Properties

Label 2.36.a
Level 2
Weight 36
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 2
Sturm bound 9
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(9\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(2))\).

Total New Old
Modular forms 10 2 8
Cusp forms 8 2 6
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.
\(+\)\(1\)
\(-\)\(1\)

Trace form

\(2q \) \(\mathstrut +\mathstrut 196428600q^{3} \) \(\mathstrut +\mathstrut 34359738368q^{4} \) \(\mathstrut -\mathstrut 2449671719940q^{5} \) \(\mathstrut +\mathstrut 16179410239488q^{6} \) \(\mathstrut -\mathstrut 911971236453200q^{7} \) \(\mathstrut -\mathstrut 73152386550858006q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 196428600q^{3} \) \(\mathstrut +\mathstrut 34359738368q^{4} \) \(\mathstrut -\mathstrut 2449671719940q^{5} \) \(\mathstrut +\mathstrut 16179410239488q^{6} \) \(\mathstrut -\mathstrut 911971236453200q^{7} \) \(\mathstrut -\mathstrut 73152386550858006q^{9} \) \(\mathstrut -\mathstrut 423075962872135680q^{10} \) \(\mathstrut +\mathstrut 1208140651886483304q^{11} \) \(\mathstrut +\mathstrut 3374617651996262400q^{12} \) \(\mathstrut +\mathstrut 40791498044702598700q^{13} \) \(\mathstrut -\mathstrut 85536603828582875136q^{14} \) \(\mathstrut -\mathstrut 439811992502079968880q^{15} \) \(\mathstrut +\mathstrut 590295810358705651712q^{16} \) \(\mathstrut -\mathstrut 450394234990716966300q^{17} \) \(\mathstrut +\mathstrut 3178098902168292556800q^{18} \) \(\mathstrut +\mathstrut 24090543446664463121560q^{19} \) \(\mathstrut -\mathstrut 42085039692313484328960q^{20} \) \(\mathstrut -\mathstrut 129846333190886892893376q^{21} \) \(\mathstrut +\mathstrut 35240478040671859507200q^{22} \) \(\mathstrut +\mathstrut 318844759828801718883600q^{23} \) \(\mathstrut +\mathstrut 277960151388673951137792q^{24} \) \(\mathstrut +\mathstrut 2389069469849272571022350q^{25} \) \(\mathstrut -\mathstrut 2135372076751279695593472q^{26} \) \(\mathstrut -\mathstrut 15515724648599139357637200q^{27} \) \(\mathstrut -\mathstrut 15667546541836708138188800q^{28} \) \(\mathstrut +\mathstrut 103136265778777057861830540q^{29} \) \(\mathstrut -\mathstrut 61369231394803503448719360q^{30} \) \(\mathstrut +\mathstrut 1267091736301409346752704q^{31} \) \(\mathstrut +\mathstrut 135250823840210691758402400q^{33} \) \(\mathstrut -\mathstrut 1472068537957672521056649216q^{34} \) \(\mathstrut +\mathstrut 2170238489924558529494259360q^{35} \) \(\mathstrut -\mathstrut 1256748431441141506039087104q^{36} \) \(\mathstrut +\mathstrut 4510080673130800660020864700q^{37} \) \(\mathstrut -\mathstrut 5804806241808838609285939200q^{38} \) \(\mathstrut +\mathstrut 3000798417177225993466212048q^{39} \) \(\mathstrut -\mathstrut 7268389697038131901098885120q^{40} \) \(\mathstrut +\mathstrut 3342165486432022025327807604q^{41} \) \(\mathstrut -\mathstrut 15778496049996305117321625600q^{42} \) \(\mathstrut +\mathstrut 22515021111766694100414005800q^{43} \) \(\mathstrut +\mathstrut 20755698355282265980520103936q^{44} \) \(\mathstrut +\mathstrut 50467317878576885543888651820q^{45} \) \(\mathstrut +\mathstrut 86409895677896633258625466368q^{46} \) \(\mathstrut -\mathstrut 449685903278907577700646271200q^{47} \) \(\mathstrut +\mathstrut 57975489807313024488937881600q^{48} \) \(\mathstrut -\mathstrut 128853132594303315270853664814q^{49} \) \(\mathstrut +\mathstrut 1036397221634256193526641459200q^{50} \) \(\mathstrut -\mathstrut 737406927859920616012398544656q^{51} \) \(\mathstrut +\mathstrut 700792600227382429850848460800q^{52} \) \(\mathstrut +\mathstrut 