
// Make newform 5780.2.a.q in Magma, downloaded from the LMFDB on 10 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5780_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5780_2_a_q();".
// 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_5780_2_a_q();".
// 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 := [34, 200, -588, -384, 1226, 152, -776, -16, 206, 0, -24, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10567812, -8406370, -50183456, 18566183, 32286419, -11574938, -7105653, 2960146, 631844, -329634, -19736, 13053], [-5397551, -7957212, 731206, 45171104, -2563205, -30302363, 1445810, 6896805, -271228, -631844, 16362, 19736], [24875675, 6356831, -101997298, -98175463, 113198195, 82213269, -42160734, -24896785, 5961993, 3134166, -271796, -135425], [-11744552, -52410078, 52936266, 129224583, -53599545, -93354948, 19037077, 26952296, -2605786, -3297561, 112715, 140626], [19717433, -17172418, -51042497, 111396373, 23320619, -94227260, -3640828, 28463791, 283535, -3565979, -18945, 155784], [-7548, -296336, 207128, 700436, -281262, -473264, 111029, 132402, -16049, -16054, 724, 684], [-44931, 144348, -83127, -469784, 138148, 385230, -57520, -121973, 8502, 16049, -362, -724], [-45985, -144348, 105856, 469784, -138148, -385230, 57520, 121973, -8502, -16049, 362, 724], [-44070528, -99507487, 133696747, 215666624, -104459311, -139125412, 33102697, 37600762, -4289273, -4443205, 180989, 186258], [6853805, -134990445, 35180080, 357327573, -58431782, -238047587, 22627434, 64266723, -3056274, -7526903, 120812, 313440]];
    Rf_basisdens := [1, 1, 5841353, 5841353, 5841353, 5841353, 5841353, 22729, 22729, 22729, 5841353, 5841353];
    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_5780_a();"
function MakeCharacter_5780_a()
    N := 5780;
    order := 1;
    char_gens := [2891, 1157, 581];
    v := [1, 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;

function MakeCharacter_5780_a_Hecke(Kf)
    return MakeCharacter_5780_a();
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, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0], [1, 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, -1, 0, -1, 0, 0, 0, 0, 0, -1], [0, 0, 1, 1, 0, -1, 0, -1, -1, 0, 1, 0], [0, 1, 1, 1, 0, -1, -1, -3, -2, 0, 1, 0], [-2, 0, 0, 1, -1, 2, 0, -1, -1, -1, 0, 0], [-3, -1, -2, -1, -1, 0, 1, 1, 0, -1, -1, -1], [-3, -1, -2, -2, 1, 1, 1, 2, 2, 0, -1, -1], [2, -1, 1, 0, 0, 0, -1, -1, 0, 2, 0, -1], [0, -1, -2, 0, -1, 2, 1, -2, -1, -2, 1, -1], [1, 0, 1, 1, 0, -2, 1, -1, 1, 0, 0, 0], [2, -2, -2, 2, 0, 0, -2, 0, 0, 0, 0, 0], [-4, -1, 0, 1, 0, -1, 1, 3, 0, 0, 1, 2], [1, -2, 0, 0, 2, 0, 1, 0, 0, -2, 0, 0], [-3, -2, -2, -3, 1, -2, 0, 4, 2, 0, 1, 1], [-4, -1, -1, -3, 2, -1, 2, 1, 1, 0, 1, 