# Properties

 Label 1155.2.a.u.1.3 Level 1155 Weight 2 Character 1155.1 Self dual yes Analytic conductor 9.223 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1155.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13448.1 Defining polynomial: $$x^{4} - 7 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.546295$$ of defining polynomial Character $$\chi$$ $$=$$ 1155.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.546295 q^{2} +1.00000 q^{3} -1.70156 q^{4} -1.00000 q^{5} +0.546295 q^{6} -1.00000 q^{7} -2.02214 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.546295 q^{2} +1.00000 q^{3} -1.70156 q^{4} -1.00000 q^{5} +0.546295 q^{6} -1.00000 q^{7} -2.02214 q^{8} +1.00000 q^{9} -0.546295 q^{10} -1.00000 q^{11} -1.70156 q^{12} -0.568438 q^{13} -0.546295 q^{14} -1.00000 q^{15} +2.29844 q^{16} +2.70156 q^{17} +0.546295 q^{18} +6.62049 q^{19} +1.70156 q^{20} -1.00000 q^{21} -0.546295 q^{22} +8.93103 q^{23} -2.02214 q^{24} +1.00000 q^{25} -0.310535 q^{26} +1.00000 q^{27} +1.70156 q^{28} +5.27000 q^{29} -0.546295 q^{30} +9.66103 q^{31} +5.29991 q^{32} -1.00000 q^{33} +1.47585 q^{34} +1.00000 q^{35} -1.70156 q^{36} -2.75362 q^{37} +3.61674 q^{38} -0.568438 q^{39} +2.02214 q^{40} -10.9831 q^{41} -0.546295 q^{42} -0.0520550 q^{43} +1.70156 q^{44} -1.00000 q^{45} +4.87897 q^{46} -10.1567 q^{47} +2.29844 q^{48} +1.00000 q^{49} +0.546295 q^{50} +2.70156 q^{51} +0.967233 q^{52} +1.56469 q^{53} +0.546295 q^{54} +1.00000 q^{55} +2.02214 q^{56} +6.62049 q^{57} +2.87897 q^{58} +6.36259 q^{59} +1.70156 q^{60} -3.71308 q^{61} +5.27777 q^{62} -1.00000 q^{63} -1.70156 q^{64} +0.568438 q^{65} -0.546295 q^{66} -10.4146 q^{67} -4.59688 q^{68} +8.93103 q^{69} +0.546295 q^{70} +2.56844 q^{71} -2.02214 q^{72} +2.26625 q^{73} -1.50429 q^{74} +1.00000 q^{75} -11.2652 q^{76} +1.00000 q^{77} -0.310535 q^{78} -2.56844 q^{79} -2.29844 q^{80} +1.00000 q^{81} -6.00000 q^{82} +17.1605 q^{83} +1.70156 q^{84} -2.70156 q^{85} -0.0284374 q^{86} +5.27000 q^{87} +2.02214 q^{88} -10.7772 q^{89} -0.546295 q^{90} +0.568438 q^{91} -15.1967 q^{92} +9.66103 q^{93} -5.54857 q^{94} -6.62049 q^{95} +5.29991 q^{96} +17.9952 q^{97} +0.546295 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 6q^{4} - 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 6q^{4} - 4q^{5} - 4q^{7} + 4q^{9} - 4q^{11} + 6q^{12} + 8q^{13} - 4q^{15} + 22q^{16} - 2q^{17} + 10q^{19} - 6q^{20} - 4q^{21} - 2q^{23} + 4q^{25} + 20q^{26} + 4q^{27} - 6q^{28} - 2q^{29} + 24q^{31} - 4q^{33} + 4q^{35} + 6q^{36} + 8q^{37} + 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} - 4q^{45} - 12q^{46} + 4q^{47} + 22q^{48} + 4q^{49} - 2q^{51} + 12q^{52} + 14q^{53} + 4q^{55} + 10q^{57} - 20q^{58} - 2q^{59} - 6q^{60} + 6q^{61} + 8q^{62} - 4q^{63} + 6q^{64} - 8q^{65} - 8q^{67} - 44q^{68} - 2q^{69} + 4q^{73} - 36q^{74} + 4q^{75} + 56q^{76} + 4q^{77} + 20q^{78} - 22q^{80} + 4q^{81} - 24q^{82} + 6q^{83} - 6q^{84} + 2q^{85} - 36q^{86} - 2q^{87} + 18q^{89} - 8q^{91} - 44q^{92} + 24q^{93} - 36q^{94} - 10q^{95} - 6q^{97} - 4q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.546295 0.386289 0.193144 0.981170i $$-0.438131\pi$$
0.193144 + 0.981170i $$0.438131\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.70156 −0.850781
$$5$$ −1.00000 −0.447214
$$6$$ 0.546295 0.223024
$$7$$ −1.00000 −0.377964
$$8$$ −2.02214 −0.714936
$$9$$ 1.00000 0.333333
$$10$$ −0.546295 −0.172754
$$11$$ −1.00000 −0.301511
$$12$$ −1.70156 −0.491199
$$13$$ −0.568438 −0.157656 −0.0788282 0.996888i $$-0.525118\pi$$
−0.0788282 + 0.996888i $$0.525118\pi$$
$$14$$ −0.546295 −0.146003
$$15$$ −1.00000 −0.258199
$$16$$ 2.29844 0.574609
$$17$$ 2.70156 0.655225 0.327613 0.944812i $$-0.393756\pi$$
0.327613 + 0.944812i $$0.393756\pi$$
$$18$$ 0.546295 0.128763
$$19$$ 6.62049 1.51885 0.759423 0.650598i $$-0.225482\pi$$
0.759423 + 0.650598i $$0.225482\pi$$
$$20$$ 1.70156 0.380481
$$21$$ −1.00000 −0.218218
$$22$$ −0.546295 −0.116470
$$23$$ 8.93103 1.86225 0.931124 0.364703i $$-0.118829\pi$$
0.931124 + 0.364703i $$0.118829\pi$$
$$24$$ −2.02214 −0.412768
$$25$$ 1.00000 0.200000
$$26$$ −0.310535 −0.0609009
$$27$$ 1.00000 0.192450
$$28$$ 1.70156 0.321565
$$29$$ 5.27000 0.978615 0.489307 0.872111i $$-0.337250\pi$$
0.489307 + 0.872111i $$0.337250\pi$$
$$30$$ −0.546295 −0.0997393
$$31$$ 9.66103 1.73517 0.867586 0.497287i $$-0.165671\pi$$
0.867586 + 0.497287i $$0.165671\pi$$
$$32$$ 5.29991 0.936901
$$33$$ −1.00000 −0.174078
$$34$$ 1.47585 0.253106
$$35$$ 1.00000 0.169031
$$36$$ −1.70156 −0.283594
$$37$$ −2.75362 −0.452692 −0.226346 0.974047i $$-0.572678\pi$$
−0.226346 + 0.974047i $$0.572678\pi$$
$$38$$ 3.61674 0.586713
$$39$$ −0.568438 −0.0910230
$$40$$ 2.02214 0.319729
$$41$$ −10.9831 −1.71527 −0.857635 0.514259i $$-0.828067\pi$$
−0.857635 + 0.514259i $$0.828067\pi$$
$$42$$ −0.546295 −0.0842951
$$43$$ −0.0520550 −0.00793831 −0.00396916 0.999992i $$-0.501263\pi$$
−0.00396916 + 0.999992i $$0.501263\pi$$
$$44$$ 1.70156 0.256520
$$45$$ −1.00000 −0.149071
$$46$$ 4.87897 0.719365
$$47$$ −10.1567 −1.48151 −0.740756 0.671774i $$-0.765533\pi$$
−0.740756 + 0.671774i $$0.765533\pi$$
$$48$$ 2.29844 0.331751
$$49$$ 1.00000 0.142857
$$50$$ 0.546295 0.0772577
$$51$$ 2.70156 0.378294
$$52$$ 0.967233 0.134131
$$53$$ 1.56469 0.214926 0.107463 0.994209i $$-0.465727\pi$$
