
// Make newform 6003.2.a.p in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6003_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_6003_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_6003_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 := [-64, -128, 352, 604, -717, -939, 750, 647, -420, -216, 124, 34, -18, -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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-128, 256, 604, -701, -939, 750, 647, -420, -216, 124, 34, -18, -2, 1], [112, -336, -604, 717, 939, -750, -647, 420, 216, -124, -34, 18, 2, -1], [8496, 10560, -37648, -30667, 63135, 25206, -46019, -7004, 15640, 108, -2450, 210, 142, -21], [-2124, -800, 12241, 3544, -21091, -1989, 15225, -664, -5152, 686, 808, -156, -47, 11], [-1604, -2620, 7661, 9877, -9660, -12843, 4986, 7444, -1196, -2090, 130, 278, -5, -14], [48, 3144, 5168, -16273, -19251, 25826, 19119, -17452, -7728, 5636, 1362, -858, -86, 49], [-6544, -3040, 49860, 2763, -97451, 15014, 74967, -19396, -26680, 8412, 4386, -1522, -266, 97], [-1266, -31, 7283, -1282, -12121, 3486, 8530, -2912, -2852, 1046, 446, -169, -26, 10], [-544, 961, 4020, -4510, -8027, 6727, 6257, -4419, -2231, 1407, 365, -212, -22, 12], [5088, 18072, -17164, -77057, 4761, 106310, 12435, -65388, -8832, 19748, 2002, -2858, -146, 157], [-6144, -18608, 8992, 85233, 21091, -119906, -33175, 74340, 15640, -22484, -2986, 3242, 198, -177]];
    Rf_basisdens := [1, 1, 1, 16, 16, 368, 92, 92, 184, 368, 46, 46, 368, 368];
    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_6003_a();"
function MakeCharacter_6003_a()
    N := 6003;
    order := 1;
    char_gens := [668, 3133, 4555];
    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_6003_a_Hecke(Kf)
    return MakeCharacter_6003_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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, -1, 0, 1, -1, 0, -1, 0, -1, 0, 0, -1, 0, -1], [0, 1, -1, 1, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0], [1, 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], [-3, 0, 0, -1, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0], [-1, -1, -1, 0, -2, 0, -1, 1, 0, -1, 0, 0, 0, 0], [3, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 1, 0], [1, 0, 1, 0, -1, -1, -1, 1, 1, 0, -1, -1, 0, -1], [2, 1, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, -1, 