# Properties

 Label 324.9.g.a Level $324$ Weight $9$ Character orbit 324.g Analytic conductor $131.991$ 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] = [324,9,Mod(53,324)]

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, 5]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("324.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 324.g (of order $$6$$, 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: $$131.990669660$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (4034 \zeta_{6} - 4034) q^{7}+O(q^{10})$$ q + (4034*z - 4034) * q^7 $$q + (4034 \zeta_{6} - 4034) q^{7} + 35806 \zeta_{6} q^{13} - 258526 q^{19} + (390625 \zeta_{6} - 390625) q^{25} + 1809406 \zeta_{6} q^{31} + 503522 q^{37} + (3492194 \zeta_{6} - 3492194) q^{43} - 10508355 \zeta_{6} q^{49} + ( - 23826526 \zeta_{6} + 23826526) q^{61} + 5421406 \zeta_{6} q^{67} + 16169282 q^{73} + ( - 18887038 \zeta_{6} + 18887038) q^{79} - 144441404 q^{91} + (176908034 \zeta_{6} - 176908034) q^{97} +O(q^{100})$$ q + (4034*z - 4034) * q^7 + 35806*z * q^13 - 258526 * q^19 + (390625*z - 390625) * q^25 + 1809406*z * q^31 + 503522 * q^37 + (3492194*z - 3492194) * q^43 - 10508355*z * q^49 + (-23826526*z + 23826526) * q^61 + 5421406*z * q^67 + 16169282 * q^73 + (-18887038*z + 18887038) * q^79 - 144441404 * q^91 + (176908034*z - 176908034) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4034 q^{7}+O(q^{10})$$ 2 * q - 4034 * q^7 $$2 q - 4034 q^{7} + 35806 q^{13} - 517052 q^{19} - 390625 q^{25} + 1809406 q^{31} + 1007044 q^{37} - 3492194 q^{43} - 10508355 q^{49} + 23826526 q^{61} + 5421406 q^{67} + 32338564 q^{73} + 18887038 q^{79} - 288882808 q^{91} - 176908034 q^{97}+O(q^{100})$$ 2 * q - 4034 * q^7 + 35806 * q^13 - 517052 * q^19 - 390625 * q^25 + 1809406 * q^31 + 1007044 * q^37 - 3492194 * q^43 - 10508355 * q^49 + 23826526 * q^61 + 5421406 * q^67 + 32338564 * q^73 + 18887038 * q^79 - 288882808 * q^91 - 176908034 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −2017.00 3493.55i 0 0 0
269.1 0 0 0 0 0 −2017.00 + 3493.55i 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 324.9.g.a 2
3.b odd 2 1 CM 324.9.g.a 2
9.c even 3 1 12.9.c.a 1
9.c even 3 1 inner 324.9.g.a 2
9.d odd 6 1 12.9.c.a 1
9.d odd 6 1 inner 324.9.g.a 2
36.f odd 6 1 48.9.e.a 1
36.h even 6 1 48.9.e.a 1
45.h odd 6 1 300.9.g.a 1
45.j even 6 1 300.9.g.a 1
45.k odd 12 2 300.9.b.b 2
45.l even 12 2 300.9.b.b 2
72.j odd 6 1 192.9.e.a 1
72.l even 6 1 192.9.e.b 1
72.n even 6 1 192.9.e.a 1
72.p odd 6 1 192.9.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.a 1 9.c even 3 1
12.9.c.a 1 9.d odd 6 1
48.9.e.a 1 36.f odd 6 1
48.9.e.a 1 36.h even 6 1
192.9.e.a 1 72.j odd 6 1
192.9.e.a 1 72.n even 6 1
192.9.e.b 1 72.l even 6 1
192.9.e.b 1 72.p odd 6 1
300.9.b.b 2 45.k odd 12 2
300.9.b.b 2 45.l even 12 2
300.9.g.a 1 45.h odd 6 1
300.9.g.a 1 45.j even 6 1
324.9.g.a 2 1.a even 1 1 trivial
324.9.g.a 2 3.b odd 2 1 CM
324.9.g.a 2 9.c even 3 1 inner
324.9.g.a 2 9.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 4034T_{7} + 16273156$$ T7^2 + 4034*T7 + 16273156

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4034 T + 16273156$$
$11$ $$T^{2}$$
$13$ $$T^{2} + \cdots + 1282069636$$
$17$ $$T^{2}$$
$19$ $$(T + 258526)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + \cdots + 3273950072836$$
$37$ $$(T - 503522)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + \cdots + 12195418933636$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + \cdots + 567703341228676$$
$67$ $$T^{2} + \cdots + 29391643016836$$
$71$ $$T^{2}$$
$73$ $$(T - 16169282)^{2}$$
$79$ $$T^{2} + \cdots + 356720204413444$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + \cdots + 31\!\cdots\!56$$