  We prove new large sieve inequalities for the Fourier coefficients
$\rho_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted
by sequences $(a_n)$ with sparse Fourier transforms - including two key types
of sequences that arise in the dispersion method. These give the first savings
in the exceptional spectrum for the critical case of sequences as long as the
level, and lead to improved bounds for various multilinear forms of Kloosterman
sums.
  We give three applications and suggest other likely consequences. We show
that both primes and smooth numbers are equidistributed in arithmetic
progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable
weights for the primes; this completely eliminates the dependency on Selberg's
eigenvalue conjecture in previous results of Lichtman and the author.
Separately, we show that the greatest prime factor of $n^2+1$ is infinitely
often greater than $n^{1.3}$, improving Merikoski's $n^{1.279}$.
