# Properties

 Label 160.6.c Level 160 Weight 6 Character orbit c Rep. character $$\chi_{160}(129,\cdot)$$ Character field $$\Q$$ Dimension 30 Newform subspaces 4 Sturm bound 144 Trace bound 9

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$144$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(160, [\chi])$$.

Total New Old
Modular forms 128 30 98
Cusp forms 112 30 82
Eisenstein series 16 0 16

## Trace form

 $$30q - 38q^{5} - 2430q^{9} + O(q^{10})$$ $$30q - 38q^{5} - 2430q^{9} - 1640q^{21} - 1946q^{25} - 16524q^{29} - 2212q^{41} - 20634q^{45} - 48006q^{49} - 91580q^{61} + 29744q^{65} - 264232q^{69} + 83702q^{81} + 186272q^{85} - 143828q^{89} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(160, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
160.6.c.a $$2$$ $$25.661$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$82$$ $$0$$ $$q+(41+19i)q^{5}+3^{5}q^{9}-122iq^{13}+\cdots$$
160.6.c.b $$4$$ $$25.661$$ $$\Q(i, \sqrt{5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(5\beta _{1}+\beta _{2})q^{3}+5\beta _{3}q^{5}+(46\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
160.6.c.c $$12$$ $$25.661$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$-60$$ $$0$$ $$q+\beta _{4}q^{3}+(-5+\beta _{5}+\beta _{8})q^{5}-\beta _{9}q^{7}+\cdots$$
160.6.c.d $$12$$ $$25.661$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$-60$$ $$0$$ $$q+\beta _{2}q^{3}+(-5+\beta _{5})q^{5}+(-5\beta _{2}-3\beta _{3}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(160, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(160, [\chi]) \cong$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$( 1 - 243 T^{2} )^{2}$$)($$( 1 - 38 T + 722 T^{2} - 9234 T^{3} + 59049 T^{4} )( 1 + 38 T + 722 T^{2} + 9234 T^{3} + 59049 T^{4} )$$)($$( 1 - 310 T^{2} + 120679 T^{4} - 26699028 T^{6} + 7125974271 T^{8} - 1080903164310 T^{10} + 205891132094649 T^{12} )^{2}$$)($$( 1 - 662 T^{2} + 270567 T^{4} - 78508116 T^{6} + 15976710783 T^{8} - 2308251273462 T^{10} + 205891132094649 T^{12} )^{2}$$)
$5$ ($$1 - 82 T + 3125 T^{2}$$)($$( 1 - 3125 T^{2} )^{2}$$)($$( 1 + 30 T + 75 T^{2} - 123500 T^{3} + 234375 T^{4} + 292968750 T^{5} + 30517578125 T^{6} )^{2}$$)($$( 1 + 30 T + 4555 T^{2} + 273300 T^{3} + 14234375 T^{4} + 292968750 T^{5} + 30517578125 T^{6} )^{2}$$)
$7$ ($$( 1 - 16807 T^{2} )^{2}$$)($$( 1 - 366 T + 66978 T^{2} - 6151362 T^{3} + 282475249 T^{4} )( 1 + 366 T + 66978 T^{2} + 6151362 T^{3} + 282475249 T^{4} )$$)($$( 1 - 42334 T^{2} + 748334111 T^{4} - 10497411086308 T^{6} + 211385864339918639 T^{8} -$$$$33\!\cdots\!34$$$$T^{10} +$$$$22\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 54910 T^{2} + 1199559327 T^{4} - 18786317442980 T^{6} + 338845819584597423 T^{8} -$$$$43\!\cdots\!10$$$$T^{10} +$$$$22\!\cdots\!49$$$$T^{12} )^{2}$$)
