
// Make newform 1682.2.b.k in Magma, downloaded from the LMFDB on 17 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1682_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1682_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_1682_2_b_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_1682_2_b_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 := [961, 0, 10537, 0, 30128, 0, 33099, 0, 16415, 0, 4119, 0, 548, 0, 37, 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], [6710203, 0, -40149880, 0, -83894220, 0, -44272376, 0, -8639916, 0, -607273, 0, 2104, 0, 1284, 0], [-465992667, 0, -567920218, 0, 344485276, 0, 356706384, 0, 104632823, 0, 14477445, 0, 991285, 0, 27407, 0], [-272137079, 0, -1360564353, 0, -1083527461, 0, -260579504, 0, 19187494, 0, 15594884, 0, 2028858, 0, 83062, 0], [563561575, 0, 2317036540, 0, 2929200301, 0, 1234571776, 0, 170890658, 0, -2234878, 0, -2075146, 0, -111310, 0], [73501, 0, 571977, 0, 1071058, 0, 759007, 0, 249235, 0, 40629, 0, 3198, 0, 97, 0], [372298774, 0, 2487776512, 0, 3913685887, 0, 2301318694, 0, 648806909, 0, 95469962, 0, 7093859, 0, 210476, 0], [167012926, 0, 136745381, 0, -910206261, 0, -1147603841, 0, -496888416, 0, -96829366, 0, -8735498, 0, -297522, 0], [0, 6710203, 0, -40149880, 0, -83894220, 0, -44272376, 0, -8639916, 0, -607273, 0, 2104, 0, 1284], [0, -7632738708, 0, -32225636488, 0, -32809597349, 0, -14938958478, 0, -3538203628, 0, -456007590, 0, -30535640, 0, -836497], [0, -16490385368, 0, -36805700960, 0, -29467116938, 0, -9396663905, 0, -793255129, 0, 126772847, 0, 25028414, 0, 1126355], [0, 4712034027, 0, 54847293839, 0, 89294419090, 0, 49540927182, 0, 12441794609, 0, 1564112238, 0, 95943751, 0, 2276840], [0, 7251940, 0, 38230690, 0, 54948512, 0, 31193680, 0, 8506650, 0, 1200170, 0, 84720, 0, 2371], [0, 68484475212, 0, 240357553952, 0, 265298765335, 0, 124791972708, 0, 28921496975, 0, 3491805708, 0, 210639250, 0, 5002886], [0, 45353098205, 0, 179487884122, 0, 236091715791, 0, 130305161843, 0, 35594343402, 0, 5113668292, 0, 371223239, 0, 10751328]];
    Rf_basisdens := [1, 1, 6652257, 73174827, 73174827, 73174827, 35299, 73174827, 73174827, 6652257, 2268419637, 2268419637, 2268419637, 1094269, 2268419637, 2268419637];
    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_1682_b();"
function MakeCharacter_1682_b()
    N := 1682;
    order := 2;
    char_gens := [843];
    v := [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_1682_b_Hecke();"
function MakeCharacter_1682_b_Hecke(Kf)
    N := 1682;
    order := 2;
    char_gens := [843];
    char_values := [[-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]];
    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, -1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1], [-3, 0, 1, 0, 0, -1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, -1, 2, 0, -1], [2, 0, 0, 1, -1, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, -1, 0, 3, -1, -2, 0, 1], [0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 2, 0, 1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -4, 0, -1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -2, -3, 5, -1, -1, 1, 0], [-1, 0, 1, -1, 1, -1, 1, -1, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -2, 1, -1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -2, -5, 2, 1, 1, 0, 0], [-6, 0, 2, 1, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -4, -2, 0, 0, -1, 2, 3, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 