
// Make newform 768.4.f.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_768_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_f_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_f_c();".
// 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_f_c();".
// 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, -3, 0, 8, 0, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 1, 0], [6, 0, -28, 0, 12, 0, -4, 0], [10, 31, -48, -44, 16, 16, -6, -5], [10, -31, -48, 44, 16, -16, -6, 5], [0, 1, 0, -20, 0, 8, 0, -3], [0, 14, 0, -20, 0, 8, 0, -2], [0, -2, 0, 44, 0, -16, 0, 6]];
    Rf_basisdens := [1, 1, 1, 4, 4, 2, 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_f();"
function MakeCharacter_768_f()
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    v := [1, 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_f_Hecke();"
function MakeCharacter_768_f_Hecke(Kf)
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -5, -5, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4], [0, 0, 0, -9, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 0], [0, 17, 0, 0, 0, 0, 0, 0], [0, 0, -29, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 65, 0, 0], [0, 0, 0, 0, 0, 0, 0, -14], [0, 0, 0, 29, -29, 0, 0, 0], [0, 0, 0, 0, 0, 0, -28, 0], [0, 61, 0, 0, 0, 0, 0, 0], [0, 0, 0, -25, -25, 0, 0, 0], [0, 0, 0, 0, 0, -221, 0, 0], [0, 0, 0, 95, -95, 0, 0, 0], [0, 0, 0, 0, 0, 0, -150, 0], [-410, 0, 0, 0, 0, 0, 0, 0], [0, 0, 11, 0, 0, 0, 0, 0], [0, 0, 0, -181, -181, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 94], [770, 0, 0, 0, 0, 0, 0, 0], [0, -167, 0, 0, 0, 0, 0, 0], [0, 0, -175, 0, 0, 0, 0, 0], [0, 0, 0, -93, -93, 0, 0, 0], [0, 0, 0, 0, 0, -533, 0, 0], [0, 0, 0, 0, 0, 0, 0, 116], [0, 0, -237, 0, 0, 0, 0, 0], [0, 0, 0, -65, -65, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10], [0, 0, 0, 253, -253, 0, 0, 0], [0, 181, 0, 0, 0, 0, 0, 0], [0, 0, 73, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -365, 0, 0], [0, 0, 0, -33, 33, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, 125, 0, 0, 0, 0, 0, 0], [0, 0, 0, 195, 195, 0, 0, 0], [0, 0, 0, 0, 0, -631, 0, 0], [0, 0, 0, 0, 0, 0, -400, 0], [-190, 0, 0, 0, 0, 0, 0, 0], [0, -239, 0, 0, 0, 0, 0, 0], [0, 0, 33, 0, 0, 0, 0, 0], [0, 0, 0, 71, -71, 0, 0, 0], [0, 0, -61, 0, 0, 0, 0, 0], [0, 0, 0, -73, -73, 0, 0, 0], [0, 0, 0, 0, 0, -2047, 0, 0], [0, 0, 0, 0, 0, 0, 0, -590], [0, 0, 0, 0, 0, 0, 820, 0], [-1198, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 615, 615, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -352], [0, 0, 0, 0, 0, 0, -614, 0], [0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 1131, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3175, 0, 0], [0, 0, 0, 0, 0, 0, 0, 622], [0, 0, 0, -875, 875, 0, 0, 0], [0, 73, 0, 0, 0, 0, 0, 0], [0, 0, 0, -201, 201, 0, 0, 0], [0, 0, 0, 0, 0, 0, 470, 0], [7030, 0, 0, 0, 0, 0, 0, 0], [0, -55, 0, 0, 0, 0, 0, 0], [0, 0, 0, -547, 547, 0, 0, 0], [1490, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 287, 287, 0, 0, 0], [0, 0, 0, 0, 0, -1037, 0, 0], [0, 0, 0, 0, 0, 0, 0, 968], [0, 0, 0, 0, 0, 0, 1170, 0], [0, 0, 1019, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2455, 0, 0], [0, 0, 0, 405, -405, 0, 0, 0], [0, 0, 0, 0, 0, 0, -896, 0], [0, 1057, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5285, 0, 0], [0, 0, 0, 0, 0, 0, 0, 188], [3574, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2215, 2215, 0, 0, 0], [0, 0, 0, 0, 0, -1759, 0, 0], [0, 0, 0, 0, 0, 0, 1860, 0], [14450, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1945, 0, 0, 0, 0, 0], [0, 0, 