# Properties

 Label 5070.2.b.z.1351.5 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## 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: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.5 Root $$0.445042i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.z.1351.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.35690i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.35690i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.69202i q^{11} -1.00000 q^{12} +3.35690 q^{14} -1.00000i q^{15} +1.00000 q^{16} -0.939001 q^{17} +1.00000i q^{18} -4.85086i q^{19} +1.00000i q^{20} -3.35690i q^{21} +1.69202 q^{22} -4.04892 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +3.35690i q^{28} -8.12498 q^{29} +1.00000 q^{30} +4.08815i q^{31} +1.00000i q^{32} -1.69202i q^{33} -0.939001i q^{34} -3.35690 q^{35} -1.00000 q^{36} -11.6310i q^{37} +4.85086 q^{38} -1.00000 q^{40} +3.86294i q^{41} +3.35690 q^{42} -4.02177 q^{43} +1.69202i q^{44} -1.00000i q^{45} -4.04892i q^{46} +1.27413i q^{47} +1.00000 q^{48} -4.26875 q^{49} -1.00000i q^{50} -0.939001 q^{51} +5.74094 q^{53} +1.00000i q^{54} -1.69202 q^{55} -3.35690 q^{56} -4.85086i q^{57} -8.12498i q^{58} -0.417895i q^{59} +1.00000i q^{60} +0.198062 q^{61} -4.08815 q^{62} -3.35690i q^{63} -1.00000 q^{64} +1.69202 q^{66} +8.93900i q^{67} +0.939001 q^{68} -4.04892 q^{69} -3.35690i q^{70} -5.15883i q^{71} -1.00000i q^{72} +11.5308i q^{73} +11.6310 q^{74} -1.00000 q^{75} +4.85086i q^{76} -5.67994 q^{77} -4.94869 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.86294 q^{82} +13.1739i q^{83} +3.35690i q^{84} +0.939001i q^{85} -4.02177i q^{86} -8.12498 q^{87} -1.69202 q^{88} +9.47650i q^{89} +1.00000 q^{90} +4.04892 q^{92} +4.08815i q^{93} -1.27413 q^{94} -4.85086 q^{95} +1.00000i q^{96} +15.4547i q^{97} -4.26875i q^{98} -1.69202i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} + 6q^{10} - 6q^{12} + 12q^{14} + 6q^{16} + 14q^{17} - 6q^{23} - 6q^{25} + 6q^{27} + 6q^{30} - 12q^{35} - 6q^{36} + 2q^{38} - 6q^{40} + 12q^{42} - 18q^{43} + 6q^{48} - 10q^{49} + 14q^{51} + 6q^{53} - 12q^{56} + 10q^{61} - 32q^{62} - 6q^{64} - 14q^{68} - 6q^{69} + 40q^{74} - 6q^{75} + 14q^{77} + 34q^{79} + 6q^{81} - 34q^{82} + 6q^{90} + 6q^{92} + 14q^{94} - 2q^{95} + O(q^{100})$$

## 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
$$7$$ − 3.35690i − 1.26879i −0.773010 0.634394i $$-0.781250\pi$$
0.773010 0.634394i $$-0.218750\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ − 1.69202i − 0.510164i −0.966919 0.255082i $$-0.917898\pi$$
0.966919 0.255082i $$-0.0821024\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 3.35690 0.897168
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −0.939001 −0.227741 −0.113871 0.993496i $$-0.536325\pi$$
−0.113871 + 0.993496i $$0.536325\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ − 4.85086i − 1.11286i −0.830894 0.556431i $$-0.812170\pi$$
0.830894 0.556431i $$-0.187830\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 3.35690i − 0.732535i
$$22$$ 1.69202 0.360740
$$23$$ −4.04892 −0.844258 −0.422129 0.906536i $$-0.638717\pi$$
−0.422129 + 0.906536i $$0.638717\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 3.35690i 0.634394i
$$29$$ −8.12498 −1.50877 −0.754386 0.656432i $$-0.772065\pi$$
−0.754386 + 0.656432i $$0.772065\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 4.08815i 0.734253i 0.930171 + 0.367126i $$0.119658\pi$$
−0.930171 + 0.367126i $$0.880342\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 1.69202i − 0.294543i
$$34$$ − 0.939001i − 0.161037i
$$35$$ −3.35690 −0.567419
$$36$$ −1.00000 −0.166667
$$37$$ − 11.6310i − 1.91213i −0.293157 0.956064i $$-0.594706\pi$$
0.293157 0.956064i $$-0.405294\pi$$
$$38$$ 4.85086 0.786913
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 3.86294i 0.603289i 0.953420 + 0.301645i $$0.0975356\pi$$
−0.953420 + 0.301645i $$0.902464\pi$$
$$42$$ 3.35690 0.517980
$$43$$ −4.02177 −0.613314 −0.306657 0.951820i $$-0.599210\pi$$
−0.306657 + 0.951820i $$0.599210\pi$$
$$44$$ 1.69202i 0.255082i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 4.04892i − 0.596980i
$$47$$ 1.27413i 0.185850i 0.995673 + 0.0929252i $$0.0296218\pi$$
−0.995673 + 0.0929252i $$0.970378\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −4.26875 −0.609821
$$50$$ − 1.00000i − 0.141421i
$$51$$ −0.939001 −0.131486
$$52$$ 0 0
$$53$$ 5.74094 0.788579 0.394289 0.918986i $$-0.370991\pi$$
0.394289 + 0.918986i $$0.370991\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ −1.69202 −0.228152
$$56$$ −3.35690 −0.448584
$$57$$ − 4.85086i − 0.642511i
$$58$$ − 8.12498i − 1.06686i
$$59$$ − 0.417895i − 0.0544053i −0.999630 0.0272026i $$-0.991340\pi$$
0.999630 0.0272026i $$-0.00865994\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 0.198062 0.0253593 0.0126796 0.999920i $$-0.495964\pi$$
0.0126796 + 0.999920i $$0.495964\pi$$
$$62$$ −4.08815 −0.519195
$$63$$ − 3.35690i − 0.422929i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 1.69202 0.208273
$$67$$ 8.93900i 1.09207i 0.837761 + 0.546036i $$0.183864\pi$$
−0.837761 + 0.546036i $$0.816136\pi$$
$$68$$ 0.939001 0.113871
$$69$$ −4.04892 −0.487432
$$70$$ − 3.35690i − 0.401226i
$$71$$ − 5.15883i − 0.612241i −0.951993 0.306120i $$-0.900969\pi$$
