
// Make newform 1600.2.f.i in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_f_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_f_i();".
// 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_f_i();".
// 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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [0, 0, 0, 2], [-1, 0, 2, -1]];
    Rf_basisdens := [1, 1, 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_f();"
function MakeCharacter_1600_f()
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_f_Hecke();"
function MakeCharacter_1600_f_Hecke(Kf)
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    char_values := [[1, 0, 0, 0], [-1, 0, 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, -1, 0, 0], [0, 0, 0, 0], [0, 0, -1, 1], [0, 0, -1, -2], [0, -2, 0, 0], [0, 0, 1, 2], [0, 0, 1, 0], [0, 0, 3, 3], [0, 0, 0, 0], [6, 2, 0, 0], [-6, 0, 0, 0], [6, -2, 0, 0], [5, 3, 0, 0], [0, 0, 3, 3], [0, -6, 0, 0], [0, 0, -3, 0], [0, 0, 5, 4], [-5, -3, 0, 0], [-6, 6, 0, 0], [0, 0, 5, 6], [-12, 0, 0, 0], [3, 1, 0, 0], [6, -4, 0, 0], [0, 0, -1, -6], [0, 0, 6, 0], [0, 0, 3, 3], [15, 1, 0, 0], [0, 0, 1, -4], [0, 0, 1, -4], [0, 0, 3, -3], [0, 0, -3, -6], [0, 0, 5, 4], [0, 0, -5, 0], [0, 0, 9, 0], [-6, 2, 0, 0], [-6, 4, 0, 0], [-11, 3, 0, 0], [0, 0, 3, 3], [-6, 0, 0, 0], [0, 0, 7, 8], [0, 0, -2, -4], [-6, -6, 0, 0], [0, 0, -1, -6], [0, -6, 0, 0], [0, -4, 0, 0], [0, 0, 1, 6], [0, 0, -5, -7], [-3, -9, 0, 0], [0, 0, -8, -4], [0, 0, 3, -6], [0, 12, 0, 0], [-10, -6, 0, 0], [0, 0, 7, 2], [0, 0, 3, 0], [0, 0, -9, -3], [0, 0, 3, 12], [6, -6, 0, 0], [6, 8, 0, 0], [-18, -6, 0, 0], [-5, 9, 0, 0], [30, 0, 0, 0], [-29, -3, 0, 0], [18, -6, 0, 0], [0, 0, -11, 0], [-12, -6, 0, 0], [0, 0, 5, -6], [0, 0, -7, 0], [9, 11, 0, 0], [0, 0, 0, -12], [0, 0, 3, 12], [-12, 12, 0, 0], [0, 0, -1, -11], [-6, 8, 0, 0], [0, 0, -1, 0], [0, 0, 3, -3], [0, 0, 9, 12], [24, 2, 0, 0], [-18, 8, 0, 0], [-14, 6, 0, 0], [0, 0, -1, 4], [0, 0, 3, 0], [-18, -6, 0, 0], [0, 0, -11, -6], [12, 4, 0, 0], [-9, 5, 0, 0], [6, -2, 0, 0], [0, 0, 7, 12], [0, 0, 6, 0], [0, 0, -9, 3], [-3, 15, 0, 0], [0, 0, 0, 0], [0, 0, -17, -7], [0, 0, 9, 6], [0, 0, 11, 24], [0, 0, 3, 3], [0, 0, 0, 12], [18, 8, 0, 0], [11, -15, 0, 0], [0, 0, -2, -16], [-7, 3, 0, 0], [6, -12, 0, 0], [21, 3, 0, 0], [18, 14, 0, 0], [0, 0, -13, -6], [0, 0, -1, 0], [9, -13, 0, 0], [0, 0, -1, 4], [12, 0, 0, 0], [-10, -6, 0, 0], [0, 0, 3, 3], [-24, 6, 0, 0], [0, 0, 9, -6], [0, 0, 1, -12], [-18, -2, 0, 0], [6, -6, 0, 0], [-5, 9, 0, 0], [0, 0, 15, 9], [12, -18, 0, 0], [0, 0, -13, -20], [0, 0, 9, 24], [0, 0, 13, 18], [-12, -18, 0, 0], [-3, -5, 0, 0], [0, 0, 5, -6], [0, 0, 3, 12], [0, 0, 8, -8], [24, 12, 0, 0], [0, 0, -13, -11], [-18, -12, 0, 0], [0, 0, -1, 0], [0, 0, 15, 9], [30, 6, 0, 0], [-30, 4, 0, 0], [6, -16, 0, 0], [-14, 12, 0, 0], [-36, -6, 0, 0], [11, -3, 0, 0], [12, -6, 0, 0], [6, -24, 0, 0], [0, 0, -17, -6], [0, 0, 21, 12], [0, 0, 7, 23], [-3, -1, 0, 0], [0, 0, -21, -12], [-12, 