
// Make newform 1617.4.a.be in Magma, downloaded from the LMFDB on 25 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1617_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1617_4_a_be();".
// 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_1617_4_a_be();".
// 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 := [5479424, 13240576, -14390128, -17983904, 10859648, 8337264, -3723569, -1796196, 660294, 199850, -63875, -11756, 3385, 346, -92, -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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, -20, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9630703939388416, 37017658353878272, -10300295833389648, -45825753562255312, 574162167671312, 18033783275202208, 829512193402045, -3229224006127669, -184065256692461, 300500943649911, 15168259928300, -15099788066784, -547562267557, 388992582015, 7255221737, -4027995121], [214041015049216, 525060580337664, -286871578572048, -659591363409616, 71593587266640, 258303073857008, -626010908919, -45794015420365, -1345664662673, 4215922142063, 144068729796, -209323762792, -5694929753, 5319432119, 78635421, -54262353], [-1242062796110848, -2647471392332032, 1560475730155056, 3716983020587824, -293888153416688, -1566878835332032, -35842143064171, 292286727196795, 13507830034379, -27874756315065, -1237901741732, 1420170306384, 46538244979, -36820853553, -628293551, 381781615], [-6619938566656, -20198948905408, 7568232151920, 27254311080352, -243324059456, -11258456325580, -711053883019, 2087177151358, 151272150209, -199519449918, -12176757350, 10230984858, 431149045, -267400230, -5618357, 2795704], [21296583500008448, 59555902868913920, -27420768220066416, -79791262692166640, 3425889044810416, 32764854602013536, 1513978746778607, -6044729428052999, -383523032057695, 575737373752413, 32390321788036, -29446840232064, -1172631236999, 768283063029, 15476858995, -8023224827], [22251063264406528, 73972522392547072, -26638428350045040, -104431354211470192, 35588852716976, 43970719025448928, 2953676468875711, -8226316021702255, -609784418541215, 789894391127829, 48558877488596, -40579685473392, -1707341357863, 1060822477005, 22125566003, -11081280307], [15364666790912, 43636669852160, -20031589504704, -57845904500336, 3106050271840, 23551846373312, 848459578004, -4310578458965, -241190574382, 407449294191, 20943856918, -20689538874, -767526818, 536164539, 10195462, -5564459], [8590480461773824, 25970584179886080, -11504296953763888, -34388060959960240, 1706497092431920, 14054435191808240, 539518458203707, -2586147056208235, -148055114864379, 245706085438137, 12760026165220, -12529279643136, -466078170595, 325732912737, 6179746815, -3388235695], [-18568193484018688, -51083223921821952, 20932756278998288, 68993659183820816, -1070181809015248, -28430136603110304, -1756312357599049, 5267408792102185, 382420228736393, -504070437236339, -30993116798188, 25898064005424, 1101143360129, -678407064507, -14380308693, 7108831141], [-4659172946797568, -13299435287382272, 6071808740684304, 18028586485611536, -780298127638480, -7413500474000288, -326933325082265, 1365988258441769, 83663590680121, -129826456946547, -7078069422028, 6622997801232, 256252076273, -172310431995, -3380120773, 1794240773], [14075467325440, 39712394285760, -18279597237456, -52587853864352, 2894839552384, 21419122136044, 744871052281, -3923633005590, -215093767571, 371266063830, 18738240426, -18871646942, -687784703, 489497542, 9146495, -5083956], [18824035877759488, 54559404530690752, -23170062378341712, -74265559182463216, 2246766208625888, 30653974573764652, 1540394214062665, -5662318025060545, -364308805460609, 539136829736871, 30248035657964, -27544807601532, -1087003703977, 717580634331, 14288645669, -7481000833]];
