
// Make newform 912.2.bn.n in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_912_bn();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_912_bn_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_912_2_bn_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_912_2_bn_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 := [6561, -2187, -4374, 1215, 1701, -108, -846, -18, 364, -6, -94, -4, 21, 5, -6, -1, 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], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2187, 18954, -1782, -7911, 2862, 2850, -1012, -1050, 106, 316, -30, -50, 33, 16, -4, 3], [-19683, 8748, 32076, -5427, -13014, 3186, 5388, -958, -2142, 124, 598, -18, -113, 18, 34, -1], [203391, 15309, -95499, -8748, 42066, 13932, -19056, -4534, 5904, 1576, -1226, -612, 385, 153, -53, -4], [111537, -71442, 2430, 8667, 12474, -4122, -20796, 2962, 4734, -364, -1162, -54, 467, 0, -76, 25], [2187, 4374, -1215, -1701, 108, 846, 18, -364, 6, 94, 4, -21, -5, 6, 1, -1], [13122, 729, -6318, -1377, 2646, 900, -1074, -346, 288, 106, -59, -36, 13, 9, -2, -1], [247131, 137052, -113724, -33777, 29430, 20502, -8820, -11786, 3054, 3020, -310, -438, -31, 174, 62, -35], [22599, 18225, -9315, -3564, 2844, 3018, -514, -1508, 154, 450, -16, -86, -25, 31, 13, -2], [-13122, -7290, 10206, 2754, -3240, -1008, 948, 956, -348, -293, 88, 66, -2, -24, -2, 5], [9477, 20412, -14742, -20601, 3294, 7986, -1676, -3210, 350, 1232, -30, -322, 15, 62, -8, -15], [-107163, 729, 83349, -7128, -17874, -2196, 6924, 6274, -3852, -1300, 746, 252, -1, -171, 23, 52], [183708, 12393, -74601, -32643, 16956, 25254, -13644, -10184, 5178, 3056, -508, -1014, 176, 327, -73, -53], [-925101, -239112, 593892, 112995, -176634, -83970, 67308, 49774, -27354, -15868, 5738, 3570, -679, -1398, 86, 289]];
    Rf_basisdens := [1, 1, 1, 23328, 23328, 34992, 69984, 2187, 2187, 69984, 5832, 4374, 11664, 34992, 34992, 69984];
    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_912_bn();"
function MakeCharacter_912_bn()
    N := 912;
    order := 6;
    char_gens := [799, 229, 305, 97];
    v := [6, 6, 3, 1];
    // 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_912_bn_Hecke();"
function MakeCharacter_912_bn_Hecke(Kf)
    N := 912;
    order := 6;
    char_gens := [799, 229, 305, 97];
