
// Make newform 280.10.a.b in Magma, downloaded from the LMFDB on 17 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_280_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_280_10_a_b();".
// 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_280_10_a_b();".
// 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 := [62307905700, -21289610540, 1028441937, -918604, -72814, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [36298363219215, -2970136023274, 14028279698, 112277988, -359195, -542], [-32533058547510, 7675998097306, -18491226687, -568944497, 151755, 7873], [-27479111238690, 1751507221384, -2316392893, -146515858, -7955, 2172], [529339123643970, -33547602918578, -6193091174, 2503630761, 1055780, -36349]];
    Rf_basisdens := [1, 1, 6739423425, 13478846850, 1925549550, 8087308110];
    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_280_a();"
function MakeCharacter_280_a()
    N := 280;
    order := 1;
    char_gens := [71, 141, 57, 241];
    v := [1, 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_280_a_Hecke( : Kf := Kf)
    return MakeCharacter_280_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 := [[0, 0, 0, 0, 0, 0], [-14, -1, 0, 0, 0, 0], [625, 0, 0, 0, 0, 0], [-2401, 0, 0, 0, 0, 0], [-1204, 22, -2, -8, -1, 1], [-21396, 48, -2, 27, 0, 1], [1299, -994, 20, 1, -24, -10], [181084, 1949, 59, 18, 2, -3], [-72374, -1, -41, 130, -4, -27], [-839067, -13520, 546, -20, -163, 56], [1334769, -3682, 83, 927, 342, 45], [-529168, 3264, -494, -323, -361, 38], [-7183860, 73425, 605, -2830, -216, 283], [2853793, 49199, -2210, 1440, 1375, 18], [2658461, 66779, 1836, -3774, 662, -435], [-33170318, 69631, -2001, 1111, -4015, -921], [-2923042, 160838, -854, -19414, -114, -792], [-7619077, 386272, -10545, 8511, 1826, -435], [61415694, 651151, -13803, 24415, 8793, -2009], [-19658808, 437767, -12179, 14271, -7787, 1053], [-82869458, 706637, 11567, 23931, 3825, 4119], [61181732, 1869644, 24833, -70062, -31218, 1049], [-19049024, -234059, -6267, 58507, -25785, 10123], [-87026172, -1280615, -57701, -168102, -19354, -7337], [-168869901, 1385518, -90580, 52217, 79036, 14682], [198476869, -3013734, -53589, -246897, 59750, -16025], [122331971, 4216159, -1248, 143930, 25218, 28327], [-566446258, -695441, 74707, 12737, -65177, -20655], [-20032421, -608340, 182946, -163054, 92673, 20104], [-398870267, -3104156, 46209, 284543, -25154, -16379], [-1411461514, -1552800, 104944, 378864, 83804, -18578], [-88502368, -3445993, -12773, -881816, 153342, -25525], [-515718253, -3923460, -418503, 441511, 300484, 69325], [-3137091114, -916502, 229324, -46730, -121094, 17274], [-3497261200, -16774860, -486452, -771470, -321194, 26942], [-76294108, -3479632, -1089727, -1054836, 147196, -107531], [-2121670714, -11908050, 32904, -892346, 885944, -3706], [-2121554656, -4373213, -303153, 1697447, 59365, 97187], [-1443341201, 3057653, 246364, -2053136, -836686, -12987], [3254564560, -20691394, -388806, -2762535, 1264154, -89415], [-1236293108, -12007467, 1333613, 1750857, 285605, -254737], [-109499122, -4897406, -347496, 3069666, 107226, -211900], [-394122490, -14949314, 1041257, -233850, -864486, 364371], [-7450901879, -14772544, 1663771, -3636561, -433754, -409109], [-9425390080, 97335104, -577696, -4778137, -65187, -37128], [-5081713182, -18476716, -1399468, 4758474, 200358, -414786], [-13467867192, -31657506, 1766024, -1142290, 227893, 298655], [2855034139, 59696257, -689920, -2229070, 2915644, 43197], [-9415700436, 10147779, 1461808, 