
// Make newform 8048.2.a.q in Magma, downloaded from the LMFDB on 20 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_q();".
// 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_q();".
// 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 := [-208, 576, 1284, -1556, -1463, 1643, 368, -616, 15, 94, -13, -5, 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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1864, -79464, 42460, 172813, -116237, -82700, 57076, 14883, -9770, -917, 543, 1], [56136, -184320, -325906, 244729, 283161, -124940, -87062, 28617, 10988, -2969, -485, 115], [65052, -127588, -348940, 245943, 406595, -221932, -157540, 77557, 24612, -11285, -1339, 577], [-193432, 396344, 1023786, -729423, -1096351, 568804, 402142, -183731, -60212, 25539, 3163, -1269], [-131744, 471292, 882834, -869847, -1062087, 663784, 419298, -214779, -66456, 30291, 3651, -1533], [72944, -136116, -523112, 302929, 683499, -319484, -276728, 120335, 44290, -18341, -2445, 969], [85748, -199010, -713098, 351689, 924969, -370074, -369564, 141625, 58386, -21833, -3185, 1163], [209308, -452248, -1550579, 887759, 1969696, -882020, -783065, 325094, 123571, -48993, -6743, 2572], [-53424, 125805, 391294, -249300, -490727, 237562, 192876, -85018, -30126, 12527, 1630, -646]];
    Rf_basisdens := [1, 1, 1, 7192, 7192, 3596, 7192, 7192, 3596, 3596, 3596, 899];
    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, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, -2, 1, 1, 0, 1, 1, 1, 1, -1, -1], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 1, -1, -1, 0, 0], [2, 0, 1, -1, -1, 1, -1, -1, -1, 1, 1, 1], [-1, 1, 0, -1, -1, 0, 0, 0, -1, 0, 1, 0], [2, 0, 1, 0, -2, 1, 0, 1, -2, 1, 0, 0], [0, -1, 1, -1, 0, -1, -1, 0, 0, 0, 1, 1], [0, 1, 1, 0, 0, 0, -2, 0, 0, 0, 2, 1], [-1, 1, -1, 2, 1, -1, 0, -1, 2, 1, -1, -2], [1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 1, 0], [-1, 3, 2, -1, -4, 2, -1, 0, -4, 0, 1, 1], [-1, -1, -2, -1, 2, -2, 2, 1, 0, -1, -1, -2], [-1, 1, 1, -1, 0, 0, -1, 0, -1, 0, 1, 2], [-2, 0, -3, 0, 1, 0, 2, 1, 1, 1, -2, 0], [-2, 0, -2, 1, 2, 0, 1, 0, 2, 0, -2, -1], [-2, 0, 0, 1, 0, 0, 0, -1, 2, 1, 0, 1], [1, -2, -2, 1, 4, -2, 1, 0, 2, 0, -1, -1], [-1, 1, -1, -1, 0, 2, 1, -2, 0, 0, 1, 0], [-7, 1, 0, -1, -2, 0, -2, 1, 1, -1, 0, 0], [1, 0, 0, 2, 1, -1, 1, 2, 1, 1, 0, 0], [3, -1, 0, 1, 1, 0, 1, 1, 0, 1, 1, -1], [2, 2, 1, 2, 0, 0, -2, 2, -3, -1, 0, 0], [-1, 2, -2, -1, 1, 1, 2, -3, -1, 1, -1, -2], [-3, 0, 0, 2, 0, -2, 0, 2, 1, 0, -1, 1], [2, 1, -1, 0, 0, 2, 1, 1, 0, 1, 0, -2], [0, -1, -3, -1, 3, -3, 3, 1, 0, -1, 1, 0], [1, -1, 0, 1, 1, -2, 0, 0, -1, 0, 1, 0], [1, 1, -4, 2, 4, 1, 3, -1, 3, 