# Properties

 Label 4235.2.a.y.1.2 Level $4235$ Weight $2$ Character 4235.1 Self dual yes Analytic conductor $33.817$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4235 = 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4235.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.8166452560$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.173513.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.52979$$ of defining polynomial Character $$\chi$$ $$=$$ 4235.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +O(q^{10})$$ $$q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +1.65369 q^{10} -0.838578 q^{12} +5.60081 q^{13} -1.65369 q^{14} +1.14142 q^{15} -4.92960 q^{16} -4.57876 q^{17} +2.80656 q^{18} +1.88149 q^{19} -0.734678 q^{20} -1.14142 q^{21} +2.08099 q^{23} -2.38836 q^{24} +1.00000 q^{25} -9.26199 q^{26} +5.36144 q^{27} +0.734678 q^{28} -5.51585 q^{29} -1.88755 q^{30} -5.46081 q^{31} +3.96713 q^{32} +7.57184 q^{34} -1.00000 q^{35} -1.24686 q^{36} -9.34694 q^{37} -3.11139 q^{38} -6.39289 q^{39} -2.09245 q^{40} +11.0136 q^{41} +1.88755 q^{42} -8.43344 q^{43} +1.69716 q^{45} -3.44131 q^{46} +1.60936 q^{47} +5.62676 q^{48} +1.00000 q^{49} -1.65369 q^{50} +5.22630 q^{51} +4.11479 q^{52} +6.21331 q^{53} -8.86613 q^{54} +2.09245 q^{56} -2.14757 q^{57} +9.12149 q^{58} +13.5862 q^{59} +0.838578 q^{60} -11.1145 q^{61} +9.03046 q^{62} -1.69716 q^{63} +3.29883 q^{64} -5.60081 q^{65} -5.60924 q^{67} -3.36392 q^{68} -2.37529 q^{69} +1.65369 q^{70} +0.696594 q^{71} -3.55121 q^{72} +13.8873 q^{73} +15.4569 q^{74} -1.14142 q^{75} +1.38229 q^{76} +10.5718 q^{78} -8.47388 q^{79} +4.92960 q^{80} -1.02820 q^{81} -18.2131 q^{82} -9.50230 q^{83} -0.838578 q^{84} +4.57876 q^{85} +13.9463 q^{86} +6.29592 q^{87} +1.49863 q^{89} -2.80656 q^{90} +5.60081 q^{91} +1.52886 q^{92} +6.23309 q^{93} -2.66137 q^{94} -1.88149 q^{95} -4.52817 q^{96} +5.93697 q^{97} -1.65369 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + O(q^{10})$$ $$5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 11 q^{12} - 12 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 7 q^{18} - 9 q^{19} - 4 q^{20} - 2 q^{21} + 17 q^{23} - 6 q^{24} + 5 q^{25} - 11 q^{26} - 11 q^{27} + 4 q^{28} - 3 q^{29} + 5 q^{30} + 2 q^{31} + 5 q^{32} + 16 q^{34} - 5 q^{35} - 15 q^{36} + 4 q^{37} - 11 q^{38} - 2 q^{39} - 6 q^{40} + 15 q^{41} - 5 q^{42} - 4 q^{43} - 3 q^{45} + 10 q^{46} - 2 q^{47} - 10 q^{48} + 5 q^{49} - 2 q^{50} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{58} - 6 q^{59} - 11 q^{60} - 20 q^{61} - 21 q^{62} + 3 q^{63} - 26 q^{64} + 12 q^{65} + 3 q^{67} - 5 q^{68} + 2 q^{70} - 6 q^{71} - 34 q^{72} - 11 q^{73} + 15 q^{74} - 2 q^{75} - 47 q^{76} + 31 q^{78} - 19 q^{79} - 2 q^{80} + 33 q^{81} - 8 q^{83} + 11 q^{84} + 14 q^{85} + 27 q^{86} + 30 q^{87} + q^{89} + 7 q^{90} - 12 q^{91} + 44 q^{92} + 3 q^{93} - 28 q^{94} + 9 q^{95} + 4 q^{96} - 7 q^{97} - 2 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.65369 −1.16933 −0.584666 0.811274i $$-0.698775\pi$$
−0.584666 + 0.811274i $$0.698775\pi$$
$$3$$ −1.14142 −0.659000 −0.329500 0.944156i $$-0.606880\pi$$
−0.329500 + 0.944156i $$0.606880\pi$$
$$4$$ 0.734678 0.367339
$$5$$ −1.00000 −0.447214
$$6$$ 1.88755 0.770591
$$7$$ 1.00000 0.377964
$$8$$ 2.09245 0.739791
$$9$$ −1.69716 −0.565718
$$10$$ 1.65369 0.522941
$$11$$ 0 0
$$12$$ −0.838578 −0.242076
$$13$$ 5.60081 1.55339 0.776693 0.629879i $$-0.216896\pi$$
0.776693 + 0.629879i $$0.216896\pi$$
$$14$$ −1.65369 −0.441966
$$15$$ 1.14142 0.294714
$$16$$ −4.92960 −1.23240
$$17$$ −4.57876 −1.11051 −0.555257 0.831679i $$-0.687380\pi$$
−0.555257 + 0.831679i $$0.687380\pi$$
$$18$$ 2.80656 0.661513
$$19$$ 1.88149 0.431642 0.215821 0.976433i $$-0.430757\pi$$
0.215821 + 0.976433i $$0.430757\pi$$
$$20$$ −0.734678 −0.164279
$$21$$ −1.14142 −0.249079
$$22$$ 0 0
$$23$$ 2.08099 0.433917 0.216958 0.976181i $$-0.430386\pi$$
0.216958 + 0.976181i $$0.430386\pi$$
$$24$$ −2.38836 −0.487523
$$25$$ 1.00000 0.200000
$$26$$ −9.26199 −1.81642
$$27$$ 5.36144 1.03181
$$28$$ 0.734678 0.138841
$$29$$ −5.51585 −1.02427 −0.512134 0.858905i $$-0.671145\pi$$
−0.512134 + 0.858905i $$0.671145\pi$$
$$30$$ −1.88755 −0.344619
$$31$$ −5.46081 −0.980790 −0.490395 0.871500i $$-0.663148\pi$$
−0.490395 + 0.871500i $$0.663148\pi$$
$$32$$ 3.96713 0.701296
$$33$$ 0 0
$$34$$ 7.57184 1.29856
$$35$$ −1.00000 −0.169031
$$36$$ −1.24686 −0.207810
$$37$$ −9.34694 −1.53663 −0.768314 0.640073i $$-0.778904\pi$$
−0.768314 + 0.640073i $$0.778904\pi$$
$$38$$ −3.11139 −0.504734
$$39$$ −6.39289 −1.02368
$$40$$ −2.09245 −0.330845
$$41$$ 11.0136 1.72004 0.860020 0.510261i $$-0.170451\pi$$
0.860020 + 0.510261i $$0.170451\pi$$
$$42$$ 1.88755 0.291256
$$43$$ −8.43344 −1.28609 −0.643044 0.765829i $$-0.722329\pi$$
−0.643044 + 0.765829i $$0.722329\pi$$
$$44$$ 0 0
$$45$$ 1.69716 0.252997
$$46$$ −3.44131 −0.507393
$$47$$ 1.60936 0.234749 0.117375 0.993088i $$-0.462552\pi$$
0.117375 + 0.993088i $$0.462552\pi$$
$$48$$ 5.62676 0.812153
$$49$$ 1.00000 0.142857
$$50$$ −1.65369 −0.233867
$$51$$ 5.22630 0.731829
