
// Make newform 7448.2.a.bm in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7448_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_7448_2_a_bm();".
// 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_7448_2_a_bm();".
// 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, -9, 8, 15, -6, -3, 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], [-5, 3, 13, -5, -3, 1], [-4, 19, 10, -9, -2, 1], [11, -13, -17, 7, 3, -1], [5, -9, -14, 6, 3, -1]];
    Rf_basisdens := [1, 1, 2, 2, 2, 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_7448_a();"
function MakeCharacter_7448_a()
    N := 7448;
    order := 1;
    char_gens := [1863, 3725, 3041, 3137];
    v := [1, 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_7448_a_Hecke(Kf)
    return MakeCharacter_7448_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, 0, 0], [-1, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [1, -1, 1, 0, 0, 1], [-2, 1, -2, 0, 1, -1], [0, 0, 0, 1, -1, 1], [1, 0, 0, 0, 0, 0], [0, 0, 0, -2, -1, 0], [0, -3, -2, -2, 0, 0], [-2, 0, -1, -1, 0, -2], [-2, 1, 0, 1, 0, 1], [4, 2, 0, 0, 0, -3], [0, 0, 1, -1, 0, 1], [-5, 0, -2, -1, 1, 0], [-3, -2, -3, 0, 1, 0], [-4, 1, 1, 1, 1, 0], [-3, -4, -2, -3, -1, 0], [2, 1, 3, 1, 2, -1], [3, -1, 0, -1, -1, -1], [2, 2, 3, 1, 1, -2], [-6, 2, -1, -1, 0, 3], [0, 2, 3, 1, 1, 0], [3, -4, 1, -1, -3, -1], [3, -3, 1, -1, 0, 2], [-4, 2, 3, 0, 2, 1], [-2, 3, 4, 0, 1, -1], [-4, 1, -3, -4, 0, 0], [1, -1, 0, 3, 1, 1], [-4, -2, -2, -3, 4, -1], [-3, -2, 1, 1, -1, 5], [-4, 4, 2, 5, 1, -1], [6, 0, 3, 3, -2, 3], [2, -6, 1, 0, 0, 3], [0, 0, -2, 0, 1, -2], [4, -11, -1, -5, -2, 1], [-5, -4, -3, -4, -2, 0], [-1, 5, 3, 2, 1, -1], [2, -5, -2, -2, -5, 1], [-10, 2, 2, 4, 0, 2], [2, -3, -5, -3, 2, -3], [4, 0, 1, -1, 6, -2], [2, 1, 4, -3, -4, 0], [4, -3, 3, 3, -4, 1], [-2, 1, 0, 3, 4, -4], [-2, -4, 6, -1, -4, 2], [-4, 8, 0, 3, -2, -1], [-2, 11, 4, 4, 1, -3], [0, -2, 3, 3, -4, 2], [-14, 0, -3, -1, 0, -1], [-3, 1, 3, 0, 2, 5], [0, 6, 4, 6, 1, -2], [-8, 0, -7, 1, 6, -3], [0, 0, -4, -3, 1, -5], [-6, -1, 0, 0, 1, -2], [2, -3, -2, -3, 4, -2], [8, -3, 5, 3, -3, 1], [0, -2, 5, 0, -3, 8], [-2, 0, 2, 0, -4, 4], [6, -5, -3, -2, -4, -2], [-16, 2, 1, 3, -1, 2], [6, 1, -8, -2, 5, -7], [-10, 4, 2, -4, 0, -1], [-7, 0, 3, 2, -4, 2], [-2, 4, 2, 2, 0, -2], [-7, 2, -5, -1, 3, 1], [2, -6, -4, 3, -2, -1], [4, 3, -3, 5, -2, -1], [2, 7, 2, 1, -4, 0], [4, -8, -7, -3, -3, 0], [0, 