
// Make newform 171.10.a.e in Magma, downloaded from the LMFDB on 15 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_171_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_171_10_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_171_10_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 := [-684004608, -2326290816, -1128074928, 1877896, 3632756, 2146, -3446, -1, 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], [-862, 0, 1, 0, 0, 0, 0, 0], [919755537257088, 229380805494336, -10516709500824, -728351550520, 16477812710, 755626284, -6574249, -248545], [-1162756962541440, -816972857783616, -2156404550472, 2584283625688, 8005981330, -2400722940, -3581003, 680653], [45613645265664, 91193057121384, 365663431800, -267845543218, -526583290, 236289207, 144686, -64873], [1910286827680896, 1916839481246400, -1691017215432, -6422673217384, 1122039890, 6239657508, -1545259, -1841779], [-15190761446784, -32302372338192, -232176331656, 111824975932, 334949170, -114994362, -85625, 35797]];
    Rf_basisdens := [1, 1, 1, 328794036480, 328794036480, 13699751520, 328794036480, 6322962240];
    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_171_a();"
function MakeCharacter_171_a()
    N := 171;
    order := 1;
    char_gens := [20, 154];
    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_171_a_Hecke(Kf)
    return MakeCharacter_171_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 := 10;
    raw_aps := [[-2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-485, -15, 1, 0, -1, 0, 0, 0], [1181, 34, 0, 1, 2, 0, 0, 1], [-4828, 312, -5, 16, -6, -4, 7, -3], [29907, -531, 20, 20, -11, 16, 3, -7], [-42157, -3202, 188, 81, 150, -8, -24, -43], [-130321, 0, 0, 0, 0, 0, 0, 0], [-2258, -13061, 1473, -71, 113, 4, 304, 25], [46967, -13318, 2585, 345, -824, 240, 93, 76], [1878418, 14719, 2543, -291, 217, 68, 1370, 195], [3702882, -110361, 1145, 895, 1541, -336, -48, -1193], [-2558504, 189690, 23138, -476, 1158, -1448, 2032, 652], [13493715, -384872, -8336, 609, -5976, -1512, 426, -453], [-9690826, 131251, 19930, 2393, -9793, 648, -6619, 1316], [-12409148, 120808, -27278, 9882, 7284, 7928, 8358, 1564], [-24209827, -791382, -80139, -14635, 41112, -3108, 5513, 14540], [33595016, -189822, 32441, 15544, -2248, -6108, 7299, 15301], [-18958405, 568928, 20193, 1257, -4674, -19760, 29991, -15366], [15184321, 47654, 17981, -34075, -4068, -13112, 39869, 33064], [-12352192, -2928624, 45569, 20374, -28634, 7916, -4295, 9601], [-16267919, -7740093, 74558, 6826, -59521, -3420, 53931, 28903], [57314112, 6656716, -85796, -72280, -135124, 18924, 101192, 27832], [153999242, -1394260, -235828, 148080, 204668, 62228, -109368, -11640], [-64337701, -6393176, 375943, -7809, -339638, 26200, -99895, -21354], [661885814, 23269105, -133863, 122325, -277209, -25140, 38418, -15549], [199168678, 5608561, 239505, -176621, 289671, -109580, -85306, 3757], [298767700, 9549924, 651662, 126928, -535008, -16820, -479702, 78814], [674787917, -7339961, 26174, 375740, -13109, -49752, -16269, 16433], [969236423, 9204824, 895035, -136005, -59854, -64088, 76965, -19026], [808301057, 77509761, -1512918, -459280, -818395, 123348, 356573, -428293], [-191177258, -56045802, 695977, 532548, -484676, 105772, 416991, 119449], [-2614205981, 46386326, -3217996, -512239, -400146, 214864, 62600, -616451], [2273864485, 1125940, -286998, 637671, -1785560, 49764, -67132, 559335], [-1110169617, -26762229, -6631719, 698766, 572073, -217008, -249870, 23340], [-742016089, 59011147, -5980904, 438684, -412477, -573988, 807577, 538827], [707311394, -39107882, 7129354, -2044086, -3255694, 308840, 926344, 350202], [-874382802, 71502460, -78310, 2245558, -538480, 1136132, -471606, -725540], [-1163426949, -25079356, -5167021, -23295, 2066446, 571800, -3182547, 88116], [3276468024, 3479066, -1696460, -1264096, 6178882, -453320, -533998, -313082], [-1928330721, 138238418, -6673063, -3184697, -1321136, -1484088, 1949041, 1080734], [7230843646, 258201541, 7756941, -2765903, -100245, -1699304, -503086, -1224473], [-293626306, -307247495, -1119754, -499017, 9942045, -1764032, 1380735, -8616], [7718056665, 