3, -2, -3], [2, 2, -1, 2, 1, 0, 3, 0, 0, 3, 0, -1], [2, 1, -2, 2, 3, -2, -1, 0, 1, 0, 2, -2], [-1, 2, 5, 1, -1, 4, -2, -2, -1, 0, 1, 2], [2, -1, 1, 1, -1, -1, -2, 0, 0, 0, 1, -1], [0, 1, 2, -2, -1, 0, -2, -1, -6, -1, 4, 3], [2, 1, 1, -2, -2, 2, 0, 2, -3, 2, -4, 0], [-5, 2, 3, 2, -1, 1, 0, -3, 1, 0, 0, 1], [-3, 1, 2, 1, 2, 0, -1, -4, 4, 2, 1, -1], [-5, 2, 4, 1, -4, 4, -3, -4, -1, -2, 3, 3], [-8, 0, -2, 2, 0, -4, 0, 2, 5, 0, -4, -1], [5, -2, 3, 2, 1, 1, 0, -1, 1, 1, 0, 0], [3, 0, 2, -1, 2, 2, -1, 0, -2, -4, 2, 0], [-3, -2, 0, -1, -4, 0, 0, 1, 1, 0, -3, 1], [1, -3, 1, 0, 0, 0, 1, -1, -1, -3, 1, 1], [-1, -1, 3, 2, 1, 2, 1, 1, 1, -3, -2, 1], [-4, -4, -1, 0, 0, -4, -3, 1, 6, -1, -2, 0], [-2, 5, 0, -2, -3, 1, -1, -4, -3, 0, 1, -1], [-8, 0, 0, -3, -2, -3, -3, 0, 2, 0, -1, -1], [-7, 0, -1, -1, 1, 0, 3, 5, -3, -3, -1, 0], [-1, -2, 0, -1, 1, -1, -2, -1, 4, 3, -1, 1], [-8, -1, 0, -3, -3, -1, -3, -3, 0, -1, -2, 1], [-7, -3, 0, -1, 0, 0, 3, -2, 0, 0, 0, 2], [2, 2, 10, -2, -6, 5, -5, -4, -4, -2, 2, 3], [-2, 1, 8, -3, -5, 1, -8, -4, -1, 0, 6, 2], [2, -1, -2, 3, 4, -3, -2, 1, 3, -2, 3, -1], [3, -2, 0, 0, 0, 0, 1, 1, -1, -1, -5, -1], [1, -3, -3, 4, 3, -3, 0, -1, 6, 3, -6, -3], [-4, 2, -1, -3, -1, 1, -2, -2, 2, 0, 1, 1], [1, 2, 4, 2, -2, 3, -5, -1, 0, 0, 4, 3], [2, 0, 0, 0, 2, 2, -1, -1, -1, -1, 4, 1], [-4, -1, -4, 1, 2, -3, 2, 1, 3, -1, -1, -5], [6, -4, -1, 0, 0, -2, 2, 2, -2, 0, -4, -2], [-4, -3, -6, -1, 5, -1, 9, 1, 2, -2, -5, -4], [-7, -2, -2, -3, -3, 0, -3, -1, 1, -1, -1, -1], [-3, 0, -5, 2, 7, -5, 3, 0, 2, -3, -2, -6], [4, -1, 9, -1, -4, 2, -4, -2, -5, -4, 7, 1], [0, 6, 1, 1, 1, 1, -1, -3, 1, 1, -1, -3], [-4, -1, -1, 2, 2, 0, 2, 2, -2, 2, 2, 2], [-2, -1, 0, 1, 3, 1, 3, -1, -1, -5, 0, 0], [5, -2, 6, 2, -1, 1, -3, -4, 3, 0, 4, 1], [-9, 5, -2, 0, -3, 1, 3, 0, -1, -2, -4, -2], [2, -3, 2, 2, -1, 1, -1, 0, -1, 4, -3, 0], [10, 2, -3, 1, 3, 1, 2, 2, 2, 4, -1, -2], [3, -1, -4, 1, 2, 4, 1, 0, 0, 2, 1, -4], [2, 2, 1, 0, 2, 2, -2, 0, 1, -2, 1, -1], [8, 0, -1, 5, 7, 1, 5, -3, 3, 3, 3, -1], [3, -4, 7, 3, -1, 0, -1, 3, -3, 0, 3, 2], [0, 0, 2, -3, -5, 1, -1, 1, -6, 3, 5, 5], [-6, -1, 6, 0, -4, 2, -4, -4, 0, 2, 0, -1], [0, 4, 3, 2, -1, 1, -5, 0, 0, 4, 1, 1], [-3, 0, 4, -1, -6, 2, -2, 3, -7, -1, 3, 5], [-8, 0, 1, -3, -1, 3, 1, -1, -5, -1, 3, 4], [2, -2, -5, 0, -2, 0, 2, 2, -1, -2, -6, 0], [6, 2, 1, 5, 3, 5, 0, -2, 2, 6, -1, 0], [1, 8, 8, 0, -10, 2, -9, -2, -6, 0, 10, 6], [-1, -5, 0, 2, 4, -2, 1, 1, -1, -3, 3, 0], [-3, -1, -4, -3, 2, -2, 3, 0, -1, 4, -5, 0], [-2, -3, -1, 4, 5, 0, 4, 3, -1, 1, 4, 1], [-3, -4, -3, 5, 3, -2, 1, 5, 4, -1, -5, -3], [3, 1, -1, 1, -3, 3, -1, 2, -1, 8, 3, 0], [-8, 2, -4, 4, 5, -1, 0, -6, 6, 2, 