
// Make newform 531.6.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_531_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_531_6_a_d();".
// 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_531_6_a_d();".
// 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 := [-50564480, 120303168, -16576192, -53008480, 15496192, 3566264, -1130486, -94393, 27968, 1045, -283, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-49, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-44540621931975981150016, 2309661345958546715392, 22682859078888124275168, -1871642618466862238976, -1621232990873406107520, 29283065204018815204, 38651364669125368297, 673646677911366614, -373612354185694543, -15069489363041309, 1260174342721206, 70995163178963], [252251457838879227840, 578535839854472803200, -234904833095610043232, -169212860659841880416, 18576113861191679248, 11187289331975452540, -197118219957298639, -237603629271932978, -1017645976025407, 2025127413667451, 12704329665294, -6070866593397], [75277099833303083684032, 17671121328252462456832, -27970911891489684321440, 2114165735599360806336, 1630276694691093230368, -227080010856454277612, -37431432449168562815, 6536419220059167446, 374909234235669753, -73035502710430869, -1348141595753418, 276443349871579], [-11260185469833302312896, 5255893016983507189504, 4029036932992469914912, -2151351397113547096704, -172745082388610069312, 154658714920895602524, 1830350641272830531, -4033264827299187998, 2124163599740043, 42942157372889537, -62814114617374, -157574427250607], [143671022417105045696, 1080506509259373478272, 258422640881088540704, 65838396490605035936, -48722748546299885392, -12479805914223073028, 1580859950161255549, 370390697832800582, -18209905676036451, -3979652032884737, 68921999442182, 14329992266831], [-55526849730687927914944, 59192065957194059700480, 20115068195709835534496, -18796017885994915956480, -671477905131811469056, 1309121973508228341708, 9721228830858762571, -32984685428531840318, -83240015220961021, 341906398675822169, 339054598749730, -1232144514507415], [86575115576869461415616, -62943275448972877997824, -36921974083861151814688, 23223995518001361140352, 2326930855637581854912, -1616015194601249637852, -70689929145009309215, 38852118588478225030, 887888971518781257, -382838896328099701, -3701116547366298, 1322194336654907], [41000179717318874316608, -59477569094551464042496, -30623812468319425334880, 24379050756287078102336, 1979508163645875132640, -1705284178220637720180, -55447889370230709313, 42190034547873270282, 659727714848877735, -429352011744410923, -2687442487401494, 1524851944712805], [-47343784358882143011392, 45134300183044023212160, 31263066652870809123104, -18595208530012418629600, -2198841964368832503824, 1320885247542407023692, 66543336099983428489, -32292352253443995538, -819809368020587799, 323268347553059587, 3369846601217454, -1131993494702445]];
    Rf_basisdens := [1, 1, 1, 376207599370663553024, 13435985691809412608, 376207599370663553024, 188103799685331776512, 13435985691809412608, 376207599370663553024, 376207599370663553024, 376207599370663553024, 188103799685331776512];
    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_531_a();"
function MakeCharacter_531_a()
    N := 531;
    order := 1;
    char_gens := [119, 415];
