# Properties

 Label 1568.3.c.g Level 1568 Weight 3 Character orbit 1568.c Analytic conductor 42.725 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1568.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.7249054517$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{28}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + \beta_{1} q^{5} + ( -5 - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} + \beta_{1} q^{5} + ( -5 - \beta_{4} ) q^{9} + ( \beta_{5} - \beta_{12} ) q^{11} + ( -\beta_{1} - \beta_{8} - \beta_{10} ) q^{13} + ( \beta_{5} + \beta_{12} - \beta_{13} ) q^{15} + ( 2 \beta_{1} + 2 \beta_{8} + \beta_{14} ) q^{17} + ( -2 \beta_{6} + \beta_{15} ) q^{19} + ( -2 \beta_{5} + \beta_{11} + \beta_{12} ) q^{23} + ( -10 - 2 \beta_{4} + 4 \beta_{7} - 2 \beta_{9} ) q^{25} + ( -2 \beta_{2} - 6 \beta_{6} - \beta_{15} ) q^{27} + ( -1 + \beta_{4} - \beta_{7} + 2 \beta_{9} ) q^{29} + ( -\beta_{3} - \beta_{6} ) q^{31} + ( 2 \beta_{1} - 11 \beta_{8} - 2 \beta_{10} - 2 \beta_{14} ) q^{33} + ( -9 - \beta_{4} + 4 \beta_{7} - 3 \beta_{9} ) q^{37} + ( -5 \beta_{5} - \beta_{11} + \beta_{13} ) q^{39} + ( -5 \beta_{8} - 2 \beta_{10} + 3 \beta_{14} ) q^{41} + ( 2 \beta_{11} + 2 \beta_{12} ) q^{43} + ( -13 \beta_{1} - 13 \beta_{8} - 7 \beta_{10} + 2 \beta_{14} ) q^{45} + ( -2 \beta_{2} + \beta_{3} + 9 \beta_{6} - 2 \beta_{15} ) q^{47} + ( 7 \beta_{5} + \beta_{12} - 2 \beta_{13} ) q^{51} + ( -5 + \beta_{4} + 6 \beta_{7} + 3 \beta_{9} ) q^{53} + ( -3 \beta_{2} + \beta_{3} + 7 \beta_{6} + \beta_{15} ) q^{55} + ( 23 + 6 \beta_{7} - 4 \beta_{9} ) q^{57} + ( -2 \beta_{2} + 2 \beta_{3} - 5 \beta_{6} ) q^{59} + ( -6 \beta_{1} + 9 \beta_{8} + \beta_{10} - 4 \beta_{14} ) q^{61} + ( 21 + 3 \beta_{4} + 4 \beta_{7} ) q^{65} + ( 9 \beta_{5} + 2 \beta_{12} ) q^{67} + ( -7 \beta_{1} + 22 \beta_{8} - 6 \beta_{10} + 6 \beta_{14} ) q^{69} + ( 4 \beta_{5} + 2 \beta_{11} - 4 \beta_{12} ) q^{71} + ( -2 \beta_{1} + 13 \beta_{8} - 8 \beta_{10} + 8 \beta_{14} ) q^{73} + ( 2 \beta_{2} - 2 \beta_{3} - 22 \beta_{6} - 2 \beta_{15} ) q^{75} + ( -2 \beta_{5} + \beta_{11} - 3 \beta_{12} - 2 \beta_{13} ) q^{79} + ( 48 + 5 \beta_{4} + 20 \beta_{7} - 4 \beta_{9} ) q^{81} + ( 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{6} + 4 \beta_{15} ) q^{83} + ( -67 - 3 \beta_{4} + \beta_{9} ) q^{85} + ( -\beta_{2} + 2 \beta_{3} + 4 \beta_{6} + \beta_{15} ) q^{87} + ( 4 \beta_{1} + \beta_{8} + 14 \beta_{10} ) q^{89} + ( 21 + 5 \beta_{4} + 7 \beta_{9} ) q^{93} + ( -12 \beta_{5} + \beta_{11} + 3 \beta_{12} ) q^{95} + ( -8 \beta_{1} - 37 \beta_{8} - 6 \beta_{10} + 7 \beta_{14} ) q^{97} + ( -12 \beta_{5} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 80q^{9} + O(q^{10})$$ $$16q - 80q^{9} - 160q^{25} - 16q^{29} - 144q^{37} - 80q^{53} + 368q^{57} + 336q^{65} + 768q^{81} - 1072q^{85} + 336q^{93} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-8161932957484190343665273 \nu^{15} - 44439187972352536048517534 \nu^{14} + 285301886851739906742316053 \nu^{13} + 1112658279033969221325555216 \nu^{12} - 4814066789420379222535386834 \nu^{11} - 19373728230501465036814065604 \nu^{10} + 62897025351342307275446286156 \nu^{9} + 109397774130481521364257407586 \nu^{8} - 727155270139808660498290086006 \nu^{7} - 358917417736567570954159885656 \nu^{6} + 2663256405889607525415415377260 \nu^{5} - 3729807910919652247800742026214 \nu^{4} - 5104731955019441877585892054673 \nu^{3} + 3653225388259983436664143334390 \nu^{2} - 8403237557696221261342105089459 \nu + 7385860188895574436656992849363$$$$)/$$$$38\!