# Properties

 Label 3700.1 Level 3700 Weight 1 Dimension 380 Nonzero newspaces 27 Newform subspaces 41 Sturm bound 820800 Trace bound 26

## Defining parameters

 Level: $$N$$ = $$3700 = 2^{2} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$27$$ Newform subspaces: $$41$$ Sturm bound: $$820800$$ Trace bound: $$26$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(3700))$$.

Total New Old
Modular forms 5582 1814 3768
Cusp forms 542 380 162
Eisenstein series 5040 1434 3606

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 370 0 10 0

## Trace form

 $$380 q + 3 q^{2} - q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8} - q^{9} + O(q^{10})$$ $$380 q + 3 q^{2} - q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8} - q^{9} + 2 q^{10} + 6 q^{13} + 11 q^{16} + 8 q^{17} - 7 q^{18} - 2 q^{19} - 8 q^{20} - 8 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{26} + 8 q^{29} + 4 q^{31} - 7 q^{32} - 2 q^{33} - 4 q^{34} + 20 q^{36} - 2 q^{37} + 2 q^{40} - 18 q^{41} + 2 q^{45} - 8 q^{46} - 2 q^{47} - 11 q^{49} + 2 q^{50} + 6 q^{51} + 6 q^{52} - 6 q^{53} + 2 q^{57} + 6 q^{58} + 3 q^{61} - q^{64} - 6 q^{65} + 6 q^{68} - 2 q^{69} + 2 q^{71} + 3 q^{72} + 6 q^{73} + 7 q^{74} - 2 q^{79} + 2 q^{80} - 7 q^{81} + 6 q^{82} + 2 q^{83} + 8 q^{84} - 6 q^{85} - 8 q^{86} + 2 q^{87} + 11 q^{89} + 2 q^{90} + 6 q^{97} + 3 q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(3700))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
3700.1.b $$\chi_{3700}(3551, \cdot)$$ 3700.1.b.a 1 1
3700.1.b.b 1
3700.1.b.c 1
3700.1.b.d 1
3700.1.b.e 2
3700.1.b.f 4
3700.1.c $$\chi_{3700}(1851, \cdot)$$ None 0 1
3700.1.f $$\chi_{3700}(1999, \cdot)$$ None 0 1
3700.1.g $$\chi_{3700}(3699, \cdot)$$ 3700.1.g.a 2 1
3700.1.g.b 2
3700.1.j $$\chi_{3700}(401, \cdot)$$ 3700.1.j.a 2 2
3700.1.j.b 2
3700.1.j.c 2
3700.1.p $$\chi_{3700}(43, \cdot)$$ 3700.1.p.a 2 2
3700.1.q $$\chi_{3700}(593, \cdot)$$ None 0 2
3700.1.r $$\chi_{3700}(1257, \cdot)$$ None 0 2
3700.1.s $$\chi_{3700}(2707, \cdot)$$ 3700.1.s.a 2 2
3700.1.t $$\chi_{3700}(549, \cdot)$$ 3700.1.t.a 2 2
3700.1.t.b 2
3700.1.w $$\chi_{3700}(899, \cdot)$$ None 0 2
3700.1.x $$\chi_{3700}(1099, \cdot)$$ 3700.1.x.a 4 2
3700.1.z $$\chi_{3700}(951, \cdot)$$ 3700.1.z.a 2 2
3700.1.ba $$\chi_{3700}(751, \cdot)$$ 3700.1.ba.a 4 2
3700.1.bf $$\chi_{3700}(739, \cdot)$$ 3700.1.bf.a 4 4
3700.1.bf.b 4
3700.1.bg $$\chi_{3700}(519, \cdot)$$ None 0 4
3700.1.bj $$\chi_{3700}(371, \cdot)$$ None 0 4
3700.1.bk $$\chi_{3700}(591, \cdot)$$ None 0 4
3700.1.bl $$\chi_{3700}(2049, \cdot)$$ None 0 4
3700.1.br $$\chi_{3700}(643, \cdot)$$ 3700.1.br.a 4 4
3700.1.bs $$\chi_{3700}(693, \cdot)$$ None 0 4
3700.1.bt $$\chi_{3700}(1157, \cdot)$$ None 0 4
