
// Make newform 6027.2.a.s in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6027_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_2_a_s();".
// 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_6027_2_a_s();".
// 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, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 1]];
    Rf_basisdens := [1, 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_6027_a();"
function MakeCharacter_6027_a()
    N := 6027;
    order := 1;
    char_gens := [4019, 493, 2794];
    v := [1, 1, 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;

function MakeCharacter_6027_a_Hecke(Kf)
    return MakeCharacter_6027_a();
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, 1, 0], [1, 0, 0], [-1, -1, 1], [0, 0, 0], [-1, -1, 0], [-3, 1, -1], [-1, 1, -2], [-1, 1, -1], [-3, -1, 1], [-1, -3, 0], [2, -4, -1], [6, 2, -1], [-1, 0, 0], [4, -2, 5], [-1, -1, -2], [6, -4, 2], [2, 2, 2], [2, 2, -1], [2, 6, -4], [-11, 1, 0], [-2, 2, 3], [-8, 4, -2], [5, -1, 3], [-6, 4, 0], [3, 3, -1], [7, -3, -2], [-4, -2, 3], [0, 8, -2], [3, -1, 1], [-7, 5, -5], [2, -6, 0], [-3, -5, 1], [-9, 5, 2], [-6, 6, 4], [2, -4, 8], [12, -4, 0], [-2, -12, 4], [2, 8, -1], [-6, 2, 0], [-11, 1, 3], [-7, -3, -2], [-6, -4, 2], [-6, 6, -6], [-1, -9, 1], [-5, -1, -7], [-3, -1, -3], [-6, 2, 2], [-8, 8, -8], [-7, 5, -2], [7, -5, -1], [-2, 12, -4], [-6, -2, -2], [6, -6, -1], [2, -10, 0], [-9, 9, -2], [-17, 3, 0], [-4, 2, 0], [-16, 2, -1], [-4, 8, 3], [-3, -1, 4], [8, -2, -5], [-17, -7, 4], [-2, -8, 1], [-14, 6, -6], [7, 3, 5], [3, 5, 6], [13, 3, -7], [-4, 0, 1], [3, -1, -6], [0, 8, -1], [-10, 8, -8], [-7, -5, 1], [-12, 6, -1], [12, -12, 5], [-14, 6, -4], [-5, 7, 8], [-3, -3, 7], [-3, -7, 3], [-2, -8, 2], [20, 0, -5], [9, -9, 1], [-2, 4, 2], [5, -1, 1], [10, -2, -9], [8, 4, 2], [24, 4, -4], [-28, 2, -4], [-22, -4, 2], [6, -12, 8], [-9, 9, 5], [8, 0, 6], [-5, -5, -10], [18, 0, 9], [-3, -9, 3], [-10, 6, -2], [13, -3, -10], [-15, 3, -8], [5, -17, 10], [2, 6, 8], [18, -12, 8], [-6, -14, 2], [11, -3, -10], [-3, 5, 8], [-5, 7, -13], [16, -4, -8], [-16, 2, 4], [-3, -3, -8], [23, -3, -4], [-7, 19, -3], [8, -2, 14], [-12, -8, -2], [-4, -2, 0], [25, -3, 3], [-10, 12, -9], [4, -2, -1], [-7, 11, -16], [-2, -2, 10], [9, -5, 9], [-11, -5, 8], [2, -6, 2], [-18, 8, 6], [8, 2, -10], [-7, 21, -1], [-6, -2, 8], [9, -5, 3], [-12, -6, 2], [12, -6, 18], [-5, 3, 2], [23, -3, 7], [-20, 4, -1], [-12, -14, -1], [4, -16, 6], [20, -8, -6], [-17, 7, -9], [-24, 10, -8], [20, 0, -1], [-13, 17, 2], [12, -6, -5], [-12, -10, -6], [8, 18, -6], [2, 4, 1], [-7, 5, 1], [6, 6, -12], [-17, 3, 14], [2, -14, 7], [22, -6, 2], [-20, 10, -8], [8, 10, -8], [16, -2, 7], [27, 5, 5], [-8, 0, 17], [34, 0, 2], [-19, -5, 1], [-5, 15, 0], [-2, 10, 8], [-21, 1, 7], [1, -17, 7], [-17, -11, 4], [-2, 4, -14], [-10, -12, 0], [-16, -8, 10], [7, -13, 11], [-16, -4, -6], [-39, 9, -8], [25, -9, 2], [5, -5, 15], [-6, -18, -2], [35, 3, 1], [4, -8, 9], [-4, -2, 14], [6, -14, 20], [22, -8, 6], [-2, 6, 0], [18, 18, -5], [-12, -4, 8], [-7, -21, 6], [7, 33, -13], [-32, 6, 6], [21, -9, -11], [-18, 12, -8], [-42, 8, -13], [0, 8, -10], [-18, 20, -14], [-6, -28, 8], [0, -4, 10], [-11, 17, -1], [-4, -10, 8], [2, -18, 0], [33, -11, 11], [-11, 1, 12], [-30, 6, -9], [10, -22, 10], [20, -4, 18], [-4, -26, 6], [-15, -17, 5], [2, -24, 10], [0, -22, 6], [5, -11, 11], [-2, 24, 4], [-1, 7, -2], [-34, -4, 10], [6, 24, -13], [-21, 15, 7], [-6, 4, -12], [-33, 7, 0], [-11, 7, -16], [4, -16, 10], [-5, -1, -22], [29, -11, -15], [28, 14, -11], [-19, 25, -13], [-31, 11, 8], [4, -16, 14], [-50, 6, 