# Properties

 Label 8.14.b Level 8 Weight 14 Character orbit b Rep. character $$\chi_{8}(5,\cdot)$$ Character field $$\Q$$ Dimension 12 Newform subspaces 2 Sturm bound 14 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$14$$ Character orbit: $$[\chi]$$ = 8.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$8$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$14$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{14}(8, [\chi])$$.

Total New Old
Modular forms 14 14 0
Cusp forms 12 12 0
Eisenstein series 2 2 0

## Trace form

 $$12q - 2q^{2} - 8556q^{4} + 58444q^{6} + 235296q^{7} + 1076872q^{8} - 5314412q^{9} + O(q^{10})$$ $$12q - 2q^{2} - 8556q^{4} + 58444q^{6} + 235296q^{7} + 1076872q^{8} - 5314412q^{9} - 7752648q^{10} - 8536664q^{12} + 21101872q^{14} - 42373856q^{15} - 62847216q^{16} - 49714600q^{17} + 262778834q^{18} + 381237904q^{20} + 196775484q^{22} - 149850784q^{23} + 1739438480q^{24} - 1654349124q^{25} - 685452248q^{26} - 978617568q^{28} + 539961392q^{30} - 10881769344q^{31} - 8956909792q^{32} + 3397961328q^{33} + 8857451100q^{34} + 18075363660q^{36} - 23710846420q^{38} + 79057948000q^{39} - 6505554528q^{40} + 12231942520q^{41} - 3443304224q^{42} - 28647791928q^{44} - 51061300848q^{46} - 146392008000q^{47} - 161587694304q^{48} + 50764315692q^{49} + 246819442102q^{50} + 441779820720q^{52} - 639572716168q^{54} - 4023386208q^{55} - 699100837568q^{56} - 351867765712q^{57} + 992195908104q^{58} + 1614284565792q^{60} - 1019628353344q^{62} + 389587709920q^{63} - 1464077986752q^{64} + 173779028800q^{65} + 2946347707608q^{66} + 2850998874984q^{68} - 5505591528768q^{70} - 1892582134752q^{71} - 8538615449032q^{72} + 324055378296q^{73} + 9399657781240q^{74} + 10892359157736q^{76} - 16235130104944q^{78} + 4716009102144q^{79} - 16787669526720q^{80} + 410958020828q^{81} + 14910525285612q^{82} + 26784826987072q^{84} - 24718490913284q^{86} + 352970975712q^{87} - 26046521626416q^{88} + 5301429525304q^{89} + 43218339424520q^{90} + 34108342295904q^{92} - 37727942061792q^{94} - 15785048234464q^{95} - 62733552180160q^{96} - 10954301425896q^{97} + 61714328751438q^{98} + O(q^{100})$$

## Decomposition of $$S_{14}^{\mathrm{new}}(8, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
8.14.b.a $$2$$ $$8.578$$ $$\Q(\sqrt{-79})$$ None $$-112$$ $$0$$ $$0$$ $$-351664$$ $$q+(-56-4\beta )q^{2}+129\beta q^{3}+(-1920+\cdots)q^{4}+\cdots$$
