Properties

Label 140.1.p.a
Level 140140
Weight 11
Character orbit 140.p
Analytic conductor 0.0700.070
Analytic rank 00
Dimension 22
Projective image D3D_{3}
CM discriminant -20
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [140,1,Mod(39,140)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(140, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("140.39"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: N N == 140=2257 140 = 2^{2} \cdot 5 \cdot 7
Weight: k k == 1 1
Character orbit: [χ][\chi] == 140.p (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.06986910176860.0698691017686
Analytic rank: 00
Dimension: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D3D_{3}
Projective field: Galois closure of 3.1.980.1
Artin image: C3×S3C_3\times S_3
Artin field: Galois closure of 6.0.392000.1

qq-expansion

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

The qq-expansion and trace form are shown below.

f(q)f(q) == qζ6q2ζ62q3+ζ62q4ζ6q5q6+ζ62q7+q8+ζ62q10+ζ6q12+q14q15ζ6q16++ζ62q98+O(q100) q - \zeta_{6} q^{2} - \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{6} + \zeta_{6}^{2} q^{7} + q^{8} + \zeta_{6}^{2} q^{10} + \zeta_{6} q^{12} + q^{14} - q^{15} - \zeta_{6} q^{16} + \cdots + \zeta_{6}^{2} q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2qq2+q3q4q52q6q7+2q8q10+q12+2q142q15q16+2q20+q21+q23+q24q25+2q27q282q29+q98+O(q100) 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} - q^{7} + 2 q^{8} - q^{10} + q^{12} + 2 q^{14} - 2 q^{15} - q^{16} + 2 q^{20} + q^{21} + q^{23} + q^{24} - q^{25} + 2 q^{27} - q^{28} - 2 q^{29}+ \cdots - q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/140Z)×\left(\mathbb{Z}/140\mathbb{Z}\right)^\times.

nn 5757 7171 101101
χ(n)\chi(n) 1-1 1-1 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
39.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i
79.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by Q(5)\Q(\sqrt{-5})
7.c even 3 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.1.p.a 2
3.b odd 2 1 1260.1.ci.b 2
4.b odd 2 1 140.1.p.b yes 2
5.b even 2 1 140.1.p.b yes 2
5.c odd 4 2 700.1.u.a 4
7.b odd 2 1 980.1.p.a 2
7.c even 3 1 inner 140.1.p.a 2
7.c even 3 1 980.1.f.c 1
7.d odd 6 1 980.1.f.d 1
7.d odd 6 1 980.1.p.a 2
8.b even 2 1 2240.1.bt.a 2
8.d odd 2 1 2240.1.bt.b 2
12.b even 2 1 1260.1.ci.a 2
15.d odd 2 1 1260.1.ci.a 2
20.d odd 2 1 CM 140.1.p.a 2
20.e even 4 2 700.1.u.a 4
21.h odd 6 1 1260.1.ci.b 2
28.d even 2 1 980.1.p.b 2
28.f even 6 1 980.1.f.a 1
28.f even 6 1 980.1.p.b 2
28.g odd 6 1 140.1.p.b yes 2
28.g odd 6 1 980.1.f.b 1
35.c odd 2 1 980.1.p.b 2
35.i odd 6 1 980.1.f.a 1
35.i odd 6 1 980.1.p.b 2
35.j even 6 1 140.1.p.b yes 2
35.j even 6 1 980.1.f.b 1
35.l odd 12 2 700.1.u.a 4
40.e odd 2 1 2240.1.bt.a 2
40.f even 2 1 2240.1.bt.b 2
56.k odd 6 1 2240.1.bt.b 2
56.p even 6 1 2240.1.bt.a 2
60.h even 2 1 1260.1.ci.b 2
84.n even 6 1 1260.1.ci.a 2
105.o odd 6 1 1260.1.ci.a 2
140.c even 2 1 980.1.p.a 2
140.p odd 6 1 inner 140.1.p.a 2
140.p odd 6 1 980.1.f.c 1
140.s even 6 1 980.1.f.d 1
140.s even 6 1 980.1.p.a 2
140.w even 12 2 700.1.u.a 4
280.bf even 6 1 2240.1.bt.b 2
280.bi odd 6 1 2240.1.bt.a 2
420.ba even 6 1 1260.1.ci.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 1.a even 1 1 trivial
140.1.p.a 2 7.c even 3 1 inner
140.1.p.a 2 20.d odd 2 1 CM
140.1.p.a 2 140.p odd 6 1 inner
140.1.p.b yes 2 4.b odd 2 1
140.1.p.b yes 2 5.b even 2 1
140.1.p.b yes 2 28.g odd 6 1
140.1.p.b yes 2 35.j even 6 1
700.1.u.a 4 5.c odd 4 2
700.1.u.a 4 20.e even 4 2
700.1.u.a 4 35.l odd 12 2
700.1.u.a 4 140.w even 12 2
980.1.f.a 1 28.f even 6 1
980.1.f.a 1 35.i odd 6 1
980.1.f.b 1 28.g odd 6 1
980.1.f.b 1 35.j even 6 1
980.1.f.c 1 7.c even 3 1
980.1.f.c 1 140.p odd 6 1
980.1.f.d 1 7.d odd 6 1
980.1.f.d 1 140.s even 6 1
980.1.p.a 2 7.b odd 2 1
980.1.p.a 2 7.d odd 6 1
980.1.p.a 2 140.c even 2 1
980.1.p.a 2 140.s even 6 1
980.1.p.b 2 28.d even 2 1
980.1.p.b 2 28.f even 6 1
980.1.p.b 2 35.c odd 2 1
980.1.p.b 2 35.i odd 6 1
1260.1.ci.a 2 12.b even 2 1
1260.1.ci.a 2 15.d odd 2 1
1260.1.ci.a 2 84.n even 6 1
1260.1.ci.a 2 105.o odd 6 1
1260.1.ci.b 2 3.b odd 2 1
1260.1.ci.b 2 21.h odd 6 1
1260.1.ci.b 2 60.h even 2 1
1260.1.ci.b 2 420.ba even 6 1
2240.1.bt.a 2 8.b even 2 1
2240.1.bt.a 2 40.e odd 2 1
2240.1.bt.a 2 56.p even 6 1
2240.1.bt.a 2 280.bi odd 6 1
2240.1.bt.b 2 8.d odd 2 1
2240.1.bt.b 2 40.f even 2 1
2240.1.bt.b 2 56.k odd 6 1
2240.1.bt.b 2 280.bf even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T32T3+1 T_{3}^{2} - T_{3} + 1 acting on S1new(140,[χ])S_{1}^{\mathrm{new}}(140, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
33 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
55 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
77 T2+T+1 T^{2} + T + 1 Copy content Toggle raw display
1111 T2 T^{2} Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T2 T^{2} Copy content Toggle raw display
2323 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
2929 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
3131 T2 T^{2} Copy content Toggle raw display
3737 T2 T^{2} Copy content Toggle raw display
4141 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
4343 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
4747 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
5353 T2 T^{2} Copy content Toggle raw display
5959 T2 T^{2} Copy content Toggle raw display
6161 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
6767 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 T2 T^{2} Copy content Toggle raw display
7979 T2 T^{2} Copy content Toggle raw display
8383 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
8989 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
9797 T2 T^{2} Copy content Toggle raw display
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