The following construction of a Grade 2 seal modular form has arisen in my research. I'm outside the worlds of automorphic forms and number theory, wondering if they resemble anything familiar or interesting.

To let $ F ( tau, z, sigma) $ Be a (meromorphic) grade 2 seal modular weight form $ k> 0 $, and let $ r in mathbb {Z} _ {> 0} $, I was forced to look at the following meromorphic function on the seal hemisphere $ mathbb {H} _ {2} $:

$$ Phi_ {r} ( tau, z, sigma): = prod_ {d | r} F large (d tau, z, tfrac {6} {d} sigma large). $$

Has such a design been investigated before or has it been created somewhere? Maybe there is a context or interpretation that someone can offer? As I briefly outline below, the congruence subset is almost a seal form $ Gamma_ {0} ^ {(2)} (r) subset Sp_ {4} ( mathbb {Z}) $, Note in particular that we maintain the symmetry between $ tau $ and $ sigma $…

From the Fourier-Jacobi extension of $ F $it is easy to write down the corresponding extent of $ Phi_ {r} $, Where $ Q = e ^ {2 pi i sigma} $:

$$ Phi_ {r} ( tau, z, sigma) = sum_ {m = 0} ^ { infty} Q ^ {m} varphi_ {k, m / r} ( tau, z). $$

You can show that $ varphi_ {k, m / r} $ behave like Jacobi weight forms $ k $ and rational index $ m / r in mathbb {Q} $ on the group $ Gamma_ {0} (r) rtimes (r mathbb {Z} times mathbb {Z}) $, So that's almost the Fourier-Jacobi extension of a seal form, but the index and exponent of $ Q $ differ by the factor of $ 1 / r $, In order to $ Phi_ {r} $ Changes properly under the following conditions:

$$ ( tau, z, sigma) mapsto bigg ( frac {a tau + b} {c tau + d}, frac {z} {c tau + d}, sigma – frac {cz ^ {2}} { textbf {r} (c tau + d)} bigg) $$

to the $ (a, b, c, d) in gamma_ {0} (r), $ and for $ lambda in textbf {r} mathbb {Z} $ and $ mu in mathbb {Z} $:

$$ ( tau, z, sigma) mapsto bigg ( tau, z + lambda tau + mu, sigma + frac {1} { textbf {r}} ( lambda ^ {2 } tau + 2 lambda z) bigg). $$

The factors of $ 1 / r $ which I've made bold above, seem to spoil this action that really comes from the action of $ Gamma_ {0} ^ {(2)} (r) $ on $ mathbb {H} _ {2} $, Is there a closely related group? $ Sp_ {4} ( mathbb {Z}) $ which I should think about instead, like similarities or something?