## Defining parameters

 Level: $$N$$ = $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$32$$ Sturm bound: $$1204224$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(2352))$$.

Total New Old
Modular forms 454944 183253 271691
Cusp forms 448224 182381 265843
Eisenstein series 6720 872 5848

## Trace form

 $$182381q - 50q^{3} - 144q^{4} + 2q^{5} - 32q^{6} - 108q^{7} + 84q^{8} + 42q^{9} + O(q^{10})$$ $$182381q - 50q^{3} - 144q^{4} + 2q^{5} - 32q^{6} - 108q^{7} + 84q^{8} + 42q^{9} + 8q^{10} - 60q^{11} - 164q^{12} - 176q^{13} + 69q^{15} - 424q^{16} - 26q^{17} - 44q^{18} - 1138q^{19} + 80q^{20} - 270q^{21} + 448q^{22} + 256q^{23} + 48q^{24} + 1765q^{25} - 20q^{26} + 124q^{27} - 144q^{28} + 1258q^{29} - 800q^{30} + 630q^{31} - 960q^{32} + 281q^{33} - 304q^{34} - 1044q^{35} - 604q^{36} - 2060q^{37} + 1256q^{38} - 181q^{39} + 2256q^{40} + 414q^{41} + 2520q^{42} + 2662q^{43} + 14584q^{44} + 3819q^{45} + 7512q^{46} + 648q^{47} + 852q^{48} - 1572q^{49} - 7692q^{50} - 2727q^{51} - 17368q^{52} - 8174q^{53} - 16168q^{54} - 4062q^{55} - 9240q^{56} - 7923q^{57} - 15072q^{58} - 4756q^{59} - 12244q^{60} - 5636q^{61} - 6612q^{62} - 24q^{63} + 3384q^{64} + 7068q^{65} + 11104q^{66} + 782q^{67} + 22672q^{68} + 14011q^{69} + 14736q^{70} + 3152q^{71} + 7912q^{72} - 4852q^{73} - 2740q^{74} + 998q^{75} - 2568q^{76} - 3348q^{78} + 5942q^{79} + 712q^{80} - 7162q^{81} + 608q^{82} + 5868q^{83} - 72q^{84} + 4902q^{85} - 1712q^{86} - 5061q^{87} + 4808q^{88} - 66q^{89} + 3260q^{90} + 10182q^{91} - 7616q^{92} + 3109q^{93} - 45392q^{94} + 47024q^{95} - 20132q^{96} - 902q^{97} - 11928q^{98} + 17216q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(2352))$$

