
// Make newform 825.6.a.m in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_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_825_6_a_m();".
// 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_825_6_a_m();".
// 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 := [92576, -51316, -13444, 5771, 358, -152, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-44, 1, 1, 0, 0, 0, 0], [464, 2642, -57, -123, 1, 1, 0], [21136, -5642, -1275, 171, 11, -1, 0], [14048, -9276, -3716, 351, 133, -3, -1], [-37040, -2754, 5315, 66, -148, 0, 1]];
    Rf_basisdens := [1, 1, 1, 48, 192, 192, 192];
    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_825_a();"
function MakeCharacter_825_a()
    N := 825;
    order := 1;
    char_gens := [551, 727, 376];
    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_825_a_Hecke(Kf)
    return MakeCharacter_825_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 := [[-1, -1, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-11, 4, 1, 2, 1, 0, 0], [121, 0, 0, 0, 0, 0, 0], [4, 42, -2, -1, -2, 1, 5], [-139, -11, -15, -5, 9, 0, 0], [-539, -23, 5, 2, 3, -9, 13], [-107, 111, -10, 11, -12, 36, -14], [-364, 21, -50, -30, 6, -19, -43], [-2434, -142, 4, 43, 4, 1, 37], [-3151, -580, 46, -49, 32, -17, 9], [-2467, 715, -167, -57, -51, 110, -38], [456, 256, 292, -55, 42, -43, 59], [-995, -88, -214, -63, -12, -27, 79], [-3152, 756, -70, -6, -278, -62, 186], [423, -1028, -472, -33, -66, -241, -51], [-3718, -563, -242, -199, -106, 60, -352], [2946, 129, 554, 109, 226, -252, -352], [-24285, -3429, -338, 99, -16, -32, 378], [-10924, -950, 266, -374, -546, 92, -552], [-26911, 241, -357, 517, 111, 70, 590], [6948, -789, 262, -1102, -596, -301, -1587], [-21606, 405, -58, -36, 348, -149, -435], [7877, 2782, -1480, -530, -664, -504, -1120], [-24241, -7166, 87, -409, -601, 211, 7], [-38872, 2935, 554, 768, 274, 465, 1321], [29048, 12427, -848, 670, 54, 363, -83], [-56144, 2272, -732, 161, -8, 225, -791], [-29084, 8586, -666, -422, 1374, -980, 436], [8429, 11401, 331, 645, -3241, 438, 1598], [9872, -12223, 858, -774, -652, 1533, -317], [46824, 4797, 2910, -3424, 692, 119, -3143], [2706, 17885, 3408, 1102, 570, -1639, 1431], [-118125, 21431, 3841, 825, 217, 344, 1092], [-31354, -8145, 5160, 1347, 308, 1448, 264], [45468, 2853, -4352, -2313, 904, -2146, 2314], [-85478, -2579, 1454, -81, -70, -358, 538], [94620, -5812, 2162, 20, 806, -852, 2456], [-151427, 2332, 2069, -147, 3837, -1501, 4039], [-219821, 5904, 9060, 221, 6022, 825, 1507], [12947, -22443, 1394, -1962, -4444, -1375, -2641], [-111085, -50277, -4050, -4469, -3048, 2128, -2926], [128846, 3933, 11092, 567, 4652, -3922, -3814], [34621, -32238, -13271, 3333, -1911, 173, 145], [-188980, 36124, 7618, 2355, 2506, -1757, 3759], [111682, 17227, -3720, -1985, -5188, -2112, -320], [-13106, -32753, 6386, -955, -8766, 1078, -2478], [-41116, -92231, 3566, 5380, -4780, -3329, 673], [-478497, 18913, 978, 9230, 4200, 2837, -801], [-149141, -2843, -3457, -5477, -11389, -740, 3432], [-511664, -50610, 4960, 3066, 2804, 668, 1232], [255328, 53367, 6714, -3147, -2458, 3926, 3230], [-271348, 6646, -17992, 2250, 5516, 3956, 11832], [139016, -18053, 18268, 2768, 13486, 3853, 2331], [5126, 32690, 3860, 14248, 10324, 12974, 3854], [-417750, -22538, 16748, 5864, 9940, 8378, 6882], [-262451, -38183, 1203, -3603, 14143, -10050, -3258], [330126, -52243, -13318, -13923, -4822, 8360, 452], [-159967, -21137, -1055, 