
// Make newform 100.4.c.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_100_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_100_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_100_4_c_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_100_4_c_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 := [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_100_c();"
function MakeCharacter_100_c()
    N := 100;
    order := 2;
    char_gens := [51, 77];
    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_100_c_Hecke();"
function MakeCharacter_100_c_Hecke(Kf)
    N := 100;
    order := 2;
    char_gens := [51, 77];
    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 := [[0, 0], [0, 1], [0, 0], [0, -26], [45, 0], [0, 44], [0, -117], [91, 0], [0, -18], [-144, 0], [26, 0], [0, 214], [-459, 0], [0, -460], [0, 468], [0, 558], [72, 0], [-118, 0], [0, -251], [108, 0], [0, 299], [898, 0], [0, 927], [-351, 0], [0, -386], [-954, 0], [0, -772], [0, -1197], [802, 0], [0, 1143], [0, 2374], [-1260, 0], [0, 891], [-389, 0], [-1296, 0], [-2710, 0], [0, 1846], [0, 1475], [0, 1476], [0, 1368], [1503, 0], [3770, 0], [-4122, 0], [0, -1963], [0, -2934], [-1412, 0], [3419, 0], [0, -100], [0, -4212], [3484, 0], [0, 918], [-3744, 0], [-4231, 0], [-2925, 0], [0, 18], [0, -6786], [-7632, 0], [650, 0], [0, 3232], [4446, 0], [0, 2483], [0, -4050], [0, -2321], [-3258, 0], [0, 3626], [0, -3852], [7553, 0], [0, 109], [0, 2835], [-2990, 0], [0, 9126], [9594, 0], [0, -9764], [0, 6722], [13537, 0], [0, -8658], [8874, 0], [0, -5876], [-1755, 0], [-4589, 0], [-5409, 0], [12116, 0], [9126, 0], [0, 629], [-4472, 0], [0, -3393], [5031, 0], [0, 6487], [2700, 0], [0, -2932], [0, 15660], [-10134, 0], [0, -1898], [-6300, 0], [-18044, 0], [0, -6876], [4806, 0], [7749, 0], [0, 8153], [-10576, 0], [0, -7553], [0, -13500], [0, 23184], [8055, 0], [3068, 0], [0, -12419], [0, 12393], [0, 23751], [11610, 0], [26675, 0], [0, -17264], [0, -26026], [0, -5022], [-7820, 0], [15002, 0], [-918, 0], [0, -23452], [0, -20556], [0, 30654], [-8919, 0], [-22912, 0], [0, -1222], [0, -144], [0, -12519], [11873, 0], [-3060, 0], [-4004, 0], [-10314, 0], [0, -9872], [0, 7436], [16900, 0], [0, -23058], [-8224, 0], [0, 7696], [-6363, 0], [-8333, 0], [0, 32760], [0, 43732], [0, -16866], [-16146, 0], [32444, 0], [-2574, 0], [0, -27604], [0, -11655], [-33428, 0], [17712, 0], [0, -10270], [0, -38709], [-15509, 0], [0, 15912], [0, 10972], [-18738, 0], [0, -21367], [0, -20124], [0, -23132], [31212, 0], [6994, 0], [-19422, 0], [0, -11699], [-42948, 0], [0, 3816], [0, -43407], [0, 43216], [-47619, 0], [0, 4671], [0, 9054], [8126, 0], [0, -38468]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_100_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_100_4_c_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_100_4_c_b(:prec:=2)
    chi := MakeCharacter_100_c();
    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_100_4_c_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_100_4_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_100_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![1, 0, 1]>],Snew);
    return Vf;
end function;