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
2352.4.a $$\chi_{2352}(1, \cdot)$$ 2352.4.a.a 1 1
2352.4.a.b 1
2352.4.a.c 1
2352.4.a.d 1
2352.4.a.e 1
2352.4.a.f 1
2352.4.a.g 1
2352.4.a.h 1
2352.4.a.i 1
2352.4.a.j 1
2352.4.a.k 1
2352.4.a.l 1
2352.4.a.m 1
2352.4.a.n 1
2352.4.a.o 1
2352.4.a.p 1
2352.4.a.q 1
2352.4.a.r 1
2352.4.a.s 1
2352.4.a.t 1
2352.4.a.u 1
2352.4.a.v 1
2352.4.a.w 1
2352.4.a.x 1
2352.4.a.y 1
2352.4.a.z 1
2352.4.a.ba 1
2352.4.a.bb 1
2352.4.a.bc 1
2352.4.a.bd 1
2352.4.a.be 1
2352.4.a.bf 1
2352.4.a.bg 1
2352.4.a.bh 1
2352.4.a.bi 1
2352.4.a.bj 1
2352.4.a.bk 1
2352.4.a.bl 2
2352.4.a.bm 2
2352.4.a.bn 2
2352.4.a.bo 2
2352.4.a.bp 2
2352.4.a.bq 2
2352.4.a.br 2
2352.4.a.bs 2
2352.4.a.bt 2
2352.4.a.bu 2
2352.4.a.bv 2
2352.4.a.bw 2
2352.4.a.bx 2
2352.4.a.by 2
2352.4.a.bz 2
2352.4.a.ca 2
2352.4.a.cb 2
2352.4.a.cc 2
2352.4.a.cd 2
2352.4.a.ce 2
2352.4.a.cf 2
2352.4.a.cg 3
2352.4.a.ch 3
2352.4.a.ci 3
2352.4.a.cj 3
2352.4.a.ck 4
2352.4.a.cl 4
2352.4.a.cm 4
2352.4.a.cn 4
2352.4.a.co 4
2352.4.a.cp 4
2352.4.a.cq 4
2352.4.a.cr 4
2352.4.b $$\chi_{2352}(1567, \cdot)$$ n/a 120 1
2352.4.c $$\chi_{2352}(1177, \cdot)$$ None 0 1
2352.4.h $$\chi_{2352}(2255, \cdot)$$ n/a 246 1
2352.4.i $$\chi_{2352}(2057, \cdot)$$ None 0 1
2352.4.j $$\chi_{2352}(1079, \cdot)$$ None 0 1
2352.4.k $$\chi_{2352}(881, \cdot)$$ n/a 236 1
2352.4.p $$\chi_{2352}(391, \cdot)$$ None 0 1
2352.4.q $$\chi_{2352}(961, \cdot)$$ n/a 240 2
2352.4.s $$\chi_{2352}(491, \cdot)$$ n/a 1948 2
2352.4.u $$\chi_{2352}(979, \cdot)$$ n/a 960 2
2352.4.w $$\chi_{2352}(589, \cdot)$$ n/a 984 2
2352.4.y $$\chi_{2352}(293, \cdot)$$ n/a 1904 2
2352.4.bb $$\chi_{2352}(1207, \cdot)$$ None 0 2
2352.4.bc $$\chi_{2352}(1697, \cdot)$$ n/a 472 2
2352.4.bd $$\chi_{2352}(263, \cdot)$$ None 0 2
2352.4.bi $$\chi_{2352}(521, \cdot)$$ None 0 2
2352.4.bj $$\chi_{2352}(863, \cdot)$$ n/a 480 2
2352.4.bk $$\chi_{2352}(361, \cdot)$$ None 0 2
2352.4.bl $$\chi_{2352}(31, \cdot)$$ n/a 240 2
2352.4.bo $$\chi_{2352}(337, \cdot)$$ n/a 1008 6
2352.4.bp $$\chi_{2352}(509, \cdot)$$ n/a 3808 4
2352.4.br $$\chi_{2352}(373, \cdot)$$ n/a 1920 4
2352.4.bt $$\chi_{2352}(19, \cdot)$$ n/a 1920 4
2352.4.bv $$\chi_{2352}(275, \cdot)$$ n/a 3808 4
2352.4.bx $$\chi_{2352}(55, \cdot)$$ None 0 6
2352.4.cc $$\chi_{2352}(209, \cdot)$$ n/a 2004 6
2352.4.cd $$\chi_{2352}(71, \cdot)$$ None 0 6
2352.4.ce $$\chi_{2352}(41, \cdot)$$ None 0 6
2352.4.cf $$\chi_{2352}(239, \cdot)$$ n/a 2016 6
2352.4.ck $$\chi_{2352}(169, \cdot)$$ None 0 6
2352.4.cl $$\chi_{2352}(223, \cdot)$$ n/a 1008 6
2352.4.cm $$\chi_{2352}(193, \cdot)$$ n/a 2016 12
2352.4.cn $$\chi_{2352}(125, \cdot)$$ n/a 16080 12
2352.4.cp $$\chi_{2352}(85, \cdot)$$ n/a 8064 12
2352.4.cr $$\chi_{2352}(139, \cdot)$$ n/a 8064 12
2352.4.ct $$\chi_{2352}(155, \cdot)$$ n/a 16080 12
2352.4.cx $$\chi_{2352}(271, \cdot)$$ n/a 2016 12
2352.4.cy $$\chi_{2352}(25, \cdot)$$ None 0 12
2352.4.cz $$\chi_{2352}(95, \cdot)$$ n/a 4032 12
2352.4.da $$\chi_{2352}(89, \cdot)$$ None 0 12
2352.4.df $$\chi_{2352}(23, \cdot)$$ None 0 12
2352.4.dg $$\chi_{2352}(17, \cdot)$$ n/a 4008 12
2352.4.dh $$\chi_{2352}(103, \cdot)$$ None 0 12
2352.4.dl $$\chi_{2352}(11, \cdot)$$ n/a 32160 24
2352.4.dn $$\chi_{2352}(115, \cdot)$$ n/a 16128 24
2352.4.dp $$\chi_{2352}(37, \cdot)$$ n/a 16128 24
2352.4.dr $$\chi_{2352}(5, \cdot)$$ n/a 32160 24

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(2352))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(2352)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(294))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(392))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(784))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1176))$$$$^{\oplus 2}$$