
// Make newform 585.2.i.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_585_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_585_i_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_585_2_i_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_585_2_i_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 := [268, 770, 1103, 1215, 1118, 859, 551, 326, 231, -19, 178, -107, 96, -44, 20, -5, 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], [194918, -45495, 176473, -207308, -224641, -48005, -267712, 151587, -259593, 190360, -147579, 72320, -31264, 7576, -1135, -89], [6293, 3336, 4420, 3205, 2087, 1228, 677, 351, 120, 139, -39, 68, -28, 16, -4, 1], [125592, 436375, 567013, 509676, 354893, 312111, 42206, 152463, -18231, 23394, 24185, -18634, 14280, -4880, 1221, -53], [-1138394, -2196081, -2836237, -2561554, -1542137, -1628113, -12878, -1096911, 600027, -630706, 267189, -122780, 18478, -2404, -851, -265], [-568472, -2176341, -3007411, -3222844, -2132363, -2073673, -271472, -1222965, 363147, -417232, 20721, 9478, -42716, 11696, -2825, -547], [752246, 1281639, 1686079, 1546546, 955763, 1218265, -182266, 877113, -423375, 406510, -164457, 90950, -25438, 10492, -2275, 685], [188851, -28941, -274471, -330217, -501563, -213625, -315041, 10989, -222348, 137726, -136806, 57808, -22988, 1274, 910, -556], [1087838, 2521281, 3272689, 3032698, 2184713, 1766167, 197588, 1367433, -865473, 991078, -524601, 290714, -92848, 28834, -4885, 1099], [3587, 8634, 5971, -1982, 1643, -4508, -3007, 2448, -6243, 4663, -3357, 1253, -331, -53, 44, -14], [-138346, -216913, -151197, -86552, -79347, 43807, -29046, 21477, 5595, -3464, 2953, 3582, -3688, 2748, -931, 195], [-220840, -454875, -371333, -648728, -287764, -375791, -141799, -150696, 16041, -119705, 83322, -78373, 40454, -19022, 5045, -938], [-177730, -435703, -340359, -271589, -167943, -74402, 31020, -108444, 120006, -89195, 40360, -3732, -7450, 6369, -2212, 405], [-53165, 19619, 101233, 277010, 203561, 180167, 172772, -14142, 139773, -68467, 56326, -10177, -650, 5068, -1973, 523], [-512798, -1943709, -3279607, -3468346, -3449927, -1944757, -1812884, -383103, -750819, 63962, -188259, -46970, 32740, -42610, 14221, -4021], [-191036, -493101, 7385, 482192, 597901, 839147, 92026, 951039, -643377, 915602, -668505, 501802, -251234, 107414, -27797, 4721]];