0.107463 + 0.994209i $$0.465727\pi$$
$$54$$ 0.546295 0.0743413
$$55$$ 1.00000 0.134840
$$56$$ 2.02214 0.270220
$$57$$ 6.62049 0.876906
$$58$$ 2.87897 0.378028
$$59$$ 6.36259 0.828339 0.414169 0.910200i $$-0.364072\pi$$
0.414169 + 0.910200i $$0.364072\pi$$
$$60$$ 1.70156 0.219671
$$61$$ −3.71308 −0.475412 −0.237706 0.971337i $$-0.576395\pi$$
−0.237706 + 0.971337i $$0.576395\pi$$
$$62$$ 5.27777 0.670277
$$63$$ −1.00000 −0.125988
$$64$$ −1.70156 −0.212695
$$65$$ 0.568438 0.0705061
$$66$$ −0.546295 −0.0672442
$$67$$ −10.4146 −1.27235 −0.636176 0.771544i $$-0.719485\pi$$
−0.636176 + 0.771544i $$0.719485\pi$$
$$68$$ −4.59688 −0.557453
$$69$$ 8.93103 1.07517
$$70$$ 0.546295 0.0652947
$$71$$ 2.56844 0.304818 0.152409 0.988318i $$-0.451297\pi$$
0.152409 + 0.988318i $$0.451297\pi$$
$$72$$ −2.02214 −0.238312
$$73$$ 2.26625 0.265244 0.132622 0.991167i $$-0.457660\pi$$
0.132622 + 0.991167i $$0.457660\pi$$
$$74$$ −1.50429 −0.174870
$$75$$ 1.00000 0.115470
$$76$$ −11.2652 −1.29220
$$77$$ 1.00000 0.113961
$$78$$ −0.310535 −0.0351612
$$79$$ −2.56844 −0.288972 −0.144486 0.989507i $$-0.546153\pi$$
−0.144486 + 0.989507i $$0.546153\pi$$
$$80$$ −2.29844 −0.256973
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 17.1605 1.88361 0.941804 0.336161i $$-0.109129\pi$$
0.941804 + 0.336161i $$0.109129\pi$$
$$84$$ 1.70156 0.185656
$$85$$ −2.70156 −0.293026
$$86$$ −0.0284374 −0.00306648
$$87$$ 5.27000 0.565003
$$88$$ 2.02214 0.215561
$$89$$ −10.7772 −1.14238 −0.571192 0.820816i $$-0.693519\pi$$
−0.571192 + 0.820816i $$0.693519\pi$$
$$90$$ −0.546295 −0.0575845
$$91$$ 0.568438 0.0595885
$$92$$ −15.1967 −1.58437
$$93$$ 9.66103 1.00180
$$94$$ −5.54857 −0.572292
$$95$$ −6.62049 −0.679248
$$96$$ 5.29991 0.540920
$$97$$ 17.9952 1.82713 0.913567 0.406689i $$-0.133317\pi$$
0.913567 + 0.406689i $$0.133317\pi$$
$$98$$ 0.546295 0.0551841
$$99$$ −1.00000 −0.100504
$$100$$ −1.70156 −0.170156
$$101$$ 0.826342 0.0822241 0.0411120 0.999155i $$-0.486910\pi$$
0.0411120 + 0.999155i $$0.486910\pi$$
$$102$$ 1.47585 0.146131
$$103$$ 10.5921 1.04367 0.521833 0.853048i $$-0.325248\pi$$
0.521833 + 0.853048i $$0.325248\pi$$
$$104$$ 1.14946 0.112714
$$105$$ 1.00000 0.0975900
$$106$$ 0.854779 0.0830235
$$107$$ −13.7864 −1.33278 −0.666390 0.745603i $$-0.732161\pi$$
−0.666390 + 0.745603i $$0.732161\pi$$
$$108$$ −1.70156 −0.163733
$$109$$ 19.8905 1.90516 0.952582 0.304282i $$-0.0984166\pi$$
0.952582 + 0.304282i $$0.0984166\pi$$
$$110$$ 0.546295 0.0520872
$$111$$ −2.75362 −0.261362
$$112$$ −2.29844 −0.217182
$$113$$ 6.43531 0.605383 0.302692 0.953089i $$-0.402115\pi$$
0.302692 + 0.953089i $$0.402115\pi$$
$$114$$ 3.61674 0.338739
$$115$$ −8.93103 −0.832823
$$116$$ −8.96723 −0.832587
$$117$$ −0.568438 −0.0525521
$$118$$ 3.47585 0.319978
$$119$$ −2.70156 −0.247652
$$120$$ 2.02214 0.184596
$$121$$ 1.00000 0.0909091
$$122$$ −2.02844 −0.183646
$$123$$ −10.9831 −0.990311
$$124$$ −16.4388 −1.47625
$$125$$ −1.00000 −0.0894427
$$126$$ −0.546295 −0.0486678
$$127$$ 4.05206 0.359562 0.179781 0.983707i $$-0.442461\pi$$
0.179781 + 0.983707i $$0.442461\pi$$
$$128$$ −11.5294 −1.01906
$$129$$ −0.0520550 −0.00458319
$$130$$ 0.310535 0.0272357
$$131$$ −5.69781 −0.497820 −0.248910 0.968527i $$-0.580072\pi$$
−0.248910 + 0.968527i $$0.580072\pi$$
$$132$$ 1.70156 0.148102
$$133$$ −6.62049 −0.574070
$$134$$ −5.68947 −0.491495
$$135$$ −1.00000 −0.0860663
$$136$$ −5.46295 −0.468444
$$137$$ −3.40312 −0.290749 −0.145374 0.989377i $$-0.546439\pi$$
−0.145374 + 0.989377i $$0.546439\pi$$
$$138$$ 4.87897 0.415326
$$139$$ −8.04724 −0.682558 −0.341279 0.939962i $$-0.610860\pi$$
−0.341279 + 0.939962i $$0.610860\pi$$
$$140$$ −1.70156 −0.143808
$$141$$ −10.1567 −0.855352
$$142$$ 1.40312 0.117748
$$143$$ 0.568438 0.0475352
$$144$$ 2.29844 0.191536
$$145$$ −5.27000 −0.437650
$$146$$ 1.23804 0.102461
$$147$$ 1.00000 0.0824786
$$148$$ 4.68545 0.385142
$$149$$ 7.50723 0.615017 0.307508 0.951545i $$-0.400505\pi$$
0.307508 + 0.951545i $$0.400505\pi$$
$$150$$ 0.546295 0.0446048
$$151$$ 3.21795 0.261873 0.130936 0.991391i $$-0.458202\pi$$
0.130936 + 0.991391i $$0.458202\pi$$
$$152$$ −13.3876 −1.08588
$$153$$ 2.70156 0.218408
$$154$$ 0.546295 0.0440217
$$155$$ −9.66103 −0.775992
$$156$$ 0.967233 0.0774406
$$157$$ −3.35107 −0.267444 −0.133722 0.991019i $$-0.542693\pi$$
−0.133722 + 0.991019i $$0.542693\pi$$
$$158$$ −1.40312 −0.111627
$$159$$ 1.56469 0.124088
$$160$$ −5.29991 −0.418995
$$161$$ −8.93103 −0.703864
$$162$$ 0.546295 0.0429210
$$163$$ 10.9600 0.858457 0.429228 0.903196i $$-0.358786\pi$$
0.429228 + 0.903196i $$0.358786\pi$$
$$164$$ 18.6884 1.45932
$$165$$ 1.00000 0.0778499
$$166$$ 9.37469 0.727617
$$167$$ −7.58830 −0.587201 −0.293600 0.955928i $$-0.594853\pi$$
−0.293600 + 0.955928i $$0.594853\pi$$
$$168$$ 2.02214 0.156012
$$169$$ −12.6769 −0.975144
$$170$$ −1.47585 −0.113192
$$171$$ 6.62049 0.506282
$$172$$ 0.0885748 0.00675377
$$173$$ −18.2094 −1.38443 −0.692216 0.721690i $$-0.743366\pi$$
−0.692216 + 0.721690i $$0.743366\pi$$
$$174$$ 2.87897 0.218254
$$175$$ −1.00000 −0.0755929
$$176$$ −2.29844 −0.173251
$$177$$ 6.36259 0.478242
$$178$$ −5.88755 −0.441290
$$179$$ 4.75362 0.355302 0.177651 0.984094i $$-0.443150\pi$$
0.177651 + 0.984094i $$0.443150\pi$$
$$180$$ 1.70156 0.126827
$$181$$ 25.2652 1.87795 0.938973 0.343991i $$-0.111779\pi$$
