# Properties

 Label 847.2.b.e Level $847$ Weight $2$ Character orbit 847.b Analytic conductor $6.763$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} - 8 x^{14} + 48 x^{13} + 26 x^{12} - 288 x^{11} + 32 x^{10} + 968 x^{9} - 462 x^{8} - 1936 x^{7} + 1664 x^{6} + 1800 x^{5} - 2612 x^{4} + 16 x^{3} + 2112 x^{2} - 2032 x + 1252$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{10} ) q^{2} + ( -\beta_{5} + \beta_{14} + \beta_{15} ) q^{3} + ( -1 - \beta_{6} ) q^{4} -\beta_{5} q^{5} + ( -2 \beta_{2} - \beta_{9} + \beta_{12} ) q^{6} + ( -\beta_{3} + \beta_{7} ) q^{7} -\beta_{4} q^{8} + ( -2 + \beta_{6} - \beta_{8} - 2 \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{10} ) q^{2} + ( -\beta_{5} + \beta_{14} + \beta_{15} ) q^{3} + ( -1 - \beta_{6} ) q^{4} -\beta_{5} q^{5} + ( -2 \beta_{2} - \beta_{9} + \beta_{12} ) q^{6} + ( -\beta_{3} + \beta_{7} ) q^{7} -\beta_{4} q^{8} + ( -2 + \beta_{6} - \beta_{8} - 2 \beta_{11} ) q^{9} + ( -\beta_{2} + \beta_{7} - \beta_{9} + \beta_{12} ) q^{10} + ( 3 \beta_{5} - \beta_{14} ) q^{12} + ( 2 \beta_{2} - \beta_{7} - \beta_{9} - 2 \beta_{12} ) q^{13} + ( -2 + \beta_{1} - \beta_{11} + 2 \beta_{14} ) q^{14} + ( -1 - \beta_{6} - \beta_{11} ) q^{15} + ( -1 - \beta_{6} - \beta_{8} - \beta_{11} ) q^{16} + ( 2 \beta_{2} - \beta_{7} - 2 \beta_{9} + \beta_{12} ) q^{17} + ( 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{10} + \beta_{13} ) q^{18} + ( \beta_{2} + \beta_{7} - 2 \beta_{9} + 2 \beta_{12} ) q^{19} + ( 2 \beta_{1} + 2 \beta_{5} - \beta_{15} ) q^{20} + ( -3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{13} ) q^{21} + ( 3 - \beta_{6} - 2 \beta_{8} ) q^{23} + ( -2 \beta_{2} - \beta_{7} + 2 \beta_{9} ) q^{24} + ( 2 + \beta_{8} ) q^{25} + ( -2 \beta_{5} - \beta_{14} + 2 \beta_{15} ) q^{26} + ( -7 \beta_{1} - \beta_{5} - 2 \beta_{14} - 3 \beta_{15} ) q^{27} + ( \beta_{2} + \beta_{3} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{28} + ( \beta_{3} - \beta_{4} + 2 \beta_{10} + \beta_{13} ) q^{29} + ( 3 \beta_{3} - \beta_{4} - 2 \beta_{10} + \beta_{13} ) q^{30} + ( -3 \beta_{1} - 3 \beta_{5} - \beta_{15} ) q^{31} + ( 3 \beta_{3} - 4 \beta_{10} ) q^{32} + ( 4 \beta_{1} + 3 \beta_{5} - 2 \beta_{14} + 2 \beta_{15} ) q^{34} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{35} + ( 1 + \beta_{8} + 2 \beta_{11} ) q^{36} + ( 5 + 2 \beta_{6} - \beta_{8} - \beta_{11} ) q^{37} + ( 6 \beta_{1} + 5 \beta_{5} + \beta_{14} + \beta_{15} ) q^{38} + ( -4 \beta_{3} - \beta_{10} - \beta_{13} ) q^{39} + ( -\beta_{7} + \beta_{9} - \beta_{12} ) q^{40} + ( 3 \beta_{2} - 2 \beta_{7} - 3 \beta_{9} - 2 \beta_{12} ) q^{41} + ( -1 - 2 \beta_{1} + 3 \beta_{5} - 3 \beta_{8} - 2 \beta_{14} - 3 \beta_{15} ) q^{42} + ( \beta_{3} + 