
// Make newform 420.2.l.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_420_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_420_l_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_420_2_l_h();".
// 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_420_2_l_h();".
// 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 := [256, -512, 576, -512, 288, -32, -144, 204, -170, 102, -36, -4, 18, -16, 9, -4, 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], [2048, -1472, 192, 224, -704, 528, -368, 42, 134, -100, 72, -18, 2, 17, -10, -3], [-3712, 6976, -8256, 6496, -2752, -528, 2512, -2786, 2002, -972, 152, 202, -266, 179, -78, 23], [8576, -10048, 3392, 1568, -4352, 4432, -3056, 866, 718, -1012, 744, -330, 58, 93, -98, 25], [-1472, 3392, -2528, 848, 376, -1104, 1136, -762, 218, 92, -164, 134, -66, 15, 14, -9], [-13952, 25920, -18240, 6368, 3168, -8336, 8608, -5602, 1634, 596, -1192, 954, -498, 107, 90, -49], [176, -144, -336, 572, -470, 252, -16, -142, 194, -142, 63, -9, -16, 19, -13, 4], [1856, -4608, 5856, -4640, 2040, 328, -1776, 2038, -1486, 700, -108, -158, 202, -137, 58, -13], [-22656, 43968, -40512, 23200, -3584, -10256, 15712, -13254, 7014, -1828, -904, 1582, -1254, 593, -114, -3], [-8192, 18368, -19584, 12960, -4032, -3200, 6736, -6534, 4070, -1492, -88, 662, -630, 361, -114, 13], [-19584, 45632, -46656, 29152, -7616, -9040, 16592, -15370, 8954, -2876, -584, 1682, -1458, 783, -198, 3], [4608, -10304, 10560, -6784, 1872, 1960, -3748, 3522, -2094, 716, 100, -374, 336, -185, 52, -5], [12928, -32064, 36544, -25888, 9376, 4560, -12112, 12506, -8202, 3324, -104, -1170, 1218, -743, 262, -43], [12928, -27712, 32704, -24544, 9824, 2928, -10384, 11154, -7778, 3468, -392, -922, 1098, -699, 286, -63], [-2176, -9920, 24512, -25056, 15840, -4208, -4736, 8958, -8254, 4756, -1512, -342, 910, -789, 410, -97], [5120, -20032, 31104, -26784, 13824, -656, -8200, 10890, -8658, 4396, -992, -706, 1086, -803, 370, -79]];
    Rf_basisdens := [1, 128, 256, 256, 64, 256, 16, 64, 256, 128, 256, 64, 256, 256, 256, 128];
    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_420_l();"
function MakeCharacter_420_l()
    N := 420;
    order := 2;
    char_gens := [211, 281, 337, 241];
    v := [1, 1, 1, 2];
    // 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_420_l_Hecke();"
function MakeCharacter_420_l_Hecke(Kf)
    N := 420;
    order := 2;
    char_gens := [211, 281, 337, 241];
