
// Make newform 900.2.w.b in Magma, downloaded from the LMFDB on 07 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_w();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_w_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_900_2_w_b();".
// 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_900_2_w_b();".
// 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 := [625, 0, -375, 0, 225, 0, -160, 0, 161, 0, 2, 0, -16, 0, -2, 0, 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, 0, 0, 0, 0], [-65470000, 0, -8261800, 0, 23271280, 0, 7126104, 0, -1162872, 0, -475624, 0, -27053, 0, 424, 0], [0, -154461375, 0, -8261800, 0, 23271280, 0, 7126104, 0, -1162872, 0, -475624, 0, -27053, 0, 424], [-70231500, 0, 6018075, 0, -21242385, 0, -8400729, 0, 2743022, 0, 1114449, 0, -55722, 0, -33864, 0], [16012750, 0, -7455600, 0, 32426760, 0, 14968229, 0, -6084772, 0, -2459824, 0, 175022, 0, 129364, 0], [-380000, 0, 2695300, 0, -96005, 0, 60391, 0, -74713, 0, -29946, 0, -1887, 0, 2721, 0], [-29255125, 0, -74017925, 0, -65737880, 0, 36109891, 0, 2754462, 0, -2875471, 0, -378462, 0, 188171, 0], [0, 159222875, 0, -6018075, 0, 21242385, 0, 8400729, 0, -2743022, 0, -1114449, 0, 55722, 0, 33864], [-97130250, 0, 6668850, 0, -77346460, 0, 61495891, 0, 7283237, 0, -5047396, 0, -1033862, 0, 278181, 0], [-1358751, 0, 1228125, 0, -292500, 0, 425000, 0, -46875, 0, -50748, 0, -3120, 0, 3470, 0], [0, 202542125, 0, -507691175, 0, -165296805, 0, 42404064, 0, 34311773, 0, 1839391, 0, -1535348, 0, -273046], [0, 96167375, 0, 17961125, 0, -57726935, 0, -20819708, 0, 5667494, 0, 2296623, 0, -65194, 0, -96348], [0, -256343375, 0, -158863075, 0, -311390520, 0, 212837346, 0, 28464847, 0, -16463101, 0, -2979772, 0, 913556], [0, 500401625, 0, -629976575, 0, 19419905, 0, -127479609, 0, 8757412, 0, 15500754, 0, 1593038, 0, -954849], [0, 14684625, 0, 17154925, 0, -66882415, 0, -28661833, 0, 10589394, 0, 4280823, 0, -267269, 0, -225288], [0, 646677250, 0, -259886950, 0, 348109305, 0, -308029064, 0, -5014898, 0, 29878109, 0, 3485348, 0, -1895704]];
    Rf_basisdens := [1, 88991375, 88991375, 88991375, 88991375, 1618025, 88991375, 88991375, 88991375, 711931, 444956875, 88991375, 444956875, 444956875, 88991375, 444956875];
    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_900_w();"
function MakeCharacter_900_w()
    N := 900;
    order := 10;
    char_gens := [451, 101, 577];
    v := [10, 10, 7];
    // 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_900_w_Hecke();"
function MakeCharacter_900_w_Hecke( : Kf := false)
    N := 900;
    order := 10;
    char_gens := [451, 101, 577];
    char_values := [[1, 0, 0, 0, 0, 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, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 0, 0, 0, 0, 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, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, -1, 0, -1, 0, -1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 1, 1, 0, 0, 1], [0, -2, 0, -1, -1, -1, -1, 0, 1, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, -1, -1, -1], [1, -1, 0, 0, 0, -1, -1, 0, 1, 2, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, -1, -1, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, -1, 0, -2], [1, 1, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 1, 1, 1], [0, 0, 1, 0, 0, 0, 0, -2, 0, 0, 1, 1, -2, -3, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, -3, -2], [-2, 0, 0, -5, -4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [1, -2, 0, -1, 2, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 2, 0, 0, -3, -1, 3, 1, 0, 4], [1, -4, 0, -2, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 5, 0, 1, 2, 3, 2, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -1, 1, 4, 2, -2, 0], [0, 0, 1, 0, 0, 0, 0, 2, 0, 0, -3, 2, 2, 2, -4, 4], [2, 2, 0, 4, 1, 2, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, -3, 2, -1, 0, 0], [2, 4, 0, 6, 4, 1, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 5, -1, -3, -2, 3, -4], [-1, 7, 0, 3, 1, 2, 1, 0, -4, -3, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 3, 1, -1], [-6, -4, 0, -2, 3, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 2, 0, 0, -3, -3, -2, -1, -1, -2], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 5, -5, -5, -1, -4], [-2, 0, 0, 4, 2, -2, 2, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 4, 3, -2, 1, -5, 2], [2, 7, 0, 3, 2, 8, 6, 0, -2, -4, 0, 0, 0, 0, 0, 0], [-4, 5, 0, -8, -6, 0, 3, 0, -3, -5, 0, 0, 0, 0, 0, 0], [1, 1, 0, -5, 8, -1, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 3, 0, 0, -2, 2, 1, -2, 2, 3], [0, 0, 5, 0, 0, 0, 0, -2, 0, 0, -6, 2, -1, 3, 1, 1], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 2, -6, -1, -5, -3], [5, 7, 0, 8, 4, 3, 3, 0, -3, -6, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, -4, -4, -6, 1, -3], [0, 6, 0, 0, 3, 0, 3, 0, -3, -6, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 3, 2, -1, 2, -2, -1], [0, -1, 0, -3, 1, 4, 3, 0, -1, -2, 0, 0, 0, 0, 0, 0], [-6, 0, 0, -9, -4, -3, 3, 0, 2, -2, 0, 0, 0, 0, 0, 0], [-1, -6, 0, -3, -9, 7, 7, 0, -4, -4, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 1, 0, 0, -2, 1, 5, 5, 4, 4], [-3, -5, 0, -2, 7, -5, -4, 0, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -3, 3, -4], [0, 0, -1, 0, 0, 0, 0, -2, 0, 0, -4, 0, 5, 3, -3, 3], [-1, -8, 0, 4, 4, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 5, 0, 0, -2, -2, 7, 5, -3, 0], [0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 1, 6, 2, -4, -3, 2], [0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 4, 3, -3, -1, -4, 1], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -4, -4, -6], [8, 9, 0, 1, 2, 3, 2, 0, -1, 1, 0, 0, 0, 0, 0, 0], [-13, -16, 0, -8, -1, -1, -1, 0, 2, 2, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, -3, 0, 0, -7, -7, 4, 4, 4, 3], [-9, 1, 0, 6, -4, -5, -3, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, -4, 0, 0, -2, 0, 4, 4, 2, 8], [-6, 3, 0, -12, -8, 0, -1, 0, 1, -3, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 3, -3, -2, 3, 0, 0], [14, 7, 0, -1, 13, 2, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, -4, 4, 0, 3, -3, 3], [9, -2, 0, 7, 0, -3, -2, 0, 1, 4, 0, 0, 0, 0, 0, 0], [13, 4, 0, 7, 5, -4, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, -3, 0, 0, -4, -1, -4, 1, 8, 3], [13, 3, 0, 0, 1, 6, 4, 