Properties

Label 224.3.s.b
Level 224
Weight 3
Character orbit 224.s
Analytic conductor 6.104
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
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: yes
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 \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−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
33.2 0 −3.45151 + 1.99273i 0 −7.80961 4.50888i 0 5.54917 4.26693i 0 3.44195 5.96164i 0
33.3 0 −2.20101 + 1.27075i 0 3.56697 + 2.05939i 0 −6.98814 + 0.407289i 0 −1.27038 + 2.20036i 0
33.4 0 −0.729881 + 0.421397i 0 −1.21685 0.702550i 0 1.56553 6.82269i 0 −4.14485 + 7.17909i 0
33.5 0 0.729881 0.421397i 0 −1.21685 0.702550i 0 −1.56553 + 6.82269i 0 −4.14485 + 7.17909i 0
33.6 0 2.20101 1.27075i 0 3.56697 + 2.05939i 0 6.98814 0.407289i 0 −1.27038 + 2.20036i 0
33.7 0 3.45151 1.99273i 0 −7.80961 4.50888i 0 −5.54917 + 4.26693i 0 3.44195 5.96164i 0
33.8 0 4.97091 2.86995i 0 5.45949 + 3.15204i 0 −5.64993 4.13259i 0 11.9733 20.7383i 0
129.1 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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
Significant digits:
Format:

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 224.3.s.b 16
4.b odd 2 1 inner 224.3.s.b 16
7.c even 3 1 1568.3.c.g 16
7.d odd 6 1 inner 224.3.s.b 16
7.d odd 6 1 1568.3.c.g 16
8.b even 2 1 448.3.s.h 16
8.d odd 2 1 448.3.s.h 16
28.f even 6 1 inner 224.3.s.b 16
28.f even 6 1 1568.3.c.g 16
28.g odd 6 1 1568.3.c.g 16
56.j odd 6 1 448.3.s.h 16
56.m even 6 1 448.3.s.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.b 16 1.a even 1 1 trivial
224.3.s.b 16 4.b odd 2 1 inner
224.3.s.b 16 7.d odd 6 1 inner
224.3.s.b 16 28.f even 6 1 inner
448.3.s.h 16 8.b even 2 1
448.3.s.h 16 8.d odd 2 1
448.3.s.h 16 56.j odd 6 1
448.3.s.h 16 56.m even 6 1
1568.3.c.g 16 7.c even 3 1
1568.3.c.g 16 7.d odd 6 1
1568.3.c.g 16 28.f even 6 1
1568.3.c.g 16 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

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