9497, -10279, 1700, -4140], [167467, -107328, 21613, 10706, 3021, 6728, 1204], [-609567, 62717, 9963, -2629, 9907, 2888, 5268], [-215234, 10229, -17224, 12625, 2260, 16992, 976], [-43222, -142743, 4884, 3588, 2444, -185, -21665], [-779369, -197475, 9140, 994, -13202, -3779, -5909], [-227936, -107160, -39546, 12882, 5494, 18638, 38], [-346848, 216822, 13620, -5540, -12028, 3430, -8266], [570586, -52565, 8902, -10819, 17154, -17614, 5514], [-460640, -248462, 33926, 5616, 22802, 530, 3134], [-34576, -34289, 1566, 8722, -30450, 9139, -4885], [-1555702, -65649, 19134, 5168, -1848, -19819, -1693], [-689218, 144728, -12446, 22230, 4930, -8046, 36426], [-496918, 22208, -22992, 8891, -23104, 3763, 21527], [-461076, 26753, -57080, 13273, -4544, 26412, 17984], [257652, 157623, 18464, 21009, 9656, 20238, -15134], [444578, 13873, 25336, -19680, -19472, -13085, -30037], [-372658, -5298, 5262, -56636, -17490, -12454, -29862], [672116, -54435, 10180, 5031, -22132, -10034, -27974], [729526, -264575, 28138, -14320, 24752, -8613, -15699], [283910, 157583, 51206, 1123, 44286, -20084, 29904], [189441, 78128, 5356, 14229, -8082, 16033, -10293], [-2459897, -79486, -30626, 46589, -5008, 16847, 8753], [1390382, -188270, 81882, 1288, -10018, 5834, -26826], [-552418, -87706, -37996, 46145, 59950, 42967, 28505], [622267, 59068, 68391, 27026, 29643, -28076, -29400], [1133473, -171060, 60292, 24615, 21590, -35081, -1559], [-833546, 18063, 15148, -21202, 1450, -40049, 5817], [1833490, -6064, -778, -21260, -3594, 14680, -29232], [1488058, -146218, 76666, 3476, 47986, -6270, 61506], [534100, -514849, -35032, -8048, 39364, -39227, -11191], [-1286288, 554070, -20808, 32210, -20752, 11268, 7716], [-953440, 433338, -16918, 21812, 7878, 16022, -24470], [540806, 611710, -14790, -26361, -25802, -6211, -10571], [117720, -198459, 11482, -5118, -65380, 67201, -11073], [-559522, -3903, -74124, 47673, -36716, 15708, 62504], [535272, -61746, -44724, 17754, -45552, 36052, 69256], [-517296, -81580, -63078, 7348, -11926, 15132, -836], [1165996, -359875, -129282, 2450, -67120, 45953, 831], [2652307, -61479, -110325, 13702, -23799, 18935, 31693], [-1238936, -491960, 134024, 30433, 7988, -62527, -48079], [158769, 96172, -185637, 18121, 42173, 1851, 26425], [-5236196, 421233, -78950, -4976, -57894, 1671, 15343], [2709250, 238258, -6076, 40084, 6960, 33482, 39054], [513827, 405265, -3049, 14569, 66703, 51126, 117238], [-1295472, -593926, -83820, -48551, -36610, 28171, -37635], [217801, 21909, -160260, -30936, 2562, -70701, 19197], [-415035, 106628, 142228, -21473, 81482, 4207, -55491], [3552114, -326260, -45682, -60160, 20438, -81912, 9416], [57679, 458244, -126078, -21393, -42816, 18175, -102115], [-1035410, -125941, 70646, -19491, 73182, 34426, 21654], [1734284, -399186, 151948, 48068, 59652, 3294, 1230], [7254068, -457516, -1656, -91694, -21824, -39954, -119874], [-3143680, -539967, 45196, -63220, -182, -54941, -21851], [2804856, 111197, 6558, 75895, 142318, 1034, 104774], [-2655132, -506799, 51650, 12053, -42466, -52868, -17316], [-6055550, -689079, -70912, 40824, 48358, -18657, 28313], [8705550, -50340, -112114, 39014, 11050, -15738, 39898], [-3900639, -884284, -12366, 106197, -11876, 31093, -44225], [-2861850, -42926, -79596, 24822, -60020, -34692, 50044], [1443630, 1391261, 90726, 41406, -84760, 23965, 77691], [-5496731, -229190, 18006, -88433, -142056, -23343, -18837], [3664124, -721766, -61872, 