Fundamental Data

• Level q = 307


• (q,∞)-Quaternion Algebra D = Q[I,J] with I² = -1, J² = -307, K = IJ = -JI


• Class Number H = 26, Type Number T = 16


• 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+1J+14K, 1I+2J+1K, 16J+0K, 16K > ( norm = 32 )
  A₆ = < 1+0I+1J+6K, 1I+10J+1K, 16J+0K, 16K > ( norm = 32 )
  A₇ = < 1+0I+1J+10K, 1I+6J+1K, 16J+0K, 16K > ( norm = 32 )
  A₈ = < 1+0I+1J+2K, 1I+14J+1K, 16J+0K, 16K > ( norm = 32 )
  A₉ = < 1+0I+1J+14K, 1I+18J+1K, 32J+0K, 32K > ( norm = 64 )
  A₁₀ = < 1+0I+17J+22K, 1I+10J+17K, 32J+0K, 32K > ( norm = 64 )
  A₁₁ = < 1+0I+17J+10K, 1I+22J+17K, 32J+0K, 32K > ( norm = 64 )
  A₁₂ = < 1+0I+1J+18K, 1I+14J+1K, 32J+0K, 32K > ( norm = 64 )
  A₁₃ = < 1+0I+33J+46K, 1I+18J+33K, 64J+0K, 64K > ( norm = 128 )
  A₁₄ = < 1+0I+33J+14K, 1I+50J+33K, 64J+0K, 64K > ( norm = 128 )
  A₁₅ = < 1+0I+17J+54K, 1I+10J+17K, 64J+0K, 64K > ( norm = 128 )
  A₁₆ = < 1+0I+17J+10K, 1I+54J+17K, 64J+0K, 64K > ( norm = 128 )
  A₁₇ = < 1+0I+33J+50K, 1I+14J+33K, 64J+0K, 64K > ( norm = 128 )
  A₁₈ = < 1+0I+97J+110K, 1I+18J+97K, 128J+0K, 128K > ( norm = 256 )
  A₁₉ = < 1+0I+33J+78K, 1I+50J+33K, 128J+0K, 128K > ( norm = 256 )
  A₂₀ = < 1+0I+33J+14K, 1I+114J+33K, 128J+0K, 128K > ( norm = 256 )
  A₂₁ = < 1+0I+81J+118K, 1I+10J+81K, 128J+0K, 128K > ( norm = 256 )
  A₂₂ = < 1+0I+81J+10K, 1I+118J+81K, 128J+0K, 128K > ( norm = 256 )
  A₂₃ = < 1+0I+33J+114K, 1I+14J+33K, 128J+0K, 128K > ( norm = 256 )
  A₂₄ = < 1+0I+33J+50K, 1I+78J+33K, 128J+0K, 128K > ( norm = 256 )
  A₂₅ = < 1+0I+97J+238K, 1I+18J+97K, 256J+0K, 256K > ( norm = 512 )
  A₂₆ = < 1+0I+97J+110K, 1I+146J+97K, 256J+0K, 256K > ( norm = 512 )
  


• 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+1K, 8J+16K, 32K >
  O₄ = (1/8) < 4+0I+4J+8K, 1I+6J+17K, 8J+16K, 32K >
  O₅ = (1/16) < 8+0I+8J+112K, 1I+2J+1K, 16J+96K, 128K >
  O₆ = (1/16) < 8+0I+8J+48K, 1I+10J+81K, 16J+96K, 128K >
  O₇ = (1/16) < 8+0I+8J+80K, 1I+6J+113K, 16J+32K, 128K >
  O₈ = (1/16) < 8+0I+8J+16K, 1I+14J+33K, 16J+32K, 128K >
  O₉ = (1/32) < 16+0I+16J+224K, 1I+18J+97K, 32J+448K, 512K >
  O₁₀ = (1/32) < 16+0I+16J+352K, 1I+10J+81K, 32J+192K, 512K >
  O₁₁ = (1/32) < 16+0I+16J+160K, 1I+22J+401K, 32J+320K, 512K >
  O₁₂ = (1/32) < 16+0I+16J+288K, 1I+14J+161K, 32J+64K, 512K >
  O₁₃ = (1/64) < 32+0I+32J+1472K, 1I+18J+1121K, 64J+896K, 2048K >
  O₁₄ = (1/64) < 32+0I+32J+448K, 1I+50J+33K, 64J+896K, 2048K >
  O₁₅ = (1/64) < 32+0I+32J+704K, 1I+10J+593K, 64J+1408K, 2048K >
  O₁₆ = (1/64) < 32+0I+32J+1344K, 1I+54J+1233K, 64J+640K, 2048K >
  O₁₇ = (1/64) < 32+0I+32J+1600K, 1I+14J+1185K, 64J+1152K, 2048K >
  O₁₈ = (1/128) < 64+0I+64J+2944K, 1I+18J+7265K, 128J+5888K, 8192K >
  O₁₉ = (1/128) < 64+0I+64J+896K, 1I+50J+2081K, 128J+1792K, 8192K >
  O₂₀ = (1/128) < 64+0I+64J+4992K, 1I+114J+929K, 128J+1792K, 8192K >
  O₂₁ = (1/128) < 64+0I+64J+1408K, 1I+10J+593K, 128J+2816K, 8192K >
  O₂₂ = (1/128) < 64+0I+64J+6784K, 1I+118J+5969K, 128J+5376K, 8192K >
  O₂₃ = (1/128) < 64+0I+64J+3200K, 1I+14J+7329K, 128J+6400K, 8192K >
  O₂₄ = (1/128) < 64+0I+64J+7296K, 1I+78J+289K, 128J+6400K, 8192K >
  O₂₅ = (1/256) < 128+0I+128J+22272K, 1I+18J+31841K, 256J+11776K, 32768K >
  O₂₆ = (1/256) < 128+0I+128J+5888K, 1I+146J+13153K, 256J+11776K, 32768K >
  


• Isomorphic Relations Among O_j's :
  O₃ is isomorphic to O₄
  O₆ is isomorphic to O₇
  O₅ is isomorphic to O₈
  O₁₀ is isomorphic to O₁₁
  O₉ is isomorphic to O₁₂
  O₁₅ is isomorphic to O₁₆
  O₁₄ is isomorphic to O₁₇
  O₂₁ is isomorphic to O₂₂
  O₂₀ is isomorphic to O₂₃
  O₁₉ is isomorphic to O₂₄
  


• 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(11) 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 ( 1.1MB )
  • 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)