
// Make newform 4235.2.a.bf in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4235_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_4235_2_a_bf();".
// 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_4235_2_a_bf();".
// 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, 7, 7, -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], [-2, -1, 1, 0, 0], [1, -3, -1, 1, 0], [1, 4, -3, -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_4235_a();"
function MakeCharacter_4235_a()
    N := 4235;
    order := 1;
    char_gens := [2542, 1816, 2906];
    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_4235_a_Hecke(Kf)
    return MakeCharacter_4235_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, 1, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 0], [2, 1, -1, -1, 0], [1, 0, -2, 3, 1], [1, 1, 2, -2, 0], [0, 2, 1, -2, -1], [4, 0, 0, -1, 1], [3, 1, 1, -3, 1], [0, -1, -1, 2, 2], [0, 0, 3, 0, 1], [-1, 3, 1, 1, 1], [-1, -2, 2, 2, -1], [2, -2, -1, 3, 2], [-1, -2, -2, -1, 1], [5, 3, -2, -1, 1], [-3, 2, 1, -1, -2], [-2, -5, 4, 0, -2], [0, 1, 3, 1, -1], [3, 1, -1, -4, -2], [4, -1, 2, -7, -2], [0, -1, -4, 1, 1], [4, 0, -3, -3, -3], [6, -1, 0, -3, 2], [0, 0, -2, -3, 3], [-6, 4, 3, 0, -4], [3, -5, -4, 0, -1], [3, -4, -5, 3, 1], [6, 1, -4, 0, -2], [3, -1, -6, -1, -3], [5, 0, 1, -1, 4], [6, -8, -3, -1, 0], [-2, 0, 1, 6, -5], [3, 5, 2, -2, 4], [-2, 5, 2, 1, -2], [6, -6, 0, -1, -5], [-3, -1, 5, 3, -5], [-6, -1, 2, 6, 1], [1, 3, 2, 0, -3], [1, -4, 1, 2, -5], [-7, 1, 4, -9, -4], [8, -8, -3, 3, 0], [-9, 0, -6, 5, 1], [4, 3, 3, -2, 3], [8, -1, -1, 0, 6], [-2, -4, 0, 3, 2], [-9, -5, 2, 2, -7], [-6, -5, 0, 3, -1], [12, -9, -4, -3, 1], [2, -4, -4, 10, -2], [-1, -6, 3, 1, -1], [12, -9, -2, -2, -1], [-5, 10, 11, -4, -1], [-1, 5, -6, -6, 1], [-8, 4, 9, 1, 6], [0, -10, -13, 9, -2], [9, -4, 5, -4, -2], [13, 4, -5, 4, -2], [4, 4, 0, -4, 5], [4, -6, 0, -2, -1], [9, 4, 4, -8, 0], [-8, 8, -3, -4, -2], [10, -4, -8, -5, -1], [-9, 4, 2, 6, -5], [2, 8, -8, -3, -2], [11, -8, 2, 7, 3], [3, -5, 4, 0, -2], [8, -3, 7, -1, -4], [8, 6, -5, 4, 3], [0, 4, -3, 3, 9], [2, 3, 2, 4, 9], [12, 6, 1, 1, 2], [-6, -3, -4, 0, -2], [12, -7, 0, 6, -1], [15, 1, -6, 1, 3], [-5, -9, -7, 4, -4], [-4, -10, -6, 6, -5], [-6, -6, -4, 14, -3], [-5, 1, -1, 10, -1], [-10, 4, -11, 0, 4], [15, -14, -7, 0, 1], [-6, 1, -12, 8, 6], [8, 0, -6, 13, -1], [0, 4, -13, 1, 0], [-14, 5, 6, -7, 3], [9, 9, -1, -4, 3], [17, -16, -4, -1, 0], [1, 4, -6, 9, -2], [-17, 0, 3, 6, 3], [2, 1, 5, -7, -3], [-11, 3, 14, -8, 3], [13, -6, -13, 5, 1], [1, -9, -11, 7, -1], [-16, 5, 4, 5, -1], [1, 3, -4, 