
// Make newform 115.6.a.e in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_115_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_115_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_115_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 := [4039776, 161487612, -601032676, -59034365, 63665212, 3889937, -2392947, -99023, 41059, 1059, -329, -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], [-56, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-78837120671241253152, -9973540126581890998868, 2310607254078894757376, 814125703926345855447, -202392023883377112555, -22287502440199711304, 5487401419700630721, 285069043201750828, -57123507018210037, -1747669161726452, 201360999557039, 4105114832381], [423876864099055285152, -5161906837584838387612, -77542357985677360256, 500704883569605868413, 6356297675972836135, -16690673740108037016, -234291308372143221, 247495996934561892, 3091513118585657, -1703399413180668, -13077055877179, 4485970522399], [3805798205999910690336, -5586869010975213647756, -596275883222611175168, 547032575104081637649, 7815586069326361475, -16755529475994001848, 233030772691900247, 207749222095847636, -5012916412515619, -1022751051852844, 24571239904313, 1306807350427], [-1476103745153076797472, -982073169755037199028, -45200787875631820224, -256543429236019693313, 17077783140135571085, 16858476224499811256, -480649728622820839, -375325923182222132, 3904511635700883, 3580709282032108, -7270386628201, -12427339108299], [-432995753145411746784, 57171008563448559174964, -97234094359407843968, -5650802509660139356031, 53810523898777793395, 185936117773058775432, -2036619212814702553, -2636716769618199884, 26652972873454701, 16486972979012436, -115989961322647, -36432839572533], [5045513173477993758048, -17781162815309080507588, -216490431747785696384, 2036611790913507510027, -14131513763130468815, -79367663917109370024, 527243880332643501, 1380792393561619068, -5635141008118417, -11007486445446372, 19321474848899, 32723488673401], [11146178496316896, -3405308342741499646196, -304748304232215437568, 360276134121258431759, 22973417002742903485, -13565781930455928968, -572775246934296503, 236455618552698636, 5885813270714211, -1935818227653524, -21549568159577, 6009477378277], [5965699480149219172128, -43811409151598334178028, -4803202769895274402944, 4800299988420576521817, 409096494568307342075, -193871900239432404664, -11928544374514462449, 3604570380833758228, 137552082893511653, -31094732549142732, -543196049549311, 100656069415571], [-8766257679568538761248, 35650354015264008351308, 4607438223694890082304, -4361698825275638211937, -362228332936090527635, 194207505208215953144, 10498662300524534329, -3860230233329078708, -123560033384938573, 34849352286845612, 502501164538551, -116538806587371]];
    Rf_basisdens := [1, 1, 1, 49721745522234040320, 49721745522234040320, 12430436380558510080, 24860872761117020160, 49721745522234040320, 24860872761117020160, 3824749655556464640, 49721745522234040320, 49721745522234040320];
    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_115_a();"
function MakeCharacter_115_a()
    N := 115;
    order := 1;
    char_gens := [47, 51];
    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_115_a_Hecke(Kf)
