Properties

 Label 2850.2.d.w.799.4 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 799.4 Root $$-1.22474 - 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.w.799.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.44949i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.44949i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.44949 q^{11} +1.00000i q^{12} +2.44949i q^{13} -4.44949 q^{14} +1.00000 q^{16} +4.44949i q^{17} -1.00000i q^{18} -1.00000 q^{19} +4.44949 q^{21} +3.44949i q^{22} +1.00000i q^{23} -1.00000 q^{24} -2.44949 q^{26} +1.00000i q^{27} -4.44949i q^{28} +4.34847 q^{29} -3.00000 q^{31} +1.00000i q^{32} -3.44949i q^{33} -4.44949 q^{34} +1.00000 q^{36} -7.79796i q^{37} -1.00000i q^{38} +2.44949 q^{39} -0.898979 q^{41} +4.44949i q^{42} +2.44949i q^{43} -3.44949 q^{44} -1.00000 q^{46} -7.79796i q^{47} -1.00000i q^{48} -12.7980 q^{49} +4.44949 q^{51} -2.44949i q^{52} +7.44949i q^{53} -1.00000 q^{54} +4.44949 q^{56} +1.00000i q^{57} +4.34847i q^{58} -6.44949 q^{59} -9.44949 q^{61} -3.00000i q^{62} -4.44949i q^{63} -1.00000 q^{64} +3.44949 q^{66} +15.2474i q^{67} -4.44949i q^{68} +1.00000 q^{69} -1.55051 q^{71} +1.00000i q^{72} +1.00000i q^{73} +7.79796 q^{74} +1.00000 q^{76} +15.3485i q^{77} +2.44949i q^{78} +5.00000 q^{79} +1.00000 q^{81} -0.898979i q^{82} -8.34847i q^{83} -4.44949 q^{84} -2.44949 q^{86} -4.34847i q^{87} -3.44949i q^{88} -2.10102 q^{89} -10.8990 q^{91} -1.00000i q^{92} +3.00000i q^{93} +7.79796 q^{94} +1.00000 q^{96} +1.55051i q^{97} -12.7980i q^{98} -3.44949 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 8 q^{14} + 4 q^{16} - 4 q^{19} + 8 q^{21} - 4 q^{24} - 12 q^{29} - 12 q^{31} - 8 q^{34} + 4 q^{36} + 16 q^{41} - 4 q^{44} - 4 q^{46} - 12 q^{49} + 8 q^{51} - 4 q^{54} + 8 q^{56} - 16 q^{59} - 28 q^{61} - 4 q^{64} + 4 q^{66} + 4 q^{69} - 16 q^{71} - 8 q^{74} + 4 q^{76} + 20 q^{79} + 4 q^{81} - 8 q^{84} - 28 q^{89} - 24 q^{91} - 8 q^{94} + 4 q^{96} - 4 q^{99} + O(q^{100})$$

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
$$7$$ 4.44949i 1.68175i 0.541230 + 0.840875i $$0.317959\pi$$
−0.541230 + 0.840875i $$0.682041\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 3.44949 1.04006 0.520030 0.854148i $$-0.325921\pi$$
0.520030 + 0.854148i $$0.325921\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 2.44949i 0.679366i 0.940540 + 0.339683i $$0.110320\pi$$
−0.940540 + 0.339683i $$0.889680\pi$$
$$14$$ −4.44949 −1.18918
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.44949i 1.07916i 0.841934 + 0.539580i $$0.181417\pi$$
−0.841934 + 0.539580i $$0.818583\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 4.44949 0.970958
$$22$$ 3.44949i 0.735434i
$$23$$ 1.00000i 0.208514i 0.994550 + 0.104257i $$0.0332465\pi$$
−0.994550 + 0.104257i $$0.966753\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.44949 −0.480384
$$27$$ 1.00000i 0.192450i
$$28$$ − 4.44949i − 0.840875i
$$29$$ 4.34847 0.807490 0.403745 0.914871i $$-0.367708\pi$$
0.403745 + 0.914871i $$0.367708\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 3.44949i − 0.600479i
$$34$$ −4.44949 −0.763081
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 7.79796i − 1.28198i −0.767551 0.640988i $$-0.778525\pi$$
0.767551 0.640988i $$-0.221475\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ 2.44949 0.392232
$$40$$ 0 0
$$41$$ −0.898979 −0.140397 −0.0701985 0.997533i $$-0.522363\pi$$
−0.0701985 + 0.997533i $$0.522363\pi$$
$$42$$ 4.44949i 0.686571i
$$43$$ 2.44949i 0.373544i 0.982403 + 0.186772i $$0.0598025\pi$$
−0.982403 + 0.186772i $$0.940197\pi$$
$$44$$ −3.44949 −0.520030
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ − 7.79796i − 1.13745i −0.822528 0.568725i $$-0.807437\pi$$
0.822528 0.568725i $$-0.192563\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −12.7980 −1.82828
$$50$$ 0 0
$$51$$ 4.44949 0.623053
$$52$$ − 2.44949i − 0.339683i
$$53$$ 7.44949i 1.02327i 0.859204 + 0.511633i $$0.170959\pi$$
−0.859204 + 0.511633i $$0.829041\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 4.44949 0.594588
$$57$$ 1.00000i 0.132453i
$$58$$ 4.34847i 0.570982i
$$59$$ −6.44949 −0.839652 −0.419826 0.907605i $$-0.637909\pi$$
−0.419826 + 0.907605i $$0.637909\pi$$
$$60$$ 0 0
$$61$$ −9.44949 −1.20988 −0.604942 0.796270i $$-0.706804\pi$$
−0.604942 + 0.796270i $$0.706804\pi$$
$$62$$ − 3.00000i − 0.381000i
$$63$$ − 4.44949i − 0.560583i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.44949 0.424603
$$67$$ 15.2474i 1.86277i 0.364033 + 0.931386i $$0.381400\pi$$
−0.364033 + 0.931386i $$0.618600\pi$$
$$68$$ − 4.44949i − 0.539580i
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −1.55051 −0.184012 −0.0920059 0.995758i $$-0.529328\pi$$
−0.0920059 + 0.995758i $$0.529328\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 1.00000i 0.117041i 0.998286 + 0.0585206i $$0.0186383\pi$$
−0.998286 + 0.0585206i $$0.981362\pi$$
