
// Make newform 5054.2.a.bm in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5054_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_5054_2_a_bm();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5054_2_a_bm();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 105, 570, -2326, 1005, 1911, -814, -555, 213, 68, -24, -3, 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], [-530962, 2788715, -1063803, -2660099, 277033, 941476, 67730, -158096, -20343, 13139, 1225, -447], [-2181, 105206, 1508292, -941236, -2103093, -9086, 693188, 104083, -82762, -17654, 3252, 839], [-293578, -4295959, 8749053, 1467335, -6538931, 131584, 1731830, -119612, -193073, 18299, 7731, -895], [-40633, 1544497, -2734811, -1556147, 2380126, 950562, -625458, -262179, 62419, 30793, -2027, -1286], [895, -199603, -3785809, 6667283, 2366810, -4828586, -596946, 1235105, 71023, -132213, -3181, 5046], [-768532, -7197671, 15008101, 2095789, -11448535, 1392036, 3270814, -712334, -409517, 108087, 18833, -5411], [205005, -167712, 3294301, -4674049, -4277477, 3329926, 1540074, -877749, -215396, 101459, 10413, -4331], [443995, 5859056, -18793910, 4734210, 13815345, -6220622, -3867760, 1947439, 480540, -240300, -22014, 10457], [701627, 3574889, -29133113, 23736689, 24101664, -15853538, -6913710, 3775997, 830535, -393225, -35795, 15220], [1189669, 16250289, -58736565, 24957905, 50488138, -20339674, -14977174, 5308691, 1855779, -600369, -82415, 25258]];
    Rf_basisdens := [1, 1, 162716, 162716, 162716, 162716, 162716, 162716, 81358, 162716, 162716, 162716];
    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_5054_a();"
function MakeCharacter_5054_a()
    N := 5054;
    order := 1;
    char_gens := [1445, 1807];
    v := [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_5054_a_Hecke(Kf)
    return MakeCharacter_5054_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 := [[1, 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, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0], [-1, 1, 0, 1, 0, 1, 2, 0, 1, -1, -1, 1], [0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0], [1, -1, 0, 0, -1, -2, -1, 1, 0, 0, 0, 1], [0, 0, -1, 0, 1, -1, -1, -1, 0, -1, 1, 0], [2, 0, -1, 0, -1, 3, 1, -1, 0, -1, -1, 0], [-2, -1, 0, -2, 0, -1, -1, 0, -1, 0, 1, -1], [3, -1, 0, -2, -1, -2, -2, 0, 0, 1, 0, -1], [1, 1, 1, 2, 0, 1, 2, 0, 0, -1, -1, 1], [1, 0, -1, 1, -1, 2, 0, 1, 1, 0, -1, 0], [-2, 0, -1, -1, -3, -3, -1, 0, 0, 0, 1, 0], [1, 1, 0, 1, 3, 1, 1, -1, -1, 0, 0, -1], [1, -1, -1, 1, 1, 3, 1, -1, 0, 0, 0, 1], [3, -2, 0, -3, 2, -1, -3, 1, 0, 0, 2, 0], [-1, 1, -1, 0, 2, 2, -2, -1, -1, 0, 1, 1], [-1, 0, 0, -2, 2, 0, -1, -1, 0, 1, 2, 0], [1, -2, 0, 1, -3, -2, 2, 1, 0, -2, -1, 0], [-1, -1, 0, -2, -1, -4, 0, 