# Properties

 Label 90.3.h.a Level $90$ Weight $3$ Character orbit 90.h Analytic conductor $2.452$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,3,Mod(11,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("90.11");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 90.h (of order $$6$$, degree $$2$$, 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: $$2.45232237924$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476$$ x^16 - 12*x^14 + 138*x^12 - 1040*x^10 + 5541*x^8 - 26220*x^6 + 99328*x^4 - 202728*x^2 + 181476 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{13} - 2 \beta_{12} + \cdots - 2 \beta_{2}) q^{9}+O(q^{10})$$ q - b5 * q^2 + (-b13 + b12 - b10 - b4 - b2 - 1) * q^3 + (-2*b9 + 2) * q^4 + b10 * q^5 + (b15 + b12 - b10 - b9 + b8 + b5 - b2 + 1) * q^6 + (b15 - b14 + b13 + 3*b12 - b11 - b8 + b5 - b4 - b3 - b2 - b1 - 1) * q^7 + 2*b12 * q^8 + (-2*b13 - 2*b12 - 3*b10 + b8 + 2*b7 + b6 - b5 + 2*b4 + b3 - 2*b2) * q^9 $$q - \beta_{5} q^{2} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{15} + 4 \beta_{14} + \cdots + 58) q^{99}+O(q^{100})$$ q - b5 * q^2 + (-b13 + b12 - b10 - b4 - b2 - 1) * q^3 + (-2*b9 + 2) * q^4 + b10 * q^5 + (b15 + b12 - b10 - b9 + b8 + b5 - b2 + 1) * q^6 + (b15 - b14 + b13 + 3*b12 - b11 - b8 + b5 - b4 - b3 - b2 - b1 - 1) * q^7 + 2*b12 * q^8 + (-2*b13 - 2*b12 - 3*b10 + b8 + 2*b7 + b6 - b5 + 2*b4 + b3 - 2*b2) * q^9 + b1 * q^10 + (-b15 + b14 - b13 - 4*b12 + 2*b11 + 3*b10 - 3*b9 - b8 + 2*b7 + 2*b6 - 4*b5 + 2*b4 + 4*b2 + 2*b1 + 6) * q^11 + (2*b12 - 2*b2 - 2) * q^12 + (-b15 + b14 + b13 - 3*b12 + b11 + b10 - 4*b9 - 2*b8 + 3*b7 + b6 + 2*b5 + b3 - b2 + b1 + 4) * q^13 + (2*b14 - 2*b13 + b12 + b11 + b10 - 2*b9 + b8 + b7 - b6 + b5 - b4 + b2 - 1) * q^14 + (-b14 - b10 + 3*b9 + b6 - b5 + b3 - b2 + b1 - 3) * q^15 - 4*b9 * q^16 + (-b15 + 3*b13 + b12 + 7*b10 + b9 - 3*b8 - 2*b6 - 2*b5 + b4 - 3*b3 + 5*b2 - b1 + 2) * q^17 + (b15 + b14 - 3*b13 + b11 - 4*b10 + 3*b9 - b7 + b5 - b4 + b3 - 2*b2 - b1 - 2) * q^18 + (-3*b14 - b13 + 6*b12 - 3*b11 - 8*b10 + 5*b9 - 3*b7 - 2*b6 + 6*b5 - 4*b4 - 2*b2 - 3*b1 - 2) * q^19 + (2*b13 + 2*b10) * q^20 + (-5*b15 - b14 - b13 - 7*b12 + b11 + 2*b10 + 3*b9 + 4*b8 - b7 - b6 - 2*b5 + 2*b4 + b3 + b2 + 3*b1 - 4) * q^21 + (b15 + 3*b14 + b13 - 3*b12 - 3*b11 - 3*b9 - b7 - 4*b5 + 3*b4 - b3 + 2*b2 - b1 + 4) * q^22 + (-3*b15 - 2*b14 + 2*b13 - 5*b12 + 2*b11 + 5*b10 + 5*b9 - 4*b8 - b7 + b6 - 2*b5 + b4 + 3*b3 + 5*b2 + 3*b1 + 7) * q^23 + (2*b11 - 4*b9 + 2*b5 + 2*b1 + 2) * q^24 + 5*b9 * q^25 + (-2*b15 + 3*b12 - b10 + 14*b9 + 3*b8 - 3*b7 - b6 + 2*b5 - b4 + b2 + b1 - 8) * q^26 + (5*b15 - 2*b14 - 2*b12 - 4*b11 - 4*b10 - 3*b9 + 3*b8 - 6*b7 - 4*b6 - 2*b5 - 2*b4 - 3*b3 + b2 - 9*b1 - 10) * q^27 + (2*b15 - 4*b14 + 4*b13 + 6*b12 - 4*b11 - 2*b10 + 4*b9 + 2*b8 - 4*b7 - 2*b6 + 6*b5 - 2*b4 - 2*b3 - 4*b2 - 2*b1 - 8) * q^28 + (4*b15 + 5*b14 + 7*b13 - 5*b12 + b11 + 9*b10 - 5*b8 - 2*b7 + b6 - 5*b5 + 4*b4 - 3*b3 + 5*b2 - 5*b1) * q^29 + (-b15 + 2*b14 - 2*b13 - 5*b12 + b11 - 4*b9 + 2*b7 + 2*b6 - 2*b5 + 2*b4 + 2*b3 + 2*b1 + 5) * q^30 + (b15 - 4*b14 + 8*b13 - b11 - b10 + 8*b9 + 5*b8 - 3*b7 + 2*b6 + 10*b5 - 6*b4 + 2*b3 - 2*b2 - b1 - 8) * q^31 + (4*b12 + 4*b5) * q^32 + (5*b15 + 4*b14 + 3*b13 - 5*b12 - 4*b11 - 3*b10 + 3*b9 + 5*b8 - 2*b7 - 2*b6 - 4*b5 + 3*b4 - b3 - b2 + 3*b1 - 20) * q^33 + (-b14 + 5*b13 - 3*b12 - b11 + 7*b10 - 5*b9 - b8 - 2*b7 - 5*b6 - b5 - 2*b4 - 3*b3 + 7*b2 - 3*b1) * q^34 + (-2*b15 - 3*b14 + 2*b13 - b12 + 3*b11 + 2*b10 + 2*b9 - 3*b7 - b6 + 5*b5 - 4*b4 + b2 + b1 - 2) * q^35 + (2*b15 - 2*b14 - 2*b13 - 4*b12 + 2*b11 + 2*b6 - 2*b5 + 2*b4 + 4) * q^36 + (-b15 + 2*b14 - 9*b13 + 15*b12 + 2*b11 - 13*b10 + 3*b9 - b8 + 2*b7 - 4*b6 + 18*b5 - 9*b4 + b3 - 3*b2 - 5*b1 - 12) * q^37 + (b15 - b14 - 5*b13 + b12 + b11 - 3*b10 + 3*b9 + b8 + 4*b7 + b6 - 4*b5 - 2*b4 + 3*b3 - b2 - 5*b1 - 6) * q^38 + (-5*b15 - b14 - 3*b13 - 6*b12 - 8*b11 - 3*b10 + 3*b9 + b8 - 4*b7 + 2*b6 - 8*b5 + 6*b4 - 2*b3 - 2) * q^39 + (-2*b15 + 2*b1) * q^40 + (5*b15 + 7*b14 - 7*b13 - 9*b12 + 5*b11 + 2*b10 - 6*b9 + 2*b8 + 7*b7 + 7*b6 - 4*b5 + 14*b4 + 5*b3 + 3*b2 + 15*b1 + 20) * q^41 + (-2*b14 + 6*b13 - 4*b12 - 2*b11 - 7*b10 + 4*b9 - 3*b8 + b7 + b6 - b5 + 3*b4 + 2*b3 - 5*b2 + 2*b1 + 4) * q^42 + (b14 - 8*b13 + 18*b12 - 5*b11 + 2*b10 + 15*b9 + b8 - b7 - b6 + 7*b5 - 7*b4 - 6*b3 - 4*b2 - 6*b1 - 6) * q^43 + (-2*b14 + 2*b13 + 8*b12 + 2*b11 + 4*b10 + 2*b9 - 4*b8 - 2*b7 + 4*b5 - 8*b4 - 4*b3 + 4*b2 - 2*b1) * q^44 + (b15 + 3*b14 - 6*b13 + b12 + b11 - 3*b10 - 6*b9 + 3*b8 + b7 + 2*b6 + 3*b5 + 2*b4 + 2*b3 + 3*b2 + b1 + 1) * q^45 + (-5*b15 + b14 - b13 - 18*b12 + b11 + 8*b10 - 4*b9 - 2*b8 + 4*b7 + 5*b6 - 18*b5 + 8*b4 + 5*b3 + 10*b2 + 5*b1 + 13) * q^46 + (6*b15 + 3*b14 - 6*b13 + 2*b12 + 3*b11 - 5*b10 - 12*b9 - 3*b8 + b7 - 2*b6 + 6*b5 - 2*b3 - 2*b2 - 10*b1 + 19) * q^47 + (4*b13 + 4*b10 + 4*b4) * q^48 + (-3*b15 + 2*b14 + 10*b13 + 15*b12 + 2*b11 + 2*b10 + 6*b9 - 4*b8 + 10*b7 - b6 - 13*b5 + 3*b4 + 6*b3 - 13*b2 + 3*b1 - 13) * q^49 + (-5*b12 - 5*b5) * q^50 + (-5*b15 - 4*b14 + 12*b13 - 16*b12 + b11 + 15*b10 - 6*b9 + b8 - b7 - 4*b6 - 14*b5 - 2*b3 + 4*b2 + 9*b1 + 14) * q^51 + (-2*b14 + 4*b13 - 12*b12 - 2*b11 - 4*b10 - 6*b9 - 2*b8 + 2*b7 + 2*b6 - 8*b5 + 2*b4 - 4*b2) * q^52 + (-16*b15 - 4*b14 + 4*b13 - 4*b12 + 4*b11 + 6*b10 - 40*b9 - 2*b8 + 2*b7 - 2*b6 + 6*b4 + 4*b3 - 2*b2 + 10*b1 + 20) * q^53 + (3*b15 - 3*b14 - 9*b13 + 12*b12 - 2*b10 - 4*b8 + 4*b7 - b6 + 13*b5 - 4*b4 - b3 - 2*b2 - 6*b1 + 8) * q^54 + (3*b14 - 5*b13 + 3*b11 - b10 - 5*b9 + 3*b8 + 6*b7 - b6 + b4 - b2 + 3*b1 + 4) * q^55 + (-2*b15 + 2*b14 - 2*b13 + 2*b12 + 4*b11 + 2*b10 - 2*b8 + 6*b7 + 2*b5 - 2*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^56 + (-13*b15 + b14 - 5*b13 + 15*b12 + 5*b11 + 7*b10 - 24*b9 - 4*b8 + 7*b7 - 5*b6 + 8*b5 - 8*b4 - b3 + 3*b2 - 3*b1 + 14) * q^57 + (-4*b15 - 4*b13 - 9*b12 + 6*b11 + 12*b10 - 6*b9 - 6*b8 - 2*b7 + b5 - 8*b3 + 16*b2 + 4*b1 + 14) * q^58 + (16*b15 - 6*b14 + 6*b13 + 10*b12 - 6*b11 - 12*b10 + 14*b9 - 8*b7 + 2*b5 - 10*b4 - 8*b3 - 16*b2 - 10) * q^59 + (2*b12 - 2*b10 + 6*b9 - 2*b7 + 2*b6 + 2*b1) * q^60 + (4*b15 + b14 - 15*b13 + 18*b12 + 13*b11 - 5*b10 - 5*b9 + b8 - 5*b7 + 7*b6 + 18*b5 - 9*b4 + 8*b3 - 6*b2 + 8*b1 - 4) * q^61 + (-10*b15 + 6*b14 - 12*b13 - 4*b12 - 6*b11 - 17*b10 + 16*b9 + 3*b8 + 9*b7 + 7*b6 - 8*b5 + 7*b4 + 6*b3 - 13*b2 + 8*b1 - 10) * q^62 + (15*b15 + 6*b14 + 8*b13 - 4*b12 - 3*b10 - 6*b9 - 7*b8 + 4*b7 + 5*b6 - 29*b5 + b4 - b3 + 5*b2 - 9*b1 + 3) * q^63 - 8 * q^64 + (4*b15 - 4*b14 + 5*b13 - b12 - 2*b11 + 3*b10 - 6*b9 + 4*b8 - 3*b7 + b5 + b4 - b3 + b2 - 7*b1 + 14) * q^65 + (-2*b15 - 7*b14 + 15*b13 + 11*b12 + b11 + 15*b10 + 9*b9 - 5*b8 - 4*b7 - b6 + 25*b5 - 12*b4 - 5*b3 - 5*b2 - 3*b1 - 4) * q^66 + (12*b15 - 3*b14 - 3*b13 + 15*b12 - 3*b11 - 6*b10 - b9 + 6*b8 - 6*b7 - 9*b5 - 3*b4 - 12*b1 + 1) * q^67 + (-2*b15 - 6*b14 + 6*b13 + 10*b12 - 6*b11 + 6*b10 + 8*b9 - 2*b7 - 6*b6 + 8*b5 - 4*b4 - 2*b3 + 2*b2 - 6*b1 - 4) * q^68 + (-5*b15 + 5*b14 - 15*b13 + 26*b12 + b11 + 6*b10 - 33*b9 - 2*b8 + 8*b7 + 5*b6 + 7*b5 - 3*b4 + 4*b3 + 4*b2 + 12*b1 + 17) * q^69 + (-2*b15 + b14 + b13 - 