Properties

Label 2.88.a
Level 2
Weight 88
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 2
Sturm bound 22
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 88 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(22\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{88}(\Gamma_0(2))\).

Total New Old
Modular forms 23 7 16
Cusp forms 21 7 14
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(2\)Dim.
\(+\)\(4\)
\(-\)\(3\)

Trace form

\(7q \) \(\mathstrut -\mathstrut 8796093022208q^{2} \) \(\mathstrut +\mathstrut 80974745693182072212q^{3} \) \(\mathstrut +\mathstrut 541598767187353870268366848q^{4} \) \(\mathstrut -\mathstrut 2920508893103043329722952304510q^{5} \) \(\mathstrut -\mathstrut 6381576261198159352072644593713152q^{6} \) \(\mathstrut +\mathstrut 5685733200018651170793947092255103816q^{7} \) \(\mathstrut -\mathstrut 680564733841876926926749214863536422912q^{8} \) \(\mathstrut +\mathstrut 879093243194986986690731015132457684762339q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 8796093022208q^{2} \) \(\mathstrut +\mathstrut 80974745693182072212q^{3} \) \(\mathstrut +\mathstrut 541598767187353870268366848q^{4} \) \(\mathstrut -\mathstrut 2920508893103043329722952304510q^{5} \) \(\mathstrut -\mathstrut 6381576261198159352072644593713152q^{6} \) \(\mathstrut +\mathstrut 5685733200018651170793947092255103816q^{7} \) \(\mathstrut -\mathstrut 680564733841876926926749214863536422912q^{8} \) \(\mathstrut +\mathstrut 879093243194986986690731015132457684762339q^{9} \) \(\mathstrut +\mathstrut 45346814034776642044414320718694217500590080q^{10} \) \(\mathstrut +\mathstrut 1045718858653864846400722202389139493224706204q^{11} \) \(\mathstrut +\mathstrut 6265117491533843230876941959545501178520403968q^{12} \) \(\mathstrut -\mathstrut 2671899427313212817950921193897595638829922521718q^{13} \) \(\mathstrut -\mathstrut 100967058322828401682216463293916492742918286606336q^{14} \) \(\mathstrut +\mathstrut 4845183259634109809863249173665448129055865803433880q^{15} \) \(\mathstrut +\mathstrut 41904174945551648470736051523641266739574897872207872q^{16} \) \(\mathstrut -\mathstrut 911843101929464476520354640649817347730389523078087394q^{17} \) \(\mathstrut -\mathstrut 6695389209765332488782046017891867662712758472441593856q^{18} \) \(\mathstrut +\mathstrut 28924509410078824981374512978553416734376012822976487460q^{19} \) \(\mathstrut -\mathstrut 225963430866330245085937808931676908407220053845097840640q^{20} \) \(\mathstrut -\mathstrut 6592029819864249529543089444323273550331678016715264344736q^{21} \) \(\mathstrut -\mathstrut 43061396618197167843252134244216093933200210184878828814336q^{22} \) \(\mathstrut +\mathstrut 286160503294493310726157265691368539698399154450063308833112q^{23} \) \(\mathstrut -\mathstrut 493750547968143722671467529194448041338254084819611316912128q^{24} \) \(\mathstrut +\mathstrut 16773411111251446550161743540865958027446772523897993540208225q^{25} \) \(\mathstrut +\mathstrut 7590058159920279723504669770694252396564881791202346294837248q^{26} \) \(\mathstrut -\mathstrut 176541606068591278219279589633249340635185932685702156017987960q^{27} \) \(\mathstrut +\mathstrut 439912298812329995920278403244976484632759548258090763287527424q^{28} \) \(\mathstrut -\mathstrut 596105450597790504763677619332294088233825086729323088019355110q^{29} \) \(\mathstrut -\mathstrut 16684169114283099118084213828160605969035928215602441898468311040q^{30} \) \(\mathstrut +\mathstrut 203456212626134666254474285706188851196255130095897048531164029984q^{31} \) \(\mathstrut -\mathstrut 52656145834278593348959013841835216159447547700274555627155488768q^{32} \) \(\mathstrut +\mathstrut 6611172792710809377309680424239520760642205214674854003605044304q^{33} \) \(\mathstrut +\mathstrut 4335268461612023802292419723184381313338650045913048973830734741504q^{34} \) \(\mathstrut +\mathstrut 1404000317055518859792186972116908136345047884535988124862456089840q^{35} \) \(\mathstrut +\mathstrut 