0, -5], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -9, -4, 0, 6, -6, 3, 5, 7, -1, -4, -1], [-6, 6, 9, -2, -8, 2, -4, -2, -6, -2, 6, 2], [0, -2, -3, -5, -2, 1, 2, 1, -6, -1, 5, 4], [-6, 9, -9, -3, -2, -2, 2, -2, -2, 4, -1, -2], [4, 2, 1, -1, -3, 3, -2, 2, -4, 0, 5, -2], [3, -9, -5, 4, 5, -7, 5, 4, 10, 0, -4, -2], [-5, -7, 1, -3, -4, 2, -1, -6, 2, 1, 1, 2], [2, -3, 3, -3, -5, -1, -6, 0, 2, 2, 2, 3], [1, 3, 9, -4, -6, 3, -5, -1, -9, 0, 8, 4], [2, -3, -2, 2, 3, -2, 6, 5, -3, 5, -4, 2], [-2, -1, -8, 0, 2, 0, -1, 3, 1, 1, 0, -4], [5, 1, 4, 0, -2, 3, 0, -2, -2, 0, 0, 0], [-13, -6, 4, -5, -1, -1, 1, -3, 5, 1, 0, 1], [-3, -6, 4, -2, 1, -3, -5, 2, 1, -6, 8, 5], [-8, -2, -10, 2, 5, -3, 6, 1, 7, -1, 1, -6], [1, -2, -6, 2, 7, -7, 7, 2, 6, -2, -2, -5], [6, -7, -1, -2, 1, 4, -1, 2, -3, -4, 4, 5], [-4, -5, -5, 4, 4, -6, 3, 1, 9, -4, -10, -7], [6, -8, -6, 1, 5, -5, 7, 3, 3, -1, -5, -1], [10, -3, -1, 2, 3, -1, 1, -4, 5, 6, 1, -4], [-1, 2, 14, -6, -6, 5, -9, -5, -10, -2, 8, 9], [10, -3, 6, 7, 2, -2, 0, 1, -1, 3, 0, 1], [0, -4, -11, 2, 8, -6, 3, 1, 8, 3, -6, -4], [-3, 6, 8, -4, -1, 5, 0, -3, -10, 1, 8, 8], [0, -1, 3, 0, 2, 2, -2, -2, -1, -4, 4, 0], [-14, -1, 2, -3, -3, 1, 0, -4, 2, -4, -2, 0], [1, -3, 3, -7, -3, -1, -2, 5, -6, -5, 3, 3], [-5, -4, 0, -1, 0, 3, -1, -1, 4, 1, -3, -3], [3, -3, -7, -1, 2, -2, 3, 6, 8, 2, -7, -2], [6, -7, -6, 1, 1, 1, 2, 2, 1, -4, 1, -2], [4, -3, -12, 2, 11, -4, 7, 6, 6, -1, -4, -7], [-7, -4, -7, -3, 8, -4, 6, -1, 6, 5, -8, -5], [7, 0, 6, -1, -8, 4, 0, -3, -7, 7, 2, 4], [2, -5, -1, 0, -4, 1, -1, 4, 2, -4, 0, -1], [-14, 10, 1, -4, 0, 4, 2, -4, -3, 2, 0, -1], [-3, 5, 0, -4, -2, 3, -7, -1, -5, -1, 4, 1], [-9, 4, 7, -5, -4, -2, -7, -4, 3, 0, 3, 0], [-18, -3, 7, -2, -8, 1, -2, -1, -5, -7, 0, 6], [-6, -3, 8, -2, -1, -3, -1, -2, 3, -6, 3, -3], [-12, 1, 1, -7, -8, 1, 1, -8, 1, 4, 3, -2], [10, -1, -1, 2, 4, 0, -2, 4, 5, 5, 0, -5], [5, -2, 1, -1, -4, -3, 0, 0, -2, 4, 1, -2], [-5, 10, 4, 4, -7, 7, -5, -2, -7, 10, 4, 2], [9, -3, 10, -3, -2, 0, -3, 0, -1, 2, 1, 6], [16, -2, -9, 3, 3, -3, 4, 2, 1, 0, -11, -6], [16, 0, 2, 5, 5, 3, 1, 1, 1, -2, 3, 4], [-15, -3, -9, -5, -5, -2, 1, 3, 3, 1, -5, -3], [14, -4, 1, -2, -4, 2, -2, -4, -4, 6, 4, 1], [-18, 4, -6, -1, 0, -5, 0, 3, 0, -6, 1, -3], [-17, 2, -3, -6, -6, 0, 3, -1, 0, -3, -5, 1], [4, 10, 0, 4, 2, 2, 2, 2, -8, 2, -4, 1], [-5, 5, 6, 4, -7, 3, -2, 1, -3, 3, 3, -2], [-1, 2, 0, 7, 9, -1, 0, -7, 10, 3, -1, -5], [-1, -6, 7, -3, -6, 0, 0, 3, -6, -1, 1, 3], [0, 0, 1, 1, 2, -5, -1, 0, 1, -1, 1, -5], [-8, 0, -1, -6, 0, 0, 5, -5, 3, -1, -5, 0], [-2, 5, 3, 6, 2, 0, -1, -1, 3, 1, 2, 7], [6, -2, -5, 2, 5, 1, 10, 7, 0, 1, -5, -3], [1, -1, -6, -3, 5, -6, 0, 2, 5, -5, -7, -2], [15, 