
// Make newform 2601.2.a.bi in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2601_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_2601_2_a_bi();".
// 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_2601_2_a_bi();".
// 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 := [-1, 12, 18, -4, -9, 0, 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], [1, -4, -1, 1, 0, 0], [-3, -1, 1, 0, 0, 0], [4, 3, -6, -1, 1, 0], [2, 14, 3, -8, -1, 1]];
    Rf_basisdens := [1, 1, 2, 1, 2, 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_2601_a();"
function MakeCharacter_2601_a()
    N := 2601;
    order := 1;
    char_gens := [290, 2026];
    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_2601_a_Hecke(Kf)
    return MakeCharacter_2601_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, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 1, 1], [1, -2, -1, -1, 1, -1], [1, 1, -1, 0, -1, -2], [2, -1, -1, -1, 1, -2], [0, 0, 0, 0, 0, 0], [2, 1, -2, 1, 1, 0], [2, 0, -2, -2, 1, -2], [1, 1, 1, 0, 0, 2], [5, 1, 0, 0, 2, 1], [0, 2, 0, 0, 1, -1], [3, -2, 2, 1, 0, 1], [0, -3, 2, 1, 0, -1], [-5, -1, 3, 0, -2, 0], [-4, 1, 1, -1, 0, 3], [1, 1, 3, 2, -1, 0], [2, 2, 1, 2, -3, 3], [-1, 3, -1, 0, 0, 5], [4, 3, 3, 1, -1, 1], [1, 1, 3, -1, -4, -1], [3, 4, 4, 4, -2, 4], [-3, 1, 7, 2, -4, 2], [0, 1, -5, 3, 0, 1], [-7, 2, 0, -1, -3, -2], [-5, 2, -4, -3, -2, -1], [5, -4, 1, -1, 2, -1], [-1, -4, 6, -3, -4, 1], [2, 5, -1, 2, 0, 4], [3, -5, 3, -2, 2, -4], [6, 1, 5, -1, 0, 6], [-6, 4, 0, -2, 0, 0], [0, -4, -4, -2, 1, -6], [4, -2, -1, 0, 5, -3], [4, -6, 0, -2, 5, -8], [-6, -3, 0, 3, 0, 1], [7, 3, 4, 0, -6, 1], [1, -5, 1, 1, 0, 1], [-1, -4, -6, -3, -2, -9], [9, -4, -2, -3, 1, -3], [-1, 6, -8, 5, 1, -3], [4, -7, -2, -1, -1, -5], [4, -1, -3, -3, -3, 5], [-11, -3, 4, 2, -1, -4], [4, 4, 6, -2, -6, 0], [10, 2, -4, 1, -1, 3], [-1, -2, -10, 1, -5, -6], [-2, 1, -7, 2, 3, -4], [-4, -2, 2, 4, 1, 2], [-6, 3, -1, 4, -3, 2], [7, -3, 3, 4, -1, 0], [-1, 3, 11, 0, 1, 10], [-4, 2, -6, -7, -4, -9], [-1, 10, 2, 3, 2, 7], [1, -3, -15, -4, 0, -6], [-3, -1, -5, 4, -2, -4], [3, -4, -6, -3, -5, 1], [0, 0, -9, -4, 2, 1], [-2, 3, 8, 1, 0, 12], [-2, 2, 4, -2, 6, 8], [7, 3, 1, -2, -3, -5], [-16, 2, -8, 0, 0, -4], [0, 9, -1, -2, 2, -4], [6, -5, 1, 3, 4, 15], [6, 2, 7, 8, -2, 5], [-8, 4, 8, 4, -5, -2], [4, 1, 12, 7, -3, 5], [15, -7, -1, -4, 2, -2], [10, 1, -3, 1, 2, 3], [1, 5, 7, -1, -12, -5], [-9, -4, -2, -11, 1, -3], [0, 