# Properties

 Label 8820.2.a.bg Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 980) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5}+O(q^{10})$$ q - q^5 $$q - q^{5} + (2 \beta + 1) q^{11} + (\beta - 5) q^{13} + (\beta + 5) q^{17} + ( - 4 \beta + 2) q^{19} + (\beta - 2) q^{23} + q^{25} + ( - 4 \beta - 1) q^{29} + (\beta - 6) q^{31} + ( - \beta - 2) q^{37} + ( - \beta + 2) q^{41} + (4 \beta + 6) q^{43} + ( - 7 \beta + 1) q^{47} + ( - 3 \beta + 8) q^{53} + ( - 2 \beta - 1) q^{55} + (\beta + 2) q^{59} + (2 \beta - 8) q^{61} + ( - \beta + 5) q^{65} + ( - 5 \beta - 4) q^{67} + ( - 6 \beta + 2) q^{71} + (2 \beta - 8) q^{73} + (10 \beta - 1) q^{79} - 8 q^{83} + ( - \beta - 5) q^{85} + 12 \beta q^{89} + (4 \beta - 2) q^{95} + ( - 9 \beta - 3) q^{97}+O(q^{100})$$ q - q^5 + (2*b + 1) * q^11 + (b - 5) * q^13 + (b + 5) * q^17 + (-4*b + 2) * q^19 + (b - 2) * q^23 + q^25 + (-4*b - 1) * q^29 + (b - 6) * q^31 + (-b - 2) * q^37 + (-b + 2) * q^41 + (4*b + 6) * q^43 + (-7*b + 1) * q^47 + (-3*b + 8) * q^53 + (-2*b - 1) * q^55 + (b + 2) * q^59 + (2*b - 8) * q^61 + (-b + 5) * q^65 + (-5*b - 4) * q^67 + (-6*b + 2) * q^71 + (2*b - 8) * q^73 + (10*b - 1) * q^79 - 8 * q^83 + (-b - 5) * q^85 + 12*b * q^89 + (4*b - 2) * q^95 + (-9*b - 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} + 2 q^{11} - 10 q^{13} + 10 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 4 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{47} + 16 q^{53} - 2 q^{55} + 4 q^{59} - 16 q^{61} + 10 q^{65} - 8 q^{67} + 4 q^{71} - 16 q^{73} - 2 q^{79} - 16 q^{83} - 10 q^{85} - 4 q^{95} - 6 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 2 * q^11 - 10 * q^13 + 10 * q^17 + 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^29 - 12 * q^31 - 4 * q^37 + 4 * q^41 + 12 * q^43 + 2 * q^47 + 16 * q^53 - 2 * q^55 + 4 * q^59 - 16 * q^61 + 10 * q^65 - 8 * q^67 + 4 * q^71 - 16 * q^73 - 2 * q^79 - 16 * q^83 - 10 * q^85 - 4 * q^95 - 6 * 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
 −1.41421 1.41421
0 0 0 −1.00000 0 0 0 0 0
1.2 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.bg 2
3.b odd 2 1 980.2.a.j 2
7.b odd 2 1 8820.2.a.bl 2
12.b even 2 1 3920.2.a.bx 2
15.d odd 2 1 4900.2.a.z 2
15.e even 4 2 4900.2.e.q 4
21.c even 2 1 980.2.a.k yes 2
21.g even 6 2 980.2.i.k 4
21.h odd 6 2 980.2.i.l 4
84.h odd 2 1 3920.2.a.bo 2
105.g even 2 1 4900.2.a.x 2
105.k odd 4 2 4900.2.e.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 3.b odd 2 1
980.2.a.k yes 2 21.c even 2 1
980.2.i.k 4 21.g even 6 2
980.2.i.l 4 21.h odd 6 2
3920.2.a.bo 2 84.h odd 2 1
3920.2.a.bx 2 12.b even 2 1
4900.2.a.x 2 105.g even 2 1
4900.2.a.z 2 15.d odd 2 1
4900.2.e.q 4 15.e even 4 2
4900.2.e.r 4 105.k odd 4 2
8820.2.a.bg 2 1.a even 1 1 trivial
8820.2.a.bl 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8820))$$:

 $$T_{11}^{2} - 2T_{11} - 7$$ T11^2 - 2*T11 - 7 $$T_{13}^{2} + 10T_{13} + 23$$ T13^2 + 10*T13 + 23 $$T_{17}^{2} - 10T_{17} + 23$$ T17^2 - 10*T17 + 23 $$T_{31}^{2} + 12T_{31} + 34$$ T31^2 + 12*T31 + 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 7$$
$13$ $$T^{2} + 10T + 23$$
$17$ $$T^{2} - 10T + 23$$
$19$ $$T^{2} - 4T - 28$$
$23$ $$T^{2} + 4T + 2$$
$29$ $$T^{2} + 2T - 31$$
$31$ $$T^{2} + 12T + 34$$
$37$ $$T^{2} + 4T + 2$$
$41$ $$T^{2} - 4T + 2$$
$43$ $$T^{2} - 12T + 4$$
$47$ $$T^{2} - 2T - 97$$
$53$ $$T^{2} - 16T + 46$$
$59$ $$T^{2} - 4T + 2$$
$61$ $$T^{2} + 16T + 56$$
$67$ $$T^{2} + 8T - 34$$
$71$ $$T^{2} - 4T - 68$$
$73$ $$T^{2} + 16T + 56$$
$79$ $$T^{2} + 2T - 199$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} - 288$$
$97$ $$T^{2} + 6T - 153$$