
// Make newform 1380.2.i.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1380_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1380_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_1380_2_i_b();".
// 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_1380_2_i_b();".
// 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, 0, -729, -972, -162, 324, 153, 48, -110, 16, 17, 12, -2, -4, -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], [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], [1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6561, -2187, 1944, 1215, 756, -837, 192, 43, 124, -1, -24, -11, -4, 8, 0, 1], [-43740, -5832, -2187, 10368, 3429, -2484, -1209, 164, 794, -116, -21, -124, 13, 64, 9, 8], [-729, -972, -162, 324, 153, 48, -110, 16, 17, 12, -2, -4, -1, 0, 1, 0], [0, 729, 972, 162, -324, -153, -48, 110, -16, -17, -12, 2, 4, 1, 0, -1], [-17496, 0, 8019, -5184, -3483, -2988, 1653, -340, -88, -188, -93, 92, -5, -8, -27, 8], [-8748, 1458, 2916, 2349, -3348, -1062, 396, 343, 196, -115, -36, -38, 20, 11, -12, -2], [54675, -75816, -17739, 2268, 18009, 1044, -1212, -892, -517, 820, -105, -184, -191, 64, 24, 8], [-34992, 11664, 8019, -1296, -4023, -36, 3873, 76, -176, -484, -105, 76, 71, -16, -27, 16], [-52488, -38637, 25272, 24138, -3672, -4734, -1380, 383, 752, 133, -444, -130, 136, 70, 0, -7], [0, -35721, 23328, 11745, 6372, -6183, 636, 1144, 436, 143, -372, 31, 56, 89, -24, -8], [8748, 4374, -10935, 9153, -729, -468, -1545, 377, 68, 109, 129, -112, 25, 1, 15, -10], [52488, 23328, 1944, -14499, 1296, 4851, 3828, -389, -668, 80, 228, 205, -76, -19, 0, 19]];
    Rf_basisdens := [1, 1, 1, 1, 2187, 17496, 729, 2187, 17496, 8748, 17496, 17496, 17496, 17496, 8748, 17496];
    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_1380_i();"
function MakeCharacter_1380_i()
    N := 1380;
    order := 2;
    char_gens := [691, 461, 277, 1201];
    v := [2, 1, 2, 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_1380_i_Hecke();"
function MakeCharacter_1380_i_Hecke(Kf)
    N := 1380;
    order := 2;
    char_gens := [691, 461, 277, 1201];
