
// Make newform 2541.2.a.bq in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2541_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_2541_2_a_bq();".
// 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_2541_2_a_bq();".
// 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 := [-11, -70, 253, 17, -316, 6, 124, -1, -19, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0], [385, -8301, -4152, 11069, 2537, -4093, -505, 558, 31, -25], [-2409, -3843, 4152, -9045, -2537, 4093, 505, -558, -31, 25], [275, 1365, 1976, 3727, -3169, -2687, 993, 530, -83, -31], [-5995, -29941, 43072, 43393, -27479, -16569, 5327, 2246, -325, -97], [-7403, -13617, 5848, 34681, 4861, -19033, -2197, 3654, 203, -229], [9449, -24601, -12520, 49477, 6253, -25813, -1709, 4742, 151, -285], [-1331, 16260, -8412, -22543, 5850, 8981, -1116, -1360, 65, 70]];
    Rf_basisdens := [1, 1, 1, 2024, 2024, 2024, 2024, 2024, 2024, 506];
    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_2541_a();"
function MakeCharacter_2541_a()
    N := 2541;
    order := 1;
    char_gens := [848, 1816, 2059];
    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_2541_a_Hecke(Kf)
    return MakeCharacter_2541_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, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1, 0, 0, 0, 1], [-1, 1, -1, 0, 0, -1, 0, 0, 0, 0], [-1, 0, 0, -1, -1, 0, 0, 1, 0, 0], [0, 0, -1, 1, 1, 0, -1, 0, 0, 0], [1, 1, 1, -1, 0, 0, 0, 0, 1, -1], [2, 0, 1, -1, 0, -1, -1, -1, -1, -1], [3, 1, -1, 2, 1, 0, 0, 1, -1, -1], [-2, 0, 0, 1, 0, 1, -1, 0, 0, -1], [1, -1, -1, 1, 0, 0, 0, 0, 1, 1], [1, -1, -2, -1, 0, 1, 1, 1, 0, 2], [0, 0, -1, 1, 0, -2, -1, -1, 1, 0], [1, 1, 2, -2, -2, 0, -1, -1, 0, -1], [0, -2, 0, -1, 0, -1, 2, 2, 0, 1], [5, -1, -1, 2, 0, -1, 2, 2, 1, 0], [5, 1, -1, 5, 0, 2, 0, 0, 1, 1], [-1, -3, 0, -2, 0, 2, -1, -1, 0, 1], [0, -2, -1, 1, 0, 0, 1, 1, 1, 0], [-2, 0, 0, -4, 0, -2, 0, -2, 0, 0], [-2, 0, 1, -2, 0, -3, -1, -1, 0, -1], [2, -2, 0, -1, 0, -1, 0, 0, 0, 1], [-7, -1, 0, -3, 0, 3, -2, 1, 0, 0], [1, 2, -2, 0, -1, -3, 0, 1, 0, -1], [0, -2, -1, 0, 1, 3, -1, 0, 0, 1], [2, -4, -1, 4, 1, 2, 1, 2, -1, 0], [-1, 3, -1, -3, 0, -2, 2, 0, 1, -1], [0, 2, 0, -1, 1, -3, 2, -1, 0, 1], [1, 1, 0, 3, 2, -1, -1, -1, -2, -2], [-3, -1, 0, 0, 