# Properties

 Label 1045.4.a.b Level $1045$ Weight $4$ Character orbit 1045.a Self dual yes Analytic conductor $61.657$ Analytic rank $1$ Dimension $20$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1045.a (trivial)

## 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: yes Analytic conductor: $$61.6569959560$$ Analytic rank: $$1$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + 623613 x^{12} - 5673747 x^{11} - 4539454 x^{10} + 37893109 x^{9} + \cdots + 17756160$$ x^20 - 8*x^19 - 82*x^18 + 700*x^17 + 2826*x^16 - 25467*x^15 - 53768*x^14 + 498499*x^13 + 623613*x^12 - 5673747*x^11 - 4539454*x^10 + 37893109*x^9 + 19879768*x^8 - 143049638*x^7 - 45064360*x^6 + 280461480*x^5 + 43097920*x^4 - 240447168*x^3 - 36068096*x^2 + 74703872*x + 17756160 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + 5 q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{6} + ( - \beta_{7} - 7) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1 - 7) q^{8} + (\beta_{9} - \beta_{5} + \beta_1 + 8) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b5 - 1) * q^3 + (b2 - b1 + 4) * q^4 + 5 * q^5 + (-b5 + b3 - b2 - 2*b1 - 2) * q^6 + (-b7 - 7) * q^7 + (-b5 + b4 - b2 + 4*b1 - 7) * q^8 + (b9 - b5 + b1 + 8) * q^9 $$q + (\beta_1 - 1) q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + 5 q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{6} + ( - \beta_{7} - 7) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1 - 7) q^{8} + (\beta_{9} - \beta_{5} + \beta_1 + 8) q^{9} + (5 \beta_1 - 5) q^{10} + 11 q^{11} + ( - \beta_{19} - \beta_{17} - \beta_{14} - \beta_{13} - \beta_{10} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \cdots - 10) q^{12}+ \cdots + (11 \beta_{9} - 11 \beta_{5} + 11 \beta_1 + 88) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b5 - 1) * q^3 + (b2 - b1 + 4) * q^4 + 5 * q^5 + (-b5 + b3 - b2 - 2*b1 - 2) * q^6 + (-b7 - 7) * q^7 + (-b5 + b4 - b2 + 4*b1 - 7) * q^8 + (b9 - b5 + b1 + 8) * q^9 + (5*b1 - 5) * q^10 + 11 * q^11 + (-b19 - b17 - b14 - b13 - b10 - b8 - 2*b7 - b6 + 5*b5 - 2*b2 - 2*b1 - 10) * q^12 + (b17 - b5 - b2 - b1 - 11) * q^13 + (-b11 - b8 + b7 - b6 - 2*b5 - 9*b1 + 4) * q^14 + (5*b5 - 5) * q^15 + (b19 - b18 - b17 + b16 + b12 - b9 + 2*b8 + 2*b7 + 2*b6 - 2*b5 - b4 - 3*b3 + 5*b2 - 6*b1 + 23) * q^16 + (b19 + b11 + b10 - b9 + 2*b7 + b6 - b5 - b4 - b3 - 23) * q^17 + (b19 + b18 + 2*b17 + 2*b14 + 3*b13 - b12 + b11 + 2*b10 - b9 + 3*b8 + 4*b7 + 2*b6 - 2*b5 - 3*b4 - 3*b3 + 4*b2 + 7*b1 + 6) * q^18 + 19 * q^19 + (5*b2 - 5*b1 + 20) * q^20 + (-b18 - b17 - 2*b16 + 2*b14 + b11 - 3*b9 + b8 + 4*b7 - 12*b5 + b4 - 2*b3 - 8*b1 + 1) * q^21 + (11*b1 - 11) * q^22 + (-b19 + b18 - b16 - b15 - 2*b12 - b11 - b10 - b9 - 3*b8 + 2*b7 - 2*b6 + b3 - 2*b2 - 3*b1 - 30) * q^23 + (2*b18 + 3*b17 - b16 + b15 + 2*b14 + 2*b13 - b11 + 2*b10 - b9 + 2*b7 - 7*b5 - 3*b4 + b3 - 5*b2 - 17*b1 - 18) * q^24 + 25 * q^25 + (b18 - b17 - b16 - b14 - 2*b13 + b12 - b11 + b9 - 2*b8 - 2*b7 - 3*b6 + 7*b5 - 2*b4 - b3 + b2 - 21*b1 + 1) * q^26 + (b18 + b16 - b15 + b14 + b13 + b11 - b10 - 2*b9 + 2*b8 + 3*b7 + 2*b6 + 7*b5 - 4*b4 - 3*b3 + 3*b2 - 3*b1 - 5) * q^27 + (-2*b19 + b16 + b15 - 3*b14 - b13 + b12 + 2*b11 + 2*b9 - 4*b7 + 2*b6 + 7*b5 - b4 + b3 - 11*b2 + 8*b1 - 45) * q^28 + (b19 + 2*b17 - b16 + 2*b15 + 2*b14 + 2*b13 - 2*b12 - 2*b11 + b10 + 3*b8 + 3*b7 + b6 - 4*b5 - 3*b4 - 4*b3 + 3*b2 - 4*b1 + 8) * q^29 + (-5*b5 + 5*b3 - 5*b2 - 10*b1 - 10) * q^30 + (-b19 - 4*b17 + 5*b16 + b15 - 4*b14 - 3*b13 + 3*b12 - b10 + b9 - 3*b8 - 5*b7 - 2*b6 - b3 + 3*b2 - 3*b1 - 5) * q^31 + (b19 - b18 + b17 - 2*b16 - b15 + 2*b14 + 4*b13 - 3*b12 + 3*b11 + b10 - b9 + 7*b7 + 2*b6 - 11*b5 + 3*b4 - b3 - 7*b2 + 21*b1 - 23) * q^32 + (11*b5 - 11) * q^33 + (b19 - 3*b18 - b16 - 2*b15 + 3*b14 - 2*b13 - b12 - 3*b10 - 2*b9 + 4*b7 - 2*b6 - 3*b5 + 3*b4 - 6*b3 - 2*b2 - 22*b1 + 41) * q^34 + (-5*b7 - 35) * q^35 + (2*b19 - 3*b18 - 3*b17 - b16 - 3*b15 + b14 - 2*b13 - b12 - 2*b10 + b9 - 2*b8 + 3*b7 - b6 - 4*b5 + 4*b4 - 8*b3 + 8*b2 + 11*b1 + 48) * q^36 + (-b19 - b17 + 3*b16 - b15 - 5*b14 - 4*b13 + 5*b12 - b11 + 2*b10 + 5*b9 - 6*b7 - b6 - 5*b5 + 3*b4 + 2*b3 + b2 - 6*b1 - 8) * q^37 + (19*b1 - 19) * q^38 + (b19 - 2*b18 + b17 + 3*b16 + b15 - 3*b14 + 2*b13 + 4*b12 - 3*b11 + 2*b10 + b9 + 3*b8 + 2*b7 + 3*b6 - 26*b5 + 4*b4 + 13*b1 - 2) * q^39 + (-5*b5 + 5*b4 - 5*b2 + 20*b1 - 35) * q^40 + (b19 - 2*b18 + b17 - 2*b16 + b15 - b14 - b13 - b12 - 2*b11 - 2*b10 + 3*b7 + b6 - 5*b5 - b4 - 5*b3 - 6*b2 - 18*b1 - 54) * q^41 + (2*b19 - 2*b18 + 4*b16 + b15 - 6*b14 - 4*b13 + 3*b12 + b11 - b10 - b9 + 2*b8 - 5*b7 + 3*b6 + 4*b5 + 5*b4 - 11*b3 + 10*b2 + 26*b1 - 43) * q^42 + (-2*b18 - b17 + 2*b16 + b15 - 4*b14 + b13 - 4*b11 - 2*b10 - 7*b7 - 3*b6 - 6*b5 + 2*b4 + b3 - 7*b2 + 7*b1 - 85) * q^43 + (11*b2 - 11*b1 + 44) * q^44 + (5*b9 - 5*b5 + 5*b1 + 40) * q^45 + (-2*b19 + b18 + 3*b16 + 3*b15 - 5*b14 - 4*b13 + 3*b12 - 2*b11 + b10 + 6*b9 + 2*b8 - 18*b7 + 3*b6 + 10*b5 - 6*b4 + 6*b3 - 5*b2 - 31*b1 - 3) * q^46 + (-4*b19 + b18 - 4*b17 - b16 + 2*b14 - b13 - 4*b12 - 4*b10 - 7*b9 + 3*b8 + 6*b7 - 4*b6 + 8*b5 + b4 + b3 - 10*b2 + 4*b1 - 102) * q^47 + (-4*b19 - b18 - 2*b17 + 