488672473436065298395261532700q^{53} \) \(\mathstrut -\mathstrut 1089127370066902327014960660480q^{54} \) \(\mathstrut -\mathstrut 1913694620239307930912622812880q^{55} \) \(\mathstrut -\mathstrut 1469507664218687354939086209024q^{56} \) \(\mathstrut -\mathstrut 367347627926763431339321359200q^{57} \) \(\mathstrut +\mathstrut 1602368024408844513712904601600q^{58} \) \(\mathstrut +\mathstrut 17202707160809275797256490073480q^{59} \) \(\mathstrut -\mathstrut 7555912496740122713265190993920q^{60} \) \(\mathstrut +\mathstrut 9325615309462610491924196925004q^{61} \) \(\mathstrut -\mathstrut 50503267112815883532252060057600q^{62} \) \(\mathstrut +\mathstrut 25444740726646678062344071206000q^{63} \) \(\mathstrut +\mathstrut 10141204801825835211973625643008q^{64} \) \(\mathstrut -\mathstrut 23669773272076187476693289096280q^{65} \) \(\mathstrut +\mathstrut 13234620499366895847478369714176q^{66} \) \(\mathstrut +\mathstrut 8383444218281090239140729225400q^{67} \) \(\mathstrut -\mathstrut 7737714038368272935403336499200q^{68} \) \(\mathstrut +\mathstrut 72004049599893441412052154967488q^{69} \) \(\mathstrut +\mathstrut 297684854196360356545093875793920q^{70} \) \(\mathstrut -\mathstrut 480139416215308096104875156857296q^{71} \) \(\mathstrut +\mathstrut 54599323393065280078464889651200q^{72} \) \(\mathstrut -\mathstrut 575428513832910713571828858308300q^{73} \) \(\mathstrut +\mathstrut 197969520780653215575049475457024q^{74} \) \(\mathstrut +\mathstrut 722662424222941813825529613592200q^{75} \) \(\mathstrut +\mathstrut 413872384985163957569993090007040q^{76} \) \(\mathstrut -\mathstrut 638623916934338661545063875060800q^{77} \) \(\mathstrut +\mathstrut 120267116816584768674325541683200q^{78} \) \(\mathstrut +\mathstrut 868756140162620235596402706665440q^{79} \) \(\mathstrut -\mathstrut 723015476517393271090766822768640q^{80} \) \(\mathstrut +\mathstrut 1623209532275970217164566535260562q^{81} \) \(\mathstrut -\mathstrut 3526031359697737982401933895270400q^{82} \) \(\mathstrut -\mathstrut 5632023915387843463455844090689000q^{83} \) \(\mathstrut -\mathstrut 2230743018241514120848418940125184q^{84} \) \(\mathstrut +\mathstrut 18677431896998104720787509075727160q^{85} \) \(\mathstrut -\mathstrut 6896184153461153365432717805617152q^{86} \) \(\mathstrut +\mathstrut 10883983710070248948451510541067600q^{87} \) \(\mathstrut +\mathstrut 605426802720367177803722706124800q^{88} \) \(\mathstrut -\mathstrut 25894399503218766762798011514281580q^{89} \) \(\mathstrut +\mathstrut 11581858686292446089528908751831040q^{90} \) \(\mathstrut -\mathstrut 13284449779048054657037283388639456q^{91} \) \(\mathstrut +\mathstrut 5477711263862711765894590522982400q^{92} \) \(\mathstrut -\mathstrut 23656673930604870524334510402374400q^{93} \) \(\mathstrut +\mathstrut 22399861743825587584762263417913344q^{94} \) \(\mathstrut +\mathstrut 41968378729570576408039912200238800q^{95} \) \(\mathstrut +\mathstrut 4775319039222254419775674518601728q^{96} \) \(\mathstrut -\mathstrut 53574180310510263037889481274797500q^{97} \) \(\mathstrut +\mathstrut 78006922355560245621325347107635200q^{98} \) \(\mathstrut -\mathstrut 40929623199791851537753077758145912q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.36.a.a \(1\) \(15.519\) \(\Q\) None \(-131072\) \(36494748\) \(389070858750\) \(-1\!\cdots\!56\) \(+\) \(q-2^{17}q^{2}+36494748q^{3}+2^{34}q^{4}+\cdots\)
2.36.a.b \(1\) \(15.519\) \(\Q\) None \(131072\) \(159933852\) \(-2\!\cdots\!90\) \(-7\!\cdots\!44\) \(-\) \(q+2^{17}q^{2}+159933852q^{3}+2^{34}q^{4}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)