# Properties

 Label 5220.2.a.b Level $5220$ Weight $2$ Character orbit 5220.a Self dual yes Analytic conductor $41.682$ 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] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5220.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: $$41.6819098551$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1740) 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)

 $$f(q)$$ $$=$$ $$q - q^{5} - 3 q^{7}+O(q^{10})$$ q - q^5 - 3 * q^7 $$q - q^{5} - 3 q^{7} + 3 q^{11} + q^{13} + 3 q^{17} - 6 q^{19} + 4 q^{23} + q^{25} - q^{29} - 4 q^{31} + 3 q^{35} - 4 q^{37} - 2 q^{41} + 4 q^{43} + 3 q^{47} + 2 q^{49} - 6 q^{53} - 3 q^{55} + 10 q^{59} - 4 q^{61} - q^{65} - 9 q^{67} + 12 q^{71} + 4 q^{73} - 9 q^{77} + 14 q^{79} - 6 q^{83} - 3 q^{85} - 7 q^{89} - 3 q^{91} + 6 q^{95} - 14 q^{97}+O(q^{100})$$ q - q^5 - 3 * q^7 + 3 * q^11 + q^13 + 3 * q^17 - 6 * q^19 + 4 * q^23 + q^25 - q^29 - 4 * q^31 + 3 * q^35 - 4 * q^37 - 2 * q^41 + 4 * q^43 + 3 * q^47 + 2 * q^49 - 6 * q^53 - 3 * q^55 + 10 * q^59 - 4 * q^61 - q^65 - 9 * q^67 + 12 * q^71 + 4 * q^73 - 9 * q^77 + 14 * q^79 - 6 * q^83 - 3 * q^85 - 7 * q^89 - 3 * q^91 + 6 * q^95 - 14 * 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 −3.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$29$$ $$+1$$

## 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 5220.2.a.b 1
3.b odd 2 1 1740.2.a.g 1
12.b even 2 1 6960.2.a.u 1
15.d odd 2 1 8700.2.a.h 1
15.e even 4 2 8700.2.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1740.2.a.g 1 3.b odd 2 1
5220.2.a.b 1 1.a even 1 1 trivial
6960.2.a.u 1 12.b even 2 1
8700.2.a.h 1 15.d odd 2 1
8700.2.g.f 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5220))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} - 3$$ T11 - 3 $$T_{19} + 6$$ T19 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 3$$
$11$ $$T - 3$$
$13$ $$T - 1$$
$17$ $$T - 3$$
$19$ $$T + 6$$
$23$ $$T - 4$$
$29$ $$T + 1$$
$31$ $$T + 4$$
$37$ $$T + 4$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T - 3$$
$53$ $$T + 6$$
$59$ $$T - 10$$
$61$ $$T + 4$$
$67$ $$T + 9$$
$71$ $$T - 12$$
$73$ $$T - 4$$
$79$ $$T - 14$$
$83$ $$T + 6$$
$89$ $$T + 7$$
$97$ $$T + 14$$