0.938973 + 0.343991i $$0.111779\pi$$
$$182$$ 0.310535 0.0230184
$$183$$ −3.71308 −0.274479
$$184$$ −18.0598 −1.33139
$$185$$ 2.75362 0.202450
$$186$$ 5.27777 0.386985
$$187$$ −2.70156 −0.197558
$$188$$ 17.2823 1.26044
$$189$$ −1.00000 −0.0727393
$$190$$ −3.61674 −0.262386
$$191$$ −17.4504 −1.26266 −0.631332 0.775513i $$-0.717491\pi$$
−0.631332 + 0.775513i $$0.717491\pi$$
$$192$$ −1.70156 −0.122800
$$193$$ 22.9546 1.65231 0.826156 0.563442i $$-0.190523\pi$$
0.826156 + 0.563442i $$0.190523\pi$$
$$194$$ 9.83067 0.705801
$$195$$ 0.568438 0.0407067
$$196$$ −1.70156 −0.121540
$$197$$ −3.36634 −0.239842 −0.119921 0.992783i $$-0.538264\pi$$
−0.119921 + 0.992783i $$0.538264\pi$$
$$198$$ −0.546295 −0.0388235
$$199$$ 9.54402 0.676557 0.338279 0.941046i $$-0.390155\pi$$
0.338279 + 0.941046i $$0.390155\pi$$
$$200$$ −2.02214 −0.142987
$$201$$ −10.4146 −0.734592
$$202$$ 0.451426 0.0317622
$$203$$ −5.27000 −0.369882
$$204$$ −4.59688 −0.321846
$$205$$ 10.9831 0.767092
$$206$$ 5.78638 0.403156
$$207$$ 8.93103 0.620749
$$208$$ −1.30652 −0.0905909
$$209$$ −6.62049 −0.457949
$$210$$ 0.546295 0.0376979
$$211$$ −7.73375 −0.532413 −0.266207 0.963916i $$-0.585770\pi$$
−0.266207 + 0.963916i $$0.585770\pi$$
$$212$$ −2.66241 −0.182855
$$213$$ 2.56844 0.175986
$$214$$ −7.53143 −0.514838
$$215$$ 0.0520550 0.00355012
$$216$$ −2.02214 −0.137589
$$217$$ −9.66103 −0.655833
$$218$$ 10.8661 0.735943
$$219$$ 2.26625 0.153139
$$220$$ −1.70156 −0.114719
$$221$$ −1.53567 −0.103300
$$222$$ −1.50429 −0.100961
$$223$$ −13.9141 −0.931758 −0.465879 0.884848i $$-0.654262\pi$$
−0.465879 + 0.884848i $$0.654262\pi$$
$$224$$ −5.29991 −0.354115
$$225$$ 1.00000 0.0666667
$$226$$ 3.51558 0.233853
$$227$$ 0.597452 0.0396543 0.0198271 0.999803i $$-0.493688\pi$$
0.0198271 + 0.999803i $$0.493688\pi$$
$$228$$ −11.2652 −0.746055
$$229$$ 21.1895 1.40024 0.700121 0.714024i $$-0.253129\pi$$
0.700121 + 0.714024i $$0.253129\pi$$
$$230$$ −4.87897 −0.321710
$$231$$ 1.00000 0.0657952
$$232$$ −10.6567 −0.699647
$$233$$ −20.2821 −1.32872 −0.664362 0.747411i $$-0.731297\pi$$
−0.664362 + 0.747411i $$0.731297\pi$$
$$234$$ −0.310535 −0.0203003
$$235$$ 10.1567 0.662553
$$236$$ −10.8263 −0.704735
$$237$$ −2.56844 −0.166838
$$238$$ −1.47585 −0.0956651
$$239$$ 2.23723 0.144715 0.0723573 0.997379i $$-0.476948\pi$$
0.0723573 + 0.997379i $$0.476948\pi$$
$$240$$ −2.29844 −0.148364
$$241$$ 8.93045 0.575261 0.287630 0.957741i $$-0.407133\pi$$
0.287630 + 0.957741i $$0.407133\pi$$
$$242$$ 0.546295 0.0351172
$$243$$ 1.00000 0.0641500
$$244$$ 6.31804 0.404471
$$245$$ −1.00000 −0.0638877
$$246$$ −6.00000 −0.382546
$$247$$ −3.76334 −0.239456
$$248$$ −19.5360 −1.24054
$$249$$ 17.1605 1.08750
$$250$$ −0.546295 −0.0345507
$$251$$ 3.35884 0.212008 0.106004 0.994366i $$-0.466194\pi$$
0.106004 + 0.994366i $$0.466194\pi$$
$$252$$ 1.70156 0.107188
$$253$$ −8.93103 −0.561489
$$254$$ 2.21362 0.138895
$$255$$ −2.70156 −0.169178
$$256$$ −2.89531 −0.180957
$$257$$ 5.80943 0.362382 0.181191 0.983448i $$-0.442005\pi$$
0.181191 + 0.983448i $$0.442005\pi$$
$$258$$ −0.0284374 −0.00177043
$$259$$ 2.75362 0.171101
$$260$$ −0.967233 −0.0599853
$$261$$ 5.27000 0.326205
$$262$$ −3.11268 −0.192302
$$263$$ −23.3221 −1.43810 −0.719050 0.694959i $$-0.755423\pi$$
−0.719050 + 0.694959i $$0.755423\pi$$
$$264$$ 2.02214 0.124454
$$265$$ −1.56469 −0.0961179
$$266$$ −3.61674 −0.221757
$$267$$ −10.7772 −0.659556
$$268$$ 17.7212 1.08249
$$269$$ 29.1246 1.77576 0.887878 0.460080i $$-0.152179\pi$$
0.887878 + 0.460080i $$0.152179\pi$$
$$270$$ −0.546295 −0.0332464
$$271$$ 31.2416 1.89779 0.948895 0.315592i $$-0.102203\pi$$
0.948895 + 0.315592i $$0.102203\pi$$
$$272$$ 6.20937 0.376499
$$273$$ 0.568438 0.0344035
$$274$$ −1.85911 −0.112313
$$275$$ −1.00000 −0.0603023
$$276$$ −15.1967 −0.914734
$$277$$ −15.8988 −0.955269 −0.477634 0.878559i $$-0.658506\pi$$
−0.477634 + 0.878559i $$0.658506\pi$$
$$278$$ −4.39616 −0.263664
$$279$$ 9.66103 0.578391
$$280$$ −2.02214 −0.120846
$$281$$ −14.1041 −0.841381 −0.420690 0.907204i $$-0.638212\pi$$
−0.420690 + 0.907204i $$0.638212\pi$$
$$282$$ −5.54857 −0.330413
$$283$$ −1.32745 −0.0789088 −0.0394544 0.999221i $$-0.512562\pi$$
−0.0394544 + 0.999221i $$0.512562\pi$$
$$284$$ −4.37036 −0.259333
$$285$$ −6.62049 −0.392164
$$286$$ 0.310535 0.0183623
$$287$$ 10.9831 0.648311
$$288$$ 5.29991 0.312300
$$289$$ −9.70156 −0.570680
$$290$$ −2.87897 −0.169059
$$291$$ 17.9952 1.05490
$$292$$ −3.85616 −0.225665
$$293$$ 18.3784 1.07368 0.536840 0.843684i $$-0.319618\pi$$
0.536840 + 0.843684i $$0.319618\pi$$
$$294$$ 0.546295 0.0318606
$$295$$ −6.36259 −0.370444
$$296$$ 5.56821 0.323646
$$297$$ −1.00000 −0.0580259
$$298$$ 4.10116 0.237574
$$299$$ −5.07674 −0.293595
$$300$$ −1.70156 −0.0982397
$$301$$ 0.0520550 0.00300040
$$302$$ 1.75795 0.101158
$$303$$ 0.826342 0.0474721
$$304$$ 15.2168 0.872743
$$305$$ 3.71308 0.212611
$$306$$ 1.47585 0.0843687
$$307$$ 6.45143 0.368202 0.184101 0.982907i $$-0.441063\pi$$
0.184101 + 0.982907i $$0.441063\pi$$
$$308$$ −1.70156 −0.0969555
$$309$$ 10.5921 0.602561
$$310$$ −5.27777 −0.299757
$$311$$ −29.4136 −1.66789 −0.833946 0.551847i $$-0.813923\pi$$
−0.833946 + 0.551847i $$0.813923\pi$$
$$312$$ 1.14946 0.0650756
$$313$$ −24.7542 −1.39919 −0.699595 0.714540i $$-0.746636\pi$$
