
// Make newform 637.2.r.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_637_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_637_r_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_r_f();".
// 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_r_f();".
// 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, 0, -93, 0, 853, 0, -1050, 0, 952, 0, -334, 0, 85, 0, -11, 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], [0, -55035781, 0, -690027, 0, 6321845, 0, -17193085, 0, 6411671, 0, -1852321, 0, 246060, 0, -24498], [-40872159, 0, -690027, 0, 6321845, 0, -17193085, 0, 6411671, 0, -1852321, 0, 246060, 0, -24498, 0], [-23829231, 0, 218482087, 0, -377649654, 0, 236538400, 0, -78484018, 0, 17340370, 0, -2170865, 0, 172099, 0], [-20454191, 0, -2790528, 0, 25566080, 0, -66564370, 0, 25929344, 0, -7490944, 0, 1000291, 0, -99072, 0], [0, 61326350, 0, 3480555, 0, -31887925, 0, 83757455, 0, -32341015, 0, 9343265, 0, -1246351, 0, 123570], [-5618970, 0, 441286822, 0, -514968195, 0, 494434675, 0, -174659083, 0, 45125185, 0, -5861765, 0, 539569, 0], [0, 5618970, 0, -441286822, 0, 514968195, 0, -494434675, 0, 174659083, 0, -45125185, 0, 5861765, 0, -539569], [23341327, 0, 2881788, 0, -26402180, 0, 67022271, 0, -26777324, 0, 7735924, 0, -1048444, 0, 102312, 0], [0, 238534757, 0, 13414713, 0, -122902055, 0, 318752537, 0, -124648349, 0, 36010699, 0, -4835650, 0, 476262], [46491129, 0, -426433173, 0, 508646350, 0, -477241590, 0, 168247412, 0, -43272864, 0, 5615705, 0, -515071, 0], [366555, 0, -3361729, 0, 4460568, 0, -3722089, 0, 1287475, 0, -317920, 0, 40712, 0, -3598, 0], [0, -187583223, 0, 1720586341, 0, -2040907245, 0, 1926159445, 0, -679401319, 0, 174943777, 0, -22708880, 0, 2084782], [0, -19219314, 0, 176278948, 0, -219868809, 0, 196608895, 0, -68898667, 0, 17480377, 0, -2261795, 0, 205171], [0, -106397226, 0, 7579585292, 0, -9751108731, 0, 9362299115, 0, -3317989973, 0, 854461673, 0, -110994637, 0, 10171607]];
    Rf_basisdens := [1, 1, 14163622, 14163622, 42490866, 14163622, 14163622, 42490866, 42490866, 7081811, 14163622, 14163622, 95271, 42490866, 3862806, 42490866];
    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_r();"
function MakeCharacter_637_r()
    N := 637;
    order := 6;
    char_gens := [248, 197];
    v := [4, 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_637_r_Hecke();"
function MakeCharacter_637_r_Hecke(Kf)
    N := 637;
    order := 6;
    char_gens := [248, 197];
    char_values := [[0, 0, 0, 0, 0, 0, 0, -1, 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+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, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 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, -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, 1, 0, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -2, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, -1, -1], [0, 0, -1, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 1, -2, -1], [2, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, -2, 2, 1], [-3, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0], [0, -3, 2, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, -3, 1, 0], [0, 2, 0, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, -1, 0, 3, 0, -1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, -1, 0, 1], [0, -5, 0, 0, 0, 0, 0, 0, -5, 0, -1, 0, 0, 3, -1, 1], [0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1], [-1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, 0], [1, 0, 0, -2, -1, 1, 0, -5, 0, -2, 0, -2, 2, 0, 0, 0], [5, 0, 0, 2, -5, 5, 0, -3, 0, 1, 0, 2, -1, 0, 0, 0], [0, -1, 3, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [3, 0, 0, 4, 0, 5, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, -2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [1, 0, 0, 6, -1, 1, 0, 5, 0, 1, 0, 6, -1, 0, 0, 0], [0, -4, 2, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0], [-4, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, -3, 