# Properties

 Label 128.5.d Level $128$ Weight $5$ Character orbit 128.d Rep. character $\chi_{128}(63,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $4$ Sturm bound $80$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 128.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$80$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(128, [\chi])$$.

Total New Old
Modular forms 72 16 56
Cusp forms 56 16 40
Eisenstein series 16 0 16

## Trace form

 $$16q + 432q^{9} + O(q^{10})$$ $$16q + 432q^{9} - 480q^{17} - 2672q^{25} + 1984q^{33} + 2976q^{41} + 528q^{49} + 12736q^{57} - 13440q^{65} - 30560q^{73} + 9168q^{81} - 8544q^{89} + 31776q^{97} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(128, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
128.5.d.a $$2$$ $$13.231$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{5}-3^{4}q^{9}+5iq^{13}-322q^{17}+\cdots$$
128.5.d.b $$2$$ $$13.231$$ $$\Q(\sqrt{2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{3}+47q^{9}+21\beta q^{11}+574q^{17}+\cdots$$
128.5.d.c $$4$$ $$13.231$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}-\beta _{1}q^{5}+\beta _{3}q^{7}+15q^{9}+\cdots$$
128.5.d.d $$8$$ $$13.231$$ 8.0.1871773696.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+(55-\beta _{6}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{5}^{\mathrm{old}}(128, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(128, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(8, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$( 1 + 81 T^{2} )^{2}$$)($$1 + 34 T^{2} + 6561 T^{4}$$)($$( 1 + 66 T^{2} + 6561 T^{4} )^{2}$$)($$( 1 + 52 T^{2} + 486 T^{4} + 341172 T^{6} + 43046721 T^{8} )^{2}$$)
$5$ ($$( 1 - 14 T + 625 T^{2} )( 1 + 14 T + 625 T^{2} )$$)($$( 1 - 25 T )^{2}( 1 + 25 T )^{2}$$)($$( 1 - 1186 T^{2} + 390625 T^{4} )^{2}$$)($$( 1 - 548 T^{2} + 377094 T^{4} - 214062500 T^{6} + 152587890625 T^{8} )^{2}$$)
$7$ ($$( 1 - 49 T )^{2}( 1 + 49 T )^{2}$$)($$( 1 - 49 T )^{2}( 1 + 49 T )^{2}$$)($$( 1 + 1342 T^{2} + 5764801 T^{4} )^{2}$$)($$( 1 - 3138 T^{2} + 5764801 T^{4} )^{4}$$)
$11$ ($$( 1 + 14641 T^{2} )^{2}$$)($$1 - 27166 T^{2} + 214358881 T^{4}$$)($$( 1 + 17666 T^{2} + 214358881 T^{4} )^{2}$$)($$( 1 + 45876 T^{2} + 934622054 T^{4} + 9833928024756 T^{6} + 45949729863572161 T^{8} )^{2}$$)
$13$ ($$( 1 - 238 T + 28561 T^{2} )( 1 + 238 T + 28561 T^{2} )$$)($$( 1 - 169 T )^{2}( 1 + 169 T )^{2}$$)($$( 1 - 10466 T^{2} + 815730721 T^{4} )^{2}$$)($$( 1 - 79268 T^{2} + 2902795398 T^{4} - 64661342792228 T^{6} + 665416609183179841 T^{8} )^{2}$$)
$17$ ($$( 1 + 322 T + 83521 T^{2} )^{2}$$)($$( 1 - 574 T + 83521 T^{2} )^{2}$$)($$( 1 + 162 T + 83521 T^{2} )^{4}$$)($$( 1 + 84 T + 155494 T^{2} + 7015764 T^{3} + 6975757441 T^{4} )^{4}$$)
$19$ ($$( 1 + 130321 T^{2} )^{2}$$)($$1 - 72286 T^{2} + 16983563041 T^{4}$$)($$( 1 + 66242 T^{2} + 16983563041 T^{4} )^{2}$$)($$( 1 + 191412 T^{2} + 35457974246 T^{4} + 3250857768803892 T^{6} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$)
$23$ ($$( 1 - 529 T )^{2}( 1 + 529 T )^{2}$$)($$( 1 - 529 T )^{2}( 1 + 529 T )^{2}$$)($$( 1 - 62018 T^{2} + 78310985281 T^{4} )^{2}$$)($$( 1 - 108420 T^{2} - 36732732538 T^{4} - 8490477024166020 T^{6} +$$$$61\!\cdots\!61$$$$T^{8} )^{2}$$)
$29$ ($$( 1 - 82 T + 707281 T^{2} )( 1 + 82 T + 707281 T^{2} )$$)($$( 1 - 841 T )^{2}( 1 + 841 T )^{2}$$)($$( 1 + 285854 T^{2} + 500246412961 T^{4} )^{2}$$)($$( 1 - 1076900 T^{2} + 796704441222 T^{4} - 538715362117700900 T^{6} +$$$$25\!\cdots\!21$$$$T^{8} )^{2}$$)
$31$ ($$( 1 - 961 T )^{2}( 1 + 961 T )^{2}$$)($$( 1 - 961 T )^{2}( 1 + 961 T )^{2}$$)($$( 1 - 1453826 T^{2} + 852891037441 T^{4} )^{2}$$)($$( 1 - 667394 T^{2} + 852891037441 T^{4} )^{4}$$)
