
// Make newform 684.6.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_684_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_684_6_a_e();".
// 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_684_6_a_e();".
// 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 := [3103, -84, -140, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-139, -91, 1, 1], [-287, 127, 5, -1], [1127, -55, -17, 1]];
    Rf_basisdens := [1, 12, 12, 12];
    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_684_a();"
function MakeCharacter_684_a()
    N := 684;
    order := 1;
    char_gens := [343, 533, 325];
    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_684_a_Hecke(Kf)
    return MakeCharacter_684_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, 0, 0, 0], [0, 0, 0, 0], [26, -3, -1, 0], [9, 3, -1, -1], [-172, 9, 3, 4], [199, 4, -14, -3], [-69, -18, 36, 6], [-361, 0, 0, 0], [-1497, -52, -4, -47], [-1318, -16, 67, 64], [-153, 124, -5, 21], [-4969, 428, 133, 35], [2663, 224, -13, 83], [-3042, 193, -81, 104], [1368, 323, 299, -56], [19064, -114, -53, -190], [-5549, 851, 661, 230], [-8498, -755, -611, -426], [1648, 785, 1426, -19], [-30336, -22, -376, -224], [-26577, -804, -1382, -482], [29641, 718, 17, 625], [-24002, -1588, -1344, -590], [-931, 1132, 953, -1673], [5064, -1358, -908, 1554], [14232, -3472, -1694, -534], [31400, -308, -1358, 486], [-50567, -2951, 819, 66], [-14458, -1430, -1467, -1036], [-118971, 206, -4123, 1327], [-149935, -2420, -2101, 2221], [-43876, 4009, 2277, 1874], [-108163, -1438, -3386, 1288], [-161358, 5613, 2923, 1852], [31606, 3209, 8783, -1482], [-153506, 3172, -1896, -4024], [-166994, -3402, 6554, 662], [86642, 1602, 6408, -6588], [-67583, 1220, 5549, -603], [-34712, 6518, -172, 1824], [-135913, -152, -11053, -4109], [-20132, -246, 2148, -3456], [-50019, -15059, 507, 8743], [317483, 20662, -5097, 467], [-256964, 9130, -1282, -3388], [110115, 1343, 8745, 3685], [-383039, 13169, -8321, -10], [56829, 18938, 12061, 4811], [-48061, -22291, 5879, 650], [-268680, -18121, 9909, 3460], [-284242, -33191, 753, -1612], [-558247, 19153, 22915, 7011], [-76851, -7304, -16143, -16435], [-108604, -13755, -33421, -12270], [288514, 442, -9648, 1536], [-211506, -38625, 1581, 10410], [-621119, -20386, -16411, -4263], [-227469, -25088, 15178, -7239], [-825316, -5947, 6753, 1486], [-211816, -21238, -39358, -25084], [-628094, -10261, -6729, -4550], [-34839, 37188, -18948, -237], [633411, -12318, -14499, 19959], [254981, 10175, -3075, 20619], [533027, 10251, -42743, -24854], [1008579, -8082, -12332, 12889], [1059874, -36437, 31304, -683], [1216824, 61078, 28338, 31070], [-445152, 51841, 64535, 18954], [-427030, 65387, 67983, 27202], [-1697515, 16441, 6807, -14098], [-477767, -41923, 14757, 37135], [-1935556, 73436, 41532, -17936], [1843267, 76684, 25124, 14089], [2093763, -42147, -28103, -2600], [-3431083, 33308, -1269, -20217], [-108674, -16015, -114079, -32844], [-2520432, -75571, -18927, -27866], [1480471, 35862, 11041, 2559], [-948663, 79936, 96701, -9305], [1815346, 51410, 12136, -4324], [1158260, 101274, -39563, -16586], [139900, 83972, 36026, 38064], [1743109, -5900, 79915, 59181], [1846679, -61860, 66571, -36533], [646580, 21191, -64563, 28260], [-2119992, -95540, 18354, 4996], [-3574299, -75382, -66562, -88656], [6327632, -13119, -4395, -2288], [80128, -210975, -74083, 22424], [1851976, 12405, -74293, -16850], [-1772052, -35120, -75240, 