    Rf_basisdens := [1, 205578, 2187, 68526, 205578, 205578, 205578, 102789, 205578, 2187, 22842, 102789, 34263, 34263, 205578, 205578];
    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_585_i();"
function MakeCharacter_585_i()
    N := 585;
    order := 3;
    char_gens := [326, 352, 496];
    v := [2, 3, 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_585_i_Hecke();"
function MakeCharacter_585_i_Hecke(Kf)
    N := 585;
    order := 3;
    char_gens := [326, 352, 496];
    char_values := [[0, 0, 0, 1, 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 := 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 2, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1], [-2, 2, 0, -1, 0, 0, 0, -1, -1, -1, 0, 1, -1, 0, 1, -1], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, -1, 0, -1, 1, 0, -1, 0, -2, 0, 0, 0, 0, 2, -1], [-2, -1, 1, 0, 0, 1, 0, -2, 0, 0, 0, 0, -1, -1, 1, 1], [0, 1, 1, 0, 0, 1, 0, 1, -1, 1, 1, -2, -1, 1, -1, 0], [-2, -1, 0, -2, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, -2, 0, -1, -1, 0, -2, 0, 2, -2, -2, 3, 1, 0, 1, -1], [0, 1, -1, 0, 2, -1, 0, 1, 0, -1, 1, 0, -1, 0, -1, 0], [0, 2, 2, -2, -1, 0, -1, 1, -1, 1, 1, 0, -2, 3, -1, 0], [1, -1, 0, -2, 3, -1, 0, 0, -1, 3, 0, -3, 1, -1, 0, 0], [3, 0, 0, 3, -2, 2, -2, 2, 2, 0, 0, 0, -2, 2, -2, 2], [-1, 2, 0, 0, 3, -2, 0, 3, 0, -1, 2, 0, -1, 1, -1, -2], [0, -2, -2, 5, -1, 2, 1, 1, 1, -1, -1, 2, 0, 1, 1, 0], [-3, 2, 0, 2, -2, 1, 0, -4, 0, -5, 0, 5, -2, 1, 0, -4], [0, 2, 2, -6, 2, -5, 1, 2, -3, 3, 3, -4, -1, -1, -3, 0], [2, -1, 1, 0, 2, 0, 0, 0, 0, 2, 3, 0, -2, 1, -2, 2], [2, -2, 0, 0, -1, 0, 0, 1, 0, 2, 2, 0, 1, 3, -1, 0], [0, -3, 0, 0, 1, -1, 6, 2, -1, 0, 0, 0, 1, -1, -2, 2], [0, 2, 0, 1, -2, 0, -1, -1, -2, -1, 0, 1, -2, 0, 0, -1], [0, 1, -3, 0, -1, -2, 0, 2, 0, 0, -2, 0, 3, 1, 1, -3], [3, 0, 0, 5, -1, 2, -3, -3, 5, -2, 0, 2, 1, 2, 1, -3], [-3, 1, 0, -6, 3, 1, 2, 0, 2, 3, 0, -3, 0, 1, 1, 0], [0, 0, -3, -5, -1, 1, 0, -4, 3, -3, -3, -1, 2, -1, 2, -1], [2, -2, -1, 0, 0, -3, 0, -3, 0, -1, -4, 0, 3, -1, 1, 2], [0, 