0, -2, -3, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 3, -4, -6, -2, -3], [0, 0, -3, 0, 0, 0, 0, -6, 0, 0, -4, -2, -2, 1, -1, -3], [7, 4, 0, -2, 1, 7, 5, 0, -7, 0, 0, 0, 0, 0, 0, 0], [-3, -26, 0, -13, -18, -6, -6, 0, 5, 5, 0, 0, 0, 0, 0, 0], [14, 13, 0, -1, 10, -1, -2, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, -2, 0, 0, 1, -1, -7, -1, 1, -2], [0, 0, -2, 0, 0, 0, 0, 3, 0, 0, 2, 1, -7, -5, 4, -4], [12, -6, 0, 6, 3, -3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, -2, 0, 0, -3, 0, -2, -2, 5, 2], [3, 17, 0, 0, 1, -1, 2, 0, 2, 4, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 4, 4, -1, -4, -4, 1], [7, 4, 0, -3, -9, 1, -6, 0, 3, 7, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 5, 0, 0, 5, 5, -5, -3, -3, -5], [-5, 11, 0, 11, -5, 3, 6, 0, -3, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 1, -11, 2, -2, 0, -6, 6, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, -1, 0, 0, -4, -3, -2, 1, -3, 3], [0, 0, 1, 0, 0, 0, 0, 5, 0, 0, -3, -1, 6, 5, -2, 4], [-4, 4, 0, -22, -13, 0, -5, 0, 5, 10, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, 0, -5, 0, 0, -3, -4, 2, 1, 3, -3], [16, 5, 0, -4, 16, -2, -4, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, -3, 0, 0, 7, -7, -3, -6, 6, -3], [0, 0, -1, 0, 0, 0, 0, -9, 0, 0, 1, -6, 4, 8, 5, 5], [22, 12, 0, 20, 1, 3, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 3, 0, 0, 1, 2, 2, -3, -4, -2], [-4, -22, 0, -16, -2, -12, -8, 0, 4, 6, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, 3, 0, 0, -7, -4, 7, 10, 0, 3], [0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 4, -3, -3, -1, 3, 0], [0, 0, 1, 0, 0, 0, 0, 5, 0, 0, -1, 7, 0, -4, -1, -6], [-13, -38, 0, -19, -22, -8, -8, 0, 3, 3, 0, 0, 0, 0, 0, 0], [6, 1, 0, 3, 1, 2, 8, 0, -4, 4, 0, 0, 0, 0, 0, 0], [7, 1, 0, 8, 3, -6, 0, 0, -2, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -6, -5, -1, 4, -2, 4], [0, 0, -3, 0, 0, 0, 0, 5, 0, 0, -12, -7, 7, 4, 4, 3], [0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 1, -1, 7, 7, -2], [16, 18, 0, 2, 0, -3, 4, 0, -2, -7, 0, 0, 0, 0, 0, 0], [-17, -12, 0, -6, -1, 7, 7, 0, -6, -6, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 0, 0, -2, 0, 0, 0, -2, 2, -7, 6, -4], [0, 0, -6, 0, 0, 0, 0, -2, 0, 0, 5, -4, -2, 2, 7, -4], [0, 0, -3, 0, 0, 0, 0, -8, 0, 0, 8, -5, -3, -8, 0, -5], [-15, -13, 0, -11, 6, -4, 4, 0, 8, 2, 0, 0, 0, 0, 0, 0], [-17, -10, 0, -30, -38, 0, -2, 0, 2, 6, 0, 0, 0, 0, 0, 0], [18, 14, 0, 0, 23, 5, 5, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, -6, 0, 0, 5, -5, -4, 3, -3, -6], [-14, -19, 0, -12, -21, -4, 2, 0, -8, -4, 0, 0, 0, 0, 0, 0], [10, -11, 0, 7, -8, -1, -3, 0, 5, 6, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 0, 0, -11, 0, 0, -9, -2, -2, 7, 7, 5], [0, 13, 0, -2, 9, 0, 2, 0, -2, -11, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, -5, 0, 0, 2, -6, -4, -7, 2, -8], [0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 5, 2, 0, -4, 3], [0, 0, 3, 0, 0, 0, 0, -4, 0, 0, -6, 2, 2, -1, 9, 11], [0, 0, 0, -6, 6, 2, -2, 0, 6, -6, 0, 0, 0, 0, 0, 0], [-3, -26, 0, -13, -22, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, -7, 0, 0, 9, 2, -13, -4, -6, -6], [0, 0, -4, 0, 0, 0, 0, 6, 0, 0, 