Properties

Label 2.86.a
Level 2
Weight 86
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newforms 2
Sturm bound 21
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 86 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(21\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{86}(\Gamma_0(2))\).

Total New Old
Modular forms 22 8 14
Cusp forms 20 8 12
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.
\(+\)\(4\)
\(-\)\(4\)

Trace form

\(8q \) \(\mathstrut +\mathstrut 269553027322002295200q^{3} \) \(\mathstrut +\mathstrut 154742504910672534362390528q^{4} \) \(\mathstrut +\mathstrut 1484419974003616205488098222000q^{5} \) \(\mathstrut +\mathstrut 826842279871891938749146151780352q^{6} \) \(\mathstrut -\mathstrut 615523597339907391982903341468497600q^{7} \) \(\mathstrut +\mathstrut 118828885947059966775185311634015921096424q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 269553027322002295200q^{3} \) \(\mathstrut +\mathstrut 154742504910672534362390528q^{4} \) \(\mathstrut +\mathstrut 1484419974003616205488098222000q^{5} \) \(\mathstrut +\mathstrut 826842279871891938749146151780352q^{6} \) \(\mathstrut -\mathstrut 615523597339907391982903341468497600q^{7} \) \(\mathstrut +\mathstrut 118828885947059966775185311634015921096424q^{9} \) \(\mathstrut +\mathstrut 467765378440194361291224013224855522508800q^{10} \) \(\mathstrut -\mathstrut 101797481860206574106717351656968700225898784q^{11} \) \(\mathstrut +\mathstrut 5213913831757698498672213602352359321842483200q^{12} \) \(\mathstrut +\mathstrut 479112641772940819648401363972751216889009628400q^{13} \) \(\mathstrut -\mathstrut 4970712848829318407030506490397860087671403053056q^{14} \) \(\mathstrut -\mathstrut 271745842789897203677825843886989463236276342644800q^{15} \) \(\mathstrut +\mathstrut 2993155353253689176481146537402947624255349848014848q^{16} \) \(\mathstrut +\mathstrut 55754425565484547496180681745937601424459350789331600q^{17} \) \(\mathstrut -\mathstrut 252682171279403821081304108731828847541902253647462400q^{18} \) \(\mathstrut -\mathstrut 2929591165061188546903035525219403377807083997751247840q^{19} \) \(\mathstrut +\mathstrut 28712858139594372059529236779806969160727870048305152000q^{20} \) \(\mathstrut -\mathstrut 177837756224578674727338565916857153704267407366365634304q^{21} \) \(\mathstrut +\mathstrut 1731453221265625162743138887790712836702697145195141529600q^{22} \) \(\mathstrut -\mathstrut 8326608631124368674563983949658834032240169517280962804800q^{23} \) \(\mathstrut +\mathstrut 15993455694178489043546292624851516862727062593102237663232q^{24} \) \(\mathstrut +\mathstrut 50440983628024394799954236936815371007845526315736871515000q^{25} \) \(\mathstrut -\mathstrut 1740356260062666646895169756863700990277016303011003197554688q^{26} \) \(\mathstrut +\mathstrut 13715447878465367754004449232495137063430285698294322855310400q^{27} \) \(\mathstrut -\mathstrut 11905957910500680414116629208459082705745212431581442578841600q^{28} \) \(\mathstrut -\mathstrut 80804886823838884071775456652113469944063607328695210326972560q^{29} \) \(\mathstrut +\mathstrut 571083860688419042422776990904949923888548001591164854599680000q^{30} \) \(\mathstrut +\mathstrut 1633567803274849645661476153182778402652823922406345625782646016q^{31} \) \(\mathstrut +\mathstrut 51629154124411494241455192195981789763524293220492860915597116800q^{33} \) \(\mathstrut +\mathstrut 76810738067911368171154268692467615587532838104833686639805988864q^{34} \) \(\mathstrut -\mathstrut 303907195718456817012846584882539686080918551354269325209569385600q^{35} \) \(\mathstrut +\mathstrut 2298484933399084177227191587423576584872237130977009353119429033984q^{36} \) \(\mathstrut -\mathstrut 3570318342481673237747139074956703063851617532078736009959140760400q^{37} \) \(\mathstrut +\mathstrut 