2014-12-07 06:18:04 -0600 answered a question Newforms and basis of new subspace [explanation] Thank you. How do we get a polynomial defining a newform? 2014-12-04 16:12:37 -0600 asked a question Newforms and basis of new subspace [explanation] Hi, 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() [ q + 2*q^3 - 2*q^5 - 4*q^6 - 3*q^7 + 5*q^8 + 9*q^9 - q^10 + 2*q^11 - 10*q^12 - 9*q^13 + O(q^14), q^2 - 2*q^3 + 2*q^5 + 2*q^6 + 2*q^7 - 5*q^8 - 8*q^9 + q^10 - 2*q^11 + 6*q^12 + 8*q^13 + O(q^14), q^4 - q^5 - 2*q^6 - q^7 + 3*q^8 + 4*q^9 - q^10 + q^11 - 4*q^12 - 3*q^13 + O(q^14) ] sage: CuspForms(Gamma0(55),2).newforms(names='a') [q + q^2 - q^4 + q^5 + O(q^6), q + a1*q^2 + (-2*a1 + 2)*q^3 + (2*a1 - 1)*q^4 - q^5 + O(q^6)]  What is the difference between those 2 and how to find a coefficient of a1? Can someone tell me how to find newforms for some S_k if the second function is not correct. Thank you.