# Properties

 Label 80.6.c Level 80 Weight 6 Character orbit c Rep. character $$\chi_{80}(49,\cdot)$$ Character field $$\Q$$ Dimension 14 Newform subspaces 4 Sturm bound 72 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$72$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(80, [\chi])$$.

Total New Old
Modular forms 66 16 50
Cusp forms 54 14 40
Eisenstein series 12 2 10

## Trace form

 $$14q + 18q^{5} - 974q^{9} + O(q^{10})$$ $$14q + 18q^{5} - 974q^{9} + 728q^{11} + 1192q^{15} - 984q^{19} + 1304q^{21} - 586q^{25} + 8260q^{29} - 9344q^{31} - 12712q^{35} + 10544q^{39} + 3580q^{41} - 10578q^{45} - 31222q^{49} + 73088q^{51} + 22952q^{55} - 69896q^{59} + 21748q^{61} + 4016q^{65} - 21096q^{69} - 47792q^{71} - 85584q^{75} + 82912q^{79} + 141254q^{81} - 97920q^{85} + 110188q^{89} + 105680q^{91} - 195752q^{95} - 152024q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(80, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
80.6.c.a $$2$$ $$12.831$$ $$\Q(\sqrt{-11})$$ None $$0$$ $$0$$ $$-90$$ $$0$$ $$q-3\beta q^{3}+(-45-5\beta )q^{5}-9\beta q^{7}+\cdots$$
80.6.c.b $$2$$ $$12.831$$ $$\Q(\sqrt{-31})$$ None $$0$$ $$0$$ $$-10$$ $$0$$ $$q-\beta q^{3}+(-5+5\beta )q^{5}-11\beta q^{7}+119q^{9}+\cdots$$
80.6.c.c $$2$$ $$12.831$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$110$$ $$0$$ $$q+7iq^{3}+(55-5i)q^{5}-79iq^{7}+47q^{9}+\cdots$$
80.6.c.d $$8$$ $$12.831$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\beta _{1}q^{3}+(1-\beta _{2})q^{5}+(-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(80, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(80, [\chi]) \cong$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$( 1 - 24 T + 243 T^{2} )( 1 + 24 T + 243 T^{2} )$$)($$1 - 362 T^{2} + 59049 T^{4}$$)($$1 - 290 T^{2} + 59049 T^{4}$$)($$1 - 472 T^{2} + 69724 T^{4} - 20119464 T^{6} + 8439593958 T^{8} - 1188034229736 T^{10} + 243112555575324 T^{12} - 97180614348674328 T^{14} + 12157665459056928801 T^{16}$$)
$5$ ($$1 + 90 T + 3125 T^{2}$$)($$1 + 10 T + 3125 T^{2}$$)($$1 - 110 T + 3125 T^{2}$$)($$1 - 8 T + 1100 T^{2} + 113000 T^{3} - 438250 T^{4} + 353125000 T^{5} + 10742187500 T^{6} - 244140625000 T^{7} + 95367431640625 T^{8}$$)
$7$ ($$1 - 30050 T^{2} + 282475249 T^{4}$$)($$1 - 18610 T^{2} + 282475249 T^{4}$$)($$1 - 8650 T^{2} + 282475249 T^{4}$$)($$1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} -$$$$10\!\cdots\!84$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{12} -$$$$10\!\cdots\!72$$$$T^{14} +$$$$63\!\cdots\!01$$$$T^{16}$$)
$11$ ($$( 1 + 252 T + 161051 T^{2} )^{2}$$)($$( 1 - 100 T + 161051 T^{2} )^{2}$$)($$( 1 - 148 T + 161051 T^{2} )^{2}$$)($$( 1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 8239191359696 T^{5} + 4570277964394604 T^{6} - 1537227326344959568 T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$)
$13$ ($$1 - 728330 T^{2} + 137858491849 T^{4}$$)($$1 - 202442 T^{2} + 137858491849 T^{4}$$)($$1 - 274730 T^{2} + 137858491849 T^{4}$$)($$1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} +$$$$15\!\cdots\!86$$$$T^{8} -$$$$60\!\cdots\!16$$$$T^{10} +$$$$20\!\cdots\!56$$$$T^{12} -$$$$41\!\cdots\!28$$$$T^{14} +$$$$36\!\cdots\!01$$$$T^{16}$$)