2678, -3360, -58556, -18484], [-7093772, -580647, 266290, 95162, 45258, 29433, -56367], [-3839237, -342294, -173900, -36795, 57326, 69295, 53473], [-6068760, 646788, -42772, -74300, 22928, 70464, 25836], [-347415, 889130, -243615, -136651, -7695, 47757, -23431], [1967859, 674595, -282646, -8934, -131396, 10463, 65725], [-838172, 416622, -156060, -55506, -30732, 63208, 141776], [1129628, -676270, 225072, 130075, 167172, 99737, 217881], [3437962, 208171, 147970, 189064, 126670, -42555, 143305], [720807, -2005243, -34033, 58203, -30429, -65924, -73000], [3138664, 336528, -132686, 11262, 108886, 75538, 22406], [-3036410, 224865, 353172, 31574, -43780, 125129, -46935], [448092, -2152249, -155912, 9389, -11148, 68978, 52018], [-2700382, 1316892, 214710, 7676, -69730, -64540, 135924], [6302034, -338479, 287620, -30571, 67576, 99280, 1480], [-6227952, 236604, 230972, 101302, 50940, 118594, -37462], [7644249, -523138, -110981, 35140, -116317, -18424, 12], [-2589654, 1372608, 39574, -212822, 66970, 46062, -6670], [2246711, 57991, 252803, -48393, -85453, -86074, -119586], [-5242029, -946834, -48543, -126248, -12375, -235264, -163860], [-7901816, -1010473, -43138, -34472, 69718, -112007, -265719], [11011626, -818345, 348726, 2861, -116170, 34398, 144186], [-452882, -1255883, 347936, 94788, -35962, 91135, -122279], [6928346, -484812, -233336, -54196, -110964, 246784, 155340], [527732, -246318, 244832, -162762, -94552, -148060, -144140], [9235936, -747585, -95514, 107609, 25002, 26418, -114470], [-3648927, -742161, 182941, -92589, -80863, -144274, 63706], [883164, -2047028, -165974, -114608, -107546, -32792, -271452], [10188878, 982031, 53276, 52938, 64308, 79071, -54625], [-1845870, -910838, -177514, -47093, -25886, 210493, -26875], [-5419550, 2417709, 38312, -17180, 37046, -96825, -143559], [-396080, 1166015, 241646, 285903, 138614, 39208, 341180], [-3418320, 2633022, -48082, -31674, 10718, -72924, -75012], [9093762, 1610392, -143266, 184566, -6066, 82146, 150546], [-6929527, 3263430, -425570, -29577, -250888, -137187, -188417], [-15773113, 425144, -469081, -72384, -118445, 238574, 26506], [-12858934, 2583657, -183292, -193470, -16562, -62567, -129113], [51564, -2209847, -394368, 97061, -88736, 77096, 10596], [-6026483, 1811135, -352637, 257829, -247469, 171032, -59996], [-1339859, -807162, 257568, -125517, -11854, 108993, 68147], [3559233, 792095, 618233, -214625, 4641, -140518, -225806], [18958032, 3228192, -767320, 1940, -101940, 9476, 63304], [2295697, 2453512, -12936, -295373, 172462, -46137, 184117], [-7467914, -1397658, -676370, -51164, 271826, 12290, -72746], [-3008509, 673173, 695018, 435191, 3420, -141774, 368732], [-11565250, 2103393, 197190, 6319, 68454, -150018, -274894], [5467758, 4343124, 95888, 66252, -69068, -169176, -217932], [-8187070, 109045, -193076, -68297, -309484, -314730, -150078], [-7190772, 1590024, -248352, 122866, -95648, 276970, 94754], [2884352, 3223165, 205838, 58734, -65160, -108323, 119299], [-12993304, 1559749, -45704, 247077, 287160, -76244, 371752], [-3228670, 286178, -252622, -141258, 134246, 11368, -169700], [-5888272, 1093149, -830716, -261157, -29796, -283550, 20750], [-19894762, 1116730, 628422, -236015, -178926, -217709, -104413], [-8096530, 1533977, -331160, -128444, -97738, -4229, -166435], [2099485, 470003, -275219, 21034, -31401, 234645, 213903], [-14999782, 575554, 171814, -130882, 242758, 217600, 190536], [3757199, 2281910, -458537, 