Properties

Label 1.74
Level 1
Weight 74
Dimension 5
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1\( 1 \) \)
Weight: \( k \) = \( 74 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{74}(\Gamma_1(1))\).

Total New Old
Modular forms 6 6 0
Cusp forms 5 5 0
Eisenstein series 1 1 0

Trace form

\( 5q - 92089333488q^{2} - 129195798226305804q^{3} + 891110479259563516160q^{4} + 23099529720469471562826750q^{5} - 3335695451789843173883221440q^{6} - 4356178487147579439806268090008q^{7} - 382321348565178801062509850849280q^{8} + 32335028924861937033188022998788065q^{9} + O(q^{10}) \) \( 5q - 92089333488q^{2} - 129195798226305804q^{3} + 891110479259563516160q^{4} + 23099529720469471562826750q^{5} - 3335695451789843173883221440q^{6} - 4356178487147579439806268090008q^{7} - 382321348565178801062509850849280q^{8} + 32335028924861937033188022998788065q^{9} - 1082726120403847340426026649690244000q^{10} + 50083756421782317855880849016778496860q^{11} + 1389386159501111619787899997887234696192q^{12} + 4753748397471720291047553799357192278886q^{13} + 263363595637460329420141722180437171694720q^{14} - 5053973124818591694478703469878924549517000q^{15} + 57509595349357883169785087518099233197588480q^{16} + 663279725175909884624200257401822351394256602q^{17} + 6936746909593304334038759770389897835631806416q^{18} + 31387402430958534058855312636541530201995765700q^{19} + 68535205735011202400715856751699460073169856000q^{20} + 873650260100519441483759434874628458149545804960q^{21} - 9492394168845783423791120540922405517833204177216q^{22} - 41173446223173693726617088414497261153026772718024q^{23} + 16216566841440134675070566439222517257116824780800q^{24} + 321010940201155914585034686800814006311220324546875q^{25} + 4496438035379975943261998606627087983483848085006560q^{26} - 12482045195529499122248281309176768796522105943881080q^{27} - 37814301174111882879202704910382387341459683517577216q^{28} - 217397685843681573854388291537651452827057882148450250q^{29} - 809277931163833776121134244746469476931174350662544000q^{30} - 3959918978904396097569386510196715694840286709585401440q^{31} - 9479144378484593489541015426974249650207814007382867968q^{32} - 73769578315518021607552488579011739564089308117835328528q^{33} - 324638192385754362369074018224584035205208569018875155680q^{34} - 1051811671165262926972759981500117019527196204743409914000q^{35} - 3440544904928519792841567448764696150680442862608580965120q^{36} - 6710426612911460903226679780416929578822322356442579950178q^{37} - 20075776649866586500887218371135359952534984707306414947520q^{38} - 53266590853912928667191550341591883996636524676049287303720q^{39} - 87443314425210154156114784714128373503740284779633935360000q^{40} - 89999556000464227434562342749528590851939245788450434103790q^{41} + 151860645080061390785566939100205618848419219581171296592384q^{42} + 1171421579861687300375905115748875459141635816468653452284156q^{43} + 3620387401482413711753562750789250834104238410455247119723520q^{44} + 8637496301209289669216900677161766462676265859857821184867750q^{45} + 13122337792181100518478081314861771936057558300377044573644160q^{46} + 26407117391117866585989236437476333636637017745173062895096432q^{47} - 305063509192507618357183882504465545852009436962512246145024q^{48} - 47859785088079990485612756297590896342589260563105328803969315q^{49} - 198341627273982740927873511630518057650356858808871192258250000q^{50} - 666572032421535161197734372534298884381369595464382997470670040q^{51} - 1819820853930882517149424007699825989256997486995920851867734528q^{52} - 2253672757277210109476755074185818948452790752453662372858468754q^{53} - 981929879310881011928466994898412465282666955980051795540412800q^{54} + 5272301841223840031246441919708126997046467528986443289345081000q^{55} + 17369101267271748189623122493455314696373137673860233497989120000q^{56} + 39672259194438733138452388186436605652387179030172301666478212240q^{57} + 63080485469034924979732075223792805715626817023343777138220605920q^{58} + 49827032054541030737892137385762950528020446432538215007741437900q^{59} + 91369267546356596727702377377074204109693912478731602288473856000q^{60} - 200207946356755779993110124671040230418699916707275462827300034090q^{61} - 456512716932753516995161459627410738703514050956609499700325060096q^{62} - 1444699606935245693297879065147738830138012519879166909023399465144q^{63} - 2671988611100492125931167910213768824231319631401339943695669002240q^{64} - 2221078323698894670419341126197735507928898751783246157619800369500q^{65} - 797490624045567852983392967901985951789407586519264116965780839680q^{66} + 1726934463658866670044856129309631728944059099751189118466533017652q^{67} + 25844159932565717701710366021885253281928511625890180327915132834304q^{68} + 43307862567803975621848996603714756131647359083362619530437754803680q^{69} + 60019715162941233136874683443010604187189046779871920798680662752000q^{70} + 29528228383509762365399718501357560736875772117852859701209862331560q^{71} + 5937319857025079492793067138582418624443592391466592842045028290560q^{72} - 237248424072934328617284197336037660641948064767277176027665176375374q^{73} - 385668396032745357868624366253781435784895053179379214963132226802080q^{74} - 785444151134296088966742788487546108897483082972592363003091781312500q^{75} - 402469771116557587325960702574461860644965601604695023105081798528000q^{76} - 1160921420345569675442649510235573702285949327395662717704239852301856q^{77} + 339124438335685564147183055565832879915531514801135920231578753382272q^{78} + 1210019475236668608258558597507004925943189831192425236151129201347600q^{79} + 7695510793835037324754830224890315360926193938062866122926593916928000q^{80} + 12895807280425851869767179003361520763349212513521199742652083367867805q^{81} + 19869905655996015892436375229312420723635630923691072274009372051144864q^{82} + 10633758535693793129462935403233565634399957806616398467042339222483716q^{83} - 17104473919241589848125111473450344991721950406193981672669373028884480q^{84} - 28641392301460491393930553180957463692521548910270676165772874742366500q^{85} - 130748824105220136322401247193488038601024385677355691179590030869522240q^{86} - 158256246334086183408933559981662943488738078279070353132629771124728040q^{87} - 221808067516055560593893382793377773024661552295403957235825845052456960q^{88} - 44230399340933247556508628049428956972940692248797881370004235332026750q^{89} + 205005728265438963974181523583877066710560749255482389735945594174668000q^{90} + 503963419362898329804217905595235878617892804641236823955454638331741360q^{91} + 1056276182251025158869277711594662138415929570173882076378175681365399552q^{92} + 2087551224100544830404247844376209582022159740935531745376318713047458432q^{93} + 1211599009021421323842808603980852406277919665987296133682388929757489920q^{94} + 1134809355692277446299742482002998127744674957590385906379174039024255000q^{95} - 1239624362529508037215561089866383709629049658259836228855181445997527040q^{96} - 4705643796144250410051227741367571092615369303801441364082016351450032918q^{97} - 7645702282951838206479981702299956043322685777549671658293752892775157616q^{98} - 15823110517274977002648456256699492623727665140588002701017544901953092820q^{99} + O(q^{100}) \)

