
// Make newform 256.12.a.o in Magma, downloaded from the LMFDB on 11 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_256_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_256_12_a_o();".
// 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_256_12_a_o();".
// 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 := [-20440645894144, 0, 760691642368, 0, -5521716928, 0, 8617108, 0, -5011, 0, 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, 2656124071129088, 0, -27251267768256, 0, 31671582116, 0, -9960207, 0, 149], [29780898726656, 0, -450735578944, 0, 1624201260, 0, -1593437, 0, 463, 0], [55181414575357952, 0, -183379598326848, 0, -3210195094180, 0, 4129573071, 0, -1283029, 0], [0, -46348590293426176, 0, 654642405138624, 0, -1317203062660, 0, 932839527, 0, -220573], [296011345849181312, 0, -593318829165888, 0, 3286505129420, 0, -3552349749, 0, 1004951, 0], [87870602277173632, 0, -1355729145183168, 0, 3364324782820, 0, -2625370239, 0, 642661, 0], [0, 86419862121950457856, 0, -652427290245764544, 0, 1264120761051460, 0, -892317094887, 0, 209202013], [0, 293686461068406784, 0, 29559906168792384, 0, -79676872043660, 0, 62583530757, 0, -15204743], [0, 40986995813025353728, 0, -206611317028721472, 0, 1552888043716780, 0, -1650307867581, 0, 474068719]];
    Rf_basisdens := [1, 16750327136256, 412137600, 233682019200, 26172386150400, 77894006400, 16691572800, 418758178406400, 23264343244800, 418758178406400];
    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_256_a();"
function MakeCharacter_256_a()
    N := 256;
    order := 1;
    char_gens := [255, 5];
    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_256_a_Hecke(Kf)
    return MakeCharacter_256_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 := 12;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 1, 0, 0, 0, 0, 0], [-3362, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 30, 0, 0, 11, 0, 0, -1, 0, 0], [0, 191, 0, 0, -2, 0, 0, 1, 1, 0], [263978, 0, 10, 4, 0, -2, 1, 0, 0, 0], [0, -410, 0, 0, -128, 0, 0, -4, -7, 1], [-4535757, 0, 67, -131, 0, -1, -8, 0, 0, 0], [0, 32018, 0, 0, -1789, 0, 0, 114, -2, 22], [-44203047, 0, 1146, -699, 0, -25, 72, 0, 0, 0], [0, -1693, 0, 0, 1286, 0, 0, 37, -11, -88], [65496232, 0, 13855, -581, 0, 42, -245, 0, 0, 0], [0, -314817, 0, 0, -103472, 0, 0, 408, 270, -130], [56022463, 0, -432, 2295, 0, 277, 96, 0, 0, 0], [0, 2871273, 0, 0, 19258, 0, 0, -2507, -639, 246], [0, -592655, 0, 0, -675151, 0, 0, -3535, -1745, 871], [0, 6440927, 0, 0, -81444, 0, 0, 5275, -2585, -1034], [0, -3629086, 0, 0, -837765, 0, 0, -7817, -290, -1586], [-1176909099, 0, 90225, -8781, 0, 1473, 2808, 0, 0, 0], [-3230568932, 0, -201289, -35991, 0, 1696, 2400, 0, 0, 0], [-5729139552, 0, 72682, -91020, 0, 6580, -648, 0, 0, 0], [0, 4777099, 0, 0, 544657, 0, 0, 48881, 4887, 2319], [10827480068, 0, -291653, 170125, 0, -3320, -12788, 0, 0, 0], [6099829522, 0, 732638, 163272, 0, 726, 20805, 0, 0, 0], [0, 203423837, 0, 0, 13948, 0, 0, -26085, 48003, -12012], [6231694393, 0, 