# Properties

 Label 1176.2.a.h Level $1176$ Weight $2$ Character orbit 1176.a Self dual yes Analytic conductor $9.390$ Analytic rank $0$ 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] = [1176,2,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1176.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: $$9.39040727770$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9}+O(q^{10})$$ q + q^3 + q^9 $$q + q^{3} + q^{9} + 4 q^{13} + 4 q^{17} - 4 q^{19} + 4 q^{23} - 5 q^{25} + q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{37} + 4 q^{39} + 12 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{51} + 6 q^{53} - 4 q^{57} + 12 q^{59} + 4 q^{61} - 4 q^{67} + 4 q^{69} - 12 q^{71} + 8 q^{73} - 5 q^{75} - 16 q^{79} + q^{81} - 4 q^{83} + 2 q^{87} + 4 q^{89} + 8 q^{93} + 16 q^{97}+O(q^{100})$$ q + q^3 + q^9 + 4 * q^13 + 4 * q^17 - 4 * q^19 + 4 * q^23 - 5 * q^25 + q^27 + 2 * q^29 + 8 * q^31 - 6 * q^37 + 4 * q^39 + 12 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^51 + 6 * q^53 - 4 * q^57 + 12 * q^59 + 4 * q^61 - 4 * q^67 + 4 * q^69 - 12 * q^71 + 8 * q^73 - 5 * q^75 - 16 * q^79 + q^81 - 4 * q^83 + 2 * q^87 + 4 * q^89 + 8 * q^93 + 16 * 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 1.00000 0 0 0 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 1176.2.a.h yes 1
3.b odd 2 1 3528.2.a.n 1
4.b odd 2 1 2352.2.a.h 1
7.b odd 2 1 1176.2.a.b 1
7.c even 3 2 1176.2.q.c 2
7.d odd 6 2 1176.2.q.h 2
8.b even 2 1 9408.2.a.u 1
8.d odd 2 1 9408.2.a.ck 1
12.b even 2 1 7056.2.a.bc 1
21.c even 2 1 3528.2.a.m 1
21.g even 6 2 3528.2.s.m 2
21.h odd 6 2 3528.2.s.n 2
28.d even 2 1 2352.2.a.r 1
28.f even 6 2 2352.2.q.h 2
28.g odd 6 2 2352.2.q.t 2
56.e even 2 1 9408.2.a.v 1
56.h odd 2 1 9408.2.a.cl 1
84.h odd 2 1 7056.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.b 1 7.b odd 2 1
1176.2.a.h yes 1 1.a even 1 1 trivial
1176.2.q.c 2 7.c even 3 2
1176.2.q.h 2 7.d odd 6 2
2352.2.a.h 1 4.b odd 2 1
2352.2.a.r 1 28.d even 2 1
2352.2.q.h 2 28.f even 6 2
2352.2.q.t 2 28.g odd 6 2
3528.2.a.m 1 21.c even 2 1
3528.2.a.n 1 3.b odd 2 1
3528.2.s.m 2 21.g even 6 2
3528.2.s.n 2 21.h odd 6 2
7056.2.a.ba 1 84.h odd 2 1
7056.2.a.bc 1 12.b even 2 1
9408.2.a.u 1 8.b even 2 1
9408.2.a.v 1 56.e even 2 1
9408.2.a.ck 1 8.d odd 2 1
9408.2.a.cl 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1176))$$:

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T - 4$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T - 8$$
$37$ $$T + 6$$
$41$ $$T - 12$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T - 4$$
$67$ $$T + 4$$
$71$ $$T + 12$$
$73$ $$T - 8$$
$79$ $$T + 16$$
$83$ $$T + 4$$
$89$ $$T - 4$$
$97$ $$T - 16$$