Properties

 Label 2850.2.d.s.799.1 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 799.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.s.799.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +6.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +1.00000i q^{38} +10.0000 q^{41} +4.00000i q^{42} +12.0000i q^{43} -4.00000 q^{44} +2.00000 q^{46} +10.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -1.00000i q^{57} -6.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} -6.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +2.00000i q^{68} -2.00000 q^{69} -16.0000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -8.00000 q^{74} +1.00000 q^{76} +16.0000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +16.0000i q^{83} +4.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -2.00000i q^{92} +6.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{11} + 8q^{14} + 2q^{16} - 2q^{19} - 8q^{21} - 2q^{24} + 12q^{29} + 12q^{31} - 4q^{34} + 2q^{36} + 20q^{41} - 8q^{44} + 4q^{46} - 18q^{49} + 4q^{51} - 2q^{54} - 8q^{56} - 8q^{59} - 20q^{61} - 2q^{64} + 8q^{66} - 4q^{69} - 32q^{71} - 16q^{74} + 2q^{76} - 20q^{79} + 2q^{81} + 8q^{84} + 24q^{86} + 4q^{89} + 20q^{94} + 2q^{96} - 8q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ − 4.00000i − 0.852803i
$$23$$ 2.00000i 0.417029i 0.978019 + 0.208514i $$0.0668628\pi$$
−0.978019 + 0.208514i $$0.933137\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 4.00000i 0.617213i
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ 10.0000i 1.45865i 0.684167 + 0.729325i $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 6.00000i − 0.787839i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ − 6.00000i − 0.762001i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 16.0000i 1.82337i
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ 16.0000i 1.75623i 0.478451 + 0.878114i $$0.341198\pi$$
−0.478451 + 0.878114i $$0.658802\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 6.00000i 0.643268i
$$88$$ 4.00000i 0.426401i
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 2.00000i − 0.208514i
$$93$$ 6.00000i 0.622171i
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 6.00000i 0.591198i 0.955312 + 0.295599i $$0.0955191\pi$$
−0.955312 + 0.295599i $$0.904481\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 4.00000i 0.377964i
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 4.00000i 0.368230i
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ 10.0000i 0.901670i
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 22.0000i 1.95218i 0.217357 + 0.976092i $$0.430256\pi$$
−0.217357 + 0.976092i $$0.569744\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ − 4.00000i − 0.346844i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 2.00000i 0.170251i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −10.0000 −0.842152
$$142$$ 16.0000i 1.34269i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ − 9.00000i − 0.742307i
$$148$$ 8.00000i 0.657596i
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 2.00000i 0.161690i
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ − 4.00000i − 0.308607i
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 12.0000i − 0.914991i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 4.00000i − 0.300658i
$$178$$ − 2.00000i − 0.149906i
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ − 10.0000i − 0.739221i
$$184$$ −2.00000 −0.147442
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ − 8.00000i − 0.585018i
$$188$$ − 10.0000i − 0.729325i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −2.00000 −0.144715 −0.0723575 0.997379i $$-0.523052\pi$$
−0.0723575 + 0.997379i $$0.523052\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 20.0000i − 1.42494i −0.701702 0.712470i $$-0.747576\pi$$
0.701702 0.712470i $$-0.252424\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 8.00000i − 0.562878i
$$203$$ 24.0000i 1.68447i
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ − 2.00000i − 0.139010i
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ − 16.0000i − 1.09630i
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 24.0000i 1.62923i
$$218$$ − 4.00000i − 0.270914i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 8.00000i − 0.536925i
