
// Make newform 1452.2.b.e in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1452_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1452_b_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_1452_2_b_e();".
// 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_1452_2_b_e();".
// 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, -8748, 5832, -2916, 1863, -1944, 1314, -528, 223, -176, 146, -72, 23, -12, 8, -4, 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], [-41380227, 24436080, -4273398, 3477168, -5389848, 5169285, -1436082, -76448, -551984, 457526, -204435, 44762, -8526, 29672, -8014, 937], [-5272128, 7884864, -2370303, 1016496, -1441737, 1315818, -644111, 46471, -112495, 158097, -66493, 18621, -7402, 8285, -5006, 153], [9073134, -578583, -2405376, -549477, 1088856, -882699, -332195, 208905, 185803, -72605, -11623, 16426, 1811, -9030, -643, 692], [-14465547, 13157721, -3801087, 2240973, -3215745, 2922270, -1095100, 127548, -325840, 325478, -133517, 35729, -14687, 20985, -9107, 1000], [-72639018, 32015493, -7923744, 11321127, -13359816, 9470385, -1744791, 531101, -1471201, 975847, -338955, 70138, -67737, 68626, -18839, 3116], [-4406076, 2005965, -566109, 669465, -766413, 664635, -166754, 16320, -95240, 67468, -30562, 5695, -3637, 5697, -1777, 353], [-14438574, -38636271, 16023663, 4037607, 267381, -4565007, 4963584, 155962, -821234, -692686, 470466, -146995, -44319, -6295, 43211, -4919], [3440151, -1521423, 1441152, -751329, 559422, -614072, 214319, -94149, 69683, -46189, 33866, -10258, 4891, -4308, 1477, -615], [-24855255, 5617674, -9696186, 6022350, -2468448, 2844567, -1068648, 991390, -470258, 142016, -216117, 91394, -35004, 14816, -4852, 6901], [-25811703, 18912933, -5596614, 1874151, -3470256, 4132242, -1419913, 202593, -296329, 381137, -216176, 46784, -17447, 19758, -9485, 2893], [53217, -67554, 35640, -9882, 11574, -14525, 8762, -1992, 680, -1426, 1061, -316, 64, -66, 64, -21], [-2193318, 2256255, -1181304, 444789, -414096, 490349, -243759, 70559, -36601, 43441, -32155, 9634, -2889, 2734, -1751, 618], [-287955, 363042, -185328, 26082, -62442, 78159, -39706, 8044, -3360, 7534, -5339, 1432, -368, 374, -324, 107], [56599560, -72752742, 38362167, -13617882, 14405553, -14610996, 8518131, -2360825, 1069627, -1444819, 1003413, -356977, 114318, -78061, 57002, -19865], [17060787, -20335212, 16407198, -4410828, 3418848, -4756521, 2894098, -1058160, 226168, -387470, 373175, -122678, 34454, -20316, 17150, -9409]];