0.951993 0.306120i $$-0.0990310\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 11.5308i 1.34958i 0.738011 + 0.674789i $$0.235765\pi$$
−0.738011 + 0.674789i $$0.764235\pi$$
$$74$$ 11.6310 1.35208
$$75$$ −1.00000 −0.115470
$$76$$ 4.85086i 0.556431i
$$77$$ −5.67994 −0.647289
$$78$$ 0 0
$$79$$ −4.94869 −0.556771 −0.278386 0.960469i $$-0.589799\pi$$
−0.278386 + 0.960469i $$0.589799\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −3.86294 −0.426590
$$83$$ 13.1739i 1.44602i 0.690836 + 0.723012i $$0.257243\pi$$
−0.690836 + 0.723012i $$0.742757\pi$$
$$84$$ 3.35690i 0.366267i
$$85$$ 0.939001i 0.101849i
$$86$$ − 4.02177i − 0.433679i
$$87$$ −8.12498 −0.871089
$$88$$ −1.69202 −0.180370
$$89$$ 9.47650i 1.00451i 0.864720 + 0.502254i $$0.167496\pi$$
−0.864720 + 0.502254i $$0.832504\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 4.04892 0.422129
$$93$$ 4.08815i 0.423921i
$$94$$ −1.27413 −0.131416
$$95$$ −4.85086 −0.497687
$$96$$ 1.00000i 0.102062i
$$97$$ 15.4547i 1.56919i 0.620009 + 0.784595i $$0.287129\pi$$
−0.620009 + 0.784595i $$0.712871\pi$$
$$98$$ − 4.26875i − 0.431209i
$$99$$ − 1.69202i − 0.170055i
$$100$$ 1.00000 0.100000
$$101$$ −5.38404 −0.535732 −0.267866 0.963456i $$-0.586318\pi$$
−0.267866 + 0.963456i $$0.586318\pi$$
$$102$$ − 0.939001i − 0.0929750i
$$103$$ −4.08575 −0.402581 −0.201291 0.979532i $$-0.564514\pi$$
−0.201291 + 0.979532i $$0.564514\pi$$
$$104$$ 0 0
$$105$$ −3.35690 −0.327599
$$106$$ 5.74094i 0.557609i
$$107$$ −16.1196 −1.55834 −0.779171 0.626812i $$-0.784359\pi$$
−0.779171 + 0.626812i $$0.784359\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ − 14.2325i − 1.36323i −0.731713 0.681613i $$-0.761279\pi$$
0.731713 0.681613i $$-0.238721\pi$$
$$110$$ − 1.69202i − 0.161328i
$$111$$ − 11.6310i − 1.10397i
$$112$$ − 3.35690i − 0.317197i
$$113$$ −13.3274 −1.25373 −0.626866 0.779127i $$-0.715663\pi$$
−0.626866 + 0.779127i $$0.715663\pi$$
$$114$$ 4.85086 0.454324
$$115$$ 4.04892i 0.377563i
$$116$$ 8.12498 0.754386
$$117$$ 0 0
$$118$$ 0.417895 0.0384703
$$119$$ 3.15213i 0.288955i
$$120$$ −1.00000 −0.0912871
$$121$$ 8.13706 0.739733
$$122$$ 0.198062i 0.0179317i
$$123$$ 3.86294i 0.348309i
$$124$$ − 4.08815i − 0.367126i
$$125$$ 1.00000i 0.0894427i
$$126$$ 3.35690 0.299056
$$127$$ 12.8509 1.14033 0.570164 0.821531i $$-0.306879\pi$$
0.570164 + 0.821531i $$0.306879\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.02177 −0.354097
$$130$$ 0 0
$$131$$ −0.570024 −0.0498032 −0.0249016 0.999690i $$-0.507927\pi$$
−0.0249016 + 0.999690i $$0.507927\pi$$
$$132$$ 1.69202i 0.147272i
$$133$$ −16.2838 −1.41199
$$134$$ −8.93900 −0.772212
$$135$$ − 1.00000i − 0.0860663i
$$136$$ 0.939001i 0.0805187i
$$137$$ − 21.5405i − 1.84033i −0.391533 0.920164i $$-0.628055\pi$$
0.391533 0.920164i $$-0.371945\pi$$
$$138$$ − 4.04892i − 0.344667i
$$139$$ 7.09783 0.602030 0.301015 0.953619i $$-0.402674\pi$$
0.301015 + 0.953619i $$0.402674\pi$$
$$140$$ 3.35690 0.283709
$$141$$ 1.27413i 0.107301i
$$142$$ 5.15883 0.432920
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 8.12498i 0.674743i
$$146$$ −11.5308 −0.954295
$$147$$ −4.26875 −0.352081
$$148$$ 11.6310i 0.956064i
$$149$$ − 1.14914i − 0.0941416i −0.998892 0.0470708i $$-0.985011\pi$$
0.998892 0.0470708i $$-0.0149886\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ − 0.0217703i − 0.00177164i −1.00000 0.000885819i $$-0.999718\pi$$
1.00000 0.000885819i $$-0.000281965\pi$$
$$152$$ −4.85086 −0.393456
$$153$$ −0.939001 −0.0759137
$$154$$ − 5.67994i − 0.457703i
$$155$$ 4.08815 0.328368
$$156$$ 0 0
$$157$$ −10.1588 −0.810763 −0.405382 0.914148i $$-0.632861\pi$$
−0.405382 + 0.914148i $$0.632861\pi$$
$$158$$ − 4.94869i − 0.393697i
$$159$$ 5.74094 0.455286
$$160$$ 1.00000 0.0790569
$$161$$ 13.5918i 1.07118i
$$162$$ 1.00000i 0.0785674i
$$163$$ − 19.1511i − 1.50003i −0.661422 0.750014i $$-0.730047\pi$$
0.661422 0.750014i $$-0.269953\pi$$
$$164$$ − 3.86294i − 0.301645i
$$165$$ −1.69202 −0.131724
$$166$$ −13.1739 −1.02249
$$167$$ 15.2174i 1.17756i 0.808293 + 0.588780i $$0.200392\pi$$
−0.808293 + 0.588780i $$0.799608\pi$$
$$168$$ −3.35690 −0.258990
$$169$$ 0 0
$$170$$ −0.939001 −0.0720181
$$171$$ − 4.85086i − 0.370954i
$$172$$ 4.02177 0.306657
$$173$$ −21.8213 −1.65904 −0.829522 0.558474i $$-0.811387\pi$$
−0.829522 + 0.558474i $$0.811387\pi$$
$$174$$ − 8.12498i − 0.615953i
$$175$$ 3.35690i 0.253757i
$$176$$ − 1.69202i − 0.127541i
$$177$$ − 0.417895i − 0.0314109i
$$178$$ −9.47650 −0.710294
$$179$$ 20.2513 1.51365 0.756826 0.653616i $$-0.226749\pi$$
0.756826 + 0.653616i $$0.226749\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −14.4969 −1.07755 −0.538775 0.842450i $$-0.681113\pi$$
−0.538775 + 0.842450i $$0.681113\pi$$
$$182$$ 0 0
$$183$$ 0.198062 0.0146412
$$184$$ 4.04892i 0.298490i
$$185$$ −11.6310 −0.855130
$$186$$ −4.08815 −0.299757
$$187$$ 1.58881i 0.116185i
$$188$$ − 1.27413i − 0.0929252i
$$189$$ − 3.35690i − 0.244178i
$$190$$ − 4.85086i − 0.351918i
$$191$$ −10.1806 −0.736643 −0.368321 0.929699i $$-0.620067\pi$$
−0.368321 + 0.929699i $$0.620067\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 27.6407i − 1.98962i −0.101741 0.994811i $$-0.532441\pi$$
0.101741 0.994811i $$-0.467559\pi$$
$$194$$ −15.4547 −1.10958
$$195$$ 0 0
