
// Make newform 289.4.a.i in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_289_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_289_4_a_i();".
// 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_289_4_a_i();".
// 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 := [29352, 21960, -117822, -4607, 93255, -5163, -20922, 627, 1872, -17, -72, 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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [668244208, 1460605688, -2629068300, -1204403168, 1702105407, 23301493, -270879424, 23350245, 15911043, -2002585, -314662, 46055], [-475178864, -1347214760, 1028876492, 1246083148, 153419951, -467836487, -77589892, 58746693, 6477251, -2946597, -153738, 52035], [-950170632, -134465856, 2859296454, -374899947, -1670888088, 366397543, 269286443, -59915226, -16175630, 3458675, 325629, -66901], [1294454224, 1860851624, -4205166484, -1028613184, 2593563441, -215264909, -415569128, 61818003, 24658509, -4158023, -491090, 86489], [-449192696, -750496792, -1403390782, 2338748637, 1871083949, -1193754170, -377038701, 164685213, 25329267, -8625496, -543997, 155568], [-883660304, -508116152, 4208802084, -1333374804, -3023861471, 1015904647, 522294276, -154376285, -32568915, 8518605, 671066, -159467], [-2412870848, -1316688056, 6613484696, -212334910, -3992868859, 822543781, 656266478, -146548157, -39994179, 8687451, 814200, -170061], [2009725648, 847224856, -6438965716, 134588220, 4199199459, -830447331, -694924972, 150736233, 42191223, -8984721, -854122, 176055], [1514531544, -2848924336, -6573286898, 5819146261, 5315185878, -2803100967, -978360605, 382772712, 63469256, -20053311, -1341287, 363289]];
    Rf_basisdens := [1, 1, 1, 47215936, 47215936, 47215936, 47215936, 47215936, 47215936, 47215936, 47215936, 47215936];
    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_289_a();"
function MakeCharacter_289_a()
    N := 289;
    order := 1;
    char_gens := [3];
    v := [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_289_a_Hecke(Kf)
    return MakeCharacter_289_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 := [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [2, -2, 0, 2, -1, 0, -1, 0, 0, 0, -1, 1], [14, -4, 2, 0, 1, -2, 0, -2, 1, 0, 1, -1], [-5, -4, 0, -3, -1, -5, -1, -2, 1, 2, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 3, 1, 2, 2, 4, -9, 0, -2, -1, 0], [18, -10, 2, -12, 1, 4, -4, 0, 0, 3, 3, 2], [54, 0, -3, 1, 2, 2, 4, -9, -5, -7, -1, 0], [44, -14, -10, -5, 6, 1, 20, 10, -1, -1, 0, -2], [-2, -2, -7, -15, 1, 15, 18, 5, -6, -8, -4, -4], [22, 28, 9, 0, 5, 14, 0, -1, 10, -5, 8, -7], [42, -16, -2, 17, 14, 3, -19, 16, -16, 1, -2, -5], [-90, -6, 7, -36, -9, 23, 16, 19, 14, 7, -2, 17], [98, 28, 4, 15, 0, -26, 13, 0, -5, 11, 6, -13], [-73, 42, -23, 0, 4, 12, 0, -15, -10, -3, 7, -8], [106, 72, -3, -5, 18, -1, -12, -19, -2, -14, 7, -17], [74, 24, -46, -21, -7, -59, 6, 12, 12, 26, 23, -13], [206, 96, 18, 20, -12, 33, -48, -24, -20, 2, -22, 25], [75, 46, -20, 30, -28, -18, 17, 48, 1, 23, -12, 10], [-17, -6, -7, -2, -12, -30, -11, -11, 17, -14, 3, 3], [67, 30, 24, 10, 27, 12, 35, 50, -27, -14, -25, 4], [-51, 0, -37, -4, -11, -32, 