5, -9, 7, 11, 0, 2, -2, 1, 8, 1, -7], [4, 1, -11, 2, 4, -2, 5, 9, 0, 7, -1, -4], [20, -4, 5, 1, 0, -1, -1, 0, -3, 4, 5, 1], [-16, 7, 5, -3, -4, -3, -6, -7, -6, 1, 1, 1], [10, -7, -5, -2, 2, -2, 4, 2, -7, 4, 0, -4], [9, 9, 7, -4, -8, 9, -3, -5, -14, 6, 6, 3], [1, 6, 4, -1, -2, 6, -5, 0, -15, -10, 9, -2], [4, 5, -3, -2, 0, 4, 6, 0, 1, -2, -6, 1], [3, 8, 2, -2, -7, 5, 2, -7, -2, 7, -4, 0], [-28, 5, 0, -1, -2, 0, 6, -3, 3, -3, -4, 1], [-1, 2, 0, -5, -7, 0, -5, -1, 0, -5, 1, -6], [-13, 3, 1, -2, 1, -3, 0, -5, 4, 5, 1, 2], [-6, -2, 1, 4, -3, 4, 3, 0, 6, 3, -6, -1], [-3, 5, 2, 5, 4, 0, 1, 0, 1, 6, 4, 0], [2, -9, -1, 5, 7, -5, 1, -3, 9, -3, -5, 0], [-11, -1, 11, -1, -14, 2, -8, -5, -1, -1, 3, 4], [-1, -3, 4, 5, 2, -2, -8, -3, 1, 6, 3, 0], [-2, 5, -3, -4, 0, 0, 6, 0, 1, 6, -10, -5], [-7, 5, 9, -1, -5, 1, -6, 2, -12, -4, 10, 13], [6, 5, -5, 2, -3, -3, -1, 4, 2, 4, -3, -3], [-11, -7, 2, 1, -4, -2, 4, 3, 5, -5, -12, -1], [3, -4, 4, -2, -1, -5, -4, -5, 3, -7, 0, -2], [4, 7, 2, 0, -6, 8, -3, 3, -7, 0, 2, 8], [8, -3, 8, 1, 1, -5, 2, 4, -4, -2, 1, 2], [2, 5, 1, 1, -5, -3, -11, -3, 1, -3, 5, -3], [-1, -6, 0, -1, 6, -6, -1, 6, 2, -12, 3, 3], [-2, 12, 12, -1, -17, 7, -7, -3, -13, -1, 7, 5], [-7, 2, 5, -1, 2, 1, -2, -6, -2, -6, 7, 6], [-13, -8, -6, 6, 7, -7, 1, 2, 6, 0, -5, -2], [11, -2, -7, 8, 13, -1, 7, 2, 9, 8, -7, -4], [9, 0, 4, 2, 2, -5, -3, -1, 2, 0, 4, 1], [-1, -4, 3, -7, 1, 0, 1, -1, -4, 0, -5, 1], [6, 1, 3, 1, 1, 5, -4, -2, -2, 2, 11, 6], [12, 1, 13, 0, -10, 4, -11, 5, -11, -7, 6, 5], [4, -2, 2, 5, -1, 5, -3, 3, 7, -5, -5, 1], [-12, -2, -14, 3, 9, -3, 10, 6, 11, 12, -8, -3], [3, -1, 1, -5, -5, 7, -4, -5, -1, 1, 5, 4], [8, -7, -8, 2, 4, 1, 10, 9, 3, 1, -2, -5], [7, -1, -7, -9, -4, 0, -2, 1, -2, 3, -3, -3], [3, 1, -1, 8, 4, 0, -4, -3, 9, -3, 2, -1], [-3, 1, 6, -1, -1, 4, -7, -1, -7, -3, 9, -4], [8, -7, -9, 6, 14, -10, 11, -1, 14, -1, -4, -4], [10, -6, -6, 2, 4, 0, -2, -2, 2, 2, 8, 5], [-5, 6, 10, 7, -8, 2, -10, -7, 6, 9, -1, 2], [2, -10, 1, -5, -1, -9, -5, -3, 4, 3, 7, 2], [-5, 12, 2, 1, -4, 0, -3, -4, -5, 2, -7, 4], [1, 12, -5, 0, 3, -3, -4, 5, 0, -1, 4, -1], [4, -5, 2, -7, -1, -1, 0, 10, -5, 4, -3, 4], [4, 2, 2, -2, 4, -2, 2, 0, -7, -8, 7, -4], [-13, 1, 6, 6, 7, 1, -1, -6, 8, -4, 8, 2], [5, 3, 4, 11, 2, -2, -4, -1, 4, 3, 5, 3], [-5, 3, 0, -1, 5, -5, -1, 1, 5, -5, 0, 1], [-4, 2, 2, 11, 4, -2, 5, 3, 6, 3, -9, -12], [6, -4, -11, 1, 7, -7, 2, 6, 11, -6, -8, -6], [-17, 6, -2, 4, -5, 1, -6, -1, 5, -11, 0, 2], [-6, -12, -1, 3, 2, 0, 2, -1, 2, -9, 2, 1], [-6, -6, -14, -1, 2, -4, 3, 3, 9, -3, -5, -3], [-17, 11, 9, -1, -10, 4, -3, -2, -3, -2, 1, 2], [12, 7, -6, -1, -5, -1, 1, 1, -3, 5, 7, -1], [-2, 5, 