Properties

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

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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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} -\beta_{9} q^{5} + ( -1 - \beta_{2} ) q^{7} + ( \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} -\beta_{9} q^{5} + ( -1 - \beta_{2} ) q^{7} + ( \beta_{1} - \beta_{2} ) q^{9} + ( \beta_{7} + \beta_{9} ) q^{11} + ( -\beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{13} ) q^{13} + ( 2 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{12} - \beta_{15} ) q^{15} + ( -\beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{11} + \beta_{15} ) q^{17} + ( -\beta_{4} - \beta_{5} + \beta_{13} - \beta_{14} ) q^{19} + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{13} - \beta_{14} ) q^{21} + ( -2 \beta_{1} + \beta_{2} - \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{23} + ( 5 + \beta_{1} + 3 \beta_{2} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{25} + ( -2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - 3 \beta_{9} - \beta_{14} ) q^{27} + ( 2 \beta_{3} + 6 \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{13} ) q^{29} + ( -7 + 2 \beta_{1} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{31} + ( 1 - 2 \beta_{1} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{33} + ( -4 \beta_{3} + 3 \beta_{5} - \beta_{7} + 2 \beta_{8} + 7 \beta_{9} + \beta_{13} ) q^{35} + ( \beta_{4} - 6 \beta_{5} + \beta_{13} + 2 \beta_{14} ) q^{37} + ( 13 - 3 \beta_{1} + \beta_{2} + 3 \beta_{6} - 2 \beta_{10} - \beta_{11} - \beta_{15} ) q^{39} + ( 2 \beta_{1} - \beta_{6} + 5 \beta_{10} - 2 \beta_{12} - 3 \beta_{15} ) q^{41} + ( 3 \beta_{4} + \beta_{5} + 6 \beta_{8} - \beta_{13} - \beta_{14} ) q^{43} + ( -5 \beta_{3} + 3 \beta_{5} - \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{13} - \beta_{14} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{6} - 3 \beta_{10} + 2 \beta_{11} ) q^{47} + ( 9 - 3 \beta_{1} + 3 \beta_{2} - \beta_{6} - \beta_{10} - \beta_{11} - 4 \beta_{12} ) q^{49} + ( 6 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{7} - 10 \beta_{8} - 5 \beta_{9} + \beta_{13} ) q^{51} + ( -8 \beta_{3} + 12 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + 4 \beta_{13} ) q^{53} + ( -24 - 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{55} + ( 5 - \beta_{1} + 3 \beta_{2} + 3 \beta_{6} - 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{15} ) q^{57} + ( 8 \beta_{3} - 3 \beta_{5} - 4 \beta_{8} - \beta_{13} ) q^{59} + ( -\beta_{4} - 10 \beta_{5} + 2 \beta_{8} + 3 \beta_{13} + 2 \beta_{14} ) q^{61} + ( 33 + 3 \beta_{2} - 3 \beta_{6} + 5 \beta_{10} - 4 \beta_{12} - 2 \beta_{15} ) q^{63} + ( 2 \beta_{2} + 7 \beta_{6} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{65} + ( 5 \beta_{5} + 10 \beta_{8} - 3 \beta_{13} + 4 \beta_{14} ) q^{67} + ( 9 \beta_{3} + \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{13} + 3 \beta_{14} ) q^{69} + ( 4 \beta_{1} - \beta_{2} - 3 \beta_{6} + 7 \beta_{10} + \beta_{11} - 3 \beta_{12} + 2 \beta_{15} ) q^{71} + ( 10 - 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{73} + ( -8 