Metaplectic cusp forms and the large sieve

We prove a power saving upper bound for the sum of Fourier coefficients ${\rho }_f(\cdot )$ of a fixed cubic metaplectic cusp form $f$ over primes. Our result is the cubic analogue of a celebrated 1990 theorem of Duke and Iwaniec, and the cuspidal analogue of a theorem due to the author and Radziwiłł for the bias in cubic Gauss sums.

The proof has two main inputs, both of independent interest. Firstly, we prove a new large sieve estimate for a bilinear form whose kernel function is ${\rho }_f(\cdot )$. The proof of the bilinear estimate uses a number field version of circle method due to Browning and Vishe, Voronoi summation, and Gauss–Ramanujan sums. Secondly, we use Voronoi summation and the cubic large sieve of Heath-Brown to prove an estimate for a linear form involving ${\rho }_f(\cdot )$. Our linear estimate overcomes a bottleneck occurring at level of distribution $\frac{2}{3}$.