# Properties

 Label 5070.2.b.j.1351.1 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ 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 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.j.1351.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} -2.00000i q^{21} +3.00000 q^{22} -3.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +3.00000 q^{29} -1.00000 q^{30} +5.00000i q^{31} -1.00000i q^{32} +3.00000i q^{33} +6.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} +2.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} -2.00000 q^{42} +1.00000 q^{43} -3.00000i q^{44} -1.00000i q^{45} +3.00000i q^{46} +3.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} -1.00000i q^{54} +3.00000 q^{55} +2.00000 q^{56} +2.00000i q^{57} -3.00000i q^{58} +9.00000i q^{59} +1.00000i q^{60} +2.00000 q^{61} +5.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} +6.00000 q^{68} -3.00000 q^{69} +2.00000i q^{70} -12.0000i q^{71} +1.00000i q^{72} -14.0000i q^{73} +7.00000 q^{74} -1.00000 q^{75} -2.00000i q^{76} +6.00000 q^{77} +5.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -6.00000i q^{83} +2.00000i q^{84} +6.00000i q^{85} -1.00000i q^{86} +3.00000 q^{87} -3.00000 q^{88} +18.0000i q^{89} -1.00000 q^{90} +3.00000 q^{92} +5.00000i q^{93} +3.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +14.0000i q^{97} -3.00000i q^{98} +3.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} - 12 q^{17} + 6 q^{22} - 6 q^{23} - 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} + 4 q^{38} + 2 q^{40} - 4 q^{42} + 2 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{51} - 12 q^{53} + 6 q^{55} + 4 q^{56} + 4 q^{61} + 10 q^{62} - 2 q^{64} + 6 q^{66} + 12 q^{68} - 6 q^{69} + 14 q^{74} - 2 q^{75} + 12 q^{77} + 10 q^{79} + 2 q^{81} + 12 q^{82} + 6 q^{87} - 6 q^{88} - 2 q^{90} + 6 q^{92} + 6 q^{94} + 4 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 3.00000i 0.904534i 0.891883 + 0.452267i $$0.149385\pi$$
−0.891883 + 0.452267i $$0.850615\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −2.00000 −0.534522
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 2.00000i − 0.436436i
$$22$$ 3.00000 0.639602
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 2.00000i 0.377964i
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 5.00000i 0.898027i 0.893525 + 0.449013i $$0.148224\pi$$
−0.893525 + 0.449013i $$0.851776\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 3.00000i 0.522233i
$$34$$ 6.00000i 1.02899i
$$35$$ −2.00000 −0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 7.00000i 1.15079i 0.817875 + 0.575396i $$0.195152\pi$$
−0.817875 + 0.575396i $$0.804848\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ − 3.00000i − 0.452267i
$$45$$ − 1.00000i − 0.149071i
$$46$$ 3.00000i 0.442326i
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 3.00000 0.428571
$$50$$ 1.00000i 0.141421i
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 3.00000 0.404520
$$56$$ 2.00000 0.267261
$$57$$ 2.00000i 0.264906i
$$58$$ − 3.00000i − 0.393919i
$$59$$ 9.00000i 1.17170i 0.810419 + 0.585850i $$0.199239\pi$$
−0.810419 + 0.585850i $$0.800761\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 5.00000 0.635001
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −3.00000 −0.361158
$$70$$ 2.00000i 0.239046i
$$71$$ − 12.0000i − 1.42414i −0.702109 0.712069i $$-0.747758\pi$$
0.702109 0.712069i $$-0.252242\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 7.00000 0.813733
$$75$$ −1.00000 −0.115470
$$76$$ − 2.00000i − 0.229416i
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 2.00000i 0.218218i
$$85$$ 6.00000i 0.650791i
$$86$$ − 1.00000i − 0.107833i
$$87$$ 3.00000 0.321634
$$88$$ −3.00000 −0.319801
$$89$$ 18.0000i 1.90800i 0.299813 + 0.953998i $$0.403076\pi$$
−0.299813 + 0.953998i $$0.596924\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 3.00000 0.312772
$$93$$ 5.00000i 0.518476i
$$94$$ 3.00000 0.309426
$$95$$ 2.00000 0.205196
$$96$$ − 1.00000i − 0.102062i
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 3.00000i 0.301511i
$$100$$ 1.00000 0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 6.00000i 0.582772i
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000i 1.34096i 0.741929 + 0.670478i $$0.233911\pi$$
