
// Make newform 4508.2.a.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4508_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_4508_2_a_f();".
// 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_4508_2_a_f();".
// 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 := [10, 29, 10, -10, -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, -1, 1, 0, 0], [-5, -7, 0, 1, 0], [9, 9, -7, -2, 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_4508_a();"
function MakeCharacter_4508_a()
    N := 4508;
    order := 1;
    char_gens := [2255, 1473, 1569];
    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_4508_a_Hecke(Kf)
    return MakeCharacter_4508_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], [-1, 0, 1, -1, 0], [0, 0, 0, 0, 0], [1, -1, 0, 0, 1], [-2, 0, -1, 0, 1], [-1, -1, 1, 0, -1], [-2, 0, 0, 2, 0], [-1, 0, 0, 0, 0], [3, 1, -1, 2, 0], [1, 0, -2, -1, -1], [-2, 2, 0, -2, 0], [0, -2, 1, 0, -1], [-2, 2, 0, 2, 0], [-1, 2, -2, -1, 1], [-2, 2, -2, -2, 0], [-1, -1, -1, 2, 1], [-3, -2, -1, -1, 0], [-1, -1, -2, 2, -1], [2, 2, -3, 0, -1], [2, -2, 1, -2, 1], [-1, -3, 2, 0, 1], [6, 0, -2, -2, 0], [-5, -1, -1, 4, -1], [-3, 2, 3, -1, 0], [6, -4, -2, 0, 0], [-4, 2, 2, -2, 0], [-6, 2, 0, 2, 0], [0, -2, 4, 0, 0], [-2, 0, -2, 0, -2], [-6, 0, 1, -2, -1], [3, -2, 2, 3, 1], [-6, -2, 2, -2, -2], [1, -1, -2, 4, 0], [-2, -2, -2, 2, 0], [-2, 2, -1, 0, -1], [-3, 1, -1, 4, -3], [0, -4, 3, 4, -1], [-5, 0, 1, 1, -2], [4, 0, 0, 2, 0], [2, -2, 5, 0, 1], [-9, -1, 1, -4, 1], [-10, 6, 0, -2, 0], [0, 2, 7, -2, -1], [4, 4, -1, -2, 3], [0, 0, 2, -4, -2], [-8, 8, 2, 0, 0], [1, 1, -1, -2, 3], [4, -2, 0, 6, 2], [-13, 3, -1, -4, -3], [7, -3, -3, -2, -2], [2, 2, 3, 0, -1], [-9, -1, -1, 0, 1], [-14, -2, 2, 0, 0], [4, 0, 5, -2, 1], [-5, -1, 0, -6, 1], [2, 2, -1, 2, 1], [-9, -1, -3, 2, 1], [-1, -1, -5, 4, -2], [-4, 2, 2, 0, -2], [-12, -2, 6, -2, 0], [3, 6, 1, 1, -4], [-13, 0, 1, -3, -2], [-1, 5, 0, 0, 4], [3, -5, -3, 0, 1], [5, -1, -4, 2, -1], [24, 2, -3, 2, 1], [12, -2, -2, -4, 2], [12, 0, 6, -4, 0], [8, 6, -1, 0, 1], [8, 0, -7, 6, -1], [-5, -1, -6, 2, -1], [-8, 8, 2, -4, 0], [-6, -6, 4, 2, -2], [7, 3, -4, -2, 1], [0, -4, 4, 0, -2], [4, -2, 0, -4, 0], [-10, -2, 3, 2, -1], [0, 0, -2, 6, 4], [4, -6, 5, 4, 1], [-2, -2, 4, 8, -2], [18, 2, -6, 2, 0], [-4, 8, -2, 4, 2], [9, 4, 5, -3, -2], [7, -3, 4, 4, 6], [-8, 0, 1, 0, -3], [11, -9, 4, 4, -1], [8, 2, 2, 0, 0], [14, 2, -1, 2, 1], [-6, -4, 6, -2, -4], [-26, 4, -2, 2, 2], [18, -4, 2, 2, 2], [-18, -6, 7, -4, 1], [6, 6, -1, 