
// Make newform 1620.3.t.a in Magma, downloaded from the LMFDB on 19 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_a();".
// 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_a();".
// 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 := [16, 0, -60, 0, 221, 0, -15, 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, -3034, 0, 0, 0, 0, 0, -1], [16, 0, -3315, 0, 221, 0, -15, 0], [-1708, -64, 0, 13039, 0, -884, -1, 60], [-1740, 3476, 6409, 0, -442, 0, 29, 1], [1740, 3476, -6409, 0, 442, 0, -29, 1], [0, -6180, 0, 22763, 0, -1547, 0, 103], [32, 3540, -6409, -13039, 442, 884, -30, -59]];
    Rf_basisdens := [1, 221, 884, 442, 442, 442, 442, 442];
    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], [0, 0, -1, 0, 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, 1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, -2, -2, 0, 0], [0, 0, 0, -3, 3, 0, 3, 3], [1, 0, 0, -1, 0, 1, 0, -1], [3, 0, 0, 3, 0, -3, 0, 3], [0, 0, -7, -1, 1, 0, 0, -1], [0, 9, 0, 0, 3, 3, 0, 0], [0, 0, 7, -6, 6, 0, 0, -6], [0, 2, 0, 6, 0, 6, -2, -6], [0, 0, 0, 1, -1, 0, -3, -1], [0, -1, 0, 0, -9, -9, 0, 0], [-14, 0, -14, 0, -4, 4, 0, 0], [-51, 0, 0, -6, 0, 6, 0, -6], [0, 0, 0, -11, 11, 0, 9, 11], [-11, 0, -11, 0, -3, 3, 0, 0], [0, 0, 0, -6, 6, 0, -8, 6], [0, -3, 0, -15, 0, -15, 3, 15], [0, -17, 0, -12, 0, -12, 17, 12], [-3, 0, -3, 0, -9, 9, 0, 0], [67, 0, 67, 0, -10, 10, 0, 0], [0, 9, 0, -11, 0, -11, -9, 11], [0, -28, 0, 0, -3, -3, 0, 0], [0, 6, 0, 0, -13, -13, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0], [-6, 0, 0, -21, 0, 21, 0, -21], [-61, 0, 0, -9, 0, 9, 0, -9], [0, 0, -138, -6, 6, 0, 0, -6], [0, -16, 0, -9, 0, -9, 16, 9], [0, 0, 0, -15, 15, 0, 6, 15], [-129, 0, -129, 0, -15, 15, 0, 0], [0, 0, 98, 6, -6, 0, 0, 6], [0, 0, 0, 4, -4, 0, 3, -4], [2, 0, 2, 0, 3, -3, 0, 0], [0, 0, 0, 33, -33, 0, -23, -33], [0, 56, 0, 30, 0, 30, -56, -30], [0, 0, 57, -27, 27, 0, 0, -27], [203, 0, 203, 0, -14, 14, 0, 0], [0, -6, 0, -31, 0, -31, 6, 31], [-13, 0, 0, -21, 0, 21, 0, -21], [0, -24, 0, 0, -18, -18, 0, 0], [0, 0, 0, 24, -24, 0, 19, -24], [125, 0, 0, -20, 0, 20, 0, -20], [-138, 0, 0, 3, 0, -3, 0, 3], [0, 0, -69, -3, 3, 0, 0, -3], [0, 62, 0, 0, 30, 30, 0, 0], [-25, 0, -25, 0, 1, -1, 0, 0], [0, 0, 235, 15, -15, 0, 0, 15], [-338, 0, 0, 8, 0, -8, 0, 8], [0, 0, 0, 8, -8, 0, -60, -8], [-19, 0, -19, 0, 27, -27, 0, 0], [0, -30, 0, -54, 0, -54, 30, 54], [0, 0, 201, 33, -33, 0, 0, 33], [202, 0, 202, 0, 2, -2, 0, 0], [0, -54, 0, -6, 0, -6, 54, 6], [151, 0, 0, 12, 0, -12, 0, 12], [0, 43, 0, 0, -3, -3, 0, 0], [0, 81, 0, 0, 57, 57, 0, 0], [0, 0, 0, 3, -3, 0, -49, -3], [0, 0, -327, 33, -33, 0, 0, 33], [0, 0, 0, 30, 0, 30, 0, -30], [0, 0, 0, -57, 57, 0, 75, 57], [0, -34, 0, 0, -3, -3, 0, 0], [-229, 0, -229, 0, -20, 20, 0, 