0, -6, 3, 3, -1], [-2, 9, 8, 3, 5, -2], [-13, -2, 0, -5, -3, 0], [2, 6, -3, 4, -2, -6], [-18, 5, -3, -4, 0, 0], [-10, 5, 2, -2, -1, 3], [6, 7, 0, -5, -1, -2], [-13, -2, -1, 5, 1, -1], [-8, 1, -11, -7, 4, -7], [-1, 1, -4, 0, -4, 0], [4, -6, 3, 2, 2, -1], [-16, 1, 3, 2, 0, 6], [-5, -2, -7, 1, -3, -1], [-2, 5, 1, 7, 7, -4], [2, -2, -2, 2, 4, -8], [10, -5, 7, 4, 0, 7], [-4, 10, 6, 1, 2, -1], [-1, -1, -1, -2, -7, 9], [9, 8, -6, 2, 2, -1], [4, 4, 6, 0, -3, -2], [-6, 4, 1, 5, -1, -4], [-8, -6, -5, -4, -3, 2], [-8, 1, 2, -3, 4, 3], [0, 12, 6, 6, -2, 0], [-2, -1, -1, 0, 6, -5], [-12, -2, 3, 1, 2, 5], [-20, 2, -2, -6, 2, 2], [4, -5, -10, -8, -1, 1], [14, -13, 4, -4, -1, 3], [-4, 2, 6, -2, 8, 0], [-14, 0, -2, -3, -2, 1], [-12, -1, -8, -9, -3, 0], [-15, -5, -5, -1, 4, 4], [16, -1, 5, -6, -6, 0], [5, 4, 9, 5, -4, 3], [8, 10, 9, 7, 3, 0], [-4, 0, 2, -4, 2, 0], [4, 6, 7, -4, 0, -1], [11, -14, 1, -4, 1, 2], [-8, 7, 5, 5, -3, 7], [4, 7, -4, 6, 1, -3], [-12, 7, -2, 5, 6, -2], [-2, 7, -1, -2, 1, 5], [-18, 4, -1, 11, 5, -4], [4, 2, -4, -2, -1, 6], [8, 1, 1, 3, 6, -3], [-2, 2, 13, -2, -6, 3], [18, 4, 8, 5, -6, -2], [0, 6, 2, -6, 1, -4], [6, -3, 15, 1, -8, 9], [-6, 0, 0, 2, -4, 4], [-2, 6, 6, -1, 2, 3], [-12, 9, 3, -7, -7, -1], [-2, 3, 11, -4, -4, 2], [14, -6, 6, 4, 0, 6], [-8, 4, -2, 2, 0, 4], [2, 0, 4, -4, -5, 2], [-12, 18, 8, 10, 2, 0], [7, -10, 6, 3, -3, -6], [-2, 8, 13, 15, -2, 3], [-2, 0, -15, 1, 3, -5], [-8, -3, -10, -1, 0, -1], [16, 4, 19, 6, -6, 4], [2, -2, -6, -6, 0, -6], [20, -10, 3, -2, -2, 5], [-2, -8, 2, -8, -12, 2], [4, 0, -6, 4, -2, -6], [-2, -4, -9, -5, -4, -7], [16, -6, 7, -1, -6, 8], [-4, 0, 5, -1, 1, -3], [-9, -10, -19, -5, -1, 1], [-18, 4, 10, -4, -2, 8], [0, -2, 4, 0, 4, -6], [0, 9, -7, 6, 2, 0], [-6, -2, -4, -12, 6, -2], [-8, 2, 12, 2, -8, 14], [0, 4, 2, 2, 9, -1], [14, 4, -3, 3, -4, 2], [10, 4, 2, 7, -1, 5], [0, -2, 7, 0, -6, 8], [-5, 6, 1, -3, 1, -1], [20, -4, -1, 1, 5, -4], [7, 12, 7, 5, 6, -1], [-2, 7, -2, 8, 5, -5], [-2, -3, -3, -8, -6, 8], [6, -2, 11, 5, -2, -5], [-2, -2, -8, 4, 5, -2], [30, -10, 7, 7, 3, -2], [-12, -4, 2, -1, 1, -5], [-6, 8, 0, 2, 0, 0], [-11, 7, 3, 2, 1, 7], [-2, -17, 3, -11, -8, -1], [20, -3, -8, -3, 9, -10], [11, -13, -5, 3, 8, -2], [20, -8, 8, -2, -8, 2], [12, -11, -8, -10, -5, -1], [-23, -12, -13, -3, 1, 1], [-8, -10, -4, -8, 7, -3], [-18, 1, -5, 0, -12, 2], [2, -24, -6, -10, 4, 4], [4, 6, 2, 5, 10, -1], [-4, 14, 3, 