117480328, 8755543, 281081, -3674998, 877464, -74131, -1988908], [-8694325578, -130155129, -15217965, 3912995, -27975, 217236, 2910626, 234289], [2704974263, 246500022, -9262406, -5176557, 14455010, -1022016, 1181026, 1552833], [-5909336875, 267845202, -26376379, 9082169, 2204712, 421920, 4399057, 520928], [-1127402024, 307541331, -8236513, 1942283, -19700243, -3695564, -3544686, 2890073], [-3075493957, 708248770, 1810721, 7515811, 9561008, 1336904, -3819743, -435462], [6273131622, -32659294, 3530565, -7701702, 1482592, -551692, 10221491, 616071], [-14841012709, 262011752, -34045788, 6008075, -12638608, 827272, -3386782, -1824275], [-10418765900, 888321263, -20741614, 6090687, 23825867, -1627384, -4146915, 2907514], [7481045444, -67339362, -48410250, 4582060, -11543118, 1624592, -3513472, 390580], [-16869641398, 1381139130, 51608417, 498644, 4457920, -2683404, -831613, 7907485], [-22477926288, 148022764, -16854600, -26444734, -7675276, -1547876, 2195480, 3072034], [11022305122, -73537557, -61293786, 17671725, 13766287, 2433632, -5342367, -930020], [-39894915658, 852212730, 30065898, -2953918, -19159722, 925704, 10855076, 11923142], [-10650576006, -317170834, 85961284, 7862200, -45286082, 6601532, -12950954, -1658774], [-26467884202, -420364702, 14112449, -17259062, -5184008, 2140764, -5500673, 3039475], [4482333988, 299882274, 138251620, -5293336, -21799990, 6703848, -19568806, -13737842], [-26882056655, -1874482056, 3398178, 13676943, 28308124, 3188884, 6242408, -5975853], [54216646824, 371216966, -5464854, 589412, -10569678, -12251088, 1677356, -1594192], [48045687279, 205186884, 54905877, -26955351, 53960010, 8895672, -21532857, 21292794], [-18396050648, 3576411951, -20462616, -40568341, -10980881, 15184784, -6186493, 4680416], [-60509210460, -264818930, 137180540, -56567528, -38540946, -12593988, 17095502, 12331090], [-14189242715, -2987846628, 18538487, 31678373, -41980818, -11019392, 3436481, 6182916], [-54886181079, 2246177768, -28021873, -52492357, -113104742, -16616344, 70003257, 13482314], [-52670046585, -852505154, -82840659, 62764073, -15132764, 2459056, -36620499, -4651588], [179038349592, 1920232648, 158362777, 72852212, -87266706, 9759692, -21181243, -2602845], [-52685222542, 675977872, -213906113, -37249740, -34928086, -20015188, 33633379, -30375423], [35349750514, -6193426428, -84067412, -43809776, -104761524, 9530672, 36480376, 5252560], [31996924438, 944208977, -316835430, 21911167, 116638157, -22845296, 11132683, -3483084], [91464776026, 1210428098, -20690492, -44546884, 57987634, -13181972, -60997926, -2467510], [-52360174120, -3991318229, 154254775, 36514141, -200981331, 38248832, -12675442, -25898969], [193445173964, 1403515894, -1805332, -111470154, 216570478, -13983496, 9290522, 34846788], [43170626812, -1033888180, 68715338, 1213264, 153419672, -11490736, -76277114, 41292474], [180714420133, -3162835855, -53057169, 5469974, 49479971, -30893888, -15049774, 29082980], [49399617030, -11359330108, 261093863, 59190220, 131600622, -40846852, 23630759, -53976131], [184345805537, -8532114528, 280964839, -87807799, -51708126, 20862632, 19777493, 1267076], [-7023150432, 16117279862, 44762952, 19168984, -314487242, 47763640, 19648810, -7638866], [90154176228, -10365203114, 189606338, -46606634, -278443094, 85156668, 54155920, 11831862], [-24858635996, -368800669, -524297819, 21017633, -6488551, -60820216, -6958492, -33500443], [-19886804049, -4012530956, 8903443, -60558435, 96056414, -61617432, 115027605, -17586504], [-297873503387, -5690165292, 492531633, 86234673, 119018354, 12912416, 43143955, 47659710], [335157078937, 7801492513, 426574870, 4564480, 341262077, 11290972, -59393367, 59110015], [-50392795786, -11015782212, -981426781, -151385366, 105158606, -41042068, 49598431, 22543135], [282296330030, 18591530072, -452905998, 131898120, 31073404, 15273284, 71783706, 227350], [-178940577624, -3425337350, 13625917, -177758756, -471007072, -26881952, 107387703, 31150801], [322519752519, -3185538393, 96038605, -296495542, 67647021, 44339672, -47341438, -20550224], [334505445615, -21213784726, -910829724, 59968271, 7787570, 