# Properties

 Label 8.7.d Level 8 Weight 7 Character orbit d Rep. character $$\chi_{8}(3,\cdot)$$ Character field $$\Q$$ Dimension 5 Newform subspaces 2 Sturm bound 7 Trace bound 1

# Related objects

## Defining parameters

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

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(8, [\chi])$$.

Total New Old
Modular forms 7 7 0
Cusp forms 5 5 0
Eisenstein series 2 2 0

## Trace form

 $$5q - 6q^{2} - 2q^{3} + 20q^{4} + 28q^{6} - 264q^{8} + 727q^{9} + O(q^{10})$$ $$5q - 6q^{2} - 2q^{3} + 20q^{4} + 28q^{6} - 264q^{8} + 727q^{9} - 1920q^{10} - 1362q^{11} + 4312q^{12} + 5760q^{14} - 10480q^{16} + 2442q^{17} - 21506q^{18} - 3938q^{19} + 31680q^{20} + 43132q^{22} - 61808q^{24} - 8275q^{25} - 59520q^{26} + 32860q^{27} + 59520q^{28} + 90240q^{30} - 81696q^{32} - 23500q^{33} - 68108q^{34} - 49920q^{35} + 75868q^{36} + 46428q^{38} - 13440q^{40} + 16698q^{41} + 38400q^{42} + 122542q^{43} - 112488q^{44} - 213120q^{46} + 326368q^{48} + 119765q^{49} + 232650q^{50} - 465412q^{51} - 254400q^{52} - 495368q^{54} + 349440q^{56} + 12500q^{57} + 516480q^{58} + 846990q^{59} - 716160q^{60} - 407040q^{62} + 726080q^{64} - 205440q^{65} + 717608q^{66} - 1386818q^{67} - 324312q^{68} - 360960q^{70} + 95656q^{72} - 149222q^{73} - 32640q^{74} + 2483950q^{75} - 88232q^{76} + 324480q^{78} - 1032960q^{80} - 186839q^{81} - 672428q^{82} - 2786082q^{83} + 602880q^{84} + 1588860q^{86} - 753392q^{88} + 403962q^{89} - 1296000q^{90} + 3398400q^{91} + 2743680q^{92} + 971520q^{94} - 1760192q^{96} + 895978q^{97} - 3332454q^{98} - 5702086q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(8, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
8.7.d.a $$1$$ $$1.840$$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$-8$$ $$46$$ $$0$$ $$0$$ $$q-8q^{2}+46q^{3}+2^{6}q^{4}-368q^{6}+\cdots$$
8.7.d.b $$4$$ $$1.840$$ 4.0.3803625.2 None $$2$$ $$-48$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-13+2\beta _{1}+\beta _{2})q^{3}+(-12+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 8 T$$)($$1 - 2 T + 24 T^{2} - 128 T^{3} + 4096 T^{4}$$)
$3$ ($$1 - 46 T + 729 T^{2}$$)($$( 1 + 24 T + 1182 T^{2} + 17496 T^{3} + 531441 T^{4} )^{2}$$)
$5$ ($$( 1 - 125 T )( 1 + 125 T )$$)($$1 - 19300 T^{2} + 256155750 T^{4} - 4711914062500 T^{6} + 59604644775390625 T^{8}$$)
$7$ ($$( 1 - 343 T )( 1 + 343 T )$$)($$1 - 236356 T^{2} + 28021959366 T^{4} - 3271471277679556 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$)
$11$ ($$1 + 2338 T + 1771561 T^{2}$$)($$( 1 - 488 T + 2238078 T^{2} - 864521768 T^{3} + 3138428376721 T^{4} )^{2}$$)
$13$ ($$( 1 - 2197 T )( 1 + 2197 T )$$)($$1 - 4994596 T^{2} + 30260873415846 T^{4} -$$$$11\!\cdots\!76$$$$T^{6} +$$$$54\!\cdots\!61$$$$T^{8}$$)
$17$ ($$1 + 1726 T + 24137569 T^{2}$$)($$( 1 - 2084 T + 32560902 T^{2} - 50302693796 T^{3} + 582622237229761 T^{4} )^{2}$$)
