
// Make newform 6048.2.a.bo in Magma, downloaded from the LMFDB on 19 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_bo();".
// 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_bo();".
// 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 := [3, -4, -9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-9, -5, 1, 1], [-9, -11, 1, 1], [-2, 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], [0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 1, -1, 0], [-3, -1, 0, -1], [-2, -1, 0, -1], [-1, -1, -1, 1], [-1, 0, 1, 1], [2, 0, 1, 1], [-1, 1, -1, 0], [-1, 0, -2, 1], [-2, -1, 1, 0], [-3, 0, 1, -1], [-4, 0, 0, 0], [-3, 0, 1, -1], [-4, -2, -2, 0], [2, 1, -3, 2], [4, -2, 1, 0], [0, -1, -1, -2], [2, 1, 3, 0], [-1, 2, -5, 1], [-5, 0, -2, 1], [2, 0, 2, -2], [-11, -1, 0, -1], [-2, 0, 1, -3], [1, -3, 0, -1], [-6, -2, -1, 1], [-4, 2, 0, 0], [-4, 1, -5, 0], [-5, 0, 3, 1], [-5, 4, -3, 1], [-2, 2, 0, -2], [-10, -2, 2, -2], [6, 2, 2, 0], [-2, -1, 3, 2], [10, 3, -1, 2], [-8, 2, 0, 0], [-11, -1, 1, -1], [11, 1, -2, 1], [0, 4, 2, 2], [1, 1, 1, 3], [4, -2, 4, -2], [-10, 0, -4, 2], [1, -1, 5, 0], [0, -2, -4, -2], [-1, -1, 1, 0], [-2, -2, -2, -2], [-4, -2, 6, 0], [-6, -2, -2, 2], [-1, -1, -4, -3], [12, -1, -1, 2], [2, 2, -4, -2], [-14, -1, 2, -2], [1, -3, 3, -1], [-1, 1, 3, 1], [-2, 4, -2, 2], [-1, 3, 1, 0], [-5, -2, 1, -3], [-4, 0, 4, 0], [-9, 1, -8, 1], [8, -7, 0, 1], [-3, 0, -5, 3], [-4, -1, 1, -4], [-7, 4, 1, 3], [-2, 0, 6, -2], [11, -4, 0, 0], [-3, 0, -2, 3], [-18, -2, 2, 0], [-11, -2, 4, 3], [1, 1, 0, -5], [11, -3, 9, -2], [-12, -2, -7, -1], [14, 3, -3, 0], [-22, -2, -6, 2], [-2, 0, -6, -2], [-12, 6, 0, -2], [5, 6, 5, 3], [-2, -3, 7, 0], [-22, 0, -4, 2], [9, 3, -1, 6], [19, 1, 3, 1], [18, -3, 1, 2], [20, 4, 0, 4], [-6, 7, 2, 2], [-16, -4, 2, 0], [5, -2, 6, 0], [5, 1, -1, -1], [2, 2, -2, 0], [9, 6, 3, 3], [8, -4, 8, 0], [-2, -5, -3, 0], [3, 0, 8, 1], [14, -3, -1, 0], [1, -6, 9, -1], [1, -3, 2, 3], [18, 3, 0, 2], [9, 2, 4, 2], [-11, -1, -1, 2], [-4, 2, -6, 4], [-16, -4, -10, 0], [3, 2, 3, -3], [-12, -2, 0, -4], [0, 6, 0, 2], [16, 4, -2, 2], [-12, -6, 0, 0], [14, 3, 8, 2], [-11, -1, 7, -1], [4, -1, 1, 0], [10, 2, -4, 2], [-2, 0, -1, -1], [-3, -2, -1, 3], [-21, 0, -6, 2], [-14, 3, 5, 0], [20, 4, 2, 0], [16, -1, -2, 1], [27, 0, 7, -3], [-5, -4, -1, -3], [14, 1, -6, -2], [-6, 3, 3, -6], [8, -2, 4, -2], [-2, 2, -5, 2], [30, -1, 2, -2], [-18, 4, -2, -2], [1, 0, 7, -5], [0, -6, 13, -5], [-9, -6, 1, -3], [4, 0, -8, -4], [-18, 1, 3, 0], [-6, -4, 12, -4], [24, 4, -1, 4], [-16, 1, 1, -2], [10, -6, 6, 0], [9, -5, 0, -1], [6, 5, -7, -2], [22, 0, 1, -2], [-14, -4, 6, -2], [-8, 4, -1, -4], [1, 2, 1, 3], [25, 2, -2, 0], [11, -2, -3, 5], [-6, 