
// Make newform 9075.2.a.dr in Magma, downloaded from the LMFDB on 28 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9075_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_9075_2_a_dr();".
// 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_9075_2_a_dr();".
// 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, -22, 23, 9, -10, -1, 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], [-4, 0, 1, 0, 0, 0], [4, -5, -1, 1, 0, 0], [-2, 16, -1, -9, 0, 1], [4, 20, -8, -10, 1, 1]];
    Rf_basisdens := [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_9075_a();"
function MakeCharacter_9075_a()
    N := 9075;
    order := 1;
    char_gens := [3026, 727, 5326];
    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_9075_a_Hecke( : Kf := Kf)
    return MakeCharacter_9075_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]];
    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], [-1, 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, -1, 1], [0, 0, 0, 1, 0, -1], [1, -2, 0, 0, 1, 0], [-2, 1, -1, -1, 1, -1], [-2, 0, -2, -1, 2, 0], [3, -1, -1, -1, 0, 1], [-5, 0, -2, 0, 0, 1], [2, -1, 1, -1, 0, -1], [2, 2, 0, -2, 0, 0], [-2, 0, 0, 1, -1, 0], [-2, -1, 1, -1, 0, 0], [-2, -3, 1, 0, 0, 1], [1, -1, 1, -1, 2, 1], [-3, 0, 0, 2, -1, 0], [0, 3, 1, 0, 0, -1], [1, -2, -4, 0, 1, 0], [-1, -3, -3, 1, 2, -1], [-2, 1, 3, 0, 0, 1], [-4, 2, 0, 1, -2, -2], [-1, -4, 2, 2, 0, -1], [6, 1, 3, 0, -2, 1], [-5, 4, 2, 2, -1, 0], [2, 0, 0, -1, 3, -2], [1, -1, 3, 1, -1, -1], [-2, 0, 0, 1, -2, -1], [3, -4, 0, 0, 1, 2], [2, 0, -4, 2, 2, 0], [-8, -3, -3, -1, 3, 1], [6, 3, 1, -1, 1, -1], [-8, -2, 0, -1, -2, 2], [-4, -1, 1, 3, -3, 3], [-5, 2, -2, -2, -1, 0], [-7, -4, 0, 0, 1, 0], [-6, 5, -1, -1, 1, -1], [-8, -1, 1, -2, 0, -1], [2, -7, 1, 2, 0, 1], [1, 3, -5, -1, 3, -1], [-8, -2, 0, 0, -6, 2], [-3, 0, 2, -2, -2, -1], [0, -4, -2, 0, 5, -1], [3, 1, -3, -1, 6, -1], [-3, 5, 1, 1, 2, -1], [5, 0, -6, 0, 5, -2], [8, -3, -5, -1, 3, -1], [-4, 4, -2, 0, -3, -2], [2, 0, 0, 1, 0, 1], [6, -5, 3, 4, -2, -1], [0, 3, -1, -3, 3, -1], [-2, 2, 0, 2, -4, 2], [-2, -3, 3, -3, 0, 0], [2, -5, -3, -1, 3, -1], [2, 2, 2, -3, 2, 2], [2, 3, 1, 3, -3, 3], [3, -8, 4, 2, 0, 1], [-8, -1, -1, 4, 2, -1], [-1, -2, 4, 0, 1, 2], [12, -2, -2, 3, 2, 1], [-7, 2, -4, 0, -1, 2], [4, -5, 3, 2, 2, -1], [-8, -2, 2, -2, 3, 1], [-12, 6, 0, 1, -1, 2], [9, 5, 5, -1, -2, -1], [5, -6, 2, -2, 0, -3], [2, 