0], [0, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 0, -2, 0, 0, 7, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -7, -2, 2], [0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, -1, -2], [0, 0, 0, -2, 0, 0, 0, -7, 0, 2, 0, -2, -2, 0, 0, 0], [1, 0, 0, 0, 5, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0], [9, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 2, 7, -7, 0, -1, 0, -3, 0, 2, 3, 0, 0, 0], [0, 5, -3, 0, 0, 0, -5, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, -1, 0, 1], [-13, 0, 0, 0, 5, 0, 0, 13, 0, 0, 0, -4, 2, 0, 0, 0], [6, 0, 0, 2, 0, -2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, -3, 0, -5, 0, 0, 0, 0, -4, -3, 1], [0, -7, 5, 0, 0, 0, -1, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 3, -3, -3], [-4, 0, 0, -4, 4, -4, 0, 15, 0, -2, 0, -4, 2, 0, 0, 0], [0, 0, 8, 0, 0, 0, -3, 0, -1, 0, 0, 0, 0, -8, -3, -1], [0, 1, -1, 0, 0, 0, 5, 0, 0, 0, -1, 0, 0, 0, 0, 0], [8, 0, 0, -4, 0, -2, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -2, 3, -3, 0, -3, 0, -3, 0, -2, 3, 0, 0, 0], [3, 0, 0, 0, 5, 0, 0, -3, 0, 0, 0, -4, -2, 0, 0, 0], [19, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -3, 0, -4], [7, 0, 0, 0, -4, 0, 0, -7, 0, 0, 0, -4, -2, 0, 0, 0], [0, 0, -3, 0, 0, 0, -4, 0, -3, 0, 0, 0, 0, 3, -4, -3], [1, 0, 0, 0, 3, 0, 0, -1, 0, 0, 0, 0, 3, 0, 0, 0], [0, 0, 5, 0, 0, 0, 2, 0, 5, 0, 0, 0, 0, -5, 2, 3], [0, 0, 6, 0, 0, 0, -1, 0, -3, 0, 0, 0, 0, -6, -1, -1], [-13, 0, 0, 0, -3, 0, 0, 13, 0, 0, 0, -2, -4, 0, 0, 0], [-5, 0, 0, -8, 5, -5, 0, -11, 0, -1, 0, -8, 1, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -2, 0, 4, 0, 0, 4, -5, -4], [0, 3, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 3, -3, -3], [5, 0, 0, -2, 0, 7, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [13, 0, 0, 0, -5, 0, 0, -13, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, -2, 0, -5, 0, 0, 0, 0, 3, -2, 5], [0, -1, 1, 0, 0, 0, -7, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -9, 0, 0, 0, 0, 0, 0, -9, 0, 1, 0, 0, 7, 9, -1], [5, 0, 0, 0, -1, 0, 0, -5, 0, 0, 0, 2, -5, 0, 0, 0], [8, 0, 0, 4, -8, 8, 0, -11, 0, 3, 0, 4, -3, 0, 0, 0], [0, 0, 1, 0, 0, 0, -6, 0, -5, 0, 0, 0, 0, -1, -6, -3], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 7, 3, -3], [-9, 0, 0, 0, -8, 0, 0, 9, 0, 0, 0, 2, -1, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -4, 0, 2, 0, 0, -6, -1, -2], [0, 9, 0, 0, 0, 0, 0, 0, 9, 0, -1, 0, 0, -1, -3, 1], [0, 14, -4, 0, 0, 0, 9, 0, 0, 0, 2, 0, 0, 0, 0, 0], [2, 0, 0, 12, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, -4, 0, 0, 0, 0, 0, 8, 2], [0, 3, -3, 0, 0, 0, 5, 0, 0, 0, -3, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -3, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -2, 1, -1, 0, -13, 0, 1, 0, -2, -1, 0, 0, 0], [3, 0, 0, 4, -3, 3, 0, 5, 0, 5, 0, 4, -5, 0, 0, 0], [0, 0, -8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 4, -6], [0, -9, 0, 0, 0, 0, 0, 0, -9, 0, -1, 0, 0, -3, 3, 1], [0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1], [0, 0, 3, 0, 0, 0, -2, 0, 7, 0, 0, 0, 0, -3, -2, 1], [7, 0, 0, 10, -7, 7, 0, -1, 0, 1, 0, 10, -1, 0, 0, 0], [0, -3, -4, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0], [0, -1, -6, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-6, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, -17, 0, 0, 0, 0, 0, 0, -17, 0, -3, 0, 0, 3, 7, 3], [-16, 0, 0, -6, 0, -8, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, -5, 0, -1, 0, 0, 7, 9, 1], [3, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, -10, -1, 0, 0, 0], [-3, 0, 0, 2, 3, -3, 0, 5, 0, 6, 0, 2, -6, 0, 0, 0], [0, 14, 0, 0, 0, 0, 0, 0, 14, 0, 4, 0, 0, -2, -3, -4], [-12, 0, 0, -4, 0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, -4, 8, 0, 0, 0, -3, 0, 0, 0, 6, 0, 0, 0, 0, 0], [-17, 0, 0, 0, -5, 0, 0, 17, 0, 0, 0, 10, 3, 0, 0, 0], [14, 0, 0, 6, -14, 14, 0, -19, 0, 0, 0, 6, 0, 0, 0, 0], [1, 0, 0, 0, -7, 0, 0, -1, 0, 0, 0, 8, -6, 0, 0, 0], [0, -11, 5, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, -3, 