
// Make newform 4020.2.q.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4020_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4020_q_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_4020_2_q_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_4020_2_q_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [4, -12, 54, -30, 189, -101, 458, 17, 270, -57, 88, -8, 11, -1, 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, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [970249939441, -76957145272, 320958040428, -308324347672, 984764699106, -934856019808, 250820920885, -708369775075, 266952713308, -234940121607, 35610122429, -29747186705, 4118856992, -2680061185], [-194090246132, 171410592976, -717088584314, 2602300698723, -2207859665815, 2035024146712, -1359823613067, 1437864427346, -660716032257, 365843919170, -86278343802, 45065673273, -8708118669, 3347996299], [861786952374, -167893036528, 695977413972, 2831612352657, 2120608715454, -2130418252432, -908790740124, -1814707096630, 455144021782, -815634907035, 64289364146, -105683331260, 9758640678, -10892153560], [410860145420, -1903129738468, -1146623852733, -6962904463922, 2865754568121, -20863509569047, -2262747973407, -12440375581653, 2399942088211, -4183707071102, 343114818991, -525040058194, 45174565234, -48522561533], [-7944658082, -43501677200, -54504988027, -160280487010, -353375346337, -473551890151, -266207177033, -184014082317, -42770485013, -65999965440, -4713678201, -8122293536, -252390382, -809674271], [-841108812233, 898967472816, -3744294132514, -2133066776665, -11470995916307, 11026504067640, -1664041174406, 8589017027582, -2969882527405, 3098569859104, -400339609502, 395084796401, -47919288537, 37194535803], [974875365383, -1422383351760, 5935414947940, -5135681407065, 18222273890390, -17209864321680, 6408314158958, -12888553943330, 5030477527030, -4117338117621, 668330017670, -518899664240, 74034574851, -45711237420], [2201747399207, -2013572275960, 8407638741310, -12925259206893, 25830617586941, -24249792781624, 10738605919044, -17910517000443, 7279578337109, -5449809996738, 962775171505, -683843870764, 104277787850, -58435287502], [903683328390, 2242764960296, 5457745711665, 21347976231450, 22467398449373, 63503082991395, 