54, -27, 40, -35, -38, 29], [-308, -82, 19, -76, 8, 86, 53, 25, -17, -18, 13, 9], [-4, -4, 87, 135, 28, 30, -68, 5, 3, -57, 15, 4], [-194, 78, 20, -79, 40, 20, -38, 34, -33, 21, 20, -11], [56, -68, -37, -33, -7, -85, 34, -49, -15, 14, -12, -61], [-231, -4, 42, -167, 41, 71, 11, -24, -21, -7, 51, 15], [120, 22, 12, -62, 46, -34, 15, 16, 25, 5, -6, 10], [-356, 104, 6, 24, -8, 86, 115, 4, 38, 19, 10, 21], [255, 18, 62, -47, 25, -35, -7, -18, 8, -18, -3, 32], [-721, 12, -18, 129, -6, 0, -37, 130, -62, -36, -24, 4], [787, -22, 111, 71, -2, 42, -84, 21, 25, 15, 53, 22], [661, 78, -44, -47, 60, 41, 99, 58, 15, -11, -22, 0], [367, -172, 0, 32, -19, 181, 56, -34, 69, -8, -29, -13], [362, -186, -46, -68, -33, -260, 16, -22, -68, 49, 63, -101], [-315, -6, 37, 245, 33, 88, -89, -41, 35, -12, -20, -52], [884, 172, -7, 66, 60, -75, 85, 73, -60, 39, 101, -114], [693, -238, 5, -139, 20, -156, -202, 59, -10, -26, 13, -30], [-560, -76, -100, -129, 19, -126, 43, -78, 83, 21, 79, -116], [149, -38, 31, 22, -52, 12, 137, -83, 127, 48, 121, 75], [-122, 0, -12, 49, 73, 181, 106, 218, -45, -60, 9, -2], [244, -96, 107, -344, 48, -88, 60, 81, 64, -77, 57, -7], [407, -184, -140, -44, 81, 92, 186, 20, -19, -51, -109, 37], [809, -126, -95, -235, 35, -27, 227, 67, 62, -70, 66, -107], [337, -322, 79, 50, 91, -4, -178, 95, -147, -51, -94, -109], [-350, 48, -88, 1, 65, -39, 119, -144, -33, 149, 177, -137], [1224, -92, -266, 234, -214, 167, -205, 30, 9, 51, -220, 21], [-668, 50, 106, 206, 83, 62, 98, -146, 70, -189, -145, -61], [-503, 136, -201, 119, 25, -94, 180, 25, -94, 143, 114, -173], [-353, -608, 54, -34, -17, 109, 244, -20, -181, -14, -81, -113], [443, -618, -14, 364, -80, 96, -10, -46, -65, 47, -226, 46], [-102, 76, 107, -282, 45, 313, -269, -67, 101, -51, 22, 20], [972, -182, -53, 501, -70, -178, -183, 13, -155, -180, -217, 9], [1594, -592, -25, -19, -81, 213, -108, 93, 17, -158, -6, 186], [1396, 142, -9, 543, 175, 113, -172, -73, -78, -219, -12, -88], [-191, 122, -82, -125, 73, 230, 226, 108, -182, -275, -223, -167], [-21, -118, 0, -82, 196, -442, -50, 180, -169, 121, 252, -238], [-741, 32, -227, -153, -86, 144, 389, 347, 48, 35, -175, 39], [-1042, -696, 274, 125, -70, 170, -182, -160, -75, 124, -162, -18], [2624, 194, 269, 243, 123, -138, 98, -13, -139, -5, 222, -67], [-285, -102, -270, -3, -86, -53, -255, -38, -87, -38, -244, 159], [2860, -790, -167, -98, -57, 232, 71, -83, 230, 24, -28, 187], [-2101, -776, 17, 331, -150, 34, -331, -101, 148, 193, 29, 3], [500, -708, -364, 261, 176, 233, -15, 198, -72, -326, -122, -98], [-627, 150, 89, 90, -127, 355, -384, -143, -207, 119, -54, -57], [-2077, 304, 203, -489, 162, 70, 76, 97, 234, 3, 279, 167], [2926, 816, -300, -582, 74, -521, 421, -296, -207, 229, 18, -113], [95, -234, 49, -453, 8, -375, 122, -35, -105, 227, 317, -315], [825, -52, -352, 139, -380, -508, -245, -392, 150, 226, 16, 22], [748, 580, -73, 367, -291, -295, -762, -163, 71, 344, 172, 348], [-1016, 810, 163, 43, 212, -102, 95, -1, 69, 26, 255, 90], [1858, -364, 231, 2, 380, -627, -425, 71, -169, -72, 37, -196], [-3379, 1494, 156, 385, 24, 403, 195, -248, -143, -162, -172, -63], [-1432, 200, 