
// Make newform 1456.2.r.p in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1456_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1456_r_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_1456_2_r_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_1456_2_r_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 := [4, 16, 64, 28, 58, 18, 41, 7, 8, -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], [22604, 98928, 392048, 39747, 287624, 36640, 87336, -9160, 7328, 983], [-348592, -132894, -526654, 25194, -386377, -49220, -109053, 12305, -9844, -1159], [-180842, -41094, -162854, -129531, -119477, -15220, -3978, 3805, -3044, 916], [-95852, -42714, -169274, 64764, -124187, -15820, -40293, 3955, -3164, -629], [-33244, -136908, -643378, -196692, -117514, -166865, -70821, -61840, 13967, -8713], [-2692, -957744, -258154, -871206, -272902, -637445, -107253, -121720, 14531, -15259], [362216, 1445112, 580892, 1240338, 331046, 750010, 104769, 137960, -15688, 16307], [136, -40590, -11648, -53832, -2993, -39580, -3726, -9041, 1552, -1058], [115064, 461448, 272618, 440202, 153734, 346240, 67401, 71240, -9652, 9203]];
    Rf_basisdens := [1, 118350, 118350, 59175, 39450, 236700, 236700, 118350, 4734, 39450];
    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_1456_r();"
function MakeCharacter_1456_r()
    N := 1456;
    order := 3;
    char_gens := [911, 1093, 1249, 561];
    v := [3, 3, 2, 3];
    // 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_1456_r_Hecke();"
function MakeCharacter_1456_r_Hecke(Kf)
    N := 1456;
    order := 3;
    char_gens := [911, 1093, 1249, 561];
    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], [1, 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, -1], [0, 0, 0, -1, 0, 0, 0, -1, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 1], [2, 0, 0, 0, 0, 1, -2, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, -1, -1, 1, 0], [0, 1, 0, -1, 0, 1, 2, -1, 1, 0], [0, 0, 1, 0, 0, 0, 2, 0, 0, 1], [-1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 2, -2, 0, -1], [0, 2, -2, 0, 0, 2, 0, 0, -1, -2], [4, 2, 2, -1, 1, 0, 0, 0, 0, 0], [0, 2, 2, -1, -1, 0, 0, 0, 0, 0], [0, -1, -4, 0, 0, -1, 0, 0, 0, -4], [-3, 0, 0, 0, 2, 2, 3, 0, 2, 0], [2, 0, 0, 0, 0, 1, -2, 0, 0, 0], [0, 4, 2, 0, 0, 