
// Make newform 456.6.a.a in Magma, downloaded from the LMFDB on 23 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_456_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_456_6_a_a();".
// 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_456_6_a_a();".
// 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 := [26148, -916, -431, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-214, -5, 1, 0], [1172, -262, -11, 1]];
    Rf_basisdens := [1, 1, 2, 10];
    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_456_a();"
function MakeCharacter_456_a()
    N := 456;
    order := 1;
    char_gens := [343, 229, 305, 97];
    v := [1, 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_456_a_Hecke(Kf)
    return MakeCharacter_456_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 := 6;
    raw_aps := [[0, 0, 0, 0], [-9, 0, 0, 0], [-5, 0, 1, 0], [19, -4, 1, -1], [33, 17, -2, 1], [-138, -15, -3, 4], [-325, 12, -5, -5], [-361, 0, 0, 0], [-80, -62, -8, 20], [2282, -113, -71, -2], [2172, 92, -42, 73], [886, 238, -24, 11], [-1722, 164, 0, -3], [4879, 532, 5, -77], [4201, -701, 54, -26], [-1264, -994, -100, 39], [-1818, 1035, 147, -221], [-3135, -1305, -22, -110], [382, 663, -53, 227], [-14774, 759, 319, 119], [-34981, 2555, 332, 358], [-12636, -2979, 57, 604], [16354, 244, -110, -917], [-30516, 2072, -838, 641], [-57588, 4085, -43, 571], [-57800, 3972, -660, -527], [-38580, 644, -342, -1297], [9110, -7394, -888, 754], [-78492, -7699, 1135, -1568], [-80454, -857, -1023, 88], [-65942, 4507, -607, 450], [-92429, -1963, 404, -351], [-38021, 13068, -2573, -505], [-41613, -8902, -989, -1297], [-106683, -6718, 2001, -3104], [-121080, 171, 519, -1562], [-132034, 16224, 4620, -1332], [142540, 12770, -4530, 3066], [73960, -2521, 1165, -1115], [-33820, -18122, 2292, 1963], [-548, 3195, -2187, 3097], [-32252, -12196, 2336, -1361], [-53377, 9113, 7806, -3826], [-150426, 8741, -3717, -4713], [237900, -5264, -720, 2085], [-65333, -39718, 2267, 5619], [-93118, 19677, -7091, -3501], [26558, 4812, 2404, -4285], [298660, -43645, -3491, 4831], [-617065, 8119, 1756, 5450], [201841, 13382, 4079, -2489], [602555, 13523, -1252, 1402], [-151172, -11864, -17974, 9546], [316911, 21937, -10156, 2151], [-581610, 9568, 6048, 3093], [568589, 36323, 14258, -16416], [-9740, -860, -13154, -2351], [185464, -10154, 13286, -4978], [-92397, 78647, -7248, 4230], [281752, -27078, 352, -837], [-295833, 1886, 491, 12313], [158414, 54784, -7120, 11783], [707042, 58375, -6977, -12301], [827329, -95759, -8, 5404], [-703030, 53894, 26950, -10250], [299688, 17531, -7557, 10680], [-319578, -48623, 19337, 11953], [-1370764, -56571, -3767, -3303], [1425323, 72467, -7678, 14617], [-1125997, 59223, 29660, -18724], [-339758, -31628, 37868, -6746], [837907, -45243, -24474, 6990], [296004, -8454, -18, -2158], [4066, -85012, 10030, 16105], [-70230, -1278, 39572, -16714], [1315366, -202946, -18572, 2914], [1762351, -201758, 16579, 1222], [-1240341, -141465, 10748, -7160], [-110972, 79759, 19223, -21162], [-1534266, -93994, -7334, 7062], [861786, 148244, -22218, 14187], [216772, 34954, -40122, 15527], [871408, 111663, -45787, 12771], [-1114212, 98175, -44653, -13829], [-1677842, 113839, 13569, -30198], [3281219, 86993, 9324, -10529], [921942, 113742, -25262, 14023], [-1938727, -9815, -59068, -41640], [507293, -166082, 56301, -24772], [475405, -23884, -54785, -4449], [278591, -18593, 15062, 68249], [-4042296, -199326, 