973909, 176847, 0, 6325, -18216, 0, 0, 0], [0, -8001132, 0, 0, 11284349, 0, 0, 65705, -80016, -4368], [0, 263837438, 0, 0, 3651941, 0, 0, 102808, 9904, -724], [-14853350834, 0, 2849106, -479430, 0, 38764, -21222, 0, 0, 0], [-38542315111, 0, 2062966, 56109, 0, 86847, -123672, 0, 0, 0], [0, -106755584, 0, 0, -11856653, 0, 0, -177049, -6600, -47112], [-125908358516, 0, -2203827, -2127423, 0, -63362, 85905, 0, 0, 0], [0, -102644313, 0, 0, -43202506, 0, 0, 80190, 169176, 49816], [0, -893449392, 0, 0, 42653221, 0, 0, -774752, 551436, -56058], [-456048704197, 0, -16149533, -2016507, 0, 101015, -61320, 0, 0, 0], [0, -1826027755, 0, 0, -100434830, 0, 0, 76249, 167365, 67782], [0, -289352395, 0, 0, -100217338, 0, 0, 893566, -29300, 103532], [-448218578878, 0, 25529747, 1614908, 0, 111828, -481696, 0, 0, 0], [0, -2500617651, 0, 0, 201970568, 0, 0, -268861, -60693, 241140], [0, 893453413, 0, 0, 74065795, 0, 0, -2906581, -3549, 207867], [0, -3456798590, 0, 0, 62811523, 0, 0, 4533734, -162982, 59242], [-1390322957899, 0, 3902150, 11912873, 0, -739245, 31688, 0, 0, 0], [119420383202, 0, 25475494, -4087056, 0, -44986, -1035795, 0, 0, 0], [0, -2470377836, 0, 0, -174825941, 0, 0, -1428406, 1448970, 192480], [-2558890668920, 0, 48209171, 9805746, 0, 170454, 1508064, 0, 0, 0], [0, 1263771361, 0, 0, 393485333, 0, 0, -3256051, -747619, -1151163], [-6648763112381, 0, -11834226, -10070841, 0, 231261, -2353128, 0, 0, 0], [0, -3243499857, 0, 0, 276722074, 0, 0, -5953606, -4275298, 557582], [0, 9900873084, 0, 0, -557782589, 0, 0, 6150522, 1498890, 870536], [1475507144084, 0, -61907573, 26579629, 0, 1143160, -2079020, 0, 0, 0], [-10677482202796, 0, 43097958, 10154352, 0, 1427336, 2995608, 0, 0, 0], [-196671969886, 0, -192224226, 5512296, 0, -663498, -806619, 0, 0, 0], [0, 15488168093, 0, 0, -652472239, 0, 0, -2317311, 114019, -2644805], [-192235068890, 0, 174574494, 1003542, 0, -3047468, 6286758, 0, 0, 0], [-14639448394233, 0, -346336459, 81306509, 0, -825905, -5185768, 0, 0, 0], [0, 29548870844, 0, 0, 1840395935, 0, 0, -998262, 575818, 2643040], [-15709167115541, 0, -31534204, 95146107, 0, -6944063, -4165056, 0, 0, 0], [0, 26355414425, 0, 0, 612405154, 0, 0, 11667589, 10948273, 4012518], [598044460372, 0, 629618643, -31523403, 0, 2184904, -2360388, 0, 0, 0], [0, 51654443888, 0, 0, -2148163628, 0, 0, -21104000, -6847751, 6422913], [0, -5612488628, 0, 0, -563038865, 0, 0, -40157226, -22266130, 1544492], [0, 15302730486, 0, 0, -928615494, 0, 0, 53247318, -22789065, -11253969], [-39835925506374, 0, 453405865, -36466856, 0, -11090440, 2761264, 0, 0, 0], [6790676045400, 0, -429887505, 267342603, 0, -5574070, 25095867, 0, 0, 0], [0, -48802167662, 0, 0, 496039259, 0, 0, -16065486, 38354910, 472230], [0, -132081414720, 0, 0, 147817027, 0, 0, 58912959, 162294, 8212646], [-399469190106, 0, -1391083018, 78206910, 0, -1256620, -12171018, 0, 0, 0], [0, 186197535859, 0, 0, 6484070663, 0, 0, 85724847, 20112351, -15234537], [0, -124174429704, 0, 0, 4834651297, 0, 0, 38208336, -45984516, -809050], [2966576144570, 