# Properties

 Label 5070.2.b.q.1351.1 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.1 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.q.1351.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.82843i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.82843i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.65685i q^{11} -1.00000 q^{12} -2.82843 q^{14} -1.00000i q^{15} +1.00000 q^{16} -0.828427 q^{17} -1.00000i q^{18} -2.82843i q^{19} +1.00000i q^{20} -2.82843i q^{21} +5.65685 q^{22} +8.48528 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.82843i q^{28} -8.82843 q^{29} -1.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} +5.65685i q^{33} +0.828427i q^{34} -2.82843 q^{35} -1.00000 q^{36} -11.6569i q^{37} -2.82843 q^{38} +1.00000 q^{40} +7.65685i q^{41} -2.82843 q^{42} -9.65685 q^{43} -5.65685i q^{44} -1.00000i q^{45} -8.48528i q^{46} -8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000i q^{50} -0.828427 q^{51} +13.3137 q^{53} -1.00000i q^{54} +5.65685 q^{55} +2.82843 q^{56} -2.82843i q^{57} +8.82843i q^{58} -2.34315i q^{59} +1.00000i q^{60} +6.00000 q^{61} -4.00000 q^{62} -2.82843i q^{63} -1.00000 q^{64} +5.65685 q^{66} -5.65685i q^{67} +0.828427 q^{68} +8.48528 q^{69} +2.82843i q^{70} +5.65685i q^{71} +1.00000i q^{72} -14.4853i q^{73} -11.6569 q^{74} -1.00000 q^{75} +2.82843i q^{76} +16.0000 q^{77} +2.34315 q^{79} -1.00000i q^{80} +1.00000 q^{81} +7.65685 q^{82} -6.34315i q^{83} +2.82843i q^{84} +0.828427i q^{85} +9.65685i q^{86} -8.82843 q^{87} -5.65685 q^{88} -15.6569i q^{89} -1.00000 q^{90} -8.48528 q^{92} -4.00000i q^{93} -8.00000 q^{94} -2.82843 q^{95} -1.00000i q^{96} +3.17157i q^{97} +1.00000i q^{98} +5.65685i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} - 4q^{12} + 4q^{16} + 8q^{17} - 4q^{25} + 4q^{27} - 24q^{29} - 4q^{30} - 4q^{36} + 4q^{40} - 16q^{43} + 4q^{48} - 4q^{49} + 8q^{51} + 8q^{53} + 24q^{61} - 16q^{62} - 4q^{64} - 8q^{68} - 24q^{74} - 4q^{75} + 64q^{77} + 32q^{79} + 4q^{81} + 8q^{82} - 24q^{87} - 4q^{90} - 32q^{94} + O(q^{100})$$

## 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
$$7$$ − 2.82843i − 1.06904i −0.845154 0.534522i $$-0.820491\pi$$
0.845154 0.534522i $$-0.179509\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 5.65685i 1.70561i 0.522233 + 0.852803i $$0.325099\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −2.82843 −0.755929
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ − 2.82843i − 0.648886i −0.945905 0.324443i $$-0.894823\pi$$
0.945905 0.324443i $$-0.105177\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 2.82843i − 0.617213i
$$22$$ 5.65685 1.20605
$$23$$ 8.48528 1.76930 0.884652 0.466252i $$-0.154396\pi$$
0.884652 + 0.466252i $$0.154396\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 2.82843i 0.534522i
$$29$$ −8.82843 −1.63940 −0.819699 0.572795i $$-0.805859\pi$$
−0.819699 + 0.572795i $$0.805859\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 4.00000i − 0.718421i −0.933257 0.359211i $$-0.883046\pi$$
0.933257 0.359211i $$-0.116954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 5.65685i 0.984732i
$$34$$ 0.828427i 0.142074i
$$35$$ −2.82843 −0.478091
$$36$$ −1.00000 −0.166667
$$37$$ − 11.6569i − 1.91638i −0.286141 0.958188i $$-0.592373\pi$$
0.286141 0.958188i $$-0.407627\pi$$
$$38$$ −2.82843 −0.458831
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 7.65685i 1.19580i 0.801571 + 0.597900i $$0.203998\pi$$
−0.801571 + 0.597900i $$0.796002\pi$$
$$42$$ −2.82843 −0.436436
$$43$$ −9.65685 −1.47266 −0.736328 0.676625i $$-0.763442\pi$$
−0.736328 + 0.676625i $$0.763442\pi$$
$$44$$ − 5.65685i − 0.852803i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 8.48528i − 1.25109i
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −1.00000 −0.142857
$$50$$ 1.00000i 0.141421i
$$51$$ −0.828427 −0.116003
$$52$$ 0 0
$$53$$ 13.3137 1.82878 0.914389 0.404836i $$-0.132671\pi$$
0.914389 + 0.404836i $$0.132671\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 5.65685 0.762770
$$56$$ 2.82843 0.377964
$$57$$ − 2.82843i − 0.374634i
$$58$$ 8.82843i 1.15923i
$$59$$ − 2.34315i − 0.305052i −0.988299 0.152526i $$-0.951259\pi$$
0.988299 0.152526i $$-0.0487407\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ − 2.82843i − 0.356348i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 5.65685 0.696311
$$67$$ − 5.65685i − 0.691095i −0.938401 0.345547i $$-0.887693\pi$$
0.938401 0.345547i $$-0.112307\pi$$
$$68$$ 0.828427 0.100462
$$69$$ 8.48528 1.02151
$$70$$ 2.82843i 0.338062i
$$71$$ 5.65685i 0.671345i 0.941979 + 0.335673i $$0.108964\pi$$
−0.941979 + 0.335673i $$0.891036\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 14.4853i − 1.69537i −0.530497 0.847687i $$-0.677995\pi$$
0.530497 0.847687i $$-0.322005\pi$$
$$74$$ −11.6569 −1.35508
$$75$$ −1.00000 −0.115470
$$76$$ 2.82843i 0.324443i
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 2.34315 0.263624 0.131812 0.991275i $$-0.457920\pi$$
0.131812 + 0.991275i $$0.457920\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 7.65685 0.845558
$$83$$ − 6.34315i − 0.696251i −0.937448 0.348125i $$-0.886818\pi$$
0.937448 0.348125i $$-0.113182\pi$$
$$84$$ 2.82843i 0.308607i
