Properties

Label 2793.1.n.b
Level $2793$
Weight $1$
Character orbit 2793.n
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(410,2793)]
 
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, 4, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2793.410");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2793.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: \(1.39388858028\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
410.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 1.00000 0 0 0 0 −0.500000 + 0.866025i 0
1451.1 0 0.500000 0.866025i 1.00000 0 0 0 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 \(\Q(\sqrt{-3}) \)
133.h even 3 1 inner
399.n 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.n.b 2
3.b odd 2 1 CM 2793.1.n.b 2
7.b odd 2 1 2793.1.n.a 2
7.c even 3 1 2793.1.bf.a 2
7.c even 3 1 2793.1.bi.a 2
7.d odd 6 1 57.1.h.a 2
7.d odd 6 1 2793.1.bi.b 2
19.c even 3 1 2793.1.bi.a 2
21.c even 2 1 2793.1.n.a 2
21.g even 6 1 57.1.h.a 2
21.g even 6 1 2793.1.bi.b 2
21.h odd 6 1 2793.1.bf.a 2
21.h odd 6 1 2793.1.bi.a 2
28.f even 6 1 912.1.bl.a 2
35.i odd 6 1 1425.1.t.a 2
35.k even 12 2 1425.1.o.a 4
56.j odd 6 1 3648.1.bl.b 2
56.m even 6 1 3648.1.bl.a 2
57.h odd 6 1 2793.1.bi.a 2
63.i even 6 1 1539.1.j.a 2
63.k odd 6 1 1539.1.n.a 2
63.s even 6 1 1539.1.n.a 2
63.t odd 6 1 1539.1.j.a 2
84.j odd 6 1 912.1.bl.a 2
105.p even 6 1 1425.1.t.a 2
105.w odd 12 2 1425.1.o.a 4
133.g even 3 1 2793.1.bf.a 2
133.h even 3 1 inner 2793.1.n.b 2
133.i even 6 1 1083.1.b.a 1
133.k odd 6 1 57.1.h.a 2
133.m odd 6 1 2793.1.bi.b 2
133.o even 6 1 1083.1.h.a 2
133.s even 6 1 1083.1.h.a 2
133.t odd 6 1 1083.1.b.b 1
133.t odd 6 1 2793.1.n.a 2
133.x odd 18 3 1083.1.l.a 6
133.z odd 18 3 1083.1.l.a 6
133.bb even 18 3 1083.1.l.b 6
133.bf even 18 3 1083.1.l.b 6
168.ba even 6 1 3648.1.bl.b 2
168.be odd 6 1 3648.1.bl.a 2
399.n odd 6 1 inner 2793.1.n.b 2
399.p even 6 1 1083.1.b.b 1
399.p even 6 1 2793.1.n.a 2
399.r odd 6 1 1083.1.h.a 2
399.s odd 6 1 1083.1.h.a 2
399.z even 6 1 2793.1.bi.b 2
399.bd even 6 1 57.1.h.a 2
399.bi odd 6 1 2793.1.bf.a 2
399.bn odd 6 1 1083.1.b.a 1
399.bs odd 18 3 1083.1.l.b 6
399.by odd 18 3 1083.1.l.b 6
399.cb even 18 3 1083.1.l.a 6
399.ci even 18 3 1083.1.l.a 6
532.bn even 6 1 912.1.bl.a 2
665.bo odd 6 1 1425.1.t.a 2
665.cg even 12 2 1425.1.o.a 4
1064.u even 6 1 3648.1.bl.a 2
1064.cw odd 6 1 3648.1.bl.b 2
1197.x even 6 1 1539.1.j.a 2
1197.bm odd 6 1 1539.1.n.a 2
1197.dt even 6 1 1539.1.n.a 2
