Properties

Label 2.50
Level 2
Weight 50
Dimension 5
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2\( 2 \) \)
Weight: \( k \) = \( 50 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{50}(\Gamma_1(2))\).

Total New Old
Modular forms 13 5 8
Cusp forms 11 5 6
Eisenstein series 2 0 2

Trace form

\( 5q + 16777216q^{2} + 264847461204q^{3} + 1407374883553280q^{4} - 18638485219296930q^{5} - 4987106140807495680q^{6} - 557868444516202015832q^{7} + 4722366482869645213696q^{8} + 280721402774176601867265q^{9} + O(q^{10}) \) \( 5q + 16777216q^{2} + 264847461204q^{3} + 1407374883553280q^{4} - 18638485219296930q^{5} - 4987106140807495680q^{6} - 557868444516202015832q^{7} + 4722366482869645213696q^{8} + 280721402774176601867265q^{9} - 3103576043133487420538880q^{10} - 28365608284701363831431940q^{11} + 74547932974272268493389824q^{12} + 1271483866889126024941566214q^{13} + 5149096806336651063449354240q^{14} + 214847871537777285741720697080q^{15} + 396140812571321687967719751680q^{16} + 5152941537749680842191457808218q^{17} + 8357477209072966959936703561728q^{18} + 46341407445428881648983974068900q^{19} - 5246267193023509460628359086080q^{20} - 559284133080226526186327023974240q^{21} - 391822908885270015944302880882688q^{22} - 7831329242647530058437600484227336q^{23} - 1403745584837359368749795329966080q^{24} + 47629982895347058295233852912669275q^{25} + 77489996734861184268108829105848320q^{26} + 206445919322502956875507659031005000q^{27} - 157026007427807851323569879795105792q^{28} - 247527321733942898985506970467111850q^{29} - 656919347782328483126789725193502720q^{30} - 4690501690750749324289013431841855840q^{31} + 1329227995784915872903807060280344576q^{32} + 54050705662548577707614642759829170928q^{33} + 44771168724957977082517808188584099840q^{34} + 146807633005118450883143794431190245360q^{35} + 79016050308044041640235218237223075840q^{36} - 141224859266832750691437704043300925442q^{37} - 287911479833086098582315297230647459840q^{38} - 1483694045958777701643328472443999407720q^{39} - 873578994460748273001554791397358305280q^{40} + 5837281266020750863591249937583403824210q^{41} + 5696800652343028796496744041099613437952q^{42} + 5426330416373199601603436204868309699484q^{43} - 7984208931319907272792305772123536752640q^{44} - 86925253652175935161144662252858764431290q^{45} - 34812502120267569161491740843751589806080q^{46} - 230419494120900556639230369093924230606352q^{47} + 20983377697760831247406890107482662764544q^{48} + 410460682658525880199903254150977318928285q^{49} + 524670684231693022835584659437082011238400q^{50} + 1840814094647166636543757567809498045302760q^{51} + 357890891820591581439361495868934543376384q^{52} - 3255741187077740704432998366691597498508466q^{53} + 3117137442451018211300844150422584374067200q^{54} - 7413119005014717463507128836848112026350360q^{55} + 1449341903644522046008760560066162526781440q^{56} - 10972460577016464801645288735793062528386160q^{57} + 35603587276128039447232465468509208612700160q^{58} - 31299442004599717113735073038141116234855700q^{59} + 60474299637429873593046630185364533784084480q^{60} - 66894309067789985792204509754926988828519690q^{61} + 159721137676069834981339855431435216886956032q^{62} - 554022586770393686777096994052605277017064056q^{63} + 111503725992653115707678591363241807529902080q^{64} - 1052628489073426957113152396992241527143651420q^{65} + 1928998656113819566256453087528516456168816640q^{66} - 20885003468135713186598920647668702204221292q^{67} + 1450424099329463330514537562561586792924971008q^{68} - 1257559858360555896437195834297656389818725920q^{69} + 7618037059227102853981604103256516677347573760q^{70} - 6594142656484316264086959029054209943412047640q^{71} + 2352420702783651680787935004437281324891373568q^{72} - 19756885769059370139676576183477802818928731406q^{73} + 12750362298794385505728320817555046789063639040q^{74} - 27905622673610296942656094011894237718545426900q^{75} + 13043946581441115021393311842568604922308198400q^{76} + 12602774717208971429421128542194190666655134176q^{77} + 37001104240479571914234863709295328137781968896q^{78} + 19096525780528455865201933661363886028057286800q^{79} - 1476692935974170951191455457926057088757268480q^{80} - 70934640664141770240050219142683596980815080995q^{81} - 136289723431352802248020881663065807260628287488q^{82} + 126217178168945828921481068490657733955840008164q^{83} - 157424488333396192411121232794771046167677501440q^{84} + 417930567475700218907549622522904206857671893660q^{85} - 680892727632315740413194786983907995215703572480q^{86} + 1321368679037303347007178017036700024540174119640q^{87} - 110288344153182865628212719184099971732055523328q^{88} + 1559139292710419737771563637847468513168982096450q^{89} - 2394128797123032286624071503358166980557946224640q^{90} + 795244717410814649701589752966368909666334406960q^{91} - 2204323216187692813920925315004509188862997692416q^{92} - 2186942403209429777679591366091800352812220382592q^{93} - 3468326649069504737796404475680414179263051202560q^{94} + 308019764153508454748351919334293296300623632600q^{95} - 395119255799781914560402250065368998826454548480q^{96} + 5206218436371069170636818419182500514721889141738q^{97} + 5650488057759347948652183551009121211852806684672q^{98} + 9268770207435901369155123842136775797278733317580q^{99} + O(q^{100}) \)