$$52$$ 4.11479 0.570619
$$53$$ 6.21331 0.853463 0.426732 0.904378i $$-0.359665\pi$$
0.426732 + 0.904378i $$0.359665\pi$$
$$54$$ −8.86613 −1.20653
$$55$$ 0 0
$$56$$ 2.09245 0.279615
$$57$$ −2.14757 −0.284453
$$58$$ 9.12149 1.19771
$$59$$ 13.5862 1.76877 0.884386 0.466756i $$-0.154577\pi$$
0.884386 + 0.466756i $$0.154577\pi$$
$$60$$ 0.838578 0.108260
$$61$$ −11.1145 −1.42307 −0.711533 0.702653i $$-0.751999\pi$$
−0.711533 + 0.702653i $$0.751999\pi$$
$$62$$ 9.03046 1.14687
$$63$$ −1.69716 −0.213821
$$64$$ 3.29883 0.412353
$$65$$ −5.60081 −0.694695
$$66$$ 0 0
$$67$$ −5.60924 −0.685277 −0.342639 0.939467i $$-0.611321\pi$$
−0.342639 + 0.939467i $$0.611321\pi$$
$$68$$ −3.36392 −0.407935
$$69$$ −2.37529 −0.285951
$$70$$ 1.65369 0.197653
$$71$$ 0.696594 0.0826705 0.0413353 0.999145i $$-0.486839\pi$$
0.0413353 + 0.999145i $$0.486839\pi$$
$$72$$ −3.55121 −0.418514
$$73$$ 13.8873 1.62539 0.812694 0.582690i $$-0.198000\pi$$
0.812694 + 0.582690i $$0.198000\pi$$
$$74$$ 15.4569 1.79683
$$75$$ −1.14142 −0.131800
$$76$$ 1.38229 0.158559
$$77$$ 0 0
$$78$$ 10.5718 1.19702
$$79$$ −8.47388 −0.953386 −0.476693 0.879070i $$-0.658165\pi$$
−0.476693 + 0.879070i $$0.658165\pi$$
$$80$$ 4.92960 0.551146
$$81$$ −1.02820 −0.114244
$$82$$ −18.2131 −2.01130
$$83$$ −9.50230 −1.04301 −0.521506 0.853247i $$-0.674630\pi$$
−0.521506 + 0.853247i $$0.674630\pi$$
$$84$$ −0.838578 −0.0914963
$$85$$ 4.57876 0.496637
$$86$$ 13.9463 1.50386
$$87$$ 6.29592 0.674993
$$88$$ 0 0
$$89$$ 1.49863 0.158854 0.0794272 0.996841i $$-0.474691\pi$$
0.0794272 + 0.996841i $$0.474691\pi$$
$$90$$ −2.80656 −0.295838
$$91$$ 5.60081 0.587125
$$92$$ 1.52886 0.159395
$$93$$ 6.23309 0.646341
$$94$$ −2.66137 −0.274500
$$95$$ −1.88149 −0.193036
$$96$$ −4.52817 −0.462154
$$97$$ 5.93697 0.602808 0.301404 0.953497i $$-0.402545\pi$$
0.301404 + 0.953497i $$0.402545\pi$$
$$98$$ −1.65369 −0.167048
$$99$$ 0 0
$$100$$ 0.734678 0.0734678
$$101$$ −3.15826 −0.314259 −0.157129 0.987578i $$-0.550224\pi$$
−0.157129 + 0.987578i $$0.550224\pi$$
$$102$$ −8.64266 −0.855751
$$103$$ 1.24686 0.122857 0.0614285 0.998111i $$-0.480434\pi$$
0.0614285 + 0.998111i $$0.480434\pi$$
$$104$$ 11.7194 1.14918
$$105$$ 1.14142 0.111391
$$106$$ −10.2749 −0.997983
$$107$$ 10.2684 0.992687 0.496343 0.868126i $$-0.334676\pi$$
0.496343 + 0.868126i $$0.334676\pi$$
$$108$$ 3.93893 0.379024
$$109$$ −5.67907 −0.543956 −0.271978 0.962303i $$-0.587678\pi$$
−0.271978 + 0.962303i $$0.587678\pi$$
$$110$$ 0 0
$$111$$ 10.6688 1.01264
$$112$$ −4.92960 −0.465804
$$113$$ 17.9797 1.69139 0.845694 0.533669i $$-0.179187\pi$$
0.845694 + 0.533669i $$0.179187\pi$$
$$114$$ 3.55141 0.332620
$$115$$ −2.08099 −0.194053
$$116$$ −4.05238 −0.376254
$$117$$ −9.50545 −0.878779
$$118$$ −22.4673 −2.06828
$$119$$ −4.57876 −0.419734
$$120$$ 2.38836 0.218027
$$121$$ 0 0
$$122$$ 18.3799 1.66404
$$123$$ −12.5712 −1.13351
$$124$$ −4.01194 −0.360282
$$125$$ −1.00000 −0.0894427
$$126$$ 2.80656 0.250028
$$127$$ −12.0909 −1.07289 −0.536447 0.843934i $$-0.680234\pi$$
−0.536447 + 0.843934i $$0.680234\pi$$
$$128$$ −13.3895 −1.18347
$$129$$ 9.62612 0.847533
$$130$$ 9.26199 0.812330
$$131$$ −1.04441 −0.0912502 −0.0456251 0.998959i $$-0.514528\pi$$
−0.0456251 + 0.998959i $$0.514528\pi$$
$$132$$ 0 0
$$133$$ 1.88149 0.163146
$$134$$ 9.27592 0.801317
$$135$$ −5.36144 −0.461439
$$136$$ −9.58081 −0.821548
$$137$$ −1.43820 −0.122874 −0.0614368 0.998111i $$-0.519568\pi$$
−0.0614368 + 0.998111i $$0.519568\pi$$
$$138$$ 3.92798 0.334372
$$139$$ 21.6489 1.83623 0.918117 0.396310i $$-0.129709\pi$$
0.918117 + 0.396310i $$0.129709\pi$$
$$140$$ −0.734678 −0.0620916
$$141$$ −1.83696 −0.154700
$$142$$ −1.15195 −0.0966694
$$143$$ 0 0
$$144$$ 8.36630 0.697192
$$145$$ 5.51585 0.458067
$$146$$ −22.9653 −1.90062
$$147$$ −1.14142 −0.0941429
$$148$$ −6.86699 −0.564463
$$149$$ 12.6540 1.03666 0.518329 0.855181i $$-0.326554\pi$$
0.518329 + 0.855181i $$0.326554\pi$$
$$150$$ 1.88755 0.154118
$$151$$ −2.48546 −0.202264 −0.101132 0.994873i $$-0.532246\pi$$
−0.101132 + 0.994873i $$0.532246\pi$$
$$152$$ 3.93691 0.319325
$$153$$ 7.77087 0.628238
$$154$$ 0 0
$$155$$ 5.46081 0.438623
$$156$$ −4.69672 −0.376038
$$157$$ −14.2773 −1.13945 −0.569727 0.821834i $$-0.692951\pi$$
−0.569727 + 0.821834i $$0.692951\pi$$
$$158$$ 14.0131 1.11483
$$159$$ −7.09201 −0.562433
$$160$$ −3.96713 −0.313629
$$161$$ 2.08099 0.164005
$$162$$ 1.70032 0.133589
$$163$$ −9.75758 −0.764272 −0.382136 0.924106i $$-0.624811\pi$$
−0.382136 + 0.924106i $$0.624811\pi$$
$$164$$ 8.09147 0.631837
$$165$$ 0 0
$$166$$ 15.7138 1.21963
$$167$$ −11.3890 −0.881308 −0.440654 0.897677i $$-0.645253\pi$$
−0.440654 + 0.897677i $$0.645253\pi$$
$$168$$ −2.38836 −0.184266
$$169$$ 18.3691 1.41301
$$170$$ −7.57184 −0.580733
$$171$$ −3.19317 −0.244188
$$172$$ −6.19586 −0.472430
$$173$$ −10.3544 −0.787227 −0.393613 0.919276i $$-0.628775\pi$$
−0.393613 + 0.919276i $$0.628775\pi$$
$$174$$ −10.4115 −0.789292
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −15.5076 −1.16562
$$178$$ −2.47826 −0.185754
$$179$$ 18.8195 1.40664 0.703318 0.710875i $$-0.251701\pi$$
