
// Make newform 2151.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2151_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_2151_2_a_f();".
// 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_2151_2_a_f();".
// 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 := [2, -7, -5, 22, 8, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [3, -5, -7, 1, 1, 0, 0], [-3, 2, 21, 7, -10, -1, 1], [3, 0, -21, -7, 10, 1, -1], [-6, 1, 22, 7, -10, -1, 1], [-6, -4, 22, 8, -10, -1, 1], [-12, 11, 51, 6, -21, -1, 2]];
    Rf_basisdens := [1, 1, 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_2151_a();"
function MakeCharacter_2151_a()
    N := 2151;
    order := 1;
    char_gens := [479, 1441];
    v := [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_2151_a_Hecke(Kf)
    return MakeCharacter_2151_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, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1], [1, 1, 0, 0, 1, 0, 0], [-2, -1, -1, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0], [-2, -1, -2, -1, -1, 1, 0], [-1, -1, 0, 0, 0, -1, -1], [-3, 0, 0, -1, 0, 0, -1], [-2, 1, 0, 1, -1, -1, 3], [-1, -1, 0, 0, -1, 1, 0], [2, 0, 0, -2, -1, 1, 0], [-4, 1, 2, 1, 2, -1, 2], [1, 0, 0, -1, -3, 2, 0], [-2, 2, 0, 2, 0, 0, 3], [0, -1, 0, -1, -1, 2, -2], [1, 2, 0, -1, 1, -1, -1], [0, -1, -1, -2, 0, 2, -3], [2, -2, 0, 0, 0, -2, 2], [4, 2, -1, 1, 1, -1, 1], [-2, 1, 2, 1, 2, -2, 1], [1, 0, 0, 1, -1, 0, 3], [-5, 1, 1, -1, 0, 0, 1], [-5, -2, -2, 1, -1, 0, -1], [-2, -3, 2, -1, -3, 1, -3], [-3, 0, 2, -1, 3, 1, 0], [-4, -5, -2, -3, -5, 4, -3], [-3, 1, -4, 2, 1, -1, 1], [3, 2, -1, 2, 2, -2, 0], [-3, 1, 2, 2, 1, 1, 3], [1, -4, -4, -1, 0, -2, -3], [0, 0, 2, 0, 0, 4, -1], [6, 0, -2, -2, 0, -1, 0], [0, 5, 6, 3, 4, 1, 2], [-1, -1, 2, -2, -1, 0, -1], [-4, 0, -2, 0, -3, -2, 2], [-6, 0, 2, 2, 3, -1, 3], [5, 3, 2, 2, 5, 1, 1], [-6, -2, 2, 2, -2, 0, -2], [4, -1, 2, -1, -2, 1, 4], [-4, -1, 2, 1, -1, 3, -3], [-3, -1, -4, -2, 1, -1, -3], [-4, -5, 0, -1, -5, 5, -3], [1, -2, 0, -1, -2, -2, -3], [-3, -4, 2, 1, 3, -3, 1], [-6, 5, 6, 1, 1, 0, 5], [-3, -2, 0, -5, -3, 3, -7], [-1, -2, 0, 5, 3, -2, -1], [2, 0, 0, 2, 1, -3, 1], [-5, -1, -4, 0, 1, -1, 7], [0, 4, 2, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0], [-10, -2, 