0, -2, 0, 0, 0, 0, 7, -3, 6], [0, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 1, 0], [-3, 0, 0, 0, 7, 0, 0, 3, 0, 0, 0, -8, -1, 0, 0, 0], [1, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [7, 0, 0, 2, -7, 7, 0, 5, 0, 5, 0, 2, -5, 0, 0, 0], [0, 2, 10, 0, 0, 0, -1, 0, 0, 0, 6, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 8, 0, -13, 0, 0, 0, 0, 1, 8, 5], [0, 5, -1, 0, 0, 0, -9, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, -6, 0, 3, 0, 0, 0, 0, 1, -6, 5], [7, 0, 0, 0, 1, 0, 0, -7, 0, 0, 0, 8, 3, 0, 0, 0], [0, 0, 5, 0, 0, 0, 7, 0, 6, 0, 0, 0, 0, -5, 7, -4], [-13, 0, 0, 0, -5, 0, 0, 13, 0, 0, 0, 8, 1, 0, 0, 0], [-1, 0, 0, -10, 1, -1, 0, 1, 0, -1, 0, -10, 1, 0, 0, 0], [28, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-20, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 12, 8, -8, 0, 1, 0, 0, 0, 12, 0, 0, 0, 0], [0, -7, 3, 0, 0, 0, -9, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, -3, 0, 3, 0, 0, 8, -14, -3], [10, 0, 0, 10, 0, 2, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 0, -14, 0, 4, 0, 0, 4, 9, -4], [1, 0, 0, -6, -1, 1, 0, -19, 0, -2, 0, -6, 2, 0, 0, 0], [20, 0, 0, 6, 0, 16, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, 0, 9, 0, 3, 0, 0, 5, 3, -3], [0, -3, -3, 0, 0, 0, -3, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, -5, 0, -5, 0, 0, 0, 0, 6, -5, -3], [3, 0, 0, 12, -3, 3, 0, 11, 0, -2, 0, 12, 2, 0, 0, 0], [31, 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, -4, 0, 0, -12, -5, 4], [0, 0, -8, 0, 0, 0, -2, 0, -16, 0, 0, 0, 0, 8, -2, 2], [0, -1, 1, 0, 0, 0, -5, 0, 0, 0, 7, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 0, -14, 0, 0, 0, -2, 0, 0, 0, 0, 0], [12, 0, 0, -4, 0, -6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [-25, 0, 0, 0, -2, 0, 0, 25, 0, 0, 0, 8, 1, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -5, 0, -3], [0, 4, 0, 0, 0, 0, 0, 0, 4, 0, -6, 0, 0, -10, -11, 6], [-15, 0, 0, 0, 15, 0, 0, 15, 0, 0, 0, -12, 8, 0, 0, 0], [0, 0, -5, 0, 0, 0, 7, 0, -12, 0, 0, 0, 0, 5, 7, -2], [-1, 0, 0, 0, -4, 0, 0, 1, 0, 0, 0, -2, -10, 0, 0, 0], [0, 0, 4, 0, 0, 0, 7, 0, 1, 0, 0, 0, 0, -4, 7, -5], [0, 0, 8, 0, 0, 0, -12, 0, 6, 0, 0, 0, 0, -8, -12, -2], [3, 0, 0, 0, -12, 0, 0, -3, 0, 0, 0, 8, -6, 0, 0, 0], [1, 0, 0, -2, -1, 1, 0, -7, 0, 9, 0, -2, -9, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, 0, 9, 0, -3, 0, 0, -3, 1, 3], [0, -9, 0, 0, 0, 0, 0, 0, -9, 0, -7, 0, 0, 3, 7, 7], [8, 0, 0, 2, 0, 16, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [6, 0, 0, -2, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-5, 0, 0, -6, 5, -5, 0, 31, 0, -6, 0, -6, 6, 0, 0, 0], [-3, 0, 0, 0, -1, 0, 0, 3, 0, 0, 0, -2, -1, 0, 0, 0], [-10, 0, 0, 2, 0, -4, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0], [-5, 0, 0, -2, 5, -5, 0, 11, 0, 8, 0, -2, -8, 0, 0, 0], [0, -17, 0, 0, 0, 0, 0, 0, -17, 0, 3, 0, 0, -1, 1, -3], [-3, 0, 0, -14, 0, -7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 7, 3, 0, 0, 0, 1, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, -12, 12, -3], [-3, 0, 0, -8, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 13, 0, 0, 0, -7, 0, 8, 0, 0, 0, 0, -13, -7, 0], [-9, 0, 0, 6, 9, -9, 0, 25, 0, 1, 0, 6, -1, 0, 0, 0], [0, 16, 4, 0, 0, 0, -7, 0, 0, 0, -4, 0, 0, 0, 0, 0], [0, -6, -3, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0], [5, 0, 0, 0, 3, 0, 0, -5, 0, 0, 0, 4, -2, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_637_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_637_2_r_f();".
// 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_r_f(:prec:=16)
    chi := MakeCharacter_637_r();
    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_r_f();".
// 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_r_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_637_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, -93, 0, 853, 0, -1050, 0, 952, 0, -334, 0, 85, 0, -11, 0, 1]>,<3,R![4, 20, 114, -62, 67, -6, 11, -2, 1]>],Snew);
    return Vf;
end function;