253, -23, 371, 85, 588, -209, -127, -470, 0, -136], [41, -174, 151, -1014, -62, -224, -471, -295, -37, -130, -357, 91], [-4028, 138, -119, 74, 397, 99, 27, 237, -339, -133, -2, -307], [-1115, -150, 57, -1212, 8, 228, 553, -573, -69, -220, -93, -127], [365, -26, 350, 989, -3, 438, -882, -470, -61, 175, -213, 242], [2893, 742, 320, -846, -291, 539, -117, 426, 251, 340, -127, 308], [718, -32, 52, -601, 184, 100, 675, -388, 225, 173, 654, -41], [4129, -364, -62, 1145, 42, -433, 197, 6, 152, -119, -122, -183], [-2218, 812, 121, -216, 31, 588, 85, 159, 98, 363, 518, 345], [520, 1790, -208, -772, -359, -768, 90, 290, 324, 275, 17, 58], [4317, 1162, 139, -289, -316, 427, 443, -323, 100, -26, -249, -115], [-2211, -902, 201, 813, -133, 976, 453, -109, -281, -602, -630, 199], [1879, -162, -244, -241, 17, 258, -460, -4, 15, 367, 73, -92], [-1660, 1182, 172, -780, 635, 637, 518, 206, -237, 163, 105, -310], [-5217, 444, -472, -312, 1, -786, -170, 516, -425, 236, -201, -380], [2768, -136, -287, -1303, 129, 271, -22, 269, 313, -328, 154, -221], [4221, 1850, -212, 60, -62, -673, 301, 478, -93, -26, 92, 150], [-155, 1240, -61, -313, 270, 127, 169, 535, -64, -33, -69, 217], [-2911, 1314, 131, -989, 654, -176, 785, -55, 52, 22, 849, -71], [-1142, 1692, 106, 903, -214, 467, 0, 292, 115, -20, -122, 416], [3548, -778, 281, -226, 69, -626, -365, 333, -312, 548, 62, 293], [-440, 1186, 76, 409, -791, 313, -961, -126, -20, 747, -3, 107], [7136, -1058, -149, 15, 219, -672, 957, -327, -9, 136, 72, -3], [-3783, -2978, 20, 278, 394, -408, -462, -70, -77, -53, 64, -380], [-2973, 456, -563, 545, -339, 688, -1506, -69, -170, -174, -788, 633], [1871, 1274, -263, -577, 177, -1642, 1010, -549, -194, 155, -140, -675], [-5103, -984, -397, -534, -637, -366, -493, 209, 279, -64, 178, 506], [1250, -2152, 27, 1097, 483, -561, 603, 95, -374, -337, 126, -884], [-4417, -580, -317, -716, 263, 1028, 225, 189, 104, 85, 244, -311], [1608, 1004, -69, 535, -59, 1111, -110, 167, -469, -524, 92, 267], [-746, -1106, 83, -730, 334, 1084, -82, -387, -145, -1042, -449, 113], [4423, -474, 90, -1782, -80, 352, 89, 192, -104, 316, 324, -97], [-3477, -298, 1080, -832, -92, 1656, 97, 792, -347, -441, -572, 416], [-487, -1286, 333, -492, 458, -727, -662, -479, -321, -443, -175, -355], [1079, -1202, 610, -504, 3, 560, -1327, 306, -29, -479, -47, 418], [5422, -82, 128, 922, -141, 1587, -863, -306, 142, -230, -949, -296], [-1445, 1452, -891, 1228, -645, 525, 497, 217, 536, -678, -968, 637], [23, -768, -216, 766, 179, -150, -675, 584, -6, 152, 351, -210], [5498, 1440, -698, 174, -40, -320, -1090, 182, -318, 250, 514, -132], [-6579, 1284, 656, 1726, -311, -372, -531, -206, 153, 63, -841, 289], [4075, 58, -129, -1672, -643, -512, 921, 317, 635, 752, 838, 241], [-2269, 1278, 235, -754, 61, -385, 2260, 115, -543, 27, 684, -683], [5298, -456, 48, 705, 689, -913, -584, 94, -619, -778, -491, -748], [443, 1436, 302, 1612, 595, -1143, -1200, -616, 0, -265, 271, -128], [-907, 2434, 105, 878, 441, -75, -1051, 129, 326, 348, -62, 176], [934, -1400, 133, 235, 240, -153, -498, 469, 198, -204, 713, 261], [-6376, -634, 1031, -515, 1130, 1292, 406, 1329, -567, -1192, -625, 