    v := [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_531_a_Hecke(Kf)
    return MakeCharacter_531_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 := 6;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-35, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-35, -9, 2, -1, 1, 1, 4, 2, 2, -1, -3, -1], [-85, -2, -10, 1, -1, -1, 1, 1, 3, -2, 0, 1], [136, -31, 0, 6, -4, 2, -12, -9, 0, -6, -4, 1], [-280, 43, -3, -9, 0, 7, 6, -8, -2, 3, -1, 2], [280, -238, 2, -13, -18, -17, 7, -2, -7, 26, 7, 10], [295, -337, -14, 3, 22, 23, -31, 8, 13, -10, -28, -14], [-939, -55, 24, -41, 55, -21, -11, -2, 3, 20, 0, 12], [-2348, -323, 5, 6, -103, -26, 54, 20, -26, 56, 69, 43], [549, 47, -122, 62, -13, 44, -17, -4, -39, -27, -86, 9], [-2607, -510, 65, 79, 5, 37, -31, 34, 37, -26, 99, -5], [6599, 122, 18, 23, 89, -23, -46, 25, -42, 5, -115, 34], [1061, -255, 12, -90, 184, -110, 102, -36, -120, 214, 27, 56], [3481, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4507, 408, 93, -239, 279, -119, 80, 20, -266, 151, -146, -54], [2589, -981, 313, 336, 269, 306, -210, -3, 36, -156, -164, -24], [-4965, 143, -163, -63, 283, 11, -204, 123, 82, -157, 32, -33], [43, -2367, 117, -293, 100, -147, 113, 1, 161, -286, -257, -176], [-2095, -2359, 122, 95, -716, -293, 84, 72, -510, 33, 401, 283], [8495, 2167, 110, 550, -827, 148, -59, -160, -121, 167, -68, 299], [3021, 2491, 199, -461, 615, -451, 49, 277, 391, -418, 71, -314], [-15969, 98, 534, -830, -458, -188, 365, 213, 279, -137, 366, -298], [-12980, -1961, -70, 1001, -728, 483, 407, -33, -313, -186, -113, 379], [-35686, 875, -650, -8, 807, -538, 205, 187, -471, -749, 203, 272], [11530, -2524, 226, 859, -47, 851, -153, -542, 273, -994, -545, 464], [-27074, 1972, -553, 335, 509, -63, 503, -667, -851, 94, -470, 550], [-36083, 1991, 1061, -1002, 1360, 68, 202, -57, -152, 762, -696, -954], [-80013, -2694, -1716, 324, 204, -342, -589, -419, 253, 611, -936, 148], [13976, 5782, 1378, 498, -2656, -194, -256, -198, 862, 338, 544, -400], [-40541, -3747, 960, -59, 119, -349, 1506, 1104, 1486, -185, -1073, -231], [-81235, 12720, -848, -314, 178, -1260, 1425, 599, 969, 1011, -794, -724], [-120018, -5325, -1318, -1190, -1047, 220, 242, -93, -540, 2390, -1468, 153], [-66095, -11927, -152, 129, 1977, -139, -1093, -224, 1771, -1678, -98, -938], [-16367, 7895, 880, 2406, 1697, 1090, -686, 234, -112, 114, -3351, -240], [-73469, -1939, 657, 1243, 558, 2081, -825, 1373, -439, -1788, -311, -24], [40197, 14770, -323, -2543, -886, -1595, -1548, -892, -1610, 143, 222, 884], [23805, -1594, 3843, 548, 1353, 3596, 586, -498, 2446, 70, -1026, -383], [84425, 3755, -3288, 1559, -353, -2467, -654, 650, 538, -787, -533, 2121], [-55555, -5706, 3174, 499, -1892, -227, 94, 2297, 1578, -907, 1735, -248], [112117, 22976, 4084, -2948, -6753, 82, -517, -1491, -415, 4695, 3208, 822], [-280916, 127, 526, -1613, -1508, 1547, -4007, -1155, -1505, -344, -1939, -317], [153444, 47321, 3244, -4674, -1535, -840, 3307, 2139, -141, -1445, 3103, -722], [-242547, 30399, -1765, 917, 198, -1697, -529, -519, -1953, 1322, -1, 2750], [-161390, 21752, 4643, 2842, 3203, -818, 531, -1999, -633, 1551, -1388, 1677], [-406296, 1861, 3596, -2059, -7730, 1831, -3575, -2007, 2065, 1372, 3233, 913], [232936, -9685, 5576, 1202, -2922, 5170, 576, 3003, 3024, -3518, 