
// Make newform 6048.2.a.bk in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6048_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_6048_2_a_bk();".
// 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_6048_2_a_bk();".
// 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 := [12, -2, -9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [-5, 0, 1, 0], [3, -7, -1, 1]];
    Rf_basisdens := [1, 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_6048_a();"
function MakeCharacter_6048_a()
    N := 6048;
    order := 1;
    char_gens := [4159, 3781, 3809, 2593];
    v := [1, 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_6048_a_Hecke(Kf)
    return MakeCharacter_6048_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, 0, 0, 0], [0, 0, 0, 0], [-1, 0, -1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [2, 1, 0, 1], [-1, 0, -1, -1], [-2, -1, 0, 0], [0, 0, 0, 1], [-2, 0, -1, 0], [-2, 0, -1, 0], [1, -1, -1, 0], [1, 0, -1, 0], [1, 0, 1, -1], [-1, -1, -1, -2], [-2, 1, -1, 2], [-1, 0, -2, 0], [0, -1, -2, 0], [0, 0, -2, 1], [-4, 0, 0, -1], [4, -2, 2, 0], [1, 0, 1, -2], [5, 1, 1, 0], [2, 2, 0, 1], [6, -1, 0, -2], [-2, 2, 0, 0], [-8, -1, -3, -2], [-4, 0, 0, 0], [7, -1, -1, 0], [4, -1, 0, 0], [3, -2, -1, 0], [0, 3, -3, 0], [6, 1, 0, 0], [0, 3, -4, 2], [0, -2, 1, 0], [7, 0, 3, 2], [2, -1, -4, -1], [-1, -2, 3, -1], [-5, -3, 1, -2], [-6, 2, 2, 0], [-4, -1, 4, 0], [6, 0, -2, -1], [14, 2, -2, 0], [7, 2, -2, 0], [-2, -1, 2, -2], [6, 1, -1, -2], [8, -2, -2, -3], [6, -3, 4, 2], [4, -1, -5, 0], [14, 0, 2, 2], [10, -1, 4, 2], [4, -3, 6, -2], [6, -1, -2, 2], [-3, 1, 5, 0], [10, -2, 2, -2], [14, -1, 4, 1], [-13, -2, 3, 0], [-6, -1, 1, 2], [9, -3, 1, 0], [4, -2, 2, 0], [12, 1, -2, 2], [-3, 0, 3, 4], [-16, 2, -6, 0], [-5, 3, -5, 0], [10, 2, 0, -4], [-12, -2, 2, 0], [-15, 2, 1, 1], [-1, -4, -4, -4], [-2, -5, 0, -4], [8, 4, 0, 1], [7, 0, 3, 1], [-6, 2, -6, 3], [14, -1, 1, -2], [15, -3, 7, 2], [19, -2, 5, 2], [-9, -5, 1, -2], [0, -1, 4, -2], [8, -4, 4, -2], [14, -2, 0, -4], [8, 2, -4, -2], [-7, 2, 0, 4], [2, -4, 6, -2], [-12, 2, 0, 0], [-4, 1, -6, 2], [-10, 3, -5, -2], [6, 5, -4, 0], [14, -1, -2, 0], [2, -3, -1, -4], [-11, 2, -3, 2], [3, 0, -5, 2], [0, 2, 4, 4], [1, -3, -9, -2], [8, -2, 6, 0], [-2, 5, 6, 4], [11, -2, 9, 2], [-13, -1, 1, -2], [-21, -2, -3, -4], [-1, -4, -5, -5], [-8, 1, 0, 4], [-11, 3, -9, -2], [17, 6, -1, 2], [-4, 5, -5, 6], [0, 1, -9, 0], [-4, 6, -2, -2], [5, -6, -3, -3], [6, -2, 2, -2], [12, -5, -1, -4], [19, 0, 1, 0], [-12, 1, -2, -3], [-10, -5, -2, 0], [-16, 4, 5, 4], [10, -2, 10, 2], [-8, 4, -2, 0], [-10, -3, -6, -6], [27, 0, -5, 0], [8, -3, -10, -2], [-10, -4, -4, -6], [-20, 0, -2, 0], [-12, 4, -1, 4], [10, -6, 2, -4], [-12, 3, -6, 0], [2, 7, -1, 4], [-6, -8, 0, -2], [-2, 1, 2, 0], [-24, 1, -4, 0], [-16, -1, 0, -2], [15, 1, 9, 2], [-29, -1, -1, 4], [-6, -8, 