# Properties

 Label 363.3 Level 363 Weight 3 Dimension 7000 Nonzero newspaces 8 Newform subspaces 49 Sturm bound 29040 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$8$$ Newform subspaces: $$49$$ Sturm bound: $$29040$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(363))$$.

Total New Old
Modular forms 10000 7282 2718
Cusp forms 9360 7000 2360
Eisenstein series 640 282 358

## Trace form

 $$7000 q - 45 q^{3} - 90 q^{4} - 5 q^{6} - 30 q^{7} + 80 q^{8} - 25 q^{9} + O(q^{10})$$ $$7000 q - 45 q^{3} - 90 q^{4} - 5 q^{6} - 30 q^{7} + 80 q^{8} - 25 q^{9} - 70 q^{10} - 10 q^{11} - 165 q^{12} - 150 q^{13} - 20 q^{14} - 25 q^{15} + 70 q^{16} + 20 q^{17} - 65 q^{18} + 30 q^{19} + 100 q^{20} - 35 q^{21} - 160 q^{22} - 80 q^{23} - 505 q^{24} - 410 q^{25} - 500 q^{26} - 195 q^{27} - 790 q^{28} - 320 q^{29} - 435 q^{30} - 490 q^{31} + 125 q^{33} + 550 q^{34} + 640 q^{35} + 795 q^{36} + 390 q^{37} + 500 q^{38} + 485 q^{39} + 650 q^{40} + 240 q^{41} + 665 q^{42} + 370 q^{43} + 330 q^{44} - 155 q^{45} + 50 q^{46} + 100 q^{47} + 45 q^{48} - 310 q^{49} - 660 q^{50} - 705 q^{51} - 670 q^{52} - 740 q^{53} - 1795 q^{54} - 620 q^{55} - 1360 q^{56} - 1225 q^{57} - 1030 q^{58} - 300 q^{59} - 1115 q^{60} - 350 q^{61} - 80 q^{62} + 5 q^{63} - 810 q^{64} + 175 q^{66} - 590 q^{67} - 160 q^{68} + 235 q^{69} + 970 q^{70} + 1040 q^{71} + 1785 q^{72} + 1770 q^{73} + 1460 q^{74} + 1695 q^{75} + 1690 q^{76} + 570 q^{77} + 2135 q^{78} + 1330 q^{79} + 740 q^{80} - 145 q^{81} - 530 q^{82} + 380 q^{83} - 1115 q^{84} - 1110 q^{85} - 1040 q^{86} - 1135 q^{87} - 1580 q^{88} - 760 q^{89} - 1315 q^{90} - 2070 q^{91} - 380 q^{92} - 565 q^{93} - 870 q^{94} - 860 q^{95} - 175 q^{96} - 510 q^{97} + 250 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(363))$$

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
363.3.b $$\chi_{363}(122, \cdot)$$ 363.3.b.a 1 1
363.3.b.b 1
363.3.b.c 2
363.3.b.d 2
363.3.b.e 2
363.3.b.f 4
363.3.b.g 4
363.3.b.h 4
363.3.b.i 4
363.3.b.j 6
363.3.b.k 6
363.3.b.l 8
363.3.b.m 8
363.3.b.n 12
363.3.c $$\chi_{363}(241, \cdot)$$ 363.3.c.a 4 1
363.3.c.b 4
363.3.c.c 4
363.3.c.d 8
363.3.c.e 16
363.3.g $$\chi_{363}(40, \cdot)$$ 363.3.g.a 16 4
363.3.g.b 16
363.3.g.c 16
363.3.g.d 16
363.3.g.e 16
363.3.g.f 16
363.3.g.g 16
363.3.g.h 32
363.3.h $$\chi_{363}(245, \cdot)$$ 363.3.h.a 4 4
363.3.h.b 4
363.3.h.c 8
363.3.h.d 8
363.3.h.e 8
363.3.h.f 8
363.3.h.g 8
363.3.h.h 8
363.3.h.i 8
363.3.h.j 16
363.3.h.k 16
363.3.h.l 16
363.3.h.m 16
363.3.h.n 16
363.3.h.o 16
363.3.h.p 24
363.3.h.q 24
363.3.h.r 48
363.3.k $$\chi_{363}(10, \cdot)$$ 363.3.k.a 440 10
363.3.l $$\chi_{363}(23, \cdot)$$ 363.3.l.a 860 10
363.3.n $$\chi_{363}(5, \cdot)$$ 363.3.n.a 3440 40
363.3.o $$\chi_{363}(7, \cdot)$$ 363.3.o.a 1760 40

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(363))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(363)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 2}$$