# Properties

 Label 448.3.s.h Level 448 Weight 3 Character orbit 448.s Analytic conductor 12.207 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 448.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2071158433$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{20}\cdot 7$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{5} q^{5} + ( -\beta_{2} - \beta_{12} ) q^{7} + ( -5 \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{5} q^{5} + ( -\beta_{2} - \beta_{12} ) q^{7} + ( -5 \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{9} + ( \beta_{2} - \beta_{4} - \beta_{6} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{11} + ( -1 - 2 \beta_{7} + \beta_{10} + \beta_{13} ) q^{13} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{14} + \beta_{15} ) q^{15} + ( -2 - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{17} + ( -\beta_{2} - 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{19} + ( 6 - 3 \beta_{3} + \beta_{5} - 5 \beta_{7} + 3 \beta_{8} + 2 \beta_{10} - \beta_{13} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{23} + ( 10 + 2 \beta_{3} + 2 \beta_{5} + 10 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 4 \beta_{11} - 4 \beta_{13} ) q^{25} + ( -2 \beta_{1} + \beta_{4} - 6 \beta_{6} + 2 \beta_{9} + 3 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{27} + ( 1 + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{8} - \beta_{10} + \beta_{13} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{9} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{31} + ( -22 - 4 \beta_{5} - 11 \beta_{7} - 2 \beta_{11} + 4 \beta_{13} ) q^{33} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{35} + ( 6 \beta_{3} - 3 \beta_{5} + 9 \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{37} + ( -2 \beta_{1} + 6 \beta_{2} - 9 \beta_{6} + \beta_{9} + 3 \beta_{12} + \beta_{15} ) q^{39} + ( 5 + 2 \beta_{3} + 10 \beta_{7} + 3 \beta_{8} + \beta_{10} - 6 \beta_{11} + \beta_{13} ) q^{41} + ( -2 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{9} - 4 \beta_{14} + 2 \beta_{15} ) q^{43} + ( -13 - 6 \beta_{3} + 6 \beta_{5} + 13 \beta_{7} - 7 \beta_{8} - 5 \beta_{10} - 2 \beta_{11} ) q^{45} + ( -2 \beta_{1} - 5 \beta_{2} - 7 \beta_{4} + 12 \beta_{6} + \beta_{9} + 6 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{47} + ( 4 - 2 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} - 5 \beta_{8} - \beta_{10} - 3 \beta_{13} ) q^{49} + ( 7 \beta_{2} - 2 \beta_{4} + 5 \beta_{6} - 2 \beta_{9} - 4 \beta_{12} + \beta_{15} ) q^{51} + ( -5 + 3 \beta_{3} + 3 \beta_{5} - 5 \beta_{7} + 9 \beta_{8} - \beta_{10} - 8 \beta_{11} - 2 \beta_{13} ) q^{53} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 9 \beta_{6} - 3 \beta_{9} - 4 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{55} + ( 23 + 4 \beta_{3} - 8 \beta_{5} + 2 \beta_{8} ) q^{57} + ( -2 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{12} + 4 \beta_{15} ) q^{59} + ( -18 - 7 \beta_{5} - 9 \beta_{7} - 5 \beta_{8} + 4 \beta_{11} - 3 \beta_{13} ) q^{61} + ( \beta_{1} - 20 \beta_{2} - 2 \beta_{4} + \beta_{6} + 2 \beta_{9} + 3 \beta_{12} - 5 \beta_{14} + 3 \beta_{15} ) q^{63} + ( 21 \beta_{7} - 3 \beta_{8} - 6 \beta_{10} - \beta_{11} - 3 \beta_{13} ) q^{65} + ( 9 \beta_{2} + 2 \beta_{4} - 20 \beta_{6} - 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{67} + ( 22 + \beta_{3} + 44 \beta_{7} - 6 \beta_{8} + 12 \beta_{11} ) q^{69} + ( 10 \beta_{2} - 2 \beta_{4} - 2 \beta_{9} + 6 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{71} + ( -13 - 6 \beta_{3} + 6 \beta_{5} + 13 \beta_{7} + 8 \beta_{8} + 8 \beta_{11} ) q^{73} + ( -2 \beta_{1} - 22 \beta_{2} - 2 \beta_{4} + 22 \beta_{6} + 2 \beta_{9} - 2 \beta_{12} + 6 \beta_{14} ) q^{75} + ( -23 + \beta_{3} - 5 \beta_{5} - 38 \beta_{7} + 6 \beta_{8} - 3 \beta_{10} + 5 \beta_{13} ) q^{77} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{9} + 3 \beta_{12} - \beta_{14} + 4 \beta_{15} ) q^{79} + ( -48 + 4 \beta_{3} + 4 \beta_{5} - 48 \beta_{7} - 16 \beta_{8} + 5 \beta_{10} + 11 \beta_{11} + 10 \beta_{13} ) q^{81} + ( -2 \beta_{2} + 6 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} - 10 \beta_{12} + 2 \beta_{14} - 6 \beta_{15} ) q^{83} + ( 67 + \beta_{3} - 2 \beta_{5} + 2 \beta_{8} + 3 \beta_{10} - 3 \beta_{13} ) q^{85} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} - \beta_{9} + 5 \beta_{12} - 5 \beta_{15} ) q^{87} + ( 2 + 10 \beta_{5} + \beta_{7} + 14 \beta_{8} - 14 \beta_{13} ) q^{89} + ( -8 \beta_{2} - \beta_{4} - 5 \beta_{6} + 2 \beta_{9} + 3 \beta_{12} - \beta_{14} ) q^{91} + ( -14 \beta_{3} + 7 \beta_{5} - 21 \beta_{7} + 5 \beta_{8} + 10 \beta_{10} + 2 \beta_{11} + 5 \beta_{13} ) q^{93} + ( \beta_{1} + 12 \beta_{2} - 5 \beta_{4} - 20 \beta_{6} + 3 \beta_{12} - 3 \beta_{14} + 3 \beta_{15} ) q^{95} + ( 37 - 2 \beta_{3} + 74 \beta_{7} + 7 \beta_{8} + \beta_{10} - 14 \beta_{11} + \beta_{13} ) q^{97} + ( 2 \beta_{1} + 24 \beta_{2} + 11 \beta_{4} - 17 \beta_{6} - 2 \beta_{9} - 3 \beta_{12} + 5 \beta_{14} - 4 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 40q^{9} + O(q^{10})$$ $$16q + 40q^{9} - 48q^{17} + 136q^{21} + 80q^{25} + 16q^{29} - 264q^{33} - 72q^{37} - 312q^{45} + 128q^{49} - 40q^{53} + 368q^{57} - 216q^{61} - 168q^{65} - 312q^{73} - 64q^{77} - 384q^{81} + 1072q^{85} + 24q^{89} + 168q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} - 33728 x^{7} - 49760 x^{6} + 203528 x^{5} + 27401 x^{4} - 156928 x^{3} + 114964 x^{2} - 248608 x + 208849$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2920417443677474293913902 \nu^{15} - 744327595600429275881449294 \nu^{14} - 175722177961496578640865646 \nu^{13} + 17937804992802510950044866537 \nu^{12} + 18466990664549970723854697428 \nu^{11} - 310573020706495203943514945672 \nu^{10} - 218500635534359350729438131136 \nu^{9} + 2682209474043646001676873266628 \nu^{8} - 1919852721388660753134380819828 \nu^{7} - 19786405682174364109141729822900 \nu^{6} + 6665949738835295403921831086648 \nu^{5} + 14160472018443853370010318486950 \nu^{4} - 83096953370417874484163105963710 \nu^{3} - 88457341854902593335381060785800 \nu^{2} - 3080762339053380854480283054134 \nu + 19593464920057236434122950881109$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{2}$$ $$=$$ $$($$$$-3515304493583606735135952 \nu^{15} + 24764143983284374176395421 \nu^{14} + 27595582130743566079841576 \nu^{13} - 543969869770112203177631988 \nu^{12} - 607867478339680085525445660 \nu^{11} + 11148510041927868533000295192 \nu^{10} - 6332471382800274132497845056 \nu^{9} - 114659322642441135223486994952 \nu^{8} + 202745664387160665881013574208 \nu^{7} + 473191360106891395382899928664 \nu^{6} - 1855079890192905098726607507180 \nu^{5} + 394717378852979064578167551616 \nu^{4} + 2129928673592806212368161833504 \nu^{3} - 9349541338079118005059857391953 \nu^{2} + 3232418508440853343831275449960 \nu - 3286263117449365410102954785844$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{3}$$ $$=$$ $$($$$$-8161932957484190343665273 \nu^{15} - 44439187972352536048517534 \nu^{14} + 285301886851739906742316053 \nu^{13} + 1112658279033969221325555216 \nu^{12} - 4814066789420379222535386834 \nu^{11} - 19373728230501465036814065604 \nu^{10} + 62897025351342307275446286156 \nu^{9} + 109397774130481521364257407586 \nu^{8} - 727155270139808660498290086006 \nu^{7} - 358917417736567570954159885656 \nu^{6} + 2663256405889607525415415377260 \nu^{5} - 3729807910919652247800742026214 \nu^{4} - 5104731955019441877585892054673 \nu^{3} + 3653225388259983436664143334390 \nu^{2} - 8403237557696221261342105089459 \nu + 7385860188895574436656992849363$$$$)/$$$$38\!