
// Make newform 6012.2.a.k in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6012_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_6012_2_a_k();".
// 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_6012_2_a_k();".
// 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 := [9, 126, 423, 192, -593, -340, 211, 82, -26, -4, 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], [66348, 48114, -469017, 319314, 584470, -197274, -136919, 32441, 6681, -1438], [-151056, 435294, 837054, -561669, -710756, 234612, 148276, -34945, -6987, 1499], [-239175, -1880118, -463230, 2390865, 832712, -752934, -204046, 81184, 9945, -2840], [-92196, -276453, -257619, 436038, 438577, -188846, -105787, 27571, 5185, -1167], [29004, 168813, 119881, -254713, -154395, 91532, 35355, -11178, -1700, 426], [60750, 460620, 651792, -667089, -775366, 291942, 174848, -43331, -8415, 1849], [204129, 1098198, 595053, -1490655, -846911, 521801, 198654, -63438, -9622, 2419], [679113, 4422843, 4419333, -6555282, -5573576, 2574117, 1274200, -349141, -61455, 14219]];
    Rf_basisdens := [1, 1, 44982, 44982, 44982, 14994, 4998, 14994, 14994, 44982];
    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_6012_a();"
function MakeCharacter_6012_a()
    N := 6012;
    order := 1;
    char_gens := [3007, 3341, 4681];
    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_6012_a_Hecke(Kf)
    return MakeCharacter_6012_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, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, -1, 0, 1, 0, 1, 0, -1], [0, -1, 0, 0, 0, 1, -1, 1, 0, 0], [3, 0, 0, 0, -1, -1, 1, 1, -1, -1], [1, 0, -1, 0, 0, 0, -1, 1, 1, 0], [0, -1, 1, 0, 0, 0, 1, 0, -1, -1], [0, -1, 0, 0, -1, 0, 0, -1, 0, 1], [-3, 1, -1, 0, 2, 2, 0, 0, 1, 1], [2, -1, -1, 0, 0, 1, -1, 0, 2, 0], [5, 1, -1, 1, 0, 0, 0, -1, 0, 1], [3, 0, 0, -1, -1, -1, 0, 0, -2, -1], [1, 0, -2, 1, -1, 1, 0, 0, 0, 1], [-2, 2, 1, 0, 2, -1, 0, -3, -1, 1], [2, -2, -2, 2, 0, 0, 0, 2, 0, -1], [4, -3, -1, -1, -1, -1, 0, 0, 1, -2], [2, 1, 0, 2, 1, 0, 0, -2, -1, 2], [0, -1, -1, 0, 0, 1, -1, 0, 0, 0], [4, 1, 1, 0, 1, -1, 1, -1, 0, 0], [1, 0, 0, 0, -3, -2, 3, 1, -1, -1], [-1, -3, -1, -1, -2, -2, -1, 1, 1, -1], [4, 1, 3, -1, 0, -1, 1, -2, -2, 2], [-6, 3, 1, 2, 2, 1, 1, -1, 1, 