M.Newman raised several questions in his 1957 [paper](http://dx.doi.org/10.1112/plms/s3-7.1.334) on modular forms.



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> **Definition:** $H\_n$ is the subclass of all zero-free weakly modular forms of weight 0 on $\Gamma\_0(n)$, where $n$ is a **composite** number. Additionally, any $h\in H\_n$ is holomorphic at cusps other than $i\infty$ and meromorphic at $i\infty$.
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One of the most interesting conjecture is:



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> **Conjecture:** Every weakly modular form of weight 0 on $\Gamma\_0(n)$ holomorphic at cusps other than $i\infty$ and meromorphic at $i\infty$ is a linear combination of functions in $H\_n$.
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**Question:** Does this conjecture have a positive answer?



An old conjecture of M.Newman
First, I think your definition of $H\_{n}$ does not agree with Newman's definition. Newman says the following: "Let $H\_{n} \subset G\_{n}$ be the set of functions of $G\_{n}$ with non-negative valence at all parabolic points of $Q\_{n}$ other than $\tau = i\infty$." Here $G\_{n}$ is the set of modular functions that are expressible in terms of the Dedekind eta-function (which I will refer to as eta-quotients). So Newman is asking if the set of level $n$ eta-quotients of weight $0$ span the space of modular functions holomorphic everywhere except at $i\infty$.


If $n$ is squarefree, any weight zero modular function that is non-vanishing on the upper half plane is a constant multiple of an eta quotient (see [Winfried Kohnen's paper](http://www.ams.org/journals/proc/2005-133-01/S0002-9939-04-07450-7/home.html)). However, this is not true when $n$ is not squarefree. The reason is that any weight zero eta-quotient has rational Fourier coefficients, and hence corresponds to an element of $\mathbb{Q}(X\_{0}(n))$. However, when $n$ is not squarefree, the cusps of $X\_{0}(n)$ need not be rational. Given any two cusps $p\_{1}$ and $p\_{2}$, the divisor class $p\_{1} - p\_{2}$ is torsion in $J\_{0}(n)$ (by a Theorem of Drinfeld and Manin) and as a consequence, there is a modular function all of whose zeroes are at $p\_{1}$ and all of whose poles are at $p\_{2}$. In general, this modular function will not have rational Fourier coefficients, and this modular function would be included in $H\_{n}$ (via your definition), but not via Newman's definition.


I've written a paper with John Webb ([on arXiv here](http://arxiv.org/abs/1311.1460)) where we study some related questions to Newman's conjecture and do some more computations. We show that if $n$ is composite, the span of $H\_{n}$ has finite codimension in the space of all modular functions holomorphic everywhere except infinity.
Also, Newman's conjecture is true for all composite $n \leq 300$ with the possible exceptions of $n = 121$ and $n = 209$. 


However, it seems likely that $n = 121$ may be a genuine exception to his conjecture. The form $f(z) = \frac{\eta(121z)^{22}}{\eta(11z)^{2}}$ has weight $10$ and has a zero of order $110$ at $i \infty$ and is nonzero everywhere else. As a consequence, if $g(z)$ is a modular function holomorphic everywhere except at infinity, then $f(z)^{r} g(z)$ is a holomorphic modular form of weight $10r$ provided $110r$ is $\geq$ the order of pole of $g(z)$ at infinity. 
For $2 \leq r \leq 8$, the subspace of $M\_{10r}(\Gamma\_{0}(121))$ generated by eta quotients has codimension $90$ - this suggests that $H\_{121}$ may have codimension $90$ in the space of all modular functions with poles only at $i\infty$.