Decomposition of \(S_{74}^{\mathrm{new}}(\Gamma_1(1))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1.74.a \(\chi_{1}(1, \cdot)\) 1.74.a.a 5 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 92089333488 T + \)\(27\!\cdots\!72\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!88\)\( T^{4} + \)\(23\!\cdots\!64\)\( T^{5} + \)\(32\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!36\)\( T^{8} + \)\(73\!\cdots\!48\)\( T^{9} + \)\(75\!\cdots\!32\)\( T^{10} \)
$3$ \( 1 + 129195798226305804 T + \)\(16\!\cdots\!83\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} + \)\(17\!\cdots\!32\)\( T^{5} + \)\(70\!\cdots\!74\)\( T^{6} + \)\(97\!\cdots\!60\)\( T^{7} + \)\(49\!\cdots\!61\)\( T^{8} + \)\(26\!\cdots\!64\)\( T^{9} + \)\(14\!\cdots\!43\)\( T^{10} \)
$5$ \( 1 - \)\(23\!\cdots\!50\)\( T + \)\(27\!\cdots\!25\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!50\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!50\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!25\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 + \)\(43\!\cdots\!08\)\( T + \)\(15\!\cdots\!07\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(70\!\cdots\!84\)\( T^{5} + \)\(51\!\cdots\!86\)\( T^{6} + \)\(66\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} + \)\(25\!\cdots\!08\)\( T^{9} + \)\(28\!\cdots\!07\)\( T^{10} \)
$11$ \( 1 - \)\(50\!\cdots\!60\)\( T + \)\(30\!\cdots\!95\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!10\)\( T^{4} - \)\(73\!\cdots\!52\)\( T^{5} + \)\(46\!\cdots\!10\)\( T^{6} - \)\(83\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!45\)\( T^{8} - \)\(61\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 - \)\(47\!\cdots\!86\)\( T + \)\(33\!\cdots\!73\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(68\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!34\)\( T^{6} - \)\(32\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!21\)\( T^{8} - \)\(88\!\cdots\!66\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \)
$17$ \( 1 - \)\(66\!\cdots\!02\)\( T + \)\(11\!\cdots\!77\)\( T^{2} - \)\(48\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!18\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{5} + \)\(47\!\cdots\!66\)\( T^{6} - \)\(21\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} - \)\(12\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!57\)\( T^{10} \)
$19$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(41\!\cdots\!95\)\( T^{2} - \)\(41\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!05\)\( T^{8} - \)\(78\!\cdots\!00\)\( T^{9} + \)\(55\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + \)\(41\!\cdots\!24\)\( T + \)\(74\!\cdots\!63\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!18\)\( T^{4} + \)\(65\!\cdots\!72\)\( T^{5} + \)\(72\!\cdots\!94\)\( T^{6} + \)\(14\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} + \)\(17\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!43\)\( T^{10} \)
$29$ \( 1 + \)\(21\!\cdots\!50\)\( T + \)\(20\!\cdots\!45\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!10\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!90\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!05\)\( T^{8} + \)\(22\!\cdots\!50\)\( T^{9} + \)\(59\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 + \)\(39\!\cdots\!40\)\( T + \)\(31\!\cdots\!95\)\( T^{2} + \)\(94\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!10\)\( T^{4} + \)\(98\!\cdots\!48\)\( T^{5} + \)\(32\!\cdots\!10\)\( T^{6} + \)\(51\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(22\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + \)\(67\!\cdots\!78\)\( T + \)\(24\!\cdots\!17\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} + \)\(42\!\cdots\!26\)\( T^{6} + \)\(57\!\cdots\!40\)\( T^{7} + \)\(66\!\cdots\!41\)\( T^{8} + \)\(55\!\cdots\!18\)\( T^{9} + \)\(24\!\cdots\!57\)\( T^{10} \)