1, 0, -1, 0, 0, -3], [4, 1, 0, 3, -1, 0, -1, -2, 2, 1], [5, -1, 0, 0, 0, 2, -1, 1, -2, -3], [-2, 0, 0, 0, 3, 0, -2, 1, 0, -2], [-1, -1, -4, -4, 0, 0, 1, 1, 0, 1], [2, 0, 0, -8, 1, 0, 2, -1, 0, 2], [-1, 1, 2, -2, 0, 0, -1, -3, 0, 1], [-4, 0, 3, -1, 0, 0, -3, -3, 0, -4], [4, -4, 2, -2, 1, 2, -1, -1, -2, 0], [4, -4, -2, 1, 0, 3, 2, 2, -2, -1], [2, 0, -1, 0, 2, 3, 0, 1, 1, -1], [3, 3, 2, -2, 0, 1, 3, 1, 0, 0], [-1, -3, 2, -6, -1, 4, -1, -2, 0, -1], [-6, -1, 2, -5, -1, -2, -3, 0, 0, -1], [1, -3, -2, 7, 1, 3, 1, 2, -2, -2], [3, 4, 2, 0, -1, -3, -2, -1, 2, -1], [-3, -1, -2, -2, 0, -4, -1, 1, -2, -3], [8, 2, -2, 6, 2, -2, 2, 0, 2, 0], [2, -4, 2, -9, 0, 1, -2, 0, -2, -3], [-7, 1, -2, -6, 0, -2, 3, 3, 1, 1], [-2, 0, -2, 1, 0, -1, -2, 2, 2, -1], [10, -2, 0, 3, 0, 1, 2, 0, 2, 3], [-1, -5, 1, 5, -2, 2, 4, 2, 0, 3], [8, -2, 4, 6, -2, 0, 0, -2, 1, 4], [-4, 0, 4, -5, -2, -1, -2, -4, 0, 1], [8, 0, 1, -3, 0, 1, 3, 1, 1, 3], [-8, -3, 0, -7, 0, 2, 1, -1, 0, 3], [8, -2, -3, 7, -2, 0, 3, 1, -1, 0], [-2, 2, 1, -5, 2, -1, -7, -1, -3, -3], [-9, 1, 0, 3, 2, -1, -4, -1, 0, -4], [6, -4, -1, 5, 0, -1, -1, -1, -3, -1], [1, 7, 0, -8, 0, -2, 1, 1, -2, 1], [3, 3, -6, 3, 4, -1, 5, 3, 2, 4], [1, -5, 1, 5, -2, 2, 4, 4, 1, 3], [7, -3, 0, 3, 1, -1, -1, 0, -2, -2], [-1, -1, 1, -3, 3, -1, -4, 1, -3, -4], [4, 0, 1, 5, 0, 0, -2, -1, -1, 0], [1, 1, -2, 5, 0, 1, 5, -1, 2, 4], [-4, 8, 0, -3, 0, -1, 4, 2, 0, 3], [13, -1, -2, 6, -1, -2, -2, 2, 0, -1], [6, 2, -1, 11, 4, 1, -1, 3, -1, -1], [8, 6, -5, 2, 2, -2, 5, 1, -1, 2], [6, 8, -2, 5, -1, 3, 2, 5, 2, 1], [-3, -1, -2, 5, 0, -1, -1, 3, 4, -2], [-5, -1, 6, -8, -1, 4, 1, 0, 0, -3], [5, 1, 2, -9, 2, -1, 3, 1, 0, 0], [-2, 0, 6, -8, 1, 2, 2, -3, 0, -4], [7, 5, 2, -7, -4, 1, 3, -3, 2, 2], [-1, -1, 0, -11, 2, 1, 1, 3, 0, 0], [-3, 4, 4, -7, -2, 6, 2, -2, -2, -2], [-1, -5, 4, -12, 1, 4, 2, 2, 0, -3], [10, 2, -2, 8, -2, 0, 6, 2, 0, 0], [0, -2, 1, 7, -2, 7, -1, 5, -1, -3], [-2, 0, 6, -1, -2, 3, 0, -4, 1, -3], [3, 3, 4, 3, 1, 1, -1, -2, -2, -2], [6, 0, 0, 13, 1, -5, -2, -1, 2, -1], [-6, 4, -2, -11, 2, -7, 2, 0, 0, -1], [-4, 4, -3, -2, -2, -3, 3, -3, 1, 3], [4, 2, 4, 3, 2, -5, -2, 2, 0, -1], [9, 3, 6, 5, -2, -1, -1, 3, 2, 0], [-2, -4, 5, -10, 0, 3, -3, -5, -1, -3], [7, -1, -5, 5, 4, 2, 0, 0, 2, 1], [9, 7, -5, -5, 0, -8, 