
// Make newform 4020.2.q.j in Magma, downloaded from the LMFDB on 20 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_j();".
// 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_j();".
// 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 := [81, -675, 4869, -7038, 10392, -915, 4060, -1096, 798, -53, 30, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-213693056523513, -1517742210897, 10811165099478, 104356504552668, 14682367453768, 13095529592739, 5330805726143, 2805124258771, 172229907007, 117178408376, 10214206017, 3882223435], [52425010656, 16205642379, -115195094163, -685298055296, -141488390232, -166492823853, 11206364358, -39224741894, 1015516512, -1527067746, -8330484, -52241027], [-55721026347906, 5516033458671, -39200281496859, -185567304670144, -47555282937464, -57724581657177, 6580973669456, -13717814681058, 458507766964, -532350662473, 1146781944, -18208180335], [4077726580611, -29398686612363, 34828675835328, -61806807816216, 3831440668683, -23548647527818, 6031178620618, -4639211296224, 294981453038, -174825009876, 5354831941, -5825001184], [-37541561931450, 5214052586664, -37054233616944, -181135313239308, -44952548682560, -54563120045688, 6193265819117, -12966377092280, 433270280680, -502794995191, 1079200536, -17210840888], [136395667228734, -29776415210874, 211593294448401, 958346242038351, 255696353386256, 313376622706938, -40028044437344, 74666534698007, -2664598755331, 2892833484967, -12877775961, 99006849920], [-31631837094135, 1769719192585947, -2018885081453712, 3583136534671908, -191204974749472, 1366034458141293, -349406047914461, 270520940621483, -17126290059559, 10196741983762, -310530618669, 340088114552], [136216640185254, -983103667365429, 1617132043932120, -2130621634155120, 243087280642482, -858137457348736, 241497537648025, -167293976034885, 11552472669332, -6274826024079, 225445934212, -208917470212], [96443745926355, -4264311775658751, 6156279135530136, -10930594128742824, 478464801171081, -4175717123058529, 1063455151268233, -840619144671909, 52500424408727, -31728339975486, 934960473697, -1061664495556], [31816447355970, -1184548981579686, 2031316101387126, -3608703960512079, 78133074301936, -1382628324291494, 349950039715648, -284857156713719, 17453392390817, -10768625400391, 304890086297, -361920964801]];
    Rf_basisdens := [1, 1, 41206785731685, 392445578397, 13735595243895, 3532010205573, 8241357146337, 41206785731685, 41206785731685, 24724071439011, 123620357195055, 41206785731685];
