Properties

Label 384.7
Level 384
Weight 7
Dimension 10320
Nonzero newspaces 10
Sturm bound 57344
Trace bound 25

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Defining parameters

Level: \( N \) = \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 10 \)
Sturm bound: \(57344\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(384))\).

Total New Old
Modular forms 24896 10416 14480
Cusp forms 24256 10320 13936
Eisenstein series 640 96 544

Trace form

\( 10320 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10}) \) \( 10320 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 16 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} - 2932 q^{21} - 32 q^{22} + 26240 q^{23} - 16 q^{24} + 82328 q^{25} - 68652 q^{27} - 32 q^{28} - 132800 q^{29} - 16 q^{30} - 16 q^{31} + 176288 q^{33} - 32 q^{34} + 324672 q^{35} - 16 q^{36} + 14368 q^{37} - 254412 q^{39} - 32 q^{40} - 434240 q^{41} - 16 q^{42} + 290856 q^{43} + 62484 q^{45} - 32 q^{46} - 16 q^{48} + 941144 q^{49} + 2821728 q^{50} - 157800 q^{51} - 5100512 q^{52} - 1774720 q^{53} - 629872 q^{54} - 465432 q^{55} + 1306144 q^{56} + 1088620 q^{57} + 7080448 q^{58} + 1772288 q^{59} + 5023280 q^{60} + 2611936 q^{61} + 2166720 q^{62} + 8 q^{63} - 4620704 q^{64} - 2982656 q^{65} - 6205264 q^{66} - 3019224 q^{67} - 7389120 q^{68} - 2164012 q^{69} - 9088160 q^{70} - 534016 q^{71} - 16 q^{72} + 4112600 q^{73} + 11696608 q^{74} + 2208664 q^{75} + 18260704 q^{76} + 3731840 q^{77} + 1905104 q^{78} - 1721888 q^{79} - 16713312 q^{80} - 2421464 q^{81} - 32 q^{82} - 16 q^{84} - 250032 q^{85} + 2029876 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} + 2349584 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} - 5970700 q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(384))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
384.7.b \(\chi_{384}(319, \cdot)\) 384.7.b.a 4 1
384.7.b.b 4
384.7.b.c 8
384.7.b.d 8
384.7.b.e 24
384.7.e \(\chi_{384}(257, \cdot)\) 384.7.e.a 24 1
384.7.e.b 24
384.7.e.c 24
384.7.e.d 24
384.7.g \(\chi_{384}(127, \cdot)\) 384.7.g.a 24 1
384.7.g.b 24
384.7.h \(\chi_{384}(65, \cdot)\) 384.7.h.a 2 1
384.7.h.b 2
384.7.h.c 2
384.7.h.d 2
384.7.h.e 8
384.7.h.f 8
384.7.h.g 24
384.7.h.h 48
384.7.i \(\chi_{384}(161, \cdot)\) n/a 184 2
384.7.l \(\chi_{384}(31, \cdot)\) 384.7.l.a 48 2
384.7.l.b 48
384.7.m \(\chi_{384}(79, \cdot)\) n/a 192 4
384.7.p \(\chi_{384}(17, \cdot)\) n/a 376 4
384.7.q \(\chi_{384}(41, \cdot)\) None 0 8
384.7.t \(\chi_{384}(7, \cdot)\) None 0 8
384.7.u \(\chi_{384}(19, \cdot)\) n/a 3072 16
384.7.x \(\chi_{384}(5, \cdot)\) n/a 6112 16

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

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(384))\) into lower level spaces

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