5, -2, -2], [-6, 1, -2, -7, 5, -4], [-8, 2, -13, 9, 7, -5], [-4, -6, 4, -4, -12, 8], [16, 6, 8, 16, -2, 6], [-6, -13, -19, -12, 0, 5], [-8, 6, 2, -2, 13, 0], [-32, -3, -12, -8, 5, -3], [2, -2, 3, 4, 3, 2], [10, 5, 8, 5, -7, -2], [11, 3, 5, 5, 0, -10], [17, -2, 3, 6, 1, 0], [1, 13, 4, -4, 4, 2], [-10, 0, -4, 12, 7, 2], [-5, 10, 1, 5, -3, 1], [6, -3, 2, 5, -4, -7], [4, 1, 2, 8, 3, -10], [-8, -4, -10, -17, 2, -3], [13, -4, 4, 16, 2, 3], [14, -10, -4, -2, 6, 4], [24, -1, 5, 8, 1, -1], [-8, -2, -2, 0, 10, 2], [4, -3, 0, -6, -5, 13], [13, 12, -3, 3, 3, -5], [26, 5, 2, 2, 1, -4], [13, -10, -9, 1, 12, -5], [4, -2, -8, 10, -5, -6], [16, -20, -3, 2, 8, -3], [8, -6, 9, 9, -2, 3], [-14, -2, -6, -4, 12, -12], [-20, 3, -15, -5, 9, -3], [-10, -3, -6, 2, -7, -7], [17, 4, 1, 7, -11, 9], [-6, -4, 4, 7, 5, -3], [-3, 6, -1, 6, 8, -4], [29, 6, -1, -3, -1, -7], [6, -24, -6, -14, -2, 2], [10, 24, 14, 12, 6, -2], [-7, 5, 2, 0, 4, -8], [-6, 24, 7, 5, 0, -1], [-7, 8, 2, -2, 10, 7], [-16, 2, 6, -4, 0, 8], [20, -7, 5, -3, 3, 0], [20, -8, 8, 0, 2, 14], [18, 5, -7, 9, 3, 4], [26, -16, -2, -4, -5, -4], [20, -6, 1, -15, -4, 7], [-6, 6, -1, 3, 0, -5], [6, -10, -8, -6, -1, -12], [5, 25, 10, 11, 5, -3], [-22, 15, 13, 13, -9, 7], [4, -2, 0, 10, 4, 0], [-4, -12, -4, -12, -3, 0], [-29, -6, -6, -8, 5, -3], [-8, 7, -4, -7, 4, -4], [10, 17, -11, 1, 11, -12], [18, 0, -2, 5, 8, -11], [-18, 22, -2, -4, -6, 0], [16, -11, 5, 2, 3, 9], [22, -13, -4, 9, 10, -2], [13, -12, 11, -11, -9, 13], [-19, 3, -6, 11, 5, -1], [-8, -13, -7, -12, -14, -4], [-12, -20, -9, -5, 5, 4], [-2, 22, -4, 12, 1, -5], [36, -4, -8, -5, -1, -5], [0, 12, 6, -4, 0, 0], [18, -2, -9, -5, 5, -7], [-6, 2, 3, 11, 6, 6], [-12, -8, 12, 2, -6, 0], [16, 10, 3, 16, 4, 7], [-2, -4, 9, -12, 2, 9], [-13, -10, 1, -1, -5, 5], [-18, 22, 5, 9, 4, -6], [-1, -14, -3, 6, 6, 4], [4, -4, -1, 5, -10, 1], [6, -1, 1, -13, -7, 9], [-18, 2, -2, -6, 8, -8], [0, 26, 6, 18, -1, -2], [4, 2, -4, 0, -7, 6], [36, -24, -10, -5, 7, -5], [-22, 9, 16, 3, -3, 2], [-42, -4, -1, 1, 4, -2], [-26, 12, -2, 12, 16, -2], [26, -4, -5, -1, -5, -11], [12, 4, 8, 12, -9, 0], [11, -2, -1, 11, 3, -9], [-2, 25, 7, 15, -5, -3], [-12, 4, -19, -8, 6, 5], [10, -6, -5, -7, 0, 11], [24, 0, 6, 10, 2, -2], [-3, 0, -10, -3, 3, -12], [2, -18, -6, -2, -4, -4], [20, -4, -10, -6, 6, -4], [-13, -6, -5, -11, 7, 5], [-16, 17, -5, -4, -6, 4], [32, 16, 15, 6, -4, -1], [-2, 2, 10, 8, -8, 0], [18, 11, -6, -5, 