
// Make newform 4235.2.a.bd in Magma, downloaded from the LMFDB on 18 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_bd();".
// 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_bd();".
// 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, 10, -2, -7, 0, 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], [-3, 0, 1, 0, 0], [2, -4, -1, 1, 0], [4, 2, -5, -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_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, 1, 0, 0, 0], [0, 1, 0, 0, 1], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 1, -1, 1, -1], [0, 1, 0, -1, 2], [1, -1, 0, 0, 1], [2, -1, 1, 0, -2], [3, -1, 2, 1, -2], [0, -1, -2, 1, -3], [0, 0, -4, 0, -1], [2, -1, 1, 2, 0], [2, -1, -4, -1, -1], [1, -2, -1, 0, -4], [-3, 0, 1, -1, 2], [0, -1, 0, 3, 0], [4, 0, 0, 1, -1], [0, 0, -1, -5, 0], [-2, 1, 2, -2, 3], [3, -3, 0, -1, 1], [3, 6, -2, 2, 1], [3, 0, -1, -5, 1], [3, 0, 4, -1, -1], [5, 0, 2, 1, -2], [9, 2, 1, -1, -1], [-1, -1, -4, -3, 6], [-2, -1, 4, -4, 4], [9, -5, 1, 2, 1], [-2, 4, 2, 5, -2], [4, 1, 2, -2, -1], [-2, 0, 4, 5, -5], [0, 10, -3, -1, 2], [-3, 2, -3, 1, 0], [10, 1, -1, 2, -2], [-5, -3, 2, -4, 3], [-9, 2, -7, 1, -3], [-3, 3, -2, 1, 6], [2, -3, -2, 3, -5], [-2, -1, -1, 6, -1], [-11, -5, 1, -2, 3], [-1, 1, 1, 2, 2], [2, -4, 1, -7, 3], [1, 4, -3, 1, -4], [4, 5, -4, 1, -6], [6, -10, 3, 0, -1], [10, -4, 6, -2, -1], [5, 1, -5, -5, 4], [3, -3, 3, -4, -1], [11, 2, 4, -3, 3], [1, 0, -2, -5, 9], [8, -6, -4, 2, 0], [10, -2, -4, -1, -6], [-10, 9, -1, 4, 5], [5, 3, -3, -2, -4], [-7, 3, -1, 2, 9], [3, 4, 7, -3, 12], [5, 0, -3, -9, 6], [3, -13, 0, -4, -2], [-3, -7, 2, -4, 2], [-6, 2, 5, 0, 2], [-8, 6, -11, -2, 0], [-7, 10, -4, -6, 0], [-8, 1, 6, -4, 10], [3, -5, 10, -1, -2], [-3, -2, -1, 0, 6], [-15, 3, 1, 1, 6], [-6, 7, 0, 11, -2], [-9, 1, -8, 2, -1], [17, 3, 5, -7, -1], [0, -7, 1, 0, 0], [1, 2, -2, 1, -12], [8, 3, 3, 4, -11], [-1, -8, 9, -1, 2], [-2, -9, 2, -6, 1], [-8, -3, 3, -2, 3], [2, -8, 6, 7, -1], [7, 0, 4, 10, -1], [10, 6, -7, 6, -10], [18, 2, -1, -4, 4], [-5, -16, 9, 2, -5], [8, -11, -4, -4, -10], [-1, 1, -1, -4, 12], [2, -7, -6, -8, 1], [-1, -7, 6, 1, 4], [13, 0, 1, -3, 8], [-15, 4, -6, -1, -1], [9, 2, 3, -4, 1], [0, 11, 3, -1, -4], [-2, -9, 9, -3, -2], [7, 11, -9, 4, -6], [7, -7, 16, 5, 1], [11, -3, 5, 8, 5], [-10, 4, -4, -9, -2], [-4, 3, 0, 5, -1], [-11, -2, -4, -5, -3], [2, -20, 10, 1, -7], [-1, 10, 3, 2, 6], [8, 0, 8, -6, 0], [15, -3, 9, 12, -1], [8, 8, 4, 7, -5], [-1, -15, 2, -7, -6], [17, 1, 