$T^{20} + 12 T^{18} + \cdots + 48681166341721$$