
// Make newform 8048.2.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8048_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_8048_2_a_p();".
// 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_8048_2_a_p();".
// 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, 1, 17, 14, -34, -27, 27, 14, -9, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 12, 65, -34, -113, 46, 57, -22, -8, 3], [-8, 20, 104, -57, -181, 78, 92, -37, -13, 5], [13, -44, -139, 108, 221, -113, -108, 47, 15, -6], [-21, 60, 230, -165, -368, 184, 180, -78, -25, 10], [22, -63, -244, 165, 402, -191, -200, 84, 28, -11], [38, -102, -394, 278, 645, -321, -321, 139, 45, -18], [45, -127, -475, 331, 770, -375, -380, 162, 53, -21], [42, -130, -473, 346, 785, -394, -392, 170, 55, -22]];
    Rf_basisdens := [1, 1, 1, 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_8048_a();"
function MakeCharacter_8048_a()
    N := 8048;
    order := 1;
    char_gens := [1007, 6037, 2017];
    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_8048_a_Hecke(Kf)
    return MakeCharacter_8048_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, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, 1, 1, 0, 0, 0, -1], [1, 1, -1, 1, -1, 0, 0, 1, -1, -1], [-2, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, -1, 1, 2, -2, 0, 1, 0, -1, 2], [0, 0, 2, 0, -1, -1, 0, 0, 1, 1], [1, -2, 2, 0, -1, -2, -1, -3, 1, 3], [-2, 0, 1, -1, 0, 0, -2, -2, 2, 1], [2, 1, 0, -1, -1, 0, 0, 1, 1, -1], [-6, 4, -3, -1, 3, 1, -1, 2, 1, -3], [-2, 1, 0, -2, 2, 0, 0, 0, 3, -1], [2, 1, 0, 2, 1, 1, 2, 0, -1, 1], [-2, 1, -3, 2, 1, 0, 0, 2, 1, 1], [0, -1, 2, 1, -3, -2, 2, 2, -3, 1], [-1, 0, 1, -1, -1, 2, 2, -3, 1, 1], [3, -3, 4, -1, -1, -2, 1, -6, 1, 1], [-4, -1, -2, -3, 3, 1, -5, 0, 2, 1], [0, 2, 3, -4, 3, 2, 3, 2, -3, -6], [-4, 3, -3, -3, 1, 1, 1, 3, 0, -4], [4, -2, 6, 0, -2, -3, 2, -2, -6, 4], [-5, 0, -2, 3, 0, -2, -5, 1, 4, 3], [-4, 4, -1, -1, 3, 0, -5, 3, 0, -3], [-7, -2, 1, 0, -3, -2, -5, -2, 2, 5], [5, -1, 3, -1, 0, 1, 3, -2, -3, 0], [2, -4, 3, 2, -6, 0, 2, -1, -3, 6], [0, 1, 4, -4, 2, 1, 2, 1, -2, -5], [-1, -1, 1, 0, -2, 0, -2, -2, -1, 3], [1, 3, -1, 2, 2, -1, 1, -2, 0, -1], [1, -6, -4, 2, 0, -3, -2, 3, 3, 0], [-5, -1, 2, -5, 5, 6, 4, 0, -3, -5], [-1, 1, 2, 2, -4, 3, 5, 2, -5, 0], [5, -5, -3, 2, -4, -2, 1, -4, 1, 4], [-3, 3, 1, -1, 6, 3, -1, 0, -5, -3], [1, 1, -1, -3, 1, 0, -3, 2, 