    return MakeCharacter_115_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 := [[1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0], [12, 0, 0, 1, -2, 3, 0, -1, -1, -1, 0, 0], [-22, 1, 0, 4, -7, 1, 3, -1, 3, 1, -4, 0], [162, -79, -4, 6, 9, -1, 1, 5, 2, -1, 0, 6], [53, -31, -22, 1, -13, -4, -3, -2, -8, -5, 1, -5], [-529, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1954, 164, 16, 6, 90, 15, 25, 9, 4, 27, 2, -4], [1559, -51, -58, -23, 29, -3, -14, 1, -10, -6, 29, 9], [812, 33, -8, 29, -169, 6, -17, 4, 23, 4, 2, 32], [3909, 157, -46, -27, -355, -42, -11, -16, -23, 1, 59, 3], [911, 63, -14, 14, -43, 62, 34, 18, 12, 51, 33, 27], [-2695, 524, -94, -23, -338, 39, 30, -5, -81, -30, 9, 43], [2941, -46, 14, -69, -386, -50, -29, 12, 41, 11, 25, -67], [-2507, 1272, -126, 5, -112, -116, -65, -198, 27, -101, -185, -81], [1938, 1172, -184, 55, -48, -179, 0, -3, 129, -213, -40, 92], [-5696, -27, 388, -29, 667, 78, -3, 168, -77, 166, 12, 86], [13578, 1554, 484, 67, 254, 110, 69, 220, -107, 244, 220, -38], [4940, -1875, 48, 115, 605, -55, 92, -87, 295, 119, -220, -108], [-9731, -1549, -10, 136, 397, 44, 90, 72, -182, -121, 211, -43], [8503, -2018, 1070, -105, 514, -108, 153, -74, -57, -79, -63, 53], [-4729, 3217, 470, 158, 179, 270, 338, 50, 110, 201, 271, -87], [-7906, 2838, -160, -361, 1624, 485, -144, -151, -457, 145, 376, -40], [51813, 8800, -2, -205, 1276, -96, -267, -278, -123, 301, -15, -471], [-11549, -1347, -1510, -51, -833, -29, 234, -369, -269, -466, -367, -105], [-1656, 8093, 428, -111, 1587, 402, 379, 440, -325, 682, 584, -150], [-3529, -739, -466, 121, 2525, 120, -141, 366, 416, -251, 417, 197], [12188, -3329, -752, 191, 1981, -58, -787, -448, -353, -356, -202, 684], [-21723, 9486, 2258, 219, 860, -313, 124, -457, 319, -898, -963, -57], [96308, 27, -1692, -23, 1291, -689, 164, -1, 289, -329, -618, -86], [16796, 8008, -768, -435, 6280, 346, 353, -276, 42, -64, 252, 280], [-30453, -18995, -1398, -866, -3255, -257, -33, -99, 186, 504, -169, -325], [87220, 3494, 332, 305, 948, 284, 265, 86, 392, 796, -480, -14], [180221, -18772, 1066, -802, 5788, 1024, -128, -436, -342, -505, 121, 247], [54202, -23613, 1836, -301, 5253, 976, 1051, 630, -1037, 1132, 2092, 1086], [23729, -4958, 26, 518, -11006, 991, 451, -219, 731, 388, -353, -105], [-16944, -3664, -3192, -1864, -1772, -1004, -396, 596, 884, -472, 32, 320], [7623, -4109, -3474, -981, -1433, 1369, -570, 1369, -2805, 772, 1607, 41], [102880, -18071, -36, -391, -2587, 435, 2024, 835, -1287, -1121, 94, 874], [91817, -16665, -2726, 1361, 12623, -150, -95, -628, -572, 597, -1141, 103], [17164, 7998, -2600, 1292, -5682, 1812, -380, 420, 736, -316, 1208, 1524], [12264, 13959, 4584, 49, 8887, -743, 10, -979, 1653, -137, -2096, -1084], [1489, 4570, 9270, 664, 1826, 1904, -1466, -1848, -162, -619, -561, 85], [-168115, -35427, -4474, -206, 331, -70, -1546, -2314, 994, 945, -745, -1667], [110387, -4262, 11534, 1027, 9542, 664, 1397, 1610, 1451, 1993, -3, -1883], [-96702, 3848, 10276, -966, -12624, 1688, 2074, 284, 374, -514, 366, -822], [-46531, -727, -5874, 312, -2133, -2602, -1796, -306, -212, -2339, 1527, -1939], [19413, 10527, -646, 490, 3045, -2692, 1228, 3332, 1664, -1887, -447, -2113], [-23264, -13163, -8768, 3603, -14635, -845, 274, 695, 2023, 633, -2060, 780], [122559, -25623, 2386, 1668, 24717, 279, 985, 473, 500, 976, 2075, 2079], [172338, 16872, 5000, -1080, 25364, -94, -4466, 526, 1514, 330, -948, 2308], [179762, 20380, -12576, -5581, 13564, 20, -3419, -514, -4938, -4838, 1584, -2840], [-343754, 18475, -9428, -105, 499, -2497, 1214, -3005, -181, -1917, -5218, -2934], [-14679, -62178, 5418, -1652, -11030, -2269, -1635, 1717, -786, -1916, -1733, 703], [337756, -47212, -11100, -124, 7526, -1489, 3219, -1883, 4810, -955, -7160, -4110], [158685, -3082, 10350, -1134, -19842, 1200, 2514, -1780, 3301, 1771, -5763, -4931], [-463302, 31603, -15188, 3149, -12413, 2341, 2734, 417, 227, -3299, -506, 1374], [166985, -32911, 8586, 3606, -17741, 3042, 1266, 582, -2780, 1155, 1713, -1017], [-52337, 86938, -5274, -7649, 12390, 1646, -329, 988, -5829, -2537, 4629, 2721], [437977, -48950, 12366, -3641, 2574, -920, -3483, -1878, 