Decomposition of \(S_{50}^{\mathrm{new}}(\Gamma_1(2))\)

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
2.50.a \(\chi_{2}(1, \cdot)\) 2.50.a.a 2 1
2.50.a.b 3

Decomposition of \(S_{50}^{\mathrm{old}}(\Gamma_1(2))\) into lower level spaces

\( S_{50}^{\mathrm{old}}(\Gamma_1(2)) \cong \) \(S_{50}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 16777216 T )^{2} \))(\( ( 1 - 16777216 T )^{3} \))
$3$ (\( 1 - 281051075592 T + \)\(33\!\cdots\!82\)\( T^{2} - \)\(67\!\cdots\!36\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 + 16203614388 T + \)\(16\!\cdots\!97\)\( T^{2} - \)\(64\!\cdots\!56\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} + \)\(92\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!87\)\( T^{6} \))
$5$ (\( 1 - 83174531182543500 T + \)\(17\!\cdots\!50\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!25\)\( T^{4} \))(\( 1 + 101813016401840430 T + \)\(12\!\cdots\!75\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!75\)\( T^{4} + \)\(32\!\cdots\!50\)\( T^{5} + \)\(56\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 + \)\(43\!\cdots\!36\)\( T + \)\(33\!\cdots\!38\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(66\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(12\!\cdots\!96\)\( T + \)\(20\!\cdots\!93\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(53\!\cdots\!51\)\( T^{4} + \)\(82\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \))
$11$ (\( 1 + \)\(25\!\cdots\!36\)\( T + \)\(76\!\cdots\!06\)\( T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(25\!\cdots\!04\)\( T + \)\(96\!\cdots\!45\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(29\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \))
$13$ (\( 1 + \)\(16\!\cdots\!28\)\( T + \)\(44\!\cdots\!42\)\( T^{2} + \)\(64\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(29\!\cdots\!42\)\( T + \)\(13\!\cdots\!07\)\( T^{2} - \)\(22\!\cdots\!76\)\( T^{3} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(43\!\cdots\!18\)\( T^{5} + \)\(56\!\cdots\!17\)\( T^{6} \))
$17$ (\( 1 - \)\(12\!\cdots\!64\)\( T + \)\(38\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!08\)\( T^{3} + \)\(38\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(39\!\cdots\!54\)\( T + \)\(70\!\cdots\!63\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!11\)\( T^{4} - \)\(15\!\cdots\!86\)\( T^{5} + \)\(75\!\cdots\!73\)\( T^{6} \))
$19$ (\( 1 - \)\(31\!\cdots\!20\)\( T + \)\(11\!\cdots\!58\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(14\!\cdots\!80\)\( T + \)\(12\!\cdots\!37\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!23\)\( T^{4} - \)\(30\!\cdots\!80\)\( T^{5} + \)\(94\!\cdots\!39\)\( T^{6} \))
$23$ (\( 1 + \)\(28\!\cdots\!28\)\( T + \)\(86\!\cdots\!22\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(49\!\cdots\!08\)\( T + \)\(23\!\cdots\!77\)\( T^{2} + \)\(55\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!51\)\( T^{4} + \)\(13\!\cdots\!52\)\( T^{5} + \)\(14\!\cdots\!47\)\( T^{6} \))
$29$ (\( 1 + \)\(11\!\cdots\!80\)\( T + \)\(99\!\cdots\!38\)\( T^{2} + \)\(53\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(93\!\cdots\!30\)\( T + \)\(95\!\cdots\!07\)\( T^{2} - \)\(86\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!83\)\( T^{4} - \)\(19\!\cdots\!30\)\( T^{5} + \)\(93\!\cdots\!09\)\( T^{6} \))
$31$ (\( 1 + \)\(71\!\cdots\!96\)\( T + \)\(34\!\cdots\!46\)\( T^{2} + \)\(84\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(24\!\cdots\!56\)\( T + \)\(16\!\cdots\!25\)\( T^{2} - \)\(71\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!75\)\( T^{4} - \)\(34\!\cdots\!96\)\( T^{5} + \)\(16\!\cdots\!11\)\( T^{6} \))
$37$ (\( 1 + \)\(45\!\cdots\!16\)\( T + \)\(12\!\cdots\!18\)\( T^{2} + \)\(31\!\cdots\!32\)\( T^{3} + \)\(48\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(30\!\cdots\!74\)\( T + \)\(21\!\cdots\!23\)\( T^{2} - \)\(40\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(14\!\cdots\!46\)\( T^{5} + \)\(33\!\cdots\!33\)\( T^{6} \))