0, -4, 0, 1, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, -2, 0, 1, 0, 0, 0, 1, -1, 2, -1, -1], [-4, 2, 0, 0, -1, 0, 0, 3, 0, -2, 0, 0, 1, 1, -2, -1], [0, 0, -4, -3, -2, -1, 0, -5, 4, -4, -4, -3, 2, 0, 3, -1], [2, 0, 0, 1, 2, -2, 2, -1, -4, 1, 0, -1, 0, -2, 0, -1], [0, -2, 3, -5, 2, 0, -2, 1, -1, 1, 1, 0, 1, -3, 0, 1], [0, 1, 6, -1, 3, -4, -2, 3, -4, 4, 4, 0, -1, -2, -3, 1], [6, 2, 0, 8, 1, 1, -4, -4, 2, -2, 0, 2, 0, 1, 2, -4], [0, 0, 1, -7, 3, -3, -2, -7, 1, -1, -1, 1, 4, -7, 3, 2], [-5, -1, 4, 0, 1, 1, 0, -4, 0, 0, -1, 0, -2, -3, 1, 3], [0, -4, 3, 3, 0, 1, -6, -4, 3, -3, -3, 3, 3, -3, 5, 2], [4, -2, 0, 9, 2, 2, -1, 2, 3, -5, 0, 5, -1, 2, -3, 2], [5, -2, -6, 0, -4, 2, 0, -1, 0, -4, -1, 0, 2, 1, 0, 1], [-8, 3, 4, 0, 3, 1, 0, 6, 0, 2, 7, 0, -4, 3, -7, 1], [-4, -6, 0, -2, -1, -3, -2, -2, 1, -2, 0, 2, 7, -3, 1, -2], [0, 5, 3, -5, 2, -5, 3, 4, -6, 6, 6, -3, -4, 2, -5, 1], [-3, 0, 1, 0, -1, 1, 0, 0, 0, -2, 0, 0, 0, 0, -1, 1], [2, -1, -1, 0, 1, -4, 0, -4, 0, 0, -6, 0, 3, -3, 0, 4], [0, 3, 2, -3, 2, 0, 4, 9, -6, 6, 6, -2, -4, 2, -8, -2], [5, -4, 0, 8, -7, 0, -5, 1, 3, -3, 0, 3, 3, 0, -1, 1], [4, -5, 0, 7, 4, 0, 4, 5, 1, -3, 0, 3, 1, 0, -4, 5], [0, 2, -1, 0, 0, 0, 4, 4, -3, 3, 3, -2, -3, 3, -3, 0], [-1, -1, 5, 0, 0, 0, 0, -2, 0, 0, -1, 0, 0, -1, 5, -3], [0, -9, -5, 3, 2, 1, 5, 4, 0, 0, 0, -4, 2, -4, 3, 3], [4, -1, 0, -3, 6, -5, 9, 2, -4, 7, 0, -7, 6, -5, 5, 2], [1, 1, 2, 0, 2, -3, 0, -6, 0, -3, -8, 0, 1, -7, 6, 0], [0, -1, 4, 1, 2, -2, -2, 3, -2, 2, 2, -2, 0, -2, -2, 0], [-5, -9, 0, -5, 0, -2, -1, 2, 0, 0, 0, 0, 4, -2, -5, 2], [-11, 1, 2, 0, -2, 5, 0, -3, 0, 1, -1, 0, -3, -4, 4, -1], [-8, 1, -1, 0, 1, -1, 0, -2, 0, -2, -3, 0, 0, -3, 7, -5], [-5, 4, 0, -10, 4, 0, 8, -2, 1, 5, 0, -5, 1, 0, 5, -2], [2, 0, 0, -5, 0, -3, 1, 4, -4, 7, 0, -7, 2, -3, 2, 4], [0, -3, -7, 4, 1, 0, 4, -1, 3, -3, -3, 3, 4, -5, 0, -3], [0, 5, -3, -9, -1, 4, 3, 3, 0, 0, 0, 0, -1, 2, -7, -7], [1, 0, -1, 0, -3, 0, 0, 3, 0, 5, 0, 0, 3, 3, -3, 0], [0, 4, 2, 9, 1, -4, 0, 3, -2, 2, 2, 5, -1, 0, -6, -4], [4, 2, 0, 3, -3, 4, -7, 1, 5, 1, 0, -1, -3, 4, -1, 1], [-11, 0, 0, -6, -1, 3, -3, 1, 0, -5, 0, 5, -6, 3, -6, 1], [4, 4, 0, 4, -1, 6, 4, -5, 4, 0, 0, 0, -8, 6, -4, -5], [0, 4, 7, 