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We determine the exact shape of the G2 equidistribution law for the one parameter family of exponential sums over Ép ≠, ‡x mod p, x ± 0 ç2(x)exp(2πi(x 7 + tx)/p). Here ç2(x) denotes the quadratic character (x/p), t in Ép is the parameter, and p is any prime other than 2 or 7. This answers a question raised in [K-L-R, page 2943, footnote 3] and in [Se]. We also analyze the analogous families when 7 is replaced by any odd integer n ≥ 3. Table of contents Introduction (1) Determinant calculations (2) Proof of Theorem 1.7 (3) Monodromy of Ìn for general odd n (4) Monodromy of Ì7: the group G2 and its finite subgroups (5) Application to explicit G2 equidistribution (6) Application to explicit SO(n) equidistribution (7) Application to the Katz-Sarnak measures √(-, c) Introduction The present work grew out of independent email exchanges with Rudnick and with Serre about the exact shape of the G2 equidistribution law for the sums in the abstract, and for their natural generalization to finite extensions of Ép. One knew that in any characteristic p > 15, after dividing these sums by a suitable normalizing factor, they were distributed like the traces of random elements of the compact form UG2 of the exceptional group G2. The initial problem was to determine precisely this normalizing factor. We carry out this determination (in sections 1 and 2) using a method which goes back to Davenport and Hasse [Dav-Ha], and which ultimately comes down to exploiting the exact shape of the relations between elementary symmetric functions and Newton