
// Make newform 4730.2.a.v in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4730_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_4730_2_a_v();".
// 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_4730_2_a_v();".
// 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 := [-1, -2, 11, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [1, -5, 0, 1, 0], [-1, 6, -5, -1, 1], [-2, -6, 6, 1, -1]];
    Rf_basisdens := [1, 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_4730_a();"
function MakeCharacter_4730_a()
    N := 4730;
    order := 1;
    char_gens := [947, 431, 1981];
    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_4730_a_Hecke(Kf)
    return MakeCharacter_4730_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 := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-1, 0, 0, 0, 0], [-1, 0, 0, 0, -1], [1, 0, 0, 0, 0], [-2, -1, 1, 0, 0], [-1, 0, -2, -1, 1], [-2, 0, 1, -3, 0], [0, -3, -1, 0, 0], [-5, 2, 0, -1, 0], [0, -2, -3, 1, -1], [1, -2, 2, 1, 0], [0, 1, 3, 2, 0], [-1, 0, 0, 0, 0], [3, -1, 2, 3, 0], [-2, -1, -2, -3, 0], [-1, 0, 3, 1, 2], [1, -2, 0, 3, 0], [-5, 3, -2, 2, 3], [-6, 3, 2, 0, 2], [0, 4, 2, 0, 2], [-1, 0, -1, 3, 2], [-1, -2, -1, 5, 0], [-4, -1, 0, -1, -1], [-2, 6, 4, -4, 0], [0, -4, -4, -1, -5], [0, 1, -3, 2, 3], [8, -1, 3, 4, -1], [-3, 2, 1, 4, 0], [-2, -1, 1, 0, -5], [-3, 3, 4, 0, 1], [7, -2, -6, 3, 2], [-7, 4, 1, -6, -1], [-1, 1, 0, 4, 3], [-4, 3, 2, 7, -1], [-7, 2, -3, -4, -3], [3, 4, -2, 5, 4], [-4, 0, -5, -1, -4], [-1, -3, -1, 1, 2], [12, -4, -6, -2, -2], [-8, 4, 2, -6, -4], [-4, -4, -4, 0, -4], [-13, 4, 5, -8, 0], [1, -5, 1, 6, 3], [8, 4, 2, 0, 2], [-2, -4, 3, -3, -4], [-1, -6, -2, -2, 1], [-1, -2, -3, -4, 1], [2, 10, 4, -4, 4], [-2, 2, -2, 0, 2], [-5, 0, 0, -5, 0], [-8, -5, -3, -2, -1], [2, 1, 3, -3, 5], [0, -2, 2, 0, -7], [-10, 11, -1, -2, 7], [6, 0, 7, 3, 2], [0, -1, -5, -2, -6], [-2, 3, -5, -4, -2], [-1, -3, -4, -2, 2], [-4, 1, 3, -6, 0], [-2, -3, 1, -2, -1], [-6, 5, -7, 0, 0], [1, 3, -6, 5, 6], [1, -3, 5, -5, -6], [4, -11, 1, 4, -1], [-5, 1, -3, -6, -1], [-2, -4, 4, -8, 4], [-12, 7, -4, -5, -2], [-2, 0, 0, -6, 2], [-8, -2, -4, 0, -6], [-8, -3, 9, -4, 2], [2, 0, 4, 14, 0], [-2, 4, -6, 0, 5], [4, -1, 5, -7, -5], [5, 2, -3, 7, -2], [-3, -2, 3, -2, -5], [-8, 1, 1, 9, 1], [-6, 0, -2, -5, -9], [-12, 3, 2, -1, 2], [8, 7, -1, -2, -3], [-1, 0, -7, -8, -1], [4, 1, -1, 2, -1], [-1, 2, -5, -1, 2], [16, -4, 2, 8, -2], [8, -2, -8, 0, 9], [1, -3, -10, 6, -1], [0, 2, -8, 2, -2], [14, -7, -4, 11, -3], [14, 3, 8, 1, 6], [-14, -6, 2, -6, -2], [24, -4, -1, -5, -2], [-4, -6, 4, -6, -2], [-1, -2, 11, -1, -4], [6, -1, -4, -4, 4], [-27, 