0, 811758516, 66498612, 0, 8564648, 27455004, 0, 0, 0], [-26277895298227, 0, -740815715, 230982403, 0, 14136769, -39464120, 0, 0, 0], [-28023391611707, 0, -439415812, 221764677, 0, 38808367, -18239184, 0, 0, 0], [0, -90543786009, 0, 0, -3115805216, 0, 0, -148499781, -44127417, 12816278], [0, 281612667959, 0, 0, -16126567555, 0, 0, -21890987, -7403267, 9632165], [3371031154292, 0, 1177525368, 361152276, 0, -18287284, 8730768, 0, 0, 0], [0, 13292278723, 0, 0, 9215952346, 0, 0, 71622209, 99854745, -7341492], [0, -7651438114, 0, 0, -10340304595, 0, 0, -326574392, 23074144, -27245332], [36003486333226, 0, 3080592714, 690531108, 0, 14549086, -27128655, 0, 0, 0], [-52109086495336, 0, 2030090239, -843968517, 0, -18323382, 53977371, 0, 0, 0], [0, -143173648274, 0, 0, 2920474825, 0, 0, -153244779, 224024828, -11257636], [0, 290033818406, 0, 0, 5790814717, 0, 0, -161618540, 66287476, 33390192], [68461425689850, 0, -1753818174, -928417602, 0, -58798110, -17044056, 0, 0, 0], [-64525807179246, 0, -300518970, -842755440, 0, 44253574, -56938851, 0, 0, 0], [87273713698802, 0, -5865859463, -736756416, 0, 55157216, -28005216, 0, 0, 0], [0, 531494141126, 0, 0, 18766298222, 0, 0, -163757358, -136126559, 11299561], [9889973004234, 0, 2296279314, -329850132, 0, 97863918, 37830681, 0, 0, 0], [89800546967592, 0, -5195134929, 1688971371, 0, 141184490, -91898805, 0, 0, 0], [0, 288811827372, 0, 0, -14894796373, 0, 0, 257984390, 49965774, 19964292], [296846319232460, 0, -1035134030, 1004323104, 0, 2492040, 65629512, 0, 0, 0], [0, 319891210184, 0, 0, 43698108772, 0, 0, 31113000, -212109549, 34594155], [337927397310687, 0, -4340482106, 463208611, 0, -44578015, 127036360, 0, 0, 0], [417405243653852, 0, -2950946697, -146766018, 0, -57651174, -129883200, 0, 0, 0], [0, -334227852343, 0, 0, 65669427080, 0, 0, -503264648, -285704676, 12950652], [0, 215377712303, 0, 0, -33658729365, 0, 0, 726554683, 555611137, 85955401], [464321140295211, 0, -6311358681, 740431053, 0, 127756623, 92202456, 0, 0, 0], [0, -787202237583, 0, 0, -32459087458, 0, 0, 361969565, -225349767, -1552386], [84823121273580, 0, 632995869, 1494668721, 0, 68075070, 221199825, 0, 0, 0], [0, -117348412272, 0, 0, -87177471693, 0, 0, 914988399, -36666090, 23664966], [0, -1636048249853, 0, 0, -36226383500, 0, 0, 1155664427, 321796799, -24812678], [0, -90335365726, 0, 0, -142377728123, 0, 0, 89516257, -203672516, 25279644], [0, -1883646258681, 0, 0, 46307433782, 0, 0, -801329119, 392907153, -140557320], [0, -1027874472990, 0, 0, -21901938074, 0, 0, -695922830, -1257723633, -165266361], [172053790864540, 0, -10782324867, 6931366353, 0, 132657598, -65892975, 0, 0, 0], [0, -1534617101646, 0, 0, -177772804978, 0, 0, -334083678, -532557387, -107455331], [485047253632654, 0, 10767748426, 2613347682, 0, -207645428, 418788666, 0, 0, 0], [0, 1856423598434, 0, 0, 9242062283, 0, 0, -1418386097, -114918660, 6112860], [-361150082746230, 0, -29267804076, 3410508036, 0, -88407528, -175604748, 0, 0, 0], [682396098104784, 0, 38163330313, 