0, -4, -4], [35, 2, 4, -3], [-10, 1, 3, 4], [-9, 2, 3, 5], [-6, -4, 6, -2], [-23, 2, -2, -1], [-20, -5, -6, 1], [23, 3, -8, 5], [-8, 0, -14, 6], [15, 2, -10, 5], [4, 2, -2, 4], [-25, 4, 1, -3], [-32, -4, 2, -2], [29, 3, -6, -5], [-14, 0, -6, 6], [-3, -1, 4, -5], [4, 2, 10, 0], [24, 0, 1, -4], [-17, -2, 4, 1], [-1, 6, -9, 1], [-16, -6, -4, 6], [-9, 0, -7, -3], [2, 0, -10, -2], [-5, -2, -11, -3], [-2, 3, 3, 6], [6, -4, 0, -4], [14, 7, 2, -3], [15, 7, 7, 1], [25, 7, 10, 3], [-8, -5, 7, -6], [-39, -5, -10, -1], [12, 1, 4, 5], [29, 3, -7, 0], [16, 3, 6, 0], [-10, -2, -4, -6], [14, 6, -10, 2], [-12, 0, 9, 1], [-12, -2, -2, 4], [-6, 6, -6, 4], [-28, -4, -10, 0], [5, 9, -1, 4], [-1, 1, 8, -7], [20, 4, 1, 4], [-29, -5, 7, -3], [-10, -2, 2, 0], [-32, 0, -4, 4], [0, -3, -8, -7], [0, 2, 2, -8], [0, -7, 7, -4], [15, -9, 12, -3], [38, 2, -2, 4], [1, -5, 0, -5], [-4, -13, -6, -8], [15, 4, 8, -7], [8, -3, 18, -1], [6, 2, 6, 8], [4, 5, -10, -4], [2, -4, -4, 2], [23, 8, 5, 9], [13, 1, 5, 2], [4, -1, 15, -2], [2, 2, -4, -2], [-23, 8, -5, -1], [23, -9, 10, -3], [-32, -6, 2, -4], [-23, 0, 4, -5], [19, -2, 1, -3], [4, 8, -12, 0], [-3, 0, 8, -4], [16, -8, 9, -4], [-22, -3, 11, -8], [-23, -6, 7, 7], [-2, -6, 0, -6], [-2, -2, -8, 2], [30, 3, 15, -2], [-6, 1, -2, 2], [-13, 9, -3, 5], [10, -10, -4, 6], [-8, -4, -6, -6], [-26, -6, -8, 2], [25, 4, -5, 3], [-40, 3, -9, 2], [-10, 8, -12, -2], [1, 1, 9, -10], [10, -1, -10, 10], [20, 4, -15, 0], [34, -2, 15, -3], [-7, 0, 10, -9], [-6, 4, 5, -7], [-24, -5, -16, -1], [0, -3, -5, -4], [24, 8, -2, 0], [-14, 3, 4, 3], [29, 0, 3, 7], [20, -8, 6, -2], [13, -6, 5, -5], [21, 5, 4, -9], [-7, 4, 11, 3], [-1, 6, -6, 5], [-5, -6, -8, -2], [-1, -3, -13, 8], [-15, 3, 9, -8], [8, 7, 12, 0], [7, 2, -15, 9], [-48, -4, -1, -1], [7, 2, 3, 5], [-10, -9, 3, 2], [20, -12, 5, -4], [-10, -8, -8, 0], [27, -9, -2, 1], [-5, 1, 2, 1], [22, 8, 4, 0], [-42, 2, -13, 2], [11, 6, 12, 1], [27, 7, -1, -2], [-24, -7, 6, -9], [-21, -1, 2, -7], [-44, 3, -1, -2], [20, -1, 15, -6], [54, -3, 8, -2], [24, -1, -7, 12], [-16, -3, -7, 10], [6, -1, 8, 2], [-19, 0, -8, 4], [-39, 0, -9, -1], [35, 7, 2, 1], [20, 2, 4, 6], [-15, 8, -1, -5], [-8, -7, 5, 2], [-18, -2, -4, 6], [-30, -1, -1, -6], [-26, 7, -9, 2], [-30, 3, -3, -4], [16, 0, -15, -3], [-33, -4, -6, 1], [-2, 0, 0, 0], [0, 8, -14, 6], [-21, -2, -9, -3], [23, 6, 13, -3], [-8, 8, 2, 6], [-32, 2, 0, -8], [4, 5, 1, 10], [-23, 10, -12, 2], [-18, -4, 5, -2], [29, 2, 2, 4], [-27, -6, 7, -1], [36, 0, -3, 5], [-33, 2, 4, 1], [42, -4, -4, 2], [-15, -6, 15, -9], [-11, 4, -1, -9], [0, -4, 2, 8], [0, 0, -1, -1], [-1, -7, -7, 1], [9, 7, 7, -2], [15, 1, -4, 5], [-17, -8, 5, 1], [-11, -6, 12, -10], [8, 6, 8, -2], [19, 4, -11, 