3, 7, -4], [20, -6, 1, 0, 4], [19, -14, 16, 0, -5], [3, 4, -2, 0, 8], [-5, 0, -2, -4, -11], [-4, -14, 5, 0, -11], [11, 10, -6, -2, 2], [11, 1, -5, 0, -15], [-14, -11, 4, -4, 0], [6, -4, -2, 3, 2], [0, 11, 2, 0, 9], [16, -6, 6, -3, -2], [-10, 2, -3, -7, 14], [-10, -13, 7, 8, -6], [10, -13, 10, 9, 1], [-17, 8, 6, 4, -6], [7, -8, -8, -9, 8], [-8, 8, 1, -6, 2], [2, -14, 16, 4, -5], [13, 1, -4, -10, 3], [7, -3, 7, 5, -2], [-3, -2, -10, 3, -12], [13, 1, -2, 9, -5], [-4, 1, -2, -6, 19], [2, -5, 0, -14, 10], [-10, 13, 3, 2, 12], [-4, -6, 4, 0, -9], [3, -13, 0, -2, 0], [-8, 0, -12, 6, 1], [-19, -10, -6, 10, -1], [12, 12, -3, -1, 13], [-1, -8, 13, 15, -12], [-3, 16, -4, -3, 12], [-6, -1, 0, 7, -3], [3, 2, -1, -7, 2], [-12, -1, 6, 13, -6], [15, 6, 9, -16, 5], [-5, -5, 5, -7, 7], [-2, -6, -3, -4, -3], [-8, 18, 3, 4, 12], [-17, -6, 4, 7, -4], [27, -2, 9, 1, -5], [-2, -2, 9, 11, -3], [-5, 16, -1, -2, -6], [6, 16, -8, 1, -9], [0, 7, 5, 0, -9], [-18, -2, -5, 3, -10], [-3, 5, -4, -16, 14], [6, 4, 0, -11, 6], [-7, 0, -4, -4, -9], [15, 16, -3, 2, -8], [-3, 9, 6, 14, 8], [11, 3, 4, -4, -7], [-11, 4, 10, -1, 1], [4, 8, -12, 3, -17], [17, 7, -14, -5, 2], [18, -10, 9, -4, -5], [-1, 0, 3, 10, 5], [-13, 21, -10, 1, 2], [-15, -11, 4, 7, -11], [-1, 1, -5, 1, 4], [9, 5, -19, -6, 3], [-4, -4, 11, -4, 0], [28, 4, 6, 6, -4], [-13, -7, 5, 14, -22], [4, -1, 8, -5, 11], [22, -1, 3, -3, 10], [-37, 4, 3, 4, 5], [5, -15, 12, 0, -2], [14, -5, -2, -19, 2], [-22, -6, 2, 5, -3], [-13, -17, 3, -10, 2], [-2, 2, -11, -7, 9], [-21, -6, 3, -9, 15], [-39, -7, 5, 3, -4], [-14, 6, -5, -2, -9], [27, -6, 2, 9, -4], [-1, 10, 11, -3, 3], [-3, 0, -4, 5, 4], [2, 7, 4, -2, -10], [-11, -3, 8, 13, -4], [26, -16, 7, -7, -6], [2, -12, 8, 1, 2], [9, 4, -10, 5, -1], [2, 12, 12, -7, 14], [16, -6, 4, -7, 3], [11, 0, -14, -13, 2], [-1, 1, -15, 5, -6], [-5, 2, -13, 6, -17], [-26, 2, 3, 13, -12], [3, 0, 12, -2, -6], [0, 0, -5, -1, 5], [-32, 12, -5, 5, 16], [19, -9, -5, 0, -1], [-14, 5, -19, 8, 0], [5, 12, -7, -4, -13], [0, 9, 0, -8, -2], [-1, -5, 18, 19, -19], [18, -7, 5, -19, 2], [4, 18, -3, 9, 10], [-14, -6, 8, -7, 12], [16, -20, 4, -9, -11], [25, 15, -5, 12, -25], [-15, 12, -12, 6, -3], [1, 11, 8, 0, 28], [-7, 6, 1, -7, 6], [18, 0, -16, -7, 5], [2, 20, -16, -4, -1], [-9, -3, -9, -3, 5], [2, 1, 5, -13, 18], [-23, 2, -8, -14, -1], [6, 3, -15, 8, -10], [3, 11, -7, -5, 1], [-5, 6, -11, -11, 5], [21, 3, 3, -2, -3], [-24, -7, 4, -14, 22], [8, 16, -6, 17, -9], [5, -10, 16, 17, -15], [-1, -10, 10, 