# Properties

 Label 224.2.x Level 224 Weight 2 Character orbit x Rep. character $$\chi_{224}(27,\cdot)$$ Character field $$\Q(\zeta_{8})$$ Dimension 120 Newform subspaces 2 Sturm bound 64 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 224.x (of order $$8$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$224$$ Character field: $$\Q(\zeta_{8})$$ Newform subspaces: $$2$$ 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 136 136 0
Cusp forms 120 120 0
Eisenstein series 16 16 0

## Trace form

 $$120q - 8q^{2} - 8q^{4} - 4q^{7} - 8q^{8} - 8q^{9} + O(q^{10})$$ $$120q - 8q^{2} - 8q^{4} - 4q^{7} - 8q^{8} - 8q^{9} - 8q^{11} - 20q^{14} - 16q^{15} + 12q^{16} - 28q^{18} - 4q^{21} - 4q^{22} - 16q^{23} - 8q^{25} - 24q^{28} - 8q^{29} + 16q^{30} - 8q^{32} + 20q^{35} + 32q^{36} - 8q^{37} - 8q^{39} + 16q^{42} - 16q^{43} - 92q^{44} - 8q^{46} + 16q^{50} - 32q^{51} + 8q^{53} + 36q^{56} - 8q^{57} - 72q^{58} - 88q^{60} + 112q^{64} - 16q^{65} - 48q^{67} - 40q^{70} + 56q^{71} - 8q^{72} - 124q^{74} - 4q^{77} + 192q^{78} - 16q^{79} - 24q^{84} - 48q^{85} - 8q^{86} - 96q^{88} - 52q^{91} + 68q^{92} - 32q^{93} + 88q^{98} + 16q^{99} + O(q^{100})$$

## 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.x.a $$8$$ $$1.789$$ 8.0.157351936.1 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{2}+\beta _{4})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(2\beta _{2}+\cdots)q^{7}+\cdots$$
224.2.x.b $$112$$ $$1.789$$ None $$-8$$ $$0$$ $$0$$ $$-4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4} + 16 T^{8}$$)
$3$ ($$( 1 + 81 T^{8} )^{2}$$)
$5$ ($$( 1 + 625 T^{8} )^{2}$$)
$7$ ($$( 1 + 49 T^{4} )^{2}$$)
$11$ ($$( 1 - 6 T^{2} + 121 T^{4} )^{2}( 1 - 206 T^{4} + 14641 T^{8} )$$)
$13$ ($$( 1 + 28561 T^{8} )^{2}$$)
$17$ ($$( 1 + 17 T^{2} )^{8}$$)
$19$ ($$( 1 + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 - 8 T + 23 T^{2} )^{4}( 1 + 18 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 - 54 T^{2} + 841 T^{4} )^{2}( 1 + 1234 T^{4} + 707281 T^{8} )$$)
$31$ ($$( 1 + 31 T^{2} )^{8}$$)
$37$ ($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}( 1 - 1294 T^{4} + 1874161 T^{8} )$$)
$41$ ($$( 1 + 1681 T^{4} )^{4}$$)
$43$ ($$( 1 + 12 T + 43 T^{2} )^{4}( 1 - 334 T^{4} + 3418801 T^{8} )$$)
$47$ ($$( 1 - 47 T^{2} )^{8}$$)
$53$ ($$( 1 - 10 T + 53 T^{2} )^{4}( 1 - 5582 T^{4} + 7890481 T^{8} )$$)
$59$ ($$( 1 + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4946 T^{4} + 20151121 T^{8} )$$)
$71$ ($$( 1 + 2914 T^{4} + 25411681 T^{8} )^{2}$$)
$73$ ($$( 1 + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 - 3646 T^{4} + 38950081 T^{8} )^{2}$$)
$83$ ($$( 1 + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 7921 T^{4} )^{4}$$)
$97$ ($$( 1 - 97 T^{2} )^{8}$$)