Properties

Label 2.52
Level 2
Weight 52
Dimension 4
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 13
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 52 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{52}(\Gamma_1(2))\).

Total New Old
Modular forms 14 4 10
Cusp forms 12 4 8
Eisenstein series 2 0 2

Trace form

\( 4q + 1076910157680q^{3} + 4503599627370496q^{4} - 1728221286058331400q^{5} + 23566263829760311296q^{6} - 4743731945229335500960q^{7} - 344685286256376531720492q^{9} + O(q^{10}) \) \( 4q + 1076910157680q^{3} + 4503599627370496q^{4} - 1728221286058331400q^{5} + 23566263829760311296q^{6} - 4743731945229335500960q^{7} - 344685286256376531720492q^{9} + 24067037337411361387315200q^{10} - 503283207719762471713841712q^{11} + 1212493046209787522784952320q^{12} - 28484059957235542963114051240q^{13} + 115998429688917348708262084608q^{14} - 2357572472449894831293451864800q^{15} + 5070602400912917605986812821504q^{16} - 66957264261194588657058982614840q^{17} - 246876127194650288020644440309760q^{18} - 869748432963213352285742458413520q^{19} - 1945804184976515166720463837593600q^{20} - 5093817921452276758554806649489792q^{21} + 17466776719635728212207100661596160q^{22} + 119657163236851406032563771003163680q^{23} + 26533254250555833983985090721480704q^{24} + 900991418956550511536063903285897500q^{25} - 1116141067339894740881674831537373184q^{26} - 2189911054317690273692361734054720160q^{27} - 5340967355220088375590250506920919040q^{28} - 56628527352633433398115232701026791400q^{29} + 44329981091378950988692387650758246400q^{30} + 179034100795922800997472564654744910208q^{31} + 1124019418901191172838399702604173577920q^{33} + 314883060660212086067995886977378418688q^{34} - 5368043532532976928236436729945674270400q^{35} - 388081131686077523611200979738199851008q^{36} - 15265672052970893454332023330692419215240q^{37} - 20049232627260367344196116645232257269760q^{38} + 112298706011079222790203570883623101632416q^{39} + 27097075096169405338709124249996283084800q^{40} - 6960934485448215957075941109057514591512q^{41} + 282668114618678623296525804067341069189120q^{42} - 469653324620434983852385243851358221046960q^{43} - 566646516687137550866599175405083230732288q^{44} - 1754411155811556679591723210661480452357800q^{45} - 1512116644615693857213603896905559571431424q^{46} + 3734342884373247807437702776124836428632640q^{47} + 1365145807774929168753026676901629583687680q^{48} + 26311063451396939040248793375412476487714212q^{49} + 14060826718418251703818076793128846622720000q^{50} - 45861823176912346817134807122304935143726112q^{51} - 32070200452351214379434713912146197752053760q^{52} + 6430935838510918233902531023920383809755960q^{53} - 166437288411003248430248044561928015912632320q^{54} - 384735931965066266899208980959235423422160800q^{55} + 130602621180642712970595794521461877628731392q^{56} + 2099409993604303785753126389689663592211743040q^{57} + 343569551276402382190114938731598747826913280q^{58} - 393261081156292712066842062865213820776546800q^{59} - 2654390627106071327288803495317658292925235200q^{60} + 4583289504442590280280739455157269513630336408q^{61} - 14718183564255876087612649865036462577030266880q^{62} - 5407995184668264720661183245953455827414976800q^{63} + 5708990770823839524233143877797980545530986496q^{64} + 26127444917167465091234674975297565631168229200q^{65} - 31109775509133356340081393044025403414715301888q^{66} + 111687415433379073229359040810024313293536926320q^{67} - 75387177594115944657516203835069658371886940160q^{68} + 210790926233651280942860049477489832371023227776q^{69} - 443004349647078529808369255966459288074636492800q^{70} + 118627152317219793230373328791717450073972542048q^{71} - 277957808610124552786581505697288895754131210240q^{72} + 256551366596744482030223570337144337722595040360q^{73} - 1218459858904371078263219296836623109660755361792q^{74} + 2573484232883199198642501430666213841191820970000q^{75} - 979249679649800118373657006824412530407353876480q^{76} + 3215379700223068479800586152548499209136015832960q^{77} - 5381353871199398107280618735790894250700140707840q^{78} + 6511846631302480883132560584341714698499815701440q^{79} - 2190780750599046343979795678865697626890069606400q^{80} - 3926978279246148602386210736313391367423564581596q^{81} - 13448749469721454290024974867604253614482883870720q^{82} - 706645826183335638122247029341433368493041506000q^{83} - 5735129123236407018188644753909217158979238494208q^{84} + 34473768910536215586645582250398147758709478335600q^{85} + 13512951600269375614913310629113068976810770825216q^{86} + 7352017476133073268675742668093725232197354347680q^{87} + 19665842281478780014971858536506450341837762723840q^{88} + 2456371107694382397799993708050822767582025478440q^{89} + 106672537836478010732514036445336686807021413990400q^{90} - 438169224493737047176033946142315161532663009117632q^{91} + 134721988941423651303320061868250527992383872696320q^{92} - 228270611573293179380566604544731575680098796049920q^{93} + 465675684802606586586515408411091535205827344334848q^{94} - 353933468443020501512525306750097873431772020988000q^{95} + 29873788488932470759940778203638049898510780727296q^{96} + 487975172072055083966044164369503680570203380952200q^{97} + 1658570891451662370921067147388279973432651216322560q^{98} - 628900793767032953986801082923115272529683182115824q^{99} + O(q^{100}) \)

