Let F be a primitive form. We can write with a fundamental discriminant. Then if and only if either , or if
An explicit computation show that is a square modulo 4. Hence, it can always be written as . Assume first . Then and so we can write with integral.Assume . A computation shows that
Since we have hence if , either , i.e , or , i.e .Assume now that p=2. If , we have
Since , . Since the square of an integer is congruent to 0 or 1 modulo 4, it follows that in other words , hence once again either or as before.Conversely, assume that either or . If , then since F is primitive, by Proposition 3.3 (2), we have .
Assume first p>3. Then , hence , so .
Assume now that p=2. Then for some other quadratic form , hence , hence the same is true for and hence .
Finally, assume that p=3. From the explicit formulas, is equivalent to and , from which it follows by the formula for the discriminant that , and in particular .
Assume now that , and . We may assume that , otherwise we are in the case which we have just dealt with.
By the same reasoning as before, writing , we have
Hence is equivalent to which implies , hence if p>2 we have .For p=2 and , we have as before
and since , we deduce that and hence , thus finishing the proof of the proposition.