68016545251019659144585255350697211368608776233487887607853792362496q^{36} \) \(\mathstrut +\mathstrut 415614429276011774504094954290616884839739198968825236706124661649826q^{37} \) \(\mathstrut -\mathstrut 414981295954429455801366187408002568554185302345422214438418309447680q^{38} \) \(\mathstrut -\mathstrut 4848075917433942437390150341067585145352417589100388240306020681209352q^{39} \) \(\mathstrut +\mathstrut 3508539796729889366487207908624035246843099907589385799724199701381120q^{40} \) \(\mathstrut -\mathstrut 6418618697764445378474998422072543198713839893055268002100719980540426q^{41} \) \(\mathstrut -\mathstrut 5897709362042313573225652906823647681445624592679859944441622179086336q^{42} \) \(\mathstrut -\mathstrut 256517552495093346533878881868115114874049551075704932806208510128442948q^{43} \) \(\mathstrut +\mathstrut 80908577810214279425242339449244072997228603775418484308777319956217856q^{44} \) \(\mathstrut -\mathstrut 1008136421053650594053909266821883015329870913151316988628297230599950070q^{45} \) \(\mathstrut -\mathstrut 2187246920712917002550544397108607274991072632429297116085248698406141952q^{46} \) \(\mathstrut -\mathstrut 16733188737624868265198708846801005986602496658340465177406132772864641744q^{47} \) \(\mathstrut +\mathstrut 484739987099808063191307956121587154538073135694033668175423182368407552q^{48} \) \(\mathstrut -\mathstrut 31666208429464795334783441124742945856803380727775507539289480402564382849q^{49} \) \(\mathstrut -\mathstrut 325107647412956537138891644142795977272301081819324164650185390975405260800q^{50} \) \(\mathstrut -\mathstrut 1488436923534378108307311683424482833828549344295074188287164039294686211096q^{51} \) \(\mathstrut -\mathstrut 206728205125918983383682533740981187243326403800086050617337009128238743552q^{52} \) \(\mathstrut -\mathstrut 3873744239282753468251342153387956686347628071807147461758111421873328064718q^{53} \) \(\mathstrut -\mathstrut 3283102625509781351631282772901147721175145116931081799392401284803938222080q^{54} \) \(\mathstrut -\mathstrut 21516706422211156955719465983343803591388253093419924345604153423284373980920q^{55} \) \(\mathstrut -\mathstrut 7811947759168217064294711991727617178670217755407559258544295859774715592704q^{56} \) \(\mathstrut +\mathstrut 14364804172733276799058201186557640544958360114702437484884205381355873721520q^{57} \) \(\mathstrut +\mathstrut 11229524416113640225861387908623839197226088206078531963629322358384398172160q^{58} \) \(\mathstrut +\mathstrut 65252100640619404855593799009539940476388245196788574888464805291977191707980q^{59} \) \(\mathstrut +\mathstrut 374877897173519797145213143903455880516931011042190085628882375527733993144320q^{60} \) \(\mathstrut +\mathstrut 1457525223204824945702563864576412154462115021937726844158273060443664370877754q^{61} \) \(\mathstrut -\mathstrut 312004483985782428273513783935283410429626502239991458777327019581334005743616q^{62} \) \(\mathstrut +\mathstrut 7772486446481763873885338030021004419412092616480944679915890933796224970961512q^{63} \) \(\mathstrut +\mathstrut 3242178498644853471859987580243261419891559570637935793104812352221567629918208q^{64} \) \(\mathstrut +\mathstrut 17501981465693343534469350932038425739742668860449414641736688344740487354991180q^{65} \) \(\mathstrut +\mathstrut 17227271677828493488011609718768884360090428630829891729822058112954320086368256q^{66} \) \(\mathstrut -\mathstrut 55063608285362556627500287200293476247490508122231737660319354361952125877519404q^{67} \) \(\mathstrut -\mathstrut 70550442839041516525388945659662118556561341681799789207271681940686335970902016q^{68} \) \(\mathstrut -\mathstrut 400760622833636458358612993092732200019663896789093288336500926158559894015798752q^{69} \) \(\mathstrut -\mathstrut 235789562607816141906138248062224322967296448509229261105547772039335525451038720q^{70} \) \(\mathstrut -\mathstrut 809919070647168245287984778020801750754647514600106512071674917136154437663707256q^{71} \) \(\mathstrut -\mathstrut 