
// Make newform 6036.2.a.f in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6036_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_6036_2_a_f();".
// 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_6036_2_a_f();".
// 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 := [1, -17, -29, 480, 623, -1918, -830, 1850, 281, -710, 7, 112, -12, -6, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-90550432649, 889051980100, 2294729878823, -4540099016410, -3375561656235, 4893856962964, 1364696185591, -2044944803430, -80387457108, 340361208007, -30186210676, -18771968543, 3028708645, -7952245], [-60056755878, 293945891681, 605489812423, -960664064613, -351047224710, 294369123865, 189473079974, 108603473694, -141373459510, -26925118609, 33379320665, -1385989722, -2221309782, 322100129], [-56412665837, 248097278534, 1787405514567, -761666677339, -4444849279339, 140577439895, 3716454436865, -13103462483, -1366694869203, 90447398538, 212995039129, -27653630613, -11372279696, 2007881369], [3219219179, 143731895017, 318919205885, 1894425132298, -2938481582867, -3826100152528, 4044469895292, 1863340381380, -1975075093259, -190637263243, 357762732425, -24374867588, -20759780039, 3152210972], [-119337497143, 26632417227, -784133180722, -3328805948384, 7304963685496, 3968036094497, -7679322381514, -1214872247346, 2972385841950, -40614512801, -465960412783, 51258093215, 24780738655, -4165887700], [176609367247, -796815072181, -3695267323010, 2908141882017, 8566098010242, -1288294144460, -7251378248568, 273075199697, 2742795204711, -211668844498, -436114908343, 58572637137, 23573883354, -4212643376], [2484730916, 458125175223, -805240090037, -5143052518658, 6442164287423, 6870213497832, -7619196456661, -2757968746381, 3320329763540, 217855681025, -564543134100, 44138491000, 31618495960, -4905000419], [82980122956, 61012448226, -2568116450402, -3793738276432, 11365564243223, 4255215809352, -10794875184923, -1104496521124, 3977671773047, -142566061996, -605726146495, 72188909248, 31641095191, -5439301880], [-146248145768, 53467299082, 3743508421935, 3237196035944, -13313274652841, -3525560595104, 11827134969219, 729167452279, -4186680361883, 220244655572, 621759608457, -78495590618, -31970020417, 5590186212], [29960854643, 30920955088, 1933119775716, 