# Properties

 Label 33.3.g.a Level $33$ Weight $3$ Character orbit 33.g Analytic conductor $0.899$ 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] = [33,3,Mod(7,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("33.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.g (of order $$10$$, degree $$4$$, 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: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521$$ x^16 - 2*x^15 + 3*x^14 - 4*x^13 + 77*x^12 + 88*x^11 - 577*x^10 + 578*x^9 + 1520*x^8 + 1868*x^7 - 1619*x^6 - 16804*x^5 + 32427*x^4 + 43316*x^3 - 71672*x^2 + 83521 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{8} + \beta_{7} - \beta_{5} + \cdots + 1) q^{2}+ \cdots + 3 \beta_{8} q^{9}+O(q^{10})$$ q + (b8 + b7 - b5 + b2 + 1) * q^2 - b14 * q^3 + (-b15 - 2*b8 - 2*b7 - b6 + 2*b5 + b4 - b3 - b1) * q^4 + (-3*b15 - 2*b14 + b13 - 2*b12 - b9 + 2*b8 + b7 - b6 + b4 - b3 + b2 + 1) * q^5 + (b15 + b14 - b13 - b10 - b8 - b7 + b6 - b4 + b3 - b2 - 1) * q^6 + (2*b12 + 2*b11 + b10 - 2*b8 - b7 - b6 + 2*b4 + 2*b1 - 3) * q^7 + (3*b15 + b14 + b12 - b11 + b10 + b9 + 4*b8 + 5*b7 + 2*b6 - 3*b5 - 2*b4 - b2) * q^8 + 3*b8 * q^9 $$q + (\beta_{8} + \beta_{7} - \beta_{5} + \cdots + 1) q^{2}+ \cdots + ( - 9 \beta_{15} - 9 \beta_{14} + \cdots + 6) q^{99}+O(q^{100})$$ q + (b8 + b7 - b5 + b2 + 1) * q^2 - b14 * q^3 + (-b15 - 2*b8 - 2*b7 - b6 + 2*b5 + b4 - b3 - b1) * q^4 + (-3*b15 - 2*b14 + b13 - 2*b12 - b9 + 2*b8 + b7 - b6 + b4 - b3 + b2 + 1) * q^5 + (b15 + b14 - b13 - b10 - b8 - b7 + b6 - b4 + b3 - b2 - 1) * q^6 + (2*b12 + 2*b11 + b10 - 2*b8 - b7 - b6 + 2*b4 + 2*b1 - 3) * q^7 + (3*b15 + b14 + b12 - b11 + b10 + b9 + 4*b8 + 5*b7 + 2*b6 - 3*b5 - 2*b4 - b2) * q^8 + 3*b8 * q^9 + (6*b15 + 6*b14 - 2*b11 + 2*b9 - 2*b8 + b7 + 2*b6 + 3*b5 - 2*b4 + 2*b3 - 2*b2 - 2) * q^10 + (b14 - 2*b13 + b12 - 9*b8 - 10*b7 + b6 + 2*b5 - 4*b4 + 4*b3 - 3*b2 - b1 - 6) * q^11 + (-b15 - 2*b14 + b13 - b12 - b11 + b10 - b9 + 3*b7 - 3*b5 - b4 + b2 - b1 + 1) * q^12 + (-2*b15 + 2*b14 + 2*b13 - 2*b11 - b10 + 2*b8 - b5 - b3 + b1 + 3) * q^13 + (8*b15 + 4*b14 - 2*b13 + 2*b12 + b11 - 6*b10 - 2*b8 - b7 + 3*b6 + 3*b5 + b3 - 3*b2 + 3*b1 - 4) * q^14 + (-b15 - b14 + b13 - b12 + b9 + 5*b8 + 5*b7 + 2*b6 - 3*b5 - b4 - b3 + b2 + 2*b1 + 1) * q^15 + (-7*b15 - 4*b14 + 4*b11 + 2*b10 - 2*b9 - 4*b8 - b7 - 4*b6 - 3*b5 - b4 - 2*b3 - 6*b1 + 3) * q^16 + (3*b15 - b13 - 2*b10 + 3*b9 + 6*b8 + b7 + 5*b4 + 3*b3 - 1) * q^17 + (-3*b8 - 3*b7 - 3*b6 + 3*b5 - 3) * q^18 + (-5*b15 - 5*b14 - 2*b12 + 5*b10 - 2*b9 + 2*b8 - 4*b7 - 3*b6 + 5*b5 + 6*b4 - 8*b3 + 5*b2 + 3*b1 - 1) * q^19 + (-14*b15 - 12*b14 + 2*b13 - 2*b12 + 2*b11 + 8*b10 - 4*b9 - 9*b8 - b7 - 8*b6 + 2*b5 + 2*b4 - 8*b3 + 3*b2 - 3*b1 + 4) * q^20 + (b15 + 2*b14 + b11 + b10 - b9 + 5*b8 + 5*b7 - b6 + 6*b4 + b3 + 6*b2 + 3) * q^21 + (-2*b15 + b14 - 4*b12 - 2*b11 - 2*b10 + 20*b8 + 16*b7 + 7*b6 - 6*b5 + 3*b4 + 7*b3 + 3*b2 + 2*b1 + 10) * q^22 + (-b15 - 6*b14 + 2*b13 - b11 + 5*b10 - b9 - 3*b7 + 3*b5 - b4 + b2 - b1 + 8) * q^23 + (-b15 + 3*b14 - b13 + 2*b12 + b11 - 2*b10 - 4*b8 - 7*b7 - 2*b3 - 4*b2 - b1 + 2) * q^24 + (15*b15 + 2*b14 - 4*b13 + 2*b12 - 11*b10 + 2*b9 - 5*b8 - 5*b7 - 3*b6 - 3*b5 - 5*b4 + 2*b3 - 2*b1) * q^25 + (-10*b15 - b14 + 2*b13 - b12 - b11 - 2*b10 + 2*b9 + 2*b8 + 3*b7 + 4*b5 + 3*b4 - 11*b3 + 4*b2 + b1 + 3) * q^26 - 3*b15 * q^27 + (2*b15 - 7*b14 + 2*b13 - 6*b10 - 7*b8 + 2*b7 - b6 - 3*b5 - 7*b4 + 11*b3 + b2 - 1) * q^28 + (7*b15 + 15*b14 + b13 - b12 - 8*b10 - b9 + 2*b8 + 7*b7 - b6 - 12*b5 - 2*b4 + 7*b3 + b2 - 3*b1 + 17) * q^29 + (b15 - b14 + 2*b12 + 3*b10 + 2*b9 - 22*b8 - 16*b7 + 6*b5 - 4*b4 - 4*b3 - 6*b2 - 4*b1) * q^30 + (-4*b15 - 4*b14 - 2*b13 - 2*b11 + 7*b10 + 4*b8 + b7 + 18*b6 - 7*b5 - 12*b4 - 7*b3 - 6*b2 + 3*b1 - 2) * q^31 + (13*b15 + 14*b14 - b11 + b10 + b9 + 17*b8 + b6 - 17*b5 + b4 + 6*b3 + b2 + 8) * q^32 + (5*b15 + 4*b14 - b13 + 2*b12 - b11 + b10 + b9 + 3*b7 - 4*b6 + 12*b5 + 2*b4 + 5*b3 - b2 + 2*b1 - 1) * q^33 + (-13*b14 + 13*b10 + 4*b8 + 3*b7 - 4*b6 + b5 + 5*b4 - b3 + 3*b2 + 4*b1 - 21) * q^34 + (-6*b15 + 8*b14 + 2*b12 + 4*b10 - 7*b8 + 9*b7 + 17*b5 + 4*b3 + 3*b2 + 2*b1 - 23) * q^35 + (3*b15 + 3*b14 - 3*b10 + 6*b8 + 3*b6 - 3*b2 + 3*b1 + 3) * q^36 + (b15 + 2*b14 - 2*b13 + 2*b12 - 7*b10 - 2*b9 - 5*b8 - 6*b7 - 6*b6 - 9*b5 - b4 + b3 - 12*b2 - 11*b1 - 7) * q^37 + (11*b15 + 4*b14 - 3*b13 + 6*b12 + 3*b9 - 6*b8 - 11*b7 + 4*b6 + 20*b5 + 5*b4 + b3 - 2*b2 + 16*b1 - 23) * q^38 + (-2*b14 + 2*b13 - b10 + b9 - 3*b8 + 9*b7 + 4*b6 - 3*b5 + 3*b4 + b3 - 4*b2 - 1) * q^39 + (5*b15 + 12*b14 - 2*b13 - 2*b12 - 4*b11 - 7*b10 + 2*b9 + 32*b8 + 9*b7 + 12*b6 + 14*b5 + 5*b4 + 9*b3 - 2*b2 + 7*b1 + 1) * q^40 + (-12*b15 - 12*b14 - 2*b12 + 5*b11 + 12*b10 - 2*b9 + 4*b8 + 8*b7 - 10*b5 - 7*b4 - 17*b3 - 5*b2 - 7*b1 - 1) * q^41 + (-6*b15 - 5*b14 + 2*b13 - b12 + 2*b11 + 5*b10 - 2*b9 - 6*b8 - 18*b7 - 10*b6 + 3*b5 + 8*b4 - 5*b3 + 5*b2 - b1 - 16) * q^42 + (6*b15 + 5*b14 + 2*b13 - 2*b12 + 4*b11 - b10 - 4*b9 - 40*b8 - 23*b7 - 6*b6 + 17*b5 - 11*b4 + 5*b3 - 11*b2 - 2*b1 - 17) * q^43 + (-12*b15 - 7*b14 + 5*b13 - b12 + 3*b10 + 2*b9 - 17*b8 - 8*b7 - 11*b6 - 20*b5 - b4 + 2*b3 - b2 - 8*b1 - 9) * q^44 + (3*b15 + 3*b14 - 3*b13 + 3*b12 + 3*b11 + 3*b9 - 3*b8 + 3*b6 - 3*b5 - 3*b4 - 3*b2 - 3) * q^45 + (-7*b15 + 5*b14 - 6*b13 - 2*b12 + 6*b11 - 6*b10 + 5*b8 + b7 - 8*b5 - 6*b3 + 5*b2 + b1 + 6) * q^46 + (b15 - 9*b14 + 8*b13 - 7*b12 - 3*b11 - 9*b10 - b9 + 10*b7 + 4*b6 - 3*b5 + 7*b4 - 4*b3 + 2*b2 - b1 - 8) * q^47 + (2*b15 + 3*b14 - 3*b13 + 3*b12 - 3*b10 - 3*b9 + 12*b8 + 15*b7 + 3*b6 - 6*b5 + 2*b3 + 9*b2 + 9*b1 + 3) * q^48 + (-b15 + b14 + 2*b13 - 4*b12 - 12*b11 - 6*b10 + 4*b9 - 2*b8 + 5*b6 - 15*b5 - 8*b4 - 3*b3 - 3*b2 - 8*b1 + 17) * q^49 + (-b15 - 4*b14 - 11*b10 - 13*b9 + 10*b8 - 25*b7 - b6 + 17*b5 - 2*b4 + 15*b3 + b2 + 17) * q^50 + (-5*b15 - b14 + 2*b13 - 3*b12 - b11 - b10 - 2*b9 - 9*b8 - 9*b7 - 4*b6 + 9*b5 - 10*b4 + 2*b3 + 2*b2 - 12*b1 - 4) * q^51 + (12*b15 + 5*b14 + 4*b12 - 2*b11 + 2*b10 + 4*b9 + 10*b8 + 9*b7 + 2*b6 + 2*b5 + 11*b4 + 4*b3 + 9*b2 + 13*b1 + 8) * q^52 + (18*b15 + 10*b14 - 8*b13 + 8*b12 - 8*b11 + 2*b10 + 16*b9 + 10*b8 + 36*b7 - 2*b6 + 9*b5 + 8*b4 - 2*b3 - 4*b2 + 3*b1 + 25) * q^53 + (3*b15 + 3*b14 - 3*b11 + 3*b9 + 3*b8 + 3*b7 + 3*b6) * q^54 + (-12*b15 - 15*b14 + 2*b13 + 12*b12 + 10*b11 + 23*b10 - 2*b9 - 8*b8 + 7*b7 + b6 + 5*b5 + 3*b4 - 6*b3 + 7*b2 + 3*b1 + 15) * q^55 + (2*b15 - 3*b14 - 7*b13 - 3*b12 + 2*b11 + 5*b10 + 2*b9 - 10*b8 - 30*b7 + 10*b6 + 20*b5 - 10*b4 - 10*b3 - 10*b2 - 8*b1 + 13) * q^56 + (-4*b15 - 4*b14 + 3*b13 - 8*b12 - 3*b11 + 2*b10 + 18*b8 + 12*b7 - 24*b5 + 2*b3 + 3*b2 - 2*b1 + 18) * q^57 + (-22*b15 - 9*b14 + 16*b13 - 8*b12 + 6*b10 - 8*b9 + 27*b8 + 13*b7 - 20*b6 - 16*b5 + 13*b4 - 8*b3 + 25*b2 - 17*b1 + 39) * q^58 + (11*b15 + b14 - 7*b13 + b12 + 6*b11 - 8*b10 - 7*b9 - 3*b8 - 20*b7 + 6*b6 + 3*b5 - 6*b4 + 17*b3 + 7*b2 + 13*b1) * q^59 + (7*b15 - b14 + b13 - 2*b12 - 4*b11 - 2*b10 + b9 + 44*b8 + 38*b7 + 9*b6 - 24*b5 + 4*b4 - 2*b3 + 7*b2 + 10*b1 + 25) * q^60 + (3*b15 + 13*b14 - 4*b13 - 2*b10 - 12*b8 + 14*b7 - 17*b6 + 6*b5 + 6*b4 - 7*b3 + 17*b2 + 2) * q^61 + (-12*b15 - 11*b14 - 4*b13 + 5*b12 + b11 + 8*b10 + 4*b9 - 49*b8 - 5*b7 - 39*b5 - 10*b4 + 5*b3 - 4*b2 - 6*b1) * q^62 + (-6*b15 - 3*b14 - 6*b12 - 6*b9 - 9*b8 - 9*b7 + 3*b5 - 3*b4 + 3*b2 - 3*b1 + 3) * q^63 + (-4*b15 - 6*b14 + 6*b13 + 2*b12 + 6*b11 + 8*b10 + 4*b9 + 3*b8 - 6*b7 - 4*b6 - 4*b5 + 12*b4 - 8*b3 + 4*b2 + 4*b1 - 12) * q^64 + (-2*b15 - 9*b14 - b13 + b12 + 6*b11 - 7*b10 - 6*b9 + 60*b8 + 33*b7 - 4*b6 - 27*b5 + 9*b4 + 2*b3 + 9*b2 + 2*b1 + 32) * q^65 + (-15*b15 - 11*b14 + 2*b13 - 5*b12 + 5*b11 - b10 - 4*b9 - 17*b8 - 8*b7 - 7*b6 + 21*b5 - 9*b4 - 7*b3 - 4*b2 - 5*b1 - 4) * q^66 + (4*b15 + 15*b14 - 2*b13 + 6*b12 + 4*b11 - 11*b10 + 4*b9 + 9*b8 - 9*b6 + 9*b5 + 25*b4 - 7*b2 + 13*b1 - 5) * q^67 + (2*b15 - 9*b14 - b13 - 7*b12 + b11 - 15*b10 - 26*b8 - 29*b7 + 29*b5 - 15*b3 - 20*b2 - 9*b1 - 21) * q^68 + (-4*b15 - 10*b14 + b12 + b11 + 4*b10 - b9 + 20*b8 + 2*b7 + 5*b6 + 3*b4 - 3*b2 + 4*b1 + 15) * q^69 + (-15*b15 - 4*b14 + 6*b13 - 4*b12 - 2*b11 + 4*b10 + 6*b9 - 48*b8 - 34*b7 + 3*b6 + 28*b5 - 13*b4 - 17*b3 - 26*b2 - 13*b1 - 10) * q^70 + (-13*b15 - 2*b13 + 4*b12 - 2*b11 - b10 + 3*b9 - 40*b8 - 33*b7 + 10*b6 + 41*b5 - 8*b4 - 3*b3 + 4*b2 - 10*b1 - 43) * q^71 + (-3*b15 - 6*b14 + 3*b13 - 3*b9 - 3*b8 + 9*b7 - 6*b6 - 6*b5 + 6*b4 + 3*b3 + 6*b2 - 3) * q^72 + (2*b15 + 12*b14 + 6*b13 - 14*b12 - 8*b11 - 13*b10 - 6*b9 + 10*b8 - 6*b7 + 7*b6 + 22*b5 + 10*b4 + 4*b3 + 6*b2 + 4*b1 - 29) * q^73 + (19*b15 + 15*b14 - 3*b12 - 8*b11 - 11*b10 - 3*b9 + 76*b8 + 72*b7 - 4*b6 - 29*b5 + 6*b4 + 25*b3 + 5*b2 + 2*b1 + 1) * q^74 + (16*b15 + 13*b14 - 7*b13 + 3*b12 - 7*b11 - 10*b10 + 6*b9 - 7*b8 - 40*b7 + 2*b6 + 6*b5 - 8*b4 + 10*b3 - 7*b2 - 3*b1 - 40) * q^75 + (-18*b15 - 7*b14 - 10*b13 + 10*b12 + 8*b11 + 11*b10 - 8*b9 - 60*b8 - 23*b7 - 2*b6 + 37*b5 - 11*b4 - 5*b3 - 11*b2 + 6*b1 - 29) * q^76 + (5*b15 - 7*b14 + 9*b13 - 9*b12 - 3*b11 + 10*b10 - 9*b9 + 41*b8 - 8*b7 - 6*b6 - 21*b5 + 23*b4 - 10*b3 + 14*b2 + 22*b1 - 11) * q^77 + (b15 - 4*b14 - 2*b13 + b11 + 5*b10 + b9 - 7*b8 + 26*b7 + 7*b6 - 33*b5 - 9*b4 - b3 - 5*b2 - 6*b1 - 8) * q^78 + (5*b15 - b14 + 2*b13 + 4*b12 - 2*b11 + 18*b10 + 16*b8 + 10*b7 - 38*b5 + 18*b3 + b2 - 5*b1 + 31) * q^79 + (-19*b15 + 5*b14 - 2*b13 - 3*b12 - 4*b11 + 21*b10 + 5*b9 - 88*b8 - 12*b7 - 13*b6 + 30*b5 - 16*b4 + b3 + 2*b2 - 7*b1 - 74) * q^80 + 9*b7 * q^81 + (59*b15 + 31*b14 - 10*b13 + 20*b12 + 10*b9 + 4*b8 + 14*b7 - b6 - 24*b5 - 10*b4 + 21*b3 - 21*b2 + 14) * q^82 + (-12*b15 - 8*b14 + 10*b13 + 26*b10 + 12*b9 + 32*b8 - 30*b7 + 19*b6 - 24*b5 - 20*b4 - 28*b3 - 19*b2 - 14) * q^83 + (19*b15 + 13*b14 - 8*b13 + 9*b12 + b11 - 2*b10 + 8*b9 + 21*b8 - 6*b7 + 13*b6 + 33*b5 + b4 + 2*b3 - 8*b2 + 9*b1 - 26) * q^84 + (-5*b15 - 2*b14 + 8*b12 + 8*b11 - b10 + 8*b9 - 38*b8 - 49*b7 + 7*b6 + 15*b5 - 11*b4 - 4*b3 - 12*b2 - 4*b1 - 23) * q^85 + (18*b15 + 23*b14 + 7*b13 - 5*b12 + 7*b11 - 19*b10 - 10*b9 + 46*b8 + 75*b7 + 24*b6 - 19*b5 - 18*b4 + 19*b3 - 4*b2 + 3*b1 + 77) * q^86 + (10*b15 - 2*b14 - 2*b13 + 2*b12 - 12*b10 - 47*b8 - 26*b7 + 5*b6 + 21*b5 - 2*b4 + 5*b3 - 2*b2 + 5*b1 - 26) * q^87 + (32*b15 + 36*b14 - 6*b13 - 12*b11 - 40*b10 + 6*b9 + 31*b8 + 11*b7 + 11*b6 + 17*b5 + 8*b4 - b3 - 4*b2 - 5*b1 + 8) * q^88 + (3*b15 + 22*b14 + 2*b13 + 8*b12 + 3*b11 - 19*b10 + 3*b9 + 10*b8 + 31*b7 - 10*b6 - 21*b5 + 3*b4 + 26*b3 + 17*b2 + 13*b1 + 26) * q^89 + (12*b15 - 6*b14 + 6*b12 + 6*b10 + 3*b8 + 3*b7 - 12*b5 + 6*b3 + 9) * q^90 + (-5*b15 - 12*b13 + 6*b12 + 17*b10 + 6*b9 + 2*b8 - 17*b7 + 9*b6 + 33*b5 - 17*b4 + 6*b3 - 20*b2 + 14*b1 - 1) * q^91 + (19*b15 + 8*b14 + 3*b13 + 8*b12 - 11*b11 + 10*b10 + 3*b9 + 18*b8 + 21*b7 + 4*b6 - 10*b5 + 8*b3 + 5*b2 + 5*b1 + 4) * q^92 + (-6*b15 - 3*b14 - 4*b13 + 8*b12 + 14*b11 + 7*b10 - 3*b9 + 5*b8 + 5*b7 - 16*b6 - 21*b5 + 7*b4 - 10*b2 + 8*b1 + 17) * q^93 + (-2*b15 + 3*b14 - 2*b13 + 4*b10 - 2*b9 - 25*b8 + 8*b7 + 15*b6 - 24*b5 - 13*b4 - 5*b3 - 15*b2 - 26) * q^94 + (7*b15 - 16*b14 + 5*b13 - 12*b12 - 7*b11 + 2*b10 - 5*b9 - 6*b8 + 17*b7 - 16*b6 - 40*b5 + 2*b4 - 28*b3 + 5*b2 - 3*b1 + 48) * q^95 + (b15 + 9*b14 - b12 - 17*b10 - b9 - 46*b8 - 40*b7 + 18*b5 - 4*b4 + 19*b3 - 3*b2 - 4*b1 + 3) * q^96 + (-8*b15 - 2*b14 - 6*b12 - 23*b10 - 12*b9 - b8 - 33*b7 + 6*b6 - 3*b5 - 4*b4 + 23*b3 + 4*b2 + b1 - 28) * q^97 + (-47*b15 - 25*b14 + 5*b13 - 5*b12 - 5*b11 + 22*b10 + 5*b9 + 58*b8 + 31*b7 + 4*b6 - 27*b5 + 23*b4 - 26*b3 + 23*b2 - b1 + 27) * q^98 + (-9*b15 - 9*b14 + 3*b13 - 3*b12 - 6*b11 + 6*b10 + 3*b9 - 3*b8 - 3*b7 + 15*b5 + 3*b4 - 15*b3 - 6*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9}+O(q^{10})$$ 16 * q + 20 * q^4 - 4 * q^5 - 30 * q^7 - 40 * q^8 - 12 * q^9 $$16 q + 20 q^{4} - 4 q^{5} - 30 q^{7} - 40 q^{8} - 12 q^{9} - 10 q^{11} - 24 q^{12} + 30 q^{13} - 2 q^{14} - 24 q^{15} + 16 q^{16} - 10 q^{17} - 