# Properties

 Label 1152.1.n.b Level $1152$ Weight $1$ Character orbit 1152.n Analytic conductor $0.575$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,1,Mod(65,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("1152.65");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1152.n (of order $$6$$, degree $$2$$, 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: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.20155392.5

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{9}+O(q^{10})$$ q + z * q^3 + z^2 * q^9 $$q + \zeta_{6} q^{3} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{11} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{17} + (\zeta_{6}^{2} + \zeta_{6}) q^{19} + \zeta_{6} q^{25} - q^{27} + \zeta_{6}^{2} q^{33} + ( - \zeta_{6} - 1) q^{41} + ( - \zeta_{6}^{2} + 1) q^{43} - \zeta_{6}^{2} q^{49} + ( - \zeta_{6}^{2} + 1) q^{51} + (\zeta_{6}^{2} - 1) q^{57} + \zeta_{6}^{2} q^{59} + ( - \zeta_{6} - 1) q^{67} + q^{73} + \zeta_{6}^{2} q^{75} - \zeta_{6} q^{81} - \zeta_{6} q^{83} + \zeta_{6} q^{97} - q^{99} +O(q^{100})$$ q + z * q^3 + z^2 * q^9 + z * q^11 + (-z^2 - z) * q^17 + (z^2 + z) * q^19 + z * q^25 - q^27 + z^2 * q^33 + (-z - 1) * q^41 + (-z^2 + 1) * q^43 - z^2 * q^49 + (-z^2 + 1) * q^51 + (z^2 - 1) * q^57 + z^2 * q^59 + (-z - 1) * q^67 + q^73 + z^2 * q^75 - z * q^81 - z * q^83 + z * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{9}+O(q^{10})$$ 2 * q + q^3 - q^9 $$2 q + q^{3} - q^{9} + q^{11} + q^{25} - 2 q^{27} - q^{33} - 3 q^{41} + 3 q^{43} + q^{49} + 3 q^{51} - 3 q^{57} - q^{59} - 3 q^{67} + 2 q^{73} - q^{75} - q^{81} - 2 q^{83} + q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^9 + q^11 + q^25 - 2 * q^27 - q^33 - 3 * q^41 + 3 * q^43 + q^49 + 3 * q^51 - 3 * q^57 - q^59 - 3 * q^67 + 2 * q^73 - q^75 - q^81 - 2 * q^83 + q^97 - 2 * q^99

## Character values

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

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
833.1 0 0.500000 + 0.866025i 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
36.h even 6 1 inner
72.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.1.n.b yes 2
3.b odd 2 1 3456.1.n.a 2
4.b odd 2 1 1152.1.n.a 2
8.b even 2 1 1152.1.n.a 2
8.d odd 2 1 CM 1152.1.n.b yes 2
9.c even 3 1 3456.1.n.b 2
9.d odd 6 1 1152.1.n.a 2
12.b even 2 1 3456.1.n.b 2
16.e even 4 2 2304.1.q.c 4
16.f odd 4 2 2304.1.q.c 4
24.f even 2 1 3456.1.n.a 2
24.h odd 2 1 3456.1.n.b 2
36.f odd 6 1 3456.1.n.a 2
36.h even 6 1 inner 1152.1.n.b yes 2
72.j odd 6 1 inner 1152.1.n.b yes 2
72.l even 6 1 1152.1.n.a 2
72.n even 6 1 3456.1.n.a 2
72.p odd 6 1 3456.1.n.b 2
144.u even 12 2 2304.1.q.c 4
144.w odd 12 2 2304.1.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.1.n.a 2 4.b odd 2 1
1152.1.n.a 2 8.b even 2 1
1152.1.n.a 2 9.d odd 6 1
1152.1.n.a 2 72.l even 6 1
1152.1.n.b yes 2 1.a even 1 1 trivial
1152.1.n.b yes 2 8.d odd 2 1 CM
1152.1.n.b yes 2 36.h even 6 1 inner
1152.1.n.b yes 2 72.j odd 6 1 inner
2304.1.q.c 4 16.e even 4 2
2304.1.q.c 4 16.f odd 4 2
2304.1.q.c 4 144.u even 12 2
2304.1.q.c 4 144.w odd 12 2
3456.1.n.a 2 3.b odd 2 1
3456.1.n.a 2 24.f even 2 1
3456.1.n.a 2 36.f odd 6 1
3456.1.n.a 2 72.n even 6 1
3456.1.n.b 2 9.c even 3 1
3456.1.n.b 2 12.b even 2 1
3456.1.n.b 2 24.h odd 2 1
3456.1.n.b 2 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - T_{11} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1152, [\chi])$$.

## Hecke characteristic polynomials

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