2, 0, 1, 1], [-18, -4, 0, -5, -2, 1, 2, 2, 0, 1], [1, -3, -2, 10, 0, 2, 4, 1, 2, -1], [-9, 3, 0, 4, 0, -2, -2, 1, 0, -3], [-7, 8, 6, -10, -1, -1, 0, -1, 0, -3], [-4, 0, -5, -5, 1, -3, -3, -4, -1, -1], [4, 6, 0, 1, -2, 3, 4, 6, 2, 1], [-1, -5, 3, 0, 0, -5, 4, -2, 1, 4], [-6, 8, 4, -5, 2, -7, -2, -2, 0, -1], [4, 4, -4, -1, -2, -5, 4, 0, 0, 7], [-13, -1, 8, -5, -1, 1, -3, -4, 2, -6], [3, -3, 2, -7, -2, 3, 5, -1, 2, 6], [2, 0, 2, -6, 0, 2, 6, 0, 2, 2], [3, 3, 2, 5, 0, -5, -6, -3, 4, 0], [-11, -3, -5, 8, -3, 3, 0, 5, -2, 4], [-10, -4, -4, -2, -4, 8, 4, 2, 0, 6], [-8, 3, 2, 1, -1, 0, -1, -4, 0, 3], [1, 4, 1, 2, -3, 6, 7, 0, 1, 0], [-10, 2, 1, 1, 4, 2, -1, -1, -3, -2], [6, -6, -1, 0, 4, 4, -1, 3, 1, -4], [5, 3, -2, 3, -5, 1, 5, 6, 2, 6], [-16, 0, -3, -5, 0, 4, 1, 5, 1, -2], [5, 4, -2, 7, 1, 0, -4, -5, 0, 0], [2, 4, 4, 7, -2, 5, -2, 0, -2, -5], [-13, 1, -4, -1, 1, -3, -1, 0, 0, 4], [5, 5, 2, 0, -4, 0, 0, -7, 2, -1], [6, -2, 4, 3, 4, -3, 0, -4, -4, 1], [7, 9, 1, -7, -4, -6, 0, -2, 3, 1], [-1, 1, -4, 0, -2, 8, 1, 1, 0, 3], [-4, -2, -4, -2, -1, 6, 5, 1, 2, 2], [10, -5, -2, 2, -5, -3, 5, 6, 0, 4], [11, 1, -6, 3, 1, -5, 1, 2, 0, -2], [-13, 1, 0, -2, 2, -1, -1, 3, 0, 4], [-3, -1, -4, 8, 2, -2, 1, 5, 0, 3], [-6, -5, 6, -9, -3, 2, -9, 0, -4, -3], [-10, 0, -2, -3, 0, 1, -2, 4, -4, -1], [-14, 2, -1, -10, -2, 1, -3, 1, -3, -3], [7, -3, -5, 12, 2, 1, -3, 0, -3, -2], [4, -4, 0, 3, -1, 5, 4, 3, 2, 3], [10, 5, -3, 7, -3, 1, 6, 5, 5, 8], [3, -9, -2, 7, 0, -1, -5, -5, -2, 0], [8, 2, 10, -6, -4, 0, -6, -6, 2, 0], [-10, 2, -4, 4, 0, -6, -6, -2, -2, 0], [9, 1, -3, -7, 2, -1, 2, 0, -3, 0], [-14, 6, 2, 6, 0, -2, -7, -2, 2, -6], [-14, 8, 0, 12, 3, -4, 0, -3, 0, -2], [8, 6, -2, 0, 6, 0, -2, -4, 0, -2], [-6, 2, 4, 4, -2, -4, -8, -4, 2, -6], [-3, -1, 3, -2, 4, -1, -8, 2, 3, -4], [15, 1, -1, 0, 2, -3, 4, 2, -2, -2], [9, 3, 2, 9, -2, -9, 5, 1, -2, 2], [6, -2, 4, 9, 0, -3, 2, -2, 0, 1], [-1, 7, -2, 2, -2, -6, 1, -1, 0, -3], [-8, -4, -2, 7, 2, 1, 0, 2, -6, -1], [2, -2, -6, 1, 2, 1, 2, 4, -4, 1], [-3, 15, -4, 4, -2, -8, 4, -1, 2, 1], [4, 4, -3, 8, 2, 1, 5, 3, -1, 3], [7, -1, -1, 20, -2, -1, 2, 2, 2, 0], [-5, 1, -2, -9, -2, -7, 1, 5, 0, 2], [0, 8, 2, -5, -2, -13, -2, -8, 6, -1], [6, -2, -1, -2, 2, -7, 3, 1, -1, 5], [-15, 3, 2, -9, 2, 3, -7, -1, -6, -6], [-15, 