
// Make newform 1110.4.a.p in Magma, downloaded from the LMFDB on 02 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1110_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_1110_4_a_p();".
// 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_1110_4_a_p();".
// 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 := [-80360, -61726, -14747, -1063, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-1312220, -691196, -71401, -384, 69], [-1708112, -709918, -57417, -102, 53], [-7096432, -3069610, -279957, -1100, 267]];
    Rf_basisdens := [1, 1, 8876, 8876, 8876];
    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_1110_a();"
function MakeCharacter_1110_a()
    N := 1110;
    order := 1;
    char_gens := [371, 667, 631];
    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_1110_a_Hecke(Kf)
    return MakeCharacter_1110_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+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 := 4;
    raw_aps := [[2, 0, 0, 0, 0], [-3, 0, 0, 0, 0], [5, 0, 0, 0, 0], [4, -1, 0, 0, 0], [14, 0, 0, -1, 1], [15, -1, -1, -1, 0], [10, 0, -2, 0, 1], [26, 0, 2, 0, -1], [24, -1, 0, 0, -2], [9, -1, 5, 4, -1], [-12, -2, 2, 7, 2], [-37, 0, 0, 0, 0], [10, -4, 10, 5, -4], [103, 11, -7, -2, -1], [67, 6, -7, -8, -2], [49, -7, 3, 2, -4], [141, 27, -11, -1, -9], [88, 0, 26, 5, 0], [44, 13, 8, 29, 7], [236, 13, -32, -1, -1], [200, 4, -10, -21, -3], [62, -7, 16, -16, 3], [73, -20, -9, 20, 5], [33, -15, 49, 7, 2], [276, 32, -26, 13, -14], [558, 9, -6, 23, 19], [531, 34, 25, 26, -2], [577, -11, 33, -22, -6], [232, 10, 38, 20, -5], [-76, -17, 52, 62, -11], [400, -38, 8, 21, -31], [1074, 8, 24, -15, -2], [655, -5, -75, -28, 5], [713, 63, -69, 0, -3], [330, 96, -48, -3, -52], [435, -23, 61, 103, -22], [-12, -122, 72, -27, 26], [44, 29, 14, 59, -38], [1270, 28, 32, 22, -5], [-1023, -43, 41, 17, -52], [-243, -18, 29, -58, -20], [1133, 41, -69, -63, 15], [644, -15, -10, 35, -8], [-763, -6, -89, -19, 44], [740, 29, 26, 28, 80], [-1096, 38, -42, 115, 8], [1905, 41, 17, -123, 13], [418, 50, 128, -29, 24], [1292, 84, -96, -120, -12], [-1017, 82, -65, -35, -22], [-1332, 64, -102, -3, 40], [531, -118, 167, 32, -32], [532, -81, 36, 100, -15], [1693, 65, 95, -18, 67], [828, -186, 60, 42, 59], [-2331, -7, -83, -6, -69], [-876, 219, -148, 76, -66], [-487, -218, 19, -63, 26], [-342, 167, -166, -129, -16], [-345, -140, 69, -29, -21], [-2160, -72, 78, -118, 89], [-3124, -95, -78, -186, -18], [-1590, 20, 176, 69, 88], [1219, 232, -339, -41, 12], [-3627, 129, 11, 98, -7], [-2391, 6, 5, 3, 22], [-200, 9, -138, 79, -21], [-1954, 195, 4, -16, -20], [-3568, -158, 196, 30, 86], [-2457, -5, 199, 74, 73], [-5622, -136, 90, -91, -82], [438, -214, 180, -33, 172], [-688, 232, -258, 82, -149], [-5474, -169, -308, -217, 99], [737, -21, -197, -224, 189], [-2446, 96, 254, 93, -15], [1801, 24, -255, 27, -24], [-828, 144, -52, -93, 120], [-1749, -29, -375, 68, 50], [-2870, 173, 156, 