2*b16 - 6*b14 - 13*b13 - b12 - 3*b11 - 8*b10 + 9*b9 - 16*b8 - 26*b7 - 10*b6 + 24*b5 + 3*b4 + 3*b3 - 13*b2 - 26*b1 - 70) * q^48 + (-2*b19 + 2*b18 + 2*b17 - 2*b15 + 6*b14 + 7*b13 + 2*b12 + 4*b11 + 2*b10 + 3*b9 + 4*b8 + 12*b7 + 4*b6 - 12*b5 - 2*b4 + 2*b3 - 11*b2 - 12*b1 + 49) * q^49 + (25*b1 - 25) * q^50 + (b18 - 2*b17 + b16 + b15 - 2*b14 + 5*b13 + 2*b12 + 2*b10 - 3*b8 - 7*b7 - 3*b6 - 46*b5 + b4 + 10*b3 - 11*b2 - 8*b1 - 15) * q^51 + (-3*b19 + 2*b18 + 6*b17 - 6*b16 + 6*b14 + 5*b13 - 5*b12 + 5*b11 + b10 - 4*b9 + b8 - 15*b5 - 4*b4 + 9*b3 - 43*b2 - 4*b1 - 168) * q^52 + (b19 + 7*b18 + 2*b17 - 8*b16 - 2*b15 + 8*b14 + 7*b13 - 3*b12 + 2*b11 + 7*b10 + 6*b9 - 2*b8 + 10*b7 - b6 - 17*b5 - 3*b4 + 3*b3 - 9*b2 - 11*b1 - 35) * q^53 + (-2*b19 + 3*b18 + b17 - 3*b16 + 2*b15 - 2*b14 - 4*b13 - 7*b12 - 3*b11 + 9*b9 - 9*b8 - 4*b7 - 6*b6 - 10*b5 + 7*b4 + 16*b3 - 36*b2 - 2*b1 - 40) * q^54 + 55 * q^55 + (3*b18 - 2*b17 - 5*b16 - 3*b15 + 9*b14 - 2*b12 + b11 - 2*b10 - 5*b9 + 8*b7 - 6*b6 + 6*b5 - 6*b4 + 9*b3 - 9*b2 - 72*b1 + 75) * q^56 + (19*b5 - 19) * q^57 + (2*b18 - b17 + b16 - 3*b15 + 2*b14 + 3*b13 + 8*b11 - 4*b10 + 2*b9 - 4*b8 - 2*b7 + 3*b6 - 5*b5 - 6*b3 - 7*b2 + 10*b1 - 23) * q^58 + (-b19 + 5*b18 + b17 + 2*b16 - 5*b14 - 4*b13 + b12 + b11 - 4*b10 - 8*b8 - 2*b7 - 2*b6 - 23*b5 + 8*b4 + 8*b3 - 4*b2 - 41*b1 - 16) * q^59 + (-5*b19 - 5*b17 - 5*b14 - 5*b13 - 5*b10 - 5*b8 - 10*b7 - 5*b6 + 25*b5 - 10*b2 - 10*b1 - 50) * q^60 + (-4*b19 - 4*b18 - 7*b17 - 5*b16 - b15 + 6*b14 + b13 + 3*b12 + 9*b11 - 4*b10 - 5*b9 + 4*b8 + 4*b7 + 7*b6 - 17*b5 + 5*b4 + 6*b3 - 15*b2 + 15) * q^61 + (-3*b19 - b18 + 2*b17 - 3*b16 + b15 + 9*b14 + 4*b13 - 5*b12 + 5*b11 - b10 - 4*b9 - 4*b8 - b7 + 4*b6 - 37*b5 + 7*b4 + 6*b3 - 25*b2 - 5*b1 - 40) * q^62 + (-5*b19 - b18 - 5*b17 - 4*b14 - 10*b13 - 2*b12 - 8*b11 - b10 - 6*b9 - 15*b8 - 18*b7 - 13*b6 - 7*b5 + 6*b4 + b3 - 12*b1 - 204) * q^63 + (6*b19 + b18 + 3*b17 + 5*b16 + b15 - b14 + 2*b12 - 8*b11 + 9*b10 + 10*b9 + 9*b8 - 6*b7 - b6 + 4*b5 - 10*b4 - 7*b3 + 46*b2 - 5*b1 + 134) * q^64 + (5*b17 - 5*b5 - 5*b2 - 5*b1 - 55) * q^65 + (-11*b5 + 11*b3 - 11*b2 - 22*b1 - 22) * q^66 + (10*b19 + 2*b18 + 5*b17 + b16 + b15 + 3*b14 + 5*b13 + 3*b12 - 10*b11 + b10 - 5*b9 + 13*b8 + 4*b7 + 8*b6 - 15*b5 - 2*b4 + b3 + 11*b2 - 13*b1 - 80) * q^67 + (6*b19 + 8*b18 + 13*b17 + 2*b16 + 6*b15 - 2*b14 + 13*b13 + 5*b12 - 2*b11 + 14*b10 + 9*b9 + 21*b8 - 2*b7 + 11*b6 - 27*b5 - 13*b4 - 2*b3 + 4*b2 + 5*b1 - 98) * q^68 + (-2*b19 - 4*b17 + 2*b16 + 3*b15 - 9*b14 - b13 + 5*b11 - b10 + 3*b9 + 7*b8 - 3*b7 - 42*b5 + b4 + 4*b3 + 22*b2 + 14*b1) * q^69 + (-5*b11 - 5*b8 + 5*b7 - 5*b6 - 10*b5 - 45*b1 + 20) * q^70 + (-b19 - 4*b18 - 2*b17 + 9*b16 + 2*b15 - 11*b14 - 9*b13 - 5*b12 - 11*b11 - 2*b10 + 3*b9 - 6*b8 - 16*b7 - 16*b5 - b4 + 11*b3 - 4*b2 - 17*b1 - 9) * q^71 + (13*b19 - 6*b18 + 8*b17 + 4*b16 + 4*b15 - 3*b14 + 7*b13 + 9*b12 - 2*b11 + 15*b10 + 20*b8 + 12*b7 + 19*b6 - 97*b5 + 7*b4 - 2*b3 + 18*b2 + 25*b1 + 36) * q^72 + (-5*b19 - b18 + 5*b17 - 8*b16 + 5*b15 + 6*b14 - 3*b13 - 3*b12 - 2*b11 - 3*b10 - 7*b9 + 9*b8 - 4*b6 - 21*b5 - 3*b4 - 10*b3 + 12*b2 - 23*b1 - 183) * q^73 + (-3*b18 + 8*b17 - 19*b16 - 8*b15 + 16*b14 + 20*b13 - 3*b12 + 9*b11 + 7*b10 - 6*b9 + 9*b8 + 37*b7 + 2*b6 - 31*b5 + 7*b4 - 11*b3 - 5*b2 - 17*b1 - 39) * q^74 + (25*b5 - 25) * q^75 + (19*b2 - 19*b1 + 76) * q^76 + (-11*b7 - 77) * q^77 + (-4*b19 - 10*b18 - 16*b17 - 2*b16 - 4*b15 - 7*b14 + 7*b12 + 4*b11 - 3*b10 + b9 - 13*b7 + 3*b6 + 24*b5 + 18*b4 - 22*b3 + 50*b2 + 16*b1 + 211) * q^78 + (b19 - 4*b18 - 5*b17 + 6*b16 - 5*b15 + 7*b14 - 6*b13 - 6*b12 + 10*b11 - 6*b10 - 8*b9 - b8 + b7 + 7*b6 + 10*b5 + 4*b4 - 5*b3 + 20*b2 + 6) * q^79 + (5*b19 - 5*b18 - 5*b17 + 5*b16 + 5*b12 - 5*b9 + 10*b8 + 10*b7 + 10*b6 - 10*b5 - 5*b4 - 15*b3 + 25*b2 - 30*b1 + 115) * q^80 + (-4*b19 - 6*b18 - 5*b17 + 3*b16 + 2*b15 - 15*b14 - 15*b13 - 6*b12 - 6*b11 - 7*b10 + 9*b9 - 12*b8 - 23*b7 - 5*b6 - 15*b5 + 15*b4 + 32*b3 - 7*b2 - 17*b1 - 3) * q^81 + (4*b19 + 7*b18 + 9*b17 - b16 + b15 + 5*b14 + 7*b13 + b12 + 3*b11 + 6*b10 - 10*b9 + 6*b8 + 5*b7 + 2*b6 - 15*b5 - 13*b4 - 4*b3 - 22*b2 - 117*b1 - 147) * q^82 + (8*b19 + 3*b18 - 10*b17 + 8*b16 - 3*b15 + 4*b14 - 10*b13 - 2*b12 + 9*b11 - 6*b10 - 8*b9 - 11*b8 + 3*b7 - 11*b6 + 16*b5 - 7*b4 + 5*b3 + 2*b2 - 61*b1 - 140) * q^83 + (19*b19 + 2*b18 + 14*b17 + 3*b16 - 7*b15 + 14*b14 + 17*b13 + 3*b12 - 6*b11 + 8*b10 + 2*b9 + 8*b8 + 42*b7 + 9*b6 - 64*b5 + 4*b4 - 6*b3 + 57*b2 - 5*b1 + 323) * q^84 + (5*b19 + 5*b11 + 5*b10 - 5*b9 + 10*b7 + 5*b6 - 5*b5 - 5*b4 - 5*b3 - 115) * q^85 + (b19 - 2*b18 - b17 - 2*b16 + b15 + b14 + 13*b13 + 11*b12 + 3*b11 + 10*b10 + 3*b9 + 9*b8 - 3*b7 + 14*b6 - 12*b5 - 12*b4 + 3*b3 + 13*b2 - 161*b1 + 151) * q^86 + (2*b19 - 3*b17 + b16 - 7*b15 + 10*b14 - 6*b13 - 2*b12 + 3*b11 - 12*b9 - 4*b8 + b7 - b6 + 8*b5 + 16*b4 + 4*b3 - 20*b2 - 24*b1 - 126) * q^87 + (-11*b5 + 11*b4 - 11*b2 + 44*b1 - 77) * q^88 + (-2*b19 - 7*b18 - 7*b17 + 6*b16 + b14 - 13*b13 + 5*b12 - 12*b11 + 5*b10 - 4*b9 - 16*b8 + 7*b7 - 11*b6 - 20*b5 + 17*b4 + b3 - 7*b2 - 77*b1 + 120) * q^89 + (5*b19 + 5*b18 + 10*b17 + 10*b14 + 15*b13 - 5*b12 + 5*b11 + 10*b10 - 5*b9 + 15*b8 + 20*b7 + 10*b6 - 10*b5 - 15*b4 - 15*b3 + 20*b2 + 35*b1 + 30) * q^90 + (-2*b19 - 9*b17 - 6*b16 - 3*b15 + 6*b14 + 11*b13 - 4*b12 + 20*b11 - 8*b9 + 11*b8 + 15*b7 + 13*b6 - 15*b5 - 8*b4 + 11*b3 + 16*b2 - 8*b1 + 98) * q^91 + (-10*b19 + 2*b18 - 2*b17 - 12*b16 - 2*b15 + 15*b14 + 5*b13 + 6*b11 - 10*b10 - 3*b9 - 20*b8 + 7*b7 - 25*b6 + 43*b5 + 16*b4 + 20*b3 - 97*b2 - 39*b1 - 178) * q^92 + (16*b19 - 8*b18 + 3*b17 + 6*b16 - 4*b15 - 10*b14 - 13*b13 - b12 - 4*b11 - 10*b10 - 3*b9 - 16*b8 + 10*b7 + 6*b6 - 3*b5 + 15*b4 - 18*b3 + 38*b2 - 89*b1 - 60) * q^93 + (b19 + 2*b18 - 4*b17 + b16 + 2*b15 - 4*b14 + 2*b13 + 7*b12 + 14*b11 - 2*b10 + 11*b9 + 5*b8 + 23*b6 + 13*b5 - 2*b4 + 26*b3 - 4*b2 - 158*b1 + 181) * q^94 + 95 * q^95 + (2*b19 + 16*b18 + 6*b17 + 5*b16 + 4*b15 + 3*b14 + 2*b13 - 6*b12 + b11 + 5*b10 - 16*b9 - 5*b8 - 11*b6 + 13*b5 - 33*b4 + 21*b3 - 27*b2 - 96*b1 - 287) * q^96 + (10*b19 - 10*b18 + 7*b17 - 3*b16 - 2*b15 + 21*b14 + 8*b13 - 14*b12 + 3*b10 - 22*b9 + 17*b8 + 30*b7 + 5*b6 - 17*b5 - 17*b4 - 2*b3 + 23*b2 - 78*b1 - 51) * q^97 + (-5*b19 + 2*b18 - 11*b17 + 12*b16 + 7*b15 - 24*b14 - 16*b13 - 8*b12 - 9*b11 + 3*b10 + 25*b9 - 4*b8 - 41*b7 - 5*b6 + 79*b5 - 11*b4 - 3*b3 + 8*b2 + 10*b1 - 130) * q^98 + (11*b9 - 11*b5 + 11*b1 + 88) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10})$$ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 $$20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100})$$ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + 623613 x^{12} - 5673747 x^{11} - 4539454 x^{10} + 37893109 x^{9} + \cdots + 17756160$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 11$$ v^2 - v - 11 $$\beta_{3}$$ $$=$$ $$( - 39\!\cdots\!97 \nu^{19} + \cdots - 10\!\cdots\!64 ) / 34\!\cdots\!88$$ (-3994758687856231406973990197*v^19 - 2068542207323588132054066868*v^18 + 625066262857449266832597516650*v^17 - 923107893843079783060498828404*v^16 - 31977884292236989195472482000258*v^15 + 77339322142504523651326528848543*v^14 + 746589678647592933774296964151868*v^13 - 2448536271935896202403179600966319*v^12 - 8171520444160025742823813980127157*v^11 + 39413166042518481211660739504734195*v^10 + 27434935899659633171073838280947898*v^9 - 344301464212946926615494487263102241*v^8 + 198085568359096724337640351591040988*v^7 + 1578521028488970797364761785995709262*v^6 - 1790791735729018198746659363302980112*v^5 - 3271777394076768207204843345982328264*v^4 + 4252188966502451270525469251950234080*v^3 + 2196886235169306852566914883296940608*v^2 - 2329726121906236985086313708643948032*v - 1017159496214166664214486016191601664) / 34582852139001987446892726703244288 $$\beta_{4}$$ $$=$$ $$( 16\!\cdots\!09 \nu^{19} + \cdots + 46\!\cdots\!92 ) / 13\!\cdots\!52$$ (166814599350794488141432427409*v^19 - 1350495829557780830759355380060*v^18 - 13687071315594442380125675315010*v^17 + 119270484596985938766333089252900*v^16 + 467725626189972904355446044544218*v^15 - 4376178938835631186279749556826035*v^14 - 8659930089323499943783232641532940*v^13 + 86143269676362073279113111487566563*v^12 + 94233607657203418323730381949943441*v^11 - 979149914399412277680283066635040151*v^10 - 599594536111287517524935040312557906*v^9 + 6430863539589623308426801741067616173*v^8 + 1939029677254957333468449895515352148*v^7 - 23070425776809799643676640135954563990*v^6 - 1203309044444085910162194680450206192*v^5 + 39621902476614788525001950457908784872*v^4 - 5759415910098472092671399035237446624*v^3 - 23378004946055384041090275439684645696*v^2 + 142563194527083089747765934440353280*v + 4664437484124174047741111094178184192) / 138331408556007949787570906812977152 $$\beta_{5}$$ $$=$$ $$( 16\!\cdots\!09 \nu^{19} + \cdots + 31\!\cdots\!20 ) / 13\!\cdots\!52$$ (166814599350794488141432427409*v^19 - 1350495829557780830759355380060*v^18 - 13687071315594442380125675315010*v^17 + 119270484596985938766333089252900*v^16 + 467725626189972904355446044544218*v^15 - 4376178938835631186279749556826035*v^14 - 8659930089323499943783232641532940*v^13 + 86143269676362073279113111487566563*v^12 + 94233607657203418323730381949943441*v^11 - 979149914399412277680283066635040151*v^10 - 599594536111287517524935040312557906*v^9 + 6430863539589623308426801741067616173*v^8 + 1939029677254957333468449895515352148*v^7 - 23070425776809799643676640135954563990*v^6 - 1203309044444085910162194680450206192*v^5 + 39621902476614788525001950457908784872*v^4 - 5897747318654480042458969942050423776*v^3 - 23101342128943368141515133626058691392*v^2 + 2632528548535226185924042257073942016*v + 3142791990008086600077831119235435520) / 138331408556007949787570906812977152 $$\beta_{6}$$ $$=$$ $$( - 26\!\cdots\!93 \nu^{19} + \cdots - 13\!\cdots\!08 ) / 98\!