158104, -63129, 291264, -183088], [4831174, -2886664, 296240, -389699, 113592, -132131, -481903], [18919174, 682836, 11022, 44693, 92234, 212373, 127541], [-1904494, 610802, -275150, -216298, -129862, 14920, -416000], [3159411, 3278171, -326522, 163416, -208812, 518541, -231737], [-5052746, 1317334, 91674, -107702, -2574, -137944, -82976], [-7951800, 3167964, -117844, 242568, -66236, -200468, -229468], [-13218610, -3447072, 745366, 46484, -47306, -4864, -321904], [-7210202, 1733150, -114976, -184727, 319800, -294549, -122449], [-12078746, -4844939, 198184, -626405, -42016, -336062, -340226], [15050068, 1293487, -581312, 153781, 9784, -9622, 416078], [-8378006, 605938, -159818, 655946, 222962, 277988, 340208], [6316474, -974750, 96196, 283031, 510930, -78823, -100977], [15789418, 2012977, -1041804, -508794, -26570, 133365, -483157], [-407888, -3878181, 474648, 198825, -55764, -321270, 97298], [-2394650, 4594046, 531482, 36618, -84466, 251360, -689612], [13892158, 1220057, -211750, -657458, -46596, -297607, -581737], [-4207348, 5205365, 16590, -48468, -341308, -162085, 28173], [-8652232, -766782, -718134, -994430, -188478, -182988, -283492], [-859154, 3125564, 251256, 391532, 286196, -231176, 268012], [-20087343, 8255449, -239471, -315447, -142567, 11354, -588174], [21338208, 2118072, 810320, -344142, -238616, 140106, -2774], [9636041, 2812350, 239853, 202243, 134893, 203495, 76931], [-8997884, -1737562, 429356, 549712, 380960, 355962, -224002], [-14443965, 6536212, -270202, -228999, -305032, -491023, -30045], [-8083800, -6095438, 524924, -272678, 28820, -283720, 173944], [-5550145, 5866006, 395512, 156269, 648818, -410797, 621813], [-40617586, -2321896, -296410, 317106, -405678, 395118, -175798], [-37947432, 214637, 149996, 33747, -318204, 145602, 144430], [734208, 5072533, -22090, 445066, 578040, -150399, 477855], [12488447, -1027459, 51975, 891525, 311015, 439134, 204670], [3445438, 4520301, -153704, 524431, 788676, -12138, 503534], [7344780, 1757845, 469668, 659641, 74592, 130720, 351568], [-21045600, 1334394, 46950, -330242, -99922, -187664, -45680], [7511756, 1141877, -1163892, -392725, -472196, -112270, -456506], [7714603, -7097992, 1504892, 711407, 156494, 141675, 233565], [-41722426, -664660, 411676, 4810, 359336, -594138, -10382], [-19835091, 3593173, -1268704, -897533, -335528, -77830, 163910], [-22366647, 4863288, 25249, -173934, -583747, -311364, -85496], [-4534499, 73535, -587759, -107171, 73765, 139304, 345012], [-503419, -4371809, -108406, -284441, -107976, 606888, 383318], [-15113654, -2110490, 108854, 228784, -204686, 113718, -76230], [-9637952, -5502393, 40058, -550675, 496510, -351994, -337890], [-15716286, -7518931, -625212, 589563, -39604, -102246, 1236030], [20092466, 4325143, -631872, 659876, 221426, 442141, -95669], [33883538, 3691540, 262740, 534842, 414048, 172174, 1349690], [-42232862, -5218950, -1357530, -212490, 723558, 274280, 272856], [1564662, 2237662, -72014, -308747, -283546, -308213, -160621], [-23208896, 7501340, -618248, -104294, 67436, 198070, -674294], [4850895, -2689341, -518822, -616253, -430772, -57416, 226386], [-23996123, 636490, 352197, -374588, -913071, -144500, -192256], [-25477150, 561133, -1747346, 232694, 529544, 350501, -432837], [27753243, -2572441, 1215614, 1437748, 141540, -219851, 744119], [17897173, 3695914, -188629, -516819, 73009, -340055, -774693], [53405197, -2375552, 