
// Make newform 1728.4.c.b in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_c_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_1728_4_c_b();".
// 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_1728_4_c_b();".
// 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_1728_c();"
function MakeCharacter_1728_c()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [1, 2, 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_1728_c_Hecke();"
function MakeCharacter_1728_c_Hecke( : Kf := false)
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    char_values := [[-1, 0], [1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, 0], [0, 1], [0, 0], [-19, 0], [0, 0], [0, -17], [0, 0], [0, 0], [0, 90], [-323, 0], [0, 0], [0, -126], [0, 0], [0, 0], [0, 0], [-719, 0], [0, -629], [0, 0], [919, 0], [0, -127], [0, 0], [0, 0], [523, 0], [0, 0], [0, 613], [0, 0], [-646, 0], [0, 0], [0, -1638], [0, 0], [0, 0], [0, -1873], [0, 0], [0, -1819], [3850, 0], [0, 1007], [0, 0], [0, 0], [0, 0], [1241, 0], [0, 0], [-3961, 0], [0, 0], [0, -3203], [0, -2701], [0, -3366], [0, 0], [4466, 0], [0, 0], [0, 0], [-2609, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -2971], [-4030, 0], [0, 0], [0, -4446], [0, 0], [0, -918], [0, 0], [-901, 0], [0, 0], [0, -3961], [7363, 0], [0, 0], [-10547, 0], [0, 0], [0, 0], [0, -1691], [-251, 0], [0, -7757], [0, 0], [0, 0], [-1190, 0], [0, 0], [16543, 0], [0, 0], [-10459, 0], [0, 0], [-2590, 0], [0, 6210], [0, 0], [0, 0], [-12710, 0], [0, 0], [0, -10583], [0, 0], [0, 0], [0, 9001], [0, 0], [0, 9450], [0, 0], [0, 0], [0, 0], [0, -53], [20789, 0], [0, 6551], [0, 0], [0, 0], [0, 0], [0, 15751], [-9613, 0], [0, 0], [0, 0], [0, 0], [-29302, 0], [0, -11519], [30241, 0], [0, 0], [0, -8873], [0, 8189], [0, 0], [0, 17226], [0, 0], [0, 0], [0, 0], [13249, 0], [-33949, 0], [0, 0], [0, 0], [0, 18810], [0, 0], [-27523, 0], [0, 0], [0, 21762], [15050, 0], [0, 14490], [0, 0], [0, 1961], [17333, 0], [0, 0], [39023, 0], [0, 0], [0, 19351], [0, 0], [0, 0], [0, -13950], [0, 0], [0, 19277], [0, 0], [-47143, 0], [0, 0], [-38411, 0], [0, 0], [0, -27037], [0, 0], [13357, 0], [0, 0], [0, 24263], [0, 0], [0, 13411], [0, 0], [0, 32130], [0, 0], [40283, 0], [0, 0], [0, 0], [0, 0], [0, 34649], [0, 0], [0, 0], [0, 0], [0, -35099], [-28910, 0], [59347, 0], [0, 0], [0, 0], [40138, 0], [0, 0], [-63629, 0], [0, 29053], [0, 0], [0, -18270], [0, 0], [0, -28927], [-39494, 0], [0, -26839], [0, 0], [61490, 0], [0, 0], [0, 0], [0, 0], [13553, 0], [0, 39203], [-21797, 0], [0, 0], [-52649, 0], [0, 0], [0, 39131], [0, 0], [0, 0], [0, 0], [-68221, 0], [82529, 0], [0, 0], [0, 0], [0, 0], [0, -48619], [63593, 0], [38323, 0], [0, 0], [0, 0], [0, -52147], [0, 0], [0, 0], [0, -44621], [-80137, 0], [0, 0], [0, -1063], [0, 0], [0, 0], [-88489, 0], [0, -55062], [0, 0], [0, 0], [0, 0], [-24751, 0], [0, 58050], [0, 0], [0, -42966], [0, 0], [-54503, 0], [0, 0], [0, 0], [0, -58859], [0, 0], [99989, 0], [0, -50310], [0, -25830], [0, 0], [0, -54757], [0, 0], [76267, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -17441], [0, -69787], [-121283, 0], [0, 0], [0, 0], [0, 67841], [0, 0], [0, -23310], [0, 0], [-114550, 0], [0, 0], [0, 0], [126667, 0], [0, 0], [0, 0], [116801, 0], [0, -28638], [0, 0], [4970, 0], [0, -50903], [0, 0], [-131563, 0], [-97631, 0], [0, 0], [0, -71910], [0, 0], [0, 0], [0, -5166], [0, 0], [30529, 0], [0, -13789], [-51590, 0], [0, -48943], [71890, 0], [0, 40933], [0, 0], [111274, 0], [5402, 0], [0, 0], [0, 0], [0, -83790], [0, 0], [44749, 0], [0, 93131], [0, 0], [-161729, 0], [0, 0], [0, 57817], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [121210, 0], [0, 0], [0, -46799], [0, 0], [0, 0], [0, 78319], [-177301, 0], [0, 0], [0, 29610]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_4_c_b();".
// 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_1728_4_c_b(:prec:=2)
    chi := MakeCharacter_1728_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1728_4_c_b();".
// 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_1728_4_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![3, 0, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;