I would like to classify the integers $m \geq 2$ for which the four quadratic polynomials $3k^2$, $3k^2+2k$, $3k^2+3k+1$, and $3k^2+5k+2$ together represent all integers modulo $m$. That is, every integer modulo $m$ should be in the range of at least one of these polynomials (where all operations are carried out modulo $m$). Computer evidence suggests that this holds if and only if $m$ is one of the following: $7, 10, 19, 2^j, 3^j, 5^j, 11^j, 13^j, 41^j, 2\cdot3^j, 5\cdot3^j$, where $j \geq 1$.
Does someone see how to prove this? Thank you.