
// Make newform 637.2.h.m in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_637_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_637_h_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_637_2_h_m();".
// 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_637_2_h_m();".
// 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, 0, 14, 0, 177, 0, 250, 0, 248, 0, 124, 0, 45, 0, 8, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1483013, 0, 41292, 0, 521544, 0, 1213590, 0, 645048, 0, 276768, 0, 50635, 0, 7068, 0], [0, -2256735, 0, 41292, 0, 521544, 0, 1213590, 0, 645048, 0, 276768, 0, 50635, 0, 7068], [408609, 0, 81918, 0, 1034676, 0, 1796116, 0, 1279692, 0, 549072, 0, 106693, 0, 14022, 0], [0, -5036536, 0, 213231, 0, 2693242, 0, 6036084, 0, 3331014, 0, 1429224, 0, 262518, 0, 36499], [-1597470, 0, 253857, 0, 3206374, 0, 6618610, 0, 3965658, 0, 1701528, 0, 318576, 0, 43453, 0], [0, -3854205, 0, 295149, 0, 3727918, 0, 7832200, 0, 4610706, 0, 1978296, 0, 369211, 0, 50521], [0, -860742, 0, -10882743, 0, -14894160, 0, -15333840, 0, -7712676, 0, -2848760, 0, -508380, 0, -64431], [-87020, 0, -10882743, 0, -14894160, 0, -15333840, 0, -7712676, 0, -2848760, 0, -508380, 0, -64431, 0], [-1305850, 0, -16512183, 0, -22175928, 0, -23561464, 0, -12008036, 0, -4592694, 0, -828304, 0, -107927, 0], [-1814963, 0, -22943719, 0, -34019270, 0, -31705946, 0, -15617428, 0, -5418506, 0, -952041, 0, -114457, 0], [-1657053, 0, -20950472, 0, -29266776, 0, -29454090, 0, -14780304, 0, -5420752, 0, -966125, 0, -121794, 0], [0, 2517795, 0, 31833215, 0, 44160936, 0, 44787930, 0, 22492980, 0, 8269512, 0, 1474505, 0, 186225], [0, 1691, 0, 194601, 0, 289428, 0, 297972, 0, 150958, 0, 55358, 0, 9879, 0, 1241], [0, 5638608, 0, 71289117, 0, 100356134, 0, 100055340, 0, 50118444, 0, 18280712, 0, 3254850, 0, 408609]];
    Rf_basisdens := [1, 1, 773722, 773722, 773722, 773722, 773722, 773722, 773722, 773722, 773722, 773722, 773722, 773722, 2734, 773722];
    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_637_h();"
function MakeCharacter_637_h()
    N := 637;
    order := 3;
    char_gens := [248, 197];
    v := [2, 2];
    // 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_637_h_Hecke();"
function MakeCharacter_637_h_Hecke(Kf)
    N := 637;
    order := 3;
    char_gens := [248, 197];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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+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, -1, 0, 0, 0, 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, 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, 1, 0, 0, 0, 0, 0], [0, -1, 0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, 2, 1], [0, -1, 0, 1, 0, 2, 0, 0, -1, 0, 0, 0, 0, 1, 0, 2], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 2, -2], [-1, 0, 1, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, -2, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 3, 1], [3, 0, 3, 0, 4, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, -2, -2, 2], [5, 0, 0, 0, 0, 0, 0, 0, 0, -5, 2, 0, -5, 0, 0, 0], [0, -4, 0, -2, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 4, 0, 1, 0, 1, 0, 0, 0, 1, -1, -4, 0, 0, 0], [0, 3, 0, -1, 0, -4, 0, 0, 3, 0, 0, 0, 0, -1, -2, -4], [0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, -2, -1], [3, 0, 0, 0, 0, 0, 0, 0, 0, -3, 1, -2, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 1, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -1, 0, 0, 0, 0, -3, 4, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -7, 0, 0, 0], [0, 1, 0, 3, 0, 2, 0, 0, 1, 0, 0, 0, 0, 3, 3, 2], [0, -4, 0, -3, 0, 2, 0, 0, -4, 0, 0, 0, 0, -3, 10, 2], [0, 6, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 2, 0, 3, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, -1, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, -2, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, -1, 0, -2, 0, 0, 5, -1, 2, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 3, -2, 3, 0, 0, 0], [4, 0, -1, 0, 4, 0, 0, 0, 0, 1, 4, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -1, 0, -3, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, -6, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, -6, 0, 5, 0, 0, 0, 0, -2, 5, 0, 6, 0, 