# Properties

 Label 37.2.h.a Level $37$ Weight $2$ Character orbit 37.h Analytic conductor $0.295$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(3,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([13]))

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

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

chi := DirichletCharacter("37.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.h (of order $$18$$, degree $$6$$, 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: $$0.295446487479$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} + 30x^{16} + 333x^{14} + 1826x^{12} + 5490x^{10} + 9432x^{8} + 9385x^{6} + 5316x^{4} + 1584x^{2} + 192$$ x^18 + 30*x^16 + 333*x^14 + 1826*x^12 + 5490*x^10 + 9432*x^8 + 9385*x^6 + 5316*x^4 + 1584*x^2 + 192 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 30x^{16} + 333x^{14} + 1826x^{12} + 5490x^{10} + 9432x^{8} + 9385x^{6} + 5316x^{4} + 1584x^{2} + 192$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 69 \nu^{16} - 2010 \nu^{14} - 21225 \nu^{12} - 107422 \nu^{10} - 284274 \nu^{8} - 398496 \nu^{6} - 289165 \nu^{4} - 101208 \nu^{2} + 64 \nu - 13392 ) / 128$$ (-69*v^16 - 2010*v^14 - 21225*v^12 - 107422*v^10 - 284274*v^8 - 398496*v^6 - 289165*v^4 - 101208*v^2 + 64*v - 13392) / 128 $$\beta_{3}$$ $$=$$ $$( 21 \nu^{17} - 57 \nu^{16} + 600 \nu^{15} - 1678 \nu^{14} + 6125 \nu^{13} - 18037 \nu^{12} + 29284 \nu^{11} - 93906 \nu^{10} + 70238 \nu^{9} - 259794 \nu^{8} + 82076 \nu^{7} - 390328 \nu^{6} + \cdots - 19808 ) / 128$$ (21*v^17 - 57*v^16 + 600*v^15 - 1678*v^14 + 6125*v^13 - 18037*v^12 + 29284*v^11 - 93906*v^10 + 70238*v^9 - 259794*v^8 + 82076*v^7 - 390328*v^6 + 41101*v^5 - 313697*v^4 + 4902*v^3 - 125860*v^2 - 1192*v - 19808) / 128 $$\beta_{4}$$ $$=$$ $$( - 153 \nu^{17} + 234 \nu^{16} - 4530 \nu^{15} + 6868 \nu^{14} - 49165 \nu^{13} + 73458 \nu^{12} - 259878 \nu^{11} + 379484 \nu^{10} - 735738 \nu^{9} + 1037220 \nu^{8} + \cdots + 65824 ) / 512$$ (-153*v^17 + 234*v^16 - 4530*v^15 + 6868*v^14 - 49165*v^13 + 73458*v^12 - 259878*v^11 + 379484*v^10 - 735738*v^9 + 1037220*v^8 - 1142912*v^7 + 1528384*v^6 - 957761*v^5 + 1190394*v^4 - 399688*v^3 + 454736*v^2 - 64208*v + 65824) / 512 $$\beta_{5}$$ $$=$$ $$( 279 \nu^{17} + 8094 \nu^{15} + 84867 \nu^{13} + 424554 \nu^{11} + 1102022 \nu^{9} + 1494432 \nu^{7} + 1024431 \nu^{5} + 326504 \nu^{3} + 37104 \nu - 256 ) / 512$$ (279*v^17 + 8094*v^15 + 84867*v^13 + 424554*v^11 + 1102022*v^9 + 1494432*v^7 + 1024431*v^5 + 326504*v^3 + 37104*v - 256) / 512 $$\beta_{6}$$ $$=$$ $$( - 691 \nu^{17} - 