5, -7], [-15, -2, 4, -2, 1], [-22, 4, -6, 2, 6], [-5, -3, -2, 18, 1], [-7, 3, 6, 5, 5], [-4, 14, -10, 1, 5], [-22, 16, 8, 1, -2], [20, -4, 6, -4, -1], [5, 3, -5, -10, 2], [-17, 0, 8, 6, -4], [1, -7, 1, 2, -4], [-8, -3, -7, -4, 1], [-23, 2, -6, 2, 0], [3, 3, -2, 8, 5], [12, 4, 5, 4, 4], [-5, 8, 3, -5, 5], [12, -3, 2, 10, 4], [5, 2, -9, 5, 9], [9, 2, -3, -7, 6], [-8, 0, 11, 0, 3], [-3, 7, 5, 3, -5], [7, 2, -10, -2, 0], [10, 0, 3, -3, -1], [-10, 10, 12, -14, 1], [20, -3, 5, -4, 0], [5, 7, 4, -20, 2], [-2, 6, 12, -5, 8], [12, 8, -15, 5, 1], [24, -11, -3, -1, -3], [-8, 5, 16, -8, -6], [-14, -12, 9, -6, -2], [8, 4, -9, 0, -3], [10, -13, -7, 8, 4], [1, -8, 5, -4, 0], [18, 9, -3, -12, 4], [1, -1, -11, 0, 0], [-27, -1, -4, -3, 2], [-12, 4, 5, -1, 10], [10, 2, 1, -7, -3], [13, -5, 5, 1, -5], [8, 2, 3, -7, 8], [7, -4, -14, 9, 9], [-9, -9, 3, 2, 1], [28, -3, -5, -7, 8], [-8, 11, 7, 2, -5], [-12, -4, 6, 0, -7], [2, 8, 5, 5, -5], [-18, -7, -8, 5, 6], [11, -15, -2, -3, -10], [9, -10, -18, 2, -7], [-19, -5, 10, 1, 7], [-22, 7, -4, 4, 9], [9, 12, -9, 9, 4], [5, 20, -15, 4, 6], [17, -12, -15, 9, 5], [11, -1, 11, -10, -4], [-37, 6, 12, -4, 3], [-13, 8, 3, -12, -2], [-9, 7, -6, 2, 0], [-2, -5, -20, 9, -1], [-5, -5, 2, 3, 3], [-24, -2, 0, -3, -1], [-2, -13, -1, 8, 1], [-19, 1, 7, 6, -9], [-16, 22, 18, -5, -1], [0, -13, 5, -7, -17], [-18, 20, 6, -1, -2], [1, -9, 6, 10, 7], [-2, 0, -6, 10, 1], [-6, -4, -6, -6, -12], [-1, -8, -9, 2, -9], [19, -5, -21, 7, -7], [21, -12, 8, -5, -8], [-17, 11, 17, 2, 11], [-23, -4, -1, 4, 4], [-1, 4, -2, 13, -3], [-1, 5, 24, -7, -1], [-7, -20, -3, 12, -5], [1, 5, -12, 11, 4], [20, -11, 12, 15, 0], [-4, -18, 6, -3, -4], [0, -3, 11, 12, 7], [6, -2, 19, 3, -5], [22, -1, -10, 11, 6], [10, 12, 13, -5, -3], [4, 0, -21, 14, 2], [-26, 4, 8, 11, 9], [1, -6, 7, -3, -2], [-21, -4, 15, -5, -1], [-20, 7, -6, 11, 9], [-19, -12, 1, 5, -15], [17, 1, 20, 3, 1], [-12, -2, 15, -7, -1], [12, 12, -2, -1, 0], [13, 9, -9, -8, -3], [3, -18, -9, 3, 0], [29, 10, 0, -6, 10], [5, -15, -16, 17, 0], [-23, -2, -7, 15, 4], [-15, -11, 12, -2, -9], [20, -10, -3, 0, -1], [-17, 1, 13, -8, -1], [-34, 2, 7, 12, 4], [8, -3, 1, 1, -3], [5, -16, 6, 3, -10], [19, -8, 7, 1, 0], [-7, 18, 13, -5, -11], [3, -3, -14, 9, -9], [-5, -5, -10, 0, 3], [-11, -23, -1, 12, -8], [-15, -4, 17, 6, 0], [0, -14, 5, -5, -12], [-17, -3, -2, -3, 13], [12, 25, -1, -12, 4], [-6, 3, 13, -7, -16], [9, 2, 0, 5, -2], [-1, -3, -9, -4, -6], [22, -4, -9, 16, -1], [-20, -5, -1, -11, -2], [2, -9, 8, -3, -20], [19, 17, -6, -2, 13], [26, 4, -3, 10, 6], [-7, 1, -4, -11, -17], [-30, 9, -4, -7, -1], [3, 