1857, 4221, 3689, -4051], [13313, 116441, 4118, -2799, -537, -373, -4190, -2361, -203, 2392, 4683, 5189], [-271346, 69197, -17848, 1209, -20903, -559, -10922, -5467, -5613, -1057, 1260, -84], [-112669, 18020, 310, -7020, 21532, 2908, -3916, 3316, -3391, 2407, 14529, 4041], [569446, -12274, -8572, -8121, 34162, -3738, -1301, -3908, -1412, -586, -2158, -4682], [-319567, 92085, 430, 1184, 28753, -6613, 3317, 429, -576, -834, 4409, -3175], [-685747, 70745, -29346, -1717, 52303, 2232, -5145, -6594, -2462, 4865, 3303, 3673], [-35509, 85285, -10102, -6571, -16301, -3195, -3252, -4419, -9251, 5684, 9105, -1181], [82728, 29442, 13308, -5508, 13376, -5451, 941, 8679, -3926, -9645, 292, 2326], [-169976, 9443, -39272, 6039, 21135, -1237, 2518, -129, 12223, 97, -13512, -2872], [155597, 66403, 15398, 8324, 7957, -11114, -4248, -834, 7828, 925, -6385, 5701], [-1101227, 134290, 5362, -3367, -7994, -4622, -5515, 644, 4783, 3785, 507, 2427], [-769060, -210866, 9668, -5618, -32906, 4516, 4326, -3952, -808, 8434, 7982, -4074], [-497837, 151087, 25474, 3717, 26607, 4913, 2848, 677, -9931, 1586, -2247, -633], [240158, -206113, 10488, 4615, 39793, -4, 6439, 7162, -2225, -5978, -5682, 420], [737227, -129999, -2130, 1611, -80713, -581, 260, -4829, 4983, 8680, -2937, -4011], [298059, 202988, -18110, 7092, -32604, 334, -14732, 5370, 598, 5417, 4225, 1583], [-326204, -30182, -7220, 5814, -14762, 140, 2366, -4720, 6460, -8878, -21766, -2206], [1637032, -84954, -3940, 6473, 32406, 447, -4116, 6951, 11784, -2533, -17214, -3410], [-446155, -228453, 51046, 1512, 49861, -2726, -9532, -5798, -4786, 3301, -1125, 4605], [11703, 111681, 26702, -3039, -114847, -6748, -10649, -11334, -9660, -5975, 5769, 6725], [-430863, -114679, 28478, -13388, -24393, 2810, 6800, 1298, 18024, 12933, -1225, -12647], [-2167433, 103664, -3754, -15346, 16308, -12462, -6184, -7486, 1485, 2961, -23003, -6735], [-536182, 28747, -62684, 4856, -33525, 465, -7085, -193, -469, -11217, 12566, -2402], [130841, -8726, -1294, 11512, 123994, -15775, -9769, -10153, 7387, -3198, -3217, 6343], [847867, 111992, 5202, 9726, -35222, -2212, 6908, 3092, 9583, -8983, -5251, 17159], [318639, 185032, -23054, 14083, -81036, 6724, 14125, 8006, -3895, -6749, 13735, 6515], [330297, -386790, 33702, 11325, 54108, -2397, -5418, 9847, 4247, 4064, 2439, 24041], [72346, -6320, 79244, 9250, 24912, 16486, 804, -9570, -2124, -5208, 18290, 2678], [910597, 207270, -46670, 17639, 23054, -9326, 3671, 3392, 653, -12079, -22365, 2019], [-615542, 54806, -39420, 11168, 31398, 746, 506, 10214, 6822, -1784, 15794, -5586], [-1667181, 92676, 13838, 12363, 42846, 1439, 1644, 3927, 3351, 8416, -21017, -3639], [45575, -6967, -22198, -30582, 61877, 149, -10043, -2625, 7320, -17690, 14895, 5459], [985609, -164459, -30730, -4958, 107945, 4179, 2143, -13359, -6054, 5806, 4589, -14807], [-398623, 264506, -26598, 11166, -70056, -10756, 19048, 12116, -6721, 20513, 9921, -16881], [1777339, -241728, -73758, 6837, -83874, -1771, -6680, -15779, 8589, 1284, -495, 18583], [1741132, -36288, 20300, 42414, -115344, 14838, 5508, -1866, 1650, -4056, -42342, -54], [-1344637, 103939, -20606, -31738, 3201, -15000, -30272, -23176, -19522, -12723, -14047, -17885], [1423599, -248320, 83538, -6329, -74854, -5227, -15898, -7839, 1585, 14060, 3353, -12945], [-339651, 186476, -4486, -6958, -62644, 22663, -361, -14107, 24053, -10036, 2007, 4407], [77516, -526865, -91096, -6953, 45673, -19960, -3817, 11766, -8127, -1270, 1310, -35472], [-1272709, 571880, -23674, -20839, 244508, 5978, -19683, -676, -12753, 9403, 53737, 31889], [974184, -31582, -6708, -22404, -237290, -4176, -22484, -31328, 4756, -32158, -3262, -4886], [-2841799, -339987, -1322, 40519, 51573, -14025, 5420, 