$$196$$ 4.26875 0.304911
$$197$$ − 13.0261i − 0.928070i −0.885817 0.464035i $$-0.846401\pi$$
0.885817 0.464035i $$-0.153599\pi$$
$$198$$ 1.69202 0.120247
$$199$$ −6.67563 −0.473223 −0.236611 0.971604i $$-0.576037\pi$$
−0.236611 + 0.971604i $$0.576037\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 8.93900i 0.630509i
$$202$$ − 5.38404i − 0.378820i
$$203$$ 27.2747i 1.91431i
$$204$$ 0.939001 0.0657432
$$205$$ 3.86294 0.269799
$$206$$ − 4.08575i − 0.284668i
$$207$$ −4.04892 −0.281419
$$208$$ 0 0
$$209$$ −8.20775 −0.567742
$$210$$ − 3.35690i − 0.231648i
$$211$$ −11.3056 −0.778309 −0.389154 0.921173i $$-0.627233\pi$$
−0.389154 + 0.921173i $$0.627233\pi$$
$$212$$ −5.74094 −0.394289
$$213$$ − 5.15883i − 0.353477i
$$214$$ − 16.1196i − 1.10191i
$$215$$ 4.02177i 0.274282i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 13.7235 0.931611
$$218$$ 14.2325 0.963947
$$219$$ 11.5308i 0.779179i
$$220$$ 1.69202 0.114076
$$221$$ 0 0
$$222$$ 11.6310 0.780623
$$223$$ 18.0911i 1.21147i 0.795666 + 0.605736i $$0.207121\pi$$
−0.795666 + 0.605736i $$0.792879\pi$$
$$224$$ 3.35690 0.224292
$$225$$ −1.00000 −0.0666667
$$226$$ − 13.3274i − 0.886523i
$$227$$ − 9.37196i − 0.622039i −0.950404 0.311019i $$-0.899330\pi$$
0.950404 0.311019i $$-0.100670\pi$$
$$228$$ 4.85086i 0.321256i
$$229$$ − 21.2664i − 1.40532i −0.711526 0.702660i $$-0.751995\pi$$
0.711526 0.702660i $$-0.248005\pi$$
$$230$$ −4.04892 −0.266978
$$231$$ −5.67994 −0.373713
$$232$$ 8.12498i 0.533431i
$$233$$ −2.03146 −0.133085 −0.0665427 0.997784i $$-0.521197\pi$$
−0.0665427 + 0.997784i $$0.521197\pi$$
$$234$$ 0 0
$$235$$ 1.27413 0.0831149
$$236$$ 0.417895i 0.0272026i
$$237$$ −4.94869 −0.321452
$$238$$ −3.15213 −0.204322
$$239$$ − 11.9903i − 0.775589i −0.921746 0.387794i $$-0.873237\pi$$
0.921746 0.387794i $$-0.126763\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 21.7627i − 1.40186i −0.713230 0.700930i $$-0.752769\pi$$
0.713230 0.700930i $$-0.247231\pi$$
$$242$$ 8.13706i 0.523070i
$$243$$ 1.00000 0.0641500
$$244$$ −0.198062 −0.0126796
$$245$$ 4.26875i 0.272720i
$$246$$ −3.86294 −0.246292
$$247$$ 0 0
$$248$$ 4.08815 0.259598
$$249$$ 13.1739i 0.834862i
$$250$$ −1.00000 −0.0632456
$$251$$ −13.7506 −0.867932 −0.433966 0.900929i $$-0.642886\pi$$
−0.433966 + 0.900929i $$0.642886\pi$$
$$252$$ 3.35690i 0.211465i
$$253$$ 6.85086i 0.430710i
$$254$$ 12.8509i 0.806334i
$$255$$ 0.939001i 0.0588025i
$$256$$ 1.00000 0.0625000
$$257$$ 24.1782 1.50820 0.754098 0.656762i $$-0.228075\pi$$
0.754098 + 0.656762i $$0.228075\pi$$
$$258$$ − 4.02177i − 0.250384i
$$259$$ −39.0441 −2.42608
$$260$$ 0 0
$$261$$ −8.12498 −0.502924
$$262$$ − 0.570024i − 0.0352162i
$$263$$ 0.960771 0.0592437 0.0296218 0.999561i $$-0.490570\pi$$
0.0296218 + 0.999561i $$0.490570\pi$$
$$264$$ −1.69202 −0.104137
$$265$$ − 5.74094i − 0.352663i
$$266$$ − 16.2838i − 0.998425i
$$267$$ 9.47650i 0.579952i
$$268$$ − 8.93900i − 0.546036i
$$269$$ 10.0761 0.614348 0.307174 0.951653i $$-0.400617\pi$$
0.307174 + 0.951653i $$0.400617\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 14.9584i 0.908657i 0.890834 + 0.454328i $$0.150121\pi$$
−0.890834 + 0.454328i $$0.849879\pi$$
$$272$$ −0.939001 −0.0569353
$$273$$ 0 0
$$274$$ 21.5405 1.30131
$$275$$ 1.69202i 0.102033i
$$276$$ 4.04892 0.243716
$$277$$ 13.0465 0.783890 0.391945 0.919989i $$-0.371802\pi$$
0.391945 + 0.919989i $$0.371802\pi$$
$$278$$ 7.09783i 0.425700i
$$279$$ 4.08815i 0.244751i
$$280$$ 3.35690i 0.200613i
$$281$$ − 16.2881i − 0.971668i −0.874051 0.485834i $$-0.838516\pi$$
0.874051 0.485834i $$-0.161484\pi$$
$$282$$ −1.27413 −0.0758731
$$283$$ 12.3773 0.735756 0.367878 0.929874i $$-0.380084\pi$$
0.367878 + 0.929874i $$0.380084\pi$$
$$284$$ 5.15883i 0.306120i
$$285$$ −4.85086 −0.287340
$$286$$ 0 0
$$287$$ 12.9675 0.765446
$$288$$ 1.00000i 0.0589256i
$$289$$ −16.1183 −0.948134
$$290$$ −8.12498 −0.477115
$$291$$ 15.4547i 0.905972i
$$292$$ − 11.5308i − 0.674789i
$$293$$ 1.73663i 0.101455i 0.998713 + 0.0507274i $$0.0161540\pi$$
−0.998713 + 0.0507274i $$0.983846\pi$$
$$294$$ − 4.26875i − 0.248959i
$$295$$ −0.417895 −0.0243308
$$296$$ −11.6310 −0.676039
$$297$$ − 1.69202i − 0.0981810i
$$298$$ 1.14914 0.0665682
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 13.5007i 0.778165i
$$302$$ 0.0217703 0.00125274
$$303$$ −5.38404 −0.309305
$$304$$ − 4.85086i − 0.278216i
$$305$$ − 0.198062i − 0.0113410i
$$306$$ − 0.939001i − 0.0536791i
$$307$$ − 25.4403i − 1.45195i −0.687720 0.725976i $$-0.741388\pi$$
0.687720 0.725976i $$-0.258612\pi$$
$$308$$ 5.67994 0.323645
$$309$$ −4.08575 −0.232430
$$310$$ 4.08815i 0.232191i
$$311$$ 10.0325 0.568892 0.284446 0.958692i $$-0.408190\pi$$
0.284446 + 0.958692i $$0.408190\pi$$
$$312$$ 0 0
$$313$$ −22.3274 −1.26202 −0.631008 0.775776i $$-0.717359\pi$$
−0.631008 + 0.775776i $$0.717359\pi$$
$$314$$ − 10.1588i − 0.573296i
$$315$$ −3.35690 −0.189140
$$316$$ 4.94869 0.278386
$$317$$ − 5.33811i − 0.299818i −0.988700 0.149909i $$-0.952102\pi$$
0.988700 0.149909i $$-0.0478981\pi$$
$$318$$ 5.74094i 0.321936i
$$319$$ 13.7476i 0.769720i
$$320$$ 1.00000i 0.0559017i
$$321$$ −16.1196 −0.899709
$$322$$ −13.5918 −0.757441
$$323$$ 4.55496i 0.253445i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 19.1511 1.06068
$$327$$ − 14.2325i − 0.787059i
