# Properties

 Label 273.3.w.c Level $273$ Weight $3$ Character orbit 273.w Analytic conductor $7.439$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 273.w (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.43871121704$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 16 x^{14} - 176 x^{13} + 344 x^{12} + 4576 x^{11} + 11040 x^{10} - 37664 x^{9} - 313120 x^{8} - 230912 x^{7} + 2040576 x^{6} + 9332224 x^{5} + 33838912 x^{4} + 73579264 x^{3} + 95390464 x^{2} + 117266688 x + 97900608$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{10} ) q^{2} + ( \beta_{4} - \beta_{9} ) q^{3} + ( \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{9} ) q^{4} + ( 2 \beta_{1} - \beta_{8} + 2 \beta_{10} + \beta_{12} ) q^{5} + ( -\beta_{3} - 3 \beta_{8} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{6} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{7} + ( 3 \beta_{1} - 3 \beta_{8} ) q^{8} + ( -2 - 2 \beta_{5} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{10} ) q^{2} + ( \beta_{4} - \beta_{9} ) q^{3} + ( \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{9} ) q^{4} + ( 2 \beta_{1} - \beta_{8} + 2 \beta_{10} + \beta_{12} ) q^{5} + ( -\beta_{3} - 3 \beta_{8} + 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{6} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{7} + ( 3 \beta_{1} - 3 \beta_{8} ) q^{8} + ( -2 - 2 \beta_{5} - \beta_{15} ) q^{9} + ( -3 \beta_{4} - 12 \beta_{5} + 3 \beta_{7} + 3 \beta_{9} ) q^{10} + ( 2 \beta_{1} + 2 \beta_{8} - \beta_{10} + \beta_{12} ) q^{11} + ( -11 - 3 \beta_{4} - 11 \beta_{5} - \beta_{15} ) q^{12} -13 q^{13} + ( \beta_{2} - 5 \beta_{4} - 5 \beta_{7} + 2 \beta_{9} - 3 \beta_{11} ) q^{14} + ( 2 \beta_{1} + \beta_{3} - \beta_{6} + 7 \beta_{8} - 7 \beta_{10} + 5 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{15} + ( -3 - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{11} ) q^{16} + ( -\beta_{2} + 5 \beta_{4} + 5 \beta_{7} - 5 \beta_{9} ) q^{17} + ( 8 \beta_{1} + 4 \beta_{3} - 4 \beta_{6} + 4 \beta_{8} - 6 \beta_{10} + 6 \beta_{12} + 3 \beta_{13} - \beta_{14} ) q^{18} + ( -\beta_{3} - \beta_{6} - \beta_{13} - \beta_{14} ) q^{19} + ( -5 \beta_{1} + 16 \beta_{8} - 11 \beta_{10} + 11 \beta_{12} ) q^{20} + ( 2 \beta_{3} - \beta_{6} - 5 \beta_{8} - 2 \beta_{12} + 4 \beta_{13} ) q^{21} + 9 q^{22} + ( -4 \beta_{4} - 4 \beta_{7} - 4 \beta_{11} - 2 \beta_{15} ) q^{23} + ( 15 \beta_{1} + 3 \beta_{3} - 3 \beta_{6} + 12 \beta_{8} - 6 \beta_{10} + 12 \beta_{12} + 3 \beta_{13} ) q^{24} + ( 6 \beta_{4} + 5 \beta_{5} - 6 \beta_{7} - 6 \beta_{9} ) q^{25} + ( 13 \beta_{1} + 13 \beta_{10} ) q^{26} + ( 4 \beta_{9} + 9 \beta_{11} ) q^{27} + ( 13 \beta_{1} + 7 \beta_{3} - 5 \beta_{6} + 7 \beta_{8} - 13 \beta_{10} + 6 \beta_{12} + 8 \beta_{13} - 6 \beta_{14} ) q^{28} + ( -\beta_{2} - 8 \beta_{9} - 8 \beta_{11} - \beta_{15} ) q^{29} + ( 33 + 12 \beta_{4} + 33 \beta_{5} + 3 \beta_{15} ) q^{30} + ( -3 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - \beta_{8} + 3 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} ) q^{31} + ( 5 \beta_{1} + 5 \beta_{8} - 6 \beta_{10} - \beta_{12} ) q^{32} + ( -\beta_{1} + \beta_{3} + 2 \beta_{6} + 4 \beta_{8} - \beta_{10} - 4 \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{33} + ( -21 \beta_{1} - 11 \beta_{3} + 3 \beta_{6} - 14 \beta_{8} + 21 \beta_{10} - 7 \beta_{12} - 14 \beta_{13} + 14 \beta_{14} ) q^{34} + ( -2 \beta_{2} + 7 \beta_{4} + 7 \beta_{7} - 5 \beta_{9} + 2 \beta_{11} + 2 \beta_{15} ) q^{35} + ( 6 + 3 \beta_{2} + 13 \beta_{9} + 9 \beta_{11} + 3 \beta_{15} ) q^{36} + ( -4 \beta_{3} - 4 \beta_{6} - 4 \beta_{13} - 4 \beta_{14} ) q^{37} + ( -\beta_{2} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{38} + ( -13 \beta_{4} + 13 \beta_{9} ) q^{39} + ( 54 + 9 \beta_{4} + 54 \beta_{5} - 9 \beta_{7} - 9 \beta_{11} ) q^{40} + ( -8 \beta_{1} + 12 \beta_{8} - 4 \beta_{10} + 4 \beta_{12} ) q^{41} + ( -35 + 3 \beta_{2} + 2 \beta_{4} - 14 \beta_{5} + 9 \beta_{7} + 9 \beta_{11} + 5 \beta_{15} ) q^{42} + ( -8 - \beta_{9} + \beta_{11} ) q^{43} + ( -5 \beta_{1} + 4 \beta_{8} - 5 \beta_{10} - 4 \beta_{12} ) q^{44} + ( -17 \beta_{1} - 10 \beta_{3} + 10 \beta_{6} - 7 \beta_{8} + 13 \beta_{10} - 14 \beta_{12} - 2 \beta_{13} + 8 \beta_{14} ) q^{45} + ( 24 \beta_{1} + 16 \beta_{3} - 16 \beta_{6} + 8 \beta_{8} - 24 \beta_{10} + 16 \beta_{12} + 6 \beta_{13} - 10 \beta_{14} ) q^{46} + ( 3 \beta_{1} - 16 \beta_{8} + 3 \beta_{10} + 16 \beta_{12} ) q^{47} + ( 22 + 2 \beta_{2} + 3 \beta_{9} + 2 \beta_{15} ) q^{48} + ( -42 - 7 \beta_{4} - 21 \beta_{5} + 7 \beta_{7} + 7 \beta_{9} ) q^{49} + ( 18 \beta_{1} - 29 \beta_{8} + 11 \beta_{10} - 11 \beta_{12} ) q^{50} + ( 35 - 2 \beta_{4} + 35 \beta_{5} - 9 \beta_{7} - 9 \beta_{11} - 5 \beta_{15} ) q^{51} + ( -13 \beta_{4} - 39 \beta_{5} + 13 \beta_{7} + 13 \beta_{9} ) q^{52} + ( -9 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} - 3 \beta_{9} ) q^{53} + ( 10 \beta_{1} + 13 \beta_{6} - \beta_{8} + 10 \beta_{10} + \beta_{12} + 13 \beta_{13} ) q^{54} + ( -6 - 3 \beta_{9} + 3 \beta_{11} ) q^{55} + ( 6 \beta_{2} + 9 \beta_{4} + 9 \beta_{7} - 3 \beta_{9} + 6 \beta_{11} + 9 \beta_{15} ) q^{56} + ( -5 \beta_{1} - 3 \beta_{3} - 2 \beta_{6} - \beta_{8} + 5 \beta_{10} - 4 \beta_{12} - \beta_{13} + \beta_{14} ) q^{57} + ( 3 \beta_{3} - 17 \beta_{6} - 10 \beta_{8} + 10 \beta_{12} - 17 \beta_{13} + 3 \beta_{14} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{8} + 7 \beta_{10} + 9 \beta_{12} ) q^{59} + ( -47 \beta_{1} - 16 \beta_{3} + 16 \beta_{6} - 31 \beta_{8} + 10 \beta_{10} - 53 \beta_{12} - 5 \beta_{13} + 11 \beta_{14} ) q^{60} + ( 23 + 9 \beta_{4} + 23 \beta_{5} - 9 \beta_{7} - 9 \beta_{11} ) q^{61} + ( -4 \beta_{2} - 5 \beta_{9} - 5 \beta_{11} - 4 \beta_{15} ) q^{62} + ( 12 \beta_{1} + 4 \beta_{6} - 7 \beta_{8} + 9 \beta_{10} + 26 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{63} + ( -7 - 15 \beta_{9} + 15 \beta_{11} ) q^{64} + ( -26 \beta_{1} + 13 \beta_{8} - 26 \beta_{10} - 13 \beta_{12} ) q^{65} + ( 9 \beta_{4} - 9 \beta_{9} ) q^{66} + ( 6 \beta_{1} + 4 \beta_{3} - 4 \beta_{6} + 2 \beta_{8} - 6 \beta_{10} + 4 \beta_{12} - 11 \beta_{13} - 15 \beta_{14} ) q^{67} + ( -26 \beta_{4} - 26 \beta_{7} - 26 \beta_{11} - 13 \beta_{15} ) q^{68} + ( -28 + 4 \beta_{2} + 4 \beta_{9} + 18 \beta_{11} + 4 \beta_{15} ) q^{69} + ( -45 \beta_{1} - 24 \beta_{3} + 18 \beta_{6} - 24 \beta_{8} + 45 \beta_{10} - 21 \beta_{12} - 24 \beta_{13} + 24 \beta_{14} ) q^{70} + ( 3 \beta_{1} - 27 \beta_{8} + 24 \beta_{10} - 24 \beta_{12} ) q^{71} + ( -6 \beta_{1} + 3 \beta_{3} + 21 \beta_{6} + 15 \beta_{8} - 6 \beta_{10} - 15 \beta_{12} + 21 \beta_{13} + 3 \beta_{14} ) q^{72} + ( 27 \beta_{1} + 18 \beta_{3} - 18 \beta_{6} + 9 \beta_{8} - 27 \beta_{10} + 18 \beta_{12} - 6 \beta_{13} - 24 \beta_{14} ) q^{73} + ( -4 \beta_{2} - 4 \beta_{4} - 4 \beta_{7} + 4 \beta_{9} ) q^{74} + ( -66 - 5 \beta_{4} - 66 \beta_{5} - 6 \beta_{15} ) q^{75} + ( -3 \beta_{1} - 3 \beta_{3} - \beta_{6} - 2 \beta_{8} + 3 \beta_{10} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{76} + ( \beta_{2} + 7 \beta_{4} + 7 \beta_{7} - 8 \beta_{9} - \beta_{11} - \beta_{15} ) q^{77} + ( 13 \beta_{3} + 39 \beta_{8} - 26 \beta_{10} + 13 \beta_{12} + 13 \beta_{13} - 13 \beta_{14} ) q^{78} + ( 10 - 10 \beta_{4} + 10 \beta_{5} + 10 \beta_{7} + 10 \beta_{11} ) q^{79} + ( -19 \beta_{1} - 19 \beta_{8} - 7 \beta_{10} - 26 \beta_{12} ) q^{80} + ( -4 \beta_{2} - 73 \beta_{5} ) q^{81} + ( 68 + 20 \beta_{4} + 68 \beta_{5} - 20 \beta_{7} - 20 \beta_{11} ) q^{82} + ( -23 \beta_{1} - 2 \beta_{8} + 25 \beta_{10} - 25 \beta_{12} ) q^{83} + ( 6 \beta_{1} - 12 \beta_{3} + 22 \beta_{6} + 2 \beta_{8} + 36 \beta_{10} + 11 \beta_{12} + 5 \beta_{13} + 8 \beta_{14} ) q^{84} + ( 45 \beta_{1} + 18 \beta_{3} - 12 \beta_{6} + 30 \beta_{8} - 45 \beta_{10} + 15 \beta_{12} + 30 \beta_{13} - 30 \beta_{14} ) q^{85} + ( 12 \beta_{1} - 3 \beta_{8} + 12 \beta_{10} + 3 \beta_{12} ) q^{86} + ( 8 \beta_{2} - 2 \beta_{4} + 56 \beta_{5} - 9 \beta_{7} + 2 \beta_{9} ) q^{87} + ( 9 \beta_{4} - 9 \beta_{5} - 9 \beta_{7} - 9 \beta_{9} ) q^{88} + ( -46 \beta_{1} - 4 \beta_{8} - 46 \beta_{10} + 4 \beta_{12} ) q^{89} + ( -24 - 12 \beta_{2} - 39 \beta_{9} - 27 \beta_{11} - 12 \beta_{15} ) q^{90} + ( 13 \beta_{1} + 13 \beta_{3} + 13 \beta_{6} + 13 \beta_{8} - 13 \beta_{10} + 13 \beta_{13} - 13 \beta_{14} ) q^{91} + ( 14 \beta_{2} + 34 \beta_{9} + 34 \beta_{11} + 14 \beta_{15} ) q^{92} + ( -7 \beta_{1} - \beta_{3} - 7 \beta_{6} + 4 \beta_{8} - 7 \beta_{10} - 4 \beta_{12} - 7 \beta_{13} - \beta_{14} ) q^{93} + ( -19 \beta_{4} + 11 \beta_{5} + 19 \beta_{7} + 19 \beta_{9} ) q^{94} + ( -3 \beta_{4} - 3 \beta_{7} + 3 \beta_{9} ) q^{95} + ( 8 \beta_{1} + 6 \beta_{3} + 5 \beta_{6} + 10 \beta_{8} + 8 \beta_{10} - 10 \beta_{12} + 5 \beta_{13} + 6 \beta_{14} ) q^{96} + ( -6 \beta_{1} - 6 \beta_{3} - 2 \beta_{6} - 4 \beta_{8} + 6 \beta_{10} - 2 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} ) q^{97} + ( 21 \beta_{1} + 49 \beta_{8} + 14 \beta_{10} + 28 \beta_{12} ) q^{98} + ( -5 \beta_{1} + 5 \beta_{3} + 7 \beta_{6} - 4 \beta_{8} + 7 \beta_{10} - 5 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 24q^{4} - 16q^{9} + O(q^{10})$$ $$16q - 24q^{4} - 16q^{9} + 96q^{10} - 88q^{12} - 208q^{13} - 24q^{16} + 144q^{22} - 40q^{25} + 264q^{30} + 96q^{36} + 432q^{40} - 448q^{42} - 128q^{43} + 352q^{48} - 504q^{49} + 280q^{51} + 312q^{52} - 96q^{55} + 184q^{61} - 112q^{64} - 448q^{69} - 528q^{75} + 80q^{79} + 584q^{81} + 544q^{82} - 448q^{87} + 72q^{88} - 384q^{90} - 88q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 16 x^{14} - 176 x^{13} + 344 x^{12} + 4576 x^{11} + 11040 x^{10} - 37664 x^{9} - 313120 x^{8} - 230912 x^{7} + 2040576 x^{6} + 9332224 x^{5} + 33838912 x^{4} + 73579264 x^{3} + 95390464 x^{2} + 117266688 x + 97900608$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2740895842426384334323 \nu^{15} - 49967287626060626106349 \nu^{14} + 89252266904147475866667 \nu^{13} - 48449432461756037882516 \nu^{12} + 8647546076300132843391454 \nu^{11} - 24597753479665640207756874 \nu^{10} - 100176291550565202922361604 \nu^{9} - 327041066724359620287318452 \nu^{8} + 1506138120434746759577131792 \nu^{7} + 9287180463535317317881291968 \nu^{6} - 6770552631624178940428704360 \nu^{5} - 40231184743717672303913106512 \nu^{4} - 250702834169499907896072954960 \nu^{3} - 734740665799470203008796312240 \nu^{2} - 731911366955753452583269155840 \nu - 954051123230626270291234117728$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{2}$$ $$=$$ $$($$$$-242923652655874 \nu^{15} - 447584982412244 \nu^{14} + 8913392580059259 \nu^{13} + 22340518099993166 \nu^{12} - 19999365309098170 \nu^{11} - 1832368498536779334 \nu^{10} + 949621789253240568 \nu^{9} + 7772748772651806992 \nu^{8} + 64812307182049095080 \nu^{7} - 61302073465293789672 \nu^{6} - 660627626735960851608 \nu^{5} + 1228955731454532181040 \nu^{4} - 11947009070900420968944 \nu^{3} - 28409259444620330853040 \nu^{2} - 36349592470129001215392 \nu - 133501831405325877366912$$$$)/$$$$82\!