    char_values := [[-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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 1, 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, 0, 0], [0, 0, 0, 0, -1, 1, -1, 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, 1, 0, 1, 0, 1, -1, 0, -1, 2, 1, -1], [1, 1, 1, 0, 1, 0, 1, 0, -1, 0, 1, 1, -1, 0, 1, 0], [0, 0, 0, -2, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0], [-1, -1, -1, 0, -1, 0, -1, 0, 1, 0, -1, -1, 1, 0, -1, 0], [0, 0, 0, 2, 1, 0, 1, 2, 0, 1, 0, 0, -1, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, -1], [1, -1, 1, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, -1, 1], [0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, -2, -1, 0, -1, 0], [0, 0, 0, -2, -1, 0, -1, 0, 0, 2, 0, 0, 1, 0, -1, -2], [-4, 0, 0, 0, 1, 0, -3, 0, 2, 1, 0, 4, -1, 0, -3, 1], [0, 0, 0, 2, 1, 0, 1, -4, 0, -1, 0, 0, -1, 0, 1, 1], [0, 0, 3, 2, -1, 3, -1, 3, 0, 2, -3, 0, -2, 2, 2, -2], [0, 0, -2, -2, 2, -4, 2, -2, 0, -1, 2, 0, 1, 0, -1, 1], [4, 0, 0, 0, 1, 0, 1, 0, -2, -1, 0, -2, 0, 0, 2, -1], [2, 0, 0, 0, 2, 0, 0, 0, -2, -1, 0, 2, -2, 0, 0, -1], [0, 0, 5, 2, -1, 5, -1, 5, 0, 3, -5, 0, -4, 2, 4, -3], [-1, -5, 1, 0, -2, 0, -4, 0, 1, 2, 1, 3, 1, 0, -5, 2], [-4, -2, -2, 0, -2, 0, -4, 0, 4, -1, -2, 0, 4, 0, -2, -1], [0, 0, 0, 4, -3, 0, -3, -2, 0, 1, 0, 0, -2, 0, 2, -1], [0, 0, 0, -2, 1, 0, 1, 2, 0, -3, 0, 0, 1, 0, -1, 3], [-3, -3, -5, 0, -2, 0, -4, 0, 3, 0, -5, -3, 5, 0, -1, 0], [0, 0, 0, 4, -3, 0, -3, -2, 0, 1, 0, 0, -2, 0, 2, -1], [-12, 0, 0, 0, 0, 0, -4, 0, 4, 2, 0, 4, 0, 0, -4, 2], [0, 0, 0, -4, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, -2, 0], [-4, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, -2, 1, 0, 1, 0], [0, 0, 1, 2, 1, -3, 1, 1, 0, 0, -1, 0, 0, -2, 0, 0], [-4, 0, 0, 0, -2, 0, 0, 0, 2, 1, 0, 2, 0, 0, -2, 1], [0, 0, -4, 2, -2, -2, -2, -4, 0, -3, 4, 0, 5, -4, -5, 3], [0, 0, -3, 0, -2, 1, -2, -3, 0, -3, 3, 0, 3, -6, -3, 3], [-1, -1, -5, 0, -1, 0, -1, 0, 1, 0, -5, -5, 3, 0, 1, 0], [0, 0, 0, 4, 4, 0, 4, -4, 0, -1, 0, 0, -2, 0, 2, 1], [2, 2, -2, 0, 1, 0, 3, 0, -2, 0, -2, -4, -1, 0, 3, 0], [-1, -1, -1, 0, -1, 0, -1, 0, 1, 0, -1, -1, 1, 0, -1, 0], [4, 0, 0, 0, -1, 0, 7, 0, -6, -3, 0, 0, -3, 0, 3, -3], [0, 0, 0, -2, -1, 0, -1, 6, 0, 0, 0, 0, 1, 0, -1, 0], [0, 0, -2, 4, 1, -8, 1, -2, 0, -2, 2, 0, 4, -4, -4, 2], [0, 0, -3, -4, 4, -7, 4, -3, 0, -1, 3, 0, 1, 2, -1, 1], [-12, 0, 0, 0, -1, 0, -1, 0, 2, 1, 0, 6, -2, 0, -4, 1], [0, 0, -1, -2, 1, -1, 1, -1, 0, -1, 1, 0, 0, -2, 0, 1], [-4, -6, 2, 0, -3, 0, -7, 0, 4, 1, 2, 6, 3, 0, -7, 1], [0, 0, 1, 2, -1, 1, -1, 1, 0, 2, -1, 0, 0, 6, 0, -2], [3, 1, 3, 0, 2, 