6, -8], [-16, 20, 7, -7, 0, 2], [-4, 12, 16, 18, 0, 8], [22, -12, -3, -11, -7, 4], [14, -1, 2, -1, 2, 4], [-26, -5, 10, 4, -5, 10], [-4, 6, 5, -3, -6, 9], [-32, 15, -2, 1, 5, 8], [-12, -16, 3, -1, -10, 9], [24, 2, 2, -8, -6, -8], [0, 0, 5, -12, 0, 9], [12, -5, -15, 3, 1, -9], [0, 4, 6, 8, -11, 16], [-25, -4, 5, -5, -7, 5], [12, -2, 6, 0, -12, 2], [44, -8, 6, -10, -10, 6], [-30, 6, -16, 2, 4, -12], [-21, -12, -5, -9, 12, -7], [-18, -9, -14, -13, 12, 3], [-13, 10, -4, 3, -1, -12], [0, 3, 2, -2, -15, 7], [-10, -2, 0, 16, 3, 4], [38, -6, -13, -7, 3, -4], [-2, -11, -13, -3, -8, 3], [2, 26, -1, -1, 0, -7], [20, 0, -2, -11, -6, -3], [8, -6, 4, 2, -9, -9], [22, 7, -14, 4, 5, -15], [2, -6, -13, -13, -4, 5], [-33, 24, -2, 8, -1, -5], [-18, 8, 23, 5, -5, 12], [12, -14, -21, -2, 8, -11], [8, -7, 1, -2, -4, 12], [44, 0, 4, 4, -4, -10], [-16, 8, -6, -1, -2, -3], [12, -23, 3, -4, -13, -1], [-24, -9, 9, 4, -6, -2], [-40, -3, -19, -12, -10, 2], [-20, 13, -11, 11, 5, -21], [22, -2, -16, -4, 6, -14], [8, -3, -18, -9, 11, -10], [2, 0, 8, -11, 6, -2], [16, 15, -8, 5, -3, -2], [-4, -1, -5, 6, -6, 0], [14, 0, -10, 9, -8, -7], [0, 2, -2, -8, -8, 0], [-8, -24, -11, -5, -2, 15], [-24, -8, -31, -11, 3, -10], [-24, 9, 6, 21, 8, -7], [-56, 8, -7, -3, -5, -2], [3, -8, 11, -3, 7, 9], [-13, -7, -7, -7, -2, -4], [-18, 7, -1, -14, -9, -1], [16, 0, 3, -5, 4, -6], [-13, 16, 9, 0, -5, -4], [-22, 16, 0, -2, 2, -2], [42, 3, 4, -6, -13, 6], [4, -12, -20, -11, -8, 3], [6, 24, 4, 10, 2, -9], [11, 16, 3, 0, 11, -6], [38, -21, -16, -21, 2, -2], [-22, -4, -10, -6, 0, -6], [-16, 6, -19, -9, 6, 1], [-5, 26, 5, 18, 4, -4], [14, 8, -19, 4, 10, -5], [50, -14, 1, -19, -7, 11], [-6, -2, -2, 9, 19, -15], [1, -7, -20, -21, 1, -9], [15, 8, -8, 6, -3, 7], [8, -20, 1, -2, 16, -9], [1, 0, -8, 8, 18, -1], [-23, -1, 10, 4, 3, 0], [9, 7, 15, 6, 0, 5], [8, -17, -10, 2, 5, -7], [22, 23, 15, -2, -8, -4], [34, -13, 12, 5, 8, 13], [-18, -11, -15, 7, 8, -3], [-7, -12, 21, 5, -11, 13], [-36, 4, -12, -10, 4, 4], [10, 23, -10, 6, 11, -7], [-36, 12, 10, 0, -1, 13], [-20, 7, -31, 3, 15, -12], [-14, 9, 19, -4, -16, 12], [-20, 13, 2, -3, -16, 0], [-18, -12, -7, -3, -4, -6], [-12, -15, -16, -10, 4, 10], [-15, -20, -26, -12, -9, 1], [-51, 6, -25, -13, 3, -9], [-4, 2, 13, -12, -2, 13], [-17, -2, 1, 11, -3, 11], [33, 3, -4, -16, 3, -6], [18, -20, 4, -18, -5, 4], [18, 20, -9, 11, 13, -6], [2, 13, 5, 12, 2, 2], [-8, 6, 8, 12, 