\cdots\!33$$ $$\beta_{4}$$ $$=$$ $$($$$$4687764179358667903827490 \nu^{15} + 10008573052443251558227112 \nu^{14} - 176973453028403935236629214 \nu^{13} - 259187322942024133059057463 \nu^{12} + 3276954039482169644025472164 \nu^{11} + 4524753164058537579368230272 \nu^{10} - 42812468468772730417302451304 \nu^{9} + 1199461520202643096021191028 \nu^{8} + 385877809481960472865171774452 \nu^{7} - 341285354834289306388036193488 \nu^{6} - 1538010994203696118079073494136 \nu^{5} + 3148129401873306054350610561234 \nu^{4} + 410971772490892892566561077754 \nu^{3} - 8610918321458095981929337389994 \nu^{2} + 13844783294622013114953545762 \nu - 2510289020758883104864126050023$$$$)/$$$$11\!\cdots\!38$$ $$\beta_{5}$$ $$=$$ $$($$$$-33282229045960332120189538 \nu^{15} - 38472491138586635287049351 \nu^{14} + 958767424475648734860770778 \nu^{13} + 1381335733635573158052227528 \nu^{12} - 16821636988084766687601118704 \nu^{11} - 22048454616262823264454109000 \nu^{10} + 177952670383067575630013044528 \nu^{9} - 5636154129460707453505177806 \nu^{8} - 1432416807939321269362752830100 \nu^{7} + 450738148978600311581503338592 \nu^{6} + 3124297095707150083277879952764 \nu^{5} - 6294626994688832428341199761172 \nu^{4} - 1643834893522645703950405785162 \nu^{3} + 143990978934001807227407665421 \nu^{2} - 5422442965226334273104713972494 \nu + 15530645078820716406525243786690$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{6}$$ $$=$$ $$($$$$-87428286339522377943258 \nu^{15} + 19379552843836720073999 \nu^{14} + 2106362162310834996966530 \nu^{13} + 1203281739031657890480621 \nu^{12} - 37567134568520370754247220 \nu^{11} - 8063335103197233251269908 \nu^{10} + 327897110740258740307119480 \nu^{9} - 362422523161540239768290176 \nu^{8} - 2202446682825009641121408292 \nu^{7} + 1724782533273118586212671000 \nu^{6} + 1423302805080590158619506672 \nu^{5} - 8021116023302597726209967294 \nu^{4} - 9591582687049399777297302042 \nu^{3} - 10981892368336939613161395337 \nu^{2} + 9341327103233175018294227090 \nu + 3686481877316456037642549777$$$$)/$$$$17\!\cdots\!38$$ $$\beta_{7}$$ $$=$$ $$($$$$-248197947118550681400 \nu^{15} - 60099576570060185087 \nu^{14} + 6110172991187791613532 \nu^{13} + 5610165377095918036704 \nu^{12} - 107123109897527970241392 \nu^{11} - 59136687103919879272608 \nu^{10} + 966301607286958942241280 \nu^{9} - 861897353924990552955686 \nu^{8} - 7059892149907340387084928 \nu^{7} + 4980438225173244013489056 \nu^{6} + 5677044395320575039843008 \nu^{5} - 40095443257835374287069696 \nu^{4} - 13407128082587646721614288 \nu^{3} + 5618854490528863822096597 \nu^{2} - 33962809459441010571298260 \nu + 11417432982080716379376990$$$$)/$$$$48\!\cdots\!06$$ $$\beta_{8}$$ $$=$$ $$($$$$-351675045716698 \nu^{15} + 84134512548244 \nu^{14} + 8997681016250754 \nu^{13} + 4445925380328116 \nu^{12} - 160496373722473788 \nu^{11} - 33874814941456464 \nu^{10} + 1467376262123572708 \nu^{9} - 1553113751898048416 \nu^{8} - 9401792267255930460 \nu^{7} + 10200336255773301928 \nu^{6} + 6044728226531394560 \nu^{5} - 34262171840112781800 \nu^{4} + 46812769807129599650 \nu^{3} - 46922766637244959676 \nu^{2} + 39937799358222111906 \nu + 240280225958040430804$$$$)/ 45742705762379025631$$ $$\beta_{9}$$ $$=$$ $$($$$$63639144463555960591721974 \nu^{15} + 6414494618062250722833405 \nu^{14} - 1594389811523605380678926738 \nu^{13} - 2078568130378060934191309418 \nu^{12} + 29217098080882129982745649904 \nu^{11} + 31896157981372677800461070740 \nu^{10} - 264172546670815086094319045696 \nu^{9} - 115466418759491887857358509212 \nu^{8} + 2031688006206730749761713738884 \nu^{7} + 1393217405592119165630488565872 \nu^{6} - 6902930136209520467640495576284 \nu^{5} - 3745588134230839817513260298772 \nu^{4} + 18961486930358497597998978789294 \nu^{3} - 19525905518592239978294821230153 \nu^{2} + 13601908630252322638109559396486 \nu - 14533939374009833087029953469310$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{10}$$ $$=$$ $$($$$$-70111144892064841956500440 \nu^{15} - 14634704533463944714948623 \nu^{14} + 1649016860697231478334790588 \nu^{13} + 1481577348717818925767409120 \nu^{12} - 28661777910076909183262856616 \nu^{11} - 13276930719192805530130032752 \nu^{10} + 247279220502483540515161299120 \nu^{9} - 281779664556308665645602840258 \nu^{8} - 1811051396750949056384208440312 \nu^{7} + 1339247919214851606512552225104 \nu^{6} + 454959023222686822309815272616 \nu^{5} - 12957304972252303076939512340488 \nu^{4} - 9551699356684231544304330269744 \nu^{3} + 6607831348977807077678926088293 \nu^{2} - 16747421159312610875411267144572 \nu - 58462958599992042806209428082540$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{11}$$ $$=$$ $$($$$$60827347242699394445343576 \nu^{15} + 87079907287736416336762704 \nu^{14} - 1611655880642224052917094404 \nu^{13} - 2999093933608546370064748072 \nu^{12} + 27806563838614510724187553632 \nu^{11} + 44524432609998588989374596888 \nu^{10} - 278528711106182749022865537984 \nu^{9} - 28915162462218712367220940608 \nu^{8} + 2490757642557419927641111076204 \nu^{7} - 180105560603671271291170131328 \nu^{6} - 5579452416269829664084613518320 \nu^{5} + 14259256397530350047124855724312 \nu^{4} + 13808957934494828563854816716056 \nu^{3} - 9481277531835610000306716967200 \nu^{2} + 23879833636573578975503954997536 \nu - 10633248927716884130454269230408$$$$)/$$$$38\!\cdots\!33$$ $$\beta_{12}$$ $$=$$ $$($$$$-121738657247652783765789184 \nu^{15} - 355138569385413589648032772 \nu^{14} + 3039311297013383670434188692 \nu^{13} + 11164156593633746237684672865 \nu^{12} - 48030780904789997927635935828 \nu^{11} - 181003040957991941067297226416 \nu^{10} + 435573479950636950805498066904 \nu^{9} + 1025813084779868491212154221300 \nu^{8} - 4887351733396643936936710328208 \nu^{7} - 7654605427314589762379655284652 \nu^{6} + 12573298901280221526822142798796 \nu^{5} - 5113904201372731045639409256930 \nu^{4} - 58340171134673012396525541961512 \nu^{3} - 14667955760828431672788846547954 \nu^{2} - 3204558575958688980395837064956 \nu + 27467563155200666421329751733813$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{13}$$ $$=$$ $$($$$$-160761995294534487157365352 \nu^{15} + 6636949758964923720295999 \nu^{14} + 3940902050635990653869263764 \nu^{13} + 2684642974495777820589183692 \nu^{12} - 69540748591711531657786821928 \nu^{11} - 21972480516256830512530933400 \nu^{10} + 617038112727940613753063509032 \nu^{9} - 676803797285059604462147245826 \nu^{8} - 4206616565364615728098865010200 \nu^{7} + 3897557035774817798654835114960 \nu^{6} + 1998806027831375158237887887624 \nu^{5} - 21684711784008114382880006260324 \nu^{4} + 2401574208024055886806248985832 \nu^{3} - 5342930036735067248094069235549 \nu^{2} - 6578572297763928167919523468228 \nu + 84532328347093586797598177342770$$$$)/$$$$77\!\cdots\!66$$ $$\beta_{14}$$ $$=$$ $$($$$$29649830917054226444680776 \nu^{15} - 36528720565211174405552059 \nu^{14} - 790132698906394841922174584 \nu^{13} + 344757146604194862076515043 \nu^{12} + 15165493386490648690889378900 \nu^{11} - 9500439822074708222757066260 \nu^{10} - 149802516474780429570759711648 \nu^{9} + 237465373791342638086163220512 \nu^{8} + 933857833471067471618577274704 \nu^{7} - 1807166925510726355173003430176 \nu^{6} - 1815092118953849364446744169332 \nu^{5} + 6191114890646005602837878433702 \nu^{4} - 826011000274627583269874467184 \nu^{3} - 11295940470559589579471527293907 \nu^{2} - 2383988750917451285403829020232 \nu - 2983335376820692431243826356249$$$$)/$$$$11\!