
// Make newform 930.2.bg.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_930_bg();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_930_bg_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_930_2_bg_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_930_2_bg_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 := [1, 0, 16, 0, 90, 0, 239, 0, 329, 0, 239, 0, 90, 0, 16, 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], [-4, 1, -33, 0, -16, 0, 166, 0, 253, 0, 135, 0, 29, 0, 2, 0], [-6, 0, -29, 16, 45, 74, 269, 165, 314, 164, 149, 75, 30, 15, 2, 1], [13, 21, 156, 166, 528, 597, 760, 1092, 539, 956, 194, 404, 33, 77, 2, 5], [8, 6, 126, 60, 623, 284, 1244, 598, 1119, 553, 479, 239, 92, 46, 6, 3], [-2, 21, -17, 282, -74, 1151, -165, 2004, -164, 1658, -75, 673, -15, 125, -1, 8], [-21, 15, -282, 106, -1151, 313, -2004, 494, -1658, 403, -673, 165, -125, 31, -8, 2], [2, 21, 17, 282, 74, 1151, 165, 2004, 164, 1658, 75, 673, 15, 125, 1, 8], [6, 2, 60, 97, 284, 668, 598, 1513, 553, 1433, 239, 628, 46, 122, 3, 8], [22, 1, 317, 0, 1410, 0, 2652, 0, 2313, 0, 973, 0, 185, 0, 12, 0], [27, -17, 342, -203, 1435, -981, 2602, -2007, 2211, -1836, 912, -793, 171, -153, 11, -10], [-15, -20, -192, -313, -787, -1480, -1408, -2871, -1194, -2525, -494, -1061, -93, -201, -6, -13], [27, 17, 342, 203, 1435, 981, 2602, 2007, 2211, 1836, 912, 793, 171, 153, 11, 10], [-1, 35, 0, 570, 0, 2606, 0, 4925, 0, 4285, 0, 1795, 0, 340, 0, 22], [26, 48, 342, 547, 1435, 2103, 2602, 3576, 2211, 2927, 912, 1182, 171, 219, 11, 14]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2];
    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_930_bg();"
function MakeCharacter_930_bg()
    N := 930;
    order := 15;
    char_gens := [311, 187, 871];
    v := [15, 15, 8];
    // 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_930_bg_Hecke();"
function MakeCharacter_930_bg_Hecke(Kf)
    N := 930;
    order := 15;
    char_gens := [311, 187, 871];
    char_values := [[1, 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, 0, 0, 0, 0, -1, 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 := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, -1, -1, 0, 0, -1, 0, 1, 0, 2, 0, 1, 0, 0, 0], [1, -1, -1, -2, -1, 0, -3, 1, 1, -2, 1, 2, 1, 2, 2, -1], [-1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, -1, -1, 0], [2, 2, 0, 2, 0, 0, 2, -1, -2, 1, -2, -2, -2, -3, 1, 2], [0, -1, -2, -2, 1, -1, -1, 1, 2, 1, 2, 2, 1, -1, 0, -1], [1, 0, -1, -2, -1, -2, -1, 1, 3, -1, 2, 2, 2, 0, 2, 0], [-3, -1, 3, -1, 2, -2, 0, -1, -2, 0, 2, 0, -5, 