In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight $p$ at maximal ideals of residue characteristic $p$ such that the attached mod-$p$ Galois representation is unramified at $p$ and the Frobenius at $p$ acts by scalars. The results lead us to ask the question whether the Gorenstein defect and the multiplicity of the attached Galois representation are always equal to $2$. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular-symbols algorithm over finite fields, and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations.
"On the Failure of the Gorenstein Property for Hecke Algebras of Prime Weight." Experiment. Math. 17 (1) 37 - 52, 2008.