14896], [843004, -72192, -53506, 3218], [-3529190, -35564, -125526, -41202], [2775688, 27627, -71153, -36710], [2058605, -56300, 133018, -46425], [1517532, 76200, 66114, 43020], [800909, -118026, -125213, -22003], [-2697347, 58099, -45461, 69732], [6409754, 44787, 42627, 75672], [715440, -369116, 496, -77376], [6660644, 144671, 110763, 7446], [-1225427, 3968, 247237, 87221], [-6321452, -261684, 23680, 61968], [-1399460, 6436, 44740, -100452], [3882381, -330964, -117098, -56758], [3426956, 135787, -91949, -78936], [5210984, -95224, 20906, -6238], [1567181, 170194, -76125, -46651], [6042488, -366856, -321852, -47246], [-151654, -306528, 99924, -1698], [129212, -159055, -316101, -124934], [5720388, 128799, -120491, -6578], [-3695868, 381202, 122404, -6846], [5607694, 24403, 135177, -131326], [993512, 252844, -82654, 96244], [-1305670, 139229, 299449, 144044], [11032273, -81263, 37867, 95431], [-680654, -247261, -31731, -55472], [-687819, 253661, 180103, -117618], [450345, -164936, -179198, 53451], [-140792, 159130, 137118, -52288], [7242926, -195100, -144111, -47424], [-1854422, -300000, -246466, -27902], [-11649430, -222437, 28747, 2700], [5620490, -379566, -736, 35826], [7756510, 182054, 264470, 30210], [-8418013, 60677, 377469, 41805], [-2690915, -144159, -391347, -77925], [1942202, 54136, -23494, -138152], [6156326, -591597, -350623, 145478], [-12492519, -69748, -259443, -53809], [3457502, 432540, 421330, -11546], [-2219072, 426357, 217203, 222828], [3543513, -150762, 48490, 98860], [2164357, -200598, -395270, -43556], [-12125473, 232402, -82048, 190573], [-929969, -345009, -98689, 263720], [-2360280, -180690, -265897, 233882], [10835577, -97266, -138280, -170770], [8022461, 726037, 98813, -119468], [1578228, 268611, 631247, -67906], [-3018317, 278191, 35879, -313823], [-10247502, -353569, 16092, -160135], [-5446601, -138676, -429452, -343159], [-11981702, 139000, -460598, 102856], [13579440, -551844, -132518, -115106], [18148718, 241858, 304956, 142898], [-8402042, -191289, -580853, 218806], [-18088659, -299556, 103655, 48533], [-9007447, 179762, -459656, -36673], [-9753980, -426445, -683475, -55058], [4386396, -687347, 293343, 30542], [-13159894, 203822, 125514, 94974], [-27580766, 387715, 547932, 122489], [3062836, -334072, 854540, 152238], [11096325, -755536, -846014, -206293], [207317, 416781, -476725, 140766], [5285127, 914636, 252962, 187614], [-6779809, 784934, 623958, -326835], [-10998564, 481626, 160678, 23818], [12023491, 640426, 450455, 509603], [2694396, -1175904, -799092, -391500], [-21661812, -948928, 503056, 371616], [5941175, -621012, 83635, 177737], [23424790, 391196, 1004808, 472810], [-2922523, -801238, 193807, -461797], [-12563652, -120719, 423053, 422734], [26181003, 106396, 923471, -231359], [-21462798, -791469, -138403, -541840], [5715412, 1452131, 15116, -110211], [-10693248, 1340258, -74107, 154338], [-20921836, -1436072, -374548, -311612], [14859251, 341133, -240495, 236544], [-1721616, 574296, 218272, 53068], [15551956, -496423, -357693, -407282], [4640306, -718422, -831356, -161564], [15553016, 945571, 91089, 369218], [-19297548, -517382, -681298, -146210], [-13200930, -327163, 328075, 13886], [34068819, -1312007, -916307, 143085], [-24266063, -392973, -188515, -361082], [49765838, -324226, -260278, 367854], [17024474, 1190276, -302666, -476540], [-15174081, -905643, -703619, -181775], [18087846, -528981, 1204835, -261940], [29255953, 211354, -375812, 