    Rf_basisdens := [1, 5677452, 630828, 1892484, 1892484, 5677452, 630828, 5677452, 630828, 5677452, 1892484, 10692, 210276, 32076, 5677452, 1892484];
    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_1452_b();"
function MakeCharacter_1452_b()
    N := 1452;
    order := 2;
    char_gens := [727, 485, 1333];
    v := [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_1452_b_Hecke();"
function MakeCharacter_1452_b_Hecke( : Kf := false)
    N := 1452;
    order := 2;
    char_gens := [727, 485, 1333];
    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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 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, 0, 0, 1], [0, 1, 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, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0], [1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 1, 0, -1, 0], [3, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, -1, -1, 0, 1, 0, 1, 0, 0, -1, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, -1, 1, 2, 0], [0, -3, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0], [-1, 0, 0, -1, 1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1], [-1, 0, 0, -2, -3, 0, -3, 0, -1, 0, -1, -1, 0, 0, 0, 1], [0, 0, 0, 3, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 0, -1], [0, 3, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0], [2, 0, 0, -3, -1, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 2, 0, 2, 0, 2, 0, -2, 2, 0, 0, 0, -1], [0, 0, -1, 0, 0, 3, 0, 0, 0, -1, 0, 0, 3, 0, -1, 0], [0, -1, -1, 0, 0, -1, 0, 0, 0, -6, 0, 0, -1, 0, -1, 0], [0, 1, 1, 0, 0, -1, 0, 1, 0, 1, 0, 0, 0, -1, 0, 0], [-3, 0, 0, 0, -3, 0, -3, 0, 3, 0, -3, 3, 0, 0, 0, 0], [2, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, -1, -1, 0, 0, 1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 0], [6, 0, 0, -2, -1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0], [0, -3, -1, 0, 0, 2, 0, -3, 0, -1, 0, 0, 1, -1, -2, 0], [0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0], [1, 0, 0, 0, 2, 0, 2, 0, -1, 0, 1, -1, 0, 0, 0, 2], [0, 4, 1, 0, 0, 2, 0, 0, 0, -1, 0, 0, 2, 0, 1, 0], [0, -2, -1, 0, 0, 1, 0, -2, 0, -1, 0, 0, 1, -3, -1, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2], [0, 6, 2, 0, 0, -2, 0, 0, 0, -1, 0, 0, -2, 0, 2, 0], [0, -1, -1, 0, 0, 1, 0, -1, 0, -1, 0, 0, 0, -3, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0, 0, 3, 0, 0, -1, 0, 1, 0], [6, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, -12, 0, 0, 0, 0], [5, 0, 0, -5, 0, 0, 0, 0, 5, 0, 0, -2, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 3, -4, -3, 