814777, -104794, -769687, -105760, -450168], [-15255557, -8045849, 1066535, 282689, 692407, -278136, 906440], [-4760812, -7642350, 204366, 485808, 119462, 562602, -444550], [5574864, -10591822, 490114, -503118, 514926, 178268, -59192], [17942949, -10333521, -338608, 234655, 779828, -58028, 430660], [10512363, 3881190, -1354517, 81025, 991875, -76719, 12981], [-28300323, -4014696, 785776, -805441, 245618, -68501, 302653], [6259229, 2773842, -292346, -372025, 257144, -499271, 115539], [28371283, 2174462, 1139004, -102707, -516414, -774733, -457671], [17323503, 3172756, -1241887, 492062, -113047, -113560, -75972], [20610656, -1865280, 2504482, -742127, 565322, -210723, -704095], [-33833161, 575595, 202530, 1057508, 143316, 995681, 663447], [47415607, 2184175, 756327, -426713, -39825, -360346, 246406], [63753844, 2165786, 181734, 233334, 192406, -75272, 329360], [-13678680, 12378986, 83742, -626043, -246538, 58791, -662741], [50057303, 2935732, 1933016, -573589, -76766, 240307, -1401587], [-35082760, -1181938, -693492, 211487, 1062602, 142357, 199099], [-10207888, 5819248, 146664, 1338812, -202612, 672500, 1150016], [-39398416, -3483713, 1433396, -481454, 354336, -137953, 122283], [-19413575, -5949261, -1878155, 299685, 284133, -482932, -278708], [-2458286, -9524065, -1453836, -414844, 222964, -201191, -479799], [-18288858, 1704868, 1966280, -500883, -267554, 659917, -701293], [175976, 1367443, -3503730, 353988, 253406, -689623, 134545], [6237200, -8128377, -2436032, -927316, -364462, 593365, -327885], [49812357, -2129723, -467230, 1348652, 83996, 204935, 639861], [-20054264, 2337056, -2071780, -651808, -437092, 640184, -1164008], [18584854, -14442654, 1741406, 954278, 484078, -61016, -290624], [-30302760, 3578282, 2725764, -129986, -22096, -841956, -302232], [26507209, -1578532, -3724775, -270754, -850459, 866860, 650192], [14800372, -3108991, -697556, -448198, -400098, 667209, -977729], [-7574794, -14998031, 147814, 560144, -637966, 361927, 175755], [1320190, -2726326, 2854966, 356091, 765426, -1401875, -195723], [59837120, -7363141, -1754268, -1163566, 18138, -155001, -1354583], [5542494, 7905838, -1401368, 851522, -1290672, 224460, -226852], [33822270, -6115878, 660532, 272266, 1505152, -973520, 1221180], [41291340, -4525717, 373368, -336448, -1646494, 657461, 702251], [81854121, 289094, -917577, 362409, -1187795, 499225, -972237], [-50593017, 8399853, -611711, -57953, 1363617, 362856, -721140], [37495423, 189991, 231086, 565560, 511196, 195881, -692397], [27280104, -7942592, 3502162, 1760560, -78382, 294508, 1053780], [5396712, 2615153, 188384, 1423682, 630546, 1047769, -287873], [-152769617, 352510, -1529037, -342686, -706009, 422658, 492550], [-45614964, 8879085, -2402092, -862893, -1047736, 887218, -583182], [13288842, -5354378, 598368, -648307, 302616, -97321, -633221], [280171, 2889574, 221628, 1484467, 52954, 51505, 33671], [25375988, -2185323, -1867730, 45891, 849250, -1092274, 902702], [115961188, 1338695, 1187542, -621821, -842890, -438364, 656208], [14889364, 8074222, -1798690, 801436, 617490, 1436434, -247450], [-81309598, 68504, -2375322, 390596, 747078, 49912, 1212296], [26639802, 2153981, 263858, 611986, 753958, 568133, 703657], [22449060, 7436976, 1125572, 1265404, 594104, -499316, 2426040], [12338342, -9384308, -148532, 649869, 459468, -880783, 190253], [-32168244, 6687762, 318404, -685601, -1540966, -119999, -873745], [-23153953, -12941505, -1392326, 603759, 