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{7} - 8 \beta_{8} - \beta_{9} - \beta_{13} + 5 \beta_{14} ) q^{75} + ( 8 \beta_{3} + 12 \beta_{5} + 4 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{13} ) q^{77} + ( -51 - 2 \beta_{1} + 3 \beta_{6} + 3 \beta_{10} + 3 \beta_{11} + \beta_{12} ) q^{79} + ( 9 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{6} + 5 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{81} + ( -4 \beta_{3} - 6 \beta_{5} - \beta_{7} + 2 \beta_{8} - 9 \beta_{9} - 2 \beta_{13} ) q^{83} + ( -16 \beta_{5} + 2 \beta_{8} + 8 \beta_{13} - 8 \beta_{14} ) q^{85} + ( 46 + 7 \beta_{1} - 5 \beta_{2} + 2 \beta_{6} - 3 \beta_{10} + 4 \beta_{11} - \beta_{12} - \beta_{15} ) q^{87} + ( -\beta_{1} + 3 \beta_{2} + \beta_{6} + 9 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 8 \beta_{15} ) q^{89} + ( -7 \beta_{4} - 4 \beta_{5} + 6 \beta_{8} + 2 \beta_{13} + 5 \beta_{14} ) q^{91} + ( -8 \beta_{3} - \beta_{4} - 10 \beta_{5} - 4 \beta_{8} - 5 \beta_{9} + 3 \beta_{13} + 2 \beta_{14} ) q^{93} + ( -2 \beta_{1} + 5 \beta_{2} + 16 \beta_{6} - 5 \beta_{11} - 3 \beta_{12} + 2 \beta_{15} ) q^{95} + ( 12 + 5 \beta_{1} + \beta_{2} - 3 \beta_{6} - 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} ) q^{97} + ( 10 \beta_{3} + 5 \beta_{4} + \beta_{5} - 16 \beta_{8} + 4 \beta_{9} - \beta_{13} - \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 \(x^{16} - 4 x^{15} + 14 x^{14} - 28 x^{13} + 50 x^{12} - 104 x^{11} - 66 x^{10} + 640 x^{9} + 555 x^{8} - 7060 x^{7} + 17714 x^{6} - 25496 x^{5} + 24840 x^{4} - 17932 x^{3} + 11724 x^{2} - 7056 x + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-566862203411123630952 \nu^{15} - 24596850319687148065498 \nu^{14} + 31986679955180037165596 \nu^{13} - 175989961158120001904648 \nu^{12} + 94575406157735378286932 \nu^{11} - 302395664081805849187938 \nu^{10} + 1360236687693649749902812 \nu^{9} + 6318318031153194229138628 \nu^{8} - 4995708608434525709605928 \nu^{7} - 41036849046439204403549354 \nu^{6} + 86101668205869548404416256 \nu^{5} - 90516260002504758373545068 \nu^{4} + 66382956995526699323432768 \nu^{3} + 3348941406271578947371772 \nu^{2} + 8432842812409739274374800 \nu - 16770232351270099881643730\)\()/ \)\(48\!\cdots\!27\)\( \)
\(\beta_{2}\)\(=\)\((\)\(73605237118565390 \nu^{15} + 47633572835668896 \nu^{14} + 139688548619286130 \nu^{13} + 1233457767323771308 \nu^{12} - 529018760445806044 \nu^{11} + 1261831727497275602 \nu^{10} - 24100324820579854794 \nu^{9} - 7093423907199410714 \nu^{8} + 202037586301504503744 \nu^{7} - 61203334305725712986 \nu^{6} - 615363780535269695166 \nu^{5} + 1135776337355918990048 \nu^{4} - 1101140289291794144046 \nu^{3} + 795703763476158680914 \nu^{2} - 402147901328092287564 \nu + 544249893260957625489\)\()/ 43737612461167616781 \)
\(\beta_{3}\)\(=\)\((\)\(99362376719317588592644 \nu^{15} - 157643615961293517755580 \nu^{14} + 654015690334010596895884 \nu^{13} - 213271506955874822687428 \nu^{12} + 995377489052373457680032 \nu^{11} - 3156489507901330068044614 \nu^{10} - 22834363346293880559522940 \nu^{9} + 29651658206534911182424006 \nu^{8} + 185396532173326788843206856 \nu^{7} - 434141249588399178636341586 \nu^{6} + 249135055907033040587373936 \nu^{5} + 192281761863540202775765666 \nu^{4} - 394936329964996076653983336 \nu^{3} + 333975898645644202702191118 \nu^{2} + 104408253736098716785228332 \nu + 88262663720799637363681324\)\()/ \)\(53\!\cdots\!97\)\( \)
