# Properties

 Label 169.4.c.k Level $169$ Weight $4$ Character orbit 169.c Analytic conductor $9.971$ Analytic rank $0$ Dimension $18$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(22,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("169.22");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.c (of order $$3$$, degree $$2$$, not 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: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + \cdots + 16777216$$ x^18 - 4*x^17 + 62*x^16 - 106*x^15 + 2016*x^14 - 2731*x^13 + 39895*x^12 - 21896*x^11 + 490851*x^10 - 318018*x^9 + 3391249*x^8 - 990441*x^7 + 13815033*x^6 - 7349264*x^5 + 28218112*x^4 - 9093120*x^3 + 32440320*x^2 - 14680064*x + 16777216 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$13^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{14} + \beta_{11}) q^{3} + (\beta_{12} - \beta_{10} - \beta_{6} + \cdots - 5) q^{4}+ \cdots + ( - 2 \beta_{14} + \beta_{10} + \cdots - 7) q^{9}+O(q^{10})$$ q + (-b3 + b2 - b1) * q^2 + (-b14 + b11) * q^3 + (b12 - b10 - b6 + 5*b3 - 5) * q^4 + (b17 - b16 - b12 - b11 + b9 - b7 - b5 + 3*b2 + 2) * q^5 + (b14 - 2*b10 + b7 - 2*b6 + 6*b3 - b1 - 6) * q^6 + (b14 + b12 - b7 + 2*b6 + 2*b4 + 4*b3 - b1 - 4) * q^7 + (b17 - b12 - b11 - 4*b9 + 4*b7 - 4*b2 + 9) * q^8 + (-2*b14 + b10 - b8 + 3*b7 + b6 - b4 + 7*b3 + b1 - 7) * q^9 $$q + ( - \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{14} + \beta_{11}) q^{3} + (\beta_{12} - \beta_{10} - \beta_{6} + \cdots - 5) q^{4}+ \cdots + ( - 18 \beta_{17} - 100 \beta_{16} + \cdots + 121) q^{99}+O(q^{100})$$ q + (-b3 + b2 - b1) * q^2 + (-b14 + b11) * q^3 + (b12 - b10 - b6 + 5*b3 - 5) * q^4 + (b17 - b16 - b12 - b11 + b9 - b7 - b5 + 3*b2 + 2) * q^5 + (b14 - 2*b10 + b7 - 2*b6 + 6*b3 - b1 - 6) * q^6 + (b14 + b12 - b7 + 2*b6 + 2*b4 + 4*b3 - b1 - 4) * q^7 + (b17 - b12 - b11 - 4*b9 + 4*b7 - 4*b2 + 9) * q^8 + (-2*b14 + b10 - b8 + 3*b7 + b6 - b4 + 7*b3 + b1 - 7) * q^9 + (b17 + b16 + 2*b15 - b14 + 2*b13 + b11 - 2*b10 - 2*b9 - 2*b8 + b6 + 3*b5 + 3*b4 + 16*b3 + 6*b2 - 6*b1) * q^10 + (3*b17 + b16 - 2*b14 - 2*b13 + 2*b11 + 2*b10 - 2*b9 + b6 - 21*b3 + 2*b2 - 2*b1) * q^11 + (-2*b17 - 3*b13 + 2*b12 + 2*b11 - 3*b9 + 3*b7 + 3*b5 - 14*b2 + 10) * q^12 + (-3*b17 + 4*b16 + 4*b15 + 4*b13 + 3*b12 + b11 - 2*b9 + 2*b7 + 4*b5 - 9*b2 - 10) * q^14 + (b17 - b16 - b15 - b14 + 9*b13 + b11 - 9*b10 - 3*b9 + b8 - b6 + 4*b5 + 4*b4 - 20*b3 + 3*b2 - 3*b1) * q^15 + (-b17 + 2*b16 + 4*b14 - 7*b13 - 4*b11 + 7*b10 + 8*b9 + 2*b6 + 5*b5 + 5*b4 - 37*b3 + 7*b2 - 7*b1) * q^16 + (2*b14 + b12 + 7*b10 + 5*b8 - 2*b7 + 7*b6 + 5*b4 - 7*b3 + 8*b1 + 7) * q^17 + (-b17 + 5*b16 - 2*b15 - 9*b13 + b12 + 2*b11 + 6*b9 - 6*b7 + 5*b5 - 3*b2 + 5) * q^18 + (-b14 - 3*b12 - b10 - b8 + 10*b6 + 14*b3 - 4*b1 - 14) * q^19 + (-8*b14 + 3*b12 - 5*b10 + 6*b8 + 6*b7 - 9*b6 - 10*b4 + 36*b3 - 15*b1 - 36) * q^20 + (-10*b17 + 2*b16 + 3*b15 + b13 + 10*b12 + 2*b11 - 14*b9 + 14*b7 + 5*b5 - 4*b2 + 24) * q^21 + (10*b14 - 3*b12 + 4*b10 + 3*b7 - 3*b6 + 8*b4 + 43*b3 + 15*b1 - 43) * q^22 + (-14*b17 - 3*b16 + 6*b14 - 2*b13 - 6*b11 + 2*b10 - 3*b6 - b5 - b4 + 22*b3 - 8*b2 + 8*b1) * q^23 + (-9*b17 - 6*b16 - 6*b15 + 11*b14 - 7*b13 - 11*b11 + 7*b10 + 5*b9 + 6*b8 - 6*b6 + b5 + b4 - 94*b3 - 3*b2 + 3*b1) * q^24 + (3*b17 - 3*b16 + 7*b15 + b13 - 3*b12 - 9*b11 + 3*b9 - 3*b7 - 4*b5 + 23*b2 + 25) * q^25 + (-6*b17 - 19*b16 + 6*b13 + 6*b12 - 6*b11 - b9 + b7 - 4*b5 + 29*b2 - 83) * q^27 + (12*b17 + 5*b16 - 8*b15 + 4*b14 - 4*b11 + 10*b9 + 8*b8 + 5*b6 - 25*b5 - 25*b4 - 46*b3 + b2 - b1) * q^28 + (5*b17 - 8*b16 - 3*b14 - 6*b13 + 3*b11 + 6*b10 - 3*b9 - 8*b6 + 6*b5 + 6*b4 - 16*b3 - 29*b2 + 29*b1) * q^29 + (-9*b14 + 6*b12 - 11*b10 + 8*b8 + 34*b7 - 17*b6 - 16*b4 + 50*b3 - 22*b1 - 50) * q^30 + (b17 - 4*b16 - 8*b15 - 12*b13 - b12 + 7*b11 + b9 - b7 - 2*b5 - 21*b2 + 82) * q^31 + (-17*b14 + 18*b12 + b10 + 10*b8 - 10*b7 - 12*b6 - 7*b4 + 89*b3 + 18*b1 - 89) * q^32 + (23*b14 + 2*b12 + 3*b10 + 5*b8 + 22*b7 + 5*b6 + 4*b4 + 59*b3 + 24*b1 - 59) * q^33 + (6*b17 - 9*b16 + 10*b15 + 29*b13 - 6*b12 + 10*b11 + 23*b9 - 23*b7 - 25*b5 + 37*b2 + 19) * q^34 + (-12*b14 - 4*b12 + 17*b10 + b8 - 12*b7 - 8*b6 - b4 - 20*b3 - 40*b1 + 20) * q^35 + (12*b17 - 3*b16 - 2*b15 - 12*b14 - 12*b13 + 12*b11 + 12*b10 - 16*b9 + 2*b8 - 3*b6 - 2*b5 - 2*b4 + 115*b3 - 31*b2 + 31*b1) * q^36 + (4*b17 + 12*b16 + 11*b15 - 16*b14 + 13*b13 + 16*b11 - 13*b10 + 14*b9 - 11*b8 + 12*b6 + 9*b5 + 9*b4 - 12*b3 - 28*b2 + 28*b1) * q^37 + (-8*b17 + 18*b16 - 21*b13 + 8*b12 + 7*b11 - 2*b9 + 2*b7 + 20*b5 - 4*b2 - 90) * q^38 + (16*b17 + 15*b16 - 4*b15 - 35*b13 - 16*b12 + 9*b11 - 24*b9 + 24*b7 - 22*b5 - 18*b2 - 8) * q^40 + (-21*b17 - 7*b16 - 2*b15 + 3*b14 - 12*b13 - 3*b11 + 12*b10 + 6*b9 + 2*b8 - 7*b6 - 2*b5 - 2*b4 - 150*b3 + 14*b2 - 14*b1) * q^41 + (5*b17 - 2*b16 - 10*b15 + 2*b14 - 19*b13 - 2*b11 + 19*b10 + 26*b9 + 10*b8 - 2*b6 - 25*b5 - 25*b4 - 20*b3 + 6*b2 - 6*b1) * q^42 + (3*b14 + 21*b12 + 5*b10 - 5*b8 - 2*b7 + 