
// Make newform 3456.2.i.k in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3456_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3456_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_3456_2_i_k();".
// 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_3456_2_i_k();".
// 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 := [729, -486, 243, -216, 198, -126, 51, -42, 22, -8, 3, -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, 0], [-972, 1701, -27, 117, -270, 141, -3, 25, -26, -3, 1, 1], [-162, 135, -54, 81, -108, 51, -24, 13, -8, 3, -2, 1], [1458, -1620, 567, -630, 252, -240, 63, -34, 20, 0, -1, 2], [2430, -1377, 621, -549, 720, -195, 123, -109, 56, -3, 5, -7], [81, -216, 162, 9, 18, -15, 0, 4, -2, 0, -2, 1], [5346, 0, 540, -315, 810, -120, 42, -131, 16, -12, -2, -11], [4374, -567, 1404, -693, 954, -327, 192, -179, 52, -33, 4, -11], [7290, -81, 2025, -1233, 1314, -453, 243, -287, 64, -45, 13, -17], [12393, -1458, 3375, -1089, 2313, -708, 357, -421, 95, -66, -1, -31], [-19197, 2025, -5589, 2925, -3735, 1311, -729, 745, -173, 117, -17, 43], [-27945, 2349, -7506, 3816, -5229, 1617, -960, 1030, -215, 165, -20, 64]];
    Rf_basisdens := [1, 486, 162, 486, 486, 81, 486, 486, 486, 486, 486, 486];
    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_3456_i();"
function MakeCharacter_3456_i()
    N := 3456;
    order := 3;
    char_gens := [2431, 2053, 2945];
    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_3456_i_Hecke();"
function MakeCharacter_3456_i_Hecke(Kf)
    N := 3456;
    order := 3;
    char_gens := [2431, 2053, 2945];
    char_values := [[1, 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]];
    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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0], [0, -1, 0, -1, 0, 0, 0, 2, 0, -1, -1, 0], [-1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, -1, 1, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, 0, 0, 1, 1, -1, -1, 1, 0, 0, -1], [0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0, 1, 1, 0, -1, -1, 0], [1, 0, -1, 0, 0, 0, -1, -1, 0, 0, -1, 1], [-3, 0, 2, 0, 0, 0, 2, 3, 0, 0, -1, 0], [-2, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -1, -1, 0, -1, 1, 0, -2, -1, 1], [2, 0, 0, 0, 0, 0, 0, -2, 0, -1, 1, 0], [0, 0, 0, 0, -2, 1, 0, -1, 1, 0, 0, 2], [-2, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0], [5, 2, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 1, 3, 1, 2, 2, -1], [3, 0, 0, 0, 0, 0, 0, -3, 1, -1, -1, -1], [4, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -3, 