18, -12], [15, -14, 9, 8, 4], [20, 4, 3, -7, -6], [-5, -5, 2, -6, 19], [-22, -4, -7, 3, -2], [21, -4, -12, -7, -5], [-28, 5, -8, -9, 10], [-26, 9, 5, 4, -3], [7, -2, 0, 9, -5], [18, 6, -4, -3, 9], [-24, 5, -15, -15, 2], [-15, -12, -3, 20, -20], [1, 15, -18, 1, -7], [13, 6, -4, -3, 14], [-16, 5, 5, 12, -6], [-16, -2, -13, 10, -19], [-14, 16, -19, -7, -9], [1, 24, -10, -5, 3], [0, 0, 6, 23, -7], [4, 6, -9, -14, 6], [-32, -21, 3, -1, 1], [-11, -7, 0, -6, 22], [37, -10, 12, -12, -8], [23, -13, 1, -3, -15], [-18, 1, -7, -14, 4], [4, 7, 13, -2, 12], [9, -19, 19, 2, -8], [-11, 5, 3, 5, 7], [-14, -24, 5, -8, 8], [-1, -8, 0, -18, 4], [29, 3, -5, 5, -13], [25, -7, 3, 11, 1], [21, -9, 19, -2, 1], [5, 7, -19, 0, -15], [-33, 6, -13, 4, 13], [-6, 9, 12, -1, 2], [-20, 10, 1, -7, 6], [-3, -8, -11, -13, 13], [35, -23, -8, -9, -2], [8, -1, 2, -18, -6], [-4, -6, -16, -11, -20], [6, -22, 8, 5, -7], [-11, 5, -15, -19, 0], [7, -5, -13, -15, -2], [-42, -9, 6, -4, 11], [4, -2, 4, 26, -3], [-11, 18, 4, 17, -5], [0, -2, 22, 0, 9], [34, -12, 16, -3, -2], [-24, 6, 2, 18, -6], [17, -3, -4, -4, 14], [-32, -2, -4, -8, -4], [-1, 6, -16, -7, -7], [5, 15, 3, 7, -2], [30, -2, -3, -9, 11], [-28, 15, 12, 9, 21], [30, -7, -3, 4, 1], [22, -9, -11, 1, -13], [-7, 20, -4, 4, 4], [13, -8, -5, -1, -5], [-8, 11, -10, -3, 14], [-21, -23, -3, 2, -14], [-22, -11, 16, 5, 3], [7, -6, -8, 6, 7], [17, 0, -12, 12, -15], [-17, 10, -6, 3, -1], [-22, 21, 2, 13, 4], [-41, 15, -17, 4, 12], [-19, -3, -3, -15, -6], [35, 18, -14, 13, -10], [40, 4, -12, 1, -15], [8, -5, 25, 12, -2], [-2, -7, 13, 14, -20], [9, 0, -24, -6, -12], [7, 9, -18, 0, -11], [3, -6, 6, 2, 14], [24, -8, -2, -9, -22], [9, 14, -7, -4, -7], [-22, 14, 19, 17, 0], [-2, 27, -17, 12, 1], [-28, 10, 7, 11, 5], [-12, -16, -14, 8, -16], [-23, 21, -19, -4, 18], [-10, -6, -22, 2, -9], [-11, 22, 9, 5, 21], [-7, -6, -1, 5, -13], [-33, -6, 2, 1, 4], [7, 15, 5, -8, -13], [7, 11, -27, 5, -17], [30, -3, -5, -13, 16], [40, -1, 5, 0, 17], [10, -18, 24, 7, -12], [18, 6, 0, -4, 2], [31, 16, -6, 7, -16], [-3, -3, 13, 18, 0], [-2, -5, -24, 5, 0], [-2, -38, 11, -2, -6], [6, 12, -19, -8, -11], [14, 9, 8, -4, -19], [-20, -1, -17, -17, 6], [-20, 22, -1, 4, 17], [-28, -11, 0, 13, -11], [-5, 0, 4, -21, 2], [-22, -16, -2, -9, 6], [35, 13, -12, -9, 10], [-8, 2, 3, 0, -8], [0, 4, 4, 10, -27], [11, 8, -7, -13, -1], [-2, 18, 4, -5, 3], [0, -17, 5, 6, 0], [21, -6, -7, 0, -10], [-30, 11, -14, 5, 11], [-6, -22, -1, 1, 27], [-9, 1, 12, -7, 20], [6, 15, -18, -11, 3], [4, -11, 