On Sp_4 modularity of Picard--Fuchs differential equations for   Calabi--Yau threefolds (with an appendix by Vicentiu Pasol)
  Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5. 
Yifan Yang, Wadim Zudilin