$$74$$ 7.79796 0.906494
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 15.3485i 1.74912i
$$78$$ 2.44949i 0.277350i
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 0.898979i − 0.0992757i
$$83$$ − 8.34847i − 0.916364i −0.888859 0.458182i $$-0.848501\pi$$
0.888859 0.458182i $$-0.151499\pi$$
$$84$$ −4.44949 −0.485479
$$85$$ 0 0
$$86$$ −2.44949 −0.264135
$$87$$ − 4.34847i − 0.466205i
$$88$$ − 3.44949i − 0.367717i
$$89$$ −2.10102 −0.222708 −0.111354 0.993781i $$-0.535519\pi$$
−0.111354 + 0.993781i $$0.535519\pi$$
$$90$$ 0 0
$$91$$ −10.8990 −1.14252
$$92$$ − 1.00000i − 0.104257i
$$93$$ 3.00000i 0.311086i
$$94$$ 7.79796 0.804298
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 1.55051i 0.157430i 0.996897 + 0.0787152i $$0.0250818\pi$$
−0.996897 + 0.0787152i $$0.974918\pi$$
$$98$$ − 12.7980i − 1.29279i
$$99$$ −3.44949 −0.346687
$$100$$ 0 0
$$101$$ −1.55051 −0.154282 −0.0771408 0.997020i $$-0.524579\pi$$
−0.0771408 + 0.997020i $$0.524579\pi$$
$$102$$ 4.44949i 0.440565i
$$103$$ − 1.89898i − 0.187112i −0.995614 0.0935560i $$-0.970177\pi$$
0.995614 0.0935560i $$-0.0298234\pi$$
$$104$$ 2.44949 0.240192
$$105$$ 0 0
$$106$$ −7.44949 −0.723558
$$107$$ 17.3485i 1.67714i 0.544794 + 0.838570i $$0.316608\pi$$
−0.544794 + 0.838570i $$0.683392\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 2.89898 0.277672 0.138836 0.990315i $$-0.455664\pi$$
0.138836 + 0.990315i $$0.455664\pi$$
$$110$$ 0 0
$$111$$ −7.79796 −0.740150
$$112$$ 4.44949i 0.420437i
$$113$$ 16.7980i 1.58022i 0.612966 + 0.790110i $$0.289976\pi$$
−0.612966 + 0.790110i $$0.710024\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ −4.34847 −0.403745
$$117$$ − 2.44949i − 0.226455i
$$118$$ − 6.44949i − 0.593724i
$$119$$ −19.7980 −1.81488
$$120$$ 0 0
$$121$$ 0.898979 0.0817254
$$122$$ − 9.44949i − 0.855517i
$$123$$ 0.898979i 0.0810583i
$$124$$ 3.00000 0.269408
$$125$$ 0 0
$$126$$ 4.44949 0.396392
$$127$$ 0.101021i 0.00896412i 0.999990 + 0.00448206i $$0.00142669\pi$$
−0.999990 + 0.00448206i $$0.998573\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 2.44949 0.215666
$$130$$ 0 0
$$131$$ 9.24745 0.807953 0.403977 0.914769i $$-0.367628\pi$$
0.403977 + 0.914769i $$0.367628\pi$$
$$132$$ 3.44949i 0.300240i
$$133$$ − 4.44949i − 0.385820i
$$134$$ −15.2474 −1.31718
$$135$$ 0 0
$$136$$ 4.44949 0.381541
$$137$$ − 4.89898i − 0.418548i −0.977857 0.209274i $$-0.932890\pi$$
0.977857 0.209274i $$-0.0671101\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ 22.2474 1.88700 0.943502 0.331367i $$-0.107510\pi$$
0.943502 + 0.331367i $$0.107510\pi$$
$$140$$ 0 0
$$141$$ −7.79796 −0.656707
$$142$$ − 1.55051i − 0.130116i
$$143$$ 8.44949i 0.706582i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −1.00000 −0.0827606
$$147$$ 12.7980i 1.05556i
$$148$$ 7.79796i 0.640988i
$$149$$ −5.79796 −0.474987 −0.237494 0.971389i $$-0.576326\pi$$
−0.237494 + 0.971389i $$0.576326\pi$$
$$150$$ 0 0
$$151$$ −23.7980 −1.93665 −0.968325 0.249692i $$-0.919671\pi$$
−0.968325 + 0.249692i $$0.919671\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 4.44949i − 0.359720i
$$154$$ −15.3485 −1.23681
$$155$$ 0 0
$$156$$ −2.44949 −0.196116
$$157$$ − 4.89898i − 0.390981i −0.980706 0.195491i $$-0.937370\pi$$
0.980706 0.195491i $$-0.0626299\pi$$
$$158$$ 5.00000i 0.397779i
$$159$$ 7.44949 0.590783
$$160$$ 0 0
$$161$$ −4.44949 −0.350669
$$162$$ 1.00000i 0.0785674i
$$163$$ − 19.7980i − 1.55070i −0.631534 0.775348i $$-0.717575\pi$$
0.631534 0.775348i $$-0.282425\pi$$
$$164$$ 0.898979 0.0701985
$$165$$ 0 0
$$166$$ 8.34847 0.647967
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ − 4.44949i − 0.343286i
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 2.44949i − 0.186772i
$$173$$ 1.65153i 0.125564i 0.998027 + 0.0627818i $$0.0199972\pi$$
−0.998027 + 0.0627818i $$0.980003\pi$$
$$174$$ 4.34847 0.329657
$$175$$ 0 0
$$176$$ 3.44949 0.260015
$$177$$ 6.44949i 0.484773i
$$178$$ − 2.10102i − 0.157478i
$$179$$ −25.7980 −1.92823 −0.964115 0.265485i $$-0.914468\pi$$
−0.964115 + 0.265485i $$0.914468\pi$$
$$180$$ 0 0
$$181$$ −14.4495 −1.07402 −0.537011 0.843575i $$-0.680447\pi$$
−0.537011 + 0.843575i $$0.680447\pi$$
$$182$$ − 10.8990i − 0.807886i
$$183$$ 9.44949i 0.698526i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ −3.00000 −0.219971
$$187$$ 15.3485i 1.12239i
$$188$$ 7.79796i 0.568725i
$$189$$ −4.44949 −0.323653
$$190$$ 0 0
$$191$$ −0.101021 −0.00730959 −0.00365479 0.999993i $$-0.501163\pi$$
−0.00365479 + 0.999993i $$0.501163\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 15.3485i 1.10481i 0.833577 + 0.552403i $$0.186289\pi$$
−0.833577 + 0.552403i $$0.813711\pi$$
$$194$$ −1.55051 −0.111320
$$195$$ 0 0
$$196$$ 12.7980 0.914140
$$197$$ 27.3485i 1.94850i 0.225475 + 0.974249i $$0.427606\pi$$
−0.225475 + 0.974249i $$0.572394\pi$$
$$198$$ − 3.44949i − 0.245145i
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 15.2474 1.07547