\cdots\!38$$ $$\beta_{15}$$ $$=$$ $$($$$$-15210254090489262517236326 \nu^{15} - 27373945785170343714276959 \nu^{14} + 339297524978299406620694046 \nu^{13} + 956286745043098251144238692 \nu^{12} - 5325245369078152301533506712 \nu^{11} - 14166657318955342136358878618 \nu^{10} + 40848949176919293801541657144 \nu^{9} + 50481760520350222075367733290 \nu^{8} - 402878647740215817621185243704 \nu^{7} - 653418190278837048505212207972 \nu^{6} + 213067903825452474960571126140 \nu^{5} - 149185748967100920283355108324 \nu^{4} - 5427825086915057975768278901462 \nu^{3} - 7702171767715200461868482215343 \nu^{2} + 1557726429555851573246291360570 \nu + 1031469506213893538816888075262$$$$)/$$$$55\!\cdots\!19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} - \beta_{12} + \beta_{11} + \beta_{6} - 2 \beta_{2} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{13} + \beta_{11} - \beta_{10} + 13 \beta_{7} - 2 \beta_{2} + 13$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} - 6 \beta_{13} - 17 \beta_{12} + 6 \beta_{10} - 11 \beta_{9} + 19 \beta_{8} + 33 \beta_{6} - 12 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 9 \beta_{1} + 24$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} - 13 \beta_{13} - 8 \beta_{12} + 5 \beta_{11} - 26 \beta_{10} - 13 \beta_{8} + 131 \beta_{7} + 44 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 52 \beta_{2} + 4 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{15} + 35 \beta_{14} - 170 \beta_{13} - 5 \beta_{12} - 22 \beta_{11} - 85 \beta_{10} - 48 \beta_{9} + 107 \beta_{8} + 680 \beta_{7} + 4 \beta_{6} - 65 \beta_{5} - 79 \beta_{4} - 65 \beta_{3} + 351 \beta_{2} - 9 \beta_{1} + 680$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-80 \beta_{15} + 37 \beta_{14} + 135 \beta_{13} - 177 \beta_{12} - 135 \beta_{10} - 122 \beta_{9} - 250 \beta_{8} + 909 \beta_{6} + 96 \beta_{5} - 67 \beta_{4} - 48 \beta_{3} - 6 \beta_{2} + 116 \beta_{1} - 1261$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$368 \beta_{15} + 397 \beta_{14} - 3780 \beta_{13} + 1469 \beta_{12} + \beta_{11} - 7560 \beta_{10} + 100 \beta_{9} - 3780 \beta_{8} + 34104 \beta_{7} - 5505 \beta_{6} + 2156 \beta_{5} - 836 \beta_{4} - 4312 \beta_{3} + 7074 \beta_{2} - 933 \beta_{1}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-340 \beta_{15} + 1600 \beta_{14} + 1558 \beta_{13} + 340 \beta_{12} + 55 \beta_{11} + 779 \beta_{10} - 2388 \beta_{9} - 834 \beta_{8} - 6799 \beta_{7} + 564 \beta_{6} + 478 \beta_{5} - 3624 \beta_{4} + 478 \beta_{3} + 20008 \beta_{2} - 224 \beta_{1} - 6799$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$1506 \beta_{15} - 1016 \beta_{14} + 70938 \beta_{13} - 931 \beta_{12} - 70938 \beta_{10} - 369 \beta_{9} - 135313 \beta_{8} - 10989 \beta_{6} + 67140 \beta_{5} + 193 \beta_{4} - 33570 \beta_{3} - 526 \beta_{2} - 157 \beta_{1} - 660288$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$17857 \beta_{15} + 20318 \beta_{14} - 12464 \beta_{13} + 72554 \beta_{12} + 4297 \beta_{11} - 24928 \beta_{10} + 4798 \beta_{9} - 12464 \beta_{8} + 125022 \beta_{7} - 262424 \beta_{6} + 2292 \beta_{5} - 40512 \beta_{4} - 4584 \beta_{3} + 339776 \beta_{2} - 46436 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$4253 \beta_{15} + 83314 \beta_{14} + 1157706 \beta_{13} - 4253 \beta_{12} - 74343 \beta_{11} + 578853 \beta_{10} - 115927 \beta_{9} - 504510 \beta_{8} - 5445440 \beta_{7} + 14180 \beta_{6} + 249249 \beta_{5} - 185406 \beta_{4} + 249249 \beta_{3} + 857981 \beta_{2} - 18433 \beta_{1} - 5445440$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$154836 \beta_{15} - 57704 \beta_{14} + 308517 \beta_{13} + 531899 \beta_{12} - 308517 \beta_{10} + 355911 \beta_{9} - 560120 \beta_{8} - 2012013 \beta_{6} + 224400 \beta_{5} + 179923 \beta_{4} - 112200 \beta_{3} + 