11*b6 - 25*b4 - 70*b3 - 14*b1 + 70) * q^43 + (17*b17 - 15*b16 + 16*b15 + 29*b13 - 17*b12 - 7*b11 + 15*b9 - 15*b7 + 11*b5 + 63) * q^44 + (42*b14 - 8*b12 - 20*b10 - 10*b8 + 14*b7 - 11*b6 - 13*b4 - 26*b3 - 42*b1 + 26) * q^45 + (-20*b14 + 4*b12 + 12*b10 - 2*b8 - 26*b7 + 27*b6 - 5*b4 - 82*b3 + 39*b1 + 82) * q^46 + (-7*b17 + 5*b16 + 4*b15 - 10*b13 + 7*b12 - 23*b11 - b9 + b7 + 25*b5 + 25*b2 + 92) * q^47 + (-9*b14 + 23*b12 + 3*b10 + 2*b8 - 33*b7 + 37*b6 + 10*b4 + 6*b3 + 42*b1 - 6) * q^48 + (27*b17 + 6*b16 + 7*b15 + 5*b14 + 19*b13 - 5*b11 - 19*b10 + 41*b9 - 7*b8 + 6*b6 - 39*b5 - 39*b4 - 55*b3 + 25*b2 - 25*b1) * q^49 + (22*b17 - 5*b16 + 8*b15 - 9*b14 + 11*b13 + 9*b11 - 11*b10 - 2*b9 - 8*b8 - 5*b6 - 26*b5 - 26*b4 + 113*b3 + 53*b2 - 53*b1) * q^50 + (-24*b17 - 30*b16 - 8*b15 + 40*b13 + 24*b12 + 11*b11 + 4*b9 - 4*b7 - 8*b5 + 44*b2 + 49) * q^51 + (-6*b17 + 43*b16 - 3*b15 + 13*b13 + 6*b12 + 12*b11 - 12*b9 + 12*b7 + 36*b5 - 18*b2 - 18) * q^53 + (26*b17 + 5*b16 + 8*b15 - 32*b14 + 18*b13 + 32*b11 - 18*b10 + 8*b9 - 8*b8 + 5*b6 + 10*b5 + 10*b4 + 325*b3 - 49*b2 + 49*b1) * q^54 + (-19*b17 + 37*b16 + 7*b15 - 19*b14 - 7*b13 + 19*b11 + 7*b10 - 83*b9 - 7*b8 + 37*b6 + 38*b5 + 38*b4 + 50*b3 - 37*b2 + 37*b1) * q^55 + (36*b14 - 35*b12 + b10 - 18*b8 - 4*b7 + 18*b6 + 55*b4 + 30*b3 + 4*b1 - 30) * q^56 + (-21*b17 + 18*b16 - b15 + b13 + 21*b12 - 43*b11 - 40*b9 + 40*b7 + 4*b5 + 34*b2 - 113) * q^57 + (3*b14 - 25*b12 + 60*b10 + 12*b8 - 4*b7 + 28*b6 + 16*b4 - 346*b3 + 25*b1 + 346) * q^58 + (-30*b14 + 13*b12 - 22*b10 - 22*b8 - b7 - 34*b6 - 8*b4 + 158*b3 - 49*b1 - 158) * q^59 + (-2*b17 + 20*b16 - 40*b15 - 51*b13 + 2*b12 - 23*b11 + 4*b9 - 4*b7 - 7*b5 - 85*b2 - 354) * q^60 + (16*b14 - 18*b12 - 51*b10 - 13*b8 + 52*b7 - 18*b6 + 17*b4 + 4*b3 - 6*b1 - 4) * q^61 + (-33*b17 + 34*b16 + 4*b15 + 17*b14 - 32*b13 - 17*b11 + 32*b10 - 38*b9 - 4*b8 + 34*b6 + 72*b5 + 72*b4 - 248*b3 + 27*b2 - 27*b1) * q^62 + (-33*b17 - 17*b16 - 13*b15 + 27*b14 - 15*b13 - 27*b11 + 15*b10 - 15*b9 + 13*b8 - 17*b6 + 34*b5 + 34*b4 + 86*b3 + 27*b2 - 27*b1) * q^63 + (62*b17 - 15*b16 - 14*b15 - 27*b13 - 62*b12 + 9*b11 + 22*b9 - 22*b7 - 66*b5 - 71*b2 + 49) * q^64 + (b17 - 43*b16 + 8*b15 + 33*b13 - b12 - 39*b11 + 79*b9 - 79*b7 + 2*b5 - 7*b2 + 381) * q^66 + (-3*b17 - 22*b16 + 31*b15 + 28*b14 - 9*b13 - 28*b11 + 9*b10 - 57*b9 - 31*b8 - 22*b6 + 37*b5 + 37*b4 + 102*b3 + 45*b2 - 45*b1) * q^67 + (22*b17 - 4*b16 + 10*b15 - 4*b14 + 115*b13 + 4*b11 - 115*b10 + 51*b9 - 10*b8 - 4*b6 - 26*b5 - 26*b4 + 185*b3 + 76*b2 - 76*b1) * q^68 + (-39*b14 - 57*b12 + 10*b10 - 6*b8 - 101*b7 - 27*b6 + 15*b4 - 212*b3 + 15*b1 + 212) * q^69 + (-29*b17 + 24*b16 - 2*b15 - 3*b13 + 29*b12 + 30*b11 + 20*b9 - 20*b7 - 57*b5 + 26*b2 - 576) * q^70 + (-71*b14 - 17*b12 - 27*b10 - 23*b8 + 11*b7 - 26*b6 + 27*b4 + 264*b3 + 11*b1 - 264) * q^71 + (67*b14 - 57*b12 + 4*b10 - 20*b8 - 30*b7 + 4*b6 + 29*b4 - 543*b3 - 117*b1 + 543) * q^72 + (39*b17 + 2*b16 + 15*b15 - 19*b13 - 39*b12 + 50*b11 - 5*b9 + 5*b7 + 45*b5 - 85*b2 + 52) * q^73 + (-4*b14 + 11*b12 - 57*b10 + 18*b8 + 42*b7 - 72*b6 - 121*b4 - 232*b3 - 44*b1 + 232) * q^74 + (-11*b17 - 34*b16 - 36*b15 - 36*b14 + 64*b13 + 36*b11 - 64*b10 + 3*b9 + 36*b8 - 34*b6 + 18*b5 + 18*b4 - 226*b3 - 15*b2 + 15*b1) * q^75 + (18*b17 + 35*b16 - 32*b15 + 12*b14 - 22*b13 - 12*b11 + 22*b10 - 60*b9 + 32*b8 + 35*b6 - 22*b5 - 22*b4 + 434*b3 - 120*b2 + 120*b1) * q^76 + (22*b17 + 76*b16 + 15*b15 + 65*b13 - 22*b12 - 24*b11 - 50*b9 + 50*b7 + 31*b5 - 40*b2 - 32) * q^77 + (9*b17 - 38*b16 - 21*b15 - 5*b13 - 9*b12 + 3*b11 + 77*b9 - 77*b7 + 3*b5 - 5*b2 + 44) * q^79 + (-82*b17 + 20*b16 - 4*b15 + 57*b14 - 57*b13 - 57*b11 + 57*b10 + 12*b9 + 4*b8 + 20*b6 + 79*b5 + 79*b4 + 56*b3 - 17*b2 + 17*b1) * q^80 + (-74*b17 - 81*b16 + 32*b15 + 85*b14 + 72*b13 - 85*b11 - 72*b10 + 8*b9 - 32*b8 - 81*b6 - 27*b5 - 27*b4 + 62*b3 + 124*b2 - 124*b1) * q^81 + (-23*b14 + 35*b12 - 12*b10 - 4*b8 - 74*b7 + 11*b6 + 18*b4 + 438*b3 + 237*b1 - 438) * q^82 + (16*b17 - 10*b16 + 13*b15 + 25*b13 - 16*b12 + 88*b11 - 85*b9 + 85*b7 + 10*b5 + 21*b2 + 444) * q^83 + (35*b14 + 47*b12 - 26*b10 - 26*b8 - 4*b7 + 37*b6 + 79*b4 - 100*b3 + 84*b1 + 100) * q^84 + (-21*b14 - 43*b12 + 88*b10 + 38*b8 - 65*b7 - 24*b6 + 90*b4 - 182*b3 + 167*b1 + 182) * q^85 + (54*b17 - 33*b16 - 50*b15 - 9*b13 - 54*b12 + 61*b11 - 12*b9 + 12*b7 - 47*b5 + 67*b2 - 14) * q^86 + (36*b14 - 18*b12 + 71*b10 + 11*b8 - 38*b7 + 88*b6 - 3*b4 - 132*b3 + 30*b1 + 132) * q^87 + (12*b17 - 77*b16 - 22*b15 + 20*b14 - b13 - 20*b11 + b10 + 107*b9 + 22*b8 - 77*b6 - 79*b5 - 79*b4 - 3*b3 + 127*b2 - 127*b1) * q^88 + (-46*b17 - 13*b16 + 29*b15 - 9*b14 - 17*b13 + 9*b11 + 17*b10 - 44*b9 - 29*b8 - 13*b6 - 18*b5 - 18*b4 - 210*b3 + 42*b2 - 42*b1) * q^89 + (-48*b17 - 35*b16 - 26*b15 - 36*b13 + 48*b12 - 82*b11 + 8*b9 - 8*b7 + 75*b5 - 5*b2 - 158) * q^90 + (-35*b17 - 42*b16 - 10*b15 + 31*b13 + 35*b12 + 46*b11 - 42*b9 + 42*b7 - 35*b5 + 96*b2 + 418) * q^92 + (-18*b17 + 82*b16 + 27*b15 - 96*b14 - 67*b13 + 96*b11 + 67*b10 - 68*b9 - 27*b8 + 82*b6 + 35*b5 + 35*b4 + 150*b3 - 86*b2 + 86*b1) * q^93 + (85*b17 - 27*b16 - 50*b15 - 39*b14 - 56*b13 + 39*b11 + 56*b10 - 54*b9 + 50*b8 - 27*b6 - 61*b5 - 61*b4 + 226*b3 + 92*b2 - 92*b1) * q^94 + (12*b14 + 14*b12 - 37*b10 - 55*b8 - 4*b7 - 37*b6 - 16*b4 - 206*b3 - 140*b1 + 206) * q^95 + (8*b17 - 53*b16 - 28*b15 + 36*b13 - 8*b12 + 7*b11 - 47*b9 + 47*b7 - 7*b5 - 38*b2 - 298) * q^96 + (-54*b14 + 17*b12 - 20*b10 - 6*b8 - 68*b7 + 8*b6 - 87*b4 - 157*b3 - 96*b1 + 157) * q^97 + (7*b14 - 32*b12 - 153*b10 - 78*b8 + 4*b7 - 10*b6 - 15*b4 + 33*b3 - 26*b1 - 33) * q^98 + (-18*b17 - 100*b16 - 14*b15 - 14*b13 + 18*b12 - 38*b11 + 70*b9 - 70*b7 - 20*b5 + 50*b2 + 121) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q - 5 q^{2} - q^{3} - 37 q^{4} + 60 q^{5} - 48 q^{6} - 38 q^{7} + 120 q^{8} - 66 q^{9}+O(q^{10})$$ 18 * q - 5 * q^2 - q^3 - 37 * q^4 + 60 * q^5 - 48 * q^6 - 38 * q^7 + 120 * q^8 - 66 * q^9 $$18 q - 5 q^{2} - q^{3} - 37 q^{4} + 60 q^{5} - 48 q^{6} - 38 q^{7} + 120 q^{8} - 66 q^{9} + 147 q^{10} - 181 q^{11} + 78 q^{12} - 294 q^{14} - 218 q^{15} - 269 q^{16} + 55 q^{17} + 158 q^{18} - 161 q^{19} - 370 q^{20} + 376 q^{21} - 340 q^{22} + 204 q^{23} - 798 q^{24} + 614 q^{25} - 1336 q^{27} - 344 q^{28} - 280 q^{29} - 521 q^{30} + 1412 q^{31} - 680 q^{32} - 500 q^{33} + 432 q^{34} - 20 q^{35} + 909 q^{36} - 298 q^{37} - 1478 q^{38} + 26 q^{40} - 1201 q^{41} + 4 q^{42} + 533 q^{43} + 710 q^{44} + 90 q^{45} + 840 q^{46} + 1912 q^{47} + 132 q^{48} - 403 q^{49} + 1156 q^{50} + 940 q^{51} - 556 q^{53} + 2555 q^{54} + 250 q^{55} - 250 q^{56} - 1620 q^{57} + 2877 q^{58} - 1377 q^{59} - 6314 q^{60} + 136 q^{61} - 2035 q^{62} + 944 q^{63} + 568 q^{64} + 6558 q^{66} + 931 q^{67} + 1536 q^{68} + 2050 q^{69} - 9708 q^{70} - 2046 q^{71} + 4342 q^{72} - 90 q^{73} + 1990 q^{74} - 2393 q^{75} + 3608 q^{76} - 1436 q^{77} + 824 q^{79} + 787 q^{80} + 835 q^{81} - 2757 q^{82} + 7418 q^{83} + 1539 q^{84} + 2106 q^{85} + 250 q^{86} + 786 q^{87} + 636 q^{88} - 1663 q^{89} - 2560 q^{90} + 8020 q^{92} + 1186 q^{93} + 2531 q^{94} + 1614 q^{95} - 6168 q^{96} + 1087 q^{97} + 282 q^{98} + 2714 q^{99}+O(q^{100})$$ 18 * q - 5 * q^2 - q^3 - 37 * q^4 + 60 * q^5 - 48 * q^6 - 38 * q^7 + 120 * q^8 - 66 * q^9 + 147 * q^10 - 181 * q^11 + 78 * q^12 - 294 * q^14 - 218 * q^15 - 269 * q^16 + 55 * q^17 + 158 * q^18 - 161 * q^19 - 370 * q^20 + 376 * q^21 - 340 * q^22 + 204 * q^23 - 798 * q^24 + 614 * q^25 - 1336 * q^27 - 344 * q^28 - 280 * q^29 - 521 * q^30 + 1412 * q^31 - 680 * q^32 - 500 * q^33 + 432 * q^34 - 20 * q^35 + 909 * q^36 - 298 * q^37 - 1478 * q^38 + 26 * q^40 - 1201 * q^41 + 4 * q^42 + 533 * q^43 + 710 * q^44 + 90 * q^45 + 840 * q^46 + 1912 * q^47 + 132 * q^48 - 403 * q^49 + 1156 * q^50 + 940 * q^51 - 556 * q^53 + 2555 * q^54 + 250 * q^55 - 250 * q^56 - 1620 * q^57 + 2877 * q^58 - 1377 * q^59 - 6314 * q^60 + 136 * q^61 - 2035 * q^62 + 944 * q^63 + 568 * q^64 + 6558 * q^66 + 931 * q^67 + 1536 * q^68 + 2050 * q^69 - 9708 * q^70 - 2046 * q^71 + 4342 * q^72 - 90 * q^73 + 1990 * q^74 - 2393 * q^75 + 3608 * q^76 - 1436 * q^77 + 824 * q^79 + 787 * q^80 + 835 * q^81 - 2757 * q^82 + 7418 * q^83 + 1539 * q^84 + 2106 * q^85 + 250 * q^86 + 786 * q^87 + 636 * q^88 - 1663 * q^89 - 2560 * q^90 + 8020 * q^92 + 1186 * q^93 + 2531 * q^94 + 1614 * q^95 - 6168 * q^96 + 1087 * q^97 + 282 * q^98 + 2714 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + \cdots + 16777216$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 71\!\cdots\!85 \nu^{17} + \cdots + 26\!\cdots\!56 ) / 32\!\cdots\!52$$ (-711249714637290827571578049685*v^17 + 4285297737447674527908968425076*v^16 - 50166706932882848379697710395542*v^15 + 162837708489280155176994192115058*v^14 - 1592996430633758348840090913039008*v^13 + 4696131522316015973018378408613255*v^12 - 32629038859958909128265112139223427*v^11 + 68204039289190924930485302554573576*v^10 - 385049642469343782162104038381925279*v^9 + 833443895340636287258423298714167498*v^8 - 2975259901274323318341287929256404261*v^7 + 4484831495997100413890099605106652605*v^6 - 11462451787834029748669326514957129229*v^5 + 18239349843222469179980755823126078576*v^4 - 37123889665400960006571677425372898560*v^3 + 25442412692851612676245942539730366464*v^2 - 15551524690656007872693435388538519552*v + 26532704040712808988967908407987142656) / 32233705447512037582042597281595850752 $$\beta_{3}$$ $$=$$ $$( - 12\!\cdots\!53 \nu^{17} + \cdots + 61\!\cdots\!76 ) / 25\!