
// Make newform 4600.2.a.bc in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4600_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_4600_2_a_bc();".
// 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_4600_2_a_bc();".
// 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 := [2, 9, 7, -7, -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], [4, 7, -6, -2, 1], [7, 8, -7, -2, 1], [-11, -20, 12, 5, -2]];
    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_4600_a();"
function MakeCharacter_4600_a()
    N := 4600;
    order := 1;
    char_gens := [1151, 2301, 2577, 1201];
    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_4600_a_Hecke(Kf)
    return MakeCharacter_4600_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], [-1, 1, 0, 0, 0], [0, 0, 0, 0, 0], [1, 0, -1, 0, 1], [-1, -1, 1, -1, -1], [-1, 2, 0, 0, -1], [0, -1, 0, 1, 2], [1, -1, -1, 1, -1], [-1, 0, 0, 0, 0], [-2, -2, 0, 1, -1], [1, 0, 1, 1, 2], [0, 0, -2, -2, 0], [0, -1, -1, 2, 3], [-3, 0, 3, 0, -3], [-2, -2, 3, -2, -4], [-4, 2, -2, -2, -2], [6, -3, -1, 3, 1], [-6, 0, 0, 0, -2], [-2, 1, 0, 1, -2], [0, 0, 1, 0, 2], [0, 1, 0, 4, 5], [9, 2, -1, -2, 1], [-3, -3, -1, -1, 3], [0, 1, 4, -3, -4], [-2, -4, 4, -2, -2], [-2, -3, 1, -1, 1], [3, 0, -3, -4, 1], [-4, 1, 2, -1, 0], [-2, 4, -2, 0, -4], [-2, 1, 4, -7, -2], [7, -4, 3, -1, 6], [5, 4, -4, 3, 6], [-10, -1, 0, 1, 4], [8, -3, 1, 1, 2], [-8, 2, -2, 0, -2], [-4, 4, 1, -4, -6], [-8, -2, -2, 6, 0], [-9, -1, 1, 6, -2], [-6, -3, 1, -1, -5], [-13, -1, 0, 3, -6], [3, -3, 1, 0, -6], [-2, 0, -2, 2, 2], [-7, -2, 3, 0, -3], [-1, 3, -4, 2, -2], [-6, 10, -3, -1, -1], [3, 2, -1, 0, 1], [-4, 4, -1, -6, -1], [4, 3, -9, 5, 5], [-4, -4, -4, 4, 8], [-2, 4, -2, 4, 4], [7, -2, 0, 0, -1], [-1, 10, -2, 1, 0], [-16, -3, 6, 3, -8], [-4, 3, 2, 1, 4], [-13, 4, 4, -3, -4], [-2, -6, 2, -4, 2], [1, 2, 4, -4, 5], [4, 3, 1, 1, 7], [-6, -2, -3, -1, 3], [6, 2, 2, -8, -4], [-4, 1, 4, -5, -10], [-6, -4, 4, 2, 0], [-10, 9, 0, 1, 2], [-3, 2, -3, -1, -6], [6, -4, 2, 0, 8], [-9, -2, 5, -4, -9], [1, -3, 0, 6, 8], [0, -1, 4, -7, 4], [-6, 8, -1, -6, -5], [-15, 2, 4, 0, 1], [-2, 8, 1, 1, -9], [-3, 6, -3, -2, -5], [-7, 4, -5, 4, 7], [2, -6, 0, 0, -2], [2, 7, -4, -1, 4], [-19, 2, -3, 2, 1], [-4, 4, 4, -4, -4], [11, -6, -3, 1, 8], [0, -7, 0, 5, 2], [12, -5, 1, 0, 5], [3, -3, 11, -5, -1], [-4, 0, 2, -6, 4], [6, 2, -12, 8, 2], [6, -7, 4, -5, -6], [1, -2, 6, 3, 0], [-6, 3, -7, -1, -4], [-8, -2, 1, 8, 3], [-2, -7, 8, -7, -12], [-7, 4, 0, -2, -7], [-12, 3, 5, -3, -9], [-5, 0, 5, -4, -9], [1, 10, -5, 4, 1], [2, 6, -3, 