\cdots\!16$$ $$\beta_{3}$$ $$=$$ $$($$$$23462447743403590492136 \nu^{15} + 3306707193004082851421 \nu^{14} - 389683344064490127615095 \nu^{13} - 4170115166200193783604428 \nu^{12} + 7416677331165346447484470 \nu^{11} + 112796363577198528409126850 \nu^{10} + 262684767249375455442020044 \nu^{9} - 864190244506350740161837124 \nu^{8} - 7918643353271901470073172088 \nu^{7} - 5181207386792600249260202176 \nu^{6} + 49839577857891202878867555208 \nu^{5} + 230090340592914837948917126896 \nu^{4} + 798090168937435092313498157648 \nu^{3} + 1595747351829822380586303621744 \nu^{2} + 2708676727517415314339816761536 \nu + 948894225283049421155774211360$$$$)/$$$$52\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$18152388531370159545178 \nu^{15} + 33952261669548056332982 \nu^{14} - 528703259529389045544045 \nu^{13} - 3060023906632549898673890 \nu^{12} + 2266346959660254884654668 \nu^{11} + 131136012297896802842851806 \nu^{10} + 167744686330067275251716604 \nu^{9} - 883204749087996487486642244 \nu^{8} - 7923531375196119296641377728 \nu^{7} - 2451694093620279005232275352 \nu^{6} + 70684310987927562231641812632 \nu^{5} + 169990626231809864491889862592 \nu^{4} + 595394627929118000755479582144 \nu^{3} + 1130987662135106186564523209776 \nu^{2} + 536560161415203772234130472384 \nu + 957566329405874163246296000352$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{5}$$ $$=$$ $$($$$$8720676811 \nu^{15} - 13202371445 \nu^{14} - 114600900799 \nu^{13} - 1350686662990 \nu^{12} + 4895130625946 \nu^{11} + 31628741905810 \nu^{10} + 48703616405708 \nu^{9} - 364306145125792 \nu^{8} - 2128682138608960 \nu^{7} + 950956299772576 \nu^{6} + 13969233833942024 \nu^{5} + 59522361775385264 \nu^{4} + 227272830400180048 \nu^{3} + 346388992713204336 \nu^{2} + 438847306359325248 \nu + 518469047993669760$$$$)/ 119594681293366656$$ $$\beta_{6}$$ $$=$$ $$($$$$50293628891097095553008 \nu^{15} - 71358597446208736540393 \nu^{14} - 653127460398806733629099 \nu^{13} - 7932337465992179372342078 \nu^{12} + 27729156391588200311021374 \nu^{11} + 182005797281603860751061086 \nu^{10} + 315362132650450353762139036 \nu^{9} - 2116195479020618986015226852 \nu^{8} - 12238948917348271016877329144 \nu^{7} + 4031357052647639892436028192 \nu^{6} + 81727112460422805396447039976 \nu^{5} + 342476886507414533167406839360 \nu^{4} + 1263295097586532675618310222096 \nu^{3} + 2345503333450615173582572185296 \nu^{2} + 3392199526235234166638098035648 \nu + 5450493844850303003980890274272$$$$)/$$$$52\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-37600096092988470445108 \nu^{15} + 115791625486460312097850 \nu^{14} + 396979116467037978167631 \nu^{13} + 5052533523498529374512222 \nu^{12} - 30502121402922487508466292 \nu^{11} - 101011856467479060832126842 \nu^{10} + 5517363532984253227941060 \nu^{9} + 1916377806662540824345922132 \nu^{8} + 6321687292165099278722173712 \nu^{7} - 19389172662767665805731857624 \nu^{6} - 50950528402846544152031676840 \nu^{5} - 138853207801601950025725456480 \nu^{4} - 528925691102421700084758157824 \nu^{3} - 302984729216542010949962019088 \nu^{2} - 599301483532701224669927873664 \nu - 1300193552468139004877267685216$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{8}$$ $$=$$ $$($$$$-250165761321952716759925 \nu^{15} + 666364840311140238378433 \nu^{14} + 2106329525900393947120239 \nu^{13} + 37887057075793876039321898 \nu^{12} - 182983516300869136192961026 \nu^{11} - 630123167099615200735406022 \nu^{10} - 1062921436808542257242876508 \nu^{9} + 11151594097388930258304036044 \nu^{8} + 46108642210499228070744865616 \nu^{7} - 58107385513601178994974873888 \nu^{6} - 282794440806689899101119047080 \nu^{5} - 1506332184212999042290049683456 \nu^{4} - 5145085836711832366243536740688 \nu^{3} - 6591537826757520292825373589520 \nu^{2} - 10720418944282553682717981769536 \nu - 12349597214137255435397535784608$$$$)/$$$$15\!