2], [4, -1, 1, 0, 1, 0, 1, 0, 0, -1], [3, 0, 0, 0, -1, -2, 2, 2, 0, -2], [1, -3, -1, -1, -1, 0, 0, -1, -1, 1], [3, 0, -1, 2, -2, -3, 2, 1, -1, -1], [8, -1, -1, 2, 2, -1, 0, -1, -1, -2], [0, -2, 1, 0, 2, 0, -3, 1, -1, -1], [8, 1, -4, 4, -1, 0, 4, 2, -1, 0], [8, -3, -1, -1, -2, 0, 0, 2, 1, -1], [-1, -2, 2, -4, -1, 2, -4, 0, 0, -1], [3, -1, -3, -2, -2, 1, 0, 1, -1, -1], [1, -5, -1, -2, -3, -1, 1, 3, -1, -2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 1, -1, 0, -2, 2, 1, 1], [7, -1, 1, 0, -1, -2, 1, -2, -2, 0], [1, -1, 1, -3, -2, 1, 2, -2, -3, 1], [7, -1, -2, 1, -1, 2, 2, 3, 4, 0], [-1, -3, 1, -4, -1, 1, -1, 0, 0, 0], [1, 1, -3, 0, 1, 3, -2, 0, 5, 0], [4, 0, 0, -1, -2, 0, 0, 3, 0, -2], [-2, 3, -2, 3, 4, 1, 1, 0, 2, 0], [-5, -3, 0, -3, 2, 2, -4, -3, 0, 0], [9, -3, -1, -2, -3, -1, 2, 0, -5, -2], [3, -1, 0, -4, 0, 2, 0, 2, -2, -2], [-2, -4, 0, -3, -2, 2, 2, 2, 1, -1], [2, 1, 0, 0, 3, 4, -2, -3, -1, 4], [-3, 4, -1, -2, -2, 1, 3, 0, 1, 2], [2, 2, -1, 1, -2, -2, 5, -1, -2, 0], [6, -2, 2, 0, 0, -1, 1, 0, 1, -3], [6, 3, -1, 2, 0, -2, 4, -1, -5, 1], [-11, 3, 1, 2, 4, 1, -1, -2, 2, 2], [-1, -5, 3, -6, -1, 1, -3, 0, 2, -1], [-3, 1, 2, -4, 0, -2, -2, -4, -3, 1], [7, -3, -3, 0, -3, -2, 1, 0, -2, -6], [-5, -1, -1, 0, 0, 3, -2, 3, 5, 1], [1, 0, 2, -1, 6, 2, -2, -2, 5, 2], [0, 5, 2, 4, 3, -2, 2, 1, 2, 3], [4, 0, 4, 1, 0, -2, 4, -2, -1, -1], [-6, 3, -1, -2, 2, 3, -1, -1, 3, 1], [2, 2, 4, 1, 2, 0, 0, -2, -5, 1], [5, -4, 1, -4, -8, -7, 3, 3, -6, -7], [7, -7, -2, -2, -4, -2, -2, 4, -2, -2], [11, -1, -1, 2, -2, -1, -1, 4, 2, 0], [5, -1, -1, -2, -1, -5, 5, 3, -1, -4], [-2, 8, -1, 4, 4, 4, 3, -3, 1, 5], [7, 1, 2, -2, 1, 0, 0, 1, -1, 1], [-3, 1, -2, 2, 4, 4, -2, -4, 2, 4], [1, -4, 0, -6, -3, 0, -4, 1, 1, -3], [-5, -1, -4, 0, 0, 2, 0, 3, -4, 1], [12, 4, -1, 7, 1, 0, 0, 0, 1, 4], [4, -1, 5, -4, -4, -3, 0, -1, -3, 0], [0, -3, -2, -1, -2, 2, 2, 5, 2, -2], [-3, -3, 1, -6, -1, 0, -3, 0, 0, -3], [-4, -1, -1, -1, 6, 4, 3, 3, 3, -3], [10, 0, -3, 0, 2, 4, -5, 1, 3, 1], [-1, 1, -4, 1, 0, 0, 0, -1, 2, 4], [3, -1, 3, -1, -3, 0, -3, 0, 1, 3], [-2, 1, 0, 8, 6, 4, 0, -4, 2, 6], [-3, -2, 3, -4, 0, 1, 3, 0, 3, -2], [9, 5, -1, 1, 0, 0, 1, -7, 0, 7], [-4, 1, -2, 1, -3, -4, 2, 1, 0, -3], [0, 2, 1, 2, 5, 1, -3, -1, 2, 3], [7, -3, -9, -1, 1, 4, -3, 2, 