
// Make newform 693.4.a.r in Magma, downloaded from the LMFDB on 14 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_693_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_693_4_a_r();".
// 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_693_4_a_r();".
// 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 := [1056, 384, -2246, -439, 560, 57, -43, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-11, -1, 1, 0, 0, 0, 0, 0], [15, -17, -2, 1, 0, 0, 0, 0], [1672, -8982, -1309, 2404, 103, -161, -2, 3], [1232, -2470, -121, 1172, 15, -117, -2, 3], [4312, 14094, 3553, -2764, -827, 229, 34, -7], [15752, 10570, -11165, -2980, 1119, 239, -26, -5]];
    Rf_basisdens := [1, 1, 1, 1, 352, 88, 352, 352];
    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_693_a();"
function MakeCharacter_693_a()
    N := 693;
    order := 1;
    char_gens := [155, 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_693_a_Hecke( : Kf := Kf)
    return MakeCharacter_693_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 := [[-1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0], [7, 0, 0, 0, 0, 0, 0, 0], [-11, 0, 0, 0, 0, 0, 0, 0], [2, -2, -1, -1, -2, 0, 1, -1], [-10, -5, 0, 0, 1, 1, 0, 2], [-6, -1, 2, 2, -2, -3, 1, 0], [-43, 1, -8, -2, -5, 1, -1, -2], [-64, 6, -9, -1, -4, -4, -1, -1], [-2, 3, -2, 0, -5, 3, -2, -4], [-41, -23, -5, 3, -1, -9, 6, -3], [-26, -15, 5, -3, 0, 11, -2, 3], [-3, -19, -10, 8, 7, 1, 1, 0], [-64, -9, -1, 1, -3, -3, -11, 5], [-100, -56, 3, -3, -1, -2, 0, 7], [-157, 1, 45, 15, 5, -9, 6, -5], [-39, -15, 9, 19, 4, 5, 3, 7], [72, -14, -6, -8, -3, -10, -11, -2], [-117, 33, 35, -3, -4, 17, 1, 5], [7, -50, -9, -21, -4, 8, -14, 7], [75, -67, 13, -15, -36, 19, -3, -1], [-400, 5, -12, 10, 5, -7, 2, -6], [-295, -29, 21, 13, -26, -15, -1, -1], [-244, 142, -47, 13, -41, -6, 4, 9], [-176, -94, 57, -33, 17, 4, -2, -11], [-442, 67, -53, 17, -36, -13, 20, -11], [-405, -45, 76, 28, 44, -43, 28, -18], [224, 180, 49, -9, 37, -22, 22, 1], [-119, -95, 3, -29, 36, 59, -7, 13], [-47, 207, 58, 14, 33, 41, -13, 24], [224, -105, 71, -19, -22, 27, -8, 33], [-973, 223, -110, -30, -57, 25, 19, -28], [-541, 390, -32, -32, -70, -14, -11, 0], [-401, -83, 118, 54, 50, 33, -18, 16], [-160, 360, 113, -7, -9, -36, 20, -47], [-216, 327, -123, -17, 66, -5, -20, -25], [-75, 105, -99, -33, -53, -21, -14, 31], [6, 112, 5, -47, 65, 28, -68, 37], [6, -122, -130, 0, 28, -40, 20, -22], [-969, 29, 1, 53, 76, -57, 1, 11], [-486, 551, 172, -38, 63, -13, 38, 2], [-233, 107, 124, 38, 45, 15, -19, 86], [-226, 200, -17, -57, 151, 116, -46, 23], [-965, -115, -155, -5, -54, -123, 61, -13], [804, 315, 250, -20, 53, 63, -30, 32], [-557, -189, 15, 69, 60, -105, -61, -35], [-558, 723, -93, -67, -64, 35, 24, -3], [-160, -387, -105, -63, -180, 1, -8, -107], [512, -301, 22, -44, 145, 