# Properties

 Label 729.2.e.i Level $729$ Weight $2$ Character orbit 729.e Analytic conductor $5.821$ 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] = [729,2,Mod(82,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([8]))

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

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

chi := DirichletCharacter("729.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 243) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{2} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{4} + (2 \zeta_{18}^{5} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{5} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{7} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{8}+O(q^{10})$$ q + (-z^5 - z^4 + z + 1) * q^2 + (-z^5 - 2*z^4 - 2*z^3 + z + 1) * q^4 + (2*z^5 - z^3 - 2*z^2 + z + 1) * q^5 + (-2*z^5 - z^4 - z^3 + z^2 - 1) * q^7 + (z^5 - 3*z^4 - 2*z^3 - 3*z^2 + z) * q^8 $$q + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{2} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{4} + (2 \zeta_{18}^{5} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{5} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{7} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{8} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{10} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} + 3) q^{11} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 2) q^{13} + ( - \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{14} + ( - 3 \zeta_{18}^{2} - \zeta_{18} - 3) q^{16} + (3 \zeta_{18}^{3} - 3) q^{17} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18}) q^{19} + ( - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{20} + ( - 5 \zeta_{18}^{4} + \zeta_{18}^{3} + 4 \zeta_{18} + 4) q^{22} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18} + 3) q^{23} + (4 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 4) q^{25} + (\zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} - 4) q^{26} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 6 \zeta_{18}^{2} - 6 \zeta_{18} - 4) q^{28} + ( - \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} + 1) q^{29} + (4 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{31} + (3 \zeta_{18}^{3} - 3 \zeta_{18} - 3) q^{32} + (3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{34} + ( - 4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 4 \zeta_{18}) q^{35} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 1) q^{37} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 3) q^{38} + ( - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} - 5) q^{40} + (2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} + 2) q^{41} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{43} + ( - 