0], [-58, 0, -58, 0, -36, 36, 0, 0], [0, 0, 0, 45, -45, 0, -55, -45], [0, 0, -78, -9, 9, 0, 0, -9], [109, 0, 109, 0, -51, 51, 0, 0], [314, 0, 314, 0, 10, -10, 0, 0], [0, 48, 0, 18, 0, 18, -48, -18], [0, -35, 0, 0, -72, -72, 0, 0], [0, 0, 0, -66, 66, 0, 0, 66], [-317, 0, 0, 15, 0, -15, 0, 15], [0, 0, -23, -59, 59, 0, 0, -59], [0, 3, 0, 0, 44, 44, 0, 0], [0, 77, 0, 81, 0, 81, -77, -81], [0, 0, 0, 74, -74, 0, -18, -74], [0, 0, 631, 6, -6, 0, 0, 6], [0, 0, 0, -13, 13, 0, 63, 13], [-331, 0, -331, 0, -39, 39, 0, 0], [0, -123, 0, -19, 0, -19, 123, 19], [0, -41, 0, -24, 0, -24, 41, 24], [177, 0, 177, 0, 36, -36, 0, 0], [63, 0, 63, 0, 57, -57, 0, 0], [0, 15, 0, 93, 0, 93, -15, -93], [0, 124, 0, 0, 69, 69, 0, 0], [0, -96, 0, 0, -27, -27, 0, 0], [0, 0, 0, -27, 27, 0, 66, 27], [-339, 0, 0, -12, 0, 12, 0, -12], [0, 105, 0, 0, 113, 113, 0, 0], [0, -75, 0, -69, 0, -69, 75, 69], [0, 0, 0, 33, -33, 0, -174, -33], [0, 0, 603, 3, -3, 0, 0, 3], [369, 0, 0, 57, 0, -57, 0, 57], [0, 0, 0, 9, -9, 0, 120, -9], [0, -51, 0, 1, 0, 1, 51, -1], [0, 23, 0, -81, 0, -81, -23, 81], [274, 0, 0, 42, 0, -42, 0, 42], [0, 25, 0, 0, 81, 81, 0, 0], [474, 0, 0, 87, 0, -87, 0, 87], [0, 0, 514, 19, -19, 0, 0, 19], [0, 111, 0, 0, 77, 77, 0, 0], [0, 0, -317, 75, -75, 0, 0, 75], [0, 71, 0, -3, 0, -3, -71, 3], [603, 0, 603, 0, -42, 42, 0, 0], [153, 0, 0, 135, 0, -135, 0, 135], [0, 0, 0, -171, 171, 0, 51, 171], [-263, 0, -263, 0, 78, -78, 0, 0], [0, 0, 0, 117, -117, 0, -35, -117], [0, -26, 0, 60, 0, 60, 26, -60], [0, 0, 49, 55, -55, 0, 0, 55], [326, 0, 326, 0, 84, -84, 0, 0], [143, 0, 0, -108, 0, 108, 0, -108], [0, -78, 0, 0, -98, -98, 0, 0], [0, 0, 0, 111, -111, 0, -93, -111], [-519, 0, 0, 45, 0, -45, 0, 45], [0, 0, 533, 14, -14, 0, 0, 14], [0, -117, 0, 0, -8, -8, 0, 0], [0, 0, -286, -66, 66, 0, 0, -66], [0, 33, 0, 0, 114, 114, 0, 0], [-310, 0, -310, 0, 16, -16, 0, 0], [425, 0, 0, -89, 0, 89, 0, -89], [221, 0, 221, 0, 93, -93, 0, 0], [0, -66, 0, 23, 0, 23, 66, -23], [318, 0, 318, 0, -90, 90, 0, 0], [0, -234, 0, -74, 0, -74, 234, 74], [0, -30, 0, 0, 81, 81, 0, 0], [0, 0, 0, 36, -36, 0, -52, -36], [-637, 0, 0, -3, 0, 3, 0, -3], [0, 0, 702, -48, 48, 0, 0, -48], [0, 0, -265, -30, 30, 0, 0, -30], [0, -310, 0, -120, 0, -120, 310, 120], [0, 0, 0, -57, 57, 0, 75, 57], [0, 0, 879, 30, -30, 0, 0, 30], [849, 0, 0, 15, 0, -15, 0, 15], [0, 0, 0, 93, -93, 0, -247, -93], [0, 0, -59, 88, -88, 0, 0, 88], [0, 207, 0, 147, 0, 147, -207, -147], [514, 0, 0, -66, 0, 66, 0, -66], [0, 132, 0, 0, 116, 116, 0, 0], [0, 0, 0, -120, 120, 0, 177, 120], [-695, 0, 0, -61, 0, 61, 0, -61], [-150, 0, 0, 156, 0, -156, 0, 156], [0, -216, 0, 0, -110, -110, 0, 0], [0, -92, 0, 0, -198, -198, 0, 0], [-681, 0, -681, 0, -147, 147, 0, 0], [0, 0, 739, 45, -45, 0, 0, 45], [1065, 0, 0, -27, 0, 27, 