39428 \beta_{2} - 316483 \beta_{1} - 2949878$$ $$\nu^{13}$$ $$=$$ $$($$$$3245192 \beta_{15} + 4023903 \beta_{14} + 16342196 \beta_{13} + 12940335 \beta_{12} - 2332695 \beta_{11} + 32684392 \beta_{10} + 1016084 \beta_{9} + 16342196 \beta_{8} - 154362832 \beta_{7} - 44915939 \beta_{6} - 6750900 \beta_{5} - 7506468 \beta_{4} + 13501800 \beta_{3} + 58872358 \beta_{2} - 8482119 \beta_{1}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$386420 \beta_{15} - 5675214 \beta_{14} + 32304774 \beta_{13} - 386420 \beta_{12} - 2503213 \beta_{11} + 16152387 \beta_{10} + 8156202 \beta_{9} - 13649174 \beta_{8} - 153106199 \beta_{7} - 1433704 \beta_{6} + 6433504 \beta_{5} + 12731758 \beta_{4} + 6433504 \beta_{3} - 64111286 \beta_{2} + 1047284 \beta_{1} - 153106199$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$82151478 \beta_{15} - 29923152 \beta_{14} - 186984966 \beta_{13} + 291303441 \beta_{12} + 186984966 \beta_{10} + 194464767 \beta_{9} + 346122221 \beta_{8} - 1081560513 \beta_{6} - 151670124 \beta_{5} + 97626093 \beta_{4} + 75835062 \beta_{3} + 22305174 \beta_{2} - 172159593 \beta_{1} + 1769317544$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-\beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 −3.16141 + 2.64174i 2.08703 − 2.02145i 1.20279 − 1.51093i −0.162551 + 0.910345i 0.869658 + 0.314400i −1.90990 + 0.286185i −2.79414 + 0.796701i 3.86852 − 1.41699i −3.16141 − 2.64174i 2.08703 + 2.02145i 1.20279 + 1.51093i −0.162551 − 0.910345i 0.869658 − 0.314400i −1.90990 − 0.286185i −2.79414 − 0.796701i 3.86852 + 1.41699i
0 −4.97091 2.86995i 0 −5.45949 + 3.15204i 0 −5.64993 + 4.13259i 0 11.9733 + 20.7383i 0
129.2 0 −3.45151 1.99273i 0 7.80961 4.50888i 0 −5.54917 4.26693i 0 3.44195 + 5.96164i 0
129.3 0 −2.20101 1.27075i 0 −3.56697 + 2.05939i 0 6.98814 + 0.407289i 0 −1.27038 2.20036i 0
129.4 0 −0.729881 0.421397i 0 1.21685 0.702550i 0 −1.56553 6.82269i 0 −4.14485 7.17909i 0
129.5 0 0.729881 + 0.421397i 0 1.21685 0.702550i 0 1.56553 + 6.82269i 0 −4.14485 7.17909i 0
129.6 0 2.20101 + 1.27075i 0 −3.56697 + 2.05939i 0 −6.98814 0.407289i 0 −1.27038 2.20036i 0
129.7 0 3.45151 + 1.99273i 0 7.80961 4.50888i 0 5.54917 + 4.26693i 0 3.44195 + 5.96164i 0
129.8 0 4.97091 + 2.86995i 0 −5.45949 + 3.15204i 0 5.64993 4.13259i 0 11.9733 + 20.7383i 0
257.1 0 −4.97091 + 2.86995i 0 −5.45949 3.15204i 0 −5.64993 4.13259i 0 11.9733 20.7383i 0
257.2 0 −3.45151 + 1.99273i 0 7.80961 + 4.50888i 0 −5.54917 + 4.26693i 0 3.44195 5.96164i 0
257.3 0 −2.20101 + 1.27075i 0 −3.56697 2.05939i 0 6.98814 0.407289i 0 −1.27038 + 2.20036i 0
257.4 0 −0.729881 + 0.421397i 0 1.21685 + 0.702550i 0 −1.56553 + 6.82269i 0 −4.14485 + 7.17909i 0
257.5 0 0.729881 0.421397i 0 1.21685 + 0.702550i 0 1.56553 6.82269i 0 −4.14485 + 7.17909i 0
257.6 0 2.20101 1.27075i 0 −3.56697 2.05939i 0 −6.98814 + 0.407289i 0 −1.27038 + 2.20036i 0
257.7 0 3.45151 1.99273i 0 7.80961 + 4.50888i 0 5.54917 4.26693i 0 3.44195 5.96164i 0
257.8 0 4.97091 2.86995i 0 −5.45949 3.15204i 0 5.64993 + 4.13259i 0 11.9733 20.7383i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.h 16
4.b odd 2 1 inner 448.3.s.h 16
7.d odd 6 1 inner 448.3.s.h 16
8.b even 2 1 224.3.s.b 16
8.d odd 2 1 224.3.s.b 16
28.f even 6 1 inner 448.3.s.h 16
56.j odd 6 1 224.3.s.b 16
56.j odd 6 1 1568.3.c.g 16
56.k odd 6 1 1568.3.c.g 16
56.m even 6 1 224.3.s.b 16
56.m even 6 1 1568.3.c.g 16
56.p even 6 1 1568.3.c.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.b 16 8.b even 2 1
224.3.s.b 16 8.d odd 2 1
224.3.s.b 16 56.j odd 6 1
224.3.s.b 16 56.m even 6 1
448.3.s.h 16 1.a even 1 1 trivial
448.3.s.h 16 4.b odd 2 1 inner
448.3.s.h 16 7.d odd 6 1 inner
448.3.s.h 16 28.f even 6 1 inner
1568.3.c.g 16 56.j odd 6 1
1568.3.c.g 16 56.k odd 6 1
1568.3.c.g 16 56.m even 6 1
