# Properties

 Label 5070.2.a.j.1.1 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

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

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −2.00000 −0.534522
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 2.00000 0.436436
$$22$$ 3.00000 0.639602
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −3.00000 −0.522233
$$34$$ −6.00000 −1.02899
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ −1.00000 −0.149071
$$46$$ −3.00000 −0.442326
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 3.00000 0.404520
$$56$$ −2.00000 −0.267261
$$57$$ 2.00000 0.264906
$$58$$ −3.00000 −0.393919
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 3.00000 0.361158
$$70$$ 2.00000 0.239046
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 1.00000 0.115470
$$76$$ 2.00000 0.229416
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −6.00000 −0.650791
$$86$$ 1.00000 0.107833
$$87$$ 3.00000 0.321634
$$88$$ 3.00000 0.319801
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 3.00000 0.312772
$$93$$ 5.00000 0.518476
$$94$$ 3.00000 0.309426
$$95$$ −2.00000 −0.205196
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −3.00000 −0.301511
$$100$$ 1.00000 0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 6.00000 0.582772
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ −7.00000 −0.664411
$$112$$ 2.00000 0.188982
$$113$$ 15.0000 1.41108 0.705541 0.708669i $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ −3.00000 −0.279751
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ 9.00000 0.828517
$$119$$ 12.0000 1.10004
$$120$$ 1.00000 0.0912871
$$121$$ −2.00000 −0.181818
$$122$$ −2.00000 −0.181071
$$123$$ 6.00000 0.541002
$$124$$ 5.00000 0.449013
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ 9.00000 0.786334 0.393167 0.919467i $$-0.371379\pi$$
0.393167 + 0.919467i $$0.371379\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 4.00000 0.346844
$$134$$ −8.00000 −0.691095
$$135$$ −1.00000 −0.0860663
$$136$$ −6.00000 −0.514496
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ −3.00000 −0.255377
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −3.00000 −0.252646
$$142$$ 12.0000 1.00702
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −3.00000 −0.249136
$$146$$ −14.0000 −1.15865
$$147$$ −3.00000 −0.247436
$$148$$ −7.00000 −0.575396
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 6.00000 0.485071
$$154$$ 6.00000 0.483494
$$155$$ −5.00000 −0.401610
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ −5.00000 −0.397779
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 6.00000 0.472866
$$162$$ −1.00000 −0.0785674
$$163$$ −13.0000 −1.01824 −0.509119 0.860696i $$-0.670029\pi$$
−0.509119 + 0.860696i $$0.670029\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 3.00000 0.233550
$$166$$ 6.00000 0.465690
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ 6.00000 0.460179
$$171$$ 2.00000 0.152944
$$172$$ −1.00000 −0.0762493
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ 2.00000 0.151186
$$176$$ −3.00000 −0.226134
$$177$$ −9.00000 −0.676481
$$178$$ 18.0000 1.34916
$$179$$ −3.00000 −0.224231 −0.112115 0.993695i $$-0.535763\pi$$
−0.112115 + 0.993695i $$0.535763\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ −3.00000 −0.221163
$$185$$ 7.00000 0.514650
$$186$$ −5.00000 −0.366618
$$187$$ −18.0000 −1.31629