    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], [1, 0, 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, 1, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 2, 0, 0, 1, 0, 1, 0], [0, -1, 0, 1, 0, -2, 0, 1, 0, 1, 0, 1], [0, 1, -1, -1, 0, 3, 0, 0, -1, 0, 0, 0], [1, 1, 0, 0, 0, -1, 1, 0, 1, 1, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1], [0, -2, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0], [-1, 1, 0, 0, 0, 1, 0, 0, -1, 0, -1, -1], [-1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 2], [-2, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0], [0, 0, -2, -2, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, 4, 0, -1, -1, -1, -1, -1], [0, -1, -1, 1, 1, -1, -1, -1, 0, -1, -1, 0], [4, 1, 0, 0, 0, -4, 1, 0, 0, 1, 1, 0], [0, 2, 0, -2, -1, -1, 0, 1, 0, 2, 1, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1], [0, -1, 1, 1, 1, 3, 0, 1, 1, 3, -1, 1], [1, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, -3, 0, 0, 0, -1, 0, 3, 0], [-2, -1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2], [-5, -3, 0, 0, 0, 5, 0, 0, 1, 0, 0, -1], [0, 0, -1, 1, 2, 0, 1, -2, 0, 0, 0, 0], [-4, 0, -1, -1, -2, 0, 1, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, -1, 0, -1, -1, -3, -1], [1, 0, -1, 2, -1, 0, -2, 0, 0, 0, 0, 0], [8, 0, 1, 1, 1, 0, -2, 0, 0, 0, 0, 0], [-6, 0, 1, 1, -2, 0, -2, -1, 0, 0, 0, 0], [-6, 0, 1, 3, 1, 0, -2, -3, 0, 0, 0, 0], [0, 4, -2, -4, 1, -9, 0, 1, -2, -1, -1, 1], [0, 2, -1, -2, 0, 2, 0, -3, -1, -1, 0, -3], [-6, 0, 0, 0, 0, 6, 3, 0, 1, 3, -1, 1], [-3, 3, 0, 0, 0, 3, 0, 0, -3, 0, -2, 2], [0, 2, 2, -2, -2, 1, 0, 1, 2, -2, 2, 1], [0, 0, -4, 0, 1, 0, -2, -1, 0, 0, 0, 0], [1, -2, 0, 0, 0, -1, 3, 0, 2, 3, 1, 2], [0, -2, 2, 2, -2, 1, 0, -1, 2, -1, 2, -1], [5, 0, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0], [3, 1, 0, 0, 0, -3, -2, 0, 3, -2, 2, 0], [0, -5, 3, 5, -2, -3, 0, 2, 3, 1, 2, 2], [0, 1, 2, -1, -1, 2, 0, 0, 2, 1, 1, 0], [-5, 0, -3, -2, 3, 0, 2, 0, 0, 0, 0, 0], [-6, 2, 0, 0, 0, 6, 2, 0, 3, 2, 5, 2], [0, 0, -1, 0, 1, 16, 0, 2, -1, -1, -1, 2], [-1, -1, 0, 0, 0, 1, -1, 0, 1, -1, 0, 1], [11, -2, 0, 0, 0, -11, 2, 0, 2, 2, -2, 0], [4, 0, 3, 3, -1, 0, -1, 1, 0, 0, 0, 0], [0, 4, -3, -4, 3, 6, 0, -1, -3, 1, -3, -1], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, -3], [4, 0, 1, 3, 0, 0, -1, 3, 0, 0, 0, 0], [-16, 0, 2, 0, -1, 0, -2, -1, 0, 0, 0, 0], [10, 0, 2, 0, 1, 0, -1, 1, 0, 0, 0, 0], [-4, 0, 2, 0, -1, 0, 1, -3, 0, 0, 0, 0], [0, 4, -1, -4, 1, 0, 0, 1, -1, -2, -1, 1], [7, 0, 0, 3, 0, 0, -1, 2, 0, 0, 0, 0], [-12, 0, -3, -3, 0, 0, 2, 0, 0, 0, 0, 0], [0, 1, 3, -1, -1, 3, 0, 0, 3, -2, 1, 0], [0, 0, -3, -5, -2, 0, 1, 2, 0, 0, 0, 0], [4, 0, 1, 1, -1, 0, -3, 4, 0, 0, 0, 0], [0, -2, 1, 2, 0, 8, 0, 0, 1, 5, 0, 0], [-12, -1, 0, 0, 0, 12, -1, 0, 2, -1, 4, -1], [0, -1, -1, 1, -1, 13, 0, 1, -1, 2, 1, 1], [-6, -1, 0, 0, 0, 6, 2, 0, 0, 2, -2, -3], [7, 0, 3, 2, 1, 0, 3, 2, 0, 0, 0, 0], [0, -3, -3, 3, 