1568.3.c.g 16 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 16 T^{2} + 134 T^{4} + 1120 T^{6} + 7897 T^{8} + 73856 T^{10} + 1194422 T^{12} + 13226000 T^{14} + 117489172 T^{16} + 1071306000 T^{18} + 7836602742 T^{20} + 39250106496 T^{22} + 339939955737 T^{24} + 3905198529120 T^{26} + 37845557888454 T^{28} + 366028679279376 T^{30} + 1853020188851841 T^{32}$$
$5$ $$( 1 + 30 T^{2} + 321 T^{4} - 1200 T^{5} - 930 T^{6} - 138960 T^{7} - 137884 T^{8} - 3474000 T^{9} - 581250 T^{10} - 18750000 T^{11} + 125390625 T^{12} + 7324218750 T^{14} + 152587890625 T^{16} )^{2}$$
$7$ $$1 - 64 T^{2} + 2268 T^{4} + 3136 T^{6} - 7054138 T^{8} + 7529536 T^{10} + 13074568668 T^{12} - 885842380864 T^{14} + 33232930569601 T^{16}$$
$11$ $$1 - 432 T^{2} + 84854 T^{4} - 7780896 T^{6} + 22311049 T^{8} + 70438127616 T^{10} + 144760417670 T^{12} - 2441540782042416 T^{14} + 459119675281580212 T^{16} - 35746598589883012656 T^{18} + 31030681144833827270 T^{20} +$$$$22\!\cdots\!36$$$$T^{22} +$$$$10\!\cdots\!89$$$$T^{24} -$$$$52\!\cdots\!96$$$$T^{26} +$$$$83\!\cdots\!14$$$$T^{28} -$$$$62\!\cdots\!92$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 4307359811040 T^{10} + 416424007224732 T^{12} - 24602777889339936 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 24 T + 1022 T^{2} + 19920 T^{3} + 479985 T^{4} + 8829120 T^{5} + 194168350 T^{6} + 3297192024 T^{7} + 67118725988 T^{8} + 952888494936 T^{9} + 16217134760350 T^{10} + 213113493209280 T^{11} + 3348258935318385 T^{12} + 40158598496944080 T^{13} + 595439926448815742 T^{14} + 4041067837425622296 T^{15} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$1 + 1440 T^{2} + 935910 T^{4} + 438945600 T^{6} + 186482707321 T^{8} + 56682302220480 T^{10} + 7723885307690070 T^{12} - 285644590063174560 T^{14} -$$$$25\!\cdots\!40$$$$T^{16} -$$$$37\!\cdots\!60$$$$T^{18} +$$$$13\!\cdots\!70$$$$T^{20} +$$$$12\!\cdots\!80$$$$T^{22} +$$$$53\!\cdots\!01$$$$T^{24} +$$$$16\!\cdots\!00$$$$T^{26} +$$$$45\!\cdots\!10$$$$T^{28} +$$$$91\!\cdots\!40$$$$T^{30} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$1 - 1648 T^{2} + 1017702 T^{4} - 249312032 T^{6} + 42919392953 T^{8} - 81219467468736 T^{10} + 45669118532829142 T^{12} + 12560582613209435984 T^{14} -$$$$19\!\cdots\!76$$$$T^{16} +$$$$35\!\cdots\!44$$$$T^{18} +$$$$35\!\cdots\!02$$$$T^{20} -$$$$17\!\cdots\!56$$$$T^{22} +$$$$26\!\cdots\!33$$$$T^{24} -$$$$42\!\cdots\!32$$$$T^{26} +$$$$48\!\cdots\!82$$$$T^{28} -$$$$22\!\cdots\!88$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 - 4 T + 2608 T^{2} - 8572 T^{3} + 2977102 T^{4} - 7209052 T^{5} + 1844588848 T^{6} - 2379293284 T^{7} + 500246412961 T^{8} )^{4}$$
$31$ $$1 + 4944 T^{2} + 12401142 T^{4} + 21706728288 T^{6} + 30278528813833 T^{8} + 35431200182034624 T^{10} + 36428088221729894790 T^{12} +$$$$35\!\cdots\!36$$$$T^{14} +$$$$33\!\cdots\!48$$$$T^{16} +$$$$32\!\cdots\!56$$$$T^{18} +$$$$31\!\cdots\!90$$$$T^{20} +$$$$27\!\cdots\!64$$$$T^{22} +$$$$22\!\cdots\!73$$$$T^{24} +$$$$14\!\cdots\!88$$$$T^{26} +$$$$76\!\cdots\!82$$$$T^{28} +$$$$28\!\cdots\!04$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$( 1 + 36 T - 2658 T^{2} - 179832 T^{3} + 2782297 T^{4} + 349680552 T^{5} + 4944965886 T^{6} - 248707896180 T^{7} - 11947732536204 T^{8} - 340481109870420 T^{9} + 9267662209871646 T^{10} + 897184626980097768 T^{11} + 9772761047206036537 T^{12} -$$$$86\!\cdots\!68$$$$T^{13} -$$$$17\!\cdots\!98$$$$T^{14} +$$$$32\!\cdots\!04$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 459871021169908160 T^{10} +$$$$42\!\cdots\!48$$$$T^{12} -$$$$24\!\cdots\!96$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 241961144887582552 T^{10} +$$$$29\!\cdots\!96$$$$T^{12} +$$$$23\!\cdots\!92$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$1 + 7200 T^{2} + 21086790 T^{4} + 20987252160 T^{6} - 28679941197479 T^{8} - 70284530432138880 T^{10} +$$$$21\!