\cdots\!16$$ (-12651779194218067640766100124353*v^17 + 44917119059773943942491776099932*v^16 - 750127928141938797504226460309278*v^15 + 939754939124052382883624930017082*v^14 - 24203285187629383122368504313775184*v^13 + 21808037534339475936211492135295979*v^12 - 467173678774801680744216537192156895*v^11 + 15991046356927536036093631209045872*v^10 - 5664506154947605320093798591702205795*v^9 + 943096376032091177684321322291090122*v^8 - 36237782377887737370613009890898646913*v^7 - 11271238373294049414542286460786924415*v^6 - 138906095108859210342295021658387577809*v^5 + 1281651065343614672492619944645992360*v^4 - 211094523555935431670867532527198252928*v^3 - 181946970896679484826970379040226437120*v^2 - 206888464086963362638129792868208181248*v + 61316730759741599941227875747601842176) / 257869643580096300656340778252766806016 $$\beta_{4}$$ $$=$$ $$( 11\!\cdots\!41 \nu^{17} + \cdots + 25\!\cdots\!24 ) / 20\!\cdots\!16$$ (118007779678858866612478098179893141*v^17 - 320614849714159981826179081453090700*v^16 + 6621642885397649298909767620564678134*v^15 - 2808978745738213780637911694949231426*v^14 + 217309089474140768403743756010528051344*v^13 - 15354167051679635108738493354098018695*v^12 + 4173533399952083102198762591134908177803*v^11 + 3433675407919571495164999244838388531408*v^10 + 52717491362834119015154326272039276649055*v^9 + 32514139657645504867248569306389928990958*v^8 + 335959658732320175568039706610120108797269*v^7 + 352690940965214664018349957675572965032619*v^6 + 1441813290609311053090549939103144498485989*v^5 + 690671849239127310309957336545395030292536*v^4 + 2258191324996135762071731931986683479328128*v^3 + 2712727364491795489702095019789702470600704*v^2 + 6522074138620945066585397055593793639030784*v + 2599087734976794519273894105830992156033024) / 2032592998109214065848442099382871156719616 $$\beta_{5}$$ $$=$$ $$( 10\!\cdots\!51 \nu^{17} + \cdots - 56\!\cdots\!96 ) / 16\!\cdots\!28$$ (1016307484260578505484980209366569651*v^17 - 5849589827225077299988475096330329004*v^16 + 70902572229780289878065138334737811162*v^15 - 219881584713740791043545881041180226654*v^14 + 2275152098127642585649435081355885217376*v^13 - 6345574889185206289952304908888729507729*v^12 + 46534596203765692949143700690045800184565*v^11 - 91835034505274786807365503827961305955896*v^10 + 555163656240564885925941330061194001395001*v^9 - 1138210106839484370947919698266745309812422*v^8 + 4194866195774517958733943979385199520405667*v^7 - 6321928745140232164374799256562243954827211*v^6 + 16068651465757069698985604779054391894068219*v^5 - 25320970631828034751354836484029634027801872*v^4 + 44353601303841539795601088045420380810715904*v^3 - 35393540456660728824487845661347423286874112*v^2 + 21703591305861043848356051644067338487726080*v - 56257437146803960534402804708263521415069696) / 16260743984873712526787536795062969253756928 $$\beta_{6}$$ $$=$$ $$( - 17\!\cdots\!07 \nu^{17} + \cdots + 13\!\cdots\!44 ) / 20\!\cdots\!16$$ (-177691654540904251311026442474551507*v^17 + 2413525906713471817358130585642181288*v^16 - 15838978496990500064562355075524691018*v^15 + 115442656865926015460897929129773065894*v^14 - 413235241985539394670453079377967489592*v^13 + 3649426900716668087329815617121948271857*v^12 - 7728871619479777457028918312336155090089*v^11 + 64406356339012425982780122534488643512988*v^10 - 46033455825647628850151305236526668137401*v^9 + 809107364632082763713151759601291519829754*v^8 - 217770343527850547124824440780029451613179*v^7 + 4783755701501548478332466022769625965511847*v^6 + 1958246394653877459358060165299460799138889*v^5 + 19146678982018916574314882128559135375756140*v^4 + 4322832533597781869982749758187049612587904*v^3 + 18836133515939225678523301287180653059319808*v^2 + 9471183939750438403083172112141999268732928*v + 13825405354643063030064557956352133149753344) / 2032592998109214065848442099382871156719616 $$\beta_{7}$$ $$=$$ $$( - 28\!\cdots\!09 \nu^{17} + \cdots + 74\!\cdots\!32 ) / 31\!\cdots\!64$$ (-28139812436185648679237440776410409*v^17 + 155254474042413323626490328654528356*v^16 - 1832422854018323615791356508410572654*v^15 + 5246749590736422313598667412512079738*v^14 - 56046049808463866129043206693162446176*v^13 + 151690118676422500719495266250707329379*v^12 - 1075591498650541643639794277436875437743*v^11 + 2012084543486787416423993516249225339784*v^10 - 11572290921995019040471400842368493502811*v^9 + 26647701266064860618579088398335655901906*v^8 - 72474488554776367306129550789146990158265*v^7 + 127342973514537594045025385106490861874705*v^6 - 188595237743998451832473237141185779267105*v^5 + 608089547363818674788543749726530118415312*v^4 - 245070131911008950308452147251396178347904*v^3 + 254802575768249803558222997396342205126656*v^2 + 1214589260097936271956688338238309316132864*v + 74075717290491787695880247727153237983232) / 312706615093725240899760322981980177956864 $$\beta_{8}$$ $$=$$ $$( 12\!\cdots\!01 \nu^{17} + \cdots + 90\!\cdots\!72 ) / 10\!