# Properties

 Label 2496.1.bx.b Level $2496$ Weight $1$ Character orbit 2496.bx Analytic conductor $1.246$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,1,Mod(737,2496)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 4]))

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

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

chi := DirichletCharacter("2496.737");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2496.bx (of order $$6$$, degree $$2$$, 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: $$1.24566627153$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.1579309056.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + z * q^3 + (-z^3 - z) * q^7 + z^2 * q^9 $$q + \zeta_{12} q^{3} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{2} q^{13} - \zeta_{12}^{5} q^{19} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{21} - q^{25} + \zeta_{12}^{3} q^{27} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{31} + \zeta_{12}^{3} q^{39} - \zeta_{12}^{5} q^{43} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{49} + 2 q^{57} + ( - \zeta_{12}^{4} + 1) q^{61} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{63} - \zeta_{12} q^{67} - q^{73} - \zeta_{12} q^{75} + (\zeta_{12}^{5} - \zeta_{12}) q^{79} + \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{91} + (\zeta_{12}^{2} + 1) q^{93} - \zeta_{12}^{2} q^{97} +O(q^{100})$$ q + z * q^3 + (-z^3 - z) * q^7 + z^2 * q^9 + z^2 * q^13 - z^5 * q^19 + (-z^4 - z^2) * q^21 - q^25 + z^3 * q^27 + (-z^5 + z) * q^31 + z^3 * q^39 - z^5 * q^43 + (z^4 + z^2 - 1) * q^49 + 2 * q^57 + (-z^4 + 1) * q^61 + (-z^5 - z^3) * q^63 - z * q^67 - q^73 - z * q^75 + (z^5 - z) * q^79 + z^4 * q^81 + (-z^5 - z^3) * q^91 + (z^2 + 1) * q^93 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} + 2 q^{13} - 4 q^{25} - 4 q^{49} + 8 q^{57} + 6 q^{61} - 4 q^{73} - 2 q^{81} + 6 q^{93} - 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^9 + 2 * q^13 - 4 * q^25 - 4 * q^49 + 8 * q^57 + 6 * q^61 - 4 * q^73 - 2 * q^81 + 6 * q^93 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$769$$ $$833$$ $$1093$$ $$\chi(n)$$ $$1$$ $$-\zeta_{12}^{2}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
737.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 0.500000i 0 0 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0
737.2 0 0.866025 + 0.500000i 0 0 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0
1121.1 0 −0.866025 + 0.500000i 0 0 0 0.866025 1.50000i 0 0.500000 0.866025i 0
1121.2 0 0.866025 0.500000i 0 0 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner
104.n odd 6 1 inner
104.r even 6 1 inner
312.bh odd 6 1 inner
312.bn even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.bx.b yes 4
3.b odd 2 1 CM 2496.1.bx.b yes 4
4.b odd 2 1 inner 2496.1.bx.b yes 4
8.b even 2 1 2496.1.bx.a 4
8.d odd 2 1 2496.1.bx.a 4
12.b even 2 1 inner 2496.1.bx.b yes 4
13.c even 3 1 2496.1.bx.a 4
24.f even 2 1 2496.1.bx.a 4
24.h odd 2 1 2496.1.bx.a 4
39.i odd 6 1 2496.1.bx.a 4
52.j odd 6 1 2496.1.bx.a 4
104.n odd 6 1 inner 2496.1.bx.b yes 4
104.r even 6 1 inner 2496.1.bx.b yes 4
156.p even 6 1 2496.1.bx.a 4
312.bh odd 6 1 inner 2496.1.bx.b yes 4
312.bn even 6 1 inner 2496.1.bx.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2496.1.bx.a 4 8.b even 2 1
2496.1.bx.a 4 8.d odd 2 1
2496.1.bx.a 4 13.c even 3 1
2496.1.bx.a 4 24.f even 2 1
2496.1.bx.a 4 24.h odd 2 1
2496.1.bx.a 4 39.i odd 6 1
2496.1.bx.a 4 52.j odd 6 1
2496.1.bx.a 4 156.p even 6 1
2496.1.bx.b yes 4 1.a even 1 1 trivial
2496.1.bx.b yes 4 3.b odd 2 1 CM
2496.1.bx.b yes 4 4.b odd 2 1 inner
2496.1.bx.b yes 4 12.b even 2 1 inner
2496.1.bx.b yes 4 104.n odd 6 1 inner
2496.1.bx.b yes 4 104.r even 6 1 inner
2496.1.bx.b yes 4 312.bh odd 6 1 inner
2496.1.bx.b yes 4 312.bn even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{61}^{2} - 3T_{61} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2496, [\chi])$$.

## Hecke characteristic polynomials

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