30 q^{18} + 42 q^{20} + 42 q^{22} + 132 q^{23} + 90 q^{24} - 2 q^{25} + 46 q^{26} - 50 q^{28} + 160 q^{29} + 180 q^{30} + 10 q^{31} + 12 q^{33} - 368 q^{34} - 320 q^{35} + 60 q^{36} - 126 q^{37} - 130 q^{38} + 30 q^{40} - 120 q^{41} - 204 q^{42} - 206 q^{44} - 12 q^{45} + 50 q^{46} - 150 q^{47} - 96 q^{48} + 210 q^{49} + 330 q^{50} - 60 q^{51} + 110 q^{52} + 342 q^{53} + 244 q^{55} + 524 q^{56} + 60 q^{57} + 150 q^{58} + 110 q^{59} + 36 q^{60} - 90 q^{61} + 40 q^{62} + 90 q^{63} - 168 q^{64} + 48 q^{66} + 36 q^{67} + 80 q^{68} + 210 q^{69} + 340 q^{70} - 236 q^{71} - 150 q^{72} - 350 q^{73} - 730 q^{74} - 408 q^{75} - 390 q^{77} - 312 q^{78} + 210 q^{79} - 806 q^{80} - 36 q^{81} + 114 q^{82} - 190 q^{83} - 180 q^{84} + 110 q^{85} + 736 q^{86} + 144 q^{88} + 76 q^{89} + 60 q^{90} + 306 q^{91} - 150 q^{92} + 144 q^{93} - 350 q^{94} + 430 q^{95} + 450 q^{96} - 354 q^{97} + 180 q^{99}+O(q^{100})$$ 16 * q + 20 * q^4 - 4 * q^5 - 30 * q^7 - 40 * q^8 - 12 * q^9 - 10 * q^11 - 24 * q^12 + 30 * q^13 - 2 * q^14 - 24 * q^15 + 16 * q^16 - 10 * q^17 - 30 * q^18 + 42 * q^20 + 42 * q^22 + 132 * q^23 + 90 * q^24 - 2 * q^25 + 46 * q^26 - 50 * q^28 + 160 * q^29 + 180 * q^30 + 10 * q^31 + 12 * q^33 - 368 * q^34 - 320 * q^35 + 60 * q^36 - 126 * q^37 - 130 * q^38 + 30 * q^40 - 120 * q^41 - 204 * q^42 - 206 * q^44 - 12 * q^45 + 50 * q^46 - 150 * q^47 - 96 * q^48 + 210 * q^49 + 330 * q^50 - 60 * q^51 + 110 * q^52 + 342 * q^53 + 244 * q^55 + 524 * q^56 + 60 * q^57 + 150 * q^58 + 110 * q^59 + 36 * q^60 - 90 * q^61 + 40 * q^62 + 90 * q^63 - 168 * q^64 + 48 * q^66 + 36 * q^67 + 80 * q^68 + 210 * q^69 + 340 * q^70 - 236 * q^71 - 150 * q^72 - 350 * q^73 - 730 * q^74 - 408 * q^75 - 390 * q^77 - 312 * q^78 + 210 * q^79 - 806 * q^80 - 36 * q^81 + 114 * q^82 - 190 * q^83 - 180 * q^84 + 110 * q^85 + 736 * q^86 + 144 * q^88 + 76 * q^89 + 60 * q^90 + 306 * q^91 - 150 * q^92 + 144 * q^93 - 350 * q^94 + 430 * q^95 + 450 * q^96 - 354 * q^97 + 180 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 138203247537734 \nu^{15} - 291050501651580 \nu^{14} - 300665341738367 \nu^{13} + \cdots - 10\!\cdots\!14 ) / 10\!\cdots\!99$$ (138203247537734*v^15 - 291050501651580*v^14 - 300665341738367*v^13 - 1705096652769584*v^12 + 9450527129283881*v^11 + 12120978905635508*v^10 - 128500627016425958*v^9 - 144385372422156236*v^8 + 122498246567678379*v^7 + 788622311310230084*v^6 - 133037791520559623*v^5 - 5550683119927089754*v^4 - 1059049981018273091*v^3 + 8378055018470127650*v^2 + 13020347824350349103*v - 10624181552957568214) / 10999895242048793399 $$\beta_{3}$$ $$=$$ $$( - 1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + \cdots + 87\!\cdots\!85 ) / 78\!\cdots\!34$$ (-1196620054805*v^15 + 7612440593330*v^14 - 13694873409440*v^13 + 37431650235165*v^12 - 160608059534630*v^11 + 396329540870948*v^10 + 970794484128155*v^9 - 2042076941900150*v^8 + 1143522707363375*v^7 - 3000401551565675*v^6 + 26580224433121053*v^5 + 18238786120480205*v^4 - 74422428601993325*v^3 + 43546075769649555*v^2 + 84006679985420530*v + 87548232770773285) / 78081122146535134 $$\beta_{4}$$ $$=$$ $$( 79\!\cdots\!69 \nu^{15} + \cdots + 55\!\cdots\!82 ) / 41\!\cdots\!94$$ (79900318483541295869*v^15 - 414504109012258813809*v^14 + 817712598296562884653*v^13 - 1181236305529453459044*v^12 + 9352590043947181216000*v^11 - 13643260700327505255674*v^10 - 56158309327266771570697*v^9 + 189766153484548192315027*v^8 + 95964394821698805650574*v^7 + 136183765825282943853761*v^6 - 829913842787061972319720*v^5 - 811030725865674829791365*v^4 + 6972042698248130206151734*v^3 + 1663249793505431751688693*v^2 - 10114210902968983105987135*v + 5561216538715898968492982) / 4136026610381798610784394 $$\beta_{5}$$ $$=$$ $$( - 12\!\cdots\!15 \nu^{15} + \cdots - 57\!\cdots\!30 ) / 41\!