# Properties

 Label 6480.2.a.o Level $6480$ Weight $2$ Character orbit 6480.a Self dual yes Analytic conductor $51.743$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 405) 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 2 q^{7}+O(q^{10})$$ q + q^5 - 2 * q^7 $$q + q^{5} - 2 q^{7} - 3 q^{11} - 4 q^{13} + 6 q^{17} + q^{19} - 6 q^{23} + q^{25} + 9 q^{29} + q^{31} - 2 q^{35} + 8 q^{37} - 3 q^{41} + 4 q^{43} + 12 q^{47} - 3 q^{49} - 6 q^{53} - 3 q^{55} + 3 q^{59} - 10 q^{61} - 4 q^{65} - 14 q^{67} - 3 q^{71} + 2 q^{73} + 6 q^{77} + 16 q^{79} - 12 q^{83} + 6 q^{85} - 15 q^{89} + 8 q^{91} + q^{95} - 4 q^{97}+O(q^{100})$$ q + q^5 - 2 * q^7 - 3 * q^11 - 4 * q^13 + 6 * q^17 + q^19 - 6 * q^23 + q^25 + 9 * q^29 + q^31 - 2 * q^35 + 8 * q^37 - 3 * q^41 + 4 * q^43 + 12 * q^47 - 3 * q^49 - 6 * q^53 - 3 * q^55 + 3 * q^59 - 10 * q^61 - 4 * q^65 - 14 * q^67 - 3 * q^71 + 2 * q^73 + 6 * q^77 + 16 * q^79 - 12 * q^83 + 6 * q^85 - 15 * q^89 + 8 * q^91 + q^95 - 4 * 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 −2.00000 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.o 1
3.b odd 2 1 6480.2.a.c 1
4.b odd 2 1 405.2.a.d yes 1
12.b even 2 1 405.2.a.c 1
20.d odd 2 1 2025.2.a.d 1
20.e even 4 2 2025.2.b.f 2
36.f odd 6 2 405.2.e.d 2
36.h even 6 2 405.2.e.e 2
60.h even 2 1 2025.2.a.c 1
60.l odd 4 2 2025.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.c 1 12.b even 2 1
405.2.a.d yes 1 4.b odd 2 1
405.2.e.d 2 36.f odd 6 2
405.2.e.e 2 36.h even 6 2
2025.2.a.c 1 60.h even 2 1
2025.2.a.d 1 20.d odd 2 1
2025.2.b.e 2 60.l odd 4 2
2025.2.b.f 2 20.e even 4 2
6480.2.a.c 1 3.b odd 2 1
6480.2.a.o 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6480))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 6$$ T17 - 6 $$T_{19} - 1$$ T19 - 1 $$T_{23} + 6$$ T23 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T + 3$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T - 1$$
$23$ $$T + 6$$
$29$ $$T - 9$$
$31$ $$T - 1$$
$37$ $$T - 8$$
$41$ $$T + 3$$
$43$ $$T - 4$$
$47$ $$T - 12$$
$53$ $$T + 6$$
$59$ $$T - 3$$
$61$ $$T + 10$$
$67$ $$T + 14$$
$71$ $$T + 3$$
$73$ $$T - 2$$
$79$ $$T - 16$$
$83$ $$T + 12$$
$89$ $$T + 15$$
$97$ $$T + 4$$