# Properties

 Label 224.3.h Level 224 Weight 3 Character orbit h Rep. character $$\chi_{224}(209,\cdot)$$ Character field $$\Q$$ Dimension 14 Newform subspaces 4 Sturm bound 96 Trace bound 9

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$56$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$96$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(224, [\chi])$$.

Total New Old
Modular forms 72 18 54
Cusp forms 56 14 42
Eisenstein series 16 4 12

## Trace form

 $$14q + 2q^{7} + 26q^{9} + O(q^{10})$$ $$14q + 2q^{7} + 26q^{9} + 40q^{15} + 36q^{23} + 26q^{25} + 40q^{39} + 14q^{49} - 136q^{57} - 42q^{63} - 104q^{65} - 316q^{71} - 156q^{79} - 90q^{81} - 280q^{95} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(224, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
224.3.h.a $$2$$ $$6.104$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$-14$$ $$q-7q^{7}-9q^{9}-\beta q^{11}-18q^{23}-5^{2}q^{25}+\cdots$$
224.3.h.b $$2$$ $$6.104$$ $$\Q(\sqrt{7})$$ $$\Q(\sqrt{-14})$$ $$0$$ $$0$$ $$0$$ $$-14$$ $$q+\beta q^{3}+\beta q^{5}-7q^{7}+19q^{9}+\beta q^{13}+\cdots$$
224.3.h.c $$2$$ $$6.104$$ $$\Q(\sqrt{2})$$ $$\Q(\sqrt{-14})$$ $$0$$ $$0$$ $$0$$ $$14$$ $$q+\beta q^{3}+3\beta q^{5}+7q^{7}-q^{9}-9\beta q^{13}+\cdots$$
224.3.h.d $$8$$ $$6.104$$ 8.0.$$\cdots$$.51 None $$0$$ $$0$$ $$0$$ $$16$$ $$q-\beta _{3}q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(224, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(224, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$( 1 + 9 T^{2} )^{2}$$)($$1 - 10 T^{2} + 81 T^{4}$$)($$1 + 10 T^{2} + 81 T^{4}$$)($$( 1 + 16 T^{2} + 178 T^{4} + 1296 T^{6} + 6561 T^{8} )^{2}$$)
$5$ ($$( 1 + 25 T^{2} )^{2}$$)($$1 + 22 T^{2} + 625 T^{4}$$)($$1 - 22 T^{2} + 625 T^{4}$$)($$( 1 + 56 T^{2} + 1602 T^{4} + 35000 T^{6} + 390625 T^{8} )^{2}$$)
$7$ ($$( 1 + 7 T )^{2}$$)($$( 1 + 7 T )^{2}$$)($$( 1 - 7 T )^{2}$$)($$( 1 - 8 T + 102 T^{2} - 392 T^{3} + 2401 T^{4} )^{2}$$)
$11$ ($$( 1 - 6 T + 121 T^{2} )( 1 + 6 T + 121 T^{2} )$$)($$( 1 - 11 T )^{2}( 1 + 11 T )^{2}$$)($$( 1 - 11 T )^{2}( 1 + 11 T )^{2}$$)($$( 1 - 364 T^{2} + 59334 T^{4} - 5329324 T^{6} + 214358881 T^{8} )^{2}$$)
$13$ ($$( 1 + 169 T^{2} )^{2}$$)($$1 + 310 T^{2} + 28561 T^{4}$$)($$1 - 310 T^{2} + 28561 T^{4}$$)($$( 1 + 344 T^{2} + 86658 T^{4} + 9824984 T^{6} + 815730721 T^{8} )^{2}$$)
$17$ ($$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$)($$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$)($$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$)($$( 1 - 148 T^{2} + 165606 T^{4} - 12361108 T^{6} + 6975757441 T^{8} )^{2}$$)
$19$ ($$( 1 + 361 T^{2} )^{2}$$)($$1 - 650 T^{2} + 130321 T^{4}$$)($$1 + 650 T^{2} + 130321 T^{4}$$)($$( 1 + 656 T^{2} + 327858 T^{4} + 85490576 T^{6} + 16983563041 T^{8} )^{2}$$)
$23$ ($$( 1 + 18 T + 529 T^{2} )^{2}$$)($$( 1 - 10 T + 529 T^{2} )^{2}$$)($$( 1 - 10 T + 529 T^{2} )^{2}$$)($$( 1 - 8 T + 642 T^{2} - 4232 T^{3} + 279841 T^{4} )^{4}$$)
$29$ ($$( 1 - 54 T + 841 T^{2} )( 1 + 54 T + 841 T^{2} )$$)($$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$)($$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$)($$( 1 - 1444 T^{2} + 1048038 T^{4} - 1021313764 T^{6} + 500246412961 T^{8} )^{2}$$)
