# Properties

 Label 6480.2.a.k Level $6480$ Weight $2$ Character orbit 6480.a Self dual yes Analytic conductor $51.743$ Analytic rank $0$ 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] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, 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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6480 = 2^{4} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6480.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: $$51.7430605098$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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)

 $$f(q)$$ $$=$$ $$q - q^{5} + 3 q^{7}+O(q^{10})$$ q - q^5 + 3 * q^7 $$q - q^{5} + 3 q^{7} + 2 q^{11} - 2 q^{13} + 4 q^{17} + 8 q^{19} - 3 q^{23} + q^{25} - q^{29} - 3 q^{35} - 4 q^{37} + 5 q^{41} + 8 q^{43} - 7 q^{47} + 2 q^{49} - 2 q^{53} - 2 q^{55} + 14 q^{59} + 7 q^{61} + 2 q^{65} + 3 q^{67} - 2 q^{71} + 4 q^{73} + 6 q^{77} + 6 q^{79} - 9 q^{83} - 4 q^{85} - 15 q^{89} - 6 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100})$$ q - q^5 + 3 * q^7 + 2 * q^11 - 2 * q^13 + 4 * q^17 + 8 * q^19 - 3 * q^23 + q^25 - q^29 - 3 * q^35 - 4 * q^37 + 5 * q^41 + 8 * q^43 - 7 * q^47 + 2 * q^49 - 2 * q^53 - 2 * q^55 + 14 * q^59 + 7 * q^61 + 2 * q^65 + 3 * q^67 - 2 * q^71 + 4 * q^73 + 6 * q^77 + 6 * q^79 - 9 * q^83 - 4 * q^85 - 15 * q^89 - 6 * q^91 - 8 * 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 3.00000 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 6480.2.a.k 1
3.b odd 2 1 6480.2.a.x 1
4.b odd 2 1 405.2.a.e 1
9.c even 3 2 720.2.q.d 2
9.d odd 6 2 2160.2.q.a 2
12.b even 2 1 405.2.a.b 1
20.d odd 2 1 2025.2.a.b 1
20.e even 4 2 2025.2.b.c 2
36.f odd 6 2 45.2.e.a 2
36.h even 6 2 135.2.e.a 2
60.h even 2 1 2025.2.a.e 1
60.l odd 4 2 2025.2.b.d 2
180.n even 6 2 675.2.e.a 2
180.p odd 6 2 225.2.e.a 2
180.v odd 12 4 675.2.k.a 4
180.x even 12 4 225.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 36.f odd 6 2
135.2.e.a 2 36.h even 6 2
225.2.e.a 2 180.p odd 6 2
225.2.k.a 4 180.x even 12 4
405.2.a.b 1 12.b even 2 1
405.2.a.e 1 4.b odd 2 1
675.2.e.a 2 180.n even 6 2
675.2.k.a 4 180.v odd 12 4
720.2.q.d 2 9.c even 3 2
2025.2.a.b 1 20.d odd 2 1
2025.2.a.e 1 60.h even 2 1
2025.2.b.c 2 20.e even 4 2
2025.2.b.d 2 60.l odd 4 2
2160.2.q.a 2 9.d odd 6 2
6480.2.a.k 1 1.a even 1 1 trivial
6480.2.a.x 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6480))$$:

 $$T_{7} - 3$$ T7 - 3 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 4$$ T17 - 4 $$T_{19} - 8$$ T19 - 8 $$T_{23} + 3$$ T23 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 3$$
$11$ $$T - 2$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T - 8$$
$23$ $$T + 3$$
$29$ $$T + 1$$
$31$ $$T$$
$37$ $$T + 4$$
$41$ $$T - 5$$
$43$ $$T - 8$$
$47$ $$T + 7$$
$53$ $$T + 2$$
$59$ $$T - 14$$
$61$ $$T - 7$$
$67$ $$T - 3$$
$71$ $$T + 2$$
$73$ $$T - 4$$
$79$ $$T - 6$$
$83$ $$T + 9$$
$89$ $$T + 15$$
$97$ $$T - 2$$