
// Make newform 1950.2.y.n in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_y_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_1950_2_y_n();".
// 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_1950_2_y_n();".
// 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 := [14089, -54888, 113253, -135130, 115386, -62556, 29687, -8010, 2556, -380, 87, -6, 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], [-3607897, 7317892, -9800366, 5230858, -3291723, 843982, -335904, 47836, -13279, 880, -176, 0], [-30150460, 78924115, -76354844, 103874238, -61642348, 34751213, -10618216, 3242004, -581520, 107747, -9796, 1070], [8385745, -120239919, 108949118, -81409454, 37476279, -18160751, 3281180, -1646316, 123933, -68637, 1974, -1070], [-38536205, 166006576, -185303962, 185283692, -99118627, 52911964, -13899396, 4888320, -705453, 176384, -11770, 2140], [-8896864806, 22026982222, -32665367269, 24048219416, -14204635072, 5860211676, -1815307513, 505040292, -87970594, 17298796, -1423807, 196298], [-34891536081, 58367776705, -61853002840, 31357497929, -15838418089, 1898810217, -1069324210, -114973215, -20210311, -10853189, 43670, -199579], [-39899132821, 71688382647, -111050635981, 68030262545, -45381571559, 14392851385, -5033911301, 1058307265, -211570737, 30806433, -3011773, 292669], [-13717956122, 36245295240, -53595055148, 44029180333, -25708183318, 11487231632, -3282046186, 981881669, -154475736, 32929734, -2416592, 370049], [-3683244445, -1967511735, 7276280393, -18680221561, 9813617281, -7474956287, 1695380863, -763671335, 94203315, -29116509, 1653947, -370049], [14213570375, -71180117206, 95735578536, -118312226931, 62438213831, -39075448160, 9381467744, -3738605045, 487832491, -136318210, 8187240, -1663873], [-52508450723, 168521884530, -244027871345, 228688296268, -128944017517, 63195515414, -17140264765, 5578721594, -827318553, 192374130, -13193117, 2230220]];
    Rf_basisdens := [1, 737586, 33157458, 33157458, 33157458, 2884698846, 2884698846, 2884698846, 2884698846, 2884698846, 2884698846, 2884698846];
    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_1950_y();"
function MakeCharacter_1950_y()
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    v := [6, 3, 5];
    // 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_1950_y_Hecke();"
function MakeCharacter_1950_y_Hecke(Kf)