21731699467035, 31403882537963, -585966489615, 10733169032550, -183514685193, 1335886940970, -49216475610, 126618741863], [-101131348946, -793639274980, -490070050147, -2896009242346, 946819186593, -8674402381075, -1041412251713, -5129198170687, 952274405431, -1725891567156, 136874887445, -216517971596, 18183711674, -20029651747], [1100661461810, 10832044605312, 6616282893811, 26131870142026, 23844336362505, 77748959678947, 26110314425969, 38225859873625, -942157914055, 13207909689396, -250245646301, 1644341611148, -63210384410, 156907511177], [438504860088, -2171714647326, -1496669907681, -8695369936605, 1482620818260, -36599366732449, -6897421130211, -21440487873381, 3401605949923, -7424278972227, 502153407798, -933363507764, 71009258532, -87738052524]];
    Rf_basisdens := [1, 1, 325648644013, 325648644013, 325648644013, 651297288026, 8458406338, 325648644013, 325648644013, 325648644013, 651297288026, 93042469718, 651297288026, 325648644013];
    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_4020_q();"
function MakeCharacter_4020_q()
    N := 4020;
    order := 3;
    char_gens := [2011, 2681, 3217, 1141];
    v := [3, 3, 3, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4020_q_Hecke();"
function MakeCharacter_4020_q_Hecke(Kf)
    N := 4020;
    order := 3;
    char_gens := [2011, 2681, 3217, 1141];
    char_values := [[1, 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], [1, 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]];
    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, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0], [0, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, 1], [0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, -1, 0, 0, 2, -1, 1, -1, 0, 1, -1, 0, -1], [-1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, -1, -1, 1], [1, 1, 0, 0, 0, -1, -1, 0, 0, 0, -1, -1, 1, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 0, -2, -1, 0], [-2, 0, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0, 1, 1], [1, 0, -1, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0], [2, -2, 0, 0, 1, -2, 2, 0, 0, 0, 1, 1, 1, -1], [0, 0, 1, 1, 0, 0, 0, 1, -1, -1, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0], [0, 1, 0, 1, 0, -1, 1, -2, 1, 1, -2, 0, -1, 1], [3, 