80337880, -231983656, -16594081], [17778600892, 19469510306, -67667629, 9042026, -508730044, 31034612, -196244911, 83727533], [-405892291434, -17039568801, -928508799, -256952371, 39830837, 3112660, 270678452, 15008513], [598899143827, -16181097189, -607138492, 76188096, -64232469, -64101196, 231823101, 87171291], [-765560637432, -14295713288, -38053276, -269854908, -487043976, -140966876, 108610044, -16657792], [719508089901, -5982153628, 85438136, -225095421, -220173116, -54636440, 65943578, -7240395], [-256805775518, 12427622413, 1539808499, 160425335, 505376367, -2636708, 111355868, -13199485], [955186822287, 14364248516, 216647043, -194239957, 156962750, 92085088, 28264145, -5378166], [-436512327883, 12110381022, -1448954547, 162727935, -44754276, 72258492, 77211969, -99637778], [653261816650, 8568872810, 110450966, 92759700, 504774702, -136323288, -379917372, -239345160], [1638302083154, -30853761224, -128784015, 451766748, 473538766, -30229668, -81994843, -132184689], [-136253463817, -40029129104, 1501728299, 216622645, 166760986, 40192768, -100395359, 39080068], [518894145277, 15963418603, 1144437007, -241300824, 442786461, -11856112, -6420780, 209487412], [-1501917797434, -14674525152, 1973545034, 414525330, 679194388, -44612988, 196646526, -93030204], [670753900849, -40097659378, 1477408655, -167426877, -1229760712, -4233868, -97681677, -33051752], [1643634699686, 5995673048, -1170360006, 70581630, 491728948, 26231336, -70328890, -267437328], [477320893932, -67822070650, -627014539, -121138468, 753067772, -31034208, -248104429, 104271653], [151017303142, -14308214760, 1293670987, 323670718, -640665270, 50631236, 154009603, -245327369], [-183620751344, -31664388640, 274303470, 625422430, -345248604, -43024944, -387125726, -335882500], [643351804237, -73570980100, 3541812883, -533514025, -397478186, 52541728, 227771585, 110283846], [-637883756411, 56560495454, -482166475, -507371317, -1070078060, -2974816, 127070361, -15950662], [468807328101, -44351139413, -1347724572, 605386182, 25118795, -191725316, 323292217, -214067931], [720358822614, 95620554800, 833424059, -102995080, -493981870, -264845844, 891700151, 88577489], [1221474297901, 47590727260, 3662520196, -474499939, 674010068, 76062912, -148555150, -148236105], [126901189176, 7580553528, 2059344596, -263774732, 1019521392, 190151728, 292933076, 225916096], [-644675492475, -20809905448, 1711791386, 929509287, -842221756, -211672720, -423652400, 52778247], [4126850048477, -30158178774, -2050313789, 342318363, 439298876, -285167172, 499090443, 235293320], [1833864629737, 55948990700, -429084006, -675784829, -923591600, -168040820, -455942876, -84878381], [-1016593918212, 21888385821, -6953039780, -61505795, -650737355, 118100032, -236160203, 288273928], [-1584437923743, 28926615959, -2056120693, 506894080, 100798753, -96918504, -290019980, 693659732], [503717820816, -31024259268, 401579882, 1222819840, 808727864, 180816200, -948309578, -94536846], [-1124468528338, 69749363999, 7142424729, 623116795, 759499933, 25039168, -458687408, -227523245], [-3399106568773, -69684359988, -481245891, 279636075, -848203742, -255700856, -518600677, -356686056], [53252193579, 32572626454, -135614203, -331325679, 1686019868, 423542168, -192280659, -872205036], [-1083270898861, 23022440314, 1775934233, -1129130189, -449150920, -59931160, 1400793409, 427088874], [3214119499155, 72690062772, -1228255314, -844319995, 1101988856, 501948240, 1288857788, 115105421], [4861858724560, -30271695471, -1813093953, 25478083, -36843861, 25085796, -802755860, 188583055], [-1351542878928, -67491732068, 4462716106, -1195905678, -716819240, -654408440, -50753302, 338265408], [-1799270532326, 55817967075, -6458268140, 916239885, -1210370677, 152991312, -685226289, -712937034], [13286571399, 143498549472, 6603210218, -147457691, -3074869812, 410088832, -445250188, -578082743], [1584376572250, -3500711414, -4949705110, -1053680842, 1980374294, -183903776, -1060667812, -569139958], [-297237768475, 199392439124, 1206240130, 758508383, 1301572200, 252639372, -1019404436, 362689535], [-882209870493, 44210241620, 6603793997, 