0], [2, -2, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, 1, -1], [3, -1, -1, 0, 0, -1, -1, 1, 0, -1, -1, 2, 0, 0], [-2, 0, 1, -1, 1, 2, 1, -1, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, 0, -1, 1, -2, -1, 0, -1, 1, 0, 0], [4, 1, -2, 1, -1, -1, -1, -1, 0, 0, -1, 0, 0, 1], [0, 3, -1, -1, 0, 1, 0, -1, 1, 0, -1, 1, -1, 1], [-5, 1, 0, 1, -1, 0, 0, 0, 1, 0, -1, -1, -2, -1], [0, -3, 1, 0, -2, 1, -1, -1, 0, -1, 1, -1, -1, -1], [-2, 3, -1, 0, 2, 2, 1, -3, 0, 0, 2, -2, -1, 0], [2, 3, 1, 0, 0, 1, 0, 1, 1, 1, -2, 0, 0, 2], [0, 0, 0, 0, -2, 0, -1, 1, -1, 1, 1, -1, 1, 0], [0, -2, 3, -1, -1, 2, 0, 1, 3, 0, 1, -2, -1, -1], [2, -1, 3, -2, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1], [-3, 1, 2, -2, 1, 1, 1, -4, -1, 1, 1, 0, 0, 1], [3, -1, 0, 0, -1, -1, 0, 2, -1, 0, 1, -2, 1, -1], [0, -1, -1, 1, 0, -1, 0, 1, -1, 0, -1, -1, 1, 1], [3, 5, -4, 2, 1, 1, 0, -1, 1, -2, -2, 0, -2, 0], [1, -3, -1, 0, -2, 2, 0, 0, 1, -1, 0, -3, -2, -4], [1, -3, -1, 1, 0, -3, -1, -3, -4, 1, 2, -1, 1, 0], [-1, 3, -1, 3, -2, 1, -2, 1, 2, 1, -3, -2, -3, -2], [-3, 2, -2, 0, 0, 2, 0, 0, 0, 0, 0, 1, 1, 1], [2, -2, 3, -2, -1, -1, -2, 0, -2, 0, 2, 0, 2, 1], [-1, 1, 0, 0, 3, 1, 1, -1, 0, -1, 2, -2, -2, 1], [5, 4, -4, -1, 2, -2, 2, 0, 0, 1, -1, 4, 2, 4], [0, 5, 0, -3, 1, 1, 3, 0, 1, 1, -1, 3, 1, 1], [4, 2, 2, 0, 0, 0, -2, 2, 0, 0, -1, 1, 1, 3], [4, -2, 1, 1, -1, 2, -2, -1, 1, -1, 0, 0, -2, -1], [1, -1, 1, -1, -2, 2, -1, -1, 0, 0, -1, -1, -1, -3], [0, 0, -3, 0, 1, 0, -2, -2, 0, 0, 0, 3, -1, 4], [-1, 0, 0, -2, 0, 0, 0, -1, 0, 0, -1, 0, -2, 2], [-5, 2, -4, 1, 0, 0, 0, 1, -2, 0, 0, 2, 1, 2], [-6, -2, -3, -1, -1, 0, 2, 0, 0, 1, 1, 0, 2, -1], [-1, -4, 4, 0, 2, 0, -1, -2, -1, -1, 2, 3, 1, 0], [-4, 1, 2, 1, -1, 1, -2, 3, 2, -1, 2, -2, 0, -2], [2, -1, -4, -3, -1, -2, 3, -1, 0, 0, -2, 2, -2, 1], [1, -1, -3, 2, -2, -1, -2, 3, 2, -2, -1, -3, -3, -2], [-1, 6, 1, 0, 3, 2, 1, -3, 2, -1, 1, -2, -2, -2], [2, -4, -2, 2, -4, 0, -1, 2, 1, -3, 0, 0, -2, -3], [1, 0, 2, -2, -2, 3, 0, -2, 1, 0, -1, 2, -2, 0], [-3, -1, 4, -1, 3, 3, 1, -4, 0, 0, 3, 1, 1, 2], [6, 2, 1, 1, 1, -2, -3, 0, -1, -1, 2, 1, 1, 0], [6, -1, 1, -3, 2, -3, 1, 1, -3, 0, 2, -1, 3, 1], [7, 1, 0, 2, 3, -1, 2, 0, -1, 3, -2, 0, 2, -1], [6, 3, 2, 2, 1, -1, -1, 0, 2, 0, 2, 1, 1, 1], [1, -8, 7, -2, -5, 0, -4, 2, -2, 0, 3, 0, 1, -4], [1, 0, 2, -3, 0, -2, 1, -1, -3, 