−0.741929 + 0.670478i $$0.766089\pi$$
$$110$$ − 3.00000i − 0.286039i
$$111$$ 7.00000i 0.664411i
$$112$$ − 2.00000i − 0.188982i
$$113$$ 15.0000 1.41108 0.705541 0.708669i $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 3.00000i 0.279751i
$$116$$ −3.00000 −0.278543
$$117$$ 0 0
$$118$$ 9.00000 0.828517
$$119$$ 12.0000i 1.10004i
$$120$$ 1.00000 0.0912871
$$121$$ 2.00000 0.181818
$$122$$ − 2.00000i − 0.181071i
$$123$$ 6.00000i 0.541002i
$$124$$ − 5.00000i − 0.449013i
$$125$$ 1.00000i 0.0894427i
$$126$$ −2.00000 −0.178174
$$127$$ −14.0000 −1.24230 −0.621150 0.783692i $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 9.00000 0.786334 0.393167 0.919467i $$-0.371379\pi$$
0.393167 + 0.919467i $$0.371379\pi$$
$$132$$ − 3.00000i − 0.261116i
$$133$$ 4.00000 0.346844
$$134$$ 8.00000 0.691095
$$135$$ − 1.00000i − 0.0860663i
$$136$$ − 6.00000i − 0.514496i
$$137$$ − 9.00000i − 0.768922i −0.923141 0.384461i $$-0.874387\pi$$
0.923141 0.384461i $$-0.125613\pi$$
$$138$$ 3.00000i 0.255377i
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 3.00000i 0.252646i
$$142$$ −12.0000 −1.00702
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 3.00000i − 0.249136i
$$146$$ −14.0000 −1.15865
$$147$$ 3.00000 0.247436
$$148$$ − 7.00000i − 0.575396i
$$149$$ − 9.00000i − 0.737309i −0.929567 0.368654i $$-0.879819\pi$$
0.929567 0.368654i $$-0.120181\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 8.00000i − 0.651031i −0.945537 0.325515i $$-0.894462\pi$$
0.945537 0.325515i $$-0.105538\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −6.00000 −0.485071
$$154$$ − 6.00000i − 0.483494i
$$155$$ 5.00000 0.401610
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ − 5.00000i − 0.397779i
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ 6.00000i 0.472866i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 13.0000i 1.01824i 0.860696 + 0.509119i $$0.170029\pi$$
−0.860696 + 0.509119i $$0.829971\pi$$
$$164$$ − 6.00000i − 0.468521i
$$165$$ 3.00000 0.233550
$$166$$ −6.00000 −0.465690
$$167$$ 9.00000i 0.696441i 0.937413 + 0.348220i $$0.113214\pi$$
−0.937413 + 0.348220i $$0.886786\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ 6.00000 0.460179
$$171$$ 2.00000i 0.152944i
$$172$$ −1.00000 −0.0762493
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ − 3.00000i − 0.227429i
$$175$$ 2.00000i 0.151186i
$$176$$ 3.00000i 0.226134i
$$177$$ 9.00000i 0.676481i
$$178$$ 18.0000 1.34916
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ − 3.00000i − 0.221163i
$$185$$ 7.00000 0.514650
$$186$$ 5.00000 0.366618
$$187$$ − 18.0000i − 1.31629i
$$188$$ − 3.00000i − 0.218797i
$$189$$ − 2.00000i − 0.145479i
$$190$$ − 2.00000i − 0.145095i
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 24.0000i 1.70993i 0.518686 + 0.854965i $$0.326421\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ 8.00000i 0.564276i
$$202$$ 6.00000i 0.422159i
$$203$$ − 6.00000i − 0.421117i
$$204$$ 6.00000 0.420084
$$205$$ 6.00000 0.419058
$$206$$ 14.0000i 0.975426i
$$207$$ −3.00000 −0.208514
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ 2.00000i 0.138013i
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.00000 0.412082
$$213$$ − 12.0000i − 0.822226i
$$214$$ 6.00000i 0.410152i
$$215$$ − 1.00000i − 0.0681994i
$$216$$ 1.00000i 0.0680414i
$$217$$ 10.0000 0.678844
$$218$$ 14.0000 0.948200
$$219$$ − 14.0000i − 0.946032i
$$220$$ −3.00000 −0.202260
$$221$$ 0 0
$$222$$ 7.00000 0.469809
$$223$$ − 10.0000i − 0.669650i −0.942280 0.334825i $$-0.891323\pi$$
0.942280 0.334825i $$-0.108677\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ − 15.0000i − 0.997785i
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ − 2.00000i − 0.132453i
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 6.00000 0.394771
$$232$$ 3.00000i 0.196960i
$$233$$ −21.0000 −1.37576 −0.687878 0.725826i $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$234$$ 0 0
$$235$$ 3.00000 0.195698
$$236$$ − 9.00000i − 0.585850i
$$237$$ 5.00000 0.324785
$$238$$ 12.0000 0.777844
$$239$$ 24.0000i 1.55243i 0.630468 + 0.776215i $$0.282863\pi$$
−0.630468 + 0.776215i $$0.717137\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 17.0000i − 1.09507i −0.836784 0.547533i $$-0.815567\pi$$