133, -161], [2896, -273, -72, -33, 156], [-6104, 176, -22, -123, -40], [1436, -115, -182, -305, 6], [-3388, 314, -108, -214, 7], [2358, 40, -362, -460, -52], [-3289, 64, 227, 408, 86], [2224, 251, -88, 180, 137], [-1443, 364, 235, 123, -12], [771, -86, 337, 246, -248], [-245, -262, -389, -71, -28], [4468, 331, -438, -348, 17], [8333, -543, 223, 25, -76], [889, -305, 159, -3, -231], [-468, -35, 26, 26, -146], [4162, 39, -226, -301, -120], [-54, -340, -122, -422, 40], [3308, -423, 246, -329, 84], [4078, -485, 248, 15, 205], [-1006, -249, 228, 206, -79], [-8048, 119, -188, -64, -112], [1545, -171, 389, 572, 118], [5178, 205, -182, -152, 265], [976, 101, -712, 43, 247], [-1818, 308, -30, 346, -135], [-4326, 158, 284, -45, -186], [-3658, -129, -38, -451, 169], [1828, 96, -218, -179, 326], [2607, 330, 547, 252, 176], [1226, -370, 328, 473, 184], [-6827, 449, -273, -176, 228], [6090, -630, 162, 55, -76], [-6186, 102, 268, 76, 78], [-3600, -621, -258, -184, 283], [-196, 514, 84, 10, -102], [-2020, -89, -630, -502, 373], [4866, 424, -246, -375, -248], [-6197, 652, -459, -600, 5], [6330, -41, -96, 185, -148], [-1382, 114, 416, 120, -244], [12004, -544, 484, -167, -148], [-13594, -265, 354, 298, 192], [-12460, 656, -388, -600, 83], [3793, 676, 23, -35, -251], [-3104, 430, -160, -300, -126], [-6143, 554, 133, -110, 94], [8107, 50, -277, -85, -280], [-7528, -115, 218, 361, -192], [14268, 374, -994, -183, -118], [-1904, -169, -854, 117, -133], [-2178, 448, -22, 559, -182], [-6643, -593, 459, 316, 221], [-34, 701, -194, -776, 11], [-6954, 671, 200, 400, -191], [-1655, 276, -877, -703, 393], [14630, -357, 390, -566, 213], [680, -379, 378, -183, -283], [-5518, -901, 246, 541, 96], [854, -459, 28, -172, -15], [-10476, 1140, -704, 309, 262], [1211, 458, 63, 244, 411], [-15436, 59, -1000, -703, 303], [7090, 373, -1298, -1066, 114], [-9710, 533, 716, -21, 202], [-4688, 974, -154, -751, -372], [6322, 831, -924, 57, -336], [17146, -96, 668, -227, -240], [-4257, -505, -343, 248, 2], [-3386, 29, -84, 442, 50], [8908, -73, -808, 79, 192], [12909, -555, -265, -341, 481], [3536, 117, 1236, 544, -3], [24320, -1253, 274, 343, -27], [-13558, -71, 1166, 252, -740], [5774, -716, 1022, -175, 266], [7547, -15, 195, 1136, -366], [11257, 156, 801, 192, -58], [4958, -674, 190, -862, -74], [17798, 0, -428, -354, 512], [2842, 205, 816, -343, -435], [7916, -1633, 962, 628, 315], [-7842, 1410, -1686, -166, 378], [-7154, -1204, -610, 346, 226], [24106, -177, -378, 254, -95], [10608, -528, 232, 579, 196], [2202, 254, 652, -253, -419], [6712, -547, -66, 737, -887], [8256, 825, -594, -370, 723], [17756, -22, 594, -737, 749], [1147, -1112, 249, -194, -298], [6994, -274, 398, -233, -234], [6318, 1050, -672, -1393, -78], [9584, 1679, -34, 56, -725], [12613, -33, -505, 332, 467], [-7152, 580, -1110, 797, 121], [6182, -936, -818, -23, 849], [4380, -138, 320, 969, 20], [3312, 336, 550, -810, -294], [13292, -57, 96, 582, -80], [-8012, 