Properties

Label 2.52.a
Level 2
Weight 52
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 2
Sturm bound 13
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 52 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{52}(\Gamma_0(2))\).

Total New Old
Modular forms 14 4 10
Cusp forms 12 4 8
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.
\(+\)\(2\)
\(-\)\(2\)

Trace form

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

Decomposition of \(S_{52}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.52.a.a \(2\) \(32.946\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-67108864\) \(187290382776\) \(-1\!\cdots\!00\) \(-4\!\cdots\!52\) \(+\) \(q-2^{25}q^{2}+(93645191388-17\beta )q^{3}+\cdots\)
2.52.a.b \(2\) \(32.946\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(67108864\) \(889619774904\) \(-5\!\cdots\!00\) \(-6\!\cdots\!08\) \(-\) \(q+2^{25}q^{2}+(444809887452-\beta )q^{3}+\cdots\)

Decomposition of \(S_{52}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{52}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{52}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)