2, 0, 1, 0, 1], [-2, 0, 0, -1, 1, 2, -1, -1, 0, -1, -1, -2], [-1, 2, 0, 1, 0, 1, 1, 0, -1, -1, 0, 0], [1, 0, -1, 1, 1, 0, 0, 1, 1, 0, -1, 2], [3, 2, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0], [1, 0, -2, -1, -4, -3, -1, 2, -1, 1, 0, 0], [2, 0, 1, -1, 1, 2, -2, 0, 0, 2, 2, 0], [-1, 0, 1, -2, -3, -2, -1, 2, 1, 1, 0, -2], [2, -2, 1, -2, -4, -3, -4, 1, 0, 1, 2, 0], [0, 0, 0, 3, 3, 1, 3, -1, 2, -1, -1, 0], [3, -2, -2, 2, -1, 5, 3, 0, 1, -3, -2, 1], [2, -3, 1, -3, 2, -4, -3, 0, -1, 4, 3, -3], [-2, -1, 0, -2, -1, -3, -3, -3, 0, 1, 1, -1], [-1, 0, -2, 1, -3, 1, 4, -1, -1, -2, -1, 0], [3, -3, 0, -1, -6, -5, -2, 1, -1, 0, 1, 1], [4, 2, 3, -4, -1, -1, -3, 3, -2, 3, -1, 0], [-2, 1, 0, -1, -4, 1, -1, -2, 1, 0, 1, 1], [6, -4, 2, -1, 3, -2, 1, -1, 0, 1, 1, 0], [1, 2, 1, 0, 7, 3, 0, -1, 2, 0, 1, 0], [1, -2, 1, -1, 5, 2, 0, -2, 0, 1, 1, -2], [-6, -2, -2, 2, 0, 4, 1, 0, 1, -2, -2, 0], [-2, -2, -1, 2, -3, -1, 3, -3, 2, -1, 1, -2], [0, 0, -2, 2, 6, 2, 0, 0, 0, 2, 0, 0], [0, 0, 0, 2, -2, 0, 3, 1, 0, 1, -2, 2], [-3, 2, 2, -1, -6, -5, 1, 2, 1, -1, -4, 2], [0, 1, -1, 3, 0, 1, 5, -2, 0, -2, 1, 1], [2, 2, 1, -3, -2, -2, 0, 0, -1, 0, 0, -2], [7, -2, 0, -1, -3, 0, 1, 1, 0, 0, -2, 2], [-1, -1, -2, -1, -1, 0, -2, 2, 0, 1, -3, 1], [-5, -3, -2, 1, 4, 3, 0, -1, -1, 0, -1, -1], [-2, -3, 1, -3, -2, -4, -3, 1, -3, 1, 1, -3], [0, -1, 2, -2, -6, -12, -1, 3, 0, 3, -1, -1], [7, -3, 1, 3, -2, 1, 3, 3, 0, 0, -2, -1], [0, -2, 2, 0, -2, -12, 0, 2, 0, 2, -2, 0], [-1, -1, 1, 2, 2, -7, -2, 2, 0, 1, 3, 1], [6, 3, -1, -3, -2, -4, 3, 0, 1, -2, -1, 1], [4, -1, -2, -1, -4, -3, 1, -2, 3, -1, 1, -1], [6, 2, -3, 3, 4, 5, 0, -1, 0, -1, -2, 0], [1, -2, 3, -4, -2, -3, -7, 0, 0, 3, 4, -2], [-5, 0, 0, -2, 0, 5, -1, 1, 1, 0, 0, 0], [8, 2, -4, 0, -6, 6, 2, -4, 0, -2, 0, -2], [4, 1, 1, 0, 3, -1, -4, 0, 0, 2, 2, -1], [0, 0, 0, 0, 4, 0, 2, -2, 2, 2, 2, 0], [8, -3, -2, 1, 3, -4, -2, 2, -2, 0, 0, 1], [-3, 4, -1, 3, 2, 3, 4, -2, 2, 1, -3, 4], [-3, 7, 0, 7, -2, 2, 8, -2, 2, -1, -3, 1], [8, -3, 3, -3, 6, -4, 0, 3, -3, 3, 0, 2], [-3, 1, -1, -3, -3, -5, -1, -1, 0, 2, 2, -1], [-2, 0, 2, 1, 2, -5, -2, -1, 1, 0, 0, 0], [2, 0, -1, -2, -3, -1, -1, -1, -2, 1, 3, -4], [12, -2, 1, 2, 1, 3, -1, 3, 0, -1, -3, 0], [-1, 0, 3, -1, 1, -2, 4, 1, -1, 0, 1, 2], [5, 0, 0, 1, 8, 3, -3, -2, -1, 1, 4, -2], [2, 1, -2, 1, 3, 3, -6, 1, -1, 1, 0, 1], [-12, 5, 0, -1, 1, 3, 2, -1, -1, -3, 0, 1], [-10, -4, 2, -1, -6, 1, 3, -2, 1, 0, -1, -2], [5, 0, -2, 1, -2, 6, -1, -1, 0, 0, 4, 0], [10, -4, -1, 0, 1, -3, -5, -1, -2, -1, 4, -1], [-8, -5, 0, -1, 5, 5, -2, -1, -1, -1, 4, -5], [-2, -7, 0, -2, -5, 0, 2, 4, 2, -1, -2, 3], [-13, 0, 1, 1, 9, 0, 2, -3, 1, 0, 1, 0], [6, -4, -1, 0, -11, -5, -1, -1, 0, 1, 1, -2], [1, -3, 0, -1, -6, -1, -2, -3, -1, 0, 7, -5], [0, 3, -6, 