
// Make newform 6048.2.a.bl in Magma, downloaded from the LMFDB on 24 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_bl();".
// 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_bl();".
// 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 := [9, -6, -10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [9, -1, -3, 1], [9, -7, -3, 1], [0, -7, -1, 1]];
    Rf_basisdens := [1, 3, 3, 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], [1, 1, 0, 0], [-1, 1, 0, -1], [-1, 0, 1, -1], [0, 0, -1, -1], [2, 0, 0, -1], [1, -1, 0, 1], [-3, -1, -1, 0], [-2, 1, -2, 1], [0, 1, 1, -1], [-1, 1, -2, 1], [1, -1, 0, -1], [-4, 2, -2, 0], [5, 1, -2, -1], [-2, 0, 2, -2], [-1, -1, -2, -1], [3, 0, -1, 2], [-1, -1, -4, 1], [1, 3, -2, -1], [5, -1, 0, 1], [-6, -1, 1, 1], [4, 0, 4, 0], [-1, 0, -3, -1], [-1, -3, -1, 2], [11, 0, 1, 1], [-3, -1, -1, 0], [14, 0, 2, 0], [-1, -1, 2, -1], [7, -1, -2, 1], [1, -1, 0, -3], [0, 2, 2, 0], [4, -2, -2, 2], [4, 0, 2, -2], [-7, 1, -6, 1], [5, 3, 0, 1], [14, 2, 4, 0], [2, 0, 0, -1], [3, 0, 1, -3], [-6, -4, 0, 0], [0, 2, -4, -1], [10, 2, 2, 0], [0, -2, -2, 2], [-8, 1, 2, -3], [0, 2, 4, -2], [4, -1, 2, -1], [4, -2, -2, 2], [-14, 0, -2, 2], [-12, -2, -2, -2], [7, 2, -1, -3], [-5, 1, -6, 1], [-2, 0, -6, -2], [1, -3, 0, 2], [8, 4, -2, -3], [-4, -2, 4, 3], [0, -2, 2, 4], [-2, 1, 6, 1], [11, -1, -2, 1], [6, -2, 0, 2], [-5, -2, -5, -1], [10, 0, 1, 1], [7, 1, 8, -1], [21, -1, 0, 1], [-7, 3, 4, -1], [-20, -2, -2, 0], [1, 2, 0, -2], [8, -5, -5, 5], [0, 2, 0, -2], [12, 1, 1, -3], [9, -2, 1, -1], [8, -1, 4, 1], [-17, -1, -3, -2], [3, 1, 2, -3], [8, -2, 2, 2], [10, 4, -2, 2], [4, 0, -2, 6], [-5, 1, 0, -5], [13, 1, -4, -5], [-4, 4, -8, -2], [-14, -1, 4, 1], [-2, 0, 0, 1], [-1, -5, 2, -1], [8, -4, 4, 0], [7, -1, 8, 0], [2, 4, -2, -4], [5, 0, -8, 0], [-6, -6, 6, 3], [4, 2, 0, -2], [13, -5, 4, 3], [-8, 0, 0, 4], [-17, 3, 0, -3], [-4, 3, -1, -3], [-3, -3, 8, 3], [3, -1, 6, -1], [3, 2, 7, -1], [-11, 3, -6, 4], [11, 2, -2, 4], [-10, -1, 0, -3], [-18, -2, -4, -2], [-10, 0, -2, 0], [-3, 3, -4, -3], [-10, -2, -8, 4], [12, 2, -4, -2], [6, -2, 2, 4], [6, 0, -6, 0], [-33, 1, 2, -2], [4, -4, 6, -1], [13, 1, -2, 1], [-24, 2, -6, 0], [-17, 1, 5, -4], [25, 1, -2, -1], [5, 6, 2, -4], [-17, -1, 0, 1], [-2, -2, -4, 4], [-2, -8, 5, 5], [23, -1, 6, 1], [19, 7, 2, 1], [13, 5, 0, 0], [-23, -1, -4, 1], [2, -2, -6, 4], [-13, 4, -5, 2], [-13, 7, -12, 0], [2, 4, 6, 2], [11, -1, 10, -5], [-19, 1, -3, -4], [-9, 3, -2, -7], [18, 2, -4, 2], [9, 3, 2, -5], [26, 2, -2, 0], [-17, -2, -3, 2], [9, -3, 12, -5], [-8, 6, 4, -2], [-3, 6, -11, -3], [15, 5, -4, -5], [1, 4, -3, -4], [-2, 4, -10, 2], [11, 2, -3, 10], [-1, 3, 6, -5], [-13, 2, -4, -6], [-1, -3, 