8, -3], [-4, -6, 3, 2, -1], [2, -2, 0, 0, -2], [0, 6, -5, -4, -3], [1, 6, 1, -1, -2], [8, 4, 2, 4, -2], [3, -9, 3, 0, -2], [-10, 8, -3, 6, 3], [-18, 2, 0, -6, -2], [-4, 8, -2, -4, 2], [6, -8, 0, 0, 2], [2, -6, 2, -2, -2], [-12, 0, -3, -6, 3], [-7, -5, 4, -8, -2], [0, 2, 0, 0, -2], [8, -2, -2, 6, 2], [12, -4, 1, 6, -1], [1, 6, -3, -3, -4], [-4, -4, 6, 2, -4], [-2, 4, -4, -8, 2], [-10, -2, -4, 8, -4], [-7, 1, -2, 2, 1], [6, -12, 0, 0, 2], [8, 6, -8, 6, 6], [-19, -6, 8, -7, -3], [4, 0, -7, 6, 5], [8, -8, 6, 4, -2], [-27, 0, -1, 1, -4], [-11, 13, 1, 4, -2], [9, 2, -3, -1, -6], [4, -2, 11, -2, 1], [-9, -4, 9, -11, 2], [-18, 4, -2, -8, 0], [24, -4, -4, 2, 2], [7, -2, 3, -1, 0], [0, -6, 0, -6, 2], [-9, 4, 5, 3, 4], [-2, 2, -1, 0, -3], [5, -7, 2, -2, 1], [-1, -5, 6, -14, -1], [-18, -4, -2, 0, 0], [-18, 4, -3, 4, -1], [-13, -1, -1, -4, 3], [9, 5, -9, 4, -1], [8, -12, 2, 4, 0], [-1, -1, -5, 12, -1], [10, 2, 4, 0, 4], [-29, 6, 2, -5, 1], [9, 5, -4, 4, 1], [-4, 0, -3, 0, -3], [-16, 8, 0, 4, 0], [-22, 0, 4, -4, 0], [8, -6, 0, 2, -4], [8, 0, 6, 10, -4], [16, -2, 1, 4, -1], [9, -13, 2, 4, 2], [6, 4, 3, 2, 3], [-3, 13, 4, 0, 1], [-3, -7, -9, 0, 3], [0, 0, 2, -4, 4], [-17, 9, 2, 0, 4], [-8, 4, -8, 8, 4], [-1, -11, -8, 0, 3], [-9, -5, 4, -6, -3], [8, 2, 5, 4, 3], [5, 9, -1, -8, 1], [-21, 3, 7, 4, 1], [0, -4, 3, -12, 1], [-4, 12, 8, 2, 0], [-12, -12, -1, 4, -5], [8, 6, -10, 10, 6], [8, 0, 4, 2, -6], [-28, -4, 6, 0, 0], [-20, 0, 8, 0, 4], [-12, 4, 4, -2, 4], [8, -2, -18, 8, 0], [8, -4, -1, -10, -3], [10, 6, 0, -6, 4], [-4, 8, -1, 14, -1], [12, -4, 4, 0, 4], [-12, 0, 10, 2, 0], [7, -8, 6, -7, -7], [-1, 11, -3, -4, 5], [-14, 0, -1, -6, -5], [12, 4, -3, -10, -3], [6, 0, 4, -6, 4], [7, -10, -1, -3, 10], [-10, 8, -7, 2, 7], [16, 10, -4, 10, -6], [-12, 2, -5, -10, 3], [12, -12, -2, -2, 4], [-2, 2, -6, -2, -2], [19, 6, -5, 1, 6], [-21, 9, 0, -6, -3], [-16, 0, -4, 0, 6], [41, -9, -5, 2, 2], [-21, -1, 9, 6, 1], [10, 0, -11, 0, -3], [-20, 6, 0, -6, -2], [8, -4, 12, -8, -4], [12, -6, 2, 8, -2], [5, -7, -6, 2, 1], [11, 4, -3, -5, -8], [-4, 10, -8, 8, 4], [-6, -6, 2, -10, 6], [9, -4, -3, 5, 2], [13, 7, -8, 0, 0], [-2, 16, 12, -4, 0], [19, -2, -9, 11, 8], [26, -6, -1, -6, -3], [-11, 10, -1, -9, 0], [-12, 12, -6, 4, 6], [18, 6, -9, 2, 1], [20, -12, -6, 4, 0], [2, -18, -1, -4, 7], [-17, -13, 5, 4, -8], [-17, -4, -6, 1, -3], [-15, 13, 1, 0, 6], [24, 10, -5, 4, -5], [-9, -15, -4, -4, 3], [10, 6, 0, 4, 6], [-25, 2, 7, 3, 4], [-17, 3, -5, -4, 1], [-18, 8, -1, -2, -3], [26, 2, -7, -6, 1], [-22, 6, 8, -10, 0], [12, 