# Properties

 Label 2496.1.l.a Level $2496$ Weight $1$ Character orbit 2496.l Self dual yes Analytic conductor $1.246$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,1,Mod(1793,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("2496.1793");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2496.l (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$1.24566627153$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.7488.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{9}+O(q^{10})$$ q - q^3 + q^9 $$q - q^{3} + q^{9} + q^{13} - q^{25} - q^{27} - q^{39} + 2 q^{43} + q^{49} + 2 q^{61} + q^{75} + 2 q^{79} + q^{81}+O(q^{100})$$ q - q^3 + q^9 + q^13 - q^25 - q^27 - q^39 + 2 * q^43 + q^49 + 2 * q^61 + q^75 + 2 * q^79 + q^81

## Character values

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

 $$n$$ $$703$$ $$769$$ $$833$$ $$1093$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1793.1
 0
0 −1.00000 0 0 0 0 0 1.00000 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 $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.l.a 1
3.b odd 2 1 CM 2496.1.l.a 1
4.b odd 2 1 2496.1.l.b 1
8.b even 2 1 624.1.l.a 1
8.d odd 2 1 39.1.d.a 1
12.b even 2 1 2496.1.l.b 1
13.b even 2 1 RM 2496.1.l.a 1
24.f even 2 1 39.1.d.a 1
24.h odd 2 1 624.1.l.a 1
39.d odd 2 1 CM 2496.1.l.a 1
40.e odd 2 1 975.1.g.a 1
40.k even 4 2 975.1.e.a 2
52.b odd 2 1 2496.1.l.b 1
56.e even 2 1 1911.1.h.a 1
56.k odd 6 2 1911.1.w.b 2
56.m even 6 2 1911.1.w.a 2
72.l even 6 2 1053.1.n.b 2
72.p odd 6 2 1053.1.n.b 2
104.e even 2 1 624.1.l.a 1
104.h odd 2 1 39.1.d.a 1
104.m even 4 2 507.1.c.a 1
104.n odd 6 2 507.1.h.a 2
104.p odd 6 2 507.1.h.a 2
104.u even 12 4 507.1.i.a 2
120.m even 2 1 975.1.g.a 1
120.q odd 4 2 975.1.e.a 2
156.h even 2 1 2496.1.l.b 1
168.e odd 2 1 1911.1.h.a 1
168.v even 6 2 1911.1.w.b 2
168.be odd 6 2 1911.1.w.a 2
312.b odd 2 1 624.1.l.a 1
312.h even 2 1 39.1.d.a 1
312.w odd 4 2 507.1.c.a 1
312.ba even 6 2 507.1.h.a 2
312.bn even 6 2 507.1.h.a 2
312.bq odd 12 4 507.1.i.a 2
520.b odd 2 1 975.1.g.a 1
520.bc even 4 2 975.1.e.a 2
728.b even 2 1 1911.1.h.a 1
728.bs odd 6 2 1911.1.w.b 2
728.cy even 6 2 1911.1.w.a 2
936.bs odd 6 2 1053.1.n.b 2
936.cl even 6 2 1053.1.n.b 2
1560.n even 2 1 975.1.g.a 1
1560.cs odd 4 2 975.1.e.a 2
2184.bf odd 2 1 1911.1.h.a 1
2184.de odd 6 2 1911.1.w.a 2
2184.fb even 6 2 1911.1.w.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 8.d odd 2 1
39.1.d.a 1 24.f even 2 1
39.1.d.a 1 104.h odd 2 1
39.1.d.a 1 312.h even 2 1
507.1.c.a 1 104.m even 4 2
507.1.c.a 1 312.w odd 4 2
507.1.h.a 2 104.n odd 6 2
507.1.h.a 2 104.p odd 6 2
507.1.h.a 2 312.ba even 6 2
507.1.h.a 2 312.bn even 6 2
507.1.i.a 2 104.u even 12 4
507.1.i.a 2 312.bq odd 12 4
624.1.l.a 1 8.b even 2 1
624.1.l.a 1 24.h odd 2 1
624.1.l.a 1 104.e even 2 1
624.1.l.a 1 312.b odd 2 1
975.1.e.a 2 40.k even 4 2
975.1.e.a 2 120.q odd 4 2
975.1.e.a 2 520.bc even 4 2
975.1.e.a 2 1560.cs odd 4 2
975.1.g.a 1 40.e odd 2 1
975.1.g.a 1 120.m even 2 1
975.1.g.a 1 520.b odd 2 1
975.1.g.a 1 1560.n even 2 1
1053.1.n.b 2 72.l even 6 2
1053.1.n.b 2 72.p odd 6 2
1053.1.n.b 2 936.bs odd 6 2
1053.1.n.b 2 936.cl even 6 2
1911.1.h.a 1 56.e even 2 1
1911.1.h.a 1 168.e odd 2 1
1911.1.h.a 1 728.b even 2 1
1911.1.h.a 1 2184.bf odd 2 1
1911.1.w.a 2 56.m even 6 2
1911.1.w.a 2 168.be odd 6 2
1911.1.w.a 2 728.cy even 6 2
1911.1.w.a 2 2184.de odd 6 2
1911.1.w.b 2 56.k odd 6 2
1911.1.w.b 2 168.v even 6 2
1911.1.w.b 2 728.bs odd 6 2
1911.1.w.b 2 2184.fb even 6 2
2496.1.l.a 1 1.a even 1 1 trivial
2496.1.l.a 1 3.b odd 2 1 CM
2496.1.l.a 1 13.b even 2 1 RM
2496.1.l.a 1 39.d odd 2 1 CM
2496.1.l.b 1 4.b odd 2 1
2496.1.l.b 1 12.b even 2 1
2496.1.l.b 1 52.b odd 2 1
2496.1.l.b 1 156.h even 2 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{43} - 2$$ T43 - 2

## Hecke characteristic polynomials

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