0], [0, 1, 0, 2, 1, -1, -1, 2, 1, -1, 1, 0], [1, -1, 3, -3, 0, 2, 1, 1, 2, 2, 0, 1], [-2, -1, 0, -2, 0, 3, -1, 3, 1, -2, -3, 0], [0, 1, 2, 3, 1, 0, -1, -5, -2, 0, -1, -1], [0, -2, -1, 0, -3, -1, -2, 0, -1, 1, 0, 1], [-2, 1, -1, 0, 0, -6, 1, 1, 2, 2, 0, -1], [2, -1, 0, -2, -2, 3, -1, -1, 1, 0, -2, -2], [-9, 1, 1, -1, -2, 1, 1, -2, 1, -1, 0, -1], [0, -3, 1, 1, -2, 1, -2, 1, -1, 2, 1, 0], [-2, -2, -5, -1, 0, 4, 2, 1, -1, -2, 0, -2], [0, 1, 3, 0, 1, -2, -1, -4, 1, 3, 1, 2], [3, 0, -5, -2, 0, 0, 2, 1, 1, 0, -2, 0], [-1, 2, -1, -4, 0, 4, 4, 2, 1, 3, 2, 1], [-6, 2, -5, -2, 1, -1, 2, 1, 2, 0, 2, 3], [-2, -1, 4, 1, 2, -1, 2, 2, 0, 0, 2, -1], [0, 1, -1, 3, -1, 0, -3, -3, 0, 0, -1, 1], [-1, -2, 2, 2, -2, 1, -3, 5, 1, 0, -4, -1], [-2, 0, 6, 1, 1, -2, -1, 1, 0, 0, -1, -1], [0, 1, 4, 2, 2, 3, 1, 1, 1, -3, 0, 1], [-6, 0, -1, -2, -1, 5, 6, -2, 2, 0, 2, 1], [-2, 1, 4, 2, 1, 3, 3, -1, -3, 1, 2, 2], [2, 1, 3, 1, 1, 4, 0, 0, -2, 0, 2, 0], [-10, -4, -4, -2, 0, 1, 1, 1, 1, 0, 0, 0], [-5, 0, -3, -1, -2, 5, 0, 5, -1, 0, 1, -1], [-4, 5, -2, -1, 0, 2, 1, -6, 2, 0, -2, -2], [-2, 2, 5, 3, -1, -5, -2, -5, -3, 3, 4, 2], [0, 0, 3, 3, -1, -6, -1, 1, 1, 4, -1, 0], [0, -2, -2, 0, 2, -3, 3, 1, 5, 1, 0, -1], [4, 4, 0, 5, 0, -2, 1, -8, -5, 0, 3, 1], [-1, -4, -1, 3, 0, 0, 1, -6, -2, -2, 3, -1], [-2, 1, 1, 4, 3, -4, -3, -4, 0, -1, -1, 1], [-4, -1, 3, 2, -3, -1, -1, -1, -3, -1, 2, 0], [-2, -2, -4, 1, 2, 2, -1, -7, -2, -3, 1, -1], [6, 5, 5, 0, -1, -4, 2, 1, 0, 0, 1, 1], [-4, 3, -4, -4, 0, 4, 0, 0, 1, 2, 2, 1], [-10, 0, -7, -5, -1, 4, 6, -4, 1, -1, -1, -2], [-4, 1, 0, -1, -2, 1, 2, -2, 0, -4, -2, -1], [-4, 1, 1, 2, 3, 4, 0, -1, -2, 1, 2, -2], [-2, 5, 8, -2, 2, 3, -1, 1, 0, -1, 1, 0], [3, -5, 1, -4, 1, 4, 4, 5, 0, 1, 0, -1], [5, 3, 3, 2, 1, -2, 1, 1, -2, 2, 5, 3], [11, -1, 4, 1, 1, -8, -5, 4, 1, -2, -3, -1], [6, -1, 3, 4, 1, -3, 1, -6, -3, -1, 3, 0], [-8, -1, 7, -2, 1, 0, -2, -1, 0, 3, -2, 2], [-1, -2, -2, 5, 3, -6, -5, 5, 0, -2, -1, 0], [-4, 0, -10, -4, -4, 8, 0, -3, 0, -2, -4, -2], [-11, 1, 4, -1, -3, -2, -1, -3, 0, -1, -1, -1], [2, -3, -9, -1, 0, 3, -2, -3, -1, -1, -3, -3], [-10, -1, -2, -2, 0, 1, 3, 1, 6, -5, -3, -4], [6, 3, -3, 5, -4, 4, -4, -7, -4, 0, -2, -1], [-6, 5, 5, -1, 1, -10, 0, -3, 0, 6, 2, 2], [-2, -2, -2, 1, 1, 8, -3, 1, -2, 2, 1, 1], [0, 8, 6, 1, 1, 6, 4, 0, -2, 0, 1, 2], [4, 0, -2, 5, 3, -10, -2, 5, 1, -1, 0, 2], [1, 2, 1, 4, 2, 0, -6, -3, -1, 0, -2, 2], [4, 1, 3, 8, 3, -2, 1, 1, -3, -2, 3, 1], [-9, 4, 8, -2, 0, 1, 3, -1, 3, 1, 2, 0], [-4, -5, -1, -2, 3, 4, 1, 7, 1, -1, -4, -2], [-6, 5, 4, 1, -4, -1, 0, -7, -6, 0, 0, -5], [-1, 5, 7, -2, -3, 3, -1, 5, 0, -2, -2, 3], [2, 2, 2, 4, 0, 2, -4, 1, -1, -1, 2, 0], [0, -3, -8, -3, -2, 2, 3, 4, 4, 0, -2, -2], [1, 5, -2, -5, 2, -4, -2, 4, 3, -2, -4, -3], [-1, -1, -3, 0, -1, 