4534, -23], [-405821, 21609, 1123, 892, -10005, 2906, 3039, -2211, -291, 4271, -1015, 805], [96276, -5074, -3306, -240, -8120, 268, -4418, -2124, -916, -4102, 2320, 2714], [6323, -4422, -13245, 2748, -1319, -1988, 2307, -1184, -3219, 239, 6781, 1442], [-85104, 32419, 2822, 7749, -294, 597, 1549, -2109, 2017, -1562, -2137, 1417], [-160225, 6237, -15729, 508, 3662, -1622, -1930, -4717, 532, -4256, -2834, -4918], [-80144, -3325, -2926, -4601, 8724, -1965, 1209, 805, -3273, 304, 5, -1605], [63001, 11725, -1517, 418, 9868, -5140, 2331, -1769, -305, 8351, 163, 451], [-134143, -21942, 1451, -6621, 11238, -3617, -4667, 2724, -5697, 112, -3603, -3121], [-271449, -81039, 6970, -659, 6830, -1027, 1920, -3494, 2966, 9331, -1825, -453], [325903, 11937, 31, 6098, 2299, 4836, -6114, 2881, 2292, -2302, -6834, -5976], [-297110, 28523, 1980, 12695, -2072, -1705, 7449, 3265, 667, -640, 5725, -2959], [-166096, -12730, 4869, -90, 7107, -6966, 4017, 2895, -3175, 8677, -3882, 1543], [41406, 50379, 2730, 4795, -14607, 4665, -5430, 1001, 8022, 501, 2744, -2706], [344996, -43246, -3716, -958, -17530, 3638, 5726, 5184, -1968, -296, 3220, 2982], [351440, -8620, 2062, -1486, -10408, 9102, -6839, -1294, -2955, -12307, 20829, 2441], [-148478, 51133, -8851, 3175, 16970, -2233, -1532, -5300, -8110, 13317, -9549, 2846], [-308053, -23017, 3496, 12512, -2024, 4772, 2650, 11822, 2422, -12854, 5437, 6440], [-618960, 11510, -21048, -11244, -24464, 2012, -1300, -2020, -66, -7172, 28946, -4362], [291782, 83093, -32417, 10987, -10150, 155, -5080, -4294, -4682, -7379, 13949, 13710], [364039, 32980, 15642, 4787, -11877, 11353, -7976, -5507, 5074, 11111, -5899, -2366], [260411, 5842, -19432, -23027, -4065, -17233, 5771, 7001, 4843, 354, 17616, -7679], [-199162, -85922, -342, -17761, -8447, -7701, 12755, -4130, -3583, 2954, -15551, 3216], [389122, -43474, -5768, 16399, -10375, -10125, 3830, 1642, -1304, 14583, 13010, 6147], [-295387, 101604, -6167, 7484, 2604, 308, -1297, 11474, 8005, 8651, -16621, -10508], [-288372, -46583, -13569, -4193, -4835, 1695, -4420, -3656, -818, -231, 14605, -16066], [276079, 212588, -11634, 7058, 2382, 4380, -631, -8031, -13773, 3903, 192, 4786], [100383, 9263, -18277, -21149, -927, 3145, -1870, 2323, 5882, 3993, 19972, -4399], [-535244, 234621, 16058, -14800, 23015, -18290, -5171, -101, -6193, -12463, 4951, 4200], [106677, -253919, -16622, 8608, 8379, 18020, -7720, -12124, 15276, -3464, -13567, -3720], [-351480, -50908, -10254, -109, -20023, 15647, -13145, -11132, 10295, 25058, -19683, 15780], [1063654, -78240, 11982, 3068, 2528, 3180, -16392, -6682, 4276, 10612, -12890, -2260], [-32022, 99757, -12216, 13403, 21598, -18613, 16595, 18329, -7199, -8806, -9579, 3511], [1250731, -45822, 34994, -25223, -32043, -18665, 27003, 4671, -13257, 29338, 19598, -3707], [-196947, -34262, -6276, -22329, 16769, -10911, 16934, 7921, -21106, 5125, 42233, -7056], [-274958, 67034, 11107, -9746, 7001, -17482, 2341, 4511, -20809, 14217, 25138, 6439], [-20534, 41948, 51213, -20612, -124, 8138, -15717, 18541, 13389, 14727, -24952, -8047], [1006688, 122109, 39596, -31052, -7144, 2780, -8274, -2891, -30190, 1090, 42468, -5989], [-102443, 168125, 1330, -6807, 14783, -29277, 18590, 480, 1882, -9625, -23177, -19913], [-1169279, 29071, -47331, 883, -37486, 13641, 3221, -31339, -7079, -12862, -4023, 