
// Make newform 768.4.d.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_768_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_d_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_d_g();".
// 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_d_g();".
// 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];
        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_d();"
function MakeCharacter_768_d()
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    v := [2, 1, 2];
    // 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_d_Hecke();"
function MakeCharacter_768_d_Hecke(Kf)
    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 := true, 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+1)]];
    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, 3], [0, 18], [-8, 0], [0, -36], [0, -10], [18, 0], [0, -100], [-72, 0], [0, -234], [-16, 0], [0, 226], [-90, 0], [0, -452], [432, 0], [0, -414], [0, 684], [0, 422], [0, 332], [360, 0], [-26, 0], [512, 0], [0, -1188], [630, 0], [-1054, 0], [0, -558], [-8, 0], [0, -1764], [0, 1622], [-1134, 0], [-592, 0], [0, -1908], [-954, 0], [0, -2564], [0, 738], [2440, 0], [0, -2554], [0, -820], [-1944, 0], [0, -1242], [0, 1116], [0, -1070], [-576, 0], [-1342, 0], [0, -1422], [-872, 0], [0, 1340], [4880, 0], [0, 2700], [0, -254], [-4410, 0], [-3888, 0], [5138, 0], [0, -4788], [-5886, 0], [-2232, 0], [0, -666], [-5536, 0], [0, -2126], [2934, 0], [0, -2036], [0, -2286], [0, 1244], [-1224, 0], [-1898, 0], [0, -9162], [0, 4348], [7154, 0], [0, 1836], [0, 5894], [11106, 0], [-13176, 0], [-6112, 0], [0, 13618], [0, -692], [-8064, 0], [0, -12654], [0, -106], [-4014, 0], [-3914, 0], [0, 4428], [0, 15490], [6768, 0], [1298, 0], [2248, 0], [0, 9612], [162, 0], [-1370, 0], [0, -15354], [-13024, 0], [0, -14436], [12096, 0], [-6056, 0], [0, -7524], [0, 5276], [-4968, 0], [0, 10998], [8838, 0], [0, -22436], [0, -4762], [0, -6004], [0, -5274], [0, -12420], [21366, 0], [0, -21140], [3266, 0], [0, -17028], [9522, 0], [10296, 0], [3382, 0], [-20656, 0], [0, 22114], [-19962, 0], [0, 604], [-152, 0], [4194, 0], [0, -7252], [6696, 0], [0, 28422], [0, -19908], [0, -14318], [30050, 0], [0, -22158], [0, 3132], [0, -20932], [0, -21834], [0, -12446], [-12528, 0], [-11576, 0], [0, -29338], [0, 2540], [18792, 0], [4832, 0], [0, 20818], [-12042, 0], [13058, 0], [0, 11826], [0, 11996], [0, 6966], [40806, 0], [0, 17980], [0, 12834], [37864, 0], [0, -42516], [0, 45638], [-17496, 0], [0, -32174], [38934, 0], [0, -29780], [-48096, 0], [0, 21302], [-7470, 0], [0, 764], [-32328, 0], [0, 36316], [-13392, 0], [-38072, 0], [-12798, 0], [-34874, 0], [0, 17190], [0, 40284], [-15498, 0], [-37160, 0], [0, -18468], [10386, 0], [-44136, 0], [-28432, 0], [0, 39778]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_768_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_768_4_d_g();".
// 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_d_g(:prec:=2)
    chi := MakeCharacter_768_d();
    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_d_g();".
// 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_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_768_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![324, 0, 1]>,<7,R![8, 1]>],Snew);
    return Vf;
end function;