4, 6, 2, -9, -8], [4, -9, -3, -3, 5, -8], [10, -1, 7, -2, 1, -4], [-9, 3, 0, 0, -6, 3], [-5, -2, 10, 1, 0, -5], [-11, -1, -3, 2, 3, -4], [-4, 3, -9, -1, 3, 3], [-13, 7, 5, 4, -9, 12], [-5, 1, 1, -8, -5, -3], [14, 6, -6, 0, 4, 6], [-10, 2, 11, 4, -2, -3], [3, -1, 3, -4, -4, -2], [0, -3, -6, -9, -3, -13], [3, 2, 12, -1, -12, -2], [-2, 2, 10, -2, -2, 16], [-11, -3, -3, -4, -3, 0], [5, 5, 13, 3, 1, 11], [-12, 12, 0, 2, -11, 4], [1, -8, -5, -3, -9, -14], [4, -3, -9, -13, 0, -3], [8, -1, -7, -1, 9, 1], [-2, 10, 7, 2, 1, 11], [14, -3, 1, 3, -3, 1], [11, -7, -16, -2, 8, -14], [4, -1, -15, -5, 4, 1], [12, -12, -4, -2, 1, -8], [7, -7, 5, -2, -5, 2], [2, 8, 4, 1, -4, 3], [-3, 3, -3, 5, 0, -7], [0, 9, 2, 3, -3, 17], [-13, -2, -4, 7, -1, 11], [0, -4, -6, 4, 15, 2], [2, -3, 11, -5, -5, 17], [5, -1, -2, 0, -1, -12], [-7, -12, -1, -7, -3, -13], [9, -2, -4, -1, -4, -17], [10, 0, -10, -4, 10, -12], [12, -1, 5, 9, 7, 1], [-6, -9, -1, -5, -5, -6], [10, 8, 4, 12, -1, 5], [16, 6, 7, 4, 3, 15], [-5, 7, 3, -6, -3, 0], [-2, -5, 7, -6, 1, -12], [-7, -1, 1, -4, -1, -5], [-2, -14, -8, -8, 9, -8], [-2, 2, 0, 4, 16, 1], [16, 1, -3, -9, -2, -1], [-15, -4, -10, -3, 4, -3], [1, -12, 14, 1, -4, 5], [9, -9, -24, -12, 3, -18], [12, 9, 0, 3, -9, 3], [9, 11, 9, -6, -4, 14], [8, -6, 6, 10, -4, -8], [-2, -2, 13, -4, 2, 18], [-15, 17, 3, 4, -4, 12], [-11, 2, 5, 7, 5, 9], [-4, 4, 14, 4, -4, 16], [-13, -4, -3, -11, -4, -11], [23, -3, 18, 2, 4, 8], [2, -9, -20, -3, 12, 2], [2, 0, -4, 8, -1, -6], [3, 6, 1, -1, -6, 18], [5, 1, -18, -4, 4, -6], [-1, 3, 5, 12, -1, -2], [-4, 0, 10, 5, 5, -9], [3, -5, -3, 8, 13, 12], [6, 1, -9, -3, 11, -2], [6, -6, -6, 0, -14, -8], [9, -13, -23, -10, 11, -18], [-9, -6, 9, -5, -12, -5], [-5, 9, 13, 6, -5, 0], [9, -8, -26, -5, 12, -8], [4, -9, -3, -13, -6, -3], [-21, 5, 6, -4, -9, -1], [7, 10, 10, 7, 4, 17], [-7, 13, 19, 8, -2, 9], [4, -1, 3, 1, 2, 25], [-9, 7, -1, 12, 4, 5], [9, -5, 9, -10, -15, 4], [9, 9, 13, 0, 3, 10], [0, 17, 11, 15, -12, 15], [10, -12, 2, 0, 9, 8], [10, 8, 6, 4, -5, 22], [-13, -6, 11, 3, -4, -16], [10, 4, 2, -6, -7, 12], [-4, -18, 8, 1, 3, 7], [-8, 0, 8, -8, 11, 6], [-2, 9, 14, -1, 0, 26], [10, -7, -21, -1, 11, -19], [25, 3, -3, -8, 9, -10], [-1, -10, -10, -11, 13, -5], [5, 3, 19, -4, -1, -6], [-12, -5, 11, -3, -13, 5], [5, 0, 28, 3, -4, 5], [8, 3, 17, 7, 4, 9], [9, 1, -3, 5, 13, -3], [-11, -9, 6, -2, 0, 1], [-11, 0, -15, 5, 3, -6], [1, 7, 15, 2, 8, 18], [-3, 1, -15, 2, 