
// Make newform 930.2.bg.d in Magma, downloaded from the LMFDB on 25 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_d();".
// 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_d();".
// 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, 149, 0, 777, 0, 1472, 0, 1350, 0, 653, 0, 167, 0, 21, 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], [-39009, 96303, -317523, 363860, -427777, 709604, -272115, 785743, -108351, 444458, -27571, 125756, -3799, 16820, -200, 831], [39009, 96303, 317523, 363860, 427777, 709604, 272115, 785743, 108351, 444458, 27571, 125756, 3799, 16820, 200, 831], [-48359, 779461, -371643, 3329765, -1717923, 3840543, -2840263, 1159795, -1990385, -421263, -639534, -299191, -92800, -54984, -4831, -3182], [-2484, -24516, 42548, -58629, 244291, -12102, 375106, 46842, 243052, 39051, 74046, 12405, 10382, 1740, 529, 88], [2484, -24516, -42548, -58629, -244291, -12102, -375106, 46842, -243052, 39051, -74046, 12405, -10382, 1740, -529, 88], [-649, 13610, 0, 19115, 0, -74527, 0, -160570, 0, -113585, 0, -35800, 0, -5100, 0, -262], [9899, 462755, 887173, 2039426, 4401064, 2975631, 6485029, 1992478, 4100400, 689145, 1230044, 126552, 170781, 11629, 8649, 419], [-9899, 462755, -887173, 2039426, -4401064, 2975631, -6485029, 1992478, -4100400, 689145, -1230044, 126552, -170781, 11629, -8649, 419], [-9836, -450734, 1540376, -973480, 6141173, 447420, 8222128, 2297667, 4965480, 1867137, 1454534, 622662, 199288, 91321, 10016, 4768], [-262, 649, -52648, 0, -222689, 0, -311137, 0, -193130, 0, -57501, 0, -7954, 0, -402, 0], [-262, -649, -52648, 0, -222689, 0, -311137, 0, -193130, 0, -57501, 0, -7954, 0, -402, 0], [10235, -30506, 1167568, -785551, 4709137, -3114978, 6376893, -4398281, 3874759, -2781923, 1137181, -842077, 155730, -118001, 7817, -6019], [9836, -450734, -1540376, -973480, -6141173, 447420, -8222128, 2297667, -4965480, 1867137, -1454534, 622662, -199288, 91321, -10016, 4768], [-59562, 799649, -1953119, 3647288, -7824115, 4268320, -10775174, 1431910, -6656682, -312912, -1979258, -271620, -273731, -51185, -13836, -2982], [97121, -96303, 2652659, -363860, 9846051, -709604, 13025901, -785743, 7857869, -444458, 2301933, -125756, 315259, -16820, 15834, -831]];