2, 2, 1, 1, 0], [-4, 4, 4, 0, -1, -1, 1], [-3, 5, 4, 6, 4, -4, 1], [4, -5, 5, -2, -1, -1, -4], [6, 6, 2, 2, -3, -1, 4], [3, 1, -6, 4, 4, -4, -3], [4, 3, 2, -3, 1, -5, 2], [-10, 2, -2, 0, 2, -1, -2], [6, 2, 0, -4, -3, -1, -1], [9, 2, 0, -1, 1, 1, -8], [-1, 0, 0, 3, 1, 1, -5], [10, -1, 3, -2, 2, -2, -3], [-8, -2, -4, -2, -2, 6, -1], [-6, 3, -4, 1, 6, -1, 3], [4, 2, -2, 2, 6, -3, -3], [8, -5, 1, 2, -6, 0, 2], [-3, -5, 2, -2, -7, -1, -1], [-17, 3, -1, 1, 3, -3, -2], [5, 2, 2, -5, -2, 1, 4], [-4, -4, 0, -6, -9, 7, -8], [-1, 2, 2, -1, -5, 1, -1], [-9, 2, 0, -1, -2, 6, 0], [12, -3, -6, 5, 1, -4, 0], [12, 9, 4, 5, 10, -4, 4], [-4, 2, 4, 4, 5, 1, 5], [1, -1, 0, -2, 6, -2, -4], [-5, -1, 4, 0, -1, -5, -3], [-8, 1, -1, 4, -2, 0, -5], [-3, -2, -6, -9, -4, 4, -3], [-1, -1, -1, -1, -1, -3, -3], [-1, -6, -2, -3, -4, 4, -3], [-10, -5, -8, -1, 0, 2, 0], [-10, -4, -2, -2, 0, 0, 1], [-14, -1, 4, 1, 2, -8, 2], [1, 2, 0, 3, 6, -5, -1], [12, 5, 6, 1, 10, 0, 4], [7, -4, -6, -1, -6, 7, -6], [21, 1, 4, 2, 7, -6, -1], [7, -1, -6, -4, -4, -2, -4], [3, -1, -7, -1, 1, 1, 5], [6, -3, -4, 3, -3, -5, 1], [-4, -1, 4, -1, -3, -1, 9], [1, -6, -8, -9, -3, 6, -5], [3, 3, 2, 2, 6, 4, 1], [16, -3, -4, -3, -2, 2, -1], [2, 0, -4, 2, 4, -3, -7], [15, 1, -8, -4, -5, 3, -3], [-2, 4, -8, 0, -7, 5, 1], [-3, -1, 0, 4, 6, 1, 0], [-1, 4, 4, -1, -7, 2, 6], [7, -4, 1, -6, -9, 5, 0], [13, 1, 0, -4, -2, 2, -8], [1, 1, 2, 6, 3, -7, 2], [-9, 11, 3, 9, 5, -5, 8], [-10, 6, -7, 5, 6, 0, 4], [-10, -9, -8, 1, -3, 2, -4], [-4, 4, 7, 5, -1, 1, 2], [-9, -1, -4, -2, 0, 2, -3], [-12, 0, -6, 2, -1, 2, 1], [-1, -4, -5, -2, -7, 5, 0], [-7, -3, 2, 4, 8, -1, -3], [9, 9, 8, 0, -1, -2, 2], [-6, -4, 6, -6, -2, 8, -9], [10, 9, 6, 1, 1, 1, 0], [3, -8, 0, -7, -7, 3, -2], [9, 1, -7, 7, 3, -5, 7], [-1, 0, -4, 5, 7, 0, 7], [0, 5, 0, -1, 3, 1, -2], [-4, 1, 10, 5, 4, 2, 2], [-9, 4, 0, -3, 3, 3, -1], [5, 9, -2, 6, 13, -8, 7], [-11, -5, 0, 6, 4, 0, -9], [5, 9, 0, 4, 5, -7, 10], [-4, -1, -2, -7, -9, 0, 0], [-12, -8, -4, -2, -1, 3, 3], [4, -2, -15, -3, -3, -1, -6], [-19, 3, 14, 4, 2, -2, 7], [11, 1, 10, 0, 0, -2, 0], [18, 13, -4, 1, 0, 0, 11], [3, -4, -14, -1, 1, -3, 4], [-16, -4, 4, 0, -1, -5, 1], [-14, 4, 5, 3, 5, -3, 0], [2, 4, 2, -2, 4, -6, -6], [-8, -1, 