# Properties

 Label 1089.3.c.m Level $1089$ Weight $3$ Character orbit 1089.c Analytic conductor $29.673$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1089.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.6731007888$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + 1868 x^{7} - 1619 x^{6} - 16804 x^{5} + 32427 x^{4} + 43316 x^{3} - 71672 x^{2} + 83521$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 11^{4}$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( -1 + \beta_{2} - \beta_{3} ) q^{4} + ( \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{4} + \beta_{7} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{7} + ( \beta_{4} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} + ( -1 + \beta_{2} - \beta_{3} ) q^{4} + ( \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{4} + \beta_{7} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{7} + ( \beta_{4} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{8} + ( \beta_{4} - 2 \beta_{8} - \beta_{11} + \beta_{15} ) q^{10} + ( \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{13} + \beta_{14} ) q^{13} + ( 3 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{10} ) q^{14} + ( -1 + \beta_{1} - 4 \beta_{2} + \beta_{5} - 2 \beta_{9} + \beta_{10} + \beta_{12} ) q^{16} + ( -\beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} - 2 \beta_{14} ) q^{17} + ( \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{11} - 2 \beta_{13} ) q^{19} + ( 8 + \beta_{1} - 9 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{9} + \beta_{10} + \beta_{12} ) q^{20} + ( -7 + \beta_{1} - 5 \beta_{2} + 2 \beta_{5} - \beta_{9} + \beta_{10} ) q^{23} + ( 5 - 4 \beta_{2} - 3 \beta_{5} + \beta_{9} - 3 \beta_{10} + \beta_{12} ) q^{25} + ( -1 - 9 \beta_{2} + 3 \beta_{3} + 2 \beta_{10} - \beta_{12} ) q^{26} + ( 9 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{28} + ( 4 \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{11} - \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{29} + ( 6 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + \beta_{5} - 5 \beta_{9} - \beta_{10} + \beta_{12} ) q^{31} + ( 5 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{32} + ( -22 - \beta_{1} + 9 \beta_{2} - 5 \beta_{3} - \beta_{5} - \beta_{10} + \beta_{12} ) q^{34} + ( 6 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{15} ) q^{35} + ( -2 - 3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} + \beta_{5} + 3 \beta_{10} - \beta_{12} ) q^{37} + ( -14 + \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 5 \beta_{9} - \beta_{10} + \beta_{12} ) q^{38} + ( 2 \beta_{4} + 11 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{11} + 5 \beta_{13} + \beta_{14} + \beta_{15} ) q^{40} + ( 3 \beta_{4} + 5 \beta_{6} + \beta_{7} + 5 \beta_{8} - 2 \beta_{14} - 3 \beta_{15} ) q^{41} + ( 3 \beta_{4} + 3 \beta_{6} + 4 \beta_{8} - 2 \beta_{11} + 3 \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{43} + ( -\beta_{4} - 7 \beta_{6} - 5 \beta_{7} + 8 \beta_{8} + 5 \beta_{11} + 4 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} ) q^{46} + ( -4 + 3 \beta_{1} + 6 \beta_{2} + \beta_{3} - \beta_{9} - 3 \beta_{10} - \beta_{12} ) q^{47} + ( -10 - \beta_{1} - 6 \beta_{2} - 2 \beta_{5} - 3 \beta_{9} - 4 \beta_{10} - 2 \beta_{12} ) q^{49} + ( -3 \beta_{4} + 2 \beta_{6} + 11 \beta_{7} - \beta_{11} - 8 \beta_{13} + 5 \beta_{14} + 9 \beta_{15} ) q^{50} + ( 4 \beta_{4} + 12 