Decomposition of \(S_{52}^{\mathrm{new}}(\Gamma_1(2))\)

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
2.52.a \(\chi_{2}(1, \cdot)\) 2.52.a.a 2 1
2.52.a.b 2

Decomposition of \(S_{52}^{\mathrm{old}}(\Gamma_1(2))\) into lower level spaces

\( S_{52}^{\mathrm{old}}(\Gamma_1(2)) \cong \) \(S_{52}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 33554432 T )^{2} \))(\( ( 1 - 33554432 T )^{2} \))
$3$ (\( 1 - 187290382776 T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(40\!\cdots\!72\)\( T^{3} + \)\(46\!\cdots\!09\)\( T^{4} \))(\( 1 - 889619774904 T + \)\(44\!\cdots\!98\)\( T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(46\!\cdots\!09\)\( T^{4} \))
$5$ (\( 1 + 1222737445852282500 T + \)\(10\!\cdots\!50\)\( T^{2} + \)\(54\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!25\)\( T^{4} \))(\( 1 + 505483840206048900 T + \)\(24\!\cdots\!50\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!25\)\( T^{4} \))
$7$ (\( 1 + \)\(41\!\cdots\!52\)\( T + \)\(26\!\cdots\!62\)\( T^{2} + \)\(51\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(64\!\cdots\!08\)\( T - \)\(61\!\cdots\!98\)\( T^{2} + \)\(80\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!49\)\( T^{4} \))
$11$ (\( 1 + \)\(51\!\cdots\!96\)\( T + \)\(24\!\cdots\!26\)\( T^{2} + \)\(66\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(86\!\cdots\!84\)\( T + \)\(18\!\cdots\!86\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \))
$13$ (\( 1 - \)\(23\!\cdots\!36\)\( T + \)\(10\!\cdots\!98\)\( T^{2} - \)\(15\!\cdots\!32\)\( T^{3} + \)\(41\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(30\!\cdots\!76\)\( T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(19\!\cdots\!12\)\( T^{3} + \)\(41\!\cdots\!69\)\( T^{4} \))
$17$ (\( 1 + \)\(38\!\cdots\!12\)\( T + \)\(14\!\cdots\!02\)\( T^{2} + \)\(21\!\cdots\!96\)\( T^{3} + \)\(32\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(28\!\cdots\!28\)\( T + \)\(13\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!24\)\( T^{3} + \)\(32\!\cdots\!89\)\( T^{4} \))
$19$ (\( 1 + \)\(13\!\cdots\!20\)\( T - \)\(41\!\cdots\!62\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(73\!\cdots\!00\)\( T + \)\(46\!\cdots\!38\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!61\)\( T^{4} \))
$23$ (\( 1 - \)\(82\!\cdots\!56\)\( T + \)\(46\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!12\)\( T^{3} + \)\(78\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(37\!\cdots\!24\)\( T + \)\(34\!\cdots\!98\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(78\!\cdots\!29\)\( T^{4} \))
$29$ (\( 1 + \)\(33\!\cdots\!20\)\( T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 + \)\(23\!\cdots\!80\)\( T + \)\(86\!\cdots\!58\)\( T^{2} + \)\(88\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))
$31$ (\( 1 - \)\(30\!\cdots\!24\)\( T + \)\(44\!\cdots\!06\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(12\!\cdots\!16\)\( T + \)\(26\!\cdots\!26\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))
$37$ (\( 1 - \)\(10\!\cdots\!08\)\( T + \)\(21\!\cdots\!42\)\( T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(90\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(25\!\cdots\!48\)\( T + \)\(33\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(90\!\cdots\!69\)\( T^{4} \))
$41$ (\( 1 - \)\(19\!\cdots\!24\)\( T + \)\(45\!\cdots\!26\)\( T^{2} - \)\(35\!\cdots\!84\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(20\!\cdots\!36\)\( T + \)\(30\!\cdots\!06\)\( T^{2} + \)\(36\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} \))
