# Properties

 Label 45.3.i.a Level $45$ Weight $3$ Character orbit 45.i Analytic conductor $1.226$ 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] = [45,3,Mod(11,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, 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("45.11");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 45.i (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: $$1.22616118962$$ 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: $$x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81$$ x^16 + 48*x^14 + 912*x^12 + 8704*x^10 + 43602*x^8 + 109032*x^6 + 117844*x^4 + 36000*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + (\beta_{11} - \beta_{7}) q^{3} + ( - \beta_{14} + 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{4} + \beta_{9} q^{5} + (\beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + \cdots - 2) q^{6}+ \cdots + (\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{4} - \beta_{2} + \cdots + 2) q^{9}+O(q^{10})$$ q + b3 * q^2 + (b11 - b7) * q^3 + (-b14 + 2*b7 - b3 - b2 - b1 + 2) * q^4 + b9 * q^5 + (b14 + b13 + b12 - 2*b10 - b8 - b6 + 2*b5 + b3 + b2 - 2) * q^6 + (-b15 + b14 + b13 + b12 - b11 - b9 - b8 + b6 + b4 - b3 + b2 + 2*b1) * q^7 + (-b15 + b14 - 2*b11 + b8 - 3*b7 + 2*b4 + b2 + 2*b1 - 2) * q^8 + (b15 - b13 - b12 + b11 + b10 + b9 - b8 + 2*b7 - b4 - b2 - b1 + 2) * q^9 $$q + \beta_{3} q^{2} + (\beta_{11} - \beta_{7}) q^{3} + ( - \beta_{14} + 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{4} + \beta_{9} q^{5} + (\beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + \cdots - 2) q^{6}+ \cdots + (14 \beta_{15} - 13 \beta_{14} - 22 \beta_{13} - 12 \beta_{12} + 14 \beta_{11} + \cdots - 51) q^{99}+O(q^{100})$$ q + b3 * q^2 + (b11 - b7) * q^3 + (-b14 + 2*b7 - b3 - b2 - b1 + 2) * q^4 + b9 * q^5 + (b14 + b13 + b12 - 2*b10 - b8 - b6 + 2*b5 + b3 + b2 - 2) * q^6 + (-b15 + b14 + b13 + b12 - b11 - b9 - b8 + b6 + b4 - b3 + b2 + 2*b1) * q^7 + (-b15 + b14 - 2*b11 + b8 - 3*b7 + 2*b4 + b2 + 2*b1 - 2) * q^8 + (b15 - b13 - b12 + b11 + b10 + b9 - b8 + 2*b7 - b4 - b2 - b1 + 2) * q^9 + (-b12 - b11 + b7) * q^10 + (b15 + b12 + b8 - b7 + b6 - b5 - b3 + b2 + b1 - 1) * q^11 + (3*b15 - 3*b14 - 3*b13 - b12 + 4*b11 + 2*b10 - b9 + b8 - 5*b7 - b6 - 2*b5 - 5*b4 - 3*b2 - 3*b1 - 2) * q^12 + (-2*b15 - b13 - b12 + b11 + b9 + 2*b8 - 3*b7 + b6 - 2*b5 - 2) * q^13 + (b15 - 2*b14 - b12 - 2*b11 + 4*b7 - b6 - 2*b5 - b4 - b3 + b1 - 1) * q^14 + (-b14 - b13 + b11 + b9 - b6 - b4 - b3 - b2 + 1) * q^15 + (b15 - b14 - b13 - b12 + b11 + 3*b10 - 5*b9 + 3*b7 - b5 - 2*b4 + 3*b3 - b2 - 6*b1) * q^16 + (-2*b15 + 2*b14 + 3*b13 - 4*b11 - 3*b9 + 2*b8 + 15*b7 + 3*b5 + 7*b4 + 2*b2 - 3*b1 + 5) * q^17 + (-b15 + 3*b14 + b13 - 2*b12 - 3*b11 + 2*b10 + 2*b9 + b8 - 6*b7 + 2*b4 + b2 + 4*b1 - 5) * q^18 + (-b15 + b14 + b13 + b12 - b11 + 3*b10 - b9 + b8 - 2*b7 + 2*b6 + 2*b5 - 3*b4 + 3*b3 - 2*b1 - 7) * q^19 + (-b15 - b12 + 2*b10 + 2*b8 + b7 + 2*b6 - 2*b5 + 3*b4 - 2*b3 - b2 - b1 + 1) * q^20 + (b14 + 2*b13 + 3*b12 + b11 - b10 - 5*b9 - 3*b8 + b5 - b4 - 2*b3 - b2 + 6*b1 + 5) * q^21 + (2*b15 + 2*b14 + b13 + 2*b12 - 12*b10 + 5*b9 - 4*b8 - 3*b7 - 2*b6 + 3*b5 - 2*b4 + 2*b2 - 3) * q^22 + (2*b15 - 4*b14 - 3*b13 - 3*b12 + 2*b11 + 2*b9 + b7 - b6 - b5 - 3*b4 - 2) * q^23 + (-3*b15 + 5*b14 + b13 + 2*b12 - 4*b11 - 6*b10 + 4*b9 + b8 - 4*b7 + 2*b6 + 9*b4 - b3 + 4*b2 + 3*b1 + 5) * q^24 - 5*b7 * q^25 + (-2*b14 + 2*b12 - 2*b11 - 2*b9 - 4*b8 + 6*b7 + 2*b5 - b3 - b2 - 2*b1 + 3) * q^26 + (2*b15 - 6*b14 - 2*b13 + b12 + 4*b11 + 8*b10 - b9 - 2*b8 - 4*b7 - 3*b6 - 9*b4 - 3*b3 - 2*b2 + 4*b1 + 1) * q^27 + (-3*b13 + 6*b11 + 6*b10 + 9*b9 - 3*b7 - 3*b5 + 3*b4 - 