2605762, -89558, -227834], [-27075986891, 110559468, 746243, -6062567, 1914340, 11983], [-3882541450, -43309044, 2744092, 6046800, 2776194, 53994], [-22053277132, -90261552, 335553, -3947812, -3863272, -581111], [-36886448602, 72533697, -944707, 8257346, -4426526, -503941], [-743206656, 220632206, -4588750, -10134146, 2795284, -732184], [-23602484604, 99347625, -4881151, -186425, 398819, -937807], [31325730710, -55311554, 5319062, -2716538, 3413000, 147014], [4524807022, 316563282, 1657216, -1277340, 1191386, 2875578], [-26019899345, -521067286, 3105347, 11084883, 2625238, 108859], [-21029345806, 144509583, 7330769, -9161459, 1315723, -571019], [-40304717053, -375521602, 10065717, -3450676, -6596644, 1803564], [-30467323890, -334046827, -5245502, -19684658, -4069060, 1789632], [-50401560648, -137733148, -9839022, 25721359, 2154690, 627831], [-11761013978, -629748825, 10969254, 7596792, -6044212, 2071542], [29319538677, -764793250, -4937311, 23116123, 206510, -2625973], [-59684265235, -53968172, -6734524, -8443565, 1370378, 2881632], [35226271300, 246760017, -14125945, -2464825, 16096133, -1649611], [24611991512, -471621911, 2590919, -40237431, 16727151, 3554487], [-83984392780, -161553940, -8816916, -2307588, 20475828, 3235026], [201850195235, 454878731, 26916182, 37703884, -4251965, -2446208], [-10942899347, 667474500, -2463247, 23773107, 1746074, -4245151], [62184223407, -1816067382, 18592576, -1401037, -16538666, -1206638], [94265527636, -334684309, -22532645, -22981381, -6384121, -195331], [97445560951, -237477867, -42781516, -4063008, -179570, -4457527], [-106294177302, -1375041080, 21412004, -56799895, -16985901, 1405638], [148638507114, 2181213473, -9095497, 7675021, 4445091, 1214269], [49700762092, -1626007494, 34218360, 17990824, -13014642, 2077878], [34323564769, 782460822, 47419178, -19812968, -14452531, 116524], [-196904800034, -1699810186, 31509086, 3218979, -4622384, 8501755], [188784714307, -1722479738, -32416813, 33892234, 6304414, -1134978], [-78928182552, 1731569240, -30451004, 47507230, -32183230, -5052246], [317902938162, 3767652666, 31879162, 63867954, -31739490, -5317588], [-83548666705, -1048434588, -72071040, 97671324, 14283565, -7634972], [192044586090, -1853209834, 41920415, 76038772, -19568642, 1950227], [-211576807542, -325268303, 38970557, 71055081, -44751147, -4119385], [217800987294, -1550573312, 69043680, -71922540, -2847350, 5801784], [360497105147, 269269391, 2034492, 119174204, 25763589, -8929720], [181924891869, 295060024, 54304333, 59903294, 56410980, -146900], [-334245561105, -3432599316, -10208165, 16710193, -4456476, 21776273], [23581809544, 1316588484, 81107078, -201102476, -64143728, -2080216], [489203382982, -2227873153, -37896837, -195794986, 2620322, -8826665], [427257535590, -585028641, -86716694, -106166048, 18163748, -16287846], [309382227439, 1917894480, -125533773, -104016255, 79331392, 27338267], [92420166264, -3318744994, 74453814, -40057402, -67936764, -1174136], [639611696162, -3809679198, 62322544, 24391204, -41935269, 11388525], [526708451848, -3840840926, 97335312, 169594404, -126600207, -35122659], [403392309423, 4151001611, -29464318, 124459042, -46798390, 14531445], [-428169435148, -2259648430, -132835202, 2967082, -58956758, -32529108], [-168230679724, -1329745791, -119321399, 292509786, 32648562, 38519137], [802511721712, 718321177, -32950355, 57310793, 89379313, 54280919], [-839011286059, -4489189630, 38045450, 102518010, -78844811, -19348170], [-378751511557, -6529910841, -165261824, 