# Properties

 Label 1053.1.n.b Level $1053$ Weight $1$ Character orbit 1053.n Analytic conductor $0.526$ 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] = [1053,1,Mod(350,1053)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1053, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

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

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

chi := DirichletCharacter("1053.350");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1053 = 3^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1053.n (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: $$0.525515458303$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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 12.6.623334901029867.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{4}+O(q^{10})$$ q + z * q^4 $$q + \zeta_{6} q^{4} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - \zeta_{6}^{2} q^{25} + \zeta_{6}^{2} q^{43} - \zeta_{6} q^{49} + \zeta_{6}^{2} q^{52} - \zeta_{6}^{2} q^{61} - q^{64} - \zeta_{6}^{2} q^{79} +O(q^{100})$$ q + z * q^4 + z * q^13 + z^2 * q^16 - z^2 * q^25 + z^2 * q^43 - z * q^49 + z^2 * q^52 - z^2 * q^61 - q^64 - z^2 * q^79 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{4}+O(q^{10})$$ 2 * q + q^4 $$2 q + q^{4} + q^{13} - q^{16} + q^{25} - 2 q^{43} - q^{49} - q^{52} + 2 q^{61} - 2 q^{64} + 2 q^{79}+O(q^{100})$$ 2 * q + q^4 + q^13 - q^16 + q^25 - 2 * q^43 - q^49 - q^52 + 2 * q^61 - 2 * q^64 + 2 * q^79