67, 16, -28], [-316, 470, 222, -4, 204, 12, 2, -34], [1147, -183, 0, 4, -16, 59, -70, -30], [81, 154, -249, 51, -10, -100, -26, -13], [157, -101, -59, -61, -15, -47, 98, -21], [1008, 560, -94, -14, -105, -120, -13, -94], [-282, 170, -73, 3, -102, 114, 15, 35], [-562, 527, 141, -51, -218, -3, -90, 19], [531, -501, -216, -96, -16, 121, -36, 102], [62, 126, 80, 66, 126, 52, -102, 100], [91, -291, -440, 32, 40, -111, 78, -42], [1190, -461, 210, -42, -134, 41, 53, -92], [1326, -654, -168, 26, -60, 44, -34, 84], [1175, 920, 72, -80, -152, 180, -135, 96], [-5, -498, -50, 22, 96, 98, -115, 22], [-1508, -920, 33, -55, 73, -200, -20, -115], [-2024, -4, -123, 87, -21, -6, 146, 33], [899, -803, 433, 141, 150, -33, -17, -53], [-62, -194, -327, 161, -35, -74, 80, -91], [-1860, -986, -164, -180, -102, -10, 36, -140], [2284, 374, 107, -253, 170, -36, -25, -65], [-210, -186, -352, -36, 137, 26, 3, 152], [-4023, -333, 177, 153, -72, -43, 77, -9], [2016, -145, 85, 115, 220, 171, 156, 151], [1985, -563, -246, -28, 267, -31, -121, -76], [-1529, 515, 115, -51, -467, 65, 30, -147], [-1265, 42, -928, 52, -234, -162, 29, 12], [-420, -34, 168, -206, -166, 64, -138, -28], [-1662, 803, 308, 278, 613, -21, -60, 138], [-2369, 1391, 2, -282, 27, 201, -17, 4], [2980, -1289, 533, -65, -50, 95, -72, 243], [467, 263, 523, 29, 199, -119, 152, -279], [-55, -1787, 379, 39, 233, 259, -38, 353], [-6744, 1172, 263, 79, 204, -216, -67, 59], [330, 3104, -212, -100, -80, 60, -84, -20], [1413, 29, -198, 186, 68, -53, -230, -8], [-2765, -1515, 151, 107, -20, -85, 127, 129], [2209, -487, 893, 229, 214, 339, -187, 183], [-1246, -820, 409, 17, 679, 64, 140, 17], [2654, 1212, 84, 206, 302, 86, -246, 68], [-2141, 439, -481, -99, -913, 33, 16, -151], [363, 1611, 331, -159, 141, -3, 26, 65], [-1184, -1976, 103, -177, -153, 68, -58, 135], [-1806, 580, 1080, -12, 88, -252, 54, 156], [-2281, -187, 8, 376, -98, -65, -32, 50], [5998, -2048, -598, -172, -7, -156, 97, -34], [-4914, -1756, -265, 383, 223, -312, 300, -161], [2858, -1717, -709, -329, -900, -47, 114, -243], [-6544, 1604, -216, -8, -151, -332, 191, -120], [-1256, 869, 702, -134, 326, 319, 45, -84], [497, -1519, -729, -283, -634, -67, -177, -55], [2103, 597, -357, 113, -62, -111, -73, -107], [-2942, -1724, 1301, -55, 500, 302, -3, 141], [-12122, 1159, -426, -164, -561, 191, 194, -8], [-866, -2380, 62, -212, 350, -38, 216, -466], [6483, -2283, -751, -233, 68, 377, -197, 307], [-1674, -2402, -97, -373, -333, 410, -84, -1], [-1333, 237, 325, -13, -705, 379, -92, 291], [2460, 971, -391, -183, -32, 385, 46, 19], [-8697, 1293, 1393, 123, 798, 33, 57, 107], [3539, 766, -347, 413, 386, -176, -98, 41], [-4511, -1817, -924, 120, -372, -383, 358, -390], [-1805, 1327, -568, 6, 385, -61, -9, 78], [1216, -88, 132, 132, 930, 84, -22, 300], [-3500, -528, -203, 