444261528, -29164421, 4874958], [-1179142475324, 285773959, -58480193, 379519403, -5473459, -48698387], [-529253488774, 2744606565, -194568177, -338207495, -110960365, -34198679], [-1399483151154, -12850986376, 11802998, -156844068, -33782450, 69998468], [-448079157962, 1264105063, -130470755, -550433211, -25483813, -35215485], [-1117658869417, 464453848, -210320536, -130681777, -12215622, -62305002], [-588594076398, 3940018089, 130362543, -128244461, -102938107, -42837279], [-1048949062535, 3781640558, 75362242, 316841769, -138767584, -59183802], [266500783940, 2552499296, -314513021, -363673930, 142546302, 61935559], [-521590050680, 16265102572, 47171272, -331528972, -12894512, -10600490], [-514425988635, -4211204807, -193187860, -554707240, 256680492, 32410345], [-606146302796, 12936555305, 225172807, 12670593, 73996563, 62179593], [-1447629705460, 1379068282, 113325358, -290178140, -122063468, 75014136], [1217936096998, -2394067949, -132832147, -477552698, 131824142, -7846999], [7367451860, -18027082442, -144588533, -91523094, 341342064, 32750633], [-2780466550114, 21653674912, -57991010, 518713988, 26565030, 8070052], [-325036575686, 23146152117, -8012336, 346768034, -227083424, 5691642], [-1763525673216, -11199718630, 100412606, -331345672, 123284012, 47656686], [-425696809970, -13577195966, -135004332, 806747545, 39341631, -63885830], [-693981520910, -16828311946, -91666140, 1373236874, -19402433, 16287559], [1070158097670, 13424394676, 641550088, -520717334, -478702196, -43743974], [62245623375, -7076405960, -147539091, -52451733, -149180368, 54416413], [-1576967562304, 7430684370, -68695600, -576816901, -69711832, 114168483], [607575438955, 18688729741, 401819498, 408826384, -13787529, -159016656], [-2854212894484, -15526264136, 80473360, -987772098, -233693050, 86412536], [568332283003, 13941272010, -397919104, 648523146, 509571091, -83692740], [3582926049851, -17606423784, -29120048, -2275378246, 154456613, -47664126], [1396661158809, -51552879492, -262322831, 733566683, 428978408, -90423515], [-1950644508428, -10644201678, -178270248, 397400188, -152721234, 223783474], [340081850346, 4070231984, -391802466, -1126618649, 134302912, 316064367], [-3107495364478, -33998164220, 275469200, 320594202, -134534387, 220933725], [-1101746672278, -27565649756, 374980732, 37727474, -527926718, -105632606], [2339018278948, -21931922000, 231292969, 439573106, 440393122, -67509667], [-2847156373414, -11958717705, -438826075, 1398560657, 157822141, -62797233], [-2123971890462, 40780581369, 78334981, -390029460, -482545220, 83792759], [2066827757710, -40531004894, 202636308, 1279487538, 585653296, 16937602], [3924043880738, -48108577932, -396129226, 1195439391, 162783592, -328303057], [-3339919183390, 23414309013, 707514336, 39132562, 326963348, -59686538], [-16470455536, -1311550758, 1211520884, -1492852267, -629338528, -179938315], [2197275359609, -32228135828, 128432169, -320159228, -373776000, -61420086], [2884406613288, 11499057260, -1537214520, -2219364170, 479229356, -50211758], [-137620619165, -29838432064, 225819188, 3453299172, -920214447, 277411032], [-110725682108, 38456585453, -272918591, -834489048, -52570284, -283666077], [-245077720200, 26075614449, 626005289, 1706345077, 754282139, -102514631], [9619208330149, -45995707616, -549807109, 2363896645, -1014079268, 61973171], [6956748346857, -36817813036, 458711873, -2521012871, -281544446, 265272529], [8181198450884, -21798527732, -753599878, -2101236028, -1164469342, 