13557340070278076771546516361269540434280107191837609612694624665600q^{38} \) \(\mathstrut -\mathstrut 52826197364072519061410300097668400287772548032981050331137250627392q^{39} \) \(\mathstrut +\mathstrut 9047898296290546547999898244249449210914774552627163603109989580800q^{40} \) \(\mathstrut -\mathstrut 101879485641498700390198644632399823374919799815471748501171952382384q^{41} \) \(\mathstrut +\mathstrut 3430430536466684160965519739670099420062166857006963306538602804019200q^{42} \) \(\mathstrut -\mathstrut 983017174328970333173953499578367781476596926607126329920567679034400q^{43} \) \(\mathstrut -\mathstrut 1969049667080889253984850502657705669675769736731414525891979451039744q^{44} \) \(\mathstrut +\mathstrut 129414053322130030478694471180705803462601292852690242215929564439862000q^{45} \) \(\mathstrut +\mathstrut 52217657421151487659816109775539445219348550536042728197335901349609472q^{46} \) \(\mathstrut +\mathstrut 418464261387472986972541171943537580545821035775417913504888854487318400q^{47} \) \(\mathstrut +\mathstrut 100851760839323638759435620035232536926471244341500257660581027459891200q^{48} \) \(\mathstrut +\mathstrut 4861201030765380588086678338224369843323136135029448905719677358494048456q^{49} \) \(\mathstrut +\mathstrut 3507024174205801426674718518642244780162117239187213725517524731166720000q^{50} \) \(\mathstrut +\mathstrut 14458555377434229192130842989239979469835897417393174031425068906319738176q^{51} \) \(\mathstrut +\mathstrut 9267386290289323200578772342872056249213880282578624968309635279119974400q^{52} \) \(\mathstrut +\mathstrut 79238893661066286548123396135403859078450913300218508816248269289127876400q^{53} \) \(\mathstrut +\mathstrut 127479891442271163284555108637214012971880501702375711073997768996547461120q^{54} \) \(\mathstrut -\mathstrut 115892409687490903974845422600674579019444744881898202357235531152417400000q^{55} \) \(\mathstrut -\mathstrut 96147569677439233318535669969921558315737813875377064640901250850221981696q^{56} \) \(\mathstrut -\mathstrut 708291507196858496950336584814158730962641225991642476431323602773416585600q^{57} \) \(\mathstrut -\mathstrut 1872002936537433921837667385553562778861363634278890764697018281522678988800q^{58} \) \(\mathstrut -\mathstrut 9718564654902213878699228800763515278155682743529550250614492725480982626720q^{59} \) \(\mathstrut -\mathstrut 5256329051546314319453026985277683016572662308968704521552881311019748556800q^{60} \) \(\mathstrut -\mathstrut 22502293106918750966308876870780925084969607013559207911952914084993042461584q^{61} \) \(\mathstrut -\mathstrut 512745364371250743679940949316115524515609735680232267031558635700145356800q^{62} \) \(\mathstrut -\mathstrut 33253188453235282602659365361968943040833119870915187924088567977510775768000q^{63} \) \(\mathstrut +\mathstrut 57896044618658097711785492504343953926634992332820282019728792003956564819968q^{64} \) \(\mathstrut +\mathstrut 208538001796442375993409864803085821949183354617266498590081927679837556887200q^{65} \) \(\mathstrut +\mathstrut 731160993111948540733699201721170861720074557308310226128580428775765105967104q^{66} \) \(\mathstrut +\mathstrut 618430087497352208039439690042770354046234497149407493289606954052900779447200q^{67} \) \(\mathstrut +\mathstrut 1078447433982339860563125081874542097822167715470030508613160573169686911385600q^{68} \) \(\mathstrut +\mathstrut 3749053568809890429242703488244876040784090696484290653048341072416446306222848q^{69} \) \(\mathstrut -\mathstrut 1712687631740715700813202036984572612382504801817892498537480528754292817920000q^{70} \) \(\mathstrut +\mathstrut 2914456738203768985035713583890518823624642690727941357848894790247547420996416q^{71} \) \(\mathstrut -\mathstrut 4887584016255318026061913823771859917295703698729015999704548287676928124518400q^{72} \) \(\mathstrut -\mathstrut 52943131008350183167528071295913452178374790158418012532048380044880083916135600q^{73} \) \(\mathstrut -\mathstrut 5385145831472910827359428561344620582528728363972547468922141298607300433412096q^{74} \) \(\mathstrut -\mathstrut 244065785632991520690690211282530753260324493030420362754725144723791313638020000q^{75} \) \(\mathstrut -\mathstrut 56666534405717979986979761320503763991405624446638989149707898025031509674557440q^{76} \) \(\mathstrut +\mathstrut 7111955725060403646131701118241159104315578946674911063489391126414174064505600q^{77} \) \(\mathstrut -\mathstrut 261791324240468500546822182213269410234570008409044716200403595760397197089177600q^{78} \) \(\mathstrut +\mathstrut 1025784172367515922042732760261822950646306776701865615471484090898577406684787840q^{79} \) \(\mathstrut +\mathstrut 555387448958203245946423939712902268301983657299063771266156608526982192300032000q^{80} \) \(\mathstrut +\mathstrut 6626221899858755588629003462743365911014779696849691608947928310864505539575554248q^{81} \) \(\mathstrut +\mathstrut 1237202594219701351105933788823014042420042676149637742207414672365206952004812800q^{82} \) \(\mathstrut +\mathstrut 7219217659129802650581247135505194878762696845871793297216243109006748312900372000q^{83} \) \(\mathstrut -\mathstrut 3439882483235606329833492514826381898240253814840602267894292173353667249060184064q^{84} \) \(\mathstrut +\mathstrut 3912830459186382949391201016846469773029483089249926950618811281750507636796962400q^{85} \) \(\mathstrut -\mathstrut 2342186222009889804507850275471051931820653834701123909729421408779546889769975808q^{86} \) \(\mathstrut -\mathstrut 200105984362001069230238288458032592723516056569427552223412430020167042682139643200q^{87} \) \(\mathstrut +\mathstrut 33491176074286972493521577678464668166400076298916295145398472140103501043308953600q^{88} \) \(\mathstrut -\mathstrut 85965578199760485237519211339136793968971795955905134183109192927336430636082565680q^{89} \) \(\mathstrut -\mathstrut 94533535251611151078408544072096720829273563268460968512105519589964501772507545600q^{90} \) \(\mathstrut -\mathstrut 17240791454229674589215417657881373257899663802715180384931132222938185892409821824q^{91} \) \(\mathstrut -\mathstrut 161060034623876366108872021610492466260287712176813826455663807597989574797479116800q^{92} \) \(\mathstrut +\mathstrut 1290240338494900619547839525109983192448007231583110758481049139812759993872897459200q^{93} \) \(\mathstrut +\mathstrut 1433869637842692892430214853576316577021589437801857515062390740922072334616833818624q^{94} \) \(\mathstrut +\mathstrut 3159648538409298714208978848652977445274561175020435690323251204202011070319474616000q^{95} \) \(\mathstrut +\mathstrut 309358424536879805970631109977718021156076018263400592941652229972136641831016333312q^{96} \) \(\mathstrut +\mathstrut 2815820736417570017461296806434083982402251944048464366157428863016933480912055050000q^{97} \) \(\mathstrut -\mathstrut 1287533752711804733389579391300849145998644069973215505103549747592962474281376153600q^{98} \) \(\mathstrut -\mathstrut 13821486497634205785853128483475818467151466908122344585095762540941651078331897454752q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{86}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.86.a.a \(4\) \(91.510\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-1\!\cdots\!16\) \(40\!\cdots\!56\) \(68\!\cdots\!00\) \(25\!\cdots\!32\) \(+\) \(q-2^{42}q^{2}+(10193857391958586164+\cdots)q^{3}+\cdots\)
2.86.a.b \(4\) \(91.510\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(17\!\cdots\!16\) \(22\!\cdots\!44\) \(79\!\cdots\!00\) \(-8\!\cdots\!32\) \(-\) \(q+2^{42}q^{2}+(57194399438541987636+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{86}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{86}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{86}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)