
// Make newform 6.5.b.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6_b_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_6_5_b_a();".
// 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_6_5_b_a();".
// 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 := [2, 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_6_b();"
function MakeCharacter_6_b()
    N := 6;
    order := 2;
    char_gens := [5];
    v := [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_6_b_Hecke();"
function MakeCharacter_6_b_Hecke(Kf)
    N := 6;
    order := 2;
    char_gens := [5];
    char_values := [[-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 := 5;
    raw_aps := [[0, 1], [-3, -3], [0, 6], [26, 0], [0, -42], [50, 0], [0, 72], [-358, 0], [0, 132], [0, -510], [-742, 0], [1874, 0], [0, 852], [-262, 0], [0, -600], [0, -162], [0, -642], [-1486, 0], [-4486, 0], [0, 1260], [290, 0], [9818, 0], [0, 2514], [0, -2772], [-478, 0], [0, 4854], [2138, 0], [0, -3834], [-4750, 0], [0, -1128], [8282, 0], [0, 1914], [0, -6396], [-23206, 0], [0, -3954], [14426, 0], [49010, 0], [-42982, 0], [0, 15612], [0, -7926], [0, 10818], [-13102, 0], [0, -24864], [18050, 0], [0, -9306], [-37222, 0], [15098, 0], [58778, 0], [0, 24042], [28562, 0], [0, 7956], [0, 456], [-61246, 0], [0, -16002], [0, -30144], [0, 29964], [0, -15174], [-27430, 0], [-93934, 0], [0, 8652], [-65830, 0], [0, 31830], [67322, 0], [0, -46668], [-22078, 0], [0, 41682], [167642, 0], [162914, 0], [0, 46734], [53234, 0], [0, -51216], [0, 4572], [-44326, 0], [-60718, 0], [30458, 0], [0, 83328], [0, -53178], [172658, 0], [0, 59208], [-150430, 0], [0, -63030], [-216046, 0], [0, -1944], [108002, 0], [357722, 0], [0, -30498], [0, 106560], [-399070, 0], [0, 13530], [144410, 0], [0, -52398], [0, 108048], [-196774, 0], [0, 58758], [189050, 0], [0, -121644], [0, -124878], [0, -97596], [-146950, 0], [-244942, 0], [-283366, 0], [0, 16650], [0, -30], [0, 84540], [-140710, 0], [36002, 0], [0, -111786], [0, -92664], [0, -214428], [306530, 0], [563162, 0], [111314, 0], [0, 42900], [-581158, 0], [232346, 0], [0, -231072], [-424678, 0], [0, 151308], [0, 254778], [0, -14190], [-358510, 0], [582434, 0], [0, 13206], [0, 57222], [-630118, 0], [0, -161838], [274130, 0], [0, -107496], [364058, 0], [-301198, 0], [872570, 0], [0, 309084], [916250, 0], [-691630, 0], [0, -31908], [-515326, 0], [0, -35514], [-107878, 0], [0, 173058], [0, 367956], [611066, 0], [0, -122082], [-178150, 0], [0, -402498], [100082, 0], [0, 402924], [172754, 0], [0, 105996], [8186, 0], [0, -136944], [245426, 0], [0, -214632], [759962, 0], [0, -2028], [-138502, 0], [0, -94344], [-1442662, 0], [0, 180912], [868610, 0], [0, 36042], [0, 82434], [0, -328068], [150746, 0], [0, -179658], [0, -75864], [0, -476700], [-879910, 0], [935378, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6_5_b_a();".
// 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_6_5_b_a(:prec:=2)
    chi := MakeCharacter_6_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_6_5_b_a();".
// 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_6_5_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;