0, 0, 6, -3, -2, -5, 0, 3, 1], [0, 4, 3, 2, -2, 1, -1, -2, 1, -2, -1, 0], [3, -7, -1, -5, -1, -8, -11, 3, -3, 4, 2, -1], [4, -3, 0, -5, -1, -11, -8, -2, -4, 4, 6, -3], [-20, 1, 0, 2, 0, 5, 3, -4, 1, -2, 1, -1], [-6, 2, 4, -2, 10, 4, -2, -2, 2, 0, 0, 0], [-6, -5, 1, -3, -8, -4, -5, 2, -3, 4, 3, -1], [0, 7, -4, 3, -3, 7, 3, -6, -1, 0, 0, -1], [5, -3, 2, 2, -3, 7, -1, -2, -1, -1, 2, -3], [-2, 0, 1, -2, -5, -1, -3, 5, -2, 1, 3, 2], [9, -2, -3, -1, -8, 2, -1, 2, -3, -3, 0, 0], [-11, 2, 4, 0, 1, -2, -2, 0, 2, 1, 1, -2], [-9, -1, -2, 0, 3, 10, 0, -6, -2, -3, 5, -2], [7, 2, -1, 1, 3, 2, 2, -5, -1, -4, 1, -2], [7, 6, 0, -3, 2, -9, -4, 1, -1, 0, 2, 0], [4, -3, 0, -5, -3, -7, -4, 1, -1, -1, 0, -1], [0, 1, -1, -3, 11, 7, -8, 0, -2, 2, 2, -3], [-3, 1, 2, 3, 11, 4, 7, 5, 4, -2, -9, 7], [8, 1, -2, -2, -4, -2, -1, -3, -2, 3, 3, -3], [0, -3, 1, 3, 2, 3, 11, 5, 4, -5, -5, 1], [3, -1, -5, -2, -10, -5, -4, -2, -2, -3, 0, 0], [-10, -1, 2, 1, 5, -8, -2, 2, -2, 2, 2, -1], [-2, -6, 5, -3, -9, -5, -4, 7, 3, 1, -2, 0], [-19, 6, 1, 1, -7, -9, 2, 2, -2, -1, 1, 2], [2, 1, 0, -3, -9, -7, -6, 1, 1, 3, 2, -1], [12, 3, 1, -3, -6, -6, -1, 0, 3, 0, 1, 1], [-1, 5, -4, 6, -3, -5, 3, -3, 0, 0, -4, -1], [2, 3, 3, 6, 4, 7, 4, -3, 0, 1, -3, -3], [6, 8, -1, -3, -9, -6, 1, 0, -1, -3, -1, 2], [-7, 1, 0, 3, -10, -2, 4, 1, 3, -2, -3, 3], [-2, -4, 4, -6, -8, -14, -8, 7, -2, 9, 0, 0], [-4, 8, 1, 8, 5, 15, 15, 1, 2, -3, -7, 4], [4, -1, -3, 1, -2, 6, -3, 2, 1, -2, 1, 1], [-5, 4, 1, -3, 7, -4, -5, -6, 1, 3, 1, -2], [-10, 4, -2, 0, 3, 2, 1, 1, -2, 1, -1, -2], [-3, 2, -3, -7, 5, 0, -9, 1, 0, 6, 1, 2], [-8, -1, 0, 1, -1, -1, -2, -5, 3, 1, 4, -1], [-4, 7, 4, -3, -1, -7, 3, -2, 1, 0, 0, 1], [-3, 0, 3, 3, 0, -8, 0, 1, 3, 0, -4, -4], [-14, 1, 2, 1, 7, 1, 2, -3, 3, 1, 2, -1], [6, 2, 0, -4, 14, 2, -4, -2, -4, 4, 2, -2], [-6, -2, 1, 6, -1, -1, 5, -1, 0, 1, -1, 2], [-7, 5, 1, 0, 6, 1, 2, 4, 2, -1, -7, 5], [-1, 0, 3, 0, 10, 3, -5, 0, -2, 7, 4, 0], [6, -8, 5, -5, 2, -7, -12, 0, -2, 2, 10, -2], [10, -1, -2, -2, 3, -3, -1, -5, 0, -1, 3, -1], [0, 1, -1, 4, 0, 17, -1, -3, 0, 0, -1, 1], [-3, 6, -4, 1, -4, 11, -5, -2, -3, 1, 4, 0], [-6, -1, 2, -4, 10, 6, -1, 1, 2, -1, 1, -3], [3, 4, 3, 1, -5, 4, 7, 0, 1, 1, -1, -2], [4, 6, 0, 1, -10, -13, 2, -4, -1, -2, 4, -2], [-6, 0, -7, 8, 8, 16, 6, -7, -3, -7, 4, 0], [-4, 3, 0, -1, -6, -1, -5, 6, -3, 0, 3, 1], [-1, 2, -5, 2, -3, -9, -3, 1, -2, -2, 8, -2], [4, -3, 1, 1, 10, -2, -5, -4, 1, 4, 7, -3], [3, -1, -2, -6, -1, -10, -7, 5, -2, 2, 2, -3], [7, 0, 0, -2, 4, 5, 4, 0, 0, -3, -5, 4], [2, 2, 3, 6, -3, -12, 7, 0, 3, 0, -1, 4], [5, 7, -4, 0, 2, 6, 0, -2, 2, 3, -3, -1], [6, 3, -3, 1, -14, -12, 3, 0, 1, 4, -1, 1], [-6, -8, 1, -6, 5, -5, -4, 2, 2, 2, 2, -5], [-14, 3, 4, -4, 0, -8, -7, 3, 0, 5, 3, -3], [11, -8, -6, -5, 