2*b3 - 2*b2 + 5) * q^28 + (3*b15 - b14 + 3*b12 + 8*b11 - 6*b10 + 4*b9 - 5*b8 - 7*b7 - 5*b6 + b5 - 4*b4 + 5*b3 + b2 + 2*b1 - 3) * q^29 + (-b14 - 2*b13 + b11 + b10 - b9 - 5*b7 - b5 - 2*b4 + 2*b3 + b2 + 3*b1 - 8) * q^30 + (b15 - b13 + 2*b11 - 6*b10 + 4*b9 + 2*b7 - b5 - 2*b4 + 4*b3 + 4*b1 + 4) * q^31 + (-4*b15 + 8*b14 + 3*b13 + 2*b12 - b11 - 2*b9 - 9*b7 + 6*b6 + 2*b5 + 5*b4 + 7) * q^32 + (-3*b15 - b14 + 6*b13 - b12 - 5*b11 - 8*b10 + 5*b9 + b8 + 10*b7 + 3*b6 + 2*b5 + 6*b4 - 4*b3 - 3*b2 - 6*b1 + 5) * q^33 + (-b15 + 4*b13 + b12 + 2*b11 + 6*b10 - 16*b9 + 3*b8 - 5*b7 - 3*b6 + 7*b5 + 2*b4 - 3*b3 + b2 + 5*b1 - 3) * q^34 + (-2*b15 + 4*b14 + 3*b13 + b12 + b11 - b10 + b5 + b4 + b3 + 3*b2 + 3*b1 - 1) * q^35 + (-b15 - 3*b14 + 4*b13 - 2*b12 - 7*b11 + 5*b10 - b9 + 4*b8 + 19*b7 + 3*b6 + 7*b4 + 6*b3 + b2 - 17*b1 + 1) * q^36 + (b15 - b14 + 2*b13 - 2*b12 - 6*b11 + 4*b9 - b8 + 9*b7 - 2*b6 - 2*b5 + 6*b4 + 6*b3 + 3*b2 - b1 + 8) * q^37 + (-4*b14 - 3*b13 - b11 - 2*b9 - 2*b8 + 5*b7 - 2*b6 + 4*b5 - 10*b4 - 5*b3 - 8*b2 - 4*b1 + 15) * q^38 + (-3*b15 + 6*b14 - 3*b13 - 2*b12 - 4*b11 - 2*b10 - 8*b9 + 2*b8 + 14*b7 + 4*b6 - b5 + 8*b4 + 3*b3 + 3*b2 - 6*b1 + 17) * q^39 + (2*b14 + 3*b13 + b12 - 5*b11 - 6*b10 - 2*b8 + 2*b7 - b6 + 4*b5 + 4*b4 + 3*b3 + 2*b2 + 3*b1 - 3) * q^40 + (3*b15 - 3*b13 - 2*b12 + 6*b11 - 2*b9 - 11*b7 - 4*b6 + 3*b5 - 8*b4 + 2*b3 - 6*b2 - 2*b1 + 3) * q^41 + (-2*b14 - b13 + b12 + b11 - 6*b10 + 2*b9 + 2*b8 + 16*b7 + b6 - 3*b5 - 3*b4 + 13*b3 + 2*b2 - 8) * q^42 + (2*b15 - 4*b14 - 5*b13 - 2*b12 - b11 + 6*b10 - 4*b9 + 2*b8 + 15*b7 - 2*b6 - 6*b5 - 2*b4 - b3 - 2*b2 + 3) * q^43 + (3*b15 - 5*b14 - 3*b13 - b12 + 19*b11 + 3*b10 + 7*b9 - b8 - 42*b7 - 10*b5 - 21*b4 - b3 - 4*b2 - 15) * q^44 + (b15 - 4*b13 - b12 + 3*b11 + 4*b10 + b9 + 2*b8 - 6*b7 - 3*b5 - 2*b4 + 3*b3 - b2 - 4*b1 + 8) * q^45 + (b15 - b14 - b13 + 4*b12 + 6*b11 + 6*b10 + 10*b9 - b8 - 3*b7 - 2*b6 - 2*b5 + 3*b4 - 21*b3 + 11*b1 - 3) * q^46 + (-2*b15 - 3*b13 - 2*b12 + 3*b11 + 12*b10 + 4*b8 - 13*b7 + 4*b6 - 4*b5 - 4*b3 - 2*b2 - 2*b1 - 19) * q^47 + (-2*b14 + 8*b13 - 8*b11 + 2*b10 - 11*b9 + 21*b7 + b5 + 3*b4 + 4*b3 + 8*b2 - 3*b1 - 4) * q^48 + (2*b15 - 3*b14 + b13 - 3*b12 - 14*b11 - 4*b9 + 6*b8 + 8*b7 + 3*b6 + b5 + 11*b4 - 4*b3 - 3*b2 - 4*b1 - 3) * q^49 + (5*b3 - 5*b1) * q^50 + (3*b15 - 4*b13 - 6*b12 + 3*b11 - 5*b10 + 13*b9 + 3*b8 - 20*b7 + b6 - 7*b5 - 8*b4 - 12*b3 + 2*b2 + 6*b1 - 15) * q^51 + (7*b15 - 3*b14 - 10*b13 - 7*b12 + 4*b11 + 6*b10 - 5*b9 + b8 - 11*b7 - b6 - 6*b5 - 7*b4 - 5*b3 - 7*b2 + 14*b1 + 3) * q^52 + (6*b15 + 6*b14 - 6*b13 - 6*b11 - 3*b9 - 6*b8 - 27*b7 + 3*b5 + 6*b3 + 5*b1 - 12) * q^53 + (-8*b15 + 5*b14 + 7*b12 - 5*b11 + b10 + 5*b9 + 9*b7 + 2*b6 + 12*b4 - 7*b3 + 9*b2 + 14*b1 - 14) * q^54 + (3*b13 + 3*b12 - 3*b11 - 3*b9 + 3*b5 - 3*b4 - 10*b3 + 2*b2 + 6*b1 - 3) * q^55 + (-5*b15 + 7*b14 - 5*b12 - 8*b11 - 6*b10 - 4*b9 - 3*b8 + 3*b7 - 3*b6 + 7*b5 - 2*b4 + 13*b3 + 9*b2 + 2*b1 - 7) * q^56 + (-9*b15 + 3*b14 + 9*b13 + 4*b12 - 11*b11 - 2*b10 - 14*b9 - b8 - 18*b7 - 2*b6 + 8*b5 + 10*b4 - 9*b3 + 3*b2 - 9*b1 - 31) * q^57 + (6*b15 - 3*b13 + b12 + 16*b11 - 12*b10 + 12*b9 - 2*b8 - b7 - b6 - 5*b5 - 11*b4 - 2*b3 - 2*b1 + 12) * q^58 + (-2*b15 + 4*b14 + 3*b13 + b12 - 8*b11 + 2*b9 + 27*b7 + 3*b6 - 5*b5 + 7*b4 - 3*b3 + 3*b1 - 16) * q^59 + (3*b15 - 4*b14 - 3*b13 + b11 - b10 + 17*b7 + b6 - 2*b5 - 2*b4 - 10*b3 - 3*b2 - 3*b1 + 10) * q^60 + (-7*b15 + 8*b14 + 4*b13 + 7*b12 - 10*b11 - 6*b10 + 8*b9 - b8 + b6 + 3*b5 + 16*b4 + 5*b3 + 7*b2 - 9*b1 + 3) * q^61 + (7*b15 - 11*b14 - 6*b13 - 2*b12 + 10*b11 - 6*b10 + 11*b9 - 3*b8 + 30*b7 - 5*b5 - 14*b4 - 2*b3 - 9*b2 - 4*b1 + 20) * q^62 + (10*b14 + 11*b13 + 10*b12 + 4*b11 + 6*b10 - 10*b9 - 4*b8 - 2*b7 + 4*b6 + 6*b5 + b3 + 2*b2 + 6*b1 - 11) * q^63 + (4*b15 - 4*b14 + 2*b13 - 3*b12 - 7*b11 + 3*b10 + b9 - 4*b8 + b7 - 8*b6 + 10*b5 - 18*b4 + 13*b3 - 5*b2 - 7*b1 + 7) * q^64 + (2*b15 - 2*b14 + 3*b13 + 2*b12 + b11 - 5*b10 + 2*b9 - 5*b8 - b7 - 5*b6 + 3*b5 + b4 - 2*b2 - 5) * q^65 + (-3*b15 + 2*b14 + b13 - 11*b12 - 11*b11 + 11*b10 - 3*b9 + 11*b8 - b7 + 4*b6 - 2*b5 + 9*b4 + 2*b3 - 5*b2 + 15*b1 + 6) * q^66 + (-10*b15 + 4*b14 + 4*b13 - 3*b12 - 2*b11 + 12*b10 - 7*b9 + 6*b8 + 6*b7 + 3*b6 - 2*b5 + 11*b4 - 12*b3 + 4*b2 - 12*b1 + 1) * q^67 + (-8*b15 + 4*b14 + 9*b13 + 14*b12 + 4*b11 - 21*b9 + 7*b7 + 2*b6 + 13*b5 + 8*b4 + 6*b3 + 12*b2 - 6*b1 - 19) * q^68 + (9*b15 - 11*b14 - 16*b13 - 8*b12 + 13*b11 - 3*b10 + 11*b9 - 7*b8 - 32*b7 - 5*b6 - 3*b5 - 24*b4 + 22*b3 - 10*b2 - 9*b1 - 5) * q^69 + (2*b14 + 3*b13 + 3*b11 - 6*b9 - 2*b8 + 6*b7 + 2*b6 + 4*b5 - 2*b4 + 7*b3 - 12*b1 - 3) * q^70 + (-2*b15 - 8*b14 - 5*b12 - 5*b11 + 3*b10 - b9 + 12*b8 + 9*b7 - 2*b5 + 10*b4 - 5*b3 - 3*b2 - 3*b1 + 2) * q^71 + (-3*b15 + 7*b14 + 2*b13 - 2*b12 - 5*b11 - 12*b10 + 8*b9 + 5*b8 + 4*b7 + 4*b6 + 18*b4 - 20*b3 - b2 + 18*b1 + 52) * q^72 + (b15 - b14 - b13 + 4*b12 + 6*b11 - 12*b10 - 8*b9 - b8 - 3*b7 - 2*b6 - 2*b5 + 3*b4 + 20*b3 + 5*b2 - 7*b1 - 13) * q^73 + (-7*b15 + 4*b14 + 9*b13 - 7*b12 - 11*b11 + 15*b10 - b9 + 10*b8 + 47*b7 + 10*b6 - 9*b5 + 34*b4 - 17*b3 + b2 - 3*b1 + 58) * q^74 + 5*b4 * q^75 + (-4*b15 - 3*b14 - 5*b13 - b12 + 18*b11 + 8*b9 + 2*b8 - 32*b7 + b6 - 9*b5 - 11*b4 + 16*b3 - 3*b2 + 16*b1 - 17) * q^76 + (6*b15 - 12*b14 - 6*b13 + b12 + 6*b11 + 13*b9 - 32*b7 - 13*b6 - 5*b4 - 5*b3 + 5*b1 + 33) * q^77 + (12*b15 - 12*b14 - 4*b13 + 3*b12 + 12*b11 + 4*b10 - 8*b9 - 6*b8 - 41*b7 - 11*b6 - b5 - 23*b4 + 3*b3 - 10*b2 + 6*b1 + 3) * q^78 + (3*b15 - 3*b14 + 3*b13 - 3*b12 + 9*b11 - 9*b10 + 9*b9 - 8*b8 - 4*b7 + 8*b6 + b5 - 14*b4 + b3 - 3*b2 - 2*b1 - 6) * q^79 + (6*b15 - 6*b14 - 6*b13 - 3*b12 + 9*b11 + 4*b10 + 2*b9 + 21*b7 - 6*b5 - 12*b4 - 6*b2 - 5*b1 + 15) * q^80 + (4*b15 - 8*b14 - 8*b13 + 6*b12 + 24*b11 + 4*b10 + 6*b9 - 11*b8 - 37*b7 - 8*b6 - 3*b5 - 21*b4 + 25*b3 - 8*b2 - 22*b1 - 15) * q^81 + (-3*b15 + 3*b14 - 6*b13 + 2*b12 + 14*b11 + 6*b10 - 3*b9 + 3*b8 - 20*b7 + 6*b6 + 3*b5 - 12*b4 + 36*b3 - 3*b2 - 21*b1 - 24) * q^82 + (3*b15 + 3*b14 + 6*b13 + 3*b12 - 18*b11 - 12*b10 - 6*b9 + 3*b8 + 36*b7 + 3*b6 + 3*b5 + 6*b4 - 6*b3 + 9*b2 + 6*b1 + 45) * q^83 + (9*b15 - 15*b14 - 12*b13 + 5*b12 + 18*b11 + 8*b10 + 20*b9 - 8*b8 + 35*b7 - 13*b6 - 2*b5 - 19*b4 - 21*b3 - 12*b2 + 73) * q^84 + (b15 - 6*b14 - 4*b13 - b12 - 2*b11 + 12*b10 - 5*b9 + 2*b8 + 2*b7 + b6 - 2*b5 - 3*b4 + 3*b3 - 6*b2 + 3*b1 + 3) * q^85 + (-8*b15 + 7*b14 + 3*b13 + 7*b12 - 8*b11 + 2*b9 + 6*b7 + 9*b6 - 5*b5 + 25*b4 - 18*b3 + 9*b2 + 18*b1 + 11) * q^86 + (6*b15 + 6*b14 + b13 - 3*b12 + 4*b11 + 2*b10 + 14*b9 - 6*b8 + 37*b7 - 7*b6 + 7*b5 - 6*b4 + 21*b3 - 2*b2 + 3*b1 + 24) * q^87 + (-12*b15 + 20*b14 + 9*b13 + 12*b12 - 15*b11 - 6*b10 - 8*b8 - 9*b7 + 8*b6 + 4*b5 + 22*b4 + b3 + 12*b2 + 6*b1 + 3) * q^88 + (-3*b15 + 3*b14 + 9*b13 - 12*b10 + 6*b9 + 3*b8 - 15*b7 + 6*b5 + 9*b4 + 3*b2 - 3*b1 - 12) * q^89 + (-b15 + 5*b14 + 2*b13 + 3*b12 - 5*b11 - 10*b10 - 7*b8 - 9*b7 - b6 + 6*b5 + b4 + 14*b3 + 2*b2 - 5*b1 + 9) * q^90 + (-6*b15 + 6*b14 - 6*b13 - 18*b12 - 6*b11 - 6*b10 - 12*b9 + 6*b8 + 6*b7 + 12*b6 - 6*b5 + 6*b4 - 19*b3 - b2 + 6*b1 + 7) * q^91 + (10*b14 + 12*b13 - 26*b11 - 30*b10 - 7*b9 - 4*b8 - 29*b7 - 4*b6 + 11*b5 + 13*b4 + 15*b3 + 20*b2 + 10*b1 - 108) * q^92 + (-3*b15 + 11*b14 + 4*b13 + b12 + 2*b11 - 10*b10 - b8 - 2*b7 + b6 + 13*b5 - b4 + 11*b3 + b2 - 3*b1 - 3) * q^93 + (-10*b15 + 14*b14 + 13*b13 + 5*b12 - 30*b11 - 12*b10 - 10*b9 - 10*b8 + 