## Defining parameters

 Level: $$N$$ = $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$32$$ Sturm bound: $$393216$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1344))$$.

Total New Old
Modular forms 149184 59836 89348
Cusp forms 145728 59396 86332
Eisenstein series 3456 440 3016

## Trace form

 $$59396q - 24q^{3} - 64q^{4} - 32q^{6} - 64q^{7} - 40q^{9} + O(q^{10})$$ $$59396q - 24q^{3} - 64q^{4} - 32q^{6} - 64q^{7} - 40q^{9} - 64q^{10} - 80q^{11} - 32q^{12} + 224q^{13} + 184q^{15} - 64q^{16} + 416q^{17} - 32q^{18} + 48q^{19} - 52q^{21} + 1728q^{22} + 1968q^{24} + 124q^{25} - 160q^{26} - 288q^{27} - 1600q^{28} - 1600q^{29} - 4672q^{30} - 1496q^{31} - 4960q^{32} - 1772q^{33} - 4064q^{34} - 456q^{35} - 1840q^{36} - 128q^{37} + 1760q^{38} + 1172q^{39} + 6496q^{40} + 3776q^{41} + 3120q^{42} + 1496q^{43} + 4000q^{44} - 1000q^{45} - 64q^{46} - 32q^{48} - 140q^{49} - 11424q^{50} + 6124q^{51} - 13312q^{52} - 896q^{54} + 2848q^{55} + 784q^{56} + 496q^{57} + 9440q^{58} - 13760q^{59} + 9760q^{60} - 64q^{61} + 11712q^{62} - 2556q^{63} + 24032q^{64} - 8096q^{65} + 11040q^{66} - 24144q^{67} + 8256q^{68} - 1736q^{69} + 3952q^{70} - 1792q^{71} - 32q^{72} + 4016q^{73} - 10528q^{74} + 13928q^{75} - 23872q^{76} + 3504q^{77} - 24320q^{78} + 34600q^{79} - 20064q^{80} - 2488q^{81} - 64q^{82} + 5360q^{83} - 4184q^{84} + 4544q^{85} + 2548q^{87} - 64q^{88} - 704q^{89} + 18688q^{90} + 5104q^{91} + 11392q^{93} - 64q^{94} + 7728q^{95} + 25808q^{96} + 5360q^{97} + 3776q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1344))$$