1], [0, -1, 0, 0, 0, 0, 0, 0, 0, 3, -1, -1, -2, 1, 0, -3], [-3, 0, 8, -1, -3, -3, -1, 7, -2, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 4, -2, 1, -2], [0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 6, -1, -1, -5, 0, -2], [0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 3, -10, 1, 1], [7, 0, 3, 1, -4, -2, -3, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 1, -3, 0, -1, -2, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, -2, 1, -2, 1, 3, 2, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 4, -3, 1, 1, -1, 2], [0, -4, 0, 0, 0, 0, 0, 0, 0, -3, 0, 2, 6, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -2, -1, 3, -4], [0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 7, -2, 8, -5, 2, 2], [2, 0, -10, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], [7, 0, -5, 0, 2, 1, -2, -4, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 1, 2, 1, 4, 2, 3, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, -2, -9, 5, -2, 2, -1, 1], [0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 1, -5, 3, -1], [-8, 0, 2, 1, 0, -3, -1, 4, 2, 0, 0, 0, 0, 0, 0, 0], [5, 0, -1, -2, -1, -2, 2, 5, -5, 0, 0, 0, 0, 0, 0, 0], [4, 0, -5, 0, -6, -1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 10, 1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -4, -3, 0, 6, 8, 2, -3], [0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 9, 1, 3, -2, 3], [0, 0, -2, 0, -1, -1, 3, 7, 7, 0, 0, 0, 0, 0, 0, 0], [3, 0, -7, -3, 0, 0, -2, 4, 4, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 2, -3, 3, -1, -3, -2, 1], [-1, 0, 5, 1, 0, 0, 3, 4, 1, 0, 0, 0, 0, 0, 0, 0], [2, 0, -2, 0, -6, 0, -1, 7, -7, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 5, -7, -2, -4, 3, 1, -1], [2, 0, -11, -1, 2, 4, -1, -6, 3, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 3, 1, 1, -1, -2, 3, 4, 0, 0, 0, 0, 0, 0, 0], [10, 0, -7, 2, -2, -1, -2, -6, -3, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 5, -5, 2, -7, 0, 2], [-8, 0, 7, -1, 2, -3, -1, -2, -6, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, -6, -5, 2, 7, 1, 4, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -8, -7, -11, 14, -4, 3], [0, 2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 6, -2, 2, 4], [-8, 0, 2, -3, 4, -1, 1, 8, -2, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, -2, 2, 0, -7, -2, 6, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -11, 0, -1, 4, 1, -5, 5, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 3, -6, -6, 1, -3], [0, 1, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0, 5, 5, 1, -2], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 8, -1, 1, 2, 4, 1], [10, 0, 0, 0, 3, 1, -8, -5, 1, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, -2, 7, -7, -2, 2, -3, 5], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -4, 9, -4, -2, 3, 3], [0, -2, 0, 0, 0, 0, 0, 0, 0, 1, -2, -3, -10, 6, 0, 5], [5, 0, 7, 1, 0, 2, 4, 0, 5, 0, 0, 0, 0, 0, 0, 0], [14, 0, 2, 0, -3, 3, 0, 5, -4, 0, 0, 0, 0, 0, 0, 0], [2, 0, -13, 3, 4, 7, 0, -8, 10, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 7, 5, -4, -5, -6, -1, 2], [0, 2, 0, 0, 0, 0, 0, 0, 0, -2, -4, 5, -7, 13, -3, 5], [8, 0, -8, 3, -7, -2, 6, 3, 7, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, -4, 12, -8, -2, -5, -2, 5], [-2, 0, 4, 3, -8, 1, 2, 8, -5, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, -7, -4, 6, 1, 10, 3, 1], [12, 0, 4, -1, -4, -5, 0, 6, 5, 0, 0, 0, 0, 0, 0, 0], [-9, 0, -5, 