# Properties

 Label 520.2.a.c Level $520$ Weight $2$ Character orbit 520.a Self dual yes Analytic conductor $4.152$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 2) q^{3} + q^{5} - 2 q^{7} + ( - 4 \beta + 3) q^{9}+O(q^{10})$$ q + (b - 2) * q^3 + q^5 - 2 * q^7 + (-4*b + 3) * q^9 $$q + (\beta - 2) q^{3} + q^{5} - 2 q^{7} + ( - 4 \beta + 3) q^{9} - 3 \beta q^{11} - q^{13} + (\beta - 2) q^{15} + (2 \beta - 2) q^{17} + (3 \beta - 4) q^{19} + ( - 2 \beta + 4) q^{21} + ( - 5 \beta - 2) q^{23} + q^{25} + (8 \beta - 8) q^{27} + (4 \beta - 4) q^{29} + \beta q^{31} + (6 \beta - 6) q^{33} - 2 q^{35} + ( - 2 \beta - 4) q^{37} + ( - \beta + 2) q^{39} + (2 \beta + 2) q^{41} + ( - 3 \beta - 6) q^{43} + ( - 4 \beta + 3) q^{45} + 2 q^{47} - 3 q^{49} + ( - 6 \beta + 8) q^{51} + ( - 2 \beta - 6) q^{53} - 3 \beta q^{55} + ( - 10 \beta + 14) q^{57} + (\beta - 4) q^{59} + (8 \beta + 4) q^{61} + (8 \beta - 6) q^{63} - q^{65} + ( - 2 \beta - 2) q^{67} + (8 \beta - 6) q^{69} + 7 \beta q^{71} + ( - 2 \beta + 4) q^{73} + (\beta - 2) q^{75} + 6 \beta q^{77} + ( - 2 \beta + 4) q^{79} + ( - 12 \beta + 23) q^{81} - 2 q^{83} + (2 \beta - 2) q^{85} + ( - 12 \beta + 16) q^{87} - 10 q^{89} + 2 q^{91} + ( - 2 \beta + 2) q^{93} + (3 \beta - 4) q^{95} + (4 \beta + 6) q^{97} + ( - 9 \beta + 24) q^{99} +O(q^{100})$$ q + (b - 2) * q^3 + q^5 - 2 * q^7 + (-4*b + 3) * q^9 - 3*b * q^11 - q^13 + (b - 2) * q^15 + (2*b - 2) * q^17 + (3*b - 4) * q^19 + (-2*b + 4) * q^21 + (-5*b - 2) * q^23 + q^25 + (8*b - 8) * q^27 + (4*b - 4) * q^29 + b * q^31 + (6*b - 6) * q^33 - 2 * q^35 + (-2*b - 4) * q^37 + (-b + 2) * q^39 + (2*b + 2) * q^41 + (-3*b - 6) * q^43 + (-4*b + 3) * q^45 + 2 * q^47 - 3 * q^49 + (-6*b + 8) * q^51 + (-2*b - 6) * q^53 - 3*b * q^55 + (-10*b + 14) * q^57 + (b - 4) * q^59 + (8*b + 4) * q^61 + (8*b - 6) * q^63 - q^65 + (-2*b - 2) * q^67 + (8*b - 6) * q^69 + 7*b * q^71 + (-2*b + 4) * q^73 + (b - 2) * q^75 + 6*b * q^77 + (-2*b + 4) * q^79 + (-12*b + 23) * q^81 - 2 * q^83 + (2*b - 2) * q^85 + (-12*b + 16) * q^87 - 10 * q^89 + 2 * q^91 + (-2*b + 2) * q^93 + (3*b - 4) * q^95 + (4*b + 6) * q^97 + (-9*b + 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 $$2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 2 q^{13} - 4 q^{15} - 4 q^{17} - 8 q^{19} + 8 q^{21} - 4 q^{23} + 2 q^{25} - 16 q^{27} - 8 q^{29} - 12 q^{33} - 4 q^{35} - 8 q^{37} + 4 q^{39} + 4 q^{41} - 12 q^{43} + 6 q^{45} + 4 q^{47} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 28 q^{57} - 8 q^{59} + 8 q^{61} - 12 q^{63} - 2 q^{65} - 4 q^{67} - 12 q^{69} + 8 q^{73} - 4 q^{75} + 8 q^{79} + 46 q^{81} - 4 q^{83} - 4 q^{85} + 32 q^{87} - 20 q^{89} + 4 q^{91} + 4 q^{93} - 8 q^{95} + 12 q^{97} + 48 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 - 2 * q^13 - 4 * q^15 - 4 * q^17 - 8 * q^19 + 8 * q^21 - 4 * q^23 + 2 * q^25 - 16 * q^27 - 8 * q^29 - 12 * q^33 - 4 * q^35 - 8 * q^37 + 4 * q^39 + 4 * q^41 - 12 * q^43 + 6 * q^45 + 4 * q^47 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 28 * q^57 - 8 * q^59 + 8 * q^61 - 12 * q^63 - 2 * q^65 - 4 * q^67 - 12 * q^69 + 8 * q^73 - 4 * q^75 + 8 * q^79 + 46 * q^81 - 4 * q^83 - 4 * q^85 + 32 * q^87 - 20 * q^89 + 4 * q^91 + 4 * q^93 - 8 * q^95 + 12 * q^97 + 48 * 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
 −1.41421 1.41421
0 −3.41421 0 1.00000 0 −2.00000 0 8.65685 0
1.2 0 −0.585786 0 1.00000 0 −2.00000 0 −2.65685 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.c 2
3.b odd 2 1 4680.2.a.w 2
4.b odd 2 1 1040.2.a.n 2
5.b even 2 1 2600.2.a.w 2
5.c odd 4 2 2600.2.d.i 4
8.b even 2 1 4160.2.a.bn 2
8.d odd 2 1 4160.2.a.u 2
12.b even 2 1 9360.2.a.ck 2
13.b even 2 1 6760.2.a.n 2
20.d odd 2 1 5200.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.c 2 1.a even 1 1 trivial
1040.2.a.n 2 4.b odd 2 1
2600.2.a.w 2 5.b even 2 1
2600.2.d.i 4 5.c odd 4 2
4160.2.a.u 2 8.d odd 2 1
4160.2.a.bn 2 8.b even 2 1
4680.2.a.w 2 3.b odd 2 1
5200.2.a.bl 2 20.d odd 2 1
6760.2.a.n 2 13.b even 2 1
9360.2.a.ck 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(520))$$:

 $$T_{3}^{2} + 4T_{3} + 2$$ T3^2 + 4*T3 + 2 $$T_{11}^{2} - 18$$ T11^2 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4T + 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 18$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$T^{2} + 8T - 2$$
$23$ $$T^{2} + 4T - 46$$
$29$ $$T^{2} + 8T - 16$$
$31$ $$T^{2} - 2$$
$37$ $$T^{2} + 8T + 8$$
$41$ $$T^{2} - 4T - 4$$
$43$ $$T^{2} + 12T + 18$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} + 12T + 28$$
$59$ $$T^{2} + 8T + 14$$
$61$ $$T^{2} - 8T - 112$$
$67$ $$T^{2} + 4T - 4$$
$71$ $$T^{2} - 98$$
$73$ $$T^{2} - 8T + 8$$
$79$ $$T^{2} - 8T + 8$$
$83$ $$(T + 2)^{2}$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 12T + 4$$