# Properties

 Label 350.2 Level 350 Weight 2 Dimension 1058 Nonzero newspaces 12 Newforms 48 Sturm bound 14400 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newforms: $$48$$ Sturm bound: $$14400$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(350))$$.

Total New Old
Modular forms 3936 1058 2878
Cusp forms 3265 1058 2207
Eisenstein series 671 0 671

## Trace form

 $$1058q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut +\mathstrut 10q^{3}$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 10q^{5}$$ $$\mathstrut +\mathstrut 14q^{6}$$ $$\mathstrut +\mathstrut 16q^{7}$$ $$\mathstrut +\mathstrut 2q^{8}$$ $$\mathstrut +\mathstrut 40q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$1058q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut +\mathstrut 10q^{3}$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 10q^{5}$$ $$\mathstrut +\mathstrut 14q^{6}$$ $$\mathstrut +\mathstrut 16q^{7}$$ $$\mathstrut +\mathstrut 2q^{8}$$ $$\mathstrut +\mathstrut 40q^{9}$$ $$\mathstrut +\mathstrut 10q^{10}$$ $$\mathstrut +\mathstrut 36q^{11}$$ $$\mathstrut +\mathstrut 10q^{12}$$ $$\mathstrut +\mathstrut 38q^{13}$$ $$\mathstrut +\mathstrut 14q^{14}$$ $$\mathstrut +\mathstrut 40q^{15}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut +\mathstrut 20q^{17}$$ $$\mathstrut -\mathstrut 12q^{18}$$ $$\mathstrut -\mathstrut 18q^{19}$$ $$\mathstrut +\mathstrut 18q^{21}$$ $$\mathstrut -\mathstrut 44q^{22}$$ $$\mathstrut -\mathstrut 8q^{23}$$ $$\mathstrut -\mathstrut 26q^{24}$$ $$\mathstrut -\mathstrut 94q^{25}$$ $$\mathstrut -\mathstrut 42q^{26}$$ $$\mathstrut -\mathstrut 140q^{27}$$ $$\mathstrut -\mathstrut 52q^{28}$$ $$\mathstrut -\mathstrut 92q^{29}$$ $$\mathstrut -\mathstrut 112q^{30}$$ $$\mathstrut -\mathstrut 84q^{31}$$ $$\mathstrut -\mathstrut 8q^{32}$$ $$\mathstrut -\mathstrut 176q^{33}$$ $$\mathstrut -\mathstrut 98q^{34}$$ $$\mathstrut -\mathstrut 52q^{35}$$ $$\mathstrut -\mathstrut 56q^{36}$$ $$\mathstrut -\mathstrut 38q^{37}$$ $$\mathstrut -\mathstrut 38q^{38}$$ $$\mathstrut -\mathstrut 144q^{39}$$ $$\mathstrut +\mathstrut 10q^{40}$$ $$\mathstrut -\mathstrut 4q^{41}$$ $$\mathstrut -\mathstrut 82q^{42}$$ $$\mathstrut -\mathstrut 36q^{43}$$ $$\mathstrut -\mathstrut 12q^{44}$$ $$\mathstrut -\mathstrut 102q^{45}$$ $$\mathstrut +\mathstrut 24q^{46}$$ $$\mathstrut +\mathstrut 4q^{47}$$ $$\mathstrut +\mathstrut 10q^{48}$$ $$\mathstrut +\mathstrut 90q^{49}$$ $$\mathstrut +\mathstrut 50q^{50}$$ $$\mathstrut +\mathstrut 84q^{51}$$ $$\mathstrut +\mathstrut 38q^{52}$$ $$\mathstrut +\mathstrut 78q^{53}$$ $$\mathstrut +\mathstrut 116q^{54}$$ $$\mathstrut +\mathstrut 16q^{55}$$ $$\mathstrut +\mathstrut 14q^{56}$$ $$\mathstrut -\mathstrut 20q^{57}$$ $$\mathstrut +\mathstrut 96q^{58}$$ $$\mathstrut -\mathstrut 42q^{59}$$ $$\mathstrut -\mathstrut 70q^{61}$$ $$\mathstrut -\mathstrut 20q^{62}$$ $$\mathstrut -\mathstrut 136q^{63}$$ $$\mathstrut +\mathstrut 4q^{64}$$ $$\mathstrut -\mathstrut 142q^{65}$$ $$\mathstrut +\mathstrut 24q^{66}$$ $$\mathstrut -\mathstrut 152q^{67}$$ $$\mathstrut -\mathstrut 60q^{68}$$ $$\mathstrut -\mathstrut 184q^{69}$$ $$\mathstrut -\mathstrut 40q^{70}$$ $$\mathstrut -\mathstrut 40q^{71}$$ $$\mathstrut +\mathstrut 38q^{72}$$ $$\mathstrut -\mathstrut 128q^{73}$$ $$\mathstrut -\mathstrut 88q^{74}$$ $$\mathstrut -\mathstrut 216q^{75}$$ $$\mathstrut -\mathstrut 18q^{76}$$ $$\mathstrut -\mathstrut 124q^{77}$$ $$\mathstrut -\mathstrut 40q^{78}$$ $$\mathstrut -\mathstrut 104q^{79}$$ $$\mathstrut +\mathstrut 10q^{80}$$ $$\mathstrut +\mathstrut 48q^{81}$$ $$\mathstrut -\mathstrut 40q^{82}$$ $$\mathstrut -\mathstrut 170q^{83}$$ $$\mathstrut -\mathstrut 2q^{84}$$ $$\mathstrut -\mathstrut 78q^{85}$$ $$\mathstrut -\mathstrut 44q^{86}$$ $$\mathstrut -\mathstrut 100q^{87}$$ $$\mathstrut -\mathstrut 12q^{88}$$ $$\mathstrut -\mathstrut 186q^{89}$$ $$\mathstrut -\mathstrut 86q^{90}$$ $$\mathstrut -\mathstrut 66q^{91}$$ $$\mathstrut -\mathstrut 64q^{92}$$ $$\mathstrut -\mathstrut 64q^{93}$$ $$\mathstrut -\mathstrut 36q^{94}$$ $$\mathstrut -\mathstrut 72q^{95}$$ $$\mathstrut -\mathstrut 34q^{96}$$ $$\mathstrut -\mathstrut 44q^{97}$$ $$\mathstrut -\mathstrut 190q^{98}$$ $$\mathstrut -\mathstrut 36q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(350))$$

