
// Make newform 8.4.b.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8_b_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_8_4_b_a();".
// 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_8_4_b_a();".
// 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 := [2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 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_8_b();"
function MakeCharacter_8_b()
    N := 8;
    order := 2;
    char_gens := [7, 5];
    v := [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_8_b_Hecke();"
function MakeCharacter_8_b_Hecke(Kf)
    N := 8;
    order := 2;
    char_gens := [7, 5];
    char_values := [[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 := [[-1, -1], [0, 2], [0, -4], [-8, 0], [0, -6], [0, 20], [-14, 0], [0, -14], [-152, 0], [0, -60], [224, 0], [0, 92], [-70, 0], [0, -166], [336, 0], [0, 12], [0, 202], [0, 36], [0, 66], [-72, 0], [-294, 0], [-464, 0], [0, -206], [266, 0], [994, 0], [0, 284], [1176, 0], [0, -102], [0, -716], [-1710, 0], [-1664, 0], [0, -254], [-1062, 0], [0, 1018], [0, 300], [744, 0], [0, 68], [0, -670], [-1960, 0], [0, 756], [0, 1234], [0, -884], [3904, 0], [3330, 0], [0, -452], [-1736, 0], [0, -1102], [1568, 0], [0, -478], [0, 1948], [-838, 0], [6288, 0], [-2926, 0], [0, 2058], [2562, 0], [-5896, 0], [0, -2028], [-1680, 0], [0, -596], [-2742, 0], [0, 1130], [0, -3492], [0, 978], [-2744, 0], [2282, 0], [0, 3620], [0, -1606], [6130, 0], [0, 938], [0, -124], [-10206, 0], [-3176, 0], [-11760, 0], [0, -4148], [0, 1162], [2688, 0], [0, 3964], [0, -2156], [12402, 0], [-12278, 0], [0, -3102], [0, -548], [7632, 0], [3794, 0], [-1848, 0], [0, -4662], [-3582, 0], [2714, 0], [0, 3156], [2224, 0], [0, 3890], [17696, 0], [1304, 0], [0, 6298], [0, 1170], [-19320, 0], [0, -1692], [-2982, 0], [0, 762], [0, 5828], [0, -4766], [0, -2700], [0, -2862], [15594, 0], [0, -6326], [6594, 0], [0, -8774], [14322, 0], [-16088, 0], [-21238, 0], [-13664, 0], [0, 7708], [-3782, 0], [0, 2202], [2056, 0], [11842, 0], [0, -6142], [19320, 0], [0, -876], [0, 10482], [0, 8492], [-10078, 0], [0, 6108], [0, -6182], [0, -11054], [0, 932], [0, -11972], [-9296, 0], [21672, 0], [0, -3580], [0, 2594], [17432, 0], [-11632, 0], [0, -6324], [39466, 0], [35266, 0], [0, 6140], [0, 13170], [0, 964], [-24390, 0], [0, -3622], [0, -3316], [-21688, 0], [0, -7350], [0, -7388], [-18760, 0], [0, 10860], [8778, 0], [0, -2134], [-9312, 0], [0, 52], [-31150, 0], [0, 4754], [37128, 0], [0, 13306], [-10512, 0], [-46104, 0], [-5726, 0], [1274, 0], [0, -9996], [0, 9042], [38250, 0], [4664, 0], [0, 11706], [-4814, 0], [-12376, 0], [45344, 0], [0, -9828]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8_4_b_a();".
// 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_8_4_b_a(:prec:=2)
    chi := MakeCharacter_8_b();
    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_8_4_b_a();".
// 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_8_4_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;