4, -6], [2, -9, 7, 2, -7, -6, 3, -4, -1, 12], [1, 7, -6, -3, 4, 4, 4, 5, -4, -6], [2, -1, -2, 8, -4, -2, -3, 6, -2, 0], [-8, 4, -8, 1, 4, 2, -3, 9, 1, -4], [-9, 2, -5, 0, 8, 0, -6, 1, 1, 0], [-3, 2, -2, -5, 1, 2, -1, 3, -3, -1], [1, -1, -1, 3, -4, 0, -1, 1, 6, 0], [-13, 6, -6, -1, 8, 2, -1, -2, 2, -4], [0, 3, 0, -3, 0, -1, -1, 5, 1, -4], [-6, 6, -2, -1, 4, 1, -2, 7, -3, -8], [3, -4, 9, -2, -4, -3, 0, -7, 4, 5], [-4, 1, -1, -2, 9, -2, -4, -5, 3, 3], [1, -1, 8, -5, 4, -2, 3, -5, -1, -1], [-2, 2, 0, -5, 1, 9, -1, 3, 1, -10], [-2, -5, -4, -7, 2, 1, 0, -1, 1, -6], [1, -3, 4, -9, 1, -1, 4, -6, 6, 1], [-2, 7, -3, 0, 6, 4, -3, 6, -10, -7], [-7, 11, -8, -7, 10, 6, 0, 5, 4, -13], [0, -8, 2, 3, -7, -7, -2, -4, -4, 8], [-5, -1, 0, -3, 0, 1, 0, 1, -3, 8], [-1, 2, 8, -1, 1, 5, 2, -4, 5, -1], [4, -7, 1, 3, -4, 0, 3, -2, -7, 1], [-6, -2, 4, -10, 6, 0, 1, -7, 3, 0], [5, -6, 7, -1, -11, -2, 3, -1, -1, 5], [0, 11, -9, -2, 2, 0, -4, 3, 1, -8], [-5, -2, -1, -1, 6, -1, -5, 1, 2, 2], [-1, 4, 4, -5, 5, 1, 2, -2, -6, -7], [4, 5, 1, 5, -4, 2, 4, -6, -7, -1], [-6, -1, 3, -2, 0, 6, 8, -1, 4, -4], [5, -7, 3, 4, -5, 0, 6, 2, 3, 3], [8, 0, -2, -3, -4, 0, 7, -5, 2, -1], [-18, -1, -3, 4, 4, -1, -3, 3, -3, 9], [-7, -1, 7, 1, 0, -2, -10, -3, 6, 6], [7, 0, -1, 8, -5, 2, 7, 0, -2, 1], [1, -8, 2, -2, 1, -3, 0, -8, -1, 5], [6, -8, -2, 4, -11, -2, 4, 2, 4, 4], [-1, -3, -2, -4, 8, 5, 3, -5, 2, -6], [1, 3, -6, 14, -9, -2, 4, 5, -3, 1], [8, -5, 10, 4, -9, 0, 5, 1, 1, 3], [1, -8, -1, 4, -5, -4, -2, -2, -4, 10], [9, -2, 4, -5, 6, -3, 10, -6, -3, -5], [0, 0, 8, -2, 5, -1, 3, 2, -2, 1], [8, 3, 2, -1, 3, 7, 10, -4, 4, -9], [-4, 4, -13, -3, 6, -3, -7, 1, 6, -4], [-9, 5, -3, 8, -5, -4, 5, 3, -8, 7], [14, -13, 1, 3, -10, -6, 3, -6, 5, 13], [2, -5, 9, 10, -4, -4, 0, 2, -12, 11], [-3, 7, -9, 10, -4, -3, 1, 6, -1, 3], [-9, -5, 6, -11, 6, 3, 3, 2, 3, 1], [5, 1, -2, -4, -2, 0, 3, -3, 10, 0], [-16, 2, -9, 0, 9, 1, -15, 8, 10, 0], [-2, -10, 3, 0, -7, -1, 0, 1, -1, -1], [16, -4, 1, -6, -4, 3, 5, -4, 0, -6], [12, -1, 0, 1, -8, 0, 6, -7, -1, 1], [0, -1, 7, -2, -10, -8, 0, -3, -2, 9], [-8, 5, -10, -13, 9, 6, -3, -4, 10, -9], [8, -10, 11, 8, -13, -10, 4, -5, 5, 10], [-3, -1, 2, -1, 3, 1, -4, -6, 3, 5], [-4, 0, -3, 7, -9, -1, -12, 3, 1, 5], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -10, 5, 0, 0, -4, -2, -7, 3, -3], [1, -10, 4, -3, 2, 3, -2, -8, 11, 1], [4, -4, 1, 7, -14, 3, 