A trace formula for convolution of Hecke series and its applications

A trace formula expressing the mean values of the form ($k=2,3,\dots$)
$$ \frac{\Gamma(2k-1)}{(4\pi)^{2k-1}}\sum_f\frac{\lambda_f(d)}{\langle f,f\rangle}\mathcal H_f^{(\chi)}(s_1)\overline{\mathcal H_f^{(\chi)}(\overline s_2)} $$
via certain arithmetic means on the group $\Gamma_0(N_1)$ is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight $2k$ with respect to $\Gamma_0(N_1)$. By $\mathcal H_f^{(\chi)}(s)$ we denote the Hecke series of the form $f$, twisted with the primitive character $\chi\pmod{N_2}$, and $\lambda_f(d)$, $(d,N_1,N_2)=1$, are the eigenvalues of the Hecke operators
$$ T_{2k}(d)f(z)=d^{k-1/2}\sum_{d_1d_2=d}d^{-2k}_2\cdot\sum_{m\,(\operatorname{mod}d_2)}f\Biggl(\frac{d_1z+m}{d_2}\Biggr). $$
The trace formula is used for obtaining the estimate
$$ \frac{d^l}{dt^l}\mathcal H_f^{(\chi)}(1/2+it)\ll_{\varepsilon,k,l,N_1}(1+|t|)^{1/2+\varepsilon}N_2^{1/2-1/8+\varepsilon} $$
for the newform $f$ for all $\varepsilon>0$, $l=0,1,2,\dots$. This improves the known result (Duke–Friedlander–Iwaniec, 1993) with upper bound
$$ (1+|t|)^2N_2^{1/2-1/22+\varepsilon} $$
on the right-hand side. As a corollary, we obtain the estimate
$$ c(n)\ll_\varepsilon h^{1/4-1/16+\varepsilon} $$
for the Fourier coefficients of holomorphic cusp forms of weight $k+1/2$, which improves Iwaniec' result (1987) with exponent $1/4-1/28+\varepsilon$. Bibl. 25 titles.