Properties

 Label 729.2.e.e Level $729$ Weight $2$ Character orbit 729.e Analytic conductor $5.821$ Analytic rank $0$ Dimension $6$ CM discriminant -3 Inner twists $12$

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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 243) Sato-Tate group: $\mathrm{U}(1)[D_{9}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{18}^{5} q^{4} - 4 \zeta_{18}^{4} q^{7} +O(q^{10})$$ q + 2*z^5 * q^4 - 4*z^4 * q^7 $$q + 2 \zeta_{18}^{5} q^{4} - 4 \zeta_{18}^{4} q^{7} - 7 \zeta_{18}^{2} q^{13} - 4 \zeta_{18} q^{16} + \zeta_{18}^{3} q^{19} + (5 \zeta_{18}^{4} - 5 \zeta_{18}) q^{25} + 8 q^{28} - 11 \zeta_{18}^{5} q^{31} + ( - 10 \zeta_{18}^{3} + 10) q^{37} - 5 \zeta_{18} q^{43} + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{49} + ( - 14 \zeta_{18}^{4} + 14 \zeta_{18}) q^{52} - \zeta_{18}^{4} q^{61} + ( - 8 \zeta_{18}^{3} + 8) q^{64} + 5 \zeta_{18}^{2} q^{67} + 7 \zeta_{18}^{3} q^{73} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{76} + (13 \zeta_{18}^{4} - 13 \zeta_{18}) q^{79} + (28 \zeta_{18}^{3} - 28) q^{91} - 5 \zeta_{18} q^{97} +O(q^{100})$$ q + 2*z^5 * q^4 - 4*z^4 * q^7 - 7*z^2 * q^13 - 4*z * q^16 + z^3 * q^19 + (5*z^4 - 5*z) * q^25 + 8 * q^28 - 11*z^5 * q^31 + (-10*z^3 + 10) * q^37 - 5*z * q^43 + (9*z^5 - 9*z^2) * q^49 + (-14*z^4 + 14*z) * q^52 - z^4 * q^61 + (-8*z^3 + 8) * q^64 + 5*z^2 * q^67 + 7*z^3 * q^73 + (2*z^5 - 2*z^2) * q^76 + (13*z^4 - 13*z) * q^79 + (28*z^3 - 28) * q^91 - 5*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 3 q^{19} + 48 q^{28} + 30 q^{37} + 24 q^{64} + 21 q^{73} - 84 q^{91}+O(q^{100})$$ 6 * q + 3 * q^19 + 48 * q^28 + 30 * q^37 + 24 * q^64 + 21 * q^73 - 84 * q^91

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 0 1.87939 0.684040i 0 0 3.75877 + 1.36808i 0 0 0
163.1 0 0 −1.53209 + 1.28558i 0 0 −3.06418 2.57115i 0 0 0
325.1 0 0 −0.347296 + 1.96962i 0 0 −0.694593 3.93923i 0 0 0
406.1 0 0 −0.347296 1.96962i 0 0 −0.694593 + 3.93923i 0 0 0
568.1 0 0 −1.53209 1.28558i 0 0 −3.06418 + 2.57115i 0 0 0
649.1 0 0 1.87939 + 0.684040i 0 0 3.75877 1.36808i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 82.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
9.c even 3 2 inner
9.d odd 6 2 inner
27.e even 9 3 inner
27.f odd 18 3 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.e 6
3.b odd 2 1 CM 729.2.e.e 6
9.c even 3 2 inner 729.2.e.e 6
9.d odd 6 2 inner 729.2.e.e 6
27.e even 9 1 243.2.a.a 1
27.e even 9 2 243.2.c.b 2
27.e even 9 3 inner 729.2.e.e 6
27.f odd 18 1 243.2.a.a 1
27.f odd 18 2 243.2.c.b 2
27.f odd 18 3 inner 729.2.e.e 6
108.j odd 18 1 3888.2.a.n 1
108.l even 18 1 3888.2.a.n 1
135.n odd 18 1 6075.2.a.w 1
135.p even 18 1 6075.2.a.w 1

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

Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{6} - 64T_{7}^{3} + 4096$$ T7^6 - 64*T7^3 + 4096

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 64T^{3} + 4096$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 343 T^{3} + 117649$$
$17$ $$T^{6}$$
$19$ $$(T^{2} - T + 1)^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6} + 1331 T^{3} + \cdots + 1771561$$
$37$ $$(T^{2} - 10 T + 100)^{3}$$
$41$ $$T^{6}$$
$43$ $$T^{6} + 125 T^{3} + 15625$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} - T^{3} + 1$$
$67$ $$T^{6} + 125 T^{3} + 15625$$
$71$ $$T^{6}$$
$73$ $$(T^{2} - 7 T + 49)^{3}$$
$79$ $$T^{6} - 2197 T^{3} + \cdots + 4826809$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6} + 125 T^{3} + 15625$$