3, -3, -6, 0, 5, 4, 8, -5, -2], [1, -3, -1, -7, 0, -2, 8, 2, -2, 5], [-14, 8, 0, -13, 2, 1, 0, 4, -6, -7], [-3, -7, 3, 1, -4, 0, 6, -2, 2, 1], [2, 2, -4, -3, 0, 1, -1, 0, -6, -7], [-2, -10, 0, -12, 1, 4, -2, -3, -2, 0], [-3, 1, -6, 4, 5, -2, 5, 0, 2, 5], [20, 4, -2, 7, -6, -5, 2, -2, 4, 1], [-1, -7, 0, -7, 1, 5, -1, 4, 2, 0], [6, -6, -4, 11, -4, 7, 4, 0, 0, 3], [-3, -11, 2, 8, -4, 8, -1, 1, -2, 1], [-16, 0, 2, 4, 0, 0, -4, 0, 2, -2], [26, 0, 0, 17, 1, 11, -2, -1, -2, -1], [-9, -3, -5, -10, -2, 9, 0, 2, 4, 2], [-2, 8, -4, -6, 3, -6, 1, 3, -2, 4], [-15, 9, 2, -15, -2, 5, -1, -1, -4, -2], [-10, -6, -2, -19, 3, 9, 3, 1, -4, -3], [2, 5, -7, -4, 1, -4, 0, -1, 1, -1], [5, 3, 2, 9, 8, 5, -7, -3, -2, -10], [2, 2, -6, 1, 2, -1, -5, 2, -4, -3], [19, 1, 0, 5, 2, 1, 7, -5, 4, 2], [8, 2, -5, 0, -2, -3, 3, 1, 3, 5], [16, 2, 4, 9, 6, 3, -4, -4, -4, -3], [1, 5, 2, -12, -4, 0, -1, 3, 2, 7], [3, -5, 4, -6, -1, 2, 1, -6, 2, 7], [1, 5, -2, 2, -6, -2, 3, 3, 6, -3], [14, -1, -3, 14, 7, 4, -2, -1, 1, -3], [7, 1, 4, 5, 6, 1, -6, -3, 4, 2], [-4, -2, -1, -12, -2, -3, 5, -3, 2, -1], [3, 3, -5, -1, -6, -2, 8, -2, 4, 9], [11, 4, 0, 0, -1, 7, 2, -3, -2, 3], [0, -2, -7, 2, -4, -2, -1, -1, 1, -2], [4, 0, -1, 3, -4, -1, 3, -1, -1, -3], [-4, -12, 0, -5, 2, 9, -2, 4, -2, 1], [10, 2, 8, -16, -2, -4, -8, -8, -4, -4], [1, -1, -6, -7, 3, -7, -6, 0, -2, 2], [18, -2, 4, 14, 0, -6, -4, -2, 4, 0], [-10, -10, -2, -3, -2, 9, 1, 2, 4, 1], [-10, 2, 5, -10, -1, 11, -1, -8, -2, -3], [-23, -1, -4, -14, 4, 0, -7, 1, -4, -1], [6, -10, 5, 13, 4, -3, 3, -1, 1, 1], [-22, -6, 1, -1, 3, -3, -3, -2, -3, 1], [-11, -3, 4, -3, -4, 3, -5, -5, -4, 2], [-13, 1, 0, 0, 4, -8, 1, 1, -2, -1], [4, 4, -6, 0, 1, 2, 4, 1, 2, -4], [11, -4, -10, 14, 3, -1, 6, 3, -2, 1], [10, -2, -4, 16, -4, 6, 2, 4, 2, -2], [6, -12, 10, 6, -4, 2, 0, 0, -2, 4], [15, -3, -2, 20, 2, 6, 5, 1, -2, -1], [-13, 1, -2, 3, -2, 7, 2, 3, 2, 2], [5, 5, -2, 3, 0, 1, -11, -3, -4, -8], [-5, -1, -3, 8, 0, -5, -4, 6, -3, 0], [26, 6, -9, 24, 4, -1, -3, 3, 1, 3], [-1, 4, 4, -22, 5, -9, -8, -11, 2, -5], [14, 1, 5, -11, -3, 3, 6, -3, 1, 2], [-15, 15, -1, -3, -4, -2, 2, 2, 0, 5], [14, 0, -9, 12, 4, -3, -1, 1, -1, 7], [10, 0, -2, 11, -2, -5, 12, 2, 4, 1], [2, 8, 4, 17, -2, -7, -4, -4, 2, -7], [17, 11, 0, 16, -4, -4, -5, -1, 