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Author: William A. Stein
The Modular Forms Database: Hecke eigenvalues

Eigenform Basis


This is a table of q-expansions of normalized new eigenforms (newforms) of even weight.

Format: For each Galois conjugacy class of eigenforms, only one form is given. The format of each entry is
[ g(x),   [c1(x), c2(x), c3(x), ..., c500(x)]]
where the ci(x) are elements of the field Q[x]/(g). It is possible to read these tables into PARI 2.0; to do this it is necessary to define a sufficiently large matrix E inside PARI, e.g., by issuing the command E=matrix(1000,30);

Warnings: (1) The eigenforms have been multiplied by an integer (=c1) so that the coefficients can be expressed as Z-linear combinations of some cn. Thus an(x) = cn(x)/c1(x) is the eigenvalue of the Hecke operator Tn.
(2) Sometimes there might not be exactly 500 cn's.

I've also added some files "eigen_" which give only the eigenvalues ap.
  1. Weight 2, trivial character   --   S2new(Gamma0(N))
    Every weight 2 newform of level <= 1000 (only 17 terms): an_1-1000.gp
    These give a1, ..., a500.
    k = 2, 1 <= N <= 100 an_s2g0new_1-100_hiprec.gp
    k = 2, 101 <= N <= 200 an_s2g0new_101-200_hiprec.gp
    k = 2, 201 <= N <= 300 an_s2g0new_201-300_hiprec.gp
    k = 2, 301 <= N <= 400 an_s2g0new_301-400_hiprec.gp
    k = 2, 401 <= N <= 500 an_s2g0new_401-500_hiprec.gp
    k = 2, 501 <= N <= 601 an_s2g0new_501-600_hiprec.gp
    k = 2, 601 <= N <= 700 an_s2g0new_601-650_hiprec.gp
    an_s2g0new_651-700_hiprec.gp
    k = 2, 701 <= N <= 800 an_s2g0new_701-750_hiprec.gp
    an_s2g0new_751-800_hiprec.gp
    k = 2, 801 <= N <= 900 an_s2g0new_801-850_hiprec.gp
    an_s2g0new_851-900_hiprec.gp
    k = 2, 901 <= N <= 1000 an_s2g0new_901-925_hiprec.gp
    an_s2g0new_926-950_hiprec.gp
    an_s2g0new_951-975_hiprec.gp
    an_s2g0new_976-1000_hiprec.gp
    k = 2, 1<=N<=500, precision 97 eigen_k2_N1-500_prec97.gp
    k = 2, 1<=N<=2500, precision 37 Eigenforms, to level 2500


  2. No character   --   Sknew(Gamma1(N))
    Hecke eigenvalues ap, p=1,2,3,5,7,... for a basis of newforms.
    k = 2, 1 <= N <= 51 ap_s2g1new_1-51.gp
    k = 3, 1 <= N <= 38 ap_s3g1new_1-38.gp
    k = 4, 1 <= N <= 28 ap_s4g1new_1-28.gp


  3. Weight 2, trivial character   --   S2new(Gamma0(N))
    Computed to enough precision to satisfy Sturm's bound.
    k = 2, 1 <= N <= 100 an_s2g0new_1-100.gp
    k = 2, 101 <= N <= 200 an_s2g0new_101-200.gp
    k = 2, 201 <= N <= 300 an_s2g0new_201-300.gp
    k = 2, 401 <= N <= 500 an_s2g0new_401-500.gp
    k = 2, 601 <= N <= 640 an_s2g0new_601-640.gp
    k = 2, 801 <= N <= 808 an_s2g0new_801-808.gp


  4. Weight 4, trivial character   --   S4new(Gamma0(N))
    k = 4, 1 <= N <= 100 an_s4g0new_1-100.gp
    k = 4, 1 <= N <= 197 eigen_k4_N1-197_prec97.gp


  5. Weight 6, trivial character   --   S6new(Gamma0(N))
    k = 6, 1 <= N <= 50 an_s6g0new_1-50.gp
    an_s6g0new_1-50_hiprec.gp
    k = 6, 51 <= N <= 100 an_s6g0new_51-100_hiprec.gp
    k = 6, 101 <= N <= 150 an_s6g0new_101-150_hiprec.gp
    k = 6, 151 <= N <= 200 an_s6g0new_151-200_hiprec.gp
    k = 6, 1<=N<=99, precision 97 eigen_k6_N1-99_prec97.gp


  6. Weight 8, trivial character   --   S8new(Gamma0(N))
    k = 8, 1 <= N <= 20 an_s8g0new_1-20.gp
    k = 8, 1<=N<=99, precision 97 eigen_k8_N1-99_prec97.gp (in progress)


  7. Weight 10, trivial character   --   S10new(Gamma0(N))
    k = 10, 1<=N<=99, precision 97 eigen_k10_N1-99_prec97.gp (in progress)


  8. Weight 12, trivial character   --   S12new(Gamma0(N))
    k = 12, 1 <= N <= 24 an_s12g0new_1-24.gp
    k = 12, 1<=N<=99, precision 97 eigen_k12_N1-99_prec97.gp (in progress)


  9. Weight 14, trivial character   --   S14new(Gamma0(N))
    k = 14, 1 <= N <= 36 an_s14g0new_1-36.gp