$41$ (\( 1 - \)\(69\!\cdots\!64\)\( T + \)\(31\!\cdots\!46\)\( T^{2} - \)\(74\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!54\)\( T + \)\(28\!\cdots\!55\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!55\)\( T^{4} + \)\(12\!\cdots\!34\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} \))
$43$ (\( 1 - \)\(23\!\cdots\!32\)\( T + \)\(35\!\cdots\!42\)\( T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(17\!\cdots\!48\)\( T + \)\(32\!\cdots\!97\)\( T^{2} + \)\(36\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + \)\(11\!\cdots\!96\)\( T + \)\(10\!\cdots\!38\)\( T^{2} + \)\(10\!\cdots\!32\)\( T^{3} + \)\(73\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(21\!\cdots\!56\)\( T + \)\(34\!\cdots\!13\)\( T^{2} + \)\(34\!\cdots\!12\)\( T^{3} + \)\(29\!\cdots\!71\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(62\!\cdots\!63\)\( T^{6} \))
$53$ (\( 1 + \)\(39\!\cdots\!68\)\( T + \)\(99\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!44\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(74\!\cdots\!02\)\( T + \)\(61\!\cdots\!67\)\( T^{2} - \)\(55\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} - \)\(70\!\cdots\!78\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \))
$59$ (\( 1 - \)\(21\!\cdots\!40\)\( T + \)\(46\!\cdots\!78\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(53\!\cdots\!40\)\( T + \)\(22\!\cdots\!17\)\( T^{2} + \)\(63\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!19\)\( T^{6} \))
$61$ (\( 1 - \)\(57\!\cdots\!04\)\( T + \)\(27\!\cdots\!86\)\( T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(91\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(12\!\cdots\!94\)\( T + \)\(12\!\cdots\!35\)\( T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!14\)\( T^{5} + \)\(27\!\cdots\!21\)\( T^{6} \))
$67$ (\( 1 + \)\(11\!\cdots\!16\)\( T + \)\(79\!\cdots\!58\)\( T^{2} + \)\(34\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!24\)\( T + \)\(13\!\cdots\!33\)\( T^{2} - \)\(71\!\cdots\!68\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} \))
$71$ (\( 1 + \)\(25\!\cdots\!16\)\( T + \)\(89\!\cdots\!26\)\( T^{2} + \)\(13\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(40\!\cdots\!24\)\( T + \)\(18\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(96\!\cdots\!35\)\( T^{4} + \)\(10\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!91\)\( T^{6} \))
$73$ (\( 1 + \)\(97\!\cdots\!88\)\( T + \)\(57\!\cdots\!62\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(40\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(10\!\cdots\!18\)\( T + \)\(83\!\cdots\!47\)\( T^{2} + \)\(39\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} + \)\(40\!\cdots\!42\)\( T^{5} + \)\(80\!\cdots\!97\)\( T^{6} \))
$79$ (\( 1 + \)\(16\!\cdots\!60\)\( T + \)\(12\!\cdots\!38\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(36\!\cdots\!60\)\( T + \)\(33\!\cdots\!57\)\( T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!83\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!59\)\( T^{6} \))
$83$ (\( 1 + \)\(24\!\cdots\!28\)\( T + \)\(64\!\cdots\!02\)\( T^{2} + \)\(26\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(15\!\cdots\!92\)\( T + \)\(23\!\cdots\!97\)\( T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!91\)\( T^{4} - \)\(17\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \))
$89$ (\( 1 - \)\(10\!\cdots\!80\)\( T + \)\(80\!\cdots\!18\)\( T^{2} - \)\(33\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(54\!\cdots\!70\)\( T + \)\(86\!\cdots\!27\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!43\)\( T^{4} - \)\(59\!\cdots\!70\)\( T^{5} + \)\(36\!\cdots\!29\)\( T^{6} \))
$97$ (\( 1 - \)\(86\!\cdots\!24\)\( T + \)\(29\!\cdots\!78\)\( T^{2} - \)\(19\!\cdots\!08\)\( T^{3} + \)\(50\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(43\!\cdots\!14\)\( T + \)\(35\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!48\)\( T^{3} + \)\(80\!\cdots\!11\)\( T^{4} - \)\(21\!\cdots\!46\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \))
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