$17$ ($$1 - 2363810 T^{2} + 2015993900449 T^{4}$$)($$1 - 1879458 T^{2} + 2015993900449 T^{4}$$)($$1 + 1354590 T^{2} + 2015993900449 T^{4}$$)($$1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} +$$$$45\!\cdots\!30$$$$T^{8} -$$$$63\!\cdots\!32$$$$T^{10} +$$$$71\!\cdots\!44$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{14} +$$$$16\!\cdots\!01$$$$T^{16}$$)
$19$ ($$( 1 - 220 T + 2476099 T^{2} )^{2}$$)($$( 1 + 2244 T + 2476099 T^{2} )^{2}$$)($$( 1 - 2220 T + 2476099 T^{2} )^{2}$$)($$( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 15000545402668432 T^{5} + 32546127598645797196 T^{6} +$$$$10\!\cdots\!12$$$$T^{7} +$$$$37\!\cdots\!01$$$$T^{8} )^{2}$$)
$23$ ($$1 - 6946370 T^{2} + 41426511213649 T^{4}$$)($$1 - 1185810 T^{2} + 41426511213649 T^{4}$$)($$1 - 11320170 T^{2} + 41426511213649 T^{4}$$)($$1 - 34675896 T^{2} + 569897415616828 T^{4} -$$$$59\!\cdots\!20$$$$T^{6} +$$$$44\!\cdots\!82$$$$T^{8} -$$$$24\!\cdots\!80$$$$T^{10} +$$$$97\!\cdots\!28$$$$T^{12} -$$$$24\!\cdots\!04$$$$T^{14} +$$$$29\!\cdots\!01$$$$T^{16}$$)
$29$ ($$( 1 + 6930 T + 20511149 T^{2} )^{2}$$)($$( 1 - 7854 T + 20511149 T^{2} )^{2}$$)($$( 1 - 270 T + 20511149 T^{2} )^{2}$$)($$( 1 - 2936 T + 58625996 T^{2} - 77951973928 T^{3} + 1466411094282230 T^{4} - 1598884552081323272 T^{5} +$$$$24\!\cdots\!96$$$$T^{6} -$$$$25\!\cdots\!64$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8} )^{2}$$)
$31$ ($$( 1 + 6752 T + 28629151 T^{2} )^{2}$$)($$( 1 - 2144 T + 28629151 T^{2} )^{2}$$)($$( 1 - 2048 T + 28629151 T^{2} )^{2}$$)($$( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} +$$$$65\!\cdots\!04$$$$T^{6} +$$$$49\!\cdots\!12$$$$T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$)
$37$ ($$1 + 56462470 T^{2} + 4808584372417849 T^{4}$$)($$1 - 30515770 T^{2} + 4808584372417849 T^{4}$$)($$1 - 119573530 T^{2} + 4808584372417849 T^{4}$$)($$1 - 251774632 T^{2} + 38631208311838780 T^{4} -$$$$41\!\cdots\!52$$$$T^{6} +$$$$32\!\cdots\!34$$$$T^{8} -$$$$19\!\cdots\!48$$$$T^{10} +$$$$89\!\cdots\!80$$$$T^{12} -$$$$27\!\cdots\!68$$$$T^{14} +$$$$53\!\cdots\!01$$$$T^{16}$$)
$41$ ($$( 1 + 198 T + 115856201 T^{2} )^{2}$$)($$( 1 + 7414 T + 115856201 T^{2} )^{2}$$)($$( 1 + 2398 T + 115856201 T^{2} )^{2}$$)($$( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} -$$$$34\!\cdots\!76$$$$T^{5} +$$$$45\!\cdots\!40$$$$T^{6} -$$$$18\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{2}$$)
$43$ ($$1 - 293842250 T^{2} + 21611482313284249 T^{4}$$)($$1 + 21442214 T^{2} + 21611482313284249 T^{4}$$)($$1 - 288754450 T^{2} + 21611482313284249 T^{4}$$)($$1 - 283211672 T^{2} + 48021567531024796 T^{4} -$$$$54\!\cdots\!84$$$$T^{6} +$$$$43\!\cdots\!06$$$$T^{8} -$$$$11\!\cdots\!16$$$$T^{10} +$$$$22\!\cdots\!96$$$$T^{12} -$$$$28\!\cdots\!28$$$$T^{14} +$$$$21\!\cdots\!01$$$$T^{16}$$)
$47$ ($$1 - 347593490 T^{2} + 52599132235830049 T^{4}$$)($$1 - 369731298 T^{2} + 52599132235830049 T^{4}$$)($$1 - 344584890 T^{2} + 52599132235830049 T^{4}$$)($$1 - 963352312 T^{2} + 473832864723586300 T^{4} -$$$$15\!\cdots\!72$$$$T^{6} +$$$$40\!\cdots\!54$$$$T^{8} -$$$$83\!\cdots\!28$$$$T^{10} +$$$$13\!\cdots\!00$$$$T^{12} -$$$$14\!\cdots\!88$$$$T^{14} +$$$$76\!\cdots\!01$$$$T^{16}$$)
$53$ ($$1 - 802472090 T^{2} + 174887470365513049 T^{4}$$)($$1 - 248174170 T^{2} + 174887470365513049 T^{4}$$)($$1 - 827605690 T^{2} + 174887470365513049 T^{4}$$)($$1 - 1183385640 T^{2} + 874785099161623996 T^{4} -$$$$49\!\cdots\!80$$$$T^{6} +$$$$23\!\cdots\!06$$$$T^{8} -$$$$86\!\cdots\!20$$$$T^{10} +$$$$26\!\cdots\!96$$$$T^{12} -$$$$63\!\cdots\!60$$$$T^{14} +$$$$93\!\cdots\!01$$$$T^{16}$$)
