# Properties

 Label 3240.2.q.bc 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: $$\Q(\sqrt{-3}, \sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 19x^{2} + 18x + 324$$ x^4 - x^3 + 19*x^2 + 18*x + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{5} - \beta_1 q^{7}+O(q^{10})$$ q + (-b2 + 1) * q^5 - b1 * q^7 $$q + ( - \beta_{2} + 1) q^{5} - \beta_1 q^{7} + ( - \beta_{2} + \beta_1) q^{11} + (3 \beta_{2} - 3) q^{13} + ( - \beta_{3} - 2) q^{17} + (\beta_{3} + 1) q^{19} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{23} - \beta_{2} q^{25} + (\beta_{2} - \beta_1) q^{29} + ( - \beta_{3} + 7 \beta_{2} - \beta_1 - 6) q^{31} + (\beta_{3} - 1) q^{35} + (\beta_{3} + 3) q^{37} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{41} + ( - 3 \beta_{2} + \beta_1) q^{43} + ( - \beta_{2} - \beta_1) q^{47} + (\beta_{3} + 11 \beta_{2} + \beta_1 - 12) q^{49} + (2 \beta_{3} - 2) q^{53} - \beta_{3} q^{55} + (12 \beta_{2} - 12) q^{59} + (4 \beta_{2} - \beta_1) q^{61} + 3 \beta_{2} q^{65} + (\beta_{3} + 10 \beta_{2} + \beta_1 - 11) q^{67} + ( - 2 \beta_{3} + 6) q^{71} + ( - \beta_{3} + 1) q^{73} + ( - 18 \beta_{2} + 18) q^{77} + 5 \beta_{2} q^{79} - 6 \beta_{2} q^{83} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{85} + 8 q^{89} + ( - 3 \beta_{3} + 3) q^{91} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{95} + (4 \beta_{2} + 3 \beta_1) q^{97}+O(q^{100})$$ q + (-b2 + 1) * q^5 - b1 * q^7 + (-b2 + b1) * q^11 + (3*b2 - 3) * q^13 + (-b3 - 2) * q^17 + (b3 + 1) * q^19 + (b3 + 3*b2 + b1 - 4) * q^23 - b2 * q^25 + (b2 - b1) * q^29 + (-b3 + 7*b2 - b1 - 6) * q^31 + (b3 - 1) * q^35 + (b3 + 3) * q^37 + (-2*b3 - 2*b1 + 2) * q^41 + (-3*b2 + b1) * q^43 + (-b2 - b1) * q^47 + (b3 + 11*b2 + b1 - 12) * q^49 + (2*b3 - 2) * q^53 - b3 * q^55 + (12*b2 - 12) * q^59 + (4*b2 - b1) * q^61 + 3*b2 * q^65 + (b3 + 10*b2 + b1 - 11) * q^67 + (-2*b3 + 6) * q^71 + (-b3 + 1) * q^73 + (-18*b2 + 18) * q^77 + 5*b2 * q^79 - 6*b2 * q^83 + (-b3 + 3*b2 - b1 - 2) * q^85 + 8 * q^89 + (-3*b3 + 3) * q^91 + (b3 - 2*b2 + b1 + 1) * q^95 + (4*b2 + 3*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} - q^{7}+O(q^{10})$$ 4 * q + 2 * q^5 - q^7 $$4 q + 2 q^{5} - q^{7} - q^{11} - 6 q^{13} - 10 q^{17} + 6 q^{19} - 7 q^{23} - 2 q^{25} + q^{29} - 13 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{41} - 5 q^{43} - 3 q^{47} - 23 q^{49} - 4 q^{53} - 2 q^{55} - 24 q^{59} + 7 q^{61} + 6 q^{65} - 21 q^{67} + 20 q^{71} + 2 q^{73} + 36 q^{77} + 10 q^{79} - 12 q^{83} - 5 q^{85} + 32 q^{89} + 6 q^{91} + 3 q^{95} + 11 q^{97}+O(q^{100})$$ 4 * q + 2 * q^5 - q^7 - q^11 - 6 * q^13 - 10 * q^17 + 6 * q^19 - 7 * q^23 - 2 * q^25 + q^29 - 13 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^41 - 5 * q^43 - 3 * q^47 - 23 * q^49 - 4 * q^53 - 2 * q^55 - 24 * q^59 + 7 * q^61 + 6 * q^65 - 21 * q^67 + 20 * q^71 + 2 * q^73 + 36 * q^77 + 10 * q^79 - 12 * q^83 - 5 * q^85 + 32 * q^89 + 6 * q^91 + 3 * q^95 + 11 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 19x^{2} + 18x + 324$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 19\nu^{2} - 19\nu + 324 ) / 342$$ (-v^3 + 19*v^2 - 19*v + 324) / 342 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 37 ) / 19$$ (v^3 + 37) / 19
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 18\beta_{2} + \beta _1 - 19$$ b3 + 18*b2 + b1 - 19 $$\nu^{3}$$ $$=$$ $$19\beta_{3} - 37$$ 19*b3 - 37