    Rf_basisdens := [1, 37642, 37642, 37642, 3422, 3422, 1298, 37642, 37642, 37642, 1298, 1298, 37642, 37642, 37642, 37642];
    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, 1, 0, 1, 0, -1, 1, 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 := [[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -1, 0, 0, 1, -1, 0, 1, 1, 0, 1, 0, 1, 0], [0, 0, -1, 1, -1, 0, 0, 0, 0, 1, 2, 0, 0, -1, -1, 1], [-2, -2, -1, 3, -1, -1, 2, -2, -2, 0, -2, 0, -1, 2, -2, 1], [-4, 3, 1, -2, -1, 3, -3, 2, 4, 0, 2, 0, 0, 0, 0, 0], [-2, -1, -1, 3, -1, -1, 1, -3, -5, -1, -4, 2, -1, 1, -3, 1], [3, 1, 2, 0, 3, 0, -1, 3, 3, 1, 3, -2, 2, 0, 1, -1], [0, 3, 2, -1, 1, 4, -5, 5, 4, 2, 2, 1, -1, -1, 1, 0], [0, -1, -1, -1, -2, 3, 3, -1, -1, 2, 3, 1, 0, 1, 1, 1], [8, -4, -4, -1, -1, -1, 7, -4, -3, 2, 3, -1, 3, -1, -2, -5], [-3, 3, 4, -1, 1, 1, -5, 2, 3, -1, 0, 1, 0, 0, -1, 0], [2, 1, -1, -3, 1, 0, -1, 2, 1, 0, 3, -3, 0, 0, 3, 0], [6, -1, 2, -2, 3, 5, -1, 2, 3, 5, 7, -2, 3, -3, 3, -3], [13, -5, -4, -2, 0, -5, 8, -8, -2, 6, 4, -3, 3, 1, 5, -1], [-1, 1, -3, -2, 1, -2, 1, -3, 2, 1, 3, -2, 2, -3, -1, 0], [-10, -1, 4, 6, 0, -3, 0, -4, -2, -6, -6, 0, -3, 6, -3, 6], [4, -7, -4, -1, -4, -6, 9, -7, -4, 1, 3, -4, 4, 4, 0, 0], [-7, 5, 7, 1, -2, 2, -1, 7, 0, -4, -2, -2, -2, 3, 1, 5], [-2, 2, 1, 0, -1, 3, -2, 5, 0, -2, -1, 0, -2, 2, 0, 3], [-2, 5, 1, -5, -4, 6, -7, 5, 4, -2, 1, 2, 2, -2, 0, -4], [18, -11, 0, 1, 3, -8, 10, -10, -7, 6, 6, -6, 6, 1, 6, 1], [2, -9, -4, 9, 0, -4, 10, -11, -12, 0, -5, 0, 1, 5, -1, 3], [4, -5, -8, 5, -6, -1, 10, -7, -13, -1, -8, 5, 1, 3, -6, -4], [4, 6, 6, -3, 10, 0, -9, 7, 5, 0, 1, -1, 0, -1, 4, 0], [-12, 7, 0, -4, 2, 9, -7, 9, 11, -3, -1, 3, -1, -2, -3, -2], [4, 7, 1, -3, 0, 4, -5, 3, 6, 3, 1, 0, -3, -5, 0, -7], [4, 0, -5, 3, -2, -1, 10, -8, -5, 0, -3, 1, 1, 4, 3, 4], [-4, 9, 1, -1, 5, 3, -3, 6, 0, -3, -1, 0, -2, 3, 1, 4], [9, -7, 1, 0, -3, -7, 5, -8, -2, 3, 2, -5, 5, 1, 1, -6], [-6, 9, 0, -1, 0, 7, -6, 8, 12, -1, -1, 6, -2, -6, -7, -3], [-2, -1, -1, -3, -3, 6, -1, 5, 5, 3, 3, 3, 0, -2, -1, -4], [5, -7, 5, 3, 1, -3, -1, -6, -10, 1, -2, 