4, 3, 0, 0, 2], [4, 0, 0, 0, -1, -3, -4, 3, -1, 0], [-2, 1, -1, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1, 0, 0], [0, 2, 3, 1, 0, 2, 0, 1, 1, 3], [-2, 0, 0, 3, 1, 0, 0, 0, 0, 0], [0, -2, -4, 2, 0, -2, 0, 2, -1, -4], [-2, 2, 2, -1, -1, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, -2, -3, 0, 2], [0, 2, -1, 2, 0, 2, 0, 2, -4, -1], [0, 0, -1, -1, 0, 0, 12, -1, 1, -1], [0, 0, 0, 0, -3, 2, 0, 0, -3, -2], [1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -2, 2, 0, 0, 0, 0, 0, 0, 0], [0, -2, -3, -3, 0, -2, -4, -3, -1, -3], [-6, 0, 0, 0, -2, 2, 6, -1, -2, 4], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 2, -1, 0, -2, 4, -1, 4, 2], [-10, 0, 0, 0, -1, 3, 10, -1, -1, 4], [7, 0, 0, 0, -1, 2, -7, 1, -1, 0], [0, -3, 0, -3, 0, -3, 4, -3, 1, 0], [-6, 3, 5, -2, -4, 0, 0, 0, 0, 0], [0, -4, -2, 2, 0, -4, 1, 2, 0, -2], [6, 0, 0, 0, 0, 0, -6, -2, 0, 3], [3, 2, -2, 0, 3, 0, 0, 0, 0, 0], [0, -2, -1, 1, 0, -2, 6, 1, -1, -1], [-4, 0, 0, 0, -1, -2, 4, -4, -1, 0], [0, 0, 4, -1, 1, 0, 0, 0, 0, 0], [12, 0, 0, 0, 2, 2, -12, 0, 2, 7], [0, -6, 2, 1, 1, 0, 0, 0, 0, 0], [-2, 1, 3, -1, 5, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 2, -4, 2, -2, 2, 1], [0, 0, 8, -2, 0, 0, -8, -2, 1, 8], [0, -2, 2, 2, 0, -2, 1, 2, -2, 2], [-8, -1, -3, -5, -1, 0, 0, 0, 0, 0], [2, 0, 0, 0, -3, 4, -2, 0, -3, 2], [14, 4, 0, -1, 1, 0, 0, 0, 0, 0], [0, -4, -4, 1, 0, -4, -6, 1, 0, -4], [8, 0, 0, 0, 0, 0, -8, 2, 0, 5], [7, 0, 0, 0, -3, 2, -7, 3, -3, 0], [0, -1, 0, 1, 0, -1, -4, 1, -1, 0], [5, 0, 0, 0, 4, -4, -5, 2, 4, 2], [2, -4, -8, -3, 1, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 3, 0, 10, 3, 3, 3], [-8, -4, -8, 9, 3, 0, 0, 0, 0, 0], [0, -5, -7, 3, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, -4, 0, -4, 2, -3], [0, 2, -2, -3, 0, 2, 0, -3, 2, -2], [0, 6, -2, -5, 0, 6, 8, -5, 0, -2], [0, 4, 9, -4, 0, 4, -6, -4, 0, 9], [-7, 2, -2, 6, -1, 0, 0, 0, 0, 0], [16, 0, 0, 0, -2, 0, -16, -2, -2, 3], [-2, -2, -2, -5, -1, 0, 0, 0, 0, 0], [-8, 0, 0, 0, -1, -4, 8, -4, -1, -2], [0, -8, -5, -4, 0, -8, -2, -4, 2, -5], [14, 0, 0, 0, -1, 4, -14, -3, -1, -7], [0, 6, 8, -5, 0, 6, 7, -5, 1, 8], [0, -2, -6, -1, -5, 0, 0, 0, 0, 0], [0, -2, 5, -2, 0, -2, 0, -2, 2, 5], [-5, 0, 0, 0, -3, 4, 5, -1, -3, -10], [0, -6, -2, 5, 0, -6, -4, 5, -2, -2], [0, 0, -2, 4, 