# Properties

 Label 819.2.s.e Level $819$ Weight $2$ Character orbit 819.s Analytic conductor $6.540$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 11 x^{14} - 4 x^{13} + 87 x^{12} - 35 x^{11} + 326 x^{10} - 205 x^{9} + 895 x^{8} - 481 x^{7} + 1005 x^{6} - 544 x^{5} + 811 x^{4} - 312 x^{3} + 195 x^{2} + 13 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} ) q^{2} + ( 1 + \beta_{3} - \beta_{4} ) q^{4} + \beta_{14} q^{5} + ( -\beta_{10} - \beta_{15} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{12} + \beta_{13} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} ) q^{2} + ( 1 + \beta_{3} - \beta_{4} ) q^{4} + \beta_{14} q^{5} + ( -\beta_{10} - \beta_{15} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{12} + \beta_{13} ) q^{8} + ( -1 + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{10} -\beta_{8} q^{11} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{15} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{15} ) q^{14} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} + \beta_{11} + \beta_{15} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{7} + \beta_{15} ) q^{17} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{19} + ( 3 - 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{20} + ( 1 + \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{22} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{12} - \beta_{13} ) q^{23} + ( -\beta_{1} + \beta_{6} + \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{25} + ( -3 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{26} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} ) q^{28} + ( 2 \beta_{2} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{29} + ( -\beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{31} + ( \beta_{4} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{32} + ( -5 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{34} + ( -\beta_{3} + 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{35} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{37} + ( -2 \beta_{2} + \beta_{3} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{38} + ( -3 + 3 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{40} + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{41} + ( 1 - 2 \beta_{1} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{12} ) q^{43} + ( 2 + \beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - \beta_{14} ) q^{44} + ( -1 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{46} + ( -2 - 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{12} - 3 \beta_{14} ) q^{47} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{49} + ( -\beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{50} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{52} + ( -2 \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{9} - \beta_{10} + \beta_{13} ) q^{53} + ( 2 \beta_{3} + \beta_{7} - 2 \beta_{9} + \beta_{10} - 2 \beta_{14} + \beta_{15} ) q^{55} + ( 3 + \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{56} + ( 