2 \beta_{4} - 3 \beta_{10} + 2 \beta_{13} ) q^{43} + ( -2 \beta_{1} + \beta_{5} - \beta_{14} - \beta_{15} ) q^{45} + ( -2 \beta_{3} + 5 \beta_{4} - 2 \beta_{10} - 2 \beta_{13} ) q^{46} + ( -4 \beta_{14} - 3 \beta_{15} ) q^{47} + ( -5 \beta_{1} + 4 \beta_{5} - 4 \beta_{14} - 2 \beta_{15} ) q^{48} + ( 1 + 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{11} + 2 \beta_{14} + 2 \beta_{15} ) q^{49} + ( -2 \beta_{3} - 3 \beta_{4} + 4 \beta_{10} + \beta_{13} ) q^{50} + ( -6 \beta_{4} + 3 \beta_{10} - 4 \beta_{13} ) q^{51} + ( -3 \beta_{2} + 4 \beta_{7} - \beta_{9} + \beta_{12} ) q^{52} + ( -2 + 3 \beta_{6} + 3 \beta_{8} ) q^{53} + ( 10 \beta_{2} + 2 \beta_{7} - 9 \beta_{9} ) q^{54} + ( -\beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} + \beta_{15} ) q^{56} + ( \beta_{3} - 9 \beta_{4} + 4 \beta_{10} - 2 \beta_{13} ) q^{57} + ( 1 - 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{11} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{5} + \beta_{14} + 3 \beta_{15} ) q^{59} + ( 7 - 2 \beta_{8} ) q^{60} + ( 2 \beta_{2} - \beta_{7} + 4 \beta_{9} + 2 \beta_{12} ) q^{61} + ( 2 \beta_{2} + 2 \beta_{7} - 7 \beta_{9} + 2 \beta_{12} ) q^{62} + ( -4 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} - \beta_{7} + 2 \beta_{9} - 3 \beta_{10} + \beta_{12} + \beta_{13} ) q^{63} + ( 8 + 2 \beta_{6} - 2 \beta_{8} - 3 \beta_{11} ) q^{64} + ( -2 \beta_{3} + \beta_{4} + 3 \beta_{10} - 2 \beta_{13} ) q^{65} + ( 1 - \beta_{6} + \beta_{8} ) q^{67} + ( -3 \beta_{2} + \beta_{7} + 7 \beta_{9} + 3 \beta_{12} ) q^{68} + ( -6 \beta_{1} - \beta_{5} - 3 \beta_{14} + 2 \beta_{15} ) q^{69} + ( 3 + \beta_{1} + 3 \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{15} ) q^{70} + ( -3 + 5 \beta_{6} - \beta_{8} - 3 \beta_{11} ) q^{71} + ( 3 \beta_{3} + 5 \beta_{4} - 3 \beta_{10} + \beta_{13} ) q^{72} + ( \beta_{2} + 4 \beta_{7} + \beta_{9} ) q^{73} + ( -6 \beta_{3} + 5 \beta_{4} + 5 \beta_{10} ) q^{74} + ( 3 \beta_{1} - 2 \beta_{5} + 5 \beta_{14} + 3 \beta_{15} ) q^{75} + ( -4 \beta_{7} + 7 \beta_{9} - \beta_{12} ) q^{76} + ( -7 + 2 \beta_{6} + \beta_{8} - 7 \beta_{11} ) q^{78} + ( -\beta_{3} + 5 \beta_{4} + 2 \beta_{10} + 2 \beta_{13} ) q^{79} + ( \beta_{1} + \beta_{5} - \beta_{14} - 2 \beta_{15} ) q^{80} + ( -3 - 4 \beta_{6} + 6 \beta_{8} + 11 \beta_{11} ) q^{81} + ( 2 \beta_{1} + \beta_{5} - 4 \beta_{14} + 3 \beta_{15} ) q^{82} + ( -2 \beta_{2} - \beta_{7} + \beta_{9} - 3 \beta_{12} ) q^{83} + ( 3 \beta_{2} - \beta_{3} + 5 \beta_{4} - 2 \beta_{7} + 2 \beta_{9} - 5 \beta_{10} - \beta_{13} ) q^{84} + ( -\beta_{3} - 3 \beta_{10} + \beta_{13} ) q^{85} + ( 3 - \beta_{6} + 4 \beta_{11} ) q^{86} + ( 5 \beta_{2} - 3 \beta_{7} - 4 \beta_{9} - 4 \beta_{12} ) q^{87} + ( -5 \beta_{1} + \beta_{5} + \beta_{14} - 4 \beta_{15} ) q^{89} + ( 4 \beta_{2} - \beta_{9} - \beta_{12} ) q^{90} + ( -2 - \beta_{1} - 2 \beta_{5} - 4 \beta_{6} + \beta_{8} + 2 \beta_{11} + \beta_{14} + 3 \beta_{15} ) q^{91} + ( -1 - 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{11} ) q^{92} + ( -3 - 5 \beta_{6} + 3 \beta_{8} + 5 \beta_{11} ) q^{93} + ( 2 \beta_{2} + 5 \beta_{7} + \beta_{9} + \beta_{12} ) q^{94} + ( 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{10} + 2 \beta_{13} ) q^{95} + ( 5 \beta_{2} + 5 \beta_{9} - 2 \beta_{12} ) q^{96} + ( -5 \beta_{1} + \beta_{5} - \beta_{14} - 3 \beta_{15} ) q^{97} + ( -2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{7} + 4 \beta_{9} + 3 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} - 40q^{9} + O(q^{10})$$ $$16q - 16q^{4} - 40q^{9} - 32q^{14} - 16q^{15} - 24q^{16} + 32q^{23} + 40q^{25} + 24q^{36} + 72q^{37} - 40q^{42} + 16q^{49} - 8q^{53} - 8q^{56} + 96q^{60} + 112q^{64} + 24q^{67} + 32q^{70} - 56q^{71} - 104q^{78} + 48q^{86} - 24q^{91} + 8q^{92} - 24q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 8 x^{14} + 48 x^{13} + 26 x^{12} - 288 x^{11} + 32 x^{10} + 968 x^{9} - 462 x^{8} - 1936 x^{7} + 1664 x^{6} + 1800 x^{5} - 2612 x^{4} + 16 x^{3} + 2112 x^{2} - 2032 x + 1252$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1806236810424 \nu^{15} + 33412641339072 \nu^{14} - 21271288918721 \nu^{13} - 426120789560307 \nu^{12} + 124186019484892 \nu^{11} + 3420001305687324 \nu^{10} - 791857644457854 \nu^{9} - 15020743812702819 \nu^{8} + 3309641403668932 \nu^{7} + 39948241880711572 \nu^{6} - 17882135837919002 \nu^{5} - 46968185611381616 \nu^{4} + 45804279696326656 \nu^{3} + 22376006410770642 \nu^{2} - 68019234201392832 \nu + 45100255204451442$$$$)/ 28235118674856242$$ $$\beta_{2}$$ $$=$$ $$($$$$-118561841142667 \nu^{15} - 258387126282021 \nu^{14} + 394285081798624 \nu^{13} + 11075219895323697 \nu^{12} - 3376523454252486 \nu^{11} - 110409382788319495 \nu^{10} + 34344959360432404 \nu^{9} + 559574244918250357 \nu^{8} - 186901941452851238 \nu^{7} - 1638942080886189696 \nu^{6} + 559753195558809430 \nu^{5} + 2794748912367024884 \nu^{4} - 999711612316333720 \nu^{3} - 2515081103771166870 \nu^{2} + 1553865884087417224 \nu + 625351269035571198$$$$)/ 1044699390969680954$$ $$\beta_{3}$$ $$=$$ $$($$$$-147861519361595 \nu^{15} - 1052658662623185 \nu^{14} + 4032523449595915 \nu^{13} + 15827200815118652 \nu^{12} - 43490965664325910 \nu^{11} - 109588363761678783 \nu^{10} + 243073021746734110 \nu^{9} + 425990905546002689 \nu^{8} - 839981856233667904 \nu^{7} - 913678225587381786 \nu^{6} + 1846906049735482844 \nu^{5} + 842688297290520002 \nu^{4} - 2995782885963161152 \nu^{3} + 427019367761917416 \nu^{2} + 