
// Make newform 6045.2.a.bh in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6045_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_6045_2_a_bh();".
// 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_6045_2_a_bh();".
// 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 := [8, 306, 760, -2198, -4238, 5469, 6915, -6304, -5201, 3854, 2056, -1319, -437, 252, 47, -25, -2, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-133114498, -324483746, 1669490717, 1089799149, -4695220286, -898215335, 4898277165, 78954606, -2421255639, 171782910, 620969061, -70627697, -83048126, 10382037, 5270067, -530768, -113557], [160614960, 1966684368, 511972718, -10936962026, -5497485944, 17031291127, 7965510540, -11993289690, -4694027971, 4400726869, 1349656637, -869010094, -196134497, 87408009, 13349902, -3509363, -312195], [64950898, 1899638378, 2699073926, -10518791456, -12298440348, 16241045978, 16639475285, -11241989735, -10164598845, 4005715774, 3189061955, -761838312, -532269412, 73542695, 44873866, -2833680, -1500797], [-18135454, -548614148, -1097688346, 3045081072, 5667993268, -4549058777, -8866269156, 2678609842, 6243488261, -657336163, -2247822761, 53714652, 429886899, 2341871, -41605574, -384325, 1604091], [154119101, 919914264, -1746027224, -5292910107, 4905261150, 8451012595, -5865410604, -6016496297, 3574397427, 2223167706, -1194843988, -442474917, 221201292, 44900786, -21236683, -1818804, 823629], [88911066, -1170041948, -1580860484, 4729675172, 6902239911, -6045124051, -10492827800, 3310199705, 7282350710, -820692828, -2598840645, 81684169, 494622034, -450507, -47761999, -267610, 1839987], [58213626, 286418488, -1568781388, -321519062, 7967091845, 850560935, -12239294860, -1437669135, 8505862278, 961805100, -3057841575, -284971261, 589221704, 38112665, -57777861, -1882134, 2263647], [37102250, 471493319, 702952785, -1815548007, -4635908462, 2220420630, 7401279035, -1015367914, -5167779456, 109116630, 1846126566, 36045591, -352121239, -9391330, 34139372, 593343, -1322580], [-30204858, -1255967342, -2435890434, 6958870154, 14149820915, -9857570968, -22073709939, 5644078754, 15091413240, -1412534597, -5254617039, 133423725, 973882157, 1155319, -91652439, -591283, 3448071], [117339748, 1167845648, -1864907048, -6141699664, 10351984080, 11237279022, -16642435131, -9680674593, 11973831065, 4327778728, -4451885089, -1014336732, 886622524, 117460815, -89703162, -5281842, 3616735], [10580640, 441811730, 1476654119, -2204078055, -7420642047, 2597281558, 11366061330, -936265285, -7894106494, -70556136, 2828341110, 101643546, -541921869, -18831079, 52769930, 1074872, -2051817], [1853237, 604301492, 2267966656, -2314760375, -11325212214, 2515907309, 16640955208, -780657571, -11069214955, -161031477, 3820473522, 126249210, -709445968, -21959060, 67288514, 1224081, -2558766], [-88198496, 602412121, 2845878583, -2753575910, -12321599225, 2491934088, 17850119443, -28301554, -12062740125, -740835504, 4272901584, 300324502, -816697554, -45045593, 79728849, 2351568, -3116457]];
