# Properties

 Label 3240.2.a.e Level $3240$ Weight $2$ Character orbit 3240.a Self dual yes Analytic conductor $25.872$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.a (trivial)

## 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: yes Analytic conductor: $$25.8715302549$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5}+O(q^{10})$$ q + q^5 $$q + q^{5} - q^{11} - 7 q^{19} - 6 q^{23} + q^{25} + 7 q^{29} + q^{31} - 2 q^{37} - 9 q^{41} - 6 q^{43} + 2 q^{47} - 7 q^{49} - q^{55} - 3 q^{59} - 10 q^{61} - 2 q^{67} - q^{71} + 4 q^{79} + 6 q^{83} + 7 q^{89} - 7 q^{95} + 2 q^{97}+O(q^{100})$$ q + q^5 - q^11 - 7 * q^19 - 6 * q^23 + q^25 + 7 * q^29 + q^31 - 2 * q^37 - 9 * q^41 - 6 * q^43 + 2 * q^47 - 7 * q^49 - q^55 - 3 * q^59 - 10 * q^61 - 2 * q^67 - q^71 + 4 * q^79 + 6 * q^83 + 7 * q^89 - 7 * q^95 + 2 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 1.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.2.a.e yes 1
3.b odd 2 1 3240.2.a.c 1
4.b odd 2 1 6480.2.a.s 1
9.c even 3 2 3240.2.q.f 2
9.d odd 6 2 3240.2.q.r 2
12.b even 2 1 6480.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.c 1 3.b odd 2 1
3240.2.a.e yes 1 1.a even 1 1 trivial
3240.2.q.f 2 9.c even 3 2
3240.2.q.r 2 9.d odd 6 2
6480.2.a.d 1 12.b even 2 1
6480.2.a.s 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3240))$$:

 $$T_{7}$$ T7 $$T_{11} + 1$$ T11 + 1 $$T_{17}$$ T17

## Hecke characteristic polynomials

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