## Defining parameters

 Level: $$N$$ = $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Newform subspaces: $$61$$ Sturm bound: $$18816$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(245))$$.

Total New Old
Modular forms 7296 6351 945
Cusp forms 6816 6061 755
Eisenstein series 480 290 190

## Trace form

 $$6061 q - 34 q^{2} - 52 q^{3} - 22 q^{4} - 26 q^{5} + 34 q^{6} + 12 q^{7} - 162 q^{8} - 221 q^{9} + O(q^{10})$$ $$6061 q - 34 q^{2} - 52 q^{3} - 22 q^{4} - 26 q^{5} + 34 q^{6} + 12 q^{7} - 162 q^{8} - 221 q^{9} - 79 q^{10} - 58 q^{11} + 166 q^{12} - 56 q^{13} + 12 q^{14} + 53 q^{15} + 86 q^{16} + 308 q^{17} - 10 q^{18} - 554 q^{19} - 415 q^{20} - 564 q^{21} - 206 q^{22} + 732 q^{23} + 2070 q^{24} + 1744 q^{25} + 1898 q^{26} + 1586 q^{27} + 732 q^{28} - 248 q^{29} - 2147 q^{30} - 1782 q^{31} - 5078 q^{32} - 3590 q^{33} - 3914 q^{34} - 1338 q^{35} - 2638 q^{36} + 1076 q^{37} + 2726 q^{38} + 2510 q^{39} + 3225 q^{40} + 4528 q^{41} + 1302 q^{42} + 244 q^{43} - 1298 q^{44} + 724 q^{45} - 4566 q^{46} - 2272 q^{47} - 10928 q^{48} - 7800 q^{49} - 5716 q^{50} - 6926 q^{51} - 10882 q^{52} - 4228 q^{53} + 910 q^{54} + 5543 q^{55} + 5784 q^{56} + 14162 q^{57} + 19310 q^{58} + 13286 q^{59} + 16345 q^{60} + 10120 q^{61} + 11166 q^{62} + 7080 q^{63} + 358 q^{64} - 5813 q^{65} - 16534 q^{66} - 12672 q^{67} - 19154 q^{68} - 22686 q^{69} - 12849 q^{70} - 12686 q^{71} - 21774 q^{72} - 10196 q^{73} - 14786 q^{74} - 4687 q^{75} - 5542 q^{76} - 504 q^{77} + 7570 q^{78} + 6786 q^{79} + 25376 q^{80} + 43855 q^{81} + 47276 q^{82} + 33624 q^{83} + 54780 q^{84} + 22313 q^{85} + 39836 q^{86} + 29426 q^{87} + 32268 q^{88} + 16188 q^{89} + 10646 q^{90} - 6774 q^{91} - 1632 q^{92} - 38550 q^{93} - 34436 q^{94} - 22721 q^{95} - 81004 q^{96} - 28678 q^{97} - 63936 q^{98} - 47080 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(245))$$

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
245.4.a $$\chi_{245}(1, \cdot)$$ 245.4.a.a 1 1
245.4.a.b 1
245.4.a.c 1
245.4.a.d 1
245.4.a.e 1
245.4.a.f 1
245.4.a.g 2
245.4.a.h 2
245.4.a.i 2
245.4.a.j 2
245.4.a.k 2
245.4.a.l 3
245.4.a.m 5
245.4.a.n 5
245.4.a.o 6
245.4.a.p 6
245.4.b $$\chi_{245}(99, \cdot)$$ 245.4.b.a 2 1
245.4.b.b 4
245.4.b.c 8
245.4.b.d 10
245.4.b.e 10
245.4.b.f 10
245.4.b.g 12
245.4.e $$\chi_{245}(116, \cdot)$$ 245.4.e.a 2 2
245.4.e.b 2
245.4.e.c 2
245.4.e.d 2
245.4.e.e 2
245.4.e.f 2
245.4.e.g 2
245.4.e.h 4
245.4.e.i 4
245.4.e.j 4
245.4.e.k 4
245.4.e.l 4
245.4.e.m 6
245.4.e.n 6
245.4.e.o 10
245.4.e.p 12
245.4.e.q 12
245.4.f $$\chi_{245}(48, \cdot)$$ 245.4.f.a 40 2
245.4.f.b 72
245.4.j $$\chi_{245}(79, \cdot)$$ 245.4.j.a 4 2
245.4.j.b 8
245.4.j.c 16
245.4.j.d 20
245.4.j.e 20
245.4.j.f 20
245.4.j.g 24
245.4.k $$\chi_{245}(36, \cdot)$$ 245.4.k.a 162 6
245.4.k.b 174
245.4.l $$\chi_{245}(68, \cdot)$$ 245.4.l.a 8 4
245.4.l.b 32
245.4.l.c 40
245.4.l.d 144
245.4.p $$\chi_{245}(29, \cdot)$$ 245.4.p.a 492 6
245.4.q $$\chi_{245}(11, \cdot)$$ 245.4.q.a 324 12
245.4.q.b 348
245.4.s $$\chi_{245}(13, \cdot)$$ 245.4.s.a 984 12
245.4.t $$\chi_{245}(4, \cdot)$$ 245.4.t.a 984 12
245.4.x $$\chi_{245}(3, \cdot)$$ 245.4.x.a 1968 24

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(245))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(245)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 2}$$