
// Make newform 4032.2.a.bb in Magma, downloaded from the LMFDB on 25 April 2024.
// 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_bb();".
// 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_bb();".
// 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)
    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+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 := [[0], [0], [2], [-1], [-4], [-2], [6], [-8], [0], [6], [8], [2], [-2], [4], [8], [6], [0], [6], [4], [8], [10], [16], [8], [6], [-6], [2], [-16], [-12], [10], [-2], [-8], [8], [6], [8], [6], [0], [-18], [12], [-16], [18], [-4], [-10], [16], [-14], [6], [0], [-12], [0], [-8], [-10], [22], [-8], [10], [24], [-2], [-24], [-14], [0], [-22], [6], [-16], [18], [8], [-8], [-14], [30], [4], [2], [12], [30], [14], [0], [16], [26], [4], [24], [-2], [14], [-18], [-14], [-16], [26], [8], [10], [24], [36], [-34], [-38], [-30], [-16], [-8], [24], [-16], [-12], [4], [-40], [34], [-18], [32], [-14], [-36], [14], [32], [-26], [-28], [-14], [-24], [-34], [24], [26], [32], [18], [38], [32], [24], [30], [-16], [32], [-26], [-12], [-2], [-14], [-6], [36], [8], [-42], [-30], [-8], [16], [-26], [-52], [-32], [-8], [2], [30], [-22], [50], [40], [-30], [-26], [-16], [-10], [24], [-28], [-26], [48], [-26], [-18], [40], [-16], [-6], [-18], [20], [16], [28], [-24], [-40], [-10], [10], [-54], [52], [6], [-16], [48], [-2], [0], [-16], [14], [2], [-6], [-60], [14], [-24], [-22], [40], [-42], [-4], [46], [-24], [22], [-56], [-48], [-14], [6], [40], [-54], [18], [8], [-38], [40], [50], [4], [20], [18], [-36], [6], [2], [-6], [-2], [24], [-2], [8], [-66], [-30], [32], [58], [24], [-28], [6], [-24], [50], [18], [-64], [48], [24], [-46], [48], [6], [24], [22], [-62], [-8], [62], [72], [-56], [18], [6], [-56], [-8], [28], [-38], [0], [-64], [-10], [-16], [64], [26], [38], [-60], [-8], [-28], [16], [56], [34], [-18], [48], [32], [-56], [-76], [0], [50], [-18], [-40], [-22], [-54], [-36], [50], [40], [2], [18], [-8], [36], [-34], [70], [-58], [-64], [-34], [70], [-36], [14], [14], [60], [74], [-8], [18], [56], [-4], [2], [58], [72], [0], [40], [-16], [46], [-48], [40], [-46], [22], [-16], [-82], [30], [-24], [-10], [48], [-46], [34], [16], [-54], [-40], [64], [66], [-74], [-16], [-12], [4], [-14], [-68], [-58], [-48], [-38], [-48], [46], [46], [-4], [24], [10], [64], [0], [34], [-50], [-32], [-86], [58], [64], [38], [-86], [28], [-32], [-72], [-34], [-14], [22], [40], [32], [-52], [-80], [90], [78], [-38], [8], [10], [6], [-6], [-56], [22], [-44], [38], [-12], [16], [-62], [96], [42], [94], [-24], [10], [86], [-40], [48], [-18], [32], [-54], [22], [-40], [-76], [-88], [90], [18], [16], [42], [-36], [40], [32], [38], [8], [-14], [72], [-24], [66], [-42], [-14], [66], [6], [24], [30], [-48], [24], [16], [78], [-4], [-88], [-46], [2], [-12], [-32], [32], [-54], [-88], [14], [44], [-34], [-58], [30], [-48], [-82], [-22], [16], [18], [30], [32], [-48], [-46], [78], [-36], [-12], [74], [98], [32], [-32], [-42], [-72], [-90], [-82], [-96], [40], [-30], [26], [-28], [-90], [24], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4032_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_a_bb();".
// 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_bb(: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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>,<19,R![8, 1]>],Snew);
    return Vf;
end function;