# Properties

 Label 3240.2.q.x 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 40) 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 $$q + \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 4 \zeta_{6} + 4) q^{11} + 2 \zeta_{6} q^{13} - 2 q^{17} + 4 q^{19} + 4 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (2 \zeta_{6} - 2) q^{29} + 8 \zeta_{6} q^{31} + 4 q^{35} + 6 q^{37} - 6 \zeta_{6} q^{41} + ( - 8 \zeta_{6} + 8) q^{43} + ( - 4 \zeta_{6} + 4) q^{47} - 9 \zeta_{6} q^{49} - 6 q^{53} + 4 q^{55} - 4 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + (2 \zeta_{6} - 2) q^{65} - 8 \zeta_{6} q^{67} - 6 q^{73} - 16 \zeta_{6} q^{77} + (16 \zeta_{6} - 16) q^{83} - 2 \zeta_{6} q^{85} + 6 q^{89} + 8 q^{91} + 4 \zeta_{6} q^{95} + ( - 14 \zeta_{6} + 14) q^{97} +O(q^{100})$$ q + z * q^5 + (-4*z + 4) * q^7 + (-4*z + 4) * q^11 + 2*z * q^13 - 2 * q^17 + 4 * q^19 + 4*z * q^23 + (z - 1) * q^25 + (2*z - 2) * q^29 + 8*z * q^31 + 4 * q^35 + 6 * q^37 - 6*z * q^41 + (-8*z + 8) * q^43 + (-4*z + 4) * q^47 - 9*z * q^49 - 6 * q^53 + 4 * q^55 - 4*z * q^59 + (-2*z + 2) * q^61 + (2*z - 2) * q^65 - 8*z * q^67 - 6 * q^73 - 16*z * q^77 + (16*z - 16) * q^83 - 2*z * q^85 + 6 * q^89 + 8 * q^91 + 4*z * q^95 + (-14*z + 14) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + q^5 + 4 * q^7 $$2 q + q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} + 4 q^{23} - q^{25} - 2 q^{29} + 8 q^{31} + 8 q^{35} + 12 q^{37} - 6 q^{41} + 8 q^{43} + 4 q^{47} - 9 q^{49} - 12 q^{53} + 8 q^{55} - 4 q^{59} + 2 q^{61} - 2 q^{65} - 8 q^{67} - 12 q^{73} - 16 q^{77} - 16 q^{83} - 2 q^{85} + 12 q^{89} + 16 q^{91} + 4 q^{95} + 14 q^{97}+O(q^{100})$$ 2 * q + q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 + 4 * q^23 - q^25 - 2 * q^29 + 8 * q^31 + 8 * q^35 + 12 * q^37 - 6 * q^41 + 8 * q^43 + 4 * q^47 - 9 * q^49 - 12 * q^53 + 8 * q^55 - 4 * q^59 + 2 * q^61 - 2 * q^65 - 8 * q^67 - 12 * q^73 - 16 * q^77 - 16 * q^83 - 2 * q^85 + 12 * q^89 + 16 * q^91 + 4 * q^95 + 14 * 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.x 2
3.b odd 2 1 3240.2.q.k 2
9.c even 3 1 360.2.a.a 1
9.c even 3 1 inner 3240.2.q.x 2
9.d odd 6 1 40.2.a.a 1
9.d odd 6 1 3240.2.q.k 2
36.f odd 6 1 720.2.a.e 1
36.h even 6 1 80.2.a.a 1
45.h odd 6 1 200.2.a.c 1
45.j even 6 1 1800.2.a.v 1
45.k odd 12 2 1800.2.f.a 2
45.l even 12 2 200.2.c.b 2
63.i even 6 1 1960.2.q.i 2
63.j odd 6 1 1960.2.q.h 2
63.n odd 6 1 1960.2.q.h 2
63.o even 6 1 1960.2.a.g 1
63.s even 6 1 1960.2.q.i 2
72.j odd 6 1 320.2.a.c 1
72.l even 6 1 320.2.a.d 1
72.n even 6 1 2880.2.a.t 1
72.p odd 6 1 2880.2.a.bg 1
99.g even 6 1 4840.2.a.f 1
117.n odd 6 1 6760.2.a.i 1
144.u even 12 2 1280.2.d.a 2
144.w odd 12 2 1280.2.d.j 2
180.n even 6 1 400.2.a.e 1
180.p odd 6 1 3600.2.a.h 1
180.v odd 12 2 400.2.c.d 2
180.x even 12 2 3600.2.f.t 2
252.s odd 6 1 3920.2.a.s 1
315.z even 6 1 9800.2.a.x 1
360.bd even 6 1 1600.2.a.k 1
360.bh odd 6 1 1600.2.a.o 1
360.br even 12 2 1600.2.c.k 2
360.bt odd 12 2 1600.2.c.m 2
396.o odd 6 1 9680.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 9.d odd 6 1
80.2.a.a 1 36.h even 6 1
200.2.a.c 1 45.h odd 6 1
200.2.c.b 2 45.l even 12 2
320.2.a.c 1 72.j odd 6 1
320.2.a.d 1 72.l even 6 1
360.2.a.a 1 9.c even 3 1
400.2.a.e 1 180.n even 6 1
400.2.c.d 2 180.v odd 12 2
720.2.a.e 1 36.f odd 6 1
1280.2.d.a 2 144.u even 12 2
1280.2.d.j 2 144.w odd 12 2
1600.2.a.k 1 360.bd even 6 1
1600.2.a.o 1 360.bh odd 6 1
1600.2.c.k 2 360.br even 12 2
1600.2.c.m 2 360.bt odd 12 2
1800.2.a.v 1 45.j even 6 1
1800.2.f.a 2 45.k odd 12 2
1960.2.a.g 1 63.o even 6 1
1960.2.q.h 2 63.j odd 6 1
1960.2.q.h 2 63.n odd 6 1
1960.2.q.i 2 63.i even 6 1
1960.2.q.i 2 63.s even 6 1
2880.2.a.t 1 72.n even 6 1
2880.2.a.bg 1 72.p odd 6 1
3240.2.q.k 2 3.b odd 2 1
3240.2.q.k 2 9.d odd 6 1
3240.2.q.x 2 1.a even 1 1 trivial
3240.2.q.x 2 9.c even 3 1 inner
3600.2.a.h 1 180.p odd 6 1
3600.2.f.t 2 180.x even 12 2
3920.2.a.s 1 252.s odd 6 1
4840.2.a.f 1 99.g even 6 1
6760.2.a.i 1 117.n odd 6 1
9680.2.a.q 1 396.o odd 6 1
9800.2.a.x 1 315.z 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 $$T_{11}^{2} - 4T_{11} + 16$$ T11^2 - 4*T11 + 16 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} - 4T + 16$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} + 2T + 4$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} - 4T + 16$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 8T + 64$$
$71$ $$T^{2}$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16T + 256$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 14T + 196$$