Existence of a Hilbert modular form of parallel weight 6 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? 
 A: 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).
A: 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.
A: 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.
