
// Make newform 1110.2.i.p in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1110_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1110_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_1110_2_i_p();".
// 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_1110_2_i_p();".
// 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 := [144, -312, 844, 100, 458, -62, 123, -6, 15, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-57589572, 20573460, -53999802, 68253733, -22088817, 9390789, -3088861, 1382625, -298647, 52206], [1346033064, -3507217508, 2363790868, -1318853566, 696971194, -91335981, 105828882, -8787045, 4535866, 1026853], [-204447812, 123286380, -323593606, 354916248, -132367151, 56274267, -21741437, 8285375, -1789641, 223413], [709537978, -61874760, 162404612, -127474135, 66432202, -28242834, 6058312, -4158250, 898182, -246441], [-1250447352, 3402468940, 1298957896, 1932936194, -184856654, 553226781, -12203286, 68403201, -8971790, 4799131], [-2060561952, 483624180, -1269383466, 850207980, -519245961, 220751037, -63953489, 32501625, -7020351, 1710145], [320551920, 4213451412, 1332828340, 1655609196, 141819053, 448160006, 24599493, 47030658, -3300175, 3473146], [6943311624, -18841944788, -4264073288, -10483725766, 1189349074, -2878823373, 98082762, -358607505, 48442714, -24622127]];
    Rf_basisdens := [1, 1, 131893214, 1582718568, 263786428, 131893214, 1582718568, 263786428, 263786428, 1582718568];
    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_1110_i();"
function MakeCharacter_1110_i()
    N := 1110;
    order := 3;
    char_gens := [371, 667, 631];
    v := [3, 3, 2];
    // 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_1110_i_Hecke();"
function MakeCharacter_1110_i_Hecke(Kf)
    N := 1110;
    order := 3;
    char_gens := [371, 667, 631];
    char_values := [[1, 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]];