9*b12 - 2*b11 - b10 + 2*b9 + b8 + 5*b7 + 2*b6 - 8*b5 + 11*b4 + 3*b3 - b2 + 3*b1 + 6) * q^70 + (-12*b15 + 9*b14 - 21*b13 - 9*b11 - 24*b10 + 9*b9 + 15*b7 + 12*b6 - 12*b5 + 6*b4 + 6*b3 - 18*b2 + 9*b1 - 6) * q^71 + (2*b15 + 6*b14 - 6*b13 - 10*b12 + 2*b11 + 2*b10 - 6*b9 - 2*b8 + 4*b7 + 2*b6 - 10*b5 + 8*b4 + 2*b3 + 2*b2 + 2*b1 + 12) * q^72 + (8*b15 + 8*b14 - 6*b13 - 18*b12 + 8*b11 - 4*b10 - 24*b9 + 8*b8 + 8*b7 + 8*b6 - 12*b5 + 24*b4 - 8*b3 + 10*b1 + 54) * q^73 + (8*b15 - 11*b14 - b13 + 9*b12 - 7*b11 - 13*b10 + 21*b9 + 11*b8 - 8*b7 - 5*b6 + 15*b5 - 4*b4 - b3 - 9*b2 - 15*b1 - 40) * q^74 + (-5*b13 - 5*b10 - 5*b4) * q^75 + (2*b15 - 16*b13 - 6*b11 - 18*b10 - 4*b9 + 6*b8 - 2*b7 + 4*b5 + 4*b3 - 8*b2 - 2*b1) * q^76 + (17*b15 - b14 + b13 + 13*b12 - 5*b11 - 17*b10 - 22*b9 + 4*b8 - 3*b7 - 7*b6 + 6*b5 - 12*b4 - 7*b3 - 7*b2 + 3*b1 - 42) * q^77 + (5*b15 + 7*b14 - 3*b13 + b12 + 9*b11 - 6*b10 - 5*b9 - 10*b8 + 7*b7 + 4*b6 - 9*b4 - b3 + 8*b2 + b1 + 14) * q^78 + (-4*b15 + 8*b14 - 6*b13 - 54*b12 + 2*b11 + 2*b10 - 2*b9 + 8*b8 + 8*b7 + 2*b6 - 30*b5 + 24*b4 + 10*b3 + 6*b2 + 10*b1 + 16) * q^79 + 4*b13 * q^80 + (2*b15 - 11*b14 + 27*b13 + 34*b12 - b11 + 20*b10 + 30*b9 - 3*b7 - 16*b6 + 25*b5 - 11*b4 - 3*b3 - 2*b2 - 18*b1 - 7) * q^81 + (2*b15 + 2*b14 + 18*b13 - 9*b12 + 2*b11 + 29*b10 - 7*b8 - 7*b7 + 5*b6 - 12*b5 + 3*b4 - 2*b3 + 3*b2 + 4*b1 - 8) * q^82 + (15*b15 - 21*b14 + 9*b13 + 25*b12 - 21*b11 - 22*b10 + 12*b9 + 21*b8 - 16*b7 - 10*b6 + 21*b5 - 15*b4 + 5*b3 - 25*b2 - 35*b1 - 13) * q^83 + (-8*b15 + 2*b14 - 2*b13 - 8*b12 + 10*b11 + 10*b10 + 6*b9 - 2*b8 + 2*b7 + 2*b6 - 8*b5 - 2*b4 + 4*b3 + 8*b2 + 6) * q^84 + (-3*b15 - 2*b14 + 8*b13 - 3*b12 + 4*b11 + 13*b10 + 3*b9 - 2*b8 - 4*b7 - 5*b6 + b5 + 3*b4 - 6*b3 + 7*b2 + 3*b1 - 2) * q^85 + (14*b15 + 3*b14 - 3*b13 + 4*b12 - 3*b11 - 18*b10 - 5*b9 + 6*b8 - b7 - 6*b6 - 3*b5 - 11*b4 - 7*b3 - 8*b2 - 20) * q^86 + (4*b15 - 16*b14 + 15*b13 + 20*b12 - 2*b11 + 30*b10 + 54*b9 - 5*b8 - 13*b7 - 7*b6 + 40*b5 - 12*b4 - 14*b3 + 7*b2 - 4) * q^87 + (2*b15 + 6*b14 - 2*b13 - 6*b12 - 6*b11 - 2*b10 - 10*b9 + 6*b8 + 4*b7 - 2*b6 - 8*b5 + 8*b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^88 + (-14*b15 + 8*b14 + 10*b13 - 42*b12 - 8*b11 - 2*b10 - 48*b9 + 10*b8 - 4*b7 - 4*b6 - 8*b5 + 10*b4 - 2*b3 + 6*b2 + 6*b1 + 25) * q^89 + (4*b15 - b14 - b13 + 7*b12 - 5*b11 - 5*b10 + 3*b9 - b8 - 2*b7 + 3*b6 + 6*b5 - b3 - 5*b2 - 3*b1 - 14) * q^90 + (-3*b14 + 13*b13 - 24*b12 - 3*b11 + 8*b10 + 13*b9 - 12*b8 - 15*b7 + 2*b6 - 54*b5 - 8*b4 + 2*b2 - 15*b1 - 52) * q^91 + (-4*b15 + 4*b14 - 4*b13 - 8*b12 - 4*b11 - 2*b10 - 6*b9 - 4*b8 + 6*b6 - 10*b5 + 2*b4 + 4*b3 + 8*b2 + 4*b1 + 22) * q^92 + (-4*b15 + 19*b14 - 12*b13 + 32*b12 - b11 - 30*b10 + 51*b9 - b8 - 5*b7 + 7*b6 + 8*b5 - 3*b4 + 2*b3 + 10*b2 - 42) * q^93 + (8*b15 - 5*b14 - 11*b13 + 9*b12 + 4*b11 + b10 + 9*b9 + b8 - 6*b7 - 5*b6 - 4*b5 - 5*b3 + 5*b2 - 8*b1 - 9) * q^94 + (-4*b15 - 7*b14 + 7*b13 + 7*b12 - 2*b11 + 9*b10 - 7*b9 - 5*b8 - 6*b7 - 4*b6 + 6*b5 - 9*b4 - b3 + 2*b2 - 6*b1 - 24) * q^95 + (-4*b15 - 4*b12 + 4*b11 + 4*b10 - 4*b9 - 4*b8 + 4*b2 + 4*b1) * q^96 + (-14*b15 - 6*b14 - 4*b13 - 18*b12 + 18*b11 + 20*b10 - 4*b9 - 6*b8 - 4*b7 + 2*b6 + 2*b5 + 4*b4 + 14*b3 + 10*b2 + 14*b1 + 8) * q^97 + (-16*b15 - 10*b14 + 4*b13 - 4*b12 + 10*b11 + 3*b10 - 40*b9 + 7*b8 - 13*b7 - 5*b6 + 18*b5 - 9*b4 + 4*b3 + b2 + 10*b1 + 14) * q^98 + (-b15 + 4*b14 + 18*b13 - 26*b12 + 5*b11 + 15*b10 + 6*b9 - 7*b8 + b7 + 10*b6 + 26*b5 + 6*b4 + 8*b3 + 8*b2 - 3*b1 + 58) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{3} + 16 q^{4} + 