2, 0, -4, 4, -2, 1, -3, 1], [6, -1, 6, -2, 7, 0, 0, 2, 0, -1, 2, -3], [0, 1, 2, 1, 3, -5, -2, -1, -3, 3, 4, -1], [1, 0, -1, 0, 4, 16, -3, 2, -1, 5, 2, -2], [-2, 3, -4, 2, 6, 15, -3, 0, -1, -4, -1, 5], [3, -1, -7, 2, 4, 5, 2, 2, -2, -5, -5, 1], [5, -4, 1, 5, -5, 7, 4, 2, 2, -3, -5, 2], [-8, 2, -1, 3, -3, -1, 12, 4, 0, -4, -10, 4], [0, -2, 1, -1, -5, -13, -2, -3, -1, 1, 2, -4], [-2, -3, 0, -7, -3, -6, -8, 4, 2, 4, -4, 3], [2, 0, 3, -7, -4, -26, -8, 3, -5, 5, 6, -2], [11, -5, -1, -8, -3, -1, -13, -1, -6, 6, 6, -5], [-1, 1, -7, 4, 8, 3, 6, -4, 2, -5, 5, 1], [6, -3, -4, 0, -3, 1, -10, 3, -3, 2, 4, -3], [-17, 3, 3, 1, 6, -5, -3, -2, -3, 1, 2, 1], [5, -6, -3, 9, -1, 8, 10, -2, 3, -9, -5, 4], [-2, 1, -4, 7, 6, 12, 4, -4, -2, 0, 6, -1], [15, -3, -1, -6, 2, -1, -6, -4, 2, 3, 5, -5], [-7, 0, -3, 4, -11, 2, 7, -5, -2, 0, -2, -2], [7, 2, -2, 2, 5, 0, 6, -1, 4, -4, -3, 4], [-1, 2, -7, 2, 3, 23, 6, -6, 0, -2, -3, -6], [-4, 1, -4, -1, -9, -5, -7, 0, -3, -2, 1, 0], [-8, 9, 3, 2, 3, 15, 10, 0, 4, -6, -4, 3], [2, -2, -1, -4, -3, 7, 4, -4, 0, 0, 6, -1], [4, 1, -11, 12, -6, 27, 9, -3, 4, -8, -5, 1], [10, 4, 0, 6, 11, 4, -3, 1, -2, 0, -7, -2], [10, 3, 0, -5, 3, -9, -4, 5, -1, 1, -2, -1], [-13, 3, -4, 0, 1, 7, -5, -4, -7, 1, 4, 1], [8, 4, 4, -2, 0, 1, 2, 3, -3, -1, -2, 0], [3, 3, -1, -8, 0, -3, -4, 4, -6, 1, 1, -3], [-5, 1, -5, 6, -14, 1, 12, -6, 4, -3, -3, -1], [-4, -2, -2, -1, 14, 15, -2, 4, -1, 2, -4, 0], [-4, 10, 0, 3, -6, 0, 1, -1, 0, 1, -3, 2], [17, 2, -3, 6, -1, 9, -3, 2, 2, 3, -5, 2], [-5, 1, -6, 2, 9, 10, 5, 3, 6, 0, -2, 3], [-4, -4, -1, 2, 19, 20, -5, -4, -7, -2, 5, -4], [-10, -4, 2, -4, -16, -24, -4, 5, -1, 1, 0, -2], [3, -7, 4, -2, -15, -6, -6, 2, 0, 3, 3, 0], [-11, 9, -6, 2, -1, -2, 5, -1, 4, -4, -6, -1], [19, 3, -12, 8, 0, 18, 11, -1, 0, -6, -3, 1], [0, 10, 2, 1, 5, 12, 1, -2, 1, 2, -1, -2], [-6, -2, -2, 4, 6, 2, -4, 0, -4, -2, 4, -4], [15, -2, -2, 2, -1, 1, 8, 3, 5, -6, -3, 4], [-6, 3, -1, 7, 2, 4, 13, -4, 9, -4, -1, -1], [5, 0, 2, 2, -13, -14, 0, -2, -2, 3, 7, 0], [-4, -3, 4, -4, -1, -6, 2, -1, 2, -1, 4, 5], [-13, 0, 5, -8, 4, 3, -9, 3, -2, 0, 6, -2], [-14, 5, -2, 7, -7, 5, 11, -6, 3, -6, -2, 5], [9, 10, 2, 4, -17, -5, 4, -2, 0, 5, -5, 2], [6, -3, -4, 3, -3, -3, -6, -3, -3, 1, -2, -3], [-12, 2, -6, 4, -16, 1, 2, -5, -1, -3, 2, -2], [-21, 4, 2, -5, 0, -1, 3, 6, 3, 1, 0, 4], [9, -3, 1, -8, -16, -15, -14, -2, -4, 5, 7, -5], [8, -10, 2, -6, 6, -8, -6, 2, -6, 2, 10, -6], [18, -5, 4, -8, 4, 2, -1, 0, 2, 8, 3, 3], [-3, 3, -6, 9, 4, 7, 12, -1, 1, -10, -11, 5], [-13, -6, -2, -2, 11, 7, -1, 2, 1, 3, -2, -3], [-5, 2, -3, 8, -1, -9, 2, -3, -4, -4, 1, 2], [-12, -3, -2, -1, -11, 1, -2, -7, -3, 1, 8, -5], [22, -4, 2, -5, 3, 1, -9, -2, 1, 6, 7, 0], [-14, 4, 3, 1, 10, 1, -2, 0, 6, 2, -2, 