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Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a toric arrangement can be computed in terms of its  arithmetic  Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems  A   n  , B   n  , C   n  , and  D   n   with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a  finite field method  for arithmetic Tutte polynomials and by enumerating signed graphs with respect to six parameters.

The Arithmetic Tutte Polynomials of the Classical Root Systems