// Make newform 2850.2.a.s in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2850_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_s();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2850_2_a_s();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2850_a();" function MakeCharacter_2850_a() N := 2850; order := 1; char_gens := [1901, 1027, 1351]; v := [1, 1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; function MakeCharacter_2850_a_Hecke(Kf) return MakeCharacter_2850_a(); end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 2; raw_aps := [[1], [-1], [0], [0], [-4], [-4], [6], [-1], [0], [-2], [0], [-4], [-12], [6], [0], [-14], [-10], [-6], [-4], [12], [8], [-8], [12], [-8], [-10], [12], [2], [20], [-14], [-10], [-10], [-20], [-6], [-12], [0], [16], [6], [-10], [12], [-2], [6], [-10], [6], [-10], [-10], [-4], [-8], [10], [-12], [-10], [14], [10], [26], [-24], [-2], [16], [-2], [-28], [-14], [12], [-2], [14], [0], [-18], [4], [18], [8], [18], [32], [34], [22], [34], [-12], [12], [12], [36], [-4], [18], [28], [-34], [8], [-14], [-12], [-18], [16], [-12], [-32], [-32], [24], [16], [-24], [38], [-18], [36], [-4], [16], [18], [-12], [0], [2], [24], [-30], [-36], [32], [4], [-8], [32], [-22], [-12], [-42], [30], [-46], [6], [-28], [28], [36], [-26], [-24], [18], [-26], [-2], [26], [-30], [-12], [-4], [20], [22], [6], [-40], [-26], [20], [-44], [-8], [-2], [-30], [50], [-38], [48], [-42], [26], [-40], [-4], [-24], [12], [50], [8], [-50], [-42], [-28], [-24], [-28], [42], [-2], [0], [12], [-48], [4], [-42], [-20], [58], [40], [42], [-44], [38], [34], [56], [48], [-2], [-14], [22], [10], [-2], [-14], [44], [-56], [38], [28], [0], [38], [-10], [28], [-34], [-8], [10], [64], [-24], [-40], [8], [-42], [-54], [22], [28], [8], [42], [-32], [-38], [18], [-22], [-54], [-24], [42], [0], [20], [-54], [-48], [-34], [-12], [68], [-6], [20], [-4], [36], [-4], [-12], [-56], [-34], [52], [-60], [-32], [54], [-70], [-40], [20], [28], [-28], [-74], [-22], [-20], [50], [-48], [58], [-4], [-40], [0], [-46], [56], [-14], [66], [-48], [-8], [-44], [-28], [-12], [-2], [-62], [42], [-24], [-10], [32], [-56], [46], [-30], [-24], [22], [14], [24], [-38], [-32], [-54], [64], [26], [-12], [-2], [16], [-38], [-8], [-30], [-30], [-48], [-6], [-18], [32], [-40], [-56], [-38], [68], [-32], [46], [70], [-36], [-24], [40], [-48], [10], [70], [54], [16], [-22], [24], [-60], [16], [-52], [38], [-46], [-16], [-4], [-64], [-86], [-42], [-26], [0], [-6], [-56], [-4], [-4], [38], [36], [-30], [-2], [26], [80], [-40], [-52], [-8], [-16], [10], [56], [36], [-28], [6], [0], [-40], [-2], [66], [-34], [18], [-80], [-4], [8], [-54], [30], [34], [40], [36], [36], [-4], [62], [-22], [6], [-16], [-28], [-50], [18], [-32], [34], [18], [-34], [-8], [-80], [22], [4], [-14], [-88], [42], [-22], [34], [38], [-68], [-46], [56], [-90], [78], [68], [10], [38], [74], [50], [-70], [-50], [-32], [24], [-16], [58], [32], [-14], [-42], [14], [-48], [6], [46], [14], [-6], [-76], [54], [4], [-12], [88], [58], [74], [-60], [30], [-6], [84], [-26], [-92], [26], [88], [-88], [-40], [100], [-14], [82], [-82], [-6], [70], [-84], [58], [48], [92], [-80], [78], [22], [-36], [-60], [-8], [-84], [-12], [46], [82], [16], [54], [92], [72], [-22], [-26], [-82], [36], [-38], [20], [-42]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2850_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_s();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_2850_2_a_s(:prec:=1) chi := MakeCharacter_2850_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2850_2_a_s();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_2850_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2850_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![4, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;