# Properties

 Label 8624.2.a.u Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^9 $$q + q^{3} - 2 q^{9} + q^{11} + q^{13} + 6 q^{17} + 2 q^{19} + 6 q^{23} - 5 q^{25} - 5 q^{27} + 9 q^{29} - 4 q^{31} + q^{33} + 2 q^{37} + q^{39} + 6 q^{41} + 4 q^{43} - 6 q^{47} + 6 q^{51} + 2 q^{57} - 3 q^{59} - 11 q^{61} - 11 q^{67} + 6 q^{69} - 2 q^{73} - 5 q^{75} - 5 q^{79} + q^{81} - 6 q^{83} + 9 q^{87} + 18 q^{89} - 4 q^{93} + 13 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^9 + q^11 + q^13 + 6 * q^17 + 2 * q^19 + 6 * q^23 - 5 * q^25 - 5 * q^27 + 9 * q^29 - 4 * q^31 + q^33 + 2 * q^37 + q^39 + 6 * q^41 + 4 * q^43 - 6 * q^47 + 6 * q^51 + 2 * q^57 - 3 * q^59 - 11 * q^61 - 11 * q^67 + 6 * q^69 - 2 * q^73 - 5 * q^75 - 5 * q^79 + q^81 - 6 * q^83 + 9 * q^87 + 18 * q^89 - 4 * q^93 + 13 * q^97 - 2 * 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 1.00000 0 0 0 0 0 −2.00000 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.u 1
4.b odd 2 1 1078.2.a.c 1
7.b odd 2 1 8624.2.a.k 1
7.d odd 6 2 1232.2.q.d 2
12.b even 2 1 9702.2.a.bs 1
28.d even 2 1 1078.2.a.e 1
28.f even 6 2 154.2.e.c 2
28.g odd 6 2 1078.2.e.k 2
84.h odd 2 1 9702.2.a.br 1
84.j odd 6 2 1386.2.k.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 28.f even 6 2
1078.2.a.c 1 4.b odd 2 1
1078.2.a.e 1 28.d even 2 1
1078.2.e.k 2 28.g odd 6 2
1232.2.q.d 2 7.d odd 6 2
1386.2.k.e 2 84.j odd 6 2
8624.2.a.k 1 7.b odd 2 1
8624.2.a.u 1 1.a even 1 1 trivial
9702.2.a.br 1 84.h odd 2 1
9702.2.a.bs 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5}$$ T5 $$T_{13} - 1$$ T13 - 1 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T - 1$$
$17$ $$T - 6$$
$19$ $$T - 2$$
$23$ $$T - 6$$
$29$ $$T - 9$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T + 6$$
$53$ $$T$$
$59$ $$T + 3$$
$61$ $$T + 11$$
$67$ $$T + 11$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T + 5$$
$83$ $$T + 6$$
$89$ $$T - 18$$
$97$ $$T - 13$$