# Properties

 Label 1152.2.a.f Level $1152$ Weight $2$ Character orbit 1152.a Self dual yes Analytic conductor $9.199$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(1,1152)]

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, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1152.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.a (trivial)

## 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: $$9.19876631285$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{5} + 2 q^{7}+O(q^{10})$$ q - 2 * q^5 + 2 * q^7 $$q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{19} - 8 q^{23} - q^{25} - 6 q^{29} - 6 q^{31} - 4 q^{35} + 2 q^{37} + 12 q^{41} - 12 q^{43} - 8 q^{47} - 3 q^{49} - 6 q^{53} + 8 q^{55} - 8 q^{59} + 10 q^{61} + 4 q^{65} + 8 q^{67} + 2 q^{73} - 8 q^{77} - 14 q^{79} - 12 q^{83} - 8 q^{85} - 8 q^{89} - 4 q^{91} - 8 q^{95} - 2 q^{97}+O(q^{100})$$ q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^19 - 8 * q^23 - q^25 - 6 * q^29 - 6 * q^31 - 4 * q^35 + 2 * q^37 + 12 * q^41 - 12 * q^43 - 8 * q^47 - 3 * q^49 - 6 * q^53 + 8 * q^55 - 8 * q^59 + 10 * q^61 + 4 * q^65 + 8 * q^67 + 2 * q^73 - 8 * q^77 - 14 * q^79 - 12 * q^83 - 8 * q^85 - 8 * q^89 - 4 * q^91 - 8 * q^95 - 2 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −2.00000 0 2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.a.f yes 1
3.b odd 2 1 1152.2.a.p yes 1
4.b odd 2 1 1152.2.a.d 1
8.b even 2 1 1152.2.a.q yes 1
8.d odd 2 1 1152.2.a.o yes 1
12.b even 2 1 1152.2.a.n yes 1
16.e even 4 2 2304.2.d.g 2
16.f odd 4 2 2304.2.d.p 2
24.f even 2 1 1152.2.a.e yes 1
24.h odd 2 1 1152.2.a.g yes 1
48.i odd 4 2 2304.2.d.e 2
48.k even 4 2 2304.2.d.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.a.d 1 4.b odd 2 1
1152.2.a.e yes 1 24.f even 2 1
1152.2.a.f yes 1 1.a even 1 1 trivial
1152.2.a.g yes 1 24.h odd 2 1
1152.2.a.n yes 1 12.b even 2 1
1152.2.a.o yes 1 8.d odd 2 1
1152.2.a.p yes 1 3.b odd 2 1
1152.2.a.q yes 1 8.b even 2 1
2304.2.d.e 2 48.i odd 4 2
2304.2.d.g 2 16.e even 4 2
2304.2.d.n 2 48.k even 4 2
2304.2.d.p 2 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1152))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T - 2$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T - 4$$
$23$ $$T + 8$$
$29$ $$T + 6$$
$31$ $$T + 6$$
$37$ $$T - 2$$
$41$ $$T - 12$$
$43$ $$T + 12$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T + 8$$
$61$ $$T - 10$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 14$$
$83$ $$T + 12$$
$89$ $$T + 8$$
$97$ $$T + 2$$
show more
show less