1197.eg odd 6 1 1539.1.j.a 2
1596.bj odd 6 1 912.1.bl.a 2
1995.bw even 6 1 1425.1.t.a 2
1995.ei odd 12 2 1425.1.o.a 4
3192.ei even 6 1 3648.1.bl.b 2
3192.ex 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 7.d odd 6 1
57.1.h.a 2 21.g even 6 1
57.1.h.a 2 133.k odd 6 1
57.1.h.a 2 399.bd even 6 1
912.1.bl.a 2 28.f even 6 1
912.1.bl.a 2 84.j odd 6 1
912.1.bl.a 2 532.bn even 6 1
912.1.bl.a 2 1596.bj odd 6 1
1083.1.b.a 1 133.i even 6 1
1083.1.b.a 1 399.bn odd 6 1
1083.1.b.b 1 133.t odd 6 1
1083.1.b.b 1 399.p even 6 1
1083.1.h.a 2 133.o even 6 1
1083.1.h.a 2 133.s even 6 1
1083.1.h.a 2 399.r odd 6 1
1083.1.h.a 2 399.s odd 6 1
1083.1.l.a 6 133.x odd 18 3
1083.1.l.a 6 133.z odd 18 3
1083.1.l.a 6 399.cb even 18 3
1083.1.l.a 6 399.ci even 18 3
1083.1.l.b 6 133.bb even 18 3
1083.1.l.b 6 133.bf even 18 3
1083.1.l.b 6 399.bs odd 18 3
1083.1.l.b 6 399.by odd 18 3
1425.1.o.a 4 35.k even 12 2
1425.1.o.a 4 105.w odd 12 2
1425.1.o.a 4 665.cg even 12 2
1425.1.o.a 4 1995.ei odd 12 2
1425.1.t.a 2 35.i odd 6 1
1425.1.t.a 2 105.p even 6 1
1425.1.t.a 2 665.bo odd 6 1
1425.1.t.a 2 1995.bw even 6 1
1539.1.j.a 2 63.i even 6 1
1539.1.j.a 2 63.t odd 6 1
1539.1.j.a 2 1197.x even 6 1
1539.1.j.a 2 1197.eg odd 6 1
1539.1.n.a 2 63.k odd 6 1
1539.1.n.a 2 63.s even 6 1
1539.1.n.a 2 1197.bm odd 6 1
1539.1.n.a 2 1197.dt even 6 1
2793.1.n.a 2 7.b odd 2 1
2793.1.n.a 2 21.c even 2 1
2793.1.n.a 2 133.t odd 6 1
2793.1.n.a 2 399.p even 6 1
2793.1.n.b 2 1.a even 1 1 trivial
2793.1.n.b 2 3.b odd 2 1 CM
2793.1.n.b 2 133.h even 3 1 inner
2793.1.n.b 2 399.n odd 6 1 inner
2793.1.bf.a 2 7.c even 3 1
2793.1.bf.a 2 21.h odd 6 1
2793.1.bf.a 2 133.g even 3 1
2793.1.bf.a 2 399.bi odd 6 1
2793.1.bi.a 2 7.c even 3 1
2793.1.bi.a 2 19.c even 3 1
2793.1.bi.a 2 21.h odd 6 1
2793.1.bi.a 2 57.h odd 6 1
2793.1.bi.b 2 7.d odd 6 1
2793.1.bi.b 2 21.g even 6 1
2793.1.bi.b 2 133.m odd 6 1
2793.1.bi.b 2 399.z even 6 1
3648.1.bl.a 2 56.m even 6 1
3648.1.bl.a 2 168.be odd 6 1
3648.1.bl.a 2 1064.u even 6 1
3648.1.bl.a 2 3192.ex odd 6 1
3648.1.bl.b 2 56.j odd 6 1
3648.1.bl.b 2 168.ba even 6 1
3648.1.bl.b 2 1064.cw odd 6 1
3648.1.bl.b 2 3192.ei even 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
$3$ \( T^{2} - T + 1 \) 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
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( (T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) 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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