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    char_values := [[1, 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, 1, 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 := [[1, 0, 0, 0, -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, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0], [-1, 1, 1, -2, -2, -1, 0, -1, 0, 2, 1, 1], [1, -1, -1, 1, 0, 1, 0, 1, 0, -1, -1, 0], [0, 0, 2, -1, -1, -1, -1, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 2, 0, 0, -1], [1, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0], [0, 0, 2, -1, -1, 0, -1, 0, 0, 0, 1, 0], [1, -2, 0, 1, 0, 2, -1, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 2, 0, -1, 0, 2, -1, 1, 0], [0, 1, 1, -1, -1, -1, 0, -1, 0, 1, 1, 1], [1, -1, 1, 0, -1, -1, 0, 1, -2, 1, 0, 1], [1, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 0, 1, 0, 2, -1, 1, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, -1], [1, -1, -1, 2, 5, 1, 0, 1, 1, -3, 0, -1], [5, 1, -3, 1, -3, 1, 1, -1, 0, -1, 0, -1], [3, 2, -1, 0, -1, 2, 0, 0, 3, -2, 0, 0], [-3, 0, 1, 0, -1, -1, 1, 1, -1, 0, 1, 1], [-5, -1, -1, 0, 1, 1, -1, -1, -3, -4, -1, -1], [2, -1, 2, -3, -2, -2, -1, -1, 3, 5, 2, 2], [-2, 1, 1, -3, -3, -1, 0, -1, 0, -3, 1, 1], [-1, 1, 1, -3, 2, 0, 1, -1, -1, 2, 1, 0], [-2, 2, -4, 1, 3, 0, 3, -2, 2, -2, -1, -2], [7, -2, -2, 3, -9, 1, -1, 1, 2, -1, -1, 1], [-4, -4, -2, 4, -3, 3, 1, 2, 1, -3, -1, -2], [0, -1, 1, 0, 1, 1, -1, 1, 1, -2, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 7, 3, -5, -3, -5, 1, -3, -2, 1, 4, 3], [5, 2, -1, 0, 1, 1, -1, -1, 0, -1, -1, -1], [0, -1, 0, 0, -6, 1, 0, 1, 6, -13, 0, -1], [-2, 4, 2, -5, 1, -3, 1, -1, -2, 1, 1, 0], [7, -1, -1, -1, -3, -2, -1, 0, 5, 2, 1, 1], [5, 5, -1, -2, -13, -3, 3, -3, -5, 4, 1, -1], [-1, 4, 0, -3, -2, 0, 3, -3, -7, 5, 2, -2], [0, -3, 0, 1, 11, 2, -1, 1, 0, 2, -1, 0], [-1, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0], [2, 0, 6, -2, -4, -2, -2, 0, -8, 2, 2, 2], [-9, -1, 1, 0, 8, 1, 0, 1, 8, -4, -1, 1], [1, 2, 1, -1, -1, -1, 0, 0, -2, -1, 1, 1], [-3, 4, 3, -8, 1, -2, 2, -4, 3, -4, 2, 2], [6, 0, -4, 2, -3, 1, 1, 0, 11, -6, -1, 0], [6, 0, 2, -1, -9, -2, 1, 0, -14, 8, -1, 0], [-1, 1, 1, -2, 0, -1, 0, 1, -7, 11, 0, -1], [-6, 1, -1, 0, 6, -1, 1, -1, 14, -7, 0, -1], [-5, -2, 4, -1, 4, -1, -3, 2, 1, 1, 1, 2], [-2, 2, 2, -5, 6, -1, 1, -1, -2, 3, 1, 0], [-1, 2, -3, 1, 7, 3, 2, -2, 4, -7, 3, -3], [2, -2, 0, 0, -4, 4, 0, 0, 4, -4, 0, 0], [-1, 3, 2, -4, -8, -4, 