
// Make newform 1989.2.a.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1989_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_1989_2_a_e();".
// 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_1989_2_a_e();".
// 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_1989_a();"
function MakeCharacter_1989_a()
    N := 1989;
    order := 1;
    char_gens := [443, 613, 1873];
    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_1989_a_Hecke(Kf)
    return MakeCharacter_1989_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<X, Y> := 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<T> := 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], [0], [2], [0], [-4], [1], [-1], [-4], [0], [2], [-8], [-2], [-2], [-4], [-8], [10], [-4], [14], [-4], [0], [-14], [-8], [4], [6], [-6], [-6], [-8], [12], [-10], [-2], [0], [4], [22], [20], [10], [-16], [-2], [-4], [-16], [-14], [12], [6], [-16], [-22], [2], [16], [12], [8], [-12], [-10], [6], [24], [10], [-12], [30], [-24], [18], [8], [-10], [6], [20], [26], [-20], [0], [-22], [10], [4], [18], [12], [30], [14], [0], [8], [22], [4], [24], [-38], [-26], [6], [26], [-28], [-26], [-24], [-14], [16], [20], [22], [-22], [-30], [24], [-4], [24], [32], [-28], [-4], [-16], [2], [-26], [-4], [38], [-36], [-30], [-20], [6], [4], [-14], [-36], [14], [-24], [10], [-8], [-42], [-2], [-12], [-32], [-34], [44], [8], [-14], [-36], [-10], [34], [-22], [12], [-20], [-30], [30], [24], [-40], [-34], [12], [-16], [24], [-42], [38], [-14], [-6], [-4], [34], [38], [4], [-30], [-16], [12], [-2], [0], [-50], [-42], [-4], [8], [38], [30], [20], [32], [-28], [-8], [-24], [22], [42], [-6], [-12], [6], [-32], [-28], [-18], [-48], [-40], [22], [-22], [-22], [-36], [-34], [0], [-38], [-64], [6], [20], [-30], [-56], [14], [-8], [36], [22], [-42], [24], [42], [14], [28], [18], [-56], [18], [28], [52], [-30], [68], [38], [-54], [46], [-42], [24], [-22], [-56], [-58], [-30], [-36], [-30], [24], [-4], [14], [-20], [66], [42], [16], [-52], [64], [18], [64], [-18], [-48], [-46], [6], [16], [-2], [40], [-36], [54], [6], [24], [-72], [-36], [46], [-52], [-24], [54], [-20], [-24], [-38], [-14], [-20], [-40], [68], [-52], [40], [30], [-74], [0], [56], [-12], [-12], [-16], [-18], [70], [48], [42], [18], [-60], [54], [-28], [74], [26], [-24], [-4], [-18], [46], [-42], [68], [-30], [54], [-44], [26], [-82], [-4], [-54], [-32], [18], [48], [60], [-2], [26], [12], [-48], [16], [0], [-58], [68], [32], [-14], [82], [48], [-2], [-30], [36], [22], [76], [-82], [18], [48], [10], [60], [-68], [-22], [50], [-40], [68], [-28], [-6], [-12], [30], [-16], [38], [-24], [66], [-50], [-44], [0], [74], [68], [-8], [58], [46], [-68], [2], [-30], [8], [62], [-78], [76], [-60], [72], [-70], [-42], [66], [56], [-44], [-60], [-68], [30], [-42], [-30], [64], [-10], [6], [18], [72], [18], [-76], [70], [4], [-24], [-38], [12], [50], [50], [56], [54], [-26], [-40], [60], [30], [-16], [62], [-10], [-32], [12], [52], [74], [10], [16], [58], [-68], [68], [8], [-54], [-24], [46], [52], [40], [66], [-50], [-70], [-38], [54], [-16], [38], [-84], [-32], [-88], [22], [-76], [-16], [26], [-62], [-100], [68], [48], [-102], [24], [-42], [-12], [58], [54], [-66], [-88], [-18], [-70], [-80], [62], [-18], [60], [-76], [-94], [-54], [36], [-36], [-22], [-62], [-24], [0], [-42], [8], [-30], [-98], [-24], [-44], [2], [-46], [36], [-66], [-108], [80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1989_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1989_2_a_e();".
// 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_1989_2_a_e(:prec:=1)
    chi := MakeCharacter_1989_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_1989_2_a_e();".
// 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_1989_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1989_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![-2, 1]>],Snew);
    return Vf;
end function;