# Properties

 Label 90.3.j.b Level $90$ Weight $3$ Character orbit 90.j Analytic conductor $2.452$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,3,Mod(29,90)]

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, 3]))

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

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

chi := DirichletCharacter("90.29");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 90.j (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: $$x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721$$ x^16 + 2*x^14 - 230*x^12 - 96*x^10 + 25551*x^8 - 7776*x^6 - 1509030*x^4 + 1062882*x^2 + 43046721 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} + \beta_{13} q^{3} + 2 \beta_{6} q^{4} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 2) q^{5}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{12} + \cdots - 1) q^{9}+O(q^{10})$$ q - b7 * q^2 + b13 * q^3 + 2*b6 * q^4 + (-b8 + b6 + b5 - b1 + 2) * q^5 + (b4 - 1) * q^6 + (b11 + b10 - 2*b7 - b5 + b3 - b2 - b1) * q^7 + (2*b7 + 2*b5) * q^8 + (-b15 + b14 - b12 + b11 - b10 + b6 - 2*b4 + 2*b3 + 2*b2 - 1) * q^9 $$q - \beta_{7} q^{2} + \beta_{13} q^{3} + 2 \beta_{6} q^{4} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots + 2) q^{5}+ \cdots + (2 \beta_{15} - 2 \beta_{14} + \cdots + 34) q^{99}+O(q^{100})$$ q - b7 * q^2 + b13 * q^3 + 2*b6 * q^4 + (-b8 + b6 + b5 - b1 + 2) * q^5 + (b4 - 1) * q^6 + (b11 + b10 - 2*b7 - b5 + b3 - b2 - b1) * q^7 + (2*b7 + 2*b5) * q^8 + (-b15 + b14 - b12 + b11 - b10 + b6 - 2*b4 + 2*b3 + 2*b2 - 1) * q^9 + (-b14 - b13 + b12 - b7 - b6 + b5 + b1 - 2) * q^10 + (-2*b12 + b11 - b10 - b9 - b8 - 2*b6 + 2*b4 - b3 - b2 + 2) * q^11 - 2*b1 * q^12 + (-2*b13 + b9 - b8 - 6*b7 - 2*b5 + 2*b1) * q^13 + (b12 + b11 - b10 + b9 + b8 + b6 - b4 + 2) * q^14 + (-b15 + 3*b13 + b10 - 2*b9 + b7 + 2*b6 - 3*b5 - 6) * q^15 + (-4*b6 - 4) * q^16 + (-2*b13 + 12*b7 + 10*b5 - 2*b3 + 2*b2 + 6*b1) * q^17 + (-b15 - b14 - 2*b11 - 2*b10 + 4*b7 - b6 - b5 - 2*b3 + 2*b2 + 2*b1 - 1) * q^18 + (2*b15 - 2*b14 + 4*b12 - 2*b11 + 2*b10 + 2*b9 + 2*b8 - 2*b6 - 2*b4 + 10) * q^19 + (-2*b13 + 2*b6 - 2*b2 - 2) * q^20 + (-b15 + b14 + b12 - 2*b11 + 2*b10 + 3*b9 + 3*b8 - 4*b6 - b4 + 2*b3 + 2*b2 - 4) * q^21 + (b15 + b14 + 6*b13 + b11 + b10 - 3*b9 + 3*b8 - 8*b7 + b6 - 6*b5 - 4*b1 + 1) * q^22 + (3*b15 + 3*b14 + 5*b13 - 3*b11 - 3*b10 - b9 + b8 + b7 + 3*b6 + 3*b5 - b3 + b2 - 4*b1 + 3) * q^23 + (2*b12 - 2*b6 - 2*b4) * q^24 + (b15 - b14 - 5*b13 - b12 + b11 + 3*b10 - b9 - 3*b8 - 4*b7 - b6 - 4*b5 + 2*b4 - b3 - 2*b2 - 1) * q^25 + (b15 - b14 - 2*b12 + 15*b6 + 2*b4 + 7) * q^26 + (-2*b15 - 2*b14 - 7*b13 - b11 - b10 + 4*b9 - 4*b8 - 3*b7 - 2*b6 - 7*b5 + 4*b3 - 4*b2 - b1 - 2) * q^27 + (2*b15 + 2*b14 + 2*b13 - 2*b9 + 2*b8 - 2*b7 + 2*b6 - 2*b1 + 2) * q^28 + (-4*b15 + 4*b14 - 8*b12 + 2*b11 - 2*b10 - 2*b9 - 2*b8 + b6 - 4*b4 + 7) * q^29 + (-2*b15 - b14 - 3*b13 + 2*b12 + 2*b9 + 10*b7 - 6*b6 + 4*b5 + b4 - 2*b3 + 3*b1) * q^30 + (b15 - b14 + 2*b12 - b11 + b10 - 