
// Make newform 60.4.d.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_60_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_60_d_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_60_4_d_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_60_4_d_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 := [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_60_d();"
function MakeCharacter_60_d()
    N := 60;
    order := 2;
    char_gens := [31, 41, 37];
    v := [2, 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_60_d_Hecke();"
function MakeCharacter_60_d_Hecke(Kf)
    N := 60;
    order := 2;
    char_gens := [31, 41, 37];
    char_values := [[1, 0], [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 := 4;
    raw_aps := [[0, 0], [0, 3], [-10, 5], [0, 22], [-14, 0], [0, 30], [0, 62], [120, 0], [0, -188], [-96, 0], [184, 0], [0, 406], [130, 0], [0, -148], [0, 448], [0, 414], [-266, 0], [-838, 0], [0, 248], [1020, 0], [0, -484], [48, 0], [0, -548], [650, 0], [0, -1816], [1688, 0], [0, 298], [0, 276], [-322, 0], [0, 486], [0, 502], [-18, 0], [0, -2334], [1676, 0], [-3452, 0], [1720, 0], [0, -1246], [0, -1760], [0, -2724], [0, 2378], [3094, 0], [-310, 0], [-516, 0], [0, 188], [0, 2578], [528, 0], [-3660, 0], [0, -2350], [0, 3260], [466, 0], [0, 3170], [-292, 0], [-842, 0], [5838, 0], [0, 4626], [0, -5468], [2976, 0], [-56, 0], [0, -4106], [-2274, 0], [0, -5504], [0, -8034], [0, 996], [-8676, 0], [0, -1732], [0, -2938], [-128, 0], [0, 5596], [0, -924], [6206, 0], [0, 3506], [7104, 0], [0, 2902], [0, 4542], [-5852, 0], [0, 8936], [-1924, 0], [0, -9194], [-10714, 0], [12346, 0], [13562, 0], [5230, 0], [-11552, 0], [0, -7972], [-7152, 0], [0, -2652], [-4542, 0], [0, 16000], [2076, 0], [0, 5406], [0, -444], [1724, 0], [0, -17046], [8814, 0], [-5256, 0], [0, -5232], [-4128, 0], [-538, 0], [0, 10336], [-8942, 0], [0, 11404], [0, 3114], [0, 3620], [6506, 0], [-17600, 0], [0, 4864], [0, -18348], [0, -16218], [24576, 0], [6578, 0], [0, 25346], [0, -27214], [0, 22478], [-3356, 0], [19952, 0], [26434, 0], [0, -10632], [0, -9384], [0, 3922], [-12542, 0], [-8662, 0], [0, -5820], [0, -778], [0, 5548], [5488, 0], [1216, 0], [-20406, 0], [-19672, 0], [0, 1194], [0, 21802], [-15280, 0], [0, -6672], [-11008, 0], [0, -3242], [17982, 0], [30462, 0], [0, 17394], [0, -20336], [0, -19546], [-3954, 0], [-3860, 0], [37016, 0], [0, -25794], [0, -44116], [11622, 0], [2584, 0], [0, 39754], [0, 18534], [11140, 0], [0, -28356], [0, -34942], [-13890, 0], [0, -11496], [0, -29988], [0, -16764], [-19152, 0], [-35712, 0], [16098, 0], [0, 24776], [-39500, 0], [0, 22836], [0, 28478], [0, 24830], [37038, 0], [0, -26346], [0, -11464], [-28952, 0], [0, -8022]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_60_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_60_4_d_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_60_4_d_a(:prec:=2)
    chi := MakeCharacter_60_d();
    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_60_4_d_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_60_4_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_60_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;