
// Make newform 476.2.q.a in Magma, downloaded from the LMFDB on 14 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_476_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_476_q_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_476_2_q_a();".
// 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_476_2_q_a();".
// 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 := [6561, 0, 0, 0, -1296, 0, 0, 0, 175, 0, 0, 0, -16, 0, 0, 0, 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, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 127575, 0, 80784, 0, 0, 0, -5425, 0, 0, 0, 1225, 0, 0, 0, -4], [20736, 0, 0, 0, -2800, 0, 0, 0, 175, 0, 0, 0, -16, 0, 0, 0], [0, 0, -40176, 0, 0, 0, 5425, 0, 0, 0, -1225, 0, 0, 0, 31, 0], [60912, 0, 0, 0, -8225, 0, 0, 0, 1400, 0, 0, 0, -47, 0, 0, 0], [0, 0, 0, 6561, 0, 0, 0, 2870, 0, 0, 0, 175, 0, 0, 0, -16], [0, 0, 0, 106272, 0, 0, 0, -14350, 0, 0, 0, -875, 0, 0, 0, -82], [0, 0, 6561, 0, 0, 0, -775, 0, 0, 0, 175, 0, 0, 0, -16, 0], [-104, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 639, 0, 1504, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 1], [0, 0, -1296, 0, 0, 0, 175, 0, 0, 0, -16, 0, 0, 0, 1, 0], [0, 2187, 0, -1296, 0, 0, 0, 175, 0, 0, 0, -16, 0, 0, 0, 1], [0, -361584, 0, 226800, 0, 48825, 0, -30625, 0, -11025, 0, 2800, 0, 279, 0, -175], [0, 559872, 0, -104976, 0, -75600, 0, 30625, 0, 4725, 0, -2800, 0, -432, 0, 256], [0, 559872, 0, 331776, 0, -75600, 0, -61250, 0, 4725, 0, 5600, 0, -432, 0, -431]];
    Rf_basisdens := [1, 1, 127575, 14175, 127575, 14175, 76545, 382725, 18225, 175, 4725, 729, 2187, 382725, 382725, 382725];
    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_476_q();"
function MakeCharacter_476_q()
    N := 476;
    order := 6;
    char_gens := [239, 409, 309];
    v := [3, 5, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_476_q_Hecke();"
function MakeCharacter_476_q_Hecke(Kf)
    N := 476;
    order := 6;
    char_gens := [239, 409, 309];
    char_values := [[-1, 0, 0, 0, 0, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, -1, 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, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0], [0, -1, 0, 0, 4, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 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, -1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, -2, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 2, -1, 0, 0, 2, 0, 2, 0, -2, 2], [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, 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], [6, 0, 0, -6, 0, -1, 0, 0, 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, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, -4, 0, 0, 1, 0, -2, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, -2, -1, 0, 0, -2, 0, 4, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -1, 0, 0, 0, 4, 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, 3, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, -4, 2, 0, 0, 0, 0, -2, -4, 1, -2], [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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 7, 0, 0, -4, 0, 2, 4, -2, -3], [0, 6, 0, 0, 1, 0, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, -1, -4, 0, 0, 1, 0, -3, -1, 2, 5], [0, 0, 0, -9, 0, -2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 2, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, -3, 0, 0, -5, 0, -3, 5, 2, -1], [0, 0, 4, 0, 0, 0, -1, 4, 0, 0, 1, 0, 3, -1, -2, -5], [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, 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, 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, -1, 0, 0, 0, 0, -1, 2, 3, 4], [0, 0, 2, 0, 0, 0, -4, -2, 0, 0, 0, 0, -2, 4, 6, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, -1, -5, 3, 4], [0, 2, 0, 0, -1, 0, 0, 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, 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, -3, 0, 0, -8, 0, 0, 0, 11, 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, 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], [-3, 0, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, -7, -3, 0, 0, -7, 0, -3, 7, 4, -5], [15, 0, 0, -30, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 7, 3, 0, 0, 7, 0, 5, -7, -6, 5], [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, 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, 5, 0, 0, 0, 1, 5, 0, 0, -1, 0, 6, 0, 3, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0], [9, 0, 0, 9, 0, -4, 0, 0, 0, -4, 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, 