
// Make newform 450.6.c.h in Magma, downloaded from the LMFDB on 19 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_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_450_6_c_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
        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, 86], [-132, 0], [0, -473], [0, -111], [-500, 0], [0, -1782], [2190, 0], [2312, 0], [0, 5621], [-1242, 0], [0, 10312], [0, 3294], [0, 10533], [7980, 0], [16622, 0], [0, -904], [24528, 0], [0, 10237], [46240, 0], [0, 25788], [-110310, 0], [0, 39191], [-141942, 0], [0, -218], [0, 116484], [174850, 0], [0, -91497], [0, 61226], [241908, 0], [0, 138549], [193540, 0], [140550, 0], [433952, 0], [0, 277961], [0, -33308], [0, -102846], [0, -216927], [-489180, 0], [719462, 0], [185928, 0], [0, -295703], [0, 224739], [-157160, 0], [253052, 0], [0, 536722], [0, -313416], [116650, 0], [0, 371523], [978720, 0], [-1132798, 0], [-905652, 0], [0, 969969], [0, 402738], [-858690, 0], [-383608, 0], [0, -1005379], [-202602, 0], [0, -110768], [0, 161253], [0, -721504], [-171312, 0], [0, -513443], [0, 376479], [1994132, 0], [0, 493511], [0, 1103004], [-2741870, 0], [0, 1194783], [-279480, 0], [0, 1238186], [0, 1372627], [1189060, 0], [0, -1628802], [1983510, 0], [0, -2487079], [1342638, 0], [1094230, 0], [-954060, 0], [-1593898, 0], [2646648, 0], [0, 1861777], [2583400, 0], [0, -3781032], [4307730, 0], [0, 1121771], [-1656702, 0], [0, -1445798], [0, -3263496], [-5962320, 0], [0, -1495954], [1204188, 0], [-9205460, 0], [0, 1679778], [-2537010, 0], [9315798, 0], [0, -2514008], [134222, 0], [0, -302824], [0, -3531201], [0, 5151468], [1047690, 0], [14076452, 0], [0, -813409], [0, 3480444], [0, 5681943], [14868840, 0], [-1236118, 0], [0, 6224906], [0, -4367453], [0, 6274749], [14665780, 0], [-196288, 0], [11159598, 0], [0, -1125788], [0, 4026594], [0, 208233], [13172100, 0], [-1694938, 0], [0, -4458023], [0, -7144701], [0, 2666568], [698252, 0], [-17987982, 0], [13946410, 0], [6222720, 0], [0, 3883646], [0, 12104167], [-12684980, 0], [0, -9881622], [-9017608, 0], [0, 562781], [-22574682, 0], [632350, 0], [0, 6293373], [0, -10789624], [0, -15488001], [4249290, 0], [3423332, 0], [-31057062, 0], [0, -15595178], [0, -4142436], [18168850, 0], [-10274280, 0], [0, 3142987], [0, 7702509], [-14352620, 0], [0, -6663882], [0, -16242319], [-15460002, 0], [0, -847388], [0, -1436286], [0, -19771084], [-11317752, 0], [-8513480, 0], [-7545870, 0], [0, 9225011], [-6750462, 0], [0, 3228384], [0, 19832403], [0, 17150726], [5771148, 0], [0, 3543729], [0, -22968102], [-45029848, 0], [0, 11868221]];
    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_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_450_6_c_h(: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_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_450_6_c_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_450_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![29584, 0, 1]>,<11,R![132, 1]>],Snew);
    return Vf;
end function;