473, 571, -512, -240, -79], [7480, 398, -182, 42, 128, -110, 40, -274], [-4514, 442, -65, 647, -1275, -462, 496, -249], [-4340, -494, 1165, 357, 665, 498, 8, 285], [-11434, 849, -401, -367, -35, -101, -211, -331], [-12126, 1152, -890, -240, -556, 214, -14, -90], [-1171, 65, -418, -70, 16, -129, -414, -88], [-1394, -4483, -352, -170, -573, -67, 172, -294], [-742, 626, 435, -29, 867, 358, -608, 439], [3612, 2096, -1050, 736, -132, -526, 336, -170], [-10515, -345, 460, 116, 1145, 429, 73, 6], [2942, -3820, 788, 368, 1110, -176, 72, 236], [4858, -3320, -702, -220, -54, -146, -24, -410], [-10685, 553, 1077, 497, 879, 223, 132, 51], [3956, 1203, 1171, 213, 419, 29, 211, 121], [-2466, -3629, -322, -272, 99, 283, -202, 476], [2717, -2411, -883, -593, -488, -619, 187, -121], [-2023, -461, -204, -70, 1209, 31, -113, -142], [-6541, 1049, -1256, 392, -118, 479, 116, 146], [871, -3243, 517, -57, 115, 351, -92, -133], [3329, -873, -271, -719, -321, 245, -196, 91], [11092, 2481, 5, 185, 480, -317, 316, 267], [14533, -2222, -837, 251, 66, -372, 182, 99], [959, -2736, 380, 356, 41, 236, -208, 168], [-1056, -5895, 932, -324, 320, -77, -199, -2], [4651, 2028, 406, 754, 569, 20, -38, -46], [959, 31, 1578, 254, 16, -777, 434, -16], [-4250, -2685, -1464, 16, -726, -643, 763, -686], [13583, -2959, 940, -36, 566, 429, 20, 410], [-2065, 1865, 913, 807, 471, 83, 192, 171], [175, 2069, 1386, -350, 32, 87, 108, 100], [7588, -1239, 455, 713, -128, -227, -90, -179], [2490, 2949, -441, 1097, 671, 211, 147, -299], [-5665, 51, 1203, 229, -464, 331, 65, 29], [9816, 5291, 1171, 419, 1450, -171, 38, -355], [-11850, 1876, -1814, -454, 324, 836, -56, 186], [-3561, -2285, 1292, -502, -179, 131, -207, -82], [-1200, 1000, -1580, -186, 572, 406, -148, 364], [11026, 1200, -2054, -74, 503, 216, -391, 254], [-2833, 4187, 2709, 311, 1959, -415, 226, 439], [2444, 172, -1765, -313, 193, 784, -452, -141], [-19425, 69, -855, -367, 258, -285, -277, -373], [5059, -673, -1699, 659, 494, -569, 655, -157], [-15904, 1932, -1721, -57, -1707, 56, 444, -1], [26588, 1876, 92, -332, -229, 76, -123, -76], [4706, -2943, -1129, -31, -574, 569, -34, 485], [14890, -38, 590, -276, -520, 664, 60, -390], [-6712, 2164, -1166, -218, -240, -136, -134, -214], [2657, 2497, -1368, 216, -1996, -387, -80, 94], [14038, 4230, -1544, -360, 346, 126, -314, 20], [6725, -75, -1521, -107, -1351, -1129, 242, -751], [8826, -4734, -293, -465, -1739, -426, 604, -317], [3443, -134, 580, 292, -1338, -898, 789, -532], [-7995, -963, 1247, 925, 605, 483, -562, 373], [13839, 65, 1099, -155, -218, 277, -599, -335], [-2304, 6030, 96, -136, -854, 270, -390, -92], [8077, 3796, -1094, 6, -205, -200, 582, -470], [4685, -219, -1186, -426, 1228, 315, -1006, 148], [7464, 269, 589, 551, 2070, -231, 206, -493], [2668, -3008, 558, 226, 1746, 