$$188$$ −3.00000 −0.218797
$$189$$ 2.00000 0.145479
$$190$$ 2.00000 0.145095
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 24.0000 1.70993 0.854965 0.518686i $$-0.173579\pi$$
0.854965 + 0.518686i $$0.173579\pi$$
$$198$$ 3.00000 0.213201
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 8.00000 0.564276
$$202$$ −6.00000 −0.422159
$$203$$ 6.00000 0.421117
$$204$$ 6.00000 0.420084
$$205$$ −6.00000 −0.419058
$$206$$ −14.0000 −0.975426
$$207$$ 3.00000 0.208514
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ 2.00000 0.138013
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −12.0000 −0.822226
$$214$$ 6.00000 0.410152
$$215$$ 1.00000 0.0681994
$$216$$ −1.00000 −0.0680414
$$217$$ 10.0000 0.678844
$$218$$ −14.0000 −0.948200
$$219$$ 14.0000 0.946032
$$220$$ 3.00000 0.202260
$$221$$ 0 0
$$222$$ 7.00000 0.469809
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 1.00000 0.0666667
$$226$$ −15.0000 −0.997785
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 2.00000 0.132453
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 3.00000 0.197814
$$231$$ −6.00000 −0.394771
$$232$$ −3.00000 −0.196960
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 3.00000 0.195698
$$236$$ −9.00000 −0.585850
$$237$$ 5.00000 0.324785
$$238$$ −12.0000 −0.777844
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 3.00000 0.191663
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ −5.00000 −0.317500
$$249$$ −6.00000 −0.380235
$$250$$ 1.00000 0.0632456
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −9.00000 −0.565825
$$254$$ −14.0000 −0.878438
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ −21.0000 −1.30994 −0.654972 0.755653i $$-0.727320\pi$$
−0.654972 + 0.755653i $$0.727320\pi$$
$$258$$ 1.00000 0.0622573
$$259$$ −14.0000 −0.869918
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ −9.00000 −0.556022
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 6.00000 0.368577
$$266$$ −4.00000 −0.245256
$$267$$ −18.0000 −1.10158
$$268$$ 8.00000 0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ −3.00000 −0.180907
$$276$$ 3.00000 0.180579
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 5.00000 0.299342
$$280$$ 2.00000 0.119523
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 3.00000 0.178647
$$283$$ −31.0000 −1.84276 −0.921379 0.388664i $$-0.872937\pi$$
−0.921379 + 0.388664i $$0.872937\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 3.00000 0.176166
$$291$$ 14.0000 0.820695
$$292$$ 14.0000 0.819288
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 9.00000 0.524000
$$296$$ 7.00000 0.406867
$$297$$ −3.00000 −0.174078
$$298$$ 9.00000 0.521356
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ −2.00000 −0.115278
$$302$$ −8.00000 −0.460348
$$303$$ 6.00000 0.344691
$$304$$ 2.00000 0.114708
$$305$$ −2.00000 −0.114520
$$306$$ −6.00000 −0.342997
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ 14.0000 0.796432
$$310$$ 5.00000 0.283981
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 13.0000 0.733632
$$315$$ −2.00000 −0.112687
$$316$$ 5.00000 0.281272
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −9.00000 −0.503903
$$320$$ −1.00000 −0.0559017
$$321$$ −6.00000 −0.334887
$$322$$ −6.00000 −0.334367
$$323$$ 12.0000 0.667698
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 13.0000 0.720003
$$327$$ 14.0000 0.774202
$$328$$ −6.00000 −0.331295
