Properties

Label 1083.1.l.a
Level $1083$
Weight $1$
Character orbit 1083.l
Analytic conductor $0.540$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -3
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1083,1,Mod(62,1083)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1083, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1083.62");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1083.l (of order \(18\), degree \(6\), 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: \(0.540487408682\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1083.1
Artin image: $S_3\times C_9$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{54} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{18}^{8} q^{3} + \zeta_{18}^{4} q^{4} + \zeta_{18}^{3} q^{7} - \zeta_{18}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{8} q^{3} + \zeta_{18}^{4} q^{4} + \zeta_{18}^{3} q^{7} - \zeta_{18}^{7} q^{9} - \zeta_{18}^{3} q^{12} + \zeta_{18} q^{13} + \zeta_{18}^{8} q^{16} - \zeta_{18}^{2} q^{21} - \zeta_{18} q^{25} + \zeta_{18}^{6} q^{27} + \zeta_{18}^{7} q^{28} + \zeta_{18}^{3} q^{31} + \zeta_{18}^{2} q^{36} - q^{37} - q^{39} + \zeta_{18}^{5} q^{43} - \zeta_{18}^{7} q^{48} + \zeta_{18}^{5} q^{52} - \zeta_{18}^{4} q^{61} + \zeta_{18} q^{63} - \zeta_{18}^{3} q^{64} + \zeta_{18}^{7} q^{67} - \zeta_{18}^{8} q^{73} + q^{75} - \zeta_{18}^{8} q^{79} - \zeta_{18}^{5} q^{81} - \zeta_{18}^{6} q^{84} + \zeta_{18}^{4} q^{91} - \zeta_{18}^{2} q^{93} + 2 \zeta_{18}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{7} - 3 q^{12} - 3 q^{27} + 3 q^{31} - 6 q^{37} - 6 q^{39} - 3 q^{64} + 6 q^{75} + 3 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1083\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(724\)
\(\chi(n)\) \(-1\) \(\zeta_{18}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
−0.173648 0.984808i
−0.766044 + 0.642788i
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 + 0.984808i
0.939693 0.342020i
0 0.173648 0.984808i 0.766044 0.642788i 0 0 0.500000 + 0.866025i 0 −0.939693 0.342020i 0
245.1 0 0.766044 + 0.642788i −0.939693 0.342020i 0 0 0.500000 + 0.866025i 0 0.173648 + 0.984808i 0
389.1 0 0.766044 0.642788i −0.939693 + 0.342020i 0 0 0.500000 0.866025i 0 0.173648 0.984808i 0
776.1 0 −0.939693 + 0.342020i 0.173648 + 0.984808i 0 0 0.500000 + 0.866025i 0 0.766044 0.642788i 0
821.1 0 0.173648 + 0.984808i 0.766044 + 0.642788i 0 0 0.500000 0.866025i 0 −0.939693 + 0.342020i 0
956.1 0 −0.939693 0.342020i 0.173648 0.984808i 0 0 0.500000 0.866025i 0 0.766044 + 0.642788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.c even 3 2 inner
19.e even 9 3 inner
57.h odd 6 2 inner
57.l odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1083.1.l.a 6
3.b odd 2 1 CM 1083.1.l.a 6
19.b odd 2 1 1083.1.l.b 6
19.c even 3 2 inner 1083.1.l.a 6
19.d odd 6 2 1083.1.l.b 6
19.e even 9 2 57.1.h.a 2
19.e even 9 1 1083.1.b.b 1
19.e even 9 3 inner 1083.1.l.a 6
19.f odd 18 1 1083.1.b.a 1
19.f odd 18 2 1083.1.h.a 2
19.f odd 18 3 1083.1.l.b 6
57.d even 2 1 1083.1.l.b 6
57.f even 6 2 1083.1.l.b 6
57.h odd 6 2 inner 1083.1.l.a 6
57.j even 18 1 1083.1.b.a 1
57.j even 18 2 1083.1.h.a 2
