
// Make newform 147.4.a.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_147_a();"
// 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_a_h();".
// 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_a_h();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_147_a();"
function MakeCharacter_147_a()
    N := 147;
    order := 1;
    char_gens := [50, 52];
    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;

function MakeCharacter_147_a_Hecke(Kf)
    return MakeCharacter_147_a();
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 := [[4], [3], [18], [0], [-50], [-36], [126], [-72], [14], [158], [-36], [-162], [-270], [-324], [-72], [-22], [468], [792], [232], [-734], [180], [236], [36], [234], [468], [-666], [-252], [670], [162], [-1390], [916], [2268], [806], [2628], [-2390], [3240], [3024], [-1784], [-3024], [1566], [3802], [-468], [482], [-810], [-2462], [-4536], [2916], [-1080], [-1332], [1620], [6718], [-3578], [-756], [-6516], [-6030], [590], [-990], [-3420], [-2734], [598], [3600], [7902], [-10224], [3888], [5112], [-10102], [5508], [-9234], [-6494], [10080], [-738], [194], [-4752], [-2306], [-7452], [-1152], [1894], [9216], [-11650], [-7524], [-3852], [10402], [-10390], [11232], [14616], [11938], [8186], [2106], [9486], [-12652], [-3708], [-8064], [-11664], [-9814], [-15228], [-11088], [-5814], [-11682], [-2988], [7130], [-5488], [5746], [13068], [-1130], [16864], [-2088], [10260], [3582], [7034], [-18072], [28584], [-10910], [-5522], [2412], [24676], [-27482], [-22752], [14832], [2822], [-15826], [-23832], [13770], [8334], [-18598], [8964], [3542], [-486], [-26928], [20628], [-9756], [19064], [-3766], [-11664], [-34182], [-4734], [-30240], [27702], [-22644], [30150], [-11318], [-29628], [17770], [7868], [35726], [27108], [23256], [35280], [-19710], [3888], [-36634], [1226], [-38538], [-37260], [26640], [-12636], [-33638], [36936], [-22302], [13824], [13554], [44878], [38362], [26444], [17820], [34438], [26064], [33696], [36072]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_147_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_147_4_a_h();".
// 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_a_h(:prec:=1)
    chi := MakeCharacter_147_a();
    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_a_h();".
// 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_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_147_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-4, 1]>,<5,R![-18, 1]>],Snew);
    return Vf;
end function;