2 \beta_{1} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{58} + ( -3 - \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{59} + ( \beta_{1} + 3 \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{9} + \beta_{10} ) q^{61} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{6} - 4 \beta_{7} - \beta_{10} + 3 \beta_{13} + \beta_{14} ) q^{62} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{64} + ( 4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{14} ) q^{65} + ( 4 - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{67} + ( 1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + 2 \beta_{12} - 3 \beta_{13} ) q^{68} + ( 6 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 4 \beta_{9} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{70} + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{71} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{73} + ( 1 - 3 \beta_{3} + \beta_{4} + 2 \beta_{7} - \beta_{10} - 2 \beta_{11} + \beta_{13} - 3 \beta_{15} ) q^{74} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{76} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{77} + ( 2 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{12} - \beta_{14} ) q^{79} + ( 4 + \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - 4 \beta_{9} + \beta_{10} - \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{80} + ( -2 \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{82} + ( -2 + 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} - 4 \beta_{15} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{14} ) q^{85} + ( -2 + 2 \beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{86} + ( 4 + \beta_{1} - 3 \beta_{4} + 3 \beta_{5} + \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - \beta_{12} + 4 \beta_{14} ) q^{88} + ( 2 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{7} + \beta_{11} + \beta_{15} ) q^{89} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 5 \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{91} + ( 7 - 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{92} + ( -2 - 4 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{94} + ( 1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{12} ) q^{95} + ( 2 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 4 \beta_{12} - \beta_{13} + 5 \beta_{14} - \beta_{15} ) q^{97} + ( -2 + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 12q^{4} + q^{7} - 12q^{8} + O(q^{10})$$ $$16q + 12q^{4} + q^{7} - 12q^{8} - 4q^{10} + 2q^{11} + 5q^{13} + 7q^{14} + 12q^{16} - 4q^{17} - 11q^{19} + 20q^{20} + 7q^{22} + 8q^{23} + 2q^{25} - 33q^{26} - q^{28} - 15q^{29} + 3q^{31} + 6q^{32} - 68q^{34} - 8q^{37} - 2q^{38} - 25q^{40} - 19q^{41} + 11q^{43} + 16q^{44} - 4q^{46} - 5q^{47} + 7q^{49} + 7q^{50} - 18q^{52} - 36q^{53} - 15q^{55} + 51q^{56} + 20q^{58} - 34q^{59} - 22q^{61} + 6q^{62} - 20q^{64} + 24q^{65} + 26q^{67} + 10q^{68} + 46q^{70} - 9q^{71} - 6q^{73} + 30q^{74} - 16q^{76} + 36q^{77} + 16q^{79} + 28q^{80} - q^{82} - 36q^{83} - 4q^{85} - 16q^{86} + 