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In this paper, we study the number of real roots of random trigonometricpolynomials with iid coefficients. When the coefficients have zero mean, unitvariance and some finite high moments, we show that the variance of the numberof real roots is asymptotically linear in terms of the expectation;furthermore, the multiplicative constant in this linear relationship dependsonly on the kurtosis of the common distribution of the polynomial'scoefficients. This result is in sharp contrast to the classical Kac polynomialswhose corresponding variance depends only on the first two moments. Our resultis perhaps the first paper to establish the variance for general distributionof the coefficients including discrete ones, for a model of random polynomialsoutside the family of the Kac polynomials. Our method gives a fine comparisonframework throughout Edgeworth expansion, asymptotic Kac-Rice formula and adetailed analysis of characteristic functions.

Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots