
// Make newform 5290.2.a.be in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5290_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_5290_2_a_be();".
// 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_5290_2_a_be();".
// 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, -20, 19, 18, -11, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-13, 25, 9, -12, -1, 1], [-16, 19, 18, -11, -2, 1], [50, -113, -66, 57, 8, -5], [56, -127, -68, 59, 8, -5]];
    Rf_basisdens := [1, 1, 3, 2, 6, 2];
    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_5290_a();"
function MakeCharacter_5290_a()
    N := 5290;
    order := 1;
    char_gens := [2117, 2121];
    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_5290_a_Hecke(Kf)
    return MakeCharacter_5290_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 := [[-1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 1, 0], [-1, -1, 0, 0, 0, 1], [0, -1, 0, -1, 0, -1], [1, 1, -1, 0, -1, 0], [0, 1, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, 1], [-1, 1, 0, 1, -3, 0], [1, 0, 0, 0, -1, 1], [0, -1, -3, 0, -3, -1], [-3, 0, -2, 0, 0, -1], [3, -2, 1, -2, -2, 0], [-6, 2, 1, 0, 1, 1], [4, -2, 1, 0, 1, 1], [1, 1, 0, 0, 1, -1], [-1, -4, 1, 0, -1, 0], [-1, 3, -1, -2, 2, 0], [1, 2, 1, 0, 4, 2], [3, 2, 0, -2, 1, -1], [-3, 0, -1, 2, 5, -2], [-3, 4, 1, 0, 1, 0], [-1, 3, 1, -2, -3, -2], [7, -4, 1, -2, -2, 2], [1, -1, 3, 0, -4, 0], [-2, 0, 3, -2, 0, 1], [10, 1, 0, 1, 2, -1], [-5, 2, -2, 4, 4, -1], [-3, 0, -2, 2, 1, -1], [-1, -2, 2, -2, 5, 1], [-3, -1, 3, -1, -3, -1], [0, 2, 4, 4, 4, 0], [-2, 3, 2, 3, 2, -1], [-1, 3, -1, 0, 8, 2], [2, -2, 0, 0, 4, 0], [5, 3, 1, 1, 10, 1], [-6, -2, -1, 2, -1, -1], [-5, 3, 0, 4, -3, 1], [-2, 2, 0, 0, -6, 0], [8, 1, 2, -1, 6, 1], [-15, 4, 0, 2, 3, 1], [-3, 2, -2, 0, -5, -3], [-7, 1, -2, -2, 1, -1], [-5, 2, -4, 4, 1, 1], [-5, 2, -1, -2, 2, -2], [-13, 4, -1, 0, 2, 0], [-2, 0, -2, -2, -5, 0], [-10, 0, -2, 0, -4, -2], [-13, 6, 4, 2, 1, 1], [2, -8, 0, 2, 0, 0], [-5, 4, 0, -2, -8, -1], [10, -3, 1, -3, -8, 0], [9, -6, 2, 0, -1, 1], [-7, -5, -1, -2, -4, -2], [4, 2, -4, -4, -4, -2], [3, -7, -2, -3, -11, -2], [6, 2, 6, 0, 6, 2], [-4, -2, 1, 2, -8, -1], [-10, 2, -2, 2, 3, 0], [2, -4, 2, -4, 2, 2], [-4, 3, 0, 6, 0, 2], [8, -2, -6, 2, 0, 0], [-11, -1, -1, -3, 7, 1], [12, -1, 6, -3, 0, -1], [13, 4, 0, 0, -3, -1], [3, -5, 1, 1, -9, -3], [-6, -5, 2, 2, 