2123303416347802048 \nu - 1445454391303932166$$$$)/ 1044699390969680954$$ $$\beta_{4}$$ $$=$$ $$($$$$335015184264499 \nu^{15} - 2641187613864247 \nu^{14} + 1259101271762677 \nu^{13} + 30207096081366034 \nu^{12} - 48485068224744562 \nu^{11} - 149976475235078035 \nu^{10} + 355041013077341522 \nu^{9} + 358286836648461350 \nu^{8} - 1292414113275723678 \nu^{7} - 257422080833817172 \nu^{6} + 2569097164393335486 \nu^{5} - 654781641764962126 \nu^{4} - 2658605579917768088 \nu^{3} + 1297866861892009338 \nu^{2} + 900669272610036840 \nu - 657582084604824454$$$$)/ 1044699390969680954$$ $$\beta_{5}$$ $$=$$ $$($$$$-11845355524648 \nu^{15} - 8185822681324 \nu^{14} + 283299612034374 \nu^{13} - 124417974421783 \nu^{12} - 2069849602607900 \nu^{11} + 930762133108183 \nu^{10} + 9007677824243690 \nu^{9} - 4643591183144488 \nu^{8} - 21715539074948568 \nu^{7} + 11950173123349290 \nu^{6} + 32116177816861078 \nu^{5} - 27880380362172282 \nu^{4} - 15944799250036648 \nu^{3} + 9950766815793644 \nu^{2} + 6162494629111028 \nu - 3431350478782242$$$$)/ 28235118674856242$$ $$\beta_{6}$$ $$=$$ $$($$$$11917111215328 \nu^{15} + 23807655997484 \nu^{14} - 453609661960810 \nu^{13} + 283428795844833 \nu^{12} + 3861654034082500 \nu^{11} - 3714091342323348 \nu^{10} - 16826031129469600 \nu^{9} + 18089878556151433 \nu^{8} + 37335803035848092 \nu^{7} - 40751494583154584 \nu^{6} - 35688154451700508 \nu^{5} + 47139720384856296 \nu^{4} - 12430545418044632 \nu^{3} - 15906229879009536 \nu^{2} + 28223231587510280 \nu - 16953126444816472$$$$)/ 28235118674856242$$ $$\beta_{7}$$ $$=$$ $$($$$$793929381913919 \nu^{15} - 5858565368540612 \nu^{14} - 606290786085145 \nu^{13} + 67256821086151432 \nu^{12} - 37806769628355878 \nu^{11} - 394724375485896057 \nu^{10} + 302891721614703524 \nu^{9} + 1334154943952591089 \nu^{8} - 1109160008265371288 \nu^{7} - 2775319981535112788 \nu^{6} + 2424010582687315882 \nu^{5} + 3123323012897178608 \nu^{4} - 2753924372503515628 \nu^{3} - 1235815225180964784 \nu^{2} + 599130875550547332 \nu - 981825396029938292$$$$)/ 1044699390969680954$$ $$\beta_{8}$$ $$=$$ $$($$$$27011392152086 \nu^{15} - 94322220582592 \nu^{14} - 225696122558276 \nu^{13} + 864208450258039 \nu^{12} + 1411845781359376 \nu^{11} - 4041812925203160 \nu^{10} - 6269728099143920 \nu^{9} + 10112854118207502 \nu^{8} + 20631359194590520 \nu^{7} - 18267893574259336 \nu^{6} - 35752779922795052 \nu^{5} + 30814487259973294 \nu^{4} + 23554149694989280 \nu^{3} - 57802642839937912 \nu^{2} + 47000942610281696 \nu + 41355316935864360$$$$)/ 28235118674856242$$ $$\beta_{9}$$ $$=$$ $$($$$$-1041050542786427 \nu^{15} + 1883112160309661 \nu^{14} + 12326969218594081 \nu^{13} - 17511147516168121 \nu^{12} - 79828976495748052 \nu^{11} + 