\cdots\!68$$ (-26152855439014240173295064393*v^19 + 744644081528315025676207590020*v^18 - 3218159341213545867236191384110*v^17 - 50417106465790611531663074990932*v^16 + 356586203117742551888711227503318*v^15 + 1244743144907356472200649976769643*v^14 - 12658017693772790992567986909535916*v^13 - 11811100299285459166347631362155691*v^12 + 223459282065681754221609041313385807*v^11 - 18025503401113937512314347452176345*v^10 - 2149997278392931104706938314303505974*v^9 + 1164867671037673598327653015324640235*v^8 + 11132734379886535172119459065236117876*v^7 - 8047507698795493216902414469786217658*v^6 - 27988242192264774218813060363520818496*v^5 + 19854755213702739095329327187805049176*v^4 + 26739538941256078877668668087072672032*v^3 - 10034170681597074030736142591149411776*v^2 - 10144907672843579159114893303871769600*v - 1329244071393430309523537521953195008) / 9880814896857710699112207629498368 $$\beta_{7}$$ $$=$$ $$( 12\!\cdots\!65 \nu^{19} + \cdots + 15\!\cdots\!32 ) / 19\!\cdots\!36$$ (123525159950971870551890158465*v^19 - 1190773904247417592817979932036*v^18 - 7966235422166388747313871075746*v^17 + 97335502882183415209384026183988*v^16 + 177055479878767121877216584110554*v^15 - 3257211061780901147762200403967507*v^14 - 1086723356047383297337294684510676*v^13 + 57395254796601216420525272419658419*v^12 - 16310541738680620271054746129520823*v^11 - 569797159612921949834815581264606639*v^10 + 307584326086485916512104733722100038*v^9 + 3158320267917432831310031782110492109*v^8 - 1930037663951864385956752721731255668*v^7 - 9099812817933363722157414892016663094*v^6 + 5368341915983778265270891041088581632*v^5 + 11993682506515188099848864466897739304*v^4 - 5948126867410425429233267277639761184*v^3 - 7138104342338140829232890637267942208*v^2 + 2683680119996150974228672074304516096*v + 1521267749819177792137684795876447232) / 19761629793715421398224415258996736 $$\beta_{8}$$ $$=$$ $$( 26\!\cdots\!21 \nu^{19} + \cdots + 37\!\cdots\!12 ) / 34\!\cdots\!88$$ (260360902712202528391180952921*v^19 - 3914936949461812342108126806600*v^18 - 2942805648054103984962522981362*v^17 + 290883149420540625425402243790620*v^16 - 736580371944334180771692721878950*v^15 - 8509922019006161965735985242328835*v^14 + 33881456868452938910175506114755672*v^13 + 122151233154196149396081148356995739*v^12 - 649033748390459326151768888199421003*v^11 - 843629500783099673286229847671112171*v^10 + 6435928273463147005514946681670478466*v^9 + 1782540497724091382340341526010228413*v^8 - 33553728922783085584400759347850092680*v^7 + 7677927161047544448763676929001684762*v^6 + 83897021834069761829070172734935327576*v^5 - 36358297946051292134739217134654305400*v^4 - 78328469315135693884048098191217004992*v^3 + 20573387140105406671025972809331013312*v^2 + 27124024119619926581672445451504713984*v + 3724432510963413575686242205536678912) / 34582852139001987446892726703244288 $$\beta_{9}$$ $$=$$ $$( - 17\!\cdots\!43 \nu^{19} + \cdots + 91\!\cdots\!52 ) / 13\!\cdots\!52$$ (-1788373734203393270729163483443*v^19 + 17620625610680817554726180833956*v^18 + 109817102890800573327380021906566*v^17 - 1416458647639749555548518336823052*v^16 - 2145333590005402777557331432325326*v^15 + 46317554304015108272222429045970617*v^14 + 3143932966026059038087650389450228*v^13 - 789270710872714718685700904947670249*v^12 + 424819230910228461772661565935728765*v^11 + 7437940801693670996697949514596624213*v^10 - 5845884607147256909761407330672796154*v^9 - 37672282502285085103167767439726811431*v^8 + 31315665438121986549723857270260197332*v^7 + 90050947670258750205026205821265800194*v^6 - 64229392528317324467544228882665564560*v^5 - 70787462678325229420290484609570954168*v^4 + 10398608918078701231943840352877346336*v^3 + 12018941177854988117823521471997361088*v^2 + 20623346724848320991524703380622993920*v + 9159915562842141515496671134866762752) / 138331408556007949787570906812977152 $$\beta_{10}$$ $$=$$ $$( 27\!\cdots\!79 \nu^{19} + \cdots - 40\!\cdots\!20 ) / 19\!\cdots\!36$$ (270631979056161931538377927979*v^19 - 1532994348055985889852408388988*v^18 - 27762295847733549523072724546806*v^17 + 144567089182818611607456436064732*v^16 + 1220387755625786830143037930548574*v^15 - 5632642022576766085663414823154385*v^14 - 29775333237570180853781011942974124*v^13 + 116830919776915455522710931276841681*v^12 + 435621330276460378770608046219302739*v^11 - 1383850153329613567319711367995462069*v^10 - 3843190529847116593055288243543609102*v^9 + 9316857742809739607682145589208458383*v^8 + 19441714532741249898070168076960390132*v^7 - 33384763213679149364602165595163349714*v^6 - 49878906261970738325712844078405770400*v^5 + 54425421573622519590226523969991369144*v^4 + 52345939195684171199645047395635759648*v^3 - 22261615337626343840943998258305178560*v^2 - 22250699430089036423923158006959744000*v - 4040234531058161861380181304955294720) / 19761629793715421398224415258996736 $$\beta_{11}$$ $$=$$ $$( - 60\!\cdots\!03 \nu^{19} + \cdots - 45\!\cdots\!08 ) / 34\!\cdots\!88$$ (-606735301470424531560367103803*v^19 + 5768879043064883284018041288732*v^18 + 40068708103979170900708632185606*v^17 - 475105107838455376976991782443996*v^16 - 940276825354377113746643638748638*v^15 + 16062621008592056323542871974760129*v^14 + 7433220755266536899691579940661676*v^13 - 287233282239453651371099585493855121*v^12 + 49152812977964960852651264525094461*v^11 + 2915855658058193305228109859042936213*v^10 - 1275396675139807482309801321556223362*v^9 - 16749965191545894588045659758802987951*v^8 + 8617872799206939005076759533053393548*v^7 + 51159680011017506364198762472799039842*v^6 - 24974661312420451876773029926241065312*v^5 - 72669954115903765261340127309533153624*v^4 + 27174505133151770345542711129734800864*v^3 + 38590126843121565972847276832453133888*v^2 - 6488722222104081019675877650164081152*v - 4585554694708791960516092041655262208) / 34582852139001987446892726703244288 $$\beta_{12}$$ $$=$$ $$( - 25\!