306 \nu^{16} - 20262 \nu^{15} - 9060 \nu^{14} - 216367 \nu^{13} - 98330 \nu^{12} - 1114850 \nu^{11} - 519756 \nu^{10} - 3034622 \nu^{9} - 1471476 \nu^{8} + \cdots - 128416 ) / 1024$$ (-691*v^17 - 306*v^16 - 20262*v^15 - 9060*v^14 - 216367*v^13 - 98330*v^12 - 1114850*v^11 - 519756*v^10 - 3034622*v^9 - 1471476*v^8 - 4443072*v^7 - 2285824*v^6 - 3428267*v^5 - 1915522*v^4 - 1292568*v^3 - 799376*v^2 - 185072*v - 128416) / 1024 $$\beta_{7}$$ $$=$$ $$( - 691 \nu^{17} + 306 \nu^{16} - 20262 \nu^{15} + 9060 \nu^{14} - 216367 \nu^{13} + 98330 \nu^{12} - 1114850 \nu^{11} + 519756 \nu^{10} - 3034622 \nu^{9} + 1471476 \nu^{8} + \cdots + 128416 ) / 1024$$ (-691*v^17 + 306*v^16 - 20262*v^15 + 9060*v^14 - 216367*v^13 + 98330*v^12 - 1114850*v^11 + 519756*v^10 - 3034622*v^9 + 1471476*v^8 - 4443072*v^7 + 2285824*v^6 - 3428267*v^5 + 1915522*v^4 - 1292568*v^3 + 799376*v^2 - 185072*v + 128416) / 1024 $$\beta_{8}$$ $$=$$ $$( - 127 \nu^{17} - 229 \nu^{16} - 3758 \nu^{15} - 6714 \nu^{14} - 40747 \nu^{13} - 71689 \nu^{12} - 215066 \nu^{11} - 369470 \nu^{10} - 607606 \nu^{9} - 1006898 \nu^{8} + \cdots - 65488 ) / 256$$ (-127*v^17 - 229*v^16 - 3758*v^15 - 6714*v^14 - 40747*v^13 - 71689*v^12 - 215066*v^11 - 369470*v^10 - 607606*v^9 - 1006898*v^8 - 941568*v^7 - 1479584*v^6 - 788471*v^5 - 1152173*v^4 - 331624*v^3 - 443000*v^2 - 54448*v - 65488) / 256 $$\beta_{9}$$ $$=$$ $$( 667 \nu^{17} + 554 \nu^{16} + 19846 \nu^{15} + 16052 \nu^{14} + 217111 \nu^{13} + 167986 \nu^{12} + 1161122 \nu^{11} + 838268 \nu^{10} + 3341230 \nu^{9} + 2170596 \nu^{8} + \cdots + 91552 ) / 1024$$ (667*v^17 + 554*v^16 + 19846*v^15 + 16052*v^14 + 217111*v^13 + 167986*v^12 + 1161122*v^11 + 838268*v^10 + 3341230*v^9 + 2170596*v^8 + 5301344*v^7 + 2944576*v^6 + 4549715*v^5 + 2046138*v^4 + 1942632*v^3 + 690096*v^2 + 319792*v + 91552) / 1024 $$\beta_{10}$$ $$=$$ $$( - 681 \nu^{17} + 754 \nu^{16} - 20018 \nu^{15} + 22180 \nu^{14} - 214685 \nu^{13} + 238170 \nu^{12} - 1114278 \nu^{11} + 1238668 \nu^{10} - 3071322 \nu^{9} + 3424628 \nu^{8} + \cdots + 259232 ) / 1024$$ (-681*v^17 + 754*v^16 - 20018*v^15 + 22180*v^14 - 214685*v^13 + 238170*v^12 - 1114278*v^11 + 1238668*v^10 - 3071322*v^9 + 3424628*v^8 - 4597888*v^7 + 5147392*v^6 - 3690001*v^5 + 4143426*v^4 - 1486312*v^3 + 1661904*v^2 - 235344*v + 259232) / 1024 $$\beta_{11}$$ $$=$$ $$( 169 \nu^{17} + 343 \nu^{16} + 4958 \nu^{15} + 10070 \nu^{14} + 52997 \nu^{13} + 107763 \nu^{12} + 273634 \nu^{11} + 557282 \nu^{10} + 748082 \nu^{9} + 1526486 \nu^{8} + \cdots + 105104 ) / 256$$ (169*v^17 + 343*v^16 + 4958*v^15 + 10070*v^14 + 52997*v^13 + 107763*v^12 + 273634*v^11 + 557282*v^10 + 748082*v^9 + 1526486*v^8 + 1105720*v^7 + 2260240*v^6 + 870673*v^5 + 1779567*v^4 + 341428*v^3 + 694720*v^2 + 52064*v + 105104) / 256 $$\beta_{12}$$ $$=$$ $$( 169 \nu^{17} - 343 \nu^{16} + 4958 \nu^{15} - 10070 \nu^{14} + 52997 \nu^{13} - 107763 \nu^{12} + 273634 \nu^{11} - 557282 \nu^{10} + 748082 \nu^{9} - 1526486 \nu^{8} + \cdots - 105104 ) / 256$$ (169*v^17 - 343*v^16 + 4958*v^15 - 10070*v^14 + 52997*v^13 - 107763*v^12 + 273634*v^11 - 557282*v^10 + 748082*v^9 - 1526486*v^8 + 1105720*v^7 - 2260240*v^6 + 870673*v^5 - 1779567*v^4 + 341428*v^3 - 694720*v^2 + 52064*v - 105104) / 256 $$\beta_{13}$$ $$=$$ $$( 1037 \nu^{17} - 2286 \nu^{16} + 30218 \nu^{15} - 67068 \nu^{14} + 319281 \nu^{13} - 716870 \nu^{12} + 1617550 \nu^{11} - 3699860 \nu^{10} + 4287330 \nu^{9} - 10100748 \nu^{8} + \cdots - 657888 ) / 1024$$ (1037*v^17 - 2286*v^16 + 30218*v^15 - 67068*v^14 + 319281*v^13 - 716870*v^12 + 1617550*v^11 - 3699860*v^10 + 4287330*v^9 - 10100748*v^8 + 6021536*v^7 - 14870336*v^6 + 4372501*v^5 - 11594014*v^4 + 1526936*v^3 - 4454544*v^2 + 202576*v - 657888) / 1024 $$\beta_{14}$$ $$=$$ $$( - 691 \nu^{17} - 3330 \nu^{16} - 20262 \nu^{15} - 97604 \nu^{14} - 216367 \nu^{13} - 1041578 \nu^{12} - 1114850 \nu^{11} - 5362028 \nu^{10} - 3034622 \nu^{9} + \cdots - 920992 ) / 1024$$ (-691*v^17 - 3330*v^16 - 20262*v^15 - 97604*v^14 - 216367*v^13 - 1041578*v^12 - 1114850*v^11 - 5362028*v^10 - 3034622*v^9 - 14581140*v^8 - 4443072*v^7 - 21341312*v^6 - 3428267*v^5 - 16510930*v^4 - 1292568*v^3 - 6289744*v^2 - 185072*v - 920992) / 1024 $$\beta_{15}$$ $$=$$ $$( 2523 \nu^{17} + 1818 \nu^{16} + 73766 \nu^{15} + 53332 \nu^{14} + 783895 \nu^{13} + 569954 \nu^{12} + 4008898 \nu^{11} + 2940892 \nu^{10} + 10791022 \nu^{9} + 8026308 \nu^{8} + \cdots + 524704 ) / 1024$$ (2523*v^17 + 1818*v^16 + 73766*v^15 + 53332*v^14 + 783895*v^13 + 569954*v^12 + 4008898*v^11 + 2940892*v^10 + 10791022*v^9 + 8026308*v^8 + 15553184*v^7 + 11813568*v^6 + 11781779*v^5 + 9213226*v^4 + 4380808*v^3 + 3544560*v^2 + 624304*v + 524704) / 1024 $$\beta_{16}$$ $$=$$ $$( 837 \nu^{17} - 106 \nu^{16} + 24538 \nu^{15} - 3076 \nu^{14} + 261961 \nu^{13} - 32274 \nu^{12} + 1349598 \nu^{11} - 161708 \nu^{10} + 3675442 \nu^{9} - 421252 \nu^{8} + \cdots - 16224 ) / 256$$ (837*v^17 - 106*v^16 + 24538*v^15 - 3076*v^14 + 261961*v^13 - 32274*v^12 + 1349598*v^11 - 161708*v^10 + 3675442*v^9 - 421252*v^8 + 5395296*v^7 - 575744*v^6 + 4198509*v^5 - 401242*v^4 + 1617784*v^3 - 132512*v^2 + 242064*v - 