108, -300, 886], [9082, 9786, -2525, 19, -681, -222, 328, -153], [-9562, -2109, 287, -375, -68, -1045, 332, -447], [9197, 1057, -771, 1317, -404, -1021, 763, -13], [2111, 2473, 682, -1092, -63, 197, -891, -284], [3696, -2318, -3370, 96, -1432, 100, 18, -14], [-1171, 1997, 140, 792, 292, -261, 608, 406], [-10524, 2654, 1391, 763, -132, -532, 597, 427], [-5483, 1669, 609, -529, -1901, -405, -586, 599], [16362, 1765, 196, 298, 1047, -555, 430, 346], [7024, 9468, -754, -686, -1062, 892, -650, 314], [12406, -829, 531, 83, 1126, -551, -284, -643], [661, -1, -1, -203, -1460, -29, 501, 549], [23919, -1360, -48, 140, 273, -232, -224, 796], [-18129, -4813, -1975, -377, -1724, 351, -421, 999], [-7488, 28, 1925, 201, 1169, 4, 682, -183], [15846, 3970, 3537, 713, 2731, 526, -182, 669], [6512, 4466, 1077, -271, -2435, 70, 712, -455], [-12904, 6048, 262, 618, -298, 696, 238, 174], [-5381, -1741, -1250, -1450, -276, 269, 874, -256], [-6600, -4154, 96, 430, -1293, -290, 473, 64], [24605, -5312, -1220, -1008, 407, -268, -60, 324], [-15839, 1623, -92, 716, -3980, -195, 366, 18], [3060, -3112, 1238, -74, -1083, -400, 453, -946], [-7860, -520, -33, 1143, -2101, -516, 280, -89], [580, 3900, 1520, -1184, 552, 1240, -554, 284], [22818, -1663, 1130, -482, -1593, -1017, 694, -424], [-11902, -276, -135, 705, 2911, -16, -130, -255], [8726, 680, 2030, 444, 2838, -398, 408, -726], [23619, -953, -734, 606, -1032, 1353, -750, 216], [-4484, -1343, 414, 264, -1089, 473, -568, -288], [-9479, -2550, -1633, -313, -2140, 24, -56, 519], [-2062, -5434, -1838, 610, -940, -2, -70, 266], [5496, 7839, 1735, -1029, 648, -175, 20, 533], [-17156, 25, -519, 151, 1006, 957, -764, 551], [-27109, 8793, 2072, 1536, 836, -341, -110, -462], [-1614, 4674, 426, 264, 892, 752, -580, 482], [15580, -5840, -2785, 19, 465, 428, -520, 163], [14364, -1476, -898, 42, -732, -100, 1316, -1002], [22377, 4520, 2656, 344, 2651, -44, -536, -164], [-15827, 5573, 3241, 311, -1091, -477, 736, 27], [-21536, -2014, 1232, 1328, -1346, -602, 476, -180], [-9300, 4258, 1736, -932, -142, 1238, -1112, 580], [-28268, -4904, -3694, -434, -732, 1036, -132, 370], [15901, -5513, 723, -353, -2404, -739, 967, -1247], [18802, 5317, -3172, -276, -1179, -2033, 566, -1582], [-8171, 131, 2536, -978, -439, 623, 851, -1422], [-12418, -7584, -957, 569, -1529, -1258, 1378, -777], [-5877, -5137, -6, -464, 3083, -453, 159, -728], [10037, -1895, 1024, -1132, 1508, 1171, -516, 758], [-16309, 5691, 1784, -264, -1387, 581, 1307, 714], [20907, 591, -3939, 39, -3193, -431, -234, -489], [-15792, 3537, 3633, 1479, 2, -643, 520, 79], [10700, 5792, 2468, 1138, 2320, -1122, 824, -668], [2848, -2206, -478, 122, -689, 2062, -843, 1566], [9715, -11423, -1431, -209, -64, 1377, 595, 1163], [21329, -7203, 6012, 2216, 1900, -837, 452, 110], [-23867, 8117, -2116, -236, -1032, 