8, -3, -4, -6, -4], [-1, 1, -8, 2, 1, -8, 1], [19, -8, 0, 1, -9, 2, -6], [0, 0, -12, -2, 1, -3, -3], [6, -7, -4, -11, -10, 9, -12], [1, 5, 8, 4, 3, -2, 8], [0, -2, 8, 4, 0, -7, -2], [13, 0, 0, -1, 4, -5, 5], [-2, -3, 0, -9, 1, 7, -1], [-21, 1, 4, 0, 0, 2, 5], [-5, 2, 2, 3, 1, -11, 1], [-15, 0, 9, 0, -2, -6, 4], [-6, -6, -10, -6, -10, 1, -9], [-8, -1, 4, 1, -5, 3, 8], [-13, 1, 0, -8, -11, 5, 5], [-11, -6, 3, -4, -9, 3, -3], [-10, 6, 10, 8, 4, -9, 4], [18, 6, 6, 2, 1, 5, 7], [7, -4, 3, -6, 0, 10, -14], [23, 9, -4, -2, -6, 1, -5], [-20, 3, 8, 3, 2, -2, 0], [8, -2, -10, -10, 0, 0, -17], [-1, 3, 0, 2, 0, -5, -3], [-5, -6, -5, -4, 3, -1, -6], [-4, -6, -6, -12, -5, 10, 2], [5, -4, 6, 1, 7, -1, -6], [2, 1, 12, 3, 8, -1, 5], [18, 5, 0, 1, 4, 4, -1], [4, 2, 4, -2, 0, 4, 0], [2, -9, 0, 1, -9, 4, -12], [-18, -18, -8, -4, -19, 5, -5], [-9, -9, -4, -6, -4, 7, -8], [-2, -3, -12, 3, 6, -10, -11], [-7, 3, -6, 2, -2, 6, 3], [-9, 12, 12, 9, 5, -6, 5], [10, 9, 10, 5, 7, 3, 15], [-14, -3, -4, -3, -15, -3, 3], [21, -9, 6, -8, 0, -2, -3], [1, 10, 2, 1, 7, -3, 6], [1, -9, 0, -6, 1, 5, 2], [1, -3, -4, 0, -7, 1, 9], [-2, 9, 6, 9, 10, 1, 10], [-1, -14, -8, -7, -7, -2, -10], [9, -2, 4, 11, 5, -8, 3], [3, -3, -17, -5, 0, 6, -5], [15, 10, -4, -1, 13, 2, -6], [-8, -4, 5, 3, -11, 3, 12], [2, 14, 10, 6, 16, -8, 11], [-1, 0, -14, 1, 3, -2, 4], [7, 10, 10, 3, 11, -1, 5], [-1, 13, -2, 6, 12, 0, 13], [10, -2, 4, 8, -4, -4, 7], [21, 1, 0, -2, 4, 9, 4], [-23, -1, -10, -2, -8, -4, 3], [-19, -2, 8, 7, 0, -2, 2], [-9, 5, -13, 9, -2, -4, 12], [-29, 0, 4, 3, -2, 2, 10], [30, 10, 6, 4, 8, -1, 3], [-18, -4, 10, 10, 4, -4, 12], [11, 5, 2, -10, 9, 1, -9], [-6, 4, 14, -2, -4, 11, 8], [27, 0, 6, 1, 2, 2, -15], [-25, 5, -8, 8, 3, 1, 8], [-7, 5, 6, -2, 9, 7, -9], [19, 3, 10, 2, 13, 1, 6], [-3, -5, -4, 2, -4, 4, -8], [-18, 5, 10, 5, 4, 6, 4], [15, 12, 4, 3, -3, 1, 6], [-6, 12, 5, 5, 7, -1, 7], [-19, 4, -3, 4, 5, -9, -2], [-9, -3, 2, 4, 0, -9, 6], [-13, 15, 0, 4, 9, 2, 2], [-22, 2, 0, -6, 10, 2, -3], [-2, -3, -20, -11, -2, 1, -6], [-5, 13, 14, 4, 14, 0, 9], [9, -4, -13, -14, -12, 0, -10], [-7, -5, -10, 2, 1, 2, 16], [3, 6, -12, 3, 9, -5, -4], [-7, 3, -2, 4, -5, -1, -6], [14, 18, 0, -2, 13, 1, 0], [-17, -16, -4, -5, -5, 7, 0], [2, 5, -2, 1, 9, 1, 