19624, -28606], [-3157408, 250359, 10611, 26074], [-1899494, -239848, 102190, -62015], [-120023, 204908, 34515, 13827], [-2107516, 174926, 40452, -38668], [-596238, -22225, 24461, 41530], [-516792, 159543, 59975, 78772], [-1265038, 2900, 116938, -68600], [-349283, -101139, 153412, -84842], [-5669040, -189563, -37125, 35027], [6174517, 149596, 55063, 26598], [-5967488, 169042, 72170, 29758], [523274, -48523, 112463, -107110], [-6485840, -208174, 8658, 61700], [-4778879, 261945, -10724, 69792], [-1231805, -323339, 82040, -109459], [2141654, -172722, -35386, -16658], [-7632354, -296953, 50351, 4563], [-4333426, 334877, 107055, -46371], [-5343486, -30325, -89251, 134240], [168903, -338417, 88152, 73066], [2893903, -45462, 20293, 95495], [-1560460, 496532, -78372, -62360], [-5456093, 22480, -15335, -108299], [1753954, 116613, 32959, -39688], [-3153789, 294366, -32705, -39067], [-6807537, -342305, -28568, -123882], [-1494327, 269688, -12593, 32038], [-1287058, 721554, -100876, 26260], [3408250, -90082, -94764, -96973], [860990, -822861, 61685, -16523], [4653150, 215063, -40075, -68222], [-7154328, 436427, -73071, -16547], [3177861, -592090, -31059, -83721], [3827076, -390290, 150950, 37523], [5084566, -215946, 193810, -9040], [-8693881, 33613, -8114, 50170], [3990257, -113612, 215247, 14665], [-969078, -118400, 338976, -108820], [10744971, 501062, -55769, -20985], [-6389226, -471271, -51763, -10885], [2699982, 431984, 94148, 693], [898157, 574401, 21872, -15580], [9946135, 94202, -95823, 56225], [1732177, 316207, 54266, -118268], [6845612, -228303, -116199, -67538], [13225444, -408582, 17614, -166902], [4022728, -774049, -3593, 164276], [-666179, 529918, 366247, -214663], [10513204, -879867, 168679, 71231], [3856149, 117284, -150997, -11840], [11694471, -346924, -165487, 64957], [-3948298, 253157, 156961, -21971], [808686, -1568884, 168158, 109445], [-1621362, 752540, 126770, 79478], [176214, 1561142, -29254, 131870], [-3141520, -1596467, -8231, 55058], [14250999, 108824, -64407, -44555], [-568696, -52106, -150050, -250302], [9313168, -629274, -126738, 199125], [12300011, 31248, -57821, -8803], [14362275, 1027466, -28117, 281895], [6301418, -1287221, 159863, -140821], [7573874, -129456, -232418, 122348], [-17225074, 258012, -13214, 150464], [-946640, 471032, 225992, 125132], [13235218, -1174956, 83920, -250678], [21475235, -393191, 32182, -90758], [3188156, 1195436, 70198, -292727], [7482690, -1096750, 305996, -242121], [11352320, -752323, 62733, 147802], [17162932, -984828, -360540, 93238], [1650796, 2746324, -154356, -64096], [6506368, 939450, 278128, 53317], [-11207260, 1374976, 68028, 204876], [5031002, 87984, 500644, -257741], [11677139, 175093, -156226, 31728], [-13248410, 474259, -3539, -261597], [15835533, 73150, 438269, -52768], [-14492852, -1719742, 155934, -151492], [-1197684, 420556, -133896, 86881], [16397416, 194198, 181224, -257866], [13049966, 71202, 514366, -156916], [-2902194, 1359943, 241621, 184960], [-14824671, 1514944, 311325, 237193], [37449432, 933510, -268610, 221474], [-2097227, 629350, -270339, 424828], [-5006258, -1098976, 538648, -295517], [-12520591, -381717, -434978, 50500], [18049425, -84708, 691179, -334949], [-10360152, 350293, 303855, 165053], [3815084, -1789664, -235500, -259397], [-6900652, 1756835, 743083, -148752], [-30112373, -1331975, -93466, 38122], [-22034567, -714196, -195541, 324770], [-13258764, 2662932, 