# Properties

 Label 520.2.d.b Level $520$ Weight $2$ Character orbit 520.d 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(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: $$6$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_{3}) q^{3} + (\beta_{4} + \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + (b4 - b3) * q^3 + (b4 + b1) * q^5 + (b5 + b4 - b3) * q^7 + (-b2 - b1) * q^9 $$q + (\beta_{4} - \beta_{3}) q^{3} + (\beta_{4} + \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{2} - \beta_1) q^{9} - \beta_1 q^{11} + \beta_{3} q^{13} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 2) q^{15}+ \cdots + (2 \beta_{2} - \beta_1 + 4) q^{99}+O(q^{100})$$ q + (b4 - b3) * q^3 + (b4 + b1) * q^5 + (b5 + b4 - b3) * q^7 + (-b2 - b1) * q^9 - b1 * q^11 + b3 * q^13 + (b5 + b4 - b3 - b1 - 2) * q^15 + (-2*b5 + 2*b3) * q^17 + (2*b2 - b1 + 4) * q^19 + (-2*b2 - 2*b1 - 4) * q^21 + (2*b5 - 3*b4 + 5*b3) * q^23 + (2*b4 - 2*b3 + 2*b2 - 2*b1 + 1) * q^25 + (-2*b5 + 2*b4) * q^27 + (b2 - b1 + 5) * q^29 + (2*b2 - 3*b1 - 2) * q^31 + (-b5 - b4 + b3) * q^33 + (2*b4 - 4*b3 - b2 - b1 - 3) * q^35 + (-3*b5 + b4 + b3) * q^37 + (b2 + 1) * q^39 + (-2*b2 + 2*b1 - 2) * q^41 + (b4 + b3) * q^43 + (-b5 + 3*b3 - 2*b2 + b1 - 4) * q^45 + (7*b5 - b4 - 3*b3) * q^47 + (-4*b2 - 2*b1 - 1) * q^49 + (4*b2 + 2*b1 + 4) * q^51 + (-4*b5 + 2*b4 - 6*b3) * q^53 + (-b4 + b3 - b2 + b1 - 3) * q^55 + (b5 + 3*b4 - 7*b3) * q^57 + (-4*b2 + 3*b1 - 2) * q^59 + (-3*b2 + b1 - 3) * q^61 + (-b5 - 3*b4 + 7*b3) * q^63 + (-b5 + b2) * q^65 + (-b5 - b4 + 5*b3) * q^67 + (3*b2 + b1 + 9) * q^69 + (4*b2 + 3*b1 - 2) * q^71 + (5*b5 - 3*b4 - 3*b3) * q^73 + (-b4 - 3*b3 - 2*b2 - 2*b1 - 6) * q^75 + (-2*b4 + 4*b3) * q^77 + (2*b2 + 10) * q^79 + (-b2 - 3*b1 - 2) * q^81 + (-b5 - 3*b4 + 7*b3) * q^83 + (-2*b4 + 6*b3 + 4*b2 + 2) * q^85 + (4*b4 - 6*b3) * q^87 + (-4*b2 + 4*b1 + 6) * q^89 + (b2 + b1 + 1) * q^91 + (-b5 - 5*b4 + b3) * q^93 + (2*b5 + b4 - 3*b3 + b2 + 5*b1 - 1) * q^95 + (6*b5 - 2*b3) * q^97 + (2*b2 - b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{9}+O(q^{10})$$ 6 * q + 2 * q^9 $$6 q + 2 q^{9} - 12 q^{15} + 20 q^{19} - 20 q^{21} + 2 q^{25} + 28 q^{29} - 16 q^{31} - 16 q^{35} + 4 q^{39} - 8 q^{41} - 20 q^{45} + 2 q^{49} + 16 q^{51} - 16 q^{55} - 4 q^{59} - 12 q^{61} - 2 q^{65} + 48 q^{69} - 20 q^{71} - 32 q^{75} + 56 q^{79} - 10 q^{81} + 4 q^{85} + 44 q^{89} + 4 q^{91} - 8 q^{95} + 20 q^{99}+O(q^{100})$$ 6 * q + 2 * q^9 - 12 * q^15 + 20 * q^19 - 20 * q^21 + 2 * q^25 + 28 * q^29 - 16 * q^31 - 16 * q^35 + 4 * q^39 - 8 * q^41 - 20 * q^45 + 2 * q^49 + 16 * q^51 - 16 * q^55 - 4 * q^59 - 12 * q^61 - 2 * q^65 + 48 * q^69 - 20 * q^71 - 32 * q^75 + 56 * q^79 - 10 * q^81 + 4 * q^85 + 44 * q^89 + 4 * q^91 - 8 * q^95 + 20 * q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0.403032 + 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 − 0.403032i
0 2.48119i 0 1.67513 1.48119i 0 4.15633i 0 −3.15633 0
209.2 0 1.17009i 0 0.539189 2.17009i 0 0.630898i 0 1.63090 0
209.3 0 0.688892i 0 −2.21432 + 0.311108i 0 1.52543i 0 2.52543 0
209.4 0 0.688892i 0 −2.21432 0.311108i 0 1.52543i 0 2.52543 0
209.5 0 1.17009i 0 0.539189 + 2.17009i 0 0.630898i 0 1.63090 0
209.6 0 2.48119i 0 1.67513 + 1.48119i 0 4.15633i 0 −3.15633 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.b 6
3.b odd 2 1 4680.2.l.d 6
4.b odd 2 1 1040.2.d.e 6
5.b even 2 1 inner 520.2.d.b 6
5.c odd 4 1 2600.2.a.x 3
5.c odd 4 1 2600.2.a.y 3
15.d odd 2 1 4680.2.l.d 6
20.d odd 2 1 1040.2.d.e 6
20.e even 4 1 5200.2.a.cc 3
20.e even 4 1 5200.2.a.ch 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.b 6 1.a even 1 1 trivial
520.2.d.b 6 5.b even 2 1 inner
1040.2.d.e 6 4.b odd 2 1
1040.2.d.e 6 20.d odd 2 1
2600.2.a.x 3 5.c odd 4 1
2600.2.a.y 3 5.c odd 4 1
4680.2.l.d 6 3.b odd 2 1
4680.2.l.d 6 15.d odd 2 1
5200.2.a.cc 3 20.e even 4 1
5200.2.a.ch 3 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 8T_{3}^{4} + 12T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 8 T^{4} + \cdots + 4$$
$5$ $$T^{6} - T^{4} + \cdots + 125$$
$7$ $$T^{6} + 20 T^{4} + \cdots + 16$$
$11$ $$(T^{3} - 4 T - 2)^{2}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 44 T^{4} + \cdots + 1600$$
$19$ $$(T^{3} - 10 T^{2} + \cdots + 26)^{2}$$
$23$ $$T^{6} + 116 T^{4} + \cdots + 17956$$
$29$ $$(T^{3} - 14 T^{2} + \cdots - 76)^{2}$$
$31$ $$(T^{3} + 8 T^{2} + \cdots - 130)^{2}$$
$37$ $$T^{6} + 72 T^{4} + \cdots + 13456$$
$41$ $$(T^{3} + 4 T^{2} - 16 T - 32)^{2}$$
$43$ $$T^{6} + 12 T^{4} + \cdots + 4$$
$47$ $$T^{6} + 404 T^{4} + \cdots + 2085136$$
$53$ $$T^{6} + 208 T^{4} + \cdots + 256$$
$59$ $$(T^{3} + 2 T^{2} + \cdots - 178)^{2}$$
$61$ $$(T^{3} + 6 T^{2} + \cdots - 100)^{2}$$
$67$ $$T^{6} + 84 T^{4} + \cdots + 4624$$
$71$ $$(T^{3} + 10 T^{2} + \cdots - 802)^{2}$$
$73$ $$T^{6} + 248 T^{4} + \cdots + 364816$$
$79$ $$(T^{3} - 28 T^{2} + \cdots - 688)^{2}$$
$83$ $$T^{6} + 188 T^{4} + \cdots + 2704$$
$89$ $$(T^{3} - 22 T^{2} + \cdots + 184)^{2}$$
$97$ $$T^{6} + 300 T^{4} + \cdots + 506944$$