3700.1.bu $$\chi_{3700}(2043, \cdot)$$ 3700.1.bu.a 4 4
3700.1.bv $$\chi_{3700}(1901, \cdot)$$ None 0 4
3700.1.cb $$\chi_{3700}(599, \cdot)$$ 3700.1.cb.a 12 6
3700.1.cc $$\chi_{3700}(151, \cdot)$$ 3700.1.cc.a 12 6
3700.1.cd $$\chi_{3700}(99, \cdot)$$ None 0 6
3700.1.ce $$\chi_{3700}(451, \cdot)$$ 3700.1.ce.a 6 6
3700.1.cg $$\chi_{3700}(709, \cdot)$$ None 0 8
3700.1.ch $$\chi_{3700}(487, \cdot)$$ 3700.1.ch.a 8 8
3700.1.ci $$\chi_{3700}(73, \cdot)$$ None 0 8
3700.1.cj $$\chi_{3700}(297, \cdot)$$ None 0 8
3700.1.ck $$\chi_{3700}(327, \cdot)$$ 3700.1.ck.a 8 8
3700.1.cq $$\chi_{3700}(561, \cdot)$$ None 0 8
3700.1.ct $$\chi_{3700}(11, \cdot)$$ None 0 8
3700.1.cu $$\chi_{3700}(211, \cdot)$$ 3700.1.cu.a 8 8
3700.1.cu.b 8
3700.1.cw $$\chi_{3700}(359, \cdot)$$ None 0 8
3700.1.cx $$\chi_{3700}(159, \cdot)$$ 3700.1.cx.a 8 8
3700.1.cx.b 8
3700.1.cy $$\chi_{3700}(607, \cdot)$$ 3700.1.cy.a 12 12
3700.1.da $$\chi_{3700}(657, \cdot)$$ None 0 12
3700.1.db $$\chi_{3700}(157, \cdot)$$ None 0 12
3700.1.dd $$\chi_{3700}(143, \cdot)$$ 3700.1.dd.a 12 12
3700.1.df $$\chi_{3700}(449, \cdot)$$ None 0 12
3700.1.di $$\chi_{3700}(301, \cdot)$$ None 0 12
3700.1.dm $$\chi_{3700}(341, \cdot)$$ None 0 16
3700.1.dn $$\chi_{3700}(23, \cdot)$$ 3700.1.dn.a 16 16
3700.1.do $$\chi_{3700}(137, \cdot)$$ None 0 16
3700.1.dp $$\chi_{3700}(233, \cdot)$$ None 0 16
3700.1.dq $$\chi_{3700}(103, \cdot)$$ 3700.1.dq.a 16 16
3700.1.dw $$\chi_{3700}(29, \cdot)$$ None 0 16
3700.1.dx $$\chi_{3700}(71, \cdot)$$ 3700.1.dx.a 24 24
3700.1.dx.b 24
3700.1.dy $$\chi_{3700}(139, \cdot)$$ 3700.1.dy.a 24 24
3700.1.dy.b 24
3700.1.dz $$\chi_{3700}(391, \cdot)$$ None 0 24
3700.1.ea $$\chi_{3700}(219, \cdot)$$ None 0 24
3700.1.ef $$\chi_{3700}(61, \cdot)$$ None 0 48
3700.1.ei $$\chi_{3700}(69, \cdot)$$ None 0 48
3700.1.ek $$\chi_{3700}(87, \cdot)$$ 3700.1.ek.a 48 48
3700.1.em $$\chi_{3700}(33, \cdot)$$ None 0 48
3700.1.en $$\chi_{3700}(77, \cdot)$$ None 0 48
3700.1.ep $$\chi_{3700}(183, \cdot)$$ 3700.1.ep.a 48 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(3700))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(3700)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(37))$$$$^{\oplus 9}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(74))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(148))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(185))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(370))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(740))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(925))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1850))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(3700))$$$$^{\oplus 1}$$