Properties

Label 2793.1.bi.b
Level $2793$
Weight $1$
Character orbit 2793.bi
Analytic conductor $1.394$
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] = [2793,1,Mod(1892,2793)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2793, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2793.1892");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2793.bi (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: \(1.39388858028\)
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 6.0.23402547.2

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + \zeta_{6}^{2} q^{4} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \zeta_{6}^{2} q^{4} + q^{9} + \zeta_{6}^{2} q^{12} + \zeta_{6} q^{13} - \zeta_{6} q^{16} + \zeta_{6}^{2} q^{19} + \zeta_{6}^{2} q^{25} + q^{27} + \zeta_{6} q^{31} + \zeta_{6}^{2} q^{36} - \zeta_{6}^{2} q^{37} + \zeta_{6} q^{39} - \zeta_{6}^{2} q^{43} - \zeta_{6} q^{48} - q^{52} + \zeta_{6}^{2} q^{57} - q^{61} + q^{64} - \zeta_{6}^{2} q^{67} - q^{73} + \zeta_{6}^{2} q^{75} - \zeta_{6} q^{76} + \zeta_{6} q^{79} + q^{81} + \zeta_{6} q^{93} + 2 \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{4} + 2 q^{9} - q^{12} + q^{13} - q^{16} - q^{19} - q^{25} + 2 q^{27} + q^{31} - q^{36} + q^{37} + q^{39} + q^{43} - q^{48} - 2 q^{52} - q^{57} - 2 q^{61} + 2 q^{64} + q^{67} - 2 q^{73} - q^{75} - q^{76} + q^{79} + 2 q^{81} + q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times\).