    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, 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, 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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, -1, -1, -1, 1, 0, -1, 1, -2, 0, 0, -1, 0, -1, -1, -1], [0, 1, -1, 1, 1, -1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 1, 1, -1, 0, 1, -1, 2, 0, 1, 0, 0, 1, 2, 1], [0, 0, 1, 1, -1, 0, 1, 0, 3, 0, 1, -1, 0, 1, 1, 1], [0, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, 1, 0], [0, 0, -1, 0, 0, -1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0], [0, -1, -1, 0, 0, 0, -1, 1, -2, 0, 1, 0, 0, -1, -1, -1], [0, 0, -1, -1, 1, 0, -1, 0, -2, 0, -1, -1, 0, 0, -1, 0], [0, -1, 1, 1, -1, 0, 1, 1, 0, 0, -1, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 1], [0, 0, -1, 1, 1, -2, 1, 0, 0, 0, 0, 0, 1, -1, 0, 1], [0, 1, 1, 1, -1, 0, 1, -1, 2, 0, -1, -1, 0, 1, 0, 1], [0, 0, -1, 0, 0, -1, 1, 0, 0, 2, 0, 0, 0, 1, 0, -1], [2, -1, 1, -1, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -3, 0, 0, 0, -3, 2, -1, 0, 1, -2, 0, 0, 0, 0], [0, -2, -2, -1, 1, 0, -2, 2, -1, 0, -1, -1, 0, -1, -1, -1], [0, 1, 0, 1, -1, 0, 0, -1, 3, 0, 1, 0, 0, 2, 2, 2], [0, -1, -1, 1, 1, -1, 1, -1, 0, 1, 0, 0, -1, 1, 0, -1], [0, 0, 2, 0, 0, 0, 2, 0, 2, 0, -1, -1, 0, 1, -1, 1], [0, 1, 0, -1, -1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 1, 0, 0, 2, 0, 0, -1, 1, 0, -1], [0, -2, -2, -1, 1, 0, -2, 2, -2, 0, -1, -1, 0, 0, -2, 0], [-6, 2, 1, 2, 2, 0, -1, 2, 0, -1, 0, 0, 1, 0, 0, 0], [0, 1, 1, 0, 0, 3, -1, 1, 0, -1, 0, 0, 0, -1, 0, 1], [2, -1, 1, -1, -1, 0, -1, -1, 0, -1, 0, 0, -2, 0, 0, 0], [-2, 2, -1, 1, 1, -2, 1, 2, 0, 1, 0, 0, 1, -1, 0, 1], [0, -4, -2, -2, 2, 0, -2, 4, -3, 0, 0, -2, 0, 0, -2, 0], [0, 1, -1, 1, 1, 0, 1, 1, 0, -2, 0, 0, 0, 1, 0, -1], [0, 0, 2, 1, -1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 3, 0], [0, 4, 1, 0, 0, 0, 1, -4, 2, 0, 0, 2, 0, 1, 0, 1], [0, -3, -2, -3, 3, 0, -2, 3, -3, 0, 0, -1, 0, -2, -2, -2], [0, 3, 0, 2, -2, 0, 0, -3, 2, 0, 2, 0, 0, 1, 0, 1], [0, -1, -1, -1, -1, 2, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, -1, 2, 0, -1, 0, -2, 0, 0, 0, -2, 0, 2], [0, 1, 3, 1, -1, 0, 3, -1, 0, 0, -2, 2, 0, 1, 0, 1], [0, 5, 2, 0, 0, 0, 2, -5, 2, 0, 1, 1, 0, 0, 2, 0], [2, -3, 0, 0, 0, -1, 0, -3, 0, -1, 0, 0, -2, -1, 0, 1], [6, 0, 0, -1, -1, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [2, -3, 4, -1, -1, 2, -4, -3, 0, 0, 0, 0, 0, 1, 0, -1], [0, -1, -3, -1, 1, 0, -3, 1, 2, 0, 0, -2, 0, 0, -2, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -2, -2, -2], [0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, -1, -3, -1], [0, -1, -3, -1, 1, 0, -3, 1, 0, 0, 1, -5, 0, -1, -1, -1], [0, 2, -1, -1, -1, 2, 1, 2, 0, 0, 0, 0, 0, -1, 0, 1], [0, -1, 1, -1, 