920, -1124, 510, 56], [3919, 364, 1173, 784, 278], [15817, 743, -519, -662, 93], [6006, -31, 248, -670, -319], [-25911, 499, -67, 123, 528], [-22992, -1032, -246, -796, 440], [20932, -1061, 26, 209, -143], [6526, -10, 56, -262, 111], [-18504, -917, -118, 957, 329], [-6453, 1398, -1097, -1251, 553], [8266, -1713, 2162, 256, -593], [15702, 650, -432, 125, -632], [-5553, -665, -309, -131, -777], [-20567, -154, 1291, 41, -566], [1592, 224, 1890, 667, -851], [-1676, -591, 1526, 1963, -486], [-22999, 495, 481, 1005, 521], [-16697, 280, -1147, -957, -168], [5866, 231, -2640, -635, 509], [4179, 847, -117, -1038, 359], [-31223, 689, -1449, -1260, 141], [1758, -924, 1198, 1309, 285], [12791, 711, -1499, -321, -363], [26764, -1773, 354, -597, -378], [17190, 1836, -1766, -1095, -464], [25083, 1504, -3017, -1529, -84], [22188, 712, -772, 271, 356], [-15846, 2054, -1456, -306, 42], [40465, -501, -791, -1330, -190], [-6421, -1818, -1937, -295, 539], [-32856, 1167, 1686, 618, 456], [-513, 390, -1801, -1440, 1240], [-4415, -888, 1777, 1566, -484], [-8623, -597, 1747, 1161, -349], [-2169, -2270, 1827, 359, 624], [-20873, 1942, 985, 1168, 649], [6628, -720, 350, 560, -917], [17893, -1751, 47, 182, 446], [18504, 66, 1302, 1764, -1014], [3837, 730, -773, -80, -100], [-7090, 425, 636, 477, -550], [-41045, 1057, 1203, -57, -416], [395, -971, -1093, 450, -783], [25235, 1147, -411, 921, 516], [-48092, -837, 808, -602, -939], [-35083, 748, -45, 396, -490], [-10791, -677, -757, -662, 611], [18692, -1048, 1012, -1815, 613], [-29691, 1122, 1317, 644, -208], [-12492, 3449, -1700, -547, -827], [52330, 877, -118, -221, 849], [-9711, 246, -361, -949, -390], [-7748, -1102, -790, -2004, 178], [-33141, 2049, 759, 965, 599], [-36547, -1616, -397, -432, 591], [9358, -2234, 1714, 291, 722], [-45868, -796, 1830, 797, -930], [-30715, -928, 1395, -36, 98], [-9805, -2388, 1721, 1885, 605], [-27144, 1448, -432, -547, 198], [-19776, 163, 1528, 2347, 233], [-28117, -402, 215, -120, -1352], [-42462, 1280, -1994, -1180, 608], [3168, 1941, 1030, 1938, 550], [-26888, -1721, -674, 781, 764], [-33346, 1372, -2666, 1507, 113], [15619, 464, -733, -1116, 1238], [4805, 2541, -2123, -1612, -680], [15579, 639, 1777, -551, -7], [-17405, -207, 1329, -38, -1301], [-12054, -1187, -2180, 568, 733], [-9506, -439, -1366, -2680, 366], [6453, 554, 2145, 992, 387], [27722, 353, -580, 67, 31], [-24860, -520, 2818, -188, 740], [-8024, 962, 180, 573, 366], [-30054, -2, -1178, 39, -208], [-41844, -2515, 4470, 1237, 279], [30047, 1345, 1851, -570, -179], [36350, 729, 676, 824, -291], [-8374, 2756, -662, 1372, -1501], [46351, 1219, -3949, -2373, -433], [4379, 404, -2509, -1773, 424], [-22870, 694, -400, 517, -1480], [-14385, 1313, 217, -927, 269], [-34958, -812, -110, 929, -1846], [-11666, -2172, -404, -2093, -1017], [-24939, -1146, 559, 1085, -810], [-56756, 4311, -1882, 13, 364], [3857, 453, -1589, -5, 403], [-27092, -872, 1270, 