## Defining parameters

 Level: $$N$$ = $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$28$$ Sturm bound: $$196608$$ Trace bound: $$22$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(960))$$.

Total New Old
Modular forms 74880 28644 46236
Cusp forms 72576 28380 44196
Eisenstein series 2304 264 2040

## Trace form

 $$28380 q - 12 q^{3} - 32 q^{4} - 48 q^{6} - 32 q^{7} - 20 q^{9} + O(q^{10})$$ $$28380 q - 12 q^{3} - 32 q^{4} - 48 q^{6} - 32 q^{7} - 20 q^{9} - 48 q^{10} - 80 q^{11} - 16 q^{12} + 256 q^{13} + 104 q^{15} - 96 q^{16} + 416 q^{17} - 16 q^{18} + 72 q^{19} - 72 q^{21} + 1856 q^{22} + 1984 q^{24} + 292 q^{25} - 160 q^{26} - 276 q^{27} - 3072 q^{28} - 1600 q^{29} - 2344 q^{30} - 1520 q^{31} - 4960 q^{32} - 1760 q^{33} - 4032 q^{34} - 456 q^{35} - 1808 q^{36} - 96 q^{37} + 1760 q^{38} + 1192 q^{39} + 3232 q^{40} + 3776 q^{41} + 6304 q^{42} + 1592 q^{43} + 4000 q^{44} - 508 q^{45} - 96 q^{46} - 16 q^{48} + 1316 q^{49} - 5712 q^{50} + 6112 q^{51} - 13280 q^{52} - 880 q^{54} + 1400 q^{55} + 1568 q^{56} + 792 q^{57} + 9472 q^{58} - 13760 q^{59} + 4872 q^{60} - 96 q^{61} + 11712 q^{62} - 2320 q^{63} + 24160 q^{64} + 4336 q^{65} + 11024 q^{66} - 16728 q^{67} + 8256 q^{68} + 1256 q^{69} + 3984 q^{70} - 2688 q^{71} - 16 q^{72} - 680 q^{73} - 10528 q^{74} + 600 q^{75} - 23904 q^{76} - 7008 q^{77} - 24256 q^{78} + 21968 q^{79} - 10032 q^{80} - 12372 q^{81} - 32 q^{82} - 5360 q^{83} - 8304 q^{84} - 7104 q^{85} - 16 q^{87} - 32 q^{88} + 704 q^{89} + 9336 q^{90} + 576 q^{91} + 10208 q^{93} - 32 q^{94} + 7728 q^{95} + 25792 q^{96} + 904 q^{97} + 7496 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(960))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
960.4.a $$\chi_{960}(1, \cdot)$$ 960.4.a.a 1 1
960.4.a.b 1
960.4.a.c 1
960.4.a.d 1
960.4.a.e 1
960.4.a.f 1
960.4.a.g 1
960.4.a.h 1
960.4.a.i 1
960.4.a.j 1
960.4.a.k 1
960.4.a.l 1
960.4.a.m 1
960.4.a.n 1
960.4.a.o 1
960.4.a.p 1
960.4.a.q 1
960.4.a.r 1
960.4.a.s 1
960.4.a.t 1
960.4.a.u 1
960.4.a.v 1
960.4.a.w 1
960.4.a.x 1
960.4.a.y 1
960.4.a.z 1
960.4.a.ba 1
960.4.a.bb 1
960.4.a.bc 1
960.4.a.bd 1
960.4.a.be 1
960.4.a.bf 1
960.4.a.bg 1
960.4.a.bh 1
960.4.a.bi 1
960.4.a.bj 1
960.4.a.bk 2
960.4.a.bl 2
960.4.a.bm 2
960.4.a.bn 2
960.4.a.bo 2
960.4.a.bp 2
960.4.b $$\chi_{960}(671, \cdot)$$ 960.4.b.a 16 1