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{11} + 3 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{52} + ( 10 + 4 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} - 6 \beta_{5} - \beta_{9} - 6 \beta_{10} - 2 \beta_{12} ) q^{53} + ( -33 + 2 \beta_{1} + 15 \beta_{2} - 6 \beta_{3} - 4 \beta_{5} + \beta_{9} - 2 \beta_{10} - 3 \beta_{12} ) q^{56} + ( 9 + 2 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} + 5 \beta_{5} + 5 \beta_{9} + 5 \beta_{10} + 3 \beta_{12} ) q^{58} + ( 17 - 3 \beta_{1} - \beta_{2} - 11 \beta_{3} + 6 \beta_{5} - 2 \beta_{9} - \beta_{10} ) q^{59} + ( -3 \beta_{4} + 15 \beta_{6} + 5 \beta_{7} - 4 \beta_{8} - 5 \beta_{11} + 4 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{61} + ( 5 \beta_{4} + 9 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 7 \beta_{11} + 6 \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{62} + ( -3 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 6 \beta_{5} + 4 \beta_{9} + 2 \beta_{10} ) q^{64} + ( 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 3 \beta_{11} + 3 \beta_{13} + 2 \beta_{14} + 5 \beta_{15} ) q^{65} + ( 8 + 4 \beta_{1} - 11 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 10 \beta_{9} - 2 \beta_{10} ) q^{67} + ( 2 \beta_{4} - 22 \beta_{6} + 8 \beta_{7} - 8 \beta_{8} - 3 \beta_{11} - 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{68} + ( -7 - 2 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} + 4 \beta_{5} + 8 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{70} + ( -43 - 2 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} + 2 \beta_{5} + 4 \beta_{9} + \beta_{12} ) q^{71} + ( 9 \beta_{4} + 16 \beta_{6} - 3 \beta_{7} - 6 \beta_{8} + 11 \beta_{13} + 3 \beta_{14} - 7 \beta_{15} ) q^{73} + ( 12 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} - 13 \beta_{11} + 3 \beta_{13} - 7 \beta_{15} ) q^{74} + ( 11 \beta_{4} + \beta_{6} + 2 \beta_{7} + 8 \beta_{8} - 3 \beta_{13} + 7 \beta_{14} ) q^{76} + ( 10 \beta_{4} - \beta_{6} + 7 \beta_{7} - 6 \beta_{8} - 6 \beta_{11} - 2 \beta_{13} + 7 \beta_{15} ) q^{79} + ( -13 + \beta_{1} - 2 \beta_{2} - 18 \beta_{3} + 5 \beta_{5} + 11 \beta_{9} + \beta_{10} ) q^{80} + ( -34 + 35 \beta_{2} + 5 \beta_{3} - 12 \beta_{5} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} ) q^{82} + ( -19 \beta_{4} + 2 \beta_{6} - 9 \beta_{7} - 10 \beta_{8} + 15 \beta_{13} + 3 \beta_{14} - 4 \beta_{15} ) q^{83} + ( -3 \beta_{4} - 5 \beta_{6} - 8 \beta_{8} - 6 \beta_{11} + 7 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{85} + ( -5 + 2 \beta_{1} + 35 \beta_{2} - 3 \beta_{3} + 6 \beta_{5} + 16 \beta_{9} + 4 \beta_{10} + \beta_{12} ) q^{86} + ( -5 - \beta_{1} - 17 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} - \beta_{9} - \beta_{10} + 6 \beta_{12} ) q^{89} + ( -36 + 5 \beta_{1} - \beta_{2} - 5 \beta_{3} - 3 \beta_{5} - 6 \beta_{9} - 3 \beta_{10} - 3 \beta_{12} ) q^{91} + ( -12 - \beta_{1} + 26 \beta_{2} - 3 \beta_{3} - 8 \beta_{5} + 5 \beta_{9} - 5 \beta_{10} - 3 \beta_{12} ) q^{92} + ( -7 \beta_{4} - 7 \beta_{6} - 2 \beta_{8} + 8 \beta_{11} - 3 \beta_{13} - 5 \beta_{14} + 6 \beta_{15} ) q^{94} + ( 17 \beta_{4} + 11 \beta_{6} - 7 \beta_{7} + 5 \beta_{8} + \beta_{11} - 7 \beta_{13} + 6 \beta_{15} ) q^{95} + ( 17 + 2 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - \beta_{5} + \beta_{9} - 7 \beta_{10} + \beta_{12} ) q^{97} + ( 8 \beta_{4} + 2 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} - 9 \beta_{11} - 2 \beta_{13} + 3 \beta_{14} + 10 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 20q^{4} + 4q^{5} + O(q^{10})$$ $$16q - 20q^{4} + 4q^{5} + 52q^{14} - 44q^{16} + 108q^{20} - 132q^{23} + 88q^{25} + 4q^{26} + 40q^{31} - 368q^{34} - 16q^{37} - 280q^{38} - 80q^{47} - 140q^{49} + 128q^{53} - 524q^{56} + 140q^{58} + 220q^{59} - 8q^{64} + 36q^{67} - 100q^{70} - 644q^{71} - 264q^{80} - 476q^{82} - 76q^{86} - 76q^{89} - 624q^{91} - 120q^{92} + 216q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + 1868 x^{7} - 1619 x^{6} - 16804 x^{5} + 32427 x^{4} + 43316 x^{3} - 71672 x^{2} + 83521$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5118046671123271072 \nu^{15} - 489076068461122451283 \nu^{14} + 603665761668368640859 \nu^{13} - 3142889137307303128175 \nu^{12} + 5505899135438262627052 \nu^{11} - 48123502702325648748962 \nu^{10} - 49085688765268181850758 \nu^{9} + 49506865108677586484433 \nu^{8} - 232289772416918157487813 \nu^{7} + 173417624055188422449022 \nu^{6} - 2916466161575039349931577 \nu^{5} - 1285772555995251420573024 \nu^{4} + 2784696249939055343599243 \nu^{3} - 7881075579364323988884836 \nu^{2} + 5765640005295629289418429 \nu - 43153595576226309686495647$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{2}$$ $$=$$ $$($$$$-1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + 37431650235165 \nu^{12} - 160608059534630 \nu^{11} + 396329540870948 \nu^{10} + 970794484128155 \nu^{9} - 2042076941900150 \nu^{8} + 1143522707363375 \nu^{7} - 3000401551565675 \nu^{6} + 26580224433121053 \nu^{5} + 18238786120480205 \nu^{4} - 74422428601993325 \nu^{3} + 43546075769649555 \nu^{2} + 84006679985420530 \nu + 87548232770773285$$$$)/ 78081122146535134$$ $$\beta_{3}$$ $$=$$ $$($$$$110772223292616559581 \nu^{15} - 170195189005621991522 \nu^{14} + 11340540325899380678 \nu^{13} - 1501040587522108900439 \nu^{12} + 7130001391847941303754 \nu^{11} + 11075544227701304479400 \nu^{10} - 70486258246526842691755 \nu^{9} - 98929721650791598640652 \nu^{8} - 61124185334789030582257 \nu^{7} + 359689551904902004493183 \nu^{6} + 188739032677142301868697 \nu^{5} - 2899284905688159414026383 \nu^{4} - 3454853169668206970039381 \nu^{3} + 2725367713713220402774851 \nu^{2} + 6239620310303427315170932 \nu - 3016502091276299502163853$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{4}$$ $$=$$ $$($$$$-62663748280134159380 \nu^{15} + 94556952146973365752 \nu^{14} - 421946922746206711963 \nu^{13} + 502957093333845754237 \nu^{12} - 5204447702405357388148 \nu^{11} - 6413049280226468225533 \nu^{10} + 13969796006764590417448 \nu^{9} - 61964156582906808486062 \nu^{8} + 32204090145478587048213 \nu^{7} - 161638787120145941052387 \nu^{6} - 111270235594930389931339 \nu^{5} + 451761595671053956556583 \nu^{4} - 2380391092219957518630737 \nu^{3} + 487204295145717053117397 \nu^{2} + 1768617021879347636164951 \nu - 3852111289316697053463740$$$$)/$$$$20\!\cdots\!97$$ $$\beta_{5}$$ $$=$$ $$($$$$-91103608012569946247 \nu^{15} + 97800578201340651463 \nu^{14} - 896288267467192974544 \nu^{13} + 1990070191856487308145 \nu^{12} - 10550450155024607104739 \nu^{11} - 5443134041278837144176 \nu^{10} - 26097423577090155234685 \nu^{9} - 35961847172466452881418 \nu^{8} + 189815808746797129337721 \nu^{7} - 728437164666172783463975 \nu^{6} - 318924716822197154520370 \nu^{5} - 1445132485956622528706109 \nu^{4} - 331867058370836509261413 \nu^{3} + 5683566660645371578911573 \nu^{2} - 11854270036204790508484676 \nu - 10358907379526901455792970$$$$)/$$$$20\!