1, 2, 2, 4, 2], [4, -3, 4, 3, -1, -2, 0, 3, 0, 5, 7, -5, 2, 3, 6, 4], [-6, 9, 7, 3, 2, -4, -11, 5, 3, -4, -4, 8, -11, -4, 3, 10], [-2, -10, -10, 6, -8, -12, 15, -14, -12, -4, -4, -7, -1, 7, -5, 0], [-1, 6, 7, 2, 6, -2, -5, 3, 9, -1, -2, 1, -2, -3, -2, 1], [17, -8, -2, -6, 2, -6, -3, -2, 1, 8, 15, -2, 4, -9, 7, -4], [-7, -4, -4, 8, -4, -5, 3, 4, -9, -3, -6, -3, -2, 6, -3, 5], [11, -1, 6, -7, 12, -3, -1, 5, 6, 9, 9, -3, 6, -2, 12, -1], [4, -1, 3, 0, 6, -4, -7, 2, 2, 0, 3, -6, 3, 3, 7, 0], [6, 3, 1, -8, -4, -5, 2, -2, 5, 8, 8, 0, -1, -1, 7, 0], [7, 3, 1, 2, 2, -6, 1, -4, 3, 3, 3, -4, -1, 2, 7, 1], [-8, 2, -2, 4, 3, -7, -7, 0, 0, -2, -9, 0, -6, 2, -4, -1], [-2, -2, 6, -4, 6, 1, -3, 12, 7, 0, 2, 0, 2, -3, 3, -1], [-9, 10, 10, 1, -2, 0, -8, 12, 5, -5, -1, 0, -6, 1, -1, 5], [-3, 8, 10, -5, 3, -4, -10, 3, 1, -3, -3, 4, 2, -5, 3, 1], [14, 2, 3, 0, 7, -16, 6, -8, -1, 1, 1, 0, 1, 1, 1, 2], [-11, -3, 0, 9, -2, -7, -3, -11, -15, -8, -12, 3, -6, 3, -4, 6], [-13, 5, 2, 3, 10, 1, -17, 8, 16, -4, -7, -6, -10, 1, -1, 1], [-7, 6, 14, 7, 2, 1, -3, 6, -3, -4, -2, -5, -2, 12, 10, 14], [-6, 1, 3, -1, 6, 2, -11, 11, 7, -1, -1, -1, -2, -6, 0, -3], [-13, 3, -6, 3, -8, 11, 4, 4, -1, -3, -7, 7, -7, -6, -9, 3], [7, 0, 0, 0, -3, -4, 3, -4, 0, 0, 4, 0, 0, 0, 4, 4], [-11, 12, -4, 3, -2, -2, 6, 0, 2, -8, -11, 0, -8, 3, -5, 3], [-2, 8, 0, -7, 4, 1, -4, 6, 10, -2, -2, 4, -2, -5, -2, -3], [-7, 14, 16, 2, 12, -7, -20, 7, 10, -5, -5, 0, -10, -2, 3, 5], [-3, 7, 6, -3, 3, 7, -11, 9, 7, 0, 2, 3, 1, -3, 3, 2], [7, -2, -5, -6, 1, -8, 4, -14, -8, 4, 4, 1, 7, -1, 11, 0], [8, -10, -10, -7, 4, -6, 7, -6, -1, 2, 5, -1, -1, -4, 2, -1], [8, -7, 0, 3, 0, -3, 13, -3, -14, -2, -1, -2, 2, 4, 0, 4], [-3, -9, -14, 3, -16, 1, 18, -14, -14, 3, -4, -2, 2, -1, -4, -9], [0, -11, 13, -6, 6, -3, 0, 4, 2, 1, 2, -4, 3, -1, 3, -1], [9, 1, 10, -10, -6, 4, -1, 6, 2, 8, 10, -8, 8, 0, 10, -6], [-3, -3, 2, 8, 4, -15, 9, -5, -8, -3, -3, -10, -1, 14, 7, 7], [-1, 1, 3, 9, -4, -9, 1, -7, -11, 1, -2, -1, -2, 6, 1, 6], [6, -12, -12, -5, 0, 6, 12, -5, 0, 5, -2, 2, 5, 0, -7, -12], [-26, 16, 20, 0, 5, 15, -30, 30, 25, -6, -2, 1, -6, 0, -6, 0], [-10, 18, 8, -4, 8, 10, -18, 25, 