# Properties

 Label 8.6 Level 8 Weight 6 Dimension 5 Nonzero newspaces 2 Newform subspaces 2 Sturm bound 24 Trace bound 1

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## Defining parameters

 Level: $$N$$ = $$8\( 8 = 2^{3}$$ \) Weight: $$k$$ = $$6$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$24$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(8))$$.

Total New Old
Modular forms 13 7 6
Cusp forms 7 5 2
Eisenstein series 6 2 4

## Trace form

 $$5q - 2q^{2} + 20q^{3} + 20q^{4} - 74q^{5} - 116q^{6} + 72q^{7} - 248q^{8} - 7q^{9} + O(q^{10})$$ $$5q - 2q^{2} + 20q^{3} + 20q^{4} - 74q^{5} - 116q^{6} + 72q^{7} - 248q^{8} - 7q^{9} + 632q^{10} + 124q^{11} + 1576q^{12} + 478q^{13} - 2384q^{14} - 1896q^{15} - 3312q^{16} - 998q^{17} + 4754q^{18} + 3044q^{19} + 4624q^{20} - 480q^{21} - 5636q^{22} + 2520q^{23} - 7792q^{24} + 3907q^{25} + 5608q^{26} - 1720q^{27} + 5152q^{28} - 3282q^{29} - 2128q^{30} - 18656q^{31} + 5408q^{32} + 128q^{33} - 4772q^{34} + 1776q^{35} - 10164q^{36} + 10326q^{37} + 15980q^{38} + 44664q^{39} + 16032q^{40} - 13454q^{41} - 26144q^{42} - 9188q^{43} - 29112q^{44} - 11618q^{45} + 29200q^{46} - 31056q^{47} + 35616q^{48} - 6403q^{49} - 47498q^{50} - 23960q^{51} - 36560q^{52} + 11686q^{53} + 23288q^{54} + 76296q^{55} + 40768q^{56} + 58848q^{57} + 3784q^{58} + 16876q^{59} + 2592q^{60} - 18482q^{61} + 34496q^{62} - 157208q^{63} - 41920q^{64} - 54892q^{65} + 43224q^{66} - 15532q^{67} + 10344q^{68} + 3680q^{69} - 68928q^{70} + 174728q^{71} - 83272q^{72} + 35090q^{73} + 17464q^{74} + 47020q^{75} + 99944q^{76} - 2976q^{77} - 174064q^{78} - 203312q^{79} - 35520q^{80} - 42867q^{81} + 161132q^{82} + 67364q^{83} + 196672q^{84} + 88652q^{85} - 18500q^{86} + 242232q^{87} - 167216q^{88} - 12638q^{89} + 142280q^{90} - 11472q^{91} - 49056q^{92} - 114560q^{93} - 98784q^{94} - 485000q^{95} - 115648q^{96} - 50710q^{97} - 117042q^{98} + 19468q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(8))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
8.6.a $$\chi_{8}(1, \cdot)$$ 8.6.a.a 1 1
8.6.b $$\chi_{8}(5, \cdot)$$ 8.6.b.a 4 1