$$202$$ − 1.55051i − 0.109094i
$$203$$ 19.3485i 1.35800i
$$204$$ −4.44949 −0.311527
$$205$$ 0 0
$$206$$ 1.89898 0.132308
$$207$$ − 1.00000i − 0.0695048i
$$208$$ 2.44949i 0.169842i
$$209$$ −3.44949 −0.238606
$$210$$ 0 0
$$211$$ −3.65153 −0.251382 −0.125691 0.992069i $$-0.540115\pi$$
−0.125691 + 0.992069i $$0.540115\pi$$
$$212$$ − 7.44949i − 0.511633i
$$213$$ 1.55051i 0.106239i
$$214$$ −17.3485 −1.18592
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 13.3485i − 0.906153i
$$218$$ 2.89898i 0.196344i
$$219$$ 1.00000 0.0675737
$$220$$ 0 0
$$221$$ −10.8990 −0.733145
$$222$$ − 7.79796i − 0.523365i
$$223$$ − 16.1010i − 1.07820i −0.842240 0.539102i $$-0.818764\pi$$
0.842240 0.539102i $$-0.181236\pi$$
$$224$$ −4.44949 −0.297294
$$225$$ 0 0
$$226$$ −16.7980 −1.11738
$$227$$ 29.5959i 1.96435i 0.187969 + 0.982175i $$0.439810\pi$$
−0.187969 + 0.982175i $$0.560190\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −7.24745 −0.478925 −0.239462 0.970906i $$-0.576971\pi$$
−0.239462 + 0.970906i $$0.576971\pi$$
$$230$$ 0 0
$$231$$ 15.3485 1.00986
$$232$$ − 4.34847i − 0.285491i
$$233$$ − 6.24745i − 0.409284i −0.978837 0.204642i $$-0.934397\pi$$
0.978837 0.204642i $$-0.0656030\pi$$
$$234$$ 2.44949 0.160128
$$235$$ 0 0
$$236$$ 6.44949 0.419826
$$237$$ − 5.00000i − 0.324785i
$$238$$ − 19.7980i − 1.28331i
$$239$$ 8.69694 0.562558 0.281279 0.959626i $$-0.409241\pi$$
0.281279 + 0.959626i $$0.409241\pi$$
$$240$$ 0 0
$$241$$ −4.44949 −0.286617 −0.143308 0.989678i $$-0.545774\pi$$
−0.143308 + 0.989678i $$0.545774\pi$$
$$242$$ 0.898979i 0.0577886i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 9.44949 0.604942
$$245$$ 0 0
$$246$$ −0.898979 −0.0573168
$$247$$ − 2.44949i − 0.155857i
$$248$$ 3.00000i 0.190500i
$$249$$ −8.34847 −0.529063
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 4.44949i 0.280292i
$$253$$ 3.44949i 0.216868i
$$254$$ −0.101021 −0.00633859
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 21.4949i − 1.34081i −0.741993 0.670407i $$-0.766119\pi$$
0.741993 0.670407i $$-0.233881\pi$$
$$258$$ 2.44949i 0.152499i
$$259$$ 34.6969 2.15596
$$260$$ 0 0
$$261$$ −4.34847 −0.269163
$$262$$ 9.24745i 0.571309i
$$263$$ 6.79796i 0.419180i 0.977789 + 0.209590i $$0.0672129\pi$$
−0.977789 + 0.209590i $$0.932787\pi$$
$$264$$ −3.44949 −0.212301
$$265$$ 0 0
$$266$$ 4.44949 0.272816
$$267$$ 2.10102i 0.128580i
$$268$$ − 15.2474i − 0.931386i
$$269$$ 2.89898 0.176754 0.0883769 0.996087i $$-0.471832\pi$$
0.0883769 + 0.996087i $$0.471832\pi$$
$$270$$ 0 0
$$271$$ 22.9444 1.39377 0.696886 0.717182i $$-0.254568\pi$$
0.696886 + 0.717182i $$0.254568\pi$$
$$272$$ 4.44949i 0.269790i
$$273$$ 10.8990i 0.659636i
$$274$$ 4.89898 0.295958
$$275$$ 0 0
$$276$$ −1.00000 −0.0601929
$$277$$ 21.0454i 1.26450i 0.774766 + 0.632248i $$0.217868\pi$$
−0.774766 + 0.632248i $$0.782132\pi$$
$$278$$ 22.2474i 1.33431i
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 7.00000 0.417585 0.208792 0.977960i $$-0.433047\pi$$
0.208792 + 0.977960i $$0.433047\pi$$
$$282$$ − 7.79796i − 0.464362i
$$283$$ − 29.7980i − 1.77130i −0.464349 0.885652i $$-0.653712\pi$$
0.464349 0.885652i $$-0.346288\pi$$
$$284$$ 1.55051 0.0920059
$$285$$ 0 0
$$286$$ −8.44949 −0.499629
$$287$$ − 4.00000i − 0.236113i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −2.79796 −0.164586
$$290$$ 0 0
$$291$$ 1.55051 0.0908925
$$292$$ − 1.00000i − 0.0585206i
$$293$$ 23.2474i 1.35813i 0.734078 + 0.679065i $$0.237615\pi$$
−0.734078 + 0.679065i $$0.762385\pi$$
$$294$$ −12.7980 −0.746392
$$295$$ 0 0
$$296$$ −7.79796 −0.453247
$$297$$ 3.44949i 0.200160i
$$298$$ − 5.79796i − 0.335867i
$$299$$ −2.44949 −0.141658
$$300$$ 0 0
$$301$$ −10.8990 −0.628207
$$302$$ − 23.7980i − 1.36942i
$$303$$ 1.55051i 0.0890745i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 4.44949 0.254360
$$307$$ − 16.3485i − 0.933056i −0.884506 0.466528i $$-0.845505\pi$$
0.884506 0.466528i $$-0.154495\pi$$
$$308$$ − 15.3485i − 0.874560i
$$309$$ −1.89898 −0.108029
$$310$$ 0 0
$$311$$ −23.7980 −1.34946 −0.674729 0.738065i $$-0.735740\pi$$
−0.674729 + 0.738065i $$0.735740\pi$$
$$312$$ − 2.44949i − 0.138675i
$$313$$ − 31.8990i − 1.80304i −0.432741 0.901518i $$-0.642453\pi$$
0.432741 0.901518i $$-0.357547\pi$$
$$314$$ 4.89898 0.276465
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ 15.2474i 0.856382i 0.903688 + 0.428191i $$0.140849\pi$$
−0.903688 + 0.428191i $$0.859151\pi$$
$$318$$ 7.44949i 0.417747i
$$319$$ 15.0000 0.839839
$$320$$ 0 0
$$321$$ 17.3485 0.968297
$$322$$ − 4.44949i − 0.247960i
$$323$$ − 4.44949i − 0.247576i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 19.7980 1.09651
$$327$$ − 2.89898i − 0.160314i
$$328$$ 0.898979i 0.0496378i
$$329$$ 34.6969 1.91290
$$330$$ 0 0
$$331$$ −22.3485 −1.22838 −0.614191 0.789157i $$-0.710518\pi$$
−0.614191 + 0.789157i $$0.710518\pi$$
$$332$$ 8.34847i 0.458182i