8, -2, -9, 6], [9, -13, 0, 15, -8, -6, -1, -1, -3, 13], [-10, 6, -7, -7, 8, 11, -5, 11, 0, -17], [1, 5, -11, 7, -3, 5, 4, 1, -1, -6], [17, -15, 20, 1, -17, -1, 11, -10, -7, 10], [-9, 5, 4, -2, -4, -3, -11, 3, 1, 3], [0, 4, -14, 1, 0, -3, -7, 1, 2, -6], [0, -10, 14, 1, -13, -10, 9, -9, 5, 15], [2, -3, -2, 1, 4, -12, -8, -6, 2, 14], [8, 2, 2, 10, -6, -6, 3, -7, -4, 6], [-1, -1, -9, -2, 7, 7, 1, 3, 1, -4], [-2, -4, 10, 3, -6, 3, 13, -4, -4, 0], [-7, -4, 3, -1, 3, 1, -14, -12, 11, 4], [-2, 1, 6, -5, 0, -3, 9, 13, -8, -8], [-5, -9, -7, -10, 7, -2, -6, -5, 6, -5], [4, -6, -5, 12, -5, 0, 8, -1, -7, 7], [-5, 17, -13, 8, 5, 9, -1, 17, -13, -19], [-20, 15, -23, -4, 11, 4, -15, 5, 17, -4], [9, 1, 10, 1, 8, 1, 7, 1, -7, -2], [16, -13, 21, 4, -11, -3, 12, -8, 1, 9], [16, 5, -3, 0, -4, 7, 13, -7, -3, -9], [3, -3, -4, 7, -2, -8, 3, 1, -8, 5], [-3, 13, -7, 7, -2, -1, -5, -2, -1, 2], [-9, 4, 19, -6, 0, 2, 1, -5, -11, -6], [-14, 17, -9, -4, 16, 13, -1, 7, 1, -18], [-2, -6, 16, -4, 2, -7, 6, -15, 10, 6], [-9, 6, 0, -6, 7, 0, -4, -6, 4, -10], [-2, -17, 6, 1, -6, -5, -3, 1, -4, 25], [6, -2, 12, 4, -13, -3, 1, -6, -8, 2], [-7, 8, -16, -9, 10, 5, -14, 11, 3, -14], [-4, 10, -10, -2, 8, -1, -7, -1, 4, -8], [-2, -5, 7, -2, 0, 1, 6, -12, 8, 16], [2, -8, -5, 8, 1, 2, -5, 3, 2, -4], [-3, -4, -6, 3, 5, 2, -11, 4, 7, -7], [17, -13, 15, 2, -11, -2, 20, -2, -3, 8], [-7, -5, 1, 8, 5, 6, -2, 3, -4, 2], [19, -15, -4, 10, -4, -3, -4, 8, -1, -2], [2, -4, 10, -8, 2, 2, 14, -11, -1, -5], [2, 0, 5, 12, -16, -11, -3, 8, 4, 16], [-13, 10, -20, -2, 6, -2, -13, 9, 10, -10], [-2, 12, -23, 5, 15, 7, -5, 8, 3, -12], [4, 5, -5, 1, -3, -5, -6, -11, 4, 4], [-5, -4, 0, 3, -3, -4, 8, 14, -13, -1], [4, 9, -2, 10, -6, -4, 4, 9, -1, -5], [-1, 6, 2, -17, 10, 10, 8, -6, 4, -20], [23, -8, 6, 8, -9, -13, 9, -6, 3, 12], [23, -2, -1, 1, 2, 0, 0, 3, -8, -12], [4, 7, 12, -10, 4, 2, -1, 4, -5, -17], [-18, 9, -7, -10, 6, 10, -5, -8, 11, -7], [7, -7, 8, -11, 6, -4, -3, -12, 1, -10], [6, 3, 12, 3, 0, -3, 9, -5, 4, -1], [2, 23, -11, 13, 8, 4, -3, 8, 2, -8], [-1, -7, -5, -6, 4, -9, -12, 0, 13, 9], [-8, 11, -9, -1, 1, 0, 6, 9, -3, -6], [11, 3, 3, 3, -4, 11, 16, -9, -7, -5], [2, 9, 2, 3, 4, 5, 4, 14, -9, -2], [-9, -6, 6, -5, 1, -5, -8, -9, 5, 24], [3, 6, -3, -1, -2, 8, -8, 9, -7, -16], [0, 14, -2, 0, -2, 5, 3, -1, 4, 5], [-30, 17, -20, -3, 7, 15, -3, 15, 3, -11], [4, -26, 0, 8, -6, -10, 0, -4, 0, 