$41$ \( 1 + \)\(89\!\cdots\!90\)\( T + \)\(21\!\cdots\!45\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!10\)\( T^{6} + \)\(46\!\cdots\!80\)\( T^{7} + \)\(33\!\cdots\!45\)\( T^{8} + \)\(77\!\cdots\!90\)\( T^{9} + \)\(46\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(12\!\cdots\!43\)\( T^{2} - \)\(85\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!88\)\( T^{5} + \)\(90\!\cdots\!14\)\( T^{6} - \)\(26\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!01\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(16\!\cdots\!43\)\( T^{10} \)
$47$ \( 1 - \)\(26\!\cdots\!32\)\( T + \)\(64\!\cdots\!87\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!78\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!06\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(99\!\cdots\!21\)\( T^{8} - \)\(47\!\cdots\!12\)\( T^{9} + \)\(20\!\cdots\!07\)\( T^{10} \)
$53$ \( 1 + \)\(22\!\cdots\!54\)\( T + \)\(53\!\cdots\!33\)\( T^{2} + \)\(71\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!38\)\( T^{4} + \)\(81\!\cdots\!32\)\( T^{5} + \)\(68\!\cdots\!74\)\( T^{6} + \)\(39\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!61\)\( T^{8} + \)\(69\!\cdots\!14\)\( T^{9} + \)\(22\!\cdots\!93\)\( T^{10} \)
$59$ \( 1 - \)\(49\!\cdots\!00\)\( T + \)\(80\!\cdots\!95\)\( T^{2} - \)\(32\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!10\)\( T^{4} - \)\(87\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!90\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!05\)\( T^{8} - \)\(61\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + \)\(20\!\cdots\!90\)\( T + \)\(10\!\cdots\!45\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!10\)\( T^{4} + \)\(49\!\cdots\!48\)\( T^{5} + \)\(99\!\cdots\!10\)\( T^{6} + \)\(72\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!45\)\( T^{8} + \)\(41\!\cdots\!90\)\( T^{9} + \)\(44\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 - \)\(17\!\cdots\!52\)\( T + \)\(45\!\cdots\!27\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(28\!\cdots\!76\)\( T^{5} + \)\(29\!\cdots\!66\)\( T^{6} - \)\(53\!\cdots\!20\)\( T^{7} + \)\(37\!\cdots\!81\)\( T^{8} - \)\(28\!\cdots\!72\)\( T^{9} + \)\(32\!\cdots\!07\)\( T^{10} \)
$71$ \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(44\!\cdots\!95\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!10\)\( T^{4} - \)\(13\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} - \)\(16\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!45\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(51\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 + \)\(23\!\cdots\!74\)\( T + \)\(55\!\cdots\!13\)\( T^{2} + \)\(77\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!94\)\( T^{6} + \)\(85\!\cdots\!80\)\( T^{7} + \)\(64\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!54\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \)
$79$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!10\)\( T^{4} - \)\(42\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!05\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 - \)\(10\!\cdots\!16\)\( T + \)\(23\!\cdots\!03\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!08\)\( T^{5} + \)\(57\!\cdots\!54\)\( T^{6} - \)\(22\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} - \)\(24\!\cdots\!76\)\( T^{9} + \)\(29\!\cdots\!43\)\( T^{10} \)
$89$ \( 1 + \)\(44\!\cdots\!50\)\( T + \)\(62\!\cdots\!45\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!90\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!05\)\( T^{8} + \)\(73\!\cdots\!50\)\( T^{9} + \)\(33\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + \)\(47\!\cdots\!18\)\( T + \)\(59\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} + \)\(32\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!06\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!21\)\( T^{8} + \)\(64\!\cdots\!38\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} \)
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