\(\beta_{4}\)\(=\)\((\)\(138581113697361730513259 \nu^{15} - 44937924854953644044843 \nu^{14} + 1482403774729084229555855 \nu^{13} + 579773633522180478212664 \nu^{12} + 6868193341519926157187790 \nu^{11} + 1831837206646334661600338 \nu^{10} - 19352196198084852843899749 \nu^{9} - 7369356273369079815842629 \nu^{8} + 108365273150190878544815738 \nu^{7} - 322960765421213409507762294 \nu^{6} + 940566927299476506274613855 \nu^{5} - 1146505501287901455850762181 \nu^{4} + 772337936533064908032680371 \nu^{3} + 182330596849811683155607769 \nu^{2} - 161547730542490666137459340 \nu - 113537031763145533567771768\)\()/ \)\(53\!\cdots\!97\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-168205981222649866459993 \nu^{15} + 722283452909459301095925 \nu^{14} - 2200838562375756723362047 \nu^{13} + 4632646557071839349088766 \nu^{12} - 6619724633870462883487076 \nu^{11} + 16474709494587523434622366 \nu^{10} + 14166933236368567892115763 \nu^{9} - 130623547662795020307028957 \nu^{8} - 129084446355814662641230164 \nu^{7} + 1332938102861827597833812202 \nu^{6} - 2849430799189762299032510721 \nu^{5} + 3465449247703949276546491501 \nu^{4} - 2786498808998386784147595327 \nu^{3} + 1596024018956010006398663969 \nu^{2} - 1098557546541429389083110714 \nu + 531388868455846783559913470\)\()/ \)\(53\!\cdots\!97\)\( \)
\(\beta_{6}\)\(=\)\((\)\(6790401932708182938854 \nu^{15} - 27059868000455474824030 \nu^{14} + 97830933106830087692206 \nu^{13} - 179996270019397229483176 \nu^{12} + 356693056433770088067712 \nu^{11} - 605829474546069837819814 \nu^{10} - 336593765301731954566586 \nu^{9} + 4607042345588424627234784 \nu^{8} + 3057864094911744963227876 \nu^{7} - 50091103347213961609340462 \nu^{6} + 123778239142807290270260226 \nu^{5} - 165259818508272519185846968 \nu^{4} + 150694128699661383327386734 \nu^{3} - 88811409075594967963341452 \nu^{2} + 53436556064592025450989376 \nu - 33270747811493876570369357\)\()/ \)\(16\!\cdots\!09\)\( \)
\(\beta_{7}\)\(=\)\((\)\(6513008970418506155080 \nu^{15} - 11018753819774313163018 \nu^{14} + 55151477852948657886232 \nu^{13} - 41860490565361010185488 \nu^{12} + 155209392964191379095072 \nu^{11} - 306098914916847928635740 \nu^{10} - 1309499576510442259114844 \nu^{9} + 1496073740349176369527222 \nu^{8} + 9450316606705609400250244 \nu^{7} - 25988677324848921357649788 \nu^{6} + 36954355156331010797118136 \nu^{5} - 42416375610402556040640946 \nu^{4} + 38235057495590049928335908 \nu^{3} - 24438696818464476779749826 \nu^{2} + 11578823514892955026274212 \nu + 2973846310014844840669276\)\()/ \)\(10\!\cdots\!53\)\( \)
\(\beta_{8}\)\(=\)\((\)\(72684469693760 \nu^{15} - 224840478635056 \nu^{14} + 808003344635480 \nu^{13} - 1282221816062488 \nu^{12} + 2428767128904824 \nu^{11} - 5282208866934684 \nu^{10} - 9590853089188752 \nu^{9} + 38143824965723228 \nu^{8} + 76037812918776760 \nu^{7} - 448764552639400932 \nu^{6} + 870266063458707304 \nu^{5} - 1015163612568399884 \nu^{4} + 845369308904828096 \nu^{3} - 555948001648106716 \nu^{2} + 399142132155581296 \nu - 187269373357533880\)\()/ 11795069338202997 \)
