
// Make newform 1950.2.bc.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_bc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_bc_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_bc_i();".
// 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_bc_i();".
// 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 := [2197, -338, -1196, 190, 104, 154, 98, -134, 8, 34, -8, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [81183629852, 64277898945, -33307875431, -8165110694, -5740233327, 4655211661, 6489146722, -4513282461, 97338749, 591783880, -163110633, -203419], [1269272187404, -833086363785, -554153957987, 414518069862, -272345430479, 135156205247, -372787906, -55879543947, 17147379373, 2300362460, -2097063441, 60968787], [618192439839, 98318215515, 28687118858, 8309942667, 35262141211, -18436259198, 29967292429, 2513656023, -7036618932, 2051209685, 454030419, -120243408], [20321135952, 44426935995, -3499453881, 3964105106, 1601173623, 4136894311, -511723478, 1065744289, 267354099, -226795620, 83789417, 16426431], [11867687, -4439955, -7481036, 77961, -940887, 1844066, 35557, -817691, 233294, 57555, -37023, 4036], [21764327, 2476201, -12957312, -7615595, -2792113, 3193024, -1390803, -291763, -91862, -18833, 28887, -20638], [-40106339, 8732837145, 3156976042, -1963928377, -1228577871, 1496635908, 32851611, -247897203, 110201552, -9226575, -7145579, 5665538], [1108736562037, 550784663520, -633066289511, -58320816489, -85505629812, -46377269759, 138431742257, -56431077566, -20291622981, 11881898255, -845264698, -680246389], [-1182431604497, 783010043855, 859741244016, -574822497041, 206023103197, -93428783546, -22116401667, 75140023171, -35058222714, -1280213355, 4870077513, -1202455216], [16572567908, 99935137380, -15605355549, -26424465301, -3795941083, -1904719281, 7168384238, -695335944, -2253760279, 556127095, 161812093, -78524401]];
    Rf_basisdens := [1, 1, 63907274600, 830794569800, 830794569800, 63907274600, 11796200, 47610004, 12781454920, 830794569800, 830794569800, 31953637300];
    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_bc();"
function MakeCharacter_1950_bc()
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    v := [6, 6, 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_bc_Hecke();"