1, 0, 1, 1, -1, 0, -4, 1, 0, -1, -1, -1], [0, -1, 1, -1, -1, 1, -1, -1, 0, -3, 0, 0, -3, 1, 0, -1], [0, 3, -1, -1, 1, 0, -1, -3, 4, 0, 2, -1, 0, 1, 1, 1], [2, 1, 2, -2, -2, 1, -2, 1, 0, -3, 0, 0, -2, 1, 0, -1], [4, 3, -3, 0, 0, -2, 3, 3, 0, 6, 0, 0, 0, 1, 0, -1], [6, 2, 3, -1, -1, 0, -3, 2, 0, -1, 0, 0, 0, -1, 0, 1], [0, 1, 3, 1, -1, 0, 3, -1, 3, 0, 0, -2, 0, 0, 0, 0], [4, 2, -2, 0, 0, 1, 2, 2, 0, -4, 0, 0, 2, 0, 0, 0], [2, 3, 2, 0, 0, 0, -2, 3, 0, 0, 0, 0, -1, 2, 0, -2], [0, 2, 0, 1, -1, 0, 0, -2, 8, 0, 1, -3, 0, 2, 2, 2], [0, 6, 0, -2, 2, 0, 0, -6, -2, 0, -1, -4, 0, 1, -4, 1], [0, 0, 1, 2, -2, 0, 1, 0, 5, 0, -2, 3, 0, 1, 1, 1], [-4, 1, -1, 1, 1, 1, 1, 1, 0, -2, 0, 0, -1, 0, 0, 0], [0, 3, 0, 1, -1, 0, 0, -3, 0, 0, 2, -2, 0, 1, -2, 1], [0, 2, -3, 0, 0, 0, -3, -2, -6, 0, 2, 2, 0, -1, 2, -1], [6, -1, -1, -2, -2, -1, 1, -1, 0, 1, 0, 0, 0, -1, 0, 1], [0, -1, -1, -3, 3, 0, -1, 1, -3, 0, 0, 1, 0, -3, -1, -3], [-4, -4, 0, -2, -2, 4, 0, -4, 0, -3, 0, 0, -2, 0, 0, 0], [0, 0, 0, 2, -2, 0, 0, 0, 2, 0, 3, 0, 0, 1, 2, 1], [0, -3, -2, 1, -1, 0, -2, 3, 5, 0, -1, 0, 0, 2, 0, 2], [0, -2, -2, -2, 2, 0, -2, 2, -6, 0, -3, 5, 0, -3, -2, -3], [4, -7, -2, 1, 1, -1, 2, -7, 0, -2, 0, 0, 0, -3, 0, 3], [-2, 6, -3, 1, 1, -4, 3, 6, 0, 2, 0, 0, 0, 1, 0, -1], [-4, 3, -1, 0, 0, -2, 1, 3, 0, 4, 0, 0, 2, 1, 0, -1], [6, -4, 0, 0, 0, -2, 0, -4, 0, 2, 0, 0, -2, -2, 0, 2], [-2, 1, -3, 2, 2, -3, 3, 1, 0, -1, 0, 0, 0, -1, 0, 1], [0, 4, 3, -1, -1, 4, -3, 4, 0, -1, 0, 0, 0, 1, 0, -1], [0, 2, 5, 1, -1, 0, 5, -2, 6, 0, 1, -1, 0, 4, 3, 4], [8, 3, -4, 0, 0, -2, 4, 3, 0, 4, 0, 0, -2, -1, 0, 1], [0, -3, 3, 1, -1, 0, 3, 3, 7, 0, 0, -2, 0, 0, 0, 0], [-2, 0, -1, -1, -1, 2, 1, 0, 0, -5, 0, 0, -1, 1, 0, -1], [0, -1, -1, 1, -1, 0, -1, 1, -5, 0, 0, 3, 0, -1, 1, -1], [0, 1, -2, 1, -1, 0, -2, -1, 4, 0, -2, 1, 0, 1, -1, 1], [0, 0, -4, -2, 2, 0, -4, 0, -4, 0, -1, -2, 0, 1, 2, 1], [0, 2, -3, -2, 2, 0, -3, -2, -8, 0, -2, -2, 0, 0, -2, 0], [2, -3, 2, -2, -2, 2, -2, -3, 0, 0, 0, 0, -1, 0, 0, 0], [-4, 0, -4, 2, 2, -3, 4, 0, 0, -1, 0, 0, 3, 1, 0, -1], [-8, 3, 2, -2, -2, 0, -2, 3, 0, -1, 0, 0, -2, 1, 0, -1], [-4, 0, 4, -1, -1, 4, -4, 0, 0, -2, 0, 0, 1, 2, 0, -2], [0, 2, 0, 0, 0, 0, 0, -2, 3, 0, 2, -4, 0, 3, 1, 3], [0, 1, 1, 3, 3, -5, -1, 1, 0, 4, 0, 0, -1, 0, 0, 0], [4, -8, 3, 0, 0, 3, -3, -8, 0, -2, 0, 0, 2, -3, 0, 3], [0, 1, 1, 0, 0, 0, 1, -1, -1, 0, -1, 4, 0, 0, -2, 0], [14, -5, 1, -3, -3, 2, -1, -5, 0, 1, 0, 0, -4, 2, 0, -2], [0, -3, 4, 