# Properties

 Label 8624.2.a.y Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ 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] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8624 = 2^{4} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8624.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: $$68.8629867032$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1078) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 2 q^{5} + q^{9}+O(q^{10})$$ q + 2 * q^3 - 2 * q^5 + q^9 $$q + 2 q^{3} - 2 q^{5} + q^{9} - q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{19} - q^{25} - 4 q^{27} + 6 q^{29} - 4 q^{31} - 2 q^{33} + 2 q^{37} + 4 q^{39} + 8 q^{41} - 12 q^{43} - 2 q^{45} - 12 q^{47} - 2 q^{53} + 2 q^{55} + 4 q^{57} - 10 q^{59} - 10 q^{61} - 4 q^{65} + 12 q^{67} - 4 q^{71} + 12 q^{73} - 2 q^{75} - 11 q^{81} + 18 q^{83} + 12 q^{87} - 8 q^{93} - 4 q^{95} - 12 q^{97} - q^{99}+O(q^{100})$$ q + 2 * q^3 - 2 * q^5 + q^9 - q^11 + 2 * q^13 - 4 * q^15 + 2 * q^19 - q^25 - 4 * q^27 + 6 * q^29 - 4 * q^31 - 2 * q^33 + 2 * q^37 + 4 * q^39 + 8 * q^41 - 12 * q^43 - 2 * q^45 - 12 * q^47 - 2 * q^53 + 2 * q^55 + 4 * q^57 - 10 * q^59 - 10 * q^61 - 4 * q^65 + 12 * q^67 - 4 * q^71 + 12 * q^73 - 2 * q^75 - 11 * q^81 + 18 * q^83 + 12 * q^87 - 8 * q^93 - 4 * q^95 - 12 * q^97 - q^99

## 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 2.00000 0 −2.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 8624.2.a.y 1
4.b odd 2 1 1078.2.a.h 1
7.b odd 2 1 8624.2.a.g 1
12.b even 2 1 9702.2.a.t 1
28.d even 2 1 1078.2.a.l yes 1
28.f even 6 2 1078.2.e.a 2
28.g odd 6 2 1078.2.e.e 2
84.h odd 2 1 9702.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.h 1 4.b odd 2 1
1078.2.a.l yes 1 28.d even 2 1
1078.2.e.a 2 28.f even 6 2
1078.2.e.e 2 28.g odd 6 2
8624.2.a.g 1 7.b odd 2 1
8624.2.a.y 1 1.a even 1 1 trivial
9702.2.a.e 1 84.h odd 2 1
9702.2.a.t 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{5} + 2$$ T5 + 2 $$T_{13} - 2$$ T13 - 2 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T - 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T - 8$$
$43$ $$T + 12$$
$47$ $$T + 12$$
$53$ $$T + 2$$
$59$ $$T + 10$$
$61$ $$T + 10$$
$67$ $$T - 12$$
$71$ $$T + 4$$
$73$ $$T - 12$$
$79$ $$T$$
$83$ $$T - 18$$
$89$ $$T$$
$97$ $$T + 12$$