
// Make newform 150.6.c.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_150_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_150_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_150_6_c_e();".
// 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_150_6_c_e();".
// 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_150_c();"
function MakeCharacter_150_c()
    N := 150;
    order := 2;
    char_gens := [101, 127];
    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_150_c_Hecke();"
function MakeCharacter_150_c_Hecke(Kf)
    N := 150;
    order := 2;
    char_gens := [101, 127];
    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 := 6;
    raw_aps := [[0, -4], [0, 9], [0, 0], [0, 233], [-498, 0], [0, -809], [0, -1002], [1705, 0], [0, -1554], [-7830, 0], [977, 0], [0, -4822], [-8148, 0], [0, -19469], [0, 8418], [0, -17664], [-35910, 0], [3527, 0], [0, 57473], [-7548, 0], [0, 646], [22720, 0], [0, -11574], [78960, 0], [0, 54593], [105552, 0], [0, 177436], [0, -183792], [-169685, 0], [0, -263484], [0, -256912], [-219048, 0], [0, 228678], [-339740, 0], [-306450, 0], [198827, 0], [0, 361283], [0, -338159], [0, 430248], [0, -603354], [374370, 0], [-232423, 0], [-846198, 0], [0, 155581], [0, -103482], [140425, 0], [-462673, 0], [0, 735271], [0, 967188], [83695, 0], [0, 873876], [1060560, 0], [-756823, 0], [-635148, 0], [0, 30708], [0, 189516], [-1109970, 0], [211952, 0], [0, 1047413], [34002, 0], [0, 281101], [0, 1558506], [0, -839917], [-292698, 0], [0, -127859], [0, 648048], [-3393148, 0], [0, -1620847], [0, -1286382], [2730550, 0], [0, 2136486], [-3264060, 0], [0, -4150777], [0, 2142421], [1946995, 0], [0, -2650524], [-282540, 0], [0, -2620657], [-286248, 0], [-4120685, 0], [2379480, 0], [-741298, 0], [-187398, 0], [0, -6551099], [-270065, 0], [0, -2779344], [-7866300, 0], [0, 6233558], [-4683048, 0], [0, 382816], [0, -1935402], [5569170, 0], [0, 4224503], [6962952, 0], [9295825, 0], [0, -6341364], [7383090, 0], [4196202, 0], [0, 3579421], [4955477, 0], [0, -5318032], [0, -4257942], [0, 4376166], [-3577170, 0], [1946927, 0], [0, 6955763], [0, 2418528], [0, 9583956], [1520700, 0], [3882827, 0], [0, -107992], [0, -7499234], [0, -1226952], [-9513395, 0], [-15644923, 0], [-9603948, 0], [0, 8243956], [0, -4073532], [0, -16819314], [-2878260, 0], [13338602, 0], [0, -6373454], [0, -22720962], [0, 15361176], [8050352, 0], [7278402, 0], [-13749785, 0], [9055830, 0], [0, -22097887], [0, -2073374], [-14884940, 0], [0, -25885644], [-7986448, 0], [0, -11556967], [3167052, 0], [-8215595, 0], [0, -11970804], [0, -17115397], [0, -28075332], [-9908160, 0], [5725727, 0], [-5965698, 0], [0, 19777771], [0, 20404488], [27718270, 0], [7798410, 0], [0, -6225539], [0, 33975708], [14728240, 0], [0, 18859416], [0, -18211237], [-7654248, 0], [0, 597451], [0, 2068878], [0, 7833308], [40813302, 0], [-6215795, 0], [4648470, 0], [0, -13360297], [33256902, 0], [0, 42153018], [0, -22469064], [0, -17499952], [54292002, 0], [0, 25592538], [0, 4824876], [1184477, 0], [0, -25817782]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_150_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_150_6_c_e();".
// 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_150_6_c_e(:prec:=2)
    chi := MakeCharacter_150_c();
    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_150_6_c_e();".
// 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_150_6_c_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_150_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![54289, 0, 1]>],Snew);
    return Vf;
end function;