1, 0, 1, 1, -1, 1, 0, 0, 1, -1, 2, 1, 0], [4, -1, 0, 0, 2, -4, 3, 0, 0, 0, 1, 2, 1, -2], [0, 0, -1, 0, 0, -1, -1, 1, -1, 1, 1, -1, -1, 0], [1, 1, 0, 0, -2, -1, -2, 0, 0, 0, 0, -1, 1, 2], [0, -2, 0, -2, 0, -1, 2, 2, 2, -2, 2, 0, 2, -1], [-1, 0, 1, 1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0], [0, -2, 1, -2, 0, 2, 2, 2, 2, 0, 2, 1, 0, -1], [-3, 0, 0, 0, 2, 3, -1, 0, 0, 0, 0, 2, -1, -2], [4, -3, 0, 0, 1, -4, 2, 0, 0, 0, -1, 0, 0, -1], [0, 0, -1, 0, -3, 0, 0, 1, -1, 1, 0, 0, 0, 0], [0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0], [-3, 0, 0, 0, 4, 3, 1, 0, 0, 0, 3, 1, 1, -4], [1, 0, 0, 0, 3, -1, -1, 0, 0, 0, 2, 0, 1, -3], [-1, 0, 3, -3, -3, 0, 0, 0, 2, -2, 0, 0, 0, 0], [1, 0, -1, 1, 3, 0, 0, -2, 1, 1, 0, 0, 0, 0], [5, 0, -2, 0, -1, 0, 0, 1, -2, 2, 0, 0, 0, 0], [-1, 0, 2, 0, -1, 0, 0, 2, 2, 1, 0, 0, 0, 0], [0, -2, 3, -2, 0, 2, 1, 1, 1, 2, 1, 3, -2, 0], [0, -2, 0, -2, 0, 9, -1, 1, -1, 2, 1, 0, -2, 1], [3, -4, 0, 0, 1, -3, 1, 0, 0, 0, 2, 0, 1, -1], [-4, 2, 0, 0, -1, 4, -1, 0, 0, 0, -3, -1, 2, 1], [0, 0, -1, 0, 0, -3, -1, 2, -1, 0, 2, -1, 0, -4], [2, 0, -3, 1, 0, 0, 0, -1, -3, 3, 0, 0, 0, 0], [-4, 3, 0, 0, -2, 4, -2, 0, 0, 0, -1, -2, -2, 2], [0, 0, 5, 0, 0, 0, -1, 0, -1, -1, 0, 5, 1, 0], [8, 0, -2, 2, 2, 0, 0, 1, -3, 2, 0, 0, 0, 0], [6, 0, 0, 0, -3, -6, -1, 0, 0, 0, -1, -5, 0, 3], [0, -2, -5, -2, 0, -3, -3, 1, -3, -1, 1, -5, 1, 1], [0, -1, -2, -1, 0, -2, -1, 5, -1, -1, 5, -2, 1, -2], [-3, 0, -1, -3, 1, 0, 0, 2, 0, 4, 0, 0, 0, 0], [6, -4, 0, 0, 1, -6, 4, 0, 0, 0, -2, -1, -2, -1], [0, -2, -3, -2, 0, -1, -2, -2, -2, 5, -2, -3, -5, -1], [-5, 0, 0, 0, -1, 5, -6, 0, 0, 0, -1, 0, -1, 1], [5, 0, 0, 0, 2, -5, 2, 0, 0, 0, -4, 0, -2, -2], [4, 0, 0, 1, 1, 0, 0, -2, 0, 2, 0, 0, 0, 0], [0, -1, 4, -1, 0, -1, 5, 2, 5, -1, 2, 4, 1, -3], [-3, 4, 0, 0, -2, 3, -3, 0, 0, 0, 0, -3, -2, 2], [6, 0, -4, -1, 1, 0, 0, 3, -2, 3, 0, 0, 0, 0], [10, 0, -1, 3, 2, 0, 0, -1, -3, 3, 0, 0, 0, 0], [-5, 0, -1, 0, -4, 0, 0, 1, 1, 0, 0, 0, 0, 0], [7, 0, -3, 2, 0, 0, 0, 5, -1, -2, 0, 0, 0, 0], [0, 2, 2, 2, 0, -6, -1, 0, -1, -4, 0, 2, 4, 1], [1, 0, -6, 1, 4, 0, 0, 0, -3, 6, 0, 0, 0, 0], [9, 0, 3, -3, 1, 0, 0, -1, 5, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, -12, -1, -2, -1, 1, -2, -3, -1, 5], [-2, 0, -1, 4, 1, 0, 0, -1, -3, -1, 0, 