8, -7, -2, -3], [4, 6, -3, 6, -7], [42, -8, 0, 6, 4], [-28, -6, 7, -6, -1], [-14, -10, 6, -2, 6], [-15, -9, -8, 12, -6], [-15, -9, 5, -2, 8], [-9, 0, 7, -5, 2], [2, -2, -7, 12, 5], [30, -12, -10, 6, 4], [-12, 4, -12, -8, 0], [-18, 20, 12, 0, 0], [0, -6, 2, 2, 4], [-7, -1, -12, -4, 1], [-9, 9, -8, 2, 9], [4, 2, 6, 6, -8], [-8, 18, 8, -6, 4], [-9, 3, 3, 20, -1], [8, 8, 4, 2, 8], [-24, 12, -5, 4, 5], [5, 8, 0, 3, 1], [6, -6, -2, 6, 6], [-13, 10, -2, 7, -7], [23, 0, -8, 9, 3], [32, -8, 1, 6, 1], [-14, 0, 4, -8, -4], [27, -12, -4, -7, -1], [9, -1, 4, 8, -2], [18, 4, -16, -2, -2], [12, -16, 8, 0, -4], [0, -12, -2, 4, 2], [-28, 2, 4, 0, -2], [7, -8, -1, -5, -2], [31, -7, 10, 4, -7], [-3, -7, 11, 16, 3], [-34, 8, -2, -4, -8], [-6, -2, 9, -12, -7], [-14, -4, 8, 0, 2], [-18, -20, 10, -2, 2], [-44, 6, -5, 0, 1], [14, 2, 1, 14, -3], [-11, -11, 0, 2, 3], [-28, 0, -2, 8, -6], [-22, -4, -12, 8, -8], [1, -6, 9, -17, -2], [14, 10, 0, -2, 2], [-18, 4, -10, 2, 2], [0, -6, -12, 0, 6], [-21, -8, -1, -1, 2], [-1, -15, 10, -8, 1], [12, -20, 3, -6, -1], [-30, 0, 3, 0, -9], [3, 5, 8, 0, -1], [7, 7, -1, 8, -3], [19, 1, -12, 8, -1], [24, -6, -3, 4, -3], [-21, 2, -10, 3, 1], [-20, 2, 4, -2, 2], [-5, 13, -8, -2, -7], [10, 12, -4, 0, 8], [-24, 8, 0, 0, -8], [35, -11, -12, 8, 2], [-2, -6, -8, 6, 0], [-12, -10, -14, -2, 2], [13, 1, -11, -4, 7], [5, -3, 7, -14, -5], [42, 2, -2, -8, 2], [10, 2, 6, -6, -8], [10, -6, 4, -14, -10], [-1, 15, -9, 6, -7], [-4, 8, -6, -2, 0], [-14, 4, 4, 0, 0], [-12, -8, 14, -4, 6], [13, 15, -4, -2, -1], [28, 8, 12, -4, -2], [-8, -12, -1, -6, 11], [-15, -15, -1, -8, 7], [-14, -14, 0, 4, -6], [-14, -4, 15, -8, 5], [-35, -9, -2, -8, 8], [-9, -7, 8, -20, 2], [-7, -7, -3, 12, -3], [10, 6, -6, 6, 14], [-4, -4, 2, -24, 2], [0, 16, 5, 8, -9], [20, -12, -6, 12, 2], [16, -10, 17, 2, -3], [-10, -12, 12, -10, 0], [-19, 0, 1, 1, 2], [-4, -12, 10, 0, -6], [-6, 2, -8, 12, 0], [-13, 2, -2, 3, 5], [-18, -10, 8, -14, -4], [-36, 12, -8, -14, -2], [-2, 0, 3, -2, -5], [27, 9, 8, 8, -9], [-37, 8, -7, -9, 0], [-3, 23, 0, -4, 0], [-2, -4, 13, 10, -3], [-17, -22, 7, -3, 6], [-44, -14, 9, 2, 1], [26, -10, -8, 16, 4], [-30, 2, -10, 2, 6], [4, 6, 11, -12, 7], [10, 24, -6, 2, 4], [2, 12, -12, 4, 0], [-21, -18, 15, -3, -6], [14, 2, -10, -6, -2], [-35, 12, 8, 3, -3], [-3, 5, -4, -6, 11], [-32, 12, 8, -18, -2], [25, -5, 3, -22, -2], [-12, -18, 1, 6, -3], [-19, 0, 0, -5, 5], [-5, 1, 24, -16, -2], [-31, -3, 0, -14, 3], [-29, 17, -2, -4, -6], [16, -10, -16, 16, 4], [-57, 7, 7, 0, -1], [10, 4, 9, -16, 