6*b7 - 5*b6 + 21*b5 + 19*b4 - 3*b3 + 14*b2 - 3*b1 - 21) * q^94 + (-2*b15 - 2*b14 + 6*b13 + 2*b12 - 11*b11 - 4*b9 + b7 + 2*b6 - 5*b5 + 5*b4 - 11*b3 + 6*b2 + 11*b1 + 11) * q^95 + (-9*b15 + 9*b14 + 16*b13 + 9*b12 - 10*b11 + 14*b10 - 28*b9 + 3*b7 + 5*b6 + 10*b5 + 25*b4 - 45*b3 + 4*b2 + 18*b1 + 6) * q^96 + (-11*b15 + b14 + 17*b13 + 11*b12 - 5*b11 - 12*b10 + 19*b9 + 3*b8 + 22*b7 - 3*b6 + 8*b5 + 7*b4 - 18*b3 + 11*b2 + 26*b1 - 6) * q^97 + (-4*b15 - 4*b14 + 3*b13 - 4*b12 - 10*b11 + 6*b10 - 8*b9 + 12*b8 + 24*b7 + 2*b5 + 17*b4 - 4*b3 - 4*b1 + 7) * q^98 + (14*b15 - 13*b14 - 22*b13 - 12*b12 + 14*b11 + 8*b10 - 12*b9 + 11*b8 - 13*b7 - 4*b6 - 9*b5 - 19*b4 - 16*b3 - 19*b2 + b1 - 51) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{3} + 16 q^{4} - 22 q^{6} + 2 q^{7} + 8 q^{9}+O(q^{10})$$ 16 * q + 4 * q^3 + 16 * q^4 - 22 * q^6 + 2 * q^7 + 8 * q^9 $$16 q + 4 q^{3} + 16 q^{4} - 22 q^{6} + 2 q^{7} + 8 q^{9} - 18 q^{11} - 22 q^{12} - 10 q^{13} - 54 q^{14} + 10 q^{15} - 32 q^{16} - 8 q^{18} - 52 q^{19} + 72 q^{21} - 24 q^{22} - 54 q^{23} + 108 q^{24} + 40 q^{25} + 34 q^{27} + 32 q^{28} - 54 q^{29} - 100 q^{30} + 32 q^{31} + 216 q^{32} + 62 q^{33} + 54 q^{34} - 86 q^{36} + 44 q^{37} + 252 q^{38} + 160 q^{39} - 30 q^{40} + 144 q^{41} - 270 q^{42} - 124 q^{43} + 140 q^{45} - 108 q^{46} - 216 q^{47} - 172 q^{48} - 54 q^{49} - 106 q^{51} + 62 q^{52} - 316 q^{54} - 18 q^{56} - 236 q^{57} + 90 q^{58} - 486 q^{59} - 10 q^{60} + 62 q^{61} - 132 q^{63} + 256 q^{64} - 90 q^{65} + 208 q^{66} + 14 q^{67} - 288 q^{68} + 90 q^{69} - 60 q^{70} + 804 q^{72} - 268 q^{73} + 540 q^{74} - 20 q^{75} - 106 q^{76} + 702 q^{77} + 290 q^{78} - 40 q^{79} - 112 q^{81} - 204 q^{82} + 522 q^{83} + 714 q^{84} + 30 q^{85} + 54 q^{86} + 106 q^{87} + 144 q^{88} + 250 q^{90} + 136 q^{91} - 1332 q^{92} + 90 q^{93} - 150 q^{94} + 180 q^{95} + 166 q^{96} - 142 q^{97} - 824 q^{99}+O(q^{100})$$ 16 * q + 4 * q^3 + 16 * q^4 - 22 * q^6 + 2 * q^7 + 8 * q^9 - 18 * q^11 - 22 * q^12 - 10 * q^13 - 54 * q^14 + 10 * q^15 - 32 * q^16 - 8 * q^18 - 52 * q^19 + 72 * q^21 - 24 * q^22 - 54 * q^23 + 108 * q^24 + 40 * q^25 + 34 * q^27 + 32 * q^28 - 54 * q^29 - 100 * q^30 + 32 * q^31 + 216 * q^32 + 62 * q^33 + 54 * q^34 - 86 * q^36 + 44 * q^37 + 252 * q^38 + 160 * q^39 - 30 * q^40 + 144 * q^41 - 270 * q^42 - 124 * q^43 + 140 * q^45 - 108 * q^46 - 216 * q^47 - 172 * q^48 - 54 * q^49 - 106 * q^51 + 62 * q^52 - 316 * q^54 - 18 * q^56 - 236 * q^57 + 90 * q^58 - 486 * q^59 - 10 * q^60 + 62 * q^61 - 132 * q^63 + 256 * q^64 - 90 * q^65 + 208 * q^66 + 14 * q^67 - 288 * q^68 + 90 * q^69 - 60 * q^70 + 804 * q^72 - 268 * q^73 + 540 * q^74 - 20 * q^75 - 106 * q^76 + 702 * q^77 + 290 * q^78 - 40 * q^79 - 112 * q^81 - 204 * q^82 + 522 * q^83 + 714 * q^84 + 30 * q^85 + 54 * q^86 + 106 * q^87 + 144 * q^88 + 250 * q^90 + 136 * q^91 - 1332 * q^92 + 90 * q^93 - 150 * q^94 + 180 * q^95 + 166 * q^96 - 142 * q^97 - 824 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{14} + 43 \nu^{12} + 715 \nu^{10} + 5777 \nu^{8} + 23051 \nu^{6} + 39821 \nu^{4} + 25605 \nu^{2} - 2376 \nu + 28755 ) / 4752$$ (v^14 + 43*v^12 + 715*v^10 + 5777*v^8 + 23051*v^6 + 39821*v^4 + 25605*v^2 - 2376*v + 28755) / 4752 $$\beta_{3}$$ $$=$$ $$( - \nu^{14} - 43 \nu^{12} - 715 \nu^{10} - 5777 \nu^{8} - 23051 \nu^{6} - 39821 \nu^{4} - 20853 \nu^{2} + 2376 \nu - 243 ) / 4752$$ (-v^14 - 43*v^12 - 715*v^10 - 5777*v^8 - 23051*v^6 - 39821*v^4 - 20853*v^2 + 2376*v - 243) / 4752 $$\beta_{4}$$ $$=$$ $$( 7 \nu^{15} - 68 \nu^{14} + 369 \nu^{13} - 3056 \nu^{12} + 7299 \nu^{11} - 53504 \nu^{10} + 65653 \nu^{9} - 460024 \nu^{8} + 241317 \nu^{7} - 2004520 \nu^{6} + 46971 \nu^{5} + \cdots - 532116 ) / 175824$$ (7*v^15 - 68*v^14 + 369*v^13 - 3056*v^12 + 7299*v^11 - 53504*v^10 + 65653*v^9 - 460024*v^8 + 241317*v^7 - 2004520*v^6 + 46971*v^5 - 4117852*v^4 - 1228523*v^3 - 3357348*v^2 - 1171917*v - 532116) / 175824 $$\beta_{5}$$ $$=$$ $$( 53 \nu^{15} + 155 \nu^{14} + 2741 \nu^{13} + 6659 \nu^{12} + 54989 \nu^{11} + 108827 \nu^{10} + 534013 \nu^{9} + 837547 \nu^{8} + 2524279 \nu^{7} + 2999593 \nu^{6} + \cdots + 1364877 ) / 351648$$ (53*v^15 + 155*v^14 + 2741*v^13 + 6659*v^12 + 54989*v^11 + 108827*v^10 + 534013*v^9 + 837547*v^8 + 2524279*v^7 + 2999593*v^6 + 4869775*v^5 + 4105945*v^4 + 1174971*v^3 + 1556541*v^2 - 2812581*v + 1364877) / 351648 $$\beta_{6}$$ $$=$$ $$( 215 \nu^{15} + 195 \nu^{14} + 10139 \nu^{13} + 9423 \nu^{12} + 187691 \nu^{11} + 177387 \nu^{10} + 1720855 \nu^{9} + 1631991 \nu^{8} + 8064517 \nu^{7} + 7502193 \nu^{6} + \cdots + 1868049 ) / 351648$$ (215*v^15 + 195*v^14 + 10139*v^13 + 9423*v^12 + 187691*v^11 + 177387*v^10 + 1720855*v^9 + 1631991*v^8 + 8064517*v^7 + 7502193*v^6 + 17804641*v^5 + 15610701*v^4 + 14745141*v^3 + 11599245*v^2 + 2590497*v + 1868049) / 351648 $$\beta_{7}$$ $$=$$ $$( - 3 \nu^{15} - 143 \nu^{13} - 2693 \nu^{11} - 25397 \nu^{9} - 125029 \nu^{7} - 304045 \nu^{5} - 313711 \nu^{3} - 87147 \nu - 2376 ) / 4752$$ (-3*v^15 - 143*v^13 - 2693*v^11 - 25397*v^9 - 125029*v^7 - 304045*v^5 - 313711*v^3 - 87147*v - 2376) / 4752 $$\beta_{8}$$ $$=$$ $$( - 261 \nu^{15} + 145 \nu^{14} - 12289 \nu^{13} + 6301 \nu^{12} - 227389 \nu^{11} + 106117 \nu^{10} - 2091313 \nu^{9} + 878585 \nu^{8} - 9940775 \nu^{7} + \cdots + 1084239 ) / 351648$$ (-261*v^15 + 145*v^14 - 12289*v^13 + 6301*v^12 - 227389*v^11 + 106117*v^10 - 2091313*v^9 + 878585*v^8 - 9940775*v^7 + 3758723*v^6 - 23140931*v^5 + 8039495*v^4 - 23654123*v^3 + 7282731*v^2 - 9140967*v + 1084239) / 351648 $$\beta_{9}$$ $$=$$ $$( 247 \nu^{15} - 77 \nu^{14} + 11847 \nu^{13} - 3245 \nu^{12} + 225075 \nu^{11} - 52613 \nu^{10} + 2150779 \nu^{9} - 413677 \nu^{8} + 10811157 \nu^{7} - 1622335 \nu^{6} + \cdots + 678645 ) / 351648$$ (247*v^15 - 77*v^14 + 11847*v^13 - 3245*v^12 + 225075*v^11 - 52613*v^10 + 2150779*v^9 - 413677*v^8 + 10811157*v^7 - 1622335*v^6 + 27181221*v^5 - 2822743*v^4 + 29303677*v^3 - 1029171*v^2 + 8323965*v + 678645) / 351648 $$\beta_{10}$$ $$=$$ $$( - 247 \nu^{15} - 77 \nu^{14} - 11847 \nu^{13} - 3245 \nu^{12} - 225075 \nu^{11} - 52613 \nu^{10} - 2150779 \nu^{9} - 413677 \nu^{8} - 10811157 \nu^{7} - 1622335 \nu^{6} + \cdots + 678645 ) / 351648$$ (-247*v^15 - 77*v^14 - 11847*v^13 - 3245*v^12 - 225075*v^11 - 52613*v^10 - 2150779*v^9 - 413677*v^8 - 10811157*v^7 - 1622335*v^6 - 27181221*v^5 - 2822743*v^4 - 29303677*v^3 - 1029171*v^2 - 8323965*v + 678645) / 351648 $$\beta_{11}$$ $$=$$ $$( 67 \nu^{15} - 5 \nu^{14} + 3257 \nu^{13} - 290 \nu^{12} + 62816 \nu^{11} - 6572 \nu^{10} + 610152 \nu^{9} - 73609 \nu^{8} + 3120355 \nu^{7} - 424519 \nu^{6} + 7992463 \nu^{5} + \cdots - 202257 ) / 87912$$ (67*v^15 - 5*v^14 + 3257*v^13 - 290*v^12 + 62816*v^11 - 6572*v^10 + 610152*v^9 - 73609*v^8 + 3120355*v^7 - 424519*v^6 + 7992463*v^5 - 1192840*v^4 + 8872982*v^3 - 1355460*v^2 + 2899188*v - 202257) / 87912 $$\beta_{12}$$ $$=$$ $$( - 245 \nu^{15} + 19 \nu^{14} - 11805 \nu^{13} + 769 \nu^{12} - 225273 \nu^{11} + 12253 \nu^{10} - 2159993 \nu^{9} + 105755 \nu^{8} - 10866783 \nu^{7} + 598721 \nu^{6} + \cdots + 336609 ) / 175824$$ (-245*v^15 + 19*v^14 - 11805*v^13 + 769*v^12 - 225273*v^11 + 12253*v^10 - 2159993*v^9 + 105755*v^8 - 10866783*v^7 + 598721*v^6 - 27234591*v^5 + 2189471*v^4 - 29353271*v^3 + 3278955*v^2 - 9022815*v + 336609) / 175824 $$\beta_{13}$$ $$=$$ $$( 161 \nu^{15} - 16 \nu^{14} + 7747 \nu^{13} - 632 \nu^{12} + 147749 \nu^{11} - 9072 \nu^{10} + 1418111 \nu^{9} - 55196 \nu^{8} + 7164083 \nu^{7} - 102148 \nu^{6} + 18140737 \nu^{5} + \cdots + 92304 ) / 58608$$ (161*v^15 - 16*v^14 + 7747*v^13 - 632*v^12 + 147749*v^11 - 9072*v^10 + 1418111*v^9 - 55196*v^8 + 7164083*v^7 - 102148*v^6 + 18140737*v^5 + 217836*v^4 + 19978355*v^3 + 688164*v^2 + 6405849*v + 92304) / 58608 $$\beta_{14}$$ $$=$$ $$( - 17 \nu^{15} - 815 \nu^{13} - 15443 \nu^{11} - 146605 \nu^{9} - 727123 \nu^{7} - 1784449 \nu^{5} - 1861413 \nu^{3} - 2376 \nu^{2} - 527391 \nu - 14256 ) / 4752$$ (-17*v^15 - 815*v^13 - 15443*v^11 - 146605*v^9 - 727123*v^7 - 1784449*v^5 - 1861413*v^3 - 2376*v^2 - 527391*v - 14256) / 4752 $$\beta_{15}$$ $$=$$ $$( - 1361 \nu^{15} - 13 \nu^{14} - 65433 \nu^{13} - 421 \nu^{12} - 1245657 \nu^{11} - 2413 \nu^{10} - 11920553 \nu^{9} + 54859 \nu^{8} - 59989011 \nu^{7} + 842569 \nu^{6} + \cdots + 1470933 ) / 351648$$ (-1361*v^15 - 13*v^14 - 65433*v^13 - 421*v^12 - 1245657*v^11 - 2413*v^10 - 11920553*v^9 + 54859*v^8 - 59989011*v^7 + 842569*v^6 - 151443987*v^5 + 4057561*v^4 - 167301143*v^3 + 6415965*v^2 - 53361783*v + 1470933) / 351648
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} - 6$$ b3 + b2 - 6 $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{14} + 2\beta_{11} - \beta_{8} + 3\beta_{7} - 2\beta_{4} - \beta_{2} - 10\beta _1 + 2$$ b15 - b14 + 2*b11 - b8 + 3*b7 - 2*b4 - b2 - 10*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{12} + \beta_{11} + 3\beta_{10} + 3\beta_{9} - \beta_{7} - 18\beta_{3} - 12\beta_{2} + 3\beta _1 + 60$$ b12 + b11 + 3*b10 + 3*b9 - b7 - 18*b3 - 12*b2 + 3*b1 + 60 $$\nu^{5}$$ $$=$$ $$- 18 \beta_{15} + 26 \beta_{14} + 6 \beta_{13} + 4 \beta_{12} - 34 \beta_{11} - 6 \beta_{10} - \beta_{9} + 10 \beta_{8} - 63 \beta_{7} + 7 \beta_{5} + 36 \beta_{4} + 4 \beta_{3} + 22 \beta_{2} + 116 \beta _1 - 42$$ -18*b15 + 26*b14 + 6*b13 + 4*b12 - 34*b11 - 6*b10 - b9 + 10*b8 - 63*b7 + 7*b5 + 36*b4 + 4*b3 + 22*b2 + 116*b1 - 42 $$\nu^{6}$$ $$=$$ $$4 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 23 \beta_{12} - 27 \beta_{11} - 57 \beta_{10} - 59 \beta_{9} - 4 \beta_{8} + 21 \beta_{7} - 8 \beta_{6} + 10 \beta_{5} - 18 \beta_{4} + 277 \beta_{3} + 139 \beta_{2} - 67 \beta _1 - 681$$ 4*b15 - 4*b14 + 2*b13 - 23*b12 - 27*b11 - 57*b10 - 59*b9 - 4*b8 + 21*b7 - 8*b6 + 10*b5 - 18*b4 + 277*b3 + 139*b2 - 67*b1 - 681 $$\nu^{7}$$ $$=$$ $$267 \beta_{15} - 471 \beta_{14} - 144 \beta_{13} - 96 \beta_{12} + 504 \beta_{11} + 186 \beta_{10} - 9 \beta_{9} - 75 \beta_{8} + 1020 \beta_{7} - 177 \beta_{5} - 552 \beta_{4} - 102 \beta_{3} - 369 \beta_{2} - 1421 \beta _1 + 684$$ 267*b15 - 471*b14 - 144*b13 - 96*b12 + 504*b11 + 186*b10 - 9*b9 - 75*b8 + 1020*b7 - 177*b5 - 552*b4 - 102*b3 - 369*b2 - 1421*b1 + 684 $$\nu^{8}$$ $$=$$ $$- 108 \beta_{15} + 108 \beta_{14} - 48 \beta_{13} + 432 \beta_{12} + 528 \beta_{11} + 900 \beta_{10} + 942 \beta_{9} + 108 \beta_{8} - 378 \beta_{7} + 216 \beta_{6} - 258 \beta_{5} + 468 \beta_{4} - 4022 \beta_{3} + \cdots + 8187$$ -108*b15 + 108*b14 - 48*b13 + 432*b12 + 528*b11 + 900*b10 + 942*b9 + 108*b8 - 378*b7 + 216*b6 - 258*b5 + 468*b4 - 4022*b3 - 1646*b2 + 1134*b1 + 8187 $$\nu^{9}$$ $$=$$ $$- 3806 \beta_{15} + 7550 \beta_{14} + 2568 \beta_{13} + 1746 \beta_{12} - 7258 \beta_{11} - 3834 \beta_{10} + 570 \beta_{9} + 314 \beta_{8} - 15072 \beta_{7} + 3264 \beta_{5} + 8080 \beta_{4} + 1872 \beta_{3} + \cdots - 10198$$ -3806*b15 + 7550*b14 + 2568*b13 + 1746*b12 - 7258*b11 - 3834*b10 + 570*b9 + 314*b8 - 15072*b7 + 3264*b5 + 8080*b4 + 1872*b3 + 5678*b2 + 18017*b1 - 10198 $$\nu^{10}$$ $$=$$ $$2088 \beta_{15} - 2088 \beta_{14} + 924 \beta_{13} - 7298 \beta_{12} - 9146 \beta_{11} - 13506 \beta_{10} - 14112 \beta_{9} - 2088 \beta_{8} + 6452 \beta_{7} - 4176 \beta_{6} + 4782 \beta_{5} - 8640 \beta_{4} + \cdots - 102042$$ 2088*b15 - 2088*b14 + 924*b13 - 7298*b12 - 9146*b11 - 13506*b10 - 14112*b9 - 2088*b8 + 6452*b7 - 4176*b6 + 4782*b5 - 8640*b4 + 56835*b3 + 20139*b2 - 17304*b1 - 102042 $$\nu^{11}$$ $$=$$ $$53703 \beta_{15} - 114319 \beta_{14} - 41040 \beta_{13} - 28598 \beta_{12} + 103604 \beta_{11} + 67170 \beta_{10} - 13732 \beta_{9} + 3493 \beta_{8} + 213993 \beta_{7} - 53438 \beta_{5} + \cdots + 146268$$ 53703*b15 - 114319*b14 - 41040*b13 - 28598*b12 + 103604*b11 + 67170*b10 - 13732*b9 + 3493*b8 + 213993*b7 - 53438*b5 - 116046*b4 - 30308*b3 - 84011*b2 - 234208*b1 + 146268 $$\nu^{12}$$ $$=$$ $$- 35528 \beta_{15} + 35528 \beta_{14} - 16156 \beta_{13} + 115609 \beta_{12} + 147921 \beta_{11} + 198303 \beta_{10} + 205525 \beta_{9} + 35528 \beta_{8} - 105171 \beta_{7} + 71056 \beta_{6} + \cdots + 1306266$$ -35528*b15 + 35528*b14 - 16156*b13 + 115609*b12 + 147921*b11 + 198303*b10 + 205525*b9 + 35528*b8 - 105171*b7 + 71056*b6 - 78278*b5 + 140400*b4 - 792812*b3 - 254066*b2 + 251609*b1 + 1306266 $$\nu^{13}$$ $$=$$ $$- 755544 \beta_{15} + 1678848 \beta_{14} + 622506 \beta_{13} + 443874 \beta_{12} - 1472856 \beta_{11} - 1083540 \beta_{10} + 258213 \beta_{9} - 132204 \beta_{8} - 2981067 \beta_{7} + \cdots - 2059032$$ -755544*b15 + 1678848*b14 + 622506*b13 + 443874*b12 - 1472856*b11 - 1083540*b10 + 258213*b9 - 132204*b8 - 2981067*b7 + 825327*b5 + 1651488*b4 + 461652*b3 + 1217196*b2 + 3101428*b1 - 2059032 $$\nu^{14}$$ $$=$$ $$566496 \beta_{15} - 566496 \beta_{14} + 265242 \beta_{13} - 1758429 \beta_{12} - 2288913 \beta_{11} - 2875095 \beta_{10} - 2948883 \beta_{9} - 566496 \beta_{8} + 1648629 \beta_{7} + \cdots - 17072361$$ 566496*b15 - 566496*b14 + 265242*b13 - 1758429*b12 - 2288913*b11 - 2875095*b10 - 2948883*b9 - 566496*b8 + 1648629*b7 - 1132992*b6 + 1206780*b5 - 2148318*b4 + 10995031*b3 + 3287305*b2 - 3570615*b1 - 17072361 $$\nu^{15}$$ $$=$$ $$10619317 \beta_{15} - 24221449 \beta_{14} - 9179076 \beta_{13} - 6672042 \beta_{12} + 20879918 \beta_{11} + 16666128 \beta_{10} - 4330389 \beta_{9} + 2724767 \beta_{8} + \cdots + 28720238$$ 10619317*b15 - 24221449*b14 - 9179076*b13 - 6672042*b12 + 20879918*b11 + 16666128*b10 - 4330389*b9 + 2724767*b8 + 41157462*b7 - 12335739*b5 - 23386952*b4 - 6801066*b3 - 17420383*b2 - 41637619*b1 + 28720238

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1 + \beta_{7}$$ $$1$$

## 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
 − 3.73655i − 3.27064i − 1.39204i − 0.0476108i 0.692902i 1.83391i 2.82877i 3.09125i 3.73655i 3.27064i 1.39204i 0.0476108i − 0.692902i − 1.83391i − 2.82877i − 3.09125i
−3.23594 1.86827i 2.62117 + 1.45927i 4.98088 + 8.62715i 1.93649 1.11803i −5.75562 9.61919i 3.63061 6.28840i 22.2764i 4.74103 + 7.65001i −8.35517
11.2 −2.83245 1.63532i −2.26486 1.96733i 3.34853 + 5.79983i −1.93649 + 1.11803i 3.19790 + 9.27615i −3.16931 + 5.48940i 8.82112i 1.25919 + 8.91148i 7.31337
11.3 −1.20554 0.696021i 2.33148 1.88791i −1.03111 1.78593i −1.93649 + 1.11803i −4.12473 + 0.653207i 4.41004 7.63842i 8.43886i 1.87156 8.80325i 3.11270
11.4 −0.0412321 0.0238054i −1.76580 2.42528i −1.99887 3.46214i 1.93649 1.11803i 0.0150729 + 0.142035i 1.90756 3.30399i 0.380778i −2.76393 + 8.56508i −0.106461
11.5 0.600071 + 0.346451i 2.96601 + 0.450312i −1.75994 3.04831i 1.93649 1.11803i 1.62381 + 1.29780i −6.14112 + 10.6367i 5.21055i 8.59444 + 2.67126i 1.54938