    Rf_basisdens := [1, 1, 1, 1, 14023831, 28047662, 28047662, 28047662, 14023831, 28047662, 28047662, 14023831, 28047662, 28047662, 14023831, 14023831, 14023831];
    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_6045_a();"
function MakeCharacter_6045_a()
    N := 6045;
    order := 1;
    char_gens := [4031, 4837, 1861, 2731];
    v := [1, 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_6045_a_Hecke(Kf)
    return MakeCharacter_6045_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+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, 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, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1], [-1, 2, -2, 1, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, -1, 0, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0], [0, 2, 0, 0, 0, 0, 1, 0, -1, -1, -1, 1, 0, 0, 0, -1, -1], [2, -1, 1, 0, -1, 0, 0, 1, -1, -1, -1, 0, 1, 1, 0, 0, -1], [1, 2, -2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, -1, 1, -1, 2], [-1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 2, 0, -1, -1, 1, -2, 1], [1, 0, 0, 1, -1, 1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 1, 0], [0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -2, 2, 1, 1, 0, 1, -1], [3, 2, 0, 0, 0, -1, 1, 0, 0, -2, 0, -1, 1, 0, -1, -1, 0], [1, 2, -1, 0, 0, -1, 2, 0, 1, -2, -2, 0, 1, -1, -1, -1, -1], [2, -1, 1, 0, 1, 1, -1, 0, 0, 1, 2, 0, -1, 0, 0, 0, 0], [-2, 0, -1, 0, -1, 1, -1, 1, -1, 2, 1, 2, 0, -1, 0, 0, 2], [-1, -3, 1, 2, 0, -2, 0, 2, 1, 0, -1, -1, 0, 0, -1, 2, 0], [-1, -1, 1, -2, -1, 0, 0, 0, -2, 1, 0, 1, 0, 0, 0, 0, 0], [4, -1, 2, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 1, -2, 1, -1], [-1, -1, 2, 0, 1, -3, 1, 1, 1, 1, -3, 1, 0, 1, 0, 0, 0], [4, 0, 0, -1, 0, 0, 0, -1, -1, 1, -1, 0, 0, 1, -1, 0, 1], [3, 2, 0, -2, -1, 1, 1, -2, -1, 1, -2, 1, 1, 0, -1, 0, -1], [3, 0, 1, -2, 0, 2, 0, -2, -1, -1, 1, 1, 0, 0, 0, -1, -2], [-1, 0, 0, 1, 0, 0, 0, -1, 0, -1, 1, -1, -1, -1, -2, 1, 1], [1, -1, 1, 0, 1, 1, -2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0], [1, 1, 0, 2, 0, 1, -1, -1, 1, 1, 3, -3, 0, 0, 0, 2, 2], [-2, 2, 0, -1, 0, -1, -1, -1, 0, 1, 1, 0, -2, -2, -1, 0, 2], [2, -3, 1, -1, 0, -2, 1, 1, 0, -1, -2, 1, 0, 0, -1, 1, -2], [0, 4, -2, -3, 0, 2, 2, -1, 0, 1, 2, 2, 1, 0, 1, -4, 1], [0, 0, 0, 0, 0, 0, -2, 0, -1, 1, 2, -2, 0, 2, 1, 0, 2], [3, 1, 3, -1, 1, -1, 1, -1, 1, -2, -4, -1, 0, 1, -2, 1, -2], [5, -4, 0, 0, 1, 1, -1, 1, 1, 2, 1, -1, 0, -1, 4, 0, 0], [4, 1, -2, -2, 0, 3, -1, -2, 0, 1, 4, -2, -1, 0, 3, -1, 1], [4, 0, -1, 0, 1, -1, 0, 0, -1, -3, 0, -1, 0, 1, -2, -1, 