1, -1], [7, 1, 2, 2, 4, -4, -4, 0, 1, -3], [-4, 1, 3, -1, 4, 0, 0, -3, 4, 2], [-5, 5, 3, 1, 6, 0, 1, 1, 2, -1], [3, 3, -3, 2, -1, 1, -3, 0, 2, 2], [2, 2, 0, -2, 0, 2, -2, 0, 2, 4], [-5, 6, -6, 6, 1, 0, 0, 1, 3, 2], [3, 1, -2, 0, -1, 2, 2, 7, -3, -3], [-9, 7, 0, 3, 0, -4, 5, -2, 1, 2], [-10, -3, 3, -2, -5, 1, -1, 5, -4, -2], [-8, 6, 3, -2, 2, 0, 1, -2, 5, -1], [-3, 3, -3, 4, 3, 5, 3, 2, 4, 1], [5, 1, 0, -2, -2, -2, 4, 5, -2, 0], [7, -3, -3, 5, -1, 2, 1, 2, -3, 3], [5, -5, -4, -6, -3, 2, -4, 5, 1, -5], [-11, 3, -1, 0, 4, 7, 1, 0, 2, 0], [-5, 7, 2, 6, 6, 0, -2, -9, -2, 6], [22, -4, 1, 1, -2, -5, 2, -4, -5, -5], [7, -4, 2, -7, 2, 5, -7, 2, 1, -1], [5, -2, 4, -1, -3, -6, -1, 0, -3, -6], [1, -11, 0, -3, -1, 2, -2, 1, -2, -4], [4, 2, -5, 11, 2, 0, -1, 3, -1, 3], [-1, -2, 8, -4, -1, -4, 2, -4, -2, 0], [13, -9, -1, 0, -4, 0, -5, 4, -1, -7], [-16, 2, 2, 2, 3, -2, 5, -3, -3, 0], [-2, -1, 3, 0, 0, 1, -1, 6, -2, -4], [-9, 2, 3, -5, 4, 2, -3, -3, -1, 1], [-9, 5, 1, 2, 2, -2, -6, -2, 5, 3], [-5, 5, 5, 4, 3, -3, -3, -8, -2, 2], [5, -3, 1, -2, 0, -2, 2, 4, 1, -3], [12, -4, -1, -1, 3, 0, -4, 4, 5, 0], [-3, 2, -4, 2, 1, 0, 2, 1, 8, -2], [3, -4, -2, 0, 1, 2, -4, 4, 8, -1], [1, -5, 3, -4, -5, 1, 3, 3, -3, -3], [0, -12, -1, -3, -3, 4, -4, 2, 1, -6], [25, -4, -2, 0, -8, -8, 1, 5, -3, -4], [8, -1, 2, -6, -3, -4, 0, -4, 1, -5], [-9, 6, 3, -2, 4, 2, -3, -5, -3, 2], [-6, 5, 0, -4, 1, 2, 0, 3, -2, 5], [-10, -4, 0, -1, -3, 1, -1, -1, -5, 1], [-4, -2, -2, 0, -6, 0, 4, 0, -8, 1], [-11, 10, 5, 2, 4, -2, 3, -4, -7, 3], [-8, 2, 2, -4, 6, 0, 6, -4, -2, -1], [-2, 5, 0, 1, 11, 4, 2, -1, 6, 1], [4, 0, 7, -2, 1, -3, -3, -6, -5, 0], [-11, 4, -4, 5, 2, 1, -1, -8, 4, 6], [-9, 8, 2, 3, 7, 4, -1, -7, 4, 6], [-5, 2, 5, 2, 2, -1, -5, 0, 3, 2], [-15, -2, 2, -3, 1, 2, -1, 0, 5, 4], [-15, 1, -3, -2, 3, -1, 3, 0, 0, -2], [10, 1, 3, 3, -3, -4, 7, -2, -8, 1], [-13, -6, 7, -3, 0, -4, -7, -3, 0, -3], [-1, 5, 9, -2, -1, -5, 3, -2, -6, -1], [8, 3, -3, 4, -3, -5, 3, 5, 0, 2], [6, -3, 7, -6, -6, -1, -3, 1, -3, 1], [-7, -6, -4, 4, 2, 11, -2, -1, 0, 5], [16, -8, -2, 0, -2, 2, -4, 3, -1, -3], [-8, -5, 1, -2, 1, 6, -11, 2, 4, -3], [6, 5, 5, 2, 4, 0, -1, -2, 3, 3], [1, -15, 2, -4, -3, -2, -6, -1, -3, -3], [-2, 0, 5, -2, 2, 1, -2, -11, -3, 1], [1, 3, 3, 8, -1, -8, 5, -4, -2, 