5, -3, 0, -1, -3, -3, -5, -1], [1, 0, 0, -3, 0, 0, -2, -2, 0, 0, 0, 0], [10, 4, 0, 0, 0, -10, -3, 0, -3, -3, -4, 0], [0, 1, 0, 0, 0, 0, -1, 0, -6, -1, -1, 1], [0, 7, -2, -7, 3, 8, 0, -4, -2, -6, -3, -4], [0, -6, 5, 6, 0, -8, 0, 2, 5, -1, 0, 2], [0, 2, -3, -2, 0, 8, 0, 0, -3, -4, 0, 0], [10, 0, -1, 1, 2, 0, 1, 3, 0, 0, 0, 0], [3, 0, -2, 3, 2, 0, -1, -2, 0, 0, 0, 0], [1, 1, 0, 0, 0, -1, -4, 0, 1, -4, 1, -3], [0, 6, -1, -6, 3, -2, 0, -1, -1, 1, -3, -1], [0, -1, 2, 1, 3, -8, 0, -1, 2, 1, -3, -1], [-3, -5, 0, 0, 0, 3, 0, 0, 1, 0, 0, 1], [-4, -5, 0, 0, 0, 4, -1, 0, 0, -1, -2, 0], [0, -6, 4, 6, -6, -1, 0, 3, 4, 1, 6, 3], [2, 3, 0, 0, 0, -2, 1, 0, -4, 1, 0, -4], [2, -4, 0, 0, 0, -2, 4, 0, 3, 4, -1, 3], [1, -5, 0, 0, 0, -1, 4, 0, 3, 4, -1, 3], [-11, 0, -1, -4, 3, 0, 0, -2, 0, 0, 0, 0], [0, -3, 0, 3, 1, -8, 0, 1, 0, 0, -1, 1], [0, 1, 3, -1, -4, -1, 0, 3, 3, 1, 4, 3], [0, 0, 1, 0, -2, 6, 0, 0, 1, -1, 2, 0], [0, 0, 1, 0, -1, -6, 0, 3, 1, 5, 1, 3], [-5, 0, 1, -2, -2, 0, 5, -4, 0, 0, 0, 0], [0, -4, 2, 4, -1, 7, 0, 4, 2, 5, 1, 4], [3, -4, 0, 0, 0, -3, -2, 0, 0, -2, -3, 1], [-2, 0, -1, 1, 2, 0, -3, 2, 0, 0, 0, 0], [-16, 0, -6, -4, 3, 0, 5, -2, 0, 0, 0, 0], [0, -2, -3, 2, 5, -12, 0, -3, -3, -3, -5, -3], [-6, 0, -4, 4, 3, 0, -5, 3, 0, 0, 0, 0], [-10, -1, 0, 0, 0, 10, -3, 0, 0, -3, 1, -1], [-13, -4, 0, 0, 0, 13, 0, 0, 2, 0, -2, 2], [3, 0, 3, 4, -3, 0, -2, 7, 0, 0, 0, 0], [16, -3, 0, 0, 0, -16, 4, 0, 2, 4, 0, 0], [13, 1, 0, 0, 0, -13, 2, 0, -5, 2, 0, 1], [-1, -3, 0, 0, 0, 1, 2, 0, -1, 2, 0, 3], [0, 4, 2, -4, 2, 7, 0, 0, 2, -3, -2, 0], [0, -1, -6, 1, 4, -12, 0, -4, -6, -1, -4, -4], [-9, 1, 0, 0, 0, 9, 2, 0, 5, 2, 4, 2], [0, 1, -5, -1, 4, -9, 0, -5, -5, -1, -4, -5], [4, -1, 0, 0, 0, -4, -3, 0, 2, -3, -2, -2], [0, -1, -4, 1, 4, -6, 0, -4, -4, -8, -4, -4], [10, 0, 2, 2, -1, 0, 0, 0, 0, 0, 0, 0], [0, -2, -1, 2, 3, -2, 0, 0, -1, 0, -3, 0], [0, 3, 1, -3, -5, 11, 0, 1, 1, -3, 5, 1], [1, -3, 0, 0, 0, -1, -2, 0, 1, -2, 2, -1], [0, 0, -5, -7, 7, 0, 1, -3, 0, 0, 0, 0], [0, 12, -5, -12, 3, 4, 0, -3, -5, -4, -3, -3], [0, -3, 0, 3, 3, -4, 0, -4, 0, 0, -3, -4], [-22, 0, 0, 0, 0, 22, 2, 0, 1, 2, 3, 3], [-8, 0, 1, 1, -3, 0, -3, 5, 0, 0, 0, 0], [4, 0, 5, 5, -3, 0, 0, -1, 0, 0, 0, 0], [-16, -2, 0, 0, 0, 16, -4, 0, -5, -4, 0, 2], [0, 4, 0, -4, -1, -11, 0, 5, 0, -2, 1, 5], [-6, 6, 0, 0, 0, 6, -9, 0, -1, -9, -3, -1], [15, -3, 0, 0, 0, -15, 6, 0, -1, 6, 1, 6], [0, -5, 5, 5, -1, -1, 0, 3, 5, -2, 1, 3], [0, -4, 4, 4, 0, -27, 0, 0, 4, 1, 0, 0], [0, -2, 1, 2, -3, -12, 0, 3, 1, 3, 3, 3], [-5, 8, 0, 0, 0, 5, 0, 0, -6, 0, 2, -1], [0, -1, 6, 1, -2, 12, 0, 1, 6, 1, 2, 1], [0, -2, 2, 2, 0, 25, 0, 4, 2, 1, 0, 4], [1, 0, -4, -3, -2, 0, 1, -8, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -4, 0, -2, -4, 