2, -7, 9, -2, 2, -2, 1], [-5, 4, 3, -4, -5, 6, 1, -3, -1, -2, -1, 0], [-3, 6, 9, 1, 5, 3, 0, -1, 0, -2, 2, 3], [5, 1, -7, 1, 2, 2, -1, -3, 4, 0, 0, 1], [6, 1, 5, 3, 1, -4, -6, -2, 0, 4, 0, -2], [-2, 0, -7, 1, -2, 10, -4, 1, -1, 0, -6, -4], [5, 2, -5, -2, 1, 0, 1, 9, -1, 2, 3, 2], [-1, 3, 3, -1, -2, 2, 1, 3, -3, 5, -1, 1], [1, 0, -8, -2, -4, -4, 0, -4, -2, 2, 2, 1], [4, 3, 10, 2, -2, 5, -1, -5, -2, -1, 1, 4], [-2, 6, -6, -3, -3, 4, 6, -13, -1, -1, 2, 0], [2, -3, 5, 2, 5, -11, -1, -2, -3, 0, 4, 1], [-8, 3, -1, -4, 1, 4, 1, 9, -2, -1, -1, 1], [5, -2, 3, 1, -1, -4, -2, 3, 4, 8, 0, -1], [5, 6, 9, 6, 6, -4, -2, 1, 1, 2, 2, 2], [-10, -2, -4, -6, 0, -1, 4, 2, 4, 4, 2, 0], [7, 4, 3, 1, 2, -4, -3, -2, 2, 1, 1, 4], [5, -1, -1, -3, 4, -6, 1, 5, 0, -4, 0, 1], [2, -2, -6, 1, 1, -4, -2, -8, 0, 0, -3, 2], [2, 4, 7, 3, -7, -1, -3, -5, -4, -3, -3, -2], [7, 4, 6, 4, 0, -2, -2, -8, -3, -5, 4, -3], [8, 2, 5, 3, -7, 8, 1, -15, -3, -2, 1, -2], [-2, -3, -5, -2, -3, 0, 0, 11, -2, -4, -1, -3], [3, 4, -4, -1, -2, -6, 0, 10, 0, 2, -2, -2], [-2, 5, 3, 3, -5, 2, -4, 4, 2, 6, 0, 4], [-8, -9, -6, -3, 4, -9, 1, 7, 8, 2, 1, 0], [-9, -4, -1, 4, -1, -8, -4, -3, 0, -2, 4, 2], [-9, -7, -6, -6, 5, -8, 0, 8, 5, 2, 1, 3], [-3, 3, 3, -3, 0, 3, 4, 3, 5, 7, 1, 6], [-4, -1, 8, -3, 0, 5, 5, 2, 2, -4, -2, -2], [4, 6, 0, -2, -4, 10, 6, -1, 2, 4, 0, 0], [8, -3, 7, 3, -4, 10, -1, 1, -6, -4, -1, -4], [6, -3, -8, -1, -1, 12, -1, 9, 2, -5, -9, 0], [10, 6, 10, 7, -1, -2, 0, -6, -8, 0, 7, 6], [5, 5, 12, 3, -1, 1, 4, -2, -3, 3, 5, -1], [-10, 5, -2, -1, -2, -3, 4, 0, 4, 0, -2, -1], [-4, -6, -15, -6, 1, -1, -2, 3, 4, 2, -2, 3], [-5, -3, -3, 2, -5, -4, 0, 3, -2, 0, -4, -2], [-1, 7, 11, -3, 2, -4, 5, -6, 2, 1, 5, 2], [-1, -4, -5, -6, -1, -5, 3, 10, 5, 2, 2, -4], [16, -4, -2, 2, 0, 0, -4, -3, 0, -2, 3, 0], [-6, -1, -5, -6, 1, -4, 1, 0, 7, -2, -1, -3], [-9, 1, -3, -5, -2, 0, -3, 4, -5, 1, -2, -3], [-4, 2, -1, -6, 0, -5, 4, -3, 0, -2, -1, 4], [5, -1, 2, 1, 5, 6, 7, 6, 3, -6, 5, 3], [-1, 0, 1, 5, -3, 6, -4, 7, -3, 6, 1, 4], [0, -2, -7, 1, 3, 3, 3, -9, 0, -1, 1, 0], [3, -4, -9, -2, 3, 6, 0, 1, 0, -4, -6, -6], [-9, 2, 8, -4, 6, 1, 2, -2, -1, -3, 2, 1], [12, -2, 4, 5, 3, -8, 0, 6, -4, 2, 7, 8], [-2, 3, 5, 3, -5, -4, 0, -7, -5, -1, -2, 4], [2, 0, -7, -2, 5, -15, 0, 13, 5, 5, 2, 5], [8, 0, 2, -1, 5, -6, 8, 0, 2, 0, -1, -4], [-6, 1, 3, -4, 1, 9, 1, 12, -3, -2, 2, 3], [0, -5, -3, -3, -4, 6, 4, -9, -4, 0, 0, -1], [8, 4, 10, 7, -1, -14, -10, -6, 1, 1, -3, 0], [4, 1, 4, 2, -5, 2, -5, -8, -11, 1, 1, -2], [-2, -1, -3, 0, 1, -15, -7, -1, 1, 3, 0, -2], [0, 1, -4, -8, 2, -3, 1, 15, 5, 8, 0, 4], [9, -7, -8, 2, 2, -6, 4, -8, -6, -7, 2, -2], [-16, -1, 4, 1, 2, 5, -2, -1, 4, -4, -2, 1], [13, -2, -11, 9, 1, -1, -8, 6, 0, -2, -3, 5], [-14, 1, -6, -5, -3, 9, 6, 0, 1, -2, 1, 1], [5, 1, -17, -5, 0, -7, -2, 3, 1, 3, 1, 6], [4, 1, -7, -1, 3, -12, 0, -8, 6, 6, -4, -2], [-13, 0, 3, -2, 2, -2, 0, 13, 1, -2, 0, -2], [-7, -1, -13, -3, -2, 22, 5, 0, 0, -7, 1, -2], [13, -8, -4, 6, 2, -1, 2, -3, 1, -1, -1, -5], [-5, 5, 13, 3, -1, 2, -2, 7, 0, 2, 1, 1], [-6, 7, -2, -2, -6, -3, 3, -3, 1, 2, -2, -4], [10, 2, 8, 7, 1, -9, -4, -14, 1, 4, -1, -1], [9, 1, -4, 8, -5, -5, -7, -13, 2, -4, -4, 1], [0, 10, 14, 1, 0, -8, -3, -12, -2, 7, 3, 3], [-7, 3, 3, 2, 1, 10, 7, -6, -2, -4, 3, 5], [9, -7, -9, -4, 6, 10, 6, 13, 8, 0, 0, 1], [-2, -2, 2, 7, 1, -6, 8, 4, -5, -5, 7, 2], [5, -2, 8, 0, 2, 4, -4, 8, -4, -4, 2, 1], [11, 8, -1, 2, -2, 14, 2, -5, -3, 2, 0, 0], [-6, 8, 0, -3, 0, -4, 5, -6, -2, 2, 4, 6], [0, 1, -13, 3, -3, -4, 2, 11, 0, -2, 4, 0], [-16, -4, 4, -6, 1, -4, -1, 12, 4, -4, -7, -2], [17, -5, -13, 4, -1, -13, -3, 6, 1, 1, 4, -3], [-8, -6, -9, -6, -4, 1, -8, 7, 6, 4, -7, -4], [4, 9, -4, 7, 1, -3, -4, -14, 1, 0, -3, 3], [-7, -1, -13, -6, 1, 13, 5, -2, -2, -10, -1, -5], [2, 3, -4, 0, 2, -9, 7, -5, 3, -5, -2, 3], [-8, 0, 1, 4, 1, 5, 6, 4, 0, 2, 6, 1], [-6, -9, 4, 8, -4, 5, -3, -6, -5, -6, -9, -2], [7, 6, 3, 5, -5, -8, -6, -2, -6, 2, -1, -5], [-4, 1, 15, -6, 3, -2, 6, -7, 0, 0, 1, -1], [-14, -9, -6, -3, 2, 15, -6, 3, 4, -2, -4, -7], [10, -2, 4, 9, 1, 0, -8, 4, -5, -3, 1, 2], [15, 0, -6, 0, -2, -5, 0, 6, -3, 3, 6, -1], [2, -6, -11, -2, -4, 2, 0, -2, -4, -4, -2, 0], [-11, -1, -15, 1, -8, -14, 3, -9, -2, -2, 2, 1], [-4, -7, -16, 3, -4, -3, -4, 2, -1, 3, 2, 5], [16, 6, 16, 7, 2, -4, -3, -18, -2, -2, -4, -2], [1, -8, -7, -5, -5, 13, 2, 0, -2, -8, 1, -1], [-15, 6, 10, 2, 0, -10, 0, 8, -6, -1, 2, 4], [18, -1, 6, -1, 2, 4, -7, 18, 1, 4, -8, -3], [4, 4, 9, 3, -2, -12, 2, -3, -5, 4, 2, 6], [0, 1, 3, 3, -3, 0, -6, 4, -7, -5, -6, -4], [-11, 5, 9, -5, 2, 8, -1, 12, 0, 3, -1, 0], [-12, -2, 13, 11, 3, -16, -5, -1, 1, 2, 5, 2], [-3, 3, 6, 3, -2, 0, -9, -2, 1, 7, -8, -5], [21, 3, 6, 0, 1, 2, 5, 1, -2, 6, -2, -1], [2, 9, 16, 8, -2, 0, 2, -6, -14, -1, 10, 5], [19, -5, 2, -3, 1, -4, -7, 5, 6, -4, -5, -2], [10, 3, -4, 8, 2, 11, -1, -5, -14, -7, -1, -2], [-12, -4, 7, 7, 3, -3, 7, 7, 3, 0, -3, 0], [2, 2, 14, 2, -6, -6, 2, 4, -2, -1, -2, 1], [4, 2, 2, -5, 3, 2, 3, -7, 8, 5, 3, 0], [-4, -5, 0, -4, 3, 4, 3, -3, 9, 3, -2, 0], [-7, 8, -6, 5, -3, -3, -6, -6, -3, 4, 3, 10], [6, 6, 6, 4, 4, 13, -3, -7, -11, -2, 8, 8], [-10, 3, 1, -6, -3, 2, 0, 1, -4, -2, -5, 1], [4, 2, 10, 5, 7, -10, -4, -8, -7, 1, 3, 8], [-12, 1, -2, -10, 1, 3, 1, 18, 9, -3, -9, -4], [1, 1, 8, -6, -3, 10, 4, 6, 3, -4, -9, -5], [-18, 2, -1, -2, 3, -19, 0, 4, 9, 3, 6, 1], [-10, 2, 6, -3, 8, 4, 3, 12, 4, 5, -1, -5], [10, -3, 0, -2, -5, 8, 3, 2, -5, 4, 1, -3], [8, 5, -5, 0, 5, 12, 5, 10, -2, -5, 1, -3], [8, 1, -1, 6, 7, 0, -2, -11, 8, 4, 1, 1], [8, 8, -2, 2, -6, 17, -4, -22, -8, -2, 0, 2], [4, -3, 8, -10, -2, 15, -1, 0, 4, 1, -7, -4], [-23, -4, -9, -1, -2, 10, 1, 18, 6, 0, -1, 1], [-20, -10, -8, -10, 0, -4, 6, 11, 7, -5, -2, -4], [3, 5, 12, 5, -9, 3, -10, -18, -5, 0, -3, 2], [1, 2, -9, -6, 5, -6, -3, 5, 7, 0, 1, -6], [8, -1, 4, 3, 4, -18, -1, -4, 6, 6, -6, -2], [-9, -5, -10, 3, 2, -4, 1, 8, 5, 3, 4, -5], [5, -9, -2, -5, -3, -4, 1, 6, -3, -6, -5, -5], [-6, 1, 0, 3, 6, 3, 0, -2, -6, 2, 8, -3], [9, -2, -11, 2, -1, -7, 7, 8, 3, -4, 0, -2], [4, 2, -5, 1, 1, -16, 7, 1, 3, 2, 7, 6], [-20, 6, -5, -5, 3, -9, 5, 5, 2, -1, -3, 6], [0, -3, 6, 7, -4, -5, -9, -6, -6, 0, 0, -2], [-18, -8, -16, -13, 0, 10, 1, 11, 6, -9, -13, -3], [-1, 9, 5, 7, -6, 15, 0, -9, -5, -2, 7, 7], [20, 7, 3, 10, -2, -2, 1, -23, -10, 4, 4, 9], [-20, -4, 5, -5, 3, -5, 11, 7, 5, -4, 5, 8], [1, -5, -10, -4, 0, 1, 7, 5, 11, 4, -4, -5], [5, -9, 0, -4, 9, 5, 7, 5, 8, 4, -2, -1], [16, 5, 4, -2, -1, 16, -5, 4, -3, -4, 1, 3], [12, 5, 1, -1, 10, -1, -1, 6, -4, -2, -2, -2], [9, 3, 12, 8, -1, -4, -2, 2, -3, 4, 3, 13], [6, -3, -12, -5, 0, 8, 5, 10, 4, -6, 2, -8], [4, -1, -12, -2, 0, -13, 3, -8, -2, -5, 5, 2], [-14, -7, -13, 0, -3, 8, -5, -5, -3, 0, -7, -3], [-5, 6, -6, -4, 0, 6, 1, -8, 2, 6, 0, 0], [-4, 3, 11, -2, 0, 14, 9, 3, -2, 0, 2, -1], [-1, -5, -4, -1, -4, -8, -3, -8, 1, -5, -2, -3], [5, -1, -9, -13, 4, -4, -3, 10, 6, 7, -5, -6], [8, -6, -3, 3, 1, 13, -2, -7, -7, -9, 8, 2], [4, 9, 16, 17, -6, -15, -14, -16, -12, 0, 2, 1], [11, 8, 19, 5, -6, 8, -3, -28, -10, -1, 1, -2], [10, 10, 8, -3, 2, -14, 7, 0, 0, 5, 11, 9], [-21, 4, 1, -6, 0, 22, 8, 3, 1, -6, -4, 4], [-8, 6, -4, -1, 2, 10, 11, -3, 5, 8, 3, 9], [-3, -4, -13, -6, 1, 12, 12, 11, 0, -5, 6, 3], [-1, -9, -6, -5, -2, -4, 1, 5, 3, 1, 3, 1], [-5, -4, -7, 4, 3, -4, 8, -11, -4, 2, 4, 2], [13, 2, 1, 0, -2, -9, 4, 0, 1, 3, 12, 7], [-16, 11, -18, -4, -2, -1, 5, 7, 6, 3, 5, 4], [-5, 10, 12, 2, -2, -4, -7, -18, -2, 0, -2, -4], [3, 1, 5, 0, 6, -2, -6, -7, -2, -2, 2, 5], [4, -12, -14, -2, -2, 1, -2, 6, -2, 0, 2, -8], [-19, 7, -7, -5, 2, 6, 5, -1, -4, -8, 6, 11], [-11, 2, 6, -13, 1, 2, 6, 1, 8, 10, 0, -1], [16, -10, 12, 3, -1, 2, -6, 6, -6, -2, 9, 4], [0, -6, 8, 0, 2, -6, 0, 16, 6, 0, -6, -2], [-9, 0, 6, -2, -2, 2, -2, -3, 5, -3, -9, 5], [-13, -3, 4, -6, -8, 9, 7, -1, 3, -9, -4, -4], [20, -2, 3, 14, 1, 2, -12, -15, -8, -2, -4, -5], [5, 6, 8, -6, 4, 0, -7, 10, 2, -4, -8, 2], [-24, -11, -15, -10, 1, 8, 10, 23, 2, -4, 2, 3], [5, 6, 10, -4, 0, 22, 5, -2, -6, -8, -4, -2], [-5, 0, -3, 6, 4, -9, 4, 7, 1, -7, 3, 1], [5, 5, 5, 0, 1, -8, -16, 7, -2, 0, -10, 0], [15, 14, 7, 3, 0, 1, -12, -5, 3, 2, -13, -5], [-10, 8, 16, 2, 0, 14, 4, 7, 4, 2, 2, 2], [2, 7, 15, 0, -3, 8, -5, -30, -10, 1, 5, -1], [-12, -3, 0, -6, -6, -3, 7, -7, -8, 5, 9, 