
// Make newform 483.4.a.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_483_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_483_4_a_c();".
// 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_483_4_a_c();".
// 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 := [-288, -524, 84, 295, 7, -34, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-10, -1, 1, 0, 0, 0, 0], [-824, -1360, 926, 149, -112, -4, 3], [784, -646, -1175, 69, 164, 1, -5], [1504, 340, -1062, -81, 112, 4, -3], [-1616, -1972, 2092, 501, -320, -26, 11]];
    Rf_basisdens := [1, 1, 1, 136, 136, 68, 136];
    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_483_a();"
function MakeCharacter_483_a()
    N := 483;
    order := 1;
    char_gens := [323, 346, 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_483_a_Hecke(Kf)
    return MakeCharacter_483_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 := [[-1, 1, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0], [-6, 0, -1, 0, -1, 0, -1], [7, 0, 0, 0, 0, 0, 0], [-20, 3, 0, 0, 4, -3, 2], [-9, -7, 2, 1, 1, 4, -2], [-27, -17, 3, -5, 0, -2, 7], [-41, -11, -3, 3, 6, 0, 3], [23, 0, 0, 0, 0, 0, 0], [-48, 34, -17, 0, -1, 17, -10], [43, -5, 1, -5, -4, 2, 13], [-98, -24, 1, 14, 13, -3, -2], [-46, -16, 1, -6, -29, -1, 0], [-26, 15, 27, 6, -19, -13, 5], [-183, -13, 16, -5, 11, 23, -3], [-146, -25, 16, -14, 14, -26, -9], [-188, -77, -6, -30, -22, -10, -7], [-4, -12, 38, 20, 16, -15, -1], [-162, 53, 14, 32, 4, 8, -47], [-244, 102, 14, 2, -38, 1, 45], [-124, -37, -50, -2, 66, -3, -4], [106, 5, 10, 54, 12, 51, -10], [-291, 45, -17, -67, -38, -60, 27], [-83, 136, 29, -27, -30, -50, 26], [-227, 160, -6, 21, -77, 64, 13], [-505, 361, -39, -25, 34, -19, -46], [392, 71, -61, -58, 23, -2, -68], [-214, -22, -46, 8, 88, -3, 17], [-196, 58, 5, 30, -159, -30, -55], [-760, 392, 59, -2, 59, -32, -1], [206, 141, 55, -70, -19, -103, 91], [-518, 119, 82, 106, -98, 29, -62], [-302, -100, 135, 114, -13, -5, 98], [-488, -38, -19, 8, 17, -90, 23], [780, 156, 140, 70, -26, -130, 44], [176, -351, -223, -90, -147, 164, -102], [-434, 214, -230, 24, 102, 126, -54], [758, 185, 8, 112, 36, -28, -121], [-781, -479, -97, 103, -150, 102, -199], [185, -355, -21, 163, 34, 256, -143], [-420, -82, -213, -152, 269, 136, -11], [766, -443, -144, -68, 276, 5, -32], [-1168, -196, -120, -28, -20, -90, 190], [730, 87, 195, 98, 31, -136, 90], [-1250, -331, -129, 220, -51, 51, -89], [852, -106, 195, -152, -143, -134, 129], [124, 609, 126, -164, 36, -95, 76], [-133, 481, -317, 173, 160, 233, -364], [-219, -660, -38, -27, -237, -53, -150], [50, 105, -146, 138, -238, 234, -155], [152, 17, -282, -236, 24, -96, 135], [932, 333, -18, -58, -4, -100, -345], [803, -16, -382, -17, 37, 24, 63], [-1933, 157, 158, 185, -97, 75, -181], [350, -306, 560, 88, 48, -140, 322], [0, -24, 337, 38, -387, -103, 176], [-5, -102, -521, 171, -198, 120, -324], [127, 881, -549, -113, 62, -12, -5], [-504, -576, -78, 88, 334, 