33], [-1233, 830, -188, 2042, -740, 778, -1334, 672, 195, 321, -424, 562], [-677, -2294, -61, -1040, -110, 279, 422, 143, -212, -787, -1105, 594], [-335, 1430, -1587, -401, -497, -333, 403, 971, 621, 652, -306, -520], [6633, 1992, -463, 1445, -75, 202, 986, 967, 262, -616, -524, 177], [-2970, 2776, -905, 1294, -400, -1033, 53, -605, 209, 311, -699, 300], [5428, -1162, 298, 987, -20, 360, 718, 36, -393, 549, -726, -291], [5333, -1692, 321, 615, 126, 2179, 80, 571, 182, -1460, -627, 278], [7347, 176, -663, 372, -97, 622, -635, -589, 297, -658, -596, -50], [6993, 2274, -198, -2113, 850, -2103, 1054, -226, -16, 741, 1164, -536], [-5772, -1506, -387, -2064, -779, -88, -836, -1137, 870, 299, 290, 340], [5864, 438, 440, 2098, 625, -479, -540, -304, -156, 289, 261, -53], [-5517, 2124, -721, -1265, -67, -1036, 738, -643, -480, -435, 282, -641], [-1503, 354, -270, 1834, 957, 10, -139, -322, -1257, -341, -717, -1044], [-1535, 1336, -271, -927, 693, 1842, 535, -493, -208, -729, 1352, -985], [-10650, 1046, 4, 1941, -687, -205, -1907, -42, -468, 472, -1031, 517], [5098, 1612, 870, 2112, 246, 115, -127, 1226, -449, -749, -792, 489], [11790, 2034, -1, -209, -437, -75, -1712, -493, -410, 442, -580, -158], [-2406, -758, -979, -335, -36, -1508, -1158, -209, 231, 1148, 2015, -353], [3707, 722, 90, -1164, 124, 1262, -136, -104, 949, 881, -272, 880], [-4275, -1034, 1602, 1628, 274, -837, -1138, -860, -1294, -53, 48, 19], [-69, -802, 771, 316, -660, 1025, -2210, 267, -345, -311, -71, -137], [564, -4224, 599, -655, -806, -1813, -1061, -993, 73, -457, 155, 263], [9639, 1750, 953, -2964, 1464, -492, 4193, -165, 137, 147, 1939, -735], [-47, -400, -998, 1518, -459, 1143, -971, -1350, 567, -429, 13, -282], [-1664, -2058, 547, -44, -57, -83, -1165, 323, 797, -71, -1898, 951], [1733, 2612, -413, 1467, -107, 388, 42, 1909, -882, -117, -1030, 765], [5349, -166, -180, 2828, -927, 398, -2238, 262, -408, 289, -683, 1015], [2492, 3820, 1488, 604, -700, 253, 979, -1212, -12, 387, 480, 80], [1439, -1308, 243, -676, -701, 770, 2331, -231, 1029, 126, 842, -542], [-989, -4460, 718, -512, -292, 286, -745, 1214, 602, -408, 86, 722], [6955, 3522, 924, 11, 5, -1336, -2079, -1616, 152, 988, 383, -254], [18228, -6, 1066, 726, 411, 194, 298, -588, -266, 1079, 1109, -1182], [12409, -898, 281, 386, -563, -680, -206, 709, 879, 997, 726, 17], [5751, 1558, 174, 3472, 304, 949, -2693, 188, -335, -546, -1404, -182], [-2345, 1162, 347, 519, 531, 2112, -390, -1411, 380, -533, -1324, 955], [-4325, -16, -7, 1933, -48, 1230, 1486, -1369, 110, -813, -2221, 27], [1424, -2328, 1619, -1001, 1049, 972, 310, 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-1404, 6171, -5914], [28524, -5522, -700, 7290, -4885, -6231, -14242, 6662, 2691, 995, -1575, 3918], [65674, -16848, 7405, 457, -167, -536, -6651, -729, 2626, 2953, 1780, 5595], [43912, 3050, 376, 527, -4766, -14062, -12746, -4010, 17, 2373, 2590, -2951], [38164, -1516, -1771, 2668, 1274, -7128, 3623, 2339, -4282, 2701, 4279, -2785], [24516, 7628, 2934, -17411, -3128, 10272, 10736, 3630, 3190, 1872, 94, 3298], [-455, 2080, -4687, -2424, -375, 5959, 9820, -3049, 1047, -1418, -3668, 4175], [-8497, 13434, -905, 