\cdots\!33$$ $$\beta_{2}$$ $$=$$ $$($$$$-48938175240500379766654 \nu^{15} + 1018957400875668923115623 \nu^{14} + 835513112149775321093486 \nu^{13} - 23340323391993423584168283 \nu^{12} - 31382921689458755274762972 \nu^{11} + 422624529274148148210626700 \nu^{10} + 197405447555300005513129088 \nu^{9} - 3858789009588295178665052992 \nu^{8} + 3862874530793152901565835124 \nu^{7} + 24367659029785269951151662472 \nu^{6} - 25736669438053538668038727184 \nu^{5} - 2151112781037815870259768302 \nu^{4} + 101958581123719058935672801610 \nu^{3} - 13157434736815645928245008145 \nu^{2} + 29704651808335598695839562814 \nu - 53023773050719004012849091831$$$$)/$$$$17\!\cdots\!38$$ $$\beta_{3}$$ $$=$$ $$($$$$-27397357799034800546743 \nu^{15} + 1558767510707882111058699 \nu^{14} - 57160957226276830090407 \nu^{13} - 36625662950316359863762512 \nu^{12} - 24393580635427761438745686 \nu^{11} + 656801861543589915205908300 \nu^{10} + 82184390631076727554594640 \nu^{9} - 5718427399084193023996909296 \nu^{8} + 7363052759282056504592478762 \nu^{7} + 34759620118464264079239940438 \nu^{6} - 40592915588254877698413087792 \nu^{5} + 1806702359859996318952592874 \nu^{4} + 155317510002317643741095004221 \nu^{3} - 13272903298917380376827852085 \nu^{2} + 39835236593548766642892768057 \nu - 81435337105865920777942382070$$$$)/$$$$85\!\cdots\!69$$ $$\beta_{4}$$ $$=$$ $$($$$$175897262115818474166 \nu^{15} - 39700912768241219437 \nu^{14} - 4343208416441587651818 \nu^{13} - 2550483807844585241654 \nu^{12} + 77457246760524776820756 \nu^{11} + 16735111331723529813756 \nu^{10} - 685333887138161379193776 \nu^{9} + 746345850366636811491008 \nu^{8} + 4542541386163315725553572 \nu^{7} - 4695086562310980701080952 \nu^{6} - 2941504366413742418249552 \nu^{5} + 16539384619686044550131514 \nu^{4} - 22763625087679702581672874 \nu^{3} + 22641835053978957045423419 \nu^{2} - 19254675737460801888584922 \nu - 584502903330940031442410309$$$$)/$$$$44\!\cdots\!59$$ $$\beta_{5}$$ $$=$$ $$($$$$172377580471175462040822 \nu^{15} + 212941530455158990989389 \nu^{14} - 4750870910547457913371838 \nu^{13} - 8575075886270639069659203 \nu^{12} + 83520657283426435935288420 \nu^{11} + 136031226314224464198067740 \nu^{10} - 850858232324077448090833992 \nu^{9} - 333463624084075292528090992 \nu^{8} + 7392719700656286585102150460 \nu^{7} + 828047657109777994066016808 \nu^{6} - 22830414462343664397603819704 \nu^{5} + 23325051206048404659865393490 \nu^{4} + 45252411702446011114980056022 \nu^{3} - 71624910281823322549240613167 \nu^{2} + 11496599637194480127236068690 \nu - 43231667292315975023081438847$$$$)/$$$$40\!