5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 3 \zeta_{18} + 5) q^{44} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + ( - 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{47} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18} - 2) q^{49} + (8 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18} - 3) q^{50} + (3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 4 \zeta_{18} - 3) q^{52} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 6) q^{53} + (4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{55} + (5 \zeta_{18}^{5} + 11 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 11 \zeta_{18} - 5) q^{56} + ( - 3 \zeta_{18}^{5} - 8 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + 4 \zeta_{18} + 4) q^{58} + ( - 7 \zeta_{18}^{5} - \zeta_{18}^{3} + 7 \zeta_{18}^{2} + \zeta_{18} + 1) q^{59} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4) q^{61} + (4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18}) q^{62} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18} - 4) q^{64} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 3) q^{65} + (4 \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 4) q^{67} + (3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{68} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{70} + (9 \zeta_{18}^{5} + 9 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 6 \zeta_{18}^{2} - 3 \zeta_{18} + 3) q^{71} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 3 \zeta_{18}) q^{73} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{74} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{76} + ( - 8 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18} - 3) q^{77} + ( - 5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 2 \zeta_{18} + 5) q^{79} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{80} + ( - 8 \zeta_{18}^{5} + \zeta_{18}^{4} + 7 \zeta_{18}^{2} + 7 \zeta_{18} + 6) q^{82} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{83} + ( - 6 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}) q^{85} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} + 1) q^{86} + (2 \zeta_{18}^{5} - 12 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 7) q^{88} + (3 \zeta_{18}^{5} + 6 \zeta_{18}^{4} + 6 \zeta_{18}^{2} + 3 \zeta_{18}) q^{89} + (5 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 4 \zeta_{18} + 2) q^{91} + (5 \zeta_{18}^{5} - 5 \zeta_{18}^{3} - 11 \zeta_{18}^{2} - 8 \zeta_{18} - 6) q^{92} + ( - 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 12 \zeta_{18}^{2} - 5 \zeta_{18} - 2) q^{94} + ( - 8 \zeta_{18}^{4} + 6 \zeta_{18}^{3} + \zeta_{18}^{2} + 6 \zeta_{18} - 8) q^{95} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 8 \zeta_{18}^{2} - 5 \zeta_{18} - 4) q^{97} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18} - 