70202500555950795 \nu^{10} + 309113289528073360 \nu^{9} - 102679862645010379 \nu^{8} - 743890408721857828 \nu^{7} - 167587573181230524 \nu^{6} + 965203421611402430 \nu^{5} + 790598965925279948 \nu^{4} - 516066212605296160 \nu^{3} - 967356194834958860 \nu^{2} + 241690618524009152 \nu - 665557302469533080$$$$)/ 1044699390969680954$$ $$\beta_{10}$$ $$=$$ $$($$$$-1137180982991043 \nu^{15} + 2325108353514161 \nu^{14} + 15178169896398695 \nu^{13} - 28525635810104525 \nu^{12} - 115348535984880472 \nu^{11} + 197563567893022495 \nu^{10} + 488542619069434468 \nu^{9} - 792030723087262350 \nu^{8} - 1274513591566924010 \nu^{7} + 2035761231703905550 \nu^{6} + 1590717249785072770 \nu^{5} - 2899284516772344726 \nu^{4} - 817379137488037320 \nu^{3} + 1757957122926958226 \nu^{2} - 1705583514667140808 \nu + 1021745261695028818$$$$)/ 1044699390969680954$$ $$\beta_{11}$$ $$=$$ $$($$$$27586196 \nu^{15} - 60484224 \nu^{14} - 429812678 \nu^{13} + 1098954297 \nu^{12} + 2710209516 \nu^{11} - 8222343940 \nu^{10} - 8477282916 \nu^{9} + 33880570978 \nu^{8} + 10209749788 \nu^{7} - 81131009856 \nu^{6} + 13487030188 \nu^{5} + 107312843204 \nu^{4} - 53956458496 \nu^{3} - 60722092488 \nu^{2} + 47341082576 \nu - 30791568858$$$$)/ 22992599138$$ $$\beta_{12}$$ $$=$$ $$($$$$-1549128975238964 \nu^{15} + 6314718842342630 \nu^{14} + 10337141022496773 \nu^{13} - 62122077139526184 \nu^{12} - 42649308628937256 \nu^{11} + 316475238016347160 \nu^{10} + 90713215492777030 \nu^{9} - 862239054546140538 \nu^{8} - 188471301222042982 \nu^{7} + 1497740867080358982 \nu^{6} - 60977283517438792 \nu^{5} - 1380488957184649542 \nu^{4} + 486077504004089848 \nu^{3} + 1051063137176443622 \nu^{2} - 943713616354830420 \nu + 1440089191177976230$$$$)/ 1044699390969680954$$ $$\beta_{13}$$ $$=$$ $$($$$$-1747168649395759 \nu^{15} + 7724437466959080 \nu^{14} + 14104831421626899 \nu^{13} - 100304130203271990 \nu^{12} - 38066114907204286 \nu^{11} + 597078317822983527 \nu^{10} - 30913172570058186 \nu^{9} - 1966222733092638378 \nu^{8} + 344943688319938954 \nu^{7} + 3551279443421674138 \nu^{6} - 665155168648250026 \nu^{5} - 2924922837458881732 \nu^{4} + 486865719770756060 \nu^{3} - 218354411962649438 \nu^{2} - 410107163164174620 \nu + 813898617538197770$$$$)/ 1044699390969680954$$ $$\beta_{14}$$ $$=$$ $$($$$$65540689755620 \nu^{15} - 229996243725450 \nu^{14} - 657627068184028 \nu^{13} + 2686380243178753 \nu^{12} + 3821157528032188 \nu^{11} - 16314115840737316 \nu^{10} - 12261418571569996 \nu^{9} + 55516738907773282 \nu^{8} + 25363499237715320 \nu^{7} - 115953298850201170 \nu^{6} - 20250999264187344 \nu^{5} + 120450959564895102 \nu^{4} - 5003877115303152 \nu^{3} - 53908970791502632 \nu^{2} + 63603018581143996 \nu - 