2], [-9, 6, 10, -9, 0, -5, 1, -6, -2, 2], [-16, 4, 0, 8, 6, -1, 1, -4, -1, 5], [5, -6, -2, -4, -4, -2, 0, 0, -2, 4], [-1, -6, 0, -10, 1, 7, -1, 1, 1, -3], [-10, 2, -1, 1, 2, 0, -1, -3, 3, -3], [19, -3, -3, 9, -3, -12, 5, 2, -9, -3], [-12, 5, -3, 5, 8, 1, 2, 6, 7, 1], [-9, -7, 2, -7, -4, 2, 3, 2, -7, -6], [-9, -5, 1, 2, 1, 5, -5, 3, 3, 1], [5, -1, 1, 2, -6, 0, 2, 10, 1, -5], [8, -2, -1, -1, -2, 0, -5, 1, -2, 2], [-19, 4, 2, 4, 3, -4, 2, 1, 2, 4], [7, 4, 3, -2, 0, -3, -4, 3, -1, -3], [4, -4, 2, 0, 1, 1, -2, -8, 0, 5], [-16, 11, 3, -2, 5, 0, 0, -1, 1, 1], [5, 7, -7, 3, 2, 3, 0, -5, 7, 4], [-8, 8, 5, 0, 9, 5, 1, -5, 2, 7], [-13, 6, -3, -3, 6, 3, 6, -4, -1, 5], [2, 5, -8, 2, -4, 0, 2, 0, -2, -2], [1, -2, 6, -8, -1, 0, 5, 1, -1, -5], [-5, -4, 6, -6, 0, -1, -2, 3, -10, -3], [-2, 1, -10, 12, 8, 5, -3, 3, 10, 0], [13, -3, -11, 2, 1, 6, 3, 4, 0, 0], [-1, -6, -3, -2, -4, 3, 6, 7, 1, -7], [0, 0, 0, 2, 0, -2, 8, -4, -4, 1], [-15, -6, -6, 4, -3, 2, -5, 3, 1, 5], [-2, 4, -2, -3, -4, -4, 2, 5, -4, 0], [3, 5, 1, 11, 5, -5, 1, -11, 0, 8], [1, -2, -7, -6, 0, 2, 1, 4, 3, 1], [-13, -3, 4, -2, 0, 4, 2, 2, -5, 1], [11, -9, -1, -1, 2, 0, -1, 3, 1, -1], [2, 4, 4, 6, 3, -4, -5, 1, -1, -6], [-8, -8, 7, -10, -1, 1, -1, -4, -3, 0], [-21, -4, 10, -8, -3, -9, -1, -7, -5, -5], [-2, 6, 6, -4, 4, 4, -10, -8, 6, 4], [-26, 12, 4, 9, 10, 2, 2, -10, -1, 5], [-14, -8, 7, -5, -6, 0, -1, 1, -9, -3], [-5, 3, 3, -2, 6, -2, -1, 2, -3, 1], [-11, -1, -3, -5, 0, 4, 1, 3, -5, -1], [9, -4, 0, 4, -4, -8, -1, -5, -7, 2], [10, 1, 3, 6, -8, -5, 1, 3, 1, 0], [10, 8, 4, -4, -5, -6, 5, 4, -3, -4], [-12, 8, 2, -4, 4, -3, -3, -9, 2, 8], [8, -7, 0, -4, -9, 2, -6, 9, 5, -4], [9, -8, -3, -2, -13, -6, 2, 4, 0, -2], [12, -1, -7, 16, 0, -4, 6, 1, -5, 1], [2, 7, 6, -1, -2, -7, -1, 0, -2, -4], [-4, -1, -9, -5, 0, 2, 7, 3, 5, -3], [-18, -1, 3, 1, -2, 2, 0, -5, 4, 4], [-15, 3, -8, -3, 6, 2, -3, 6, 3, 0], [2, 4, 4, -7, -7, -1, 1, 5, -5, -5], [-3, -2, 1, 6, 0, -4, -1, 8, 3, -9], [3, -5, 11, -6, -1, -5, 3, -10, 0, 0], [19, 8, -4, 8, 5, 2, 0, -3, 5, 10], [-7, -8, 6, -4, -6, -1, 0, 3, -6, -7], [-8, 3, 1, -6, -8, -4, 2, -1, -1, 1], [13, 2, 9, 0, -4, -15, 7, -3, -8, 0], [5, 4, 10, 0, -3, 2, 1, -1, -7, 1], [-9, 0, -6, 1, 0, 2, -2, 0, 4, 0], [1, 11, 6, 8, 8, -1, -1, -7, 8, 8], [-1, 5, 