// Make newform 320.10.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_320_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_320_10_a_l();". // 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_320_10_a_l();". // 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 := [-79, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [0, 16]]; Rf_basisdens := [1, 1]; 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_320_a();" function MakeCharacter_320_a() N := 320; order := 1; char_gens := [191, 261, 257]; 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_320_a_Hecke(Kf) return MakeCharacter_320_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 := 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 := 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 := 10; raw_aps := [[0, 0], [-130, 1], [625, 0], [190, -69], [51360, 30], [-89570, -564], [158010, 348], [68636, 5160], [332730, 6393], [3446874, 5880], [-145916, 26250], [-5630690, 59976], [14886726, 72180], [-5854090, -99615], [-31246650, -42573], [-4708890, 212532], [-46465428, -373980], [-97836962, -721440], [-109883710, 203139], [-155603508, 1397610], [-49612030, -2047716], [-271130888, -70020], [-628457850, 192957], [-231145926, 5421960], [835858370, -2902956], [593410074, 3512040], [221674930, -960723], [-249042630, 5435043], [726810718, -1177080], [-763442310, -13181592], [-4425461570, 7684191], [438215520, -29723790], [-4562549550, 16951536], [3276221276, -21103380], [3893692326, 15977160], [-4241962580, 52919370], [1802716150, 631704], [5200327670, -33654435], [3622830, 109829019], [-6033752730, -39181368], [-20120817132, -31184580], [-1868195030, -9437520], [-4045319028, -211723530], [14961338330, 71205372], [-10284296370, 4522884], [9530496952, -90525120], [7052087504, -173409870], [17392057930, 229646085], [-8107988550, -4030497], [21212864146, 58986240], [-26817249030, -333620172], [-11554367976, 197678220], [16175404814, 140156580], [50380749600, -190455450], [-6281712390, -277620312], [-100072707630, 4792821], [-41240171058, -817339440], [15073810540, -302388570], [11884468870, 1099932648], [44344622406, -910845060], [-101670486130, -712207707], [-97046310210, -242777520], [-102274523710, -608327541], [45587739828, 179012910], [137326658090, 1390226376], [161227146150, 259723932], [-137606920792, -161514630], [-126530990590, 1506593784], [-74126232030, 1857192975], [-360018204950, -883612080], [203891824530, -1071014904], [-79341904344, 1817532960], [-54375776810, 576658179], [-372062485370, 962419128], [-107446254268, -1438135800], [380822692410, -2341372227], [-183838550874, 452220240], [-387667343330, 2227520076], [140179190370, 2969575200], [569175787166, -2431842180], [772137282756, -2939060100], [-636913820666, 756762120], [-459921283236, -5679437490], [-242441264830, 4017285204], [413950061752, -4805148960], [243183984390, 1246459137], [167496460806, -335582700], [1042843024730, 2155836672], [513733206210, -5087981880], [698658521650, -269391303], [-1026679377630, 1390279275], [-235494243816, 6217550220], [992561459110, 5394966303], [70709287200, 1627339350], [658472549276, -1031668560], [-3599220270, -11873375979], [-1106148875046, 2784123360], [17965062714, -6376147200], [486139326350, 3686895693], [-1170694100654, -15672561000], [-2077320826150, -7212961233], [442620804990, 351145080], [2020367856750, -2408422911], [450217464030, 20787710700], [-3522141156952, -3728230950], [-381299545750, -525114864], [-2823403235910, -106720317], [-2718204624750, -9624184224], [-3930607558080, -3829718340], [2072229737270, -17974239060], [88903639510, 34295369859], [1731049021150, 4947253596], [-1352323390470, 22350324708], [-2279799676300, 8081101440], [778435665628, -19319414430], [-2156581307538, 34090131780], [-3295186212850, 7139276073], [3859099092630, 7099379751], [-2119116417930, -1314857004], [7993719098436, -4170063180], [446412433462, 10977961680], [-1269128866030, -62205321492], [6998619399630, 13452098124], [-4723454537130, -31960769439], [-381560883352, 3410779230], [-4976157968130, 1440325320], [2322921561154, -69700159560], [4840236023064, -44637103620], [-9620591428370, -11798915733], [-5985918236810, 34563357504], [-6468463294540, -56373978780], [-9198204807750, -22752569367], [-9355741693436, 3971363130], [-10156025871050, 15591317904], [1377068364522, 69609472800], [3401690820482, 2853109320], [1624023880830, 15951738900], [2674204206170, -37808296713], [6362802246390, -33579090444], [6207288287034, -75692678400], [-7264217059696, 31601446230], [-3896141278506, 112998109560], [-9044140698470, -59709310791], [-12427495861470, 23036954655], [-2412995306402, 57518556480], [-106883870352, -5360973660], [-5840557301330, 105758149860], [-14242078014510, 53900734296], [-1639572880804, -115064028180], [-14062835400390, -118210764507], [-4729044128570, 13737984528], [-1099248393618, -107851512420], [-14995451074930, 67606481793], [-3496414089810, -88760050341], [24130089529490, -64091125377], [8197545214068, -42672479670], [-2028548491184, 