# Properties

 Label 3696.2.a.bo.1.3 Level $3696$ Weight $2$ Character 3696.1 Self dual yes Analytic conductor $29.513$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3696.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.5127085871$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.254102$$ of defining polynomial Character $$\chi$$ $$=$$ 3696.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +4.18953 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +4.18953 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.17313 q^{13} -4.18953 q^{15} +6.85446 q^{17} +0.318669 q^{19} -1.00000 q^{21} +1.87086 q^{23} +12.5522 q^{25} -1.00000 q^{27} -3.17313 q^{29} -9.23353 q^{31} -1.00000 q^{33} +4.18953 q^{35} -7.55220 q^{37} +3.17313 q^{39} +9.36266 q^{41} +10.8873 q^{43} +4.18953 q^{45} +8.06040 q^{47} +1.00000 q^{49} -6.85446 q^{51} +0.508203 q^{53} +4.18953 q^{55} -0.318669 q^{57} +7.04399 q^{59} -2.00000 q^{61} +1.00000 q^{63} -13.2939 q^{65} +2.66492 q^{67} -1.87086 q^{69} +5.01641 q^{71} -4.82687 q^{73} -12.5522 q^{75} +1.00000 q^{77} -5.01641 q^{79} +1.00000 q^{81} -3.52461 q^{83} +28.7170 q^{85} +3.17313 q^{87} -1.74173 q^{89} -3.17313 q^{91} +9.23353 q^{93} +1.33508 q^{95} -12.2499 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 4.18953 1.87362 0.936808 0.349843i $$-0.113765\pi$$
0.936808 + 0.349843i $$0.113765\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −3.17313 −0.880067 −0.440034 0.897981i $$-0.645033\pi$$
−0.440034 + 0.897981i $$0.645033\pi$$
$$14$$ 0 0
$$15$$ −4.18953 −1.08173
$$16$$ 0 0
$$17$$ 6.85446 1.66245 0.831225 0.555936i $$-0.187640\pi$$
0.831225 + 0.555936i $$0.187640\pi$$
$$18$$ 0 0
$$19$$ 0.318669 0.0731078 0.0365539 0.999332i $$-0.488362\pi$$
0.0365539 + 0.999332i $$0.488362\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 1.87086 0.390102 0.195051 0.980793i $$-0.437513\pi$$
0.195051 + 0.980793i $$0.437513\pi$$
$$24$$ 0 0
$$25$$ 12.5522 2.51044
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −3.17313 −0.589235 −0.294617 0.955615i $$-0.595192\pi$$
−0.294617 + 0.955615i $$0.595192\pi$$
$$30$$ 0 0
$$31$$ −9.23353 −1.65839 −0.829195 0.558959i $$-0.811201\pi$$
−0.829195 + 0.558959i $$0.811201\pi$$
$$32$$ 0 0
$$33$$ −1.00000 −0.174078
$$34$$ 0 0
$$35$$ 4.18953 0.708161
$$36$$ 0 0
$$37$$ −7.55220 −1.24157 −0.620787 0.783980i $$-0.713187\pi$$
−0.620787 + 0.783980i $$0.713187\pi$$
$$38$$ 0 0
$$39$$ 3.17313 0.508107
$$40$$ 0 0
$$41$$ 9.36266 1.46220 0.731101 0.682269i $$-0.239007\pi$$
0.731101 + 0.682269i $$0.239007\pi$$
$$42$$ 0 0
$$43$$ 10.8873 1.66029 0.830147 0.557545i $$-0.188257\pi$$
0.830147 + 0.557545i $$0.188257\pi$$
$$44$$ 0 0
$$45$$ 4.18953 0.624539
$$46$$ 0 0
$$47$$ 8.06040 1.17573 0.587865 0.808959i $$-0.299969\pi$$
0.587865 + 0.808959i $$0.299969\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.85446 −0.959816
$$52$$ 0 0
$$53$$ 0.508203 0.0698071 0.0349036 0.999391i $$-0.488888\pi$$
0.0349036 + 0.999391i $$0.488888\pi$$
$$54$$ 0 0
$$55$$ 4.18953 0.564917
$$56$$ 0 0
$$57$$ −0.318669 −0.0422088
$$58$$ 0 0
$$59$$ 7.04399 0.917050 0.458525 0.888682i $$-0.348378\pi$$
0.458525 + 0.888682i $$0.348378\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −13.2939 −1.64891
$$66$$ 0 0
$$67$$ 2.66492 0.325572 0.162786 0.986661i $$-0.447952\pi$$
0.162786 + 0.986661i $$0.447952\pi$$
$$68$$ 0 0
$$69$$ −1.87086 −0.225226
$$70$$ 0 0
$$71$$ 5.01641 0.595338 0.297669 0.954669i $$-0.403791\pi$$
0.297669 + 0.954669i $$0.403791\pi$$