16, -1, 0, -6, 0, -1, 1, 1, 3, -2, 3, -2, -1], [0, -4, -2, -3, 0, 5, 0, -4, 2, -2, -2, -1, 2, -2, 6, 4], [4, 2, 0, 0, 0, 0, -1, -2, -1, 4, 0, -4, -1, 0, 1, -2], [-5, 11, 0, -6, 4, 2, 3, -7, -6, 1, 0, -1, -10, 2, 1, -7], [17, 4, 2, 0, 0, -5, 0, 8, 0, 7, -1, 0, 5, 4, -7, -1], [10, 6, 0, 5, 2, -3, 2, -6, -6, 5, 0, -5, 0, -3, 6, -6], [0, 0, 2, 3, 2, 3, 4, 7, -6, 6, 6, -1, -4, 2, -3, 3], [5, -1, 0, 0, 5, -7, 0, 3, 0, 4, 2, 0, 2, 4, 0, -3], [0, -1, -3, 8, -5, 7, 2, 8, 1, -1, -1, 3, -4, 9, -3, -4], [8, 2, 0, 5, -5, 1, -5, 3, 0, 3, 0, -3, -2, 1, 0, 3], [-12, 2, -4, 0, 4, -2, 0, 3, 0, -2, 3, 0, -2, 1, -5, 2], [0, -8, -3, -2, -2, -1, -1, 3, 4, -4, -4, 8, 2, 0, 3, -1], [0, -5, -2, -12, 1, -4, 4, 4, -2, 2, 2, -4, -1, 0, 2, 4], [0, -3, 6, 10, 2, -9, -3, 4, -3, 3, 3, 3, -1, -1, 0, 3], [6, 10, 0, 9, 1, 0, -6, -10, -2, -3, 0, 3, -2, 0, 8, -10], [2, -2, 13, 0, 5, -5, 0, 0, 0, 4, 2, 0, 0, 2, 4, -4], [10, -2, 1, 0, 1, 0, 0, -5, 0, -3, -2, 0, -1, -3, -5, 10], [11, -1, 0, 11, 3, -2, -4, 0, -5, 0, 0, 0, -1, -2, -2, 0], [9, 7, 0, 8, 3, 2, -1, -3, -2, 1, 0, -1, -6, 2, 1, -3], [10, -2, 1, 0, -2, -1, 0, -3, 0, 6, -4, 0, 3, -1, 11, -8], [-1, 3, 0, 4, 2, 2, -1, -5, 5, -5, 0, 5, 1, 2, 4, -5], [4, -4, 0, -4, -5, -4, 1, 5, -3, 8, 0, -8, 5, -4, 1, 5], [0, -4, -4, 0, -2, 8, 2, -1, 2, -2, -2, 3, 0, 2, 5, 3], [-1, -3, 9, 0, -2, 0, 0, -2, 0, 2, -1, 0, 2, 1, -3, 5], [-7, 0, 0, 2, -1, 3, 4, -8, 6, -9, 0, 9, 0, 3, 0, -8], [7, 2, -8, 0, 9, -13, 0, 0, 0, -1, -6, 0, 4, -2, 7, -7], [0, 3, -3, 5, -2, 5, 2, 0, 1, -1, -1, 3, -1, 3, -2, -3], [0, 1, -10, -1, -3, 3, 9, 3, 1, -1, -1, -5, -2, 5, -2, -3], [-8, 2, 0, 0, 0, 3, 0, 7, 0, 8, 8, 0, -3, 5, -11, 4], [0, 4, 0, -8, 4, 1, 4, -1, -4, 4, 4, -8, 0, -4, -3, 1], [4, 4, -2, 0, -2, 1, 0, 11, 0, 2, 6, 0, 1, 7, -7, -4], [5, -2, -6, 0, -7, 4, 0, -4, 0, -1, -5, 0, 3, -2, 11, -7], [-7, 2, 1, 0, -5, 5, 0, -1, 0, 1, -3, 0, 0, -3, -2, 3], [-3, 8, 0, -5, 2, 1, 4, -1, -4, 2, 0, -2, -6, 1, 2, -1], [-5, 0, 1, 0, 0, -2, 0, -4, 0, -8, -6, 0, 2, -4, -2, 6], [0, 2, -5, 10, 0, 4, -1, -12, 6, -6, -6, 5, 6, -6, 4, -2], [1, -12, 0, 5, 0, -4, -6, 1, 3, -4, 0, 4, 11, -4, -1, 1], [0, -2, -2, -5, -5, 12, -3, -6, 5, -5, -5, 1, 0, 5, 8, 3], [-2, 4, -3, 0, 0, 6, 0, 5, 0, 