12, 0, 0], [0, -2, 0, 0], [0, 0, -11, 8], [0, 0, 17, 24], [0, 0, 15, 15], [6, -8, 0, 0], [-18, 6, 0, 0], [-11, 3, 0, 0], [0, 0, 3, 21], [-17, 9, 0, 0], [6, 18, 0, 0], [12, 16, 0, 0], [-18, 6, 0, 0], [0, 0, -7, 6], [0, 0, -6, -12], [51, -3, 0, 0], [0, 0, -11, -4], [0, 0, 3, 3], [0, 0, -7, -26], [0, 0, -1, 10], [0, 0, -9, 9], [-6, -22, 0, 0], [36, -6, 0, 0], [2, 12, 0, 0], [-24, 6, 0, 0], [0, 0, 7, -16], [0, 0, 22, 20], [-42, 6, 0, 0], [0, 0, 13, 18], [36, -4, 0, 0], [-6, -20, 0, 0], [0, 0, -1, 6], [0, 0, 6, 12], [0, 0, -13, -17], [0, 0, 5, 16], [0, 0, 7, 5], [0, 0, 13, 14], [-12, -10, 0, 0], [0, 0, 23, 22], [0, 0, 3, 3], [0, 0, -12, 0], [18, 16, 0, 0], [-1, -3, 0, 0], [-22, 6, 0, 0], [-42, 6, 0, 0], [0, 0, 5, 12], [-21, 1, 0, 0], [0, 0, 13, 6], [0, 0, -9, -24], [27, 5, 0, 0], [0, 0, 15, 18], [-26, 6, 0, 0], [48, -6, 0, 0], [0, 0, -23, -28], [0, 0, -21, 3], [0, 0, -12, -24], [6, -18, 0, 0], [-12, -6, 0, 0], [2, 0, 0, 0], [0, 0, 21, 12], [30, 0, 0, 0], [24, -20, 0, 0], [39, -3, 0, 0], [42, 12, 0, 0], [0, 0, 1, 18], [0, 0, -23, -18], [0, 0, -9, 0], [0, 0, -13, 1], [-15, -29, 0, 0], [12, 0, 0, 0], [38, -18, 0, 0], [0, 0, -5, -7], [6, -8, 0, 0], [0, 0, -9, 15], [-24, 6, 0, 0], [0, 0, -19, -8], [-12, -28, 0, 0], [-18, -4, 0, 0], [0, 0, -9, -27], [15, 1, 0, 0], [0, 0, -13, -8], [0, 0, -23, 2], [60, 0, 0, 0], [0, 0, -1, 7], [0, 0, 29, 10], [-24, 14, 0, 0], [0, 0, -17, -12], [6, 14, 0, 0], [6, 10, 0, 0], [17, -9, 0, 0], [0, 0, -9, -15], [-10, 6, 0, 0], [0, -6, 0, 0], [0, 0, 5, 28], [-18, 6, 0, 0], [3, 17, 0, 0], [0, 0, 23, -6], [0, 0, -13, 13], [0, 0, 16, -4], [0, 0, -25, 4], [36, -12, 0, 0], [0, 0, -13, -23], [0, 0, -21, 6], [0, 0, 25, 0], [0, 0, 3, -21], [-36, -18, 0, 0], [6, 16, 0, 0], [0, 0, -21, -27], [-10, 6, 0, 0], [42, 0, 0, 0], [0, 0, -19, -8], [0, 0, -10, -8], [-7, -21, 0, 0], [-6, 36, 0, 0], [0, 0, 7, 0], [0, 0, -5, -13], [-3, 31, 0, 0], [0, 0, 12, 12], [6, 40, 0, 0], [0, 0, -11, -10], [0, 0, -1, 0], [0, 0, 12, -12], [-6, 0, 0, 0], [35, 9, 0, 0], [-6, 12, 0, 0], [0, 0, -22, 4], [13, -33, 0, 0], [0, 0, -17, -18], [-36, 16, 0, 0], [0, 0, 7, 12], [0, 0, -13, -23], [-21, 5, 0, 0], [0, 0, -20, 8], [-10, -6, 0, 0], [0, 0, -13, 10], [0, 0, -21, 15], [18, 6, 0, 0], [0, 0, -33, -21], [0, 0, 38, 4], [13, -9, 0, 0], [-6, 18, 0, 0], [0, 0, -13, 18], [24, -6, 0, 0], [-48, -12, 0, 0], [-30, 6, 0, 0], [0, 0, 21, 24], [15, 9, 0, 0], [0, 0, 1, -10], [0, 0, 29, 22], [-42, -20, 0, 0], [0, 0, -9, -24], [-30, 10, 0, 0], [-6, 12, 0, 0], [0, 0, 9, 0], [-43, 15, 0, 0], [0, 0, 31, 0], [-30, 12, 0, 0], [-24, -8, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_f_i();".
// 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_f_i(:prec:=4)
    chi := MakeCharacter_1600_f();
    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_f_i();".
// 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_f_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -2, 1]>,<31,R![24, -12, 1]>],Snew);
    return Vf;
end function;