# Properties

 Label 3850.2.c.y.1849.2 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 770) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.2 Root $$-2.37228i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.y.1849.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +6.74456i q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456i q^{17} +1.00000i q^{18} +6.74456 q^{19} +2.00000 q^{21} +1.00000i q^{22} -6.74456i q^{23} -2.00000 q^{24} +6.74456 q^{26} +4.00000i q^{27} +1.00000i q^{28} -8.74456 q^{29} +4.74456 q^{31} -1.00000i q^{32} -2.00000i q^{33} +6.74456 q^{34} +1.00000 q^{36} -0.744563i q^{37} -6.74456i q^{38} -13.4891 q^{39} -4.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} -6.74456 q^{46} -4.74456i q^{47} +2.00000i q^{48} -1.00000 q^{49} -13.4891 q^{51} -6.74456i q^{52} -12.7446i q^{53} +4.00000 q^{54} +1.00000 q^{56} +13.4891i q^{57} +8.74456i q^{58} +8.74456 q^{59} +1.25544 q^{61} -4.74456i q^{62} +1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +4.00000i q^{67} -6.74456i q^{68} +13.4891 q^{69} -4.00000 q^{71} -1.00000i q^{72} +10.7446i q^{73} -0.744563 q^{74} -6.74456 q^{76} +1.00000i q^{77} +13.4891i q^{78} -6.74456 q^{79} -11.0000 q^{81} +4.00000i q^{82} +8.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -17.4891i q^{87} -1.00000i q^{88} -15.4891 q^{89} +6.74456 q^{91} +6.74456i q^{92} +9.48913i q^{93} -4.74456 q^{94} +2.00000 q^{96} +16.7446i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 4 q^{16} + 4 q^{19} + 8 q^{21} - 8 q^{24} + 4 q^{26} - 12 q^{29} - 4 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{39} - 16 q^{41} + 4 q^{44} - 4 q^{46} - 4 q^{49} - 8 q^{51} + 16 q^{54} + 4 q^{56} + 12 q^{59} + 28 q^{61} - 4 q^{64} - 8 q^{66} + 8 q^{69} - 16 q^{71} + 20 q^{74} - 4 q^{76} - 4 q^{79} - 44 q^{81} - 8 q^{84} + 16 q^{86} - 16 q^{89} + 4 q^{91} + 4 q^{94} + 8 q^{96} + 4 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ − 2.00000i − 0.577350i
$$13$$ 6.74456i 1.87061i 0.353849 + 0.935303i $$0.384873\pi$$
−0.353849 + 0.935303i $$0.615127\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.74456i 1.63580i 0.575363 + 0.817898i $$0.304861\pi$$
−0.575363 + 0.817898i $$0.695139\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 6.74456 1.54731 0.773654 0.633608i $$-0.218427\pi$$
0.773654 + 0.633608i $$0.218427\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 1.00000i 0.213201i
$$23$$ − 6.74456i − 1.40634i −0.711022 0.703169i $$-0.751768\pi$$
0.711022 0.703169i $$-0.248232\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 0 0
$$26$$ 6.74456 1.32272
$$27$$ 4.00000i 0.769800i
$$28$$ 1.00000i 0.188982i
$$29$$ −8.74456 −1.62382 −0.811912 0.583779i $$-0.801573\pi$$
−0.811912 + 0.583779i $$0.801573\pi$$
$$30$$ 0 0
$$31$$ 4.74456 0.852149 0.426074 0.904688i $$-0.359896\pi$$
0.426074 + 0.904688i $$0.359896\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ 6.74456 1.15668
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 0.744563i − 0.122405i −0.998125 0.0612027i $$-0.980506\pi$$
0.998125 0.0612027i $$-0.0194936\pi$$
$$38$$ − 6.74456i − 1.09411i
$$39$$ −13.4891 −2.15999
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −6.74456 −0.994432
$$47$$ − 4.74456i − 0.692066i −0.938222 0.346033i $$-0.887529\pi$$
0.938222 0.346033i $$-0.112471\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −13.4891 −1.88886
$$52$$ − 6.74456i − 0.935303i
$$53$$ − 12.7446i − 1.75060i −0.483580 0.875300i $$-0.660664\pi$$
0.483580 0.875300i $$-0.339336\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 13.4891i 1.78668i
$$58$$ 8.74456i 1.14822i
$$59$$ 8.74456 1.13845 0.569223 0.822183i $$-0.307244\pi$$
0.569223 + 0.822183i $$0.307244\pi$$
$$60$$ 0 0
$$61$$ 1.25544 0.160742 0.0803711 0.996765i $$-0.474389\pi$$
0.0803711 + 0.996765i $$0.474389\pi$$
$$62$$ − 4.74456i − 0.602560i
$$63$$ 1.00000i 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 6.74456i − 0.817898i
$$69$$ 13.4891 1.62390
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 10.7446i 1.25756i 0.777585 + 0.628778i $$0.216445\pi$$
−0.777585 + 0.628778i $$0.783555\pi$$
$$74$$ −0.744563 −0.0865536
$$75$$ 0 0
$$76$$ −6.74456 −0.773654
$$77$$ 1.00000i 0.113961i
$$78$$ 13.4891i 1.52734i
$$79$$ −6.74456 −0.758823 −0.379411 0.925228i $$-0.623873\pi$$
−0.379411 + 0.925228i $$0.623873\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 4.00000i 0.441726i
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 17.4891i − 1.87503i
$$88$$ − 1.00000i − 0.106600i
$$89$$ −15.4891 −1.64184 −0.820922 0.571040i $$-0.806540\pi$$
−0.820922 + 0.571040i $$0.806540\pi$$