1], [-32, 10, 2, 10, 6], [51, -15, -3, 18, 6], [6, -6, -8, -6, 0], [22, -2, -2, -2, -10], [4, 12, 2, 8, -6], [-24, -12, 0, 10, -10], [12, 4, -4, -8, 4], [20, -6, 7, -12, 7], [10, 12, -14, -6, 4], [-10, -12, 18, 2, -2], [47, -1, -3, 4, -3], [-11, -10, 16, 1, -3], [-12, 10, 7, -14, -1], [49, -1, -4, -2, 7], [-62, -2, 4, 0, -4], [-14, 6, -2, 14, 8], [-10, 4, -10, -16, 4], [16, 0, -6, 16, 0], [-26, 18, 0, 0, 2], [5, -27, 3, 4, -8], [-8, -6, -16, -2, 2], [-28, 2, -2, 0, 0], [0, -8, -3, -6, 11], [-2, -6, -9, 0, 1], [-31, -3, 4, -6, -1], [27, 14, 4, -5, 7], [-20, 0, -9, -2, 3], [18, -12, -9, -8, -7], [-29, -5, 16, -10, -3], [2, 4, -6, -8, 6], [-28, -2, 7, -6, 1], [-17, 3, -11, -6, -3], [-10, 4, 3, -14, -11], [12, -6, 2, -16, -4], [28, -20, -4, -12, 6], [-20, 14, 5, -6, -7], [1, 8, 20, -5, 5], [6, 2, 16, -14, 8], [-7, -20, 9, -11, -2], [9, -10, 19, -1, -8], [-22, 2, -3, 2, -1], [35, -2, 3, 1, 14], [32, -16, 0, 2, 8], [-18, 4, -8, 10, 0], [36, -8, 9, 10, 1], [-6, -12, -14, 10, -2], [-1, -7, 0, 8, -10], [-38, -10, -13, 4, 1], [-32, -2, 0, 4, 6], [-21, 11, -10, -22, 3], [18, 4, -4, -2, 2], [12, -12, 4, -2, -10], [-10, 10, -8, 18, -2], [6, 20, 3, 6, 1], [-29, 11, -5, 2, 1], [19, -7, 14, 0, 1], [12, -6, -14, 16, 2], [-8, 10, -2, -18, 6], [-51, 5, 7, 4, 3], [-6, 2, -6, -2, 6], [25, -1, 16, -10, -1], [-18, -4, -1, -20, 1], [-35, -7, -5, 8, -4], [-3, 9, -18, 6, 13], [-35, -23, 9, -8, 3], [28, 2, -5, 12, 7], [9, 3, -8, 20, -6], [12, -4, 8, 6, 16], [24, 8, 3, 6, -5], [8, 2, 8, -14, 0], [33, 0, 15, -9, 2], [-6, -24, 5, 8, -7], [-12, -14, 13, 2, 5], [20, -8, -4, 24, 0], [16, 16, -2, -4, -6], [2, -8, 8, -12, -14], [10, -14, -5, 2, 1], [12, -2, -10, 18, -6], [17, 6, 21, -3, 0], [-7, -6, 23, 3, -4], [-28, 6, 2, -6, 0], [-60, 0, 6, -2, 4], [-13, -1, 19, 0, 5], [-4, 12, -2, 8, 0], [21, 13, 15, -14, 3], [16, -22, 9, 0, 1], [10, 6, 11, -6, -9], [-17, 13, 8, -20, 3], [26, -10, -8, -12, 4], [-39, -5, 4, -16, 2], [13, 15, -4, -12, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4508_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4508_2_a_f();".
// 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_4508_2_a_f(:prec:=5)
    chi := MakeCharacter_4508_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_4508_2_a_f();".
// 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_4508_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4508_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![38, 16, -22, -8, 3, 1]>,<5,R![160, 94, -34, -20, 2, 1]>],Snew);
    return Vf;
end function;