# Properties

 Label 5220.2.a.w Level $5220$ Weight $2$ Character orbit 5220.a Self dual yes Analytic conductor $41.682$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 580) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + ( - \beta_{2} + 1) q^{7}+O(q^{10})$$ q - q^5 + (-b2 + 1) * q^7 $$q - q^{5} + ( - \beta_{2} + 1) q^{7} + (\beta_{2} - 1) q^{11} + (\beta_{2} + \beta_1) q^{13} + ( - \beta_{2} - 3) q^{17} + ( - \beta_1 + 1) q^{19} + ( - \beta_1 + 1) q^{23} + q^{25} - q^{29} + (\beta_1 - 1) q^{31} + (\beta_{2} - 1) q^{35} + ( - \beta_{2} + 5) q^{37} + (\beta_{2} - 3 \beta_1) q^{41} + (2 \beta_{2} + 3 \beta_1 - 5) q^{43} + ( - 3 \beta_1 - 1) q^{47} + ( - 3 \beta_{2} - \beta_1 + 1) q^{49} + (3 \beta_{2} - 3 \beta_1 - 2) q^{53} + ( - \beta_{2} + 1) q^{55} + ( - \beta_{2} + 3 \beta_1 - 2) q^{59} + (3 \beta_{2} + 5 \beta_1 - 6) q^{61} + ( - \beta_{2} - \beta_1) q^{65} + (\beta_1 - 1) q^{67} + ( - 3 \beta_{2} + \beta_1 + 2) q^{71} + ( - \beta_1 + 1) q^{73} + (3 \beta_{2} + \beta_1 - 8) q^{77} + ( - 3 \beta_{2} - 9) q^{79} + (3 \beta_{2} + 2 \beta_1 + 3) q^{83} + (\beta_{2} + 3) q^{85} + ( - 4 \beta_{2} + 2 \beta_1) q^{89} + (2 \beta_{2} - 6) q^{91} + (\beta_1 - 1) q^{95} + (2 \beta_{2} - 5 \beta_1 + 3) q^{97}+O(q^{100})$$ q - q^5 + (-b2 + 1) * q^7 + (b2 - 1) * q^11 + (b2 + b1) * q^13 + (-b2 - 3) * q^17 + (-b1 + 1) * q^19 + (-b1 + 1) * q^23 + q^25 - q^29 + (b1 - 1) * q^31 + (b2 - 1) * q^35 + (-b2 + 5) * q^37 + (b2 - 3*b1) * q^41 + (2*b2 + 3*b1 - 5) * q^43 + (-3*b1 - 1) * q^47 + (-3*b2 - b1 + 1) * q^49 + (3*b2 - 3*b1 - 2) * q^53 + (-b2 + 1) * q^55 + (-b2 + 3*b1 - 2) * q^59 + (3*b2 + 5*b1 - 6) * q^61 + (-b2 - b1) * q^65 + (b1 - 1) * q^67 + (-3*b2 + b1 + 2) * q^71 + (-b1 + 1) * q^73 + (3*b2 + b1 - 8) * q^77 + (-3*b2 - 9) * q^79 + (3*b2 + 2*b1 + 3) * q^83 + (b2 + 3) * q^85 + (-4*b2 + 2*b1) * q^89 + (2*b2 - 6) * q^91 + (b1 - 1) * q^95 + (2*b2 - 5*b1 + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + 2 q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 + 2 * q^7 $$3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61} - 2 q^{65} - 2 q^{67} + 4 q^{71} + 2 q^{73} - 20 q^{77} - 30 q^{79} + 14 q^{83} + 10 q^{85} - 2 q^{89} - 16 q^{91} - 2 q^{95} + 6 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2

## 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
 0.311108 2.17009 −1.48119
0 0 0 −1.00000 0 −1.52543 0 0 0
1.2 0 0 0 −1.00000 0 −0.630898 0 0 0
1.3 0 0 0 −1.00000 0 4.15633 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.w 3
3.b odd 2 1 580.2.a.d 3
12.b even 2 1 2320.2.a.o 3
15.d odd 2 1 2900.2.a.f 3
15.e even 4 2 2900.2.c.g 6
24.f even 2 1 9280.2.a.bt 3
24.h odd 2 1 9280.2.a.bh 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.a.d 3 3.b odd 2 1
2320.2.a.o 3 12.b even 2 1
2900.2.a.f 3 15.d odd 2 1
2900.2.c.g 6 15.e even 4 2
5220.2.a.w 3 1.a even 1 1 trivial
9280.2.a.bh 3 24.h odd 2 1
9280.2.a.bt 3 24.f even 2 1

## Hecke kernels

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

 $$T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4$$ T7^3 - 2*T7^2 - 8*T7 - 4 $$T_{11}^{3} + 2T_{11}^{2} - 8T_{11} + 4$$ T11^3 + 2*T11^2 - 8*T11 + 4 $$T_{19}^{3} - 2T_{19}^{2} - 4T_{19} + 4$$ T19^3 - 2*T19^2 - 4*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots - 4$$
$11$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$13$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$17$ $$T^{3} + 10 T^{2} + \cdots - 4$$
$19$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$23$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$29$ $$(T + 1)^{3}$$
$31$ $$T^{3} + 2 T^{2} + \cdots - 4$$
$37$ $$T^{3} - 14 T^{2} + \cdots - 68$$
$41$ $$T^{3} + 2 T^{2} + \cdots - 200$$
$43$ $$T^{3} + 10 T^{2} + \cdots - 388$$
$47$ $$T^{3} + 6 T^{2} + \cdots - 68$$
$53$ $$T^{3} + 6 T^{2} + \cdots - 712$$
$59$ $$T^{3} + 4 T^{2} + \cdots + 80$$
$61$ $$T^{3} + 10 T^{2} + \cdots - 1432$$
$67$ $$T^{3} + 2 T^{2} + \cdots - 4$$
$71$ $$T^{3} - 4 T^{2} + \cdots - 16$$
$73$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$79$ $$T^{3} + 30 T^{2} + \cdots - 108$$
$83$ $$T^{3} - 14 T^{2} + \cdots + 548$$
$89$ $$T^{3} + 2 T^{2} + \cdots - 200$$
$97$ $$T^{3} - 6 T^{2} + \cdots - 548$$