Properties

Label 336.4.q.d
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,-3,0,15] 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,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 42)
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)q3+15ζ6q5+(7ζ621)q79ζ6q9+(9ζ69)q1188q1345q15+(84ζ6+84)q17+104ζ6q19++81q99+O(q100) q + (3 \zeta_{6} - 3) q^{3} + 15 \zeta_{6} q^{5} + (7 \zeta_{6} - 21) q^{7} - 9 \zeta_{6} q^{9} + (9 \zeta_{6} - 9) q^{11} - 88 q^{13} - 45 q^{15} + ( - 84 \zeta_{6} + 84) q^{17} + 104 \zeta_{6} q^{19} + \cdots + 81 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q3q3+15q535q79q99q11176q1390q15+84q17+104q19+21q2184q23100q25+54q27+102q29+185q3127q33420q35++162q99+O(q100) 2 q - 3 q^{3} + 15 q^{5} - 35 q^{7} - 9 q^{9} - 9 q^{11} - 176 q^{13} - 90 q^{15} + 84 q^{17} + 104 q^{19} + 21 q^{21} - 84 q^{23} - 100 q^{25} + 54 q^{27} + 102 q^{29} + 185 q^{31} - 27 q^{33} - 420 q^{35}+ \cdots + 162 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 7.50000 + 12.9904i 0 −17.5000 + 6.06218i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 7.50000 12.9904i 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.d 2
4.b odd 2 1 42.4.e.b 2
7.c even 3 1 inner 336.4.q.d 2
7.c even 3 1 2352.4.a.u 1
7.d odd 6 1 2352.4.a.q 1
12.b even 2 1 126.4.g.a 2
28.d even 2 1 294.4.e.e 2
28.f even 6 1 294.4.a.g 1
28.f even 6 1 294.4.e.e 2
28.g odd 6 1 42.4.e.b 2
28.g odd 6 1 294.4.a.a 1
84.h odd 2 1 882.4.g.l 2
84.j odd 6 1 882.4.a.h 1
84.j odd 6 1 882.4.g.l 2
84.n even 6 1 126.4.g.a 2
84.n even 6 1 882.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 4.b odd 2 1
42.4.e.b 2 28.g odd 6 1
126.4.g.a 2 12.b even 2 1
126.4.g.a 2 84.n even 6 1
294.4.a.a 1 28.g odd 6 1
294.4.a.g 1 28.f even 6 1
294.4.e.e 2 28.d even 2 1
294.4.e.e 2 28.f even 6 1
336.4.q.d 2 1.a even 1 1 trivial
336.4.q.d 2 7.c even 3 1 inner
882.4.a.h 1 84.j odd 6 1
882.4.a.r 1 84.n even 6 1
882.4.g.l 2 84.h odd 2 1
882.4.g.l 2 84.j odd 6 1
2352.4.a.q 1 7.d odd 6 1
2352.4.a.u 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5215T5+225 T_{5}^{2} - 15T_{5} + 225 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 T215T+225 T^{2} - 15T + 225 Copy content Toggle raw display
77 T2+35T+343 T^{2} + 35T + 343 Copy content Toggle raw display
1111 T2+9T+81 T^{2} + 9T + 81 Copy content Toggle raw display
1313 (T+88)2 (T + 88)^{2} Copy content Toggle raw display
1717 T284T+7056 T^{2} - 84T + 7056 Copy content Toggle raw display
1919 T2104T+10816 T^{2} - 104T + 10816 Copy content Toggle raw display
2323 T2+84T+7056 T^{2} + 84T + 7056 Copy content Toggle raw display
2929 (T51)2 (T - 51)^{2} Copy content Toggle raw display
3131 T2185T+34225 T^{2} - 185T + 34225 Copy content Toggle raw display
3737 T2+44T+1936 T^{2} + 44T + 1936 Copy content Toggle raw display
4141 (T+168)2 (T + 168)^{2} Copy content Toggle raw display
4343 (T+326)2 (T + 326)^{2} Copy content Toggle raw display
4747 T2+138T+19044 T^{2} + 138T + 19044 Copy content Toggle raw display
5353 T2+639T+408321 T^{2} + 639T + 408321 Copy content Toggle raw display
5959 T2159T+25281 T^{2} - 159T + 25281 Copy content Toggle raw display
6161 T2+722T+521284 T^{2} + 722T + 521284 Copy content Toggle raw display
6767 T2+166T+27556 T^{2} + 166T + 27556 Copy content Toggle raw display
7171 (T+1086)2 (T + 1086)^{2} Copy content Toggle raw display
7373 T2+218T+47524 T^{2} + 218T + 47524 Copy content Toggle raw display
7979 T2+583T+339889 T^{2} + 583T + 339889 Copy content Toggle raw display
8383 (T597)2 (T - 597)^{2} Copy content Toggle raw display
8989 T21038T+1077444 T^{2} - 1038 T + 1077444 Copy content Toggle raw display
9797 (T+169)2 (T + 169)^{2} Copy content Toggle raw display
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