8, 2], [0, -2, 1, -4, 1, -1], [-1, 4, -3, 2, 0, 0], [-8, -8, 2, 2, -4, 0], [-8, 2, 2, 6, -8, 4], [6, -2, -2, -2, 2, 0], [-1, -5, 2, -6, 0, 1], [-6, -4, 4, -4, -2, -2], [-5, -3, -4, 2, -1, -1], [-5, -3, -4, 2, -1, 3], [-4, -2, -4, -2, -13, -2], [0, -1, 3, -1, 3, 0], [-13, -2, 0, -4, 0, 3], [2, -3, 4, -1, 8, 1], [-12, -4, 1, 0, 7, 1], [-1, -5, 1, 1, -1, 5], [-3, 3, 2, -1, -1, 2], [-17, 3, -1, -1, 2, -1], [-6, 3, 1, -1, -5, -4], [17, -4, 0, 2, 2, 1], [18, -6, 6, -6, 0, 0], [-11, 0, -3, -2, 0, -6], [-19, 2, 2, -4, -10, -1], [-9, -4, 5, 2, 8, 0], [13, -8, -4, -2, -1, 1], [-8, 2, 4, 0, 4, 2], [9, -6, -6, -2, -9, -3], [-5, -7, -2, 1, -5, 2], [2, 10, 0, 8, 6, 0], [-1, -4, 0, 4, 12, 1], [-3, 4, 3, 6, 1, -2], [18, 4, 2, 2, 8, 4], [5, -1, 1, 1, -2, 3], [1, 4, 0, 0, 9, 5], [6, -6, 2, 0, -7, -2], [-15, 4, -3, 2, -9, -2], [-15, 5, 6, -2, -4, -1], [-16, 0, 1, -6, -5, -3], [-12, 5, 2, 0, 2, 0], [-4, 6, -1, 6, 11, -3], [1, -5, 5, -2, -4, 0], [18, 3, -1, 1, -1, 6], [-10, 2, 1, 6, -5, 1], [5, -4, 4, 2, -7, 3], [-5, 1, -1, -10, -1, 0], [-6, -5, -5, 3, -12, 0], [-14, -2, -8, 4, -6, -2], [-12, -4, -1, 0, 14, 3], [-13, 2, 2, -2, 6, -1], [8, -2, -5, 4, -5, -1], [4, 3, 4, -1, 16, 3], [-5, 0, -6, 4, 6, 1], [1, -3, 2, -6, 3, -1], [-9, 4, -3, 4, 0, -2], [-3, 4, 4, 2, 7, 3], [1, 3, -1, 11, 8, -1], [10, -2, -2, 2, -8, -2], [0, -11, -2, -5, -14, -3], [3, -5, -6, 0, -1, -5], [-3, -3, 0, -1, 3, 0], [-5, 3, 3, -6, 4, 4], [18, -6, -5, -4, -13, -1], [-19, 10, -7, -2, 2, -4], [-11, 5, 0, 5, 9, -6], [-3, -4, 2, 2, 7, 3], [-13, -2, 3, 0, 2, -6], [16, 3, -1, 5, -1, 2], [18, 0, 4, -4, 3, 2], [8, 10, 3, 2, 3, 3], [-8, 16, 2, 8, 15, 0], [4, -2, -2, -6, -2, -2], [-2, 1, -7, 4, -1, -3], [19, 4, 1, -2, 4, 4], [-11, -6, -5, 2, -2, -6], [-20, -4, -2, -6, -14, -4], [-9, -6, -4, -8, -10, -5], [-8, 4, -2, -4, -14, 0], [21, 4, -4, 4, -7, -1], [-4, -11, -4, -5, -16, 3], [-15, 0, -2, -4, 1, -1], [6, -2, 6, -4, 12, 2], [5, -2, -1, 0, -4, -2], [-17, -1, -4, -2, 5, 7], [-11, 0, -6, -2, 0, -5], [-7, 1, 3, -3, 0, 9], [5, 2, 2, 2, -7, -1], [-18, 0, -3, 2, 10, -9], [-8, -4, -1, -4, 5, -5], [19, -16, 1, -4, -10, 2], [0, -2, -2, 10, 22, 0], [11, 5, 1, -3, -13, -3], [5, -9, -8, 6, 7, 1], [11, -7, 1, -7, -8, 3], [-9, 11, -3, -8, -3, -4], [10, -8, 1, -6, -17, 1], [-21, 5, 0, -2, -12, -1], [1, 3, 3, -11, 9, 1], [-27, -1, -5, 6, 10, -2], [13, -7, 3, -8, -18, 0], [-6, 2, 8, -8, 8, 4], [5, 