4], [-8, -3, -13, -7, 6, -11, -2, -7, 7, -3, -5, 1], [3, 4, 4, 7, -8, 4, -10, -12, -4, 4, -2, 2], [-5, 6, 10, 13, 5, -1, -2, -11, 2, 2, 6, 11], [13, -1, 6, 7, -5, -8, 1, -2, -5, -4, -1, 5], [-10, -4, -11, -16, 1, 4, 10, 11, 5, 4, 7, -4], [4, 6, 2, -9, 2, 4, 3, 7, 0, 5, -1, 7], [-4, -1, -6, 5, 3, -4, 2, 10, -7, -12, 3, 3], [1, 6, -2, 4, -2, 0, -5, -10, -2, -6, -6, 2], [12, 14, -9, -1, 2, -12, 2, 1, -5, -2, 2, 0], [-2, -2, 5, 12, -5, -3, -8, -17, -11, 1, 4, 7], [-6, 5, 1, -8, 1, 6, 5, 19, 13, 0, -7, 1], [-4, 10, 3, 1, 7, 5, 9, -2, 4, -5, -5, -2], [-15, -7, -18, -5, -1, -4, 3, 16, 9, 8, 1, -5], [-8, 3, -3, -2, 1, -17, 1, 8, 3, 11, 1, 2], [-9, 3, -22, -11, 1, -11, -4, 2, 3, 2, 1, -4], [-2, -7, -9, 5, -7, 2, -2, -5, 4, -4, -8, -2], [-4, -7, 4, -1, 3, -3, 2, 14, -3, -11, -1, -6], [-16, -1, -6, 8, -8, 11, 1, -4, -8, -13, 1, -4], [-19, 6, -2, -3, -3, 11, -6, 2, -1, 0, -9, -4], [18, -1, -4, 9, -1, -2, -11, -7, -12, -8, -1, -3], [-8, 3, 3, 4, 5, -18, -11, -3, -8, -3, -5, 3], [-6, 2, 19, 15, 1, -7, 3, -12, -5, 0, 13, 4], [0, -10, -4, -2, -2, 4, -6, 2, 0, 8, 0, 4], [28, 4, 5, 8, 7, -10, -6, 11, -8, -5, 2, -4], [9, 6, 2, -4, -6, -4, 5, 2, 6, 4, -6, 6], [-9, 8, -4, 0, 0, 3, -4, 4, -7, -9, -6, 5], [-19, 3, 3, -7, -2, -6, 1, -2, 2, -3, -5, -4], [-9, -14, 2, 6, -12, 7, -4, 1, -5, 5, 5, 5], [18, 10, 6, 1, 2, 16, 1, -15, 0, 3, -9, 1], [19, -19, -1, 5, -6, 2, -9, -3, -8, -4, -2, -5], [12, -8, 2, -4, 0, 4, 6, -11, 6, 2, -9, -2], [-8, -7, -13, -10, -3, 22, 10, 9, 0, -2, -1, 3], [28, 1, 4, 4, 3, -10, -5, 6, -3, 4, 3, 1], [-5, 1, 4, 5, 2, -2, -8, -10, -11, 0, 10, -1], [-12, 7, 11, 6, -1, -12, -1, -10, -7, 9, 7, -4], [2, 13, -3, -4, -5, -11, -7, -12, 3, 10, -6, -1], [3, 11, 7, 3, -2, -6, 3, 11, -3, 7, 1, 5], [-10, -6, -3, 2, 3, -19, -7, -4, 5, 0, 8, -1], [7, 5, 23, 5, -3, 8, -2, -11, -8, -6, -5, -11], [0, -5, -12, 5, -6, -2, -1, -7, 2, 8, -9, -10], [-18, -1, -18, 0, 4, -3, 3, 11, -2, -3, -3, 0], [-12, -11, 0, -5, -1, 14, 0, -4, 7, -4, -9, -1], [-10, 10, 1, -3, -2, -1, 2, 13, 5, 8, -11, 0], [16, -7, -8, -9, -2, 14, -3, 7, 8, 4, -7, 0], [0, -13, -12, 4, -1, 3, -11, -3, -5, -9, -8, -10], [-10, 2, 4, -5, 9, -2, 12, 12, 14, -2, 1, 0], [-7, 5, -8, 5, -3, 14, 9, -4, -1, -6, 1, 3], [-16, 1, -5, -17, 3, 14, 8, 6, -2, -6, 2, 2], [-20, 5, 1, 0, -9, -10, -1, -11, -4, 3, -1, 3], [-14, 3, 0, 8, 5, -6, -9, 6, 3, 11, -1, 8], [-5, -2, 23, 10, 1, -14, 0, 11, -2, 3, 0, -1], [-14, -4, -1, 1, -1, -5, 1, 5, 4, 5, -1, 10], [-9, -2, -11, -1, -5, 7, 6, -26, -6, -4, -1, -3], [-31, -3, -14, -6, -4, 11, 7, -5, 3, 4, 0, 1], [-4, -9, -6, -10, -8, -1, 9, 1, -2, 5, 1, -8], [-10, 5, 6, -2, 0, -18, -4, -14, -3, -4, -4, 1], [-4, 6, -9, -1, 10, 1, 2, 25, 11, -3, 3, 5], [12, 6, 6, 2, -2, 0, -12, -14, 0, 2, -6, -10], [1, -6, 0, 0, 0, 7, -6, -4, -11, -11, -8, -9], [13, 