$$333$$ 7.79796i 0.427326i
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 4.44949 0.242740
$$337$$ − 24.8990i − 1.35633i −0.734908 0.678167i $$-0.762775\pi$$
0.734908 0.678167i $$-0.237225\pi$$
$$338$$ 7.00000i 0.380750i
$$339$$ 16.7980 0.912340
$$340$$ 0 0
$$341$$ −10.3485 −0.560401
$$342$$ 1.00000i 0.0540738i
$$343$$ − 25.7980i − 1.39296i
$$344$$ 2.44949 0.132068
$$345$$ 0 0
$$346$$ −1.65153 −0.0887868
$$347$$ − 23.5959i − 1.26670i −0.773867 0.633348i $$-0.781680\pi$$
0.773867 0.633348i $$-0.218320\pi$$
$$348$$ 4.34847i 0.233102i
$$349$$ 25.9444 1.38877 0.694386 0.719603i $$-0.255676\pi$$
0.694386 + 0.719603i $$0.255676\pi$$
$$350$$ 0 0
$$351$$ −2.44949 −0.130744
$$352$$ 3.44949i 0.183858i
$$353$$ 18.2474i 0.971214i 0.874177 + 0.485607i $$0.161401\pi$$
−0.874177 + 0.485607i $$0.838599\pi$$
$$354$$ −6.44949 −0.342787
$$355$$ 0 0
$$356$$ 2.10102 0.111354
$$357$$ 19.7980i 1.04782i
$$358$$ − 25.7980i − 1.36346i
$$359$$ 22.8990 1.20856 0.604281 0.796771i $$-0.293460\pi$$
0.604281 + 0.796771i $$0.293460\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 14.4495i − 0.759448i
$$363$$ − 0.898979i − 0.0471842i
$$364$$ 10.8990 0.571262
$$365$$ 0 0
$$366$$ −9.44949 −0.493933
$$367$$ − 5.55051i − 0.289734i −0.989451 0.144867i $$-0.953725\pi$$
0.989451 0.144867i $$-0.0462755\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 0.898979 0.0467990
$$370$$ 0 0
$$371$$ −33.1464 −1.72088
$$372$$ − 3.00000i − 0.155543i
$$373$$ 22.4495i 1.16239i 0.813764 + 0.581195i $$0.197415\pi$$
−0.813764 + 0.581195i $$0.802585\pi$$
$$374$$ −15.3485 −0.793650
$$375$$ 0 0
$$376$$ −7.79796 −0.402149
$$377$$ 10.6515i 0.548582i
$$378$$ − 4.44949i − 0.228857i
$$379$$ 1.30306 0.0669338 0.0334669 0.999440i $$-0.489345\pi$$
0.0334669 + 0.999440i $$0.489345\pi$$
$$380$$ 0 0
$$381$$ 0.101021 0.00517544
$$382$$ − 0.101021i − 0.00516866i
$$383$$ − 16.2474i − 0.830206i −0.909774 0.415103i $$-0.863746\pi$$
0.909774 0.415103i $$-0.136254\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −15.3485 −0.781217
$$387$$ − 2.44949i − 0.124515i
$$388$$ − 1.55051i − 0.0787152i
$$389$$ −21.5959 −1.09496 −0.547478 0.836820i $$-0.684412\pi$$
−0.547478 + 0.836820i $$0.684412\pi$$
$$390$$ 0 0
$$391$$ −4.44949 −0.225020
$$392$$ 12.7980i 0.646395i
$$393$$ − 9.24745i − 0.466472i
$$394$$ −27.3485 −1.37780
$$395$$ 0 0
$$396$$ 3.44949 0.173343
$$397$$ 23.9444i 1.20173i 0.799349 + 0.600867i $$0.205178\pi$$
−0.799349 + 0.600867i $$0.794822\pi$$
$$398$$ 10.0000i 0.501255i
$$399$$ −4.44949 −0.222753
$$400$$ 0 0
$$401$$ 11.2020 0.559403 0.279702 0.960087i $$-0.409764\pi$$
0.279702 + 0.960087i $$0.409764\pi$$
$$402$$ 15.2474i 0.760474i
$$403$$ − 7.34847i − 0.366053i
$$404$$ 1.55051 0.0771408
$$405$$ 0 0
$$406$$ −19.3485 −0.960248
$$407$$ − 26.8990i − 1.33333i
$$408$$ − 4.44949i − 0.220283i
$$409$$ 22.8990 1.13228 0.566141 0.824309i $$-0.308436\pi$$
0.566141 + 0.824309i $$0.308436\pi$$
$$410$$ 0 0
$$411$$ −4.89898 −0.241649
$$412$$ 1.89898i 0.0935560i
$$413$$ − 28.6969i − 1.41208i
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ −2.44949 −0.120096
$$417$$ − 22.2474i − 1.08946i
$$418$$ − 3.44949i − 0.168720i
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ − 3.65153i − 0.177754i
$$423$$ 7.79796i 0.379150i
$$424$$ 7.44949 0.361779
$$425$$ 0 0
$$426$$ −1.55051 −0.0751225
$$427$$ − 42.0454i − 2.03472i
$$428$$ − 17.3485i − 0.838570i
$$429$$ 8.44949 0.407945
$$430$$ 0 0
$$431$$ −5.10102 −0.245708 −0.122854 0.992425i $$-0.539205\pi$$
−0.122854 + 0.992425i $$0.539205\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 4.65153i − 0.223538i −0.993734 0.111769i $$-0.964348\pi$$
0.993734 0.111769i $$-0.0356517\pi$$
$$434$$ 13.3485 0.640747
$$435$$ 0 0
$$436$$ −2.89898 −0.138836
$$437$$ − 1.00000i − 0.0478365i
$$438$$ 1.00000i 0.0477818i
$$439$$ 12.1010 0.577550 0.288775 0.957397i $$-0.406752\pi$$
0.288775 + 0.957397i $$0.406752\pi$$
$$440$$ 0 0
$$441$$ 12.7980 0.609427
$$442$$ − 10.8990i − 0.518412i
$$443$$ − 21.2474i − 1.00950i −0.863267 0.504748i $$-0.831585\pi$$
0.863267 0.504748i $$-0.168415\pi$$
$$444$$ 7.79796 0.370075
$$445$$ 0 0
$$446$$ 16.1010 0.762405
$$447$$ 5.79796i 0.274234i
$$448$$ − 4.44949i − 0.210219i
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ −3.10102 −0.146021
$$452$$ − 16.7980i − 0.790110i
$$453$$ 23.7980i 1.11813i
$$454$$ −29.5959 −1.38901
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ − 24.8990i − 1.16473i −0.812929 0.582363i $$-0.802128\pi$$
0.812929 0.582363i $$-0.197872\pi$$
$$458$$ − 7.24745i − 0.338651i
$$459$$ −4.44949 −0.207684
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 15.3485i 0.714075i
$$463$$ 14.6969i 0.683025i 0.939877 + 0.341512i $$0.110939\pi$$
−0.939877 + 0.341512i $$0.889061\pi$$
$$464$$ 4.34847 0.201873
$$465$$ 0 0
$$466$$ 6.24745 0.289407
$$467$$ − 6.34847i − 0.293772i −0.989153 0.146886i $$-0.953075\pi$$
