
// Make newform 80.6.c.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_80_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_80_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_80_6_c_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_80_6_c_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, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_80_c();"
function MakeCharacter_80_c()
    N := 80;
    order := 2;
    char_gens := [31, 21, 17];
    v := [2, 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_80_c_Hecke();"
function MakeCharacter_80_c_Hecke(Kf)
    N := 80;
    order := 2;
    char_gens := [31, 21, 17];
    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 := 6;
    raw_aps := [[0, 0], [0, 7], [55, -5], [0, -79], [148, 0], [0, -342], [0, 1024], [2220, 0], [0, -623], [270, 0], [2048, 0], [0, -2186], [-2398, 0], [0, 1147], [0, 5341], [0, -1482], [-39740, 0], [-42298, 0], [0, -16049], [4248, 0], [0, -15052], [35280, 0], [0, -13913], [85210, 0], [0, -48616], [-4298, 0], [0, 62057], [0, 21171], [35990, 0], [0, 114408], [0, -87619], [-299652, 0], [0, 53964], [-196460, 0], [-138850, 0], [-416152, 0], [0, 216554], [0, 74567], [0, 279801], [0, -171902], [23980, 0], [-651898, 0], [-202752, 0], [0, 226328], [0, 168734], [-561000, 0], [805548, 0], [0, 609277], [0, -282169], [-560330, 0], [0, 146788], [584240, 0], [-563798, 0], [1019748, 0], [0, 328704], [0, -281183], [-366570, 0], [-1160752, 0], [0, -1257506], [2086002, 0], [0, -1119013], [0, 487878], [0, -43929], [-599352, 0], [0, 1046708], [0, -1208126], [1640948, 0], [0, 1093864], [0, -1372509], [2651150, 0], [0, -1528832], [3793560, 0], [0, 1555301], [0, 707598], [-3902620, 0], [0, 347837], [-498290, 0], [0, 547834], [-2491598, 0], [3633490, 0], [-3643780, 0], [-1825298, 0], [-2854352, 0], [0, 293888], [6116040, 0], [0, -1178853], [-5497350, 0], [0, -580196], [-2303298, 0], [0, 1356717], [0, -2025249], [5605280, 0], [0, -3565839], [-5881452, 0], [1757100, 0], [0, 2457057], [5754990, 0], [-1619798, 0], [0, 5955827], [4076302, 0], [0, -6184009], [0, 65154], [0, -2958433], [9030130, 0], [10709348, 0], [0, -610256], [0, 7355211], [0, -4260072], [2901000, 0], [5727602, 0], [0, 4399621], [0, -517042], [0, 6492724], [7920020, 0], [-16821752, 0], [-15515398, 0], [0, 5290047], [0, -6894159], [0, 794618], [-9124340, 0], [6503102, 0], [0, 1089048], [0, 1992094], [0, -2957813], [15547148, 0], [-22710298, 0], [-6298410, 0], [21191120, 0], [0, -6780519], [0, -13470662], [2774140, 0], [0, 9276897], [2192848, 0], [0, -4743746], [9695802, 0], [-9327870, 0], [0, 4840398], [0, 2761511], [0, -8555966], [14530890, 0], [21354548, 0], [36796702, 0], [0, -16533423], [0, -8865469], [-12937530, 0], [33181240, 0], [0, 2586118], [0, -5265996], [-11474140, 0], [0, 9636117], [0, 11526194], [22669002, 0], [0, 1836687], [0, -16982439], [0, 10666371], [10351248, 0], [-25901880, 0], [31322970, 0], [0, -10423036], [-38292898, 0], [0, 21254391], [0, -19592732], [0, 9241801], [-39503052, 0], [0, 16452144], [0, -13289863], [-19296352, 0], [0, -25600866]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_80_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_80_6_c_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_80_6_c_c(:prec:=2)
    chi := MakeCharacter_80_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_80_6_c_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_80_6_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_80_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![196, 0, 1]>],Snew);
    return Vf;
end function;