16224) / 256 $$\beta_{17}$$ $$=$$ $$( - 890 \nu^{17} - 601 \nu^{16} - 26096 \nu^{15} - 17578 \nu^{14} - 278666 \nu^{13} - 186909 \nu^{12} - 1436200 \nu^{11} - 956734 \nu^{10} - 3913276 \nu^{9} - 2578570 \nu^{8} + \cdots - 148720 ) / 256$$ (-890*v^17 - 601*v^16 - 26096*v^15 - 17578*v^14 - 278666*v^13 - 186909*v^12 - 1436200*v^11 - 956734*v^10 - 3913276*v^9 - 2578570*v^8 - 5747896*v^7 - 3721872*v^6 - 4474666*v^5 - 2820865*v^4 - 1722444*v^3 - 1046304*v^2 - 256240*v - 148720) / 256
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{17} - \beta_{15} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} - \beta _1 - 4$$ -b17 - b15 + b10 - b9 + b6 + b5 - 2*b4 - b3 - b1 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{16} - 3 \beta_{15} - \beta_{14} - \beta_{13} + 3 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + 6 \beta_{5} + 2 \beta_{4} - \beta_{3} - 8 \beta _1 + 3$$ b16 - 3*b15 - b14 - b13 + 3*b12 + b11 + b10 + b9 + b7 + 6*b5 + 2*b4 - b3 - 8*b1 + 3 $$\nu^{4}$$ $$=$$ $$8 \beta_{17} - 3 \beta_{16} + 11 \beta_{15} - \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + \beta_{11} - 11 \beta_{10} + 11 \beta_{9} + 4 \beta_{8} + 5 \beta_{7} - 12 \beta_{6} - 14 \beta_{5} + 22 \beta_{4} + 7 \beta_{3} + 4 \beta_{2} + 9 \beta _1 + 29$$ 8*b17 - 3*b16 + 11*b15 - b14 + 3*b13 + 3*b12 + b11 - 11*b10 + 11*b9 + 4*b8 + 5*b7 - 12*b6 - 14*b5 + 22*b4 + 7*b3 + 4*b2 + 9*b1 + 29 $$\nu^{5}$$ $$=$$ $$- 5 \beta_{17} - 9 \beta_{16} + 40 \beta_{15} + 13 \beta_{14} + 19 \beta_{13} - 57 \beta_{12} - 21 \beta_{11} - 14 \beta_{10} - 14 \beta_{9} + 8 \beta_{8} - 5 \beta_{7} + 3 \beta_{6} - 101 \beta_{5} - 28 \beta_{4} + 22 \beta_{3} + 80 \beta _1 - 53$$ -5*b17 - 9*b16 + 40*b15 + 13*b14 + 19*b13 - 57*b12 - 21*b11 - 14*b10 - 14*b9 + 8*b8 - 5*b7 + 3*b6 - 101*b5 - 28*b4 + 22*b3 + 80*b1 - 53 $$\nu^{6}$$ $$=$$ $$- 75 \beta_{17} + 54 \beta_{16} - 129 \beta_{15} + 26 \beta_{14} - 54 \beta_{13} - 68 \beta_{12} - 10 \beta_{11} + 121 \beta_{10} - 129 \beta_{9} - 70 \beta_{8} - 98 \beta_{7} + 139 \beta_{6} + 175 \beta_{5} - 250 \beta_{4} - 51 \beta_{3} + \cdots - 278$$ -75*b17 + 54*b16 - 129*b15 + 26*b14 - 54*b13 - 68*b12 - 10*b11 + 121*b10 - 129*b9 - 70*b8 - 98*b7 + 139*b6 + 175*b5 - 250*b4 - 51*b3 - 78*b2 - 90*b1 - 278 $$\nu^{7}$$ $$=$$ $$93 \beta_{17} + 79 \beta_{16} - 514 \beta_{15} - 171 \beta_{14} - 265 \beta_{13} + 819 \beta_{12} + 311 \beta_{11} + 164 \beta_{10} + 172 \beta_{9} - 172 \beta_{8} + 23 \beta_{7} - 63 \beta_{6} + 1429 \beta_{5} + 336 \beta_{4} + \cdots + 765$$ 93*b17 + 79*b16 - 514*b15 - 171*b14 - 