
// Make newform 1638.2.m.k in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_m_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_1638_2_m_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_1638_2_m_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 := [16, 76, 345, 164, 233, 36, 110, 14, 15, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [438544, -66732, -301041, 1003571, -156870, 19422, -67528, 8466, -2241, -503], [3620228, 274164, 1236807, 579550, 644490, -79794, -28346, -34782, 9207, -3064], [-3118196, -3391749, -3196425, -478621, -1723536, -625302, -334006, -12699, -10833, -3977], [-2149816, -9757146, -3491505, -6543167, -967302, -3082518, -375260, -413376, 54315, -27409], [-1074620, -3963994, -2349679, -2887781, -440214, -1521222, -200758, -216152, 29941, -14545], [165664, 16896855, 5556402, 10649492, 744876, 5037264, 561428, 695481, -96426, 45928], [5377120, 29957226, 17738031, 20509619, 4653846, 10555542, 1662752, 1399728, -178473, 97465], [742892, 14145267, 5109630, 11604142, 367938, 5520312, 484042, 805587, -117222, 51344]];
    Rf_basisdens := [1, 1, 2044008, 1022004, 1022004, 2044008, 340668, 1022004, 2044008, 1022004];
    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_1638_m();"
function MakeCharacter_1638_m()
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    v := [3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_m_Hecke();"
function MakeCharacter_1638_m_Hecke(Kf)
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    char_values := [[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, 1, 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+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, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 2, -1, 1, 1, -1], [0, 1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0, 1, -1], [1, 2, 0, 0, 2, 1, -1, 0, 1, 1], [0, 0, 1, 1, 1, 0, 1, 0, 2, -1], [0, 0, 0, 0, 1, 0, 0, -1, 1, 1], [-2, -1, 0, 0, -1, -2, 1, 1, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, -1, 1], [2, 0, 0, 0, 1, 2, -1, 1, 0, 0], [0, 3, 3, 1, 0, -1, -1, 1, 1, 0], [0, -1, -1, 1, 1, -1, -2, 1, 2, -1], [3, -1, 0, 0, -2, 3, 1, 1, -1, -1], [0, 0, 4, 2, -1, 0, -1, 0, 1, -2], [3, -1, 0, 0, 0, 