$$85$$ 0.828427i 0.0898555i
$$86$$ 9.65685i 1.04133i
$$87$$ −8.82843 −0.946507
$$88$$ −5.65685 −0.603023
$$89$$ − 15.6569i − 1.65962i −0.558044 0.829812i $$-0.688448\pi$$
0.558044 0.829812i $$-0.311552\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −8.48528 −0.884652
$$93$$ − 4.00000i − 0.414781i
$$94$$ −8.00000 −0.825137
$$95$$ −2.82843 −0.290191
$$96$$ − 1.00000i − 0.102062i
$$97$$ 3.17157i 0.322024i 0.986952 + 0.161012i $$0.0514759\pi$$
−0.986952 + 0.161012i $$0.948524\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 5.65685i 0.568535i
$$100$$ 1.00000 0.100000
$$101$$ −16.1421 −1.60620 −0.803101 0.595843i $$-0.796818\pi$$
−0.803101 + 0.595843i $$0.796818\pi$$
$$102$$ 0.828427i 0.0820265i
$$103$$ 1.65685 0.163255 0.0816274 0.996663i $$-0.473988\pi$$
0.0816274 + 0.996663i $$0.473988\pi$$
$$104$$ 0 0
$$105$$ −2.82843 −0.276026
$$106$$ − 13.3137i − 1.29314i
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ − 8.82843i − 0.845610i −0.906221 0.422805i $$-0.861046\pi$$
0.906221 0.422805i $$-0.138954\pi$$
$$110$$ − 5.65685i − 0.539360i
$$111$$ − 11.6569i − 1.10642i
$$112$$ − 2.82843i − 0.267261i
$$113$$ 6.48528 0.610084 0.305042 0.952339i $$-0.401330\pi$$
0.305042 + 0.952339i $$0.401330\pi$$
$$114$$ −2.82843 −0.264906
$$115$$ − 8.48528i − 0.791257i
$$116$$ 8.82843 0.819699
$$117$$ 0 0
$$118$$ −2.34315 −0.215704
$$119$$ 2.34315i 0.214796i
$$120$$ 1.00000 0.0912871
$$121$$ −21.0000 −1.90909
$$122$$ − 6.00000i − 0.543214i
$$123$$ 7.65685i 0.690395i
$$124$$ 4.00000i 0.359211i
$$125$$ 1.00000i 0.0894427i
$$126$$ −2.82843 −0.251976
$$127$$ −9.65685 −0.856907 −0.428454 0.903564i $$-0.640941\pi$$
−0.428454 + 0.903564i $$0.640941\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −9.65685 −0.850239
$$130$$ 0 0
$$131$$ −6.14214 −0.536641 −0.268320 0.963330i $$-0.586469\pi$$
−0.268320 + 0.963330i $$0.586469\pi$$
$$132$$ − 5.65685i − 0.492366i
$$133$$ −8.00000 −0.693688
$$134$$ −5.65685 −0.488678
$$135$$ − 1.00000i − 0.0860663i
$$136$$ − 0.828427i − 0.0710370i
$$137$$ − 17.3137i − 1.47921i −0.673041 0.739605i $$-0.735012\pi$$
0.673041 0.739605i $$-0.264988\pi$$
$$138$$ − 8.48528i − 0.722315i
$$139$$ 6.34315 0.538019 0.269009 0.963138i $$-0.413304\pi$$
0.269009 + 0.963138i $$0.413304\pi$$
$$140$$ 2.82843 0.239046
$$141$$ − 8.00000i − 0.673722i
$$142$$ 5.65685 0.474713
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 8.82843i 0.733161i
$$146$$ −14.4853 −1.19881
$$147$$ −1.00000 −0.0824786
$$148$$ 11.6569i 0.958188i
$$149$$ − 3.65685i − 0.299581i −0.988718 0.149791i $$-0.952140\pi$$
0.988718 0.149791i $$-0.0478599\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 12.0000i 0.976546i 0.872691 + 0.488273i $$0.162373\pi$$
−0.872691 + 0.488273i $$0.837627\pi$$
$$152$$ 2.82843 0.229416
$$153$$ −0.828427 −0.0669744
$$154$$ − 16.0000i − 1.28932i
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 5.31371 0.424080 0.212040 0.977261i $$-0.431989\pi$$
0.212040 + 0.977261i $$0.431989\pi$$
$$158$$ − 2.34315i − 0.186411i
$$159$$ 13.3137 1.05585
$$160$$ −1.00000 −0.0790569
$$161$$ − 24.0000i − 1.89146i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 11.3137i 0.886158i 0.896483 + 0.443079i $$0.146114\pi$$
−0.896483 + 0.443079i $$0.853886\pi$$
$$164$$ − 7.65685i − 0.597900i
$$165$$ 5.65685 0.440386
$$166$$ −6.34315 −0.492324
$$167$$ 8.97056i 0.694163i 0.937835 + 0.347081i $$0.112827\pi$$
−0.937835 + 0.347081i $$0.887173\pi$$
$$168$$ 2.82843 0.218218
$$169$$ 0 0
$$170$$ 0.828427 0.0635375
$$171$$ − 2.82843i − 0.216295i
$$172$$ 9.65685 0.736328
$$173$$ 9.31371 0.708108 0.354054 0.935225i $$-0.384803\pi$$
0.354054 + 0.935225i $$0.384803\pi$$
$$174$$ 8.82843i 0.669281i
$$175$$ 2.82843i 0.213809i
$$176$$ 5.65685i 0.426401i
$$177$$ − 2.34315i − 0.176122i
$$178$$ −15.6569 −1.17353
$$179$$ −7.51472 −0.561676 −0.280838 0.959755i $$-0.590612\pi$$
−0.280838 + 0.959755i $$0.590612\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 7.65685 0.569129 0.284565 0.958657i $$-0.408151\pi$$
0.284565 + 0.958657i $$0.408151\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 8.48528i 0.625543i
$$185$$ −11.6569 −0.857029
$$186$$ −4.00000 −0.293294
$$187$$ − 4.68629i − 0.342696i
$$188$$ 8.00000i 0.583460i
$$189$$ − 2.82843i − 0.205738i
$$190$$ 2.82843i 0.205196i
$$191$$ 11.3137 0.818631 0.409316 0.912393i $$-0.365768\pi$$
0.409316 + 0.912393i $$0.365768\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 2.48528i 0.178894i 0.995992 + 0.0894472i $$0.0285100\pi$$
−0.995992 + 0.0894472i $$0.971490\pi$$
$$194$$ 3.17157 0.227706
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 13.3137i 0.948562i 0.880373 + 0.474281i $$0.157292\pi$$
−0.880373 + 0.474281i $$0.842708\pi$$
$$198$$ 5.65685 0.402015
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ − 5.65685i − 0.399004i
$$202$$ 16.1421i 1.13576i
$$203$$ 24.9706i 1.75259i
$$204$$ 0.828427 0.0580015
$$205$$ 7.65685 0.534778
$$206$$ − 1.65685i − 0.115439i
$$207$$ 8.48528 0.589768
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 2.82843i 0.195180i
$$211$$ 0.686292 0.0472463 0.0236231 0.999721i $$-0.492480\pi$$
0.0236231 + 0.999721i $$0.492480\pi$$
$$212$$ −13.3137 −0.914389