2], [5, 1, -6, 4, -17, -3, 7, 4, -1, -5, -4, 7], [17, 0, 4, 4, 5, 7, 8, -2, 4, 1, -5, -4], [7, -3, -3, -4, -3, -3, -7, 1, 2, 0, 4, 5], [9, 0, 5, -8, 9, 1, 0, 0, 4, 2, -1, -6], [19, 3, -7, 1, 6, 9, 2, -6, 0, -3, 1, 1], [16, -4, -4, -8, 4, 12, -14, 0, -4, 6, 8, -4], [-2, 1, 3, -1, 8, -4, 6, 5, 3, -5, -7, 7], [-17, 10, -6, 4, 6, 10, 7, -13, 4, -3, -4, -2], [0, -5, 5, -3, 8, 2, 0, 7, -1, -1, -3, 1], [6, 2, -1, -1, 3, 3, -6, 6, -6, 0, -2, 2], [-9, 7, 6, 0, 5, 6, 11, -1, 2, -2, 0, 3], [25, -9, 3, 0, -4, -5, -4, 2, -2, 1, 1, -3], [-4, -6, -5, 4, -19, -7, 1, -1, 4, -3, 3, 0], [9, -3, 6, -3, -17, -15, -4, -1, -3, 2, 1, -9], [15, 6, 0, 4, 4, -1, 0, -5, -1, -4, 5, -2], [22, -6, -5, -7, 7, -11, -17, 3, -6, 9, 7, -4], [9, -6, 4, 5, -10, -11, 1, 6, -1, -3, -4, -2], [-19, -3, -2, 7, -8, 16, 8, -2, 0, -3, -11, 1], [-7, 3, 7, 0, 0, 5, 6, 3, 3, 6, -1, 1], [14, -2, 3, 2, 17, 23, 7, -1, 6, 1, -11, 0], [11, 9, 0, 6, -17, 8, 9, 1, 0, -2, -2, 3], [16, 2, 2, 4, 14, 6, 8, 6, 8, -4, -10, 4], [4, 3, 6, -9, -8, -16, -5, 5, -2, 3, 7, -1], [-6, 1, -5, 0, 0, -6, 0, -11, 3, -3, 4, -5], [8, -10, 1, -1, 2, -8, -2, 0, 2, 6, 4, -4], [-4, -1, 13, -9, 14, -1, 0, 2, 4, 4, -2, 2], [-5, 4, 2, -1, 2, 1, 7, -4, -3, 1, 8, 2], [-13, 0, -6, -8, -5, -3, -6, -1, 3, 4, 7, -4], [-4, -1, 4, 5, -9, 5, 12, 1, 3, -3, -10, 1], [-7, 2, -1, 2, -12, -6, 7, 0, 7, 1, -4, 6], [3, -1, -1, 2, 2, -13, -2, -2, -2, -1, 5, -1], [9, -7, -3, -3, -13, 0, 3, -2, -1, -3, -2, -1], [-1, -11, 6, -5, 11, 5, -7, 4, -1, 1, 4, 1], [-17, -4, -3, 5, -13, 0, -2, 1, -3, -2, -5, -6], [3, -8, 5, 1, -3, -10, 4, 7, 3, 4, -3, 0], [-17, 3, -3, 6, 6, 1, 9, -5, 4, -2, -1, 1], [22, 9, -7, 5, 22, 22, 7, -8, 1, -4, -3, 7], [-7, 1, -5, 10, -4, 3, 4, 2, 2, -3, -9, -1], [14, 0, -2, 1, -11, -10, 3, 5, -2, -5, -7, 8], [-3, 9, 6, 5, 14, 7, -2, 1, -5, 0, 3, 3], [-14, 0, 9, 0, -1, -5, 11, 6, 3, 2, -11, 4], [-3, 10, 3, 5, 1, 2, 4, -1, -1, 2, 1, -2], [1, -10, 1, -3, 15, 6, -7, 0, -1, 7, 5, -4], [6, 6, 2, -7, -14, -7, -10, -1, 3, 4, 0, 2], [-16, -4, 0, -4, 12, 1, -2, 3, 1, 5, -2, 2], [17, 2, -15, 7, 13, 26, 11, -2, 1, -7, -2, 3], [2, 2, 0, 0, 6, 6, 3, 5, 4, 1, -12, 4], [6, 9, -1, 3, 18, 8, 1, -8, 1, -4, 13, 1], [6, 0, -5, 14, 4, 16, 8, -2, -1, -6, 0, 8], [-4, 1, -2, 1, -1, -5, -9, 0, -5, 2, 5, -8], [3, 6, -6, 1, -16, -15, -4, 1, -1, -4, 6, 2], [5, 6, 0, 6, 4, 5, -4, -4, -2, 5, 5, 4], [1, 6, -7, 7, 3, 21, 8, -10, 4, -7, -1, -4], [20, -10, -4, -12, 1, -6, -13, 7, -6, 7, 5, -2], [1, 1, -8, 16, -7, 22, 14, -10, -2, -11, -1, 0], [14, -9, -1, 2, -7, 6, 4, 3, 3, -3, -12, 3], [-15, 2, 5, 6, -8, 5, 14, -8, 4, -7, -5, 3], [6, 0, -3, -4, 3, 17, 1, -3, 0, -3, 3, -4], [17, -12, -4, -6, -1, -8, -2, 4, 4, 7, -3, 0], [9, 9, -3, 2, 10, 21, -3, -1, -2, -6, -3, 3], [2, 5, 7, -3, 16, 10, -3, 