2, -2, -5, 7, -2, 2], [-5, 1, -1, -1, 3, 0, -1, -1, 3, 0, 1, 0], [-1, 0, 1, -1, 2, -1, -1, -2, -7, 19, -1, 3], [4, -3, -3, 7, 7, 3, 0, 3, 0, -1, -3, -3], [-5, 0, 2, -4, -5, -1, -2, -2, 1, -10, 0, 2], [1, -2, -1, 3, -5, 1, -1, -1, 1, 3, -1, 2], [0, -1, -3, 4, 4, 2, -1, 3, -1, -11, -4, -3], [-3, 3, -3, 1, 3, 4, 1, -4, -8, -4, -1, -5], [6, -3, 1, 2, -8, 0, -3, 3, 2, -2, 0, 0], [-2, 1, 1, -4, -4, -1, 1, -1, 0, 5, 2, 1], [-1, -7, 1, 3, 9, 2, -5, 2, -1, 10, -5, 3], [2, 1, 0, -3, -1, -1, -1, -1, 2, 3, 1, 1], [-10, -4, 1, 3, 2, 2, -2, -2, 5, 3, -2, -2], [-2, 4, 2, -4, -4, -4, 2, -2, -15, 15, 0, 0], [7, -1, 2, -2, -1, -1, -2, -2, 4, 5, 1, 1], [-12, 1, 0, 0, 6, 1, 0, -2, -16, -1, 0, -2], [-3, 8, 1, -6, -4, -4, 5, -5, 0, -2, 2, -2], [-9, 0, -3, 3, 6, 3, 3, 0, -3, -3, -3, -3], [7, 7, 1, -5, 3, -4, 3, -1, -3, -4, 4, 1], [-6, 1, -5, 2, 6, 5, 5, -1, 4, -5, -4, -1], [-4, -4, -2, 4, 12, 0, -2, 2, -14, 18, -4, 4], [-13, 5, 5, -10, -16, -5, -2, -5, 0, 5, 3, 5], [-2, -2, -4, 1, 3, -1, 1, 2, -3, 1, -1, 1], [-2, 5, 3, -7, -7, -4, 2, -3, -1, 9, 5, 3], [-1, 6, 1, -2, 1, -6, 0, -2, -1, 0, 0, 2], [0, -1, -2, 1, 1, 1, 1, 1, 0, -1, -1, -1], [-3, -3, 3, -3, -8, 0, -3, 1, 2, 0, -3, 2], [-8, 0, 0, -4, -10, 0, 2, 0, 0, -4, 2, 0], [11, 0, 1, -2, -6, -2, -2, 1, 7, 2, 2, 3], [5, -1, -3, 2, -4, -3, 2, 1, 12, -4, -3, 1], [-1, 8, 1, -7, -7, 1, 6, -6, 0, -5, 5, -5], [11, 6, 2, -1, 7, -4, 2, -2, -2, 7, 4, 2], [-4, 1, -4, -1, 4, 0, -1, 0, -4, 0, 1, 1], [-6, -1, -5, 3, 9, 0, 2, 1, -1, 1, -3, 1], [7, -3, 1, 1, -13, 1, -2, 2, -7, 8, -1, 1], [8, -1, -2, 1, -3, 0, 1, 5, 4, 0, -1, 4], [18, -2, 6, -2, -18, -6, -6, 2, -2, 5, 4, 2], [9, -3, 5, -3, -7, -6, -3, 2, -3, 6, 3, 5], [-3, 2, -4, 3, 3, 3, 1, 1, 2, -1, -3, -3], [-5, 1, 1, -3, 1, -1, 2, -2, -1, 4, -3, 3], [-6, 0, 0, 2, -4, 0, -2, 0, 0, -12, -2, 0], [2, 3, -2, 2, 3, -1, 2, -4, 1, 1, 2, 2], [5, 3, -3, 0, 0, 1, -2, -3, 12, -8, 5, -3], [2, 4, -4, -1, -3, 5, 5, -5, 1, -4, 3, -3], [11, 2, 3, -2, -7, 0, -2, -2, 7, 0, 2, 0], [-8, 3, 1, -5, -10, -2, 1, -1, -1, 20, 2, 1], [12, 0, 4, -2, -4, -4, 2, 2, -10, -6, 4, 4], [5, -13, -5, 9, 11, 9, -1, 5, 4, -11, -6, -5], [14, -3, -5, 6, -15, 5, -1, 1, -7, 6, 1, -1], [-4, -6, -3, 7, 18, 0, -4, 4, -6, 10, -4, 4], [-8, 2, -2, 0, 6, -2, 4, -2, 24, -12, -2, -2], [0, 2, -8, 3, 4, 5, 3, -1, -18, -5, -3, -4], [-14, -4, 6, -1, 11, -2, -3, 4, 6, 0, -1, 4], [-10, 1, -3, 5, 3, 3, 2, 2, -5, -8, -3, -3], [11, -2, 1, 