37, 157], [-1634, 669, -63, -230, 365, 96, -54], [-2601, 307, 317, -251, 232, -53, 178], [314, 174, -100, -220, -80, 130, -68], [-558, 112, -85, -284, -177, -285, 198], [-1387, -526, 25, -117, -562, -240, 10], [-1441, -56, -401, 249, -314, 1, -229], [-832, -468, -89, -34, -15, 158, 437], [1908, -986, 180, -232, 276, -258, 384], [180, -308, -450, -568, 602, 73, -133], [2230, 84, -59, -236, 337, -59, 512], [2047, 393, 469, 159, 74, 1, 142], [-14, -342, 327, -442, -499, -141, 448], [-144, -532, -61, 264, -285, 742, -611], [-1208, -1105, 479, -36, -271, 49, -95], [-608, 399, -174, 182, -340, -199, 48], [172, -573, -69, 526, 287, 674, -408], [-2448, -1187, 180, -184, 84, -401, 44], [-1674, -399, -184, -80, 144, 427, -532], [763, 576, 743, -3, -122, -189, -383], [-124, -753, -118, 850, 320, 63, -218], [-4426, 109, 504, -300, 282, -581, 176], [-323, -1647, 117, -505, -162, -635, 444], [-1928, -19, -235, 250, -379, 303, 17], [-4134, -753, -865, 234, -143, 33, 367], [3227, 585, -258, 7, 309, 531, 101], [-2043, 819, 535, 533, 368, -44, -277], [-1146, -397, 153, 224, -53, 580, -554], [-4004, 1583, 42, 100, 1004, -368, 453], [1018, 757, -200, 38, -1322, 290, -811], [2271, 73, 913, 287, 146, 225, -414], [-1114, -1460, -1109, -660, -617, 407, -144], [-4219, 178, 658, 147, 261, 216, -85], [-3074, -2557, -72, 502, 434, 367, -784], [-2590, -494, -17, 784, -161, -43, 156], [8206, 2554, 90, 228, -368, -271, -437], [1866, -2547, 90, 536, 826, -286, 383], [2224, 815, -136, -958, -706, 658, -109], [-7155, -2790, 1011, -309, 660, -863, 1375], [-5951, -354, -431, 715, 154, 449, -507], [-7525, 1950, -845, -225, -168, 671, -415], [1438, -468, -466, -600, -766, -192, -498], [3350, -2095, -234, -164, -358, 6, -313], [-3752, 770, 900, 274, -238, -592, -320], [3879, 1643, -224, 859, -569, 628, -682], [3090, 443, 1341, 154, 513, -92, 68], [7810, 1455, 626, -278, -208, -547, 582], [-8724, 2408, -374, -694, 496, -1130, 498], [636, 4282, -1446, -158, -16, 1023, -1037], [-3980, 1239, 717, -748, -1593, -262, -120], [426, -486, 1882, -36, -204, -567, -1], [6330, 1613, 1076, 66, -526, -260, -991], [1967, -272, 1705, 981, 916, -568, 358], [-9984, -1528, 785, -576, -757, -481, 466], [-1364, 108, 80, -1028, 1508, -465, 591], [-6299, 430, 39, 243, 536, 418, 368], [15834, -813, -370, -142, -56, 519, 340], [-2718, 1331, -473, 564, -775, 269, -1271], [-11383, -1982, 463, 1103, -308, 738, -1228], [-3043, 437, -65, 1319, 110, 1887, -1128], [-6954, 660, -1361, 652, 323, 576, -191], [5618, 655, -794, 926, -1248, -101, -492], [-6514, 1917, -576, 44, -640, 21, 820], [5860, 104, 219, 12, -837, -883, 496], [-3714, -1404, -36, -414, 388, -671, 1419], [6872, -547, -54, 484, 412, 1347, -802], [16314, -2849, -5, -42, 267, -1011, 249], [9904, 669, -166, 280, -658, -460, -573], [-6040, 1410, -1536, 628, -4, 1297, -1133], [-8019, -790, -1350, -113, 391, 692, 393], [13034, 861, -1002, -402, -596, 783, 712], [-2986, 1634, 1398, -840, 1882, -490, 