0.989153 0.146886i $$-0.0469251\pi$$
$$468$$ 2.44949i 0.113228i
$$469$$ −67.8434 −3.13272
$$470$$ 0 0
$$471$$ −4.89898 −0.225733
$$472$$ 6.44949i 0.296862i
$$473$$ 8.44949i 0.388508i
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ 19.7980 0.907438
$$477$$ − 7.44949i − 0.341089i
$$478$$ 8.69694i 0.397789i
$$479$$ −16.5959 −0.758287 −0.379143 0.925338i $$-0.623781\pi$$
−0.379143 + 0.925338i $$0.623781\pi$$
$$480$$ 0 0
$$481$$ 19.1010 0.870932
$$482$$ − 4.44949i − 0.202669i
$$483$$ 4.44949i 0.202459i
$$484$$ −0.898979 −0.0408627
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 32.0000i − 1.45006i −0.688718 0.725029i $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 9.44949i 0.427758i
$$489$$ −19.7980 −0.895295
$$490$$ 0 0
$$491$$ 43.5959 1.96746 0.983728 0.179664i $$-0.0575009\pi$$
0.983728 + 0.179664i $$0.0575009\pi$$
$$492$$ − 0.898979i − 0.0405291i
$$493$$ 19.3485i 0.871411i
$$494$$ 2.44949 0.110208
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ − 6.89898i − 0.309462i
$$498$$ − 8.34847i − 0.374104i
$$499$$ 23.8434 1.06738 0.533688 0.845682i $$-0.320806\pi$$
0.533688 + 0.845682i $$0.320806\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ − 18.0000i − 0.803379i
$$503$$ − 29.7980i − 1.32863i −0.747455 0.664313i $$-0.768724\pi$$
0.747455 0.664313i $$-0.231276\pi$$
$$504$$ −4.44949 −0.198196
$$505$$ 0 0
$$506$$ −3.44949 −0.153349
$$507$$ − 7.00000i − 0.310881i
$$508$$ − 0.101021i − 0.00448206i
$$509$$ 30.1464 1.33622 0.668108 0.744064i $$-0.267104\pi$$
0.668108 + 0.744064i $$0.267104\pi$$
$$510$$ 0 0
$$511$$ −4.44949 −0.196834
$$512$$ 1.00000i 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 21.4949 0.948099
$$515$$ 0 0
$$516$$ −2.44949 −0.107833
$$517$$ − 26.8990i − 1.18302i
$$518$$ 34.6969i 1.52450i
$$519$$ 1.65153 0.0724942
$$520$$ 0 0
$$521$$ 15.6969 0.687695 0.343848 0.939025i $$-0.388270\pi$$
0.343848 + 0.939025i $$0.388270\pi$$
$$522$$ − 4.34847i − 0.190327i
$$523$$ 33.3939i 1.46021i 0.683334 + 0.730106i $$0.260529\pi$$
−0.683334 + 0.730106i $$0.739471\pi$$
$$524$$ −9.24745 −0.403977
$$525$$ 0 0
$$526$$ −6.79796 −0.296405
$$527$$ − 13.3485i − 0.581468i
$$528$$ − 3.44949i − 0.150120i
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 6.44949 0.279884
$$532$$ 4.44949i 0.192910i
$$533$$ − 2.20204i − 0.0953810i
$$534$$ −2.10102 −0.0909200
$$535$$ 0 0
$$536$$ 15.2474 0.658589
$$537$$ 25.7980i 1.11326i
$$538$$ 2.89898i 0.124984i
$$539$$ −44.1464 −1.90152
$$540$$ 0 0
$$541$$ 22.1464 0.952149 0.476075 0.879405i $$-0.342059\pi$$
0.476075 + 0.879405i $$0.342059\pi$$
$$542$$ 22.9444i 0.985546i
$$543$$ 14.4495i 0.620087i
$$544$$ −4.44949 −0.190770
$$545$$ 0 0
$$546$$ −10.8990 −0.466433
$$547$$ − 46.6413i − 1.99424i −0.0758461 0.997120i $$-0.524166\pi$$
0.0758461 0.997120i $$-0.475834\pi$$
$$548$$ 4.89898i 0.209274i
$$549$$ 9.44949 0.403294
$$550$$ 0 0
$$551$$ −4.34847 −0.185251
$$552$$ − 1.00000i − 0.0425628i
$$553$$ 22.2474i 0.946058i
$$554$$ −21.0454 −0.894134
$$555$$ 0 0
$$556$$ −22.2474 −0.943502
$$557$$ 27.3485i 1.15879i 0.815046 + 0.579396i $$0.196711\pi$$
−0.815046 + 0.579396i $$0.803289\pi$$
$$558$$ 3.00000i 0.127000i
$$559$$ −6.00000 −0.253773
$$560$$ 0 0
$$561$$ 15.3485 0.648013
$$562$$ 7.00000i 0.295277i
$$563$$ − 6.24745i − 0.263299i −0.991296 0.131649i $$-0.957973\pi$$
0.991296 0.131649i $$-0.0420273\pi$$
$$564$$ 7.79796 0.328353
$$565$$ 0 0
$$566$$ 29.7980 1.25250
$$567$$ 4.44949i 0.186861i
$$568$$ 1.55051i 0.0650580i
$$569$$ 33.1918 1.39147 0.695737 0.718297i $$-0.255078\pi$$
0.695737 + 0.718297i $$0.255078\pi$$
$$570$$ 0 0
$$571$$ −10.2474 −0.428842 −0.214421 0.976741i $$-0.568787\pi$$
−0.214421 + 0.976741i $$0.568787\pi$$
$$572$$ − 8.44949i − 0.353291i
$$573$$ 0.101021i 0.00422019i
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 0.101021i 0.00420554i 0.999998 + 0.00210277i $$0.000669333\pi$$
−0.999998 + 0.00210277i $$0.999331\pi$$
$$578$$ − 2.79796i − 0.116380i
$$579$$ 15.3485 0.637861
$$580$$ 0 0
$$581$$ 37.1464 1.54109
$$582$$ 1.55051i 0.0642707i
$$583$$ 25.6969i 1.06426i
$$584$$ 1.00000 0.0413803
$$585$$ 0 0
$$586$$ −23.2474 −0.960343
$$587$$ − 33.4495i − 1.38061i −0.723519 0.690304i $$-0.757477\pi$$
0.723519 0.690304i $$-0.242523\pi$$
$$588$$ − 12.7980i − 0.527779i
$$589$$ 3.00000 0.123613
$$590$$ 0 0
$$591$$ 27.3485 1.12497
$$592$$ − 7.79796i − 0.320494i
$$593$$ − 8.49490i − 0.348844i −0.984671 0.174422i $$-0.944194\pi$$
0.984671 0.174422i $$-0.0558056\pi$$
$$594$$ −3.44949 −0.141534
$$595$$ 0 0
$$596$$ 5.79796 0.237494
$$597$$ − 10.0000i − 0.409273i
$$598$$ − 2.44949i − 0.100167i
$$599$$ −16.4495 −0.672108 −0.336054 0.941843i $$-0.609092\pi$$
−0.336054 + 0.941843i $$0.609092\pi$$
$$600$$ 0 0
$$601$$ 17.1464 0.699417 0.349709 0.936858i $$-0.386281\pi$$
0.349709 + 0.936858i $$0.386281\pi$$
$$602$$ − 10.8990i − 0.444209i
$$603$$ − 15.2474i − 0.620924i
$$604$$ 23.7980 0.968325
