# Properties

 Label 210.2 Level 210 Weight 2 Dimension 249 Nonzero newspaces 12 Newforms 32 Sturm bound 4608 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newforms: $$32$$ Sturm bound: $$4608$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(210))$$.

Total New Old
Modular forms 1344 249 1095
Cusp forms 961 249 712
Eisenstein series 383 0 383

## Trace form

 $$249q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut 9q^{3}$$ $$\mathstrut +\mathstrut 9q^{4}$$ $$\mathstrut +\mathstrut 21q^{5}$$ $$\mathstrut +\mathstrut 17q^{6}$$ $$\mathstrut +\mathstrut 33q^{7}$$ $$\mathstrut +\mathstrut q^{8}$$ $$\mathstrut +\mathstrut 17q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$249q$$ $$\mathstrut +\mathstrut q^{2}$$ $$\mathstrut +\mathstrut 9q^{3}$$ $$\mathstrut +\mathstrut 9q^{4}$$ $$\mathstrut +\mathstrut 21q^{5}$$ $$\mathstrut +\mathstrut 17q^{6}$$ $$\mathstrut +\mathstrut 33q^{7}$$ $$\mathstrut +\mathstrut q^{8}$$ $$\mathstrut +\mathstrut 17q^{9}$$ $$\mathstrut +\mathstrut 5q^{10}$$ $$\mathstrut +\mathstrut 28q^{11}$$ $$\mathstrut -\mathstrut 7q^{12}$$ $$\mathstrut +\mathstrut 14q^{13}$$ $$\mathstrut -\mathstrut 15q^{14}$$ $$\mathstrut -\mathstrut 19q^{15}$$ $$\mathstrut +\mathstrut q^{16}$$ $$\mathstrut -\mathstrut 6q^{17}$$ $$\mathstrut -\mathstrut 39q^{18}$$ $$\mathstrut +\mathstrut 20q^{19}$$ $$\mathstrut -\mathstrut 7q^{20}$$ $$\mathstrut -\mathstrut 27q^{21}$$ $$\mathstrut -\mathstrut 4q^{22}$$ $$\mathstrut -\mathstrut 7q^{24}$$ $$\mathstrut -\mathstrut 7q^{25}$$ $$\mathstrut -\mathstrut 10q^{26}$$ $$\mathstrut -\mathstrut 63q^{27}$$ $$\mathstrut -\mathstrut 23q^{28}$$ $$\mathstrut -\mathstrut 66q^{29}$$ $$\mathstrut -\mathstrut 31q^{30}$$ $$\mathstrut -\mathstrut 16q^{31}$$ $$\mathstrut +\mathstrut q^{32}$$ $$\mathstrut -\mathstrut 76q^{33}$$ $$\mathstrut -\mathstrut 62q^{34}$$ $$\mathstrut -\mathstrut 111q^{35}$$ $$\mathstrut -\mathstrut 31q^{36}$$ $$\mathstrut -\mathstrut 122q^{37}$$ $$\mathstrut -\mathstrut 52q^{38}$$ $$\mathstrut -\mathstrut 106q^{39}$$ $$\mathstrut -\mathstrut 11q^{40}$$ $$\mathstrut -\mathstrut 38q^{41}$$ $$\mathstrut -\mathstrut 27q^{42}$$ $$\mathstrut -\mathstrut 20q^{43}$$ $$\mathstrut -\mathstrut 4q^{44}$$ $$\mathstrut -\mathstrut 51q^{45}$$ $$\mathstrut -\mathstrut 24q^{46}$$ $$\mathstrut -\mathstrut 24q^{47}$$ $$\mathstrut -\mathstrut 7q^{48}$$ $$\mathstrut +\mathstrut 25q^{49}$$ $$\mathstrut +\mathstrut 33q^{50}$$ $$\mathstrut -\mathstrut 30q^{51}$$ $$\mathstrut -\mathstrut 34q^{52}$$ $$\mathstrut +\mathstrut 30q^{53}$$ $$\mathstrut -\mathstrut 31q^{54}$$ $$\mathstrut -\mathstrut 32q^{55}$$ $$\mathstrut +\mathstrut 17q^{56}$$ $$\mathstrut +\mathstrut 28q^{57}$$ $$\mathstrut -\mathstrut 26q^{58}$$ $$\mathstrut -\mathstrut 20q^{59}$$ $$\mathstrut +\mathstrut q^{60}$$ $$\mathstrut -\mathstrut 66q^{61}$$ $$\mathstrut +\mathstrut 32q^{62}$$ $$\mathstrut -\mathstrut 55q^{63}$$ $$\mathstrut +\mathstrut 9q^{64}$$ $$\mathstrut -\mathstrut 46q^{65}$$ $$\mathstrut -\mathstrut 36q^{66}$$ $$\mathstrut -\mathstrut 172q^{67}$$ $$\mathstrut -\mathstrut 6q^{68}$$ $$\mathstrut -\mathstrut 56q^{69}$$ $$\mathstrut +\mathstrut q^{70}$$ $$\mathstrut -\mathstrut 72q^{71}$$ $$\mathstrut +\mathstrut 41q^{72}$$ $$\mathstrut -\mathstrut 126q^{73}$$ $$\mathstrut +\mathstrut 30q^{74}$$ $$\mathstrut -\mathstrut 111q^{75}$$ $$\mathstrut -\mathstrut 12q^{76}$$ $$\mathstrut -\mathstrut 12q^{77}$$ $$\mathstrut +\mathstrut 38q^{78}$$ $$\mathstrut -\mathstrut 48q^{79}$$ $$\mathstrut +\mathstrut 21q^{80}$$ $$\mathstrut +\mathstrut 49q^{81}$$ $$\mathstrut +\mathstrut 122q^{82}$$ $$\mathstrut +\mathstrut 36q^{83}$$ $$\mathstrut +\mathstrut 45q^{84}$$ $$\mathstrut +\mathstrut 66q^{85}$$ $$\mathstrut +\mathstrut 52q^{86}$$ $$\mathstrut +\mathstrut 110q^{87}$$ $$\mathstrut +\mathstrut 20q^{88}$$ $$\mathstrut +\mathstrut 26q^{89}$$ $$\mathstrut +\mathstrut 85q^{90}$$ $$\mathstrut +\mathstrut 110q^{91}$$ $$\mathstrut +\mathstrut 24q^{92}$$ $$\mathstrut +\mathstrut 160q^{93}$$ $$\mathstrut +\mathstrut 128q^{94}$$ $$\mathstrut +\mathstrut 80q^{95}$$ $$\mathstrut +\mathstrut q^{96}$$ $$\mathstrut +\mathstrut 74q^{97}$$ $$\mathstrut +\mathstrut q^{98}$$ $$\mathstrut +\mathstrut 188q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$