9], [1, -14, -6, -9, -5, 7, -19], [14, 6, 10, 6, 8, -6, 8], [-5, -2, -2, -3, 10, 5, -4], [-11, 0, 24, 9, 3, -1, 8], [-19, 8, 8, -5, -7, 3, 12], [9, -7, -4, -8, 1, 2, -9], [11, -8, 20, -3, -12, -3, -5], [25, 6, -9, 2, -3, -1, -1], [2, -2, 2, -6, 7, -3, 2], [-12, 2, 2, 8, 1, -14, 4], [-8, 1, 16, 9, 5, 3, 3], [-1, 13, 4, 6, 6, -11, 13], [-3, -4, -13, 4, 3, 1, 7], [5, -2, -8, -5, 2, -2, 3], [11, 4, -14, 5, 2, -2, 2], [-10, 7, 2, 5, -5, 0, 2], [8, 0, -12, 0, 3, -4, 2], [-6, 5, -18, 1, 3, -5, 9], [22, 1, 10, 3, 0, 4, -13], [-6, -7, 3, -6, 2, 4, -17], [-11, -4, -4, -5, -10, 1, 2], [12, 6, 18, 10, 3, 3, 12], [11, 6, 14, 1, 16, -4, 7], [-24, -14, -6, -2, -13, -1, -14], [-13, 7, 16, 10, 14, -5, 16], [-7, -1, 10, -4, -10, 0, -11], [8, -3, 6, 9, 15, -5, -1], [21, -3, -6, -2, -3, 4, -7], [11, 5, -10, 2, 8, -6, -18], [-25, -1, -12, 2, -1, 1, 9], [8, -1, 14, 5, 6, 2, -4], [6, -12, -4, 0, -7, 5, -4], [13, 9, 0, -4, -6, 0, -5], [9, -1, -7, 5, 4, -10, -5], [1, -6, 2, 1, 1, 8, -8], [-5, -3, -10, -6, 6, -2, -8], [14, -4, 12, 6, -5, 1, -16], [11, -1, -10, 0, 3, -12, 6], [-18, 10, 8, 12, 13, -17, 15], [11, 2, 3, 10, -1, -5, -7], [32, 3, -2, 11, 6, -12, 7], [24, 8, 4, 0, 4, -7, -5], [16, 5, -14, 3, 4, -14, 9], [15, 6, 0, -3, 5, -2, 11], [-48, 8, 2, 4, 4, 4, 1], [-3, -8, -6, 3, -4, 0, 15], [-19, -23, -1, -9, -9, 9, -15], [-28, -3, 14, 1, 0, -2, -4], [10, -6, -10, 2, 8, -3, -14], [2, 3, -6, -5, 8, 0, 9], [-26, -7, 0, -3, 8, 4, -2], [19, -1, 0, -4, 0, 4, -7], [2, -7, 6, -11, -8, 4, -15], [4, 0, 16, 2, -1, -9, -5], [25, 0, 0, -3, 12, 1, -20], [-6, -3, -6, -1, 8, -6, 4], [9, 21, 16, 2, 11, -2, 7], [10, 13, 12, 3, 7, 7, 12], [1, -1, -6, -10, -14, 8, -16], [8, -11, -18, -3, -8, 2, -15], [15, 2, 7, 6, -9, 3, 11], [-25, 19, 9, 7, 14, -4, 11], [12, -4, -4, -10, -17, 5, -6], [-3, -15, 2, -4, 0, -10, -10], [-4, -4, -6, 14, -2, -3, 8], [30, -4, 5, -7, -16, 2, 3], [-7, 9, -6, 0, -10, 6, 9], [2, 6, -2, 6, 12, -1, 14], [16, -5, -4, -1, 9, 11, -1], [34, 1, -12, 1, -1, 1, -5], [7, 19, 0, 8, 5, -3, 3], [19, 1, 8, 2, 5, 14, -8], [-12, 4, -2, 6, -4, 2, -15], [-13, -7, 0, 4, 1, 6, -2], [-43, 2, 2, -1, 4, -2, -1], [-23, 9, -2, 14, 11, 1, 7], [11, 5, -8, 0, 1, -9, 9], [18, 6, -10, -8, -4, 6, 4], [7, 8, 14, 5, -2, 1, -1], [-8, 