0, 1, 1, -3, -3, 11, 10, -1, -1], [17, 1, -5, 2, -13, 2, 1, -1, 3, -3, 0, -1], [2, 0, 3, -5, -5, -5, 2, 2, -2, 3, 5, 5], [9, 1, -4, 0, 2, 2, 0, 0, -5, -7, -2, -2], [4, -1, -1, 1, -3, 4, 0, 1, 31, -17, -1, 1], [-30, -1, 2, 0, 14, -1, 0, 3, -8, 1, 0, 3], [-11, 3, 1, -2, 7, 2, 3, -3, 7, -6, 0, -3], [2, -8, -2, 10, 14, 2, -6, 4, -4, 14, -6, 2], [-8, 2, 5, -6, 2, -4, 1, -1, -1, 3, -2, 2], [-16, -3, 2, -1, 7, -4, -1, 2, 14, 4, 1, 3], [4, -13, -4, 8, -8, 13, 0, 5, 4, -5, 0, -5], [-1, 13, 1, -7, -6, -8, 5, -5, -6, 4, 5, 0], [-3, 8, 2, -6, -3, -5, -2, -2, -3, 8, 0, 2], [-2, 7, 5, -11, -4, -6, -3, -5, -1, 0, 2, 5], [-1, 0, 1, -1, 0, -1, -1, 2, -1, -1, -1, -1], [-4, 5, 3, -1, -5, -4, -2, -3, -1, -11, 1, 3], [6, -8, -6, 6, 11, 7, 1, 6, 1, 2, -5, -6], [-6, 2, 2, -4, 6, 2, 2, -2, -14, 12, 2, -2], [3, 6, -3, -1, 0, 1, 7, -2, -3, -3, 7, -5], [10, 7, 5, -5, 7, -6, -2, -5, -1, 11, 3, 5], [11, 0, 0, 1, -2, -2, 5, 5, -8, -6, 2, 2], [-1, -2, 9, -3, -4, -5, -3, 1, 11, 5, 3, 4], [-6, 9, 6, -12, -7, -9, 0, -4, -12, 23, 0, 4], [6, 3, -2, -1, -14, 0, 3, -3, 0, 0, 0, 0], [-6, -3, 3, 4, 0, 0, 1, 1, -3, -3, 0, 0], [-8, -8, 0, 1, -3, 4, -5, 0, 4, 6, -5, 0], [-24, 2, 2, -2, 22, -4, 0, -2, 6, -2, 2, -2], [12, -2, 0, 1, -13, -2, 1, 2, 8, -2, -3, 2], [-7, -7, -5, 6, -7, 6, 5, 5, 1, -20, 0, -5], [-16, -5, 6, 3, 5, -2, 3, 3, 4, 2, -3, 0], [-2, -9, -5, 11, 2, 7, 1, 5, 2, 4, -4, -5], [6, -12, -6, 12, 4, 12, 0, 4, 12, -20, 0, -4], [2, 7, -2, -1, -7, -2, 5, -3, -10, 14, 5, -2], [30, 1, -2, -2, -14, -1, -2, -1, 12, 1, 2, 1], [6, 1, 3, -2, -12, -3, 3, -1, 8, -3, -2, -1], [8, -10, -1, 8, -6, 0, -7, 7, 5, 1, -6, 6], [8, 2, -2, 6, 8, 0, 2, 2, -2, 18, 0, -2], [-16, -2, -2, 0, 9, -2, 0, 1, -9, 2, 0, 1], [20, 1, -6, 1, 2, 2, 3, 3, -3, -5, -2, -2], [-2, 3, 3, -3, -3, -2, 2, -3, 1, -1, 1, -3], [-13, -2, -9, 8, 11, 6, 8, -2, 13, -6, -8, -10], [-12, -4, 0, 0, 2, 2, -4, -4, -2, -4, -2, -2], [-2, -9, -2, 7, 19, 9, -5, 5, 16, -17, 1, -1], [10, 1, 4, -2, -2, -2, -2, -2, -5, -3, 2, 2], [-13, -7, 0, 2, 0, 0, 2, 2, 3, 3, 0, 0], [4, 0, -4, 7, 3, 1, 1, 4, -3, -3, 1, -5], [8, -9, 1, 4, -4, -1, -5, 9, 4, 3, -4, 9], [8, 2, 0, 2, -10, -6, -2, 2, 4, -6, 2, -2], [-12, 6, -2, 0, -14, -2, 4, 2, -4, -12, 2, -2], [0, -7, -3, 7, -1, 5, 4, 3, 2, -15, 1, -3], [9, -9, -8, 10, 8, 8, 2, 2, -7, -15, -8, -8], [1, -12, -1, 8, 4, 6, -6, 6, 13, -20, -6, 0], [18, -2, 2, 0, -10, -2, 0, -2, -12, 2, 0, -2], [-12, -4, 0, 4, 32, 0, -4, 