1, -1, -3, 1, -3], [-1, 2, 6, -2, -4, -2], [-8, -4, 0, 3, -4, 1], [0, 1, 3, 4, -4, 1], [2, 2, 0, -6, 0, 0], [-3, 0, -2, 0, -3, 2], [4, 1, -5, -3, 1, -3], [-6, 4, 0, 0, 0, 2], [-10, 1, 5, 3, 4, -3], [7, -8, 4, -2, 6, -1], [4, 4, 4, -1, 4, 0], [-11, -4, -4, 2, -2, 0], [6, 1, -1, 0, 2, 5], [-8, -8, -2, -4, 1, 2], [-2, 2, 4, 0, 2, -2], [1, 4, 6, 2, -8, 1], [-9, 5, 1, 3, -4, -1], [8, -1, 5, 0, -4, 5], [4, 2, 2, 1, -2, -2], [10, -2, 4, 4, -1, -1], [-6, 13, 1, 3, -2, 1], [10, -2, 6, 4, 0, 0], [0, -2, 6, -1, -3, -4], [-2, -5, -9, -2, 6, -1], [2, 4, 6, 2, 0, -2], [6, 2, -2, -2, 0, 2], [-7, -2, -8, -2, 1, 4], [4, 1, 3, 1, 5, -1], [-2, -1, -1, -2, 2, 3], [-22, 5, 1, 2, 2, 1], [-7, 2, 6, 6, -3, 0], [0, -2, -2, -4, -2, 2], [-14, 0, 2, 0, -4, 4], [12, 0, 2, -5, 1, 4], [-4, 2, 2, 0, -6, 0], [12, 8, -2, 0, -4, 0], [-1, 7, 3, -1, -2, 1], [-1, 8, 4, 0, 0, -1], [4, -5, 9, 1, 3, -1], [10, 0, 2, 0, -3, 1], [-20, -7, -3, 4, 0, 1], [1, 7, -1, -3, 3, -5], [-10, -8, 6, 2, -2, 0], [-13, 4, -4, -4, 6, 1], [-8, 9, 1, 4, -7, 2], [6, 4, 4, 6, -2, -4], [6, 5, -1, -1, -3, 1], [4, 0, 6, 3, -8, 0], [-11, 2, 0, 4, 1, 0], [-12, 3, 5, -3, -5, 3], [-14, -2, -6, -2, 8, -2], [14, 8, -2, -4, -8, 4], [5, 5, -3, -7, -1, 1], [5, 0, 4, 8, -5, -2], [8, 2, -4, 0, 5, 1], [-2, 10, -2, 0, 0, 0], [1, -11, 5, 1, -2, -1], [-6, 3, -9, -1, 0, 3], [0, 7, 3, 3, -3, 3], [-10, 1, 3, -2, 0, -3], [6, -14, -6, 2, 2, 0], [-8, 4, 2, 6, -1, -3], [19, 5, 5, -7, 2, -1], [12, -3, 3, -1, -1, -5], [9, -5, -1, -1, -4, -3], [-19, -12, -4, 8, 2, -1], [-12, 16, 2, 1, -2, -4], [-23, 6, -2, 0, 6, 2], [-8, -7, -3, -3, 5, -3], [14, -4, 0, -2, 4, 0], [-6, 2, 2, 10, 0, 2], [-16, 9, 7, -2, -6, -3], [-5, 17, 1, -5, 0, -1], [10, 2, 2, 2, -6, 2], [-5, -6, 0, 0, 1, -6], [-14, 1, -5, -4, -2, 3], [7, 12, -4, 0, 6, -4], [0, 4, 0, -10, -2, 0], [-1, 12, 0, 0, 0, -5], [4, -2, 0, -5, -7, 2], [19, -2, 4, 6, -1, 0], [6, -3, -5, 4, 2, 3], [9, 8, 0, -2, 8, -5], [-6, -6, -2, 7, 4, -2], [-1, 10, 2, 0, -3, -4], [-2, 12, 4, 8, -8, -2], [-11, 8, -10, -6, 9, -4], [0, 0, 2, 2, 2, 0], [-21, 6, -8, 2, -1, 0], [-2, 1, 3, -1, 4, -3], [-35, -2, 0, -2, 0, 3], [-14, 1, 1, -1, -2, -3], [18, -6, 2, -1, -5, 0], [-10, -6, 4, 4, -1, 1], [-11, 0, -6, 0, 11, 0], [6, -12, -10, -6, 12, -4], [-14, -7, -9, -5, 10, 0], [-6, 5, -1, -1, -5, 5], [-2, 7, -7, -1, 11, -1], [-7, -4, -10, 2, 4, -3], [4, 8, 6, -2, -9, 2], [-4, 1, -3, -5, -1, -3], [-8, 7, 11, 0, -6, 