$$90$$ 0 0
$$91$$ 6.74456 0.707022
$$92$$ 6.74456i 0.703169i
$$93$$ 9.48913i 0.983976i
$$94$$ −4.74456 −0.489364
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ 16.7446i 1.70015i 0.526659 + 0.850076i $$0.323444\pi$$
−0.526659 + 0.850076i $$0.676556\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 2.74456 0.273094 0.136547 0.990634i $$-0.456399\pi$$
0.136547 + 0.990634i $$0.456399\pi$$
$$102$$ 13.4891i 1.33562i
$$103$$ − 0.744563i − 0.0733639i −0.999327 0.0366820i $$-0.988321\pi$$
0.999327 0.0366820i $$-0.0116789\pi$$
$$104$$ −6.74456 −0.661359
$$105$$ 0 0
$$106$$ −12.7446 −1.23786
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ 18.2337 1.74647 0.873235 0.487299i $$-0.162018\pi$$
0.873235 + 0.487299i $$0.162018\pi$$
$$110$$ 0 0
$$111$$ 1.48913 0.141342
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 10.0000i 0.940721i 0.882474 + 0.470360i $$0.155876\pi$$
−0.882474 + 0.470360i $$0.844124\pi$$
$$114$$ 13.4891 1.26337
$$115$$ 0 0
$$116$$ 8.74456 0.811912
$$117$$ − 6.74456i − 0.623535i
$$118$$ − 8.74456i − 0.805002i
$$119$$ 6.74456 0.618273
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 1.25544i − 0.113662i
$$123$$ − 8.00000i − 0.721336i
$$124$$ −4.74456 −0.426074
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −2.74456 −0.239794 −0.119897 0.992786i $$-0.538256\pi$$
−0.119897 + 0.992786i $$0.538256\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ − 6.74456i − 0.584828i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −6.74456 −0.578341
$$137$$ − 3.48913i − 0.298096i −0.988830 0.149048i $$-0.952379\pi$$
0.988830 0.149048i $$-0.0476209\pi$$
$$138$$ − 13.4891i − 1.14827i
$$139$$ −14.7446 −1.25062 −0.625309 0.780377i $$-0.715027\pi$$
−0.625309 + 0.780377i $$0.715027\pi$$
$$140$$ 0 0
$$141$$ 9.48913 0.799129
$$142$$ 4.00000i 0.335673i
$$143$$ − 6.74456i − 0.564009i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.7446 0.889226
$$147$$ − 2.00000i − 0.164957i
$$148$$ 0.744563i 0.0612027i
$$149$$ 0.744563 0.0609969 0.0304985 0.999535i $$-0.490291\pi$$
0.0304985 + 0.999535i $$0.490291\pi$$
$$150$$ 0 0
$$151$$ −9.25544 −0.753197 −0.376598 0.926377i $$-0.622906\pi$$
−0.376598 + 0.926377i $$0.622906\pi$$
$$152$$ 6.74456i 0.547056i
$$153$$ − 6.74456i − 0.545266i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ 13.4891 1.07999
$$157$$ − 0.510875i − 0.0407722i −0.999792 0.0203861i $$-0.993510\pi$$
0.999792 0.0203861i $$-0.00648955\pi$$
$$158$$ 6.74456i 0.536569i
$$159$$ 25.4891 2.02142
$$160$$ 0 0
$$161$$ −6.74456 −0.531546
$$162$$ 11.0000i 0.864242i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −32.4891 −2.49916
$$170$$ 0 0
$$171$$ −6.74456 −0.515770
$$172$$ − 4.00000i − 0.304997i
$$173$$ 20.2337i 1.53834i 0.639045 + 0.769169i $$0.279330\pi$$
−0.639045 + 0.769169i $$0.720670\pi$$
$$174$$ −17.4891 −1.32585
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 17.4891i 1.31456i
$$178$$ 15.4891i 1.16096i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −19.4891 −1.44862 −0.724308 0.689477i $$-0.757840\pi$$
−0.724308 + 0.689477i $$0.757840\pi$$
$$182$$ − 6.74456i − 0.499940i
$$183$$ 2.51087i 0.185609i
$$184$$ 6.74456 0.497216
$$185$$ 0 0
$$186$$ 9.48913 0.695776
$$187$$ − 6.74456i − 0.493211i
$$188$$ 4.74456i 0.346033i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 16.7446 1.20219
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 10.0000i − 0.712470i −0.934396 0.356235i $$-0.884060\pi$$
0.934396 0.356235i $$-0.115940\pi$$
$$198$$ − 1.00000i − 0.0710669i
$$199$$ −3.25544 −0.230772 −0.115386 0.993321i $$-0.536810\pi$$
−0.115386 + 0.993321i $$0.536810\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ − 2.74456i − 0.193107i
$$203$$ 8.74456i 0.613748i
$$204$$ 13.4891 0.944428
$$205$$ 0 0
$$206$$ −0.744563 −0.0518761
$$207$$ 6.74456i 0.468780i
$$208$$ 6.74456i 0.467651i
$$209$$ −6.74456 −0.466531
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 12.7446i 0.875300i
$$213$$ − 8.00000i − 0.548151i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ − 4.74456i − 0.322082i
$$218$$ − 18.2337i − 1.23494i
$$219$$ −21.4891 −1.45210
$$220$$ 0 0
$$221$$ −45.4891 −3.05993
$$222$$ − 1.48913i − 0.0999435i
$$223$$ 15.2554i 1.02158i 0.859706 + 0.510790i $$0.170647\pi$$
−0.859706 + 0.510790i $$0.829353\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ − 13.4891i − 0.893339i
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ − 8.74456i − 0.574109i
$$233$$ − 24.9783i − 1.63638i −0.574948 0.818190i $$-0.694978\pi$$
0.574948 0.818190i $$-0.305022\pi$$
$$234$$ −6.74456 −0.440906
$$235$$ 0 0
