# Properties

 Label 3240.1.bh.d Level $3240$ Weight $1$ Character orbit 3240.bh Analytic conductor $1.617$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -120 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,1,Mod(269,3240)]

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, 3, 1, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3240.269");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3240.bh (of order $$6$$, 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: $$1.61697064093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1080.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6}^{2} q^{5} - q^{8} +O(q^{10})$$ q + z * q^2 + z^2 * q^4 - z^2 * q^5 - q^8 $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6}^{2} q^{5} - q^{8} + q^{10} - \zeta_{6} q^{11} - \zeta_{6}^{2} q^{13} - \zeta_{6} q^{16} + q^{17} + \zeta_{6} q^{20} - \zeta_{6}^{2} q^{22} + \zeta_{6}^{2} q^{23} - \zeta_{6} q^{25} + q^{26} - \zeta_{6} q^{29} - \zeta_{6}^{2} q^{31} - \zeta_{6}^{2} q^{32} + \zeta_{6} q^{34} + 2 q^{37} + \zeta_{6}^{2} q^{40} + \zeta_{6} q^{43} + q^{44} - q^{46} - \zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} - \zeta_{6}^{2} q^{50} + \zeta_{6} q^{52} - q^{55} - \zeta_{6}^{2} q^{58} - 2 \zeta_{6}^{2} q^{59} + q^{62} + q^{64} - \zeta_{6} q^{65} + 2 \zeta_{6}^{2} q^{67} + \zeta_{6}^{2} q^{68} + 2 \zeta_{6} q^{74} + \zeta_{6} q^{79} - q^{80} - \zeta_{6}^{2} q^{85} + \zeta_{6}^{2} q^{86} + \zeta_{6} q^{88} - \zeta_{6} q^{92} - \zeta_{6}^{2} q^{94} - q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 - z^2 * q^5 - q^8 + q^10 - z * q^11 - z^2 * q^13 - z * q^16 + q^17 + z * q^20 - z^2 * q^22 + z^2 * q^23 - z * q^25 + q^26 - z * q^29 - z^2 * q^31 - z^2 * q^32 + z * q^34 + 2 * q^37 + z^2 * q^40 + z * q^43 + q^44 - q^46 - z * q^47 + z^2 * q^49 - z^2 * q^50 + z * q^52 - q^55 - z^2 * q^58 - 2*z^2 * q^59 + q^62 + q^64 - z * q^65 + 2*z^2 * q^67 + z^2 * q^68 + 2*z * q^74 + z * q^79 - q^80 - z^2 * q^85 + z^2 * q^86 + z * q^88 - z * q^92 - z^2 * q^94 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 + q^5 - 2 * q^8 $$2 q + q^{2} - q^{4} + q^{5} - 2 q^{8} + 2 q^{10} - q^{11} + q^{13} - q^{16} + 2 q^{17} + q^{20} + q^{22} - q^{23} - q^{25} + 2 q^{26} - q^{29} + q^{31} + q^{32} + q^{34} + 4 q^{37} - q^{40} + q^{43} + 2 q^{44} - 2 q^{46} - q^{47} - q^{49} + q^{50} + q^{52} - 2 q^{55} + q^{58} + 2 q^{59} + 2 q^{62} + 2 q^{64} - q^{65} - 2 q^{67} - q^{68} + 2 q^{74} + q^{79} - 2 q^{80} + q^{85} - q^{86} + q^{88} - q^{92} + q^{94} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + q^5 - 2 * q^8 + 2 * q^10 - q^11 + q^13 - q^16 + 2 * q^17 + q^20 + q^22 - q^23 - q^25 + 2 * q^26 - q^29 + q^31 + q^32 + q^34 + 4 * q^37 - q^40 + q^43 + 2 * q^44 - 2 * q^46 - q^47 - q^49 + q^50 + q^52 - 2 * q^55 + q^58 + 2 * q^59 + 2 * q^62 + 2 * q^64 - q^65 - 2 * q^67 - q^68 + 2 * q^74 + q^79 - 2 * q^80 + q^85 - q^86 + q^88 - q^92 + q^94 - 2 * q^98

## Character values

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

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-\zeta_{6}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 0 −1.00000 0 1.00000
1349.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 0 −1.00000 0 1.00000
 $$n$$: 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
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$
9.c even 3 1 inner
360.bh odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.1.bh.d 2
3.b odd 2 1 3240.1.bh.a 2
5.b even 2 1 3240.1.bh.b 2
8.b even 2 1 3240.1.bh.c 2
9.c even 3 1 1080.1.i.a 1
9.c even 3 1 inner 3240.1.bh.d 2
9.d odd 6 1 1080.1.i.d yes 1
9.d odd 6 1 3240.1.bh.a 2
15.d odd 2 1 3240.1.bh.c 2
24.h odd 2 1 3240.1.bh.b 2
40.f even 2 1 3240.1.bh.a 2
45.h odd 6 1 1080.1.i.b yes 1
45.h odd 6 1 3240.1.bh.c 2
45.j even 6 1 1080.1.i.c yes 1
45.j even 6 1 3240.1.bh.b 2
72.j odd 6 1 1080.1.i.c yes 1
72.j odd 6 1 3240.1.bh.b 2
72.n even 6 1 1080.1.i.b yes 1
72.n even 6 1 3240.1.bh.c 2
120.i odd 2 1 CM 3240.1.bh.d 2
360.bh odd 6 1 1080.1.i.a 1
360.bh odd 6 1 inner 3240.1.bh.d 2
360.bk even 6 1 1080.1.i.d yes 1
360.bk even 6 1 3240.1.bh.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.1.i.a 1 9.c even 3 1
1080.1.i.a 1 360.bh odd 6 1
1080.1.i.b yes 1 45.h odd 6 1
1080.1.i.b yes 1 72.n even 6 1
1080.1.i.c yes 1 45.j even 6 1
1080.1.i.c yes 1 72.j odd 6 1
1080.1.i.d yes 1 9.d odd 6 1
1080.1.i.d yes 1 360.bk even 6 1
3240.1.bh.a 2 3.b odd 2 1
3240.1.bh.a 2 9.d odd 6 1
3240.1.bh.a 2 40.f even 2 1
3240.1.bh.a 2 360.bk even 6 1
3240.1.bh.b 2 5.b even 2 1
3240.1.bh.b 2 24.h odd 2 1
3240.1.bh.b 2 45.j even 6 1
3240.1.bh.b 2 72.j odd 6 1
3240.1.bh.c 2 8.b even 2 1
3240.1.bh.c 2 15.d odd 2 1
3240.1.bh.c 2 45.h odd 6 1
3240.1.bh.c 2 72.n even 6 1
3240.1.bh.d 2 1.a even 1 1 trivial
3240.1.bh.d 2 9.c even 3 1 inner
3240.1.bh.d 2 120.i odd 2 1 CM
3240.1.bh.d 2 360.bh odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3240, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} + T_{11} + 1$$ T11^2 + T11 + 1 $$T_{13}^{2} - T_{13} + 1$$ T13^2 - T13 + 1 $$T_{17} - 1$$ T17 - 1 $$T_{61}$$ T61

## Hecke characteristic polynomials

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