7, 7, 0, 7, -1, 3, 14, 3, 7, 6, 7], [-12, -6, 8, -8, 2, 13, 1, -1, 7, -3, 2, -1], [14, -6, 18, 0, -4, 11, 3, 13, -9, 8, 0, 2], [-13, 7, 9, 6, -1, 12, 5, 0, -6, 2, -5, -7], [-22, -1, -14, -8, 6, 16, 2, 0, 10, -1, -8, -7], [-15, -13, 3, 6, 1, -8, 5, -13, -4, -6, 6, -3], [-4, -8, -15, -5, -5, 4, -7, 3, -1, 0, -7, -8], [-21, 6, 4, -3, -1, -3, -2, -4, 6, -2, -11, -1], [24, 11, 10, 9, 1, 9, 2, -6, -7, 3, 5, 8], [8, -9, -3, -8, 3, 6, -3, 2, 4, 7, -7, -7], [10, 10, -4, 7, -1, -19, -6, -2, -1, 7, -9, 6], [-6, -9, 6, -8, 5, -4, -3, 17, 15, 5, -8, 6], [21, -6, -2, -8, -5, -2, -2, -2, 2, 6, 3, -2], [2, -4, -1, 2, 0, -4, -9, 5, -3, 4, -8, 0], [-7, -6, -11, -13, -5, -4, 4, 11, 9, -4, 1, 0], [-9, -1, 8, 6, -5, 7, -3, -8, -2, -16, -7, 3], [-8, 9, 8, 1, -6, -19, 2, -17, -6, 2, 12, 5], [1, 4, 4, -2, 0, 14, 5, 0, -4, -10, 2, -6], [26, -7, -8, -9, 4, 8, -1, 22, 14, 6, -6, -4], [-19, -1, -3, -2, -1, 24, 3, -22, -6, -4, 5, 7], [23, 2, 4, 2, 3, -18, -4, -12, -6, 4, 1, 2], [7, 5, 17, 1, -4, -20, -11, -19, -5, 7, -5, 5], [21, 14, 3, 0, -5, 1, 3, -8, 1, 3, 6, -1], [24, -4, -12, -1, 4, -18, -5, 10, -4, 7, 1, 13], [-21, 5, 8, 9, -3, -25, -4, 10, 3, 9, 9, -3], [-2, 4, 12, 12, -8, -4, -14, 0, -6, 10, -2, -2], [1, -4, -4, 0, -6, -7, -11, -7, -1, 10, 0, -1], [0, -11, -12, -10, 6, 3, 3, 3, -1, 2, 7, -2], [16, -10, -16, -3, 5, 4, 2, 12, 4, 0, 3, -4], [-4, -6, 15, 4, 5, -17, 4, 3, -2, -2, 10, 9], [0, -2, 12, 15, -1, 1, 2, -11, 1, 5, 4, 0], [5, 4, -6, -1, 1, 16, 13, -11, -4, 0, 1, 10], [9, 4, -5, -6, 1, -4, 1, 15, 3, 14, 5, -2], [-2, 4, 7, 2, 7, -10, -6, -11, -4, -4, 8, 3], [-18, -7, 13, 7, 2, 3, 9, -5, 4, 2, 11, 0], [-6, -7, -4, 2, 0, -2, -4, -8, 1, 2, -2, -13], [-7, -3, 6, 1, 12, -9, -1, 11, -3, 3, 3, 7], [5, -8, 8, 6, 4, 6, -7, -2, -4, -12, -4, -12], [9, -12, -2, 15, -3, -4, -17, -6, -14, -6, 3, -4], [5, -8, 12, 1, 1, -4, -5, -25, -4, 0, 11, -6], [-15, 6, -9, -4, 5, 10, 5, 5, 5, 2, 5, 6], [-25, -15, -29, -7, 0, 1, -1, 20, 3, -3, -2, 1], [-8, -11, 1, 8, -7, 6, -13, -5, -8, 9, 1, -1], [-5, -7, 4, 10, 3, 3, -17, -29, -8, -2, 0, -1], [-26, 2, -1, -1, 6, -3, 6, 9, 9, 4, 1, -2], [4, -10, -13, -5, 3, 1, 7, 0, 13, -2, -1, -16], [22, 1, 3, -5, 1, -18, 4, 6, 6, 0, -2, -8], [-5, -2, -9, 7, 2, -7, -6, -7, 5, 8, -3, -5], [-7, -2, -4, 3, 5, -6, 8, -4, 4, -4, 1, -7], [1, 19, 9, 4, 5, 2, -3, 1, 8, 2, -6, 3], [-9, -10, -7, -10, 6, 8, 4, 5, 5, -6, -4, -6], [-14, -8, -34, -15, -2, 6, 7, 2, 10, -7, 1, 5], [11, -15, -15, 6, 3, 6, -5, -1, 0, 2, -7, -7], [17, 6, -3, -11, 5, -4, 8, -1, 6, 10, 2, -1], [-6, -12, 19, 6, -7, 22, 4, -3, -11, -14, -3, -8], [-9, 13, 2, -2, 0, -7, -7, -19, 1, 5, -4, -6], [-4, 13, 6, 1, -1, -1, 6, -20, -13, -6, 11, 1], [-5, -2, 6, -3, 9, -35, 0, 6, 2, 8, 15, 5], [-9, 1, 26, -7, 3, 1, -4, 2, 7, 10, 3, 