1, 2, -2, 1, -3], [0, 4, 1, -6, -1, 1, 3, 0, 1, 1, 0, 1, 0, -2], [1, 5, 2, -3, 3, 3, 3, -1, 2, 2, 1, -1, -1, 4], [11, -2, 2, 2, -2, -2, -2, 8, 2, -1, 0, -1, 1, 0], [-1, -2, -4, 5, -2, -2, -1, 1, -1, 2, -3, 0, 0, 1], [-6, 0, 6, -2, 0, 2, 0, -4, 2, -1, 1, -3, -3, -2], [-4, -3, 4, -1, 1, 1, -1, 0, 1, -1, 0, 2, -2, 0], [-3, -2, -2, 1, 0, 1, -2, 3, -1, 2, 2, 1, 1, 2], [-2, 0, -2, 2, -2, 0, 0, -2, 0, 0, -1, -1, -1, -1], [8, -1, -2, 2, -3, -3, -4, 5, 0, -2, -2, 0, -2, -2], [-4, -4, 3, 0, 1, 2, 0, -1, -1, -1, 2, 0, 0, -2], [6, -5, 4, 0, -3, 2, -2, 3, 4, -2, 1, -4, -2, -2], [10, -2, -5, 0, -3, 0, -3, 4, 1, 0, -5, 3, 2, -1], [0, 1, 2, -3, 1, 1, 2, 1, 1, -2, 1, 4, 0, -1], [-7, -4, -2, -3, 0, 1, 2, -3, 2, 1, 5, 2, 1, 3], [2, 5, -1, 3, 0, 5, 2, 3, 8, -1, -6, -1, -3, 0], [4, -5, -1, 3, -2, -2, -3, 0, -4, 3, 0, -2, 1, -2], [-2, 1, 0, 2, -1, 3, 2, -2, 2, 0, 0, -4, -6, -4], [0, 9, 0, -2, 3, 1, -1, -1, 2, -1, 1, -3, 2, 3], [-3, 1, 0, 0, 1, 3, 0, -2, 2, 0, -3, 0, -4, 0], [4, -3, -4, 2, -1, 1, 1, -2, 0, -1, 0, -1, -3, -6], [-1, 0, 2, 5, -4, 1, -3, 3, 1, 1, -2, -6, 0, -6], [7, 0, -4, 5, 0, 0, -1, 0, 0, 2, 0, -6, 0, 1], [0, -4, 2, -1, 0, 0, -2, -1, 2, -1, 4, -4, -2, -2], [-1, 2, -2, 2, -2, -4, -2, 1, 0, -1, -2, 1, -1, 2], [3, -2, -4, 1, -4, 0, -3, 2, -2, -1, -1, -1, -1, -1], [6, -2, -2, -1, 2, 2, 1, -4, -4, -1, 0, 3, -1, -1], [1, 0, 6, -4, -2, 2, -1, 3, 2, -2, 1, -3, 1, -3], [-3, -1, -1, 0, -4, 3, -3, 3, 6, -1, -2, 1, -5, -1], [1, -3, 0, 1, 1, 3, 0, 0, 2, -4, 4, 3, -1, -2], [2, -2, 2, 4, 0, -3, -3, -3, -4, -1, -1, 1, 1, 0], [3, 1, -9, 5, 2, 3, 0, 1, 1, 2, -2, -3, 1, 3], [-5, 6, -3, -3, 3, 0, 6, -2, 0, 1, 2, -4, 0, -1], [5, -17, 3, 1, -4, -1, -4, -3, -6, -1, 3, -3, 1, -6], [3, -2, -5, 0, -3, -2, 1, 4, 3, -3, -6, -2, -4, 0], [-1, -6, 0, 0, 2, -3, 1, -1, 2, 2, -3, 2, 0, 0], [3, -7, -1, -1, -2, -5, -3, 6, 0, 0, -1, 4, 2, 2], [-4, 1, -4, 0, 1, 3, 3, -4, 2, -1, 0, 0, -3, -2], [-4, -1, 0, 1, 1, 1, -1, -6, 0, 1, -2, 2, -1, 3], [7, -4, -2, 6, 0, 2, -2, -3, -2, 0, -4, -3, -3, -3], [-3, 8, -5, 3, -1, 2, 0, -3, 1, -3, -2, 5, -1, 0], [-2, 2, 3, 6, 1, 0, -4, -2, -2, 0, -2, 4, -1, 2], [-1, 0, 2, 3, 2, 3, 2, -4, -2, 0, 1, -5, -3, -2], [6, 4, 1, -3, -1, 2, 5, -2, -1, 0, 1, 1, -2, 2], [-2, -7, 3, -1, -4, 1, -5, 3, -4, 2, -2, -1, 0, -2], [-1, -1, 1, -6, -6, 1, -4, 2, 2, -2, -1, -5, -1, 0], [1, 6, -6, 1, 0, 0, -2, 2, 0, -1, -1, 