# Properties

 Label 363.4 Level 363 Weight 4 Dimension 10570 Nonzero newspaces 8 Sturm bound 38720 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Sturm bound: $$38720$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(363))$$.

Total New Old
Modular forms 14840 10850 3990
Cusp forms 14200 10570 3630
Eisenstein series 640 280 360

## Trace form

 $$10570 q - 45 q^{3} - 90 q^{4} - 145 q^{6} - 130 q^{7} + 160 q^{8} + 115 q^{9} + O(q^{10})$$ $$10570 q - 45 q^{3} - 90 q^{4} - 145 q^{6} - 130 q^{7} + 160 q^{8} + 115 q^{9} + 290 q^{10} + 100 q^{11} + 235 q^{12} - 10 q^{13} - 780 q^{14} - 665 q^{15} - 1610 q^{16} - 600 q^{17} - 325 q^{18} + 810 q^{19} + 1900 q^{20} + 785 q^{21} + 1160 q^{22} + 760 q^{23} + 1975 q^{24} + 630 q^{25} + 100 q^{26} - 105 q^{27} - 2190 q^{28} - 1400 q^{29} - 3915 q^{30} - 2850 q^{31} - 3560 q^{32} - 1855 q^{33} - 3930 q^{34} - 2320 q^{35} - 2485 q^{36} - 10 q^{37} + 1300 q^{38} + 2095 q^{39} + 2970 q^{40} + 4440 q^{41} + 4825 q^{42} + 210 q^{43} + 770 q^{44} + 5255 q^{45} + 3490 q^{46} + 2360 q^{47} + 4085 q^{48} + 3870 q^{49} + 2860 q^{50} + 2045 q^{51} + 4690 q^{52} - 640 q^{53} + 1025 q^{54} - 1500 q^{55} - 720 q^{56} - 1735 q^{57} + 3290 q^{58} + 2040 q^{59} - 6435 q^{60} - 2410 q^{61} - 5520 q^{62} - 8305 q^{63} - 12090 q^{64} - 6720 q^{65} - 7195 q^{66} - 6850 q^{67} - 9760 q^{68} - 6675 q^{69} - 10310 q^{70} - 3720 q^{71} + 165 q^{72} - 11010 q^{73} - 3900 q^{74} + 2105 q^{75} + 7890 q^{76} + 900 q^{77} + 14575 q^{78} + 7110 q^{79} + 19620 q^{80} + 15595 q^{81} + 25430 q^{82} + 15440 q^{83} + 23105 q^{84} + 24710 q^{85} + 11920 q^{86} + 10385 q^{87} + 11180 q^{88} + 7000 q^{89} - 15255 q^{90} - 4490 q^{91} - 22100 q^{92} - 12265 q^{93} - 26190 q^{94} - 7200 q^{95} - 28335 q^{96} - 12030 q^{97} - 10120 q^{98} - 4920 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(363))$$

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
363.4.a $$\chi_{363}(1, \cdot)$$ 363.4.a.a 1 1
363.4.a.b 1
363.4.a.c 1
363.4.a.d 1
363.4.a.e 1
363.4.a.f 1
363.4.a.g 1
363.4.a.h 1
363.4.a.i 2
363.4.a.j 2
363.4.a.k 2
363.4.a.l 2
363.4.a.m 2
363.4.a.n 2
363.4.a.o 2
363.4.a.p 4
363.4.a.q 4
363.4.a.r 4
363.4.a.s 4
363.4.a.t 4
363.4.a.u 6
363.4.a.v 6
363.4.d $$\chi_{363}(362, \cdot)$$ 363.4.d.a 4 1
363.4.d.b 4
363.4.d.c 12
363.4.d.d 40
363.4.d.e 40
363.4.e $$\chi_{363}(124, \cdot)$$ n/a 216 4
363.4.f $$\chi_{363}(161, \cdot)$$ n/a 400 4
363.4.i $$\chi_{363}(34, \cdot)$$ n/a 660 10
363.4.j $$\chi_{363}(32, \cdot)$$ n/a 1300 10
363.4.m $$\chi_{363}(4, \cdot)$$ n/a 2640 40
363.4.p $$\chi_{363}(2, \cdot)$$ n/a 5200 40

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(363))$$ into lower level spaces

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