0], [-1, 1, -1, 1, 0, 2, -2, 0, -2, 3, 4, 1, 0, 0, -1, 1, 0], [-3, 3, -3, 0, -1, 2, 0, 2, 0, -1, 4, 1, 0, -4, 0, -2, 2], [0, 1, 0, -3, 1, -2, 0, 0, 0, 1, -1, 0, 0, 1, 0, -2, 2], [5, -5, 1, 0, 1, 0, -2, 1, 0, 2, 2, 1, 0, 1, 1, 2, 0], [0, 1, 0, -1, 2, 1, -1, -1, 0, 0, 1, 0, -2, 0, 2, -1, 1], [3, 0, 1, -2, -1, 1, 1, -1, 0, -1, -2, 1, -1, 1, -1, 0, -2], [2, 1, 3, -1, 1, -1, 1, 0, -2, -1, -1, 2, 1, 0, 1, -1, -2], [3, 0, -1, 2, 0, -1, -1, 2, 1, -1, 0, -1, -1, -1, 1, 0, -2], [3, 3, -1, 1, 1, -1, 0, -1, 0, 1, 4, -2, -1, -2, -1, 1, 2], [-1, 1, -1, -4, 1, 1, 0, 0, -1, 2, 0, 4, 0, -1, 2, -1, 0], [-1, -3, -1, 1, 1, -1, -1, 1, 3, 0, -1, -1, -1, 1, 1, 2, -2], [0, -3, 0, 0, 1, -2, 1, 0, 2, 1, -4, -1, 0, 1, 1, 3, -2], [-1, -2, 3, 1, 0, -2, 0, 0, -1, 1, -1, -1, -1, 2, 1, 2, 0], [1, 4, -1, -3, 0, 1, 2, -2, 0, 1, 3, 2, -1, -1, 0, -3, 3], [0, 7, -2, -1, -2, 3, -1, -1, -3, 0, 5, 0, -1, -1, 0, -3, 1], [0, 0, -3, 1, -1, -2, 1, 4, 0, -3, -3, 3, 2, 0, -1, 0, 0], [-4, -2, -3, 3, 2, -2, -1, 3, 0, 2, 5, -1, -1, 0, 3, 1, 2], [0, 3, -5, 2, -1, 1, 1, -1, 2, -2, 1, 0, 0, -1, 1, -2, 2], [0, -3, -1, 0, 2, -1, -1, -1, 2, 3, 0, 2, -2, -2, 2, 2, 1], [-3, 0, -1, -2, -1, 1, -2, -2, -1, 1, 1, 0, 1, 2, -1, 0, 0], [4, 0, -2, 2, -1, 2, 1, 0, -2, -1, -2, 5, 3, -1, 0, -1, -1], [-2, 2, 0, -4, 0, -1, 0, -3, 0, 4, 2, 1, -2, 0, 1, -1, 2], [7, 2, 1, 0, 1, 2, 1, -2, 0, 1, 0, 0, 0, -1, 0, 1, 1], [-3, -1, 0, 1, 0, 3, -2, 0, 0, 1, 2, 2, -1, -5, 1, 1, 1], [4, -4, -1, 4, 3, -1, -1, -1, 5, -2, -1, -4, 0, 0, 1, 3, -2], [4, -2, 0, 0, 1, 2, -1, -1, 0, 3, 0, 1, 1, 2, -1, 0, 0], [2, -4, 0, 0, 1, -3, 2, 0, 2, -3, -7, 1, 1, 1, -2, -1, -2], [-1, -4, -2, 4, 2, -2, -3, 3, 2, -1, 2, -4, -1, 1, 0, -1, 2], [6, -2, 2, 0, 1, -3, 4, -1, 2, -3, -6, -1, 2, 5, -4, -1, -4], [1, -2, -5, 3, 1, 0, -1, 1, 2, 2, -1, 0, 3, 0, 0, 4, 3], [-3, -1, -1, 3, -2, 2, -1, 0, -1, 0, -1, -2, 0, -1, -2, 5, -1], [0, -2, -2, 0, -1, -1, 1, 3, -1, 0, -1, 2, 2, 1, 2, 2, -2], [8, -1, -1, 1, 0, 0, 1, -3, 0, 1, 2, -3, -1, -2, -1, 1, 1], [2, -5, -5, -2, 0, -1, 0, 1, -1, 2, 0, 1, -2, 0, 2, 1, 2], [7, 2, 0, 0, -2, 0, -2, -1, -1, 0, -1, -3, 1, 0, -3, 1, 3], [2, -1, 0, -2, 1, 0, 1, 0, 0, 1, 2, 1, 0, -1, 2, -4, 3], [6, 2, -1, -2, 1, 2, -1, -3, 1, 0, 0, -3, 1, 0, 1, -3, 0], [4, 0, 4, 0, 1, -1, 2, 2, 1, -1, -2, 1, 0, 1, 0, -3, -4], [-2, -10, 3, 1, -1, 0, -3, 3, -1, 0, -3, -1, 1, 2, 0, 4, -3], [0, 0, -4, 3, 2, -1, 2, 0, 2, -3, 2, -1, 0, -1, 0, -1, 1], [11, 2, -4, 0, 1, 0, -1, -3, -1, -1, 4, -2, -1, -1, -1, -2, 2], [-4, -2, 1, -1, -1, 1, 2, 1, -2, 1, -1, 3, 0, 0, 1, 0, 0], [1, -1, 0, 2, -1, -2, 0, 3, 2, 0, 2, -1, -1, -1, 2, -1, 0], [0, 0, -1, -5, -2, 0, 1, 0, -3, -1, -2, 1, 0, 0, 1, -2, 0], [0, -5, 2, 0, 2, 1, -4, 2, -1, 3, 0, 3, -1, 1, 0, 2, -3], [2, -1, 3, 1, 1, -3, 1, 2, 0, -3, -8, -2, 3, 2, -3, 3, -2], [1, -2, 2, -5, 3, -2, -1, -3, 1, 0, -8, 0, -1, 