2, 7, -14, -5], [-6, 5, 7, 2, 3], [-2, 6, -2, 14, -2], [-6, -2, -2, 4, -2], [1, 2, 11, 2, -9], [-4, 8, -5, -12, -2], [5, -5, -12, 5, 2], [6, 0, -8, -10, -2], [10, 5, -5, -10, -4], [12, -1, -7, 6, 2], [15, -7, -17, 7, 3], [-5, -3, 4, 7, -4], [-5, 8, 6, -4, 5], [16, 4, -2, 4, -2], [4, -8, -1, -1, -5], [16, -13, -3, -7, -5], [-1, -5, -7, 7, 1], [2, -5, 5, 2, 4], [6, 1, -1, 4, -4], [-8, 20, 12, -12, 1], [-8, 5, 6, 4, 0], [0, -9, -2, -3, 9], [10, -2, 15, 1, -7], [11, 2, -1, 6, -7], [-2, 3, -4, 3, -9], [14, -4, 7, 11, -7], [-4, -12, -6, 8, -8], [-5, -6, 2, 5, 0], [2, -11, 1, -1, -1], [-15, -5, 9, -3, -4], [17, 2, 11, 2, 1], [-1, 4, -10, -1, -3], [-8, -9, -5, -2, 6], [-19, 3, 3, 13, 8], [0, 18, 2, -6, -4], [-19, 1, 0, -16, 4], [4, -8, 8, -5, -1], [6, -21, 4, 0, -4], [-17, 14, 10, -12, 1], [2, 8, 8, -6, 2], [-11, -4, 5, 6, -1], [-14, -4, 6, -2, -8], [15, -8, 6, 5, -2], [-20, -1, -1, -4, -5], [2, 4, 18, -3, -7], [-8, 15, 3, -2, 8], [18, -7, 16, 4, 0], [12, -2, -10, -9, 3], [-10, -5, -9, 0, -10], [20, -10, -2, 4, 4], [-5, -8, 1, 12, 7], [-14, 8, -6, -12, -2], [-32, 7, -1, -14, -4], [2, 6, -15, -16, -12], [6, 2, -4, 18, -2], [10, -4, 0, 8, -12], [-6, 6, 2, -8, 4], [24, -4, -8, -5, 1], [0, -12, -11, 11, 7], [6, 11, -3, 6, -5], [14, -8, -7, -1, -1], [18, -2, 1, 9, -4], [-17, -1, -8, -7, 4], [-13, 10, 18, 3, -2], [-3, 3, 12, 2, 5], [7, -12, 8, 17, 3], [9, -3, -9, -15, -10], [30, -5, -10, 19, 3], [-14, 2, -7, -7, -11], [6, 9, -3, 8, 15], [-12, -10, 12, 2, 6], [-11, -4, -7, -12, -1], [-10, 28, 6, -4, 10], [9, -1, 8, 13, 6], [-24, 8, 2, -12, 2], [2, -2, 11, -6, 8], [-9, 3, -13, -7, 1], [4, 9, 5, 3, -5], [13, 2, 5, 14, 15], [-23, 19, -1, -11, 12], [-28, 5, -3, -20, -5], [-12, -10, 11, -12, -12], [-16, 8, 8, 0, 4], [-9, -7, 12, 11, 8], [11, -3, 5, 23, 2], [-16, 19, 6, -17, 3], [18, -8, 10, -6, 6], [22, -12, 5, -3, 3], [-23, 6, 16, -9, -2], [-4, 9, -4, -3, 1], [-17, 2, -5, 6, 15], [-5, -3, -10, 7, 12], [-1, -3, -4, -10, -6], [8, 3, -5, 8, 5], [-10, -16, 0, 2, -4], [-6, 2, 2, -16, -12], [10, -3, -15, 6, 10], [13, 4, 19, -4, -7], [-9, 6, 11, 13, 6], [-9, 5, -8, 4, 10], [-5, 14, -7, 6, -5], [31, 9, 4, -5, 8], [10, -6, 0, 28, -2], [-7, -8, -1, 2, 0], [6, -4, 10, 3, 5], [2, -2, -3, -5, -4], [0, -10, 11, 1, -3], [-18, 12, 12, -18, -4], [0, -15, 0, -15, -3], [-4, 2, -4, 20, -2], [-7, -14, -9, -4, -1], [22, 7, -6, -3, 1], [22, 4, -2, 17, 1], [0, 12, -4, 8, 16], [-2, -7, 5, -3, 9], [-2, -5, 9, 2, 7], [-8, -3, 2, -19, -15], [-30, 10, -4, -28, 4], [-27, 15, 10, -21, 2], [24, -12, 1, 9, -13], [10, -14, 4, -4, -4], [27, 9, 3, 5, -6], [9, 7, -13, -5, -11], [15, -21, -4, 13, 0], [7, 14, 15, 2, 5], [3, 