# Properties

 Label 507.1.i.a Level $507$ Weight $1$ Character orbit 507.i Analytic conductor $0.253$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,1,Mod(146,507)]

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([3, 2]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("507.146");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 507.i (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: $$0.253025961405$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $C_3\times D_4$ Artin field: Galois closure of 12.6.48388330638999.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^3 - z * q^4 - z * q^9 $$q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{9} - q^{12} + \zeta_{6}^{2} q^{16} + q^{25} - q^{27} + \zeta_{6}^{2} q^{36} + \zeta_{6} q^{43} + \zeta_{6} q^{48} - \zeta_{6}^{2} q^{49} + \zeta_{6} q^{61} + q^{64} - \zeta_{6}^{2} q^{75} - q^{79} + \zeta_{6}^{2} q^{81} +O(q^{100})$$ q - z^2 * q^3 - z * q^4 - z * q^9 - q^12 + z^2 * q^16 + q^25 - q^27 + z^2 * q^36 + z * q^43 + z * q^48 - z^2 * q^49 + z * q^61 + q^64 - z^2 * q^75 - q^79 + z^2 * q^81 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{4} - q^{9}+O(q^{10})$$ 2 * q + q^3 - q^4 - q^9 $$2 q + q^{3} - q^{4} - q^{9} - 2 q^{12} - q^{16} + 2 q^{25} - 2 q^{27} - q^{36} + 2 q^{43} + q^{48} + q^{49} + 2 q^{61} + 2 q^{64} + q^{75} - 4 q^{79} - q^{81}+O(q^{100})$$ 2 * q + q^3 - q^4 - q^9 - 2 * q^12 - q^16 + 2 * q^25 - 2 * q^27 - q^36 + 2 * q^43 + q^48 + q^49 + 2 * q^61 + 2 * q^64 + q^75 - 4 * q^79 - q^81

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0 0 −0.500000 + 0.866025i 0
191.1 0 0.500000 0.866025i −0.500000 0.866025i 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
13.c even 3 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
39.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.i.a 2
3.b odd 2 1 CM 507.1.i.a 2
13.b even 2 1 RM 507.1.i.a 2
13.c even 3 1 507.1.c.a 1
13.c even 3 1 inner 507.1.i.a 2
13.d odd 4 2 507.1.h.a 2
13.e even 6 1 507.1.c.a 1
13.e even 6 1 inner 507.1.i.a 2
13.f odd 12 2 39.1.d.a 1
13.f odd 12 2 507.1.h.a 2
39.d odd 2 1 CM 507.1.i.a 2
39.f even 4 2 507.1.h.a 2
39.h odd 6 1 507.1.c.a 1
39.h odd 6 1 inner 507.1.i.a 2
39.i odd 6 1 507.1.c.a 1
39.i odd 6 1 inner 507.1.i.a 2
39.k even 12 2 39.1.d.a 1
39.k even 12 2 507.1.h.a 2
52.l even 12 2 624.1.l.a 1
65.o even 12 2 975.1.e.a 2
65.s odd 12 2 975.1.g.a 1
65.t even 12 2 975.1.e.a 2
91.w even 12 2 1911.1.w.a 2
91.x odd 12 2 1911.1.w.b 2
91.ba even 12 2 1911.1.w.a 2
91.bc even 12 2 1911.1.h.a 1
91.bd odd 12 2 1911.1.w.b 2
104.u even 12 2 2496.1.l.a 1
104.x odd 12 2 2496.1.l.b 1
117.w odd 12 2 1053.1.n.b 2
117.x even 12 2 1053.1.n.b 2
117.bb odd 12 2 1053.1.n.b 2
117.bc even 12 2 1053.1.n.b 2
156.v odd 12 2 624.1.l.a 1
195.bc odd 12 2 975.1.e.a 2
195.bh even 12 2 975.1.g.a 1
195.bn odd 12 2 975.1.e.a 2
273.bs odd 12 2 1911.1.w.a 2
273.bv even 12 2 1911.1.w.b 2
273.bw even 12 2 1911.1.w.b 2
273.ca odd 12 2 1911.1.h.a 1
273.ch odd 12 2 1911.1.w.a 2
312.bo even 12 2 2496.1.l.b 1
312.bq odd 12 2 2496.1.l.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 13.f odd 12 2
39.1.d.a 1 39.k even 12 2
507.1.c.a 1 13.c even 3 1
507.1.c.a 1 13.e even 6 1
507.1.c.a 1 39.h odd 6 1
507.1.c.a 1 39.i odd 6 1
507.1.h.a 2 13.d odd 4 2
507.1.h.a 2 13.f odd 12 2
507.1.h.a 2 39.f even 4 2
507.1.h.a 2 39.k even 12 2
507.1.i.a 2 1.a even 1 1 trivial
507.1.i.a 2 3.b odd 2 1 CM
507.1.i.a 2 13.b even 2 1 RM
507.1.i.a 2 13.c even 3 1 inner
507.1.i.a 2 13.e even 6 1 inner
507.1.i.a 2 39.d odd 2 1 CM
507.1.i.a 2 39.h odd 6 1 inner
507.1.i.a 2 39.i odd 6 1 inner
624.1.l.a 1 52.l even 12 2
624.1.l.a 1 156.v odd 12 2
975.1.e.a 2 65.o even 12 2
975.1.e.a 2 65.t even 12 2
975.1.e.a 2 195.bc odd 12 2
975.1.e.a 2 195.bn odd 12 2
975.1.g.a 1 65.s odd 12 2
975.1.g.a 1 195.bh even 12 2
1053.1.n.b 2 117.w odd 12 2
1053.1.n.b 2 117.x even 12 2
1053.1.n.b 2 117.bb odd 12 2
1053.1.n.b 2 117.bc even 12 2
1911.1.h.a 1 91.bc even 12 2
1911.1.h.a 1 273.ca odd 12 2
1911.1.w.a 2 91.w even 12 2
1911.1.w.a 2 91.ba even 12 2
1911.1.w.a 2 273.bs odd 12 2
1911.1.w.a 2 273.ch odd 12 2
1911.1.w.b 2 91.x odd 12 2
1911.1.w.b 2 91.bd odd 12 2
1911.1.w.b 2 273.bv even 12 2
1911.1.w.b 2 273.bw even 12 2
2496.1.l.a 1 104.u even 12 2
2496.1.l.a 1 312.bq odd 12 2
2496.1.l.b 1 104.x odd 12 2
2496.1.l.b 1 312.bo even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(507, [\chi])$$.

## Hecke characteristic polynomials

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