11.6 1.58822 + 0.916957i 0.822398 + 2.88508i −0.318381 0.551452i −1.93649 + 1.11803i −1.33934 + 5.33623i 3.23398 5.60142i 8.50342i −7.64732 + 4.74536i −4.10076
11.7 2.44978 + 1.41438i 0.110987 2.99795i 2.00096 + 3.46576i −1.93649 + 1.11803i 4.51214 7.18734i −3.97472 + 6.88441i 0.00541780i −8.97536 0.665467i −6.32531
11.8 2.67710 + 1.54563i −2.82138 + 1.01971i 2.77793 + 4.81151i 1.93649 1.11803i −9.12922 1.63094i 1.10296 1.91037i 4.80953i 6.92040 5.75396i 6.91225
41.1 −3.23594 + 1.86827i 2.62117 1.45927i 4.98088 8.62715i 1.93649 + 1.11803i −5.75562 + 9.61919i 3.63061 + 6.28840i 22.2764i 4.74103 7.65001i −8.35517
41.2 −2.83245 + 1.63532i −2.26486 + 1.96733i 3.34853 5.79983i −1.93649 1.11803i 3.19790 9.27615i −3.16931 5.48940i 8.82112i 1.25919 8.91148i 7.31337
41.3 −1.20554 + 0.696021i 2.33148 + 1.88791i −1.03111 + 1.78593i −1.93649 1.11803i −4.12473 0.653207i 4.41004 + 7.63842i 8.43886i 1.87156 + 8.80325i 3.11270
41.4 −0.0412321 + 0.0238054i −1.76580 + 2.42528i −1.99887 + 3.46214i 1.93649 + 1.11803i 0.0150729 0.142035i 1.90756 + 3.30399i 0.380778i −2.76393 8.56508i −0.106461
41.5 0.600071 0.346451i 2.96601 0.450312i −1.75994 + 3.04831i 1.93649 + 1.11803i 1.62381 1.29780i −6.14112 10.6367i 5.21055i 8.59444 2.67126i 1.54938
41.6 1.58822 0.916957i 0.822398 2.88508i −0.318381 + 0.551452i −1.93649 1.11803i −1.33934 5.33623i 3.23398 + 5.60142i 8.50342i −7.64732 4.74536i −4.10076
41.7 2.44978 1.41438i 0.110987 + 2.99795i 2.00096 3.46576i −1.93649 1.11803i 4.51214 + 7.18734i −3.97472 6.88441i 0.00541780i −8.97536 + 0.665467i −6.32531
41.8 2.67710 1.54563i −2.82138 1.01971i 2.77793 4.81151i 1.93649 + 1.11803i −9.12922 + 1.63094i 1.10296 + 1.91037i 4.80953i 6.92040 + 5.75396i 6.91225
 $$n$$: 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 45.3.i.a 16
3.b odd 2 1 135.3.i.a 16
4.b odd 2 1 720.3.bs.c 16
5.b even 2 1 225.3.j.b 16
5.c odd 4 2 225.3.i.b 32
9.c even 3 1 135.3.i.a 16
9.c even 3 1 405.3.c.a 16
9.d odd 6 1 inner 45.3.i.a 16
9.d odd 6 1 405.3.c.a 16
12.b even 2 1 2160.3.bs.c 16
15.d odd 2 1 675.3.j.b 16
15.e even 4 2 675.3.i.c 32
36.f odd 6 1 2160.3.bs.c 16
36.h even 6 1 720.3.bs.c 16
45.h odd 6 1 225.3.j.b 16
45.j even 6 1 675.3.j.b 16
45.k odd 12 2 675.3.i.c 32
45.l even 12 2 225.3.i.b 32

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 24 T^{14} + 408 T^{12} + \cdots + 81$$
$3$ $$T^{16} - 4 T^{15} + 4 T^{14} + \cdots + 43046721$$
$5$ $$(T^{4} - 5 T^{2} + 25)^{4}$$
$7$ $$T^{16} - 2 T^{15} + \cdots + 4654875290256$$
$11$ $$T^{16} + 18 T^{15} + \cdots + 112356358416$$
$13$ $$T^{16} + 10 T^{15} + \cdots + 69380636886016$$
$17$ $$T^{16} + 2790 T^{14} + \cdots + 17\!\cdots\!00$$
$19$ $$(T^{8} + 26 T^{7} - 1415 T^{6} + \cdots - 3133657244)^{2}$$
$23$ $$T^{16} + 54 T^{15} + \cdots + 64\!\cdots\!16$$
$29$ $$T^{16} + 54 T^{15} + \cdots + 21\!\cdots\!56$$
$31$ $$T^{16} - 32 T^{15} + \cdots + 396718580736$$
$37$ $$(T^{8} - 22 T^{7} + \cdots - 143779124336)^{2}$$
$41$ $$T^{16} - 144 T^{15} + \cdots + 44\!\cdots\!61$$
$43$ $$T^{16} + 124 T^{15} + \cdots + 36\!\cdots\!56$$
$47$ $$T^{16} + 216 T^{15} + \cdots + 28\!\cdots\!76$$
$53$ $$T^{16} + 30792 T^{14} + \cdots + 32\!\cdots\!00$$
$59$ $$T^{16} + 486 T^{15} + \cdots + 25\!\cdots\!76$$
$61$ $$T^{16} - 62 T^{15} + \cdots + 18\!\cdots\!56$$
$67$ $$T^{16} - 14 T^{15} + \cdots + 39\!\cdots\!61$$
$71$ $$T^{16} + 24048 T^{14} + \cdots + 34\!\cdots\!00$$
$73$ $$(T^{8} + 134 T^{7} + \cdots - 3873480104384)^{2}$$
$79$ $$T^{16} + 40 T^{15} + \cdots + 15\!\cdots\!16$$
$83$ $$T^{16} - 522 T^{15} + \cdots + 23\!\cdots\!96$$
$89$ $$T^{16} + 18666 T^{14} + \cdots + 23\!\cdots\!00$$
$97$ $$T^{16} + 142 T^{15} + \cdots + 17\!\cdots\!76$$