    Rf_basisdens := [1, 1, 1, 1, 60298062483456, 591157475328, 3546944851968, 22909598208, 60298062483456, 60298062483456, 22909598208, 10049677080576, 20099354161152, 3546944851968, 7636532736, 7537257810432];
    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_1617_a();"
function MakeCharacter_1617_a()
    N := 1617;
    order := 1;
    char_gens := [1079, 199, 442];
    v := [1, 1, 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;

function MakeCharacter_1617_a_Hecke( : Kf := Kf)
    return MakeCharacter_1617_a();
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]];
    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 := 4;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 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, 0, 0, 0, 0], [11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-10, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1], [-10, -4, -1, -2, 1, 2, 0, 2, 0, 2, 1, 0, -1, 0, -1, -1], [0, -1, 1, 0, 0, 0, 1, -1, 1, 0, -1, -1, 0, -1, 2, 0], [23, 3, -3, -3, -2, 0, -1, -1, 0, 0, 1, 0, -1, 0, -2, 0], [-41, -4, -3, 1, -2, -2, -1, -4, -1, 1, -2, -2, -1, 1, 3, 1], [6, 7, -7, -2, 1, 3, -3, 3, 2, -1, 0, 4, 2, -2, 0, 0], [-41, 3, 4, 3, -3, -5, 2, -3, 5, -5, -1, 0, 4, 6, 4, 0], [-29, 16, -14, -1, 1, -1, 2, -4, 2, -1, 1, -4, 1, -2, -2, -4], [-148, 0, -11, -2, 2, 4, -3, 6, 10, 3, 2, -3, -2, -4, 1, -5], [2, -3, -13, -4, 3, -4, 2, 3, 9, -4, -1, 4, 1, 2, 1, 3], [4, -43, -11, -2, -1, -2, -2, -8, -5, 6, 3, -6, -2, 2, -3, -5], [-92, 3, -1, 6, -5, -2, 0, -4, -9, -3, -2, -2, -3, 3, 4, 4], [-3, -1, -13, -5, 6, 1, -4, 10, 11, 5, -6, 4, -9, -2, 3, -3], [26, -43, 1, -10, 2, -3, -6, -6, 10, 3, -6, -3, -4, 1, -1, -3], [-26, -33, -10, 8, 6, 12, 2, -4, 5, -1, 1, 14, 0, -5, -7, 1], [57, 10, -10, 13, -19, -29, 2, -4, 2, -20, 12, -14, 7, 7, 9, 9], [-187, -20, 13, 21, -9, -14, 10, -6, -7, -17, -8, -1, 4, 2, 10, 10], [-171, 21, -27, 9, -4, -6, 5, 11, 12, 1, 12, -10, 8, -3, -5, 5], [-209, 41, 11, 11, 1, 17, -7, 14, -13, 0, 11, 3, 3, -8, -15, -1], [8, -132, -10, -10, -9, -4, -9, -15, 28, 1, -9, 13, -9, -6, -2, 2], [-301, 48, -7, 1, 14, 2, 13, 16, -13, 4, -21, 10, -13, -6, -2, 0], [11, -15, -31, 7, 5, 17, -7, 40, 5, -8, 13, 7, -4, -10, -13, -1], [-43, 57, 35, 13, 1, -13, 17, -9, -25, -18, 17, -15, 7, 14, 9, 9], [214, -117, 15, -2, 12, -5, 8, -14, 2, -23, -18, 17, 4, 3, -13, 9], [-143, -86, -22, -9, 19, 39, -10, -10, -12, 41, -23, -2, -15, -10, -6, -6], [-176, 2, 99, 4, 3, 10, 2, -13, 76, 14, -7, -2, -6, -10, 7, -13], [191, -231, 77, 7, 12, 21, -2, -31, -23, 6, 17, 8, 6, -7, 2, 0], [-450, -185, 76, 10, 10, 6, 2, 16, 35, -3, 1, -26, 7, -17, -1, -5], [195, -145, 13, 17, 9, 6, -6, 67, -8, 30, -25, 5, 10, 3, 15, 11], [-323, 61, -39, -5, -17, 11, -11, -77, -43, 2, 25, -7, -25, 2, -7, -11], [-321, -182, 57, -17, -8, -38, 7, -3, 15, -20, 11, 18, 17, 18, 18, 20], [-80, 83, 51, -4, 1, 50, 0, 70, 65, 27, -4, 6, 0, 1, 14, -8], [-563, -167, 47, -67, 23, 7, 13, 37, 75, 34, -7, -17, -23, 4, 5, -9], [183, -137, 73, -43, 20, 71, -18, -28, 47, 53, 18, 10, -20, -44, -23, -33], [-21, -40, -16, 75, -37, -69, 26, -64, 58, -60, 54, -12, 57, 41, 3, 23], [-369, -108, 107, 47, -50, -96, 27, -31, -1, -59, -18, -24, 25, 55, 19, 29], [502, -156, 76, -4, -19, -39, 26, 127, 43, -37, 11, -7, -9, 37, -2, -8], [-679, -75, -27, 11, -33, -51, 7, -133, -41, -38, 47, -41, 59, 36, 9, 21], [227, 64, -86, -49, 27, -24, -11, 35, -61, 32, -18, 18, -17, -14, -15, 