931108, -822960, 2110418], [-16201283, -5445302, 1802080, -85410, 580128, -1543108, -104092], [-35698923, 11383308, -269987, 1181157, 1251413, -731757, 799343], [-45905512, 2515773, 62646, -1900568, -1581114, -1009677, -1749277], [174397973, 6586615, 791361, 1302977, 12833, 29012, 230300], [-68195364, 7822678, 2392280, 1137604, 295424, -1035258, 1285774], [17730819, -1208686, 2371392, -1955867, -663062, -1473857, -2309487], [-59826717, -7738975, 896419, -2201851, -701461, -685070, -2403670], [-40821960, -9350479, -1207002, 314236, -571956, 919151, 188993], [12483854, 811184, 43824, 109959, -1788848, 624015, -956045], [45364540, 4684441, -1440054, 1470166, 582530, -89659, 2044013], [61917152, -6186880, -616930, -1065495, 232430, -1414371, -1068407], [23949818, 59818, 4151454, 1499310, 215878, 541320, 301416], [-67754234, 17584763, -213982, -610352, 380788, -149247, -887785], [64096019, 12977246, 2226481, -592909, 832491, -673449, -197195], [28445562, -16273856, 1004048, -684196, 1714876, 99108, -575592], [-33801553, -3341704, -832285, 122483, 889403, 356169, 1730869], [-58520463, 7375510, -767895, 1099864, 1249865, 961492, 2278516], [2025876, -9706728, 2514500, -161104, -209892, -1711168, 1654328], [-35901800, 1860878, 933118, 2262464, -798818, 672862, 1654742], [-29860504, 2310189, -3544112, -649647, -1860596, 1261176, 1387432], [50066961, -3614654, -2317876, 993659, 279030, -454507, 672319], [69478833, -521681, 4087850, -1131628, 1403628, -1097523, 281043], [25943903, 5138297, -566746, 532875, 2928600, 1127546, 409972], [94643086, -388889, -304572, 1202240, 448268, -772923, -479219], [-117457982, -7264916, 2451876, -24836, 1097988, -1173936, 678128], [-39394864, -9161678, 613306, -886704, 61630, -1730438, 261974], [-54306327, -15933101, 890925, -1287531, 695485, -8204, -1649380], [5853864, -16101093, 453910, -683449, -336438, -3384820, -1597548], [109125202, 6271120, 1969376, -618628, 622216, -280332, -675604], [-18160190, 16739273, 1245256, 1360383, -642980, 2737402, -10310], [-37444293, 15928924, 1352092, 1801083, 791414, -328173, 2942317], [-138614438, -164436, 613002, -698408, -334902, -1828192, -62272], [81106271, -12871717, -188168, -705464, -450454, 2330189, -491557], [120740274, 21702814, 717254, 172478, 772834, -757868, 1153640], [30327215, -16468071, 2398371, 803615, 2551007, -889716, 3373064], [98929470, 5989958, -1855500, -1690271, 1052670, 630679, -863319], [20148059, -492403, -226987, 1561331, -736699, -602508, 251560], [-118687062, -14218424, 141828, -1457479, 657868, -1082047, -563307], [-81614762, -17412378, -1914490, -833258, 156542, -1557684, 1614660], [47957783, 15149901, -1384124, -524755, 2769920, -486468, -579636], [-73903324, 26003019, 3190188, 264799, 355168, -505880, 352904], [-72327368, 10886033, 1209146, 1455269, -1465366, 2674188, 1315328], [58525906, 3731452, 1221568, -1236028, -2469028, -218056, -742900], [123064105, -339426, -101847, -926641, -1691175, -1265573, -456681], [47748204, -34161794, -320104, -1217711, -79292, -116957, -2022317], [-69824670, -6467842, 968780, -66624, 622044, -1286, -2726822], [-104203597, 3517726, -116937, 1660378, -464657, 1273442, -1106278], [-231151288, -26173747, 3167126, -374040, -1208484, -696161, -581199], [115914565, -8479993, -2641031, -804770, -341397, -1235947, -848161], [70633691, 3811174, -4942992, 1944119, 641434, 1480101, 3672751], [-191666772, -19088245, 