1, -1, 0, 4, 3, 2, 0, 2, 4, 0, 1, 4, 1], [-4, -2, 1, 1, 1, 0, -1, -2, 0, 1, 0, 0, 3, 1, 0, -1], [0, -1, 4, 0, 0, 2, -4, -1, 0, -2, 0, 0, 0, 0, 0, 0], [0, 7, 4, 1, 1, -1, -4, 7, 0, 1, 0, 0, 1, 2, 0, -2], [8, 4, 0, 2, 2, 1, 0, 4, 0, 3, 0, 0, 3, 1, 0, -1], [10, 4, -3, -1, -1, -2, 3, 4, 0, 4, 0, 0, 4, 1, 0, -1], [0, -1, 2, 4, -4, 0, 2, 1, 4, 0, -2, 4, 0, 3, 1, 3], [4, -7, 4, -1, -1, 0, -4, -7, 0, 2, 0, 0, -4, 0, 0, 0], [0, -5, 2, -2, 2, 0, 2, 5, -2, 0, 2, 0, 0, -4, 0, -4], [0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, -3, -5, -3], [0, -6, -3, -3, 3, 0, -3, 6, -6, 0, -3, 1, 0, -4, -6, -4], [10, 5, -1, 1, 1, -1, 1, 5, 0, 0, 0, 0, -3, 0, 0, 0], [-2, 1, -1, 1, 1, -6, 1, 1, 0, 4, 0, 0, 3, 1, 0, -1], [0, 2, 1, 0, 0, 0, 1, -2, 0, 0, -4, 2, 0, -3, 0, -3], [-8, 5, 1, -1, -1, 3, -1, 5, 0, 0, 0, 0, 0, 2, 0, -2], [0, -4, 5, 3, -3, 0, 5, 4, 12, 0, 1, 1, 0, 4, 6, 4], [0, 2, 4, 0, 0, 0, 4, -2, 2, 0, 2, 6, 0, -4, 3, -4], [2, 2, -4, 4, 4, -6, 4, 2, 0, 3, 0, 0, 0, 2, 0, -2], [0, -4, -2, 0, 0, 0, -2, 4, -6, 0, 1, -2, 0, -3, 0, -3], [0, 3, -1, -3, 3, 0, -1, -3, -7, 0, 0, 1, 0, -3, 1, -3], [0, -3, 2, 0, 0, 0, 2, 3, 3, 0, 0, -1, 0, 0, -3, 0], [8, -9, -1, -1, -1, 2, 1, -9, 0, 1, 0, 0, -2, 2, 0, -2], [0, -4, 1, 1, -1, 0, 1, 4, -8, 0, -2, 2, 0, -3, -2, -3], [8, 2, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 1, 0, 0, 0], [-4, -8, 3, 0, 0, 1, -3, -8, 0, 3, 0, 0, -3, 0, 0, 0], [0, -1, -2, 4, -4, 0, -2, 1, 3, 0, 4, 1, 0, 0, 5, 0], [4, -6, 2, -1, -1, 2, -2, -6, 0, 1, 0, 0, 1, -1, 0, 1], [10, -4, 1, -3, -3, 6, -1, -4, 0, -3, 0, 0, -2, -3, 0, 3], [0, -8, 1, -1, 1, 0, 1, 8, -6, 0, -1, 1, 0, 0, 3, 0], [0, 3, 1, 0, 0, 0, 1, -3, 2, 0, 1, 2, 0, 1, -1, 1], [0, -1, -3, -5, 5, 0, -3, 1, -9, 0, -2, -4, 0, -6, -6, -6], [0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, 0, -3, 0, -3], [-4, 1, 1, 3, 3, -3, -1, 1, 0, 0, 0, 0, 3, 4, 0, -4], [-10, -6, -1, 0, 0, 2, 1, -6, 0, 2, 0, 0, -2, -3, 0, 3], [0, 14, 3, 1, -1, 0, 3, -14, 4, 0, 0, 0, 0, 3, 1, 3], [0, -8, 0, -2, 2, 0, 0, 8, -7, 0, 0, 6, 0, -4, -2, -4], [0, -7, -1, -1, 1, 0, -1, 7, -4, 0, -2, -3, 0, -1, 3, -1], [0, 3, -5, -3, 3, 0, -5, -3, -8, 0, -1, -1, 0, -1, 1, -1], [-14, -5, 1, -1, -1, 0, -1, -5, 0, 6, 0, 0, 0, 4, 0, -4], [0, -6, 2, 0, 0, 0, 2, 6, 2, 0, -1, 2, 0, -1, 4, -1], [8, -3, 2, -1, -1, -1, -2, -3, 0, -1, 0, 0, 1, 2, 0, -2], [0, -2, 0, 3, -3, 0, 0, 2, 1, 0, 1, -1, 0, 3, 1, 3], [-16, -5, 3, 0, 0, -3, -3, -5, 0, -1, 0, 0, -4, 3, 0, -3], [0, -1, 