3, -3, -4, -1, 1, 6], [-3, 8, 0, 13, 9, 4, 1, 0, -8, -7, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 8, -8, 0, 0, 2], [2, 9, 0, 7, 13, -6, -4, 0, 12, 8, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, -12, 0, 0, -9, -9, 4, 6, 6, 12], [-26, -2, 0, -1, 13, 7, 7, 0, -11, -11, 0, 0, 0, 0, 0, 0], [-12, -1, 0, 18, 7, -3, 6, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-23, 0, 0, 2, -18, 5, -5, 0, 5, -5, 0, 0, 0, 0, 0, 0], [0, 0, -10, 0, 0, 0, 0, 4, 0, 0, 1, -1, -5, -4, -3, -14], [-27, -17, 0, -18, -8, 2, -7, 0, -9, -1, 0, 0, 0, 0, 0, 0], [3, 23, 0, -17, 4, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, -2, 0, 0, -6, 4, 1, -5, 1, -1], [-5, 7, 0, 5, -3, 11, 10, 0, -9, -20, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, -3, 0, 0, -3, -5, 7, 5, 10, 10], [4, -2, 0, 7, -1, -10, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, -4, 0, 0, -2, -7, 3, -2, -1, -5], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 10, -4, -4, -4, -2, -6], [-23, 0, 0, -5, -15, 1, -1, 0, 8, -8, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, -12, 0, 0, 7, -3, 0, 5, 7, -4], [-25, -42, 0, -21, -21, 0, 0, 0, -4, -4, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, -8, 0, 0, -6, -13, 3, -2, 11, 11], [8, 14, 0, 6, 26, 3, 12, 0, -6, -9, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, 0, -10, 0, 0, 9, -10, -5, -5, 15, -15], [-14, 13, 0, -1, 3, 7, 0, 0, 7, -7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 0, -12, 2, 7, 5, -7, 5], [1, 26, 0, 8, 6, 2, 3, 0, -4, -1, 0, 0, 0, 0, 0, 0], [0, 0, -12, 0, 0, 0, 0, 3, 0, 0, -5, -2, 2, -10, -2, 4], [-14, -6, 0, -3, 2, 11, 11, 0, -6, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 8, 8, 0, 2, 2, -2], [-8, 24, 0, 7, -25, 2, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, -10, 0, 0, 7, -3, 3, 5, -6, 9], [-17, -8, 0, -14, -21, 0, -1, 0, 1, -12, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, -7, 0, 0, 5, -4, -1, -4, 8, -8], [3, -14, 0, 1, -12, -4, -2, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 6, 0, 0, -2, 1, 14, -4, 12, 1], [25, 19, 0, 19, 17, 2, 11, 0, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, 3, 0, 0, -3, 6, 6, 11, -5, 1], [0, 0, -14, 0, 0, 0, 0, -5, 0, 0, -14, -6, 9, 9, 5, 8], [0, 0, -14, 0, 0, 0, 0, -6, 0, 0, -14, -8, -2, 8, 16, 6], [-1, -21, 0, -11, 9, 1, -12, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 5, 0, 0, -1, 9, -9, -4, -10, -8], [0, 0, 7, 0, 0, 0, 0, 4, 0, 0, -1, 0, -2, -6, -6, -6], [0, 0, 2, 0, 0, 0, 0, 8, 0, 0, -5, 5, 2, 11, -11, 8], [2, 18, 0, 19, 17, 2, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0], [11, 1, 0, 12, 11, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_w_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_2_w_b();".
// 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_900_2_w_b(:prec:=16)
    chi := MakeCharacter_900_w();
    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_900_2_w_b();".
// 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_900_2_w_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_w();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![121, 0, 359, 0, 166, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;