4220854652683, -11953799699222, -5968092819620, 12373242334648, 2096077938535, -4874086179798, 20346448125, 778020971724, -81776887223, -41934570665, 6957537558], [-77123318663, -114830934896, 4026197625691, 5441706575046, -16474469130588, -7907306353990, 16231443971053, 2906083325035, -6366193544144, -15375661377, 1019970626770, -103938633805, -55197598148, 9079109911], [70165343107, 662798462431, -6531867947174, -16212111998511, 34756741871282, 22136370008464, -36417219408842, -8258332157603, 14701619372219, 278650380659, -2390891047375, 228630777256, 130464228246, -21130993205]];
    Rf_basisdens := [1, 1, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729, 27475685729];
    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_6036_a();"
function MakeCharacter_6036_a()
    N := 6036;
    order := 1;
    char_gens := [3019, 4025, 2017];
    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_6036_a_Hecke(Kf)
    return MakeCharacter_6036_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, 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, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, -1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1], [-1, 2, 1, 1, -1, -1, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, -1, -1, -1, 0, -1, 0, 1, 0, 0, 0, -1], [1, -1, -1, -1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0, 1, 0, 1, 0, -1, -1], [-1, 0, -1, 0, 0, 1, -1, 1, -1, -1, 0, 0, 0, 1], [-2, 1, 0, 1, 0, 0, -1, 0, 0, 1, -1, -1, -1, 0], [-2, 1, -1, -2, -1, 1, -1, -1, -2, 1, 2, 1, 0, 3], [0, 2, 0, 2, -1, -1, 1, 1, -1, -1, 0, 1, -2, 0], [0, -2, -1, -1, 2, 0, 0, -1, 0, 0, 0, -2, -1, -1], [0, 1, 1, -1, 1, -1, 1, 1, 1, 0, 0, 1, 1, -2], [1, -1, -3, -3, 0, 1, 0, -2, 1, -1, 1, 1, 2, 1], [-3, 3, 2, 3, 1, 0, 0, 2, -2, -1, -4, -1, -1, 0], [2, 2, -1, 0, -1, 1, 1, 1, 0, 0, 0, 2, 1, 2], [-2, 3, 0, 0, 2, 0, 2, 1, -1, 1, 0, 0, 2, 2], [-1, 0, -1, 1, -3, -1, 0, -1, 1, 1, -1, -2, -1, -3], [-2, 2, 1, 0, 1, -2, 1, 1, -1, 2, -1, -1, 2, -1], [-1, 1, -1, -1, -3, -2, -2, -3, -1, 2, 1, -2, 1, 0], [2, 1, 1, 1, -1, 0, 3, 1, 2, -3, -1, 2, -1, -1], [-2, 1, -2, 2, -1, -1, 1, 0, 0, -1, 2, 4, 2, 0], [-1, 1, 3, 1, 1, -1, -1, 0, -1, 2, 0, 1, 0, 1], [0, -2, 0, -1, 1, 3, -1, -1, 1, -1, 0, 2, 1, 2], [1, -1, -1, 0, 2, 3, -1, 1, -3, -1, 1, 1, -2, 2], [-2, -2, 0, 0, 2, 0, -1, 0, 