# Properties

 Label 324.3.g.b Level $324$ Weight $3$ Character orbit 324.g Analytic conductor $8.828$ 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] = [324,3,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, 3, names="a")

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

chi := DirichletCharacter("324.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ 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: $$8.82836056527$$ 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{7}+O(q^{10})$$ q + (2*z - 2) * q^7 $$q + (2 \zeta_{6} - 2) q^{7} + 22 \zeta_{6} q^{13} + 26 q^{19} + (25 \zeta_{6} - 25) q^{25} + 46 \zeta_{6} q^{31} + 26 q^{37} + ( - 22 \zeta_{6} + 22) q^{43} + 45 \zeta_{6} q^{49} + (74 \zeta_{6} - 74) q^{61} - 122 \zeta_{6} q^{67} - 46 q^{73} + ( - 142 \zeta_{6} + 142) q^{79} - 44 q^{91} + (2 \zeta_{6} - 2) q^{97} +O(q^{100})$$ q + (2*z - 2) * q^7 + 22*z * q^13 + 26 * q^19 + (25*z - 25) * q^25 + 46*z * q^31 + 26 * q^37 + (-22*z + 22) * q^43 + 45*z * q^49 + (74*z - 74) * q^61 - 122*z * q^67 - 46 * q^73 + (-142*z + 142) * q^79 - 44 * q^91 + (2*z - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} + 22 q^{13} + 52 q^{19} - 25 q^{25} + 46 q^{31} + 52 q^{37} + 22 q^{43} + 45 q^{49} - 74 q^{61} - 122 q^{67} - 92 q^{73} + 142 q^{79} - 88 q^{91} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 22 * q^13 + 52 * q^19 - 25 * q^25 + 46 * q^31 + 52 * q^37 + 22 * q^43 + 45 * q^49 - 74 * q^61 - 122 * q^67 - 92 * q^73 + 142 * q^79 - 88 * q^91 - 2 * 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 −1.00000 1.73205i 0 0 0
269.1 0 0 0 0 0 −1.00000 + 1.73205i 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 324.3.g.b 2
3.b odd 2 1 CM 324.3.g.b 2
4.b odd 2 1 1296.3.q.b 2
9.c even 3 1 12.3.c.a 1
9.c even 3 1 inner 324.3.g.b 2
9.d odd 6 1 12.3.c.a 1
9.d odd 6 1 inner 324.3.g.b 2
12.b even 2 1 1296.3.q.b 2
36.f odd 6 1 48.3.e.a 1
36.f odd 6 1 1296.3.q.b 2
36.h even 6 1 48.3.e.a 1
36.h even 6 1 1296.3.q.b 2
45.h odd 6 1 300.3.g.b 1
45.j even 6 1 300.3.g.b 1
45.k odd 12 2 300.3.b.a 2
45.l even 12 2 300.3.b.a 2
63.g even 3 1 588.3.p.c 2
63.h even 3 1 588.3.p.c 2
63.i even 6 1 588.3.p.b 2
63.j odd 6 1 588.3.p.c 2
63.k odd 6 1 588.3.p.b 2
63.l odd 6 1 588.3.c.c 1
63.n odd 6 1 588.3.p.c 2
63.o even 6 1 588.3.c.c 1
63.s even 6 1 588.3.p.b 2
63.t odd 6 1 588.3.p.b 2
72.j odd 6 1 192.3.e.b 1
72.l even 6 1 192.3.e.a 1
72.n even 6 1 192.3.e.b 1
72.p odd 6 1 192.3.e.a 1
99.g even 6 1 1452.3.e.b 1
99.h odd 6 1 1452.3.e.b 1
144.u even 12 2 768.3.h.b 2
144.v odd 12 2 768.3.h.b 2
144.w odd 12 2 768.3.h.a 2
144.x even 12 2 768.3.h.a 2
180.n even 6 1 1200.3.l.b 1
180.p odd 6 1 1200.3.l.b 1
180.v odd 12 2 1200.3.c.c 2
180.x even 12 2 1200.3.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 9.c even 3 1
12.3.c.a 1 9.d odd 6 1
48.3.e.a 1 36.f odd 6 1
48.3.e.a 1 36.h even 6 1
192.3.e.a 1 72.l even 6 1
192.3.e.a 1 72.p odd 6 1
192.3.e.b 1 72.j odd 6 1
192.3.e.b 1 72.n even 6 1
300.3.b.a 2 45.k odd 12 2
300.3.b.a 2 45.l even 12 2
300.3.g.b 1 45.h odd 6 1
300.3.g.b 1 45.j even 6 1
324.3.g.b 2 1.a even 1 1 trivial
324.3.g.b 2 3.b odd 2 1 CM
324.3.g.b 2 9.c even 3 1 inner
324.3.g.b 2 9.d odd 6 1 inner
588.3.c.c 1 63.l odd 6 1
588.3.c.c 1 63.o even 6 1
588.3.p.b 2 63.i even 6 1
588.3.p.b 2 63.k odd 6 1
588.3.p.b 2 63.s even 6 1
588.3.p.b 2 63.t odd 6 1
588.3.p.c 2 63.g even 3 1
588.3.p.c 2 63.h even 3 1
588.3.p.c 2 63.j odd 6 1
588.3.p.c 2 63.n odd 6 1
768.3.h.a 2 144.w odd 12 2
768.3.h.a 2 144.x even 12 2
768.3.h.b 2 144.u even 12 2
768.3.h.b 2 144.v odd 12 2
1200.3.c.c 2 180.v odd 12 2
1200.3.c.c 2 180.x even 12 2
1200.3.l.b 1 180.n even 6 1
1200.3.l.b 1 180.p odd 6 1
1296.3.q.b 2 4.b odd 2 1
1296.3.q.b 2 12.b even 2 1
1296.3.q.b 2 36.f odd 6 1
1296.3.q.b 2 36.h even 6 1
1452.3.e.b 1 99.g even 6 1
1452.3.e.b 1 99.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 22T + 484$$
$17$ $$T^{2}$$
$19$ $$(T - 26)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 46T + 2116$$
$37$ $$(T - 26)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 22T + 484$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 74T + 5476$$
$67$ $$T^{2} + 122T + 14884$$
$71$ $$T^{2}$$
$73$ $$(T + 46)^{2}$$
$79$ $$T^{2} - 142T + 20164$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 2T + 4$$