\cdots\!08$$ (126543990587135484431278081497488601*v^17 + 208762668878921528318117901981632781*v^16 + 5970867959345733675058440589482295642*v^15 + 27459915676943125788530355723516869956*v^14 + 236508807988146513892700680252435625206*v^13 + 1025633998810678629429714589510037231437*v^12 + 4918371168908491010320706212565092574660*v^11 + 24042261272325099248793588819535593364335*v^10 + 80644233898562337402479881898454561037523*v^9 + 306532638805414133251070522221220719124545*v^8 + 600724040878286286259192867449174011907063*v^7 + 2126333990222684040290325022056423568217568*v^6 + 3267428414261633479044627729446492827709384*v^5 + 8439830767309468805557333858306074830537033*v^4 + 5516515428197706065710942696333710451645440*v^3 + 10857074857319097234363075911204336655482880*v^2 + 788047500277438524211730672670244345495552*v + 9097772444703935900792378768610487877763072) / 1016296499054607032924221049691435578359808 $$\beta_{9}$$ $$=$$ $$( 11\!\cdots\!13 \nu^{17} + \cdots - 33\!\cdots\!56 ) / 81\!\cdots\!64$$ (1174879701303891867766139357917140013*v^17 - 5708009226111751553667517509771323284*v^16 + 78038453009644682719692071792251152998*v^15 - 190428205265845402485314659171011695842*v^14 + 2538464679783797345560143763278040040224*v^13 - 5275704376245447799859936703981860050959*v^12 + 51635044008805966411747083060418263545963*v^11 - 66572803851034804539013281741347473826568*v^10 + 635518063228994775555517404603644648440167*v^9 - 845369517032290016120165082415172976483898*v^8 + 4743082101359712049371294322411979819684797*v^7 - 4505551816399528272767732367965737307768277*v^6 + 19652173586414008578555301429370187925338533*v^5 - 20938810689915491486132037300875465314673328*v^4 + 51409467699972493238562180643771722320105728*v^3 - 45233608156891968874802524856536244262379520*v^2 + 63907562749483982777268337802484624061169664*v - 33369315757511301070812726371968962218950656) / 8130371992436856263393768397531484626878464 $$\beta_{10}$$ $$=$$ $$( - 38\!\cdots\!47 \nu^{17} + \cdots - 14\!\cdots\!88 ) / 20\!\cdots\!16$$ (-384071564307516721592547461227563447*v^17 + 549126297426276748488848303735312082*v^16 - 20688679229576576947600577417633267786*v^15 - 17538951417583583968934665454829256742*v^14 - 718209994068326062119446034878717794492*v^13 - 875353039359605776741669412581871995459*v^12 - 14193726199362127888903410059943838516995*v^11 - 29286431284736725064973198428459284626126*v^10 - 196663888416885237906004605055053833487957*v^9 - 353392775576513420512249746699519779260848*v^8 - 1341852249996864486539973456932659426203123*v^7 - 2755989465187598295503495963177192997012187*v^6 - 6356294989556584074344479434767795936427525*v^5 - 9912478331455532601740400780271678901435178*v^4 - 10319285628213579530710248568389862967276928*v^3 - 16230860036109278134481850505909962348267520*v^2 - 5749855363898295696247885635724054809960448*v - 14369516879043468653025303486590610729074688) / 2032592998109214065848442099382871156719616 $$\beta_{11}$$ $$=$$ $$( - 28\!\cdots\!69 \nu^{17} + \cdots + 11\!\cdots\!84 ) / 12\!\cdots\!56$$ (-283386522853573915493505895465138069*v^17 + 922917945235906094328772575013491572*v^16 - 15851895151229034138888112700673379734*v^15 + 14243038048673131656517216561759800690*v^14 - 499147698784537477280674978463382204064*v^13 + 309969775247531561339975353595730218119*v^12 - 9132915465743580982422430022566338937219*v^11 - 2877782553640734297079972844909505921272*v^10 - 104389771652272883099542926166433106541983*v^9 + 2564799438334923276865049303356877986250*v^8 - 534193159827217602254681604792661366994725*v^7 - 294306646182844919104799500940817593077827*v^6 - 1600619114644585114880807083406416314554637*v^5 + 1466760687146123292328737178580157974312048*v^4 + 898878422583056771408729893412350759195648*v^3 + 2320528285141117853699148319743340293206016*v^2 - 1710540694122861681881504124250024053178368*v + 11153916621750214171391831799148523168989184) / 1250826460374900963599041291927920711827456 $$\beta_{12}$$ $$=$$ $$( 54\!\cdots\!69 \nu^{17} + \cdots + 17\!\cdots\!56 ) / 20\!\cdots\!16$$ (544108253089501070672719752439766769*v^17 - 903210169940456487025043771200964538*v^16 + 28910570748270878212749344219370285398*v^15 + 19048705181033080143211010801245792802*v^14 + 984227546139020585727656098582400720764*v^13 + 979548815676939611745034392641506027141*v^12 + 18947183886667039934824006641942726304953*v^11 + 35873180562787452489345431329271645277950*v^10 + 254799096655607790054337375718611570618563*v^9 + 402026405885752536360289742783667585513660*v^8 + 1629617287281676494108938659688170253675469*v^7 + 3197074129171623690606273159159324335901457*v^6 + 7920139677272588326558062736761574808552559*v^5 + 10188349469789344194362566389348438197108362*v^4 + 12773823236491175899930875360103644176822400*v^3 + 19340791810620906362792721102302808911878144*v^2 + 26340775116206097293322381133251734017064960*v + 17294780962891408816821048740807625716269056) / 2032592998109214065848442099382871156719616 $$\beta_{13}$$ $$=$$ $$( 25\!\cdots\!97 \nu^{17} + \cdots - 63\!\cdots\!36 ) / 81\!