Conjectures Relating to a Generalization of the Ramanujan Tau Function

While working with Hecke transforms of weight 12 modular forms over Q(√2) and Q(√3) a peculiar phenomenom was observed. The characteristic polynomial for the Hecke transform factored into a linear and an irreducible part. The linear factor was related to the Ramanujan tau function in an intrinsic fashion that implied this phenomenom may be more than an accident. For instance, over Q(√2) the Hecke transform of index 2 + √2 had -24 as an eigenvalue, while the Hecke transform of index 3 + √2 had eigenvalue -16744. One notes τ ( N ( 2 + √ 2 ) ) = τ ( 2 ) = − 24 τ ( N ( 3 + √ 2 ) ) = τ ( 7 ) = − 16744 $$ \begin{array}{*{20}{c}} {\tau \left( {N\left( {2 + \surd 2} \right)} \right) = \tau \left( 2 \right) = - 24} \\ {\tau \left( {N\left( {3 + \surd 2} \right)} \right) = \tau \left( 7 \right) = - 16744} \end{array} $$ where N is the field norm of Q(√2) over Q. Unfortunately some of the eigenvalues of the corresponding modular eigenform are not Ramanujan tau function values. An explanation of this phenomenom is given by Doi-Naganuma lifting of modular forms. The author would like to thank Harvey Cohn and Carlos Moreno for illuminating discussions.