1752609022, 0, 58240106, -491432432, 0, 0, 0], [-92267635091220, 0, -24563399769, 1121118153, 0, -389192080, -505486056, 0, 0, 0], [122770275530706, 0, -36196286408, 1858906482, 0, -89376458, 673863552, 0, 0, 0], [0, -1267189659002, 0, 0, -204123637883, 0, 0, 1693846316, 1146637724, -235303304], [-124600550500740, 0, -20143262377, -6917322247, 0, 10208464, -52872136, 0, 0, 0], [0, 1638897970466, 0, 0, -80872257267, 0, 0, 2979563377, 354543682, -608622638], [-383332678635029, 0, -11533477465, -4949599011, 0, 407192175, -408819576, 0, 0, 0], [787946645893282, 0, -59430529946, -774715376, 0, 21382182, -374075171, 0, 0, 0], [0, 1480145553922, 0, 0, 39779323151, 0, 0, 621004619, -349921606, -37716342], [-771486830265597, 0, 15148473405, -4839028023, 0, -621982317, 445961256, 0, 0, 0], [0, 1375589017176, 0, 0, 325362074483, 0, 0, 2427943454, 1295886798, -542811624], [0, -556529209881, 0, 0, 221129234869, 0, 0, 1539645253, 314187591, 65646015], [0, 1188671683174, 0, 0, -20576174341, 0, 0, 2543889446, -1494319678, 192361950], [1486196658094266, 0, 39012069834, -5725874268, 0, 104018398, -1084833759, 0, 0, 0], [0, 1504695303415, 0, 0, -284456199662, 0, 0, -1447888719, 997675697, 417691136], [0, 5777971852981, 0, 0, 155133193736, 0, 0, 639096872, -462536528, 702560560], [0, -4332426133375, 0, 0, 90323472439, 0, 0, -1601860721, -1230761693, -422814597], [0, -1351420255073, 0, 0, -234293447128, 0, 0, 342210951, -3352135077, -396403302], [0, -993558342025, 0, 0, 329469942610, 0, 0, 2148358729, -1660655303, -382181448], [-2870926910447330, 0, -23205769666, 9146768954, 0, -175113754, 1505295416, 0, 0, 0], [-2811870475506880, 0, -6649831559, 6348236334, 0, 783104858, -246493200, 0, 0, 0], [0, -3036313977598, 0, 0, 469086052111, 0, 0, 2686717678, 2869455874, -653776710], [0, 4295797106579, 0, 0, -177288919642, 0, 0, 1706567446, 853380574, 971314382], [-1783596022718632, 0, -24656653969, 41947971386, 0, 26480766, 413529776, 0, 0, 0], [-4171505168420418, 0, -14197560366, -6436030062, 0, 2189425742, -961888872, 0, 0, 0], [0, -7857592758768, 0, 0, 124198759065, 0, 0, -7511409636, 1335590928, 73957290], [-3068260978860744, 0, -129879079393, -21626984005, 0, 938661418, -819831349, 0, 0, 0], [3191720282146810, 0, 12346715930, -12065407980, 0, -1410685026, 2679990657, 0, 0, 0], [0, -10134487509717, 0, 0, 844451168734, 0, 0, 803488037, -3680875099, 148512320], [0, 1012823704193, 0, 0, 160186419045, 0, 0, 1852483933, -2875297523, -494326859], [0, -13949304100747, 0, 0, 980049993806, 0, 0, 1574783317, -5940737175, 583927098], [3436602132721608, 0, -11156404401, 2235248427, 0, -1415671350, -2981426949, 0, 0, 0], [0, 1066220441842, 0, 0, 524952385621, 0, 0, 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-13101015271205, -7876657049667, 0, 229814359768, -147827160348, 0, 0, 0], [0, -1452357095588217, 0, 0, -22135714543037, 0, 0, -483272280461, -370216898607, -64184942055], [-1655765426287252114, 0, -28890974925780, 1342698429798, 0, -248465524942, 71750198112, 0, 0, 