
// Make newform 1600.2.l.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_l_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_l_d();".
// 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_1600_2_l_d();".
// 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 := [9, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -2, 0, 1], [-3, 1, 1, 0], [3, 1, -1, 0]];
    Rf_basisdens := [1, 3, 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_1600_l();"
function MakeCharacter_1600_l()
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    v := [4, 3, 4];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_l_Hecke();"
function MakeCharacter_1600_l_Hecke(Kf)
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    char_values := [[1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 1], [0, 0, 0, 0], [1, 1, 1, -1], [-1, -2, 1, 0], [-2, 0, 0, 2], [-2, 1, -1, -1], [-1, 2, 0, -1], [1, 3, 1, -1], [2, -2, 0, 0], [-1, 1, -1, -1], [5, 3, 2, 0], [0, 5, 0, 0], [-1, 3, -4, 0], [8, 0, 0, 0], [1, -1, 2, 0], [-3, -1, -2, 0], [5, -1, 0, -4], [-8, 7, 0, 1], [-2, 2, -2, 2], [1, 10, 1, -1], [-1, -1, 1, 1], [6, -5, 0, -1], [-3, -6, -3, 3], [0, -2, 2, 2], [1, 3, -2, 0], [2, 2, 2, -2], [5, 4, 1, 0], [1, -7, 0, 6], [9, -2, 2, 2], [-3, -1, 1, 1], [-1, 1, 0, 0], [-2, 5, -2, 2], [7, 8, -1, 0], [8, 12, -4, 0], [1, -1, 1, -1], [9, -7, 0, -2], [6, -5, 0, -1], [0, -18, 0, 0], [6, 0, 0, -6], [3, 2, 0, -5], [1, -1, 2, 0], [11, 3, -3, -3], [-8, 1, -1, -1], [5, -5, 10, 0], [-2, 6, -2, 2], [-6, 1, 0, 5], [-6, 0, 0, 0], [9, -13, 0, 4], [-2, -2, 0, 0], [-6, -8, -6, 6], [12, 2, -2, -2], [2, 5, -5, -5], [-6, 1, -7, 0], [-10, -2, 2, 2], [-5, -1, -5, 5], [4, -2, 0, -2], [-9, 3, -3, -3], [-10, -12, 2, 0], [-4, -6, -4, 4], [-6, -5, -1, 0], [1, 9, -8, 0], [-13, 12, 0, 1], [3, -3, 3, -3], [0, 16, 0, 0], [8, -2, 0, -6], [-1, 0, -1, 0], [3, 6, -6, -6], [5, -4, 9, 0], [15, -7, 0, -8], [20, 4, -4, -4], [1, 13, 1, -1], [20, 2, -2, -2], [11, 7, 4, 0], [-8, 1, -9, 0], [7, 3, -3, -3], [13, 5, 8, 0], [-16, 8, 0, 8], [6, -1, 1, 1], [-2, 13, -2, 2], [15, -18, 0, 3], [-4, 0, -4, 0], [1, 3, -3, -3], [-4, 5, -5, -5], [2, 12, 2, -2], [4, 11, -7, 0], [24, -1, 1, 1], [0, -17, 0, 0], [-22, 24, 0, -2], [-18, -2, 2, 2], [-5, -3, 0, 8], [13, -7, 7, 7], [-7, 9, -7, 7], [-21, -25, 4, 0], [-9, -3, 0, 12], [6, -12, 6, -6], [-11, 3, 0, 8], [2, -7, 2, -2], [-1, -4, 3, 0], [14, -16, 0, 2], [-3, -8, 0, 11], [24, -22, 0, -2], [-11, 5, 0, 6], [0, -9, 0, 0], [19, 11, 8, 0], [9, 2, -2, -2], [-10, -1, -9, 0], [9, 8, -8, -8], [7, -13, 7, -7], [3, 32, 3, -3], [-12, 10, -10, -10], [1, -7, 8, 0], [4, 14, 4, -4], [17, 13, 4, 0], [-1, 21, -1, 1], [-4, -6, 6, 6], [7, -15, 0, 8], [10, 12, 10, -10], [21, -25, 0, 4], [24, -23, 0, -1], [-14, -24, 10, 0], [-36, -2, 2, 2], [20, 16, 4, 0], [-1, 2, -3, 0], [25, -24, 0, -1], [-12, 4, 0, 8], [-22, -22, 0, 0], [7, -3, 3, 3], [8, 24, 8, -8], [8, -10, 0, 2], [19, -23, 0, 4], [1, -17, 1, -1], [4, -14, 14, 14], [30, 32, -2, 0], [-4, 19, -4, 4], [4, -11, 11, 11], [26, 22, 4, 0], [-7, 7, 0, 0], [11, -7, 0, -4], [8, -12, 8, -8], [29, 33, -4, 0], [-19, -11, -8, 0], [-6, -10, -6, 6], [10, 9, 1, 0], [-10, 18, 0, -8], [-2, -4, -2, 