1, -2, -1], [-1, 1, -4, 0, -1, 3, 0, 1, 5, -1, 1, 3, 1, 2, -1, 2], [1, -1, 1, -1, 4, 0, 4, 2, 3, 4, 0, -1, -1, -3, 2, 1], [-1, 0, 4, -3, 3, -4, 1, 0, -2, 4, -1, -3, -1, -3, -1, -3], [2, 4, 1, -1, -4, 3, -1, 2, -1, -2, -5, 1, -3, 4, 0, -2], [1, 1, -1, -3, 1, 2, 3, 3, 1, -5, -1, 2, -2, 2, 3, -3], [0, -1, 2, -4, 1, -1, 0, 2, -1, 0, -1, 0, 0, -4, 0, -3], [3, 5, 0, -3, 0, -4, 0, 1, -5, -2, -4, 0, 0, -2, 2, -1], [-1, 2, -5, -6, -3, 3, -2, 3, 2, -6, 1, 5, 0, 8, 3, -3], [8, 0, 0, 1, 1, 5, 5, 3, 3, 0, -1, 1, -1, -4, 7, 2], [0, 3, -2, 1, 2, -2, 0, 0, 1, 3, -1, 0, 5, -2, 0, 0], [4, 6, -4, -5, -3, 0, 0, 1, 0, -4, -6, 2, -2, 4, 6, 0], [0, 0, 1, 1, 0, -3, -4, -5, -1, 2, 5, -5, -1, -1, 0, 2], [-2, 1, 6, 3, 1, -6, 0, -7, -6, 3, -1, -4, -3, 2, -5, -4], [-6, 5, -1, 1, 0, -8, -5, -4, -3, 2, 0, -2, 5, 0, -3, -4], [-2, 3, 0, 4, -4, 0, 1, -4, -3, -2, 0, -2, -3, 8, 0, 0], [3, 3, 4, -2, 1, 0, 1, 3, -7, -3, -7, 3, -7, 5, 6, -5], [-2, -2, 0, -4, 3, 0, 1, 2, 4, 5, 1, -2, 2, -5, 0, -2], [0, -3, 2, -3, 0, -2, 2, -6, -4, -2, 0, -6, 1, 3, 0, -2], [0, 0, 0, -3, 0, 0, -3, 5, 0, -6, 0, 3, 3, 3, 1, -4], [1, -1, 5, 2, 4, 0, 7, -3, -1, -3, 1, -6, -3, -7, -3, 2], [5, -1, -3, 4, 3, 4, 6, 3, 4, -3, 1, -4, 6, -9, 2, 10], [3, 0, 2, 4, -1, 4, 2, 0, 1, -3, -5, 0, 1, 3, -1, 1], [6, 2, 0, 0, -5, 1, 0, 4, 3, -3, -3, -3, 5, 1, 5, 2], [-4, -2, -2, -4, 2, 0, -3, 11, 7, -6, 7, 8, 5, 2, -3, 0], [3, 1, 5, -3, 0, -2, 2, -3, -5, 1, -3, -4, -5, -6, -5, 3], [-3, -6, 0, 10, 1, 7, 5, -12, 6, 7, 0, -6, -1, -6, -2, 3], [-11, -3, 0, 0, 0, 0, -3, 0, 0, -1, 3, 2, 0, 11, -1, -2], [-4, -1, 0, 8, 1, -2, 1, -4, 3, 1, 4, -6, 4, -2, -8, 5], [2, 1, 1, 11, -1, 0, 5, -3, -1, 5, 4, -11, 0, -3, -4, 3], [-4, 6, 1, -1, -2, -7, 1, -6, -1, 0, 0, -6, 2, -2, -2, 2], [-6, 0, -8, -1, -5, 6, -5, -6, 3, -10, 5, 6, 3, 16, 1, -7], [9, 1, 0, 2, -1, 3, 3, 7, 0, -1, -3, 5, -4, -8, 7, 4], [2, -1, 0, -1, -3, 0, 0, 8, 3, -8, -1, 4, 0, 1, 3, -2], [-2, 0, -4, -3, 5, 1, 3, -3, -4, 8, -1, 3, -4, -5, 4, -7], [1, 0, 2, 6, 1, 0, 2, -7, -4, 5, 0, -14, 2, -8, 1, 14], [4, 3, -3, -3, -4, 4, -3, 4, 7, -4, -4, 10, 4, 4, -6, 0], [3, -9, -1, -3, 1, -1, 2, -6, 2, 8, 5, -10, 1, -10, 1, 3], [-2, -1, 15, 2, 3, -3, 6, 6, -12, -2, -7, 7, -15, 5, 3, -2], [3, 1, -7, 1, -8, 7, 2, -7, -4, -11, -3, -6, -2, 11, 0, -6], [2, 1, 0, -12, -1, 1, -2, 10, 0, -1, -3, 12, -1, 3, 2, -10], [-2, -1, 2, -8, -2, -3, -1, 8, 2, -18, 5, 10, -1, 10, 8, -9], [1, -4, 1, -7, -1, 4, 0, 3, 1, -3, 3, -1, -5, -5, -1, -4], [-5, 0, -4, 4, -7, -6, -9, -5, 1, -2, 11, 5, 1, 7, -4, -1], [-6, 2, 5, 5, 2, -9, -3, 3, -7, 3, -4, -1, -3, -1, -2, 4], [-1, -2, 