0, 0, 0, -3, 1, -2, -1, 1, 0], [2, 0, 1, 0, 0, 0, 1, -2, 0, 0, -2, 0], [0, 0, 0, 1, 0, 0, 0, -7, 0, 0, 1, 0], [2, 1, -3, 0, -2, 1, 0, 0, 0, 0, 0, 0], [-2, -1, -1, -1, -2, -2, 0, 0, 0, 0, 0, 0], [0, -3, 0, -2, -3, -1, 1, 0, -1, -3, -2, 3], [4, -2, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, -1, 0, -3, -1, -1, -2, 1, -1, -1, -3, 1], [-1, 0, -1, 0, 0, 0, -1, 1, 0, 1, 3, 0], [0, 0, 0, -1, -1, 0, 5, -3, 0, 0, -1, 1], [0, 1, 0, 3, 0, 0, 0, 0, 0, 1, 3, 0], [-7, 0, -1, 0, 0, 0, -1, 7, -1, 2, 0, -1], [0, 0, 0, 0, -2, 2, -1, 2, 2, 0, 0, 2], [10, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 2, 2, 7, 2, 2, -1, 0], [6, 0, -3, 0, 0, 0, -3, -6, 0, -2, 2, 0], [-6, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 3, -3, 1, 2, 2, 0, 0, 0, 0, 0, 0], [-7, 0, 2, 0, 0, 0, 2, 7, -2, 0, 1, 2], [0, -3, 0, -1, 0, -2, -1, -5, -2, -3, -1, 0], [-2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -2, 1, -3, -3, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, -2, 0, -2, -3, -5, -2, -1, -2, 0], [-5, 0, 2, 0, 0, 0, 2, 5, -2, 0, -1, -2], [7, 0, -4, 0, 0, 0, -4, -7, -1, 0, 4, 0], [0, 2, 0, 2, 0, 4, 1, 14, 4, 2, 2, 0], [-7, 3, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 3, 1, -3, -3, 1, 0, 2, -3], [-5, 0, 4, 0, 0, 0, 4, 5, 4, 0, 0, -2], [12, -1, 3, -2, 0, 1, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, -2, 2, -2, 1, 2, -4, 0, 2], [7, 0, -1, 0, 0, 0, -1, -7, -1, -4, -4, 3], [-4, 3, -1, -3, -2, 2, 0, 0, 0, 0, 0, 0], [8, -2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0], [4, 0, 5, 0, 0, 0, 5, -4, 2, -2, -2, 2], [-8, 0, 2, 0, 0, 0, 2, 8, 1, -2, -3, 1], [0, -1, 0, -5, -1, -1, 0, -9, -1, -1, -5, 1], [0, 0, 0, 0, 2, -2, 3, 8, -2, 0, 0, -2], [6, 1, -3, 0, -2, -3, 0, 0, 0, 0, 0, 0], [0, 2, 0, 8, 3, 1, -3, 1, 1, 2, 8, -3], [-5, 0, 4, 0, 0, 0, 4, 5, -2, 0, -6, 2], [-6, 0, -6, 0, 0, 0, -6, 6, 0, -3, -3, -4], [-3, 0, 1, 0, 0, 0, 1, 3, -2, 3, 2, 0], [0, -4, 0, 2, 0, 0, -4, -5, 0, -4, 2, 0], [0, 0, 0, 0, 2, 3, 4, -5, 3, 0, 0, -2], [10, 0, -3, 0, 0, 0, -3, -10, 4, 2, 4, 0], [-7, 0, 1, 0, 0, 0, 1, 7, 0, -3, 3, 0], [0, 2, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 3, 0, 0, 0, 3, 11, 3, 0, -2, 1], [0, -1, 0, 1, 0, 0, 0, 16, 0, -1, 1, 0], [-2, 3, 6, 5, 1, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -7, -2, 0, 4, -5, 0, -2, -7, 2], [4, 0, 1, 0, 0, 0, 1, -4, 0, 2, 0, 0], [-6, -5, -3, -3, -4, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, 2, -3, 3, 2, -1, -1, 0], [0, 4, 0, 6, 2, 2, -4, -5, 2, 4, 6, -2], [0, 3, 0, -4, 0, 2, 7, -3, 2, 3, -4, 0], [6, 0, 4, 0, 0, 0, 4, -6, -2, 1, 1, 2], [0, 