4, 1], [17, 13, -1, 19, 2, -4, -10, -4, -1, -1], [18, 4, 4, 4, -6, 2, 0, 0, 0, 0], [4, -4, 0, 4, 8, 2, -4, -2, -3, -8], [-11, -11, 4, -1, -5, 7, -1, -10, 4, 6], [-6, -9, -12, -13, 2, 10, 9, 11, 2, 9], [6, 8, 1, 11, 6, -3, -7, -7, -1, -1], [10, 6, 1, 2, -4, 3, -1, -3, 1, -1], [-5, -11, -8, 3, 6, -3, -7, 3, -2, -2], [7, 9, 4, 7, -2, 9, 3, 1, -2, -4], [1, -2, -7, -19, 3, 1, 3, 6, 1, 5], [-6, -4, -8, 26, 0, 0, -5, 0, -6, -4], [21, -7, -7, 3, 0, -4, 7, 2, -1, 5], [-10, 3, 0, -17, -1, -2, -5, 0, -6, -1], [-19, 5, 5, -4, 4, -1, -6, -4, -3, -10], [-7, -10, -3, -5, -5, 5, 1, -2, 3, 9], [15, 1, -2, 13, 0, 11, 5, 3, 2, 4], [18, -2, -1, -5, 2, -4, 11, -1, 3, 8], [-19, 7, 4, -18, 3, -8, 1, -2, 2, 1], [12, 16, 1, 12, 4, -4, 3, -3, -5, -4], [-11, 7, 0, -7, -8, -1, -3, -1, 2, 2], [4, 2, -6, -3, 2, 7, 4, 0, 0, 11], [9, -13, -4, -3, -4, -3, -1, -1, -2, 4], [20, -4, 5, 5, 4, 0, -6, -5, -5, -8], [-8, -6, 0, -16, 2, -2, -4, -8, -8, 0], [17, 7, 6, -4, 6, -8, 5, -3, 4, -9], [-3, 1, 1, 21, 0, -1, 2, -2, 5, 0], [25, -1, -1, 11, -4, -11, 2, -6, 5, 4], [-5, -6, -5, -10, 1, -2, 5, 4, -3, -2], [21, 1, 6, 18, 0, -4, -10, -7, 2, -1], [-23, 9, -8, -27, 4, 5, 3, 5, -4, -6], [3, 7, 5, 5, -8, -1, 4, 0, 1, 2], [28, 2, -6, 31, 8, -3, 4, 2, -4, -1], [11, 7, -7, 18, 2, -5, -2, -6, 9, 8], [11, -3, 0, 6, -5, -6, 3, 2, -2, -9], [-10, -2, 4, -3, 0, -4, -2, 2, 0, 2], [-7, -3, -2, -10, 0, -2, -2, 5, -2, -5], [-5, 9, 8, 0, 3, -8, -5, 2, -2, -3], [0, 0, 12, 1, -4, 1, -10, -10, 2, -3], [-17, 19, -4, -12, 4, -2, 6, 1, 4, -3], [-4, -2, -5, 4, 3, -1, -9, -2, -4, -7], [-1, -5, 2, 17, 0, -2, 7, -1, 2, 5], [-10, 10, 0, -33, 0, -7, 4, 0, 6, 1], [-10, -8, 2, -12, -4, 14, 0, -4, -6, 2], [-5, -11, -12, 9, 0, -1, 1, 11, 2, 0], [-8, -10, -7, 7, 4, 4, -7, 3, 1, 0], [45, 3, 0, 13, -2, 5, 0, 1, 4, 2], [8, 4, -4, 1, 4, -3, 6, 0, -4, -1], [-13, 1, -4, 5, -2, -3, -1, -5, -2, -6], [-8, 2, 11, -8, -4, -3, -3, -11, 4, -1], [8, 7, 8, -20, -5, 7, 1, -6, 4, -2], [-9, 11, 0, -20, 1, -4, -3, 8, -6, -15], [-14, -8, -2, -12, -2, 8, 2, 6, 6, 2], [-10, -8, 7, -18, -12, 5, 5, -3, 1, -1], [15, 5, 20, -11, -7, 5, 5, -4, 2, 0], [11, 9, 6, 4, -6, 4, 11, -3, -2, 3], [-12, 0, 10, 11, -4, -5, -8, -8, -4, -9], [24, -4, 10, 7, 2, -5, -8, -2, -4, -1], [-21, -3, 2, 6, -1, 8, 5, 2, 0, 3], [0, 