# Properties

 Label 768.3.h.h Level $768$ Weight $3$ Character orbit 768.h Analytic conductor $20.926$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 768.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.9264843029$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} - 7060 x^{7} + 17714 x^{6} - 25496 x^{5} + 24840 x^{4} - 17932 x^{3} + 11724 x^{2} - 7056 x + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{34}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 384) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{8} q^{3} -\beta_{11} q^{5} + ( 1 - \beta_{1} ) q^{7} + ( \beta_{1} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q -\beta_{8} q^{3} -\beta_{11} q^{5} + ( 1 - \beta_{1} ) q^{7} + ( \beta_{1} - \beta_{13} ) q^{9} + ( -\beta_{2} - \beta_{5} + \beta_{10} - \beta_{11} ) q^{11} + ( \beta_{3} - \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{13} + ( -2 - \beta_{1} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{15} + ( -\beta_{1} - \beta_{6} - 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{17} + ( -\beta_{3} + \beta_{8} - \beta_{9} - \beta_{14} ) q^{19} + ( 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{14} ) q^{21} + ( \beta_{1} + \beta_{6} + \beta_{7} - 2 \beta_{13} + 2 \beta_{15} ) q^{23} + ( 5 - 3 \beta_{1} - \beta_{4} - \beta_{6} + \beta_{12} - \beta_{13} ) q^{25} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{10} + 3 \beta_{11} - \beta_{14} ) q^{27} + ( -\beta_{5} + 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{29} + ( 7 + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{12} + 2 \beta_{13} ) q^{31} + ( 1 + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} + 2 \beta_{13} - \beta_{15} ) q^{33} + ( 5 \beta_{2} + 3 \beta_{5} - 3 \beta_{8} - \beta_{9} - \beta_{10} - 7 \beta_{11} ) q^{35} + ( -\beta_{3} - 6 \beta_{8} + \beta_{9} - 2 \beta_{14} ) q^{37} + ( -13 + \beta_{1} + 2 \beta_{4} + \beta_{6} + 3 \beta_{12} - 3 \beta_{13} - \beta_{15} ) q^{39} + ( 5 \beta_{4} + 2 \beta_{7} + \beta_{12} - 2 \beta_{13} + 3 \beta_{15} ) q^{41} + ( 3 \beta_{3} - 6 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{14} ) q^{43} + ( -6 \beta_{2} - 2 \beta_{5} + 3 \beta_{8} + 4 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{45} + ( -2 \beta_{1} + 3 \beta_{4} - 2 \beta_{6} - \beta_{12} + 2 \beta_{13} ) q^{47} + ( 9 - 3 \beta_{1} - \beta_{4} - \beta_{6} + 4 \beta_{7} + \beta_{12} + 3 \beta_{13} ) q^{49} + ( -5 \beta_{2} - 2 \beta_{3} + 5 \beta_{5} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 5 \beta_{11} ) q^{51} + ( -4 \beta_{2} + 12 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} ) q^{53} + ( 24 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{12} - 2 \beta_{13} ) q^{55} + ( 5 - 3 \beta_{1} - 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{12} + \beta_{13} + 3 \beta_{15} ) q^{57} + ( -8 \beta_{2} - 4 \beta_{5} + 3 \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{3} + 2 \beta_{5} - 10 \beta_{8} + 3 \beta_{9} - 2 \beta_{14} ) q^{61} + ( -33 + 3 \beta_{1} - 5 \beta_{4} - 4 \beta_{7} - 3 \beta_{12} - 2 \beta_{15} ) q^{63} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 7 \beta_{12} + \beta_{15} ) q^{65} + ( -10 \beta_{5} - 5 \beta_{8} + 3 \beta_{9} + 4 \beta_{14} ) q^{67} + ( 6 \beta_{2} + 3 \beta_{5} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{14} ) q^{69} + ( -\beta_{1} - 7 \beta_{4} - \beta_{6} - 3 \beta_{7} - 3 \beta_{12} + 4 \beta_{13} + 2 \beta_{15} ) q^{71} + ( 10 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{12} ) q^{73} + ( 9 \beta_{2} - 3 \beta_{3} + 17 \beta_{5} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + 5 \beta_{14} ) q^{75} + ( 12 \beta_{2} + 8 \beta_{5} + 12 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{77} + ( 51 - 3 \beta_{4} - 3 \beta_{6} + \beta_{7} + 3 \beta_{12} - 2 \beta_{13} ) q^{79} + ( 9 - 3 \beta_{1} + 5 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{12} - 3 \beta_{13} + 2 \beta_{15} ) q^{81} + ( 5 \beta_{2} + 3 \beta_{5} + 6 \beta_{8} + 2 \beta_{9} - \beta_{10} + 9 \beta_{11} ) q^{83} + ( 2 \beta_{5} - 16 \beta_{8} + 8 \beta_{9} + 8 \beta_{14} ) q^{85} + ( -46 - 5 \beta_{1} + 3 \beta_{4} - 4 \beta_{6} - \beta_{7} + 2 \beta_{12} + 7 \beta_{13} - \beta_{15} ) q^{87} + ( -3 \beta_{1} + 9 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} - \beta_{12} + \beta_{13} - 8 \beta_{15} ) q^{89} + ( -7 \beta_{3} - 6 \beta_{5} + 4 \beta_{8} - 2 \beta_{9} + 5 \beta_{14} ) q^{91} + ( -8 \beta_{2} + \beta_{3} - 12 \beta_{5} - 10 \beta_{8} + 3 \beta_{9} - 5 \beta_{11} - 2 \beta_{14} ) q^{93} + ( 5 \beta_{1} + 5 \beta_{6} - 3 \beta_{7} + 16 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{95} + ( 12 - \beta_{1} - 3 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{12} - 5 \beta_{13} ) q^{97} + ( -10 \beta_{2} + 5 \beta_{3} + 6 \beta_{5} - \beta_{8} + \beta_{9} - 4 \beta_{11} - \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{7} + O(q^{10})$$ $$16q + 16q^{7} - 32q^{15} + 80q^{25} + 112q^{31} + 16q^{33} - 208q^{39} + 144q^{49} + 384q^{55} + 80q^{57} - 528q^{63} + 160q^{73} + 816q^{79} + 144q^{81} - 736q^{87} + 192q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} - 7060 x^{7} + 17714 x^{6} - 25496 x^{5} + 24840 x^{4} - 17932 x^{3} + 11724 x^{2} - 7056 x + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-73605237118565390 \nu^{15} - 47633572835668896 \nu^{14} - 139688548619286130 \nu^{13} - 1233457767323771308 \nu^{12} + 529018760445806044 \nu^{11} - 1261831727497275602 \nu^{10} + 24100324820579854794 \nu^{9} + 7093423907199410714 \nu^{8} - 202037586301504503744 \nu^{7} + 61203334305725712986 \nu^{6} + 615363780535269695166 \nu^{5} - 1135776337355918990048 \nu^{4} + 1101140289291794144046 \nu^{3} - 795703763476158680914 \nu^{2} + 402147901328092287564 \nu - 544249893260957625489$$$$)/ 43737612461167616781$$ $$\beta_{2}$$ $$=$$ $$($$$$-99362376719317588592644 \nu^{15} + 157643615961293517755580 \nu^{14} - 654015690334010596895884 \nu^{13} + 213271506955874822687428 \nu^{12} - 995377489052373457680032 \nu^{11} + 3156489507901330068044614 \nu^{10} + 22834363346293880559522940 \nu^{9} - 29651658206534911182424006 \nu^{8} - 185396532173326788843206856 \nu^{7} + 434141249588399178636341586 \nu^{6} - 249135055907033040587373936 \nu^{5} - 192281761863540202775765666 \nu^{4} + 394936329964996076653983336 \nu^{3} - 333975898645644202702191118 \nu^{2} - 104408253736098716785228332 \nu - 88262663720799637363681324$$$$)/$$$$53\!\cdots\!97$$ $$\beta_{3}$$ $$=$$ $$($$$$189224219934601032264355 \nu^{15} - 640202263351630099843971 \nu^{14} + 1553120990278286891368081 \nu^{13} - 3621706423247909764322422 \nu^{12} + 1682269426299819388064648 \nu^{11} - 18840296358896891889846508 \nu^{10} - 40628742994151317163457289 \nu^{9} + 110448656540655743708472337 \nu^{8} + 320568277591574100193142280 \nu^{7} - 1216821803698994193762710484 \nu^{6} + 1574248391414505437517732471 \nu^{5} - 1486026968627483519601442321 \nu^{4} + 1132201301739918183217506765 \nu^{3} - 1417240917087734151396424793 \nu^{2} + 998815185235190819052860430 \nu - 303265474878736101234979238$$$$)/$$$$53\!\cdots\!97$$ $$\beta_{4}$$ $$=$$ $$($$$$3640490453456954097858 \nu^{15} - 16696296122407843524494 \nu^{14} + 51123108622447450476286 \nu^{13} - 115611452811381285277960 \nu^{12} + 161505766591708053420364 \nu^{11} - 415306479428912573409582 \nu^{10} - 265512122939865771157942 \nu^{9} + 2864130346548477692928604 \nu^{8} + 2134124510400861462739424 \nu^{7} - 29166355503786286734932230 \nu^{6} + 69392925277420838243005046 \nu^{5} - 93214027840038103299879880 \nu^{4} + 78452027971271711503230598 \nu^{3} - 44569897710476397997173428 \nu^{2} + 27020900329861475554330964 \nu - 16533045201889393898502397$$$$)/$$$$69\!\cdots\!61$$ $$\beta_{5}$$ $$=$$ $$($$$$72684469693760 \nu^{15} - 224840478635056 \nu^{14} + 808003344635480 \nu^{13} - 1282221816062488 \nu^{12} + 2428767128904824 \nu^{11} - 5282208866934684 \nu^{10} - 9590853089188752 \nu^{9} + 38143824965723228 \nu^{8} + 76037812918776760 \nu^{7} - 448764552639400932 \nu^{6} + 870266063458707304 \nu^{5} - 1015163612568399884 \nu^{4} + 845369308904828096 \nu^{3} - 555948001648106716 \nu^{2} + 399142132155581296 \nu - 187269373357533880$$$$)/ 11795069338202997$$ $$\beta_{6}$$ $$=$$ $$($$$$-30964682511934783503810 \nu^{15} + 122223497851645985487134 \nu^{14} - 351087091130861083128466 \nu^{13} + 722317551925352307477868 \nu^{12} - 830174841462470676732352 \nu^{11} + 2767985707332668051783076 \nu^{10} + 4213575685143668590694674 \nu^{9} - 22382276527890369452376370 \nu^{8} - 32569021563057742201550024 \nu^{7} + 234941060542798650808228840 \nu^{6} - 435187087035768768804873974 \nu^{5} + 449020567409998748155182748 \nu^{4} - 249729491763515335269157846 \nu^{3} + 96943521582465721562442746 \nu^{2} - 112158128752047556736724332 \nu + 73635341491264992735750559$$$$)/$$$$48\!