6, 7, 0, -6, 1, -4, -1], [12, -3, 0, 9, 1, 1, -7, 3, 6, 3, 0, -3, 4, 1, 1, 3], [-16, -1, 8, 0, 2, 2, 0, 0, 0, -4, 5, 0, -4, 1, 3, -3], [0, -1, -11, -1, -5, -2, 4, -7, 7, -7, -7, 8, 2, 3, 6, -1], [-18, 14, 0, -13, -7, 6, 2, -8, 3, -5, 0, 5, -9, 6, 5, -8], [0, -2, -3, -4, 6, 4, 4, -1, -1, 1, 1, -3, 5, -11, -2, -1], [-3, 0, 5, 0, -7, 4, 0, 1, 0, 4, -2, 0, 3, 1, -3, 2], [0, 3, -1, -1, 1, -6, -2, -5, 3, -3, -3, 1, 4, -5, -2, -5], [19, -16, 0, 17, 3, -4, 1, -1, 8, 2, 0, -2, 16, -4, 0, -1], [-15, -2, -5, 0, -3, 3, 0, 0, 0, 6, 2, 0, 0, 2, 9, -9], [-10, 10, 0, -5, -1, 2, -4, -7, -5, -5, 0, 5, -9, 2, 1, -7], [0, 7, 9, -1, 6, -2, 4, 11, -13, 13, 13, -1, -7, 1, -11, 2], [-14, -1, 6, 0, 2, 0, 0, -2, 0, 3, 1, 0, -2, -1, 1, 1], [0, 0, 2, -12, 4, -4, 3, 2, -5, 5, 5, -8, -1, -3, -1, 4], [4, -3, 0, -6, -3, -6, 2, -1, 0, 10, 0, -10, 12, -6, 9, -1], [0, -3, -16, -3, -3, 7, 5, -13, 11, -11, -11, 4, 8, -5, 8, -3], [4, 0, 0, 3, 4, -2, 5, 6, -7, 1, 0, -1, -3, -2, -3, 6], [-1, -1, 0, 3, -2, -2, -15, -4, 4, -4, 0, 4, 8, -2, 7, -4], [3, -3, -11, 0, -5, 4, 0, -4, 0, -16, -2, 0, 1, -1, 1, 3], [23, -21, 0, 19, 0, -7, 3, 11, 2, 4, 0, -4, 16, -7, -5, 11], [2, 7, 1, 0, -2, 5, 0, 7, 0, -1, 3, 0, -3, 0, -2, -5], [1, 3, 0, 10, -13, 1, -6, 3, -4, -9, 0, 9, -6, 1, -3, 3], [18, 1, -13, 0, -4, -3, 0, -4, 0, -1, -12, 0, 7, -5, -4, 8], [20, -5, 0, 23, -5, -4, 2, 3, -6, -3, 0, 3, 2, -4, -3, 3], [0, 1, -5, -3, -2, -3, -2, -8, 7, -7, -7, 10, 5, -3, 2, -5], [-3, 0, 12, 0, 15, -10, 0, -1, 0, 3, 4, 0, -5, -1, 3, -2], [0, -14, 16, 17, 8, -10, -5, 20, -11, 11, 11, -5, -3, -5, -3, 8], [0, 16, 1, -8, 0, 1, 8, 8, -9, 9, 9, 2, -9, 9, -15, -6], [8, -1, 1, 0, -2, 3, 0, 3, 0, 7, 5, 0, -1, 4, 7, -10], [0, 19, 3, 8, -5, 6, 0, 5, -3, 3, 3, 5, -8, 13, -16, -13], [0, 3, -3, -10, 2, 3, 4, -5, -1, 1, 1, -2, 1, -3, 1, 2], [7, 4, 1, 0, 3, -9, 0, 1, 0, 0, -9, 0, 6, -3, 6, -7], [0, 8, -6, -22, -6, 6, 3, -7, 3, -3, -3, -3, -3, 9, 2, -1], [0, 1, -7, -15, -4, 3, 3, -8, 4, -4, -4, -8, 0, 4, 7, 3], [3, -5, 8, 0, 13, -7, 0, -1, 0, 2, 10, 0, -6, 4, 0, 1], [-7, 10, 14, 0, 2, -2, 0, 20, 0, 11, 10, 0, 0, 10, -12, -8], [-15, 0, 0, 2, -8, 4, -2, -7, 9, -17, 0, 17, 1, 4, 1, -7], [0, 7, 3, -3, 5, -1, 7, 6, -10, 10, 10, 