# Properties

 Label 8.6.b Level 8 Weight 6 Character orbit b Rep. character $$\chi_{8}(5,\cdot)$$ Character field $$\Q$$ Dimension 4 Newform subspaces 1 Sturm bound 6 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 8.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$6$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(8, [\chi])$$.

Total New Old
Modular forms 6 6 0
Cusp forms 4 4 0
Eisenstein series 2 2 0

## Trace form

 $$4q - 2q^{2} + 20q^{4} - 116q^{6} + 96q^{7} - 248q^{8} - 164q^{9} + O(q^{10})$$ $$4q - 2q^{2} + 20q^{4} - 116q^{6} + 96q^{7} - 248q^{8} - 164q^{9} + 632q^{10} + 1576q^{12} - 2384q^{14} - 416q^{15} - 3312q^{16} + 200q^{17} + 4754q^{18} + 4624q^{20} - 5636q^{22} + 2336q^{23} - 7792q^{24} + 1556q^{25} + 5608q^{26} + 5152q^{28} - 2128q^{30} - 12928q^{31} + 5408q^{32} - 2352q^{33} - 4772q^{34} - 10164q^{36} + 15980q^{38} + 35104q^{39} + 16032q^{40} - 4568q^{41} - 26144q^{42} - 29112q^{44} + 29200q^{46} - 54720q^{47} + 35616q^{48} + 9828q^{49} - 47498q^{50} - 36560q^{52} + 23288q^{54} + 85472q^{55} + 40768q^{56} - 2032q^{57} + 3784q^{58} + 2592q^{60} + 34496q^{62} - 153440q^{63} - 41920q^{64} - 19520q^{65} + 43224q^{66} + 10344q^{68} - 68928q^{70} + 206688q^{71} - 83272q^{72} + 39976q^{73} + 17464q^{74} + 99944q^{76} - 174064q^{78} - 247872q^{79} - 35520q^{80} + 29684q^{81} + 161132q^{82} + 196672q^{84} - 18500q^{86} + 307872q^{87} - 167216q^{88} - 84632q^{89} + 142280q^{90} - 49056q^{92} - 98784q^{94} - 259744q^{95} - 115648q^{96} - 99576q^{97} - 117042q^{98} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(8, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
8.6.b.a $$4$$ $$1.283$$ 4.0.218489.1 None $$-2$$ $$0$$ $$0$$ $$96$$ $$q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(5+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - 8 T^{2} + 64 T^{3} + 1024 T^{4}$$
$3$ $$1 - 404 T^{2} + 84150 T^{4} - 23855796 T^{6} + 3486784401 T^{8}$$
$5$ $$1 - 7028 T^{2} + 24404246 T^{4} - 68632812500 T^{6} + 95367431640625 T^{8}$$
$7$ $$( 1 - 48 T + 15502 T^{2} - 806736 T^{3} + 282475249 T^{4} )^{2}$$
$11$ $$1 - 296436 T^{2} + 49128544726 T^{4} - 7688786399022036 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$1 - 894228 T^{2} + 396323515894 T^{4} - 123276923449147572 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$( 1 - 100 T + 2767462 T^{2} - 141985700 T^{3} + 2015993900449 T^{4} )^{2}$$
$19$ $$1 - 6794580 T^{2} + 21506967947254 T^{4} - 41658020173929518580 T^{6} +$$$$37\!\cdots\!01$$$$T^{8}$$
$23$ $$( 1 - 1168 T + 10055470 T^{2} - 7517648624 T^{3} + 41426511213649 T^{4} )^{2}$$
$29$ $$1 - 31255380 T^{2} + 976386653995702 T^{4} -$$$$13\!\cdots\!80$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$31$ $$( 1 + 6464 T + 65013054 T^{2} + 185058832064 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 241262580 T^{2} + 24149916431784598 T^{4} -$$$$11\!\cdots\!20$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 + 2284 T + 146603254 T^{2} + 264615563084 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 - 466346868 T^{2} + 96250708269010006 T^{4} -$$$$10\!\cdots\!32$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$( 1 + 27360 T + 591338206 T^{2} + 6274879391520 T^{3} + 52599132235830049 T^{4} )^{2}$$
$53$ $$1 - 1039152180 T^{2} + 595616955270391126 T^{4} -$$$$18\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$1 - 1537424180 T^{2} + 1399694789142612374 T^{4} -$$$$78\!\cdots\!80$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$1 + 741098540 T^{2} + 1478044222094100534 T^{4} +$$$$52\!\cdots\!40$$$$T^{6} +$$$$50\!\cdots\!01$$$$T^{8}$$
$67$ $$1 - 1366835860 T^{2} + 3274116308996825526 T^{4} -$$$$24\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 - 103344 T + 6217736974 T^{2} - 186456278049744 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 - 19988 T + 2541602870 T^{2} - 41436555000884 T^{3} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$( 1 + 123936 T + 9855929374 T^{2} + 381358061866464 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 10047855188 T^{2} + 55381071937674414326 T^{4} -$$$$15\!\cdots\!12$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 + 42316 T + 4292401174 T^{2} + 236295059643884 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 49788 T + 16391371462 T^{2} + 427546496715516 T^{3} + 73742412689492826049 T^{4} )^{2}$$