
// Make newform 1728.4.f.b in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_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_1728_4_f_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_f_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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 2, 0], [0, 2, 0, -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_1728_f();"
function MakeCharacter_1728_f()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_f_Hecke();"
function MakeCharacter_1728_f_Hecke( : Kf := false)
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    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 := 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, 0], [0, 0, 0, 0], [0, 17, 0, 0], [0, 0, 0, 0], [0, 0, 17, 0], [0, 0, 0, 0], [0, 0, 0, -73], [0, 0, 0, 0], [0, 0, 0, 0], [0, -308, 0, 0], [0, 0, -71, 0], [0, 0, 0, 0], [0, 0, 0, 126], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 179, 0], [0, 0, 0, -251], [0, 0, 0, 0], [271, 0, 0, 0], [0, -503, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1853, 0, 0, 0], [0, 0, 0, 0], [0, 19, 0, 0], [0, 0, 0, 0], [0, 0, -1260, 0], [0, 0, 0, 0], [0, 380, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 703], [0, 0, 0, 0], [0, -3709, 0, 0], [0, 0, 468, 0], [0, 0, 0, -2393], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 739, 0], [0, 0, 0, 0], [5111, 0, 0, 0], [0, 0, 0, 0], [0, -4373, 0, 0], [0, 0, 0, -3331], [0, -3220, 0, 0], [0, 0, 0, 0], [0, 0, -3060, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-4769, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -8101, 0, 0], [0, 0, -4788, 0], [0, 0, 0, 0], [0, 0, 0, -4446], [0, 0, 0, 0], [0, 0, 0, 918], [0, 0, 0, 0], [9109, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -2969], [12293, 0, 0, 0], [0, 0, 0, 0], [0, 0, -7487, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -9413, 0, 0], [0, 0, 4033, 0], [0, 0, 0, -827], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 9108, 0], [0, 0, 0, 0], [-8297, 0, 0, 0], [0, 0, 0, 0], [0, 0, -9199, 0], [0, 0, 0, 0], [2590, 0, 0, 0], [0, -14924, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-12710, 0, 0, 0], [0, 0, 0, 0], [0, 11969, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -6103, 0, 0], [0, 0, 0, 0], [0, 0, 0, -9450], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -11987], [0, 0, 14489, 0], [0, 0, 0, 8191], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -7561], [-27323, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-29302, 0, 0, 0], [0, 6137, 0, 0], [0, 0, -15877, 0], [0, 0, 0, 0], [0, 0, 0, 17783], [0, 26459, 0, 0], [0, 0, 0, 0], [0, 0, 0, 17226], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 2411, 0], [-9899, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 18810], [0, 0, 0, 0], [0, 0, -21223, 0], [0, 0, 0, 0], [0, -10780, 0, 0], [0, 0, 21204, 0], [0, 0, 0, -14490], [0, 0, 0, 0], [0, 17569, 0, 0], [0, 0, 19601, 0], [0, 0, 0, 0], [-34417, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -24049], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 13950], [0, 0, 0, 0], [0, 45611, 0, 0], [0, 0, 0, 0], [0, 0, 21403, 0], [0, 0, 0, 0], [0, 0, -28367, 0], [0, 0, 0, 0], [0, 0, 0, -4267], [0, 0, 0, 0], [0, 0, 21169, 0], [0, 0, 0, 0], [0, 0, 0, 3583], [0, 0, 0, 0], [0, 0, 0, 31429], [0, 0, 0, 0], [0, 2756, 0, 0], [0, 0, 0, 0], [-15227, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -53927, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -59669, 0, 0], [0, 0, -32292, 0], [8693, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 29700, 0], [0, 0, 0, 0], [-48259, 0, 0, 0], [0, 65683, 0, 0], [0, 0, 0, 0], [0, 0, 0, -18270], [0, 0, 0, 0], [0, 19441, 0, 0], [0, 0, 33300, 0], [0, 67537, 0, 0], [0, 0, 0, 0], [0, 0, -21924, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 14419, 0], [0, 0, 0, 35837], [52037, 0, 0, 0], [0, 0, 0, 0], [76519, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1819], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-75419, 0, 0, 0], [0, 0, -46493, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -82549, 0, 0], [0, 0, -48941, 0], [88037, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -70957, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 3349], [-1513, 0, 0, 0], [0, 0, 0, 0], [0, -45431, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [11951, 0, 0, 0], [0, -15860, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 41131, 0], [0, -29036, 0, 0], [0, 0, 0, 0], [0, -77380, 0, 0], [0, 0, 0, 0], [0, 0, 323, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -67517, 0, 0], [0, 0, 0, 0], [0, 0, 36233, 0], [0, 0, 0, -50310], [0, -103588, 0, 0], [0, 0, 0, 0], [0, 0, 0, -59203], [0, 0, 0, 0], [-36307, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -49249], [0, -109261, 0, 0], [0, 0, -57023, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -81863, 0, 0], [0, 0, 0, 0], [0, 0, 0, -23310], [0, 0, 0, 0], [0, 0, -32508, 0], [0, 0, 0, 0], [0, 0, 0, 0], [41813, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -75221, 0], [0, 0, 0, -28638], [0, 0, 0, 0], [4970, 0, 0, 0], [0, 127889, 0, 0], [0, 0, 0, 0], [0, 0, -55423, 0], [0, 0, 77507, 0], [0, 0, 0, 0], [0, 0, 0, 71910], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 5166], [0, 0, 0, 0], [0, 0, -56269, 0], [0, 0, 0, -78931], [51590, 0, 0, 0], [0, 13033, 0, 0], [71890, 0, 0, 0], [0, -5021, 0, 0], [0, 0, 0, 0], [0, 0, 59220, 0], [5402, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 59092, 0, 0], [0, 0, 0, 0], [0, 0, 66889, 0], [0, 0, 0, 44909], [0, 0, 0, 0], [71119, 0, 0, 0], [0, 0, 0, 0], [0, -150967, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -68796, 0], [0, 0, 0, 0], [0, 146249, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -96121], [-101771, 0, 0, 0], [0, 0, 0, 0], [0, -171236, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_f_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_4_f_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_f_b(:prec:=4)
    chi := MakeCharacter_1728_f();
    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_f_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_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![289, 0, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;