$$605$$ 0 0
$$606$$ −1.55051 −0.0629852
$$607$$ 46.1918i 1.87487i 0.348162 + 0.937434i $$0.386806\pi$$
−0.348162 + 0.937434i $$0.613194\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 19.3485 0.784040
$$610$$ 0 0
$$611$$ 19.1010 0.772745
$$612$$ 4.44949i 0.179860i
$$613$$ − 37.1918i − 1.50216i −0.660209 0.751082i $$-0.729532\pi$$
0.660209 0.751082i $$-0.270468\pi$$
$$614$$ 16.3485 0.659771
$$615$$ 0 0
$$616$$ 15.3485 0.618407
$$617$$ 28.2929i 1.13903i 0.821982 + 0.569514i $$0.192868\pi$$
−0.821982 + 0.569514i $$0.807132\pi$$
$$618$$ − 1.89898i − 0.0763882i
$$619$$ −45.1464 −1.81459 −0.907294 0.420497i $$-0.861856\pi$$
−0.907294 + 0.420497i $$0.861856\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ − 23.7980i − 0.954211i
$$623$$ − 9.34847i − 0.374539i
$$624$$ 2.44949 0.0980581
$$625$$ 0 0
$$626$$ 31.8990 1.27494
$$627$$ 3.44949i 0.137759i
$$628$$ 4.89898i 0.195491i
$$629$$ 34.6969 1.38346
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ − 5.00000i − 0.198889i
$$633$$ 3.65153i 0.145135i
$$634$$ −15.2474 −0.605554
$$635$$ 0 0
$$636$$ −7.44949 −0.295391
$$637$$ − 31.3485i − 1.24207i
$$638$$ 15.0000i 0.593856i
$$639$$ 1.55051 0.0613372
$$640$$ 0 0
$$641$$ 37.7980 1.49293 0.746465 0.665425i $$-0.231750\pi$$
0.746465 + 0.665425i $$0.231750\pi$$
$$642$$ 17.3485i 0.684689i
$$643$$ 0.202041i 0.00796772i 0.999992 + 0.00398386i $$0.00126811\pi$$
−0.999992 + 0.00398386i $$0.998732\pi$$
$$644$$ 4.44949 0.175334
$$645$$ 0 0
$$646$$ 4.44949 0.175063
$$647$$ 25.8990i 1.01819i 0.860709 + 0.509097i $$0.170021\pi$$
−0.860709 + 0.509097i $$0.829979\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −22.2474 −0.873289
$$650$$ 0 0
$$651$$ −13.3485 −0.523168
$$652$$ 19.7980i 0.775348i
$$653$$ 14.6969i 0.575136i 0.957760 + 0.287568i $$0.0928467\pi$$
−0.957760 + 0.287568i $$0.907153\pi$$
$$654$$ 2.89898 0.113359
$$655$$ 0 0
$$656$$ −0.898979 −0.0350993
$$657$$ − 1.00000i − 0.0390137i
$$658$$ 34.6969i 1.35263i
$$659$$ 18.0454 0.702949 0.351475 0.936197i $$-0.385680\pi$$
0.351475 + 0.936197i $$0.385680\pi$$
$$660$$ 0 0
$$661$$ 23.5959 0.917775 0.458887 0.888494i $$-0.348248\pi$$
0.458887 + 0.888494i $$0.348248\pi$$
$$662$$ − 22.3485i − 0.868598i
$$663$$ 10.8990i 0.423281i
$$664$$ −8.34847 −0.323983
$$665$$ 0 0
$$666$$ −7.79796 −0.302165
$$667$$ 4.34847i 0.168373i
$$668$$ − 18.0000i − 0.696441i
$$669$$ −16.1010 −0.622501
$$670$$ 0 0
$$671$$ −32.5959 −1.25835
$$672$$ 4.44949i 0.171643i
$$673$$ 45.3485i 1.74806i 0.485876 + 0.874028i $$0.338501\pi$$
−0.485876 + 0.874028i $$0.661499\pi$$
$$674$$ 24.8990 0.959073
$$675$$ 0 0
$$676$$ −7.00000 −0.269231
$$677$$ 22.3485i 0.858921i 0.903086 + 0.429461i $$0.141296\pi$$
−0.903086 + 0.429461i $$0.858704\pi$$
$$678$$ 16.7980i 0.645122i
$$679$$ −6.89898 −0.264759
$$680$$ 0 0
$$681$$ 29.5959 1.13412
$$682$$ − 10.3485i − 0.396263i
$$683$$ 35.6413i 1.36378i 0.731456 + 0.681889i $$0.238841\pi$$
−0.731456 + 0.681889i $$0.761159\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 25.7980 0.984971
$$687$$ 7.24745i 0.276507i
$$688$$ 2.44949i 0.0933859i
$$689$$ −18.2474 −0.695172
$$690$$ 0 0
$$691$$ 9.75255 0.371005 0.185502 0.982644i $$-0.440609\pi$$
0.185502 + 0.982644i $$0.440609\pi$$
$$692$$ − 1.65153i − 0.0627818i
$$693$$ − 15.3485i − 0.583040i
$$694$$ 23.5959 0.895689
$$695$$ 0 0
$$696$$ −4.34847 −0.164828
$$697$$ − 4.00000i − 0.151511i
$$698$$ 25.9444i 0.982010i
$$699$$ −6.24745 −0.236300
$$700$$ 0 0
$$701$$ −32.4949 −1.22732 −0.613658 0.789572i $$-0.710302\pi$$
−0.613658 + 0.789572i $$0.710302\pi$$
$$702$$ − 2.44949i − 0.0924500i
$$703$$ 7.79796i 0.294106i
$$704$$ −3.44949 −0.130008
$$705$$ 0 0
$$706$$ −18.2474 −0.686752
$$707$$ − 6.89898i − 0.259463i
$$708$$ − 6.44949i − 0.242387i
$$709$$ 12.7526 0.478932 0.239466 0.970905i $$-0.423028\pi$$
0.239466 + 0.970905i $$0.423028\pi$$
$$710$$ 0 0
$$711$$ −5.00000 −0.187515
$$712$$ 2.10102i 0.0787391i
$$713$$ − 3.00000i − 0.112351i
$$714$$ −19.7980 −0.740920
$$715$$ 0 0
$$716$$ 25.7980 0.964115
$$717$$ − 8.69694i − 0.324793i
$$718$$ 22.8990i 0.854582i
$$719$$ 19.2020 0.716115 0.358058 0.933699i $$-0.383439\pi$$
0.358058 + 0.933699i $$0.383439\pi$$
$$720$$ 0 0
$$721$$ 8.44949 0.314675
$$722$$ 1.00000i 0.0372161i
$$723$$ 4.44949i 0.165478i
$$724$$ 14.4495 0.537011
$$725$$ 0 0
$$726$$ 0.898979 0.0333643
$$727$$ 22.4949i 0.834290i 0.908840 + 0.417145i $$0.136969\pi$$
−0.908840 + 0.417145i $$0.863031\pi$$
$$728$$ 10.8990i 0.403943i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −10.8990 −0.403113
$$732$$ − 9.44949i − 0.349263i
$$733$$ − 4.14643i − 0.153152i −0.997064 0.0765759i $$-0.975601\pi$$
0.997064 0.0765759i $$-0.0243988\pi$$
$$734$$ 5.55051 0.204873
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 52.5959i 1.93740i
$$738$$ 0.898979i 0.0330919i
$$739$$ −32.8990 −1.21021 −0.605104 0.796146i $$-0.706869\pi$$
−0.605104 + 0.796146i $$0.706869\pi$$
