# Properties

 Label 1224.1.bb Level $1224$ Weight $1$ Character orbit 1224.bb Rep. character $\chi_{1224}(67,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $4$ Newform subspaces $2$ Sturm bound $216$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1224.bb (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$1224$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$216$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(1224, [\chi])$$.

Total New Old
Modular forms 12 12 0
Cusp forms 4 4 0
Eisenstein series 8 8 0

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

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

## Trace form

 $$4q + 2q^{2} - 2q^{4} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} - 4q^{8} - 2q^{9} - 2q^{16} + 2q^{17} - 4q^{18} + 4q^{19} + 2q^{25} + 2q^{32} - 6q^{33} + q^{34} - 2q^{36} + 2q^{38} + 2q^{43} + 2q^{49} - 2q^{50} + 3q^{51} + 2q^{59} + 4q^{64} - 2q^{67} - q^{68} + 2q^{72} - 2q^{76} - 2q^{81} - 4q^{83} - 2q^{86} - 8q^{89} + 4q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(1224, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1224.1.bb.a $$2$$ $$0.611$$ $$\Q(\sqrt{-3})$$ $$D_{6}$$ $$\Q(\sqrt{-2})$$ None $$1$$ $$-1$$ $$0$$ $$0$$ $$q-\zeta_{6}^{2}q^{2}+\zeta_{6}^{2}q^{3}-\zeta_{6}q^{4}+\zeta_{6}q^{6}+\cdots$$
1224.1.bb.b $$2$$ $$0.611$$ $$\Q(\sqrt{-3})$$ $$D_{6}$$ $$\Q(\sqrt{-2})$$ None $$1$$ $$1$$ $$0$$ $$0$$ $$q-\zeta_{6}^{2}q^{2}-\zeta_{6}^{2}q^{3}-\zeta_{6}q^{4}-\zeta_{6}q^{6}+\cdots$$

## Hecke characteristic polynomials

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