4, 5, 5, 0, -10, 4, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 5, 0, -2, 3, 3, 1, 0], [2, 0, 4, 5, -4, 0, 4, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 0, 2, -8, 5, 0, 12, 1, -4], [-1, 0, -5, -1, -1, 1, -3, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 1, -6, 4, 3, 3, 7, -3], [-9, 0, 15, 0, 3, 0, -3, -3, 8, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, -8, 8, -4, 2, 1, -2, -4], [0, -3, 0, 0, 0, 0, 0, 0, 0, 1, 6, -6, -1, -4, 0, -2], [-4, 0, 7, -2, 3, -5, -3, -9, -1, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 10, -12, 0, 11, -3, -2], [-6, 0, -2, -8, 6, -8, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -6, 14, 1, 7, -5, 0, 2], [0, 7, 0, 0, 0, 0, 0, 0, 0, 3, 17, -12, 6, -15, -7, 1], [-9, 0, 11, -7, -3, -7, 3, 13, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 2, -5, 3, 3, 2, 3], [4, 0, -4, -2, 3, 11, -2, -2, -4, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 3, -1, -18, -8, 5, 1, -5], [-4, 0, 0, -4, -5, -5, 1, 1, -3, 0, 0, 0, 0, 0, 0, 0], [-8, 0, -2, -4, 7, 7, 0, -1, 5, 0, 0, 0, 0, 0, 0, 0], [20, 0, -8, -4, -1, -3, 4, -2, -8, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, 0, 0, -2, -1, 20, 0, -6, 8, -5], [-2, 0, 6, 2, -8, 0, 3, 4, -6, 0, 0, 0, 0, 0, 0, 0], [16, 0, -12, 3, -2, 5, 1, -10, 4, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -7, -5, 7, 3, -2, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -5, 5, 6, -3, -2, 3], [11, 0, -8, -3, -2, -1, -8, -12, -5, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, -7, 4, 4, 10, -13, 3, 4], [19, 0, -8, -1, -4, 3, 4, -6, -7, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 17, 5, -5, -2, 9, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, -6, -6, 3, 16, -1, 1], [0, 3, 0, 0, 0, 0, 0, 0, 0, -4, -8, 15, 3, -11, 3, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 7, -8, 18, -7, -2], [-10, 0, 2, 2, 1, -7, 3, 15, -3, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 12, 8, -9, -1, -7, -1, -3], [0, -6, 0, 0, 0, 0, 0, 0, 0, -4, 6, -1, 6, 3, 1, 1], [-15, 0, 3, 2, 4, -1, -4, -12, 9, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 11, -7, -9, -13, -3, 3], [-17, 0, 2, 1, 4, 2, -3, -14, 1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -14, 1, -1, -4, 0, 3, 17, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, 0, 0, -6, -11, 25, 2, -14, -1, 7], [0, -3, 0, 0, 0, 0, 0, 0, 0, -4, -19, 3, -8, 7, 1, -1], [19, 0, -5, 4, 5, 0, 2, -17, 7, 0, 0, 0, 0, 0, 0, 0], [-20, 0, -1, 1, -1, -2, -4, 3, 5, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, -3, 4, 1, 7, -16, 8, -3], [8, 0, -10, -9, 6, 0, -2, 3, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -4, -1, -3, -1, 9, -6, 0, 0, 0, 0, 0, 0, 0], [13, 0, -17, 7, 4, 2, 3, -16, 13, 0, 0, 0, 0, 0, 0, 0], [-15, 0, 3, -7, 0, -4, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -17, -7, 6, 2, -4, -10, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -7, -4, 14, 10, 1, 5, 0], [0, 9, 0, 0, 0, 0, 0, 0, 0, 1, 7, -7, 6, 3, -2, 7], [0, 4, 0, 0, 0, 0, 0, 0, 0, -4, -5, 6, -10, 16, 0, 8], [0, -8, 0, 0, 0, 0, 0, 0, 0, 7, 0, 2, -1, 2, -3, -1], [0, -3, 0, 0, 0, 0, 0, 0, 0, 5, 3, -2, -2, -13, -3, -3], [0, -3, 0, 0, 0, 0, 0, 0, 0, -6, 1, 0, 13, -15, 4, 0], [5, 0, -3, 2, -6, -1, -13, -7, -4, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, -7, 9, -7, 2, 11, 0, -3], [0, 12, 0, 0, 0, 0, 0, 0, 0, -3, 8, -2, 2, -15, 1, -2], [-16, 0, 2, -4, -3, -2, -4, 6, -17, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 