4, -1, -2, -5, -5, 0, -5, -4, 9, 5], [19, -1, -5, 4, 0, 1, 2, 8, -7, 10, -6, -6], [7, -10, 2, -2, 11, -1, 5, -2, -2, 2, -4, 3], [9, -6, 9, -3, -13, 9, -2, 6, -9, 2, -2, 8], [8, 3, -3, -3, 7, -3, 0, 4, 0, -2, -3, 0], [-3, 0, 0, 5, 2, 2, 5, -6, 5, -4, 1, -4], [-2, 5, 6, -6, -16, 5, -8, -3, -4, -7, 2, 8], [4, 5, 18, -5, -17, 7, -12, -6, -17, -2, 8, 4], [-1, 7, -1, 2, 1, -7, -6, 1, -1, -9, 0, 2], [-11, 9, 9, -8, -7, 5, -8, -3, -3, -3, 8, 0], [1, 0, -8, 2, 5, -1, 5, -8, 5, -4, -8, -12], [-2, -2, 3, 0, 2, 0, 0, 4, 0, -14, -4, 4], [-8, -4, 2, -5, 0, -2, 0, -5, -2, -15, 4, 6], [-8, -5, -2, 6, 4, 0, 1, 11, 7, -7, -3, -5], [-9, 10, -2, 14, 3, -3, -5, -2, 6, 2, -2, -6], [-5, 2, 2, -5, -6, 2, -3, 0, -8, 0, 9, 5], [17, -6, 4, -11, -5, 0, 3, 3, -4, -6, 3, 2], [17, 0, -7, -2, 1, -1, -2, 5, -6, 9, 6, -2], [-3, -3, -7, -5, -4, -6, 3, -2, -1, 2, 5, 4], [5, -3, -7, -1, 1, 0, 5, 3, -3, -5, 3, -5], [-11, 6, 1, -4, -1, 1, 0, 1, -4, -8, -4, -4], [9, 1, -1, 8, 2, 1, 3, -5, 15, 9, -10, -10], [1, 3, -1, 2, -4, 0, -4, 4, 10, 8, -1, 6], [7, -4, -9, 5, 3, -6, -4, -2, 13, -3, -11, -8], [-15, -4, 8, -3, -4, 0, -2, 1, 5, -11, -9, 1], [-7, 5, 6, -2, -16, 6, -12, -4, -3, -4, -1, 2], [1, 2, 9, -4, 1, -1, -3, -4, -8, -6, 14, 5], [9, 4, 10, -1, -8, 4, 1, 2, -12, 8, 3, 9], [-6, 2, -6, 0, 8, -2, 4, 0, 4, 2, -10, 0], [0, 10, 2, -3, -12, 5, -1, 0, -4, 5, -1, 3], [5, 9, -1, -3, 3, 3, -1, -3, -4, -1, -2, -1], [-18, -6, -5, 9, 10, -8, 6, 5, 9, -5, -8, -4], [-6, 4, 15, 4, -10, 8, -11, -7, 6, -7, 4, 11], [5, 4, -17, 1, 6, -7, 11, 5, 13, 9, -11, -11], [-7, -16, 9, -2, 3, 1, -6, 1, -3, -8, 0, 3], [-21, 6, 5, -5, -6, -4, -5, -2, 2, -8, 3, 9], [8, 5, 16, 4, -8, 2, -11, 5, -5, 3, 8, 6], [13, 0, 5, 8, 6, -2, -2, -4, 9, -2, 1, -5], [7, 10, -3, -1, -1, -1, -7, -1, 6, 7, 6, 2], [14, -1, -5, -3, 2, -2, -2, 3, -9, 3, 14, -1], [2, 8, -9, 3, 7, 1, 11, -5, 4, 7, -5, 0], [-17, -6, -17, -3, 6, -5, 15, 5, 3, 1, -7, -9], [14, -5, -4, 1, 3, -5, 3, 7, -3, -5, -3, 1], [6, 1, 7, 1, -5, -1, -5, 1, -3, -9, -1, 2], [-4, 0, -5, 0, 0, -4, 5, 1, -2, -1, 2, -2], [9, 0, 3, -2, -8, -1, -3, -5, -2, 13, 0, -2], [-6, -13, 3, -3, 1, -3, -2, 2, 2, -6, 10, 3], [-14, 0, -6, -7, -1, 1, 4, 0, 6, 0, -5, 3], [4, 9, 0, -5, -2, -4, -6, -4, 2, -4, 13, 5], [18, 4, -3, 7, -5, 11, 1, 7, -10, 3, 7, 1], [-11, -13, 3, -7, -4, -2, -4, 3, 1, 1, 8, 10], [-1, -4, 14, -2, -7, 7, -2, -3, -4, 5, 4, 6], [-14, 6, 19, -8, -10, 8, -12, -18, 0, -8, 12, 6], [9, -1, -10, 1, 12, -6, 8, 3, 4, -3, -1, -5], [-1, -2, -3, -1, -2, 3, 0, 4, -5, -10, 3, -1], [2, 1, -6, 6, 2, -2, 5, -1, 3, -5, -2, -2], [6, -16, -10, 0, 4, -3, 12, 3, 7, 3, -14, -8], [-17, 12, -7, 8, 0, 1, -4, 4, 10, -2, -2, -2], [-20, 2, -9, -1, 1, -5, -5, -1, 