8912], [-2282236, -124673, -7006, -8867, -28685, -6685, 23728, 26917, 15926, -7065, 38476, -14458], [-1369972, 183986, 33305, -18243, 4822, -7821, 4843, 17861, 9661, -1876, -6074, -2376], [-441936, 126829, 900, -7537, 50290, 7799, -8107, -7795, 19745, 23796, 15387, -19405], [1426394, 179163, 24368, 20199, -848, 17973, -22162, -22903, 28834, -13167, -25890, 19146], [-1360622, -177938, -68933, 22650, 18024, -18656, 22319, -5455, -9907, -17347, -45642, 18457], [-365557, 315059, -14133, -10827, -13993, 19579, 13603, -23353, 2031, 13606, -15089, 26454], [-334823, -31592, 36920, 24220, -4290, 19966, 21663, -6119, 10397, -30457, 28862, -4068], [1657633, -192716, -57406, 12331, 19643, -14355, -1164, 17913, 16192, -47613, 21025, 2796], [-817647, 141573, -14377, 17589, -59795, -8613, 1767, 4005, -14373, 27190, 16795, 20836], [-456103, 477824, -7160, -6305, -40647, 141, -13348, -2047, 4680, 25439, 22421, -354], [-590957, -154564, 40064, 18848, -20870, 32394, -13051, 2555, 371, -427, -48750, 20220], [-809267, -488734, -6122, -6603, 2853, 26999, -9029, -6057, -119, -3068, 17668, -2655], [-1157810, -114947, -32831, -18393, 38906, -18521, 42600, 4954, -13198, 44581, -66947, 8260], [-893648, 160477, 20936, -41490, 86393, -1040, 45157, 19477, 11671, 38839, -32067, -11030], [2057297, -491238, 97924, 28222, 21375, -23380, 47247, 19517, -12797, 62657, -10406, 28702], [-5837529, -268129, -19809, 4655, 74139, 6333, 19105, -7043, 45349, 1754, -44065, -49968], [-577717, 754849, -87291, -18354, 20115, -24832, -66776, -10483, -23302, -19748, 82124, -29960], [-477179, -454246, 4555, -35863, 40795, -53977, 34177, -19064, -16969, 58540, -14563, 29343], [-1369283, -59055, 76666, -45973, 15503, -16739, 51332, 18132, 21970, -77717, -50179, -3119], [-942499, 78596, -99178, -40729, -3509, -76807, 10968, 32387, -6456, -27521, 101015, -8534], [-2729524, -346784, 39108, -11027, -31359, -36175, -5780, 9770, -40494, -13189, 29606, 24451], [-2676084, 454639, -114309, -53381, -51624, -7905, 47990, -2646, 10252, 20747, 65597, -4496], [-2098601, -226189, 38742, 2093, -8806, 16577, -2050, -13992, 46460, -3011, 87849, -14725], [1024532, 467090, -33348, 49755, 90395, 47075, -77956, -1638, 3978, -3047, -8160, -20539], [-2869552, 121191, -86906, 46394, -20182, 47986, -45128, -18265, -60528, -21510, -51830, 49459], [86505, 308632, -82832, 4650, -142154, -17448, -8059, -8855, -48381, -47929, 51840, 34444], [-2428361, 51751, -152524, -17057, -68898, -8193, -26564, 21390, -27194, -14459, 46833, 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[-255127913, 8396573, 1143281, -3232364, 102859, -2925350, 2105634, -392145, -959288, -135020, -554952, 267082], [-223302926, -4171006, -475778, 1699562, -2940336, -947386, -1764586, -1265014, -307004, 2493376, -887158, 216218], [-16354640, 12400685, -1075300, -226869, 3070766, 1301397, 2011895, 203433, -329137, 1647588, -3580629, 28773], [-143798337, -328719, -1061826, 805220, -319627, 1175342, -2027849, -1375436, 2460169, -965421, 163268, 1863475], [-239301844, -16239167, 3780242, -548527, -3177487, 2107079, 292978, 217693, 633530, 1163335, 2703314, -314140], [-117211239, -27491502, 3799463, 599220, 1749422, -1470348, 2183111, 1153894, -1149023, 1553359, -2010403, 1985562], [-134945556, -16320421, -5310466, -1271772, -55101, -3879682, 