
// Make newform 29.2.e.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_29_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_29_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_29_2_e_a();".
// 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_29_2_e_a();".
// 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 := [64, -96, 0, 104, -72, -28, 57, -14, -18, 13, 0, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [96, 32, -176, 68, 102, -83, -14, 40, -7, -10, 7, 1], [-288, 256, 96, -196, 38, 59, -58, 4, 7, -10, 5, -1], [-32, 192, -128, -100, 134, -5, -58, 36, 7, -10, 5, -1], [64, -80, 40, 44, -52, 17, 22, -18, 1, 4, -3, 1], [224, 0, -288, 220, 134, -165, 6, 68, -25, -10, 5, -1], [96, 0, -128, 12, 94, -49, -50, 36, 11, -18, 1, 3], [16, -40, 20, 16, -29, 6, 10, -9, 0, 3, -1, 0], [32, -64, 96, -12, -86, 53, 26, -36, 9, 10, -5, 1], [16, 8, -12, 0, 31, -5, -16, 9, 5, -5, 0, 1], [-352, 352, 176, -420, 82, 199, -122, -40, 59, -14, -11, 3], [480, -416, -368, 740, -146, -435, 290, 80, -151, 30, 31, -15]];
    Rf_basisdens := [1, 128, 128, 128, 64, 128, 128, 16, 64, 32, 128, 128];
    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_29_e();"
function MakeCharacter_29_e()
    N := 29;
    order := 14;
    char_gens := [2];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_29_e_Hecke();"
function MakeCharacter_29_e_Hecke(Kf)
    N := 29;
    order := 14;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 1, 0, 0, 0, 1, 0, 1, 1, 0], [1, 1, 0, 0, 0, 1, -1, 1, -1, -1, 0, 0], [-1, -1, 0, 1, 1, -1, 0, 0, 0, 1, -1, 1], [-1, 1, 0, 0, 0, -1, 1, -1, -1, -1, 0, 0], [1, 1, 0, 0, 0, 1, -1, 0, 1, -1, 1, -1], [3, 3, 2, -2, -2, 2, -2, 0, 0, -1, 0, -2], [-2, -3, 1, 0, 2, 0, 2, 0, 1, 1, 0, 0], [-3, -4, -1, 2, 0, -1, 1, 1, 1, 2, 1, 2], [0, 2, 0, -1, 1, 0, 2, -1, 0, -1, -2, 1], [-2, 1, 3, -1, 3, -1, -2, -1, 1, 0, -1, -3], [1, 2, -1, -3, -2, 0, 0, -2, -1, -2, -2, -3], [2, 2, 1, 3, -6, 2, -2, 1, 1, 0, 2, -3], [1, 2, -1, -2, -2, 0, 0, -3, -1, -1, 0, -1], [-4, -6, 0, 5, 5, -4, 0, 3, 0, 3, 0, 5], [-1, 1, 0, 0, -2, 5, 3, 0, 5, 1, -1, 3], [1, 0, -2, -2, 0, 0, 0, 2, -2, -2, 2, 1], [4, 0, -2, -2, -2, -4, 0, 0, -2, 2, 0, -2], [-4, -8, -2, 4, 5, -3, 0, 4, -4, 3, 2, 8], [-3, -3, 3, -1, 0, 3, 5, 0, 2, -2, -3, -1], [3, 5, -2, -2, -6, -1, -7, 2, -1, 1, 1, -5], [3, 2, -2, -6, -1, -2, -2, -2, 0, -2, 0, -2], [5, 1, 3, 1, -2, 1, -3, -2, -4, 0, 1, -1], [-1, -2, 1, 6, 