12, -3, -9, 1], [-17, 10, 4, 1, -10, 4, -6, -1, -1, -7, 1, -1], [2, -4, -1, -1, -3, 7, -6, -2, -3, 2, 10, 0], [5, 8, -9, 14, 10, -5, 3, -1, 3, 1, -8, -7], [15, -5, -5, 1, 4, -3, -1, -3, 5, 3, 13, 2], [15, 4, -4, -3, 0, 0, 5, 8, 3, -3, -5, -11], [-2, -4, 10, -3, -6, -2, -9, -10, 0, 0, 10, 8], [3, 2, -6, -1, -2, 2, 4, 7, -9, 0, -3, 5], [5, 6, -6, -3, 2, 4, 4, -3, 0, -11, -1, -4], [-6, 0, -12, 2, 1, -1, 3, 6, 1, 8, 5, 4], [-6, 12, 17, -4, -10, 6, -6, -6, -5, -6, 10, 5], [-6, -1, 6, 0, -4, -2, 7, 1, 5, -5, -4, 4], [-14, -19, -8, -1, 9, -14, 2, 7, 14, -5, -11, -7], [0, 7, 5, -5, -10, 9, -10, -5, -9, -9, 5, 0], [-22, -7, 7, -2, 0, 0, -5, -5, 12, -3, -5, -1], [-7, -10, -16, 1, 10, -8, 7, 0, 13, 4, -11, -11], [28, 11, -1, -7, -12, 6, -3, -4, -12, 2, 6, 2], [0, -4, -1, 0, -3, 1, 7, 4, -6, -6, 5, 0], [4, -8, -23, 3, 13, -5, 16, 12, 16, 0, -13, -9], [12, 4, -7, -5, 1, 1, 0, 0, 7, -6, -7, -6], [-19, 0, -7, 5, 13, -6, 4, -6, 10, 10, 1, -7], [0, 10, -1, -12, -2, 2, 0, -4, -6, -5, 4, 9], [15, -14, 1, 10, 16, -2, 7, 4, 4, 2, -16, -3], [3, 0, 11, 5, -7, 11, -5, -6, -8, 10, 7, 4], [9, 3, 1, 2, -1, -1, -10, 7, -12, -11, -2, 3], [0, -3, 3, -5, 2, 6, 7, -3, -15, 3, 7, -1], [-2, 11, 18, -9, -13, 7, -9, -5, -14, 4, 3, 4], [4, 0, 8, 0, -7, 8, -4, 3, -5, 4, 4, 9], [9, -8, 9, -2, -12, 14, -4, -2, -7, 6, 5, 6], [16, -7, 2, 0, -2, 0, -5, -5, 1, -3, 0, -4], [-22, 19, -1, 0, -5, -2, 6, 1, -5, -1, -8, -3], [15, -3, 0, -1, -3, 6, -3, -9, 7, 8, 3, -4], [-21, -2, 0, 1, 7, -8, 8, 2, 7, -4, -3, 1], [-15, -2, -6, -2, -1, 5, -1, 8, -6, 2, 12, -1], [10, 11, -4, 6, 5, -6, -9, -8, 10, -2, 0, -5], [1, -2, -21, 4, 22, -7, 7, 5, 12, 4, -2, -8], [3, -14, 11, 1, 0, 0, -1, 10, -3, -14, 6, 5], [-2, 5, -9, 3, 3, -9, 0, 14, 4, 3, -5, 1], [-10, 10, 6, 0, -1, -2, -5, -4, 6, -4, 2, 0], [3, -12, 4, -5, -2, -9, -6, 6, -1, -10, 9, 8], [17, -6, -15, 0, 11, -7, 9, 12, 0, -2, -5, -4], [-14, 6, 3, -2, -4, 6, -2, -8, -13, 5, 0, 2], [24, 5, 11, 2, -7, 3, -4, -3, 4, 1, 5, 3], [0, 1, -2, 5, 12, -2, 4, -9, -5, -3, 8, -4], [3, 0, -11, 4, 14, -1, 7, -11, 14, 7, -2, -6], [2, -7, -15, -1, 5, -7, 11, 11, 3, -7, -6, -11], [5, 0, 3, 0, 2, -3, -3, -3, 0, -7, -2, -6], [3, 2, 1, 3, 0, 4, 3, -2, -6, 0, -8, 0], [5, -3, 0, 3, 5, -9, 8, 9, -3, -3, -11, -7], [11, -14, 7, 1, -3, 12, -3, -3, -1, 11, 1, 6], [-17, 0, -13, 3, 16, -10, -4, -3, 25, -3, -9, -1], [17, 0, -8, -1, -2, 4, -3, -6, 14, 6, -2, 4], [-11, 4, 11, -3, -8, 4, -4, -7, 4, -7, 4, 1], [-12, 14, -4, -7, 2, 1, 4, -1, -10, 5, -1, 1], [-9, -4, -2, 9, -1, 3, 2, 5, 0, 5, -3, 1], [-26, 5, -15, -7, 3, 2, 10, -7, 1, -5, -7, -7], [-9, -8, -14, 4, 13, -7, 11, -8, 8, 4, 4, -7], [-7, 7, 5, -1, 2, 2, 2, 1, -2, -19, 7, -3], [19, -8, -9, 