$$328$$ 3.86294 0.213295
$$329$$ 4.27711 0.235805
$$330$$ − 1.69202i − 0.0931427i
$$331$$ − 4.49827i − 0.247247i −0.992329 0.123624i $$-0.960548\pi$$
0.992329 0.123624i $$-0.0394516\pi$$
$$332$$ − 13.1739i − 0.723012i
$$333$$ − 11.6310i − 0.637376i
$$334$$ −15.2174 −0.832661
$$335$$ 8.93900 0.488390
$$336$$ − 3.35690i − 0.183134i
$$337$$ 11.5676 0.630129 0.315064 0.949070i $$-0.397974\pi$$
0.315064 + 0.949070i $$0.397974\pi$$
$$338$$ 0 0
$$339$$ −13.3274 −0.723843
$$340$$ − 0.939001i − 0.0509245i
$$341$$ 6.91723 0.374589
$$342$$ 4.85086 0.262304
$$343$$ − 9.16852i − 0.495054i
$$344$$ 4.02177i 0.216839i
$$345$$ 4.04892i 0.217986i
$$346$$ − 21.8213i − 1.17312i
$$347$$ 27.1226 1.45602 0.728008 0.685568i $$-0.240446\pi$$
0.728008 + 0.685568i $$0.240446\pi$$
$$348$$ 8.12498 0.435545
$$349$$ − 12.8562i − 0.688178i −0.938937 0.344089i $$-0.888188\pi$$
0.938937 0.344089i $$-0.111812\pi$$
$$350$$ −3.35690 −0.179434
$$351$$ 0 0
$$352$$ 1.69202 0.0901850
$$353$$ 24.6262i 1.31072i 0.755316 + 0.655361i $$0.227484\pi$$
−0.755316 + 0.655361i $$0.772516\pi$$
$$354$$ 0.417895 0.0222109
$$355$$ −5.15883 −0.273802
$$356$$ − 9.47650i − 0.502254i
$$357$$ 3.15213i 0.166828i
$$358$$ 20.2513i 1.07031i
$$359$$ 23.8629i 1.25944i 0.776823 + 0.629719i $$0.216830\pi$$
−0.776823 + 0.629719i $$0.783170\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −4.53079 −0.238463
$$362$$ − 14.4969i − 0.761942i
$$363$$ 8.13706 0.427085
$$364$$ 0 0
$$365$$ 11.5308 0.603549
$$366$$ 0.198062i 0.0103529i
$$367$$ 33.1933 1.73267 0.866337 0.499459i $$-0.166468\pi$$
0.866337 + 0.499459i $$0.166468\pi$$
$$368$$ −4.04892 −0.211064
$$369$$ 3.86294i 0.201096i
$$370$$ − 11.6310i − 0.604668i
$$371$$ − 19.2717i − 1.00054i
$$372$$ − 4.08815i − 0.211960i
$$373$$ −10.8465 −0.561613 −0.280806 0.959764i $$-0.590602\pi$$
−0.280806 + 0.959764i $$0.590602\pi$$
$$374$$ −1.58881 −0.0821554
$$375$$ 1.00000i 0.0516398i
$$376$$ 1.27413 0.0657081
$$377$$ 0 0
$$378$$ 3.35690 0.172660
$$379$$ − 6.09485i − 0.313071i −0.987672 0.156536i $$-0.949967\pi$$
0.987672 0.156536i $$-0.0500326\pi$$
$$380$$ 4.85086 0.248844
$$381$$ 12.8509 0.658369
$$382$$ − 10.1806i − 0.520885i
$$383$$ − 7.95407i − 0.406434i −0.979134 0.203217i $$-0.934860\pi$$
0.979134 0.203217i $$-0.0651397\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 5.67994i 0.289477i
$$386$$ 27.6407 1.40688
$$387$$ −4.02177 −0.204438
$$388$$ − 15.4547i − 0.784595i
$$389$$ −1.50365 −0.0762380 −0.0381190 0.999273i $$-0.512137\pi$$
−0.0381190 + 0.999273i $$0.512137\pi$$
$$390$$ 0 0
$$391$$ 3.80194 0.192272
$$392$$ 4.26875i 0.215604i
$$393$$ −0.570024 −0.0287539
$$394$$ 13.0261 0.656245
$$395$$ 4.94869i 0.248996i
$$396$$ 1.69202i 0.0850273i
$$397$$ − 34.9627i − 1.75473i −0.479826 0.877364i $$-0.659300\pi$$
0.479826 0.877364i $$-0.340700\pi$$
$$398$$ − 6.67563i − 0.334619i
$$399$$ −16.2838 −0.815210
$$400$$ −1.00000 −0.0500000
$$401$$ 31.1148i 1.55380i 0.629624 + 0.776900i $$0.283209\pi$$
−0.629624 + 0.776900i $$0.716791\pi$$
$$402$$ −8.93900 −0.445837
$$403$$ 0 0
$$404$$ 5.38404 0.267866
$$405$$ − 1.00000i − 0.0496904i
$$406$$ −27.2747 −1.35362
$$407$$ −19.6799 −0.975498
$$408$$ 0.939001i 0.0464875i
$$409$$ − 32.7318i − 1.61849i −0.587474 0.809243i $$-0.699878\pi$$
0.587474 0.809243i $$-0.300122\pi$$
$$410$$ 3.86294i 0.190777i
$$411$$ − 21.5405i − 1.06251i
$$412$$ 4.08575 0.201291
$$413$$ −1.40283 −0.0690287
$$414$$ − 4.04892i − 0.198993i
$$415$$ 13.1739 0.646681
$$416$$ 0 0
$$417$$ 7.09783 0.347582
$$418$$ − 8.20775i − 0.401454i
$$419$$ 13.0694 0.638480 0.319240 0.947674i $$-0.396572\pi$$
0.319240 + 0.947674i $$0.396572\pi$$
$$420$$ 3.35690 0.163800
$$421$$ 16.4789i 0.803132i 0.915830 + 0.401566i $$0.131534\pi$$
−0.915830 + 0.401566i $$0.868466\pi$$
$$422$$ − 11.3056i − 0.550347i
$$423$$ 1.27413i 0.0619502i
$$424$$ − 5.74094i − 0.278805i
$$425$$ 0.939001 0.0455482
$$426$$ 5.15883 0.249946
$$427$$ − 0.664874i − 0.0321755i
$$428$$ 16.1196 0.779171
$$429$$ 0 0
$$430$$ −4.02177 −0.193947
$$431$$ − 41.0640i − 1.97798i −0.147974 0.988991i $$-0.547275\pi$$
0.147974 0.988991i $$-0.452725\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 6.04593 0.290549 0.145275 0.989391i $$-0.453593\pi$$
0.145275 + 0.989391i $$0.453593\pi$$
$$434$$ 13.7235i 0.658748i
$$435$$ 8.12498i 0.389563i
$$436$$ 14.2325i 0.681613i
$$437$$ 19.6407i 0.939543i
$$438$$ −11.5308 −0.550963
$$439$$ 26.2543 1.25305 0.626524 0.779402i $$-0.284477\pi$$
0.626524 + 0.779402i $$0.284477\pi$$
$$440$$ 1.69202i 0.0806640i
$$441$$ −4.26875 −0.203274
$$442$$ 0 0
$$443$$ −39.3846 −1.87122 −0.935610 0.353035i $$-0.885150\pi$$
−0.935610 + 0.353035i $$0.885150\pi$$
$$444$$ 11.6310i 0.551984i
$$445$$ 9.47650 0.449229
$$446$$ −18.0911 −0.856640
$$447$$ − 1.14914i − 0.0543527i
$$448$$ 3.35690i 0.158598i
$$449$$ 9.39075i 0.443177i 0.975140 + 0.221588i $$0.0711241\pi$$
−0.975140 + 0.221588i $$0.928876\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 6.53617 0.307776
$$452$$ 13.3274 0.626866
$$453$$ − 0.0217703i − 0.00102286i
$$454$$ 9.37196 0.439848
$$455$$ 0 0
$$456$$ −4.85086 −0.227162
$$457$$ − 2.66487i − 0.124658i −0.998056 0.0623288i $$-0.980147\pi$$
0.998056 0.0623288i $$-0.0198527\pi$$
$$458$$ 21.2664 0.993712
$$459$$ −0.939001 −0.0438288