582], [9040, 203, 1960, -84, -350, -1083, -348], [-5666, 3333, -1582, 896, 84, 2058, -861], [3050, -295, 574, 1468, 2096, -1002, -69], [-4250, 2397, -1492, 700, 180, 949, -814], [14986, -650, 32, -224, 1744, -46, 556], [-3175, 7811, 462, -671, -221, -115, 51], [-6340, -1530, 749, 56, 2239, -1105, 1982], [8417, 519, 1537, -1141, 2386, -1685, 2652], [-3578, -2823, 1311, -310, 675, 990, 1786], [4506, -824, -1460, -430, 518, 1073, 665], [2612, 1286, 619, -1194, 85, -133, 782], [-26, -7742, -1188, 1356, 1268, 816, -294], [2342, 506, 1353, 1378, 121, 1196, 1225], [-2028, -4575, -246, 294, 510, 750, -2295], [5271, 3695, -179, -431, 486, -192, 907], [-4016, 2824, -724, 222, 2630, -705, 1721], [-15275, 2347, 2138, 73, 195, 569, 39], [13658, -509, 131, 2450, -217, 1144, -1062], [3664, -4369, 125, -754, -1835, -1316, 768], [-4054, 3834, 838, -666, 1712, -2518, 3044], [-17196, 512, -784, -30, -6, 160, -166], [16411, 118, 1943, 17, 950, -599, 13], [19402, 316, -784, -328, -2082, 1671, -1503], [-1417, -162, 987, 2533, -1282, 46, -292], [-2274, -3156, -99, 1098, -1049, 1588, -1261], [4670, 463, -2035, -1128, 1175, -1502, 1136], [1894, 651, -2686, -858, -440, -377, -652], [-2048, -1554, 435, 1706, 115, 856, -2477], [-11533, -482, 2328, 31, 1087, -846, 1417], [9944, 1413, 1921, -960, 1275, -1622, 2202], [3386, -4747, -2915, 764, -2613, 526, -1454], [6974, -1840, 3462, 1190, 24, 229, 2037], [2188, -3438, -2020, 764, 2836, -402, -640], [-12111, -5042, -555, 1487, -1122, 914, -2282], [2426, 2689, 485, 1912, -721, 2513, -2755], [29380, 787, 1818, -592, 150, -1297, 1256], [1700, 784, -920, 682, 1536, -425, 427], [-2613, -7669, -855, -1123, -1046, 696, -693], [-19326, 7999, 2402, -78, -200, -558, 113], [-15271, 2816, 51, 1549, -1500, 206, -1978], [-3298, -7098, 312, -1772, 34, -235, 2659], [-16357, -5695, 1426, -649, -383, -2391, 699], [3874, -2857, 611, -2494, 861, -1995, 2387], [-19328, -5327, 1490, -902, -216, -693, -688], [4907, 5542, 1805, -2121, 290, -3807, 3845], [-11969, -4795, -3193, 1077, 1022, 1278, 1415], [21918, -6478, -2048, -900, -1246, 769, -915], [-2820, -2577, -314, 528, -1316, 1256, 1553], [-16322, 520, 3273, 2190, -3023, 404, -2375], [14212, 1839, -1966, 888, -2848, 2632, -2849], [6504, 2399, -2614, -258, -2244, 3560, -443], [-6281, 927, 1234, 2047, 133, -858, 682], [-1808, 5011, 1056, 1116, 514, -272, -1349], [244, -1983, -1745, 810, 1153, -149, -239], [-33744, 488, -737, 1230, -1253, 637, -2398], [3770, -4923, 2139, -988, 2863, -1863, 651], [-1262, -7068, 1168, -100, 228, -824, 2852], [12465, 1067, -1993, -1815, -170, 3166, 79], [-2874, -3328, -945, -2416, 3135, -2356, 3965], [-2925, 5972, -2944, 2217, 179, 2862, -1527], [10416, 587, 2488, 440, 1904, -1096, 2853], [3128, 13032, -2075, -908, -2029, 1090, -1965], [-8650, 5042, 2270, -650, -444, -366, -92], [3179, 6481, -2935, -723, 586, 238, -2845], [892, -4308, -862, 506, -1394, -3407, 1409], [-4755, -606, -2361, -2495, -372, -74, -610], [-8008, -6818, -1256, 