−0.699595 + 0.714540i $$0.746636\pi$$
$$314$$ −1.83067 −0.103311
$$315$$ 1.00000 0.0563436
$$316$$ 4.37036 0.245852
$$317$$ −15.0955 −0.847850 −0.423925 0.905697i $$-0.639348\pi$$
−0.423925 + 0.905697i $$0.639348\pi$$
$$318$$ 0.854779 0.0479336
$$319$$ −5.27000 −0.295063
$$320$$ 1.70156 0.0951202
$$321$$ −13.7864 −0.769481
$$322$$ −4.87897 −0.271895
$$323$$ 17.8857 0.995186
$$324$$ −1.70156 −0.0945312
$$325$$ −0.568438 −0.0315313
$$326$$ 5.98741 0.331612
$$327$$ 19.8905 1.09995
$$328$$ 22.2094 1.22631
$$329$$ 10.1567 0.559959
$$330$$ 0.546295 0.0300725
$$331$$ 6.78263 0.372807 0.186404 0.982473i $$-0.440317\pi$$
0.186404 + 0.982473i $$0.440317\pi$$
$$332$$ −29.1996 −1.60254
$$333$$ −2.75362 −0.150897
$$334$$ −4.14545 −0.226829
$$335$$ 10.4146 0.569013
$$336$$ −2.29844 −0.125390
$$337$$ −31.9509 −1.74048 −0.870238 0.492631i $$-0.836035\pi$$
−0.870238 + 0.492631i $$0.836035\pi$$
$$338$$ −6.92531 −0.376687
$$339$$ 6.43531 0.349518
$$340$$ 4.59688 0.249301
$$341$$ −9.66103 −0.523174
$$342$$ 3.61674 0.195571
$$343$$ −1.00000 −0.0539949
$$344$$ 0.105263 0.00567538
$$345$$ −8.93103 −0.480830
$$346$$ −9.94768 −0.534791
$$347$$ −13.6694 −0.733810 −0.366905 0.930258i $$-0.619583\pi$$
−0.366905 + 0.930258i $$0.619583\pi$$
$$348$$ −8.96723 −0.480694
$$349$$ 16.1720 0.865668 0.432834 0.901474i $$-0.357514\pi$$
0.432834 + 0.901474i $$0.357514\pi$$
$$350$$ −0.546295 −0.0292007
$$351$$ −0.568438 −0.0303410
$$352$$ −5.29991 −0.282486
$$353$$ −9.94313 −0.529219 −0.264610 0.964356i $$-0.585243\pi$$
−0.264610 + 0.964356i $$0.585243\pi$$
$$354$$ 3.47585 0.184739
$$355$$ −2.56844 −0.136319
$$356$$ 18.3381 0.971919
$$357$$ −2.70156 −0.142982
$$358$$ 2.59688 0.137249
$$359$$ −12.5845 −0.664187 −0.332094 0.943246i $$-0.607755\pi$$
−0.332094 + 0.943246i $$0.607755\pi$$
$$360$$ 2.02214 0.106576
$$361$$ 24.8309 1.30689
$$362$$ 13.8022 0.725429
$$363$$ 1.00000 0.0524864
$$364$$ −0.967233 −0.0506968
$$365$$ −2.26625 −0.118621
$$366$$ −2.02844 −0.106028
$$367$$ 36.9699 1.92981 0.964907 0.262592i $$-0.0845772\pi$$
0.964907 + 0.262592i $$0.0845772\pi$$
$$368$$ 20.5274 1.07007
$$369$$ −10.9831 −0.571756
$$370$$ 1.50429 0.0782041
$$371$$ −1.56469 −0.0812344
$$372$$ −16.4388 −0.852314
$$373$$ −13.4109 −0.694390 −0.347195 0.937793i $$-0.612866\pi$$
−0.347195 + 0.937793i $$0.612866\pi$$
$$374$$ −1.47585 −0.0763143
$$375$$ −1.00000 −0.0516398
$$376$$ 20.5384 1.05919
$$377$$ −2.99567 −0.154285
$$378$$ −0.546295 −0.0280984
$$379$$ −0.805672 −0.0413846 −0.0206923 0.999786i $$-0.506587\pi$$
−0.0206923 + 0.999786i $$0.506587\pi$$
$$380$$ 11.2652 0.577892
$$381$$ 4.05206 0.207593
$$382$$ −9.53304 −0.487753
$$383$$ −22.3494 −1.14200 −0.571001 0.820949i $$-0.693445\pi$$
−0.571001 + 0.820949i $$0.693445\pi$$
$$384$$ −11.5294 −0.588356
$$385$$ −1.00000 −0.0509647
$$386$$ 12.5400 0.638269
$$387$$ −0.0520550 −0.00264610
$$388$$ −30.6199 −1.55449
$$389$$ −29.4863 −1.49501 −0.747507 0.664253i $$-0.768750\pi$$
−0.747507 + 0.664253i $$0.768750\pi$$
$$390$$ 0.310535 0.0157245
$$391$$ 24.1277 1.22019
$$392$$ −2.02214 −0.102134
$$393$$ −5.69781 −0.287416
$$394$$ −1.83902 −0.0926482
$$395$$ 2.56844 0.129232
$$396$$ 1.70156 0.0855067
$$397$$ 4.45143 0.223411 0.111705 0.993741i $$-0.464369\pi$$
0.111705 + 0.993741i $$0.464369\pi$$
$$398$$ 5.21384 0.261346
$$399$$ −6.62049 −0.331439
$$400$$ 2.29844 0.114922
$$401$$ 22.7123 1.13420 0.567099 0.823650i $$-0.308066\pi$$
0.567099 + 0.823650i $$0.308066\pi$$
$$402$$ −5.68947 −0.283765
$$403$$ −5.49170 −0.273561
$$404$$ −1.40607 −0.0699547
$$405$$ −1.00000 −0.0496904
$$406$$ −2.87897 −0.142881
$$407$$ 2.75362 0.136492
$$408$$ −5.46295 −0.270456
$$409$$ 5.09259 0.251812 0.125906 0.992042i $$-0.459816\pi$$
0.125906 + 0.992042i $$0.459816\pi$$
$$410$$ 6.00000 0.296319
$$411$$ −3.40312 −0.167864
$$412$$ −18.0230 −0.887932
$$413$$ −6.36259 −0.313083
$$414$$ 4.87897 0.239788
$$415$$ −17.1605 −0.842376
$$416$$ −3.01267 −0.147708
$$417$$ −8.04724 −0.394075
$$418$$ −3.61674 −0.176901
$$419$$ −2.86286 −0.139860 −0.0699300 0.997552i $$-0.522278\pi$$
−0.0699300 + 0.997552i $$0.522278\pi$$
$$420$$ −1.70156 −0.0830277
$$421$$ −16.1047 −0.784894 −0.392447 0.919775i $$-0.628371\pi$$
−0.392447 + 0.919775i $$0.628371\pi$$
$$422$$ −4.22491 −0.205665
$$423$$ −10.1567 −0.493838
$$424$$ −3.16402 −0.153658
$$425$$ 2.70156 0.131045
$$426$$ 1.40312 0.0679816
$$427$$ 3.71308 0.179689
$$428$$ 23.4584 1.13390
$$429$$ 0.568438 0.0274445
$$430$$ 0.0284374 0.00137137
$$431$$ −4.67794 −0.225329 −0.112664 0.993633i $$-0.535938\pi$$
−0.112664 + 0.993633i $$0.535938\pi$$
$$432$$ 2.29844 0.110584
$$433$$ −27.4337 −1.31838 −0.659189 0.751977i $$-0.729100\pi$$
−0.659189 + 0.751977i $$0.729100\pi$$
$$434$$ −5.27777 −0.253341
$$435$$ −5.27000 −0.252677
$$436$$ −33.8449 −1.62088
$$437$$ 59.1278 2.82847
$$438$$ 1.23804 0.0591558
$$439$$ 5.73433 0.273685 0.136842 0.990593i $$-0.456305\pi$$
0.136842 + 0.990593i $$0.456305\pi$$
$$440$$ −2.02214 −0.0964019
$$441$$ 1.00000 0.0476190
$$442$$ −0.838929 −0.0399038
$$443$$ −9.58536 −0.455414 −0.227707 0.973730i $$-0.573123\pi$$
−0.227707 + 0.973730i $$0.573123\pi$$
$$444$$ 4.68545 0.222362
$$445$$ 10.7772 0.510890
$$446$$ −7.60121 −0.359927
$$447$$ 7.50723 0.355080
$$448$$ 1.70156 0.0803913
$$449$$ 30.9463 1.46045 0.730223 0.683209i $$-0.239416\pi$$