1], [-458, 87, 52, 82, -11, -29, 14, -15, -38, -56, 4, 31, 18, -32, 25, -5], [-369, -140, 14, -39, 25, 19, 8, 34, 44, 20, -86, 70, -35, -9, 45, -5], [-1420, -53, 40, -38, 1, 33, -50, 51, -20, -27, -27, 31, -2, -47, -12, -2], [696, 264, -49, -34, 3, 22, 10, 105, 88, 66, -69, 36, 8, -4, -3, -3], [-336, 411, 116, 4, 13, -89, 22, -138, -48, -26, 16, -37, -22, 56, 1, 39], [421, -248, 198, 29, -20, 16, -28, 53, -33, -22, 48, -21, 26, -33, -24, -18], [802, 57, -27, -16, -13, -14, -34, -54, 29, -6, -23, -2, -36, -12, 27, -23], [-711, 209, -102, -67, 34, 54, -60, 75, -72, 55, 11, -1, -56, -60, -53, -57], [-510, 225, 189, 36, 13, -18, -10, 62, 41, -4, -11, 22, 12, -30, 1, -11], [-1311, 416, -164, -47, 1, -20, 7, -27, -47, 1, 49, -42, -8, -3, -8, -40], [382, 97, 73, -30, -4, 33, -36, 25, -40, 52, -49, 19, -49, -16, 24, -16], [-399, 592, 89, 71, -67, 16, -12, 3, -23, -14, -3, -59, 38, 13, 3, 27], [-338, 589, 1, -30, 41, 10, -34, -88, 5, -1, -8, 20, -16, 9, -44, 2], [-805, 129, -7, -63, 55, -25, -1, -15, 83, 29, -10, 19, -43, -17, -40, 24], [880, 33, 243, 90, -35, 16, 24, -76, 35, -4, -23, -20, 42, 26, 95, 41], [233, 418, -59, 49, -61, -106, 14, -5, 111, -32, 53, -89, 1, 93, 11, 21], [-671, 545, 89, -13, 10, 23, -34, 32, 115, 63, -20, 42, -44, -56, 9, -7], [-1965, 299, 238, 57, 0, -108, 18, 81, 66, -51, -11, -39, 25, -24, 115, -15], [-862, 255, -98, -174, -15, 44, -21, 50, -49, -31, -19, 53, -14, 44, -26, 4], [-1356, 370, -122, -16, 40, 2, 36, -63, -20, 21, -77, 28, -58, -43, 23, -23], [1664, 94, 24, 242, -98, 5, 21, -197, -55, -75, -37, 36, 132, 22, -1, 79], [729, 140, -80, -125, -15, 51, -76, 178, 282, 32, -16, 2, 13, -9, -1, -5], [-118, 267, 63, 92, 17, 22, 42, 93, -7, 30, -33, 32, 13, -34, 19, 1], [370, -438, -78, 84, 14, 67, -13, -3, -107, 68, 48, -16, -56, -25, -66, -42], [-49, 196, -106, 17, -70, -42, 34, 241, -73, -72, 86, 37, 45, 37, -18, 30], [-1288, 784, 122, 196, -17, -50, -15, -96, -48, -29, -95, 77, 80, -22, 44, 12], [2072, -470, 246, 70, -67, 97, -54, 3, 215, 94, -48, -5, -9, -72, 29, -89], [-2604, 819, -374, -176, 45, -59, 32, -179, 30, 23, -17, -9, -74, -29, -20, 18], [-964, 550, 378, 174, -79, 24, -19, -250, -286, -117, -29, 9, 57, 2, 74, 42], [481, -815, 21, 33, -35, -66, 4, 171, 146, 21, 68, -83, 2, -2, -4, -72], [-1912, 693, -94, 32, 31, -22, -29, -100, -71, -87, -39, 133, 40, -58, -16, 10], [586, 235, 17, 42, -65, -97, 21, -312, -140, -95, 168, -168, 25, 80, 48, 62], [-1654, 145, -58, 60, -21, 55, 28, -174, 6, 28, -6, -153, -64, -24, 51, -33], [1125, -774, -74, 149, -53, -42, 73, -181, -199, -103, 7, 0, 107, 99, 18, -22], [-1570, -416, 49, -258, 65, 102, -38, -20, -52, 76, 33, 56, -28, -18, 27, -61], [-1507, 443, -35, 87, -23, -4, -16, -59, 70, -14, -21, -63, -84, 143, -1, 59], [483, -999, 239, -123, 49, 64, -62, 197, -226, 43, -114, -15, -136, -48, 62, -100], [-47, -269, -385, -215, 174, -33, 16, 4, -271, 116, 89, -74, -75, 49, -130, -36], [-3259, 313, -319, 1, -43, 63, -19, 170, -101, 17, 106, -7, -17, 55, 0, 32], [1272, -171, 153, -88, -117, 72, -110, 8, 45, 47, -96, 88, 18, -45, 66, -88], [-11, -735, -429, -177, 98, 176, -71, 166, 172, 15, -50, 160, -40, -127, -143, -95], [3196, -207, -311, -68, 136, 91, 78, 472, -278, 55, 62, 93, -90, -47, -155, -45], [-95, -1033, 19, -289, 75, -131, 65, 53, -587, 54, -111, -67, -105, 158, -15, 33], [1333, 152, 554, -29, 45, -4, 97, -372, -17, -156, 116, 72, 113, -38, 71, 105], [-1149, 341, 517, 75, 21, -34, 16, 73, 96, -29, 82, 63, 2, -142, -78, -26], [-1094, 613, 35, 8, -21, -46, 66, -68, -147, 22, 85, 98, 20, 40, 5, -11], [-905, 700, -366, -123, 