2, -2, -8, -14, -5, 4], [6, -3, 6, 7, 10, -2, 2], [14, -6, -19, 3, 4, -6, -9], [-5, 5, -2, -2, 4, -8, 6], [-5, 9, -13, -5, 12, -2, 5], [21, 1, 2, 2, 8, 12, -18], [2, 3, 6, 5, 1, -9, -2], [22, -12, 3, -5, -13, 5, -4], [7, 1, 6, -12, 9, 5, -6], [14, -5, -4, 3, 11, -1, -7], [11, -2, 6, -7, -1, 5, -6], [-13, 2, 0, 3, 17, -11, 9], [28, -3, 0, -5, -4, 6, 7], [-7, -4, 14, -1, -2, 4, -7], [9, 4, -4, 3, 10, -14, 13], [-13, 4, -8, -5, -5, -1, 2], [0, -7, 16, -7, 4, 4, 1], [6, -10, -2, -8, -6, 9, -9], [-11, 19, 2, 16, 21, -11, 28], [-5, -16, 4, -3, -10, 0, 10], [-5, -14, -2, -9, -18, 6, -2], [21, -7, -18, -10, -4, 0, -5], [-15, 14, 15, -2, 0, -4, 7], [7, 2, -22, -5, -3, 10, -2], [27, 7, 12, 4, -3, -10, 2], [4, -1, 11, 0, 8, -2, -2], [-24, -15, 14, 5, 3, -6, -3], [5, -7, 0, -4, 5, -4, -10], [0, 6, 8, 14, 12, -16, 9], [4, 7, 12, -9, 7, 1, 0], [-3, -9, 24, -4, -11, 2, -9], [-3, -9, 16, -2, -9, 0, -12], [7, -15, 16, -6, -15, 5, -2], [10, -17, -12, -11, -6, 14, -15], [2, -1, 8, -7, 0, 6, -3], [-29, -9, -12, -8, -11, 0, 1], [26, 7, 26, 9, 7, -3, -2], [6, 3, 18, 9, 16, -2, 11], [22, -16, -4, -12, -5, 15, -14], [-29, -2, 2, 5, -2, 6, 11], [27, -4, -12, 7, 10, -4, -11], [16, 5, 6, 1, -4, -4, -6], [2, -11, 2, -5, -3, 13, -19], [-14, -8, 0, 0, -13, 7, 18], [-10, -11, 11, -2, -3, 11, -8], [-18, -5, -10, -3, -12, -7, 18], [-4, 9, 14, 5, 12, 9, 6], [-31, 3, 2, 4, 4, -10, 9], [4, 1, 2, -9, -12, 2, -17], [4, 1, 0, 9, 6, -4, -6], [-10, -4, 22, -8, -6, 6, -1], [26, 11, -4, 7, 0, -10, 7], [-10, -4, 9, 1, 19, -3, 6], [-18, 6, -6, 4, -7, -10, 7], [-21, -16, -4, -7, -14, 14, -5], [38, -5, 5, -12, -8, -2, -8], [21, -7, 4, -2, -13, 4, 2], [-11, 21, 16, -4, 7, -1, 12], [21, 1, -16, 4, 7, -5, -5], [-14, 10, 4, 0, -1, 7, 6], [15, 8, 11, 4, 0, -6, -1], [29, -18, -2, -13, -12, 4, -15], [-2, 7, -6, -5, -8, 4, -13], [-10, -7, 5, 12, 4, 0, 16], [9, -5, -8, -2, -7, 5, 3], [9, -2, -16, 3, 5, -7, 2], [1, -21, -12, -10, -8, 6, 5], [13, 11, -10, 10, 17, -13, 11], [-20, -19, 6, -11, -17, 20, -4], [-29, 11, 5, 5, 3, -3, 23], [26, 5, 2, 11, 17, -7, -11], [3, -11, 8, -12, 7, 7, -25], [0, -8, -12, 8, 2, -2, 7], [-6, -3, 10, 17, 0, -2, -2], [-23, -4, -4, -5, -3, -4, -10], [7, 19, -2, 0, 9, 3, 15], [24, -1, -4, 7, -6, -3, 1], [-17, 4, 18, 1, 10, 0, 16], [14, 