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Properties

Label 9.20
Level 9
Weight 20
Dimension 44
Nonzero newspaces 2
Newform subspaces 5
Sturm bound 120
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 20 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 5 \)
Sturm bound: \(120\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_1(9))\).

Total New Old
Modular forms 61 49 12
Cusp forms 53 44 9
Eisenstein series 8 5 3

Trace form

\( 44 q - 567 q^{2} - 30972 q^{3} - 2509261 q^{4} - 10625580 q^{5} + 3458439 q^{6} + 104279428 q^{7} + 716841738 q^{8} - 3758981886 q^{9} + O(q^{10}) \) \( 44 q - 567 q^{2} - 30972 q^{3} - 2509261 q^{4} - 10625580 q^{5} + 3458439 q^{6} + 104279428 q^{7} + 716841738 q^{8} - 3758981886 q^{9} - 75846528 q^{10} - 9449221536 q^{11} - 7927816620 q^{12} + 21106342552 q^{13} - 201580100496 q^{14} + 58370157252 q^{15} - 776226560881 q^{16} + 1647246311172 q^{17} - 2879454169764 q^{18} + 4045935692392 q^{19} - 8465535862668 q^{20} - 364239724704 q^{21} + 18078822369351 q^{22} - 13398906854988 q^{23} - 39244342881699 q^{24} + 14159726652326 q^{25} - 189642627431244 q^{26} + 29908210918608 q^{27} + 144760806759292 q^{28} - 435385335105576 q^{29} + 76784250318048 q^{30} + 204641803065964 q^{31} - 570208078633617 q^{32} - 208469810575818 q^{33} - 262935964472715 q^{34} + 1646807894438616 q^{35} + 2309133358204659 q^{36} - 3287518705060832 q^{37} + 4411020385643427 q^{38} + 2816907211218684 q^{39} - 3413497635608904 q^{40} - 1376184071772810 q^{41} - 6395584030167438 q^{42} - 11378055023420120 q^{43} + 50622508857302442 q^{44} - 5622607373550300 q^{45} - 19730613657053592 q^{46} - 21530991857113836 q^{47} + 56857867798863075 q^{48} + 30231494836803978 q^{49} - 110326339393653231 q^{50} - 81971095169485116 q^{51} + 78168625110533314 q^{52} + 75942108694479672 q^{53} + 67265891843699961 q^{54} - 40552673608539576 q^{55} - 332919437158029906 q^{56} + 160753722236498514 q^{57} + 305187018308084196 q^{58} - 274953987693366912 q^{59} - 945471231091892916 q^{60} + 31746631244043844 q^{61} + 990978688797763428 q^{62} + 46965586362630228 q^{63} - 121252145109421438 q^{64} - 329723211065669460 q^{65} + 1298352235728159018 q^{66} + 37096660177810240 q^{67} + 11224171958396535 q^{68} - 923568025096911636 q^{69} - 537963077026019358 q^{70} + 2142338966500029264 q^{71} - 217228785836560533 q^{72} - 114212879336398124 q^{73} + 1252111652240608836 q^{74} + 678581144120511492 q^{75} - 608952817625806211 q^{76} - 589456777413071880 q^{77} + 385169643643537494 q^{78} + 2274280328481797764 q^{79} - 8790665232554162208 q^{80} + 987132598109473266 q^{81} - 2014058973403223802 q^{82} - 81705176594781228 q^{83} - 8998163637139816998 q^{84} + 6354278652907272096 q^{85} + 185120945112641697 q^{86} + 17006073983138531556 q^{87} - 3920454008482028661 q^{88} - 17007193347579455616 q^{89} - 15190939518331939956 q^{90} + 26318504661488432744 q^{91} + 20707052681786889186 q^{92} - 24981147079282834404 q^{93} - 33934514052748155624 q^{94} + 21454968802546390032 q^{95} + 29448407083070043000 q^{96} - 5283778474613340242 q^{97} - 14570393279984391096 q^{98} + 12119790365774365572 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.20.a \(\chi_{9}(1, \cdot)\) 9.20.a.a 1 1
9.20.a.b 1
9.20.a.c 2
9.20.a.d 4
9.20.c \(\chi_{9}(4, \cdot)\) 9.20.c.a 36 2

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)