\(n\) \(932\) \(2110\) \(2206\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1892.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 −0.500000 + 0.866025i 0 0 0 0 1.00000 0
2762.1 0 1.00000 −0.500000 0.866025i 0 0 0 0 1.00000 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 \(\Q(\sqrt{-3}) \)
133.g even 3 1 inner
399.bi odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2793.1.bi.b 2
3.b odd 2 1 CM 2793.1.bi.b 2
7.b odd 2 1 2793.1.bi.a 2
7.c even 3 1 57.1.h.a 2
7.c even 3 1 2793.1.n.a 2
7.d odd 6 1 2793.1.n.b 2
7.d odd 6 1 2793.1.bf.a 2
19.c even 3 1 2793.1.n.a 2
21.c even 2 1 2793.1.bi.a 2
21.g even 6 1 2793.1.n.b 2
21.g even 6 1 2793.1.bf.a 2
21.h odd 6 1 57.1.h.a 2
21.h odd 6 1 2793.1.n.a 2
28.g odd 6 1 912.1.bl.a 2
35.j even 6 1 1425.1.t.a 2
35.l odd 12 2 1425.1.o.a 4
56.k odd 6 1 3648.1.bl.a 2
56.p even 6 1 3648.1.bl.b 2
57.h odd 6 1 2793.1.n.a 2
63.g even 3 1 1539.1.j.a 2
63.h even 3 1 1539.1.n.a 2
63.j odd 6 1 1539.1.n.a 2
63.n odd 6 1 1539.1.j.a 2
84.n even 6 1 912.1.bl.a 2
105.o odd 6 1 1425.1.t.a 2
105.x even 12 2 1425.1.o.a 4
133.g even 3 1 1083.1.b.b 1
133.g even 3 1 inner 2793.1.bi.b 2
133.h even 3 1 57.1.h.a 2
133.j odd 6 1 1083.1.h.a 2
133.k odd 6 1 2793.1.bi.a 2
133.m odd 6 1 2793.1.n.b 2
133.n odd 6 1 1083.1.b.a 1
133.r odd 6 1 1083.1.h.a 2
133.t odd 6 1 2793.1.bf.a 2
133.u even 9 3 1083.1.l.a 6
133.w even 9 3 1083.1.l.a 6
133.bd odd 18 3 1083.1.l.b 6
133.be odd 18 3 1083.1.l.b 6
168.s odd 6 1 3648.1.bl.b 2
168.v even 6 1 3648.1.bl.a 2
399.n odd 6 1 57.1.h.a 2
399.p even 6 1 2793.1.bf.a 2
399.w even 6 1 1083.1.h.a 2
399.x even 6 1 1083.1.b.a 1
399.z even 6 1 2793.1.n.b 2
399.bd even 6 1 2793.1.bi.a 2
399.bi odd 6 1 1083.1.b.b 1
399.bi odd 6 1 inner 2793.1.bi.b 2
399.bm even 6 1 1083.1.h.a 2
399.br even 18 3 1083.1.l.b 6
399.bv even 18 3 1083.1.l.b 6
399.ca odd 18 3 1083.1.l.a 6
399.ch odd 18 3 1083.1.l.a 6
532.bk odd 6 1 912.1.bl.a 2
665.t even 6 1 1425.1.t.a 2
665.bw odd 12 2 1425.1.o.a 4
1064.bd odd 6 1 3648.1.bl.a 2
1064.ca even 6 1 3648.1.bl.b 2
1197.l even 3 1 1539.1.j.a 2
1197.n even 3 1 1539.1.n.a 2
1197.dk odd 6 1 1539.1.n.a 2
1197.do odd 6 1 1539.1.j.a 2
1596.v even 6 1 912.1.bl.a 2
1995.di odd 6 1 1425.1.t.a 2
1995.dt even 12 2 1425.1.o.a 4
3192.fu odd 6 1 3648.1.bl.b 2
3192.gd even 6 1 3648.1.bl.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 7.c even 3 1
57.1.h.a 2 21.h odd 6 1
57.1.h.a 2 133.h even 3 1
57.1.h.a 2 399.n odd 6 1
912.1.bl.a 2 28.g odd 6 1
912.1.bl.a 2 84.n even 6 1
912.1.bl.a 2 532.bk odd 6 1
912.1.bl.a 2 1596.v even 6 1
1083.1.b.a 1 133.n odd 6 1
1083.1.b.a 1 399.x even 6 1
1083.1.b.b 1 133.g even 3 1
1083.1.b.b 1 399.bi odd 6 1
1083.1.h.a 2 133.j odd 6 1
1083.1.h.a 2 133.r odd 6 1
1083.1.h.a 2 399.w even 6 1
1083.1.h.a 2 399.bm even 6 1
1083.1.l.a 6 133.u even 9 3
1083.1.l.a 6 133.w even 9 3
1083.1.l.a 6 399.ca odd 18 3
1083.1.l.a 6 399.ch odd 18 3
1083.1.l.b 6 133.bd odd 18 3
1083.1.l.b 6 133.be odd 18 3
1083.1.l.b 6 399.br even 18 3
1083.1.l.b 6 399.bv even 18 3
1425.1.o.a 4 35.l odd 12 2
1425.1.o.a 4 105.x even 12 2
1425.1.o.a 4 665.bw odd 12 2
1425.1.o.a 4 1995.dt even 12 2
1425.1.t.a 2 35.j even 6 1
1425.1.t.a 2 105.o odd 6 1
1425.1.t.a 2 665.t even 6 1
1425.1.t.a 2 1995.di odd 6 1
1539.1.j.a 2 63.g even 3 1
1539.1.j.a 2 63.n odd 6 1
1539.1.j.a 2 1197.l even 3 1
1539.1.j.a 2 1197.do odd 6 1
1539.1.n.a 2 63.h even 3 1
1539.1.n.a 2 63.j odd 6 1
1539.1.n.a 2 1197.n even 3 1
1539.1.n.a 2 1197.dk odd 6 1
2793.1.n.a 2 7.c even 3 1
2793.1.n.a 2 19.c even 3 1
2793.1.n.a 2 21.h odd 6 1
2793.1.n.a 2 57.h odd 6 1
2793.1.n.b 2 7.d odd 6 1
2793.1.n.b 2 21.g even 6 1
2793.1.n.b 2 133.m odd 6 1
2793.1.n.b 2 399.z even 6 1
2793.1.bf.a 2 7.d odd 6 1
2793.1.bf.a 2 21.g even 6 1
2793.1.bf.a 2 133.t odd 6 1
2793.1.bf.a 2 399.p even 6 1
2793.1.bi.a 2 7.b odd 2 1
2793.1.bi.a 2 21.c even 2 1
2793.1.bi.a 2 133.k odd 6 1
2793.1.bi.a 2 399.bd even 6 1
2793.1.bi.b 2 1.a even 1 1 trivial
2793.1.bi.b 2 3.b odd 2 1 CM
2793.1.bi.b 2 133.g even 3 1 inner
2793.1.bi.b 2 399.bi odd 6 1 inner
3648.1.bl.a 2 56.k odd 6 1
3648.1.bl.a 2 168.v even 6 1
3648.1.bl.a 2 1064.bd odd 6 1
3648.1.bl.a 2 3192.gd even 6 1
3648.1.bl.b 2 56.p even 6 1
3648.1.bl.b 2 168.s odd 6 1
3648.1.bl.b 2 1064.ca even 6 1
3648.1.bl.b 2 3192.fu odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2793, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) 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
$19$ \( T^{2} + T + 1 \) 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
$37$ \( T^{2} - T + 1 \) 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
$73$ \( (T + 1)^{2} \) 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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