\(\beta_{9}\)\(=\)\((\)\(-11913187010808341876886 \nu^{15} + 23975944496966615836928 \nu^{14} - 106642177371262500986628 \nu^{13} + 98301957290644270684438 \nu^{12} - 290568605512493599763378 \nu^{11} + 573606376298445507057866 \nu^{10} + 2228428785917414061095450 \nu^{9} - 3750630518098402823426118 \nu^{8} - 16446624534719448243157154 \nu^{7} + 54982088196114388725677778 \nu^{6} - 84288446412857750839274966 \nu^{5} + 81148429500991178855466900 \nu^{4} - 52304577766421834673743632 \nu^{3} + 33176743048971708020631918 \nu^{2} - 26181132013428199701150194 \nu + 1355563446291770005494690\)\()/ \)\(10\!\cdots\!53\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-8590963939173287204922 \nu^{15} + 24782518091559641357842 \nu^{14} - 89745678480652141506314 \nu^{13} + 135476168681173667015816 \nu^{12} - 249499837561607796846344 \nu^{11} + 591703019575590569469378 \nu^{10} + 1325660672206617260076806 \nu^{9} - 4155937925896538946657008 \nu^{8} - 9905665874716843874452804 \nu^{7} + 50146307848111421366757962 \nu^{6} - 93083033745883153746014782 \nu^{5} + 103889695914473851097068424 \nu^{4} - 73178237160681092916555914 \nu^{3} + 39673385329533537320290348 \nu^{2} - 31292553550792027697485096 \nu + 13684821395007567982090991\)\()/ \)\(69\!\cdots\!61\)\( \)
\(\beta_{11}\)\(=\)\((\)\(24715216064579606069486 \nu^{15} - 79460875658277059697770 \nu^{14} + 277024202851992353422070 \nu^{13} - 474786203606817420148460 \nu^{12} + 816380125847849561708776 \nu^{11} - 1969395595180132411073940 \nu^{10} - 3352397998641378076052014 \nu^{9} + 13273079500814614399892122 \nu^{8} + 25709033962120106517407536 \nu^{7} - 153191271875729544128575552 \nu^{6} + 307218073267831130526672458 \nu^{5} - 380664176575104821593938596 \nu^{4} + 309754167628716953270813298 \nu^{3} - 185137221179726910144009926 \nu^{2} + 105733915211096336853554044 \nu - 46686519020778643421797319\)\()/ \)\(16\!\cdots\!09\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-130106052556950797355780 \nu^{15} + 321223890579875640949646 \nu^{14} - 1291793086665807162924508 \nu^{13} + 1571156493433276621191208 \nu^{12} - 3679775915956289968074784 \nu^{11} + 7429238745089648407420266 \nu^{10} + 21354439084280983922373904 \nu^{9} - 52470279276469307261421184 \nu^{8} - 157562063031922873646387540 \nu^{7} + 692555336564099776580177818 \nu^{6} - 1207738878885580826721033956 \nu^{5} + 1261789034942960358060536764 \nu^{4} - 881528882580169504570534216 \nu^{3} + 530318426368218242023283456 \nu^{2} - 393155335888932187970393348 \nu + 145148953596669786547801528\)\()/ \)\(48\!\cdots\!27\)\( \)