0.703318 + 0.710875i $$0.251701\pi$$
$$180$$ 1.24686 0.0929356
$$181$$ −7.75369 −0.576327 −0.288164 0.957581i $$-0.593045\pi$$
−0.288164 + 0.957581i $$0.593045\pi$$
$$182$$ −9.26199 −0.686544
$$183$$ 12.6863 0.937801
$$184$$ 4.35436 0.321008
$$185$$ 9.34694 0.687201
$$186$$ −10.3076 −0.755788
$$187$$ 0 0
$$188$$ 1.18236 0.0862325
$$189$$ 5.36144 0.389987
$$190$$ 3.11139 0.225724
$$191$$ 15.9410 1.15345 0.576723 0.816939i $$-0.304331\pi$$
0.576723 + 0.816939i $$0.304331\pi$$
$$192$$ −3.76535 −0.271741
$$193$$ −8.52230 −0.613448 −0.306724 0.951798i $$-0.599233\pi$$
−0.306724 + 0.951798i $$0.599233\pi$$
$$194$$ −9.81789 −0.704883
$$195$$ 6.39289 0.457804
$$196$$ 0.734678 0.0524770
$$197$$ 19.0556 1.35765 0.678826 0.734299i $$-0.262489\pi$$
0.678826 + 0.734299i $$0.262489\pi$$
$$198$$ 0 0
$$199$$ −12.7118 −0.901118 −0.450559 0.892747i $$-0.648775\pi$$
−0.450559 + 0.892747i $$0.648775\pi$$
$$200$$ 2.09245 0.147958
$$201$$ 6.40251 0.451598
$$202$$ 5.22277 0.367473
$$203$$ −5.51585 −0.387137
$$204$$ 3.83965 0.268829
$$205$$ −11.0136 −0.769225
$$206$$ −2.06192 −0.143661
$$207$$ −3.53177 −0.245475
$$208$$ −27.6098 −1.91439
$$209$$ 0 0
$$210$$ −1.88755 −0.130254
$$211$$ 28.4506 1.95862 0.979311 0.202363i $$-0.0648620\pi$$
0.979311 + 0.202363i $$0.0648620\pi$$
$$212$$ 4.56478 0.313510
$$213$$ −0.795108 −0.0544799
$$214$$ −16.9808 −1.16078
$$215$$ 8.43344 0.575156
$$216$$ 11.2185 0.763323
$$217$$ −5.46081 −0.370704
$$218$$ 9.39140 0.636066
$$219$$ −15.8513 −1.07113
$$220$$ 0 0
$$221$$ −25.6448 −1.72506
$$222$$ −17.6429 −1.18411
$$223$$ −23.9632 −1.60470 −0.802348 0.596856i $$-0.796416\pi$$
−0.802348 + 0.596856i $$0.796416\pi$$
$$224$$ 3.96713 0.265065
$$225$$ −1.69716 −0.113144
$$226$$ −29.7328 −1.97779
$$227$$ −20.6007 −1.36732 −0.683658 0.729803i $$-0.739612\pi$$
−0.683658 + 0.729803i $$0.739612\pi$$
$$228$$ −1.57777 −0.104490
$$229$$ 0.903360 0.0596957 0.0298479 0.999554i $$-0.490498\pi$$
0.0298479 + 0.999554i $$0.490498\pi$$
$$230$$ 3.44131 0.226913
$$231$$ 0 0
$$232$$ −11.5416 −0.757745
$$233$$ −23.2459 −1.52289 −0.761444 0.648231i $$-0.775509\pi$$
−0.761444 + 0.648231i $$0.775509\pi$$
$$234$$ 15.7190 1.02759
$$235$$ −1.60936 −0.104983
$$236$$ 9.98148 0.649739
$$237$$ 9.67228 0.628282
$$238$$ 7.57184 0.490809
$$239$$ −8.21214 −0.531199 −0.265600 0.964083i $$-0.585570\pi$$
−0.265600 + 0.964083i $$0.585570\pi$$
$$240$$ −5.62676 −0.363206
$$241$$ 8.33040 0.536608 0.268304 0.963334i $$-0.413537\pi$$
0.268304 + 0.963334i $$0.413537\pi$$
$$242$$ 0 0
$$243$$ −14.9107 −0.956522
$$244$$ −8.16557 −0.522747
$$245$$ −1.00000 −0.0638877
$$246$$ 20.7888 1.32545
$$247$$ 10.5378 0.670507
$$248$$ −11.4264 −0.725580
$$249$$ 10.8461 0.687346
$$250$$ 1.65369 0.104588
$$251$$ −9.77541 −0.617018 −0.308509 0.951221i $$-0.599830\pi$$
−0.308509 + 0.951221i $$0.599830\pi$$
$$252$$ −1.24686 −0.0785450
$$253$$ 0 0
$$254$$ 19.9946 1.25457
$$255$$ −5.22630 −0.327284
$$256$$ 15.5443 0.971521
$$257$$ 4.10056 0.255786 0.127893 0.991788i $$-0.459179\pi$$
0.127893 + 0.991788i $$0.459179\pi$$
$$258$$ −15.9186 −0.991047
$$259$$ −9.34694 −0.580791
$$260$$ −4.11479 −0.255189
$$261$$ 9.36126 0.579448
$$262$$ 1.72712 0.106702
$$263$$ −25.6453 −1.58136 −0.790678 0.612232i $$-0.790272\pi$$
−0.790678 + 0.612232i $$0.790272\pi$$
$$264$$ 0 0
$$265$$ −6.21331 −0.381680
$$266$$ −3.11139 −0.190771
$$267$$ −1.71057 −0.104685
$$268$$ −4.12098 −0.251729
$$269$$ 13.7027 0.835470 0.417735 0.908569i $$-0.362824\pi$$
0.417735 + 0.908569i $$0.362824\pi$$
$$270$$ 8.86613 0.539576
$$271$$ 8.60468 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$272$$ 22.5715 1.36860
$$273$$ −6.39289 −0.386915
$$274$$ 2.37833 0.143680
$$275$$ 0 0
$$276$$ −1.74507 −0.105041
$$277$$ −28.8920 −1.73595 −0.867975 0.496607i $$-0.834579\pi$$
−0.867975 + 0.496607i $$0.834579\pi$$
$$278$$ −35.8005 −2.14717
$$279$$ 9.26784 0.554851
$$280$$ −2.09245 −0.125048
$$281$$ −7.98397 −0.476284 −0.238142 0.971230i $$-0.576538\pi$$
−0.238142 + 0.971230i $$0.576538\pi$$
$$282$$ 3.03775 0.180895
$$283$$ 7.39498 0.439586 0.219793 0.975547i $$-0.429462\pi$$
0.219793 + 0.975547i $$0.429462\pi$$
$$284$$ 0.511772 0.0303681
$$285$$ 2.14757 0.127211
$$286$$ 0 0
$$287$$ 11.0136 0.650114
$$288$$ −6.73283 −0.396736
$$289$$ 3.96507 0.233239
$$290$$ −9.12149 −0.535632
$$291$$ −6.77659 −0.397251
$$292$$ 10.2027 0.597068
$$293$$ −13.7881 −0.805508 −0.402754 0.915308i $$-0.631947\pi$$
−0.402754 + 0.915308i $$0.631947\pi$$
$$294$$ 1.88755 0.110084
$$295$$ −13.5862 −0.791019
$$296$$ −19.5580 −1.13678
$$297$$ 0 0
$$298$$ −20.9258 −1.21220
$$299$$ 11.6552 0.674040
$$300$$ −0.838578 −0.0484153
$$301$$ −8.43344 −0.486096
$$302$$ 4.11017 0.236514
$$303$$ 3.60491 0.207097
$$304$$ −9.27498 −0.531957
$$305$$ 11.1145 0.636414
$$306$$ −12.8506 −0.734619
$$307$$ 5.89759 0.336593 0.168297 0.985736i $$-0.446173\pi$$
0.168297 + 0.985736i $$0.446173\pi$$
$$308$$ 0 0
$$309$$ −1.42320 −0.0809628
$$310$$ −9.03046 −0.512896
$$311$$ −26.0053 −1.47462 −0.737311 0.675553i $$-0.763905\pi$$
−0.737311 + 0.675553i $$0.763905\pi$$
