
// Make newform 5760.2.a.bx in Magma, downloaded from the LMFDB on 12 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5760_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_5760_2_a_bx();".
// 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_5760_2_a_bx();".
// 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 := [-4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [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_5760_a();"
function MakeCharacter_5760_a()
    N := 5760;
    order := 1;
    char_gens := [2431, 901, 641, 3457];
    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_5760_a_Hecke( : Kf := Kf)
    return MakeCharacter_5760_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]];
    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], [-1, 0], [-1, -1], [2, 0], [1, 1], [-3, 1], [-1, -1], [-1, -1], [0, 2], [3, 1], [-1, -1], [2, 0], [-2, 2], [-9, -1], [-10, 0], [6, 0], [-2, 0], [-2, 2], [-8, 0], [4, -2], [9, -1], [-4, 0], [10, 0], [10, 0], [-14, 0], [-1, 3], [-6, 2], [-4, 2], [-11, 1], [11, -1], [0, -2], [7, -1], [-3, -3], [-12, -2], [-5, -3], [-7, -3], [8, -4], [-15, 1], [-14, 0], [-4, -2], [8, -2], [-4, 0], [16, 2], [-2, 0], [-1, 5], [-15, 1], [-19, 1], [0, 0], [-10, 0], [-5, -5], [2, -2], [-16, 2], [4, -2], [-3, -7], [3, 3], [0, 2], [3, 1], [-19, 1], [8, -2], [-22, -2], [4, 6], [-6, -2], [-22, -2], [-4, 6], [-2, 0], [-3, 1], [0, -6], [2, 2], [-6, 4], [-1, 7], [4, 0], [5, -3], [13, 1], [-9, 3], [1, 1], [6, -4], [-3, -3], [20, -2], [2, 4], [-8, 2], [-2, 8], [0, 4], [-8, 2], [15, -3], [-20, 4], [14, 0], [-10, 4], [32, 2], [-33, -1], [6, -6], [20, 4], [-7, 5], [-18, 4], [17, -3], [19, -5], [-14, 0], [-10, 0], [6, -6], [0, -2], [18, -2], [-18, 4], [2, -6], [0, -6], [-23, 5], [10, 8], [-22, -6], [-11, -7], [-8, -4], [2, -8], [9, 1], [5, -3], [11, -5], [17, 1], [-15, 7], [28, 2], [16, 4], [9, 1], [-20, -6], [26, 0], [-2, 0], [-12, -2], [20, -6], [-18, 2], [9, 1], [18, 0], [-18, -8], [14, 2], [9, 5], [17, 5], [29, 1], [-9, 7], [15, -3], [-7, 5], [-42, 0], [40, -2], [2, -12], [-30, -2], [-20, 2], [4, 2], [13, 1], [0, -6], [-33, 3], [40, -4], [4, -6], [44, 0], [17, 1], [-35, 1], [-21, -1], [-35, -3], [33, 5], [-2, 4], [-16, 4], [-27, 5], [4, -8], [34, -6], [35, 1], [2, -4], [-14, 0], [6, -12], [-2, -6], [-41, -1], [-39, -3], [24, 2], [-35, 1], [35, 3], [-11, -1], [15, -5], [-12, 2], [16, 6], [26, 4], [0, -2], [-24, -8], [-40, 2], [33, -1], [-18, -4], [-1, -9], [8, 2], [45, -3], [28, 2], [35, 7], [-38, -4], [-13, -1], [9, 5], [17, 9], [10, 0], [15, -9], [-12, 0], [42, -4], [-4, -8], [8, -6], [-30, 6], [-11, 13], [-20, 6], [-24, 0], [-7, 5], [-34, 0], [25, -7], [-3, 1], [3, -13], [6, -12], [5, -1], [-11, 5], [40, -6], [24, -2], [-12, -2], [15, -7], [42, 2], [30, -4], [-5, 3], [-12, 6], [-24, -6], [7, 