53446090927838356$$$$)/ 28235118674856242$$ $$\beta_{15}$$ $$=$$ $$($$$$-75710870805340 \nu^{15} + 246212044566059 \nu^{14} + 725706845851137 \nu^{13} - 2917945396268176 \nu^{12} - 3246497926276112 \nu^{11} + 16534921958206198 \nu^{10} + 5380772843062716 \nu^{9} - 51675862451587651 \nu^{8} + 6644012212648884 \nu^{7} + 93540366552406226 \nu^{6} - 52157579306454274 \nu^{5} - 82967315362663904 \nu^{4} + 89490152468439968 \nu^{3} - 2263846659510274 \nu^{2} - 85982174975796068 \nu + 59846398429104566$$$$)/ 28235118674856242$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{9} + \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{15} + 2 \beta_{12} + \beta_{11} - 7 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \beta_{7} + 7 \beta_{4} - 4 \beta_{2} + 3 \beta_{1} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{15} + \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 8 \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{15} - 2 \beta_{12} + \beta_{11} - 17 \beta_{10} - \beta_{9} - \beta_{8} + 17 \beta_{4} + 5 \beta_{3} + 10 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$16 \beta_{15} + 3 \beta_{14} + \beta_{12} - 29 \beta_{10} - 11 \beta_{9} + 10 \beta_{5} + 19 \beta_{4} + 26 \beta_{3} + 9 \beta_{2} + 36 \beta_{1} - 22$$ $$\nu^{7}$$ $$=$$ $$42 \beta_{15} + 7 \beta_{14} - 27 \beta_{12} - 12 \beta_{11} - 59 \beta_{10} - 39 \beta_{9} - 29 \beta_{8} - 12 \beta_{7} - \beta_{6} + 59 \beta_{4} + 35 \beta_{3} + 52 \beta_{2} + 49 \beta_{1} - 15$$ $$\nu^{8}$$ $$=$$ $$68 \beta_{15} + 24 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 42 \beta_{11} - 92 \beta_{10} - 132 \beta_{9} - 44 \beta_{8} - 2 \beta_{6} + 16 \beta_{5} + 88 \beta_{4} + 72 \beta_{3} + 140 \beta_{2} + 100 \beta_{1} - 170$$ $$\nu^{9}$$ $$=$$ $$84 \beta_{15} + 48 \beta_{14} + 9 \beta_{13} - 142 \beta_{12} - 149 \beta_{11} - 89 \beta_{10} - 295 \beta_{9} - 211 \beta_{8} - 69 \beta_{7} + 6 \beta_{6} + 98 \beta_{4} + 108 \beta_{3} + 438 \beta_{2} + 132 \beta_{1} - 213$$ $$\nu^{10}$$ $$=$$ $$68 \beta_{15} + 74 \beta_{14} + 30 \beta_{13} - 104 \beta_{12} - 400 \beta_{11} + 4 \beta_{10} - 776 \beta_{9} - 380 \beta_{8} - 2 \beta_{7} + 20 \beta_{6} - 90 \beta_{5} + 122 \beta_{4} - 38 \beta_{3} + 980 \beta_{2} - 48 \beta_{1} - 848$$ $$\nu^{11}$$ $$=$$ $$-418 \beta_{15} + 77 \beta_{14} + 55 \beta_{13} - 521 \beta_{12} - 1008 \beta_{11} + 709 \beta_{10} - 1476 \beta_{9} - 1033 \beta_{8} - 179 \beta_{7} + 152 \beta_{6} - 22 \beta_{5} - 654 \beta_{4} - 220 \beta_{3} + 2329 \beta_{2} - 374 \beta_{1} - 1356$$ $$\nu^{12}$$ $$=$$ $$-1520 \beta_{15} - 304 \beta_{14} + 12 \beta_{13} - 736 \beta_{12} - 2270 \beta_{11} + 2564 \beta_{10} - 2992 \beta_{9} - 1900 \beta_{8} + 36 \beta_{7} + 372 \beta_{6} - 