# Properties

 Label 65.2.t.a Level $65$ Weight $2$ Character orbit 65.t Analytic conductor $0.519$ 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] = [65,2,Mod(7,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 11]))

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

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

chi := DirichletCharacter("65.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 65.t (of order $$12$$, degree $$4$$, 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: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1$$ x^20 + 26*x^18 + 279*x^16 + 1604*x^14 + 5353*x^12 + 10466*x^10 + 11441*x^8 + 6176*x^6 + 1263*x^4 + 78*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( - 20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + \cdots - 409 ) / 5992$$ (-20*v^18 - 389*v^16 - 2695*v^14 - 7125*v^12 + 1214*v^10 + 39860*v^8 + 68102*v^6 + 46015*v^4 + 16571*v^2 + 2996*v - 409) / 5992 $$\beta_{2}$$ $$=$$ $$( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} + \cdots + 409 ) / 5992$$ (20*v^18 + 389*v^16 + 2695*v^14 + 7125*v^12 - 1214*v^10 - 39860*v^8 - 68102*v^6 - 46015*v^4 - 16571*v^2 + 2996*v + 409) / 5992 $$\beta_{3}$$ $$=$$ $$( - 69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} + \cdots - 539 ) / 1712$$ (-69*v^18 - 1647*v^16 - 15798*v^14 - 78322*v^12 - 214723*v^10 - 324081*v^8 - 257858*v^6 - 105030*v^4 - 19961*v^2 - 539) / 1712 $$\beta_{4}$$ $$=$$ $$( 321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + \cdots - 3773 ) / 23968$$ (321*v^19 - 483*v^18 + 9951*v^17 - 11529*v^16 + 127330*v^15 - 110586*v^14 + 869910*v^13 - 548254*v^12 + 3425391*v^11 - 1503061*v^10 + 7807469*v^9 - 2268567*v^8 + 9753906*v^7 - 1805006*v^6 + 5800042*v^5 - 735210*v^4 + 1183313*v^3 - 139727*v^2 + 39483*v - 3773) / 23968 $$\beta_{5}$$ $$=$$ $$( 700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + \cdots + 26961 ) / 11984$$ (700*v^19 + 617*v^18 + 17360*v^17 + 16607*v^16 + 174468*v^15 + 185192*v^14 + 912240*v^13 + 1108682*v^12 + 2625448*v^11 + 3846937*v^10 + 3934784*v^9 + 7752327*v^8 + 2192820*v^7 + 8524592*v^6 - 974624*v^5 + 4331072*v^4 - 1268316*v^3 + 669321*v^2 - 170688*v + 26961) / 11984 $$\beta_{6}$$ $$=$$ $$( - 409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} + \cdots - 2996 ) / 5992$$ (-409*v^19 - 10614*v^17 - 113722*v^15 - 653341*v^13 - 2182252*v^11 - 4281808*v^9 - 4719229*v^7 - 2594086*v^5 - 562582*v^3 - 48473*v - 2996) / 5992 $$\beta_{7}$$ $$=$$ $$( 1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + \cdots - 30699 ) / 23968$$ (1635*v^19 - 309*v^18 + 41725*v^17 - 7171*v^16 + 436590*v^15 - 66542*v^14 + 2423698*v^13 - 319614*v^12 + 7689981*v^11 - 870831*v^10 + 13910023*v^9 - 1440917*v^8 + 13309334*v^7 - 1613590*v^6 + 5445918*v^5 - 1272982*v^4 + 481307*v^3 - 501629*v^2 + 71073*v - 30699) / 23968 $$\beta_{8}$$ $$=$$ $$( 700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + \cdots - 26961 ) / 11984$$ (700*v^19 - 617*v^18 + 17360*v^17 - 16607*v^16 + 174468*v^15 - 185192*v^14 + 912240*v^13 - 1108682*v^12 + 2625448*v^11 - 3846937*v^10 + 3934784*v^9 - 7752327*v^8 + 2192820*v^7 - 8524592*v^6 - 974624*v^5 - 4331072*v^4 - 1268316*v^3 - 669321*v^2 - 170688*v - 26961) / 11984 $$\beta_{9}$$ $$=$$ $$( - 419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} + \cdots + 83703 ) / 23968$$ (-419*v^19 + 4593*v^18 - 12681*v^17 + 118807*v^16 - 158886*v^15 + 1266230*v^14 - 1064734*v^13 + 7212374*v^12 - 4108073*v^11 + 23752795*v^10 - 9110155*v^9 + 45500681*v^8 - 10820986*v^7 + 47979462*v^6 - 5625126*v^5 + 23917926*v^4 - 518719*v^3 + 3775361*v^2 + 159919*v + 83703) / 23968 $$\beta_{10}$$ $$=$$ $$( - 3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} + \cdots + 81509 ) / 23968$$ (-3773*v^19 - 565*v^18 - 97615*v^17 - 11551*v^16 - 1041138*v^15 - 83062*v^14 - 5941306*v^13 - 198098*v^12 - 19648615*v^11 + 414413*v^10 - 37985157*v^9 + 3110895*v^8 - 40898326*v^7 + 5871486*v^6 - 21497042*v^5 + 4145562*v^4 - 4030089*v^3 + 861543*v^2 - 166551*v + 81509) / 23968 $$\beta_{11}$$ $$=$$ $$( 3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + \cdots + 39483 ) / 23968$$ (3773*v^19 + 321*v^18 + 97615*v^17 + 9951*v^16 + 1041138*v^15 + 127330*v^14 + 5941306*v^13 + 869910*v^12 + 19648615*v^11 + 3425391*v^10 + 37985157*v^9 + 7807469*v^8 + 40898326*v^7 + 9753906*v^6 + 21497042*v^5 + 5800042*v^4 + 4030089*v^3 + 1183313*v^2 + 154567*v + 39483) / 23968 $$\beta_{12}$$ $$=$$ $$( 3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + \cdots - 39483 ) / 23968$$ (3773*v^19 - 321*v^18 + 97615*v^17 - 9951*v^16 + 1041138*v^15 - 127330*v^14 + 5941306*v^13 - 869910*v^12 + 19648615*v^11 - 3425391*v^10 + 37985157*v^9 - 7807469*v^8 + 40898326*v^7 - 9753906*v^6 + 21497042*v^5 - 5800042*v^4 + 4030089*v^3 - 1183313*v^2 + 154567*v - 39483) / 23968 $$\beta_{13}$$ $$=$$ $$( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984$$ (-2753*v^19 - 867*v^18 - 71035*v^17 - 22593*v^16 - 754642*v^15 - 243222*v^14 - 4279914*v^13 - 1404094*v^12 - 14010639*v^11 - 4705845*v^10 - 26598933*v^9 - 9214959*v^8 - 27637370*v^7 - 9971998*v^6 - 13384022*v^5 - 5100714*v^4 - 1923401*v^3 - 798859*v^2 - 7127*v - 15221) / 11984 $$\beta_{14}$$ $$=$$ $$( 2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984$$ (2753*v^19 - 867*v^18 + 71035*v^17 - 22593*v^16 + 754642*v^15 - 243222*v^14 + 4279914*v^13 - 1404094*v^12 + 14010639*v^11 - 4705845*v^10 + 26598933*v^9 - 9214959*v^8 + 27637370*v^7 - 9971998*v^6 + 13384022*v^5 - 5100714*v^4 + 1923401*v^3 - 798859*v^2 + 7127*v - 15221) / 11984 $$\beta_{15}$$ $$=$$ $$( - 6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} + \cdots + 1285 ) / 23968$$ (-6965*v^19 + 1863*v^18 - 180971*v^17 + 49605*v^16 - 1940134*v^15 + 545958*v^14 - 11139534*v^13 + 3217222*v^12 - 37111487*v^11 + 10951069*v^10 - 72399061*v^9 + 21535831*v^8 - 78924566*v^7 + 22849098*v^6 - 42433790*v^5 + 10804742*v^4 - 8585045*v^3 + 1254563*v^2 - 504763*v + 1285) / 23968 $$\beta_{16}$$ $$=$$ $$( 8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + \cdots + 23947 ) / 23968$$ (8297*v^19 - 175*v^18 + 216915*v^17 - 2093*v^16 + 2342074*v^15 + 9562*v^14 + 13551722*v^13 + 273770*v^12 + 45488043*v^11 + 1753171*v^10 + 89206401*v^9 + 5197801*v^8 + 97040830*v^7 + 7462350*v^6 + 50921018*v^5 + 4439554*v^4 + 9271077*v^3 + 591213*v^2 + 419203*v + 23947) / 23968 $$\beta_{17}$$ $$=$$ $$( - 4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} + \cdots + 11984 ) / 11984$$ (-4979*v^19 - 129835*v^17 - 1398012*v^15 - 8068552*v^13 - 27041293*v^11 - 53105203*v^9 - 58320582*v^7 - 31671424*v^5 - 6571905*v^3 + 5992*v^2 - 381235*v + 11984) / 11984 $$\beta_{18}$$ $$=$$ $$( 10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + \cdots + 20195 ) / 23968$$ (10417*v^19 - 483*v^18 + 268635*v^17 - 11529*v^16 + 2853942*v^15 - 110586*v^14 + 16210930*v^13 - 548254*v^12 + 53334711*v^11 - 1503061*v^10 + 102576749*v^9 - 2268567*v^8 + 110068986*v^7 - 1805006*v^6 + 58130790*v^5 - 735210*v^4 + 11443805*v^3 - 127743*v^2 + 679767*v + 20195) / 23968 $$\beta_{19}$$ $$=$$ $$( - 16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} + \cdots - 15757 ) / 23968$$ (-16083*v^19 - 483*v^18 - 419809*v^17 - 11529*v^16 - 4525598*v^15 - 110586*v^14 - 26151990*v^13 - 548254*v^12 - 87742737*v^11 - 1503061*v^10 - 172338195*v^9 - 2268567*v^8 - 188614114*v^7 - 1805006*v^6 - 100694134*v^5 - 735210*v^4 - 19274543*v^3 - 139727*v^2 - 820129*v - 15757) / 23968
 $$\nu$$ $$=$$ $$\beta_{2} + \beta_1$$ b2 + b1 $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{10} - \beta_{3} + \beta _1 - 2$$ b12 + b10 - b3 + b1 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 7 \beta_1$$ b19 + 2*b18 - b13 - 3*b12 - 2*b11 - 2*b10 - b9 - b8 - b7 - b6 + b5 + 2*b4 - b3 - 6*b2 - 7*b1 $$\nu^{4}$$ $$=$$ $$\beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} + \cdots + 8$$ b19 - b18 + b17 + 2*b16 - 2*b15 + 3*b14 - 6*b12 - 2*b11 - 7*b10 + b9 - 3*b8 - b7 + b6 + b5 - b4 + 7*b3 - b2 - 7*b1 + 8 $$\nu^{5}$$ $$=$$ $$- 9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + \cdots - 2$$ -9*b19 - 17*b18 - b17 - 3*b14 + 11*b13 + 26*b12 + 17*b11 + 17*b10 + 8*b9 + 10*b8 + 8*b7 + 5*b6 - 6*b5 - 18*b4 + 9*b3 + 37*b2 + 46*b1 - 2 $$\nu^{6}$$ $$=$$ $$- 13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + \cdots - 39$$ -13*b19 + 13*b18 - 13*b17 - 26*b16 + 26*b15 - 32*b14 + 4*b13 + 38*b12 + 25*b11 + 53*b10 - 10*b9 + 37*b8 + 10*b7 - 13*b6 - 11*b5 + 16*b4 - 53*b3 + 18*b2 + 51*b1 - 39 $$\nu^{7}$$ $$=$$ $$66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} + \cdots + 21$$ 66*b19 + 122*b18 + 14*b17 + 37*b14 - 89*b13 - 189*b12 - 121*b11 - 120*b10 - 52*b9 - 76*b8 - 52*b7 - 24*b6 + 28*b5 + 138*b4 - 69*b3 - 236*b2 - 304*b1 + 21 $$\nu^{8}$$ $$=$$ $$120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} + \cdots + 213$$ 120*b19 - 116*b18 + 116*b17 + 232*b16 - 240*b15 + 260*b14 - 56*b13 - 252*b12 - 229*b11 - 397*b10 + 76*b9 - 332*b8 - 84*b7 + 112*b6 + 100*b5 - 152*b4 + 398*b3 - 185*b2 - 368*b1 + 213 $$\nu^{9}$$ $$=$$ $$- 462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + \cdots - 170$$ -462*b19 - 840*b18 - 136*b17 - 332*b14 + 657*b13 + 1306*b12 + 818*b11 + 813*b10 + 325*b9 + 537*b8 + 325*b7 + 122*b6 - 113*b5 - 1011*b4 + 506*b3 + 1544*b2 + 2032*b1 - 170 $$\nu^{10}$$ $$=$$ $$- 974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + \cdots - 1255$$ -974*b19 + 914*b18 - 914*b17 - 1828*b16 + 1948*b15 - 1941*b14 + 546*b13 + 1707*b12 + 1862*b11 + 2910*b10 - 539*b9 + 2648*b8 + 659*b7 - 854*b6 - 820*b5 + 1229*b4 - 2933*b3 + 1578*b2 + 2621*b1 - 1255 $$\nu^{11}$$ $$=$$ $$3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} + \cdots + 1267$$ 3210*b19 + 5733*b18 + 1135*b17 + 2648*b14 - 4690*b13 - 8890*b12 - 5456*b11 - 5476*b10 - 2042*b9 - 3716*b8 - 2042*b7 - 676*b6 + 368*b5 + 7277*b4 - 3655*b3 - 10296*b2 - 13730*b1 + 1267 $$\nu^{12}$$ $$=$$ $$7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} + \cdots + 7797$$ 7438*b19 - 6832*b18 + 6832*b17 + 13664*b16 - 14876*b15 + 14032*b14 - 4610*b13 - 11688*b12 - 14278*b11 - 20988*b10 + 3766*b9 - 19984*b8 - 4978*b7 + 6226*b6 + 6320*b5 - 9292*b4 + 21288*b3 - 12378*b2 - 18508*b1 + 7797 $$\nu^{13}$$ $$=$$ $$- 22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + \cdots - 9130$$ -22356*b19 - 39188*b18 - 8780*b17 - 19984*b14 + 33058*b13 + 60422*b12 + 36438*b11 + 37058*b10 + 13074*b9 + 25620*b8 + 13074*b7 + 4096*b6 - 528*b5 - 51934*b4 + 26218*b3 + 69645*b2 + 93629*b1 - 9130 $$\nu^{14}$$ $$=$$ $$- 54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} + \cdots - 50290$$ -54982*b19 + 49778*b18 - 49778*b17 - 99556*b16 + 109964*b15 - 100084*b14 + 36202*b13 + 80555*b12 + 105990*b11 + 149815*b10 - 26322*b9 + 146494*b8 + 36730*b7 - 44574*b6 - 46938*b5 + 68030*b4 - 152877*b3 + 92972*b2 + 130077*b1 - 50290 $$\nu^{15}$$ $$=$$ $$156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} + \cdots + 64782$$ 156271*b19 + 269080*b18 + 65190*b17 + 146494*b14 - 231975*b13 - 412141*b12 - 245006*b11 - 252616*b10 - 85481*b9 - 176993*b8 - 85481*b7 - 26707*b6 - 6031*b5 + 368826*b4 - 187213*b3 - 476096*b2 - 643231*b1 + 64782 $$\nu^{16}$$ $$=$$ $$398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + \cdots + 333034$$ 398897*b19 - 357747*b18 + 357747*b17 + 715494*b16 - 797794*b15 + 709463*b14 - 272912*b13 - 557750*b12 - 771768*b11 - 1062637*b10 + 184581*b9 - 1056441*b8 - 266881*b7 + 316597*b6 + 340947*b5 - 489763*b4 + 1090029*b3 - 681429*b2 - 912121*b1 + 333034 $$\nu^{17}$$ $$=$$ $$- 1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + \cdots - 456258$$ -1095413*b19 - 1856511*b18 - 472963*b17 - 1056441*b14 + 1625819*b13 + 2825568*b12 + 1660831*b11 + 1734115*b10 + 569378*b9 + 1226922*b8 + 569378*b7 + 182897*b6 + 88166*b5 - 2610994*b4 + 1332033*b3 + 3279739*b2 + 4444476*b1 - 456258 $$\nu^{18}$$ $$=$$ $$- 2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} + \cdots - 2246355$$ -2861437*b19 + 2550751*b18 - 2550751*b17 - 5101502*b16 + 5722874*b15 - 5013336*b14 + 2007386*b13 + 3875098*b12 + 5552425*b11 + 7508303*b10 - 1297848*b9 + 7545155*b8 + 1919220*b7 - 2240065*b6 - 2443653*b5 + 3492968*b4 - 7734751*b3 + 4921700*b2 + 6390257*b1 - 2246355 $$\nu^{19}$$ $$=$$ $$7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} + \cdots + 3202109$$ 7691530*b19 + 12862874*b18 + 3385870*b17 + 7545155*b14 - 11394317*b13 - 19469181*b12 - 11344809*b11 - 11973534*b10 - 3849162*b9 - 8532880*b8 - 3849162*b7 - 1287312*b6 - 834556*b5 + 18442656*b4 - 9449577*b3 - 22718962*b2 - 30843334*b1 + 3202109

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$\beta_{11} + \beta_{12}$$ $$-\beta_{11}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 − 2.64975i − 1.83163i − 0.493902i − 0.274809i 1.51805i 2.64975i 1.83163i 0.493902i 0.274809i − 1.51805i 2.25081i 1.02262i 0.131303i − 1.58474i − 2.08794i − 2.25081i − 1.02262i − 0.131303i 1.58474i 2.08794i
−2.29475 1.32488i 0.335680 + 1.25278i 2.51060 + 4.34849i 1.81654 1.30391i 0.889471 3.31955i 0.0561740 + 0.0972962i 8.00544i 1.14131 0.658935i −5.89604 + 0.585458i
7.2 −1.58624 0.915816i −0.512942 1.91432i 0.677439 + 1.17336i −1.69810 + 1.45480i −0.939520 + 3.50634i −1.76945 3.06478i 1.18163i −0.803451 + 0.463873i 4.02593 0.752519i
7.3 −0.427732 0.246951i −0.243392 0.908353i −0.878030 1.52079i −0.284413 2.21791i −0.120212 + 0.448637i 1.83775 + 3.18307i 1.85513i 1.83221 1.05783i −0.426062 + 1.01890i
7.4 −0.237991 0.137404i 0.611610 + 2.28256i −0.962240 1.66665i 1.45395 + 1.69883i 0.168076 0.627267i −0.193052 0.334376i 1.07848i −2.23793 + 1.29207i −0.112600 0.604086i
7.5 1.31467 + 0.759023i 0.175069 + 0.653367i 0.152233 + 0.263675i −2.15400 + 0.600231i −0.265763 + 0.991842i −1.29744 2.24723i 2.57390i 2.20184 1.27123i −3.28738 0.845834i