3224952, -1859667, -1739716, -2204690, -393338], [107001718, -13740890, -7360534, -1583640, -1745306, 1018670, 521706], [150134539, -22121195, 1522884, 624758, 612454, 740961, 1111543], [96126683, -28026669, 78889, 864691, 418985, 1540542, 2419586], [-37289283, -1800803, -3117574, 627764, 1150232, 2330183, 1877185], [160582054, 20265, -1284268, -457298, 606758, -163963, 2137403], [-29695186, 21102292, 1132970, -3409722, -1376854, -1229906, 988670], [-60559308, 25503248, -389352, 1546663, -771268, 1628335, 1272975], [-79969218, -17875610, -1803614, -2410134, 1572246, -17144, -1135604], [108814196, 6091420, 1700690, 3614882, -471666, 1379002, 1395614], [74587379, 16467425, 657806, 1404998, 2108260, 416213, -727585], [38011756, -21720821, 308536, -322235, -1698908, 491670, -3042746], [74226492, 36251808, -1842046, -2501586, -890014, -1377718, -3212782], [173793732, 7314810, -2917734, -4102, -329642, 362508, 12072], [54536235, -19516133, 1525471, 241440, 2335429, -1737473, 650725], [131807515, -13237746, 1861880, -2594011, -241654, -1490093, 551069], [16814378, 14204462, 2265956, 3150240, 1271820, -965558, 3420210], [110298497, -2866486, 3608037, -4444, 3590949, -2626972, 1363028], [88066416, -17284314, 954184, 266975, -1199284, 2397753, -904295], [-54555611, -11960637, 769045, 2405231, -1121615, 2993870, -1418534], [-253613513, 1962618, 2629314, -1119983, -922328, 124643, -399791], [37070686, 10144237, 1576090, 2569752, -2722928, 2184199, 813337], [-18422647, -10925097, -3612727, 524773, 1902725, 88656, -602132], [-134883706, 27244651, 1412924, 1518986, 2819396, 568723, -2259165], [-33042358, -18873286, 816976, 1104479, -3878096, -203435, 1044025], [78321305, 35206235, 3279201, -2025005, -1642747, -2490390, -642522], [178618253, -12882795, 5134578, -489747, 1250264, -269760, 2120382], [-40949242, -2184397, 6206400, 1059464, 1947362, -1447051, -451997], [308242384, -4569912, 1779392, 2844868, 1574128, 1303020, -177924], [61299759, -10749622, -4180528, -3384246, -2098368, -561656, -977520], [-56166555, -16177587, 3725063, -594817, -2375169, -1010388, -1753248], [-144370580, -13026198, -943488, -937612, -465596, -948766, 184550], [81257378, -11923325, 1700500, 1147057, 1591220, 315674, 91094], [-19932968, -15666622, -2463880, -889946, -815120, 835404, -3161188], [-64360378, 19811051, 2822490, -1071845, -506910, -3227440, 2889572], [-171457670, -10758760, -1835912, 2902608, -1878292, 1080040, -523900], [247725961, -3969803, -468413, -1441313, 2409691, -1124084, -1421712], [56638913, -7000283, 7307311, 2273409, 629839, 1560622, -1084470], [-146034749, -15526901, -2139986, 1417816, 2320832, 570829, 1964659], [25597628, -6137218, -4818512, 172670, 1330872, 832984, 3100824], [172080686, 4092124, 4261736, -3106336, 1964420, -2940156, 501656], [244031994, 28400321, 4927968, -1500037, -1827564, -1764530, 704846], [97387610, 10579916, -4132614, 3731274, 609150, -273470, 1461838], [311210478, -27545613, 2030042, 617525, 1744358, -761894, 3164962], [27991468, -4739416, 3811572, 5060628, 2316600, 502252, 188760], [122914200, 28522396, -1062154, -1402172, -61866, 977740, 2185540], [35147899, -18612321, 3064481, 3955707, 2452209, 3155114, 409486], [-22755754, 14216626, 4190698, -3601762, 1127498, -1729216, -166176], [44761496, -3366460, -3287868, -1555788, 2288428, 3129720, -1621344], [-123847729, 2803897, -4961858, 1006137, 1026456, 758096, -2321030], [-44218458, -41383720, -6366828, 766407, 972492, 1766575, -767493], [-264984843, 