\cdots\!27$$ $$\beta_{7}$$ $$=$$ $$($$$$35220176603425455380400 \nu^{15} - 32006703464375436767938 \nu^{14} + 238371309052252800044600 \nu^{13} + 36936950069573329739656 \nu^{12} + 521383090631562825694928 \nu^{11} - 932380116944940123230634 \nu^{10} - 8781276532560910172334860 \nu^{9} + 2724335024283234546960200 \nu^{8} + 59396498158646402591599588 \nu^{7} - 105822817363836738019230770 \nu^{6} + 79729091073366660116651896 \nu^{5} + 15786583481454523793225260 \nu^{4} - 103299492669661029430155556 \nu^{3} + 30926645260328445790809788 \nu^{2} + 21468729734104125727704124 \nu + 36707798999442881296523572$$$$)/$$$$48\!\cdots\!27$$ $$\beta_{8}$$ $$=$$ $$($$$$-496011314854612629237607 \nu^{15} + 1407423641116043044984739 \nu^{14} - 5236363327383127844285983 \nu^{13} + 7674579346797568635198524 \nu^{12} - 15170187401690208428739514 \nu^{11} + 33483168646838080662868536 \nu^{10} + 74147872428604737899472801 \nu^{9} - 233702847930081684199658665 \nu^{8} - 558017997097579641379188182 \nu^{7} + 2872720671982035201104284980 \nu^{6} - 5364246117903744242824857047 \nu^{5} + 6097981717619334251998696003 \nu^{4} - 4691038047271369875397782463 \nu^{3} + 2830934339193932474639480993 \nu^{2} - 1935825001234129541998511804 \nu + 948191375097728418362664476$$$$)/$$$$53\!\cdots\!97$$ $$\beta_{9}$$ $$=$$ $$($$$$168205981222649866459993 \nu^{15} - 722283452909459301095925 \nu^{14} + 2200838562375756723362047 \nu^{13} - 4632646557071839349088766 \nu^{12} + 6619724633870462883487076 \nu^{11} - 16474709494587523434622366 \nu^{10} - 14166933236368567892115763 \nu^{9} + 130623547662795020307028957 \nu^{8} + 129084446355814662641230164 \nu^{7} - 1332938102861827597833812202 \nu^{6} + 2849430799189762299032510721 \nu^{5} - 3465449247703949276546491501 \nu^{4} + 2786498808998386784147595327 \nu^{3} - 1596024018956010006398663969 \nu^{2} + 1098557546541429389083110714 \nu - 531388868455846783559913470$$$$)/$$$$17\!\cdots\!99$$ $$\beta_{10}$$ $$=$$ $$($$$$551433367755288482572036 \nu^{15} - 1408219547486122305150558 \nu^{14} + 5735315644397360890020964 \nu^{13} - 7688654266282923636617572 \nu^{12} + 17692316089595233177136120 \nu^{11} - 35945008775855893630086130 \nu^{10} - 85094055795357376986267568 \nu^{9} + 217705740186354083996814700 \nu^{8} + 624628548695658880225303860 \nu^{7} - 2887010849285275188911969358 \nu^{6} + 5532641255779921708628630832 \nu^{5} - 6902862737843796856509056704 \nu^{4} + 6125863483899854452295642124 \nu^{3} - 4068255658692241415900074508 \nu^{2} + 2284234538899424073865200552 \nu - 797051948299789731534756680$$$$)/$$$$53\!\cdots\!97$$ $$\beta_{11}$$ $$=$$ $$($$$$11913187010808341876886 \nu^{15} - 23975944496966615836928 \nu^{14} + 106642177371262500986628 \nu^{13} - 98301957290644270684438 \nu^{12} + 290568605512493599763378 \nu^{11} - 573606376298445507057866 \nu^{10} - 2228428785917414061095450 \nu^{9} + 3750630518098402823426118 \nu^{8} + 16446624534719448243157154 \nu^{7} - 54982088196114388725677778 \nu^{6} + 84288446412857750839274966 \nu^{5} - 81148429500991178855466900 \nu^{4} + 52304577766421834673743632 \nu^{3} - 33176743048971708020631918 \nu^{2} + 26181132013428199701150194 \nu - 1355563446291770005494690$$$$)/$$$$10\!