Properties

Label 9.36
Level 9
Weight 36
Dimension 82
Nonzero newspaces 2
Newform subspaces 5
Sturm bound 216
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 36 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 5 \)
Sturm bound: \(216\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_1(9))\).

Total New Old
Modular forms 109 87 22
Cusp forms 101 82 19
Eisenstein series 8 5 3

Trace form

\( 82 q - 122487 q^{2} + 234015468 q^{3} - 318146910413 q^{4} - 3006266728710 q^{5} - 59737352657913 q^{6} - 546425100068272 q^{7} + 5026374864186378 q^{8} + 9175446088051032 q^{9} + O(q^{10}) \) \( 82 q - 122487 q^{2} + 234015468 q^{3} - 318146910413 q^{4} - 3006266728710 q^{5} - 59737352657913 q^{6} - 546425100068272 q^{7} + 5026374864186378 q^{8} + 9175446088051032 q^{9} - 418890507067100928 q^{10} - 4008638332447214712 q^{11} - 31462691461602808620 q^{12} + 12280176140362217942 q^{13} - 101117176350999039696 q^{14} - 1502594729236581301728 q^{15} - 5540759341157124032369 q^{16} + 3894612577658083019952 q^{17} + 19460296241711379237276 q^{18} + 82475346186796916299664 q^{19} - 144181344879794916135948 q^{20} - 330955212778594258387074 q^{21} + 555160052615633570790471 q^{22} - 2673572164965010248128688 q^{23} - 137937839221873113991011 q^{24} - 17799635365631512230690794 q^{25} + 67510910095656638186579892 q^{26} - 12375816765062910766127352 q^{27} + 16413262591251515592391292 q^{28} + 13762044597831451722184026 q^{29} + 98718670988962752986893728 q^{30} - 102240922079426152493681464 q^{31} - 1266105798391160601449666577 q^{32} + 883201725288483631865060382 q^{33} + 1345161867202672681267291509 q^{34} + 2169857373798278188424604936 q^{35} - 1962671796384605428076904717 q^{36} + 2231877843193534368800820428 q^{37} + 11975628090949829356903205667 q^{38} - 9643572094080771002856073776 q^{39} - 19754779485473185483278825864 q^{40} + 13126699067957096308643657562 q^{41} + 54285209381827478945858262642 q^{42} - 18465511606158794066426250880 q^{43} + 469268093642709756998063269290 q^{44} - 436080598210402069493271658590 q^{45} + 581154432567301320981520331112 q^{46} - 836966339401756813425874818816 q^{47} + 563818004771093572754348128995 q^{48} - 1093266219275673601506065717754 q^{49} - 2253213623196748507202013521391 q^{50} + 3907551541613170750588163276244 q^{51} - 3441498285148082554283019366526 q^{52} + 7689659135802507948509832556512 q^{53} - 8307939901225665375764250473223 q^{54} - 1821691678310928329689970497776 q^{55} + 9460330167648258047158662620334 q^{56} - 15799776815126603441273084932236 q^{57} + 28479997403267953407320120255076 q^{58} - 13890630075929221290787861423968 q^{59} - 69542565806461699737703476812916 q^{60} + 86511375303486816682256918259998 q^{61} - 371310351260584733399423253252252 q^{62} + 113405854196301115920224919706008 q^{63} + 443837993371020603742533970799746 q^{64} - 41611059834440717030439239675790 q^{65} + 148158153003049493197795996750506 q^{66} + 127434260506629475857697522079000 q^{67} - 608061234298823197052441789601225 q^{68} + 251562318490432573574028853323942 q^{69} + 169025581140336875451524263153122 q^{70} - 86627525399150444182812548449728 q^{71} - 1175469110954554915098265582908693 q^{72} + 1481811291938044651505725413580916 q^{73} - 4567512961177021274454106161038076 q^{74} + 1407533923532394059050366584753372 q^{75} + 5870175088111753387484422543251581 q^{76} - 6529173841036306281700489985180850 q^{77} + 1528029915669170439303268806262614 q^{78} + 2326963790803273255235512345478960 q^{79} - 7256418691048378243650744850951968 q^{80} - 2345699154663621834598908899742096 q^{81} + 9218059187637909161255795024902278 q^{82} - 21426040178915188671295317186644508 q^{83} - 5558512390117261416737143493916390 q^{84} + 10849178393693187295052744390321316 q^{85} - 36618167368626134103096440998788831 q^{86} + 1271722254930240884178874809772176 q^{87} + 54499518674773882698050965100829579 q^{88} - 29039649726088969666647366798407736 q^{89} + 56337530460325129801879060296769164 q^{90} - 51425022610218687657887352273469256 q^{91} - 48599347419890821229482754308105374 q^{92} - 23177521164064087873609979339051394 q^{93} + 64337788470564756500133943232824152 q^{94} - 2513352015570027613898746862387088 q^{95} - 74493469509195252570888626594008776 q^{96} + 175868235095855304431892697012765838 q^{97} - 361385200492703339427978658024033656 q^{98} + 110575121588324698113939336537275244 q^{99} + O(q^{100}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.36.a \(\chi_{9}(1, \cdot)\) 9.36.a.a 2 1
9.36.a.b 3
9.36.a.c 3
9.36.a.d 6
9.36.c \(\chi_{9}(4, \cdot)\) 9.36.c.a 68 2

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)