$$223$$ − 6.00000i − 0.401790i −0.979613 0.200895i $$-0.935615\pi$$
0.979613 0.200895i $$-0.0643850\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ 6.00000i 0.393919i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ − 10.0000i − 0.649570i
$$238$$ − 8.00000i − 0.518563i
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 0 0
$$248$$ 6.00000i 0.381000i
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 8.00000i 0.502956i
$$254$$ 22.0000 1.38040
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 12.0000i 0.747087i
$$259$$ 32.0000 1.98838
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 6.00000i 0.369976i 0.982741 + 0.184988i $$0.0592246\pi$$
−0.982741 + 0.184988i $$0.940775\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ 2.00000i 0.122398i
$$268$$ 0 0
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 2.00000 0.120386
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 10.0000i 0.595491i
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 40.0000i 2.36113i
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ − 4.00000i − 0.232104i
$$298$$ 20.0000i 1.15857i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −48.0000 −2.76667
$$302$$ − 10.0000i − 0.575435i
$$303$$ 8.00000i 0.459588i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ − 16.0000i − 0.911685i
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ 34.0000 1.92796 0.963982 0.265969i $$-0.0856919\pi$$
0.963982 + 0.265969i $$0.0856919\pi$$
$$312$$ 0 0
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 8.00000i 0.445823i
$$323$$ 2.00000i 0.111283i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 4.00000i 0.221201i
$$328$$ 10.0000i 0.552158i
$$329$$ −40.0000 −2.20527
$$330$$ 0 0
$$331$$ 24.0000 1.31916 0.659580 0.751635i $$-0.270734\pi$$
0.659580 + 0.751635i $$0.270734\pi$$
$$332$$ − 16.0000i − 0.878114i
$$333$$ 8.00000i 0.438397i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 26.0000i 1.41631i 0.706057 + 0.708155i $$0.250472\pi$$
−0.706057 + 0.708155i $$0.749528\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4.00000i − 0.213201i
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 8.00000i 0.423405i
$$358$$ − 20.0000i − 1.05703i
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 12.0000i − 0.630706i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ 2.00000i 0.104257i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 8.00000 0.415339
$$372$$ − 6.00000i − 0.311086i
$$373$$ − 8.00000i − 0.414224i −0.978317 0.207112i $$-0.933593\pi$$
0.978317 0.207112i $$-0.0664065\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ −10.0000 −0.515711
$$377$$ 0 0
$$378$$ − 4.00000i − 0.205738i
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −22.0000 −1.12709
$$382$$ 2.00000i 0.102329i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ − 12.0000i − 0.609994i
$$388$$ 10.0000i 0.507673i
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ − 9.00000i − 0.454569i
$$393$$ 0 0
$$394$$ −20.0000 −1.00759
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ − 30.0000i − 1.50566i −0.658217 0.752828i $$-0.728689\pi$$
0.658217 0.752828i $$-0.271311\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −8.00000 −0.398015
$$405$$ 0 0
$$406$$ 24.0000 1.19110
$$407$$ − 32.0000i − 1.58618i
$$408$$ 2.00000i 0.0990148i
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ − 6.00000i − 0.295599i
$$413$$ − 16.0000i − 0.787309i
$$414$$ −2.00000 −0.0982946
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000i 0.195881i
$$418$$ 4.00000i 0.195646i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ − 10.0000i − 0.486217i
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ − 40.0000i − 1.93574i
$$428$$ 4.00000i 0.193347i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 18.0000i 0.865025i 0.901628 + 0.432512i $$0.142373\pi$$
−0.901628 + 0.432512i $$0.857627\pi$$
$$434$$ 24.0000 1.15204
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ − 2.00000i − 0.0956730i
$$438$$ 2.00000i 0.0955637i
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ − 20.0000i − 0.945968i
$$448$$ − 4.00000i − 0.188982i
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ − 14.0000i − 0.658505i
$$453$$ 10.0000i 0.469841i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ − 2.00000i − 0.0934539i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 16.0000i 0.744387i