2*b9 - 2*b8 - 9*b6 + 2*b4 - 6*b3 - 6*b2 - 1) * q^31 - 4*b5 * q^32 + (2*b15 + 2*b14 + 8*b13 - 2*b11 - 2*b10 - 2*b9 + 2*b8 + 8*b7 + 2*b6 - 12*b5 - 3*b3 + 3*b2 + 2*b1 + 2) * q^33 + (2*b15 - 2*b14 - 2*b12 - 2*b11 + 2*b10 + 2*b9 + 2*b8 - 16*b6 + 4*b4 - 16) * q^34 + (4*b15 - b14 + 3*b13 - b12 + b11 + 5*b10 + 3*b8 - 2*b7 - 5*b6 + 4*b5 + 7*b4 + 4*b3 - 5*b2 - 9*b1 - 3) * q^35 + (-2*b11 + 2*b10 - 2*b9 - 2*b8 - 4*b6 + 2*b4 - 2*b3 - 2*b2) * q^36 + (-2*b15 - 2*b14 - 6*b13 - 2*b9 + 2*b8 + 2*b7 - 2*b6 - 4*b5 + 6*b1 - 2) * q^37 + (-8*b13 + 4*b9 - 4*b8 - 10*b7 + 4*b5 + 4*b3 - 4*b2 + 4*b1) * q^38 + (b15 - b14 - 2*b12 + b11 - b10 + 2*b9 + 2*b8 + 15*b6 + 6*b4 - 11) * q^39 + (2*b14 + 2*b13 - 2*b12 + 2*b11 - 2*b9 + 4*b7 + 2*b5 - 2*b4 + 2) * q^40 + (3*b15 - 3*b14 + 5*b12 + 2*b11 - 2*b10 - 9*b6 - 5*b4 - 15) * q^41 + (-b15 - b14 - 4*b13 - 2*b11 - 2*b10 + 6*b9 - 6*b8 + 7*b7 - b6 + 2*b5 - 2*b3 + 2*b2 - 1) * q^42 + (-2*b13 - 4*b11 - 4*b10 + 2*b9 - 2*b8 + 7*b7 + 6*b5 - 3*b3 + 3*b2) * q^43 + (-2*b15 + 2*b14 + 4*b12 + 2*b9 + 2*b8 + 10*b6 - 4*b4 + 2*b3 + 2*b2 + 6) * q^44 + (-5*b15 + 3*b14 - 2*b13 - 3*b12 + 2*b11 - 2*b9 + 5*b8 - 10*b7 - 4*b6 - 9*b5 - 7*b4 + 5*b3 + 5*b2 - b1 + 14) * q^45 + (-3*b15 + 3*b14 - 6*b12 - b11 + b10 - 5*b9 - 5*b8 + 3*b6 - b4 + 6*b3 + 6*b2 - 8) * q^46 + (10*b13 - 5*b9 + 5*b8 - 9*b7 - 5*b5 - 5*b3 + 5*b2 - 5*b1) * q^47 + (-4*b13 + 4*b1) * q^48 + (3*b15 - 3*b14 + 9*b12 - 4*b11 + 4*b10 + 4*b9 + 4*b8 - 6*b6 + 5*b4 - 3) * q^49 + (b15 - 4*b14 - b13 + 3*b12 + 2*b11 + b10 + 4*b9 + 3*b8 - 2*b7 + 5*b6 - b5 - 3*b4 + 2*b3 - 2*b2 + 2*b1 + 14) * q^50 + (2*b15 - 2*b14 + 12*b12 + 2*b11 - 2*b10 + 4*b9 + 4*b8 + 32*b6 - 10*b4 + 20) * q^51 + (4*b13 + 4*b7 + 12*b5 + 2*b3 - 2*b2 - 4*b1) * q^52 + (8*b13 - 4*b9 + 4*b8 + 10*b7 + 2*b5 - 4*b3 + 4*b2 + 8*b1) * q^53 + (-b15 + b14 - 5*b12 + 4*b11 - 4*b10 - 6*b9 - 6*b8 + 2*b6 - 4*b3 - 4*b2 + 16) * q^54 + (-5*b15 + 2*b13 - 5*b12 + 5*b11 - 5*b10 - 7*b8 + 7*b5 + 5*b4 - 5*b3 + 2*b2 - 2*b1 - 4) * q^55 + (-2*b15 + 2*b14 - 2*b12 + 4*b6 - 4*b4 + 4*b3 + 4*b2) * q^56 + (2*b13 + 6*b11 + 6*b10 + 2*b9 - 2*b8 + 8*b7 + 38*b5 + 4*b3 - 4*b2) * q^57 + (16*b13 - 4*b9 + 4*b8 + 2*b7 - 7*b5 - 8*b3 + 8*b2 + 8*b1) * q^58 + (-5*b15 + 5*b14 - 4*b12 + b11 - b10 - 8*b9 - 8*b8 - 11*b6 + 4*b4 - 27) * q^59 + (4*b15 + 2*b12 + 2*b10 + 2*b8 - 6*b7 - 16*b6 - 2*b5 + 2*b4 - 4*b3 - 2*b2 - 4*b1 - 4) * q^60 + (-5*b15 + 5*b14 + 5*b12 + 9*b11 - 9*b10 + 3*b9 + 3*b8 + 15*b6 - 18*b4 + 6*b3 + 6*b2 + 15) * q^61 + (3*b15 + 3*b14 + 2*b13 + 6*b11 + 6*b10 - 4*b9 + 4*b8 - 10*b7 + 3*b6 - 6*b5 + 2*b3 - 2*b2 + 2*b1 + 3) * q^62 + (-b15 - b14 - 11*b13 - 2*b11 - 2*b10 + 3*b9 - 3*b8 - 29*b7 - b6 + 20*b5 + 4*b3 - 4*b2 - 1) * q^63 + 8 * q^64 + (-5*b15 - 7*b14 - 17*b13 + b12 - 3*b11 - 3*b10 + 7*b9 - 7*b8 - 4*b7 + 4*b5 + 2*b4 + 7*b3 + 2*b2 + 10*b1 - 14) * q^65 + (-b15 + b14 - 4*b12 - 3*b11 + 3*b10 - b9 - b8 - 7*b6 + 6*b4 + 4*b3 + 4*b2 + 19) * q^66 + (-2*b15 - 2*b14 - 7*b13 - 2*b11 - 2*b10 + 2*b9 - 2*b8 + b7 - 2*b6 + 2*b5 + 3*b3 - 3*b2 - 6*b1 - 2) * q^67 + (4*b13 + 4*b9 - 4*b8 - 4*b7 - 16*b5 + 4*b3 - 4*b2 - 8*b1) * q^68 + (8*b15 - 8*b14 - 6*b12 + 5*b11 - 5*b10 - 14*b9 - 14*b8 - 13*b6 + 9*b4 - 3*b3 - 3*b2 - 20) * q^69 + (-3*b15 + 3*b14 + 2*b13 + 3*b12 + 5*b11 - 4*b10 + 5*b9 - 2*b8 - 10*b7 - 11*b6 - 6*b5 - 9*b4 + 8*b3 - 2*b2 - 13*b1 - 11) * q^70 + (4*b15 - 4*b14 + 4*b12 - 6*b11 + 6*b10 + 6*b9 + 6*b8 + 12*b6 + 14*b4 + 4) * q^71 + (-2*b15 - 2*b14 - 4*b13 + 2*b11 + 2*b10 + 2*b9 - 2*b8 - 2*b7 - 2*b6 + 4*b1 - 2) * q^72 + (2*b15 + 2*b14 + 4*b13 + 6*b9 - 6*b8 + 4*b7 + 2*b6 - 4*b5 + 6*b3 - 6*b2 - 4*b1 + 2) * q^73 + (-2*b15 + 2*b14 - 2*b12 - 4*b6 - 4*b4 - 4*b3 - 4*b2 + 8) * q^74 + (5*b15 + b14 + 10*b13 + 3*b12 - 3*b11 + 12*b10 - 3*b9 + 16*b8 + 4*b7 + 31*b6 - 2*b5 + 7*b4 - 3*b3 - 4*b2 + 5) * q^75 + (-12*b12 + 4*b11 - 4*b10 - 4*b9 - 4*b8 + 24*b6 + 4*b4) * q^76 + (-6*b13 + 5*b9 - 5*b8 + 6*b7 - 24*b5 - 6*b3 + 6*b2 - 10*b1) * q^77 + (3*b15 + 3*b14 + 10*b13 - 2*b9 + 2*b8 + 16*b7 + 3*b6 + 10*b5 + 2*b3 - 2*b2 - 18*b1 + 3) * q^78 + (4*b15 - 4*b14 - 4*b12 - 11*b11 + 11*b10 + 5*b9 + 5*b8 + 50*b6 + 22*b4 - 3*b3 - 3*b2 + 50) * q^79 + (4*b13 + 4*b8 - 8*b6 - 4*b5 + 4*b2 + 4*b1 - 4) * q^80 + (-2*b15 + 2*b14 - 10*b12 + 2*b11 - 2*b10 - 53*b6 - 5*b3 - 5*b2 - 52) * q^81 + (2*b15 + 2*b14 - 6*b13 - 4*b9 + 4*b8 + b7 + 2*b6 - 11*b5 + 6*b3 - 6*b2 + 6*b1 + 2) * q^82 + (-6*b15 - 6*b14 - 2*b13 - 3*b11 - 3*b10 - 4*b9 + 4*b8 - 6*b7 - 6*b6 - 7*b5 - 5*b3 + 5*b2 - b1 - 6) * q^83 + (6*b15 - 6*b14 + 2*b12 - 2*b11 + 2*b10 - 2*b9 - 2*b8 - 6*b6 + 8*b4 - 2*b3 - 2*b2 + 4) * q^84 + (-10*b15 + 8*b14 - 8*b13 - 18*b12 + 8*b11 - 10*b10 + 2*b9 - 8*b8 + 46*b7 + 34*b6 + 26*b5 - 18*b4 + 10*b3 + 14*b2 + 12*b1 + 8) * q^85 + (b15 - b14 - 2*b12 - 3*b11 + 3*b10 - 5*b9 - 5*b8 - 5*b6 + 2*b4 - 9) * q^86 + (12*b15 + 12*b14 + 15*b13 + 6*b11 + 6*b10 - 6*b9 + 6*b8 - 72*b7 + 12*b6 - 48*b5 + 12*b3 - 12*b2 - 21*b1 + 12) * q^87 + (-8*b13 - 2*b11 - 2*b10 + 2*b9 - 2*b8 + 12*b7 + 16*b5 - 4*b3 + 4*b2 + 4*b1) * q^88 + (-6*b15 + 6*b14 + 24*b12 - 6*b11 + 6*b10 - b9 - b8 - 68*b6 - 6*b4 - 7*b3 - 7*b2 - 31) * q^89 + (-5*b15 + b13 + 5*b12 - 5*b11 - 5*b10 + 5*b9 - b8 - 5*b7 + 10*b6 - 4*b5 - 10*b4 - 10*b3 + 6*b2 + 9*b1 + 13) * q^90 + (-b15 + b14 - 2*b12 + 6*b11 - 6*b10 + 3*b9 + 3*b8 + b6 + 6*b4 - 9*b3 - 9*b2 - 5) * q^91 + (-12*b15 - 12*b14 - 12*b13 - 6*b11 - 6*b10 + 8*b9 - 8*b8 + 20*b7 - 12*b6 + 2*b5 - 6*b3 + 6*b2 + 14*b1 - 12) * q^92 + (6*b15 + 6*b14 + 16*b13 + 6*b11 + 6*b10 - 11*b9 + 11*b8 - 14*b7 + 6*b6 - 14*b5 - 10*b3 + 10*b2 + 8*b1 + 6) * q^93 + (15*b12 - 5*b11 + 5*b10 + 5*b9 + 5*b8 + 13*b6 - 5*b4) * q^94 + (12*b15 - 6*b14 + 10*b13 + 4*b12 - 8*b11 + 2*b10 - 8*b8 - 4*b7 + 8*b6 - 14*b5 - 4*b4 + 4*b3 - 4*b2 - 14*b1 + 22) * q^95 + (-4*b12 + 4*b6 + 4) * q^96 + (18*b13 - 4*b11 - 4*b10 - 10*b9 + 10*b8 + 18*b7 + 12*b5 - 5*b3 + 5*b2 + 16*b1) * q^97 + (-18*b13 + 8*b9 - 8*b8 - 6*b7 + 8*b5 + 6*b3 - 6*b2 - 10*b1) * q^98 + (2*b15 - 2*b14 - 20*b12 + 9*b11 - 9*b10 - 7*b9 - 7*b8 + 42*b6 - 8*b4 + 7*b3 + 7*b2 + 34) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4} + 