8.14.b.b $$10$$ $$8.578$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$110$$ $$0$$ $$0$$ $$586960$$ $$q+(11+\beta _{1})q^{2}+(3\beta _{1}+\beta _{3})q^{3}+(-472+\cdots)q^{4}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 112 T + 8192 T^{2}$$)($$1 - 110 T + 8408 T^{2} - 390976 T^{3} - 935936 T^{4} + 3612475392 T^{5} - 7667187712 T^{6} - 26237955211264 T^{7} + 4622346883170304 T^{8} - 495395959010754560 T^{9} + 36893488147419103232 T^{10}$$)
$3$ ($$1 + 2069910 T^{2} + 2541865828329 T^{4}$$)($$1 - 8978642 T^{2} + 41923886773365 T^{4} -$$$$13\!\cdots\!32$$$$T^{6} +$$$$31\!\cdots\!22$$$$T^{8} -$$$$56\!\cdots\!92$$$$T^{10} +$$$$78\!\cdots\!38$$$$T^{12} -$$$$85\!\cdots\!12$$$$T^{14} +$$$$68\!\cdots\!85$$$$T^{16} -$$$$37\!\cdots\!02$$$$T^{18} +$$$$10\!\cdots\!49$$$$T^{20}$$)
$5$ ($$1 - 1931729850 T^{2} + 1490116119384765625 T^{4}$$)($$1 - 4565314338 T^{2} + 10147469641592691461 T^{4} -$$$$14\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!50$$$$T^{8} -$$$$17\!\cdots\!00$$$$T^{10} +$$$$23\!\cdots\!50$$$$T^{12} -$$$$32\!\cdots\!00$$$$T^{14} +$$$$33\!\cdots\!25$$$$T^{16} -$$$$22\!\cdots\!50$$$$T^{18} +$$$$73\!\cdots\!25$$$$T^{20}$$)
$7$ ($$( 1 + 175832 T + 96889010407 T^{2} )^{2}$$)($$( 1 - 293480 T + 239610643203 T^{2} - 95417056996483936 T^{3} +$$$$32\!\cdots\!74$$$$T^{4} -$$$$13\!\cdots\!28$$$$T^{5} +$$$$31\!\cdots\!18$$$$T^{6} -$$$$89\!\cdots\!64$$$$T^{7} +$$$$21\!\cdots\!29$$$$T^{8} -$$$$25\!\cdots\!80$$$$T^{9} +$$$$85\!\cdots\!07$$$$T^{10} )^{2}$$)
$11$ ($$1 - 62100824999962 T^{2} +$$$$11\!\cdots\!61$$$$T^{4}$$)($$1 - 179340783809858 T^{2} +$$$$16\!\cdots\!41$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{6} +$$$$50\!\cdots\!78$$$$T^{8} -$$$$19\!\cdots\!64$$$$T^{10} +$$$$60\!\cdots\!58$$$$T^{12} -$$$$14\!\cdots\!00$$$$T^{14} +$$$$27\!\cdots\!21$$$$T^{16} -$$$$36\!\cdots\!78$$$$T^{18} +$$$$24\!\cdots\!01$$$$T^{20}$$)
$13$ ($$1 + 360392330876150 T^{2} +$$$$91\!\cdots\!09$$$$T^{4}$$)($$1 - 1881700477395890 T^{2} +$$$$17\!\cdots\!13$$$$T^{4} -$$$$10\!\cdots\!00$$$$T^{6} +$$$$48\!\cdots\!02$$$$T^{8} -$$$$16\!\cdots\!20$$$$T^{10} +$$$$44\!\cdots\!18$$$$T^{12} -$$$$91\!\cdots\!00$$$$T^{14} +$$$$13\!\cdots\!77$$$$T^{16} -$$$$13\!\cdots\!90$$$$T^{18} +$$$$64\!\cdots\!49$$$$T^{20}$$)
