# Properties

 Label 300.1.b Level $300$ Weight $1$ Character orbit 300.b Rep. character $\chi_{300}(149,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $1$ Sturm bound $60$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$60$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(300, [\chi])$$.

Total New Old
Modular forms 20 2 18
Cusp forms 2 2 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 2 0 0 0

## Trace form

 $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{19} - 2q^{21} - 2q^{31} + 2q^{39} - 2q^{61} - 4q^{79} + 2q^{81} + 2q^{91} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(300, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
300.1.b.a $$2$$ $$0.150$$ $$\Q(\sqrt{-1})$$ $$D_{3}$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{3}-iq^{7}-q^{9}+iq^{13}+q^{19}+\cdots$$

## Hecke characteristic polynomials

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