0, 0, -4, 4, -3, -2], [10, 0, 0, 0, 2, 6, -10, -1, 2, 4], [6, -4, -4, 2, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 0, 0, 2, 12, 0, 0, 5], [-7, -4, 0, 1, -2, 0, 0, 0, 0, 0], [0, -2, -1, 3, 0, -2, -20, 3, 3, -1], [0, 2, -3, -1, 0, 2, 22, -1, -5, -3], [-8, -4, -4, -4, -2, 0, 0, 0, 0, 0], [0, 0, -10, -2, 0, 0, 0, -2, -1, -10], [-8, -2, -10, -3, -1, 0, 0, 0, 0, 0], [-2, -4, 4, -3, -1, 0, 0, 0, 0, 0], [0, 2, -3, -1, 0, 2, 8, -1, -3, -3], [10, 0, 0, 0, -4, 7, -10, 6, -4, 0], [-12, 0, 0, 0, -6, -5, 12, 6, -6, -4], [-10, -8, -8, 4, 2, 0, 0, 0, 0, 0], [0, 0, 1, -4, 0, 0, -6, -4, 4, 1], [-2, 2, -6, 7, 5, 0, 0, 0, 0, 0], [0, -2, 2, -4, 0, -2, -16, -4, 3, 2], [8, 0, 0, 0, -2, -10, -8, 3, -2, -6], [0, 4, -1, -6, 0, 4, 14, -6, -4, -1], [0, 0, 0, -3, 0, 0, 12, -3, -2, 0], [6, 4, 0, 5, 7, 0, 0, 0, 0, 0], [2, 0, 0, 0, -4, -4, -2, 7, -4, -4], [-8, 0, 0, 0, -5, -4, 8, -3, -5, -13], [0, -6, 6, -4, 0, -6, 17, -4, 4, 6], [2, 0, 0, 0, 2, 6, -2, -4, 2, 1], [-2, 0, 0, 0, 1, -6, 2, 4, 1, -16], [10, -5, 5, -6, -4, 0, 0, 0, 0, 0], [0, -12, -2, 1, 0, -12, -12, 1, -4, -2], [6, 0, 0, 0, -3, -2, -6, 11, -3, -5], [13, -8, 8, 0, -3, 0, 0, 0, 0, 0], [0, 8, -3, 7, 0, 8, -14, 7, -3, -3], [8, 0, 0, 0, -2, 4, -8, 7, -2, 4], [4, 4, -8, 7, 1, 0, 0, 0, 0, 0], [16, 0, 0, 0, -6, 6, -16, 4, -6, -1], [6, -4, -12, 1, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, -7, 0, 18], [10, 7, -7, -6, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -10, 0, 1, 1, 7], [0, 6, -12, -1, 0, 6, -4, -1, -2, -12], [4, 4, 4, -8, -8, 0, 0, 0, 0, 0], [12, 0, 0, 0, -1, 2, -12, 2, -1, -10], [-12, -10, -6, -2, -4, 0, 0, 0, 0, 0], [0, 8, 10, -4, 0, 8, -1, -4, -4, 10], [28, 0, 0, 0, 6, 2, -28, -4, 6, 3], [0, 5, 8, -4, 0, 5, -6, -4, 8, 8], [2, 4, 12, 2, -2, 0, 0, 0, 0, 0], [0, -2, -2, 7, 0, -2, 8, 7, -4, -2], [0, 0, 5, -4, 0, 0, -18, -4, 4, 5], [-6, 2, 2, -1, -7, 0, 0, 0, 0, 0], [0, 2, 6, 4, 0, 2, -14, 4, 5, 6], [10, 0, 0, 0, 2, -8, -10, -2, 2, 1], [-6, -11, -9, 11, 5, 0, 0, 0, 0, 0], [0, -10, 7, 0, 0, -10, 6, 0, 0, 7], [11, 2, 2, -7, 0, 0, 0, 0, 0, 0], [0, 4, 10, -3, 0, 4, -4, -3, 4, 10], [-2, -4, 4, -6, -2, 0, 0, 0, 0, 0], [4, 0, 0, 0, -3, -4, -4, 0, -3, 0], [26, 0, 0, 0, 5, 1, -26, 3, 5, 8], [8, 10, -2, -2, -4, 0, 0, 0, 0, 0], [3, 0, 0, 0, 1, -4, -3, 5, 1, 4], [26, 0, 8, -5, -3, 0, 0, 0, 