3, -1, 0, -1, -1], [0, 0, 0, -2, 0, -1, 1, -2, -1, 0], [0, 3, 3, 0, 2, 4, -2, 0, 2, -2], [2, 3, 0, 0, 0, 2, 1, 3, 1, 1], [0, 1, 1, 0, 2, 2, -2, 0, 2, -2], [0, 0, 0, -1, -1, 0, -1, 0, 0, -1], [3, 0, -7, -2, 1, 0, 1, 0, -4, 5], [0, 0, 0, 2, -1, 2, -1, 2, 1, 1], [0, 1, 1, -2, -2, -4, 4, -2, -4, 2], [0, -3, -3, -1, 0, -3, 1, -1, -1, 0], [-1, 0, -2, 0, 0, 0, 0, 0, -3, 3], [5, 0, 0, 0, 2, 5, 0, -2, 2, 2], [0, 2, 2, 1, -1, -6, 1, 1, -1, 1], [-4, 1, 0, 0, 3, -4, -2, 0, 1, 1], [0, 1, 1, 2, 1, -5, 1, 2, -1, -1], [-1, 0, -5, -2, 1, 0, 1, 0, -2, 3], [0, 0, 0, 3, 1, -1, 0, 3, 0, -1], [-9, 2, 0, 0, -1, -9, 2, -4, 1, 1], [-2, -6, 0, 0, -4, -2, 4, -2, 0, 0], [3, 4, 0, 0, -1, 3, 0, 3, -1, -1], [10, 3, 0, 0, 0, 10, -1, -1, -1, -1], [-2, -2, 0, 0, -2, -2, 5, -2, 3, 3], [-3, 1, 0, 0, -4, -3, 0, 1, -4, -4], [0, 1, 1, -1, -1, -6, 3, -1, -3, 1], [-9, 0, -1, 2, 0, 0, 0, 0, 2, -2], [5, -4, 0, 0, -1, 5, -2, 0, -3, -3], [0, 4, 4, -2, 1, -5, -2, -2, 2, -1], [-5, 1, 0, 0, -5, -5, 2, -2, -3, -3], [2, 0, -5, -1, 2, 0, 2, 0, -2, 4], [-4, 3, 0, 0, -1, -4, -2, 0, -3, -3], [-5, 6, 0, 0, 0, -5, 0, 3, 0, 0], [-5, 0, -5, 1, -1, 0, -1, 0, 1, -2], [0, 5, 5, 0, 1, 1, 2, 0, -2, -1], [0, 4, 4, 0, -2, -5, 1, 0, -1, 2], [1, 0, 0, 5, 0, 0, 0, 0, 4, -4], [-7, 0, 1, 3, -4, 0, -4, 0, 2, -6], [0, 1, 1, 0, 3, -6, 0, 0, 0, -3], [-17, 0, 3, 0, 1, 0, 1, 0, 4, -3], [0, -4, -4, 4, 0, 0, -2, 4, 2, 0], [7, 0, -5, -1, -2, 0, -2, 0, -2, 0], [1, 0, -3, -1, 1, 0, 1, 0, -4, 5], [-9, 0, 2, -2, 4, 0, 4, 0, 2, 2], [4, 0, 2, 3, -2, 0, -2, 0, -5, 3], [0, 3, 3, 0, -3, -1, 0, 0, 0, 3], [0, -9, -9, -3, -3, 5, 4, -3, -4, 3], [11, 0, -6, -5, 1, 0, 1, 0, -1, 2], [13, 0, 0, 0, 4, 13, -5, 5, -1, -1], [0, -8, -8, -1, -1, 8, 2, -1, -2, 1], [0, 3, 3, 2, 0, 12, 2, 2, -2, 0], [1, 0, 0, 0, 3, 1, 0, -3, 3, 3], [8, 0, 0, -4, -2, 0, -2, 0, -7, 5], [-10, 0, -1, 0, -4, 0, -4, 0, 0, -4], [-4, 0, 0, 0, -5, -4, 9, 0, 4, 4], [0, 5, 5, -1, 4, -1, -4, -1, 4, -4], [0, -4, -4, -2, -3, -8, 0, -2, 0, 3], [0, 3, 3, -1, 0, -10, 0, -1, 0, 0], [-17, -10, 0, 0, -7, -17, 4, -6, -3, -3], [3, -10, 0, 0, -7, 3, 3, 0, -4, -4], [17, -6, 0, 0, 2, 17, -3, -1, -1, -1], [-5, 2, 0, 0, -1, -5, -1, 2, -2, -2], [-14, -2, 0, 0, 0, -14, -4, -1, -4, -4], [9, 0, 7, 6, 2, 0, 2, 0, 