$17$ ($$( 1 + 133520302 T + 9904578032905937 T^{2} )^{2}$$)($$( 1 - 108663002 T + 32276400633367229 T^{2} -$$$$23\!\cdots\!84$$$$T^{3} +$$$$50\!\cdots\!42$$$$T^{4} -$$$$29\!\cdots\!76$$$$T^{5} +$$$$49\!\cdots\!54$$$$T^{6} -$$$$23\!\cdots\!96$$$$T^{7} +$$$$31\!\cdots\!37$$$$T^{8} -$$$$10\!\cdots\!22$$$$T^{9} +$$$$95\!\cdots\!57$$$$T^{10} )^{2}$$)
$19$ ($$1 - 82896428399326282 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$)($$1 - 162780863064290354 T^{2} +$$$$13\!\cdots\!97$$$$T^{4} -$$$$80\!\cdots\!36$$$$T^{6} +$$$$39\!\cdots\!66$$$$T^{8} -$$$$17\!\cdots\!68$$$$T^{10} +$$$$69\!\cdots\!46$$$$T^{12} -$$$$25\!\cdots\!96$$$$T^{14} +$$$$75\!\cdots\!77$$$$T^{16} -$$$$15\!\cdots\!34$$$$T^{18} +$$$$17\!\cdots\!01$$$$T^{20}$$)
$23$ ($$( 1 + 35585416 T + 504036361936467383 T^{2} )^{2}$$)($$( 1 + 39339976 T + 1242751119340586771 T^{2} +$$$$42\!\cdots\!92$$$$T^{3} +$$$$76\!\cdots\!02$$$$T^{4} +$$$$38\!\cdots\!08$$$$T^{5} +$$$$38\!\cdots\!66$$$$T^{6} +$$$$10\!\cdots\!88$$$$T^{7} +$$$$15\!\cdots\!77$$$$T^{8} +$$$$25\!\cdots\!96$$$$T^{9} +$$$$32\!\cdots\!43$$$$T^{10} )^{2}$$)
$29$ ($$1 - 18002148940349036842 T^{2} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 - 60047671171855484498 T^{2} +$$$$18\!\cdots\!41$$$$T^{4} -$$$$38\!\cdots\!60$$$$T^{6} +$$$$58\!\cdots\!78$$$$T^{8} -$$$$67\!\cdots\!84$$$$T^{10} +$$$$61\!\cdots\!38$$$$T^{12} -$$$$42\!\cdots\!60$$$$T^{14} +$$$$21\!\cdots\!01$$$$T^{16} -$$$$73\!\cdots\!38$$$$T^{18} +$$$$12\!\cdots\!01$$$$T^{20}$$)
$31$ ($$( 1 + 5765001568 T + 24417546297445042591 T^{2} )^{2}$$)($$( 1 - 324116896 T + 94759416869201467419 T^{2} -$$$$76\!\cdots\!28$$$$T^{3} +$$$$40\!\cdots\!58$$$$T^{4} -$$$$32\!\cdots\!48$$$$T^{5} +$$$$98\!\cdots\!78$$$$T^{6} -$$$$45\!\cdots\!68$$$$T^{7} +$$$$13\!\cdots\!49$$$$T^{8} -$$$$11\!\cdots\!56$$$$T^{9} +$$$$86\!\cdots\!51$$$$T^{10} )^{2}$$)
$37$ ($$1 -$$$$31\!\cdots\!70$$$$T^{2} +$$$$59\!\cdots\!09$$$$T^{4}$$)($$1 -$$$$85\!\cdots\!14$$$$T^{2} +$$$$35\!\cdots\!29$$$$T^{4} -$$$$83\!\cdots\!24$$$$T^{6} +$$$$11\!\cdots\!78$$$$T^{8} -$$$$14\!\cdots\!64$$$$T^{10} +$$$$67\!\cdots\!02$$$$T^{12} -$$$$29\!\cdots\!44$$$$T^{14} +$$$$73\!\cdots\!41$$$$T^{16} -$$$$10\!\cdots\!54$$$$T^{18} +$$$$73\!\cdots\!49$$$$T^{20}$$)