0, -2], [-9, -11, 14, -4, -5, -6], [13, -3, -27, -8, 11, -14], [-4, 14, 1, 6, -12, 10], [-18, 16, 9, 0, -17, 2], [27, -4, -14, -11, 12, -19], [-19, -4, -1, 9, -1, 10], [-5, 9, 5, 2, -14, -2], [9, -8, -20, -5, 4, -15], [-18, 0, -18, -4, -3, -15], [1, 10, -18, 4, 4, -12], [16, -5, -13, -5, -1, -13], [4, -1, -9, -3, -13, -14], [21, 2, -12, 3, 4, 7], [-4, -2, 2, 0, -8, 4], [10, -2, -16, -4, 5, -26], [-17, 8, 9, -3, -6, -5], [-15, 4, 10, -3, -23, -1], [-8, 15, 8, 1, -3, 9], [1, 20, 20, 9, -3, 25], [2, -10, -4, 2, -7, 3], [0, 20, -8, 2, 6, 2], [11, -12, 15, -9, 1, -1], [-1, -5, -9, -8, -3, -16], [-16, -11, -3, -5, 0, -13], [7, 10, 20, 3, 2, 3], [-8, -7, -1, -7, 8, -8], [12, 0, 16, 3, -6, -7], [-14, -13, -11, -1, -3, -15], [0, 12, 18, -2, -5, 16], [24, -8, -22, -4, 20, -20], [26, 2, -12, -2, -1, -2], [35, 3, 5, -4, 0, 5], [-5, 7, 1, 16, -8, 5], [-25, 23, 9, 6, -8, 12], [-18, 1, 21, 15, -8, 25], [0, 15, -14, -7, 9, 3], [-4, 6, -2, 8, -1, 22], [-28, -1, -33, 5, 3, -13], [-29, 2, -1, -11, -21, -3], [-1, -14, -10, -9, 10, -8], [0, -1, 7, 5, -3, 9], [22, 0, -28, -2, 12, 6], [21, 27, 17, 6, 0, 34], [-2, -6, 6, 14, -3, 0], [-18, 8, 3, 2, -20, 10], [14, 6, -16, 4, 10, -1], [-3, -1, -1, -6, -15, 8], [33, -2, -20, 1, 23, -1], [2, 19, -9, 7, 16, 7], [21, 8, -21, -5, 14, -11], [28, -8, -12, 0, 16, -8], [7, 16, -20, 3, 0, -1], [-12, -16, -24, 2, 17, -19], [-19, -18, -14, 3, 8, -9], [15, 28, -7, 15, -3, 16], [-13, -7, -23, -8, 2, -26], [-2, -15, 17, 1, -13, -11], [-8, -7, -12, -15, -12, -8], [19, -3, 7, 14, 10, 14], [27, 7, -16, -4, 10, 10], [22, -3, 3, 18, 20, 4], [18, -10, -7, -16, 0, 2], [25, 0, -16, -3, 1, 23], [1, 11, 30, 4, -15, 16], [-1, 9, 11, -8, 6, -20], [6, -28, 16, -12, 8, -4], [-8, 6, 4, 8, -7, 20], [13, 17, -3, -12, 8, 6], [-7, -21, -3, -4, 10, -4], [-15, 12, -26, -3, -6, -13], [-27, -3, 5, -4, 4, 3], [2, -19, -5, 3, -4, 3], [-4, 19, -15, 10, -2, 8], [-42, 0, -10, -4, 0, 6], [2, 2, -2, 0, -4, 0], [-4, 19, 47, 11, 0, 48], [-10, -18, -10, -4, 7, -36], [-6, 33, 12, 19, -14, 22], [-8, 5, 1, 5, 2, -11], [36, 8, 12, 8, -4, 7], [14, 2, 2, -4, -8, 10], [24, 3, 9, -7, -4, 1], [-1, -1, 0, 4, 14, 25], [-14, -5, 21, 1, -6, 1], [13, -15, -13, 2, 7, -8], [8, -10, -24, -8, 9, 5], [22, 19, -6, 1, 4, 3], [-17, -2, 40, 9, -18, 7], [-6, 5, -4, -9, 6, 3], [18, -23, -15, -11, 1, -7], [-33, -4, -4, 7, 6, 9], [8, -6, 31, -14, -2, 14], [-6, 5, -40, -5, 12, -4], [-7, 12, 2, 5, 11, -3], [8, -19, -9, -18, 4, -12], [-20, -10, 