0, 0], [0, -2, -6, 7, 0, -2, 2, 7, 2, -6], [-12, 0, 0, 0, -4, 4, 12, -10, -4, -3], [-28, -1, -7, 2, 2, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 6, -2, 1, -6, 6, -4], [-20, -7, 3, 7, -1, 0, 0, 0, 0, 0], [0, -2, -6, -4, -6, 0, 0, 0, 0, 0], [3, 0, 0, 0, 4, 12, -3, -6, 4, -4], [0, 4, 11, -3, 0, 4, -20, -3, 3, 11], [0, 16, 11, -2, 0, 16, 18, -2, -4, 11], [0, -2, 10, -2, 0, -2, -16, -2, -7, 10], [0, -14, -2, 4, 6, 0, 0, 0, 0, 0], [-16, 4, -8, 1, 1, 0, 0, 0, 0, 0], [0, -16, -11, 4, 0, -16, -14, 4, 0, -11], [2, 0, 0, 0, -1, 0, -2, -13, -1, -5], [16, 2, 2, 5, 1, 0, 0, 0, 0, 0], [0, 8, 5, -4, 0, 8, 10, -4, -8, 5], [0, 2, -8, -4, 0, 2, -22, -4, -1, -8], [-9, 12, 8, 1, 0, 0, 0, 0, 0, 0], [12, 0, 0, 0, -5, -6, -12, 6, -5, -10], [0, -7, -8, -2, 0, -7, 16, -2, 0, -8], [-17, -10, -10, -1, -2, 0, 0, 0, 0, 0], [-8, 11, 5, 8, 0, 0, 0, 0, 0, 0], [0, -2, 1, 5, 0, -2, -6, 5, 9, 1], [6, 0, 0, 0, 4, 0, -6, -3, 4, 14], [0, 0, 0, 0, 3, 1, 0, -7, 3, -2], [-8, 0, 0, 0, -2, 4, 8, 0, -2, 5], [-3, 0, 0, 0, -4, -14, 3, -2, -4, -8], [12, 4, -4, 6, -4, 0, 0, 0, 0, 0], [0, 6, 0, -10, 0, 6, -13, -10, 10, 0], [10, 0, 0, 0, -1, 11, -10, 1, -1, -2], [18, 2, -10, -10, 10, 0, 0, 0, 0, 0], [0, 0, -7, 14, 0, 0, 6, 14, -2, -7], [-2, 0, 0, 0, -7, 0, 2, 8, -7, -2], [12, 0, 0, 0, 4, 0, -12, -6, 4, -3], [-29, -8, -12, 5, 0, 0, 0, 0, 0, 0], [8, 7, 13, -5, -5, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -1, -4, 2, 8, -1, 4], [6, -2, -22, 6, 0, 0, 0, 0, 0, 0], [0, 0, 10, 11, 0, 0, 5, 11, -5, 10], [0, 5, -2, 6, 0, 5, 10, 6, -4, -2], [10, 8, -4, -13, 1, 0, 0, 0, 0, 0], [-9, -2, -2, 1, 4, 0, 0, 0, 0, 0], [0, 4, -2, 3, 0, 4, -40, 3, -2, -2], [20, 0, 0, 0, -3, -3, -20, -1, -3, 14], [-6, 0, 0, 0, -2, -8, 6, 7, -2, -6], [11, 0, 0, 0, -5, -12, -11, -3, -5, -6], [4, 0, 0, 0, 0, -6, -4, 10, 0, -13], [0, -2, -4, 6, 0, -2, -16, 6, -5, -4], [-16, 0, 0, 0, 3, -3, 16, 5, 3, -10], [0, 0, 4, 11, 0, 0, 35, 11, -5, 4], [-10, -13, -7, -1, -7, 0, 0, 0, 0, 0], [0, 4, 19, -2, 0, 4, 2, -2, 0, 19], [0, 0, -8, 7, 0, 0, 18, 7, 2, -8], [12, 0, 0, 0, -5, -14, -12, 9, -5, -3], [-19, 0, 0, 0, -4, 6, 19, 8, -4, 18], [-12, 0, 0, 0, 2, 8, 12, -7, 2, 14], [0, -10, -8, 3, 0, -10, -20, 3, -2, -8], [-24, -6, 6, 0, -6, 0, 0, 0, 0, 0], [0, -6, -9, -5, 0, -6, -20, -5, -1, -9], [-6, 0, 0, 0, 11, -4, 6, -12, 