0, 0], [-6, 0, -3, 0, -6, 0, 5, 0, 0, -2, -6, -5, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 0, -4, 3, 2], [-2, 0, 3, 0, -2, 0, -1, 0, 0, 5, -2, 1, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 7, -2, -8], [0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, 0, 0, 5, 2, 2], [9, 0, 0, 0, 0, 0, 0, 0, 0, -9, 3, 2, -1, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -9, 0], [-8, 0, -2, 0, -8, 0, 5, 0, 0, 1, -8, -5, 2, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 6, 5, -1, 0, 0, 0], [1, 0, 1, 0, 1, 0, 7, 0, 0, -5, 1, -7, -1, 0, 0, 0], [0, 5, 0, -5, 0, 2, 0, 0, 5, 0, 0, 0, 0, -5, 0, 2], [-5, 0, 1, 0, -5, 0, -8, 0, 0, 9, -5, 8, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, -7, 0, 0, 0, 0, 0, 10, -2], [0, -7, 0, 7, 0, -2, 0, 0, -7, 0, 0, 0, 0, 7, -2, -2], [0, -6, 0, 2, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, -5, -3, 3, 5, 0, 0, 0], [-1, 0, 2, 0, -7, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -10, 0, -3, 0, 5, 0, 0, -10, 0, 0, 0, 0, -3, 9, 5], [0, 4, 0, -6, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -1, 0, 11, 0, 0, 4, 0, 0, 0, 0, -1, 2, 11], [15, 0, 0, 0, 0, 0, 0, 0, 0, -15, -6, 2, 6, 0, 0, 0], [0, -8, 0, 1, 0, -1, 0, 0, -8, 0, 0, 0, 0, 1, -5, -1], [0, -4, 0, 6, 0, 9, 0, 0, -4, 0, 0, 0, 0, 6, -3, 9], [-5, 0, -4, 0, -5, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 10, 0, 2, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -7, 0, 5, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 7, 0, -9, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 2, 0, -2, 0, 0, -7, 0, 0, 0, 0, 2, -7, -2], [0, 0, 0, 0, 0, 0, 0, 10, 9, 0, 0, 0, 0, 0, -10, -6], [0, -9, 0, 13, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-15, 0, 0, 0, 0, 0, 0, 0, 0, 15, -5, -6, 15, 0, 0, 0], [3, 0, 5, 0, 3, 0, -1, 0, 0, -5, 3, 1, -5, 0, 0, 0], [13, 0, 6, 0, 2, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-11, 0, -7, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 10, 0, 0, 0, 0, -4, 4, 1], [0, -4, 0, -5, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, -10, 6, 5, -8, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 1, 0, 2, 0, 6, 0, 0, -7, 2, -6, -1, 0, 0, 0], [-2, 0, 3, 0, -2, 0, 4, 0, 0, 22, -2, -4, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 11, -4, 0, 0, 0, 0, 1, -11, -5], [-2, 0, -1, 0, -2, 0, -5, 0, 0, 11, -2, 5, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, 5, 0, 0, 0, 0, 11, -10, -15], [3, 0, -8, 0, -1, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 10, 0, 1, 0, 2, 0, 0, 10, 0, 0, 0, 0, 1, -5, 2], [0, 0, 0, 0, 0, 0, 0, -5, -14, 0, 0, 0, 0, -4, 5, -1], [-13, 0, -7, 0, -12, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, -3, 6, 0, 0, 0], [0, 2, 0, 10, 0, -13, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 15, 0, -10, 0, -4, 0, 0, 15, 0, 0, 0, 0, -10, -8, -4], [12, 0, 0, 0, 0, 0, 0, 0, 0, -12, -4, -10, 8, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 0, 0, -14, -2, 4, 6, 0, 0, 0], [-17, 0, 7, 0, -2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, -10, 0, 11, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -11, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, -3, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 12, 0, -11, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 11, 0, 3, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, -5, 0, 5, 0, -7, 0, 0, -2, 5, 7, 5, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 5, -4, -4, 1, 0, 0, 0], [0, -5, 0, -12, 0, -1, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -8, 0, -3, 0, 0, -2, 0, 0, 0, 0, -8, -1, -3], [0, 0, 0, 0, 0, 0, 0, -1, 4, 0, 0, 0, 0, -6, 1, -8], [0, 6, 0, 2, 0, -14, 0, 0, 6, 0, 0, 0, 0, 2, -1, -14], [-14, 0, 0, 0, 0, 0, 0, 0, 0, 14, -6, -14, -3, 0, 0, 0], [-7, 0, 4, 0, -14, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 3, -2, 4, 5, 0, 0, 0], [0, -15, 0, -5, 0, 1, 0, 0, -15, 0, 0, 0, 0, -5, 21, 1], [-35, 0, -3, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-15, 0, -4, 0, -15, 0, 2, 0, 0, 6, -15, -2, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 9, 0, 0, 0, 0, 1, 5, -12], [0, 0, 0, 0, 0, 0, 0, 4, 12, 0, 0, 0, 0, 1, -4, -1], [0, -2, 0, 3, 0, 9, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0], [13, 0, 9, 0, 13, 0, 3, 0, 