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Properties

Label 1539.1.n.a
Level $1539$
Weight $1$
Character orbit 1539.n
Analytic conductor $0.768$
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] = [1539,1,Mod(539,1539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1539, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1539.539");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1539 = 3^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1539.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: \(0.768061054442\)
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 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1083.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.7105563.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{4} + \zeta_{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{4} + \zeta_{6} q^{7} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} + q^{19} + q^{25} - \zeta_{6}^{2} q^{28} - \zeta_{6}^{2} q^{31} - q^{37} - \zeta_{6}^{2} q^{43} - \zeta_{6}^{2} q^{52} - q^{61} + q^{64} + \zeta_{6} q^{67} + \zeta_{6} q^{73} - \zeta_{6} q^{76} - \zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{91} + \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{4} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{4} + q^{7} + q^{13} - q^{16} + 2 q^{19} + 2 q^{25} + q^{28} + q^{31} - 2 q^{37} + q^{43} + q^{52} - 2 q^{61} + 2 q^{64} + q^{67} + q^{73} - q^{76} + q^{79} - q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1539\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
539.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 −0.500000 − 0.866025i 0 0 0.500000 + 0.866025i 0 0 0
1322.1 0 0 −0.500000 + 0.866025i 0 0 0.500000 − 0.866025i 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 \(\Q(\sqrt{-3}) \)
171.g even 3 1 inner
171.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1539.1.n.a 2
3.b odd 2 1 CM 1539.1.n.a 2
9.c even 3 1 57.1.h.a ✓ 2
9.c even 3 1 1539.1.j.a 2
9.d odd 6 1 57.1.h.a ✓ 2
9.d odd 6 1 1539.1.j.a 2
19.c even 3 1 1539.1.j.a 2
36.f odd 6 1 912.1.bl.a 2
36.h even 6 1 912.1.bl.a 2
45.h odd 6 1 1425.1.t.a 2
45.j even 6 1 1425.1.t.a 2
45.k odd 12 2 1425.1.o.a 4
45.l even 12 2 1425.1.o.a 4
57.h odd 6 1 1539.1.j.a 2
63.g even 3 1 2793.1.n.a 2
63.h even 3 1 2793.1.bi.b 2
63.i even 6 1 2793.1.bi.a 2
63.j odd 6 1 2793.1.bi.b 2
63.k odd 6 1 2793.1.n.b 2
63.l odd 6 1 2793.1.bf.a 2
63.n odd 6 1 2793.1.n.a 2
63.o even 6 1 2793.1.bf.a 2
63.s even 6 1 2793.1.n.b 2
63.t odd 6 1 2793.1.bi.a 2
72.j odd 6 1 3648.1.bl.b 2
72.l even 6 1 3648.1.bl.a 2
72.n even 6 1 3648.1.bl.b 2
72.p odd 6 1 3648.1.bl.a 2
171.g even 3 1 1083.1.b.b 1
171.g even 3 1 inner 1539.1.n.a 2
171.h even 3 1 57.1.h.a ✓ 2
171.i odd 6 1 1083.1.h.a 2
171.j odd 6 1 57.1.h.a ✓ 2
171.k even 6 1 1083.1.b.a 1
171.l even 6 1 1083.1.h.a 2
171.n odd 6 1 1083.1.b.b 1
171.n odd 6 1 inner 1539.1.n.a 2
171.o odd 6 1 1083.1.h.a 2
171.s odd 6 1 1083.1.b.a 1
171.t even 6 1 1083.1.h.a 2
171.v even 9 3 1083.1.l.a 6
171.w even 9 3 1083.1.l.a 6
171.x even 18 3 1083.1.l.b 6
171.z odd 18 3 1083.1.l.a 6
171.bc odd 18 3 1083.1.l.b 6
