Fundamental Data

• Level q = 67


• (q,∞)-Quaternion Algebra D = Q[I,J] with I² = -1, J² = -67, K = IJ = -JI


• Class Number H = 6, Type Number T = 4


• A Maximal Order O in D :
  O = (1/2) < 1+0I+1J+0K, 1I+0J+1K, 2J+0K, 2K >


• Representatives of Left O-Ideal Classes :
  A₁ = < 1+0I+1J+0K, 1I+0J+1K, 2J+0K, 2K > ( norm = 4 )
  A₂ = < 1+0I+1J+2K, 1I+2J+1K, 4J+0K, 4K > ( norm = 8 )
  A₃ = < 1+0I+1J+6K, 1I+2J+1K, 8J+0K, 8K > ( norm = 16 )
  A₄ = < 1+0I+1J+2K, 1I+6J+1K, 8J+0K, 8K > ( norm = 16 )
  A₅ = < 1+0I+9J+6K, 1I+10J+9K, 16J+0K, 16K > ( norm = 32 )
  A₆ = < 1+0I+9J+10K, 1I+6J+9K, 16J+0K, 16K > ( norm = 32 )
  


• Maximal Orders O_j = (A_j)^-1(A_j) :
  O₁ = (1/2) < 1+0I+1J+0K, 1I+0J+1K, 2J+0K, 2K >
  O₂ = (1/4) < 2+0I+2J+4K, 1I+2J+1K, 4J+0K, 8K >
  O₃ = (1/8) < 4+0I+4J+24K, 1I+2J+9K, 8J+16K, 32K >
  O₄ = (1/8) < 4+0I+4J+8K, 1I+6J+25K, 8J+16K, 32K >
  O₅ = (1/16) < 8+0I+8J+48K, 1I+10J+121K, 16J+96K, 128K >
  O₆ = (1/16) < 8+0I+8J+80K, 1I+6J+25K, 16J+32K, 128K >
  


• Isomorphic Relations Among O_j's :
  O₃ is isomorphic to O₄
  O₅ is isomorphic to O₆
  


• Group Order e_j of the Unit Group of O_j :
  e₁ = 4, e_j = 2 ( 2 <= j <= H )

Further Data ( Text Files )

• Factorization of the Characteristic Polynomials of the Hecke Operators on S₂⁺(Γ₀(q))


• Factorization of the Characteristic Polynomials of the Hecke Operators on M₂^-(Γ₀(q)) ( Including the Eisenstein Series )


• The eigenvalues of B(2) and the common eigenvectors v_j ( j = 1, ..., H ) of B(n)'s
  • The j-th line of this text file indicates the associated eigenvalue and the transposed vector of v_j = ( v_ij ), i.e., those data are stored in the order
         for j = 1 to H
             the eigenvalue, v_1j, v_2j, ... , v_H,j


• Representatives { A_ij } _{i = 1,...,H} of Left O_j-Ideal Classes


• Theta Series θ_ij Associated to A_ij
  • The range of the Fourier expansion = 1024
  • The Fourier coefficients of e_j θ_ij(z) = Σ_n=0,1,2,... a_ij(n) exp(2πinz) are stored in the order
         for j = 1 to H
             for i = j to H
                 a_ij(0), a_ij(1), ..., a_ij(1024)