\cdots\!97$$ $$\beta_{6}$$ $$=$$ $$($$$$-220022054371202978027 \nu^{15} + 693153953445206321738 \nu^{14} - 1459197633946223459032 \nu^{13} + 2137121814851796241745 \nu^{12} - 17779780045102206213782 \nu^{11} - 1301619909458847211258 \nu^{10} + 138539668212486981227629 \nu^{9} - 331179575316578516127764 \nu^{8} + 104237825940233572715185 \nu^{7} - 257224713787740556175171 \nu^{6} + 621551985943438654559695 \nu^{5} + 3152774566201858360094701 \nu^{4} - 11617108568865974295515255 \nu^{3} + 8942645006085142477615377 \nu^{2} + 16005136245018598052513174 \nu - 20796741235259030368136243$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{7}$$ $$=$$ $$($$$$-24369863836783695 \nu^{15} + 96595086067747539 \nu^{14} - 137757484292541855 \nu^{13} + 342271797869305646 \nu^{12} - 2352734303043296426 \nu^{11} + 1877030649542568202 \nu^{10} + 18481461281240971491 \nu^{9} - 25835636527821345653 \nu^{8} - 18495311978420991834 \nu^{7} - 52622614087140895119 \nu^{6} + 173666200235241080852 \nu^{5} + 419562897878523918533 \nu^{4} - 1021878055471572811098 \nu^{3} + 8569288085647399849 \nu^{2} + 1220427399955036859005 \nu - 771347781839850762230$$$$)/$$$$37\!\cdots\!66$$ $$\beta_{8}$$ $$=$$ $$($$$$331214275066397396578 \nu^{15} - 2327854046290114537485 \nu^{14} + 5581753805286988008981 \nu^{13} - 11083341203195379407635 \nu^{12} + 46462638240783443711556 \nu^{11} - 126557445128392548474094 \nu^{10} - 187815706484451573137606 \nu^{9} + 1001168229918578084609439 \nu^{8} - 759106120403616486898307 \nu^{7} + 69150836603796213280496 \nu^{6} - 4805720479241895464721969 \nu^{5} + 875488142057279707187468 \nu^{4} + 33332366741511498366923325 \nu^{3} - 36354372847836567070595786 \nu^{2} - 31106880041888624278187303 \nu + 62448816337649207684063087$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{9}$$ $$=$$ $$($$$$-346846663817408475336 \nu^{15} + 396271293857112362775 \nu^{14} - 241460383366094726593 \nu^{13} + 3007124239661137238919 \nu^{12} - 23327421685018967669212 \nu^{11} - 47181706777028885775924 \nu^{10} + 180615363363583496231158 \nu^{9} + 187122049375273083182761 \nu^{8} - 91927305527432793814873 \nu^{7} - 1144252658555549170678220 \nu^{6} - 712773826444467637729457 \nu^{5} + 6811586276607505895145280 \nu^{4} + 3600075300459734885770527 \nu^{3} - 10109472841637565639504916 \nu^{2} - 919831796446951625112659 \nu + 2326700697046292559758031$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{10}$$ $$=$$ $$($$$$27841001391058287711 \nu^{15} - 56734590384175806489 \nu^{14} + 36957112534807647629 \nu^{13} - 203129756460236237424 \nu^{12} + 2018079866614404196466 \nu^{11} + 2099731799037360256404 \nu^{10} - 19282799540413774394449 \nu^{9} - 83683565215716791601 \nu^{8} + 31440215226417970130710 \nu^{7} + 56518782117191086056949 \nu^{6} - 96903887929313182641538 \nu^{5} - 621951148353291269884475 \nu^{4} + 489546973039043406909558 \nu^{3} + 1096602478426018247274321 \nu^{2} - 2302612646300234391423585 \nu - 1528524838084480278967976$$$$)/$$$$24\!