960.4.b.b 16
960.4.b.c 32
960.4.b.d 32
960.4.d $$\chi_{960}(289, \cdot)$$ 960.4.d.a 12 1
960.4.d.b 12
960.4.d.c 24
960.4.d.d 24
960.4.f $$\chi_{960}(769, \cdot)$$ 960.4.f.a 2 1
960.4.f.b 2
960.4.f.c 2
960.4.f.d 2
960.4.f.e 2
960.4.f.f 2
960.4.f.g 2
960.4.f.h 2
960.4.f.i 2
960.4.f.j 2
960.4.f.k 2
960.4.f.l 2
960.4.f.m 4
960.4.f.n 4
960.4.f.o 4
960.4.f.p 4
960.4.f.q 4
960.4.f.r 6
960.4.f.s 6
960.4.f.t 8
960.4.f.u 8
960.4.h $$\chi_{960}(191, \cdot)$$ 960.4.h.a 8 1
960.4.h.b 16
960.4.h.c 24
960.4.h.d 24
960.4.h.e 24
960.4.k $$\chi_{960}(481, \cdot)$$ 960.4.k.a 2 1
960.4.k.b 2
960.4.k.c 6
960.4.k.d 6
960.4.k.e 8
960.4.k.f 8
960.4.k.g 8
960.4.k.h 8
960.4.m $$\chi_{960}(479, \cdot)$$ n/a 144 1
960.4.o $$\chi_{960}(959, \cdot)$$ n/a 140 1
960.4.s $$\chi_{960}(241, \cdot)$$ 960.4.s.a 44 2
960.4.s.b 52
960.4.t $$\chi_{960}(239, \cdot)$$ n/a 280 2
960.4.v $$\chi_{960}(257, \cdot)$$ n/a 280 2
960.4.w $$\chi_{960}(127, \cdot)$$ n/a 144 2
960.4.y $$\chi_{960}(847, \cdot)$$ n/a 144 2
960.4.bb $$\chi_{960}(497, \cdot)$$ n/a 280 2
960.4.bc $$\chi_{960}(367, \cdot)$$ n/a 144 2
960.4.bf $$\chi_{960}(17, \cdot)$$ n/a 280 2
960.4.bh $$\chi_{960}(223, \cdot)$$ n/a 144 2
960.4.bi $$\chi_{960}(353, \cdot)$$ n/a 288 2
960.4.bk $$\chi_{960}(431, \cdot)$$ n/a 192 2
960.4.bl $$\chi_{960}(49, \cdot)$$ n/a 144 2
960.4.bo $$\chi_{960}(103, \cdot)$$ None 0 4
960.4.br $$\chi_{960}(233, \cdot)$$ None 0 4
960.4.bs $$\chi_{960}(119, \cdot)$$ None 0 4
960.4.bv $$\chi_{960}(121, \cdot)$$ None 0 4
960.4.bx $$\chi_{960}(71, \cdot)$$ None 0 4
960.4.by $$\chi_{960}(169, \cdot)$$ None 0 4
960.4.cb $$\chi_{960}(137, \cdot)$$ None 0 4
960.4.cc $$\chi_{960}(7, \cdot)$$ None 0 4
960.4.cf $$\chi_{960}(173, \cdot)$$ n/a 4576 8
960.4.cg $$\chi_{960}(43, \cdot)$$ n/a 2304 8
960.4.ci $$\chi_{960}(61, \cdot)$$ n/a 1536 8
960.4.ck $$\chi_{960}(109, \cdot)$$ n/a 2304 8
960.4.cn $$\chi_{960}(11, \cdot)$$ n/a 3072 8
960.4.cp $$\chi_{960}(59, \cdot)$$ n/a 4576 8
960.4.cr $$\chi_{960}(53, \cdot)$$ n/a 4576 8
960.4.cs $$\chi_{960}(163, \cdot)$$ n/a 2304 8

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(960))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(960)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 14}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(480))$$$$^{\oplus 2}$$