2, -2, 0, 0], [6, 2, 3, 0, -3, 6, -1, 0, -6, 1, 0, 3, 2, 1, -1, 0, -3], [-5, -2, 1, 3, -3, 3, -1, 0, -1, 1, 3, -1, 1, -2, 3, 0, -2], [10, -3, -1, -2, 0, -2, 0, -1, 1, 4, 0, 0, 1, 3, 0, -1, 0], [-4, 0, -3, 2, 0, -2, -2, 2, 0, 2, 6, 0, -2, -2, 0, 0, 6], [7, -3, 5, -3, 1, -4, 1, 1, 1, 1, -4, -4, 1, 4, 2, 0, -1], [2, -2, 5, -1, -1, 0, 3, 0, 1, 0, -5, 1, 0, 2, -3, 5, -4], [-2, -7, 10, -1, 0, -2, 2, 0, -1, 1, -8, 4, 0, 4, -2, 6, -3], [0, 2, 1, 1, -1, 0, -1, -2, 2, 1, 2, -1, 0, 1, 2, 1, 3], [6, -6, -1, 1, 0, 2, -1, 0, 0, 0, 1, -3, 0, -3, 3, 1, -3], [-3, 3, -6, 4, -1, 1, -1, 2, 3, -2, 4, -1, 1, -4, 0, 1, 1], [-1, 3, 4, -1, 1, 0, -2, -3, 0, -1, 0, -3, 1, 3, 0, -2, 1], [7, -6, -1, 1, -1, 2, -4, -2, 0, 2, 5, -3, 0, -2, 3, -1, -1], [-5, -4, 5, -2, 0, -3, 0, -1, -1, 1, -5, -2, 2, 2, -1, 4, -4], [2, 4, 0, -1, -2, 3, -2, 0, -2, 1, 3, 1, 4, -3, -1, -2, 2], [-3, -1, -3, 2, -1, 4, -1, 0, 0, 1, 9, 0, 0, -2, 3, -1, 4], [3, -5, 1, -2, 1, 6, -1, -1, -2, 3, 7, 4, -3, -2, 4, -2, -1], [-8, 5, -6, 3, 1, 1, -3, 2, 1, -2, 7, -2, -1, -4, 1, 0, 4], [10, -6, -1, 1, 2, -3, -2, 0, 4, -2, -1, -5, 2, 3, 1, 1, -1], [1, 1, 2, -2, 3, -6, 3, 0, 4, 1, -9, 0, -1, 1, -2, 2, 0], [10, -3, -2, 1, -1, 2, -4, 1, 0, -2, 6, -2, 1, 1, 1, -4, 1], [1, -8, 1, -2, -1, -4, 2, 4, 0, 1, -5, 3, 2, 4, 3, 2, -4], [-1, -2, 5, -1, 2, 2, -5, 0, -2, 2, 0, -2, -2, 0, -1, 3, 0], [3, -8, -3, 3, 1, 2, -3, 1, 0, 6, 2, 4, 1, 2, 2, 0, 2], [2, 5, -7, 0, -1, 2, 0, -1, 0, -1, 0, -2, 1, -1, 2, 0, 4], [2, 8, -7, 2, -3, 2, 1, 0, -2, -1, 2, 1, 2, -3, -1, -4, 0], [2, 0, 5, -2, 0, -2, 2, -3, -1, -1, -2, 0, 1, 3, 0, -1, 1], [3, 4, -2, -2, -3, 0, 2, -2, -1, -4, 1, -5, 0, 2, -3, -5, 1], [0, -6, 4, 1, 1, 2, -1, 0, 1, -3, -3, -1, -2, -1, -1, 2, -1], [-5, 3, -3, 3, -1, 2, 2, 1, -1, -6, -1, 1, 2, -1, -2, -5, 1], [3, 2, -2, 3, 0, 3, 0, 1, -3, -3, 2, 2, 2, 2, -2, -3, 0], [-3, -5, -1, 3, 1, -1, 2, 0, 0, -1, 5, -2, -3, 0, 2, -2, 0], [3, -3, -2, -3, -1, 1, 0, 1, -2, 0, 0, 2, -1, 2, 0, 1, 0], [-8, -4, -1, -2, 0, 2, 0, 1, -1, 3, -2, 2, 1, 3, 4, -1, -2], [3, -5, 0, -1, -1, -1, 7, 0, 0, -1, -4, 1, 1, 2, 0, -2, -5], [-6, -6, 2, 2, -2, -2, -1, 3, -3, 2, 2, -1, 1, 0, 2, 0, 2], [-7, 8, 0, -2, -2, -2, 4, 3, -2, -1, 1, 3, 1, -1, -1, -2, 3], [6, -4, 6, -4, 0, -3, 3, -3, -2, -3, -6, -2, 1, 3, -3, 2, -5], [10, -6, 3, -2, 3, -1, 0, 3, 0, 2, -4, 3, 0, 2, 4, 2, -7], [4, 5, -2, -3, -1, -3, 1, -4, 3, -1, -2, -2, 3, 2, -4, 2, 4], [-10, -5, 0, 6, 1, -6, 6, 3, 2, -5, -3, -6, 4, 1, -3, 1, -2], [-3, 7, 2, -2, 3, 0, -1, -1, -1, 3, 3, -4, 1, 0, -1, 1, 2], [8, 2, 0, 0, 2, -1, 3, 3, -3, -1, 1, 2, 0, -1, 2, -1, -2], [3, -3, 4, 3, 1, 1, 0, -1, 1, -1, 0, 2, -1, -2, 0, 2, 1], [4, -11, 5, 2, -1, 3, -4, 5, -2, 1, 1, -1, 0, 0, 6, 0, -6], [1, 1, 1, -3, -1, 3, 0, -2, -5, 4, 0, 4, -1, 1, -1, 0, 5], [-6, -4, 3, -7, 2, -2, 1, -1, 0, 1, -4, 5, -1, 2, 1, 2, 2], [11, -7, 4, 0, 1, 