7, -4], [-6, 2, -6, -3], [-4, 4, -8, 4], [-2, -1, -4, 3], [-8, 8, 4, 2], [-5, 3, -3, 6], [5, 2, 5, 5], [-2, 0, 0, -2], [5, 0, 5, 0], [24, 0, 2, -4], [-10, 6, 2, -2], [10, -2, 6, 4], [-20, 1, 0, -2], [0, 5, 5, 2], [3, -2, 9, -4], [-24, -1, 0, 2], [-2, -5, 4, 4], [19, -1, 5, -4], [-10, 0, -2, 1], [-15, 4, 9, 1], [-6, -6, 4, 4], [10, -4, 2, -5], [5, 1, 9, -4], [18, 0, 0, -1], [23, -4, -1, -3], [-9, -5, -1, 4], [-17, 0, -3, 2], [16, 5, -2, -2], [-1, -2, -1, 2], [21, 4, 3, 4], [-10, 1, -2, 0], [19, 0, 5, 0], [-8, -2, 2, 4], [-16, -4, -2, -2], [20, -4, 2, -2], [-7, -4, -6, -8], [6, -2, 6, 2], [11, 1, 5, -2], [-7, -2, -1, -4], [-14, -5, 0, -3], [1, 0, -2, -4], [21, 0, 3, -2], [20, -2, -4, -2], [-22, 5, 4, 3], [12, 7, -6, 1], [-13, 0, 0, -4], [-10, -7, 8, -2], [-25, 0, 3, -3], [20, -2, 10, 3], [10, -1, -1, -2], [-2, -2, 2, -4], [-6, -4, -2, 3], [1, -4, -5, -2], [1, 8, 4, 8], [6, -2, 4, 2], [-5, -4, 3, -7], [-16, 1, 2, 1], [-11, -2, -1, 0], [10, 6, 4, 2], [10, 1, 0, 2], [-10, -11, 9, -4], [-23, 1, 11, 0], [14, -1, 12, 2], [-6, 2, -8, 4], [-17, 6, 7, 5], [-3, 2, 11, 2], [10, 6, -8, 4], [27, -4, 3, -7], [-6, 1, 5, 8], [-17, 5, -1, 2], [2, -10, 4, -3], [-7, 9, 3, -2], [-22, 5, 1, 6], [2, 3, -1, -2], [-18, 0, -14, -2], [-12, -2, -4, -6], [-33, 2, -6, -4], [-3, -2, 7, 2], [-28, -2, -3, -4], [-20, 4, 6, 2], [28, -4, 10, 2], [32, 5, -6, 0], [-3, 6, -10, 0], [10, 0, 4, 4], [33, -4, -1, -8], [13, -2, 0, -4], [-11, -7, 1, -2], [-30, -5, 2, 4], [-23, 2, -11, -2], [-26, -8, 2, -6], [12, -1, 10, 7], [-24, -1, 1, 6], [-5, -3, 7, -2], [-24, 3, 2, 6], [28, 3, 6, -2], [-15, 4, 5, 8], [1, 5, -7, -4], [-7, 1, -5, -8], [-47, -2, -7, -5], [-4, 2, 0, 0], [-22, -1, -1, 2], [-18, 1, 0, -4], [-30, -4, -4, -6], [18, -5, 16, -6], [-11, -4, -9, -8], [-10, 10, 6, 4], [-26, 6, 8, 6], [-8, 0, 12, 0], [-14, -4, -6, -6], [4, -6, -11, 0], [-20, -2, 4, -6], [7, -11, 3, -2], [6, 2, -8, 4], [2, -4, 6, -2], [4, 8, -13, 4], [-10, 10, 2, 8], [-2, -8, 0, -3], [7, 7, 3, 6], [-12, 1, 5, 10], [29, -3, 13, 4], [39, -2, -1, -1], [-28, -2, -8, -1], [21, -5, 5, 4], [-6, -2, 4, -1], [-22, 0, -14, -7], [-16, -6, -4, -2], [-51, 4, -7, -2], [-26, -1, -2, -2], [-17, 7, -9, -8], [-12, 8, -6, 6], [30, -6, 0, 2], [14, 0, 2, 0], [40, 4, -2, 4], [22, -4, 6, 4], [2, -2, 2, -6], [0, -7, -6, -8], [30, 10, -4, 1], [-8, -12, 14, -6], [-4, -8, 2, -4], [-6, 2, 6, 2], [39, -4, -1, -3], [2, 8, -8, 0], [-26, -7, 4, -1], [4, -2, 4, 0], [-10, 5, -8, 2], [-35, -2, -9, -6], [34, 0, 6, -6], [2, 10, 5, -4], [-16, 3, 10, 4], [-5, -5, 5, -4], [-5, 5, -15, -4], [-3, -6, 2, -4], [-27, 1, 9, 2], [-3, 0, -3, 2], [-21, 3, -5, -8], [-20, -8, 4, 3], [-20, 0, -14, -2], [28, -8, -4, -12], [-15, -13, 7, -4], [10, 10, -5, -4], [-10, -3, -9, -6], [-3, 2, -19, 3], [18, -2, 8, -8], [21, 8, 4, 4], [-12, -1, 0, 8], [9, 4, 8, 12], [15, 3, 11, -6], [33, -6, 9, 2], [24, 2, 