30 q^{5} - 8 q^{6} - 16 q^{9}+O(q^{10})$$ 16 * q - 16 * q^4 + 30 * q^5 - 8 * q^6 - 16 * q^9 $$16 q - 16 q^{4} + 30 q^{5} - 8 q^{6} - 16 q^{9} - 16 q^{10} + 60 q^{11} + 12 q^{14} - 98 q^{15} - 32 q^{16} + 144 q^{19} - 60 q^{20} - 56 q^{21} + 8 q^{24} + 6 q^{25} + 72 q^{29} + 52 q^{30} + 28 q^{31} - 136 q^{34} + 40 q^{36} - 276 q^{39} + 16 q^{40} - 180 q^{41} + 242 q^{45} - 56 q^{46} + 12 q^{49} + 144 q^{50} - 8 q^{51} + 260 q^{54} + 20 q^{55} - 24 q^{56} - 228 q^{59} + 20 q^{60} + 68 q^{61} + 128 q^{64} - 102 q^{65} + 440 q^{66} - 16 q^{69} - 112 q^{70} + 72 q^{74} - 274 q^{75} - 144 q^{76} + 420 q^{79} - 500 q^{81} + 176 q^{84} - 136 q^{85} - 48 q^{86} + 40 q^{90} - 168 q^{91} - 164 q^{94} + 276 q^{95} + 16 q^{96} + 268 q^{99}+O(q^{100})$$ 16 * q - 16 * q^4 + 30 * q^5 - 8 * q^6 - 16 * q^9 - 16 * q^10 + 60 * q^11 + 12 * q^14 - 98 * q^15 - 32 * q^16 + 144 * q^19 - 60 * q^20 - 56 * q^21 + 8 * q^24 + 6 * q^25 + 72 * q^29 + 52 * q^30 + 28 * q^31 - 136 * q^34 + 40 * q^36 - 276 * q^39 + 16 * q^40 - 180 * q^41 + 242 * q^45 - 56 * q^46 + 12 * q^49 + 144 * q^50 - 8 * q^51 + 260 * q^54 + 20 * q^55 - 24 * q^56 - 228 * q^59 + 20 * q^60 + 68 * q^61 + 128 * q^64 - 102 * q^65 + 440 * q^66 - 16 * q^69 - 112 * q^70 + 72 * q^74 - 274 * q^75 - 144 * q^76 + 420 * q^79 - 500 * q^81 + 176 * q^84 - 136 * q^85 - 48 * q^86 + 40 * q^90 - 168 * q^91 - 164 * q^94 + 276 * q^95 + 16 * q^96 + 268 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - \nu^{15} + 98 \nu^{14} - 1694 \nu^{13} - 11468 \nu^{12} + 16043 \nu^{11} - 2128 \nu^{10} + \cdots + 3632930676 ) / 1084139640$$ (-v^15 + 98*v^14 - 1694*v^13 - 11468*v^12 + 16043*v^11 - 2128*v^10 + 79917*v^9 + 1816008*v^8 - 3084327*v^7 + 1239912*v^6 + 4500927*v^5 - 106945272*v^4 + 215730054*v^3 - 174299526*v^2 - 1145609649*v + 3632930676) / 1084139640 $$\beta_{3}$$ $$=$$ $$( \nu^{15} + 98 \nu^{14} + 1694 \nu^{13} - 11468 \nu^{12} - 16043 \nu^{11} - 2128 \nu^{10} + \cdots + 3632930676 ) / 1084139640$$ (v^15 + 98*v^14 + 1694*v^13 - 11468*v^12 - 16043*v^11 - 2128*v^10 - 79917*v^9 + 1816008*v^8 + 3084327*v^7 + 1239912*v^6 - 4500927*v^5 - 106945272*v^4 - 215730054*v^3 - 174299526*v^2 + 1145609649*v + 3632930676) / 1084139640 $$\beta_{4}$$ $$=$$ $$( 5 \nu^{14} + 55 \nu^{12} - 2032 \nu^{10} + 15657 \nu^{8} + 213588 \nu^{6} - 1070253 \nu^{4} + \cdots + 65426292 ) / 20076660$$ (5*v^14 + 55*v^12 - 2032*v^10 + 15657*v^8 + 213588*v^6 - 1070253*v^4 - 6519447*v^2 + 65426292) / 20076660 $$\beta_{5}$$ $$=$$ $$( - 256 \nu^{15} + 703 \nu^{13} + 72245 \nu^{11} - 469200 \nu^{9} - 2736405 \nu^{7} + \cdots - 1856323413 \nu ) / 4878628380$$ (-256*v^15 + 703*v^13 + 72245*v^11 - 469200*v^9 - 2736405*v^7 + 53892540*v^5 + 126240201*v^3 - 1856323413*v) / 4878628380 $$\beta_{6}$$ $$=$$ $$( - 212 \nu^{14} + 1034 \nu^{12} + 27619 \nu^{10} - 245004 \nu^{8} - 1736091 \nu^{6} + \cdots - 865185948 ) / 271034910$$ (-212*v^14 + 1034*v^12 + 27619*v^10 - 245004*v^8 - 1736091*v^6 + 19717506*v^4 + 37207431*v^2 - 865185948) / 271034910 $$\beta_{7}$$ $$=$$ $$( 1225 \nu^{15} - 16666 \nu^{13} - 256559 \nu^{11} + 3638289 \nu^{9} + 19138311 \nu^{7} + \cdots + 12751926795 \nu ) / 4878628380$$ (1225*v^15 - 16666*v^13 - 256559*v^11 + 3638289*v^9 + 19138311*v^7 - 331210701*v^5 - 582314994*v^3 + 12751926795*v) / 4878628380 $$\beta_{8}$$ $$=$$ $$( - 644 \nu^{15} + 5394 \nu^{14} + 13265 \nu^{13} - 7680 \nu^{12} + 167749 \nu^{11} + \cdots - 376260228 ) / 3252418920$$ (-644*v^15 + 5394*v^14 + 13265*v^13 - 7680*v^12 + 167749*v^11 - 923262*v^10 - 1493484*v^9 + 1604052*v^8 - 10810521*v^7 + 63067518*v^6 + 101392236*v^5 - 104255748*v^4 + 451875753*v^3 - 1418986836*v^2 - 3018053439*v - 376260228) / 3252418920 $$\beta_{9}$$ $$=$$ $$( 644 \nu^{15} + 5394 \nu^{14} - 13265 \nu^{13} - 7680 \nu^{12} - 167749 \nu^{11} + \cdots - 376260228 ) / 3252418920$$ (644*v^15 + 5394*v^14 - 13265*v^13 - 7680*v^12 - 167749*v^11 - 923262*v^10 + 1493484*v^9 + 1604052*v^8 + 10810521*v^7 + 63067518*v^6 - 101392236*v^5 - 104255748*v^4 - 451875753*v^3 - 1418986836*v^2 + 3018053439*v - 376260228) / 3252418920 $$\beta_{10}$$ $$=$$ $$( 4495 \nu^{15} - 18027 \nu^{14} - 29809 \nu^{13} + 278145 \nu^{12} - 779024 \nu^{11} + \cdots - 179743975020 ) / 9757256760$$ (4495*v^15 - 18027*v^14 - 29809*v^13 + 278145*v^12 - 779024*v^11 + 3475530*v^10 + 3783639*v^9 - 57721545*v^8 + 55286856*v^7 - 239281290*v^6 - 453675411*v^5 + 4783351725*v^4 - 1580341509*v^3 + 3736797867*v^2 + 15081232698*v - 179743975020) / 9757256760 $$\beta_{11}$$ $$=$$ $$( 4495 \nu^{15} + 18027 \nu^{14} - 29809 \nu^{13} - 278145 \nu^{12} - 779024 \nu^{11} + \cdots + 179743975020 ) / 9757256760$$ (4495*v^15 + 18027*v^14 - 29809*v^13 - 278145*v^12 - 779024*v^11 - 3475530*v^10 + 3783639*v^9 + 57721545*v^8 + 55286856*v^7 + 239281290*v^6 - 453675411*v^5 - 4783351725*v^4 - 1580341509*v^3 - 3736797867*v^2 + 15081232698*v + 179743975020) / 9757256760 $$\beta_{12}$$ $$=$$ $$( - 2413 \nu^{14} + 6352 \nu^{12} + 417695 \nu^{10} - 1418565 \nu^{8} - 33448095 \nu^{6} + \cdots - 5822999037 ) / 542069820$$ (-2413*v^14 + 6352*v^12 + 417695*v^10 - 1418565*v^8 - 33448095*v^6 + 151232265*v^4 + 1170600498*v^2 - 5822999037) / 542069820 $$\beta_{13}$$ $$=$$ $$( - 212 \nu^{15} + 1034 \nu^{13} + 27619 \nu^{11} - 245004 \nu^{9} - 1736091 \nu^{7} + \cdots - 594151038 \nu ) / 271034910$$ (-212*v^15 + 1034*v^13 + 27619*v^11 - 245004*v^9 - 1736091*v^7 + 19717506*v^5 + 37207431*v^3 - 594151038*v) / 271034910 $$\beta_{14}$$ $$=$$ $$( 1561 \nu^{15} - 578 \nu^{14} - 8533 \nu^{13} + 16745 \nu^{12} - 279308 \nu^{11} + \cdots - 7814839905 ) / 1084139640$$ (1561*v^15 - 578*v^14 - 8533*v^13 + 16745*v^12 - 279308*v^11 + 57205*v^10 + 1916733*v^9 - 2277150*v^8 + 20331252*v^7 - 7898625*v^6 - 155171457*v^5 + 194528790*v^4 - 586371879*v^3 + 153809523*v^2 + 5950013436*v - 7814839905) / 1084139640 $$\beta_{15}$$ $$=$$ $$( 1561 \nu^{15} + 1426 \nu^{14} - 8533 \nu^{13} - 20881 \nu^{12} - 279308 \nu^{11} + \cdots + 10191444057 ) / 1084139640$$ (1561*v^15 + 1426*v^14 - 8533*v^13 - 20881*v^12 - 279308*v^11 - 167681*v^10 + 1916733*v^9 + 3257166*v^8 + 20331252*v^7 + 14842989*v^6 - 155171457*v^5 - 273398814*v^4 - 586371879*v^3 - 302639247*v^2 + 5950013436*v + 10191444057) / 1084139640