695, -1268], [-18122, -1038, -1988, -1120, 1130], [5660, 1750, -1316, -657, -1253], [32299, 85, -915, 945, 984], [-2487, -2178, 1817, 2012, 1196], [-31460, 3244, 36, -484, -1556], [8752, 416, 600, 2138, 53], [-34324, 1721, -538, -1987, -162], [-16505, -2007, -1371, -435, 382], [-9938, 94, -970, -3690, 1816], [-19203, 1927, 795, 184, 134], [16937, 342, 571, 2135, 388], [18178, 1648, 2066, 2593, 1110], [-14832, -342, 1394, 3205, -1394], [42499, 2906, -1587, -892, -512], [-33949, 841, -2175, 25, -695], [-13025, -1880, 19, 462, -1210], [52884, 772, 1126, 1523, -920], [40240, 2905, -2150, -1510, 237], [4657, -357, -2689, -229, -1583], [-15216, -3723, 5068, 501, -261], [-20522, -2478, 754, -1008, -435], [-9904, 506, -1660, -3435, 2104], [17876, -5305, 3102, 900, 623], [13860, 1909, 414, 924, -2143], [-3520, -3000, -2084, -1618, -67], [-37982, -1248, 2660, 572, 1702], [-4374, 1410, 48, -181, -1346], [32339, -1337, -679, 1255, 396], [-7155, 373, 2597, -603, 1642], [42538, -2578, 964, -2221, -536], [53490, 1344, -1270, 1034, 628], [26733, 809, 173, -1691, -1199], [-2307, -723, -1023, 2278, 457], [-37874, -1700, 5452, 1249, 338], [52128, 1540, 3308, 3122, -54], [-22386, -2359, -1168, -978, 1780], [37627, 1103, 2429, 388, 551], [38830, 2787, 536, 3390, -647], [-48628, -2387, -456, -1559, -581], [3900, 1512, -4400, -2572, 1270], [31600, 2155, 6, -204, 512], [40438, 993, -4306, -674, -2043], [-19966, 2639, 1870, 444, 1452], [6278, -1630, 1372, 852, -1730], [-93062, 926, -1822, -2880, 1402], [55284, -556, 406, 2060, -1121], [-61910, -2639, -1232, -4447, -1235], [20302, -577, 4298, -967, -524], [29534, -209, 1266, -1082, 2938], [-37758, -1687, -910, 1674, -2077], [12897, -1316, -1843, -110, 2207], [50646, -1209, 3196, 1470, -82], [10692, 1055, 552, -449, 2786], [14093, 2226, 3839, 1414, -2170], [-72854, -1618, -2586, -3423, 475], [-20525, -577, 4167, 1557, -483], [-12578, -3584, 5018, 3163, -976], [-81055, 1494, -419, 604, 4], [27461, 106, -3129, 1401, -2982], [-1032, 4357, -5330, -491, 1100], [25046, 689, 604, 3915, -1531], [21390, 140, -5824, -450, 1034], [-101580, -545, 654, 801, -547], [-24654, 4476, -3614, 2772, -50], [85440, 2551, -2078, -3240, -369], [40294, -2126, -3272, -453, 1253], [60387, 230, 1969, -2859, 2166], [-68030, -1395, -2180, 3177, -1175], [19213, -4855, 2569, -2179, 249], [1548, 742, -5320, -879, -1276], [68968, -35, 420, 1829, 269], [27660, -5024, 3272, 587, 400], [-74721, 2223, -1151, -2262, -619], [-57421, -3971, -4537, -2124, 1751], [-65836, -1450, -134, 1159, 792], [10556, 1523, -5078, -4323, -187], [-8325, -738, 1705, 691, 176], [41622, -2448, 2148, 2484, -634], [1506, -1639, 4632, -240, -379], [36647, -733, 1925, -762, -2025], [28231, 195, -2523, -5449, -519], [17339, -4151, 1163, -457, 361], [89036, 4463, -2844, -771, -1488], [-44724, 504, -1104, 861, -1788], [44228, -122, 1258, -762, 3170], [504, -3915, -500, 362, -1117], [73289, -859, -3373, 454, 3181], [-40195, 319, -2505, 