3, -10, -10, 7, 10, -5, -1], [0, -6, 0, 0, 0, 0, 0, 0, 0, -7, -22, 11, -9, 12, 1, 5], [3, 0, 2, 8, -5, 2, 2, 9, 0, 0, 0, 0, 0, 0, 0, 0], [-29, 0, 11, 0, 8, 5, -8, -6, 7, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, -8, -14, 2, -7, 29, -2, 9], [0, -7, 0, 0, 0, 0, 0, 0, 0, -3, -15, -5, -15, -1, -4, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 16, 11, 12, -18, 7, 2], [0, 3, 0, 0, 0, 0, 0, 0, 0, -5, 7, 12, -3, -14, -1, 8], [0, 9, 0, 0, 0, 0, 0, 0, 0, 4, 3, 3, -7, 13, -5, 8], [3, 0, -4, -1, -1, 0, -1, -9, 12, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, -8, -11, -4, 4, 18, -4, 5], [3, 0, 7, 1, -4, 0, -3, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 8, 4, 0, -8, 0, 6, -8, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 4, -13, -1, -4, 2, -1, 3], [-5, 0, -24, -5, 2, -6, -11, -8, 7, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 9, -11, -22, 17, 0, 1], [0, -2, 0, 0, 0, 0, 0, 0, 0, 1, -6, 1, 17, -9, -2, -4], [0, -9, 0, 0, 0, 0, 0, 0, 0, 2, 27, -7, -3, -17, -5, 0], [-3, 0, -11, 4, 10, 7, 4, -16, -1, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 4, 8, 3, 6, -7, -16, 16, 0, 0, 0, 0, 0, 0, 0], [0, 0, -13, -3, 12, 9, 7, -12, 5, 0, 0, 0, 0, 0, 0, 0], [-35, 0, 9, 3, 5, -3, -4, 5, 8, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 10, -3, -15, -7, -14, -3, -5], [-10, 0, 16, 7, 1, 6, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, -8, 3, 13, 1, -7, -1, 2], [0, 3, 0, 0, 0, 0, 0, 0, 0, 15, 0, -3, -3, -15, 0, -6], [-1, 0, -7, -3, 8, 3, 3, 4, 6, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 3, 21, -4, 9, -24, 1, -4], [-1, 0, 13, -2, -4, 1, 11, 4, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 16, 1, 6, 4, 5, 2], [6, 0, -1, 0, -13, -2, -7, 11, -12, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 6, 0, -11, 11, -15, 1, -7], [-7, 0, 22, -4, 6, -7, -7, -4, 3, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 23, -7, -1, -1, 0, 9, -12, 0, 0, 0, 0, 0, 0, 0], [0, 0, 16, 3, 4, 9, 5, 6, -2, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -2, 9, 1, -4, 0, 0, 9], [-4, 0, 4, -3, -8, -1, 7, 16, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -4, -5, 6, 1, 6, 6, -1, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, -3, 13, -4, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, 0, 0, 0, -3, 7, -9, -5, -4, -6, -4], [0, 4, 0, 0, 0, 0, 0, 0, 0, 10, 8, -11, -18, 4, -1, -8], [0, 0, 4, -2, 12, 6, -6, -4, -2, 0, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, 0, 0, 0, 0, 0, -12, -3, 15, 3, 1, 3, -4], [0, 12, 0, 0, 0, 0, 0, 0, 0, 6, 5, 0, -18, 20, -3, 8], [-8, 0, 4, 4, 2, 8, 10, 2, 4, 0, 0, 0, 0, 0, 0, 0], [12, 0, -20, -1, -1, 8, -2, -17, 2, 0, 0, 0, 0, 0, 0, 0], [-6, 0, -8, 6, -1, 9, 9, -7, 3, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 5, -2, 2, -1, 13, 28, 4, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, -2, -13, -10, 21, 2, -6], [0, -1, 0, 0, 0, 0, 0, 0, 0, 4, 15, -6, 11, -18, -4, 1], [0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 16, -4, 2, 16, -8, -2], [-7, 0, 1, 3, -12, 2, -9, 4, 7, 0, 0, 0, 0, 0, 0, 0], [15, 0, -9, -4, -14, -8, -1, 3, -5, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 17, -2, 8, -4, -21, -3, -1], [0, -2, 0, 0, 0, 0, 0, 0, 0, -8, 1, -7, 18, 5, -1, -4], [0, -5, 0, 0, 0, 0, 0, 0, 0, 3, -1, -10, 7, -12, 11, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, -12, 7, -12, -2, -1], [-9, 0, 1, -6, -1, -2, 7, -5, -3, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, -4, -20, 