\cdots\!26$$ $$\beta_{6}$$ $$=$$ $$($$$$87428286339522377943258 \nu^{15} - 19379552843836720073999 \nu^{14} - 2106362162310834996966530 \nu^{13} - 1203281739031657890480621 \nu^{12} + 37567134568520370754247220 \nu^{11} + 8063335103197233251269908 \nu^{10} - 327897110740258740307119480 \nu^{9} + 362422523161540239768290176 \nu^{8} + 2202446682825009641121408292 \nu^{7} - 1724782533273118586212671000 \nu^{6} - 1423302805080590158619506672 \nu^{5} + 8021116023302597726209967294 \nu^{4} + 9591582687049399777297302042 \nu^{3} + 10981892368336939613161395337 \nu^{2} - 9341327103233175018294227090 \nu - 3686481877316456037642549777$$$$)/$$$$17\!\cdots\!38$$ $$\beta_{7}$$ $$=$$ $$($$$$-351675045716698 \nu^{15} + 84134512548244 \nu^{14} + 8997681016250754 \nu^{13} + 4445925380328116 \nu^{12} - 160496373722473788 \nu^{11} - 33874814941456464 \nu^{10} + 1467376262123572708 \nu^{9} - 1553113751898048416 \nu^{8} - 9401792267255930460 \nu^{7} + 10200336255773301928 \nu^{6} + 6044728226531394560 \nu^{5} - 34262171840112781800 \nu^{4} + 46812769807129599650 \nu^{3} - 46922766637244959676 \nu^{2} + 39937799358222111906 \nu + 240280225958040430804$$$$)/ 45742705762379025631$$ $$\beta_{8}$$ $$=$$ $$($$$$248197947118550681400 \nu^{15} + 60099576570060185087 \nu^{14} - 6110172991187791613532 \nu^{13} - 5610165377095918036704 \nu^{12} + 107123109897527970241392 \nu^{11} + 59136687103919879272608 \nu^{10} - 966301607286958942241280 \nu^{9} + 861897353924990552955686 \nu^{8} + 7059892149907340387084928 \nu^{7} - 4980438225173244013489056 \nu^{6} - 5677044395320575039843008 \nu^{5} + 40095443257835374287069696 \nu^{4} + 13407128082587646721614288 \nu^{3} - 5618854490528863822096597 \nu^{2} + 33962809459441010571298260 \nu - 35592347497900753218122943$$$$)/$$$$24\!\cdots\!53$$ $$\beta_{9}$$ $$=$$ $$($$$$-631161306625356779617 \nu^{15} + 150505234538692630725 \nu^{14} + 16500767577615813975981 \nu^{13} + 7419045073782499938700 \nu^{12} - 294271138216409917948542 \nu^{11} - 63717891799250964885804 \nu^{10} + 2750860404617240971368368 \nu^{9} - 2834324749199366123952240 \nu^{8} - 17259606953295549222511902 \nu^{7} + 19235282507483501622825296 \nu^{6} + 11183900242770749268101232 \nu^{5} - 62836954235436549118005834 \nu^{4} + 85357101295984988485619803 \nu^{3} - 86018275869553972168815051 \nu^{2} + 73144181174155227325195629 \nu + 327917369601661120754045677$$$$)/$$$$44\!\cdots\!59$$ $$\beta_{10}$$ $$=$$ $$($$$$-107274637135815474213267623 \nu^{15} + 40440310585103025551191222 \nu^{14} + 2509657568814871159359711123 \nu^{13} + 970451882572829151852741190 \nu^{12} - 44287196461473841197989452438 \nu^{11} + 1749022612776647015483582528 \nu^{10} + 369261641263869769858666117920 \nu^{9} - 588689505051165656418132450628 \nu^{8} - 2281678710917973731743246639250 \nu^{7} + 2977319895231402273537853555688 \nu^{6} - 1436373880362576535141563797140 \nu^{5} - 13591200467210556482109017274192 \nu^{4} + 1529669380689354048836851412717 \nu^{3} - 3020774732138613521871714908018 \nu^{2} - 3259759170842048260323290216941 \nu + 5648824684655197559037381780752$$$$)/$$$$38\!