$$329$$ −6.00000 −0.330791
$$330$$ −3.00000 −0.165145
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ −7.00000 −0.383598
$$334$$ 9.00000 0.492458
$$335$$ −8.00000 −0.437087
$$336$$ 2.00000 0.109109
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 15.0000 0.814688
$$340$$ −6.00000 −0.325396
$$341$$ −15.0000 −0.812296
$$342$$ −2.00000 −0.108148
$$343$$ −20.0000 −1.07990
$$344$$ 1.00000 0.0539164
$$345$$ −3.00000 −0.161515
$$346$$ −12.0000 −0.645124
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 3.00000 0.160817
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 9.00000 0.478345
$$355$$ 12.0000 0.636894
$$356$$ −18.0000 −0.953998
$$357$$ 12.0000 0.635107
$$358$$ 3.00000 0.158555
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ 16.0000 0.840941
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −14.0000 −0.732793
$$366$$ −2.00000 −0.104542
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 6.00000 0.312348
$$370$$ −7.00000 −0.363913
$$371$$ −12.0000 −0.623009
$$372$$ 5.00000 0.259238
$$373$$ −25.0000 −1.29445 −0.647225 0.762299i $$-0.724071\pi$$
−0.647225 + 0.762299i $$0.724071\pi$$
$$374$$ 18.0000 0.930758
$$375$$ −1.00000 −0.0516398
$$376$$ 3.00000 0.154713
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 38.0000 1.95193 0.975964 0.217930i $$-0.0699304\pi$$
0.975964 + 0.217930i $$0.0699304\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ 14.0000 0.717242
$$382$$ 12.0000 0.613973
$$383$$ −21.0000 −1.07305 −0.536525 0.843884i $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 6.00000 0.305788
$$386$$ 4.00000 0.203595
$$387$$ −1.00000 −0.0508329
$$388$$ 14.0000 0.710742
$$389$$ 3.00000 0.152106 0.0760530 0.997104i $$-0.475768\pi$$
0.0760530 + 0.997104i $$0.475768\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 3.00000 0.151523
$$393$$ 9.00000 0.453990
$$394$$ −24.0000 −1.20910
$$395$$ −5.00000 −0.251577
$$396$$ −3.00000 −0.150756
$$397$$ −31.0000 −1.55585 −0.777923 0.628360i $$-0.783727\pi$$
−0.777923 + 0.628360i $$0.783727\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 4.00000 0.200250
$$400$$ 1.00000 0.0500000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ −1.00000 −0.0496904
$$406$$ −6.00000 −0.297775
$$407$$ 21.0000 1.04093
$$408$$ −6.00000 −0.297044
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 6.00000 0.296319
$$411$$ 9.00000 0.443937
$$412$$ 14.0000 0.689730
$$413$$ −18.0000 −0.885722
$$414$$ −3.00000 −0.147442
$$415$$ 6.00000 0.294528
$$416$$ 0 0
$$417$$ 14.0000 0.685583
$$418$$ 6.00000 0.293470
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ −16.0000 −0.779792 −0.389896 0.920859i $$-0.627489\pi$$
−0.389896 + 0.920859i $$0.627489\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ −3.00000 −0.145865
$$424$$ 6.00000 0.291386
$$425$$ 6.00000 0.291043
$$426$$ 12.0000 0.581402
$$427$$ 4.00000 0.193574
$$428$$ −6.00000 −0.290021
$$429$$ 0 0
$$430$$ −1.00000 −0.0482243
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −40.0000 −1.92228 −0.961139 0.276066i $$-0.910969\pi$$
−0.961139 + 0.276066i $$0.910969\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ −3.00000 −0.143839
$$436$$ 14.0000 0.670478
$$437$$ 6.00000 0.287019
$$438$$ −14.0000 −0.668946
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ −3.00000 −0.143019
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −7.00000 −0.332205
$$445$$ 18.0000 0.853282