724, 4364, 1651, 2585], [-5946, -1376, 1484, -1516, 3320, -4366, 3514], [-9245, -1150, -205, 165, 934, 2267, 159], [3135, 11904, -204, 863, 4237, 559, 862], [-2642, 3244, -1828, -384, -568, -1008, -2976], [-2119, -8330, -1889, -555, -1998, 1931, -4273], [-7353, 886, 120, -79, -3807, -2345, -432], [6283, -2726, -153, 3595, 1542, 2358, 1244], [-18295, 758, -1942, -1677, -2899, -2566, -1663], [-27069, 1544, -279, -531, -236, -2413, 3607], [-7514, -544, -2049, 2840, -3019, 1025, -3202], [10918, -355, -168, -1572, 1428, -3522, -467], [7306, 3021, -1000, -2, -2040, 2214, 1701], [13956, -3625, 4154, 2396, 892, 671, 3170], [-18520, -9019, 425, 1232, 771, 516, -2500], [4139, 5795, 3821, -2573, -394, -2525, 3422], [23270, 4859, -2058, 2930, -2360, 2749, -2710], [13791, 1634, 3525, -1665, -3202, -2359, 3009], [11159, -3627, 1542, -409, 3041, 522, 1230], [19661, 4977, -1267, -79, 2594, 3239, -870], [22212, -2939, -4786, -80, 1016, 457, -2956], [21858, -1583, -273, -3076, -213, -1961, 1347], [-4020, -7104, -663, 4266, 1721, 4377, -1464], [-19844, 6738, -445, 416, 4303, -33, 2548], [6838, -806, -7632, 566, 430, 309, -1653], [-7516, -3762, 3800, 540, -264, -90, -572], [-23964, 3096, -3172, -3936, 2568, -287, 51], [-3432, 4260, 408, -4168, -736, -2108, -1716], [3902, 2865, -1188, 814, -1324, 1051, -1856], [-2816, -6489, 1122, -1608, 1600, -3802, 1231], [-2073, 4877, 5081, -311, -2604, -2272, 2347], [23834, 2964, -4550, 1762, 344, 2046, -1882], [-10556, 29, -1775, 2084, 3003, 3461, -4107], [11344, -4123, 5750, 3604, -1336, 1727, -510], [-422, 3037, 2134, -544, -6112, 2141, -334], [21345, 9198, 2535, -293, -3156, -328, -2910], [-3786, -4015, -266, -860, -1040, -2080, -279], [26244, -5384, -5184, -5082, 2102, 776, 3426], [-13853, -12498, 1435, -303, 3718, -3912, 3252], [31854, -708, 2604, 900, 1856, -111, 2665], [14337, -11561, 2264, 861, 3849, 975, 203], [-36850, -6707, 4784, 380, 1644, -1975, 3030], [22566, -3696, 255, -176, 8179, -1621, 2084], [-4355, -1253, 2857, 3257, 2654, 4886, 259], [1818, 9637, 520, -254, 894, 3031, -2000], [3556, -3675, -3256, 2610, -4080, 3128, -3825], [16133, 4752, -4293, -1851, 3790, -677, -939], [-18048, 1863, 4464, -1310, 2088, 811, 3434], [-23296, -7164, 945, -4484, -1519, -2193, 3242], [-24133, 6621, -2834, 487, -3363, 3100, -1588], [-38375, -7940, -2847, -1199, -3560, 663, -989], [58131, 14267, -1690, -3083, 267, 1511, 1153], [69204, 470, 1225, -22, 609, 1065, -918], [-14272, 5998, -847, -2114, -2335, 1379, 1800], [8976, 590, 1501, 244, -1993, -2417, 4780], [-1863, 3906, -1291, -1109, -2762, -1299, 1193], [-3052, -10926, 641, -666, -3531, 193, -1318], [-50245, -918, 4722, 1067, 1281, -1900, 2237], [-5091, 8493, 6669, 3359, 4960, -2756, 1211], [22002, -781, 2927, -2244, -1415, -1272, 4524], [29484, -1662, 683, 604, 1661, -219, 220], [-10735, -13423, -851, -2575, -2576, -772, 1409], [34681, -9184, -1639, -2103, -1658, -272, 3202], [20946, -10871, -356, -112, 