\(\beta_{13}\)\(=\)\((\)\(496011314854612629237607 \nu^{15} - 1407423641116043044984739 \nu^{14} + 5236363327383127844285983 \nu^{13} - 7674579346797568635198524 \nu^{12} + 15170187401690208428739514 \nu^{11} - 33483168646838080662868536 \nu^{10} - 74147872428604737899472801 \nu^{9} + 233702847930081684199658665 \nu^{8} + 558017997097579641379188182 \nu^{7} - 2872720671982035201104284980 \nu^{6} + 5364246117903744242824857047 \nu^{5} - 6097981717619334251998696003 \nu^{4} + 4691038047271369875397782463 \nu^{3} - 2830934339193932474639480993 \nu^{2} + 1935825001234129541998511804 \nu - 948191375097728418362664476\)\()/ \)\(17\!\cdots\!99\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-1616704394158454953703273 \nu^{15} + 5429645526017051627942601 \nu^{14} - 18468527778844345582183349 \nu^{13} + 32464050124625463133335476 \nu^{12} - 54517597014187311567946522 \nu^{11} + 130941855412527438772606814 \nu^{10} + 205473073451029032122966111 \nu^{9} - 926528130988712994027392201 \nu^{8} - 1635974600572180699345050870 \nu^{7} + 10489955360444520150293545662 \nu^{6} - 20884504322289839082585178749 \nu^{5} + 25289442332085468235646887427 \nu^{4} - 20597844140725828676136780801 \nu^{3} + 13285935170750536715201328361 \nu^{2} - 9566286276612433994012056188 \nu + 4540575104313619678991709844\)\()/ \)\(53\!\cdots\!97\)\( \)
\(\beta_{15}\)\(=\)\((\)\(212457904015996007722308 \nu^{15} - 685391664748231303553084 \nu^{14} + 2377104280783034795722072 \nu^{13} - 3994202810179077315449392 \nu^{12} + 6956242868611039002966124 \nu^{11} - 16326080972690283552664140 \nu^{10} - 28253227622269491985415116 \nu^{9} + 116809191218979853253523988 \nu^{8} + 221342755050324522216294260 \nu^{7} - 1345616710449714098644922284 \nu^{6} + 2627341790001511721947146452 \nu^{5} - 3102265819152201974954549176 \nu^{4} + 2471016010147884355794089104 \nu^{3} - 1540230428741537897930332472 \nu^{2} + 1082580752062076026913195444 \nu - 518021587644632004999733864\)\()/ \)\(48\!\cdots\!27\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-6 \beta_{15} - 6 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - 8 \beta_{11} - 5 \beta_{10} - 6 \beta_{9} - 3 \beta_{8} + 6 \beta_{7} + \beta_{6} - 18 \beta_{5} - 4 \beta_{2} + 24\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(-6 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} - 21 \beta_{8} + 20 \beta_{6} - 6 \beta_{4} + 24 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} - 36\)\()/48\)
\(\nu^{3}\)\(=\)\((\)\(21 \beta_{15} + 18 \beta_{14} - 14 \beta_{13} - 25 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} + 75 \beta_{9} - 3 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} - 45 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 55 \beta_{2} - 48 \beta_{1} - 60\)\()/48\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{15} - 4 \beta_{13} + 4 \beta_{12} + 12 \beta_{11} - 3 \beta_{10} + 60 \beta_{8} - 29 \beta_{6} + 12 \beta_{4} - 12 \beta_{2} + 8 \beta_{1}\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(336 \beta_{15} + 258 \beta_{14} + 112 \beta_{13} + 706 \beta_{12} - 110 \beta_{11} - 35 \beta_{10} - 1320 \beta_{9} + 129 \beta_{8} - 96 \beta_{7} - 161 \beta_{6} + 756 \beta_{5} + 90 \beta_{4} - 150 \beta_{3} - 790 \beta_{2} + 528 \beta_{1} + 1584\)\()/96\)
\(\nu^{6}\)\(=\)\((\)\(-24 \beta_{15} - 102 \beta_{14} + 546 \beta_{13} + 122 \beta_{12} - 580 \beta_{11} - 481 \beta_{10} - 90 \beta_{9} - 924 \beta_{8} - 42 \beta_{7} + 869 \beta_{6} + 2034 \beta_{5} - 558 \beta_{4} - 1308 \beta_{3} + 376 \beta_{2} + 618 \beta_{1} + 5724\)\()/48\)
