## Defining parameters

 Level: $$N$$ = $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$705600$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2450))$$.

Total New Old
Modular forms 179760 52006 127754
Cusp forms 173041 52006 121035
Eisenstein series 6719 0 6719

## Trace form

 $$52006q - q^{2} - 8q^{3} - 5q^{4} - 5q^{5} - 16q^{6} - 8q^{7} - q^{8} - 41q^{9} + O(q^{10})$$ $$52006q - q^{2} - 8q^{3} - 5q^{4} - 5q^{5} - 16q^{6} - 8q^{7} - q^{8} - 41q^{9} - 5q^{10} - 36q^{11} - 8q^{12} - 34q^{13} - 6q^{14} - 20q^{15} - 5q^{16} - 46q^{17} - 12q^{18} - 24q^{19} - 26q^{21} + 4q^{22} - 32q^{23} + 4q^{24} + 83q^{25} + 66q^{26} + 220q^{27} + 40q^{28} + 154q^{29} + 164q^{30} + 144q^{31} + 4q^{32} + 376q^{33} + 175q^{34} + 72q^{35} + 151q^{36} + 231q^{37} + 178q^{38} + 434q^{39} - 5q^{40} + 158q^{41} + 198q^{42} + 168q^{43} + 102q^{44} + 159q^{45} + 150q^{46} + 100q^{47} + 6q^{48} + 118q^{49} - 25q^{50} + 36q^{51} - 20q^{52} - 45q^{53} + 14q^{54} + 28q^{55} + 36q^{56} + 124q^{57} - 18q^{58} + 168q^{59} + 244q^{61} - 2q^{62} + 156q^{63} - 5q^{64} + 179q^{65} - 84q^{66} + 268q^{67} - 6q^{68} + 236q^{69} + 56q^{71} - 37q^{72} + 238q^{73} - 10q^{74} + 252q^{75} - 24q^{76} + 60q^{77} - 52q^{78} + 208q^{79} - 5q^{80} + 111q^{81} - 34q^{82} + 196q^{83} - 26q^{84} + 111q^{85} + 40q^{86} + 308q^{87} + 60q^{88} + 411q^{89} + 187q^{90} + 182q^{91} + 140q^{92} + 584q^{93} + 216q^{94} + 324q^{95} + 80q^{96} + 406q^{97} + 240q^{98} + 528q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2450))$$

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
2450.2.a $$\chi_{2450}(1, \cdot)$$ 2450.2.a.a 1 1
2450.2.a.b 1
2450.2.a.c 1
2450.2.a.d 1
2450.2.a.e 1
2450.2.a.f 1
2450.2.a.g 1
2450.2.a.h 1
2450.2.a.i 1
2450.2.a.j 1
2450.2.a.k 1
2450.2.a.l 1
2450.2.a.m 1
2450.2.a.n 1
2450.2.a.o 1
2450.2.a.p 1
2450.2.a.q 1
2450.2.a.r 1
2450.2.a.s 1
2450.2.a.t 1
2450.2.a.u 1
2450.2.a.v 1
2450.2.a.w 1
2450.2.a.x 1
2450.2.a.y 1
2450.2.a.z 1
2450.2.a.ba 1
2450.2.a.bb 1
2450.2.a.bc 1
2450.2.a.bd 1
2450.2.a.be 1
2450.2.a.bf 1
2450.2.a.bg 1
2450.2.a.bh 1
2450.2.a.bi 2
2450.2.a.bj 2
2450.2.a.bk 2
2450.2.a.bl 2
2450.2.a.bm 2
2450.2.a.bn 2
2450.2.a.bo 2
2450.2.a.bp 2
2450.2.a.bq 2
2450.2.a.br 2
2450.2.a.bs 2
2450.2.a.bt 4
2450.2.a.bu 4
2450.2.c $$\chi_{2450}(99, \cdot)$$ 2450.2.c.a 2 1
2450.2.c.b 2
2450.2.c.c 2
2450.2.c.d 2
2450.2.c.e 2
2450.2.c.f 2
2450.2.c.g 2
2450.2.c.h 2
2450.2.c.i 2
2450.2.c.j 2
2450.2.c.k 2
2450.2.c.l 2
2450.2.c.m 2
2450.2.c.n 2
2450.2.c.o 2
2450.2.c.p 2
2450.2.c.q 2
2450.2.c.r 2
2450.2.c.s 2
2450.2.c.t 4
2450.2.c.u 4
2450.2.c.v 4
2450.2.c.w 4
2450.2.c.x 8
2450.2.e $$\chi_{2450}(851, \cdot)$$ n/a 128 2
2450.2.g $$\chi_{2450}(293, \cdot)$$ n/a 120 2
2450.2.h $$\chi_{2450}(491, \cdot)$$ n/a 412 4
2450.2.j $$\chi_{2450}(949, \cdot)$$ n/a 120 2
2450.2.l $$\chi_{2450}(351, \cdot)$$ n/a 540 6
2450.2.n $$\chi_{2450}(589, \cdot)$$ n/a 408 4
2450.2.p $$\chi_{2450}(607, \cdot)$$ n/a 240 4
2450.2.t $$\chi_{2450}(449, \cdot)$$ n/a 504 6
2450.2.u $$\chi_{2450}(361, \cdot)$$ n/a 800 8
2450.2.v $$\chi_{2450}(97, \cdot)$$ n/a 800 8
2450.2.x $$\chi_{2450}(51, \cdot)$$ n/a 1056 12
2450.2.z $$\chi_{2450}(307, \cdot)$$ n/a 1008 12
2450.2.bb $$\chi_{2450}(79, \cdot)$$ n/a 800 8
2450.2.bd $$\chi_{2450}(71, \cdot)$$ n/a 3360 24
2450.2.be $$\chi_{2450}(149, \cdot)$$ n/a 1008 12
2450.2.bi $$\chi_{2450}(117, \cdot)$$ n/a 1600 16
2450.2.bj $$\chi_{2450}(29, \cdot)$$ n/a 3360 24
2450.2.bm $$\chi_{2450}(143, \cdot)$$ n/a 2016 24
2450.2.bo $$\chi_{2450}(11, \cdot)$$ n/a 6720 48
2450.2.bp $$\chi_{2450}(13, \cdot)$$ n/a 6720 48
2450.2.bt $$\chi_{2450}(9, \cdot)$$ n/a 6720 48
2450.2.bv $$\chi_{2450}(3, \cdot)$$ n/a 13440 96

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2450))$$ into lower level spaces

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