3, -1], [-9, 0, 3, 3, -2, 0, -2, 0, -5, 3], [11, 4, 0, 0, 8, 11, -9, 1, -1, -1], [-16, 0, 0, 4, -3, 0, -3, 0, 3, -6], [0, 0, 0, -1, 0, 0, 3, -1, -3, 0], [0, -3, -3, 3, -5, 14, 9, 3, -9, 5], [4, 0, 5, 4, -3, 0, -3, 0, 2, -5], [0, -2, -2, -3, 2, -4, -1, -3, 1, -2], [0, 10, 10, 7, -1, 2, 1, 7, -1, 1], [8, 0, 8, 5, 4, 0, 4, 0, 3, 1], [0, -1, -1, 0, 1, -23, -3, 0, 3, -1], [-8, 0, -10, 2, 6, 0, 6, 0, -3, 9], [0, 0, 0, -4, -2, -13, 1, -4, -1, 2], [0, -2, -2, 6, -1, 2, 2, 6, -2, 1], [5, 0, 0, -5, -2, 0, -2, 0, -7, 5], [8, 0, 0, 0, -2, 8, 4, 4, 2, 2], [0, 8, 8, 3, 4, -4, -8, 3, 8, -4], [0, 8, 8, 0, 7, -1, -7, 0, 7, -7], [6, 0, 0, 1, 1, 0, 1, 0, 6, -5], [-17, -4, 0, 0, -1, -17, -1, 2, -2, -2], [-10, 0, 10, -1, 0, 0, 0, 0, 6, -6], [0, 5, 5, 4, 3, 3, 0, 4, 0, -3], [8, 0, 7, 5, 1, 0, 1, 0, 3, -2], [0, -9, -9, -4, -6, 5, 7, -4, -7, 6], [7, 0, 1, 1, 2, 0, 2, 0, 2, 0], [9, 0, 5, 6, 4, 0, 4, 0, 3, 1], [3, -4, 0, 0, 2, 3, -2, 6, 0, 0], [-6, 4, 0, 0, 7, -6, -9, 0, -2, -2], [23, 0, 0, 0, -2, 23, -2, -1, -4, -4], [0, 7, 7, 1, 3, 5, -4, 1, 4, -3], [33, -5, 0, 0, 1, 33, -3, 0, -2, -2], [12, 1, 0, 0, 2, 12, -4, 5, -2, -2], [15, 8, 0, 0, 5, 15, -3, 1, 2, 2], [0, -1, -1, 0, 1, 6, 1, 0, -1, -1], [0, -6, -6, 0, -4, 10, 0, 0, 0, 4], [13, -13, 0, 0, -3, 13, 4, -9, 1, 1], [0, 11, 11, 4, 0, 4, -3, 4, 3, 0], [20, 0, 2, -1, -3, 0, -3, 0, 2, -5], [0, -8, -8, 3, 0, 3, -3, 3, 3, 0], [0, -2, -2, 3, -7, -6, 7, 3, -7, 7], [-4, 0, 13, 4, -6, 0, -6, 0, 2, -8], [0, 10, 10, 1, 6, -9, -6, 1, 6, -6], [0, -3, -3, -1, -10, -9, 6, -1, -6, 10], [9, 4, 0, 0, -1, 9, -1, 3, -2, -2], [0, 3, 3, -7, -1, -12, -3, -7, 3, 1], [15, 0, -1, -7, 1, 0, 1, 0, -3, 4], [-21, 0, 1, -1, -1, 0, -1, 0, 0, -1], [-5, 0, -2, -6, 0, 0, 0, 0, 3, -3], [28, 6, 0, 0, 4, 28, -5, -1, -1, -1], [-11, 5, 0, 0, 8, -11, -4, 1, 4, 4], [-2, 0, -1, -6, -2, 0, -2, 0, -2, 0], [0, -1, -1, -4, 2, -14, -4, -4, 4, -2], [0, -7, -7, 1, -9, -13, 9, 1, -9, 9], [8, 5, 0, 0, 2, 8, -1, 6, 1, 1], [0, 0, 0, 8, 2, 0, 2, 0, -1, 3], [-5, 2, 0, 0, 2, -5, -5, 3, -3, -3], [13, -7, 0, 0, -6, 13, 3, -1, -3, -3], [-23, 2, 0, 0, -3, -23, 2, 1, -1, -1], [3, 0, -3, -2, -3, 0, -3, 0, -14, 11], [-2, 0, 1, 4, 3, 0, 3, 0, 0, 3], [0, -9, -9, 0, -8, -7, 7, 0, -7, 8], [0, 5, 5, -7, 3, 