    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, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, -1, 0, 0, 0, 0], [0, 2, 2, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, 0, 0, 0, 1, 0, 0, 1], [-1, 0, -2, 0, 1, 0, 0, -1, 0, 0], [2, 0, 1, 0, -1, 1, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, 0, 0, -1, 0, 0, 1, 0], [0, 2, 2, 0, 0, 0, 1, 0, 0, -1], [1, 0, 1, 0, -2, 0, 0, -1, 0, 0], [4, 0, 2, 0, 1, -1, 0, 0, 0, 0], [-1, 2, 0, 0, 0, -1, -1, 0, 0, 1], [-2, 2, 0, 1, 1, -1, -2, 0, 2, 1], [0, 2, 2, -2, 0, 0, 1, 0, 0, 3], [0, 1, 1, 2, 0, 0, 1, 0, 0, 2], [0, 1, 1, 1, 0, 0, -1, -1, -1, 0], [-1, 0, 1, 0, -2, -2, 0, -1, 0, 0], [0, 1, 1, 0, 0, 0, 3, 1, 1, 2], [-1, 1, 0, -1, -1, -2, -1, 0, -1, 2], [-5, 0, 0, -1, -1, 0, -5, 0, 1, 0], [1, 0, 1, 0, -2, 0, 0, 1, 0, 0], [-2, 0, -4, 0, 2, 0, 0, -1, 0, 0], [0, 0, -1, 0, 2, 0, 0, -2, 0, 0], [0, 0, 0, -2, 0, 0, -3, -2, -2, -1], [10, -5, 0, -2, -2, 1, 10, 0, -1, -1], [0, -3, 0, -1, -1, -1, 0, 0, -2, 1], [-5, 2, 0, 2, 2, 2, -5, 0, 1, -2], [1, -2, 0, -4, -4, -3, 1, 0, -1, 3], [-2, 0, -5, 0, 1, -1, 0, 2, 0, 0], [4, -3, 0, 3, 3, 0, 4, 0, 0, 0], [4, 0, 1, 0, -3, 1, 0, 0, 0, 0], [0, 3, 3, 0, 0, 0, 5, 1, 1, 0], [-9, 2, 0, 2, 2, -2, -9, 0, 0, 2], [-4, -2, 0, 0, 0, -2, -4, 0, -1, 2], [0, 0, 0, 0, 0, 0, -5, -1, -1, -1], [0, 4, 0, 3, 3, 3, 0, 0, 1, -3], [-5, 0, 0, 0, -1, 0, 0, -1, 0, 0], [0, -1, -1, -4, 0, 0, -1, -1, -1, -2], [9, 0, 4, 0, -4, -1, 0, 0, 0, 0], [2, 0, -3, 0, 4, 1, 0, 1, 0, 0], [-4, 2, 0, -2, -2, 0, -4, 0, -2, 0], [-3, 0, -3, 0, 4, 2, 0, 0, 0, 0], [-4, 0, 2, 0, -4, -1, 0, -1, 0, 0], [8, 0, -1, 0, 1, -2, 0, 2, 0, 0], [0, 5, 5, 3, 0, 0, 11, -1, -1, -2], [0, 3, 3, 0, 0, 0, -10, 1, 1, 3], [2, 0, -3, 0, -1, 1, 0, 1, 0, 0], [-4, -1, 0, -3, -3, -1, -4, 0, 1, 1], [0, -5, -5, -1, 0, 0, -13, -1, -1, 1], [0, 0, -8, 0, 1, 1, 0, -2, 0, 0], [2, 1, 0, -3, -3, -1, 2, 0, 0, 1], [0, -4, -4, -5, 0, 0, -3, 1, 1, 2], [4, 0, 7, 0, 1, -1, 0, 1, 0, 0], [-1, -1, 0, 2, 2, 0, -1, 0, 2, 0], [0, -3, -3, -1, 0, 0, 3, -2, -2, 0], [-1, -2, 0, -3, -3, 2, -1, 0, -1, -2], [0, 3, 3, -2, 0, 0, 1, -2, -2, 0], [0, -2, -2, 5, 0, 0, -1, 1, 1, -4], [17, 0, 5, 0, 2, 2, 0, -1, 0, 0], [-5, 6, 0, 6, 6, 1, -5, 0, -2, -1], [-1, -1, 0, 2, 2, 6, -1, 0, -1, -6], [-8, 6, 0, 3, 3, 1, -8, 0, 3, -1], [13, 2, 0, -2, -2, -2, 13, 0, 1, 2], [0, -5, -5, -2, 0, 0, 10, -1, -1, 3], [6, 0, 5, 0, -1, -1, 0, -1, 0, 0], [-18, 2, 0, 4, 4, 0, -18, 0, -2, 0], [6, -9, 0, 3, 3, 3, 6, 0, 1, -3], [14, 0, 1, 0, 1, 3, 0, -1, 0, 0], [0, 0, 0, -6, 0, 0, 8, -1, -1, 3], [0, -4, -4, 0, 