3817, 1180, 8405, 3050, 5675, 3075, 4204, 1795, 3709], [40180, -15420, 2595, 5533, -3382, 6284, -825, 3333, 4132, -4383, -6221, 4673], [-30839, -11836, 7241, -9881, 670, -666, 6693, -1505, 164, 725, 925, -178], [25190, 14288, -8202, 6488, 1468, -13810, 13080, 3674, 478, 5350, 4842, -3918], [13797, 17806, -10484, 3909, -5164, -2575, 415, 2764, -4401, 474, -2876, -973], [-36665, 5432, 1607, 9937, 2153, -1198, -793, 2989, -3028, -2801, -5066, 1639], [48154, 9380, -2553, -3618, -1648, -13110, -2997, 2653, 2008, 5257, 3189, -3381], [19071, 14532, -4144, -8900, -4101, -88, 1278, -3220, -2521, 3106, 1757, -2518], [58258, -3414, -3952, -5183, -6618, -17262, 3491, 880, 3491, 5997, 6270, 424], [-24350, 7444, 3217, 6506, -2734, -3693, -12159, -7821, -4384, 481, 5347, 2837], [8753, 13466, -2397, -4897, -5454, 8308, 4335, 945, 3804, -2232, -4959, 7385], [46323, 3100, -5341, -5969, 6548, 7780, 5291, -2379, -3632, -1057, -1283, -944], [33186, 5158, 1006, 2317, 1625, 89, -5313, 5380, -886, 409, 3239, -6749], [-7022, -5626, -10700, -1655, -8222, 4274, -6042, -4986, 1255, -6253, -7570, 2521], [32221, -20292, 10385, 6392, 7857, 8250, -1781, 4539, -5557, -4610, -1062, -156], [39103, 18866, 5127, 3239, -1457, -4042, -2173, -8503, 1427, -1156, 324, 2696], [56308, -21818, -2506, 14310, -4517, -13353, -6813, -5516, -1126, -420, 685, -2042], [7012, -11838, -1412, -5569, -2906, 8480, -5716, -2016, 5683, -229, -1126, 2682], [-6859, 6818, -4029, -6066, 775, -15044, -1527, -2575, -33, 2982, 88, -49], [-22607, -7734, 230, 3011, -4461, -4688, -3837, -5590, 2149, 1777, -5759, 5326], [-1262, -12038, 835, -10696, -1787, -1837, 2245, -2217, 129, 2595, 2812, 2671], [-16682, -21852, 2095, -4960, 1140, 2074, 3670, -3199, 1752, 6185, 2221, -539], [-13498, -7802, -4890, -14688, -3737, -8336, 10392, -6672, 1922, 1021, 4669, -4368], [10426, 19066, -1147, 7193, -2747, 5388, -320, -4077, -485, 1574, 2668, 3085], [-13025, -7110, -4151, -22579, 5363, 306, 4620, 7623, -2964, -2349, 4094, -403], [49494, 7990, -683, -1765, 562, 10892, 14814, 2663, 2139, -493, 119, -1641], [56095, 4302, 2811, 3917, 7413, -7583, -581, -2299, -1962, -5606, 4654, -6955], [-17968, -10338, -2086, -2696, 3221, 13486, 5674, -2038, 1282, -4810, -5667, 4273], [19482, -21788, 5201, -10565, 1192, 6959, 8697, 5985, 3895, 7071, 3139, 5211], [-20707, 9748, -6216, 12377, -1432, 374, 2000, 6016, 2394, -1632, -762, 199], [23948, -452, -8974, 8537, -4553, 3354, 4644, 8342, -1306, 3978, -1403, 2674], [-9892, 14014, -5780, -4859, -3495, 10078, -1399, 2328, -3881, 2417, 1165, -4006], [10627, -2302, -2305, -2628, 4006, -4815, -5413, -6541, 171, -638, -793, -51], [-23635, -10266, -1686, -7653, 472, 9749, 1757, 2822, -4567, -1432, -2034, 6745], [-1301, -24960, 8465, 4367, 1470, 10091, -6764, 2831, 1088, -3004, -2443, -1310], [-7177, -722, 10673, 9679, 860, 3018, 10666, -6273, -3067, -1966, -6321, -1239], [-22522, 10156, 785, -15455, -1409, -841, -4852, 5531, 2197, -1028, 4420, 5434], [10604, 840, 3044, 15267, 263, -2690, -3075, -4156, 1852, 4082, 5103, -2974], [-113483, -11946, 6869, 5907, 6611, 1058, -1972, 915, 3226, -3345, -1106, 2453], [-43526, 13040, -11346, 