2, 6, 9, 12], [-3, -2, 22, 3, -5, 15], [0, -12, 39, 0, -6, 13], [-9, 14, -2, 7, -3, -9], [-4, 5, 1, 1, -1, 25], [0, -9, 2, 15, -8, 14], [-14, -12, 6, 0, -15, -13], [-16, -16, 13, -12, -4, -9], [-12, -8, 18, 7, -11, 21], [11, 13, -1, 0, -5, 21], [-21, 10, 6, 7, -7, -9], [8, -22, 5, -2, 14, 13], [-7, 2, -9, 19, 9, 8], [2, -9, -5, -13, -6, -9], [4, 0, 14, -6, 10, 22], [21, -16, 14, -3, -9, -5], [6, 0, -8, -8, -1, -26], [-1, 21, 14, 2, -16, 19], [4, 33, 9, 4, -9, 30], [-2, -3, 5, -7, -6, 1], [-25, 6, -18, -6, 0, -6], [-3, -9, 21, 2, 1, 26], [16, -3, 24, -9, -13, 4], [24, 19, 25, 3, -3, 31], [15, -14, -32, -13, 5, -35], [-10, -20, -16, 2, 27, 16], [30, -9, -7, -3, -3, -5], [-12, -6, 16, 0, -7, 6], [2, 0, -8, -1, -24, -15], [-49, 4, -4, -1, 3, 1], [-30, -8, 0, -7, 6, -1], [-9, -2, 0, -3, -2, -29], [-20, 5, 3, 5, 1, 15], [-4, 19, 5, -11, -5, 32], [13, -13, -9, 7, -1, -17], [-46, 6, -8, 0, 5, 2], [-1, -7, -16, 2, -4, 14], [-2, 4, 4, -10, 6, 8], [-13, 3, -2, -4, -10, 8], [-14, -3, 13, 15, 1, -9], [-20, 27, 19, 17, -20, 15], [4, -15, -12, 9, 10, -35], [-23, -2, -6, -19, 5, 1], [37, 2, 3, 13, 12, 11], [28, 0, -2, -2, -5, -10], [21, -15, -1, -8, 11, 2], [-18, -1, 31, 11, -9, 17], [19, -7, -21, -23, 14, -23], [-58, 4, 2, 0, -21, 8], [1, 9, -4, -18, 9, 0], [-4, 6, -10, 0, 15, 24], [14, -22, -12, 0, -1, 4], [9, -4, 33, 15, -10, 1], [-8, -3, -9, -3, -12, -11], [2, 7, 6, 3, 13, 37], [-14, -6, -10, -3, 8, 19], [13, 2, -20, 17, 30, -3], [-30, 10, 21, 10, -19, 21], [-7, 9, 19, -12, -19, 24], [1, -16, -10, 2, 23, -30], [23, 6, -32, -8, 15, -20], [18, 3, 1, 2, -2, 8], [11, 6, 6, -17, 1, 3], [-46, -1, 9, 3, 1, -7], [-11, 1, 20, 12, -8, 38], [0, 1, 5, 1, 5, 5], [23, -9, -13, -14, 3, -20], [-10, -8, -18, 4, -1, -34], [17, -8, -36, -11, 11, -20], [-14, 8, 22, 0, 7, 36], [0, 11, 41, 7, -4, 28], [-14, 17, -7, -3, -17, -27], [-11, -5, -2, 16, 5, 5], [-10, 7, 9, -2, -28, 8], [-2, 10, 14, 27, 4, 9], [-6, -11, 25, -11, -15, -5], [13, 0, -22, 7, -11, -5], [28, -3, 5, 7, 17, 11], [2, -7, -1, 3, 2, -19], [-36, 5, -11, -9, 16, 13], [25, -3, 17, 9, 15, 1], [28, -4, -22, 0, 4, -21], [-10, -36, -16, 0, 2, -42], [-9, -26, 8, 11, -6, -1], [-5, -10, -28, 2, 15, -30], [0, 6, 29, 8, -7, 20], [-34, -30, -24, -20, -7, -56], [-4, -4, -17, -8, -4, -10], [6, -5, -10, 15, 17, 18], [-1, -10, 6, 3, 19, 3], [30, -8, -26, -6, 17, -36], [14, 8, 16, 18, -16, 0], [-32, 24, -8, -4, -19, 14], [-21, -18, 28, -11, -18, 9], [-4, -24, 14, 