$$236$$ −8.74456 −0.569223
$$237$$ − 13.4891i − 0.876213i
$$238$$ − 6.74456i − 0.437185i
$$239$$ 14.7446 0.953746 0.476873 0.878972i $$-0.341770\pi$$
0.476873 + 0.878972i $$0.341770\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ − 10.0000i − 0.641500i
$$244$$ −1.25544 −0.0803711
$$245$$ 0 0
$$246$$ −8.00000 −0.510061
$$247$$ 45.4891i 2.89440i
$$248$$ 4.74456i 0.301280i
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ −26.2337 −1.65586 −0.827928 0.560835i $$-0.810480\pi$$
−0.827928 + 0.560835i $$0.810480\pi$$
$$252$$ − 1.00000i − 0.0629941i
$$253$$ 6.74456i 0.424027i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 10.2337i 0.638360i 0.947694 + 0.319180i $$0.103407\pi$$
−0.947694 + 0.319180i $$0.896593\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −0.744563 −0.0462649
$$260$$ 0 0
$$261$$ 8.74456 0.541275
$$262$$ 2.74456i 0.169560i
$$263$$ 26.9783i 1.66355i 0.555113 + 0.831775i $$0.312675\pi$$
−0.555113 + 0.831775i $$0.687325\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ −6.74456 −0.413536
$$267$$ − 30.9783i − 1.89584i
$$268$$ − 4.00000i − 0.244339i
$$269$$ −20.9783 −1.27907 −0.639533 0.768763i $$-0.720872\pi$$
−0.639533 + 0.768763i $$0.720872\pi$$
$$270$$ 0 0
$$271$$ 14.9783 0.909864 0.454932 0.890526i $$-0.349664\pi$$
0.454932 + 0.890526i $$0.349664\pi$$
$$272$$ 6.74456i 0.408949i
$$273$$ 13.4891i 0.816399i
$$274$$ −3.48913 −0.210786
$$275$$ 0 0
$$276$$ −13.4891 −0.811950
$$277$$ − 15.4891i − 0.930651i −0.885140 0.465326i $$-0.845937\pi$$
0.885140 0.465326i $$-0.154063\pi$$
$$278$$ 14.7446i 0.884320i
$$279$$ −4.74456 −0.284050
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ − 9.48913i − 0.565069i
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ −6.74456 −0.398814
$$287$$ 4.00000i 0.236113i
$$288$$ 1.00000i 0.0589256i
$$289$$ −28.4891 −1.67583
$$290$$ 0 0
$$291$$ −33.4891 −1.96317
$$292$$ − 10.7446i − 0.628778i
$$293$$ 13.2554i 0.774391i 0.921998 + 0.387195i $$0.126556\pi$$
−0.921998 + 0.387195i $$0.873444\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 0.744563 0.0432768
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 0.744563i − 0.0431314i
$$299$$ 45.4891 2.63070
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 9.25544i 0.532591i
$$303$$ 5.48913i 0.315342i
$$304$$ 6.74456 0.386827
$$305$$ 0 0
$$306$$ −6.74456 −0.385561
$$307$$ 1.48913i 0.0849889i 0.999097 + 0.0424944i $$0.0135305\pi$$
−0.999097 + 0.0424944i $$0.986470\pi$$
$$308$$ − 1.00000i − 0.0569803i
$$309$$ 1.48913 0.0847134
$$310$$ 0 0
$$311$$ −20.7446 −1.17632 −0.588158 0.808746i $$-0.700147\pi$$
−0.588158 + 0.808746i $$0.700147\pi$$
$$312$$ − 13.4891i − 0.763671i
$$313$$ − 2.23369i − 0.126256i −0.998005 0.0631278i $$-0.979892\pi$$
0.998005 0.0631278i $$-0.0201076\pi$$
$$314$$ −0.510875 −0.0288303
$$315$$ 0 0
$$316$$ 6.74456 0.379411
$$317$$ − 2.23369i − 0.125456i −0.998031 0.0627282i $$-0.980020\pi$$
0.998031 0.0627282i $$-0.0199801\pi$$
$$318$$ − 25.4891i − 1.42936i
$$319$$ 8.74456 0.489602
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 6.74456i 0.375860i
$$323$$ 45.4891i 2.53108i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 36.4674i 2.01665i
$$328$$ − 4.00000i − 0.220863i
$$329$$ −4.74456 −0.261576
$$330$$ 0 0
$$331$$ 14.9783 0.823279 0.411640 0.911347i $$-0.364956\pi$$
0.411640 + 0.911347i $$0.364956\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ 0.744563i 0.0408018i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ 32.4891i 1.76718i
$$339$$ −20.0000 −1.08625
$$340$$ 0 0
$$341$$ −4.74456 −0.256932
$$342$$ 6.74456i 0.364704i
$$343$$ 1.00000i 0.0539949i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 20.2337 1.08777
$$347$$ 22.9783i 1.23354i 0.787145 + 0.616769i $$0.211559\pi$$
−0.787145 + 0.616769i $$0.788441\pi$$
$$348$$ 17.4891i 0.937516i
$$349$$ −30.7446 −1.64572 −0.822859 0.568245i $$-0.807623\pi$$
−0.822859 + 0.568245i $$0.807623\pi$$
$$350$$ 0 0
$$351$$ −26.9783 −1.43999
$$352$$ 1.00000i 0.0533002i
$$353$$ 8.74456i 0.465426i 0.972545 + 0.232713i $$0.0747603\pi$$
−0.972545 + 0.232713i $$0.925240\pi$$
$$354$$ 17.4891 0.929537
$$355$$ 0 0
$$356$$ 15.4891 0.820922
$$357$$ 13.4891i 0.713920i
$$358$$ − 4.00000i − 0.211407i
$$359$$ 1.25544 0.0662594 0.0331297 0.999451i $$-0.489453\pi$$
0.0331297 + 0.999451i $$0.489453\pi$$
$$360$$ 0 0
$$361$$ 26.4891 1.39416
$$362$$ 19.4891i 1.02433i
$$363$$ 2.00000i 0.104973i
$$364$$ −6.74456 −0.353511
$$365$$ 0 0
$$366$$ 2.51087 0.131246
$$367$$ 8.74456i 0.456462i 0.973607 + 0.228231i $$0.0732942\pi$$
−0.973607 + 0.228231i $$0.926706\pi$$
$$368$$ − 6.74456i − 0.351585i