555, 1032, 442], [-6880, 1840, 4956, 216, -544, -1280, -306, 0], [11858, -5275, -3532, -732, 1808, -201, -677, -666], [21202, 430, 2466, 282, -1374, 802, 736, 222], [15219, 5775, -1612, 360, 3286, -209, -1322, -294], [-24804, 9040, -403, 133, 1315, 200, -12, -215], [-9362, -9592, -4766, 650, -846, 316, -306, -526], [-9808, -1250, -6407, -895, -2331, -694, -242, -191], [17607, -4903, 2776, -708, 2536, 1153, -770, 1514], [-29386, 6434, 845, -1371, 1605, -162, -1498, 405], [17098, 609, -2859, -769, -1767, 2223, -761, 467], [-6943, -12513, -564, 104, -985, -595, -717, -1330], [-9249, -5964, 2064, 484, 3400, -720, -35, 508], [-15128, -4887, 2412, 762, 97, -235, 1218, 798], [-9318, -6284, 170, 666, 1758, 664, 1356, 1214], [11118, 4313, 3369, 693, -1714, -1413, 1416, -1705], [-31052, -322, -2004, -2472, -1868, 1666, -1706, 248], [4242, 6591, 2231, 673, -936, -1625, -192, -447], [-24217, -2851, -3907, -1469, -2727, -533, -366, -33], [-24747, 6727, -1237, 605, -2918, -1393, 871, 73], [-24286, 3446, 1839, 1319, -1674, -2442, -485, -961], [-5411, -2119, 891, -1547, -1414, 697, 575, -1515], [-4960, 2741, -1816, -600, -3241, -301, -510, -530], [7299, -7171, 1805, -297, -1214, -2375, 109, -233], [-11382, 2476, -3000, -2028, 1596, 1132, -752, 328], [33150, -1290, -1904, -1118, -324, 2236, -454, 1292], [-14126, -9292, 886, 594, 1962, 168, 522, -106], [-805, -3799, -425, -173, -937, 403, 294, 1249], [10490, -2481, 190, 262, 2952, 433, 1087, 800], [27185, -12812, 1522, 1366, 3421, -1620, -644, 718], [-10211, 5878, -7075, -927, -326, 368, 716, 261], [-3312, 13556, 2388, 780, -1726, -400, -666, 1540], [-52976, 13896, 5974, 206, 762, 1344, 156, 166], [515, 3059, -1173, -851, -396, -865, 1025, -2063], [32014, 14884, -341, 99, 2850, -170, 351, -169], [5800, -7842, -162, -214, 2771, -494, 321, -210], [17176, -6866, 6121, 121, 49, -630, 100, -1079], [-14092, 368, 1894, 2056, 1822, -2870, 846, -1674], [48370, -9720, 3532, 1460, 321, -1000, 805, -188], [10031, -1515, -901, -3411, 84, 989, -607, 301], [35665, 4755, 2655, -1171, -2338, 1819, -109, -483], [-14870, -10083, -2349, -1655, -2556, 1037, 396, 1417], [-31841, -4597, -4601, 105, 579, 69, 1364, -959], [-24774, 5518, -1159, -1207, 2279, -1718, 282, -303], [-8563, -4421, -2790, 2254, -2533, -831, 305, -264], [-2135, -2182, 7367, 139, 4508, 960, -560, 691], [-32356, 6521, -5990, -2128, -1849, -1211, 1058, -308], [-39624, -7084, -1870, 1086, 5136, -8, -30, -70], [39763, 7187, 4245, -221, 68, 1487, -273, 2043], [23408, -20346, 685, -699, 1175, -182, -192, -183], [-20814, -3995, 3460, 98, 2959, -1075, -1226, 162], [1850, -3578, 1505, 273, -2383, -506, 246, 269], [15474, 3590, -1608, 108, 1230, 342, 1302, 1572], [-2856, 2120, 8623, 783, 3583, 1596, 1110, 647], [5964, 10226, 1185, 1277, 4190, -1624, 133, -707], [29536, -14727, 