We define various twisted \theta-series, which we denote by X_i, where i is just an index, and the letter is that of twice the weight (so B would be weight 1, and E would be weight 5/2).
See here for some notes about computation about some theory about \eta-quotients.
We have weighted lattice sums for
Weight 1/2
A_1(q)=\eta(q)^1=q^{1/24}\sum_n (-1)^n q^{n(3n+1)/2}
A_2(q)=\eta(q)^2\cdot\eta(q^2)^{-1}=\sum_n (-1)^n q^{n^2}
A_3(q)=\eta(q)^{-1}\cdot\eta(q^2)^2=q^{1/8}\sum_n q^{n(2n+1)}=q^{1/8}\sum_{n=0}^\infty q^{n(n+1)/2}
A_4(q)=\eta(q)^1\cdot\eta(q^2)^{-1}\cdot\eta(q^4)^1=q^{1/8}\sum_n (-1)^n q^{n(2n+1)}
A_5(q)=\eta(q)^{-1}\cdot\eta(q^2)^3\cdot\eta(q^4)^{-1}=q^{1/24}\sum_n (-1)^{n(n+1)/2} q^{n(3n+1)/2}
A_6(q)=\eta(q)^{-2}\cdot\eta(q^2)^5\cdot\eta(q^4)^{-2}=\sum_n q^{n^2}
A_7(q)=\eta(q)^1\cdot\eta(q^2)^{-1}\cdot\eta(q^3)^{-1}\cdot\eta(q^6)^2=\sum_n (-1)^nq^{3n^2+2n}
Weight 3/2
C_1(q)=\eta(q)^3=q^{1/8}\sum_n (4n+1)q^{n(2n+1)}=q^{1/8}\sum_{n=0}^\infty (-1)^n(2n+1)q^{n(n+1)/2}
C_2(q)=\eta(q)^{-2}\cdot\eta(q^2)^5=q^{1/3}\sum_n(-1)^n (3n+1)q^{3n^2+2n}
C_3(q)=\eta(q)^5\cdot\eta(q^2)^{-2}=q^{1/24}\sum_n (-1)^n (6n+1)q^{n(3n+1)/2}
C_4(q)=\eta(q)^2\cdot\eta(q^2)^{-1}\cdot\eta(q^4)^2=q^{1/3}\sum_n (3n+1)q^{3n^2+2n}
C_5(q)=\eta(q)^{-3}\cdot\eta(q^2)^9\cdot\eta(q^4)^{-3}=q^{1/8}\sum_n (-1)^n(4n+1)q^{n(2n+1)}=q^{1/8}\sum_{n=0}^\infty (-1)^{n(n-1)/2} (2n+1) q^{n(n+1)/2}
C_6(q)=\eta(q)^{-5}\cdot\eta(q^2)^{13}\cdot\eta(q^4)^{-5}=q^{1/24}\sum_n (-1)^{n(n-1)/2} (6n+1) q^{n(3n+1)/2}
D_1(q)=\eta(q)^{-4}\cdot\eta(q^2)^9\cdot\eta(q^4)^1\cdot\eta(q^8)^{-2}= \Biggl[\mathop{\sum\sum}_{m\>\rm{odd},n\equiv 2 (4)\atop \chi_{24}(m)=(-1)^{(n-2)/4}} nq^{(m^2+3n^2)/12}+\mathop{\sum\sum}_{m\>\rm{odd},n\equiv 0 (4)\atop \chi_{-24}(m)=(-1)^{n/4}} mq^{(m^2+3n^2)/12} \Biggr]
D_2(q)=\eta(q)^{-4}\cdot\eta(q^2)^{11}\cdot\eta(q^4)^{-5}\cdot\eta(q^8)^2= {1\over 2}\Biggl[\mathop{\sum\sum}_{n\>\rm{odd}, m\equiv 2(4)\atop \chi_{-8}(n)=\chi(12,m/2)} nq^{(m^2+3n^2)/12}+\mathop{\sum\sum}_{n\>\rm{odd}, m\equiv 0(4)\atop \chi_8(n)=\psi_1(m/4)} mq^{(m^2+3n^2)/12}\Biggr]
D_3(q)=\eta(q)^{-1}\cdot\eta(q^2)^2\cdot\eta(q^3)^1\cdot\eta(q^4)^1\cdot\eta(q^6)^{-1}\cdot\eta(q^{12})^2
D_4(q)=\eta(q)^1\cdot\eta(q^2)^{-1}\cdot\eta(q^3)^{-3}\cdot\eta(q^4)^2\cdot\eta(q^6)^2\cdot\eta(q^{12})
F_1(q)=\eta(q)^{-1}\cdot\eta(q^2)^7={1\over 4} \mathop{\sum\sum}_{m,n, 3\not\mid m, 4\not\mid m,n\atop m+n\equiv 3,5\pmod{8}} mn q^{(m^2+9n^2)/24}
F_2(q)=\eta(q)^{-7}\cdot\eta(q^2)^{20}\eta(q^4)^{-7}={1\over 3}\mathop{\sum\sum}_{\chi_{-24}(m)=1\atop n\>\rm{even},\> \chi_{-36}(n/2)=1} \chi_{12}(m)\chi_{-3}(n)(m^2-n^2)q^{(m^2+n^2)/24}
The most useful expressions for some of the D,F functions seem to be GP/PARI scripts.
We should be able to compute the product of two forms of half-integral weight faster than a general convolution, as both are sparse.
These then give an efficient way of computing some of our Shimura lifts.