# Properties

 Label 162.2.e.a Level $162$ Weight $2$ Character orbit 162.e Analytic conductor $1.294$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,2,Mod(19,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("162.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.e (of order $$9$$, degree $$6$$, not 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: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{18}^{4} + \zeta_{18}) q^{2} - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3}) q^{7} - \zeta_{18}^{3} q^{8} +O(q^{10})$$ q + (-z^4 + z) * q^2 - z^5 * q^4 + (-z^5 - z^4 + z^3 + z^2) * q^5 + (z^5 - 2*z^4 + z^3) * q^7 - z^3 * q^8 $$q + ( - \zeta_{18}^{4} + \zeta_{18}) q^{2} - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3}) q^{7} - \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} + 1) q^{10} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} - 1) q^{11} + ( - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 2) q^{13} + (\zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{14} - \zeta_{18} q^{16} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{17} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 3 \zeta_{18}) q^{19} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{20} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{22} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2) q^{23} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{25} + (2 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 3) q^{26} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{28} + (3 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} - 3) q^{29} + (4 \zeta_{18}^{5} + 3 \zeta_{18} - 3) q^{31} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{32} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{34} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{35} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} + 5) q^{37} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18} + 4) q^{38} + (\zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} + 1) q^{40} + (3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - \zeta_{18}^{2} - 5 \zeta_{18} + 3) q^{41} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 4) q^{43} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18}^{2} - \zeta_{18} + 1) q^{44} + ( - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{46} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{47} + ( - \zeta_{18}^{5} - 4 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{49} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18} + 1) q^{50} + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18} - 2) q^{52} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} + 2) q^{53} + ( - \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 5 \zeta_{18}^{2} + 5 \zeta_{18} - 3) q^{55} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{56} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18} - 1) q^{58} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} - 