1, 0, 1, -2, -2, 0], [-1, -1, 1, -1, 1, 0, -2, -3, 0, 1, 2, -1, 0, 0], [-2, -4, -2, 0, 1, 0, -1, -1, -1, -1, 3, 1, -1, 0], [5, 0, 1, 0, 1, 1, 1, 0, 4, -2, 0, 1, 2, 0], [-5, 3, -1, -2, -1, 2, -2, -1, -2, 2, 0, -2, 0, 3], [0, -1, 1, 0, -2, -4, 1, -1, 0, 0, 0, -1, -2, -1], [-1, 1, 0, 1, -2, 1, 0, 3, 2, -1, 0, -1, 2, -1], [-3, 0, 2, 4, -3, -1, -1, 0, 0, -1, -4, -4, -3, -3], [-1, -1, 0, -1, 0, 0, 2, 0, 1, 0, 0, 0, 0, 3], [0, 2, 1, 3, -4, -4, 1, -1, 3, -3, -2, 2, 1, -3], [3, -1, -2, -1, 0, 1, -2, 1, 1, 1, 2, 1, 3, 2], [-3, 2, 1, 1, -3, -1, 1, -1, -1, -1, -4, -1, -3, -3], [3, 1, 5, 5, 0, -2, 0, 3, 0, -1, -3, -1, -5, -1], [-2, -4, -1, -4, 2, 4, -1, -1, 1, 0, 2, -4, 0, 0], [0, -1, 2, 3, 1, 0, -1, 2, -2, 1, -2, -2, -3, -1], [-1, -3, -2, -6, 2, 2, 1, -1, 2, 2, 3, 2, 2, -1], [1, -4, 1, 0, 2, 0, 1, 0, 5, -2, 3, 2, -1, -2], [-3, -7, -2, -6, 4, 4, -2, -1, 2, 1, 3, -2, 2, -2], [-3, 1, 0, 0, 1, 4, 0, 2, -2, -1, -2, 3, -1, 3], [-4, -3, -1, -2, -1, 1, 1, -1, 3, 0, 2, 0, 2, -2], [4, 2, 3, 2, -2, -5, 3, 2, 2, 1, -5, -2, -1, -5], [-7, 2, 1, 2, -3, -4, 1, 1, -3, 0, -3, 1, -1, 0], [-7, 4, 0, 2, -4, -6, 2, -1, 0, -2, -2, 1, 2, -1], [0, -1, -1, 1, 2, 2, 1, 2, 1, -4, -2, 0, -2, -1], [1, 0, 2, 1, 0, -3, 5, 0, 5, -3, -4, 0, 1, -4], [-2, 5, 1, 3, -3, -1, -2, 1, 0, -1, -2, 0, 3, 1], [-4, 2, 0, 6, -2, -4, 0, 3, 0, -3, -3, -1, 0, -3], [5, -1, 1, -4, 2, 2, 1, -1, 4, 4, 1, 0, 1, -2], [-6, -4, -1, -3, 2, 0, 0, -2, 3, 0, 0, -4, 1, -3], [-4, -3, 1, 3, -1, -3, -2, -2, 1, -2, 1, 3, 0, 1], [-3, -7, -6, -7, 6, 8, -3, -1, 1, 0, 6, 0, 3, 2], [-3, 4, 2, 4, -2, -5, 4, -2, 0, 1, -2, 2, 1, -2], [-5, 1, 1, -2, -2, -4, -2, -3, 0, 5, 0, -2, 0, -1], [-6, 1, 3, 0, 2, -2, -2, 1, -7, 2, -3, -4, -3, 0], [-3, -1, 0, -1, 4, 5, 2, 1, -4, 2, 2, 0, -3, 3], [-4, 0, -2, -2, -4, -4, 1, -1, -1, 4, 4, 2, 2, 2], [-7, 7, 5, 8, -5, -7, 2, 1, -2, -3, -6, 1, -3, -4], [1, 0, 1, 8, -2, -2, -2, 0, 0, -6, -5, 3, -2, 1], [-8, 3, 4, 1, -1, -5, -5, -6, -3, 4, -2, -3, -2, 1], [-7, 1, -5, 0, -4, -2, 1, -1, -2, 0, 5, 4, 2, 3], [0, 3, -6, -3, -3, 0, -3, -4, -3, 3, 3, 2, 4, 3], [-12, 4, -3, -3, -2, -2, -3, -1, -2, 4, 2, 1, 4, 6], [-6, -1, 5, 4, 0, 0, -7, -3, -5, 3, -4, -4, -5, 1], [2, 2, -2, 2, 0, 1, 4, 2, 2, -3, 2, 5, -1, 2], [-1, -3, -3, -3, -1, 3, 1, -4, 1, -1, 5, 4, 2, 6], [-2, -2, 1, 2, 3, 6, -1, 6, 1, -5, -2, 2, 2, 2], [0, 2, 3, 1, 