2024625, 490419, -264999, 5791, 1793371, -1537676], [20313852, 8528097, -4016114, 3278601, 4509604, 1487841, -2763071, -1455025, -1302629, -1412798, -3628827, 1325759], [-40862801, 19968170, 3589403, -718379, 1723369, 1607083, -678111, 1405266, 1193425, 1426308, -1073577, 122523], [14674921, -10914599, -5919855, -4469014, -3438906, -1330516, 2484798, -857413, -1115224, 2760386, 2030170, 216666], [102260350, 12262568, 175143, -885760, -1785761, 887608, 1769779, 1116789, 1778965, -1386047, -2630912, -568935], [33987335, -3190061, -4008210, 1269813, -318820, 96581, -272327, 787126, -176433, -3969378, 3245076, -607550], [-108818500, 9161296, 2847880, 4659859, 1741254, 68791, -2993745, -1611066, -1407707, -678106, -2969919, 1044928], [45829161, 4916742, -4407178, -2106718, 2165818, 263636, 475207, 162383, -73649, -224491, -2277334, -1595384], [58757521, -21479870, -2243279, 933985, -480667, 3281007, -371401, -441846, 1210629, -2227122, 717437, 51749], [-70788012, -4196323, -2305308, 928716, 3137729, -1736802, -1139207, -1391501, 117245, 1712999, 739337, 1532290], [234875580, -11195744, -6033828, 2357432, -1172272, 1409092, 64464, -755940, -420540, -1409192, 192492, 1486168], [-217295557, 20256513, 342988, -626819, 5671403, -1999031, 931263, -271098, 1565883, -509650, 2465420, 765932], [-131710809, 19498927, 1880833, -1358674, 3741696, -1444820, 2497348, 605761, -871486, 705088, -4336404, 33842], [146071205, -18984937, 3816553, -2497086, 2068484, 179460, -811093, 1576039, 381851, 1802387, -1467751, 266425], [90777804, -20986108, -1857510, -415557, 4181803, 981315, 2410721, 1952938, -271913, -430228, -1004895, -3625098], [-213582953, -20448249, 567877, 2587174, 194547, 1815780, -3957694, -1123323, 376392, 1491802, -1976180, 1274838], [139097380, 2583337, 3523402, 342985, 6225858, 1787603, -518678, 1014509, 2510704, -410327, -2879142, -1892914], [6109598, 7667184, 1440138, -1450382, 1919442, 695390, -921484, -744464, 42924, -1580902, -45072, -543800], [-34861644, 38382501, -788696, 1001387, -3125028, -1575189, 1107015, -537379, -2380115, 5848402, 2220949, 2446647], [10478148, 11341927, -7365878, 1599688, -2354773, -1179778, -1308513, -763721, 165751, -3217815, -1597679, 1106322], [-5360426, 41643562, -3830594, -1227237, 4551348, 1834611, -3267177, -2151288, -3151535, -1447422, -3141695, 613248], [-170139371, -47403695, -42472, 3365208, 2605205, 1554762, -39235, 1494804, 402987, -4633975, -4152950, 378391], [228991237, -2101971, 1921157, -479651, 1484526, 472835, 2513439, 2406909, -357545, 846766, -332715, -3241060], [192704917, -16018742, 1778725, 79910, 82627, -3194058, 1583843, 758608, -2157363, 2384601, -500955, 1553934]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_531_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_531_6_a_d();".
// 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_531_6_a_d(:prec:=12)
    chi := MakeCharacter_531_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_531_6_a_d();".
// 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_531_6_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_531_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-50564480, 120303168, -16576192, -53008480, 15496192, 3566264, -1130486, -94393, 27968, 1045, -283, -4, 1]>],Snew);
    return Vf;
end function;