15, -4, -7, 4, -3], [3, 0, 12, -2, -3, 0, -3, 0, 7, -10], [-24, 0, -1, 7, -1, 0, -1, 0, 6, -7], [-25, 0, 0, -4, 4, 0, 4, 0, 7, -3], [10, 0, 0, -8, 2, 0, 2, 0, 0, 2], [0, -11, -11, -6, -7, -6, 9, -6, -9, 7], [0, 3, 3, -1, 4, 4, -3, -1, 3, -4], [10, 0, 5, 4, 1, 0, 1, 0, 1, 0], [20, 10, 0, 0, 3, 20, -4, 1, -1, -1], [0, 6, 6, 6, 7, 31, -8, 6, 8, -7], [0, 10, 10, -4, 5, 5, -9, -4, 9, -5], [7, 15, 0, 0, 8, 7, -10, 1, -2, -2], [11, 11, 0, 0, 2, 11, 0, 3, 2, 2], [-11, 0, 6, 10, 0, 0, 0, 0, 10, -10], [6, 0, 3, -2, -7, 0, -7, 0, -1, -6], [0, 4, 4, 5, 0, 10, -2, 5, 2, 0], [-13, 0, 9, 3, -3, 0, -3, 0, 9, -12], [-27, 7, 0, 0, 1, -27, -1, 3, 0, 0], [7, -15, 0, 0, 1, 7, 2, -2, 3, 3], [2, 0, -1, -6, 2, 0, 2, 0, -1, 3], [1, -20, 0, 0, -4, 1, 4, -11, 0, 0], [17, 0, -5, -10, 5, 0, 5, 0, -2, 7], [0, -1, -1, 4, 2, 18, -2, 4, 2, -2], [-4, 0, -8, -1, -5, 0, -5, 0, -7, 2], [6, 13, 0, 0, 5, 6, -10, 3, -5, -5], [0, -6, -6, -1, 7, 23, -4, -1, 4, -7], [3, -3, 0, 0, -4, 3, -3, 1, -7, -7], [0, 6, 6, 1, 5, -12, 0, 1, 0, -5], [-15, 0, 7, 0, -2, 0, -2, 0, 2, -4], [-6, 0, 5, -7, -2, 0, -2, 0, 0, -2], [0, 6, 6, 5, 5, -13, -4, 5, 4, -5], [-14, -4, 0, 0, -1, -14, -2, 3, -3, -3], [0, 0, 0, 4, -5, 0, 2, 4, -2, 5], [-9, -1, 0, 0, 6, -9, 1, 1, 7, 7], [-1, 0, 12, 3, -6, 0, -6, 0, 7, -13], [23, 6, 0, 0, 7, 23, -8, -3, -1, -1], [0, -12, -12, -2, -4, 2, 8, -2, -8, 4], [28, 10, 0, 0, 11, 28, -1, 7, 10, 10], [-11, -1, 0, 0, 5, -11, 0, -2, 5, 5], [8, 7, 0, 0, 3, 8, 4, 0, 7, 7], [-6, 0, -1, -2, 0, 0, 0, 0, -9, 9], [0, -4, -4, 3, -10, -21, 5, 3, -5, 10], [-17, 0, 1, 3, -6, 0, -6, 0, -4, -2], [16, 0, 4, 5, -4, 0, -4, 0, 3, -7], [0, 1, 1, -1, -6, 12, 7, -1, -7, 6], [0, -3, -3, -2, -5, 3, 5, -2, -5, 5], [17, 0, -5, -2, -1, 0, -1, 0, -8, 7], [7, 13, 0, 0, 7, 7, -5, -7, 2, 2], [-15, -4, 0, 0, 1, -15, -1, -7, 0, 0], [-8, 0, 0, -1, 8, 0, 8, 0, -3, 11], [-11, 0, -10, -4, 2, 0, 2, 0, 3, -1], [15, -10, 0, 0, -10, 15, -2, 3, -12, -12], [0, -13, -13, 0, -5, 0, 8, 0, -8, 5], [0, -6, -6, 9, 0, 12, -2, 9, 2, 0], [2, 0, -16, -5, 5, 0, 5, 0, -4, 9], [37, 0, 1, -10, 2, 0, 2, 0, -9, 11], [0, 9, 9, -3, 10, 7, -10, -3, 10, -10], [31, 4, 0, 0, 0, 31, -1, 5, -1, -1], [5, -1, 0, 0, -12, 5, 11, -4, -1, -1], [-7, 0, -14, -5, 4, 0, 4, 0, 0, 