3) q^{98}+O(q^{100})$$ q + (-z^5 - z^4 + z + 1) * q^2 + (-z^5 - 2*z^4 - 2*z^3 + z + 1) * q^4 + (2*z^5 - z^3 - 2*z^2 + z + 1) * q^5 + (-2*z^5 - z^4 - z^3 + z^2 - 1) * q^7 + (z^5 - 3*z^4 - 2*z^3 - 3*z^2 + z) * q^8 + (z^5 + z^4 - 2*z^2 + z) * q^10 + (z^5 - z^3 + 2*z^2 - z + 3) * q^11 + (-2*z^4 + z^3 - z^2 + z - 2) * q^13 + (-z^4 - 3*z^3 - z^2 - 3*z - 1) * q^14 + (-3*z^2 - z - 3) * q^16 + (3*z^3 - 3) * q^17 + (3*z^5 - 3*z^4 + z^3 - 3*z^2 + 3*z) * q^19 + (-z^4 + 2*z^3 + 2*z^2 - 2) * q^20 + (-5*z^4 + z^3 + 4*z + 4) * q^22 + (-2*z^5 - 2*z^4 - 2*z^3 - z + 3) * q^23 + (4*z^5 - z^4 - z^3 + z^2 + z - 4) * q^25 + (z^4 - z^2 - z - 4) * q^26 + (3*z^5 + 3*z^4 - 6*z^2 - 6*z - 4) * q^28 + (-z^5 - 4*z^4 - 3*z^3 + 3*z^2 + 4*z + 1) * q^29 + (4*z^5 + z^4 + z^3 + z - 2) * q^31 + (3*z^3 - 3*z - 3) * q^32 + (3*z^4 + 3*z^3 + 3*z^2 - 3) * q^34 + (-4*z^5 + 3*z^4 + 2*z^3 + 3*z^2 - 4*z) * q^35 + (3*z^5 + 3*z^4 - z^3 - 3*z^2 + 1) * q^37 + (-z^5 + z^3 - 2*z^2 + 4*z - 3) * q^38 + (-5*z^4 + 4*z^3 + 3*z^2 + 4*z - 5) * q^40 + (2*z^4 + 3*z^3 - z^2 + 3*z + 2) * q^41 + (z^5 - z^3 - 2*z^2 + 4*z - 1) * q^43 + (-5*z^5 - 5*z^4 - 5*z^3 + 2*z^2 + 3*z + 5) * q^44 + (-2*z^5 - 5*z^4 - 3*z^3 - 5*z^2 - 2*z) * q^46 + (-2*z^4 - 5*z^3 - 5*z^2 + 5) * q^47 + (z^4 + z^3 - 2*z - 2) * q^49 + (8*z^5 + 2*z^4 + 2*z^3 + z - 3) * q^50 + (3*z^5 + 4*z^4 + 3*z^3 - 3*z^2 - 4*z - 3) * q^52 + (3*z^5 - 3*z^4 + 6) * q^53 + (4*z^5 - 2*z^4 - 2*z^2 - 2*z + 3) * q^55 + (5*z^5 + 11*z^4 + 3*z^3 - 3*z^2 - 11*z - 5) * q^56 + (-3*z^5 - 8*z^4 - 8*z^3 + 4*z + 4) * q^58 + (-7*z^5 - z^3 + 7*z^2 + z + 1) * q^59 + (-5*z^5 + 2*z^4 - z^3 + 4*z^2 - 4) * q^61 + (4*z^5 + 3*z^4 + 4*z^3 + 3*z^2 + 4*z) * q^62 + (-3*z^5 - 3*z^4 + 4*z^3 + 3*z - 4) * q^64 + (-5*z^5 + 5*z^3 + 2*z^2 - z - 3) * q^65 + (4*z^4 + z^3 + 2*z^2 + z + 4) * q^67 + (3*z^4 + 3*z^3 + 3*z^2 + 3*z + 3) * q^68 + (-2*z^5 + 2*z^3 + 4*z^2 - 3*z + 2) * q^70 + (9*z^5 + 9*z^4 - 3*z^3 - 6*z^2 - 3*z + 3) * q^71 + (3*z^5 + 3*z^4 - 2*z^3 + 3*z^2 + 3*z) * q^73 + (3*z^5 + 5*z^4 + 2*z^3 - z^2 + 1) * q^74 + (-z^5 - z^4 + 2*z^3 + z^2 - z - 1) * q^76 + (-8*z^5 - 2*z^4 - 2*z^3 + 5*z - 3) * q^77 + (-5*z^5 - 2*z^4 - 4*z^3 + 4*z^2 + 2*z + 5) * q^79 + (-3*z^5 + z^4 + 2*z^2 + 2*z - 1) * q^80 + (-8*z^5 + z^4 + 7*z^2 + 7*z + 6) * q^82 + (-z^5 + 2*z^4 + 3*z^3 - 3*z^2 - 2*z + 1) * q^83 + (-6*z^5 + 3*z^4 + 3*z^3 - 3*z) * q^85 + (-z^5 + 3*z^4 - 4*z^3 + z^2 + z + 1) * q^86 + (2*z^5 - 12*z^4 - 5*z^3 - 7*z^2 + 7) * q^88 + (3*z^5 + 6*z^4 + 6*z^2 + 3*z) * q^89 + (5*z^5 + 5*z^4 - 2*z^3 - z^2 - 4*z + 2) * q^91 + (5*z^5 - 5*z^3 - 11*z^2 - 8*z - 6) * q^92 + (-2*z^4 - 5*z^3 - 12*z^2 - 5*z - 2) * q^94 + (-8*z^4 + 6*z^3 + z^2 + 6*z - 8) * q^95 + (4*z^5 - 4*z^3 - 8*z^2 - 5*z - 4) * q^97 + (3*z^5 + 3*z^4 + 3*z^3 - 3*z - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10})$$ 6 * q + 6 * q^2 + 3 * q^5 - 9 * q^7 - 6 * q^8 $$6 q + 6 q^{2} + 3 q^{5} - 9 q^{7} - 6 q^{8} + 15 q^{11} - 9 q^{13} - 15 q^{14} - 18 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 27 q^{22} + 12 q^{23} - 27 q^{25} - 24 q^{26} - 24 q^{28} - 3 q^{29} - 9 q^{31} - 9 q^{32} - 9 q^{34} + 6 q^{35} + 3 q^{37} - 15 q^{38} - 18 q^{40} + 21 q^{41} - 9 q^{43} + 15 q^{44} - 9 q^{46} + 15 q^{47} - 9 q^{49} - 12 q^{50} - 9 q^{52} + 36 q^{53} + 18 q^{55} - 21 q^{56} + 3 q^{59} - 27 q^{61} + 12 q^{62} - 12 q^{64} - 3 q^{65} + 27 q^{67} + 27 q^{68} + 18 q^{70} + 9 q^{71} - 6 q^{73} + 12 q^{74} - 24 q^{77} + 18 q^{79} - 6 q^{80} + 36 q^{82} + 15 q^{83} + 9 q^{85} - 6 q^{86} + 27 q^{88} + 6 q^{91} - 51 q^{92} - 27 q^{94} - 30 q^{95} - 36 q^{97} - 9 q^{98}+O(q^{100})$$ 6 * q + 6 * q^2 + 3 * q^5 - 9 * q^7 - 6 * q^8 + 15 * q^11 - 9 * q^13 - 15 * q^14 - 18 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 27 * q^22 + 12 * q^23 - 27 * q^25 - 24 * q^26 - 24 * q^28 - 3 * q^29 - 9 * q^31 - 9 * q^32 - 9 * q^34 + 6 * q^35 + 3 * q^37 - 15 * q^38 - 18 * q^40 + 21 * q^41 - 9 * q^43 + 15 * q^44 - 9 * q^46 + 15 * q^47 - 9 * q^49 - 12 * q^50 - 9 * q^52 + 36 * q^53 + 18 * q^55 - 21 * q^56 + 3 * q^59 - 27 * q^61 + 12 * q^62 - 12 * q^64 - 3 * q^65 + 27 * q^67 + 27 * q^68 + 18 * q^70 + 9 * q^71 - 6 * q^73 + 12 * q^74 - 24 * q^77 + 18 * q^79 - 6 * q^80 + 36 * q^82 + 15 * q^83 + 9 * q^85 - 6 * q^86 + 27 * q^88 + 6 * q^91 - 51 * q^92 - 27 * q^94 - 30 * q^95 - 36 * q^97 - 9 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 −0.766044 + 0.642788i −0.173648 + 0.984808i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.173648 − 0.984808i −0.766044 − 0.642788i
0.233956 + 1.32683i 0 0.173648 0.0632028i 1.26604 + 1.06234i 0 −2.26604 0.824773i 1.47178 + 2.54920i 0 −1.11334 + 1.92836i
163.1 0.826352 0.300767i 0 −0.939693 + 0.788496i 0.673648 + 3.82045i 0 −1.67365 1.40436i −1.41875 + 2.45734i 0 1.70574 + 2.95442i
325.1 1.93969 1.62760i 0 0.766044 4.34445i −0.439693 + 0.160035i 0 −0.560307 3.17766i −3.05303 5.28801i 0 −0.592396 + 1.02606i
406.1 1.93969 + 1.62760i 0 0.766044 + 4.34445i −0.439693 0.160035i 0 −0.560307 + 3.17766i −3.05303 + 5.28801i 0 −0.592396 1.02606i
568.1 0.826352 + 0.300767i 0 −0.939693 0.788496i 0.673648 3.82045i 0 −1.67365 + 1.40436i −1.41875 2.45734i 0 1.70574 2.95442i
649.1 0.233956 1.32683i 0 0.173648 + 0.0632028i 1.26604 1.06234i 0 −2.26604 + 0.824773i 1.47178 2.54920i 0 −1.11334 1.92836i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.1 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 729.2.e.i 6
3.b odd 2 1 729.2.e.a 6
9.c even 3 1 729.2.e.b 6
9.c even 3 1 729.2.e.c 6
9.d odd 6 1 729.2.e.g 6
9.d odd 6 1 729.2.e.h 6
27.e even 9 1 243.2.a.f yes 3
27.e even 9 2 243.2.c.e 6