212982755940], [-1832576801322, 11657513940], [-4383092796310, 73293953676], [-30239177319822, 62261948640], [34401042140370, 80160703491], [-24533966036550, 61531654848], [-5188274037290, -77116045761], [-24836809734264, 139111339290], [3398550777810, -250832562804], [9610292513490, 81369589341], [32636549761924, -75535414110], [22163381320390, 136204402428], [30625281169970, -89867580840], [-21207743820210, 163400851884], [26128491235716, -66125203320], [-17490276011054, 106710632640], [-3455053091532, 25077782310], [-12231852028030, -170686481196], [-8243616827504, 302057325120], [50053220385642, 98731851000], [41422813207208, -165272122950], [31901388853866, -182183386560], [-42677406707510, -137900907735], [14835089699086, -98239727040], [30395016451030, 335996999259], [31157451124368, 123651451770], [24855820340470, -202142883636], [-4581422810910, -294044126340], [-27839512604310, -266968724091], [1005676329090, -70738374840], [-11630702700530, -96868562328], [5673550323470, -94310609847], [29084714057342, 151762398060], [87321039934068, -73915658310], [31579182655850, 397079072976], [-47432124374730, 402974073393], [24063191084600, 295995050070], [-16099571947650, 628623196560], [-14970403410630, -29361950577], [77614766653050, 44566827708], [48852038463374, -101247236820], [10931243918470, 353660104908], [45896240317410, 102612865920], [729068940450, -592964602635], [-20579015868966, -653801070840], [1997989751068, 683973265590], [-17796474190730, 465083454204], [-39195900384046, -278613000000], [-82542615397860, -2683088460], [9394884465510, -925179794448], [-117611314892192, -260932128600], [-41301335089890, -376486985535], [-35116225312326, -236645927640], [106132065670616, 506116985250], [-55930798491430, -132835190748], [-109241977642122, 478642200360], [-76999323418190, -347087554083], [51406932378810, -104196950037], [24549745160688, -164841880260], [103816960564790, -152304626700], [-35297591146610, -47733283905], [137819331580818, 84483759840], [56556293869230, 1072671078819], [113816898489870, -450544722156], [188267108457094, -62630680080], [97346488255648, -738165302460], [-63128176740462, 161268404520], [8704364855650, 361745981433], [118033586938890, -632092710753], [-106657462813898, 413339813760], [-97408201996830, 423762495300], [183759942268800, -91019658600], [147767945231830, -618602486961], [-3423858545784, 1229691479370], [-7396455279410, -814147391148], [22423375458668, 1242541046520], [116221954043452, 1441091469630], [104966750977014, 1738185238140], [-189427783759810, 563893105509], [-128260333162530, -1557998751705], [82077748886342, 831808367460], [50986312375350, -1463403424644], [55874230126500, -1716039753120], [54105184956420, 1206416856990], [239981169987510, 473223071253], [-52017142756384, -283532392170], [-61301154143030, -924079597695], [175091409917290, -702843244560], [-398865434910, -1869355362840], [-50841275906904, -2122883516520], [-132215010193010, 1872973253895], [16348208949528, -1484252627370], [117071843823836, -1239011519820], [350371599486450, 7857834009], [-67086932857490, 583506696780], [-60006170380782, 2607209220000], [-182096437816890, 1625202216447], [24005150921294, -1775364440580], [63631508565990, -2205459743568], [-278369461482708, -641917854480], [30483397775578, -176866131960], [-37404789607150, -694350226377], [88308757930230, -2374428913944], [270508924946810, -148490400960], [122051903646490, -2721999031131], [61641301261290, 2785250957319], [-413529073726574, -260330531040], [75372762311470, 2367317591760], [93047539499370, -28001344476], [-77291305835956, -1553039402400], [-171523792440486, -2778054488160], [-295473981199638, 586967780880], [-274896732778810, -764149025151], [323613311895150, -2602889715384], [-359581550793182, -2099564725200], [-172930174633150, 3174899972859], [109955603351450, 3404339625492], [16325187738256, 1126453771440], [121811344972010, -494099035584], [-55635635089670, 1850725705041], [-203467924949550, 3643917738711], [-178477808107670, 4178256515160], [446498098998758, 384619549260], [195664609569072, 768352115370], [-30079824952470, -751975145331], [-19638203035196, 3715695377910], [339582611577750, 3949039230711], [-86535176193062, -2204219121120], [37306970483210, 3392726608683], [299340594320676, -708805955670], [400802098292570, -2378880677028], [-456814151175810, 900928037580], [-457766674622480, -469011217500], [339873415215318, -4834698581580], [-42528166907058, 4040656989480], [32188344529770, -3557144918721], [-186327360830190, 496309684116], [47876568483408, -1693193214450], [100590726049990, -3682807863552], [-368171391643698, 2014597247160], [-780491819871308, -2631054313830], [-380726511261330, -904595852304], [416661329887140, 2635715142000], [-7349263836910, -2196684848997], [-701112066876430, -149374326240], [238893029629590, 4599714978888], [386288447971288, -3824113533900], [882555941265390, -1412483031711], [-621033446281408, -1976608048410], [515857465282370, 