4], [0, -1, -1, 3, -6, -22, 5, 3, -5, 6], [15, 0, 0, -3, -5, 0, -5, 0, -1, -4], [0, 0, 0, -4, -2, 12, -3, -4, 3, 2], [-47, 0, 0, 1, -1, 0, -1, 0, 2, -3], [-8, 1, 0, 0, 9, -8, -8, 2, 1, 1], [-15, 0, 0, 0, 6, -15, -1, -9, 5, 5], [0, -6, -6, -4, -6, -27, 13, -4, -13, 6], [0, 6, 6, -2, -4, -17, 2, -2, -2, 4], [16, -14, 0, 0, -3, 16, 4, -9, 1, 1], [3, 5, 0, 0, 1, 3, 1, -4, 2, 2], [1, 0, 10, 6, -1, 0, -1, 0, 10, -11], [-14, 0, 15, 2, 2, 0, 2, 0, 3, -1], [-18, 0, 6, -2, 1, 0, 1, 0, -3, 4], [-9, -9, 0, 0, -8, -9, 12, -2, 4, 4], [0, -1, 0, 0, 8, 0, -7, -4, 1, 1], [-13, 0, 0, 2, 5, 0, 5, 0, 8, -3], [0, -12, -12, -3, -4, 3, -1, -3, 1, 4], [-18, 15, 0, 0, 6, -18, -7, 5, -1, -1], [-4, 0, 14, 11, -1, 0, -1, 0, 14, -15], [-16, 0, 0, -10, 4, 0, 4, 0, 4, 0], [8, 0, -2, 0, -2, 0, -2, 0, 12, -14], [-6, 0, -2, -3, -5, 0, -5, 0, 7, -12], [15, 0, 3, 3, 2, 0, 2, 0, -1, 3], [0, 5, 5, 10, -3, 19, 1, 10, -1, 3], [-14, 16, 0, 0, 1, -14, 0, 6, 1, 1], [5, 10, 0, 0, 5, 5, -9, 0, -4, -4], [-22, 0, -11, -7, -5, 0, -5, 0, -7, 2], [14, 0, 20, 4, 0, 0, 0, 0, 2, -2], [-37, 0, -2, 9, 1, 0, 1, 0, 5, -4], [-14, -3, 0, 0, -13, -14, 4, -3, -9, -9], [0, -1, -1, 0, -1, 5, -1, 0, 1, 1], [0, -4, -4, -6, -12, -16, 15, -6, -15, 12], [0, 2, 2, -14, -8, -15, 6, -14, -6, 8], [24, 2, 0, 0, 6, 24, -11, -5, -5, -5], [0, -5, 0, 0, -9, 0, 0, 1, -9, -9], [3, 0, -8, 1, 1, 0, 1, 0, -6, 7], [15, -2, 0, 0, 7, 15, -10, 2, -3, -3], [-25, 21, 0, 0, 3, -25, -2, 4, 1, 1], [6, 0, 10, 6, -2, 0, -2, 0, -2, 0], [0, -1, -1, 3, -10, 18, 13, 3, -13, 10], [42, -1, 0, 0, -1, 42, -2, -5, -3, -3], [0, -1, -1, 8, -2, 24, 0, 8, 0, 2], [22, 0, -15, -7, 7, 0, 7, 0, -11, 18], [-31, 0, -7, 2, -1, 0, -1, 0, -1, 0], [23, 0, -10, -7, -2, 0, -2, 0, 1, -3], [0, 4, 4, 2, -9, -26, -2, 2, 2, 9], [0, -11, -11, -10, 4, -10, 4, -10, -4, -4], [0, -6, -6, -15, -8, -27, 13, -15, -13, 8], [0, -21, -21, -8, -2, 11, 7, -8, -7, 2], [15, 0, -16, -1, 3, 0, 3, 0, -12, 15], [-11, 3, 0, 0, 5, -11, 2, 2, 7, 7], [-18, -7, 0, 0, -6, -18, 5, -1, -1, -1], [32, 0, -1, 9, 2, 0, 2, 0, 1, 1], [-34, 14, 0, 0, -3, -34, 1, 5, -2, -2], [-9, 9, 0, 0, 0, -9, 1, -7, 1, 1], [5, -10, 0, 0, -3, 5, 6, -18, 3, 3], [0, -15, 0, 0, -16, 0, 6, 4, -10, -10], [12, 0, 8, 3, -2, 0, -2, 0, 