# Properties

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

## $q$-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 $$-0.682041\pi$$
0.541230 0.840875i $$-0.317959\pi$$
$$8$$ 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 $$-0.889680\pi$$
0.940540 0.339683i $$-0.110320\pi$$
$$14$$ −4.44949 −1.18918
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 4.44949i − 1.07916i −0.841934 0.539580i $$-0.818583\pi$$
0.841934 0.539580i $$-0.181417\pi$$
$$18$$ 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 $$-0.966753\pi$$
0.994550 0.104257i $$-0.0332465\pi$$
$$24$$ −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 $$0.221475\pi$$
−0.767551 + 0.640988i $$0.778525\pi$$
$$38$$ 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 $$-0.940197\pi$$
0.982403 0.186772i $$-0.0598025\pi$$
$$44$$ −3.44949 −0.520030
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 7.79796i 1.13745i 0.822528 + 0.568725i $$0.192563\pi$$
−0.822528 + 0.568725i $$0.807437\pi$$
$$48$$ 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 $$-0.829041\pi$$
0.859204 0.511633i $$-0.170959\pi$$
$$54$$ −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 $$-0.618600\pi$$
0.364033 0.931386i $$-0.381400\pi$$
$$68$$ 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 $$-0.981362\pi$$
0.998286 0.0585206i $$-0.0186383\pi$$
$$74$$ 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 $$0.151499\pi$$
−0.888859 + 0.458182i $$0.848501\pi$$
$$84$$ −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 $$-0.974918\pi$$
0.996897 0.0787152i $$-0.0250818\pi$$
$$98$$ 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 $$0.0298234\pi$$
−0.995614 + 0.0935560i $$0.970177\pi$$
$$104$$ 2.44949 0.240192
$$105$$ 0 0
$$106$$ −7.44949 −0.723558
$$107$$ − 17.3485i − 1.67714i −0.544794 0.838570i $$-0.683392\pi$$
0.544794 0.838570i $$-0.316608\pi$$
$$108$$ 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 $$-0.710024\pi$$
0.612966 0.790110i $$-0.289976\pi$$
$$114$$ −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 $$-0.998573\pi$$
0.999990 0.00448206i $$-0.00142669\pi$$
$$128$$ 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 $$0.0671101\pi$$
−0.977857 + 0.209274i $$0.932890\pi$$
$$138$$ − 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 $$0.0626299\pi$$
−0.980706 + 0.195491i $$0.937370\pi$$
$$158$$ − 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 $$0.282425\pi$$
−0.631534 + 0.775348i $$0.717575\pi$$
$$164$$ 0.898979 0.0701985
$$165$$ 0 0
$$166$$ 8.34847 0.647967
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ 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 $$-0.980003\pi$$
0.998027 0.0627818i $$-0.0199972\pi$$
$$174$$ 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 $$-0.813711\pi$$
0.833577 0.552403i $$-0.186289\pi$$
$$194$$ −1.55051 −0.111320
$$195$$ 0 0
$$196$$ 12.7980 0.914140
$$197$$ − 27.3485i − 1.94850i −0.225475 0.974249i $$-0.572394\pi$$
0.225475 0.974249i $$-0.427606\pi$$
$$198$$ 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 $$0.181236\pi$$
−0.842240 + 0.539102i $$0.818764\pi$$
$$224$$ −4.44949 −0.297294
$$225$$ 0 0
$$226$$ −16.7980 −1.11738
$$227$$ − 29.5959i − 1.96435i −0.187969 0.982175i $$-0.560190\pi$$
0.187969 0.982175i $$-0.439810\pi$$
$$228$$ 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 $$0.0656030\pi$$
−0.978837 + 0.204642i $$0.934397\pi$$
$$234$$ 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 $$0.233881\pi$$
−0.741993 + 0.670407i $$0.766119\pi$$
$$258$$ − 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 $$-0.932787\pi$$
0.977789 0.209590i $$-0.0672129\pi$$
$$264$$ −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 $$-0.782132\pi$$
0.774766 0.632248i $$-0.217868\pi$$