0], [0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 2, -2, 1, 0], [3, 0, 0, 4, 0, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0], [1, 0, 0, 4, 5, 0, -5, 0, -4, 0, 0, -8, 0, 0, 0, 0], [-3, 0, 0, -2, -4, 0, -4, 0, 1, 0, -3, 1, 0, 0, 0, 2], [0, 8, 0, 0, 0, -1, 0, 0, 0, 7, 0, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, -3, 0, 0, 0, -1, 0, 0, 3, -1, 1, 0], [3, 0, 0, 3, -6, 0, 6, 0, -3, 0, 0, 1, 0, 0, 0, 0], [0, -2, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0], [-5, 0, 0, 5, 1, 0, -1, 0, -5, 0, 0, -3, 0, 0, 0, 0], [0, 3, 1, 0, 0, 2, 0, 3, 0, 1, 0, 0, -5, 1, 2, 0], [10, 0, 0, 5, -1, 0, 1, 0, -5, 0, 0, -1, 0, 0, 0, 0], [0, 3, 2, 0, 0, -4, 0, 3, 0, 2, 0, 0, 1, 4, 1, 0], [0, 4, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, -4, 1, 0, 0], [0, -10, -4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, -4, 0], [3, 0, 0, 4, 3, 0, 3, 0, 1, 0, 3, 1, 0, 0, 0, 3], [2, 0, 0, -3, 0, 0, 0, 0, -5, 0, 2, -5, 0, 0, 0, 3], [0, 1, -3, 0, 0, 0, 0, 1, 0, -3, 0, 0, -1, 0, 4, 0], [0, 0, 0, -7, -5, 0, -5, 0, -7, 0, 0, -7, 0, 0, 0, -2], [0, 1, -6, 0, 0, 1, 0, 0, 0, -6, 0, 0, 1, 0, -6, 0], [0, 10, -2, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, -2, 0], [0, -2, -3, 0, 0, 0, 0, -2, 0, -3, 0, 0, 2, 0, 1, 0], [0, 6, 3, 0, 0, -1, 0, 0, 0, 5, 0, 0, -1, 0, 3, 0], [0, -3, -3, 0, 0, 2, 0, -3, 0, -3, 0, 0, 1, 3, 0, 0], [0, 2, -1, 0, 0, -1, 0, 0, 0, 4, 0, 0, -1, 0, -1, 0], [-3, 0, 0, -4, -6, 0, -6, 0, -1, 0, -3, -1, 0, 0, 0, 4], [1, 0, 0, -2, 3, 0, -3, 0, 2, 0, 0, 8, 0, 0, 0, 0], [1, 0, 0, -1, -2, 0, -2, 0, -2, 0, 1, -2, 0, 0, 0, 2], [11, 0, 0, 9, 0, 0, 0, 0, -9, 0, 0, -3, 0, 0, 0, 0], [0, 2, -3, 0, 0, -1, 0, 0, 0, 5, 0, 0, -1, 0, -3, 0], [0, 0, 1, 0, 0, -4, 0, 0, 0, 1, 0, 0, 4, -4, -1, 0], [0, -3, 1, 0, 0, 2, 0, 0, 0, -6, 0, 0, 2, 0, 1, 0], [-2, 0, 0, -6, -8, 0, -8, 0, -4, 0, -2, -4, 0, 0, 0, 6], [0, 0, -1, 0, 0, 3, 0, 0, 0, -1, 0, 0, -3, -2, 1, 0], [-6, 0, 0, 5, -2, 0, 2, 0, -5, 0, 0, 12, 0, 0, 0, 0], [0, 7, 0, 0, 0, -1, 0, 0, 0, 11, 0, 0, -1, 0, 0, 0], [-3, 0, 0, 2, 7, 0, -7, 0, -2, 0, 0, -10, 0, 0, 0, 0], [-3, 0, 0, -1, 2, 0, 2, 0, 2, 0, -3, 2, 0, 0, 0, -6], [-3, 0, 0, -3, -1, 0, -1, 0, 0, 0, -3, 0, 0, 0, 0, -5], [-1, 0, 0, 1, -2, 0, 2, 0, -1, 0, 0, -5, 0, 0, 0, 0], [-5, 0, 0, -2, -3, 0, -3, 0, 3, 0, -5, 3, 0, 0, 0, -1], [0, -4, -3, 0, 0, 2, 0, 0, 0, 7, 0, 0, 2, 0, -3, 0], [6, 0, 0, 7, 9, 0, 9, 0, 1, 0, 6, 1, 0, 0, 0, -3], [-2, 0, 0, 7, 6, 0, -6, 0, -7, 0, 0, 2, 0, 0, 0, 0], [0, -2, -2, 0, 0, 1, 0, -2, 0, -2, 0, 0, 1, 1, 0, 0], [8, 0, 0, 4, 1, 0, -1, 0, -4, 0, 0, -10, 0, 0, 0, 0], [0, -11, 2, 0, 0, 1, 0, 0, 0, -7, 