# Properties

 Label 507.2.j.b Level $507$ Weight $2$ Character orbit 507.j Analytic conductor $4.048$ 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] = [507,2,Mod(316,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("507.316");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.j (of order $$6$$, degree $$2$$, not 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: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - 2*z * q^4 + (-4*z + 2) * q^5 + (-z + 2) * q^7 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} - 2 q^{12} + ( - 2 \zeta_{6} - 2) q^{15} + (4 \zeta_{6} - 4) q^{16} + (2 \zeta_{6} - 4) q^{19} + (4 \zeta_{6} - 8) q^{20} + ( - 2 \zeta_{6} + 1) q^{21} + ( - 6 \zeta_{6} + 6) q^{23} - 7 q^{25} - q^{27} + ( - 2 \zeta_{6} - 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + ( - 2 \zeta_{6} + 1) q^{31} + ( - 2 \zeta_{6} + 4) q^{33} - 6 \zeta_{6} q^{35} + (2 \zeta_{6} - 2) q^{36} + (4 \zeta_{6} + 4) q^{41} + \zeta_{6} q^{43} + ( - 8 \zeta_{6} + 4) q^{44} + (2 \zeta_{6} - 4) q^{45} + ( - 4 \zeta_{6} + 2) q^{47} + 4 \zeta_{6} q^{48} + (4 \zeta_{6} - 4) q^{49} + 12 q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + (4 \zeta_{6} - 2) q^{57} + ( - 2 \zeta_{6} + 4) q^{59} + (8 \zeta_{6} - 4) q^{60} - \zeta_{6} q^{61} + ( - \zeta_{6} - 1) q^{63} + 8 q^{64} + ( - 5 \zeta_{6} - 5) q^{67} - 6 \zeta_{6} q^{69} + ( - 6 \zeta_{6} + 12) q^{71} + ( - 2 \zeta_{6} + 1) q^{73} + (7 \zeta_{6} - 7) q^{75} + (4 \zeta_{6} + 4) q^{76} + 6 q^{77} - 11 q^{79} + (8 \zeta_{6} + 8) q^{80} + (\zeta_{6} - 1) q^{81} + (16 \zeta_{6} - 8) q^{83} + (2 \zeta_{6} - 4) q^{84} + 6 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} - 12 q^{92} + ( - \zeta_{6} - 1) q^{93} + 12 \zeta_{6} q^{95} + ( - 3 \zeta_{6} + 6) q^{97} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 - 2*z * q^4 + (-4*z + 2) * q^5 + (-z + 2) * q^7 - z * q^9 + (2*z + 2) * q^11 - 2 * q^12 + (-2*z - 2) * q^15 + (4*z - 4) * q^16 + (2*z - 4) * q^19 + (4*z - 8) * q^20 + (-2*z + 1) * q^21 + (-6*z + 6) * q^23 - 7 * q^25 - q^27 + (-2*z - 2) * q^28 + (6*z - 6) * q^29 + (-2*z + 1) * q^31 + (-2*z + 4) * q^33 - 6*z * q^35 + (2*z - 2) * q^36 + (4*z + 4) * q^41 + z * q^43 + (-8*z + 4) * q^44 + (2*z - 4) * q^45 + (-4*z + 2) * q^47 + 4*z * q^48 + (4*z - 4) * q^49 + 12 * q^53 + (-12*z + 12) * q^55 + (4*z - 2) * q^57 + (-2*z + 4) * q^59 + (8*z - 4) * q^60 - z * q^61 + (-z - 1) * q^63 + 8 * q^64 + (-5*z - 5) * q^67 - 6*z * q^69 + (-6*z + 12) * q^71 + (-2*z + 1) * q^73 + (7*z - 7) * q^75 + (4*z + 4) * q^76 + 6 * q^77 - 11 * q^79 + (8*z + 8) * q^80 + (z - 1) * q^81 + (16*z - 8) * q^83 + (2*z - 4) * q^84 + 6*z * q^87 + (-4*z - 4) * q^89 - 12 * q^92 + (-z - 1) * q^93 + 12*z * q^95 + (-3*z + 6) * q^97 + (-4*z + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{4} + 3 q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^4 + 3 * q^7 - q^9 $$2 q + q^{3} - 2 q^{4} + 3 q^{7} - q^{9} + 6 q^{11} - 4 q^{12} - 6 q^{15} - 4 q^{16} - 6 q^{19} - 12 q^{20} + 6 