20], [-16, 18, -18, 2, 10, 8, -2, 11, -2, -10], [8, -19, 8, 9, -19, -8, -3, 5, 6, 8], [-4, -4, 2, -3, 2, -4, 8, -1, -5, 8], [-13, 0, -1, -9, 12, 8, -4, 9, 2, -5], [3, 9, -1, 7, -10, 9, -3, 10, -10, -1], [29, -17, 5, 13, -19, -4, 14, 0, -4, 14], [-11, 8, -3, -6, 5, -2, -6, 13, -7, -12], [4, 7, 0, 0, -2, 4, -11, 5, -3, -9], [-4, -11, 4, -9, 9, -16, -18, -8, 15, 16], [-8, 7, 0, 5, -1, -3, -6, 10, 5, -1], [-3, 2, 11, 0, -4, -1, -2, -5, 1, 16], [-7, 8, -14, -12, 16, -2, -7, 1, 5, -11], [21, 7, 1, -5, -6, -7, 3, 1, -1, -17], [0, 18, -15, 7, 2, 14, 10, 5, 0, -21], [0, -14, -6, -3, 5, 4, 6, 4, -5, -1], [-13, -8, 11, 0, 4, 2, -10, -4, -4, 0], [-12, 4, 8, -24, 13, 9, 2, -3, 0, -8], [11, -9, 17, -5, -1, -5, 2, -13, -11, -3], [-2, 0, -4, -3, -1, 11, -5, -10, -1, 3], [22, -6, 6, 7, -12, -12, 4, -8, 4, -4], [12, 2, -11, -4, 4, 5, -6, 5, 10, -14], [10, -13, -7, 18, -19, -7, -1, 4, 2, 23], [9, 14, -7, 0, 1, 5, -2, 7, 0, -12], [2, -5, 16, -2, 7, -5, 11, -8, 4, 10], [11, 2, 6, -4, -1, 8, -3, 7, -6, -9], [-8, 2, -4, -1, -4, -2, 11, -14, 5, 0], [18, -12, 5, 4, -11, -4, 2, -6, -4, 14], [0, 15, -23, -11, 4, 14, -7, -1, 10, -17], [-4, 6, -14, 7, 9, -2, -18, -4, 11, 9], [-4, 2, 11, 8, -8, -1, -8, -8, 2, -1], [-3, 4, -4, 5, -15, 8, -3, 10, -10, 5], [-18, -2, 0, -6, 14, 1, -16, 2, 8, -7], [-7, 5, -6, -18, 9, 8, -5, -7, 6, -18], [14, 4, -6, 10, -2, -10, 14, 4, 2, 4], [47, -18, 10, 3, -13, -1, 23, -7, -15, 2], [-18, -14, 4, -1, 2, 0, 8, -3, -3, 7], [-9, 10, -21, 3, 12, 0, -10, 4, 8, -10], [-2, -8, 11, 22, -20, -8, 1, -2, -9, 14], [7, 7, -12, 4, 7, 3, 1, 4, -17, -10], [-1, 7, -13, 7, 5, -3, 7, 3, -5, 9], [-1, 7, 6, 10, -3, 2, 16, -8, -5, -1], [27, 10, -5, 10, -10, -2, 8, 4, -8, -8], [-5, -2, -5, 4, -1, -8, -3, -11, 14, 20], [-4, -19, 6, -4, 2, -2, -5, -10, 5, 0], [-16, 17, -9, -6, 11, -3, -12, -7, 9, 7], [16, -1, -5, -6, -6, 12, 2, 1, -4, -13], [-2, -15, 5, -7, -6, -11, -14, 4, -1, 13], [-16, 4, -23, 12, -7, -4, -2, 10, 12, -12], [24, -5, 15, 0, -19, 10, 22, -6, -9, -13], [-5, -7, 1, -13, 8, 2, 3, -3, 10, -10], [4, -12, 25, 4, -11, -13, -19, -7, 8, 22], [-36, 7, -6, 1, 5, 0, -13, -1, 12, -1], [10, 10, -7, 5, 0, 0, 7, 1, -9, -10], [8, 12, -7, 17, -7, -3, -7, 3, 8, 10], [3, -1, 12, 10, 0, -5, 8, -9, 3, 14], [-19, 18, -35, -14, 25, 7, -27, 11, 16, -33], [13, -5, 8, 12, -8, 1, 22, 0, -31, -4], [-6, 22, -1, 3, -2, -2, -1, 6, -2, 0], [-16, 5, -2, -9, 