$$446$$ 10.0000 0.473514
$$447$$ −9.00000 −0.425685
$$448$$ 2.00000 0.0944911
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −18.0000 −0.847587
$$452$$ 15.0000 0.705541
$$453$$ 8.00000 0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 6.00000 0.280056
$$460$$ −3.00000 −0.139876
$$461$$ 15.0000 0.698620 0.349310 0.937007i $$-0.386416\pi$$
0.349310 + 0.937007i $$0.386416\pi$$
$$462$$ 6.00000 0.279145
$$463$$ −34.0000 −1.58011 −0.790057 0.613033i $$-0.789949\pi$$
−0.790057 + 0.613033i $$0.789949\pi$$
$$464$$ 3.00000 0.139272
$$465$$ −5.00000 −0.231869
$$466$$ −21.0000 −0.972806
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ −3.00000 −0.138380
$$471$$ −13.0000 −0.599008
$$472$$ 9.00000 0.414259
$$473$$ 3.00000 0.137940
$$474$$ −5.00000 −0.229658
$$475$$ 2.00000 0.0917663
$$476$$ 12.0000 0.550019
$$477$$ −6.00000 −0.274721
$$478$$ −24.0000 −1.09773
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −17.0000 −0.774329
$$483$$ 6.00000 0.273009
$$484$$ −2.00000 −0.0909091
$$485$$ −14.0000 −0.635707
$$486$$ −1.00000 −0.0453609
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ −13.0000 −0.587880
$$490$$ −3.00000 −0.135526
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 18.0000 0.810679
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 5.00000 0.224507
$$497$$ −24.0000 −1.07655
$$498$$ 6.00000 0.268866
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −9.00000 −0.402090
$$502$$ −15.0000 −0.669483
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ −6.00000 −0.266996
$$506$$ 9.00000 0.400099
$$507$$ 0 0
$$508$$ 14.0000 0.621150
$$509$$ 3.00000 0.132973 0.0664863 0.997787i $$-0.478821\pi$$
0.0664863 + 0.997787i $$0.478821\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 28.0000 1.23865
$$512$$ −1.00000 −0.0441942
$$513$$ 2.00000 0.0883022
$$514$$ 21.0000 0.926270
$$515$$ −14.0000 −0.616914
$$516$$ −1.00000 −0.0440225
$$517$$ 9.00000 0.395820
$$518$$ 14.0000 0.615125
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ −3.00000 −0.131306
$$523$$ 11.0000 0.480996 0.240498 0.970650i $$-0.422689\pi$$
0.240498 + 0.970650i $$0.422689\pi$$
$$524$$ 9.00000 0.393167
$$525$$ 2.00000 0.0872872
$$526$$ 15.0000 0.654031
$$527$$ 30.0000 1.30682
$$528$$ −3.00000 −0.130558
$$529$$ −14.0000 −0.608696
$$530$$ −6.00000 −0.260623
$$531$$ −9.00000 −0.390567
$$532$$ 4.00000 0.173422
$$533$$ 0 0
$$534$$ 18.0000 0.778936
$$535$$ 6.00000 0.259403
$$536$$ −8.00000 −0.345547
$$537$$ −3.00000 −0.129460
$$538$$ −18.0000 −0.776035
$$539$$ 9.00000 0.387657
$$540$$ −1.00000 −0.0430331
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ −16.0000 −0.686626
$$544$$ −6.00000 −0.257248
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 9.00000 0.384461
$$549$$ 2.00000 0.0853579
$$550$$ 3.00000 0.127920
$$551$$ 6.00000 0.255609
$$552$$ −3.00000 −0.127688
$$553$$ 10.0000 0.425243
$$554$$ 1.00000 0.0424859
$$555$$ 7.00000 0.297133
$$556$$ 14.0000 0.593732
$$557$$ −6.00000 −0.254228 −0.127114 0.991888i $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ −5.00000 −0.211667
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ −18.0000 −0.759961
$$562$$ −18.0000 −0.759284
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ −15.0000 −0.631055
$$566$$ 31.0000 1.30303
$$567$$ 2.00000 0.0839921
$$568$$ 12.0000 0.503509
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 2.00000 0.0837708