$$312$$ −13.3768 −0.757311
$$313$$ −12.1886 −0.688938 −0.344469 0.938798i $$-0.611941\pi$$
−0.344469 + 0.938798i $$0.611941\pi$$
$$314$$ 23.6102 1.33240
$$315$$ 1.69716 0.0956239
$$316$$ −6.22557 −0.350216
$$317$$ 13.5894 0.763258 0.381629 0.924316i $$-0.375363\pi$$
0.381629 + 0.924316i $$0.375363\pi$$
$$318$$ 11.7280 0.657671
$$319$$ 0 0
$$320$$ −3.29883 −0.184410
$$321$$ −11.7206 −0.654181
$$322$$ −3.44131 −0.191777
$$323$$ −8.61488 −0.479345
$$324$$ −0.755394 −0.0419663
$$325$$ 5.60081 0.310677
$$326$$ 16.1360 0.893689
$$327$$ 6.48222 0.358468
$$328$$ 23.0454 1.27247
$$329$$ 1.60936 0.0887268
$$330$$ 0 0
$$331$$ 18.3402 1.00807 0.504034 0.863684i $$-0.331849\pi$$
0.504034 + 0.863684i $$0.331849\pi$$
$$332$$ −6.98113 −0.383139
$$333$$ 15.8632 0.869299
$$334$$ 18.8338 1.03054
$$335$$ 5.60924 0.306465
$$336$$ 5.62676 0.306965
$$337$$ −25.2973 −1.37803 −0.689015 0.724747i $$-0.741957\pi$$
−0.689015 + 0.724747i $$0.741957\pi$$
$$338$$ −30.3767 −1.65228
$$339$$ −20.5224 −1.11462
$$340$$ 3.36392 0.182434
$$341$$ 0 0
$$342$$ 5.28051 0.285537
$$343$$ 1.00000 0.0539949
$$344$$ −17.6465 −0.951437
$$345$$ 2.37529 0.127881
$$346$$ 17.1228 0.920530
$$347$$ −14.4944 −0.778098 −0.389049 0.921217i $$-0.627196\pi$$
−0.389049 + 0.921217i $$0.627196\pi$$
$$348$$ 4.62547 0.247951
$$349$$ −2.97717 −0.159364 −0.0796822 0.996820i $$-0.525391\pi$$
−0.0796822 + 0.996820i $$0.525391\pi$$
$$350$$ −1.65369 −0.0883932
$$351$$ 30.0284 1.60280
$$352$$ 0 0
$$353$$ 14.0782 0.749304 0.374652 0.927165i $$-0.377762\pi$$
0.374652 + 0.927165i $$0.377762\pi$$
$$354$$ 25.6447 1.36300
$$355$$ −0.696594 −0.0369714
$$356$$ 1.10101 0.0583534
$$357$$ 5.22630 0.276605
$$358$$ −31.1216 −1.64483
$$359$$ −15.3313 −0.809154 −0.404577 0.914504i $$-0.632581\pi$$
−0.404577 + 0.914504i $$0.632581\pi$$
$$360$$ 3.55121 0.187165
$$361$$ −15.4600 −0.813685
$$362$$ 12.8222 0.673918
$$363$$ 0 0
$$364$$ 4.11479 0.215674
$$365$$ −13.8873 −0.726896
$$366$$ −20.9792 −1.09660
$$367$$ 13.7180 0.716072 0.358036 0.933708i $$-0.383447\pi$$
0.358036 + 0.933708i $$0.383447\pi$$
$$368$$ −10.2585 −0.534759
$$369$$ −18.6918 −0.973058
$$370$$ −15.4569 −0.803567
$$371$$ 6.21331 0.322579
$$372$$ 4.57931 0.237426
$$373$$ −12.6631 −0.655670 −0.327835 0.944735i $$-0.606319\pi$$
−0.327835 + 0.944735i $$0.606319\pi$$
$$374$$ 0 0
$$375$$ 1.14142 0.0589428
$$376$$ 3.36750 0.173665
$$377$$ −30.8933 −1.59108
$$378$$ −8.86613 −0.456025
$$379$$ 12.0373 0.618314 0.309157 0.951011i $$-0.399953\pi$$
0.309157 + 0.951011i $$0.399953\pi$$
$$380$$ −1.38229 −0.0709098
$$381$$ 13.8008 0.707038
$$382$$ −26.3613 −1.34876
$$383$$ −9.96182 −0.509025 −0.254513 0.967069i $$-0.581915\pi$$
−0.254513 + 0.967069i $$0.581915\pi$$
$$384$$ 15.2830 0.779910
$$385$$ 0 0
$$386$$ 14.0932 0.717325
$$387$$ 14.3129 0.727564
$$388$$ 4.36176 0.221435
$$389$$ 22.1252 1.12179 0.560895 0.827887i $$-0.310457\pi$$
0.560895 + 0.827887i $$0.310457\pi$$
$$390$$ −10.5718 −0.535326
$$391$$ −9.52837 −0.481870
$$392$$ 2.09245 0.105684
$$393$$ 1.19211 0.0601339
$$394$$ −31.5119 −1.58755
$$395$$ 8.47388 0.426367
$$396$$ 0 0
$$397$$ −26.4968 −1.32984 −0.664918 0.746916i $$-0.731533\pi$$
−0.664918 + 0.746916i $$0.731533\pi$$
$$398$$ 21.0214 1.05371
$$399$$ −2.14757 −0.107513
$$400$$ −4.92960 −0.246480
$$401$$ −2.98212 −0.148920 −0.0744600 0.997224i $$-0.523723\pi$$
−0.0744600 + 0.997224i $$0.523723\pi$$
$$402$$ −10.5877 −0.528068
$$403$$ −30.5850 −1.52355
$$404$$ −2.32031 −0.115439
$$405$$ 1.02820 0.0510915
$$406$$ 9.12149 0.452692
$$407$$ 0 0
$$408$$ 10.9358 0.541400
$$409$$ −1.71895 −0.0849966 −0.0424983 0.999097i $$-0.513532\pi$$
−0.0424983 + 0.999097i $$0.513532\pi$$
$$410$$ 18.2131 0.899480
$$411$$ 1.64159 0.0809738
$$412$$ 0.916042 0.0451302
$$413$$ 13.5862 0.668533
$$414$$ 5.84043 0.287042
$$415$$ 9.50230 0.466449
$$416$$ 22.2191 1.08938
$$417$$ −24.7105 −1.21008
$$418$$ 0 0
$$419$$ −24.6056 −1.20206 −0.601031 0.799226i $$-0.705243\pi$$
−0.601031 + 0.799226i $$0.705243\pi$$
$$420$$ 0.838578 0.0409184
$$421$$ −29.8732 −1.45593 −0.727965 0.685615i $$-0.759533\pi$$
−0.727965 + 0.685615i $$0.759533\pi$$
$$422$$ −47.0484 −2.29028
$$423$$ −2.73133 −0.132802
$$424$$ 13.0010 0.631385
$$425$$ −4.57876 −0.222103
$$426$$ 1.31486 0.0637052
$$427$$ −11.1145 −0.537868
$$428$$ 7.54399 0.364652
$$429$$ 0 0
$$430$$ −13.9463 −0.672549
$$431$$ 15.5900 0.750943 0.375472 0.926834i $$-0.377481\pi$$
0.375472 + 0.926834i $$0.377481\pi$$
$$432$$ −26.4298 −1.27160
$$433$$ 10.6888 0.513671 0.256835 0.966455i $$-0.417320\pi$$
0.256835 + 0.966455i $$0.417320\pi$$
$$434$$ 9.03046 0.433476
$$435$$ −6.29592 −0.301866
$$436$$ −4.17229 −0.199816
$$437$$ 3.91536 0.187297
$$438$$ 26.2131 1.25251
$$439$$ −18.1537 −0.866427 −0.433214 0.901291i $$-0.642620\pi$$
−0.433214 + 0.901291i $$0.642620\pi$$
$$440$$ 0 0
$$441$$ −1.69716 −0.0808169
$$442$$ 42.4084 2.01716
$$443$$ −9.93668 −0.472106 −0.236053 0.971740i $$-0.575854\pi$$
−0.236053 + 0.971740i $$0.575854\pi$$
$$444$$ 7.83814 0.371982
$$445$$ −1.49863 −0.0710418
$$446$$ 39.6277 1.87642
$$447$$ −14.4436 −0.683158
