Properties

Label 2.36.a
Level 2
Weight 36
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

\(2\)Dim.
\(+\)\(1\)
\(-\)\(1\)

Trace form

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

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 131072 T \))(\( 1 - 131072 T \))
$3$ (\( 1 - 36494748 T + 50031545098999707 T^{2} \))(\( 1 - 159933852 T + 50031545098999707 T^{2} \))
$5$ (\( 1 - 389070858750 T + \)\(29\!\cdots\!25\)\( T^{2} \))(\( 1 + 2838742578690 T + \)\(29\!\cdots\!25\)\( T^{2} \))
$7$ (\( 1 + 129689369490856 T + \)\(37\!\cdots\!43\)\( T^{2} \))(\( 1 + 782281866962344 T + \)\(37\!\cdots\!43\)\( T^{2} \))
$11$ (\( 1 - 469638570722172852 T + \)\(28\!\cdots\!51\)\( T^{2} \))(\( 1 - 738502081164310452 T + \)\(28\!\cdots\!51\)\( T^{2} \))
$13$ (\( 1 - 28541547044626383638 T + \)\(97\!\cdots\!57\)\( T^{2} \))(\( 1 - 12249951000076215062 T + \)\(97\!\cdots\!57\)\( T^{2} \))
$17$ (\( 1 - \)\(53\!\cdots\!14\)\( T + \)\(11\!\cdots\!93\)\( T^{2} \))(\( 1 + \)\(58\!\cdots\!14\)\( T + \)\(11\!\cdots\!93\)\( T^{2} \))
$19$ (\( 1 - \)\(34\!\cdots\!80\)\( T + \)\(57\!\cdots\!99\)\( T^{2} \))(\( 1 + \)\(10\!\cdots\!20\)\( T + \)\(57\!\cdots\!99\)\( T^{2} \))
$23$ (\( 1 + \)\(17\!\cdots\!72\)\( T + \)\(45\!\cdots\!07\)\( T^{2} \))(\( 1 - \)\(48\!\cdots\!72\)\( T + \)\(45\!\cdots\!07\)\( T^{2} \))
$29$ (\( 1 - \)\(45\!\cdots\!70\)\( T + \)\(15\!\cdots\!49\)\( T^{2} \))(\( 1 - \)\(57\!\cdots\!70\)\( T + \)\(15\!\cdots\!49\)\( T^{2} \))
$31$ (\( 1 - \)\(19\!\cdots\!52\)\( T + \)\(15\!\cdots\!51\)\( T^{2} \))(\( 1 + \)\(19\!\cdots\!48\)\( T + \)\(15\!\cdots\!51\)\( T^{2} \))
$37$ (\( 1 - \)\(14\!\cdots\!54\)\( T + \)\(77\!\cdots\!93\)\( T^{2} \))(\( 1 - \)\(30\!\cdots\!46\)\( T + \)\(77\!\cdots\!93\)\( T^{2} \))
$41$ (\( 1 - \)\(15\!\cdots\!02\)\( T + \)\(28\!\cdots\!01\)\( T^{2} \))(\( 1 + \)\(11\!\cdots\!98\)\( T + \)\(28\!\cdots\!01\)\( T^{2} \))
$43$ (\( 1 - \)\(37\!\cdots\!08\)\( T + \)\(14\!\cdots\!07\)\( T^{2} \))(\( 1 + \)\(15\!\cdots\!08\)\( T + \)\(14\!\cdots\!07\)\( T^{2} \))
$47$ (\( 1 + \)\(31\!\cdots\!76\)\( T + \)\(33\!\cdots\!43\)\( T^{2} \))(\( 1 + \)\(13\!\cdots\!24\)\( T + \)\(33\!\cdots\!43\)\( T^{2} \))
$53$ (\( 1 - \)\(65\!\cdots\!98\)\( T + \)\(22\!\cdots\!57\)\( T^{2} \))(\( 1 + \)\(16\!\cdots\!98\)\( T + \)\(22\!\cdots\!57\)\( T^{2} \))
$59$ (\( 1 + \)\(12\!\cdots\!60\)\( T + \)\(95\!\cdots\!99\)\( T^{2} \))(\( 1 - \)\(18\!\cdots\!40\)\( T + \)\(95\!\cdots\!99\)\( T^{2} \))
$61$ (\( 1 - \)\(16\!\cdots\!02\)\( T + \)\(30\!\cdots\!01\)\( T^{2} \))(\( 1 - \)\(91\!\cdots\!02\)\( T + \)\(30\!\cdots\!01\)\( T^{2} \))
$67$ (\( 1 - \)\(11\!\cdots\!64\)\( T + \)\(81\!\cdots\!43\)\( T^{2} \))(\( 1 + \)\(10\!\cdots\!64\)\( T + \)\(81\!\cdots\!43\)\( T^{2} \))
$71$ (\( 1 + \)\(39\!\cdots\!48\)\( T + \)\(62\!\cdots\!51\)\( T^{2} \))(\( 1 + \)\(82\!\cdots\!48\)\( T + \)\(62\!\cdots\!51\)\( T^{2} \))
$73$ (\( 1 + \)\(33\!\cdots\!22\)\( T + \)\(16\!\cdots\!57\)\( T^{2} \))(\( 1 + \)\(24\!\cdots\!78\)\( T + \)\(16\!\cdots\!57\)\( T^{2} \))
$79$ (\( 1 - \)\(13\!\cdots\!20\)\( T + \)\(26\!\cdots\!99\)\( T^{2} \))(\( 1 + \)\(51\!\cdots\!80\)\( T + \)\(26\!\cdots\!99\)\( T^{2} \))
$83$ (\( 1 + \)\(82\!\cdots\!32\)\( T + \)\(14\!\cdots\!07\)\( T^{2} \))(\( 1 + \)\(48\!\cdots\!68\)\( T + \)\(14\!\cdots\!07\)\( T^{2} \))
$89$ (\( 1 + \)\(13\!\cdots\!90\)\( T + \)\(16\!\cdots\!49\)\( T^{2} \))(\( 1 + \)\(12\!\cdots\!90\)\( T + \)\(16\!\cdots\!49\)\( T^{2} \))
$97$ (\( 1 - \)\(12\!\cdots\!74\)\( T + \)\(34\!\cdots\!93\)\( T^{2} \))(\( 1 + \)\(65\!\cdots\!74\)\( T + \)\(34\!\cdots\!93\)\( T^{2} \))
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