4, -7, 7, 0, 0], [-2, 0, 4, -2, 2, -2, -2, 2, -16, 14, 2, -2], [-19, -3, -5, 4, 27, 0, -3, 3, 17, -7, 0, 3], [-12, -4, 2, 1, 5, -3, 1, 2, -30, 3, -1, 1], [24, -4, -8, 8, 4, 4, 8, 8, -2, -6, -4, -4], [-9, 10, 9, -22, -12, -6, 4, -9, 3, -2, 4, 5], [16, 0, 8, -4, -20, 4, -4, 0, -4, 0, 4, 0], [4, 0, -2, -4, -2, -2, 2, 2, 6, 8, 2, 2], [-10, -8, 12, -2, 5, 2, -7, 8, 20, -7, -1, 8], [-37, 1, 5, 1, 16, 2, 1, 1, -16, -2, -1, 0], [-3, -10, 0, 8, 16, -2, -8, 8, 12, -6, -6, 6], [-7, -8, 6, -3, 2, 4, -3, 3, 10, -4, -6, 6], [10, -6, 6, 2, -16, -4, -8, 8, -2, 2, 0, 0], [-5, 0, -3, 3, 4, 3, 3, 1, 19, -3, -3, -2], [-2, 16, 2, -10, -8, -10, 6, -6, 4, -18, 6, 0], [5, -16, -5, 14, -4, 12, -4, 11, -1, 0, -4, -7], [-20, 5, 9, -11, -7, -7, -6, -6, -1, 6, 7, 7], [-2, -4, 2, -1, -1, 1, -3, 6, 10, -25, -3, -3], [12, 4, 0, 0, 8, -2, 4, 0, -2, -20, 4, 0], [-2, 7, -6, 1, 7, -3, 5, -5, 4, -5, 1, -1], [2, 0, 4, -3, 1, -2, -3, -4, 2, 8, 1, 4], [-9, -7, 15, -4, -1, -5, -5, 7, 10, 1, -2, 7], [12, -3, 4, -1, -4, -4, 3, 3, 2, 6, 4, 4], [-1, 13, 9, -11, -7, -7, -6, -6, -3, 4, 7, 7], [-1, -1, 1, 1, 6, 0, -2, 2, -18, 16, 2, -2], [0, -7, -4, 13, 9, 9, -1, -1, -2, -11, -9, -9], [-4, 4, 4, -4, -2, -2, 2, -4, -34, 16, 2, -4], [-1, -2, 1, -3, 7, 3, 1, 1, 5, -11, 1, -2], [10, -3, 1, 4, -11, -3, -5, 5, 9, -10, 1, -1], [-14, 8, 7, -19, -12, -12, -2, -2, -4, 8, 12, 12], [11, 5, 3, -7, 6, -4, 2, -3, -1, 9, 5, 3], [-20, 3, -2, -2, 11, 1, -2, 2, 2, -1, 2, 4], [-24, -1, -4, -1, 14, -2, -1, 4, -1, 2, 1, 5], [1, 0, -1, 5, 10, -3, -3, 4, 7, -19, -3, -1], [8, -1, -10, 6, 6, 6, 4, 4, 3, -3, -6, -6], [-10, -7, -5, 1, -6, 6, 1, 5, 1, 12, -4, -5], [24, 2, -4, -3, -10, -1, -3, -3, 6, 1, 3, 0], [0, 16, 4, -11, -5, -10, 1, -4, -6, 16, 5, 4], [-3, -8, -4, 8, 15, 1, -4, 4, 9, -2, -7, 7], [-29, 1, -1, 1, 15, 2, 1, -4, 6, -2, -1, -5], [8, 7, 9, -4, -4, -4, -5, -5, -8, -4, 4, 4], [3, -3, 6, 0, -2, -2, -6, 6, 7, -11, 4, -4], [-13, 5, 7, -7, 3, -2, -7, -8, 13, 2, 7, -1], [2, -3, -2, 5, -3, 2, -1, -2, -10, 28, -1, 3], [39, -1, -1, 1, -41, -1, 3, 1, -8, 5, -4, 1], [-11, -8, -8, 12, 9, 9, 2, 2, 3, -6, -9, -9], [-8, 4, 0, -6, -16, -2, 8, 0, -2, 0, 8, 0], [-20, -9, -3, 4, -11, 6, -6, 3, 3, 2, -9, -3], [4, 14, -6, -6, -16, 0, 12, -12, -24, 20, 4, -4], [23, 0, -11, 7, -6, 7, 7, 4, -5, -7, -7, -3], [-8, 11, 3, -11, -13, -7, 2, -3, -4, 7, 5, 3], [-2, 5, 2, -8, -2, -1, 4, -8, -8, 23, 4, 4], [-17, -9, 1, 1, -16, 4, -2, -1, 