2, -1, 4, 9, 2], [-11, -7, 0, 0, -3, 3], [1, -12, 2, -12, -10, -1], [14, -4, -5, 4, 1, 3], [-4, 2, 6, 2, 6, 4], [-9, -7, -1, 0, -3, 6], [13, -9, 9, -2, 6, 4], [-1, 2, 4, 0, 4, 9], [-18, 8, -2, -4, -8, -6], [3, 0, -6, 0, 15, -3], [3, 1, 5, 0, 4, 8], [6, -9, -2, 5, 6, 5], [-1, 0, 2, -6, -7, -3], [-19, -3, -2, 4, -20, 1], [-16, -7, 6, 3, 12, 1], [6, 6, -4, 6, 2, 4], [3, 7, -5, 8, -8, -6], [2, 13, 4, 9, 10, -1], [6, -1, 8, -9, 2, 5], [-3, -12, -3, 0, -5, -6], [-4, -7, 1, 6, 7, 3], [-5, 7, -3, 6, 0, 2], [6, -10, 6, -8, 12, 0], [-22, 2, -6, 4, 0, 4], [1, -8, 2, -2, 12, -3], [9, 8, 4, 8, 7, 1], [-7, 12, -1, 6, 5, 6], [-29, 1, -3, -5, 3, -1], [25, -10, 11, 2, -9, 4], [22, -2, 9, -4, 5, 5], [35, 2, 8, 0, 8, 5], [11, 0, 0, -4, -9, 5], [-9, -3, -2, 12, 13, -1], [9, 1, -2, -11, 7, 0], [13, -3, -2, -8, 1, -3], [21, -8, -4, 2, -8, -7], [-17, -2, 2, 10, -4, 3], [-4, 2, 0, 6, -18, 4], [11, 8, -1, 2, 12, 2], [-4, 3, 0, -2, 0, 8], [-2, 13, -1, 9, -7, 0], [13, 4, 4, 16, 1, 1], [-14, 6, 3, 4, -3, -3], [10, 0, -4, 8, -16, -4], [21, -1, 8, -3, -3, -6], [9, 2, 8, 2, 2, 3], [6, -6, -6, -2, 0, 6], [4, -2, -10, -8, -5, -6], [11, 0, 6, -2, 9, -3], [-8, 2, 0, 8, 0, 0], [-25, -7, -6, -3, -3, -2], [14, 2, -12, 2, -6, -2], [-1, -10, -4, -2, -7, -7], [-7, -2, 0, -8, -7, 3], [18, 15, 7, 6, 25, 5], [-7, -1, -4, -3, 9, 2], [-8, 0, 2, 0, -10, -4], [-5, -6, -4, -2, -7, -7], [9, 1, -1, -1, -13, -1], [-15, 15, 2, 9, 11, 4], [-9, 3, -4, -3, -9, 4], [0, 8, 3, -2, 15, -1], [-28, 3, -4, 1, 2, 1], [-13, 3, 4, 4, -10, 1], [9, 4, -5, 8, 8, 6], [2, -1, 1, -2, 13, -3], [10, 16, -6, -2, -11, -4], [14, 0, -4, 10, 8, 0], [-7, -2, -16, 4, -12, -5], [7, -4, 7, 6, -14, -4], [14, -6, -2, 6, 4, -2], [-21, 7, -3, 2, 12, -2], [-19, 6, -8, 6, -2, -5], [6, 4, -5, 8, 25, 3], [-4, 0, 12, 6, -12, 2], [11, -12, 1, 0, -14, -8], [4, -14, -1, 6, -7, 3], [-19, -13, -4, -3, -13, -10], [-13, 12, 8, 4, 17, -3], [-30, -3, -3, 1, -12, 0], [-5, 11, 0, 0, 28, -3], [5, -11, 1, 4, -2, 2], [15, -2, -2, 4, 17, 9], [33, -14, -1, -4, -7, 2], [-31, -7, -6, -1, -9, -2], [25, 4, 4, -10, 6, 1], [-14, 8, 4, 0, -2, -4], [0, 6, 2, 2, 14, -2], [-25, -9, -10, -2, -23, -3], [-19, -18, 5, -4, 3, -6], [10, 15, -4, 11, -4, 5], [-7, 8, -10, 6, -1, 3], [33, -1, 7, 0, 8, 4], [7, 4, 9, -6, 27, 2], [-14, 0, -10, 10, 8, -4], [-15, 6, -16, 2, -11, -5], [3, 0, -7, 2, 26, -2], [-13, -7, 2, -10, 16, 5], [-31, 3, -12, 0, 