\cdots\!30$$$$T^{12} +$$$$13\!\cdots\!80$$$$T^{14} +$$$$38\!\cdots\!20$$$$T^{16} +$$$$66\!\cdots\!80$$$$T^{18} +$$$$50\!\cdots\!30$$$$T^{20} -$$$$81\!\cdots\!80$$$$T^{22} -$$$$16\!\cdots\!59$$$$T^{24} +$$$$58\!\cdots\!60$$$$T^{26} +$$$$28\!\cdots\!90$$$$T^{28} +$$$$47\!\cdots\!00$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$( 1 + 20 T - 6098 T^{2} - 355352 T^{3} + 15866345 T^{4} + 1398217832 T^{5} + 8057716910 T^{6} - 2171403120260 T^{7} - 80650058761388 T^{8} - 6099471364810340 T^{9} + 63579262181733710 T^{10} + 30990604965455452328 T^{11} +$$$$98\!\cdots\!45$$$$T^{12} -$$$$62\!\cdots\!48$$$$T^{13} -$$$$29\!\cdots\!18$$$$T^{14} +$$$$27\!\cdots\!80$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$59$ $$1 + 13056 T^{2} + 66899190 T^{4} + 216438091392 T^{6} + 1041428149652041 T^{8} + 6415617182341109568 T^{10} +$$$$27\!\cdots\!66$$$$T^{12} +$$$$85\!\cdots\!44$$$$T^{14} +$$$$25\!\cdots\!80$$$$T^{16} +$$$$10\!\cdots\!84$$$$T^{18} +$$$$41\!\cdots\!86$$$$T^{20} +$$$$11\!\cdots\!08$$$$T^{22} +$$$$22\!\cdots\!81$$$$T^{24} +$$$$56\!\cdots\!92$$$$T^{26} +$$$$21\!\cdots\!90$$$$T^{28} +$$$$50\!\cdots\!76$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 + 108 T + 15678 T^{2} + 1273320 T^{3} + 112514745 T^{4} + 8281698840 T^{5} + 635104006110 T^{6} + 41916162290292 T^{7} + 2814046950849908 T^{8} + 155970039882176532 T^{9} + 8793549087062088510 T^{10} +$$$$42\!\cdots\!40$$$$T^{11} +$$$$21\!\cdots\!45$$$$T^{12} +$$$$90\!\cdots\!20$$$$T^{13} +$$$$41\!\cdots\!38$$$$T^{14} +$$$$10\!\cdots\!28$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$1 - 19248 T^{2} + 171825062 T^{4} - 1105279125792 T^{6} + 6322631625605113 T^{8} - 28716244420307792256 T^{10} +$$$$95\!\cdots\!42$$$$T^{12} -$$$$31\!\cdots\!16$$$$T^{14} +$$$$13\!\cdots\!84$$$$T^{16} -$$$$63\!\cdots\!36$$$$T^{18} +$$$$38\!\cdots\!22$$$$T^{20} -$$$$23\!\cdots\!16$$$$T^{22} +$$$$10\!\cdots\!53$$$$T^{24} -$$$$36\!\cdots\!92$$$$T^{26} +$$$$11\!\cdots\!02$$$$T^{28} -$$$$25\!\cdots\!68$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 67143192723001800472 T^{10} +$$$$20\!\cdots\!60$$$$T^{12} +$$$$39\!\cdots\!12$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 + 156 T + 22270 T^{2} + 2208648 T^{3} + 203639977 T^{4} + 17676576216 T^{5} + 1406019151294 T^{6} + 115719264813828 T^{7} + 8350708700464372 T^{8} + 616667962192889412 T^{9} + 39928470709062473854 T^{10} +$$$$26\!\cdots\!24$$$$T^{11} +$$$$16\!\cdots\!37$$$$T^{12} +$$$$94\!\cdots\!52$$$$T^{13} +$$$$51\!\cdots\!70$$$$T^{14} +$$$$19\!\cdots\!04$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$1 - 34640 T^{2} + 602988550 T^{4} - 7689396100960 T^{6} + 82922855327705561 T^{8} -$$$$77\!\cdots\!00$$$$T^{10} +$$$$63\!\cdots\!50$$$$T^{12} -$$$$46\!\cdots\!20$$$$T^{14} +$$$$30\!\cdots\!00$$$$T^{16} -$$$$17\!\cdots\!20$$$$T^{18} +$$$$95\!\cdots\!50$$$$T^{20} -$$$$45\!\cdots\!00$$$$T^{22} +$$$$19\!\cdots\!81$$$$T^{24} -$$$$68\!\cdots\!60$$$$T^{26} +$$$$21\!\cdots\!50$$$$T^{28} -$$$$47\!\cdots\!40$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} -$$$$29\!\cdots\!28$$$$T^{10} +$$$$13\!\cdots\!20$$$$T^{12} -$$$$37\!\cdots\!48$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 12 T + 8678 T^{2} - 103560 T^{3} + 5715441 T^{4} + 150778776 T^{5} - 475047477962 T^{6} + 26903544978684 T^{7} - 3925857799931356 T^{8} + 213102979776155964 T^{9} - 29805543348733992842 T^{10} + 74934230745999443736 T^{11} +$$$$22\!\cdots\!21$$$$T^{12} -$$$$32\!\cdots\!60$$$$T^{13} +$$$$21\!\cdots\!38$$$$T^{14} -$$$$23\!\cdots\!92$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} -$$$$10\!\cdots\!56$$$$T^{10} +$$$$61\!\cdots\!40$$$$T^{12} -$$$$27\!\cdots\!20$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$