function MakeCharacter_1950_bc_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], [1, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 1, 0, 0, 1, 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], [1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0], [0, 1, 0, -1, 0, 1, 1, 1, -1, 1, 0, -1], [0, 0, 0, 2, -1, 1, -1, -2, 0, 0, 1, 1], [-3, 2, -1, -3, -1, 1, -1, -1, -2, 0, -1, 0], [0, 2, 2, 1, 2, 0, -2, -1, 0, -1, -1, 0], [0, -1, -3, 1, -3, 3, 3, 0, -1, 2, 2, 1], [0, 0, 2, 1, 0, 1, 1, -1, 0, -2, 0, 0], [-1, 2, 1, 1, -2, 3, -1, 1, 1, 1, 2, -3], [2, -2, -1, 1, 6, -3, -2, 3, 2, -3, -2, 4], [-4, 3, -1, -2, -3, 2, 0, -1, -3, 2, 0, -1], [-3, 1, -3, -1, -2, 1, -2, -3, -2, 0, 0, 2], [3, 0, 2, 2, 0, 1, 4, -3, 2, -1, 0, -2], [2, -1, 0, 0, 7, -2, -3, 2, 4, -2, -3, 3], [2, -2, -1, -1, -6, -1, 0, 1, -1, 0, 1, 0], [2, -1, 1, 2, -2, 1, 1, -4, 2, -1, 1, -2], [0, -2, 2, 4, 0, -2, -2, -2, 2, 0, 2, 2], [0, -2, -2, -2, 6, -2, 2, 2, 2, 0, 0, 0], [1, 0, -2, -6, 2, -3, 6, 6, -4, -1, -2, 4], [-3, 4, -3, -3, 4, 1, -1, 3, -3, 1, -4, 1], [-2, 2, -4, 0, -4, 4, 0, 6, -2, 4, 4, 0], [4, -8, 5, 4, 5, -4, -2, 1, 8, -3, 0, 0], [0, 0, -6, -2, -10, 2, 0, 2, -2, 4, -2, 0], [0, -1, -1, 4, -1, 4, -4, -3, -1, 0, 0, 1], [-1, 1, 0, 0, 7, -1, -3, 2, 2, -1, -3, 3], [0, -2, 0, 2, 0, 2, -4, -2, -2, -2, 0, 2], [2, 2, -4, 0, -4, 6, 8, 0, -2, 2, 4, 0], [1, 0, -4, -4, 3, -2, 1, 2, -3, -2, -4, 6], [0, -1, 0, -2, 5, -3, 8, -2, -1, -1, -1, 1], [2, -3, -1, -1, -8, 0, 1, -3, 2, 0, 0, -1], [3, -2, 0, 0, -1, 0, -1, 0, 5, 0, 2, 0], [2, 2, -6, -3, 2, 2, 4, 5, -4, -1, -1, 4], [2, -1, -2, 1, 2, 4, -1, -1, -5, 3, 2, 0], [4, -4, -2, 6, 0, 0, 2, -6, 2, 0, 0, 2], [-9, 1, -1, -1, 1, -1, 0, 1, -1, -1, -1, 0], [1, -3, -5, -1, -10, 3, 2, 1, 0, 4, 2, 0], [2, 4, -3, -5, 12, 0, 1, 12, -4, 1, -5, 4], [3, -2, 6, 2, -2, 0, 1, -6, 5, -2, 0, -6], [0, -3, -7, -1, -1, 1, 3, 0, -3, 4, 2, 3], [8, -4, -4, -4, 7, -1, -1, 5, 5, -1, -5, 1], [2, -6, 2, 2, 8, -6, -4, 6, 4, -4, -4, 4], [10, 4, 5, 3, -7, 4, 9, -3, -4, 9, 7, -4], [-6, 5, -4, -4, 3, 3, 0, 6, -5, 1, -1, -1], [-2, -4, -2, 8, -6, 2, -6, -12, 4, -2, 4, 0], [0, 0, -7, -2, -9, 2, 4, 7, 0, 7, 4, 0], [-6, -1, 0, 2, -3, 1, -2, -2, 1, -1, 3, -3], [4, 0, -8, -2, -4, 4, -4, 2, -2, 6, -2, 0], [0, 2, -2, 2, -4, 8, 2, -6, 0, 0, 4, -2], [-8, 1, 0, 0, 1, -2, 1, 1, -2, -2, 1, -1], [0, 2, 4, 0, 0, -4, 0, -8, 2, 4, 0, -4], [1, 4, 0, 4, -12, 8, -5, -4, -4, 4, 0, 0], [10, 2, -1, 4, -2, 3, 12, 4, -1, 5, 6, -2], [0, 1, 2, 2, -6, 7, 4, -1, 1, -1, 5, -1], [0, 2, -3, -3, 6, -2, 3, -4, 2, 5, 1, -2], [-8, 6, 6, -4, 0, 6, -2, 8, 0, 2, 2, -12], [-18, 