10, -4, 6, 3, 1, -7, 4, 0], [3, -2, 4, -9, 1, 2, -1, 9, 4, -6], [3, 4, 11, 4, 2, -11, 9, -4, -13, 0], [-17, 2, 2, 4, 8, 0, 1, -5, 3, 6], [0, 8, -5, 3, -2, 0, 7, 1, 2, -4], [5, -7, -4, 7, -2, 0, 0, 5, 4, 4], [-1, 3, -1, 4, -1, -3, -3, 3, -1, -1], [0, -2, -8, 1, -3, -2, 1, 11, -4, -9], [13, -8, 10, -4, -11, -10, -3, -5, -7, -3], [10, 0, -2, 0, 4, 1, -7, -2, 1, -7], [-12, 9, 1, 2, 0, 1, 7, 0, -4, 10], [11, -9, -3, 3, -8, -4, 5, -3, -9, -3], [-4, 0, -8, 2, 0, 3, 5, 8, 1, 5], [-18, 12, 6, 8, 4, -7, 3, -7, 2, 10], [-5, -2, -6, -6, -6, 3, 6, 5, 0, -9], [1, -11, 9, -8, -4, -8, -1, -8, -5, -3], [-4, 1, 3, 0, -2, -6, 10, -3, -7, -7], [-9, -2, -10, 3, 0, 8, 0, 8, 8, -4], [-22, 11, -6, -4, 3, 8, 4, -2, 1, 4], [6, 8, 7, 13, 8, 0, 7, -3, 0, 12], [1, -9, -5, 0, 1, 7, 1, 1, -5, -1], [-14, -2, 8, 2, 2, -6, -8, -7, 5, 5], [-9, 0, -7, 0, 10, 9, -5, 0, 3, 3], [-7, -3, -5, -1, -7, 5, -3, 13, 6, 0], [5, -1, -6, 8, -6, -3, 11, 3, -2, -4], [4, 8, 2, 7, 5, -1, 7, -1, 3, 9], [-8, 9, 0, -4, 9, 6, -6, 5, 8, 5], [6, -1, -5, 2, 0, 0, -6, 3, 3, -3], [6, 19, -5, 8, 4, 1, 0, 3, 3, 4], [-6, 2, 4, 5, 9, -4, -3, 1, 2, 1], [30, -7, -1, 3, -2, -2, 6, -4, -9, -5], [-32, 10, 4, -7, 6, 14, -4, -7, 2, 4], [-14, 2, 1, -2, 5, 3, -5, -2, 5, -2], [5, -4, -1, -10, -4, -1, -9, 1, -2, 0], [-3, -11, 4, -10, -4, 0, -1, -6, -4, 3], [8, -7, -6, 13, -6, -4, 1, 10, 5, -3], [-13, 2, -4, 1, 8, 8, 6, -1, 6, 6], [-2, -8, 10, -11, 0, 8, -5, 1, 1, 0], [-17, 5, 10, -1, 4, 0, -4, -11, 10, 6], [-15, 7, 3, 7, 0, -3, -4, -5, -1, 12], [-4, 0, 2, 4, 4, 5, -3, -1, 6, 0], [-17, 2, -2, 1, -4, 4, -8, 6, 1, -2], [5, -6, -4, 10, 7, -2, 0, -7, 0, 8], [21, -2, 4, 6, -1, 0, -2, -4, -4, 3], [2, -8, -2, 2, 2, 9, -9, 4, 9, 7], [-7, 2, 7, 0, -1, -8, 8, -8, -4, 4], [12, -6, 6, 2, 4, -4, -10, 0, 0, -2], [5, 3, -8, 1, 2, 1, 5, 4, -6, -6], [-8, -4, 4, -7, -2, 2, -7, 5, 9, 2], [5, 1, 5, -10, 0, 1, -8, -1, 1, 5], [0, -1, 7, 2, 6, 7, -11, -9, 1, 4], [-9, -4, -2, -3, 1, 4, 5, 2, -5, 0], [-10, 0, -9, 2, -3, 1, -5, 1, 6, -3], [-1, 7, -5, 12, 7, 5, 5, 2, 10, -3], [-7, 7, 0, -4, -2, 0, -5, 2, 0, 3], [9, 2, 10, -7, 0, -5, -3, -6, -2, -2], [-16, 3, 17, -7, 6, -3, -5, -14, 4, 0], [9, -15, 3, -7, -1, -3, -5, -3, 6, 0], [-4, 5, 6, -2, 3, -2, -2, -6, 3, 5], [26, 6, -5, 9, 1, -4, 12, 8, -9, -6], [15, 3, -10, 