Hi I have the following problems concerning quadratic and ternary forms. Any help would be greatly appreciated.

$3\displaystyle\sum_{x, y\in\mathbb{Z}}q^{x^2+xy+7y^2}=3\displaystyle\sum_{x, y\in\mathbb{Z}}q^{9(x^2+xy+y^2)}+P_{3,1}\left(\displaystyle\sum_{x,y\in\mathbb{Z}}q^{x^2+xy+y^2}\right)$, where $P_{3,1}$ is the operator which takes only the exponents of $q$ which are congruent to 1 modulo 3.

Which of the integers are not represented by the ternary form (5, 8, 11, -4, 1, 2) assuming that it is regular, though its not yet known whether its regular or not. Its genus has 2 elements.

If $F$ is a ternary positive quadratic form with Hessian, $H(F)=5$. Then min (F)=1.

Comment: Though the upper bound for such minimum value turns out to be 2 inclusive, I am not able to eliminate the case when it is 2.

Any help would be greatly appreciated.

Thanks.