# Properties

 Label 348.2 Level 348 Weight 2 Dimension 1414 Nonzero newspaces 12 Newform subspaces 22 Sturm bound 13440 Trace bound 6

## Defining parameters

 Level: $$N$$ = $$348 = 2^{2} \cdot 3 \cdot 29$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$22$$ Sturm bound: $$13440$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(348))$$.

Total New Old
Modular forms 3640 1518 2122
Cusp forms 3081 1414 1667
Eisenstein series 559 104 455

## Trace form

 $$1414q - 28q^{4} - 14q^{6} - 28q^{9} + O(q^{10})$$ $$1414q - 28q^{4} - 14q^{6} - 28q^{9} - 28q^{10} - 14q^{12} - 56q^{13} - 28q^{16} - 14q^{18} - 28q^{22} + 28q^{23} - 14q^{24} + 42q^{27} - 56q^{28} + 56q^{29} - 28q^{30} + 56q^{31} + 14q^{33} - 28q^{34} + 56q^{35} - 14q^{36} - 28q^{37} + 28q^{39} - 28q^{40} - 14q^{42} - 28q^{44} - 42q^{45} - 168q^{46} - 56q^{47} - 126q^{48} - 168q^{49} - 196q^{50} - 56q^{51} - 252q^{52} - 126q^{53} - 14q^{54} - 224q^{55} - 196q^{56} - 112q^{57} - 364q^{58} - 56q^{59} - 182q^{60} - 168q^{61} - 196q^{62} - 28q^{63} - 280q^{64} - 126q^{65} - 126q^{66} - 112q^{67} - 196q^{68} - 84q^{69} - 252q^{70} - 56q^{71} - 42q^{72} - 126q^{73} - 28q^{74} - 70q^{75} - 28q^{76} - 14q^{78} - 140q^{81} - 28q^{82} + 28q^{84} - 56q^{85} - 70q^{87} - 56q^{88} - 14q^{90} - 84q^{93} - 28q^{94} + 28q^{96} - 210q^{97} + 140q^{98} - 126q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(348))$$

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
348.2.a $$\chi_{348}(1, \cdot)$$ 348.2.a.a 1 1
348.2.a.b 1
348.2.a.c 1
348.2.a.d 1
348.2.b $$\chi_{348}(347, \cdot)$$ 348.2.b.a 8 1
348.2.b.b 12
348.2.b.c 12
348.2.b.d 24
348.2.c $$\chi_{348}(59, \cdot)$$ 348.2.c.a 16 1
348.2.c.b 40
348.2.h $$\chi_{348}(289, \cdot)$$ 348.2.h.a 2 1
348.2.h.b 2
348.2.h.c 2
348.2.i $$\chi_{348}(307, \cdot)$$ 348.2.i.a 60 2
348.2.l $$\chi_{348}(17, \cdot)$$ 348.2.l.a 20 2
348.2.m $$\chi_{348}(25, \cdot)$$ 348.2.m.a 12 6
348.2.m.b 12
348.2.n $$\chi_{348}(13, \cdot)$$ 348.2.n.a 36 6
348.2.s $$\chi_{348}(23, \cdot)$$ 348.2.s.a 336 6
348.2.t $$\chi_{348}(35, \cdot)$$ 348.2.t.a 336 6
348.2.u $$\chi_{348}(77, \cdot)$$ 348.2.u.a 120 12
348.2.x $$\chi_{348}(19, \cdot)$$ 348.2.x.a 360 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(348))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(348)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(29))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(58))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(87))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(116))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(174))$$$$^{\oplus 2}$$