# Properties

 Label 360.1.p Level 360 Weight 1 Character orbit p Rep. character $$\chi_{360}(19,\cdot)$$ Character field $$\Q$$ Dimension 2 Newform subspaces 2 Sturm bound 72 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 360.p (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$40$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$72$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(360, [\chi])$$.

Total New Old
Modular forms 10 4 6
Cusp forms 2 2 0
Eisenstein series 8 2 6

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

 $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{10} + 2q^{16} - 4q^{19} + 2q^{25} - 2q^{40} - 4q^{46} - 2q^{49} + 2q^{64} - 4q^{76} + 4q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(360, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
360.1.p.a $$1$$ $$0.180$$ $$\Q$$ $$D_{2}$$ $$\Q(\sqrt{-15})$$, $$\Q(\sqrt{-10})$$ $$\Q(\sqrt{6})$$ $$-1$$ $$0$$ $$1$$ $$0$$ $$q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}+q^{16}+\cdots$$
360.1.p.b $$1$$ $$0.180$$ $$\Q$$ $$D_{2}$$ $$\Q(\sqrt{-15})$$, $$\Q(\sqrt{-10})$$ $$\Q(\sqrt{6})$$ $$1$$ $$0$$ $$-1$$ $$0$$ $$q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}+q^{16}+\cdots$$

## Hecke characteristic polynomials

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