
// Make newform 1620.3.t.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1620_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1620_t_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_1620_3_t_b();".
// 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_1620_3_t_b();".
// 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, 0, -3, 0, 8, 0, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, -1], [-9, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 24, 0, -8, 0, 3, 0], [0, 15, 0, -40, 0, 16, 0, -5], [21, 58, -56, 0, 24, 0, -7, 2], [9, 4, 0, -88, 0, 32, 1, -12], [21, 0, -56, 0, 24, 0, -7, 0]];
    Rf_basisdens := [1, 2, 2, 8, 2, 8, 4, 4];
    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_1620_t();"
function MakeCharacter_1620_t()
    N := 1620;
    order := 6;
    char_gens := [811, 1541, 1297];
    v := [6, 1, 3];
    // 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_1620_t_Hecke();"
function MakeCharacter_1620_t_Hecke(Kf)
    N := 1620;
    order := 6;
    char_gens := [811, 1541, 1297];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0], [-1, 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 := 3;
    raw_aps := [[0, 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, 0, -2, 0, 0, 0], [0, 0, 1, 0, 0, 2, 2, -1], [0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 7, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 3, 0, 0, 6, 6, -3], [-34, 0, 0, 34, 0, 0, 0, 0], [0, 11, 0, 0, 11, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, -2], [0, 0, 0, 0, -7, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1], [0, 0, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 22, 0, -11], [0, 0, 0, -74, 0, 0, 0, 0], [0, 23, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, 0, 12, 0], [0, -14, 0, 0, -14, 0, 0, 0], [0, 0, 0, 78, 0, 0, 0, 0], [0, 0, -23, 0, 0, 0, 0, -23], [0, 0, -2, 0, 0, 0, -4, 0], [0, 0, 0, 0, 8, 0, 0, 0], [0, 0, 17, 0, 0, 34, 34, -17], [0, -26, 0, 0, 0, 0, 0, 0], [0, 0, -33, 0, 0, 0, 0, 0], [74, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9], [0, -4, 0, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, 18, 0, -9], [0, 0, -39, 0, 0, 0, 0, -39], [118, 0, 0, -118, 0, 0, 0, 0], [0, 0, 0, 0, 0, 22, 0, -11], [0, 0, 0, -34, 0, 0, 0, 0], [0, 23, 0, 0, 0, 0, 0, 0], [0, 17, 0, 0, 17, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 15], [0, 0, 17, 0, 0, 0, 0, 17], [0, 0, 29, 0, 0, 0, 58, 0], [-166, 0, 0, 0, 0, 0, 0, 0], [0, 0, 24, 0, 0, 48, 48, -24], [0, 8, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [114, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, 68, 0, 0, 0], [0, 0, -55, 0, 0, 0, 0, -55], [-154, 0, 0, 154, 0, 0, 0, 0], [0, 0, -41, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -40, 0, 20], [0, 0, 0, 206, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, 