24q^{88} + 40q^{89} - 10q^{91} + 94q^{92} - 20q^{94} + 7q^{97} - 18q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 11 x^{14} - 4 x^{13} + 87 x^{12} - 35 x^{11} + 326 x^{10} - 205 x^{9} + 895 x^{8} - 481 x^{7} + 1005 x^{6} - 544 x^{5} + 811 x^{4} - 312 x^{3} + 195 x^{2} + 13 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-850102445244 \nu^{15} - 602738721141 \nu^{14} - 9256957678228 \nu^{13} - 3074932976676 \nu^{12} - 70844833728257 \nu^{11} - 21460338818148 \nu^{10} - 251702286046476 \nu^{9} - 11860911508749 \nu^{8} - 637132974569996 \nu^{7} - 98337615982400 \nu^{6} - 589939014330730 \nu^{5} - 26247556294384 \nu^{4} - 577279503454940 \nu^{3} - 108368220879957 \nu^{2} - 208029345221468 \nu - 13972229088464$$$$)/ 200652098581830$$ $$\beta_{3}$$ $$=$$ $$($$$$945658852056 \nu^{15} + 5453415441339 \nu^{14} + 13517776750957 \nu^{13} + 52870958416314 \nu^{12} + 88760072093213 \nu^{11} + 389923212586737 \nu^{10} + 338939264634954 \nu^{9} + 1187742608006541 \nu^{8} + 384615393503879 \nu^{7} + 2746301731062440 \nu^{6} + 187320602242210 \nu^{5} + 1195526676592186 \nu^{4} - 1373541222944320 \nu^{3} + 455677150451133 \nu^{2} + 30629905626407 \nu - 611267573740024$$$$)/ 200652098581830$$ $$\beta_{4}$$ $$=$$ $$($$$$1548397573197 \nu^{15} + 5359246221883 \nu^{14} + 19993119508609 \nu^{13} + 49756879408343 \nu^{12} + 139973996494901 \nu^{11} + 364492101483669 \nu^{10} + 525071177418723 \nu^{9} + 1064033894083157 \nu^{8} + 891852285648643 \nu^{7} + 2481887787922950 \nu^{6} + 676023888749330 \nu^{5} + 1083373096954242 \nu^{4} - 999941039148235 \nu^{3} + 497936518850021 \nu^{2} + 33550802926699 \nu - 10161380439778$$$$)/ 200652098581830$$ $$\beta_{5}$$ $$=$$ $$($$$$-3810848648686 \nu^{15} + 2398500018441 \nu^{14} - 35957350192522 \nu^{13} + 44493471781581 \nu^{12} - 278712020050663 \nu^{11} + 344198532927168 \nu^{10} - 856384219169819 \nu^{9} + 1557997436445829 \nu^{8} - 2334814734982064 \nu^{7} + 3362003460236605 \nu^{6} - 1249677488024080 \nu^{5} + 3339064567965244 \nu^{4} - 1980977600835720 \nu^{3} + 766323242696737 \nu^{2} - 136810746763792 \nu - 8612982866581$$$$)/ 200652098581830$$ $$\beta_{6}$$ $$=$$ $$($$$$22240023182504 \nu^{15} + 33095488971866 \nu^{14} + 219131096234958 \nu^{13} + 229934387339641 \nu^{12} + 1505862005807122 \nu^{11} + 1718479268465178 \nu^{10} + 3895919201439001 \nu^{9} + 3400337677941704 \nu^{8} + 5141508670414336 \nu^{7} + 11075617584988655 \nu^{6} - 11233975942651870 \nu^{5} + 399705759957354 \nu^{4} - 15416083486559320 \nu^{3} + 6039092233128032 \nu^{2} - 10163858567310982 \nu - 2454346762052891$$$$)/ 601956295745490$$ $$\beta_{7}$$ $$=$$ $$($$$$-23960698558757 \nu^{15} + 80722315132867 \nu^{14} - 248272416035934 \nu^{13} + 945357317741162 \nu^{12} - 2278547346806461 \nu^{11} + 7369609606825686 \nu^{10} - 9561210057401608 \nu^{9} + 27440315802224893 \nu^{8} - 34791569261503438 \nu^{7} + 68922263214064465 \nu^{6} - 51995180624536250 \nu^{5} + 58806711807132708 \nu^{4} - 56537190221514755 \nu^{3} + 39140928712744339 \nu^{2} - 