1064, 764, -1070, 175, -753, 586], [10483, 23715, -2856, -254, 1055, 783, -509, 234], [-7738, 6066, 4990, 2110, -591, -2246, 3075, -2494], [-11740, -34, 198, -1400, -656, -152, -1614, 214], [-11882, -6901, -314, -1484, 1399, -429, -1106, -156], [33334, -2223, -2238, -1578, -1401, -21, 1066, -376], [63644, -15912, 233, 539, -733, 306, 174, -243], [-12090, 2338, -5818, 674, -1624, -1202, -1170, 550], [5847, 16733, 2594, -422, 3726, 305, -654, 392], [16338, 10033, 3532, 2078, 1343, -1407, 110, 794], [8316, -10060, 1296, 602, 3706, -102, 714, -832], [-2472, -22986, -6679, 523, -6533, -1184, 1404, -1251], [7692, -11119, -5671, -789, -5382, 397, -4, -1265], [20606, -697, 1311, 1191, 1252, -283, 1062, -707], [-51460, 1641, -638, -1044, -5401, 2005, -914, 376], [13208, -5915, -2986, -50, -1096, -333, 1607, -356], [-11231, -4714, 4151, 419, -282, -736, -1170, -309], [25859, 5001, 3564, 2448, 1274, 721, 1424, -166], [-23385, -3763, -1336, 280, -1791, -3161, -167, 34], [-60127, -7793, -5294, 1074, 577, 701, -81, 1012], [-20239, -10103, -83, 3199, -1465, -1693, 2298, -1753], [-20016, -9636, -298, 1906, 2963, -3180, 1189, -914], [-14814, 4010, -2819, -2587, -4841, -110, 368, 53], [33818, -14355, -1700, -3054, -167, 325, 250, -122], [-47431, 2603, -6890, 912, -333, 1663, 731, 1168], [6080, -16676, -1307, -165, -1321, -4054, 388, -1603], [-61338, 5242, 608, -1324, -50, 1190, 562, -388], [17994, 15224, -5556, 2426, 678, -1918, 270, -908], [16577, -9937, -7702, -762, -3552, -881, 2656, -2688], [-27373, -8907, -2397, -1149, -3636, 651, -77, -147], [-23644, 8632, 2680, 1956, -1304, -508, 292, -1784], [41252, -3101, 2565, 571, 3258, 1223, -1164, 1427], [-32614, -13812, -2876, 580, 1664, -1140, 1526, -1412], [36309, 8647, -4155, 1433, -2444, -2635, 1721, -773], [-49086, 11782, -779, 685, -1463, 2230, -316, 1589], [-51282, 2564, -4330, -2168, -1678, -294, 1020, 1050], [30470, -4895, 1196, -2588, 2282, 1287, 293, 1018], [-6611, -24407, -2629, 2219, 3893, -2137, 1058, -487], [31434, 13962, -6554, 268, -3218, -1232, 2766, -2246], [-45762, -6636, -862, -1638, -1510, 3820, -2698, 2098], [44185, -6475, 8854, 854, 2272, -563, -1678, 2788], [40050, 6350, 3294, -406, -5512, -1030, 880, -1526], [18162, -7514, 5824, 3296, -1806, -1790, 3026, -1212], [9492, -7892, -6317, 3255, -2317, -2868, -610, 151], [-25750, -14998, 325, -2747, -1638, 1166, -999, 1009], [-32983, 17739, 2844, -320, -1290, -2825, 1436, 1546], [4129, -7055, -2313, -4151, -2836, 753, -2235, 1945], [6138, 13248, -2378, -476, 3802, 1230, -512, -578], [-12831, 9413, 6716, 588, -1458, -3429, 1858, 658], [-23246, 24037, 4118, -2724, -3719, 765, -2318, 508], [17629, 11351, 1938, -352, -1339, 51, 685, -3216], [10291, 7319, 5963, -1631, 1701, 2433, -482, -527], [393, 18207, 3941, 1601, 2017, 1173, 