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ −12.0000 −0.500870
$$575$$ 3.00000 0.125109
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −4.00000 −0.166234
$$580$$ −3.00000 −0.124568
$$581$$ −12.0000 −0.497844
$$582$$ −14.0000 −0.580319
$$583$$ 18.0000 0.745484
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 10.0000 0.412043
$$590$$ −9.00000 −0.370524
$$591$$ 24.0000 0.987228
$$592$$ −7.00000 −0.287698
$$593$$ 27.0000 1.10876 0.554379 0.832265i $$-0.312956\pi$$
0.554379 + 0.832265i $$0.312956\pi$$
$$594$$ 3.00000 0.123091
$$595$$ −12.0000 −0.491952
$$596$$ −9.00000 −0.368654
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 2.00000 0.0815139
$$603$$ 8.00000 0.325785
$$604$$ 8.00000 0.325515
$$605$$ 2.00000 0.0813116
$$606$$ −6.00000 −0.243733
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 6.00000 0.243132
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ −31.0000 −1.25208 −0.626039 0.779792i $$-0.715325\pi$$
−0.626039 + 0.779792i $$0.715325\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ −6.00000 −0.241943
$$616$$ 6.00000 0.241747
$$617$$ 21.0000 0.845428 0.422714 0.906263i $$-0.361077\pi$$
0.422714 + 0.906263i $$0.361077\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ −46.0000 −1.84890 −0.924448 0.381308i $$-0.875474\pi$$
−0.924448 + 0.381308i $$0.875474\pi$$
$$620$$ −5.00000 −0.200805
$$621$$ 3.00000 0.120386
$$622$$ 12.0000 0.481156
$$623$$ −36.0000 −1.44231
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ −6.00000 −0.239617
$$628$$ −13.0000 −0.518756
$$629$$ −42.0000 −1.67465
$$630$$ 2.00000 0.0796819
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −5.00000 −0.198889
$$633$$ 20.0000 0.794929
$$634$$ −12.0000 −0.476581
$$635$$ −14.0000 −0.555573
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 9.00000 0.356313
$$639$$ −12.0000 −0.474713
$$640$$ 1.00000 0.0395285
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 6.00000 0.236801
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 1.00000 0.0393750
$$646$$ −12.0000 −0.472134
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 27.0000 1.05984
$$650$$ 0 0
$$651$$ 10.0000 0.391931
$$652$$ −13.0000 −0.509119
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ −9.00000 −0.351659
$$656$$ 6.00000 0.234261
$$657$$ 14.0000 0.546192
$$658$$ 6.00000 0.233904
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 3.00000 0.116775
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ −32.0000 −1.24372
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ −4.00000 −0.155113
$$666$$ 7.00000 0.271244
$$667$$ 9.00000 0.348481
$$668$$ −9.00000 −0.348220
$$669$$ −10.0000 −0.386622
$$670$$ 8.00000 0.309067
$$671$$ −6.00000 −0.231627
$$672$$ −2.00000 −0.0771517
$$673$$ −4.00000 −0.154189 −0.0770943 0.997024i $$-0.524564\pi$$
−0.0770943 + 0.997024i $$0.524564\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ −15.0000 −0.576072
$$679$$ 28.0000 1.07454
$$680$$ 6.00000 0.230089
$$681$$ 0 0
$$682$$ 15.0000 0.574380
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ −9.00000 −0.343872
$$686$$ 20.0000 0.763604
$$687$$ 14.0000 0.534133
$$688$$ −1.00000 −0.0381246
$$689$$ 0 0
$$690$$ 3.00000 0.114208
$$691$$ −46.0000 −1.74992 −0.874961 0.484193i $$-0.839113\pi$$
−0.874961 + 0.484193i $$0.839113\pi$$
$$692$$ 12.0000 0.456172
$$693$$ −6.00000 −0.227921
$$694$$ 30.0000 1.13878
$$695$$ −14.0000 −0.531050
$$696$$ −3.00000 −0.113715