$41$ ($$( 1 + 23546348918 T +$$$$92\!\cdots\!21$$$$T^{2} )^{2}$$)($$( 1 - 29662320178 T +$$$$14\!\cdots\!89$$$$T^{2} -$$$$15\!\cdots\!16$$$$T^{3} +$$$$10\!\cdots\!62$$$$T^{4} -$$$$29\!\cdots\!76$$$$T^{5} +$$$$97\!\cdots\!02$$$$T^{6} -$$$$13\!\cdots\!56$$$$T^{7} +$$$$11\!\cdots\!29$$$$T^{8} -$$$$21\!\cdots\!18$$$$T^{9} +$$$$67\!\cdots\!01$$$$T^{10} )^{2}$$)
$43$ ($$1 -$$$$32\!\cdots\!30$$$$T^{2} +$$$$29\!\cdots\!49$$$$T^{4}$$)($$1 -$$$$61\!\cdots\!14$$$$T^{2} +$$$$15\!\cdots\!05$$$$T^{4} -$$$$25\!\cdots\!04$$$$T^{6} +$$$$52\!\cdots\!62$$$$T^{8} -$$$$10\!\cdots\!24$$$$T^{10} +$$$$15\!\cdots\!38$$$$T^{12} -$$$$22\!\cdots\!04$$$$T^{14} +$$$$39\!\cdots\!45$$$$T^{16} -$$$$46\!\cdots\!14$$$$T^{18} +$$$$22\!\cdots\!49$$$$T^{20}$$)
$47$ ($$( 1 + 68107736592 T +$$$$54\!\cdots\!27$$$$T^{2} )^{2}$$)($$( 1 + 5088267408 T +$$$$10\!\cdots\!43$$$$T^{2} +$$$$16\!\cdots\!28$$$$T^{3} +$$$$79\!\cdots\!70$$$$T^{4} +$$$$90\!\cdots\!04$$$$T^{5} +$$$$43\!\cdots\!90$$$$T^{6} +$$$$48\!\cdots\!12$$$$T^{7} +$$$$17\!\cdots\!69$$$$T^{8} +$$$$45\!\cdots\!28$$$$T^{9} +$$$$48\!\cdots\!07$$$$T^{10} )^{2}$$)
$53$ ($$1 -$$$$24\!\cdots\!30$$$$T^{2} +$$$$67\!\cdots\!29$$$$T^{4}$$)($$1 -$$$$15\!\cdots\!98$$$$T^{2} +$$$$12\!\cdots\!49$$$$T^{4} -$$$$63\!\cdots\!52$$$$T^{6} +$$$$24\!\cdots\!06$$$$T^{8} -$$$$72\!\cdots\!20$$$$T^{10} +$$$$16\!\cdots\!74$$$$T^{12} -$$$$29\!\cdots\!32$$$$T^{14} +$$$$37\!\cdots\!61$$$$T^{16} -$$$$32\!\cdots\!38$$$$T^{18} +$$$$14\!\cdots\!49$$$$T^{20}$$)
$59$ ($$1 -$$$$19\!\cdots\!22$$$$T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$)($$1 -$$$$54\!\cdots\!82$$$$T^{2} +$$$$16\!\cdots\!77$$$$T^{4} -$$$$32\!\cdots\!52$$$$T^{6} +$$$$49\!\cdots\!86$$$$T^{8} -$$$$58\!\cdots\!80$$$$T^{10} +$$$$54\!\cdots\!26$$$$T^{12} -$$$$39\!\cdots\!12$$$$T^{14} +$$$$21\!\cdots\!17$$$$T^{16} -$$$$80\!\cdots\!02$$$$T^{18} +$$$$16\!\cdots\!01$$$$T^{20}$$)
$61$ ($$1 -$$$$14\!\cdots\!62$$$$T^{2} +$$$$26\!\cdots\!61$$$$T^{4}$$)($$1 -$$$$10\!\cdots\!62$$$$T^{2} +$$$$47\!\cdots\!37$$$$T^{4} -$$$$13\!\cdots\!12$$$$T^{6} +$$$$29\!\cdots\!66$$$$T^{8} -$$$$51\!\cdots\!60$$$$T^{10} +$$$$77\!\cdots\!26$$$$T^{12} -$$$$96\!\cdots\!52$$$$T^{14} +$$$$86\!\cdots\!97$$$$T^{16} -$$$$48\!\cdots\!42$$$$T^{18} +$$$$12\!\cdots\!01$$$$T^{20}$$)
$67$ ($$1 -$$$$95\!\cdots\!30$$$$T^{2} +$$$$30\!\cdots\!69$$$$T^{4}$$)($$1 -$$$$19\!\cdots\!42$$$$T^{2} +$$$$23\!\cdots\!49$$$$T^{4} -$$$$18\!\cdots\!88$$$$T^{6} +$$$$12\!\cdots\!86$$$$T^{8} -$$$$71\!\cdots\!60$$$$T^{10} +$$$$37\!\cdots\!34$$$$T^{12} -$$$$17\!\cdots\!68$$$$T^{14} +$$$$64\!\cdots\!41$$$$T^{16} -$$$$16\!\cdots\!82$$$$T^{18} +$$$$24\!\cdots\!49$$$$T^{20}$$)