3], [-11, -3, -5, -1, 2, 7], [-26, 5, -1, -2, -6, -3], [4, 4, 0, 6, 5, -3], [18, -11, -5, -1, -1, 1], [-14, 10, 10, -3, -6, 2], [11, 1, 1, -1, -8, 7], [-10, -11, -5, 9, 0, 1], [8, -6, -4, -6, -2, 0], [-1, 17, 1, -3, -7, 1], [-13, 0, -2, -6, 1, 2], [-20, -5, -3, 2, 4, -1], [-4, 22, 8, -4, -2, -2], [-14, 5, -3, -4, -1, 2], [2, -4, -8, -3, -1, 4], [0, -10, 6, 2, -8, 2], [1, 0, -8, 2, -4, -1], [13, 4, 10, 6, -17, 2], [4, -2, 4, 10, -7, -4], [8, 5, -3, -2, 10, -3], [-1, -14, -2, 0, 1, 4], [6, 0, -4, 3, 1, -4], [1, 5, 5, -5, -10, -1], [-2, -4, -10, -3, 10, -8], [8, -10, 6, -1, 1, -4], [-14, 6, 0, 0, -11, 3], [-8, 8, -2, 4, -3, -2], [-35, 12, -4, -4, -1, -2], [-8, 15, -7, -5, 4, -2], [-18, 2, 2, -4, -2, -2], [6, 3, -5, 2, -6, -1], [-9, 14, 0, -2, -5, 0], [-13, -4, -4, 6, -2, -1], [-5, 13, 5, 5, -3, -7], [-32, 14, 6, 0, -4, -2], [-12, 4, 2, -1, -11, 4], [7, -3, -3, -7, -6, 3], [8, -8, 4, -5, 9, -2], [-12, -5, -3, 1, 0, -9], [3, 7, -5, 3, 4, 1], [-3, 6, 14, -4, -12, -1], [4, -9, 1, 3, -10, 7], [19, -18, -8, -4, 11, 0], [-10, 12, 0, 1, 1, -10], [4, -1, -11, -8, 6, -5], [-20, 12, 2, 0, -5, -2], [11, 2, 8, 6, -3, 4], [2, 12, -4, -2, -6, 4], [-12, -4, 4, 5, -13, 2], [-26, 11, -1, -8, -7, -2], [-19, -2, 6, 0, -12, 2], [-2, 14, -6, -6, 2, 6], [-8, -7, 9, 1, 0, 1], [-13, 10, -6, 4, -5, -2], [14, -6, -2, 0, 2, 2], [0, -3, -5, -7, 6, 3], [-28, -13, 1, 9, 8, -8], [-8, -10, -8, -6, 12, -2], [18, 8, 0, -10, 4, 0], [36, -1, 3, 4, 4, -1], [-34, 2, -6, 4, -1, -3], [-17, 7, 3, -1, -12, -3], [7, 5, 5, 3, -4, -5], [2, -10, 10, 15, -6, 2], [-13, 2, -2, 2, -11, 2], [-18, -9, 1, 8, -2, 1], [3, 5, 3, 5, -6, 11], [-4, -15, -11, -4, -1, 8], [-8, 5, 11, 8, -8, 1], [2, -6, 0, 0, 0, -6], [20, 7, -3, -1, 3, -7], [20, -12, 4, 8, 0, 4], [15, 2, 6, -10, -1, 4], [15, 8, 12, -6, -6, 0], [14, 3, 3, -1, 1, 7], [-26, -12, 0, 4, -8, 8], [18, -2, 8, -2, -6, 6], [-12, 11, 1, -10, -4, -5], [22, -16, 0, -4, 6, 0], [-22, -18, -2, 4, 0, -4], [15, 6, -6, 6, -1, 0], [-16, -11, -5, -5, 4, -3], [8, 1, 3, -1, 15, -5], [-1, -6, -2, 6, 6, -2], [-18, 16, -2, -4, -1, 3], [-24, 8, -4, -4, 6, -4], [-12, -9, -3, 3, 6, -3], [-36, -6, -10, -2, 10, -2], [-16, 12, 8, 7, -6, -5], [5, 6, 8, 6, -1, 0], [-12, 6, 0, -6, -4, 2], [2, 2, -2, 0, -12, 0], [11, -9, -9, -9, 5, -1], [-1, 8, 0, 8, 5, -6], [0, 4, 6, -8, 5, -1], [-27, -6, -4, -2, 5, 4], [-6, 2, 6, -12, 0, 2], [-42, -3, -11, 0, 8, 1], [17, -12, -2, -6, 7, 