265*b13 + 819*b12 + 311*b11 + 164*b10 + 172*b9 - 172*b8 + 23*b7 - 63*b6 + 1429*b5 + 336*b4 - 336*b3 - 921*b1 + 765 $$\nu^{8}$$ $$=$$ $$828 \beta_{17} - 772 \beta_{16} + 1600 \beta_{15} - 436 \beta_{14} + 772 \beta_{13} + 1078 \beta_{12} + 92 \beta_{11} - 1420 \beta_{10} + 1600 \beta_{9} + 990 \beta_{8} + 1444 \beta_{7} - 1656 \beta_{6} - 2192 \beta_{5} + \cdots + 3100$$ 828*b17 - 772*b16 + 1600*b15 - 436*b14 + 772*b13 + 1078*b12 + 92*b11 - 1420*b10 + 1600*b9 + 990*b8 + 1444*b7 - 1656*b6 - 2192*b5 + 3020*b4 + 430*b3 + 1186*b2 + 1007*b1 + 3100 $$\nu^{9}$$ $$=$$ $$- 1365 \beta_{17} - 766 \beta_{16} + 6639 \beta_{15} + 2254 \beta_{14} + 3496 \beta_{13} - 10992 \beta_{12} - 4234 \beta_{11} - 1927 \beta_{10} - 2131 \beta_{9} + 2700 \beta_{8} - 94 \beta_{7} + 999 \beta_{6} + \cdots - 10402$$ -1365*b17 - 766*b16 + 6639*b15 + 2254*b14 + 3496*b13 - 10992*b12 - 4234*b11 - 1927*b10 - 2131*b9 + 2700*b8 - 94*b7 + 999*b6 - 19235*b5 - 4058*b4 + 4627*b3 + 11339*b1 - 10402 $$\nu^{10}$$ $$=$$ $$- 9974 \beta_{17} + 10377 \beta_{16} - 20351 \beta_{15} + 6319 \beta_{14} - 10377 \beta_{13} - 15207 \beta_{12} - 919 \beta_{11} + 17471 \beta_{10} - 20351 \beta_{9} - 13246 \beta_{8} - 19631 \beta_{7} + \cdots - 37415$$ -9974*b17 + 10377*b16 - 20351*b15 + 6319*b14 - 10377*b13 - 15207*b12 - 919*b11 + 17471*b10 - 20351*b9 - 13246*b8 - 19631*b7 + 20406*b6 + 27848*b5 - 37822*b4 - 4225*b3 - 16542*b2 - 12080*b1 - 37415 $$\nu^{11}$$ $$=$$ $$18648 \beta_{17} + 8283 \beta_{16} - 86009 \beta_{15} - 29539 \beta_{14} - 45579 \beta_{13} + 144461 \beta_{12} + 56031 \beta_{11} + 23431 \beta_{10} + 26931 \beta_{9} - 38068 \beta_{8} + 167 \beta_{7} + \cdots + 138011$$ 18648*b17 + 8283*b16 - 86009*b15 - 29539*b14 - 45579*b13 + 144461*b12 + 56031*b11 + 23431*b10 + 26931*b9 - 38068*b8 + 167*b7 - 14224*b6 + 253874*b5 + 50362*b4 - 61499*b3 - 143775*b1 + 138011 $$\nu^{12}$$ $$=$$ $$125127 \beta_{17} - 136617 \beta_{16} + 261744 \beta_{15} - 86255 \beta_{14} + 136617 \beta_{13} + 204829 \beta_{12} + 10119 \beta_{11} - 220796 \beta_{10} + 261744 \beta_{9} + 174000 \beta_{8} + \cdots + 469379$$ 125127*b17 - 136617*b16 + 261744*b15 - 86255*b14 + 136617*b13 + 204829*b12 + 10119*b11 - 220796*b10 + 261744*b9 + 174000*b8 + 259615*b7 - 257539*b6 - 357413*b5 + 482540*b4 + 46796*b3 + 222160*b2 + 150664*b1 + 469379 $$\nu^{13}$$ $$=$$ $$- 247697 \beta_{17} - 97158 \beta_{16} + 1115619 \beta_{15} + 385382 \beta_{14} + 592552 \beta_{13} - 1884676 \beta_{12} - 733382 \beta_{11} - 293195 \beta_{10} - 344855 \beta_{9} + \cdots - 1810496$$ -247697*b17 - 