24906954], [1213758796542, -11535421423, -1467505571, -2820769415, 762144101, 283904811], [3436805114292, 16232703422, -215154594, 987444166, 614436182, -20178902], [240161187424, 81652094671, -570926645, 271214194, -460379446, 62895745], [4498302333656, 26000207408, 791326228, 1686224159, -461342005, -304996238], [11213196711298, -14669329364, -419565604, 307209344, 401298946, 43951366], [477895633359, 28822382439, -1331793020, 1831619910, 258100361, 115823750], [-11026567798434, -46070699216, -1162337322, -201650920, 501595274, 197169866], [-11676387721419, -5482310601, -758178028, 3699455726, -446799353, -50680842], [3892519821346, -14418550473, -2213607485, -5061300607, -35137231, 60278939], [966992126270, 104541397882, 1797353387, 5086697758, -864363926, 399325989], [5896029993828, 127659109886, -101536090, 742596716, 888520118, -440960334], [-1968727717833, -35053676186, 2302614810, 4014025399, -694113628, -529585826], [8950926667971, -7625150012, 817511003, -2175262909, -1973199850, -180319191], [-4561980760803, 116575450489, -1333741298, -1002349248, 1900717275, -31121890], [970621627467, 79098366, 391032673, 2506040237, -2236627540, 328085803], [-5133901862360, 73979427838, 2878796608, -4746222724, 219233640, 553335832], [-4705332050348, -119348924397, 1351731529, 4698968092, 761777846, 265236521], [-22850593740577, -11260595214, 636498595, -3410243187, 120955908, 320782657], [8634035185827, 6206988003, -1521806910, 1774366770, 2018739642, 837262017], [-14334808932616, -90971654557, 2717897547, -364875977, -369088605, 81355633], [865111309540, 10761931028, -1412129598, -1905599741, 2551957734, 94684395], [19058320068825, 109664732962, -2507320653, 2780239704, -802923068, 316923352], [274221121880, -83079357090, 371323200, 5189406224, -104649746, -657784784], [-82853530946, -74079765703, 2786007759, 846813761, -614480621, 118030445], [21462815494126, -47582060754, 677074222, 3578681136, 2557565616, 289909226], [-5266376327530, 17070734298, -52596735, 3681415476, -3881849004, -429007921], [3325664153886, -99692590248, -2044945622, 6689311234, 1775675788, 752221016], [37245343400805, -33810894156, -395099309, -4736008329, 734873798, -419511411], [4388224489588, 25730781850, 1215147496, -3961827818, -2949477154, -95475286], [-5152051743966, -152861930880, 293187040, -9049839548, 1452544167, 490286149], [-18212903773033, -31853703412, -2800972216, -7200740442, 4129896899, 330064696], [7893359944252, -124534562764, 791064628, 3066176330, -227441618, -405346268], [8328473474565, 222333817452, 885509917, -6016954275, -723003762, 600913911], 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-76236484999], [-1137651169611530, 7838723149405, 118558669673, 69474664023, -181885943545, 10113336685], [-242980518771175, 11007694841481, -102542195008, -323988145786, 181311060034, -47160730007], [-18192091193983, 483242614412, -204474310875, -456992654296, -13018825024, 64774461410], [-894088555140794, 5425919147088, 176043946938, 69926580585, 130517380168, 26165250373], [-127336101744186, 6931142360004, 139932192568, -83363795396, -225773039599, 32895854739], [936940496773300, -9839344145493, -201996725909, 225331074835, 283824455991, -25457806511], [-281498919087694, -4639656757812, -243655249249, 58413298684, 106072383364, -54980217331], [2807624242065656, 1090606664194, 42899190908, -48844866008, -10672127984, -21167878126], [-77420811454855, 3733725611492, -282950935993, -276171909655, 77458875454, -31507143925], [-290555948775024, 