11, 0], [-14, 2, 2, -7, 3, 0, 0, 0, 0, 0], [0, -4, 0, -2, 0, -4, 14, -2, -5, 0], [14, 0, 0, 0, 12, 8, -14, -5, 12, 12], [-4, -10, 10, 9, 1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -2, -8, 2, 3, -2, -14], [0, -1, -4, 5, 0, -1, -18, 5, -7, -4], [0, -2, 5, 7, 0, -2, -12, 7, 5, 5], [18, -12, -12, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 0, -1, 12, 14, 3, -1, 15], [0, 2, 18, -2, 0, 2, -15, -2, 6, 18], [21, -12, 0, 3, 0, 0, 0, 0, 0, 0], [2, -2, -6, -4, -6, 0, 0, 0, 0, 0], [0, 5, -4, 3, 0, 5, 16, 3, 11, -4], [6, 0, 0, 0, -8, 0, -6, 2, -8, 11], [0, -8, -6, -4, 0, -8, -4, -4, 9, -6], [-12, 0, 0, 0, 7, 4, 12, 7, 7, 1], [-1, 0, 0, 0, 4, 10, 1, -10, 4, 14], [0, -15, -18, 1, 0, -15, 8, 1, -5, -18], [10, -6, 18, 9, 1, 0, 0, 0, 0, 0], [0, -2, -4, 6, 0, -2, -7, 6, 4, -4], [-14, -16, 0, -5, 1, 0, 0, 0, 0, 0], [0, -4, -2, -4, 0, -4, 40, -4, -3, -2], [0, -8, 3, 14, 0, -8, -22, 14, 4, 3], [12, 6, -10, 1, 5, 0, 0, 0, 0, 0], [15, 4, 8, -15, -4, 0, 0, 0, 0, 0], [0, -10, -6, -9, 0, -10, 8, -9, 0, -6], [-8, 0, 0, 0, -2, 0, 8, -8, -2, -15], [0, 2, -5, -9, 0, 2, -20, -9, -1, -5], [0, 11, -2, 1, 0, 11, -22, 1, -1, -2], [-27, 0, 0, 0, -5, 2, 27, -5, -5, -12], [-12, 0, 0, 0, 0, -10, 12, 4, 0, -15], [22, 9, 15, 12, 6, 0, 0, 0, 0, 0], [27, 0, 0, 0, -1, 14, -27, 3, -1, 6], [12, -10, -6, -8, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 2, 8, -2, 0, 2, -1], [0, -6, 4, -4, 0, -6, 14, -4, 7, 4], [0, 2, 10, -1, 0, 2, 4, -1, 0, 10], [0, -2, -2, 1, 9, 0, 0, 0, 0, 0], [4, -10, 6, 1, -7, 0, 0, 0, 0, 0], [24, 0, 0, 0, 8, -3, -24, -6, 8, 4], [0, -16, -17, 0, 0, -16, 12, 0, 0, -17], [-22, 0, 0, 0, -6, 6, 22, 6, -6, 15], [0, -6, 2, -11, 0, -6, 12, -11, 2, 2], [-26, -10, -22, 2, -2, 0, 0, 0, 0, 0], [0, 8, -3, -7, 0, 8, -2, -7, 7, -3], [10, 7, -7, -6, -8, 0, 0, 0, 0, 0], [-20, 0, 0, 0, 2, -5, 20, 0, 2, -6], [4, 0, 0, 0, -3, 5, -4, 9, -3, 4], [16, -4, 8, -7, -11, 0, 0, 0, 0, 0], [10, -2, -14, -2, 2, 0, 0, 0, 0, 0], [0, -10, 0, 3, 0, -10, 10, 3, 4, 0], [14, 0, 0, 0, -4, -9, -14, -2, -4, -12], [19, -12, -12, 12, 11, 0, 0, 0, 0, 0], [3, 0, 0, 0, -11, 14, -3, -1, -11, 4], [0, 4, 9, 0, 0, 4, 18, 0, 4, 9], [25, 0, 0, 0, 9, -4, -25, -7, 9, 2], [-26, 0, 0, 0, 0, -13, 26, 0, 0, -14], [-9, 6, -6, 0, 15, 0, 0, 0, 0, 0], [0, 10, -8, 1, 0, 10, 0, 1, -8, -8], [4, 