0, 4, -4, -2, -4, 0, 0, 0, 0, 0, 0], [7, -4, -4, -5, 1, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 9, 0, 0, 0, 9, 3, 1, 4, 0, 3], [-13, 0, 1, 0, 0, 0, 1, 13, 0, 2, 3, -1], [-3, 0, -2, 3, 1, -3, 0, 0, 0, 0, 0, 0], [-13, 0, 1, 0, 0, 0, 1, 13, 0, -1, 3, 0], [4, 0, -2, 0, 0, 0, -2, -4, 2, -1, 3, 2], [0, 0, 0, 2, -1, 3, 7, -1, 3, 0, 2, 1], [-14, -1, 1, -4, -2, -1, 0, 0, 0, 0, 0, 0], [-1, 0, -5, 0, 0, 0, -5, 1, 5, -2, -4, -3], [16, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 3, 1, -4, 3, -1, -4, 2, 3, -1], [0, 2, 0, 1, 1, 4, -1, -3, 4, 2, 1, -1], [-10, -6, 3, 1, -1, 5, 0, 0, 0, 0, 0, 0], [0, 1, 0, 3, 0, 4, 6, 2, 4, 1, 3, 0], [5, 1, -1, 4, 3, 1, 0, 0, 0, 0, 0, 0], [6, -2, 5, 1, 1, -1, 0, 0, 0, 0, 0, 0], [-6, -2, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 0, 2, 0, 0, 0, 2, 9, 5, 0, 0, -2], [0, -1, 5, -1, 2, 2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 2, -2, -1, 2, 17, -1, 2, 2, 2], [-7, 0, 6, 0, 0, 0, 6, 7, 0, 4, 0, 0], [0, 0, 0, 8, 4, 1, -2, -1, 1, 0, 8, -4], [7, 0, 4, 5, -1, 1, 0, 0, 0, 0, 0, 0], [-5, 0, -5, 0, 0, 0, -5, 5, 0, 2, 3, -5], [-2, -1, -1, 1, 2, 2, 0, 0, 0, 0, 0, 0], [0, -4, 0, -3, 4, -2, -4, 11, -2, -4, -3, -4], [-5, 0, -1, 0, 0, 0, -1, 5, 2, 3, 3, 0], [0, 4, 0, -1, -4, 2, 2, -23, 2, 4, -1, 4], [-6, -1, -7, 1, 0, 4, 0, 0, 0, 0, 0, 0], [0, -1, 0, 5, 0, -2, 3, 7, -2, -1, 5, 0], [-15, 0, 2, 0, 0, 0, 2, 15, 1, -2, -2, 2], [8, 0, 0, 6, 6, 2, 0, 0, 0, 0, 0, 0], [13, 0, 2, 0, 0, 0, 2, -13, 2, 4, -2, 0], [0, 0, 0, 0, 4, 1, -4, -7, 1, 0, 0, -4], [-8, 6, -10, -2, -2, 4, 0, 0, 0, 0, 0, 0], [0, 4, 0, 4, 4, 4, 1, -2, 4, 4, 4, -4], [3, 0, -3, 0, 0, 0, -3, -3, 0, 1, -4, -2], [0, 2, 0, 2, 2, -2, -3, -2, -2, 2, 2, -2], [2, 0, 0, 0, 0, 0, 0, -2, 3, 8, -1, -1], [14, 0, -1, 0, 0, 0, -1, -14, -2, 4, 8, -2], [14, 3, 7, 2, 6, 1, 0, 0, 0, 0, 0, 0], [-11, 0, -4, 0, 0, 0, -4, 11, -1, -1, -1, -3], [10, -3, 1, -3, 4, 4, 0, 0, 0, 0, 0, 0], [4, 0, 1, 0, 0, 0, 1, -4, 0, 6, 6, -4], [12, -8, 0, -4, -4, 0, 0, 0, 0, 0, 0, 0], [-19, 0, 1, 0, 0, 0, 1, 19, -3, 0, -4, 1], [0, 1, 0, 9, 2, 2, 4, 4, 2, 1, 9, -2], [-8, -1, 8, 1, 1, 4, 0, 0, 0, 0, 0, 0], [0, -6, 0, -3, -8, 0, -2, 11, 0, -6, -3, 8], [0, 2, 0, -5, 2, 0, 0, 5, 0, 2, -5, -2], [-2, -4, 0, -4, -4, 4, 0, 0, 0, 0, 0, 0], [0, 2, 0, 1, 3, 5, 4, 8, 5, 2, 1, -3], [0, 3, 0, 4, -1, 1, 11, -6, 1, 3, 4, 1], [-2, 1, -1, 3, 2, -2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 7, 1, -3, 6, -17, -3, 1, 7, -1], [0, -3, 0, 5, 2, -2, 0, 18, -2, -3, 5, -2], [-7, -2, 4, -5, 1, -5, 0, 0, 0, 0, 0, 0], [-16, 3, 1, -4, 0, -1, 0, 0, 0, 0, 0, 0], [-16, 0, 0, 0, 