7, -1, -2, -9, 12, -9], [18, 5, -18, -1, 0, 6, 4, 11, -15, -6], [2, 23, -1, 0, 5, 2, 1, -9, 6, 2], [-8, -7, -10, 8, -12, 3, -7, 10, 10, -3], [-4, 21, 4, -13, 11, 4, 11, -7, -13, -6], [8, 12, 4, 5, 3, -2, -5, 19, -3, 1], [-7, -4, 5, 8, -1, 0, -1, -2, -3, 8], [7, 8, 6, 13, -12, -7, 14, 4, -8, -7], [18, 1, -18, 1, -2, 5, 11, 4, 2, -23], [-5, 0, 19, -3, 2, 2, -9, 3, 5, -10], [2, -4, -17, 11, -3, -11, 2, 14, -5, -5], [24, -25, 18, -13, -7, -3, 8, -8, -4, 3], [-4, 14, 0, 0, 4, -7, -7, 2, 12, -1], [-4, 23, -16, 5, 4, 21, 6, 12, -11, -28], [-25, 2, -11, -5, 15, 5, -19, -4, 12, 17], [3, -3, -21, 9, -6, -6, -7, 4, -3, 8], [13, 10, 8, -2, -3, -1, 9, -6, -14, 1], [12, -14, 2, 13, -5, 6, 6, 4, -2, 10], [14, -10, 19, 15, -17, -10, 17, 6, -15, 14], [13, 15, -21, 4, 14, 6, -7, 10, -17, -14], [4, 0, -9, 8, -4, -8, -16, -3, 13, 19], [-12, 7, -28, 12, 7, -4, -6, 25, 1, 2], [5, 5, 12, -1, -3, -11, -10, -2, -8, 19], [23, -6, 15, 1, -12, -2, 25, 6, -15, 7], [7, 7, -18, 17, 8, 12, 9, 3, -3, 1], [6, 15, 8, -4, -4, 10, 8, 12, -12, -7], [23, -6, 5, 9, -21, -9, 16, -6, 5, 23], [9, 8, 13, -4, 0, 7, 8, -5, -4, 2], [12, -10, 13, -7, -14, -19, -3, -9, 10, 3], [-9, 10, -5, 0, -8, -5, 6, 10, -1, -5], [-15, -13, -8, -6, 7, 9, -13, 5, 0, -4], [-6, -4, -4, 21, -13, -6, -3, -6, 0, 22], [3, 8, 3, -3, 1, 19, -4, 4, 15, -12], [5, -18, 16, -6, -6, -6, -7, -6, -9, 2], [15, -14, 0, 9, 1, -2, 6, 5, 3, -2], [7, 8, -9, 3, 12, 19, 12, 13, -15, -16], [-18, -8, 4, -10, 17, 2, 0, 4, -4, -21], [15, -18, -4, 20, -10, 0, 3, -3, -6, 15], [2, 19, -2, -4, 9, 11, -5, 7, -19, -16], [0, -11, 29, -7, -12, -4, -3, -13, -8, -1], [5, -3, -6, -10, -4, 5, -8, 6, 10, -17], [-21, 11, -9, -9, 16, 4, 1, 19, 5, -8], [3, -3, 9, 8, -5, 4, 3, 7, -14, 15], [-17, 18, -19, 9, -7, 4, -7, 12, 4, 9], [-3, -19, 18, 4, -5, 0, -3, -5, 0, 22], [8, 7, -29, 16, -5, -17, -11, 10, 5, 7], [5, 13, 0, -3, -8, 4, 4, 8, -6, -6], [11, 6, 11, -6, 8, -2, 7, -8, 3, 12], [20, -17, 2, 10, -22, -10, 4, -2, -6, 21], [11, 3, 4, 7, 10, 1, -4, 0, -4, -4], [-9, 22, 4, 1, -9, -4, 0, 9, 1, -10], [-4, -3, -6, 6, 3, -1, -4, 10, -4, 11], [-10, -3, 4, -18, 16, -6, -25, -1, 21, 2], [11, -1, -30, 0, 8, 3, -2, 1, -2, -6], [3, -5, -8, -7, 18, 5, 5, -10, 8, -19], [-7, -13, -2, -13, 6, -3, 0, 11, 10, 2], [-13, 7, -24, 3, 10, 20, 10, 31, -20, -29], [-13, 19, 5, -19, -2, 8, 1, 4, 3, -15], [-34, 2, -4, 6, 3, 0, -19, 9, 5, 27], [-14, 18, -7, -2, 6, 