6, 3, 3, 5, 7, -4, -2, 1, 2, -6], [2, 3, 4, 1, 0, -4, 5, 2, 0, -1, -5, -4, -2, -7], [-2, 6, 1, 1, -3, -7, 3, 1, -5, -2, -1, 6, 2, 3], [-6, 5, -3, 0, 1, 4, 2, 4, -1, -4, -1, -1, 1, 2], [-6, 2, 2, 2, 0, -7, -1, -2, 1, 2, -4, -3, 2, -2], [-5, -6, -3, -5, 4, 2, -1, -4, 3, 2, 4, 2, 3, 4], [2, 0, -6, 3, -1, 5, -1, -2, -1, -4, 2, 2, -2, 5], [-5, 1, 2, -2, 0, 2, -2, -1, 1, 1, -1, -7, 1, -2], [-3, -1, 0, 2, -1, 4, -2, 1, -2, 2, 1, -2, -4, 1], [-10, 2, -5, -4, -5, -2, -2, -3, 2, 3, -1, -6, 2, -1], [-11, -2, -2, -3, 2, -2, 0, -3, 0, -1, 2, 0, 3, 2], [1, 1, -3, -1, -1, 1, 0, -3, 1, -2, 0, -4, 0, -4], [-7, 1, 0, 1, 2, 0, 0, 2, 3, 1, -1, -5, 3, 0], [-3, -10, -1, -1, 6, 8, -3, 0, 4, -3, 4, 0, -3, -2], [7, -1, 1, -2, 3, 2, 7, 6, 10, -7, -1, 3, 3, -6], [-3, -4, 2, 3, 8, 7, -4, 5, -4, 1, 4, 1, -2, 8], [6, 3, 3, 8, -1, 0, 1, 3, 5, -6, -4, 0, 1, -5], [-4, -4, -7, -14, 8, 9, -6, -3, 0, 3, 8, -3, 7, 5], [-3, -2, -1, 4, -1, -5, 4, -1, 2, 0, -3, -5, -3, -6], [3, 1, -2, 3, -2, 2, 2, 2, -6, -1, 0, -1, -5, 1], [1, 1, 2, 4, 4, 2, 6, 4, 2, -3, -1, 6, 2, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, -4, 1, 5, 3, -1, 2, 1, -1, -1, -2, 2, -3, -1], [-2, -10, -6, -6, -2, 4, -2, -5, 3, 3, 9, 4, 4, 6], [-6, 8, 2, -3, -6, 0, -1, 1, -4, 6, -1, -5, -2, -2], [-8, 8, 4, 1, 7, -3, 4, 3, -5, 2, 0, 5, 4, 4], [0, 8, 0, 5, -5, -1, -3, -4, -4, 1, -4, 1, -3, 3], [6, -9, 0, -5, 10, 4, 0, -2, 4, 0, 4, 4, -2, 0], [2, -1, -1, 10, -1, -8, 3, 3, 2, -2, -2, -3, 1, -4], [-3, 2, -6, 1, 1, 7, -1, 5, 0, -4, -4, 3, 3, 3], [5, 6, -2, 4, -4, -1, 4, -1, 5, -8, -3, 6, 3, -1], [-8, 2, 4, 8, 2, -1, -2, 4, -6, -5, -7, -5, -5, 3], [11, 11, 5, 2, -1, -1, -1, 4, 1, 5, -4, -1, 4, -1], [4, 0, -5, 1, -11, -3, 3, -2, -1, 3, 2, 4, 0, -3], [-3, 1, 2, -2, 0, -1, 0, -2, -3, 4, -7, -10, -3, -3], [-4, 2, 0, 4, -6, -4, 1, -1, -1, -7, -2, 0, -4, 0], [-2, 0, 8, 4, 3, -9, -1, 0, 4, 0, -5, -7, 4, -10], [7, -9, -2, -10, 1, 5, -1, -6, 8, 0, 9, 6, 2, 3], [5, -2, -5, -3, -1, 0, -1, -3, 6, 1, 1, 2, 1, -3], [-3, -2, 0, 1, 2, 0, -1, -5, -5, 0, -3, -2, -7, 3], [-6, 6, 4, 2, -2, -1, 1, 4, -6, -1, -1, 4, -1, 3], [-3, -2, -1, 1, -2, 2, -5, 0, -2, -4, 1, 6, 0, 0], [-4, 5, 1, -5, 5, 1, 0, -1, -2, 2, -1, -7, -1, 1], [-5, 0, 8, 1, 3, 0, -6, 2, -3, 4, -6, -7, 1, 1], [12, -3, 0, -4, -5, -1, -1, 1, -2, 0, 1, -6, -5, 2], [13, -11, -3, -1, 6, 4, -1, -1, 