0, 0, 0], [4, 0, -5, -2, 1, 0, 0, -2, 3, 2, 0, 0, 0, 0], [0, 3, -1, 3, 0, 9, -2, -3, -2, 4, -3, -1, -4, 1], [2, -2, 0, 0, 2, -2, -2, 0, 0, 0, -2, 3, -4, -2], [0, -1, 2, -1, 0, -10, -2, 2, -2, 1, 2, 2, -1, -3], [3, 4, 0, 0, -4, -3, -6, 0, 0, 0, 0, -1, 1, 4], [-2, 0, 1, -4, -1, 0, 0, 2, 4, 1, 0, 0, 0, 0], [0, -4, 2, -4, 0, 1, 2, -2, 2, 2, -2, 2, -2, -1], [-6, 0, -2, -4, 0, 0, 0, 0, 3, -3, 0, 0, 0, 0], [1, 1, 0, 0, 5, -1, 0, 0, 0, 0, -1, -3, 2, -5], [4, 3, 0, 0, 2, -4, -4, 0, 0, 0, -3, -7, -1, -2], [0, 2, -4, 2, 0, 9, -5, 3, -5, -2, 3, -4, 2, 0], [0, 3, 1, 3, 0, 11, -4, 1, -4, 2, 1, 1, -2, 1], [0, 1, 1, 1, 0, -2, 2, 3, 2, 1, 3, 1, -1, -1], [-6, 0, 2, -1, -1, 0, 0, 5, 6, -7, 0, 0, 0, 0], [-3, 0, 2, 1, -4, 0, 0, -2, 1, -6, 0, 0, 0, 0], [0, -2, 0, 0, 3, 0, 6, 0, 0, 0, -2, 0, 1, -3], [0, -1, -2, -1, 0, 1, -1, 0, -1, 3, 0, -2, -3, -5], [0, 0, 3, 0, 0, 2, -2, -3, -2, -3, -3, 3, 3, 0], [-6, 6, 0, 0, -3, 6, -4, 0, 0, 0, -3, -2, -2, 3], [-2, -2, 0, 0, 4, 2, -3, 0, 0, 0, 2, 4, 0, -4], [0, 0, 3, 0, 0, -8, 9, 0, 9, 1, 0, 3, -1, 0], [-4, 0, 0, 0, 8, 4, 1, 0, 0, 0, 6, 0, 2, -8], [-2, 0, 0, 0, -1, 2, -3, 0, 0, 0, 4, -4, 2, 1], [-2, 3, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, -1, -3], [-13, 0, 1, -3, -3, 0, 0, 0, 6, -6, 0, 0, 0, 0], [0, -4, -5, -4, 0, 5, -2, 1, -2, 4, 1, -5, -4, -2], [0, 4, -4, 4, 0, 7, -2, 1, -2, -5, 1, -4, 5, 3], [0, 0, 3, 0, 0, -7, 0, 7, 0, -4, 7, 3, 4, -3], [0, 0, 2, 0, 0, 3, -5, -4, -5, -1, -4, 2, 1, -1], [2, 0, 9, 0, -3, 0, 0, -1, 2, -4, 0, 0, 0, 0], [0, 3, -4, 3, 0, 4, -6, -1, -6, 2, -1, -4, -2, 6], [5, -2, 0, 0, -4, -5, -3, 0, 0, 0, -5, 1, -1, 4], [-10, 0, 1, 4, 3, 0, 0, 1, -5, -5, 0, 0, 0, 0], [3, 0, 0, 1, -4, 0, 0, -1, -6, 0, 0, 0, 0, 0], [0, 2, 0, 2, 0, -1, -1, 2, -1, -5, 2, 0, 5, -1], [11, 0, 3, -4, -6, 0, 0, 1, -1, 0, 0, 0, 0, 0], [18, -1, 0, 0, 4, -18, 0, 0, 0, 0, -3, 1, -1, -4], [1, -3, 0, 0, -6, -1, -2, 0, 0, 0, 0, -2, -6, 6], [1, 0, -4, 2, 1, 0, 0, -2, -1, -2, 0, 0, 0, 0], [-15, 0, 0, 0, 6, 15, 3, 0, 0, 0, 2, 0, 3, -6], [-8, -2, 0, 0, 3, 8, 2, 0, 0, 0, 1, -2, -4, -3], [-10, -3, 0, 0, -1, 10, 6, 0, 0, 0, 1, 2, 0, 1], [0, -2, 4, -2, 0, 0, 0, 2, 0, -5, 2, 4, 5, 0], [0, -4, 2, -4, 0, -16, 9, 0, 9, -2, 0, 2, 2, 2], [24, 1, 0, 0, -3, -24, 5, 0, 0, 0, -5, 7, -2, 3], [0, 1, 6, 1, 0, 3, 2, -1, 