4, -6, -12, 0, 2, 0, -4, -4, -2], [-16, 7, 4, -12, 3, 3, -3, -6, 0, 6], [-10, -12, -12, -2, 3, 14, 5, 11, -2, 4], [-15, 10, -4, -5, -1, -8, 2, 1, -2, 10], [-15, 6, 3, -15, 3, -5, -3, 0, -1, -13], [-6, 4, 8, 12, 0, -2, 4, -4, -4, -2], [-7, 1, 0, -4, 2, 4, -1, 3, -4, 3], [-9, 7, -10, 11, 1, -7, 3, 8, 0, -2], [8, 4, 0, -1, -4, 1, 6, 4, 4, -1], [-12, -10, -6, -2, 0, 2, 6, -2, 0, 10], [-25, 5, 1, -28, 4, -4, -6, 2, -3, -5], [0, -2, 1, -13, -2, -2, 3, 1, 8, -4], [1, 5, 7, 4, 1, 2, -8, -11, -3, -3], [7, 9, 5, -5, -2, -2, -16, -2, -1, -11], [-8, -5, 0, -12, -5, 7, 1, 10, 4, 6], [-4, 10, -2, 7, 4, -3, -11, -2, -2, 1], [-21, 7, -4, -5, 5, -17, -1, -2, 0, -4], [6, 4, 8, -10, -4, 4, 6, -6, -2, 6], [-5, 1, -2, 12, 4, 6, -1, 5, 4, -5], [-3, -7, 10, -2, -6, 4, -7, -9, -2, 7], [2, 5, 3, 8, 1, 0, -12, -11, 5, -1], [14, -2, 2, 26, 2, -4, 7, 0, -2, -2], [1, 2, 4, -11, 3, 12, 6, 1, -2, -8], [-15, 1, 5, -32, -8, 5, 10, 0, 6, 8], [-7, -7, -4, 17, -4, 5, 9, 1, 2, 2], [-13, -12, 2, 14, -1, -1, -10, -5, -4, -5], [-4, -10, 6, 8, 2, 14, -10, -4, -8, -12], [-6, 8, -2, -27, -3, -5, 0, 3, -6, -7], [13, -15, 8, -20, -4, 16, 5, 7, -4, 1], [3, -21, -2, -17, -4, 3, 6, 1, 4, 12], [-6, -6, 7, -7, -6, -6, -1, -9, 3, -2], [6, 7, -3, 16, -1, 0, 8, 11, 3, 5], [12, -4, -8, 13, 3, 1, 16, 5, 6, 17], [-1, -7, -2, 1, 4, 7, 11, -3, -2, 2], [14, 4, 0, 5, -6, 3, -2, -2, 5, 1], [-1, -5, 8, -5, 0, 2, 1, -7, -4, 5], [-10, -6, 4, -5, -2, 11, 4, 0, -6, 1], [-17, -13, 6, 0, 3, -2, -1, -4, -2, -5], [7, 1, -8, 7, -3, 3, 17, 4, 2, 14], [1, -13, 4, -35, -1, 9, 3, 4, -6, -2], [20, -4, 1, -20, 0, 11, 14, 1, 1, 11], [1, 7, 0, -4, 0, -10, 5, 1, -4, -5], [44, 2, -6, 11, 6, 7, 4, 6, -2, -5], [-25, -11, -2, -14, -6, 6, -1, 7, 0, 7], [11, 13, 0, 0, -2, 8, 7, 7, -4, 1], [-6, 6, 6, 14, 3, -16, -4, -7, 2, -4], [37, -13, -3, 20, 2, 0, 2, 0, -3, 1], [10, -9, 1, -4, 3, 6, -2, -1, 5, 3], [-4, -18, -3, 14, -2, 7, 7, -1, -2, 9], [1, -19, -2, 15, 0, 3, 3, -1, -4, 8], [13, -9, 5, 0, 6, -3, -4, -6, -5, 2], [-19, -1, -10, 29, 4, 1, -3, 7, -2, -2], [21, -3, 11, -1, -6, -2, 0, -4, -1, 1], [8, 9, 6, 14, -7, -5, 1, 2, 2, 4], [6, 4, -1, 0, 2, -5, 4, 5, -3, -1], [15, 5, -7, -4, 2, -17, 2, 4, -1, -2], [1, -9, 3, 3, 4, -14, -8, -6, 1, -1], [-9, 3, 0, 18, 7, -4, -1, -2, -8, 1], [-11, 0, -4, -23, 1, -6, 