0.836784 0.547533i $$-0.184433\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ − 3.00000i − 0.191663i
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ −5.00000 −0.317500
$$249$$ − 6.00000i − 0.380235i
$$250$$ 1.00000 0.0632456
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 9.00000i − 0.565825i
$$254$$ 14.0000i 0.878438i
$$255$$ 6.00000i 0.375735i
$$256$$ 1.00000 0.0625000
$$257$$ 21.0000 1.30994 0.654972 0.755653i $$-0.272680\pi$$
0.654972 + 0.755653i $$0.272680\pi$$
$$258$$ − 1.00000i − 0.0622573i
$$259$$ 14.0000 0.869918
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ − 9.00000i − 0.556022i
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 6.00000i 0.368577i
$$266$$ − 4.00000i − 0.245256i
$$267$$ 18.0000i 1.10158i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 11.0000i − 0.668202i −0.942537 0.334101i $$-0.891567\pi$$
0.942537 0.334101i $$-0.108433\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ − 3.00000i − 0.180907i
$$276$$ 3.00000 0.180579
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ − 14.0000i − 0.839664i
$$279$$ 5.00000i 0.299342i
$$280$$ − 2.00000i − 0.119523i
$$281$$ − 18.0000i − 1.07379i −0.843649 0.536895i $$-0.819597\pi$$
0.843649 0.536895i $$-0.180403\pi$$
$$282$$ 3.00000 0.178647
$$283$$ 31.0000 1.84276 0.921379 0.388664i $$-0.127063\pi$$
0.921379 + 0.388664i $$0.127063\pi$$
$$284$$ 12.0000i 0.712069i
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 19.0000 1.11765
$$290$$ −3.00000 −0.176166
$$291$$ 14.0000i 0.820695i
$$292$$ 14.0000i 0.819288i
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ 9.00000 0.524000
$$296$$ −7.00000 −0.406867
$$297$$ 3.00000i 0.174078i
$$298$$ −9.00000 −0.521356
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ − 2.00000i − 0.115278i
$$302$$ −8.00000 −0.460348
$$303$$ −6.00000 −0.344691
$$304$$ 2.00000i 0.114708i
$$305$$ − 2.00000i − 0.114520i
$$306$$ 6.00000i 0.342997i
$$307$$ − 8.00000i − 0.456584i −0.973593 0.228292i $$-0.926686\pi$$
0.973593 0.228292i $$-0.0733141\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ −14.0000 −0.796432
$$310$$ − 5.00000i − 0.283981i
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 13.0000i 0.733632i
$$315$$ −2.00000 −0.112687
$$316$$ −5.00000 −0.281272
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 9.00000i 0.503903i
$$320$$ 1.00000i 0.0559017i
$$321$$ −6.00000 −0.334887
$$322$$ 6.00000 0.334367
$$323$$ − 12.0000i − 0.667698i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 13.0000 0.720003
$$327$$ 14.0000i 0.774202i
$$328$$ −6.00000 −0.331295
$$329$$ 6.00000 0.330791
$$330$$ − 3.00000i − 0.165145i
$$331$$ 32.0000i 1.75888i 0.476011 + 0.879440i $$0.342082\pi$$
−0.476011 + 0.879440i $$0.657918\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 7.00000i 0.383598i
$$334$$ 9.00000 0.492458
$$335$$ 8.00000 0.437087
$$336$$ − 2.00000i − 0.109109i
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 15.0000 0.814688
$$340$$ − 6.00000i − 0.325396i
$$341$$ −15.0000 −0.812296
$$342$$ 2.00000 0.108148
$$343$$ − 20.0000i − 1.07990i
$$344$$ 1.00000i 0.0539164i
$$345$$ 3.00000i 0.161515i
$$346$$ 12.0000i 0.645124i
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ −3.00000 −0.160817
$$349$$ − 8.00000i − 0.428230i −0.976808 0.214115i $$-0.931313\pi$$
0.976808 0.214115i $$-0.0686868\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ 9.00000 0.478345
$$355$$ −12.0000 −0.636894
$$356$$ − 18.0000i − 0.953998i
$$357$$ 12.0000i 0.635107i
$$358$$ − 3.00000i − 0.158555i
$$359$$ − 24.0000i − 1.26667i −0.773877 0.633336i $$-0.781685\pi$$
0.773877 0.633336i $$-0.218315\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 15.0000 0.789474
$$362$$ − 16.0000i − 0.840941i
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ −14.0000 −0.732793
$$366$$ − 2.00000i − 0.104542i
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 6.00000i 0.312348i
$$370$$ − 7.00000i − 0.363913i
$$371$$ 12.0000i 0.623009i
$$372$$ − 5.00000i − 0.259238i
$$373$$ −25.0000 −1.29445 −0.647225 0.762299i $$-0.724071\pi$$
−0.647225 + 0.762299i $$0.724071\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 1.00000i 0.0516398i
$$376$$ −3.00000 −0.154713
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 38.0000i 1.95193i 0.217930 + 0.975964i $$0.430070\pi$$
