
// Make newform 128.5.d.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_128_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_128_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_128_5_d_b();".
// 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_128_5_d_b();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 8]];
    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_128_d();"
function MakeCharacter_128_d()
    N := 128;
    order := 2;
    char_gens := [127, 5];
    v := [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_128_d_Hecke();"
function MakeCharacter_128_d_Hecke(Kf)
    N := 128;
    order := 2;
    char_gens := [127, 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 := 5;
    raw_aps := [[0, 0], [0, 1], [0, 0], [0, 0], [0, 21], [0, 0], [574, 0], [0, -51], [0, 0], [0, 0], [0, 0], [0, 0], [-1246, 0], [0, 105], [0, 0], [0, 0], [0, -615], [0, 0], [0, -651], [0, 0], [-9506, 0], [0, 0], [0, -711], [-5474, 0], [9982, 0], [0, 0], [0, 0], [0, 1869], [0, 0], [-15934, 0], [0, 0], [0, -1395], [19106, 0], [0, -3399], [0, 0], [0, 0], [0, 0], [0, 1449], [0, 0], [0, 0], [0, -1071], [0, 0], [0, 0], [64894, 0], [0, 0], [0, 0], [0, -7119], [0, 0], [0, 3345], [0, 0], [-79778, 0], [0, 0], [78526, 0], [0, 7689], [16898, 0], [0, 0], [0, 0], [0, 0], [0, 0], [101278, 0], [0, -4059], [0, 0], [0, 13821], [0, 0], [80738, 0], [0, 0], [0, -819], [1346, 0], [0, -12831], [0, 0], [-211582, 0], [0, 0], [0, 0], [0, 0], [0, -19551], [0, 0], [0, 0], [0, 0], [-265154, 0], [-223006, 0], [0, 30069], [0, 0], [0, 0], [40894, 0], [0, 0], [0, -9219], [-346754, 0], [-361054, 0], [0, 0], [0, 0], [0, 2805], [0, 0], [0, 0], [0, -41055], [0, 42105], [0, 0], [0, 0], [469154, 0], [0, 34029], [0, 0], [0, -35259], [0, 0], [0, -22035], [-255646, 0], [0, 54285], [-665854, 0], [0, -29019], [39746, 0], [0, 0], [-112994, 0], [0, 0], [0, 0], [649822, 0], [0, -54795], [0, 0], [589438, 0], [0, 45525], [0, 0], [0, 0], [0, 76041], [0, 0], [-646658, 0], [0, 0], [0, 45969], [0, 75969], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -41055], [0, 0], [0, 0], [0, 0], [784994, 0], [71806, 0], [0, 0], [0, -104175], [0, 0], [1193762, 0], [0, 25365], [0, 0], [0, 0], [0, -119259], [0, 0], [0, 0], [0, 0], [-24094, 0], [0, 71601], [0, 0], [0, 0], [515522, 0], [0, 18501], [0, 0], [0, 89985], [0, 0], [0, 0], [606718, 0], [1240414, 0], [0, 0], [0, 141645], [-1796254, 0], [0, 0], [0, -144639], [-1769666, 0], [0, 0], [0, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_128_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_128_5_d_b();".
// 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_128_5_d_b(:prec:=2)
    chi := MakeCharacter_128_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_128_5_d_b();".
// 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_128_5_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_128_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<3,R![-128, 0, 1]>],Snew);
    return Vf;
end function;