# Properties

 Label 400.1.b Level 400 Weight 1 Character orbit b Rep. character $$\chi_{400}(351,\cdot)$$ Character field $$\Q$$ Dimension 1 Newform subspaces 1 Sturm bound 60 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 400.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$60$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(400, [\chi])$$.

Total New Old
Modular forms 19 1 18
Cusp forms 1 1 0
Eisenstein series 18 0 18

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 1 0 0 0

## Trace form

 $$q + q^{9} + O(q^{10})$$ $$q + q^{9} - 2q^{29} - 2q^{41} + q^{49} - 2q^{61} + q^{81} - 2q^{89} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(400, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
400.1.b.a $$1$$ $$0.200$$ $$\Q$$ $$D_{2}$$ $$\Q(\sqrt{-1})$$, $$\Q(\sqrt{-5})$$ $$\Q(\sqrt{5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+q^{9}-2q^{29}-2q^{41}+q^{49}-2q^{61}+\cdots$$

## Hecke characteristic polynomials

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