0.730223 + 0.683209i $$0.239416\pi$$
$$450$$ 0.546295 0.0257526
$$451$$ 10.9831 0.517173
$$452$$ −10.9501 −0.515049
$$453$$ 3.21795 0.151192
$$454$$ 0.326385 0.0153180
$$455$$ −0.568438 −0.0266488
$$456$$ −13.3876 −0.626931
$$457$$ −42.2246 −1.97519 −0.987593 0.157036i $$-0.949806\pi$$
−0.987593 + 0.157036i $$0.949806\pi$$
$$458$$ 11.5757 0.540898
$$459$$ 2.70156 0.126098
$$460$$ 15.1967 0.708550
$$461$$ 9.87464 0.459908 0.229954 0.973201i $$-0.426142\pi$$
0.229954 + 0.973201i $$0.426142\pi$$
$$462$$ 0.546295 0.0254159
$$463$$ 2.56009 0.118978 0.0594888 0.998229i $$-0.481053\pi$$
0.0594888 + 0.998229i $$0.481053\pi$$
$$464$$ 12.1128 0.562321
$$465$$ −9.66103 −0.448019
$$466$$ −11.0800 −0.513271
$$467$$ −19.6924 −0.911256 −0.455628 0.890170i $$-0.650585\pi$$
−0.455628 + 0.890170i $$0.650585\pi$$
$$468$$ 0.967233 0.0447104
$$469$$ 10.4146 0.480904
$$470$$ 5.54857 0.255937
$$471$$ −3.35107 −0.154409
$$472$$ −12.8661 −0.592209
$$473$$ 0.0520550 0.00239349
$$474$$ −1.40312 −0.0644476
$$475$$ 6.62049 0.303769
$$476$$ 4.59688 0.210697
$$477$$ 1.56469 0.0716420
$$478$$ 1.22219 0.0559016
$$479$$ 15.3672 0.702144 0.351072 0.936348i $$-0.385817\pi$$
0.351072 + 0.936348i $$0.385817\pi$$
$$480$$ −5.29991 −0.241907
$$481$$ 1.56526 0.0713698
$$482$$ 4.87866 0.222217
$$483$$ −8.93103 −0.406376
$$484$$ −1.70156 −0.0773437
$$485$$ −17.9952 −0.817119
$$486$$ 0.546295 0.0247804
$$487$$ −1.52848 −0.0692621 −0.0346310 0.999400i $$-0.511026\pi$$
−0.0346310 + 0.999400i $$0.511026\pi$$
$$488$$ 7.50839 0.339889
$$489$$ 10.9600 0.495630
$$490$$ −0.546295 −0.0246791
$$491$$ −40.7359 −1.83839 −0.919193 0.393808i $$-0.871157\pi$$
−0.919193 + 0.393808i $$0.871157\pi$$
$$492$$ 18.6884 0.842538
$$493$$ 14.2372 0.641213
$$494$$ −2.05589 −0.0924990
$$495$$ 1.00000 0.0449467
$$496$$ 22.2053 0.997046
$$497$$ −2.56844 −0.115210
$$498$$ 9.37469 0.420090
$$499$$ −20.3312 −0.910150 −0.455075 0.890453i $$-0.650388\pi$$
−0.455075 + 0.890453i $$0.650388\pi$$
$$500$$ 1.70156 0.0760962
$$501$$ −7.58830 −0.339020
$$502$$ 1.83491 0.0818963
$$503$$ −23.5952 −1.05206 −0.526030 0.850466i $$-0.676320\pi$$
−0.526030 + 0.850466i $$0.676320\pi$$
$$504$$ 2.02214 0.0900734
$$505$$ −0.826342 −0.0367717
$$506$$ −4.87897 −0.216897
$$507$$ −12.6769 −0.563000
$$508$$ −6.89482 −0.305908
$$509$$ 38.0993 1.68872 0.844361 0.535775i $$-0.179981\pi$$
0.844361 + 0.535775i $$0.179981\pi$$
$$510$$ −1.47585 −0.0653517
$$511$$ −2.26625 −0.100253
$$512$$ 21.4771 0.949161
$$513$$ 6.62049 0.292302
$$514$$ 3.17366 0.139984
$$515$$ −10.5921 −0.466742
$$516$$ 0.0885748 0.00389929
$$517$$ 10.1567 0.446693
$$518$$ 1.50429 0.0660945
$$519$$ −18.2094 −0.799303
$$520$$ −1.14946 −0.0504073
$$521$$ 33.9952 1.48936 0.744678 0.667424i $$-0.232603\pi$$
0.744678 + 0.667424i $$0.232603\pi$$
$$522$$ 2.87897 0.126009
$$523$$ 10.9517 0.478884 0.239442 0.970911i $$-0.423035\pi$$
0.239442 + 0.970911i $$0.423035\pi$$
$$524$$ 9.69518 0.423536
$$525$$ −1.00000 −0.0436436
$$526$$ −12.7407 −0.555522
$$527$$ 26.0999 1.13693
$$528$$ −2.29844 −0.100027
$$529$$ 56.7633 2.46797
$$530$$ −0.854779 −0.0371292
$$531$$ 6.36259 0.276113
$$532$$ 11.2652 0.488408
$$533$$ 6.24321 0.270423
$$534$$ −5.88755 −0.254779
$$535$$ 13.7864 0.596037
$$536$$ 21.0599 0.909650
$$537$$ 4.75362 0.205134
$$538$$ 15.9106 0.685954
$$539$$ −1.00000 −0.0430730
$$540$$ 1.70156 0.0732236
$$541$$ 13.4971 0.580285 0.290143 0.956983i $$-0.406297\pi$$
0.290143 + 0.956983i $$0.406297\pi$$
$$542$$ 17.0671 0.733095
$$543$$ 25.2652 1.08423
$$544$$ 14.3180 0.613881
$$545$$ −19.8905 −0.852015
$$546$$ 0.310535 0.0132897
$$547$$ −42.0182 −1.79657 −0.898285 0.439414i $$-0.855186\pi$$
−0.898285 + 0.439414i $$0.855186\pi$$
$$548$$ 5.79063 0.247363
$$549$$ −3.71308 −0.158471
$$550$$ −0.546295 −0.0232941
$$551$$ 34.8900 1.48636
$$552$$ −18.0598 −0.768677
$$553$$ 2.56844 0.109221
$$554$$ −8.68545 −0.369009
$$555$$ 2.75362 0.116885
$$556$$ 13.6929 0.580707
$$557$$ −27.9461 −1.18411 −0.592057 0.805896i $$-0.701684\pi$$
−0.592057 + 0.805896i $$0.701684\pi$$
$$558$$ 5.27777 0.223426
$$559$$ 0.0295901 0.00125153
$$560$$ 2.29844 0.0971267
$$561$$ −2.70156 −0.114060
$$562$$ −7.70500 −0.325016
$$563$$ −0.266247 −0.0112210 −0.00561050 0.999984i $$-0.501786\pi$$
−0.00561050 + 0.999984i $$0.501786\pi$$
$$564$$ 17.2823 0.727717
$$565$$ −6.43531 −0.270736
$$566$$ −0.725180 −0.0304816
$$567$$ −1.00000 −0.0419961
$$568$$ −5.19375 −0.217925
$$569$$ −35.3983 −1.48397 −0.741987 0.670414i $$-0.766116\pi$$
−0.741987 + 0.670414i $$0.766116\pi$$
$$570$$ −3.61674 −0.151489
$$571$$ −0.440134 −0.0184191 −0.00920953 0.999958i $$-0.502932\pi$$
−0.00920953 + 0.999958i $$0.502932\pi$$
$$572$$ −0.967233 −0.0404421
$$573$$ −17.4504 −0.728999
$$574$$ 6.00000 0.250435
$$575$$ 8.93103 0.372450
$$576$$ −1.70156 −0.0708984
$$577$$ −25.1030 −1.04505 −0.522527 0.852623i $$-0.675010\pi$$
−0.522527 + 0.852623i $$0.675010\pi$$
$$578$$ −5.29991 −0.220447
$$579$$ 22.9546 0.953963
$$580$$ 8.96723 0.372344
$$581$$ −17.1605 −0.711937
$$582$$ 9.83067 0.407494
$$583$$ −1.56469 −0.0648026
$$584$$ −4.58268 −0.189633
$$585$$ 0.568438 0.0235020
$$586$$ 10.0400 0.414750
$$587$$ 3.24638 0.133993 0.0669963 0.997753i $$-0.478658\pi$$
0.0669963 + 0.997753i $$0.478658\pi$$
$$588$$ −1.70156 −0.0701712