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ −12.7446 −0.661665
$$372$$ − 9.48913i − 0.491988i
$$373$$ − 16.9783i − 0.879100i −0.898218 0.439550i $$-0.855138\pi$$
0.898218 0.439550i $$-0.144862\pi$$
$$374$$ −6.74456 −0.348753
$$375$$ 0 0
$$376$$ 4.74456 0.244682
$$377$$ − 58.9783i − 3.03753i
$$378$$ − 4.00000i − 0.205738i
$$379$$ 37.4891 1.92569 0.962844 0.270060i $$-0.0870435\pi$$
0.962844 + 0.270060i $$0.0870435\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000i 0.818631i
$$383$$ 19.7228i 1.00779i 0.863765 + 0.503894i $$0.168100\pi$$
−0.863765 + 0.503894i $$0.831900\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 16.7446i − 0.850076i
$$389$$ 7.48913 0.379714 0.189857 0.981812i $$-0.439198\pi$$
0.189857 + 0.981812i $$0.439198\pi$$
$$390$$ 0 0
$$391$$ 45.4891 2.30048
$$392$$ − 1.00000i − 0.0505076i
$$393$$ − 5.48913i − 0.276890i
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ − 35.4891i − 1.78115i −0.454838 0.890574i $$-0.650303\pi$$
0.454838 0.890574i $$-0.349697\pi$$
$$398$$ 3.25544i 0.163180i
$$399$$ 13.4891 0.675301
$$400$$ 0 0
$$401$$ 23.4891 1.17299 0.586495 0.809953i $$-0.300507\pi$$
0.586495 + 0.809953i $$0.300507\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ 32.0000i 1.59403i
$$404$$ −2.74456 −0.136547
$$405$$ 0 0
$$406$$ 8.74456 0.433985
$$407$$ 0.744563i 0.0369066i
$$408$$ − 13.4891i − 0.667811i
$$409$$ 21.4891 1.06257 0.531284 0.847194i $$-0.321710\pi$$
0.531284 + 0.847194i $$0.321710\pi$$
$$410$$ 0 0
$$411$$ 6.97825 0.344212
$$412$$ 0.744563i 0.0366820i
$$413$$ − 8.74456i − 0.430292i
$$414$$ 6.74456 0.331477
$$415$$ 0 0
$$416$$ 6.74456 0.330679
$$417$$ − 29.4891i − 1.44409i
$$418$$ 6.74456i 0.329887i
$$419$$ 0.744563 0.0363743 0.0181871 0.999835i $$-0.494211\pi$$
0.0181871 + 0.999835i $$0.494211\pi$$
$$420$$ 0 0
$$421$$ 4.51087 0.219847 0.109923 0.993940i $$-0.464939\pi$$
0.109923 + 0.993940i $$0.464939\pi$$
$$422$$ 12.0000i 0.584151i
$$423$$ 4.74456i 0.230689i
$$424$$ 12.7446 0.618931
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ − 1.25544i − 0.0607549i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 13.4891 0.651261
$$430$$ 0 0
$$431$$ −17.2554 −0.831165 −0.415583 0.909555i $$-0.636422\pi$$
−0.415583 + 0.909555i $$0.636422\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 36.7446i 1.76583i 0.469532 + 0.882915i $$0.344423\pi$$
−0.469532 + 0.882915i $$0.655577\pi$$
$$434$$ −4.74456 −0.227746
$$435$$ 0 0
$$436$$ −18.2337 −0.873235
$$437$$ − 45.4891i − 2.17604i
$$438$$ 21.4891i 1.02679i
$$439$$ −14.9783 −0.714873 −0.357436 0.933937i $$-0.616349\pi$$
−0.357436 + 0.933937i $$0.616349\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 45.4891i 2.16370i
$$443$$ − 29.4891i − 1.40107i −0.713618 0.700535i $$-0.752945\pi$$
0.713618 0.700535i $$-0.247055\pi$$
$$444$$ −1.48913 −0.0706708
$$445$$ 0 0
$$446$$ 15.2554 0.722366
$$447$$ 1.48913i 0.0704332i
$$448$$ 1.00000i 0.0472456i
$$449$$ 4.97825 0.234938 0.117469 0.993077i $$-0.462522\pi$$
0.117469 + 0.993077i $$0.462522\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ − 10.0000i − 0.470360i
$$453$$ − 18.5109i − 0.869717i
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −13.4891 −0.631686
$$457$$ − 3.48913i − 0.163214i −0.996665 0.0816072i $$-0.973995\pi$$
0.996665 0.0816072i $$-0.0260053\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ −26.9783 −1.25924
$$460$$ 0 0
$$461$$ 33.7228 1.57063 0.785314 0.619098i $$-0.212502\pi$$
0.785314 + 0.619098i $$0.212502\pi$$
$$462$$ 2.00000i 0.0930484i
$$463$$ 1.25544i 0.0583451i 0.999574 + 0.0291726i $$0.00928723\pi$$
−0.999574 + 0.0291726i $$0.990713\pi$$
$$464$$ −8.74456 −0.405956
$$465$$ 0 0
$$466$$ −24.9783 −1.15710
$$467$$ − 16.9783i − 0.785660i −0.919611 0.392830i $$-0.871496\pi$$
0.919611 0.392830i $$-0.128504\pi$$
$$468$$ 6.74456i 0.311768i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 1.02175 0.0470797
$$472$$ 8.74456i 0.402501i
$$473$$ − 4.00000i − 0.183920i
$$474$$ −13.4891 −0.619576
$$475$$ 0 0
$$476$$ −6.74456 −0.309137
$$477$$ 12.7446i 0.583533i
$$478$$ − 14.7446i − 0.674401i
$$479$$ 41.4891 1.89569 0.947843 0.318737i $$-0.103259\pi$$
0.947843 + 0.318737i $$0.103259\pi$$
$$480$$ 0 0
$$481$$ 5.02175 0.228972
$$482$$ − 20.0000i − 0.910975i
$$483$$ − 13.4891i − 0.613776i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ − 9.25544i − 0.419404i −0.977765 0.209702i $$-0.932751\pi$$
0.977765 0.209702i $$-0.0672494\pi$$
$$488$$ 1.25544i 0.0568310i
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ 30.9783 1.39803 0.699014 0.715108i $$-0.253622\pi$$
0.699014 + 0.715108i $$0.253622\pi$$