13, 0, -3, 3, 0, 2, 1, 0], [-12, -4, 0, 7, 0, -5, 0, -5, -15, -11, -18, 6, -1, 11, -7, 1], [-2, -12, -7, -4, -1, 0, 8, -3, -7, -2, -1, -2, 5, 4, -2, -6], [-6, 8, 6, 4, -9, -7, -2, -7, 2, -4, -4, 0, 2, 2, 0, 0], [20, -7, 4, -16, 0, 8, 0, 7, 9, 9, 13, -3, 8, -9, 8, -11], [-17, 21, -1, -5, -2, 20, -14, 17, 23, -2, -2, 6, 2, -6, -8, -3], [2, -9, -9, -5, -12, 6, 10, -4, 0, -1, 6, 0, 4, 1, 5, 3], [5, -4, 1, 0, -5, 1, -4, 0, -2, -3, -1, -7, 3, 2, 0, -6], [-7, 12, 8, -4, 8, 11, -7, 11, 15, -8, -16, 10, -4, -4, -11, -7], [0, -1, -9, 2, 2, -5, 2, -15, -5, -2, -8, -1, 5, 1, -3, -6], [13, -15, -4, 10, 3, -16, 12, -23, -18, 13, 0, -5, 3, 10, 1, -2], [-13, 4, -7, -2, -7, 25, -4, 6, 6, -9, -7, 8, -1, -2, -11, -2], [4, 4, -12, 2, -14, 1, -1, -2, -11, 4, 2, 4, -4, -3, -1, 1], [-2, -5, -17, 3, 7, -3, 9, -8, 2, -2, -4, 3, -3, -7, -10, 1], [0, 2, -8, 10, 6, -11, 0, -19, 9, -2, -13, -3, -5, 2, -5, -1], [6, -10, -1, 3, 1, -7, 6, -8, -10, -3, -3, 0, 4, 4, 5, 0], [16, -3, -1, -1, -1, 7, 5, 2, 7, 8, 2, -1, 4, 0, 2, -6], [-22, 14, 7, -7, -3, 23, -8, 18, 8, -7, 0, 10, 3, -3, -14, -3], [32, -18, -4, -16, 1, -1, 24, -10, -6, 13, 22, -9, 26, 3, 12, -13], [14, -10, -7, -5, -1, 3, 19, -1, -3, 7, 9, -13, 13, 4, 0, -9], [-8, 7, 6, 1, -4, 9, -6, 9, 9, -7, -9, 11, -4, -11, -13, -2], [-18, 2, -6, 4, -7, 10, -12, 15, 10, -5, -8, 11, -11, -11, -19, -3], [-3, -12, 4, -2, 5, 6, 3, 2, 2, 1, 0, 7, -1, -3, 0, 2], [0, -1, 1, -3, 16, 4, -17, 7, 19, 2, 0, 2, 1, -5, -5, -7], [5, 8, 3, -10, -13, 4, 8, -8, -7, 10, 14, -8, 10, 10, 8, 0], [-18, 13, 17, 1, -21, 6, -9, 4, -1, -3, 1, 1, -1, -1, -2, 5], [-6, 17, 5, -10, 10, 9, -9, 18, 26, 3, 3, 3, 6, -8, 0, -4], [-9, -7, -14, 12, -14, -5, 7, -26, -19, -2, -6, -2, -6, 9, -7, 5], [7, 0, -2, -4, -10, 7, 5, 15, 6, -1, -2, 3, 3, -4, -1, -4], [-4, -9, 3, 4, 4, 6, 3, 5, 3, -8, 4, -12, 12, 8, 0, -4], [18, -9, -5, 15, 19, -20, 24, -15, -13, 9, -8, -1, -9, -2, 0, 4], [1, 12, 14, 2, 0, 15, 8, 3, 0, -2, 6, 4, 0, -2, -2, 12], [-16, 3, 3, -2, 7, -6, 0, -11, 13, -7, -2, -4, 1, 7, 1, 5], [-2, 12, 4, -6, -8, 7, -7, 0, 1, 6, 7, -2, 0, 0, 2, 0], [9, -4, 3, 3, -1, 4, 0, -1, 0, 2, -2, 5, 4, -7, -3, -3], [-3, -4, -1, -5, -1, 