
// Make newform 4032.2.a.z in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4032_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_4032_2_a_z();".
// 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_4032_2_a_z();".
// 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_4032_a();"
function MakeCharacter_4032_a()
    N := 4032;
    order := 1;
    char_gens := [127, 3781, 1793, 577];
    v := [1, 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_4032_a_Hecke( : Kf := Kf)
    return MakeCharacter_4032_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]];
    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 := [[0], [0], [0], [1], [4], [4], [2], [-6], [8], [2], [4], [-10], [10], [4], [4], [-2], [-10], [8], [-8], [0], [-6], [16], [-2], [-18], [-2], [0], [4], [16], [-10], [-6], [16], [6], [14], [-10], [6], [16], [-12], [4], [20], [-12], [0], [12], [0], [14], [6], [-4], [8], [-24], [22], [20], [6], [24], [14], [10], [-2], [-16], [-12], [16], [-14], [6], [10], [8], [10], [-24], [-26], [-34], [-12], [-18], [12], [-20], [-18], [-16], [-8], [-6], [28], [20], [-6], [28], [2], [-2], [-6], [18], [8], [22], [-24], [24], [30], [22], [28], [-24], [-18], [12], [-24], [-8], [-8], [-16], [20], [-38], [-14], [-30], [-20], [30], [-46], [-6], [-4], [-14], [18], [6], [24], [-46], [-48], [22], [-22], [-22], [16], [34], [-34], [-12], [-38], [-12], [-24], [-6], [36], [24], [-22], [-6], [-26], [-36], [-28], [-8], [4], [16], [-24], [-10], [-30], [6], [-24], [-46], [20], [26], [34], [-2], [-8], [0], [-24], [-52], [36], [18], [-26], [48], [14], [38], [-16], [28], [32], [16], [40], [18], [-22], [-40], [4], [22], [8], [-42], [-26], [-36], [40], [-8], [-2], [-4], [-24], [36], [-48], [-6], [-28], [-26], [-16], [-54], [32], [8], [8], [18], [14], [-34], [0], [-60], [-22], [-6], [-14], [-12], [2], [24], [-32], [-20], [-32], [6], [-18], [-2], [6], [-8], [54], [28], [56], [-22], [-66], [40], [-4], [12], [-30], [-2], [-50], [32], [16], [54], [40], [-2], [-40], [-54], [24], [10], [-50], [-40], [30], [-24], [-26], [62], [-50], [40], [0], [4], [6], [-42], [-8], [22], [2], [-32], [30], [-42], [8], [-32], [36], [18], [48], [10], [-26], [-28], [-16], [6], [-60], [24], [2], [-34], [-16], [50], [-56], [-64], [14], [62], [64], [-18], [24], [28], [-16], [56], [50], [-10], [14], [-18], [-48], [66], [-36], [76], [-38], [64], [-14], [16], [60], [50], [-2], [-50], [56], [-40], [-20], [8], [2], [0], [-46], [38], [52], [30], [70], [-18], [-6], [46], [-22], [-44], [-80], [-12], [-38], [46], [46], [-6], [56], [44], [-76], [58], [-4], [32], [40], [-30], [24], [-54], [-10], [68], [32], [2], [2], [-24], [-70], [-34], [-42], [-78], [-88], [-32], [-54], [22], [28], [34], [-64], [-46], [50], [-6], [60], [74], [32], [14], [-74], [78], [-14], [92], [-38], [42], [-28], [-48], [74], [84], [-52], [12], [72], [56], [-50], [26], [54], [12], [58], [-42], [-84], [-42], [-66], [72], [-22], [6], [-48], [-48], [10], [-26], [20], [56], [-26], [80], [22], [40], [86], [24], [2], [70], [40], [10], [50], [-66], [20], [-38], [-40], [-82], [94], [-24], [16], [16], [88], [-92], [-34], [16], [-96], [22], [48], [34], [4], [-54], [-80], [-22], [-72], [-66], [-24], [-14], [76], [52], [10], [66], [38], [-14], [90], [-46], [-92], [-88], [74], [-84], [-96], [-12], [66], [0], [26], [-56], [24], [-30], [54], [-64], [-76], [86], [-18], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4032_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_a_z();".
// 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_4032_2_a_z(:prec:=1)
    chi := MakeCharacter_4032_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_4032_2_a_z();".
// 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_4032_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-4, 1]>,<13,R![-4, 1]>,<17,R![-2, 1]>,<19,R![6, 1]>],Snew);
    return Vf;
end function;