$$589$$ 63.9608 2.63546
$$590$$ −3.47585 −0.143098
$$591$$ −3.36634 −0.138473
$$592$$ −6.32902 −0.260121
$$593$$ 10.7252 0.440430 0.220215 0.975451i $$-0.429324\pi$$
0.220215 + 0.975451i $$0.429324\pi$$
$$594$$ −0.546295 −0.0224147
$$595$$ 2.70156 0.110753
$$596$$ −12.7740 −0.523244
$$597$$ 9.54402 0.390611
$$598$$ −2.77340 −0.113413
$$599$$ 36.3607 1.48566 0.742829 0.669481i $$-0.233483\pi$$
0.742829 + 0.669481i $$0.233483\pi$$
$$600$$ −2.02214 −0.0825537
$$601$$ −39.7034 −1.61954 −0.809769 0.586749i $$-0.800407\pi$$
−0.809769 + 0.586749i $$0.800407\pi$$
$$602$$ 0.0284374 0.00115902
$$603$$ −10.4146 −0.424117
$$604$$ −5.47553 −0.222796
$$605$$ −1.00000 −0.0406558
$$606$$ 0.451426 0.0183379
$$607$$ −9.91893 −0.402597 −0.201299 0.979530i $$-0.564516\pi$$
−0.201299 + 0.979530i $$0.564516\pi$$
$$608$$ 35.0880 1.42301
$$609$$ −5.27000 −0.213551
$$610$$ 2.02844 0.0821290
$$611$$ 5.77348 0.233570
$$612$$ −4.59688 −0.185818
$$613$$ 39.2681 1.58602 0.793012 0.609206i $$-0.208512\pi$$
0.793012 + 0.609206i $$0.208512\pi$$
$$614$$ 3.52438 0.142232
$$615$$ 10.9831 0.442881
$$616$$ −2.02214 −0.0814745
$$617$$ 31.3462 1.26195 0.630976 0.775802i $$-0.282655\pi$$
0.630976 + 0.775802i $$0.282655\pi$$
$$618$$ 5.78638 0.232762
$$619$$ −12.3916 −0.498061 −0.249030 0.968496i $$-0.580112\pi$$
−0.249030 + 0.968496i $$0.580112\pi$$
$$620$$ 16.4388 0.660200
$$621$$ 8.93103 0.358390
$$622$$ −16.0685 −0.644287
$$623$$ 10.7772 0.431781
$$624$$ −1.30652 −0.0523027
$$625$$ 1.00000 0.0400000
$$626$$ −13.5231 −0.540491
$$627$$ −6.62049 −0.264397
$$628$$ 5.70205 0.227537
$$629$$ −7.43907 −0.296615
$$630$$ 0.546295 0.0217649
$$631$$ 33.2233 1.32260 0.661299 0.750123i $$-0.270006\pi$$
0.661299 + 0.750123i $$0.270006\pi$$
$$632$$ 5.19375 0.206596
$$633$$ −7.73375 −0.307389
$$634$$ −8.24661 −0.327515
$$635$$ −4.05206 −0.160801
$$636$$ −2.66241 −0.105571
$$637$$ −0.568438 −0.0225223
$$638$$ −2.87897 −0.113980
$$639$$ 2.56844 0.101606
$$640$$ 11.5294 0.455739
$$641$$ 18.9179 0.747211 0.373605 0.927588i $$-0.378121\pi$$
0.373605 + 0.927588i $$0.378121\pi$$
$$642$$ −7.53143 −0.297242
$$643$$ −5.82554 −0.229737 −0.114868 0.993381i $$-0.536645\pi$$
−0.114868 + 0.993381i $$0.536645\pi$$
$$644$$ 15.1967 0.598834
$$645$$ 0.0520550 0.00204966
$$646$$ 9.77085 0.384429
$$647$$ 8.30759 0.326605 0.163302 0.986576i $$-0.447785\pi$$
0.163302 + 0.986576i $$0.447785\pi$$
$$648$$ −2.02214 −0.0794373
$$649$$ −6.36259 −0.249753
$$650$$ −0.310535 −0.0121802
$$651$$ −9.66103 −0.378646
$$652$$ −18.6492 −0.730359
$$653$$ 44.4182 1.73822 0.869109 0.494621i $$-0.164693\pi$$
0.869109 + 0.494621i $$0.164693\pi$$
$$654$$ 10.8661 0.424897
$$655$$ 5.69781 0.222632
$$656$$ −25.2439 −0.985610
$$657$$ 2.26625 0.0884147
$$658$$ 5.54857 0.216306
$$659$$ −13.4965 −0.525750 −0.262875 0.964830i $$-0.584671\pi$$
−0.262875 + 0.964830i $$0.584671\pi$$
$$660$$ −1.70156 −0.0662332
$$661$$ 31.7450 1.23474 0.617370 0.786673i $$-0.288198\pi$$
0.617370 + 0.786673i $$0.288198\pi$$
$$662$$ 3.70532 0.144011
$$663$$ −1.53567 −0.0596405
$$664$$ −34.7010 −1.34666
$$665$$ 6.62049 0.256732
$$666$$ −1.50429 −0.0582899
$$667$$ 47.0665 1.82242
$$668$$ 12.9120 0.499579
$$669$$ −13.9141 −0.537951
$$670$$ 5.68947 0.219803
$$671$$ 3.71308 0.143342
$$672$$ −5.29991 −0.204449
$$673$$ 5.81295 0.224073 0.112036 0.993704i $$-0.464263\pi$$
0.112036 + 0.993704i $$0.464263\pi$$
$$674$$ −17.4546 −0.672326
$$675$$ 1.00000 0.0384900
$$676$$ 21.5705 0.829634
$$677$$ −23.3699 −0.898177 −0.449088 0.893487i $$-0.648251\pi$$
−0.449088 + 0.893487i $$0.648251\pi$$
$$678$$ 3.51558 0.135015
$$679$$ −17.9952 −0.690592
$$680$$ 5.46295 0.209494
$$681$$ 0.597452 0.0228944
$$682$$ −5.27777 −0.202096
$$683$$ 13.4705 0.515433 0.257716 0.966221i $$-0.417030\pi$$
0.257716 + 0.966221i $$0.417030\pi$$
$$684$$ −11.2652 −0.430735
$$685$$ 3.40312 0.130027
$$686$$ −0.546295 −0.0208576
$$687$$ 21.1895 0.808430
$$688$$ −0.119645 −0.00456143
$$689$$ −0.889427 −0.0338845
$$690$$ −4.87897 −0.185739
$$691$$ 9.79358 0.372565 0.186283 0.982496i $$-0.440356\pi$$
0.186283 + 0.982496i $$0.440356\pi$$
$$692$$ 30.9844 1.17785
$$693$$ 1.00000 0.0379869
$$694$$ −7.46751 −0.283463
$$695$$ 8.04724 0.305249
$$696$$ −10.6567 −0.403941
$$697$$ −29.6715 −1.12389
$$698$$ 8.83469 0.334398
$$699$$ −20.2821 −0.767139
$$700$$ 1.70156 0.0643130
$$701$$ −9.27000 −0.350123 −0.175062 0.984557i $$-0.556012\pi$$
−0.175062 + 0.984557i $$0.556012\pi$$
$$702$$ −0.310535 −0.0117204
$$703$$ −18.2303 −0.687569
$$704$$ 1.70156 0.0641300
$$705$$ 10.1567 0.382525
$$706$$ −5.43188 −0.204431
$$707$$ −0.826342 −0.0310778
$$708$$ −10.8263 −0.406879
$$709$$ −19.9898 −0.750732 −0.375366 0.926877i $$-0.622483\pi$$
−0.375366 + 0.926877i $$0.622483\pi$$
$$710$$ −1.40312 −0.0526583
$$711$$ −2.56844 −0.0963240
$$712$$ 21.7931 0.816732
$$713$$ 86.2829 3.23132
$$714$$ −1.47585 −0.0552323
$$715$$ −0.568438 −0.0212584
$$716$$ −8.08857 −0.302284
$$717$$ 2.23723 0.0835510
$$718$$ −6.87487 −0.256568
$$719$$ −26.7174 −0.996391 −0.498196 0.867065i $$-0.666004\pi$$
−0.498196 + 0.867065i $$0.666004\pi$$
$$720$$ −2.29844 −0.0856577
$$721$$ −10.5921 −0.394469
$$722$$ 13.5650 0.504837
$$723$$ 8.93045 0.332127
$$724$$ −42.9903 −1.59772
$$725$$ 5.27000 0.195723
$$726$$ 0.546295 0.0202749
$$727$$ 16.4955 0.611782 0.305891 0.952066i $$-0.401046\pi$$
