
// Make newform 147.4.e.f in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_147_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_147_e_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_147_4_e_f();".
// 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_147_4_e_f();".
// 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, -1, 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_147_e();"
function MakeCharacter_147_e()
    N := 147;
    order := 3;
    char_gens := [50, 52];
    v := [3, 2];
    // 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_147_e_Hecke();"
function MakeCharacter_147_e_Hecke(Kf)
    N := 147;
    order := 3;
    char_gens := [50, 52];
    char_values := [[1, 0], [0, -1]];
    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, 1], [3, -3], [0, -12], [0, 0], [-20, 20], [-84, 0], [96, -96], [0, -12], [0, 176], [58, 0], [264, -264], [0, -258], [0, 0], [156, 0], [0, 408], [722, -722], [-492, 492], [0, 492], [-412, 412], [296, 0], [-240, 240], [0, -776], [924, 0], [0, 744], [-168, 0], [1524, -1524], [0, 408], [0, 820], [918, -918], [-110, 0], [16, 0], [0, -1692], [-1126, 1126], [1092, 0], [0, -1070], [120, -120], [-1836, 1836], [0, -916], [504, 0], [0, 1836], [-2372, 2372], [-1092, 0], [0, -2512], [2430, -2430], [-1762, 0], [-3096, 3096], [156, 0], [5040, 0], [-2172, 2172], [0, -2700], [0, 3802], [-4408, 0], [-3096, 3096], [-924, 0], [0, 2760], [2360, -2360], [-4020, 4020], [0, -4800], [-6446, 6446], [-2602, 0], [6900, -6900], [-4452, 0], [-2436, 0], [7488, -7488], [0, 1752], [0, 1562], [0, 7092], [366, 0], [6364, -6364], [-10500, 0], [-408, 408], [0, 11936], [2448, -2448], [0, -11374], [-5892, 0], [0, 10488], [-4514, 4514], [0, 6036], [0, 6770], [-12504, 12504], [9492, 0], [5182, 0], [5720, -5720], [13608, 0], [0, -12864], [0, 13252], [226, 0], [0, 11334], [-1596, 0], [12728, 0], [0, 3012], [4296, -4296], [8184, -8184], [-12164, 0], [0, -972], [7728, 0], [0, -11604], [10848, -10848], [0, 18132], [0, -6950], [17012, 0], [-3926, 3926], [18828, -18828], [0, -11990], [15716, -15716], [13872, -13872], [8820, 0], [0, 16872], [6056, -6056], [10752, 0], [0, -20256], [28190, -28190], [29318, 0], [-24348, 24348], [-25184, 0], [-32318, 32318], [-3948, 0], [-13848, 13848], [0, 3158], [-24596, 0], [15468, -15468], [13470, 0], [0, 9564], [-13852, 13852], [0, 324], [24922, 0], [0, 17886], [0, 6792], [1512, 0], [0, 11244], [1996, -1996], [-656, 0], [0, -1056], [-18702, 0], [0, -17904], [-7560, 0], [14292, -14292], [-26364, 26364], [17220, 0], [-16442, 16442], [-31332, 0], [0, 25810], [-12368, 12368], [6316, 0], [23868, -23868], [-48216, 0], [-27300, 0], [-8640, 8640], [0, -24372], [0, -2176], [0, 27574], [16968, 0], [-1860, 0], [0, -2280], [-36084, 36084], [24152, 0], [0, -36336], [0, -432], [22176, 0], [43524, -43524], [0, -1868], [-9238, 0], [-30616, 0], [0, 27540], [16402, -16402], [-55176, 55176], [-27096, 27096], [16812, -16812]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_147_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_147_4_e_f();".
// 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_147_4_e_f(:prec:=2)
    chi := MakeCharacter_147_e();
    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_147_4_e_f();".
// 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_147_4_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_147_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, -1, 1]>,<5,R![144, 12, 1]>],Snew);
    return Vf;
end function;