3506, 1395, 1228], [8780, 8280, 1924, 2594, -3400, -493, 1405], [18276, 668, 1402, -4252, -5942, -1450, -2376], [-23993, -3912, 72, -1483, -239, 1319, -2378], [-51470, -4025, 876, 2236, 2920, 9, 1102], [-69978, 10438, -1628, 1422, -1980, 3515, -2455], [-8635, 7343, 3337, -4599, 3900, -5624, 6911], [23400, -1724, -1374, -2606, -3156, -2342, -5358], [-22773, 4375, -5267, -451, 2836, 4178, 723], [38720, -796, 1533, -2198, 1297, -783, 3856], [43752, -5591, 5454, 2928, 2796, -2620, 1811], [-8477, 1894, -24, -6757, -3267, -2226, 2895], [-13312, 6325, 1164, -5052, -372, -3309, 234], [-30385, -17357, -1652, 3261, -401, 2546, -2990], [2528, -6138, 242, 624, 494, 2926, 934], [-19240, -5891, -724, -2598, 2976, -621, -894], [-32283, 1606, -2363, 5053, -2820, 3845, -3145], [-17894, 296, 10281, -800, 1621, -455, -452], [-31184, 5183, 4476, -1038, -2228, 515, -146], [33838, 9131, 3788, -3656, 284, -557, 3022], [2005, 9900, -5308, 2431, -6501, 1608, -3345], [-10403, 10807, 824, -705, 1147, -1257, -2211], [-41052, -2074, -222, -3854, -686, -3663, 1151], [12096, -18557, -5774, -3412, 2854, 3555, -2236], [-33806, -423, -2426, 2820, -2438, -1949, -4920], [-18280, -4150, -915, -3812, 7769, -4152, 8245], [5254, 10205, -1982, -5622, -3698, -2049, 4120], [-76522, 14385, 1254, -1956, 5818, -4629, 2410], [-12116, -4728, -1494, -3786, -6792, 3046, 378], [14673, 11932, 135, -5045, -4056, 2560, 1946], [-2886, 5597, 4651, 4934, -4081, -1620, -48], [2996, -7770, -6330, 5616, 2242, 2441, -5023], [-17319, -1597, 2536, 2681, -3443, 2508, -4168], [-3096, -12398, -2594, -296, -5950, 4325, -2077], [-56973, -1924, 7204, 181, 5429, -4738, 6287], [-21094, 8670, -3672, 1972, -2108, -746, -1428], [7624, -18489, -2158, -3138, 10428, -143, 6850], [-25618, -3435, 1595, -1784, 2525, -2768, 6694], [-32103, -4996, -1369, -843, -5332, -2258, -2338], [26604, -5067, 6616, 5864, 2906, 3951, 4866], [-34956, -13762, -2985, 1642, 5377, 8729, 476], [-29481, 3347, -4530, -3927, -1715, -3608, -4938], [18946, -3338, 4002, -6664, 1434, -3480, 5936], [-386, 6293, -2680, 670, 682, 2173, -7354], [16504, -17546, -4387, 4256, -7089, 1158, -4287], [3314, 3061, 2651, -2696, -7047, -1616, 5326], [-10706, 17750, -229, -4186, -1775, -5359, 1424], [-25080, -15270, 2767, -3208, 3645, -3354, 6915], [17454, 9723, -998, -7164, -2590, -4085, -1318], [26098, -10147, 4653, 3356, -4313, 3692, -328], [-7628, 20966, 3492, 1652, -9540, 383, -3745], [54953, 6294, -876, 3087, -1395, 1538, -6839], [24058, -1253, 6964, 1192, -7688, 4663, 1262], [35772, -11239, -4900, 5220, 7688, -2278, -3121], [-3644, -5437, 6170, 5896, -12, 8433, 22], [-4578, -5141, -5470, 378, 2796, 1397, -5606], [41481, -14236, 233, -2349, -10982, 611, -2195], [-35488, 19297, 1530, 1368, 1540, 1277, -566], [-18390, -7909, -2868, 4570, 7460, 1671, -184], [-39595, -2086, 2418, -7351, 8495, -5060, 6051], [91606, -2180, 7590, -1804, 818, -7828, 4110], [-51637, 11374, -3845, -9, 2020, -537, 577], [-14972, 