0, 0, 16, -1, -1, -5], [-19, 1, 0, -5, -5, 2, -19, 0, -1, -2], [0, -2, -2, -3, 0, 0, -4, -4, -4, 1], [0, -5, -5, 3, 0, 0, -8, 1, 1, -1], [7, 0, 9, 0, 3, -1, 0, 1, 0, 0], [-6, 0, 8, 0, 1, 1, 0, 3, 0, 0], [18, 4, 0, -1, -1, -2, 18, 0, -1, 2], [-1, 2, 0, -3, -3, 2, -1, 0, -1, -2], [-1, 0, -8, 0, 2, -1, 0, 0, 0, 0], [0, 1, 1, -1, 0, 0, 21, -1, -1, -2], [-8, 0, -4, 0, 0, 2, 0, 2, 0, 0], [0, 3, 3, -4, 0, 0, -5, 2, 2, -4], [-15, 0, -5, 0, 5, 2, 0, -1, 0, 0], [0, 0, 0, -2, 0, 0, 26, 2, 2, 0], [0, 8, 8, -2, 0, 0, 4, 0, 0, 0], [-2, -1, 0, -5, -5, -1, -2, 0, -2, 1], [0, 0, 0, 1, 0, 0, -17, -2, -2, -3], [-10, 0, -5, 0, -1, 1, 0, -3, 0, 0], [2, 4, 0, -4, -4, -2, 2, 0, 2, 2], [-6, 0, 1, 0, 1, 5, 0, 1, 0, 0], [7, 0, -8, 0, 7, 2, 0, -3, 0, 0], [0, 5, 5, -5, 0, 0, -5, -3, -3, 0], [9, 0, 0, -2, -2, -5, 9, 0, -3, 5], [0, 1, 0, -1, -1, 5, 0, 0, -1, -5], [0, 6, 6, -1, 0, 0, 3, 3, 3, 4], [0, 1, 1, 4, 0, 0, 17, -1, -1, 2], [0, 0, 3, 0, 0, -5, 0, -5, 0, 0], [-9, 0, 7, 0, 4, 2, 0, -1, 0, 0], [9, 2, 0, -3, -3, 0, 9, 0, -1, 0], [1, 0, 1, 0, 5, 0, 0, -1, 0, 0], [-14, 0, 2, 0, -1, 3, 0, -3, 0, 0], [-4, -2, 0, 4, 4, -1, -4, 0, -1, 1], [-1, 2, 0, -4, -4, -3, -1, 0, -2, 3], [0, -5, -5, -5, 0, 0, 3, 3, 3, 2], [4, 0, -6, 0, 6, 4, 0, -3, 0, 0], [0, -7, -7, 3, 0, 0, -3, 1, 1, -6], [-17, 3, 0, -1, -1, -1, -17, 0, -1, 1], [-8, 3, 0, 1, 1, -5, -8, 0, 3, 5], [-18, -1, 0, -8, -8, 0, -18, 0, -3, 0], [20, -1, 0, 1, 1, -3, 20, 0, 2, 3], [4, 0, -1, 0, 0, -7, 0, 1, 0, 0], [17, -3, 0, -2, -2, -4, 17, 0, 0, 4], [-8, -6, 0, 1, 1, 5, -8, 0, 1, -5], [5, 0, 3, 0, -2, -2, 0, 4, 0, 0], [0, 4, 4, 7, 0, 0, 18, 0, 0, -1], [0, -4, -4, -3, 0, 0, 11, 1, 1, -2], [0, 2, 2, -1, 0, 0, -12, 4, 4, 1], [0, -1, -1, -2, 0, 0, -11, -3, -3, 0], [0, -8, -8, 2, 0, 0, 1, 2, 2, -1], [22, 0, 0, 0, -3, 1, 0, 1, 0, 0], [2, 4, 0, -2, -2, 0, 2, 0, 0, 0], [15, -4, 0, -1, -1, -5, 15, 0, 2, 5], [14, 0, 0, -6, -6, -4, 14, 0, 3, 4], [-14, 0, -9, 0, 4, -3, 0, 3, 0, 0], [-13, 1, 0, 5, 5, -2, -13, 0, -1, 2], [0, -6, -6, 2, 0, 0, -10, 3, 3, 1], [0, 9, 9, 3, 0, 0, 1, 1, 1, 1], [15, 0, -5, 0, 0, 1, 0, 4, 0, 0], [0, 6, 6, -1, 0, 0, 4, 2, 2, -3], [7, 0, -7, 0, 4, 0, 0, -2, 0, 0], [2, 10, 0, 2, 2, -3, 2, 0, 3, 3], [6, -2, 0, -7, -7, 5, 6, 0, -1, -5], [-20, 