$$460$$ − 4.04892i − 0.188782i
$$461$$ − 24.5284i − 1.14240i −0.820810 0.571201i $$-0.806478\pi$$
0.820810 0.571201i $$-0.193522\pi$$
$$462$$ − 5.67994i − 0.264255i
$$463$$ 4.50498i 0.209364i 0.994506 + 0.104682i $$0.0333825\pi$$
−0.994506 + 0.104682i $$0.966618\pi$$
$$464$$ −8.12498 −0.377193
$$465$$ 4.08815 0.189583
$$466$$ − 2.03146i − 0.0941055i
$$467$$ −21.3317 −0.987112 −0.493556 0.869714i $$-0.664303\pi$$
−0.493556 + 0.869714i $$0.664303\pi$$
$$468$$ 0 0
$$469$$ 30.0073 1.38561
$$470$$ 1.27413i 0.0587711i
$$471$$ −10.1588 −0.468094
$$472$$ −0.417895 −0.0192352
$$473$$ 6.80492i 0.312891i
$$474$$ − 4.94869i − 0.227301i
$$475$$ 4.85086i 0.222572i
$$476$$ − 3.15213i − 0.144478i
$$477$$ 5.74094 0.262860
$$478$$ 11.9903 0.548424
$$479$$ 32.9517i 1.50560i 0.658249 + 0.752800i $$0.271297\pi$$
−0.658249 + 0.752800i $$0.728703\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 21.7627 0.991264
$$483$$ 13.5918i 0.618448i
$$484$$ −8.13706 −0.369867
$$485$$ 15.4547 0.701763
$$486$$ 1.00000i 0.0453609i
$$487$$ − 9.33034i − 0.422798i −0.977400 0.211399i $$-0.932198\pi$$
0.977400 0.211399i $$-0.0678020\pi$$
$$488$$ − 0.198062i − 0.00896586i
$$489$$ − 19.1511i − 0.866041i
$$490$$ −4.26875 −0.192842
$$491$$ 20.4349 0.922213 0.461107 0.887345i $$-0.347453\pi$$
0.461107 + 0.887345i $$0.347453\pi$$
$$492$$ − 3.86294i − 0.174155i
$$493$$ 7.62937 0.343609
$$494$$ 0 0
$$495$$ −1.69202 −0.0760507
$$496$$ 4.08815i 0.183563i
$$497$$ −17.3177 −0.776804
$$498$$ −13.1739 −0.590337
$$499$$ 3.54958i 0.158901i 0.996839 + 0.0794505i $$0.0253166\pi$$
−0.996839 + 0.0794505i $$0.974683\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 15.2174i 0.679865i
$$502$$ − 13.7506i − 0.613721i
$$503$$ −19.8146 −0.883490 −0.441745 0.897141i $$-0.645640\pi$$
−0.441745 + 0.897141i $$0.645640\pi$$
$$504$$ −3.35690 −0.149528
$$505$$ 5.38404i 0.239587i
$$506$$ −6.85086 −0.304558
$$507$$ 0 0
$$508$$ −12.8509 −0.570164
$$509$$ 32.4325i 1.43754i 0.695246 + 0.718772i $$0.255296\pi$$
−0.695246 + 0.718772i $$0.744704\pi$$
$$510$$ −0.939001 −0.0415797
$$511$$ 38.7077 1.71233
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.85086i − 0.214170i
$$514$$ 24.1782i 1.06646i
$$515$$ 4.08575i 0.180040i
$$516$$ 4.02177 0.177049
$$517$$ 2.15585 0.0948142
$$518$$ − 39.0441i − 1.71550i
$$519$$ −21.8213 −0.957849
$$520$$ 0 0
$$521$$ 7.87694 0.345095 0.172547 0.985001i $$-0.444800\pi$$
0.172547 + 0.985001i $$0.444800\pi$$
$$522$$ − 8.12498i − 0.355621i
$$523$$ −23.4276 −1.02442 −0.512208 0.858861i $$-0.671172\pi$$
−0.512208 + 0.858861i $$0.671172\pi$$
$$524$$ 0.570024 0.0249016
$$525$$ 3.35690i 0.146507i
$$526$$ 0.960771i 0.0418916i
$$527$$ − 3.83877i − 0.167220i
$$528$$ − 1.69202i − 0.0736358i
$$529$$ −6.60627 −0.287229
$$530$$ 5.74094 0.249370
$$531$$ − 0.417895i − 0.0181351i
$$532$$ 16.2838 0.705993
$$533$$ 0 0
$$534$$ −9.47650 −0.410088
$$535$$ 16.1196i 0.696911i
$$536$$ 8.93900 0.386106
$$537$$ 20.2513 0.873908
$$538$$ 10.0761i 0.434410i
$$539$$ 7.22282i 0.311109i
$$540$$ 1.00000i 0.0430331i
$$541$$ 1.21552i 0.0522593i 0.999659 + 0.0261297i $$0.00831828\pi$$
−0.999659 + 0.0261297i $$0.991682\pi$$
$$542$$ −14.9584 −0.642517
$$543$$ −14.4969 −0.622123
$$544$$ − 0.939001i − 0.0402593i
$$545$$ −14.2325 −0.609654
$$546$$ 0 0
$$547$$ −15.1371 −0.647214 −0.323607 0.946192i $$-0.604896\pi$$
−0.323607 + 0.946192i $$0.604896\pi$$
$$548$$ 21.5405i 0.920164i
$$549$$ 0.198062 0.00845309
$$550$$ −1.69202 −0.0721480
$$551$$ 39.4131i 1.67905i
$$552$$ 4.04892i 0.172333i
$$553$$ 16.6122i 0.706424i
$$554$$ 13.0465i 0.554294i
$$555$$ −11.6310 −0.493709
$$556$$ −7.09783 −0.301015
$$557$$ − 0.147817i − 0.00626321i −0.999995 0.00313160i $$-0.999003\pi$$
0.999995 0.00313160i $$-0.000996822\pi$$
$$558$$ −4.08815 −0.173065
$$559$$ 0 0
$$560$$ −3.35690 −0.141855
$$561$$ 1.58881i 0.0670796i
$$562$$ 16.2881 0.687073
$$563$$ −40.6945 −1.71507 −0.857535 0.514426i $$-0.828005\pi$$
−0.857535 + 0.514426i $$0.828005\pi$$
$$564$$ − 1.27413i − 0.0536504i
$$565$$ 13.3274i 0.560686i
$$566$$ 12.3773i 0.520258i
$$567$$ − 3.35690i − 0.140976i
$$568$$ −5.15883 −0.216460
$$569$$ 26.2433 1.10017 0.550087 0.835107i $$-0.314594\pi$$
0.550087 + 0.835107i $$0.314594\pi$$
$$570$$ − 4.85086i − 0.203180i
$$571$$ −2.31575 −0.0969110 −0.0484555 0.998825i $$-0.515430\pi$$
−0.0484555 + 0.998825i $$0.515430\pi$$
$$572$$ 0 0
$$573$$ −10.1806 −0.425301
$$574$$ 12.9675i 0.541252i
$$575$$ 4.04892 0.168852
$$576$$ −1.00000 −0.0416667
$$577$$ − 14.5241i − 0.604646i −0.953206 0.302323i $$-0.902238\pi$$
0.953206 0.302323i $$-0.0977621\pi$$
$$578$$ − 16.1183i − 0.670432i
$$579$$ − 27.6407i − 1.14871i
$$580$$ − 8.12498i − 0.337372i
$$581$$ 44.2234 1.83470
$$582$$ −15.4547 −0.640619
$$583$$ − 9.71379i − 0.402304i
$$584$$ 11.5308 0.477148
$$585$$ 0 0
$$586$$ −1.73663 −0.0717394
$$587$$ − 32.9028i − 1.35804i −0.734119 0.679021i $$-0.762404\pi$$
0.734119 0.679021i $$-0.237596\pi$$
$$588$$ 4.26875 0.176040
$$589$$ 19.8310 0.817122
$$590$$ − 0.417895i − 0.0172045i
$$591$$ − 13.0261i − 0.535821i
$$592$$ − 11.6310i − 0.478032i
$$593$$ 36.0146i 1.47894i 0.673188 + 0.739471i $$0.264924\pi$$
−0.673188 + 0.739471i $$0.735076\pi$$
$$594$$ 1.69202 0.0694245
$$595$$ 3.15213 0.129225
$$596$$ 1.14914i 0.0470708i