4, -2, 4], [6, -4, 1, 4, -3, 0, 0, 4, 2, 2, -1, -6, 0, -1], [12, -7, 6, -2, 1, -5, 1, 0, -4, 2, 3, 4, 4, 2], [4, 0, -2, 4, 0, -2, -3, 3, -1, -2, -6, -2, 2, -6], [-1, -2, -5, 2, 1, 2, 0, -1, 1, 3, -5, 4, 0, 2], [2, 4, 2, 0, 4, 6, 6, -3, 2, 3, 2, -2, 0, 3], [2, -8, 4, -3, 0, 0, 1, -1, 1, 1, 1, -4, -1, -1], [2, 8, -3, 0, 5, 1, 1, -6, -3, 0, 1, 2, 1, 2], [-12, -2, -2, 1, -8, 2, -4, 4, 1, -1, 1, -7, -2, -2], [6, 7, 2, -2, -1, 2, 2, -1, 4, 1, -4, 0, -1, 6], [-6, 2, -4, 4, 2, 4, 1, -5, -1, 1, 1, 0, 1, 2], [3, -3, -1, -1, 0, 3, 3, 4, 10, 0, -5, 0, -4, 0], [6, 5, -1, -2, 6, -5, 3, -2, 0, 0, 1, 7, 1, 4], [13, -9, 0, -1, 1, -2, 0, -4, -6, 0, 3, 1, 1, 0], [-1, -1, -1, 0, -2, -1, -3, 6, 3, -1, -1, -3, 3, -3], [1, 1, -4, -6, 3, -3, -1, -4, -1, -1, 2, 7, 1, 2], [10, -6, 2, 4, 0, -4, -5, 6, 0, 0, -1, -5, 2, 0], [0, 7, -7, 3, 4, 3, 2, -3, 4, -3, -2, 1, -3, 4], [6, -4, -2, 1, 4, 0, 4, -2, 0, 0, -2, 0, -2, 1], [-14, -5, 1, -3, -2, -1, -2, 1, 4, -1, 2, 0, 1, -2], [-4, 0, 0, 0, 0, 2, 3, 3, 4, 1, -1, 0, 2, 3], [13, 1, -8, -2, -7, -1, 0, 4, 1, 1, -9, 1, -1, 4], [-8, -4, 3, -8, -1, -2, 4, -4, -7, 3, 3, 6, 3, 3], [3, -5, 5, -4, 4, 3, 1, -5, 0, 3, 0, 5, -1, 3], [6, 1, 3, 6, -6, -1, -3, 10, 2, -1, -8, -2, -2, -4], [9, -4, -5, 2, 1, -3, -3, 4, 1, -1, 3, 0, 0, 2], [-14, -4, 4, 2, 0, 1, -2, 0, 1, -3, 3, -2, -1, -4], [-3, -1, 1, 2, 2, 2, 2, -2, 3, 3, 0, 2, 0, 2], [-3, 4, 10, -4, -2, -2, 0, 2, 3, -1, -1, 1, -3, 4], [-1, 11, -2, 6, 3, 0, 0, 2, -1, 1, -6, 3, 1, 0], [15, 1, -2, -1, 3, -3, 3, -5, -2, 2, -1, 8, 3, 6], [5, -4, -6, 2, -8, -4, -2, 6, 0, -2, 1, 1, 2, -6], [-2, -6, 5, -2, -7, -2, -7, 7, 4, 1, -3, 4, -3, 1], [4, -3, 0, -4, -3, -1, 3, -1, 0, 0, 2, 3, -1, 1], [3, -5, 2, -3, -1, -3, -3, 5, 0, -1, 2, 6, 4, 4], [0, 1, -1, -3, -4, -3, -5, 2, -1, 1, -3, -2, 4, 1], [7, -2, -4, 4, -4, -2, -4, 7, 3, -1, -5, 1, -3, -2], [1, -3, -7, 2, 4, -4, 4, -3, -1, 1, -1, 1, -1, 3], [-1, -2, 0, 6, -2, 2, -6, -1, -2, 0, 2, 1, -3, -3], [-3, -11, 5, -7, -10, -1, -8, 1, -2, -2, 2, 4, 2, 0], [13, 1, 6, 1, -3, -3, 1, 1, 3, 1, -2, -1, -1, 5], [-10, 4, 4, -1, 8, 4, 3, -3, 2, 4, 4, 3, 3, 4], [4, -3, 2, -2, -1, -1, 2, -3, -4, 0, 5, 1, 1, -1], [5, 0, 5, 1, 3, -2, 4, 1, 1, -2, 4, -1, 1, -3], [1, 1, 2, 0, 1, 5, 3, -3, 3, -2, 4, -1, -5, -7], [3, -5, 5, 2, 2, 2, 1, 5, -3, 1, 0, -4, 1, -3], [2, -15, 6, 3, -5, -1, -3, 0, -6, 0, 2, -1, -3, -8], [-3, 7, -3, 