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I am new in the field of Hilbert modular forms and I could not find any Hilbert modular form of this specific weight for the 2-fold upper half-plane.

Does anybody know an example of any level?

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  • $\begingroup$ What fields were you looking over? $\endgroup$
    – Kimball
    Commented Mar 18, 2017 at 19:46

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In this paper, van der Geer and Zagier come across a concrete Hilbert cusp form of weight $6$ on $\mathrm{SL}_2(\mathcal{O})$, namely

$$f=-16(x+x^{-1})^3 (q+(27x^3-39x-39x^{-1}+27x^{-3})q^2+(285x^6-792x^4-45x^2+1356-45^{-2}-792^{-4}+285^{-6})q^3+\cdots )$$

Not sure if it is the kind of thing you are looking for.

LMFDB doesn't have any Hilbert modular form of that weight (see here).

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    $\begingroup$ My understanding is that LMFDB just has parallel weight 2. Searching for weight 2 gives all 368356 examples. $\endgroup$
    – post.as.a.guest
    Commented Mar 17, 2017 at 22:38
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You can construct some by (pluri-) harmonic theta series on even-dimensional quadratic spaces, imposing congruence conditions to assure not-identical-vanishing. For example, $$ \theta(z_1,z_2) \;=\; \sum_{m,n\in \mathfrak o,\,m=m_o,\,n=n_o\!\!\!\mod\!\! N} P_1(m_1,n_1)\,P_2(m_2,n_2)\,e^{2\pi i ((m_1^2+n_1^2)z_1+(m_2^2+n_2^2)z_2)} $$ where $P_1,P_2$ are homogeneous, harmonic, both of degree $d$, $m_1,m_2$ are the two real imbeddings of $m\in\mathfrak o$, is a cuspform of weight $(1+d,1+d)$. Sufficiently increasing $N$ will guarantee non-vanishing on general principles. Specific numerical choices are surely computable, if not by hand, in Sage or similar. Likewise, with quadratic spaces of dimension $2n$ and harmonic polynomials of degrees $d,d,\ldots,d$ (with $d>0$), you can make cuspforms of weight $(n+d,n+d,\ldots,n+d)$, with non-vanishing assured by a sufficiently strong congruence condition.

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There is functionality for computing Hilbert modular forms in MAGMA, based on the algorithms of Dembele, Donnelly, Greenberg, and Voight. If you don't have access to a copy, try the online MAGMA calculator at http://magma.maths.usyd.edu.au/calc/.

Here is a sample script:

F<w> := QuadraticField(5);
S := HilbertCuspForms(F,1*Integers(F),[6,6]);
fs := NewformDecomposition(S);
f := Eigenform(fs[1]);
pps := PrimesUpTo(100,F);
for pp in pps do
  printf "a(<%o,%o>) = %o\n", F!Basis(pp)[1], F!Basis(pp)[2],    HeckeEigenvalue(f, pp);
end for;

This computes the space of Hilbert modular forms over $\mathbf{Q}(\sqrt{5})$ of level 1 and parallel weight 6, and prints out the Hecke eigenvalues for the unique eigenform in this space for all primes of norm up to 100.

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