# Properties

 Label 324.2.e.a Level $324$ Weight $2$ Character orbit 324.e Analytic conductor $2.587$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,2,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, 2, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$2.58715302549$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} +O(q^{10})$$ q - 3*z * q^5 + (2*z - 2) * q^7 $$q - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + ( - 6 \zeta_{6} + 6) q^{11} - 5 \zeta_{6} q^{13} - 3 q^{17} + 2 q^{19} - 6 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + (3 \zeta_{6} - 3) q^{29} + 4 \zeta_{6} q^{31} + 6 q^{35} + 5 q^{37} + 6 \zeta_{6} q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + 3 \zeta_{6} q^{49} - 6 q^{53} - 18 q^{55} + 12 \zeta_{6} q^{59} + (5 \zeta_{6} - 5) q^{61} + (15 \zeta_{6} - 15) q^{65} - 2 \zeta_{6} q^{67} + 6 q^{71} - q^{73} + 12 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} + 9 \zeta_{6} q^{85} - 3 q^{89} + 10 q^{91} - 6 \zeta_{6} q^{95} + ( - 10 \zeta_{6} + 10) q^{97} +O(q^{100})$$ q - 3*z * q^5 + (2*z - 2) * q^7 + (-6*z + 6) * q^11 - 5*z * q^13 - 3 * q^17 + 2 * q^19 - 6*z * q^23 + (4*z - 4) * q^25 + (3*z - 3) * q^29 + 4*z * q^31 + 6 * q^35 + 5 * q^37 + 6*z * q^41 + (-10*z + 10) * q^43 + 3*z * q^49 - 6 * q^53 - 18 * q^55 + 12*z * q^59 + (5*z - 5) * q^61 + (15*z - 15) * q^65 - 2*z * q^67 + 6 * q^71 - q^73 + 12*z * q^77 + (-10*z + 10) * q^79 + 9*z * q^85 - 3 * q^89 + 10 * q^91 - 6*z * q^95 + (-10*z + 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - 2 * q^7 $$2 q - 3 q^{5} - 2 q^{7} + 6 q^{11} - 5 q^{13} - 6 q^{17} + 4 q^{19} - 6 q^{23} - 4 q^{25} - 3 q^{29} + 4 q^{31} + 12 q^{35} + 10 q^{37} + 6 q^{41} + 10 q^{43} + 3 q^{49} - 12 q^{53} - 36 q^{55} + 12 q^{59} - 5 q^{61} - 15 q^{65} - 2 q^{67} + 12 q^{71} - 2 q^{73} + 12 q^{77} + 10 q^{79} + 9 q^{85} - 6 q^{89} + 20 q^{91} - 6 q^{95} + 10 q^{97}+O(q^{100})$$ 2 * q - 3 * q^5 - 2 * q^7 + 6 * q^11 - 5 * q^13 - 6 * q^17 + 4 * q^19 - 6 * q^23 - 4 * q^25 - 3 * q^29 + 4 * q^31 + 12 * q^35 + 10 * q^37 + 6 * q^41 + 10 * q^43 + 3 * q^49 - 12 * q^53 - 36 * q^55 + 12 * q^59 - 5 * q^61 - 15 * q^65 - 2 * q^67 + 12 * q^71 - 2 * q^73 + 12 * q^77 + 10 * q^79 + 9 * q^85 - 6 * q^89 + 20 * q^91 - 6 * q^95 + 10 * 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 −1.50000 + 2.59808i 0 −1.00000 1.73205i 0 0 0
217.1 0 0 0 −1.50000 2.59808i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.e.a 2
3.b odd 2 1 324.2.e.d 2
4.b odd 2 1 1296.2.i.d 2
9.c even 3 1 324.2.a.d yes 1
9.c even 3 1 inner 324.2.e.a 2
9.d odd 6 1 324.2.a.b 1
9.d odd 6 1 324.2.e.d 2
12.b even 2 1 1296.2.i.p 2
36.f odd 6 1 1296.2.a.j 1
36.f odd 6 1 1296.2.i.d 2
36.h even 6 1 1296.2.a.a 1
36.h even 6 1 1296.2.i.p 2
45.h odd 6 1 8100.2.a.f 1
45.j even 6 1 8100.2.a.a 1
45.k odd 12 2 8100.2.d.a 2
45.l even 12 2 8100.2.d.j 2
72.j odd 6 1 5184.2.a.bc 1
72.l even 6 1 5184.2.a.z 1
72.n even 6 1 5184.2.a.g 1
72.p odd 6 1 5184.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 9.d odd 6 1
324.2.a.d yes 1 9.c even 3 1
324.2.e.a 2 1.a even 1 1 trivial
324.2.e.a 2 9.c even 3 1 inner
324.2.e.d 2 3.b odd 2 1
324.2.e.d 2 9.d odd 6 1
1296.2.a.a 1 36.h even 6 1
1296.2.a.j 1 36.f odd 6 1
1296.2.i.d 2 4.b odd 2 1
1296.2.i.d 2 36.f odd 6 1
1296.2.i.p 2 12.b even 2 1
1296.2.i.p 2 36.h even 6 1
5184.2.a.d 1 72.p odd 6 1
5184.2.a.g 1 72.n even 6 1
5184.2.a.z 1 72.l even 6 1
5184.2.a.bc 1 72.j odd 6 1
8100.2.a.a 1 45.j even 6 1
8100.2.a.f 1 45.h odd 6 1
8100.2.d.a 2 45.k odd 12 2
8100.2.d.j 2 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T - 5)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T + 1)^{2}$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$T^{2}$$
$89$ $$(T + 3)^{2}$$
$97$ $$T^{2} - 10T + 100$$