57.j even 18 3 1083.1.l.b 6
57.l odd 18 2 57.1.h.a 2
57.l odd 18 1 1083.1.b.b 1
57.l odd 18 3 inner 1083.1.l.a 6
76.l odd 18 2 912.1.bl.a 2
95.p even 18 2 1425.1.t.a 2
95.q odd 36 4 1425.1.o.a 4
133.u even 9 1 2793.1.n.a 2
133.u even 9 1 2793.1.bi.b 2
133.w even 9 1 2793.1.n.a 2
133.w even 9 1 2793.1.bi.b 2
133.x odd 18 1 2793.1.n.b 2
133.x odd 18 1 2793.1.bi.a 2
133.y odd 18 2 2793.1.bf.a 2
133.z odd 18 1 2793.1.n.b 2
133.z odd 18 1 2793.1.bi.a 2
152.t even 18 2 3648.1.bl.b 2
152.u odd 18 2 3648.1.bl.a 2
171.v even 9 1 1539.1.j.a 2
171.v even 9 1 1539.1.n.a 2
171.w even 9 1 1539.1.j.a 2
171.w even 9 1 1539.1.n.a 2
171.z odd 18 1 1539.1.j.a 2
171.z odd 18 1 1539.1.n.a 2
171.bf odd 18 1 1539.1.j.a 2
171.bf odd 18 1 1539.1.n.a 2
228.v even 18 2 912.1.bl.a 2
285.bd odd 18 2 1425.1.t.a 2
285.bi even 36 4 1425.1.o.a 4
399.ca odd 18 1 2793.1.n.a 2
399.ca odd 18 1 2793.1.bi.b 2
399.cb even 18 1 2793.1.n.b 2
399.cb even 18 1 2793.1.bi.a 2
399.ch odd 18 1 2793.1.n.a 2
399.ch odd 18 1 2793.1.bi.b 2
399.ci even 18 1 2793.1.n.b 2
399.ci even 18 1 2793.1.bi.a 2
399.cj even 18 2 2793.1.bf.a 2
456.bh odd 18 2 3648.1.bl.b 2
456.bu even 18 2 3648.1.bl.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 19.e even 9 2
57.1.h.a 2 57.l odd 18 2
912.1.bl.a 2 76.l odd 18 2
912.1.bl.a 2 228.v even 18 2
1083.1.b.a 1 19.f odd 18 1
1083.1.b.a 1 57.j even 18 1
1083.1.b.b 1 19.e even 9 1
1083.1.b.b 1 57.l odd 18 1
1083.1.h.a 2 19.f odd 18 2
1083.1.h.a 2 57.j even 18 2
1083.1.l.a 6 1.a even 1 1 trivial
1083.1.l.a 6 3.b odd 2 1 CM
1083.1.l.a 6 19.c even 3 2 inner
1083.1.l.a 6 19.e even 9 3 inner
1083.1.l.a 6 57.h odd 6 2 inner
1083.1.l.a 6 57.l odd 18 3 inner
1083.1.l.b 6 19.b odd 2 1
1083.1.l.b 6 19.d odd 6 2
1083.1.l.b 6 19.f odd 18 3
1083.1.l.b 6 57.d even 2 1
1083.1.l.b 6 57.f even 6 2
1083.1.l.b 6 57.j even 18 3
1425.1.o.a 4 95.q odd 36 4
1425.1.o.a 4 285.bi even 36 4
1425.1.t.a 2 95.p even 18 2
1425.1.t.a 2 285.bd odd 18 2
1539.1.j.a 2 171.v even 9 1
1539.1.j.a 2 171.w even 9 1
1539.1.j.a 2 171.z odd 18 1
1539.1.j.a 2 171.bf odd 18 1
1539.1.n.a 2 171.v even 9 1
1539.1.n.a 2 171.w even 9 1
1539.1.n.a 2 171.z odd 18 1
1539.1.n.a 2 171.bf odd 18 1
2793.1.n.a 2 133.u even 9 1
2793.1.n.a 2 133.w even 9 1
2793.1.n.a 2 399.ca odd 18 1
2793.1.n.a 2 399.ch odd 18 1
2793.1.n.b 2 133.x odd 18 1
2793.1.n.b 2 133.z odd 18 1
2793.1.n.b 2 399.cb even 18 1
2793.1.n.b 2 399.ci even 18 1
2793.1.bf.a 2 133.y odd 18 2
2793.1.bf.a 2 399.cj even 18 2
2793.1.bi.a 2 133.x odd 18 1
2793.1.bi.a 2 133.z odd 18 1
2793.1.bi.a 2 399.cb even 18 1
2793.1.bi.a 2 399.ci even 18 1
2793.1.bi.b 2 133.u even 9 1
2793.1.bi.b 2 133.w even 9 1
2793.1.bi.b 2 399.ca odd 18 1
2793.1.bi.b 2 399.ch odd 18 1
3648.1.bl.a 2 152.u odd 18 2
3648.1.bl.a 2 456.bu even 18 2
3648.1.bl.b 2 152.t even 18 2
3648.1.bl.b 2 456.bh odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} - T_{13}^{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1083, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$67$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$79$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
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