1, -1, -3, 0, 5, 4, -1, -2, 2, 2, -2, 0], [-6, -7, 7, 6, -2, 1, -3, 4, -1, -3, 6, -6, 1, 0, 1, 1, 0], [-8, 7, 4, -2, 0, 3, -2, 0, -3, 5, 0, 1, -3, -1, 1, 1, 2], [-9, -11, 2, 2, -2, -2, 0, 2, 1, 3, -4, 5, 4, -1, -1, 3, 0], [5, -1, 3, 2, 2, 2, 1, 3, 0, -1, 0, 0, 1, 3, 2, 0, -6], [-2, -3, 3, -2, 1, 0, 1, -4, 0, -4, -9, 5, 0, -1, -3, 2, -4], [0, 0, 0, 0, 3, -2, -1, -6, 4, 0, -1, -3, -4, 1, 0, 2, -1], [2, -10, 4, -2, 3, -4, 3, 0, 4, -1, -11, 1, 0, 5, -2, 3, -3], [2, 0, 4, -1, 0, -1, 1, -4, 0, -4, 2, -10, -1, 0, -3, -1, -1], [11, -15, 7, 0, 7, -7, -3, 1, 4, 0, -7, -1, 0, 5, 0, 5, -2], [-6, 5, -2, 0, 0, -1, 2, -4, 2, 0, -2, -3, -3, 1, -4, -1, -1], [-10, 2, 3, 3, 0, 0, 0, 1, 1, 0, -1, -4, -1, 1, -1, 3, 2], [-9, 11, -7, -1, -3, 4, 0, -3, 0, 1, 6, -3, -1, -1, -2, 0, 7], [6, 2, -1, -2, 1, -5, 1, -2, 0, 0, -6, 0, 0, 0, 1, -3, 3], [-3, -6, 2, 3, -2, -4, -2, 5, 0, 0, -4, -3, 5, 5, 0, 2, 1], [0, -6, -2, 0, 0, -4, -2, 0, 5, 2, 1, -6, -2, -2, 1, 5, -2], [0, -1, -4, 5, 1, 0, -1, -2, 1, -3, 4, -5, 2, -1, -5, 1, 0], [0, 0, 0, -2, 3, -2, -3, 0, 2, -1, -2, -1, 1, 1, 2, 5, 0], [-2, 3, 2, -4, 2, 5, 2, -3, -3, -1, 5, 3, -1, -2, 3, -3, 1], [2, 4, 3, -1, 0, -2, -3, -2, 0, 0, 1, -1, -2, 1, -3, 1, 2], [2, 5, -3, 3, 0, 2, -3, 3, -1, 3, 4, 1, -3, -4, 3, -1, 3], [-3, 11, -7, -2, 1, 0, -5, -3, -1, 2, 7, 2, -1, 0, 0, -3, 3], [-10, 7, -1, 1, -3, 2, 2, 2, -2, -3, 10, -4, 0, -4, 2, -4, 1], [-9, 4, 0, -3, -3, -1, 2, 1, -3, -1, 3, -2, 4, 2, -1, -2, 1], [-10, 3, 0, 0, 1, -3, 0, 3, 0, -3, -2, 3, 2, 0, -1, -3, 1], [7, 7, 4, -3, 3, 1, -2, -3, 0, -2, -2, 0, 2, 0, -3, -2, -4], [1, 0, 3, 1, -2, -1, 1, -4, -2, -7, 3, -5, 1, 0, -6, -1, 1], [-7, 2, -3, 2, 2, -1, 1, 3, -2, 0, -2, 5, 2, -1, -2, -2, 0], [-4, -3, 4, 2, -1, -4, 0, 0, 1, 4, 1, 0, 4, 0, 0, 1, 4], [-1, 8, -7, -3, -2, 2, -1, -1, -2, -2, 1, 4, -1, -5, -4, -2, 3], [4, -6, 6, 0, 1, -2, 0, 0, 1, 2, 4, -6, -2, 2, 1, -1, 2], [1, -10, 4, 2, -3, 3, -5, -1, -1, 5, -1, -1, 0, -1, -3, 6, 0], [-1, 7, -3, 1, -1, -1, 0, 0, 1, 1, -1, 0, 5, 5, -2, -2, 3], [2, -9, -5, 5, 3, -2, -1, 2, 6, 2, 2, -1, -3, 1, 1, 4, -1], [-9, 2, -1, 5, -6, 5, -5, 2, -2, 4, 7, -7, -3, -2, 3, 5, 3], [3, -7, 1, 2, -1, 4, -4, 1, 2, -2, 2, -1, 1, -3, -3, 2, 3], [8, 0, 8, -1, 4, -3, 2, 1, -1, 2, -11, 3, 0, 6, 0, 6, -4], [-3, -2, -2, 4, 0, 2, 2, 2, 0, -1, 1, 1, 2, -4, 4, -5, 1], [-4, -4, 3, -2, 2, -1, 1, -5, 2, 0, -9, -2, -2, 5, -2, 2, -5], [2, -3, 6, 6, -1, 2, -7, -2, 0, 1, 5, -9, 2, 0, -3, 5, 0], [0, -7, -5, 2, -2, 2, -4, 2, 1, 0, 7, -8, 0, -4, -1, 0, 7], [5, -3, 7, -6, 1, 3, -2, -2, -3, 6, -3, 10, -3, -1, 1, 2, -2], [-14, -5, -1, 6, 0, -1, 0, 2, -1, 1, -6, 5, 4, -1, -1, 5, 0], [11, -3, -4, -1, -1, 3, -2, 3, -5, 1, 7, 8, 0, -5, 1, -3, 1], [-4, 0, -6, 3, 1, 5, 1, -3, 1, -2, 4, -4, 0, -2, -2, -4, -3], [-6, 12, -3, 4, -3, 5, 2, 0, -2, 3, 11, 1, 0, -5, 4, -3, 1], [10, -6, 4, 2, -1, 