$$597$$ −6.67563 −0.273215
$$598$$ 0 0
$$599$$ −29.4644 −1.20388 −0.601942 0.798540i $$-0.705606\pi$$
−0.601942 + 0.798540i $$0.705606\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ −28.4004 −1.15848 −0.579239 0.815158i $$-0.696650\pi$$
−0.579239 + 0.815158i $$0.696650\pi$$
$$602$$ −13.5007 −0.550246
$$603$$ 8.93900i 0.364024i
$$604$$ 0.0217703i 0 0.000885819i
$$605$$ − 8.13706i − 0.330819i
$$606$$ − 5.38404i − 0.218712i
$$607$$ 15.6450 0.635012 0.317506 0.948256i $$-0.397155\pi$$
0.317506 + 0.948256i $$0.397155\pi$$
$$608$$ 4.85086 0.196728
$$609$$ 27.2747i 1.10523i
$$610$$ 0.198062 0.00801931
$$611$$ 0 0
$$612$$ 0.939001 0.0379569
$$613$$ − 17.6329i − 0.712188i −0.934450 0.356094i $$-0.884108\pi$$
0.934450 0.356094i $$-0.115892\pi$$
$$614$$ 25.4403 1.02669
$$615$$ 3.86294 0.155769
$$616$$ 5.67994i 0.228851i
$$617$$ 10.8582i 0.437133i 0.975822 + 0.218566i $$0.0701380\pi$$
−0.975822 + 0.218566i $$0.929862\pi$$
$$618$$ − 4.08575i − 0.164353i
$$619$$ 34.2083i 1.37495i 0.726208 + 0.687475i $$0.241281\pi$$
−0.726208 + 0.687475i $$0.758719\pi$$
$$620$$ −4.08815 −0.164184
$$621$$ −4.04892 −0.162477
$$622$$ 10.0325i 0.402268i
$$623$$ 31.8116 1.27451
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 22.3274i − 0.892381i
$$627$$ −8.20775 −0.327786
$$628$$ 10.1588 0.405382
$$629$$ 10.9215i 0.435470i
$$630$$ − 3.35690i − 0.133742i
$$631$$ − 31.9571i − 1.27219i −0.771611 0.636095i $$-0.780549\pi$$
0.771611 0.636095i $$-0.219451\pi$$
$$632$$ 4.94869i 0.196848i
$$633$$ −11.3056 −0.449357
$$634$$ 5.33811 0.212003
$$635$$ − 12.8509i − 0.509971i
$$636$$ −5.74094 −0.227643
$$637$$ 0 0
$$638$$ −13.7476 −0.544274
$$639$$ − 5.15883i − 0.204080i
$$640$$ −1.00000 −0.0395285
$$641$$ 45.8297 1.81016 0.905082 0.425238i $$-0.139810\pi$$
0.905082 + 0.425238i $$0.139810\pi$$
$$642$$ − 16.1196i − 0.636190i
$$643$$ 20.6418i 0.814032i 0.913421 + 0.407016i $$0.133431\pi$$
−0.913421 + 0.407016i $$0.866569\pi$$
$$644$$ − 13.5918i − 0.535592i
$$645$$ 4.02177i 0.158357i
$$646$$ −4.55496 −0.179212
$$647$$ −1.40389 −0.0551928 −0.0275964 0.999619i $$-0.508785\pi$$
−0.0275964 + 0.999619i $$0.508785\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −0.707087 −0.0277556
$$650$$ 0 0
$$651$$ 13.7235 0.537866
$$652$$ 19.1511i 0.750014i
$$653$$ −4.54229 −0.177753 −0.0888767 0.996043i $$-0.528328\pi$$
−0.0888767 + 0.996043i $$0.528328\pi$$
$$654$$ 14.2325 0.556535
$$655$$ 0.570024i 0.0222727i
$$656$$ 3.86294i 0.150822i
$$657$$ 11.5308i 0.449859i
$$658$$ 4.27711i 0.166739i
$$659$$ 17.4161 0.678435 0.339217 0.940708i $$-0.389838\pi$$
0.339217 + 0.940708i $$0.389838\pi$$
$$660$$ 1.69202 0.0658618
$$661$$ − 43.3682i − 1.68683i −0.537263 0.843415i $$-0.680542\pi$$
0.537263 0.843415i $$-0.319458\pi$$
$$662$$ 4.49827 0.174830
$$663$$ 0 0
$$664$$ 13.1739 0.511246
$$665$$ 16.2838i 0.631459i
$$666$$ 11.6310 0.450693
$$667$$ 32.8974 1.27379
$$668$$ − 15.2174i − 0.588780i
$$669$$ 18.0911i 0.699443i
$$670$$ 8.93900i 0.345344i
$$671$$ − 0.335126i − 0.0129374i
$$672$$ 3.35690 0.129495
$$673$$ 3.49875 0.134867 0.0674334 0.997724i $$-0.478519\pi$$
0.0674334 + 0.997724i $$0.478519\pi$$
$$674$$ 11.5676i 0.445568i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −13.3284 −0.512253 −0.256126 0.966643i $$-0.582446\pi$$
−0.256126 + 0.966643i $$0.582446\pi$$
$$678$$ − 13.3274i − 0.511834i
$$679$$ 51.8799 1.99097
$$680$$ 0.939001 0.0360090
$$681$$ − 9.37196i − 0.359134i
$$682$$ 6.91723i 0.264874i
$$683$$ 24.1739i 0.924989i 0.886622 + 0.462494i $$0.153045\pi$$
−0.886622 + 0.462494i $$0.846955\pi$$
$$684$$ 4.85086i 0.185477i
$$685$$ −21.5405 −0.823020
$$686$$ 9.16852 0.350056
$$687$$ − 21.2664i − 0.811362i
$$688$$ −4.02177 −0.153329
$$689$$ 0 0
$$690$$ −4.04892 −0.154140
$$691$$ − 37.8810i − 1.44106i −0.693423 0.720530i $$-0.743898\pi$$
0.693423 0.720530i $$-0.256102\pi$$
$$692$$ 21.8213 0.829522
$$693$$ −5.67994 −0.215763
$$694$$ 27.1226i 1.02956i
$$695$$ − 7.09783i − 0.269236i
$$696$$ 8.12498i 0.307977i
$$697$$ − 3.62730i − 0.137394i
$$698$$ 12.8562 0.486616
$$699$$ −2.03146 −0.0768368
$$700$$ − 3.35690i − 0.126879i
$$701$$ 21.5967 0.815696 0.407848 0.913050i $$-0.366279\pi$$
0.407848 + 0.913050i $$0.366279\pi$$
$$702$$ 0 0
$$703$$ −56.4204 −2.12794
$$704$$ 1.69202i 0.0637705i
$$705$$ 1.27413 0.0479864
$$706$$ −24.6262 −0.926821
$$707$$ 18.0737i 0.679730i
$$708$$ 0.417895i 0.0157054i
$$709$$ − 41.3062i − 1.55129i −0.631172 0.775643i $$-0.717426\pi$$
0.631172 0.775643i $$-0.282574\pi$$
$$710$$ − 5.15883i − 0.193608i
$$711$$ −4.94869 −0.185590
$$712$$ 9.47650 0.355147
$$713$$ − 16.5526i − 0.619898i
$$714$$ −3.15213 −0.117965
$$715$$ 0 0
$$716$$ −20.2513 −0.756826
$$717$$ − 11.9903i − 0.447786i
$$718$$ −23.8629 −0.890557
$$719$$ 35.5338 1.32519 0.662593 0.748980i $$-0.269456\pi$$
0.662593 + 0.748980i $$0.269456\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ 13.7154i 0.510790i
$$722$$ − 4.53079i − 0.168619i
$$723$$ − 21.7627i − 0.809364i
$$724$$ 14.4969 0.538775
$$725$$ 8.12498 0.301754
$$726$$ 8.13706i 0.301995i
$$727$$ −19.6635 −0.729281 −0.364640 0.931148i $$-0.618808\pi$$
−0.364640 + 0.931148i $$0.618808\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 11.5308i 0.426774i
$$731$$ 3.77645 0.139677
