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Properties

Label 2793.1.bf.a
Level 27932793
Weight 11
Character orbit 2793.bf
Analytic conductor 1.3941.394
Analytic rank 00
Dimension 22
Projective image D3D_{3}
CM discriminant -3
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2793,1,Mod(197,2793)]
 
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, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2793.197");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 2793=3⋅72⋅19 2793 = 3 \cdot 7^{2} \cdot 19
Weight: k k == 1 1
Character orbit: [χ][\chi] == 2793.bf (of order 66, degree 22, 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: 1.393888580281.39388858028
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2−x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 57)
Projective image: D3D_{3}
Projective field: Galois closure of 3.1.1083.1
Artin image: C6×S3C_6\times S_3
Artin field: Galois closure of Q[x]/(x12−⋯ )\mathbb{Q}[x]/(x^{12} - \cdots)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ62q3−ζ6q4−ζ6q9−q12−ζ6q13+ζ62q16−q19−ζ6q25−q27+q31+ζ62q36−q37−q39+⋯−2ζ62q97+O(q100) q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{9} - q^{12} - \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - q^{19} - \zeta_{6} q^{25} - q^{27} + q^{31} + \zeta_{6}^{2} q^{36} - q^{37} - q^{39} + \cdots - 2 \zeta_{6}^{2} q^{97} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q+q3−q4−q9−2q12−q13−q16−2q19−q25−2q27+2q31−q36−2q37−2q39+q43+q48−q52−q57−q61+2q64+q67+⋯+2q97+O(q100) 2 q + q^{3} - q^{4} - q^{9} - 2 q^{12} - q^{13} - q^{16} - 2 q^{19} - q^{25} - 2 q^{27} + 2 q^{31} - q^{36} - 2 q^{37} - 2 q^{39} + q^{43} + q^{48} - q^{52} - q^{57} - q^{61} + 2 q^{64} + q^{67}+ \cdots + 2 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 932932 21102110 22062206
χ(n)\chi(n) −1-1 11 −ζ6-\zeta_{6}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
197.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0 0 −0.500000 + 0.866025i 0
638.1 0 0.500000 − 0.866025i −0.500000 − 0.866025i 0 0 0 0 −0.500000 − 0.866025i 0
nn: 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(−3)\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 2793.1.bf.a 2
3.b odd 2 1 CM 2793.1.bf.a 2
7.b odd 2 1 57.1.h.a ✓ 2
7.c even 3 1 2793.1.n.b 2
7.c even 3 1 2793.1.bi.a 2
7.d odd 6 1 2793.1.n.a 2
7.d odd 6 1 2793.1.bi.b 2
19.c even 3 1 inner 2793.1.bf.a 2
21.c even 2 1 57.1.h.a ✓ 2
21.g even 6 1 2793.1.n.a 2
21.g even 6 1 2793.1.bi.b 2
21.h odd 6 1 2793.1.n.b 2
21.h odd 6 1 2793.1.bi.a 2
28.d even 2 1 912.1.bl.a 2
35.c odd 2 1 1425.1.t.a 2
35.f even 4 2 1425.1.o.a 4
56.e even 2 1 3648.1.bl.a 2
56.h odd 2 1 3648.1.bl.b 2
57.h odd 6 1 inner 2793.1.bf.a 2
63.l odd 6 1 1539.1.j.a 2
63.l odd 6 1 1539.1.n.a 2
63.o even 6 1 1539.1.j.a 2
63.o even 6 1 1539.1.n.a 2
84.h odd 2 1 912.1.bl.a 2
105.g even 2 1 1425.1.t.a 2
105.k odd 4 2 1425.1.o.a 4
133.c even 2 1 1083.1.h.a 2
133.g even 3 1 2793.1.n.b 2
133.h even 3 1 2793.1.bi.a 2
133.k odd 6 1 2793.1.n.a 2
133.m odd 6 1 57.1.h.a ✓ 2
133.m odd 6 1 1083.1.b.b 1
133.p even 6 1 1083.1.b.a 1
133.p even 6 1 1083.1.h.a 2
133.t odd 6 1 2793.1.bi.b 2
133.y odd 18 6 1083.1.l.a 6
133.ba even 18 6 1083.1.l.b 6
168.e odd 2 1 3648.1.bl.a 2
168.i even 2 1 3648.1.bl.b 2