9, -1], [27, -4, -5, 2, 7, -2], [-8, 4, -7, 12, 16, 3], [20, 1, 8, 3, -10, 7], [-15, -2, -2, 2, -5, 9], [-14, -10, 4, 12, -9, 8], [-5, -19, -1, -5, -15, 1], [15, -10, 0, -14, -17, -1], [-11, 8, 10, 6, -5, -1], [-22, 12, 7, 2, 19, -1], [-6, -3, 0, -2, 18, 6], [5, 8, 7, 14, 22, 2], [30, -16, 6, -12, -5, -4], [-20, -1, 3, -7, 12, 0], [21, 8, 5, 4, 14, 0], [7, 6, 4, -6, -9, 3], [-25, 15, -5, -2, -8, -8], [14, 10, 4, 14, 30, 2], [7, -5, 1, -9, -32, 1], [25, -7, -5, -4, -14, -4], [5, 6, 0, -2, 4, 5], [15, 0, 17, -16, -4, 4], [21, -1, 9, 4, 6, 6], [-10, -7, 5, 8, -7, -1], [-4, 10, 0, 8, 18, 6], [10, 0, -18, -12, -3, -2], [-8, 10, 3, 4, -3, 5], [0, -1, -3, -1, -6, -4], [-15, -3, 6, -10, -3, -1], [-13, -9, -6, 8, -9, -1], [-6, -16, 1, -4, -27, -5], [-5, 19, 4, -2, 31, 7], [-26, -8, -4, 8, 14, 0], [-24, -7, 0, -2, -12, 0], [-23, 19, 3, 11, -5, -1], [30, -8, 15, 4, -3, 7], [-43, 6, 1, -8, -24, -8], [0, -20, -4, -14, -16, 2], [-4, -3, -1, -11, -18, 0], [-21, 2, -8, 10, 13, 1], [-8, 11, -10, 8, -22, -2], [10, 13, 4, -5, 14, 1], [-37, 10, -5, -6, -26, -10], [25, -1, 6, -4, -9, -3], [44, 2, 8, -2, 8, 2], [-22, 3, 12, 9, 0, -1], [-8, -6, 11, 8, -10, -1], [-2, 19, 4, -4, 10, 0], [5, 13, 4, 9, 7, 2], [24, 2, 3, 2, -4, 5], [5, -10, -2, -8, 7, 3], [-7, -15, 4, 5, 1, 10], [23, -4, -4, 6, -14, -1], [0, 3, -5, -10, -17, -1], [-13, -13, -4, 6, 0, -1], [-15, 9, 3, 4, 19, 0], [22, -2, -5, -8, 19, -5], [3, 0, -4, 0, 11, -3], [5, 1, 3, -2, -3, 6], [-15, 16, 4, 6, 26, -3], [-16, 3, -1, -3, 22, 0], [-1, 15, -11, 9, 4, 5], [13, -3, -6, -5, -9, -2], [-26, 6, -12, 8, 6, 4], [-15, 10, -6, -12, -9, -5], [-21, 9, 0, 8, 39, -3], [-2, -4, -14, 12, 16, 4], [-7, -1, -9, -3, -24, 1], [-1, -6, 4, 0, -22, -5], [21, -9, 15, 1, 0, 3], [-5, 1, 2, 2, 27, -3], [5, 12, 12, 2, 18, 5], [20, -15, -13, -6, -19, -3], [-13, 17, 3, 8, -10, -2], [-22, 6, 2, 6, 8, 0], [-13, -17, -1, -6, -19, 4], [-2, 2, -15, -14, -3, -9], [-27, 6, -7, -2, -8, -14], [27, -10, -5, -2, -14, 4], [-4, 8, 7, 2, 19, 3], [-9, -2, -11, -2, -2, -6], [21, -11, -3, -11, -18, 1], [3, 2, -12, 4, -15, -7], [-22, -12, 7, 4, 7, -5], [8, 22, 5, 8, 17, -1], [22, -24, -4, -4, -28, 0], [-15, 14, 6, -4, 15, 11], [-23, -3, -3, 6, -20, 4], [28, 6, 6, 0, 2, -8], [-8, 2, 2, -6, 26, 4], [-24, -12, -1, -10, 11, 1], [-8, 3, -1, -13, -4, 2], [12, 0, 5, -4, 29, 1], [-2, 2, 1, -4, 1, -1], [4, -10, 4, 10, -4, 8], [-8, 12, 