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(8))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(8)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T - 8 T^{2} + 64 T^{3} + 1024 T^{4}$$)
$3$ ($$1 - 20 T + 243 T^{2}$$)($$1 - 404 T^{2} + 84150 T^{4} - 23855796 T^{6} + 3486784401 T^{8}$$)
$5$ ($$1 + 74 T + 3125 T^{2}$$)($$1 - 7028 T^{2} + 24404246 T^{4} - 68632812500 T^{6} + 95367431640625 T^{8}$$)
$7$ ($$1 + 24 T + 16807 T^{2}$$)($$( 1 - 48 T + 15502 T^{2} - 806736 T^{3} + 282475249 T^{4} )^{2}$$)
$11$ ($$1 - 124 T + 161051 T^{2}$$)($$1 - 296436 T^{2} + 49128544726 T^{4} - 7688786399022036 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$)
$13$ ($$1 - 478 T + 371293 T^{2}$$)($$1 - 894228 T^{2} + 396323515894 T^{4} - 123276923449147572 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$)
$17$ ($$1 + 1198 T + 1419857 T^{2}$$)($$( 1 - 100 T + 2767462 T^{2} - 141985700 T^{3} + 2015993900449 T^{4} )^{2}$$)
$19$ ($$1 - 3044 T + 2476099 T^{2}$$)($$1 - 6794580 T^{2} + 21506967947254 T^{4} - 41658020173929518580 T^{6} +$$$$37\!\cdots\!01$$$$T^{8}$$)
$23$ ($$1 - 184 T + 6436343 T^{2}$$)($$( 1 - 1168 T + 10055470 T^{2} - 7517648624 T^{3} + 41426511213649 T^{4} )^{2}$$)
$29$ ($$1 + 3282 T + 20511149 T^{2}$$)($$1 - 31255380 T^{2} + 976386653995702 T^{4} -$$$$13\!\cdots\!80$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$)
$31$ ($$1 + 5728 T + 28629151 T^{2}$$)($$( 1 + 6464 T + 65013054 T^{2} + 185058832064 T^{3} + 819628286980801 T^{4} )^{2}$$)
$37$ ($$1 - 10326 T + 69343957 T^{2}$$)($$1 - 241262580 T^{2} + 24149916431784598 T^{4} -$$$$11\!\cdots\!20$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$)
$41$ ($$1 + 8886 T + 115856201 T^{2}$$)($$( 1 + 2284 T + 146603254 T^{2} + 264615563084 T^{3} + 13422659310152401 T^{4} )^{2}$$)
$43$ ($$1 + 9188 T + 147008443 T^{2}$$)($$1 - 466346868 T^{2} + 96250708269010006 T^{4} -$$$$10\!\cdots\!32$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$)
$47$ ($$1 - 23664 T + 229345007 T^{2}$$)($$( 1 + 27360 T + 591338206 T^{2} + 6274879391520 T^{3} + 52599132235830049 T^{4} )^{2}$$)
$53$ ($$1 - 11686 T + 418195493 T^{2}$$)($$1 - 1039152180 T^{2} + 595616955270391126 T^{4} -$$$$18\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$)
$59$ ($$1 - 16876 T + 714924299 T^{2}$$)($$1 - 1537424180 T^{2} + 1399694789142612374 T^{4} -$$$$78\!\cdots\!80$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$)
$61$ ($$1 + 18482 T + 844596301 T^{2}$$)($$1 + 741098540 T^{2} + 1478044222094100534 T^{4} +$$$$52\!\cdots\!40$$$$T^{6} +$$$$50\!\cdots\!01$$$$T^{8}$$)
$67$ ($$1 + 15532 T + 1350125107 T^{2}$$)($$1 - 1366835860 T^{2} + 3274116308996825526 T^{4} -$$$$24\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$)
$71$ ($$1 + 31960 T + 1804229351 T^{2}$$)($$( 1 - 103344 T + 6217736974 T^{2} - 186456278049744 T^{3} + 3255243551009881201 T^{4} )^{2}$$)
$73$ ($$1 + 4886 T + 2073071593 T^{2}$$)($$( 1 - 19988 T + 2541602870 T^{2} - 41436555000884 T^{3} + 4297625829703557649 T^{4} )^{2}$$)
$79$ ($$1 - 44560 T + 3077056399 T^{2}$$)($$( 1 + 123936 T + 9855929374 T^{2} + 381358061866464 T^{3} + 9468276082626847201 T^{4} )^{2}$$)
$83$ ($$1 - 67364 T + 3939040643 T^{2}$$)($$1 - 10047855188 T^{2} + 55381071937674414326 T^{4} -$$$$15\!\cdots\!12$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$)
$89$ ($$1 - 71994 T + 5584059449 T^{2}$$)($$( 1 + 42316 T + 4292401174 T^{2} + 236295059643884 T^{3} + 31181719929966183601 T^{4} )^{2}$$)
$97$ ($$1 - 48866 T + 8587340257 T^{2}$$)($$( 1 + 49788 T + 16391371462 T^{2} + 427546496715516 T^{3} + 73742412689492826049 T^{4} )^{2}$$)
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