2, -1, -1, 3, -3, 4, 3, -4], [3, -3, 2, -1, -3, 3, -6, 2, 2, -1, -1, -1], [2, 5, 0, 3, 3, 0, -5, 4, 0, 0, 4, -1], [-2, -1, -3, 6, -1, 5, 4, -2, 0, -2, 0, 5], [-3, -5, -2, -4, 0, -5, 3, -6, 3, 1, -3, 1], [-3, -3, -1, 3, 4, -3, -1, 2, -4, -2, 3, -1], [-3, -2, -2, 0, 4, 0, 5, 0, 0, 0, -2, 2], [-1, 4, 5, -4, -4, 4, 4, -9, 9, -4, -5, -9], [-3, 3, 0, -2, -8, 1, 7, 0, 1, 1, 1, 3], [6, 8, 6, -5, -5, 6, -10, 1, 0, 1, 0, -1], [3, -3, -4, 0, 2, 0, 0, 4, -2, 0, 0, 2], [-8, 2, 0, 3, 1, -4, -2, -1, 0, 7, 6, 1], [-5, 8, 5, -3, -8, -5, -6, -4, 1, 4, 2, -1], [10, 6, -2, -5, -1, 0, 2, -3, -4, -7, -10, -1], [9, 6, -5, -1, -3, 0, -8, 6, -10, -5, 5, 1], [-3, -7, 0, -2, 6, -3, -3, 0, -3, 1, -3, -3], [-1, -3, -6, 2, 14, -3, 7, -3, -3, -3, -6, 8], [4, 5, 1, -4, -6, 2, 0, 0, -3, -1, -4, -6], [-7, -6, -3, 11, 0, 0, 6, 0, 3, 0, 0, 7], [-3, 0, 5, 8, 0, 6, -2, 5, 5, 6, 5, 3], [-4, 0, 0, 14, 6, 0, 8, 6, -2, 0, 2, 8], [0, -2, -4, 0, 2, -4, 0, -2, 3, 7, -2, 2], [4, -3, 5, -2, -5, 7, 8, 0, 10, -4, -4, 1], [7, 0, -7, -4, 0, 7, -7, 7, -5, -2, 3, 4], [-11, 5, 5, -7, 4, -3, 3, -8, 6, 0, -3, -3], [5, 5, -5, -5, 0, 1, -1, -6, -4, -10, -1, -5], [-1, 10, 3, -3, -6, 6, -4, -2, 1, -4, 4, -1], [10, 0, -12, 10, 1, -3, -4, 14, -14, 3, 12, 4], [3, 8, 1, -9, 1, -2, 0, -2, 4, -1, 3, -1], [-1, -1, 2, -6, -10, 1, -1, 3, 3, 1, 2, -8], [-18, -18, 1, 9, 3, -4, 1, 5, 2, 12, 4, 6], [-3, 2, 3, -4, 2, 3, 7, -7, -5, -8, -7, 0], [-2, -5, 2, -1, 5, -2, 13, -8, 8, -6, -8, -7], [-11, -10, 7, 7, 8, -6, -2, 0, 7, 6, 6, 3], [1, -5, 0, -10, 0, -10, 2, -10, 4, 0, -10, -10], [13, 14, 3, 0, -2, 3, -13, 3, 3, 0, 3, -8], [2, 14, 8, 1, -6, 4, -9, 6, 4, 3, 3, -14], [-4, -4, 5, 3, 2, 2, -2, 3, 3, 2, -2, 1], [-3, -7, 4, 2, -2, 7, -1, 11, 11, 7, 4, -2], [13, 6, -4, -2, 0, 4, -3, 10, -10, -4, 4, 6], [2, 2, -2, -8, -2, 0, 0, -8, 0, -2, -2, -6], [3, -5, 5, -5, 2, 1, 1, -4, 6, 0, 1, -3], [3, -2, -6, -4, 5, -8, 0, -6, -12, -2, -8, 2], [9, 4, -2, -6, -20, 4, -16, 6, -2, 2, 2, -9], [-6, -8, 0, -6, 0, 0, 12, -10, 6, 0, -6, 2], [3, 9, 0, -2, -12, 4, 8, 0, 4, -5, 4, 0], [-8, -10, -10, 2, 12, -16, 0, 2, -10, 10, 0, 14], [-4, 13, 3, -14, -2, -2, 0, -4, 3, -3, 0, -6], [-9, 13, 4, 2, -2, 2, -11, -3, 2, 1, -1, 6], [-3, -7, 0, 8, 4, 1, 1, 0, 1, 11, -5, 9], [15, 6, 5, -2, -14, 5, -13, 3, 3, -2, 3, -10], [4, 6, -4, 2, -6, 4, 4, 4, -1, -3, -2, 0], [3, 6, -3, -1, 6, 4, 12, -2, -3, -6, -6, 3], [-6, -10, 4, 1, -1, 2, -8, -1, 8, 3, 2, -11], [8, -1, -7, 