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Properties

Label 16.9
Level 16
Weight 9
Dimension 34
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 144
Trace bound 1

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Defining parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(144\)
Trace bound: \(1\)

Dimensions

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

Total New Old
Modular forms 71 38 33
Cusp forms 57 34 23
Eisenstein series 14 4 10

Trace form

\( 34 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 506 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8} - 14460 q^{9} + O(q^{10}) \) \( 34 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 506 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8} - 14460 q^{9} - 1860 q^{10} - 19778 q^{11} + 14068 q^{12} - 17146 q^{13} - 58900 q^{14} + 245336 q^{16} - 16636 q^{17} - 223730 q^{18} + 167550 q^{19} - 135380 q^{20} + 621244 q^{21} + 985700 q^{22} - 845572 q^{23} - 800592 q^{24} - 909172 q^{25} - 184760 q^{26} + 38656 q^{27} + 94136 q^{28} + 2860486 q^{29} + 1881700 q^{30} - 3395192 q^{32} - 6084868 q^{33} + 1634900 q^{34} + 426620 q^{35} + 1877324 q^{36} + 9044934 q^{37} - 3010352 q^{38} - 7650052 q^{39} + 3256936 q^{40} - 8799480 q^{41} - 3731304 q^{42} + 6314814 q^{43} - 10360940 q^{44} + 17951754 q^{45} + 1510780 q^{46} + 4591848 q^{48} + 3067998 q^{49} + 14752710 q^{50} - 28969860 q^{51} + 4335652 q^{52} + 27396166 q^{53} + 18667040 q^{54} + 46326780 q^{55} - 36346408 q^{56} - 26616576 q^{57} - 1034696 q^{58} - 46004162 q^{59} - 20687200 q^{60} - 55299706 q^{61} - 64043472 q^{62} + 50827456 q^{64} + 23168588 q^{65} + 65035780 q^{66} + 59474558 q^{67} + 74339312 q^{68} - 33968324 q^{69} + 39082496 q^{70} - 79832068 q^{71} + 929812 q^{72} + 38504968 q^{73} - 168043460 q^{74} + 88519490 q^{75} - 166219388 q^{76} - 18785860 q^{77} + 5059788 q^{78} - 12904856 q^{80} + 116973410 q^{81} + 261305296 q^{82} - 34471682 q^{83} + 456781480 q^{84} - 189400628 q^{85} + 9841444 q^{86} + 149712636 q^{87} - 173775752 q^{88} + 139985928 q^{89} - 650366536 q^{90} - 76429060 q^{91} - 449169832 q^{92} - 224712320 q^{93} - 239278032 q^{94} + 378359728 q^{96} + 148649220 q^{97} + 630040554 q^{98} + 184838270 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

We only show spaces with odd 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
16.9.c \(\chi_{16}(15, \cdot)\) 16.9.c.a 2 1
16.9.c.b 2
16.9.d \(\chi_{16}(7, \cdot)\) None 0 1
16.9.f \(\chi_{16}(3, \cdot)\) 16.9.f.a 30 2

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

\( S_{9}^{\mathrm{old}}(\Gamma_1(16)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)