171.bd even 18 3 1083.1.l.b 6
171.be odd 18 3 1083.1.l.b 6
171.bf odd 18 3 1083.1.l.a 6
684.x odd 6 1 912.1.bl.a 2
684.bi even 6 1 912.1.bl.a 2
855.s even 6 1 1425.1.t.a 2
855.br odd 6 1 1425.1.t.a 2
855.bw odd 12 2 1425.1.o.a 4
855.bx even 12 2 1425.1.o.a 4
1197.n even 3 1 2793.1.bi.b 2
1197.p even 3 1 2793.1.n.a 2
1197.ba even 6 1 2793.1.bf.a 2
1197.bm odd 6 1 2793.1.n.b 2
1197.cc odd 6 1 2793.1.bi.a 2
1197.cq odd 6 1 2793.1.n.a 2
1197.cs even 6 1 2793.1.bi.a 2
1197.dk odd 6 1 2793.1.bi.b 2
1197.dt even 6 1 2793.1.n.b 2
1197.ec odd 6 1 2793.1.bf.a 2
1368.bh odd 6 1 3648.1.bl.b 2
1368.bn even 6 1 3648.1.bl.a 2
1368.cb even 6 1 3648.1.bl.b 2
1368.cj odd 6 1 3648.1.bl.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a ✓ 2 9.c even 3 1
57.1.h.a ✓ 2 9.d odd 6 1
57.1.h.a ✓ 2 171.h even 3 1
57.1.h.a ✓ 2 171.j odd 6 1
912.1.bl.a 2 36.f odd 6 1
912.1.bl.a 2 36.h even 6 1
912.1.bl.a 2 684.x odd 6 1
912.1.bl.a 2 684.bi even 6 1
1083.1.b.a 1 171.k even 6 1
1083.1.b.a 1 171.s odd 6 1
1083.1.b.b 1 171.g even 3 1
1083.1.b.b 1 171.n odd 6 1
1083.1.h.a 2 171.i odd 6 1
1083.1.h.a 2 171.l even 6 1
1083.1.h.a 2 171.o odd 6 1
1083.1.h.a 2 171.t even 6 1
1083.1.l.a 6 171.v even 9 3
1083.1.l.a 6 171.w even 9 3
1083.1.l.a 6 171.z odd 18 3
1083.1.l.a 6 171.bf odd 18 3
1083.1.l.b 6 171.x even 18 3
1083.1.l.b 6 171.bc odd 18 3
1083.1.l.b 6 171.bd even 18 3
1083.1.l.b 6 171.be odd 18 3
1425.1.o.a 4 45.k odd 12 2
1425.1.o.a 4 45.l even 12 2
1425.1.o.a 4 855.bw odd 12 2
1425.1.o.a 4 855.bx even 12 2
1425.1.t.a 2 45.h odd 6 1
1425.1.t.a 2 45.j even 6 1
1425.1.t.a 2 855.s even 6 1
1425.1.t.a 2 855.br odd 6 1
1539.1.j.a 2 9.c even 3 1
1539.1.j.a 2 9.d odd 6 1
1539.1.j.a 2 19.c even 3 1
1539.1.j.a 2 57.h odd 6 1
1539.1.n.a 2 1.a even 1 1 trivial
1539.1.n.a 2 3.b odd 2 1 CM
1539.1.n.a 2 171.g even 3 1 inner
1539.1.n.a 2 171.n odd 6 1 inner
2793.1.n.a 2 63.g even 3 1
2793.1.n.a 2 63.n odd 6 1
2793.1.n.a 2 1197.p even 3 1
2793.1.n.a 2 1197.cq odd 6 1
2793.1.n.b 2 63.k odd 6 1
2793.1.n.b 2 63.s even 6 1
2793.1.n.b 2 1197.bm odd 6 1
2793.1.n.b 2 1197.dt even 6 1
2793.1.bf.a 2 63.l odd 6 1
2793.1.bf.a 2 63.o even 6 1
2793.1.bf.a 2 1197.ba even 6 1
2793.1.bf.a 2 1197.ec odd 6 1
2793.1.bi.a 2 63.i even 6 1
2793.1.bi.a 2 63.t odd 6 1
2793.1.bi.a 2 1197.cc odd 6 1
2793.1.bi.a 2 1197.cs even 6 1
2793.1.bi.b 2 63.h even 3 1
2793.1.bi.b 2 63.j odd 6 1
2793.1.bi.b 2 1197.n even 3 1
2793.1.bi.b 2 1197.dk odd 6 1
3648.1.bl.a 2 72.l even 6 1
3648.1.bl.a 2 72.p odd 6 1
3648.1.bl.a 2 1368.bn even 6 1
3648.1.bl.a 2 1368.cj odd 6 1
3648.1.bl.b 2 72.j odd 6 1
3648.1.bl.b 2 72.n even 6 1
3648.1.bl.b 2 1368.bh odd 6 1
3648.1.bl.b 2 1368.cb even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1539, [\chi])\).

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
$7$ \( T^{2} - T + 1 \) 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
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) 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
$61$ \( (T + 1)^{2} \) 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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