
// Make newform 1764.3.z.d in Magma, downloaded from the LMFDB on 20 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1764_z();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1764_z_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_1764_3_z_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_1764_3_z_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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_1764_z();"
function MakeCharacter_1764_z()
    N := 1764;
    order := 6;
    char_gens := [883, 785, 1081];
    v := [6, 6, 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_1764_z_Hecke();"
function MakeCharacter_1764_z_Hecke( : Kf := false)
    N := 1764;
    order := 6;
    char_gens := [883, 785, 1081];
    char_values := [[1, 0], [1, 0], [0, 1]];
    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 := 3;
    raw_aps := [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-8, 16], [0, 0], [32, -16], [0, 0], [0, 0], [-24, -24], [0, 26], [0, 0], [22, 0], [0, 0], [0, 0], [0, 0], [-112, 56], [-122, 122], [0, 0], [80, 80], [0, 142], [0, 0], [0, 0], [112, -224], [0, 0], [80, -40], [0, 0], [-214, 214], [0, 0], [146, 0], [0, 0], [0, 0], [-160, 320], [0, 0], [-286, 286], [-168, -168], [0, 262], [0, 0], [0, 0], [0, 0], [-104, 208], [0, 0], [-382, 382], [0, 0], [-56, -56], [166, 0], [168, -336], [0, 0], [528, -264], [0, 0], [0, 0], [224, 224], [0, 0], [0, 0], [0, 0], [0, 0], [-560, 280], [-122, 122], [0, 0], [192, 192], [0, 0], [288, -576], [0, 0], [-704, 352], [0, 0], [0, 362], [-482, 0], [0, 0], [-280, 560], [0, 0], [0, 0], [88, 88], [0, -698], [694, 0], [0, 0], [0, 0], [816, -408], [0, 0], [-304, -304], [0, 0], [-358, 0], [0, 0], [48, -96], [1008, -504], [0, 0], [0, 0], [0, -814], [0, 0], [-526, 0], [0, 0], [0, 0], [962, -962], [0, 0], [0, 26], [0, 0], [0, 0], [0, 0], [-416, 208], [0, -1034], [506, 0], [0, 0], [0, 0], [0, 0], [886, -886], [368, 368], [0, 0], [0, 0], [0, 0], [624, -1248], [-1040, 520], [1126, -1126], [0, 0], [-704, -704], [674, 0], [0, 0], [720, -1440], [0, 0], [0, 0], [0, 0], [760, 760], [-1154, 0], [0, 0], [0, 0], [480, -240], [0, 0], [0, -934], [0, 0], [-792, 1584], [-1200, 600], [1222, -1222], [0, 0], [0, 1202], [838, 0], [0, 0], [-64, 128], [0, 0], [112, 112], [0, 0], [0, 0], [-336, 672], [0, 0], [-1058, 1058], [0, 0], [-920, -920], [0, 0], [232, -464], [0, 0], [1120, -560], [0, 0], [0, 598], [0, 0], [1702, 0], [0, 0], [-214, 214], [0, 0], [0, 866], [0, 0], [-832, 1664], [0, 0], [0, 0], [0, 0], [-1534, 0], [0, 0], [0, 0], [0, 0], [-46, 46], [360, 360], [1826, 0], [0, 0], [0, 0], [504, -1008], [0, 0], [-1298, 1298], [1144, 1144], [0, 0], [1994, 0], [0, 0], [952, -1904], [-1488, 744], [0, 2158], [0, 0], [-1702, 0], [0, 0], [0, 0], [0, 0], [-118, 118], [1088, 1088], [0, 814], [0, 0], [1984, -992], [0, 0], [0, 1754], [0, 0], [0, 0], [0, 0], [-2398, 2398], [0, -74], [0, 0], [0, 0], [0, 0], [680, -1360], [560, -280], [-560, -560], [0, 0], [0, 0], [-2128, 1064], [0, 0], [0, 0], [-1456, -1456], [0, 478], [0, 0], [-2018, 0], [0, 0], [0, 0], [1760, -880], [-2638, 2638], [0, 0], [0, 0], [0, 0], [0, 2714], [-1224, 2448], [0, 0], [0, -2446], [0, 0], [-506, 0], [0, 0], [0, 0], [-304, 152], [0, 0], [-2902, 2902], [1680, 1680], [-2834, 0], [0, 0], [-1520, 3040], [0, 0], [2720, -1360], [0, 0], [0, 0], [0, 0], [0, 0], [3328, -1664], [-1768, -1768], [0, 746], [0, 0], [0, 0], [1672, -3344], [0, 0], [2134, -2134], [0, 0], [694, 0], [0, 0], [0, 0], [1856, -3712], [0, 0], [0, 0], [2566, -2566], [480, 480], [0, 0], [-3360, 1680], [-1298, 1298], [0, 0], [1480, 1480], [328, -656], [0, 0], [-3840, 1920], [0, 0], [0, 0], [-1846, 0], [0, 0], [-2992, 1496], [2906, -2906], [-480, -480], [0, -814], [-672, 1344], [2576, -1288], [0, 0], [3146, -3146], [0, 1874], [0, 0], [0, 0], [-962, 962], [0, 0], [-344, 688], [4256, -2128], [0, 0], [2978, -2978], [0, 0], [200, 200], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-22, 0], [0, 0], [3952, -1976], [0, 0], [0, 0], [1840, -3680], [4480, -2240], [0, 0], [1294, -1294]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1764_z_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1764_3_z_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_1764_3_z_d(:prec:=2)
    chi := MakeCharacter_1764_z();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_1764_3_z_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_1764_3_z_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_z();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![0, 1]>,<13,R![192, 0, 1]>,<19,R![768, -48, 1]>],Snew);
    return Vf;
end function;