  Chebyshev famously observed empirically that more often than not, there are
more primes of the form $3 \bmod 4$ up to $x$ than of the form $1 \bmod 4$.
This was confirmed theoretically much later by Rubinstein and Sarnak in a
logarithmic density sense. Our understanding of this is conditional on the
generalized Riemann hypothesis as well as on the linear independence of the
zeros of $L$-functions.
  We investigate similar questions for sums of two squares in arithmetic
progressions. We find a significantly stronger bias than in primes, which
happens for almost all integers in a \emph{natural density} sense. Because the
bias is more pronounced, we do not need to assume linear independence of zeros,
only a Chowla-type conjecture on nonvanishing of $L$-functions at $1/2$. To
illustrate, we have under GRH that the number of sums of two squares up to $x$
that are $1 \bmod 3$ is greater than those that are $2 \bmod 3$ 100% of the
time in natural density sense.