2, 3, 6, -10], [-24, -2, 16, -25, 0, -1, -12, -10, -2, -3], [9, -13, 6, 26, -1, -6, -1, 4, -2, 7], [4, 2, 4, -13, -2, -13, 2, -6, 4, 9], [-32, 1, 1, -18, -1, -10, 2, 1, 3, -7], [13, 5, 5, -11, -2, 0, 14, 10, 0, 3], [17, 5, -4, 15, 0, 7, -3, 3, -2, 6], [-8, -10, -21, 8, 0, 12, 7, 5, 7, 10], [3, 3, 3, -11, 1, -9, 6, -3, 3, 0], [-12, 0, 8, 9, 3, 3, -2, -5, 6, 9], [-13, 13, -5, 8, 2, -1, 2, 4, 2, 2], [-19, -1, 10, 3, 3, -3, -3, -10, -2, -6], [-24, 6, -9, -2, 2, -5, 9, 5, -5, 7], [-4, -10, -11, -3, 6, 10, 0, 3, -3, 8], [-27, 3, -2, -10, 0, 14, -1, 9, -4, -1], [-11, -1, 11, -27, 0, 4, 2, 8, -3, -7], [-14, -2, 0, 17, 0, -3, 2, 2, 8, -1], [-26, -12, 3, -10, 0, 5, 5, 7, 4, -1], [6, -2, 11, 9, -2, -13, 1, 1, -1, 3], [-33, 0, 0, -15, 4, 0, -10, -8, -10, -4], [0, -14, -7, -7, 2, 0, 3, -3, 3, 12], [26, 4, 3, -14, 2, -2, -1, 3, -1, -6], [-16, -3, -4, -8, 3, 5, 1, 8, 4, -10], [-26, -18, 3, 6, -6, 3, -3, 5, 2, -1], [30, 2, 16, -15, -2, 1, -6, -12, 2, -5], [17, 1, -10, 8, 8, -2, 7, 1, 2, 1], [4, -8, -7, 26, 4, 7, 11, 7, 5, 9], [-17, -7, -8, 5, -5, -3, 3, 8, 4, -6], [0, -9, -5, 18, 1, -6, -14, 7, 1, -5], [32, -8, 5, -4, 0, 5, 15, 7, -2, 5], [-21, -7, 7, 12, -2, -3, -8, -2, 2, -2], [4, 0, 8, 19, 0, -11, 4, 0, 1, 3], [-9, -2, 0, 2, -1, 3, 4, -7, -4, 5], [15, -7, 1, 4, 1, 18, -6, -1, -3, 1], [-34, 10, 6, -7, 0, 9, -4, 10, -2, -5], [-1, -13, 8, 6, -4, 18, -15, -3, -4, 1], [-4, -23, -8, 20, 4, 5, -7, 1, -2, 2], [18, 14, -14, 17, -4, 5, 0, 4, 1, 1], [-16, 15, -16, 1, 5, -12, 5, 2, 0, -3], [-23, 11, 0, -14, 2, 0, -7, -7, 1, -3], [3, 3, 12, -13, -3, -1, -21, -6, -4, 0], [14, -11, 0, 8, -1, -5, -1, -8, -4, 0], [-13, 15, -8, 8, 3, -6, 3, -8, 2, -3], [40, -4, -2, 15, -2, 9, -2, -4, 0, 1], [16, -6, -3, -3, -3, -6, -3, 0, 6, -4], [-6, -12, -16, -9, 2, 17, 8, 14, 0, 7], [14, -8, -4, 7, -4, -9, -13, -2, 4, 1], [-6, -24, 2, 5, -2, 5, 0, 0, -2, 5], [-15, 11, -5, -12, 6, -13, -2, -4, 0, -6], [-8, -18, 7, -2, 6, 3, -9, 1, -3, -5], [-1, -17, 10, -25, -3, 1, -3, 0, -8, -6], [4, 0, -10, 8, 5, -14, 4, 5, -2, 8], [7, 7, 3, -15, 2, -7, -2, -12, -1, 4], [-37, 3, 6, -26, 0, -4, -5, -1, -2, -1], [7, -7, 6, -4, -1, 4, 3, -14, 2, 7], [-2, -10, 6, 11, -8, 3, -6, -10, -4, 3], [-30, -2, 4, -14, -1, -14, 0, 3, -2, -2], [-6, 20, 6, -20, -6, 4, 4, -4, 6, 4], [-8, -9, 