1, -2], [-16, 0, 3, -1, 0, 0, -2, -2, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, -3, 0, -4, -3, -2, -6], [-24, -5, 0, 0, 0, 24, -1, 0, -4, -1, -1, 1], [0, -3, 4, 3, -1, -4, 0, 2, 4, 1, 1, 2], [-10, 6, 0, 0, 0, 10, -5, 0, -3, -5, 1, -3], [11, 0, -3, 4, 3, 0, 1, -3, 0, 0, 0, 0], [8, 1, 0, 0, 0, -8, 5, 0, 0, 5, 2, 2], [0, 2, 0, -2, -1, 19, 0, 4, 0, -3, 1, 4], [0, 1, 4, -1, -3, 12, 0, 6, 4, 5, 3, 6], [0, -3, -6, 3, 0, -10, 0, -1, -6, 0, 0, -1], [8, 0, 2, 0, -1, 0, 1, 6, 0, 0, 0, 0], [-5, 3, 0, 0, 0, 5, -1, 0, 3, -1, 3, -1], [0, 3, -2, -3, -2, 8, 0, 1, -2, -1, 2, 1], [9, 0, 6, -3, -2, 0, 4, 1, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 3, 4, 0, 2, 4, -2, 7], [-18, 0, 3, -1, 3, 0, -1, 2, 0, 0, 0, 0], [0, -9, 2, 9, 0, 4, 0, -2, 2, -2, 0, -2], [0, 7, -1, -7, 0, 9, 0, -3, -1, -4, 0, -3], [5, 0, 0, 0, 0, -5, -3, 0, 6, -3, 2, 3], [0, -5, 3, 5, 0, 17, 0, -2, 3, 0, 0, -2], [-12, -5, 0, 0, 0, 12, 9, 0, 2, 9, -2, 0], [-17, 0, 1, 6, -3, 0, -2, 3, 0, 0, 0, 0], [0, 0, -4, 0, 5, 17, 0, 1, -4, -8, -5, 1], [27, 0, 0, -3, -2, 0, -2, 4, 0, 0, 0, 0], [1, 0, 4, 7, -2, 0, -2, 9, 0, 0, 0, 0], [-1, 0, -4, 1, -1, 0, -4, 0, 0, 0, 0, 0], [8, 0, 0, 8, 0, 0, -1, 4, 0, 0, 0, 0], [0, 0, -3, 1, 6, 0, -1, 1, 0, 0, 0, 0], [-6, -10, 0, 0, 0, 6, 1, 0, 3, 1, -1, -6], [2, -1, 0, 0, 0, -2, 4, 0, 4, 4, -1, -4], [0, 6, -3, -6, -3, -4, 0, 2, -3, -7, 3, 2], [2, 0, 1, -7, -2, 0, 4, 1, 0, 0, 0, 0], [10, 0, 4, -2, -3, 0, 2, 5, 0, 0, 0, 0], [-7, 0, 3, 8, 2, 0, -3, -2, 0, 0, 0, 0], [8, -9, 0, 0, 0, -8, 0, 0, -4, 0, -6, -3], [8, 0, -1, -9, 1, 0, 5, 1, 0, 0, 0, 0], [-4, 0, -7, 3, 4, 0, -5, 4, 0, 0, 0, 0], [0, -3, -5, 3, 1, 7, 0, 0, -5, -1, -1, 0], [-11, 8, 0, 0, 0, 11, -3, 0, -2, -3, 4, 4], [0, 3, -8, -3, 5, -14, 0, -5, -8, -7, -5, -5], [16, 1, 0, 0, 0, -16, -12, 0, -4, -12, 1, -4], [0, 7, 1, -7, -1, -5, 0, -5, 1, -5, 1, -5], [17, -4, 0, 0, 0, -17, -5, 0, 2, -5, 5, -1], [11, -3, 0, 0, 0, -11, -2, 0, -1, -2, -2, -2], [-6, 0, 6, 12, -6, 0, -1, 4, 0, 0, 0, 0], [10, 0, 3, 1, 2, 0, 3, -4, 0, 0, 0, 0], [23, 0, 1, -2, -5, 0, -5, 1, 0, 0, 0, 0], [0, 2, -4, -2, 4, 3, 0, -3, -4, -10, -4, -3], [-15, 1, 0, 0, 0, 15, -4, 0, 1, -4, -2, 1], [-9, 0, 4, 5, -2, 0, -6, 5, 0, 0, 0, 0], [-8, 1, 0, 0, 0, 8, -6, 0, -4, -6, -11, -5], [0, 2, 0, -2, 3, 11, 0, -3, 0, 0, -3, -3], [-18, 0, 3, 1, 5, 0, -1, 1, 0, 0, 0, 0], [0, 4, -2, -4, -3, 11, 0, 8, -2, -3, 3, 8], [0, -2, -2, 2, -1, 1, 0, -6, -2, 0, 1, -6], [5, -3, 0, 0, 0, -5, 4, 0, 7, 4, -1, 4], [4, 0, -4, -6, 4, 0, 5, -5, 0, 0, 0, 0], [-8, 0, -4, -4, 2, 0, 9, 0, 0, 0, 0, 0], [-3, 4, 0, 0, 0, 3, -9, 0, -2, -9, 0, -1], [15, 0, 7, 6, -4, 0, -13, 4, 0, 0, 0, 0], [0, -5, -1, 5, -1, 1, 0, 3, -1, -3, 1, 3], [0, -9, 8, 9, -2, 14, 0, 4, 8, 9, 2, 4], [-26, 0, 2, 