6], [7, -5, -8], [38, 6, -11], [11, -19, 13], [-1, 7, -5], [35, -1, 12], [-4, 10, -18], [-17, -9, -9], [16, -14, 19], [-7, 7, 18], [37, 23, -7], [-39, 19, 3], [-21, 3, 15], [-6, -4, 4], [-13, 5, 9], [22, 18, -2], [21, 3, 9], [10, -12, -4], [27, 1, -11], [6, 12, -7], [37, -11, 5], [17, 7, 5], [-38, 2, -12], [-3, 13, 11], [4, 2, 18], [24, -12, -20], [9, -7, 14], [10, 2, -6], [12, -16, 0], [-14, 2, 20], [26, 14, 7], [30, 12, -14], [-1, -39, 9], [8, -12, 18], [-7, 9, -28], [-4, -14, 7], [30, -18, -12], [-26, -18, 13], [-13, -5, 11], [-2, 22, -4], [-6, -6, -19], [36, -18, 22], [20, -28, 22], [-25, 15, -17], [-3, 35, -19], [-27, 3, 14], [-19, 1, -3], [-10, 28, -15], [-25, 19, -10], [-8, -6, 14], [-28, 26, 8], [-9, 19, 9], [-10, -28, 7], [33, -1, -12], [36, -6, 6], [38, -14, 24], [-27, 3, 0], [14, 0, -18], [-16, -2, -9], [-32, 20, -1], [16, -26, 21], [14, -16, 8], [4, -12, 4], [21, -11, -12], [-28, -2, 0], [-31, -7, -7], [-7, -3, 2], [-21, 15, -20], [-1, -7, -3], [20, 16, -18], [10, -12, 14], [30, -10, 10], [53, -7, 8], [-3, 13, 1], [12, 10, -22], [-9, 33, -11], [-33, 13, -16], [-23, -21, 2], [28, -20, -6], [-27, -21, 2], [39, -15, 3], [11, 19, -5], [53, 7, 0], [22, 18, -14], [8, -6, -26], [20, -24, 32], [5, -1, -1], [-2, 28, 6], [18, 36, -16], [40, -18, 5], [-19, -15, 6], [52, -6, 7], [50, 8, 4], [-39, 15, -3], [-12, -28, 11], [-62, -6, 0], [-11, 1, -23], [-33, -1, 6], [-8, -10, -22], [-16, 6, -8], [0, -30, 13], [1, 23, -9], [12, -2, 32], [-32, 16, 2], [-19, -1, -19], [-48, 6, -14], [41, 23, -6], [-56, -8, 10], [-30, 4, -14], [-19, -7, 11], [21, 11, -1], [35, -9, 7], [22, 0, -12], [47, 5, -15], [77, -9, -5], [-25, 15, -30], [-36, 10, -6], [-8, 26, -20], [14, 32, -8], [-44, 28, 4], [-34, 6, 6], [50, 16, -1], [-3, 9, -22], [3, -1, 9], [3, -25, 13], [15, -29, 1], [22, 16, 7], [18, -32, 14], [-10, 16, -2], [-42, -8, 2], [6, 14, 0], [18, -32, 2], [36, -8, 12], [38, 8, 10], [-42, -20, 11], [-4, 4, -24], [-76, 10, -4], [-45, -7, 5], [-18, 18, 17], [-41, 1, 2], [-68, 10, 1], [-8, 2, 4], [14, -20, -18], [31, -7, 5], [36, 0, 0], [-15, -11, -17], [-3, 7, -19], [-12, 4, 25], [34, -28, 12], [5, 37, 2], [6, -34, 2], [6, 2, -4], [31, -9, -3], [28, -26, 22], [6, 12, -27], [-24, 4, 9], [-32, 20, -16], [28, 16, -8], [38, 2, 14], [19, 1, -12], [-10, -10, 24], [-23, 5, 13], [29, -29, 13], [13, -37, 15], [-10, 28, -18], [-47, -5, 6], [17, 5, 5], [-43, 11, -18], [29, -19, 21], [-18, 4, 13], [-30, 28, -14], [-30, -2, 12], [-16, -12, -6], [43, -19, -19], [20, -30, 8], [-26, 0, 1], [36, -20, -16], [20, 14, -18], [5, -9, -12], [-14, -14, 18], [-12, 8, 16], [-12, 4, -4], [-22, 4, -10], [31, 21, -13], [-4, 18, 14], [-18, 8, 9], [4, 34, -14], [8, -40, 27], [-24, 10, 2], [34, 8, -23], [-20, 2, -26], [5, 13, 1], [37, -9, -19], [26, 2, -3], [-3, 15, -16], [29, -5, 5], [-3, 19, -19], [14, 2, -1], [14, -4, -2], [-14, 10, 6], [42, 18, -6], [-51, -11, 9], [-15, -31, -7], [7, -15, -19], [51, 1, 11], [49, -21, -18], [48, 20, -14], [-19, -23, 13], [6, -4, -24], [-55, -9, 7], [3, 43, -16], [36, -16, 3], [-75, 13, -19], [-43, 21, 0], [-11, -13, 22], [-57, 9, -13], [13, 25, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6027_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_2_a_s();".
// 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_6027_2_a_s(:prec:=3)
    chi := MakeCharacter_6027_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6027_2_a_s();".
// 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_6027_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, -4, -1, 1]>,<5,R![-4, -2, 4, 1]>,<13,R![-4, 14, 8, 1]>],Snew);
    return Vf;
end function;