# Properties

 Label 3800.1 Level 3800 Weight 1 Dimension 306 Nonzero newspaces 10 Newform subspaces 47 Sturm bound 864000 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$10$$ Newform subspaces: $$47$$ Sturm bound: $$864000$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(3800))$$.

Total New Old
Modular forms 6656 1700 4956
Cusp forms 608 306 302
Eisenstein series 6048 1394 4654

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 294 0 12 0

## Trace form

 $$306 q + q^{2} + q^{4} + 4 q^{6} + 2 q^{7} + q^{8} + 13 q^{9} + O(q^{10})$$ $$306 q + q^{2} + q^{4} + 4 q^{6} + 2 q^{7} + q^{8} + 13 q^{9} - 6 q^{11} + 2 q^{12} - 14 q^{14} + 25 q^{16} + 4 q^{17} - 6 q^{18} + 3 q^{19} - 7 q^{22} + 4 q^{23} - 4 q^{24} - 28 q^{26} - 5 q^{27} + 2 q^{28} - 12 q^{31} + q^{32} + 2 q^{33} + 2 q^{34} + 5 q^{36} + 4 q^{37} - q^{38} - 2 q^{39} - 2 q^{41} - 2 q^{42} + 4 q^{43} + 5 q^{44} - 4 q^{46} - 4 q^{47} - 7 q^{48} - q^{49} - 29 q^{51} - 2 q^{53} + 10 q^{54} - 4 q^{56} + 8 q^{57} + 2 q^{58} + 6 q^{59} + 6 q^{61} + q^{64} - 12 q^{66} + 2 q^{67} - 5 q^{68} + 2 q^{71} - 6 q^{72} - 7 q^{73} - q^{76} + 18 q^{81} + 2 q^{82} + 6 q^{83} - 22 q^{86} - 6 q^{87} + 2 q^{88} - 12 q^{89} - 8 q^{91} + 2 q^{92} + 2 q^{93} + 4 q^{94} + 12 q^{96} + q^{98} + q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(3800))$$