$$697$$ 36.0000 1.36360
$$698$$ −8.00000 −0.302804
$$699$$ 21.0000 0.794293
$$700$$ 2.00000 0.0755929
$$701$$ −21.0000 −0.793159 −0.396580 0.918000i $$-0.629803\pi$$
−0.396580 + 0.918000i $$0.629803\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ −3.00000 −0.113067
$$705$$ 3.00000 0.112987
$$706$$ −30.0000 −1.12906
$$707$$ 12.0000 0.451306
$$708$$ −9.00000 −0.338241
$$709$$ 32.0000 1.20179 0.600893 0.799330i $$-0.294812\pi$$
0.600893 + 0.799330i $$0.294812\pi$$
$$710$$ −12.0000 −0.450352
$$711$$ 5.00000 0.187515
$$712$$ 18.0000 0.674579
$$713$$ 15.0000 0.561754
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ −3.00000 −0.112115
$$717$$ 24.0000 0.896296
$$718$$ −24.0000 −0.895672
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 28.0000 1.04277
$$722$$ 15.0000 0.558242
$$723$$ 17.0000 0.632237
$$724$$ −16.0000 −0.594635
$$725$$ 3.00000 0.111417
$$726$$ 2.00000 0.0742270
$$727$$ −4.00000 −0.148352 −0.0741759 0.997245i $$-0.523633\pi$$
−0.0741759 + 0.997245i $$0.523633\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 14.0000 0.518163
$$731$$ −6.00000 −0.221918
$$732$$ 2.00000 0.0739221
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −20.0000 −0.738213
$$735$$ 3.00000 0.110657
$$736$$ −3.00000 −0.110581
$$737$$ −24.0000 −0.884051
$$738$$ −6.00000 −0.220863
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 7.00000 0.257325
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ −5.00000 −0.183309
$$745$$ 9.00000 0.329734
$$746$$ 25.0000 0.915315
$$747$$ −6.00000 −0.219529
$$748$$ −18.0000 −0.658145
$$749$$ −12.0000 −0.438470
$$750$$ 1.00000 0.0365148
$$751$$ 41.0000 1.49611 0.748056 0.663636i $$-0.230988\pi$$
0.748056 + 0.663636i $$0.230988\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 15.0000 0.546630
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 2.00000 0.0727393
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −38.0000 −1.38022
$$759$$ −9.00000 −0.326679
$$760$$ 2.00000 0.0725476
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ −14.0000 −0.507166
$$763$$ 28.0000 1.01367
$$764$$ −12.0000 −0.434145
$$765$$ −6.00000 −0.216930
$$766$$ 21.0000 0.758761
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −13.0000 −0.468792 −0.234396 0.972141i $$-0.575311\pi$$
−0.234396 + 0.972141i $$0.575311\pi$$
$$770$$ −6.00000 −0.216225
$$771$$ −21.0000 −0.756297
$$772$$ −4.00000 −0.143963
$$773$$ −48.0000 −1.72644 −0.863220 0.504828i $$-0.831556\pi$$
−0.863220 + 0.504828i $$0.831556\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 5.00000 0.179605
$$776$$ −14.0000 −0.502571
$$777$$ −14.0000 −0.502247
$$778$$ −3.00000 −0.107555
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −18.0000 −0.643679
$$783$$ 3.00000 0.107211
$$784$$ −3.00000 −0.107143
$$785$$ 13.0000 0.463990
$$786$$ −9.00000 −0.321019
$$787$$ 35.0000 1.24762 0.623808 0.781578i $$-0.285585\pi$$
0.623808 + 0.781578i $$0.285585\pi$$
$$788$$ 24.0000 0.854965
$$789$$ −15.0000 −0.534014
$$790$$ 5.00000 0.177892
$$791$$ 30.0000 1.06668
$$792$$ 3.00000 0.106600
$$793$$ 0 0
$$794$$ 31.0000 1.10015
$$795$$ 6.00000 0.212798
$$796$$ 8.00000 0.283552
$$797$$ −24.0000 −0.850124 −0.425062 0.905164i $$-0.639748\pi$$
−0.425062 + 0.905164i $$0.639748\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ −18.0000 −0.636794
$$800$$ −1.00000 −0.0353553
$$801$$ −18.0000 −0.635999
$$802$$ 12.0000 0.423735