6, -6, -12, -2], [-8, -28, -7, -6, -37, -5], [39, -11, -12, -2, -6, -5], [-6, -3, -18, 10, 0, -8], [-24, -8, 0, -4, 18, 10], [-13, 15, 12, -10, -9, -1], [-4, 18, 2, 2, -8, 0], [-8, -10, -3, -6, -20, 3], [-17, 10, -1, 0, 2, -6], [6, -22, -11, 6, -17, -5], [-3, 12, 13, 6, 4, -10], [21, 7, -2, -6, 7, -9], [-13, 0, -3, 0, 8, -10], [20, 7, -2, 1, -2, -5], [-4, -6, -1, -10, -37, -3], [-19, -8, -4, -2, -25, 1], [5, 5, -11, -6, -21, 0], [-4, 30, 5, 6, 32, 3], [16, -4, -14, 10, 16, 2], [-5, 3, -10, 8, -7, -1], [-19, -23, 5, 10, 5, 4], [-18, -8, -13, 10, 11, -1], [13, -4, 11, 0, -28, -2], [-41, 2, -6, -4, 8, -3], [18, 16, 2, 16, 20, -4], [-39, 13, -16, -1, 17, -4], [-2, 9, -6, 5, -6, -11], [-15, 8, 15, -2, 17, 14], [-17, -1, -8, -5, -11, -10], [19, -12, 3, 2, -35, -2], [-18, -9, -4, 9, 32, -5], [23, -18, -4, -8, -25, -3], [-32, 4, 6, 0, 18, 2], [31, 0, 11, -2, 8, 2], [-3, 5, -7, 13, 19, -9], [-7, 2, 14, -8, 7, 5], [-5, 12, -7, 4, -13, 2], [-42, 15, -13, 7, 0, -8], [-4, -20, -8, -8, -26, 2], [21, 10, -2, 0, 13, -5], [39, 6, -1, -4, 8, -2], [32, -12, 2, -2, 4, 0], [3, -17, 1, -19, -11, -3], [45, -2, 15, 4, 6, 4], [1, 5, 19, -2, -2, 10], [-14, -10, -8, -4, -10, 8], [-22, 1, -3, 1, 33, 0], [-17, 2, 5, -6, 15, 10], [1, -10, 2, -14, -1, -11], [-50, -2, -12, 12, -6, 2], [-37, -3, 6, 0, -3, -1], [-38, 2, -20, 4, -11, -4], [-3, -3, -10, -6, 10, -5], [13, -7, -3, 1, -19, 3], [5, -16, 8, -10, -19, -1], [-39, -7, -4, -5, -19, -12], [32, 0, -5, 4, 7, -5], [5, 5, -7, -18, -1, -4], [-9, 3, 0, -7, -27, 0], [-17, -1, 0, -2, -33, 3], [19, 4, 3, 10, 16, 8], [-51, 12, 0, 8, -3, 3], [-9, -8, 7, 2, 22, -6], [-2, -6, 2, -2, -4, 14], [12, 4, 6, 4, -6, 4], [9, 0, -5, 0, -23, -4], [-2, -5, 11, -3, 23, 6], [-18, -6, -7, 8, 23, 7], [25, -11, -9, -10, -36, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5290_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5290_2_a_be();".
// 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_5290_2_a_be(:prec:=6)
    chi := MakeCharacter_5290_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_5290_2_a_be();".
// 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_5290_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5290_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 20, 19, -18, -11, 2, 1]>,<7,R![-8, 8, 127, -26, -23, 2, 1]>,<11,R![5022, 2376, -439, -314, -9, 10, 1]>],Snew);
    return Vf;
end function;