2, -3, -1, 6, 3, -1], [-4, 0, 0, 0, 2, 4, 3, 0, 0, 0, 8, 4, 8, -2], [0, -1, -8, -1, 0, 9, -5, 6, -5, -1, 6, -8, 1, 2], [-8, 0, 6, -2, -4, 0, 0, -5, 10, -1, 0, 0, 0, 0], [0, 1, 1, 1, 0, -6, 4, 2, 4, 2, 2, 1, -2, 1], [0, 7, -2, 7, 0, 9, -6, -8, -6, 6, -8, -2, -6, 3], [-10, 4, 0, 0, 1, 10, 0, 0, 0, 0, 3, 4, -4, -1], [2, 0, -2, -5, -5, 0, 0, 4, 8, -2, 0, 0, 0, 0], [0, -1, 4, -1, 0, 6, -11, -4, -11, 4, -4, 4, -4, 1], [0, 1, 4, 1, 0, 5, -1, -3, -1, -2, -3, 4, 2, 0], [4, 0, 0, 0, 5, -4, -3, 0, 0, 0, 2, -2, -2, -5], [-8, 0, 2, -3, -1, 0, 0, 2, 10, -2, 0, 0, 0, 0], [-5, 0, -6, 2, 3, 0, 0, -2, 0, 11, 0, 0, 0, 0], [10, 0, 0, 0, -5, -10, 4, 0, 0, 0, -3, -1, -5, 5], [0, 6, 1, 6, 0, -9, -3, -5, -3, 1, -5, 1, -1, 4], [-3, 3, 0, 0, -5, 3, -9, 0, 0, 0, 0, -4, 3, 5], [-4, 7, 0, 0, -5, 4, -3, 0, 0, 0, 0, -5, 5, 5], [0, 4, 2, 4, 0, -12, -2, -2, -2, -5, -2, 2, 5, 5], [0, -7, 4, -7, 0, 4, 8, -2, 8, -1, -2, 4, 1, -2], [0, 4, 2, 4, 0, -7, 1, -6, 1, 5, -6, 2, -5, 3], [13, -2, 0, 0, -6, -13, -1, 0, 0, 0, 0, -7, -2, 6], [0, 0, 5, 0, 0, -20, 4, -8, 4, 4, -8, 5, -4, 2], [0, 2, 4, 2, 0, 4, 2, -4, 2, -1, -4, 4, 1, -4], [-1, 0, 0, 1, -2, 0, 0, -2, 1, -8, 0, 0, 0, 0], [-3, 1, 0, 0, -8, 3, 7, 0, 0, 0, -3, -2, -2, 8], [5, 0, 3, -3, 1, 0, 0, 1, -3, -6, 0, 0, 0, 0], [8, 5, 0, 0, -6, -8, -1, 0, 0, 0, -2, 2, 2, 6], [-18, 7, 0, 0, -4, 18, -10, 0, 0, 0, 3, -5, -3, 4], [0, 8, -4, 8, 0, 4, -7, 1, -7, 1, 1, -4, -1, 0], [-7, 0, 0, 0, 3, 7, 3, 0, 0, 0, 4, 6, 1, -3], [8, 0, 0, -5, -3, 0, 0, 2, 7, -1, 0, 0, 0, 0], [-2, 1, 0, 0, 2, 2, 9, 0, 0, 0, -2, -2, 0, -2], [0, 3, -2, 3, 0, -4, 2, -7, 2, 4, -7, -2, -4, 2], [0, 4, -8, 4, 0, 8, -11, -5, -11, 3, -5, -8, -3, 4], [0, -2, 7, -2, 0, -15, 6, -4, 6, 3, -4, 7, -3, -2], [15, 0, 2, -3, 2, 0, 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[-3, 6, 0, 0, 5, 3, -15, 0, 0, 0, -2, -10, -5, -5], [0, 4, -5, 4, 0, -20, -11, -3, -11, 0, -3, -5, 0, -1], [0, 3, 9, 3, 0, 4, -13, 0, -13, -1, 0, 9, 1, -5], [-7, 3, 0, 0, 9, 7, 7, 0, 0, 0, 5, 0, 10, -9], [21, 0, -1, -11, -3, 0, 0, 3, 4, -3, 0, 0, 0, 0], [-15, 0, 0, -7, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0], [0, 0, -12, 2, 10, 0, 0, -3, -8, 13, 0, 0, 0, 0], [-19, -4, 0, 0, 7, 19, -3, 0, 0, 0, 0, -3, 0, -7], [0, 4, -11, 4, 0, 25, -10, 2, -10, 5, 2, -11, -5, 10], [0, -7, 