3, -2, -1, 3, 1, 1, 2, -5, -1, -1, 3], [-9, 3, -7, 1, 0, -2, -1, -1, 0, -2, -3, 4, -1, 3], [23, 0, 2, -3, 6, -1, 4, -6, -2, 2, -1, 5, 1, 6], [-1, -1, 0, 3, -1, -5, -3, -1, -11, 0, 3, -3, 4, -1], [4, 7, 0, 6, -5, 3, -5, 9, 5, -4, -1, -1, -1, -3], [-2, 8, -3, -2, 1, 0, 7, 2, 5, -1, -2, 5, 1, 5], [-1, -6, 6, 3, -4, -2, -5, -5, -4, 0, 3, -4, -3, -8], [-4, 3, 5, -9, -4, -3, -1, 3, -3, -2, 0, 5, 3, 1], [8, -6, -3, 5, -7, -2, -8, 4, 3, -2, -4, -1, -2, 0], [18, 3, -4, 7, -5, -1, 1, 10, 4, -4, -3, -4, -4, -1], [-5, 3, 6, -2, 1, -3, -2, -1, 2, -6, 6, 5, 1, -3], [5, 2, 3, -6, -3, -4, -2, 1, -2, 2, -4, 7, 6, 7], [7, 8, -2, 1, -4, -1, -5, -1, -2, -4, -1, 6, -2, 3], [-16, -1, 11, -3, 4, 1, 1, -3, -3, -2, 2, 7, 1, -1], [16, -1, 3, -3, -4, -5, -4, 5, -2, -3, -2, 1, 1, -4], [-10, 4, 6, 0, 0, 4, 2, -1, 4, -1, 2, -6, -4, 3], [4, 2, 7, -3, 5, 0, 1, -5, -4, 3, 5, -1, -1, 6], [-18, 0, 1, -3, 5, 6, 8, -5, 1, 2, 5, -3, -5, -1], [22, -7, 3, -3, 6, -1, 1, 0, -7, 2, 0, 3, 5, 1], [-2, -7, 3, 3, -2, -5, -2, 2, -8, 1, -5, -10, -2, -7], [10, 0, 5, -1, 7, -3, 3, 0, -1, -1, 0, 4, 0, 3], [12, 11, -8, 9, 1, 1, -1, 3, 8, -2, -8, -1, -5, 0], [3, -2, -7, 0, -1, 0, -1, 3, -1, 2, -3, 8, -1, 4], [-11, 3, 3, 0, 4, 1, -1, -4, -1, 3, 2, 8, 2, 8], [-10, -8, 5, -2, -7, 4, -4, -2, 4, -3, 1, 0, -2, 0], [2, -3, 0, 9, -1, 3, -3, -5, -3, 1, -1, -2, 1, 1], [7, -1, -3, 5, -4, 3, -6, 3, 0, 0, -4, -6, 0, -8], [3, 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-1], [-3, -3, 4, 2, 1, 7, -4, 9, 6, -2, 0, 7, 1, 9], [4, 0, -7, -3, -7, -4, 2, 5, 4, -4, -7, 6, 0, 2], [-4, 9, 1, 13, 2, 3, 3, -7, 0, 7, -3, -8, 1, -4], [25, -6, 4, -3, -10, -4, -10, 3, -2, -1, -3, 6, -6, 10], [-4, 9, 4, 10, -1, 3, -6, 5, 2, -3, -4, -7, 0, -5], [10, -20, 11, -4, 3, -2, -4, 2, -5, -1, 7, -3, 3, -3], [5, 15, -4, 5, 3, 1, 4, -1, 4, -1, 1, -7, -4, -4], [-4, 7, -9, 9, 0, 7, 0, -4, 6, 2, -2, -5, -5, -3], [-2, -11, -7, -8, -10, -9, -8, -1, -5, -2, -1, 14, 1, 8], [13, 9, 1, -2, 10, 7, 4, -6, -7, 1, 8, 2, 5, -3], [25, -20, 16, 7, -2, -6, -10, 7, -9, 2, 6, -6, 8, -7], [-10, 6, 7, 2, -1, 2, -3, -11, -3, 0, 10, -11, -3, -4], [10, -4, 10, 0, 10, -2, 5, -5, 0, 5, -3, -6, -4, 3], [3, 22, -5, -2, -5, -6, 4, 5, 3, -2, -12, 15, 3, 2], [19, -5, 3, 6, -8, -8, -7, 8, 6, 0, -6, 5, 2, -5], [4, 17, 5, 1, 4, 5, 1, -1, 5, -3, 4, -5, -7, 2], [3, 10, -10, 9, 0, 0, -3, 12, 8, -3, -12, -2, -4, -6], [12, 