$$213$$ 5.65685i 0.387601i
$$214$$ 4.00000i 0.273434i
$$215$$ 9.65685i 0.658592i
$$216$$ 1.00000i 0.0680414i
$$217$$ −11.3137 −0.768025
$$218$$ −8.82843 −0.597937
$$219$$ − 14.4853i − 0.978825i
$$220$$ −5.65685 −0.381385
$$221$$ 0 0
$$222$$ −11.6569 −0.782357
$$223$$ − 10.8284i − 0.725125i −0.931959 0.362563i $$-0.881902\pi$$
0.931959 0.362563i $$-0.118098\pi$$
$$224$$ −2.82843 −0.188982
$$225$$ −1.00000 −0.0666667
$$226$$ − 6.48528i − 0.431394i
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 2.82843i 0.187317i
$$229$$ 4.14214i 0.273720i 0.990590 + 0.136860i $$0.0437011\pi$$
−0.990590 + 0.136860i $$0.956299\pi$$
$$230$$ −8.48528 −0.559503
$$231$$ 16.0000 1.05272
$$232$$ − 8.82843i − 0.579615i
$$233$$ −5.51472 −0.361281 −0.180641 0.983549i $$-0.557817\pi$$
−0.180641 + 0.983549i $$0.557817\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 2.34315i 0.152526i
$$237$$ 2.34315 0.152204
$$238$$ 2.34315 0.151884
$$239$$ − 16.0000i − 1.03495i −0.855697 0.517477i $$-0.826871\pi$$
0.855697 0.517477i $$-0.173129\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ 5.31371i 0.342286i 0.985246 + 0.171143i $$0.0547460\pi$$
−0.985246 + 0.171143i $$0.945254\pi$$
$$242$$ 21.0000i 1.34993i
$$243$$ 1.00000 0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 1.00000i 0.0638877i
$$246$$ 7.65685 0.488183
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ − 6.34315i − 0.401981i
$$250$$ 1.00000 0.0632456
$$251$$ −10.8284 −0.683484 −0.341742 0.939794i $$-0.611017\pi$$
−0.341742 + 0.939794i $$0.611017\pi$$
$$252$$ 2.82843i 0.178174i
$$253$$ 48.0000i 3.01773i
$$254$$ 9.65685i 0.605925i
$$255$$ 0.828427i 0.0518781i
$$256$$ 1.00000 0.0625000
$$257$$ 4.82843 0.301189 0.150595 0.988596i $$-0.451881\pi$$
0.150595 + 0.988596i $$0.451881\pi$$
$$258$$ 9.65685i 0.601209i
$$259$$ −32.9706 −2.04869
$$260$$ 0 0
$$261$$ −8.82843 −0.546466
$$262$$ 6.14214i 0.379462i
$$263$$ 16.4853 1.01653 0.508263 0.861202i $$-0.330288\pi$$
0.508263 + 0.861202i $$0.330288\pi$$
$$264$$ −5.65685 −0.348155
$$265$$ − 13.3137i − 0.817855i
$$266$$ 8.00000i 0.490511i
$$267$$ − 15.6569i − 0.958184i
$$268$$ 5.65685i 0.345547i
$$269$$ −14.4853 −0.883183 −0.441592 0.897216i $$-0.645586\pi$$
−0.441592 + 0.897216i $$0.645586\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 7.31371i 0.444276i 0.975015 + 0.222138i $$0.0713036\pi$$
−0.975015 + 0.222138i $$0.928696\pi$$
$$272$$ −0.828427 −0.0502308
$$273$$ 0 0
$$274$$ −17.3137 −1.04596
$$275$$ − 5.65685i − 0.341121i
$$276$$ −8.48528 −0.510754
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ − 6.34315i − 0.380437i
$$279$$ − 4.00000i − 0.239474i
$$280$$ − 2.82843i − 0.169031i
$$281$$ 8.34315i 0.497710i 0.968541 + 0.248855i $$0.0800543\pi$$
−0.968541 + 0.248855i $$0.919946\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 17.6569 1.04959 0.524796 0.851228i $$-0.324142\pi$$
0.524796 + 0.851228i $$0.324142\pi$$
$$284$$ − 5.65685i − 0.335673i
$$285$$ −2.82843 −0.167542
$$286$$ 0 0
$$287$$ 21.6569 1.27836
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −16.3137 −0.959630
$$290$$ 8.82843 0.518423
$$291$$ 3.17157i 0.185921i
$$292$$ 14.4853i 0.847687i
$$293$$ − 16.6274i − 0.971384i −0.874130 0.485692i $$-0.838568\pi$$
0.874130 0.485692i $$-0.161432\pi$$
$$294$$ 1.00000i 0.0583212i
$$295$$ −2.34315 −0.136423
$$296$$ 11.6569 0.677541
$$297$$ 5.65685i 0.328244i
$$298$$ −3.65685 −0.211836
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 27.3137i 1.57434i
$$302$$ 12.0000 0.690522
$$303$$ −16.1421 −0.927341
$$304$$ − 2.82843i − 0.162221i
$$305$$ − 6.00000i − 0.343559i
$$306$$ 0.828427i 0.0473580i
$$307$$ − 21.6569i − 1.23602i −0.786169 0.618011i $$-0.787939\pi$$
0.786169 0.618011i $$-0.212061\pi$$
$$308$$ −16.0000 −0.911685
$$309$$ 1.65685 0.0942551
$$310$$ 4.00000i 0.227185i
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −30.9706 −1.75056 −0.875280 0.483617i $$-0.839323\pi$$
−0.875280 + 0.483617i $$0.839323\pi$$
$$314$$ − 5.31371i − 0.299870i
$$315$$ −2.82843 −0.159364
$$316$$ −2.34315 −0.131812
$$317$$ − 25.3137i − 1.42176i −0.703314 0.710880i $$-0.748297\pi$$
0.703314 0.710880i $$-0.251703\pi$$
$$318$$ − 13.3137i − 0.746596i
$$319$$ − 49.9411i − 2.79617i
$$320$$ 1.00000i 0.0559017i
$$321$$ −4.00000 −0.223258
$$322$$ −24.0000 −1.33747
$$323$$ 2.34315i 0.130376i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 11.3137 0.626608
$$327$$ − 8.82843i − 0.488213i
$$328$$ −7.65685 −0.422779
$$329$$ −22.6274 −1.24749
$$330$$ − 5.65685i − 0.311400i
$$331$$ 8.48528i 0.466393i 0.972430 + 0.233197i $$0.0749186\pi$$
−0.972430 + 0.233197i $$0.925081\pi$$
$$332$$ 6.34315i 0.348125i
$$333$$ − 11.6569i − 0.638792i
$$334$$ 8.97056 0.490847
$$335$$ −5.65685 −0.309067
$$336$$ − 2.82843i − 0.154303i
$$337$$ −10.9706 −0.597605 −0.298802 0.954315i $$-0.596587\pi$$
−0.298802 + 0.954315i $$0.596587\pi$$
$$338$$ 0 0
$$339$$ 6.48528 0.352232
$$340$$ − 0.828427i − 0.0449278i
$$341$$ 22.6274 1.22534
$$342$$ −2.82843 −0.152944
$$343$$ − 16.9706i − 0.916324i
$$344$$ − 9.65685i − 0.520663i
$$345$$ − 8.48528i − 0.456832i
$$346$$ − 9.31371i − 0.500708i
$$347$$ −9.65685 −0.518407 −0.259204 0.965823i $$-0.583460\pi$$
−0.259204 + 0.965823i $$0.583460\pi$$