    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], [-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, 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, 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, 0, 0, 0, 0, -1, 0, 0, 1, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, -1, -2, 0, -1, 0, -1, 1, 0, 1, 1, -1, 0, 1, 0], [0, 0, -1, 0, -1, -1, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0], [0, 1, 0, 0, 0, 1, 1, 1, 0, 0, -1, 0, 0, -1, -1, -1], [-1, 2, 0, 0, 2, 1, 1, 1, 0, -2, 0, 0, 1, -1, -1, 0], [0, 0, 0, 0, 2, 0, 0, 2, 0, -2, 0, 0, 0, 0, -1, 0], [0, 2, 0, -1, 2, 1, 2, 0, 1, 0, -1, 0, 1, -2, -1, -1], [0, -1, -2, 0, 0, -1, 1, 1, 2, -1, 1, 1, -1, -1, 1, 0], [1, -1, 0, 2, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [1, 2, 1, 1, 0, -1, 0, -3, 0, 2, 1, 0, 0, 0, 1, 0], [-1, 1, 0, -1, -2, -3, 0, -4, 1, 1, 1, 1, -1, 0, 2, 0], [-2, 2, 2, -1, 2, -1, 0, 0, 1, 0, 1, 0, 1, 0, -1, -1], [0, -1, 0, 1, 0, 0, 0, 0, -1, 1, 0, -2, -1, 2, 1, 0], [1, -1, 1, 1, 0, 2, -1, 2, -2, 0, -2, -1, 2, 1, -3, -1], [0, 1, -1, 1, 1, -1, 0, -2, 1, 0, 1, -2, 0, 1, 2, 0], [2, 2, 0, 1, 0, 0, 0, 0, -1, 2, -2, 0, 1, 1, -2, -2], [-4, 1, 0, -4, 2, 1, 1, 1, 0, 0, 1, 0, 0, -1, 1, 1], [1, 0, 0, 1, 0, -2, 0, -2, 2, 0, 2, 0, -2, 0, 2, 0], [1, 3, 2, -1, 2, 1, 0, 2, 1, -3, -1, 1, 1, 0, -2, 0], [0, 1, 0, 0, 1, -2, -1, -2, 0, 2, 0, 2, 0, -1, 0, 0], [0, 1, 0, -3, -2, 0, 0, 2, 1, -3, 0, 2, 1, -2, -1, 0], [-3, 0, -2, 3, 1, -1, -1, -2, -1, -1, 2, 1, -1, -1, 1, 1], [2, -4, 0, 1, -2, -1, 0, -2, 1, -2, 1, 0, -1, 0, 1, 1], [1, 1, -1, 2, -1, 0, 0, -2, 1, -1, 1, 2, -1, 1, 1, 0], [4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, -2, 2, 0], [0, 3, 1, 0, 1, 1, 2, 2, 1, -1, -2, -3, 1, 0, 1, -2], [-2, -1, 0, 0, -1, -2, 0, -1, 0, 1, 0, -1, 0, 2, 2, 0], [-4, -3, -4, -2, -6, -1, 0, -2, 0, 3, -3, 0, -4, 2, 3, 1], [2, 0, 2, -2, 3, 0, 0, -3, 2, 0, 0, 0, 2, 0, 0, 0], [2, -2, 0, 1, 2, 2, 0, 2, 1, -4, 2, 0, -1, -2, 2, 2], [0, 3, 1, 1, 5, 2, 2, -2, 2, 2, -2, -2, 3, -4, -1, -1], [0, -2, 0, 0, 0, -2, 0, -4, 2, 2, 2, 1, 2, 0, 1, 0], [-1, -3, 1, -2, -3, -1, -2, 0, -1, -1, 1, 0, -1, 2, 1, 0], [2, 1, -1, 2, 0, -4, 1, -2, 2, 0, 2, 2, -2, -1, 4, 1], [1, 4, 3, 0, 6, 0, 1, 0, 3, -5, 1, 0, 3, -2, 1, -2], [0, -3, -2, 5, -2, -2, 0, 0, 3, 1, 2, -2, 1, 0, 3, 0], [-2, 3, 4, -2, 0, 1, -1, -3, -4, 4, -1, -2, 4, 1, -3, -1], [9, -2, 1, 0, 0, 3, 0, -1, 1, 0, -1, 1, -1, -2, -3, 0], [2, -1, -1, 1, -5, 1, 0, -4, 0, 4, -3, 0, -1, 1, -2, -2], [3, 1, -1, 6, -1, -1, 1, -1, 1, -2, 1, 1, -1, -1, 1, 0], [0, -5, 0, 0, -1, 0, 0, -5, 1, 5, 1, -1, 1, -2, 2, 0], [2, -4, 0, 4, -2, 0, -1, 3, 0, -1, 0, 3, 0, 1, 0, 0], [0, -3, 1, 3, -1, 1, 0, 2, 0, 2, -1, -2, 1, 2, 0, 0], [-3, 3, 2, 3, 2, 3, 0, 4, -1, -3, -3, -3, -1, 0, 0, 0], [2, 3, -1, -2, 1, 0, 2, 0, 1, -1, 0, 7, 1, -2, -7, -2], [5, 2, -1, 0, 2, -1, 0, 3, -1, -2, -1, -1, -1, 2, 1, 