6, 5, 0, -4, -4, 0, 0, 0, 0], [-2, -2, 0, 0, 0, 2, 3, 0, 1, 3, 0, -5], [3, -4, 0, 0, 0, -3, 5, 0, 2, 5, -4, 3], [0, -2, 5, 2, 6, 28, 0, 1, 5, -7, -6, 1], [0, -8, 0, 8, 1, -19, 0, 5, 0, 7, -1, 5], [9, 0, 2, 3, 2, 0, -1, 1, 0, 0, 0, 0], [4, 2, 0, 0, 0, -4, -3, 0, -1, -3, -12, -1], [-24, 0, -3, -11, 4, 0, 4, 0, 0, 0, 0, 0], [-2, 0, 8, 10, -5, 0, -6, 1, 0, 0, 0, 0], [26, -7, 0, 0, 0, -26, 5, 0, 4, 5, -5, 0], [0, 1, -4, -1, -5, -20, 0, 1, -4, -1, 5, 1], [0, 2, -1, -2, 2, 6, 0, 1, -1, -6, -2, 1], [0, 1, -2, -1, -2, 2, 0, -8, -2, -6, 2, -8], [0, 4, -3, -4, -1, 2, 0, 5, -3, 3, 1, 5], [-3, 0, 1, -2, 9, 0, -5, 1, 0, 0, 0, 0], [0, 2, 10, -2, -9, -7, 0, -1, 10, 1, 9, -1], [0, -6, 7, 6, -4, 6, 0, -3, 7, 2, 4, -3], [9, -1, 0, 0, 0, -9, 6, 0, 1, 6, -3, 3], [-2, -2, 0, 0, 0, 2, -4, 0, 7, -4, -2, 8], [0, 14, -6, -14, 2, 5, 0, -2, -6, -9, -2, -2], [0, -2, 1, 2, 10, -8, 0, 5, 1, 5, -10, 5], [1, 8, 0, 0, 0, -1, -2, 0, 0, -2, 2, 1], [6, 0, 3, 7, -3, 0, 3, -8, 0, 0, 0, 0], [6, 5, 0, 0, 0, -6, -1, 0, 0, -1, -2, 3], [3, -8, 0, 0, 0, -3, 9, 0, 6, 9, -2, -1], [5, 0, 2, -1, -3, 0, 0, 6, 0, 0, 0, 0], [0, 4, 4, -4, -1, 13, 0, 1, 4, 10, 1, 1], [0, -10, 3, 10, 4, -22, 0, 3, 3, 7, -4, 3], [-9, 4, 0, 0, 0, 9, 5, 0, 0, 5, -2, -2], [-8, 0, -2, 4, 5, 0, -4, 1, 0, 0, 0, 0], [-19, -12, 0, 0, 0, 19, 6, 0, 4, 6, 7, 2], [3, -3, 0, 0, 0, -3, 2, 0, 5, 2, 4, 6], [26, 0, -1, -5, 9, 0, 9, -7, 0, 0, 0, 0], [0, -2, -2, 2, 7, 13, 0, 3, -2, -3, -7, 3], [18, 4, 0, 0, 0, -18, 1, 0, -5, 1, -2, -5], [-10, 0, 1, -7, -3, 0, -5, 10, 0, 0, 0, 0], [19, 0, -6, -7, -4, 0, -1, -6, 0, 0, 0, 0], [-4, 1, 0, 0, 0, 4, 0, 0, -6, 0, 8, 1], [4, 0, 3, 9, -2, 0, 5, -3, 0, 0, 0, 0], [0, -5, 6, 5, 8, -12, 0, 4, 6, 4, -8, 4], [-6, 0, -8, 4, 2, 0, 1, -3, 0, 0, 0, 0], [0, 9, -3, -9, -6, 25, 0, -5, -3, -9, 6, -5], [12, 0, -7, -9, -2, 0, -1, -4, 0, 0, 0, 0], [0, -2, -8, 2, 2, 3, 0, -3, -8, 3, -2, -3], [0, -2, -1, 2, 0, 10, 0, -5, -1, -7, 0, -5], [0, -1, 11, 1, -8, 9, 0, 5, 11, 0, 8, 5], [0, 14, -4, -14, -5, 27, 0, 2, -4, -6, 5, 2], [-15, 0, 0, 11, 3, 0, 3, 3, 0, 0, 0, 0], [-20, 4, 0, 0, 0, 20, -5, 0, -7, -5, -1, -3], [0, 8, -8, -8, 0, 17, 0, 2, -8, -6, 0, 2], [15, -5, 0, 0, 0, -15, 6, 0, -1, 6, 6, 1], [0, -5, 0, 5, 5, 2, 0, -9, 0, -1, -5, -9], [6, 3, 0, 0, 0, -6, 5, 0, 8, 5, 1, 1], [1, 0, 6, -1, -1, 0, -4, 2, 0, 0, 0, 0], [-13, 3, 0, 0, 0, 13, -1, 0, 1, -1, 11, 5], [-32, 10, 0, 0, 0, 32, 2, 0, 3, 2, 7, 3], [-18, 0, -1, -9, 6, 0, 0, -2, 0, 0, 0, 0], [5, 0, -3, -2, 2, 0, 3, -3, 0, 0, 0, 0], [-19, 0, -2, -11, -3, 0, 0, 8, 0, 0, 0, 0], [10, 9, 0, 0, 0, -10, -11, 0, -2, -11, 3, -2], [0, -3, 9, 3, -13, 5, 0, 6, 9, 3, 13, 6], [0, 9, -1, -9, 9, -9, 0, -9, -1, -2, -9, -9], [1, 7, 0, 0, 0, -1, 4, 0, -1, 4, 2, 9], [0, 0, 6, 0, -1, 3, 0, 