−0.217930 + 0.975964i $$0.569930\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −14.0000 −0.717242
$$382$$ 12.0000i 0.613973i
$$383$$ − 21.0000i − 1.07305i −0.843884 0.536525i $$-0.819737\pi$$
0.843884 0.536525i $$-0.180263\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ − 6.00000i − 0.305788i
$$386$$ 4.00000 0.203595
$$387$$ 1.00000 0.0508329
$$388$$ − 14.0000i − 0.710742i
$$389$$ −3.00000 −0.152106 −0.0760530 0.997104i $$-0.524232\pi$$
−0.0760530 + 0.997104i $$0.524232\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 3.00000i 0.151523i
$$393$$ 9.00000 0.453990
$$394$$ 24.0000 1.20910
$$395$$ − 5.00000i − 0.251577i
$$396$$ − 3.00000i − 0.150756i
$$397$$ 31.0000i 1.55585i 0.628360 + 0.777923i $$0.283727\pi$$
−0.628360 + 0.777923i $$0.716273\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ 12.0000i 0.599251i 0.954057 + 0.299626i $$0.0968618\pi$$
−0.954057 + 0.299626i $$0.903138\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ − 1.00000i − 0.0496904i
$$406$$ −6.00000 −0.297775
$$407$$ −21.0000 −1.04093
$$408$$ − 6.00000i − 0.297044i
$$409$$ − 10.0000i − 0.494468i −0.968956 0.247234i $$-0.920478\pi$$
0.968956 0.247234i $$-0.0795217\pi$$
$$410$$ − 6.00000i − 0.296319i
$$411$$ − 9.00000i − 0.443937i
$$412$$ 14.0000 0.689730
$$413$$ 18.0000 0.885722
$$414$$ 3.00000i 0.147442i
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ 14.0000 0.685583
$$418$$ 6.00000i 0.293470i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ − 16.0000i − 0.779792i −0.920859 0.389896i $$-0.872511\pi$$
0.920859 0.389896i $$-0.127489\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 3.00000i 0.145865i
$$424$$ − 6.00000i − 0.291386i
$$425$$ 6.00000 0.291043
$$426$$ −12.0000 −0.581402
$$427$$ − 4.00000i − 0.193574i
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ −1.00000 −0.0482243
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 40.0000 1.92228 0.961139 0.276066i $$-0.0890309\pi$$
0.961139 + 0.276066i $$0.0890309\pi$$
$$434$$ − 10.0000i − 0.480015i
$$435$$ − 3.00000i − 0.143839i
$$436$$ − 14.0000i − 0.670478i
$$437$$ − 6.00000i − 0.287019i
$$438$$ −14.0000 −0.668946
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 3.00000i 0.143019i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ − 7.00000i − 0.332205i
$$445$$ 18.0000 0.853282
$$446$$ −10.0000 −0.473514
$$447$$ − 9.00000i − 0.425685i
$$448$$ 2.00000i 0.0944911i
$$449$$ − 36.0000i − 1.69895i −0.527633 0.849473i $$-0.676920\pi$$
0.527633 0.849473i $$-0.323080\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ −18.0000 −0.847587
$$452$$ −15.0000 −0.705541
$$453$$ − 8.00000i − 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −6.00000 −0.280056
$$460$$ − 3.00000i − 0.139876i
$$461$$ 15.0000i 0.698620i 0.937007 + 0.349310i $$0.113584\pi$$
−0.937007 + 0.349310i $$0.886416\pi$$
$$462$$ − 6.00000i − 0.279145i
$$463$$ 34.0000i 1.58011i 0.613033 + 0.790057i $$0.289949\pi$$
−0.613033 + 0.790057i $$0.710051\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 5.00000 0.231869
$$466$$ 21.0000i 0.972806i
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ − 3.00000i − 0.138380i
$$471$$ −13.0000 −0.599008
$$472$$ −9.00000 −0.414259
$$473$$ 3.00000i 0.137940i
$$474$$ − 5.00000i − 0.229658i
$$475$$ − 2.00000i − 0.0917663i
$$476$$ − 12.0000i − 0.550019i
$$477$$ −6.00000 −0.274721
$$478$$ 24.0000 1.09773
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −17.0000 −0.774329
$$483$$ 6.00000i 0.273009i
$$484$$ −2.00000 −0.0909091
$$485$$ 14.0000 0.635707
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 13.0000i 0.587880i
$$490$$ −3.00000 −0.135526
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ −18.0000 −0.810679
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 5.00000i 0.224507i
$$497$$ −24.0000 −1.07655
$$498$$ −6.00000 −0.268866
$$499$$ 32.0000i 1.43252i 0.697835 + 0.716258i $$0.254147\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 9.00000i 0.402090i
$$502$$ 15.0000i 0.669483i
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 6.00000i 0.266996i
$$506$$ −9.00000 −0.400099
$$507$$ 0 0
$$508$$ 14.0000 0.621150
$$509$$ 3.00000i 0.132973i 0.997787 + 0.0664863i $$0.0211789\pi$$
