## Defining parameters

 Level: $$N$$ = $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$235200$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1225))$$.

Total New Old
Modular forms 60480 52999 7481
Cusp forms 57121 51010 6111
Eisenstein series 3359 1989 1370

## Trace form

 $$51010q - 202q^{2} - 199q^{3} - 190q^{4} - 245q^{5} - 303q^{6} - 230q^{7} - 328q^{8} - 172q^{9} + O(q^{10})$$ $$51010q - 202q^{2} - 199q^{3} - 190q^{4} - 245q^{5} - 303q^{6} - 230q^{7} - 328q^{8} - 172q^{9} - 235q^{10} - 303q^{11} - 127q^{12} - 169q^{13} - 216q^{14} - 422q^{15} - 262q^{16} - 167q^{17} - 119q^{18} - 171q^{19} - 250q^{20} - 353q^{21} - 305q^{22} - 159q^{23} - 293q^{24} - 279q^{25} - 639q^{26} - 265q^{27} - 284q^{28} - 399q^{29} - 374q^{30} - 359q^{31} - 345q^{32} - 327q^{33} - 314q^{34} - 324q^{35} - 818q^{36} - 274q^{37} - 295q^{38} - 372q^{39} - 369q^{40} - 371q^{41} - 381q^{42} - 419q^{43} - 347q^{44} - 337q^{45} - 405q^{46} - 197q^{47} - 452q^{48} - 272q^{49} - 793q^{50} - 577q^{51} - 373q^{52} - 136q^{53} - 395q^{54} - 278q^{55} - 498q^{56} - 447q^{57} - 479q^{58} - 315q^{59} - 490q^{60} - 530q^{61} - 507q^{62} - 366q^{63} - 750q^{64} - 379q^{65} - 591q^{66} - 431q^{67} - 577q^{68} - 425q^{69} - 468q^{70} - 613q^{71} - 612q^{72} - 413q^{73} - 507q^{74} - 422q^{75} - 917q^{76} - 297q^{77} - 683q^{78} - 395q^{79} - 547q^{80} - 424q^{81} - 538q^{82} - 345q^{83} - 608q^{84} - 563q^{85} - 498q^{86} - 389q^{87} - 774q^{88} - 390q^{89} - 601q^{90} - 463q^{91} - 732q^{92} - 545q^{93} - 512q^{94} - 422q^{95} - 874q^{96} - 450q^{97} - 672q^{98} - 860q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1225))$$

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
1225.2.a $$\chi_{1225}(1, \cdot)$$ 1225.2.a.a 1 1
1225.2.a.b 1
1225.2.a.c 1
1225.2.a.d 1
1225.2.a.e 1
1225.2.a.f 1
1225.2.a.g 1
1225.2.a.h 1
1225.2.a.i 1
1225.2.a.j 1
1225.2.a.k 2
1225.2.a.l 2
1225.2.a.m 2
1225.2.a.n 2
1225.2.a.o 2
1225.2.a.p 2
1225.2.a.q 2
1225.2.a.r 2
1225.2.a.s 2
1225.2.a.t 2
1225.2.a.u 2
1225.2.a.v 2
1225.2.a.w 3
1225.2.a.x 3
1225.2.a.y 3
1225.2.a.z 3
1225.2.a.ba 4
1225.2.a.bb 4
1225.2.a.bc 4
1225.2.b $$\chi_{1225}(99, \cdot)$$ 1225.2.b.a 2 1
1225.2.b.b 2
1225.2.b.c 2
1225.2.b.d 2
1225.2.b.e 4
1225.2.b.f 4
1225.2.b.g 4
1225.2.b.h 4
1225.2.b.i 4
1225.2.b.j 4
1225.2.b.k 4
1225.2.b.l 6
1225.2.b.m 6
1225.2.b.n 8
1225.2.e $$\chi_{1225}(226, \cdot)$$ n/a 114 2
1225.2.f $$\chi_{1225}(293, \cdot)$$ n/a 112 2
1225.2.h $$\chi_{1225}(246, \cdot)$$ n/a 388 4
1225.2.k $$\chi_{1225}(324, \cdot)$$ n/a 112 2
1225.2.l $$\chi_{1225}(176, \cdot)$$ n/a 510 6
1225.2.o $$\chi_{1225}(344, \cdot)$$ n/a 392 4
1225.2.p $$\chi_{1225}(68, \cdot)$$ n/a 224 4
1225.2.t $$\chi_{1225}(274, \cdot)$$ n/a 492 6
1225.2.u $$\chi_{1225}(116, \cdot)$$ n/a 768 8
1225.2.w $$\chi_{1225}(48, \cdot)$$ n/a 768 8
1225.2.x $$\chi_{1225}(51, \cdot)$$ n/a 1032 12
1225.2.z $$\chi_{1225}(118, \cdot)$$ n/a 984 12
1225.2.ba $$\chi_{1225}(79, \cdot)$$ n/a 768 8
1225.2.bd $$\chi_{1225}(36, \cdot)$$ n/a 3312 24
1225.2.be $$\chi_{1225}(74, \cdot)$$ n/a 984 12
1225.2.bi $$\chi_{1225}(117, \cdot)$$ n/a 1536 16
1225.2.bj $$\chi_{1225}(29, \cdot)$$ n/a 3312 24
1225.2.bn $$\chi_{1225}(82, \cdot)$$ n/a 1968 24
1225.2.bo $$\chi_{1225}(11, \cdot)$$ n/a 6624 48
1225.2.bp $$\chi_{1225}(13, \cdot)$$ n/a 6624 48
1225.2.bt $$\chi_{1225}(4, \cdot)$$ n/a 6624 48
1225.2.bu $$\chi_{1225}(3, \cdot)$$ n/a 13248 96

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1225))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1225)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(245))$$$$^{\oplus 2}$$