784 \beta_{5} - 1820 \beta_{4} - 2160 \beta_{3} + 4536 \beta_{2} - 2948 \beta_{1} - 2998$$ $$\nu^{13}$$ $$=$$ $$-5928 \beta_{15} - 1404 \beta_{14} - 130 \beta_{13} - 1168 \beta_{12} - 4444 \beta_{11} + 8314 \beta_{10} - 4622 \beta_{9} - 3268 \beta_{8} + 222 \beta_{7} + 1154 \beta_{6} - 104 \beta_{5} - 8418 \beta_{4} - 5252 \beta_{3} + 7982 \beta_{2} - 7488 \beta_{1} - 5116$$ $$\nu^{14}$$ $$=$$ $$-15354 \beta_{15} - 5886 \beta_{14} - 1422 \beta_{13} - 2548 \beta_{12} - 7476 \beta_{11} + 21738 \beta_{10} - 4898 \beta_{9} - 4984 \beta_{8} + 582 \beta_{7} + 2436 \beta_{6} - 2664 \beta_{5} - 20224 \beta_{4} - 17484 \beta_{3} + 11160 \beta_{2} - 24134 \beta_{1} - 4436$$ $$\nu^{15}$$ $$=$$ $$-40066 \beta_{15} - 16672 \beta_{14} - 4230 \beta_{13} + 2008 \beta_{12} - 7726 \beta_{11} + 52246 \beta_{10} + 688 \beta_{9} - 302 \beta_{8} + 4506 \beta_{7} + 4200 \beta_{6} + 1086 \beta_{5} - 56406 \beta_{4} - 39140 \beta_{3} + 3822 \beta_{2} - 55094 \beta_{1} - 3728$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
846.1
 −1.81128 − 0.707107i −1.31120 − 0.707107i 2.06986 + 0.707107i 0.845980 + 0.707107i 0.160107 + 0.707107i 2.23033 + 0.707107i −1.67036 − 0.707107i 1.48657 − 0.707107i 1.48657 + 0.707107i −1.67036 + 0.707107i 2.23033 − 0.707107i 0.160107 − 0.707107i 0.845980 − 0.707107i 2.06986 − 0.707107i −1.31120 + 0.707107i −1.81128 + 0.707107i
2.20793i 1.86190i −2.87496 2.47437i −4.11094 −2.03541 1.69029i 1.93185i −0.466655 −5.46323
846.2 2.20793i 1.86190i −2.87496 2.47437i 4.11094 2.03541 1.69029i 1.93185i −0.466655 5.46323
846.3 2.06181i 1.78904i −2.25106 0.290104i −3.68867 2.64256 0.129958i 0.517638i −0.200678 −0.598140
846.4 2.06181i 1.78904i −2.25106 0.290104i 3.68867 −2.64256 0.129958i 0.517638i −0.200678 0.598140
846.5 1.69029i 3.31624i −0.857091 0.780746i −5.60542 1.45775 2.20793i 1.93185i −7.99745 −1.31969
846.6 1.69029i 3.31624i −0.857091 0.780746i 5.60542 −1.45775 2.20793i 1.93185i −7.99745 1.31969
846.7 0.129958i 2.08212i 1.98311 1.78432i −0.270589 1.65799 2.06181i 0.517638i −1.33522 0.231887
846.8 0.129958i 2.08212i 1.98311 1.78432i 0.270589 −1.65799 2.06181i 0.517638i −1.33522 −0.231887
846.9 0.129958i 2.08212i 1.98311 1.78432i 0.270589 −1.65799 + 2.06181i 0.517638i −1.33522 −0.231887
846.10 0.129958i 2.08212i 1.98311 1.78432i −0.270589 1.65799 + 2.06181i 0.517638i −1.33522 0.231887
846.11 1.69029i 3.31624i −0.857091 0.780746i 5.60542 −1.45775 + 2.20793i 1.93185i −7.99745 1.31969
846.12 1.69029i 3.31624i −0.857091 0.780746i −5.60542 1.45775 + 2.20793i 1.93185i −7.99745 −1.31969
846.13 2.06181i 1.78904i −2.25106 0.290104i 3.68867 −2.64256 + 0.129958i 0.517638i −0.200678 0.598140
846.14 2.06181i 1.78904i −2.25106 0.290104i −3.68867 2.64256 + 0.129958i 0.517638i −0.200678 −0.598140