8, -2, -4, 10, 0, -3, 0], [-27, -4, 4, -2, 2, 6, 0, 0, 6, 4], [-8, -2, 9, 0, -5, 1, -1, 7, -2, 1], [18, -6, -7, -4, -6, 4, 5, 10, -1, -9], [-2, 3, 1, 6, -6, -1, -4, -1, -3, 7], [6, 1, -1, 6, 0, 5, 9, 7, 5, 5], [12, -2, 0, -6, 5, 4, -7, 4, 3, 2], [28, -6, -9, 4, -9, 3, 7, 5, -6, -8], [6, 3, -7, 2, -6, -7, 9, 3, -7, -11], [-2, -4, 2, -4, -5, 4, -11, 3, 3, 6], [-5, 9, -3, 2, 3, -5, 8, 6, 1, -2], [-15, 0, 0, -4, -3, 2, 8, 5, 7, 1], [15, 0, 14, -4, 3, 0, 2, -7, -2, 4], [-5, -7, -3, 8, -10, -6, 9, 8, -1, -7], [12, 3, 3, 4, 0, 1, 3, 6, -8, -6], [-16, 6, -2, 2, 13, 9, 0, -10, 8, 17], [16, 5, 3, 1, 4, -7, -4, -10, -1, 7], [28, -7, 1, 0, 3, 5, -7, 1, 0, -6], [-5, 2, -1, 5, 4, 4, -9, -3, 1, 3], [-32, 0, 10, -17, 2, 1, -6, -5, 2, -1], [7, 13, 9, 14, 16, -4, 4, -17, 3, 14], [-19, 0, 0, 5, 13, 6, 3, 0, 5, 6], [31, 8, -7, 20, -1, -6, 8, 8, 6, 2], [-4, 4, 0, -2, 11, 6, 1, 3, 13, 0], [-11, 8, -5, 13, 6, 2, 3, 3, -7, 7], [22, -8, 6, -5, -16, -14, 2, -6, -13, -1], [5, -4, 4, 8, -2, -3, 5, 1, -6, 0], [20, -1, 3, 4, -6, 0, 3, -5, -12, 1], [-23, -3, -5, -7, 0, 9, 2, -3, 5, 2], [4, -4, 2, -8, -12, 0, 0, 14, -4, -14], [25, -14, -7, 2, -4, -11, 5, 10, -5, -6], [5, 0, -8, 7, -7, 4, 13, 10, -5, -8], [5, 4, -1, 4, -12, -5, 4, 1, 1, 7], [2, 10, 2, 2, 10, 1, -5, -18, 3, 15], [-2, 10, -2, 3, 8, 8, 6, 3, 6, 2], [-19, 6, -6, 0, 7, 10, 0, -1, -1, 8], [24, -1, -1, -11, -5, -6, 17, 2, -8, -3], [1, -7, 1, 5, -2, -8, -3, -5, -1, 3], [-7, 1, -3, -5, 2, 3, -8, -13, 1, 12], [3, 11, 2, 1, 10, 2, 0, -5, 4, 10], [-10, -2, 8, 7, 5, 1, 1, -5, -3, 1], [-8, 11, -4, 3, 0, -10, 3, 0, 1, 1], [20, -2, 4, 3, -3, -4, 5, 1, -8, -7], [-3, 4, 5, 4, 4, -5, 3, -1, 2, 4], [1, 3, 3, 3, 2, -1, -8, 7, 7, -6], [6, -9, -6, -3, -8, -3, -3, 4, 0, 8], [2, -4, -8, -9, -4, 8, -5, 1, 7, 0], [10, -5, -11, 11, -7, 4, 9, 8, 1, -6], [1, 20, -1, 10, 14, 7, 5, -6, -1, 5], [-20, 4, -2, 7, 3, -2, 0, -5, 8, -2], [14, -3, -6, 3, 2, 8, -6, -1, 12, 10], [-10, 15, -2, -1, 2, -12, 12, 1, -4, -2], [5, 8, -7, 10, -3, -6, 0, -8, -10, 12], [0, 8, -10, 6, -8, -4, 8, 1, 1, 3], [-1, -2, -12, 6, -3, 1, 13, 9, -3, -1], [-2, -6, -10, 3, -8, 4, 10, 13, 0, -4], [8, -1, -6, 2, -8, -9, -1, -3, -8, -8], [-13, 9, 0, 3, 0, -3, -3, -12, 4, 6], [-3, 16, -7, 2, 7, 10, 4, 8, 6, 8], [9, -19, 8, -10, -4, 8, -9, 