
// Make newform 768.3.h.d in Magma, downloaded from the LMFDB on 05 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_768_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_h_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_3_h_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_3_h_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, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 2, -1, -2], [0, 4, 0, 4]];
    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_768_h();"
function MakeCharacter_768_h()
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    v := [2, 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_768_h_Hecke();"
function MakeCharacter_768_h_Hecke( : Kf := false)
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, -1, 0], [0, -1, -2, 0], [6, 0, 0, 0], [0, -1, -2, 0], [0, 5, 0, 0], [0, 0, 0, 4], [0, 1, 0, 0], [0, 0, 0, -2], [0, 3, 6, 0], [-22, 0, 0, 0], [0, 3, 0, 0], [0, 0, 0, 6], [0, -41, 0, 0], [0, 0, 0, -12], [0, 11, 22, 0], [0, -13, -26, 0], [0, -43, 0, 0], [0, 1, 0, 0], [0, 0, 0, -22], [-82, 0, 0, 0], [10, 0, 0, 0], [0, -13, -26, 0], [0, 0, 0, -6], [-94, 0, 0, 0], [0, -9, -18, 0], [134, 0, 0, 0], [0, -9, -18, 0], [0, 5, 0, 0], [0, 0, 0, 12], [106, 0, 0, 0], [0, -1, -2, 0], [0, 0, 0, -18], [0, 39, 0, 0], [0, -29, -58, 0], [-218, 0, 0, 0], [0, -43, 0, 0], [0, -111, 0, 0], [0, 0, 0, -30], [0, -33, -66, 0], [0, 27, 54, 0], [0, -45, 0, 0], [0, 0, 0, 48], [2, 0, 0, 0], [0, 15, 30, 0], [-250, 0, 0, 0], [0, 17, 0, 0], [-278, 0, 0, 0], [0, 39, 78, 0], [0, -29, 0, 0], [0, 0, 0, 70], [0, 0, 0, 4], [-30, 0, 0, 0], [0, 19, 38, 0], [0, 0, 0, -32], [0, 0, 0, -38], [0, 71, 142, 0], [266, 0, 0, 0], [0, -173, 0, 0], [0, 0, 0, -22], [0, 23, 0, 0], [0, 39, 78, 0], [0, -15, 0, 0], [0, 0, 0, 102], [-210, 0, 0, 0], [0, 27, 54, 0], [0, -217, 0, 0], [-510, 0, 0, 0], [0, 27, 54, 0], [0, 213, 0, 0], [0, 0, 0, -8], [0, 0, 0, -78], [-566, 0, 0, 0], [0, -109, 0, 0], [0, 71, 0, 0], [0, 0, 0, 0], [0, -97, -194, 0], [0, -155, 0, 0], [0, 0, 0, -60], [270, 0, 0, 0], [0, -9, -18, 0], [0, 227, 0, 0], [0, 0, 0, -44], [706, 0, 0, 0], [486, 0, 0, 0], [0, -125, -250, 0], [0, 0, 0, 128], [-338, 0, 0, 0], [0, -137, -274, 0], [74, 0, 0, 0], [0, -141, -282, 0], [0, 0, 0, -56], [134, 0, 0, 0], [0, -9, -18, 0], [0, -15, 0, 0], [0, 0, 0, -42], [0, -21, -42, 0], [0, 0, 0, 14], [0, 247, 0, 0], [0, 117, 0, 0], [0, 145, 0, 0], [0, 31, 62, 0], [0, 43, 86, 0], [0, 0, 0, -110], [0, -201, 0, 0], [98, 0, 0, 0], [0, -97, -194, 0], [0, 0, 0, -124], [0, 0, 0, -114], [398, 0, 0, 0], [170, 0, 0, 0], [0, 515, 0, 0], [0, 0, 0, -186], [0, 7, 0, 0], [-1114, 0, 0, 0], [0, 0, 0, 80], [0, -399, 0, 0], [0, 0, 0, 26], [0, -57, -114, 0], [0, -141, -282, 0], [0, -493, 0, 0], [34, 0, 0, 0], [0, 71, 142, 0], [0, 23, 46, 0], [0, 289, 0, 0], [0, 219, 438, 0], [0, 611, 0, 0], [0, 0, 0, 44], [870, 0, 0, 0], [0, -107, 0, 0], [0, -479, 0, 0], [0, 0, 0, 178], [-630, 0, 0, 0], [0, -301, 0, 0], [0, 0, 0, 194], [770, 0, 0, 0], [0, -33, -66, 0], [0, 257, 0, 0], [0, -125, -250, 0], [0, 0, 0, 134], [0, 727, 0, 0], [0, 171, 342, 0], [166, 0, 0, 0], [0, -173, -346, 0], [0, 629, 0, 0], [0, 0, 0, 234], [0, 371, 0, 0], [0, 0, 0, 42], [0, 615, 0, 0], [0, 0, 0, 8], [0, -411, 0, 0], [0, 0, 0, -116], [0, 481, 0, 0], [0, 0, 0, -202], [0, -521, 0, 0], [0, 0, 0, 284], [614, 0, 0, 0], [0, 0, 0, -8], [462, 0, 0, 0], [0, 63, 126, 0], [0, 123, 246, 0], [0, 0, 0, -270], [70, 0, 0, 0], [0, 111, 222, 0], [0, 0, 0, 180], [0, 0, 0, -114], [-854, 0, 0, 0], [0, 259, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_768_h_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_768_3_h_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_3_h_d(:prec:=4)
    chi := MakeCharacter_768_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_h_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_3_h_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_768_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-32, 0, 1]>,<7,R![-6, 1]>],Snew);
    return Vf;
end function;