$$348$$ 8.82843 0.473253
$$349$$ 12.1421i 0.649954i 0.945722 + 0.324977i $$0.105356\pi$$
−0.945722 + 0.324977i $$0.894644\pi$$
$$350$$ 2.82843 0.151186
$$351$$ 0 0
$$352$$ 5.65685 0.301511
$$353$$ − 5.31371i − 0.282820i −0.989951 0.141410i $$-0.954836\pi$$
0.989951 0.141410i $$-0.0451636\pi$$
$$354$$ −2.34315 −0.124537
$$355$$ 5.65685 0.300235
$$356$$ 15.6569i 0.829812i
$$357$$ 2.34315i 0.124012i
$$358$$ 7.51472i 0.397165i
$$359$$ − 28.2843i − 1.49279i −0.665505 0.746393i $$-0.731784\pi$$
0.665505 0.746393i $$-0.268216\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 11.0000 0.578947
$$362$$ − 7.65685i − 0.402435i
$$363$$ −21.0000 −1.10221
$$364$$ 0 0
$$365$$ −14.4853 −0.758194
$$366$$ − 6.00000i − 0.313625i
$$367$$ 25.6569 1.33928 0.669638 0.742687i $$-0.266449\pi$$
0.669638 + 0.742687i $$0.266449\pi$$
$$368$$ 8.48528 0.442326
$$369$$ 7.65685i 0.398600i
$$370$$ 11.6569i 0.606011i
$$371$$ − 37.6569i − 1.95505i
$$372$$ 4.00000i 0.207390i
$$373$$ −2.68629 −0.139091 −0.0695455 0.997579i $$-0.522155\pi$$
−0.0695455 + 0.997579i $$0.522155\pi$$
$$374$$ −4.68629 −0.242322
$$375$$ 1.00000i 0.0516398i
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ −2.82843 −0.145479
$$379$$ 7.51472i 0.386005i 0.981198 + 0.193003i $$0.0618226\pi$$
−0.981198 + 0.193003i $$0.938177\pi$$
$$380$$ 2.82843 0.145095
$$381$$ −9.65685 −0.494736
$$382$$ − 11.3137i − 0.578860i
$$383$$ 29.6569i 1.51539i 0.652606 + 0.757697i $$0.273676\pi$$
−0.652606 + 0.757697i $$0.726324\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ − 16.0000i − 0.815436i
$$386$$ 2.48528 0.126497
$$387$$ −9.65685 −0.490885
$$388$$ − 3.17157i − 0.161012i
$$389$$ 6.48528 0.328817 0.164408 0.986392i $$-0.447429\pi$$
0.164408 + 0.986392i $$0.447429\pi$$
$$390$$ 0 0
$$391$$ −7.02944 −0.355494
$$392$$ − 1.00000i − 0.0505076i
$$393$$ −6.14214 −0.309830
$$394$$ 13.3137 0.670735
$$395$$ − 2.34315i − 0.117896i
$$396$$ − 5.65685i − 0.284268i
$$397$$ 30.2843i 1.51992i 0.649968 + 0.759962i $$0.274782\pi$$
−0.649968 + 0.759962i $$0.725218\pi$$
$$398$$ 10.3431i 0.518455i
$$399$$ −8.00000 −0.400501
$$400$$ −1.00000 −0.0500000
$$401$$ − 26.9706i − 1.34685i −0.739258 0.673423i $$-0.764823\pi$$
0.739258 0.673423i $$-0.235177\pi$$
$$402$$ −5.65685 −0.282138
$$403$$ 0 0
$$404$$ 16.1421 0.803101
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 24.9706 1.23927
$$407$$ 65.9411 3.26858
$$408$$ − 0.828427i − 0.0410133i
$$409$$ 3.65685i 0.180820i 0.995905 + 0.0904099i $$0.0288177\pi$$
−0.995905 + 0.0904099i $$0.971182\pi$$
$$410$$ − 7.65685i − 0.378145i
$$411$$ − 17.3137i − 0.854022i
$$412$$ −1.65685 −0.0816274
$$413$$ −6.62742 −0.326114
$$414$$ − 8.48528i − 0.417029i
$$415$$ −6.34315 −0.311373
$$416$$ 0 0
$$417$$ 6.34315 0.310625
$$418$$ − 16.0000i − 0.782586i
$$419$$ −10.8284 −0.529003 −0.264502 0.964385i $$-0.585207\pi$$
−0.264502 + 0.964385i $$0.585207\pi$$
$$420$$ 2.82843 0.138013
$$421$$ 24.1421i 1.17662i 0.808637 + 0.588308i $$0.200206\pi$$
−0.808637 + 0.588308i $$0.799794\pi$$
$$422$$ − 0.686292i − 0.0334081i
$$423$$ − 8.00000i − 0.388973i
$$424$$ 13.3137i 0.646571i
$$425$$ 0.828427 0.0401846
$$426$$ 5.65685 0.274075
$$427$$ − 16.9706i − 0.821263i
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ 9.65685 0.465695
$$431$$ 16.0000i 0.770693i 0.922772 + 0.385346i $$0.125918\pi$$
−0.922772 + 0.385346i $$0.874082\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 22.9706 1.10389 0.551947 0.833879i $$-0.313885\pi$$
0.551947 + 0.833879i $$0.313885\pi$$
$$434$$ 11.3137i 0.543075i
$$435$$ 8.82843i 0.423291i
$$436$$ 8.82843i 0.422805i
$$437$$ − 24.0000i − 1.14808i
$$438$$ −14.4853 −0.692134
$$439$$ −22.6274 −1.07995 −0.539974 0.841682i $$-0.681566\pi$$
−0.539974 + 0.841682i $$0.681566\pi$$
$$440$$ 5.65685i 0.269680i
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ 30.3431 1.44165 0.720823 0.693119i $$-0.243764\pi$$
0.720823 + 0.693119i $$0.243764\pi$$
$$444$$ 11.6569i 0.553210i
$$445$$ −15.6569 −0.742206
$$446$$ −10.8284 −0.512741
$$447$$ − 3.65685i − 0.172963i
$$448$$ 2.82843i 0.133631i
$$449$$ 26.2843i 1.24043i 0.784431 + 0.620216i $$0.212955\pi$$
−0.784431 + 0.620216i $$0.787045\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ −43.3137 −2.03956
$$452$$ −6.48528 −0.305042
$$453$$ 12.0000i 0.563809i
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 2.82843 0.132453
$$457$$ − 20.8284i − 0.974313i −0.873315 0.487156i $$-0.838034\pi$$
0.873315 0.487156i $$-0.161966\pi$$
$$458$$ 4.14214 0.193549
$$459$$ −0.828427 −0.0386677
$$460$$ 8.48528i 0.395628i
$$461$$ − 14.0000i − 0.652045i −0.945362 0.326023i $$-0.894291\pi$$
0.945362 0.326023i $$-0.105709\pi$$
$$462$$ − 16.0000i − 0.744387i
$$463$$ − 3.79899i − 0.176554i −0.996096 0.0882770i $$-0.971864\pi$$
0.996096 0.0882770i $$-0.0281361\pi$$
$$464$$ −8.82843 −0.409849
$$465$$ −4.00000 −0.185496
$$466$$ 5.51472i 0.255464i
$$467$$ −7.31371 −0.338438 −0.169219 0.985578i $$-0.554125\pi$$
−0.169219 + 0.985578i $$0.554125\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 8.00000i 0.369012i
$$471$$ 5.31371 0.244843
$$472$$ 2.34315 0.107852
$$473$$ − 54.6274i − 2.51177i
$$474$$ − 2.34315i − 0.107624i
$$475$$ 2.82843i 0.129777i