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
210.2.a $$\chi_{210}(1, \cdot)$$ 210.2.a.a 1 1
210.2.a.b 1
210.2.a.c 1
210.2.a.d 1
210.2.a.e 1
210.2.b $$\chi_{210}(41, \cdot)$$ 210.2.b.a 4 1
210.2.b.b 4
210.2.d $$\chi_{210}(209, \cdot)$$ 210.2.d.a 8 1
210.2.d.b 8
210.2.g $$\chi_{210}(169, \cdot)$$ 210.2.g.a 2 1
210.2.g.b 2
210.2.i $$\chi_{210}(121, \cdot)$$ 210.2.i.a 2 2
210.2.i.b 2
210.2.i.c 2
210.2.i.d 2
210.2.j $$\chi_{210}(113, \cdot)$$ 210.2.j.a 12 2
210.2.j.b 12
210.2.m $$\chi_{210}(13, \cdot)$$ 210.2.m.a 8 2
210.2.m.b 8
210.2.n $$\chi_{210}(79, \cdot)$$ 210.2.n.a 4 2
210.2.n.b 12
210.2.r $$\chi_{210}(101, \cdot)$$ 210.2.r.a 12 2
210.2.r.b 12
210.2.t $$\chi_{210}(59, \cdot)$$ 210.2.t.a 4 2
210.2.t.b 4
210.2.t.c 4
210.2.t.d 4
210.2.t.e 8
210.2.t.f 8
210.2.u $$\chi_{210}(73, \cdot)$$ 210.2.u.a 16 4
210.2.u.b 16
210.2.x $$\chi_{210}(23, \cdot)$$ 210.2.x.a 64 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(210))$$ into lower level spaces

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