8, -3], [-18, 0, -4, -4], [-20, -9, -1, -1], [1, 1, 4, 3], [-3, -1, 0, 1], [-8, 0, 0, 4], [6, 1, 3, -3], [22, -4, -3, 1], [-5, -2, -17, 1], [12, 0, -6, 2], [-20, -3, -3, 1], [2, 4, -2, -2], [1, 1, -6, 5], [10, 2, 14, -4], [-1, -2, 11, -3], [4, -8, 4, 8], [-15, -6, 5, -3], [-2, 0, 2, -10], [5, 2, -9, -6], [-24, -5, -1, -5], [9, -1, -4, 5], [4, -8, 10, 2], [-15, -5, 4, -7], [14, -2, -4, -2], [-25, 1, 4, -3], [-3, -3, 6, 1], [2, 0, -4, -4], [0, -4, 5, 1], [2, 0, 8, -5], [-7, -8, -5, 3], [-9, 1, -6, 1], [-13, -2, -1, 1], [2, 0, -7, 1], [-12, 1, 2, -3], [-7, -1, 10, 0], [-14, 0, 2, 2], [-24, -4, -16, 2], [-17, 7, 7, -8], [2, -2, 8, 2], [12, 0, -6, -2], [-14, 2, -12, 4], [-14, -1, -6, -1], [-13, 2, 11, -1], [-21, 0, -5, 0], [22, 4, 0, 7], [12, -2, 4, -2], [-8, 0, 12, -4], [-12, 2, -3, 3], [0, -2, -14, 4], [-13, -3, 8, -5], [15, 2, 3, 9], [0, -4, 10, -6], [-37, -4, 5, 3], [-9, 1, 10, 6], [30, 5, 3, -1], [-12, -6, -9, 5], [4, 2, -4, 2], [-21, 1, 2, -4], [12, 4, 8, 6], [21, -1, 4, -7], [24, -3, -8, 7], [3, -3, 2, -11], [-6, -4, 14, 2], [-25, 3, -6, 3], [9, -6, 9, -3], [-34, 2, 4, -6], [-4, -3, 9, -5], [19, -1, 6, -7], [14, 6, -8, -6], [-17, -2, -4, 6], [-11, 2, -1, -6], [5, 1, 8, -9], [-7, 1, 6, -9], [2, -8, -2, 2], [2, -10, 8, -2], [1, -5, 0, 9], [13, 1, -4, 4], [-2, 2, 2, -5], [26, -8, -2, 6], [8, 0, 10, -6], [22, 4, 6, -6], [-1, 5, -12, 7], [35, 5, -6, -3], [16, 0, 8, -2], [-28, -3, -4, 3], [3, 7, 4, 4], [-27, -2, 3, -2], [11, 1, 3, 2], [-40, -1, -1, 5], [-7, 7, -13, -4], [40, 2, -5, 1], [23, 7, 2, -1], [2, 8, -6, 4], [-26, -6, -1, 5], [-41, 3, -6, -1], [-6, 2, 10, 0], [-25, -5, 6, 3], [-7, 2, -7, 7], [-33, -5, 2, -1], [12, -7, -7, 5], [1, -6, 6, 4], [-22, -3, -10, 5], [28, -3, 2, 1], [-21, 3, 0, -4], [-1, -11, 8, 1], [7, 3, -3, -6], [-23, 1, -2, 5], [-7, 3, -16, 1], [-21, -4, 7, 4], [34, 0, -4, 4], [-19, -2, -3, 1], [-9, 8, -7, -7], [-18, 2, 10, -4], [-1, 4, -1, -2], [30, 1, 3, 1], [-18, 1, -20, 7], [24, -10, 7, 1], [9, 2, 9, -7], [19, -7, 6, 5], [11, -5, 12, 5], [23, -3, -6, -4], [-3, 5, 6, -3], [25, 5, -8, -1], [27, 1, 6, -2], [3, -6, 4, 2], [19, 13, 0, -5], [-31, -2, 1, 1], [28, 4, -2, -6], [31, -5, 10, -1], [29, 9, 0, 3], [14, -6, -4, -2], [37, 1, 14, -3], [13, 3, 8, 1], [-37, -5, -4, -7], [-35, -7, 7, 6], [24, 1, 1, 11], [34, -8, 8, 0], [-12, -4, 2, -2], [-15, -1, 4, 1], [11, -9, -2, 5], [-2, 0, -4, 8], [34, 2, -8, -4], [29, -7, -4, 7], [3, 2, -10, 4], [-7, -8, 9, 0], [-49, 6, -4, -2], [45, 5, 6, 3], [-9, 3, -1, -4], [-48, 7, -13, 3], [12, -4, 0, 14], [49, -3, -6, 7], [-45, -3, 0, 3], [-14, -8, 2, 4], [21, -3, 13, 0], [18, -6, -10, 3], [-16, 1, -4, -1], [5, 0, 15, 5], [17, -7, -6, 13], [-23, 12, -18, 2], [-32, -4, -2, 2], [-19, 