$$803$$ −42.0000 −1.48215
$$804$$ 8.00000 0.282138
$$805$$ −6.00000 −0.211472
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ −6.00000 −0.211079
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 6.00000 0.210559
$$813$$ 11.0000 0.385787
$$814$$ −21.0000 −0.736050
$$815$$ 13.0000 0.455370
$$816$$ 6.00000 0.210042
$$817$$ −2.00000 −0.0699711
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ 27.0000 0.942306 0.471153 0.882051i $$-0.343838\pi$$
0.471153 + 0.882051i $$0.343838\pi$$
$$822$$ −9.00000 −0.313911
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ −3.00000 −0.104447
$$826$$ 18.0000 0.626300
$$827$$ 6.00000 0.208640 0.104320 0.994544i $$-0.466733\pi$$
0.104320 + 0.994544i $$0.466733\pi$$
$$828$$ 3.00000 0.104257
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ −6.00000 −0.208263
$$831$$ −1.00000 −0.0346896
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ −14.0000 −0.484780
$$835$$ 9.00000 0.311458
$$836$$ −6.00000 −0.207514
$$837$$ 5.00000 0.172825
$$838$$ 12.0000 0.414533
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ −20.0000 −0.689655
$$842$$ 16.0000 0.551396
$$843$$ 18.0000 0.619953
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ −4.00000 −0.137442
$$848$$ −6.00000 −0.206041
$$849$$ −31.0000 −1.06392
$$850$$ −6.00000 −0.205798
$$851$$ −21.0000 −0.719871
$$852$$ −12.0000 −0.411113
$$853$$ −19.0000 −0.650548 −0.325274 0.945620i $$-0.605456\pi$$
−0.325274 + 0.945620i $$0.605456\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ −2.00000 −0.0683986
$$856$$ 6.00000 0.205076
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −46.0000 −1.56950 −0.784750 0.619813i $$-0.787209\pi$$
−0.784750 + 0.619813i $$0.787209\pi$$
$$860$$ 1.00000 0.0340997
$$861$$ 12.0000 0.408959
$$862$$ −12.0000 −0.408722
$$863$$ −45.0000 −1.53182 −0.765909 0.642949i $$-0.777711\pi$$
−0.765909 + 0.642949i $$0.777711\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −12.0000 −0.408012
$$866$$ 40.0000 1.35926
$$867$$ 19.0000 0.645274
$$868$$ 10.0000 0.339422
$$869$$ −15.0000 −0.508840
$$870$$ 3.00000 0.101710
$$871$$ 0 0
$$872$$ −14.0000 −0.474100
$$873$$ 14.0000 0.473828
$$874$$ −6.00000 −0.202953
$$875$$ −2.00000 −0.0676123
$$876$$ 14.0000 0.473016
$$877$$ 23.0000 0.776655 0.388327 0.921521i $$-0.373053\pi$$
0.388327 + 0.921521i $$0.373053\pi$$
$$878$$ 4.00000 0.134993
$$879$$ −30.0000 −1.01187
$$880$$ 3.00000 0.101130
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −19.0000 −0.639401 −0.319700 0.947519i $$-0.603582\pi$$
−0.319700 + 0.947519i $$0.603582\pi$$
$$884$$ 0 0
$$885$$ 9.00000 0.302532
$$886$$ 36.0000 1.20944
$$887$$ −57.0000 −1.91387 −0.956936 0.290298i $$-0.906246\pi$$
−0.956936 + 0.290298i $$0.906246\pi$$
$$888$$ 7.00000 0.234905
$$889$$ 28.0000 0.939090
$$890$$ −18.0000 −0.603361
$$891$$ −3.00000 −0.100504
$$892$$ −10.0000 −0.334825
$$893$$ −6.00000 −0.200782
$$894$$ 9.00000 0.301005
$$895$$ 3.00000 0.100279
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ 15.0000 0.500278
$$900$$ 1.00000 0.0333333
$$901$$ −36.0000 −1.19933
$$902$$ 18.0000 0.599334
$$903$$ −2.00000 −0.0665558
$$904$$ −15.0000 −0.498893
$$905$$ 16.0000 0.531858
$$906$$ −8.00000 −0.265782
$$907$$ 17.0000 0.564476 0.282238 0.959344i $$-0.408923\pi$$
0.282238 + 0.959344i $$0.408923\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 18.0000 0.595713
$$914$$ −2.00000 −0.0661541