1, 6, -3, 1, 1], [-6, -4, 0, 0, 0, 6, 0, 0, 1, 0, 4, 3], [8, 0, -1, 1, 5, 0, -4, 6, 0, 0, 0, 0], [0, -11, 7, 11, 2, -5, 0, -6, 7, 1, -2, -6], [0, -1, 5, 1, -5, -13, 0, -2, 5, 9, 5, -2], [13, 0, 4, -1, -3, 0, -2, -4, 0, 0, 0, 0], [25, 0, -2, -9, 0, 0, 9, -2, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 3, -5, 0, 8, -5, 7, -3], [18, 3, 0, 0, 0, -18, 11, 0, -2, 11, 2, 6], [0, 6, -3, -6, -5, 44, 0, -1, -3, -1, 5, -1], [-16, 0, 9, 5, 5, 0, -6, -6, 0, 0, 0, 0], [19, 0, 0, 9, 0, 0, -9, -4, 0, 0, 0, 0], [7, 0, -2, 5, 8, 0, -2, -4, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, -11, 0, -2, -11, 2, 6], [0, -1, 2, 1, -10, -12, 0, 3, 2, 3, 10, 3], [3, 1, 0, 0, 0, -3, 0, 0, -1, 0, 11, 8], [13, 6, 0, 0, 0, -13, -11, 0, 0, -11, -3, 0], [21, 0, 5, 8, 4, 0, -14, 1, 0, 0, 0, 0], [-4, -4, 0, 0, 0, 4, -6, 0, -3, -6, -5, -1], [26, 0, -4, 0, 4, 0, -5, -7, 0, 0, 0, 0], [0, 0, -2, 0, 3, 7, 0, -6, -2, 5, -3, -6], [31, 0, 4, 3, 0, 0, -3, 0, 0, 0, 0, 0], [-12, 0, 3, 1, -7, 0, -9, 4, 0, 0, 0, 0], [14, -15, 0, 0, 0, -14, 5, 0, 10, 5, -3, -3], [-15, 0, -11, -10, 6, 0, 4, -1, 0, 0, 0, 0], [4, 0, -10, -10, -1, 0, 1, -5, 0, 0, 0, 0], [-3, 0, 4, 5, -2, 0, 2, 8, 0, 0, 0, 0], [8, 0, 7, 15, 0, 0, -9, 2, 0, 0, 0, 0], [-3, 0, -5, -4, -1, 0, 2, -6, 0, 0, 0, 0], [13, 0, -2, -9, 5, 0, -5, -6, 0, 0, 0, 0], [0, 14, -14, -14, 4, -35, 0, -4, -14, -8, -4, -4], [28, 0, -3, 5, 1, 0, 2, 1, 0, 0, 0, 0], [2, 10, 0, 0, 0, -2, -11, 0, -5, -11, -2, -2], [0, 6, 0, -6, -10, -5, 0, 6, 0, -3, 10, 6], [-15, 3, 0, 0, 0, 15, -3, 0, -3, -3, -3, 6], [0, 5, -14, -5, 5, -10, 0, 0, -14, 4, -5, 0], [0, 4, -4, -4, 7, 17, 0, 0, -4, 5, -7, 0], [-1, 0, -7, 2, 6, 0, 1, -9, 0, 0, 0, 0], [0, 3, 6, -3, -5, -16, 0, 3, 6, 0, 5, 3], [-1, -8, 0, 0, 0, 1, 8, 0, 8, 8, 9, -2], [0, 5, -1, -5, -7, 43, 0, -2, -1, -4, 7, -2], [0, -8, 7, 8, 0, 8, 0, 8, 7, 12, 0, 8], [0, 8, -1, -8, -1, -36, 0, 1, -1, -5, 1, 1], [14, 12, 0, 0, 0, -14, -10, 0, -3, -10, -2, -5], [-6, 3, 0, 0, 0, 6, 8, 0, 4, 8, -6, -1], [-6, 0, -5, -9, 7, 0, 7, -8, 0, 0, 0, 0], [-2, 6, 0, 0, 0, 2, -1, 0, 9, -1, -1, -5], [0, 11, -9, -11, 8, -15, 0, -12, -9, -6, -8, -12], [0, -7, -11, 7, 9, -15, 0, -4, -11, -6, -9, -4], [-26, 10, 0, 0, 0, 26, -1, 0, 3, -1, 4, 3], [-1, 0, 1, 0, -6, 0, 10, -10, 0, 0, 0, 0], [-18, 0, -4, 2, -5, 0, 1, 4, 0, 0, 0, 0], [0, 0, -6, 8, 7, 0, 0, -12, 0, 0, 0, 0], [27, -9, 0, 0, 0, -27, 3, 0, 1, 3, -3, -4], [0, -11, -4, 11, 0, 16, 0, -3, -4, 10, 0, -3], [0, -4, -1, 4, -11, 12, 0, 3, -1, 0, 11, 3], [20, -6, 0, 0, 0, -20, -9, 0, -5, -9, -4, -1], [-7, 0, -7, -16, 6, 0, 4, -4, 0, 0, 0, 0], [29, 10, 0, 0, 0, -29, -4, 0, -4, -4, -3, -5], [-6, 0, 0, 0, 0, 6, 6, 0, -1, 6, 10, 1], [-16, 0, 3, -1, -2, 0, 1, -1, 0, 0, 0, 0], [0, 0, 2, 0, 3, -29, 0, -2, 