\cdots\!31 \nu^{19} + \cdots + 30\!\cdots\!16 ) / 13\!\cdots\!52$$ (-2526990315991479089613411173931*v^19 + 26667155422505892509686888565940*v^18 + 138516335443058192084806522373238*v^17 - 2117402209616253534347091463202380*v^16 - 1686174618301662166841336836378814*v^15 + 68211031844784043497652849153653761*v^14 - 39935024725118885708686490013354172*v^13 - 1141277687554219765450319462867812945*v^12 + 1369116854255621981106733954404625109*v^11 + 10514206951398952337561196792996274893*v^10 - 15745772516642533368318557269282927962*v^9 - 51742504702700706730303205824261627743*v^8 + 84917062152136417536129689123716303012*v^7 + 118842757478767702707146612522384627538*v^6 - 206228610661151742702250237747011448368*v^5 - 86480460564257256804979291227186852856*v^4 + 164578565496505962916066614172484308896*v^3 + 19085173842419991668351144677263064000*v^2 - 37744287474427449805916379618904297984*v + 3028833422376758239915819722763836416) / 138331408556007949787570906812977152 $$\beta_{13}$$ $$=$$ $$( - 32\!\cdots\!39 \nu^{19} + \cdots + 16\!\cdots\!72 ) / 17\!\cdots\!44$$ (-322110730455045326351781591139*v^19 + 2523430745321182509741525497485*v^18 + 26226833805703484327121799515942*v^17 - 215378220151877471120335459560238*v^16 - 906342286093101609257379218832354*v^15 + 7584536471876917095136911563918787*v^14 + 17638617012993194849076977133867889*v^13 - 142020285167817803035246324972266521*v^12 - 215985527950874961354236415166376600*v^11 + 1517024326433960104853370853714683586*v^10 + 1723117424565402392873800594726283875*v^9 - 9196654933759586094171566144269490493*v^8 - 8548891525163274223746464224468704511*v^7 + 29588314024106391902157517427781421090*v^6 + 22790290853551431471783371834774037402*v^5 - 43414785106996972475387319139399627056*v^4 - 26420895231700353389990207881691066232*v^3 + 18506734721934176294206901861211219328*v^2 + 12616556194272009725631699511159088832*v + 1686516275898058668146233210685871872) / 17291426069500993723446363351622144 $$\beta_{14}$$ $$=$$ $$( 27\!\cdots\!13 \nu^{19} + \cdots + 32\!\cdots\!48 ) / 13\!\cdots\!52$$ (2734152877994688274310085418013*v^19 - 25960975634296432951773789470284*v^18 - 180534291368196176802780946225306*v^17 + 2139027652052384986936135482936852*v^16 + 4215271811969510885322664893037842*v^15 - 72360823327143083125584087136181527*v^14 - 31805510278174905588040903723339964*v^13 + 1295125238580121506290799808583585159*v^12 - 276596555524681255603009881706843715*v^11 - 13166806452301108599035940705207235403*v^10 + 6688304822622065981151389474842671734*v^9 + 75828841172607309470084358969969321225*v^8 - 47687572364152184089631805094650619484*v^7 - 232673117757990695207356487502292907870*v^6 + 156547691847078294963028803307203841040*v^5 + 333106525975707064139691062251794023240*v^4 - 220048506363736353752904209181598042720*v^3 - 177894154458912902726902379542509481536*v^2 + 83245588353134266419335019750371993088*v + 32130619130711692779488453151696002048) / 138331408556007949787570906812977152 $$\beta_{15}$$ $$=$$ $$( - 19\!\cdots\!71 \nu^{19} + \cdots + 41\!\cdots\!76 ) / 69\!\cdots\!76$$ (-1919313306468480675248340463471*v^19 + 16287040824113915720418958816672*v^18 + 141106858120813943537320812242814*v^17 - 1329892414657902353036207406768468*v^16 - 4252050758864421185836519934535958*v^15 + 44269001827933218141611786689127397*v^14 + 70649464444231661736192823107424608*v^13 - 769486358710604031469242161845589277*v^12 - 774249552990611950560226063352144251*v^11 + 7404080636894780700052916364411170133*v^10 + 6497714981597840600274478844794621018*v^9 - 38191876369926948550330412419396272315*v^8 - 41085242382295321735141271248561303296*v^7 + 91605745515222116795029612773332628330*v^6 + 159832150835194479332006249082043704488*v^5 - 65332774450185673166824998924649224760*v^4 - 303057496372158469040224203351886861184*v^3 - 15460043563048985235383592413689688128*v^2 + 167168282198202823223642312275894786816*v + 41404525471547362382781934010256050176) / 69165704278003974893785453406488576 $$\beta_{16}$$ $$=$$ $$( 58\!\cdots\!27 \nu^{19} + \cdots + 25\!\cdots\!88 ) / 13\!\cdots\!52$$ (5860862776188473328983662662227*v^19 - 61455534139715025644526163315244*v^18 - 323184664283558102681155459577862*v^17 + 4870548366671082701532424860106908*v^16 + 4078178482968792209399747248640558*v^15 - 156401216132856434257004328875922793*v^14 + 86576312434407992012429187608312420*v^13 + 2602274091016702038009725681506144777*v^12 - 3058531810236390439132199567749105141*v^11 - 23731187734056638811482300307705084061*v^10 + 35215687580361853970131096562530006866*v^9 + 114458037507572873325837443002465580215*v^8 - 188507364007700683283051246246075921276*v^7 - 250978141154510968053896759010267872514*v^6 + 447945430523716351548517140775104032576*v^5 + 159143692118104562221003527005901843704*v^4 - 329605989292145423043887524251492424288*v^3 - 63268263686124659224791415378002972096*v^2 + 74301629338761126769418014138538850304*v + 25985517162189879743030209058957093888) / 138331408556007949787570906812977152 $$\beta_{17}$$ $$=$$ $$( 62\!