
// Make newform 600.2.y.b in Magma, downloaded from the LMFDB on 23 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_600_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_600_y_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_600_2_y_b();".
// 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_600_2_y_b();".
// 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, -1, 1, -1, 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_600_y();"
function MakeCharacter_600_y()
    N := 600;
    order := 5;
    char_gens := [151, 301, 401, 577];
    v := [5, 5, 5, 3];
    // 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_600_y_Hecke();"
function MakeCharacter_600_y_Hecke(Kf)
    N := 600;
    order := 5;
    char_gens := [151, 301, 401, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1]];
    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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 1], [0, -2, 1, -2], [0, 0, -2, 2], [-2, -2, 2, 2], [1, 1, 1, 0], [0, 1, 4, 1], [0, 4, 2, 4], [-4, 2, -2, 4], [5, -5, 0, 1], [0, -4, 2, -4], [5, -6, 5, 0], [-3, -1, -3, 0], [6, 0, 4, -4], [-6, 6, 0, 6], [9, -9, 0, -2], [-8, 4, -8, 0], [-10, 7, -7, 10], [0, 2, 8, 2], [0, 0, 0, -10], [4, 7, -7, -4], [0, 0, 0, 4], [0, 6, -10, 6], [-13, 16, -16, 13], [-1, 1, 0, 5], [11, 0, 5, -5], [6, -6, 0, 6], [-8, 0, 2, -2], [-3, 9, -3, 0], [-5, -4, -5, 0], [2, 10, -10, -2], [0, -6, 4, -6], [9, -5, 9, 0], [4, 10, -10, -4], [2, 0, 1, -1], [-8, 0, -6, 6], [17, 0, -1, 1], [-2, 2, -2, 0], [0, 4, -4, 4], [13, 2, -2, -13], [-18, 18, 0, 6], [0, -5, 14, -5], [-14, 4, -14, 0], [-8, 0, -3, 3], [-11, 11, 0, 18], [-6, 0, -12, 12], [16, -20, 20, -16], [14, -20, 20, -14], [4, -12, 12, -4], [-13, 13, 0, 18], [0, -5, 25, -5], [-12, 20, -20, 12], [9, -4, 9, 0], [12, 0, 10, -10], [12, 0, 11, -11], [10, -14, 10, 0], [0, -7, 5, -7], [6, -6, 0, -6], [19, -26, 26, -19], [0, 19, -15, 19], [0, -2, 12, -2], [7, 0, -11, 11], [-14, 0, -20, 20], [2, 4, -4, -2], [-12, 2, -12, 0], [0, 8, 14, 8], [0, -18, 2, -18], [-12, 6, -12, 0], [8, -8, 0, 10], [33, 0, 3, -3], [-4, 4, 0, 14], [2, 26, 2, 0], [0, -18, 6, -18], [-18, 18, -18, 18], [-4, 4, 0, -12], [0, -2, -22, -2], [10, -7, 7, -10], [16, -16, 0, -6], [24, 0, -5, 5], [-3, -8, -3, 0], [0, 0, 4, 0], [-17, 17, 0, -5], [0, 16, 10, 16], [0, 5, 15, 5], [10, -24, 24, -10], [-6, 0, 2, -2], [-11, 0, 1, -1], [22, 0, 8, -8], [9, 18, -18, -9], [-8, 18, -8, 0], [0, 10, 0, 10], [-6, 6, 0, -28], [2, -12, 2, 0], [0, 20, 0, 0], [8, 0, 6, -6], [0, 0, 0, 20], [7, -30, 7, 0], [-1, 1, 0, -28], [-10, 28, -28, 10], [-3, -26, -3, 0], [-26, 26, 0, 12], [-9, 0, 9, -9], [0, -4, 0, 0], [0, 1, -32, 1], [18, -18, 0, -14], [18, 10, -10, -18], [-2, 22, -2, 0], [-7, 0, 5, -5], [4, 0, -18, 18], [40, 0, 9, -9], [-34, 0, -12, 12], [1, -22, 1, 0], [0, 19, 9, 19], [0, -24, 4, -24], [0, 0, 10, 0], [0, -18, 0, 0], [-4, 0, 20, -20], [-10, 10, 0, 8], [-7, 7, 0, 9], [26, -22, 26, 0], [-22, -6, 6, 22], [-3, 16, -16, 3], [22, -6, 6, -22], [0, 0, 44, 0], [2, 40, 2, 0], [-30, 0, 1, -1], [-7, 29, -7, 0], [0, 2, 2, 2], [26, 4, -4, -26], [0, 4, -6, 4], [-26, 2, -2, 26], [-6, 0, 18, -18], [34, 0, 4, -4], [-14, 0, -29, 29], [28, -27, 27, -28], [0, 4, 38, 4], [-2, 29, -29, 2], [-28, 6, -28, 0], [1, -1, 0, -14], [11, -32, 11, 0], [-38, 48, -48, 38], [-28, 28, 0, 14], [-14, -16, 16, 14], [22, -18, 18, -22], [-21, 21, 0, -10], [10, 0, 0, -10], [-5, 5, 0, 10], [22, 0, 0, 0], [4, 32, 4, 0], [-22, 12, -22, 0], [-30, 15, -15, 30], [0, 28, 2, 28], [0, -16, 10, -16], [4, -18, 4, 0], [22, 0, 20, -20], [18, -6, 6, -18], [0, -18, 14, -18], [-23, 23, 0, 35], [29, -16, 29, 0], [-13, -22, -13, 0], [18, -18, 0, 12], [-7, 7, 0, 23], [0, 32, -16, 32], [6, -6, 0, -12], [-40, 25, -25, 40], [0, -16, -22, -16], [-2, 20, -2, 0], [4, -4, 0, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_600_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_600_2_y_b();".
// 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_600_2_y_b(:prec:=4)
    chi := MakeCharacter_600_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_600_2_y_b();".
// 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_600_2_y_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_600_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, -2, 1]>],Snew);
    return Vf;
end function;