16 q^{6} - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 16 * q - 4 * q^3 + 16 * q^4 + 16 * q^6 - 4 * q^7 - 4 * q^9 $$16 q - 4 q^{3} + 16 q^{4} + 16 q^{6} - 4 q^{7} - 4 q^{9} - 16 q^{12} + 20 q^{13} - 36 q^{14} - 20 q^{15} - 32 q^{16} + 16 q^{18} + 80 q^{19} - 44 q^{21} + 24 q^{22} + 108 q^{23} - 8 q^{24} + 40 q^{25} - 124 q^{27} - 16 q^{28} - 72 q^{29} + 20 q^{30} - 16 q^{31} - 264 q^{33} - 48 q^{34} + 32 q^{36} - 88 q^{37} - 72 q^{38} - 8 q^{39} + 108 q^{41} + 128 q^{42} + 92 q^{43} - 80 q^{45} + 24 q^{46} + 216 q^{47} - 16 q^{48} - 84 q^{49} + 168 q^{51} - 40 q^{52} + 148 q^{54} - 72 q^{56} + 28 q^{57} + 144 q^{59} + 40 q^{60} - 76 q^{61} - 104 q^{63} - 128 q^{64} + 180 q^{65} + 96 q^{66} + 56 q^{67} + 72 q^{68} - 72 q^{69} + 60 q^{70} + 64 q^{72} + 416 q^{73} - 288 q^{74} + 20 q^{75} + 80 q^{76} - 684 q^{77} + 56 q^{78} + 80 q^{79} + 320 q^{81} - 192 q^{82} + 396 q^{83} + 40 q^{84} - 60 q^{85} - 216 q^{86} + 420 q^{87} - 48 q^{88} - 160 q^{90} - 656 q^{91} + 216 q^{92} - 356 q^{93} - 84 q^{94} - 360 q^{95} - 80 q^{96} - 16 q^{97} + 816 q^{99}+O(q^{100})$$ 16 * q - 4 * q^3 + 16 * q^4 + 16 * q^6 - 4 * q^7 - 4 * q^9 - 16 * q^12 + 20 * q^13 - 36 * q^14 - 20 * q^15 - 32 * q^16 + 16 * q^18 + 80 * q^19 - 44 * q^21 + 24 * q^22 + 108 * q^23 - 8 * q^24 + 40 * q^25 - 124 * q^27 - 16 * q^28 - 72 * q^29 + 20 * q^30 - 16 * q^31 - 264 * q^33 - 48 * q^34 + 32 * q^36 - 88 * q^37 - 72 * q^38 - 8 * q^39 + 108 * q^41 + 128 * q^42 + 92 * q^43 - 80 * q^45 + 24 * q^46 + 216 * q^47 - 16 * q^48 - 84 * q^49 + 168 * q^51 - 40 * q^52 + 148 * q^54 - 72 * q^56 + 28 * q^57 + 144 * q^59 + 40 * q^60 - 76 * q^61 - 104 * q^63 - 128 * q^64 + 180 * q^65 + 96 * q^66 + 56 * q^67 + 72 * q^68 - 72 * q^69 + 60 * q^70 + 64 * q^72 + 416 * q^73 - 288 * q^74 + 20 * q^75 + 80 * q^76 - 684 * q^77 + 56 * q^78 + 80 * q^79 + 320 * q^81 - 192 * q^82 + 396 * q^83 + 40 * q^84 - 60 * q^85 - 216 * q^86 + 420 * q^87 - 48 * q^88 - 160 * q^90 - 656 * q^91 + 216 * q^92 - 356 * q^93 - 84 * q^94 - 360 * q^95 - 80 * q^96 - 16 * q^97 + 816 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476$$ :

 $$\beta_{1}$$ $$=$$ $$( 41 \nu^{14} - 225 \nu^{12} + 1333 \nu^{10} + 31 \nu^{8} - 114072 \nu^{6} + 798716 \nu^{4} + \cdots + 15616332 ) / 4341600$$ (41*v^14 - 225*v^12 + 1333*v^10 + 31*v^8 - 114072*v^6 + 798716*v^4 - 1960260*v^2 + 15616332) / 4341600 $$\beta_{2}$$ $$=$$ $$( 289974 \nu^{15} - 348681 \nu^{14} + 3618395 \nu^{13} - 9875745 \nu^{12} - 23841698 \nu^{11} + \cdots - 355528882692 ) / 93092587200$$ (289974*v^15 - 348681*v^14 + 3618395*v^13 - 9875745*v^12 - 23841698*v^11 + 28678107*v^10 + 428493699*v^9 - 977371161*v^8 - 3224449208*v^7 + 4536751752*v^6 + 12138565484*v^5 - 25359169116*v^4 - 82255565400*v^3 + 146208299220*v^2 + 280246144428*v - 355528882692) / 93092587200 $$\beta_{3}$$ $$=$$ $$( - 399973 \nu^{15} - 1285313 \nu^{14} + 10390855 \nu^{13} + 16249060 \nu^{12} + \cdots + 398535722064 ) / 93092587200$$ (-399973*v^15 - 1285313*v^14 + 10390855*v^13 + 16249060*v^12 - 112973589*v^11 - 166716449*v^10 + 955405567*v^9 + 1377483212*v^8 - 6286525684*v^7 - 8086497004*v^6 + 26772305892*v^5 + 39887171792*v^4 - 124475395860*v^3 - 196264014900*v^2 + 460060749324*v + 398535722064) / 93092587200 $$\beta_{4}$$ $$=$$ $$( 334523 \nu^{15} + 505875 \nu^{14} - 4891765 \nu^{13} - 10514390 \nu^{12} + 54769019 \nu^{11} + \cdots - 217467572760 ) / 31030862400$$ (334523*v^15 + 505875*v^14 - 4891765*v^13 - 10514390*v^12 + 54769019*v^11 + 115051595*v^10 - 450373037*v^9 - 