$71$ ($$( 1 + 1309471657368 T +$$$$11\!\cdots\!11$$$$T^{2} )^{2}$$)($$( 1 - 363180589992 T +$$$$34\!\cdots\!95$$$$T^{2} -$$$$14\!\cdots\!16$$$$T^{3} +$$$$61\!\cdots\!94$$$$T^{4} -$$$$22\!\cdots\!64$$$$T^{5} +$$$$71\!\cdots\!34$$$$T^{6} -$$$$20\!\cdots\!36$$$$T^{7} +$$$$53\!\cdots\!45$$$$T^{8} -$$$$66\!\cdots\!72$$$$T^{9} +$$$$21\!\cdots\!51$$$$T^{10} )^{2}$$)
$73$ ($$( 1 - 478647871914 T +$$$$16\!\cdots\!33$$$$T^{2} )^{2}$$)($$( 1 + 316620182766 T +$$$$64\!\cdots\!53$$$$T^{2} +$$$$24\!\cdots\!84$$$$T^{3} +$$$$18\!\cdots\!54$$$$T^{4} +$$$$60\!\cdots\!52$$$$T^{5} +$$$$31\!\cdots\!82$$$$T^{6} +$$$$67\!\cdots\!76$$$$T^{7} +$$$$29\!\cdots\!61$$$$T^{8} +$$$$24\!\cdots\!86$$$$T^{9} +$$$$13\!\cdots\!93$$$$T^{10} )^{2}$$)
$79$ ($$( 1 + 364547231600 T +$$$$46\!\cdots\!39$$$$T^{2} )^{2}$$)($$( 1 - 2722551782672 T +$$$$12\!\cdots\!19$$$$T^{2} -$$$$22\!\cdots\!64$$$$T^{3} +$$$$55\!\cdots\!14$$$$T^{4} -$$$$94\!\cdots\!68$$$$T^{5} +$$$$25\!\cdots\!46$$$$T^{6} -$$$$49\!\cdots\!44$$$$T^{7} +$$$$12\!\cdots\!61$$$$T^{8} -$$$$12\!\cdots\!52$$$$T^{9} +$$$$22\!\cdots\!99$$$$T^{10} )^{2}$$)
$83$ ($$1 -$$$$16\!\cdots\!70$$$$T^{2} +$$$$78\!\cdots\!69$$$$T^{4}$$)($$1 -$$$$43\!\cdots\!74$$$$T^{2} +$$$$87\!\cdots\!65$$$$T^{4} -$$$$12\!\cdots\!64$$$$T^{6} +$$$$14\!\cdots\!62$$$$T^{8} -$$$$14\!\cdots\!44$$$$T^{10} +$$$$11\!\cdots\!78$$$$T^{12} -$$$$76\!\cdots\!04$$$$T^{14} +$$$$42\!\cdots\!85$$$$T^{16} -$$$$16\!\cdots\!54$$$$T^{18} +$$$$30\!\cdots\!49$$$$T^{20}$$)
$89$ ($$( 1 + 102457641350 T +$$$$21\!\cdots\!69$$$$T^{2} )^{2}$$)($$( 1 - 2753172404002 T +$$$$62\!\cdots\!29$$$$T^{2} -$$$$14\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!54$$$$T^{4} -$$$$47\!\cdots\!48$$$$T^{5} +$$$$47\!\cdots\!26$$$$T^{6} -$$$$70\!\cdots\!24$$$$T^{7} +$$$$66\!\cdots\!61$$$$T^{8} -$$$$64\!\cdots\!42$$$$T^{9} +$$$$51\!\cdots\!49$$$$T^{10} )^{2}$$)
$97$ ($$( 1 + 6157717373342 T +$$$$67\!\cdots\!77$$$$T^{2} )^{2}$$)($$( 1 - 680566660394 T +$$$$17\!\cdots\!53$$$$T^{2} +$$$$11\!\cdots\!08$$$$T^{3} +$$$$15\!\cdots\!70$$$$T^{4} +$$$$20\!\cdots\!28$$$$T^{5} +$$$$10\!\cdots\!90$$$$T^{6} +$$$$53\!\cdots\!32$$$$T^{7} +$$$$54\!\cdots\!49$$$$T^{8} -$$$$13\!\cdots\!54$$$$T^{9} +$$$$13\!\cdots\!57$$$$T^{10} )^{2}$$)