3, -3, 2, 1, -2, 5, -2, 1, 0, 2, 0, 0], [9, 5, -1, -1, 2, -1, 1, 2, 2, 2, -1, -3, -1, -2, 3, 0, -2], [6, 4, -4, 0, 3, 0, -4, -1, 3, 2, 2, 2, 1, -1, 3, -1, 8], [-4, 2, -4, 1, -5, -1, 0, 4, 2, -1, 4, -3, -1, -1, 3, 2, 0], [-5, -11, 2, 3, 1, -2, 0, 1, -2, 1, -6, 1, 1, 3, -2, 4, -4], [17, 5, -3, 2, 2, 0, 1, 0, 3, -3, 4, 0, 1, -1, 0, -4, -1], [-1, 0, 5, 2, 1, -3, 4, 0, 2, -3, -9, -4, 3, 7, -2, 3, -6], [8, 9, 5, -2, 1, 1, 3, -5, -3, -2, -8, 6, 2, 5, -5, 0, 0], [11, -4, 3, -1, -1, 4, 2, -5, 1, 2, -5, 5, 1, -1, -3, 1, -2], [-4, 2, -8, 3, -3, 5, -1, -2, -1, 2, 9, 0, -1, -6, 2, -4, 1], [5, 2, 4, -4, 1, 3, 5, -6, 0, 1, -4, 5, -1, -2, -2, -7, 0], [-1, 5, 1, -2, 0, 2, -2, 0, 1, 0, -5, 5, -2, 1, 3, 2, -7], [1, -4, 3, 3, 0, 0, 4, -5, 1, 0, 0, -1, 0, 3, -2, 0, -3], [-9, -3, 2, 0, 3, -2, -3, 3, -2, 4, -1, 10, 4, 2, 4, 3, 0], [-11, 1, 7, -3, 1, 0, 2, 1, 1, -1, -9, 1, 3, -3, 0, 4, -3], [-2, 1, -8, -2, -1, -3, 3, -2, 1, 5, 3, -4, -5, -2, 3, 1, 1], [5, 9, 4, -3, -1, -3, 6, 0, -3, -3, -5, 4, 3, 3, -6, -6, 4], [0, -2, 3, 5, -3, 3, -2, 0, -4, 1, 11, -3, 0, -3, -2, 5, 3], [9, 0, -6, -4, -2, 0, 2, -3, 2, 1, -2, 5, 3, 1, 1, -2, 3], [-8, -2, -2, 1, 1, -1, 1, 1, -1, -2, -2, -4, 0, -1, 1, 1, -5], [4, 1, -3, -4, 2, 2, 3, 0, -2, -3, 4, 3, -4, -5, 2, -4, -1], [-4, -5, -6, 7, 0, 5, -6, 4, -1, 1, 11, 3, 3, 1, 0, -2, 1], [-3, 2, 1, 3, 0, -2, 0, 1, 0, -2, 0, 3, -1, 1, -3, 3, 0], [15, -1, 5, -4, 1, -7, 6, 1, -1, -4, -16, 2, 5, 10, -5, 0, -3], [-5, 16, -11, 1, -1, 5, 0, -3, 1, 0, 11, -6, 2, 1, -2, -8, 7], [6, -3, 9, -4, 2, -3, 0, 0, 1, -1, -7, -1, -3, 3, -2, 0, -1], [-2, 2, -11, 5, 1, -1, -5, 1, 4, 7, 10, -4, 2, -1, 2, 3, 6], [-3, 10, -3, 1, -2, 2, -7, -3, -1, -2, 7, -4, 3, 1, -5, 1, 4], [-10, -7, 1, -3, 6, -5, -3, 0, 4, 0, 1, 0, -5, 3, 3, -2, 4], [5, -16, 8, -3, 5, -7, 2, -2, 2, 2, -14, 2, -1, 5, -5, 3, -2], [4, -1, 5, -2, 0, -4, -4, 1, -1, 0, -8, -4, -1, 3, -2, 7, -2], [-5, 3, 9, -2, 3, -5, 8, -2, 5, -4, -8, -2, -1, 3, -1, 0, -11], [-11, -1, 4, -6, 4, -8, 6, -3, 2, 4, -7, 3, 0, 1, 0, 1, 2], [3, -8, 3, -1, 0, 0, -1, -4, 1, 3, -8, 2, 1, -1, -5, 8, -2], [9, -8, 0, 0, 0, 1, -2, -4, 3, 1, -3, -4, -6, 1, 3, 6, -4], [-2, -9, 4, 3, -1, 2, -3, -1, 1, -1, 7, -13, 2, 2, 1, 1, -1], [4, -1, -2, 3, -1, 4, -5, 3, 1, 3, 8, -7, -3, -8, 5, 1, 0], [-5, 18, -12, -1, 2, 5, -3, -1, 0, 2, 10, 5, 0, -3, -1, -5, 11], [9, 4, 5, -4, -3, 3, 5, -3, -4, -2, -2, -1, 2, 1, 0, -4, -6], [-2, -9, 1, 6, 1, 2, -3, 2, 4, -1, 2, -1, 4, -3, 1, 3, 1], [11, -2, -1, 0, 0, 7, -5, -1, 1, 9, 11, -7, -4, -5, 8, 3, 1], [-5, -7, -2, -1, -1, 3, -4, 0, -1, 2, 0, 0, 1, -1, -1, 0, 1], [-5, -7, -1, 4, -1, 1, -2, 2, -1, 4, 1, 4, -1, 2, -4, 11, 1], [0, -11, 5, 6, -4, 6, -2, 5, -2, -1, 1, -6, -1, -2, 1, 5, -7], [16, -14, 4, -6, 3, -2, -4, 3, 2, 2, -5, -4, -3, 1, 7, 3, -5], [-6, -3, 8, -2, 2, -7, 5, -2, 4, -5, -7, -2, -5, 2, -7, 3, -7], [0, 1, 