
// Make newform 1155.2.i.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1155_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1155_i_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_1155_2_i_c();".
// 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_1155_2_i_c();".
// 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 := [625, 0, 1125, 0, 900, 0, 395, 0, 11, 0, -49, 0, -4, 0, 1, 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], [-33766125, 0, 1900400, 0, 1397920, 0, 584304, 0, 1664, 0, -53806, 0, 6064, 0, -4976, 0], [0, 5720425875, 0, 2011459400, 0, -61141995, 0, -220485308, 0, -23218303, 0, 7571087, 0, 3769822, 0, 52907], [32055875, 0, 47891850, 0, 34025725, 0, 2776621, 0, -3371514, 0, -423219, 0, 26336, 0, 75631, 0], [-101116250, 0, -92662550, 0, 254483540, 0, 145662708, 0, -38951322, 0, -3507762, 0, -1820672, 0, 1098448, 0], [0, -1014860875, 0, -72330250, 0, 341075305, 0, 104639904, 0, -26926811, 0, -6858931, 0, -254886, 0, 739864], [30499350, 0, 40690320, 0, 28658755, 0, 2128291, 0, -2994144, 0, -365249, 0, 44136, 0, 64321, 0], [-1320364000, 0, -1461240500, 0, -1049398975, 0, -302484579, 0, 157291286, 0, 13453081, 0, 2186986, 0, -3708119, 0], [434987125, 0, 477661630, 0, 310606859, 0, 81866729, 0, -46979616, 0, -3783171, 0, -529596, 0, 1073735, 0], [0, 955700, 0, 1102430, 0, 906528, 0, 117086, 0, -101834, 0, -12284, 0, 486, 0, 2293], [-730658725, 0, -798277730, 0, -437603572, 0, -95756546, 0, 65045254, 0, 4966964, 0, 355734, 0, -1446284, 0], [0, 379881750, 0, 509713400, 0, 226234550, 0, 8880998, 0, -27002382, 0, -2369422, 0, 520818, 0, 515378], [0, 249500, 0, 287550, 0, 237486, 0, 31110, 0, -26110, 0, -3000, 0, 90, 0, 599], [0, -722601875, 0, -843562400, 0, -466969935, 0, -90901286, 0, 66369199, 0, 5426879, 0, 201274, 0, -1468861], [0, 977629125, 0, 1106504500, 0, 639251405, 0, 141150094, 0, -94005021, 0, -7374841, 0, -547346, 0, 2100604], [0, -1087173250, 0, -1128269700, 0, -742870540, 0, -219278916, 0, 116269094, 0, 8983874, 0, 1648844, 0, -2699561]];
    Rf_basisdens := [1, 15616375, 858900625, 15616375, 171780125, 858900625, 3123275, 171780125, 34356025, 633875, 34356025, 78081875, 74525, 78081875, 78081875, 78081875];
    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_1155_i();"
function MakeCharacter_1155_i()
    N := 1155;
    order := 2;
    char_gens := [386, 232, 661, 211];
    v := [2, 2, 1, 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_1155_i_Hecke();"
function MakeCharacter_1155_i_Hecke(Kf)
    N := 1155;
    order := 2;
    char_gens := [386, 232, 661, 211];
    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], [-1, 0, 0, 0, 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, -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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [1, 1, 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, -1, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, 1], [3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, -1, 0, 0, 0, -2, 0, 0, -1, 0, 0, 0], [-2, 1, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2, 2], [0, 0, 0, 0, -3, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, -1, 0, 0, 0, 8, 0, 0, -1, 0, 0, 0], [-1, -2, 0, -3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, -3, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, -1, 3], [4, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -3, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, -2, 2], [0, 0, 0, 0, 1, 0, 0, 2, -2, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 1, -5], [0, 0, 1, 0, 0, 0, 0, 0, 0, -3, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 2, 0, 0, 0, 5, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0], [0, 0, 1, 0, 0, -2, 0, 0, 0, 5, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0], [-5, -2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, 4, 6], [0, -4, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, -2, 0], [0, 0, 0, 0, -2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, -4, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0], [-6, -1, 0, -3, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 6, 6, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, -6, -2], [-4, 1, 0, -3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, -6, 0, 0, -4, 0, 0, 0], [-4, -2, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 6, 0, -6, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, -4, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 8, -4, 0, -2, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 6, 1, 3], [0, 0, 3, 0, 0, 3, 0, 0, 0, -4, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 4, 8, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 6, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 6, -6], [0, 0, -2, 0, 0, -6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, -3, 0, 0, 0, -10, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -6, 0, 0, 0, 3, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, -3, 3], [0, 0, 0, 0, -4, 0, 0, 4, -2, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, 2, -2, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -6, -4, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -6, -7, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 6, 0], [0, 0, 2, 0, 0, 2, 0, 0, 0, -2, 0, 0, -8, 0, 0, 0], [0, 0, 3, 0, 0, 2, 0, 0, 0, -5, 0, 0, -1, 0, 0, 0], [8, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [13, -4, 0, 5, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, -6, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 6, 2, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -6, 3, -1], [0, 0, 4, 0, 0, 2, 0, 0, 0, 8, 0, 0, -4, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -6, 9, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 6, 0, 0, 0, -9, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 2, -1, 0, -5, 0, 0, 0, 0, 0], [5, 4, 0, 5, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -3, 0, 0, 0, -8, 0, 0, 5, 0, 0, 0], [8, 5, 0, -1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 4, 0, 0, 0, -4, 