# Properties

 Label 350.10 Level 350 Weight 10 Dimension 9538 Nonzero newspaces 12 Sturm bound 72000 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$10$$ Nonzero newspaces: $$12$$ Sturm bound: $$72000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_1(350))$$.

Total New Old
Modular forms 32736 9538 23198
Cusp forms 32064 9538 22526
Eisenstein series 672 0 672

## Trace form

 $$9538 q + 32 q^{2} - 1196 q^{3} + 512 q^{4} - 4670 q^{5} - 5408 q^{6} - 6048 q^{7} + 8192 q^{8} - 149440 q^{9} + O(q^{10})$$ $$9538 q + 32 q^{2} - 1196 q^{3} + 512 q^{4} - 4670 q^{5} - 5408 q^{6} - 6048 q^{7} + 8192 q^{8} - 149440 q^{9} + 117920 q^{10} - 424782 q^{11} - 306176 q^{12} + 697266 q^{13} + 438752 q^{14} + 487040 q^{15} + 2228224 q^{16} - 852838 q^{17} - 5103808 q^{18} + 1178100 q^{19} + 2365440 q^{20} - 2075632 q^{21} - 2122944 q^{22} - 7740182 q^{23} - 8429568 q^{24} - 16905374 q^{25} + 4783968 q^{26} - 10886720 q^{27} - 8590848 q^{28} + 52661884 q^{29} + 10791168 q^{30} + 14409418 q^{31} - 8388608 q^{32} - 110635422 q^{33} - 35887648 q^{34} - 45397972 q^{35} + 117279744 q^{36} + 123621752 q^{37} + 74046112 q^{38} + 233478348 q^{39} - 26173440 q^{40} - 99066076 q^{41} - 408532768 q^{42} - 256378412 q^{43} + 57217536 q^{44} + 979266938 q^{45} + 444780160 q^{46} - 431033546 q^{47} - 46530560 q^{48} - 249760550 q^{49} - 157285600 q^{50} - 439157818 q^{51} + 88960512 q^{52} + 1496040924 q^{53} + 1419450432 q^{54} + 960102176 q^{55} - 3481600 q^{56} - 1182399944 q^{57} - 1119727104 q^{58} - 2805736416 q^{59} + 367063040 q^{60} - 240834044 q^{61} + 292544704 q^{62} + 3123707600 q^{63} + 335544320 q^{64} - 1827852502 q^{65} + 893090176 q^{66} - 293771994 q^{67} + 930273792 q^{68} - 4342900428 q^{69} + 431771520 q^{70} + 88875944 q^{71} + 37117952 q^{72} + 2922961422 q^{73} - 1053846400 q^{74} - 4012494256 q^{75} - 65401344 q^{76} + 2611653926 q^{77} - 2318604928 q^{78} - 3955142886 q^{79} - 306053120 q^{80} - 2770267978 q^{81} + 6347320704 q^{82} + 3654166402 q^{83} - 702262272 q^{84} - 3284017598 q^{85} - 7634673344 q^{86} - 11476737704 q^{87} + 3135651840 q^{88} + 13105694700 q^{89} + 5303688864 q^{90} - 11620150854 q^{91} - 3674942464 q^{92} - 21471690182 q^{93} + 3676727488 q^{94} + 10261496568 q^{95} + 2262827008 q^{96} + 11635959084 q^{97} + 16832873504 q^{98} + 4850311676 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_1(350))$$

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
350.10.a $$\chi_{350}(1, \cdot)$$ 350.10.a.a 1 1
350.10.a.b 1
350.10.a.c 1
350.10.a.d 1
350.10.a.e 2
350.10.a.f 2
350.10.a.g 2
350.10.a.h 2
350.10.a.i 2
350.10.a.j 2
350.10.a.k 3
350.10.a.l 3
350.10.a.m 4
350.10.a.n 4
350.10.a.o 4
350.10.a.p 4
350.10.a.q 5
350.10.a.r 5
350.10.a.s 5
350.10.a.t 5
350.10.a.u 6
350.10.a.v 6
350.10.a.w 8
350.10.a.x 8
350.10.c $$\chi_{350}(99, \cdot)$$ 350.10.c.a 2 1
350.10.c.b 2
350.10.c.c 2
350.10.c.d 2
350.10.c.e 4
350.10.c.f 4
350.10.c.g 4
350.10.c.h 4
350.10.c.i 4
350.10.c.j 4
350.10.c.k 6
350.10.c.l 6
350.10.c.m 8
350.10.c.n 8
350.10.c.o 10
350.10.c.p 10
350.10.e $$\chi_{350}(51, \cdot)$$ n/a 228 2
350.10.g $$\chi_{350}(293, \cdot)$$ n/a 216 2
350.10.h $$\chi_{350}(71, \cdot)$$ n/a 536 4
350.10.j $$\chi_{350}(149, \cdot)$$ n/a 216 2
350.10.m $$\chi_{350}(29, \cdot)$$ n/a 544 4
350.10.o $$\chi_{350}(143, \cdot)$$ n/a 432 4
350.10.q $$\chi_{350}(11, \cdot)$$ n/a 1440 8
350.10.r $$\chi_{350}(13, \cdot)$$ n/a 1440 8
350.10.u $$\chi_{350}(9, \cdot)$$ n/a 1440 8
350.10.x $$\chi_{350}(3, \cdot)$$ n/a 2880 16

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

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_1(350))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_1(350)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 6}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 2}$$