0, -27], [0, 0, 0, -243, 243, 0, 115, 243], [0, -135, 0, -51, 0, -51, 135, 51], [0, -45, 0, 21, 0, 21, 45, -21], [0, 0, -711, -27, 27, 0, 0, -27], [0, -42, 0, 0, -54, -54, 0, 0], [0, -15, 0, 0, -15, -15, 0, 0], [-738, 0, 0, 105, 0, -105, 0, 105], [0, -237, 0, 0, -87, -87, 0, 0], [0, 22, 0, -21, 0, -21, -22, 21], [0, 0, 0, 121, -121, 0, -270, -121], [483, 0, 483, 0, -126, 126, 0, 0], [-170, 0, 0, -202, 0, 202, 0, -202], [0, 0, 0, 87, -87, 0, -98, -87], [0, -36, 0, -93, 0, -93, 36, 93], [0, 0, 1086, -72, 72, 0, 0, -72], [-361, 0, -361, 0, 97, -97, 0, 0], [343, 0, 0, 180, 0, -180, 0, 180], [0, 261, 0, 0, 33, 33, 0, 0], [-497, 0, 0, -72, 0, 72, 0, -72], [0, 0, 945, 93, -93, 0, 0, 93], [0, 78, 0, 0, -171, -171, 0, 0], [0, 0, -741, -33, 33, 0, 0, -33], [0, 0, 0, -161, 161, 0, -87, 161], [0, 106, 0, 0, 18, 18, 0, 0], [0, 0, 647, -108, 108, 0, 0, -108], [0, 0, 0, -15, 15, 0, 45, 15], [973, 0, 973, 0, -27, 27, 0, 0], [0, -153, 0, -312, 0, -312, 153, 312], [0, -95, 0, -114, 0, -114, 95, 114], [614, 0, 614, 0, 72, -72, 0, 0], [0, -41, 0, 0, -156, -156, 0, 0], [0, 42, 0, 0, -43, -43, 0, 0], [0, 0, 0, -78, 78, 0, 110, 78], [630, 0, 0, -102, 0, 102, 0, -102], [0, 0, 721, -47, 47, 0, 0, -47], [0, 102, 0, 0, 75, 75, 0, 0], [0, -49, 0, 69, 0, 69, 49, -69], [0, 130, 0, 0, -12, -12, 0, 0], [0, 0, 631, 72, -72, 0, 0, 72], [0, 327, 0, 181, 0, 181, -327, -181], [0, -168, 0, -300, 0, -300, 168, 300], [-1061, 0, -1061, 0, -154, 154, 0, 0], [403, 0, 0, 153, 0, -153, 0, 153], [0, 24, 0, 0, -243, -243, 0, 0], [-591, 0, 0, -117, 0, 117, 0, -117], [0, 0, -711, -165, 165, 0, 0, -165], [0, 0, -369, 126, -126, 0, 0, 126], [0, -126, 0, 0, -36, -36, 0, 0], [-198, 0, -198, 0, -30, 30, 0, 0], [-122, 0, 0, -118, 0, 118, 0, -118], [0, 0, 0, -162, 162, 0, 243, 162], [402, 0, 402, 0, 15, -15, 0, 0], [0, 0, 0, 234, -234, 0, 60, -234], [769, 0, 769, 0, 36, -36, 0, 0], [0, 123, 0, 94, 0, 94, -123, -94], [198, 0, 0, -228, 0, 228, 0, -228], [-606, 0, 0, -84, 0, 84, 0, -84], [0, 0, -835, 224, -224, 0, 0, 224], [0, 36, 0, 0, -134, -134, 0, 0], [0, 0, 154, -36, 36, 0, 0, -36], [0, 73, 0, -78, 0, -78, -73, 78], [0, 0, 0, 15, -15, 0, -141, -15], [0, -368, 0, 0, -105, -105, 0, 0], [-18, 0, -18, 0, -69, 69, 0, 0], [0, 0, 0, -134, 134, 0, 90, 134], [185, 0, 185, 0, 240, -240, 0, 0], [0, 0, 0, 39, -39, 0, -197, -39], [0, -30, 0, 0, 152, 152, 0, 0], [-319, 0, 0, 199, 0, -199, 0, 199], [0, 0, 281, -115, 115, 0, 0, -115], [0, 0, -894, -54, 54, 0, 0, -54], [0, 0, -325, 207, -207, 0, 0, 207], [0, 0, 0, 87, -87, 0, -285, -87], [0, 125, 0, 411, 0, 411, -125, -411], [0, 0, 141, 300, -300, 0, 0, 300], [429, 0, 429, 0, 285, -285, 0, 0], [2359, 0, 2359, 0, 23, -23, 0, 0], [0, 369, 0, 191, 0, 191, -369, -191], [0, 152, 0, 0, 180, 180, 0, 0], [0, -162, 0, 0, -290, -290, 