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A248395 q-Expansion of the modular form of weight 3/2, g*theta(4) in Tunnell's notation (see Comments).

%I A248395 #27 May 17 2025 07:31:06
%S A248395 0,1,0,0,0,2,0,0,0,-1,0,0,0,-2,0,0,0,0,0,0,0,-4,0,0,0,-1,0,0,0,2,0,0,
%T A248395 0,-4,0,0,0,2,0,0,0,4,0,0,0,2,0,0,0,1,0,0,0,-2,0,0,0,4,0,0,0,2,0,0,0,
%U A248395 4,0,0,0,0,0,0,0,-4,0,0,0,0
%N A248395 q-Expansion of the modular form of weight 3/2, g*theta(4) in Tunnell's notation (see Comments).
%C A248395 g = Product_{m>=1} (1-q^(8*m))*(1-q^(16*m)),
%C A248395 theta(t) = Sum_{n=-oo..oo} (q^(t*n^2)).
%C A248395 Although the OEIS does not normally include sequences in which only every fourth term is nonzero, this one is important enough to warrant an exception.
%H A248395 G. C. Greubel, <a href="/A248395/b248395.txt">Table of n, a(n) for n = 0..5000</a>
%H A248395 J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334.
%F A248395 From _G. C. Greubel_, Jul 02 2018: (Start)
%F A248395 Expansion of eta(q^8)*eta(q^16)*theta_{3}(0, q^4)/q in powers of q.
%F A248395 Expansion of eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)). (End)
%p A248395 # This produces a list of the first 100 terms:
%p A248395 g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
%p A248395 g:=series(g,q,100);
%p A248395 th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
%p A248395 series(g*th(4),q,100);
%p A248395 seriestolist(%);
%t A248395 QP := QPochhammer; a:= CoefficientList[Series[QP[q^8]*QP[q^16]* EllipticTheta[3, 0, q^4], {q, 0, 60}], q]; Join[{0}, Table[a[[n]], {n, 1, 50}]] (* _G. C. Greubel_, Jul 02 2018 *)
%o A248395 (PARI) my(q='q+O('q^80)); A = eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)); concat([0], Vec(A)) \\ _G. C. Greubel_, Jul 02 2018
%Y A248395 The nonzero quadrisection is A080966, which has further information and references.
%Y A248395 Cf. A248394.
%Y A248395 Used in A248397-A248406.
%K A248395 sign
%O A248395 0,6
%A A248395 _N. J. A. Sloane_, Oct 18 2014