0], [-670905269188759630, 0, -15347868753912, 4983570930408, 0, 298091774064, -310702681464, 0, 0, 0], [1250388811374439658, 0, -32437371454492, 1778789792276, 0, 456718052280, -26470419676, 0, 0, 0], [0, -2060699000111537, 0, 0, -26526562825323, 0, 0, 801726655301, -1148141419489, -204935707177], [417480764807544008, 0, 12197318829319, -4664497997565, 0, 507733196250, 495527785587, 0, 0, 0], [0, 269199779027092, 0, 0, -20466336003319, 0, 0, -4169727133224, -1095068739156, 352589733402], [667186179303585918, 0, 9637994493840, 5397406602942, 0, 375623916458, -964889884656, 0, 0, 0], [-716866173794027700, 0, -33254815861667, 1320510843889, 0, -267914448706, 741044922385, 0, 0, 0], [506483385595050714, 0, -12359666890974, -10853563833732, 0, 171148597006, -539387424423, 0, 0, 0], [0, 1798864382211639, 0, 0, -81736695522780, 0, 0, -1938692649660, -403988864172, -98181755532], [0, -1365960989484106, 0, 0, -81015187436916, 0, 0, -4345833679664, -523660144181, 361529272675], [-1566683274349070717, 0, 8222524982934, -5751132139545, 0, 462509509645, -391689520728, 0, 0, 0], [0, 357311656880384, 0, 0, 65393377982359, 0, 0, -4700104521282, 1080689970798, -122844683880], [0, -1080635701404481, 0, 0, 2775822359166, 0, 0, 1539951216457, -16580941031, -203697762392], [0, -3192897483730398, 0, 0, -202561905787249, 0, 0, 4212915852742, 418456182922, 211774570450], [669160701482541974, 0, -13169030574932, -5144645114418, 0, -441979080726, 747399688032, 0, 0, 0], [0, 2487302369450326, 0, 0, 179037430510339, 0, 0, 87253563471, 953285556386, -190295603278], [0, 440129402447733, 0, 0, -228567772919370, 0, 0, -777233308554, 612520965018, -21995259414], [0, -1444519561992971, 0, 0, 226144568748215, 0, 0, 1944235125351, 117566496765, -588033420507], [0, -6668108742042870, 0, 0, 11503116742011, 0, 0, -415383556710, 47982787398, 187787635878], [-1735269848696905174, 0, 9866352650876, -2644297662436, 0, -412658955816, 427903966148, 0, 0, 0], [-251565855607272364, 0, 32890774549771, 5074938540189, 0, -433876615704, 399827047788, 0, 0, 0], [1704687175111661377, 0, 27589728064964, -4400662028127, 0, 118406624179, -205966786272, 0, 0, 0], [0, 1160177315159070, 0, 0, 367076003828291, 0, 0, 2935442434998, -34343006526, 162348516310], [-417549495123221160, 0, 23158014909007, -3404088351125, 0, 620302491146, 555323283163, 0, 0, 0], [0, -6716205568203873, 0, 0, 173466105547042, 0, 0, 1019873138569, 287367633801, -653820527616], [1239084631034908853, 0, -34940116779447, -8092293491661, 0, 691991892657, 584501673768, 0, 0, 0], [0, 2111861651132587, 0, 0, -59519071987312, 0, 0, 982442134215, -1017522170925, -751389668394], [0, 3467324936710321, 0, 0, 91177733414938, 0, 0, 55221435330, 1509794923916, 753735974636], [0, -746897264848486, 0, 0, -376293472649071, 0, 0, 679671350632, -1110489128984, 572079565376], [0, 2431245169136814, 0, 0, -347176410466463, 0, 0, 556228148661, -1476099261102, 443376164450], [-1158459198852805730, 0, 16114807625610, -11241494817198, 0, 901043331070, -1291379932728, 0, 0, 