## Defining parameters

 Level: $$N$$ = $$224\( 224 = 2^{5} \cdot 7$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$3072$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(224))$$.

Total New Old
Modular forms 208 59 149
Cusp forms 16 9 7
Eisenstein series 192 50 142

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 5 4 0 0

## Trace form

 $$9q - 2q^{5} + q^{7} - q^{9} + O(q^{10})$$ $$9q - 2q^{5} + q^{7} - q^{9} - 4q^{16} - 2q^{17} - 4q^{18} - 2q^{21} + 4q^{22} - 6q^{23} - q^{25} - 2q^{33} + 2q^{37} - 4q^{43} + 4q^{44} - 3q^{49} - 2q^{53} + 4q^{56} - 4q^{57} + 2q^{61} + 3q^{63} + 4q^{67} + 4q^{69} + 2q^{71} + 2q^{73} + 4q^{74} - 2q^{77} + 2q^{79} + 3q^{81} + 4q^{85} + 2q^{89} - 4q^{92} - 2q^{93} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(224))$$

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
224.1.c $$\chi_{224}(97, \cdot)$$ None 0 1
224.1.d $$\chi_{224}(127, \cdot)$$ None 0 1
224.1.g $$\chi_{224}(15, \cdot)$$ None 0 1
224.1.h $$\chi_{224}(209, \cdot)$$ 224.1.h.a 1 1
224.1.k $$\chi_{224}(71, \cdot)$$ None 0 2
224.1.l $$\chi_{224}(41, \cdot)$$ None 0 2
224.1.n $$\chi_{224}(17, \cdot)$$ None 0 2
224.1.o $$\chi_{224}(79, \cdot)$$ None 0 2
224.1.r $$\chi_{224}(95, \cdot)$$ 224.1.r.a 4 2
224.1.s $$\chi_{224}(33, \cdot)$$ None 0 2
224.1.v $$\chi_{224}(13, \cdot)$$ 224.1.v.a 4 4
224.1.w $$\chi_{224}(43, \cdot)$$ None 0 4
224.1.y $$\chi_{224}(23, \cdot)$$ None 0 4
224.1.bb $$\chi_{224}(73, \cdot)$$ None 0 4
224.1.bc $$\chi_{224}(5, \cdot)$$ None 0 8
224.1.bf $$\chi_{224}(11, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(224))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(224)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4}$$)
$3$ ($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)
$5$ ($$1 + T^{2}$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$1 + T^{8}$$)
$7$ ($$1 - T$$)($$( 1 + T^{2} )^{2}$$)($$1 + T^{4}$$)
$11$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)
$13$ ($$1 + T^{2}$$)($$( 1 + T^{2} )^{4}$$)($$1 + T^{8}$$)
$17$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)
$19$ ($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)
$23$ ($$( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)
$29$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)
$31$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)
$37$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)
$41$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{4} )^{2}$$)
$43$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T )^{4}( 1 + T^{4} )$$)
$47$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{2} )^{4}$$)
$53$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 + T^{4} )$$)
$59$ ($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)
$61$ ($$1 + T^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$1 + T^{8}$$)
$67$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T^{4} )$$)
$71$ ($$( 1 - T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)
$73$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$79$ ($$( 1 - T )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{4} )^{2}$$)
$83$ ($$1 + T^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 + T^{8}$$)
$89$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$97$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)