# Properties

 Label 6480.2.a.bi Level $6480$ Weight $2$ Character orbit 6480.a Self dual yes Analytic conductor $51.743$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (\beta + 3) q^{7}+O(q^{10})$$ q - q^5 + (b + 3) * q^7 $$q - q^{5} + (\beta + 3) q^{7} + ( - \beta - 4) q^{11} + (2 \beta - 2) q^{13} + (\beta + 1) q^{17} + ( - 2 \beta - 1) q^{19} - 2 \beta q^{23} + q^{25} + ( - 3 \beta + 2) q^{29} + 3 q^{31} + ( - \beta - 3) q^{35} + ( - \beta - 1) q^{37} + (3 \beta + 2) q^{41} + ( - 3 \beta + 5) q^{43} + ( - \beta - 7) q^{47} + (6 \beta + 5) q^{49} + ( - \beta - 5) q^{53} + (\beta + 4) q^{55} + (\beta - 10) q^{59} + 4 q^{61} + ( - 2 \beta + 2) q^{65} - 2 \beta q^{67} + ( - \beta - 2) q^{71} + ( - 5 \beta + 1) q^{73} + ( - 7 \beta - 15) q^{77} + ( - 2 \beta - 12) q^{79} + (3 \beta - 3) q^{83} + ( - \beta - 1) q^{85} - 3 \beta q^{89} + 4 \beta q^{91} + (2 \beta + 1) q^{95} + ( - 5 \beta - 1) q^{97} +O(q^{100})$$ q - q^5 + (b + 3) * q^7 + (-b - 4) * q^11 + (2*b - 2) * q^13 + (b + 1) * q^17 + (-2*b - 1) * q^19 - 2*b * q^23 + q^25 + (-3*b + 2) * q^29 + 3 * q^31 + (-b - 3) * q^35 + (-b - 1) * q^37 + (3*b + 2) * q^41 + (-3*b + 5) * q^43 + (-b - 7) * q^47 + (6*b + 5) * q^49 + (-b - 5) * q^53 + (b + 4) * q^55 + (b - 10) * q^59 + 4 * q^61 + (-2*b + 2) * q^65 - 2*b * q^67 + (-b - 2) * q^71 + (-5*b + 1) * q^73 + (-7*b - 15) * q^77 + (-2*b - 12) * q^79 + (3*b - 3) * q^83 + (-b - 1) * q^85 - 3*b * q^89 + 4*b * q^91 + (2*b + 1) * q^95 + (-5*b - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 6 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 6 * q^7 $$2 q - 2 q^{5} + 6 q^{7} - 8 q^{11} - 4 q^{13} + 2 q^{17} - 2 q^{19} + 2 q^{25} + 4 q^{29} + 6 q^{31} - 6 q^{35} - 2 q^{37} + 4 q^{41} + 10 q^{43} - 14 q^{47} + 10 q^{49} - 10 q^{53} + 8 q^{55} - 20 q^{59} + 8 q^{61} + 4 q^{65} - 4 q^{71} + 2 q^{73} - 30 q^{77} - 24 q^{79} - 6 q^{83} - 2 q^{85} + 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 6 * q^7 - 8 * q^11 - 4 * q^13 + 2 * q^17 - 2 * q^19 + 2 * q^25 + 4 * q^29 + 6 * q^31 - 6 * q^35 - 2 * q^37 + 4 * q^41 + 10 * q^43 - 14 * q^47 + 10 * q^49 - 10 * q^53 + 8 * q^55 - 20 * q^59 + 8 * q^61 + 4 * q^65 - 4 * q^71 + 2 * q^73 - 30 * q^77 - 24 * q^79 - 6 * q^83 - 2 * q^85 + 2 * 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
 −1.73205 1.73205
0 0 0 −1.00000 0 1.26795 0 0 0
1.2 0 0 0 −1.00000 0 4.73205 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.bi 2
3.b odd 2 1 6480.2.a.br 2
4.b odd 2 1 405.2.a.h yes 2
12.b even 2 1 405.2.a.g 2
20.d odd 2 1 2025.2.a.g 2
20.e even 4 2 2025.2.b.h 4
36.f odd 6 2 405.2.e.i 4
36.h even 6 2 405.2.e.l 4
60.h even 2 1 2025.2.a.m 2
60.l odd 4 2 2025.2.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 12.b even 2 1
405.2.a.h yes 2 4.b odd 2 1
405.2.e.i 4 36.f odd 6 2
405.2.e.l 4 36.h even 6 2
2025.2.a.g 2 20.d odd 2 1
2025.2.a.m 2 60.h even 2 1
2025.2.b.g 4 60.l odd 4 2
2025.2.b.h 4 20.e even 4 2
6480.2.a.bi 2 1.a even 1 1 trivial
6480.2.a.br 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6480))$$:

 $$T_{7}^{2} - 6T_{7} + 6$$ T7^2 - 6*T7 + 6 $$T_{11}^{2} + 8T_{11} + 13$$ T11^2 + 8*T11 + 13 $$T_{13}^{2} + 4T_{13} - 8$$ T13^2 + 4*T13 - 8 $$T_{17}^{2} - 2T_{17} - 2$$ T17^2 - 2*T17 - 2 $$T_{19}^{2} + 2T_{19} - 11$$ T19^2 + 2*T19 - 11 $$T_{23}^{2} - 12$$ T23^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 6T + 6$$
$11$ $$T^{2} + 8T + 13$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$T^{2} - 2T - 2$$
$19$ $$T^{2} + 2T - 11$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} - 4T - 23$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 2T - 2$$
$41$ $$T^{2} - 4T - 23$$
$43$ $$T^{2} - 10T - 2$$
$47$ $$T^{2} + 14T + 46$$
$53$ $$T^{2} + 10T + 22$$
$59$ $$T^{2} + 20T + 97$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} - 12$$
$71$ $$T^{2} + 4T + 1$$
$73$ $$T^{2} - 2T - 74$$
$79$ $$T^{2} + 24T + 132$$
$83$ $$T^{2} + 6T - 18$$
$89$ $$T^{2} - 27$$
$97$ $$T^{2} + 2T - 74$$