$31$ ($$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$)($$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$)($$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$)($$( 1 + 140 T^{2} + 1206054 T^{4} + 129292940 T^{6} + 852891037441 T^{8} )^{2}$$)
$37$ ($$( 1 - 38 T + 1369 T^{2} )( 1 + 38 T + 1369 T^{2} )$$)($$( 1 - 37 T )^{2}( 1 + 37 T )^{2}$$)($$( 1 - 37 T )^{2}( 1 + 37 T )^{2}$$)($$( 1 - 4612 T^{2} + 8955366 T^{4} - 8643630532 T^{6} + 3512479453921 T^{8} )^{2}$$)
$41$ ($$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$)($$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$)($$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$)($$( 1 - 5380 T^{2} + 12875334 T^{4} - 15202594180 T^{6} + 7984925229121 T^{8} )^{2}$$)
$43$ ($$( 1 - 58 T + 1849 T^{2} )( 1 + 58 T + 1849 T^{2} )$$)($$( 1 - 43 T )^{2}( 1 + 43 T )^{2}$$)($$( 1 - 43 T )^{2}( 1 + 43 T )^{2}$$)($$( 1 - 1324 T^{2} + 5788998 T^{4} - 4526492524 T^{6} + 11688200277601 T^{8} )^{2}$$)
$47$ ($$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$)($$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$)($$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$)($$( 1 - 4852 T^{2} + 14998950 T^{4} - 23676212212 T^{6} + 23811286661761 T^{8} )^{2}$$)
$53$ ($$( 1 - 6 T + 2809 T^{2} )( 1 + 6 T + 2809 T^{2} )$$)($$( 1 - 53 T )^{2}( 1 + 53 T )^{2}$$)($$( 1 - 53 T )^{2}( 1 + 53 T )^{2}$$)($$( 1 - 7780 T^{2} + 30664230 T^{4} - 61387942180 T^{6} + 62259690411361 T^{8} )^{2}$$)
$59$ ($$( 1 + 3481 T^{2} )^{2}$$)($$1 - 1130 T^{2} + 12117361 T^{4}$$)($$1 + 1130 T^{2} + 12117361 T^{4}$$)($$( 1 + 9200 T^{2} + 44845170 T^{4} + 111479721200 T^{6} + 146830437604321 T^{8} )^{2}$$)
$61$ ($$( 1 + 3721 T^{2} )^{2}$$)($$1 - 7370 T^{2} + 13845841 T^{4}$$)($$1 + 7370 T^{2} + 13845841 T^{4}$$)($$( 1 + 14840 T^{2} + 82747650 T^{4} + 205472280440 T^{6} + 191707312997281 T^{8} )^{2}$$)
$67$ ($$( 1 - 118 T + 4489 T^{2} )( 1 + 118 T + 4489 T^{2} )$$)($$( 1 - 67 T )^{2}( 1 + 67 T )^{2}$$)($$( 1 - 67 T )^{2}( 1 + 67 T )^{2}$$)($$( 1 - 11884 T^{2} + 74122758 T^{4} - 239475921964 T^{6} + 406067677556641 T^{8} )^{2}$$)
$71$ ($$( 1 - 114 T + 5041 T^{2} )^{2}$$)($$( 1 + 110 T + 5041 T^{2} )^{2}$$)($$( 1 - 110 T + 5041 T^{2} )^{2}$$)($$( 1 + 136 T + 12678 T^{2} + 685576 T^{3} + 25411681 T^{4} )^{4}$$)
$73$ ($$( 1 - 73 T )^{2}( 1 + 73 T )^{2}$$)($$( 1 - 73 T )^{2}( 1 + 73 T )^{2}$$)($$( 1 - 73 T )^{2}( 1 + 73 T )^{2}$$)($$( 1 - 9364 T^{2} + 72904614 T^{4} - 265921128724 T^{6} + 806460091894081 T^{8} )^{2}$$)
$79$ ($$( 1 + 94 T + 6241 T^{2} )^{2}$$)($$( 1 - 130 T + 6241 T^{2} )^{2}$$)($$( 1 + 130 T + 6241 T^{2} )^{2}$$)($$( 1 - 8 T + 11526 T^{2} - 49928 T^{3} + 38950081 T^{4} )^{4}$$)
$83$ ($$( 1 + 6889 T^{2} )^{2}$$)($$1 - 13130 T^{2} + 47458321 T^{4}$$)($$1 + 13130 T^{2} + 47458321 T^{4}$$)($$( 1 + 6608 T^{2} - 3756750 T^{4} + 313604585168 T^{6} + 2252292232139041 T^{8} )^{2}$$)
$89$ ($$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$)($$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$)($$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$)($$( 1 - 17428 T^{2} + 180509478 T^{4} - 1093471776148 T^{6} + 3936588805702081 T^{8} )^{2}$$)
$97$ ($$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$)($$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$)($$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$)($$( 1 - 13588 T^{2} + 180804198 T^{4} - 1202935870228 T^{6} + 7837433594376961 T^{8} )^{2}$$)