2878007521193, -439772123399, -175597901414, 238653069344, -13170783453], [288599578774750, -3397191508040, -115529478910, -283374172034, 363469282955, 5839605533], [592392270014062, -1524082831638, 36261178774, -253028881778, -174452687858, -33424882390], [-1362489346953659, -9236044445082, 177871056183, 146232485517, -168644471158, 35459042481], [576409696569496, 10996931116746, -5206496750, -101061048990, -153360847804, 46460805042], [178454648998196, 3379869119179, 177773380117, -265083031302, -43521300998, 69593337175], [-795776542736557, 8240510177098, 19866162224, 712733489185, 224202643916, 8165013750], [220530325034828, 2745680411144, 500823449474, -96984196724, 65877344052, 29208872572], [1087045823649166, -2519654554658, -336086756484, 104293484690, -54498000690, 26854253294], [779499440985342, 1958670279853, -162631755765, -159749275987, 11566309787, 15489128679], [-95453914306217, -6428585745636, -111238373291, -168881945600, 5101225426, 96101267454], [-917890022698098, 3879237346935, -242921327727, -22600272789, 122078836437, 30318726019], [925276363990058, 1041537591600, -10118640290, -342361644652, 64563147230, 118447239626], [-1948321126861989, -8177250337314, -252740847970, -338068589261, -180691436508, -48559749670], [-1784089343842914, -4161767906539, -271934305367, 489264254787, 65508635595, -23799016897], [-2978924002778963, -4686837326117, -290160633480, 233249503668, 145289725441, 11993915862], [251586903157288, -8339570851831, 446314900475, 829200521549, -158003921801, 21702372423], [272796996683531, -559408739112, 88439374179, 346067685851, -6725843276, 7541317347], [848111905455834, -4147028256814, -55946126542, -257518711334, 58995691032, -75126180352], [848158412194526, -7171077257906, -379198507941, 635291858354, 115572141128, 14580825855], [-1246396920411026, 14085006237714, 10083289728, 756688365612, 190638222450, 28318009036], [537214741606438, 13490546487991, -2842798571, 214953284349, 108919400615, 116635746557], [548853133563927, -27809485769623, -39602163594, 592389170740, 183014113920, 10001652169], [1110470149930717, -3422180127504, -60643598922, 254623965532, -73944688373, -69379954638], [-319016595894292, 2023767644690, -158299169464, -580718540538, 326180943828, 34749097316], [-3418454364477556, -4880360890108, 28998791966, 704921146008, 95584008820, -29224345754], [-2549223803404534, 15077545074012, 167301265990, -242206059814, -376684986656, -22820329626], [889676495750105, 18384414590574, 74679698252, 550972822115, -67901368644, 32276715394], [-183856661495716, 23421830507728, -40384581262, -27049665169, 29821889042, 14138210675], [2189039429175372, 18127796204459, -7150077821, 688326149201, -122212279803, -46036864823], [4175172518410450, 8597327932582, 38229756220, -243225357240, -181607869286, 31294471730], [-1137421853026848, 22430015570659, -375204462003, -958019755984, 94539017846, -42172091427], [-270029500856253, 2209083772722, 330352289087, 91368461033, 77542652544, -24155059897]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_280_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_280_10_a_b();".
// 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_280_10_a_b(:prec:=6)
    chi := MakeCharacter_280_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_280_10_a_b();".
// 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_280_10_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![561668029200, 49830144408, 981970281, -3104100, -69874, 84, 1]>],Snew);
    return Vf;
end function;