11906, 9148, -2740, -10612, -995, 1223], [22167, -18341, 2137, -1297, 7962, -6013, 5162], [41846, -22038, 7381, 3280, -2315, 2159, -756], [13220, 20894, -1635, -3056, -6993, 3253, -6126], [-36171, 9651, -3451, 837, 1722, 1343, 3696], [33022, -8371, -4594, 110, -2676, -1749, -4390], [7216, 28772, 2809, -6200, -1383, -5984, 2963], [-7890, 12283, 673, -6328, -1561, -8650, 8132], [516, -17870, -11112, 990, -6722, 2296, 3908], [-16286, 8044, -406, -4312, -3762, -2542, 2888], [-15444, -17888, -1949, -4896, -3161, 2249, 1508], [-6305, -2758, -6147, 1481, 3384, 9144, -5756], [12708, -19525, 10736, 5238, 2544, -5040, 3417], [-14642, 15261, 271, -5208, 3073, -9362, 6548], [16254, 1282, 2230, -2198, -5560, -2292, -294], [-38978, -1679, -923, 4286, 8133, -6856, 5234], [-36331, 7796, 3793, 2579, -130, -123, 1099], [66324, 6577, 1440, 7912, 832, 6312, -5509], [-16806, 33824, -4409, 270, -1017, 9514, -4771], [-30420, 13445, -8188, -7278, 2400, -4433, -1392], [81294, -8135, 5237, 10774, 5805, 1706, 488], [56557, -4358, 5933, 6713, 998, 5392, -7732], [21550, 11785, -10766, -2970, 2040, 5709, 76], [-32026, 16206, -6378, -3676, -3402, -6832, 6148], [7976, -12717, -4367, 36, -5211, -1221, 3939], [-26321, 9078, 2988, 969, 6269, -3089, 9240], [40274, 16597, 5526, 1556, -766, -408, -2827], [-33828, -16219, -902, -10906, -836, -4740, 8737], [-31814, 26178, -7403, -10306, -3591, -4811, -1474], [242, 916, -10584, 280, -3832, -2013, -9443], [-23110, -35074, 2399, 1278, 501, -1649, 5130], [-17166, -2710, -309, 8186, 2917, 9335, -6610], [-35316, -4371, 3326, -5162, -11088, -2724, -2755], [-70162, -7048, 6697, -142, 8629, -3641, 13428], [85998, 6259, -8835, -6714, 3357, 1256, -1528], [-59080, -3220, 4798, 1952, 194, 7460, 3662], [-48969, -2799, -339, -6805, 4054, -3020, 7523], [-44466, -15037, -1124, 4424, -3838, -3653, -7036], [31244, -4875, 2988, 9952, -7782, -243, 1038], [-858, -26645, -6160, -2912, -568, 7113, 280], [-71010, 3432, -9139, -4978, -12189, -2457, -820], [-36572, -12935, 974, -2860, -2464, -10217, 3854], [-13484, -3684, -2312, -368, 700, 5548, -4024], [-39698, -8899, 4399, 880, 865, -354, 3036], [16154, -787, -7421, 1750, 11279, 576, 1568], [17417, 18830, -3785, -51, -676, -2188, -6742], [-57960, 15728, 636, -3028, -1736, -3626, 4590], [21326, 5680, -12884, -6190, 3442, 7311, -5223], [-3994, -11692, 5122, -5908, -13808, 1187, -4145], [27178, 29344, -2761, -3804, -2253, 2155, -426], [-47440, 8170, -11006, -340, 8086, 7456, -7368], [27676, -11138, -3636, -2506, -8234, 5860, 6192], [-23654, -15760, 2930, -878, 1060, -6514, 3406], [-11132, 6387, -8358, -3410, 4898, -85, 1268], [23894, 9469, 14444, -2144, -3810, -3860, -143], [41532, 13810, -6529, -2188, 8631, 5704, -2623], [-26291, -21675, 10282, 557, 1683, 1899, 8303], [-39539, 17710, 4693, 951, 13350, -13111, 8861], [102756, -22673, -2438, -5558, -522, -6057, -1198], [10194, -11111, 2900, 6460, 2140, 3215, -5230], [-56738, 9800, -6062, 