112, 152, -82, 1, 111, 151, 39, -163, -134, -68, -153, -39], [-1459, -364, 568, 149, -90, -83, 33, 14, -52, -110, 234, -13, -12, -34, 10, -20], [1562, 833, -199, -68, -119, -160, -78, -36, -9, -116, -73, -36, -110, 254, 55, 109], [1755, 599, -105, -291, 360, 165, 22, 382, 13, 289, -182, 116, -365, -54, -235, -37], [-221, 104, 80, 37, 48, 156, -164, 221, 427, 115, -19, 95, -50, -42, 27, -35], [386, 928, 181, -188, 101, 64, -134, 248, 148, 96, 103, -10, -222, -32, 5, -157], [-895, 554, 104, -13, -123, -62, 23, -75, 507, -93, -15, -36, 90, 101, -40, 6], [-1898, 610, -489, 14, 29, -6, 60, -88, 98, -66, -3, 176, 103, -136, -33, -61], [-3951, -89, -117, 83, -232, -263, 0, -801, -421, -173, 156, -206, 156, 42, 65, -35], [-183, 783, 457, 175, -44, -195, 96, -175, 255, -241, -186, 116, 118, 34, 163, 153], [1845, -737, -543, -133, -38, 48, 63, 115, -22, 30, 241, -166, -55, -136, 86, -116], [-3591, 43, 268, 225, -90, -367, 91, -166, -129, -316, -48, -21, -55, 162, -32, 156], [1605, -204, 118, -163, 254, 222, -96, -131, -631, 232, 38, 85, -178, -139, -24, -118], [-6613, 1833, 219, 145, -12, -173, 68, -85, -119, -11, 42, -236, 50, 54, 145, 51], [-3314, -1106, 86, -262, 158, 18, -154, 453, -298, 291, 21, -72, -236, -9, -161, -47], [-5465, 371, -1305, 89, -99, -196, 88, -311, 244, -98, 17, -107, 2, 69, 103, 17], [3627, 357, 216, 105, 139, -59, 36, 1029, -25, 45, -9, -40, 114, -116, -30, 24], [232, -2092, -154, -18, 81, 240, 5, 52, -140, 240, 54, -149, -45, -245, 11, -127], [2851, -83, 554, 121, -8, -178, 140, 259, -440, -361, 293, 113, 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-608, -1447, 1110, -650, -1170, 3829, 695, 963, -1643, 761, 340, 472, -1279, 481], [27933, 1691, 3013, 1099, 141, -793, -1389, -3293, 2879, -1268, 1699, -1285, -683, -1068, -1473, 543], [36193, 13934, 2260, -1373, 556, 1528, -1690, -1255, 4173, 2830, -1350, -1381, -1011, -1975, -1462, -1536], [-23533, -13758, -934, -625, -219, 446, -1703, -2447, 2667, 2047, -2117, -606, -2347, 1193, 234, -104], [4077, -4884, 4575, 5889, -19, -2200, 1560, 3290, 1431, -3945, 638, 379, 2404, 464, 156, 1560], [-40173, -27877, 1201, -4605, 925, 3963, -2075, 66, 1855, 2171, -2656, 3299, -1829, -2825, -2294, -966], [170, -4329, -896, 172, 1884, -140, 1356, -5050, 1247, -1138, -398, 644, 1596, -402, 1214, -218], [-42204, -2350, 6772, -1326, -307, -1559, 628, -169, 831, -1511, 845, 1227, 441, 725, 670, 1198], [29855, -2459, -2357, 2429, 1581, 342, 288, -3291, -1418, -652, -4355, 2191, -338, -1405, 127, 247], [68163, -8779, 7549, 507, -1949, -849, -1025, 1337, 9293, -871, -1776, 1657, 465, 471, 1242, 574], [31140, -15517, -347, -766, 441, 1661, -1263, -9342, -1390, 1921, -812, 1106, -607, -1334, 730, -956]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1617_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1617_4_a_be();".
// 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_1617_4_a_be(:prec:=16)
    chi := MakeCharacter_1617_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1617_4_a_be();".
// 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_1617_4_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1617_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![5479424, 13240576, -14390128, -17983904, 10859648, 8337264, -3723569, -1796196, 660294, 199850, -63875, -11756, 3385, 346, -92, -4, 1]>,<5,R![2109136224221248, -611464983053120, -606955741501840, 267592651087520, -1754180133936, -10748273654248, 517223965058, 198271516252, -8318793823, -2044166348, 44768284, 11833372, -17890, -34892, -494, 40, 1]>],Snew);
    return Vf;
end function;