# Properties

 Label 729.2.e.f 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{18}^{5} q^{4} - \zeta_{18}^{4} q^{7} +O(q^{10})$$ q + 2*z^5 * q^4 - z^4 * q^7 $$q + 2 \zeta_{18}^{5} q^{4} - \zeta_{18}^{4} q^{7} + 5 \zeta_{18}^{2} q^{13} - 4 \zeta_{18} q^{16} + 7 \zeta_{18}^{3} q^{19} + (5 \zeta_{18}^{4} - 5 \zeta_{18}) q^{25} + 2 q^{28} + 4 \zeta_{18}^{5} q^{31} + (11 \zeta_{18}^{3} - 11) q^{37} - 8 \zeta_{18} q^{43} + ( - 6 \zeta_{18}^{5} + 6 \zeta_{18}^{2}) q^{49} + (10 \zeta_{18}^{4} - 10 \zeta_{18}) q^{52} - \zeta_{18}^{4} q^{61} + ( - 8 \zeta_{18}^{3} + 8) q^{64} + 5 \zeta_{18}^{2} q^{67} + 7 \zeta_{18}^{3} q^{73} + (14 \zeta_{18}^{5} - 14 \zeta_{18}^{2}) q^{76} + ( - 17 \zeta_{18}^{4} + 17 \zeta_{18}) q^{79} + ( - 5 \zeta_{18}^{3} + 5) q^{91} + 19 \zeta_{18} q^{97} +O(q^{100})$$ q + 2*z^5 * q^4 - z^4 * q^7 + 5*z^2 * q^13 - 4*z * q^16 + 7*z^3 * q^19 + (5*z^4 - 5*z) * q^25 + 2 * q^28 + 4*z^5 * q^31 + (11*z^3 - 11) * q^37 - 8*z * q^43 + (-6*z^5 + 6*z^2) * q^49 + (10*z^4 - 10*z) * q^52 - z^4 * q^61 + (-8*z^3 + 8) * q^64 + 5*z^2 * q^67 + 7*z^3 * q^73 + (14*z^5 - 14*z^2) * q^76 + (-17*z^4 + 17*z) * q^79 + (-5*z^3 + 5) * q^91 + 19*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 21 q^{19} + 12 q^{28} - 33 q^{37} + 24 q^{64} + 21 q^{73} + 15 q^{91}+O(q^{100})$$ 6 * q + 21 * q^19 + 12 * q^28 - 33 * q^37 + 24 * q^64 + 21 * q^73 + 15 * 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 0.939693 + 0.342020i 0 0 0
163.1 0 0 −1.53209 + 1.28558i 0 0 −0.766044 0.642788i 0 0 0
325.1 0 0 −0.347296 + 1.96962i 0 0 −0.173648 0.984808i 0 0 0
406.1 0 0 −0.347296 1.96962i 0 0 −0.173648 + 0.984808i 0 0 0
568.1 0 0 −1.53209 1.28558i 0 0 −0.766044 + 0.642788i 0 0 0
649.1 0 0 1.87939 + 0.684040i 0 0 0.939693 0.342020i 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.f 6
3.b odd 2 1 CM 729.2.e.f 6
9.c even 3 2 inner 729.2.e.f 6
9.d odd 6 2 inner 729.2.e.f 6
27.e even 9 1 27.2.a.a 1
27.e even 9 2 81.2.c.a 2
27.e even 9 3 inner 729.2.e.f 6
27.f odd 18 1 27.2.a.a 1
27.f odd 18 2 81.2.c.a 2
27.f odd 18 3 inner 729.2.e.f 6
108.j odd 18 1 432.2.a.e 1
108.j odd 18 2 1296.2.i.i 2
108.l even 18 1 432.2.a.e 1
108.l even 18 2 1296.2.i.i 2
135.n odd 18 1 675.2.a.e 1
135.p even 18 1 675.2.a.e 1
135.q even 36 2 675.2.b.f 2
135.r odd 36 2 675.2.b.f 2
189.y odd 18 1 1323.2.a.i 1
189.be even 18 1 1323.2.a.i 1
216.r odd 18 1 1728.2.a.o 1
216.t even 18 1 1728.2.a.n 1
216.v even 18 1 1728.2.a.o 1
216.x odd 18 1 1728.2.a.n 1
297.o even 18 1 3267.2.a.f 1
297.q odd 18 1 3267.2.a.f 1
351.bi odd 18 1 4563.2.a.e 1
351.bl even 18 1 4563.2.a.e 1
459.r even 18 1 7803.2.a.k 1
459.t odd 18 1 7803.2.a.k 1
513.bw even 18 1 9747.2.a.f 1
513.ca odd 18 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 27.e even 9 1
27.2.a.a 1 27.f odd 18 1
81.2.c.a 2 27.e even 9 2
81.2.c.a 2 27.f odd 18 2
432.2.a.e 1 108.j odd 18 1
432.2.a.e 1 108.l even 18 1
675.2.a.e 1 135.n odd 18 1
675.2.a.e 1 135.p even 18 1
675.2.b.f 2 135.q even 36 2
675.2.b.f 2 135.r odd 36 2
729.2.e.f 6 1.a even 1 1 trivial
729.2.e.f 6 3.b odd 2 1 CM
729.2.e.f 6 9.c even 3 2 inner
729.2.e.f 6 9.d odd 6 2 inner
729.2.e.f 6 27.e even 9 3 inner
729.2.e.f 6 27.f odd 18 3 inner
1296.2.i.i 2 108.j odd 18 2
1296.2.i.i 2 108.l even 18 2
1323.2.a.i 1 189.y odd 18 1
1323.2.a.i 1 189.be even 18 1
1728.2.a.n 1 216.t even 18 1
1728.2.a.n 1 216.x odd 18 1
1728.2.a.o 1 216.r odd 18 1
1728.2.a.o 1 216.v even 18 1
3267.2.a.f 1 297.o even 18 1
3267.2.a.f 1 297.q odd 18 1
4563.2.a.e 1 351.bi odd 18 1
4563.2.a.e 1 351.bl even 18 1
7803.2.a.k 1 459.r even 18 1
7803.2.a.k 1 459.t odd 18 1
9747.2.a.f 1 513.bw even 18 1
9747.2.a.f 1 513.ca odd 18 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{6} - T_{7}^{3} + 1$$ T7^6 - T7^3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - T^{3} + 1$$
$11$ $$T^{6}$$
$13$ $$T^{6} + 125 T^{3} + 15625$$
$17$ $$T^{6}$$
$19$ $$(T^{2} - 7 T + 49)^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6} - 64T^{3} + 4096$$
$37$ $$(T^{2} + 11 T + 121)^{3}$$
$41$ $$T^{6}$$
$43$ $$T^{6} + 512 T^{3} + 262144$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} - T^{3} + 1$$
$67$ $$T^{6} + 125 T^{3} + 15625$$
$71$ $$T^{6}$$
$73$ $$(T^{2} - 7 T + 49)^{3}$$
$79$ $$T^{6} + 4913 T^{3} + \cdots + 24137569$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6} - 6859 T^{3} + \cdots + 47045881$$