$$740$$ 0 0
$$741$$ −2.44949 −0.0899843
$$742$$ − 33.1464i − 1.21684i
$$743$$ 53.3939i 1.95883i 0.201854 + 0.979416i $$0.435303\pi$$
−0.201854 + 0.979416i $$0.564697\pi$$
$$744$$ 3.00000 0.109985
$$745$$ 0 0
$$746$$ −22.4495 −0.821934
$$747$$ 8.34847i 0.305455i
$$748$$ − 15.3485i − 0.561196i
$$749$$ −77.1918 −2.82053
$$750$$ 0 0
$$751$$ −23.7980 −0.868400 −0.434200 0.900817i $$-0.642969\pi$$
−0.434200 + 0.900817i $$0.642969\pi$$
$$752$$ − 7.79796i − 0.284362i
$$753$$ 18.0000i 0.655956i
$$754$$ −10.6515 −0.387906
$$755$$ 0 0
$$756$$ 4.44949 0.161826
$$757$$ 48.1464i 1.74991i 0.484203 + 0.874956i $$0.339110\pi$$
−0.484203 + 0.874956i $$0.660890\pi$$
$$758$$ 1.30306i 0.0473293i
$$759$$ 3.44949 0.125209
$$760$$ 0 0
$$761$$ 51.3485 1.86138 0.930690 0.365808i $$-0.119207\pi$$
0.930690 + 0.365808i $$0.119207\pi$$
$$762$$ 0.101021i 0.00365959i
$$763$$ 12.8990i 0.466974i
$$764$$ 0.101021 0.00365479
$$765$$ 0 0
$$766$$ 16.2474 0.587044
$$767$$ − 15.7980i − 0.570431i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 7.89898 0.284844 0.142422 0.989806i $$-0.454511\pi$$
0.142422 + 0.989806i $$0.454511\pi$$
$$770$$ 0 0
$$771$$ −21.4949 −0.774120
$$772$$ − 15.3485i − 0.552403i
$$773$$ − 18.4949i − 0.665215i −0.943065 0.332608i $$-0.892072\pi$$
0.943065 0.332608i $$-0.107928\pi$$
$$774$$ 2.44949 0.0880451
$$775$$ 0 0
$$776$$ 1.55051 0.0556601
$$777$$ − 34.6969i − 1.24475i
$$778$$ − 21.5959i − 0.774251i
$$779$$ 0.898979 0.0322093
$$780$$ 0 0
$$781$$ −5.34847 −0.191383
$$782$$ − 4.44949i − 0.159113i
$$783$$ 4.34847i 0.155402i
$$784$$ −12.7980 −0.457070
$$785$$ 0 0
$$786$$ 9.24745 0.329846
$$787$$ − 33.4495i − 1.19235i −0.802856 0.596173i $$-0.796687\pi$$
0.802856 0.596173i $$-0.203313\pi$$
$$788$$ − 27.3485i − 0.974249i
$$789$$ 6.79796 0.242014
$$790$$ 0 0
$$791$$ −74.7423 −2.65753
$$792$$ 3.44949i 0.122572i
$$793$$ − 23.1464i − 0.821954i
$$794$$ −23.9444 −0.849755
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 3.50510i 0.124157i 0.998071 + 0.0620786i $$0.0197729\pi$$
−0.998071 + 0.0620786i $$0.980227\pi$$
$$798$$ − 4.44949i − 0.157510i
$$799$$ 34.6969 1.22749
$$800$$ 0 0
$$801$$ 2.10102 0.0742359
$$802$$ 11.2020i 0.395558i
$$803$$ 3.44949i 0.121730i
$$804$$ −15.2474 −0.537736
$$805$$ 0 0
$$806$$ 7.34847 0.258839
$$807$$ − 2.89898i − 0.102049i
$$808$$ 1.55051i 0.0545468i
$$809$$ 3.55051 0.124829 0.0624146 0.998050i $$-0.480120\pi$$
0.0624146 + 0.998050i $$0.480120\pi$$
$$810$$ 0 0
$$811$$ −19.4495 −0.682964 −0.341482 0.939888i $$-0.610929\pi$$
−0.341482 + 0.939888i $$0.610929\pi$$
$$812$$ − 19.3485i − 0.678998i
$$813$$ − 22.9444i − 0.804695i
$$814$$ 26.8990 0.942809
$$815$$ 0 0
$$816$$ 4.44949 0.155763
$$817$$ − 2.44949i − 0.0856968i
$$818$$ 22.8990i 0.800644i
$$819$$ 10.8990 0.380841
$$820$$ 0 0
$$821$$ −23.1464 −0.807816 −0.403908 0.914800i $$-0.632348\pi$$
−0.403908 + 0.914800i $$0.632348\pi$$
$$822$$ − 4.89898i − 0.170872i
$$823$$ 31.7980i 1.10841i 0.832381 + 0.554204i $$0.186977\pi$$
−0.832381 + 0.554204i $$0.813023\pi$$
$$824$$ −1.89898 −0.0661541
$$825$$ 0 0
$$826$$ 28.6969 0.998494
$$827$$ 0.247449i 0.00860463i 0.999991 + 0.00430232i $$0.00136947\pi$$
−0.999991 + 0.00430232i $$0.998631\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ 39.6413 1.37680 0.688400 0.725331i $$-0.258313\pi$$
0.688400 + 0.725331i $$0.258313\pi$$
$$830$$ 0 0
$$831$$ 21.0454 0.730057
$$832$$ − 2.44949i − 0.0849208i
$$833$$ − 56.9444i − 1.97301i
$$834$$ 22.2474 0.770366
$$835$$ 0 0
$$836$$ 3.44949 0.119303
$$837$$ − 3.00000i − 0.103695i
$$838$$ 0 0
$$839$$ −10.6515 −0.367732 −0.183866 0.982951i $$-0.558861\pi$$
−0.183866 + 0.982951i $$0.558861\pi$$
$$840$$ 0 0
$$841$$ −10.0908 −0.347959
$$842$$ 22.0000i 0.758170i
$$843$$ − 7.00000i − 0.241093i
$$844$$ 3.65153 0.125691
$$845$$ 0 0
$$846$$ −7.79796 −0.268099
$$847$$ 4.00000i 0.137442i
$$848$$ 7.44949i 0.255817i
$$849$$ −29.7980 −1.02266
$$850$$ 0 0
$$851$$ 7.79796 0.267311
$$852$$ − 1.55051i − 0.0531196i
$$853$$ − 1.10102i − 0.0376982i −0.999822 0.0188491i $$-0.994000\pi$$
0.999822 0.0188491i $$-0.00600021\pi$$
$$854$$ 42.0454 1.43876
$$855$$ 0 0
$$856$$ 17.3485 0.592958
$$857$$ − 36.4949i − 1.24664i −0.781966 0.623321i $$-0.785783\pi$$
0.781966 0.623321i $$-0.214217\pi$$
$$858$$ 8.44949i 0.288461i
$$859$$ 4.20204 0.143372 0.0716859 0.997427i $$-0.477162\pi$$
0.0716859 + 0.997427i $$0.477162\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ − 5.10102i − 0.173741i
$$863$$ 17.3031i 0.589003i 0.955651 + 0.294502i $$0.0951536\pi$$
−0.955651 + 0.294502i $$0.904846\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 4.65153 0.158065
$$867$$ 2.79796i 0.0950237i
$$868$$ 13.3485i 0.453077i
$$869$$ 17.2474 0.585080
$$870$$ 0 0
$$871$$ −37.3485 −1.26550
$$872$$ − 2.89898i − 0.0981718i
$$873$$ − 1.55051i − 0.0524768i
$$874$$ 1.00000 0.0338255
$$875$$ 0 0
$$876$$ −1.00000 −0.0337869
