# Properties

 Label 520.2.a.f 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: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ q + b * q^3 + q^5 + 2 * q^7 + 3 * q^9 $$q + \beta q^{3} + q^{5} + 2 q^{7} + 3 q^{9} + (\beta - 2) q^{11} - q^{13} + \beta q^{15} + ( - 2 \beta + 2) q^{17} + ( - \beta - 2) q^{19} + 2 \beta q^{21} + ( - \beta + 4) q^{23} + q^{25} + 4 q^{29} + (\beta - 2) q^{31} + ( - 2 \beta + 6) q^{33} + 2 q^{35} - 2 \beta q^{37} - \beta q^{39} + ( - 2 \beta + 6) q^{41} + ( - 3 \beta + 4) q^{43} + 3 q^{45} - 2 q^{47} - 3 q^{49} + (2 \beta - 12) q^{51} + (2 \beta + 6) q^{53} + (\beta - 2) q^{55} + ( - 2 \beta - 6) q^{57} + ( - 3 \beta + 2) q^{59} + (4 \beta + 4) q^{61} + 6 q^{63} - q^{65} + ( - 2 \beta - 10) q^{67} + (4 \beta - 6) q^{69} + ( - \beta - 6) q^{71} + 6 \beta q^{73} + \beta q^{75} + (2 \beta - 4) q^{77} - 2 \beta q^{79} - 9 q^{81} + 2 q^{83} + ( - 2 \beta + 2) q^{85} + 4 \beta q^{87} + 6 q^{89} - 2 q^{91} + ( - 2 \beta + 6) q^{93} + ( - \beta - 2) q^{95} + ( - 4 \beta - 2) q^{97} + (3 \beta - 6) q^{99} +O(q^{100})$$ q + b * q^3 + q^5 + 2 * q^7 + 3 * q^9 + (b - 2) * q^11 - q^13 + b * q^15 + (-2*b + 2) * q^17 + (-b - 2) * q^19 + 2*b * q^21 + (-b + 4) * q^23 + q^25 + 4 * q^29 + (b - 2) * q^31 + (-2*b + 6) * q^33 + 2 * q^35 - 2*b * q^37 - b * q^39 + (-2*b + 6) * q^41 + (-3*b + 4) * q^43 + 3 * q^45 - 2 * q^47 - 3 * q^49 + (2*b - 12) * q^51 + (2*b + 6) * q^53 + (b - 2) * q^55 + (-2*b - 6) * q^57 + (-3*b + 2) * q^59 + (4*b + 4) * q^61 + 6 * q^63 - q^65 + (-2*b - 10) * q^67 + (4*b - 6) * q^69 + (-b - 6) * q^71 + 6*b * q^73 + b * q^75 + (2*b - 4) * q^77 - 2*b * q^79 - 9 * q^81 + 2 * q^83 + (-2*b + 2) * q^85 + 4*b * q^87 + 6 * q^89 - 2 * q^91 + (-2*b + 6) * q^93 + (-b - 2) * q^95 + (-4*b - 2) * q^97 + (3*b - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9 $$2 q + 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} + 12 q^{33} + 4 q^{35} + 12 q^{41} + 8 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 24 q^{51} + 12 q^{53} - 4 q^{55} - 12 q^{57} + 4 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{77} - 18 q^{81} + 4 q^{83} + 4 q^{85} + 12 q^{89} - 4 q^{91} + 12 q^{93} - 4 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^29 - 4 * q^31 + 12 * q^33 + 4 * q^35 + 12 * q^41 + 8 * q^43 + 6 * q^45 - 4 * q^47 - 6 * q^49 - 24 * q^51 + 12 * q^53 - 4 * q^55 - 12 * q^57 + 4 * q^59 + 8 * q^61 + 12 * q^63 - 2 * q^65 - 20 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^77 - 18 * q^81 + 4 * q^83 + 4 * q^85 + 12 * q^89 - 4 * q^91 + 12 * q^93 - 4 * q^95 - 4 * q^97 - 12 * 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
 −2.44949 2.44949
0 −2.44949 0 1.00000 0 2.00000 0 3.00000 0
1.2 0 2.44949 0 1.00000 0 2.00000 0 3.00000 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.f 2
3.b odd 2 1 4680.2.a.z 2
4.b odd 2 1 1040.2.a.k 2
5.b even 2 1 2600.2.a.q 2
5.c odd 4 2 2600.2.d.h 4
8.b even 2 1 4160.2.a.bg 2
8.d odd 2 1 4160.2.a.ba 2
12.b even 2 1 9360.2.a.cc 2
13.b even 2 1 6760.2.a.s 2
20.d odd 2 1 5200.2.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.f 2 1.a even 1 1 trivial
1040.2.a.k 2 4.b odd 2 1
2600.2.a.q 2 5.b even 2 1
2600.2.d.h 4 5.c odd 4 2
4160.2.a.ba 2 8.d odd 2 1
4160.2.a.bg 2 8.b even 2 1
4680.2.a.z 2 3.b odd 2 1
5200.2.a.bv 2 20.d odd 2 1
6760.2.a.s 2 13.b even 2 1
9360.2.a.cc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(520))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 6$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 4T - 2$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 20$$
$19$ $$T^{2} + 4T - 2$$
$23$ $$T^{2} - 8T + 10$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} + 4T - 2$$
$37$ $$T^{2} - 24$$
$41$ $$T^{2} - 12T + 12$$
$43$ $$T^{2} - 8T - 38$$
$47$ $$(T + 2)^{2}$$
$53$ $$T^{2} - 12T + 12$$
$59$ $$T^{2} - 4T - 50$$
$61$ $$T^{2} - 8T - 80$$
$67$ $$T^{2} + 20T + 76$$
$71$ $$T^{2} + 12T + 30$$
$73$ $$T^{2} - 216$$
$79$ $$T^{2} - 24$$
$83$ $$(T - 2)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4T - 92$$