2, 12, 21, 9], [-24, -6, -4, 2, -18], [20, -18, -14, 20, 4], [-16, 1, -2, -7, -13], [-9, 21, -8, -3, 2], [7, -14, 3, -10, -19], [-15, 15, 1, -14, 3], [-14, 20, -6, 2, 8], [-30, -2, -7, 5, -8], [-34, 27, -7, -6, 13], [-2, 9, 19, 2, -1], [-16, 8, 14, 12, -6], [20, -10, -1, -5, 5], [-21, 13, -6, -17, -2], [24, -5, 3, 18, -9], [2, -18, -4, 6, 2], [11, 11, -5, -21, -12], [-20, 3, -22, -5, -7], [-10, 15, 13, -2, -6], [43, 0, -11, 22, 13], [16, -6, 17, 5, -15], [-12, 2, -9, -19, 2], [0, -21, -17, 2, -13], [-45, 21, -8, -11, 10], [-16, 29, 5, -8, 6], [24, -8, 14, 0, -10], [-5, -3, -16, 9, 0], [8, 18, 18, -4, -6], [-8, -4, 0, 0, 19], [1, 3, -13, 17, 4], [14, 12, 11, 7, -9], [0, 0, 18, -10, -8], [-14, 22, 0, 10, 18], [-2, 11, 19, -4, 15], [-2, 3, -8, 15, 13], [17, 11, 0, -13, -4], [2, -1, 12, -16, 4], [-8, 7, -13, 0, 11], [-9, 13, 10, 15, 2], [4, -16, -18, 0, -8], [20, -1, 20, 7, 6], [-23, 27, -2, 7, 18], [18, 4, 0, 24, 4], [-11, 7, -1, -15, -4], [4, -3, 5, -12, -8], [3, -3, 12, 7, 12], [16, -18, -2, -12, 2], [-22, -17, -9, 6, -5], [-19, -12, -14, 7, -7], [9, 28, -1, -14, -5], [-4, -24, -6, 10, 0], [12, -17, 11, 16, 8], [28, -19, 6, -3, -17], [-36, 13, -1, -1, 3], [-35, 0, 5, 2, -9], [-35, 5, -9, -12, -11], [-5, -8, -13, 16, 11], [13, 15, -4, 10, -11], [-4, -8, -10, 8, 6], [-30, 25, 7, -20, 6], [-21, 22, 8, -6, 9], [42, 4, 5, 7, 3], [-5, -9, -6, 4, 21], [-11, 17, -6, 6, -7], [-13, 6, -10, -9, -14], [7, 8, -7, 20, 9], [-13, -2, 14, 11, 10], [-12, -18, 0, -8, -6], [8, -7, -15, 22, -1], [-6, -16, 11, -11, 2], [-20, 0, -2, -22, -4], [-44, 8, -6, 6, 0], [-25, 13, 16, -14, 3], [9, -29, 11, -3, -8], [-3, -13, -13, 19, 5], [-30, -16, -2, -2, -2], [17, 11, -4, 11, 6], [-3, -11, -30, 0, 3], [3, -17, -15, 9, 14], [-17, 9, -16, 0, 9], [-11, 15, 20, -3, 6], [10, 10, 11, -14, 14], [-2, -18, -4, 2, -18], [-24, 0, 4, -14, -7], [12, -6, 6, 8, -2], [-8, -18, -6, 10, 6], [-11, 13, -10, -12, -2], [8, -8, 20, 4, 8], [-1, -13, 28, -10, 2], [-10, 4, -3, 7, 4], [-26, 11, -7, 6, 11], [-30, 22, -8, -17, 1], [-9, -11, -6, 15, -16], [-8, 10, 6, 10, -2], [-19, -2, 0, -6, -3], [8, 10, -11, -11, -3], [30, -24, 2, 14, -4], [-10, 29, 7, -6, 11], [18, 1, -9, -6, 8], [12, 5, -11, 12, 4], [1, -7, 26, 18, 8], [-15, 1, 19, -3, 12], [-13, -11, -30, 5, -10], [18, 6, 10, -8, -14], [19, 24, 9, -3, 2], [28, -11, 6, 13, -11], [-23, 8, 16, -11, 7], [-7, -16, -11, -2, 11], [5, -5, 12, -12, 5], [27, 1, -1, -17, 14], [10, 21, 1, 7, 15], [16, -29, -8, 11, -9], [-55, -2, 5, 10, 0], [27, -18, 7, 2, 3], [23, 4, -7, 20, -5], [6, -4, -11, -19, -7], [8, 16, 15, -21, -3], [-10, -5, -11, -5, -3], [29, -2, 24, 11, -20], [-34, 23, 10, -17, 