
// Make newform 768.4.c.d in Magma, downloaded from the LMFDB on 28 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_768_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_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_768_4_c_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_768_4_c_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 := [26, 0, 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_768_c();"
function MakeCharacter_768_c()
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    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_768_c_Hecke();"
function MakeCharacter_768_c_Hecke( : Kf := false)
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    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], [-1, -1], [0, 2], [0, -2], [46, 0], [-44, 0], [0, -4], [0, 2], [-88, 0], [0, -50], [0, 42], [-332, 0], [0, 96], [0, -46], [-384, 0], [0, 90], [-630, 0], [-236, 0], [0, -10], [680, 0], [-422, 0], [0, 146], [186, 0], [0, 188], [-1062, 0], [0, 2], [0, 390], [234, 0], [724, 0], [0, 200], [0, 442], [1742, 0], [0, -472], [0, 150], [0, -94], [0, 294], [-1948, 0], [0, 666], [2008, 0], [0, 238], [2382, 0], [-2652, 0], [-4784, 0], [-3074, 0], [0, -950], [0, -378], [0, -674], [0, 242], [-1270, 0], [-4556, 0], [0, 172], [1920, 0], [-1618, 0], [-1586, 0], [0, -776], [-4344, 0], [0, -538], [0, -638], [-4956, 0], [0, 388], [0, 1566], [0, 362], [0, -1626], [10104, 0], [-1198, 0], [0, -986], [0, -1202], [390, 0], [4366, 0], [5492, 0], [0, 16], [2216, 0], [0, -1398], [-2028, 0], [0, 258], [-2128, 0], [0, 378], [-6220, 0], [0, -380], [1270, 0], [-9126, 0], [-3436, 0], [1920, 0], [-15766, 0], [0, -1434], [8190, 0], [0, 2700], [-10682, 0], [0, 606], [0, 522], [-12246, 0], [15008, 0], [0, -1122], [5018, 0], [0, -3154], [-10632, 0], [0, 702], [0, 312], [0, -806], [-9484, 0], [0, 1074], [0, 214], [17898, 0], [0, -4136], [0, 654], [-5066, 0], [20042, 0], [0, 5168], [-4344, 0], [18250, 0], [0, 1034], [11748, 0], [0, -764], [0, -1410], [0, 5262], [0, -4636], [0, 3186], [7384, 0], [0, 2958], [-20434, 0], [12148, 0], [12730, 0], [0, 3858], [6558, 0], [0, -5974], [0, -6170], [25396, 0], [-7888, 0], [0, 1270], [14772, 0], [0, -4386], [-19080, 0], [0, 3986], [-4188, 0], [0, 6032], [-10370, 0], [0, 2114], [0, 4586], [0, -634], [0, 6640], [0, 2058], [0, 2962], [0, -3522], [-6598, 0], [41028, 0], [-17816, 0], [28308, 0], [0, 6616], [0, 66], [-48208, 0], [-42844, 0], [0, 1368], [0, -942], [37336, 0], [0, 3234], [35360, 0], [0, 862], [0, 356], [-34034, 0], [0, -7250], [24862, 0], [0, -4472], [0, -7858], [-43522, 0], [0, 3468], [42360, 0], [0, -4942], [-5356, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_768_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_768_4_c_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_768_4_c_d(:prec:=2)
    chi := MakeCharacter_768_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(997) - 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_768_4_c_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_768_4_c_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_768_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![104, 0, 1]>,<7,R![104, 0, 1]>,<11,R![-46, 1]>,<13,R![44, 1]>,<23,R![88, 1]>],Snew);
    return Vf;
end function;