0], [6, -3, -1, 0, 0, 5, 1, -2, -1, 1, -4, -3, -2, -2, 2, -2], [-3, 6, -1, 3, 2, -1, -2, 3, 0, -8, 3, 2, 0, -2, 1, 2], [-3, -1, -1, -6, -3, 1, -1, -1, 1, -2, 1, 3, -1, 3, 1, 0], [-2, 2, -2, -1, 8, 2, 0, 10, -2, -6, -2, 0, 0, 0, -6, 0], [8, -3, -4, 4, -6, 0, 0, -2, 0, 3, 0, 0, -4, 0, 4, 4], [0, -8, 4, 8, 0, 3, -2, 2, -4, 8, -3, -6, 0, 2, 0, 1], [1, 7, 1, -1, 5, 2, 3, -1, 4, -4, 0, 6, 4, -7, -6, -3], [5, 6, 1, 5, 3, -5, -2, -4, 0, 8, -1, 1, 0, 2, 0, -2], [1, -5, 1, 1, -1, 0, 1, 3, 0, -6, 2, -2, 0, -1, 0, 1], [-3, -2, 1, 0, 3, -1, -1, -1, 1, -4, 2, 1, 0, 0, 0, 2], [0, 0, -4, -4, 4, -3, 2, -6, 6, 4, 3, 2, 2, -4, 2, -1], [2, 0, -2, 2, -5, 2, 2, -3, 0, -2, 0, 0, 0, -2, 0, 2], [-1, 5, 1, 0, 5, 3, 0, 0, 1, -5, 5, 6, 5, -8, -7, 0], [0, 4, -1, 0, 2, 2, -1, 4, -3, -5, -1, -1, -3, -1, 0, 1], [0, 0, 2, 0, 6, 2, 1, 1, -2, 5, 0, -3, 0, 1, 0, 0], [0, -3, -1, -1, -3, -4, -6, -4, -2, 0, 4, 2, -3, 3, 3, 3], [0, -5, 2, -15, -6, 2, 0, 8, -3, -1, -2, -2, -1, 4, -1, 0], [-1, -1, -1, -2, -2, 2, -2, 1, 1, 0, -2, -1, 2, 2, -2, 0], [1, 2, -3, -1, -1, -4, -3, 1, -1, -5, 5, 1, -1, 1, 4, 3], [-5, 7, -3, 5, 6, 0, 3, -4, 2, -4, 2, 1, 2, -7, 1, -3], [-2, 5, -1, 0, -4, 1, 1, 2, -1, 5, 0, 1, 2, -2, -2, -2], [2, -1, 0, 4, 3, 2, 2, 1, 0, 1, 2, -1, -2, 2, 2, 0], [5, -1, 1, -5, 5, 2, 1, -3, -2, 2, -4, -6, -2, 3, 2, -1], [0, -10, -3, -4, 1, -5, -4, -8, 1, 11, 5, -2, -2, 2, 6, 2], [-4, 3, -6, -8, -2, 0, 5, -1, 6, -7, 3, 1, -3, -2, 3, 0], [-3, 0, -1, -6, -1, 2, 3, 5, 1, -4, -2, -1, 2, -3, -2, 0], [4, -8, 2, 4, -2, -2, 0, 2, 2, -8, 4, 1, -2, 0, 5, 0], [10, -5, -4, 5, 0, -4, 0, 4, 1, -5, 0, 0, -5, 2, 1, 4], [5, 1, -3, 5, 4, -3, 2, 3, 3, -1, 2, 1, -3, -2, 3, 2], [-12, -3, -2, -12, 1, 1, 1, 6, 0, -4, -1, 1, 0, -1, 0, 1], [-10, 3, -1, -5, -5, -2, 0, -4, 0, 8, 0, 0, -1, 1, 1, 1], [3, 3, 3, 6, -3, 3, -1, -1, -3, -2, -3, 1, 3, 1, -3, 0], [3, -11, 2, -3, -8, 0, -2, 0, -1, 9, -1, -5, -1, 4, 4, 2], [0, 1, 6, 8, 4, 7, 2, 4, -2, 3, -7, -8, 4, -2, -3, -1], [12, 6, 1, 6, 3, 1, 0, 2, -2, 3, 1, 0, 3, -1, -6, 0], [-6, 3, -6, 6, 2, -2, -1, -3, -2, -4, 5, 8, -2, -5, -3, 1], [-4, 3, 6, -2, 0, 0, 0, -6, 0, 3, 4, 0, 6, -2, 0, -2], [3, 6, 3, 0, 6, -3, 3, -2, 3, -3, -1, -2, 5, 1, 3, -6], [-9, 1, 0, 0, 6, 0, -2, 2, 0, -8, 5, 5, 1, -3, -5, 4], [3, -6, -1, -3, -9, -1, 0, 4, -2, 6, -1, -3, -2, 4, 2, 0], [-3, -5, -1, 0, -9, -2, -2, 0, -1, 7, 1, -2, -3, 3, 5, 4], [-4, -3, 0, -8, -1, 0, 3, 6, 0, -4, 0, 0, 0, -3, 0, 0], [0, 3, 4, 0, -2, -3, 0, -6, 4, 5, -1, 0, 0, 2, 3, -5], [-5, -3, 1, -10, -3, -1, -2, 4, -1, -1, -3, 4, 3, -2, -3, 0], [-2, -7, -2, 2, -8, -5, -1, -1, 0, 6, 3, -2, 