5, -33, -7, 7, 6, 3, 3, 4], [-8, 2, -1, 5, 2, 12, 3, 5, -4, -8], [36, -4, 2, -8, -8, 4, 2, 4, 1, 10], [34, 9, 10, 6, -1, 1, -3, -8, 6, -6], [22, -6, 1, -2, 4, 5, 14, 3, -3, 3], [-9, 4, 4, -26, 2, -13, -24, -8, -6, -9], [8, -8, -7, 16, -8, -2, 1, 1, 7, 8], [7, -3, 8, 20, 2, -2, 2, -3, 10, 5], [15, 15, 1, 0, 10, -1, -6, -6, -1, -12], [1, -17, -3, 19, -2, 10, 8, -4, 5, -1], [4, -14, 8, 3, 0, 15, -8, 0, 0, -1], [-3, 3, -8, 12, -1, 10, 11, 10, 2, -1], [-11, -11, 0, -8, 9, 6, -7, -6, -8, -11], [-5, -15, 4, -3, -8, 5, 10, 5, 8, 6], [0, 4, -1, 3, 2, -12, 11, 1, 6, 6], [-17, -17, -1, -17, 2, -13, 0, -2, -1, -2], [-2, -18, -5, 13, 2, 4, 9, 13, -1, 8], [-1, 1, -7, 8, -2, -17, 0, 8, 5, 2], [21, -19, 5, -6, -2, 14, -6, -6, -1, 3], [31, -1, 2, 25, 4, 7, -3, -7, 8, 4], [6, 4, 8, 2, -8, 8, 18, 0, 6, 12], [18, 6, -2, 27, -9, -3, 14, 7, 4, 1], [-16, -12, 3, -38, -3, 1, -1, 2, -2, -3], [5, -15, 1, 15, -6, -2, 4, 2, -5, 5], [14, -5, 12, 4, -6, -3, -9, -11, 6, 8], [13, 3, -17, 22, -2, -1, 10, 10, 6, 2], [-1, -1, -1, -9, 0, -6, 7, 2, 5, -3], [0, -8, -3, -5, 4, -7, -3, 13, -7, -11], [-15, 11, -2, 7, -2, -3, 13, 11, 8, 6], [-11, -1, -4, 27, 6, -13, -1, 3, -4, -2], [17, 3, 13, 17, 2, 4, -10, -4, -7, -3], [7, 3, -16, 8, 2, 14, -5, 11, -2, 1], [-2, 10, -9, 0, -4, 1, 12, 5, -1, 7], [9, -7, -4, -9, 8, 5, 7, 15, 0, 0], [-12, -2, -2, -12, 2, 6, -10, 6, 4, -4], [-24, 8, 13, -11, 0, -4, -13, -9, 2, -12], [-5, -5, 3, 6, 8, 15, -4, -2, -4, -2], [8, 2, -4, -1, 5, 13, 16, 15, 4, 1], [7, 1, -5, 26, -2, 2, 10, 14, -1, 7], [14, -6, 0, -21, 6, 11, 0, 2, -4, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2541_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2541_2_a_bq();".
// 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_2541_2_a_bq(:prec:=10)
    chi := MakeCharacter_2541_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_2541_2_a_bq();".
// 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_2541_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2541_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-11, -70, 253, 17, -316, 6, 124, -1, -19, 0, 1]>,<5,R![4336, -2392, -6768, 4812, 1751, -1873, 108, 179, -28, -5, 1]>,<13,R![2816, -74752, 90048, -12448, -19424, 4808, 1540, -334, -65, 6, 1]>],Snew);
    return Vf;
end function;