0, -9], [0, 0, -47, 0, 0, 0, 0, -47], [0, 0, -15, 0, 0, 0, -30, 0], [-398, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 73, 0, 0, 0], [0, 0, -6, 0, 0, -12, -12, 6], [0, 13, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 87], [0, -123, 0, 0, -123, 0, 0, 0], [0, 0, 0, 0, 0, -108, 0, 54], [0, 0, 0, 0, -142, 0, 0, 0], [0, 0, 137, 0, 0, 0, 0, 137], [0, 0, 0, -202, 0, 0, 0, 0], [0, -92, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -57], [0, 0, 0, 118, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, -7], [0, 0, 6, 0, 0, 0, 12, 0], [0, 0, 0, 0, 88, 0, 0, 0], [0, 33, 0, 0, 0, 0, 0, 0], [394, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -17], [0, 0, -31, 0, 0, -62, -62, 31], [0, -163, 0, 0, -163, 0, 0, 0], [0, 0, 0, 0, 0, -40, 0, 20], [206, 0, 0, -206, 0, 0, 0, 0], [0, 0, 0, 0, 0, -46, 0, 23], [0, 0, 0, 38, 0, 0, 0, 0], [0, 0, 68, 0, 0, 0, 136, 0], [0, -68, 0, 0, -68, 0, 0, 0], [0, 0, 0, 366, 0, 0, 0, 0], [0, 0, 129, 0, 0, 0, 0, 129], [0, 0, -48, 0, 0, 0, -96, 0], [0, 0, 0, 0, -26, 0, 0, 0], [0, 0, -57, 0, 0, -114, -114, 57], [0, 24, 0, 0, 0, 0, 0, 0], [0, 0, -33, 0, 0, 0, 0, 0], [0, 0, -64, 0, 0, -128, -128, 64], [0, 162, 0, 0, 162, 0, 0, 0], [0, 0, 0, 0, 0, 30, 0, -15], [486, 0, 0, -486, 0, 0, 0, 0], [0, 0, -177, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, 3], [0, 0, -110, 0, 0, 0, -220, 0], [0, -73, 0, 0, -73, 0, 0, 0], [-86, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -41, 0, 0, 0], [0, 0, -81, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 223], [0, 0, -70, 0, 0, -140, -140, 70], [-394, 0, 0, 394, 0, 0, 0, 0], [0, 152, 0, 0, 152, 0, 0, 0], [0, 0, -159, 0, 0, 0, 0, -159], [0, 0, 135, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -12, 0, 6], [0, 0, 0, -434, 0, 0, 0, 0], [0, 164, 0, 0, 0, 0, 0, 0], [0, 211, 0, 0, 211, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 103], [0, 0, 0, 1046, 0, 0, 0, 0], [-46, 0, 0, 0, 0, 0, 0, 0], [0, 0, -128, 0, 0, -256, -256, 128], [0, -201, 0, 0, 0, 0, 0, 0], [0, 0, -129, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -241], [0, 0, 91, 0, 0, 182, 182, -91], [-1082, 0, 0, 1082, 0, 0, 0, 0], [0, 0, 0, 0, 264, 0, 0, 0], [0, 0, 137, 0, 0, 0, 0, 137], [0, 0, 71, 0, 0, 0, 0, 0], [0, 0, 0, -922, 0, 0, 0, 0], [0, 0, 53, 0, 0, 0, 106, 0], [0, 0, 0, 966, 0, 0, 0, 0], [0, 0, 124, 0, 0, 0, 248, 0], [0, 0, 0, 0, -102, 0, 0, 0], [0, -41, 0, 0, 0, 0, 0, 0], [1082, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 255], [958, 0, 0, -958, 0, 0, 0, 0], [0, -193, 0, 0, -193, 0, 0, 0], [0, 0, 0, 0, 0, 252, 0, -126], [1326, 0, 0, -1326, 0, 0, 0, 0], [0, 0, -33, 0, 0, 0, 0, 0], [0, 283, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -137], [0, 0, 138, 0, 0, 0, 276, 0], [874, 0, 0, 0, 0, 0, 0, 0], [0, 0, 29, 0, 0, 58, 58, -29], [0, -342, 0, 0, 0, 0, 0, 