# Properties

 Label 507.2.a.e Level $507$ Weight $2$ Character orbit 507.a Self dual yes 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(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$4.04841538248$$ 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: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
0 −1.00000 −2.00000 −3.46410 0 −1.73205 0 1.00000 0
1.2 0 −1.00000 −2.00000 3.46410 0 1.73205 0 1.00000 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
$$3$$ $$+1$$
$$13$$ $$-1$$

## 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 507.2.a.e 2
3.b odd 2 1 1521.2.a.h 2
4.b odd 2 1 8112.2.a.bu 2
13.b even 2 1 inner 507.2.a.e 2
13.c even 3 2 507.2.e.f 4
13.d odd 4 2 507.2.b.c 2
13.e even 6 2 507.2.e.f 4
13.f odd 12 2 39.2.j.a 2
13.f odd 12 2 507.2.j.b 2
39.d odd 2 1 1521.2.a.h 2
39.f even 4 2 1521.2.b.f 2
39.k even 12 2 117.2.q.a 2
52.b odd 2 1 8112.2.a.bu 2
52.l even 12 2 624.2.bv.b 2
65.o even 12 2 975.2.w.d 4
65.s odd 12 2 975.2.bc.c 2
65.t even 12 2 975.2.w.d 4
156.v odd 12 2 1872.2.by.f 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

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

## Hecke characteristic polynomials

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