Bimagic squares

[KW Heuer]

Some definitions.

An n x n array has 2n+2 _lines_: n rows, n columns, and two full
diagonals. A square is _magic_ if the sum of the n elements in
a line is constant (i.e. independent of the line). A square is
_bimagic_ if it is magic and the sum of the squares of each line
is also constant. A square is _normal_ if its elements are the
first n^2 positive integers. (Since magic and bimagic squares
preserved under affine transformations, you may substitute your
favorite arithmetic progression. I often use [0,n) instead of
(0,n].)

What is the smallest normal bimagic square (after the trivial
case n = 1)?

More specifically, I've proved to my satisfaction that there are
none for 2 <= n <= 6, and I have an example for n = 8. Is there
a bimagic square of order 7? (The magic constant is 175 and the
bimagic constant is 5775.)

Karl W. Z. Heuer (ihnp4!bentley!kwh), The Walking Lint