27.e even 9 1 729.2.e.b 6
27.e even 9 1 729.2.e.c 6
27.e even 9 1 inner 729.2.e.i 6
27.f odd 18 1 243.2.a.e 3
27.f odd 18 2 243.2.c.f 6
27.f odd 18 1 729.2.e.a 6
27.f odd 18 1 729.2.e.g 6
27.f odd 18 1 729.2.e.h 6
108.j odd 18 1 3888.2.a.bk 3
108.l even 18 1 3888.2.a.bd 3
135.n odd 18 1 6075.2.a.bv 3
135.p even 18 1 6075.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 27.f odd 18 1
243.2.a.f yes 3 27.e even 9 1
243.2.c.e 6 27.e even 9 2
243.2.c.f 6 27.f odd 18 2
729.2.e.a 6 3.b odd 2 1
729.2.e.a 6 27.f odd 18 1
729.2.e.b 6 9.c even 3 1
729.2.e.b 6 27.e even 9 1
729.2.e.c 6 9.c even 3 1
729.2.e.c 6 27.e even 9 1
729.2.e.g 6 9.d odd 6 1
729.2.e.g 6 27.f odd 18 1
729.2.e.h 6 9.d odd 6 1
729.2.e.h 6 27.f odd 18 1
729.2.e.i 6 1.a even 1 1 trivial
729.2.e.i 6 27.e even 9 1 inner
3888.2.a.bd 3 108.l even 18 1
3888.2.a.bk 3 108.j odd 18 1
6075.2.a.bq 3 135.p even 18 1
6075.2.a.bv 3 135.n odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(729, [\chi])$$:

 $$T_{2}^{6} - 6T_{2}^{5} + 18T_{2}^{4} - 30T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9$$ T2^6 - 6*T2^5 + 18*T2^4 - 30*T2^3 + 36*T2^2 - 27*T2 + 9 $$T_{5}^{6} - 3T_{5}^{5} + 18T_{5}^{4} - 24T_{5}^{3} + 9T_{5}^{2} + 27T_{5} + 9$$ T5^6 - 3*T5^5 + 18*T5^4 - 24*T5^3 + 9*T5^2 + 27*T5 + 9 $$T_{7}^{6} + 9T_{7}^{5} + 45T_{7}^{4} + 152T_{7}^{3} + 342T_{7}^{2} + 459T_{7} + 289$$ T7^6 + 9*T7^5 + 45*T7^4 + 152*T7^3 + 342*T7^2 + 459*T7 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 3 T^{5} + 18 T^{4} - 24 T^{3} + \cdots + 9$$
$7$ $$T^{6} + 9 T^{5} + 45 T^{4} + 152 T^{3} + \cdots + 289$$
$11$ $$T^{6} - 15 T^{5} + 99 T^{4} - 300 T^{3} + \cdots + 9$$
$13$ $$T^{6} + 9 T^{5} + 45 T^{4} + 152 T^{3} + \cdots + 289$$
$17$ $$(T^{2} + 3 T + 9)^{3}$$
$19$ $$T^{6} - 3 T^{5} + 33 T^{4} + 74 T^{3} + \cdots + 1$$
$23$ $$T^{6} - 12 T^{5} + 99 T^{4} + \cdots + 2601$$
$29$ $$T^{6} + 3 T^{5} - 36 T^{4} + \cdots + 3249$$
$31$ $$T^{6} + 9 T^{5} + 72 T^{4} + 278 T^{3} + \cdots + 361$$
$37$ $$T^{6} - 3 T^{5} + 33 T^{4} + 74 T^{3} + \cdots + 1$$
$41$ $$T^{6} - 21 T^{5} + 207 T^{4} + \cdots + 47961$$
$43$ $$T^{6} + 9 T^{5} + 72 T^{4} + 278 T^{3} + \cdots + 361$$
$47$ $$T^{6} - 15 T^{5} + 180 T^{4} + \cdots + 71289$$
$53$ $$(T^{3} - 18 T^{2} + 81 T - 81)^{2}$$
$59$ $$T^{6} - 3 T^{5} - 36 T^{4} + \cdots + 103041$$
$61$ $$T^{6} + 27 T^{5} + 270 T^{4} + \cdots + 2809$$
$67$ $$T^{6} - 27 T^{5} + 324 T^{4} + \cdots + 11881$$
$71$ $$T^{6} - 9 T^{5} + 243 T^{4} + \cdots + 998001$$
$73$ $$T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 157609$$
$79$ $$T^{6} - 18 T^{5} + 153 T^{4} + \cdots + 2809$$
$83$ $$T^{6} - 15 T^{5} + 72 T^{4} + \cdots + 2601$$
$89$ $$T^{6} + 189 T^{4} + 1998 T^{3} + \cdots + 998001$$
$97$ $$T^{6} + 36 T^{5} + 396 T^{4} + \cdots + 361$$