158355], [-13505260, -1802230, -616890, -614876], [10977533, -707262, 637613, 378899], [16484517, -253992, 330294, 926379], [7372313, -35668, -12685, 531837], [5049456, -655171, -360039, -63328], [-37000775, -52110, -234033, -75579], [40306286, 269380, -932219, -24792], [29494812, -945362, 276334, -288238], [22528074, 1065354, -34064, 323380], [-29041003, 372000, -39236, 160720], [11023462, -1649734, -593996, -7642], [-8058497, -2118893, 337917, -131742], [37997135, 1577047, -507843, -213127], [31421026, 752072, 455824, 332762], [8580493, -1562118, 551195, 303371], [-31503954, 696142, -505513, -103400], [-76446813, -137596, -572697, 267547], [-18873008, -880484, 120426, -10920], [-4740402, 929645, -765749, 745434], [-1312093, 2757849, 452103, 295281], [33940799, -667235, -1633757, -293646], [27285995, -2059652, -787756, -271952], [35674217, 375993, 545671, -475274], [50424593, 722919, 398987, -780148], [58936378, 1139296, 907434, 375252], [-25261012, 1579471, 101135, -634700], [-41555515, 1214287, 724289, -686126], [-6567029, 1153478, -1114807, 648021], [2011613, 2021380, 770747, -684647], [26401511, 13864, -151342, 923725], [-12839790, -54804, -470428, 305150], [-29944593, -141110, -2582365, 290067], [11386896, 1025445, -685113, 4950], [-398468, -1326838, -761408, 111260], [-28573377, 171542, 2029335, 536341], [-37014997, 1282530, -444069, 380761], [51454403, 1298661, 1964247, 858429], [-27581285, 1620745, 1425641, -538960], [-90159616, -991805, 317541, -109486], [13957268, -1004892, -235694, -439666], [-4538719, -1973552, -2077441, -648965], [-18638486, 3632854, 2042390, 196390], [12136632, -974421, 219625, 1002778], [-23238736, -3026315, -88985, -376536], [-54170868, -1976563, -1097006, -452419], [-399161, 3357482, -363483, -584645], [-5845904, 2017932, 734974, 366210], [-25718572, 412827, 892617, 802746], [-68350043, -461575, -106039, 222387], [-8843336, 1851605, 2877903, 510274], [23438752, -2298365, -1173013, -632128], [-6740406, 2974688, 1797640, -74154], [35566139, 1385230, 156921, 12803], [26479030, 2623986, -212500, 1114774], [53788834, 1831119, 1250779, -113576], [-41484669, -641396, -2260452, -467875], [19110823, -2191514, -1909260, 366089], [-57393901, 2108968, 430227, -997177], [62973690, 177375, -1899795, -747582], [86780426, 956189, 21551, -73218], [-43816185, -1474402, -2221275, -402349], [65757716, -1010628, -687636, 1582056], [96141729, 581451, 692921, 972683], [-6389978, -3000972, 89612, 1550480], [-21901877, -905648, -972546, 632658], [-110258811, 873656, -743110, -150653], [-29052587, -3054266, 12049, -62331], [9777978, 1165487, -2099067, -1545658], [-26474188, -1377148, -1059570, -2313780], [31076400, -2583645, -3555479, -763064], [-52726045, 597132, -1827881, -988589], [-47744847, 1205676, 3673839, 866865], [-25263076, 1179812, 2119170, 1488394], [-60411359, -5293686, 634395, 1007403], [1466446, -3267898, -462752, -381790], [79543704, -2954091, 447205, -57416], [57223251, 1090326, 3322796, -491791], [-53721690, -2460435, -2232491, -1556972], [-119349366, -2669303, 796504, 198255], [-84102598, 55366, 2204339, 373288], [-45286633, -4257620, -537074, -1103754], [-19858487, 367699, -784929, -2200732], [-38716650, -18483, 414637, -1119764], [73477284, 3280434, 1002681, -1078170], [-28376400, -1454975, -4269980, -698253], [-74800949, 5509412, 4348014, -64838], [-15348179, 2598847, -1919115, 812135], [23564267, -2537772, -425931, 1279557], [-17442937, 