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$730$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
350.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0.500000 + 0.866025i 0 0 0 0 0 0
701.1 0 0 0.500000 0.866025i 0 0 0 0 0 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 $$\Q(\sqrt{-39})$$
9.c even 3 1 inner
9.d odd 6 1 inner
117.n odd 6 1 inner
117.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1053.1.n.b 2
3.b odd 2 1 CM 1053.1.n.b 2
9.c even 3 1 39.1.d.a 1
9.c even 3 1 inner 1053.1.n.b 2
9.d odd 6 1 39.1.d.a 1
9.d odd 6 1 inner 1053.1.n.b 2
13.b even 2 1 RM 1053.1.n.b 2
36.f odd 6 1 624.1.l.a 1
36.h even 6 1 624.1.l.a 1
39.d odd 2 1 CM 1053.1.n.b 2
45.h odd 6 1 975.1.g.a 1
45.j even 6 1 975.1.g.a 1
45.k odd 12 2 975.1.e.a 2
45.l even 12 2 975.1.e.a 2
63.g even 3 1 1911.1.w.b 2
63.h even 3 1 1911.1.w.b 2
63.i even 6 1 1911.1.w.a 2
63.j odd 6 1 1911.1.w.b 2
63.k odd 6 1 1911.1.w.a 2
63.l odd 6 1 1911.1.h.a 1
63.n odd 6 1 1911.1.w.b 2
63.o even 6 1 1911.1.h.a 1
63.s even 6 1 1911.1.w.a 2
63.t odd 6 1 1911.1.w.a 2
72.j odd 6 1 2496.1.l.b 1
72.l even 6 1 2496.1.l.a 1
72.n even 6 1 2496.1.l.b 1
72.p odd 6 1 2496.1.l.a 1
117.f even 3 1 507.1.h.a 2
117.h even 3 1 507.1.h.a 2
117.k odd 6 1 507.1.h.a 2
117.l even 6 1 507.1.h.a 2
117.m odd 6 1 507.1.h.a 2
117.n odd 6 1 39.1.d.a 1
117.n odd 6 1 inner 1053.1.n.b 2
117.r even 6 1 507.1.h.a 2
117.t even 6 1 39.1.d.a 1
117.t even 6 1 inner 1053.1.n.b 2
117.u odd 6 1 507.1.h.a 2
117.v odd 6 1 507.1.h.a 2
117.w odd 12 2 507.1.i.a 2
117.x even 12 2 507.1.i.a 2
117.y odd 12 2 507.1.c.a 1
117.z even 12 2 507.1.c.a 1
117.bb odd 12 2 507.1.i.a 2
117.bc even 12 2 507.1.i.a 2
468.x even 6 1 624.1.l.a 1
468.bg odd 6 1 624.1.l.a 1
585.be even 6 1 975.1.g.a 1
585.bo odd 6 1 975.1.g.a 1
585.cs even 12 2 975.1.e.a 2
585.dm odd 12 2 975.1.e.a 2
819.bf odd 6 1 1911.1.w.a 2
819.bk even 6 1 1911.1.w.b 2
819.bt even 6 1 1911.1.w.a 2
819.ce even 6 1 1911.1.h.a 1
819.cp odd 6 1 1911.1.w.b 2
819.cq even 6 1 1911.1.w.b 2
819.cy odd 6 1 1911.1.h.a 1
819.dv odd 6 1 1911.1.w.a 2
819.ed odd 6 1 1911.1.w.b 2
819.eg even 6 1 1911.1.w.a 2
936.bs odd 6 1 2496.1.l.a 1
936.bx even 6 1 2496.1.l.b 1
936.cl even 6 1 2496.1.l.a 1
936.cv odd 6 1 2496.1.l.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 9.c even 3 1
39.1.d.a 1 9.d odd 6 1
39.1.d.a 1 117.n odd 6 1
39.1.d.a 1 117.t even 6 1
507.1.c.a 1 117.y odd 12 2
507.1.c.a 1 117.z even 12 2
507.1.h.a 2 117.f even 3 1
507.1.h.a 2 117.h even 3 1
507.1.h.a 2 117.k odd 6 1
507.1.h.a 2 117.l even 6 1
507.1.h.a 2 117.m odd 6 1
507.1.h.a 2 117.r even 6 1
507.1.h.a 2 117.u odd 6 1
507.1.h.a 2 117.v odd 6 1
507.1.i.a 2 117.w odd 12 2
507.1.i.a 2 117.x even 12 2
507.1.i.a 2 117.bb odd 12 2
507.1.i.a 2 117.bc even 12 2
624.1.l.a 1 36.f odd 6 1
624.1.l.a 1 36.h even 6 1
624.1.l.a 1 468.x even 6 1
624.1.l.a 1 468.bg odd 6 1
975.1.e.a 2 45.k odd 12 2
975.1.e.a 2 45.l even 12 2
975.1.e.a 2 585.cs even 12 2
975.1.e.a 2 585.dm odd 12 2
975.1.g.a 1 45.h odd 6 1
975.1.g.a 1 45.j even 6 1
975.1.g.a 1 585.be even 6 1
975.1.g.a 1 585.bo odd 6 1
1053.1.n.b 2 1.a even 1 1 trivial
1053.1.n.b 2 3.b odd 2 1 CM
1053.1.n.b 2 9.c even 3 1 inner
1053.1.n.b 2 9.d odd 6 1 inner
1053.1.n.b 2 13.b even 2 1 RM
1053.1.n.b 2 39.d odd 2 1 CM
1053.1.n.b 2 117.n odd 6 1 inner
1053.1.n.b 2 117.t even 6 1 inner
1911.1.h.a 1 63.l odd 6 1
1911.1.h.a 1 63.o even 6 1
1911.1.h.a 1 819.ce even 6 1
1911.1.h.a 1 819.cy odd 6 1
1911.1.w.a 2 63.i even 6 1
1911.1.w.a 2 63.k odd 6 1
1911.1.w.a 2 63.s even 6 1
1911.1.w.a 2 63.t odd 6 1
1911.1.w.a 2 819.bf odd 6 1
1911.1.w.a 2 819.bt even 6 1
1911.1.w.a 2 819.dv odd 6 1
1911.1.w.a 2 819.eg even 6 1
1911.1.w.b 2 63.g even 3 1
1911.1.w.b 2 63.h even 3 1
1911.1.w.b 2 63.j odd 6 1
1911.1.w.b 2 63.n odd 6 1
1911.1.w.b 2 819.bk even 6 1
1911.1.w.b 2 819.cp odd 6 1
1911.1.w.b 2 819.cq even 6 1
1911.1.w.b 2 819.ed odd 6 1
2496.1.l.a 1 72.l even 6 1
2496.1.l.a 1 72.p odd 6 1
2496.1.l.a 1 936.bs odd 6 1
2496.1.l.a 1 936.cl even 6 1
2496.1.l.b 1 72.j odd 6 1
2496.1.l.b 1 72.n even 6 1
2496.1.l.b 1 936.bx even 6 1
2496.1.l.b 1 936.cv odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(1053, [\chi])$$.

## Hecke characteristic polynomials

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