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We present the exact closed-form expressions to the symbol error rate (SER) of coherent M-ary phase-shift keying(MPSK) and M-ary quardrature amplitude modulation (MQAM), when these modulations schemes are used along with space-time block codes in spatially correlated MIMO Nakagami fading channels with arbitrary fading parameters. Using the well-known moment generating function-based analysis approach, we express the average SERs in terms of Lauricella's multivariate hypergeometric functions. Using exponential bounds for the Gaussian Q-function and for MPSK error probability over additive Gaussian noise channels, we also present new bounds and simple approximate solutions for the exact closed- form expressions to the SERs of MPSK and MQAM, respectively. The new bounds and approximate solutions can be used to compute the average SER of MPSK and MQAM. Numerical results are presented to verify the accuracy of the solutions.