## Defining parameters

 Level: $$N$$ = $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$464640$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1815))$$.

Total New Old
Modular forms 118720 79388 39332
Cusp forms 113601 77700 35901
Eisenstein series 5119 1688 3431

## Trace form

 $$77700q - 4q^{2} - 92q^{3} - 188q^{4} - 252q^{6} - 148q^{7} + 68q^{8} - 50q^{9} + O(q^{10})$$ $$77700q - 4q^{2} - 92q^{3} - 188q^{4} - 252q^{6} - 148q^{7} + 68q^{8} - 50q^{9} - 214q^{10} + 20q^{11} - 96q^{12} - 156q^{13} + 96q^{14} - 97q^{15} - 332q^{16} + 104q^{17} - 34q^{18} - 76q^{19} + 92q^{20} - 178q^{21} - 80q^{22} + 56q^{23} - 8q^{24} - 190q^{25} + 160q^{26} - 152q^{27} - 36q^{28} + 48q^{29} - 127q^{30} - 492q^{31} + 12q^{32} - 120q^{33} - 236q^{34} + 72q^{35} - 258q^{36} - 68q^{37} + 136q^{38} - 2q^{39} - 382q^{40} + 208q^{41} - 54q^{42} - 100q^{43} + 60q^{44} - 165q^{45} - 412q^{46} - 140q^{48} - 204q^{49} - 104q^{50} - 230q^{51} - 396q^{52} - 24q^{53} - 152q^{54} - 380q^{55} - 280q^{56} - 294q^{57} - 548q^{58} - 184q^{59} - 411q^{60} - 724q^{61} - 176q^{62} - 238q^{63} - 540q^{64} - 136q^{65} - 390q^{66} - 436q^{67} - 48q^{68} - 154q^{69} - 154q^{70} + 160q^{71} - 122q^{72} + 116q^{73} + 256q^{74} + 3q^{75} + 76q^{76} + 180q^{77} - 14q^{78} + 260q^{79} + 388q^{80} - 230q^{81} + 124q^{82} + 448q^{83} - 46q^{84} - 26q^{85} + 424q^{86} + 142q^{87} + 120q^{88} + 224q^{89} - 369q^{90} - 372q^{91} + 112q^{92} - 42q^{93} - 292q^{94} - 36q^{95} - 628q^{96} - 276q^{97} - 164q^{98} - 240q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1815))$$

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
1815.2.a $$\chi_{1815}(1, \cdot)$$ 1815.2.a.a 1 1
1815.2.a.b 1
1815.2.a.c 1
1815.2.a.d 1
1815.2.a.e 1
1815.2.a.f 2
1815.2.a.g 2
1815.2.a.h 2
1815.2.a.i 2
1815.2.a.j 2
1815.2.a.k 2
1815.2.a.l 3
1815.2.a.m 3
1815.2.a.n 3
1815.2.a.o 4
1815.2.a.p 4
1815.2.a.q 4
1815.2.a.r 4
1815.2.a.s 4
1815.2.a.t 4
1815.2.a.u 4
1815.2.a.v 4
1815.2.a.w 4
1815.2.a.x 4
1815.2.a.y 6
1815.2.c $$\chi_{1815}(364, \cdot)$$ 1815.2.c.a 2 1
1815.2.c.b 2
1815.2.c.c 4
1815.2.c.d 6
1815.2.c.e 6
1815.2.c.f 8
1815.2.c.g 8
1815.2.c.h 12
1815.2.c.i 12
1815.2.c.j 24
1815.2.c.k 24
1815.2.d $$\chi_{1815}(1814, \cdot)$$ n/a 200 1
1815.2.f $$\chi_{1815}(1451, \cdot)$$ n/a 144 1
1815.2.j $$\chi_{1815}(967, \cdot)$$ n/a 216 2
1815.2.k $$\chi_{1815}(122, \cdot)$$ n/a 400 2
1815.2.m $$\chi_{1815}(511, \cdot)$$ n/a 288 4
1815.2.p $$\chi_{1815}(161, \cdot)$$ n/a 576 4
1815.2.r $$\chi_{1815}(239, \cdot)$$ n/a 800 4
1815.2.s $$\chi_{1815}(124, \cdot)$$ n/a 432 4
1815.2.u $$\chi_{1815}(166, \cdot)$$ n/a 880 10
1815.2.w $$\chi_{1815}(323, \cdot)$$ n/a 1600 8
1815.2.x $$\chi_{1815}(112, \cdot)$$ n/a 864 8
1815.2.bb $$\chi_{1815}(131, \cdot)$$ n/a 1760 10
1815.2.bd $$\chi_{1815}(164, \cdot)$$ n/a 2600 10
1815.2.be $$\chi_{1815}(34, \cdot)$$ n/a 1320 10
1815.2.bh $$\chi_{1815}(23, \cdot)$$ n/a 5200 20
1815.2.bi $$\chi_{1815}(43, \cdot)$$ n/a 2640 20
1815.2.bk $$\chi_{1815}(16, \cdot)$$ n/a 3520 40
1815.2.bm $$\chi_{1815}(4, \cdot)$$ n/a 5280 40
1815.2.bn $$\chi_{1815}(29, \cdot)$$ n/a 10400 40
1815.2.bp $$\chi_{1815}(41, \cdot)$$ n/a 7040 40
1815.2.bt $$\chi_{1815}(7, \cdot)$$ n/a 10560 80
1815.2.bu $$\chi_{1815}(38, \cdot)$$ n/a 20800 80

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1815))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1815)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(165))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(363))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(605))$$$$^{\oplus 2}$$