\cdots\!53$$ $$\beta_{12}$$ $$=$$ $$($$$$55024520198138880566010 \nu^{15} - 137783157785429009305526 \nu^{14} + 563850788327923100286814 \nu^{13} - 679041821146738360614520 \nu^{12} + 1686037666090608468184996 \nu^{11} - 3052264204664955485878014 \nu^{10} - 8430821140772456286131806 \nu^{9} + 22863780092201702657803180 \nu^{8} + 63574381834947111771677288 \nu^{7} - 297132976451917827250801510 \nu^{6} + 537165476707658079331848830 \nu^{5} - 570509132045867792137860712 \nu^{4} + 415165850424849819875437414 \nu^{3} - 232158640560184879151842796 \nu^{2} + 190211240740289941355047052 \nu - 79874676786308848296228229$$$$)/$$$$48\!\cdots\!27$$ $$\beta_{13}$$ $$=$$ $$($$$$31817579385645488535444 \nu^{15} - 88206778931937685372070 \nu^{14} + 340478365886124775238104 \nu^{13} - 477367827448243316342072 \nu^{12} + 1021272473055663921364308 \nu^{11} - 2064820988020967478333898 \nu^{10} - 4644466413137907833313952 \nu^{9} + 14475875407010959495107452 \nu^{8} + 34387091160570332254797960 \nu^{7} - 181898556717941278052465898 \nu^{6} + 347302706535448206066655268 \nu^{5} - 395686452807303374493405652 \nu^{4} + 306148472751434611559085668 \nu^{3} - 188198004344939422253821672 \nu^{2} + 121084587780806107656104808 \nu - 55028840081614189320893790$$$$)/$$$$16\!\cdots\!09$$ $$\beta_{14}$$ $$=$$ $$($$$$1944509727790417716480887 \nu^{15} - 6114785714223635371831415 \nu^{14} + 21504052543851716703107285 \nu^{13} - 35505982914351192419445234 \nu^{12} + 63068059782007057113198960 \nu^{11} - 147950314564777996000852984 \nu^{10} - 265454012643265202130323149 \nu^{9} + 1029607431255999657920021909 \nu^{8} + 2064908151313945678083008888 \nu^{7} - 12029737929564727753564018440 \nu^{6} + 23399319641003821026377525075 \nu^{5} - 27921974802000853211099091929 \nu^{4} + 22502383378998811767386967937 \nu^{3} - 14520845490988459183442145385 \nu^{2} + 10403553731305134146927457278 \nu - 4957377610955501313794460850$$$$)/$$$$53\!\cdots\!97$$ $$\beta_{15}$$ $$=$$ $$($$$$212457904015996007722308 \nu^{15} - 685391664748231303553084 \nu^{14} + 2377104280783034795722072 \nu^{13} - 3994202810179077315449392 \nu^{12} + 6956242868611039002966124 \nu^{11} - 16326080972690283552664140 \nu^{10} - 28253227622269491985415116 \nu^{9} + 116809191218979853253523988 \nu^{8} + 221342755050324522216294260 \nu^{7} - 1345616710449714098644922284 \nu^{6} + 2627341790001511721947146452 \nu^{5} - 3102265819152201974954549176 \nu^{4} + 2471016010147884355794089104 \nu^{3} - 1540230428741537897930332472 \nu^{2} + 1082580752062076026913195444 \nu - 518021587644632004999733864$$$$)/$$$$48\!