\cdots\!33$$ $$\beta_{11}$$ $$=$$ $$($$$$1384364485533756987372876 \nu^{15} - 728768123650168136784649 \nu^{14} - 35908961898282986992714064 \nu^{13} - 472622045688405467009363 \nu^{12} + 658611626622471479423905864 \nu^{11} - 200247607529712853033798708 \nu^{10} - 6422914281900997227071722896 \nu^{9} + 10421619794726874885163185080 \nu^{8} + 42733803632554989723902755304 \nu^{7} - 75737920268645070599379844900 \nu^{6} - 46248813080430974509909364096 \nu^{5} + 348113674580553310579782267406 \nu^{4} - 10929072089602277258238994884 \nu^{3} - 188529834431551320226184211949 \nu^{2} - 272111133428858037728504337072 \nu - 74244525959635624072959944923$$$$)/$$$$40\!\cdots\!26$$ $$\beta_{12}$$ $$=$$ $$($$$$150357722430704850757448 \nu^{15} + 60075198633659217891900 \nu^{14} - 4028295388456629283530112 \nu^{13} - 4156719365271510392861673 \nu^{12} + 72661216673303491288979384 \nu^{11} + 56042699511608033717055244 \nu^{10} - 723159070367915974211775172 \nu^{9} + 303091942964013600088722988 \nu^{8} + 5658945163844004844545788056 \nu^{7} - 3056389735529669058506661454 \nu^{6} - 13723062855387011820116672024 \nu^{5} + 27731587236969783024247115660 \nu^{4} + 22370325291012881548177717004 \nu^{3} - 44749936787945170476365911956 \nu^{2} - 6602631145691352621161837336 \nu - 25283341868433551078267787519$$$$)/$$$$29\!\cdots\!09$$ $$\beta_{13}$$ $$=$$ $$($$$$1239986586879849068452267 \nu^{15} + 446611647240208516332506 \nu^{14} - 33172367614068978380700341 \nu^{13} - 31579135335561447776626614 \nu^{12} + 595924841732233384601337874 \nu^{11} + 409213280791450479744422696 \nu^{10} - 5938580497464899800282006892 \nu^{9} + 3285672381231400343517307320 \nu^{8} + 45850802484701684033452880102 \nu^{7} - 29875274922222088196649094936 \nu^{6} - 104669895774889538889453816484 \nu^{5} + 237551379027465268422197142796 \nu^{4} + 162608902664302276970871278667 \nu^{3} - 346649214519037359708573123958 \nu^{2} - 75641461576722423956309523053 \nu - 192223036278949580449972700050$$$$)/$$$$20\!\cdots\!63$$ $$\beta_{14}$$ $$=$$ $$($$$$-267063145461629828893524862 \nu^{15} - 170988238389309380274226828 \nu^{14} + 6785108178874501898339214806 \nu^{13} + 8460206761577540248730427138 \nu^{12} - 118392865570954626911808600220 \nu^{11} - 109560588073441856744426807204 \nu^{10} + 1114274795933677419493107459288 \nu^{9} - 553827182119669552363089749314 \nu^{8} - 8791627040505418243644692523444 \nu^{7} + 3847948136217721768781922320392 \nu^{6} + 12900955768739878151945366278840 \nu^{5} - 48759551313770278706482388246430 \nu^{4} - 27203310203993899440149864599118 \nu^{3} + 15595962883292813976143238673104 \nu^{2} - 56018922452006827133376345549914 \nu + 54779321279479592533993047161903$$$$)/$$$$38\!\cdots\!33$$ $$\beta_{15}$$ $$=$$ $$($$$$-1724264633383462570433182 \nu^{15} - 2048070062977697491721035 \nu^{14} + 42252483719801659728501638 \nu^{13} + 81709401945744643690593333 \nu^{12} - 713861670067369782746122716 \nu^{11} - 1187832908714722814839439772 \nu^{10} + 6406234029560720562724506800 \nu^{9} + 1681944271704283897258555520 \nu^{8} - 55593224006113528318853699116 \nu^{7} - 23875599108767871968486476460 \nu^{6} + 91927619257102208322535794256 \nu^{5} - 163382527833887830227336488522 \nu^{4} - 453859953990828211998061613734 \nu^{3} - 199082612062867058370460165843 \nu^{2} + 124732580360973654215585228342 \nu + 207448752980846201883934977429$$$$)/$$$$17\!