# Properties

 Label 403.1 Level 403 Weight 1 Dimension 4 Nonzero newspaces 1 Newform subspaces 1 Sturm bound 13440 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$403 = 13 \cdot 31$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$13440$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(403))$$.

Total New Old
Modular forms 366 322 44
Cusp forms 6 4 2
Eisenstein series 360 318 42

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

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

## Trace form

 $$4q - 2q^{2} + 2q^{7} - 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 2q^{7} - 4q^{8} - 4q^{14} + 2q^{16} + 2q^{19} - 4q^{25} - 2q^{33} - 4q^{38} + 2q^{39} + 2q^{41} + 2q^{50} + 4q^{51} - 2q^{56} + 2q^{59} + 4q^{64} + 4q^{66} + 2q^{67} - 2q^{69} - 2q^{71} + 2q^{78} + 2q^{81} + 2q^{82} + 2q^{87} - 2q^{93} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(403))$$

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
403.1.b $$\chi_{403}(402, \cdot)$$ None 0 1
403.1.d $$\chi_{403}(92, \cdot)$$ None 0 1
403.1.j $$\chi_{403}(125, \cdot)$$ None 0 2
403.1.m $$\chi_{403}(181, \cdot)$$ None 0 2
403.1.n $$\chi_{403}(347, \cdot)$$ None 0 2
403.1.o $$\chi_{403}(68, \cdot)$$ None 0 2
403.1.p $$\chi_{403}(61, \cdot)$$ 403.1.p.a 4 2
403.1.q $$\chi_{403}(88, \cdot)$$ None 0 2
403.1.t $$\chi_{403}(30, \cdot)$$ None 0 2
403.1.u $$\chi_{403}(192, \cdot)$$ None 0 2
403.1.w $$\chi_{403}(274, \cdot)$$ None 0 2
403.1.x $$\chi_{403}(27, \cdot)$$ None 0 4
403.1.z $$\chi_{403}(77, \cdot)$$ None 0 4
403.1.bb $$\chi_{403}(5, \cdot)$$ None 0 4
403.1.bc $$\chi_{403}(98, \cdot)$$ None 0 4
403.1.bd $$\chi_{403}(32, \cdot)$$ None 0 4
403.1.bh $$\chi_{403}(67, \cdot)$$ None 0 4
403.1.bm $$\chi_{403}(8, \cdot)$$ None 0 8
403.1.bo $$\chi_{403}(53, \cdot)$$ None 0 8
403.1.bq $$\chi_{403}(23, \cdot)$$ None 0 8
403.1.br $$\chi_{403}(127, \cdot)$$ None 0 8
403.1.bu $$\chi_{403}(17, \cdot)$$ None 0 8
403.1.bv $$\chi_{403}(3, \cdot)$$ None 0 8
403.1.bw $$\chi_{403}(29, \cdot)$$ None 0 8
403.1.bx $$\chi_{403}(42, \cdot)$$ None 0 8
403.1.by $$\chi_{403}(12, \cdot)$$ None 0 8
403.1.ca $$\chi_{403}(7, \cdot)$$ None 0 16
403.1.ce $$\chi_{403}(2, \cdot)$$ None 0 16
403.1.cf $$\chi_{403}(20, \cdot)$$ None 0 16
403.1.cg $$\chi_{403}(18, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(403))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(403)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(31))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

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