0], [0, 0, 103, 0, 0, 0, 0, 0], [1002, 0, 0, 0, 0, 0, 0, 0], [0, 0, 34, 0, 0, 68, 68, -34], [0, 0, 0, 0, 193, 0, 0, 0], [0, 0, 201, 0, 0, 0, 0, 201], [278, 0, 0, -278, 0, 0, 0, 0], [0, 0, -129, 0, 0, 0, 0, 0], [0, -157, 0, 0, 0, 0, 0, 0], [0, 0, -60, 0, 0, 0, -120, 0], [0, 237, 0, 0, 237, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 63], [0, 0, 0, 0, 129, 0, 0, 0], [0, 0, 36, 0, 0, 72, 72, -36], [1314, 0, 0, 0, 0, 0, 0, 0], [0, 0, 168, 0, 0, 336, 336, -168], [0, 322, 0, 0, 322, 0, 0, 0], [0, 0, 0, 0, 0, 82, 0, -41], [0, 0, -375, 0, 0, 0, 0, -375], [0, 0, -185, 0, 0, 0, 0, 0], [0, 134, 0, 0, 0, 0, 0, 0], [0, 0, 15, 0, 0, 0, 30, 0], [0, 0, 0, 0, 0, 0, 0, 39], [0, 0, -415, 0, 0, 0, 0, -415], [-1646, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -291, 0, 0, 0], [34, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 159], [0, 0, 21, 0, 0, 42, 42, -21], [246, 0, 0, -246, 0, 0, 0, 0], [0, 0, 0, 0, 0, 316, 0, -158], [0, 0, 0, 0, -122, 0, 0, 0], [-1394, 0, 0, 1394, 0, 0, 0, 0], [0, 0, 0, 0, 0, 156, 0, -78], [0, 0, 0, 406, 0, 0, 0, 0], [0, 0, -105, 0, 0, 0, -210, 0], [0, -158, 0, 0, -158, 0, 0, 0], [0, 0, 0, 1046, 0, 0, 0, 0], [0, 0, 0, 0, -92, 0, 0, 0], [0, 0, 239, 0, 0, 478, 478, -239], [0, 133, 0, 0, 0, 0, 0, 0], [0, 0, -345, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -449], [0, 0, 117, 0, 0, 234, 234, -117], [0, -103, 0, 0, -103, 0, 0, 0], [0, 0, 0, 0, -497, 0, 0, 0], [-514, 0, 0, 514, 0, 0, 0, 0], [0, 0, 232, 0, 0, 0, 464, 0], [0, 432, 0, 0, 432, 0, 0, 0], [0, 0, 97, 0, 0, 0, 0, 97], [-326, 0, 0, 0, 0, 0, 0, 0], [0, 0, -69, 0, 0, -138, -138, 69], [0, 0, -417, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 327], [-882, 0, 0, 882, 0, 0, 0, 0], [0, 0, 0, 0, 279, 0, 0, 0], [0, 0, 105, 0, 0, 0, 0, 105], [0, 0, 175, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 162, 0, -81], [0, 0, 0, -642, 0, 0, 0, 0], [0, 129, 0, 0, 0, 0, 0, 0], [0, 0, 0, 958, 0, 0, 0, 0], [0, 0, 7, 0, 0, 0, 14, 0], [0, 0, 15, 0, 0, 0, 0, 0], [1554, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -289], [0, 0, 118, 0, 0, 236, 236, -118], [566, 0, 0, -566, 0, 0, 0, 0], [0, -548, 0, 0, -548, 0, 0, 0], [0, 0, 0, 0, 0, 54, 0, -27], [0, 0, 0, 0, -62, 0, 0, 0], [0, 0, 57, 0, 0, 0, 0, 57], [0, 0, 0, 0, 0, -164, 0, 82], [0, 0, 0, 1598, 0, 0, 0, 0], [0, -472, 0, 0, 0, 0, 0, 0], [0, 0, 164, 0, 0, 328, 328, -164], [0, 0, -49, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 23], [-1914, 0, 0, 1914, 0, 0, 0, 0], [46, 0, 0, -46, 0, 0, 0, 0], [0, 0, 0, 0, 0, 144, 0, -72], [0, -424, 0, 0, -424, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 63], [0, 0, 0, -1434, 0, 0, 0, 0], [0, 0, -311, 0, 0, 0, 0, -311], [0, 0, -88, 0, 0, 0, -176, 0], [0, 0, 0, 0, 38, 0, 0, 0], [0, 0, -47, 0, 0, -94, -94, 47], [0, -197, 0, 0, 0, 0, 0, 0], [-2278, 0, 0, 0, 0, 0, 0, 0], [1966, 