28.1 −2.29475 + 1.32488i 0.335680 1.25278i 2.51060 4.34849i 1.81654 + 1.30391i 0.889471 + 3.31955i 0.0561740 0.0972962i 8.00544i 1.14131 + 0.658935i −5.89604 0.585458i
28.2 −1.58624 + 0.915816i −0.512942 + 1.91432i 0.677439 1.17336i −1.69810 1.45480i −0.939520 3.50634i −1.76945 + 3.06478i 1.18163i −0.803451 0.463873i 4.02593 + 0.752519i
28.3 −0.427732 + 0.246951i −0.243392 + 0.908353i −0.878030 + 1.52079i −0.284413 + 2.21791i −0.120212 0.448637i 1.83775 3.18307i 1.85513i 1.83221 + 1.05783i −0.426062 1.01890i
28.4 −0.237991 + 0.137404i 0.611610 2.28256i −0.962240 + 1.66665i 1.45395 1.69883i 0.168076 + 0.627267i −0.193052 + 0.334376i 1.07848i −2.23793 1.29207i −0.112600 + 0.604086i
28.5 1.31467 0.759023i 0.175069 0.653367i 0.152233 0.263675i −2.15400 0.600231i −0.265763 0.991842i −1.29744 + 2.24723i 2.57390i 2.20184 + 1.27123i −3.28738 + 0.845834i
37.1 −1.94926 + 1.12540i −1.91913 0.514229i 1.53307 2.65535i 0.247944 2.22228i 4.31958 1.15743i 0.638592 1.10607i 2.39966i 0.820542 + 0.473740i 2.01765 + 4.61083i
37.2 −0.885613 + 0.511309i 2.69193 + 0.721300i −0.477126 + 0.826407i −1.45744 1.69584i −2.75281 + 0.737614i −0.481787 + 0.834479i 3.02107i 4.12812 + 2.38337i 2.15782 + 0.756660i
37.3 −0.113711 + 0.0656513i 0.332179 + 0.0890070i −0.991380 + 1.71712i 2.08297 + 0.813169i −0.0436159 + 0.0116869i 1.39069 2.40874i 0.522947i −2.49566 1.44087i −0.290243 + 0.0442830i
37.4 1.37242 0.792369i 0.190588 + 0.0510678i 0.255697 0.442881i −2.23506 + 0.0672627i 0.302032 0.0809291i 0.274164 0.474866i 2.35905i −2.56436 1.48053i −3.01415 + 1.86330i
37.5 1.80821 1.04397i −2.66159 0.713171i 1.17974 2.04338i 2.22760 0.194361i −5.55724 + 1.48906i −1.45563 + 2.52122i 0.750585i 3.97738 + 2.29634i 3.82506 2.67700i
58.1 −1.94926 1.12540i −1.91913 + 0.514229i 1.53307 + 2.65535i 0.247944 + 2.22228i 4.31958 + 1.15743i 0.638592 + 1.10607i 2.39966i 0.820542 0.473740i 2.01765 4.61083i
58.2 −0.885613 0.511309i 2.69193 0.721300i −0.477126 0.826407i −1.45744 + 1.69584i −2.75281 0.737614i −0.481787 0.834479i 3.02107i 4.12812 2.38337i 2.15782 0.756660i
58.3 −0.113711 0.0656513i 0.332179 0.0890070i −0.991380 1.71712i 2.08297 0.813169i −0.0436159 0.0116869i 1.39069 + 2.40874i 0.522947i −2.49566 + 1.44087i −0.290243 0.0442830i
58.4 1.37242 + 0.792369i 0.190588 0.0510678i 0.255697 + 0.442881i −2.23506 0.0672627i 0.302032 + 0.0809291i 0.274164 + 0.474866i 2.35905i −2.56436 + 1.48053i −3.01415 1.86330i
58.5 1.80821 + 1.04397i −2.66159 + 0.713171i 1.17974 + 2.04338i 2.22760 + 0.194361i −5.55724 1.48906i −1.45563 2.52122i 0.750585i 3.97738 2.29634i 3.82506 + 2.67700i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.t.a yes 20
3.b odd 2 1 585.2.dp.a 20
5.b even 2 1 325.2.x.b 20
5.c odd 4 1 65.2.o.a 20
5.c odd 4 1 325.2.s.b 20
13.b even 2 1 845.2.t.g 20
13.c even 3 1 845.2.f.e 20