$$278$$ − 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 $$0.346288\pi$$
−0.464349 + 0.885652i $$0.653712\pi$$
$$284$$ 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 $$-0.762385\pi$$
0.734078 0.679065i $$-0.237615\pi$$
$$294$$ −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 $$0.154495\pi$$
−0.884506 + 0.466528i $$0.845505\pi$$
$$308$$ 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 $$0.357547\pi$$
−0.432741 + 0.901518i $$0.642453\pi$$
$$314$$ 4.89898 0.276465
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ − 15.2474i − 0.856382i −0.903688 0.428191i $$-0.859151\pi$$
0.903688 0.428191i $$-0.140849\pi$$
$$318$$ − 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 $$0.237225\pi$$
−0.734908 + 0.678167i $$0.762775\pi$$
$$338$$ − 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 $$0.218320\pi$$
−0.773867 + 0.633348i $$0.781680\pi$$
$$348$$ − 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 $$-0.838599\pi$$
0.874177 0.485607i $$-0.161401\pi$$
$$354$$ −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 $$0.0462755\pi$$
−0.989451 + 0.144867i $$0.953725\pi$$
$$368$$ − 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 $$-0.802585\pi$$
0.813764 0.581195i $$-0.197415\pi$$
$$374$$ −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 $$0.136254\pi$$
−0.909774 + 0.415103i $$0.863746\pi$$
$$384$$ −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 $$-0.794822\pi$$
0.799349 0.600867i $$-0.205178\pi$$
$$398$$ − 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 $$0.0356517\pi$$
−0.993734 + 0.111769i $$0.964348\pi$$
$$434$$ 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 $$0.168415\pi$$
−0.863267 + 0.504748i $$0.831585\pi$$
$$444$$ 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 $$0.197872\pi$$
−0.812929 + 0.582363i $$0.802128\pi$$
$$458$$ 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 $$-0.889061\pi$$
0.939877 0.341512i $$-0.110939\pi$$
$$464$$ 4.34847 0.201873
$$465$$ 0 0
$$466$$ 6.24745 0.289407
$$467$$ 6.34847i 0.293772i 0.989153 + 0.146886i $$0.0469251\pi$$
−0.989153 + 0.146886i $$0.953075\pi$$
$$468$$ − 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 $$0.258174\pi$$
−0.688718 + 0.725029i $$0.741826\pi$$
$$488$$ − 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 $$0.231276\pi$$
−0.747455 + 0.664313i $$0.768724\pi$$
$$504$$ −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 $$-0.739471\pi$$
0.683334 0.730106i $$-0.260529\pi$$
$$524$$ −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 $$0.475834\pi$$
−0.0758461 + 0.997120i $$0.524166\pi$$
$$548$$ − 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 $$-0.803289\pi$$
0.815046 0.579396i $$-0.196711\pi$$
$$558$$ − 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 $$0.0420273\pi$$
−0.991296 + 0.131649i $$0.957973\pi$$
$$564$$ 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 $$-0.999331\pi$$
0.999998 0.00210277i $$-0.000669333\pi$$
$$578$$ 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 $$0.242523\pi$$
−0.723519 + 0.690304i $$0.757477\pi$$
$$588$$ 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 $$0.0558056\pi$$
−0.984671 + 0.174422i $$0.944194\pi$$
$$594$$ −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 $$-0.613194\pi$$
0.348162 0.937434i $$-0.386806\pi$$
$$608$$ 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 $$0.270468\pi$$
−0.660209 + 0.751082i $$0.729532\pi$$
$$614$$ 16.3485 0.659771
$$615$$ 0 0
$$616$$ 15.3485 0.618407
$$617$$ − 28.2929i − 1.13903i −0.821982 0.569514i $$-0.807132\pi$$
0.821982 0.569514i $$-0.192868\pi$$
$$618$$ 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 $$-0.998732\pi$$
0.999992 0.00398386i $$-0.00126811\pi$$
$$644$$ 4.44949 0.175334
$$645$$ 0 0
$$646$$ 4.44949 0.175063