2], [4, -1, -9, 3, 3, -5], [8, 2, -12, 2, 1, -4], [-5, -4, -8, -10, 15, 2], [9, 14, 0, 4, -2, 3], [-24, 5, -1, 9, -5, 5], [23, 8, -6, 6, 5, -2], [-5, -12, -18, 0, 9, 0], [-8, 3, -7, -5, 9, 3], [-2, -12, -10, -2, 7, 2], [23, -9, 5, 3, -4, 5], [4, -2, -4, -4, 4, -6], [18, -14, -2, 3, -1, 8], [21, 6, -16, 6, 9, -4], [-24, 2, -2, -13, 3, 6], [28, 4, -6, -6, 7, 2], [27, 4, -8, -4, 11, -4], [4, 2, 16, 4, -14, 2], [-7, -4, 12, 2, -7, 6], [6, 6, 16, -5, -5, 2], [11, 10, -8, 2, 15, 0], [-18, -3, 5, -1, 0, 1], [4, -3, -3, 3, 4, 5], [-8, 1, -1, -1, 13, -11], [-38, 3, 1, 9, -4, 2], [12, 3, 3, 0, 8, -1], [2, 14, -4, -2, 10, -2], [22, 10, 8, -2, -8, -2], [9, -14, -4, 10, -3, -4], [12, -9, -15, -11, 8, 0], [14, -13, 3, 8, -2, 3], [17, 8, 4, -6, -3, -2], [5, 8, 8, -2, -7, 2], [18, -14, 14, -3, 2, -3], [-15, -5, 3, 1, 2, -7], [-4, 12, -8, -1, 5, 2], [6, 12, -4, -4, -14, 4], [19, -10, -2, 2, 11, 0], [-2, 11, 13, 9, -21, 9], [13, 18, 2, 0, -10, 2], [2, -10, 4, 8, 6, -2], [3, -2, 12, -2, 4, -7], [-10, 14, -2, 9, -17, -2], [-16, 15, -3, -9, 2, 3], [4, -9, 9, 17, -2, -5], [-18, -4, 4, 4, 10, 0], [-8, -7, -5, 1, -9, 5], [1, -12, 8, -2, 4, -4], [-30, 16, -12, -8, 4, -4], [-12, -18, -2, 4, -8, -2], [15, 10, 14, 2, -13, 4], [-10, 4, 6, 5, -4, -8], [-1, -8, 10, -4, 3, -6], [-3, -8, 6, 0, 4, -1], [22, 5, 1, -15, -2, 1], [19, -6, 4, -2, 7, 6], [0, 11, -5, -12, 1, 0], [19, -17, 5, 3, 0, 5], [-18, -3, -1, 7, -3, -3], [35, 14, -4, 0, 17, -10], [36, -3, 11, 0, -8, 3], [20, -6, -18, -9, -2, 1], [5, -14, 10, 16, -4, 6], [-12, -19, 9, 5, -3, -3], [19, -11, -3, -7, 8, -5], [22, 7, 1, -9, 7, -5], [-26, -11, -5, 11, 1, -7], [0, 0, -20, -1, 7, 4], [-11, -2, 18, 6, 2, -2], [-20, 6, -4, -1, 5, 6], [-11, 5, -7, -13, -1, 1], [11, -22, -2, 4, -1, 2], [-4, -6, 6, -6, 5, 3], [-24, 8, 12, 6, -10, 0], [18, 9, 15, 9, -18, 1], [27, 2, 0, 6, -5, 4], [-18, -14, 14, 6, -12, 10], [8, -7, -5, 14, 0, -7], [5, 4, 8, 8, 2, -4], [21, -4, 10, -2, -5, 2], [-48, 1, 3, 4, -10, 1], [10, -16, 4, 9, 2, 7], [34, -3, 7, -3, 15, -7], [35, 12, 2, -2, -11, 10], [-2, 3, -9, -13, 10, -1], [8, 8, 8, -4, 4, 8], [-13, -5, -1, 15, -7, 7], [40, -17, -7, -11, 12, -4], [-24, 10, -12, -2, 0, 2], [-32, 14, 2, 4, -8, 6], [4, -7, 5, 0, -1, 0], [16, -25, 1, 4, 2, -1], [19, -10, 0, 0, -1, 0], [28, -12, 0, 0, -2, 16], [-12, 4, -4, -4, -4, -8], [24, 5, 9, 10, 0, -7], [-17, 6, 0, 4, -3, -6], [-28, -14, -18, 10, 3, -3], [14, 7, 9, 