$$463$$ − 36.0000i − 1.67306i −0.547920 0.836531i $$-0.684580\pi$$
0.547920 0.836531i $$-0.315420\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 8.00000i 0.370196i 0.982720 + 0.185098i $$0.0592602\pi$$
−0.982720 + 0.185098i $$0.940740\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ − 4.00000i − 0.184115i
$$473$$ 48.0000i 2.20704i
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ 2.00000i 0.0915737i
$$478$$ − 6.00000i − 0.274434i
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 2.00000i 0.0910975i
$$483$$ − 8.00000i − 0.364013i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ − 64.0000i − 2.87079i
$$498$$ 16.0000i 0.716977i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 24.0000i 1.07117i
$$503$$ − 34.0000i − 1.51599i −0.652263 0.757993i $$-0.726180\pi$$
0.652263 0.757993i $$-0.273820\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ 13.0000i 0.577350i
$$508$$ − 22.0000i − 0.976092i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 40.0000i 1.75920i
$$518$$ − 32.0000i − 1.40600i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ − 12.0000i − 0.522728i
$$528$$ 4.00000i 0.174078i
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 4.00000i 0.173422i
$$533$$ 0 0
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 20.0000i 0.863064i
$$538$$ 14.0000i 0.603583i
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 4.00000i 0.171815i
$$543$$ 12.0000i 0.514969i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 16.0000i − 0.684111i −0.939680 0.342055i $$-0.888877\pi$$
0.939680 0.342055i $$-0.111123\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ − 2.00000i − 0.0851257i
$$553$$ − 40.0000i − 1.70097i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 4.00000i 0.169485i 0.996403 + 0.0847427i $$0.0270068\pi$$
−0.996403 + 0.0847427i $$0.972993\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ − 10.0000i − 0.421825i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 10.0000 0.421076
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 4.00000i 0.167984i
$$568$$ − 16.0000i − 0.671345i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ − 2.00000i − 0.0835512i
$$574$$ 40.0000 1.66957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 14.0000i − 0.582828i −0.956597 0.291414i $$-0.905874\pi$$
0.956597 0.291414i $$-0.0941257\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −64.0000 −2.65517
$$582$$ − 10.0000i − 0.414513i
$$583$$ − 8.00000i − 0.331326i
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ −6.00000 −0.247226
$$590$$ 0 0
$$591$$ 20.0000 0.822690
$$592$$ − 8.00000i − 0.328798i
$$593$$ − 10.0000i − 0.410651i −0.978694 0.205325i $$-0.934175\pi$$
0.978694 0.205325i $$-0.0658253\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ 4.00000i 0.163709i
$$598$$ 0 0
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 48.0000i 1.95633i
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ 8.00000 0.324978
$$607$$ − 22.0000i − 0.892952i −0.894795 0.446476i $$-0.852679\pi$$
0.894795 0.446476i $$-0.147321\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 6.00000i 0.241355i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 2.00000 0.0802572
$$622$$ − 34.0000i − 1.36328i
$$623$$ 8.00000i 0.320513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ − 4.00000i − 0.159745i
$$628$$ − 18.0000i − 0.718278i
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ − 20.0000i − 0.794929i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ − 24.0000i − 0.950169i
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 46.0000 1.81689 0.908445 0.418004i $$-0.137270\pi$$
0.908445 + 0.418004i $$0.137270\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ − 28.0000i − 1.10421i −0.833774 0.552106i $$-0.813824\pi$$
0.833774 0.552106i $$-0.186176\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ − 18.0000i − 0.707653i −0.935311 0.353827i $$-0.884880\pi$$
0.935311 0.353827i $$-0.115120\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ −24.0000 −0.940634
$$652$$ 20.0000i 0.783260i
$$653$$ 36.0000i 1.40879i 0.709809 + 0.704394i $$0.248781\pi$$
−0.709809 + 0.704394i $$0.751219\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 40.0000i 1.55936i