\(\nu^{7}\)\(=\)\((\)\(-5622 \beta_{15} - 4230 \beta_{14} + 4304 \beta_{13} - 3248 \beta_{12} - 824 \beta_{11} + 1807 \beta_{10} + 4938 \beta_{9} - 9111 \beta_{8} - 666 \beta_{7} + 4885 \beta_{6} - 10530 \beta_{5} - 2436 \beta_{4} + 2268 \beta_{3} + 5012 \beta_{2} - 3828 \beta_{1} - 3168\)\()/96\)
\(\nu^{8}\)\(=\)\((\)\(-664 \beta_{13} - 470 \beta_{12} + 241 \beta_{11} + 241 \beta_{10} + 204 \beta_{9} - 537 \beta_{8} + 240 \beta_{7} + 241 \beta_{6} - 1992 \beta_{5} + 1074 \beta_{3} - 15 \beta_{2} - 711 \beta_{1} - 7826\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(25887 \beta_{15} + 19854 \beta_{14} - 27580 \beta_{13} - 6163 \beta_{12} + 11513 \beta_{11} - 871 \beta_{10} + 22821 \beta_{9} + 36831 \beta_{8} - 1305 \beta_{7} - 19285 \beta_{6} + 22509 \beta_{5} + 13011 \beta_{4} + 10095 \beta_{3} + 6679 \beta_{2} - 11550 \beta_{1} - 29760\)\()/48\)
\(\nu^{10}\)\(=\)\((\)\(9960 \beta_{15} + 15954 \beta_{14} + 32808 \beta_{13} + 61634 \beta_{12} + 4874 \beta_{11} - 18259 \beta_{10} - 29322 \beta_{9} + 212361 \beta_{8} - 9402 \beta_{7} - 130369 \beta_{6} + 133362 \beta_{5} + 50208 \beta_{4} - 102708 \beta_{3} - 36782 \beta_{2} + 75564 \beta_{1} + 559224\)\()/48\)
\(\nu^{11}\)\(=\)\((\)\(-213516 \beta_{15} - 164130 \beta_{14} + 439808 \beta_{13} + 483922 \beta_{12} - 159926 \beta_{11} - 99125 \beta_{10} - 1017084 \beta_{9} - 128085 \beta_{8} - 58836 \beta_{7} + 88753 \beta_{6} + 634104 \beta_{5} - 115278 \beta_{4} - 360150 \beta_{3} - 587734 \beta_{2} + 651480 \beta_{1} + 2317416\)\()/96\)
\(\nu^{12}\)\(=\)\((\)\(-26632 \beta_{15} - 39942 \beta_{14} + 54690 \beta_{13} - 42390 \beta_{12} - 76622 \beta_{11} - 2299 \beta_{10} - 393759 \beta_{8} + 240323 \beta_{6} - 11016 \beta_{5} - 113112 \beta_{4} + 76622 \beta_{2} - 34232 \beta_{1}\)\()/8\)
\(\nu^{13}\)\(=\)\((\)\(-1932090 \beta_{15} - 1515762 \beta_{14} - 1313744 \beta_{13} - 5732648 \beta_{12} + 564880 \beta_{11} + 1316833 \beta_{10} + 9514074 \beta_{9} - 5468937 \beta_{8} + 591966 \beta_{7} + 3306931 \beta_{6} - 10701882 \beta_{5} - 1192080 \beta_{4} + 4010472 \beta_{3} + 6509636 \beta_{2} - 6476340 \beta_{1} - 25007400\)\()/96\)
\(\nu^{14}\)\(=\)\((\)\(1690992 \beta_{15} + 2026332 \beta_{14} - 7129998 \beta_{13} - 2696008 \beta_{12} + 5100122 \beta_{11} + 2434406 \beta_{10} + 4451004 \beta_{9} + 11033373 \beta_{8} + 1069020 \beta_{7} - 7604350 \beta_{6} - 13939452 \beta_{5} + 4700778 \beta_{4} + 8969736 \beta_{3} - 1047458 \beta_{2} - 5658090 \beta_{1} - 52309692\)\()/48\)
\(\nu^{15}\)\(=\)\((\)\(20908563 \beta_{15} + 16832388 \beta_{14} - 13166462 \beta_{13} + 16387733 \beta_{12} + 6187211 \beta_{11} - 5332504 \beta_{10} - 18616767 \beta_{9} + 51872070 \beta_{8} - 824673 \beta_{7} - 30015688 \beta_{6} + 43923357 \beta_{5} + 14701467 \beta_{4} - 9638301 \beta_{3} - 20410547 \beta_{2} + 16271508 \beta_{1} + 52459200\)\()/48\)

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:

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

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

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