# Properties

 Label 3600.1 Level 3600 Weight 1 Dimension 82 Nonzero newspaces 14 Newforms 23 Sturm bound 691200 Trace bound 49

## Defining parameters

 Level: $$N$$ = $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$14$$ Newforms: $$23$$ Sturm bound: $$691200$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(3600))$$.

Total New Old
Modular forms 7192 912 6280
Cusp forms 920 82 838
Eisenstein series 6272 830 5442

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 74 0 8 0

## Trace form

 $$82q$$ $$\mathstrut -\mathstrut q^{5}$$ $$\mathstrut -\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$82q$$ $$\mathstrut -\mathstrut q^{5}$$ $$\mathstrut -\mathstrut 2q^{9}$$ $$\mathstrut -\mathstrut 2q^{13}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut -\mathstrut 4q^{19}$$ $$\mathstrut +\mathstrut 8q^{21}$$ $$\mathstrut +\mathstrut q^{25}$$ $$\mathstrut +\mathstrut 2q^{29}$$ $$\mathstrut -\mathstrut 8q^{31}$$ $$\mathstrut +\mathstrut 4q^{34}$$ $$\mathstrut +\mathstrut 7q^{37}$$ $$\mathstrut +\mathstrut 10q^{41}$$ $$\mathstrut +\mathstrut 12q^{46}$$ $$\mathstrut +\mathstrut 4q^{49}$$ $$\mathstrut +\mathstrut 5q^{53}$$ $$\mathstrut +\mathstrut 8q^{61}$$ $$\mathstrut +\mathstrut 5q^{65}$$ $$\mathstrut +\mathstrut 6q^{69}$$ $$\mathstrut +\mathstrut 14q^{73}$$ $$\mathstrut +\mathstrut 4q^{76}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut +\mathstrut 3q^{85}$$ $$\mathstrut +\mathstrut 3q^{89}$$ $$\mathstrut +\mathstrut 20q^{91}$$ $$\mathstrut -\mathstrut 4q^{94}$$ $$\mathstrut +\mathstrut 14q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(3600))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
3600.1.c $$\chi_{3600}(449, \cdot)$$ 3600.1.c.a 4 1
3600.1.e $$\chi_{3600}(3151, \cdot)$$ 3600.1.e.a 1 1
3600.1.e.b 1
3600.1.e.c 2
3600.1.e.d 2
3600.1.g $$\chi_{3600}(1351, \cdot)$$ None 0 1
3600.1.i $$\chi_{3600}(2249, \cdot)$$ None 0 1
3600.1.j $$\chi_{3600}(1999, \cdot)$$ 3600.1.j.a 2 1
3600.1.j.b 4
3600.1.l $$\chi_{3600}(1601, \cdot)$$ 3600.1.l.a 2 1
3600.1.l.b 2
3600.1.n $$\chi_{3600}(3401, \cdot)$$ None 0 1
3600.1.p $$\chi_{3600}(199, \cdot)$$ None 0 1
3600.1.r $$\chi_{3600}(1099, \cdot)$$ None 0 2
3600.1.s $$\chi_{3600}(701, \cdot)$$ None 0 2
3600.1.v $$\chi_{3600}(1943, \cdot)$$ None 0 2
3600.1.y $$\chi_{3600}(793, \cdot)$$ None 0 2
3600.1.ba $$\chi_{3600}(107, \cdot)$$ None 0 2
3600.1.bb $$\chi_{3600}(757, \cdot)$$ 3600.1.bb.a 2 2
3600.1.bb.b 2
3600.1.be $$\chi_{3600}(1907, \cdot)$$ None 0 2
3600.1.bf $$\chi_{3600}(2557, \cdot)$$ 3600.1.bf.a 2 2
3600.1.bf.b 2
3600.1.bh $$\chi_{3600}(2593, \cdot)$$ 3600.1.bh.a 2 2
3600.1.bh.b 4
3600.1.bk $$\chi_{3600}(143, \cdot)$$ 3600.1.bk.a 4 2
3600.1.bn $$\chi_{3600}(1349, \cdot)$$ None 0 2
3600.1.bo $$\chi_{3600}(451, \cdot)$$ 3600.1.bo.a 4 2
3600.1.bq $$\chi_{3600}(1399, \cdot)$$ None 0 2
3600.1.br $$\chi_{3600}(1001, \cdot)$$ None 0 2
3600.1.bt $$\chi_{3600}(401, \cdot)$$ None 0 2
3600.1.bv $$\chi_{3600}(799, \cdot)$$ None 0 2
3600.1.by $$\chi_{3600}(1049, \cdot)$$ None 0 2