0, -10, 13, -3, -9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -9, 0, 0, 0, 0, 2, 1, 15], [0, 0, 0, 0, 0, 0, 0, -3, -13, 0, 0, 0, 0, -1, 3, 20], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 5, -12, -8, 17, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 1, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, -23, 0, 0, 0, 0, 7, -9, -7], [-21, 0, 0, 0, 0, 0, 0, 0, 0, 21, -5, 1, 13, 0, 0, 0], [-16, 0, 4, 0, 9, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, -8, 0, 15, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 16, 0, 12, 0, -4, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -6, 0, 13, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [15, 0, 0, 0, 0, 0, 0, 0, 0, -15, 1, -1, -10, 0, 0, 0], [0, -5, 0, -1, 0, -7, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -4, 0, 8, 0, -12, 0, 0, -3, 8, 12, 4, 0, 0, 0], [0, -7, 0, -2, 0, -15, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-28, 0, 1, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 12, 0, 7, 0, 0, 0, 0, 12, 0, 0, 0, 0, 7, -6, 0], [-8, 0, 6, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, -10, 0, 5, 0, -4, 0, 0, 8, 5, 4, 10, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, -3, 0, 0, 0, 0, -7, 13, 8], [0, 5, 0, -1, 0, -4, 0, 0, 5, 0, 0, 0, 0, -1, -7, -4], [0, 4, 0, 5, 0, -5, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 0, 0, 0, 0, 0, 0, 14, -7, -2, -5, 0, 0, 0], [11, 0, -4, 0, 11, 0, 0, 0, 0, -7, 11, 0, 4, 0, 0, 0], [-17, 0, -6, 0, -13, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, -3, 0, -21, 0, 16, 0, 0, 6, -21, -16, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 15, 0, 0, 0, 0, 0, 3, -2], [0, 0, 0, 0, 0, 0, 0, 14, -5, 0, 0, 0, 0, 7, -14, 4], [0, 2, 0, -4, 0, -23, 0, 0, 2, 0, 0, 0, 0, -4, 4, -23], [0, 6, 0, 2, 0, 4, 0, 0, 6, 0, 0, 0, 0, 2, 19, 4], [0, 3, 0, -9, 0, -11, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, -5, 11, -10, -15, 0, 0, 0], [0, 6, 0, -9, 0, -14, 0, 0, 6, 0, 0, 0, 0, -9, 2, -14], [13, 0, 2, 0, -2, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 0, -12, 0, -3, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, -2, 0, -2, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -17, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -16, 0, 5, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -13, 0, 0, 0, 0, 0, 0, 0, 2, -9, -13], [0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 3, -1, 13], [0, -6, 0, -4, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 0, 0, 0, 0, 0, 0, 0, 18, -9, -8, -5, 0, 0, 0], [1, 0, -2, 0, 1, 0, 9, 0, 0, -16, 1, -9, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -26, -4, 0, 0, 0, 0, -12, 26, 11], [28, 0, 9, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 11, 0, 17, 0, -11, 0, 0, 11, 0, 0, 0, 0, 17, -14, -11], [-9, 0, 0, 0, 0, 0, 0, 0, 0, 9, -9, -11, -4, 0, 0, 0], [-29, 0, 6, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 1, 0, -4, 0, 0, 6, 1, 4, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -29, 0, 0, 0, 0, 1, -1, -2], [0, -7, 0, 15, 0, 0, 0, 0, -7, 0, 0, 0, 0, 15, -10, 0], [0, 0, 0, 0, 0, 0, 0, -9, -7, 0, 0, 0, 0, 11, 9, 2], [-16, 0, 11, 0, -1, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0, 0, -8, 14, 7, -7, 0, 0, 0], [16, 0, 1, 0, 3, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, -6, 0, 0, 0, 0, -14, -3, 21], [5, 0, 0, 0, 0, 0, 0, 0, 0, -5, -5, 8, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 22, 12, 0, 0, 0, 0, 5, -22, -3], [6, 0, 0, 0, 0, 0, 0, 0, 0, -6, 15, -2, 8, 0, 0, 0], [0, 15, 0, -5, 0, 14, 0, 0, 15, 0, 0, 0, 0, -5, 11, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_637_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_637_2_h_m();".
// 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_637_2_h_m(:prec:=16)
    chi := MakeCharacter_637_h();
    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_637_2_h_m();".
// 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_637_2_h_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_637_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -12, -5, 2, 1]>,<3,R![2401, 0, 5880, 0, 11117, 0, 6668, 0, 2760, 0, 698, 0, 129, 0, 14, 0, 1]>,<5,R![1, 0, 12, 0, 113, 0, 344, 0, 792, 0, 410, 0, 165, 0, 14, 0, 1]>],Snew);
    return Vf;
end function;