See the Products page for the various products that are used.
Computation
It need not be immediately clear how useful these are for efficient computation of the q-coefficients. However, we can note that various \eta-quotients (of weight 3/2) can be written as linearly weighted \theta-functions (of rank 1). For instance, in other work (see the 8k4 page) we have exploited this for \eta(q^8)^3. It turns out that there is a similar formula for
These can be used for the above. For instance, the first weight 7/2 lift above has exponent vector
In fact we find that each of the 12 forms can be written as a product \varphi_1 \varphi_2 of \eta-quotients where \varphi_1 is sparse of weight 3/2 and \varphi_2 is CM of weight 1, 2, or 3. This \varphi_2 can be expressed as a weighted \theta-function of a rank-2 lattice. Thus all the q-expansions should now be about as easy to compute as was the case for the forms of weight 5/2. The factorizations noted in earlier paragraphs as the products of three forms of weight 3/2 or 1/2 include the \varphi_1 \varphi_2 form as a special case, because \varphi_2 can be taken to be the product of two of the factors, and indeed that is how the fast computation was organized in the weight 5/2 case. For the remaining forms (two of weight 7/2 and all four of weight 9/2), \varphi_2 is \eta(q^8)^a\eta(q^{16})^b\eta(q^{32})^c\eta(q^{64})^d with exponent vector
Eta-product transformations
The exponent vectors that arise aren't quite as random as they appear: they form orbits under some transformations that preserve both the multiplicative span of \{ \eta(q^{2^n}) \; | \; n \in {\bf Z} \} and the subset of this span consisting of weighted theta functions of lattices of a given rank. Suppose a_n is the exponent of \eta(q^{2^n}). Usually a_n=0 for n<0. Then we can:
(i) Most obviously, replace q by q^{2^m} for some integer m; this translates the indices so a_n becomes a_{n+m}. We can use this to remove leading zeros from the coefficient vector.
(ii) Apply a Fricke involution w taking \tau to -2^m/\tau for some integer m; this reverses the order of the indices, so a_n becomes a_{m-n} (and multiplies the form by some power of 2^{1/2}). Weighted thetas are preserved thanks to Poisson.
(iii) Replace q by -q (i.e. translate \tau by 1/2). This clearly takes weighted thetas to weighted thetas. It also preserves the factor \eta(q^{2^n}) for each n>0, but takes \eta(q) to \eta(q^2)^3 / (\eta(q)\eta(q^4)). Since we assume that a_n=0 for n<0, we thus have an involution that takes [a_0, a_1, a_2, a_3, a_4, \ldots] to [-a_0, a_1+3a_0, a_2-a_0, a_3, a_4, \ldots].
Our eta products with weighted-theta expansions then group as follows under these transformations.
\bullet Weight 1/2: Our ten exponent vectors [a_0,a_1,a_2] form two orbits:
\bullet Weight 3/2: Again ten vectors but only two orbits, one obtained by tripling our first weight-1/2 orbit to
\bullet Weight 3: The involution (iii) takes [-1,7,0,0] to [-7,20,-7,0], and its conjugate by Fricke (ii) takes [-4,9,1,-2] via [-2,1,9,-4] and [2,-5,11,-4] to [-4,11,-5,2].
Note too that some of our target forms of half-integral weight are also related by the same transformations; e.g. the conjugate of (iii) by (ii) pairs [-1,12,-2] with [1,6,2] and [-7,20,-2,-2] with [-7,22,-8,2].
Other eta-quotients
However, the above does not tell us much about A_7(q), or D_3(q) or D_4(q), or the S_1(q) and S_2(q) for 27A.
As references, we might give http://www.math.wisc.edu/~ono/reprints/030.pdf which shows that the only half-integral weights that we should obtain are 1/2 and 3/2. Also related is a paper of Ono and Robins: http://www.math.wisc.edu/~ono/reprints/004.pdf which lists some formulae for integral weights. The two-term examples of integral weight were catalogued by Gordon and Robins in "Lacunarity of Dedekind eta-products", Glasgow Math. J. 37 (1995), 1--14, MR 1316958 96d:11044, and they claim that their work can be generalised. A more recent REU has also explored some of these questions: http://www.math.wisc.edu/~ono/reu08lacunarity.pdf