
// Make newform 1764.3.z.f in Magma, downloaded from the LMFDB on 27 November 2025.
// 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_f();".
// 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_f();".
// 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], [2, -1], [0, 0], [15, -15], [-8, 16], [17, 17], [-18, 9], [0, -9], [6, 0], [7, 7], [0, -31], [-32, 64], [10, 0], [50, -25], [-57, 57], [-47, -47], [94, -47], [49, -49], [126, 0], [15, 15], [0, 73], [-8, 16], [66, -33], [-16, 32], [57, 57], [-82, 41], [0, 39], [-103, 103], [78, 0], [50, 0], [114, -57], [63, -63], [-136, 272], [0, -177], [-55, 55], [-57, -57], [0, -23], [-160, 320], [162, -81], [63, -63], [-72, 144], [0, -57], [209, -209], [150, 0], [119, 119], [346, 0], [192, -384], [81, 81], [-194, 97], [0, -273], [222, 0], [127, 127], [56, -112], [-222, 111], [-153, 153], [-263, -263], [-82, 41], [-359, 359], [222, 0], [-1, -1], [136, -272], [-56, 112], [281, 281], [-258, 129], [0, 447], [0, -151], [274, 0], [543, -543], [-104, 208], [-175, -175], [0, 39], [-201, -201], [0, -383], [-230, 0], [370, -185], [-201, 201], [62, -31], [0, -177], [-193, -193], [344, -688], [-22, 0], [327, -327], [-16, 32], [-690, 345], [0, -729], [270, 0], [0, -799], [-248, 496], [-814, 0], [-462, 231], [-311, -311], [857, -857], [-570, 0], [0, 425], [112, -224], [-446, 223], [-143, -143], [-914, 457], [0, 721], [-118, 0], [-633, 633], [-159, -159], [0, 447], [-527, 527], [-97, -97], [552, -1104], [-350, 175], [807, -807], [-224, 448], [846, -423], [-503, 503], [-930, 0], [-305, -305], [194, 0], [-33, 33], [200, -400], [441, 441], [0, -33], [870, 0], [-585, -585], [-14, 0], [-318, 159], [-273, 273], [-178, 89], [-906, 0], [0, -703], [-686, 343], [-16, 32], [1118, -559], [577, -577], [-162, 0], [0, -775], [-950, 0], [226, -113], [528, -1056], [-103, -103], [-193, -193], [-504, 1008], [-465, 465], [-72, 144], [0, -753], [25, -25], [246, 0], [855, 855], [-208, 416], [-696, 1392], [113, 113], [1646, -823], [0, 855], [0, 241], [-768, 1536], [-1286, 0], [-1486, 743], [737, -737], [-1266, 0], [0, 473], [386, -193], [-496, 992], [-71, -71], [0, -201], [-66, 0], [194, 0], [1714, -857], [-1521, 1521], [649, 649], [-823, 823], [-73, -73], [914, 0], [482, -241], [1119, -1119], [696, -1392], [0, 663], [-1775, 1775], [-665, -665], [416, -832], [-934, 0], [1959, -1959], [-1168, 2336], [382, -191], [0, 745], [664, -1328], [2138, 0], [1858, -929], [-1545, 1545], [553, 553], [-199, 199], [-1041, -1041], [0, -527], [377, 377], [318, -159], [1782, 0], [0, 1625], [-766, 383], [-609, 609], [-31, -31], [881, -881], [0, -1007], [-48, 96], [-686, 343], [-249, 249], [608, -1216], [-482, 241], [623, 623], [-1144, 2288], [-263, -263], [-1490, 745], [0, -1881], [-1170, 0], [463, 463], [0, 2353], [-1128, 2256], [-62, 0], [2482, -1241], [553, 553], [1502, -751], [1049, -1049], [177, 177], [0, -2265], [822, 0], [0, -1807], [-1088, 2176], [0, -33], [0, 1241], [856, -1712], [-1430, 0], [-478, 239], [-777, 777], [-1266, 633], [0, -873], [521, -521], [159, 159], [2770, 0], [2847, -2847], [856, -1712], [-87, -87], [2718, -1359], [0, 1071], [-522, 0], [640, -1280], [-561, 561], [-18, 9], [-1401, -1401], [0, 449], [1056, -2112], [-2158, 1079], [-704, 1408], [-655, -655], [-815, 815], [1758, 0], [-614, 0], [1410, -705], [1719, -1719], [896, -1792], [-1351, -1351], [0, 1095], [697, -697], [-353, -353], [1464, -2928], [-2274, 1137], [-2327, 2327], [774, 0], [-1529, -1529], [-104, 208], [81, 81], [2542, -1271], [-1530, 0], [1344, -2688], [3146, 0], [1239, -1239], [-2946, 1473], [-415, 415], [1439, 1439], [0, -1735], [1296, -2592], [-3218, 1609], [0, -9], [-919, 919], [0, 1697], [-2926, 1463], [-1351, -1351], [2761, -2761], [-2000, 4000], [808, -1616], [1582, -791], [0, 2727], [1409, -1409], [-1482, 0], [-1081, -1081], [272, -544], [-345, 345], [-1967, -1967], [0, 351], [824, -1648], [2906, 0], [1673, 1673], [526, -263], [760, -1520], [-526, 263], [-8, 16], [638, -319], [0, -3345], [-263, 263]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_z();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![3, -3, 1]>,<11,R![225, -15, 1]>,<13,R![192, 0, 1]>,<19,R![243, 27, 1]>],Snew);
    return Vf;
end function;