11, -6, 3, -1, -1, -3, -2, 2, -3, 10, -2, -3, -2, -3, -2], [4, 0, -1, 1, 6, -8, -1, 3, 2, 0, 5, -1, 0, 4, 1, -3, 2], [-12, -3, -9, 3, -2, 0, -2, 3, -3, 1, 11, -6, 1, -1, 2, -4, 4], [0, 12, -3, 1, -2, 6, 2, -5, -2, 0, 2, 6, -5, -3, -3, -3, -1], [1, -6, 5, 2, -1, 0, -2, 7, -1, 4, 7, 1, -5, -3, 5, 3, -3], [-7, -1, -3, 3, -6, 3, 1, -3, 0, -1, -3, -11, -4, -2, 0, 4, 0], [-1, 2, -2, 0, -3, 2, 0, 9, 0, -1, 0, 5, 3, -5, 6, -5, -4], [-2, 7, -4, 6, -1, 1, -2, 1, 1, 2, 12, -3, 2, -5, 1, 2, 10], [1, -8, 3, 2, 3, 2, -3, -5, 4, 1, 2, 0, -4, 0, 0, 7, -1], [1, -6, 0, 0, 2, -2, 2, 2, 1, -6, -5, 3, -2, -4, -1, 1, 0], [8, 3, -6, 0, 0, 3, 5, 2, 0, -6, -3, 2, 11, 2, 0, -5, -3], [-10, -1, 2, 1, 6, 1, -3, 0, 1, -3, 0, -4, -3, 2, -2, 0, -7], [-1, 3, -4, 1, 0, 0, 1, -5, 1, -2, 1, 4, 1, 2, 1, -4, 0], [-1, -6, -2, 7, 0, -1, 4, 7, 2, -11, -6, -2, 2, 2, -2, -1, -8], [6, 0, 3, 2, -3, 6, 0, 1, -7, -2, -2, 4, 5, 1, 0, -2, -4], [10, -8, 1, 0, 0, 2, -6, 2, 0, -1, 0, 2, -1, -2, 0, -3, -4], [5, -1, 5, 5, 0, -3, 2, 3, -1, -5, -5, 6, -1, 0, -4, -2, -1], [6, 5, -10, -3, 3, -3, 3, -2, 4, 2, 2, 4, -1, -4, 3, -2, -1], [-10, -3, 1, 3, 1, 0, -6, 9, 0, 6, 2, 2, 3, 1, 6, 5, -1], [-16, -4, 1, 5, 1, -3, 2, 6, 3, -2, 0, -7, 1, -3, 5, 1, -1], [7, 8, 8, -4, -1, -5, 9, 1, -1, -9, -9, -3, 2, 1, -5, 0, -7], [6, 10, 3, -8, 1, -2, 6, -1, -4, 4, 3, 6, -5, 0, 2, -6, 2], [5, -8, 8, 0, -3, -5, 2, 4, 3, -6, -3, -6, 4, 5, -2, 2, -3], [-13, 0, 2, 6, -1, 0, -5, 0, 0, -1, 0, 0, 0, -1, 2, 1, 3], [-1, -2, 1, 1, 3, -2, 1, 2, 2, -3, -1, 4, 2, -1, -2, 5, -6], [5, -6, 2, 4, -3, 0, -5, 4, 1, 5, 4, -4, 1, 1, 8, 5, -1], [-2, -9, -2, 0, 1, -5, 0, 8, 3, 3, -2, 7, 7, 2, 3, 1, -1], [-5, 0, -4, 6, 0, -1, -3, 2, 5, 4, 4, -5, -3, 0, 4, -2, 7], [-3, 7, 0, -2, 0, -2, 8, -3, -4, -3, -10, 3, 3, 4, -5, -3, -1], [5, 5, 3, -2, -1, 1, -5, -3, 0, -5, 6, -5, 1, 1, -5, -5, -1], [3, 12, -11, -2, -9, 5, 0, -1, -5, 0, 1, 0, 4, -4, 0, -4, 4], [12, -1, 0, 1, 7, -4, 0, -1, 1, -5, -9, -4, -1, 4, -3, 1, -5], [0, -3, 4, 0, 6, -2, 0, 4, 3, -2, -11, 4, 0, 4, -2, 5, -5], [3, -5, -2, -1, 0, 1, 0, 4, -3, -3, 0, 0, -1, -1, -2, -2, -3], [20, -2, 7, -7, -4, -1, 8, 2, -2, -13, -14, -1, 7, 3, -3, -4, -14], [15, 2, 4, -6, 7, -6, 4, 1, 3, -4, -8, 5, 2, 3, 1, -7, -8], [2, 1, -4, -4, -2, 5, 3, -7, -3, 8, 3, -2, -2, -4, 4, 1, 5], [-6, 4, 1, -3, 0, 1, 9, 2, -1, -4, -4, 10, -1, -2, 0, -8, -1], [10, 7, -5, 1, 2, -3, 0, 6, 2, 2, -3, 7, -1, 3, 4, -1, 1], [-4, 2, 9, -5, -1, -3, 0, 0, 0, -3, -8, 5, 3, 6, 0, -1, -3], [3, 6, -4, -8, 3, -2, 0, -3, 0, -2, 6, -3, 1, 1, -4, -4, 6], [-6, -1, 6, -1, 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6, 3, -6, 3, -4, 2, -7, -3, -1, 4, -13, -5, 7, -1, -4, 0], [-3, 16, -1, -11, -6, 6, 11, -5, -8, -4, -10, 14, 1, 4, -3, -7, -7], [16, 15, -6, -1, 1, 0, 7, 1, -1, -4, 1, 7, -1, 0, 2, -13, -2], [-22, 8, 6, -8, -1, -3, 10, -1, -8, 