$11$ ($$( 1 + 161051 T^{2} )^{2}$$)($$( 1 + 161051 T^{2} )^{4}$$)($$( 1 + 220194 T^{2} + 46554422967 T^{4} + 4986677846505148 T^{6} +$$$$12\!\cdots\!67$$$$T^{8} +$$$$14\!\cdots\!94$$$$T^{10} +$$$$17\!\cdots\!01$$$$T^{12} )^{2}$$)($$( 1 + 311650 T^{2} + 88422679991 T^{4} + 16030422126529084 T^{6} +$$$$22\!\cdots\!91$$$$T^{8} +$$$$20\!\cdots\!50$$$$T^{10} +$$$$17\!\cdots\!01$$$$T^{12} )^{2}$$)
$13$ ($$( 1 - 1194 T + 371293 T^{2} )( 1 + 1194 T + 371293 T^{2} )$$)($$( 1 - 371293 T^{2} )^{4}$$)($$( 1 - 921822 T^{2} + 378030377607 T^{4} - 125995996576493956 T^{6} +$$$$52\!\cdots\!43$$$$T^{8} -$$$$17\!\cdots\!22$$$$T^{10} +$$$$26\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 183262 T^{2} + 260173900423 T^{4} - 19831011157232516 T^{6} +$$$$35\!\cdots\!27$$$$T^{8} -$$$$34\!\cdots\!62$$$$T^{10} +$$$$26\!\cdots\!49$$$$T^{12} )^{2}$$)
$17$ ($$( 1 - 2242 T + 1419857 T^{2} )( 1 + 2242 T + 1419857 T^{2} )$$)($$( 1 - 1419857 T^{2} )^{4}$$)($$( 1 - 7508390 T^{2} + 24616526523247 T^{4} - 45351621021996027220 T^{6} +$$$$49\!\cdots\!03$$$$T^{8} -$$$$30\!\cdots\!90$$$$T^{10} +$$$$81\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 + 1156442 T^{2} + 502148271983 T^{4} - 1980542974409409364 T^{6} +$$$$10\!\cdots\!67$$$$T^{8} +$$$$47\!\cdots\!42$$$$T^{10} +$$$$81\!\cdots\!49$$$$T^{12} )^{2}$$)
$19$ ($$( 1 + 2476099 T^{2} )^{2}$$)($$( 1 + 2476099 T^{2} )^{4}$$)($$( 1 + 8862994 T^{2} + 40251921150215 T^{4} +$$$$12\!\cdots\!80$$$$T^{6} +$$$$24\!\cdots\!15$$$$T^{8} +$$$$33\!\cdots\!94$$$$T^{10} +$$$$23\!\cdots\!01$$$$T^{12} )^{2}$$)($$( 1 + 3143250 T^{2} + 17341223849991 T^{4} + 36134402435803900316 T^{6} +$$$$10\!\cdots\!91$$$$T^{8} +$$$$11\!\cdots\!50$$$$T^{10} +$$$$23\!\cdots\!01$$$$T^{12} )^{2}$$)
$23$ ($$( 1 - 6436343 T^{2} )^{2}$$)($$( 1 - 4838 T + 11703122 T^{2} - 31139027434 T^{3} + 41426511213649 T^{4} )( 1 + 4838 T + 11703122 T^{2} + 31139027434 T^{3} + 41426511213649 T^{4} )$$)($$( 1 - 27906430 T^{2} + 367961637958719 T^{4} -$$$$29\!\cdots\!08$$$$T^{6} +$$$$15\!\cdots\!31$$$$T^{8} -$$$$47\!\cdots\!30$$$$T^{10} +$$$$71\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 2071710 T^{2} + 29683977204927 T^{4} -$$$$44\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!23$$$$T^{8} -$$$$35\!\cdots\!10$$$$T^{10} +$$$$71\!\cdots\!49$$$$T^{12} )^{2}$$)