\cdots\!38$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} - 2 \beta_{10} - 13 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{1} + 13$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} - 6 \beta_{9} + 19 \beta_{7} - 37 \beta_{6} + 6 \beta_{4} + 2 \beta_{3} - 11 \beta_{2} + 24$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} - 5 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 4 \beta_{11} - 34 \beta_{10} + 2 \beta_{9} - 131 \beta_{8} - 10 \beta_{7} - 46 \beta_{6} - 48 \beta_{5} - 13 \beta_{4} - 4 \beta_{2} - 40 \beta_{1} - 131$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{15} + 22 \beta_{14} - 66 \beta_{13} + 27 \beta_{12} + 39 \beta_{11} - 277 \beta_{10} - 65 \beta_{9} - 680 \beta_{8} + 172 \beta_{7} - 290 \beta_{6} + 303 \beta_{5} + 85 \beta_{4} + 9 \beta_{3} - 57 \beta_{2} - 472 \beta_{1} + 680$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-43 \beta_{15} + 48 \beta_{9} - 163 \beta_{7} - 921 \beta_{6} - 135 \beta_{4} + 6 \beta_{3} - 122 \beta_{2} - 1261$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$268 \beta_{15} - \beta_{14} - 1033 \beta_{13} + 229 \beta_{12} + 733 \beta_{11} - 11339 \beta_{10} + 2156 \beta_{9} - 34104 \beta_{8} - 5935 \beta_{7} + 5873 \beta_{6} + 6141 \beta_{5} - 3780 \beta_{4} - 100 \beta_{3} + 933 \beta_{2} - 17807 \beta_{1} - 34104$$$$)/16$$ $$\nu^{8}$$ $$=$$ $$($$$$-788 \beta_{15} - 55 \beta_{14} - 2836 \beta_{13} + 472 \beta_{12} + 2164 \beta_{11} + 2392 \beta_{10} + 478 \beta_{9} + 6799 \beta_{8} - 1312 \beta_{7} - 16832 \beta_{6} + 17620 \beta_{5} - 779 \beta_{4} + 224 \beta_{3} - 2612 \beta_{2} + 3826 \beta_{1} - 6799$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$490 \beta_{15} + 33570 \beta_{9} - 97945 \beta_{7} + 9937 \beta_{6} - 70938 \beta_{4} + 526 \beta_{3} - 369 \beta_{2} - 660288$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$13059 \beta_{15} - 4297 \beta_{14} - 51234 \beta_{13} + 12057 \beta_{12} + 36840 \beta_{11} - 33095 \beta_{10} + 2292 \beta_{9} - 125022 \beta_{8} - 10459 \beta_{7} + 280281 \beta_{6} + 293340 \beta_{5} - 12464 \beta_{4} - 4798 \beta_{3} + 46436 \beta_{2} - 39971 \beta_{1} - 125022$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-32613 \beta_{15} + 74343 \beta_{14} - 152793 \beta_{13} + 54954 \beta_{12} + 97494 \beta_{11} + 1662216 \beta_{10} + 249249 \beta_{9} + 5445440 \beta_{8} - 753759 \beta_{7} - 709441 \beta_{6} + 742054 \beta_{5} - 578853 \beta_{4} + 18433 \beta_{3} - 134360 \beta_{2} + 2409963 \beta_{1} - 5445440$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$97132 \beta_{15} + 112200 \beta_{9} - 363803 \beta_{7} + 2090869 \beta_{6} - 308517 \beta_{4} - 39428 \beta_{3} + 355911 \beta_{2} - 2949878$$ $$\nu^{13}$$ $$=$$ $$($$$$2229108 \beta_{15} + 2332695 \beta_{14} - 9498203 \beta_{13} + 2810879 \beta_{12} + 6449951 \beta_{11} + 46693893 \beta_{10} - 6750900 \beta_{9} + 154362832 \beta_{8} + 20760401 \beta_{7} + 48161131 \beta_{6} + 50390239 \beta_{5} + 