1006107630*v^8 + 2453681884*v^7 + 5860915100*v^6 - 11752459772*v^5 - 24442809320*v^4 + 43393300140*v^3 + 99082262220*v^2 - 99395821764*v - 217467572760) / 31030862400 $$\beta_{5}$$ $$=$$ $$( 72336 \nu^{15} - 137953 \nu^{14} - 727168 \nu^{13} + 1003727 \nu^{12} + 8534536 \nu^{11} + \cdots + 238211532 ) / 3723703488$$ (72336*v^15 - 137953*v^14 - 727168*v^13 + 1003727*v^12 + 8534536*v^11 - 13552693*v^10 - 57472056*v^9 + 78257407*v^8 + 281714968*v^7 - 359476976*v^6 - 1256883688*v^5 + 1821417244*v^4 + 4360916928*v^3 - 4552848588*v^2 - 5137962768*v + 238211532) / 3723703488 $$\beta_{6}$$ $$=$$ $$( - 389063 \nu^{15} - 2906172 \nu^{14} + 3910476 \nu^{13} + 32198997 \nu^{12} + \cdots + 254978122644 ) / 18618517440$$ (-389063*v^15 - 2906172*v^14 + 3910476*v^13 + 32198997*v^12 - 41355367*v^11 - 368265072*v^10 + 289966484*v^9 + 2652848757*v^8 - 1232178324*v^7 - 13072707156*v^6 + 5887829920*v^5 + 58963177164*v^4 - 24253950828*v^3 - 209378180376*v^2 + 8327803248*v + 254978122644) / 18618517440 $$\beta_{7}$$ $$=$$ $$( 1043836 \nu^{15} + 64681 \nu^{14} - 7299935 \nu^{13} + 2483580 \nu^{12} + 85117948 \nu^{11} + \cdots - 1432070568 ) / 46546293600$$ (1043836*v^15 + 64681*v^14 - 7299935*v^13 + 2483580*v^12 + 85117948*v^11 - 12112387*v^10 - 426869119*v^9 + 289135856*v^8 + 1167814088*v^7 - 777599952*v^6 - 6927687844*v^5 + 6260614996*v^4 + 8263330320*v^3 - 8681432700*v^2 + 67820635332*v - 1432070568) / 46546293600 $$\beta_{8}$$ $$=$$ $$( - 2122022 \nu^{15} - 3591180 \nu^{14} + 26082035 \nu^{13} + 40328710 \nu^{12} + \cdots + 527986862280 ) / 93092587200$$ (-2122022*v^15 - 3591180*v^14 + 26082035*v^13 + 40328710*v^12 - 288538566*v^11 - 497484220*v^10 + 2261008643*v^9 + 3418102590*v^8 - 11341843376*v^7 - 19909524640*v^6 + 52045731108*v^5 + 79293396400*v^4 - 196982993160*v^3 - 326286554880*v^2 + 246821929596*v + 527986862280) / 93092587200 $$\beta_{9}$$ $$=$$ $$( 215 \nu^{15} - 2296 \nu^{13} + 25907 \nu^{11} - 183272 \nu^{9} + 859816 \nu^{7} - 4023896 \nu^{5} + \cdots + 4110048 ) / 8220096$$ (215*v^15 - 2296*v^13 + 25907*v^11 - 183272*v^9 + 859816*v^7 - 4023896*v^5 + 13619076*v^3 - 16167456*v + 4110048) / 8220096 $$\beta_{10}$$ $$=$$ $$( 881988 \nu^{15} + 2003904 \nu^{14} - 8761925 \nu^{13} - 15591600 \nu^{12} + 105064844 \nu^{11} + \cdots + 1201238208 ) / 31030862400$$ (881988*v^15 + 2003904*v^14 - 8761925*v^13 - 15591600*v^12 + 105064844*v^11 + 197558352*v^10 - 710437917*v^9 - 1105357536*v^8 + 3568588304*v^7 + 4671785232*v^6 - 16500347012*v^5 - 22447088496*v^4 + 57910093920*v^3 + 55660376160*v^2 - 68432605524*v + 1201238208) / 31030862400 $$\beta_{11}$$ $$=$$ $$( - 2939307 \nu^{15} + 1272107 \nu^{14} + 15171655 \nu^{13} - 28863275 \nu^{12} + \cdots - 476058785436 ) / 93092587200$$ (-2939307*v^15 + 1272107*v^14 + 15171655*v^13 - 28863275*v^12 - 240053431*v^11 + 280160391*v^10 + 985073223*v^9 - 2776174763*v^8 - 3725084656*v^7 + 14389821056*v^6 + 18339758908*v^5 - 69129029868*v^4 - 13131190020*v^3 + 217571972580*v^2 - 258558675444*v - 476058785436) / 93092587200 $$\beta_{12}$$ $$=$$ $$( - 9042 \nu^{15} + 90896 \nu^{13} - 1066817 \nu^{11} + 7184007 \nu^{9} - 35214371 \nu^{7} + \cdots + 642245346 \nu ) / 232731468$$ (-9042*v^15 + 90896*v^13 - 1066817*v^11 + 7184007*v^9 - 35214371*v^7 + 157110461*v^5 - 545114616*v^3 + 642245346*v) / 232731468 $$\beta_{13}$$ $$=$$ $$( - 13164 \nu^{15} + 130775 \nu^{13} - 1568132 \nu^{11} + 10603551 \nu^{9} + \cdots + 1021382172 \nu ) / 231573600$$ (-13164*v^15 + 130775*v^13 - 1568132*v^11 + 10603551*v^9 - 