$$476$$ − 2.34315i − 0.107398i
$$477$$ 13.3137 0.609593
$$478$$ −16.0000 −0.731823
$$479$$ − 11.3137i − 0.516937i −0.966020 0.258468i $$-0.916782\pi$$
0.966020 0.258468i $$-0.0832177\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 5.31371 0.242033
$$483$$ − 24.0000i − 1.09204i
$$484$$ 21.0000 0.954545
$$485$$ 3.17157 0.144014
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 16.4853i − 0.747019i −0.927626 0.373510i $$-0.878154\pi$$
0.927626 0.373510i $$-0.121846\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ 11.3137i 0.511624i
$$490$$ 1.00000 0.0451754
$$491$$ 38.1421 1.72133 0.860665 0.509171i $$-0.170048\pi$$
0.860665 + 0.509171i $$0.170048\pi$$
$$492$$ − 7.65685i − 0.345198i
$$493$$ 7.31371 0.329393
$$494$$ 0 0
$$495$$ 5.65685 0.254257
$$496$$ − 4.00000i − 0.179605i
$$497$$ 16.0000 0.717698
$$498$$ −6.34315 −0.284243
$$499$$ − 0.485281i − 0.0217242i −0.999941 0.0108621i $$-0.996542\pi$$
0.999941 0.0108621i $$-0.00345758\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 8.97056i 0.400775i
$$502$$ 10.8284i 0.483296i
$$503$$ −23.5147 −1.04847 −0.524235 0.851574i $$-0.675649\pi$$
−0.524235 + 0.851574i $$0.675649\pi$$
$$504$$ 2.82843 0.125988
$$505$$ 16.1421i 0.718316i
$$506$$ 48.0000 2.13386
$$507$$ 0 0
$$508$$ 9.65685 0.428454
$$509$$ 37.3137i 1.65390i 0.562275 + 0.826951i $$0.309926\pi$$
−0.562275 + 0.826951i $$0.690074\pi$$
$$510$$ 0.828427 0.0366834
$$511$$ −40.9706 −1.81243
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 2.82843i − 0.124878i
$$514$$ − 4.82843i − 0.212973i
$$515$$ − 1.65685i − 0.0730097i
$$516$$ 9.65685 0.425119
$$517$$ 45.2548 1.99031
$$518$$ 32.9706i 1.44864i
$$519$$ 9.31371 0.408826
$$520$$ 0 0
$$521$$ 26.9706 1.18160 0.590801 0.806817i $$-0.298812\pi$$
0.590801 + 0.806817i $$0.298812\pi$$
$$522$$ 8.82843i 0.386410i
$$523$$ −10.6274 −0.464704 −0.232352 0.972632i $$-0.574642\pi$$
−0.232352 + 0.972632i $$0.574642\pi$$
$$524$$ 6.14214 0.268320
$$525$$ 2.82843i 0.123443i
$$526$$ − 16.4853i − 0.718792i
$$527$$ 3.31371i 0.144347i
$$528$$ 5.65685i 0.246183i
$$529$$ 49.0000 2.13043
$$530$$ −13.3137 −0.578311
$$531$$ − 2.34315i − 0.101684i
$$532$$ 8.00000 0.346844
$$533$$ 0 0
$$534$$ −15.6569 −0.677538
$$535$$ 4.00000i 0.172935i
$$536$$ 5.65685 0.244339
$$537$$ −7.51472 −0.324284
$$538$$ 14.4853i 0.624505i
$$539$$ − 5.65685i − 0.243658i
$$540$$ 1.00000i 0.0430331i
$$541$$ 14.4853i 0.622771i 0.950284 + 0.311385i $$0.100793\pi$$
−0.950284 + 0.311385i $$0.899207\pi$$
$$542$$ 7.31371 0.314151
$$543$$ 7.65685 0.328587
$$544$$ 0.828427i 0.0355185i
$$545$$ −8.82843 −0.378168
$$546$$ 0 0
$$547$$ 0.686292 0.0293437 0.0146719 0.999892i $$-0.495330\pi$$
0.0146719 + 0.999892i $$0.495330\pi$$
$$548$$ 17.3137i 0.739605i
$$549$$ 6.00000 0.256074
$$550$$ −5.65685 −0.241209
$$551$$ 24.9706i 1.06378i
$$552$$ 8.48528i 0.361158i
$$553$$ − 6.62742i − 0.281826i
$$554$$ 26.0000i 1.10463i
$$555$$ −11.6569 −0.494806
$$556$$ −6.34315 −0.269009
$$557$$ 10.6863i 0.452793i 0.974035 + 0.226396i $$0.0726945\pi$$
−0.974035 + 0.226396i $$0.927306\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 0 0
$$560$$ −2.82843 −0.119523
$$561$$ − 4.68629i − 0.197855i
$$562$$ 8.34315 0.351934
$$563$$ 30.3431 1.27881 0.639406 0.768870i $$-0.279181\pi$$
0.639406 + 0.768870i $$0.279181\pi$$
$$564$$ 8.00000i 0.336861i
$$565$$ − 6.48528i − 0.272838i
$$566$$ − 17.6569i − 0.742173i
$$567$$ − 2.82843i − 0.118783i
$$568$$ −5.65685 −0.237356
$$569$$ −31.6569 −1.32712 −0.663562 0.748121i $$-0.730956\pi$$
−0.663562 + 0.748121i $$0.730956\pi$$
$$570$$ 2.82843i 0.118470i
$$571$$ −20.9706 −0.877591 −0.438795 0.898587i $$-0.644595\pi$$
−0.438795 + 0.898587i $$0.644595\pi$$
$$572$$ 0 0
$$573$$ 11.3137 0.472637
$$574$$ − 21.6569i − 0.903940i
$$575$$ −8.48528 −0.353861
$$576$$ −1.00000 −0.0416667
$$577$$ 23.4558i 0.976480i 0.872710 + 0.488240i $$0.162361\pi$$
−0.872710 + 0.488240i $$0.837639\pi$$
$$578$$ 16.3137i 0.678561i
$$579$$ 2.48528i 0.103285i
$$580$$ − 8.82843i − 0.366580i
$$581$$ −17.9411 −0.744323
$$582$$ 3.17157 0.131466
$$583$$ 75.3137i 3.11918i
$$584$$ 14.4853 0.599405
$$585$$ 0 0
$$586$$ −16.6274 −0.686872
$$587$$ − 2.62742i − 0.108445i −0.998529 0.0542226i $$-0.982732\pi$$
0.998529 0.0542226i $$-0.0172680\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −11.3137 −0.466173
$$590$$ 2.34315i 0.0964658i
$$591$$ 13.3137i 0.547653i
$$592$$ − 11.6569i − 0.479094i
$$593$$ − 0.343146i − 0.0140913i −0.999975 0.00704565i $$-0.997757\pi$$
0.999975 0.00704565i $$-0.00224272\pi$$
$$594$$ 5.65685 0.232104
$$595$$ 2.34315 0.0960596
$$596$$ 3.65685i 0.149791i
$$597$$ −10.3431 −0.423317
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 29.3137 1.19573 0.597866 0.801596i $$-0.296016\pi$$
0.597866 + 0.801596i $$0.296016\pi$$
$$602$$ 27.3137 1.11322
$$603$$ − 5.65685i − 0.230365i
$$604$$ − 12.0000i − 0.488273i
$$605$$ 21.0000i 0.853771i
$$606$$ 16.1421i 0.655729i
$$607$$ 28.9706 1.17588 0.587939 0.808905i $$-0.299939\pi$$
0.587939 + 0.808905i $$0.299939\pi$$
$$608$$ −2.82843 −0.114708
$$609$$ 24.9706i 1.01186i
$$610$$ −6.00000 −0.242933
$$611$$ 0 0
$$612$$ 0.828427 0.0334872
$$613$$ − 22.2843i − 0.900053i −0.893015 0.450027i $$-0.851414\pi$$