4, -4], [27, -2, -5, -6], [53, -2, 12, 4], [4, 11, 4, -2], [0, 7, -14, -4], [4, -5, 5, 6], [16, -2, -2, 0], [-4, 9, 4, 6], [15, 8, -5, 5], [14, 1, -11, 4], [-8, 10, -10, 2], [-2, -2, 10, -3], [28, 7, -6, -2], [-10, -10, 0, -2], [-12, 4, 12, 4], [2, -10, -5, 0], [-22, -8, 6, 2], [11, 2, -17, 1], [18, 0, -10, 5], [6, 8, -4, -2], [-9, 6, 2, 0], [-24, 0, -12, 0], [24, 11, 2, 10], [-20, -10, 4, 4], [26, -4, 16, 0], [1, -1, -5, 8], [1, -6, 7, 10], [8, -10, -4, -6], [8, 3, -14, -2], [-14, -10, -4, -6], [-31, -2, -3, -7], [-34, 6, 0, 4], [24, 3, -14, 3], [44, -3, 4, 2], [-6, -4, 10, 0], [-4, 9, 8, 6], [-6, 3, -4, 10], [-3, -11, -3, -12], [-5, -2, 7, -1], [-5, -2, -10, -4], [-6, -14, 4, -8], [5, 12, -3, -1], [6, 6, -8, -8], [-34, 4, 6, 8], [-5, -1, -3, 4], [-16, 11, 10, 12], [-13, 0, 9, -6], [28, 2, 12, 6], [-22, 1, 1, 6], [10, 9, -4, 8], [24, 8, 8, 11], [-7, 10, 9, 15], [-3, -15, 7, 2], [22, -8, -10, -8], [6, 3, 10, 4], [5, 6, -14, 8], [-32, -11, 17, -6], [56, 0, 1, 4], [47, 7, -1, -4], [-37, 0, -3, -4], [32, -2, 14, -2], [31, -7, 11, -8], [24, -8, 6, 4], [44, -1, -2, -6], [43, 1, 5, 2], [1, 4, 5, 3], [-46, 3, -12, -1], [-38, 8, -2, 2], [12, 4, 14, 4], [17, 5, -1, -8], [39, -10, 3, 0], [13, -4, 3, -10], [3, -4, -6, 4], [20, 2, 10, 8], [-6, 13, 0, 6], [-2, -5, -4, 5], [42, -8, -3, -4], [-12, 8, -13, 0], [6, 4, -4, -8], [28, -5, -1, 4], [4, -13, 2, 3], [2, -9, -6, 4], [30, -2, 22, 2], [28, -8, -10, 0], [-35, 6, -7, -6], [-18, 3, 12, -2], [48, 4, 10, 2], [-14, 6, -8, 8], [14, -2, 16, -2], [-16, 0, -8, -2], [16, 14, -6, 4], [36, 3, 2, -1], [19, 18, -5, 5], [12, -8, 0, -12], [3, -12, 8, -4], [2, -2, 12, 2], [48, 12, -8, 2], [-36, 3, -12, 2], [-10, 1, -8, 5], [18, -5, 5, -8], [-36, -15, 7, -6], [39, -2, 11, -13], [24, -12, -6, -7], [-12, 1, 1, 10], [24, -4, 4, -5], [-14, -2, -6, -4], [-9, -2, 13, 0], [-31, -14, 13, -3], [10, -2, 8, 2], [32, 1, 6, 10], [25, 3, -7, 0], [6, -8, -6, -8], [20, 8, 18, 14], [-40, -1, -3, 8], [28, 2, 0, 6], [-18, -18, 1, -12], [4, -7, 16, 0], [23, 2, -9, 5], [23, 2, 4, -8], [2, 6, 0, -4], [32, -7, -2, 3], [-22, 9, -16, -6], [-17, 0, -9, 7], [-52, 2, 2, -4], [44, -2, 23, 8], [18, 1, -4, -3], [-24, 11, 18, 2], [11, -2, -4, -8], [-28, -9, -4, -12], [65, -2, 1, 0], [-20, 0, 2, -2], [-28, -5, 4, -8], [26, -4, -8, 0], [-40, 10, 2, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6048_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6048_2_a_bk();".
// 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_6048_2_a_bk(:prec:=4)
    chi := MakeCharacter_6048_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_6048_2_a_bk();".
// 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_6048_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![48, -12, -15, 2, 1]>,<11,R![192, 16, -36, 0, 1]>,<13,R![88, 184, -21, -8, 1]>,<17,R![-89, -154, -48, 2, 1]>],Snew);
    return Vf;
end function;