2], [-29, -25, -4, 0], [-2, 15, -2, 2], [-3, -12, 9, 0], [6, -8, 8, 8], [-18, 28, 0, -10], [18, 6, -6, -6], [-21, 20, 0, 1], [-8, 20, -8, 8], [-5, -9, 4, 0], [8, 10, -10, -10], [-5, -21, -5, 5], [-20, 0, 0, 0], [-3, 26, -3, 3], [2, 4, 0, -6], [-17, 27, 0, -10], [2, 9, 2, -2], [2, -20, 2, -2], [-31, -34, 3, 0], [0, -17, 17, 17], [-1, 25, -1, 1], [-13, -11, 11, 11], [-10, -4, -6, 0], [-8, -3, 3, 3], [26, 26, 0, 0], [-18, -3, -15, 0], [1, -21, 0, 20], [-19, -1, -19, 19], [1, -40, 1, -1], [10, 10, -10, -10], [-15, -4, -15, 15], [4, 11, -7, 0], [21, 9, 12, 0], [4, -20, 4, -4], [-5, 3, 0, 2], [8, 0, 0, 0], [-9, -4, 0, 13], [1, -5, 6, 0], [-7, -20, -7, 7], [24, 10, -10, -10], [-12, -10, -2, 0], [-11, 23, 0, -12], [5, 0, 0, -5], [-6, 37, -6, 6], [0, 2, -2, -2], [3, -8, 8, 8], [14, 7, 7, 0], [-14, 11, 0, 3], [-11, 9, 0, 2], [17, -28, 0, 11], [8, -12, 8, -8], [-21, -8, 8, 8], [19, -15, 0, -4], [-7, 16, -16, -16], [1, -7, 1, -1], [9, -7, 0, -2], [-26, -4, 4, 4], [0, 6, -6, 0], [-45, 0, 0, 0], [17, 17, 0, 0], [-17, -7, 0, 24], [-13, -3, 3, 3], [-27, 15, 0, 12], [-6, -8, -6, 6], [9, 11, -2, 0], [-22, -4, 4, 4], [31, 35, -4, 0], [6, 28, 6, -6], [25, 20, 5, 0], [5, -11, 5, -5], [6, 39, 6, -6], [33, 5, -5, -5], [-54, -1, 1, 1], [7, 25, 7, -7], [-11, 5, 0, 6], [-9, 1, -10, 0], [2, -48, 2, -2], [-10, 15, -15, -15], [-15, 1, -1, -1], [15, -8, 0, -7], [13, -3, 16, 0], [-7, -2, -7, 7], [2, -8, 8, 8], [-10, -28, -10, 10], [-6, -15, 9, 0], [9, -25, 0, 16], [28, -13, 0, -15], [-47, 5, -5, -5], [3, -28, 3, -3], [-21, 5, -26, 0], [-11, -9, 9, 9], [18, 13, -13, -13], [-4, -12, 8, 0], [-8, -23, 15, 0], [12, 28, 12, -12], [45, -40, 0, -5], [24, 15, 9, 0], [3, -29, 3, -3], [-8, 8, 0, 0], [-11, 8, -8, -8], [14, 10, 14, -14], [5, -3, 3, 3], [9, 11, 0, -20], [2, 13, -11, 0], [50, -4, 4, 4], [-24, 18, 0, 6], [-27, -4, 4, 4], [-5, -43, -5, 5], [-9, 30, -9, 9], [-2, -20, 0, 22], [12, -3, 0, -9], [16, 18, -2, 0], [-10, -5, -5, 0], [5, -5, 10, 0], [16, 27, 16, -16], [-11, 5, -5, -5], [-6, 27, 0, -21], [8, 24, -16, 0], [14, -30, 0, 16], [-10, 3, -3, -3], [-7, -8, 0, 15], [-1, 23, 0, -22], [6, -16, 6, -6], [9, 22, -13, 0], [-9, 13, -22, 0], [32, -16, 0, -16], [-8, 27, 0, -19], [1, -10, 1, -1], [40, -10, 10, 10], [-14, 4, -4, -4], [2, 52, 2, -2], [45, 36, 9, 0], [-6, -12, 0, 18], [0, 10, 0, 0], [1, 6, 0, -7], [-40, -4, 4, 4], [-3, -47, -3, 3], [8, -26, 8, -8], [-4, 16, -20, 0], [-30, -37, 7, 0], [32, -6, 6, 6], [28, 12, -12, -12], [-10, 16, -26, 0], [-8, 2, -8, 8], [22, 12, -12, -12], [7, -1, 0, -6], [15, -23, 0, 8], [-7, 38, -7, 7], [-6, 13, -19, 0], [28, -30, 0, 2], [3, -7, 0, 4], [-65, -3, 3, 3], [-14, -20, 6, 0], [-3, -19, 16, 0], [-8, -3, 0, 11], [18, 23, 18, -18], [-3, 23, 0, -20], [-24, 6, -6, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_l_d();".
// 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_1600_2_l_d(:prec:=4)
    chi := MakeCharacter_1600_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1600_2_l_d();".
// 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_1600_2_l_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, -10, 2, 2, 1]>,<7,R![100, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;