−0.997787 + 0.0664863i $$0.978821\pi$$
$$510$$ 6.00000 0.265684
$$511$$ −28.0000 −1.23865
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 2.00000i 0.0883022i
$$514$$ − 21.0000i − 0.926270i
$$515$$ 14.0000i 0.616914i
$$516$$ −1.00000 −0.0440225
$$517$$ −9.00000 −0.395820
$$518$$ − 14.0000i − 0.615125i
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ − 3.00000i − 0.131306i
$$523$$ 11.0000 0.480996 0.240498 0.970650i $$-0.422689\pi$$
0.240498 + 0.970650i $$0.422689\pi$$
$$524$$ −9.00000 −0.393167
$$525$$ 2.00000i 0.0872872i
$$526$$ 15.0000i 0.654031i
$$527$$ − 30.0000i − 1.30682i
$$528$$ 3.00000i 0.130558i
$$529$$ −14.0000 −0.608696
$$530$$ 6.00000 0.260623
$$531$$ 9.00000i 0.390567i
$$532$$ −4.00000 −0.173422
$$533$$ 0 0
$$534$$ 18.0000 0.778936
$$535$$ 6.00000i 0.259403i
$$536$$ −8.00000 −0.345547
$$537$$ 3.00000 0.129460
$$538$$ − 18.0000i − 0.776035i
$$539$$ 9.00000i 0.387657i
$$540$$ 1.00000i 0.0430331i
$$541$$ − 20.0000i − 0.859867i −0.902861 0.429934i $$-0.858537\pi$$
0.902861 0.429934i $$-0.141463\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 16.0000 0.686626
$$544$$ 6.00000i 0.257248i
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 9.00000i 0.384461i
$$549$$ 2.00000 0.0853579
$$550$$ −3.00000 −0.127920
$$551$$ 6.00000i 0.255609i
$$552$$ − 3.00000i − 0.127688i
$$553$$ − 10.0000i − 0.425243i
$$554$$ − 1.00000i − 0.0424859i
$$555$$ 7.00000 0.297133
$$556$$ −14.0000 −0.593732
$$557$$ 6.00000i 0.254228i 0.991888 + 0.127114i $$0.0405714\pi$$
−0.991888 + 0.127114i $$0.959429\pi$$
$$558$$ 5.00000 0.211667
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ − 18.0000i − 0.759961i
$$562$$ −18.0000 −0.759284
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ − 3.00000i − 0.126323i
$$565$$ − 15.0000i − 0.631055i
$$566$$ − 31.0000i − 1.30303i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 12.0000 0.503509
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ − 2.00000i − 0.0837708i
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ − 12.0000i − 0.500870i
$$575$$ 3.00000 0.125109
$$576$$ −1.00000 −0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 4.00000i 0.166234i
$$580$$ 3.00000i 0.124568i
$$581$$ −12.0000 −0.497844
$$582$$ 14.0000 0.580319
$$583$$ − 18.0000i − 0.745484i
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −10.0000 −0.412043
$$590$$ − 9.00000i − 0.370524i
$$591$$ 24.0000i 0.987228i
$$592$$ 7.00000i 0.287698i
$$593$$ − 27.0000i − 1.10876i −0.832265 0.554379i $$-0.812956\pi$$
0.832265 0.554379i $$-0.187044\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 12.0000 0.491952
$$596$$ 9.00000i 0.368654i
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ −2.00000 −0.0815139
$$603$$ 8.00000i 0.325785i
$$604$$ 8.00000i 0.325515i
$$605$$ − 2.00000i − 0.0813116i
$$606$$ 6.00000i 0.243733i
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ − 6.00000i − 0.243132i
$$610$$ −2.00000 −0.0809776
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ − 31.0000i − 1.25208i −0.779792 0.626039i $$-0.784675\pi$$
0.779792 0.626039i $$-0.215325\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 6.00000 0.241943
$$616$$ 6.00000i 0.241747i
$$617$$ 21.0000i 0.845428i 0.906263 + 0.422714i $$0.138923\pi$$
−0.906263 + 0.422714i $$0.861077\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ 46.0000i 1.84890i 0.381308 + 0.924448i $$0.375474\pi$$
−0.381308 + 0.924448i $$0.624526\pi$$
$$620$$ −5.00000 −0.200805
$$621$$ −3.00000 −0.120386
$$622$$ − 12.0000i − 0.481156i
$$623$$ 36.0000 1.44231
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 8.00000i − 0.319744i
$$627$$ −6.00000 −0.239617
$$628$$ 13.0000 0.518756
$$629$$ − 42.0000i − 1.67465i
$$630$$ 2.00000i 0.0796819i
$$631$$ 16.0000i 0.636950i 0.947931 + 0.318475i $$0.103171\pi$$
−0.947931 + 0.318475i $$0.896829\pi$$
$$632$$ 5.00000i 0.198889i
$$633$$ 20.0000 0.794929
$$634$$ 12.0000 0.476581
$$635$$ 14.0000i 0.555573i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 9.00000 0.356313
$$639$$ − 12.0000i − 0.474713i
$$640$$ 1.00000 0.0395285
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 6.00000i 0.236801i
$$643$$ − 16.0000i − 0.630978i −0.948929 0.315489i $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ − 6.00000i − 0.236433i