$$448$$ 3.29883 0.155855
$$449$$ −18.0920 −0.853813 −0.426906 0.904296i $$-0.640397\pi$$
−0.426906 + 0.904296i $$0.640397\pi$$
$$450$$ 2.80656 0.132303
$$451$$ 0 0
$$452$$ 13.2093 0.621312
$$453$$ 2.83696 0.133292
$$454$$ 34.0671 1.59885
$$455$$ −5.60081 −0.262570
$$456$$ −4.49367 −0.210436
$$457$$ 15.9901 0.747987 0.373993 0.927431i $$-0.377988\pi$$
0.373993 + 0.927431i $$0.377988\pi$$
$$458$$ −1.49387 −0.0698041
$$459$$ −24.5487 −1.14584
$$460$$ −1.52886 −0.0712834
$$461$$ −27.9612 −1.30228 −0.651140 0.758957i $$-0.725709\pi$$
−0.651140 + 0.758957i $$0.725709\pi$$
$$462$$ 0 0
$$463$$ −30.8389 −1.43320 −0.716602 0.697482i $$-0.754304\pi$$
−0.716602 + 0.697482i $$0.754304\pi$$
$$464$$ 27.1910 1.26231
$$465$$ −6.23309 −0.289053
$$466$$ 38.4414 1.78076
$$467$$ −8.41735 −0.389509 −0.194754 0.980852i $$-0.562391\pi$$
−0.194754 + 0.980852i $$0.562391\pi$$
$$468$$ −6.98344 −0.322810
$$469$$ −5.60924 −0.259011
$$470$$ 2.66137 0.122760
$$471$$ 16.2964 0.750900
$$472$$ 28.4284 1.30852
$$473$$ 0 0
$$474$$ −15.9949 −0.734671
$$475$$ 1.88149 0.0863285
$$476$$ −3.36392 −0.154185
$$477$$ −10.5449 −0.482820
$$478$$ 13.5803 0.621149
$$479$$ −29.6865 −1.35641 −0.678205 0.734873i $$-0.737242\pi$$
−0.678205 + 0.734873i $$0.737242\pi$$
$$480$$ 4.52817 0.206682
$$481$$ −52.3505 −2.38698
$$482$$ −13.7759 −0.627474
$$483$$ −2.37529 −0.108079
$$484$$ 0 0
$$485$$ −5.93697 −0.269584
$$486$$ 24.6576 1.11849
$$487$$ 37.8313 1.71430 0.857150 0.515067i $$-0.172233\pi$$
0.857150 + 0.515067i $$0.172233\pi$$
$$488$$ −23.2565 −1.05277
$$489$$ 11.1375 0.503656
$$490$$ 1.65369 0.0747059
$$491$$ −18.6985 −0.843850 −0.421925 0.906631i $$-0.638645\pi$$
−0.421925 + 0.906631i $$0.638645\pi$$
$$492$$ −9.23578 −0.416381
$$493$$ 25.2558 1.13746
$$494$$ −17.4263 −0.784046
$$495$$ 0 0
$$496$$ 26.9196 1.20873
$$497$$ 0.696594 0.0312465
$$498$$ −17.9361 −0.803736
$$499$$ 22.8758 1.02406 0.512030 0.858967i $$-0.328894\pi$$
0.512030 + 0.858967i $$0.328894\pi$$
$$500$$ −0.734678 −0.0328558
$$501$$ 12.9997 0.580782
$$502$$ 16.1655 0.721499
$$503$$ 11.6087 0.517607 0.258804 0.965930i $$-0.416672\pi$$
0.258804 + 0.965930i $$0.416672\pi$$
$$504$$ −3.55121 −0.158183
$$505$$ 3.15826 0.140541
$$506$$ 0 0
$$507$$ −20.9669 −0.931173
$$508$$ −8.88292 −0.394116
$$509$$ −13.8562 −0.614163 −0.307082 0.951683i $$-0.599353\pi$$
−0.307082 + 0.951683i $$0.599353\pi$$
$$510$$ 8.64266 0.382704
$$511$$ 13.8873 0.614339
$$512$$ 1.07349 0.0474422
$$513$$ 10.0875 0.445373
$$514$$ −6.78105 −0.299099
$$515$$ −1.24686 −0.0549433
$$516$$ 7.07210 0.311332
$$517$$ 0 0
$$518$$ 15.4569 0.679138
$$519$$ 11.8187 0.518783
$$520$$ −11.7194 −0.513930
$$521$$ −25.2028 −1.10415 −0.552077 0.833793i $$-0.686165\pi$$
−0.552077 + 0.833793i $$0.686165\pi$$
$$522$$ −15.4806 −0.677567
$$523$$ 28.9229 1.26471 0.632355 0.774679i $$-0.282089\pi$$
0.632355 + 0.774679i $$0.282089\pi$$
$$524$$ −0.767303 −0.0335198
$$525$$ −1.14142 −0.0498157
$$526$$ 42.4093 1.84913
$$527$$ 25.0038 1.08918
$$528$$ 0 0
$$529$$ −18.6695 −0.811716
$$530$$ 10.2749 0.446311
$$531$$ −23.0579 −1.00063
$$532$$ 1.38229 0.0599297
$$533$$ 61.6852 2.67188
$$534$$ 2.82874 0.122412
$$535$$ −10.2684 −0.443943
$$536$$ −11.7370 −0.506962
$$537$$ −21.4810 −0.926974
$$538$$ −22.6600 −0.976943
$$539$$ 0 0
$$540$$ −3.93893 −0.169505
$$541$$ −13.9092 −0.598003 −0.299001 0.954253i $$-0.596653\pi$$
−0.299001 + 0.954253i $$0.596653\pi$$
$$542$$ −14.2294 −0.611207
$$543$$ 8.85023 0.379800
$$544$$ −18.1645 −0.778798
$$545$$ 5.67907 0.243265
$$546$$ 10.5718 0.452433
$$547$$ −12.4603 −0.532766 −0.266383 0.963867i $$-0.585829\pi$$
−0.266383 + 0.963867i $$0.585829\pi$$
$$548$$ −1.05661 −0.0451363
$$549$$ 18.8630 0.805054
$$550$$ 0 0
$$551$$ −10.3780 −0.442118
$$552$$ −4.97017 −0.211544
$$553$$ −8.47388 −0.360346
$$554$$ 47.7783 2.02990
$$555$$ −10.6688 −0.452866
$$556$$ 15.9050 0.674520
$$557$$ −37.5067 −1.58921 −0.794605 0.607127i $$-0.792322\pi$$
−0.794605 + 0.607127i $$0.792322\pi$$
$$558$$ −15.3261 −0.648806
$$559$$ −47.2341 −1.99779
$$560$$ 4.92960 0.208314
$$561$$ 0 0
$$562$$ 13.2030 0.556934
$$563$$ 24.5527 1.03477 0.517387 0.855752i $$-0.326905\pi$$
0.517387 + 0.855752i $$0.326905\pi$$
$$564$$ −1.34957 −0.0568272
$$565$$ −17.9797 −0.756411
$$566$$ −12.2290 −0.514022
$$567$$ −1.02820 −0.0431802
$$568$$ 1.45759 0.0611590
$$569$$ 39.6609 1.66267 0.831336 0.555771i $$-0.187577\pi$$
0.831336 + 0.555771i $$0.187577\pi$$
$$570$$ −3.55141 −0.148752
$$571$$ −31.3014 −1.30992 −0.654961 0.755663i $$-0.727315\pi$$
−0.654961 + 0.755663i $$0.727315\pi$$
$$572$$ 0 0
$$573$$ −18.1954 −0.760122
$$574$$ −18.2131 −0.760199
$$575$$ 2.08099 0.0867834
$$576$$ −5.59862 −0.233276
$$577$$ −33.8941 −1.41103 −0.705516 0.708694i $$-0.749285\pi$$
−0.705516 + 0.708694i $$0.749285\pi$$
$$578$$ −6.55698 −0.272734
$$579$$ 9.72754 0.404263
$$580$$ 4.05238 0.168266
$$581$$ −9.50230 −0.394222
$$582$$ 11.2064 0.464518
$$583$$ 0 0
$$584$$ 29.0585 1.20245
$$585$$ 9.50545 0.393002
$$586$$ 22.8011 0.941907
$$587$$ −32.3331 −1.33453 −0.667264 0.744821i $$-0.732535\pi$$