\cdots\!27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-6 \beta_{15} + 6 \beta_{14} - 12 \beta_{13} + 9 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} - 6 \beta_{8} - 9 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} + 12 \beta_{1} + 24$$$$)/96$$ $$\nu^{2}$$ $$=$$ $$($$$$6 \beta_{13} + 12 \beta_{12} + 2 \beta_{9} + 24 \beta_{8} + 6 \beta_{6} - 21 \beta_{5} + 12 \beta_{4} + 6 \beta_{3} - 24 \beta_{2} + 6 \beta_{1} - 36$$$$)/48$$ $$\nu^{3}$$ $$=$$ $$($$$$21 \beta_{15} - 18 \beta_{14} + 18 \beta_{13} - 75 \beta_{11} + 9 \beta_{10} + 8 \beta_{9} + 21 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} - 12 \beta_{5} - 3 \beta_{3} - 12 \beta_{2} - 39 \beta_{1} - 60$$$$)/48$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{15} + 8 \beta_{13} - 29 \beta_{12} - 4 \beta_{9} + 4 \beta_{7} - 12 \beta_{6} + 60 \beta_{5} + 3 \beta_{4} - 12 \beta_{3} - 12 \beta_{1}$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$336 \beta_{15} - 258 \beta_{14} - 816 \beta_{13} - 45 \beta_{12} + 1320 \beta_{11} - 96 \beta_{10} - 368 \beta_{9} - 684 \beta_{8} - 522 \beta_{7} - 6 \beta_{6} + 225 \beta_{5} - 81 \beta_{4} - 90 \beta_{3} + 246 \beta_{2} + 894 \beta_{1} + 1584$$$$)/96$$ $$\nu^{6}$$ $$=$$ $$($$$$-24 \beta_{15} + 102 \beta_{14} - 702 \beta_{13} + 1485 \beta_{12} + 90 \beta_{11} - 42 \beta_{10} - 458 \beta_{9} - 978 \beta_{8} - 582 \beta_{7} - 36 \beta_{6} - 882 \beta_{5} - 135 \beta_{4} + 558 \beta_{3} + 1350 \beta_{2} + 168 \beta_{1} + 5724$$$$)/48$$ $$\nu^{7}$$ $$=$$ $$($$$$-5622 \beta_{15} + 4230 \beta_{14} + 2424 \beta_{13} + 3285 \beta_{12} - 4938 \beta_{11} - 666 \beta_{10} + 5732 \beta_{9} - 6246 \beta_{8} + 1404 \beta_{7} + 2424 \beta_{6} - 8445 \beta_{5} - 207 \beta_{4} + 2436 \beta_{3} - 1602 \beta_{2} - 1764 \beta_{1} - 3168$$$$)/96$$ $$\nu^{8}$$ $$=$$ $$($$$$711 \beta_{13} - 241 \beta_{12} - 204 \beta_{11} + 240 \beta_{10} + 664 \beta_{9} + 1992 \beta_{8} + 470 \beta_{7} + 241 \beta_{6} - 777 \beta_{5} + 241 \beta_{4} - 1314 \beta_{2} + 15 \beta_{1} - 7826$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$25887 \beta_{15} - 19854 \beta_{14} + 17676 \beta_{13} - 25875 \beta_{12} - 22821 \beta_{11} - 1305 \beta_{10} - 18458 \beta_{9} + 49875 \beta_{8} + 16473 \beta_{7} - 4923 \beta_{6} + 38136 \beta_{5} + 7461 \beta_{4} - 13011 \beta_{3} - 8790 \beta_{2} - 23115 \beta_{1} - 29760$$$$)/48$$ $$\nu^{10}$$ $$=$$ $$($$$$9960 \beta_{15} - 15954 \beta_{14} - 56760 \beta_{13} - 81057 \beta_{12} + 29322 \beta_{11} - 9402 \beta_{10} - 66508 \beta_{9} - 164586 \beta_{8} - 21378 \beta_{7} - 54186 \beta_{6} + 221763 \beta_{5} - 31053 \beta_{4} - 50208 \beta_{3} + 112110 \beta_{2} - 22278 \beta_{1} + 559224$$$$)/48$$ $$\nu^{11}$$ $$=$$ $$($$$$-213516 \beta_{15} + 164130 \beta_{14} - 643848 \beta_{13} + 307989 \beta_{12} + 1017084 \beta_{11} - 58836 \beta_{10} - 118232 \beta_{9} - 1040016 \beta_{8} - 592170 \beta_{7} - 59310 \beta_{6} - 69249 \beta_{5} - 120111 \beta_{4} + 115278 \beta_{3} + 418986 \beta_{2} + 688350 \beta_{1} + 2317416$$$$)/96$$ $$\nu^{12}$$ $$=$$ $$($$$$-26632 \beta_{15} + 39942 \beta_{14} - 34232 \beta_{13} + 240323 \beta_{12} + 54690 \beta_{9} - 11016 \beta_{8} - 42390 \beta_{7} + 76622 \beta_{6} - 393759 \beta_{5} + 2299 \beta_{4} + 113112 \beta_{3} + 76622 \beta_{1}$$$$)/8$$ $$\nu^{13}$$ $$=$$ $$($$$$-1932090 \beta_{15} + 1515762 \beta_{14} + 6297528 \beta_{13} + 952899 \beta_{12} - 9514074 \beta_{11} + 591966 \beta_{10} + 4469908 \beta_{9} + 6649074 \beta_{8} + 4687188 \beta_{7} + 1789152 \beta_{6} - 6060903 \beta_{5} + 1037199 \beta_{4} + 1192080 \beta_{3} - 4602438 \beta_{2} - 5285364 \beta_{1} - 25007400$$$$)/96$$ $$\nu^{14}$$ $$=$$ $$($$$$1690992 \beta_{15} - 2026332 \beta_{14} + 7796130 \beta_{13} - 11968518 \beta_{12} - 4451004 \beta_{11} + 1069020 \beta_{10} + 2404114 \beta_{9} + 14662884 \beta_{8} + 6394044 \beta_{7} - 735954 \beta_{6} + 9964353 \beta_{5} + 1929762 \beta_{4} - 4700778 \beta_{3} - 10038756 \beta_{2} - 4788618 \beta_{1} - 52309692$$$$)/48$$ $$\nu^{15}$$ $$=$$ $$($$$$20908563 \beta_{15} - 16832388 \beta_{14} - 10200522 \beta_{13} - 24458910 \beta_{12} + 18616767 \beta_{11} - 824673 \beta_{10} - 25152404 \beta_{9} + 7965531 \beta_{8} - 4527519 \beta_{7} - 11743989 \beta_{6} + 52696743 \beta_{5} - 224274 \beta_{4} - 14701467 \beta_{3} + 10462974 \beta_{2} + 2479347 \beta_{1} + 52459200$$$$)/48$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 1.38361 − 0.573111i 1.38361 + 0.573111i 0.463678 + 1.11942i 0.463678 − 1.11942i −0.280847 + 0.678024i −0.280847 − 0.678024i 0.926870 − 0.383922i 0.926870 + 0.383922i −1.09921 − 2.65372i −1.09921 + 2.65372i −2.18318 − 0.904303i −2.18318 + 0.904303i 1.57980 − 0.654376i 1.57980 + 0.654376i 1.20927 + 2.91944i 1.20927 − 2.91944i
0 −2.98985 0.246559i 0 6.63641 0 0.578158 0 8.87842 + 1.47435i 0
641.2 0 −2.98985 + 0.246559i 0 6.63641 0 0.578158 0 8.87842 1.47435i 0
641.3 0 −2.55118 1.57844i 0 −1.31534 0 −10.2329 0 4.01705 + 8.05378i 0
641.4 0 −2.55118 + 1.57844i 0 −1.31534 0 −10.2329 0 4.01705 8.05378i 0
641.5 0 −1.32750 2.69031i 0 −0.640013 0 2.72077 0 −5.47550 + 7.14275i 0
641.6 0 −1.32750 + 2.69031i 0 −0.640013 0 2.72077 0 −5.47550 7.14275i 0
641.7 0 −0.888828 2.86531i 0 −8.59176 0 10.9340 0 −7.41997 + 5.09353i 0
641.8 0 −0.888828 + 2.86531i 0 −8.59176 0 10.9340 0 −7.41997 5.09353i 0
641.9 0 0.888828 2.86531i 0 8.59176 0 10.9340 0 −7.41997 5.09353i 0
641.10 0 0.888828 + 2.86531i 0 8.59176 0 10.9340 0 −7.41997 + 5.09353i 0
641.11 0 1.32750 2.69031i 0 0.640013 0 2.72077 0 −5.47550 7.14275i 0
641.12 0 1.32750 + 2.69031i 0 0.640013 0 2.72077 0 −5.47550 + 7.14275i 0
641.13 0 2.55118 1.57844i 0 1.31534 0 −10.2329 0 4.01705 8.05378i 0
641.14 0 2.55118 + 1.57844i 0 1.31534 0 −10.2329 0 4.01705 + 8.05378i 0
641.15 0 2.98985 0.246559i 0 −6.63641 0 0.578158 0 8.87842 1.47435i 0
641.16 0 2.98985 + 0.246559i 0 −6.63641 0 0.578158 0 8.87842 + 1.47435i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 641.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.h.h 16
3.b odd 2 1 inner 768.3.h.h 16
4.b odd 2 1 768.3.h.g 16
8.b even 2 1 inner 768.3.h.h 16
8.d odd 2 1 768.3.h.g 16
12.b even 2 1 768.3.h.g 16
16.e even 4 1 384.3.e.a 8