9, -6, -4, -6, 6, -3, 0, -9, 3, 2, -3], [7, -9, 2, 4, 9, -6, -2, 4, 7, -8, -5, 4], [-2, 4, 4, -2, -1, 3, -1, -13, 1, -1, 1, -5], [0, -1, 6, 0, 2, -4, 1, 2, -1, -7, -4, 1], [0, 4, 1, 1, -3, 1, -4, -1, -7, 0, -3, 0], [6, 4, 2, -2, -2, 4, 16, 0, 0, 0, 2, -4], [-8, 0, -2, -2, 8, 0, 2, 6, -2, 0, 0, -2], [-4, 6, -2, -2, 18, 0, 0, 10, -6, 0, -2, 0], [8, -7, 6, 6, 7, -10, 3, 2, 6, -4, -7, 1], [-6, -2, 2, 0, 8, -2, 8, 0, 0, -2, 2, 0], [1, 6, -2, -2, -6, 5, 0, -2, -6, 5, -2, 0], [2, -10, 4, 10, -4, 0, -8, -18, 12, -10, 0, -4], [-4, 4, -8, -14, 8, 2, 0, 10, -4, -4, -12, 0], [-6, 0, 1, 3, -9, 2, -1, -7, 0, 1, 5, -4], [-2, -6, -4, -4, 6, -10, -6, -2, -4, -2, -6, 10], [0, -4, -4, -2, 2, -4, 0, 2, -4, 0, -2, 4], [-14, 3, 8, -8, 1, -2, -11, 0, 0, 0, -5, -6], [8, -2, 1, 4, -21, -2, 2, -23, 2, -1, 2, 2], [4, 0, -3, -5, 6, -7, -4, 5, -5, -2, -5, 0], [10, 3, 4, 4, -6, 9, 2, -4, -5, 9, 6, -2], [4, 5, 7, 5, 8, 3, -9, -5, -7, -2, 0, 0], [12, 7, 2, 0, -8, 8, 14, 0, -7, 9, 8, -9], [-4, 7, 7, 7, 10, 7, -3, -7, 2, 0, 0, 0], [0, 2, 6, -2, -2, -10, 6, 12, 2, -4, -8, -2], [8, 0, 0, 14, -7, 3, 9, -3, 7, 11, 7, -7], [-8, 0, 0, 0, -12, -2, -6, -14, 0, 0, -2, 4], [13, 0, 5, 3, -3, 3, 13, -3, 4, 0, 3, -8], [0, 4, 2, 2, -10, 8, -10, 2, 4, 2, 6, -4], [-6, -4, -6, -6, 8, 2, 6, 10, -2, 2, 0, -6], [0, -2, -4, 1, -24, 6, 2, -1, -6, 5, 3, 0], [-2, -2, 1, -5, -3, -2, -3, -5, 2, -1, -7, 2], [-6, -1, -8, -3, -6, 2, 7, 3, 1, 5, -2, 0], [8, 2, 8, 4, 0, -2, 12, 12, 6, 2, 0, -8], [8, -6, -6, -6, 8, -2, 2, 8, 4, -2, -4, -2], [12, -2, -2, 8, 10, -4, 2, 6, 2, -4, 4, 6], [5, 6, 8, 4, -12, 0, -11, -4, 1, -4, -2, 0], [0, -2, -5, -4, -1, -8, -6, 9, -2, 3, -4, 2], [-2, -2, 0, -4, 4, -10, -4, -4, -2, 2, -4, 4], [-5, 8, -6, -4, -6, 4, 3, 0, -9, 8, 4, 2], [4, -4, 8, 7, -18, -3, -1, -21, 4, 4, 4, 2], [-10, 2, 1, 1, -2, 7, 1, -1, -3, 7, 2, -1], [0, -1, 2, 4, 4, -6, -5, 2, -1, -3, -10, 1], [2, 2, 6, -8, 5, -9, 7, 11, -1, -1, -5, -1], [-10, -6, -4, -4, 16, 4, 8, 14, -2, 4, 4, -8], [20, -1, -1, 7, -2, -5, 5, -7, 1, -2, 2, 9], [8, -6, -10, 2, 4, -6, 2, -2, -2, -4, -4, 16], [0, 8, 17, -2, 5, -8, 0, 3, 8, -9, -6, -8], [-4, 6, -9, -9, -13, -4, 0, -2, -6, -4, -9, 0], [-11, 0, -4, 0, -2, 9, -20, -16, -2, -5, 2, 2], [-32, 8, -12, -12, 2, 2, -14, 4, -8, -4, -10, 4], [11, -5, 2, 2, 5, -9, 15, -7, 2, -1, -5, 3], [8, -10, 6, 6, 20, -8, -2, 6, 12, -8, 4, 2], [0, -2, 4, 0, -4, 0, -7, 4, -2, -6, 0, 2], [-13, 10, -8, -6, 6, 2, -3, 26, -13, 12, 4, 6], [-18, 0, 