10, -7, 0, -26, 15, 10, 15, -10, 10, 10, 10, -3], [1, 3, 0, 0, 1, -1, -1, 0, 0, 0, -7, 8, -2, -1], [23, 0, -5, 7, -1, 0, 0, 1, -8, 7, 0, 0, 0, 0], [-19, 11, 0, 0, 4, 19, -9, 0, 0, 0, 5, 3, 5, -4], [6, -15, 0, 0, 9, -6, 15, 0, 0, 0, 5, 4, 7, -9], [-6, 0, -2, 3, -5, 0, 0, -1, -8, 5, 0, 0, 0, 0], [0, 3, -6, 3, 0, 6, 4, 3, 4, -8, 3, -6, 8, 6], [-8, -6, 0, 0, 1, 8, 14, 0, 0, 0, -6, 11, 2, -1], [-19, 0, 0, -7, 2, 0, 0, 9, 13, 1, 0, 0, 0, 0], [-11, 0, -2, 6, 9, 0, 0, -5, 5, 5, 0, 0, 0, 0], [14, 0, 7, -2, 5, 0, 0, 1, 2, 2, 0, 0, 0, 0], [0, -6, -5, -6, 0, -14, -12, 2, -12, 6, 2, -5, -6, -8], [-13, 4, 0, 0, -4, 13, -2, 0, 0, 0, 3, -4, 5, 4], [25, 0, 5, 10, -8, 0, 0, -3, -4, -3, 0, 0, 0, 0], [16, -7, 0, 0, 10, -16, 7, 0, 0, 0, 6, 11, 13, -10], [0, -3, 1, -3, 0, 24, 7, 1, 7, 2, 1, 1, -2, -16], [0, -2, 2, -2, 0, -11, 6, -5, 6, -7, -5, 2, 7, -10], [-33, 1, 0, 0, 1, 33, 2, 0, 0, 0, 10, -1, -9, -1], [0, 7, 6, 7, 0, 15, -5, -6, -5, 1, -6, 6, -1, 3], [19, 0, 0, 0, -2, -19, -8, 0, 0, 0, 2, 7, 5, 2], [31, 0, 2, 2, 5, 0, 0, 5, 7, 3, 0, 0, 0, 0], [1, -4, 0, 0, 5, -1, -4, 0, 0, 0, 3, -4, 1, -5], [15, 0, -18, 7, 10, 0, 0, -3, -7, 1, 0, 0, 0, 0], [-22, 0, 7, -10, -9, 0, 0, 10, 11, -4, 0, 0, 0, 0], [8, 0, 8, -1, 5, 0, 0, 4, 6, 6, 0, 0, 0, 0], [19, 0, 5, -8, -8, 0, 0, 10, 1, -11, 0, 0, 0, 0], [-21, 0, 1, -6, -8, 0, 0, 4, 19, -9, 0, 0, 0, 0], [0, -8, -2, -8, 0, 22, 8, -1, 8, 6, -1, -2, -6, -2], [5, -7, 0, 0, 7, -5, 5, 0, 0, 0, 1, 2, 0, -7], [-33, 1, 0, 0, -6, 33, 4, 0, 0, 0, 0, 5, 3, 6], [9, 4, 0, 0, -6, -9, -11, 0, 0, 0, 1, -5, -7, 6], [0, -13, 0, -13, 0, 1, 2, 5, 2, 5, 5, 0, -5, 1], [27, 2, 0, 0, -6, -27, -7, 0, 0, 0, -18, 4, -7, 6], [0, 11, -15, 11, 0, -4, -16, 2, -16, 14, 2, -15, -14, 10], [0, -5, -2, -5, 0, 5, 16, -11, 16, 9, -11, -2, -9, -5], [19, -8, 0, 0, 5, -19, 10, 0, 0, 0, 14, 6, 12, -5], [8, -13, 0, 0, 1, -8, 6, 0, 0, 0, -8, -7, -12, -1], [4, 7, 0, 0, -8, -4, -12, 0, 0, 0, -4, -17, -4, 8], [14, 4, 0, 0, -4, -14, 13, 0, 0, 0, 5, -1, 6, 4], [13, 6, 0, 0, -12, -13, -6, 0, 0, 0, -5, -13, -16, 12], [0, 0, -1, 0, 0, 2, 6, 1, 6, -15, 1, -1, 15, -8], [0, -4, -7, -4, 0, 5, -6, 9, -6, -8, 9, -7, 8, -4], [0, 2, 0, 2, 0, 17, -4, -1, -4, 11, -1, 0, -11, 6], [-30, 0, 1, 3, 14, 0, 0, -1, -8, -8, 0, 0, 0, 0], [-8, 0, -1, 13, 6, 0, 0, -4, 10, -5, 0, 0, 