0.893015 0.450027i $$-0.148586\pi$$
$$614$$ −21.6569 −0.874000
$$615$$ 7.65685 0.308754
$$616$$ 16.0000i 0.644658i
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ − 1.65685i − 0.0666485i
$$619$$ − 34.8284i − 1.39987i −0.714205 0.699936i $$-0.753212\pi$$
0.714205 0.699936i $$-0.246788\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 8.48528 0.340503
$$622$$ − 24.0000i − 0.962312i
$$623$$ −44.2843 −1.77421
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 30.9706i 1.23783i
$$627$$ 16.0000 0.638978
$$628$$ −5.31371 −0.212040
$$629$$ 9.65685i 0.385044i
$$630$$ 2.82843i 0.112687i
$$631$$ − 33.6569i − 1.33986i −0.742425 0.669929i $$-0.766324\pi$$
0.742425 0.669929i $$-0.233676\pi$$
$$632$$ 2.34315i 0.0932053i
$$633$$ 0.686292 0.0272776
$$634$$ −25.3137 −1.00534
$$635$$ 9.65685i 0.383221i
$$636$$ −13.3137 −0.527923
$$637$$ 0 0
$$638$$ −49.9411 −1.97719
$$639$$ 5.65685i 0.223782i
$$640$$ 1.00000 0.0395285
$$641$$ 4.62742 0.182772 0.0913860 0.995816i $$-0.470870\pi$$
0.0913860 + 0.995816i $$0.470870\pi$$
$$642$$ 4.00000i 0.157867i
$$643$$ − 39.5980i − 1.56159i −0.624786 0.780796i $$-0.714814\pi$$
0.624786 0.780796i $$-0.285186\pi$$
$$644$$ 24.0000i 0.945732i
$$645$$ 9.65685i 0.380238i
$$646$$ 2.34315 0.0921898
$$647$$ −8.48528 −0.333591 −0.166795 0.985992i $$-0.553342\pi$$
−0.166795 + 0.985992i $$0.553342\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 13.2548 0.520298
$$650$$ 0 0
$$651$$ −11.3137 −0.443419
$$652$$ − 11.3137i − 0.443079i
$$653$$ −42.2843 −1.65471 −0.827356 0.561678i $$-0.810156\pi$$
−0.827356 + 0.561678i $$0.810156\pi$$
$$654$$ −8.82843 −0.345219
$$655$$ 6.14214i 0.239993i
$$656$$ 7.65685i 0.298950i
$$657$$ − 14.4853i − 0.565125i
$$658$$ 22.6274i 0.882109i
$$659$$ −7.51472 −0.292732 −0.146366 0.989231i $$-0.546758\pi$$
−0.146366 + 0.989231i $$0.546758\pi$$
$$660$$ −5.65685 −0.220193
$$661$$ − 8.14214i − 0.316692i −0.987384 0.158346i $$-0.949384\pi$$
0.987384 0.158346i $$-0.0506162\pi$$
$$662$$ 8.48528 0.329790
$$663$$ 0 0
$$664$$ 6.34315 0.246162
$$665$$ 8.00000i 0.310227i
$$666$$ −11.6569 −0.451694
$$667$$ −74.9117 −2.90059
$$668$$ − 8.97056i − 0.347081i
$$669$$ − 10.8284i − 0.418651i
$$670$$ 5.65685i 0.218543i
$$671$$ 33.9411i 1.31028i
$$672$$ −2.82843 −0.109109
$$673$$ −32.6274 −1.25769 −0.628847 0.777529i $$-0.716473\pi$$
−0.628847 + 0.777529i $$0.716473\pi$$
$$674$$ 10.9706i 0.422570i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 12.3431 0.474386 0.237193 0.971463i $$-0.423773\pi$$
0.237193 + 0.971463i $$0.423773\pi$$
$$678$$ − 6.48528i − 0.249066i
$$679$$ 8.97056 0.344259
$$680$$ −0.828427 −0.0317687
$$681$$ − 4.00000i − 0.153280i
$$682$$ − 22.6274i − 0.866449i
$$683$$ 33.6569i 1.28784i 0.765091 + 0.643922i $$0.222694\pi$$
−0.765091 + 0.643922i $$0.777306\pi$$
$$684$$ 2.82843i 0.108148i
$$685$$ −17.3137 −0.661523
$$686$$ −16.9706 −0.647939
$$687$$ 4.14214i 0.158032i
$$688$$ −9.65685 −0.368164
$$689$$ 0 0
$$690$$ −8.48528 −0.323029
$$691$$ 27.7990i 1.05752i 0.848770 + 0.528762i $$0.177343\pi$$
−0.848770 + 0.528762i $$0.822657\pi$$
$$692$$ −9.31371 −0.354054
$$693$$ 16.0000 0.607790
$$694$$ 9.65685i 0.366569i
$$695$$ − 6.34315i − 0.240609i
$$696$$ − 8.82843i − 0.334641i
$$697$$ − 6.34315i − 0.240264i
$$698$$ 12.1421 0.459587
$$699$$ −5.51472 −0.208586
$$700$$ − 2.82843i − 0.106904i
$$701$$ −0.142136 −0.00536839 −0.00268419 0.999996i $$-0.500854\pi$$
−0.00268419 + 0.999996i $$0.500854\pi$$
$$702$$ 0 0
$$703$$ −32.9706 −1.24351
$$704$$ − 5.65685i − 0.213201i
$$705$$ −8.00000 −0.301297
$$706$$ −5.31371 −0.199984
$$707$$ 45.6569i 1.71710i
$$708$$ 2.34315i 0.0880608i
$$709$$ − 7.17157i − 0.269334i −0.990891 0.134667i $$-0.957004\pi$$
0.990891 0.134667i $$-0.0429965\pi$$
$$710$$ − 5.65685i − 0.212298i
$$711$$ 2.34315 0.0878748
$$712$$ 15.6569 0.586765
$$713$$ − 33.9411i − 1.27111i
$$714$$ 2.34315 0.0876900
$$715$$ 0 0
$$716$$ 7.51472 0.280838
$$717$$ − 16.0000i − 0.597531i
$$718$$ −28.2843 −1.05556
$$719$$ 29.6569 1.10601 0.553007 0.833177i $$-0.313480\pi$$
0.553007 + 0.833177i $$0.313480\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 4.68629i − 0.174527i
$$722$$ − 11.0000i − 0.409378i
$$723$$ 5.31371i 0.197619i
$$724$$ −7.65685 −0.284565
$$725$$ 8.82843 0.327880
$$726$$ 21.0000i 0.779383i
$$727$$ 45.9411 1.70386 0.851931 0.523654i $$-0.175432\pi$$
0.851931 + 0.523654i $$0.175432\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 14.4853i 0.536124i
$$731$$ 8.00000 0.295891
$$732$$ −6.00000 −0.221766
$$733$$ 0.343146i 0.0126744i 0.999980 + 0.00633719i $$0.00201720\pi$$
−0.999980 + 0.00633719i $$0.997983\pi$$
$$734$$ − 25.6569i − 0.947012i
$$735$$ 1.00000i 0.0368856i
$$736$$ − 8.48528i − 0.312772i
$$737$$ 32.0000 1.17874
$$738$$ 7.65685 0.281853
$$739$$ − 14.1421i − 0.520227i −0.965578 0.260113i $$-0.916240\pi$$
0.965578 0.260113i $$-0.0837600\pi$$
$$740$$ 11.6569 0.428514
$$741$$ 0 0
$$742$$ −37.6569 −1.38243
$$743$$ 36.2843i 1.33114i 0.746335 + 0.665570i $$0.231812\pi$$
−0.746335 + 0.665570i $$0.768188\pi$$
$$744$$ 4.00000 0.146647
$$745$$ −3.65685 −0.133977
$$746$$ 2.68629i 0.0983521i
$$747$$ − 6.34315i − 0.232084i
$$748$$ 4.68629i 0.171348i
$$749$$ 11.3137i 0.413394i