18100020514647914 \nu - 1527844957759087$$$$)/ 601956295745490$$ $$\beta_{8}$$ $$=$$ $$($$$$-1778153235146 \nu^{15} + 3078673249492 \nu^{14} - 23056845757699 \nu^{13} + 36844479871649 \nu^{12} - 206484663504642 \nu^{11} + 300268949144295 \nu^{10} - 984482545211503 \nu^{9} + 1154069266885630 \nu^{8} - 3294396074768529 \nu^{7} + 3159979917916907 \nu^{6} - 5797955983317930 \nu^{5} + 2782805001313508 \nu^{4} - 5401830167261236 \nu^{3} + 2544041682468796 \nu^{2} - 2398400014950427 \nu + 13468531477239$$$$)/ 40130419716366$$ $$\beta_{9}$$ $$=$$ $$($$$$13972229088464 \nu^{15} - 850102445244 \nu^{14} + 153091781251963 \nu^{13} - 65145874032084 \nu^{12} + 1212508997719692 \nu^{11} - 559872851824497 \nu^{10} + 4533486344021116 \nu^{9} - 3116009249181596 \nu^{8} + 12493284122666531 \nu^{7} - 7357775166121180 \nu^{6} + 13943752617923920 \nu^{5} - 8190831638455146 \nu^{4} + 11305230234449920 \nu^{3} - 4936614979055708 \nu^{2} + 2616216451370523 \nu + 174261731510394$$$$)/ 200652098581830$$ $$\beta_{10}$$ $$=$$ $$($$$$-78334216662179 \nu^{15} + 42305988951004 \nu^{14} - 796493782907553 \nu^{13} + 800506448243009 \nu^{12} - 6287052669219262 \nu^{11} + 6369277178514687 \nu^{10} - 21667564253989861 \nu^{9} + 29202146424854866 \nu^{8} - 60056970145125211 \nu^{7} + 67366680956630260 \nu^{6} - 51818201522647850 \nu^{5} + 68348199564129876 \nu^{4} - 46941064229416025 \nu^{3} + 32439278838552808 \nu^{2} - 2452125901935503 \nu - 1561887871205524$$$$)/ 601956295745490$$ $$\beta_{11}$$ $$=$$ $$($$$$40582993882149 \nu^{15} + 45466225490261 \nu^{14} + 457612339093228 \nu^{13} + 320116699841551 \nu^{12} + 3451826421856817 \nu^{11} + 2289567725966748 \nu^{10} + 12469185555804981 \nu^{9} + 4596778428280669 \nu^{8} + 29626192865873556 \nu^{7} + 13380711590415220 \nu^{6} + 26853299881601610 \nu^{5} + 5262660243488864 \nu^{4} + 13462802422151435 \nu^{3} + 6376257529840357 \nu^{2} + 432729870487758 \nu + 541126055004554$$$$)/ 200652098581830$$ $$\beta_{12}$$ $$=$$ $$($$$$-24469220650027 \nu^{15} + 15238766802188 \nu^{14} - 263471571349674 \nu^{13} + 261485413803346 \nu^{12} - 2133804675429128 \nu^{11} + 2112117989573874 \nu^{10} - 8069708080242212 \nu^{9} + 9430098598560608 \nu^{8} - 23566610847554492 \nu^{7} + 22912974402719057 \nu^{6} - 27864296349778072 \nu^{5} + 22807330510966992 \nu^{4} - 25099987842740767 \nu^{3} + 13749352221620660 \nu^{2} - 6383574622903984 \nu - 273843217222007$$$$)/ 120391259149098$$ $$\beta_{13}$$ $$=$$ $$($$$$-131547920533543 \nu^{15} + 66691456016783 \nu^{14} - 1384852130591721 \nu^{13} + 1273004752646143 \nu^{12} - 11058467161934039 \nu^{11} + 10269404382982179 \nu^{10} - 40207214376187487 \nu^{9} + 47455714076985947 \nu^{8} - 114175121032835207 \nu^{7} + 112908211013582750 \nu^{6} - 120446275827543100 \nu^{5} + 117538497500186772 \nu^{4} - 107879081650263685 \nu^{3} + 65375118976888151 \nu^{2} - 20971641574908571 \nu + 20950999529842$$$$)/ 601956295745490$$ $$\beta_{14}$$ $$=$$ $$($$$$46673194766134 \nu^{15} + 504608087166 \nu^{14} + 508750470699368 \nu^{13} - 187060135461519 \nu^{12} + 4004999085302952 \nu^{11} - 1633893875813922 \nu^{10} + 14795782515868121 \nu^{9} - 9718282575984076 \nu^{8} + 40199282496485536 \nu^{7} - 22689027227537705 \nu^{6} + 43268255944038050 \nu^{5} - 26716032806738496 \nu^{4} + 34523127081241160 \nu^{3} - 14919838930830898 \nu^{2} + 8016090006501768 \nu - 79869396342261$$$$)/ 200652098581830$$ $$\beta_{15}$$ $$=$$ $$($$$$-140712016641263 \nu^{15} - 74541657903737 \nu^{14} - 1577680530611646 \nu^{13} - 227311479417322 \nu^{12} - 12231961189578409 \nu^{11} - 1099348556784906 \nu^{10} - 45544873806547732 \nu^{9} + 8153767607643457 \nu^{8} - 118092162473133742 \nu^{7} + 16417888475821585 \nu^{6} - 126589546885231730 \nu^{5} + 37652742907899012 \nu^{4} - 88243990870548815 \nu^{3} + 19109720516331511 \nu^{2} - 19457791048108706 \nu + 431512462062377$$$$)/ 601956295745490$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14} - 3 \beta_{9} + \beta_{5} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} + \beta_{12} - \beta_{6} + \beta_{3} - 4 \beta_{2} - 3 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} - 7 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + 15 \beta_{9} - \beta_{8} - 6 \beta_{5} + 6 \beta_{4} + \beta_{1} - 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} + 8 \beta_{14} + 9 \beta_{13} - \beta_{12} - \beta_{11} - 8 \beta_{9} + \beta_{8} - 8 \beta_{7} + 8 \beta_{6} + \beta_{5} - 8 \beta_{3} + 20 \beta_{2} - 8 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{15} - \beta_{13} - \beta_{12} + 10 \beta_{11} + 2 \beta_{10} - 9 \beta_{7} + \beta_{6} - 34 \beta_{4} + 45 \beta_{3} - 10 \beta_{2} - 11 \beta_{1} + 84$$ $$\nu^{7}$$ $$=$$ $$-\beta_{15} - 56 \beta_{14} - \beta_{13} - 41 \beta_{12} - 13 \beta_{10} + 56 \beta_{9} - 12 \beta_{8} + 54 \beta_{7} - 10 \beta_{5} + 10 \beta_{4} + 111 \beta_{1} - 56$$ $$\nu^{8}$$ $$=$$ $$-24 \beta_{15} + 286 \beta_{14} + 81 \beta_{13} - 78 \beta_{12} - 78 \beta_{11} + 65 \beta_{10} - 494 \beta_{9} + 78 \beta_{8} + 62 \beta_{7} - 8 \beta_{6} + 198 \beta_{5} - 286 \beta_{3} + 77 \beta_{2} + 8 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$118 \beta_{15} - 437 \beta_{13} + 335 \beta_{12} + 105 \beta_{11} + 102 \beta_{10} - 16 \beta_{7} - 335 \beta_{6} - 79 \beta_{4} + 382 \beta_{3} - 651 \beta_{2} - 316 \beta_{1} + 382$$ $$\nu^{10}$$ $$=$$ $$-437 \beta_{15} - 1818 \beta_{14} - 437 \beta_{13} + 594 \beta_{12} - 644 \beta_{10} + 2986 \beta_{9} - 555 \beta_{8} + 50 \beta_{7} - 1184 \beta_{5} + 1184 \beta_{4} + 547 \beta_{1} - 2986$$ $$\nu^{11}$$ $$=$$ $$-762 \beta_{15} + 2583 \beta_{14} + 3017 \beta_{13} - 812 \beta_{12} - 812 \beta_{11} + 168 \beta_{10} - 2596 \beta_{9} + 812 \beta_{8} - 2037 \beta_{7} + 2087 \beta_{6} + 587 \beta_{5} - 2583 \beta_{3} + 3940 \beta_{2} - 2087 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$4423 \beta_{15} - 1476 \beta_{13} - 98 \beta_{12} + 3779 \beta_{11} + 1574 \beta_{10} - 2849 \beta_{7} + 98 \beta_{6} - 7211 \beta_{4} + 11577 \beta_{3} - 3777 \beta_{2} - 