## Defining parameters

 Level: $$N$$ = $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$1411200$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(2450))$$.

Total New Old
Modular forms 532560 156015 376545
Cusp forms 525840 156015 369825
Eisenstein series 6720 0 6720

## Trace form

 $$156015 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 5 q^{5} + 88 q^{6} + 48 q^{7} - 16 q^{8} - 250 q^{9} + O(q^{10})$$ $$156015 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 5 q^{5} + 88 q^{6} + 48 q^{7} - 16 q^{8} - 250 q^{9} - 50 q^{10} + 4 q^{11} - 32 q^{12} - 16 q^{13} - 132 q^{14} + 80 q^{15} - 96 q^{16} + 808 q^{17} + 638 q^{18} + 320 q^{19} - 80 q^{20} + 72 q^{21} - 336 q^{22} - 556 q^{23} - 192 q^{24} + 1373 q^{25} + 2648 q^{26} + 4264 q^{27} + 264 q^{28} - 1540 q^{29} - 1656 q^{30} - 3716 q^{31} + 96 q^{32} - 12108 q^{33} - 4838 q^{34} - 2448 q^{35} - 5736 q^{36} - 2841 q^{37} - 1436 q^{38} - 492 q^{39} + 120 q^{40} + 4160 q^{41} + 2340 q^{42} + 6472 q^{43} + 3320 q^{44} + 8669 q^{45} + 4388 q^{46} + 6572 q^{47} - 224 q^{48} - 7974 q^{49} + 350 q^{50} - 10576 q^{51} - 664 q^{52} - 7065 q^{53} - 2460 q^{54} + 5668 q^{55} + 288 q^{56} + 17056 q^{57} + 4896 q^{58} + 17648 q^{59} + 2720 q^{60} + 11776 q^{61} + 9700 q^{62} + 3936 q^{63} - 640 q^{64} - 3971 q^{65} + 1480 q^{66} - 19468 q^{67} - 1248 q^{68} - 28344 q^{69} - 12592 q^{71} + 752 q^{72} - 27184 q^{73} - 8400 q^{74} - 19528 q^{75} - 4048 q^{76} - 4098 q^{77} - 10360 q^{78} - 916 q^{79} + 80 q^{80} + 29150 q^{81} + 2280 q^{82} + 34236 q^{83} + 912 q^{84} + 23021 q^{85} + 21624 q^{86} + 87544 q^{87} + 13248 q^{88} + 68635 q^{89} + 29022 q^{90} + 9450 q^{91} + 14432 q^{92} + 4336 q^{93} + 6560 q^{94} - 7436 q^{95} - 2432 q^{96} - 31420 q^{97} - 15216 q^{98} - 69580 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(2450))$$

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
2450.4.a $$\chi_{2450}(1, \cdot)$$ 2450.4.a.a 1 1
2450.4.a.b 1
2450.4.a.c 1
2450.4.a.d 1
2450.4.a.e 1
2450.4.a.f 1
2450.4.a.g 1
2450.4.a.h 1
2450.4.a.i 1
2450.4.a.j 1
2450.4.a.k 1
2450.4.a.l 1
2450.4.a.m 1
2450.4.a.n 1
2450.4.a.o 1
2450.4.a.p 1
2450.4.a.q 1
2450.4.a.r 1
2450.4.a.s 1
2450.4.a.t 1
2450.4.a.u 1
2450.4.a.v 1
2450.4.a.w 1
2450.4.a.x 1
2450.4.a.y 1
2450.4.a.z 1
2450.4.a.ba 1
2450.4.a.bb 1
2450.4.a.bc 1
2450.4.a.bd 1
2450.4.a.be 1
2450.4.a.bf 1
2450.4.a.bg 1
2450.4.a.bh 1
2450.4.a.bi 1
2450.4.a.bj 1
2450.4.a.bk 1
2450.4.a.bl 1
2450.4.a.bm 1
2450.4.a.bn 1
2450.4.a.bo 1
2450.4.a.bp 1
2450.4.a.bq 1
2450.4.a.br 2
2450.4.a.bs 2
2450.4.a.bt 2
2450.4.a.bu 2
2450.4.a.bv 2
2450.4.a.bw 2
2450.4.a.bx 2
2450.4.a.by 2
2450.4.a.bz 2
2450.4.a.ca 2
2450.4.a.cb 3
2450.4.a.cc 3
2450.4.a.cd 3
2450.4.a.ce 3
2450.4.a.cf 3
2450.4.a.cg 3
2450.4.a.ch 3
2450.4.a.ci 3
2450.4.a.cj 4
2450.4.a.ck 4
2450.4.a.cl 4
2450.4.a.cm 4
2450.4.a.cn 4
2450.4.a.co 4
2450.4.a.cp 4
2450.4.a.cq 4
2450.4.a.cr 4
2450.4.a.cs 4
2450.4.a.ct 4
2450.4.a.cu 4
2450.4.a.cv 6
2450.4.a.cw 6
2450.4.a.cx 6
2450.4.a.cy 6
2450.4.a.cz 8
2450.4.a.da 8
2450.4.a.db 10
2450.4.a.dc 10
2450.4.c $$\chi_{2450}(99, \cdot)$$ n/a 184 1
2450.4.e $$\chi_{2450}(851, \cdot)$$ n/a 380 2
2450.4.g $$\chi_{2450}(293, \cdot)$$ n/a 360 2
2450.4.h $$\chi_{2450}(491, \cdot)$$ n/a 1228 4
2450.4.j $$\chi_{2450}(949, \cdot)$$ n/a 360 2
2450.4.l $$\chi_{2450}(351, \cdot)$$ n/a 1596 6
2450.4.n $$\chi_{2450}(589, \cdot)$$ n/a 1232 4
2450.4.p $$\chi_{2450}(607, \cdot)$$ n/a 720 4
2450.4.t $$\chi_{2450}(449, \cdot)$$ n/a 1512 6
2450.4.u $$\chi_{2450}(361, \cdot)$$ n/a 2400 8
2450.4.v $$\chi_{2450}(97, \cdot)$$ n/a 2400 8
2450.4.x $$\chi_{2450}(51, \cdot)$$ n/a 3192 12
2450.4.z $$\chi_{2450}(307, \cdot)$$ n/a 3024 12
2450.4.bb $$\chi_{2450}(79, \cdot)$$ n/a 2400 8
2450.4.bd $$\chi_{2450}(71, \cdot)$$ n/a 10080 24
2450.4.be $$\chi_{2450}(149, \cdot)$$ n/a 3024 12
2450.4.bi $$\chi_{2450}(117, \cdot)$$ n/a 4800 16
2450.4.bj $$\chi_{2450}(29, \cdot)$$ n/a 10080 24
2450.4.bm $$\chi_{2450}(143, \cdot)$$ n/a 6048 24
2450.4.bo $$\chi_{2450}(11, \cdot)$$ n/a 20160 48
2450.4.bp $$\chi_{2450}(13, \cdot)$$ n/a 20160 48
2450.4.bt $$\chi_{2450}(9, \cdot)$$ n/a 20160 48
2450.4.bv $$\chi_{2450}(3, \cdot)$$ n/a 40320 96

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(2450))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(2450)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(245))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(350))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(490))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1225))$$$$^{\oplus 2}$$