4, 8, -6, 10, 0, 0, 0, 6, -2, 1, -8, 3, 1, -5], [-1, -9, -8, -6, 8, -1, 7, 7, 2, -4, 3, 7, 1, -2, 5, -2], [-3, 1, 0, 4, -3, 0, -10, -3, 6, 0, 9, 1, 3, 4, -4, 1], [1, -2, 7, -7, 9, -8, 9, 1, 3, 14, 0, 5, 0, -14, 2, -1], [2, -5, -5, 10, -4, 8, -2, 10, 8, -3, -2, 3, 13, 2, 1, 7], [-10, 9, 4, 0, 4, -2, 11, 0, -2, 2, -7, -3, -6, 6, -2, 3], [1, -8, 2, 1, 2, -3, 6, -4, 4, 4, 0, -2, 7, -12, -3, 3], [4, 6, -9, -7, 0, 7, 2, -3, 3, -3, -11, -5, 3, 5, 2, -1], [-2, -11, -1, -6, 6, 8, 7, 8, 12, 3, 7, 0, 7, -9, 3, -3], [-3, 5, 1, -14, 7, -7, 4, -2, -4, -9, -11, -2, 0, 0, 7, -7], [2, -6, 0, 10, 2, 0, 2, 5, 6, 2, 0, 7, 2, -2, 7, 10], [-10, 0, 6, -5, -1, -7, 5, -8, -12, 8, -11, 8, -12, 0, 3, -11], [-8, 0, 0, 12, -13, 0, 1, -14, -14, -5, -1, -2, -7, 10, -5, -2], [8, 2, -7, -7, 2, 9, -4, 24, 12, -10, 0, 12, 3, 4, 17, -4], [-1, 4, -12, 1, -1, 6, -7, 9, 11, -8, 4, 8, 12, 11, 5, -1], [-3, -2, 4, 8, 6, 5, 6, -3, -7, 4, -4, -5, -4, -8, -6, 7], [8, -3, 2, 8, 3, -1, -1, 0, -2, 16, -2, -8, 1, -8, -5, 0], [-1, 4, -9, -6, -2, 0, -9, -3, 4, 0, 5, 2, 2, -2, -7, 2], [-15, 0, 0, 1, 12, 4, 4, 0, -4, 8, 4, -5, 4, -9, -16, 2], [-22, -7, 6, 2, 3, -3, 0, -7, 0, -2, 8, -5, 0, -2, -1, 1], [-5, 0, -9, -4, -1, 3, -10, 3, 13, -4, 19, -3, 13, 1, -4, 7], [-4, -19, -5, 2, 5, 0, -9, 2, 15, -6, 19, 4, 9, -3, 2, 2], [10, 16, -10, -10, 0, 10, 3, 8, 7, 1, -13, 3, -6, 5, 5, 6], [4, 2, -10, -4, 4, 9, -5, 14, 14, -2, 4, 2, 10, 6, -1, 4], [-6, 1, -4, -6, -2, 0, -3, -6, 0, -1, 1, 4, 4, 8, 1, -10], [-3, 8, -6, 3, -11, -1, -10, 9, -5, -12, -1, 4, 6, 23, -6, -3], [1, -4, -2, -4, 2, -2, 2, -8, -3, -6, -1, 3, 0, 2, 16, 4], [-2, 1, 1, -3, 8, -4, 1, -8, -3, 14, 2, -12, -8, -20, -2, 6], [-5, -9, 11, -6, 8, -1, -3, 12, -2, 8, 10, 7, -3, -8, 7, -6], [-6, 2, 7, -20, 13, -13, 5, 14, -5, 7, -11, 18, 0, 1, 10, -10], [1, -4, 0, -6, 6, 1, 1, 8, -6, 7, -5, 15, -1, -9, 7, -12], [-3, 2, 1, -5, 1, -6, 0, 2, -3, 4, -3, 3, 3, -6, 3, -11], [-10, -1, -2, -6, 2, 1, -1, 21, -1, -10, -2, 16, 3, 4, -9, 0], [1, 3, 8, -8, 5, 0, 5, 1, -11, 11, -14, 14, -14, 4, 2, -16], [-14, -10, 7, 10, 5, 3, 5, -6, -4, 8, 0, -3, -7, -6, 4, -11], [1, 19, -5, -8, 5, -8, 6, -8, -8, 2, -1, -14, -3, -6, -3, 6], [-14, -17, -1, 5, 4, 9, -9, 5, 9, -5, 13, -6, 10, 10, -9, 11], [-6, -5, -1, -4, 0, 0, -18, 8, 17, -2, 17, 11, 6, 7, -3, -5], [1, 5, -5, -12, 14, -11, 0, 15, 4, -4, 0, -1, 19, -9, -1, 7], [-8, 4, 0, -3, -6, -2, -2, 1, 0, 4, 0, 14, -4, -2, -5, -6], [3, 7, 0, 17, -1, 2, -1, -13, -7, -4, 2, -16, -1, -2, -13, 20], [-13, 0, 7, 8, 2, 1, 3, -11, 4, 15, -4, 11, 4, -4, -10, -1], [1, 1, 3, 5, 6, 2, 3, -1, 3, 