$$492$$ 8.00000i 0.360668i
$$493$$ − 58.9783i − 2.65625i
$$494$$ 45.4891 2.04665
$$495$$ 0 0
$$496$$ 4.74456 0.213037
$$497$$ 4.00000i 0.179425i
$$498$$ 16.0000i 0.716977i
$$499$$ 13.4891 0.603856 0.301928 0.953331i $$-0.402370\pi$$
0.301928 + 0.953331i $$0.402370\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 26.2337i 1.17087i
$$503$$ 6.51087i 0.290306i 0.989409 + 0.145153i $$0.0463674\pi$$
−0.989409 + 0.145153i $$0.953633\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 6.74456 0.299832
$$507$$ − 64.9783i − 2.88579i
$$508$$ 0 0
$$509$$ 35.4891 1.57303 0.786514 0.617573i $$-0.211884\pi$$
0.786514 + 0.617573i $$0.211884\pi$$
$$510$$ 0 0
$$511$$ 10.7446 0.475311
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 26.9783i 1.19112i
$$514$$ 10.2337 0.451389
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 4.74456i 0.208666i
$$518$$ 0.744563i 0.0327142i
$$519$$ −40.4674 −1.77632
$$520$$ 0 0
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ − 8.74456i − 0.382739i
$$523$$ − 17.4891i − 0.764746i −0.924008 0.382373i $$-0.875107\pi$$
0.924008 0.382373i $$-0.124893\pi$$
$$524$$ 2.74456 0.119897
$$525$$ 0 0
$$526$$ 26.9783 1.17631
$$527$$ 32.0000i 1.39394i
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −22.4891 −0.977788
$$530$$ 0 0
$$531$$ −8.74456 −0.379482
$$532$$ 6.74456i 0.292414i
$$533$$ − 26.9783i − 1.16856i
$$534$$ −30.9783 −1.34056
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 8.00000i 0.345225i
$$538$$ 20.9783i 0.904437i
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 1.76631 0.0759397 0.0379698 0.999279i $$-0.487911\pi$$
0.0379698 + 0.999279i $$0.487911\pi$$
$$542$$ − 14.9783i − 0.643371i
$$543$$ − 38.9783i − 1.67272i
$$544$$ 6.74456 0.289171
$$545$$ 0 0
$$546$$ 13.4891 0.577281
$$547$$ − 14.9783i − 0.640424i −0.947346 0.320212i $$-0.896246\pi$$
0.947346 0.320212i $$-0.103754\pi$$
$$548$$ 3.48913i 0.149048i
$$549$$ −1.25544 −0.0535808
$$550$$ 0 0
$$551$$ −58.9783 −2.51256
$$552$$ 13.4891i 0.574135i
$$553$$ 6.74456i 0.286808i
$$554$$ −15.4891 −0.658070
$$555$$ 0 0
$$556$$ 14.7446 0.625309
$$557$$ 0.978251i 0.0414498i 0.999785 + 0.0207249i $$0.00659741\pi$$
−0.999785 + 0.0207249i $$0.993403\pi$$
$$558$$ 4.74456i 0.200853i
$$559$$ −26.9783 −1.14106
$$560$$ 0 0
$$561$$ 13.4891 0.569511
$$562$$ − 14.0000i − 0.590554i
$$563$$ 5.48913i 0.231339i 0.993288 + 0.115670i $$0.0369014\pi$$
−0.993288 + 0.115670i $$0.963099\pi$$
$$564$$ −9.48913 −0.399564
$$565$$ 0 0
$$566$$ 28.0000 1.17693
$$567$$ 11.0000i 0.461957i
$$568$$ − 4.00000i − 0.167836i
$$569$$ −16.5109 −0.692172 −0.346086 0.938203i $$-0.612489\pi$$
−0.346086 + 0.938203i $$0.612489\pi$$
$$570$$ 0 0
$$571$$ −17.4891 −0.731897 −0.365949 0.930635i $$-0.619255\pi$$
−0.365949 + 0.930635i $$0.619255\pi$$
$$572$$ 6.74456i 0.282004i
$$573$$ − 32.0000i − 1.33682i
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 8.74456i 0.364041i 0.983295 + 0.182020i $$0.0582637\pi$$
−0.983295 + 0.182020i $$0.941736\pi$$
$$578$$ 28.4891i 1.18499i
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ 8.00000 0.331896
$$582$$ 33.4891i 1.38817i
$$583$$ 12.7446i 0.527826i
$$584$$ −10.7446 −0.444613
$$585$$ 0 0
$$586$$ 13.2554 0.547577
$$587$$ 4.97825i 0.205474i 0.994709 + 0.102737i $$0.0327601\pi$$
−0.994709 + 0.102737i $$0.967240\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 20.0000 0.822690
$$592$$ − 0.744563i − 0.0306013i
$$593$$ − 40.2337i − 1.65220i −0.563524 0.826100i $$-0.690555\pi$$
0.563524 0.826100i $$-0.309445\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −0.744563 −0.0304985
$$597$$ − 6.51087i − 0.266472i
$$598$$ − 45.4891i − 1.86019i
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −33.4891 −1.36605 −0.683025 0.730395i $$-0.739336\pi$$
−0.683025 + 0.730395i $$0.739336\pi$$
$$602$$ − 4.00000i − 0.163028i
$$603$$ − 4.00000i − 0.162893i
$$604$$ 9.25544 0.376598
$$605$$ 0 0
$$606$$ 5.48913 0.222980
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ − 6.74456i − 0.273528i
$$609$$ −17.4891 −0.708695
$$610$$ 0 0
$$611$$ 32.0000 1.29458
$$612$$ 6.74456i 0.272633i
$$613$$ 11.4891i 0.464041i 0.972711 + 0.232021i $$0.0745337\pi$$
−0.972711 + 0.232021i $$0.925466\pi$$
$$614$$ 1.48913 0.0600962
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ 10.0000i 0.402585i 0.979531 + 0.201292i $$0.0645141\pi$$
−0.979531 + 0.201292i $$0.935486\pi$$
$$618$$ − 1.48913i − 0.0599014i
$$619$$ −0.744563 −0.0299265 −0.0149632 0.999888i $$-0.504763\pi$$
−0.0149632 + 0.999888i $$0.504763\pi$$
$$620$$ 0 0
$$621$$ 26.9783 1.08260
$$622$$ 20.7446i 0.831781i
$$623$$ 15.4891i 0.620559i
