
// Make newform 450.6.c.o in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_450_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_450_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_450_6_c_o();".
// 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_450_6_c_o();".
// 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;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_450_c();"
function MakeCharacter_450_c()
    N := 450;
    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_450_c_Hecke();"
function MakeCharacter_450_c_Hecke(Kf)
    N := 450;
    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, -2], [0, 0], [0, 0], [0, 11], [768, 0], [0, -23], [0, 189], [-1100, 0], [0, 993], [-5610, 0], [-3988, 0], [0, 71], [-1542, 0], [0, -2513], [0, 12369], [0, 7083], [28380, 0], [5522, 0], [0, 12371], [-42372, 0], [0, -26063], [39640, 0], [0, 29913], [57690, 0], [0, 72191], [141258, 0], [0, 69907], [0, 43209], [-218450, 0], [0, 14403], [0, 108251], [244608, 0], [0, -119751], [-30860, 0], [-100950, 0], [12452, 0], [0, 3011], [0, -250433], [0, 277629], [0, -208677], [-52380, 0], [546662, 0], [452028, 0], [0, 242797], [0, 505089], [807640, 0], [149552, 0], [0, -221753], [0, 210009], [-1058750, 0], [0, 636723], [-370680, 0], [-561298, 0], [-577152, 0], [0, -325731], [0, -458787], [-735390, 0], [-1121308, 0], [0, 830171], [-1452102, 0], [0, 154507], [0, 796503], [0, -623629], [665988, 0], [0, -295643], [0, -258171], [-3295768, 0], [0, -955489], [0, 1210029], [-2507270, 0], [0, 206583], [1737720, 0], [0, -580489], [0, 171877], [-573140, 0], [0, 1440273], [-3085590, 0], [0, -442729], [3753438, 0], [1946530, 0], [-2991660, 0], [3966602, 0], [5171148, 0], [0, -2267423], [1082200, 0], [0, 540393], [2617830, 0], [0, -795229], [-4254702, 0], [0, 1633027], [0, -300771], [-4579320, 0], [0, -3526129], [2623488, 0], [3612340, 0], [0, -4578147], [7261590, 0], [-5810202, 0], [0, -4085333], [-817378, 0], [0, 1753751], [0, 4807449], [0, -1009707], [1378590, 0], [8542952, 0], [0, 1157291], [0, 464169], [0, 454743], [-8511360, 0], [6124982, 0], [0, 2258231], [0, 4819897], [0, -4963251], [-7633220, 0], [18031412, 0], [-9301902, 0], [0, -6916613], [0, -7449831], [0, 9665283], [-4061100, 0], [-13515238, 0], [0, 7175977], [0, 947649], [0, -1458207], [14427752, 0], [15867918, 0], [301810, 0], [21267720, 0], [0, -7750429], [0, -6092483], [29028220, 0], [0, -8057247], [-2924308, 0], [0, -13016269], [-16326282, 0], [-25813250, 0], [0, 9529623], [0, 8670551], [0, -12908451], [8884890, 0], [-24639568, 0], [-11376762, 0], [0, -6539153], [0, -17859411], [-16188050, 0], [-25549680, 0], [0, -11147663], [0, 9843609], [-6775820, 0], [0, 13190193], [0, -14758069], [14856498, 0], [0, -7267013], [0, -8601411], [0, 17221691], [983748, 0], [-30885680, 0], [-32087370, 0], [0, -7625989], [-3481662, 0], [0, -12700491], [0, 24867603], [0, -15258649], [-31985352, 0], [0, 1453929], [0, -17481027], [3004652, 0], [0, -16039429]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_450_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_450_6_c_o();".
// 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_450_6_c_o(:prec:=2)
    chi := MakeCharacter_450_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_450_6_c_o();".
// 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_450_6_c_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_450_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![484, 0, 1]>,<11,R![-768, 1]>],Snew);
    return Vf;
end function;