2, -1, 8, -3, 3], [7, -2, 1, 7, -5, 4, 19, 0, 1, -3, -8, -5], [3, 10, -3, 9, -13, 2, 9, -2, -3, -3, -5, -2], [-2, -4, -5, 3, -14, -9, 8, 0, -4, -6, -2, -2], [11, 4, 4, 0, 14, 12, 5, -5, 0, 3, 8, 0], [16, 2, -2, -2, 0, -6, -3, -3, 0, -3, 0, -2], [-11, 3, -4, 0, -5, 14, -7, -3, -6, 4, 2, -3], [4, 3, -8, 4, 8, 16, -5, 1, 0, -5, -1, -1], [-18, -3, -12, 1, 3, 9, -10, -7, -3, -1, 12, -7], [-5, -2, -4, 3, -8, -4, -3, 2, -5, -5, 8, -2], [8, -5, 0, 10, -2, 6, -1, 2, 0, 0, -7, -3], [15, -11, 4, -10, -3, -8, -19, 7, -10, 4, 10, -11], [18, -6, -5, 1, -13, -3, 6, 4, 0, -6, 4, 2], [0, 7, -7, 7, -2, 0, -7, 0, -7, 4, 3, 1], [-22, -1, 4, 5, -5, 1, 5, 2, 5, -6, -9, 6], [1, -2, -4, -6, 5, 0, -1, 10, -6, -1, -5, 0], [-4, 3, -17, -2, 4, 26, -4, -3, -3, -1, 4, -1], [13, 0, -1, 14, -5, -1, 5, 0, 1, -1, -4, 6], [6, 14, 3, 0, -5, 5, -3, -1, 4, 1, -3, 2], [-1, 12, 5, 7, -15, -30, 12, -1, -3, 0, -3, 6], [-16, -2, 13, -4, -13, -23, 2, 4, -2, 0, 7, 0], [-2, 6, -3, -2, -1, -11, -1, 3, 4, 1, 7, 4], [23, -3, -6, -1, -6, 7, -1, 1, 1, -4, -3, 1], [-6, 2, 8, -10, -11, -18, -2, -3, 3, 5, -2, 0], [-2, 4, -2, 19, 2, 23, 19, -1, 3, -9, -8, 6], [8, 0, 0, -6, -2, -16, -10, 10, 0, 8, 2, 6], [8, 0, -1, -7, 3, -6, -5, -5, -1, 3, 3, 2], [14, -11, 0, -3, 21, 7, -12, -5, -1, 13, 10, -7], [4, -10, -8, 9, -9, 9, 11, -5, 11, -7, -5, 4], [-15, 8, 7, -2, 4, -1, 4, 1, -2, 6, 2, 0], [5, 1, 0, -3, 0, -6, -5, 1, 2, 6, 2, -2], [4, -1, -6, 2, 14, 2, 1, 3, 0, -1, -3, -1], [-10, 4, -7, 6, 1, -3, 1, -1, 2, 1, -1, 6], [13, -3, 3, -9, -22, -17, -13, -5, -4, 2, 10, -9], [7, -5, 8, -3, 4, 12, 4, 4, -2, 1, -1, -5], [24, -5, 5, -6, -11, -12, -8, 3, 5, 1, -2, -1], [-40, 9, 1, 5, 1, 9, 3, 0, -1, 0, -9, 1], [4, 3, 2, -1, 7, 16, 6, -2, 8, -2, 0, -3], [15, -2, -3, 9, -5, 2, 14, -1, 7, -4, -5, 6], [0, 3, 1, -2, -2, 3, 5, -9, -4, 0, 7, -9], [-3, -2, 1, 7, -17, 8, 14, -7, -1, -4, 1, 2], [-4, 2, 0, 2, 6, 16, -6, 0, -4, 2, -2, -2], [-7, -1, 2, -4, 9, 16, -5, 3, 8, 0, -8, 1], [4, 1, 8, 0, 5, -6, 9, 1, 8, -1, -3, 3], [12, 10, 2, -15, -7, -28, -13, 5, -2, 9, 5, 0], [21, -3, 3, -9, 10, -7, -17, 0, -1, 11, 2, -7], [-3, 8, 7, -13, 15, -4, -3, 2, 1, 3, -5, 4], [-18, -2, -14, 2, 3, 7, -1, -9, 5, -3, 1, -2], [6, 5, -3, -6, 12, -22, -14, -3, -3, 9, 8, -5], [-10, -16, -4, -1, 5, 2, -5, -9, 2, 1, 7, -2], [16, -14, -6, -7, 2, -10, 0, 3, 11, -3, -1, 4], [1, 7, -5, 3, 2, 9, 8, 2, 2, -1, -7, 5], [18, -4, 8, 2, -3, -4, -1, -1, -2, 4, 3, -6], [10, 2, -3, 12, 3, 19, -2, -4, -4, -8, 6, -3], [3, 8, 3, 15, 1, -22, 8, -4, 2, -3, -1, 8], [-16, 12, -1, 0, -1, 13, 8, -5, 3, 1, -5, 0], [-1, -1, 13, -10, -10, -7, -4, 6, 0, -1, -7, 3], [2, -3, 6, -14, 6, -9, -21, 14, -3, 4, 1, -1], [14, -3, -4, 4, 2, 16, -4, -8, -2, -2, -2, -8], [18, 0, -8, -3, -22, 0, -3, 0, 3, 2, -1, -6], [6, -6, -3, 