0, 2, 0, -3, 1, 3, 3, -3, 0, 1, 1], [0, 8, 0, 0, 3, 0, 5, 0, 0, -4, 0, -2, -1, 0, 7, -4], [0, 0, 0, 0, 0, 0, -4, 0, 4, 2, 0, 0, 2, 0, -2, 2], [0, 0, 0, 4, -1, 0, -1, 6, 0, -1, 0, 0, -2, 0, 2, 1], [-8, 0, 0, 0, 1, 0, 1, 0, -2, -1, 0, 2, -2, 0, 0, -1], [0, 0, 7, 0, -2, 11, -2, 7, 0, 3, -7, 0, -7, -2, 7, -3], [0, 0, -3, 0, -4, 5, -4, -3, 0, -2, 3, 0, 3, -2, -3, 2], [-2, 0, 0, 0, 0, 0, 4, 0, -4, -2, 0, -4, 0, 0, 4, -2], [0, 0, 6, 6, 0, 0, 0, 6, 0, 3, -6, 0, -3, 0, 3, -3], [0, 0, -6, -2, 1, -6, 1, -6, 0, -4, 6, 0, 5, -4, -5, 4], [0, 0, 0, -8, 6, 0, 6, -2, 0, -2, 0, 0, 4, 0, -4, 2], [0, 0, 0, 8, 3, 0, 3, -2, 0, 1, 0, 0, -4, 0, 4, -1], [-5, -3, -5, 0, -2, 0, -6, 0, 5, -1, -5, -1, 7, 0, -1, -1], [2, -2, -2, 0, -3, 0, 3, 0, -2, 2, -2, -8, -3, 0, -3, 2], [0, 0, 0, -4, 2, 0, 2, 0, 0, -2, 0, 0, 2, 0, -2, 2], [-8, 0, 0, 0, 7, 0, 1, 0, -8, -4, 0, 0, -4, 0, 4, -4], [0, 0, 4, 4, -1, 2, -1, 4, 0, 4, -4, 0, -2, 8, 2, -4], [-4, 0, 0, 0, -5, 0, 9, 0, -4, -2, 0, -4, 0, 0, 4, -2], [0, 0, 2, 4, -4, 6, -4, 2, 0, 0, -2, 0, 0, -4, 0, 0], [-5, -1, -3, 0, 0, 0, -6, 0, 5, -2, -3, 3, 7, 0, 1, -2], [0, 0, -3, -2, 1, -3, 1, -3, 0, 0, 3, 0, 2, 6, -2, 0], [0, -8, -4, 0, -3, 0, -5, 0, 0, 4, -4, -2, 3, 0, -5, 4], [0, -2, 6, 0, -1, 0, -1, 0, 0, 1, 6, 6, -3, 0, -5, 1], [0, 0, 0, -8, 0, 0, 0, -2, 0, 0, 0, 0, 4, 0, -4, 0], [8, 0, 0, 0, 5, 0, -3, 0, -2, -1, 0, -2, 0, 0, 2, -1], [0, 0, 6, 2, 3, -2, 3, 6, 0, 4, -6, 0, -5, 4, 5, -4], [0, 0, -1, -8, 4, -1, 4, -1, 0, 0, 1, 0, -3, 2, 3, 0], [4, 0, 0, 0, -4, 0, 4, 0, 0, 0, 0, 4, -2, 0, -2, 0], [12, 8, 4, 0, 7, 0, 13, 0, -12, 2, 4, -2, -11, 0, 9, 2], [2, -2, 6, 0, 1, 0, -1, 0, -2, 2, 6, 8, -3, 0, -3, 2], [0, 0, 0, -6, 3, 0, 3, -2, 0, -4, 0, 0, 3, 0, -3, 4], [0, 0, 0, 8, 4, 0, 4, 8, 0, 5, 0, 0, -4, 0, 4, -5], [7, 11, 3, 0, 7, 0, 11, 0, -7, -2, 3, -1, -7, 0, 11, -2], [0, 0, 0, -10, 1, 0, 1, -4, 0, -3, 0, 0, 5, 0, -5, 3], [14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 2, 0, 2, 0], [0, 0, 2, -10, 2, 8, 2, 2, 0, 1, -2, 0, -7, 0, 7, -1], [-6, 0, 0, 0, -2, 0, 14, 0, -12, -6, 0, -8, -2, 0, 10, -6], [0, 0, -15, -12, 2, -7, 2, -15, 0, -8, 15, 0, 9, -2, -9, 8], [7, 11, 5, 0, 6, 0, 12, 0, -7, -2, 5, -1, -9, 0, 9, -2], [5, 7, -3, 0, 2, 0, 10, 0, -5, -1, -3, -11, -5, 0, 7, -1], [0, 0, 0, 16, -6, 0, -6, -2, 0, 6, 0, 0, -8, 0, 8, -6], [0, 0, 0, 12, -6, 0, -6, 4, 0, 4, 0, 0, -6, 0, 6, -4], [-2, -4, 0, 0, -3, 0, -3, 0, 2, 1, 0, 0, 1, 0, -5, 1], [0, 0, 0, 8, 1, 0, 1, 8, 0, -2, 0, 0, -4, 0, 4, 2], [-10, 0, 0, 0, -7, 0, 3, 0, 4, 2, 0, 2, 1, 0, -3, 2], [0, 0, 0, 0, 3, 0, 3, -14, 0, -5, 0, 