1898, -1669], [-83720, 5569, -2102, -918, 4095, -89, -236, -1056], [-83058, -1900, 8541, 1345, 6163, 1500, 984, 1217], [7036, 4908, 5890, -438, 2762, 1984, -264, -1058], [-14428, 24162, -1646, 546, -1840, -366, 1334, -510], [-49424, -22044, 479, -837, -375, 600, -2744, 267], [25468, -2564, 1263, -671, 5361, 522, -3078, 1063], [-29671, -2117, -3369, 913, 2157, -1567, 2160, 1253], [-8208, 6090, 3401, 651, -6195, -816, -1038, -1439], [-35401, -7227, 1768, -536, -1510, 1479, 1188, 538], [-15784, -5721, -8583, 279, -2208, -1801, -1902, 1707], [11818, 8238, 7618, 370, 640, -182, 402, 830], [6128, -7968, -5978, -2834, -6692, -3028, -890, -1762], [-19240, -10283, 7783, 511, -1646, -785, 836, -1295], [53021, 4576, 9680, 2964, 5444, -1012, 359, -1244], [-11582, -1034, 1966, 306, -985, 1726, -2651, 3462], [-37960, -13124, -3476, 1726, -388, 218, -2342, 660], [40569, 5159, 11172, 420, 4572, 1569, -186, 2814], [-51232, -5164, -637, -711, 9201, 686, 148, 563], [12052, 1429, -7336, -52, -3973, 2943, -818, -642], [-69322, 17831, 6423, 1233, 3953, 4077, -781, 2181], [26116, -25021, 1137, 4067, 608, -873, 340, 1099], [-48447, -12127, -2398, 1402, 1046, 285, 116, -1320], [-118203, 5263, 1608, 1322, -373, -1225, -1687, 570], [44420, -10668, -2512, 1618, -2122, -4714, -538, -496], [-44508, 12311, 10237, 185, 202, -1075, -2128, 767], [27153, -6129, 3297, 2919, 2900, -177, 3079, -2657], [-39583, -2061, 982, -5550, 930, 385, -2230, -648], [34542, 18710, 6864, 244, 7104, -3138, 1316, -2940], [-49949, 15555, 10055, 2303, 5343, 3025, -788, 869], [-53830, 20262, 5427, -275, -3893, 1044, -488, 1071], [14109, -3565, -5693, -2589, 7112, 1965, -269, 2385], [-20208, 19814, 624, -4064, -1648, 2382, -2568, 1420], [-16262, -26298, -6279, 201, 1311, 2286, 424, 1241], [-65210, 22638, 3387, -2057, 4491, 766, -442, -29], [-38500, 2184, 6631, 863, -4406, 2904, -2533, -1], [-66978, -9532, 8101, -71, -734, 4018, -1133, 2261], [-48252, 11460, -3174, 182, 1826, 236, 1324, -4290], [68331, -9525, -1001, -1593, 3067, -3, -3112, 1221], [-59973, 683, 6443, -4643, 1487, 1633, -1712, 477], [21850, 3290, -843, -2415, 1718, -2898, -339, -827], [-49461, 8158, 1077, -199, -4786, 1920, 2242, 1349], [-6278, 9699, 5701, 2213, 6170, 1897, 842, 551], [378, -10035, -1845, -2031, -1662, -967, -338, -751], [75803, -41104, -3940, 1348, 1507, -72, 678, -292], [-42986, -2322, -3545, -2033, 6403, 718, 430, 1143], [-38753, 5277, -11455, -1715, -7258, -4653, -1533, -861], [-96379, 21189, 4494, 1658, -2882, -791, -344, 1464], [-27229, 2100, -13360, -3684, -1972, 80, -2379, -3028], [1508, 41223, 5795, -1823, 4409, -323, 857, 549], [-41115, 20791, -4695, -1545, -2663, 1253, -130, 231], [-71460, -12969, 10057, 1187, 2362, 2579, -1856, 2835], [-49001, -2245, 1988, 1766, 4317, -3573, 673, 930], [28598, -4313, -2701, -4111, 