
// Make newform 224.2.q.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_q_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_224_2_q_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_224_2_q_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 := [4, -12, 96, -340, 624, -654, 431, -189, 48, 1, 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], [-9570816, -2665028, 58499210, -118229724, 103674982, -53823631, 19540967, -865016, -1014429, -701704, 407977, -6789], [-7508164, -19439928, 23756054, 15569686, -20646084, 8507441, -4710591, -2520360, 1428477, 509986, -171179, -76837], [-178398, 26100310, -80402588, 115305627, -91525793, 49851815, -17461376, 2261772, 184300, 439891, -302383, 54041], [-2572044, 70472, -12968760, 16374455, 3585319, -14441625, 9850368, -5748844, 592838, 563109, 82977, -127195], [15787424, -31162314, 4078592, 41121960, -75428613, 71188533, -34529048, 12614519, -877952, -714509, -410251, 295642], [27316, -142154, 208368, 212358, -648191, 580379, -323728, 113135, 1316, -6307, -4785, 2236], [3770204, 7943010, 33607448, -178977012, 255549677, -193796667, 96716674, -29122539, -594282, 959613, 1471277, -588552], [-5832856, 19832586, 1135708, 45177602, -131228611, 131465595, -72465116, 30777883, -1848624, -2171303, -892121, 658600], [-2329388, -29719866, 211570982, -478822416, 520339833, -342514582, 152603387, -35847041, -2200105, -349445, 2509218, -721311], [938824, -33967380, 36475726, 136789536, -285355566, 248926171, -132130403, 45642708, -304551, -2371448, -1864021, 944653], [76880, -452518, 1622152, -3042430, 3227207, -2085585, 920210, -224935, -9766, -1205, 15163, -4510]];
    Rf_basisdens := [1, 5500348, 5500348, 2750174, 2750174, 5500348, 41356, 5500348, 5500348, 5500348, 5500348, 26068];
    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_224_q();"
function MakeCharacter_224_q()
    N := 224;
    order := 6;
    char_gens := [127, 197, 129];
    v := [3, 3, 5];
    // 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_224_q_Hecke();"
function MakeCharacter_224_q_Hecke(Kf)
    N := 224;
    order := 6;
    char_gens := [127, 197, 129];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]];