10, 1, 8, 2, -22], [4, -3, 23, 1, 8, -11, 9, -6, -15, 29], [-12, 3, 5, -8, 12, 13, 13, -6, 13, 3], [2, -24, -19, 13, 2, -19, -11, -6, 16, 16], [-18, 6, -6, -8, 23, -6, -1, -2, 6, -13], [-24, 9, -3, -25, 5, 1, -16, -1, 14, -13], [-24, 6, -12, -19, 10, 12, 4, 10, 1, -25], [-1, 17, 15, -17, 20, 19, 6, 11, -10, -22], [-5, -21, 4, -10, -2, 9, -1, -3, 7, -7], [-8, 25, 15, -14, 11, 22, 5, 5, 2, -22], [-25, 5, 5, 19, -14, 1, -5, 8, -12, 16], [0, 16, 2, -16, 23, 15, -9, 11, 2, -23], [-15, -14, 3, -1, 0, -11, 7, -5, -14, 18], [16, -17, 16, -6, -8, -11, -15, -6, 1, 36], [2, -10, 14, 13, -9, -7, -20, -11, -2, 15], [16, -26, 2, 18, -18, -14, 2, 1, -9, 35], [16, 0, -23, -2, 4, 2, -13, -7, 0, -4], [1, -14, -27, -5, 14, -2, -11, 1, 9, -17], [-6, -13, 0, 5, 4, -11, -11, -5, 14, 9], [21, -6, 3, -1, -12, -5, 11, 4, -14, -10], [14, -11, 23, -12, -6, -6, 20, -19, -4, 14], [-2, 32, -7, -18, 14, 12, 3, 11, -2, -21], [-9, 1, 5, -6, 25, -1, -19, -17, 10, 3], [-7, 2, -10, 2, 13, 1, 10, 5, -7, -12], [20, -1, 2, 13, -9, 7, 9, 6, 0, -11], [3, -13, 5, -2, -6, -16, 4, -17, 4, 26], [-1, -20, 1, -6, -4, -5, -3, -17, 24, 6], [8, -17, 23, -16, -12, 3, 5, -13, 10, 0], [22, -14, 8, -15, 1, -19, 13, -18, -2, 19], [11, -2, 25, -2, -8, 2, 24, 0, -17, 5], [-8, -15, 9, 3, 2, -1, 11, 16, -8, 14], [-7, 14, -40, -9, 21, 8, -13, 13, 20, -18], [25, 4, -4, 4, 5, -4, 5, -3, -18, -10], [6, -4, -15, 22, -12, -12, 2, 6, -13, 15], [0, 3, -6, 4, 6, -4, -2, -10, -14, -4], [-24, 6, -3, -7, 21, 22, -2, -4, 6, 2], [-1, 3, 17, -16, 19, 4, 17, -8, -21, -6], [6, 0, -12, 18, -7, -6, -8, -3, -9, -1], [-4, 9, 0, 1, 12, 0, -9, -7, 20, -10], [-1, -16, -9, 1, -3, 8, -22, 7, 18, 12], [13, 21, -15, -16, 22, 24, 27, 1, 0, -52], [-18, 6, -13, 9, -1, 17, -7, 3, 12, 7], [7, -14, 17, 24, -16, -12, 2, 0, -6, 14], [-18, 10, -18, 8, 11, 11, 4, 2, 13, -5], [8, 7, -1, 17, -1, 2, -16, 23, -15, -3], [-2, 19, -11, -2, 6, 8, 25, 14, -11, -42], [5, 15, 3, 2, -12, 6, 6, 11, 3, -1], [-20, -10, 1, 2, 7, -18, -10, -4, 12, 19], [-9, -15, -1, 20, -16, 6, 11, -3, 3, 29], [-5, 4, -1, -2, -3, 11, -5, 7, 9, -23], [2, 24, -3, -8, 13, 6, 1, 5, -8, -22], [-12, 3, -22, -1, 16, 2, -21, 21, 1, -13], [19, -22, 12, -2, -2, 5, 3, -6, -21, -2], [-3, -31, 5, 4, 3, -1, -12, -2, 17, -1], [-10, 1, -12, -9, 11, 18, 0, 12, -14, -5], [-2, -8, 4, 0, -1, -10, 0, -4, 25, -11], [4, -2, -3, 10, -4, -8, 7, -1, 9, 24], [26, -18, 11, 