# Properties

 Label 408.1 Level 408 Weight 1 Dimension 20 Nonzero newspaces 2 Newform subspaces 3 Sturm bound 9216 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$408 = 2^{3} \cdot 3 \cdot 17$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ Sturm bound: $$9216$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(408))$$.

Total New Old
Modular forms 434 80 354
Cusp forms 50 20 30
Eisenstein series 384 60 324

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 16 0 4 0

## Trace form

 $$20q - 4q^{7} + O(q^{10})$$ $$20q - 4q^{7} - 8q^{12} - 4q^{13} - 8q^{24} + 4q^{33} + 4q^{37} - 16q^{43} - 4q^{51} + 8q^{54} + 4q^{55} - 8q^{57} - 4q^{61} + 4q^{63} + 8q^{66} - 4q^{69} + 4q^{79} - 12q^{81} - 4q^{85} + 4q^{91} + 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(408))$$

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
408.1.b $$\chi_{408}(101, \cdot)$$ None 0 1
408.1.d $$\chi_{408}(307, \cdot)$$ None 0 1
408.1.g $$\chi_{408}(137, \cdot)$$ None 0 1
408.1.i $$\chi_{408}(271, \cdot)$$ None 0 1
408.1.k $$\chi_{408}(103, \cdot)$$ None 0 1
408.1.m $$\chi_{408}(305, \cdot)$$ None 0 1
408.1.n $$\chi_{408}(67, \cdot)$$ None 0 1
408.1.p $$\chi_{408}(341, \cdot)$$ None 0 1
408.1.r $$\chi_{408}(115, \cdot)$$ None 0 2
408.1.t $$\chi_{408}(149, \cdot)$$ None 0 2
408.1.u $$\chi_{408}(89, \cdot)$$ 408.1.u.a 4 2
408.1.w $$\chi_{408}(55, \cdot)$$ None 0 2
408.1.y $$\chi_{408}(161, \cdot)$$ None 0 4
408.1.z $$\chi_{408}(127, \cdot)$$ None 0 4
408.1.be $$\chi_{408}(53, \cdot)$$ None 0 4
408.1.bf $$\chi_{408}(19, \cdot)$$ None 0 4
408.1.bg $$\chi_{408}(11, \cdot)$$ 408.1.bg.a 8 8
408.1.bg.b 8
408.1.bj $$\chi_{408}(37, \cdot)$$ None 0 8
408.1.bk $$\chi_{408}(73, \cdot)$$ None 0 8
408.1.bn $$\chi_{408}(23, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(408))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(408)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(68))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(136))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(204))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{8}$$)($$1 + T^{8}$$)
$3$ ($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{8}$$)
$5$ ($$1 - T^{4} + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$7$ ($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$11$ ($$1 - T^{4} + T^{8}$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)
$13$ ($$( 1 + T + T^{2} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)
$17$ ($$1 + T^{4}$$)($$1 + T^{8}$$)($$1 + T^{8}$$)
$19$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$23$ ($$1 - T^{4} + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$29$ ($$( 1 + T^{4} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$31$ ($$( 1 + T^{4} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$37$ ($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$41$ ($$1 - T^{4} + T^{8}$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)
$43$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T )^{8}( 1 + T^{4} )^{2}$$)($$( 1 + T )^{8}( 1 + T^{4} )^{2}$$)
$47$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)
$53$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$59$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}( 1 + T^{4} )^{2}$$)
$61$ ($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$67$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$71$ ($$( 1 + T^{4} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$73$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)
$79$ ($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$83$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T )^{8}( 1 + T^{4} )^{2}$$)($$( 1 - T )^{8}( 1 + T^{4} )^{2}$$)
$89$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$97$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)