$59$ ($$( 1 - 24660 T + 714924299 T^{2} )^{2}$$)($$( 1 - 25972 T + 714924299 T^{2} )^{2}$$)($$( 1 + 39740 T + 714924299 T^{2} )^{2}$$)($$( 1 + 45840 T + 3064286732 T^{2} + 94721285480976 T^{3} + 3348109683185502486 T^{4} +$$$$67\!\cdots\!24$$$$T^{5} +$$$$15\!\cdots\!32$$$$T^{6} +$$$$16\!\cdots\!60$$$$T^{7} +$$$$26\!\cdots\!01$$$$T^{8} )^{2}$$)
$61$ ($$( 1 + 5698 T + 844596301 T^{2} )^{2}$$)($$( 1 + 3058 T + 844596301 T^{2} )^{2}$$)($$( 1 + 42298 T + 844596301 T^{2} )^{2}$$)($$( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} -$$$$12\!\cdots\!84$$$$T^{5} +$$$$27\!\cdots\!64$$$$T^{6} -$$$$37\!\cdots\!28$$$$T^{7} +$$$$50\!\cdots\!01$$$$T^{8} )^{2}$$)
$67$ ($$1 - 795787610 T^{2} + 1822837804551761449 T^{4}$$)($$1 + 755362070 T^{2} + 1822837804551761449 T^{4}$$)($$1 - 1669968610 T^{2} + 1822837804551761449 T^{4}$$)($$1 - 9281919064 T^{2} + 39492482666681482588 T^{4} -$$$$10\!\cdots\!40$$$$T^{6} +$$$$16\!\cdots\!62$$$$T^{8} -$$$$18\!\cdots\!60$$$$T^{10} +$$$$13\!\cdots\!88$$$$T^{12} -$$$$56\!\cdots\!36$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16}$$)
$71$ ($$( 1 + 53352 T + 1804229351 T^{2} )^{2}$$)($$( 1 + 37608 T + 1804229351 T^{2} )^{2}$$)($$( 1 - 4248 T + 1804229351 T^{2} )^{2}$$)($$( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} -$$$$33\!\cdots\!96$$$$T^{5} +$$$$11\!\cdots\!56$$$$T^{6} -$$$$36\!\cdots\!16$$$$T^{7} +$$$$10\!\cdots\!01$$$$T^{8} )^{2}$$)
$73$ ($$1 + 883886830 T^{2} + 4297625829703557649 T^{4}$$)($$1 - 3569749522 T^{2} + 4297625829703557649 T^{4}$$)($$1 - 3239892370 T^{2} + 4297625829703557649 T^{4}$$)($$1 - 9140679496 T^{2} + 46078306824990298588 T^{4} -$$$$15\!\cdots\!60$$$$T^{6} +$$$$37\!\cdots\!02$$$$T^{8} -$$$$66\!\cdots\!40$$$$T^{10} +$$$$85\!\cdots\!88$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$34\!\cdots\!01$$$$T^{16}$$)
$79$ ($$( 1 + 51920 T + 3077056399 T^{2} )^{2}$$)($$( 1 - 79728 T + 3077056399 T^{2} )^{2}$$)($$( 1 - 35280 T + 3077056399 T^{2} )^{2}$$)($$( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} +$$$$10\!\cdots\!12$$$$T^{5} +$$$$67\!\cdots\!04$$$$T^{6} +$$$$63\!\cdots\!68$$$$T^{7} +$$$$89\!\cdots\!01$$$$T^{8} )^{2}$$)
$83$ ($$1 - 4053674810 T^{2} + 15516041187205853449 T^{4}$$)($$1 - 7612675530 T^{2} + 15516041187205853449 T^{4}$$)($$1 - 7103795010 T^{2} + 15516041187205853449 T^{4}$$)($$1 - 6759897816 T^{2} + 40943120759345365468 T^{4} -$$$$86\!\cdots\!80$$$$T^{6} +$$$$39\!\cdots\!42$$$$T^{8} -$$$$13\!\cdots\!20$$$$T^{10} +$$$$98\!\cdots\!68$$$$T^{12} -$$$$25\!\cdots\!84$$$$T^{14} +$$$$57\!\cdots\!01$$$$T^{16}$$)
$89$ ($$( 1 + 9990 T + 5584059449 T^{2} )^{2}$$)($$( 1 - 826 T + 5584059449 T^{2} )^{2}$$)($$( 1 - 85210 T + 5584059449 T^{2} )^{2}$$)($$( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} +$$$$11\!\cdots\!30$$$$T^{4} +$$$$27\!\cdots\!08$$$$T^{5} +$$$$50\!\cdots\!96$$$$T^{6} +$$$$36\!\cdots\!48$$$$T^{7} +$$$$97\!\cdots\!01$$$$T^{8} )^{2}$$)
$97$ ($$1 - 6923133890 T^{2} + 73742412689492826049 T^{4}$$)($$1 - 15761405890 T^{2} + 73742412689492826049 T^{4}$$)($$1 - 7720618690 T^{2} + 73742412689492826049 T^{4}$$)($$1 - 45263915272 T^{2} +$$$$10\!\cdots\!40$$$$T^{4} -$$$$14\!\cdots\!12$$$$T^{6} +$$$$15\!\cdots\!94$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$55\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$29\!\cdots\!01$$$$T^{16}$$)