M.Newman raised several questions in his 1957 paper on modular forms.

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 $i\infty$.

One of the most interesting conjecture is:

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 $H_n$.

Question: Does this conjecture have a positive answer?

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    $\begingroup$ I believe Newman states his conjecture and defines the relevant notation in his 1957 PLMS paper (instead of the 1959 one). In particular, $H_{n}$ is defined on page 342 and the conjecture is stated on page 343. I've taken the liberty of editing the question and changing the link. $\endgroup$ – Jeremy Rouse Jan 11 '15 at 17:45

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). 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) 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 $i\infty$.

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