
// Make newform 1600.2.n.s in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_n_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_n_s();".
// 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_n_s();".
// 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, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 3, 3];
    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_n();"
function MakeCharacter_1600_n()
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    v := [2, 4, 3];
    // 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_n_Hecke();"
function MakeCharacter_1600_n_Hecke(Kf)
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 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, 1, 1, 0], [0, 0, 0, 0], [2, 0, -2, 0], [0, 1, 1, 1], [-2, 0, 2, 2], [-2, 1, -2, 0], [-5, 1, 0, -1], [-2, 2, -2, 0], [0, -2, 2, -2], [0, -2, 2, -2], [6, 0, 6, 0], [5, 0, 0, 0], [-2, 4, -2, 0], [0, 0, 0, 6], [4, 0, -4, -2], [-2, 4, 0, -4], [-4, 2, 0, -2], [3, 0, -3, -1], [0, 0, -2, 0], [2, 0, -2, -3], [-4, 2, 0, -2], [1, 3, 1, 0], [0, -2, -7, -2], [0, 4, 0, 0], [0, -6, 0, 6], [0, 2, 0, 0], [3, 0, -3, 7], [0, 2, -8, 2], [8, 0, -8, -3], [-6, 0, 6, 2], [0, -4, -6, -4], [4, -5, 4, 0], [3, -5, 0, 5], [0, -2, -14, -2], [0, -6, -8, -6], [-4, 10, -4, 0], [-9, 1, -9, 0], [-8, 0, 8, 2], [4, 0, -4, 8], [-3, -5, 0, 5], [14, 4, 0, -4], [0, 2, -18, 2], [10, 0, -10, 5], [4, -2, 4, 0], [0, 4, 0, -4], [0, 5, 1, 5], [12, 4, 12, 0], [10, 0, -10, 0], [0, 4, 4, 4], [-12, 0, 12, 4], [-6, -8, 0, 8], [1, 2, 0, -2], [0, -1, -15, -1], [4, 12, 4, 0], [6, -10, 6, 0], [0, 0, 0, 0], [0, 2, 22, 2], [10, -2, 10, 0], [22, 4, 0, -4], [-7, -7, -7, 0], [2, 0, -2, -4], [-9, 0, 9, -7], [0, 2, -20, 2], [12, 0, -12, -4], [-8, 12, -8, 0], [0, 3, 3, 3], [-16, 7, -16, 0], [-17, 0, 17, 3], [0, -6, 4, -6], [-12, 0, 12, -4], [24, -2, 0, 2], [4, 0, -4, -12], [0, 0, 0, 2], [-11, -9, 0, 9], [-2, -8, -2, 0], [0, -4, -2, -4], [4, -8, 4, 0], [-9, 2, 0, -2], [0, 4, 1, 4], [5, 7, 0, -7], [8, 8, 0, -8], [0, 2, 8, 2], [10, 0, -10, 3], [26, 0, 0, 0], [-1, -13, -1, 0], [0, 2, 1, 2], [12, 9, 12, 0], [-12, 4, 0, -4], [12, -12, 12, 0], [-2, 0, 2, 20], [2, -2, 0, 2], [10, 0, -10, -2], [0, 8, -18, 8], [-2, 4, 0, -4], [4, 8, 4, 0], [0, -4, 30, -4], [-19, 0, 0, 0], [-7, -1, -7, 0], [0, 0, 0, 0], [-5, 0, 5, -3], [-6, -2, -6, 0], [-6, -4, -6, 0], [0, -8, -25, -8], [0, 0, 14, 0], [8, 1, 8, 0], [-7, 0, 7, 7], [-4, 0, 4, 1], [-16, 6, 0, -6], [1, -6, 0, 6], [-4, 0, 4, -8], [-12, 0, 12, -14], [-4, -16, -4, 0], [-18, 0, 0, 0], [0, -6, 10, -6], [6, 4, 0, -4], [-14, 12, -14, 0], [-8, 0, 8, 16], [26, 0, -26, 0], [13, -7, 0, 7], [-6, -14, 0, 14], [-4, 0, 4, 8], [-28, 0, -28, 0], [-11, -1, -11, 0], [0, -11, -13, -11], [18, -2, 0, 2], [0, 0, 12, 0], [-36, 2, 0, -2], [0, 0, 0, -14], [-18, 0, 18, -2], [-14, -8, 0, 8], [2, -6, 2, 0], [0, 16, 2, 16], [0, 12, 0, 0], [-17, 0, 0, 0], [0, 6, 11, 6], [-2, 0, 2, 22], [2, 0, -2, -4], [4, 6, 4, 0], [0, -4, 10, -4], [0, 0, 26, 0], [-6, -4, 0, 4], [-4, 12, -4, 0], [19, 0, -19, 13], [0, -2, 42, -2], [0, 16, 0, -16], [12, 0, -12, 6], [16, -7, 16, 0], [35, 3, 0, -3], [8, -2, 