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{14} + \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 1$$ b15 - b14 + b12 + b9 + b8 + b6 + b4 - b3 - b2 + 1 $$\nu^{3}$$ $$=$$ $$2 \beta_{15} + 2 \beta_{14} + 7 \beta_{13} + \beta_{11} + \beta_{10} - 4 \beta_{9} + 4 \beta_{8} + \cdots + 2$$ 2*b15 + 2*b14 + 7*b13 + b11 + b10 - 4*b9 + 4*b8 + 3*b7 + 2*b6 + 7*b5 - 4*b3 + 4*b2 + b1 + 2 $$\nu^{4}$$ $$=$$ $$4 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} + \beta_{6} + 6 \beta_{4} + \cdots + 55$$ 4*b12 - 2*b11 + 2*b10 + 7*b9 + 7*b8 + b6 + 6*b4 + 7*b3 + 7*b2 + 55 $$\nu^{5}$$ $$=$$ $$7 \beta_{15} + 7 \beta_{14} - \beta_{13} - 19 \beta_{11} - 19 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} + \cdots + 7$$ 7*b15 + 7*b14 - b13 - 19*b11 - 19*b10 + 10*b9 - 10*b8 - 24*b7 + 7*b6 + 26*b5 - 2*b3 + 2*b2 + 38*b1 + 7 $$\nu^{6}$$ $$=$$ $$114 \beta_{15} - 114 \beta_{14} + 38 \beta_{12} - 31 \beta_{11} + 31 \beta_{10} + 47 \beta_{9} + \cdots - 76$$ 114*b15 - 114*b14 + 38*b12 - 31*b11 + 31*b10 + 47*b9 + 47*b8 + 146*b6 + 159*b4 - 46*b3 - 46*b2 - 76 $$\nu^{7}$$ $$=$$ $$116 \beta_{15} + 116 \beta_{14} + 376 \beta_{13} + 49 \beta_{11} + 49 \beta_{10} - 199 \beta_{9} + \cdots + 116$$ 116*b15 + 116*b14 + 376*b13 + 49*b11 + 49*b10 - 199*b9 + 199*b8 + 261*b7 + 116*b6 + 1366*b5 - 76*b3 + 76*b2 - 233*b1 + 116 $$\nu^{8}$$ $$=$$ $$- 108 \beta_{15} + 108 \beta_{14} + 340 \beta_{12} - 122 \beta_{11} + 122 \beta_{10} + 265 \beta_{9} + \cdots - 518$$ -108*b15 + 108*b14 + 340*b12 - 122*b11 + 122*b10 + 265*b9 + 265*b8 - 1223*b6 + 930*b4 + 535*b3 + 535*b2 - 518 $$\nu^{9}$$ $$=$$ $$571 \beta_{15} + 571 \beta_{14} - 553 \beta_{13} - 2401 \beta_{11} - 2401 \beta_{10} + 424 \beta_{9} + \cdots + 571$$ 571*b15 + 571*b14 - 553*b13 - 2401*b11 - 2401*b10 + 424*b9 - 424*b8 + 24*b7 + 571*b6 + 734*b5 - 44*b3 + 44*b2 - 535*b1 + 571 $$\nu^{10}$$ $$=$$ $$8337 \beta_{15} - 8337 \beta_{14} + 1685 \beta_{12} - 4243 \beta_{11} + 4243 \beta_{10} - 1396 \beta_{9} + \cdots - 27157$$ 8337*b15 - 8337*b14 + 1685*b12 - 4243*b11 + 4243*b10 - 1396*b9 - 1396*b8 - 12919*b6 + 10626*b4 - 1951*b3 - 1951*b2 - 27157 $$\nu^{11}$$ $$=$$ $$- 4480 \beta_{15} - 4480 \beta_{14} - 18467 \beta_{13} + 1090 \beta_{11} + 1090 \beta_{10} + \cdots - 4480$$ -4480*b15 - 4480*b14 - 18467*b13 + 1090*b11 + 1090*b10 - 2503*b9 + 2503*b8 + 16746*b7 - 4480*b6 + 110107*b5 + 8804*b3 - 8804*b2 - 10514*b1 - 4480 $$\nu^{12}$$ $$=$$ $$- 18420 \beta_{15} + 18420 \beta_{14} + 5920 \beta_{12} - 6440 \beta_{11} + 6440 \beta_{10} + \cdots - 319289$$ -18420*b15 + 18420*b14 + 5920*b12 - 6440*b11 + 6440*b10 - 35030*b9 - 35030*b8 - 230780*b6 + 93000*b4 - 18170*b3 - 18170*b2 - 319289 $$\nu^{13}$$ $$=$$ $$47170 \beta_{15} + 47170 \beta_{14} + 6140 \beta_{13} - 115510 \beta_{11} - 115510 \beta_{10} + \cdots + 47170$$ 47170*b15 + 47170*b14 + 6140*b13 - 115510*b11 - 115510*b10 - 124460*b9 + 124460*b8 + 3780*b7 + 47170*b6 - 451420*b5 + 24340*b3 - 24340*b2 - 149329*b1 + 47170 $$\nu^{14}$$ $$=$$ $$363051 \beta_{15} - 363051 \beta_{14} + 91811 \beta_{12} - 375340 \beta_{11} + 375340 \beta_{10} + \cdots - 2664409$$ 363051*b15 - 363051*b14 + 91811*b12 - 375340*b11 + 375340*b10 - 217309*b9 - 217309*b8 - 3600829*b6 + 194631*b4 - 108841*b3 - 108841*b2 - 2664409 $$\nu^{15}$$ $$=$$ $$- 961348 \beta_{15} - 961348 \beta_{14} - 4958843 \beta_{13} + 360511 \beta_{11} + 360511 \beta_{10} + \cdots - 961348$$ -961348*b15 - 961348*b14 - 4958843*b13 + 360511*b11 + 360511*b10 + 434546*b9 - 434546*b8 - 1670667*b7 - 961348*b6 + 3754957*b5 + 1050866*b3 - 1050866*b2 + 1335451*b1 - 961348

## Character values

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

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$-\beta_{6}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 −0.173499 − 2.99498i −2.87096 + 0.870383i 2.97088 + 0.416995i −0.633522 + 2.93235i 0.633522 − 2.93235i −2.97088 − 0.416995i 2.87096 − 0.870383i 0.173499 + 2.99498i −0.173499 + 2.99498i −2.87096 − 0.870383i 2.97088 − 0.416995i −0.633522 − 2.93235i 0.633522 + 2.93235i −2.97088 + 0.416995i 2.87096 + 0.870383i 0.173499 − 2.99498i
−0.707107 + 1.22474i −2.68048 1.34723i −1.00000 1.73205i 4.82012 + 1.32909i 3.54540 2.33026i 2.75375 + 1.58988i 2.82843 5.36992 + 7.22246i −5.03613 + 4.96360i
29.2 −0.707107 + 1.22474i −0.681708 + 2.92152i −1.00000 1.73205i −0.137425 + 4.99811i −3.09608 2.90074i −4.16400 2.40409i 2.82843 −8.07055 3.98325i −6.02424 3.70251i
29.3 −0.707107 + 1.22474i 1.84657 2.36436i −1.00000 1.73205i 0.410361 4.98313i 1.59002 + 3.93343i −8.30302 4.79375i 2.82843 −2.18038 8.73189i 5.81290 + 4.02619i
29.4 −0.707107 + 1.22474i 2.22272 + 2.01482i −1.00000 1.73205i 3.82116 3.22471i −4.03934 + 1.29758i 7.59195 + 4.38322i 2.82843 0.881011 + 8.95678i 1.24747 + 6.96016i
29.5 0.707107 1.22474i −2.22272 2.01482i −1.00000 1.73205i 4.70326 1.69687i −4.03934 + 1.29758i −7.59195 4.38322i −2.82843 0.881011 + 8.95678i 1.24747 6.96016i
29.6 0.707107 1.22474i −1.84657 + 2.36436i −1.00000 1.73205i 4.52070 + 2.13618i 1.59002 + 3.93343i 8.30302 + 4.79375i −2.82843 −2.18038 8.73189i 5.81290 4.02619i
29.7 0.707107 1.22474i 0.681708 2.92152i −1.00000 1.73205i −4.39720 2.38004i −3.09608 2.90074i 4.16400 + 2.40409i −2.82843 −8.07055 3.98325i −6.02424 + 3.70251i
29.8 0.707107 1.22474i 2.68048 + 1.34723i −1.00000 1.73205i 1.25903 4.83889i 3.54540 2.33026i −2.75375 1.58988i −2.82843 5.36992 + 7.22246i −5.03613 4.96360i
59.1 −0.707107 1.22474i −2.68048 + 1.34723i −1.00000 + 1.73205i 4.82012 1.32909i 3.54540 + 2.33026i 2.75375 1.58988i 2.82843 5.36992 7.22246i −5.03613 4.96360i
59.2 −0.707107 1.22474i −0.681708 2.92152i −1.00000 + 1.73205i −0.137425 4.99811i −3.09608 + 2.90074i −4.16400 + 2.40409i 2.82843 −8.07055 + 3.98325i −6.02424 + 3.70251i