0, 0, 1, 0, 2, 0], [8, 0, 0, 2, 3, 0, 3, 0, -6, 0, 8, -6, 0, 0, 0, -2], [-4, 0, 0, -3, 3, 0, 3, 0, 1, 0, -4, 1, 0, 0, 0, -4], [0, -4, -2, 0, 0, 1, 0, 0, 0, -13, 0, 0, 1, 0, -2, 0], [0, -1, -3, 0, 0, 0, 0, -1, 0, -3, 0, 0, 1, 2, 2, 0], [-17, 0, 0, -4, 7, 0, -7, 0, 4, 0, 0, 3, 0, 0, 0, 0], [-7, 0, 0, -3, -4, 0, -4, 0, 4, 0, -7, 4, 0, 0, 0, 0], [0, 4, 2, 0, 0, 1, 0, 4, 0, 2, 0, 0, -5, 5, 2, 0], [1, 0, 0, -3, 3, 0, -3, 0, 3, 0, 0, -9, 0, 0, 0, 0], [0, -2, 2, 0, 0, -4, 0, -2, 0, 2, 0, 0, 6, -5, -4, 0], [2, 0, 0, -1, -8, 0, 8, 0, 1, 0, 0, 12, 0, 0, 0, 0], [0, 2, -1, 0, 0, -3, 0, 2, 0, -1, 0, 0, 1, -2, 3, 0], [-3, 0, 0, 1, -3, 0, -3, 0, 4, 0, -3, 4, 0, 0, 0, 3], [-8, 0, 0, -6, -3, 0, -3, 0, 2, 0, -8, 2, 0, 0, 0, -4], [0, -2, 4, 0, 0, -1, 0, 0, 0, 3, 0, 0, -1, 0, 4, 0], [0, -5, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -4, 0], [0, -8, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 4, 0, -2, 0], [0, 3, 6, 0, 0, -3, 0, 3, 0, 6, 0, 0, 0, 2, -3, 0], [0, 3, 5, 0, 0, 3, 0, 3, 0, 5, 0, 0, -6, 3, -2, 0], [0, -2, 1, 0, 0, 5, 0, -2, 0, 1, 0, 0, -3, -2, -3, 0], [0, 9, 1, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 1, 0], [8, 0, 0, -7, -3, 0, 3, 0, 7, 0, 0, -3, 0, 0, 0, 0], [-1, 0, 0, 0, 4, 0, 4, 0, 1, 0, -1, 1, 0, 0, 0, 4], [0, -1, 1, 0, 0, 8, 0, -1, 0, 1, 0, 0, -7, 2, -2, 0], [6, 0, 0, 3, 7, 0, 7, 0, -3, 0, 6, -3, 0, 0, 0, 2], [0, -13, -5, 0, 0, 2, 0, 0, 0, 14, 0, 0, 2, 0, -5, 0], [0, -20, -4, 0, 0, 5, 0, 0, 0, 1, 0, 0, 5, 0, -4, 0], [0, 4, -7, 0, 0, 5, 0, 0, 0, -7, 0, 0, 5, 0, -7, 0], [3, 0, 0, 6, 5, 0, 5, 0, 3, 0, 3, 3, 0, 0, 0, -10], [-33, 0, 0, -9, 3, 0, -3, 0, 9, 0, 0, 21, 0, 0, 0, 0], [-13, 0, 0, -8, 6, 0, -6, 0, 8, 0, 0, 11, 0, 0, 0, 0], [0, 0, 0, 1, 4, 0, 4, 0, 1, 0, 0, 1, 0, 0, 0, 3], [-3, 0, 0, -9, -8, 0, 8, 0, 9, 0, 0, -12, 0, 0, 0, 0], [3, 0, 0, 2, -3, 0, -3, 0, -1, 0, 3, -1, 0, 0, 0, 1], [3, 0, 0, 3, -5, 0, -5, 0, 0, 0, 3, 0, 0, 0, 0, 9], [0, -4, 4, 0, 0, -4, 0, -4, 0, 4, 0, 0, 8, -4, -8, 0], [-1, 0, 0, 8, -11, 0, 11, 0, -8, 0, 0, 17, 0, 0, 0, 0], [0, 17, 7, 0, 0, -3, 0, 0, 0, -8, 0, 0, -3, 0, 7, 0], [0, -4, -1, 0, 0, 4, 0, -4, 0, -1, 0, 0, 0, 4, -3, 0], [7, 0, 0, 4, -3, 0, -3, 0, -3, 0, 7, -3, 0, 0, 0, 6], [-10, 0, 0, -1, -3, 0, 3, 0, 1, 0, 0, 19, 0, 0, 0, 0], [0, 3, -2, 0, 0, -3, 0, 3, 0, -2, 0, 0, 0, 2, 5, 0], [9, 0, 0, 1, -2, 0, 2, 0, -1, 0, 0, 10, 0, 0, 0, 0], [-5, 0, 0, 2, 2, 0, 2, 0, 7, 0, -5, 7, 0, 0, 0, 6], [3, 0, 0, -2, 1, 0, -1, 0, 2, 0, 0, 15, 0, 0, 0, 0], [0, -10, 3, 0, 0, 5, 0, 0, 0, -7, 0, 0, 5, 0, 3, 0], [0, -1, 1, 0, 0, -7, 0, 0, 0, 14, 0, 0, -7, 0, 1, 0], [0, -3, -5, 