q^{23} - 14 q^{25} - 2 q^{27} - 6 q^{28} - 6 q^{29} + 6 q^{33} - 6 q^{35} - 2 q^{36} + 12 q^{41} + q^{43} - 6 q^{45} + 4 q^{48} - 4 q^{49} + 24 q^{53} + 12 q^{55} + 6 q^{59} - q^{61} - 3 q^{63} + 16 q^{64} - 15 q^{67} - 6 q^{69} + 18 q^{71} - 7 q^{75} + 12 q^{76} + 12 q^{77} - 22 q^{79} + 24 q^{80} - q^{81} - 6 q^{84} + 6 q^{87} - 12 q^{89} - 24 q^{92} - 3 q^{93} + 12 q^{95} + 9 q^{97}+O(q^{100})$$ 2 * q + q^3 - 2 * q^4 + 3 * q^7 - q^9 + 6 * q^11 - 4 * q^12 - 6 * q^15 - 4 * q^16 - 6 * q^19 - 12 * q^20 + 6 * q^23 - 14 * q^25 - 2 * q^27 - 6 * q^28 - 6 * q^29 + 6 * q^33 - 6 * q^35 - 2 * q^36 + 12 * q^41 + q^43 - 6 * q^45 + 4 * q^48 - 4 * q^49 + 24 * q^53 + 12 * q^55 + 6 * q^59 - q^61 - 3 * q^63 + 16 * q^64 - 15 * q^67 - 6 * q^69 + 18 * q^71 - 7 * q^75 + 12 * q^76 + 12 * q^77 - 22 * q^79 + 24 * q^80 - q^81 - 6 * q^84 + 6 * q^87 - 12 * q^89 - 24 * q^92 - 3 * q^93 + 12 * q^95 + 9 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i −1.00000 1.73205i 3.46410i 0 1.50000 0.866025i 0 −0.500000 0.866025i 0
361.1 0 0.500000 + 0.866025i −1.00000 + 1.73205i 3.46410i 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.b 2
13.b even 2 1 39.2.j.a 2
13.c even 3 1 39.2.j.a 2
13.c even 3 1 507.2.b.c 2
13.d odd 4 2 507.2.e.f 4
13.e even 6 1 507.2.b.c 2
13.e even 6 1 inner 507.2.j.b 2
13.f odd 12 2 507.2.a.e 2
13.f odd 12 2 507.2.e.f 4
39.d odd 2 1 117.2.q.a 2
39.h odd 6 1 1521.2.b.f 2
39.i odd 6 1 117.2.q.a 2
39.i odd 6 1 1521.2.b.f 2
39.k even 12 2 1521.2.a.h 2
52.b odd 2 1 624.2.bv.b 2
52.j odd 6 1 624.2.bv.b 2
52.l even 12 2 8112.2.a.bu 2
65.d even 2 1 975.2.bc.c 2
65.h odd 4 2 975.2.w.d 4
65.n even 6 1 975.2.bc.c 2
65.q odd 12 2 975.2.w.d 4
156.h even 2 1 1872.2.by.f 2
156.p even 6 1 1872.2.by.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 13.b even 2 1
39.2.j.a 2 13.c even 3 1
117.2.q.a 2 39.d odd 2 1
117.2.q.a 2 39.i odd 6 1
507.2.a.e 2 13.f odd 12 2
507.2.b.c 2 13.c even 3 1
507.2.b.c 2 13.e even 6 1
507.2.e.f 4 13.d odd 4 2
507.2.e.f 4 13.f odd 12 2
507.2.j.b 2 1.a even 1 1 trivial
507.2.j.b 2 13.e even 6 1 inner
624.2.bv.b 2 52.b odd 2 1
624.2.bv.b 2 52.j odd 6 1
975.2.w.d 4 65.h odd 4 2
975.2.w.d 4 65.q odd 12 2
975.2.bc.c 2 65.d even 2 1
975.2.bc.c 2 65.n even 6 1
1521.2.a.h 2 39.k even 12 2
1521.2.b.f 2 39.h odd 6 1
1521.2.b.f 2 39.i odd 6 1
1872.2.by.f 2 156.h even 2 1
1872.2.by.f 2 156.p even 6 1
8112.2.a.bu 2 52.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}^{2} + 12$$ T5^2 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 12$$
$7$ $$T^{2} - 3T + 3$$
$11$ $$T^{2} - 6T + 12$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 6T + 12$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 12T + 48$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 12$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} - 6T + 12$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 15T + 75$$
$71$ $$T^{2} - 18T + 108$$
$73$ $$T^{2} + 3$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} - 9T + 27$$