59.3 −0.707107 1.22474i 1.84657 + 2.36436i −1.00000 + 1.73205i 0.410361 + 4.98313i 1.59002 3.93343i −8.30302 + 4.79375i 2.82843 −2.18038 + 8.73189i 5.81290 4.02619i
59.4 −0.707107 1.22474i 2.22272 2.01482i −1.00000 + 1.73205i 3.82116 + 3.22471i −4.03934 1.29758i 7.59195 4.38322i 2.82843 0.881011 8.95678i 1.24747 6.96016i
59.5 0.707107 + 1.22474i −2.22272 + 2.01482i −1.00000 + 1.73205i 4.70326 + 1.69687i −4.03934 1.29758i −7.59195 + 4.38322i −2.82843 0.881011 8.95678i 1.24747 + 6.96016i
59.6 0.707107 + 1.22474i −1.84657 2.36436i −1.00000 + 1.73205i 4.52070 2.13618i 1.59002 3.93343i 8.30302 4.79375i −2.82843 −2.18038 + 8.73189i 5.81290 + 4.02619i
59.7 0.707107 + 1.22474i 0.681708 + 2.92152i −1.00000 + 1.73205i −4.39720 + 2.38004i −3.09608 + 2.90074i 4.16400 2.40409i −2.82843 −8.07055 + 3.98325i −6.02424 3.70251i
59.8 0.707107 + 1.22474i 2.68048 1.34723i −1.00000 + 1.73205i 1.25903 + 4.83889i 3.54540 + 2.33026i −2.75375 + 1.58988i −2.82843 5.36992 7.22246i −5.03613 + 4.96360i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h 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.j.b 16
3.b odd 2 1 270.3.j.b 16
5.b even 2 1 inner 90.3.j.b 16
5.c odd 4 2 450.3.i.e 16
9.c even 3 1 270.3.j.b 16
9.c even 3 1 810.3.b.b 16
9.d odd 6 1 inner 90.3.j.b 16
9.d odd 6 1 810.3.b.b 16
15.d odd 2 1 270.3.j.b 16
15.e even 4 2 1350.3.i.e 16
45.h odd 6 1 inner 90.3.j.b 16
45.h odd 6 1 810.3.b.b 16
45.j even 6 1 270.3.j.b 16
45.j even 6 1 810.3.b.b 16
45.k odd 12 2 1350.3.i.e 16
45.l even 12 2 450.3.i.e 16

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} - 202 T_{7}^{14} + 27898 T_{7}^{12} - 2058640 T_{7}^{10} + 109528039 T_{7}^{8} + \cdots + 2726544000625$$ acting on $$S_{3}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{2} + 4)^{4}$$
$3$ $$T^{16} + 8 T^{14} + \cdots + 43046721$$
$5$ $$T^{16} + \cdots + 152587890625$$
$7$ $$T^{16} + \cdots + 2726544000625$$
$11$ $$(T^{8} - 30 T^{7} + \cdots + 110923024)^{2}$$
$13$ $$T^{16} + \cdots + 1991891886336$$
$17$ $$(T^{8} - 1520 T^{6} + \cdots + 640000)^{2}$$
$19$ $$(T^{4} - 36 T^{3} + \cdots - 118800)^{4}$$
$23$ $$T^{16} + \cdots + 26\!\cdots\!61$$
$29$ $$(T^{8} - 36 T^{7} + \cdots + 334196141409)^{2}$$
$31$ $$(T^{8} - 14 T^{7} + \cdots + 27544049296)^{2}$$
$37$ $$(T^{8} + 3016 T^{6} + \cdots + 1414963456)^{2}$$
$41$ $$(T^{8} + 90 T^{7} + \cdots + 15399072649)^{2}$$
$43$ $$T^{16} + \cdots + 38\!\cdots\!16$$
$47$ $$T^{16} + \cdots + 30\!\cdots\!01$$
$53$ $$(T^{8} - 3264 T^{6} + \cdots + 81293414400)^{2}$$
$59$ $$(T^{8} + \cdots + 28666857972736)^{2}$$
$61$ $$(T^{8} - 34 T^{7} + \cdots + 116562836569)^{2}$$
$67$ $$T^{16} + \cdots + 31\!\cdots\!25$$
$71$ $$(T^{8} + 11328 T^{6} + \cdots + 39661519104)^{2}$$
$73$ $$(T^{8} + \cdots + 12985356390400)^{2}$$
$79$ $$(T^{8} + \cdots + 68\!\cdots\!00)^{2}$$
$83$ $$T^{16} + \cdots + 45\!\cdots\!21$$
$89$ $$(T^{8} + \cdots + 11\!\cdots\!00)^{2}$$
$97$ $$T^{16} + \cdots + 74\!\cdots\!00$$