0, 6, 0, -7, -2, 0, 1, 0, 0], [0, 0, 0, 5, 0, 0, -12, -1, -1, -5], [21, 0, 1, 0, -2, 2, 0, -2, 0, 0], [1, 4, 0, -3, -3, 4, 1, 0, -3, -4], [0, -6, -6, -13, 0, 0, -3, -1, -1, 4], [0, 12, 12, -3, 0, 0, 12, 1, 1, -1], [0, 1, 1, 1, 0, 0, -24, 4, 4, 5], [-16, 0, 0, -5, -5, 8, -16, 0, 0, -8], [0, 1, 1, 3, 0, 0, 0, 3, 3, 9], [-12, 5, 0, -6, -6, -1, -12, 0, -1, 1], [1, 6, 0, 2, 2, 3, 1, 0, -6, -3], [14, -12, 0, 3, 3, 1, 14, 0, 0, -1], [-12, 0, -1, 0, 5, -1, 0, 2, 0, 0], [-7, 0, -4, 0, 4, -2, 0, -7, 0, 0], [-18, 6, 0, 5, 5, -1, -18, 0, 4, 1], [-11, 0, 6, 0, 2, -2, 0, -4, 0, 0], [0, 2, 2, 3, 0, 0, -3, 3, 3, 6], [0, -2, -2, 4, 0, 0, 6, 1, 1, 4], [-22, 0, -2, 0, 4, 0, 0, 4, 0, 0], [0, -3, -3, 4, 0, 0, 3, 5, 5, -2], [-10, 0, -3, 0, -3, -1, 0, -1, 0, 0], [7, 0, -11, 0, -4, -2, 0, 0, 0, 0], [0, 8, 8, 3, 0, 0, -3, 5, 5, 0], [-4, 1, 0, 4, 4, 5, -4, 0, 5, -5], [18, 1, 0, 1, 1, 1, 18, 0, 1, -1], [-8, 3, 0, 1, 1, 5, -8, 0, 5, -5], [8, -5, 0, -11, -11, -3, 8, 0, 2, 3], [0, 0, 0, 4, 0, 0, 17, 3, 3, 2], [1, -14, 0, -3, -3, -2, 1, 0, -3, 2], [-10, -3, 0, 3, 3, 5, -10, 0, -2, -5], [19, 0, 0, -9, -9, -2, 19, 0, 3, 2], [2, 0, 4, 0, 4, 4, 0, 5, 0, 0], [0, -3, 0, 8, 8, -2, 0, 0, -3, 2], [19, 0, 0, 0, 6, -4, 0, 0, 0, 0], [-22, 0, -4, 0, -1, -5, 0, -1, 0, 0], [-14, 12, 0, 4, 4, 2, -14, 0, 2, -2], [-16, 1, 0, 6, 6, 1, -16, 0, -1, -1], [0, 2, 2, 2, 0, 0, -17, 2, 2, 3], [0, -10, -10, 2, 0, 0, 24, 2, 2, 0], [0, 2, 2, 2, 0, 0, -14, -2, -2, -4], [0, 10, 10, 9, 0, 0, -20, 3, 3, -1], [-19, 2, 0, -3, -3, 7, -19, 0, -2, -7], [20, -6, 0, -12, -12, -4, 20, 0, -2, 4], [-7, 0, -3, 0, -4, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, -6, -6, -5], [2, 0, -3, 0, 4, 4, 0, 2, 0, 0], [0, 2, 2, 3, 0, 0, 6, -4, -4, -9], [-1, 11, 0, 8, 8, 3, -1, 0, 0, -3], [0, -10, -10, 2, 0, 0, -6, -5, -5, -2], [0, -12, -12, -5, 0, 0, -22, -2, -2, 3], [12, 0, 8, 0, 6, -6, 0, -3, 0, 0], [0, 5, 5, -2, 0, 0, -12, -3, -3, 4], [0, -9, -9, -6, 0, 0, 11, 2, 2, 0], [0, 1, 1, -1, 0, 0, -3, 2, 2, 6], [0, -4, -4, 6, 0, 0, -11, 2, 2, -1], [-6, 0, 6, 0, -6, -2, 0, -4, 0, 0], [1, -1, 0, -3, -3, -2, 1, 0, 3, 2], [0, -2, -2, -5, 0, 0, 5, 0, 0, 5], [0, -6, -6, 4, 0, 0, 8, -3, -3, 6], [0, -13, -13, -9, 0, 0, -3, 5, 5, 0], [-10, 7, 0, 7, 7, 3, -10, 