3875 \beta_{1} + 18359$$ $$\nu^{13}$$ $$=$$ $$-1476 \beta_{15} - 17376 \beta_{14} - 1476 \beta_{13} - 7051 \beta_{12} - 6829 \beta_{10} + 17658 \beta_{9} - 5899 \beta_{8} + 13880 \beta_{7} - 4266 \beta_{5} + 4266 \beta_{4} + 24295 \beta_{1} - 17658$$ $$\nu^{14}$$ $$=$$ $$-11252 \beta_{15} + 73854 \beta_{14} + 30104 \beta_{13} - 25132 \beta_{12} - 25132 \beta_{11} + 18303 \beta_{10} - 114139 \beta_{9} + 25132 \beta_{8} + 13331 \beta_{7} + 549 \beta_{6} + 44456 \beta_{5} - 73854 \beta_{3} + 25801 \beta_{2} - 549 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$48185 \beta_{15} - 116740 \beta_{13} + 80356 \beta_{12} + 41356 \beta_{11} + 36384 \beta_{10} - 11801 \beta_{7} - 80356 \beta_{6} - 30616 \beta_{4} + 116428 \beta_{3} - 151544 \beta_{2} - 71188 \beta_{1} + 120172$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{9}$$ $$-1 + \beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −1.27528 + 2.20885i −1.02737 + 1.77946i −0.532778 + 0.922798i −0.0340180 + 0.0589209i 0.379240 − 0.656863i 0.415625 − 0.719884i 0.857510 − 1.48525i 1.21707 − 2.10803i −1.27528 − 2.20885i −1.02737 − 1.77946i −0.532778 − 0.922798i −0.0340180 − 0.0589209i 0.379240 + 0.656863i 0.415625 + 0.719884i 0.857510 + 1.48525i 1.21707 + 2.10803i
−2.55056 0 4.50537 1.39351 2.41363i 0 −2.46231 0.968004i −6.39011 0 −3.55423 + 6.15611i
289.2 −2.05474 0 2.22196 0.274662 0.475728i 0 2.59269 0.527227i −0.456078 0 −0.564359 + 0.977499i
289.3 −1.06556 0 −0.864591 −1.19023 + 2.06154i 0 −0.813611 2.51755i 3.05238 0 1.26826 2.19668i
289.4 −0.0680360 0 −1.99537 −1.52954 + 2.64923i 0 0.910236 + 2.48424i 0.271829 0 0.104063 0.180243i
289.5 0.758480 0 −1.42471 0.357869 0.619848i 0 −1.32400 + 2.29064i −2.59757 0 0.271437 0.470142i
289.6 0.831251 0 −1.30902 1.30847 2.26634i 0 1.78280 1.95490i −2.75063 0 1.08767 1.88389i
289.7 1.71502 0 0.941295 −1.22863 + 2.12806i 0 −2.38702 1.14112i −1.81570 0 −2.10713 + 3.64966i
289.8 2.43414 0 3.92506 0.613891 1.06329i 0 2.20121 + 1.46788i 4.68588 0 1.49430 2.58820i
802.1 −2.55056 0 4.50537 1.39351 + 2.41363i 0 −2.46231 + 0.968004i −6.39011 0 −3.55423 6.15611i
802.2 −2.05474 0 2.22196 0.274662 + 0.475728i 0 2.59269 + 0.527227i −0.456078 0 −0.564359 0.977499i
802.3 −1.06556 0 −0.864591 −1.19023 2.06154i 0 −0.813611 + 2.51755i 3.05238 0 1.26826 + 2.19668i
802.4 −0.0680360 0 −1.99537 −1.52954 2.64923i 0 0.910236 2.48424i 0.271829 0 0.104063 + 0.180243i
802.5 0.758480 0 −1.42471 0.357869 + 0.619848i 0 −1.32400 2.29064i −2.59757 0 0.271437 + 0.470142i
802.6 0.831251 0 −1.30902 1.30847 + 2.26634i 0 1.78280 + 1.95490i −2.75063 0 1.08767 + 1.88389i
802.7 1.71502 0 0.941295 −1.22863 2.12806i 0 −2.38702 + 1.14112i −1.81570 0 −2.10713 3.64966i
802.8 2.43414 0 3.92506 0.613891 + 1.06329i 0 2.20121 1.46788i 4.68588 0 1.49430 + 2.58820i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 802.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.s.e 16
3.b odd 2 1 273.2.l.b yes 16
7.c even 3 1 819.2.n.e 16
13.c even 3 1 819.2.n.e 16
21.h odd 6 1 273.2.j.b 16