3, 1, 2, 1, -8, 3, 0, 3, 2, -2], [5, -2, 4, -4, 19, -12, -3, 4, 3, -8, -2, 0], [2, 4, 3, -7, 1, -6, 10, -10, -2, 4, -1, -2], [-2, -6, -2, -2, -16, -2, 8, 2, -4, 0, 4, 0], [8, 5, 5, -7, -6, -3, 13, -9, -4, 6, -2, -2], [6, -2, 10, 2, -8, -6, -4, -2, 4, -8, 4, 0], [0, 5, 0, 4, -1, -3, 4, 4, 5, 5, -7, -5], [0, -1, -14, 2, -15, 7, 2, 8, -1, 13, 5, 1], [2, 0, -3, 11, -4, 5, 2, 3, 1, 6, 11, -2], [8, -15, 5, 6, 15, -12, -8, 3, 15, -10, -6, 9], [0, 2, -5, 5, 3, 4, -1, -7, 2, 7, -1, -2], [-10, 0, 10, 10, 0, -6, -2, -6, 2, -6, 8, 2], [4, 8, -4, 0, 4, 4, -12, 0, -4, 4, -8, 0], [-20, 4, 5, 8, -13, 4, -6, -9, -4, 9, 12, -4], [2, -3, 0, 0, 29, -10, -11, 9, 14, -10, -11, 11], [-4, 5, 1, 0, -12, -1, -1, 0, -6, -1, -5, 0], [16, 4, -10, -2, -4, -2, 20, -2, -10, 8, 2, 12], [3, 9, -4, -4, -13, 3, -3, -6, -6, 3, -7, 3], [-2, 5, 6, -6, 1, 0, 1, -10, 0, -2, -1, -5], [-8, 10, 1, 0, 7, 2, -2, 9, -10, 11, 2, 6], [-16, -11, -1, 1, 3, -7, -15, -4, 11, -12, -6, 3], [0, 3, 2, 0, 4, -8, 3, 4, 3, 1, -8, -3], [0, 1, -4, -6, 3, -5, 24, 2, 1, 5, 1, -1], [0, -2, 8, -1, 8, -10, 2, 1, 2, -9, 1, 0], [6, -3, 8, 8, -28, 2, 5, -26, 3, 5, 10, -7], [9, -4, 4, 4, 10, 4, -5, -4, 3, 0, 8, 0], [9, -5, 9, 5, 6, 1, -4, -5, 6, -4, 10, 0], [-8, 16, -6, -6, -16, 10, 8, -6, -14, 16, 10, -4], [0, -1, -9, 0, -25, 6, 6, 19, -1, 8, 6, 1], [-4, -4, -15, 7, -7, 4, -8, -8, -4, 10, 7, 8], [4, -9, 1, 9, -7, -3, -1, -10, 9, -8, 6, -3], [1, 10, 0, -18, 6, -2, 11, 14, -11, 4, -8, 2], [8, -8, -14, 2, 0, 0, 14, 4, -6, 0, 0, 14], [0, -9, -9, -2, 3, -2, 6, -1, -9, 0, 0, 9], [-8, 0, -4, -2, -24, 0, 8, 2, -8, 2, -2, 0], [2, 4, -17, -1, -12, 20, 17, 5, -9, 4, 12, 5], [1, -3, -2, -2, 13, -9, -9, 3, 12, -9, -11, 9], [24, -6, -7, -7, 33, -8, -4, 18, 10, -8, -11, 4], [-4, 1, -10, -2, -5, 6, -1, 1, -6, 4, 5, 5], [-10, 1, 4, 8, -19, 12, 9, -8, -8, 4, 7, 0], [0, 3, 10, -2, -10, 6, -22, 4, 3, -7, 8, -3], [-13, 12, -9, -9, -10, 7, 3, 1, -15, 7, -6, -3], [0, 2, -9, 5, -6, 12, -9, -6, 2, 11, 7, -2], [0, -1, -8, 8, -14, 18, 13, -4, -1, 7, 10, 1], [12, 2, 0, 0, 14, -4, -2, 6, 0, -4, -2, 2], [4, -12, 8, 12, 8, -8, -8, 4, 12, -4, 0, 0], [-6, 3, -13, -10, 6, -7, 3, 10, 4, 3, -13, 0], [-1, -2, 0, 0, 34, -13, -14, 10, 16, -13, -14, 14], [0, 2, 4, -4, 2, -4, 2, -10, 0, 0, -2, -2], [-4, 4, 2, -14, -10, 10, 8, 0, -4, 6, -4, -16], [10, -2, -3, 3, 4, 1, 8, 7, 1, -2, 1, 2], [1, -1, 12, -12, 13, -17, 1, -4, 0, -9, -13, 1], [0, -6, -20, -3, 14, -8, -6, 13, -8, -7, -9, 28], [-2, 