I would like to classify the integers m >= 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, 2.3^j, 5.3^j, where j >= 1.

Does someone see how to prove this?  Thank you.