0, 0, 0, 0, 5], [0, 0, 6, -4, -2, 14, -2, 6, 0, 4, -6, 0, -8, 4, 8, -4], [8, 0, 0, 0, -6, 0, 12, 0, -6, -3, 0, -6, 0, 0, 6, -3], [0, 0, -3, 4, 4, -15, 4, -3, 0, -3, 3, 0, 5, -6, -5, 3], [6, 6, 2, 0, 1, 0, 11, 0, -6, 0, 2, -8, -9, 0, 3, 0], [0, 0, 0, 6, 3, 0, 3, -2, 0, 0, 0, 0, -3, 0, 3, 0], [0, 0, 0, 0, 5, 0, 5, -6, 0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 10, 1, 0, 1, 8, 0, 4, 0, 0, -5, 0, 5, -4], [16, 0, 0, 0, -3, 0, 7, 0, -4, -2, 0, -16, 6, 0, 10, -2], [-14, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0, 4, -2, 0, -2, 0], [-10, 0, 0, 0, 4, 0, -6, 0, 2, 1, 0, 6, -2, 0, -4, 1], [0, 0, 13, 2, -1, 13, -1, 13, 0, 6, -13, 0, -12, -2, 12, -6], [0, 0, 0, -8, -1, 0, -1, 0, 0, 2, 0, 0, 4, 0, -4, -2], [0, 0, 0, 6, -3, 0, -3, 4, 0, 3, 0, 0, -3, 0, 3, -3], [8, 8, -4, 0, 5, 0, 11, 0, -8, 0, -4, -10, -5, 0, 11, 0], [7, 11, 5, 0, 4, 0, 14, 0, -7, -2, 5, -5, -11, 0, 7, -2], [0, 0, 0, 4, -1, 0, -1, 12, 0, 0, 0, 0, -2, 0, 2, 0], [0, 0, -6, 6, -7, 2, -7, -6, 0, -4, 6, 0, 9, -4, -9, 4], [0, 0, 9, 2, -1, 9, -1, 9, 0, 3, -9, 0, -8, -6, 8, -3], [-2, 0, 0, 0, 8, 0, -8, 0, 0, 0, 0, 12, -6, 0, -6, 0], [4, 0, 0, 0, 0, 0, 4, 0, -4, -2, 0, -8, 2, 0, 6, -2], [2, 10, -2, 0, 1, 0, 11, 0, -2, -4, -2, -12, -5, 0, 7, -4], [0, 0, 5, -2, 3, 1, 3, 5, 0, 2, -5, 0, -6, -2, 6, -2], [1, 9, 1, 0, 5, 0, 5, 0, -1, -4, 1, 1, -1, 0, 9, -4], [8, 14, 14, 0, 6, 0, 16, 0, -8, -3, 14, 4, -16, 0, 6, -3], [0, 0, 0, -10, 1, 0, 1, -4, 0, -7, 0, 0, 5, 0, -5, 7], [20, 0, 0, 0, -3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 10, 1, 0, 1, 2, 0, 0, 0, 0, -5, 0, 5, 0], [0, 0, 3, -6, 1, 7, 1, 3, 0, 0, -3, 0, -6, -6, 6, 0], [0, 0, 1, -4, 6, -7, 6, 1, 0, 1, -1, 0, -3, 2, 3, -1], [-16, 0, 0, 0, -9, 0, 3, 0, 6, 3, 0, 6, 0, 0, -6, 3], [4, -4, -4, 0, 2, 0, -2, 0, -4, 4, -4, 0, 2, 0, 2, 4], [0, 0, 8, 0, 1, 6, 1, 8, 0, 6, -8, 0, -8, 8, 8, -6], [0, 0, 0, 4, -4, 0, -4, -6, 0, -2, 0, 0, -2, 0, 2, 2], [-11, -15, -7, 0, -7, 0, -19, 0, 11, 2, -7, 5, 15, 0, -11, 2], [0, 0, 0, 8, -4, 0, -4, 0, 0, 7, 0, 0, -4, 0, 4, -7], [-10, 0, 0, 0, 8, 0, -12, 0, 4, 2, 0, 8, -2, 0, -6, 2], [0, 0, -8, -4, 2, -8, 2, -8, 0, -2, 8, 0, 6, 8, -6, 2], [4, 0, 0, 0, 6, 0, -2, 0, -4, -2, 0, 8, -6, 0, -2, -2], [-5, -13, -13, 0, -9, 0, -9, 0, 5, 4, -13, -13, 9, 0, -9, 4], [-8, 4, 0, 0, -1, 0, -3, 0, 8, -6, 0, 2, 5, 0, 1, -6], [0, 0, 0, -6, -3, 0, -3, -2, 0, -3, 0, 0, 3, 0, -3, 3], [0, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, 6, 3, 0, 3, 0], [6, 6, 10, 0, 5, 0, 7, 0, -6, 0, 10, 8, -9, 0, 3, 0], [0, 0, 0, -6, 3, 0, 3, -14, 0, -9, 0, 0, 3, 