3, 7, -2, 0, 4, 2, -10, -7, -5], [-31, 2, -7, -2, 3, 3, 8, -5, 1, 3, -10, -9], [-8, 16, 1, 3, -2, -5, -2, -1, -6, -1, 3, -2], [10, -3, -5, 1, 3, -9, -6, 4, 2, -10, 9, -4], [-7, -10, -10, 0, 10, -9, 11, -1, 4, -10, -14, -13], [12, 6, 2, 3, 7, 3, 0, -2, -2, -8, -3, -3], [-25, -2, -2, -4, -9, 5, 4, -5, 1, 3, -12, 2], [4, 7, 8, 4, 4, 6, -7, -3, -6, 3, 6, 7], [-5, -4, -9, 2, 12, -3, 14, 4, 10, -4, -22, -12], [-6, -11, -3, -2, 4, -4, -2, 2, 12, 0, -11, -2], [4, 8, 2, 0, 2, 0, -3, -9, 8, 3, -4, 3], [0, 6, 5, 6, 3, 7, 3, 0, -6, 4, 17, -1], [13, 10, 11, -5, -12, 6, -16, 3, -19, 5, 15, 13], [13, -2, -7, -6, 0, 4, 6, 0, -12, -2, -1, 9], [-3, 13, 7, -8, -5, 8, -4, -2, -9, -8, 10, 7], [-31, -2, 14, -13, -23, 5, -13, 0, -15, -20, 9, 14], [-17, 14, 0, -4, -11, 3, -6, -1, -5, 1, 10, 2], [-18, -16, 3, -4, 1, -6, -5, 0, 11, 0, 0, 2], [-9, -9, 8, -4, -10, 6, -5, 3, 1, 3, 1, 10], [-13, 7, 14, -7, -22, 12, -6, -1, -22, -3, 5, 11], [-18, -4, 1, 1, 8, 0, 6, 2, -5, -6, 3, -4], [6, 6, 1, -9, -1, -3, -3, -1, 2, 13, -1, 0], [7, 10, -4, -4, -6, -1, -1, 3, -14, 13, 2, 1], [-4, 13, 13, 3, -11, 11, -4, -8, -13, 18, 11, 10], [14, 6, -11, 1, 1, 3, 5, 3, 0, 8, -15, 2], [-16, 15, 14, -1, -12, 13, -11, 0, -8, -4, 9, 12], [-12, -3, 16, -1, -2, -4, -7, -8, 4, 4, 0, 13], [-6, -3, 5, -10, -13, -2, -12, -1, -3, 2, 10, 3], [-26, 15, -12, -7, -3, 3, 13, 3, -5, 5, -9, -7], [13, -1, 2, -3, -1, 6, 7, -3, 3, -1, -1, 6], [-1, -2, 10, -8, -13, 2, -12, -14, -6, 8, 12, 6], [15, -5, 13, -1, -8, -2, -14, -1, -9, -7, 10, 8], [-23, 18, -7, 10, 5, 1, 10, -1, 9, 5, -9, -10], [-28, -2, 7, -7, -8, 6, -14, -4, -6, -6, 9, 15], [-1, 2, 2, -3, -5, -3, 3, -8, 5, 10, -5, -6], [-25, -13, -9, 1, 18, -18, 8, 3, 23, -15, -9, -5], [-33, -3, 17, -17, -18, 0, -2, -11, -6, -9, -1, 7], [2, -12, -12, 4, 15, 0, 12, 1, 15, 11, -6, -11], [2, 13, 6, 7, 0, -2, -2, -1, -5, 13, 0, -4], [0, -13, -3, -1, -3, -3, 10, 12, -8, 2, -9, 0], [-13, -2, -2, -12, -5, -5, -2, 3, -4, -3, -2, 4], [-12, -10, -18, 4, 7, -5, 8, 3, 7, -3, -15, -6], [-9, 0, -1, 2, 8, -2, -1, -4, 10, -8, -8, 3], [-9, 6, 6, -2, -5, 3, 4, -1, -4, 15, -6, 0], [1, -9, 3, 2, 5, 5, -7, -4, 4, -12, 2, -9], [9, -11, -4, 0, 1, -5, 0, 13, -9, -9, 0, 2], [12, -13, -12, 5, 6, -13, 10, 7, 3, 7, -15, -11], [-4, 8, -5, 0, 12, -8, 1, -7, -3, -5, 1, -7], [5, 5, -17, -3, 2, 0, 13, -2, -1, 2, -12, -9], [-5, 2, 14, 3, -19, 6, -10, -8, -4, 8, -7, 4], [16, -6, -8, 1, 5, 4, 10, 1, -4, -1, 5, 2], [-14, -10, -7, 5, -3, 1, 6, 8, 7, 16, -18, -4], [13, 5, -1, -2, -7, 9, 6, -3, -18, -1, 9, -2], [33, 8, -12, 8, 4, 5, 6, 8, 2, -2, -4, -9], [-10, -6, -1, -13, 5, 1, 9, -5, -8, 5, -7, 4], [-1, 9, 2, -5, -13, 6, 5, 5, -20, -5, 11, 3], [-17, -6, -15, 