$29$ ($$( 1 - 2950 T + 20511149 T^{2} )^{2}$$)($$( 1 - 1686 T + 20511149 T^{2} )^{4}$$)($$( 1 + 5326 T + 46243827 T^{2} + 133362268948 T^{3} + 948514025927223 T^{4} + 2240686724556870526 T^{5} +$$$$86\!\cdots\!49$$$$T^{6} )^{4}$$)($$( 1 + 1966 T + 38511987 T^{2} + 51339089236 T^{3} + 789925103643063 T^{4} + 827110420668195166 T^{5} +$$$$86\!\cdots\!49$$$$T^{6} )^{4}$$)
$31$ ($$( 1 + 28629151 T^{2} )^{2}$$)($$( 1 + 28629151 T^{2} )^{4}$$)($$( 1 + 10594234 T^{2} - 153174449706673 T^{4} +$$$$19\!\cdots\!88$$$$T^{6} -$$$$12\!\cdots\!73$$$$T^{8} +$$$$71\!\cdots\!34$$$$T^{10} +$$$$55\!\cdots\!01$$$$T^{12} )^{2}$$)($$( 1 + 47574970 T^{2} + 2163796989249871 T^{4} +$$$$78\!\cdots\!04$$$$T^{6} +$$$$17\!\cdots\!71$$$$T^{8} +$$$$31\!\cdots\!70$$$$T^{10} +$$$$55\!\cdots\!01$$$$T^{12} )^{2}$$)
$37$ ($$( 1 - 12242 T + 69343957 T^{2} )( 1 + 12242 T + 69343957 T^{2} )$$)($$( 1 - 69343957 T^{2} )^{4}$$)($$( 1 - 234974254 T^{2} + 32080895329334807 T^{4} -$$$$26\!\cdots\!92$$$$T^{6} +$$$$15\!\cdots\!43$$$$T^{8} -$$$$54\!\cdots\!54$$$$T^{10} +$$$$11\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 301201198 T^{2} + 42883739211630103 T^{4} -$$$$37\!\cdots\!24$$$$T^{6} +$$$$20\!\cdots\!47$$$$T^{8} -$$$$69\!\cdots\!98$$$$T^{10} +$$$$11\!\cdots\!49$$$$T^{12} )^{2}$$)
$41$ ($$( 1 - 20950 T + 115856201 T^{2} )^{2}$$)($$( 1 - 211028098 T^{2} + 13422659310152401 T^{4} )^{2}$$)($$( 1 + 27418 T + 531273543 T^{2} + 6402679071436 T^{3} + 61551334383790143 T^{4} +$$$$36\!\cdots\!18$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6} )^{4}$$)($$( 1 - 16390 T + 318388423 T^{2} - 3510238606580 T^{3} + 36887273131161023 T^{4} -$$$$21\!\cdots\!90$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6} )^{4}$$)
$43$ ($$( 1 - 147008443 T^{2} )^{2}$$)($$( 1 - 11862 T + 70353522 T^{2} - 1743814150866 T^{3} + 21611482313284249 T^{4} )( 1 + 11862 T + 70353522 T^{2} + 1743814150866 T^{3} + 21611482313284249 T^{4} )$$)($$( 1 - 613900486 T^{2} + 182904754783784951 T^{4} -$$$$33\!\cdots\!72$$$$T^{6} +$$$$39\!\cdots\!99$$$$T^{8} -$$$$28\!\cdots\!86$$$$T^{10} +$$$$10\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 640358182 T^{2} + 187647756282389367 T^{4} -$$$$33\!\cdots\!76$$$$T^{6} +$$$$40\!\cdots\!83$$$$T^{8} -$$$$29\!\cdots\!82$$$$T^{10} +$$$$10\!\cdots\!49$$$$T^{12} )^{2}$$)
$47$ ($$( 1 - 229345007 T^{2} )^{2}$$)($$( 1 - 33334 T + 555577778 T^{2} - 7644986463338 T^{3} + 52599132235830049 T^{4} )( 1 + 33334 T + 555577778 T^{2} + 7644986463338 T^{3} + 52599132235830049 T^{4} )$$)($$( 1 - 579920494 T^{2} + 165136673035562991 T^{4} -$$$$38\!\cdots\!48$$$$T^{6} +$$$$86\!\cdots\!59$$$$T^{8} -$$$$16\!\cdots\!94$$$$T^{10} +$$$$14\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 764642190 T^{2} + 310900354010297967 T^{4} -$$$$85\!\cdots\!20$$$$T^{6} +$$$$16\!\cdots\!83$$$$T^{8} -$$$$21\!\cdots\!90$$$$T^{10} +$$$$14\!\cdots\!49$$$$T^{12} )^{2}$$)