29611120, 2713314, 747507, 4668812, 1547967, 2055233], [-118730485, 54057734, 2721981, -2157805, -1031233, 2345671, -4375699], [120602608, -24131504, -2426756, -592972, -2360260, -2716204, -2688268], [249519103, -36194400, -1801012, -1893298, -2404560, -1337450, -2513430], [-53727680, -6881818, 7977056, 3488148, 1092132, 39150, 3427594], [-34936925, 9110120, -10827608, 420169, -1343362, -631027, 2209187], [146920600, 32524447, 2697340, 2651899, 3170312, -1345904, -389384], [-1802094, 17175897, -134692, 3215814, 780196, 1766041, 264281], [383959474, -22582371, -5109162, -2756447, -1103426, 373588, -1839712], [98607591, -12841276, -9831201, -743806, 294799, -2680356, 1297484], [155369458, -8552904, 4106578, -1986222, -2765770, -1463090, -2729030], [29429908, 22933365, 790166, -3008783, 303398, -2238900, 744880], [228851959, -22489760, -636957, -381649, 2962291, -4172235, 1354449], [-63259822, -53001491, 4819138, 16604, -653658, -651457, -2010885], [-41758418, 2820461, 3722786, -2020722, 607788, -2890539, 785707], [129719646, 29320191, -5967752, -770931, -3792428, -873502, -3034550], [-49713416, 6888981, 2681578, -3590162, -3673620, 2216221, -2355301], [46961849, -15664716, 856253, -1107029, -1054603, 2641753, -2684059], [-14986353, 1409769, -9435515, 3476751, 1538525, 2270628, 1862304], [244737092, 9547251, -7604572, -765847, -4579720, 528842, -1255070], [-76858234, 22452350, 9210, 1026610, 3717414, -1428416, 4577124], [-20051246, 9549287, 9679340, 1693427, 1301092, 481928, -1694668], [-202983763, 9569602, 9306080, 4381629, 1124322, 3478031, 609833], [40160508, -43618913, 1283834, 1399425, 1864866, -1252350, 2335278], [2067291, 51742902, 3836087, 5537577, -2250809, 1650985, 5190125], [1320763, 18128232, 6382780, -1901171, 1307674, -1577679, -4615905], [115537647, 7214739, 4594039, -4124544, 732525, 1746191, -4156931], [-80732120, 27486795, 1661044, 395303, -4495328, 772528, -670312], [241668162, -14114284, 1231472, 420400, 2088616, -628508, -1787572], [-151146708, -16682474, -7443766, -793490, 1190550, -4678988, 929680], [-98647404, 7343837, -5279314, -1195859, 307378, 348512, -4019944], [-239547996, 33775267, -1369674, -135196, 875880, -2820827, -1518149], [-230718306, -12999292, -1899072, 2244464, 4512560, 914052, 1815540], [209481238, 1543942, -13318876, -117776, -588524, 3573698, -1817902], [171916158, 17851842, 2998892, 1579887, -2198908, 1450933, -5685567], [-219412849, 27077711, 2390918, 3162965, 2180464, -657674, -1787088], [278357032, 2531255, 5403216, -930688, 2596402, -2312471, -3072345], [434691931, 3162995, 1072512, -150319, 817840, -1662462, -1781378], [-154727792, 36936048, -6620514, -1382902, -1561270, -3428650, 2264594], [-190255313, 12737362, -6675576, -1984911, -1890442, -1519061, 4625321], [-19031347, -26661671, 14975485, 4300597, 536941, -1008770, 3111982], [177141500, 20042165, 6609374, -5620639, -1607510, -3114364, 236924], [286956478, -40246822, -5529424, 3401634, -816608, -1396568, 1421680]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_6_a_m();".
// 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_825_6_a_m(:prec:=7)
    chi := MakeCharacter_825_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_825_6_a_m();".
// 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_825_6_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-125184, -9288, 27140, 2894, -1053, -119, 9, 1]>],Snew);
    return Vf;
end function;