0, 0, 1, 0, -3, 0, -5, 0, 0, 2, 1, 2, 0], [4, 0, 0, 1, -2, 0, 2, 0, -1, 0, 0, -2, 0, 0, 0, 0], [3, 0, 0, 12, 11, 0, -11, 0, -12, 0, 0, -22, 0, 0, 0, 0], [0, -5, 5, 0, 0, -1, 0, -5, 0, 5, 0, 0, 6, -5, -10, 0], [0, 15, 9, 0, 0, -4, 0, 0, 0, -7, 0, 0, -4, 0, 9, 0], [5, 0, 0, -8, 2, 0, 2, 0, -13, 0, 5, -13, 0, 0, 0, -2], [0, -14, -11, 0, 0, 4, 0, 0, 0, -3, 0, 0, 4, 0, -11, 0], [-4, 0, 0, 1, 3, 0, 3, 0, 5, 0, -4, 5, 0, 0, 0, 2], [0, 1, 4, 0, 0, 4, 0, 1, 0, 4, 0, 0, -5, -4, -3, 0], [0, 2, -4, 0, 0, -4, 0, 0, 0, 15, 0, 0, -4, 0, -4, 0], [0, 3, 5, 0, 0, -4, 0, 3, 0, 5, 0, 0, 1, -7, -2, 0], [-3, 0, 0, -2, 15, 0, -15, 0, 2, 0, 0, -12, 0, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 5, 0], [7, 0, 0, -4, -12, 0, 12, 0, 4, 0, 0, 21, 0, 0, 0, 0], [-7, 0, 0, 6, 4, 0, 4, 0, 13, 0, -7, 13, 0, 0, 0, 2], [0, -11, -3, 0, 0, 5, 0, 0, 0, -10, 0, 0, 5, 0, -3, 0], [0, 3, 2, 0, 0, 1, 0, 3, 0, 2, 0, 0, -4, -5, 1, 0], [-15, 0, 0, -5, 0, 0, 0, 0, 5, 0, 0, 31, 0, 0, 0, 0], [6, 0, 0, 7, 3, 0, 3, 0, 1, 0, 6, 1, 0, 0, 0, -6], [0, -1, 3, 0, 0, -6, 0, 0, 0, 8, 0, 0, -6, 0, 3, 0], [3, 0, 0, -1, 4, 0, 4, 0, -4, 0, 3, -4, 0, 0, 0, -1], [6, 0, 0, 14, -3, 0, 3, 0, -14, 0, 0, 6, 0, 0, 0, 0], [0, -1, -5, 0, 0, 0, 0, -1, 0, -5, 0, 0, 1, 3, 4, 0], [16, 0, 0, 10, 10, 0, -10, 0, -10, 0, 0, 9, 0, 0, 0, 0], [-1, 0, 0, -9, -8, 0, -8, 0, -8, 0, -1, -8, 0, 0, 0, -6], [0, -7, -5, 0, 0, -7, 0, 0, 0, -8, 0, 0, -7, 0, -5, 0], [-1, 0, 0, 11, 4, 0, 4, 0, 12, 0, -1, 12, 0, 0, 0, 0], [0, -1, 9, 0, 0, 1, 0, 0, 0, 10, 0, 0, 1, 0, 9, 0], [0, 4, 0, 0, 0, -7, 0, 4, 0, 0, 0, 0, 3, 1, 4, 0], [0, 0, 0, 1, -3, 0, -3, 0, 1, 0, 0, 1, 0, 0, 0, -9], [0, -6, -1, 0, 0, -1, 0, -6, 0, -1, 0, 0, 7, -4, -5, 0], [0, 13, -1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, -1, 0], [-8, 0, 0, -8, 1, 0, 1, 0, 0, 0, -8, 0, 0, 0, 0, -4], [3, 0, 0, 9, 2, 0, 2, 0, 6, 0, 3, 6, 0, 0, 0, 0], [-4, 0, 0, -14, -13, 0, -13, 0, -10, 0, -4, -10, 0, 0, 0, 2], [-4, 0, 0, 7, 11, 0, -11, 0, -7, 0, 0, 6, 0, 0, 0, 0], [0, -3, 4, 0, 0, 5, 0, 0, 0, 2, 0, 0, 5, 0, 4, 0], [0, 12, 4, 0, 0, 0, 0, 0, 0, -25, 0, 0, 0, 0, 4, 0], [-16, 0, 0, -16, -8, 0, -8, 0, 0, 0, -16, 0, 0, 0, 0, 0], [0, 1, -2, 0, 0, 1, 0, 1, 0, -2, 0, 0, -2, 4, 3, 0], [5, 0, 0, 3, -16, 0, 16, 0, -3, 0, 0, 2, 0, 0, 0, 0], [0, -7, -2, 0, 0, 1, 0, -7, 0, -2, 0, 0, 6, -2, -5, 0], [0, 2, 0, 0, 0, -10, 0, 0, 0, 2, 0, 0, -10, 0, 0, 0], [-20, 0, 0, 13, 9, 0, -9, 0, -13, 0, 0, -9, 0, 0, 0, 0], [-4, 0, 0, 0, 15, 0, 15, 0, 4, 0, -4, 4, 0, 0, 0, -2], [0, -26, -5, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, -5, 0], [-2, 0, 