\cdots\!82$$ $$\beta_{11}$$ $$=$$ $$($$$$-491794533520167608490 \nu^{15} + 901188862202916401331 \nu^{14} - 3513644298666979364435 \nu^{13} + 8970360813785855460649 \nu^{12} - 55128918437669555904304 \nu^{11} - 1051882486226896470238 \nu^{10} + 24125594629075976001972 \nu^{9} - 14838493059808301198313 \nu^{8} - 64947380812895073686085 \nu^{7} - 2106615682151562836025168 \nu^{6} + 1042816552025016489581611 \nu^{5} + 1801491819505295602632238 \nu^{4} + 631002537638292209533883 \nu^{3} - 6216187971145756538445718 \nu^{2} - 5362289602753199938134797 \nu + 11655149699405640386109225$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{12}$$ $$=$$ $$($$$$-691034563587916915737 \nu^{15} + 2806954437225253898934 \nu^{14} - 8045985534020736227770 \nu^{13} + 16718554763267774358099 \nu^{12} - 84986255321267976091904 \nu^{11} + 102156951448543758372016 \nu^{10} + 194182167290765317461477 \nu^{9} - 1070825668818105880143790 \nu^{8} + 895632462416637737420655 \nu^{7} - 2601957057924215348257099 \nu^{6} + 5468585706172162446683881 \nu^{5} - 1547582697533981048296861 \nu^{4} - 28370972505874455264564597 \nu^{3} + 28218233032859070366205565 \nu^{2} + 5198924312299588660655566 \nu - 18337495305982233746732551$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{13}$$ $$=$$ $$($$$$849990483185904416025 \nu^{15} - 3730989166437327889422 \nu^{14} + 7456596902957108195684 \nu^{13} - 15788136799018423755999 \nu^{12} + 89965673367961653663664 \nu^{11} - 107019855084457027909192 \nu^{10} - 539110377434202489812473 \nu^{9} + 1319321835199372487091284 \nu^{8} - 265504945153451273789049 \nu^{7} + 1788288519805307742651413 \nu^{6} - 6235479095247208525965459 \nu^{5} - 10898837022540635238008519 \nu^{4} + 49134210651174921612242223 \nu^{3} - 20884555220267863820579845 \nu^{2} - 50510464121676799778673936 \nu + 63129557099007689485664241$$$$)/$$$$41\!\cdots\!94$$ $$\beta_{14}$$ $$=$$ $$($$$$-666110474896368591010 \nu^{15} + 2098597115474769212497 \nu^{14} - 5475750913316501230191 \nu^{13} + 10690759718530261922245 \nu^{12} - 67625028031115812524375 \nu^{11} + 30835681828234576416313 \nu^{10} + 260639738173456415348277 \nu^{9} - 763341276304184014455963 \nu^{8} + 261068453994508556908628 \nu^{7} - 1568622043815512853402511 \nu^{6} + 2722579385909269185871932 \nu^{5} + 5127081274242995649454220 \nu^{4} - 26828583686990890888136112 \nu^{3} + 14963042598215149832260602 \nu^{2} + 26185136215437696264540956 \nu - 39149589542397270244564119$$$$)/$$$$20\!\cdots\!97$$ $$\beta_{15}$$ $$=$$ $$($$$$3084335617013753438064 \nu^{15} - 9030254723203551577571 \nu^{14} + 18935602585925518420309 \nu^{13} - 36040499637255171918253 \nu^{12} + 281231162053109020442988 \nu^{11} - 7454526513370189035702 \nu^{10} - 1635736480009956866061230 \nu^{9} + 3128882321703874961336905 \nu^{8} + 748068444337856889442125 \nu^{7} + 7407138856127690604873560 \nu^{6} - 10495839243181498337480351 \nu^{5} - 40435037875517509387354522 \nu^{4} + 123142752521176196788848553 \nu^{3} - 5366727257661381628175906 \nu^{2} - 122782663782179497578784425 \nu + 129907195151503297187373233$$$$)/$$$$41\!\cdots\!94$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - 3 \beta_{14} - 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 3 \beta_{9} - \beta_{7} + 8 \beta_{6} - \beta_{5} + 3 \beta_{4} + 7 \beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/22$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + 3 \beta_{9} - 7 \beta_{7} + 12 \beta_{6} + \beta_{5} - 27 \beta_{4} + 9 \beta_{3} - 15 \beta_{2} - 6 \beta_{1} - 9$$$$)/22$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - 10 \beta_{14} + \beta_{13} + \beta_{12} + 11 \beta_{11} - 12 \beta_{10} - 45 \beta_{9} - 18 \beta_{7} + 34 \beta_{6} + 10 \beta_{5} - 39 \beta_{4} - 66 \beta_{3} + 28 \beta_{2} + 19 \beta_{1} + 