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Properties

Label 9.16
Level 9
Weight 16
Dimension 34
Nonzero newspaces 2
Newform subspaces 6
Sturm bound 96
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 16 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_1(9))\).

Total New Old
Modular forms 49 39 10
Cusp forms 41 34 7
Eisenstein series 8 5 3

Trace form

\( 34 q - 39 q^{2} + 3345 q^{3} - 63245 q^{4} - 263985 q^{5} + 2208231 q^{6} - 3833803 q^{7} + 17372874 q^{8} - 15596991 q^{9} + O(q^{10}) \) \( 34 q - 39 q^{2} + 3345 q^{3} - 63245 q^{4} - 263985 q^{5} + 2208231 q^{6} - 3833803 q^{7} + 17372874 q^{8} - 15596991 q^{9} + 72831072 q^{10} - 152523210 q^{11} + 182013108 q^{12} + 114413435 q^{13} - 870427968 q^{14} - 657143253 q^{15} + 3385752463 q^{16} - 7519760568 q^{17} + 7761499596 q^{18} + 5812908530 q^{19} - 14439690348 q^{20} + 18429715101 q^{21} + 27811138119 q^{22} - 51701220105 q^{23} - 22375042227 q^{24} - 60694221269 q^{25} + 138847810548 q^{26} + 162819945072 q^{27} - 527169172036 q^{28} + 48605645553 q^{29} + 588495022128 q^{30} + 242767446425 q^{31} - 663826164273 q^{32} - 752623919694 q^{33} - 81694659915 q^{34} + 2566441619286 q^{35} - 98909775117 q^{36} - 210563897152 q^{37} - 4857394070589 q^{38} + 3663944150079 q^{39} + 4495804779336 q^{40} - 4527337851078 q^{41} - 12459707404686 q^{42} + 6169386285512 q^{43} + 13176755222442 q^{44} + 4884357020085 q^{45} - 12187591387224 q^{46} - 7872218416407 q^{47} + 12130882824291 q^{48} - 6982268524389 q^{49} + 16460748817809 q^{50} + 5493352002093 q^{51} - 11111574488686 q^{52} - 18173629905030 q^{53} - 8013203434503 q^{54} - 7167647851026 q^{55} + 38102520844014 q^{56} - 15469760310585 q^{57} - 51684278201148 q^{58} - 23213211747054 q^{59} + 143327945409084 q^{60} + 40734510082397 q^{61} - 179472703471260 q^{62} + 41573408563611 q^{63} + 181674971889154 q^{64} - 67735045720065 q^{65} - 193311009227574 q^{66} - 18111432891340 q^{67} + 124952944189959 q^{68} + 270237544770987 q^{69} - 286550354936478 q^{70} + 10065140778312 q^{71} - 121608306427509 q^{72} + 436048220789546 q^{73} + 51576894410772 q^{74} - 327902975378103 q^{75} - 601696651619587 q^{76} - 53002926936123 q^{77} - 527565898825578 q^{78} + 455664581709407 q^{79} + 1758334990883232 q^{80} + 316134370401669 q^{81} - 668253144271578 q^{82} - 397311068428185 q^{83} - 262197855811446 q^{84} + 189823062933666 q^{85} - 1338536820409215 q^{86} - 1151998177977873 q^{87} + 1593209504920971 q^{88} + 2940672425509386 q^{89} + 1343761639138764 q^{90} - 1487743938333794 q^{91} - 4337086416010494 q^{92} + 309143584649451 q^{93} + 2511258072054792 q^{94} - 3789939941878188 q^{95} - 236708398351224 q^{96} - 2124018146111644 q^{97} + 5880039188509368 q^{98} + 5495206589055945 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_1(9))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
9.16.a \(\chi_{9}(1, \cdot)\) 9.16.a.a 1 1
9.16.a.b 1
9.16.a.c 1
9.16.a.d 1
9.16.a.e 2
9.16.c \(\chi_{9}(4, \cdot)\) 9.16.c.a 28 2

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)