\cdots\!64$$ (2526101201559938792790164790670253897*v^17 - 9779844476526703698583236030022668100*v^16 + 150915309780882846586472963768918625966*v^15 - 237176869929281004018956130692827318778*v^14 + 4812407387201767703872836279372860472096*v^13 - 6202063625651382837081810878669539923139*v^12 + 91648758037148187486624855518198847870127*v^11 - 43802060754305513023589587213698003644456*v^10 + 1073363852919683529471361518377680305807547*v^9 - 802705174307464372727970285225939186043186*v^8 + 6400760175667716899922031217061085517921945*v^7 - 2968051447120212142601522728161943368025745*v^6 + 21625205686790869768410633167428690198856609*v^5 - 27184387228200022824729924321474990615476272*v^4 + 16051344637338108828113781253477304988439424*v^3 - 39776635883462283136873623960877591603703808*v^2 + 26272566816218076696135269843448631593402368*v - 63956989119658928877595847785747344167796736) / 8130371992436856263393768397531484626878464 $$\beta_{14}$$ $$=$$ $$( 82\!\cdots\!31 \nu^{17} + \cdots + 18\!\cdots\!88 ) / 20\!\cdots\!16$$ (825753501135630129085795503232402331*v^17 - 2526049958113068269366480842665290118*v^16 + 48402340843779342979068812486389482962*v^15 - 39764153625926950265646984695248413178*v^14 + 1596081452269776566444571718893636825508*v^13 - 661007291825740054188270969943949623817*v^12 + 31231071441784520660293801761149009541771*v^11 + 13925282455253679973115434956976201332018*v^10 + 394388692053959817886675924623671819696865*v^9 + 149335915145167375983582497383119774823420*v^8 + 2612901457795379527735258171713046316795039*v^7 + 2121462789475260173023454963118714495312243*v^6 + 10584339341328188665033733337644526842984525*v^5 + 7239338401340576670162528424686958194117926*v^4 + 16465950114000161654220726253247440589132160*v^3 + 18349203984893883422860340975921296885598208*v^2 + 8864023191813075697570211578971510434906112*v + 18032768319747167911601667131812467337330688) / 2032592998109214065848442099382871156719616 $$\beta_{15}$$ $$=$$ $$( - 65\!\cdots\!29 \nu^{17} + \cdots + 10\!\cdots\!20 ) / 12\!\cdots\!56$$ (-658198158967893195258812670233283729*v^17 + 2451801148764874020658433614961260260*v^16 - 38810549945351714188409116366605955742*v^15 + 55452066070262313497576729623758776010*v^14 - 1236090985596955915904490374730454332960*v^13 + 1417072421960578955357698600769665576155*v^12 - 23372804537171037295980782611633131687847*v^11 + 7596780855236692501949644337069988453224*v^10 - 272945656294509180517913364851788213903635*v^9 + 167084364447262475113479091442675627739458*v^8 - 1583431260785808443892994573018348951246049*v^7 + 523371894659278543760765596432363241071769*v^6 - 5251787060373312155364080845829026827296681*v^5 + 6337914271649795064332715881666789122132400*v^4 - 2942720444972489420463496923859102416599552*v^3 + 9358954096265896818509466826870611292680192*v^2 - 6267849711517167260526406456563825604689920*v + 10690180481949166050526829113202256453304320) / 1250826460374900963599041291927920711827456 $$\beta_{16}$$ $$=$$ $$( 11\!\cdots\!21 \nu^{17} + \cdots - 19\!\cdots\!44 ) / 16\!\cdots\!28$$ (11256256425619544356587544350379342921*v^17 - 50568380418876469661711381365192146884*v^16 + 708594854860328328968158364573861304878*v^15 - 1503711931048240360048270630871145122042*v^14 + 22610906473944704662369582187380446101792*v^13 - 41411852123832535955401523022908455855811*v^12 + 441730065860045844550420489286177747335471*v^11 - 458640953280196289270353239326517493706280*v^10 + 5202886962510511752362507690016154014315579*v^9 - 6457603152352019366315568300090198117252274*v^8 + 34173172853494245424774812056705473057099545*v^7 - 30958681938187058069275998679417604347465489*v^6 + 122150430464187604967080714839068716272641185*v^5 - 171679206264280312183515547306181301994448944*v^4 + 185617872038882365734339330822787656564153088*v^3 - 245326051901109728761441014936126101585915904*v^2 + 156110715560274548457649667690239946328506368*v - 194053852976520331065252356819550434486124544) / 16260743984873712526787536795062969253756928 $$\beta_{17}$$ $$=$$ $$( 12\!\cdots\!63 \nu^{17} + \cdots - 14\!\cdots\!88 ) / 16\!\cdots\!28$$ (12001098732949906459355905537631920863*v^17 - 45619635032407385389370974893016086924*v^16 + 732776371491698166464198774254903206978*v^15 - 1121528874569025290395574106392407130166*v^14 + 23856277602215105834847009699508289780416*v^13 - 28055247336305069243185709373104046064373*v^12 + 469480572188092181691100091159098870313721*v^11 - 170990757103446116197351253748027599367912*v^10 + 5789377017832424998113481693311440813841853*v^9 - 2760998229088752629072493335641612136993710*v^8 + 39317479749852131149933187521174038961446895*v^7 - 4796420232910280832040013165003676105741319*v^6 + 160390105433273353505806814319362567833134455*v^5 - 62808837596537205986495947429680461009297792*v^4 + 302733271356295543973420905588837976745711616*v^3 - 49182353744158907085739985935770257056681984*v^2 + 338099740428245975139439597528701226691592192*v - 143660479853767440950495408050924542217945088) / 16260743984873712526787536795062969253756928
