# Properties

 Label 520.2.k.a Level $520$ Weight $2$ Character orbit 520.k Analytic conductor $4.152$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(441,520)]

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, 1]))

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

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

chi := DirichletCharacter("520.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.k (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: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q - b1 * q^3 - b3 * q^5 + (b5 - b4 + b3) * q^7 + (b2 - b1) * q^9 $$q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{2} - \beta_1) q^{9} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{11} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{13} - \beta_{5} q^{15} + (2 \beta_{2} - 2 \beta_1 + 2) q^{17} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{19} + (2 \beta_{5} + 2 \beta_{3}) q^{21} + (2 \beta_{2} - \beta_1 + 4) q^{23} - q^{25} + (2 \beta_1 + 2) q^{27} + (3 \beta_{2} + 3 \beta_1 - 1) q^{29} + ( - 5 \beta_{4} + 3 \beta_{3}) q^{31} + ( - 3 \beta_{5} + 3 \beta_{4} - 7 \beta_{3}) q^{33} + (\beta_{2} - \beta_1 + 1) q^{35} + (\beta_{5} + \beta_{4} - 3 \beta_{3}) q^{37} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{39}+ \cdots + ( - 4 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{99}+O(q^{100})$$ q - b1 * q^3 - b3 * q^5 + (b5 - b4 + b3) * q^7 + (b2 - b1) * q^9 + (-2*b5 - b4 - b3) * q^11 + (b4 + 2*b2 - b1 + 2) * q^13 - b5 * q^15 + (2*b2 - 2*b1 + 2) * q^17 + (2*b5 - b4 + b3) * q^19 + (2*b5 + 2*b3) * q^21 + (2*b2 - b1 + 4) * q^23 - q^25 + (2*b1 + 2) * q^27 + (3*b2 + 3*b1 - 1) * q^29 + (-5*b4 + 3*b3) * q^31 + (-3*b5 + 3*b4 - 7*b3) * q^33 + (b2 - b1 + 1) * q^35 + (b5 + b4 - 3*b3) * q^37 + (-b4 + b3 - b2 - 3*b1 + 1) * q^39 + (-4*b5 - 2*b4) * q^41 + (3*b1 - 6) * q^43 + (-b5 + b4) * q^45 + (3*b5 - 3*b4 + 3*b3) * q^47 + (2*b1 + 3) * q^49 + (-4*b1 + 4) * q^51 + (2*b2 + 4*b1) * q^53 + (b2 + 2*b1 - 1) * q^55 + (3*b5 - b4 + 5*b3) * q^57 + (b4 - b3) * q^59 + (-3*b2 - b1 - 7) * q^61 + (b5 + b4 + 3*b3) * q^63 + (-b5 + 2*b4 - 2*b3 - b2) * q^65 + (b5 + b4 + 3*b3) * q^67 + (-b2 - 5*b1 + 1) * q^69 + (6*b5 - 5*b4 + b3) * q^71 + (-5*b5 + 3*b4 - b3) * q^73 + b1 * q^75 + (2*b2 - 6*b1 + 4) * q^77 + (-6*b2 + 2*b1 + 2) * q^79 + (-5*b2 + 3*b1 - 6) * q^81 + (-b5 + 7*b4 - 7*b3) * q^83 + (-2*b5 + 2*b4 - 2*b3) * q^85 + (-6*b2 + 4*b1 - 12) * q^87 + (2*b5 - 2*b4 - 2*b3) * q^89 + (2*b5 + 6*b3 - b2 + b1 + 1) * q^91 + (3*b5 + 5*b4 - 5*b3) * q^93 + (b2 - 2*b1 + 1) * q^95 + (-6*b5 + 2*b4) * q^97 + (-4*b5 + 3*b4 - 3*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{9}+O(q^{10})$$ 6 * q - 2 * q^9 $$6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77} + 24 q^{79} - 26 q^{81} - 60 q^{87} + 8 q^{91} + 4 q^{95}+O(q^{100})$$ 6 * q - 2 * q^9 + 8 * q^13 + 8 * q^17 + 20 * q^23 - 6 * q^25 + 12 * q^27 - 12 * q^29 + 4 * q^35 + 8 * q^39 - 36 * q^43 + 18 * q^49 + 24 * q^51 - 4 * q^53 - 8 * q^55 - 36 * q^61 + 2 * q^65 + 8 * q^69 + 20 * q^77 + 24 * q^79 - 26 * q^81 - 60 * q^87 + 8 * q^91 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
