# Properties

 Label 210.3 Level 210 Weight 3 Dimension 504 Nonzero newspaces 12 Newform subspaces 17 Sturm bound 6912 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$17$$ Sturm bound: $$6912$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(210))$$.

Total New Old
Modular forms 2496 504 1992
Cusp forms 2112 504 1608
Eisenstein series 384 0 384

## Trace form

 $$504 q - 8 q^{2} - 20 q^{3} - 16 q^{4} - 12 q^{5} + 16 q^{6} + 40 q^{7} + 16 q^{8} + 44 q^{9} + O(q^{10})$$ $$504 q - 8 q^{2} - 20 q^{3} - 16 q^{4} - 12 q^{5} + 16 q^{6} + 40 q^{7} + 16 q^{8} + 44 q^{9} + 72 q^{10} + 104 q^{11} + 40 q^{12} + 112 q^{13} + 76 q^{15} - 32 q^{16} + 8 q^{17} - 40 q^{18} - 80 q^{19} - 16 q^{20} - 28 q^{21} - 64 q^{22} - 64 q^{23} - 48 q^{24} + 228 q^{25} + 144 q^{26} - 56 q^{27} + 128 q^{28} + 240 q^{29} + 156 q^{30} + 624 q^{31} + 32 q^{32} + 372 q^{33} + 320 q^{34} + 100 q^{35} + 96 q^{36} - 96 q^{37} - 304 q^{38} - 352 q^{39} - 48 q^{40} - 736 q^{41} - 352 q^{42} - 1200 q^{43} - 336 q^{44} - 666 q^{45} - 592 q^{46} - 552 q^{47} - 32 q^{48} + 24 q^{49} - 88 q^{50} - 124 q^{51} + 64 q^{52} - 8 q^{53} - 192 q^{54} + 248 q^{55} + 128 q^{56} + 680 q^{57} + 640 q^{58} + 1008 q^{59} + 252 q^{60} + 1632 q^{61} + 160 q^{62} - 116 q^{63} + 128 q^{64} + 564 q^{65} - 32 q^{66} - 32 q^{67} - 16 q^{68} - 104 q^{69} - 456 q^{70} - 704 q^{71} - 16 q^{72} - 1552 q^{73} - 480 q^{74} - 298 q^{75} - 592 q^{76} - 1312 q^{77} - 848 q^{78} - 1408 q^{79} + 48 q^{80} - 276 q^{81} - 928 q^{82} - 1728 q^{83} - 216 q^{84} - 1568 q^{85} + 336 q^{86} - 1392 q^{87} + 416 q^{88} + 144 q^{89} - 744 q^{90} + 688 q^{91} + 160 q^{92} - 628 q^{93} + 1552 q^{94} + 244 q^{95} - 32 q^{96} + 40 q^{97} + 312 q^{98} - 1288 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(210))$$

We only show spaces with odd 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
210.3.c $$\chi_{210}(29, \cdot)$$ 210.3.c.a 24 1
210.3.e $$\chi_{210}(71, \cdot)$$ 210.3.e.a 16 1
210.3.f $$\chi_{210}(181, \cdot)$$ 210.3.f.a 8 1
210.3.h $$\chi_{210}(139, \cdot)$$ 210.3.h.a 16 1
210.3.k $$\chi_{210}(83, \cdot)$$ 210.3.k.a 32 2
210.3.k.b 32
210.3.l $$\chi_{210}(43, \cdot)$$ 210.3.l.a 8 2
210.3.l.b 16
210.3.o $$\chi_{210}(31, \cdot)$$ 210.3.o.a 8 2
210.3.o.b 16
210.3.p $$\chi_{210}(19, \cdot)$$ 210.3.p.a 32 2
210.3.q $$\chi_{210}(149, \cdot)$$ 210.3.q.a 64 2
210.3.s $$\chi_{210}(11, \cdot)$$ 210.3.s.a 40 2
210.3.v $$\chi_{210}(37, \cdot)$$ 210.3.v.a 32 4
210.3.v.b 32
210.3.w $$\chi_{210}(17, \cdot)$$ 210.3.w.a 64 4
210.3.w.b 64

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(210))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(210)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 2}$$