2, -6], [0, -18, -9, 8, 4, 3], [-12, 2, -20, -14, 0, 0], [16, -22, -2, -13, 0, -2], [-10, 21, -2, 6, 16, 0], [-2, -4, -26, 10, 4, -10], [-4, 3, -17, 10, 13, -1], [42, -12, 17, 1, -10, 1], [12, 0, 14, 14, 4, 12], [-38, -10, 0, 4, -14, 10], [-38, 30, 1, 17, 7, -14], [-6, -11, -14, -9, 2, 2], [33, -2, 9, 13, 11, 13], [-8, 22, 6, 24, 11, -4], [4, -16, 0, -4, -2, 6], [16, 2, -11, 0, 12, -23], [-42, -17, -19, -5, -3, 6], [4, -11, 12, 4, -13, -3], [-19, -2, 9, 5, 11, -13], [22, 14, 14, 18, -19, 6], [-8, -20, -15, -7, -7, 17], [-13, 0, -1, -7, -1, 3], [2, 4, -4, -14, 2, 14], [-18, 7, 24, 5, -7, 2], [-12, 0, 20, 10, -7, 1], [-26, 10, -8, 24, 23, -10], [6, -11, -13, 3, -3, 1], [-4, 21, -1, 15, -8, 3], [-24, 23, 3, 1, -13, 11], [18, -20, -1, -11, 1, -5], [-20, 6, 14, -1, -13, 13], [8, 15, 22, 25, -18, 13], [-4, 1, 15, 2, -20, 18], [-34, 5, -10, 6, 21, -15], [-3, 14, 15, -5, 5, 5], [16, 0, 0, 18, 4, 12], [-24, 11, 4, -3, -6, 10], [-22, -31, -22, -12, 3, 13], [-41, -8, -1, 15, 2, 1], [29, -1, 22, -1, -19, 15], [21, -9, 23, -9, -18, 12], [-40, 6, -13, -3, 6, 0], [-2, 4, 10, 8, -1, -1], [60, 3, 2, 7, -5, -2], [40, 10, 8, 14, 6, 0], [-4, 2, 18, -2, -8, 6], [-8, -26, 8, -7, -23, 9], [-25, 9, 3, -7, 8, 10], [-4, 4, -6, -4, -11, -16], [-6, 3, -3, 9, 4, 17], [27, 9, 21, 6, -7, 5], [6, -16, 7, 21, 2, 3], [-5, 6, -19, -4, 0, -10], [-6, -23, -9, 4, 6, 0], [20, 5, -1, -9, -7, -1], [0, -2, -13, -12, -8, -3], [-8, -9, 21, 3, -3, -5], [1, 13, 8, -3, 1, -27], [-22, -11, 0, 2, 5, -3], [-34, -4, 8, 12, -3, 14], [22, -11, 1, -17, -3, 7], [-21, 2, 20, 8, -11, 11], [-16, -27, 12, -4, -15, 17], [-32, 1, -27, 3, 4, -15], [-16, -8, -13, -17, -11, -5], [-30, -10, -19, -18, 6, -9], [4, -26, -12, -28, -2, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7448_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7448_2_a_bm();".
// 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_7448_2_a_bm(:prec:=6)
    chi := MakeCharacter_7448_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_7448_2_a_bm();".
// 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_7448_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7448_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![7, 19, 2, -19, -6, 3, 1]>,<5,R![53, 123, 59, -24, -17, 1, 1]>,<11,R![575, -539, 4, 121, -28, -3, 1]>,<13,R![-3920, 1544, 639, -194, -40, 6, 1]>,<17,R![148, -322, 91, 138, -55, -2, 1]>],Snew);
    return Vf;
end function;