# Properties

 Label 2496.1.bd.a Level $2496$ Weight $1$ Character orbit 2496.bd Analytic conductor $1.246$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ 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(1919,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))

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

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

chi := DirichletCharacter("2496.1919");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2496.bd (of order $$4$$, 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: $$1.24566627153$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 624) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.105456.1 Artin image: $C_4^2:C_2^2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-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.

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

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1919.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 −1.00000 1.00000i 0 −1.00000 0
2111.1 0 1.00000i 0 0 0 −1.00000 + 1.00000i 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 $$\Q(\sqrt{-3})$$
52.f even 4 1 inner
156.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.1.bd.a 2
3.b odd 2 1 CM 2496.1.bd.a 2
4.b odd 2 1 2496.1.bd.b 2
8.b even 2 1 624.1.bd.a 2
8.d odd 2 1 624.1.bd.b yes 2
12.b even 2 1 2496.1.bd.b 2
13.d odd 4 1 2496.1.bd.b 2
24.f even 2 1 624.1.bd.b yes 2
24.h odd 2 1 624.1.bd.a 2
39.f even 4 1 2496.1.bd.b 2
52.f even 4 1 inner 2496.1.bd.a 2
104.j odd 4 1 624.1.bd.b yes 2
104.m even 4 1 624.1.bd.a 2
156.l odd 4 1 inner 2496.1.bd.a 2
312.w odd 4 1 624.1.bd.a 2
312.y even 4 1 624.1.bd.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.1.bd.a 2 8.b even 2 1
624.1.bd.a 2 24.h odd 2 1
624.1.bd.a 2 104.m even 4 1
624.1.bd.a 2 312.w odd 4 1
624.1.bd.b yes 2 8.d odd 2 1
624.1.bd.b yes 2 24.f even 2 1
624.1.bd.b yes 2 104.j odd 4 1
624.1.bd.b yes 2 312.y even 4 1
2496.1.bd.a 2 1.a even 1 1 trivial
2496.1.bd.a 2 3.b odd 2 1 CM
2496.1.bd.a 2 52.f even 4 1 inner
2496.1.bd.a 2 156.l odd 4 1 inner
2496.1.bd.b 2 4.b odd 2 1
2496.1.bd.b 2 12.b even 2 1
2496.1.bd.b 2 13.d odd 4 1
2496.1.bd.b 2 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 2T_{7} + 2$$ acting on $$S_{1}^{\mathrm{new}}(2496, [\chi])$$.

## Hecke characteristic polynomials

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