0, 4, -4, 0, 0, 1, 3, 0, 3, 0, 0, 2, 6, 2], [-4, 3, -5, 1, 1, -8, 5, 3, 0, 0, 0, 0, 2, 3, 0, -3], [-4, 0, 2, -4, -4, 1, -2, 0, 0, 1, 0, 0, -5, 3, 0, -3], [4, -1, 6, -1, -1, -1, -6, -1, 0, -2, 0, 0, -1, 2, 0, -2], [8, 5, 2, 2, 2, -6, -2, 5, 0, -2, 0, 0, 2, 1, 0, -1], [2, 0, -3, 3, 3, -4, 3, 0, 0, 0, 0, 0, 4, -3, 0, 3], [0, -7, 3, -1, 1, 0, 3, 7, -6, 0, 0, 2, 0, 1, -4, 1], [4, 3, -5, 0, 0, -1, 5, 3, 0, 5, 0, 0, 4, 1, 0, -1], [0, -9, -6, -4, 4, 0, -6, 9, -4, 0, -4, -2, 0, -4, -4, -4], [10, 2, -2, 2, 2, -4, 2, 2, 0, -2, 0, 0, 6, 2, 0, -2], [-2, 3, -2, 2, 2, 0, 2, 3, 0, -1, 0, 0, 4, 1, 0, -1], [-2, -1, 4, -2, -2, 0, -4, -1, 0, 4, 0, 0, -3, 6, 0, -6], [0, 11, 7, 1, -1, 0, 7, -11, 2, 0, 0, 4, 0, -1, 2, -1], [0, 5, 7, 3, -3, 0, 7, -5, 12, 0, -3, 2, 0, 5, 0, 5], [-4, -5, -5, 1, 1, -4, 5, -5, 0, -4, 0, 0, 0, -4, 0, 4], [0, -1, 5, 1, -1, 0, 5, 1, 4, 0, -1, 7, 0, -3, 0, -3], [0, 1, -1, -3, 3, 0, -1, -1, -2, 0, -4, -1, 0, -1, -1, -1], [0, -4, 3, 2, -2, 0, 3, 4, 7, 0, 1, -2, 0, 0, 4, 0], [-2, 7, -2, 0, 0, 4, 2, 7, 0, 5, 0, 0, 2, -1, 0, 1], [0, -3, -1, -1, 1, 0, -1, 3, 2, 0, 2, 0, 0, -1, -6, -1], [-2, 0, 0, 2, 2, -6, 0, 0, 0, -4, 0, 0, 6, 4, 0, -4], [4, 2, -2, -1, -1, -2, 2, 2, 0, -4, 0, 0, -3, 0, 0, 0], [8, 8, -5, 1, 1, -4, 5, 8, 0, -2, 0, 0, 0, -1, 0, 1], [12, -1, -3, 1, 1, -5, 3, -1, 0, 0, 0, 0, 0, 2, 0, -2], [12, 6, 1, 0, 0, 3, -1, 6, 0, -6, 0, 0, -2, 1, 0, -1], [-10, 9, 4, 0, 0, -1, -4, 9, 0, -1, 0, 0, 2, 3, 0, -3], [-2, 4, 3, 3, 3, -2, -3, 4, 0, -2, 0, 0, 2, -3, 0, 3], [0, 1, 3, -1, 1, 0, 3, -1, 2, 0, 0, 0, 0, 0, -2, 0], [0, 1, -2, 0, 0, 0, -2, -1, 1, 0, 0, 1, 0, 0, -1, 0], [-4, -3, 0, 0, 0, -2, 0, -3, 0, 2, 0, 0, -6, 3, 0, -3], [-10, 3, -1, 0, 0, -7, 1, 3, 0, 3, 0, 0, 0, 1, 0, -1], [-8, 6, -1, -1, -1, 0, 1, 6, 0, -2, 0, 0, 0, -1, 0, 1], [0, -9, 1, -1, 1, 0, 1, 9, -3, 0, -4, -2, 0, 2, 0, 2], [10, 4, -1, -2, -2, -2, 1, 4, 0, 0, 0, 0, 4, 2, 0, -2], [-10, -12, 0, 2, 2, -2, 0, -12, 0, 3, 0, 0, 2, 0, 0, 0], [-6, 4, 1, 1, 1, 0, -1, 4, 0, -7, 0, 0, 5, -3, 0, 3], [0, -9, -1, 1, -1, 0, -1, 9, 2, 0, -2, 1, 0, 5, 5, 5], [0, -9, 3, -1, 1, 0, 3, 9, -2, 0, 3, 4, 0, -3, -4, -3], [0, -1, 0, 0, 0, 0, 0, 1, 4, 0, 3, -7, 0, -2, -1, -2], [-4, 6, -1, -2, -2, 2, 1, 6, 0, -6, 0, 0, 1, 1, 0, -1], [-4, -4, -1, -1, -1, 4, 1, -4, 0, 1, 0, 0, -2, -5, 0, 5], [-12, -1, 2, 1, 1, -6, -2, -1, 0, -2, 0, 0, -4, 2, 0, -2], [0, 0, 1, -6, 6, 0, 1, 0, -7, 0, -4, 1, 0, -5, -5, -5], [-12, 0, -4, 2, 2, -3, 4, 0, 0, -3, 