0], [0, -1728054525102656, 0, 0, -70970223707051, 0, 0, -5568800912376, -1405755131740, -214605173410], [0, 1019781614621989, 0, 0, -428470486746004, 0, 0, -1410938472892, 634752336398, -497630495106], [208369607758192892, 0, -60908104420811, 15564697292841, 0, -387566717106, 1019690030793, 0, 0, 0], [0, 3247541006600867, 0, 0, 427175522923190, 0, 0, -311037762327, 1001370129313, -105290197892], [3028124751290746430, 0, -28116143781026, 5861300014698, 0, 26312363622, -873468610584, 0, 0, 0], [0, -2430080023668820, 0, 0, 300320298543213, 0, 0, -1227519415976, 824192416204, 100308207850], [253723756524711036, 0, 2229284522943, -17312824250127, 0, -531035052240, 116197759944, 0, 0, 0], [-722380633423910130, 0, 17056305846368, 3878377350398, 0, -340801457702, 514383970880, 0, 0, 0], [0, 3236086299721239, 0, 0, 48703070036936, 0, 0, -2474433766672, 1103436414078, -16459070226], [-821612407621793510, 0, 6189657278466, -20847895077636, 0, -581754053522, -139594762935, 0, 0, 0], [-1229077852072472108, 0, 81464459700445, -3963207911318, 0, -224257570818, 1873905318496, 0, 0, 0], [0, 3426445991657649, 0, 0, -31932423673408, 0, 0, 1154959624215, 2583407365551, 176095397740], [152230841783564252, 0, -37415494145929, 11421130507625, 0, 758896393056, 1135977005504, 0, 0, 0], [-701450219286382009, 0, 80484337102732, 960440175511, 0, -376566887003, 1115177501392, 0, 0, 0], [0, 4353119160178391, 0, 0, -346368127248073, 0, 0, 2864152942063, 3552968308107, 129839838627], [0, 470377714760615, 0, 0, -290996506239106, 0, 0, 7359806562622, -1674906641390, 19924921634], [-831692702961478668, 0, 7619095555099, -6607978417491, 0, 788679113224, 529626861084, 0, 0, 0], [0, 2249016688386179, 0, 0, -142814860432870, 0, 0, 3108050993733, -2121499235387, -706538950208], [1828653841492952839, 0, 29242675676285, -21679981303683, 0, 372030839199, -211825885416, 0, 0, 0], [173027483348673836, 0, -99304929372347, -14523268616199, 0, 213137179950, -390474964071, 0, 0, 0], [0, -2344547710562838, 0, 0, -541105936465751, 0, 0, -1185481334555, -1453833885408, -709602723936], [0, -367870150185794, 0, 0, -335154987078995, 0, 0, 4475683355777, 2923346710310, -755066795306], [-1575422146088938941, 0, 125260447420256, 4574432249531, 0, -970591766303, 172949405984, 0, 0, 0], [0, -4963022865885441, 0, 0, 111236745751564, 0, 0, -1121684995193, 403929742347, 250722196914], [3113929248202595217, 0, 66805276637643, 6466860413643, 0, 1651166799289, -1105986463512, 0, 0, 0], [0, 5110661845488518, 0, 0, -107987272338445, 0, 0, 3512814129370, 2076718677598, 834071497778], [0, -6962683276148599, 0, 0, 559183241638045, 0, 0, 893706827317, -1908049063123, 717680629205], [-3967857948198329899, 0, -70721488763870, -3884351584847, 0, -598178059637, 1626405538792, 0, 0, 0], [-337959632095826682, 0, -83214542817186, 1471728704790, 0, -1717133322860, 241522027014, 0, 0, 0], [-1084748721792150886, 0, 53611918509922, 23533767200956, 0, -695560225394, 1649698724761, 0, 0, 0], [-1415545962683084152, 0, 26509473720791, 10481988077043, 