2, -8], [12, 4, -1, -8, 6], [11, -4, -10, -1, 8], [-5, 6, 1, -6, -3], [-15, 2, -8, 5, 8], [-4, 13, -2, -3, 6], [9, -1, 11, -9, -7], [-10, -6, 5, -5, 7], [-4, 1, 9, 1, 2], [8, -14, 2, 4, 2], [-11, 6, -5, 6, 7], [-16, -1, 2, 5, -10], [20, 4, 0, 4, 4], [8, 9, -3, 2, 9], [2, 7, -3, 3, 0], [-9, 5, 7, 3, -1], [-10, 1, -1, -1, -3], [3, 5, -9, 6, 10], [24, -10, 0, 2, 4], [-10, 4, -4, 4, -2], [0, -6, 4, -10, -12], [0, -16, 6, 4, -4], [-7, 0, 1, 2, 1], [30, 2, -6, 0, 8], [7, 8, -5, 6, 5], [3, -8, -1, 9, 0], [-6, 0, -8, 5, 5], [5, -3, 3, 1, 5], [-2, -2, -8, 4, 10], [10, 8, -5, -5, 1], [26, -8, -6, 0, 8], [9, 9, -9, 8, 8], [10, 4, -13, 6, 3], [2, 0, 0, 2, 8], [12, 8, -6, 2, 18], [6, 8, -6, -8, 6], [6, 0, -10, 10, 4], [16, -10, 2, 4, -2], [7, 0, -1, 3, -10], [-21, 4, 3, 2, 3], [-5, -2, 3, -10, -3], [-6, 6, 8, -6, -4], [12, 7, -4, 2, -3], [-6, -3, -2, -5, 2], [-26, 10, -6, -6, -8], [23, -12, -1, 4, 7], [14, 0, -4, 4, 0], [3, 5, -4, 1, 10], [18, 8, 3, -4, 0], [13, -7, -4, -3, -6], [-5, -2, 6, 1, -12], [-11, 11, -11, -1, 7], [-9, 3, 4, 7, -2], [-13, 8, -7, -4, 5], [-23, 10, 3, 0, -3], [-7, -9, 1, -2, 2], [15, -7, -7, 2, -8], [-15, 8, 6, -3, -12], [26, -7, -1, 3, -5], [26, -2, 8, -2, -6], [-10, -12, 3, 2, -5], [-1, 4, -10, 3, 16], [-3, 10, 9, -12, -15], [-13, 0, 7, -2, -9], [-8, -2, 6, 10, -2], [-10, 4, -8, 5, 1], [4, -3, -4, -1, -2], [-28, 18, 2, -6, -6], [29, -8, 1, 1, 10], [-6, 3, 0, 1, 0], [2, -12, 5, -8, 0], [5, -7, 9, 1, -5], [-4, -9, -6, 9, -2], [7, -8, 1, 8, 1], [-8, 1, -9, 11, 5], [19, 1, 4, 9, 10], [-12, 1, 4, -9, -12], [-32, 0, 9, 8, -14], [1, -13, -3, -5, -1], [-11, -4, -4, 2, -3], [25, -8, -5, -6, 17], [12, -19, 6, 1, 0], [-7, -2, 6, -9, -20], [-16, 3, -8, 5, 4], [13, 3, -5, 6, -8], [18, -2, -9, -3, 9], [-1, 6, 7, -8, -25], [-12, 2, -4, -2, 6], [-13, 4, -4, -7, -2], [-21, 1, -1, 7, 7], [-10, 4, 1, 9, 7], [-3, -14, -4, 2, 20], [-15, 8, -1, -2, -9], [-14, 14, -4, -10, -2], [-2, 11, -3, 1, -7], [15, 1, -3, -5, -5], [3, -11, -8, 19, 11], [18, 6, -14, 2, 14], [8, -7, 0, 6, 15], [-12, -4, 3, -6, 3], [-15, 1, 3, 5, -19], [30, 3, 5, -3, -3], [11, 19, -5, -5, 3], [12, -10, 2, -4, -2], [4, -17, 4, -1, -6], [-8, 12, 2, -6, -12], [-20, 3, 4, 7, 6], [-23, 6, -6, 7, 0], [6, -10, -4, 0, 20], [14, 21, -11, -1, 11], [14, 14, 1, 11, -1], [-14, 15, 8, -11, 2], [-23, -8, 11, -16, 1], [14, 10, 4, -19, -1], [4, 10, 8, -14, -18], [20, 0, -7, -8, 2], [7, 10, -4, 0, -9], [1, -8, 5, -1, 8], [18, -12, 4, 13, 5], [3, 10, -4, -12, -5], [-9, -8, -1, 0, 15], [-27, 1, 3, -1, -11], [0, 3, -13, 9, 17], [10, -7, -2, -13, -14], [1, -2, -4, 