0, 439, 439, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1600], [3430, 0, 0, 0, 0, 0, 0, 0], [0, 437, 0, 0, 0, 0, 0, 0], [0, 0, 2347, 0, 0, 0, 0, 0], [0, 0, 0, 267, 267, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2200, 0], [0, 0, -1263, 0, 0, 0, 0, 0], [0, 0, 0, 995, 995, 0, 0, 0], [0, 0, 0, -169, 169, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1146, 0], [0, 1625, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1046], [0, 0, 0, -1299, 1299, 0, 0, 0], [0, 0, 0, 0, 0, -11981, 0, 0], [0, 0, 0, 1951, -1951, 0, 0, 0], [0, -1795, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3629, -3629, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2618], [0, 0, 0, -635, 635, 0, 0, 0], [19490, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -197, -197, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2884], [0, 0, 0, 0, 0, 0, 350, 0], [8758, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2525, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8225, 0, 0], [0, 0, 0, 0, 0, 0, 0, 946], [0, 0, 0, 2573, -2573, 0, 0, 0], [0, 0, -1575, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -784], [0, 0, 0, 2063, -2063, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2598, 0], [0, 581, 0, 0, 0, 0, 0, 0], [0, 0, 0, 875, 875, 0, 0, 0], [0, 0, 0, 0, 0, -4711, 0, 0], [-17470, 0, 0, 0, 0, 0, 0, 0], [0, -2327, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1821, -1821, 0, 0, 0], [0, 0, 0, 391, -391, 0, 0, 0], [0, -1975, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7087, 0, 0], [0, 0, 0, 0, 0, 0, -4740, 0], [0, 0, 1033, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5875, 0, 0], [0, 0, 0, -2577, 2577, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3694, 0], [0, 0, -677, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7145, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1898], [-29566, 0, 0, 0, 0, 0, 0, 0], [0, -2399, 0, 0, 0, 0, 0, 0], [0, 0, 0, 503, -503, 0, 0, 0], [0, -3439, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1166], [0, 0, 0, 1053, -1053, 0, 0, 0], [0, 3997, 0, 0, 0, 0, 0, 0], [0, 0, -2695, 0, 0, 0, 0, 0], [0, 0, 0, 1863, 1863, 0, 0, 0], [0, 0, 0, 0, 0, -7037, 0, 0], [0, 0, 0, 0, 0, 0, 2410, 0], [0, 0, 0, 0, 0, -11815, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3502], [0, 0, 0, 1765, -1765, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3624, 0], [0, 0, 0, 0, 0, 20875, 0, 0], [0, 0, 0, 0, 0, 0, 0, -268], [0, 0, 0, 5607, -5607, 0, 0, 0], [0, 0, 0, 0, 0, 0, -922, 0], [0, 0, 0, -4819, 4819, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7340, 0], [0, 0, 1833, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1144], [-37850, 0, 0, 0, 0, 0, 0, 0], [0, 1565, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1117, -1117, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5018], [0, 0, -2047, 0, 0, 0, 0, 0], [0, 0, 0, 5835, 5835, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 716], [0, 0, 0, 0, 0, 0, -3746, 0], [0, 0, 6243, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9185, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_768_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_768_4_f_c();".
// 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_f_c(:prec:=8)
    chi := MakeCharacter_768_f();
    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_f_c();".
// 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_f_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_768_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![-80, 0, 1]>,<19,R![-4860, 0, 1]>],Snew);
    return Vf;
end function;