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ −8.00000 −0.311164 −0.155582 0.987823i $$-0.549725\pi$$
−0.155582 + 0.987823i $$0.549725\pi$$
$$662$$ − 24.0000i − 0.932786i
$$663$$ 0 0
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 12.0000i 0.464642i
$$668$$ − 12.0000i − 0.464294i
$$669$$ 6.00000 0.231973
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 4.00000i 0.154303i
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ 40.0000 1.53506
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ − 24.0000i − 0.919007i
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 2.00000i 0.0763048i
$$688$$ 12.0000i 0.457496i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 16.0000i − 0.607790i
$$694$$ 28.0000 1.06287
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ − 20.0000i − 0.757554i
$$698$$ 10.0000i 0.378506i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −8.00000 −0.302156 −0.151078 0.988522i $$-0.548274\pi$$
−0.151078 + 0.988522i $$0.548274\pi$$
$$702$$ 0 0
$$703$$ 8.00000i 0.301726i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 32.0000i 1.20348i
$$708$$ 4.00000i 0.150329i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 2.00000i 0.0749532i
$$713$$ 12.0000i 0.449404i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 6.00000i 0.224074i
$$718$$ 6.00000i 0.223918i
$$719$$ −34.0000 −1.26799 −0.633993 0.773339i $$-0.718585\pi$$
−0.633993 + 0.773339i $$0.718585\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −12.0000 −0.445976
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 10.0000i 0.369611i
$$733$$ 38.0000i 1.40356i 0.712393 + 0.701781i $$0.247612\pi$$
−0.712393 + 0.701781i $$0.752388\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ 0 0
$$738$$ 10.0000i 0.368105i
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 8.00000i − 0.293689i
$$743$$ 36.0000i 1.32071i 0.750953 + 0.660356i $$0.229595\pi$$
−0.750953 + 0.660356i $$0.770405\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0 0
$$746$$ −8.00000 −0.292901
$$747$$ − 16.0000i − 0.585409i
$$748$$ 8.00000i 0.292509i
$$749$$ 16.0000 0.584627
$$750$$ 0 0
$$751$$ 10.0000 0.364905 0.182453 0.983215i $$-0.441596\pi$$
0.182453 + 0.983215i $$0.441596\pi$$
$$752$$ 10.0000i 0.364662i
$$753$$ − 24.0000i − 0.874609i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 8.00000i 0.290573i
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 22.0000i 0.796976i
$$763$$ 16.0000i 0.579239i
$$764$$ 2.00000 0.0723575
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 14.0000i 0.503871i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 32.0000i 1.14799i
$$778$$ 8.00000i 0.286814i
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ − 4.00000i − 0.143040i
$$783$$ − 6.00000i − 0.214423i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 20.0000i 0.712470i
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ − 4.00000i − 0.142134i
$$793$$ 0 0
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ − 34.0000i − 1.20434i −0.798367 0.602171i $$-0.794303\pi$$
0.798367 0.602171i $$-0.205697\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ 20.0000 0.707549
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 30.0000i 1.05934i
$$803$$ 8.00000i 0.282314i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 14.0000i − 0.492823i
$$808$$ 8.00000i 0.281439i
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ − 24.0000i − 0.842235i
$$813$$ − 4.00000i − 0.140286i
$$814$$ −32.0000 −1.12160
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ − 12.0000i − 0.419827i
$$818$$ − 18.0000i − 0.629355i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ −6.00000 −0.209020
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 2.00000i 0.0695048i
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ 0 0
$$833$$ 18.0000i 0.623663i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ − 6.00000i − 0.207390i
$$838$$ − 12.0000i − 0.414533i
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 8.00000i 0.275698i
$$843$$ 10.0000i 0.344418i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ 20.0000i 0.687208i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ 16.0000i 0.548151i
$$853$$ − 54.0000i − 1.84892i −0.381273 0.924462i $$-0.624514\pi$$
0.381273 0.924462i $$-0.375486\pi$$