1) q^{59} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{61} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18}) q^{62} + (\zeta_{18}^{3} - 1) q^{64} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 4 \zeta_{18}^{2} - \zeta_{18} - 3) q^{65} + ( - 3 \zeta_{18}^{3} - \zeta_{18}^{2} - 3 \zeta_{18}) q^{67} + (\zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} + 1) q^{68} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 2) q^{70} + ( - 5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{71} + ( - 5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 5 \zeta_{18}) q^{73} + (2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{74} + ( - 3 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 1) q^{76} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 7 \zeta_{18} - 6) q^{77} + ( - 8 \zeta_{18}^{5} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 8) q^{79} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 1) q^{80} + (5 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{82} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{83} + (5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 7 \zeta_{18} + 2) q^{85} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{86} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{88} + ( - \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - \zeta_{18}) q^{89} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 5 \zeta_{18} - 4) q^{91} + ( - 2 \zeta_{18}^{2} - \zeta_{18} - 2) q^{92} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{94} + ( - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 10 \zeta_{18}^{2} + 3 \zeta_{18} - 5) q^{95} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} - 3) q^{97} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 4 \zeta_{18} + 1) q^{98} +O(q^{100})$$ q + (-z^4 + z) * q^2 - z^5 * q^4 + (-z^5 - z^4 + z^3 + z^2) * q^5 + (z^5 - 2*z^4 + z^3) * q^7 - z^3 * q^8 + (-z^3 - z^2 + z + 1) * q^10 + (-3*z^5 + 3*z^3 + 2*z^2 + z - 1) * q^11 + (-2*z^4 - 3*z^2 - 2) * q^13 + (z^3 - 2*z^2 + z) * q^14 - z * q^16 + (4*z^5 + 4*z^4 - 2*z^3 - z^2 - 3*z + 2) * q^17 + (-3*z^5 + 4*z^4 + 3*z^3 + 4*z^2 - 3*z) * q^19 + (-z^5 - z^4 + z^2 - 1) * q^20 + (-z^5 + z^4 - 3*z^3 + z^2 + 2*z + 2) * q^22 + (-z^5 - 2*z^4 - 2*z^3 + 2) * q^23 + (z^5 + 2*z^4 - z^3 + z^2 - 2*z - 1) * q^25 + (2*z^4 - 2*z^2 - 2*z - 3) * q^26 + (-z^5 + z^2 + z - 2) * q^28 + (3*z^5 + z^4 + z^3 - z^2 - z - 3) * q^29 + (4*z^5 + 3*z - 3) * q^31 + (z^5 - z^2) * q^32 + (3*z^5 - 2*z^4 + 4*z^3 + z^2 - 1) * q^34 + (3*z^5 - 2*z^4 - z^3 - 2*z^2 + 3*z) * q^35 + (3*z^5 + 3*z^4 - 5*z^3 - z^2 - 2*z + 5) * q^37 + (3*z^5 - 3*z^3 + z^2 + 3*z + 4) * q^38 + (z^4 - z^3 - z^2 - z + 1) * q^40 + (3*z^4 - 5*z^3 - z^2 - 5*z + 3) * q^41 + (-2*z^5 + 2*z^3 - 2*z^2 + 4*z - 4) * q^43 + (-2*z^5 - 2*z^4 - z^3 + 3*z^2 - z + 1) * q^44 + (-2*z^4 - z^3 - 2*z^2) * q^46 + (-2*z^5 + 3*z^4 - z^3 + z^2 - 1) * q^47 + (-z^5 - 4*z^4 + z^3 + z^2 + 3*z + 3) * q^49 + (2*z^5 + z^4 + z^3 - 2*z + 1) * q^50 + (2*z^5 + 3*z^4 - 3*z - 2) * q^52 + (-2*z^5 - z^4 + 3*z^2 + 3*z + 2) * q^53 + (-z^5 - 