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 $8$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{4} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - \zeta_{6}^{2} q^{25} + \zeta_{6}^{2} q^{43} - \zeta_{6} q^{49} + \zeta_{6}^{2} q^{52} - \zeta_{6}^{2} q^{61} - q^{64} - \zeta_{6}^{2} q^{79} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{4} + q^{13} - q^{16} + q^{25} - 2 q^{43} - q^{49} - q^{52} + 2 q^{61} - 2 q^{64} + 2 q^{79}+O(q^{100}) \) Copy content Toggle raw display

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
0.500000 + 0.866025i
0.500000 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:

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 \(S_{1}^{\mathrm{new}}(1053, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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