5, -8, -6], [-38, 12, -12, -8, 8, 2], [11, 2, 18, -4, -4, -6], [-4, 1, -9, 4, -2, -3], [37, -1, 11, -3, -6, 1], [-62, 4, 0, 7, -3, 4], [-20, 7, 5, 7, -2, -1], [-2, -18, -6, -12, 2, 2], [-15, -4, 2, 6, 4, 1], [4, -8, -18, 2, 16, 0], [-16, -4, -14, 4, 0, 8], [-33, 10, -20, 4, 0, 7], [26, -11, 13, 5, 14, -3], [-25, 8, -6, -2, 1, -6], [-8, -7, -5, 0, 16, -3], [-6, -3, 1, 8, -19, 0], [3, 10, -4, -10, 11, -6], [34, -3, -3, 11, 5, 5], [21, -17, 19, 11, -8, 5], [44, 13, 3, -8, 12, -5], [-8, 2, 6, 2, 16, -2], [20, -4, -10, -2, 11, 1], [-40, 8, -18, -4, 2, 4], [8, 14, -6, -7, 7, 0], [-21, -5, -15, -7, 2, 1], [10, 11, -1, -19, -1, -1], [-28, -4, 8, 8, 2, -4], [1, 12, -2, -2, -11, 4], [-14, -24, 10, -2, 8, -4], [-12, -2, -2, -6, -6, 10], [43, -3, -3, -7, 8, -5], [-32, -15, 1, -1, -10, 9], [17, 5, -7, 11, 12, 3], [18, -11, 11, 5, -8, 13], [-20, 4, 2, -2, -21, 2], [-38, 10, -10, -3, 4, 7], [10, -6, -6, 2, 2, 0], [24, 14, 0, -5, 9, -2], [-8, -1, -7, 2, 6, 3], [-30, -11, -1, 1, -19, 9], [-23, 32, 12, 2, -10, -7], [-14, 8, -12, -9, 4, 4], [26, -12, 10, 0, -2, 2], [-10, -6, -4, -2, 8, -2], [-4, 26, -4, 2, -5, -1], [16, -14, 0, 8, 5, 1], [-4, -2, -10, -5, -3, 8], [-7, 15, -13, -1, 8, -11], [-22, 14, 10, 4, -1, 9], [-28, 13, -11, -1, 2, -3], [12, -18, -4, -4, -4, -6], [-19, 0, 16, 4, 1, -8], [6, 4, -8, -2, -12, 4], [4, 3, -3, -7, 1, -3], [-34, 10, -14, -1, -8, 2], [25, 8, 2, 2, -5, 14], [-10, -10, -2, 12, 10, 2], [-34, 7, -3, 0, -2, 7], [-1, -4, 14, 4, 2, -1], [-2, -27, -7, -4, 9, 0], [-18, 22, -10, 3, -1, 4], [22, -27, -9, -2, 0, 5], [7, -30, 0, 4, -13, 4], [38, 2, -2, -4, -16, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9075_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9075_2_a_dr();".
// 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_9075_2_a_dr(:prec:=6)
    chi := MakeCharacter_9075_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_9075_2_a_dr();".
// 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_9075_2_a_dr( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9075_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, -22, 23, 9, -10, -1, 1]>,<7,R![-368, 232, 136, -62, -19, 4, 1]>,<13,R![32, -432, 300, 110, -47, -2, 1]>,<17,R![46, -2036, 1025, 106, -68, -2, 1]>,<19,R![-512, -128, 368, 64, -47, -2, 1]>,<23,R![-1072, 4656, -528, -564, -21, 12, 1]>,<37,R![98540, 4710, -7687, -858, 126, 24, 1]>],Snew);
    return Vf;
end function;