8, 4, 0, -2, 18, -9, -7, 2, 12], [-2, -2, -7, -7, -9, -10, -7, 3, -8, 4, 10, 2], [3, -9, -12, 6, 8, -8, 1, 9, -6, -1, 1, 6], [-11, -22, -1, 6, 8, 4, 16, -5, 5, -4, 1, 11], [13, 7, 6, -14, -16, 3, -15, -2, 3, -1, -1, -7], [4, 0, 6, 11, 11, 4, 2, -1, 0, -1, 0, 3], [1, 16, -1, -22, -16, 1, -11, -11, 5, -6, 1, -26], [5, 16, -3, -11, -6, 0, 10, -8, -3, -8, -8, -9], [-9, -6, 9, 8, 6, -9, -7, 1, -3, 12, 5, 6], [-3, 5, 1, 3, 10, 11, -1, 10, -4, 0, 11, 13], [-11, -16, -8, 20, 11, -2, 14, 2, 0, 2, 0, 14], [-7, 4, 5, 8, 0, -6, 2, 1, 1, -6, 5, 3], [-13, -21, -4, 20, 10, -3, 5, 3, -3, 3, 4, 4], [-6, -10, 0, 4, 6, -12, 0, -6, -8, 0, -8, 0], [19, 35, 4, -20, -14, -1, -19, 3, -3, 1, -4, -12], [-2, -2, -10, -3, -1, -4, 16, -7, 0, -7, 0, -9], [6, -6, -2, 3, 1, 6, 6, 11, -4, -5, 6, 11], [-1, -10, -5, -1, 4, -2, -6, -4, -5, -2, -2, 1], [-4, 6, 2, 5, 1, 6, -12, 9, 4, 1, 10, -9], [13, 12, 11, 2, -12, 11, -15, -1, 1, -10, -1, -12], [13, -5, -2, -16, 4, 2, 10, -12, -4, -9, -6, 8], [5, 28, 15, -27, -1, 2, 0, -14, 8, -15, -7, -15], [4, -22, 0, 22, 10, 0, 0, 0, 0, 0, 0, 10], [0, 9, 2, 7, 7, 2, -7, 10, -10, -12, 10, -1], [-1, 2, 1, -10, -2, -1, -9, 5, 1, 0, -7, 6], [1, 2, -7, 3, 2, -8, 5, 2, -14, 4, 6, 15], [-2, 10, 2, 18, 6, 0, 0, 6, -6, 2, 8, 0], [-5, -15, 0, 8, 10, -6, -6, 0, -6, 13, -4, 2], [-1, -1, 1, -1, -8, 5, 7, -4, 12, 14, -5, 7], [-17, -16, 12, 14, 16, -6, -5, 4, -4, 6, -12, 6], [5, 5, 0, 0, 6, 3, -5, -5, -5, 3, 0, 0], [6, 22, -8, -3, -4, 0, -3, -2, -8, -5, -1, -18], [5, -22, -21, 0, -6, -21, -5, 9, -15, 6, 9, 8], [5, -7, -19, 15, 4, -3, 1, 16, -20, 0, -3, 19], [1, 0, 9, 11, -6, 8, -6, 18, 9, 10, 10, 5], [-2, 19, 2, -13, -19, -2, -15, -12, 0, 2, 12, -19], [13, 19, 10, -16, -18, 0, -16, -10, 2, 0, 0, -34], [-11, -10, 3, 4, -2, 3, 7, 17, 9, 6, 17, 4], [19, 13, 4, 0, -10, 3, -23, -9, -9, 3, 4, -4], [-1, 1, 10, 6, -2, -1, 11, -13, 13, 1, -10, -20], [-13, -20, -7, -1, 0, 0, 20, 0, 7, 0, 0, 13], [-13, -13, -7, 9, 4, -1, 19, -6, 4, 10, 1, 5], [9, 0, -5, -3, 2, 4, 11, 2, 0, 2, 0, -9], [-19, -7, -7, 19, 14, -1, 13, 6, -8, 0, -1, 33], [7, -14, 3, -7, -6, 0, 3, -10, 6, -4, -14, -7], [-24, -11, 1, 0, 13, -9, 10, -10, 2, 10, 0, 9], [-30, -22, 10, 28, -2, 10, 2, 4, 23, 13, 4, 10], [-31, -17, 2, 24, 6, -9, 3, 6, 11, 13, 3, 7], [17, 10, -13, -7, -2, 2, -8, 8, -3, -10, 10, -9], [3, 8, 0, -16, -2, 2, -6, -2, 4, 0, 0, -5], [-21, -18, -12, 5, 10, -14, 18, -16, 2, 0, -8, 10], [-24, 2, -4, 3, 