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{17} + \beta_{16} - \beta_{13} + \beta_{10} + \beta_{6} - 12\beta_{3} - 2\beta_{2} + 2\beta_1$$ -b17 + b16 - b13 + b10 + b6 - 12*b3 - 2*b2 + 2*b1 $$\nu^{3}$$ $$=$$ $$-2\beta_{17} + 3\beta_{16} - 3\beta_{13} + 2\beta_{12} - \beta_{11} - 4\beta_{9} + 4\beta_{7} - 23\beta_{2} - 12$$ -2*b17 + 3*b16 - 3*b13 + 2*b12 - b11 - 4*b9 + 4*b7 - 23*b2 - 12 $$\nu^{4}$$ $$=$$ $$-8\beta_{14} + 27\beta_{12} - 37\beta_{10} + 8\beta_{7} - 32\beta_{6} - 5\beta_{4} + 260\beta_{3} - 77\beta _1 - 260$$ -8*b14 + 27*b12 - 37*b10 + 8*b7 - 32*b6 - 5*b4 + 260*b3 - 77*b1 - 260 $$\nu^{5}$$ $$=$$ $$75 \beta_{17} - 128 \beta_{16} + 10 \beta_{15} - 45 \beta_{14} + 166 \beta_{13} + 45 \beta_{11} + \cdots - 636 \beta_1$$ 75*b17 - 128*b16 + 10*b15 - 45*b14 + 166*b13 + 45*b11 - 166*b10 + 138*b9 - 10*b8 - 128*b6 - 18*b5 - 18*b4 + 604*b3 + 636*b2 - 636*b1 $$\nu^{6}$$ $$=$$ $$748 \beta_{17} - 1004 \beta_{16} + 46 \beta_{15} + 1315 \beta_{13} - 748 \beta_{12} + 339 \beta_{11} + \cdots + 6752$$ 748*b17 - 1004*b16 + 46*b15 + 1315*b13 - 748*b12 + 339*b11 + 490*b9 - 490*b7 - 299*b5 + 2746*b2 + 6752 $$\nu^{7}$$ $$=$$ $$1762 \beta_{14} - 2684 \beta_{12} + 6896 \beta_{10} + 644 \beta_{8} - 4588 \beta_{7} + 4677 \beta_{6} + \cdots + 23360$$ 1762*b14 - 2684*b12 + 6896*b10 + 644*b8 - 4588*b7 + 4677*b6 + 1299*b4 - 23360*b3 + 19224*b1 + 23360 $$\nu^{8}$$ $$=$$ $$- 21858 \beta_{17} + 32278 \beta_{16} - 3242 \beta_{15} + 12178 \beta_{14} - 46550 \beta_{13} + \cdots + 95033 \beta_1$$ -21858*b17 + 32278*b16 - 3242*b15 + 12178*b14 - 46550*b13 - 12178*b11 + 46550*b10 - 21190*b9 + 3242*b8 + 32278*b6 + 12940*b5 + 12940*b4 - 192772*b3 - 95033*b2 + 95033*b1 $$\nu^{9}$$ $$=$$ $$- 93585 \beta_{17} + 164491 \beta_{16} - 29122 \beta_{15} - 260223 \beta_{13} + 93585 \beta_{12} + \cdots - 831272$$ -93585*b17 + 164491*b16 - 29122*b15 - 260223*b13 + 93585*b12 - 65220*b11 - 155738*b9 + 155738*b7 + 63424*b5 - 609880*b2 - 831272 $$\nu^{10}$$ $$=$$ $$- 423443 \beta_{14} + 667892 \beta_{12} - 1637813 \beta_{10} - 155970 \beta_{8} + 812858 \beta_{7} + \cdots - 5853008$$ -423443*b14 + 667892*b12 - 1637813*b10 - 155970*b8 + 812858*b7 - 1058555*b6 - 500294*b4 + 5853008*b3 - 3244441*b1 - 5853008 $$\nu^{11}$$ $$=$$ $$3212681 \beta_{17} - 5697380 \beta_{16} + 1156558 \beta_{15} - 2341040 \beta_{14} + \cdots - 19896385 \beta_1$$ 3212681*b17 - 5697380*b16 + 1156558*b15 - 2341040*b14 + 9404329*b13 + 2341040*b11 - 9404329*b10 + 5347886*b9 - 1156558*b8 - 5697380*b6 - 2641593*b5 - 2641593*b4 + 28645472*b3 + 19896385*b2 - 19896385*b1 $$\nu^{12}$$ $$=$$ $$21117643 \beta_{17} - 35204052 \beta_{16} + 6439744 \beta_{15} + 57218756 \beta_{13} - 21117643 \beta_{12} + \cdots + 185118648$$ 21117643*b17 - 35204052*b16 + 6439744*b15 + 57218756*b13 - 21117643*b12 + 14595321*b11 + 29575380*b9 - 29575380*b7 - 18387680*b5 + 110201064*b2 + 185118648 $$\nu^{13}$$ $$=$$ $$82492865 \beta_{14} - 109440828 \beta_{12} + 332264719 \beta_{10} + 43215104 \beta_{8} + \cdots + 974986692$$ 82492865*b14 - 109440828*b12 + 332264719*b10 + 43215104*b8 - 184233596*b7 + 195850226*b6 + 101502237*b4 - 974986692*b3 + 659993730*b1 + 974986692 $$\nu^{14}$$ $$=$$ $$- 684437956 \beta_{17} + 1181688405 \beta_{16} - 246219578 \beta_{15} + 501405476 \beta_{14} + \cdots + 3738644514 \beta_1$$ -684437956*b17 + 1181688405*b16 - 246219578*b15 + 501405476*b14 - 1986451992*b13 - 501405476*b11 + 1986451992*b10 - 1047709358*b9 + 246219578*b8 + 1181688405*b6 + 658045263*b5 + 658045263*b4 - 6013474260*b3 - 3738644514*b2 + 3738644514*b1 $$\nu^{15}$$ $$=$$ $$- 3716482924 \beta_{17} + 6705745868 \beta_{16} - 1562310104 \beta_{15} - 11588149060 \beta_{13} + \cdots - 33051801880$$ -3716482924*b17 + 6705745868*b16 - 1562310104*b15 - 11588149060*b13 + 3716482924*b12 - 2873366536*b11 - 6343709628*b9 + 6343709628*b7 + 3727539608*b5 - 22116656065*b2 - 33051801880 $$\nu^{16}$$ $$=$$ $$- 17193582024 \beta_{14} + 22567596581 \beta_{12} - 68620566389 \beta_{10} - 9017389320 \beta_{8} + \cdots - 198745293060$$ -17193582024*b14 + 22567596581*b12 - 68620566389*b10 - 9017389320*b8 + 36572897116*b7 - 39902478713*b6 - 23181789868*b4 + 198745293060*b3 - 126885353254*b1 - 198745293060 $$\nu^{17}$$ $$=$$ $$126112059954 \beta_{17} - 229119790391 \beta_{16} + 55380969056 \beta_{15} - 99346555769 \beta_{14} + \cdots - 745830358799 \beta_1$$ 126112059954*b17 - 229119790391*b16 + 55380969056*b15 - 99346555769*b14 + 400991725275*b13 + 99346555769*b11 - 400991725275*b10 + 218067369976*b9 - 55380969056*b8 - 229119790391*b6 - 133249111332*b5 - 133249111332*b4 + 1119843402916*b3 + 745830358799*b2 - 745830358799*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 −2.21418 − 3.83507i −1.91278 − 3.31303i −1.08068 − 1.87179i −0.613994 − 1.06347i 0.425471 + 0.736937i 0.695058 + 1.20388i 1.36382 + 2.36220i 2.41719 + 4.18670i 2.92009 + 5.05774i −2.21418 + 3.83507i −1.91278 + 3.31303i −1.08068 + 1.87179i −0.613994 + 1.06347i 0.425471 − 0.736937i 0.695058 − 1.20388i 1.36382 − 2.36220i 2.41719 − 4.18670i 2.92009 − 5.05774i