43$$$$)/22$$ $$\nu^{4}$$ $$=$$ $$($$$$-64 \beta_{14} - 20 \beta_{13} + 4 \beta_{12} + 61 \beta_{11} - 26 \beta_{10} - 77 \beta_{9} - 22 \beta_{8} + 19 \beta_{7} + 101 \beta_{6} - 4 \beta_{5} + 134 \beta_{4} - 92 \beta_{3} - 170 \beta_{2} - 41 \beta_{1} - 376$$$$)/22$$ $$\nu^{5}$$ $$=$$ $$($$$$58 \beta_{15} + 308 \beta_{14} + 22 \beta_{13} + 55 \beta_{12} - 228 \beta_{11} - 55 \beta_{10} - 50 \beta_{9} + 198 \beta_{8} + 165 \beta_{7} - 352 \beta_{6} - 55 \beta_{5} - 59 \beta_{4} - 5 \beta_{3} - 490 \beta_{2} - 160 \beta_{1} - 1686$$$$)/22$$ $$\nu^{6}$$ $$=$$ $$($$$$225 \beta_{15} + 909 \beta_{14} + 73 \beta_{13} - 207 \beta_{12} - 656 \beta_{11} - 145 \beta_{10} - 721 \beta_{9} + 484 \beta_{8} + 941 \beta_{7} - 1192 \beta_{6} + 339 \beta_{5} + 543 \beta_{4} - 1173 \beta_{3} + 1559 \beta_{2} + 496 \beta_{1} + 3373$$$$)/22$$ $$\nu^{7}$$ $$=$$ $$($$$$400 \beta_{15} - 785 \beta_{14} - 136 \beta_{13} + 7 \beta_{12} + 487 \beta_{11} + 1071 \beta_{10} + 2583 \beta_{9} - 429 \beta_{8} + 2943 \beta_{7} + 469 \beta_{6} - 656 \beta_{5} + 6334 \beta_{4} + 3108 \beta_{3} + 2544 \beta_{2} + 171 \beta_{1} + 3202$$$$)/22$$ $$\nu^{8}$$ $$=$$ $$($$$$321 \beta_{15} + 4293 \beta_{14} + 3 \beta_{13} + 761 \beta_{12} - 3190 \beta_{11} + 5861 \beta_{10} + 16105 \beta_{9} + 2332 \beta_{8} - 1429 \beta_{7} - 5376 \beta_{6} - 5051 \beta_{5} - 6673 \beta_{4} + 21667 \beta_{3} + 257 \beta_{2} - 2896 \beta_{1} - 6353$$$$)/22$$ $$\nu^{9}$$ $$=$$ $$($$$$-4906 \beta_{15} - 4418 \beta_{14} + 1830 \beta_{13} - 6737 \beta_{12} + 2608 \beta_{11} + 4526 \beta_{10} + 580 \beta_{9} - 4037 \beta_{8} - 26659 \beta_{7} + 6604 \beta_{6} + 8563 \beta_{5} - 40102 \beta_{4} - 4865 \beta_{3} + 76445 \beta_{2} + 19378 \beta_{1} + 153704$$$$)/22$$ $$\nu^{10}$$ $$=$$ $$-1379 \beta_{15} - 6424 \beta_{14} + 244 \beta_{13} - 10 \beta_{12} + 4709 \beta_{11} + 10 \beta_{10} + 30 \beta_{9} - 4164 \beta_{8} - 3670 \beta_{7} + 7606 \beta_{6} + 10 \beta_{5} + 1067 \beta_{4} - 20 \beta_{3} - 62 \beta_{2} + 50 \beta_{1} + 868$$ $$\nu^{11}$$ $$=$$ $$($$$$-48333 \beta_{15} - 52847 \beta_{14} + 7697 \beta_{13} + 70896 \beta_{12} + 37425 \beta_{11} - 46784 \beta_{10} + 5954 \beta_{9} - 36432 \beta_{8} - 273967 \beta_{7} + 85577 \beta_{6} - 83216 \beta_{5} - 389386 \beta_{4} + 61574 \beta_{3} - 773345 \beta_{2} - 193489 \beta_{1} - 1521849$$$$)/22$$ $$\nu^{12}$$ $$=$$ $$($$$$29622 \beta_{15} + 431982 \beta_{14} - 6082 \beta_{13} - 97516 \beta_{12} - 313729 \beta_{11} - 610994 \beta_{10} - 1603643 \beta_{9} + 270446 \beta_{8} - 158561 \beta_{7} - 478037 \beta_{6} + 535580 \beta_{5} - 759588 \beta_{4} - 2152218 \beta_{3} - 82812 \beta_{2} + 264577 \beta_{1} + 593012$$$$)/22$$ $$\nu^{13}$$ $$=$$ $$($$$$384634 \beta_{15} - 794386 \beta_{14} - 109130 \beta_{13} + 42054 \beta_{12} + 581537 \beta_{11} - 1140591 \beta_{10} - 2662309 \beta_{9} - 413281 \beta_{8} + 3121661 \beta_{7} + 656019 \beta_{6} + 643202 \beta_{5} + 5928961 \beta_{4} - 3446322 \beta_{3} - 2267171 \beta_{2} - 86463 \beta_{1} - 3205038$$$$)/22$$ $$\nu^{14}$$ $$=$$ $$($$$$2730431 \beta_{15} + 9010113 \beta_{14} - 735975 \beta_{13} + 1930823 \beta_{12} - 6572417 \beta_{11} + 1788321 \beta_{10} + 7297796 \beta_{9} + 6026944 \beta_{8} + 9967394 \beta_{7} - 11138081 \beta_{6} - 4238623 \beta_{5} + 6342833 \beta_{4} + 10919575 \beta_{3} - 16084997 \beta_{2} - 5310555 \beta_{1} - 35016379$$$$)/22$$ $$\nu^{15}$$ $$=$$ $$($$$$7132447 \beta_{15} + 32581351 \beta_{14} - 1764499 \beta_{13} - 5821200 \beta_{12} - 23856742 \beta_{11} + 5821200 \beta_{10} + 4247820 \beta_{9} + 21263880 \beta_{8} + 18765087 \beta_{7} - 38808715 \beta_{6} + 5821200 \beta_{5} - 5296790 \beta_{4} + 1573380 \beta_{3} + 67208421 \beta_{2} + 15890220 \beta_{1} + 128691549$$$$)/22$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