$19$ ($$1 + 2482 T + 47045881 T^{2}$$)($$( 1 + 728 T + 92449758 T^{2} + 34249401368 T^{3} + 2213314919066161 T^{4} )^{2}$$)
$23$ ($$( 1 - 12167 T )( 1 + 12167 T )$$)($$1 - 212117956 T^{2} + 23026671552237126 T^{4} -$$$$46\!\cdots\!76$$$$T^{6} +$$$$48\!\cdots\!41$$$$T^{8}$$)
$29$ ($$( 1 - 24389 T )( 1 + 24389 T )$$)($$1 - 1409719204 T^{2} + 1160515330165289766 T^{4} -$$$$49\!\cdots\!64$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$)
$31$ ($$( 1 - 29791 T )( 1 + 29791 T )$$)($$1 - 2758360324 T^{2} + 3362667870952277766 T^{4} -$$$$21\!\cdots\!64$$$$T^{6} +$$$$62\!\cdots\!21$$$$T^{8}$$)
$37$ ($$( 1 - 50653 T )( 1 + 50653 T )$$)($$1 - 8121202276 T^{2} + 29600495645847907686 T^{4} -$$$$53\!\cdots\!56$$$$T^{6} +$$$$43\!\cdots\!61$$$$T^{8}$$)
$41$ ($$1 - 134642 T + 4750104241 T^{2}$$)($$( 1 + 58972 T + 9857729958 T^{2} + 280123147300252 T^{3} + 22563490300366186081 T^{4} )^{2}$$)
$43$ ($$1 + 74914 T + 6321363049 T^{2}$$)($$( 1 - 98728 T + 13190101374 T^{2} - 624095531101672 T^{3} + 39959630797262576401 T^{4} )^{2}$$)
$47$ ($$( 1 - 103823 T )( 1 + 103823 T )$$)($$1 - 13578767236 T^{2} +$$$$16\!\cdots\!86$$$$T^{4} -$$$$15\!\cdots\!76$$$$T^{6} +$$$$13\!\cdots\!81$$$$T^{8}$$)
$53$ ($$( 1 - 148877 T )( 1 + 148877 T )$$)($$1 - 42194545636 T^{2} +$$$$11\!\cdots\!86$$$$T^{4} -$$$$20\!\cdots\!76$$$$T^{6} +$$$$24\!\cdots\!81$$$$T^{8}$$)
$59$ ($$1 - 304958 T + 42180533641 T^{2}$$)($$( 1 - 271016 T + 102678575166 T^{2} - 11431599505249256 T^{3} +$$$$17\!\cdots\!81$$$$T^{4} )^{2}$$)
$61$ ($$( 1 - 226981 T )( 1 + 226981 T )$$)($$1 - 160868902564 T^{2} +$$$$11\!\cdots\!46$$$$T^{4} -$$$$42\!\cdots\!44$$$$T^{6} +$$$$70\!\cdots\!41$$$$T^{8}$$)
$67$ ($$1 + 596626 T + 90458382169 T^{2}$$)($$( 1 + 395096 T + 204987839262 T^{2} + 35739744961443224 T^{3} +$$$$81\!\cdots\!61$$$$T^{4} )^{2}$$)
$71$ ($$( 1 - 357911 T )( 1 + 357911 T )$$)($$1 - 164364621124 T^{2} +$$$$21\!\cdots\!06$$$$T^{4} -$$$$26\!\cdots\!84$$$$T^{6} +$$$$26\!\cdots\!81$$$$T^{8}$$)
$73$ ($$1 + 593134 T + 151334226289 T^{2}$$)($$( 1 - 221956 T + 284381984742 T^{2} - 33589539530201284 T^{3} +$$$$22\!\cdots\!21$$$$T^{4} )^{2}$$)
$79$ ($$( 1 - 493039 T )( 1 + 493039 T )$$)($$1 - 828257339524 T^{2} +$$$$28\!\cdots\!06$$$$T^{4} -$$$$48\!\cdots\!84$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$)
$83$ ($$1 - 678926 T + 326940373369 T^{2}$$)($$( 1 + 1732504 T + 1400265667422 T^{2} + 566425504623285976 T^{3} +$$$$10\!\cdots\!61$$$$T^{4} )^{2}$$)
$89$ ($$1 + 357262 T + 496981290961 T^{2}$$)($$( 1 - 380612 T + 970422538278 T^{2} - 189157043115248132 T^{3} +$$$$24\!\cdots\!21$$$$T^{4} )^{2}$$)
$97$ ($$1 - 1822754 T + 832972004929 T^{2}$$)($$( 1 + 463388 T + 953987784774 T^{2} + 385989231420039452 T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$)