cv.complex variables – The loss of double periodicity (ellipticity)

Consider a meromorphic function $f(mathfrak{a}_1, mathfrak{a}_2)$, such that
$$
begin{align}
f(mathfrak{a}_1, mathfrak{a}_2) = f (mathfrak{a}_1 + 1, mathfrak{a}_2) = f(mathfrak{a}_1 + tau, mathfrak{a}_2) = f(mathfrak{a}_1, mathfrak{a}_2 + 1) = f(mathfrak{a}_1, mathfrak{a}_2+ tau) .
end{align}
$$

So essentially $f$ is elliptic w.r.t. both $mathfrak{a}_{1,2}$.

My question is: consider the integral
$$
I(mathfrak{a}_1) equiv int_0^1 d mathfrak{a}_2 f(mathfrak{a}_1, mathfrak{a}_2) .
$$

Question: do we still have ellipticity $I(mathfrak{a}_1 + 1) = I (mathfrak{a}_1 + tau) = I(mathfrak{a}_1)$?

I once thought that the answer is of course positive, but recently I encounter an example where it is not.

Consider
$$
f(mathfrak{a}_1, mathfrak{a}_2) equiv frac{vartheta_1(2mathfrak{a}_2)^2}{
prod_pm vartheta_4(pm 2mathfrak{a}_2 – mathfrak{a}_1)}
frac{1}{vartheta_4(mathfrak{a}_1)^4}
frac{vartheta_1(2mathfrak{a}_1)^2}{prod_pm vartheta_4(pm2mathfrak{mathfrak{b}} + mathfrak{a}_1)} .
$$

The function is elliptic w.r.t. both $mathfrak{a}_{1,2}$.

Now we can integrate over $mathfrak{a}_2$ from $0 to 1$, using the formula
$$
int_0^1 d mathfrak{a}_2 prod_pm frac{vartheta_1(2mathfrak{a}_2)^2}{
vartheta_4(pm 2mathfrak{a}_2 – mathfrak{a}_1)} = frac{1}{ pieta(tau)^3 }frac{ vartheta’_4(mathfrak{a}_1) }{ vartheta_1(2mathfrak{a}_1) } .
$$

Now, due to the presence of $vartheta’_4(mathfrak{a}_1)$, it is easy to see that
$$
int_0^1 f(mathfrak{a}_1, mathfrak{a}_2) d mathfrak{a}_2
$$

is no longer elliptic in $mathfrak{a}_1$ (it is still invariant under the unit shift, but not so for $tau$-shift).

Why does the integration destroy the periodicity ?