    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 := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0], [1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, -1, 0, 0, 0, -1], [0, 2, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, -1, 0, 0, -1, 0, 0, -2, 0, 0], [0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, -1], [0, 0, 1, 0, 1, -1, 0, 0, 0, 0, 1, 2], [0, 0, -2, 0, 2, -3, 0, 1, 0, 0, 0, 3], [2, -1, 0, -2, 0, 0, 2, 0, -3, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 2, -1, 0, 3, 0, 0, 2, -1], [0, 0, 4, 0, -1, -1, 0, 0, 0, 0, -4, 0], [-2, 4, 0, 2, 0, 0, -2, 0, -1, 0, 0, 0], [0, 0, -2, 0, 2, 1, 0, 1, 0, 0, 4, 1], [-6, 0, 0, 2, 0, 0, 4, 0, 1, -2, 0, 0], [0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2], [1, 4, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, 2, 0, -3, 0, 0, 0, -2], [2, -2, 0, -4, 0, 0, -8, 0, 2, -2, 0, 0], [2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-6, -1, 0, -2, 0, 0, -2, 0, 7, -7, 0, 0], [0, 0, 0, 0, 2, -2, 0, -1, 0, 0, 0, 4], [0, 0, -1, 0, -5, 0, 0, -5, 0, 0, 2, 0], [-2, -3, 0, -3, 0, 0, 1, 0, 4, -2, 0, 0], [0, 0, 2, 0, 2, 4, 0, -1, 0, 0, -2, 0], [2, -1, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -4, 4], [5, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, 0], [-6, 0, 0, -1, 0, 0, 0, 0, 3, -6, 0, 0], [-2, 2, 0, 4, 0, 0, 8, 0, -2, 2, 0, 0], [0, 0, 0, 0, -3, 2, 0, 1, 0, 0, 0, -2], [0, 0, -1, 0, 5, 1, 0, 2, 0, 0, 1, 0], [0, 0, 2, 0, -1, 1, 0, -2, 0, 0, 2, -2], [-3, -2, 0, -2, 0, 0, -6, 0, 6, -3, 0, 0], [0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 1], [0, 0, 0, 0, -2, -3, 0, 1, 0, 0, 0, -3], [-7, 0, 0, 1, 0, 0, 15, 0, -4, 8, 0, 0], [0, 0, 4, 0, 6, 0, 0, 0, 0, 0, -2, 0], [0, 0, 1, 0, 4, -7, 0, 3, 0, 0, 0, 7], [4, 0, 0, -1, 0, 0, 2, 0, -3, 6, 0, 0], [0, 0, 0, 0, 4, 0, 0, 5, 0, 0, 2, 3], [0, 0, -1, 0, 3, -3, 0, 6, 0, 0, -1, 6], [0, -4, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0], [0, 0, 12, 0, -4, 0, 0, 0, 0, 0, -6, 0], [9, 2, 0, 1, 0, 0, 9, 0, -12, 0, 0, 0], [0, 0, -2, 0, 1, -2, 0, 3, 0, 0, 4, -2], [-1, 3, 0, 3, 0, 0, -8, 0, 2, -1, 0, 0], [0, 0, 0, 0, -4, 0, 0, -1, 0, 0, 6, -7], [-1, -4, 0, -2, 0, 0, -1, 0, -2, 0, 0, 0], [-10, -1, 0, -2, 0, 0, 18, 0, 1, -1, 0, 0], [1, -5, 0, 5, 0, 0, 2, 0, 0, 5, 0, 0], [0, 0, -1, 0, -4, 6, 0, -5, 0, 0, 1, 0], [0, 0, -6, 0, 3, -3, 0, -2, 0, 0, -6, 6], [0, 0, -5, 0, -1, -4, 0, 3, 0, 0, 10, -4], [0, 0, -8, 0, -3, 1, 0, -2, 0, 0, 8, 0], [-2, 1, 0, 0, 0, 0, 0, 0, -5, -5, 0, 0], [-13, -6, 0, -3, 0, 0, -13, 0, 10, 0, 0, 0], [0, 0, 8, 0, -10, 12, 0, -6, 0, 0, -4, -6], [10, 6, 0, 12, 0, 0, -12, 0, -4, 4, 0, 0], [0, 0, -5, 0, -2, 2, 0, -3, 0, 0, -5, -4], [16, 0, 0, 