# Properties

 Label 520.2.a.g Level $520$ Weight $2$ Character orbit 520.a Self dual yes Analytic conductor $4.152$ Analytic rank $0$ 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] = [520,2,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.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: $$4.15222090511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} - q^{5} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 - q^5 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} - q^{5} + (2 \beta + 3) q^{9} + ( - \beta + 3) q^{11} - q^{13} + ( - \beta - 1) q^{15} + 2 q^{17} + ( - \beta + 3) q^{19} + ( - \beta - 1) q^{23} + q^{25} + (2 \beta + 10) q^{27} + (2 \beta + 4) q^{29} + ( - 3 \beta + 1) q^{31} + (2 \beta - 2) q^{33} + ( - 2 \beta - 4) q^{37} + ( - \beta - 1) q^{39} + ( - 4 \beta - 2) q^{41} + ( - \beta - 1) q^{43} + ( - 2 \beta - 3) q^{45} + (4 \beta + 4) q^{47} - 7 q^{49} + (2 \beta + 2) q^{51} + ( - 4 \beta - 2) q^{53} + (\beta - 3) q^{55} + (2 \beta - 2) q^{57} + ( - 3 \beta + 1) q^{59} - 2 \beta q^{61} + q^{65} + (2 \beta + 6) q^{67} + ( - 2 \beta - 6) q^{69} + (\beta - 3) q^{71} + (2 \beta - 12) q^{73} + (\beta + 1) q^{75} + (2 \beta + 2) q^{79} + (6 \beta + 11) q^{81} + 4 q^{83} - 2 q^{85} + (6 \beta + 14) q^{87} + 10 q^{89} + ( - 2 \beta - 14) q^{93} + (\beta - 3) q^{95} - 10 q^{97} + (3 \beta - 1) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 - q^5 + (2*b + 3) * q^9 + (-b + 3) * q^11 - q^13 + (-b - 1) * q^15 + 2 * q^17 + (-b + 3) * q^19 + (-b - 1) * q^23 + q^25 + (2*b + 10) * q^27 + (2*b + 4) * q^29 + (-3*b + 1) * q^31 + (2*b - 2) * q^33 + (-2*b - 4) * q^37 + (-b - 1) * q^39 + (-4*b - 2) * q^41 + (-b - 1) * q^43 + (-2*b - 3) * q^45 + (4*b + 4) * q^47 - 7 * q^49 + (2*b + 2) * q^51 + (-4*b - 2) * q^53 + (b - 3) * q^55 + (2*b - 2) * q^57 + (-3*b + 1) * q^59 - 2*b * q^61 + q^65 + (2*b + 6) * q^67 + (-2*b - 6) * q^69 + (b - 3) * q^71 + (2*b - 12) * q^73 + (b + 1) * q^75 + (2*b + 2) * q^79 + (6*b + 11) * q^81 + 4 * q^83 - 2 * q^85 + (6*b + 14) * q^87 + 10 * q^89 + (-2*b - 14) * q^93 + (b - 3) * q^95 - 10 * q^97 + (3*b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{9} + 6 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} + 20 q^{27} + 8 q^{29} + 2 q^{31} - 4 q^{33} - 8 q^{37} - 2 q^{39} - 4 q^{41} - 2 q^{43} - 6 q^{45} + 8 q^{47} - 14 q^{49} + 4 q^{51} - 4 q^{53} - 6 q^{55} - 4 q^{57} + 2 q^{59} + 2 q^{65} + 12 q^{67} - 12 q^{69} - 6 q^{71} - 24 q^{73} + 2 q^{75} + 4 q^{79} + 22 q^{81} + 8 q^{83} - 4 q^{85} + 28 q^{87} + 20 q^{89} - 28 q^{93} - 6 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 + 6 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 + 6 * q^19 - 2 * q^23 + 2 * q^25 + 20 * q^27 + 8 * q^29 + 2 * q^31 - 4 * q^33 - 8 * q^37 - 2 * q^39 - 4 * q^41 - 2 * q^43 - 6 * q^45 + 8 * q^47 - 14 * q^49 + 4 * q^51 - 4 * q^53 - 6 * q^55 - 4 * q^57 + 2 * q^59 + 2 * q^65 + 12 * q^67 - 12 * q^69 - 6 * q^71 - 24 * q^73 + 2 * q^75 + 4 * q^79 + 22 * q^81 + 8 * q^83 - 4 * q^85 + 28 * q^87 + 20 * q^89 - 28 * q^93 - 6 * q^95 - 20 * q^97 - 2 * q^99

## 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
 −0.618034 1.61803
0 −1.23607 0 −1.00000 0 0 0 −1.47214 0
1.2 0 3.23607 0 −1.00000 0 0 0 7.47214 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
$$2$$ $$-1$$
$$5$$ $$+1$$
$$13$$ $$+1$$

## 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 520.2.a.g 2
3.b odd 2 1 4680.2.a.bc 2
4.b odd 2 1 1040.2.a.i 2
5.b even 2 1 2600.2.a.o 2
5.c odd 4 2 2600.2.d.j 4
8.b even 2 1 4160.2.a.x 2
8.d odd 2 1 4160.2.a.bm 2
12.b even 2 1 9360.2.a.cs 2
13.b even 2 1 6760.2.a.t 2
20.d odd 2 1 5200.2.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.g 2 1.a even 1 1 trivial
1040.2.a.i 2 4.b odd 2 1
2600.2.a.o 2 5.b even 2 1
2600.2.d.j 4 5.c odd 4 2
4160.2.a.x 2 8.b even 2 1
4160.2.a.bm 2 8.d odd 2 1
4680.2.a.bc 2 3.b odd 2 1
5200.2.a.bz 2 20.d odd 2 1
6760.2.a.t 2 13.b even 2 1
9360.2.a.cs 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(520))$$:

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{11}^{2} - 6T_{11} + 4$$ T11^2 - 6*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} - 6T + 4$$
$23$ $$T^{2} + 2T - 4$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$T^{2} - 2T - 44$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} + 4T - 76$$
$43$ $$T^{2} + 2T - 4$$
$47$ $$T^{2} - 8T - 64$$
$53$ $$T^{2} + 4T - 76$$
$59$ $$T^{2} - 2T - 44$$
$61$ $$T^{2} - 20$$
$67$ $$T^{2} - 12T + 16$$
$71$ $$T^{2} + 6T + 4$$
$73$ $$T^{2} + 24T + 124$$
$79$ $$T^{2} - 4T - 16$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$(T + 10)^{2}$$