0, 5, 5, 1], [7, -4, -1, 7, -6, -4, 0, -1, 0, -4, -1, 0, 0, 0, -1, 0], [-18, 2, -4, -9, 0, 3, 0, 4, -3, 2, -3, 0, -1, 0, -9, 1], [0, -7, 2, -4, 0, 1, -2, 0, -4, 7, -1, -12, -2, 0, 5, 1], [-12, 7, 6, 0, 5, 0, 2, 1, 6, -3, 0, 1, 4, -2, -2, -4], [-10, 2, -4, -5, 0, -3, 0, 4, 3, 2, 1, 0, -7, 1, 5, 5], [0, 3, -4, -6, -4, -3, 2, 2, 1, -7, 3, 0, -3, 2, 3, -1], [2, 9, 3, 1, 9, 3, 0, 6, 2, 0, 1, 0, 1, -2, -6, -2], [-4, -5, 0, -2, -10, 3, 0, -10, 4, 5, -3, 0, -4, 0, 1, -3], [-8, 7, -2, 0, 7, 0, -1, 10, -2, -8, 0, -2, -2, 1, 4, 2], [-10, -4, -4, 0, -2, 0, -2, -2, -4, 0, 4, 0, 0, -2, 4, 4], [11, -4, -3, 0, 0, 4, -1, 4, -3, -1, -5, 2, -7, 2, -9, 2], [-5, 1, -3, 0, 0, -5, -2, -1, -3, -2, 8, 3, 4, -1, 2, 4], [1, 2, -3, -1, 3, -1, -2, 0, -4, -4, 1, 3, -4, 2, -2, 2], [-8, 2, -2, -4, 4, -5, 0, 6, -4, -2, -3, 0, 2, 4, -2, -1], [0, -6, -2, -6, -4, -6, -8, -6, 0, 2, 6, 6, -2, -2, 3, 4], [1, -15, 2, 2, -10, -2, -6, 8, -2, -4, 1, 3, -1, 5, 1, 0], [-6, -5, 4, -12, 0, 5, 3, 9, -4, -3, -1, -7, 1, 1, -1, 0], [2, 1, 1, 2, 1, 9, -2, 2, -3, 3, -4, -1, 3, 2, -5, -2], [16, -4, 0, 8, -8, 1, 0, -8, 0, 4, 3, 0, 0, -2, 2, 3], [-9, -3, -1, 0, 3, -4, -2, -4, -1, -5, 7, 4, 3, -1, -1, 4], [0, 5, -2, 6, 4, -1, 2, -4, 2, -1, 1, 0, 0, -5, 1, -1], [15, 0, 3, 15, 4, 3, -2, 5, -4, 2, -3, -4, 4, 2, -7, -2], [3, 0, -1, 3, 2, -6, 0, -5, 8, 0, 7, 0, -8, 0, 7, 0], [10, -2, -2, 20, -4, -2, -2, -4, 2, -2, 4, -2, -4, 4, 4, 0], [2, 2, -4, 4, 6, -2, 4, 2, 4, 2, 0, 4, 0, -6, 0, 0], [0, -1, -8, 17, 0, -10, -4, -12, 3, 1, 10, 10, -5, -4, 5, 2], [14, 0, -2, 7, -4, -1, 0, -2, 3, 4, 1, 0, -5, 0, 1, 3], [4, -9, 1, 8, -2, -3, -3, 4, -1, 1, 2, 5, -2, 2, 2, 0], [2, 0, 6, 0, 4, 2, -2, 0, 6, -6, 6, -2, 2, -6, 10, 4], [7, 5, 3, 0, 3, 1, -1, 5, 3, -4, 3, 1, 1, -3, 1, 2], [-20, 2, 0, 0, 4, 0, -2, 2, 0, -6, 6, 2, 2, -4, 2, 4], [-4, 14, 0, 4, 12, 2, 2, -2, 4, -12, 2, 1, 4, -10, 1, -2], [0, -1, 7, 3, 5, 5, -4, -2, -9, 6, -5, -10, -2, 5, 0, 2], [5, -14, 3, -5, -9, -1, -6, 2, -4, 8, -1, -1, -4, 10, 0, 6], [1, -7, 8, 1, 0, 9, -4, 6, -7, -3, -3, -3, 7, 4, -6, -4], [1, -8, -3, 0, -15, -5, 1, -5, -3, 16, -6, -3, -4, 10, 0, -2], [7, 4, 1, 7, -2, 2, 0, -5, -2, 4, -1, 4, 2, 0, 3, 0], [10, -8, 0, 10, 2, 0, 0, 8, 2, -8, 2, 3, -2, 0, 5, 0], [-2, 6, 4, -1, -4, 5, 0, -8, -3, 10, 1, 0, 7, -3, -7, -3], [0, 7, -6, 0, -3, -2, 0, -7, 3, 7, -3, 7, -3, 0, 4, 0], [-4, 9, 2, -8, -1, -4, -2, -13, -2, 1, 2, 5, -2, 0, 2, 0], [0, 3, 2, 2, 4, 3, 2, 0, 1, 1, -3, -8, 3, -8, 1, -1], [0, 2, 0, -3, -2, -1, -2, 0, 1, -4, 1, -4, 1, 4, 3, 1], [-7, 4, -9, -14, 0, -6, 1, -4, 9, -1, 1, 8, -1, -6, 1, 0], [-13, 3, -1, 0, -1, 