$$915$$ −2.00000 −0.0661180
$$916$$ 14.0000 0.462573
$$917$$ 18.0000 0.594412
$$918$$ −6.00000 −0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 3.00000 0.0989071
$$921$$ 8.00000 0.263609
$$922$$ −15.0000 −0.493999
$$923$$ 0 0
$$924$$ −6.00000 −0.197386
$$925$$ −7.00000 −0.230159
$$926$$ 34.0000 1.11731
$$927$$ 14.0000 0.459820
$$928$$ −3.00000 −0.0984798
$$929$$ −12.0000 −0.393707 −0.196854 0.980433i $$-0.563072\pi$$
−0.196854 + 0.980433i $$0.563072\pi$$
$$930$$ 5.00000 0.163956
$$931$$ −6.00000 −0.196642
$$932$$ 21.0000 0.687878
$$933$$ −12.0000 −0.392862
$$934$$ −18.0000 −0.588978
$$935$$ 18.0000 0.588663
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 8.00000 0.261070
$$940$$ 3.00000 0.0978492
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ 13.0000 0.423563
$$943$$ 18.0000 0.586161
$$944$$ −9.00000 −0.292925
$$945$$ −2.00000 −0.0650600
$$946$$ −3.00000 −0.0975384
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 5.00000 0.162392
$$949$$ 0 0
$$950$$ −2.00000 −0.0648886
$$951$$ 12.0000 0.389127
$$952$$ −12.0000 −0.388922
$$953$$ 9.00000 0.291539 0.145769 0.989319i $$-0.453434\pi$$
0.145769 + 0.989319i $$0.453434\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 12.0000 0.388311
$$956$$ 24.0000 0.776215
$$957$$ −9.00000 −0.290929
$$958$$ 0 0
$$959$$ 18.0000 0.581250
$$960$$ −1.00000 −0.0322749
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 17.0000 0.547533
$$965$$ 4.00000 0.128765
$$966$$ −6.00000 −0.193047
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 12.0000 0.385496
$$970$$ 14.0000 0.449513
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 28.0000 0.897639
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −57.0000 −1.82359 −0.911796 0.410644i $$-0.865304\pi$$
−0.911796 + 0.410644i $$0.865304\pi$$
$$978$$ 13.0000 0.415694
$$979$$ 54.0000 1.72585
$$980$$ 3.00000 0.0958315
$$981$$ 14.0000 0.446986
$$982$$ 0 0
$$983$$ 9.00000 0.287055 0.143528 0.989646i $$-0.454155\pi$$
0.143528 + 0.989646i $$0.454155\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −24.0000 −0.764704
$$986$$ −18.0000 −0.573237
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ −3.00000 −0.0953945
$$990$$ −3.00000 −0.0953463
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ 32.0000 1.01549
$$994$$ 24.0000 0.761234
$$995$$ −8.00000 −0.253617
$$996$$ −6.00000 −0.190117
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ −7.00000 −0.221470
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.a.j.1.1 1
13.3 even 3 390.2.i.c.61.1 2
13.5 odd 4 5070.2.b.j.1351.2 2
13.8 odd 4 5070.2.b.j.1351.1 2
13.9 even 3 390.2.i.c.211.1 yes 2
13.12 even 2 5070.2.a.v.1.1 1
39.29 odd 6 1170.2.i.d.451.1 2
39.35 odd 6 1170.2.i.d.991.1 2
65.3 odd 12 1950.2.z.k.1699.1 4
65.9 even 6 1950.2.i.n.601.1 2
65.22 odd 12 1950.2.z.k.1849.1 4
65.29 even 6 1950.2.i.n.451.1 2
65.42 odd 12 1950.2.z.k.1699.2 4
65.48 odd 12 1950.2.z.k.1849.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.c.61.1 2 13.3 even 3
390.2.i.c.211.1 yes 2 13.9 even 3
1170.2.i.d.451.1 2 39.29 odd 6
1170.2.i.d.991.1 2 39.35 odd 6
1950.2.i.n.451.1 2 65.29 even 6
1950.2.i.n.601.1 2 65.9 even 6
1950.2.z.k.1699.1 4 65.3 odd 12
1950.2.z.k.1699.2 4 65.42 odd 12
1950.2.z.k.1849.1 4 65.22 odd 12
1950.2.z.k.1849.2 4 65.48 odd 12
5070.2.a.j.1.1 1 1.1 even 1 trivial
5070.2.a.v.1.1 1 13.12 even 2
5070.2.b.j.1351.1 2 13.8 odd 4
5070.2.b.j.1351.2 2 13.5 odd 4