Properties

Label 52.2.l.a
Level 5252
Weight 22
Character orbit 52.l
Analytic conductor 0.4150.415
Analytic rank 00
Dimension 44
CM discriminant -4
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [52,2,Mod(7,52)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(52, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 11])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("52.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 52=2213 52 = 2^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 52.l (of order 1212, degree 44, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: 0.4152220905110.415222090511
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ12)\Q(\zeta_{12})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: U(1)[D12]\mathrm{U}(1)[D_{12}]

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)
 

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ12\zeta_{12}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ123ζ122ζ12)q2+2ζ12q4+(2ζ123ζ122++2)q5+(2ζ1232)q8+(3ζ1223)q9++(7ζ1227ζ127)q98+O(q100) q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots + 2) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+ \cdots + (7 \zeta_{12}^{2} - 7 \zeta_{12} - 7) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q2q2+6q58q86q96q10+4q13+8q1624q17+12q18+12q20+10q26+4q29+8q324q34+26q3724q4028q4116q50+14q98+O(q100) 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9} - 6 q^{10} + 4 q^{13} + 8 q^{16} - 24 q^{17} + 12 q^{18} + 12 q^{20} + 10 q^{26} + 4 q^{29} + 8 q^{32} - 4 q^{34} + 26 q^{37} - 24 q^{40} - 28 q^{41} - 16 q^{50}+ \cdots - 14 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/52Z)×\left(\mathbb{Z}/52\mathbb{Z}\right)^\times.

nn 2727 4141
χ(n)\chi(n) 1-1 ζ12\zeta_{12}

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}
7.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.36603 + 0.366025i 0 1.73205 1.00000i 2.36603 + 2.36603i 0 0 −2.00000 + 2.00000i −1.50000 2.59808i −4.09808 2.36603i
11.1 0.366025 1.36603i 0 −1.73205 1.00000i 0.633975 + 0.633975i 0 0 −2.00000 + 2.00000i −1.50000 + 2.59808i 1.09808 0.633975i
15.1 −1.36603 0.366025i 0 1.73205 + 1.00000i 2.36603 2.36603i 0 0 −2.00000 2.00000i −1.50000 + 2.59808i −4.09808 + 2.36603i
19.1 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 0.633975 0.633975i 0 0 −2.00000 2.00000i −1.50000 2.59808i 1.09808 + 0.633975i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by Q(1)\Q(\sqrt{-1})
13.f odd 12 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.l.a 4
3.b odd 2 1 468.2.cb.d 4
4.b odd 2 1 CM 52.2.l.a 4
8.b even 2 1 832.2.bu.d 4
8.d odd 2 1 832.2.bu.d 4
12.b even 2 1 468.2.cb.d 4
13.b even 2 1 676.2.l.d 4
13.c even 3 1 676.2.f.e 4
13.c even 3 1 676.2.l.c 4
13.d odd 4 1 676.2.l.c 4
13.d odd 4 1 676.2.l.e 4
13.e even 6 1 676.2.f.d 4
13.e even 6 1 676.2.l.e 4
13.f odd 12 1 inner 52.2.l.a 4
13.f odd 12 1 676.2.f.d 4
13.f odd 12 1 676.2.f.e 4
13.f odd 12 1 676.2.l.d 4
39.k even 12 1 468.2.cb.d 4
52.b odd 2 1 676.2.l.d 4
52.f even 4 1 676.2.l.c 4
52.f even 4 1 676.2.l.e 4
52.i odd 6 1 676.2.f.d 4
52.i odd 6 1 676.2.l.e 4
52.j odd 6 1 676.2.f.e 4
52.j odd 6 1 676.2.l.c 4
52.l even 12 1 inner 52.2.l.a 4
52.l even 12 1 676.2.f.d 4
52.l even 12 1 676.2.f.e 4
52.l even 12 1 676.2.l.d 4
104.u even 12 1 832.2.bu.d 4
104.x odd 12 1 832.2.bu.d 4
156.v odd 12 1 468.2.cb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.a 4 1.a even 1 1 trivial
52.2.l.a 4 4.b odd 2 1 CM
52.2.l.a 4 13.f odd 12 1 inner
52.2.l.a 4 52.l even 12 1 inner
468.2.cb.d 4 3.b odd 2 1
468.2.cb.d 4 12.b even 2 1
468.2.cb.d 4 39.k even 12 1
468.2.cb.d 4 156.v odd 12 1
676.2.f.d 4 13.e even 6 1
676.2.f.d 4 13.f odd 12 1
676.2.f.d 4 52.i odd 6 1
676.2.f.d 4 52.l even 12 1
676.2.f.e 4 13.c even 3 1
676.2.f.e 4 13.f odd 12 1
676.2.f.e 4 52.j odd 6 1
676.2.f.e 4 52.l even 12 1
676.2.l.c 4 13.c even 3 1
676.2.l.c 4 13.d odd 4 1
676.2.l.c 4 52.f even 4 1
676.2.l.c 4 52.j odd 6 1
676.2.l.d 4 13.b even 2 1
676.2.l.d 4 13.f odd 12 1
676.2.l.d 4 52.b odd 2 1
676.2.l.d 4 52.l even 12 1
676.2.l.e 4 13.d odd 4 1
676.2.l.e 4 13.e even 6 1
676.2.l.e 4 52.f even 4 1
676.2.l.e 4 52.i odd 6 1
832.2.bu.d 4 8.b even 2 1
832.2.bu.d 4 8.d odd 2 1
832.2.bu.d 4 104.u even 12 1
832.2.bu.d 4 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3 T_{3} acting on S2new(52,[χ])S_{2}^{\mathrm{new}}(52, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+2T3++4 T^{4} + 2 T^{3} + \cdots + 4 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T46T3++9 T^{4} - 6 T^{3} + \cdots + 9 Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 T44T3++169 T^{4} - 4 T^{3} + \cdots + 169 Copy content Toggle raw display
1717 T4+24T3++2209 T^{4} + 24 T^{3} + \cdots + 2209 Copy content Toggle raw display
1919 T4 T^{4} Copy content Toggle raw display
2323 T4 T^{4} Copy content Toggle raw display
2929 T44T3++5041 T^{4} - 4 T^{3} + \cdots + 5041 Copy content Toggle raw display
3131 T4 T^{4} Copy content Toggle raw display
3737 T426T3++3721 T^{4} - 26 T^{3} + \cdots + 3721 Copy content Toggle raw display
4141 T4+28T3++14641 T^{4} + 28 T^{3} + \cdots + 14641 Copy content Toggle raw display
4343 T4 T^{4} Copy content Toggle raw display
4747 T4 T^{4} Copy content Toggle raw display
5353 (T214T+37)2 (T^{2} - 14 T + 37)^{2} Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 T410T3++6889 T^{4} - 10 T^{3} + \cdots + 6889 Copy content Toggle raw display
6767 T4 T^{4} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 T422T3++529 T^{4} - 22 T^{3} + \cdots + 529 Copy content Toggle raw display
7979 T4 T^{4} Copy content Toggle raw display
8383 T4 T^{4} Copy content Toggle raw display
8989 T4+6T3++324 T^{4} + 6 T^{3} + \cdots + 324 Copy content Toggle raw display
9797 T4+10T3++2500 T^{4} + 10 T^{3} + \cdots + 2500 Copy content Toggle raw display
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