8, -19, -20, 13, -17, -9, 24], [2, 6, 2, 12, -2, -19, -12, -4, 14, 23], [52, -27, 20, 3, -14, 1, 7, -6, -13, 15], [-30, -7, -13, -8, 12, 7, -3, 16, 3, -9], [-22, 25, -14, -25, 28, 11, -14, -1, 16, -28], [8, -5, 21, 12, -21, 8, 2, -8, -13, 29], [-7, -10, 10, -8, -5, -8, 2, 16, -5, -10], [-25, -4, 3, 6, -2, 4, -1, 11, 10, -10], [1, -10, 9, 23, -27, -13, -9, 11, -2, 10], [-8, 16, -1, 7, 5, -3, -3, 0, -8, 12], [30, -6, 15, 6, -12, -4, 4, -6, -1, 28], [14, -17, 0, -1, -3, -11, 0, 1, 21, -5], [-21, 37, -28, 8, 24, 17, -12, 41, -19, -29], [-28, 15, 4, 12, -4, 0, -10, 14, 7, 28], [-6, 20, -23, 18, 9, 1, -10, 15, -11, -17], [3, -7, 15, 2, -16, -10, -19, -2, 26, 12], [11, -4, 31, -5, -10, -1, 4, -15, -5, -3], [-11, 1, 5, 6, 5, 7, -1, -11, 14, 3], [7, -8, -9, 14, -14, 9, 25, -7, -24, 8], [-10, 6, 5, -12, 29, 23, -3, -1, 4, -2], [16, 5, -5, -26, 22, 2, 20, -6, -3, -32], [25, -15, 24, -5, -6, -1, 31, -3, -14, 17], [18, -5, 16, -10, 2, 9, 20, -4, -22, -3], [-25, -7, 2, -1, -11, -3, -15, -6, -1, 4], [52, 0, 2, 3, -6, -7, -7, 0, 1, -5], [12, 3, -16, 7, 14, 8, -9, 7, 3, -2], [-24, 3, -19, -16, 20, -4, -18, -7, -6, 16], [-3, 9, -5, 22, -14, 2, -6, -1, -14, 19], [-23, -5, -9, 11, 0, 4, -7, 9, 1, 17], [31, -13, -6, 1, -5, -3, -10, -4, 13, 22], [23, 4, 4, 5, 4, 12, 28, 6, -5, -20], [8, -16, 6, -13, 17, 3, -3, -35, 10, -7], [-31, 22, -26, -10, 23, 10, -23, 9, 9, -46], [22, -16, 27, 14, -26, -23, 6, 0, -9, 34], [16, 0, 8, -6, -5, -6, -11, -15, -2, 10], [-38, -4, -9, -14, 9, 19, 2, 0, -1, -8], [14, 3, -17, 20, 12, -11, 9, 7, -4, 16], [-18, 5, -9, 29, -5, 10, -3, 25, 6, -11], [-44, 45, -19, -15, 41, 26, -12, -8, 1, -39], [-2, 1, 11, -30, 2, 1, 6, -12, 0, -6], [-9, -3, 0, -3, -5, 4, 4, 14, -18, 4], [-5, 2, 17, 22, -8, -7, 3, 14, -19, 20], [-17, 31, 7, -15, 11, 4, 4, 3, 4, -32], [37, -3, 19, -9, 7, -7, 20, -20, -9, -13], [0, 9, 6, -10, -1, -14, 6, -8, 13, -7], [-4, 17, -16, -15, 20, 5, 24, 9, -9, -43], [-23, 34, -17, -25, 38, 21, -2, 10, -7, -33], [6, -4, 20, 21, -11, 4, 23, -5, -16, 21], [3, -3, -11, 15, -6, -13, 3, 1, -12, 23], [1, 5, 13, 20, -14, 7, 22, 8, -18, 8], [-19, 5, -12, 7, 4, 7, 6, 6, -4, -29], [-11, 0, -1, -17, 19, 4, -12, 5, -5, -23], [9, -2, 9, -10, 8, -6, -11, 8, 11, 0], [6, 21, -24, -1, 14, 18, 14, 6, 16, -32], [-6, 11, -2, 4, 11, 13, -11, -8, -8, 4], [-33, 12, -26, 9, 2, 15, -19, 26, 5, 13], [2, -8, -11, 2, 0, -11, 13, -4, -8, 1], [-22, 