# Properties

 Label 1152.1.h.a Level $1152$ Weight $1$ Character orbit 1152.h Analytic conductor $0.575$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,1,Mod(449,1152)]

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([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("1152.449");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1152.h (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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.6912.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{3} - \zeta_{8}) q^{5}+O(q^{10})$$ q + (z^3 - z) * q^5 $$q + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + \zeta_{8}^{2} q^{13} + (\zeta_{8}^{3} + \zeta_{8}) q^{17} + q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{29} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} - q^{49} + (\zeta_{8}^{3} - \zeta_{8}) q^{53} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{65} - \zeta_{8}^{2} q^{85} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89} +O(q^{100})$$ q + (z^3 - z) * q^5 + z^2 * q^13 + (z^3 + z) * q^17 + q^25 + (-z^3 + z) * q^29 + (z^3 + z) * q^41 - q^49 + (z^3 - z) * q^53 + (-2*z^3 - 2*z) * q^65 - z^2 * q^85 + (-z^3 - z) * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{25} - 4 q^{49}+O(q^{100})$$ 4 * q + 4 * q^25 - 4 * q^49

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −1.41421 0 0 0 0 0
449.2 0 0 0 −1.41421 0 0 0 0 0
449.3 0 0 0 1.41421 0 0 0 0 0
449.4 0 0 0 1.41421 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1152, [\chi])$$.

## Hecke characteristic polynomials

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