399.h odd 2 1 1083.1.h.a 2
399.n odd 6 1 2793.1.bi.a 2
399.p even 6 1 2793.1.bi.b 2
399.q odd 6 1 1083.1.b.a 1
399.q odd 6 1 1083.1.h.a 2
399.z even 6 1 57.1.h.a ✓ 2
399.z even 6 1 1083.1.b.b 1
399.bd even 6 1 2793.1.n.a 2
399.bi odd 6 1 2793.1.n.b 2
399.bx odd 18 6 1083.1.l.b 6
399.cj even 18 6 1083.1.l.a 6
532.u even 6 1 912.1.bl.a 2
665.bh odd 6 1 1425.1.t.a 2
665.ci even 12 2 1425.1.o.a 4
1064.bg odd 6 1 3648.1.bl.b 2
1064.ch even 6 1 3648.1.bl.a 2
1197.ba even 6 1 1539.1.n.a 2
1197.bn odd 6 1 1539.1.j.a 2
1197.du even 6 1 1539.1.j.a 2
1197.ec odd 6 1 1539.1.n.a 2
1596.bw odd 6 1 912.1.bl.a 2
1995.bl even 6 1 1425.1.t.a 2
1995.dx odd 12 2 1425.1.o.a 4
3192.du odd 6 1 3648.1.bl.a 2
3192.fh 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 7.b odd 2 1
57.1.h.a ✓ 2 21.c even 2 1
57.1.h.a ✓ 2 133.m odd 6 1
57.1.h.a ✓ 2 399.z even 6 1
912.1.bl.a 2 28.d even 2 1
912.1.bl.a 2 84.h odd 2 1
912.1.bl.a 2 532.u even 6 1
912.1.bl.a 2 1596.bw odd 6 1
1083.1.b.a 1 133.p even 6 1
1083.1.b.a 1 399.q odd 6 1
1083.1.b.b 1 133.m odd 6 1
1083.1.b.b 1 399.z even 6 1
1083.1.h.a 2 133.c even 2 1
1083.1.h.a 2 133.p even 6 1
1083.1.h.a 2 399.h odd 2 1
1083.1.h.a 2 399.q odd 6 1
1083.1.l.a 6 133.y odd 18 6
1083.1.l.a 6 399.cj even 18 6
1083.1.l.b 6 133.ba even 18 6
1083.1.l.b 6 399.bx odd 18 6
1425.1.o.a 4 35.f even 4 2
1425.1.o.a 4 105.k odd 4 2
1425.1.o.a 4 665.ci even 12 2
1425.1.o.a 4 1995.dx odd 12 2
1425.1.t.a 2 35.c odd 2 1
1425.1.t.a 2 105.g even 2 1
1425.1.t.a 2 665.bh odd 6 1
1425.1.t.a 2 1995.bl even 6 1
1539.1.j.a 2 63.l odd 6 1
1539.1.j.a 2 63.o even 6 1
1539.1.j.a 2 1197.bn odd 6 1
1539.1.j.a 2 1197.du even 6 1
1539.1.n.a 2 63.l odd 6 1
1539.1.n.a 2 63.o even 6 1
1539.1.n.a 2 1197.ba even 6 1
1539.1.n.a 2 1197.ec odd 6 1
2793.1.n.a 2 7.d odd 6 1
2793.1.n.a 2 21.g even 6 1
2793.1.n.a 2 133.k odd 6 1
2793.1.n.a 2 399.bd even 6 1
2793.1.n.b 2 7.c even 3 1
2793.1.n.b 2 21.h odd 6 1
2793.1.n.b 2 133.g even 3 1
2793.1.n.b 2 399.bi odd 6 1
2793.1.bf.a 2 1.a even 1 1 trivial
2793.1.bf.a 2 3.b odd 2 1 CM
2793.1.bf.a 2 19.c even 3 1 inner
2793.1.bf.a 2 57.h odd 6 1 inner
2793.1.bi.a 2 7.c even 3 1
2793.1.bi.a 2 21.h odd 6 1
2793.1.bi.a 2 133.h even 3 1
2793.1.bi.a 2 399.n 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.t odd 6 1
2793.1.bi.b 2 399.p even 6 1
3648.1.bl.a 2 56.e even 2 1
3648.1.bl.a 2 168.e odd 2 1
3648.1.bl.a 2 1064.ch even 6 1
3648.1.bl.a 2 3192.du odd 6 1
3648.1.bl.b 2 56.h odd 2 1
3648.1.bl.b 2 168.i even 2 1
3648.1.bl.b 2 1064.bg odd 6 1
3648.1.bl.b 2 3192.fh even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(2793,[χ])S_{1}^{\mathrm{new}}(2793, [\chi]):

T2 T_{2} Copy content Toggle raw display
T132+T13+1 T_{13}^{2} + T_{13} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 (T−1)2 (T - 1)^{2} Copy content Toggle raw display
3737 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
4747 T2 T^{2} Copy content Toggle raw display
5353 T2 T^{2} Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
6767 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
7979 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
8383 T2 T^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 T2−2T+4 T^{2} - 2T + 4 Copy content Toggle raw display
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