\cdots\!94$$ (-122411618273740428915*v^15 + 390636691234340826078*v^14 - 875766486639397095366*v^13 + 1306893009198853700308*v^12 - 11122165667396999800860*v^11 + 1352851863401509576374*v^10 + 68084957445524132210249*v^9 - 180048790704766856126064*v^8 - 7207303915835648139573*v^7 - 246702387055823098949248*v^6 + 488650962940974324993039*v^5 + 1534542665221003166135310*v^4 - 6976139470347708662854381*v^3 + 1584736358268525956493486*v^2 + 6489773955290969592983712*v - 5737747263618475322503230) / 4136026610381798610784394 $$\beta_{6}$$ $$=$$ $$( - 13\!\cdots\!90 \nu^{15} + \cdots - 14\!\cdots\!47 ) / 41\!\cdots\!94$$ (-136359504183385890890*v^15 + 375227479116253831410*v^14 - 988352396932402617563*v^13 + 1181001191967106814165*v^12 - 10341194991166970946468*v^11 - 5186301744109515584546*v^10 + 72904516698863990088998*v^9 - 223430759423433745295680*v^8 + 152854650725265611821303*v^7 - 104458453399814352710637*v^6 + 131176280566363812419931*v^5 + 1643119706749150800729097*v^4 - 7289423934676201623932819*v^3 + 8681896656673390953560247*v^2 + 7009039219983139174322425*v - 14395227503745014111293247) / 4136026610381798610784394 $$\beta_{7}$$ $$=$$ $$( - 17\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!48 ) / 41\!\cdots\!94$$ (-170240978662162064119*v^15 + 434330161717510886082*v^14 - 909817832880658011498*v^13 + 1611262423238216663326*v^12 - 14163899845284185510064*v^11 - 6409586754553109280864*v^10 + 91124937595251030901909*v^9 - 159377154247336302510228*v^8 - 25618165363271255976237*v^7 - 382107707960321190084594*v^6 + 269559976623314579423811*v^5 + 2388290245026519865709064*v^4 - 6440011982471751717571965*v^3 - 88831069153588802148432*v^2 + 6767600917775046214158900*v - 10633476167661152687325848) / 4136026610381798610784394 $$\beta_{8}$$ $$=$$ $$( - 20\!\cdots\!52 \nu^{15} + \cdots - 12\!\cdots\!26 ) / 41\!\cdots\!94$$ (-206074168461557516052*v^15 + 708563165563293316406*v^14 - 1346611723653217936835*v^13 + 2263013632083543127888*v^12 - 18560750721332235068174*v^11 + 5237533698052177949662*v^10 + 133720108959147123348880*v^9 - 287797606597911626958148*v^8 - 55824128700867687245691*v^7 - 399468647443749302413782*v^6 + 979026668318049167132803*v^5 + 3044197524673710725500914*v^4 - 11303824104537481334436817*v^3 + 1845484707680277480548616*v^2 + 11349844369944629860390067*v - 12139260995132491579346226) / 4136026610381798610784394 $$\beta_{9}$$ $$=$$ $$( 24\!\cdots\!14 \nu^{15} + \cdots + 29\!\cdots\!54 ) / 41\!\cdots\!94$$ (245204915440680156914*v^15 - 776594441642987820173*v^14 + 2376052169206225011042*v^13 - 4478763958971146181708*v^12 + 24746729416159566188668*v^11 - 11798681872105894888540*v^10 - 77205981106427487557846*v^9 + 358881589762381071470923*v^8 - 401666468771928513101288*v^7 + 725446686209137214367935*v^6 - 711143038597944288083530*v^5 + 239881156011128668612579*v^4 + 11370877427156155063566520*v^3 - 16802658167616863816782085*v^2 + 2853132220575548469672140*v + 29383283378807446672223354) / 4136026610381798610784394 $$\beta_{10}$$ $$=$$ $$( 29\!\cdots\!12 \nu^{15} + \cdots + 19\!\cdots\!73 ) / 41\!\cdots\!94$$ (290375359378885003312*v^15 - 743934382221418472242*v^14 + 1767266717836305278230*v^13 - 2850753731085855890079*v^12 + 26153841247373778060042*v^11 + 7902296848980404937094*v^10 - 130800194424547620941876*v^9 + 292119947250793672950606*v^8 + 124867657205700169186062*v^7 + 701839303337308061729221*v^6 - 395644720931754611009464*v^5 - 3190848798199793757451725*v^4 + 11234369776810738073080218*v^3 - 631699645499098517354817*v^2 - 10344016932751524448182256*v + 19782979528236067552714073) / 4136026610381798610784394 $$\beta_{11}$$ $$=$$ $$( - 31\!\cdots\!70 \nu^{15} + \cdots - 62\!\cdots\!81 ) / 41\!