Properties

Label 1083.1.h.a
Level $1083$
Weight $1$
Character orbit 1083.h
Analytic conductor $0.540$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1083,1,Mod(68,1083)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1083, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1083.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1083.h (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: \(0.540487408682\)
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: \(\Z[a_1, a_2, a_3]\)
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: $C_6\times S_3$
Artin field: Galois closure of 12.0.12381017456889.1

$q$-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.

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

Character values

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

\(n\) \(362\) \(724\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i −0.500000 0.866025i 0 0 −1.00000 0 −0.500000 0.866025i 0
653.1 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 −1.00000 0 −0.500000 + 0.866025i 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 \(\Q(\sqrt{-3}) \)
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1083.1.h.a 2
3.b odd 2 1 CM 1083.1.h.a 2
19.b odd 2 1 57.1.h.a 2
19.c even 3 1 1083.1.b.a 1
19.c even 3 1 inner 1083.1.h.a 2
19.d odd 6 1 57.1.h.a 2
19.d odd 6 1 1083.1.b.b 1
19.e even 9 6 1083.1.l.b 6
19.f odd 18 6 1083.1.l.a 6
57.d even 2 1 57.1.h.a 2
57.f even 6 1 57.1.h.a 2
57.f even 6 1 1083.1.b.b 1
57.h odd 6 1 1083.1.b.a 1
57.h odd 6 1 inner 1083.1.h.a 2
57.j even 18 6 1083.1.l.a 6
57.l odd 18 6 1083.1.l.b 6
76.d even 2 1 912.1.bl.a 2
76.f even 6 1 912.1.bl.a 2
95.d odd 2 1 1425.1.t.a 2
95.g even 4 2 1425.1.o.a 4
95.h odd 6 1 1425.1.t.a 2
95.l even 12 2 1425.1.o.a 4
133.c even 2 1 2793.1.bf.a 2
133.i even 6 1 2793.1.bi.a 2
133.j odd 6 1 2793.1.bi.b 2
133.n odd 6 1 2793.1.n.a 2
133.o even 6 1 2793.1.n.b 2
133.o even 6 1 2793.1.bi.a 2
133.p even 6 1 2793.1.bf.a 2
133.r odd 6 1 2793.1.n.a 2
133.r odd 6 1 2793.1.bi.b 2
133.s even 6 1 2793.1.n.b 2
152.b even 2 1 3648.1.bl.a 2
152.g odd 2 1 3648.1.bl.b 2
152.l odd 6 1 3648.1.bl.b 2
152.o even 6 1 3648.1.bl.a 2
171.i odd 6 1 1539.1.n.a 2
171.k even 6 1 1539.1.j.a 2
171.l even 6 1 1539.1.j.a 2
171.l even 6 1 1539.1.n.a 2
171.o odd 6 1 1539.1.j.a 2
171.o odd 6 1 1539.1.n.a 2
171.s odd 6 1 1539.1.j.a 2
171.t even 6 1 1539.1.n.a 2
228.b odd 2 1 912.1.bl.a 2
228.n odd 6 1 912.1.bl.a 2
285.b even 2 1 1425.1.t.a 2
285.j odd 4 2 1425.1.o.a 4
285.q even 6 1 1425.1.t.a 2
285.w odd 12 2 1425.1.o.a 4
399.h odd 2 1 2793.1.bf.a 2
399.q odd 6 1 2793.1.bf.a 2
399.r odd 6 1 2793.1.n.b 2
399.s odd 6 1 2793.1.n.b 2
399.s odd 6 1 2793.1.bi.a 2
399.w even 6 1 2793.1.n.a 2
399.w even 6 1 2793.1.bi.b 2
399.x even 6 1 2793.1.n.a 2
399.bm even 6 1 2793.1.bi.b 2
399.bn odd 6 1 2793.1.bi.a 2
456.l odd 2 1 3648.1.bl.a 2
456.p even 2 1 3648.1.bl.b 2
456.s odd 6 1 3648.1.bl.a 2
456.v even 6 1 3648.1.bl.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 19.b odd 2 1
57.1.h.a 2 19.d odd 6 1
57.1.h.a 2 57.d even 2 1
57.1.h.a 2 57.f even 6 1
912.1.bl.a 2 76.d even 2 1
912.1.bl.a 2 76.f even 6 1
912.1.bl.a 2 228.b odd 2 1
912.1.bl.a 2 228.n odd 6 1
1083.1.b.a 1 19.c even 3 1
1083.1.b.a 1 57.h odd 6 1
1083.1.b.b 1 19.d odd 6 1
1083.1.b.b 1 57.f even 6 1
1083.1.h.a 2 1.a even 1 1 trivial
1083.1.h.a 2 3.b odd 2 1 CM
1083.1.h.a 2 19.c even 3 1 inner
1083.1.h.a 2 57.h odd 6 1 inner
1083.1.l.a 6 19.f odd 18 6
1083.1.l.a 6 57.j even 18 6
1083.1.l.b 6 19.e even 9 6
1083.1.l.b 6 57.l odd 18 6
1425.1.o.a 4 95.g even 4 2
1425.1.o.a 4 95.l even 12 2
1425.1.o.a 4 285.j odd 4 2
1425.1.o.a 4 285.w odd 12 2
1425.1.t.a 2 95.d odd 2 1
1425.1.t.a 2 95.h odd 6 1
1425.1.t.a 2 285.b even 2 1
1425.1.t.a 2 285.q even 6 1
1539.1.j.a 2 171.k even 6 1
1539.1.j.a 2 171.l even 6 1
1539.1.j.a 2 171.o odd 6 1
1539.1.j.a 2 171.s odd 6 1
1539.1.n.a 2 171.i odd 6 1
1539.1.n.a 2 171.l even 6 1
1539.1.n.a 2 171.o odd 6 1
1539.1.n.a 2 171.t even 6 1
2793.1.n.a 2 133.n odd 6 1
2793.1.n.a 2 133.r odd 6 1
2793.1.n.a 2 399.w even 6 1
2793.1.n.a 2 399.x even 6 1
2793.1.n.b 2 133.o even 6 1
2793.1.n.b 2 133.s even 6 1
2793.1.n.b 2 399.r odd 6 1
2793.1.n.b 2 399.s odd 6 1
2793.1.bf.a 2 133.c even 2 1
2793.1.bf.a 2 133.p even 6 1
2793.1.bf.a 2 399.h odd 2 1
2793.1.bf.a 2 399.q odd 6 1
2793.1.bi.a 2 133.i even 6 1
2793.1.bi.a 2 133.o even 6 1
2793.1.bi.a 2 399.s odd 6 1
2793.1.bi.a 2 399.bn odd 6 1
2793.1.bi.b 2 133.j odd 6 1
2793.1.bi.b 2 133.r odd 6 1
2793.1.bi.b 2 399.w even 6 1
2793.1.bi.b 2 399.bm even 6 1
3648.1.bl.a 2 152.b even 2 1
3648.1.bl.a 2 152.o even 6 1
3648.1.bl.a 2 456.l odd 2 1
3648.1.bl.a 2 456.s odd 6 1
3648.1.bl.b 2 152.g odd 2 1
3648.1.bl.b 2 152.l odd 6 1
3648.1.bl.b 2 456.p even 2 1
3648.1.bl.b 2 456.v even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1083, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) 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
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) 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
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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