10, -3, -4, 13, -4, 10, -10, 0, -4, 12, 3, 5, 6], [12, -18, 6, 11, -12, -6, -14, 15, 4, -3, -8, -3, 0, -4], [-13, -16, 14, 2, 0, 6, -1, -2, -7, -6, 18, -1, -3, -11], [5, -21, -1, 5, -8, 7, -9, -2, 5, -3, -5, -5, -3, -6], [-10, 8, 8, -13, 2, 5, -1, 3, 12, -1, -9, 5, -1, 11], [-9, -3, 4, -3, -1, -5, -5, 3, 5, 3, 0, 3, 1, 3], [-12, 9, 1, 0, 0, 7, 7, -2, 2, 2, 1, -13, -5, -12], [13, 9, -10, 11, -3, -9, -5, 6, 2, 0, -11, 9, 1, 8], [-19, 10, 1, -4, 17, 4, 5, -13, -4, -2, 5, 10, 2, 5], [2, -12, 10, -1, -6, -4, -9, 6, -3, -1, 0, -8, -2, -4], [-7, 18, -3, -2, 3, -6, 6, 3, 2, -3, -1, 7, 1, 11], [0, 2, -17, 4, 1, -6, 5, -2, 9, 0, -3, -4, 0, 9], [-8, 2, -4, 1, -4, -4, 1, -5, 0, 0, 4, -3, -5, -8], [-27, -3, 6, -13, 11, 7, 10, -10, -4, 7, 8, -3, 1, 9], [28, -2, -10, -2, 6, -2, 9, -4, 5, -1, -2, 2, 0, 9], [-8, 14, -12, 5, 6, 8, 13, -1, -3, 6, -4, -13, -2, -1], [-2, -18, 11, 1, -7, -1, 2, 4, -9, 4, 10, -9, 9, -9], [6, -8, -6, -1, 2, 6, 7, 2, 0, 2, 1, 8, 2, 9], [15, 0, -18, 14, -4, 2, -5, 8, -3, 1, -12, -13, 1, -2], [-10, -3, -5, -5, -4, -3, 5, -15, -8, 0, -3, 1, 1, -5], [2, -6, 6, 5, 12, 1, 8, -11, -3, -1, 6, -3, 4, 2], [-2, 23, 0, -1, 1, 5, 3, -2, 11, -2, -9, 8, -5, 6], [20, 4, -14, -4, 4, 2, 8, 1, -2, 2, -6, 8, 2, 6], [-18, 2, -2, 0, 8, 10, 1, -7, -1, -1, 1, 13, -5, 4], [9, 0, -10, 7, 2, 4, -4, 3, 0, 6, -4, -3, -5, 0], [-4, 4, 0, -1, -4, -11, -6, 7, 4, 1, 0, -1, 5, 9], [14, 5, -10, 3, -13, 1, -5, 3, 5, -7, -11, -10, -9, -15], [7, -11, 20, -3, -1, 5, -6, -10, -9, 0, 10, 8, 4, -5], [-2, -5, -5, 2, -18, -6, -2, 8, 4, 0, 0, -11, -6, 1], [21, 14, 4, 7, 10, 2, -5, 0, -4, 2, 0, -7, 2, -3], [-4, 8, -3, -5, -1, -8, -6, -6, -6, -3, -3, 8, 6, 5], [29, -18, 1, 6, -3, 2, -1, 4, -13, 3, 1, 1, 11, -3], [13, 7, 5, -3, -2, -1, 7, -8, -4, 0, 8, 2, -7, -1], [-7, 15, -3, -2, -4, 7, 0, 7, 18, 0, -9, -3, -9, 8], [-9, -1, -1, -4, 0, 7, -8, 0, 10, -12, 1, -1, -3, -2], [6, 16, -7, -3, -1, 4, -1, -5, 9, -6, -4, 1, -1, 5], [5, 12, 1, -5, -1, 3, 0, 4, 9, -2, -7, -4, -3, 7], [-8, 13, 4, -6, 9, 0, 4, -4, 4, 9, 7, -2, 2, 9], [3, 3, 3, 16, 4, 4, 5, 7, 4, 3, 0, -16, 4, -10], [-13, 8, 8, -5, 0, 8, 11, 5, 7, -1, -2, 1, -3, -3], [-11, 6, -4, 0, 16, 4, 8, -14, 2, 3, 2, -3, 3, 6], [3, 10, -11, 9, 3, 2, -4, -5, -4, -3, -5, -3, -7, -8], [6, -4, 10, -16, 0, 1, -4, -8, -8, -5, 11, 14, 0, 1], [3, -6, -3, -2, 9, -6, 8, -6, -12, 1, 0, 2, 4, 