8, 0], [4, -12, 4, 0], [26, -4, 0, 4], [1, 17, 1, 0], [40, 0, -40, 0], [6, 0, -6, -4], [0, -4, -20, -4], [-42, -6, 0, 6], [0, -8, 26, -8], [-18, 3, -18, 0], [-6, -10, 0, 10], [22, 0, -22, -4], [24, 0, -24, 5], [-12, 0, 12, 4], [0, 7, 15, 7], [-30, -3, -30, 0], [-2, 4, -2, 0], [0, 2, 28, 2], [8, 8, 8, 0], [0, -2, -35, -2], [14, 0, -14, 10], [-27, -9, 0, 9], [16, -6, 0, 6], [0, -18, -16, -18], [6, 0, -6, 7], [34, -4, 0, 4], [0, 2, -31, 2], [0, 9, -1, 9], [-40, -6, 0, 6], [8, 20, 8, 0], [0, -4, -30, -4], [-8, 0, 8, 24], [0, -5, -15, -5], [6, 0, -6, -20], [-34, -3, -34, 0], [-4, -2, -4, 0], [0, 8, 0, 8], [-22, -12, -22, 0], [39, -5, 39, 0], [0, 12, -23, 12], [0, 16, -2, 16], [-24, 0, 24, -5], [-25, -5, -25, 0], [0, -9, 19, -9], [-6, -16, 0, 16], [-11, 0, 11, -29], [4, 0, -4, 4], [17, -8, 0, 8], [0, 0, 0, -18], [12, -1, 12, 0], [-6, -8, -6, 0], [0, -8, 2, -8], [0, -8, 36, -8], [-24, -8, -24, 0], [0, 16, -5, 16], [38, -12, 0, 12], [-12, 2, -12, 0], [-6, -10, 0, 10], [38, 4, 38, 0], [0, -8, -6, -8], [0, -4, 6, -4], [4, 12, 4, 0], [-4, 22, 0, -22], [-24, -22, -24, 0], [9, 0, -9, 13], [18, -18, 0, 18], [-1, 20, 0, -20], [14, 0, -14, -22], [-9, 14, 0, -14], [-10, 0, 10, 0], [-14, 0, 14, -28], [14, -12, 0, 12], [2, -8, 0, 8], [0, 2, 5, 2], [-2, 4, -2, 0], [-31, 0, 31, -13], [0, -6, 8, -6], [2, 0, -2, 27], [10, 12, 0, -12], [-24, 0, 24, 2], [0, -3, -45, -3], [14, 0, -14, 24], [-1, -1, 0, 1], [0, 2, 12, 2], [-21, -6, 0, 6], [-26, -12, -26, 0], [-38, 0, 38, -4], [0, 22, 23, 22], [-2, 0, 2, 22], [21, 7, 0, -7], [0, -12, 4, -12], [-15, -3, -15, 0], [0, -17, -7, -17], [-6, -10, -6, 0], [0, 12, -32, 12], [0, 0, 0, 3], [24, 0, 0, 0], [-30, 0, 30, -12], [0, -8, 46, -8], [51, -7, 0, 7], [32, -6, 32, 0], [0, 12, 0, 0], [9, 0, 0, 0], [-14, 0, 14, 4], [0, -14, 25, -14], [-2, 0, 2, -6], [39, 11, 0, -11], [-26, 2, 0, -2], [-25, 0, 25, 5], [2, 24, 2, 0], [-4, 15, -4, 0], [-10, -20, -10, 0], [11, 0, -11, 19], [0, -2, 22, -2], [-6, 0, 6, -2], [-10, -5, -10, 0], [-21, -11, 0, 11], [0, 20, -6, 20], [-30, 20, 0, -20], [3, -23, 3, 0], [-30, 0, 30, 16], [14, -22, 0, 22], [-3, 0, 3, 25], [-38, 0, 38, -7], [22, -20, 0, 20], [8, -20, 8, 0], [16, 16, 16, 0], [5, 0, -5, -3], [0, -12, -32, -12], [-46, 8, 0, -8], [0, 23, 25, 23], [-24, 4, -24, 0], [0, 6, 34, 6], [8, 0, -8, -40], [8, -8, 0, 8], [33, 0, -33, -5], [0, 8, -32, 8], [12, 0, -12, 20], [-6, -34, -6, 0], [-20, -12, 0, 12], [0, -8, 18, -8], [24, 14, 0, -14], [-26, 0, 26, -8], [-6, 0, 6, 21], [0, -5, 69, -5], [8, 0, -8, -4], [0, -32, 6, -32], [0, 18, -4, 18], [18, 0, -18, -14], [-54, -8, 0, 8], [-11, 0, 11, -11], [20, 0, -20, -17], [10, 32, 10, 0], [-72, -4, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_n_s();".
// 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_n_s(:prec:=4)
    chi := MakeCharacter_1600_n();
    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_n_s();".
// 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_n_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 4, 8, -4, 1]>,<7,R![8, -4, 1]>,<11,R![25, 0, 14, 0, 1]>,<13,R![16, -32, 32, 8, 1]>],Snew);
    return Vf;
end function;