3, -2, 0, 2, 1, 8, 1, -3, -3, 2, -3], [0, -9, -9, -8, -1, -9, 7, -11, -10, -1, 1, -9, -1, 7, 2, 1], [-8, 4, -10, 0, 3, -4, -10, -11, -4, -1, 0, -7, 7, 7, 7, 1], [6, -22, -10, -1, 11, -14, 3, -15, 0, -4, 4, -5, 3, 1, 2, 0], [-12, 16, 19, -5, 27, 1, -28, 11, 38, -4, 0, 2, 0, -3, -4, -5], [14, 10, 0, -7, 11, 0, -11, 22, 14, 2, 9, -6, 1, -2, 7, -1], [-6, 5, -8, 3, -9, -7, -6, 1, -15, -9, -2, -9, -1, 4, 5, 8], [-22, -7, -6, 8, -12, -2, 1, -2, -19, -12, -12, -11, -2, 12, -1, 6], [4, 2, 0, -16, 2, -2, 8, -1, -1, 5, 10, 1, -1, 2, 18, 6], [5, -11, 12, 18, 4, -13, 0, -14, -4, -5, -14, 4, -9, 5, -9, 10], [-16, 11, 11, 1, -4, 6, -10, 21, 0, -7, 0, -5, -13, 12, -1, 15], [-9, -5, -2, 1, -5, 13, -6, 8, 3, -11, -2, 2, -2, 1, -11, -8], [13, 14, 16, -1, 25, 0, -5, 13, 20, 2, 1, 1, 4, 2, -2, -2], [-39, 21, 18, 1, 5, 23, -33, 26, 22, -16, -9, 17, -17, -8, 0, 9], [-10, -6, -17, 3, 0, -8, 4, -12, -8, -3, -5, 2, -3, -2, -8, -2], [16, -10, -11, 8, 0, 7, 19, -4, -6, 6, 5, 1, 12, 4, -15, -6], [-17, 5, -6, 13, -3, -1, 2, -1, -4, -11, -14, 9, -14, 8, -11, 6], [26, 0, 19, -9, 14, -5, -5, 9, 16, 14, 24, -9, 7, 0, 24, 10], [-9, 19, 11, 1, 2, 2, -17, 10, 16, -1, -12, 22, -8, -8, 2, 0], [-5, -4, 13, 8, 4, -8, 0, 0, -8, -11, -10, -2, -4, 11, -4, 13], [13, 5, 4, -14, -7, 2, -3, 11, 14, 4, 21, 5, 2, -19, 2, -2], [-15, 16, 5, 11, 12, -12, -18, 12, 6, -12, -17, -5, -16, 6, -5, 16], [12, -7, -4, -11, 6, 2, 13, 11, 7, 2, 14, -11, 5, -3, 6, -2], [1, -6, -4, 5, -3, -6, 4, -7, -8, 4, -15, 5, -1, 5, -12, -11], [21, -2, -2, 2, -10, -4, 3, -2, -24, -6, -11, 9, 6, -3, 0, -3], [17, -3, -1, 9, 6, 3, 6, -1, 0, 5, -8, -2, -6, -3, -9, -9], [4, -9, -5, 0, 1, -15, 8, -3, 8, 0, 3, -6, 5, 5, -1, 0], [-4, 6, -3, 6, -5, 2, 0, 2, -8, -3, -5, 1, -3, 3, -3, 6], [10, 14, 23, -9, 46, -4, -22, 23, 40, 7, 7, -6, 6, -6, 13, -3], [17, -7, -9, 1, 8, -13, 22, -17, -22, -10, -10, -10, 9, 15, -5, -4], [-6, 4, -1, 14, -10, -15, -11, -14, -5, 5, -6, 5, -5, 3, 0, 16], [15, -6, 2, 4, -3, -16, 11, -11, -15, 7, -4, 0, 3, 4, -1, -4], [13, -6, -5, 4, 4, -12, 21, -21, -12, 3, 0, -9, 1, 9, 1, -3], [13, -9, -11, -1, -7, 1, 6, -13, 4, 5, 7, 2, 3, -6, 2, -3], [-32, 12, 33, 2, 18, 5, -32, 34, 9, -10, 1, 2, -5, 0, 1, 11], [-10, 13, 17, -6, -11, 17, -10, 20, 5, 0, 1, 5, -5, -2, 4, 6], [15, -17, -2, 0, -5, -17, 23, -20, -18, 0, 9, -18, 13, 13, 13, 0], [37, -22, -24, -4, -26, -4, 28, -21, -23, 3, 7, 4, 7, -13, -1, -13], [-18, -15, -5, -7, -10, 2, -7, 7, 4, -11, 7, -8, 11, 11, 0, 3], [7, 3, -14, 1, 11, -11, 14, -5, 6, 4, -7, 2, -7, -4, 4, 2], [1, 0, -2, 2, 9, -6, 0, -23, -16, -8, -25, 17, -16, -1, -11, 8], [18, -12, -14, -9, 2, 1, -3, -3, 1, 16, -1, 15, 4, -15, 3, -17], [13, -6, -2, -2, -8, -3, 16, -9, -15, 5, 7, 8, 13, 1, 3, 1], [-13, 9, 22, -2, 0, 7, -21, 18, 0, -11, -11, 10, 1, 1, 2, 9], [-27, 10, -3, 15, 0, 34, -23, 18, 16, -12, -25, 13, -6, 2, -23, -11], [-12, 18, 13, 7, 10, 9, -42, 15, 11, -7, -10, 6, -6, -6, 1, 0], [-5, 4, -1, 5, -11, 5, 17, -1, -8, -4, -15, -10, -2, 15, 0, 4], [-14, -7, -5, -5, -11, 14, -3, -22, 9, 5, -2, 5, 7, -2, -3, -8], [-11, 6, 0, -2, 5, 2, -8, 9, 14, 0, 2, -5, -5, 2, -2, 2], [-6, 26, 13, 8, 18, 5, -19, 41, 15, -9, -16, 5, -3, 8, -7, 3], [25, -11, -10, -22, -2, 15, 8, 21, 3, 16, 33, -17, 32, 0, 23, -16], [11, -8, -18, -7, -7, 4, 10, -10, 4, -2, 2, -2, 5, 5, 4, -5], [32, -9, -10, -9, 15, -9, 14, -8, 0, 17, 19, -15, 11, -2, 9, -24], [20, -9, 6, -8, 15, -11, 0, -8, 16, -3, 4, -4, 4, 3, 4, -7], [10, -2, 8, -12, 16, 6, -14, 6, 10, 4, 4, -4, 8, -8, 8, -4], [8, -5, -15, -2, -6, 3, -13, 14, 0, 6, 15, 2, 10, -8, 2, 1], [16, -17, -11, 17, 6, -28, 24, -17, -29, -5, -11, -9, 5, 11, 0, 8], [5, -6, 11, 5, -2, -16, 0, -13, -9, 1, 10, -9, 2, 6, 3, -1], [-4, 21, 26, -8, 5, -8, -17, 13, 5, -8, 10, -1, 0, -8, 14, 14], [6, 3, 6, 13, 23, -22, 5, -13, 10, 5, 7, -9, 7, 11, 5, 7], [-30, 27, 12, -4, 11, 44, -19, 27, 26, -8, -4, 10, 2, -5, -12, -5], [23, -9, -1, -13, -9, -2, 13, 7, 1, 12, 13, 5, 6, -18, -5, -17], [-35, -10, 14, 18, -4, -5, 0, -4, -14, -20, -17, 1, -9, 20, -9, 13], [8, -12, -8, -4, 1, 20, 27, -3, -27, 4, -2, 12, 12, 12, 0, 0]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_930_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![841, 261, -2261, 386, 3906, -7042, 7726, -6337, 4307, -2623, 1556, -848, 401, -166, 54, -11, 1]>],Snew);
    return Vf;
end function;