4*z^4 + 5*z^2 + 5*z - 3) * q^55 + (-z^5 + 2*z^4 - z^3 + z^2 - 2*z + 1) * q^56 + (z^5 + 3*z^4 + 3*z^3 - 2*z - 1) * q^58 + (z^5 - 3*z^4 + 4*z^3 - z^2 - z - 1) * q^59 + (z^5 + 2*z^4 - 4*z^3 - 5*z^2 + 5) * q^61 + (-3*z^5 + 3*z^4 + 4*z^3 + 3*z^2 - 3*z) * q^62 + (z^3 - 1) * q^64 + (z^5 - z^3 - 4*z^2 - z - 3) * q^65 + (-3*z^3 - z^2 - 3*z) * q^67 + (z^4 + 3*z^3 - 2*z^2 + 3*z + 1) * q^68 + (-3*z^5 + 3*z^3 + z^2 - z - 2) * q^70 + (-5*z^5 - 5*z^4 + 4*z^3 - z^2 + 6*z - 4) * q^71 + (-5*z^5 - 2*z^4 + z^3 - 2*z^2 - 5*z) * q^73 + (2*z^5 - 5*z^4 + 3*z^3 + z^2 - 1) * q^74 + (-3*z^5 - 4*z^4 + 3*z^3 + 3*z^2 + z + 1) * q^76 + (-z^5 - z^4 - z^3 + 7*z - 6) * q^77 + (-8*z^5 - 5*z^3 + 5*z^2 + 8) * q^79 + (z^5 - z^4 - 1) * q^80 + (5*z^5 - 3*z^4 - 2*z^2 - 2*z - 1) * q^82 + (-2*z^5 + 2*z^3 - 2*z^2 + 2) * q^83 + (5*z^4 + 5*z^3 - 7*z + 2) * q^85 + (-4*z^5 + 4*z^4 - 2*z^3 + 4*z^2 - 2*z - 2) * q^86 + (z^5 - z^4 - 2*z^3 - 3*z^2 + 3) * q^88 + (-z^5 - 3*z^4 + 5*z^3 - 3*z^2 - z) * q^89 + (-z^5 - z^4 + 4*z^3 - 4*z^2 + 5*z - 4) * q^91 + (-2*z^2 - z - 2) * q^92 + (z^4 - 2*z^3 + 3*z^2 - 2*z + 1) * q^94 + (-5*z^4 + 3*z^3 + 10*z^2 + 3*z - 5) * q^95 + (-2*z^5 + 2*z^3 - z^2 + 6*z - 3) * q^97 + (-3*z^5 - 3*z^4 - z^3 - z^2 + 4*z + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10})$$ 6 * q + 3 * q^5 + 3 * q^7 - 3 * q^8 $$6 q + 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} + 3 q^{11} - 12 q^{13} + 3 q^{14} + 6 q^{17} + 9 q^{19} - 6 q^{20} + 3 q^{22} + 6 q^{23} - 9 q^{25} - 18 q^{26} - 12 q^{28} - 15 q^{29} - 18 q^{31} + 6 q^{34} - 3 q^{35} + 15 q^{37} + 15 q^{38} + 3 q^{40} + 3 q^{41} - 18 q^{43} + 3 q^{44} - 3 q^{46} - 9 q^{47} + 21 q^{49} + 9 q^{50} - 12 q^{52} + 12 q^{53} - 18 q^{55} + 3 q^{56} + 3 q^{58} + 6 q^{59} + 18 q^{61} + 12 q^{62} - 3 q^{64} - 21 q^{65} - 9 q^{67} + 15 q^{68} - 3 q^{70} - 12 q^{71} + 3 q^{73} + 3 q^{74} + 15 q^{76} - 39 q^{77} + 33 q^{79} - 6 q^{80} - 6 q^{82} + 18 q^{83} + 27 q^{85} - 18 q^{86} + 12 q^{88} + 15 q^{89} - 12 q^{91} - 12 q^{92} - 21 q^{95} - 12 q^{97} + 3 q^{98}+O(q^{100})$$ 6 * q + 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 + 3 * q^11 - 12 * q^13 + 3 * q^14 + 6 * q^17 + 9 * q^19 - 6 * q^20 + 3 * q^22 + 6 * q^23 - 9 * q^25 - 18 * q^26 - 12 * q^28 - 15 * q^29 - 18 * q^31 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 18 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 21 * q^49 + 9 * q^50 - 12 * q^52 + 12 * q^53 - 18 * q^55 + 3 * q^56 + 3 * q^58 + 6 * q^59 + 18 * q^61 + 12 * q^62 - 3 * q^64 - 21 * q^65 - 9 * q^67 + 15 * q^68 - 3 * q^70 - 12 * q^71 + 3 * q^73 + 3 * q^74 + 15 * q^76 - 39 * q^77 + 33 * q^79 - 6 * q^80 - 6 * q^82 + 18 * q^83 + 27 * q^85 - 18 * q^86 + 12 * q^88 + 15 * q^89 - 12 * q^91 - 12 * q^92 - 21 * q^95 - 12 * q^97 + 3 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 − 0.984808i
−0.939693 + 0.342020i 0 0.766044 0.642788i −0.439693 2.49362i 0 −1.79813 1.50881i −0.500000 + 0.866025i 0 1.26604 + 2.19285i
37.1 0.766044 0.642788i 0 0.173648 0.984808i 1.26604 0.460802i 0 −0.0209445 0.118782i −0.500000 0.866025i 0 0.673648 1.16679i