16342196 \beta_{4} - 1016084 \beta_{3} + 8482119 \beta_{2} + 66946593 \beta_{1} + 154362832$$$$)/16$$ $$\nu^{14}$$ $$=$$ $$($$$$2480988 \beta_{15} + 2503213 \beta_{14} + 10250770 \beta_{13} - 2807806 \beta_{12} - 7108918 \beta_{11} + 45953948 \beta_{10} + 6433504 \beta_{9} + 153106199 \beta_{8} - 20082678 \beta_{7} + 53474096 \beta_{6} - 55955084 \beta_{5} - 16152387 \beta_{4} - 1047284 \beta_{3} + 9203486 \beta_{2} + 65254460 \beta_{1} - 153106199$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$52228326 \beta_{15} - 75835062 \beta_{9} + 234972317 \beta_{7} + 1126170861 \beta_{6} + 186984966 \beta_{4} - 22305174 \beta_{3} + 194464767 \beta_{2} + 1769317544$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 3.86852 + 1.41699i −3.16141 + 2.64174i 2.08703 − 2.02145i −2.79414 − 0.796701i −1.90990 − 0.286185i 1.20279 − 1.51093i −0.162551 + 0.910345i 0.869658 − 0.314400i 0.869658 + 0.314400i −0.162551 − 0.910345i 1.20279 + 1.51093i −1.90990 + 0.286185i −2.79414 + 0.796701i 2.08703 + 2.02145i −3.16141 − 2.64174i 3.86852 − 1.41699i
0 5.73991i 0 6.30408i 0 0 0 −23.9466 0
97.2 0 5.73991i 0 6.30408i 0 0 0 −23.9466 0
97.3 0 3.98546i 0 9.01776i 0 0 0 −6.88390 0
97.4 0 3.98546i 0 9.01776i 0 0 0 −6.88390 0
97.5 0 2.54150i 0 4.11878i 0 0 0 2.54076 0
97.6 0 2.54150i 0 4.11878i 0 0 0 2.54076 0
97.7 0 0.842794i 0 1.40510i 0 0 0 8.28970 0
97.8 0 0.842794i 0 1.40510i 0 0 0 8.28970 0
97.9 0 0.842794i 0 1.40510i 0 0 0 8.28970 0
97.10 0 0.842794i 0 1.40510i 0 0 0 8.28970 0
97.11 0 2.54150i 0 4.11878i 0 0 0 2.54076 0
97.12 0 2.54150i 0 4.11878i 0 0 0 2.54076 0
97.13 0 3.98546i 0 9.01776i 0 0 0 −6.88390 0
97.14 0 3.98546i 0 9.01776i 0 0 0 −6.88390 0
97.15 0 5.73991i 0 6.30408i 0 0 0 −23.9466 0
97.16 0 5.73991i 0 6.30408i 0 0 0 −23.9466 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{8} + 56 T_{3}^{6} + 878 T_{3}^{4} + 3976 T_{3}^{2} + 2401$$ $$T_{5}^{8} + 140 T_{5}^{6} + 5558 T_{5}^{4} + 65260 T_{5}^{2} + 108241$$ $$T_{11}^{8} - 536 T_{11}^{6} + 80958 T_{11}^{4} - 2585576 T_{11}^{2} + 19018321$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 16 T^{2} + 122 T^{4} - 416 T^{6} + 331 T^{8} - 33696 T^{10} + 800442 T^{12} - 8503056 T^{14} + 43046721 T^{16} )^{2}$$
$5$ $$( 1 - 60 T^{2} + 2058 T^{4} - 53040 T^{6} + 1281491 T^{8} - 33150000 T^{10} + 803906250 T^{12} - 14648437500 T^{14} + 152587890625 T^{16} )^{2}$$
$7$ 1
$11$ $$( 1 + 432 T^{2} + 101770 T^{4} + 18091872 T^{6} + 2519133147 T^{8} + 264883097952 T^{10} + 21815303319370 T^{12} + 1355801058743472 T^{14} + 45949729863572161 T^{16} )^{2}$$
$13$ $$( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 4307359811040 T^{10} + 416424007224732 T^{12} - 24602777889339936 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 - 1468 T^{2} + 1106730 T^{4} - 537928880 T^{6} + 183685895411 T^{8} - 44928357986480 T^{10} + 7720280032677930 T^{12} - 855289444253289148 T^{14} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$( 1 - 1440 T^{2} + 1137690 T^{4} - 599664000 T^{6} + 244339668779 T^{8} - 78148812144000 T^{10} + 19322029836115290 T^{12} - 3187173483455271840 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$( 1 + 1648 T^{2} + 1698202 T^{4} + 1274662432 T^{6} + 740326951915 T^{8} + 356702809633312 T^{10} + 132987871826164762 T^{12} + 36115301063969489008 T^{14} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$( 1 + 4 T + 2608 T^{2} + 8572 T^{3} + 2977102 T^{4} + 7209052 T^{5} + 1844588848 T^{6} + 2379293284 T^{7} + 500246412961 T^{8} )^{4}$$
