## Defining parameters

 Level: $$N$$ = $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$583200$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2025))$$.

Total New Old
Modular forms 148824 107324 41500
Cusp forms 142777 105028 37749
Eisenstein series 6047 2296 3751

## Trace form

 $$105028 q - 156 q^{2} - 234 q^{3} - 262 q^{4} - 192 q^{5} - 378 q^{6} - 263 q^{7} - 168 q^{8} - 234 q^{9} + O(q^{10})$$ $$105028 q - 156 q^{2} - 234 q^{3} - 262 q^{4} - 192 q^{5} - 378 q^{6} - 263 q^{7} - 168 q^{8} - 234 q^{9} - 464 q^{10} - 261 q^{11} - 234 q^{12} - 269 q^{13} - 177 q^{14} - 288 q^{15} - 422 q^{16} - 171 q^{17} - 225 q^{18} - 371 q^{19} - 192 q^{20} - 351 q^{21} - 255 q^{22} - 123 q^{23} - 180 q^{24} - 320 q^{25} - 405 q^{26} - 207 q^{27} - 321 q^{28} - 129 q^{29} - 288 q^{30} - 411 q^{31} - 12 q^{32} - 207 q^{33} - 201 q^{34} - 156 q^{35} - 342 q^{36} - 341 q^{37} - 51 q^{38} - 234 q^{39} - 248 q^{40} - 159 q^{41} - 189 q^{42} - 167 q^{43} + 171 q^{44} - 288 q^{45} - 469 q^{46} + 27 q^{47} - 135 q^{48} - 144 q^{49} - 96 q^{50} - 675 q^{51} - 77 q^{52} + 39 q^{53} - 108 q^{54} - 440 q^{55} + 63 q^{56} - 180 q^{57} - 157 q^{58} - 63 q^{59} - 288 q^{60} - 405 q^{61} - 51 q^{62} - 180 q^{63} - 300 q^{64} - 192 q^{65} - 378 q^{66} - 251 q^{67} - 156 q^{68} - 288 q^{69} - 280 q^{70} - 357 q^{71} - 450 q^{72} - 368 q^{73} - 171 q^{74} - 288 q^{75} - 655 q^{76} - 129 q^{77} - 351 q^{78} - 119 q^{79} - 12 q^{80} - 450 q^{81} - 504 q^{82} - 33 q^{83} - 459 q^{84} - 248 q^{85} - 21 q^{86} - 378 q^{87} - 31 q^{88} + 78 q^{89} - 288 q^{90} - 422 q^{91} + 255 q^{92} - 216 q^{93} + 69 q^{94} - 72 q^{95} - 387 q^{96} - 65 q^{97} + 438 q^{98} - 162 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2025))$$

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
2025.2.a $$\chi_{2025}(1, \cdot)$$ 2025.2.a.a 1 1
2025.2.a.b 1
2025.2.a.c 1
2025.2.a.d 1
2025.2.a.e 1
2025.2.a.f 1
2025.2.a.g 2
2025.2.a.h 2
2025.2.a.i 2
2025.2.a.j 2
2025.2.a.k 2
2025.2.a.l 2
2025.2.a.m 2
2025.2.a.n 3
2025.2.a.o 3
2025.2.a.p 4
2025.2.a.q 4
2025.2.a.r 4
2025.2.a.s 4
2025.2.a.t 4
2025.2.a.u 4
2025.2.a.v 4
2025.2.a.w 4
2025.2.a.x 4
2025.2.a.y 4
2025.2.a.z 4
2025.2.b $$\chi_{2025}(649, \cdot)$$ 2025.2.b.a 2 1
2025.2.b.b 2
2025.2.b.c 2
2025.2.b.d 2
2025.2.b.e 2
2025.2.b.f 2
2025.2.b.g 4
2025.2.b.h 4
2025.2.b.i 4
2025.2.b.j 4
2025.2.b.k 4
2025.2.b.l 6
2025.2.b.m 6
2025.2.b.n 8
2025.2.b.o 8
2025.2.b.p 8
2025.2.e $$\chi_{2025}(676, \cdot)$$ n/a 146 2
2025.2.f $$\chi_{2025}(1457, \cdot)$$ n/a 136 2
2025.2.h $$\chi_{2025}(406, \cdot)$$ n/a 464 4
2025.2.k $$\chi_{2025}(1324, \cdot)$$ n/a 140 2
2025.2.l $$\chi_{2025}(226, \cdot)$$ n/a 324 6
2025.2.n $$\chi_{2025}(244, \cdot)$$ n/a 464 4
2025.2.q $$\chi_{2025}(107, \cdot)$$ n/a 280 4
2025.2.r $$\chi_{2025}(136, \cdot)$$ n/a 944 8
2025.2.u $$\chi_{2025}(199, \cdot)$$ n/a 312 6
2025.2.w $$\chi_{2025}(242, \cdot)$$ n/a 928 8
2025.2.x $$\chi_{2025}(76, \cdot)$$ n/a 3024 18
2025.2.z $$\chi_{2025}(109, \cdot)$$ n/a 944 8
2025.2.bb $$\chi_{2025}(143, \cdot)$$ n/a 624 12
2025.2.bd $$\chi_{2025}(46, \cdot)$$ n/a 2112 24
2025.2.be $$\chi_{2025}(49, \cdot)$$ n/a 2880 18
2025.2.bh $$\chi_{2025}(53, \cdot)$$ n/a 1888 16
2025.2.bk $$\chi_{2025}(19, \cdot)$$ n/a 2112 24
2025.2.bn $$\chi_{2025}(32, \cdot)$$ n/a 5760 36
2025.2.bo $$\chi_{2025}(16, \cdot)$$ n/a 19296 72
2025.2.bp $$\chi_{2025}(8, \cdot)$$ n/a 4224 48
2025.2.br $$\chi_{2025}(4, \cdot)$$ n/a 19296 72
2025.2.bv $$\chi_{2025}(2, \cdot)$$ n/a 38592 144

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2025))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(2025)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(675))$$$$^{\oplus 2}$$