▁on ▁ τ ( n ) ▁Suppose ▁that ▁ ▁is ▁a ▁weight ▁ ▁integer ▁new form ▁and ▁the ▁Fourier ▁coefficients ▁ ▁are ▁integers . ▁Consider ▁the ▁problem : ▁If ▁ ▁does ▁not ▁have ▁complex ▁multiplication , ▁prove ▁that ▁almost ▁all ▁pr imes ▁ ▁have ▁the ▁property ▁that ▁. ▁Indeed , ▁most ▁pr imes ▁should ▁have ▁this ▁property , ▁and ▁hence ▁they ▁are ▁called ▁ordinary . ▁Despite ▁the ▁big ▁adv ances ▁by ▁Del igne ▁and ▁Ser re ▁on ▁Gal ois ▁representations , ▁which ▁determine ▁ ▁for ▁ ▁cop r ime ▁to ▁, ▁we ▁do ▁not ▁have ▁any ▁clue ▁as ▁to ▁how ▁to ▁compute ▁. ▁The ▁only ▁theorem ▁in ▁this ▁regard ▁is ▁El k ies ' ▁famous ▁result ▁for ▁mod ular ▁elli ptic ▁curves , ▁which ▁indeed ▁guarante es ▁that ▁there ▁are ▁infinitely ▁many ▁pr imes ▁ ▁for ▁which ▁, ▁which ▁in ▁turn ▁is ▁obviously ▁. ▁We ▁do ▁not ▁know ▁any ▁examples ▁of ▁non - CM ▁ ▁with ▁weight ▁ ▁for ▁which ▁ ▁mod ▁ ▁for ▁infinitely ▁many ▁pr imes ▁ ▁( although ▁it ▁should ▁be ▁true ▁for ▁almost ▁all ▁). ▁We ▁also ▁do ▁not ▁know ▁any ▁examples ▁where ▁ ▁mod ▁ ▁for ▁infinitely ▁many ▁. ▁Some ▁people ▁had ▁begun ▁to ▁doubt ▁whether ▁ ▁indeed ▁for ▁infinitely ▁many ▁. ▁As ▁evidence , ▁many ▁provided ▁Ram an uj an ' s ▁ ▁( case ▁of ▁weight ▁). ▁The ▁largest ▁known ▁ ▁for ▁which ▁ ▁is ▁. ▁The ▁only ▁solutions ▁to ▁the ▁equation ▁ ▁are ▁ ▁and ▁ ▁up ▁to ▁. ▁▁ ▁conject ured ▁that ▁ ▁for