9, -6], [-9, -5, -5, -12, -10], [-5, 8, -1, 14, 21], [-4, -10, -7, 18, 6], [-6, 2, 16, 3, -11], [-9, 0, 3, -16, -5], [2, 13, -5, 13, 12], [39, -18, 3, 10, 7], [-1, 9, 1, 2, -8], [-18, 26, 3, -11, -3], [-2, -5, -12, 13, 2], [21, -12, 2, -7, 0], [6, -2, 10, -16, 10], [-14, 16, 9, -18, -7], [-2, 2, -4, -2, -12], [4, 9, 4, -5, -2], [31, 4, 7, -14, -5], [9, -18, -8, 6, 8], [20, -8, -18, 12, 8], [-15, 20, 5, 2, -11], [26, 11, -8, 17, 12], [-12, 6, -4, -4, 4], [-31, -5, -2, 14, 6], [27, -8, 4, 5, 6], [3, -17, 3, -1, 15], [-13, -13, 24, -14, -8], [28, 18, -11, 6, 10], [-1, -2, -7, 9, -12], [-32, 16, -3, -10, -5], [-16, -16, 1, 0, 2], [3, -20, -6, 3, 8], [4, -20, -4, 0, 12], [-5, -11, -3, -1, 1], [-7, -6, -7, 0, -3], [48, 2, -4, 8, -2], [24, -7, -2, 5, 12], [-15, 8, -5, 0, 5], [38, -21, -2, 3, 0], [-7, -13, -8, 3, 6], [-48, -9, 1, 3, 2], [-22, -4, -6, 0, -2], [3, -7, 7, -1, 9], [6, 10, -20, 13, -1], [-2, -3, 3, -15, 0], [24, 0, 0, -8, 16], [5, -33, 5, 5, 1], [-34, 15, -5, -7, -1], [-8, -8, 12, -6, 6], [-16, 7, -12, -1, 6], [23, -7, 5, -1, 19], [-20, 26, -4, -2, -4], [32, 0, -2, 12, -2], [31, -3, 3, -3, -7], [18, 2, -3, -9, 19], [-26, -14, 20, -3, -21], [-22, 15, 10, -13, 0], [20, 9, -4, 15, 18], [-9, -4, 1, 22, 3], [17, -10, -8, 4, -7], [15, 2, -5, -5, -12], [4, -4, -3, 10, 21], [25, -29, 0, 11, 2], [-4, 6, -16, -2, 0], [22, -1, 4, -17, -6], [23, -6, 11, -3, 10], [-10, 8, -2, -2, 14], [-11, -18, 9, -8, 11], [-22, -4, -4, 0, 18], [10, -18, 19, -8, 7], [2, 12, 2, 0, -26], [-30, 0, 8, 0, -4], [0, -8, 20, 0, 2], [16, 9, -25, 3, 25], [19, 1, 8, -15, 2], [-30, 0, -6, -6, -4], [-7, -2, -7, -4, -19], [-19, -22, 12, -2, -16], [-2, 5, 4, 1, -10], [13, -8, -5, -11, 0], [24, -18, 6, -6, 8], [-7, -16, 13, -12, -17], [-26, 4, -4, -3, 1], [-23, -12, 13, 7, -4], [-1, -1, 5, 10, -2], [-46, -3, 4, -9, -2], [34, -14, -16, 4, 14], [1, -20, -1, 0, 1], [-26, -3, 3, 13, 0], [-18, -7, 2, 1, 2], [-28, 3, 7, -10, 3], [8, -16, -8, -4, 8], [-20, 0, 0, 20, -2], [-26, 4, 4, -6, -16], [13, 4, -1, 2, 9], [-6, 2, -1, -2, 0], [30, -10, 2, 14, 14], [12, -27, 6, 13, 6], [30, 12, -3, 16, 4], [-16, 16, 0, -4, -18], [24, -8, 20, -6, -12], [-9, 17, -15, 8, -6], [-16, 8, -3, 0, 2], [10, 17, -12, -5, 8], [-19, 5, 0, 2, -26], [-44, 14, -6, 2, -6], [8, -11, -6, -5, 8], [-42, 20, 7, -5, -11], [-42, 15, -5, 1, 9], [0, -4, 6, 6, -16], [60, -14, 2, 2, 12], [0, 9, 0, 7, 2], [3, -16, 23, -9, -2], [-45, 16, 9, -2, 11], [-24, -14, 10, -14, -4], [-20, -2, -13, 14, 22], [-41, 8, 4, 6, -1], [60, -2, 3, -2, -6], [13, -19, -9, 8, 8], [-20, -5, 18, 5, -14], [-22, -16, 5, 