5, -9, -1, 1], [-7, 0, 4, 1, 0, 0, -3, -3, -4, -4, 0, 0], [3, -6, 4, -9, 4, 1, -5, -4, 5, 5, -1, 4], [-2, 16, 2, -11, -18, -9, 7, -3, -14, 11, 7, -4], [2, 0, 14, -4, -8, -4, -4, -2, -32, 4, 4, 2], [-11, -4, 1, 2, 26, 2, -3, 3, -29, 29, 0, 0], [-13, -8, 4, -1, 20, 6, -3, 3, 18, -14, -4, 4], [-2, 2, 2, 0, 0, -6, -4, 0, 22, -42, -4, 4], [17, 2, 4, -3, -23, 4, 2, -2, 16, -11, 0, -2], [-21, -11, 6, 4, 1, 1, -4, -4, 0, -1, -1, -1], [9, 3, -3, 0, -9, -1, 3, -3, 28, -15, 0, -3], [-1, 6, -2, -3, 2, 4, 5, -5, 16, -22, 6, -6], [18, -2, -1, 3, -29, 1, -2, 2, -14, 13, 1, -1], [-2, 11, 2, -10, -35, -5, 6, -10, -21, 45, 6, 4], [-17, -1, 5, -12, -21, -2, -1, -5, 3, -5, 4, 5], [-14, 5, 6, -20, -13, -13, 0, 0, 2, 15, 13, 13], [8, -10, -6, 13, 11, 8, 3, 6, 2, -6, -3, -6], [-2, 12, 2, -10, 10, -6, 6, -8, 4, -10, 6, 2], [12, 1, -12, 8, 0, 9, 8, -3, -10, -9, -8, -11], [-15, 8, 7, -12, 11, -1, 5, -5, -10, 5, 5, -5], [-7, 4, 4, -8, -8, -4, -3, -4, 0, -15, 1, 4], [-18, 0, -10, 5, 25, 3, 3, 0, 25, -14, -3, 0], [0, 11, 1, -15, -5, -6, 4, -1, -5, 1, 5, 1], [36, -7, -2, 3, 1, 1, 3, 3, 2, 1, -1, -1], [-6, 4, -6, 2, 21, 5, 4, -4, 14, -19, 5, -5], [10, -10, -5, 3, 6, 6, -4, -4, 5, -1, -6, -6], [3, -1, -1, 3, -1, 2, 3, 1, 7, -2, -3, -2], [10, -6, 4, 1, -5, -4, -9, 6, -6, 8, 3, 6], [8, -8, -8, 17, 30, 7, -1, 7, 2, -9, -1, -6], [5, -2, -12, 7, 3, 4, 4, 2, -4, 1, -6, 2], [-19, 7, 1, -4, 17, 2, 1, -7, -3, -3, 6, -7], [10, -5, 7, -1, -11, -12, -6, 5, 19, -1, 1, 5], [-13, 5, 1, -1, 6, 4, -1, 1, 8, -4, 1, 2], [9, 8, 2, -5, 10, -5, -3, -2, -3, 26, -1, 2], [-12, 15, 12, -25, -15, -14, 1, -11, -6, 18, 1, 10], [10, 4, -3, -3, 0, 0, 0, 0, 5, 5, 0, 0], [-2, 3, 2, -2, -7, -5, -2, -6, -9, 29, -2, 8], [-4, -4, 2, -6, 3, 1, -9, -2, 3, 15, -7, 2], [42, 1, 6, -4, -24, -3, -4, 2, 2, 3, 4, 6], [8, -4, -4, 10, 4, 4, 8, 4, 0, -30, 4, -4], [-6, 7, -21, 7, 13, 1, 14, -7, 4, -6, -7, -7], [-18, -13, -9, 11, -9, 11, 0, 9, 2, 5, -9, -9], [18, -4, -5, 5, 4, 4, 2, 2, -17, -21, -4, -4], [-13, -4, -4, 4, 14, 3, 3, 4, -25, 13, -7, 4], [-2, -4, 0, 0, -4, 0, 0, 0, 6, 2, -8, 8], [-12, -2, 4, -3, 4, -5, -3, 9, 0, 5, 3, 12], [1, 2, -1, -3, -4, 3, 5, -3, -11, 21, 5, -2], [18, 4, 2, -3, -21, -8, 1, -4, 12, -4, 3, -4], [-18, 9, 0, -9, 21, 3, 9, -9, 18, -21, 3, -3], [-4, 5, 4, -7, -18, -6, -1, 0, -10, 16, -1, 1], [-11, 8, 8, -11, -3, -13, 3, -3, -10, 13, -3, 3], [-7, 3, -5, 12, -2, 1, 0, 5, -4, -8, -5, -5], [-4, 2, -7, 