3, 0, 8, -1, 1, -5, -4, -5, 2, 3, 0], [8, 3, -4, 16, -3, 2, 0, -1, 4, -3, -2, 9, 2, 0, -2, 0], [-4, -3, 4, -2, -2, -1, 0, -6, 8, -1, 5, 0, -4, -2, 7, 1], [-4, -3, 4, -2, -6, -9, 0, -10, 4, 3, 1, 0, 0, 4, 3, -3], [0, 3, 8, 14, 6, 9, 2, 4, -4, 3, -9, -4, 4, 2, -7, -1], [0, -7, -2, 2, -2, -6, -8, -8, 0, 5, 6, -4, -2, 2, 8, 4], [2, 1, 1, 4, -2, -3, 1, -2, -1, -3, 2, -5, -2, -2, 2, 0], [6, 4, -8, 6, -4, -4, 0, -6, 6, 4, -2, 9, -6, 0, 7, 0], [2, -5, -1, 2, 4, 0, 1, 6, 4, -6, 4, 0, -4, -1, 4, 1], [-19, 0, -9, -19, -2, 1, 4, -1, 8, -4, 3, 0, -8, -4, 3, 4], [0, 0, 4, 13, 4, 4, 0, 0, -5, 4, -4, -16, -1, 7, 4, 0], [-1, -7, -5, 0, -7, -3, -2, -2, -5, 5, 1, -6, -3, 4, 13, 4], [5, 7, 4, -5, -1, 5, 5, 8, 4, -2, -5, 6, 4, -5, -11, -5], [4, -1, -8, 8, -3, -10, -2, -7, 8, -1, 6, 11, -6, -2, 6, 0], [0, 1, 3, -9, 1, 2, -2, 0, -2, 0, -2, -2, 1, 6, -1, 1], [0, 11, -2, 15, 2, -2, 0, -4, -1, -9, 2, 6, -3, 2, -1, 0], [6, -4, -2, -6, -5, -2, -6, 3, -4, -2, 4, -2, -4, 2, 6, 6], [21, -1, -3, 0, 7, -5, -2, 2, -3, -9, 3, 2, -1, 4, -1, 4], [-3, 9, 3, 0, -1, 4, 4, 2, 3, 5, -5, 0, 3, -3, -5, -8], [-2, 3, -6, 2, 3, -1, 5, -8, 2, 2, 3, 7, 2, -9, -4, -5], [0, -3, 7, 5, 3, 6, -2, 2, -9, 6, -6, -10, -2, 13, -1, 1], [0, -5, -6, 4, -4, -7, -2, -4, 0, 1, 7, 0, -6, 8, 7, 1], [-2, 11, -2, 2, 8, 1, 5, -3, 4, -6, 1, 4, 4, -9, -3, -5], [8, -7, -2, 8, -7, 1, 3, 2, 0, -10, 1, 1, 0, -3, 2, 3], [11, 7, -2, 11, 0, -11, -2, -10, 5, 9, 1, 9, -5, 2, 10, -2], [8, 6, -4, 8, 1, 6, 2, -3, 2, 4, 0, -4, -2, -2, -4, 2], [-13, -6, 1, 0, -6, 3, -1, -2, 1, 5, -1, -2, -3, -1, 3, 2], [-7, 2, 9, -7, 7, 9, -2, 4, -8, 4, -1, -7, 8, 2, -8, -2], [13, 3, 4, -13, 8, 3, 2, -4, 3, -1, -3, -5, 3, -2, 2, -2], [0, -6, -6, -7, -6, -5, 2, 2, 6, 0, 5, 8, 0, -8, 1, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_912_bn_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_912_2_bn_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_912_2_bn_n(:prec:=16)
    chi := MakeCharacter_912_bn();
    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_912_2_bn_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_912_2_bn_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_912_bn();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![430336, -1275264, 1443392, -544320, -252000, 242688, 25760, -72960, 9624, 10632, -2224, -1188, 326, 72, -21, -3, 1]>,<7,R![-32, -332, -278, 234, 223, -22, -30, 0, 1]>,<17,R![2945449984, 4960026624, 3228762112, 748683264, -150208512, -99373056, 2434048, 9556992, 1128960, -402432, -73808, 13344, 3260, -216, -69, 3, 1]>],Snew);
    return Vf;
end function;