2, 10, -11, -6, -13, -10, -19, 20, 11], [-3, -1, 2, -10, 7, -14, -18, -18, 13, 27], [4, 11, -32, -9, 21, 16, -16, 10, -6, -32], [35, -3, 9, 3, -3, -10, 18, -8, -34, -8], [12, -24, -1, 1, -6, 3, 16, -4, 13, 4], [-6, -14, 15, 18, -28, -17, -18, -13, 3, 52], [-6, 0, -7, 3, -10, -3, -9, 10, -6, 18], [-17, 22, -25, 13, 19, 16, -5, 10, 2, -21], [45, -22, 27, 10, -18, -14, 2, -12, -5, 3], [23, -22, 14, 13, -23, -5, 16, -10, -11, 31], [-7, 13, -12, 3, -9, -7, 1, 19, -11, -7], [-14, -9, 6, -12, 20, -16, -7, -17, 4, 21], [-7, -17, -4, 10, -27, 3, 13, 5, -8, 14], [-1, -2, -23, 0, 0, 10, 3, 29, -13, -29], [-9, 19, -37, -7, 14, 12, -14, 0, 21, -37], [7, -4, 10, -7, -4, -6, -17, -13, 8, 20], [14, -9, 0, 8, -1, 7, -1, -19, 24, 0], [-20, -10, 5, -5, 4, -6, 3, 12, 3, 16], [34, -2, 0, 12, -32, -17, 10, 2, -4, 17], [-15, -4, -5, 10, -5, 5, 3, 12, -21, -19], [-22, -27, 19, -6, -4, -3, -1, -6, -11, 10], [27, -7, -11, 23, -3, -16, -11, 5, -3, 19], [25, -23, 16, 0, -6, -13, 12, 8, -1, 9], [15, -4, 12, -10, -13, 13, 21, -16, 1, -19], [-12, 1, 3, -19, 14, -1, -9, -21, 3, -23], [7, 4, 17, -17, 14, 0, 25, -22, -8, -35], [-25, -18, -3, 27, -22, -9, -16, 9, 6, 29], [-12, 4, 7, -15, 15, 9, -7, -22, 3, -23], [28, 9, -13, 3, -3, 3, -11, -21, -12, -4], [-1, -4, -5, 17, -18, 6, 1, -3, -2, 19], [-7, 2, -12, 34, -23, -16, 4, 21, 1, 26], [17, 4, 3, 7, -6, 13, 9, 10, 13, -18], [-11, -19, 11, 38, -21, -14, 6, 5, -16, 46], [-4, -4, -27, -7, -7, 3, -20, 1, 16, 5], [-2, -16, 41, -12, -4, 2, -2, -14, 6, -9], [-15, -26, 15, -8, -7, 2, 11, -20, -5, 4], [18, -20, -8, -16, 11, -8, -7, -23, -2, 13], [7, 6, -22, 20, -17, -23, -12, 19, 1, 7], [43, -9, 17, 16, -19, 10, 36, 6, 0, -4], [-33, 25, -15, -10, 27, 20, -7, 10, 5, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8048_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8048_2_a_p();".
// 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_8048_2_a_p(:prec:=10)
    chi := MakeCharacter_8048_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_8048_2_a_p();".
// 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_8048_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 14, 5, -86, 54, 75, -85, 10, 18, -8, 1]>,<5,R![1, -9, 18, 16, -59, 7, 41, -7, -11, 1, 1]>,<7,R![-1, 4, 86, 214, 23, -228, 8, 64, -9, -5, 1]>,<13,R![-19, 4107, 5985, -481, -4882, -2777, -212, 307, 120, 18, 1]>],Snew);
    return Vf;
end function;