0, 0, -3], [8, -8, 0, -11, -11, -4, 8, 0, -5, 4], [3, 0, -10, 0, 1, 2, 0, 3, 0, 0], [0, -4, -4, -8, 0, 0, -32, -1, -1, 4], [-4, 14, 0, 11, 11, 1, -4, 0, 4, -1], [-16, 0, -18, 0, -6, -4, 0, 1, 0, 0], [15, 0, 5, 0, -6, 4, 0, 2, 0, 0], [-5, -3, 0, -9, -9, 0, -5, 0, -2, 0], [18, -5, 0, -3, -3, -4, 18, 0, 2, 4], [14, 0, -6, 0, 3, 5, 0, -4, 0, 0], [0, 0, 0, -2, 0, 0, -18, -8, -8, -4], [9, -1, 0, -4, -4, -2, 9, 0, 0, 2], [-8, -8, 0, -2, -2, 2, -8, 0, -6, -2], [21, 0, 6, 0, 0, 9, 0, 0, 0, 0], [0, 14, 14, 12, 0, 0, -13, 5, 5, 0], [-2, 11, 0, 12, 12, 3, -2, 0, -1, -3], [-6, 0, 1, 0, -3, -3, 0, -2, 0, 0], [-1, 0, -9, 0, -1, 5, 0, 4, 0, 0], [2, -5, 0, -3, -3, 5, 2, 0, -5, -5], [0, 7, 7, -1, 0, 0, 14, 5, 5, 5], [-27, 0, 8, 0, 8, 6, 0, 2, 0, 0], [0, 5, 5, 9, 0, 0, -17, -4, -4, 1], [-1, 0, -8, 0, 3, -10, 0, -7, 0, 0], [-6, -2, 0, -7, -7, 5, -6, 0, 6, -5], [0, 4, 4, -1, 0, 0, -15, -3, -3, 4], [28, -8, 0, -10, -10, -4, 28, 0, -4, 4], [0, 7, 7, -2, 0, 0, 17, -5, -5, 2], [0, 2, 2, 4, 0, 0, 32, 0, 0, 0], [11, -11, 0, 8, 8, -3, 11, 0, 2, 3], [37, -6, 0, -6, -6, -5, 37, 0, -4, 5], [-18, 0, -6, 0, -10, 6, 0, -2, 0, 0], [0, 0, -2, 0, -4, -10, 0, -1, 0, 0], [0, 3, 3, -3, 0, 0, 4, -1, -1, 5], [0, -9, -9, -3, 0, 0, -4, 3, 3, -3], [-19, 0, 4, 0, 5, 4, 0, -1, 0, 0], [22, 0, 0, 0, 2, -5, 0, -3, 0, 0], [-46, 6, 0, 5, 5, 0, -46, 0, 2, 0], [7, -11, 0, -4, -4, -4, 7, 0, 2, 4], [8, 0, 12, 0, -7, -5, 0, 3, 0, 0], [0, -4, -4, -12, 0, 0, 12, 1, 1, 2], [0, 8, 8, 7, 0, 0, 0, 0, 0, 5], [-33, 7, 0, -2, -2, -3, -33, 0, 2, 3], [0, -2, -2, -12, 0, 0, -7, -4, -4, 7], [0, -4, -4, 9, 0, 0, -16, -2, -2, -5], [4, 0, -11, 0, -3, -11, 0, -5, 0, 0], [-12, 0, -4, 0, 6, 6, 0, 4, 0, 0], [20, 0, 18, 0, 3, 6, 0, -2, 0, 0], [1, 0, -5, 0, -15, -7, 0, 0, 0, 0], [0, 1, 1, -6, 0, 0, -33, -1, -1, 2], [-10, 0, -9, 0, 7, 3, 0, -5, 0, 0], [0, 7, 7, 5, 0, 0, 27, -3, -3, -2], [0, -1, -1, 8, 0, 0, -10, 3, 3, -5], [29, -8, 0, -10, -10, -5, 29, 0, -3, 5], [-1, -1, 0, 0, 0, 1, -1, 0, 8, -1], [24, 0, 2, 0, 9, 3, 0, -6, 0, 0], [-34, 0, 5, 0, 0, -3, 0, 0, 0, 0], [-4, 0, -6, 0, -7, 7, 0, 1, 0, 0], [9, 10, 0, 7, 7, -2, 9, 0, 5, 2], [0, 9, 9, 8, 0, 0, -6, 0, 0, -3], [11, -14, 0, -5, -5, 0, 11, 0, 1, 0], [5, -4, 0, 10, 10, 3, 5, 0, -3, -3], [0, -3, -3, 