−0.667264 + 0.744821i $$0.732535\pi$$
$$588$$ −0.838578 −0.0345824
$$589$$ −10.2744 −0.423351
$$590$$ 22.4673 0.924964
$$591$$ −21.7504 −0.894693
$$592$$ 46.0767 1.89374
$$593$$ −25.6741 −1.05431 −0.527153 0.849770i $$-0.676741\pi$$
−0.527153 + 0.849770i $$0.676741\pi$$
$$594$$ 0 0
$$595$$ 4.57876 0.187711
$$596$$ 9.29663 0.380805
$$597$$ 14.5096 0.593837
$$598$$ −19.2741 −0.788177
$$599$$ −21.7518 −0.888756 −0.444378 0.895839i $$-0.646575\pi$$
−0.444378 + 0.895839i $$0.646575\pi$$
$$600$$ −2.38836 −0.0975046
$$601$$ 15.6850 0.639806 0.319903 0.947450i $$-0.396350\pi$$
0.319903 + 0.947450i $$0.396350\pi$$
$$602$$ 13.9463 0.568407
$$603$$ 9.51975 0.387674
$$604$$ −1.82601 −0.0742994
$$605$$ 0 0
$$606$$ −5.96139 −0.242165
$$607$$ −22.9458 −0.931343 −0.465671 0.884958i $$-0.654187\pi$$
−0.465671 + 0.884958i $$0.654187\pi$$
$$608$$ 7.46409 0.302709
$$609$$ 6.29592 0.255123
$$610$$ −18.3799 −0.744180
$$611$$ 9.01372 0.364656
$$612$$ 5.70909 0.230776
$$613$$ −40.0317 −1.61687 −0.808433 0.588588i $$-0.799684\pi$$
−0.808433 + 0.588588i $$0.799684\pi$$
$$614$$ −9.75276 −0.393589
$$615$$ 12.5712 0.506920
$$616$$ 0 0
$$617$$ 14.7371 0.593293 0.296647 0.954987i $$-0.404132\pi$$
0.296647 + 0.954987i $$0.404132\pi$$
$$618$$ 2.35352 0.0946725
$$619$$ 25.4079 1.02123 0.510616 0.859809i $$-0.329417\pi$$
0.510616 + 0.859809i $$0.329417\pi$$
$$620$$ 4.01194 0.161123
$$621$$ 11.1571 0.447719
$$622$$ 43.0045 1.72432
$$623$$ 1.49863 0.0600413
$$624$$ 31.5144 1.26159
$$625$$ 1.00000 0.0400000
$$626$$ 20.1560 0.805597
$$627$$ 0 0
$$628$$ −10.4892 −0.418566
$$629$$ 42.7974 1.70645
$$630$$ −2.80656 −0.111816
$$631$$ −26.1912 −1.04265 −0.521327 0.853357i $$-0.674563\pi$$
−0.521327 + 0.853357i $$0.674563\pi$$
$$632$$ −17.7311 −0.705307
$$633$$ −32.4742 −1.29073
$$634$$ −22.4726 −0.892503
$$635$$ 12.0909 0.479813
$$636$$ −5.21034 −0.206603
$$637$$ 5.60081 0.221912
$$638$$ 0 0
$$639$$ −1.18223 −0.0467683
$$640$$ 13.3895 0.529266
$$641$$ 8.22192 0.324746 0.162373 0.986729i $$-0.448085\pi$$
0.162373 + 0.986729i $$0.448085\pi$$
$$642$$ 19.3822 0.764955
$$643$$ 31.8519 1.25612 0.628059 0.778166i $$-0.283850\pi$$
0.628059 + 0.778166i $$0.283850\pi$$
$$644$$ 1.52886 0.0602455
$$645$$ −9.62612 −0.379028
$$646$$ 14.2463 0.560513
$$647$$ −22.5883 −0.888036 −0.444018 0.896018i $$-0.646447\pi$$
−0.444018 + 0.896018i $$0.646447\pi$$
$$648$$ −2.15145 −0.0845168
$$649$$ 0 0
$$650$$ −9.26199 −0.363285
$$651$$ 6.23309 0.244294
$$652$$ −7.16867 −0.280747
$$653$$ −0.763209 −0.0298667 −0.0149333 0.999888i $$-0.504754\pi$$
−0.0149333 + 0.999888i $$0.504754\pi$$
$$654$$ −10.7196 −0.419168
$$655$$ 1.04441 0.0408083
$$656$$ −54.2928 −2.11978
$$657$$ −23.5690 −0.919512
$$658$$ −2.66137 −0.103751
$$659$$ 42.2863 1.64724 0.823619 0.567143i $$-0.191951\pi$$
0.823619 + 0.567143i $$0.191951\pi$$
$$660$$ 0 0
$$661$$ −17.2730 −0.671843 −0.335921 0.941890i $$-0.609048\pi$$
−0.335921 + 0.941890i $$0.609048\pi$$
$$662$$ −30.3289 −1.17877
$$663$$ 29.2715 1.13681
$$664$$ −19.8830 −0.771612
$$665$$ −1.88149 −0.0729609
$$666$$ −26.2328 −1.01650
$$667$$ −11.4784 −0.444447
$$668$$ −8.36725 −0.323739
$$669$$ 27.3522 1.05750
$$670$$ −9.27592 −0.358360
$$671$$ 0 0
$$672$$ −4.52817 −0.174678
$$673$$ 39.3343 1.51623 0.758113 0.652124i $$-0.226122\pi$$
0.758113 + 0.652124i $$0.226122\pi$$
$$674$$ 41.8338 1.61138
$$675$$ 5.36144 0.206362
$$676$$ 13.4954 0.519053
$$677$$ −4.86816 −0.187098 −0.0935492 0.995615i $$-0.529821\pi$$
−0.0935492 + 0.995615i $$0.529821\pi$$
$$678$$ 33.9376 1.30337
$$679$$ 5.93697 0.227840
$$680$$ 9.58081 0.367407
$$681$$ 23.5141 0.901062
$$682$$ 0 0
$$683$$ 28.9705 1.10853 0.554263 0.832341i $$-0.313000\pi$$
0.554263 + 0.832341i $$0.313000\pi$$
$$684$$ −2.34595 −0.0896998
$$685$$ 1.43820 0.0549507
$$686$$ −1.65369 −0.0631380
$$687$$ −1.03112 −0.0393395
$$688$$ 41.5735 1.58498
$$689$$ 34.7996 1.32576
$$690$$ −3.92798 −0.149536
$$691$$ −15.4179 −0.586526 −0.293263 0.956032i $$-0.594741\pi$$
−0.293263 + 0.956032i $$0.594741\pi$$
$$692$$ −7.60711 −0.289179
$$693$$ 0 0
$$694$$ 23.9691 0.909855
$$695$$ −21.6489 −0.821189
$$696$$ 13.1739 0.499354
$$697$$ −50.4288 −1.91013
$$698$$ 4.92331 0.186350
$$699$$ 26.5334 1.00358
$$700$$ 0.734678 0.0277682
$$701$$ −6.78227 −0.256163 −0.128081 0.991764i $$-0.540882\pi$$
−0.128081 + 0.991764i $$0.540882\pi$$
$$702$$ −49.6576 −1.87420
$$703$$ −17.5861 −0.663274
$$704$$ 0 0
$$705$$ 1.83696 0.0691838
$$706$$ −23.2808 −0.876186
$$707$$ −3.15826 −0.118779
$$708$$ −11.3931 −0.428178
$$709$$ −21.1656 −0.794891 −0.397446 0.917626i $$-0.630103\pi$$
−0.397446 + 0.917626i $$0.630103\pi$$
$$710$$ 1.15195 0.0432319
$$711$$ 14.3815 0.539348
$$712$$ 3.13580 0.117519
$$713$$ −11.3639 −0.425581
$$714$$ −8.64266 −0.323443
$$715$$ 0 0
$$716$$ 13.8263 0.516712
$$717$$ 9.37352 0.350061
$$718$$ 25.3531 0.946170
$$719$$ 32.6186 1.21647 0.608234 0.793758i $$-0.291878\pi$$
0.608234 + 0.793758i $$0.291878\pi$$
$$720$$ −8.36630 −0.311794
$$721$$ 1.24686 0.0464356
$$722$$ 25.5660 0.951468
$$723$$ −9.50851 −0.353625
$$724$$ −5.69646 −0.211707