6, -6, 11, 1, 0, -3, -1, -1], [-10, -10, -12, 4, 3, -1, -4, 5, 7, 9, 17, -13, -4, -2, 9, 2, 7], [2, -12, 8, -2, 1, -8, -4, 6, -4, -1, -7, 2, -3, 6, 4, -5, -2], [-23, 4, -2, 5, -5, 8, -2, -7, 1, -3, 6, -11, -1, -3, -5, -6, 5], [-1, 3, -8, -5, -2, -3, 4, 3, 1, 3, -2, 8, 0, -6, 5, 1, -2], [3, -6, 13, -1, 9, -4, 2, -8, 6, 0, 6, -15, -4, 4, -2, 4, 1], [-4, 9, -8, 2, 1, 8, -9, -4, -1, 8, 23, -3, -4, -1, 5, 2, 7], [4, 16, -7, 1, -6, 9, 14, -4, -3, -2, -10, 15, 3, -3, -6, -6, 1], [33, 1, -1, -3, -1, 3, 6, -5, 1, -2, 0, 0, -3, -2, 2, -3, -1], [-28, -4, -2, 1, -1, 4, -3, -2, 2, 3, -7, 5, -2, -3, -6, 3, 4], [17, -9, -1, 9, 0, -5, 0, 6, 0, -7, -1, -8, 3, 0, 0, 6, -6], [7, -1, 4, 9, -1, 6, -5, 2, 0, -2, 2, -4, 1, 3, 5, 6, 0], [-10, 2, 10, -1, -2, 3, -1, 5, -2, 0, 2, 2, 6, 0, -2, 6, 0], [10, -13, 5, 4, -2, -1, 5, 8, -2, -5, 0, -4, -2, 2, -2, 8, -3], [-8, -4, 2, 6, 3, 2, -10, 4, 1, 5, 12, -14, -6, -7, 8, 7, 3], [-6, -9, 4, -3, 2, 3, 0, 0, 4, 2, -9, 3, 8, -3, 3, 0, -2], [-12, -3, -3, -3, -3, -1, 6, 5, 2, 3, -9, 1, 2, 2, 4, -6, 0], [-17, -7, -15, 14, 7, -5, -5, 8, 6, 6, 5, 1, -1, -2, 0, 7, 9], [-15, -9, 15, -2, -1, -10, 7, -1, 3, -3, -21, -4, -1, 10, -5, 7, -5], [-17, 4, -5, 7, 2, -5, 5, 7, 5, -2, 11, -5, 5, -3, 3, -5, 7], [3, 0, 3, -1, -4, -8, 0, 4, -3, -7, -15, -7, 8, 6, -4, 3, -8], [-10, 19, -11, -1, 0, -1, -1, -6, 1, -1, 18, -6, -9, -6, 0, -6, 14], [-3, 13, -7, -1, -3, -6, 4, -2, -2, 1, 5, -9, -2, 4, -2, 0, 1], [1, 10, 5, -7, -3, 8, -4, -1, -2, 2, 6, 5, -3, -11, 7, -1, 5], [6, -14, 5, -2, -1, -5, 1, -4, -6, 8, -11, 12, 1, 10, -6, 6, -2], [-8, 1, 2, -1, -1, 0, 6, -3, -2, -1, 4, 6, 1, 0, -13, -1, 5], [0, 1, -3, -5, 4, -7, 8, 5, -1, -6, -6, -5, 7, 0, 5, -8, -8], [-14, -12, -2, 2, -5, 6, 0, 0, -4, 2, 1, 6, 0, -1, -5, 1, 4], [-5, 14, 3, 3, -6, -1, -1, -2, -3, -8, -3, -7, 5, 0, -13, 7, 0], [-6, -4, -1, -2, -4, 0, 7, 5, -7, 9, 3, 7, -9, -6, 1, -2, 3], [-2, -1, 11, -7, -3, -9, 7, 3, -6, -1, -11, 12, 2, 5, -9, 3, -1], [-21, -19, -8, 12, -3, 4, -1, -2, 1, 3, 0, -12, -6, -1, 2, 6, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6045_2_a_bh();".
// 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_6045_2_a_bh(:prec:=17)
    chi := MakeCharacter_6045_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_6045_2_a_bh();".
// 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_6045_2_a_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![8, 306, 760, -2198, -4238, 5469, 6915, -6304, -5201, 3854, 2056, -1319, -437, 252, 47, -25, -2, 1]>,<7,R![300032, 897408, -588588, -2265596, 864749, 2220808, -1037415, -936806, 650335, 89191, -163078, 32523, 8156, -3886, 285, 87, -18, 1]>,<11,R![4723456, 44392576, 10479472, -162850784, -42358960, 79057452, 17965529, -16985131, -3185249, 2006215, 289292, -138926, -13939, 5573, 332, -118, -3, 1]>],Snew);
    return Vf;
end function;