# Properties

 Label 324.4.e.d Level $324$ Weight $4$ Character orbit 324.e Analytic conductor $19.117$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 324.e (of order $$3$$, degree $$2$$, 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: $$19.1166188419$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (17 \zeta_{6} - 17) q^{7}+O(q^{10})$$ q + (17*z - 17) * q^7 $$q + (17 \zeta_{6} - 17) q^{7} - 89 \zeta_{6} q^{13} + 107 q^{19} + ( - 125 \zeta_{6} + 125) q^{25} - 308 \zeta_{6} q^{31} - 433 q^{37} + ( - 520 \zeta_{6} + 520) q^{43} + 54 \zeta_{6} q^{49} + ( - 901 \zeta_{6} + 901) q^{61} - 1007 \zeta_{6} q^{67} - 271 q^{73} + (503 \zeta_{6} - 503) q^{79} + 1513 q^{91} + (1853 \zeta_{6} - 1853) q^{97} +O(q^{100})$$ q + (17*z - 17) * q^7 - 89*z * q^13 + 107 * q^19 + (-125*z + 125) * q^25 - 308*z * q^31 - 433 * q^37 + (-520*z + 520) * q^43 + 54*z * q^49 + (-901*z + 901) * q^61 - 1007*z * q^67 - 271 * q^73 + (503*z - 503) * q^79 + 1513 * q^91 + (1853*z - 1853) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 17 q^{7}+O(q^{10})$$ 2 * q - 17 * q^7 $$2 q - 17 q^{7} - 89 q^{13} + 214 q^{19} + 125 q^{25} - 308 q^{31} - 866 q^{37} + 520 q^{43} + 54 q^{49} + 901 q^{61} - 1007 q^{67} - 542 q^{73} - 503 q^{79} + 3026 q^{91} - 1853 q^{97}+O(q^{100})$$ 2 * q - 17 * q^7 - 89 * q^13 + 214 * q^19 + 125 * q^25 - 308 * q^31 - 866 * q^37 + 520 * q^43 + 54 * q^49 + 901 * q^61 - 1007 * q^67 - 542 * q^73 - 503 * q^79 + 3026 * q^91 - 1853 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −8.50000 14.7224i 0 0 0
217.1 0 0 0 0 0 −8.50000 + 14.7224i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.e.d 2
3.b odd 2 1 CM 324.4.e.d 2
9.c even 3 1 108.4.a.c 1
9.c even 3 1 inner 324.4.e.d 2
9.d odd 6 1 108.4.a.c 1
9.d odd 6 1 inner 324.4.e.d 2
36.f odd 6 1 432.4.a.g 1
36.h even 6 1 432.4.a.g 1
72.j odd 6 1 1728.4.a.q 1
72.l even 6 1 1728.4.a.p 1
72.n even 6 1 1728.4.a.q 1
72.p odd 6 1 1728.4.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.c 1 9.c even 3 1
108.4.a.c 1 9.d odd 6 1
324.4.e.d 2 1.a even 1 1 trivial
324.4.e.d 2 3.b odd 2 1 CM
324.4.e.d 2 9.c even 3 1 inner
324.4.e.d 2 9.d odd 6 1 inner
432.4.a.g 1 36.f odd 6 1
432.4.a.g 1 36.h even 6 1
1728.4.a.p 1 72.l even 6 1
1728.4.a.p 1 72.p odd 6 1
1728.4.a.q 1 72.j odd 6 1
1728.4.a.q 1 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 17T_{7} + 289$$ T7^2 + 17*T7 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 17T + 289$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 89T + 7921$$
$17$ $$T^{2}$$
$19$ $$(T - 107)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 308T + 94864$$
$37$ $$(T + 433)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 520T + 270400$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 901T + 811801$$
$67$ $$T^{2} + 1007 T + 1014049$$
$71$ $$T^{2}$$
$73$ $$(T + 271)^{2}$$
$79$ $$T^{2} + 503T + 253009$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 1853 T + 3433609$$