8, -6, 5, -6, -10, -8, 2], [11, 8, 2, -11, -3, -1, -2, -2, -5, 4, -5, -18, -2, -10, -5, 11], [9, -4, 2, -10, 2, -2, 2, 20, 0, -8, 0, 10, 2, -9, 10, -1], [6, 4, 9, -8, 0, -7, 7, -12, -12, -7, 2, -15, -13, 5, 3, -4], [15, 2, 2, -13, -6, 0, -10, -2, -10, -8, -2, -4, 6, 4, 4, -7], [-2, 5, -1, 24, -5, 5, 0, -9, 0, 10, -9, 6, 0, 8, -12, 12], [7, -4, 0, 5, -4, -5, -5, -9, -1, 9, -4, -1, 0, -5, 2, 10], [4, 9, 10, -1, -3, 0, 3, -3, -6, 0, -9, -6, -13, 0, -3, -1], [-5, -5, 10, 14, 13, -1, 10, -13, -6, 21, -4, -12, -13, -29, -9, 8], [-6, -1, 7, -7, 1, 8, 2, 10, -7, -12, -13, 7, -8, 6, 7, -10], [0, 9, 7, -3, -4, -7, 9, -3, -7, -9, -13, 0, -14, 4, -5, -3], [-10, -3, 1, -13, 1, 6, 2, 2, 1, 0, 0, 1, -5, 13, -11, -11], [2, -13, -16, 1, -11, 16, -4, 13, 22, -13, 17, 14, 11, 11, 2, 14], [-3, 6, -14, 6, -6, -3, 1, -9, 14, -8, 18, -6, 3, 5, -1, 9], [-3, 0, -1, -12, 3, -1, -1, -2, -1, 2, 2, -9, -3, -1, 8, -1], [10, 0, 4, 15, 6, 6, 5, -2, 6, 6, -4, 2, 6, -4, -3, 1], [10, -14, -4, 2, 0, 11, 4, 5, 18, -5, 7, 5, 4, -5, 10, 7], [-21, -5, 11, 26, 0, 0, 7, -15, -7, 11, -1, -9, 0, -7, -13, 13], [23, -2, 7, -23, 10, 0, 18, 0, 6, -11, 2, 0, -11, -10, 25, 2], [15, 1, -13, -7, 6, 13, 4, 12, 14, -9, -5, 5, 7, 4, 5, 5], [8, -8, -5, -3, 3, 9, 3, 8, 4, -10, 3, 7, 3, 5, 16, -3], [5, 2, 2, 10, -6, -9, 4, 2, -6, -7, 0, 1, 3, -18, 12, 4], [21, 6, -9, 8, -5, 0, -3, 8, 0, 1, -2, -18, 9, -16, 6, 26], [-14, -2, 0, 12, 2, 5, 2, -14, -3, 18, 6, -12, 6, -9, -28, 5], [12, 10, -12, 3, -11, 12, -11, -5, 2, -23, 1, -2, 1, 12, 11, -2], [-10, 4, -4, -1, -13, -11, 0, -6, -10, -15, 0, 10, -9, 24, 10, -9], [-11, -6, 0, -2, -3, -18, -18, -5, -4, 9, 5, 9, 11, -2, -9, -4], [-5, 9, -12, 8, -10, 10, -2, -5, 2, -7, -6, -13, 0, -2, -4, 4], [-2, 15, -11, -9, -21, 4, 0, -11, -3, -25, -4, 2, -6, 18, 2, -7], [1, 2, -1, 18, -2, -1, 5, -28, 1, 9, 7, -18, 1, -8, -10, 28], [1, 5, 3, 20, -9, 2, 4, 0, -11, -9, 0, 0, -13, 11, 20, 1], [3, -11, -9, -3, 8, 8, 0, 3, 17, 3, 10, 2, 9, -8, 4, 3], [0, 3, 1, -10, 0, -1, -15, -5, 14, 2, 16, -5, -4, 10, 0, 2], [7, 15, -5, 3, -11, 0, -7, -10, -7, -2, -15, -20, -3, 1, -14, -1], [13, -1, 0, 5, 3, 19, 19, 9, 14, -11, -3, -10, -2, -3, 8, 10], [-13, 1, -5, 22, -3, 3, -1, -26, 1, -7, 3, -15, 0, 3, -11, 11], [3, 0, 18, 3, 7, -12, 8, -1, -3, 22, -9, 1, -3, -21, 1, -2], [-2, 5, -13, 1, -3, -4, -13, -7, 1, 2, 12, 1, 3, -7, -13, 5], [36, -13, 7, -3, 0, -4, 0, 3, -1, -2, 7, -15, 6, -15, 18, 16], [-4, 6, -4, 4, -12, -5, -7, -11, -8, -4, -1, -4, 4, 5, -6, -4], [7, -15, 15, 1, 6, 5, 5, 1, 5, 12, 1, -5, -10, -13, 6, 6], [-2, -4, 3, 1, 4, -9, 0, 10, -3, -12, 