97158*b16 + 1115619*b15 + 385382*b14 + 592552*b13 - 1884676*b12 - 733382*b11 - 293195*b10 - 344855*b9 + 513244*b8 + 3566*b7 + 192911*b6 - 3321635*b5 - 638050*b4 + 806439*b3 + 1846742*b1 - 1810496 $$\nu^{14}$$ $$=$$ $$- 1599045 \beta_{17} + 1784373 \beta_{16} - 3383418 \beta_{15} + 1146323 \beta_{14} - 1784373 \beta_{13} - 2704415 \beta_{12} - 119835 \beta_{11} + 2829226 \beta_{10} - 3383418 \beta_{9} + \cdots - 5996963$$ -1599045*b17 + 1784373*b16 - 3383418*b15 + 1146323*b14 - 1784373*b13 - 2704415*b12 - 119835*b11 + 2829226*b10 - 3383418*b9 - 2270058*b8 - 3397251*b7 + 3295781*b6 + 4613599*b5 - 6212644*b4 - 559168*b3 - 2931282*b2 - 1917777*b1 - 5996963 $$\nu^{15}$$ $$=$$ $$3250014 \beta_{17} + 1197218 \beta_{16} - 14478084 \beta_{15} - 5015426 \beta_{14} - 7697246 \beta_{13} + 24519142 \beta_{12} + 9555542 \beta_{11} + 3734348 \beta_{10} + \cdots + 23623192$$ 3250014*b17 + 1197218*b16 - 14478084*b15 - 5015426*b14 - 7697246*b13 + 24519142*b12 + 9555542*b11 + 3734348*b10 + 4447232*b9 - 6782020*b8 - 78882*b7 - 2557178*b6 + 43283486*b5 + 8181580*b4 - 10516368*b3 - 23858465*b1 + 23623192 $$\nu^{16}$$ $$=$$ $$20608451 \beta_{17} - 23229040 \beta_{16} + 43837491 \beta_{15} - 15047460 \beta_{14} + 23229040 \beta_{13} + 35387360 \beta_{12} + 1483420 \beta_{11} - 36501279 \beta_{10} + \cdots + 77276212$$ 20608451*b17 - 23229040*b16 + 43837491*b15 - 15047460*b14 + 23229040*b13 + 35387360*b12 + 1483420*b11 - 36501279*b10 + 43837491*b9 + 29534568*b8 + 44261776*b7 - 42486555*b6 - 59730319*b5 + 80338770*b4 + 6966711*b3 + 38351920*b2 + 24661531*b1 + 77276212 $$\nu^{17}$$ $$=$$ $$- 42405000 \beta_{17} - 15148291 \beta_{16} + 187934249 \beta_{15} + 65190479 \beta_{14} + 99958291 \beta_{13} - 318622701 \beta_{12} - 124259615 \beta_{11} + \cdots - 307435721$$ -42405000*b17 - 15148291*b16 + 187934249*b15 + 65190479*b14 + 99958291*b13 - 318622701*b12 - 124259615*b11 - 48023055*b10 - 57553291*b9 + 88786740*b8 + 1214869*b7 + 33530584*b6 - 562936206*b5 - 105576346*b4 + 136809795*b3 + 309047952*b1 - 307435721

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 1.23399i 0.885952i 3.60322i 0.660907i − 0.834738i − 2.47983i 1.77531i 0.752039i − 1.92581i 1.23399i − 0.885952i − 3.60322i − 0.660907i 0.834738i 2.47983i − 1.77531i − 0.752039i 1.92581i
−1.61954 + 1.93010i −0.968719 + 0.812851i −0.755055 4.28213i −0.681528 + 1.87248i 3.18617i 2.99976 + 1.09182i 5.12375 + 2.95820i −0.243256 + 1.37957i −2.51031 4.34798i
3.2 −0.256873 + 0.306129i −0.0473670 + 0.0397456i 0.319565 + 1.81234i 0.853756 2.34567i 0.0247100i −4.17818 1.52073i −1.32907 0.767337i −0.520281 + 2.95066i 0.498773 + 0.863900i