2, -2, 10, 5, 4, 2], [-1, -3, -2, -9, -5, 1, -2, -5, -1, 6, 3, 1, 9, 1], [11, -1, 5, 0, 4, 3, 5, 5, 4, 1, -6, 2, -8, -5], [2, -9, 3, 0, 6, -2, 0, 0, 5, 0, -2, -6, -5, -8], [7, 6, 1, 3, -8, -8, -2, 0, -6, -1, 4, 6, -4, 0], [6, 4, 9, 2, 2, 3, -3, 3, -3, 0, -1, -1, -8, -4], [-6, -8, 4, -2, 3, -1, -3, -4, -2, 8, 2, 1, 0, -1], [-13, 10, 5, 0, 2, 0, -1, -5, -6, 4, -1, 0, 4, 2], [3, -6, 0, -3, -7, -4, -4, -7, -1, 4, 1, -3, -4, 1], [-1, 7, 0, 1, -5, -3, 6, 0, 1, 0, 5, 4, 4, 1], [-11, 2, 5, 9, -2, -7, 2, -3, -4, -3, -8, -5, -8, -8], [2, 6, 8, 5, -4, -6, -2, 2, 4, 5, 0, 1, 2, 0], [1, -5, -3, -2, 9, 5, 3, 5, 4, -8, -5, 5, 7, -2], [-2, 5, 8, 11, -1, -7, 4, 7, 3, -9, -6, 2, -2, -6], [5, -3, 3, 0, -5, -5, -1, -8, 0, 4, 6, 1, -5, -2], [-5, -1, 0, 7, -1, -4, 1, 3, 2, -5, -2, -4, 0, -4], [-2, -9, 2, -1, 6, 2, 1, 2, 5, -1, -3, -2, -7, -7], [-1, -2, 5, 5, 0, 0, 4, 7, -3, 0, -6, -2, -3, -8], [5, 1, -1, -1, -3, -6, 5, -2, 4, 0, 6, 6, 6, 1], [14, -6, 4, 0, 2, -1, 6, 1, 6, -1, 3, -3, -5, -2], [-3, 1, 0, -9, -2, 0, -1, -5, 4, 4, 8, 0, 9, 2], [6, 7, 0, 3, -10, -3, -3, -2, -3, -3, -2, 2, -4, 3], [0, -9, -2, 4, -1, 3, -2, 2, -6, -3, 1, -5, -10, -2], [1, -1, 2, -2, -3, 0, -2, 2, -6, 4, 2, -2, -7, -2], [7, 4, -1, 4, -2, 4, -1, 4, 1, -7, -7, -2, 3, 6], [-6, -3, -2, 1, 2, 0, -6, 1, -2, 9, 4, -3, 6, 2], [4, -8, 3, -3, -4, -1, -6, -3, 0, -2, 6, 3, -4, 1], [11, 2, 4, -6, 0, -8, 6, 1, 5, -1, 2, 3, 1, -4], [2, -2, 0, 1, 7, 5, 2, 7, -8, 3, 1, 0, -7, 9], [-6, 0, -2, -16, 1, -1, 0, -6, -4, 7, 7, 1, 2, -1], [6, 11, 9, 10, -8, -8, 5, 3, 2, -5, -10, 3, -1, -5], [-6, 4, 10, 12, 2, -9, 0, -2, -2, 1, -11, -8, -6, -7], [-3, 0, -4, -12, 7, 9, 2, 6, 0, 1, 3, 4, 8, 11], [9, 2, 2, -2, -9, 2, -7, -1, -2, 3, 8, 5, -1, 4], [4, -3, -6, -8, -3, 2, -2, -4, 1, 3, -6, -7, -2, -2], [-2, -5, -9, -9, 4, 4, -2, -8, -4, 6, 12, 6, 6, 5], [14, 6, 5, 5, -5, -1, 3, 0, -2, -3, -4, 2, -2, 3], [-13, -2, 1, -3, 6, 7, -1, 1, -4, 5, 2, -3, 3, 6], [-4, -2, -4, -3, 0, 2, -2, -4, 1, 2, 8, 3, 8, 7], [11, 0, 1, 4, 5, -1, 7, 2, 13, -4, -1, 6, 3, -5], 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[-13, 18, -2, 6, -1, -10, 12, 3, 2, 1, -12, 1, -2, -5], [-5, 5, -2, -1, 0, -6, 2, -14, -11, 3, 4, 8, 5, 10], [-5, -15, 15, 7, 11, 5, -4, 3, 4, -3, -7, -12, -4, -3], [-8, 6, -4, 8, -5, 1, -2, 12, -13, 0, 9, 9, -4, 5], [12, -12, 11, 