0, 0, 0, 16, 6, 1, 1, -2], [0, 4, 0, 2, 3, 3, 1, 15, 3, 4, 2, -3], [-2, 0, -8, 0, -4, 4, 0, 0, 0, 0, 0, 0], [-4, 1, -3, 3, 4, 0, 0, 0, 0, 0, 0, 0], [9, 0, -6, 0, 0, 0, -6, -9, 4, -2, 5, 2], [4, 0, 4, 0, 0, 0, 4, -4, 2, 3, 3, 2], [22, 0, -4, 0, 0, 0, -4, -22, -5, -2, -3, -1], [0, 4, 0, 4, 0, 1, -4, -5, 1, 4, 4, 0], [4, 4, 0, -10, -2, 2, 0, 0, 0, 0, 0, 0], [0, -1, 0, -5, -2, 2, -4, -2, 2, -1, -5, 2], [-8, 3, 3, 3, 2, 6, 0, 0, 0, 0, 0, 0], [-2, 3, -3, 2, 2, -3, 0, 0, 0, 0, 0, 0], [0, 2, 0, 6, 3, 7, 5, 17, 7, 2, 6, -3], [7, 0, -3, 0, 0, 0, -3, -7, 0, 2, 1, -9], [7, 0, 5, 0, 0, 0, 5, -7, -7, 0, 2, -1], [-4, 8, 4, 8, 2, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 8, 0, 0, 0, 8, 14, 1, 4, -3, 5], [6, -3, 7, -3, -4, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 2, -6, -5, 0, -6, 2, 0, -2], [13, 0, 6, 0, 0, 0, 6, -13, -1, 2, -6, 2], [17, 6, 6, 1, 3, 5, 0, 0, 0, 0, 0, 0], [0, -4, 0, -2, -3, 3, -3, -5, 3, -4, -2, 3], [6, -2, 1, 1, -3, 3, 0, 0, 0, 0, 0, 0], [0, 4, 0, 4, 8, 2, -6, 3, 2, 4, 4, -8], [-9, 0, 7, 0, 0, 0, 7, 9, 3, 0, 8, -7], [0, 0, -4, -2, -2, -6, 0, 0, 0, 0, 0, 0], [-2, 0, -1, 0, 0, 0, -1, 2, 8, -4, 2, -4], [14, 1, -3, 7, 10, 2, 0, 0, 0, 0, 0, 0], [0, -3, 0, -3, 0, 0, 2, 4, 0, -3, -3, 0], [17, 0, 1, 0, 0, 0, 1, -17, 0, 8, 5, 1], [0, 3, 0, -1, 2, 6, -4, 2, 6, 3, -1, -2], [0, -2, 0, -10, -5, 3, 1, 21, 3, -2, -10, 5], [1, 0, -9, 0, 0, 0, -9, -1, 2, -1, -3, -8], [0, -4, 0, 1, 0, 0, 2, -3, 0, -4, 1, 0], [0, -5, 0, -6, -1, -3, -3, 12, -3, -5, -6, 1], [17, 0, -5, 0, 0, 0, -5, -17, 0, 4, -5, 1], [-28, -1, -3, -1, -2, 2, 0, 0, 0, 0, 0, 0], [-8, -2, -6, -10, -8, -4, 0, 0, 0, 0, 0, 0], [10, 0, 7, 0, 0, 0, 7, -10, 2, 2, -6, 6], [-13, 0, 8, 0, 0, 0, 8, 13, 10, 2, 1, -2], [-3, 0, -3, 0, 0, 0, -3, 3, 8, 2, 3, 3], [0, -6, 0, 6, -4, 0, 1, 10, 0, -6, 6, 4], [-7, -6, 6, -9, -3, -3, 0, 0, 0, 0, 0, 0], [0, 2, 0, 5, 0, 8, -8, -5, 8, 2, 5, 0], [-8, 0, 2, 0, 0, 0, 2, 8, -6, -7, -1, 2], [-4, 2, 2, -8, -2, -6, 0, 0, 0, 0, 0, 0], [-15, 0, -2, 0, 0, 0, -2, 15, 3, 1, 1, 1], [0, 4, 0, 8, 6, 2, 0, -7, 2, 4, 8, -6], [-4, 6, 2, 10, 0, 4, 0, 0, 0, 0, 0, 0], [5, -4, 8, -5, 3, -3, 0, 0, 0, 0, 0, 0], [-19, 0, -2, 0, 0, 0, -2, 19, -1, -6, 2, 0], [-20, -1, 3, 6, 0, 5, 0, 0, 0, 0, 0, 0], [-26, 0, 3, 0, 0, 0, 3, 26, 4, 6, 8, 0], [-6, -9, 5, -4, -2, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, -8, -8, 2, 0, 5, 2, 0, -8, 8], [0, 1, 0, 3, 2, 2, -1, 21, 2, 1, 3, -2], [2, 0, -1, 0, 0, 0, -1, -2, -6, 2, -2, 2], [1, 0, 4, 0, 0, 0, 4, -1, 4, 4, 6, 0], [28, 2, 2, 4, 0, 6, 0, 0, 0, 0, 0, 0], [0, 2, 0, -2, 2, -2, 1, 32, -2, 2, -2, -2], [27, 