$$645$$ − 1.00000i − 0.0393750i
$$646$$ −12.0000 −0.472134
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −27.0000 −1.05984
$$650$$ 0 0
$$651$$ 10.0000 0.391931
$$652$$ − 13.0000i − 0.509119i
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 14.0000 0.547443
$$655$$ − 9.00000i − 0.351659i
$$656$$ 6.00000i 0.234261i
$$657$$ − 14.0000i − 0.546192i
$$658$$ − 6.00000i − 0.233904i
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ −3.00000 −0.116775
$$661$$ − 32.0000i − 1.24466i −0.782757 0.622328i $$-0.786187\pi$$
0.782757 0.622328i $$-0.213813\pi$$
$$662$$ 32.0000 1.24372
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ − 4.00000i − 0.155113i
$$666$$ 7.00000 0.271244
$$667$$ −9.00000 −0.348481
$$668$$ − 9.00000i − 0.348220i
$$669$$ − 10.0000i − 0.386622i
$$670$$ − 8.00000i − 0.309067i
$$671$$ 6.00000i 0.231627i
$$672$$ −2.00000 −0.0771517
$$673$$ 4.00000 0.154189 0.0770943 0.997024i $$-0.475436\pi$$
0.0770943 + 0.997024i $$0.475436\pi$$
$$674$$ 14.0000i 0.539260i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ − 15.0000i − 0.576072i
$$679$$ 28.0000 1.07454
$$680$$ −6.00000 −0.230089
$$681$$ 0 0
$$682$$ 15.0000i 0.574380i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ − 2.00000i − 0.0764719i
$$685$$ −9.00000 −0.343872
$$686$$ −20.0000 −0.763604
$$687$$ − 14.0000i − 0.534133i
$$688$$ 1.00000 0.0381246
$$689$$ 0 0
$$690$$ 3.00000 0.114208
$$691$$ − 46.0000i − 1.74992i −0.484193 0.874961i $$-0.660887\pi$$
0.484193 0.874961i $$-0.339113\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 6.00000 0.227921
$$694$$ 30.0000i 1.13878i
$$695$$ − 14.0000i − 0.531050i
$$696$$ 3.00000i 0.113715i
$$697$$ − 36.0000i − 1.36360i
$$698$$ −8.00000 −0.302804
$$699$$ −21.0000 −0.794293
$$700$$ − 2.00000i − 0.0755929i
$$701$$ 21.0000 0.793159 0.396580 0.918000i $$-0.370197\pi$$
0.396580 + 0.918000i $$0.370197\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ − 3.00000i − 0.113067i
$$705$$ 3.00000 0.112987
$$706$$ 30.0000 1.12906
$$707$$ 12.0000i 0.451306i
$$708$$ − 9.00000i − 0.338241i
$$709$$ − 32.0000i − 1.20179i −0.799330 0.600893i $$-0.794812\pi$$
0.799330 0.600893i $$-0.205188\pi$$
$$710$$ 12.0000i 0.450352i
$$711$$ 5.00000 0.187515
$$712$$ −18.0000 −0.674579
$$713$$ − 15.0000i − 0.561754i
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ −3.00000 −0.112115
$$717$$ 24.0000i 0.896296i
$$718$$ −24.0000 −0.895672
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ 28.0000i 1.04277i
$$722$$ − 15.0000i − 0.558242i
$$723$$ − 17.0000i − 0.632237i
$$724$$ −16.0000 −0.594635
$$725$$ −3.00000 −0.111417
$$726$$ − 2.00000i − 0.0742270i
$$727$$ 4.00000 0.148352 0.0741759 0.997245i $$-0.476367\pi$$
0.0741759 + 0.997245i $$0.476367\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 14.0000i 0.518163i
$$731$$ −6.00000 −0.221918
$$732$$ −2.00000 −0.0739221
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ − 20.0000i − 0.738213i
$$735$$ − 3.00000i − 0.110657i
$$736$$ 3.00000i 0.110581i
$$737$$ −24.0000 −0.884051
$$738$$ 6.00000 0.220863
$$739$$ 16.0000i 0.588570i 0.955718 + 0.294285i $$0.0950814\pi$$
−0.955718 + 0.294285i $$0.904919\pi$$
$$740$$ −7.00000 −0.257325
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ − 9.00000i − 0.330178i −0.986279 0.165089i $$-0.947209\pi$$
0.986279 0.165089i $$-0.0527911\pi$$
$$744$$ −5.00000 −0.183309
$$745$$ −9.00000 −0.329734
$$746$$ 25.0000i 0.915315i
$$747$$ − 6.00000i − 0.219529i
$$748$$ 18.0000i 0.658145i
$$749$$ 12.0000i 0.438470i
$$750$$ 1.00000 0.0365148
$$751$$ −41.0000 −1.49611 −0.748056 0.663636i $$-0.769012\pi$$
−0.748056 + 0.663636i $$0.769012\pi$$
$$752$$ 3.00000i 0.109399i
$$753$$ −15.0000 −0.546630
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 2.00000i 0.0727393i
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 38.0000 1.38022
$$759$$ − 9.00000i − 0.326679i
$$760$$ 2.00000i 0.0725476i
$$761$$ 12.0000i 0.435000i 0.976060 + 0.217500i $$0.0697902\pi$$
−0.976060 + 0.217500i $$0.930210\pi$$
$$762$$ 14.0000i 0.507166i
$$763$$ 28.0000 1.01367
$$764$$ 12.0000 0.434145
$$765$$ 6.00000i 0.216930i
$$766$$ −21.0000 −0.758761
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 13.0000i − 0.468792i −0.972141 0.234396i $$-0.924689\pi$$