1, -10, -8], [3, 0, 15, -3, 29], [-18, -17, -5, -2, -11], [13, 29, -6, 6, 7], [-9, 14, -5, 1, 6], [-15, -23, 9, 7, -3], [10, 3, 2, -8, 12], [16, 0, 9, 4, 3], [0, 0, 5, -5, 3], [31, -1, -7, 12, -13], [0, -26, 19, 4, -14], [22, -9, 17, 17, 0], [-15, -18, 10, -8, 21], [-29, 1, 2, -10, 19], [-5, 6, 8, 21, -1], [-11, -38, 21, -1, -3], [2, -17, -5, 19, 1], [-6, 0, 9, 16, 4], [11, -17, 4, -3, 9], [17, -11, -22, -19, 8], [13, 29, -19, 12, -7], [11, -8, 1, 12, -2], [20, -12, -5, -20, -4], [25, 13, 0, 15, 5], [-8, 3, -23, 21, -13], [10, 20, 2, 8, 18], [0, -5, 10, -4, -19], [-4, -10, 15, 17, 0], [15, 19, 3, -19, -4], [7, -23, -8, 6, -5], [5, 10, -15, -23, 25], [-8, 26, 7, -1, -11], [-19, 29, -7, 11, 2], [46, -12, 1, -1, 0], [-11, -12, 8, 3, -7], [-8, 34, -1, 6, 25], [-38, 0, 3, -4, 10], [1, -6, 13, -7, 1], [10, 3, -3, -13, 33], [-47, 7, -19, -1, -9], [16, -3, 11, -6, 17], [22, 5, 17, 2, 19], [7, -16, -3, -6, -8], [-4, 22, -7, 1, -1], [-35, 4, 2, 23, 0], [7, -27, -3, -1, -17], [-9, -10, 5, 12, 9], [-31, 7, -13, -20, -2], [-10, -14, 11, -16, 9], [-16, -4, 2, 16, -20], [3, 24, -18, -12, 3], [-50, 20, -16, 1, 2], [-11, -28, 0, -17, -4], [-6, 15, 5, 17, 4], [-32, 1, -7, -12, 11], [-9, 4, -8, 15, -1], [57, -10, 12, 3, 6], [56, -9, 3, -5, -4], [44, -15, -10, -5, 7], [-28, 38, -2, 15, 13], [-8, -6, 19, 1, 17], [-17, -23, 17, -5, 5], [26, -25, -8, -21, 7], [45, -5, 5, 0, 11], [-13, 21, 12, 9, 13], [36, 8, 4, -5, -24], [-5, 13, -17, 6, -1], [1, -12, -13, 2, -9], [22, -19, -10, -15, -7], [28, -27, -3, 1, -3], [45, -1, -7, -16, 16], [-20, 1, -6, -6, -5], [-1, -29, 8, 18, -17], [-4, 7, 6, -10, 8], [-19, 9, 7, 6, 7], [20, -3, -10, -9, -6], [24, 21, 1, 4, 19], [13, -27, 0, 1, -30], [-4, -17, 0, 6, -12], [12, 27, -4, 3, 5], [-1, -15, 5, 7, 20], [-17, 24, 12, 13, 3], [-13, 20, 8, 1, 4], [-8, -16, 7, 20, -35], [-1, 15, -26, 11, -3], [-29, 7, -17, -9, 14], [-10, 4, -26, 24, -21], [40, 26, -1, -12, 11], [-38, 3, -17, -6, 2], [29, 1, 6, -16, 5], [2, 25, -1, 0, 10], [19, -11, 21, 11, 7], [-2, 2, -5, 31, -21], [-1, -15, -14, -10, 8], [-17, -28, 8, -11, 18]];
    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_bd();".
// 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_bd(: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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4235_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 10, -2, -7, 0, 1]>,<3,R![-1, 1, 19, -9, -2, 1]>,<13,R![169, -260, 119, -8, -6, 1]>],Snew);
    return Vf;
end function;