16, -4], [20, -18, -8, 0, 14], [-6, 2, 0, -8, -4], [-4, -7, 16, -10, 15], [-28, 12, -12, 0, 2], [31, -14, 9, 5, 0], [-8, -15, 6, -1, -16], [-15, 6, 11, -12, -31], [14, -12, -6, -10, 16], [-20, 2, 6, -6, 0], [6, 0, 6, 8, 6], [-52, 2, 13, -14, -6], [-62, -4, 5, 2, -11], [-35, -6, -1, -8, -5], [-14, -16, 12, -6, 6], [35, 3, 4, -10, 4], [4, 2, -9, 6, 18], [7, -8, -7, -2, -25], [-22, -6, 10, -16, -6], [8, -8, -4, 0, 4], [49, -7, 5, -5, 21], [23, -20, -7, -2, 1], [1, -5, 5, -11, -21], [-6, -16, 16, 3, -5], [-26, 31, -7, -3, -9], [-46, -4, -2, 2, -16], [-13, -21, 0, -7, -1], [27, 4, -12, 9, 18], [21, -6, -7, -8, 1], [-8, -5, -3, -13, 6], [10, 19, -21, 7, 32], [-45, 0, -7, 3, -8], [-4, 8, 10, -8, -24], [4, 25, -4, -9, -4], [41, -20, -2, 13, 8], [-22, 4, -5, 6, 27], [-59, 10, -8, -15, -10], [-12, 16, -4, -14, -10], [7, -10, -7, 24, 7], [14, -4, 8, 13, -7], [-10, -11, 15, -11, -32], [-23, -22, 19, -22, 3], [-8, 9, -4, 13, -4], [-22, -17, 4, 23, -2], [14, -12, 16, 1, -11], [46, 6, -10, 2, 20], [-16, -11, -10, 25, 6], [-26, 3, -13, -3, 1], [-59, 9, 8, 2, -2], [37, -22, 11, 4, 21], [-29, -8, -13, 19, 10], [1, -18, 21, -15, -8], [-14, -21, 15, -7, -17], [12, -4, 14, -16, -40], [-6, 0, 14, -14, -2], [36, -9, -24, 9, 16], [27, -1, -12, -21, 10], [-10, -5, 13, -5, -8], [-22, -9, 8, 7, -26], [8, -8, 16, -16, -12], [4, -9, 6, 1, -18], [37, 6, -1, 8, 7], [-26, 11, 12, 7, -14], [31, -11, -1, 21, 9], [-14, 0, 16, -16, 0], [-22, 6, -17, -3, 9], [2, -3, 15, -3, -20], [17, 4, -17, 4, 31], [-34, -19, 18, -5, -10], [40, 0, -7, 8, -10], [37, -4, -3, -3, -10], [50, -2, -12, 0, 8], [-11, 8, -4, -13, -20], [11, -6, -9, -14, -7], [22, -14, -9, 12, -5], [-26, -5, 5, 3, 0], [-25, 16, 12, -2, -5], [-3, -17, 3, 9, 17], [19, 20, -1, -6, 1], [42, 13, -8, 13, 6], [-2, -2, 8, 13, 23], [14, 2, 2, -6, 14], [-6, 5, -17, 11, 25], [-6, 3, -4, -5, 14], [-9, 2, -15, 10, 31], [40, -20, 12, 4, -4], [22, 0, 6, -8, -8], [-2, 9, -17, 19, 9], [-28, 14, 9, -8, -19], [-29, -19, 24, -5, -21], [3, 2, -8, -4, 23], [9, 6, 3, -4, 25], [3, -4, 0, 0, 28], [-3, -27, 14, 0, -2], [-21, 30, 7, -17, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4600_2_a_bc();".
// 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_4600_2_a_bc(:prec:=5)
    chi := MakeCharacter_4600_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_4600_2_a_bc();".
// 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_4600_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![10, -1, -16, -5, 3, 1]>,<7,R![-32, 56, 12, -16, -1, 1]>,<11,R![-8, -92, -88, -17, 4, 1]>],Snew);
    return Vf;
end function;