3], [-3, -20, -22, 15, 0], [20, -16, -32, 13, -5], [44, -3, -3, 15, -1], [17, 8, 0, 5, 18], [3, -22, -8, -12, -23], [2, 2, -5, 12, -6], [23, -17, 3, 3, -22], [-3, -9, -2, 5, 12], [29, -17, 5, 23, -4], [-9, -16, 13, -8, 7], [-25, 5, -11, -18, -17], [-36, 28, 23, 3, 1], [-12, -17, -9, -9, -9], [-40, 11, 25, -14, -2], [-26, -5, -6, -11, -7], [0, -10, 7, -14, 0], [-23, 2, -6, 9, 18], [-17, 4, 0, -14, -9], [22, 1, 8, 22, 30], [-10, -12, 10, -25, 5], [-2, 9, -15, 4, 3], [-12, -8, -4, 6, 20], [5, -15, 6, 10, 4], [20, 6, -28, 16, 4], [-14, -3, 24, 8, 0], [39, -21, -7, 33, -6], [10, -1, -3, 3, -3], [14, -3, -31, 9, 1], [31, -3, -13, 7, 16], [-8, -13, 11, -6, -7], [-3, 8, -17, -20, 5], [1, 0, 9, 14, -9], [48, -6, 14, 0, -8], [-12, 3, 34, -3, 5], [-26, -14, -12, 0, 2], [0, -14, -16, 18, 14], [-21, -6, 1, -18, -31], [-1, 4, 13, 11, 18], [-28, -9, -10, 17, -10], [-19, -8, 9, -6, 7], [-21, 12, 26, -5, 6], [16, 1, 7, 0, 16], [-14, 3, -3, -34, -30], [47, 5, 16, 4, 19], [-8, 3, -8, -5, 17], [17, -25, -8, 36, 11], [1, -7, -26, -9, -10], [1, -9, 22, -13, -16], [-10, -24, -5, 16, 6], [27, 2, -5, 27, 10], [0, -11, -3, -6, 9], [-30, 10, 11, -6, -8], [13, 4, 18, 9, 8], [-46, -2, 14, -20, -12], [4, 6, -2, -20, 14], [35, -11, -26, 11, -2], [-24, 12, 4, 5, -11], [55, -23, 8, 25, -12], [-29, 10, -2, -9, -18], [3, -20, 17, 4, 5], [-26, 13, -14, 11, 9], [1, 34, 4, -3, -3], [-22, 6, -4, 0, 12], [-31, 11, 4, 16, -1], [14, -11, 25, -2, -16], [-2, -12, -16, -2, -2], [-36, 30, 14, -36, 4], [-39, 2, 0, -15, 12], [50, -30, -4, 4, -16], [35, -13, -7, -11, 5], [40, -4, -6, 2, -7], [-35, 2, 9, 28, 12], [-36, -3, 3, 11, -3], [15, 11, -16, -20, -11], [-55, 1, -2, -16, -3], [-19, -13, 16, -12, -25], [12, 23, -3, 6, 16], [-11, -4, -1, 27, 16], [6, 20, -20, 10, -3], [-40, 23, 19, 22, 14], [-2, 28, 21, -19, 10], [-55, 17, -6, -12, 6], [22, 0, -24, -10, -2], [0, -6, -4, -8, -30], [45, -11, 9, -1, -6], [-27, 0, -13, -8, -11], [12, -31, -12, 21, 1], [6, 5, -28, 13, 21], [-16, -21, -9, -10, 11], [-23, -19, 0, -17, 2], [-37, 14, 19, -28, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4730_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4730_2_a_v();".
// 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_4730_2_a_v(:prec:=5)
    chi := MakeCharacter_4730_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_4730_2_a_v();".
// 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_4730_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4730_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -2, 11, -5, -2, 1]>,<7,R![-19, -52, -33, 3, 6, 1]>,<13,R![8, -40, 30, 42, 12, 1]>],Snew);
    return Vf;
end function;