$$624$$ −13.4891 −0.539997
$$625$$ 0 0
$$626$$ −2.23369 −0.0892761
$$627$$ − 13.4891i − 0.538704i
$$628$$ 0.510875i 0.0203861i
$$629$$ 5.02175 0.200230
$$630$$ 0 0
$$631$$ −2.97825 −0.118562 −0.0592811 0.998241i $$-0.518881\pi$$
−0.0592811 + 0.998241i $$0.518881\pi$$
$$632$$ − 6.74456i − 0.268284i
$$633$$ − 24.0000i − 0.953914i
$$634$$ −2.23369 −0.0887111
$$635$$ 0 0
$$636$$ −25.4891 −1.01071
$$637$$ − 6.74456i − 0.267229i
$$638$$ − 8.74456i − 0.346201i
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ 11.4891 0.453793 0.226897 0.973919i $$-0.427142\pi$$
0.226897 + 0.973919i $$0.427142\pi$$
$$642$$ 24.0000i 0.947204i
$$643$$ 46.4674i 1.83249i 0.400613 + 0.916247i $$0.368797\pi$$
−0.400613 + 0.916247i $$0.631203\pi$$
$$644$$ 6.74456 0.265773
$$645$$ 0 0
$$646$$ 45.4891 1.78975
$$647$$ − 8.74456i − 0.343784i −0.985116 0.171892i $$-0.945012\pi$$
0.985116 0.171892i $$-0.0549880\pi$$
$$648$$ − 11.0000i − 0.432121i
$$649$$ −8.74456 −0.343254
$$650$$ 0 0
$$651$$ 9.48913 0.371908
$$652$$ 4.00000i 0.156652i
$$653$$ 15.7228i 0.615281i 0.951503 + 0.307641i $$0.0995394\pi$$
−0.951503 + 0.307641i $$0.900461\pi$$
$$654$$ 36.4674 1.42599
$$655$$ 0 0
$$656$$ −4.00000 −0.156174
$$657$$ − 10.7446i − 0.419185i
$$658$$ 4.74456i 0.184962i
$$659$$ 41.4891 1.61619 0.808093 0.589054i $$-0.200500\pi$$
0.808093 + 0.589054i $$0.200500\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ − 14.9783i − 0.582146i
$$663$$ − 90.9783i − 3.53330i
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 0.744563 0.0288512
$$667$$ 58.9783i 2.28365i
$$668$$ 0 0
$$669$$ −30.5109 −1.17962
$$670$$ 0 0
$$671$$ −1.25544 −0.0484656
$$672$$ − 2.00000i − 0.0771517i
$$673$$ 20.9783i 0.808652i 0.914615 + 0.404326i $$0.132494\pi$$
−0.914615 + 0.404326i $$0.867506\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ 32.4891 1.24958
$$677$$ − 36.2337i − 1.39257i −0.717764 0.696287i $$-0.754834\pi$$
0.717764 0.696287i $$-0.245166\pi$$
$$678$$ 20.0000i 0.768095i
$$679$$ 16.7446 0.642597
$$680$$ 0 0
$$681$$ 40.0000 1.53280
$$682$$ 4.74456i 0.181679i
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 6.74456 0.257885
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 12.0000i 0.457829i
$$688$$ 4.00000i 0.152499i
$$689$$ 85.9565 3.27468
$$690$$ 0 0
$$691$$ 1.76631 0.0671937 0.0335968 0.999435i $$-0.489304\pi$$
0.0335968 + 0.999435i $$0.489304\pi$$
$$692$$ − 20.2337i − 0.769169i
$$693$$ − 1.00000i − 0.0379869i
$$694$$ 22.9783 0.872242
$$695$$ 0 0
$$696$$ 17.4891 0.662924
$$697$$ − 26.9783i − 1.02187i
$$698$$ 30.7446i 1.16370i
$$699$$ 49.9565 1.88953
$$700$$ 0 0
$$701$$ 22.2337 0.839755 0.419877 0.907581i $$-0.362073\pi$$
0.419877 + 0.907581i $$0.362073\pi$$
$$702$$ 26.9783i 1.01823i
$$703$$ − 5.02175i − 0.189399i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 8.74456 0.329106
$$707$$ − 2.74456i − 0.103220i
$$708$$ − 17.4891i − 0.657282i
$$709$$ −0.510875 −0.0191863 −0.00959315 0.999954i $$-0.503054\pi$$
−0.00959315 + 0.999954i $$0.503054\pi$$
$$710$$ 0 0
$$711$$ 6.74456 0.252941
$$712$$ − 15.4891i − 0.580480i
$$713$$ − 32.0000i − 1.19841i
$$714$$ 13.4891 0.504818
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 29.4891i 1.10129i
$$718$$ − 1.25544i − 0.0468525i
$$719$$ −7.72281 −0.288012 −0.144006 0.989577i $$-0.545999\pi$$
−0.144006 + 0.989577i $$0.545999\pi$$
$$720$$ 0 0
$$721$$ −0.744563 −0.0277290
$$722$$ − 26.4891i − 0.985823i
$$723$$ 40.0000i 1.48762i
$$724$$ 19.4891 0.724308
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ − 14.2337i − 0.527898i −0.964537 0.263949i $$-0.914975\pi$$
0.964537 0.263949i $$-0.0850251\pi$$
$$728$$ 6.74456i 0.249970i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −26.9783 −0.997827
$$732$$ − 2.51087i − 0.0928046i
$$733$$ − 16.2337i − 0.599605i −0.954001 0.299802i $$-0.903079\pi$$
0.954001 0.299802i $$-0.0969208\pi$$
$$734$$ 8.74456 0.322768
$$735$$ 0 0
$$736$$ −6.74456 −0.248608
$$737$$ − 4.00000i − 0.147342i
$$738$$ − 4.00000i − 0.147242i
$$739$$ 30.9783 1.13955 0.569777 0.821800i $$-0.307030\pi$$
0.569777 + 0.821800i $$0.307030\pi$$
$$740$$ 0 0
$$741$$ −90.9783 −3.34217
$$742$$ 12.7446i 0.467868i
$$743$$ 26.9783i 0.989736i 0.868968 + 0.494868i $$0.164784\pi$$
−0.868968 + 0.494868i $$0.835216\pi$$
$$744$$ −9.48913 −0.347888
$$745$$ 0 0
$$746$$ −16.9783 −0.621618
$$747$$ − 8.00000i − 0.292705i
$$748$$ 6.74456i 0.246606i
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ − 4.74456i − 0.173016i
$$753$$ − 52.4674i − 1.91202i
$$754$$ −58.9783 −2.14786
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 19.2554i 0.699851i 0.936778 + 0.349925i $$0.113793\pi$$