7, -1, -11], [-5, -19, -15, -6, 0, -4], [20, 3, -7, -1, -20, 7], [7, -14, 8, 7, 18, 35], [33, 7, 5, 14, 10, 40], [6, 10, -26, -5, 8, 1], [-20, 9, 1, 19, 7, -13], [-51, -9, 11, -5, -15, -5], [-33, 10, 3, -1, 0, 18], [-13, 21, -7, -8, -8, 32], [21, 9, 9, 0, 10, 33], [12, 6, -18, 8, 10, -26], [-30, -22, 20, -6, -6, -4], [18, 2, 16, 1, 11, 27], [-30, 10, 21, 8, 6, 23], [16, 15, -27, 5, 8, -19], [-10, -15, 11, -3, -17, -1], [6, 23, -14, 7, 11, 2], [17, 19, -13, 16, -1, 32], [-5, 7, -37, 2, 1, -17], [14, 0, -18, -16, 16, -24], [-9, -5, -1, 14, -19, -6], [-24, 16, 7, -6, -17, 45], [-9, 28, -8, 1, -8, 1], [-12, 20, 41, 14, -21, 11], [53, 2, 2, 7, 7, 7], [-3, 4, -20, 5, 19, -10], [13, -28, 29, -3, -1, 7], [-11, -1, -29, -5, 3, -33], [29, -20, 16, 9, 3, -17], [-7, 21, -28, 20, 18, -6], [4, -21, 13, 3, 2, -11], [-9, -11, 51, 6, -21, 2], [6, 3, 49, -2, 10, 28], [-34, 6, 14, 12, -13, 26], [-23, 3, 11, 8, -3, 11], [-6, -7, 16, -5, 2, 2], [-10, -5, 21, 3, -9, -3], [-10, -21, -13, -3, -14, -14], [-5, 13, 37, -2, -25, -4], [9, 23, 3, 4, -13, -7], [27, 8, 4, 7, 7, 3], [4, 23, -9, 2, -7, 20], [-2, -5, 15, 3, -8, 29], [18, -17, -35, -5, -3, -25], [35, 0, 10, -7, -24, -8], [16, -19, -4, -1, 17, 3], [25, 9, 11, 4, -8, 40], [27, -20, 22, 11, -1, 5], [-26, -16, 26, -6, -5, 8], [41, -10, -20, -16, -1, 8], [-21, -30, 0, -5, 2, -5], [18, -31, -3, -1, 8, -5], [8, -7, -6, -7, -8, -10], [11, -15, -11, 0, 9, -19], [4, 2, -10, 4, -18, -2], [25, -19, -27, -24, -3, -20], [-16, -8, -19, 2, 7, -45], [16, 19, -5, 5, 0, 13], [2, -13, -43, -9, 5, -49], [0, 0, -10, -20, -10, 0], [-24, 14, -2, -10, -34, -15], [-12, 9, 7, 5, -16, 17], [-18, 19, 21, 17, -15, 25], [-14, 9, 12, -5, -6, 44], [-23, -14, 14, -7, 17, 1], [11, -27, 17, -2, -11, -12], [-18, -12, -8, 10, -11, -4], [11, 13, 18, -4, 9, 25], [-13, 27, 27, 16, -20, 26]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2601_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2601_2_a_bi();".
// 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_2601_2_a_bi(:prec:=6)
    chi := MakeCharacter_2601_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_2601_2_a_bi();".
// 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_2601_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2601_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![8, 24, 6, -19, -6, 3, 1]>,<5,R![136, -228, 60, 51, -18, -3, 1]>,<7,R![-477, -297, 171, 75, -27, -3, 1]>],Snew);
    return Vf;
end function;