0, 17, 11, -3, -5, -4, 3, 5, -8], [10, -10, -4, 10, -16, -8, -2, -2, -4, -6, 0, -8], [-14, 1, -2, -1, -11, -3, 0, -5, -1, -9, 0, -5], [3, 9, -4, 12, 15, 26, 11, -9, 4, -4, -8, 5], [3, -3, -3, -7, -6, 0, -12, -4, -9, -1, 3, -5], [-2, 10, -5, -7, 2, -11, 4, -4, 0, 0, 6, 4], [-4, 2, -10, -1, 9, 23, -6, -4, -5, -2, 9, 0], [-21, -3, -8, -4, -9, -16, 3, -1, -6, 0, 8, 1], [-21, 9, 2, -8, 15, 15, -11, 2, -9, -1, 6, -3], [28, -15, -6, -11, 3, -7, -19, 2, -9, 4, 8, -5], [6, 12, 6, 4, 8, 0, 10, -4, -4, -2, -4, 4], [-18, 3, 3, -1, 16, 18, -3, 0, 3, 2, -3, -1], [6, 0, -4, -4, -10, 7, -2, 7, -9, -1, 6, 4], [-31, -9, 1, -5, -17, -10, -1, -5, 1, -2, 0, -3], [-11, 6, -4, 8, -4, 0, 5, -15, 0, -9, 2, 2], [14, -1, 9, 1, -12, -12, 9, 4, 7, 2, -9, 1], [-10, 12, -1, -3, 4, -17, -6, 0, -4, 6, 12, 0], [3, -3, 3, -2, 8, -3, -8, 0, -2, 3, 11, -9], [6, 13, -5, 3, -10, 18, -6, -5, -3, -5, -1, -3], [6, -2, 1, -8, -9, 7, 1, 3, 2, -3, -9, 2], [5, -17, 4, -5, 8, -19, -10, 8, -11, 9, 5, -5], [36, 3, -8, 7, 6, 1, 9, 7, 3, -3, -11, 5], [-33, 5, 2, 4, -9, 7, 13, -8, 3, -9, 4, 3], [-3, -4, 0, -11, 14, 3, -17, 0, -1, 11, 4, -6], [16, 0, 6, 0, 18, 4, -6, 2, -2, 6, 2, -4], [-13, 13, -4, 12, 11, 16, 17, -19, 2, -10, -4, -1], [-8, -4, 5, -8, -1, -6, -7, 0, -7, 4, 7, 0], [-24, 12, 5, 12, 5, -3, 15, -5, 6, -9, -7, -2], [8, 3, -1, 5, 6, 21, 15, -13, 2, -5, -3, -9], [2, -3, -11, 13, -7, 9, 22, -4, 4, -5, -12, 7], [-5, 5, -5, 6, 8, 11, 8, -5, 1, 4, -7, 7], [15, 7, 1, 4, 22, 21, 12, 2, 0, -5, -15, 5], [9, -4, 11, -8, -10, -9, 1, 9, 6, 0, -14, 4], [7, 6, 1, 4, 9, -10, -11, -11, 0, 2, 8, -6], [16, 6, 2, 11, -1, 1, 21, 2, 3, -6, -5, 4], [-6, 6, -4, 16, 12, 24, 25, -5, 8, -13, -8, 4], [-21, 2, -6, 12, 2, 13, 15, -4, 3, -11, -3, 10], [-6, -3, -9, -3, -39, -6, 0, 1, 1, -1, -4, 1], [-6, 1, -1, 6, -8, -11, 7, 5, -4, -2, -1, 1], [4, 5, -5, -7, 19, 24, -13, -4, 1, 8, 7, 3], [-5, -7, 1, -12, -10, -28, -4, 1, 1, 0, -1, 5], [-7, 11, 7, 1, 12, -20, 6, -4, 3, -3, -5, 3], [-4, 11, -13, 6, -6, 18, -2, -12, -6, -6, 6, -5], [-29, 7, 4, -4, 3, 10, -1, 5, 2, 0, -4, -1], [0, 20, -4, 6, 6, 8, -2, -10, -2, 4, 0, -6], [15, -7, 0, 0, -9, 0, -5, -3, 6, 0, 0, -5], [-2, 15, -8, 11, 4, 11, 6, -9, -1, -1, 7, -1], [-6, 8, -7, -5, 11, 20, -7, -6, -5, 3, 5, -6], [9, -4, 1, 2, 16, -8, -1, 2, 1, 3, -4, -4], [10, 16, 0, 8, 2, 7, 16, 3, 3, -5, -10, 10], [10, -1, -2, 7, -21, -2, 2, 2, 2, 0, -2, -3], [-19, 3, 4, -10, 23, 10, -6, -6, -2, 9, 9, -8], [-30, 0, 2, 7, 6, 29, 15, -2, 11, -5, -8, 4], [9, 3, -7, 9, -16, -2, 2, 3, -1, -5, -5, 7], [2, 2, 11, -16, 2, -31, -15, 2, -7, 10, 7, 0], [-1, -1, 1, -13, -9, 17, -15, 8, -10, 3, 2, -5], [21, 8, 4, 3, -4, -3, 5, 2, 3, -5, -6, 6], [12, 11, -8, 9, -15, -3, 6, -8, -4, -6, 2, 3], [15, -14, 4, -22, 18, -6, -13, 0, 