# Properties

 Label 1521.4 Level 1521 Weight 4 Dimension 198986 Nonzero newspaces 30 Sturm bound 681408 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$30$$ Sturm bound: $$681408$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1521))$$.

Total New Old
Modular forms 257352 200827 56525
Cusp forms 253704 198986 54718
Eisenstein series 3648 1841 1807

## Trace form

 $$198986 q - 201 q^{2} - 267 q^{3} - 211 q^{4} - 213 q^{5} - 255 q^{6} - 113 q^{7} + 12 q^{8} - 219 q^{9} + O(q^{10})$$ $$198986 q - 201 q^{2} - 267 q^{3} - 211 q^{4} - 213 q^{5} - 255 q^{6} - 113 q^{7} + 12 q^{8} - 219 q^{9} - 762 q^{10} - 384 q^{11} - 420 q^{12} - 360 q^{13} - 678 q^{14} - 237 q^{15} - 127 q^{16} + 378 q^{17} - 48 q^{18} + 196 q^{19} + 846 q^{20} - 243 q^{21} - 945 q^{22} - 687 q^{23} - 555 q^{24} - 940 q^{25} - 966 q^{26} - 936 q^{27} - 4922 q^{28} - 4281 q^{29} - 3240 q^{30} - 1325 q^{31} + 381 q^{32} + 510 q^{33} + 4353 q^{34} + 6048 q^{35} + 5655 q^{36} + 3082 q^{37} + 8535 q^{38} + 1536 q^{39} + 13614 q^{40} + 6756 q^{41} + 2994 q^{42} + 2050 q^{43} + 1944 q^{44} - 1419 q^{45} - 6234 q^{46} - 5757 q^{47} - 7773 q^{48} - 9636 q^{49} - 18009 q^{50} - 5631 q^{51} - 4494 q^{52} - 2166 q^{53} + 951 q^{54} - 2148 q^{55} - 5340 q^{56} - 1485 q^{57} - 2190 q^{58} - 1980 q^{59} - 3300 q^{60} + 1807 q^{61} - 18210 q^{62} - 12429 q^{63} - 23428 q^{64} - 7533 q^{65} - 14694 q^{66} - 6578 q^{67} - 11655 q^{68} + 1347 q^{69} + 7608 q^{70} + 12090 q^{71} + 15291 q^{72} + 17254 q^{73} + 37518 q^{74} + 16413 q^{75} + 40171 q^{76} + 28731 q^{77} + 17268 q^{78} + 17629 q^{79} + 57582 q^{80} + 11649 q^{81} + 23280 q^{82} + 14685 q^{83} + 12138 q^{84} + 5238 q^{85} + 8493 q^{86} - 2913 q^{87} - 19551 q^{88} - 9390 q^{89} - 25668 q^{90} - 8004 q^{91} - 53760 q^{92} - 23949 q^{93} - 55026 q^{94} - 40122 q^{95} + 6312 q^{96} - 16448 q^{97} + 9636 q^{98} + 27393 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1521))$$

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
1521.4.a $$\chi_{1521}(1, \cdot)$$ 1521.4.a.a 1 1
1521.4.a.b 1
1521.4.a.c 1
1521.4.a.d 1
1521.4.a.e 1
1521.4.a.f 1
1521.4.a.g 1
1521.4.a.h 1
1521.4.a.i 1
1521.4.a.j 1
1521.4.a.k 1
1521.4.a.l 2
1521.4.a.m 2
1521.4.a.n 2
1521.4.a.o 2
1521.4.a.p 2
1521.4.a.q 2
1521.4.a.r 2
1521.4.a.s 2
1521.4.a.t 2
1521.4.a.u 3
1521.4.a.v 4
1521.4.a.w 4
1521.4.a.x 4
1521.4.a.y 4
1521.4.a.z 4
1521.4.a.ba 4
1521.4.a.bb 4
1521.4.a.bc 8
1521.4.a.bd 8
1521.4.a.be 9
1521.4.a.bf 9
1521.4.a.bg 9
1521.4.a.bh 9
1521.4.a.bi 9
1521.4.a.bj 9
1521.4.a.bk 10
1521.4.a.bl 12
1521.4.a.bm 18
1521.4.a.bn 18
1521.4.b $$\chi_{1521}(1351, \cdot)$$ n/a 188 1
1521.4.e $$\chi_{1521}(508, \cdot)$$ n/a 908 2
1521.4.f $$\chi_{1521}(529, \cdot)$$ n/a 904 2
1521.4.g $$\chi_{1521}(991, \cdot)$$ n/a 374 2
1521.4.h $$\chi_{1521}(22, \cdot)$$ n/a 904 2
1521.4.i $$\chi_{1521}(746, \cdot)$$ n/a 308 2
1521.4.l $$\chi_{1521}(823, \cdot)$$ n/a 904 2
1521.4.q $$\chi_{1521}(316, \cdot)$$ n/a 376 2
1521.4.r $$\chi_{1521}(868, \cdot)$$ n/a 904 2
1521.4.t $$\chi_{1521}(337, \cdot)$$ n/a 904 2
1521.4.x $$\chi_{1521}(587, \cdot)$$ n/a 1808 4
1521.4.z $$\chi_{1521}(239, \cdot)$$ n/a 1808 4
1521.4.ba $$\chi_{1521}(80, \cdot)$$ n/a 616 4
1521.4.bc $$\chi_{1521}(488, \cdot)$$ n/a 1808 4
1521.4.be $$\chi_{1521}(118, \cdot)$$ n/a 2724 12
1521.4.bh $$\chi_{1521}(64, \cdot)$$ n/a 2712 12
1521.4.bi $$\chi_{1521}(16, \cdot)$$ n/a 13056 24
1521.4.bj $$\chi_{1521}(55, \cdot)$$ n/a 5448 24
1521.4.bk $$\chi_{1521}(61, \cdot)$$ n/a 13056 24
1521.4.bl $$\chi_{1521}(40, \cdot)$$ n/a 13056 24
1521.4.bn $$\chi_{1521}(8, \cdot)$$ n/a 4368 24
1521.4.bq $$\chi_{1521}(25, \cdot)$$ n/a 13056 24
1521.4.bs $$\chi_{1521}(43, \cdot)$$ n/a 13056 24
1521.4.bt $$\chi_{1521}(10, \cdot)$$ n/a 5424 24
1521.4.by $$\chi_{1521}(4, \cdot)$$ n/a 13056 24
1521.4.cb $$\chi_{1521}(20, \cdot)$$ n/a 26112 48
1521.4.cd $$\chi_{1521}(71, \cdot)$$ n/a 8736 48
1521.4.ce $$\chi_{1521}(5, \cdot)$$ n/a 26112 48
1521.4.cg $$\chi_{1521}(2, \cdot)$$ n/a 26112 48

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1521))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1521)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(117))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(507))$$$$^{\oplus 2}$$