0, 0, 5, -3, 0, 3], [14, 7, 1, 1, 1, 2, -1, 7, 0, -1, 0, 0, -2, -2, 0, 2], [-6, -7, -2, 1, 1, -2, 2, -7, 0, -6, 0, 0, 1, 2, 0, -2], [0, -13, -11, -5, 5, 0, -11, 13, -10, 0, -1, 0, 0, -5, -2, -5], [2, -7, -5, -1, -1, 1, 5, -7, 0, 6, 0, 0, -3, -2, 0, 2], [0, 12, -1, 2, -2, 0, -1, -12, 9, 0, 2, -2, 0, 1, -1, 1], [-8, 2, 1, -3, -3, 2, -1, 2, 0, -4, 0, 0, -1, 1, 0, -1], [0, 4, 0, -1, 1, 0, 0, -4, -8, 0, -4, 2, 0, 0, 2, 0], [0, -9, 1, -1, 1, 0, 1, 9, -4, 0, -2, 4, 0, -4, -3, -4], [0, 8, 0, 1, -1, 0, 0, -8, 3, 0, -3, -3, 0, 1, -1, 1], [-6, -8, 2, -4, -4, -2, -2, -8, 0, 1, 0, 0, -3, -1, 0, 1], [-4, -5, -2, -1, -1, 5, 2, -5, 0, -1, 0, 0, -5, -2, 0, 2], [0, -4, 0, 4, -4, 0, 0, 4, 8, 0, 5, 1, 0, 3, 7, 3], [0, 1, 0, 1, -1, 0, 0, -1, 1, 0, -3, -2, 0, 4, -6, 4], [-10, 8, 0, 0, 0, 2, 0, 8, 0, -2, 0, 0, 0, -2, 0, 2], [0, 7, 9, 3, -3, 0, 9, -7, 12, 0, -4, 4, 0, 7, 2, 7], [10, -11, 6, -2, -2, 10, -6, -11, 0, -7, 0, 0, -4, 1, 0, -1], [4, 0, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 7, -3, 0, 3], [4, -8, 7, -5, -5, 4, -7, -8, 0, -2, 0, 0, -3, -1, 0, 1], [0, -6, -5, -1, 1, 0, -5, 6, -10, 0, -1, -5, 0, 0, -5, 0], [8, 5, 5, 1, 1, 4, -5, 5, 0, -7, 0, 0, -8, -2, 0, 2], [0, 3, 11, 1, -1, 0, 11, -3, 6, 0, 2, 0, 0, -1, -2, -1], [0, 6, 4, 2, -2, 0, 4, -6, 18, 0, 6, 0, 0, 2, 3, 2], [0, 1, 3, 1, -1, 0, 3, -1, 1, 0, 0, -5, 0, 3, 9, 3], [0, 0, -5, 1, -1, 0, -5, 0, -9, 0, -1, -3, 0, 1, 1, 1], [14, 10, -1, -1, -1, 0, 1, 10, 0, -1, 0, 0, -5, 1, 0, -1], [-16, -1, 5, 0, 0, 4, -5, -1, 0, 0, 0, 0, 2, -1, 0, 1], [0, -6, 1, 2, -2, 0, 1, 6, 10, 0, -2, -2, 0, 6, 0, 6], [0, 8, -1, -2, 2, 0, -1, -8, -3, 0, 5, 0, 0, -6, -2, -6], [-26, 4, 6, -2, -2, 2, -6, 4, 0, -5, 0, 0, 3, -1, 0, 1], [0, 1, 7, 0, 0, 0, 7, -1, -4, 0, -1, 8, 0, -5, -3, -5], [0, 4, -3, 3, -3, 0, -3, -4, 6, 0, -1, -7, 0, 2, -5, 2], [0, -7, -4, -4, -4, 6, 4, -7, 0, 0, 0, 0, -4, -2, 0, 2], [0, 6, 9, 5, -5, 0, 9, -6, 7, 0, -1, 5, 0, 5, 5, 5], [-4, 2, -2, 0, 0, -1, 2, 2, 0, -1, 0, 0, 9, 1, 0, -1], [0, 3, 0, 0, 0, 0, 0, -3, 7, 0, -6, -1, 0, 2, -3, 2], [10, -11, -1, 1, 1, -1, 1, -11, 0, 0, 0, 0, -1, 2, 0, -2], [0, -5, -3, -1, 1, 0, -3, 5, -8, 0, 5, -5, 0, -1, 0, -1], [0, -9, 1, 1, -1, 0, 1, 9, 4, 0, 0, 3, 0, 3, 5, 3], [0, 5, -1, -1, 1, 0, -1, -5, -8, 0, 1, 2, 0, -5, -2, -5], [0, -5, 5, -3, 3, 0, 5, 5, 7, 0, 0, -3, 0, -3, -5, -3], [10, -1, 1, -2, -2, 3, -1, -1, 0, 5, 0, 0, 2, -1, 0, 1], [2, -1, 3, -1, -1, -2, -3, -1, 0, -2, 0, 0, -2, -2, 0, 2], [0, 6, -4, 