0, 2, 0, 0, 0, 2, -27, 0, 8, 5, -4], [0, -6, 0, 4, -4, 4, -1, 6, 4, -6, 4, 4], [-13, 0, 0, 0, 0, 0, 0, 13, 0, -8, -10, 8], [-8, 9, -1, 4, 8, -3, 0, 0, 0, 0, 0, 0], [-18, 1, -3, -3, -4, -4, 0, 0, 0, 0, 0, 0], [16, 0, -8, -2, -2, 10, 0, 0, 0, 0, 0, 0], [0, -8, 0, -7, -3, 0, 9, 1, 0, -8, -7, 3], [19, 0, -8, 0, 0, 0, -8, -19, 2, -8, -2, 0], [0, 2, 0, 1, 7, 0, -9, 17, 0, 2, 1, -7], [1, 0, -8, -11, -3, -3, 0, 0, 0, 0, 0, 0], [0, -5, 0, 3, -4, 0, 6, -22, 0, -5, 3, 4], [23, 0, 2, 0, 0, 0, 2, -23, 2, 10, -3, -2], [-17, 0, -3, 0, 0, 0, -3, 17, -10, -5, -4, -4], [0, 0, 0, -3, 0, -10, 4, 7, -10, 0, -3, 0], [21, 0, -4, 0, 0, 0, -4, -21, -2, 4, 4, 2], [0, 12, 0, 4, 5, -3, 3, 3, -3, 12, 4, -5], [11, 0, 5, 0, 0, 0, 5, -11, 4, 5, -1, 2], [-24, -2, 6, 6, 6, 4, 0, 0, 0, 0, 0, 0], [0, 3, 0, 1, 6, 6, -6, -12, 6, 3, 1, -6], [0, -7, 0, -9, -2, 2, -4, -14, 2, -7, -9, 2], [0, -2, 0, 11, 2, 0, 2, 5, 0, -2, 11, -2], [0, 3, 0, -1, 0, 2, 9, 39, 2, 3, -1, 0], [-16, 0, -8, -4, 2, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, -7, -1, 0, -13, 3, 0, -8, -7, 1], [12, 0, -1, 0, 0, 0, -1, -12, -4, 6, -4, -4], [2, 0, 9, 0, 0, 0, 9, -2, -7, -3, 2, 1], [12, 2, -10, -8, -2, -2, 0, 0, 0, 0, 0, 0], [-7, 0, -7, 0, 0, 0, -7, 7, -9, 0, -2, -1], [-1, 0, -1, 0, 0, 0, -1, 1, -6, -9, -4, 4], [0, 6, 0, 6, -2, -2, -9, -16, -2, 6, 6, 2], [14, 8, -6, 4, 0, 8, 0, 0, 0, 0, 0, 0], [0, 2, 0, 4, 5, 11, -3, -1, 11, 2, 4, -5], [0, 0, 0, -6, 0, -2, -4, -9, -2, 0, -6, 0], [-7, 0, -1, 0, 0, 0, -1, 7, -6, -1, 6, 0], [-19, 0, 12, 0, 0, 0, 12, 19, 2, 10, -1, 2], [0, 4, 0, -1, -9, 1, 8, 0, 1, 4, -1, 9], [14, 1, -3, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -5, -1, -8, -5, -11, -8, 2, -5, 1], [8, -2, -2, -8, 4, 2, 0, 0, 0, 0, 0, 0], [-11, 0, -7, 0, 0, 0, -7, 11, -4, -2, -5, 3], [-14, 0, 4, -8, 4, -4, 0, 0, 0, 0, 0, 0], [0, -2, 0, 3, -2, 0, -2, -7, 0, -2, 3, 2], [0, 9, -1, -3, 2, 6, 0, 0, 0, 0, 0, 0], [0, 7, 0, 3, -2, 6, 1, -19, 6, 7, 3, 2], [17, 0, -6, 0, 0, 0, -6, -17, -2, 10, 15, -8], [-20, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, -4, 0, -4, -2, 3, 2, 29, 3, -4, -4, 2], [0, -3, 0, 2, -6, -4, 3, -1, -4, -3, 2, 6], [-28, 2, 10, 6, 2, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, -4, -8, 8, 2, -8, -1, -1, 4], [11, -4, -6, -9, -1, 1, 0, 0, 0, 0, 0, 0], [0, 4, 0, -10, -3, 1, 7, -3, 1, 4, -10, 3], [6, 0, 5, 0, 0, 0, 5, -6, -7, -1, -4, 1], [2, 0, -9, 0, 0, 0, -9, -2, -6, 2, 0, -2], [2, -7, 2, -7, -9, -6, 0, 0, 0, 0, 0, 0], [30, -5, -5, -9, -6, 2, 0, 0, 0, 0, 0, 0], [-7, 0, -5, 0, 0, 0, -5, 7, -2, -11, -8, 8], [-10, 2, -2, -2, -10, 6, 0, 