Odd twists
The weight 5/2 lift for odd positive twists of the newform 8k4 is given by \eta(q^8)^3\cdot G^2 where
G= 1-2q^8+2q^{32}-2q^{72}+-\dots=\sum_n (-1)^n q^{8n^2}
The series for \eta(q^8)^3 is itself lacunary, and G^2 can be written as a lattice sum simply by squaring, to get
\biggl(\sum_{n=0}^\infty (-1)^n(2n+1)q^{(2n+1)^2}\biggr)\cdot \biggl(\sum_a\sum_b (-1)^{a+b} q^{8(a^2+b^2)}\biggr)=
=q - 7q^9 + 16q^{17} - 7q^{25} - 16q^{33} + 17q^{49} + 48q^{57} - 64q^{65} - 16q^{73} + q^{81} + 16q^{89} + 16q^{97}+O(q^{100})
An alternative expression for the weight 5/2 lift is to write it as a product of a \Theta-series (weight 1/2) and the modular form associated to the congruent number curve (weight 2).
\Bigl(\sum_n q^{4n^2}\Bigr)\cdot\biggl(\mathop{\sum\sum}_{a,b\equiv 3,2 (4)\atop a,b\equiv 1,0 (4)} a\,q^{a^2+b^2}\biggr)
Note that the multiplication kills off the coefficients of all exponents that are 5 mod 8, and so the previous formula should be superior to this one.
Even twists
The lift for even twists is of level 64, and is given by \eta(q^8)^3\cdot F^2 where
F= 1 - 2q^4 + 2q^{16} - 2q^{36} + 2q^{64} - + \dots=\sum_n (-1)^n q^{4n^2}
As above, this can be reduced to one convolution as
\biggl(\sum_{n=0}^\infty (-1)^n(2n+1)q^{(2n+1)^2}\biggr)\cdot \biggl(\sum_a\sum_b (-1)^{a+b} q^{4(a^2+b^2)}\biggr)
=q-4q^5+q^9 + 12q^{13} - 8q^{17} - 8q^{21} - 7q^{25} + 4q^{29} + 24q^{33} - 4q^{37}+ 16q^{41} - 28q^{45}-31q^{49}+20q^{53}-8q^{57} +
+ 4q^{61} + 16q^{65} + 48q^{69} - 8q^{73} - 16q^{77} - 23q^{81} - 8q^{85} - 24q^{89} - 40q^{93} + 56q^{97}+O(q^{100})