$53$ ($$( 1 - 7294 T + 418195493 T^{2} )( 1 + 7294 T + 418195493 T^{2} )$$)($$( 1 - 418195493 T^{2} )^{4}$$)($$( 1 - 2026325390 T^{2} + 1866514545072595447 T^{4} -$$$$99\!\cdots\!20$$$$T^{6} +$$$$32\!\cdots\!03$$$$T^{8} -$$$$61\!\cdots\!90$$$$T^{10} +$$$$53\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 1611835534 T^{2} + 1232324270520828407 T^{4} -$$$$61\!\cdots\!84$$$$T^{6} +$$$$21\!\cdots\!43$$$$T^{8} -$$$$49\!\cdots\!34$$$$T^{10} +$$$$53\!\cdots\!49$$$$T^{12} )^{2}$$)
$59$ ($$( 1 + 714924299 T^{2} )^{2}$$)($$( 1 + 714924299 T^{2} )^{4}$$)($$( 1 + 55950786 T^{2} + 253972763776609047 T^{4} -$$$$45\!\cdots\!68$$$$T^{6} +$$$$12\!\cdots\!47$$$$T^{8} +$$$$14\!\cdots\!86$$$$T^{10} +$$$$13\!\cdots\!01$$$$T^{12} )^{2}$$)($$( 1 + 2333761794 T^{2} + 3044701823751957015 T^{4} +$$$$26\!\cdots\!80$$$$T^{6} +$$$$15\!\cdots\!15$$$$T^{8} +$$$$60\!\cdots\!94$$$$T^{10} +$$$$13\!\cdots\!01$$$$T^{12} )^{2}$$)
$61$ ($$( 1 + 18950 T + 844596301 T^{2} )^{2}$$)($$( 1 - 1041591898 T^{2} + 713342911662882601 T^{4} )^{2}$$)($$( 1 - 16138 T + 744681843 T^{2} - 29097452426876 T^{3} + 628955530019662743 T^{4} -$$$$11\!\cdots\!38$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6} )^{4}$$)($$( 1 + 29558 T + 2350852083 T^{2} + 48603849831556 T^{3} + 1985520973499944983 T^{4} +$$$$21\!\cdots\!58$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6} )^{4}$$)
$67$ ($$( 1 - 1350125107 T^{2} )^{2}$$)($$( 1 - 100434 T + 5043494178 T^{2} - 135598464996438 T^{3} + 1822837804551761449 T^{4} )( 1 + 100434 T + 5043494178 T^{2} + 135598464996438 T^{3} + 1822837804551761449 T^{4} )$$)($$( 1 - 2173336374 T^{2} + 6882868324909910631 T^{4} -$$$$82\!\cdots\!68$$$$T^{6} +$$$$12\!\cdots\!19$$$$T^{8} -$$$$72\!\cdots\!74$$$$T^{10} +$$$$60\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 3741380758 T^{2} + 9572756841213379047 T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$17\!\cdots\!03$$$$T^{8} -$$$$12\!\cdots\!58$$$$T^{10} +$$$$60\!\cdots\!49$$$$T^{12} )^{2}$$)
$71$ ($$( 1 + 1804229351 T^{2} )^{2}$$)($$( 1 + 1804229351 T^{2} )^{4}$$)($$( 1 + 942628714 T^{2} + 4622069102042239647 T^{4} +$$$$48\!\cdots\!68$$$$T^{6} +$$$$15\!\cdots\!47$$$$T^{8} +$$$$99\!\cdots\!14$$$$T^{10} +$$$$34\!\cdots\!01$$$$T^{12} )^{2}$$)($$( 1 + 9793576042 T^{2} + 41626938509482688159 T^{4} +$$$$98\!\cdots\!36$$$$T^{6} +$$$$13\!\cdots\!59$$$$T^{8} +$$$$10\!\cdots\!42$$$$T^{10} +$$$$34\!\cdots\!01$$$$T^{12} )^{2}$$)