0.305891 + 0.952066i $$0.401046\pi$$
$$728$$ −1.14946 −0.0426020
$$729$$ 1.00000 0.0370370
$$730$$ −1.23804 −0.0458219
$$731$$ −0.140630 −0.00520138
$$732$$ 6.31804 0.233522
$$733$$ −22.6838 −0.837847 −0.418923 0.908022i $$-0.637592\pi$$
−0.418923 + 0.908022i $$0.637592\pi$$
$$734$$ 20.1965 0.745465
$$735$$ −1.00000 −0.0368856
$$736$$ 47.3337 1.74474
$$737$$ 10.4146 0.383628
$$738$$ −6.00000 −0.220863
$$739$$ −41.7166 −1.53457 −0.767285 0.641306i $$-0.778393\pi$$
−0.767285 + 0.641306i $$0.778393\pi$$
$$740$$ −4.68545 −0.172241
$$741$$ −3.76334 −0.138250
$$742$$ −0.854779 −0.0313799
$$743$$ −26.4664 −0.970959 −0.485480 0.874248i $$-0.661355\pi$$
−0.485480 + 0.874248i $$0.661355\pi$$
$$744$$ −19.5360 −0.716224
$$745$$ −7.50723 −0.275044
$$746$$ −7.32630 −0.268235
$$747$$ 17.1605 0.627870
$$748$$ 4.59688 0.168078
$$749$$ 13.7864 0.503744
$$750$$ −0.546295 −0.0199479
$$751$$ −36.2802 −1.32388 −0.661942 0.749555i $$-0.730268\pi$$
−0.661942 + 0.749555i $$0.730268\pi$$
$$752$$ −23.3446 −0.851291
$$753$$ 3.35884 0.122403
$$754$$ −1.63652 −0.0595985
$$755$$ −3.21795 −0.117113
$$756$$ 1.70156 0.0618852
$$757$$ 27.3451 0.993874 0.496937 0.867786i $$-0.334458\pi$$
0.496937 + 0.867786i $$0.334458\pi$$
$$758$$ −0.440134 −0.0159864
$$759$$ −8.93103 −0.324176
$$760$$ 13.3876 0.485619
$$761$$ −16.2537 −0.589195 −0.294597 0.955621i $$-0.595186\pi$$
−0.294597 + 0.955621i $$0.595186\pi$$
$$762$$ 2.21362 0.0801909
$$763$$ −19.8905 −0.720084
$$764$$ 29.6929 1.07425
$$765$$ −2.70156 −0.0976752
$$766$$ −12.2094 −0.441143
$$767$$ −3.61674 −0.130593
$$768$$ −2.89531 −0.104476
$$769$$ −38.2698 −1.38004 −0.690022 0.723789i $$-0.742399\pi$$
−0.690022 + 0.723789i $$0.742399\pi$$
$$770$$ −0.546295 −0.0196871
$$771$$ 5.80943 0.209221
$$772$$ −39.0588 −1.40576
$$773$$ 6.60438 0.237543 0.118772 0.992922i $$-0.462104\pi$$
0.118772 + 0.992922i $$0.462104\pi$$
$$774$$ −0.0284374 −0.00102216
$$775$$ 9.66103 0.347034
$$776$$ −36.3888 −1.30628
$$777$$ 2.75362 0.0987855
$$778$$ −16.1082 −0.577507
$$779$$ −72.7134 −2.60523
$$780$$ −0.967233 −0.0346325
$$781$$ −2.56844 −0.0919060
$$782$$ 13.1808 0.471346
$$783$$ 5.27000 0.188334
$$784$$ 2.29844 0.0820871
$$785$$ 3.35107 0.119605
$$786$$ −3.11268 −0.111026
$$787$$ −23.1916 −0.826692 −0.413346 0.910574i $$-0.635640\pi$$
−0.413346 + 0.910574i $$0.635640\pi$$
$$788$$ 5.72804 0.204053
$$789$$ −23.3221 −0.830287
$$790$$ 1.40312 0.0499209
$$791$$ −6.43531 −0.228813
$$792$$ 2.02214 0.0718537
$$793$$ 2.11066 0.0749517
$$794$$ 2.43179 0.0863010
$$795$$ −1.56469 −0.0554937
$$796$$ −16.2397 −0.575602
$$797$$ 32.5872 1.15430 0.577150 0.816638i $$-0.304165\pi$$
0.577150 + 0.816638i $$0.304165\pi$$
$$798$$ −3.61674 −0.128031
$$799$$ −27.4391 −0.970724
$$800$$ 5.29991 0.187380
$$801$$ −10.7772 −0.380795
$$802$$ 12.4076 0.438127
$$803$$ −2.26625 −0.0799741
$$804$$ 17.7212 0.624977
$$805$$ 8.93103 0.314777
$$806$$ −3.00009 −0.105674
$$807$$ 29.1246 1.02523
$$808$$ −1.67098 −0.0587849
$$809$$ 13.5303 0.475699 0.237850 0.971302i $$-0.423557\pi$$
0.237850 + 0.971302i $$0.423557\pi$$
$$810$$ −0.546295 −0.0191948
$$811$$ 27.7241 0.973525 0.486763 0.873534i $$-0.338178\pi$$
0.486763 + 0.873534i $$0.338178\pi$$
$$812$$ 8.96723 0.314688
$$813$$ 31.2416 1.09569
$$814$$ 1.50429 0.0527252
$$815$$ −10.9600 −0.383914
$$816$$ 6.20937 0.217372
$$817$$ −0.344630 −0.0120571
$$818$$ 2.78205 0.0972723
$$819$$ 0.568438 0.0198628
$$820$$ −18.6884 −0.652627
$$821$$ 40.9774 1.43012 0.715061 0.699062i $$-0.246399\pi$$
0.715061 + 0.699062i $$0.246399\pi$$
$$822$$ −1.85911 −0.0648439
$$823$$ 3.27352 0.114108 0.0570539 0.998371i $$-0.481829\pi$$
0.0570539 + 0.998371i $$0.481829\pi$$
$$824$$ −21.4187 −0.746154
$$825$$ −1.00000 −0.0348155
$$826$$ −3.47585 −0.120940
$$827$$ −36.5217 −1.26998 −0.634992 0.772519i $$-0.718997\pi$$
−0.634992 + 0.772519i $$0.718997\pi$$
$$828$$ −15.1967 −0.528122
$$829$$ 21.2335 0.737469 0.368735 0.929535i $$-0.379791\pi$$
0.368735 + 0.929535i $$0.379791\pi$$
$$830$$ −9.37469 −0.325400
$$831$$ −15.8988 −0.551525
$$832$$ 0.967233 0.0335328
$$833$$ 2.70156 0.0936036
$$834$$ −4.39616 −0.152227
$$835$$ 7.58830 0.262604
$$836$$ 11.2652 0.389614
$$837$$ 9.66103 0.333934
$$838$$ −1.56397 −0.0540263
$$839$$ 20.7571 0.716616 0.358308 0.933603i $$-0.383354\pi$$
0.358308 + 0.933603i $$0.383354\pi$$
$$840$$ −2.02214 −0.0697706
$$841$$ −1.22709 −0.0423136
$$842$$ −8.79790 −0.303196
$$843$$ −14.1041 −0.485771
$$844$$ 13.1595 0.452967
$$845$$ 12.6769 0.436098
$$846$$ −5.54857 −0.190764
$$847$$ −1.00000 −0.0343604
$$848$$ 3.59633 0.123499
$$849$$ −1.32745 −0.0455580
$$850$$ 1.47585 0.0506212
$$851$$ −24.5926 −0.843025
$$852$$ −4.37036 −0.149726
$$853$$ −31.3333 −1.07283 −0.536417 0.843953i $$-0.680222\pi$$
−0.536417 + 0.843953i $$0.680222\pi$$
$$854$$ 2.02844 0.0694117
$$855$$ −6.62049 −0.226416
$$856$$ 27.8780 0.952852
$$857$$ 13.5411 0.462554 0.231277 0.972888i $$-0.425710\pi$$
0.231277 + 0.972888i $$0.425710\pi$$
$$858$$ 0.310535 0.0106015
$$859$$ −19.5644 −0.667530 −0.333765 0.942656i $$-0.608319\pi$$
−0.333765 + 0.942656i $$0.608319\pi$$
$$860$$ −0.0885748 −0.00302038
$$861$$ 10.9831 0.374302
$$862$$ −2.55554 −0.0870419
$$863$$ −8.91434 −0.303448 −0.151724 0.988423i $$-0.548482\pi$$
−0.151724 + 0.988423i $$0.548482\pi$$