$$750$$ 1.00000 0.0365148
$$751$$ −11.3137 −0.412843 −0.206422 0.978463i $$-0.566182\pi$$
−0.206422 + 0.978463i $$0.566182\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ −10.8284 −0.394610
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 2.82843i 0.102869i
$$757$$ 19.9411 0.724773 0.362386 0.932028i $$-0.381962\pi$$
0.362386 + 0.932028i $$0.381962\pi$$
$$758$$ 7.51472 0.272947
$$759$$ 48.0000i 1.74229i
$$760$$ − 2.82843i − 0.102598i
$$761$$ 27.6569i 1.00256i 0.865285 + 0.501280i $$0.167137\pi$$
−0.865285 + 0.501280i $$0.832863\pi$$
$$762$$ 9.65685i 0.349831i
$$763$$ −24.9706 −0.903995
$$764$$ −11.3137 −0.409316
$$765$$ 0.828427i 0.0299518i
$$766$$ 29.6569 1.07155
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 14.0000i 0.504853i 0.967616 + 0.252426i $$0.0812286\pi$$
−0.967616 + 0.252426i $$0.918771\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ 4.82843 0.173892
$$772$$ − 2.48528i − 0.0894472i
$$773$$ 53.3137i 1.91756i 0.284148 + 0.958780i $$0.408289\pi$$
−0.284148 + 0.958780i $$0.591711\pi$$
$$774$$ 9.65685i 0.347108i
$$775$$ 4.00000i 0.143684i
$$776$$ −3.17157 −0.113853
$$777$$ −32.9706 −1.18281
$$778$$ − 6.48528i − 0.232509i
$$779$$ 21.6569 0.775937
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 7.02944i 0.251372i
$$783$$ −8.82843 −0.315502
$$784$$ −1.00000 −0.0357143
$$785$$ − 5.31371i − 0.189654i
$$786$$ 6.14214i 0.219083i
$$787$$ − 24.0000i − 0.855508i −0.903895 0.427754i $$-0.859305\pi$$
0.903895 0.427754i $$-0.140695\pi$$
$$788$$ − 13.3137i − 0.474281i
$$789$$ 16.4853 0.586892
$$790$$ −2.34315 −0.0833654
$$791$$ − 18.3431i − 0.652207i
$$792$$ −5.65685 −0.201008
$$793$$ 0 0
$$794$$ 30.2843 1.07475
$$795$$ − 13.3137i − 0.472189i
$$796$$ 10.3431 0.366603
$$797$$ −16.6274 −0.588973 −0.294487 0.955656i $$-0.595149\pi$$
−0.294487 + 0.955656i $$0.595149\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ 6.62742i 0.234461i
$$800$$ 1.00000i 0.0353553i
$$801$$ − 15.6569i − 0.553208i
$$802$$ −26.9706 −0.952364
$$803$$ 81.9411 2.89164
$$804$$ 5.65685i 0.199502i
$$805$$ −24.0000 −0.845889
$$806$$ 0 0
$$807$$ −14.4853 −0.509906
$$808$$ − 16.1421i − 0.567878i
$$809$$ 13.3137 0.468085 0.234043 0.972226i $$-0.424804\pi$$
0.234043 + 0.972226i $$0.424804\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 1.85786i 0.0652384i 0.999468 + 0.0326192i $$0.0103849\pi$$
−0.999468 + 0.0326192i $$0.989615\pi$$
$$812$$ − 24.9706i − 0.876295i
$$813$$ 7.31371i 0.256503i
$$814$$ − 65.9411i − 2.31124i
$$815$$ 11.3137 0.396302
$$816$$ −0.828427 −0.0290008
$$817$$ 27.3137i 0.955586i
$$818$$ 3.65685 0.127859
$$819$$ 0 0
$$820$$ −7.65685 −0.267389
$$821$$ − 34.2843i − 1.19653i −0.801299 0.598265i $$-0.795857\pi$$
0.801299 0.598265i $$-0.204143\pi$$
$$822$$ −17.3137 −0.603885
$$823$$ 52.9706 1.84644 0.923219 0.384275i $$-0.125548\pi$$
0.923219 + 0.384275i $$0.125548\pi$$
$$824$$ 1.65685i 0.0577193i
$$825$$ − 5.65685i − 0.196946i
$$826$$ 6.62742i 0.230597i
$$827$$ 9.65685i 0.335802i 0.985804 + 0.167901i $$0.0536989\pi$$
−0.985804 + 0.167901i $$0.946301\pi$$
$$828$$ −8.48528 −0.294884
$$829$$ 53.3137 1.85166 0.925831 0.377938i $$-0.123367\pi$$
0.925831 + 0.377938i $$0.123367\pi$$
$$830$$ 6.34315i 0.220174i
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ 0.828427 0.0287033
$$834$$ − 6.34315i − 0.219645i
$$835$$ 8.97056 0.310439
$$836$$ −16.0000 −0.553372
$$837$$ − 4.00000i − 0.138260i
$$838$$ 10.8284i 0.374062i
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ − 2.82843i − 0.0975900i
$$841$$ 48.9411 1.68763
$$842$$ 24.1421 0.831993
$$843$$ 8.34315i 0.287353i
$$844$$ −0.686292 −0.0236231
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ 59.3970i 2.04090i
$$848$$ 13.3137 0.457195
$$849$$ 17.6569 0.605982
$$850$$ − 0.828427i − 0.0284148i
$$851$$ − 98.9117i − 3.39065i
$$852$$ − 5.65685i − 0.193801i
$$853$$ 38.2843i 1.31083i 0.755270 + 0.655414i $$0.227506\pi$$
−0.755270 + 0.655414i $$0.772494\pi$$
$$854$$ −16.9706 −0.580721
$$855$$ −2.82843 −0.0967302
$$856$$ − 4.00000i − 0.136717i
$$857$$ 20.8284 0.711486 0.355743 0.934584i $$-0.384228\pi$$
0.355743 + 0.934584i $$0.384228\pi$$
$$858$$ 0 0
$$859$$ 37.9411 1.29453 0.647267 0.762263i $$-0.275912\pi$$
0.647267 + 0.762263i $$0.275912\pi$$
$$860$$ − 9.65685i − 0.329296i
$$861$$ 21.6569 0.738064
$$862$$ 16.0000 0.544962
$$863$$ 28.2843i 0.962808i 0.876499 + 0.481404i $$0.159873\pi$$
−0.876499 + 0.481404i $$0.840127\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ − 9.31371i − 0.316676i
$$866$$ − 22.9706i − 0.780571i
$$867$$ −16.3137 −0.554043
$$868$$ 11.3137 0.384012
$$869$$ 13.2548i 0.449639i
$$870$$ 8.82843 0.299312
$$871$$ 0 0
$$872$$ 8.82843 0.298968
$$873$$ 3.17157i 0.107341i
$$874$$ −24.0000 −0.811812
$$875$$ 2.82843 0.0956183
$$876$$ 14.4853i 0.489412i
$$877$$ 51.2548i 1.73075i 0.501122 + 0.865376i $$0.332921\pi$$
−0.501122 + 0.865376i $$0.667079\pi$$
$$878$$ 22.6274i 0.763638i
$$879$$ − 16.6274i − 0.560829i
$$880$$ 5.65685 0.190693
$$881$$ 10.2843 0.346486 0.173243 0.984879i $$-0.444575\pi$$
0.173243 + 0.984879i $$0.444575\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ −31.3137 −1.05379 −0.526895 0.849930i $$-0.676644\pi$$
−0.526895 + 0.849930i $$0.676644\pi$$