4035304, -1035368, 1401333], [78772142, 3805633, -2123183, -1033254], [-49303482, -1164506, -631545, 92636], [29797738, 3663780, -377422, -1537336], [165969360, 3013983, 378215, 100514], [38074142, 6733234, 1340386, 344214], [-76421661, -1972460, 4514250, 1431761], [6111311, 3047371, 5333579, 564193], [-41704679, 4833458, 5537214, -120149], [33563720, 4686701, 1922921, 264138], [-45090464, 1155053, -2001291, -1837688], [44190517, -1265962, 1804666, 1983488], [-18109876, 7067452, -402494, 2266260], [-96290025, 2501851, 2729195, -308063], [-128397598, 1679088, 3171712, 1608302], [34288428, 206298, -3301360, -2238784], [51575252, 405397, -2691611, 143496], [-48072950, -3432832, -2450588, 116438], [-111489376, -1670046, -326058, 2271628], [32854800, 47390, 1138988, 1863576], [-113032680, 190700, -801056, 976746], [-28919818, -5568684, -3343228, 608798], [41315266, -5354057, -60785, -1136834], [-148598364, 424345, -2755594, -1389459], [-74532526, 647131, 2267639, 292624], [-1160378, 7432497, -746807, 1582696], [58018647, 1775630, 5737656, -1192747], [213253917, -3254237, -2390693, 38763], [8378146, -4435349, 2963572, -504671], [146512832, 1216957, -2330759, -1477188], [15117581, 3910224, -2009469, -2190309], [-132546723, 583653, 1450465, 471352], [-63976850, 271084, -4762925, 2179308], [17894916, -4141899, 681281, -43540], [32531756, -4021680, 2514600, 520518], [-106081958, 2001213, 2427759, 1391654], [-44140202, -684250, 2756834, 2848974], [-95006444, -2887962, -1729892, 1288434], [16847925, 1433160, 5830997, 1032569], [27437246, 3248404, -1639838, -1152666], [-40179, 9121962, 1699843, 2622193], [6860254, -3017479, 1412173, 64964], [2346039, -8539436, -1223991, 395113], [-137000227, 2571420, 989526, 1427166], [129959840, -3614581, -3381039, 1414746], [-10516522, -1722223, -9379954, -2405067], [-166713633, -3980658, -985194, -827520], [290571, 2092254, -1901378, -50027], [106509991, 3154996, 1203457, 164601], [-29816507, -3130513, -970597, -84024], [-84655851, 2455452, 2667923, -1855321], [117996405, -8032649, -3740803, 596026], [-66227966, 4199781, 876234, 2036889], [225639709, 4051248, 798329, 2096689], [135766144, 204469, -2378839, 738596], [-153595644, -4462169, -173397, -2140270], [104204931, 2319618, -2169528, -1050183], [-18782785, -1805292, -161282, -1232591], [116381916, 3595147, 3196393, 1123766], [25479906, 5271831, 5711275, 3984406], [54665460, 3404885, -4200831, 1734506], [-165605270, 7693070, 1993913, 2087160], [-106254221, 918590, 4272509, -1725147], [-14690231, 5308516, 2591926, -2530338], [-55269687, -131720, 647850, 4181939], [-51573756, -1784954, -6540677, -340902], [9133911, -237102, 974648, 1889270], [-111922820, -2575960, 429805, 824366], [-180579985, 5592414, 1427351, 869531], [115233045, 4882118, 4767108, -1726201], [77624734, -1176456, 402314, 289738], [40051216, 1125230, -1787108, -5147464], [-162719034, 142461, -2473926, -49803], [37253734, -304454, -8750806, -4165756], [-26572942, 1813225, -6099853, -2292434], [191133115, 4556568, -3553073, -2160761], [-143626478, -1697456, -5837664, 2185010], [273261798, -3725149, 441153, -2474542], [99785564, -3111152, -9084, 2143334], [4192319, 11393478, 1821999, 867387], [-10321750, 11636696, -3163702, -495634], [-152574033, -2031340, -3377304, 4413665], [-134871180, 7011690, 7706166, 1500858], [47331537, 2579434, -2078934, -3216824], [43371273, 5527770, 8164501, 5522803], [-176871570, 9320551, 4464517, 