2, 3, -3, -2], [-8, 2, 0, 0, 0, 8, -4, 0, 3, -4, -1, 2], [-13, 0, 8, 1, -13, 0, 5, 0, 0, 0, 0, 0], [11, 0, -3, -14, -2, 0, 10, -7, 0, 0, 0, 0], [-29, 0, -3, 8, 8, 0, -7, 2, 0, 0, 0, 0], [0, -5, 4, 5, -10, 14, 0, 1, 4, 9, 10, 1], [21, 4, 0, 0, 0, -21, 4, 0, -2, 4, -1, -6], [-7, 0, -8, -7, -9, 0, -5, -5, 0, 0, 0, 0], [15, -14, 0, 0, 0, -15, 3, 0, 2, 3, 0, -3], [0, -4, -2, 4, 9, 23, 0, -1, -2, -5, -9, -1], [0, 4, 0, -4, 3, -35, 0, 2, 0, -2, -3, 2], [18, -6, 0, 0, 0, -18, 3, 0, 1, 3, -3, -4], [0, -6, 1, 6, -5, -4, 0, 7, 1, 1, 5, 7], [-12, 3, 0, 0, 0, 12, 4, 0, 2, 4, 10, 3], [-13, 0, 3, 8, -6, 0, -4, -1, 0, 0, 0, 0], [-1, -13, 0, 0, 0, 1, 5, 0, -1, 5, -2, -1], [17, 0, -2, -11, -1, 0, 11, -4, 0, 0, 0, 0], [-10, 0, -5, 9, 9, 0, -7, -2, 0, 0, 0, 0], [-2, 0, -1, 9, -7, 0, -1, 5, 0, 0, 0, 0], [-2, 0, 6, 12, -2, 0, 1, 3, 0, 0, 0, 0], [14, 0, -14, -14, 0, 0, 3, 3, 0, 0, 0, 0], [0, 18, -6, -18, -3, 5, 0, 2, -6, -9, 3, 2], [-28, 8, 0, 0, 0, 28, -1, 0, 5, -1, 6, 3], [-12, -23, 0, 0, 0, 12, 18, 0, 8, 18, 6, 1], [-5, 0, 0, 0, 0, 5, -6, 0, -6, -6, -7, -3], [0, 9, 1, -9, -10, 17, 0, 9, 1, -1, 10, 9], [-8, -11, 0, 0, 0, 8, 10, 0, 10, 10, 6, 4], [0, 3, -2, -3, 4, -12, 0, -1, -2, 3, -4, -1], [0, 3, -9, -3, 8, -19, 0, 1, -9, -3, -8, 1], [21, 13, 0, 0, 0, -21, -5, 0, -3, -5, -1, -9], [-4, -2, 0, 0, 0, 4, -8, 0, -3, -8, -7, 2], [4, -1, 0, 0, 0, -4, -13, 0, 4, -13, 0, 3], [11, 4, 0, 0, 0, -11, -5, 0, -6, -5, -11, -1], [36, -9, 0, 0, 0, -36, -4, 0, -6, -4, -9, -5], [0, 11, 2, -11, 3, 4, 0, -1, 2, -7, -3, -1], [0, -1, 6, 1, 7, 40, 0, -3, 6, -8, -7, -3], [0, -2, 2, 2, -3, 41, 0, 2, 2, 4, 3, 2], [31, 0, 2, -3, 3, 0, 4, -7, 0, 0, 0, 0], [-23, 0, -6, -7, 0, 0, -4, -3, 0, 0, 0, 0], [9, -1, 0, 0, 0, -9, 4, 0, -7, 4, -2, 4], [-11, 0, -3, 2, -6, 0, 0, -8, 0, 0, 0, 0], [4, -2, 0, 0, 0, -4, -15, 0, 3, -15, -3, 4], [-2, -1, 0, 0, 0, 2, -1, 0, 0, -1, 6, 3], [20, 1, 0, 0, 0, -20, -2, 0, 8, -2, 7, 9], [28, 1, 0, 0, 0, -28, 2, 0, -4, 2, -8, -10], [0, 4, -6, -4, 1, 7, 0, -3, -6, 12, -1, -3], [0, 15, -9, -15, 6, -11, 0, -2, -9, -19, -6, -2], [-30, 0, -7, -7, -3, 0, -5, 4, 0, 0, 0, 0], [27, 0, -2, 7, 3, 0, -8, -5, 0, 0, 0, 0], [-35, 0, 3, 6, 1, 0, 4, -1, 0, 0, 0, 0], [-14, 10, 0, 0, 0, 14, -8, 0, 7, -8, 7, 3], [-8, 0, 2, 6, -3, 0, -2, 8, 0, 0, 0, 0], [18, 0, 2, 10, 6, 0, 0, -7, 0, 0, 0, 0], [-5, 0, 10, 5, -15, 0, 0, 0, 0, 0, 0, 0], [26, -3, 0, 0, 0, -26, 2, 0, 8, 2, 1, -1], [-11, -8, 0, 0, 0, 11, 0, 0, 6, 0, 0, 1], [-6, 0, 5, -3, -10, 0, -4, 0, 0, 0, 0, 0], [-12, 9, 0, 0, 0, 12, -4, 0, -2, -4, -3, -3], [9, 5, 0, 0, 0, -9, -5, 0, -11, -5, 6, 6], [34, -12, 0, 0, 0, -34, 3, 0, 3, 3, -3, 6], [1, 0, 10, -7, -6, 0, -8, 8, 0, 0, 0, 0], [4, 9, 0, 0, 0, -4, 3, 0, -6, 3, -4, 0], [-18, -3, 0, 0, 0, 18, 