$$647$$ − 25.8990i − 1.01819i −0.860709 0.509097i $$-0.829979\pi$$
0.860709 0.509097i $$-0.170021\pi$$
$$648$$ 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 $$-0.907153\pi$$
0.957760 0.287568i $$-0.0928467\pi$$
$$654$$ 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 $$-0.661499\pi$$
0.485876 0.874028i $$-0.338501\pi$$
$$674$$ 24.8990 0.959073
$$675$$ 0 0
$$676$$ −7.00000 −0.269231
$$677$$ − 22.3485i − 0.858921i −0.903086 0.429461i $$-0.858704\pi$$
0.903086 0.429461i $$-0.141296\pi$$
$$678$$ − 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 $$-0.761159\pi$$
0.731456 0.681889i $$-0.238841\pi$$
$$684$$ −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 $$-0.863031\pi$$
0.908840 0.417145i $$-0.136969\pi$$
$$728$$ − 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 $$0.0243988\pi$$
−0.997064 + 0.0765759i $$0.975601\pi$$
$$734$$ 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 $$-0.564697\pi$$
0.201854 0.979416i $$-0.435303\pi$$
$$744$$ 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 $$-0.660890\pi$$
0.484203 0.874956i $$-0.339110\pi$$
$$758$$ − 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 $$0.107928\pi$$
−0.943065 + 0.332608i $$0.892072\pi$$
$$774$$ 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 $$0.203313\pi$$
−0.802856 + 0.596173i $$0.796687\pi$$
$$788$$ 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 $$-0.980227\pi$$
0.998071 0.0620786i $$-0.0197729\pi$$
$$798$$ 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 $$-0.813023\pi$$
0.832381 0.554204i $$-0.186977\pi$$
$$824$$ −1.89898 −0.0661541
$$825$$ 0 0
$$826$$ 28.6969 0.998494
$$827$$ − 0.247449i − 0.00860463i −0.999991 0.00430232i $$-0.998631\pi$$
0.999991 0.00430232i $$-0.00136947\pi$$
$$828$$ − 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 $$0.00600021\pi$$
−0.999822 + 0.0188491i $$0.994000\pi$$
$$854$$ 42.0454 1.43876
$$855$$ 0 0
$$856$$ 17.3485 0.592958
$$857$$ 36.4949i 1.24664i 0.781966 + 0.623321i $$0.214217\pi$$
−0.781966 + 0.623321i $$0.785783\pi$$
$$858$$ − 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 $$-0.904846\pi$$
0.955651 0.294502i $$-0.0951536\pi$$
$$864$$ −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 $$0.0491060\pi$$
−0.988124 + 0.153660i $$0.950894\pi$$
$$878$$ − 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 $$0.198144\pi$$
−0.812430 + 0.583058i $$0.801856\pi$$
$$884$$ 10.8990 0.366572
$$885$$ 0 0
$$886$$ 21.2474 0.713822
$$887$$ 4.89898i 0.164492i 0.996612 + 0.0822458i $$0.0262093\pi$$
−0.996612 + 0.0822458i $$0.973791\pi$$
$$888$$ − 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 $$0.119016\pi$$
−0.930910 + 0.365249i $$0.880984\pi$$
$$908$$ 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 $$0.102603\pi$$
−0.948497 + 0.316785i $$0.897397\pi$$
$$938$$ 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 $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ − 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 $$-0.919252\pi$$
0.967996 0.250964i $$-0.0807477\pi$$
$$954$$ 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 $$0.189636\pi$$
−0.827722 + 0.561138i $$0.810364\pi$$
$$968$$ 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 $$-0.898510\pi$$
0.949600 0.313464i $$-0.101490\pi$$
$$978$$ 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 $$0.297590\pi$$
−0.593892 + 0.804544i $$0.702410\pi$$
$$984$$ 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 $$-0.785226\pi$$
0.780874 0.624689i $$-0.214774\pi$$
$$998$$ − 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.799.1 4
5.2 odd 4 2850.2.a.bj.1.2 yes 2
5.3 odd 4 2850.2.a.bc.1.1 2
5.4 even 2 inner 2850.2.d.w.799.4 4
15.2 even 4 8550.2.a.bu.1.2 2
15.8 even 4 8550.2.a.bv.1.1 2

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