0, 0, 0, -7, 0, 0, 2, 0, 0], [0, 0, -8, 0, 6, -7, 0, 1, 0, 0, 0, 7], [5, 12, 0, 12, 0, 0, 6, 0, -10, 5, 0, 0], [-2, -5, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0], [26, 0, 0, 0, 0, 0, -20, 0, -3, 6, 0, 0], [0, 0, 12, 0, -4, -6, 0, 3, 0, 0, -6, 3], [2, -6, 0, -3, 0, 0, 2, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, -1], [0, 0, 1, 0, -2, 2, 0, -7, 0, 0, 1, -4], [0, 0, 4, 0, -2, 1, 0, 1, 0, 0, 0, -1], [-2, -9, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 5, 0, -6, -1, 0, -5, 0, 0, -10, -1], [0, 0, 0, 0, 12, 10, 0, 1, 0, 0, 0, 0], [0, 0, 2, 0, -3, 2, 0, -5, 0, 0, -4, 2], [-3, -1, 0, -1, 0, 0, 6, 0, 6, -3, 0, 0], [-6, 6, 0, 3, 0, 0, -6, 0, -7, 0, 0, 0], [6, -2, 0, -4, 0, 0, -28, 0, 8, -8, 0, 0], [0, 0, 0, 0, 4, 0, 0, 13, 0, 0, -4, -5], [0, 0, -7, 0, 11, 3, 0, 4, 0, 0, 7, 0], [10, 7, 0, 14, 0, 0, -2, 0, -9, 9, 0, 0], [0, 0, -3, 0, 4, -3, 0, 7, 0, 0, 6, -3], [4, 1, 0, 1, 0, 0, 11, 0, -8, 4, 0, 0], [-2, 5, 0, 0, 0, 0, 0, 0, -3, -3, 0, 0], [0, -6, 0, -6, 0, 0, -15, 0, 0, 0, 0, 0], [0, 0, -12, 0, 4, -10, 0, 5, 0, 0, 6, 5], [0, 0, 0, 0, 2, 0, 0, 10, 0, 0, -6, -6], [-5, 0, 0, 0, 0, 0, -4, 0, 0, -13, 0, 0], [0, 0, -1, 0, 6, -6, 0, 13, 0, 0, -1, 12], [0, 0, -1, 0, 0, -6, 0, 3, 0, 0, 1, 0], [-10, -1, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0], [5, -2, 0, -2, 0, 0, -2, 0, -10, 5, 0, 0], [0, 0, -12, 0, 4, -16, 0, 8, 0, 0, 6, 8], [0, 0, 4, 0, -7, -2, 0, -5, 0, 0, -8, -2], [7, -12, 0, -6, 0, 0, 7, 0, 6, 0, 0, 0], [-4, 1, 0, -1, 0, 0, 7, 0, 0, 10, 0, 0], [0, 0, 8, 0, 1, 2, 0, -3, 0, 0, 0, -2], [2, -11, 0, 0, 0, 0, 0, 0, 11, 11, 0, 0], [0, 0, -2, 0, 2, -12, 0, 7, 0, 0, 2, 0], [8, 4, 0, 2, 0, 0, 8, 0, 9, 0, 0, 0], [8, 2, 0, 2, 0, 0, 25, 0, -16, 8, 0, 0], [15, 0, 0, -5, 0, 0, -19, 0, 2, -4, 0, 0], [5, 8, 0, 4, 0, 0, 5, 0, 4, 0, 0, 0], [-14, -2, 0, -4, 0, 0, 28, 0, 0, 0, 0, 0], [3, 3, 0, -3, 0, 0, -8, 0, 0, -13, 0, 0], [0, 0, 5, 0, -10, 4, 0, -7, 0, 0, -5, 0], [-6, -3, 0, -6, 0, 0, -22, 0, 17, -17, 0, 0], [0, 0, 1, 0, 2, 7, 0, -5, 0, 0, -2, 7], [0, 0, 6, 0, -4, -6, 0, 1, 0, 0, -6, 0], [6, 13, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [-10, 16, 0, 8, 0, 0, -10, 0, -9, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, -12, 0, 0, -4, -8], [-5, 0, 0, 6, 0, 0, 1, 0, 2, -4, 0, 0], [-6, -2, 0, -4, 0, 0, -12, 0, 12, -12, 0, 0], [0, 0, 7, 0, -4, 4, 0, -1, 0, 0, 7, -8], [0, 0, -12, 0, -4, 7, 0, -3, 0, 0, 0, -7], [12, 6, 0, 0, 0, 0, 0, 0, -8, -8, 