Triple Petersson Inner Products With Theta Functions

Our current work requires us to bound triple products of the form
$$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$ where $\langle \cdot, \cdot \rangle$ is the Petersson inner product, theta is, $$\theta(z) = \sum_{n\in\mathbb{Z}} e^{2\pi i n^2 z}$$ the weight 1/2 holomorphic form on $\Gamma_0(4)$, and $\mu_j(z)$ is a Maass form with eigenvalue $\frac{1}{4}+t_j^2$ on $\Gamma_0(4).$

We know that these have exponential decay in the $t_j$ aspect, but we're looking for a citation that gives something specific along the lines of $\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle \sim |t_j|^\alpha e^{-\frac{\pi}{2}|t_j|}$, at least on average. The smaller we can make $\alpha$ the better.

The best we can currently find comes from Corollary 20 in Kıral, 2015 which seems to give that $\alpha \leq 7/2$ on average when we sum over $t_j$. Currently that's not quite small enough for our purposes.

Edit: Yes, sorry. Our question is, anybody know a smaller bound for alpha? Thanks!

• Technically there's no question here, though I think I can guess the intended one. – Kimball Nov 19 '16 at 14:26
• I don't know Maass forms, how do you see your inner product has exponential decay with respect to the eigenvalue ? Do you expand $\mu_j$ as a Fourier series ? – reuns Nov 19 '16 at 17:57