8, 26, -4, -1, 1, -23, 21, 2, -2], [12, -3, 9, -11, 7, -3, -4, -9, 6, 22, 5, 9], [-6, -5, 6, -7, -5, 0, -5, 1, 0, 8, -5, 4], [-6, -1, 6, -9, 3, -2, -3, 0, 12, -20, -3, 3], [42, 4, -6, 0, -18, 4, 0, 2, -12, -4, 0, 2], [-16, 1, 8, -11, 5, -1, 3, -3, -12, 17, -5, 5], [4, -6, -4, 13, 7, 1, -5, 2, 10, -13, -5, 3], [-2, 6, 10, -6, -4, 0, -6, 0, -10, 0, 6, 6], [4, -4, 5, -14, -6, -6, -7, -7, -5, 1, 6, 6], [2, 14, 9, -7, -6, -6, -4, -4, -2, 4, 6, 6], [4, 6, -10, 2, -7, 11, 13, -6, -3, -7, -7, -6], [-7, 6, 7, -18, -6, -2, 4, -16, 5, 12, 4, 12], [-17, 2, -15, 4, 16, 6, 4, -1, 13, -6, -4, -5], [39, -9, -1, 5, -37, 3, -1, 9, 4, 1, -8, 9], [3, -8, -3, 4, 4, 10, 2, 5, -15, 26, 2, -7], [-8, -12, 1, 2, 16, 14, -3, 3, 1, 3, -4, 4], [14, 0, 8, 0, -11, 0, 0, 2, 24, 0, 0, 2], [-2, -2, 2, -1, 8, -1, -3, 1, -2, 7, -3, 2], [6, 4, 0, 2, 10, -2, -4, 0, -2, -38, -4, 0], [14, -19, -5, 23, 22, 12, -3, 5, 7, -36, -8, -5], [-21, 5, 10, -12, 20, -4, 2, -2, -19, 17, 2, -2], [-11, 8, 10, -19, -20, -9, -1, -10, 1, 22, 9, 10], [-18, 4, -3, 3, 4, 4, -2, -2, -18, -22, -4, -4], [9, -3, -4, 0, 4, 4, -4, -4, 23, 19, -4, -4], [-18, 6, 2, -2, 30, -14, 0, 0, -20, 22, -2, 2], [3, 8, -8, -5, -2, -2, 7, 7, -6, -4, 2, 2], [25, -5, -9, 10, -8, 5, 10, 5, -17, -5, -10, -5], [0, 12, 0, -5, 13, -7, 5, -6, -6, 7, 5, 1], [46, -6, 12, -3, -51, -4, -7, 6, 6, 2, 1, 6], [17, 6, -1, 3, -8, 9, 3, 0, 25, -9, -3, -3], [-14, -3, -9, 6, 18, 11, 5, 3, 6, -7, -8, 3], [19, 0, 5, 2, -12, 2, 2, -1, 11, -2, -2, -3], [-31, -3, 17, -4, 7, -7, -4, -5, -3, 7, 4, -1], [1, 9, 4, -14, -5, -5, -8, -8, 2, 7, 5, 5], [-33, -2, -1, 5, 4, 4, -2, -2, 15, 11, -4, -4], [-8, -3, 7, -6, -5, -2, -10, -7, 5, 14, -3, 7], [25, -9, -3, 6, -15, 2, -7, 9, 21, -7, -2, 9], [17, 8, -8, 0, -20, 10, 11, -8, -18, 0, -3, -8], [-33, -5, -11, 11, 22, 6, 11, 1, 20, -6, -11, -10], [29, 8, -4, -11, -2, -2, -3, -3, -14, -12, 2, 2], [-4, 0, 2, -1, 12, 0, -10, 0, 18, -9, 10, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_y_n();".
// 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_1950_2_y_n(:prec:=12)
    chi := MakeCharacter_1950_y();
    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_1950_2_y_n();".
// 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_1950_2_y_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![26244, -87480, 318816, 55728, 81846, 12528, 14052, 2328, 1105, 100, 45, 4, 1]>],Snew);
    return Vf;
end function;