3918, -3588, -5023, -9035], [55946, -9824, -4946, -5034, 4860, 7304, 2016], [-56, 27844, -5064, 3948, -13620, 9053, -11945], [-20815, -4991, -1743, -4449, -1814, 8379, -2002], [23639, 937, -2042, -4889, -5191, 3846, -5074], [18498, 159, -5715, -2702, -9885, 137, 2303], [-34915, -20455, -896, -3295, 10081, -1745, -3943], [-43376, -22206, -5597, -5534, -8313, 4267, 3470], [-42408, -5420, 1674, 5144, 1618, -5696, -4922], [48496, -2947, 7834, 4542, -12732, -1163, -1850], [14929, -7277, 2207, 3753, 10430, 1325, 9280], [10326, 12867, -9237, 7588, -9031, 8626, -17282], [-5832, -391, -4815, 34, -39, -8220, -4954], [20780, -10174, 4032, 5494, -4720, -4297, -1169], [-22620, -9507, -9468, 1848, -6332, -2632, -267], [21318, 13034, -447, 6976, 5543, 8615, -3504], [120123, 1851, 649, 8065, -3860, 11032, -12191], [51496, -14999, -5951, 1652, -6379, -2187, -6523], [8820, -29434, 13719, 2544, -6537, 1635, -7744], [-8713, -21972, 944, 1391, -4699, 1692, -6053], [-40890, 33876, -13058, 3338, -7532, 14063, -11127], [34036, -36404, 1734, 6228, 5998, 1644, -4898], [-3803, 7794, -1129, -8939, -3544, -14692, 2544], [68957, 8364, -1243, 9481, -3410, -6074, -296], [18805, -33154, -13861, 3359, -11684, 10194, -11014], [-89090, -10899, -8801, -6258, -1417, 3288, -416], [62980, -6006, -2168, 786, 7482, 1728, 6012], [-57677, -29290, -7287, -3069, -4770, 7573, -6863], [-106494, -8277, 4122, -4424, -8712, 1892, -3235], [-71684, 27436, 4642, -4192, 11100, -16395, 18881], [19718, 5293, -5969, -7078, 9563, 8286, -114], [9760, -6376, -15816, 3468, -880, 11324, -15964], [25150, 12336, 12482, -2656, 4144, -6991, -1549], [104224, -6040, -9424, -2996, -1292, -8523, -1619], [92793, -26544, -7501, -8213, -3620, 4041, 9553], [84712, -27519, 12088, -8680, -2602, -17441, 13392], [5493, -27389, -8957, -1171, -4044, 7408, 6183], [86735, -16381, -4769, -371, -10418, -7848, -1269], [-2044, 17670, 8410, 1716, 8738, 3814, 12166], [-33244, 12009, -4670, 7258, 380, 7927, 1644], [-88820, 23489, -9520, -4978, -13862, 857, 614], [-53338, 26865, 4544, 2320, 5500, -9398, 7817], [-5154, -2148, 3060, -6484, -7492, -2353, 689], [10180, -14414, 1088, -5824, -17138, -3525, -7737], [-56171, 57278, 1859, 4139, 17512, 5185, -489], [-58954, 19304, -10766, -1172, -2744, -8589, -11787], [43106, -10582, 7530, 2120, 3690, -542, 8638], [-73172, 17300, 7227, -11664, 7067, -11002, 9515], [-32413, -9037, -3020, 643, 14211, -140, -8588]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_483_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_483_4_a_c();".
// 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_483_4_a_c(:prec:=7)
    chi := MakeCharacter_483_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_483_4_a_c();".
// 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_483_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_483_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-460, 388, 677, -2, -143, -19, 6, 1]>],Snew);
    return Vf;
end function;