16.e even 4 1 384.3.e.c yes 8
16.f odd 4 1 384.3.e.b yes 8
16.f odd 4 1 384.3.e.d yes 8
24.f even 2 1 768.3.h.g 16
24.h odd 2 1 inner 768.3.h.h 16
48.i odd 4 1 384.3.e.a 8
48.i odd 4 1 384.3.e.c yes 8
48.k even 4 1 384.3.e.b yes 8
48.k even 4 1 384.3.e.d yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.e.a 8 16.e even 4 1
384.3.e.a 8 48.i odd 4 1
384.3.e.b yes 8 16.f odd 4 1
384.3.e.b yes 8 48.k even 4 1
384.3.e.c yes 8 16.e even 4 1
384.3.e.c yes 8 48.i odd 4 1
384.3.e.d yes 8 16.f odd 4 1
384.3.e.d yes 8 48.k even 4 1
768.3.h.g 16 4.b odd 2 1
768.3.h.g 16 8.d odd 2 1
768.3.h.g 16 12.b even 2 1
768.3.h.g 16 24.f even 2 1
768.3.h.h 16 1.a even 1 1 trivial
768.3.h.h 16 3.b odd 2 1 inner
768.3.h.h 16 8.b even 2 1 inner
768.3.h.h 16 24.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{8} - 120 T_{5}^{6} + 3504 T_{5}^{4} - 7040 T_{5}^{2} + 2304$$ $$T_{7}^{4} - 4 T_{7}^{3} - 108 T_{7}^{2} + 368 T_{7} - 176$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$43046721 - 236196 T^{4} - 41472 T^{6} + 4230 T^{8} - 512 T^{10} - 36 T^{12} + T^{16}$$
$5$ $$( 2304 - 7040 T^{2} + 3504 T^{4} - 120 T^{6} + T^{8} )^{2}$$
$7$ $$( -176 + 368 T - 108 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$11$ $$( 20214016 - 3167360 T^{2} + 69552 T^{4} - 488 T^{6} + T^{8} )^{2}$$
$13$ $$( 61214976 + 5698816 T^{2} + 145248 T^{4} + 720 T^{6} + T^{8} )^{2}$$
$17$ $$( 991494144 + 60416000 T^{2} + 480000 T^{4} + 1248 T^{6} + T^{8} )^{2}$$
$19$ $$( 9673115904 + 155767936 T^{2} + 833712 T^{4} + 1704 T^{6} + T^{8} )^{2}$$
$23$ $$( 48132849664 + 478846976 T^{2} + 1671936 T^{4} + 2336 T^{6} + T^{8} )^{2}$$
$29$ $$( 78767790336 - 896411520 T^{2} + 3074736 T^{4} - 3704 T^{6} + T^{8} )^{2}$$
$31$ $$( 50256 + 17520 T - 828 T^{2} - 28 T^{3} + T^{4} )^{4}$$
$37$ $$( 238565818624 + 4824434944 T^{2} + 14380128 T^{4} + 7312 T^{6} + T^{8} )^{2}$$
$41$ $$( 3916979306496 + 31405105152 T^{2} + 34597632 T^{4} + 10976 T^{6} + T^{8} )^{2}$$
$43$ $$( 841666795776 + 7476619392 T^{2} + 11090352 T^{4} + 5800 T^{6} + T^{8} )^{2}$$
$47$ $$( 424144797696 + 3025141760 T^{2} + 6905856 T^{4} + 5376 T^{6} + T^{8} )^{2}$$
$53$ $$( 870911869798656 - 673606710144 T^{2} + 188831664 T^{4} - 22776 T^{6} + T^{8} )^{2}$$
$59$ $$( 12745356964096 - 31548679808 T^{2} + 26455344 T^{4} - 8840 T^{6} + T^{8} )^{2}$$
$61$ $$( 21019364057344 + 48727118080 T^{2} + 37553760 T^{4} + 10768 T^{6} + T^{8} )^{2}$$
$67$ $$( 46033728153856 + 124646806144 T^{2} + 81127728 T^{4} + 17800 T^{6} + T^{8} )^{2}$$
$71$ $$( 1706597351424 + 36124434432 T^{2} + 77642496 T^{4} + 22688 T^{6} + T^{8} )^{2}$$
$73$ $$( -899312 + 192608 T - 5544 T^{2} - 40 T^{3} + T^{4} )^{4}$$
$79$ $$( -38762928 + 676400 T + 7524 T^{2} - 204 T^{3} + T^{4} )^{4}$$
$83$ $$( 56638749360384 - 102766133376 T^{2} + 59473584 T^{4} - 13416 T^{6} + T^{8} )^{2}$$
$89$ $$( 9213001971859456 + 4688908992512 T^{2} + 730308096 T^{4} + 45632 T^{6} + T^{8} )^{2}$$
$97$ $$( 8416272 + 274112 T - 8136 T^{2} - 48 T^{3} + T^{4} )^{4}$$