# Properties

 Label 3240.2.q.bd 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_1) q^{7}+O(q^{10})$$ q + (-b1 + 1) * q^5 + (b3 - b1) * q^7 $$q + ( - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_1) q^{7} - \beta_{3} q^{11} + (\beta_{3} - \beta_{2} - \beta_1) q^{13} - 2 \beta_{2} q^{17} + q^{19} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{23} - \beta_1 q^{25} - \beta_{3} q^{29} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{31}+ \cdots - 2 \beta_{3} q^{97}+O(q^{100})$$ q + (-b1 + 1) * q^5 + (b3 - b1) * q^7 - b3 * q^11 + (b3 - b2 - b1) * q^13 - 2*b2 * q^17 + q^19 + (b3 - b2 - 3*b1 + 2) * q^23 - b1 * q^25 - b3 * q^29 + (-3*b3 + 3*b2 + 4*b1 - 1) * q^31 + b2 * q^35 + 6 * q^37 + (-2*b3 + 2*b2 - 5*b1 + 7) * q^41 - 2*b3 * q^43 + (b3 - 7*b1) * q^47 + (-b3 + b2 + 2*b1 - 1) * q^49 + 3*b2 * q^53 + (-b2 - 1) * q^55 + (-5*b1 + 5) * q^59 + (2*b3 - 8*b1) * q^61 + (b3 - b1) * q^65 + (2*b3 - 2*b2 + 4*b1 - 6) * q^67 + (-b2 + 1) * q^71 + (-2*b2 + 8) * q^73 + (-8*b1 + 8) * q^77 + (-2*b3 - 2*b1) * q^79 - 10*b1 * q^83 + (2*b3 - 2*b2 - 2*b1) * q^85 + (-b2 + 1) * q^89 + (b2 - 8) * q^91 + (-b1 + 1) * q^95 - 2*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} - q^{7}+O(q^{10})$$ 4 * q + 2 * q^5 - q^7 $$4 q + 2 q^{5} - q^{7} - q^{11} + q^{13} + 4 q^{17} + 4 q^{19} + 5 q^{23} - 2 q^{25} - q^{29} - 5 q^{31} - 2 q^{35} + 24 q^{37} + 12 q^{41} - 2 q^{43} - 13 q^{47} - 3 q^{49} - 6 q^{53} - 2 q^{55} + 10 q^{59} - 14 q^{61} - q^{65} - 10 q^{67} + 6 q^{71} + 36 q^{73} + 16 q^{77} - 6 q^{79} - 20 q^{83} + 2 q^{85} + 6 q^{89} - 34 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 - q^7 - q^11 + q^13 + 4 * q^17 + 4 * q^19 + 5 * q^23 - 2 * q^25 - q^29 - 5 * q^31 - 2 * q^35 + 24 * q^37 + 12 * q^41 - 2 * q^43 - 13 * q^47 - 3 * q^49 - 6 * q^53 - 2 * q^55 + 10 * q^59 - 14 * q^61 - q^65 - 10 * q^67 + 6 * q^71 + 36 * q^73 + 16 * q^77 - 6 * q^79 - 20 * q^83 + 2 * q^85 + 6 * q^89 - 34 * q^91 + 2 * q^95 - 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

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

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

## 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
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0 0 0 0.500000 0.866025i 0 −1.68614 2.92048i 0 0 0
1081.2 0 0 0 0.500000 0.866025i 0 1.18614 + 2.05446i 0 0 0
2161.1 0 0 0 0.500000 + 0.866025i 0 −1.68614 + 2.92048i 0 0 0
2161.2 0 0 0 0.500000 + 0.866025i 0 1.18614 2.05446i 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.bd 4
3.b odd 2 1 3240.2.q.ba 4
9.c even 3 1 3240.2.a.i 2
9.c even 3 1 inner 3240.2.q.bd 4
9.d odd 6 1 3240.2.a.m yes 2
9.d odd 6 1 3240.2.q.ba 4
36.f odd 6 1 6480.2.a.bd 2
36.h even 6 1 6480.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.i 2 9.c even 3 1
3240.2.a.m yes 2 9.d odd 6 1
3240.2.q.ba 4 3.b odd 2 1
3240.2.q.ba 4 9.d odd 6 1
3240.2.q.bd 4 1.a even 1 1 trivial
3240.2.q.bd 4 9.c even 3 1 inner
6480.2.a.bd 2 36.f odd 6 1
6480.2.a.bo 2 36.h even 6 1

## Hecke kernels

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

 $$T_{7}^{4} + T_{7}^{3} + 9T_{7}^{2} - 8T_{7} + 64$$ T7^4 + T7^3 + 9*T7^2 - 8*T7 + 64 $$T_{11}^{4} + T_{11}^{3} + 9T_{11}^{2} - 8T_{11} + 64$$ T11^4 + T11^3 + 9*T11^2 - 8*T11 + 64 $$T_{17}^{2} - 2T_{17} - 32$$ T17^2 - 2*T17 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} + T^{3} + \cdots + 64$$
$11$ $$T^{4} + T^{3} + \cdots + 64$$
$13$ $$T^{4} - T^{3} + \cdots + 64$$
$17$ $$(T^{2} - 2 T - 32)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} - 5 T^{3} + \cdots + 4$$
$29$ $$T^{4} + T^{3} + \cdots + 64$$
$31$ $$T^{4} + 5 T^{3} + \cdots + 4624$$
$37$ $$(T - 6)^{4}$$
$41$ $$T^{4} - 12 T^{3} + \cdots + 9$$
$43$ $$T^{4} + 2 T^{3} + \cdots + 1024$$
$47$ $$T^{4} + 13 T^{3} + \cdots + 1156$$
$53$ $$(T^{2} + 3 T - 72)^{2}$$
$59$ $$(T^{2} - 5 T + 25)^{2}$$
$61$ $$T^{4} + 14 T^{3} + \cdots + 256$$
$67$ $$T^{4} + 10 T^{3} + \cdots + 64$$
$71$ $$(T^{2} - 3 T - 6)^{2}$$
$73$ $$(T^{2} - 18 T + 48)^{2}$$
$79$ $$T^{4} + 6 T^{3} + \cdots + 576$$
$83$ $$(T^{2} + 10 T + 100)^{2}$$
$89$ $$(T^{2} - 3 T - 6)^{2}$$
$97$ $$T^{4} + 2 T^{3} + \cdots + 1024$$