# Properties

 Label 2496.1.de.c Level $2496$ Weight $1$ Character orbit 2496.de Analytic conductor $1.246$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,1,Mod(479,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 6, 6, 7]))

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

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

chi := DirichletCharacter("2496.479");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2496.de (of order $$12$$, degree $$4$$, 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: $$4$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{2} q^{3} + (\zeta_{12}^{5} - 1) q^{7} + \zeta_{12}^{4} q^{9}+O(q^{10})$$ q + z^2 * q^3 + (z^5 - 1) * q^7 + z^4 * q^9 $$q + \zeta_{12}^{2} q^{3} + (\zeta_{12}^{5} - 1) q^{7} + \zeta_{12}^{4} q^{9} - \zeta_{12} q^{13} + (\zeta_{12}^{4} - \zeta_{12}) q^{19} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{21} - \zeta_{12}^{3} q^{25} - q^{27} + ( - \zeta_{12}^{2} + \zeta_{12}) q^{31} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{37} - \zeta_{12}^{3} q^{39} - \zeta_{12} q^{43} + ( - \zeta_{12}^{5} - \zeta_{12}^{4} + 1) q^{49} + ( - \zeta_{12}^{3} - 1) q^{57} - \zeta_{12} q^{61} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{63} + (\zeta_{12}^{4} + \zeta_{12}^{3}) q^{67} + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{73} - \zeta_{12}^{5} q^{75} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{79} - \zeta_{12}^{2} q^{81} + (\zeta_{12} + 1) q^{91} + ( - \zeta_{12}^{4} + \zeta_{12}^{3}) q^{93} + (\zeta_{12}^{3} - \zeta_{12}^{2}) q^{97} +O(q^{100})$$ q + z^2 * q^3 + (z^5 - 1) * q^7 + z^4 * q^9 - z * q^13 + (z^4 - z) * q^19 + (-z^2 - z) * q^21 - z^3 * q^25 - q^27 + (-z^2 + z) * q^31 + (z^5 + z^2) * q^37 - z^3 * q^39 - z * q^43 + (-z^5 - z^4 + 1) * q^49 + (-z^3 - 1) * q^57 - z * q^61 + (-z^4 - z^3) * q^63 + (z^4 + z^3) * q^67 + (z^5 + z^4) * q^73 - z^5 * q^75 + (z^4 + z^2) * q^79 - z^2 * q^81 + (z + 1) * q^91 + (-z^4 + z^3) * q^93 + (z^3 - z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 4 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 4 * q^7 - 2 * q^9 $$4 q + 2 q^{3} - 4 q^{7} - 2 q^{9} - 2 q^{19} - 2 q^{21} - 4 q^{27} - 2 q^{31} + 2 q^{37} + 6 q^{49} - 4 q^{57} + 2 q^{63} - 2 q^{67} - 2 q^{73} - 2 q^{81} + 4 q^{91} + 2 q^{93} - 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^3 - 4 * q^7 - 2 * q^9 - 2 * q^19 - 2 * q^21 - 4 * q^27 - 2 * q^31 + 2 * q^37 + 6 * q^49 - 4 * q^57 + 2 * q^63 - 2 * q^67 - 2 * q^73 - 2 * q^81 + 4 * q^91 + 2 * q^93 - 2 * q^97

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
479.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0.500000 + 0.866025i 0 0 0 −1.86603 + 0.500000i 0 −0.500000 + 0.866025i 0
1055.1 0 0.500000 + 0.866025i 0 0 0 −0.133975 0.500000i 0 −0.500000 + 0.866025i 0
1631.1 0 0.500000 0.866025i 0 0 0 −1.86603 0.500000i 0 −0.500000 0.866025i 0
2399.1 0 0.500000 0.866025i 0 0 0 −0.133975 + 0.500000i 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 $$\Q(\sqrt{-3})$$
104.u even 12 1 inner
312.bq odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.de.c yes 4
3.b odd 2 1 CM 2496.1.de.c yes 4
4.b odd 2 1 2496.1.de.b yes 4
8.b even 2 1 2496.1.de.a 4
8.d odd 2 1 2496.1.de.d yes 4
12.b even 2 1 2496.1.de.b yes 4
13.f odd 12 1 2496.1.de.d yes 4
24.f even 2 1 2496.1.de.d yes 4
24.h odd 2 1 2496.1.de.a 4
39.k even 12 1 2496.1.de.d yes 4
52.l even 12 1 2496.1.de.a 4
104.u even 12 1 inner 2496.1.de.c yes 4
104.x odd 12 1 2496.1.de.b yes 4
156.v odd 12 1 2496.1.de.a 4
312.bo even 12 1 2496.1.de.b yes 4
312.bq odd 12 1 inner 2496.1.de.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2496.1.de.a 4 8.b even 2 1
2496.1.de.a 4 24.h odd 2 1
2496.1.de.a 4 52.l even 12 1
2496.1.de.a 4 156.v odd 12 1
2496.1.de.b yes 4 4.b odd 2 1
2496.1.de.b yes 4 12.b even 2 1
2496.1.de.b yes 4 104.x odd 12 1
2496.1.de.b yes 4 312.bo even 12 1
2496.1.de.c yes 4 1.a even 1 1 trivial
2496.1.de.c yes 4 3.b odd 2 1 CM
2496.1.de.c yes 4 104.u even 12 1 inner
2496.1.de.c yes 4 312.bq odd 12 1 inner
2496.1.de.d yes 4 8.d odd 2 1
2496.1.de.d yes 4 13.f odd 12 1
2496.1.de.d yes 4 24.f even 2 1
2496.1.de.d yes 4 39.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2496, [\chi])$$:

 $$T_{7}^{4} + 4T_{7}^{3} + 5T_{7}^{2} + 2T_{7} + 1$$ T7^4 + 4*T7^3 + 5*T7^2 + 2*T7 + 1 $$T_{19}^{4} + 2T_{19}^{3} + 2T_{19}^{2} + 4T_{19} + 4$$ T19^4 + 2*T19^3 + 2*T19^2 + 4*T19 + 4

## Hecke characteristic polynomials

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