Properties

 Label 8712.2.a.x Level $8712$ Weight $2$ Character orbit 8712.a Self dual yes Analytic conductor $69.566$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8712 = 2^{3} \cdot 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8712.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: $$69.5656702409$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + 2 q^{7}+O(q^{10})$$ q + 3 * q^5 + 2 * q^7 $$q + 3 q^{5} + 2 q^{7} - 6 q^{17} - 4 q^{19} - q^{23} + 4 q^{25} - 8 q^{29} - 7 q^{31} + 6 q^{35} - q^{37} + 4 q^{41} - 6 q^{43} + 8 q^{47} - 3 q^{49} - 2 q^{53} + q^{59} - 4 q^{61} - 5 q^{67} - 3 q^{71} - 16 q^{73} - 2 q^{79} - 2 q^{83} - 18 q^{85} - 15 q^{89} - 12 q^{95} - 7 q^{97}+O(q^{100})$$ q + 3 * q^5 + 2 * q^7 - 6 * q^17 - 4 * q^19 - q^23 + 4 * q^25 - 8 * q^29 - 7 * q^31 + 6 * q^35 - q^37 + 4 * q^41 - 6 * q^43 + 8 * q^47 - 3 * q^49 - 2 * q^53 + q^59 - 4 * q^61 - 5 * q^67 - 3 * q^71 - 16 * q^73 - 2 * q^79 - 2 * q^83 - 18 * q^85 - 15 * q^89 - 12 * q^95 - 7 * 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 3.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

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 8712.2.a.x 1
3.b odd 2 1 968.2.a.a 1
11.b odd 2 1 792.2.a.g 1
12.b even 2 1 1936.2.a.l 1
24.f even 2 1 7744.2.a.b 1
24.h odd 2 1 7744.2.a.bk 1
33.d even 2 1 88.2.a.a 1
33.f even 10 4 968.2.i.j 4
33.h odd 10 4 968.2.i.i 4
44.c even 2 1 1584.2.a.q 1
88.b odd 2 1 6336.2.a.h 1
88.g even 2 1 6336.2.a.k 1
132.d odd 2 1 176.2.a.c 1
165.d even 2 1 2200.2.a.k 1
165.l odd 4 2 2200.2.b.a 2
231.h odd 2 1 4312.2.a.l 1
264.m even 2 1 704.2.a.l 1
264.p odd 2 1 704.2.a.b 1
528.s odd 4 2 2816.2.c.d 2
528.x even 4 2 2816.2.c.i 2
660.g odd 2 1 4400.2.a.a 1
660.q even 4 2 4400.2.b.b 2
924.n even 2 1 8624.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 33.d even 2 1
176.2.a.c 1 132.d odd 2 1
704.2.a.b 1 264.p odd 2 1
704.2.a.l 1 264.m even 2 1
792.2.a.g 1 11.b odd 2 1
968.2.a.a 1 3.b odd 2 1
968.2.i.i 4 33.h odd 10 4
968.2.i.j 4 33.f even 10 4
1584.2.a.q 1 44.c even 2 1
1936.2.a.l 1 12.b even 2 1
2200.2.a.k 1 165.d even 2 1
2200.2.b.a 2 165.l odd 4 2
2816.2.c.d 2 528.s odd 4 2
2816.2.c.i 2 528.x even 4 2
4312.2.a.l 1 231.h odd 2 1
4400.2.a.a 1 660.g odd 2 1
4400.2.b.b 2 660.q even 4 2
6336.2.a.h 1 88.b odd 2 1
6336.2.a.k 1 88.g even 2 1
7744.2.a.b 1 24.f even 2 1
7744.2.a.bk 1 24.h odd 2 1
8624.2.a.c 1 924.n even 2 1
8712.2.a.x 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8712))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{7} - 2$$ T7 - 2 $$T_{13}$$ T13 $$T_{17} + 6$$ T17 + 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T + 1$$
$29$ $$T + 8$$
$31$ $$T + 7$$
$37$ $$T + 1$$
$41$ $$T - 4$$
$43$ $$T + 6$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T - 1$$
$61$ $$T + 4$$
$67$ $$T + 5$$
$71$ $$T + 3$$
$73$ $$T + 16$$
$79$ $$T + 2$$
$83$ $$T + 2$$
$89$ $$T + 15$$
$97$ $$T + 7$$
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