$$877$$ − 9.10102i − 0.307320i −0.988124 0.153660i $$-0.950894\pi$$
0.988124 0.153660i $$-0.0491060\pi$$
$$878$$ 12.1010i 0.408390i
$$879$$ 23.2474 0.784117
$$880$$ 0 0
$$881$$ 50.6969 1.70802 0.854012 0.520254i $$-0.174163\pi$$
0.854012 + 0.520254i $$0.174163\pi$$
$$882$$ 12.7980i 0.430930i
$$883$$ − 34.6515i − 1.16612i −0.812430 0.583058i $$-0.801856\pi$$
0.812430 0.583058i $$-0.198144\pi$$
$$884$$ 10.8990 0.366572
$$885$$ 0 0
$$886$$ 21.2474 0.713822
$$887$$ − 4.89898i − 0.164492i −0.996612 0.0822458i $$-0.973791\pi$$
0.996612 0.0822458i $$-0.0262093\pi$$
$$888$$ 7.79796i 0.261682i
$$889$$ −0.449490 −0.0150754
$$890$$ 0 0
$$891$$ 3.44949 0.115562
$$892$$ 16.1010i 0.539102i
$$893$$ 7.79796i 0.260949i
$$894$$ −5.79796 −0.193913
$$895$$ 0 0
$$896$$ 4.44949 0.148647
$$897$$ 2.44949i 0.0817861i
$$898$$ 15.0000i 0.500556i
$$899$$ −13.0454 −0.435089
$$900$$ 0 0
$$901$$ −33.1464 −1.10427
$$902$$ − 3.10102i − 0.103253i
$$903$$ 10.8990i 0.362695i
$$904$$ 16.7980 0.558692
$$905$$ 0 0
$$906$$ −23.7980 −0.790634
$$907$$ − 22.0000i − 0.730498i −0.930910 0.365249i $$-0.880984\pi$$
0.930910 0.365249i $$-0.119016\pi$$
$$908$$ − 29.5959i − 0.982175i
$$909$$ 1.55051 0.0514272
$$910$$ 0 0
$$911$$ 6.49490 0.215186 0.107593 0.994195i $$-0.465686\pi$$
0.107593 + 0.994195i $$0.465686\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ − 28.7980i − 0.953073i
$$914$$ 24.8990 0.823585
$$915$$ 0 0
$$916$$ 7.24745 0.239462
$$917$$ 41.1464i 1.35877i
$$918$$ − 4.44949i − 0.146855i
$$919$$ −23.8434 −0.786520 −0.393260 0.919427i $$-0.628653\pi$$
−0.393260 + 0.919427i $$0.628653\pi$$
$$920$$ 0 0
$$921$$ −16.3485 −0.538700
$$922$$ 12.0000i 0.395199i
$$923$$ − 3.79796i − 0.125011i
$$924$$ −15.3485 −0.504928
$$925$$ 0 0
$$926$$ −14.6969 −0.482971
$$927$$ 1.89898i 0.0623707i
$$928$$ 4.34847i 0.142745i
$$929$$ 31.5959 1.03663 0.518314 0.855190i $$-0.326560\pi$$
0.518314 + 0.855190i $$0.326560\pi$$
$$930$$ 0 0
$$931$$ 12.7980 0.419436
$$932$$ 6.24745i 0.204642i
$$933$$ 23.7980i 0.779110i
$$934$$ 6.34847 0.207728
$$935$$ 0 0
$$936$$ −2.44949 −0.0800641
$$937$$ − 19.3939i − 0.633570i −0.948497 0.316785i $$-0.897397\pi$$
0.948497 0.316785i $$-0.102603\pi$$
$$938$$ − 67.8434i − 2.21516i
$$939$$ −31.8990 −1.04098
$$940$$ 0 0
$$941$$ 36.6413 1.19447 0.597237 0.802065i $$-0.296265\pi$$
0.597237 + 0.802065i $$0.296265\pi$$
$$942$$ − 4.89898i − 0.159617i
$$943$$ − 0.898979i − 0.0292748i
$$944$$ −6.44949 −0.209913
$$945$$ 0 0
$$946$$ −8.44949 −0.274717
$$947$$ 8.00000i 0.259965i 0.991516 + 0.129983i $$0.0414921\pi$$
−0.991516 + 0.129983i $$0.958508\pi$$
$$948$$ 5.00000i 0.162392i
$$949$$ −2.44949 −0.0795138
$$950$$ 0 0
$$951$$ 15.2474 0.494432
$$952$$ 19.7980i 0.641656i
$$953$$ 15.4949i 0.501929i 0.967996 + 0.250964i $$0.0807477\pi$$
−0.967996 + 0.250964i $$0.919252\pi$$
$$954$$ 7.44949 0.241186
$$955$$ 0 0
$$956$$ −8.69694 −0.281279
$$957$$ − 15.0000i − 0.484881i
$$958$$ − 16.5959i − 0.536190i
$$959$$ 21.7980 0.703893
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 19.1010i 0.615842i
$$963$$ − 17.3485i − 0.559047i
$$964$$ 4.44949 0.143308
$$965$$ 0 0
$$966$$ −4.44949 −0.143160
$$967$$ − 34.8990i − 1.12228i −0.827722 0.561138i $$-0.810364\pi$$
0.827722 0.561138i $$-0.189636\pi$$
$$968$$ − 0.898979i − 0.0288943i
$$969$$ −4.44949 −0.142938
$$970$$ 0 0
$$971$$ 41.6413 1.33633 0.668167 0.744011i $$-0.267079\pi$$
0.668167 + 0.744011i $$0.267079\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 98.9898i 3.17347i
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ −9.44949 −0.302471
$$977$$ 19.5959i 0.626929i 0.949600 + 0.313464i $$0.101490\pi$$
−0.949600 + 0.313464i $$0.898510\pi$$
$$978$$ − 19.7980i − 0.633069i
$$979$$ −7.24745 −0.231629
$$980$$ 0 0
$$981$$ −2.89898 −0.0925573
$$982$$ 43.5959i 1.39120i
$$983$$ − 50.4495i − 1.60909i −0.593892 0.804544i $$-0.702410\pi$$
0.593892 0.804544i $$-0.297590\pi$$
$$984$$ 0.898979 0.0286584
$$985$$ 0 0
$$986$$ −19.3485 −0.616181
$$987$$ − 34.6969i − 1.10442i
$$988$$ 2.44949i 0.0779287i
$$989$$ −2.44949 −0.0778892
$$990$$ 0 0
$$991$$ 44.1010 1.40092 0.700458 0.713694i $$-0.252979\pi$$
0.700458 + 0.713694i $$0.252979\pi$$
$$992$$ − 3.00000i − 0.0952501i
$$993$$ 22.3485i 0.709207i
$$994$$ 6.89898 0.218822
$$995$$ 0 0
$$996$$ 8.34847 0.264531
$$997$$ 39.4495i 1.24938i 0.780874 + 0.624689i $$0.214774\pi$$
−0.780874 + 0.624689i $$0.785226\pi$$
$$998$$ 23.8434i 0.754749i
$$999$$ 7.79796 0.246717
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.w.799.4 4
5.2 odd 4 2850.2.a.bc.1.1 2
5.3 odd 4 2850.2.a.bj.1.2 yes 2
5.4 even 2 inner 2850.2.d.w.799.1 4
15.2 even 4 8550.2.a.bv.1.1 2
15.8 even 4 8550.2.a.bu.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.1 2 5.2 odd 4
2850.2.a.bj.1.2 yes 2 5.3 odd 4
2850.2.d.w.799.1 4 5.4 even 2 inner
2850.2.d.w.799.4 4 1.1 even 1 trivial
8550.2.a.bu.1.2 2 15.8 even 4
8550.2.a.bv.1.1 2 15.2 even 4