0, 0], [0, 0, 0, 201, -201, 0, -505, -201], [251, 0, 0, 147, 0, -147, 0, 147], [0, 0, 1634, -36, 36, 0, 0, -36], [0, 0, 0, -90, 90, 0, -42, 90], [0, -169, 0, 0, 123, 123, 0, 0], [1031, 0, 1031, 0, 91, -91, 0, 0], [0, 0, -261, 219, -219, 0, 0, 219], [-839, 0, 0, -25, 0, 25, 0, -25], [0, 0, 0, -57, 57, 0, 408, 57], [0, -141, 0, 129, 0, 129, 141, -129], [954, 0, 954, 0, -60, 60, 0, 0], [-1165, 0, 0, -147, 0, 147, 0, -147], [0, 0, 0, -75, 75, 0, 148, 75], [755, 0, 0, -99, 0, 99, 0, -99], [0, 0, -1570, 56, -56, 0, 0, 56], [0, -252, 0, 0, 30, 30, 0, 0], [0, -54, 0, 21, 0, 21, 54, -21], [0, 0, 0, -300, 300, 0, 411, 300], [0, 0, -1166, -174, 174, 0, 0, -174], [-823, 0, 0, 187, 0, -187, 0, 187], [0, 0, 0, 156, -156, 0, 180, -156], [0, 360, 0, 194, 0, 194, -360, -194], [0, 0, 1025, 221, -221, 0, 0, 221], [409, 0, 409, 0, -30, 30, 0, 0], [-757, 0, -757, 0, -8, 8, 0, 0], [0, -84, 0, 99, 0, 99, 84, -99], [259, 0, 0, 201, 0, -201, 0, 201], [0, 347, 0, 0, 165, 165, 0, 0], [-355, 0, 0, 148, 0, -148, 0, 148], [0, 157, 0, 0, 0, 0, -157, 0], [0, 153, 0, 0, -72, -72, 0, 0], [179, 0, 179, 0, -134, 134, 0, 0], [0, 0, -301, -45, 45, 0, 0, -45], [0, -16, 0, -54, 0, -54, 16, 54], [0, 0, 801, -69, 69, 0, 0, -69], [-706, 0, -706, 0, -126, 126, 0, 0], [0, 6, 0, -210, 0, -210, -6, 210], [0, 474, 0, 0, 214, 214, 0, 0], [0, 0, 0, -18, 18, 0, 256, 18], [0, 0, -906, -156, 156, 0, 0, -156], [0, 0, 443, 291, -291, 0, 0, 291], [0, -404, 0, -24, 0, -24, 404, 24], [0, 301, 0, 0, -54, -54, 0, 0], [0, 0, -553, -336, 336, 0, 0, -336], [0, 0, 0, -318, 318, 0, -37, 318], [0, 371, 0, 120, 0, 120, -371, -120], [0, 0, -475, 92, -92, 0, 0, 92], [-163, 0, -163, 0, -129, 129, 0, 0], [-985, 0, 0, 156, 0, -156, 0, 156], [0, 591, 0, 0, 153, 153, 0, 0], [0, 0, -1242, 66, -66, 0, 0, 66], [0, 0, 863, -78, 78, 0, 0, -78], [247, 0, 247, 0, 191, -191, 0, 0], [269, 0, 269, 0, 33, -33, 0, 0], [0, 0, 0, 330, -330, 0, -272, -330], [0, -273, 0, 59, 0, 59, 273, -59], [0, 48, 0, 279, 0, 279, -48, -279], [0, 0, 650, -175, 175, 0, 0, -175], [-955, 0, -955, 0, 87, -87, 0, 0], [0, -144, 0, 232, 0, 232, 144, -232], [0, -180, 0, 0, 100, 100, 0, 0], [1389, 0, 0, 192, 0, -192, 0, 192], [0, 0, -1002, -30, 30, 0, 0, -30], [0, 0, 0, 561, -561, 0, -204, -561], [0, -487, 0, 0, -273, -273, 0, 0], [0, 0, 0, -330, 330, 0, 222, 330], [1929, 0, 1929, 0, -126, 126, 0, 0], [-498, 0, -498, 0, 195, -195, 0, 0], [0, -174, 0, 109, 0, 109, 174, -109], [0, -267, 0, 0, 21, 21, 0, 0], [0, 0, 0, -105, 105, 0, 443, 105], [357, 0, 0, 24, 0, -24, 0, 24], [526, 0, 0, 177, 0, -177, 0, 177]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1620_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![160000, 0, -23600, 0, 3081, 0, -59, 0, 1]>,<17,R![-50, -3, 1]>],Snew);
    return Vf;
end function;