518030648835487930953142112427925210444137937256121834226706661503466875318697984q^{72} \) \(\mathstrut -\mathstrut 219512310360407845351868489935679581335396475392666027259295788773156980304650218q^{73} \) \(\mathstrut +\mathstrut 452472924926239618943165637943161754452853007389822882038116546395265373814390784q^{74} \) \(\mathstrut +\mathstrut 6593685854768020524596274859859454921366588771690718440054536491922198823162938700q^{75} \) \(\mathstrut +\mathstrut 2237925519713958252330471211436059223523937348828262137788025067535321731107389440q^{76} \) \(\mathstrut +\mathstrut 48189115305190595783092790866014996034307724133112314071272331368300210410524125472q^{77} \) \(\mathstrut +\mathstrut 48081726017840595821209933479161246054796432520697414746736368982540680665841532928q^{78} \) \(\mathstrut +\mathstrut 41887116389531328958071612822794594017886544653464533298449747875153256542229116240q^{79} \) \(\mathstrut -\mathstrut 17483073655232760837896749167977098445596166906703884539729185449335209220594728960q^{80} \) \(\mathstrut -\mathstrut 248994518459692627779800898861327495684048603642164795554115700218871127194937232673q^{81} \) \(\mathstrut -\mathstrut 153202624298947612187137453318057981364754468717800354072092846651985747066681819136q^{82} \) \(\mathstrut +\mathstrut 27257017012339100300294357239702418965717947451107447905359873023757776672679112292q^{83} \) \(\mathstrut -\mathstrut 510033603385821707465274229300107668220346554281768662136841683924010076286226530304q^{84} \) \(\mathstrut -\mathstrut 2506886845979623636803991768917828986869790710671731032315853010962471330626470636060q^{85} \) \(\mathstrut +\mathstrut 519819449213973876349059841635819218020008559557659400228146512177608052150473064448q^{86} \) \(\mathstrut -\mathstrut 1922070520588291269199218450486529887337922429718809259552885686567174203015479399240q^{87} \) \(\mathstrut -\mathstrut 3331714188825896454625401182308883924115908264476133136131968077628398518736418504704q^{88} \) \(\mathstrut +\mathstrut 4440505247623517019185317469641406104946126401490640001257528431562302963958177912870q^{89} \) \(\mathstrut +\mathstrut 15241462304536744288696433335833522172114047020301618477837980810562148664194168258560q^{90} \) \(\mathstrut +\mathstrut 74555096827161452424559032067290863540960492854480108854649860697571114878578265254064q^{91} \) \(\mathstrut +\mathstrut 22140596543144327541314530498775923307971867425518811618874714621917340133633260781568q^{92} \) \(\mathstrut +\mathstrut 68748487413042570550647295232416513125721447243886369418131426808002365372654345636224q^{93} \) \(\mathstrut +\mathstrut 58658951178719098483906984727989597905547419204747217682011490043116799856978453069824q^{94} \) \(\mathstrut +\mathstrut 141850625202675413431502055471455983500859519069709632169030111296354263823346521217400q^{95} \) \(\mathstrut -\mathstrut 38202098296803867369516485901062753981801306524845541206691164975946195491456697761792q^{96} \) \(\mathstrut -\mathstrut 792509579329487478750020088969755231354891894671840856415934423025323544960504090993554q^{97} \) \(\mathstrut -\mathstrut 1080896543784503157734778360994413992107726045156619741783739180932200778075193269551104q^{98} \) \(\mathstrut +\mathstrut 89132395521541223284365576449202212809152812784979536786243127508244728259781595844108q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.88.a.a \(3\) \(95.867\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(26\!\cdots\!24\) \(-3\!\cdots\!16\) \(11\!\cdots\!50\) \(-2\!\cdots\!88\) \(-\) \(q+2^{43}q^{2}+(-107421079853628369972+\cdots)q^{3}+\cdots\)
2.88.a.b \(4\) \(95.867\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\!\cdots\!32\) \(40\!\cdots\!28\) \(-4\!\cdots\!60\) \(85\!\cdots\!04\) \(+\) \(q-2^{43}q^{2}+(100809496313516795532+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{88}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{88}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{88}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)