39.i odd 6 1 273.2.j.b 16
91.h even 3 1 inner 819.2.s.e 16
273.s odd 6 1 273.2.l.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.b 16 21.h odd 6 1
273.2.j.b 16 39.i odd 6 1
273.2.l.b yes 16 3.b odd 2 1
273.2.l.b yes 16 273.s odd 6 1
819.2.n.e 16 7.c even 3 1
819.2.n.e 16 13.c even 3 1
819.2.s.e 16 1.a even 1 1 trivial
819.2.s.e 16 91.h even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{8} - 11 T_{2}^{6} + 2 T_{2}^{5} + 34 T_{2}^{4} - 13 T_{2}^{3} - 26 T_{2}^{2} + 13 T_{2} + 1$$ $$T_{11}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 13 T - 26 T^{2} - 13 T^{3} + 34 T^{4} + 2 T^{5} - 11 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$3969 - 14364 T + 39195 T^{2} - 54726 T^{3} + 63667 T^{4} - 38698 T^{5} + 28885 T^{6} - 9940 T^{7} + 9537 T^{8} - 1748 T^{9} + 1785 T^{10} - 162 T^{11} + 247 T^{12} - 10 T^{13} + 19 T^{14} + T^{16}$$
$7$ $$5764801 - 823543 T - 352947 T^{2} + 33614 T^{3} + 81634 T^{4} - 5978 T^{6} - 3955 T^{7} + 3111 T^{8} - 565 T^{9} - 122 T^{10} + 34 T^{12} + 2 T^{13} - 3 T^{14} - T^{15} + T^{16}$$
$11$ $$1290496 + 3707904 T + 7798928 T^{2} + 8381920 T^{3} + 7202369 T^{4} + 3369137 T^{5} + 1631947 T^{6} + 395980 T^{7} + 214678 T^{8} + 32062 T^{9} + 19695 T^{10} + 532 T^{11} + 1213 T^{12} - 4 T^{13} + 45 T^{14} - 2 T^{15} + T^{16}$$
$13$ $$815730721 - 313742585 T + 28960854 T^{2} + 3712930 T^{3} + 7054567 T^{4} - 1581840 T^{5} + 24505 T^{6} - 59345 T^{7} + 64050 T^{8} - 4565 T^{9} + 145 T^{10} - 720 T^{11} + 247 T^{12} + 10 T^{13} + 6 T^{14} - 5 T^{15} + T^{16}$$
$17$ $$( 6257 - 6070 T - 8327 T^{2} + 1242 T^{3} + 1343 T^{4} - 92 T^{5} - 68 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$19$ $$37161216 + 112215168 T + 245091888 T^{2} + 247849800 T^{3} + 190971177 T^{4} + 87888870 T^{5} + 36450886 T^{6} + 11032792 T^{7} + 3560924 T^{8} + 863411 T^{9} + 216156 T^{10} + 36354 T^{11} + 6704 T^{12} + 832 T^{13} + 133 T^{14} + 11 T^{15} + T^{16}$$
$23$ $$( -153 + 99 T + 828 T^{2} - 1249 T^{3} + 366 T^{4} + 158 T^{5} - 50 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$29$ $$539772289 + 939263724 T + 1219783833 T^{2} + 903851100 T^{3} + 580727595 T^{4} + 269398555 T^{5} + 131500576 T^{6} + 50104329 T^{7} + 17529998 T^{8} + 4334775 T^{9} + 909121 T^{10} + 130721 T^{11} + 18034 T^{12} + 1795 T^{13} + 226 T^{14} + 15 T^{15} + T^{16}$$
$31$ $$1771652281 - 3100128423 T + 9581587387 T^{2} + 6783294460 T^{3} + 9847063780 T^{4} + 603764255 T^{5} + 817561534 T^{6} + 85846365 T^{7} + 47316923 T^{8} + 3198672 T^{9} + 1106281 T^{10} + 31198 T^{11} + 17739 T^{12} + 195 T^{13} + 168 T^{14} - 3 T^{15} + T^{16}$$
$37$ $$( -1999 - 21198 T - 25723 T^{2} + 5589 T^{3} + 3950 T^{4} - 333 T^{5} - 119 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$41$ $$8083371138129 + 1963019978388 T + 1235964296232 T^{2} + 200918881728 T^{3} + 100795848822 T^{4} + 15653125431 T^{5} + 4893908248 T^{6} + 668782307 T^{7} + 161613427 T^{8} + 20127250 T^{9} + 3510173 T^{10} + 345162 T^{11} + 46413 T^{12} + 3776 T^{13} + 383 T^{14} + 19 T^{15} + T^{16}$$