53262512*v^7 + 246273836*v^5 - 864329760*v^3 + 1021382172*v) / 231573600 $$\beta_{14}$$ $$=$$ $$( - 6407642 \nu^{15} - 1841030 \nu^{14} + 51299115 \nu^{13} + 26308340 \nu^{12} + \cdots + 264953937600 ) / 93092587200$$ (-6407642*v^15 - 1841030*v^14 + 51299115*v^13 + 26308340*v^12 - 658870066*v^11 - 270226710*v^10 + 3902295683*v^9 + 2197570700*v^8 - 18088262136*v^7 - 11611655240*v^6 + 88561363228*v^5 + 48044765640*v^4 - 244804539000*v^3 - 179193159720*v^2 + 147527614476*v + 264953937600) / 93092587200 $$\beta_{15}$$ $$=$$ $$( 48776 \nu^{15} + 2911 \nu^{14} - 509200 \nu^{13} - 15975 \nu^{12} + 5779688 \nu^{11} + \cdots + 1108759572 ) / 616507200$$ (48776*v^15 + 2911*v^14 - 509200*v^13 - 15975*v^12 + 5779688*v^11 + 94643*v^10 - 41174984*v^9 + 2201*v^8 + 196325008*v^7 - 8099112*v^6 - 944356424*v^5 + 56708836*v^4 + 3231324240*v^3 - 139178460*v^2 - 3847632048*v + 1108759572) / 616507200
 $$\nu$$ $$=$$ $$( - 2 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 4 ) / 6$$ (-2*b15 + 2*b14 - 6*b13 + 2*b11 + b10 - 2*b9 - b8 + 3*b7 + b6 - 2*b5 + b4 + 2*b3 + 3*b2 + 2*b1 + 4) / 6 $$\nu^{2}$$ $$=$$ $$( 2 \beta_{15} + 6 \beta_{14} - 3 \beta_{12} - 6 \beta_{11} - \beta_{10} - 2 \beta_{9} + 3 \beta_{8} + \cdots + 12 ) / 6$$ (2*b15 + 6*b14 - 3*b12 - 6*b11 - b10 - 2*b9 + 3*b8 + b7 - b6 - 2*b5 + 7*b4 - 2*b3 + 3*b2 + b1 + 12) / 6 $$\nu^{3}$$ $$=$$ $$( - 5 \beta_{15} - 4 \beta_{14} - 18 \beta_{13} + 45 \beta_{12} - 4 \beta_{11} - 11 \beta_{10} + \cdots - 38 ) / 6$$ (-5*b15 - 4*b14 - 18*b13 + 45*b12 - 4*b11 - 11*b10 + 34*b9 - b8 - 9*b7 - 11*b6 + 19*b5 - 23*b4 - 4*b3 - 15*b2 - 7*b1 - 38) / 6 $$\nu^{4}$$ $$=$$ $$( - 4 \beta_{15} + 6 \beta_{14} - 18 \beta_{13} - 33 \beta_{12} - 6 \beta_{11} - 37 \beta_{10} + \cdots - 90 ) / 6$$ (-4*b15 + 6*b14 - 18*b13 - 33*b12 - 6*b11 - 37*b10 - 14*b9 + 3*b8 + 13*b7 - b6 - 44*b5 + 31*b4 + 10*b3 - 9*b2 + 55*b1 - 90) / 6 $$\nu^{5}$$ $$=$$ $$( 19 \beta_{15} - 52 \beta_{14} + 144 \beta_{13} - 27 \beta_{12} - 52 \beta_{11} - 47 \beta_{10} + \cdots - 176 ) / 6$$ (19*b15 - 52*b14 + 144*b13 - 27*b12 - 52*b11 - 47*b10 + 82*b9 - 43*b8 - 129*b7 - 47*b6 + 109*b5 - 137*b4 - 34*b3 - 81*b2 - 73*b1 - 176) / 6 $$\nu^{6}$$ $$=$$ $$( - 118 \beta_{15} - 246 \beta_{14} - 90 \beta_{13} + 99 \beta_{12} + 246 \beta_{11} - 109 \beta_{10} + \cdots - 102 ) / 6$$ (-118*b15 - 246*b14 - 90*b13 + 99*b12 + 246*b11 - 109*b10 + 10*b9 - 303*b8 + 61*b7 + 71*b6 - 68*b5 - 323*b4 + 4*b3 - 75*b2 + 127*b1 - 102) / 6 $$\nu^{7}$$ $$=$$ $$( 391 \beta_{15} + 290 \beta_{14} + 396 \beta_{13} - 2367 \beta_{12} + 290 \beta_{11} + 319 \beta_{10} + \cdots + 2230 ) / 6$$ (391*b15 + 290*b14 + 396*b13 - 2367*b12 + 290*b11 + 319*b10 - 3104*b9 - 109*b8 + 399*b7 + 319*b6 - 533*b5 + 529*b4 + 218*b3 + 537*b2 + 107*b1 + 2230) / 6 $$\nu^{8}$$ $$=$$ $$( - 256 \beta_{15} - 714 \beta_{14} + 1170 \beta_{13} + 3039 \beta_{12} + 714 \beta_{11} + 2747 \beta_{10} + \cdots + 5430 ) / 6$$ (-256*b15 - 714*b14 + 1170*b13 + 3039*b12 + 714*b11 + 2747*b10 + 202*b9 - 1545*b8 + 205*b7 + 407*b6 + 4180*b5 - 1949*b4 - 626*b3 + 219*b2 - 1865*b1 + 5430) / 6 $$\nu^{9}$$ $$=$$ $$( - 2417 \beta_{15} + 4028 \beta_{14} - 9216 \beta_{13} - 6435 \beta_{12} + 4028 \beta_{11} + \cdots + 13552 ) / 6$$ (-2417*b15 + 4028*b14 - 9216*b13 - 6435*b12 + 4028*b11 + 2203*b10 - 11558*b9 + 2735*b8 + 8937*b7 + 2203*b6 - 5759*b5 + 7141*b4 + 2174*b3 + 4377*b2 + 5015*b1 + 13552) / 6 $$\nu^{10}$$ $$=$$ $$( 3614 \beta_{15} + 11814 \beta_{14} + 6534 \beta_{13} - 1815 \beta_{12} - 11814 \beta_{11} + 10925 \beta_{10} + \cdots + 24930 ) / 6$$ (3614*b15 + 11814*b14 + 6534*b13 - 1815*b12 - 11814*b11 + 10925*b10 - 4586*b9 + 12171*b8 + 2443*b7 - 2143*b6 + 11656*b5 + 21343*b4 + 2800*b3 - 657*b2 - 8027*b1 + 24930) / 6 $$\nu^{11}$$ $$=$$ $$( - 46367 \beta_{15} - 3310 \beta_{14} - 52128 \beta_{13} + 85239 \beta_{12} - 3310 \beta_{11} + \cdots - 80726 ) / 6$$ (-46367*b15 - 3310*b14 - 52128*b13 + 85239*b12 - 3310*b11 - 5015*b10 + 145540*b9 + 5609*b8 - 867*b7 - 5015*b6 + 6301*b5 - 4421*b4 - 3166*b3 - 8181*b2 + 19961*b1 - 80726) / 6 $$\nu^{12}$$ $$=$$ $$( 22112 \beta_{15} + 81330 \beta_{14} - 73674 \beta_{13} - 220923 \beta_{12} - 81330 \beta_{11} + \cdots - 212358 ) / 6$$ (22112*b15 + 81330*b14 - 73674*b13 - 220923*b12 - 81330*b11 - 168391*b10 - 37106*b9 + 105933*b8 + 16063*b7 - 21043*b6 - 299876*b5 + 180145*b4 + 40666*b3 - 19623*b2 + 75277*b1 - 212358) / 6 $$\nu^{13}$$ $$=$$ $$( 50989 \beta_{15} - 258820 \beta_{14} + 353016 \beta_{13} + 572463 \beta_{12} - 258820 \beta_{11} + \cdots - 856784 ) / 6$$ (50989*b15 - 258820*b14 + 353016*b13 + 572463*b12 - 258820*b11 - 130307*b10 + 771670*b9 - 154783*b8 - 550977*b7 - 130307*b6 + 341539*b5 - 415397*b4 - 137374*b3 - 267681*b2 - 256987*b1 - 856784) / 6 $$\nu^{14}$$ $$=$$ $$( - 150718 \beta_{15} - 451686 \beta_{14} - 517374 \beta_{13} - 380793 \beta_{12} + 451686 \beta_{11} + \cdots - 2222106 ) / 6$$ (-150718*b15 - 451686*b14 - 517374*b13 - 380793*b12 + 451686*b11 - 911653*b10 + 150250*b9 - 571227*b8 - 27155*b7 + 123095*b6 - 1455440*b5 - 871727*b4 - 146696*b3 + 23601*b2 + 640555*b1 - 2222106) / 6 $$\nu^{15}$$ $$=$$ $$( 3029575 \beta_{15} - 660730 \beta_{14} + 3995640 \beta_{13} - 3532959 \beta_{12} - 660730 \beta_{11} + \cdots + 2510854 ) / 6$$ (3029575*b15 - 660730*b14 + 3995640*b13 - 3532959*b12 - 660730*b11 - 292745*b10 - 7275548*b9 - 378145*b8 - 1375605*b7 - 292745*b6 + 796555*b5 - 963635*b4 - 336730*b3 - 629475*b2 - 2075065*b1 + 2510854) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1 - \beta_{9}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 2.11536 − 1.82514i −2.11536 + 0.410927i −2.11536 − 1.82514i 2.11536 + 0.410927i 1.42311 + 1.82514i 1.42311 − 0.410927i −1.42311 − 0.410927i −1.42311 + 1.82514i 2.11536 + 1.82514i −2.11536 − 0.410927i −2.11536 + 1.82514i 2.11536 − 0.410927i 1.42311 − 1.82514i 1.42311 + 0.410927i −1.42311 + 0.410927i −1.42311 − 1.82514i
−1.22474 0.707107i −2.99847 0.0956863i 1.00000 + 1.73205i 1.93649 1.11803i 3.60470 + 2.23743i −4.35057 + 7.53542i 2.82843i 8.98169 + 0.573826i −3.16228
11.2 −1.22474 0.707107i −2.68332 + 1.34156i 1.00000 + 1.73205i −1.93649 + 1.11803i 4.23501 + 0.254321i 6.95799 12.0516i 2.82843i 5.40042 7.19969i 3.16228
11.3 −1.22474 0.707107i −0.129213 2.99722i 1.00000 + 1.73205i 1.93649 1.11803i −1.96110 + 3.76219i 3.31598 5.74345i 2.82843i −8.96661 + 0.774559i −3.16228
11.4 −1.22474 0.707107i 1.13677 + 2.77628i 1.00000 + 1.73205i −1.93649 + 1.11803i 0.570872 4.20406i −3.24916 + 5.62771i 2.82843i −6.41550 + 6.31200i 3.16228
11.5 1.22474 + 0.707107i −2.43483 + 1.75260i 1.00000 + 1.73205i −1.93649 + 1.11803i −4.22132 + 0.424801i −5.39499 + 9.34440i 2.82843i 2.85680 8.53456i −3.16228
11.6 1.22474 + 0.707107i 0.605881 + 2.93818i 1.00000 + 1.73205i 1.93649 1.11803i −1.33556 + 4.02694i 0.531482 0.920554i 2.82843i −8.26582 + 3.56038i 3.16228
11.7 1.22474 + 0.707107i 1.52181 2.58536i 1.00000 + 1.73205i 1.93649 1.11803i 3.69195 2.09033i −0.496891 + 0.860641i 2.82843i −4.36821 7.86884i 3.16228