440968, 221303], [20752258, -612943, 203417, -338945], [-16129280, -2969357, 540643, -219483], [-21738836, -936326, -364092, 37512], [12462832, 187535, 261251, -447961], [35846157, 560165, -476982, 277583], [146410, 1904272, 113854, 181804], [-10613778, 1669482, 120562, 368209], [14533990, -25342, -780676, -42303], [-14155414, 474121, 668151, -386854], [-15793029, -138843, 401178, -348058], [15153534, -1614654, -328478, -165234], [-22163546, 2327182, -755738, -24918], [-19063359, 581143, -719778, 252342], [-14272646, 646592, 56008, 400955], [-13320858, -2034716, -624960, 354769], [-16404596, -1138135, -403257, 158766], [-11139088, 828849, -458943, 37113], [-5003250, -4348374, -4940, 124505], [-12301007, -2282910, 135537, -788874], [18811569, -829086, -676583, -69157], [11904192, 3052194, 78066, 792096], [-43814713, 3578, -856483, -244781], [-3904030, 4479040, -71550, -135270], [-29272666, 283418, -1086582, 479320], [-37585680, -183316, -4596, -701429], [-1486209, 56148, -785235, -12735], [-22939148, 1590702, 1164394, -424088], [-34987016, 1296187, -856783, 137771], [-24574476, -990172, -45066, 850656], [-24883728, 3574430, -148590, -161076], [-8183034, 768396, -1380896, 624477], [-45728266, -1853394, 322104, 359864], [-12353193, -4010824, 23541, 367496], [-5449924, 526347, 222529, 385674], [-37717902, 1118964, 138920, -133241], [39988406, -3551909, 61653, 137078], [-16588987, -580420, -505645, 201543], [57035586, -1643025, 171171, -157215], [-41118163, 746679, -225290, 1029534], [-38300612, 984551, -546653, 516902], [-9644446, -4297487, 681769, 221423], [23933080, -1577493, 191959, -199698], [2508697, 1195111, -932672, 617347], [22786545, -4159805, 1004106, 75366], [-17090054, -2889750, -709832, -695396], [-32198952, 879870, -1423268, 250357], [11465504, 4417479, -121689, -920104], [-29347519, 4091950, 653937, -321099], [28127747, 5090715, -582792, 214994], [44736975, 1839749, 702030, 335736], [14431283, 724928, 1030905, -506168], [8616642, 2068790, 986512, -253129], [13311650, 2821899, -103293, -479473], [38812450, 373910, 817380, -23114], [16696197, 203184, -1638933, 778717], [53232492, -6787002, 225670, 165416], [43033260, -1534447, -823453, 253562], [61246438, -3835069, 841373, 371916], [-7028711, -4672029, 136446, -110948], [-8627327, -5506098, -272171, -713387], [35527074, 3411937, -391751, -356265], [13518136, -6532916, 495496, 68100], [51183861, -2656935, -543160, -631030], [32622830, -423804, -231480, -637470], [-29722847, -1932832, 2338357, -1113591], [97310340, 4290590, -550972, 1118702], [-43420258, 4184247, -142171, 113837], [63555119, 4192832, -192391, 677362], [40305602, 3859380, 164686, -1540397], [96975387, -3059587, -871550, 555954], [7158304, 426523, -854027, 1495537], [17758198, -3437932, 1444864, -695917], [12797878, -1696954, -328486, -832946], [-42350930, 1046646, -942038, -629943], [25836586, -2972113, 1269211, -277543], [-36318503, -2870703, 1395116, -991594], [82237908, -5867268, -1456176, 248445], [72908877, 1125778, -429205, 1176497], [-69831878, 6206640, -159990, 207062], [-32379000, 1906396, -299994, 1120929], [-16874391, -3438342, -476805, 1744447], [-106931678, -2079025, 745283, 250537], [45555407, 3844200, 69789, 1259780], [20924668, 3309578, 843698, 1219779], [-27650112, -6794604, -829696, 423376], [-45495005, -3737243, 413762, -522314], [-85695309, -4662036, 320713, -45129], [81055314, 3004573, 712067, -1061981], [-75510456, 