9, -10, 13, 10, -16, 7], [-10, 17, 2, 13, 23, -9, 13], [19, 7, -10, -2, 5, -5, 8], [-11, 15, 6, 4, 4, -2, 22], [-34, -11, 6, -5, -16, 5, 24], [1, 4, -2, 19, 6, -12, 5], [-11, -21, -10, -8, -22, 11, -10], [7, 3, 2, 0, -7, 11, -6], [-4, 14, -8, 2, 13, -7, -1], [22, -13, -24, -5, -5, 2, -13], [9, 5, -4, -6, 4, -2, -2], [-3, -7, 9, -1, -16, 0, -4], [-2, 1, -18, 1, 2, -1, 4], [29, 7, -19, -17, -8, 6, -2], [-13, 5, -10, 2, -14, -2, 18], [-5, -12, -5, 4, -7, 7, -14], [-17, 13, -13, 11, 13, -9, 12], [-3, 3, 4, 12, -3, 3, -6], [-32, 11, 2, -1, -6, 8, -1], [4, -4, -14, -2, 3, -4, -3], [24, 4, -24, -4, -2, -4, -2], [2, 13, 4, 1, 17, -7, 7], [3, 7, -14, 6, 16, -24, -5], [-11, 3, 14, -2, 3, 9, 2], [-8, 1, -14, -7, -24, 18, -9], [14, -16, 16, -10, -6, -3, -21], [-6, -2, -17, -9, -4, -4, -23], [-16, -6, -12, 0, 2, -6, -14], [-11, -17, 0, -4, -5, 14, 5], [-2, 0, -14, -12, 5, 6, -18], [-7, -7, -2, -12, -13, 3, -26], [-16, -5, 10, -5, 8, -2, 1], [25, 3, 16, 12, 9, -13, -4], [48, -12, -4, -4, -1, 2, -22], [-26, 6, -14, 6, -12, -2, 18], [9, 5, 6, -10, -15, 9, 4], [-26, -1, -8, -5, -2, -4, 2], [9, -12, 4, 13, -5, -6, 6], [13, 1, -2, -2, -7, 13, 12], [20, -12, -10, -10, -9, 10, -11], [-17, -13, -6, -10, -18, 10, -12], [13, 28, -2, 9, 21, -1, 21], [-10, 2, 8, -6, -16, -4, 7], [-10, 3, -22, 3, -5, 12, 3], [3, -7, 10, 16, 5, -11, 15], [17, -20, 0, -7, -7, 3, -23], [-13, 6, 6, -3, 9, 0, 17], [22, 8, 8, 8, 20, -8, 16], [4, -5, 6, -7, -2, 2, -16], [4, -7, -8, 1, -4, 2, 4], [-23, 8, -7, -10, -13, 11, -6], [7, 9, 16, -2, 0, -1, -4], [-13, -13, -10, -2, -4, 12, 9], [-7, -7, -14, 4, -2, 2, -1], [-9, 5, -28, 0, -3, 0, -1], [-15, -20, 8, -5, -15, 3, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2151_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2151_2_a_f();".
// 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_2151_2_a_f(:prec:=7)
    chi := MakeCharacter_2151_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_2151_2_a_f();".
// 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_2151_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2151_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -49, -1, 44, 0, -12, 0, 1]>,<5,R![-8, -12, 38, 19, -31, -11, 3, 1]>],Snew);
    return Vf;
end function;