2, -2, -13], [-4, 9, -2, 13, 6, -7, 9, -8, -2, 4], [22, 5, 1, -2, 2, 6, -4, -5, 1, 3], [-12, 4, 3, 2, 9, 1, -5, -5, 12, 0], [7, 2, -2, -11, -2, -2, 2, -8, 3, -8], [7, -16, 0, -14, -11, -2, -8, 1, -11, -5], [25, -18, -5, -2, -1, -6, -2, 8, 4, -4], [-8, 9, 3, -8, -2, -9, 14, -5, 1, -3], [-1, 3, 11, 0, 5, 5, -12, -6, 3, 14], [5, 3, -4, 16, 10, 4, 2, -1, 10, 12], [26, -18, -10, -9, -4, 0, -8, 5, -8, -4], [12, -4, 5, -4, -12, -11, 2, 7, -1, -5], [-9, 9, -11, 3, -8, 0, 12, 9, -2, -3], [18, -1, 5, -1, 9, -5, -8, -8, 7, 4], [8, 0, 1, -10, 1, 3, -11, 6, 1, -1], [23, -2, 0, -10, -4, 0, 10, 8, 2, -5], [-14, -1, 4, -10, -9, -4, -4, -5, -11, 0], [-15, 6, 0, -1, 6, 10, -6, -1, 8, 12], [12, -2, 1, 7, -8, -14, 1, 5, 5, -1], [33, 12, -6, 8, -2, -2, 5, 7, 3, 8], [-4, -5, -4, 3, 6, 8, -9, 4, 13, -5], [-5, -10, -12, 0, -4, 11, 1, 9, 2, -6], [-11, 16, 2, 12, 17, 2, 8, -1, 5, 3], [38, 10, -4, 15, 4, -3, 14, 5, 2, -1], [7, -9, -12, 0, -8, 4, 2, 20, -3, -2], [-8, 7, 10, -6, 9, 4, -2, -16, -7, -2], [-6, -2, 2, -7, 3, -2, -8, -11, -4, -4], [-7, 3, 4, 4, 9, 8, -12, -7, 7, 9], [23, -1, -5, 15, -1, -7, 5, 13, 4, -2], [13, -7, -5, -11, 2, 15, -18, 5, 19, -2], [12, 5, 1, 6, -6, -4, 2, 5, 3, 1], [3, -13, 9, -7, -2, -3, 0, -4, 1, -3], [6, 5, 5, 1, 9, -1, -2, 4, -5, -4], [-25, -1, 3, 3, 2, 11, -18, -1, 7, 4], [33, -9, -6, 7, -9, -2, -8, 13, 2, -6], [-8, 12, 13, -12, -5, -3, -7, -2, -1, -5], [11, -8, -1, -8, -10, 5, 3, 13, -8, -18], [5, -3, -3, 12, -3, -4, 7, -2, 2, 7], [-13, 3, -3, -12, 3, 3, 3, 3, -3, 0], [-29, 17, 16, -12, 10, 9, -9, -13, 10, 12], [0, -11, 5, -11, -8, -5, -5, -2, -14, -2], [5, 6, 12, -10, 1, -4, -8, -2, -4, 7], [24, -11, -5, -1, 0, 0, 2, 15, 4, -8], [19, 1, 1, 1, -1, 2, 13, 4, -13, -1], [-7, 25, 3, 14, 19, 3, 3, -18, 6, 19], [28, 3, -13, 6, -6, -5, 5, 1, -5, -1], [-12, 6, 2, -2, 5, 2, -5, -12, 11, 4], [3, -14, 2, -14, -8, -2, -7, -7, -1, -8], [-7, -9, 11, -5, -6, -11, -4, -8, -7, 1], [-14, 12, 0, 16, 6, -12, 10, -10, -8, 10], [-19, -4, 9, -18, -2, 4, -7, 0, 5, -7], [3, 12, 1, 1, 7, -4, -4, -6, 9, 7], [17, -1, 1, 6, 9, 2, 7, -4, 10, 6], [-8, -16, -1, -20, -7, 3, -5, 7, 8, -14], [-14, -8, -1, -7, -5, -8, 0, 6, -3, -10], [3, 13, 1, -7, 6, 0, 3, -1, 1, 1], [-1, -6, 10, -5, -9, -1, -8, -2, -14, 1], [1, 7, 0, 0, 4, 2, 0, -17, -8, 