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Modular data is an important topic of study in rational conformal fieldtheory. A modular datum defines finite dimensional representations of themodular group $\mbox{SL}_2(\mathbf{Z})$. For every Fourier matrix in a modulardatum there exists an Allen matrix obtained from the Fourier matrix afterdividing each its row with the first entry of that row. In this paper, weclassify the non-homogenous Fourier matrices and non-homogenous Allen matricesup to rank $5$. The methods developed here are useful for the classification ofthe matrices of higher ranks. Also, we establish some results that are helpfulin recognizing the $C$-algebras not arising from Allen matrices by just lookingat the character table of the $C$-algebra, in particular, the first row of thecharacter table.

Classification of non-homogeneous Fourier matrices associated with modular data up to rank 5