3.3 1.48976 1.77542i −2.36330 + 1.98304i −0.585455 3.32028i −0.253479 + 0.696428i 7.15011i −1.02732 0.373914i −2.75280 1.58933i 1.13178 6.41863i 0.858832 + 1.48754i
4.1 −2.59056 0.456785i 0.354362 + 2.00969i 4.62296 + 1.68262i 0.640888 + 0.763781i 5.36808i −1.58572 + 1.33057i −6.65125 3.84010i −1.09419 + 0.398252i −1.31137 2.27137i
4.2 −1.11764 0.197069i −0.522172 2.96139i −0.669111 0.243536i 1.89897 + 2.26310i 3.41266i 1.25550 1.05349i 2.66549 + 1.53892i −5.67807 + 2.06665i −1.67637 2.90355i
4.3 0.502458 + 0.0885970i 0.199899 + 1.13369i −1.63477 0.595008i −0.986823 1.17605i 0.587340i 0.422615 0.354616i −1.65240 0.954012i 1.57379 0.572814i −0.391643 0.678346i
21.1 −0.841146 + 2.31103i 2.14040 0.779043i −3.10124 2.60225i −2.84116 0.500973i 5.60182i 0.251492 1.42628i 4.36277 2.51884i 1.67628 1.40657i 3.54759 6.14461i
21.2 −0.491168 + 1.34947i −2.59058 + 0.942893i −0.0477438 0.0400618i 2.52515 + 0.445253i 3.95904i 0.593686 3.36696i −2.40985 + 1.39133i 3.52391 2.95691i −1.84113 + 3.18893i
21.3 0.424710 1.16688i −0.702528 + 0.255699i 0.350853 + 0.294401i −2.65578 0.468285i 0.928365i −0.231838 + 1.31482i 2.64335 1.52614i −1.86997 + 1.56909i −1.67437 + 2.90009i
25.1 −1.61954 1.93010i −0.968719 0.812851i −0.755055 + 4.28213i −0.681528 1.87248i 3.18617i 2.99976 1.09182i 5.12375 2.95820i −0.243256 1.37957i −2.51031 + 4.34798i
25.2 −0.256873 0.306129i −0.0473670 0.0397456i 0.319565 1.81234i 0.853756 + 2.34567i 0.0247100i −4.17818 + 1.52073i −1.32907 + 0.767337i −0.520281 2.95066i 0.498773 0.863900i
25.3 1.48976 + 1.77542i −2.36330 1.98304i −0.585455 + 3.32028i −0.253479 0.696428i 7.15011i −1.02732 + 0.373914i −2.75280 + 1.58933i 1.13178 + 6.41863i 0.858832 1.48754i
28.1 −2.59056 + 0.456785i 0.354362 2.00969i 4.62296 1.68262i 0.640888 0.763781i 5.36808i −1.58572 1.33057i −6.65125 + 3.84010i −1.09419 0.398252i −1.31137 + 2.27137i
28.2 −1.11764 + 0.197069i −0.522172 + 2.96139i −0.669111 + 0.243536i 1.89897 2.26310i 3.41266i 1.25550 + 1.05349i 2.66549 1.53892i −5.67807 2.06665i −1.67637 + 2.90355i
28.3 0.502458 0.0885970i 0.199899 1.13369i −1.63477 + 0.595008i −0.986823 + 1.17605i 0.587340i 0.422615 + 0.354616i −1.65240 + 0.954012i 1.57379 + 0.572814i −0.391643 + 0.678346i
30.1 −0.841146 2.31103i 2.14040 + 0.779043i −3.10124 + 2.60225i −2.84116 + 0.500973i 5.60182i 0.251492 + 1.42628i 4.36277 + 2.51884i 1.67628 + 1.40657i 3.54759 + 6.14461i
30.2 −0.491168 1.34947i −2.59058 0.942893i −0.0477438 + 0.0400618i 2.52515 0.445253i 3.95904i 0.593686 + 3.36696i −2.40985 1.39133i 3.52391 + 2.95691i −1.84113 3.18893i
30.3 0.424710 + 1.16688i −0.702528 0.255699i 0.350853 0.294401i −2.65578 + 0.468285i 0.928365i −0.231838 1.31482i 2.64335 + 1.52614i −1.86997 1.56909i −1.67437 2.90009i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 30.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.2.h.a 18
3.b odd 2 1 333.2.bl.d 18
4.b odd 2 1 592.2.bq.d 18
5.b even 2 1 925.2.bb.a 18
5.c odd 4 2 925.2.ba.a 36
37.f even 9 1 1369.2.b.g 18
37.h even 18 1 inner 37.2.h.a 18
37.h even 18 1 1369.2.b.g 18
37.i odd 36 2 1369.2.a.m 18
111.n odd 18 1 333.2.bl.d 18
148.o odd 18 1 592.2.bq.d 18
185.v even 18 1 925.2.bb.a 18
185.y odd 36 2 925.2.ba.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.h.a 18 1.a even 1 1 trivial
37.2.h.a 18 37.h even 18 1 inner
333.2.bl.d 18 3.b odd 2 1
333.2.bl.d 18 111.n odd 18 1
592.2.bq.d 18 4.b odd 2 1
592.2.bq.d 18 148.o odd 18 1
925.2.ba.a 36 5.c odd 4 2
925.2.ba.a 36 185.y odd 36 2
925.2.bb.a 18 5.b even 2 1
925.2.bb.a 18 185.v even 18 1
1369.2.a.m 18 37.i odd 36 2
1369.2.b.g 18 37.f even 9 1
1369.2.b.g 18 37.h even 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} + 9 T^{17} + 42 T^{16} + 135 T^{15} + \cdots + 243$$
$3$ $$T^{18} + 9 T^{17} + 42 T^{16} + 122 T^{15} + \cdots + 64$$
$5$ $$T^{18} + 3 T^{17} - 6 T^{16} + \cdots + 110592$$
$7$ $$T^{18} + 3 T^{17} - 6 T^{16} + \cdots + 36864$$
$11$ $$T^{18} - 9 T^{17} + 78 T^{16} + \cdots + 46656$$
$13$ $$T^{18} - 9 T^{17} + 63 T^{16} - 225 T^{15} + \cdots + 27$$
$17$ $$T^{18} + 15 T^{17} + 93 T^{16} + \cdots + 23705163$$
$19$ $$T^{18} - 6 T^{17} - 33 T^{16} + \cdots + 5614272$$
$23$ $$T^{18} + 9 T^{17} + \cdots + 180910927872$$
$29$ $$T^{18} + 18 T^{17} + 51 T^{16} + \cdots + 30509163$$
$31$ $$T^{18} + 204 T^{16} + \cdots + 1142154432$$
$37$ $$T^{18} + \cdots + 129961739795077$$
$41$ $$T^{18} + 24 T^{17} + 378 T^{16} + \cdots + 6561$$
$43$ $$T^{18} + 414 T^{16} + \cdots + 74566243008$$
$47$ $$T^{18} + 36 T^{17} + \cdots + 2176782336$$
$53$ $$T^{18} + 39 T^{17} + \cdots + 1450162667529$$
$59$ $$T^{18} + 6 T^{17} + \cdots + 1084930414272$$
$61$ $$T^{18} - 42 T^{17} + \cdots + 24685456563$$
$67$ $$T^{18} - 78 T^{16} + \cdots + 12\!\cdots\!16$$
$71$ $$T^{18} + 9 T^{17} + \cdots + 20139015744$$
$73$ $$(T^{9} + 27 T^{8} - 39 T^{7} + \cdots + 8467983)^{2}$$
$79$ $$T^{18} + \cdots + 890793213988032$$
$83$ $$T^{18} + 24 T^{17} + \cdots + 4807480896$$
$89$ $$T^{18} - 18 T^{17} + \cdots + 15282295962363$$
$97$ $$T^{18} + 9 T^{17} + \cdots + 2765473961067$$