3600.1.ca $$\chi_{3600}(151, \cdot)$$ None 0 2
3600.1.cc $$\chi_{3600}(751, \cdot)$$ 3600.1.cc.a 2 2
3600.1.cc.b 2
3600.1.ce $$\chi_{3600}(1649, \cdot)$$ None 0 2
3600.1.cf $$\chi_{3600}(89, \cdot)$$ None 0 4
3600.1.ch $$\chi_{3600}(631, \cdot)$$ None 0 4
3600.1.cj $$\chi_{3600}(271, \cdot)$$ 3600.1.cj.a 8 4
3600.1.cl $$\chi_{3600}(1169, \cdot)$$ None 0 4
3600.1.cn $$\chi_{3600}(919, \cdot)$$ None 0 4
3600.1.cp $$\chi_{3600}(521, \cdot)$$ None 0 4
3600.1.cr $$\chi_{3600}(161, \cdot)$$ None 0 4
3600.1.ct $$\chi_{3600}(559, \cdot)$$ 3600.1.ct.a 4 4
3600.1.cw $$\chi_{3600}(1051, \cdot)$$ None 0 4
3600.1.cx $$\chi_{3600}(149, \cdot)$$ None 0 4
3600.1.cz $$\chi_{3600}(193, \cdot)$$ None 0 4
3600.1.da $$\chi_{3600}(1343, \cdot)$$ 3600.1.da.a 8 4
3600.1.dd $$\chi_{3600}(157, \cdot)$$ None 0 4
3600.1.de $$\chi_{3600}(443, \cdot)$$ None 0 4
3600.1.dh $$\chi_{3600}(493, \cdot)$$ None 0 4
3600.1.di $$\chi_{3600}(1307, \cdot)$$ None 0 4
3600.1.dl $$\chi_{3600}(407, \cdot)$$ None 0 4
3600.1.dm $$\chi_{3600}(457, \cdot)$$ None 0 4
3600.1.do $$\chi_{3600}(101, \cdot)$$ None 0 4
3600.1.dp $$\chi_{3600}(499, \cdot)$$ None 0 4
3600.1.dv $$\chi_{3600}(341, \cdot)$$ None 0 8
3600.1.dw $$\chi_{3600}(19, \cdot)$$ None 0 8
3600.1.dx $$\chi_{3600}(287, \cdot)$$ 3600.1.dx.a 16 8
3600.1.ea $$\chi_{3600}(433, \cdot)$$ None 0 8
3600.1.ec $$\chi_{3600}(397, \cdot)$$ None 0 8
3600.1.ed $$\chi_{3600}(467, \cdot)$$ None 0 8
3600.1.eg $$\chi_{3600}(37, \cdot)$$ None 0 8
3600.1.eh $$\chi_{3600}(323, \cdot)$$ None 0 8
3600.1.ej $$\chi_{3600}(73, \cdot)$$ None 0 8
3600.1.em $$\chi_{3600}(503, \cdot)$$ None 0 8
3600.1.en $$\chi_{3600}(91, \cdot)$$ None 0 8
3600.1.eo $$\chi_{3600}(269, \cdot)$$ None 0 8
3600.1.es $$\chi_{3600}(79, \cdot)$$ None 0 8
3600.1.eu $$\chi_{3600}(641, \cdot)$$ None 0 8
3600.1.ew $$\chi_{3600}(41, \cdot)$$ None 0 8
3600.1.ex $$\chi_{3600}(439, \cdot)$$ None 0 8
3600.1.ey $$\chi_{3600}(209, \cdot)$$ None 0 8
3600.1.fa $$\chi_{3600}(31, \cdot)$$ None 0 8
3600.1.fc $$\chi_{3600}(391, \cdot)$$ None 0 8
3600.1.fe $$\chi_{3600}(329, \cdot)$$ None 0 8
3600.1.fg $$\chi_{3600}(29, \cdot)$$ None 0 16
3600.1.fh $$\chi_{3600}(211, \cdot)$$ None 0 16
3600.1.fl $$\chi_{3600}(313, \cdot)$$ None 0 16
3600.1.fm $$\chi_{3600}(23, \cdot)$$ None 0 16
3600.1.fp $$\chi_{3600}(83, \cdot)$$ None 0 16
3600.1.fq $$\chi_{3600}(13, \cdot)$$ None 0 16
3600.1.ft $$\chi_{3600}(203, \cdot)$$ None 0 16
3600.1.fu $$\chi_{3600}(133, \cdot)$$ None 0 16
3600.1.fx $$\chi_{3600}(47, \cdot)$$ None 0 16
3600.1.fy $$\chi_{3600}(97, \cdot)$$ None 0 16
3600.1.gc $$\chi_{3600}(139, \cdot)$$ None 0 16
3600.1.gd $$\chi_{3600}(221, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(3600))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(3600)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 9}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(720))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(900))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1800))$$$$^{\oplus 2}$$