\cdots\!94$$ (-318073994808497231570*v^15 + 2293064170187332819187*v^14 - 4466190043659119653421*v^13 + 7837297929564868783317*v^12 - 37867833616479840448874*v^11 + 105998876476806512378294*v^10 + 292392729702505742733160*v^9 - 1040119147479516578145527*v^8 + 388039704245625216000627*v^7 + 506700705832317210409544*v^6 + 4387546501364479788167939*v^5 + 3561481521476851963965096*v^4 - 36719641709862948496729891*v^3 + 27844288339436115275679098*v^2 + 49425586234727022770461941*v - 62351648937184846117813681) / 4136026610381798610784394 $$\beta_{12}$$ $$=$$ $$( - 20\!\cdots\!33 \nu^{15} + \cdots - 91\!\cdots\!68 ) / 24\!\cdots\!82$$ (-20975988167062581733*v^15 + 39547953338415445145*v^14 - 176449598889577838775*v^13 + 274358583941958469564*v^12 - 2164217419437062242990*v^11 - 919597481073259125842*v^10 + 1823706335460357174691*v^9 - 20580179620448945951591*v^8 + 8451228402253354843526*v^7 - 93694473582422293608019*v^6 + 5075887138800366711766*v^5 - 35902260644185079638963*v^4 - 523468395825219908972988*v^3 + 334862658377387529702803*v^2 - 868849722103095172193017*v - 916714740028087519831068) / 243295682963635212399082 $$\beta_{13}$$ $$=$$ $$( 35\!\cdots\!76 \nu^{15} + \cdots - 19\!\cdots\!69 ) / 41\!\cdots\!94$$ (357782714812193812176*v^15 - 123523917827887574891*v^14 - 800114075158984515449*v^13 + 1607794271396129055839*v^12 + 17685313247604191574774*v^11 + 86155282281315189826046*v^10 - 231037252399480846221794*v^9 - 157743783882519768425145*v^8 + 810871220438588178709463*v^7 + 566992698905865867472240*v^6 + 1009287290081580835466493*v^5 - 9850704470596476333646446*v^4 + 1712964652569758738280733*v^3 + 26803847027910109683168612*v^2 - 22439135362612872985811991*v - 19147500634288685846910169) / 4136026610381798610784394 $$\beta_{14}$$ $$=$$ $$( 47\!\cdots\!48 \nu^{15} + \cdots + 91\!\cdots\!78 ) / 41\!\cdots\!94$$ (476964966549267156748*v^15 - 1473705395402246668575*v^14 + 2999350746100514306003*v^13 - 6596755340890207762998*v^12 + 45185336926506497857202*v^11 - 12469513850456023691368*v^10 - 254095554940103528613762*v^9 + 430971747898100629606361*v^8 + 58620806557048961231735*v^7 + 1194059572369888956687611*v^6 - 2290539345841782308758671*v^5 - 6240010475731289229690247*v^4 + 16674572537573573808687019*v^3 - 1448346950335563611661837*v^2 - 18024602065451566144299597*v + 9148560862865248236217178) / 4136026610381798610784394 $$\beta_{15}$$ $$=$$ $$( 57\!\cdots\!67 \nu^{15} + \cdots + 27\!\cdots\!46 ) / 41\!\cdots\!94$$ (570985375503210482467*v^15 - 1520250623078270404265*v^14 + 3751560263724227188644*v^13 - 6673310293034086938068*v^12 + 53067762013301206658762*v^11 + 5500764870110710645582*v^10 - 246766753353328108128263*v^9 + 564892960196155782707651*v^8 + 84055353158510742790295*v^7 + 1438278968368169116260339*v^6 - 1482438585718953558456509*v^5 - 6069367767126172146850815*v^4 + 21509477402988567555648733*v^3 - 2777676281482766127449851*v^2 - 20004927468191489674451150*v + 27664828216431640824705446) / 4136026610381798610784394
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{15} + \beta_{10} + 3\beta_{8} - 3\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_1$$ -b15 + b10 + 3*b8 - 3*b7 - 2*b5 + b4 - b3 + b1 $$\nu^{3}$$ $$=$$ $$- 2 \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} + 6 \beta_{6} + \cdots + 2$$ -2*b15 - b12 - b11 + b10 - b9 - b8 - 3*b7 + 6*b6 - 11*b5 - 9*b4 - 2*b2 - 3*b1 + 2 $$\nu^{4}$$ $$=$$ $$4 \beta_{15} - 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} - 27 \beta_{8} - 27 \beta_{7} + 13 \beta_{6} + \cdots - 42$$ 4*b15 - 9*b14 - 2*b13 - 2*b9 - 27*b8 - 27*b7 + 13*b6 + 29*b5 - 19*b4 - 9*b3 - 13*b2 - 42 $$\nu^{5}$$ $$=$$ $$- 4 \beta_{15} - 36 \beta_{14} + 13 \beta_{13} - 13 \beta_{12} - 9 \beta_{11} + 23 \beta_{10} + \cdots - 73$$ -4*b15 - 36*b14 + 13*b13 - 13*b12 - 9*b11 + 23*b10 + 9*b9 + 34*b8 - 10*b7 - 21*b6 + 61*b5 + 42*b4 - 44*b3 + 42*b2 - 30*b1 - 73 $$\nu^{6}$$ $$=$$ $$7 \beta_{15} + 7 \beta_{14} + 44 \beta_{13} - 76 \beta_{12} - 44 \beta_{11} - 142 \beta_{10} + \cdots + 209$$ 7*b15 + 7*b14 + 44*b13 - 76*b12 - 44*b11 - 142*b10 - 65*b8 + 209*b7 + 98*b3 + 105*b2 - 191*b1 + 209 $$\nu^{7}$$ $$=$$ $$449 \beta_{15} + 59 \beta_{14} - 59 \beta_{13} + 157 \beta_{12} + 98 \beta_{11} - 504 \beta_{10} + \cdots - 59$$ 449*b15 + 59*b14 - 59*b13 + 157*b12 + 98*b11 - 504*b10 + 59*b9 - 551*b8 + 489*b7 - 143*b6 + 710*b5 + 257*b4 + 390*b3 - 59*b2 + 316*b1 - 59 $$\nu^{8}$$ $$=$$ $$11 \beta_{15} + 378 \beta_{14} + 602 \beta_{12} + 390 \beta_{11} + 591 \beta_{10} + 602 \beta_{9} + \cdots + 1028$$ 11*b15 + 378*b14 + 602*b12 + 390*b11 + 591*b10 + 602*b9 + 2340*b8 + 1312*b7 - 1589*b6 - 434*b5 + 2901*b4 + 710*b2 + 1312*b1 + 1028 $$\nu^{9}$$ $$=$$ $$- 2504 \beta_{15} + 4164 \beta_{14} - 201 \beta_{13} - 367 \beta_{9} + 3660 \beta_{8} + 3660 \beta_{7} + \cdots + 11389$$ -2504*b15 + 4164*b14 - 201*b13 - 367*b9 + 3660*b8 + 3660*b7 + 635*b6 - 12024*b5 - 1320*b4 + 4164*b3 - 635*b2 + 11389 $$\nu^{10}$$ $$=$$ $$2504 \beta_{15} + 4696 \beta_{14} - 6668 \beta_{13} + 6668 \beta_{12} + 4164 \beta_{11} + 1972 \beta_{10} + \cdots - 11612$$ 2504*b15 + 4696*b14 - 6668*b13 + 6668*b12 + 4164*b11 + 1972*b10 - 4164*b9 - 15494*b8 - 18735*b7 + 9866*b6 - 4121*b5 - 18692*b4 + 3312*b3 - 18692*b2 + 14030*b1 - 11612 $$\nu^{11}$$ $$=$$ $$- 22394 \beta_{15} - 22394 \beta_{14} - 3312 \beta_{13} + 5504 \beta_{12} + 3312 \beta_{11} + \cdots - 76812$$ -22394*b15 - 22394*b14 - 3312*b13 + 5504*b12 + 3312*b11 + 67722*b10 + 42552*b8 - 76812*b7 - 64410*b3 - 7433*b2 + 12584*b1 - 76812 $$\nu^{12}$$ $$=$$ $$- 102120 \beta_{15} - 39824 \beta_{14} + 39824 \beta_{13} - 104234 \beta_{12} - 64410 \beta_{11} + \cdots + 39824$$ -102120*b15 - 39824*b14 + 39824*b13 - 104234*b12 - 64410*b11 + 35457*b10 - 39824*b9 - 2325*b8 - 69559*b7 + 93965*b6 - 109959*b5 - 176694*b4 - 62296*b3 + 39824*b2 - 216518*b1 + 39824 $$\nu^{13}$$ $$=$$ $$300533 \beta_{15} - 247371 \beta_{14} - 99867 \beta_{12} - 62296 \beta_{11} - 400400 \beta_{10} + \cdots - 695314$$ 300533*b15 - 247371*b14 - 99867*b12 - 62296*b11 - 400400*b10 - 99867*b9 - 905140*b8 - 209826*b7 + 241814*b6 + 945646*b5 - 451640*b4 - 109959*b2 - 209826*b1 - 695314 $$\nu^{14}$$ $$=$$ $$594635 \beta_{15} - 966540 \beta_{14} + 338104 \beta_{13} + 547904 \beta_{9} + 786321 \beta_{8} + \cdots - 1445411$$ 594635*b15 - 966540*b14 + 338104*b13 + 547904*b9 + 786321*b8 + 786321*b7 - 1283750*b6 + 2729161*b5 + 2417009*b4 - 966540*b3 + 1283750*b2 - 1445411 $$\nu^{15}$$ $$=$$ $$- 594635 \beta_{15} + 1832695 \beta_{14} + 1561175 \beta_{13} - 1561175 \beta_{12} - 966540 \beta_{11} + \cdots + 11063785$$ -594635*b15 + 1832695*b14 + 1561175*b13 - 1561175*b12 - 966540*b11 - 3393870*b10 + 966540*b9 + 3573048*b8 + 9449472*b7 - 2278463*b6 - 4192596*b5 + 4290336*b4 + 3933458*b3 + 4290336*b2 - 3245003*b1 + 11063785

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.24350 − 2.23726i 1.60675 − 1.36085i −0.797732 + 1.94863i −1.43448 + 2.82504i −1.95510 − 0.109518i −1.29715 + 0.104262i 0.988132 + 0.846795i 1.64608 + 1.06057i 2.24350 + 2.23726i 1.60675 + 1.36085i −0.797732 − 1.94863i −1.43448 − 2.82504i −1.95510 + 0.109518i −1.29715 − 0.104262i 0.988132 − 0.846795i 1.64608 − 1.06057i
−0.577539 + 0.794915i −0.535233 + 1.64728i 0.937730 + 2.88604i −0.321645 + 0.233689i −1.00033 1.37683i 6.87311 2.23321i −6.57364 2.13591i −2.42705 1.76336i 0.390645i
7.2 −0.184008 + 0.253266i 0.535233 1.64728i 1.20578 + 3.71102i 5.99919 4.35866i 0.318712 + 0.438669i −9.53633 + 3.09854i −2.35267 0.764430i −2.42705 1.76336i 2.32142i
7.3 1.30204 1.79211i 0.535233 1.64728i −0.280267 0.862573i −7.03442 + 5.11081i −2.25520 3.10402i 6.34535 2.06173i 6.51625 + 2.11726i −2.42705 1.76336i 19.2609i
7.4 1.69557 2.33376i −0.535233 + 1.64728i −1.33538 4.10989i 0.356879 0.259287i 2.93682 + 4.04219i −10.0641 + 3.27002i −0.881730 0.286491i −2.42705 1.76336i 1.27251i
13.1 −3.47243 1.12826i −1.40126 + 1.01807i 7.54873 + 5.48447i 1.69033 + 5.20232i 6.01443 1.95421i −4.20886 + 5.79300i −11.4402 15.7461i 0.927051 2.85317i 19.9718i