$$725$$ −5.51585 −0.204854
$$726$$ 0 0
$$727$$ 22.4245 0.831678 0.415839 0.909438i $$-0.363488\pi$$
0.415839 + 0.909438i $$0.363488\pi$$
$$728$$ 11.7194 0.434350
$$729$$ 20.1040 0.744593
$$730$$ 22.9653 0.849983
$$731$$ 38.6147 1.42822
$$732$$ 9.32037 0.344491
$$733$$ −28.7607 −1.06230 −0.531150 0.847278i $$-0.678240\pi$$
−0.531150 + 0.847278i $$0.678240\pi$$
$$734$$ −22.6852 −0.837326
$$735$$ 1.14142 0.0421020
$$736$$ 8.25556 0.304304
$$737$$ 0 0
$$738$$ 30.9104 1.13783
$$739$$ 2.12241 0.0780742 0.0390371 0.999238i $$-0.487571\pi$$
0.0390371 + 0.999238i $$0.487571\pi$$
$$740$$ 6.86699 0.252436
$$741$$ −12.0281 −0.441865
$$742$$ −10.2749 −0.377202
$$743$$ 38.6935 1.41953 0.709763 0.704440i $$-0.248802\pi$$
0.709763 + 0.704440i $$0.248802\pi$$
$$744$$ 13.0424 0.478158
$$745$$ −12.6540 −0.463607
$$746$$ 20.9408 0.766696
$$747$$ 16.1269 0.590052
$$748$$ 0 0
$$749$$ 10.2684 0.375200
$$750$$ −1.88755 −0.0689237
$$751$$ 2.01833 0.0736498 0.0368249 0.999322i $$-0.488276\pi$$
0.0368249 + 0.999322i $$0.488276\pi$$
$$752$$ −7.93350 −0.289305
$$753$$ 11.1579 0.406615
$$754$$ 51.0878 1.86051
$$755$$ 2.48546 0.0904551
$$756$$ 3.93893 0.143257
$$757$$ 39.9006 1.45021 0.725107 0.688637i $$-0.241790\pi$$
0.725107 + 0.688637i $$0.241790\pi$$
$$758$$ −19.9059 −0.723015
$$759$$ 0 0
$$760$$ −3.93691 −0.142807
$$761$$ −28.9628 −1.04990 −0.524950 0.851133i $$-0.675916\pi$$
−0.524950 + 0.851133i $$0.675916\pi$$
$$762$$ −22.8222 −0.826762
$$763$$ −5.67907 −0.205596
$$764$$ 11.7115 0.423706
$$765$$ −7.77087 −0.280956
$$766$$ 16.4737 0.595220
$$767$$ 76.0937 2.74759
$$768$$ −17.7427 −0.640233
$$769$$ −5.26499 −0.189860 −0.0949302 0.995484i $$-0.530263\pi$$
−0.0949302 + 0.995484i $$0.530263\pi$$
$$770$$ 0 0
$$771$$ −4.68047 −0.168563
$$772$$ −6.26114 −0.225344
$$773$$ −0.286828 −0.0103165 −0.00515825 0.999987i $$-0.501642\pi$$
−0.00515825 + 0.999987i $$0.501642\pi$$
$$774$$ −23.6690 −0.850764
$$775$$ −5.46081 −0.196158
$$776$$ 12.4228 0.445952
$$777$$ 10.6688 0.382741
$$778$$ −36.5881 −1.31175
$$779$$ 20.7220 0.742442
$$780$$ 4.69672 0.168169
$$781$$ 0 0
$$782$$ 15.7569 0.563467
$$783$$ −29.5729 −1.05685
$$784$$ −4.92960 −0.176057
$$785$$ 14.2773 0.509579
$$786$$ −1.97137 −0.0703166
$$787$$ −27.4622 −0.978923 −0.489462 0.872025i $$-0.662807\pi$$
−0.489462 + 0.872025i $$0.662807\pi$$
$$788$$ 13.9997 0.498718
$$789$$ 29.2721 1.04211
$$790$$ −14.0131 −0.498565
$$791$$ 17.9797 0.639284
$$792$$ 0 0
$$793$$ −62.2502 −2.21057
$$794$$ 43.8174 1.55502
$$795$$ 7.09201 0.251528
$$796$$ −9.33911 −0.331016
$$797$$ 22.6304 0.801609 0.400805 0.916164i $$-0.368731\pi$$
0.400805 + 0.916164i $$0.368731\pi$$
$$798$$ 3.55141 0.125718
$$799$$ −7.36887 −0.260692
$$800$$ 3.96713 0.140259
$$801$$ −2.54341 −0.0898668
$$802$$ 4.93149 0.174137
$$803$$ 0 0
$$804$$ 4.70378 0.165890
$$805$$ −2.08099 −0.0733453
$$806$$ 50.5779 1.78153
$$807$$ −15.6406 −0.550575
$$808$$ −6.60849 −0.232486
$$809$$ 37.2584 1.30994 0.654968 0.755656i $$-0.272682\pi$$
0.654968 + 0.755656i $$0.272682\pi$$
$$810$$ −1.70032 −0.0597430
$$811$$ 0.190250 0.00668059 0.00334030 0.999994i $$-0.498937\pi$$
0.00334030 + 0.999994i $$0.498937\pi$$
$$812$$ −4.05238 −0.142211
$$813$$ −9.82158 −0.344458
$$814$$ 0 0
$$815$$ 9.75758 0.341793
$$816$$ −25.7636 −0.901906
$$817$$ −15.8674 −0.555130
$$818$$ 2.84260 0.0993893
$$819$$ −9.50545 −0.332147
$$820$$ −8.09147 −0.282566
$$821$$ 30.0724 1.04953 0.524767 0.851246i $$-0.324152\pi$$
0.524767 + 0.851246i $$0.324152\pi$$
$$822$$ −2.71468 −0.0946853
$$823$$ 20.6546 0.719975 0.359987 0.932957i $$-0.382781\pi$$
0.359987 + 0.932957i $$0.382781\pi$$
$$824$$ 2.60899 0.0908885
$$825$$ 0 0
$$826$$ −22.4673 −0.781738
$$827$$ −25.3871 −0.882795 −0.441397 0.897312i $$-0.645517\pi$$
−0.441397 + 0.897312i $$0.645517\pi$$
$$828$$ −2.59471 −0.0901724
$$829$$ 6.53298 0.226900 0.113450 0.993544i $$-0.463810\pi$$
0.113450 + 0.993544i $$0.463810\pi$$
$$830$$ −15.7138 −0.545435
$$831$$ 32.9779 1.14399
$$832$$ 18.4761 0.640544
$$833$$ −4.57876 −0.158645
$$834$$ 40.8634 1.41498
$$835$$ 11.3890 0.394133
$$836$$ 0 0
$$837$$ −29.2778 −1.01199
$$838$$ 40.6899 1.40561
$$839$$ −13.3320 −0.460271 −0.230136 0.973159i $$-0.573917\pi$$
−0.230136 + 0.973159i $$0.573917\pi$$
$$840$$ 2.38836 0.0824064
$$841$$ 1.42465 0.0491257
$$842$$ 49.4008 1.70247
$$843$$ 9.11308 0.313871
$$844$$ 20.9020 0.719478
$$845$$ −18.3691 −0.631916
$$846$$ 4.51677 0.155290
$$847$$ 0 0
$$848$$ −30.6291 −1.05181
$$849$$ −8.44080 −0.289687
$$850$$ 7.57184 0.259712
$$851$$ −19.4509 −0.666769
$$852$$ −0.584148 −0.0200126
$$853$$ −45.3221 −1.55180 −0.775899 0.630857i $$-0.782704\pi$$
−0.775899 + 0.630857i $$0.782704\pi$$
$$854$$ 18.3799 0.628947
$$855$$ 3.19317 0.109204
$$856$$ 21.4861 0.734381
$$857$$ −26.0895 −0.891199 −0.445599 0.895232i $$-0.647009\pi$$
−0.445599 + 0.895232i $$0.647009\pi$$
$$858$$ 0 0
$$859$$ 40.0079 1.36505 0.682527 0.730861i $$-0.260881\pi$$
0.682527 + 0.730861i $$0.260881\pi$$
$$860$$ 6.19586 0.211277
$$861$$ −12.5712 −0.428425
$$862$$ −25.7809 −0.878103
$$863$$ 58.6608 1.99684 0.998418 0.0562309i $$-0.0179083\pi$$