6, -1, 9, 10, 14, -10], [6, 0, 11, -15, -4, 3, 0, 8, -6, 3, -11, -6, -4, -24, -6, -16], [-3, -8, -1, 15, 7, -2, 8, 6, 9, 5, -3, -1, 8, -9, 1, 17], [-17, 3, 0, -18, -9, -3, -3, 11, 0, -16, -3, 22, -6, 23, 1, -36], [8, 9, 3, -2, -3, 6, -3, 10, 9, -7, 3, -6, 0, 5, -6, 1], [8, -1, -4, -5, 0, 0, -14, -14, 10, 8, 10, -18, 5, 1, 4, 12], [7, 2, 0, 1, -4, 0, 4, -13, -10, -11, -6, -12, -5, 24, -13, -12], [10, -1, -5, -3, -4, 8, -4, 10, -2, -21, 2, 9, 2, 20, 20, -9], [-9, 15, 0, -4, 3, -15, 0, -4, -15, 4, 3, -18, -15, 6, 1, 14], [12, 5, -1, -2, 1, 6, 5, 0, -9, -3, -15, -10, -4, 1, 4, -14], [12, -1, -15, -28, -8, 0, -17, 16, 8, -30, 1, 32, 8, 24, 17, -27], [7, 6, 7, 26, -17, 20, 0, 9, -2, -10, -13, -7, -11, 29, -7, 26], [-12, 14, 0, 6, -6, 8, 8, -17, -9, 14, -9, -3, -23, -3, -18, 12], [-1, 15, 4, -1, -18, -18, 0, -6, -20, -12, -19, 1, -3, 12, 1, -8], [-20, -8, -5, 25, -9, 13, -5, -14, 5, -15, 0, -5, 9, 21, -3, -6], [22, 12, -1, -9, 1, 13, 3, 20, 6, -17, -7, -13, -2, 8, -2, -3], [11, -6, 2, -4, 6, 12, 3, 10, -2, 11, -11, 4, -12, 0, 6, -10], [0, -4, -14, 8, -10, 11, -10, 0, 7, -18, 8, 6, 8, 2, 0, 16], [21, -9, -5, -8, 9, 3, -6, 13, 5, 22, -4, 8, -3, -1, 8, -13], [11, 4, -7, 0, -1, 0, 2, -11, -4, 1, -4, -22, 7, -10, -13, -2], [-35, -9, 12, 20, 10, -10, -2, -18, 2, 12, 6, 5, 0, 3, -10, 10], [-4, 0, 16, -4, 8, -12, 4, -4, 4, 30, 0, 4, 4, -26, 4, -8], [-4, -2, -16, 12, 3, 0, 4, -8, -2, 6, 2, -16, 20, -11, -15, 5], [40, -5, 0, 0, 0, 1, -2, 6, 4, -9, 2, -14, 5, -20, 20, 11], [16, -2, 16, -12, 18, -1, 18, 16, -13, -2, -7, -8, -7, -13, 32, -1], [-2, 16, -8, 2, -5, -4, -1, 12, 3, -16, -4, 13, 8, 19, 0, -2], [-12, 7, 10, -5, 18, -10, 11, 2, -10, 30, -18, -3, -5, -15, -15, -3], [14, 15, 11, 11, -11, 4, 23, -24, -18, 11, -22, -25, -34, 0, 2, -17], [-12, 8, -5, -8, -9, 4, 0, 3, 0, -23, -3, 12, -1, 27, 12, -6], [-21, -6, 11, -6, 1, -1, 1, 13, -1, 8, 1, 5, 0, -6, 3, -3], [-4, 2, 0, -14, 4, -16, -16, -5, -14, 13, 8, 18, 6, 6, 10, -28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_930_bg_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_930_2_bg_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_930_2_bg_f(:prec:=16)
    chi := MakeCharacter_930_bg();
    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_930_2_bg_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_930_2_bg_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_930_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![7921, 9523, 16431, 12332, -7674, -3426, 15284, 1581, -8203, -4039, 374, 1174, 691, 272, 76, 13, 1]>],Snew);
    return Vf;
end function;