1, -10, 1], [12, -8, 10, 2], [4, -3, 9, -13], [6, 2, -2, 16], [-37, -10, -10, 0], [37, -1, -6, 6], [-29, -7, -6, -3], [27, 6, 9, -6], [-16, -9, 0, -3], [-20, -10, 6, 6], [33, -3, 2, -1], [-8, -2, 6, 0], [-5, -9, 14, 3], [1, 10, 3, -10], [-12, 4, 12, 0], [19, 3, 2, -5], [-32, 4, 14, -3], [-32, -2, 2, 4], [11, 9, -8, -7], [-22, -4, 6, 10], [-6, 6, 1, -5], [16, 8, 10, 1], [35, 1, 2, 5], [-2, 0, -6, 10], [-12, 10, -2, 0], [-20, 6, -8, 10], [-12, -6, -1, -3], [13, -2, 5, 3], [28, -6, 10, 6], [-15, 5, 3, -2], [-20, -10, 18, 10], [25, 1, -3, 12], [-41, -9, -2, 3], [15, -5, -6, 3], [-8, 14, -2, -8], [-1, -7, -11, 4], [-21, -1, -4, -8], [17, -3, -12, 15], [-13, 9, 5, -4], [-19, 9, -3, -12], [9, 3, 0, -7], [-39, 4, -9, 9], [27, 3, 2, -1], [21, 9, -2, 7], [7, -10, 7, 9], [-7, -3, 4, 11], [-25, 3, 6, 3], [-3, 11, -16, 1], [-29, 0, 9, -13], [22, -4, 3, 7], [-40, -4, 2, -10], [28, 12, -4, 0], [-16, 1, 0, 3], [3, 3, -7, 6], [-4, -5, -9, -3], [27, 5, 4, -7], [48, 2, 6, -4], [-19, 1, -18, 7], [55, 10, -1, -3], [14, 8, -4, 12], [-7, 7, -2, 6], [27, 4, 1, 5], [1, 1, 8, 0], [48, -2, 6, 0], [14, 4, 3, -1], [-27, 0, -7, 6], [-20, 4, 2, -2], [24, -2, 13, -9], [37, -3, 12, -2], [-6, 2, 7, -11], [21, 6, -3, -9], [23, 3, 18, -7], [-1, 1, 21, -4], [24, -1, 6, 3], [31, -11, -16, 17], [-11, -8, -5, 3], [10, 2, -2, 4], [33, -3, -4, 12], [-45, 1, 0, -5], [-12, 0, -2, -3], [12, -6, 2, 10], [-51, -13, -4, 1], [-5, -15, 0, 1], [-45, -6, -2, 4], [57, -5, 4, -1], [20, -2, 4, -6], [-10, 2, 0, -10], [-30, -4, 10, -6], [38, -8, 14, -2], [26, 2, -6, 8], [-33, 0, -1, -4], [-26, 3, 1, 1], [42, 12, 2, -2], [-19, -14, 9, 7], [33, 2, -8, -10], [8, 7, 12, 1], [-24, 1, -11, 1], [23, -5, 4, 5], [23, 7, 18, -5], [7, -5, -4, 1], [-7, 3, 0, -7], [22, 6, -18, -4], [-60, -5, 3, -5], [9, -12, 1, 14], [-38, -11, -6, 1], [26, -1, -8, 5], [-15, 9, -6, 1], [25, -6, -4, 10], [-12, -6, -22, 8], [-18, -10, -4, -2], [-13, 7, 14, 1], [-29, 8, -15, 9], [-32, -8, 4, -4], [-56, -8, -2, -2], [-5, -4, -9, 8], [35, -8, -13, 16], [-4, -10, 14, 0], [23, -6, -9, 7], [12, 2, 18, -2], [-18, 8, 6, -6], [8, 10, 6, 4], [54, 14, 2, 1], [49, -3, 6, -5], [-12, 0, 10, -2], [1, 2, -7, 1], [-11, 6, -23, -5], [10, 16, -10, -6], [-15, -8, -8, 8], [-21, -1, -2, -1]];
    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_bl();".
// 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_bl(: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_bl();".
// 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_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![27, -10, -12, 2, 1]>,<11,R![-73, 114, -32, -2, 1]>,<13,R![-32, -176, -40, 4, 1]>,<17,R![48, -80, -32, 4, 1]>],Snew);
    return Vf;
end function;