0, -3, 9], [-6, 0, 0, 0, -4, 0, -4, 0, 8, 4, 0, -8, 8, 0, 0, 4], [0, 0, -10, -8, 1, -4, 1, -10, 0, -4, 10, 0, 6, 4, -6, 4], [-12, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 4, -2, 0, -2, 0], [0, 0, -2, 4, -1, -4, -1, -2, 0, -2, 2, 0, 4, -4, -4, 2], [0, 0, 0, 2, -1, 0, -1, 0, 0, -7, 0, 0, -1, 0, 1, 7], [5, 1, 13, 0, 3, 0, 3, 0, -5, 2, 13, 13, -9, 0, -3, 2], [0, 0, 0, 0, -6, 0, -6, -8, 0, 3, 0, 0, 0, 0, 0, -3], [6, 0, 0, 0, -1, 0, -7, 0, 8, 4, 0, 2, 3, 0, -5, 4], [0, 0, 0, -8, -6, 0, -6, 10, 0, 4, 0, 0, 4, 0, -4, -4], [-12, 0, 0, 0, -1, 0, -1, 0, 2, 1, 0, -6, 4, 0, 2, 1], [0, 0, -2, -4, 4, -6, 4, -2, 0, -2, 2, 0, 0, -4, 0, 2], [9, 5, -3, 0, 7, 0, 7, 0, -9, 2, -3, -3, -3, 0, 11, 2], [0, 0, -8, -2, 3, -12, 3, -8, 0, -6, 8, 0, 7, -8, -7, 6], [-13, -21, -1, 0, -11, 0, -23, 0, 13, 4, -1, 11, 13, 0, -21, 4], [0, 0, 0, -4, 4, 0, 4, -22, 0, -4, 0, 0, 2, 0, -2, 4], [-10, -18, 2, 0, -5, 0, -23, 0, 10, 4, 2, 20, 13, 0, -15, 4], [0, 0, 0, 18, 1, 0, 1, 2, 0, 7, 0, 0, -9, 0, 9, -7], [14, 0, 0, 0, 0, 0, 6, 0, -6, -3, 0, -6, 0, 0, 6, -3], [0, 0, 0, 14, -3, 0, -3, 16, 0, 13, 0, 0, -7, 0, 7, -13], [-34, 0, 0, 0, -2, 0, 0, 0, 2, 1, 0, 6, -2, 0, -4, 1], [0, 0, -1, 4, -6, 7, -6, -1, 0, 0, 1, 0, 3, 2, -3, 0], [2, -8, 4, 0, -4, 0, -2, 0, -2, 5, 4, 2, -4, 0, -10, 5], [0, 0, 0, 14, -5, 0, -5, 2, 0, 7, 0, 0, -7, 0, 7, -7], [3, -9, -3, 0, 0, 0, -6, 0, -3, 6, -3, 3, 3, 0, -3, 6], [0, 0, 0, -12, 5, 0, 5, 12, 0, -2, 0, 0, 6, 0, -6, 2], [0, 0, 0, 4, -6, 0, -6, -2, 0, 6, 0, 0, -2, 0, 2, -6], [0, 0, 3, 0, 6, -9, 6, 3, 0, 3, -3, 0, -3, 6, 3, -3], [-2, 0, 0, 0, 9, 0, 1, 0, -10, -5, 0, 0, -5, 0, 5, -5], [0, 0, 0, -18, 8, 2, 8, 0, 0, -1, 0, 0, -9, -4, 9, 1], [0, 0, 9, 16, -6, 5, -6, 9, 0, 7, -9, 0, -1, 10, 1, -7], [0, 0, 0, -14, 7, 0, 7, -4, 0, -7, 0, 0, 7, 0, -7, 7], [-10, -16, -20, 0, -10, 0, -16, 0, 10, 3, -20, -14, 18, 0, -8, 3], [-3, -3, 5, 0, 1, 0, -7, 0, 3, 0, 5, 13, 3, 0, -3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_420_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_420_2_l_h();".
// 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_420_2_l_h(:prec:=16)
    chi := MakeCharacter_420_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_420_2_l_h();".
// 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_420_2_l_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_420_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![31104, 0, -11360, 0, 1400, 0, -66, 0, 1]>,<17,R![55296, 0, -19520, 0, 1968, 0, -76, 0, 1]>,<43,R![1616, -256, -96, 4, 1]>],Snew);
    return Vf;
end function;