Properties

Label 336.4.q.b
Level 336336
Weight 44
Character orbit 336.q
Analytic conductor 19.82519.825
Analytic rank 00
Dimension 22
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,4,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.193"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 336=2437 336 = 2^{4} \cdot 3 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 336.q (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 19.824641761919.8246417619
Analytic rank: 00
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,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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+(3ζ63)q32ζ6q5+(7ζ614)q79ζ6q9+(18ζ618)q11+33q13+6q15+(68ζ6+68)q17+25ζ6q19++162q99+O(q100) q + (3 \zeta_{6} - 3) q^{3} - 2 \zeta_{6} q^{5} + ( - 7 \zeta_{6} - 14) q^{7} - 9 \zeta_{6} q^{9} + (18 \zeta_{6} - 18) q^{11} + 33 q^{13} + 6 q^{15} + ( - 68 \zeta_{6} + 68) q^{17} + 25 \zeta_{6} q^{19} + \cdots + 162 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q3q32q535q79q918q11+66q13+12q15+68q17+25q19+84q21+92q23+121q25+54q27+184q29+25q3154q33+14q35++324q99+O(q100) 2 q - 3 q^{3} - 2 q^{5} - 35 q^{7} - 9 q^{9} - 18 q^{11} + 66 q^{13} + 12 q^{15} + 68 q^{17} + 25 q^{19} + 84 q^{21} + 92 q^{23} + 121 q^{25} + 54 q^{27} + 184 q^{29} + 25 q^{31} - 54 q^{33} + 14 q^{35}+ \cdots + 324 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/336Z)×\left(\mathbb{Z}/336\mathbb{Z}\right)^\times.

nn 8585 113113 127127 241241
χ(n)\chi(n) 11 11 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.

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}
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 + 2.59808i 0 −1.00000 1.73205i 0 −17.5000 6.06218i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −1.00000 + 1.73205i 0 −17.5000 + 6.06218i 0 −4.50000 + 7.79423i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.b 2
4.b odd 2 1 168.4.q.b 2
7.c even 3 1 inner 336.4.q.b 2
7.c even 3 1 2352.4.a.bc 1
7.d odd 6 1 2352.4.a.j 1
12.b even 2 1 504.4.s.d 2
28.f even 6 1 1176.4.a.k 1
28.g odd 6 1 168.4.q.b 2
28.g odd 6 1 1176.4.a.d 1
84.n even 6 1 504.4.s.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.b 2 4.b odd 2 1
168.4.q.b 2 28.g odd 6 1
336.4.q.b 2 1.a even 1 1 trivial
336.4.q.b 2 7.c even 3 1 inner
504.4.s.d 2 12.b even 2 1
504.4.s.d 2 84.n even 6 1
1176.4.a.d 1 28.g odd 6 1
1176.4.a.k 1 28.f even 6 1
2352.4.a.j 1 7.d odd 6 1
2352.4.a.bc 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T52+2T5+4 T_{5}^{2} + 2T_{5} + 4 acting on S4new(336,[χ])S_{4}^{\mathrm{new}}(336, [\chi]). 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+3T+9 T^{2} + 3T + 9 Copy content Toggle raw display
55 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
77 T2+35T+343 T^{2} + 35T + 343 Copy content Toggle raw display
1111 T2+18T+324 T^{2} + 18T + 324 Copy content Toggle raw display
1313 (T33)2 (T - 33)^{2} Copy content Toggle raw display
1717 T268T+4624 T^{2} - 68T + 4624 Copy content Toggle raw display
1919 T225T+625 T^{2} - 25T + 625 Copy content Toggle raw display
2323 T292T+8464 T^{2} - 92T + 8464 Copy content Toggle raw display
2929 (T92)2 (T - 92)^{2} Copy content Toggle raw display
3131 T225T+625 T^{2} - 25T + 625 Copy content Toggle raw display
3737 T2213T+45369 T^{2} - 213T + 45369 Copy content Toggle raw display
4141 (T94)2 (T - 94)^{2} Copy content Toggle raw display
4343 (T67)2 (T - 67)^{2} Copy content Toggle raw display
4747 T2278T+77284 T^{2} - 278T + 77284 Copy content Toggle raw display
5353 T2400T+160000 T^{2} - 400T + 160000 Copy content Toggle raw display
5959 T2744T+553536 T^{2} - 744T + 553536 Copy content Toggle raw display
6161 T2734T+538756 T^{2} - 734T + 538756 Copy content Toggle raw display
6767 T2555T+308025 T^{2} - 555T + 308025 Copy content Toggle raw display
7171 (T642)2 (T - 642)^{2} Copy content Toggle raw display
7373 T2+973T+946729 T^{2} + 973T + 946729 Copy content Toggle raw display
7979 T2+785T+616225 T^{2} + 785T + 616225 Copy content Toggle raw display
8383 (T822)2 (T - 822)^{2} Copy content Toggle raw display
8989 T2+424T+179776 T^{2} + 424T + 179776 Copy content Toggle raw display
9797 (T+734)2 (T + 734)^{2} Copy content Toggle raw display
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