12, 8, -10, 8], [5, -4, -16, 8, 3], [-1, 6, -13, 1, 4], [-7, 5, 14, -8, 0], [7, 14, 13, -9, 10], [28, -9, -4, 17, 9], [19, 20, -2, -7, 11], [-12, -3, 0, 10, 7], [-12, -9, 0, 3, -7], [3, 17, -8, -11, -3], [7, -12, 16, -11, -18], [27, -16, -14, 8, 11], [-2, -1, -23, 1, -1], [-8, -10, -13, -1, -1], [36, 0, 1, 0, 7], [-14, -7, -13, -6, 17], [1, 3, 12, 3, -8], [-4, -11, -2, 1, 17], [-3, -1, 4, 1, -13], [-14, 16, 8, -16, 5], [-3, -13, -9, 13, 0], [1, -14, 13, -4, -6], [-11, -4, 26, 2, 6], [12, 9, 11, -17, 4], [16, -14, 11, 10, -1], [-30, 20, -1, 16, 5], [-1, 6, 1, -4, -3], [-8, -9, 19, -23, -6], [18, -12, -8, 4, -1], [-45, 10, 10, 2, 8], [-28, -11, 9, 5, -12], [18, -13, -7, 5, -4], [-19, -17, 14, 2, 3], [19, 5, -24, 4, -7], [-19, -3, 1, 18, 5], [-1, 0, 4, 17, -11], [11, -34, -11, 9, 2], [-6, 11, 18, -11, 4], [6, -8, 10, 11, 1], [4, 22, -17, 6, 8], [-5, 14, 13, 5, 5], [-7, 1, 12, 7, -9], [22, 14, -24, -5, -4], [30, -10, 10, -9, -8], [-18, -17, 4, 4, -22], [8, 11, 9, 6, 2], [20, 1, -2, 3, 15], [34, 7, -1, 0, 10], [21, -12, 17, 13, 5], [2, -18, 10, 10, -12], [-23, 22, 5, -6, -12], [-4, -16, 14, 8, 8], [24, -13, -6, -11, 7], [4, 4, 10, -7, -18], [21, -15, 2, 11, 10], [-41, 3, 9, -9, -3], [0, 5, -6, 0, -7], [9, -3, -1, -7, -12], [23, 8, 8, -12, 4], [10, -9, -2, -7, 4], [-13, 0, -30, -3, 5], [19, -24, 3, 14, -1], [11, 7, -5, -19, -9], [-23, 33, 25, -8, -6], [-43, 3, 7, -18, -2], [-24, 5, 4, 13, 15], [-5, -10, 0, 15, -3], [11, 14, -3, -22, 1], [-2, 24, -6, 1, 8], [20, 20, -7, -11, 7], [-53, -1, -2, -5, -9], [-36, 8, -5, 4, -5], [10, -16, -19, 20, -1], [9, -14, 22, -4, -14], [17, 1, -16, 18, -10], [-5, -8, -6, -12, -16], [19, 6, -1, 7, -13], [5, -1, -13, -10, -15], [5, -2, -3, 9, 10], [20, -11, -18, -8, -1], [15, -9, -4, -1, -12], [4, 4, 8, 12, 14], [-39, 0, 3, 0, -5], [16, 7, -13, -28, 6], [12, -5, 0, 23, 12], [-4, 19, -16, 19, 8], [0, -4, -13, -1, 19], [21, -13, -18, 16, 1], [-28, 19, 25, -5, 8], [-15, -6, 20, -3, 2], [-10, 9, -6, 12, 17], [-37, 4, 13, 19, -1], [18, 10, 4, 0, 0], [6, -24, -5, -3, -11], [-21, 14, -11, 16, 5], [-3, 2, -12, 3, 1], [-16, 0, 20, -16, -7], [2, 33, 19, -14, -3], [-18, 9, -20, 24, 16], [17, -8, 22, 13, -2], [10, -9, -13, 10, -7], [17, -25, -19, 1, -3], [-2, 4, 13, -5, -9], [-7, 20, 25, -11, 7], [-42, 2, -8, -5, -7], [8, -18, 0, 2, -5], [-42, 15, -1, 12, 0], [-36, 27, 8, -1, -8], [-29, 11, 8, 13, -7], [-4, 12, -13, -4, 3], [35, 2, 16, 8, -9], [11, 29, -7, -7, 3], [-9, 1, 6, -15, 2], [24, -22, -22, 15, -3], [21, -23, -9, 13, 3], [-2, 6, 9, -10, -3], [18, -1, 6, -25, -6], [-38, 17, 8, -18, -19], [-7, -11, 8, 0, -4], [-12, 6, 