0, 0, -1966, 0, 0, 0, 0], [0, 0, 0, 0, 0, -60, 0, 30], [0, 0, 0, 0, 569, 0, 0, 0], [0, 0, -31, 0, 0, 0, 0, -31], [-162, 0, 0, 162, 0, 0, 0, 0], [0, 0, -329, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 54, 0, -27], [0, 0, -162, 0, 0, 0, -324, 0], [0, 0, -447, 0, 0, 0, 0, -447], [1994, 0, 0, 0, 0, 0, 0, 0], [0, 158, 0, 0, 0, 0, 0, 0], [-406, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 407], [0, 0, -258, 0, 0, -516, -516, 258], [0, 12, 0, 0, 12, 0, 0, 0], [0, 0, 0, 0, 0, 642, 0, -321], [86, 0, 0, -86, 0, 0, 0, 0], [0, 0, -241, 0, 0, 0, 0, 0], [0, -681, 0, 0, 0, 0, 0, 0], [0, 0, -112, 0, 0, 0, -224, 0], [0, 0, 0, 0, 0, 0, 0, 527], [0, 0, 0, 1966, 0, 0, 0, 0], [0, 0, 425, 0, 0, 0, 0, 425], [0, 0, -267, 0, 0, 0, -534, 0], [-326, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 29, 0, 0, 0], [0, 0, 287, 0, 0, 0, 0, 0], [0, 406, 0, 0, 406, 0, 0, 0], [0, 0, 0, 0, 108, 0, 0, 0], [0, 0, 425, 0, 0, 0, 0, 425], [-122, 0, 0, 122, 0, 0, 0, 0], [0, -103, 0, 0, -103, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -57], [0, 0, 0, -922, 0, 0, 0, 0], [0, 0, 153, 0, 0, 0, 306, 0], [0, 0, -278, 0, 0, -556, -556, 278], [0, -91, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -225], [-2954, 0, 0, 2954, 0, 0, 0, 0], [0, -179, 0, 0, -179, 0, 0, 0], [0, 0, 0, 0, -206, 0, 0, 0], [-914, 0, 0, 914, 0, 0, 0, 0], [0, 868, 0, 0, 0, 0, 0, 0], [0, 422, 0, 0, 422, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -409], [0, 0, 0, -874, 0, 0, 0, 0], [-2686, 0, 0, 0, 0, 0, 0, 0], [0, 0, -357, 0, 0, -714, -714, 357], [0, 0, 0, 0, 0, 0, 0, 735], [1918, 0, 0, -1918, 0, 0, 0, 0], [0, 0, 97, 0, 0, 0, 0, 97], [0, 0, 0, 1286, 0, 0, 0, 0], [0, 353, 0, 0, 0, 0, 0, 0], [0, 0, -316, 0, 0, 0, -632, 0], [0, -708, 0, 0, -708, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 119], [0, 0, 0, -82, 0, 0, 0, 0], [0, 0, 304, 0, 0, 0, 608, 0], [0, 0, 271, 0, 0, 542, 542, -271], [0, 0, 63, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -345], [0, 0, 0, 0, 0, 78, 0, -39], [0, 0, 0, 0, 683, 0, 0, 0], [0, 0, 0, 0, 0, -342, 0, 171], [0, 0, 0, 606, 0, 0, 0, 0], [0, 0, 225, 0, 0, 0, 0, 225], [0, 0, -365, 0, 0, 0, -730, 0], [0, 0, 0, 0, 159, 0, 0, 0], [0, -146, 0, 0, 0, 0, 0, 0], [0, 0, -585, 0, 0, 0, 0, 0], [-3326, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1620_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1620_3_t_b();".
// 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_1620_3_t_b(:prec:=8)
    chi := MakeCharacter_1620_t();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1620_3_t_b();".
// 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_1620_3_t_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1620_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![4096, 0, -64, 0, 1]>,<17,R![-980, 0, 1]>],Snew);
    return Vf;
end function;