0, -9, -5, 0, -5, 0, -7, 0, -2, -7, 0, 0, 0, 4], [0, -4, -5, 0, 0, -3, 0, 0, 0, -1, 0, 0, -3, 0, -5, 0], [0, 6, -13, 0, 0, 6, 0, 0, 0, -15, 0, 0, 6, 0, -13, 0], [-39, 0, 0, 0, 5, 0, -5, 0, 0, 0, 0, 12, 0, 0, 0, 0], [0, 1, 3, 0, 0, -5, 0, 1, 0, 3, 0, 0, 4, -1, -2, 0], [15, 0, 0, 3, 15, 0, -15, 0, -3, 0, 0, -1, 0, 0, 0, 0], [0, 8, 4, 0, 0, 2, 0, 8, 0, 4, 0, 0, -10, 3, 4, 0], [-6, 0, 0, -13, -16, 0, -16, 0, -7, 0, -6, -7, 0, 0, 0, 3], [-10, 0, 0, -10, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 5], [0, 0, -3, 0, 0, 4, 0, 0, 0, 5, 0, 0, 4, 0, -3, 0], [-5, 0, 0, 6, -7, 0, 7, 0, -6, 0, 0, 11, 0, 0, 0, 0], [0, -13, -8, 0, 0, -5, 0, 0, 0, -12, 0, 0, -5, 0, -8, 0], [0, 2, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 5, 2, 0], [10, 0, 0, 2, -8, 0, 8, 0, -2, 0, 0, 25, 0, 0, 0, 0], [0, 5, -3, 0, 0, -8, 0, 5, 0, -3, 0, 0, 3, 1, 8, 0], [19, 0, 0, 10, -15, 0, 15, 0, -10, 0, 0, 0, 0, 0, 0, 0], [10, 0, 0, 4, -3, 0, -3, 0, -6, 0, 10, -6, 0, 0, 0, -10], [0, -1, -3, 0, 0, -2, 0, -1, 0, -3, 0, 0, 3, 2, 2, 0], [-7, 0, 0, -4, -11, 0, -11, 0, 3, 0, -7, 3, 0, 0, 0, 9], [0, -4, 3, 0, 0, 4, 0, 0, 0, 11, 0, 0, 4, 0, 3, 0], [-39, 0, 0, -6, 16, 0, -16, 0, 6, 0, 0, -1, 0, 0, 0, 0], [0, -2, -5, 0, 0, 3, 0, -2, 0, -5, 0, 0, -1, 0, 3, 0], [0, -1, 6, 0, 0, -8, 0, -1, 0, 6, 0, 0, 9, -8, -7, 0], [0, 0, 5, 0, 0, 3, 0, 0, 0, 5, 0, 0, -3, 8, -5, 0], [0, -16, 4, 0, 0, -2, 0, 0, 0, -8, 0, 0, -2, 0, 4, 0], [8, 0, 0, -9, 3, 0, -3, 0, 9, 0, 0, -23, 0, 0, 0, 0], [0, -3, 5, 0, 0, 1, 0, 0, 0, -22, 0, 0, 1, 0, 5, 0], [-3, 0, 0, -5, -18, 0, -18, 0, -2, 0, -3, -2, 0, 0, 0, 10], [9, 0, 0, 5, 8, 0, 8, 0, -4, 0, 9, -4, 0, 0, 0, -3], [30, 0, 0, 5, 4, 0, -4, 0, -5, 0, 0, 4, 0, 0, 0, 0], [0, 2, -6, 0, 0, 0, 0, 2, 0, -6, 0, 0, -2, -5, 8, 0], [0, -1, -7, 0, 0, 3, 0, -1, 0, -7, 0, 0, -2, -5, 6, 0], [-2, 0, 0, -6, 7, 0, -7, 0, 6, 0, 0, -10, 0, 0, 0, 0], [0, 5, -3, 0, 0, -6, 0, 0, 0, 7, 0, 0, -6, 0, -3, 0], [1, 0, 0, -2, -4, 0, -4, 0, -3, 0, 1, -3, 0, 0, 0, -16], [2, 0, 0, 23, 5, 0, -5, 0, -23, 0, 0, 13, 0, 0, 0, 0], [-2, 0, 0, -1, 3, 0, 3, 0, 1, 0, -2, 1, 0, 0, 0, -18], [0, 9, 4, 0, 0, -9, 0, 9, 0, 4, 0, 0, 0, 7, 5, 0], [13, 0, 0, 18, 9, 0, -9, 0, -18, 0, 0, -17, 0, 0, 0, 0], [0, 16, 1, 0, 0, -5, 0, 0, 0, 3, 0, 0, -5, 0, 1, 0], [0, 3, 4, 0, 0, -1, 0, 3, 0, 4, 0, 0, -2, 4, -1, 0], [11, 0, 0, 10, 6, 0, 6, 0, -1, 0, 11, -1, 0, 0, 0, 6], [2, 0, 0, -11, -11, 0, -11, 0, -13, 0, 2, -13, 0, 0, 0, 4], [0, 10, -2, 0, 0, 7, 0, 0, 0, 9, 0, 0, 7, 0, -2, 0], [0, -10, 3, 0, 0, 6, 0, 0, 0, 1, 0, 0, 6, 0, 3, 0], [3, 0, 0, -2, 1, 0, 1, 0, -5, 0, 3, -5, 0, 0, 0, 