0, 0, 0, -4, -6, -5, 0, -2, 2, 0, -1, -1, -1], [0, -4, -3, 2, -2, 0, -3, 4, 6, 0, 2, -4, 0, 3, -4, 3], [0, -12, -3, 0, 0, 0, -3, 12, 5, 0, 4, -4, 0, 1, -1, 1], [-2, 7, 1, 2, 2, -4, -1, 7, 0, 4, 0, 0, -1, 1, 0, -1], [0, -4, 1, -1, 1, 0, 1, 4, -12, 0, 4, 4, 0, -7, 1, -7], [0, 6, 5, 1, -1, 0, 5, -6, 6, 0, 3, -5, 0, 2, 0, 2], [0, -1, 0, 3, -3, 0, 0, 1, -2, 0, 0, -1, 0, 3, 1, 3], [0, -6, -4, -2, 2, 0, -4, 6, -6, 0, 3, -8, 0, 1, -4, 1], [18, -15, 1, -1, -1, 2, -1, -15, 0, -3, 0, 0, 1, 1, 0, -1], [0, -7, -6, -4, 4, 0, -6, 7, -6, 0, 3, -1, 0, 0, -1, 0], [-6, 7, 3, -3, -3, 2, -3, 7, 0, -4, 0, 0, 6, 2, 0, -2], [0, -3, 6, 2, -2, 0, 6, 3, -2, 0, -5, 10, 0, -1, 3, -1], [0, -5, 2, -2, 2, 0, 2, 5, 4, 0, -3, -6, 0, 1, -1, 1], [8, 12, -7, 2, 2, -7, 7, 12, 0, 10, 0, 0, -2, 3, 0, -3], [-4, 9, 1, -1, -1, -1, -1, 9, 0, -2, 0, 0, -3, 2, 0, -2], [4, 1, -5, -1, -1, 4, 5, 1, 0, 6, 0, 0, -4, 3, 0, -3], [4, 2, -1, -3, -3, 2, 1, 2, 0, 3, 0, 0, -9, 3, 0, -3], [0, 2, -2, 3, -3, 0, -2, -2, 10, 0, 4, 0, 0, 2, 6, 2], [0, 8, 6, -1, 1, 0, 6, -8, 2, 0, -3, 3, 0, 0, -2, 0], [-8, -10, 4, -3, -3, 2, -4, -10, 0, -2, 0, 0, 1, 0, 0, 0], [-12, -7, 2, -4, -4, 0, -2, -7, 0, -3, 0, 0, -8, -1, 0, 1], [0, 6, -4, -4, 4, 0, -4, -6, -14, 0, -7, -3, 0, -1, -2, -1], [-16, 4, 1, 2, 2, 3, -1, 4, 0, 0, 0, 0, 0, -1, 0, 1], [0, 9, 0, -1, 1, 0, 0, -9, -1, 0, 1, 2, 0, -6, -2, -6], [-26, -16, 7, -1, -1, 4, -7, -16, 0, -6, 0, 0, -4, -1, 0, 1], [0, -8, -3, 0, 0, -5, 3, -8, 0, 2, 0, 0, -2, 3, 0, -3], [0, -7, -2, 0, 0, 0, -2, 7, 8, 0, -2, -4, 0, 5, 4, 5], [0, -2, 4, -1, 1, 0, 4, 2, -1, 0, 0, 3, 0, -2, 1, -2], [0, -3, 4, 0, 0, 0, 4, 3, -6, 0, -1, 2, 0, 1, 3, 1], [0, -2, 8, 0, 0, 0, 8, 2, -8, 0, -5, 11, 0, -1, 1, -1], [0, -2, -3, 0, 0, 0, -3, 2, 0, 0, -6, 4, 0, 3, 2, 3], [0, -9, -6, -4, 4, 0, -6, 9, -13, 0, 2, 3, 0, -8, -3, -8], [4, 9, -8, 1, 1, 0, 8, 9, 0, -6, 0, 0, 2, -2, 0, 2], [0, 2, -8, 0, 0, 0, -8, -2, 6, 0, 1, -6, 0, -1, 4, -1], [8, 1, 1, -3, -3, 7, -1, 1, 0, -1, 0, 0, 3, -3, 0, 3], [4, -3, 0, 5, 5, -3, 0, -3, 0, -6, 0, 0, 11, -2, 0, 2], [0, 10, -2, 2, -2, 0, -2, -10, -7, 0, 0, 6, 0, 0, 2, 0], [0, 2, -12, -3, 3, 0, -12, -2, -10, 0, 2, 0, 0, -6, 2, -6], [0, 8, -3, 5, -5, 0, -3, -8, 6, 0, 4, -2, 0, 3, 3, 3], [-2, 2, 2, 2, 2, 2, -2, 2, 0, -1, 0, 0, 6, -4, 0, 4], [-6, 3, 0, -4, -4, 2, 0, 3, 0, 2, 0, 0, 0, 3, 0, -3], [0, -6, -4, 4, -4, 0, -4, 6, -3, 0, 4, -2, 0, 4, 4, 4], [0, -1, 2, 2, -2, 0, 2, 1, -8, 0, 6, -2, 0, 0, -4, 0], [-10, 