I am new to Sage and modular forms. I have some conceptual questions.
When I write
sage: S = CuspForms(Gamma0(55),2,prec=14)
sage: S.new_subspace().basis()
the output is
[ q + 2q^3 - 2q^5 - 4q^6 - 3q^7 + 5q^8 + 9q^9 - q^10 + 2q^11 - 10q^12 - 9q^13 + O(q^14), q^2 - 2q^3 + 2q^5 + 2q^6 + 2q^7 - 5q^8 - 8q^9 + q^10 - 2q^11 + 6q^12 + 8q^13 + O(q^14),
q^4 - q^5 - 2q^6 - q^7 + 3q^8 + 4q^9 - q^10 + q^11 - 4q^12 - 3*q^13 + O(q^14) ]
When I write
sage: CuspForms(Gamma0(55),2).newforms(names='a')
the output is
[q + q^2 - q^4 + q^5 + O(q^6),
q + a1q^2 + (-2a1 + 2)q^3 + (2a1 - 1)*q^4 - q^5 + O(q^6)]
What is the difference between those two and how to find a coefficient of a1?
Can someone tell me how to find newforms for some $S_k(N)$? I have troubles understanding concepts of Hecke forms and cusp spaces.