$37$ ($$( 1 - 2162 T + 1874161 T^{2} )( 1 + 2162 T + 1874161 T^{2} )$$)($$( 1 - 1369 T )^{2}( 1 + 1369 T )^{2}$$)($$( 1 - 1462178 T^{2} + 3512479453921 T^{4} )^{2}$$)($$( 1 - 3128612 T^{2} + 6805143118086 T^{4} - 10989185369290687652 T^{6} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$)
$41$ ($$( 1 + 3038 T + 2825761 T^{2} )^{2}$$)($$( 1 + 1246 T + 2825761 T^{2} )^{2}$$)($$( 1 - 1890 T + 2825761 T^{2} )^{4}$$)($$( 1 - 996 T + 5420294 T^{2} - 2814457956 T^{3} + 7984925229121 T^{4} )^{4}$$)
$43$ ($$( 1 + 3418801 T^{2} )^{2}$$)($$1 + 5426402 T^{2} + 11688200277601 T^{4}$$)($$( 1 - 1630462 T^{2} + 11688200277601 T^{4} )^{2}$$)($$( 1 + 5340852 T^{2} + 29705752269926 T^{4} + 62424947829025856052 T^{6} +$$$$13\!\cdots\!01$$$$T^{8} )^{2}$$)
$47$ ($$( 1 - 2209 T )^{2}( 1 + 2209 T )^{2}$$)($$( 1 - 2209 T )^{2}( 1 + 2209 T )^{2}$$)($$( 1 - 7768706 T^{2} + 23811286661761 T^{4} )^{2}$$)($$( 1 - 11288836 T^{2} + 65631563659014 T^{4} -$$$$26\!\cdots\!96$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8} )^{2}$$)
$53$ ($$( 1 - 2482 T + 7890481 T^{2} )( 1 + 2482 T + 7890481 T^{2} )$$)($$( 1 - 2809 T )^{2}( 1 + 2809 T )^{2}$$)($$( 1 - 11876386 T^{2} + 62259690411361 T^{4} )^{2}$$)($$( 1 - 7465252 T^{2} + 129570259041798 T^{4} -$$$$46\!\cdots\!72$$$$T^{6} +$$$$38\!\cdots\!21$$$$T^{8} )^{2}$$)
$59$ ($$( 1 + 12117361 T^{2} )^{2}$$)($$1 - 24178078 T^{2} + 146830437604321 T^{4}$$)($$( 1 + 19112066 T^{2} + 146830437604321 T^{4} )^{2}$$)($$( 1 - 11608652 T^{2} + 321512309200230 T^{4} -$$$$17\!\cdots\!92$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8} )^{2}$$)
$61$ ($$( 1 - 6958 T + 13845841 T^{2} )( 1 + 6958 T + 13845841 T^{2} )$$)($$( 1 - 3721 T )^{2}( 1 + 3721 T )^{2}$$)($$( 1 - 22046306 T^{2} + 191707312997281 T^{4} )^{2}$$)($$( 1 - 16591268 T^{2} + 144825482457990 T^{4} -$$$$31\!\cdots\!08$$$$T^{6} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$)
$67$ ($$( 1 + 20151121 T^{2} )^{2}$$)($$1 - 13944286 T^{2} + 406067677556641 T^{4}$$)($$( 1 + 37495106 T^{2} + 406067677556641 T^{4} )^{2}$$)($$( 1 + 68122548 T^{2} + 1934597002567910 T^{4} +$$$$27\!\cdots\!68$$$$T^{6} +$$$$16\!\cdots\!81$$$$T^{8} )^{2}$$)
$71$ ($$( 1 - 5041 T )^{2}( 1 + 5041 T )^{2}$$)($$( 1 - 5041 T )^{2}( 1 + 5041 T )^{2}$$)($$( 1 + 9393982 T^{2} + 645753531245761 T^{4} )^{2}$$)($$( 1 - 10042500 T^{2} + 116338596303494 T^{4} -$$$$64\!\cdots\!00$$$$T^{6} +$$$$41\!\cdots\!21$$$$T^{8} )^{2}$$)
$73$ ($$( 1 + 1442 T + 28398241 T^{2} )^{2}$$)($$( 1 + 9506 T + 28398241 T^{2} )^{2}$$)($$( 1 - 2750 T + 28398241 T^{2} )^{4}$$)($$( 1 + 4916 T + 10002918 T^{2} + 139605752756 T^{3} + 806460091894081 T^{4} )^{4}$$)
$79$ ($$( 1 - 6241 T )^{2}( 1 + 6241 T )^{2}$$)($$( 1 - 6241 T )^{2}( 1 + 6241 T )^{2}$$)($$( 1 - 13977986 T^{2} + 1517108809906561 T^{4} )^{2}$$)($$( 1 - 48021378 T^{2} + 1517108809906561 T^{4} )^{4}$$)
$83$ ($$( 1 + 47458321 T^{2} )^{2}$$)($$1 + 30209954 T^{2} + 2252292232139041 T^{4}$$)($$( 1 + 7728578 T^{2} + 2252292232139041 T^{4} )^{2}$$)($$( 1 + 61584436 T^{2} + 3095005172414694 T^{4} +$$$$13\!\cdots\!76$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8} )^{2}$$)
$89$ ($$( 1 - 9758 T + 62742241 T^{2} )^{2}$$)($$( 1 + 5474 T + 62742241 T^{2} )^{2}$$)($$( 1 - 2430 T + 62742241 T^{2} )^{4}$$)($$( 1 + 6708 T + 129691750 T^{2} + 420874952628 T^{3} + 3936588805702081 T^{4} )^{4}$$)
$97$ ($$( 1 - 1918 T + 88529281 T^{2} )^{2}$$)($$( 1 - 9982 T + 88529281 T^{2} )^{2}$$)($$( 1 - 7454 T + 88529281 T^{2} )^{4}$$)($$( 1 + 5460 T + 83752934 T^{2} + 483369874260 T^{3} + 7837433594376961 T^{4} )^{4}$$)