# Properties

 Label 1152.1.b.b Level $1152$ Weight $1$ Character orbit 1152.b Analytic conductor $0.575$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -24, 24 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,1,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1152.703");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1152.b (of order $$2$$, degree $$1$$, 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: $$0.574922894553$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{6})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.191102976.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{5} +O(q^{10})$$ q - z * q^5 $$q - i q^{5} - 3 q^{25} - i q^{29} + q^{49} + i q^{53} + q^{73} + q^{97} +O(q^{100})$$ q - z * q^5 - 3 * q^25 - z * q^29 + q^49 + z * q^53 + q^73 + q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 6 q^{25} + 2 q^{49} + 4 q^{73} + 4 q^{97}+O(q^{100})$$ 2 * q - 6 * q^25 + 2 * q^49 + 4 * q^73 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 1.00000i − 1.00000i
0 0 0 2.00000i 0 0 0 0 0
703.2 0 0 0 2.00000i 0 0 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
24.f even 2 1 RM by $$\Q(\sqrt{6})$$
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.1.b.b 2
3.b odd 2 1 inner 1152.1.b.b 2
4.b odd 2 1 CM 1152.1.b.b 2
8.b even 2 1 inner 1152.1.b.b 2
8.d odd 2 1 inner 1152.1.b.b 2
12.b even 2 1 inner 1152.1.b.b 2
16.e even 4 1 2304.1.g.a 1
16.e even 4 1 2304.1.g.c 1
16.f odd 4 1 2304.1.g.a 1
16.f odd 4 1 2304.1.g.c 1
24.f even 2 1 RM 1152.1.b.b 2
24.h odd 2 1 CM 1152.1.b.b 2
48.i odd 4 1 2304.1.g.a 1
48.i odd 4 1 2304.1.g.c 1
48.k even 4 1 2304.1.g.a 1
48.k even 4 1 2304.1.g.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.1.b.b 2 1.a even 1 1 trivial
1152.1.b.b 2 3.b odd 2 1 inner
1152.1.b.b 2 4.b odd 2 1 CM
1152.1.b.b 2 8.b even 2 1 inner
1152.1.b.b 2 8.d odd 2 1 inner
1152.1.b.b 2 12.b even 2 1 inner
1152.1.b.b 2 24.f even 2 1 RM
1152.1.b.b 2 24.h odd 2 1 CM
2304.1.g.a 1 16.e even 4 1
2304.1.g.a 1 16.f odd 4 1
2304.1.g.a 1 48.i odd 4 1
2304.1.g.a 1 48.k even 4 1
2304.1.g.c 1 16.e even 4 1
2304.1.g.c 1 16.f odd 4 1
2304.1.g.c 1 48.i odd 4 1
2304.1.g.c 1 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1152, [\chi])$$.

## Hecke characteristic polynomials

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