−2.71418 + 4.70109i −0.837548 + 1.45068i −10.7335 18.5910i 7.70909 −4.54651 7.87478i 7.51249 + 13.0120i 73.1038 12.0970 + 20.9527i −20.9238 + 36.2412i
22.2 −2.41278 + 4.17905i −2.22176 + 3.84820i −7.64299 13.2380i −12.7712 −10.7212 18.5697i −13.0936 22.6787i 35.1589 3.62756 + 6.28312i 30.8140 53.3715i
22.3 −1.58068 + 2.73781i −3.54442 + 6.13911i −0.997073 1.72698i 13.6039 −11.2051 19.4079i 7.16574 + 12.4114i −18.9866 −11.6258 20.1364i −21.5034 + 37.2450i
22.4 −1.11399 + 1.92949i 4.87434 8.44260i 1.51803 + 2.62931i −8.20685 10.8600 + 18.8100i −4.17747 7.23560i −24.5882 −34.0183 58.9214i 9.14239 15.8351i
22.5 −0.0745292 + 0.129088i 3.24429 5.61927i 3.98889 + 6.90896i −10.2526 0.483588 + 0.837600i 14.8372 + 25.6987i −2.38162 −7.55083 13.0784i 0.764114 1.32348i
22.6 0.195058 0.337850i −1.80483 + 3.12606i 3.92391 + 6.79640i −7.52136 0.704093 + 1.21953i −9.77228 16.9261i 6.18247 6.98515 + 12.0986i −1.46710 + 2.54109i
22.7 0.863817 1.49617i −3.44796 + 5.97204i 2.50764 + 4.34336i 20.8281 5.95681 + 10.3175i −3.78283 6.55206i 22.4856 −10.2768 17.8000i 17.9916 31.1624i
22.8 1.91719 3.32067i 0.139581 0.241762i −3.35124 5.80452i 11.3710 −0.535209 0.927008i −15.5311 26.9007i 4.97517 13.4610 + 23.3152i 21.8004 37.7594i
22.9 2.42009 4.19172i 3.09831 5.36643i −7.71365 13.3604i 15.2399 −14.9964 25.9745i −2.15810 3.73794i −35.9495 −5.69903 9.87102i 36.8818 63.8811i
146.1 −2.71418 4.70109i −0.837548 1.45068i −10.7335 + 18.5910i 7.70909 −4.54651 + 7.87478i 7.51249 13.0120i 73.1038 12.0970 20.9527i −20.9238 36.2412i
146.2 −2.41278 4.17905i −2.22176 3.84820i −7.64299 + 13.2380i −12.7712 −10.7212 + 18.5697i −13.0936 + 22.6787i 35.1589 3.62756 6.28312i 30.8140 + 53.3715i
146.3 −1.58068 2.73781i −3.54442 6.13911i −0.997073 + 1.72698i 13.6039 −11.2051 + 19.4079i 7.16574 12.4114i −18.9866 −11.6258 + 20.1364i −21.5034 37.2450i
146.4 −1.11399 1.92949i 4.87434 + 8.44260i 1.51803 2.62931i −8.20685 10.8600 18.8100i −4.17747 + 7.23560i −24.5882 −34.0183 + 58.9214i 9.14239 + 15.8351i
146.5 −0.0745292 0.129088i 3.24429 + 5.61927i 3.98889 6.90896i −10.2526 0.483588 0.837600i 14.8372 25.6987i −2.38162 −7.55083 + 13.0784i 0.764114 + 1.32348i
146.6 0.195058 + 0.337850i −1.80483 3.12606i 3.92391 6.79640i −7.52136 0.704093 1.21953i −9.77228 + 16.9261i 6.18247 6.98515 12.0986i −1.46710 2.54109i
146.7 0.863817 + 1.49617i −3.44796 5.97204i 2.50764 4.34336i 20.8281 5.95681 10.3175i −3.78283 + 6.55206i 22.4856 −10.2768 + 17.8000i 17.9916 + 31.1624i
146.8 1.91719 + 3.32067i 0.139581 + 0.241762i −3.35124 + 5.80452i 11.3710 −0.535209 + 0.927008i −15.5311 + 26.9007i 4.97517 13.4610 23.3152i 21.8004 + 37.7594i
146.9 2.42009 + 4.19172i 3.09831 + 5.36643i −7.71365 + 13.3604i 15.2399 −14.9964 + 25.9745i −2.15810 + 3.73794i −35.9495 −5.69903 + 9.87102i 36.8818 + 63.8811i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.k 18
13.b even 2 1 169.4.c.l 18
13.c even 3 1 169.4.a.l yes 9
13.c even 3 1 inner 169.4.c.k 18
13.d odd 4 2 169.4.e.h 36
13.e even 6 1 169.4.a.k 9
13.e even 6 1 169.4.c.l 18
13.f odd 12 2 169.4.b.g 18
13.f odd 12 2 169.4.e.h 36
39.h odd 6 1 1521.4.a.bh 9
39.i odd 6 1 1521.4.a.bg 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k 9 13.e even 6 1
169.4.a.l yes 9 13.c even 3 1
169.4.b.g 18 13.f odd 12 2
169.4.c.k 18 1.a even 1 1 trivial
169.4.c.k 18 13.c even 3 1 inner
169.4.c.l 18 13.b even 2 1
169.4.c.l 18 13.e even 6 1
169.4.e.h 36 13.d odd 4 2
169.4.e.h 36 13.f odd 12 2
1521.4.a.bg 9 39.i odd 6 1
1521.4.a.bh 9 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} + 5 T_{2}^{17} + 67 T_{2}^{16} + 200 T_{2}^{15} + 2303 T_{2}^{14} + 6060 T_{2}^{13} + \cdots + 118336$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} + 5 T^{17} + \cdots + 118336$$
$3$ $$T^{18} + \cdots + 20654576089$$
$5$ $$(T^{9} - 30 T^{8} + \cdots - 3059376152)^{2}$$
$7$ $$T^{18} + \cdots + 76\!\cdots\!76$$
$11$ $$T^{18} + \cdots + 76\!\cdots\!61$$
$13$ $$T^{18}$$
$17$ $$T^{18} + \cdots + 31\!\cdots\!49$$
$19$ $$T^{18} + \cdots + 74\!\cdots\!61$$
$23$ $$T^{18} + \cdots + 46\!\cdots\!64$$
$29$ $$T^{18} + \cdots + 31\!\cdots\!16$$
$31$ $$(T^{9} + \cdots + 30\!\cdots\!56)^{2}$$
$37$ $$T^{18} + \cdots + 96\!\cdots\!04$$
$41$ $$T^{18} + \cdots + 38\!\cdots\!09$$
$43$ $$T^{18} + \cdots + 12\!\cdots\!29$$
$47$ $$(T^{9} + \cdots - 11\!\cdots\!84)^{2}$$
$53$ $$(T^{9} + \cdots - 12\!\cdots\!76)^{2}$$
$59$ $$T^{18} + \cdots + 75\!\cdots\!29$$
$61$ $$T^{18} + \cdots + 75\!\cdots\!24$$
$67$ $$T^{18} + \cdots + 10\!\cdots\!01$$
$71$ $$T^{18} + \cdots + 19\!\cdots\!84$$
$73$ $$(T^{9} + \cdots + 31\!\cdots\!21)^{2}$$
$79$ $$(T^{9} + \cdots + 63\!\cdots\!88)^{2}$$
$83$ $$(T^{9} + \cdots - 14\!\cdots\!61)^{2}$$
$89$ $$T^{18} + \cdots + 19\!\cdots\!29$$
$97$ $$T^{18} + \cdots + 43\!\cdots\!21$$