# Properties

 Label 224.2.u Level 224 Weight 2 Character orbit u Rep. character $$\chi_{224}(29,\cdot)$$ Character field $$\Q(\zeta_{8})$$ Dimension 96 Newform subspaces 3 Sturm bound 64 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 224.u (of order $$8$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$32$$ Character field: $$\Q(\zeta_{8})$$ Newform subspaces: $$3$$ Sturm bound: $$64$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(224, [\chi])$$.

Total New Old
Modular forms 136 96 40
Cusp forms 120 96 24
Eisenstein series 16 0 16

## Trace form

 $$96q + O(q^{10})$$ $$96q - 16q^{10} - 32q^{12} - 20q^{16} - 20q^{18} - 28q^{22} - 8q^{23} - 16q^{24} + 40q^{26} - 48q^{27} + 40q^{30} + 40q^{32} + 40q^{34} + 16q^{36} - 56q^{38} - 48q^{39} + 40q^{40} - 8q^{43} + 4q^{44} - 64q^{46} - 104q^{48} - 104q^{50} + 32q^{51} - 64q^{52} - 16q^{53} + 64q^{54} + 64q^{55} + 28q^{56} + 72q^{58} - 8q^{60} - 64q^{61} + 24q^{62} + 40q^{63} - 64q^{66} + 40q^{67} - 8q^{68} - 64q^{69} + 48q^{70} - 48q^{72} + 28q^{74} + 64q^{75} - 16q^{77} + 24q^{78} - 16q^{80} - 80q^{82} + 40q^{86} - 112q^{87} + 80q^{88} - 120q^{90} + 148q^{92} + 40q^{94} - 128q^{95} - 64q^{96} - 128q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(224, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
224.2.u.a $$4$$ $$1.789$$ $$\Q(\zeta_{8})$$ None $$-4$$ $$-4$$ $$-8$$ $$0$$ $$q+(-1+\zeta_{8}^{2})q^{2}+(-1-\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{3}+\cdots$$
224.2.u.b $$40$$ $$1.789$$ None $$4$$ $$4$$ $$8$$ $$0$$
224.2.u.c $$52$$ $$1.789$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(224, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(224, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T + 2 T^{2} )^{2}$$)
$3$ ($$1 + 4 T + 12 T^{2} + 28 T^{3} + 56 T^{4} + 84 T^{5} + 108 T^{6} + 108 T^{7} + 81 T^{8}$$)
$5$ ($$1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 160 T^{5} + 600 T^{6} + 1000 T^{7} + 625 T^{8}$$)
$7$ ($$1 + T^{4}$$)
$11$ ($$1 + 12 T + 86 T^{2} + 428 T^{3} + 1634 T^{4} + 4708 T^{5} + 10406 T^{6} + 15972 T^{7} + 14641 T^{8}$$)
$13$ ($$1 + 8 T^{2} + 72 T^{3} + 32 T^{4} + 936 T^{5} + 1352 T^{6} + 28561 T^{8}$$)
$17$ ($$( 1 - 17 T^{2} )^{4}$$)
$19$ ($$1 + 4 T + 36 T^{2} + 196 T^{3} + 920 T^{4} + 3724 T^{5} + 12996 T^{6} + 27436 T^{7} + 130321 T^{8}$$)
$23$ ($$( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$)
$29$ ($$1 + 16 T + 162 T^{2} + 1216 T^{3} + 7490 T^{4} + 35264 T^{5} + 136242 T^{6} + 390224 T^{7} + 707281 T^{8}$$)
$31$ ($$( 1 + 12 T + 90 T^{2} + 372 T^{3} + 961 T^{4} )^{2}$$)
$37$ ($$1 + 16 T + 162 T^{2} + 1312 T^{3} + 9026 T^{4} + 48544 T^{5} + 221778 T^{6} + 810448 T^{7} + 1874161 T^{8}$$)
$41$ ($$1 + 8 T + 32 T^{2} + 360 T^{3} + 4034 T^{4} + 14760 T^{5} + 53792 T^{6} + 551368 T^{7} + 2825761 T^{8}$$)
$43$ ($$1 + 16 T + 114 T^{2} + 632 T^{3} + 3810 T^{4} + 27176 T^{5} + 210786 T^{6} + 1272112 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 - 100 T^{2} + 5766 T^{4} - 220900 T^{6} + 4879681 T^{8}$$)
$53$ ($$1 - 20 T + 118 T^{2} + 412 T^{3} - 8478 T^{4} + 21836 T^{5} + 331462 T^{6} - 2977540 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 + 20 T + 300 T^{2} + 3180 T^{3} + 28600 T^{4} + 187620 T^{5} + 1044300 T^{6} + 4107580 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 + 16 T + 96 T^{2} + 256 T^{3} + 512 T^{4} + 15616 T^{5} + 357216 T^{6} + 3631696 T^{7} + 13845841 T^{8}$$)
$67$ ($$1 - 16 T + 226 T^{2} - 2456 T^{3} + 23362 T^{4} - 164552 T^{5} + 1014514 T^{6} - 4812208 T^{7} + 20151121 T^{8}$$)
$71$ ($$1 + 1154 T^{4} + 25411681 T^{8}$$)
$73$ ($$1 - 24 T + 288 T^{2} - 3096 T^{3} + 30146 T^{4} - 226008 T^{5} + 1534752 T^{6} - 9336408 T^{7} + 28398241 T^{8}$$)
$79$ ($$( 1 - 60 T^{2} + 6241 T^{4} )^{2}$$)
$83$ ($$1 + 4 T + 132 T^{2} + 1236 T^{3} + 11608 T^{4} + 102588 T^{5} + 909348 T^{6} + 2287148 T^{7} + 47458321 T^{8}$$)
$89$ ($$1 + 32 T + 512 T^{2} + 6432 T^{3} + 68258 T^{4} + 572448 T^{5} + 4055552 T^{6} + 22559008 T^{7} + 62742241 T^{8}$$)
$97$ ($$( 1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4} )^{2}$$)