Properties

Label 733.2.e.a
Level 733733
Weight 22
Character orbit 733.e
Analytic conductor 5.8535.853
Analytic rank 11
Dimension 22
Inner twists 22

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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] = [733,2,Mod(308,733)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("733.308"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(733, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 2, names="a")
 
Level: N N == 733 733
Weight: k k == 2 2
Character orbit: [χ][\chi] == 733.e (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 5.853034468165.85303446816
Analytic rank: 11
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
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,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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 ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(2ζ6+1)q2+(2ζ62)q3q4+(2ζ6+1)q5+(2ζ6+2)q6+(2ζ62)q7+(2ζ6+1)q8ζ6q93q10++(2ζ6+4)q99+O(q100) q + ( - 2 \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + (2 \zeta_{6} + 2) q^{6} + ( - 2 \zeta_{6} - 2) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} - \zeta_{6} q^{9} - 3 q^{10} + \cdots + ( - 2 \zeta_{6} + 4) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q32q4+6q66q7q96q10+2q126q14+6q1510q163q173q184q19+12q21+12q2212q23+6q24+4q258q27++6q99+O(q100) 2 q - 2 q^{3} - 2 q^{4} + 6 q^{6} - 6 q^{7} - q^{9} - 6 q^{10} + 2 q^{12} - 6 q^{14} + 6 q^{15} - 10 q^{16} - 3 q^{17} - 3 q^{18} - 4 q^{19} + 12 q^{21} + 12 q^{22} - 12 q^{23} + 6 q^{24} + 4 q^{25} - 8 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/733Z)×\left(\mathbb{Z}/733\mathbb{Z}\right)^\times.

nn 66
χ(n)\chi(n) 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}
308.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i −1.00000 1.73205i −1.00000 1.73205i 3.00000 1.73205i −3.00000 + 1.73205i 1.73205i −0.500000 + 0.866025i −3.00000
426.1 1.73205i −1.00000 + 1.73205i −1.00000 1.73205i 3.00000 + 1.73205i −3.00000 1.73205i 1.73205i −0.500000 0.866025i −3.00000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
733.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 733.2.e.a 2
733.e even 6 1 inner 733.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
733.2.e.a 2 1.a even 1 1 trivial
733.2.e.a 2 733.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(733,[χ])S_{2}^{\mathrm{new}}(733, [\chi]):

T22+3 T_{2}^{2} + 3 Copy content Toggle raw display
T32+2T3+4 T_{3}^{2} + 2T_{3} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+3 T^{2} + 3 Copy content Toggle raw display
33 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
55 T2+3 T^{2} + 3 Copy content Toggle raw display
77 T2+6T+12 T^{2} + 6T + 12 Copy content Toggle raw display
1111 T2+12 T^{2} + 12 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
1919 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
2323 T2+12T+48 T^{2} + 12T + 48 Copy content Toggle raw display
2929 T2 T^{2} Copy content Toggle raw display
3131 T24T+16 T^{2} - 4T + 16 Copy content Toggle raw display
3737 T2T+1 T^{2} - T + 1 Copy content Toggle raw display
4141 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
4343 T2+18T+108 T^{2} + 18T + 108 Copy content Toggle raw display
4747 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
5353 T2+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
5959 T2+6T+36 T^{2} + 6T + 36 Copy content Toggle raw display
6161 T27T+49 T^{2} - 7T + 49 Copy content Toggle raw display
6767 T2+48 T^{2} + 48 Copy content Toggle raw display
7171 T2+6T+12 T^{2} + 6T + 12 Copy content Toggle raw display
7373 T2+14T+196 T^{2} + 14T + 196 Copy content Toggle raw display
7979 T216T+256 T^{2} - 16T + 256 Copy content Toggle raw display
8383 T2+12T+144 T^{2} + 12T + 144 Copy content Toggle raw display
8989 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
9797 T2+13T+169 T^{2} + 13T + 169 Copy content Toggle raw display
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