9], [-16, -2, 8, -10], [21, 0, -8, -1], [-2, -8, -8, -4], [-11, 4, 6, 2], [-4, -3, 6, -8], [30, -1, 13, 0], [0, -8, -1, 4], [7, -3, -7, -2], [-34, 10, 2, -2], [-31, 2, -13, 3], [0, 12, -4, 4], [-20, -15, 3, -12], [32, 4, 9, -4], [-4, 0, -8, 0], [-11, 6, 5, -9], [-13, -3, -3, -7], [2, -12, -2, -6], [-33, -6, 9, -11], [-34, 2, -4, 6], [-30, 11, -10, 1], [-5, 1, 9, -7], [-6, -1, 1, 12], [-6, -14, 0, -10], [0, 4, 8, -8], [22, 6, -10, -4], [-32, 1, -2, 7], [17, 9, -8, 3], [-14, 2, -2, 10], [-8, -14, 7, -7], [-46, 0, 4, -10], [0, -2, 5, -1], [-7, 8, 3, -1], [32, 9, 19, 4], [-8, 0, 0, 12], [-10, 10, 9, -1], [-4, -1, -4, 12], [-46, -7, -9, -4], [22, 6, -7, 11], [12, 6, 5, 7], [11, -10, 23, -3], [-13, -3, -12, 1], [-10, -7, -3, 6], [-59, -8, 5, -13], [-9, 7, -12, 5], [4, 5, 9, -14], [14, 13, -15, 10], [7, 4, -7, 9], [-11, -13, -4, 7], [28, -5, 0, -5], [-16, -8, -10, 2], [0, 10, -14, -4], [21, 1, 9, -10], [24, 12, 15, 7], [-23, 8, 2, -1], [31, 10, 13, 9], [-12, 6, 10, -12], [-35, -4, 13, -1], [29, -5, 10, -5], [30, -12, 0, -4], [-6, 1, 16, -10], [-37, 13, -2, 1], [-10, -7, -10, 6], [-30, 2, -4, 2], [-12, -3, 10, 11], [-44, -10, -9, -8], [28, -2, 8, -2], [-40, -3, 8, 1], [40, -1, -4, 4], [14, 7, 0, 11], [-7, 9, -12, -5], [-9, 12, -1, -7], [-6, -10, 1, -5], [19, 7, 13, -4], [15, -2, 25, -3], [13, 7, -10, -1], [12, -4, 14, -6], [-18, 9, -2, -2], [-22, 5, -7, 6], [-31, -3, -5, 7], [-38, -2, 10, 2], [-4, 15, 1, 0], [23, 2, 1, 13], [33, -10, 0, 2], [-27, 8, -15, 7], [40, -6, 4, -6], [44, 10, 18, -4], [48, -10, 10, -4], [-10, -16, -6, -10], [16, 10, 12, 10], [-8, 12, -7, 8], [5, 4, -4, -5], [10, 2, -4, 6], [17, 9, -4, 11], [-65, -2, -2, -4], [31, 9, 17, -6], [3, 6, -6, 9], [40, -1, 11, -6], [5, 2, -3, 11], [-26, 3, -15, 4], [35, 6, -1, 5], [-6, -8, -4, 8], [-35, 10, -10, -1], [12, 6, 3, -8], [27, -13, 1, 0], [1, -15, -7, 2], [51, 8, -11, 13], [3, 4, -4, 4], [-24, 2, 10, -16], [-16, -2, -6, -8], [-61, -6, -7, 1], [15, -11, 10, -3], [8, 4, 8, -12], [12, -12, 6, -2], [-4, -16, -7, -12], [-56, -8, 1, 0], [18, 0, -2, 6], [-25, 13, -8, 1], [6, 2, -6, -2], [22, 0, 6, 6], [54, 8, -10, 14], [-5, -13, 13, -7], [13, -8, 25, -9], [10, -6, 2, -8], [-33, -5, 2, -11], [13, 5, 4, 7], [-38, 2, -12, 6], [17, -16, 8, 0], [9, -14, 1, -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_bo();".
// 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_bo(: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_bo();".
// 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_bo( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, -6, -12, 2, 1]>,<11,R![-73, -86, -24, 2, 1]>,<13,R![48, -32, -36, 0, 1]>,<17,R![-64, -224, -24, 8, 1]>],Snew);
    return Vf;
end function;