1], [3, -21, -2, 4, 2, 0, -6, 10, 1, -11], [5, 9, 6, -9, -10, -1, 13, -10, -10, 6], [-1, 19, -9, 10, 10, 12, 0, 10, 5, 5], [-8, 5, 6, -4, 5, 0, -4, -4, -5, 9], [19, 0, 10, -5, -1, 0, -1, -1, -8, 10], [-28, -2, 11, -12, 7, 0, 2, -6, -6, -3], [-15, 7, 3, 0, -3, -11, 11, -9, -7, 2], [7, -7, 11, -15, -8, -8, 7, -13, -12, -1], [-6, 5, -12, 4, -7, 0, -8, 4, 7, 9], [17, -13, -11, 5, -6, -10, 1, 5, 7, -7], [-19, 10, 16, -8, 7, -4, -3, -19, 1, 1], [-16, 8, -7, 10, 7, 5, -1, 8, 5, -5], [15, -3, 9, 5, 0, 0, 3, -5, 0, 9], [12, -7, -6, 1, 2, 5, -1, 4, -4, 0], [12, -2, -4, 6, -5, 2, 5, 3, -9, 2], [29, -6, -10, 2, -5, 4, 6, 1, -5, 1], [-15, -2, 2, -13, 9, 6, -3, -10, 5, 4], [-33, 8, -6, -3, -2, -3, 5, 6, 3, 11], [13, 1, -2, -4, 0, -10, 4, -6, -3, -7], [7, -9, 2, -2, 10, 2, -4, -12, 1, 4], [20, -1, -3, -2, -16, -8, 1, -1, -2, 1], [-14, 1, -16, 20, 5, 0, -2, 1, 6, -1], [-31, 16, -14, 15, 15, 4, 5, 5, 12, 10], [-20, 10, 8, 8, 16, -2, -12, -13, 5, 10], [20, -4, -4, -14, -2, -1, -11, 4, 3, -3], [38, -4, 15, -8, 1, -2, 2, -8, 2, -3], [18, -10, 1, 2, -2, -12, 11, 10, -15, -15], [28, 0, 2, -9, -4, -1, -7, 0, 2, 4], [16, -6, -4, 5, 1, 16, -9, 3, 12, 15], [-8, -10, -6, 4, -11, 0, 5, 12, -3, -10], [3, 7, 2, 8, 6, -1, -7, -13, 6, 20], [-4, 18, 1, 1, 13, -4, 6, -8, 1, 6], [28, -7, -11, 4, -5, -2, -1, 0, 14, -5], [-2, 0, -18, 8, -2, 3, 1, 8, 3, 3], [12, 11, -11, 10, 2, 3, 7, 10, 6, 2], [5, 10, 1, 4, -1, -2, 8, 10, -4, 2], [19, -12, -9, 5, -3, -6, 10, 6, -11, -13], [3, 6, -6, 4, 0, -6, 2, 0, -4, 0], [20, 3, -9, 9, 3, -4, 11, -2, -4, 3], [10, -2, -8, -1, 13, 12, -11, 1, 8, -7], [-7, -8, 2, 10, 6, 4, 2, -6, -4, 9], [23, -1, -1, 8, -8, 1, -6, 7, 9, -1], [22, -7, -7, 8, 1, 4, -6, 9, -1, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6012_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6012_2_a_k();".
// 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_6012_2_a_k(:prec:=10)
    chi := MakeCharacter_6012_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_6012_2_a_k();".
// 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_6012_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6012_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![399, -1196, -587, 2728, 404, -1182, 55, 152, -19, -6, 1]>],Snew);
    return Vf;
end function;