−0.936778 + 0.349925i $$0.886207\pi$$
$$758$$ − 37.4891i − 1.36167i
$$759$$ −13.4891 −0.489624
$$760$$ 0 0
$$761$$ −29.4891 −1.06898 −0.534490 0.845175i $$-0.679496\pi$$
−0.534490 + 0.845175i $$0.679496\pi$$
$$762$$ 0 0
$$763$$ − 18.2337i − 0.660104i
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 19.7228 0.712614
$$767$$ 58.9783i 2.12958i
$$768$$ 2.00000i 0.0721688i
$$769$$ 8.00000 0.288487 0.144244 0.989542i $$-0.453925\pi$$
0.144244 + 0.989542i $$0.453925\pi$$
$$770$$ 0 0
$$771$$ −20.4674 −0.737115
$$772$$ 2.00000i 0.0719816i
$$773$$ − 51.4891i − 1.85194i −0.377603 0.925968i $$-0.623252\pi$$
0.377603 0.925968i $$-0.376748\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −16.7446 −0.601095
$$777$$ − 1.48913i − 0.0534221i
$$778$$ − 7.48913i − 0.268498i
$$779$$ −26.9783 −0.966596
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ − 45.4891i − 1.62669i
$$783$$ − 34.9783i − 1.25002i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ −5.48913 −0.195791
$$787$$ − 5.48913i − 0.195666i −0.995203 0.0978331i $$-0.968809\pi$$
0.995203 0.0978331i $$-0.0311911\pi$$
$$788$$ 10.0000i 0.356235i
$$789$$ −53.9565 −1.92090
$$790$$ 0 0
$$791$$ 10.0000 0.355559
$$792$$ 1.00000i 0.0355335i
$$793$$ 8.46738i 0.300685i
$$794$$ −35.4891 −1.25946
$$795$$ 0 0
$$796$$ 3.25544 0.115386
$$797$$ 42.4674i 1.50427i 0.659008 + 0.752136i $$0.270976\pi$$
−0.659008 + 0.752136i $$0.729024\pi$$
$$798$$ − 13.4891i − 0.477510i
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ 15.4891 0.547281
$$802$$ − 23.4891i − 0.829430i
$$803$$ − 10.7446i − 0.379167i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 32.0000 1.12715
$$807$$ − 41.9565i − 1.47694i
$$808$$ 2.74456i 0.0965534i
$$809$$ −34.4674 −1.21181 −0.605904 0.795538i $$-0.707189\pi$$
−0.605904 + 0.795538i $$0.707189\pi$$
$$810$$ 0 0
$$811$$ −18.7446 −0.658211 −0.329105 0.944293i $$-0.606747\pi$$
−0.329105 + 0.944293i $$0.606747\pi$$
$$812$$ − 8.74456i − 0.306874i
$$813$$ 29.9565i 1.05062i
$$814$$ 0.744563 0.0260969
$$815$$ 0 0
$$816$$ −13.4891 −0.472214
$$817$$ 26.9783i 0.943850i
$$818$$ − 21.4891i − 0.751350i
$$819$$ −6.74456 −0.235674
$$820$$ 0 0
$$821$$ 7.72281 0.269528 0.134764 0.990878i $$-0.456972\pi$$
0.134764 + 0.990878i $$0.456972\pi$$
$$822$$ − 6.97825i − 0.243394i
$$823$$ 7.76631i 0.270717i 0.990797 + 0.135358i $$0.0432186\pi$$
−0.990797 + 0.135358i $$0.956781\pi$$
$$824$$ 0.744563 0.0259381
$$825$$ 0 0
$$826$$ −8.74456 −0.304262
$$827$$ − 4.00000i − 0.139094i −0.997579 0.0695468i $$-0.977845\pi$$
0.997579 0.0695468i $$-0.0221553\pi$$
$$828$$ − 6.74456i − 0.234390i
$$829$$ −14.4674 −0.502473 −0.251236 0.967926i $$-0.580837\pi$$
−0.251236 + 0.967926i $$0.580837\pi$$
$$830$$ 0 0
$$831$$ 30.9783 1.07462
$$832$$ − 6.74456i − 0.233826i
$$833$$ − 6.74456i − 0.233685i
$$834$$ −29.4891 −1.02112
$$835$$ 0 0
$$836$$ 6.74456 0.233266
$$837$$ 18.9783i 0.655984i
$$838$$ − 0.744563i − 0.0257205i
$$839$$ 3.25544 0.112390 0.0561951 0.998420i $$-0.482103\pi$$
0.0561951 + 0.998420i $$0.482103\pi$$
$$840$$ 0 0
$$841$$ 47.4674 1.63681
$$842$$ − 4.51087i − 0.155455i
$$843$$ 28.0000i 0.964371i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 4.74456 0.163121
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 12.7446i − 0.437650i
$$849$$ −56.0000 −1.92192
$$850$$ 0 0
$$851$$ −5.02175 −0.172143
$$852$$ 8.00000i 0.274075i
$$853$$ − 22.7446i − 0.778759i −0.921077 0.389379i $$-0.872689\pi$$
0.921077 0.389379i $$-0.127311\pi$$
$$854$$ −1.25544 −0.0429602
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 43.2119i 1.47609i 0.674751 + 0.738046i $$0.264251\pi$$
−0.674751 + 0.738046i $$0.735749\pi$$
$$858$$ − 13.4891i − 0.460511i
$$859$$ 29.2119 0.996698 0.498349 0.866976i $$-0.333940\pi$$
0.498349 + 0.866976i $$0.333940\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 17.2554i 0.587723i
$$863$$ − 40.2337i − 1.36957i −0.728745 0.684785i $$-0.759896\pi$$
0.728745 0.684785i $$-0.240104\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ 36.7446 1.24863
$$867$$ − 56.9783i − 1.93508i
$$868$$ 4.74456i 0.161041i
$$869$$ 6.74456 0.228794
$$870$$ 0 0
$$871$$ −26.9783 −0.914123
$$872$$ 18.2337i 0.617471i
$$873$$ − 16.7446i − 0.566718i
$$874$$ −45.4891 −1.53869
$$875$$ 0 0
$$876$$ 21.4891 0.726050
$$877$$ 26.4674i 0.893740i 0.894599 + 0.446870i $$0.147461\pi$$
−0.894599 + 0.446870i $$0.852539\pi$$
$$878$$ 14.9783i 0.505491i
$$879$$ −26.5109 −0.894190
$$880$$ 0 0
$$881$$ −55.4891 −1.86948 −0.934738 0.355338i $$-0.884366\pi$$
−0.934738 + 0.355338i $$0.884366\pi$$
$$882$$ − 1.00000i − 0.0336718i