13.2 −2.40785 0.782357i 1.40126 1.01807i 1.94958 + 1.41645i −2.61024 8.03348i −4.17052 + 1.35508i 1.43445 1.97435i 2.36641 + 3.25708i 0.927051 2.85317i 21.3855i
13.3 1.28981 + 0.419086i 1.40126 1.01807i −1.74808 1.27006i 0.708979 + 2.18201i 2.23402 0.725878i −5.74346 + 7.90520i −4.91103 6.75946i 0.927051 2.85317i 3.11151i
13.4 2.35440 + 0.764990i −1.40126 + 1.01807i 1.72190 + 1.25104i −0.789076 2.42853i −4.07793 + 1.32500i −0.100159 + 0.137856i −2.72337 3.74840i 0.927051 2.85317i 6.32135i
19.1 −0.577539 0.794915i −0.535233 1.64728i 0.937730 2.88604i −0.321645 0.233689i −1.00033 + 1.37683i 6.87311 + 2.23321i −6.57364 + 2.13591i −2.42705 + 1.76336i 0.390645i
19.2 −0.184008 0.253266i 0.535233 + 1.64728i 1.20578 3.71102i 5.99919 + 4.35866i 0.318712 0.438669i −9.53633 3.09854i −2.35267 + 0.764430i −2.42705 + 1.76336i 2.32142i
19.3 1.30204 + 1.79211i 0.535233 + 1.64728i −0.280267 + 0.862573i −7.03442 5.11081i −2.25520 + 3.10402i 6.34535 + 2.06173i 6.51625 2.11726i −2.42705 + 1.76336i 19.2609i
19.4 1.69557 + 2.33376i −0.535233 1.64728i −1.33538 + 4.10989i 0.356879 + 0.259287i 2.93682 4.04219i −10.0641 3.27002i −0.881730 + 0.286491i −2.42705 + 1.76336i 1.27251i
28.1 −3.47243 + 1.12826i −1.40126 1.01807i 7.54873 5.48447i 1.69033 5.20232i 6.01443 + 1.95421i −4.20886 5.79300i −11.4402 + 15.7461i 0.927051 + 2.85317i 19.9718i
28.2 −2.40785 + 0.782357i 1.40126 + 1.01807i 1.94958 1.41645i −2.61024 + 8.03348i −4.17052 1.35508i 1.43445 + 1.97435i 2.36641 3.25708i 0.927051 + 2.85317i 21.3855i
28.3 1.28981 0.419086i 1.40126 + 1.01807i −1.74808 + 1.27006i 0.708979 2.18201i 2.23402 + 0.725878i −5.74346 7.90520i −4.91103 + 6.75946i 0.927051 + 2.85317i 3.11151i
28.4 2.35440 0.764990i −1.40126 1.01807i 1.72190 1.25104i −0.789076 + 2.42853i −4.07793 1.32500i −0.100159 0.137856i −2.72337 + 3.74840i 0.927051 + 2.85317i 6.32135i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.g.a 16
3.b odd 2 1 99.3.k.c 16
4.b odd 2 1 528.3.bf.b 16
11.b odd 2 1 363.3.g.f 16
11.c even 5 1 363.3.c.e 16
11.c even 5 1 363.3.g.a 16
11.c even 5 1 363.3.g.f 16
11.c even 5 1 363.3.g.g 16
11.d odd 10 1 inner 33.3.g.a 16
11.d odd 10 1 363.3.c.e 16
11.d odd 10 1 363.3.g.a 16
11.d odd 10 1 363.3.g.g 16
33.f even 10 1 99.3.k.c 16
33.f even 10 1 1089.3.c.m 16
33.h odd 10 1 1089.3.c.m 16
44.g even 10 1 528.3.bf.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a 16 1.a even 1 1 trivial
33.3.g.a 16 11.d odd 10 1 inner
99.3.k.c 16 3.b odd 2 1
99.3.k.c 16 33.f even 10 1
363.3.c.e 16 11.c even 5 1
363.3.c.e 16 11.d odd 10 1
363.3.g.a 16 11.c even 5 1
363.3.g.a 16 11.d odd 10 1
363.3.g.f 16 11.b odd 2 1
363.3.g.f 16 11.c even 5 1
363.3.g.g 16 11.c even 5 1
363.3.g.g 16 11.d odd 10 1
528.3.bf.b 16 4.b odd 2 1
528.3.bf.b 16 44.g even 10 1
1089.3.c.m 16 33.f even 10 1
1089.3.c.m 16 33.h odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 18 T^{14} + \cdots + 3721$$
$3$ $$(T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2}$$
$5$ $$T^{16} + 4 T^{15} + \cdots + 9369721$$
$7$ $$T^{16} + \cdots + 22159001881$$
$11$ $$T^{16} + \cdots + 45\!\cdots\!61$$
$13$ $$T^{16} + \cdots + 47\!\cdots\!16$$
$17$ $$T^{16} + \cdots + 18\!\cdots\!76$$
$19$ $$T^{16} + \cdots + 99\!\cdots\!96$$
$23$ $$(T^{8} - 66 T^{7} + \cdots + 255717136)^{2}$$
$29$ $$T^{16} + \cdots + 17\!\cdots\!36$$
$31$ $$T^{16} + \cdots + 59\!\cdots\!41$$
$37$ $$T^{16} + \cdots + 30\!\cdots\!36$$
$41$ $$T^{16} + \cdots + 10\!\cdots\!76$$
$43$ $$T^{16} + \cdots + 13\!\cdots\!56$$
$47$ $$T^{16} + \cdots + 30\!\cdots\!16$$
$53$ $$T^{16} + \cdots + 41\!\cdots\!41$$
$59$ $$T^{16} + \cdots + 34\!\cdots\!41$$
$61$ $$T^{16} + \cdots + 13\!\cdots\!16$$
$67$ $$(T^{8} - 18 T^{7} + \cdots + 47006885776)^{2}$$
$71$ $$T^{16} + \cdots + 40\!\cdots\!16$$
$73$ $$T^{16} + \cdots + 63\!\cdots\!36$$
$79$ $$T^{16} + \cdots + 14\!\cdots\!81$$
$83$ $$T^{16} + \cdots + 16\!\cdots\!41$$
$89$ $$(T^{8} + \cdots - 15121642690304)^{2}$$
$97$ $$T^{16} + \cdots + 29\!\cdots\!01$$