6, -8], [0, -11, -11, -8, -8, 1, 6, -8, -6, 8], [-12, 0, 1, -7, -5, 0, -5, 0, 6, -11], [25, -10, 0, 0, -2, 25, -2, 5, -4, -4], [-26, -8, 0, 0, -11, -26, 4, -4, -7, -7], [-29, 0, 0, 0, -2, -29, -1, -3, -3, -3], [-21, -12, 0, 0, -11, -21, 8, 6, -3, -3], [-20, -9, 0, 0, -4, -20, 0, -11, -4, -4], [-26, 0, 1, -5, -5, 0, -5, 0, -8, 3], [-17, -5, 0, 0, -11, -17, 11, -5, 0, 0], [0, -4, -4, -2, -9, 36, 2, -2, -2, 9], [0, 0, -12, -1, -1, 0, -1, 0, 0, -1], [0, -4, -4, -10, -4, 18, 0, -10, 0, 4], [19, 0, -13, -6, 7, 0, 7, 0, -4, 11], [0, 5, 5, -6, -5, -31, -3, -6, 3, 5], [11, 16, 0, 0, -2, 11, -1, 0, -3, -3], [0, 3, 3, 6, 1, 3, 0, 6, 0, -1], [-9, -3, 0, 0, 2, -9, -6, -2, -4, -4], [0, -10, -10, -5, 2, -12, 12, -5, -12, -2], [-25, 0, -10, -8, -8, 0, -8, 0, -13, 5], [-20, 10, 0, 0, -2, -20, 3, 8, 1, 1], [4, 16, 0, 0, 9, 4, -7, 10, 2, 2], [10, -15, 0, 0, -7, 10, 10, -7, 3, 3], [-35, -9, 0, 0, -12, -35, 13, -9, 1, 1], [0, 4, 4, -9, 9, 0, -13, -9, 13, -9], [0, -7, -7, -5, -4, 1, -3, -5, 3, 4], [7, 0, -11, -10, -1, 0, -1, 0, -12, 11], [5, 5, 0, 0, 3, 5, -1, -11, 2, 2], [0, 5, 5, -2, 0, 9, -9, -2, 9, 0], [0, 1, 1, -2, 1, 31, -1, -2, 1, -1], [6, 6, 0, 0, -5, 6, -1, -2, -6, -6], [-3, 0, 1, 2, 0, 0, 0, 0, -3, 3], [18, 0, -5, -13, -2, 0, -2, 0, -13, 11], [0, -4, -4, 6, 12, 3, -5, 6, 5, -12], [0, -9, -9, -1, -17, 6, 19, -1, -19, 17], [13, 0, 5, -3, -13, 0, -13, 0, -9, -4], [3, 0, -16, -4, 2, 0, 2, 0, -15, 17], [0, -6, -6, -4, 9, 15, -1, -4, 1, -9], [0, -3, -3, 13, 8, 2, 0, 13, 0, -8], [-14, 5, 0, 0, -17, -14, 9, -9, -8, -8], [0, 9, 9, 7, -1, 48, 2, 7, -2, 1], [-18, -3, 0, 0, -4, -18, -1, 12, -5, -5], [-2, 0, 28, 8, -7, 0, -7, 0, 7, -14], [12, -15, 0, 0, 2, 12, -1, -14, 1, 1], [18, -5, 0, 0, 3, 18, 0, 8, 3, 3], [0, -8, -8, 7, -9, -16, 5, 7, -5, 9], [0, -15, -15, 3, -8, -18, 11, 3, -11, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_m_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_1638_2_m_k(:prec:=10)
    chi := MakeCharacter_1638_m();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1638_2_m_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_1638_2_m_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 76, 345, 164, 233, 36, 110, 14, 15, -2, 1]>],Snew);
    return Vf;
end function;