0, 0], [0, 0, -2, 0, 5, -5, 0, 0, 0, 0, -2, 10], [10, -13, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0], [0, 0, 6, 0, 7, 4, 0, 3, 0, 0, -12, 4], [21, 0, 0, 11, 0, 0, -21, 0, 0, 0, 0, 0], [17, 8, 0, -8, 0, 0, -10, 0, 0, -3, 0, 0], [0, 0, 0, 0, -8, 0, 0, -4, 0, 0, 8, -12], [0, 0, -10, 0, 4, 3, 0, -7, 0, 0, 0, -3], [0, 0, -1, 0, -5, 0, 0, -5, 0, 0, 2, 0], [0, 0, 12, 0, -4, 16, 0, -8, 0, 0, -6, -8], [0, 0, -8, 0, 8, -5, 0, 13, 0, 0, 16, -5], [-23, 0, 0, 7, 0, 0, 15, 0, 4, -8, 0, 0], [0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 8, -6], [0, 0, 1, 0, 2, -9, 0, 7, 0, 0, 0, 9], [0, 0, 0, 0, -8, 0, 0, -13, 0, 0, -10, -3], [41, 3, 0, -3, 0, 0, -18, 0, 0, 5, 0, 0], [2, 1, 0, 2, 0, 0, 22, 0, -13, 13, 0, 0], [0, 0, 0, 0, 8, -8, 0, 13, 0, 0, 0, 16], [8, 16, 0, 8, 0, 0, 8, 0, 9, 0, 0, 0], [0, 0, -12, 0, 4, -6, 0, 3, 0, 0, 6, 3], [-1, 0, 0, 12, 0, 0, 5, 0, -2, 4, 0, 0], [6, 2, 0, 4, 0, 0, 12, 0, -12, 12, 0, 0], [0, 0, 10, 0, 1, -10, 0, 9, 0, 0, 0, 10], [0, 0, -3, 0, 10, 0, 0, 5, 0, 0, 3, 0], [-16, 10, 0, 0, 0, 0, 0, 0, -6, -6, 0, 0], [0, 0, 10, 0, 2, -2, 0, 7, 0, 0, 10, 4], [0, 0, 0, 0, 0, 12, 0, -6, 0, 0, 0, -6], [0, 0, 0, 0, 0, -10, 0, 5, 0, 0, 0, 5], [-11, 12, 0, 6, 0, 0, -11, 0, -12, 0, 0, 0], [-14, 1, 0, -1, 0, 0, 5, 0, 0, -4, 0, 0], [0, 0, 7, 0, 3, 0, 0, -3, 0, 0, 0, 0], [0, 0, 4, 0, -9, 16, 0, -7, 0, 0, 0, -16], [8, -10, 0, -20, 0, 0, -8, 0, -4, 4, 0, 0], [-8, -2, 0, 0, 0, 0, 0, 0, -12, -12, 0, 0], [0, 0, 7, 0, -10, -5, 0, -5, 0, 0, -14, -5], [0, 0, 0, 10, 0, 0, -14, 0, 7, -14, 0, 0], [0, 0, 0, 0, 2, 0, 0, -6, 0, 0, 6, 10], [0, 0, 11, 0, -8, 9, 0, -1, 0, 0, 0, -9], [-26, -2, 0, 2, 0, 0, 7, 0, 0, -12, 0, 0], [-20, 4, 0, 8, 0, 0, 8, 0, 16, -16, 0, 0], [0, 0, 0, 0, -7, 7, 0, 0, 0, 0, 0, -14], [7, -14, 0, -14, 0, 0, 0, 0, -14, 7, 0, 0], [-14, -11, 0, 0, 0, 0, 0, 0, 11, 11, 0, 0], [0, 0, 0, 0, -2, 0, 0, -6, 0, 0, -8, 2], [3, 10, 0, -10, 0, 0, 0, 0, 0, 3, 0, 0], [28, 0, 0, 19, 0, 0, -6, 0, -11, 22, 0, 0], [0, 0, 1, 0, 9, -9, 0, 4, 0, 0, 1, 18], [0, 0, -7, 0, -12, 6, 0, -9, 0, 0, 7, 0], [0, 0, 2, 0, 1, -1, 0, -12, 0, 0, 2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_2_q_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_224_2_q_a(:prec:=12)
    chi := MakeCharacter_224_q();
    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_224_2_q_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_224_2_q_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;