0, 0], [28, 2, 0, 0, -5, -28, -7, 0, 0, 0, 1, -15, 2, 5], [35, 0, -3, -5, -7, 0, 0, 3, -8, -3, 0, 0, 0, 0], [1, -8, 0, 0, 1, -1, 14, 0, 0, 0, -6, 7, -7, -1], [-16, -17, 0, 0, 18, 16, 12, 0, 0, 0, 6, 17, 0, -18], [3, -7, 0, 0, 8, -3, -10, 0, 0, 0, -1, 1, -6, -8], [13, -1, 0, 0, -6, -13, 7, 0, 0, 0, 6, 10, 17, 6], [0, 4, 9, 4, 0, -5, 5, -3, 5, -7, -3, 9, 7, 6], [0, 7, -3, 7, 0, 17, -1, 5, -1, 9, 5, -3, -9, 1], [-20, 0, 11, 0, -7, 0, 0, 0, 23, 6, 0, 0, 0, 0], [36, 0, 1, -11, -10, 0, 0, 7, 6, 2, 0, 0, 0, 0], [17, 0, -6, 6, 5, 0, 0, 0, -9, 2, 0, 0, 0, 0], [-10, 6, 0, 0, -7, 10, -7, 0, 0, 0, 4, -2, -16, 7], [-8, 0, -2, 10, 4, 0, 0, -3, -14, -9, 0, 0, 0, 0], [8, 0, 11, 1, -2, 0, 0, 2, 9, 6, 0, 0, 0, 0], [-34, 0, 4, -14, -4, 0, 0, 1, 23, -6, 0, 0, 0, 0], [19, 6, 0, 0, -10, -19, 0, 0, 0, 0, -9, 9, -4, 10], [-9, -11, 0, 0, 6, 9, 7, 0, 0, 0, 0, 5, 0, -6], [19, 0, 11, -3, -1, 0, 0, -7, 5, 2, 0, 0, 0, 0], [-1, 4, 0, 0, 8, 1, 2, 0, 0, 0, 9, -3, 8, -8], [1, 0, 0, 0, -5, -1, -11, 0, 0, 0, -1, -3, 11, 5], [-19, 3, 0, 0, -5, 19, -14, 0, 0, 0, -6, -5, -1, 5], [24, 0, 2, -6, -6, 0, 0, 0, -11, -1, 0, 0, 0, 0], [-2, 0, 0, 0, -2, 2, 7, 0, 0, 0, 8, -10, -4, 2], [-27, 16, 0, 0, -8, 27, -8, 0, 0, 0, -1, -1, -2, 8], [0, 5, -8, 5, 0, 36, -18, 1, -18, 0, 1, -8, 0, 3], [14, -1, 0, 0, -8, -14, 7, 0, 0, 0, -1, -6, 3, 8], [0, -5, 4, -5, 0, 11, 7, 2, 7, -9, 2, 4, 9, -4], [-9, 0, 5, -9, 9, 0, 0, -5, 16, 3, 0, 0, 0, 0], [-32, 0, 4, -5, 3, 0, 0, -18, 2, 12, 0, 0, 0, 0], [-36, 0, 13, -5, -14, 0, 0, 7, 19, -13, 0, 0, 0, 0], [5, 5, 0, 0, 8, -5, 2, 0, 0, 0, 7, 14, 15, -8], [31, 0, 11, 4, 6, 0, 0, 5, 2, -3, 0, 0, 0, 0], [0, -11, 5, -11, 0, -10, 4, 0, 4, 0, 0, 5, 0, -12], [0, -2, -10, -2, 0, 12, -4, 15, -4, 2, 15, -10, -2, 5], [-14, 0, 12, 2, 6, 0, 0, -16, 2, 4, 0, 0, 0, 0], [27, 2, 0, 0, -14, -27, -5, 0, 0, 0, 4, 2, 11, 14], [-16, 9, 0, 0, -8, 16, -2, 0, 0, 0, 5, -13, 11, 8], [0, -5, 1, -5, 0, 17, -4, 2, -4, 3, 2, 1, -3, -12], [0, 6, 0, 6, 0, 30, -19, 1, -19, -1, 1, 0, 1, -11], [0, -10, 7, -10, 0, -23, 11, -13, 11, 7, -13, 7, -7, 9], [0, -5, -6, -5, 0, -3, 15, 5, 15, -2, 5, -6, 2, -5], [0, -7, 13, -7, 0, 8, 10, 1, 10, -13, 1, 13, 13, -13], [0, 4, -9, 4, 0, 1, -13, -12, -13, -2, -12, -9, 2, 10], [-6, -7, 0, 0, 8, 6, 15, 0, 0, 0, -15, 19, 0, -8], [0, 7, 3, 7, 