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
3800.1.b $$\chi_{3800}(949, \cdot)$$ 3800.1.b.a 2 1
3800.1.b.b 2
3800.1.b.c 6
3800.1.b.d 6
3800.1.c $$\chi_{3800}(951, \cdot)$$ None 0 1
3800.1.h $$\chi_{3800}(3001, \cdot)$$ None 0 1
3800.1.i $$\chi_{3800}(2699, \cdot)$$ None 0 1
3800.1.l $$\chi_{3800}(2851, \cdot)$$ None 0 1
3800.1.m $$\chi_{3800}(2849, \cdot)$$ None 0 1
3800.1.n $$\chi_{3800}(799, \cdot)$$ None 0 1
3800.1.o $$\chi_{3800}(1101, \cdot)$$ 3800.1.o.a 1 1
3800.1.o.b 1
3800.1.o.c 3
3800.1.o.d 3
3800.1.o.e 3
3800.1.o.f 3
3800.1.o.g 8
3800.1.r $$\chi_{3800}(1293, \cdot)$$ None 0 2
3800.1.s $$\chi_{3800}(607, \cdot)$$ None 0 2
3800.1.x $$\chi_{3800}(457, \cdot)$$ None 0 2
3800.1.y $$\chi_{3800}(1443, \cdot)$$ 3800.1.y.a 2 2
3800.1.y.b 2
3800.1.y.c 2
3800.1.y.d 2
3800.1.y.e 4
3800.1.y.f 4
3800.1.y.g 8
3800.1.bc $$\chi_{3800}(449, \cdot)$$ None 0 2
3800.1.bd $$\chi_{3800}(1451, \cdot)$$ 3800.1.bd.a 2 2
3800.1.bd.b 2
3800.1.bd.c 2
3800.1.bd.d 2
3800.1.bd.e 2
3800.1.bd.f 4
3800.1.be $$\chi_{3800}(901, \cdot)$$ None 0 2
3800.1.bf $$\chi_{3800}(999, \cdot)$$ None 0 2
3800.1.bh $$\chi_{3800}(1151, \cdot)$$ None 0 2
3800.1.bi $$\chi_{3800}(749, \cdot)$$ None 0 2
3800.1.bn $$\chi_{3800}(1299, \cdot)$$ 3800.1.bn.a 4 2
3800.1.bn.b 4
3800.1.bn.c 4
3800.1.bo $$\chi_{3800}(601, \cdot)$$ None 0 2
3800.1.bq $$\chi_{3800}(569, \cdot)$$ None 0 4
3800.1.br $$\chi_{3800}(571, \cdot)$$ None 0 4
3800.1.bv $$\chi_{3800}(341, \cdot)$$ None 0 4
3800.1.bw $$\chi_{3800}(39, \cdot)$$ None 0 4
3800.1.bz $$\chi_{3800}(191, \cdot)$$ None 0 4
3800.1.ca $$\chi_{3800}(189, \cdot)$$ None 0 4
3800.1.cb $$\chi_{3800}(419, \cdot)$$ None 0 4
3800.1.cc $$\chi_{3800}(721, \cdot)$$ None 0 4
3800.1.ch $$\chi_{3800}(957, \cdot)$$ None 0 4
3800.1.ci $$\chi_{3800}(407, \cdot)$$ None 0 4
3800.1.cj $$\chi_{3800}(657, \cdot)$$ 3800.1.cj.a 4 4
3800.1.cj.b 4
3800.1.cj.c 4
3800.1.ck $$\chi_{3800}(107, \cdot)$$ 3800.1.ck.a 8 4
3800.1.ck.b 8
3800.1.ck.c 8
3800.1.ck.d 8
3800.1.cp $$\chi_{3800}(401, \cdot)$$ None 0 6
3800.1.cq $$\chi_{3800}(99, \cdot)$$ 3800.1.cq.a 12 6
3800.1.cq.b 12
3800.1.cq.c 12
3800.1.cr $$\chi_{3800}(199, \cdot)$$ None 0 6
3800.1.cs $$\chi_{3800}(1701, \cdot)$$ None 0 6
3800.1.cv $$\chi_{3800}(251, \cdot)$$ 3800.1.cv.a 6 6
3800.1.cv.b 6
3800.1.cv.c 6
3800.1.cv.d 6
3800.1.cv.e 6
3800.1.cv.f 12
3800.1.cw $$\chi_{3800}(249, \cdot)$$ None 0 6
3800.1.db $$\chi_{3800}(1549, \cdot)$$ None 0 6
3800.1.dc $$\chi_{3800}(351, \cdot)$$ None 0 6
3800.1.dd $$\chi_{3800}(227, \cdot)$$ None 0 8
3800.1.de $$\chi_{3800}(153, \cdot)$$ None 0 8
3800.1.dj $$\chi_{3800}(303, \cdot)$$ None 0 8
3800.1.dk $$\chi_{3800}(77, \cdot)$$ None 0 8
3800.1.dn $$\chi_{3800}(69, \cdot)$$ None 0 8
3800.1.do $$\chi_{3800}(311, \cdot)$$ None 0 8
3800.1.dp $$\chi_{3800}(521, \cdot)$$ None 0 8
3800.1.dq $$\chi_{3800}(539, \cdot)$$ None 0 8
3800.1.dt $$\chi_{3800}(11, \cdot)$$ None 0 8
3800.1.du $$\chi_{3800}(369, \cdot)$$ None 0 8
3800.1.dy $$\chi_{3800}(159, \cdot)$$ None 0 8
3800.1.dz $$\chi_{3800}(141, \cdot)$$ None 0 8
3800.1.ea $$\chi_{3800}(593, \cdot)$$ None 0 12
3800.1.eb $$\chi_{3800}(243, \cdot)$$ 3800.1.eb.a 24 12
3800.1.eb.b 24
3800.1.eb.c 24
3800.1.eb.d 24
3800.1.ee $$\chi_{3800}(93, \cdot)$$ None 0 12
3800.1.ef $$\chi_{3800}(143, \cdot)$$ None 0 12
3800.1.el $$\chi_{3800}(27, \cdot)$$ None 0 16
3800.1.em $$\chi_{3800}(273, \cdot)$$ None 0 16
3800.1.en $$\chi_{3800}(103, \cdot)$$ None 0 16
3800.1.eo $$\chi_{3800}(197, \cdot)$$ None 0 16
3800.1.et $$\chi_{3800}(21, \cdot)$$ None 0 24
3800.1.eu $$\chi_{3800}(119, \cdot)$$ None 0 24
3800.1.ev $$\chi_{3800}(139, \cdot)$$ None 0 24
3800.1.ew $$\chi_{3800}(41, \cdot)$$ None 0 24
3800.1.ey $$\chi_{3800}(111, \cdot)$$ None 0 24
3800.1.ez $$\chi_{3800}(29, \cdot)$$ None 0 24
3800.1.fe $$\chi_{3800}(89, \cdot)$$ None 0 24
3800.1.ff $$\chi_{3800}(131, \cdot)$$ None 0 24
3800.1.fi $$\chi_{3800}(127, \cdot)$$ None 0 48
3800.1.fj $$\chi_{3800}(213, \cdot)$$ None 0 48
3800.1.fm $$\chi_{3800}(3, \cdot)$$ None 0 48
3800.1.fn $$\chi_{3800}(17, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(3800))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(3800)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 24}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 18}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 16}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 9}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(190))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(380))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(475))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(760))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(950))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1900))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(3800))$$$$^{\oplus 1}$$