# Properties

 Label 3240.2.q.y Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + (\beta_{3} - \beta_1 - 1) q^{7}+O(q^{10})$$ q + b1 * q^5 + (b3 - b1 - 1) * q^7 $$q + \beta_1 q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} - 2 \beta_1 - 2) q^{11} + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{13} + 2 q^{17} + q^{19} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{23} + ( - \beta_1 - 1) q^{25} + ( - \beta_{3} - 2 \beta_1 - 2) q^{29} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} + 1) q^{35} + ( - 2 \beta_{2} - 2) q^{37} + (2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{41} + ( - 4 \beta_1 - 4) q^{43} + (\beta_{3} - 3 \beta_1 - 3) q^{47} + ( - \beta_{3} + \beta_{2} + 8 \beta_1) q^{49} + ( - \beta_{2} + 5) q^{53} + (\beta_{2} + 2) q^{55} - 13 \beta_1 q^{59} - 2 \beta_{3} q^{61} + (\beta_{3} - 3 \beta_1 - 3) q^{65} + (2 \beta_{3} - 2 \beta_{2} - 8 \beta_1) q^{67} + ( - \beta_{2} - 4) q^{71} + (4 \beta_{2} - 2) q^{73} + ( - 2 \beta_{3} + 2 \beta_{2} - 12 \beta_1) q^{77} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{79} + (2 \beta_{3} - 6 \beta_1 - 6) q^{83} + 2 \beta_1 q^{85} + 3 \beta_{2} q^{89} + ( - 3 \beta_{2} + 17) q^{91} + \beta_1 q^{95} + ( - 16 \beta_1 - 16) q^{97}+O(q^{100})$$ q + b1 * q^5 + (b3 - b1 - 1) * q^7 + (-b3 - 2*b1 - 2) * q^11 + (-b3 + b2 + 3*b1) * q^13 + 2 * q^17 + q^19 + (-b3 + b2 - 3*b1) * q^23 + (-b1 - 1) * q^25 + (-b3 - 2*b1 - 2) * q^29 + (b3 - b2 + 2*b1) * q^31 + (-b2 + 1) * q^35 + (-2*b2 - 2) * q^37 + (2*b3 - 2*b2 - b1) * q^41 + (-4*b1 - 4) * q^43 + (b3 - 3*b1 - 3) * q^47 + (-b3 + b2 + 8*b1) * q^49 + (-b2 + 5) * q^53 + (b2 + 2) * q^55 - 13*b1 * q^59 - 2*b3 * q^61 + (b3 - 3*b1 - 3) * q^65 + (2*b3 - 2*b2 - 8*b1) * q^67 + (-b2 - 4) * q^71 + (4*b2 - 2) * q^73 + (-2*b3 + 2*b2 - 12*b1) * q^77 + (-2*b3 - 2*b1 - 2) * q^79 + (2*b3 - 6*b1 - 6) * q^83 + 2*b1 * q^85 + 3*b2 * q^89 + (-3*b2 + 17) * q^91 + b1 * q^95 + (-16*b1 - 16) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} - q^{7}+O(q^{10})$$ 4 * q - 2 * q^5 - q^7 $$4 q - 2 q^{5} - q^{7} - 5 q^{11} - 5 q^{13} + 8 q^{17} + 4 q^{19} + 7 q^{23} - 2 q^{25} - 5 q^{29} - 5 q^{31} + 2 q^{35} - 12 q^{37} - 8 q^{43} - 5 q^{47} - 15 q^{49} + 18 q^{53} + 10 q^{55} + 26 q^{59} - 2 q^{61} - 5 q^{65} + 14 q^{67} - 18 q^{71} + 26 q^{77} - 6 q^{79} - 10 q^{83} - 4 q^{85} + 6 q^{89} + 62 q^{91} - 2 q^{95} - 32 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 - q^7 - 5 * q^11 - 5 * q^13 + 8 * q^17 + 4 * q^19 + 7 * q^23 - 2 * q^25 - 5 * q^29 - 5 * q^31 + 2 * q^35 - 12 * q^37 - 8 * q^43 - 5 * q^47 - 15 * q^49 + 18 * q^53 + 10 * q^55 + 26 * q^59 - 2 * q^61 - 5 * q^65 + 14 * q^67 - 18 * q^71 + 26 * q^77 - 6 * q^79 - 10 * q^83 - 4 * q^85 + 6 * q^89 + 62 * q^91 - 2 * q^95 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20$$ (v^3 + 4*v^2 - 4*v - 25) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10$$ (3*v^3 + 2*v^2 + 8*v - 25) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3$$ (b3 + b2 - 2*b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3$$ (-b3 + 2*b2 + 14*b1 + 13) / 3 $$\nu^{3}$$ $$=$$ $$( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3$$ (8*b3 - 4*b2 - 4*b1 + 19) / 3

## Character values

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

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 0 0 −0.500000 + 0.866025i 0 −2.13746 3.70219i 0 0 0
1081.2 0 0 0 −0.500000 + 0.866025i 0 1.63746 + 2.83616i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −2.13746 + 3.70219i 0 0 0
2161.2 0 0 0 −0.500000 0.866025i 0 1.63746 2.83616i 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.y 4
3.b odd 2 1 3240.2.q.be 4
9.c even 3 1 3240.2.a.n yes 2
9.c even 3 1 inner 3240.2.q.y 4
9.d odd 6 1 3240.2.a.h 2
9.d odd 6 1 3240.2.q.be 4
36.f odd 6 1 6480.2.a.bm 2
36.h even 6 1 6480.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.h 2 9.d odd 6 1
3240.2.a.n yes 2 9.c even 3 1
3240.2.q.y 4 1.a even 1 1 trivial
3240.2.q.y 4 9.c even 3 1 inner
3240.2.q.be 4 3.b odd 2 1
3240.2.q.be 4 9.d odd 6 1
6480.2.a.bf 2 36.h even 6 1
6480.2.a.bm 2 36.f odd 6 1

## Hecke kernels

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

 $$T_{7}^{4} + T_{7}^{3} + 15T_{7}^{2} - 14T_{7} + 196$$ T7^4 + T7^3 + 15*T7^2 - 14*T7 + 196 $$T_{11}^{4} + 5T_{11}^{3} + 33T_{11}^{2} - 40T_{11} + 64$$ T11^4 + 5*T11^3 + 33*T11^2 - 40*T11 + 64 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + T^{3} + \cdots + 196$$
$11$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$13$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$17$ $$(T - 2)^{4}$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} - 7 T^{3} + \cdots + 4$$
$29$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$31$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$37$ $$(T^{2} + 6 T - 48)^{2}$$
$41$ $$T^{4} + 57T^{2} + 3249$$
$43$ $$(T^{2} + 4 T + 16)^{2}$$
$47$ $$T^{4} + 5 T^{3} + \cdots + 64$$
$53$ $$(T^{2} - 9 T + 6)^{2}$$
$59$ $$(T^{2} - 13 T + 169)^{2}$$
$61$ $$T^{4} + 2 T^{3} + \cdots + 3136$$
$67$ $$T^{4} - 14 T^{3} + \cdots + 64$$
$71$ $$(T^{2} + 9 T + 6)^{2}$$
$73$ $$(T^{2} - 228)^{2}$$
$79$ $$T^{4} + 6 T^{3} + \cdots + 2304$$
$83$ $$T^{4} + 10 T^{3} + \cdots + 1024$$
$89$ $$(T^{2} - 3 T - 126)^{2}$$
$97$ $$(T^{2} + 16 T + 256)^{2}$$