# Properties

 Label 3240.2.q.a Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $0$ 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: $$0$$ 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 120) 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} +O(q^{10})$$ q - z * q^5 + (4*z - 4) * q^7 $$q - \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{7} + 6 \zeta_{6} q^{13} + 2 q^{17} + 4 q^{19} - 8 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (6 \zeta_{6} - 6) q^{29} + 4 q^{35} - 6 q^{37} + 10 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + ( - 8 \zeta_{6} + 8) q^{47} - 9 \zeta_{6} q^{49} - 10 q^{53} + (6 \zeta_{6} - 6) q^{61} + ( - 6 \zeta_{6} + 6) q^{65} + 4 \zeta_{6} q^{67} - 14 q^{73} + (16 \zeta_{6} - 16) q^{79} + ( - 12 \zeta_{6} + 12) q^{83} - 2 \zeta_{6} q^{85} - 2 q^{89} - 24 q^{91} - 4 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} +O(q^{100})$$ q - z * q^5 + (4*z - 4) * q^7 + 6*z * q^13 + 2 * q^17 + 4 * q^19 - 8*z * q^23 + (z - 1) * q^25 + (6*z - 6) * q^29 + 4 * q^35 - 6 * q^37 + 10*z * q^41 + (-4*z + 4) * q^43 + (-8*z + 8) * q^47 - 9*z * q^49 - 10 * q^53 + (6*z - 6) * q^61 + (-6*z + 6) * q^65 + 4*z * q^67 - 14 * q^73 + (16*z - 16) * q^79 + (-12*z + 12) * q^83 - 2*z * q^85 - 2 * q^89 - 24 * q^91 - 4*z * q^95 + (2*z - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q - q^5 - 4 * q^7 $$2 q - q^{5} - 4 q^{7} + 6 q^{13} + 4 q^{17} + 8 q^{19} - 8 q^{23} - q^{25} - 6 q^{29} + 8 q^{35} - 12 q^{37} + 10 q^{41} + 4 q^{43} + 8 q^{47} - 9 q^{49} - 20 q^{53} - 6 q^{61} + 6 q^{65} + 4 q^{67} - 28 q^{73} - 16 q^{79} + 12 q^{83} - 2 q^{85} - 4 q^{89} - 48 q^{91} - 4 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - q^5 - 4 * q^7 + 6 * q^13 + 4 * q^17 + 8 * q^19 - 8 * q^23 - q^25 - 6 * q^29 + 8 * q^35 - 12 * q^37 + 10 * q^41 + 4 * q^43 + 8 * q^47 - 9 * q^49 - 20 * q^53 - 6 * q^61 + 6 * q^65 + 4 * q^67 - 28 * q^73 - 16 * q^79 + 12 * q^83 - 2 * q^85 - 4 * q^89 - 48 * q^91 - 4 * q^95 - 2 * 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 −2.00000 3.46410i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −2.00000 + 3.46410i 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.a 2
3.b odd 2 1 3240.2.q.m 2
9.c even 3 1 360.2.a.e 1
9.c even 3 1 inner 3240.2.q.a 2
9.d odd 6 1 120.2.a.a 1
9.d odd 6 1 3240.2.q.m 2
36.f odd 6 1 720.2.a.f 1
36.h even 6 1 240.2.a.a 1
45.h odd 6 1 600.2.a.a 1
45.j even 6 1 1800.2.a.c 1
45.k odd 12 2 1800.2.f.g 2
45.l even 12 2 600.2.f.c 2
63.o even 6 1 5880.2.a.p 1
72.j odd 6 1 960.2.a.g 1
72.l even 6 1 960.2.a.n 1
72.n even 6 1 2880.2.a.r 1
72.p odd 6 1 2880.2.a.b 1
144.u even 12 2 3840.2.k.z 2
144.w odd 12 2 3840.2.k.a 2
180.n even 6 1 1200.2.a.r 1
180.p odd 6 1 3600.2.a.bo 1
180.v odd 12 2 1200.2.f.f 2
180.x even 12 2 3600.2.f.l 2
360.bd even 6 1 4800.2.a.bh 1
360.bh odd 6 1 4800.2.a.bl 1
360.br even 12 2 4800.2.f.u 2
360.bt odd 12 2 4800.2.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 9.d odd 6 1
240.2.a.a 1 36.h even 6 1
360.2.a.e 1 9.c even 3 1
600.2.a.a 1 45.h odd 6 1
600.2.f.c 2 45.l even 12 2
720.2.a.f 1 36.f odd 6 1
960.2.a.g 1 72.j odd 6 1
960.2.a.n 1 72.l even 6 1
1200.2.a.r 1 180.n even 6 1
1200.2.f.f 2 180.v odd 12 2
1800.2.a.c 1 45.j even 6 1
1800.2.f.g 2 45.k odd 12 2
2880.2.a.b 1 72.p odd 6 1
2880.2.a.r 1 72.n even 6 1
3240.2.q.a 2 1.a even 1 1 trivial
3240.2.q.a 2 9.c even 3 1 inner
3240.2.q.m 2 3.b odd 2 1
3240.2.q.m 2 9.d odd 6 1
3600.2.a.bo 1 180.p odd 6 1
3600.2.f.l 2 180.x even 12 2
3840.2.k.a 2 144.w odd 12 2
3840.2.k.z 2 144.u even 12 2
4800.2.a.bh 1 360.bd even 6 1
4800.2.a.bl 1 360.bh odd 6 1
4800.2.f.n 2 360.bt odd 12 2
4800.2.f.u 2 360.br even 12 2
5880.2.a.p 1 63.o even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + 4T + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 10T + 100$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 6T + 36$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2}$$
$73$ $$(T + 14)^{2}$$
$79$ $$T^{2} + 16T + 256$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} + 2T + 4$$