\cdots\!40$$ $$\beta_{9}$$ $$=$$ $$($$$$-50229924220807943675926 \nu^{15} + 134174953016742483421057 \nu^{14} + 506856859295783262766728 \nu^{13} + 7316226644053250742984638 \nu^{12} - 37408613043250104801267244 \nu^{11} - 138532802239566438769224288 \nu^{10} - 139642065933408787433184900 \nu^{9} + 2428328569349261336011852472 \nu^{8} + 9366331001888042017638001424 \nu^{7} - 16015894012543913553307482216 \nu^{6} - 70856365353937996580644925232 \nu^{5} - 259490494713195787577598182320 \nu^{4} - 932866478515768684527546548448 \nu^{3} - 890725629073167797932738215712 \nu^{2} - 1164237954406702466441109946656 \nu - 2064750710812539980211384948480$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{10}$$ $$=$$ $$($$$$-377483293763027555126654 \nu^{15} + 862012978868505031108961 \nu^{14} + 4171994395858734861701820 \nu^{13} + 56700094108685111211984682 \nu^{12} - 261042208674625130509238060 \nu^{11} - 1144940905327410968966893440 \nu^{10} - 1484758180739527539361565076 \nu^{9} + 18018045208898778426380921056 \nu^{8} + 77149085414368778914674195472 \nu^{7} - 94550171310802877655316547016 \nu^{6} - 580999009253799866431046214672 \nu^{5} - 2161706316427127802221267634224 \nu^{4} - 7583266232656538326468443305952 \nu^{3} - 9624813922498572402585806709536 \nu^{2} - 11797102026518209986939058264224 \nu - 18436994687653549365122066253120$$$$)/$$$$15\!\cdots\!40$$ $$\beta_{11}$$ $$=$$ $$($$$$-76272810421984360100806 \nu^{15} + 141544925637719199156739 \nu^{14} + 959001844890376474407348 \nu^{13} + 11709791094232631362337342 \nu^{12} - 48262659980609858803277836 \nu^{11} - 259855160714949305075447232 \nu^{10} - 368367939375091739413416540 \nu^{9} + 3624547586838333584632534952 \nu^{8} + 17232379818175668144460959728 \nu^{7} - 14327102604432559058141410872 \nu^{6} - 132343923380182677748123793712 \nu^{5} - 471145620176994452224972108912 \nu^{4} - 1679492846092855117596421531872 \nu^{3} - 2465617669137961976930038852960 \nu^{2} - 2212765353490068388724361926496 \nu - 3306336067268112462602117384064$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{12}$$ $$=$$ $$($$$$-79013706264410744435129 \nu^{15} + 191512213263779825263088 \nu^{14} + 869749577986228998540681 \nu^{13} + 11758240526694387400219858 \nu^{12} - 56910206056909991646669290 \nu^{11} - 235257407235283664867690358 \nu^{10} - 268191647824526536491054936 \nu^{9} + 3951588653562693204919853404 \nu^{8} + 15726241697740921384883827936 \nu^{7} - 23614283067967876376022702840 \nu^{6} - 125573370748558498807695089352 \nu^{5} - 430914435433276779921059002400 \nu^{4} - 1428790011923355209700348576912 \nu^{3} - 1730877003338491773921242540720 \nu^{2} - 1796628691213336316868168962784 \nu - 2352284944037486192310883266336$$$$)/$$$$31\!\cdots\!28$$ $$\beta_{13}$$ $$=$$ $$($$$$-178546106847565561250739 \nu^{15} + 385033124236801847086253 \nu^{14} + 2241106475421094849784846 \nu^{13} + 25872416712835392264806704 \nu^{12} - 119414490020946878474491094 \nu^{11} - 588478189579633043213903888 \nu^{10} - 510190238200506226114896628 \nu^{9} + 8384094058083245483091168652 \nu^{8} + 37703408614015556234249294656 \nu^{7} - 53563104756714140912703607928 \nu^{6} - 283094855476030953374660658112 \nu^{5} - 918014415247160273298431277968 \nu^{4} - 3623960320583978303980811790224 \nu^{3} - 4605824056173437253705861892576 \nu^{2} - 5915515398849179149190599067808 \nu - 9953338096495597815030149019552$$$$)/$$$$52\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$187414018302707447846341 \nu^{15} - 308680043147559268424170 \nu^{14} - 2571488847115456728610317 \nu^{13} - 28753033095756810181869950 \nu^{12} + 112547124638202356962249240 \nu^{11} + 690817385388165808885509570 \nu^{10} + 906830666968159912316159496 \nu^{9} - 8875619214241179919334985920 \nu^{8} - 45884716670977003070329423160 \nu^{7} + 35189621602176013120725087048 \nu^{6} + 356842988766576546476613776616 \nu^{5} + 1229779685310697849599779294416 \nu^{4} + 4143719510800586526497017785984 \nu^{3} + 5440153711115088673082092651216 \nu^{2} + 5370548497693787984585278491168 \nu + 6675968846314845289777935197568$$$$)/$$$$52\!