0, 0, -2, 0, 0, 0], [-6, 5, 0, -7, 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, -10, -2, 2], [0, 0, 1, 0, 0, 2, 0, 0, 0, -17, 0, 0, 1, 0, 0, 0], [-19, 2, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 2, 0, 0, 0, -4, 0, 0, 8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 10, -5, 1], [-6, 8, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, -5, 0, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -14, -9, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 2, -6, 0], [-6, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 2, 2, 4], [-2, 10, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -14, 3, 0, -1, 0, 0, 0, 0, 0], [-3, 8, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 7, -3], [0, 0, 1, 0, 0, -8, 0, 0, 0, 9, 0, 0, -5, 0, 0, 0], [0, 0, -1, 0, 0, 6, 0, 0, 0, -3, 0, 0, -7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 8, -6], [0, 0, 0, 0, 7, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -16, -1, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, -10, -6, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -6, 2, 4], [0, 0, 0, 0, -7, 0, 0, 0, -7, 0, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 0, 10, 2, 0, -6, 0, 0, 0, 0, 0], [0, 0, -8, 0, 0, -4, 0, 0, 0, 4, 0, 0, -6, 0, 0, 0], [0, 0, 5, 0, 0, 3, 0, 0, 0, -22, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, -6, -6], [12, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 1, -7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -8, 0, -6], [0, 0, 0, 0, -2, 0, 0, 0, 6, 0, 8, 0, 0, 0, 0, 0], [-20, 6, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, -4, 0, 0, 0, -6, 0, 0, 8, 0, 0, 0], [1, -14, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-20, -6, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 9, 0, 0, -11, 0, 0, 0], [0, 0, 2, 0, 0, 4, 0, 0, 0, -14, 0, 0, 6, 0, 0, 0], [-33, 2, 0, -3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 10, 1, 0, -3, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 9, 0, 0, 0, 12, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 10, -3, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -10, 1, -3], [-2, -8, 0, -8, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -18, 0, 0, -8, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 4, 3, 0, -3, 0, 0, 0, 0, 0], [1, 10, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 4, 0, 0, 0, -3, 0, 0, -7, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -2, -12, 4], [0, 0, 0, 0, -10, 0, 0, -8, -6, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, -6, 0, 0, -2, 0, 0, 0, 0, 0], [-13, 6, 0, 1, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-14, 4, 0, -12, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -14, -4, -10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 14, -5, -1], [0, 0, 6, 0, 0, -2, 0, 0, 0, 6, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -20, -4, 10], [0, 0, 0, 0, 0, -8, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, -2, -10, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 10, 8], [0, 0, 0, 0, 3, 0, 0, -16, 5, 0, -5, 0, 0, 0, 0, 0], [8, -1, 0, 7, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, -8, 0, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, -1, 0, 0, -6, 0, 0, 0, 25, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -8, 4, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -18, -2, 6], [0, 0, 7, 0, 0, -1, 0, 0, 0, 2, 0, 0, -5, 0, 0, 0], [3, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 2, -9, 0, -1, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, -4, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0], [-12, -4, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -14, -11, -1], [8, 3, 0, 3, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [12, 8, 0, -4, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, -4, -8, 0, 10, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 6, 0, 0, 0, -6, 0, 0, 10, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 12, 14, 12], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 10, 10], [-21, -6, 0, 3, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, -6, -16, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 8, -7, 0, -5, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, -7, 0, 0, -11, 0, 0, 0], [11, -8, 0, 3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0, 0, 16, 0, 0, -16, 0, 0, 0], [35, -8, 0, -5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 4, -12], [0, 0, 0, 0, -1, 0, 0, -10, 6, 0, -8, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 1, 0, 0, 0, 14, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 10, 7, 0, -1, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 9, 0, 0, 0, -38, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -10, -15, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 0, 2, 3, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -18, 0, 0, 0, 24, 0, 0, 8, 0, 0, 0], [0, 0, -4, 0, 0, -8, 0, 0, 0, -10, 0, 0, -16, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 10, 12, 0, -4, 0, 0, 0, 0, 0], [-10, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -10, -2, 2], [14, -8, 0, 14, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 8, 10, 6], [-16, 4, 0, 10, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 14, 7, -21], [15, 10, 0, -5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 0, 0, -11, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 14, 7, 0, -13, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 10, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -22, 4, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 10, -4, -2], [0, 0, -4, 0, 0, -12, 0, 0, 0, -8, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 2, 22, 0, -4, 0, 0, 0, 0, 0], [-28, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 8, 0, 0, 0, -23, 0, 0, 11, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -4, 6, 0, 14, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 6, 0, 0, 0, 6, 0, 0, -10, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, -10, 0, -2, 0, 0, 0, 0, 0], [-8, -9, 0, 1, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -5, -9], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 2, -18, 0], [0, 0, 0, 0, 3, 0, 0, 16, 19, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 10, 7, 5], [0, 0, 1, 0, 0, 10, 0, 0, 0, 13, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 10, 2, -10], [0, 0, -7, 0, 0, -14, 0, 0, 0, 23, 0, 0, 9, 0, 0, 0], [0, 0, 4, 0, 0, -4, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 11, 0, 0, 0, -4, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, -2, -18, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, -12, -3, 0, 3, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -9, 0, 0, 0, -28, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, -10, -11, 0, -15, 0, 0, 0, 0, 0], [0, 0, 7, 0, 0, -6, 0, 0, 0, -19, 0, 0, 13, 0, 0, 0], [16, -2, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 5, 0, 0, 0, -2, 0, 0, 15, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 8, 28, -4], [0, 0, -8, 0, 0, -18, 0, 0, 0, 30, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, -16, -7, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, 12], [2, -16, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [47, -2, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 16, 6, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, -2, -4], [-8, 5, 0, 3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-18, -3, 0, -13, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 10, -11, 3], [0, 0, 0, 0, -1, 0, 0, -6, -18, 0, -8, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0, 0, 12, 0, 0, 13, 0, 0, 0], [-20, -9, 0, -1, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -20, -4, -6], [0, 0, 0, 0, 7, 0, 0, -18, -11, 0, -7, 0, 0, 0, 0, 0], [-12, 3, 0, 13, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 14, 14, -20], [0, 0, 0, 0, 1, 0, 0, 18, 16, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -13, 0, 0, -16, -21, 0, 13, 0, 0, 0, 0, 0], [0, 0, 7, 0, 0, 12, 0, 0, 0, 5, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 10, -13, 1], [0, 0, -3, 0, 0, 11, 0, 0, 0, -14, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, -16, 0, -20, 0, 0, 0, 0, 0], [20, 14, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 12, 0, 0, 0, -22, 0, 0, 2, 0, 0, 0], [6, 8, 0, 4, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 18, 9, 19], [0, 0, 0, 0, 0, -12, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [-5, -8, 0, -11, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -10, -1, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 2, 0], [0, 0, 4, 0, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, -9, 0, 0, -6, 0, 0, 0, 33, 0, 0, -9, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 2, 6, 0, -16, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, -4, 6, 0, -16, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 8, 10, -10], [6, 24, 0, 4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 5, 0, 0, 0, 4, 0, 0, -15, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, -7, 9], [0, 0, 0, 0, 1, 0, 0, -10, 1, 0, -19, 0, 0, 0, 0, 0], [25, -4, 0, -9, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, -2, -4, -16], [0, 0, -1, 0, 0, -1, 0, 0, 0, -26, 0, 0, 7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 6, -6, 6], [0, 0, 0, 0, 5, 0, 0, -4, -19, 0, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 2, -8, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, -14, -11, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 24, 6, -4], [0, 0, -7, 0, 0, 19, 0, 0, 0, -4, 0, 0, -3, 0, 0, 0], [0, 0, -11, 0, 0, -5, 0, 0, 0, 20, 0, 0, -15, 0, 0, 0], [-14, -20, 0, 4, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 3, 0, 0, -15, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 8, 21, 0, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 0, 2, 0, -12, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 8, 0, 0, 0, -10, 0, 0, -2, 0, 0, 0], [6, -16, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 13, 0, 0, 8, 0, 0, 0, 7, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, -6, -12, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -16, -14, 12], [0, 0, 0, 0, 0, 8, 0, 0, 0, 32, 0, 0, 16, 0, 0, 0], [-17, -8, 0, -15, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 4, 4, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 2, -26, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -8, 18, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, -5, 17], [4, 16, 0, 12, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -22, 6, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, -6, 9, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, -10, -12, -4], [2, 3, 0, 13, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16, 5, 0, 7, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 14, -8, 0, 12, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 14, 0, 0, 0, -22, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, -4, 0, 12], [-4, 8, 0, 22, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 9, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 8, 20, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 24, 22, -14], [0, 0, 0, 0, 10, 0, 0, 4, 18, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, -10, 10, 4], [0, 0, 2, 0, 0, 2, 0, 0, 0, -10, 0, 0, 2, 0, 0, 0], [0, 0, 3, 0, 0, 5, 0, 0, 0, -16, 0, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 12, 28, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, -12, -14, 20], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, -14, -10, -18], [0, 0, 0, 0, -3, 0, 0, -20, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 0, 0, -6, -12, 0, 12, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1155_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1155_2_i_c();".
// 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_1155_2_i_c(:prec:=16)
    chi := MakeCharacter_1155_i();
    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_1155_2_i_c();".
// 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_1155_2_i_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 0, 84, 0, 61, 0, 14, 0, 1]>],Snew);
    return Vf;
end function;