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Consider the space of weight 1 modular forms for $\Gamma_{1}(7)$. A basis element of this dimension 3 space is $$f(q) = q - q^{3} + 2q^{4} + 2q^{5} - 3q^{6} + q^{7} + 3q^{8} - 2q^{9} - q^{10} + \cdots .$$ Does anyone recognize this as some product/division of eta functions? The command to generate a basis for the above mentioned modular forms is ModularForms(Gamma1(7), 1).basis(). The $f(q)$ above is the second element in the basis that is output.

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Looks like what you have is a product not of eta functions but of Klein forms, which is almost as good. It's $q\prod_{n=1}^\infty (1-q^n)^{c_n}$ where $c_n = 2,0,1,-2$ according as $n \equiv 0, \pm 1, \pm 2, \pm 3 \bmod 7$. cf. the formulas on p.84 of my chapter "The Klein Quartic in Number Theory" = library.msri.org/books/Book35/files/elkies.pdf in The Eightfold Way: The Beauty of Klein's Quartic Curve = library.msri.org/books/Book35/contents.html . –  Noam D. Elkies Jun 10 '12 at 0:38
P.S. To reproduce the computation "ModularForms(Gamma1(7), 1).basis()", run that command in Sage. –  Noam D. Elkies Jun 10 '12 at 0:46
@Noam Elkies: This will actually do, thank you! (On another note, how should I accept your comment as an answer?) –  lk728 Jun 10 '12 at 15:10

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