3, 8, 8, -2, 1, 0, -1, -3, 11, 9, 1], [11, -4, -6, 6, -2, -1, 18, -13, 0, 5, 2, 4], [0, 7, 20, -8, 10, -20, -5, 10, 7, -13, -12, -7], [-14, 4, 1, -1, 13, 5, -10, 33, -1, 2, 3, -6], [-3, 2, 18, 4, 5, -3, -15, -21, 11, -7, -5, -13], [-27, 8, -12, -12, -6, -1, -2, 2, -6, -1, -14, 2], [-14, 12, -10, 2, -18, 18, -12, -12, -4, 18, 18, -8], [-2, -3, -13, -6, 12, 1, 5, 6, 0, 7, -3, 0], [14, 2, 10, 12, -20, 10, 16, -24, 8, 6, 14, -20], [-1, 2, -17, -7, -21, 3, -1, 7, -14, 10, -9, 0], [-33, 0, -1, -1, 0, -1, 1, 1, -1, -1, 0, -1], [-16, 3, 3, -9, 2, 3, -13, 1, 0, -6, -6, -6], [-28, -14, 6, 4, 10, -2, -16, 8, 14, -8, 2, -10], [0, 0, -16, 6, -11, 3, 5, 1, -5, 19, 11, 5], [0, 2, 4, -8, 10, -2, 32, -8, 2, -2, 6, -2], [0, -2, -2, -4, -18, 0, -4, 18, -2, 0, 4, 2], [6, -10, -6, 6, 4, -4, 2, 14, 0, -4, -4, 10], [14, -5, 19, 19, 2, -4, -1, -9, 6, -4, 18, 1], [-5, 7, 3, 3, -21, 15, 2, -11, -9, 15, 5, -2], [-2, -5, 2, -2, 7, -14, -9, 17, 0, 0, -7, 5], [-2, -16, 3, 11, 24, -18, -19, 6, 16, -13, -7, 20], [2, 9, 4, -7, -22, 4, 11, -33, -5, 7, 2, -8], [6, -9, -3, -3, -3, -12, -2, -1, 11, -12, -5, 2], [1, -2, -7, 17, -10, 13, -5, -17, 5, 7, 10, -3], [10, -6, 12, 8, 8, 4, 9, 12, 6, 6, 12, -14], [10, 0, 0, 8, -10, 16, -10, -8, 14, 8, 8, 0], [1, -2, -11, -1, 19, -1, -1, 33, -4, -4, -3, 12], [20, -4, 4, 6, -16, 8, -16, -6, 6, 2, 10, 0], [18, -6, 9, 9, 15, -2, 2, 4, 4, -2, 11, -2], [44, -11, 9, 3, 1, -1, 21, 0, 11, -2, 2, -9], [6, -2, 0, 2, 12, 4, 4, 22, 4, -6, 0, -4], [0, 4, 0, 4, 0, 0, 8, 0, 4, 4, -4, -4], [3, 4, 11, -3, 3, -4, 6, -4, 4, -2, -3, -8], [25, -9, -6, 12, 1, -10, 16, -4, 1, 4, 3, 16], [-5, 9, -2, 6, -13, 23, -3, -9, 2, 3, 13, -11], [-18, 17, -12, -12, -39, 4, 1, -13, -18, 4, -11, -1], [8, 5, -1, 2, -20, 5, -13, -2, 0, 3, -3, 0], [1, -5, 5, -15, 10, -15, -4, -5, 0, -10, -20, 10], [-2, 0, -1, -9, -19, 0, -1, -19, 0, -1, -9, 0], [-18, 4, -1, -1, -16, 19, 9, -3, -13, 19, 8, -9], [12, 8, 3, 4, 11, 2, 8, 13, -8, 11, 6, 4], [0, 6, 15, -1, -1, -2, 17, 3, 6, -9, -1, -6], [8, -8, -10, -2, 4, -8, 14, -14, -6, 0, -4, 14], [-2, -4, 16, 16, -8, 4, -6, -8, 14, 4, 12, -20], [-20, 17, -10, -12, -16, 8, -2, -8, -17, 7, -4, 1], [-8, -12, 10, -7, 10, -22, -20, -11, 6, -9, -19, 12], [0, -8, 9, -9, 15, -12, -1, -3, -8, -17, -3, 8], [-22, 15, -10, -10, 5, 6, 7, 11, -22, 6, -3, -7], [0, -5, -8, -4, -19, 7, 14, 12, -5, 3, 11, 5], [-2, 6, 0, 14, -12, 28, -4, -14, -8, 14, 8, 0], [-2, 10, 8, 12, 12, 16, -8, -12, 2, 4, 