883634665, -3401106486, 604237288, -1780077185, -446977974], [678147599984, -50800905411, 2332761253, -1473365571, 1216507451, -121956132, 338110402, 1228082619], [-1260864230474, 23993625860, -4609572355, -974481492, 1963029890, 652846076, 926134693, 315249711], [674025046845, -4444914346, 6654300874, 1650766655, 1094575234, -12335024, 663928418, -255875207], [3194366503902, 235719340004, 212106229, -990925172, 143433282, -1113634268, 1017553205, -203061293], [-2936109932473, 322644235820, 3360223901, -65088381, -798207294, -689831480, 982383579, 243169592], [-1903873465382, -7605888636, -3893702390, 2343944822, -2069724080, -28379184, -147435126, 668732], [3950220435241, -161638734584, -7452514569, 3213909573, -1749546614, -568215112, -486354507, -599152560], [5180276359255, -211330179610, -11486095098, -944856835, 3098200842, -223802040, -2151449254, -576765473], [4091942952723, 978225396, -3280391081, -2775066987, 1595914886, -322875648, -1576075703, 1176763596], [-960545705563, 54456215389, 2440667661, 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[1938300299015388, -35861208035120, -1001278567922, 151795901198, -442967317724, -178202743240, 162880227810, 199368933788], [-913976568747985, 16293546625749, 1563876913021, -679579248000, -878088833317, 2657058264, -365591133192, -141484293136], [778924346033572, -7651524649380, 1438334672694, -35776357792, -2117439042136, -107260033796, -5356910662, -517374317138], [-734115754653013, -65362195583844, 1294461796934, -700232933479, -276045796456, -1075860028, 735352455788, 1738386761], [718982307862164, 8770440712988, 462076177330, 800699341682, -829587987864, 561028711032, 81655053298, -407254369048], [-777052988234980, 69329713455205, -1646898331977, -392698492261, 263452919343, 77448589716, -848563012144, 308910592963], [-1260331193593883, -11062460478536, 206202877821, 989383241491, -1973849795090, 450004449176, -442925465825, -426964512892], [1559529867462463, 63916452049227, 3427202094720, 173741727948, -1408754506541, 359302069324, 448773720833, 101468391747], [-2950563287906250, 86497472115538, 1647160138962, 341395817466, 1236183959918, 32096927248, 87824544036, -32833834506], [-135998370509994, 48630451465382, -1907535876562, -1293638358080, 411261421066, 171450260400, 84570838408, -92994978424], [2484213913496199, -78011271869470, -3125492936129, 59749641577, 420127608100, 169959749200, 196836884323, -175475586442], [1636560775545217, -59629485446379, 1114784273674, 113426341262, -221457032655, -266670012592, 163410175261, -537553842535], [1795290285256142, 79807055451747, -682837682810, 1070722032865, -693165005545, -97909976568, -335228023583, -60989467712], [1731225160112309, -56486305439174, -548017765341, 356230888705, -963455396588, -198676010752, -321198088481, -127551234566], [2451079888448388, 39024295345338, -5741208480178, -254546220568, 320071425118, 140280408400, 88336338372, -231002215204], [-2703111564235041, -8276094274296, 1163796625799, 294743017959, 704728334314, -169929660360, 356305626393, -271680732130], [-147481397910237, -64600820629402, -1463331033603, -162035484517, -622370374828, -6647587772, -215361534079, -31811092014], [408749851343569, -31733171377660, -92380970142, -550449029167, 2202617455088, -263911118888, -59920485352, 619993926173], [-2867591035928415, -44132105052232, 2881887411750, -704159680265, -224053653212, -242989058756, 23755150828, -33343707761], [165826427786864, 17935490573941, -4102376177034, -636562381151, 1407878074921, 94117082024, 154398472423, 307627741738]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_171_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_171_10_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_171_10_a_e(:prec:=8)
    chi := MakeCharacter_171_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_171_10_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_171_10_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_171_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![-500910592, -2093127296, -1053152416, 26549304, 3405936, -42966, -3320, 17, 1]>],Snew);
    return Vf;
end function;