4, 0, 0, 0, 0, 1, -5, -3, -4], [0, 1, 0, 0, 12, 0, 0, 0, 13, 0, 0, 2, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, -12, 0, 0, 8, 0, -1, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 8, 0, 0, 0, -3, 0, 0, 3, 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], [16, 0, 0, 16, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, -8, -4, 0, 0, 8, 0, -3, -8, 2, 5], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, -10, 0, 0, 0, 2, -8, 0, 0, -2, 0, -1, 2, 5, 6], [0, 10, 0, 0, -10, 0, 0, 0, -5, 0, 0, 10, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 7, 1, 0, 0, 0, 0, -1, -7, -9, 4], [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, -9, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 12, 0, 0, 0, 0, 0, -5, 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, 0], [0, 0, -3, 0, 0, 0, 2, -3, 0, 0, 2, 0, -10, 0, 6, -6], [0, 0, 5, 0, 0, 0, 0, 6, 0, 0, 0, 0, -5, 1, -9, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, -4, -2, 0, 0, 0, 0, 2, -4, 3, 10], [0, 0, -5, 0, 0, 0, 0, 2, 0, 0, 7, 0, -6, 0, -1, 1], [0, 0, 0, 0, 0, 4, 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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 4, 0, 9, -4, -16, 5], [0, 6, 0, 0, -20, 0, 0, 0, -3, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 12, 0, 5, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0], [0, 0, 7, 0, 0, 0, 8, 7, 0, 0, -8, 0, 4, 0, 2, -2], [0, 0, 0, 0, -11, 0, 0, 0, -27, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [48, 0, 0, -24, 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, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, 0, -10, -15, 0, 0, 10, 0, -2, 0, -1, 1], [-1, 0, 0, 1, 0, 6, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 7, 3, 0, 0, 7, 0, -9, -7, 8, 5], [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, 4, 0, 0, 0, -8, 4, 0, 0, 8, 0, 3, -8, -2, -5], [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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 12, 0, 0, 0, 12, 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, 0], [0, 0, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, -1, 8, -7, -6], [0, 0, 2, 0, 0, 0, 0, 13, 0, 0, -11, 0, 3, 11, 2, -7], [0, 9, 0, 0, 0, 0, 0, 0, -7, 0, 0, -9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 11, 16, 0, 0, -11, 0, 2, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-22, 0, 0, 11, 0, -6, 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, 6, 0, 0, 0, -10, 6, 0, 0, -10, 0, 5, 10, -4, -7], [0, 0, 5, 0, 0, 0, -11, 5, 0, 0, 0, 0, -5, -11, 3, -4], [0, -5, 0, 0, 0, 0, 0, 0, -14, 0, 0, 5, 0, 0, 0, 0], [0, -3, 0, 0, 20, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -38, 0, 0, 0, -19, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, 0, -13, 0, 0, 9, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, -5, 1, 0, 0, -5, 0, -14, 0, 11, -11], [-27, 0, 0, 54, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, -2, 0, 0, -7, 0, -16, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 14, 0, 0, 0, 2, 16, 0, 0, -2, 0, 3, 2, -7, -10], [0, 0, -6, 0, 0, 0, 4, -6, 0, 0, 4, 0, -11, -4, 16, 1], [-26, 0, 0, -26, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 0, -1, -10, 0, 0, 0, 0, 9, -1, 5, 12], [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, 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, 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, 0], [0, 0, 5, 0, 0, 0, -10, -5, 0, 0, 10, 0, 2, 0, 1, -1], [0, 0, -7, 0, 0, 0, 1, -7, 0, 0, -1, 0, -10, 0, -5, 5], [0, 0, 3, 0, 0, 0, -11, -8, 0, 0, 0, 0, -3, 11, 9, -4], [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, 0, 0, 0, 0, -6, 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, -11, 1, 0, 0, 0, 0, -1, -11, 5, 8], [0, 9, 0, 0, 0, 0, 0, 0, 7, 0, 0, -9, 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], [24, 0, 0, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, -7, -3, 0, 0, -7, 0, -17, 7, 18, -5], [0, 0, 7, 0, 0, 0, 0, -6, 0, 0, 0, 0, -7, -13, -3, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_476_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_476_2_q_a();".
// 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_476_2_q_a(:prec:=16)
    chi := MakeCharacter_476_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_476_2_q_a();".
// 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_476_2_q_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_476_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2401, 0, -11760, 0, 49564, 0, -37008, 0, 21087, 0, -3456, 0, 412, 0, -24, 0, 1]>],Snew);
    return Vf;
end function;