9, -1, -1, -1, -5, 1, 9, 0, 6, 0, 0, 5, 2, 0, -2], [0, 4, 4, -6, 6, 0, 4, -4, -12, 0, -7, 4, 0, -3, -4, -3], [0, -7, -7, -5, 5, 0, -7, 7, -10, 0, -5, -8, 0, -3, -2, -3], [0, 3, 4, 8, -8, 0, 4, -3, 20, 0, 1, 1, 0, 12, 10, 12], [4, 8, -3, 1, 1, 8, 3, 8, 0, -2, 0, 0, 3, -5, 0, 5], [4, 4, 5, 0, 0, 4, -5, 4, 0, -8, 0, 0, 1, 1, 0, -1], [0, 10, 12, 5, -5, 0, 12, -10, 15, 0, 8, -3, 0, 4, 3, 4], [4, -1, -4, -1, -1, 3, 4, -1, 0, 10, 0, 0, -1, 0, 0, 0], [0, -1, -5, 3, -3, 0, -5, 1, -2, 0, -3, -1, 0, 1, -1, 1], [24, 5, 4, 2, 2, 2, -4, 5, 0, 3, 0, 0, 4, -3, 0, 3], [0, 1, 2, 4, -4, 0, 2, -1, 20, 0, 2, -6, 0, 8, 10, 8], [0, -1, -2, 0, 0, 0, -2, 1, -6, 0, -4, -4, 0, 1, -5, 1], [-14, -7, 6, -1, -1, 2, -6, -7, 0, -2, 0, 0, -4, -1, 0, 1], [0, -2, 3, 9, -9, 0, 3, 2, 8, 0, 2, 0, 0, 9, 10, 9], [6, 3, -4, 2, 2, -6, 4, 3, 0, 2, 0, 0, 5, -6, 0, 6], [0, 5, -3, -4, 4, 0, -3, -5, -12, 0, 3, 4, 0, -7, 3, -7], [0, 11, 5, 7, -7, 0, 5, -11, 17, 0, -2, 0, 0, 8, 2, 8], [-10, -7, 3, -1, -1, -2, -3, -7, 0, -4, 0, 0, 2, 2, 0, -2], [-4, 0, 4, -1, -1, 6, -4, 0, 0, -11, 0, 0, -3, -1, 0, 1], [0, 2, -4, 0, 0, 0, -4, -2, 8, 0, -2, -8, 0, 4, 2, 4], [14, 4, -1, 5, 5, 0, 1, 4, 0, -4, 0, 0, 10, -1, 0, 1], [0, 6, 3, -4, -4, 1, -3, 6, 0, 8, 0, 0, 4, 3, 0, -3], [0, -12, -12, -5, 5, 0, -12, 12, -22, 0, 2, -4, 0, -8, -6, -8], [-20, 0, -7, 1, 1, 0, 7, 0, 0, 5, 0, 0, -2, 1, 0, -1], [-4, 0, 4, -4, -4, 2, -4, 0, 0, 4, 0, 0, -5, 0, 0, 0], [22, -2, 5, 1, 1, 2, -5, -2, 0, 2, 0, 0, -2, 3, 0, -3], [0, -5, -2, -6, 6, 0, -2, 5, -2, 0, -6, -8, 0, -1, -5, -1], [0, -3, -1, -1, 1, 0, -1, 3, -3, 0, -2, 1, 0, -3, -5, -3], [0, -5, -4, 4, -4, 0, -4, 5, 18, 0, 5, 0, 0, 5, 9, 5], [12, -3, 3, -1, -1, 6, -3, -3, 0, -10, 0, 0, 4, -5, 0, 5], [0, 2, -4, 0, 0, 0, -4, -2, 9, 0, 4, -6, 0, 2, 6, 2], [-8, 11, -3, 1, 1, -1, 3, 11, 0, 2, 0, 0, -4, 4, 0, -4], [0, -14, -14, -6, 6, 0, -14, 14, -6, 0, -3, -1, 0, -5, -2, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1380_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1380_2_i_b();".
// 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_1380_2_i_b(:prec:=16)
    chi := MakeCharacter_1380_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_1380_2_i_b();".
// 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_1380_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1380_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-1024, -2368, -816, 924, 486, -50, -44, 0, 1]>],Snew);
    return Vf;
end function;