10], [-2, 0, 0, 5, 6, 0, -6, 0, -5, 0, 0, 16, 0, 0, 0, 0], [0, -1, 8, 0, 0, -7, 0, -1, 0, 8, 0, 0, 8, -6, -9, 0], [0, -17, -4, 0, 0, 19, 0, 0, 0, -17, 0, 0, 19, 0, -4, 0], [-20, 0, 0, -17, -2, 0, -2, 0, 3, 0, -20, 3, 0, 0, 0, -3], [15, 0, 0, 3, 8, 0, 8, 0, -12, 0, 15, -12, 0, 0, 0, 2], [0, 5, 1, 0, 0, 3, 0, 0, 0, -24, 0, 0, 3, 0, 1, 0], [0, -5, 3, 0, 0, 0, 0, -5, 0, 3, 0, 0, 5, -8, -8, 0], [12, 0, 0, 12, 2, 0, -2, 0, -12, 0, 0, -14, 0, 0, 0, 0], [0, 14, 8, 0, 0, -16, 0, 0, 0, 3, 0, 0, -16, 0, 8, 0], [0, 11, -3, 0, 0, 2, 0, 0, 0, 14, 0, 0, 2, 0, -3, 0], [0, 4, -5, 0, 0, 5, 0, 4, 0, -5, 0, 0, -9, 14, 9, 0], [6, 0, 0, 1, -1, 0, 1, 0, -1, 0, 0, 27, 0, 0, 0, 0], [0, 0, -7, 0, 0, -2, 0, 0, 0, -7, 0, 0, 2, 6, 7, 0], [1, 0, 0, -20, -10, 0, 10, 0, 20, 0, 0, -1, 0, 0, 0, 0], [0, -4, 1, 0, 0, 3, 0, -4, 0, 1, 0, 0, 1, 4, -5, 0], [4, 0, 0, 4, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, -16], [11, 0, 0, 1, 7, 0, 7, 0, -10, 0, 11, -10, 0, 0, 0, 5], [-4, 0, 0, -3, -5, 0, -5, 0, 1, 0, -4, 1, 0, 0, 0, 12], [0, -3, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0], [5, 0, 0, -11, 7, 0, -7, 0, 11, 0, 0, -21, 0, 0, 0, 0], [2, 0, 0, -1, -3, 0, 3, 0, 1, 0, 0, 23, 0, 0, 0, 0], [0, 5, 6, 0, 0, -6, 0, 5, 0, 6, 0, 0, 1, 10, -1, 0], [0, 4, 8, 0, 0, 0, 0, 4, 0, 8, 0, 0, -4, 9, -4, 0], [20, 0, 0, 4, -8, 0, 8, 0, -4, 0, 0, 17, 0, 0, 0, 0], [16, 0, 0, 13, 7, 0, 7, 0, -3, 0, 16, -3, 0, 0, 0, -8], [0, -7, 5, 0, 0, -1, 0, 0, 0, 24, 0, 0, -1, 0, 5, 0], [0, -2, -4, 0, 0, -6, 0, -2, 0, -4, 0, 0, 8, -6, 2, 0], [0, 8, -4, 0, 0, -11, 0, 0, 0, 21, 0, 0, -11, 0, -4, 0], [0, 5, 6, 0, 0, -2, 0, 5, 0, 6, 0, 0, -3, 6, -1, 0], [-2, 0, 0, -6, -2, 0, -2, 0, -4, 0, -2, -4, 0, 0, 0, 16], [-20, 0, 0, -6, -13, 0, 13, 0, 6, 0, 0, 26, 0, 0, 0, 0], [0, -3, 4, 0, 0, -3, 0, -3, 0, 4, 0, 0, 6, 2, -7, 0], [0, 3, 1, 0, 0, -1, 0, 3, 0, 1, 0, 0, -2, -5, 2, 0], [23, 0, 0, 13, 9, 0, -9, 0, -13, 0, 0, -9, 0, 0, 0, 0], [0, -12, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0], [-11, 0, 0, -11, -12, 0, -12, 0, 0, 0, -11, 0, 0, 0, 0, 4], [0, 2, 0, 0, 0, -2, 0, 0, 0, 3, 0, 0, -2, 0, 0, 0], [0, 9, 0, 0, 0, 2, 0, 0, 0, -22, 0, 0, 2, 0, 0, 0], [0, 5, -9, 0, 0, -3, 0, 5, 0, -9, 0, 0, -2, -3, 14, 0], [0, -1, -10, 0, 0, 5, 0, 0, 0, -10, 0, 0, 5, 0, -10, 0], [0, -26, -11, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, -11, 0], [-16, 0, 0, -13, -14, 0, -14, 0, 3, 0, -16, 3, 0, 0, 0, -3], [-45, 0, 0, -16, 7, 0, -7, 0, 16, 0, 0, 16, 0, 0, 0, 0], [15, 0, 0, 18, 8, 0, 8, 0, 3, 0, 15, 3, 0, 0, 0, 4], [10, 0, 0, 15, 7, 0, 7, 0, 5, 0, 10, 5, 0, 0, 0, -8], [0, -4, -1, 