22587, 20831, -1022, 1483, 12429], [1318944, 107191, -47056, 44559, 141891, 5773, 12004, 25653, 7189, 30935, 20668, -2788], [-2573042, -409647, 56592, -46166, -2219, 33532, 18070, 43200, -3418, 37032, 53836, 3280], [1926478, 97846, -18624, -18620, -54130, -26748, 15796, 5404, -9658, 12972, 27608, 2196], [-211081, 226323, 116754, -4141, 118731, 29321, -13190, 13785, -30287, -2142, 33621, 40083], [2951966, 356886, -96944, -11173, -5490, 1055, 31458, -3917, 11006, 20645, -46532, -38328], [-818586, 214194, -92124, 35832, -109106, 15292, 19740, -7468, 24680, -8754, -70510, -1766], [718274, -121558, -67152, -17530, 5502, -3506, -19796, -40930, -55546, -10618, -13952, -33464], [1409389, 50484, 154134, 35600, 68090, 39023, 22825, -29011, 31738, -2432, -36387, 5113], [1508702, -57502, -106336, -31559, 91798, 8368, 8527, 6526, -35576, -3310, 37684, 480], [672251, -340649, 70794, -41408, -48983, 6020, -17414, 24904, -29510, -19031, 26265, 12867], [3590667, -132585, 7366, 18716, -19699, -37374, -31796, -20958, 30600, -9031, -51613, 10785], [-5733787, 211162, 47690, 38579, 43878, -24156, 14705, 5278, 30091, 16027, 223, 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[-118513149, 1008441, 1801706, -370831, -2948123, 529245, 1249000, 1352377, -594911, 2570038, 1059317, -1340149], [-102143316, 14564351, 7522180, -1286878, -6168251, -2115784, -1515364, 534280, 752507, 553262, 1120292, -113270], [103639281, -18580270, -3973114, 351731, -2047648, -346229, -525872, -234037, -148445, -782476, -2066573, 1521061], [-166788026, 2409782, -515192, 1037163, 1690150, -70426, -978897, 853748, 1316246, -1481404, -1907408, 352252], [54619626, -7237702, 8132596, -1305750, -1348954, 695108, -552886, -181648, -1302628, -178366, 1740218, 564386], [-101126763, 428324, 412422, 344048, 4161726, 938909, 1290559, 607367, 1857626, -647162, -455727, -91607], [58893025, 766883, -728034, 39245, -522101, -454606, -1655627, -2300044, -624630, -762233, -1563615, -1388099], [185571336, -15041118, 795292, 852565, -3599370, -1315079, -422578, -263955, -743953, 146013, -707970, 272970], [-338347087, 31120933, 467618, 19946, 14536219, -1360194, 44182, 76394, 1150308, 67695, -715267, 188191], [185368270, -16345076, 1727548, 772888, -7179732, 1037420, 2189644, 879180, -405602, 1197470, 816438, -706922], [163338784, 15152476, 5078392, -821294, -1239964, 1244420, -911818, 469912, -518544, 2557336, 2276936, -706204], [155607842, -6387707, 6808588, 1421045, -6435815, 1459167, 1558092, -615113, 1556655, 452311, -1500318, -535566], [59698732, -15773639, -6510672, -89487, -2571183, -774461, -174148, -768933, 1028003, -1169455, -2354400, 1960288], [12793987, -8986751, -3963150, -142188, -16075797, -1712136, -1981822, -1707548, -205898, -3112675, -1179239, 275931], [-78116125, 343504, -4427270, 1550320, 1863934, -1060952, -968342, 1665948, 543935, -1010007, -1418807, -98081], [-78071228, 6880575, 3358580, -1328889, 5614387, -2240657, -151842, -1503477, 99409, -1619437, 470810, 37854], [244041948, -5904108, -2703552, -451311, 2119080, 693608, -1256871, -735946, -1391405, -105076, 1322194, -483380]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_115_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_115_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_115_6_a_e(:prec:=12)
    chi := MakeCharacter_115_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_115_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_115_6_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_115_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-429434688, 958286464, -394046000, -185549448, 46692536, 10198454, -1917514, -227565, 35620, 2231, -307, -8, 1]>],Snew);
    return Vf;
end function;