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
350.2.a $$\chi_{350}(1, \cdot)$$ 350.2.a.a 1 1
350.2.a.b 1
350.2.a.c 1
350.2.a.d 1
350.2.a.e 1
350.2.a.f 1
350.2.a.g 2
350.2.a.h 2
350.2.c $$\chi_{350}(99, \cdot)$$ 350.2.c.a 2 1
350.2.c.b 2
350.2.c.c 2
350.2.c.d 2
350.2.e $$\chi_{350}(51, \cdot)$$ 350.2.e.a 2 2
350.2.e.b 2
350.2.e.c 2
350.2.e.d 2
350.2.e.e 2
350.2.e.f 2
350.2.e.g 2
350.2.e.h 2
350.2.e.i 2
350.2.e.j 2
350.2.e.k 2
350.2.e.l 2
350.2.g $$\chi_{350}(293, \cdot)$$ 350.2.g.a 8 2
350.2.g.b 16
350.2.h $$\chi_{350}(71, \cdot)$$ 350.2.h.a 8 4
350.2.h.b 12
350.2.h.c 16
350.2.h.d 20
350.2.j $$\chi_{350}(149, \cdot)$$ 350.2.j.a 4 2
350.2.j.b 4
350.2.j.c 4
350.2.j.d 4
350.2.j.e 4
350.2.j.f 4
350.2.m $$\chi_{350}(29, \cdot)$$ 350.2.m.a 24 4
350.2.m.b 40
350.2.o $$\chi_{350}(143, \cdot)$$ 350.2.o.a 8 4
350.2.o.b 8
350.2.o.c 16
350.2.o.d 16
350.2.q $$\chi_{350}(11, \cdot)$$ 350.2.q.a 8 8
350.2.q.b 72
350.2.q.c 80
350.2.r $$\chi_{350}(13, \cdot)$$ 350.2.r.a 160 8
350.2.u $$\chi_{350}(9, \cdot)$$ 350.2.u.a 160 8
350.2.x $$\chi_{350}(3, \cdot)$$ 350.2.x.a 320 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(350))$$ into lower level spaces

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