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Properties

Label 384.9
Level 384
Weight 9
Dimension 13776
Nonzero newspaces 10
Sturm bound 73728
Trace bound 25

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Defining parameters

Level: \( N \) = \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 10 \)
Sturm bound: \(73728\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(384))\).

Total New Old
Modular forms 33088 13872 19216
Cusp forms 32448 13776 18672
Eisenstein series 640 96 544

Trace form

\( 13776 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10}) \) \( 13776 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 16 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} - 26260 q^{21} - 32 q^{22} + 1691136 q^{23} - 16 q^{24} - 2833192 q^{25} - 51084 q^{27} - 32 q^{28} + 4264704 q^{29} - 16 q^{30} - 16 q^{31} - 6610720 q^{33} - 32 q^{34} - 4831488 q^{35} - 16 q^{36} + 9440992 q^{37} + 7650036 q^{39} - 32 q^{40} - 17498880 q^{41} - 16 q^{42} - 3719448 q^{43} + 1562484 q^{45} - 32 q^{46} - 16 q^{48} + 46118360 q^{49} - 20580768 q^{50} - 27697800 q^{51} - 183792608 q^{52} + 21434880 q^{53} + 39680912 q^{54} + 92653544 q^{55} + 227499552 q^{56} + 12400108 q^{57} - 203505152 q^{58} - 89877504 q^{59} - 327416272 q^{60} - 195808288 q^{61} - 175726656 q^{62} + 8 q^{63} + 387071584 q^{64} + 239533056 q^{65} + 454631600 q^{66} + 186760936 q^{67} + 327458880 q^{68} + 34572404 q^{69} - 371559584 q^{70} - 159664128 q^{71} - 16 q^{72} - 502953000 q^{73} - 680115744 q^{74} - 24274568 q^{75} + 637215456 q^{76} + 379857408 q^{77} + 735959504 q^{78} + 144406496 q^{79} - 808638048 q^{80} - 182266136 q^{81} - 32 q^{82} - 16 q^{84} - 6250032 q^{85} + 149712628 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} - 276511888 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} + 630017012 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(384))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
384.9.b \(\chi_{384}(319, \cdot)\) 384.9.b.a 8 1
384.9.b.b 8
384.9.b.c 16
384.9.b.d 32
384.9.e \(\chi_{384}(257, \cdot)\) n/a 128 1
384.9.g \(\chi_{384}(127, \cdot)\) 384.9.g.a 32 1
384.9.g.b 32
384.9.h \(\chi_{384}(65, \cdot)\) n/a 128 1
384.9.i \(\chi_{384}(161, \cdot)\) n/a 248 2
384.9.l \(\chi_{384}(31, \cdot)\) n/a 128 2
384.9.m \(\chi_{384}(79, \cdot)\) n/a 256 4
384.9.p \(\chi_{384}(17, \cdot)\) n/a 504 4
384.9.q \(\chi_{384}(41, \cdot)\) None 0 8
384.9.t \(\chi_{384}(7, \cdot)\) None 0 8
384.9.u \(\chi_{384}(19, \cdot)\) n/a 4096 16
384.9.x \(\chi_{384}(5, \cdot)\) n/a 8160 16

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(384))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(384)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 7}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 2}\)