5365357981404], [-520112172223230, -5694203848473], [-421951843560470, 1637216629560], [-649691971583064, -4075849922040], [-407362936697570, -3256817959944], [-54439422012150, 2797002004317], [-451173837896190, 697631835120], [654843515781090, 6917169556560], [-308890864629970, -3391461534039], [67408224511710, -10488528887085], [-1329470658070006, 866318410680], [29201143379964, -2741276902200], [378775882225908, 774550644450], [296826780559130, 1399746738792], [-149036036522922, 8671392752340], [-149565551794072, 1507154337630], [62788652474690, -1959162784176], [-1525332536652930, 311574071520], [1005048172393570, -2816572104327], [-766950714544350, -6806532625560], [-1012044564737314, -886884949260], [179269860506348, 7898681346480], [-1805942567523850, 689022652905], [744346603628790, -8267249112477], [-742896818889450, 5104681674108], [431449912306990, -1100304721080], [-350280571628490, -2413228747104], [-1546976099669720, -2216885388780], [262988570143830, 6467781262221], [1247310223518224, -1206412577730], [681728326825530, 5301357861483], [820723803775150, 8115715848600], [-250742001939750, 10434736414080], [-970985206773994, -3243308988900], [-1721276859002690, 4439986434351], [-1444853424312290, 1965575373216], [397622413395690, 2965062021984], [-82119172589226, 9016922692440], [-1732096151148500, 1662024146610], [1363147435600590, -6284657411976], [943177855879212, 1522444101360], [1335734125921510, -4253802648480], [1092283307117330, 5474278158783], [-782523574079964, -8088418456950], [1497440496903630, 1750400059764], [-976090550937040, -10435772947710], [81892989660170, 14027949907848], [2183548462733358, 734212188960], [-139801706367230, 11551136100921], [-334715545793726, 326363718240], [202374405982770, -10470665182836], [-369239295212232, 5428273511220], [-2517102221762472, -1361898042510], [-2181581328424590, 5859294578364], [1294437458706570, -4170023595063], [-777326273204330, -2332035301188], [172111628404746, 2753421599520], [-144513887425290, 6626093147523], [1459770384829884, 347062733940], [-1243309860296710, -13895497232145], [61585695528650, 10386823994988], [-1808272543125570, 12255702617700], [-193884904286750, 5538604040253], [2026524768887990, 3155004708900], [-1351798800156912, -2992658078910], [-312116624674012, -15447370746000], [1102466517215730, 5144610118569], [-625067654878650, 3283906644120], [-325471865858804, 13542208546410], [2862554615859310, -4055093858820], [-170430012518868, 15234754165680], [1761133432298868, -1915442534550], [831320674277330, 2791101447168], [1695035299589766, -10216228691100], [-1409079711423550, -14497109933256], [-2178292895008782, 6877769153280], [-1364042338226190, 24332521964664], [-594813385728650, 777422768967], [1844399948374290, 7479654469296], [1489273495648796, -8582956280280], [-866082302278470, 16920554781153], [792200121216028, -12809671591770], [1839785442552430, -5538655742760], [546648175809830, 13117029228801], [-823772000812650, -24174582210909], [323151838018550, 19555440398580], [1707874464148470, 13991359739016], [-583662189720516, -4833738395940], [-2177626565649550, 7792319421075], [300925886276124, -16848861639750], [1259157358522130, -6071036199552], [-978795127265288, 12807643873740], [1256504544694074, -1120797313920], [-783996472856128, 4093026089550], [292569155849814, -3018552725040], [-1557080101278290, 16016379081480], [1581452033064090, -16354867980300], [-466963446430250, -6105915535341], [839381798344530, 3735954502656], [-1500180722191290, -20075528968800], [-1461304239900500, -21364799471190], [223774169047270, 16968972401700], [-182426884897902, -30424977366960], [1604763617096870, 22566800335581], [63673446304740, 41149405534500], [858782628596690, -19161118500792], [2395431078199950, 3592748928264], [891102660459630, -35689948517211], [-4750221121337728, -12127336371630], [-1122749614001710, 7891704486348], [-1106706556407774, -22827888062040], [3958372435543584, -13174285327320], [170171322581950, 31004988700371], [1243470965966370, -36919745654700], [-3681549141324870, 13323140830113], [1518610923189630, 12698994644400], [-30448516420130, -14337867050484], [-2081173281374010, -3792618680733], [820769795764380, -14629051112340], [6089386263091850, 8565602469408], [1282813163150790, 23231253228984], [1361812662056430, 22307506209729], [-1912290711562626, 11257885831980], [-399596421114808, 7473293176410], [1832641575950424, -20765223781800]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_320_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_320_10_a_l();". // 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_320_10_a_l(:prec:=2) chi := MakeCharacter_320_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_320_10_a_l();". // 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_320_10_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_320_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign))); Vf := Kernel([<3,R![-3324, 260, 1]>],Snew); return Vf; end function;