# Properties

 Label 3240.2.q.e Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.q (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: $$25.8715302549$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} +O(q^{10})$$ q - z * q^5 $$q - \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{11} + 3 q^{17} - q^{19} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (4 \zeta_{6} - 4) q^{29} + 5 \zeta_{6} q^{31} + 10 q^{37} - 6 \zeta_{6} q^{41} + ( - 6 \zeta_{6} + 6) q^{43} + (8 \zeta_{6} - 8) q^{47} + 7 \zeta_{6} q^{49} + 3 q^{53} + 2 q^{55} + (5 \zeta_{6} - 5) q^{61} + 2 \zeta_{6} q^{67} + 2 q^{71} + 6 q^{73} + ( - 11 \zeta_{6} + 11) q^{79} + (9 \zeta_{6} - 9) q^{83} - 3 \zeta_{6} q^{85} + 10 q^{89} + \zeta_{6} q^{95} + (8 \zeta_{6} - 8) q^{97} +O(q^{100})$$ q - z * q^5 + (2*z - 2) * q^11 + 3 * q^17 - q^19 - 3*z * q^23 + (z - 1) * q^25 + (4*z - 4) * q^29 + 5*z * q^31 + 10 * q^37 - 6*z * q^41 + (-6*z + 6) * q^43 + (8*z - 8) * q^47 + 7*z * q^49 + 3 * q^53 + 2 * q^55 + (5*z - 5) * q^61 + 2*z * q^67 + 2 * q^71 + 6 * q^73 + (-11*z + 11) * q^79 + (9*z - 9) * q^83 - 3*z * q^85 + 10 * q^89 + z * q^95 + (8*z - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5}+O(q^{10})$$ 2 * q - q^5 $$2 q - q^{5} - 2 q^{11} + 6 q^{17} - 2 q^{19} - 3 q^{23} - q^{25} - 4 q^{29} + 5 q^{31} + 20 q^{37} - 6 q^{41} + 6 q^{43} - 8 q^{47} + 7 q^{49} + 6 q^{53} + 4 q^{55} - 5 q^{61} + 2 q^{67} + 4 q^{71} + 12 q^{73} + 11 q^{79} - 9 q^{83} - 3 q^{85} + 20 q^{89} + q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - q^5 - 2 * q^11 + 6 * q^17 - 2 * q^19 - 3 * q^23 - q^25 - 4 * q^29 + 5 * q^31 + 20 * q^37 - 6 * q^41 + 6 * q^43 - 8 * q^47 + 7 * q^49 + 6 * q^53 + 4 * q^55 - 5 * q^61 + 2 * q^67 + 4 * q^71 + 12 * q^73 + 11 * q^79 - 9 * q^83 - 3 * q^85 + 20 * q^89 + q^95 - 8 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$$.

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1081.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 0 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 0 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 3240.2.q.e 2
3.b odd 2 1 3240.2.q.s 2
9.c even 3 1 1080.2.a.j yes 1
9.c even 3 1 inner 3240.2.q.e 2
9.d odd 6 1 1080.2.a.d 1
9.d odd 6 1 3240.2.q.s 2
36.f odd 6 1 2160.2.a.r 1
36.h even 6 1 2160.2.a.f 1
45.h odd 6 1 5400.2.a.w 1
45.j even 6 1 5400.2.a.x 1
45.k odd 12 2 5400.2.f.r 2
45.l even 12 2 5400.2.f.j 2
72.j odd 6 1 8640.2.a.bt 1
72.l even 6 1 8640.2.a.bs 1
72.n even 6 1 8640.2.a.o 1
72.p odd 6 1 8640.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.d 1 9.d odd 6 1
1080.2.a.j yes 1 9.c even 3 1
2160.2.a.f 1 36.h even 6 1
2160.2.a.r 1 36.f odd 6 1
3240.2.q.e 2 1.a even 1 1 trivial
3240.2.q.e 2 9.c even 3 1 inner
3240.2.q.s 2 3.b odd 2 1
3240.2.q.s 2 9.d odd 6 1
5400.2.a.w 1 45.h odd 6 1
5400.2.a.x 1 45.j even 6 1
5400.2.f.j 2 45.l even 12 2
5400.2.f.r 2 45.k odd 12 2
8640.2.a.o 1 72.n even 6 1
8640.2.a.p 1 72.p odd 6 1
8640.2.a.bs 1 72.l even 6 1
8640.2.a.bt 1 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3240, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2}$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 4T + 16$$
$31$ $$T^{2} - 5T + 25$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 6T + 36$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$(T - 2)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 8T + 64$$