# Properties

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

# Related objects

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: no (minimal twist has level 384) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{5} + 2 q^{7}+O(q^{10})$$ q - 4 * q^5 + 2 * q^7 $$q - 4 q^{5} + 2 q^{7} + 4 q^{11} - 2 q^{13} + 2 q^{17} - 8 q^{19} - 4 q^{23} + 11 q^{25} - 6 q^{31} - 8 q^{35} + 2 q^{37} - 6 q^{41} - 4 q^{47} - 3 q^{49} - 16 q^{55} - 4 q^{59} - 14 q^{61} + 8 q^{65} - 4 q^{67} - 12 q^{71} - 10 q^{73} + 8 q^{77} + 10 q^{79} - 12 q^{83} - 8 q^{85} + 14 q^{89} - 4 q^{91} + 32 q^{95} + 10 q^{97}+O(q^{100})$$ q - 4 * q^5 + 2 * q^7 + 4 * q^11 - 2 * q^13 + 2 * q^17 - 8 * q^19 - 4 * q^23 + 11 * q^25 - 6 * q^31 - 8 * q^35 + 2 * q^37 - 6 * q^41 - 4 * q^47 - 3 * q^49 - 16 * q^55 - 4 * q^59 - 14 * q^61 + 8 * q^65 - 4 * q^67 - 12 * q^71 - 10 * q^73 + 8 * q^77 + 10 * q^79 - 12 * q^83 - 8 * q^85 + 14 * q^89 - 4 * q^91 + 32 * q^95 + 10 * 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 −4.00000 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.b 1
3.b odd 2 1 384.2.a.h yes 1
4.b odd 2 1 1152.2.a.a 1
8.b even 2 1 1152.2.a.t 1
8.d odd 2 1 1152.2.a.s 1
12.b even 2 1 384.2.a.d yes 1
15.d odd 2 1 9600.2.a.e 1
16.e even 4 2 2304.2.d.f 2
16.f odd 4 2 2304.2.d.o 2
24.f even 2 1 384.2.a.e yes 1
24.h odd 2 1 384.2.a.a 1
48.i odd 4 2 768.2.d.c 2
48.k even 4 2 768.2.d.f 2
60.h even 2 1 9600.2.a.bz 1
120.i odd 2 1 9600.2.a.bk 1
120.m even 2 1 9600.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 24.h odd 2 1
384.2.a.d yes 1 12.b even 2 1
384.2.a.e yes 1 24.f even 2 1
384.2.a.h yes 1 3.b odd 2 1
768.2.d.c 2 48.i odd 4 2
768.2.d.f 2 48.k even 4 2
1152.2.a.a 1 4.b odd 2 1
1152.2.a.b 1 1.a even 1 1 trivial
1152.2.a.s 1 8.d odd 2 1
1152.2.a.t 1 8.b even 2 1
2304.2.d.f 2 16.e even 4 2
2304.2.d.o 2 16.f odd 4 2
9600.2.a.e 1 15.d odd 2 1
9600.2.a.t 1 120.m even 2 1
9600.2.a.bk 1 120.i odd 2 1
9600.2.a.bz 1 60.h even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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