$$864$$ 5.29991 0.180307
$$865$$ 18.2094 0.619137
$$866$$ −14.9869 −0.509275
$$867$$ −9.70156 −0.329482
$$868$$ 16.4388 0.557971
$$869$$ 2.56844 0.0871283
$$870$$ −2.87897 −0.0976063
$$871$$ 5.92008 0.200594
$$872$$ −40.2214 −1.36207
$$873$$ 17.9952 0.609045
$$874$$ 32.3012 1.09260
$$875$$ 1.00000 0.0338062
$$876$$ −3.85616 −0.130288
$$877$$ 6.17626 0.208557 0.104279 0.994548i $$-0.466747\pi$$
0.104279 + 0.994548i $$0.466747\pi$$
$$878$$ 3.13263 0.105721
$$879$$ 18.3784 0.619889
$$880$$ 2.29844 0.0774803
$$881$$ 19.3511 0.651954 0.325977 0.945378i $$-0.394307\pi$$
0.325977 + 0.945378i $$0.394307\pi$$
$$882$$ 0.546295 0.0183947
$$883$$ 28.7925 0.968945 0.484473 0.874806i $$-0.339012\pi$$
0.484473 + 0.874806i $$0.339012\pi$$
$$884$$ 2.61304 0.0878861
$$885$$ −6.36259 −0.213876
$$886$$ −5.23643 −0.175921
$$887$$ −43.4970 −1.46049 −0.730243 0.683187i $$-0.760593\pi$$
−0.730243 + 0.683187i $$0.760593\pi$$
$$888$$ 5.56821 0.186857
$$889$$ −4.05206 −0.135902
$$890$$ 5.88755 0.197351
$$891$$ −1.00000 −0.0335013
$$892$$ 23.6757 0.792722
$$893$$ −67.2426 −2.25019
$$894$$ 4.10116 0.137163
$$895$$ −4.75362 −0.158896
$$896$$ 11.5294 0.385169
$$897$$ −5.07674 −0.169507
$$898$$ 16.9058 0.564154
$$899$$ 50.9136 1.69806
$$900$$ −1.70156 −0.0567187
$$901$$ 4.22709 0.140825
$$902$$ 6.00000 0.199778
$$903$$ 0.0520550 0.00173228
$$904$$ −13.0131 −0.432810
$$905$$ −25.2652 −0.839843
$$906$$ 1.75795 0.0584039
$$907$$ −9.20210 −0.305551 −0.152775 0.988261i $$-0.548821\pi$$
−0.152775 + 0.988261i $$0.548821\pi$$
$$908$$ −1.01660 −0.0337371
$$909$$ 0.826342 0.0274080
$$910$$ −0.310535 −0.0102941
$$911$$ 7.98331 0.264499 0.132249 0.991216i $$-0.457780\pi$$
0.132249 + 0.991216i $$0.457780\pi$$
$$912$$ 15.2168 0.503878
$$913$$ −17.1605 −0.567929
$$914$$ −23.0671 −0.762992
$$915$$ 3.71308 0.122751
$$916$$ −36.0553 −1.19130
$$917$$ 5.69781 0.188158
$$918$$ 1.47585 0.0487103
$$919$$ −36.4176 −1.20131 −0.600653 0.799510i $$-0.705093\pi$$
−0.600653 + 0.799510i $$0.705093\pi$$
$$920$$ 18.0598 0.595415
$$921$$ 6.45143 0.212582
$$922$$ 5.39447 0.177657
$$923$$ −1.46000 −0.0480565
$$924$$ −1.70156 −0.0559773
$$925$$ −2.75362 −0.0905384
$$926$$ 1.39857 0.0459597
$$927$$ 10.5921 0.347889
$$928$$ 27.9305 0.916865
$$929$$ 1.06660 0.0349940 0.0174970 0.999847i $$-0.494430\pi$$
0.0174970 + 0.999847i $$0.494430\pi$$
$$930$$ −5.27777 −0.173065
$$931$$ 6.62049 0.216978
$$932$$ 34.5112 1.13045
$$933$$ −29.4136 −0.962957
$$934$$ −10.7579 −0.352008
$$935$$ 2.70156 0.0883505
$$936$$ 1.14946 0.0375714
$$937$$ −4.74611 −0.155049 −0.0775243 0.996990i $$-0.524702\pi$$
−0.0775243 + 0.996990i $$0.524702\pi$$
$$938$$ 5.68947 0.185768
$$939$$ −24.7542 −0.807823
$$940$$ −17.2823 −0.563687
$$941$$ 33.1988 1.08225 0.541125 0.840942i $$-0.317999\pi$$
0.541125 + 0.840942i $$0.317999\pi$$
$$942$$ −1.83067 −0.0596465
$$943$$ −98.0902 −3.19426
$$944$$ 14.6240 0.475971
$$945$$ 1.00000 0.0325300
$$946$$ 0.0284374 0.000924579 0
$$947$$ 6.30361 0.204840 0.102420 0.994741i $$-0.467341\pi$$
0.102420 + 0.994741i $$0.467341\pi$$
$$948$$ 4.37036 0.141943
$$949$$ −1.28822 −0.0418175
$$950$$ 3.61674 0.117343
$$951$$ −15.0955 −0.489506
$$952$$ 5.46295 0.177055
$$953$$ −39.4705 −1.27857 −0.639287 0.768968i $$-0.720770\pi$$
−0.639287 + 0.768968i $$0.720770\pi$$
$$954$$ 0.854779 0.0276745
$$955$$ 17.4504 0.564680
$$956$$ −3.80679 −0.123120
$$957$$ −5.27000 −0.170355
$$958$$ 8.39501 0.271230
$$959$$ 3.40312 0.109893
$$960$$ 1.70156 0.0549177
$$961$$ 62.3355 2.01082
$$962$$ 0.855094 0.0275693
$$963$$ −13.7864 −0.444260
$$964$$ −15.1957 −0.489421
$$965$$ −22.9546 −0.738936
$$966$$ −4.87897 −0.156978
$$967$$ −20.7192 −0.666285 −0.333142 0.942877i $$-0.608109\pi$$
−0.333142 + 0.942877i $$0.608109\pi$$
$$968$$ −2.02214 −0.0649942
$$969$$ 17.8857 0.574571
$$970$$ −9.83067 −0.315644
$$971$$ 0.281521 0.00903444 0.00451722 0.999990i $$-0.498562\pi$$
0.00451722 + 0.999990i $$0.498562\pi$$
$$972$$ −1.70156 −0.0545776
$$973$$ 8.04724 0.257983
$$974$$ −0.835001 −0.0267551
$$975$$ −0.568438 −0.0182046
$$976$$ −8.53429 −0.273176
$$977$$ 53.1267 1.69967 0.849836 0.527047i $$-0.176701\pi$$
0.849836 + 0.527047i $$0.176701\pi$$
$$978$$ 5.98741 0.191456
$$979$$ 10.7772 0.344442
$$980$$ 1.70156 0.0543544
$$981$$ 19.8905 0.635055
$$982$$ −22.2538 −0.710147
$$983$$ −47.2072 −1.50567 −0.752837 0.658207i $$-0.771315\pi$$
−0.752837 + 0.658207i $$0.771315\pi$$
$$984$$ 22.2094 0.708009
$$985$$ 3.36634 0.107261
$$986$$ 7.77773 0.247693
$$987$$ 10.1567 0.323293
$$988$$ 6.40356 0.203724
$$989$$ −0.464905 −0.0147831
$$990$$ 0.546295 0.0173624
$$991$$ −30.4901 −0.968549 −0.484274 0.874916i $$-0.660916\pi$$
−0.484274 + 0.874916i $$0.660916\pi$$
$$992$$ 51.2026 1.62568
$$993$$ 6.78263 0.215240
$$994$$ −1.40312 −0.0445044
$$995$$ −9.54402 −0.302566
$$996$$ −29.1996 −0.925226
$$997$$ 6.62107 0.209691 0.104846 0.994489i $$-0.466565\pi$$
0.104846 + 0.994489i $$0.466565\pi$$
$$998$$ −11.1068 −0.351581
$$999$$ −2.75362 −0.0871206
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 1155.2.a.u.1.3 4
3.2 odd 2 3465.2.a.bl.1.2 4
5.4 even 2 5775.2.a.bz.1.2 4
7.6 odd 2 8085.2.a.bn.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.u.1.3 4 1.1 even 1 trivial
3465.2.a.bl.1.2 4 3.2 odd 2
5775.2.a.bz.1.2 4 5.4 even 2
8085.2.a.bn.1.3 4 7.6 odd 2