$31$ $$( 1 - 4944 T^{2} + 12041994 T^{4} - 18914445024 T^{6} + 21218074483547 T^{8} - 17467887183009504 T^{10} + 10270508755518297354 T^{12} -$$$$38\!\cdots\!84$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$( 1 + 36 T + 3954 T^{2} + 161088 T^{3} + 7052651 T^{4} + 220529472 T^{5} + 7410432594 T^{6} + 92366150724 T^{7} + 3512479453921 T^{8} )^{4}$$
$41$ $$( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 459871021169908160 T^{10} +$$$$42\!\cdots\!48$$$$T^{12} -$$$$24\!\cdots\!96$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 241961144887582552 T^{10} +$$$$29\!\cdots\!96$$$$T^{12} +$$$$23\!\cdots\!92$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 - 7200 T^{2} + 30753210 T^{4} - 100217929920 T^{6} + 252870771077579 T^{8} - 489031528489955520 T^{10} +$$$$73\!\cdots\!10$$$$T^{12} -$$$$83\!\cdots\!00$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$( 1 + 20 T + 6498 T^{2} + 242656 T^{3} + 21504539 T^{4} + 681620704 T^{5} + 51272345538 T^{6} + 443287222580 T^{7} + 62259690411361 T^{8} )^{4}$$
$59$ $$( 1 - 13056 T^{2} + 103559946 T^{4} - 567820281792 T^{6} + 2269772666794523 T^{8} - 6880483337595390912 T^{10} +$$$$15\!\cdots\!66$$$$T^{12} -$$$$23\!\cdots\!36$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$61$ $$( 1 - 19692 T^{2} + 191982810 T^{4} - 1196247954960 T^{6} + 5243298235324451 T^{8} - 16563058980951321360 T^{10} +$$$$36\!\cdots\!10$$$$T^{12} -$$$$52\!\cdots\!32$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$( 1 + 19248 T^{2} + 198660442 T^{4} + 1359268530912 T^{6} + 6980138907036075 T^{8} + 27390784637899952352 T^{10} +$$$$80\!\cdots\!22$$$$T^{12} +$$$$15\!\cdots\!28$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 67143192723001800472 T^{10} +$$$$20\!\cdots\!60$$$$T^{12} +$$$$39\!\cdots\!12$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 20204 T^{2} + 194066938 T^{4} - 1215991652240 T^{6} + 6486410485261123 T^{8} - 34532023994299709840 T^{10} +$$$$15\!\cdots\!78$$$$T^{12} -$$$$46\!\cdots\!84$$$$T^{14} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 + 34640 T^{2} + 596941050 T^{4} + 6494320935520 T^{6} + 48452484640984139 T^{8} +$$$$25\!\cdots\!20$$$$T^{10} +$$$$90\!\cdots\!50$$$$T^{12} +$$$$20\!\cdots\!40$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} -$$$$29\!\cdots\!28$$$$T^{10} +$$$$13\!\cdots\!20$$$$T^{12} -$$$$37\!\cdots\!48$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 17212 T^{2} + 211999818 T^{4} - 2253472306160 T^{6} + 21544216480380371 T^{8} -$$$$14\!\cdots\!60$$$$T^{10} +$$$$83\!\cdots\!58$$$$T^{12} -$$$$42\!\cdots\!52$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} -$$$$10\!\cdots\!56$$$$T^{10} +$$$$61\!\cdots\!40$$$$T^{12} -$$$$27\!\cdots\!20$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$