0.972141 0.234396i $$-0.0753112\pi$$
$$770$$ −6.00000 −0.216225
$$771$$ 21.0000 0.756297
$$772$$ − 4.00000i − 0.143963i
$$773$$ − 48.0000i − 1.72644i −0.504828 0.863220i $$-0.668444\pi$$
0.504828 0.863220i $$-0.331556\pi$$
$$774$$ − 1.00000i − 0.0359443i
$$775$$ − 5.00000i − 0.179605i
$$776$$ −14.0000 −0.502571
$$777$$ 14.0000 0.502247
$$778$$ 3.00000i 0.107555i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ − 18.0000i − 0.643679i
$$783$$ 3.00000 0.107211
$$784$$ 3.00000 0.107143
$$785$$ 13.0000i 0.463990i
$$786$$ − 9.00000i − 0.321019i
$$787$$ − 35.0000i − 1.24762i −0.781578 0.623808i $$-0.785585\pi$$
0.781578 0.623808i $$-0.214415\pi$$
$$788$$ − 24.0000i − 0.854965i
$$789$$ −15.0000 −0.534014
$$790$$ −5.00000 −0.177892
$$791$$ − 30.0000i − 1.06668i
$$792$$ −3.00000 −0.106600
$$793$$ 0 0
$$794$$ 31.0000 1.10015
$$795$$ 6.00000i 0.212798i
$$796$$ 8.00000 0.283552
$$797$$ 24.0000 0.850124 0.425062 0.905164i $$-0.360252\pi$$
0.425062 + 0.905164i $$0.360252\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ − 18.0000i − 0.636794i
$$800$$ 1.00000i 0.0353553i
$$801$$ 18.0000i 0.635999i
$$802$$ 12.0000 0.423735
$$803$$ 42.0000 1.48215
$$804$$ − 8.00000i − 0.282138i
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ − 6.00000i − 0.211079i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 32.0000i 1.12367i 0.827249 + 0.561836i $$0.189905\pi$$
−0.827249 + 0.561836i $$0.810095\pi$$
$$812$$ 6.00000i 0.210559i
$$813$$ − 11.0000i − 0.385787i
$$814$$ 21.0000i 0.736050i
$$815$$ 13.0000 0.455370
$$816$$ −6.00000 −0.210042
$$817$$ 2.00000i 0.0699711i
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ 27.0000i 0.942306i 0.882051 + 0.471153i $$0.156162\pi$$
−0.882051 + 0.471153i $$0.843838\pi$$
$$822$$ −9.00000 −0.313911
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ − 14.0000i − 0.487713i
$$825$$ − 3.00000i − 0.104447i
$$826$$ − 18.0000i − 0.626300i
$$827$$ − 6.00000i − 0.208640i −0.994544 0.104320i $$-0.966733\pi$$
0.994544 0.104320i $$-0.0332667\pi$$
$$828$$ 3.00000 0.104257
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ 6.00000i 0.208263i
$$831$$ 1.00000 0.0346896
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ − 14.0000i − 0.484780i
$$835$$ 9.00000 0.311458
$$836$$ 6.00000 0.207514
$$837$$ 5.00000i 0.172825i
$$838$$ 12.0000i 0.414533i
$$839$$ − 42.0000i − 1.45000i −0.688748 0.725001i $$-0.741839\pi$$
0.688748 0.725001i $$-0.258161\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −20.0000 −0.689655
$$842$$ −16.0000 −0.551396
$$843$$ − 18.0000i − 0.619953i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ − 4.00000i − 0.137442i
$$848$$ −6.00000 −0.206041
$$849$$ 31.0000 1.06392
$$850$$ − 6.00000i − 0.205798i
$$851$$ − 21.0000i − 0.719871i
$$852$$ 12.0000i 0.411113i
$$853$$ 19.0000i 0.650548i 0.945620 + 0.325274i $$0.105456\pi$$
−0.945620 + 0.325274i $$0.894544\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 2.00000 0.0683986
$$856$$ − 6.00000i − 0.205076i
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 0 0
$$859$$ −46.0000 −1.56950 −0.784750 0.619813i $$-0.787209\pi$$
−0.784750 + 0.619813i $$0.787209\pi$$
$$860$$ 1.00000i 0.0340997i
$$861$$ 12.0000 0.408959
$$862$$ 12.0000 0.408722
$$863$$ − 45.0000i − 1.53182i −0.642949 0.765909i $$-0.722289\pi$$
0.642949 0.765909i $$-0.277711\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 12.0000i 0.408012i
$$866$$ − 40.0000i − 1.35926i
$$867$$ 19.0000 0.645274
$$868$$ −10.0000 −0.339422
$$869$$ 15.0000i 0.508840i
$$870$$ −3.00000 −0.101710
$$871$$ 0 0
$$872$$ −14.0000 −0.474100
$$873$$ 14.0000i 0.473828i
$$874$$ −6.00000 −0.202953
$$875$$ 2.00000 0.0676123
$$876$$ 14.0000i 0.473016i
$$877$$ 23.0000i 0.776655i 0.921521 + 0.388327i $$0.126947\pi$$
−0.921521 + 0.388327i $$0.873053\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ 30.0000i 1.01187i
$$880$$ 3.00000 0.101130
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 19.0000 0.639401 0.319700 0.947519i $$-0.396418\pi$$
0.319700 + 0.947519i $$0.396418\pi$$
$$884$$ 0 0
$$885$$ 9.00000 0.302532
$$886$$ 36.0000i 1.20944i
$$887$$ −57.0000 −1.91387 −0.956936 0.290298i $$-0.906246\pi$$
−0.956936 + 0.290298i $$0.906246\pi$$
$$888$$ −7.00000 −0.234905
$$889$$ 28.0000i 0.939090i