8], [-15, 14, 3, -16, 5, -4, 9, 3, -1, 2, 3, 4, 1, 12], [12, -17, 4, -7, 1, 2, 2, -6, -9, -2, 8, -5, -2, -7], [-25, 15, -10, -5, -7, 3, 2, 6, 7, 3, -8, -1, -2, 14], [11, 8, 5, 8, -1, 0, 2, 1, -10, 3, 5, -5, 5, -3], [-23, -1, -2, 2, -11, 5, -7, 1, 4, -3, 14, -10, -4, -9], [-5, -4, -19, 6, 5, -4, 9, -9, -7, 6, 3, 3, -3, 4], [-18, -3, 0, -2, -7, -4, -9, 5, -8, -3, 3, 4, 1, 4], [12, 1, -7, -3, -4, 5, 2, -5, -4, -5, 0, -8, -3, -2], [-8, -4, -9, 3, -3, -2, -3, -4, -5, -7, 6, 5, -1, 4], [2, -11, 3, -11, -6, -7, -1, -2, 0, 4, -8, 8, 0, 10], [-13, 18, -7, -5, 1, -10, 1, 9, -1, 2, -5, 11, 3, 11], [-7, -18, -6, 7, -8, -2, -6, 14, 1, -3, -4, 0, 4, 0], [1, 13, -7, 8, 6, 6, 3, -2, 16, -1, -7, 4, -7, 3], [8, 2, -16, 14, -14, 0, -14, 10, 2, -2, -11, -5, -9, -13], [4, -16, 19, -7, -5, 6, -4, 1, 2, -4, 12, -3, -1, -13], [14, -9, 7, -10, -14, -7, 2, -2, 0, 0, 0, 3, 1, -6], [-9, -17, 6, -9, -9, -9, -5, 9, -5, -1, 10, 1, 5, -1], [-13, 0, 7, -8, 7, -6, 4, -6, -10, 6, 9, -6, 5, 0], [6, 9, 0, -7, 5, 0, 9, 2, 9, 5, 0, 3, -3, 5], [8, -6, -7, 9, -9, 2, -6, 1, -1, -3, -8, 4, -4, -1], [-8, -2, -14, -4, 0, 0, 3, -2, 3, -7, 6, 14, -8, 5], [12, 6, -16, -5, -2, -4, 9, 4, 3, 0, -6, 10, -6, 1], [18, 5, -6, 4, 5, 8, 9, -7, 0, 2, -4, 1, -2, 8], [-2, 9, 10, 0, 5, 3, 10, -8, -8, 10, 2, -4, 0, 8], [-21, -9, 5, 2, -6, 11, -6, -11, -1, 5, 5, -19, -4, -14], [3, -9, 1, 5, -2, -8, -10, 1, -3, -2, 2, 12, -2, 0], [-3, -13, -2, 5, -3, 3, -6, -2, 4, 1, -2, 7, -5, 7], [10, -17, -1, -11, -6, -7, -3, 4, 1, -5, 4, -13, 1, 0], [-6, 2, -4, 5, -4, 0, 4, 2, 13, -1, -6, -15, -7, -5], [-6, 0, 5, -5, 3, 4, -6, -2, 1, 4, 12, -4, 6, 0], [-16, -19, 1, 2, -2, 3, 6, -1, -1, 3, 2, -2, -6, -4], [-18, 1, -1, 2, 2, -1, 3, 1, -6, -2, 0, -2, 6, -5], [7, 7, 4, 0, 9, 5, 4, -8, 0, 1, -6, 7, 3, 4], [20, 1, -10, 4, 1, 1, 7, -4, 0, 3, -4, 7, -1, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6003_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_6003_2_a_p(:prec:=14)
    chi := MakeCharacter_6003_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_6003_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_6003_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-64, -128, 352, 604, -717, -939, 750, 647, -420, -216, 124, 34, -18, -2, 1]>,<5,R![-128, 0, 1920, 1904, -5256, -5532, 5276, 4192, -2540, -1025, 479, 96, -37, -3, 1]>],Snew);
    return Vf;
end function;