2, 0], [9, 8, -2, -14, -2, 4, 34, 0, -8, 0, 2, 0], [-12, 10, -2, -2, 26, -8, -8, 10, -2, -8, -10, 8], [-26, 11, 5, -9, 2, 5, -15, 17, -6, 8, 2, -10], [2, -17, 18, 14, -7, 3, 4, -4, 17, 1, 17, -23], [-1, 11, 22, -10, 5, -1, 3, 2, 6, -9, -5, -17], [0, -11, -9, 8, -9, 12, -14, -3, -11, -2, 4, 11], [6, -16, -14, 10, 4, -6, -2, 2, -2, -2, -4, 18], [0, 2, 21, 1, 8, -9, -9, -7, 11, -7, -8, -13], [8, -4, 4, -4, 4, -12, -4, 4, 0, -8, 0, 0], [0, 14, 10, 12, -4, 18, -14, -14, 14, 4, 6, -14], [14, -8, -2, -2, 13, -1, 3, 9, 5, -1, 1, -3], [0, -5, 4, -12, 16, -27, 18, 11, -5, -9, -15, 5], [28, 4, 2, -24, 20, -14, 32, 24, -10, -2, -20, 12], [12, -14, 20, 4, 4, -8, -2, -16, 20, -16, -10, -12], [-14, 3, 19, 5, 4, -2, -37, -17, 12, -6, -4, -15], [0, 4, 1, 11, 6, 9, 16, -15, 4, 3, -2, -4], [0, -4, -1, -8, -27, -4, 4, 31, -4, -3, 4, 4], [20, -10, -2, 12, -2, -12, -2, -14, 10, -12, 0, 14], [0, 18, -13, -13, 4, 7, -3, 7, -15, 7, -16, 3], [35, -2, 1, 1, -12, 7, 11, -1, -9, 7, 12, -11], [1, -1, -4, -4, 1, 1, 5, 14, -4, -3, -1, 5], [19, -13, 11, 11, -10, 11, -6, -11, 6, 0, 24, 0], [38, -8, -1, -9, 13, -6, 13, 7, 8, -9, -15, 4], [4, 2, -4, -9, -14, 0, 6, -33, -4, -5, -7, 4], [0, 0, -16, -2, -20, 12, 26, 8, 0, 16, 14, 0], [4, 6, 4, -10, 26, 4, 6, 30, -6, 10, -6, -2], [0, 2, -16, 12, -20, 18, -4, 2, 2, 18, 6, -2], [-15, 12, 2, 10, -14, 24, -3, 6, 1, 10, 22, -26], [-8, -5, 9, -4, -21, 6, 2, -15, 5, 4, 2, -17], [13, 6, -3, 3, 11, 9, -19, -3, 2, 6, -3, 0], [-18, 0, -12, -12, 2, 2, 6, 10, -6, 2, -6, -6], [-26, 10, -6, -6, 4, -6, 16, 6, -12, 0, -16, 0], [8, 4, 7, 2, 3, 14, 18, 17, -4, 11, 16, -24], [-6, 0, -2, -3, 14, -4, 6, 3, -6, -1, -3, 0], [0, -8, -3, -3, 16, -8, -9, -8, -8, -5, -5, 8], [-2, -7, 12, 12, -46, 6, 5, -40, 7, 5, 18, -19], [8, -4, -3, -6, 29, -10, -6, 19, 4, -7, -16, 16], [1, 6, 10, -18, 8, -3, 12, -17, -4, -13, -8, -2], [0, 0, 7, -16, 9, -24, 25, 15, 0, -7, -8, 0], [23, 3, 6, 6, -15, -3, -7, -14, 4, -3, -1, 7], [10, 6, 14, 3, 6, -8, -16, -3, 14, -11, -3, 0], [-18, 4, -12, 2, -4, 12, -14, 2, -4, -2, 6, 0], [4, 0, -10, 8, 2, -4, -2, -2, 0, -10, 4, 8], [-15, 15, 6, 6, 1, 7, -3, -4, -12, 7, 3, 3], [11, -18, -5, -7, -39, -9, 7, 7, 6, -2, 11, 0], [-19, 1, -8, -8, -11, 7, -1, -2, 0, 7, -9, 1], [0, 0, 12, 12, 4, 0, -32, -4, 0, -12, -12, 0], [-6, 0, 10, -16, 13, -11, -9, -13, -3, -15, -13, 3], [0, 0, 9, 5, 25, -12, -1, -13, 0, -9, -17, 0], [-11, -2, 0, -12, 17, -24, 13, 12, -13, -12, -10, 0], [16, -14, 