0, -2, -1, -7, -7, -1, 11, -4, -8, -4, -2], [-4, -7, -2, -4, 19, 12, -4, -2, 2, 1, 2, -11, 11, 6], [-3, 20, 16, 22, -5, -16, 6, 7, 3, -13, -18, 4, -1, -12], [14, -12, -4, -14, -5, 1, 2, -8, 17, 1, 12, 9, 5, -11], [-19, 4, 3, 23, 5, -1, 0, 6, -10, -2, -17, 3, -8, 1], [4, 9, -3, 7, -19, -17, 2, -9, -1, -2, -13, -2, -2, -5], [10, 11, 10, -3, -13, -15, 5, -13, 1, 5, 5, 4, 3, -1], [-11, 14, -3, 14, -19, -20, 10, -7, 0, -4, -7, 3, 9, -2], [12, 1, 6, -6, -5, -8, 9, 3, 12, 4, 6, 2, 4, -6], [1, 5, 7, -8, 3, 6, 0, 7, 2, 1, 2, -4, 3, -3], [0, -6, -6, -3, -12, 8, -4, -6, 0, 5, 4, -4, 3, 6], [8, 0, 4, -7, -4, 0, -5, -2, 4, 9, 6, 0, 11, -1], [9, -7, -7, 0, -2, 0, 4, -5, 5, -5, 13, 13, -4, -1], [5, 10, 4, 8, 10, 0, -1, 4, 0, -13, -11, -5, 0, 8], [-8, 13, -5, 0, 1, 5, -4, 3, -6, -3, 8, 15, 10, 20], [3, -6, -9, 6, -5, 1, -5, 3, 1, -14, 2, 1, -6, -1], [18, -5, 2, 0, -9, -12, 8, -5, -3, 4, 2, -13, -7, -9], [2, -10, 3, -6, 3, 13, -4, -3, 11, 3, 4, -1, -8, -3], [-5, -6, 14, -4, 6, -7, 2, -3, -4, 11, -3, -15, -2, -7], [-9, 21, 4, 16, -3, -1, -1, 14, 1, -7, -13, 2, -8, -1], [-12, 0, 1, -7, 7, 4, -3, 6, -3, 17, 5, -7, 7, 0], [-4, 2, -5, -1, -1, -4, 1, 2, 2, -11, 0, 8, 6, -3], [-5, 18, 14, 15, -2, -1, 15, 18, -2, -8, -24, -3, -16, -5], [5, 2, 3, 12, -4, -19, 2, -10, 4, 2, -6, -2, 10, -7], [4, -1, 2, -11, 9, 13, 0, 12, 0, -1, 1, 12, 1, -1], [14, -5, 1, 19, 14, 2, 7, 9, 5, -16, -6, 2, -8, 0], [1, 3, 3, -9, 3, 7, -6, 1, -9, 12, 7, -2, -14, 2], [-5, -2, 0, 7, 7, 5, 0, 10, 5, -15, -15, -1, -4, -7], [7, -5, 6, 3, 3, -12, 5, 1, 4, -6, 2, 16, 9, -3], [-9, 12, 8, 15, -3, -10, -7, 4, -11, 1, -8, -4, -1, -3], [2, -21, -12, 0, 3, 4, -12, -1, 0, -3, 15, 10, 10, 8], [7, 0, 13, 0, 8, -2, -1, 5, 7, 13, 1, -7, 0, -5], [11, -11, -9, -1, -9, 0, -4, -3, 9, -11, -3, -2, 3, -7], [-4, -5, -6, 5, 11, 6, 5, 2, 5, -6, 13, 6, -9, 3], [-31, -8, -1, -6, 8, -12, -1, -4, -10, 1, 0, -12, 2, -2], [8, -11, -9, 9, 9, -2, 6, 5, -3, -8, 2, 5, -11, -11], [5, -5, -4, -3, 12, 7, -8, 11, -14, -1, -1, -6, 7, 10], [-8, 15, 3, 7, -17, 1, -4, 1, -10, -4, -3, 4, -4, 5], [-9, -13, 4, 0, 17, 6, 6, 10, 16, -8, -5, -5, 4, -17], [-1, 23, 11, 28, -16, -8, 6, 13, 0, -7, -12, 2, -6, 3], [-11, -1, -4, 8, 1, 2, 3, 1, 7, 0, 1, -1, -11, -5], [-34, -4, -7, -5, -1, 4, -7, -2, -6, 2, -2, -12, -7, -6], [-1, -14, 0, -3, 4, 6, -7, 0, 6, 9, -5, -3, 