Transcendence results obtained for each of our three non-CM eigenfunctions on congruence subgroups

We obtained the following (negative) results for each of the three non-CM eigenfunctions which we have computed. All results were proved under the assumption that our values for the eigenvalue r and the Fourier coefficients lambda_2(f), lambda_3(f), lambda_5(f), lambda_7(f), lambda_11(f) are correct to at least 495 decimal digits.

Case 1: r=5.4361... on Gamma_0(5), trivial character

The eigenvalue lambda=r^2+1/4 is NOT [2,10^162]-algebraic, nor is it [10,10^43]-, [30,10^14]- or [50,5*10^7]-algebraic. Exactly the same assertions hold for r and for each of the Hecke eigenvalues lambda_2(f), lambda_3(f), lambda_7(f), lambda_11(f). (However, lambda_5(f)=sqrt(1/5).)

Furthermore, there does NOT exist any relation of the form r=2(pi)q/log(alpha), where alpha>0 is [2,10^105]- or [10,10^35]-algebraic and q is rational with |q| <= 10 and denominator d(q) <= 30.

Furthermore, there does NOT exist any relation lambda_p(f) = e(q) alpha^(ir) + e(-q) alpha^(-ir) with
p in {2,3,7}, alpha > 0, | 2pi q + r log(alpha) | <= pi,
q rational, and alpha, q satisfying the conditions in any one line of the following table:
 alpha [2,10^103]-algebraic |q| <= 30 d(q) <= 100 alpha [10,10^29]-algebraic |q| <= 30 d(q) <= 10 alpha [30,10^12]-algebraic |q| <= 4 d(q) <= 10

Case 2: r=4.8937... on Gamma_0(5), nontrivial character (5/.)

The eigenvalue lambda=r^2+1/4 is NOT [2,10^162]-algebraic, nor is it [10,10^43]-, [30,10^14]- or [50,4*10^7]-algebraic. Exactly the same assertions hold for r and for each of the Hecke eigenvalues lambda_2(f), lambda_3(f), lambda_7(f), lambda_11(f). (In fact, for the purely imaginary numbers lambda_2(f), lambda_3(f), lambda_7(f) we checked the stronger assertions that Im(lambda_2(f)), Im(lambda_3(f)), Im(lambda_7(f)) are not [2,10^162]-, [10,10^43]-, [30,10^14]- or [50,4*10^7]-algebraic.)

We also found that lambda_5(f), which is a complex number with absolute value 1, is NOT a root of unity of the form e(a/b) with a,b integers, 0< b <=10^243; nor is it [2,10^243]-, [10,10^80]- or [30,10^28]-algebraic.

Furthermore, there does NOT exist any relation of the form r=2(pi)q/log(alpha), where alpha>0 is [2,10^104]- or [10,10^34]-algebraic and q is rational with |q| <= 10 and denominator d(q) <= 30.

Furthermore, there does NOT exist any relation lambda_p(f) = e(q) alpha^(ir) + chi_f(p) e(-q) alpha^(-ir)
p in {2,3,7,11}, alpha > 0, | 2pi q + r log(alpha) | <= pi,
q rational, and alpha, q satisfying the conditions in any one line of the following table:
 alpha [2,10^110]-algebraic |q| <= 30 d(q) <= 100 alpha [10,10^36]-algebraic |q| <= 30 d(q) <= 10 alpha [30,3*10^11]-algebraic |q| <= 4 d(q) <= 10

(Note: For this r-value there are TWO Hecke eigenfunctions, which are conjugate to each other. Note that lambda_p(f) is real in [-2,2] for p with chi_f(p)=1, and purely imaginary in i[-2,2] for p with chi_f(p)=-1. This means that all solutions z to lambda_p(f) = z+chi_f(p)/z have |z|=1. Note that in the case chi_f(p)=-1 our tests covered BOTH the Hecke eigenfunctions, i.e. +-lambda_p(f).)

Case 3: r=2.5923... on Gamma_0(6), trivial character

The eigenvalue lambda=r^2+1/4 is NOT [2,10^161]-algebraic, nor is it [10,10^43]-, [30,10^14]- or [50,4*10^7]-algebraic. Exactly the same assertions hold for r and for each of the Hecke eigenvalues lambda_5(f), lambda_7(f), lambda_11(f). (However, lambda_2(f)=-sqrt(1/2) and lambda_3=sqrt(1/3).)

Furthermore, there does NOT exist any relation of the form r=2(pi)q/log(alpha), where alpha>0 is [2,10^104]- or [10,10^33]-algebraic and q is rational with |q| <= 10 and denominator d(q) <= 30.

Furthermore, there does NOT exist any relation lambda_p(f) = e(q) alpha^(ir) + e(-q) alpha^(-ir) with
p in {5,7,11}, alpha > 0, | 2pi q + r log(alpha) | <= pi,
q rational, and alpha, q satisfying the conditions in any one line of the following table:
 alpha [2,10^103]-algebraic |q| <= 30 d(q) <= 100 alpha [10,10^30]-algebraic |q| <= 10 d(q) <= 10 alpha [30,4*10^9]-algebraic |q| <= 4 d(q) <= 10