3718508, -195364, 420668], [71218197, 5123195, -1460780, 566175], [90556180, -3429054, 11170, -538829], [-112298218, 6086851, -696287, 405979], [93696913, -217091, 837946, -503713], [-131741526, -1551237, 160243, -1234409], [-25614444, 3970560, 2109944, -1108246], [20028681, -11748017, 1166914, 804216], [9853632, 2057049, -1802057, 962902], [-49714673, 2267082, 333007, 152181], [-67002073, -3895177, 1275304, -905618], [-22874143, -9897773, 873522, 346292], [-98818754, -5844727, 1308607, -1832300], [-7333405, -349646, -2150341, 431409], [-9352330, 9546377, 59411, -333220], [-75060444, -1076840, -2030072, -384201], [65666789, 2866727, -3259952, 1622975], [-62496474, -147013, -3069463, 749654], [1952626, -5214957, 397167, -625593], [21657084, -4317251, -3565185, 733790], [86416416, 11981956, -1975852, 600475], [-121892998, -7137068, 860668, 384649], [-74609605, 4231938, -316933, 2134112], [-5189158, 12597440, -2608766, 1538], [330421, 3968852, -197945, -1780801], [63580711, -4735543, -19074, -40110], [80260016, -1519028, -1244514, 257493], [75642551, 8864366, -1680705, 208693], [50010334, -4302344, 1520878, -884112], [-95711007, 6353318, 257977, -352259], [23422434, 8245810, -1108534, 1493854], [25721692, -6803626, -2407550, -752290], [-47764258, -2676937, -3482253, 802968], [85080147, -13296125, 1875964, 388776], [5817046, 1318962, -742954, 124230], [-4745817, 8560547, 1719396, -2108628], [17412976, 4631674, 258138, 1967243], [-20481846, -3531460, 3875152, -1919381], [98520412, -371926, -535874, -1525398], [19631420, 9844532, -60666, -1513804], [-230114106, -1284980, -427136, 1935706], [-188830841, 3853635, 1522306, -98741], [240592706, 5918888, -1176338, 1208126], [53011981, 11657135, -830326, 1380766], [-83438169, 3406630, -250591, 1443547], [123669088, -9295878, -3077306, -235676], [-57961285, 8406863, -2801476, 867430], [-185771336, -415634, 562316, -116405], [-48089484, 3028632, -3069982, -276207], [30363974, -8591900, -871340, 187820], [-59329802, 3476147, -1706507, -1083893], [-7656876, -3821479, -2896637, 760827], [-249490144, -5153034, 1137750, -1751318], [206096296, -8696478, -861082, -465668], [-266236455, 4642662, 10281, 160248], [80162677, 1065859, 1242014, -3610682], [-70400540, 9518934, 816336, 1477383], [-157001940, -2768346, -1872906, 2088874], [-140988635, 9676915, -2457160, 346891], [43289767, 10780324, -492075, -1757837], [51905095, 7077109, 1905008, 2375043], [-199762868, 3899244, -2595692, -103829], [49973966, -2846651, 2975071, 845478], [-177432519, -2965665, 3130162, -311094], [231362680, -7318258, -1373846, 147792], [-99395080, 5542809, 3205531, 351368], [-10133273, 15450498, -702343, -1325671], [-37989386, -10001959, 2832307, -3342612], [240511484, -11008817, 1502723, 1143972], [-234267792, 1606535, -2844429, -1906296], [-98049128, -4877303, 974667, 2386137], [-26328770, -12823596, 401752, -1832138], [-65114620, 12198973, 2869663, -1912977], [39424880, 18132039, -76155, 745565], [-277446067, 4232250, -1971955, 21418], [-140763136, -1439307, 519003, -1119741], [180643392, 4827406, -1441992, 1439712], [-71929817, 9796680, 991953, -1759418], [-191338372, -2279372, 650510, 1674721], [71002180, -10982309, -4062843, -584370], [-251297414, -3139175, 872291, 21480], [135466096, 14454856, -2452322, 1934982], [23993538, -16790040, -825090, -1842069], [361870947, 6908056, -1680945, 301555], [-102780540, 7894641, 1021115, 1941971], [-125702601, -14699521, 