8, 10, -6, 18, 8, 6, -2, -17, -13], [-8, 11, 3, -2, -6, 4, -3, -7, -9, -11, 6, 4], [15, -13, 0, 3, 2, 0, 4, 3, -3, 9, -3, -1], [-6, 16, -8, 0, -16, 6, 1, -1, -1, 13, 2, 2], [-30, -2, -7, -2, -9, 0, 7, -4, 6, 6, -24, -5], [-12, -3, 5, -4, -6, 6, 2, -8, -8, 4, 8, 8], [5, 5, -6, 5, 4, 8, 10, 7, -1, 1, -15, -7], [-30, 5, -6, -1, 3, -3, 8, 8, -16, 4, 1, 1], [-11, 14, 7, -2, -16, 10, -2, 2, -17, 6, -9, 3], [12, 2, 0, 4, 0, 0, -9, 3, -7, -13, 9, 4], [0, -1, -4, -5, 0, 5, 1, -14, 5, 10, -3, 1], [3, 2, 6, -7, 1, 10, -2, 0, -7, -2, 5, 11], [-10, 4, 14, -2, -22, 16, -6, -10, -13, 14, 0, 8], [3, 20, -7, 7, -2, 8, 7, 2, -3, 12, -3, -4], [5, -17, 6, -10, -5, -1, -7, -4, 4, 0, 4, 9], [31, -10, 13, 14, 1, 9, 3, -8, -2, 14, -2, -2], [-9, -20, 2, -6, -3, 5, 10, 3, -17, 3, -1, 3], [1, -7, -3, 4, 9, -9, -1, 6, 5, 4, -1, -2], [-1, -3, -5, 0, 4, -6, 6, -1, -3, -1, 0, -15], [-22, -1, -1, -5, -2, -8, -4, -1, -2, -7, 0, -2], [16, -19, -17, 5, 21, -5, 8, 4, 9, 4, -11, -6], [26, 12, 1, 12, 4, 0, 1, 5, 1, -5, 9, 1], [-21, 1, 15, -17, -16, 2, -16, -3, -3, -15, -7, 8], [-24, -10, -12, -1, 6, 3, 11, 2, 6, 9, -3, -3], [-12, -4, -2, -5, 5, -15, 6, 6, 6, -12, -3, 5], [20, 4, -7, 11, 9, -7, 3, 7, -4, -1, -9, -5], [21, 7, -9, 12, 10, -15, -1, 5, 14, -1, -10, -15], [-3, -12, -1, -5, -2, -6, -2, 3, 2, -9, 5, -4], [-31, -22, 11, 9, 4, 0, 2, -3, 6, -7, 1, 5], [8, -6, -7, -1, -5, -9, 7, 9, -2, -5, -15, -2], [11, -13, -12, -4, 2, -2, 13, 15, -4, -7, -3, 0], [0, -20, -4, -1, 7, -3, -3, -3, 2, -5, -7, -5], [-8, -8, 6, 0, -3, 8, 1, 2, -15, -12, 8, 9], [3, 11, -8, -5, 5, -11, -3, 7, 3, -3, 0, -5], [7, 21, -10, 6, 9, 1, 4, 1, 4, 9, -8, -3], [-26, 16, 14, 1, -15, 7, -15, -3, -8, -7, 7, 9], [-10, 3, 12, -8, -14, 12, -8, 4, -16, 2, 18, 3], [-9, 6, 3, -14, -2, 3, -7, 3, -5, -10, 10, 11], [-35, -9, 7, -4, -1, 9, 7, 0, -6, 2, 4, 10], [3, -8, -7, -2, 9, -8, 3, 15, 13, 7, -12, -6], [12, -10, -1, -7, -5, 3, 0, 2, -11, 2, 15, 2], [14, 2, 2, -7, 10, -5, 2, -1, -6, -7, 11, 5], [25, -4, -22, 21, 19, -8, 9, 11, 15, 5, -7, -7], [-6, 1, -8, 3, 11, -9, 5, 17, 8, -13, -4, -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_q();".
// 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_q(:prec:=12)
    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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-208, -576, 1284, 1556, -1463, -1643, 368, 616, 15, -94, -13, 5, 1]>,<5,R![312, -2320, 5038, -1577, -4199, 2939, 746, -982, 41, 121, -18, -5, 1]>,<7,R![3271, 13780, 17045, -401, -13108, -4934, 3070, 1780, -195, -217, -12, 8, 1]>,<13,R![784, -3200, -3476, 34252, -57109, 31671, 1566, -5825, 859, 294, -61, -4, 1]>],Snew);
    return Vf;
end function;