$$883$$ − 8.00000i − 0.269221i −0.990899 0.134611i $$-0.957022\pi$$
0.990899 0.134611i $$-0.0429784\pi$$
$$884$$ 45.4891 1.52996
$$885$$ 0 0
$$886$$ −29.4891 −0.990707
$$887$$ 41.4891i 1.39307i 0.717524 + 0.696534i $$0.245276\pi$$
−0.717524 + 0.696534i $$0.754724\pi$$
$$888$$ 1.48913i 0.0499718i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 11.0000 0.368514
$$892$$ − 15.2554i − 0.510790i
$$893$$ − 32.0000i − 1.07084i
$$894$$ 1.48913 0.0498038
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 90.9783i 3.03768i
$$898$$ − 4.97825i − 0.166126i
$$899$$ −41.4891 −1.38374
$$900$$ 0 0
$$901$$ 85.9565 2.86363
$$902$$ − 4.00000i − 0.133185i
$$903$$ 8.00000i 0.266223i
$$904$$ −10.0000 −0.332595
$$905$$ 0 0
$$906$$ −18.5109 −0.614983
$$907$$ 14.5109i 0.481826i 0.970547 + 0.240913i $$0.0774468\pi$$
−0.970547 + 0.240913i $$0.922553\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ −2.74456 −0.0910314
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 13.4891i 0.446670i
$$913$$ − 8.00000i − 0.264761i
$$914$$ −3.48913 −0.115410
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 2.74456i 0.0906334i
$$918$$ 26.9783i 0.890415i
$$919$$ 55.2119 1.82127 0.910637 0.413207i $$-0.135592\pi$$
0.910637 + 0.413207i $$0.135592\pi$$
$$920$$ 0 0
$$921$$ −2.97825 −0.0981367
$$922$$ − 33.7228i − 1.11060i
$$923$$ − 26.9783i − 0.888000i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ 1.25544 0.0412562
$$927$$ 0.744563i 0.0244546i
$$928$$ 8.74456i 0.287054i
$$929$$ 28.9783 0.950746 0.475373 0.879784i $$-0.342313\pi$$
0.475373 + 0.879784i $$0.342313\pi$$
$$930$$ 0 0
$$931$$ −6.74456 −0.221044
$$932$$ 24.9783i 0.818190i
$$933$$ − 41.4891i − 1.35829i
$$934$$ −16.9783 −0.555545
$$935$$ 0 0
$$936$$ 6.74456 0.220453
$$937$$ 18.7446i 0.612358i 0.951974 + 0.306179i $$0.0990506\pi$$
−0.951974 + 0.306179i $$0.900949\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ 4.46738 0.145787
$$940$$ 0 0
$$941$$ 12.2337 0.398807 0.199403 0.979917i $$-0.436100\pi$$
0.199403 + 0.979917i $$0.436100\pi$$
$$942$$ − 1.02175i − 0.0332904i
$$943$$ 26.9783i 0.878533i
$$944$$ 8.74456 0.284611
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 8.00000i 0.259965i 0.991516 + 0.129983i $$0.0414921\pi$$
−0.991516 + 0.129983i $$0.958508\pi$$
$$948$$ 13.4891i 0.438106i
$$949$$ −72.4674 −2.35239
$$950$$ 0 0
$$951$$ 4.46738 0.144865
$$952$$ 6.74456i 0.218593i
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ 12.7446 0.412620
$$955$$ 0 0
$$956$$ −14.7446 −0.476873
$$957$$ 17.4891i 0.565343i
$$958$$ − 41.4891i − 1.34045i
$$959$$ −3.48913 −0.112670
$$960$$ 0 0
$$961$$ −8.48913 −0.273843
$$962$$ − 5.02175i − 0.161908i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ −13.4891 −0.434005
$$967$$ 50.9783i 1.63935i 0.572829 + 0.819675i $$0.305846\pi$$
−0.572829 + 0.819675i $$0.694154\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ −90.9783 −2.92264
$$970$$ 0 0
$$971$$ 19.7228 0.632935 0.316468 0.948603i $$-0.397503\pi$$
0.316468 + 0.948603i $$0.397503\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ 14.7446i 0.472689i
$$974$$ −9.25544 −0.296563
$$975$$ 0 0
$$976$$ 1.25544 0.0401856
$$977$$ − 12.9783i − 0.415211i −0.978213 0.207606i $$-0.933433\pi$$
0.978213 0.207606i $$-0.0665670\pi$$
$$978$$ − 8.00000i − 0.255812i
$$979$$ 15.4891 0.495035
$$980$$ 0 0
$$981$$ −18.2337 −0.582157
$$982$$ − 30.9783i − 0.988556i
$$983$$ 2.23369i 0.0712436i 0.999365 + 0.0356218i $$0.0113412\pi$$
−0.999365 + 0.0356218i $$0.988659\pi$$
$$984$$ 8.00000 0.255031
$$985$$ 0 0
$$986$$ −58.9783 −1.87825
$$987$$ − 9.48913i − 0.302042i
$$988$$ − 45.4891i − 1.44720i
$$989$$ 26.9783 0.857858
$$990$$ 0 0
$$991$$ 1.48913 0.0473036 0.0236518 0.999720i $$-0.492471\pi$$
0.0236518 + 0.999720i $$0.492471\pi$$
$$992$$ − 4.74456i − 0.150640i
$$993$$ 29.9565i 0.950641i
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ − 39.2119i − 1.24185i −0.783868 0.620927i $$-0.786756\pi$$
0.783868 0.620927i $$-0.213244\pi$$
$$998$$ − 13.4891i − 0.426991i
$$999$$ 2.97825 0.0942277
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3850.2.c.y.1849.2 4
5.2 odd 4 770.2.a.k.1.2 2
5.3 odd 4 3850.2.a.bc.1.1 2
5.4 even 2 inner 3850.2.c.y.1849.3 4
15.2 even 4 6930.2.a.bo.1.2 2
20.7 even 4 6160.2.a.r.1.2 2
35.27 even 4 5390.2.a.bq.1.1 2
55.32 even 4 8470.2.a.bu.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.2 2 5.2 odd 4
3850.2.a.bc.1.1 2 5.3 odd 4
3850.2.c.y.1849.2 4 1.1 even 1 trivial
3850.2.c.y.1849.3 4 5.4 even 2 inner
5390.2.a.bq.1.1 2 35.27 even 4
6160.2.a.r.1.2 2 20.7 even 4
6930.2.a.bo.1.2 2 15.2 even 4
8470.2.a.bu.1.1 2 55.32 even 4