$43$ $$99740169 - 331099011 T + 1632996468 T^{2} + 2074846047 T^{3} + 2322528861 T^{4} + 1036684662 T^{5} + 432047899 T^{6} + 44119592 T^{7} + 19781109 T^{8} + 697579 T^{9} + 743840 T^{10} - 17767 T^{11} + 13834 T^{12} - 738 T^{13} + 209 T^{14} - 11 T^{15} + T^{16}$$
$47$ $$67950412929 + 64504055196 T + 67571538318 T^{2} + 22682038218 T^{3} + 16191597669 T^{4} + 5376556224 T^{5} + 2673506353 T^{6} + 514371364 T^{7} + 119392286 T^{8} + 11810774 T^{9} + 2332902 T^{10} + 177165 T^{11} + 31310 T^{12} + 1249 T^{13} + 208 T^{14} + 5 T^{15} + T^{16}$$
$53$ $$9921384931329 + 4661035629471 T + 3089386115343 T^{2} + 1152619632636 T^{3} + 586267472109 T^{4} + 194363134260 T^{5} + 54294819883 T^{6} + 10692402752 T^{7} + 1750527711 T^{8} + 221890801 T^{9} + 25287588 T^{10} + 2413855 T^{11} + 216216 T^{12} + 15380 T^{13} + 932 T^{14} + 36 T^{15} + T^{16}$$
$59$ $$( 441561 + 826761 T + 375993 T^{2} + 29293 T^{3} - 13721 T^{4} - 2731 T^{5} - 66 T^{6} + 17 T^{7} + T^{8} )^{2}$$
$61$ $$124397290000 + 108222468000 T + 82641831900 T^{2} + 36352837440 T^{3} + 17180956201 T^{4} + 6451426769 T^{5} + 2268479329 T^{6} + 609480338 T^{7} + 138513968 T^{8} + 24132664 T^{9} + 3613917 T^{10} + 431426 T^{11} + 48659 T^{12} + 4472 T^{13} + 393 T^{14} + 22 T^{15} + T^{16}$$
$67$ $$2264374886656 + 1975335975936 T + 3642647613712 T^{2} - 1960759780256 T^{3} + 1444829812641 T^{4} - 229715645695 T^{5} + 64983052361 T^{6} - 10342901098 T^{7} + 2081058330 T^{8} - 249465388 T^{9} + 29353413 T^{10} - 2226010 T^{11} + 181979 T^{12} - 10308 T^{13} + 703 T^{14} - 26 T^{15} + T^{16}$$
$71$ $$83773776969 - 20561604480 T + 20967163785 T^{2} - 2744864808 T^{3} + 2780286789 T^{4} - 277973169 T^{5} + 234368050 T^{6} - 11543141 T^{7} + 12961446 T^{8} - 361137 T^{9} + 489959 T^{10} + 3745 T^{11} + 11726 T^{12} + 245 T^{13} + 182 T^{14} + 9 T^{15} + T^{16}$$
$73$ $$160073179912009 + 66600928216186 T + 36680214002748 T^{2} + 7642605269104 T^{3} + 3305063970677 T^{4} + 724196899862 T^{5} + 169601996078 T^{6} + 22208400224 T^{7} + 2922352164 T^{8} + 227292680 T^{9} + 23253165 T^{10} + 1325046 T^{11} + 132414 T^{12} + 4434 T^{13} + 419 T^{14} + 6 T^{15} + T^{16}$$
$79$ $$131635449856 + 321878745088 T + 774406233088 T^{2} + 95789479936 T^{3} + 80430074368 T^{4} - 2353172992 T^{5} + 5395275200 T^{6} - 341811840 T^{7} + 225775776 T^{8} - 25190048 T^{9} + 6455852 T^{10} - 489704 T^{11} + 65557 T^{12} - 3264 T^{13} + 401 T^{14} - 16 T^{15} + T^{16}$$
$83$ $$( -5580400 + 1407360 T + 1716687 T^{2} + 278677 T^{3} - 12714 T^{4} - 5306 T^{5} - 196 T^{6} + 18 T^{7} + T^{8} )^{2}$$
$89$ $$( 683431 + 557321 T - 47843 T^{2} - 89244 T^{3} + 627 T^{4} + 2705 T^{5} - 82 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$97$ $$575206497472369 + 79965878185767 T + 57912115603890 T^{2} + 1080834531421 T^{3} + 3177568412981 T^{4} + 41029800056 T^{5} + 91600064626 T^{6} - 390457090 T^{7} + 1798332012 T^{8} - 6201142 T^{9} + 20183034 T^{10} - 422856 T^{11} + 141437 T^{12} - 1589 T^{13} + 466 T^{14} - 7 T^{15} + T^{16}$$