# Properties

 Label 520.2.q.e Level $520$ Weight $2$ Character orbit 520.q Analytic conductor $4.152$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(81,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("520.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.q (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$4.15222090511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{3} - q^{5} - 3 \zeta_{6} q^{7} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (-3*z + 3) * q^3 - q^5 - 3*z * q^7 - 6*z * q^9 $$q + ( - 3 \zeta_{6} + 3) q^{3} - q^{5} - 3 \zeta_{6} q^{7} - 6 \zeta_{6} q^{9} + (5 \zeta_{6} - 5) q^{11} + ( - 4 \zeta_{6} + 1) q^{13} + (3 \zeta_{6} - 3) q^{15} - 3 \zeta_{6} q^{17} + 5 \zeta_{6} q^{19} - 9 q^{21} + ( - 3 \zeta_{6} + 3) q^{23} + q^{25} - 9 q^{27} + ( - 5 \zeta_{6} + 5) q^{29} + 8 q^{31} + 15 \zeta_{6} q^{33} + 3 \zeta_{6} q^{35} + ( - 9 \zeta_{6} + 9) q^{37} + ( - 3 \zeta_{6} - 9) q^{39} + (3 \zeta_{6} - 3) q^{41} + \zeta_{6} q^{43} + 6 \zeta_{6} q^{45} - 12 q^{47} + (2 \zeta_{6} - 2) q^{49} - 9 q^{51} + 2 q^{53} + ( - 5 \zeta_{6} + 5) q^{55} + 15 q^{57} + 5 \zeta_{6} q^{59} + \zeta_{6} q^{61} + (18 \zeta_{6} - 18) q^{63} + (4 \zeta_{6} - 1) q^{65} + ( - 15 \zeta_{6} + 15) q^{67} - 9 \zeta_{6} q^{69} + \zeta_{6} q^{71} - 10 q^{73} + ( - 3 \zeta_{6} + 3) q^{75} + 15 q^{77} - 4 q^{79} + (9 \zeta_{6} - 9) q^{81} + 12 q^{83} + 3 \zeta_{6} q^{85} - 15 \zeta_{6} q^{87} + ( - \zeta_{6} + 1) q^{89} + (9 \zeta_{6} - 12) q^{91} + ( - 24 \zeta_{6} + 24) q^{93} - 5 \zeta_{6} q^{95} - 7 \zeta_{6} q^{97} + 30 q^{99} +O(q^{100})$$ q + (-3*z + 3) * q^3 - q^5 - 3*z * q^7 - 6*z * q^9 + (5*z - 5) * q^11 + (-4*z + 1) * q^13 + (3*z - 3) * q^15 - 3*z * q^17 + 5*z * q^19 - 9 * q^21 + (-3*z + 3) * q^23 + q^25 - 9 * q^27 + (-5*z + 5) * q^29 + 8 * q^31 + 15*z * q^33 + 3*z * q^35 + (-9*z + 9) * q^37 + (-3*z - 9) * q^39 + (3*z - 3) * q^41 + z * q^43 + 6*z * q^45 - 12 * q^47 + (2*z - 2) * q^49 - 9 * q^51 + 2 * q^53 + (-5*z + 5) * q^55 + 15 * q^57 + 5*z * q^59 + z * q^61 + (18*z - 18) * q^63 + (4*z - 1) * q^65 + (-15*z + 15) * q^67 - 9*z * q^69 + z * q^71 - 10 * q^73 + (-3*z + 3) * q^75 + 15 * q^77 - 4 * q^79 + (9*z - 9) * q^81 + 12 * q^83 + 3*z * q^85 - 15*z * q^87 + (-z + 1) * q^89 + (9*z - 12) * q^91 + (-24*z + 24) * q^93 - 5*z * q^95 - 7*z * q^97 + 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} - 2 q^{5} - 3 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 - 2 * q^5 - 3 * q^7 - 6 * q^9 $$2 q + 3 q^{3} - 2 q^{5} - 3 q^{7} - 6 q^{9} - 5 q^{11} - 2 q^{13} - 3 q^{15} - 3 q^{17} + 5 q^{19} - 18 q^{21} + 3 q^{23} + 2 q^{25} - 18 q^{27} + 5 q^{29} + 16 q^{31} + 15 q^{33} + 3 q^{35} + 9 q^{37} - 21 q^{39} - 3 q^{41} + q^{43} + 6 q^{45} - 24 q^{47} - 2 q^{49} - 18 q^{51} + 4 q^{53} + 5 q^{55} + 30 q^{57} + 5 q^{59} + q^{61} - 18 q^{63} + 2 q^{65} + 15 q^{67} - 9 q^{69} + q^{71} - 20 q^{73} + 3 q^{75} + 30 q^{77} - 8 q^{79} - 9 q^{81} + 24 q^{83} + 3 q^{85} - 15 q^{87} + q^{89} - 15 q^{91} + 24 q^{93} - 5 q^{95} - 7 q^{97} + 60 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 - 2 * q^5 - 3 * q^7 - 6 * q^9 - 5 * q^11 - 2 * q^13 - 3 * q^15 - 3 * q^17 + 5 * q^19 - 18 * q^21 + 3 * q^23 + 2 * q^25 - 18 * q^27 + 5 * q^29 + 16 * q^31 + 15 * q^33 + 3 * q^35 + 9 * q^37 - 21 * q^39 - 3 * q^41 + q^43 + 6 * q^45 - 24 * q^47 - 2 * q^49 - 18 * q^51 + 4 * q^53 + 5 * q^55 + 30 * q^57 + 5 * q^59 + q^61 - 18 * q^63 + 2 * q^65 + 15 * q^67 - 9 * q^69 + q^71 - 20 * q^73 + 3 * q^75 + 30 * q^77 - 8 * q^79 - 9 * q^81 + 24 * q^83 + 3 * q^85 - 15 * q^87 + q^89 - 15 * q^91 + 24 * q^93 - 5 * q^95 - 7 * q^97 + 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$261$$ $$391$$ $$417$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 + 2.59808i 0 −1.00000 0 −1.50000 + 2.59808i 0 −3.00000 + 5.19615i 0
321.1 0 1.50000 2.59808i 0 −1.00000 0 −1.50000 2.59808i 0 −3.00000 5.19615i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.q.e 2
4.b odd 2 1 1040.2.q.a 2
13.c even 3 1 inner 520.2.q.e 2
13.c even 3 1 6760.2.a.a 1
13.e even 6 1 6760.2.a.b 1
52.j odd 6 1 1040.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.q.e 2 1.a even 1 1 trivial
520.2.q.e 2 13.c even 3 1 inner
1040.2.q.a 2 4.b odd 2 1
1040.2.q.a 2 52.j odd 6 1
6760.2.a.a 1 13.c even 3 1
6760.2.a.b 1 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(520, [\chi])$$:

 $$T_{3}^{2} - 3T_{3} + 9$$ T3^2 - 3*T3 + 9 $$T_{11}^{2} + 5T_{11} + 25$$ T11^2 + 5*T11 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 3T + 9$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} - 5T + 25$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - 9T + 81$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$(T + 12)^{2}$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 5T + 25$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 15T + 225$$
$71$ $$T^{2} - T + 1$$
$73$ $$(T + 10)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - T + 1$$
$97$ $$T^{2} + 7T + 49$$