# Properties

 Label 1200.1 Level 1200 Weight 1 Dimension 28 Nonzero newspaces 6 Newform subspaces 10 Sturm bound 76800 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$1200\( 1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$76800$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1200))$$.

Total New Old
Modular forms 1750 213 1537
Cusp forms 182 28 154
Eisenstein series 1568 185 1383

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 28 0 0 0

## Trace form

 $$28q + O(q^{10})$$ $$28q + 4q^{16} + 4q^{19} - 4q^{21} - 4q^{24} + 4q^{31} + 4q^{34} - 12q^{36} - 12q^{46} + 4q^{49} - 4q^{51} - 4q^{54} - 8q^{61} - 4q^{69} - 4q^{76} - 8q^{79} - 4q^{81} - 4q^{91} - 4q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1200))$$

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
1200.1.c $$\chi_{1200}(449, \cdot)$$ 1200.1.c.a 2 1
1200.1.e $$\chi_{1200}(751, \cdot)$$ None 0 1
1200.1.g $$\chi_{1200}(151, \cdot)$$ None 0 1
1200.1.i $$\chi_{1200}(1049, \cdot)$$ None 0 1
1200.1.j $$\chi_{1200}(799, \cdot)$$ None 0 1
1200.1.l $$\chi_{1200}(401, \cdot)$$ 1200.1.l.a 1 1
1200.1.l.b 1
1200.1.n $$\chi_{1200}(1001, \cdot)$$ None 0 1
1200.1.p $$\chi_{1200}(199, \cdot)$$ None 0 1
1200.1.q $$\chi_{1200}(499, \cdot)$$ None 0 2
1200.1.r $$\chi_{1200}(101, \cdot)$$ 1200.1.r.a 4 2
1200.1.u $$\chi_{1200}(407, \cdot)$$ None 0 2
1200.1.x $$\chi_{1200}(457, \cdot)$$ None 0 2
1200.1.z $$\chi_{1200}(107, \cdot)$$ 1200.1.z.a 2 2
1200.1.z.b 2
1200.1.ba $$\chi_{1200}(493, \cdot)$$ None 0 2
1200.1.bd $$\chi_{1200}(443, \cdot)$$ 1200.1.bd.a 2 2
1200.1.bd.b 2
1200.1.be $$\chi_{1200}(157, \cdot)$$ None 0 2
1200.1.bg $$\chi_{1200}(193, \cdot)$$ None 0 2
1200.1.bj $$\chi_{1200}(143, \cdot)$$ 1200.1.bj.a 4 2
1200.1.bj.b 8
1200.1.bm $$\chi_{1200}(149, \cdot)$$ None 0 2
1200.1.bn $$\chi_{1200}(451, \cdot)$$ None 0 2
1200.1.bp $$\chi_{1200}(89, \cdot)$$ None 0 4
1200.1.br $$\chi_{1200}(391, \cdot)$$ None 0 4
1200.1.bt $$\chi_{1200}(31, \cdot)$$ None 0 4
1200.1.bv $$\chi_{1200}(209, \cdot)$$ None 0 4
1200.1.bx $$\chi_{1200}(439, \cdot)$$ None 0 4
1200.1.bz $$\chi_{1200}(41, \cdot)$$ None 0 4
1200.1.cb $$\chi_{1200}(161, \cdot)$$ None 0 4
1200.1.cd $$\chi_{1200}(79, \cdot)$$ None 0 4
1200.1.cg $$\chi_{1200}(221, \cdot)$$ None 0 8
1200.1.ch $$\chi_{1200}(19, \cdot)$$ None 0 8
1200.1.ci $$\chi_{1200}(47, \cdot)$$ None 0 8
1200.1.cl $$\chi_{1200}(97, \cdot)$$ None 0 8
1200.1.cn $$\chi_{1200}(133, \cdot)$$ None 0 8
1200.1.co $$\chi_{1200}(203, \cdot)$$ None 0 8
1200.1.cr $$\chi_{1200}(13, \cdot)$$ None 0 8
1200.1.cs $$\chi_{1200}(83, \cdot)$$ None 0 8
1200.1.cu $$\chi_{1200}(73, \cdot)$$ None 0 8
1200.1.cx $$\chi_{1200}(23, \cdot)$$ None 0 8
1200.1.cy $$\chi_{1200}(91, \cdot)$$ None 0 8
1200.1.cz $$\chi_{1200}(29, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1200))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1200)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)
$3$ ($$1 + T^{2}$$)($$1 + T$$)($$1 - T$$)($$1 + T^{4}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)
$5$ 1
$7$ ($$1 - T^{2} + T^{4}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$11$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{8}$$)
$13$ ($$1 - T^{2} + T^{4}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$17$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$19$ ($$( 1 + T + T^{2} )^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)
$23$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$29$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{8}$$)
$31$ ($$( 1 - T + T^{2} )^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)
$37$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$41$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)
$43$ ($$1 - T^{2} + T^{4}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$47$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$53$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$59$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)
$61$ ($$( 1 + T + T^{2} )^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{8}$$)($$( 1 + T + T^{2} )^{8}$$)
$67$ ($$1 - T^{2} + T^{4}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$71$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{8}$$)
$73$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$79$ ($$( 1 - T )^{4}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T )^{8}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{8}$$)
$83$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$89$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{8}$$)
$97$ ($$1 - T^{2} + T^{4}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T^{2} )^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)