## Character values

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

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 + \beta_{2}$$

## 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
 2.38600 + 4.13267i −1.88600 − 3.26665i 2.38600 − 4.13267i −1.88600 + 3.26665i
0 0 0 0.500000 0.866025i 0 −2.38600 4.13267i 0 0 0
1081.2 0 0 0 0.500000 0.866025i 0 1.88600 + 3.26665i 0 0 0
2161.1 0 0 0 0.500000 + 0.866025i 0 −2.38600 + 4.13267i 0 0 0
2161.2 0 0 0 0.500000 + 0.866025i 0 1.88600 3.26665i 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.bc 4
3.b odd 2 1 3240.2.q.z 4
9.c even 3 1 1080.2.a.m 2
9.c even 3 1 inner 3240.2.q.bc 4
9.d odd 6 1 1080.2.a.n yes 2
9.d odd 6 1 3240.2.q.z 4
36.f odd 6 1 2160.2.a.z 2
36.h even 6 1 2160.2.a.bb 2
45.h odd 6 1 5400.2.a.ca 2
45.j even 6 1 5400.2.a.cb 2
45.k odd 12 2 5400.2.f.be 4
45.l even 12 2 5400.2.f.bd 4
72.j odd 6 1 8640.2.a.cq 2
72.l even 6 1 8640.2.a.cn 2
72.n even 6 1 8640.2.a.de 2
72.p odd 6 1 8640.2.a.db 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.m 2 9.c even 3 1
1080.2.a.n yes 2 9.d odd 6 1
2160.2.a.z 2 36.f odd 6 1
2160.2.a.bb 2 36.h even 6 1
3240.2.q.z 4 3.b odd 2 1
3240.2.q.z 4 9.d odd 6 1
3240.2.q.bc 4 1.a even 1 1 trivial
3240.2.q.bc 4 9.c even 3 1 inner
5400.2.a.ca 2 45.h odd 6 1
5400.2.a.cb 2 45.j even 6 1
5400.2.f.bd 4 45.l even 12 2
5400.2.f.be 4 45.k odd 12 2
8640.2.a.cn 2 72.l even 6 1
8640.2.a.cq 2 72.j odd 6 1
8640.2.a.db 2 72.p odd 6 1
8640.2.a.de 2 72.n even 6 1

## Hecke kernels

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

 $$T_{7}^{4} + T_{7}^{3} + 19T_{7}^{2} - 18T_{7} + 324$$ T7^4 + T7^3 + 19*T7^2 - 18*T7 + 324 $$T_{11}^{4} + T_{11}^{3} + 19T_{11}^{2} - 18T_{11} + 324$$ T11^4 + T11^3 + 19*T11^2 - 18*T11 + 324 $$T_{17}^{2} + 5T_{17} - 12$$ T17^2 + 5*T17 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} + T^{3} + \cdots + 324$$
$11$ $$T^{4} + T^{3} + \cdots + 324$$
$13$ $$(T^{2} + 3 T + 9)^{2}$$
$17$ $$(T^{2} + 5 T - 12)^{2}$$
$19$ $$(T^{2} - 3 T - 16)^{2}$$
$23$ $$T^{4} + 7 T^{3} + \cdots + 36$$
$29$ $$T^{4} - T^{3} + \cdots + 324$$
$31$ $$T^{4} + 13 T^{3} + \cdots + 576$$
$37$ $$(T^{2} - 7 T - 6)^{2}$$
$41$ $$T^{4} - 2 T^{3} + \cdots + 5184$$
$43$ $$T^{4} + 5 T^{3} + \cdots + 144$$
$47$ $$T^{4} + 3 T^{3} + \cdots + 256$$
$53$ $$(T^{2} + 2 T - 72)^{2}$$
$59$ $$(T^{2} + 12 T + 144)^{2}$$
$61$ $$T^{4} - 7 T^{3} + \cdots + 36$$
$67$ $$T^{4} + 21 T^{3} + \cdots + 8464$$
$71$ $$(T^{2} - 10 T - 48)^{2}$$
$73$ $$(T^{2} - T - 18)^{2}$$
$79$ $$(T^{2} - 5 T + 25)^{2}$$
$83$ $$(T^{2} + 6 T + 36)^{2}$$
$89$ $$(T - 8)^{4}$$
$97$ $$T^{4} - 11 T^{3} + \cdots + 17956$$