0, 21, -9, 8, -9, -8, 8, 3, 8, 3], [16, 0, -5, 3, 14, 0, 0, -7, -10, 14, 0, 0, 0, 0], [0, 14, -3, 14, 0, 12, -17, -12, -17, 13, -12, -3, -13, 6], [0, -5, -1, -5, 0, 14, -5, 7, -5, 2, 7, -1, -2, -14], [0, 10, -18, 10, 0, -4, -10, 0, -10, 7, 0, -18, -7, 7], [-6, 2, 0, 0, -12, 6, -3, 0, 0, 0, -3, -16, -11, 12], [-10, 0, -18, 0, 18, 0, 0, 3, -4, 13, 0, 0, 0, 0], [0, -1, 2, -1, 0, 16, -11, -9, -11, 15, -9, 2, -15, -5], [0, -5, -2, -5, 0, -25, 2, -6, 2, 2, -6, -2, -2, 14], [-18, -5, 0, 0, 1, 18, 10, 0, 0, 0, 1, 2, -12, -1], [0, 10, 1, 10, 0, -4, -18, 3, -18, 3, 3, 1, -3, 2], [41, 0, -9, 3, 9, 0, 0, 0, -6, -6, 0, 0, 0, 0], [9, -5, 0, 0, 10, -9, 13, 0, 0, 0, -10, 17, 4, -10], [0, -11, 18, -11, 0, -17, 12, 10, 12, -2, 10, 18, 2, -15], [0, -5, -4, -5, 0, 42, 13, -1, 13, -5, -1, -4, 5, -5], [0, -18, 4, -18, 0, -16, 5, 11, 5, -11, 11, 4, 11, -10], [-23, 0, 6, -17, -10, 0, 0, 2, 11, -14, 0, 0, 0, 0], [0, 0, -15, 0, 0, 18, -12, 9, -12, 3, 9, -15, -3, 4], [-3, -1, 0, 0, -4, 3, -7, 0, 0, 0, -2, -10, -3, 4], [33, -4, 0, 0, 8, -33, 3, 0, 0, 0, 1, 9, 11, -8], [17, 0, 7, 1, 5, 0, 0, -12, 4, 6, 0, 0, 0, 0], [-22, 0, 9, -14, -9, 0, 0, -1, 10, 2, 0, 0, 0, 0], [-12, 0, 4, 2, -10, 0, 0, -4, 18, 0, 0, 0, 0, 0], [-10, 0, -8, 9, 3, 0, 0, -1, 0, 9, 0, 0, 0, 0], [31, -5, 0, 0, 4, -31, -10, 0, 0, 0, -4, 5, -3, -4], [0, 0, -9, 0, 0, -6, -2, -12, -2, -2, -12, -9, 2, 14], [0, 0, -10, 0, 0, 1, -15, -12, -15, 3, -12, -10, -3, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4020_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4020_2_q_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_4020_2_q_k(:prec:=14)
    chi := MakeCharacter_4020_q();
    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_4020_2_q_k();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_4020_2_q_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4020_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![253009, -109654, 196412, -55186, 86396, -21740, 22767, -4257, 3972, -662, 443, -48, 29, -3, 1]>,<11,R![112225, -954415, 7545961, -4939116, 3345335, -1182652, 526254, -150791, 52703, -10626, 2508, -314, 66, -6, 1]>,<17,R![165649, -289784, 1646137, 3493090, 6676031, 4625169, 2356939, 755693, 197164, 36897, 6523, 900, 132, 12, 1]>],Snew);
    return Vf;
end function;