## Defining parameters

 Level: $$N$$ = $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$56448$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(490))$$.

Total New Old
Modular forms 21648 6207 15441
Cusp forms 20688 6207 14481
Eisenstein series 960 0 960

## Trace form

 $$6207 q + 2 q^{2} + 40 q^{3} - 4 q^{4} - 53 q^{5} - 152 q^{6} - 96 q^{7} + 8 q^{8} + 251 q^{9} + O(q^{10})$$ $$6207 q + 2 q^{2} + 40 q^{3} - 4 q^{4} - 53 q^{5} - 152 q^{6} - 96 q^{7} + 8 q^{8} + 251 q^{9} + 122 q^{10} + 124 q^{11} + 160 q^{12} + 206 q^{13} + 264 q^{14} - 104 q^{15} + 48 q^{16} - 534 q^{17} - 598 q^{18} - 220 q^{19} - 180 q^{20} - 144 q^{21} - 840 q^{22} - 1044 q^{23} - 288 q^{24} - 2393 q^{25} - 500 q^{26} - 944 q^{27} + 48 q^{28} + 2418 q^{29} + 2232 q^{30} + 2848 q^{31} + 32 q^{32} + 5304 q^{33} + 2020 q^{34} + 2028 q^{35} + 2364 q^{36} - 5242 q^{37} - 2864 q^{38} - 5464 q^{39} - 840 q^{40} - 5234 q^{41} + 72 q^{42} - 2656 q^{43} + 608 q^{44} - 3585 q^{45} + 5240 q^{46} + 1596 q^{47} + 832 q^{48} + 15948 q^{49} + 3562 q^{50} + 17864 q^{51} + 2024 q^{52} + 10134 q^{53} + 4200 q^{54} - 2504 q^{55} - 576 q^{56} - 10336 q^{57} - 9252 q^{58} - 14948 q^{59} - 4536 q^{60} - 16254 q^{61} - 10472 q^{62} - 15648 q^{63} + 1472 q^{64} + 2494 q^{65} - 1952 q^{66} + 2840 q^{67} - 2136 q^{68} + 7096 q^{69} - 756 q^{70} + 1440 q^{71} - 2392 q^{72} + 8930 q^{73} + 5036 q^{74} + 5156 q^{75} + 4976 q^{76} + 5388 q^{77} + 5536 q^{78} + 1448 q^{79} - 848 q^{80} - 37653 q^{81} - 4812 q^{82} - 29784 q^{83} - 1824 q^{84} - 21842 q^{85} - 6552 q^{86} - 34896 q^{87} - 2592 q^{88} - 17410 q^{89} - 14286 q^{90} + 3564 q^{91} - 6192 q^{92} + 29360 q^{93} - 5728 q^{94} + 15448 q^{95} + 640 q^{96} + 14330 q^{97} + 3648 q^{98} + 44564 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(490))$$

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
490.4.a $$\chi_{490}(1, \cdot)$$ 490.4.a.a 1 1
490.4.a.b 1
490.4.a.c 1
490.4.a.d 1
490.4.a.e 1
490.4.a.f 1
490.4.a.g 1
490.4.a.h 1
490.4.a.i 1
490.4.a.j 1
490.4.a.k 1
490.4.a.l 1
490.4.a.m 1
490.4.a.n 1
490.4.a.o 1
490.4.a.p 2
490.4.a.q 2
490.4.a.r 2
490.4.a.s 2
490.4.a.t 2
490.4.a.u 2
490.4.a.v 3
490.4.a.w 3
490.4.a.x 4
490.4.a.y 4
490.4.c $$\chi_{490}(99, \cdot)$$ 490.4.c.a 2 1
490.4.c.b 2
490.4.c.c 6
490.4.c.d 8
490.4.c.e 12
490.4.c.f 12
490.4.c.g 20
490.4.e $$\chi_{490}(361, \cdot)$$ 490.4.e.a 2 2
490.4.e.b 2
490.4.e.c 2
490.4.e.d 2
490.4.e.e 2
490.4.e.f 2
490.4.e.g 2
490.4.e.h 2
490.4.e.i 2
490.4.e.j 2
490.4.e.k 2
490.4.e.l 2
490.4.e.m 2
490.4.e.n 2
490.4.e.o 2
490.4.e.p 2
490.4.e.q 2
490.4.e.r 2
490.4.e.s 2
490.4.e.t 4
490.4.e.u 4
490.4.e.v 4
490.4.e.w 4
490.4.e.x 4
490.4.e.y 6
490.4.e.z 8
490.4.e.ba 8
490.4.g $$\chi_{490}(97, \cdot)$$ n/a 120 2
490.4.i $$\chi_{490}(79, \cdot)$$ n/a 120 2
490.4.k $$\chi_{490}(71, \cdot)$$ n/a 336 6
490.4.l $$\chi_{490}(117, \cdot)$$ n/a 240 4
490.4.p $$\chi_{490}(29, \cdot)$$ n/a 504 6
490.4.q $$\chi_{490}(11, \cdot)$$ n/a 672 12
490.4.s $$\chi_{490}(13, \cdot)$$ n/a 1008 12
490.4.t $$\chi_{490}(9, \cdot)$$ n/a 1008 12
490.4.w $$\chi_{490}(3, \cdot)$$ n/a 2016 24

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(490))$$ into lower level spaces

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