0, 0, 0, 0, -2, -4, 14, 0, -9], [-36, 10, 6, 2, 2, 0, 0, 0, 0, 0], [-28, 0, 0, 0, 3, 4, 28, -6, 3, 8], [10, 16, 16, 4, 8, 0, 0, 0, 0, 0], [-24, 0, 0, 0, 3, 4, 24, 12, 3, 10], [0, 4, 7, 1, 0, 4, 22, 1, -11, 7], [16, 0, 8, 10, 0, 0, 0, 0, 0, 0], [22, -4, 4, 0, -12, 0, 0, 0, 0, 0], [14, -8, -8, -5, 1, 0, 0, 0, 0, 0], [5, 0, 0, 0, -8, -4, -5, 6, -8, -6], [0, 4, -22, -9, 0, 4, -2, -9, -4, -22], [-6, 0, 0, 0, 10, -5, 6, -2, 10, -2], [22, 0, 0, 0, 0, -18, -22, 9, 0, 0], [0, 0, 5, 16, 0, 0, -18, 16, 0, 5], [10, -4, -12, -8, 0, 0, 0, 0, 0, 0], [0, -13, -4, -1, 0, -13, 14, -1, 3, -4], [0, -5, 2, -11, 0, -5, 28, -11, 1, 2], [-20, 0, 0, 0, 2, 4, 20, -9, 2, 20], [0, 2, 8, 2, 0, 2, -10, 2, 5, 8], [0, -6, -23, 18, 0, -6, -14, 18, -8, -23], [-34, 0, 0, 0, 1, 0, 34, 3, 1, -11], [14, 0, 0, 0, -6, 16, -14, 4, -6, 7], [2, 20, 4, 6, 4, 0, 0, 0, 0, 0], [-18, -2, 6, -7, -3, 0, 0, 0, 0, 0], [0, 17, 8, -7, 0, 17, 32, -7, -3, 8], [0, 0, 17, -1, 0, 0, 4, -1, 3, 17], [13, 0, 0, 0, 4, -16, -13, 0, 4, -32], [-6, 10, 18, -7, -7, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 3, 9, 4, -3, 3, 6], [-21, -2, 6, 5, 8, 0, 0, 0, 0, 0], [5, 0, 0, 0, -2, 2, -5, 0, -2, 12], [-8, 0, 0, 0, 7, -4, 8, -3, 7, -1], [0, -4, -10, 7, 0, -4, -18, 7, 0, -10], [-26, -7, 11, -1, 1, 0, 0, 0, 0, 0], [2, 12, 4, 2, -2, 0, 0, 0, 0, 0], [-9, 0, 0, 0, -3, 4, 9, 15, -3, 10], [0, 2, -5, -15, 0, 2, -14, -15, 11, -5], [5, 8, 0, -2, -9, 0, 0, 0, 0, 0], [0, 14, 21, -10, 0, 14, 28, -10, -2, 21], [2, 8, 0, 19, 5, 0, 0, 0, 0, 0], [0, -6, 22, 8, 0, -6, -27, 8, 12, 22], [0, 8, 0, 3, 0, 8, 24, 3, -4, 0], [-34, 0, 0, 0, 4, -10, 34, 0, 4, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1456_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1456_2_r_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_1456_2_r_p(:prec:=10)
    chi := MakeCharacter_1456_r();
    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_1456_2_r_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_1456_2_r_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1456_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 64, 256, 72, 144, 4, 65, 0, 9, 0, 1]>,<5,R![2304, 2304, 3264, 480, 1024, 156, 217, 10, 19, 2, 1]>,<11,R![1089, -1485, 2751, -1386, 2467, -1749, 1099, -352, 85, -11, 1]>],Snew);
    return Vf;
end function;