# Properties

 Label 8820.2.a.p Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ 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] = [8820,2,Mod(1,8820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8820, 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("8820.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8820.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: $$70.4280545828$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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)

 $$f(q)$$ $$=$$ $$q + q^{5}+O(q^{10})$$ q + q^5 $$q + q^{5} - 6 q^{11} + 2 q^{13} + 6 q^{17} + 8 q^{19} - 3 q^{23} + q^{25} - 3 q^{29} + 2 q^{31} + 8 q^{37} + 3 q^{41} + 5 q^{43} - 12 q^{53} - 6 q^{55} - q^{61} + 2 q^{65} - 7 q^{67} - 10 q^{73} - 4 q^{79} - 3 q^{83} + 6 q^{85} + 3 q^{89} + 8 q^{95} - 10 q^{97}+O(q^{100})$$ q + q^5 - 6 * q^11 + 2 * q^13 + 6 * q^17 + 8 * q^19 - 3 * q^23 + q^25 - 3 * q^29 + 2 * q^31 + 8 * q^37 + 3 * q^41 + 5 * q^43 - 12 * q^53 - 6 * q^55 - q^61 + 2 * q^65 - 7 * q^67 - 10 * q^73 - 4 * q^79 - 3 * q^83 + 6 * q^85 + 3 * q^89 + 8 * q^95 - 10 * 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 1.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 8820.2.a.p 1
3.b odd 2 1 980.2.a.g 1
7.b odd 2 1 8820.2.a.a 1
7.c even 3 2 1260.2.s.c 2
12.b even 2 1 3920.2.a.k 1
15.d odd 2 1 4900.2.a.i 1
15.e even 4 2 4900.2.e.m 2
21.c even 2 1 980.2.a.e 1
21.g even 6 2 980.2.i.f 2
21.h odd 6 2 140.2.i.a 2
84.h odd 2 1 3920.2.a.w 1
84.n even 6 2 560.2.q.f 2
105.g even 2 1 4900.2.a.q 1
105.k odd 4 2 4900.2.e.n 2
105.o odd 6 2 700.2.i.b 2
105.x even 12 4 700.2.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 21.h odd 6 2
560.2.q.f 2 84.n even 6 2
700.2.i.b 2 105.o odd 6 2
700.2.r.a 4 105.x even 12 4
980.2.a.e 1 21.c even 2 1
980.2.a.g 1 3.b odd 2 1
980.2.i.f 2 21.g even 6 2
1260.2.s.c 2 7.c even 3 2
3920.2.a.k 1 12.b even 2 1
3920.2.a.w 1 84.h odd 2 1
4900.2.a.i 1 15.d odd 2 1
4900.2.a.q 1 105.g even 2 1
4900.2.e.m 2 15.e even 4 2
4900.2.e.n 2 105.k odd 4 2
8820.2.a.a 1 7.b odd 2 1
8820.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8820))$$:

 $$T_{11} + 6$$ T11 + 6 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 6$$ T17 - 6 $$T_{31} - 2$$ T31 - 2

## Hecke characteristic polynomials

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