0, -1979913409894, 108435688179, 0, 0, 0], [0, -2729612195891398, 0, 0, 487305502309331, 0, 0, 1962238871338, 625888020342, -1419862165386], [815761222922237384, 0, 89580306814431, -10257589954341, 0, -1211704057078, 950377535739, 0, 0, 0], [1764751187746339257, 0, -71254089356065, 1339907840907, 0, -946432727623, -1149888750168, 0, 0, 0], [2131292463235106438, 0, 41208489681534, 20372518956342, 0, -240464447020, -3259487106330, 0, 0, 0], [0, 219184216375123, 0, 0, 408782681060162, 0, 0, 2567780562290, 1688595497618, -117902360670], [-4539692271749170909, 0, 65636843179441, 10735459886209, 0, -449388254981, 648942015208, 0, 0, 0], [1635328002319191569, 0, 20899185633620, 12252430731921, 0, 772892102339, -1548682633632, 0, 0, 0], [0, -4637837810804735, 0, 0, 526201124983610, 0, 0, -7942460289499, 557086129025, 291545309118], [0, 5528714274620825, 0, 0, -56883199630147, 0, 0, -10492847049443, 2924314917547, -438946944061], [-5169013854722020958, 0, 19921710359116, -6487068873254, 0, -271808965650, -1617182142080, 0, 0, 0], [3054059904609915918, 0, 22920545065578, -21038741085054, 0, -1712764980404, 1900200664602, 0, 0, 0], [0, 2733493015304191, 0, 0, 177299420506294, 0, 0, 312482081469, 1802463663413, 2035148372828], [0, -15950559739877956, 0, 0, -801199866199213, 0, 0, -4750897255617, 23114044906, 222551915962], [0, -11596354411419997, 0, 0, -396626505831935, 0, 0, 3682082986561, 4292344186159, 57758515047], [-4823076329279274834, 0, -62027373558091, 42206104165508, 0, -13635269492, 1683053539136, 0, 0, 0], [2677353427656044792, 0, 7734343827599, 22701893992779, 0, -644358500566, -840346736661, 0, 0, 0], [1444640926646856864, 0, 48621687436036, 35361205678680, 0, 1099976259592, -524920249728, 0, 0, 0], [-552189747526377156, 0, 20474080340293, -10842286098983, 0, -855914367730, 4772274919945, 0, 0, 0], [0, -3748419771486339, 0, 0, 120741850831856, 0, 0, -5988969596496, -5643712162272, -260901574880], [0, -2337764196459777, 0, 0, -503764666943468, 0, 0, 694494778015, -889672125453, 720060742362], [0, -11850428761207872, 0, 0, 636570275924105, 0, 0, -1889684184528, -1232174635764, 845666603134], [4398120618977493258, 0, 50096880953202, -13986941715828, 0, 1362588947694, 2140274391897, 0, 0, 0], [-978207179736851490, 0, -76908377198930, -1030096054134, 0, -907672657370, 2553903279816, 0, 0, 0], [1520895206534002700, 0, 19209815266389, -2567380184151, 0, -813844622098, -1585730233575, 0, 0, 0], [0, -13301519140339422, 0, 0, -506799482247071, 0, 0, 9831298297672, 7070004050904, -1226723473656], [-788483480414475759, 0, 31490298489957, -6571084906089, 0, -1052297998211, -2103107650680, 0, 0, 0], [0, -824901658962585, 0, 0, -476616048225182, 0, 0, -1181969511747, 475997946453, 503344786028], [-4521527763776953534, 0, -70953643447218, 29212820139576, 0, -816111784138, 60414607749, 0, 0, 0], [0, 6799051899405731, 0, 0, 367437265724595, 0, 0, -7238337205037, -1033933325627, 318789041197], [0, 3551189955319896, 0, 0, -1311974919183389, 0, 0, 