13.c even 3 1 845.2.t.f 20
13.d odd 4 1 845.2.o.e 20
13.d odd 4 1 845.2.o.f 20
13.e even 6 1 845.2.f.d 20
13.e even 6 1 845.2.t.e 20
13.f odd 12 1 65.2.o.a 20
13.f odd 12 1 845.2.k.d 20
13.f odd 12 1 845.2.k.e 20
13.f odd 12 1 845.2.o.g 20
15.e even 4 1 585.2.cf.a 20
39.k even 12 1 585.2.cf.a 20
65.f even 4 1 845.2.t.f 20
65.h odd 4 1 845.2.o.g 20
65.k even 4 1 845.2.t.e 20
65.o even 12 1 325.2.x.b 20
65.o even 12 1 845.2.f.d 20
65.o even 12 1 845.2.t.g 20
65.q odd 12 1 845.2.k.e 20
65.q odd 12 1 845.2.o.e 20
65.r odd 12 1 845.2.k.d 20
65.r odd 12 1 845.2.o.f 20
65.s odd 12 1 325.2.s.b 20
65.t even 12 1 inner 65.2.t.a yes 20
65.t even 12 1 845.2.f.e 20
195.bc odd 12 1 585.2.dp.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.o.a 20 5.c odd 4 1
65.2.o.a 20 13.f odd 12 1
65.2.t.a yes 20 1.a even 1 1 trivial
65.2.t.a yes 20 65.t even 12 1 inner
325.2.s.b 20 5.c odd 4 1
325.2.s.b 20 65.s odd 12 1
325.2.x.b 20 5.b even 2 1
325.2.x.b 20 65.o even 12 1
585.2.cf.a 20 15.e even 4 1
585.2.cf.a 20 39.k even 12 1
585.2.dp.a 20 3.b odd 2 1
585.2.dp.a 20 195.bc odd 12 1
845.2.f.d 20 13.e even 6 1
845.2.f.d 20 65.o even 12 1
845.2.f.e 20 13.c even 3 1
845.2.f.e 20 65.t even 12 1
845.2.k.d 20 13.f odd 12 1
845.2.k.d 20 65.r odd 12 1
845.2.k.e 20 13.f odd 12 1
845.2.k.e 20 65.q odd 12 1
845.2.o.e 20 13.d odd 4 1
845.2.o.e 20 65.q odd 12 1
845.2.o.f 20 13.d odd 4 1
845.2.o.f 20 65.r odd 12 1
845.2.o.g 20 13.f odd 12 1
845.2.o.g 20 65.h odd 4 1
845.2.t.e 20 13.e even 6 1
845.2.t.e 20 65.k even 4 1
845.2.t.f 20 13.c even 3 1
845.2.t.f 20 65.f even 4 1
845.2.t.g 20 13.b even 2 1
845.2.t.g 20 65.o even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 6 T^{19} + \cdots + 1$$
$3$ $$T^{20} + 2 T^{19} + \cdots + 16$$
$5$ $$T^{20} - 9 T^{18} + \cdots + 9765625$$
$7$ $$T^{20} + 2 T^{19} + \cdots + 64$$
$11$ $$T^{20} + 16 T^{19} + \cdots + 256$$
$13$ $$T^{20} + \cdots + 137858491849$$
$17$ $$T^{20} - 4 T^{19} + \cdots + 1168561$$
$19$ $$T^{20} + \cdots + 1583721616$$
$23$ $$T^{20} + 10 T^{19} + \cdots + 144$$
$29$ $$T^{20} + \cdots + 206213167449$$
$31$ $$T^{20} - 104 T^{17} + \cdots + 2166784$$
$37$ $$T^{20} + \cdots + 4508182449$$
$41$ $$T^{20} + \cdots + 3748255729$$
$43$ $$T^{20} + \cdots + 1370772640000$$
$47$ $$(T^{10} + 20 T^{9} + \cdots - 28416)^{2}$$
$53$ $$T^{20} + \cdots + 2978634160384$$
$59$ $$T^{20} + 16 T^{19} + \cdots + 33856$$
$61$ $$T^{20} + \cdots + 826457355409$$
$67$ $$T^{20} + \cdots + 15478905336976$$
$71$ $$T^{20} + \cdots + 11\!\cdots\!44$$
$73$ $$T^{20} + \cdots + 64\!\cdots\!24$$
$79$ $$T^{20} + \cdots + 75\!\cdots\!36$$
$83$ $$(T^{10} - 24 T^{9} + \cdots + 3393024)^{2}$$
$89$ $$T^{20} + \cdots + 329648222500$$
$97$ $$T^{20} + \cdots + 28\!\cdots\!36$$