2, 7, -4, 2, 2], [1, 0, -17, 0, 6, -1, 3, -18, 4, 0, 0, 0, 0, 0, 0, 0], [12, 0, 4, 8, -2, -2, -4, -15, 0, 0, 0, 0, 0, 0, 0, 0], [37, 0, -7, -2, -2, 3, 4, -14, -7, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, -7, -26, 15, -1, 12, 2, -9], [-7, 0, 19, 0, -3, 10, 3, 5, -14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 28, -6, 24, -18, 2, 2], [0, -5, 0, 0, 0, 0, 0, 0, 0, 14, -4, -17, -16, 14, -11, 1], [0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 6, -6, -11, -12, -4, -1], [18, 0, -16, -3, -3, 6, 6, 3, 11, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -29, 0, -6, 13, -7, 5], [0, 0, 6, -4, -8, -5, 12, 22, -9, 0, 0, 0, 0, 0, 0, 0], [20, 0, 6, 2, 4, 3, -5, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 16, 10, 0, 1], [0, 4, 0, 0, 0, 0, 0, 0, 0, 13, -1, -18, -23, 14, -3, -8], [-17, 0, 13, 4, 5, 4, -5, -3, 4, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, -4, 1, -22, -12, 4, 4, -3], [0, -19, 0, 0, 0, 0, 0, 0, 0, 0, -12, 15, -2, 16, 13, -1], [0, 3, 0, 0, 0, 0, 0, 0, 0, 7, -27, -5, -14, 15, -2, 3], [-2, 0, -3, -3, 2, 9, -7, -6, 5, 0, 0, 0, 0, 0, 0, 0], [2, 0, 10, 0, 11, 1, 8, -11, 2, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 0, -8, -17, 5, 7, 11, 1, -12], [5, 0, 5, 1, 2, 8, -6, -2, -2, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, -11, -10, 0, -11, 30, -3, 11], [0, 12, 0, 0, 0, 0, 0, 0, 0, -2, 16, -8, 3, -19, -4, -7], [1, 0, 9, -7, -6, -10, 6, 14, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, -1, -3, 15, 0, 8, -8, 2], [0, 7, 0, 0, 0, 0, 0, 0, 0, 4, 1, -17, -9, -15, -1, 12], [14, 0, -7, 3, 7, -1, -6, -19, 18, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 0, 2, -30, 2, -23, 35, 0, 3], [0, -5, 0, 0, 0, 0, 0, 0, 0, 1, 17, 21, 10, -12, 8, -8], [0, -1, 0, 0, 0, 0, 0, 0, 0, -6, -13, 14, 6, 4, 6, -3], [0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 24, -13, 2, -22, 3, 6], [15, 0, -9, 1, 3, -4, 5, 8, -6, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 5, 9, -3, -17, 1, -4], [12, 0, -24, -3, 7, -2, 1, 8, 10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 9, 9, 11, -9, 3], [0, -3, 0, 0, 0, 0, 0, 0, 0, 7, -7, 4, 0, -10, 9, 3], [-13, 0, -5, -7, -2, -1, -19, -4, -8, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -9, 10, -20, -22, 4, 1], [0, -5, 0, 0, 0, 0, 0, 0, 0, -5, -31, 21, -5, 20, 2, -12], [0, 12, 0, 0, 0, 0, 0, 0, 0, 3, 22, -1, 5, 4, -2, 5], [-31, 0, 22, 9, -10, -6, 7, 20, -15, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, -7, -8, 9, 5, 8, 2, 3], [0, -10, 0, 0, 0, 0, 0, 0, 0, 3, 16, -1, -14, -9, 1, -8], [-15, 0, 1, 4, -8, 12, 4, 10, 6, 0, 0, 0, 0, 0, 0, 0], [-24, 0, 6, -10, 3, -3, 0, 23, 8, 0, 0, 0, 0, 0, 0, 0], [0, 11, 0, 0, 0, 0, 0, 0, 0, -3, 3, -9, 0, 9, -18, 7], [-15, 0, 15, 3, 16, 5, 3, -12, -4, 0, 0, 0, 0, 0, 0, 0], [21, 0, -3, 0, 2, 7, -7, -16, -6, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -6, -5, 12, -5, -3, -12, -8, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -6, 1, -8, 11, 0, -4, -12, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 26, 1, 4, 2, -9, 6, 4, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 1, 20, -16, -7, 10, -5, 5], [0, 1, 0, 0, 0, 0, 0, 0, 0, 5, -9, -3, 0, -9, 0, -11], [3, 0, -6, 0, -6, 0, -6, 30, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 1, -16, 8, -4, 19, 7, -4], [0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 8, -12, -17, 23, -3, 6], [0, 2, 0, 0, 0, 0, 0, 0, 0, -4, 14, 7, -13, -21, 1, -1], [-6, 0, -3, 3, 6, 3, 