# Properties

 Label 520.2.d.a Level $520$ Weight $2$ Character orbit 520.d 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(209,520)]

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, 1, 0]))

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

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

chi := DirichletCharacter("520.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.d (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i + 2) q^{5} - 4 i q^{7} + 3 q^{9} +O(q^{10})$$ q + (i + 2) * q^5 - 4*i * q^7 + 3 * q^9 $$q + (i + 2) q^{5} - 4 i q^{7} + 3 q^{9} - 2 q^{11} + i q^{13} - 4 i q^{17} + 2 q^{19} - 2 i q^{23} + (4 i + 3) q^{25} - 6 q^{29} + 8 q^{31} + ( - 8 i + 4) q^{35} + 6 i q^{37} + 10 q^{41} + 4 i q^{43} + (3 i + 6) q^{45} - 9 q^{49} - 6 i q^{53} + ( - 2 i - 4) q^{55} - 6 q^{59} + 2 q^{61} - 12 i q^{63} + (2 i - 1) q^{65} - 4 i q^{67} - 12 q^{71} + 2 i q^{73} + 8 i q^{77} - 8 q^{79} + 9 q^{81} + 12 i q^{83} + ( - 8 i + 4) q^{85} - 14 q^{89} + 4 q^{91} + (2 i + 4) q^{95} + 10 i q^{97} - 6 q^{99} +O(q^{100})$$ q + (i + 2) * q^5 - 4*i * q^7 + 3 * q^9 - 2 * q^11 + i * q^13 - 4*i * q^17 + 2 * q^19 - 2*i * q^23 + (4*i + 3) * q^25 - 6 * q^29 + 8 * q^31 + (-8*i + 4) * q^35 + 6*i * q^37 + 10 * q^41 + 4*i * q^43 + (3*i + 6) * q^45 - 9 * q^49 - 6*i * q^53 + (-2*i - 4) * q^55 - 6 * q^59 + 2 * q^61 - 12*i * q^63 + (2*i - 1) * q^65 - 4*i * q^67 - 12 * q^71 + 2*i * q^73 + 8*i * q^77 - 8 * q^79 + 9 * q^81 + 12*i * q^83 + (-8*i + 4) * q^85 - 14 * q^89 + 4 * q^91 + (2*i + 4) * q^95 + 10*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 + 6 * q^9 $$2 q + 4 q^{5} + 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{25} - 12 q^{29} + 16 q^{31} + 8 q^{35} + 20 q^{41} + 12 q^{45} - 18 q^{49} - 8 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 24 q^{71} - 16 q^{79} + 18 q^{81} + 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 + 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^25 - 12 * q^29 + 16 * q^31 + 8 * q^35 + 20 * q^41 + 12 * q^45 - 18 * q^49 - 8 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 24 * q^71 - 16 * q^79 + 18 * q^81 + 8 * q^85 - 28 * q^89 + 8 * q^91 + 8 * q^95 - 12 * q^99

## Character values

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

 $$n$$ $$41$$ $$261$$ $$391$$ $$417$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 − 1.00000i 1.00000i
0 0 0 2.00000 1.00000i 0 4.00000i 0 3.00000 0
209.2 0 0 0 2.00000 + 1.00000i 0 4.00000i 0 3.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.d.a 2
3.b odd 2 1 4680.2.l.a 2
4.b odd 2 1 1040.2.d.a 2
5.b even 2 1 inner 520.2.d.a 2
5.c odd 4 1 2600.2.a.g 1
5.c odd 4 1 2600.2.a.i 1
15.d odd 2 1 4680.2.l.a 2
20.d odd 2 1 1040.2.d.a 2
20.e even 4 1 5200.2.a.o 1
20.e even 4 1 5200.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.a 2 1.a even 1 1 trivial
520.2.d.a 2 5.b even 2 1 inner
1040.2.d.a 2 4.b odd 2 1
1040.2.d.a 2 20.d odd 2 1
2600.2.a.g 1 5.c odd 4 1
2600.2.a.i 1 5.c odd 4 1
4680.2.l.a 2 3.b odd 2 1
4680.2.l.a 2 15.d odd 2 1
5200.2.a.o 1 20.e even 4 1
5200.2.a.u 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 100$$