0, 0, 10, 0, -1, 7], [0, -17, 13, 17, 0, -1, 0, -3, 13, -2, 0, -3], [-14, 3, 0, 0, 0, 14, -1, 0, -4, -1, 9, -2], [0, 7, -2, -7, 8, -28, 0, 2, -2, 4, -8, 2], [-43, 0, 1, 8, -2, 0, 2, 2, 0, 0, 0, 0], [30, 0, 3, 1, 1, 0, -7, 3, 0, 0, 0, 0], [16, 0, -10, -6, -5, 0, -2, 1, 0, 0, 0, 0], [29, 4, 0, 0, 0, -29, 2, 0, -10, 2, 9, -2], [45, 0, 4, 11, -3, 0, -7, 5, 0, 0, 0, 0], [0, -3, 6, 3, 2, -46, 0, 1, 6, 3, -2, 1], [0, 1, 7, -1, -12, 15, 0, 7, 7, 0, 12, 7], [2, 0, 6, 0, 2, 0, 0, 14, 0, 0, 0, 0], [-16, -13, 0, 0, 0, 16, 4, 0, 2, 4, 2, -2], [-20, -5, 0, 0, 0, 20, -3, 0, 4, -3, -7, -7], [0, -13, -2, 13, 2, -4, 0, 3, -2, -8, -2, 3], [0, -6, 3, 6, 2, -30, 0, -7, 3, 2, -2, -7], [0, -15, 3, 15, 4, -9, 0, -6, 3, 9, -4, -6], [0, -8, 3, 8, -4, 24, 0, 1, 3, -3, 4, 1], [0, 0, -17, 0, 10, 20, 0, 4, -17, 0, -10, 4], [0, -4, -8, 4, -6, -15, 0, 3, -8, 0, 6, 3], [-5, 10, 0, 0, 0, 5, 3, 0, -6, 3, -11, -5], [0, 9, -1, -9, -5, 27, 0, -4, -1, -3, 5, -4], [-15, 0, -2, -7, 1, 0, 0, 11, 0, 0, 0, 0], [0, 4, 10, -4, -4, 15, 0, -1, 10, -5, 4, -1], [0, -4, 4, 4, -5, 5, 0, 5, 4, 17, 5, 5], [0, -1, 5, 1, -5, 15, 0, -9, 5, -6, 5, -9], [-24, -7, 0, 0, 0, 24, -5, 0, 4, -5, -3, -2], [-24, 0, 6, 6, -7, 0, 8, 2, 0, 0, 0, 0], [0, -16, 7, 16, -4, 20, 0, 3, 7, 10, 4, 3], [0, 4, -2, -4, 8, -29, 0, -2, -2, -6, -8, -2], [-33, -7, 0, 0, 0, 33, 0, 0, -3, 0, 4, -9], [0, -1, -8, 1, 3, -10, 0, -5, -8, 5, -3, -5], [-17, 0, 1, -12, 1, 0, 4, 8, 0, 0, 0, 0], [-15, -2, 0, 0, 0, 15, 0, 0, -14, 0, -10, 5], [0, 19, -3, -19, 11, -9, 0, -11, -3, -3, -11, -11], [0, 2, -1, -2, -16, -10, 0, 5, -1, -8, 16, 5], [0, -9, 0, 9, -1, 46, 0, 0, 0, 5, 1, 0], [-41, 0, -6, -11, 6, 0, 1, 2, 0, 0, 0, 0], [0, 11, 6, -11, -5, 20, 0, 1, 6, -3, 5, 1], [-8, -7, 0, 0, 0, 8, 6, 0, 6, 6, 2, 8], [29, 2, 0, 0, 0, -29, 2, 0, 0, 2, -5, -11], [-13, 0, -5, 0, 3, 0, 8, -8, 0, 0, 0, 0], [-19, 0, 9, 14, -8, 0, -11, 0, 0, 0, 0, 0], [8, 0, 2, -8, -6, 0, -8, 10, 0, 0, 0, 0], [6, 0, 11, -1, -2, 0, -3, 1, 0, 0, 0, 0], [-24, 3, 0, 0, 0, 24, -4, 0, -4, -4, -6, -5], [0, -1, 6, 1, -2, -12, 0, -5, 6, -1, 2, -5], [0, 7, 0, -7, -8, 18, 0, 0, 0, -10, 8, 0]];
    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_j();".
// 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_j(:prec:=12)
    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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4020_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1155625, 618125, 856300, -130675, 234221, -11920, 22138, -1559, 1415, -56, 46, -2, 1]>,<11,R![1476225, 1705860, 1767096, 644112, 301761, 64899, 28842, 4932, 1633, 181, 56, 5, 1]>,<17,R![154449, -349377, 662989, -477462, 329050, 30097, 58966, -1082, 2470, -257, 106, -9, 1]>],Snew);
    return Vf;
end function;