\cdots\!80$$ $$\beta_{15}$$ $$=$$ $$($$$$-7673373742587874 \nu^{15} + 15728500554503692 \nu^{14} + 84627321827383995 \nu^{13} + 1199681528211152294 \nu^{12} - 5029485685832613274 \nu^{11} - 24104493692250527718 \nu^{10} - 40964385084872513256 \nu^{9} + 358606789126050089120 \nu^{8} + 1691715144102335947688 \nu^{7} - 1330876457193615635304 \nu^{6} - 12230617071925475468568 \nu^{5} - 51000763275834253943824 \nu^{4} - 164257421028687496205040 \nu^{3} - 234451448771349453099952 \nu^{2} - 291293123488915188976800 \nu - 287384130460190230237056$$$$)/$$$$82\!\cdots\!16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$-\beta_{12} + \beta_{11} - \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{15} - 2 \beta_{12} + \beta_{11} + 6 \beta_{10} - \beta_{9} + 2 \beta_{5} - 2 \beta_{3} - \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{15} - 6 \beta_{13} - 2 \beta_{12} + 7 \beta_{11} + 2 \beta_{10} + 9 \beta_{9} - 3 \beta_{8} + 6 \beta_{7} - 15 \beta_{6} - 3 \beta_{3} - 3 \beta_{2} - \beta_{1} + 33$$ $$\nu^{4}$$ $$=$$ $$-10 \beta_{15} - 16 \beta_{13} - 48 \beta_{12} + 80 \beta_{11} - 32 \beta_{10} + 56 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} - 20 \beta_{6} + 24 \beta_{5} - 4 \beta_{4} - 60 \beta_{3} + 2 \beta_{2} - 76 \beta_{1} - 42$$ $$\nu^{5}$$ $$=$$ $$-140 \beta_{15} - 100 \beta_{14} - 40 \beta_{13} + 282 \beta_{12} + 70 \beta_{11} + 90 \beta_{10} - 54 \beta_{9} + 50 \beta_{8} + 80 \beta_{7} - 150 \beta_{6} + 220 \beta_{5} + 180 \beta_{4} - 210 \beta_{3} - 120 \beta_{2} + 392 \beta_{1} - 440$$ $$\nu^{6}$$ $$=$$ $$-280 \beta_{15} - 120 \beta_{14} - 1040 \beta_{13} + 1532 \beta_{12} - 32 \beta_{11} - 988 \beta_{10} + 1472 \beta_{9} - 960 \beta_{8} + 780 \beta_{7} - 1360 \beta_{6} - 2064 \beta_{5} + 540 \beta_{4} - 424 \beta_{3} - 160 \beta_{2} + 844 \beta_{1} - 916$$ $$\nu^{7}$$ $$=$$ $$-392 \beta_{15} - 1400 \beta_{14} - 1736 \beta_{13} + 6112 \beta_{12} + 1152 \beta_{11} - 11408 \beta_{10} + 4644 \beta_{9} + 672 \beta_{8} - 1848 \beta_{7} - 420 \beta_{6} - 7700 \beta_{5} + 1512 \beta_{4} - 3528 \beta_{3} + 1428 \beta_{2} - 4124 \beta_{1} - 26180$$ $$\nu^{8}$$ $$=$$ $$-4680 \beta_{15} - 14560 \beta_{14} + 5280 \beta_{13} + 63264 \beta_{12} - 31984 \beta_{11} - 11424 \beta_{10} - 20336 \beta_{9} - 15840 \beta_{8} - 7968 \beta_{7} + 320 \beta_{6} - 18368 \beta_{5} + 32608 \beta_{4} + 2528 \beta_{3} - 4792 \beta_{2} + 55552 \beta_{1} - 102080$$ $$\nu^{9}$$ $$=$$ $$34512 \beta_{15} + 31920 \beta_{14} - 52992 \beta_{13} + 193328 \beta_{12} - 151664 \beta_{11} - 91640 \beta_{10} + 54360 \beta_{9} - 225656 \beta_{8} + 288 \beta_{7} - 5640 \beta_{6} - 544896 \beta_{5} + 53136 \beta_{4} + 73752 \beta_{3} + 21264 \beta_{2} + 9272 \beta_{1} - 332112$$ $$\nu^{10}$$ $$=$$ $$225416 \beta_{15} + 138560 \beta_{14} + 324000 \beta_{13} + 586080 \beta_{12} - 630536 \beta_{11} - 806592 \beta_{10} - 75064 \beta_{9} - 95680 \beta_{8} - 893040 \beta_{7} + 669280 \beta_{6} - 1559312 \beta_{5} - 157680 \beta_{4} + 281584 \beta_{3} + 278696 \beta_{2} - 825136 \beta_{1} - 2529224$$ $$\nu^{11}$$ $$=$$ $$912120 \beta_{15} + 132000 \beta_{14} + 3524752 \beta_{13} + 3000592 \beta_{12} - 5249976 \beta_{11} + 1805776 \beta_{10} - 4256280 \beta_{9} - 2414984 \beta_{8} - 3422672 \beta_{7} + 2731080 \beta_{6} - 1364880 \beta_{5} + 1517472 \beta_{4} + 3465176 \beta_{3} - 304920 \beta_{2} + 3742584 \beta_{1} - 2419032$$ $$\nu^{12}$$ $$=$$ $$8715152 \beta_{15} + 16343360 \beta_{14} + 5499584 \beta_{13} - 11473984 \beta_{12} - 11912352 \beta_{11} + 13264576 \beta_{10} + 1420704 \beta_{9} - 21509728 \beta_{8} - 6786144 \beta_{7} + 12232160 \beta_{6} - 45372096 \beta_{5} - 12031776 \beta_{4} + 14135904 \beta_{3} + 1767728 \beta_{2} - 18399968 \beta_{1} + 12213744$$ $$\nu^{13}$$ $$=$$ $$26048256 \beta_{15} + 60513440 \beta_{14} + 83049408 \beta_{13} - 74961680 \beta_{12} - 24939472 \beta_{11} + 10794576 \beta_{10} - 9743760 \beta_{9} + 77072944 \beta_{8} - 113015552 \beta_{7} + 96436080 \beta_{6} + 40630304 \beta_{5} - 113267232 \beta_{4} + 37066640 \beta_{3} + 13762528 \beta_{2} - 120783040 \beta_{1} + 50642592$$ $$\nu^{14}$$ $$=$$ $$62006112 \beta_{15} + 137067840 \beta_{14} + 470008448 \beta_{13} - 402492768 \beta_{12} - 132751072 \beta_{11} + 469866208 \beta_{10} - 255977632 \beta_{9} + 210892864 \beta_{8} - 187856032 \beta_{7} + 177020480 \beta_{6} + 1270042048 \beta_{5} - 200307744 \beta_{4} + 280724416 \beta_{3} - 134058848 \beta_{2} + 431560480 \beta_{1} + 1507589632$$ $$\nu^{15}$$ $$=$$ $$413475488 \beta_{15} + 2002436800 \beta_{14} + 36917888 \beta_{13} - 4235715200 \beta_{12} + 1725017184 \beta_{11} + 1960879104 \beta_{10} + 1695252672 \beta_{9} + 486664864 \beta_{8} + 1157105280 \beta_{7} + 327788160 \beta_{6} + 1890276960 \beta_{5} - 2820557376 \beta_{4} - 82527136 \beta_{3} - 244388992 \beta_{2} - 1754484928 \beta_{1} + 6623300288$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1 - \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −2.73864 − 0.330355i −2.73864 − 4.07201i 4.54639 + 2.26426i 4.54639 − 1.47739i 0.144025 + 1.47739i 0.144025 − 2.26426i −1.95177 + 4.07201i −1.95177 + 0.330355i −2.73864 + 0.330355i −2.73864 + 4.07201i 4.54639 − 2.26426i 4.54639 + 1.47739i 0.144025 − 1.47739i 0.144025 + 2.26426i −1.95177 − 4.07201i −1.95177 − 0.330355i
−1.70956 2.96105i −2.79279 + 1.09560i −3.84521 + 6.66010i 3.81256 + 6.60355i 8.01856 + 6.39660i −4.11804 + 5.66055i 12.6180 6.59934 6.11953i 13.0356 22.5784i
116.2 −1.70956 2.96105i 0.447581 + 2.96642i −3.84521 + 6.66010i 3.81256 + 6.60355i 8.01856 6.39660i 4.11804 5.66055i 12.6180 −8.59934 + 2.65543i 13.0356 22.5784i
116.3 −0.759866 1.31613i −0.447581 2.96642i 0.845208 1.46394i −0.681452 1.18031i −3.56409 + 2.84316i 0.736052 + 6.96119i −8.64790 −8.59934 + 2.65543i −1.03562 + 1.79375i
116.4 −0.759866 1.31613i 2.79279 1.09560i 0.845208 1.46394i −0.681452 1.18031i −3.56409 2.84316i −0.736052 6.96119i −8.64790 6.59934 6.11953i −1.03562 + 1.79375i
116.5 0.759866 + 1.31613i −0.447581 2.96642i 0.845208 1.46394i 0.681452 + 1.18031i 3.56409 2.84316i −0.736052 6.96119i 8.64790 −8.59934 + 2.65543i −1.03562 + 1.79375i
116.6 0.759866 + 1.31613i 2.79279 1.09560i 0.845208 1.46394i 0.681452 + 1.18031i 3.56409 + 2.84316i 0.736052 + 6.96119i 8.64790 6.59934 6.11953i −1.03562 + 1.79375i
116.7 1.70956 + 2.96105i −2.79279 + 1.09560i −3.84521 + 6.66010i −3.81256 6.60355i −8.01856 6.39660i 4.11804 5.66055i −12.6180 6.59934 6.11953i 13.0356 22.5784i
116.8 1.70956 + 2.96105i 0.447581 + 2.96642i −3.84521 + 6.66010i −3.81256 6.60355i −8.01856 + 6.39660i −4.11804 + 5.66055i −12.6180 −8.59934 + 2.65543i 13.0356 22.5784i
233.1 −1.70956 + 2.96105i −2.79279 1.09560i −3.84521 6.66010i 3.81256 6.60355i 8.01856 6.39660i −4.11804 5.66055i 12.6180 6.59934 + 6.11953i 13.0356 + 22.5784i
233.2 −1.70956 + 2.96105i 0.447581 2.96642i −3.84521 6.66010i 3.81256 6.60355i 8.01856 + 6.39660i 4.11804 + 5.66055i 12.6180 −8.59934 2.65543i 13.0356 + 22.5784i