604.1
 1.64608 − 1.06057i 2.24350 − 2.23726i 0.988132 − 0.846795i −1.95510 − 0.109518i 1.60675 − 1.36085i −1.29715 + 0.104262i −1.43448 − 2.82504i −0.797732 − 1.94863i −0.797732 + 1.94863i −1.43448 + 2.82504i −1.29715 − 0.104262i 1.60675 + 1.36085i −1.95510 + 0.109518i 0.988132 + 0.846795i 2.24350 + 2.23726i 1.64608 + 1.06057i
3.65113i 0 −9.33074 −5.47004 0 7.16054i 19.4632i 0 19.9718i
604.2 2.88468i 0 −4.32139 0.441126 0 10.5820i 0.927105i 0 1.27251i
604.3 2.53176i 0 −2.40981 8.44690 0 2.44043i 4.02597i 0 21.3855i
604.4 2.47556i 0 −2.12839 2.55350 0 0.170400i 4.63328i 0 6.32135i
604.5 2.21517i 0 −0.906963 −8.69502 0 6.67189i 6.85159i 0 19.2609i
604.6 1.35619i 0 2.16075 −2.29430 0 9.77137i 8.35515i 0 3.11151i
604.7 0.982569i 0 3.03456 −0.397576 0 7.22681i 6.91194i 0 0.390645i
604.8 0.313054i 0 3.90200 7.41540 0 10.0271i 2.47375i 0 2.32142i
604.9 0.313054i 0 3.90200 7.41540 0 10.0271i 2.47375i 0 2.32142i
604.10 0.982569i 0 3.03456 −0.397576 0 7.22681i 6.91194i 0 0.390645i
604.11 1.35619i 0 2.16075 −2.29430 0 9.77137i 8.35515i 0 3.11151i
604.12 2.21517i 0 −0.906963 −8.69502 0 6.67189i 6.85159i 0 19.2609i
604.13 2.47556i 0 −2.12839 2.55350 0 0.170400i 4.63328i 0 6.32135i
604.14 2.53176i 0 −2.40981 8.44690 0 2.44043i 4.02597i 0 21.3855i
604.15 2.88468i 0 −4.32139 0.441126 0 10.5820i 0.927105i 0 1.27251i
604.16 3.65113i 0 −9.33074 −5.47004 0 7.16054i 19.4632i 0 19.9718i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 604.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.c.m 16
3.b odd 2 1 363.3.c.e 16
11.b odd 2 1 inner 1089.3.c.m 16
11.c even 5 1 99.3.k.c 16
11.d odd 10 1 99.3.k.c 16
33.d even 2 1 363.3.c.e 16
33.f even 10 1 33.3.g.a 16
33.f even 10 1 363.3.g.a 16
33.f even 10 1 363.3.g.f 16
33.f even 10 1 363.3.g.g 16
33.h odd 10 1 33.3.g.a 16
33.h odd 10 1 363.3.g.a 16
33.h odd 10 1 363.3.g.f 16
33.h odd 10 1 363.3.g.g 16
132.n odd 10 1 528.3.bf.b 16
132.o even 10 1 528.3.bf.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a 16 33.f even 10 1
33.3.g.a 16 33.h odd 10 1
99.3.k.c 16 11.c even 5 1
99.3.k.c 16 11.d odd 10 1
363.3.c.e 16 3.b odd 2 1
363.3.c.e 16 33.d even 2 1
363.3.g.a 16 33.f even 10 1
363.3.g.a 16 33.h odd 10 1
363.3.g.f 16 33.f even 10 1
363.3.g.f 16 33.h odd 10 1
363.3.g.g 16 33.f even 10 1
363.3.g.g 16 33.h odd 10 1
528.3.bf.b 16 132.n odd 10 1
528.3.bf.b 16 132.o even 10 1
1089.3.c.m 16 1.a even 1 1 trivial
1089.3.c.m 16 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(1089, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3721 + 46518 T^{2} + 94305 T^{4} + 74898 T^{6} + 29084 T^{8} + 6102 T^{10} + 705 T^{12} + 42 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 3061 + 1032 T - 18020 T^{2} - 1738 T^{3} + 3629 T^{4} + 178 T^{5} - 120 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$7$ $$22159001881 + 768877871502 T^{2} + 197587255593 T^{4} + 14140969254 T^{6} + 474207380 T^{8} + 8613174 T^{10} + 87153 T^{12} + 462 T^{14} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$4711546245615616 + 764136753063936 T^{2} + 41398432159680 T^{4} + 1064474266464 T^{6} + 14663808209 T^{8} + 113858844 T^{10} + 495510 T^{12} + 1116 T^{14} + T^{16}$$