2], [-2, -8, 18, 5, 2, 10, 5, 7, 0, -5, 3, 9], [-14, -18, -26, -17, -3, 10, 2, 12, 14, 0, -15, 2], [-1, -1, 19, 4, 1, 7, -11, -8, -9, -3, -2, -5], [-20, -2, 3, 7, 1, -6, -13, -1, -3, -6, 9, 6], [-11, 4, 10, -1, 7, 8, 2, 17, -2, -2, 0, -1], [1, 2, 20, 1, -1, -11, 0, 14, 5, 0, -11, -6], [-14, -8, -2, 0, 0, -8, -6, 2, 6, 0, 2, 10], [15, -7, -2, 3, -6, 6, -4, 0, 5, -2, -8, -1], [-6, -21, -12, -15, 4, 13, 12, 23, 8, -2, 4, -5], [-23, -2, -4, -7, -7, -7, -2, -6, -1, 8, 5, 4], [26, -4, 12, 4, 0, -18, 4, 2, -8, 0, 6, 2], [14, -17, 18, 12, 10, -11, -9, 3, 0, -3, 5, 2], [20, 8, -2, -4, -2, 3, -10, 3, -4, 4, -3, 6], [-22, -15, -2, -7, 0, 7, 4, 9, 21, 1, -2, 1], [-13, 6, 13, 13, -3, -11, -16, -19, 0, 8, -6, -1], [-9, 4, 6, -4, 4, -16, 1, -18, 4, 2, 2, 8], [-6, -17, -26, -10, -1, 0, -3, 20, 17, 4, -7, -5], [-14, 8, 1, -3, 2, -3, 4, -1, -5, 5, 1, -9], [-14, -6, -8, -3, 4, -4, 13, 8, 0, -5, -9, -9], [-21, -11, -11, -8, 3, 1, 1, -3, 8, 2, 0, -7], [4, 0, 9, 7, 5, -8, -9, 5, -9, 4, 5, -2], [6, -7, -15, -11, -1, 14, -6, 2, 3, 9, -12, -12], [-17, 1, 5, 5, -1, 14, 2, 5, -4, 6, -3, 5], [-26, 1, 7, 0, 7, 9, 9, -4, -5, -9, -3, -12], [3, -6, -13, 2, 3, -12, -4, 1, -2, 1, -10, -1], [12, 15, 12, -2, -5, -4, 9, -10, 5, 10, 7, -5], [-22, 1, 15, -4, 4, -2, 7, 3, -4, -4, 10, 3], [3, 6, 4, 0, 0, -14, -6, -8, -3, -3, -6, -5], [3, -3, 11, 6, -6, -28, -8, 5, 2, 12, -4, -1], [7, 1, 5, -3, 4, -22, -1, 2, 11, 9, 6, -1], [-20, -14, -10, -5, -7, -3, -8, 19, -3, -9, -12, -10], [8, 15, 6, 16, 0, 8, -10, -6, -4, 0, -4, 10], [4, 13, 0, -11, -11, 12, 3, -7, 0, 14, -3, 5], [-14, -3, -3, 6, -7, 0, 0, 17, -8, -6, -3, 5], [-2, -12, -7, -11, 5, -21, -7, 11, 12, -1, -1, -4], [-9, 6, 4, -9, -1, -14, 3, 5, 6, 4, 1, 6], [-5, -11, -17, -5, 6, -3, -10, 17, 11, 16, -5, -5], [-26, -4, 7, -4, 3, 25, -4, 11, 0, -4, -6, 1], [-15, 5, -2, 6, -2, 8, 0, -24, -2, 2, -12, -7], [-2, 3, -11, 1, -7, -4, -5, 15, -2, 6, -9, 1], [7, 1, -29, -5, -8, 31, -10, 11, 3, 2, -11, -5], [2, 9, -2, 1, 8, 3, 1, -1, -12, -6, 5, -2], [10, -6, 4, 8, -6, 8, -14, -5, -18, -8, 0, -4], [-12, 7, 5, 1, 0, 28, -1, 0, 2, -4, -10, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5780_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5780_2_a_q();".
// 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_5780_2_a_q(:prec:=12)
    chi := MakeCharacter_5780_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_5780_2_a_q();".
// 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_5780_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5780_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![34, 200, -588, -384, 1226, 152, -776, -16, 206, 0, -24, 0, 1]>,<7,R![28642, -59608, -1028, 53672, -6806, -19480, 1260, 3312, 40, -264, -20, 8, 1]>],Snew);
    return Vf;
end function;