233.3 −0.759866 + 1.31613i −0.447581 + 2.96642i 0.845208 + 1.46394i −0.681452 + 1.18031i −3.56409 2.84316i 0.736052 6.96119i −8.64790 −8.59934 2.65543i −1.03562 1.79375i
233.4 −0.759866 + 1.31613i 2.79279 + 1.09560i 0.845208 + 1.46394i −0.681452 + 1.18031i −3.56409 + 2.84316i −0.736052 + 6.96119i −8.64790 6.59934 + 6.11953i −1.03562 1.79375i
233.5 0.759866 1.31613i −0.447581 + 2.96642i 0.845208 + 1.46394i 0.681452 1.18031i 3.56409 + 2.84316i −0.736052 + 6.96119i 8.64790 −8.59934 2.65543i −1.03562 1.79375i
233.6 0.759866 1.31613i 2.79279 + 1.09560i 0.845208 + 1.46394i 0.681452 1.18031i 3.56409 2.84316i 0.736052 6.96119i 8.64790 6.59934 + 6.11953i −1.03562 1.79375i
233.7 1.70956 2.96105i −2.79279 1.09560i −3.84521 6.66010i −3.81256 + 6.60355i −8.01856 + 6.39660i 4.11804 + 5.66055i −12.6180 6.59934 + 6.11953i 13.0356 + 22.5784i
233.8 1.70956 2.96105i 0.447581 2.96642i −3.84521 6.66010i −3.81256 + 6.60355i −8.01856 6.39660i −4.11804 5.66055i −12.6180 −8.59934 2.65543i 13.0356 + 22.5784i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 233.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
13.b even 2 1 inner
21.h odd 6 1 inner
39.d odd 2 1 inner
91.r even 6 1 inner
273.w odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.3.w.c 16
3.b odd 2 1 inner 273.3.w.c 16
7.c even 3 1 inner 273.3.w.c 16
13.b even 2 1 inner 273.3.w.c 16
21.h odd 6 1 inner 273.3.w.c 16
39.d odd 2 1 inner 273.3.w.c 16
91.r even 6 1 inner 273.3.w.c 16
273.w odd 6 1 inner 273.3.w.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.3.w.c 16 1.a even 1 1 trivial
273.3.w.c 16 3.b odd 2 1 inner
273.3.w.c 16 7.c even 3 1 inner
273.3.w.c 16 13.b even 2 1 inner
273.3.w.c 16 21.h odd 6 1 inner
273.3.w.c 16 39.d odd 2 1 inner
273.3.w.c 16 91.r even 6 1 inner
273.3.w.c 16 273.w odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{8} + 14 T_{2}^{6} + 169 T_{2}^{4} + 378 T_{2}^{2} + 729$$ $$T_{19}^{8} - 70 T_{19}^{6} + 4753 T_{19}^{4} - 10290 T_{19}^{2} + 21609$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 729 + 378 T^{2} + 169 T^{4} + 14 T^{6} + T^{8} )^{2}$$
$3$ $$( 6561 + 324 T^{2} - 65 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$5$ $$( 11664 + 6480 T^{2} + 3492 T^{4} + 60 T^{6} + T^{8} )^{2}$$
$7$ $$( 5764801 + 302526 T^{2} + 7693 T^{4} + 126 T^{6} + T^{8} )^{2}$$
$11$ $$( 59049 + 10206 T^{2} + 1521 T^{4} + 42 T^{6} + T^{8} )^{2}$$
$13$ $$( 13 + T )^{16}$$
$17$ $$( 5554571841 - 63647766 T^{2} + 654787 T^{4} - 854 T^{6} + T^{8} )^{2}$$
$19$ $$( 21609 - 10290 T^{2} + 4753 T^{4} - 70 T^{6} + T^{8} )^{2}$$
$23$ $$( 49787136 - 7507584 T^{2} + 1125040 T^{4} - 1064 T^{6} + T^{8} )^{2}$$
$29$ $$( 670761 + 1946 T^{2} + T^{4} )^{4}$$
$31$ $$( 830131344 - 10487568 T^{2} + 103684 T^{4} - 364 T^{6} + T^{8} )^{2}$$
$37$ $$( 1416167424 - 42147840 T^{2} + 1216768 T^{4} - 1120 T^{6} + T^{8} )^{2}$$
$41$ $$( 645888 - 2528 T^{2} + T^{4} )^{4}$$
$43$ $$( 42 + 16 T + T^{2} )^{8}$$
$47$ $$( 5132413223289 + 11014778346 T^{2} + 21373561 T^{4} + 4862 T^{6} + T^{8} )^{2}$$
$53$ $$( 1394598310951041 - 475243829046 T^{2} + 124606755 T^{4} - 12726 T^{6} + T^{8} )^{2}$$
$59$ $$( 149492809449 + 622495230 T^{2} + 2205457 T^{4} + 1610 T^{6} + T^{8} )^{2}$$
$61$ $$( 1570009 + 57638 T + 3369 T^{2} - 46 T^{3} + T^{4} )^{4}$$
$67$ $$( 841672300329 - 9620139522 T^{2} + 109038769 T^{4} - 10486 T^{6} + T^{8} )^{2}$$
$71$ $$( 6940323 - 9738 T^{2} + T^{4} )^{4}$$
$73$ $$( 13077026686909584 - 2449480415760 T^{2} + 344461572 T^{4} - 21420 T^{6} + T^{8} )^{2}$$
$79$ $$( 4410000 + 42000 T + 2500 T^{2} - 20 T^{3} + T^{4} )^{4}$$
$83$ $$( 1379052 - 11004 T^{2} + T^{4} )^{4}$$
$89$ $$( 33109016542804224 + 5716418666112 T^{2} + 805006224 T^{4} + 31416 T^{6} + T^{8} )^{2}$$
$97$ $$( 397488 + 1288 T^{2} + T^{4} )^{4}$$