$73$ ($$( 1 - 88806 T + 2073071593 T^{2} )( 1 + 88806 T + 2073071593 T^{2} )$$)($$( 1 - 2073071593 T^{2} )^{4}$$)($$( 1 - 4736672630 T^{2} + 15959781190221205247 T^{4} -$$$$37\!\cdots\!40$$$$T^{6} +$$$$68\!\cdots\!03$$$$T^{8} -$$$$87\!\cdots\!30$$$$T^{10} +$$$$79\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 5477783158 T^{2} + 15527894807038343935 T^{4} -$$$$34\!\cdots\!40$$$$T^{6} +$$$$66\!\cdots\!15$$$$T^{8} -$$$$10\!\cdots\!58$$$$T^{10} +$$$$79\!\cdots\!49$$$$T^{12} )^{2}$$)
$79$ ($$( 1 + 3077056399 T^{2} )^{2}$$)($$( 1 + 3077056399 T^{2} )^{4}$$)($$( 1 + 454164826 T^{2} + 18487132928554429487 T^{4} -$$$$50\!\cdots\!28$$$$T^{6} +$$$$17\!\cdots\!87$$$$T^{8} +$$$$40\!\cdots\!26$$$$T^{10} +$$$$84\!\cdots\!01$$$$T^{12} )^{2}$$)($$( 1 + 8854539610 T^{2} + 48426464203018517551 T^{4} +$$$$17\!\cdots\!76$$$$T^{6} +$$$$45\!\cdots\!51$$$$T^{8} +$$$$79\!\cdots\!10$$$$T^{10} +$$$$84\!\cdots\!01$$$$T^{12} )^{2}$$)
$83$ ($$( 1 - 3939040643 T^{2} )^{2}$$)($$( 1 - 163262 T + 13327240322 T^{2} - 643095653457466 T^{3} + 15516041187205853449 T^{4} )( 1 + 163262 T + 13327240322 T^{2} + 643095653457466 T^{3} + 15516041187205853449 T^{4} )$$)($$( 1 - 11599602390 T^{2} + 72197091204656028039 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$11\!\cdots\!11$$$$T^{8} -$$$$27\!\cdots\!90$$$$T^{10} +$$$$37\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 17886738486 T^{2} +$$$$14\!\cdots\!27$$$$T^{4} -$$$$73\!\cdots\!08$$$$T^{6} +$$$$22\!\cdots\!23$$$$T^{8} -$$$$43\!\cdots\!86$$$$T^{10} +$$$$37\!\cdots\!49$$$$T^{12} )^{2}$$)
$89$ ($$( 1 - 51050 T + 5584059449 T^{2} )^{2}$$)($$( 1 + 149286 T + 5584059449 T^{2} )^{4}$$)($$( 1 + 930 T + 9446079447 T^{2} + 231471892494140 T^{3} + 52747469192025044703 T^{4} +$$$$28\!\cdots\!30$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6} )^{4}$$)($$( 1 - 88734 T + 11682221271 T^{2} - 581994460142148 T^{3} + 65234218073636339679 T^{4} -$$$$27\!\cdots\!34$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6} )^{4}$$)
$97$ ($$( 1 - 92142 T + 8587340257 T^{2} )( 1 + 92142 T + 8587340257 T^{2} )$$)($$( 1 - 8587340257 T^{2} )^{4}$$)($$( 1 - 35609466630 T^{2} +$$$$64\!\cdots\!47$$$$T^{4} -$$$$68\!\cdots\!40$$$$T^{6} +$$$$47\!\cdots\!03$$$$T^{8} -$$$$19\!\cdots\!30$$$$T^{10} +$$$$40\!\cdots\!49$$$$T^{12} )^{2}$$)($$( 1 - 22620660742 T^{2} +$$$$25\!\cdots\!35$$$$T^{4} -$$$$22\!\cdots\!60$$$$T^{6} +$$$$18\!\cdots\!15$$$$T^{8} -$$$$12\!\cdots\!42$$$$T^{10} +$$$$40\!\cdots\!49$$$$T^{12} )^{2}$$)