10, -8, -11, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, -12, -8, 8, -2], [0, 16, 0, 0, 0, 0, 0, 0, 0, -1, 8, -5, -12, 28, -2, 8], [1, 0, 9, 0, 2, -13, -2, 6, 11, 0, 0, 0, 0, 0, 0, 0], [7, 0, -14, 1, 6, 6, -15, -10, 10, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, -4, 28, -29, 4, -16, -9, 4], [0, -3, 0, 0, 0, 0, 0, 0, 0, -8, 3, 17, 8, -14, 4, 3], [9, 0, -5, -13, 10, -2, -5, -6, -13, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, 0, 0, 2, -3, 6, -1, -39, -2, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 10, -6, 5, -4, -13, -9, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -3, -5, 6, -20, 6, -12], [0, -7, 0, 0, 0, 0, 0, 0, 0, -1, -5, 18, 24, -7, 7, 3], [0, 6, 0, 0, 0, 0, 0, 0, 0, -5, 17, 6, 17, -20, -9, 6], [0, -3, 0, 0, 0, 0, 0, 0, 0, -1, 7, 8, 29, -2, 9, 0], [-30, 0, 14, 2, 8, 10, 0, 8, 12, 0, 0, 0, 0, 0, 0, 0], [17, 0, -13, 1, 4, 12, -5, -30, 3, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 16, -2, 19, 4, 7, -8, 12, 0, 0, 0, 0, 0, 0, 0], [-18, 0, -6, -1, 6, 7, -17, -30, 11, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, 2, -35, 7, -2, 26, -1, -1], [0, -1, 0, 0, 0, 0, 0, 0, 0, 8, 8, 13, -1, -19, 1, -8], [0, 3, 0, 0, 0, 0, 0, 0, 0, -2, 5, -8, 2, -22, -8, 11], [0, -2, 0, 0, 0, 0, 0, 0, 0, 3, 18, -18, -11, -14, 0, -5], [13, 0, -17, 3, -8, 4, -3, -2, -19, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, -1, 20, -15, 5, -4, -2, 1], [7, 0, 5, -8, -2, -8, 1, -4, 7, 0, 0, 0, 0, 0, 0, 0], [-43, 0, 17, -6, 5, -10, 0, 21, 13, 0, 0, 0, 0, 0, 0, 0], [10, 0, -16, 0, -9, 5, 8, 1, 12, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 15, -5, 9, -13, 7, 9, 7, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -14, -6, 12, -4, -5, -10, -17, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 6, -6, 12, -12, 2, 0, -9], [0, 5, 0, 0, 0, 0, 0, 0, 0, 12, 9, -5, 3, -19, 5, 4], [0, 19, 0, 0, 0, 0, 0, 0, 0, 17, 11, -8, -18, 9, 0, 3], [0, -7, 0, 0, 0, 0, 0, 0, 0, 6, 13, -8, -4, 16, -4, 7], [7, 0, 7, 2, 4, 1, 3, -10, -18, 0, 0, 0, 0, 0, 0, 0], [-9, 0, 9, 11, -5, 12, 12, 0, -5, 0, 0, 0, 0, 0, 0, 0], [45, 0, -29, 3, -14, 2, 4, 10, -8, 0, 0, 0, 0, 0, 0, 0], [-6, 0, -18, -3, -11, -10, -8, 13, -9, 0, 0, 0, 0, 0, 0, 0], [14, 0, -27, -1, 6, 3, -5, 2, 9, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, -4, -4, 7, -9, 12, 5, -8], [0, -1, 0, 0, 0, 0, 0, 0, 0, 7, 9, -19, -4, 7, -10, 5], [4, 0, 10, 5, 8, 1, -6, -24, 3, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, -15, 6, -16, 2, -5, -7, 6], [10, 0, -36, 2, 3, -1, -14, -21, 12, 0, 0, 0, 0, 0, 0, 0], [15, 0, -11, -1, -7, 1, -7, 4, 1, 0, 0, 0, 0, 0, 0, 0], [0, 18, 0, 0, 0, 0, 0, 0, 0, 2, -4, -12, 1, 2, 0, 6], [0, 2, 0, 0, 0, 0, 0, 0, 0, -6, 24, 6, 21, -17, 8, -5], [21, 0, -9, 1, -13, -3, -8, -11, -18, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, -14, 8, -2, 14, -8, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1682_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1682_2_b_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_1682_2_b_k(:prec:=16)
    chi := MakeCharacter_1682_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_1682_2_b_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_1682_2_b_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1682_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![961, 0, 10537, 0, 30128, 0, 33099, 0, 16415, 0, 4119, 0, 548, 0, 37, 0, 1]>,<5,R![-3600, -5400, -300, 1650, 265, -160, -30, 5, 1]>],Snew);
    return Vf;
end function;