# Properties

 Label 3240.2.q.b Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} +O(q^{10})$$ q - z * q^5 + (2*z - 2) * q^7 $$q - \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + (4 \zeta_{6} - 4) q^{11} + 2 \zeta_{6} q^{13} + 5 q^{17} - 5 q^{19} - \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 2 \zeta_{6} + 2) q^{29} - 7 \zeta_{6} q^{31} + 2 q^{35} - 6 q^{37} + (4 \zeta_{6} - 4) q^{43} + (4 \zeta_{6} - 4) q^{47} + 3 \zeta_{6} q^{49} + 9 q^{53} + 4 q^{55} - 14 \zeta_{6} q^{59} + ( - 11 \zeta_{6} + 11) q^{61} + ( - 2 \zeta_{6} + 2) q^{65} - 14 \zeta_{6} q^{67} - 12 q^{73} - 8 \zeta_{6} q^{77} + ( - 3 \zeta_{6} + 3) q^{79} + ( - \zeta_{6} + 1) q^{83} - 5 \zeta_{6} q^{85} - 4 q^{91} + 5 \zeta_{6} q^{95} + (16 \zeta_{6} - 16) q^{97} +O(q^{100})$$ q - z * q^5 + (2*z - 2) * q^7 + (4*z - 4) * q^11 + 2*z * q^13 + 5 * q^17 - 5 * q^19 - z * q^23 + (z - 1) * q^25 + (-2*z + 2) * q^29 - 7*z * q^31 + 2 * q^35 - 6 * q^37 + (4*z - 4) * q^43 + (4*z - 4) * q^47 + 3*z * q^49 + 9 * q^53 + 4 * q^55 - 14*z * q^59 + (-11*z + 11) * q^61 + (-2*z + 2) * q^65 - 14*z * q^67 - 12 * q^73 - 8*z * q^77 + (-3*z + 3) * q^79 + (-z + 1) * q^83 - 5*z * q^85 - 4 * q^91 + 5*z * q^95 + (16*z - 16) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - q^5 - 2 * q^7 $$2 q - q^{5} - 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{17} - 10 q^{19} - q^{23} - q^{25} + 2 q^{29} - 7 q^{31} + 4 q^{35} - 12 q^{37} - 4 q^{43} - 4 q^{47} + 3 q^{49} + 18 q^{53} + 8 q^{55} - 14 q^{59} + 11 q^{61} + 2 q^{65} - 14 q^{67} - 24 q^{73} - 8 q^{77} + 3 q^{79} + q^{83} - 5 q^{85} - 8 q^{91} + 5 q^{95} - 16 q^{97}+O(q^{100})$$ 2 * q - q^5 - 2 * q^7 - 4 * q^11 + 2 * q^13 + 10 * q^17 - 10 * q^19 - q^23 - q^25 + 2 * q^29 - 7 * q^31 + 4 * q^35 - 12 * q^37 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 18 * q^53 + 8 * q^55 - 14 * q^59 + 11 * q^61 + 2 * q^65 - 14 * q^67 - 24 * q^73 - 8 * q^77 + 3 * q^79 + q^83 - 5 * q^85 - 8 * q^91 + 5 * q^95 - 16 * q^97

## Character values

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

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 −1.00000 1.73205i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −1.00000 + 1.73205i 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.b 2
3.b odd 2 1 3240.2.q.p 2
9.c even 3 1 1080.2.a.l yes 1
9.c even 3 1 inner 3240.2.q.b 2
9.d odd 6 1 1080.2.a.e 1
9.d odd 6 1 3240.2.q.p 2
36.f odd 6 1 2160.2.a.m 1
36.h even 6 1 2160.2.a.e 1
45.h odd 6 1 5400.2.a.j 1
45.j even 6 1 5400.2.a.q 1
45.k odd 12 2 5400.2.f.x 2
45.l even 12 2 5400.2.f.f 2
72.j odd 6 1 8640.2.a.cd 1
72.l even 6 1 8640.2.a.bi 1
72.n even 6 1 8640.2.a.t 1
72.p odd 6 1 8640.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.e 1 9.d odd 6 1
1080.2.a.l yes 1 9.c even 3 1
2160.2.a.e 1 36.h even 6 1
2160.2.a.m 1 36.f odd 6 1
3240.2.q.b 2 1.a even 1 1 trivial
3240.2.q.b 2 9.c even 3 1 inner
3240.2.q.p 2 3.b odd 2 1
3240.2.q.p 2 9.d odd 6 1
5400.2.a.j 1 45.h odd 6 1
5400.2.a.q 1 45.j even 6 1
5400.2.f.f 2 45.l even 12 2
5400.2.f.x 2 45.k odd 12 2
8640.2.a.k 1 72.p odd 6 1
8640.2.a.t 1 72.n even 6 1
8640.2.a.bi 1 72.l even 6 1
8640.2.a.cd 1 72.j odd 6 1

## Hecke kernels

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

 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{17} - 5$$ T17 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T - 5)^{2}$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$T^{2} + 7T + 49$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$T^{2} + 4T + 16$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2} + 14T + 196$$
$61$ $$T^{2} - 11T + 121$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$T^{2}$$
$73$ $$(T + 12)^{2}$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$T^{2} - T + 1$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 16T + 256$$