2492869250743, -2021165587920, -923710889808], [-2866891169434745966, 0, 68069572001840, 7798655803104, 0, -409946943216, 3160091852424, 0, 0, 0], [0, 13249286696640371, 0, 0, -11368399401536, 0, 0, -10904780969997, 865820958087, 212733653754], [0, 8540053188956581, 0, 0, -234386964450625, 0, 0, -3961817489361, -4710345643899, -2785284437907], [0, 4695665834994451, 0, 0, -226403874586230, 0, 0, 3844262585786, 3782972112842, 1058426012186], [3535628949126151356, 0, 116372876602263, 25381032493689, 0, 753017736080, -2874621012072, 0, 0, 0], [0, 9384588102808683, 0, 0, 707960120147438, 0, 0, -6352173551431, -106070234943, 1339073747172], [-9235605757178014748, 0, 41500222588160, 15795551175812, 0, 331352730572, 2250624618368, 0, 0, 0], [3138925615491136301, 0, 80999906924721, 60385652385435, 0, 393078993129, -1183895015256, 0, 0, 0], [-4442157866071443502, 0, 166397658584566, -38226396238304, 0, -2343112790714, -207576796163, 0, 0, 0], [-9298124104641055400, 0, -133278424474119, -1945572625530, 0, 1721068006498, -3281645751696, 0, 0, 0], [0, -2557516840814305, 0, 0, -427344251824600, 0, 0, -799394677425, 3523695329779, -1977885465278], [0, 6827108028976289, 0, 0, -559336866385468, 0, 0, -4581487829693, -2995357160861, 118706848784], [-6831997577609635115, 0, 101278230675036, -610637065131, 0, 3158288431903, -4458636069648, 0, 0, 0], [0, 6170290135392635, 0, 0, 864228987502001, 0, 0, 11580181740505, 5158104051093, 2707284210813], [2466760389262831800, 0, -61377797809977, -5523038724285, 0, 2849734154714, 982108495155, 0, 0, 0], [0, 5432037678699253, 0, 0, -108122028208024, 0, 0, -15095474322309, -6484298572509, -1395482018124], [0, -25018512968677460, 0, 0, -551796708692120, 0, 0, -8512645409668, 3298950586971, -1418613862413], [1544039051395618124, 0, -13907634647465, -10562021161031, 0, 1525391594512, -2366371439480, 0, 0, 0], [0, 9176347309355600, 0, 0, 995023009302176, 0, 0, 22679430080908, -3638484730979, 1144341265349], [-8670788236588928142, 0, -4451079183127, 13306770638096, 0, 2342756807392, 3666953663600, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_256_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_256_12_a_o();".
// 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_256_12_a_o(:prec:=10)
    chi := MakeCharacter_256_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 12));
    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_256_12_a_o();".
// 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_256_12_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_256_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,12,sign)));
    Vf := Kernel([<3,R![-23616501181712148139121664, 0, 2820473894112854402304, 0, -58059024148420992, 0, 415491272352, 0, -1121932, 0, 1]>,<5,R![-46208751498243071728497409638400000000, 0, 12886904566420501696672768000000, 0, -922914430681034275840000, 0, 24403540255091200, 0, -264891952, 0, 1]>,<7,R![-53374179670054702907392, 5740462575196131328, 342941398016, -5087977856, 16808, 1]>],Snew);
    return Vf;
end function;