## Defining parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$18$$ Sturm bound: $$76800$$ Trace bound: $$8$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(600))$$.

Total New Old
Modular forms 29472 11279 18193
Cusp forms 28128 11115 17013
Eisenstein series 1344 164 1180

## Trace form

 $$11115 q + 2 q^{2} - 11 q^{3} - 4 q^{4} - 22 q^{5} - 34 q^{6} + 28 q^{7} - 76 q^{8} - 35 q^{9} + O(q^{10})$$ $$11115 q + 2 q^{2} - 11 q^{3} - 4 q^{4} - 22 q^{5} - 34 q^{6} + 28 q^{7} - 76 q^{8} - 35 q^{9} - 32 q^{10} + 84 q^{11} + 148 q^{12} + 142 q^{13} + 428 q^{14} - 88 q^{15} - 584 q^{16} - 210 q^{17} - 798 q^{18} - 856 q^{19} - 760 q^{20} - 136 q^{21} - 616 q^{22} + 576 q^{24} + 2 q^{25} + 1528 q^{26} + 217 q^{27} + 1672 q^{28} + 1566 q^{29} - 332 q^{30} + 1516 q^{31} - 1608 q^{32} - 508 q^{33} - 204 q^{34} - 912 q^{35} - 616 q^{36} - 1308 q^{37} + 3368 q^{38} + 14 q^{39} + 3088 q^{40} - 1258 q^{41} + 3232 q^{42} - 3384 q^{43} + 3280 q^{44} - 1910 q^{45} + 1344 q^{46} - 600 q^{47} - 3916 q^{48} + 375 q^{49} - 2840 q^{50} + 2746 q^{51} - 5768 q^{52} - 412 q^{53} + 890 q^{54} - 1376 q^{55} - 1864 q^{56} + 2120 q^{57} - 2500 q^{58} + 1252 q^{59} + 3268 q^{60} - 362 q^{61} - 2108 q^{62} + 3568 q^{63} - 9304 q^{64} + 5586 q^{65} - 3356 q^{66} - 5456 q^{67} - 2960 q^{68} + 2232 q^{69} - 824 q^{70} + 2192 q^{71} - 1048 q^{72} - 398 q^{73} + 10192 q^{74} + 3192 q^{75} + 10360 q^{76} - 2208 q^{77} + 7940 q^{78} + 12188 q^{79} + 6320 q^{80} - 9555 q^{81} - 13716 q^{82} - 6860 q^{83} - 11308 q^{84} - 19302 q^{85} - 8 q^{86} - 6342 q^{87} - 10872 q^{88} + 1228 q^{89} - 1012 q^{90} - 8488 q^{91} + 960 q^{92} - 792 q^{93} + 18688 q^{94} + 5824 q^{95} - 3860 q^{96} + 10770 q^{97} + 23322 q^{98} - 7828 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(600))$$

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
600.4.a $$\chi_{600}(1, \cdot)$$ 600.4.a.a 1 1
600.4.a.b 1
600.4.a.c 1
600.4.a.d 1
600.4.a.e 1
600.4.a.f 1
600.4.a.g 1
600.4.a.h 1
600.4.a.i 1
600.4.a.j 1
600.4.a.k 1
600.4.a.l 1
600.4.a.m 1
600.4.a.n 1
600.4.a.o 1
600.4.a.p 1
600.4.a.q 1
600.4.a.r 2
600.4.a.s 2
600.4.a.t 2
600.4.a.u 2
600.4.a.v 2
600.4.a.w 2
600.4.b $$\chi_{600}(251, \cdot)$$ n/a 222 1
600.4.d $$\chi_{600}(349, \cdot)$$ n/a 108 1
600.4.f $$\chi_{600}(49, \cdot)$$ 600.4.f.a 2 1
600.4.f.b 2
600.4.f.c 2
600.4.f.d 2
600.4.f.e 2
600.4.f.f 2
600.4.f.g 2
600.4.f.h 2
600.4.f.i 2
600.4.f.j 4
600.4.f.k 4
600.4.h $$\chi_{600}(551, \cdot)$$ None 0 1
600.4.k $$\chi_{600}(301, \cdot)$$ n/a 114 1
600.4.m $$\chi_{600}(299, \cdot)$$ n/a 212 1
600.4.o $$\chi_{600}(599, \cdot)$$ None 0 1
600.4.r $$\chi_{600}(257, \cdot)$$ n/a 108 2
600.4.s $$\chi_{600}(7, \cdot)$$ None 0 2
600.4.v $$\chi_{600}(43, \cdot)$$ n/a 216 2
600.4.w $$\chi_{600}(293, \cdot)$$ n/a 424 2
600.4.y $$\chi_{600}(121, \cdot)$$ n/a 176 4
600.4.ba $$\chi_{600}(71, \cdot)$$ None 0 4
600.4.bc $$\chi_{600}(169, \cdot)$$ n/a 184 4
600.4.be $$\chi_{600}(109, \cdot)$$ n/a 720 4
600.4.bg $$\chi_{600}(11, \cdot)$$ n/a 1424 4
600.4.bi $$\chi_{600}(119, \cdot)$$ None 0 4
600.4.bk $$\chi_{600}(59, \cdot)$$ n/a 1424 4
600.4.bm $$\chi_{600}(61, \cdot)$$ n/a 720 4
600.4.bp $$\chi_{600}(53, \cdot)$$ n/a 2848 8
600.4.bq $$\chi_{600}(67, \cdot)$$ n/a 1440 8
600.4.bt $$\chi_{600}(103, \cdot)$$ None 0 8
600.4.bu $$\chi_{600}(17, \cdot)$$ n/a 720 8

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(600))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(600)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$