$$732$$ −0.198062 −0.00732059
$$733$$ − 46.8702i − 1.73119i −0.500743 0.865596i $$-0.666940\pi$$
0.500743 0.865596i $$-0.333060\pi$$
$$734$$ 33.1933i 1.22519i
$$735$$ 4.26875i 0.157455i
$$736$$ − 4.04892i − 0.149245i
$$737$$ 15.1250 0.557136
$$738$$ −3.86294 −0.142197
$$739$$ 8.57779i 0.315539i 0.987476 + 0.157770i $$0.0504303\pi$$
−0.987476 + 0.157770i $$0.949570\pi$$
$$740$$ 11.6310 0.427565
$$741$$ 0 0
$$742$$ 19.2717 0.707488
$$743$$ − 44.9506i − 1.64908i −0.565805 0.824539i $$-0.691435\pi$$
0.565805 0.824539i $$-0.308565\pi$$
$$744$$ 4.08815 0.149879
$$745$$ −1.14914 −0.0421014
$$746$$ − 10.8465i − 0.397120i
$$747$$ 13.1739i 0.482008i
$$748$$ − 1.58881i − 0.0580926i
$$749$$ 54.1118i 1.97720i
$$750$$ −1.00000 −0.0365148
$$751$$ −44.0374 −1.60695 −0.803474 0.595339i $$-0.797018\pi$$
−0.803474 + 0.595339i $$0.797018\pi$$
$$752$$ 1.27413i 0.0464626i
$$753$$ −13.7506 −0.501101
$$754$$ 0 0
$$755$$ −0.0217703 −0.000792301 0
$$756$$ 3.35690i 0.122089i
$$757$$ −13.2798 −0.482661 −0.241331 0.970443i $$-0.577584\pi$$
−0.241331 + 0.970443i $$0.577584\pi$$
$$758$$ 6.09485 0.221375
$$759$$ 6.85086i 0.248670i
$$760$$ 4.85086i 0.175959i
$$761$$ − 9.11231i − 0.330321i −0.986267 0.165160i $$-0.947186\pi$$
0.986267 0.165160i $$-0.0528142\pi$$
$$762$$ 12.8509i 0.465537i
$$763$$ −47.7770 −1.72964
$$764$$ 10.1806 0.368321
$$765$$ 0.939001i 0.0339497i
$$766$$ 7.95407 0.287392
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 31.7784i − 1.14596i −0.819570 0.572979i $$-0.805788\pi$$
0.819570 0.572979i $$-0.194212\pi$$
$$770$$ −5.67994 −0.204691
$$771$$ 24.1782 0.870757
$$772$$ 27.6407i 0.994811i
$$773$$ 7.11960i 0.256074i 0.991769 + 0.128037i $$0.0408677\pi$$
−0.991769 + 0.128037i $$0.959132\pi$$
$$774$$ − 4.02177i − 0.144560i
$$775$$ − 4.08815i − 0.146851i
$$776$$ 15.4547 0.554792
$$777$$ −39.0441 −1.40070
$$778$$ − 1.50365i − 0.0539084i
$$779$$ 18.7385 0.671378
$$780$$ 0 0
$$781$$ −8.72886 −0.312343
$$782$$ 3.80194i 0.135957i
$$783$$ −8.12498 −0.290363
$$784$$ −4.26875 −0.152455
$$785$$ 10.1588i 0.362584i
$$786$$ − 0.570024i − 0.0203321i
$$787$$ 44.2121i 1.57599i 0.615682 + 0.787995i $$0.288881\pi$$
−0.615682 + 0.787995i $$0.711119\pi$$
$$788$$ 13.0261i 0.464035i
$$789$$ 0.960771 0.0342044
$$790$$ −4.94869 −0.176066
$$791$$ 44.7385i 1.59072i
$$792$$ −1.69202 −0.0601234
$$793$$ 0 0
$$794$$ 34.9627 1.24078
$$795$$ − 5.74094i − 0.203610i
$$796$$ 6.67563 0.236611
$$797$$ −19.2241 −0.680954 −0.340477 0.940253i $$-0.610589\pi$$
−0.340477 + 0.940253i $$0.610589\pi$$
$$798$$ − 16.2838i − 0.576441i
$$799$$ − 1.19641i − 0.0423258i
$$800$$ − 1.00000i − 0.0353553i
$$801$$ 9.47650i 0.334836i
$$802$$ −31.1148 −1.09870
$$803$$ 19.5104 0.688505
$$804$$ − 8.93900i − 0.315254i
$$805$$ 13.5918 0.479048
$$806$$ 0 0
$$807$$ 10.0761 0.354694
$$808$$ 5.38404i 0.189410i
$$809$$ 0.0988996 0.00347713 0.00173856 0.999998i $$-0.499447\pi$$
0.00173856 + 0.999998i $$0.499447\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 6.70304i 0.235376i 0.993051 + 0.117688i $$0.0375482\pi$$
−0.993051 + 0.117688i $$0.962452\pi$$
$$812$$ − 27.2747i − 0.957155i
$$813$$ 14.9584i 0.524613i
$$814$$ − 19.6799i − 0.689782i
$$815$$ −19.1511 −0.670833
$$816$$ −0.939001 −0.0328716
$$817$$ 19.5090i 0.682534i
$$818$$ 32.7318 1.14444
$$819$$ 0 0
$$820$$ −3.86294 −0.134900
$$821$$ − 23.7450i − 0.828706i −0.910116 0.414353i $$-0.864008\pi$$
0.910116 0.414353i $$-0.135992\pi$$
$$822$$ 21.5405 0.751311
$$823$$ −18.0049 −0.627611 −0.313806 0.949487i $$-0.601604\pi$$
−0.313806 + 0.949487i $$0.601604\pi$$
$$824$$ 4.08575i 0.142334i
$$825$$ 1.69202i 0.0589086i
$$826$$ − 1.40283i − 0.0488107i
$$827$$ − 40.4566i − 1.40682i −0.710787 0.703408i $$-0.751661\pi$$
0.710787 0.703408i $$-0.248339\pi$$
$$828$$ 4.04892 0.140710
$$829$$ −8.56704 −0.297546 −0.148773 0.988871i $$-0.547532\pi$$
−0.148773 + 0.988871i $$0.547532\pi$$
$$830$$ 13.1739i 0.457273i
$$831$$ 13.0465 0.452579
$$832$$ 0 0
$$833$$ 4.00836 0.138881
$$834$$ 7.09783i 0.245778i
$$835$$ 15.2174 0.526621
$$836$$ 8.20775 0.283871
$$837$$ 4.08815i 0.141307i
$$838$$ 13.0694i 0.451474i
$$839$$ 6.75600i 0.233243i 0.993176 + 0.116622i $$0.0372065\pi$$
−0.993176 + 0.116622i $$0.962794\pi$$
$$840$$ 3.35690i 0.115824i
$$841$$ 37.0153 1.27639
$$842$$ −16.4789 −0.567900
$$843$$ − 16.2881i − 0.560993i
$$844$$ 11.3056 0.389154
$$845$$ 0 0
$$846$$ −1.27413 −0.0438054
$$847$$ − 27.3153i − 0.938564i
$$848$$ 5.74094 0.197145
$$849$$ 12.3773 0.424789
$$850$$ 0.939001i 0.0322075i
$$851$$ 47.0930i 1.61433i
$$852$$ 5.15883i 0.176739i
$$853$$ 37.1269i 1.27120i 0.772018 + 0.635600i $$0.219247\pi$$
−0.772018 + 0.635600i $$0.780753\pi$$
$$854$$ 0.664874 0.0227515
$$855$$ −4.85086 −0.165896
$$856$$ 16.1196i 0.550957i
$$857$$ 24.3937 0.833274 0.416637 0.909073i $$-0.363209\pi$$
0.416637 + 0.909073i $$0.363209\pi$$
$$858$$ 0 0
$$859$$ 43.9366 1.49910 0.749549 0.661949i $$-0.230270\pi$$
0.749549 + 0.661949i $$0.230270\pi$$
$$860$$ − 4.02177i − 0.137141i
$$861$$ 12.9675 0.441930
$$862$$ 41.0640 1.39864
$$863$$ 16.3846i 0.557739i 0.960329 + 0.278870i $$0.0899598\pi$$
−0.960329 + 0.278870i $$0.910040\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ 21.8213i 0.741947i
$$866$$ 6.04593i 0.205449i
$$867$$ −16.1183 −0.547405
$$868$$ −13.7235 −0.465805