\cdots\!67 \nu^{19} + \cdots + 46\!\cdots\!20 ) / 13\!\cdots\!52$$ (6243587762078502040184754164967*v^19 - 64412829750065273239199976483172*v^18 - 358056776142994516698966362061102*v^17 + 5154825361696526204378795230840060*v^16 + 5401809518929685296087502879429974*v^15 - 167852297007722567124040900078831989*v^14 + 59753165782527919679906368208597708*v^13 + 2852400493983402729047485902351608325*v^12 - 2753470503145951901670297015451511161*v^11 - 26930123403804907456498800174815477841*v^10 + 33427325911953030677855447873940510530*v^9 + 138532519311185431874831280983538443691*v^8 - 185679589634918657738628976580453594388*v^7 - 352058982302904859873612266450803173818*v^6 + 468391033428631407807866899845139030704*v^5 + 363886863416399985029125223169593658840*v^4 - 414280763967850314213711456002677685024*v^3 - 194784778257691164791602763440105589952*v^2 + 118488278756469740761564753031855086080*v + 46217886103058991717909125565034193920) / 138331408556007949787570906812977152 $$\beta_{18}$$ $$=$$ $$( - 35\!\cdots\!61 \nu^{19} + \cdots - 35\!\cdots\!28 ) / 69\!\cdots\!76$$ (-3543949853526097963471903435761*v^19 + 37175121062452472296264218597616*v^18 + 198340779699890366472307985246882*v^17 - 2975221374372404180002181960083660*v^16 - 2646310538798880004313874813034218*v^15 + 96937215687017315911723109259004571*v^14 - 48834303749444178998814691508173840*v^13 - 1650129410363802786557437880292585059*v^12 + 1846965064480131146812643157999681803*v^11 + 15644403771251129141597974718346723003*v^10 - 22100195742101185678405281587529904234*v^9 - 81283771947994434360723045893569299557*v^8 + 125356958142191471677724075521773470224*v^7 + 211786758996958649335043575841388152310*v^6 - 335930920639816473412237031550580438600*v^5 - 233353683049101476351614860541327197064*v^4 + 352344334799787582275312942835609298048*v^3 + 128546586646400237964351273269060650048*v^2 - 126806643885529669770104961027518524160*v - 35139485451163901040223031902672918528) / 69165704278003974893785453406488576 $$\beta_{19}$$ $$=$$ $$( - 85\!\cdots\!05 \nu^{19} + \cdots - 48\!\cdots\!48 ) / 13\!\cdots\!52$$ (-8594010954294800399321506317305*v^19 + 85410402679933920810414122333468*v^18 + 524727998844334322855700449465042*v^17 - 6896571287350775712905175358391684*v^16 - 9979254176322390402926725014205034*v^15 + 227156930967178655028896385127746987*v^14 + 153456878855132463785513764075340*v^13 - 3919110691319537738302265125159284443*v^12 + 2403094063973111432614286252550222631*v^11 + 37780024308437588173137340613069419407*v^10 - 33152112578567773088545747911580532926*v^9 - 200354969365438833189659805632203941109*v^8 + 191868205835992371865177381821778730220*v^7 + 534736019777413251578960456775440894790*v^6 - 497178628316881260665452957843906220752*v^5 - 599506804310975338461275442890470571688*v^4 + 459946410071502261473173009555896858592*v^3 + 288413892034579136974596559321102377792*v^2 - 136506169652057135709109574486510304768*v - 48947594072994128246929152723342875648) / 138331408556007949787570906812977152
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 11$$ b2 + b1 + 11 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + 2\beta_{2} + 20\beta _1 + 11$$ -b5 + b4 + 2*b2 + 20*b1 + 11 $$\nu^{4}$$ $$=$$ $$\beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + \beta_{12} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 31 \beta_{2} + 48 \beta _1 + 224$$ b19 - b18 - b17 + b16 + b12 - b9 + 2*b8 + 2*b7 + 2*b6 - 6*b5 + 3*b4 - 3*b3 + 31*b2 + 48*b1 + 224 $$\nu^{5}$$ $$=$$ $$6 \beta_{19} - 6 \beta_{18} - 4 \beta_{17} + 3 \beta_{16} - \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{10} - 6 \beta_{9} + 10 \beta_{8} + 17 \beta_{7} + 12 \beta_{6} - 63 \beta_{5} + 40 \beta_{4} - 16 \beta_{3} + 106 \beta_{2} + 514 \beta _1 + 554$$ 6*b19 - 6*b18 - 4*b17 + 3*b16 - b15 + 2*b14 + 4*b13 + 2*b12 + 3*b11 + b10 - 6*b9 + 10*b8 + 17*b7 + 12*b6 - 63*b5 + 40*b4 - 16*b3 + 106*b2 + 514*b1 + 554 $$\nu^{6}$$ $$=$$ $$67 \beta_{19} - 60 \beta_{18} - 46 \beta_{17} + 48 \beta_{16} - 5 \beta_{15} + 11 \beta_{14} + 24 \beta_{13} + 39 \beta_{12} + 10 \beta_{11} + 15 \beta_{10} - 51 \beta_{9} + 119 \beta_{8} + 146 \beta_{7} + 121 \beta_{6} - 384 \beta_{5} + 165 \beta_{4} + \cdots + 5936$$ 67*b19 - 60*b18 - 46*b17 + 48*b16 - 5*b15 + 11*b14 + 24*b13 + 39*b12 + 10*b11 + 15*b10 - 51*b9 + 119*b8 + 146*b7 + 121*b6 - 384*b5 + 165*b4 - 178*b3 + 1018*b2 + 1934*b1 + 5936 $$\nu^{7}$$ $$=$$ $$420 \beta_{19} - 372 \beta_{18} - 204 \beta_{17} + 187 \beta_{16} - 80 \beta_{15} + 161 \beta_{14} + 310 \beta_{13} + 130 \beta_{12} + 175 \beta_{11} + 137 \beta_{10} - 298 \beta_{9} + 709 \beta_{8} + 1098 \beta_{7} + 769 \beta_{6} + \cdots + 23126$$ 420*b19 - 372*b18 - 204*b17 + 187*b16 - 80*b15 + 161*b14 + 310*b13 + 130*b12 + 175*b11 + 137*b10 - 298*b9 + 709*b8 + 1098*b7 + 769*b6 - 2995*b5 + 1432*b4 - 1082*b3 + 4692*b2 + 15686*b1 + 23126 $$\nu^{8}$$ $$=$$ $$3377 \beta_{19} - 2777 \beta_{18} - 1582 \beta_{17} + 1819 \beta_{16} - 462 \beta_{15} + 962 \beta_{14} + 2014 \beta_{13} + 1367 \beta_{12} + 798 \beta_{11} + 1287 \beta_{10} - 1936 \beta_{9} + 5850 \beta_{8} + \cdots + 187075$$ 3377*b19 - 2777*b18 - 1582*b17 + 1819*b16 - 462*b15 + 962*b14 + 2014*b13 + 1367*b12 + 798*b11 + 1287*b10 - 1936*b9 + 5850*b8 + 7763*b7 + 5806*b6 - 18926*b5 + 7029*b4 - 8613*b3 + 36287*b2 + 75513*b1 + 187075 $$\nu^{9}$$ $$=$$ $$21643 \beta_{19} - 17550 \beta_{18} - 7475 \beta_{17} + 8277 \beta_{16} - 4543 \beta_{15} + 9024 \beta_{14} + 17486 \beta_{13} + 6093 \beta_{12} + 8123 \beta_{11} + 9641 \beta_{10} - 11268 \beta_{9} + \cdots + 927526$$ 21643*b19 - 17550*b18 - 7475*b17 + 8277*b16 - 4543*b15 + 9024*b14 + 17486*b13 + 6093*b12 + 8123*b11 + 9641*b10 - 11268*b9 + 37020*b8 + 54723*b7 + 37382*b6 - 133060*b5 + 51216*b4 - 55004*b3 + 196028*b2 + 535659*b1 + 927526 $$\nu^{10}$$ $$=$$ $$155005 \beta_{19} - 119346 \beta_{18} - 48702 \beta_{17} + 64603 \beta_{16} - 28242 \beta_{15} + 56925 \beta_{14} + 116132 \beta_{13} + 49187 \beta_{12} + 44159 \beta_{11} + 76038 \beta_{10} + \cdots + 6575409$$ 155005*b19 - 119346*b18 - 48702*b17 + 64603*b16 - 28242*b15 + 56925*b14 + 116132*b13 + 49187*b12 + 44159*b11 + 76038*b10 - 67755*b9 + 268431*b8 + 367689*b7 + 257613*b6 - 855328*b5 + 278022*b4 - 390486*b3 + 1367484*b2 + 2942312*b1 + 6575409 $$\nu^{11}$$ $$=$$ $$1002188 \beta_{19} - 760767 \beta_{18} - 237489 \beta_{17} + 325670 \beta_{16} - 226505 \beta_{15} + 444272 \beta_{14} + 867742 \beta_{13} + 258394 \beta_{12} + 356225 \beta_{11} + \cdots + 36909652$$ 1002188*b19 - 760767*b18 - 237489*b17 + 325670*b16 - 226505*b15 + 444272*b14 + 867742*b13 + 258394*b12 + 356225*b11 + 533366*b10 - 395006*b9 + 1731410*b8 + 2497120*b7 + 1660952*b6 - 5749689*b5 + 1868228*b4 - 2532261*b3 + 8016625*b2 + 19587974*b1 + 36909652 $$\nu^{12}$$ $$=$$ $$6818979 \beta_{19} - 4993113 \beta_{18} - 1399582 \beta_{17} + 2267171 \beta_{16} - 1454239 \beta_{15} + 2871821 \beta_{14} + 5767424 \beta_{13} + 1840975 \beta_{12} + 2121360 \beta_{11} + \cdots + 246131231$$ 6818979*b19 - 4993113*b18 - 1399582*b17 + 2267171*b16 - 1454239*b15 + 2871821*b14 + 5767424*b13 + 1840975*b12 + 2121360*b11 + 3861377*b10 - 2324714*b9 + 11860541*b8 + 16462214*b7 + 11034047*b6 - 37158178*b5 + 10769635*b4 - 17127309*b3 + 53260663*b2 + 115281671*b1 + 246131231 $$\nu^{13}$$ $$=$$ $$44172983 \beta_{19} - 31931267 \beta_{18} - 6866475 \beta_{17} + 12257988 \beta_{16} - 10557314 \beta_{15} + 20512025 \beta_{14} + 40228592 \beta_{13} + 10584947 \beta_{12} + \cdots + 1470459851$$ 44172983*b19 - 31931267*b18 - 6866475*b17 + 12257988*b16 - 10557314*b15 + 20512025*b14 + 40228592*b13 + 10584947*b12 + 15307535*b11 + 26242611*b10 - 13623343*b9 + 76850649*b8 + 109363028*b7 + 71063145*b6 - 244471932*b5 + 69688104*b4 - 111336599*b3 + 325435245*b2 + 745752558*b1 + 1470459851 $$\nu^{14}$$ $$=$$ $$292956562 \beta_{19} - 206552694 \beta_{18} - 37604907 \beta_{17} + 80298409 \beta_{16} - 68567621 \beta_{15} + 133630846 \beta_{14} + 265796356 \beta_{13} + \cdots + 9540322513$$ 292956562*b19 - 206552694*b18 - 37604907*b17 + 80298409*b16 - 68567621*b15 + 133630846*b14 + 265796356*b13 + 71030556*b12 + 95255305*b11 + 181276484*b10 - 80113203*b9 + 511718184*b8 + 714137498*b7 + 464126576*b6 - 1580171067*b5 + 416689596*b4 - 735735190*b3 + 2113622146*b2 + 4549853093*b1 + 9540322513 $$\nu^{15}$$ $$=$$ $$1895755282 \beta_{19} - 1322082746 \beta_{18} - 180712613 \beta_{17} + 455725798 \beta_{16} - 472381015 \beta_{15} + 911624935 \beta_{14} + 1790194732 \beta_{13} + \cdots + 58802396908$$ 1895755282*b19 - 1322082746*b18 - 180712613*b17 + 455725798*b16 - 472381015*b15 + 911624935*b14 + 1790194732*b13 + 428662136*b12 + 650152984*b11 + 1209292641*b10 - 473193653*b9 + 3314046503*b8 + 4681232506*b7 + 2984876407*b6 - 10280855513*b5 + 2652174471*b4 - 4773459408*b3 + 13188346944*b2 + 29080446235*b1 + 58802396908 $$\nu^{16}$$ $$=$$ $$12398941408 \beta_{19} - 8498991098 \beta_{18} - 916947006 \beta_{17} + 2894980946 \beta_{16} - 3074530997 \beta_{15} + 5940401999 \beta_{14} + 11744810552 \beta_{13} + \cdots + 377068449485$$ 12398941408*b19 - 8498991098*b18 - 916947006*b17 + 2894980946*b16 - 3074530997*b15 + 5940401999*b14 + 11744810552*b13 + 2796051752*b12 + 4126432764*b11 + 8127198734*b10 - 2803197894*b9 + 21728800737*b8 + 30387721045*b7 + 19325332347*b6 - 66340371287*b5 + 16221419329*b4 - 31157082602*b3 + 84779646240*b2 + 180840900372*b1 + 377068449485 $$\nu^{17}$$ $$=$$ $$80070537363 \beta_{19} - 54399311658 \beta_{18} - 4121592075 \beta_{17} + 16995242528 \beta_{16} - 20576464899 \beta_{15} + 39520278569 \beta_{14} + \cdots + 2361234897287$$ 80070537363*b19 - 54399311658*b18 - 4121592075*b17 + 16995242528*b16 - 20576464899*b15 + 39520278569*b14 + 77595876524*b13 + 17312985945*b12 + 27369324172*b11 + 53558709704*b10 - 16717187852*b9 + 140435089999*b8 + 197542432955*b7 + 124122672371*b6 - 428932027776*b5 + 102623940677*b4 - 201551550238*b3 + 534704936912*b2 + 1150821558901*b1 + 2361234897287 $$\nu^{18}$$ $$=$$ $$519479969859 \beta_{19} - 348726275330 \beta_{18} - 18319042043 \beta_{17} + 106513938337 \beta_{16} - 133699833268 \beta_{15} + 256854127331 \beta_{14} + \cdots + 15072375790850$$ 519479969859*b19 - 348726275330*b18 - 18319042043*b17 + 106513938337*b16 - 133699833268*b15 + 256854127331*b14 + 505880846194*b13 + 111459257695*b12 + 175114958137*b11 + 353942650543*b10 - 100059208652*b9 + 912550545267*b8 + 1277034190244*b7 + 799746522597*b6 - 2762223527248*b5 + 636710515242*b4 - 1306214168320*b3 + 3422011678922*b2 + 7231840541983*b1 + 15072375790850 $$\nu^{19}$$ $$=$$ $$3347512819449 \beta_{19} - 2231480755745 \beta_{18} - 65788200504 \beta_{17} + 640316557656 \beta_{16} - 879941448401 \beta_{15} + 1684327872066 \beta_{14} + \cdots + 95172808147204$$ 3347512819449*b19 - 2231480755745*b18 - 65788200504*b17 + 640316557656*b16 - 879941448401*b15 + 1684327872066*b14 + 3304721860176*b13 + 699646109461*b12 + 1143917844363*b11 + 2312813829456*b10 - 602918825526*b9 + 5884464268952*b8 + 8257407329996*b7 + 5130938800314*b6 - 17792617984594*b5 + 4023551267326*b4 - 8426069532731*b3 + 21704012621537*b2 + 45975789289745*b1 + 95172808147204

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.47888 −4.19424 −3.95689 −3.42835 −2.55697 −2.53684 −2.41218 −0.822685 −0.731640 −0.259637 0.999641 1.12533 1.38435 2.13989 3.57546 3.74913 4.23917 4.30367 5.47116 6.39053
−5.47888 −2.61641 22.0181 5.00000 14.3350 −5.28814 −76.8037 −20.1544 −27.3944
1.2 −5.19424 8.38824 18.9801 5.00000 −43.5706 −4.76095 −57.0335 43.3626 −25.9712
1.3 −4.95689 −6.17893 16.5708 5.00000 30.6283 −36.0727 −42.4843 11.1792 −24.7845
1.4 −4.42835 −1.78158 11.6103 5.00000 7.88948 24.1111 −15.9878 −23.8260 −22.1418
1.5 −3.55697 3.83974 4.65205 5.00000 −13.6579 −6.66228 11.9086 −12.2564 −17.7849
1.6 −3.53684 1.54358 4.50922 5.00000 −5.45941 −15.5595 12.3463 −24.6173 −17.6842
1.7 −3.41218 −9.99772 3.64299 5.00000 34.1140 −3.40924 14.8669 72.9544 −17.0609
1.8 −1.82269 5.45780 −4.67782 5.00000 −9.94786 19.7082 23.1077 2.78762 −9.11343
1.9 −1.73164 −4.39818 −5.00142 5.00000 7.61606 22.5500 22.5138 −7.65602 −8.65820
1.10 −1.25964 7.52739 −6.41331 5.00000 −9.48178 −31.9473 18.1555 29.6616 −6.29819
1.11 −0.000359199 0 −7.63742 −8.00000 5.00000 0.00274335 −26.4307 0.00574718 31.3302 −0.00179599
1.12 0.125334 −3.53244 −7.98429 5.00000 −0.442735 −25.7104 −2.00338 −14.5219 0.626671
1.13 0.384350 −2.95996 −7.85228 5.00000 −1.13766 3.33433 −6.09282 −18.2387 1.92175
1.14 1.13989 3.45749 −6.70066 5.00000 3.94115 1.89071 −16.7571 −15.0457 5.69943
1.15 2.57546 8.83286 −1.36700 5.00000 22.7487 −30.7821 −24.1243 51.0194 12.8773
1.16 2.74913 −7.66455 −0.442295 5.00000 −21.0708 25.2191 −23.2090 31.7453 13.7456
1.17 3.23917 2.30947 2.49220 5.00000 7.48077 −5.43134 −17.8407 −21.6663 16.1958
1.18 3.30367 −7.25904 2.91425 5.00000 −23.9815 −9.92986 −16.8016 25.6936 16.5184
1.19 4.47116 0.109820 11.9912 5.00000 0.491023 −7.59378 17.8455 −26.9879 22.3558
1.20 5.39053 −8.44019 21.0578 5.00000 −45.4970 −18.2352 70.3882 44.2368 26.9526
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.4.a.b 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.b 20 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 12 T_{2}^{19} - 44 T_{2}^{18} - 1004 T_{2}^{17} - 727 T_{2}^{16} + 32533 T_{2}^{15} + 74163 T_{2}^{14} - 521840 T_{2}^{13} - 1700735 T_{2}^{12} + 4319428 T_{2}^{11} + 18581116 T_{2}^{10} - 16253206 T_{2}^{9} + \cdots - 3840$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1045))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + 12 T^{19} - 44 T^{18} + \cdots - 3840$$
$3$ $$T^{20} + 21 T^{19} + \cdots - 751355754688$$
$5$ $$(T - 5)^{20}$$
$7$ $$T^{20} + 131 T^{19} + \cdots + 27\!\cdots\!72$$
$11$ $$(T - 11)^{20}$$
$13$ $$T^{20} + 223 T^{19} + \cdots - 33\!\cdots\!20$$
$17$ $$T^{20} + 471 T^{19} + \cdots + 92\!\cdots\!60$$
$19$ $$(T - 19)^{20}$$
$23$ $$T^{20} + 653 T^{19} + \cdots + 18\!\cdots\!64$$
$29$ $$T^{20} - 51 T^{19} + \cdots + 19\!\cdots\!60$$
$31$ $$T^{20} + 90 T^{19} + \cdots + 31\!\cdots\!00$$
$37$ $$T^{20} + 96 T^{19} + \cdots + 10\!\cdots\!96$$
$41$ $$T^{20} + 1284 T^{19} + \cdots + 62\!\cdots\!60$$
$43$ $$T^{20} + 1592 T^{19} + \cdots + 77\!\cdots\!20$$
$47$ $$T^{20} + 2030 T^{19} + \cdots - 48\!\cdots\!00$$
$53$ $$T^{20} + 943 T^{19} + \cdots - 13\!\cdots\!16$$
$59$ $$T^{20} + 515 T^{19} + \cdots - 18\!\cdots\!20$$
$61$ $$T^{20} - 446 T^{19} + \cdots + 99\!\cdots\!80$$
$67$ $$T^{20} + 1719 T^{19} + \cdots + 55\!\cdots\!28$$
$71$ $$T^{20} + 90 T^{19} + \cdots + 25\!\cdots\!40$$
$73$ $$T^{20} + 3763 T^{19} + \cdots - 68\!\cdots\!16$$
$79$ $$T^{20} - 2 T^{19} + \cdots - 39\!\cdots\!00$$
$83$ $$T^{20} + 3436 T^{19} + \cdots + 82\!\cdots\!00$$
$89$ $$T^{20} - 1700 T^{19} + \cdots + 43\!\cdots\!08$$
$97$ $$T^{20} + 1956 T^{19} + \cdots - 10\!\cdots\!24$$