In [this paper](http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/3071/1/PRIMS41_3_683.pdf), the following result is proved.


For any prime $p$, all the Fourier coefficients of


$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod\_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$


are non-negative.


As seeing [this sequence](https://oeis.org/A175595), I conjectured


all the Fourier coefficients of


$$\eta(q^m)^m / \eta(q) = q^{\frac{m^2-1}{12}} \prod\_{n=1}^\infty (1 - q^{mn})^m (1 - q^{n})^{-1}$$


are non-negative for any positive integer $m$, .


Is this conjecture true?



Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?
Yes this is true and these Fourier coefficients actually enumerate some combinatorial objects called $m$-cores. These are partitions with no hooklength divisible by $m$. In fact for $m\geq 4$ these coefficients are positive. This last result was proved in ["Defect zero $p-$blocks for finite simple groups"](http://www.ams.org/journals/tran/1996-348-01/S0002-9947-96-01481-X/home.html), by Granville and Ono.


I want to also point out that Saito's conjecture on nonnegativity of particular eta quotients (which was the subject of the paper you link to) has been fully proven in ["K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products"](https://arxiv.org/abs/math/0702027) by Berkovich and Garvan. 