0.998418 + 0.0562309i $$0.0179083\pi$$
$$864$$ 21.2695 0.723603
$$865$$ 10.3544 0.352059
$$866$$ −17.6759 −0.600652
$$867$$ −4.52582 −0.153705
$$868$$ −4.01194 −0.136174
$$869$$ 0 0
$$870$$ 10.4115 0.352982
$$871$$ −31.4163 −1.06450
$$872$$ −11.8832 −0.402414
$$873$$ −10.0760 −0.341020
$$874$$ −6.47477 −0.219012
$$875$$ −1.00000 −0.0338062
$$876$$ −11.6456 −0.393468
$$877$$ −33.3513 −1.12619 −0.563096 0.826392i $$-0.690390\pi$$
−0.563096 + 0.826392i $$0.690390\pi$$
$$878$$ 30.0205 1.01314
$$879$$ 15.7380 0.530830
$$880$$ 0 0
$$881$$ 21.4875 0.723933 0.361967 0.932191i $$-0.382105\pi$$
0.361967 + 0.932191i $$0.382105\pi$$
$$882$$ 2.80656 0.0945019
$$883$$ −43.7638 −1.47277 −0.736385 0.676563i $$-0.763469\pi$$
−0.736385 + 0.676563i $$0.763469\pi$$
$$884$$ −18.8407 −0.633680
$$885$$ 15.5076 0.521282
$$886$$ 16.4321 0.552049
$$887$$ 4.54756 0.152692 0.0763460 0.997081i $$-0.475675\pi$$
0.0763460 + 0.997081i $$0.475675\pi$$
$$888$$ 22.3239 0.749141
$$889$$ −12.0909 −0.405516
$$890$$ 2.47826 0.0830715
$$891$$ 0 0
$$892$$ −17.6053 −0.589467
$$893$$ 3.02799 0.101328
$$894$$ 23.8851 0.798839
$$895$$ −18.8195 −0.629067
$$896$$ −13.3895 −0.447311
$$897$$ −13.3036 −0.444193
$$898$$ 29.9184 0.998391
$$899$$ 30.1210 1.00459
$$900$$ −1.24686 −0.0415621
$$901$$ −28.4493 −0.947782
$$902$$ 0 0
$$903$$ 9.62612 0.320337
$$904$$ 37.6215 1.25127
$$905$$ 7.75369 0.257741
$$906$$ −4.69144 −0.155863
$$907$$ −17.4034 −0.577869 −0.288935 0.957349i $$-0.593301\pi$$
−0.288935 + 0.957349i $$0.593301\pi$$
$$908$$ −15.1349 −0.502268
$$909$$ 5.36006 0.177782
$$910$$ 9.26199 0.307032
$$911$$ 4.17760 0.138410 0.0692050 0.997602i $$-0.477954\pi$$
0.0692050 + 0.997602i $$0.477954\pi$$
$$912$$ 10.5867 0.350560
$$913$$ 0 0
$$914$$ −26.4427 −0.874645
$$915$$ −12.6863 −0.419397
$$916$$ 0.663678 0.0219286
$$917$$ −1.04441 −0.0344894
$$918$$ 40.5959 1.33987
$$919$$ −18.4017 −0.607018 −0.303509 0.952829i $$-0.598158\pi$$
−0.303509 + 0.952829i $$0.598158\pi$$
$$920$$ −4.35436 −0.143559
$$921$$ −6.73164 −0.221815
$$922$$ 46.2390 1.52280
$$923$$ 3.90149 0.128419
$$924$$ 0 0
$$925$$ −9.34694 −0.307326
$$926$$ 50.9978 1.67589
$$927$$ −2.11612 −0.0695025
$$928$$ −21.8821 −0.718315
$$929$$ −52.1959 −1.71249 −0.856245 0.516569i $$-0.827209\pi$$
−0.856245 + 0.516569i $$0.827209\pi$$
$$930$$ 10.3076 0.337999
$$931$$ 1.88149 0.0616632
$$932$$ −17.0782 −0.559416
$$933$$ 29.6830 0.971777
$$934$$ 13.9197 0.455465
$$935$$ 0 0
$$936$$ −19.8896 −0.650113
$$937$$ 12.3471 0.403363 0.201681 0.979451i $$-0.435359\pi$$
0.201681 + 0.979451i $$0.435359\pi$$
$$938$$ 9.27592 0.302869
$$939$$ 13.9123 0.454010
$$940$$ −1.18236 −0.0385643
$$941$$ −13.8509 −0.451525 −0.225763 0.974182i $$-0.572487\pi$$
−0.225763 + 0.974182i $$0.572487\pi$$
$$942$$ −26.9492 −0.878052
$$943$$ 22.9193 0.746354
$$944$$ −66.9746 −2.17984
$$945$$ −5.36144 −0.174408
$$946$$ 0 0
$$947$$ −21.7800 −0.707756 −0.353878 0.935292i $$-0.615137\pi$$
−0.353878 + 0.935292i $$0.615137\pi$$
$$948$$ 7.10601 0.230792
$$949$$ 77.7803 2.52486
$$950$$ −3.11139 −0.100947
$$951$$ −15.5113 −0.502988
$$952$$ −9.58081 −0.310516
$$953$$ −7.56150 −0.244941 −0.122471 0.992472i $$-0.539082\pi$$
−0.122471 + 0.992472i $$0.539082\pi$$
$$954$$ 17.4380 0.564577
$$955$$ −15.9410 −0.515837
$$956$$ −6.03328 −0.195130
$$957$$ 0 0
$$958$$ 49.0921 1.58609
$$959$$ −1.43820 −0.0464419
$$960$$ 3.76535 0.121526
$$961$$ −1.17956 −0.0380503
$$962$$ 86.5713 2.79117
$$963$$ −17.4271 −0.561581
$$964$$ 6.12016 0.197117
$$965$$ 8.52230 0.274343
$$966$$ 3.92798 0.126381
$$967$$ 24.0488 0.773357 0.386679 0.922215i $$-0.373622\pi$$
0.386679 + 0.922215i $$0.373622\pi$$
$$968$$ 0 0
$$969$$ 9.83321 0.315888
$$970$$ 9.81789 0.315233
$$971$$ 16.0049 0.513622 0.256811 0.966462i $$-0.417328\pi$$
0.256811 + 0.966462i $$0.417328\pi$$
$$972$$ −10.9546 −0.351368
$$973$$ 21.6489 0.694031
$$974$$ −62.5611 −2.00459
$$975$$ −6.39289 −0.204736
$$976$$ 54.7901 1.75379
$$977$$ 16.4515 0.526330 0.263165 0.964751i $$-0.415234\pi$$
0.263165 + 0.964751i $$0.415234\pi$$
$$978$$ −18.4180 −0.588941
$$979$$ 0 0
$$980$$ −0.734678 −0.0234684
$$981$$ 9.63827 0.307726
$$982$$ 30.9214 0.986741
$$983$$ 30.2550 0.964984 0.482492 0.875900i $$-0.339732\pi$$
0.482492 + 0.875900i $$0.339732\pi$$
$$984$$ −26.3045 −0.838558
$$985$$ −19.0556 −0.607160
$$986$$ −41.7651 −1.33007
$$987$$ −1.83696 −0.0584710
$$988$$ 7.74192 0.246303
$$989$$ −17.5499 −0.558055
$$990$$ 0 0
$$991$$ −13.0778 −0.415428 −0.207714 0.978190i $$-0.566602\pi$$
−0.207714 + 0.978190i $$0.566602\pi$$
$$992$$ −21.6637 −0.687824
$$993$$ −20.9339 −0.664317
$$994$$ −1.15195 −0.0365376
$$995$$ 12.7118 0.402992
$$996$$ 7.96841 0.252489
$$997$$ −31.8380 −1.00832 −0.504160 0.863610i $$-0.668198\pi$$
−0.504160 + 0.863610i $$0.668198\pi$$
$$998$$ −37.8294 −1.19747
$$999$$ −50.1131 −1.58551
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 4235.2.a.y.1.2 5
11.10 odd 2 4235.2.a.be.1.4 yes 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.y.1.2 5 1.1 even 1 trivial
4235.2.a.be.1.4 yes 5 11.10 odd 2