1, -5, 0, -8, 2], [-6, -3, -9, 0, 3, -11, 0, -6, 0, -6, -11, 0, 8, -3, 12, -6], [-7, -4, 5, 0, 6, -11, 0, -2, 0, 7, -3, 0, 5, 2, 9, -7], [-5, 8, 0, -4, -6, 6, -2, 4, 4, -1, 0, 1, -8, 6, 0, 4], [2, -12, 0, 2, -9, -3, -8, 10, 1, 0, 0, 0, 7, -3, -5, 10], [-2, 7, 0, 0, 1, -1, -6, -8, 4, -2, 0, 2, 6, -1, 13, -8], [0, -1, 7, -4, 0, 2, -4, 4, -3, 3, 3, -3, -3, 3, 0, 3], [-9, -10, 4, 0, -6, 8, 0, -18, 0, -5, -6, 0, -2, -8, 11, 7], [0, -17, -3, 26, -4, 16, -3, 8, 6, -6, -6, -3, 2, 2, 7, 1], [-13, 0, 6, 0, 2, 3, 0, 6, 0, 11, 11, 0, -5, 6, -4, -2], [4, -9, 4, 0, 3, -6, 0, -7, 0, -3, -1, 0, 3, 2, 8, -1], [0, -10, 12, -5, 4, -1, -9, -2, -3, 3, 3, 2, 1, -5, 11, 14], [-30, 13, 0, -28, 2, 8, 4, -11, 7, -2, 0, 2, -9, 8, 4, -11], [-13, 9, 0, -19, -7, 0, 4, -2, 0, 6, 0, -6, 0, 0, 9, -2], [-14, -4, -6, 0, -10, 8, 0, -8, 0, -15, -6, 0, 2, -4, 15, -7], [-8, 4, 0, -10, -2, 1, 6, 1, -8, 2, 0, -2, -10, 1, -6, 1], [13, -4, -15, 0, -5, 2, 0, -8, 0, -8, -7, 0, 3, -4, -2, 10], [0, 6, -6, -2, -3, 6, 2, -17, 4, -4, -4, 9, 1, 2, 10, 6], [33, -19, 0, 23, 7, -8, 2, 13, -5, 10, 0, -10, 11, -8, -8, 13], [2, -5, 8, 0, -5, 4, 0, 4, 0, 6, 8, 0, 1, 9, -9, 5], [0, -2, 6, -9, 0, 2, -1, 11, -5, 5, 5, 4, -5, 5, -4, 1], [3, -4, 8, 0, 10, -11, 0, -4, 0, -3, -1, 0, 1, 0, 9, -5], [0, -4, -5, -4, 1, -2, 1, -7, 4, -4, -4, -11, 5, -6, 6, 2], [24, -3, 0, 21, 4, 5, -6, 7, 8, 3, 0, -3, -2, 5, -5, 7], [-9, 1, 0, 0, 5, 3, 0, 4, 0, 7, 11, 0, -8, 3, 6, -10], [11, -2, 0, 8, -3, 5, 8, 2, 2, 3, 0, -3, -8, 5, -10, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_585_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_585_2_i_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_585_2_i_e(:prec:=16)
    chi := MakeCharacter_585_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_585_2_i_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_585_2_i_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_585_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 19, 295, 1114, 3027, 4627, 5539, 3873, 2812, 1368, 875, 324, 158, 38, 17, 3, 1]>,<7,R![395641, -18870, 856340, -251056, 1614395, -359094, 544052, -191361, 140146, -44604, 20644, -6158, 2127, -476, 97, -11, 1]>],Snew);
    return Vf;
end function;