4, 9, 8, -4], [-6, -6, 8, -5, -11, -3, 3, 10, -1, 0, 3, 4], [40, -8, 5, -4, 13, -11, -1, 3, 0, 7, 5, 4], [-1, 2, -2, 5, 10, -12, 7, 1, -6, 2, -4, 2], [-4, 4, -9, -6, 9, -1, -20, -4, -2, 6, 6, -5], [-14, 2, -7, 13, 10, 7, 8, -12, 2, -6, 2, 2], [-8, 19, -3, 7, -3, -5, 20, -14, -4, -4, 2, 3], [-29, 6, -2, 11, -4, 7, 13, 3, 6, -8, 2, 12], [1, 14, 6, -7, 1, 3, -3, 3, -6, 4, 0, -10], [0, -1, 14, -15, -15, -21, -14, 1, -3, 9, -4, -11], [-8, 15, -16, 2, -4, 15, -1, -4, -7, -2, 3, 5], [15, 4, -3, 11, -5, 13, 0, -12, 2, -3, 5, -6], [27, -2, 5, -3, -15, -20, -8, 1, -5, 12, -1, -2], [-5, 10, 4, 1, 12, 1, -1, 2, -5, -3, -8, 4], [15, -11, -1, -4, 15, 4, -1, -8, 1, -4, 6, 3], [-8, 5, 7, -8, 6, -10, -4, 0, -4, 6, 6, -11], [-30, -6, 2, -10, 4, -14, -11, 3, 2, 3, -4, 6], [9, -3, -13, -1, -1, -6, -15, 1, -5, 6, 4, -5], [14, 4, 6, 2, -8, -26, 6, 6, 0, 2, 2, 8], [19, 6, 4, 9, 10, -5, 3, -2, 1, -5, -2, 6], [28, 4, -4, -3, -2, -2, 0, 6, -4, 5, 6, 2], [4, -4, 14, -7, -12, -25, -9, 3, 1, 13, 10, 0], [1, -13, -2, -2, -5, -35, -11, 3, -3, -4, 8, -3], [26, -5, 4, -1, -3, -10, 4, -2, -2, -2, 0, -3], [1, 9, -2, 16, -9, 34, 13, 1, 4, -6, -12, -1], [9, 6, 0, -8, 15, -1, -20, -1, -7, 4, 11, 0], [25, -10, -6, 6, -13, -10, 0, -2, -2, -3, -5, 0], [24, -21, 1, -13, 9, 2, -13, -4, 3, 8, 11, -1], [-1, -4, -3, 10, -4, -7, 7, 0, 2, 1, 0, 4], [-16, -1, -2, -2, -13, -17, -5, -3, -4, -3, 11, 1], [27, -9, -3, -4, -2, -13, -10, 4, 4, 7, 3, 1], [12, -8, -4, 0, 4, 22, 2, 0, -2, 2, 4, 6], [-17, -13, -3, 2, -2, 17, 0, 0, 4, -1, 3, 1], [-1, 0, 2, 3, -6, 17, 11, -2, 1, -3, 0, 2], [18, 3, 12, -6, 2, -14, 1, 6, 2, 6, 1, -1], [-5, 6, -2, -14, -26, -31, -12, 15, -11, 4, 7, 4], [-21, 3, 9, -9, 22, -6, -10, 3, -1, 4, 3, 7], [-2, 5, -11, 3, 0, 0, 5, 6, 1, 0, -3, -3], [13, -14, -2, -6, -9, -4, -8, 7, 0, -2, -1, 4], [4, 0, -18, 1, 1, 16, -3, 1, 0, 3, -9, 0], [-16, 1, 7, 12, -8, -14, 22, 2, 2, -6, -6, 5], [-3, 14, -8, 9, -3, 1, 5, -18, -3, -11, 9, -2], [16, -5, 10, -5, 1, 1, 0, 7, -1, -3, 6, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5054_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5054_2_a_bm();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_5054_2_a_bm(:prec:=12)
    chi := MakeCharacter_5054_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_5054_2_a_bm();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_5054_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5054_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 105, 570, -2326, 1005, 1911, -814, -555, 213, 68, -24, -3, 1]>,<5,R![-5256, -19872, 61866, -39705, -11682, 17952, -1847, -2571, 561, 149, -42, -3, 1]>,<13,R![2483704, -2784300, -973740, 1350445, 212895, -235059, -33436, 17346, 2613, -545, -87, 6, 1]>],Snew);
    return Vf;
end function;