2675, -791, -1251, -1339], [47703, 8531, -557, 4887, 1409, -1471, 2910, 813], [-70734, -1897, -4773, -2121, -2432, 81, -1466, 777], [30052, 2788, 1263, -3477, 1055, -100, 674, -661], [-29058, -1352, -14389, -3261, -855, 2652, -2786, -241], [-53783, 25687, -2845, -707, 3882, -2189, 131, 161], [77438, -18170, -3501, -869, -1345, -5254, -520, -2361], [-16709, -10039, -3640, -108, 2342, -1317, 332, 3262], [2118, -6872, 866, 3946, -6771, -808, 567, -1086], [6947, 2053, 6535, 2467, -1816, 2807, 2439, 1481], [-4025, 33173, -7412, -1916, -2120, 1061, -946, 802], [-23648, 16572, 12589, 2989, 3434, 422, -41, 973], [-50968, -10528, -4177, 3975, -2001, -3592, 1258, -2493], [54000, -76, 4851, 2115, 2471, 2668, 2542, 1267], [-67353, 3507, 4285, -51, -2886, 4393, -483, 2399], [61444, -16268, -3981, 325, -6429, 1122, -612, -2525], [-11205, 7495, 2628, -3984, 4910, 697, -1934, -742], [19773, -19195, -2767, 1893, -5245, -3333, 2094, 455], [-78768, -15792, -2524, 3230, -3692, -3410, 438, -3000], [-61732, -952, -7070, -210, -3188, -2572, -1806, -2014], [-65992, -24008, 3416, 3240, 9556, 408, 868, 768], [79750, -451, 3313, 663, -2884, 909, 2642, -4237], [-67887, 23771, -5103, -489, 5972, 1423, -865, 1619], [-28174, -16930, -7046, -1558, -1600, 4238, -1274, 2390], [-57285, -4131, 5786, -700, -1483, -1015, 665, 2188], [2008, -13524, -1200, -3648, 4036, 2168, 1726, 2232], [34063, 32960, 1494, -2714, -4797, 1164, -1772, 762], [66506, -31134, -97, 2659, 822, -3266, -285, -313], [48020, 27981, 1494, 3620, 4995, -1447, 934, 264], [37667, -7261, 5227, -4837, 5904, 2833, -1259, 3061], [-44732, 22156, -2713, -2229, -4929, 84, 2522, -1129], [-11322, -4916, 763, -4243, -3553, -1878, 730, 1463], [-6216, 18196, 541, -5699, -5202, 1734, -2709, 1533], [-8749, -48835, 3698, -1420, -3715, 1701, 109, 2320], [39770, 33609, 882, -568, 4189, -23, -842, 2064], [15366, -11870, -3716, 5464, -1350, -5226, 420, 144], [-25961, 14008, 14140, 128, -1795, 3092, 3596, 2308], [-42117, -9367, 15286, 2234, -539, 4983, -1669, 3092], [-13197, -21379, 3026, 658, -2402, 145, 1046, -2192], [-9676, 1636, -5610, 254, 2216, -48, 826, -1482], [16008, -876, -4698, 6622, 4974, -3360, -1000, 1446]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_693_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_693_4_a_r();".
// 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_693_4_a_r(:prec:=8)
    chi := MakeCharacter_693_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_693_4_a_r();".
// 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_693_4_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_693_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-672, -3164, -292, 1497, 200, -187, -29, 6, 1]>,<5,R![-4547664, 18752580, -6144666, 133342, 100333, -3572, -572, 10, 1]>],Snew);
    return Vf;
end function;