
// Make newform 147.6.e.c in Magma, downloaded from the LMFDB on 20 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_6_e_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_147_6_e_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, -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 := 6;
    raw_aps := [[0, -5], [-9, 9], [0, -94], [0, 0], [-52, 52], [-770, 0], [2022, -2022], [0, -1732], [0, 576], [5518, 0], [-6336, 6336], [0, 7338], [-3262, 0], [5420, 0], [0, -864], [-4182, 4182], [11220, -11220], [0, 45602], [-1396, 1396], [18720, 0], [-46362, 46362], [0, -97424], [-81228, 0], [0, 3182], [4914, 0], [166354, -166354], [0, -157160], [0, 6764], [-178398, 178398], [-45134, 0], [-205056, 0], [0, -72964], [94182, -94182], [-47796, 0], [0, 124266], [446296, -446296], [159746, -159746], [0, -247252], [-684488, 0], [0, 610474], [-662252, 662252], [154630, 0], [0, -486904], [-620546, 620546], [-236570, 0], [-82104, 82104], [99892, 0], [-186704, 0], [-336372, 336372], [0, 926314], [0, -1257114], [-347016, 0], [-99170, 99170], [344428, 0], [0, -295130], [1272464, -1272464], [-276774, 276774], [0, 1289936], [-1716550, 1716550], [-1472182, 0], [-1028812, 1028812], [-1186066, 0], [1518916, 0], [-212808, 212808], [0, 1894], [0, 1578978], [0, 3394708], [2027314, 0], [-3488852, 3488852], [965566, 0], [-1153930, 1153930], [0, -1611104], [-3677472, 3677472], [0, -649766], [320700, 0], [0, 2361888], [3533898, -3533898], [0, -4048110], [0, -2076450], [-2574314, 2574314], [848148, 0], [1436822, 0], [-2354376, 2354376], [-3788078, 0], [0, 3643224], [0, -2483892], [-2631774, 0], [0, 1161302], [2813846, 0], [6842992, 0], [0, -3343140], [4282480, -4282480], [8931752, -8931752], [2753060, 0], [0, -4804084], [-6024648, 0], [0, 8429866], [-9250578, 9250578], [0, -5844940], [0, -9225326], [-6443372, 0], [-3742126, 3742126], [14638364, -14638364], [0, -14180458], [1251604, -1251604], [-5943778, 5943778], [-6461924, 0], [0, 2346054], [13495920, -13495920], [3878490, 0], [0, 533488], [-5146102, 5146102], [-2378598, 0], [-16002316, 16002316], [12345944, 0], [3437550, -3437550], [-16219084, 0], [-11992856, 11992856], [0, -1580094], [6983580, 0], [-3696022, 3696022], [1846882, 0], [0, 7685010], [-7121796, 7121796], [0, 3237868], [-7391634, 0], [0, 5333610], [0, 11456400], [-24953992, 0], [0, 14339826], [-922932, 922932], [-9389952, 0], [0, 408032], [25960454, 0], [0, -18355410], [-747166, 0], [-20269166, 20269166], [4699820, -4699820], [584710, 0], [16401270, -16401270], [-304948, 0], [0, -34342838], [-15668376, 15668376], [-29688636, 0], [23070802, -23070802], [23264248, 0], [19151510, 0], [5346830, -5346830], [0, -39585836], [0, 25028376], [0, 5025890], [-26049030, 0], [-6824620, 0], [0, -23383496], [-39595916, 39595916], [-4675704, 0], [0, 4925944], [0, 32368774], [-33233718, 0], [26642650, -26642650], [0, -31466300], [-13451590, 0], [-28496312, 0], [0, -18185772], [-32094114, 32094114], [-15615384, 15615384], [-48449936, 48449936], [45433562, -45433562]];
    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_6_e_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_147_6_e_c(:prec:=2)
    chi := MakeCharacter_147_e();
    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_147_6_e_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_147_6_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_147_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![25, 5, 1]>,<5,R![8836, 94, 1]>],Snew);
    return Vf;
end function;