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
1344.4.a $$\chi_{1344}(1, \cdot)$$ 1344.4.a.a 1 1
1344.4.a.b 1
1344.4.a.c 1
1344.4.a.d 1
1344.4.a.e 1
1344.4.a.f 1
1344.4.a.g 1
1344.4.a.h 1
1344.4.a.i 1
1344.4.a.j 1
1344.4.a.k 1
1344.4.a.l 1
1344.4.a.m 1
1344.4.a.n 1
1344.4.a.o 1
1344.4.a.p 1
1344.4.a.q 1
1344.4.a.r 1
1344.4.a.s 1
1344.4.a.t 1
1344.4.a.u 1
1344.4.a.v 1
1344.4.a.w 1
1344.4.a.x 1
1344.4.a.y 1
1344.4.a.z 1
1344.4.a.ba 1
1344.4.a.bb 1
1344.4.a.bc 2
1344.4.a.bd 2
1344.4.a.be 2
1344.4.a.bf 2
1344.4.a.bg 2
1344.4.a.bh 2
1344.4.a.bi 2
1344.4.a.bj 2
1344.4.a.bk 2
1344.4.a.bl 2
1344.4.a.bm 2
1344.4.a.bn 2
1344.4.a.bo 2
1344.4.a.bp 2
1344.4.a.bq 2
1344.4.a.br 2
1344.4.a.bs 3
1344.4.a.bt 3
1344.4.a.bu 3
1344.4.a.bv 3
1344.4.b $$\chi_{1344}(895, \cdot)$$ 1344.4.b.a 2 1
1344.4.b.b 2
1344.4.b.c 2
1344.4.b.d 2
1344.4.b.e 8
1344.4.b.f 8
1344.4.b.g 12
1344.4.b.h 12
1344.4.b.i 24
1344.4.b.j 24
1344.4.c $$\chi_{1344}(673, \cdot)$$ 1344.4.c.a 6 1
1344.4.c.b 6
1344.4.c.c 6
1344.4.c.d 6
1344.4.c.e 12
1344.4.c.f 12
1344.4.c.g 12
1344.4.c.h 12
1344.4.h $$\chi_{1344}(575, \cdot)$$ n/a 144 1
1344.4.i $$\chi_{1344}(545, \cdot)$$ n/a 192 1
1344.4.j $$\chi_{1344}(1247, \cdot)$$ n/a 144 1
1344.4.k $$\chi_{1344}(1217, \cdot)$$ n/a 188 1
1344.4.p $$\chi_{1344}(223, \cdot)$$ 1344.4.p.a 16 1
1344.4.p.b 16
1344.4.p.c 32
1344.4.p.d 32
1344.4.q $$\chi_{1344}(193, \cdot)$$ n/a 192 2
1344.4.s $$\chi_{1344}(239, \cdot)$$ n/a 288 2
1344.4.u $$\chi_{1344}(559, \cdot)$$ n/a 192 2
1344.4.w $$\chi_{1344}(337, \cdot)$$ n/a 144 2
1344.4.y $$\chi_{1344}(209, \cdot)$$ n/a 376 2
1344.4.bb $$\chi_{1344}(31, \cdot)$$ n/a 192 2
1344.4.bc $$\chi_{1344}(257, \cdot)$$ n/a 376 2
1344.4.bd $$\chi_{1344}(95, \cdot)$$ n/a 384 2
1344.4.bi $$\chi_{1344}(353, \cdot)$$ n/a 384 2
1344.4.bj $$\chi_{1344}(191, \cdot)$$ n/a 376 2
1344.4.bk $$\chi_{1344}(289, \cdot)$$ n/a 192 2
1344.4.bl $$\chi_{1344}(703, \cdot)$$ n/a 192 2
1344.4.bo $$\chi_{1344}(41, \cdot)$$ None 0 4
1344.4.bq $$\chi_{1344}(169, \cdot)$$ None 0 4
1344.4.bs $$\chi_{1344}(71, \cdot)$$ None 0 4
1344.4.bu $$\chi_{1344}(55, \cdot)$$ None 0 4
1344.4.bw $$\chi_{1344}(17, \cdot)$$ n/a 752 4
1344.4.by $$\chi_{1344}(529, \cdot)$$ n/a 384 4
1344.4.ca $$\chi_{1344}(271, \cdot)$$ n/a 384 4
1344.4.cc $$\chi_{1344}(431, \cdot)$$ n/a 752 4
1344.4.cg $$\chi_{1344}(85, \cdot)$$ n/a 2304 8
1344.4.ch $$\chi_{1344}(139, \cdot)$$ n/a 3072 8
1344.4.ci $$\chi_{1344}(155, \cdot)$$ n/a 4608 8
1344.4.cj $$\chi_{1344}(125, \cdot)$$ n/a 6112 8
1344.4.cn $$\chi_{1344}(103, \cdot)$$ None 0 8
1344.4.cp $$\chi_{1344}(23, \cdot)$$ None 0 8
1344.4.cr $$\chi_{1344}(25, \cdot)$$ None 0 8
1344.4.ct $$\chi_{1344}(89, \cdot)$$ None 0 8
1344.4.cw $$\chi_{1344}(5, \cdot)$$ n/a 12224 16
1344.4.cx $$\chi_{1344}(11, \cdot)$$ n/a 12224 16
1344.4.cy $$\chi_{1344}(19, \cdot)$$ n/a 6144 16
1344.4.cz $$\chi_{1344}(37, \cdot)$$ n/a 6144 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1344))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1344)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 14}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(448))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(672))$$$$^{\oplus 2}$$