1413, -2806, -7007, -451, -3476, -6592, 231, -842, -3102], [-8116, 8818, -4848, 15792, 279, -622, -10404, 2440, 324, 5045, -2119, 4803], [27976, 6810, 1477, 1706, 939, -2170, -7252, -3765, -3783, -488, 1158, -257], [-25384, -5102, 3759, -2770, -409, 3857, -6228, -7229, -2322, -215, 174, 2575], [-42574, 4288, -2207, -2178, -1345, -13268, 14626, -521, 3972, 2719, -1498, -7359], [40241, 4362, 4174, -13253, 3906, 15576, 1369, 3142, 868, -159, 1374, 3903], [46041, -1718, -3046, -20522, 374, -9006, 11301, 5262, 5741, 1856, 8262, -273], [1177, 13404, 1671, -3213, 2669, -6426, 14776, -883, 14, 1637, 4006, -8975], [5765, -9216, 1103, -14797, 3924, 8642, 4761, -4559, -260, 309, -135, -471], [-54975, -1406, -5512, -3981, 5629, 1406, 17418, 536, -1553, -3239, 1311, -5785], [-2306, 8160, 6594, -297, 2702, 5933, -1341, -5384, 884, -33, -3438, 6445], [86933, -18624, -9139, -20498, -1490, 11746, 9736, -1869, 5228, -711, 2697, -269], [-25564, 1570, -13590, -10705, 907, -4651, 10967, 2612, -2218, 4996, 381, -6529], [12504, -12464, 1509, 5383, 3155, -776, 555, 2269, -4182, 3915, -52, 252], [-47375, -14974, -1421, 8288, 3642, -204, -6325, -1641, -3089, 5296, 5559, 385], [-2955, -15742, 7781, -5667, 7409, 8484, 7590, 3823, 755, -6402, -1824, -85], [-14370, -18578, 4265, -11881, 1477, 9670, 9912, 10719, -1407, -2458, -7088, 1937], [13579, -2046, -3630, 9399, 343, 10954, 3149, 822, -1158, -4402, -4005, 3164], [62258, -9182, 612, 8537, 4844, 2710, -11241, -1216, -4975, -2070, -180, -1076], [-109335, 4386, -1738, 2274, -1677, 7664, 3180, -2458, 4357, 8893, 7229, 668], [21144, 20440, 4185, 7411, -891, 3681, 2906, -1753, -2385, 2856, 7954, -6264], [57099, 17122, -5428, -4730, 328, -1350, 10032, 6104, 97, 2888, -616, -1782], [-64934, 4218, -5523, -6468, -5031, -8187, -3579, -8371, 2999, 833, -930, 965], [-37099, 11828, 9576, 5788, 2365, -11449, 5084, -2454, -5221, -1316, -345, -2433], [9325, -5150, -1486, 15666, -6182, 6966, 6968, 2096, 1195, -4031, -5834, 2966], [30737, -70, 7442, 1396, -4487, 10196, -7698, 950, 2783, 2635, -4373, 2848], [-50196, 9744, -4096, -11390, 4712, 13321, 6526, 7998, 2063, -3125, -400, -109], [-47403, -4938, -7157, 9076, -2965, -20198, -10048, -3529, -19, 7145, 7702, 1841], [-40648, 3888, -2620, -12503, -2768, -4790, -1202, -11064, 1372, 148, 1004, -2322], [51923, -20018, -24, 13499, 1054, -3858, -8641, -4612, -5201, 782, -2230, -1707], [30367, 13080, -4145, -1229, -1318, 4101, -6051, -333, 6024, 1913, -1581, 6203]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_289_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_289_4_a_i();".
// 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_289_4_a_i(:prec:=12)
    chi := MakeCharacter_289_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_289_4_a_i();".
// 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_289_4_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_289_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![29352, -21960, -117822, 4607, 93255, 5163, -20922, -627, 1872, 17, -72, 0, 1]>,<3,R![-1242003, -6443010, 22610910, -10471190, -3422262, 1841532, 177392, -99708, -2097, 2228, -54, -18, 1]>],Snew);
    return Vf;
end function;