12, 24, 2, -8, 2, 0, 16, 4, 10, -4], [-12, -12, 2, 0, 24, -2, 24, 0, 6, -2, 12, 0], [-12, -4, 5, 1, 6, -5, -31, -9, 3, -7, -6, 1], [-1, 7, 9, 3, 14, -3, -6, -3, -14, -6, -4, 0], [9, 6, 11, 7, -15, 1, 15, -11, 0, 18, 13, -12], [2, -8, 6, 6, -11, -5, 11, -3, -3, -5, 17, -11], [5, -6, 5, 5, 6, -23, -1, -9, 5, 11, -6, 1], [-6, -2, -1, -11, -19, 4, 1, -15, 2, -3, -7, -10], [-27, 23, -7, -9, -34, 21, -4, -31, -16, 16, 14, -14], [12, 4, 4, 4, 20, 4, 16, 52, 0, 8, 8, -8], [0, 2, 10, 12, -22, 12, 12, -10, -2, 12, 24, -22], [-6, -8, -26, -10, -12, 6, 14, 10, -12, 16, -2, 0], [14, 2, 0, 18, -11, 9, 21, -23, 9, 13, 11, -11], [-1, 4, -6, -3, -12, 0, -3, 3, -12, 3, -7, 0], [16, 0, -1, -11, -2, 3, 16, -15, 1, -14, -11, -2], [0, -1, 6, 0, -19, -7, -4, 26, -1, -7, -7, 1], [-14, 3, -6, 0, 26, -2, -9, 24, -3, -3, -2, 7], [38, 1, -8, -8, -11, 0, 1, -1, -2, 0, -7, -1], [-32, 11, -19, -24, -1, 6, -10, 5, -11, -8, -18, -1], [5, 4, 3, -25, 10, -25, 25, 35, -11, 5, -10, 7], [-26, 6, -12, -12, -46, 14, 6, -14, -12, 14, -6, -6], [1, -12, 6, 6, -16, -11, 10, -6, 2, -11, 16, -10], [-8, 1, 2, 10, -5, 2, -21, -20, 6, 8, 5, -7], [-5, 1, -17, -3, -32, 11, 4, 3, -10, 14, -4, 0], [10, -9, -1, 29, -10, 7, 1, -9, 12, 4, 20, -6], [0, 3, -19, 9, -21, 25, -3, -4, 3, 22, 16, -3], [-6, 16, -4, -30, -4, 4, 10, -6, -16, -2, -14, 0], [-6, 5, 4, 8, 0, 3, 0, 3, -5, 9, 11, -1], [18, -4, -9, -12, 6, -15, -14, 12, -11, -3, -8, 0], [-6, -22, 14, 22, 24, -10, -28, 26, 24, -12, 0, -4], [14, 3, -2, -2, 12, -11, 2, 8, -5, -11, 0, -2], [29, -12, -7, -7, -8, 1, 7, 3, 5, 1, 0, -7], [0, 3, 21, 9, 16, 1, -19, -17, 3, -18, -8, -3], [0, -5, -15, 11, -13, 19, -13, -6, -5, 10, 8, 5], [-58, 5, -16, -22, -2, -2, -31, -4, -5, -11, -24, 9], [33, 2, 3, 17, -5, 5, 35, 11, 2, 16, 19, -8], [-19, -4, 18, -6, 16, -13, -48, -22, 6, -19, -16, -2], [0, 5, 8, -8, -6, 4, -27, 2, 5, -3, 12, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_bc_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_bc_i();".
// 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_bc_i(:prec:=12)
    chi := MakeCharacter_1950_bc();
    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_bc_i();".
// 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_bc_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_bc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1024, -1536, -384, 1728, 304, -1344, 260, 384, -107, -84, 55, -12, 1]>],Snew);
    return Vf;
end function;