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Properties

Label 1027.1.o.a
Level 10271027
Weight 11
Character orbit 1027.o
Analytic conductor 0.5130.513
Analytic rank 00
Dimension 22
Projective image D3D_{3}
CM discriminant -79
Inner twists 44

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1027,1,Mod(315,1027)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1027, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) 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("1027.315"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: N N == 1027=13⋅79 1027 = 13 \cdot 79
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1027.o (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.5125397679740.512539767974
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: x2−x+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 3.1.13351.1
Artin image: C3×S3C_3\times S_3
Artin field: Galois closure of 6.0.83323591.1

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) == q−ζ62q2−q5+q8−ζ6q9+ζ62q10−ζ62q11+q13−ζ62q16−q18+ζ6q19−ζ6q22−ζ62q23+⋯−q99+O(q100) q - \zeta_{6}^{2} q^{2} - q^{5} + q^{8} - \zeta_{6} q^{9} + \zeta_{6}^{2} q^{10} - \zeta_{6}^{2} q^{11} + q^{13} - \zeta_{6}^{2} q^{16} - q^{18} + \zeta_{6} q^{19} - \zeta_{6} q^{22} - \zeta_{6}^{2} q^{23} + \cdots - q^{99} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q+q2−2q5+2q8−q9−q10+q11+2q13+q16−2q18+q19−q22+q23+q26−2q31+2q38−2q40+q45−q46−q49−q55+⋯−2q99+O(q100) 2 q + q^{2} - 2 q^{5} + 2 q^{8} - q^{9} - q^{10} + q^{11} + 2 q^{13} + q^{16} - 2 q^{18} + q^{19} - q^{22} + q^{23} + q^{26} - 2 q^{31} + 2 q^{38} - 2 q^{40} + q^{45} - q^{46} - q^{49} - q^{55}+ \cdots - 2 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1027Z)×\left(\mathbb{Z}/1027\mathbb{Z}\right)^\times.

nn 8080 872872
χ(n)\chi(n) −ζ6-\zeta_{6} −1-1

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}
315.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0 −1.00000 0 0 1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i
789.1 0.500000 − 0.866025i 0 0 −1.00000 0 0 1.00000 −0.500000 − 0.866025i −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
79.b odd 2 1 CM by Q(−79)\Q(\sqrt{-79})
13.c even 3 1 inner
1027.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1027.1.o.a ✓ 2
13.c even 3 1 inner 1027.1.o.a ✓ 2
79.b odd 2 1 CM 1027.1.o.a ✓ 2
1027.o odd 6 1 inner 1027.1.o.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1027.1.o.a ✓ 2 1.a even 1 1 trivial
1027.1.o.a ✓ 2 13.c even 3 1 inner
1027.1.o.a ✓ 2 79.b odd 2 1 CM
1027.1.o.a ✓ 2 1027.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T22−T2+1 T_{2}^{2} - T_{2} + 1 acting on S1new(1027,[χ])S_{1}^{\mathrm{new}}(1027, [\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 T2 T^{2} Copy content Toggle raw display
55 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
1313 (T−1)2 (T - 1)^{2} Copy content Toggle raw display
1717 T2 T^{2} Copy content Toggle raw display
1919 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
2323 T2−T+1 T^{2} - T + 1 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 T2 T^{2} Copy content Toggle raw display
4141 T2 T^{2} Copy content Toggle raw display
4343 T2 T^{2} 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^{2} Copy content Toggle raw display
6767 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
7171 T2 T^{2} Copy content Toggle raw display
7373 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
7979 (T−1)2 (T - 1)^{2} Copy content Toggle raw display
8383 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
8989 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
9797 T2−T+1 T^{2} - T + 1 Copy content Toggle raw display
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