0, -5], [4, -11, -13, -24, 8, 16, 0, -6, 2, 6, 12, 6, 2, 11], [4, -15, 5, 13, -6, 2, 0, 6, 10, -7, -6, -6, -5, -11], [18, 14, 4, 2, -15, -10, 5, 0, -2, -6, -10, -9, -5, -2], [-13, -5, 2, -7, 9, 7, -5, 3, -9, 8, -4, -6, 9, 11], [-7, 11, 3, 8, -18, -19, 2, -8, -6, -3, -3, 6, 2, -5], [3, -28, -12, -27, 0, 9, -7, -23, 13, 11, 8, -7, 4, -9], [-6, 10, 4, -2, -9, 3, -11, -1, -15, 5, -1, -3, 2, 5], [-10, 20, 22, 13, 15, 1, 5, 12, -5, -5, -1, 11, 4, 3], [13, -3, -6, -2, 4, -1, -2, -11, 9, -8, 6, 1, 2, 8], [-8, -8, -1, 12, 6, 1, 1, 3, -4, -16, -6, -5, 5, 3], [24, -9, -3, -5, 5, 3, 7, -1, 17, -9, -1, 4, 3, -10], [9, -3, 0, 1, 4, 7, -1, 7, -2, -3, -10, 1, 9, 5], [-14, 10, 10, 7, 7, -1, 8, 3, -5, 3, 0, 16, 2, 3], [-8, -12, -6, -13, 1, 10, -10, -6, -15, 17, 11, -2, -6, 13], [6, 7, -12, 0, -8, -15, 10, -8, 0, 4, 5, 8, 7, 0], [-21, 19, 4, -5, 1, -14, 2, -5, -9, 10, -6, -7, 4, 2], [11, -23, -7, -10, 17, 13, 6, 7, 12, -8, 9, -2, 7, 9], [-21, -5, -1, -1, 3, 4, -9, -8, -11, 5, 9, 6, -5, 2], [3, -11, 9, -4, -2, -12, -4, -11, -6, 6, -1, -12, 5, -9], [12, -4, 5, 9, -16, -10, -5, -13, 10, -5, -2, -9, -3, -18], [-10, -19, -19, -10, -7, -6, 1, -10, 4, 5, 11, -1, 4, -1], [-8, 20, 6, 22, 3, 10, 5, 17, -12, -11, -13, 4, -1, 15], [-16, 9, -3, 9, 3, 0, -3, 8, -14, -9, -4, 4, -3, 7], [-5, 16, 3, -2, 7, 10, -5, 4, -14, 1, -1, -1, 9, 16], [-13, 4, -1, -6, 6, -16, -2, -2, -11, 10, 0, -2, -2, -1], [-8, 10, 3, -4, -2, -13, 2, -5, -4, 1, -1, 8, 4, -12], [5, 3, 16, 9, 0, -10, 1, 1, -7, -4, 0, 6, 3, 4], [-16, -16, 0, -5, 11, -2, 6, -5, 9, -7, 5, -6, -13, -5], [17, 0, -2, 8, -12, -13, 6, -7, 5, -3, -9, 0, 4, 3], [-1, -22, -2, -3, 20, 8, -6, -5, -2, 2, 4, -9, -2, 1], [7, 13, 12, -1, -2, -5, 9, -6, 5, -15, 5, 20, -5, -2], [2, 21, 9, 2, -15, -11, 1, -5, -15, 11, 1, 3, -4, 14], [2, -9, 0, -12, 14, 0, -3, -3, 6, 8, -2, -10, 13, -8], [11, -3, -16, 0, -9, 3, 2, -1, 7, -7, -10, -8, -4, -6], [-4, -12, 9, 3, 12, -9, 3, -1, -2, 5, 5, -2, -10, -14], [4, -12, 0, -6, 13, 4, 6, -1, 15, -5, 4, -2, 13, 1], [4, -11, -7, -7, -5, -1, -3, -4, 13, 2, 4, 1, 4, 5], [10, 3, 11, 10, -5, 3, 0, 15, 8, -2, -8, -6, -3, 0], [-15, -19, -6, -11, 7, 8, 3, -6, 8, -3, -1, -9, -5, -15], [-9, -8, -10, -8, -9, 12, -6, 2, -4, 3, 9, 5, -4, 11], [-3, -4, -1, -5, -2, -3, -10, 1, -2, 14, 6, -3, 3, 5], [9, -13, -12, -6, 6, 12, -6, 5, 6, 0, 20, 