73.1 0.173648 0.984808i 0 −0.939693 0.342020i 0.673648 0.565258i 0 3.31908 1.20805i −0.500000 + 0.866025i 0 −0.439693 0.761570i
91.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.673648 + 0.565258i 0 3.31908 + 1.20805i −0.500000 0.866025i 0 −0.439693 + 0.761570i
127.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i 1.26604 + 0.460802i 0 −0.0209445 + 0.118782i −0.500000 + 0.866025i 0 0.673648 + 1.16679i
145.1 −0.939693 0.342020i 0 0.766044 + 0.642788i −0.439693 + 2.49362i 0 −1.79813 + 1.50881i −0.500000 0.866025i 0 1.26604 2.19285i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.2.e.a 6
3.b odd 2 1 54.2.e.a 6
9.c even 3 1 486.2.e.a 6
9.c even 3 1 486.2.e.c 6
9.d odd 6 1 486.2.e.b 6
9.d odd 6 1 486.2.e.d 6
12.b even 2 1 432.2.u.a 6
27.e even 9 1 inner 162.2.e.a 6
27.e even 9 1 486.2.e.a 6
27.e even 9 1 486.2.e.c 6
27.e even 9 1 1458.2.a.d 3
27.e even 9 2 1458.2.c.a 6
27.f odd 18 1 54.2.e.a 6
27.f odd 18 1 486.2.e.b 6
27.f odd 18 1 486.2.e.d 6
27.f odd 18 1 1458.2.a.a 3
27.f odd 18 2 1458.2.c.d 6
108.l even 18 1 432.2.u.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.a 6 3.b odd 2 1
54.2.e.a 6 27.f odd 18 1
162.2.e.a 6 1.a even 1 1 trivial
162.2.e.a 6 27.e even 9 1 inner
432.2.u.a 6 12.b even 2 1
432.2.u.a 6 108.l even 18 1
486.2.e.a 6 9.c even 3 1
486.2.e.a 6 27.e even 9 1
486.2.e.b 6 9.d odd 6 1
486.2.e.b 6 27.f odd 18 1
486.2.e.c 6 9.c even 3 1
486.2.e.c 6 27.e even 9 1
486.2.e.d 6 9.d odd 6 1
486.2.e.d 6 27.f odd 18 1
1458.2.a.a 3 27.f odd 18 1
1458.2.a.d 3 27.e even 9 1
1458.2.c.a 6 27.e even 9 2
1458.2.c.d 6 27.f odd 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 3T_{5}^{5} + 9T_{5}^{4} - 24T_{5}^{3} + 36T_{5}^{2} - 27T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9$$
$7$ $$T^{6} - 3 T^{5} - 6 T^{4} + 8 T^{3} + \cdots + 1$$
$11$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 3249$$
$13$ $$T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 289$$
$17$ $$T^{6} - 6 T^{5} + 63 T^{4} + \cdots + 25281$$
$19$ $$T^{6} - 9 T^{5} + 93 T^{4} + \cdots + 32041$$
$23$ $$T^{6} - 6 T^{5} + 36 T^{4} - 111 T^{3} + \cdots + 9$$
$29$ $$T^{6} + 15 T^{5} + 99 T^{4} + 300 T^{3} + \cdots + 9$$
$31$ $$T^{6} + 18 T^{5} + 171 T^{4} + \cdots + 5041$$
$37$ $$T^{6} - 15 T^{5} + 171 T^{4} + \cdots + 289$$
$41$ $$T^{6} - 3 T^{5} + 36 T^{4} + \cdots + 47961$$
$43$ $$T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 64$$
$47$ $$T^{6} + 9 T^{5} + 9 T^{4} - 72 T^{3} + \cdots + 81$$
$53$ $$(T^{3} - 6 T^{2} - 9 T - 3)^{2}$$
$59$ $$T^{6} - 6 T^{5} + 36 T^{4} + \cdots + 3249$$
$61$ $$T^{6} - 18 T^{5} + 153 T^{4} + \cdots + 2809$$
$67$ $$T^{6} + 9 T^{5} + 45 T^{4} + 80 T^{3} + \cdots + 1$$
$71$ $$T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929$$
$73$ $$T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 72361$$
$79$ $$T^{6} - 33 T^{5} + 510 T^{4} + \cdots + 466489$$
$83$ $$T^{6} - 18 T^{5} + 144 T^{4} + \cdots + 5184$$
$89$ $$T^{6} - 15 T^{5} + 189 T^{4} + \cdots + 25281$$
$97$ $$T^{6} + 12 T^{5} + 51 T^{4} + \cdots + 16129$$