$$869$$ 8.37329i 0.284044i
$$870$$ −8.12498 −0.275463
$$871$$ 0 0
$$872$$ −14.2325 −0.481973
$$873$$ 15.4547i 0.523063i
$$874$$ −19.6407 −0.664357
$$875$$ 3.35690 0.113484
$$876$$ − 11.5308i − 0.389589i
$$877$$ 28.1347i 0.950040i 0.879975 + 0.475020i $$0.157559\pi$$
−0.879975 + 0.475020i $$0.842441\pi$$
$$878$$ 26.2543i 0.886039i
$$879$$ 1.73663i 0.0585750i
$$880$$ −1.69202 −0.0570380
$$881$$ −39.1454 −1.31884 −0.659421 0.751773i $$-0.729199\pi$$
−0.659421 + 0.751773i $$0.729199\pi$$
$$882$$ − 4.26875i − 0.143736i
$$883$$ −46.2833 −1.55756 −0.778779 0.627298i $$-0.784161\pi$$
−0.778779 + 0.627298i $$0.784161\pi$$
$$884$$ 0 0
$$885$$ −0.417895 −0.0140474
$$886$$ − 39.3846i − 1.32315i
$$887$$ −3.08097 −0.103449 −0.0517244 0.998661i $$-0.516472\pi$$
−0.0517244 + 0.998661i $$0.516472\pi$$
$$888$$ −11.6310 −0.390312
$$889$$ − 43.1390i − 1.44684i
$$890$$ 9.47650i 0.317653i
$$891$$ − 1.69202i − 0.0566849i
$$892$$ − 18.0911i − 0.605736i
$$893$$ 6.18060 0.206826
$$894$$ 1.14914 0.0384332
$$895$$ − 20.2513i − 0.676926i
$$896$$ −3.35690 −0.112146
$$897$$ 0 0
$$898$$ −9.39075 −0.313373
$$899$$ − 33.2161i − 1.10782i
$$900$$ 1.00000 0.0333333
$$901$$ −5.39075 −0.179592
$$902$$ 6.53617i 0.217631i
$$903$$ 13.5007i 0.449274i
$$904$$ 13.3274i 0.443261i
$$905$$ 14.4969i 0.481895i
$$906$$ 0.0217703 0.000723269 0
$$907$$ 2.61224 0.0867379 0.0433689 0.999059i $$-0.486191\pi$$
0.0433689 + 0.999059i $$0.486191\pi$$
$$908$$ 9.37196i 0.311019i
$$909$$ −5.38404 −0.178577
$$910$$ 0 0
$$911$$ 24.9302 0.825973 0.412987 0.910737i $$-0.364486\pi$$
0.412987 + 0.910737i $$0.364486\pi$$
$$912$$ − 4.85086i − 0.160628i
$$913$$ 22.2905 0.737709
$$914$$ 2.66487 0.0881462
$$915$$ − 0.198062i − 0.00654774i
$$916$$ 21.2664i 0.702660i
$$917$$ 1.91351i 0.0631897i
$$918$$ − 0.939001i − 0.0309917i
$$919$$ −54.4462 −1.79602 −0.898008 0.439980i $$-0.854986\pi$$
−0.898008 + 0.439980i $$0.854986\pi$$
$$920$$ 4.04892 0.133489
$$921$$ − 25.4403i − 0.838285i
$$922$$ 24.5284 0.807800
$$923$$ 0 0
$$924$$ 5.67994 0.186856
$$925$$ 11.6310i 0.382426i
$$926$$ −4.50498 −0.148043
$$927$$ −4.08575 −0.134194
$$928$$ − 8.12498i − 0.266716i
$$929$$ − 40.8267i − 1.33948i −0.742596 0.669740i $$-0.766405\pi$$
0.742596 0.669740i $$-0.233595\pi$$
$$930$$ 4.08815i 0.134056i
$$931$$ 20.7071i 0.678647i
$$932$$ 2.03146 0.0665427
$$933$$ 10.0325 0.328450
$$934$$ − 21.3317i − 0.697993i
$$935$$ 1.58881 0.0519596
$$936$$ 0 0
$$937$$ −46.9571 −1.53402 −0.767010 0.641635i $$-0.778256\pi$$
−0.767010 + 0.641635i $$0.778256\pi$$
$$938$$ 30.0073i 0.979773i
$$939$$ −22.3274 −0.728626
$$940$$ −1.27413 −0.0415574
$$941$$ 35.8525i 1.16876i 0.811481 + 0.584379i $$0.198662\pi$$
−0.811481 + 0.584379i $$0.801338\pi$$
$$942$$ − 10.1588i − 0.330993i
$$943$$ − 15.6407i − 0.509332i
$$944$$ − 0.417895i − 0.0136013i
$$945$$ −3.35690 −0.109200
$$946$$ −6.80492 −0.221247
$$947$$ 8.73019i 0.283693i 0.989889 + 0.141846i $$0.0453039\pi$$
−0.989889 + 0.141846i $$0.954696\pi$$
$$948$$ 4.94869 0.160726
$$949$$ 0 0
$$950$$ −4.85086 −0.157383
$$951$$ − 5.33811i − 0.173100i
$$952$$ 3.15213 0.102161
$$953$$ −10.2107 −0.330758 −0.165379 0.986230i $$-0.552885\pi$$
−0.165379 + 0.986230i $$0.552885\pi$$
$$954$$ 5.74094i 0.185870i
$$955$$ 10.1806i 0.329437i
$$956$$ 11.9903i 0.387794i
$$957$$ 13.7476i 0.444398i
$$958$$ −32.9517 −1.06462
$$959$$ −72.3092 −2.33498
$$960$$ 1.00000i 0.0322749i
$$961$$ 14.2871 0.460873
$$962$$ 0 0
$$963$$ −16.1196 −0.519447
$$964$$ 21.7627i 0.700930i
$$965$$ −27.6407 −0.889786
$$966$$ −13.5918 −0.437309
$$967$$ − 34.6907i − 1.11558i −0.829983 0.557789i $$-0.811650\pi$$
0.829983 0.557789i $$-0.188350\pi$$
$$968$$ − 8.13706i − 0.261535i
$$969$$ 4.55496i 0.146326i
$$970$$ 15.4547i 0.496221i
$$971$$ 34.8039 1.11691 0.558454 0.829535i $$-0.311395\pi$$
0.558454 + 0.829535i $$0.311395\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ − 23.8267i − 0.763849i
$$974$$ 9.33034 0.298963
$$975$$ 0 0
$$976$$ 0.198062 0.00633982
$$977$$ 23.3086i 0.745707i 0.927890 + 0.372854i $$0.121621\pi$$
−0.927890 + 0.372854i $$0.878379\pi$$
$$978$$ 19.1511 0.612383
$$979$$ 16.0344 0.512463
$$980$$ − 4.26875i − 0.136360i
$$981$$ − 14.2325i − 0.454409i
$$982$$ 20.4349i 0.652103i
$$983$$ − 41.3639i − 1.31930i −0.751571 0.659652i $$-0.770704\pi$$
0.751571 0.659652i $$-0.229296\pi$$
$$984$$ 3.86294 0.123146
$$985$$ −13.0261 −0.415045
$$986$$ 7.62937i 0.242969i
$$987$$ 4.27711 0.136142
$$988$$ 0 0
$$989$$ 16.2838 0.517795
$$990$$ − 1.69202i − 0.0537760i
$$991$$ 49.4209 1.56991 0.784953 0.619555i $$-0.212687\pi$$
0.784953 + 0.619555i $$0.212687\pi$$
$$992$$ −4.08815 −0.129799
$$993$$ − 4.49827i − 0.142748i
$$994$$ − 17.3177i − 0.549283i
$$995$$ 6.67563i 0.211632i
$$996$$ − 13.1739i − 0.417431i
$$997$$ 44.1879 1.39944 0.699722 0.714415i $$-0.253307\pi$$
0.699722 + 0.714415i $$0.253307\pi$$
$$998$$ −3.54958 −0.112360
$$999$$ − 11.6310i − 0.367989i
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.z.1351.5 6
13.5 odd 4 5070.2.a.bw.1.2 yes 3
13.8 odd 4 5070.2.a.bp.1.2 3
13.12 even 2 inner 5070.2.b.z.1351.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bp.1.2 3 13.8 odd 4
5070.2.a.bw.1.2 yes 3 13.5 odd 4
5070.2.b.z.1351.2 6 13.12 even 2 inner
5070.2.b.z.1351.5 6 1.1 even 1 trivial