11], [4, 8], [-2, -2], [-24, -10], [-23, 1], [-12, -2], [29, 5], [36, 2], [22, 8], [27, 9], [14, 12], [-21, 3], [-26, -6], [-24, -2], [5, 1], [10, -10], [51, -5], [20, 6], [-27, -3], [-19, -7], [-17, -3], [12, -6], [-10, 2], [7, -9], [4, -2], [-16, -10], [-16, -10], [0, 16], [30, -2], [53, 1], [23, -5], [-6, -4], [15, 7], [-26, 6], [-23, -7], [18, -4], [27, -9], [-51, -3], [-17, -5], [-18, 0], [13, 5], [-16, 10], [52, -2], [-54, 0], [0, -2], [40, 4], [-32, -6], [-36, -2], [5, 1], [-18, 2], [6, 8], [-49, 7], [-21, -5], [21, -7], [46, -4], [44, 6], [-38, 6], [30, -4], [-4, -6], [18, -2], [-48, -6], [-47, 7], [4, -2], [3, -1], [22, 6], [-42, -8], [24, 2], [-24, 6], [-39, 1], [-3, 11], [13, -3], [36, 2], [8, 4], [0, 12], [-42, -4], [20, -6], [53, -1], [36, -10], [-26, -12], [-44, -4], [27, 11], [-4, -2], [11, 11], [-24, 10], [-23, 11], [4, -10], [24, -2], [14, 10], [16, 10], [-18, 4], [-5, -19], [2, -14], [-33, -9], [26, -8], [-16, 4], [-62, -4], [-46, 10], [13, 17], [-27, 5], [-36, 6], [32, 2], [-52, 0], [17, 9], [12, -6], [-36, -2], [-10, -2], [28, -2], [24, 2], [-13, -9], [24, 2], [2, 0], [-35, 9], [37, 9], [10, -8], [-31, -11], [42, -10], [-41, 7], [56, -2], [18, -12], [30, 4], [31, 1], [70, -2], [35, -5], [4, 16], [-40, 6], [39, 7], [14, -4], [23, -5], [45, 1], [-23, -3], [-26, 4], [21, 11], [-64, 2], [-2, -4], [50, -4], [-40, 12], [22, -10], [54, 8], [-21, 7], [2, -16], [-2, 12], [-11, 9], [-54, -4], [51, -5], [88, 0], [32, -6], [27, 3], [19, 11], [-21, -17], [-16, 2], [19, 3], [-48, -2], [-38, -2], [26, 8], [18, -4], [25, 1], [-38, -12], [36, 2], [-21, 15], [-67, -3], [-48, 10], [-9, -15], [27, -1], [34, 8], [-6, 10], [-6, 8], [34, 12], [-26, 12], [-52, -2], [27, 3], [-33, -9], [-21, 3], [-31, -11], [-41, 15], [15, -11], [93, 1], [-46, -2], [-37, -5], [-36, -6], [-6, 0], [10, -12], [-30, -2], [-36, 4], [14, -4], [-43, 11], [-32, 6], [15, 7], [6, -4], [72, 6], [31, -9], [-29, 7], [-37, -5], [-16, -6], [61, 3], [-5, 15], [4, -2], [-60, 0], [-24, -6], [-18, -4], [-6, -16], [6, 18], [-35, 13], [48, 2], [-12, -10], [-52, 4], [7, 7], [31, -9], [19, 19], [4, -10], [19, 7], [-33, -1], [-6, 12], [32, -6], [-56, -6], [-4, -8], [2, 20], [-13, 11], [-64, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5760_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5760_2_a_bx();".
// 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_5760_2_a_bx(:prec:=2)
    chi := MakeCharacter_5760_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_5760_2_a_bx();".
// 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_5760_2_a_bx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5760_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, 2, 1]>,<11,R![-2, 1]>,<13,R![-16, -2, 1]>,<17,R![-8, 6, 1]>,<29,R![-68, 0, 1]>],Snew);
    return Vf;
end function;