$$854$$ −40.0000 −1.36877
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ −40.0000 −1.36320
$$862$$ 4.00000i 0.136241i
$$863$$ 28.0000i 0.953131i 0.879139 + 0.476566i $$0.158119\pi$$
−0.879139 + 0.476566i $$0.841881\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 18.0000 0.611665
$$867$$ 13.0000i 0.441503i
$$868$$ − 24.0000i − 0.814613i
$$869$$ −40.0000 −1.35691
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 4.00000i 0.135457i
$$873$$ 10.0000i 0.338449i
$$874$$ −2.00000 −0.0676510
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 8.00000i 0.270141i 0.990836 + 0.135070i $$0.0431261\pi$$
−0.990836 + 0.135070i $$0.956874\pi$$
$$878$$ 22.0000i 0.742464i
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ − 36.0000i − 1.21150i −0.795656 0.605748i $$-0.792874\pi$$
0.795656 0.605748i $$-0.207126\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ − 12.0000i − 0.402921i −0.979497 0.201460i $$-0.935431\pi$$
0.979497 0.201460i $$-0.0645687\pi$$
$$888$$ 8.00000i 0.268462i
$$889$$ −88.0000 −2.95143
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 6.00000i 0.200895i
$$893$$ − 10.0000i − 0.334637i
$$894$$ −20.0000 −0.668900
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 0 0
$$898$$ − 18.0000i − 0.600668i
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ − 40.0000i − 1.33185i
$$903$$ − 48.0000i − 1.59734i
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ − 32.0000i − 1.06254i −0.847202 0.531271i $$-0.821714\pi$$
0.847202 0.531271i $$-0.178286\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ −8.00000 −0.265343
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ 64.0000i 2.11809i
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 12.0000i 0.395199i
$$923$$ 0 0
$$924$$ 16.0000 0.526361
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ − 6.00000i − 0.197066i
$$928$$ − 6.00000i − 0.196960i
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ − 6.00000i − 0.196537i
$$933$$ 34.0000i 1.11311i
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 18.0000i 0.586472i
$$943$$ 20.0000i 0.651290i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 10.0000i 0.324785i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 8.00000i 0.259281i
$$953$$ 30.0000i 0.971795i 0.874016 + 0.485898i $$0.161507\pi$$
−0.874016 + 0.485898i $$0.838493\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 24.0000i 0.775810i
$$958$$ 6.00000i 0.193851i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 4.00000i 0.128898i
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 12.0000i 0.385894i 0.981209 + 0.192947i $$0.0618045\pi$$
−0.981209 + 0.192947i $$0.938195\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 16.0000i 0.512936i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ − 20.0000i − 0.639529i
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ − 24.0000i − 0.765871i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ − 40.0000i − 1.27321i
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −50.0000 −1.58830 −0.794151 0.607720i $$-0.792084\pi$$
−0.794151 + 0.607720i $$0.792084\pi$$
$$992$$ − 6.00000i − 0.190500i
$$993$$ 24.0000i 0.761617i
$$994$$ −64.0000 −2.02996
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ − 22.0000i − 0.696747i −0.937356 0.348373i $$-0.886734\pi$$
0.937356 0.348373i $$-0.113266\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ −8.00000 −0.253109
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 2850.2.d.s.799.1 2
5.2 odd 4 2850.2.a.x.1.1 1
5.3 odd 4 114.2.a.a.1.1 1
5.4 even 2 inner 2850.2.d.s.799.2 2
15.2 even 4 8550.2.a.a.1.1 1
15.8 even 4 342.2.a.f.1.1 1
20.3 even 4 912.2.a.h.1.1 1
35.13 even 4 5586.2.a.p.1.1 1
40.3 even 4 3648.2.a.j.1.1 1
40.13 odd 4 3648.2.a.bb.1.1 1
60.23 odd 4 2736.2.a.j.1.1 1
95.18 even 4 2166.2.a.i.1.1 1
285.113 odd 4 6498.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.a.1.1 1 5.3 odd 4
342.2.a.f.1.1 1 15.8 even 4
912.2.a.h.1.1 1 20.3 even 4
2166.2.a.i.1.1 1 95.18 even 4
2736.2.a.j.1.1 1 60.23 odd 4
2850.2.a.x.1.1 1 5.2 odd 4
2850.2.d.s.799.1 2 1.1 even 1 trivial
2850.2.d.s.799.2 2 5.4 even 2 inner
3648.2.a.j.1.1 1 40.3 even 4
3648.2.a.bb.1.1 1 40.13 odd 4
5586.2.a.p.1.1 1 35.13 even 4
6498.2.a.h.1.1 1 285.113 odd 4
8550.2.a.a.1.1 1 15.2 even 4