4, 0, 0, 12, 1, 1, 1], [-12, 0, -2, 0, -2, 8, 0, 7, 0, 0], [-10, 12, 0, -5, -5, -3, -10, 0, -5, 3], [-25, 0, -7, 0, -2, 4, 0, 1, 0, 0], [-6, 1, 0, 13, 13, 12, -6, 0, -2, -12], [-1, 7, 0, 9, 9, -4, -1, 0, 3, 4], [0, -2, -2, 2, 0, 0, -9, -2, -2, 9], [6, 7, 0, 4, 4, 9, 6, 0, 2, -9], [0, -1, -1, -5, 0, 0, -4, 0, 0, -9], [0, -2, -2, -8, 0, 0, -1, 1, 1, 2], [0, 13, 13, 1, 0, 0, -20, 6, 6, -5], [0, -2, -2, -14, 0, 0, -28, -8, -8, 0], [25, -21, 0, -2, -2, 4, 25, 0, -4, -4], [-2, 0, -4, 0, 11, -3, 0, 1, 0, 0], [-6, 19, 0, 8, 8, 1, -6, 0, 5, -1], [-15, 0, 1, 0, 4, 2, 0, 4, 0, 0], [-38, 0, 5, 0, -8, 3, 0, -1, 0, 0], [22, 2, 0, -2, -2, -2, 22, 0, -1, 2], [15, 0, -1, 0, -6, -2, 0, -5, 0, 0], [0, 0, 0, 12, 0, 0, 30, 4, 4, 2], [18, 0, 1, 0, -3, 3, 0, 5, 0, 0], [0, -13, -13, -8, 0, 0, 7, 5, 5, 0], [13, 5, 0, 3, 3, -6, 13, 0, -4, 6], [39, 2, 0, -2, -2, -5, 39, 0, -9, 5], [-4, 0, -2, 0, -3, -13, 0, -2, 0, 0], [0, -4, -4, 2, 0, 0, -26, 3, 3, -10], [0, -18, -18, -6, 0, 0, 17, -3, -3, -5], [-10, 0, -18, 0, -6, -6, 0, -2, 0, 0], [7, 7, 0, -4, -4, -4, 7, 0, 4, 4], [32, 2, 0, 2, 2, -8, 32, 0, 6, 8], [-20, 0, 11, 0, 2, 7, 0, -1, 0, 0], [-28, 0, -16, 0, 5, -5, 0, -5, 0, 0], [-29, 0, -13, 0, 6, 2, 0, -8, 0, 0], [14, 4, 0, -3, -3, 5, 14, 0, -1, -5], [-8, 0, -9, 0, -5, -7, 0, 3, 0, 0], [-3, -11, 0, -1, -1, -4, -3, 0, -3, 4], [28, 0, 2, 0, 2, 0, 0, 1, 0, 0], [0, -10, -10, -1, 0, 0, -19, -9, -9, -4], [-2, 2, 0, 5, 5, -4, -2, 0, -3, 4], [2, -1, 0, 1, 1, -3, 2, 0, 8, 3], [13, 0, -5, 0, -6, 2, 0, 1, 0, 0], [-17, 10, 0, 16, 16, -1, -17, 0, -2, 1], [0, 4, 4, 3, 0, 0, 11, 3, 3, -8], [-9, 0, -5, 0, 4, 2, 0, 8, 0, 0], [0, -5, -5, 4, 0, 0, 22, -1, -1, 3], [3, 0, 2, 0, -8, 7, 0, 4, 0, 0], [-2, 0, -8, 0, -6, -10, 0, 3, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_2_i_p();".
// 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_1110_2_i_p(:prec:=10)
    chi := MakeCharacter_1110_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_1110_2_i_p();".
// 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_1110_2_i_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![256, 672, 1572, 1240, 1110, -260, 487, -24, 23, 0, 1]>,<11,R![-492, 404, -1, -41, 1, 1]>],Snew);
    return Vf;
end function;