3, 4, 12], [-18, 14, 4, 6, -2, -5, 3, 0, -7, 0, -14, -13, 2, -3], [0, -21, -8, -16, 18, 19, 8, 8, 15, -2, 22, 6, 10, -2], [15, -11, -11, -10, -4, 2, 8, 5, 14, 3, 8, 3, 1, -7], [-9, 12, 13, 1, -3, -5, -7, 1, -10, 5, -7, 0, -11, -3], [-4, 12, 6, 9, -10, -1, 10, 7, -4, -7, -18, 4, -13, -3], [-7, 18, 10, 2, -7, -5, 14, 2, -5, -10, 0, 12, -18, -2], [-10, -2, 10, 2, 4, -1, -3, -6, -5, -1, -3, -13, -3, -3], [8, 8, 8, -6, -8, -6, -4, -8, -4, 5, 10, 13, 0, 13], [-8, 7, -4, 9, -1, -9, 5, -5, -8, -4, -5, -3, 4, 5], [1, -5, 4, -13, 0, 6, -1, 4, -3, -6, 2, 10, 5, 15], [14, 5, 7, 4, 1, 1, 0, 2, 3, 2, 6, -2, 5, 7], [-20, 15, 14, 21, 6, -13, 0, 0, -7, -2, -25, -5, -5, -5], [-9, 18, 13, 9, -7, -9, -5, 12, -4, 6, -16, 1, 5, -6], [-17, -1, -22, -27, 3, 18, -7, -8, -3, 0, 11, 1, 9, 14], [25, -4, -4, -19, -4, 0, 9, -2, 5, 5, 13, -5, 1, 0], [-30, 1, -8, -11, 3, 4, -4, 1, 2, 7, 2, -6, 20, 2], [-8, -2, -3, 0, 10, 8, 4, 6, 18, -16, -15, -3, -1, -15], [-3, 0, 0, 11, -17, 4, -12, -6, 0, 4, -5, -13, -18, -9], [-6, 8, -10, -18, 9, -2, 3, -1, 6, 1, 3, -2, 12, 2], [-35, 3, -3, 5, 15, 2, -5, 9, -12, -9, 0, 3, 6, 13], [-4, -15, -5, 3, 7, -6, -3, -5, 0, 1, 6, 11, 0, -10], [0, 18, 4, 11, -4, -1, 1, -3, -5, 3, -11, -8, -9, 5], [25, -4, -2, -5, 7, 12, -6, -1, -9, 17, 7, 2, 14, 15], [-2, -20, -5, 2, 5, 3, -7, 7, -3, 0, 5, -5, -3, 6], [0, 5, 8, -7, -13, -4, 1, -4, -8, 10, 8, 5, 3, 4], [32, -20, 2, -10, 10, 14, 3, 2, 12, -7, 19, 10, 2, 8], [1, 11, 12, -3, -9, 0, 5, -6, 4, 0, -1, -7, 5, -8], [-29, 2, -4, -8, 7, 2, -9, 5, -14, 6, -6, -19, 2, 6], [-1, -5, 3, 0, 7, 0, -4, 4, 9, -2, 0, -1, 2, -9], [6, 0, -4, -3, -2, 8, 0, 2, 6, 5, 4, -6, 3, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6036_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6036_2_a_f();".
// 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_6036_2_a_f(:prec:=14)
    chi := MakeCharacter_6036_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_6036_2_a_f();".
// 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_6036_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6036_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 17, -29, -480, 623, 1918, -830, -1850, 281, 710, 7, -112, -12, 6, 1]>,<7,R![-316, 1094, 547, -4382, 803, 6094, -1687, -3930, 786, 1248, -62, -177, -15, 7, 1]>],Snew);
    return Vf;
end function;