1815114], [92736906, 14823476, 2115396, -828976], [220505163, -6128382, 4887791, -1398175], [26489028, -2877877, -11374126, -3861473], [-27829346, -362381, -3215437, -1293056], [185831907, -8070052, 3027007, -1638385], [34704238, 5255858, 7204094, -3036894], [32497302, -4594962, -6471450, -1282650], [-125222687, -4032858, 2267749, 3029257], [-289466330, 1775288, -4814516, 2381054], [47697776, 1859473, -2390792, -1290537], [177012551, 1340307, -7262225, -2994223], [227083408, 8049648, -160207, -903266], [76597860, -3866341, -2061525, 2519812], [-150135186, -4344229, -1125087, -3125138], [223291227, 7889447, -197521, -542324], [-45333410, -5282336, -212706, 4040874], [222272844, 4107851, -3168135, -164402], [-22253325, -4029798, 4195658, -2066356], [-268145098, -3567236, -3671938, -5420378], [-107464206, 6943062, -4459534, -3557446], [19358300, -2044527, 171959, 189414], [248409679, -1837479, 5603045, -3342103], [259928052, -2431071, -2440533, 5780070], [-86279944, -9014129, -10347736, -3883475], [288800190, 2707670, -1197762, -3962944], [-213364255, -575716, -2547358, -715041], [247512048, -7319468, 3344536, -3922074], [-130109660, -8302431, -2239797, -2255328], [-92998743, 723280, -8409347, -3789247], [81039164, -8659410, 5678236, -21956], [149064317, -469902, -3139158, 3167527], [-90508530, 14338230, 11543786, -2363488], [229733802, -5144477, 1900851, -3806044], [-59058186, 2215194, 5907820, -1015538], [60707520, -9494626, -13170328, -5988960], [42375605, 138332, -8868625, 1624503], [135063127, -12335210, 871993, -2458355], [-37195799, 391920, -2400455, -6550607], [-30140754, 4925058, 10108120, -4064702], [102658801, 8236222, 8164461, -6881035], [56525296, -1659195, 11211839, 5505154], [-441244930, -1606196, 1008340, -803442], [-25057964, 10355718, -637934, 4016472], [17157689, -171182, -2313603, -4548739], [72548415, 7966337, 1061191, 1828217], [-339996194, -5817603, -8951651, 304588], [-371943237, 5935676, -1526977, 3978931], [360385362, 10192237, -3934686, 645893], [-306046960, -6005943, -1147223, -6035974], [207879847, 6227964, -951489, -1558215], [92250886, 13182575, 7407955, 5181086], [42372885, -4456661, -4746293, 7586204], [249798586, 7840951, 5489023, -3994188], [-195387207, 19121419, 9911895, 1294808], [-112819494, 1579296, -11431482, -4414032], [-230640380, -4488346, -6582316, -5496894], [-36487609, -11330136, -4669169, 1985257], [54644217, -16138569, -12385933, -3745840], [-169276088, 11730702, 8837838, 5228456], [132568455, 5613054, -1565725, -547477], [278672094, 12073496, 10666387, 4090412], [-385087145, 20353231, 13529201, -1338667], [-410871414, -1117433, -1444630, 3078663], [-354385194, 6393766, 1679602, -1414236], [-262725751, 2012296, -546212, 2140035], [469886892, -16594075, -8162976, 301727], [-147275107, -6762796, -1585260, 407820], [428601412, 2796415, 9261075, 2590556], [26157283, -12735941, -4356417, -5027103]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_684_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_684_6_a_e();".
// 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_684_6_a_e(:prec:=4)
    chi := MakeCharacter_684_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_684_6_a_e();".
// 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_684_6_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_684_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![-14782624, 809300, -6087, -110, 1]>],Snew);
    return Vf;
end function;