25, -4, -8, 3, -8, 3, 6, -1], [-9, 2, -3, 9, 14, 8, 16, 6, -3, -2, -2, 3], [-11, -3, 3, 1, 6, -5, 7, -8, 2, 0, -5, 7], [6, 12, 8, -7, -7, 6, 14, -3, 0, -3, 0, -29], [-9, 1, 0, -8, -18, 7, 5, 0, 7, 1, 13, -11], [33, 37, -12, -18, -18, 13, -17, 13, -13, -13, 12, -6], [-1, -10, 3, 0, 4, -6, 14, -5, 5, 6, -3, 5], [11, -14, -7, -8, -4, -4, -4, 7, 7, -4, -7, -1], [6, 19, 21, -14, -15, 7, -12, 4, 0, 4, 0, -23], [10, 12, 14, -12, -12, 14, 2, -6, 3, -11, -6, -12], [5, -12, -5, 24, 12, 5, 11, 9, -9, 4, 9, -4], [7, 6, 23, 11, 14, 4, -15, 4, 0, 4, 0, 3], [15, 6, 10, -18, -8, 18, 29, -20, 20, -18, -10, -8], [-4, -14, -4, 4, 2, -20, 0, -6, -4, 4, 0, -4], [10, 10, 9, -9, -1, 6, -21, 3, -2, -16, -6, -8], [3, 13, 0, 0, 10, 7, 3, 0, 7, -3, -13, 23], [-16, -24, 0, 15, 21, -22, -4, 11, 0, 11, 0, 19], [5, -14, -7, -4, 6, 0, 4, 13, 1, -7, -8, 11], [3, 8, 11, -21, -6, -4, 2, -20, 11, -16, -16, -19], [5, -8, -11, -19, -11, 0, 12, -12, 8, -11, -19, 11], [-1, -9, -22, 4, -8, -15, -15, 14, -7, 11, 7, 5], [9, -3, 0, 6, 6, -5, -7, 0, -5, -3, 1, -7], [11, -7, -8, -18, -8, 3, -3, -1, -1, 3, -8, -10], [1, -1, -10, 0, 34, -11, 9, -9, -9, -11, -10, 10], [0, 16, 0, 3, -2, -14, -19, 0, -14, 3, 21, -2], [-21, -19, -2, 6, 10, -9, 17, -14, 7, 1, -7, 11], [-22, -4, 16, 16, -4, 0, 12, -8, 22, 16, -6, 8], [13, 5, -5, -15, 10, 11, -9, 16, -2, 0, 11, -5], [-8, 6, 4, 2, -10, 0, 8, -16, 2, 4, 2, 2], [25, 25, 1, -27, -24, 5, -27, -4, 2, -6, -5, -3], [-9, -24, -9, 24, 22, 0, 0, 9, -7, 9, 2, 31], [19, 19, -14, -4, 2, 0, 0, -14, -6, -12, 0, -6], [-5, -15, 2, -6, 6, -9, 7, -2, 11, 9, -1, 5], [3, -2, 1, -4, -6, 1, 1, 3, 3, 2, 3, -2], [2, 7, -2, 29, -7, 2, 3, 18, -14, 12, 10, 5], [13, -2, -9, -2, -6, -9, -11, 3, -1, 8, 3, -2], [-6, -10, -6, -9, -7, -8, 0, 15, 0, 15, 0, -3], [3, 3, 0, 2, 8, 6, 6, -6, 0, -12, -6, -6], [15, 5, 2, -10, 2, -5, 3, 7, -7, 5, -2, 12], [1, -18, -7, 19, 6, -6, 12, 8, -9, -2, 2, 7], [-17, -6, 0, -7, 8, 6, -10, 0, 6, 10, 2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_29_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_29_2_e_a();".
// 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_29_2_e_a(:prec:=12)
    chi := MakeCharacter_29_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_29_2_e_a();".
// 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_29_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_29_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;