9, 15, -8], [-6, -11, -3, 29, 0], [-6, -8, 5, -20, -6], [18, 13, -25, -26, -1], [29, -3, -14, 13, 6], [41, 9, -6, 4, 7], [-28, 14, 16, 0, -15], [-41, 4, 14, 7, 12], [1, 27, 7, -6, 14], [9, 23, -12, -24, 8], [4, 13, 12, -7, 3], [10, -15, 22, 9, 6], [-45, -1, 3, -11, -12], [10, 12, 12, 0, -15], [54, 13, -9, -1, 5], [-2, 18, 16, -33, -3], [-23, -15, 12, 6, -13], [7, 10, 20, -12, -3], [-4, 22, 6, -2, 5], [-8, -5, 1, 17, -13], [39, -3, -5, -29, 8], [2, -2, 2, 32, -2], [34, 19, 6, 0, -6], [11, 10, 9, -23, 8], [-32, -6, 18, 7, 4], [-3, -5, 4, -4, -24], [-20, -7, 6, 31, 2], [-29, -21, -16, 7, -4], [26, -2, -26, 1, -18], [-15, 8, -7, 29, 16], [-22, -9, 8, -7, 17], [24, -19, -17, -8, -3], [-16, 6, 0, -2, -5], [12, 1, 0, -23, 4], [17, -11, -13, 3, 10], [14, -1, 9, -7, 12], [12, 19, -4, -4, 1], [18, -3, -4, -24, -17], [13, 2, 14, -12, 5], [-11, -7, 18, 9, -8], [14, -6, 3, -7, 1], [0, -9, 21, 7, 2], [17, 19, 21, -28, -1], [17, 4, 3, 10, 1], [4, 7, 11, 4, 21], [-36, -4, 4, 20, -14], [-3, -1, -5, -10, -4], [1, 14, -9, -11, 13], [-2, -14, -3, 7, 9], [43, -18, -16, 7, -2], [2, -15, -6, 24, 11], [-32, -11, -10, 29, -7], [-66, 0, -15, 7, 9], [7, -10, 4, 3, 18], [-15, -3, 3, 17, -15], [45, -5, -18, 1, -17], [-11, 17, 26, -15, -2], [-48, 3, 5, 1, -22], [7, -17, 9, -13, -19], [-41, -1, 8, 14, -9], [32, 11, -25, 5, -3], [-3, -20, 15, 5, -17], [35, -51, -30, 18, 3], [-1, 3, -7, -6, -15], [-11, 15, -25, -21, 7], [-15, 9, -9, -11, 4], [-33, 26, 21, 4, 19], [2, 15, -16, -16, -6], [-3, 25, 14, -4, 4], [36, -24, -5, 2, 12], [1, 19, 34, -22, -3], [-23, -20, -22, 19, 7], [-2, 17, 24, -4, 19], [18, 10, -18, 19, 9], [-16, -18, 5, 18, 2], [-19, -3, 15, 17, -3], [30, -10, -2, 27, 2], [2, -13, -12, -7, 1], [22, 10, 19, -19, -5], [44, -5, -23, -16, 1], [-8, 33, -3, -23, -3], [-14, -12, 2, 19, 6], [16, -15, 5, -8, -10], [-18, -29, -9, 20, -9], [0, 6, -11, 0, 3], [1, 1, 20, -6, -8], [-18, 0, -7, -16, -3], [-4, -1, 11, 21, -16], [9, 11, -20, 7, 16], [25, -16, -7, 16, 12], [-12, 20, 15, -11, 13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4235_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4235_2_a_bf();".
// 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_4235_2_a_bf(:prec:=5)
    chi := MakeCharacter_4235_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_4235_2_a_bf();".
// 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_4235_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4235_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 0, 10, -5, -2, 1]>,<3,R![-3, 7, 7, -5, -2, 1]>,<13,R![9, 40, 45, 4, -8, 1]>],Snew);
    return Vf;
end function;