$$890$$ − 18.0000i − 0.603361i
$$891$$ 3.00000i 0.100504i
$$892$$ 10.0000i 0.334825i
$$893$$ −6.00000 −0.200782
$$894$$ −9.00000 −0.301005
$$895$$ − 3.00000i − 0.100279i
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ 15.0000i 0.500278i
$$900$$ 1.00000 0.0333333
$$901$$ 36.0000 1.19933
$$902$$ 18.0000i 0.599334i
$$903$$ − 2.00000i − 0.0665558i
$$904$$ 15.0000i 0.498893i
$$905$$ − 16.0000i − 0.531858i
$$906$$ −8.00000 −0.265782
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 2.00000i 0.0662266i
$$913$$ 18.0000 0.595713
$$914$$ 2.00000 0.0661541
$$915$$ − 2.00000i − 0.0661180i
$$916$$ 14.0000i 0.462573i
$$917$$ − 18.0000i − 0.594412i
$$918$$ 6.00000i 0.198030i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −3.00000 −0.0989071
$$921$$ − 8.00000i − 0.263609i
$$922$$ 15.0000 0.493999
$$923$$ 0 0
$$924$$ −6.00000 −0.197386
$$925$$ − 7.00000i − 0.230159i
$$926$$ 34.0000 1.11731
$$927$$ −14.0000 −0.459820
$$928$$ − 3.00000i − 0.0984798i
$$929$$ − 12.0000i − 0.393707i −0.980433 0.196854i $$-0.936928\pi$$
0.980433 0.196854i $$-0.0630724\pi$$
$$930$$ − 5.00000i − 0.163956i
$$931$$ 6.00000i 0.196642i
$$932$$ 21.0000 0.687878
$$933$$ 12.0000 0.392862
$$934$$ 18.0000i 0.588978i
$$935$$ −18.0000 −0.588663
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ − 16.0000i − 0.522419i
$$939$$ 8.00000 0.261070
$$940$$ −3.00000 −0.0978492
$$941$$ 42.0000i 1.36916i 0.728937 + 0.684580i $$0.240015\pi$$
−0.728937 + 0.684580i $$0.759985\pi$$
$$942$$ 13.0000i 0.423563i
$$943$$ − 18.0000i − 0.586161i
$$944$$ 9.00000i 0.292925i
$$945$$ −2.00000 −0.0650600
$$946$$ 3.00000 0.0975384
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ −5.00000 −0.162392
$$949$$ 0 0
$$950$$ −2.00000 −0.0648886
$$951$$ 12.0000i 0.389127i
$$952$$ −12.0000 −0.388922
$$953$$ −9.00000 −0.291539 −0.145769 0.989319i $$-0.546566\pi$$
−0.145769 + 0.989319i $$0.546566\pi$$
$$954$$ 6.00000i 0.194257i
$$955$$ 12.0000i 0.388311i
$$956$$ − 24.0000i − 0.776215i
$$957$$ 9.00000i 0.290929i
$$958$$ 0 0
$$959$$ −18.0000 −0.581250
$$960$$ 1.00000i 0.0322749i
$$961$$ 6.00000 0.193548
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 17.0000i 0.547533i
$$965$$ 4.00000 0.128765
$$966$$ 6.00000 0.193047
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ − 12.0000i − 0.385496i
$$970$$ − 14.0000i − 0.449513i
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ − 28.0000i − 0.897639i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 57.0000i − 1.82359i −0.410644 0.911796i $$-0.634696\pi$$
0.410644 0.911796i $$-0.365304\pi$$
$$978$$ 13.0000 0.415694
$$979$$ −54.0000 −1.72585
$$980$$ 3.00000i 0.0958315i
$$981$$ 14.0000i 0.446986i
$$982$$ 0 0
$$983$$ − 9.00000i − 0.287055i −0.989646 0.143528i $$-0.954155\pi$$
0.989646 0.143528i $$-0.0458446\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 24.0000 0.764704
$$986$$ 18.0000i 0.573237i
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ −3.00000 −0.0953945
$$990$$ − 3.00000i − 0.0953463i
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 5.00000 0.158750
$$993$$ 32.0000i 1.01549i
$$994$$ 24.0000i 0.761234i
$$995$$ 8.00000i 0.253617i
$$996$$ 6.00000i 0.190117i
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 7.00000i 0.221470i
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 5070.2.b.j.1351.1 2
13.2 odd 12 390.2.i.c.61.1 2
13.5 odd 4 5070.2.a.j.1.1 1
13.6 odd 12 390.2.i.c.211.1 yes 2
13.8 odd 4 5070.2.a.v.1.1 1
13.12 even 2 inner 5070.2.b.j.1351.2 2
39.2 even 12 1170.2.i.d.451.1 2
39.32 even 12 1170.2.i.d.991.1 2
65.2 even 12 1950.2.z.k.1699.2 4
65.19 odd 12 1950.2.i.n.601.1 2
65.28 even 12 1950.2.z.k.1699.1 4
65.32 even 12 1950.2.z.k.1849.1 4
65.54 odd 12 1950.2.i.n.451.1 2
65.58 even 12 1950.2.z.k.1849.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.c.61.1 2 13.2 odd 12
390.2.i.c.211.1 yes 2 13.6 odd 12
1170.2.i.d.451.1 2 39.2 even 12
1170.2.i.d.991.1 2 39.32 even 12
1950.2.i.n.451.1 2 65.54 odd 12
1950.2.i.n.601.1 2 65.19 odd 12
1950.2.z.k.1699.1 4 65.28 even 12
1950.2.z.k.1699.2 4 65.2 even 12
1950.2.z.k.1849.1 4 65.32 even 12
1950.2.z.k.1849.2 4 65.58 even 12
5070.2.a.j.1.1 1 13.5 odd 4
5070.2.a.v.1.1 1 13.8 odd 4
5070.2.b.j.1351.1 2 1.1 even 1 trivial
5070.2.b.j.1351.2 2 13.12 even 2 inner