825, -300], [29261, -3280, 1599, 521, 1694], [-109379, -1434, 187, 5681, 202], [53388, -2463, 4394, 340, 1143], [59819, -5323, 867, -1120, -355], [54076, -7159, 2000, 1108, -25], [99386, 1296, 2412, -98, -145], [-111271, -4504, -801, -3431, 2356], [91038, 1909, -134, -2883, 1159], [104193, -573, -87, -471, 3995], [-7596, -2243, -1034, 759, -557], [3694, -3413, 2886, -3624, 477], [103356, -1898, -300, 1289, 572], [2466, 2631, -3248, 1326, -403], [50540, -3315, -2476, 976, -935], [20153, -1060, 3455, 1786, 1506], [-122776, 1896, -1926, -2242, -1243], [17320, -3636, 2470, -1751, 248], [-23516, -2023, 3600, 4150, -2289], [30730, 304, 984, 1137, -587], [72474, -2960, -2278, 882, -1102], [30852, 1395, -3258, -38, 512], [37821, 2601, 4015, -2552, 6], [121380, -3007, 1848, -207, 160], [-32708, -3186, 4820, 587, -832], [-13648, 7785, -1716, 1383, -1290], [-947, 610, -973, -1014, -172], [24773, -4585, -2413, -2188, -351], [-113598, -338, 1470, -2531, -2102], [126875, -4174, 423, -911, 64], [-74876, 2557, -2950, -123, -1348], [-44833, -1038, -995, -182, -1132], [52062, 1424, -1632, 2331, 2430], [96636, 2082, 30, 1372, -2215], [1125, 4002, 3591, 2617, 835], [-21126, 6276, -3296, 4342, -732], [32639, 3227, -4235, 1152, 175], [84045, 2777, -905, 1165, -2248], [1914, 5307, -1970, 2897, -321], [35536, 5179, 1782, -66, -412], [-65658, -2521, -114, -4193, 1041], [51250, 2649, -3444, -1497, 1867], [-20621, 3954, -325, -1774, -622], [28686, 3849, 1218, -4702, 1234], [-110580, -3296, 1792, -1437, 240], [97542, -3763, 140, -3686, -436], [-48749, -3983, 4157, 2637, -63], [-5490, 304, 5982, 5405, 2442], [-22981, -3680, 1149, 2826, 1806], [-10593, 5465, -6197, 1734, -792], [91695, -4223, 3207, 2909, 1848], [14166, -4195, 508, 2179, -645], [-74518, -1637, 1820, 3526, 1377], [-63642, 9821, -3888, 533, -841], [161646, 2547, 1500, 4232, 726], [65803, -585, 911, -3509, 748], [4852, -10127, 6642, -3648, 441], [-18726, -3467, 4420, -2078, 2500], [-61331, -2888, 2535, -2221, -2912], [-47986, 4619, -4096, -5465, 1314], [-17586, -5623, -5192, -2084, 2823], [-19242, -5611, -690, -1874, -348], [42257, -4762, 101, -26, 342], [17738, 5873, -6054, -865, 2433], [-5257, -1904, 2915, 5419, -4304], [66043, -2557, -3853, 1291, -1752], [-32054, -839, 3634, -1516, -1513], [-72334, 487, -1036, 1554, 1536], [42364, 8761, -4946, -1822, -2922], [-48819, -336, -3911, -5877, 2980], [-104470, -253, 9526, 3704, 510], [116004, -1472, 3348, -489, 670], [-13169, -969, 6833, 5144, 1509], [-2034, 1086, 1308, 4887, -944]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_4_a_p();".
// 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_1110_4_a_p(:prec:=5)
    chi := MakeCharacter_1110_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_1110_4_a_p();".
// 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_1110_4_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![630480, -229702, 26959, -919, -19, 1]>],Snew);
    return Vf;
end function;