11.8 1.22474 + 0.707107i 2.98138 + 0.333742i 1.00000 + 1.73205i −1.93649 + 1.11803i 3.41544 + 2.51690i 0.686165 1.18847i 2.82843i 8.77723 + 1.99002i −3.16228
41.1 −1.22474 + 0.707107i −2.99847 + 0.0956863i 1.00000 1.73205i 1.93649 + 1.11803i 3.60470 2.23743i −4.35057 7.53542i 2.82843i 8.98169 0.573826i −3.16228
41.2 −1.22474 + 0.707107i −2.68332 1.34156i 1.00000 1.73205i −1.93649 1.11803i 4.23501 0.254321i 6.95799 + 12.0516i 2.82843i 5.40042 + 7.19969i 3.16228
41.3 −1.22474 + 0.707107i −0.129213 + 2.99722i 1.00000 1.73205i 1.93649 + 1.11803i −1.96110 3.76219i 3.31598 + 5.74345i 2.82843i −8.96661 0.774559i −3.16228
41.4 −1.22474 + 0.707107i 1.13677 2.77628i 1.00000 1.73205i −1.93649 1.11803i 0.570872 + 4.20406i −3.24916 5.62771i 2.82843i −6.41550 6.31200i 3.16228
41.5 1.22474 0.707107i −2.43483 1.75260i 1.00000 1.73205i −1.93649 1.11803i −4.22132 0.424801i −5.39499 9.34440i 2.82843i 2.85680 + 8.53456i −3.16228
41.6 1.22474 0.707107i 0.605881 2.93818i 1.00000 1.73205i 1.93649 + 1.11803i −1.33556 4.02694i 0.531482 + 0.920554i 2.82843i −8.26582 3.56038i 3.16228
41.7 1.22474 0.707107i 1.52181 + 2.58536i 1.00000 1.73205i 1.93649 + 1.11803i 3.69195 + 2.09033i −0.496891 0.860641i 2.82843i −4.36821 + 7.86884i 3.16228
41.8 1.22474 0.707107i 2.98138 0.333742i 1.00000 1.73205i −1.93649 1.11803i 3.41544 2.51690i 0.686165 + 1.18847i 2.82843i 8.77723 1.99002i −3.16228
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.h.a 16
3.b odd 2 1 270.3.h.a 16
4.b odd 2 1 720.3.bs.d 16
5.b even 2 1 450.3.i.g 16
5.c odd 4 2 450.3.k.c 32
9.c even 3 1 270.3.h.a 16
9.c even 3 1 810.3.d.c 16
9.d odd 6 1 inner 90.3.h.a 16
9.d odd 6 1 810.3.d.c 16
12.b even 2 1 2160.3.bs.d 16
15.d odd 2 1 1350.3.i.g 16
15.e even 4 2 1350.3.k.b 32
36.f odd 6 1 2160.3.bs.d 16
36.h even 6 1 720.3.bs.d 16
45.h odd 6 1 450.3.i.g 16
45.j even 6 1 1350.3.i.g 16
45.k odd 12 2 1350.3.k.b 32
45.l even 12 2 450.3.k.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.h.a 16 1.a even 1 1 trivial
90.3.h.a 16 9.d odd 6 1 inner
270.3.h.a 16 3.b odd 2 1
270.3.h.a 16 9.c even 3 1
450.3.i.g 16 5.b even 2 1
450.3.i.g 16 45.h odd 6 1
450.3.k.c 32 5.c odd 4 2
450.3.k.c 32 45.l even 12 2
720.3.bs.d 16 4.b odd 2 1
720.3.bs.d 16 36.h even 6 1
810.3.d.c 16 9.c even 3 1
810.3.d.c 16 9.d odd 6 1
1350.3.i.g 16 15.d odd 2 1
1350.3.i.g 16 45.j even 6 1
1350.3.k.b 32 15.e even 4 2
1350.3.k.b 32 45.k odd 12 2
2160.3.bs.d 16 12.b even 2 1
2160.3.bs.d 16 36.f odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{2} + 4)^{4}$$
$3$ $$T^{16} + 4 T^{15} + \cdots + 43046721$$
$5$ $$(T^{4} - 5 T^{2} + 25)^{4}$$
$7$ $$T^{16} + \cdots + 6662640625$$
$11$ $$T^{16} + \cdots + 7901855928576$$
$13$ $$T^{16} + \cdots + 991374241321216$$
$17$ $$T^{16} + \cdots + 29\!\cdots\!00$$
$19$ $$(T^{8} - 40 T^{7} + \cdots + 4403341840)^{2}$$
$23$ $$T^{16} + \cdots + 15\!\cdots\!61$$
$29$ $$T^{16} + \cdots + 35\!\cdots\!61$$
$31$ $$T^{16} + \cdots + 12\!\cdots\!56$$
$37$ $$(T^{8} + 44 T^{7} + \cdots - 221671664)^{2}$$
$41$ $$T^{16} + \cdots + 23\!\cdots\!81$$
$43$ $$T^{16} + \cdots + 82\!\cdots\!00$$
$47$ $$T^{16} + \cdots + 64\!\cdots\!41$$
$53$ $$T^{16} + \cdots + 19\!\cdots\!76$$
$59$ $$T^{16} + \cdots + 51\!\cdots\!00$$
$61$ $$T^{16} + \cdots + 20\!\cdots\!61$$
$67$ $$T^{16} + \cdots + 52\!\cdots\!25$$
$71$ $$T^{16} + \cdots + 19\!\cdots\!76$$
$73$ $$(T^{8} + \cdots - 638114737319936)^{2}$$
$79$ $$T^{16} + \cdots + 22\!\cdots\!00$$
$83$ $$T^{16} + \cdots + 11\!\cdots\!21$$
$89$ $$T^{16} + \cdots + 95\!\cdots\!25$$
$97$ $$T^{16} + \cdots + 10\!\cdots\!56$$