$43$ (\( 1 + \)\(43\!\cdots\!24\)\( T + \)\(39\!\cdots\!58\)\( T^{2} + \)\(88\!\cdots\!68\)\( T^{3} + \)\(41\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(33\!\cdots\!36\)\( T - \)\(40\!\cdots\!62\)\( T^{2} + \)\(67\!\cdots\!52\)\( T^{3} + \)\(41\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + \)\(50\!\cdots\!12\)\( T + \)\(22\!\cdots\!42\)\( T^{2} + \)\(95\!\cdots\!36\)\( T^{3} + \)\(35\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(88\!\cdots\!52\)\( T + \)\(56\!\cdots\!82\)\( T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!09\)\( T^{4} \))
$53$ (\( 1 - \)\(10\!\cdots\!96\)\( T + \)\(18\!\cdots\!98\)\( T^{2} - \)\(91\!\cdots\!12\)\( T^{3} + \)\(75\!\cdots\!09\)\( T^{4} \))(\( 1 + \)\(99\!\cdots\!36\)\( T + \)\(18\!\cdots\!18\)\( T^{2} + \)\(85\!\cdots\!92\)\( T^{3} + \)\(75\!\cdots\!09\)\( T^{4} \))
$59$ (\( 1 + \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!18\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(71\!\cdots\!40\)\( T + \)\(41\!\cdots\!18\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!81\)\( T^{4} \))
$61$ (\( 1 - \)\(21\!\cdots\!44\)\( T + \)\(17\!\cdots\!06\)\( T^{2} - \)\(24\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(24\!\cdots\!64\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(27\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!21\)\( T^{4} \))
$67$ (\( 1 - \)\(52\!\cdots\!28\)\( T + \)\(33\!\cdots\!62\)\( T^{2} - \)\(71\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(58\!\cdots\!92\)\( T + \)\(28\!\cdots\!82\)\( T^{2} - \)\(79\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!89\)\( T^{4} \))
$71$ (\( 1 - \)\(71\!\cdots\!64\)\( T + \)\(50\!\cdots\!66\)\( T^{2} - \)\(18\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(46\!\cdots\!84\)\( T + \)\(64\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!64\)\( T^{3} + \)\(67\!\cdots\!41\)\( T^{4} \))
$73$ (\( 1 - \)\(63\!\cdots\!76\)\( T + \)\(29\!\cdots\!98\)\( T^{2} - \)\(67\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 + \)\(37\!\cdots\!16\)\( T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(40\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))
$79$ (\( 1 - \)\(19\!\cdots\!60\)\( T + \)\(10\!\cdots\!58\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(36\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(45\!\cdots\!80\)\( T + \)\(16\!\cdots\!58\)\( T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!41\)\( T^{4} \))
$83$ (\( 1 + \)\(72\!\cdots\!24\)\( T + \)\(14\!\cdots\!78\)\( T^{2} + \)\(54\!\cdots\!08\)\( T^{3} + \)\(55\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(65\!\cdots\!24\)\( T + \)\(12\!\cdots\!78\)\( T^{2} - \)\(49\!\cdots\!08\)\( T^{3} + \)\(55\!\cdots\!89\)\( T^{4} \))
$89$ (\( 1 - \)\(76\!\cdots\!20\)\( T + \)\(67\!\cdots\!78\)\( T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(68\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(74\!\cdots\!80\)\( T + \)\(62\!\cdots\!78\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!21\)\( T^{4} \))
$97$ (\( 1 + \)\(29\!\cdots\!72\)\( T + \)\(41\!\cdots\!02\)\( T^{2} + \)\(63\!\cdots\!16\)\( T^{3} + \)\(44\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(78\!\cdots\!72\)\( T + \)\(53\!\cdots\!02\)\( T^{2} - \)\(16\!\cdots\!16\)\( T^{3} + \)\(44\!\cdots\!09\)\( T^{4} \))
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