# Properties

 Label 8624.2.a.a 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 77) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + q^{5} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 + q^5 + 6 * q^9 $$q - 3 q^{3} + q^{5} + 6 q^{9} + q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 6 q^{19} + 5 q^{23} - 4 q^{25} - 9 q^{27} + 10 q^{29} + q^{31} - 3 q^{33} - 5 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 6 q^{45} + 8 q^{47} + 6 q^{51} - 6 q^{53} + q^{55} + 18 q^{57} + 3 q^{59} + 2 q^{61} + 4 q^{65} + 3 q^{67} - 15 q^{69} - q^{71} - 10 q^{73} + 12 q^{75} - 6 q^{79} + 9 q^{81} + 12 q^{83} - 2 q^{85} - 30 q^{87} + 15 q^{89} - 3 q^{93} - 6 q^{95} + 5 q^{97} + 6 q^{99}+O(q^{100})$$ q - 3 * q^3 + q^5 + 6 * q^9 + q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 6 * q^19 + 5 * q^23 - 4 * q^25 - 9 * q^27 + 10 * q^29 + q^31 - 3 * q^33 - 5 * q^37 - 12 * q^39 + 2 * q^41 + 8 * q^43 + 6 * q^45 + 8 * q^47 + 6 * q^51 - 6 * q^53 + q^55 + 18 * q^57 + 3 * q^59 + 2 * q^61 + 4 * q^65 + 3 * q^67 - 15 * q^69 - q^71 - 10 * q^73 + 12 * q^75 - 6 * q^79 + 9 * q^81 + 12 * q^83 - 2 * q^85 - 30 * q^87 + 15 * q^89 - 3 * q^93 - 6 * q^95 + 5 * q^97 + 6 * 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 −3.00000 0 1.00000 0 0 0 6.00000 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.a 1
4.b odd 2 1 539.2.a.c 1
7.b odd 2 1 1232.2.a.l 1
12.b even 2 1 4851.2.a.j 1
28.d even 2 1 77.2.a.a 1
28.f even 6 2 539.2.e.f 2
28.g odd 6 2 539.2.e.c 2
44.c even 2 1 5929.2.a.f 1
56.e even 2 1 4928.2.a.bj 1
56.h odd 2 1 4928.2.a.a 1
84.h odd 2 1 693.2.a.c 1
140.c even 2 1 1925.2.a.h 1
140.j odd 4 2 1925.2.b.e 2
308.g odd 2 1 847.2.a.b 1
308.s odd 10 4 847.2.f.h 4
308.t even 10 4 847.2.f.i 4
924.n even 2 1 7623.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 28.d even 2 1
539.2.a.c 1 4.b odd 2 1
539.2.e.c 2 28.g odd 6 2
539.2.e.f 2 28.f even 6 2
693.2.a.c 1 84.h odd 2 1
847.2.a.b 1 308.g odd 2 1
847.2.f.h 4 308.s odd 10 4
847.2.f.i 4 308.t even 10 4
1232.2.a.l 1 7.b odd 2 1
1925.2.a.h 1 140.c even 2 1
1925.2.b.e 2 140.j odd 4 2
4851.2.a.j 1 12.b even 2 1
4928.2.a.a 1 56.h odd 2 1
4928.2.a.bj 1 56.e even 2 1
5929.2.a.f 1 44.c even 2 1
7623.2.a.j 1 924.n even 2 1
8624.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3} + 3$$ T3 + 3 $$T_{5} - 1$$ T5 - 1 $$T_{13} - 4$$ T13 - 4 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T - 4$$
$17$ $$T + 2$$
$19$ $$T + 6$$
$23$ $$T - 5$$
$29$ $$T - 10$$
$31$ $$T - 1$$
$37$ $$T + 5$$
$41$ $$T - 2$$
$43$ $$T - 8$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T - 3$$
$61$ $$T - 2$$
$67$ $$T - 3$$
$71$ $$T + 1$$
$73$ $$T + 10$$
$79$ $$T + 6$$
$83$ $$T - 12$$
$89$ $$T - 15$$
$97$ $$T - 5$$