$17$ $$186934325634232576 + 52483817235178624 T^{2} + 1828587595169840 T^{4} + 25382574169816 T^{6} + 180625132729 T^{8} + 722370926 T^{10} + 1644875 T^{12} + 1994 T^{14} + T^{16}$$
$19$ $$992978640332394496 + 63623550649288704 T^{2} + 1677492711376320 T^{4} + 23467114203936 T^{6} + 187302616889 T^{8} + 852931746 T^{10} + 2088435 T^{12} + 2394 T^{14} + T^{16}$$
$23$ $$( 255717136 + 98313344 T + 5317032 T^{2} - 2384728 T^{3} - 389395 T^{4} - 14042 T^{5} + 862 T^{6} + 66 T^{7} + T^{8} )^{2}$$
$29$ $$17\!\cdots\!36$$$$+ 20760021028497076224 T^{2} + 471882464641626240 T^{4} + 3099439223509056 T^{6} + 9411372394529 T^{8} + 15156940806 T^{10} + 13218555 T^{12} + 5814 T^{14} + T^{16}$$
$31$ $$( 24355353121 + 319481400 T - 1524629402 T^{2} - 9879940 T^{3} + 4693799 T^{4} + 34660 T^{5} - 4098 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$37$ $$( 1739681856 + 3180542112 T + 1529211000 T^{2} + 232136052 T^{3} + 9129469 T^{4} - 108572 T^{5} - 6310 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$41$ $$10\!\cdots\!76$$$$+$$$$32\!\cdots\!96$$$$T^{2} + 28405413317088259760 T^{4} + 98031265321869064 T^{6} + 153820597813849 T^{8} + 123207083864 T^{10} + 52641830 T^{12} + 11456 T^{14} + T^{16}$$
$43$ $$13\!\cdots\!56$$$$+$$$$94\!\cdots\!52$$$$T^{2} +$$$$50\!\cdots\!60$$$$T^{4} + 921202724978729952 T^{6} + 823261948051849 T^{8} + 407533181588 T^{10} + 113748830 T^{12} + 16708 T^{14} + T^{16}$$
$47$ $$( -17400069104 + 26295048560 T - 3348553432 T^{2} + 63322700 T^{3} + 6275989 T^{4} - 144530 T^{5} - 4783 T^{6} + 40 T^{7} + T^{8} )^{2}$$
$53$ $$( -64358351232579 + 5238632285946 T - 6819788700 T^{2} - 4957779786 T^{3} + 55389109 T^{4} + 1102474 T^{5} - 15400 T^{6} - 64 T^{7} + T^{8} )^{2}$$
$59$ $$( -1862262005879 + 164599115940 T + 7628459908 T^{2} - 684726490 T^{3} - 4787251 T^{4} + 756130 T^{5} - 5388 T^{6} - 110 T^{7} + T^{8} )^{2}$$
$61$ $$13\!\cdots\!16$$$$+$$$$15\!\cdots\!04$$$$T^{2} +$$$$49\!\cdots\!52$$$$T^{4} + 1562482018762891752 T^{6} + 1834590096717665 T^{8} + 984487524618 T^{10} + 247162707 T^{12} + 27306 T^{14} + T^{16}$$
$67$ $$( 47006885776 + 43441331072 T + 13796874228 T^{2} + 1665816284 T^{3} + 50379185 T^{4} - 102566 T^{5} - 13937 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$71$ $$( 2000407740496 + 327101827184 T - 9562836856 T^{2} - 1604256892 T^{3} - 16280231 T^{4} + 1303696 T^{5} + 35381 T^{6} + 322 T^{7} + T^{8} )^{2}$$
$73$ $$63\!\cdots\!36$$$$+$$$$25\!\cdots\!08$$$$T^{2} +$$$$25\!\cdots\!80$$$$T^{4} + 1048213463214046048 T^{6} + 1775937000505849 T^{8} + 1095219382562 T^{10} + 278043635 T^{12} + 29402 T^{14} + T^{16}$$
$79$ $$14\!\cdots\!81$$$$+$$$$12\!\cdots\!58$$$$T^{2} +$$$$32\!\cdots\!25$$$$T^{4} + 32046510672422807358 T^{6} + 15310394162227124 T^{8} + 3895731666942 T^{10} + 532329105 T^{12} + 36702 T^{14} + T^{16}$$
$83$ $$16\!\cdots\!41$$$$+$$$$47\!\cdots\!06$$$$T^{2} +$$$$19\!\cdots\!65$$$$T^{4} +$$$$29\!\cdots\!54$$$$T^{6} + 177369580339779364 T^{8} + 30655322876954 T^{10} + 2280143945 T^{12} + 77726 T^{14} + T^{16}$$
$89$ $$( -15121642690304 + 5524647856512 T - 434456318960 T^{2} + 321809632 T^{3} + 196550429 T^{4} - 540742 T^{5} - 26250 T^{6} + 38 T^{7} + T^{8} )^{2}$$
$97$ $$( 54474630569101 + 1018131288588 T - 103383443370 T^{2} - 1639288152 T^{3} + 64849559 T^{4} + 796512 T^{5} - 14670 T^{6} - 108 T^{7} + T^{8} )^{2}$$