441.1
 0.403032 + 0.403032i 0.403032 − 0.403032i −0.854638 − 0.854638i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i
0 −1.67513 0 1.00000i 0 0.806063i 0 −0.193937 0
441.2 0 −1.67513 0 1.00000i 0 0.806063i 0 −0.193937 0
441.3 0 −0.539189 0 1.00000i 0 1.70928i 0 −2.70928 0
441.4 0 −0.539189 0 1.00000i 0 1.70928i 0 −2.70928 0
441.5 0 2.21432 0 1.00000i 0 2.90321i 0 1.90321 0
441.6 0 2.21432 0 1.00000i 0 2.90321i 0 1.90321 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 441.6 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.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.k.a 6
3.b odd 2 1 4680.2.g.j 6
4.b odd 2 1 1040.2.k.b 6
5.b even 2 1 2600.2.k.b 6
5.c odd 4 1 2600.2.f.c 6
5.c odd 4 1 2600.2.f.d 6
13.b even 2 1 inner 520.2.k.a 6
13.d odd 4 1 6760.2.a.u 3
13.d odd 4 1 6760.2.a.v 3
39.d odd 2 1 4680.2.g.j 6
52.b odd 2 1 1040.2.k.b 6
65.d even 2 1 2600.2.k.b 6
65.h odd 4 1 2600.2.f.c 6
65.h odd 4 1 2600.2.f.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.k.a 6 1.a even 1 1 trivial
520.2.k.a 6 13.b even 2 1 inner
1040.2.k.b 6 4.b odd 2 1
1040.2.k.b 6 52.b odd 2 1
2600.2.f.c 6 5.c odd 4 1
2600.2.f.c 6 65.h odd 4 1
2600.2.f.d 6 5.c odd 4 1
2600.2.f.d 6 65.h odd 4 1
2600.2.k.b 6 5.b even 2 1
2600.2.k.b 6 65.d even 2 1
4680.2.g.j 6 3.b odd 2 1
4680.2.g.j 6 39.d odd 2 1
6760.2.a.u 3 13.d odd 4 1
6760.2.a.v 3 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 4T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{3} - 4 T - 2)^{2}$$
$5$ $$(T^{2} + 1)^{3}$$
$7$ $$T^{6} + 12 T^{4} + \cdots + 16$$
$11$ $$T^{6} + 52 T^{4} + \cdots + 2116$$
$13$ $$T^{6} - 8 T^{5} + \cdots + 2197$$
$17$ $$(T^{3} - 4 T^{2} - 16 T + 32)^{2}$$
$19$ $$T^{6} + 32 T^{4} + \cdots + 100$$
$23$ $$(T^{3} - 10 T^{2} + \cdots + 26)^{2}$$
$29$ $$(T^{3} + 6 T^{2} + \cdots - 428)^{2}$$
$31$ $$T^{6} + 172 T^{4} + \cdots + 13924$$
$37$ $$T^{6} + 40 T^{4} + \cdots + 16$$
$41$ $$T^{6} + 188 T^{4} + \cdots + 40000$$
$43$ $$(T^{3} + 18 T^{2} + \cdots + 54)^{2}$$
$47$ $$T^{6} + 108 T^{4} + \cdots + 11664$$
$53$ $$(T^{3} + 2 T^{2} + \cdots - 200)^{2}$$
$59$ $$T^{6} + 8 T^{4} + \cdots + 4$$
$61$ $$(T^{3} + 18 T^{2} + \cdots + 52)^{2}$$
$67$ $$T^{6} + 52 T^{4} + \cdots + 16$$
$71$ $$T^{6} + 336 T^{4} + \cdots + 75076$$
$73$ $$T^{6} + 200 T^{4} + \cdots + 71824$$
$79$ $$(T^{3} - 12 T^{2} + \cdots - 32)^{2}$$
$83$ $$T^{6} + 372 T^{4} + \cdots + 929296$$
$89$ $$T^{6} + 64 T^{4} + \cdots + 1024$$
$97$ $$T^{6} + 268 T^{4} + \cdots + 462400$$