0, 0, 13, 0, 0, 0, -13, 0, 0, 13, 0, -1, 0], [0, 2, -4, 0, 0, -9, 0, 2, 0, -4, 0, 0, 7, -1, 6, 0], [14, 0, 0, 8, -21, 0, 21, 0, -8, 0, 0, -4, 0, 0, 0, 0], [50, 0, 0, 10, -12, 0, 12, 0, -10, 0, 0, -35, 0, 0, 0, 0], [-7, 0, 0, 11, -7, 0, 7, 0, -11, 0, 0, 25, 0, 0, 0, 0], [0, -7, 3, 0, 0, -2, 0, 0, 0, 27, 0, 0, -2, 0, 3, 0], [0, 7, 1, 0, 0, -9, 0, 0, 0, 0, 0, 0, -9, 0, 1, 0], [-10, 0, 0, 7, 5, 0, -5, 0, -7, 0, 0, -4, 0, 0, 0, 0], [3, 0, 0, 9, -2, 0, -2, 0, 6, 0, 3, 6, 0, 0, 0, -4], [0, -11, 1, 0, 0, 2, 0, 0, 0, 16, 0, 0, 2, 0, 1, 0], [0, -1, -10, 0, 0, 5, 0, 0, 0, -10, 0, 0, 5, 0, -10, 0], [0, -1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 7, 0, 0], [0, -8, -12, 0, 0, 0, 0, -8, 0, -12, 0, 0, 8, -3, 4, 0], [-14, 0, 0, -9, 1, 0, -1, 0, 9, 0, 0, -15, 0, 0, 0, 0], [0, -9, 2, 0, 0, 11, 0, -9, 0, 2, 0, 0, -2, -1, -11, 0], [0, 14, 8, 0, 0, -1, 0, 0, 0, -17, 0, 0, -1, 0, 8, 0], [0, 4, -15, 0, 0, -1, 0, 0, 0, 11, 0, 0, -1, 0, -15, 0], [15, 0, 0, 19, 6, 0, 6, 0, 4, 0, 15, 4, 0, 0, 0, -3], [-19, 0, 0, 0, -10, 0, 10, 0, 0, 0, 0, -5, 0, 0, 0, 0], [0, 8, 10, 0, 0, -8, 0, 8, 0, 10, 0, 0, 0, 5, -2, 0], [-34, 0, 0, -7, -1, 0, 1, 0, 7, 0, 0, 59, 0, 0, 0, 0], [0, -5, -3, 0, 0, 8, 0, -5, 0, -3, 0, 0, -3, -7, -2, 0], [-12, 0, 0, -6, -6, 0, -6, 0, 6, 0, -12, 6, 0, 0, 0, -11], [12, 0, 0, 4, 1, 0, 1, 0, -8, 0, 12, -8, 0, 0, 0, -12], [0, -3, 0, 0, 0, 9, 0, -3, 0, 0, 0, 0, -6, 3, -3, 0], [0, 3, 9, 0, 0, 1, 0, 3, 0, 9, 0, 0, -4, 1, -6, 0], [0, -2, -4, 0, 0, 6, 0, 0, 0, 21, 0, 0, 6, 0, -4, 0], [0, -4, -8, 0, 0, 5, 0, -4, 0, -8, 0, 0, -1, -3, 4, 0], [17, 0, 0, -1, -13, 0, 13, 0, 1, 0, 0, 15, 0, 0, 0, 0], [-7, 0, 0, -2, 6, 0, 6, 0, 5, 0, -7, 5, 0, 0, 0, -12], [0, -3, 5, 0, 0, -4, 0, -3, 0, 5, 0, 0, 7, -12, -8, 0], [0, 7, -4, 0, 0, -3, 0, 0, 0, 24, 0, 0, -3, 0, -4, 0], [0, -17, 7, 0, 0, 3, 0, 0, 0, -14, 0, 0, 3, 0, 7, 0], [0, -3, -1, 0, 0, 7, 0, -3, 0, -1, 0, 0, -4, -9, -2, 0], [0, 6, 2, 0, 0, -14, 0, 0, 0, 15, 0, 0, -14, 0, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1452_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1452_2_b_e();".
// 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_1452_2_b_e(:prec:=16)
    chi := MakeCharacter_1452_b();
    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_1452_2_b_e();".
// 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_1452_2_b_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1452_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![121, 0, 748, 0, 253, 0, 28, 0, 1]>,<7,R![3025, 0, 1975, 0, 420, 0, 35, 0, 1]>],Snew);
    return Vf;
end function;