
// Make newform 4032.2.a.bv 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_bv();".
// 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_bv();".
// 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
        poly := [-1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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], [0, 0], [1, 1], [-1, 0], [2, 2], [-3, 1], [0, -2], [1, -1], [4, 0], [0, -2], [-2, 2], [0, 2], [4, 2], [2, 2], [6, 2], [-10, 0], [7, 1], [-9, -1], [4, 0], [4, -4], [6, -4], [-4, -4], [7, 1], [6, 0], [8, 2], [5, -3], [10, -2], [0, -4], [-4, -2], [8, 2], [0, -4], [-5, -3], [-6, -4], [-1, 1], [-6, 4], [0, 4], [-3, 1], [10, -6], [10, 6], [-1, 3], [16, -4], [1, 5], [-4, 4], [-4, 2], [2, 4], [-2, -6], [-12, 0], [-4, -4], [15, 1], [1, -11], [-2, 8], [4, -8], [8, 2], [-11, 3], [14, 0], [20, -4], [7, -5], [24, 0], [10, -8], [-26, 0], [5, -5], [-15, 1], [13, -5], [8, 0], [-4, -2], [18, -4], [18, 2], [-12, 10], [-2, -2], [-11, -7], [-2, 8], [-8, -4], [-4, 12], [6, -4], [-18, 6], [-2, -6], [0, -2], [13, 1], [28, -2], [-8, -6], [-11, -5], [2, 0], [-28, 0], [-4, 6], [32, 0], [-4, 8], [-10, 8], [-12, 2], [15, 3], [0, 16], [3, 13], [-22, 6], [-20, 0], [-20, -8], [4, 8], [4, -4], [-21, -9], [20, -6], [-11, 3], [14, -4], [-18, 6], [18, -4], [29, -5], [-4, -10], [-10, -2], [-6, 0], [-3, 3], [-6, -12], [0, 8], [-6, 16], [8, -8], [-16, -6], [-12, -10], [-19, -13], [36, 4], [-28, 6], [-37, -3], [2, -10], [4, -6], [-6, -14], [7, 7], [6, -4], [31, -5], [12, -16], [-3, -5], [-28, -6], [-24, 10], [2, 22], [10, -10], [7, -1], [10, -6], [16, 4], [-36, 0], [-8, -14], [8, -2], [4, -10], [29, 5], [-19, 3], [-41, -5], [4, 6], [-15, -17], [22, -8], [-16, 16], [-24, 4], [-9, -1], [-26, 2], [-7, -3], [32, -2], [-45, 5], [-12, -4], [0, -6], [2, -12], [32, -4], [-2, -14], [-8, 4], [-8, -12], [0, 16], [16, 14], [-18, 4], [-11, -11], [18, -6], [-34, -8], [-12, -8], [-25, 1], [-14, 12], [-10, 2], [24, 0], [11, -13], [28, 2], [7, 19], [-20, -8], [-7, -11], [4, 12], [34, -8], [-18, -6], [34, -12], [-24, -4], [-32, -10], [36, 4], [-29, 3], [-36, 0], [-3, -5], [-36, -10], [-18, 16], [8, 4], [11, -1], [10, 8], [-7, -9], [38, 4], [46, 2], [34, -8], [-4, 16], [24, -12], [3, -1], [-16, 4], [58, -4], [-12, -6], [-32, -14], [-30, -12], [48, 0], [6, 24], [-14, 14], [19, 11], [46, -4], [11, -3], [45, -3], [2, -18], [14, 6], [0, 2], [15, 25], [-20, -6], [-7, -7], [-16, -12], [3, -19], [-40, 8], [-24, 2], [-16, -16], [20, 18], [16, -8], [-44, 10], [12, -2], [-40, 8], [-10, 8], [28, -8], [-9, -23], [-28, -10], [-26, -16], [-4, 16], [12, 20], [18, 2], [4, -18], [35, -3], [8, -24], [14, 8], [9, 23], [-16, -24], [24, -6], [-6, -28], [-36, 8], [12, 12], [34, 2], [53, 3], [-20, 4], [42, 8], [-6, 12], [-14, 6], [16, 0], [3, -11], [-22, 2], [-24, 0], [6, 20], [30, -8], [68, 4], [-6, 0], [17, 9], [-32, -4], [-20, 14], [3, -3], [17, 1], [-48, -14], [12, 8], [-34, 6], [-17, -1], [-25, 15], [-16, -26], [-1, -23], [-22, -4], [14, 8], [24, -4], [-12, 10], [1, -3], [38, -10], [-2, 12], [24, 4], [-22, 24], [4, -12], [38, -2], [38, 12], [16, 14], [39, -15], [-36, -20], [-24, -8], [14, -30], [39, 7], [41, 7], [72, 4], [-38, -8], [26, -20], [50, -2], [30, 24], [14, -32], [-17, 9], [36, 6], [55, 9], [-46, -8], [59, 7], [-40, 0], [11, 15], [-11, 19], [-1, -15], [-56, -6], [20, 18], [8, -8], [46, 6], [42, 18], [-42, -4], [6, -26], [-45, -13], [-12, -24], [10, 24], [12, -4], [8, 22], [-14, -20], [-54, 2], [-20, 28], [10, -16], [21, 19], [32, 0], [-2, -36], [6, -8], [11, 5], [-50, -4], [-35, -3], [-4, 12], [44, -18], [20, -2], [-26, -2], [25, 23], [32, 12], [-28, 2], [-26, -4], [12, -14], [-46, -10], [-7, -1], [-40, -20], [43, -3], [-20, -18], [14, -8], [26, 24], [10, 22], [-2, -36], [-12, 30], [55, -13], [20, 4], [28, 2], [-38, -6], [-63, -11], [22, 30], [-4, -20], [37, -3], [15, 17], [-30, 8], [30, 16], [18, -34], [22, 20], [-30, 4], [-30, -18], [-25, 1], [-18, 16], [-4, 24], [-18, 4], [-46, -4], [40, -4], [16, -12], [-55, 7], [36, 2], [-25, -5], [76, 0], [20, -14], [-8, 12], [39, 9], [24, 0], [2, 12], [0, 8], [-2, -4], [31, -15], [-60, 8], [-26, -12], [16, -10], [64, 2], [19, -1], [-44, -10], [-32, -16], [-10, -24], [11, 13], [64, -16], [28, -20], [23, -25], [12, 0], [22, -6], [52, 10], [-79, 1], [32, -4], [43, 5], [24, 4], [6, -12], [6, 26], [4, 26], [8, -4], [0, 38], [-37, 27], [22, 24], [64, 0], [-64, -2], [-17, 11], [-22, 14], [34, 24], [36, -18], [-45, -19], [-67, -5], [46, -4], [-4, 42], [42, -14], [-12, 8], [-6, -32], [-57, -5], [28, 12], [18, 14], [16, 6], [-92, -4], [-52, 10], [27, -29], [8, 16], [-23, 15], [4, -10], [5, -43], [50, 2], [-34, -24], [7, 17], [-30, -18]];
    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_bv();".
// 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_bv(:prec:=2)
    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_bv();".
// 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_bv( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, -2, 1]>,<11,R![-16, -4, 1]>,<13,R![4, 6, 1]>,<17,R![-20, 0, 1]>,<19,R![-4, -2, 1]>],Snew);
    return Vf;
end function;