0, 0, 0, 0, 0], [1, 2, -10, -3, -7, 7, 0, 0, 0, 0, 0, 0], [29, 0, 3, 0, 0, 0, 3, -29, -3, -4, -8, -1], [47, 0, -3, 0, 0, 0, -3, -47, 0, 0, 1, -3], [0, 2, 0, 0, 4, 0, 7, -6, 0, 2, 0, -4], [6, 3, 1, 1, 2, -6, 0, 0, 0, 0, 0, 0], [0, -8, 0, -5, 1, 3, -8, -2, 3, -8, -5, -1], [-11, 0, 2, 0, 0, 0, 2, 11, 1, -4, 0, 0], [-28, 3, 5, 7, 8, 0, 0, 0, 0, 0, 0, 0], [5, 0, 4, 0, 0, 0, 4, -5, 10, -4, 8, 4], [0, 2, 0, 2, 2, 1, 4, 9, 1, 2, 2, -2], [0, 5, 0, 3, -4, -4, 0, -8, -4, 5, 3, 4], [0, 3, 0, -3, -4, 0, 10, -30, 0, 3, -3, 4], [-6, -10, -10, -10, -8, 0, 0, 0, 0, 0, 0, 0], [17, 0, 0, 0, 0, 0, 0, -17, 2, 0, 2, 4], [0, -4, 0, -1, -2, -2, -10, 13, -2, -4, -1, 2], [0, -3, 0, -11, -8, 4, 9, 9, 4, -3, -11, 8], [20, -8, 8, -4, 8, -4, 0, 0, 0, 0, 0, 0], [0, -7, 0, 1, 1, 1, 4, -3, 1, -7, 1, -1], [-24, 0, 6, 0, 0, 0, 6, 24, 6, -3, -7, -2], [-11, -2, 2, -1, 3, -5, 0, 0, 0, 0, 0, 0], [-5, 0, -10, 0, 0, 0, -10, 5, -1, 2, 14, -4], [0, 0, 0, 1, 6, 2, 6, -17, 2, 0, 1, -6], [0, -2, 0, -7, -6, -2, 6, 5, -2, -2, -7, 6], [-9, 0, 9, 0, 0, 0, 9, 9, 5, 6, -8, 1], [-6, 0, -5, 0, 0, 0, -5, 6, -2, 0, -6, 6], [0, 2, 0, 1, 1, 12, -5, 3, 12, 2, 1, -1], [22, -2, -3, 1, -3, -3, 0, 0, 0, 0, 0, 0], [-12, -3, 1, 5, -6, 6, 0, 0, 0, 0, 0, 0], [0, -6, 0, -12, -4, -4, -3, -18, -4, -6, -12, 4], [-21, 0, -2, 0, 0, 0, -2, 21, 2, -2, 5, -4], [9, 13, -13, 8, 5, -1, 0, 0, 0, 0, 0, 0], [6, 0, -4, 0, 0, 0, -4, -6, 8, 1, 7, -12], [-20, 3, 8, 11, 1, -2, 0, 0, 0, 0, 0, 0], [0, -4, 0, -2, 0, -2, -12, 3, -2, -4, -2, 0], [0, -6, 0, -6, -4, -1, 2, -31, -1, -6, -6, 4], [-20, 0, 3, 0, 0, 0, 3, 20, 8, 0, -4, 4], [0, 6, 0, 4, -8, 4, 7, -20, 4, 6, 4, 8], [-11, 0, 9, 0, 0, 0, 9, 11, 11, 2, 4, 7], [-10, 0, -6, 0, 0, 0, -6, 10, 2, -1, -11, 2], [10, -2, 8, -2, -8, -8, 0, 0, 0, 0, 0, 0], [-7, 0, 6, 0, 0, 0, 6, 7, -5, -7, -15, 5], [0, 0, 0, 0, -2, 6, 4, 17, 6, 0, 0, 2], [-14, -7, 5, -11, -8, -8, 0, 0, 0, 0, 0, 0], [-32, 2, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3456_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3456_2_i_k();".
// 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_3456_2_i_k(:prec:=12)
    chi := MakeCharacter_3456_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_3456_2_i_k();".
// 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_3456_2_i_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3456_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2304, 4224, 9424, 1720, 4665, -1674, 2928, -948, 465, -60, 24, -2, 1]>,<7,R![394384, 324048, 294516, 117452, 67353, 21192, 10164, 2400, 861, 152, 48, 6, 1]>],Snew);
    return Vf;
end function;