846.15 2.20793i 1.86190i −2.87496 2.47437i 4.11094 2.03541 + 1.69029i 1.93185i −0.466655 5.46323
846.16 2.20793i 1.86190i −2.87496 2.47437i −4.11094 −2.03541 + 1.69029i 1.93185i −0.466655 −5.46323
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 846.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.b.e 16
7.b odd 2 1 inner 847.2.b.e 16
11.b odd 2 1 inner 847.2.b.e 16
11.c even 5 4 847.2.l.m 64
11.d odd 10 4 847.2.l.m 64
77.b even 2 1 inner 847.2.b.e 16
77.j odd 10 4 847.2.l.m 64
77.l even 10 4 847.2.l.m 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.b.e 16 1.a even 1 1 trivial
847.2.b.e 16 7.b odd 2 1 inner
847.2.b.e 16 11.b odd 2 1 inner
847.2.b.e 16 77.b even 2 1 inner
847.2.l.m 64 11.c even 5 4
847.2.l.m 64 11.d odd 10 4
847.2.l.m 64 77.j odd 10 4
847.2.l.m 64 77.l even 10 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 12 T_{2}^{6} + 47 T_{2}^{4} + 60 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 60 T^{2} + 47 T^{4} + 12 T^{6} + T^{8} )^{2}$$
$3$ $$( 529 + 488 T^{2} + 161 T^{4} + 22 T^{6} + T^{8} )^{2}$$
$5$ $$( 1 + 14 T^{2} + 26 T^{4} + 10 T^{6} + T^{8} )^{2}$$
$7$ $$5764801 - 941192 T^{2} + 259308 T^{4} - 56056 T^{6} + 6374 T^{8} - 1144 T^{10} + 108 T^{12} - 8 T^{14} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$( 21904 - 7664 T^{2} + 968 T^{4} - 52 T^{6} + T^{8} )^{2}$$
$17$ $$( 14641 - 24224 T^{2} + 2729 T^{4} - 94 T^{6} + T^{8} )^{2}$$
$19$ $$( 121801 - 34106 T^{2} + 2882 T^{4} - 94 T^{6} + T^{8} )^{2}$$
$23$ $$( 313 + 176 T - 25 T^{2} - 8 T^{3} + T^{4} )^{4}$$
$29$ $$( 42849 + 17604 T^{2} + 2079 T^{4} + 84 T^{6} + T^{8} )^{2}$$
$31$ $$( 36481 + 25946 T^{2} + 2714 T^{4} + 94 T^{6} + T^{8} )^{2}$$
$37$ $$( 36 - 108 T + 90 T^{2} - 18 T^{3} + T^{4} )^{4}$$
$41$ $$( 167281 - 45974 T^{2} + 4058 T^{4} - 130 T^{6} + T^{8} )^{2}$$
$43$ $$( 157609 + 44244 T^{2} + 3695 T^{4} + 108 T^{6} + T^{8} )^{2}$$
$47$ $$( 9665881 + 1003886 T^{2} + 28697 T^{4} + 304 T^{6} + T^{8} )^{2}$$
$53$ $$( 3289 - 148 T - 147 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$59$ $$( 829921 + 147758 T^{2} + 8249 T^{4} + 160 T^{6} + T^{8} )^{2}$$
$61$ $$( 13935289 - 1096022 T^{2} + 30209 T^{4} - 328 T^{6} + T^{8} )^{2}$$
$67$ $$( -143 + 96 T - 7 T^{2} - 6 T^{3} + T^{4} )^{4}$$
$71$ $$( 3373 - 826 T - 69 T^{2} + 14 T^{3} + T^{4} )^{4}$$
$73$ $$( 10896601 - 1011934 T^{2} + 29714 T^{4} - 314 T^{6} + T^{8} )^{2}$$
$79$ $$( 42849 + 1270728 T^{2} + 38655 T^{4} + 360 T^{6} + T^{8} )^{2}$$
$83$ $$( 32761 - 234590 T^{2} + 12482 T^{4} - 202 T^{6} + T^{8} )^{2}$$
$89$ $$( 45468049 + 4544528 T^{2} + 86537 T^{4} + 526 T^{6} + T^{8} )^{2}$$
$97$ $$( 966289 + 206486 T^{2} + 13418 T^{4} + 274 T^{6} + T^{8} )^{2}$$