4125960, -2276840], [-278262434, 8228692, -1978480, 1497690], [55607952, 3016966, -836530, 4286042], [-132050730, -12625571, 3174617, -2929355], [-107439807, -3973084, -3268061, 4002935], [-25468952, -5659627, -2277931, 268967], [135814272, -16990212, -5401476, 494273], [151743280, -2434188, 2724882, 2784757], [-42686670, -15541282, -1686770, -140848], [116921774, -1756526, 2401200, 736685], [-27800548, 15329337, 4154231, -2016875], [16391177, -14675891, -161568, 1188818], [-236091646, -7835144, 2760280, -926233], [91865223, -14293268, 3309717, -2096635], [315118823, -4917347, 1867950, -1480442], [225014032, -7119719, 210611, -388239], [205979520, -3105908, 1620982, 553814], [-236972179, 7345165, 2461744, 2807912], [209940137, 14710174, -5450069, 4026913], [-10922992, 14737324, -201678, -3766042], [-280540724, -7248724, 2422266, 891325], [117602085, -4101830, -629213, -2645063], [420559401, -1200986, 3045185, -948131], [223078381, -8786090, 4739627, 2113227], [-237071624, 12308207, -2723299, 1238747], [162014916, -19816091, 1057107, 391629], [346639096, 72518, -3831774, 256112], [310061770, -6526812, 5855488, 709036], [292950917, 8062936, 255011, 3884207], [134986608, 22666421, 756287, -1602805], [117227206, 8653217, 4104779, -1679449], [-143240408, 3473104, -3850150, -751611], [-111828566, 14515336, 4391718, 305965], [-7426127, -1077408, -3427553, 3881991], [-141126658, -1145425, -1204485, 2922767], [-105867282, 6530100, -401236, -1554704], [147638148, -2895349, 2186951, -1222656], [-153665706, 7670411, 2790305, -2498370], [147316988, -4016419, 8393719, 331151], [-56305124, 15162446, 5576750, 378401], [275078536, -15596001, 2713905, 757296], [317078165, -4226286, -1539973, 440085], [518999710, -12689621, -514275, 668854], [245548416, 5878542, -1492568, 5250805], [-400848284, -4819647, -3562615, 85034], [190933981, -2952846, 2124085, 166707], [360073183, -5637371, -1807354, 2129050], [-234884978, -27745008, -1148200, 3351693], [21182036, -10316531, -2899741, -3492847], [-314924761, 152621, 2437720, -966883], [165751038, -20383590, 1643554, -1250442], [-102754105, 4963692, 120177, -334456], [-23531014, 18040678, 820740, 6946060], [-248678431, 6432146, -673323, 4612803], [94441450, -22940578, -3545446, 3846460], [-54207428, -4393502, -3801102, 688951], [270781780, 20720541, -6846381, 4315795], [14245272, 13948595, 2337791, 6547194], [-31891478, -10512776, 278076, -6404320], [-169620702, 1821840, -5875342, 2432198], [68541328, -14041003, 3513957, -794646], [-57766302, 9417443, 5266823, -1655712], [12604397, 5644007, 1866398, -1470770], [-319258012, -7423747, 2223123, 3311033], [-546979580, -114948, -4213770, -1964664], [80446826, 24267663, -3806795, -1749006], [-525174212, -12253221, 1633749, 273085], [-33086043, -6007972, -551355, 1501445], [15246935, 7373450, 3806695, -574481], [-403387811, 29746261, -3085820, -1781710]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_456_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_456_6_a_a();".
// 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_456_6_a_a(:prec:=4)
    chi := MakeCharacter_456_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_456_6_a_a();".
// 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_456_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_456_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![3815176, 114930, -8127, 20, 1]>],Snew);
    return Vf;
end function;