$$884$$ 0 0
$$885$$ −2.34315 −0.0787640
$$886$$ − 30.3431i − 1.01940i
$$887$$ −40.4853 −1.35936 −0.679681 0.733508i $$-0.737882\pi$$
−0.679681 + 0.733508i $$0.737882\pi$$
$$888$$ 11.6569 0.391178
$$889$$ 27.3137i 0.916072i
$$890$$ 15.6569i 0.524819i
$$891$$ 5.65685i 0.189512i
$$892$$ 10.8284i 0.362563i
$$893$$ −22.6274 −0.757198
$$894$$ −3.65685 −0.122304
$$895$$ 7.51472i 0.251189i
$$896$$ 2.82843 0.0944911
$$897$$ 0 0
$$898$$ 26.2843 0.877117
$$899$$ 35.3137i 1.17778i
$$900$$ 1.00000 0.0333333
$$901$$ −11.0294 −0.367444
$$902$$ 43.3137i 1.44219i
$$903$$ 27.3137i 0.908943i
$$904$$ 6.48528i 0.215697i
$$905$$ − 7.65685i − 0.254522i
$$906$$ 12.0000 0.398673
$$907$$ −8.28427 −0.275075 −0.137537 0.990497i $$-0.543919\pi$$
−0.137537 + 0.990497i $$0.543919\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ −16.1421 −0.535401
$$910$$ 0 0
$$911$$ 24.9706 0.827312 0.413656 0.910433i $$-0.364252\pi$$
0.413656 + 0.910433i $$0.364252\pi$$
$$912$$ − 2.82843i − 0.0936586i
$$913$$ 35.8823 1.18753
$$914$$ −20.8284 −0.688943
$$915$$ − 6.00000i − 0.198354i
$$916$$ − 4.14214i − 0.136860i
$$917$$ 17.3726i 0.573693i
$$918$$ 0.828427i 0.0273422i
$$919$$ 41.9411 1.38351 0.691755 0.722132i $$-0.256838\pi$$
0.691755 + 0.722132i $$0.256838\pi$$
$$920$$ 8.48528 0.279751
$$921$$ − 21.6569i − 0.713618i
$$922$$ −14.0000 −0.461065
$$923$$ 0 0
$$924$$ −16.0000 −0.526361
$$925$$ 11.6569i 0.383275i
$$926$$ −3.79899 −0.124843
$$927$$ 1.65685 0.0544182
$$928$$ 8.82843i 0.289807i
$$929$$ 33.5980i 1.10231i 0.834402 + 0.551157i $$0.185813\pi$$
−0.834402 + 0.551157i $$0.814187\pi$$
$$930$$ 4.00000i 0.131165i
$$931$$ 2.82843i 0.0926980i
$$932$$ 5.51472 0.180641
$$933$$ 24.0000 0.785725
$$934$$ 7.31371i 0.239312i
$$935$$ −4.68629 −0.153258
$$936$$ 0 0
$$937$$ 16.6274 0.543194 0.271597 0.962411i $$-0.412448\pi$$
0.271597 + 0.962411i $$0.412448\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ −30.9706 −1.01069
$$940$$ 8.00000 0.260931
$$941$$ − 54.9706i − 1.79199i −0.444065 0.895995i $$-0.646464\pi$$
0.444065 0.895995i $$-0.353536\pi$$
$$942$$ − 5.31371i − 0.173130i
$$943$$ 64.9706i 2.11573i
$$944$$ − 2.34315i − 0.0762629i
$$945$$ −2.82843 −0.0920087
$$946$$ −54.6274 −1.77609
$$947$$ 30.3431i 0.986020i 0.870024 + 0.493010i $$0.164103\pi$$
−0.870024 + 0.493010i $$0.835897\pi$$
$$948$$ −2.34315 −0.0761018
$$949$$ 0 0
$$950$$ 2.82843 0.0917663
$$951$$ − 25.3137i − 0.820853i
$$952$$ −2.34315 −0.0759418
$$953$$ 27.8579 0.902405 0.451202 0.892422i $$-0.350995\pi$$
0.451202 + 0.892422i $$0.350995\pi$$
$$954$$ − 13.3137i − 0.431047i
$$955$$ − 11.3137i − 0.366103i
$$956$$ 16.0000i 0.517477i
$$957$$ − 49.9411i − 1.61437i
$$958$$ −11.3137 −0.365529
$$959$$ −48.9706 −1.58134
$$960$$ 1.00000i 0.0322749i
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ − 5.31371i − 0.171143i
$$965$$ 2.48528 0.0800040
$$966$$ −24.0000 −0.772187
$$967$$ 7.51472i 0.241657i 0.992673 + 0.120829i $$0.0385551\pi$$
−0.992673 + 0.120829i $$0.961445\pi$$
$$968$$ − 21.0000i − 0.674966i
$$969$$ 2.34315i 0.0752727i
$$970$$ − 3.17157i − 0.101833i
$$971$$ 15.5147 0.497891 0.248946 0.968517i $$-0.419916\pi$$
0.248946 + 0.968517i $$0.419916\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ − 17.9411i − 0.575166i
$$974$$ −16.4853 −0.528222
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 8.34315i 0.266921i 0.991054 + 0.133460i $$0.0426089\pi$$
−0.991054 + 0.133460i $$0.957391\pi$$
$$978$$ 11.3137 0.361773
$$979$$ 88.5685 2.83066
$$980$$ − 1.00000i − 0.0319438i
$$981$$ − 8.82843i − 0.281870i
$$982$$ − 38.1421i − 1.21716i
$$983$$ − 2.34315i − 0.0747347i −0.999302 0.0373674i $$-0.988103\pi$$
0.999302 0.0373674i $$-0.0118972\pi$$
$$984$$ −7.65685 −0.244092
$$985$$ 13.3137 0.424210
$$986$$ − 7.31371i − 0.232916i
$$987$$ −22.6274 −0.720239
$$988$$ 0 0
$$989$$ −81.9411 −2.60558
$$990$$ − 5.65685i − 0.179787i
$$991$$ −42.9117 −1.36313 −0.681567 0.731755i $$-0.738701\pi$$
−0.681567 + 0.731755i $$0.738701\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 8.48528i 0.269272i
$$994$$ − 16.0000i − 0.507489i
$$995$$ 10.3431i 0.327900i
$$996$$ 6.34315i 0.200990i
$$997$$ 61.3137 1.94182 0.970912 0.239435i $$-0.0769623\pi$$
0.970912 + 0.239435i $$0.0769623\pi$$
$$998$$ −0.485281 −0.0153613
$$999$$ − 11.6569i − 0.368807i
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.q.1351.1 4
13.5 odd 4 5070.2.a.bc.1.2 2
13.8 odd 4 390.2.a.h.1.1 2
13.12 even 2 inner 5070.2.b.q.1351.4 4
39.8 even 4 1170.2.a.o.1.1 2
52.47 even 4 3120.2.a.bc.1.2 2
65.8 even 4 1950.2.e.o.1249.2 4
65.34 odd 4 1950.2.a.bd.1.2 2
65.47 even 4 1950.2.e.o.1249.3 4
156.47 odd 4 9360.2.a.ch.1.2 2
195.8 odd 4 5850.2.e.bk.5149.4 4
195.47 odd 4 5850.2.e.bk.5149.1 4
195.164 even 4 5850.2.a.cl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 13.8 odd 4
1170.2.a.o.1.1 2 39.8 even 4
1950.2.a.bd.1.2 2 65.34 odd 4
1950.2.e.o.1249.2 4 65.8 even 4
1950.2.e.o.1249.3 4 65.47 even 4
3120.2.a.bc.1.2 2 52.47 even 4
5070.2.a.bc.1.2 2 13.5 odd 4
5070.2.b.q.1351.1 4 1.1 even 1 trivial
5070.2.b.q.1351.4 4 13.12 even 2 inner
5850.2.a.cl.1.2 2 195.164 even 4
5850.2.e.bk.5149.1 4 195.47 odd 4
5850.2.e.bk.5149.4 4 195.8 odd 4
9360.2.a.ch.1.2 2 156.47 odd 4