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Let f_i be an infinite sequence of elliptic Hecke eigenforms such that the individual weights and levels are unbounded as i goes to infinity. When does one expect that f_i's have a common zero? Any guess based on heuristics/ conjectures is welcome! Any non-trivial example of this phenoemena?

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By "nontrivial" do you mean the list should not admit a finite partition into subsets, each having a common multiple with a zero in the same place? –  S. Carnahan Mar 17 '13 at 7:16
    
If you'll allow me to use oldforms (which you probably won't) then there's a trick to ensure a common zero -- because of elementary considerations involving elliptic points (which can be dressed up via a "stacky" argument to look much more fancy), if the weight of a level 1 form (eigenform or not) is not 0 mod 4 then it will have a zero at i, and if it's not 0 mod 6 it will have a zero at $\rho$. However this trick goes away if you force the conductors to go to infinity, and in this case why would one expect common zeros at all? –  user30035 Mar 17 '13 at 9:14
    
Yes, by non-trivial, I mean what Carnahan said and also to avoid oldforms. It seems that this might be a very rare phenomena. However, I do not know of any heuristics or conjectures even hinting this. –  Basic Mar 17 '13 at 14:08
    
Does Borcherd product type consideration say something? –  Basic Mar 19 '13 at 6:12
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