$$72$$ 0 0
$$73$$ −4.82687 −0.564943 −0.282471 0.959276i $$-0.591154\pi$$
−0.282471 + 0.959276i $$0.591154\pi$$
$$74$$ 0 0
$$75$$ −12.5522 −1.44940
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ −5.01641 −0.564390 −0.282195 0.959357i $$-0.591062\pi$$
−0.282195 + 0.959357i $$0.591062\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −3.52461 −0.386876 −0.193438 0.981112i $$-0.561964\pi$$
−0.193438 + 0.981112i $$0.561964\pi$$
$$84$$ 0 0
$$85$$ 28.7170 3.11479
$$86$$ 0 0
$$87$$ 3.17313 0.340195
$$88$$ 0 0
$$89$$ −1.74173 −0.184623 −0.0923115 0.995730i $$-0.529426\pi$$
−0.0923115 + 0.995730i $$0.529426\pi$$
$$90$$ 0 0
$$91$$ −3.17313 −0.332634
$$92$$ 0 0
$$93$$ 9.23353 0.957472
$$94$$ 0 0
$$95$$ 1.33508 0.136976
$$96$$ 0 0
$$97$$ −12.2499 −1.24379 −0.621896 0.783100i $$-0.713637\pi$$
−0.621896 + 0.783100i $$0.713637\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 4.88727 0.486302 0.243151 0.969988i $$-0.421819\pi$$
0.243151 + 0.969988i $$0.421819\pi$$
$$102$$ 0 0
$$103$$ 0.637339 0.0627988 0.0313994 0.999507i $$-0.490004\pi$$
0.0313994 + 0.999507i $$0.490004\pi$$
$$104$$ 0 0
$$105$$ −4.18953 −0.408857
$$106$$ 0 0
$$107$$ −0.956008 −0.0924208 −0.0462104 0.998932i $$-0.514714\pi$$
−0.0462104 + 0.998932i $$0.514714\pi$$
$$108$$ 0 0
$$109$$ −7.61259 −0.729154 −0.364577 0.931173i $$-0.618786\pi$$
−0.364577 + 0.931173i $$0.618786\pi$$
$$110$$ 0 0
$$111$$ 7.55220 0.716823
$$112$$ 0 0
$$113$$ −7.70892 −0.725194 −0.362597 0.931946i $$-0.618110\pi$$
−0.362597 + 0.931946i $$0.618110\pi$$
$$114$$ 0 0
$$115$$ 7.83805 0.730902
$$116$$ 0 0
$$117$$ −3.17313 −0.293356
$$118$$ 0 0
$$119$$ 6.85446 0.628347
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −9.36266 −0.844203
$$124$$ 0 0
$$125$$ 31.6402 2.82998
$$126$$ 0 0
$$127$$ 5.49180 0.487318 0.243659 0.969861i $$-0.421652\pi$$
0.243659 + 0.969861i $$0.421652\pi$$
$$128$$ 0 0
$$129$$ −10.8873 −0.958571
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0.318669 0.0276321
$$134$$ 0 0
$$135$$ −4.18953 −0.360578
$$136$$ 0 0
$$137$$ 15.6126 1.33387 0.666937 0.745114i $$-0.267605\pi$$
0.666937 + 0.745114i $$0.267605\pi$$
$$138$$ 0 0
$$139$$ 9.01641 0.764762 0.382381 0.924005i $$-0.375104\pi$$
0.382381 + 0.924005i $$0.375104\pi$$
$$140$$ 0 0
$$141$$ −8.06040 −0.678808
$$142$$ 0 0
$$143$$ −3.17313 −0.265350
$$144$$ 0 0
$$145$$ −13.2939 −1.10400
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −5.20594 −0.426487 −0.213244 0.976999i $$-0.568403\pi$$
−0.213244 + 0.976999i $$0.568403\pi$$
$$150$$ 0 0
$$151$$ −6.24993 −0.508612 −0.254306 0.967124i $$-0.581847\pi$$
−0.254306 + 0.967124i $$0.581847\pi$$
$$152$$ 0 0
$$153$$ 6.85446 0.554150
$$154$$ 0 0
$$155$$ −38.6842 −3.10719
$$156$$ 0 0
$$157$$ −18.1208 −1.44620 −0.723099 0.690745i $$-0.757283\pi$$
−0.723099 + 0.690745i $$0.757283\pi$$
$$158$$ 0 0
$$159$$ −0.508203 −0.0403031
$$160$$ 0 0
$$161$$ 1.87086 0.147445
$$162$$ 0 0
$$163$$ 2.66492 0.208733 0.104366 0.994539i $$-0.466719\pi$$
0.104366 + 0.994539i $$0.466719\pi$$
$$164$$ 0 0
$$165$$ −4.18953 −0.326155
$$166$$ 0 0
$$167$$ 11.1455 0.862468 0.431234 0.902240i $$-0.358078\pi$$
0.431234 + 0.902240i $$0.358078\pi$$
$$168$$ 0 0
$$169$$ −2.93126 −0.225482
$$170$$ 0 0
$$171$$ 0.318669 0.0243693
$$172$$ 0 0
$$173$$ −24.8461 −1.88902 −0.944508 0.328489i $$-0.893461\pi$$
−0.944508 + 0.328489i $$0.893461\pi$$
$$174$$ 0 0
$$175$$ 12.5522 0.948857
$$176$$ 0 0
$$177$$ −7.04399 −0.529459
$$178$$ 0 0
$$179$$ 12.7581 0.953588 0.476794 0.879015i $$-0.341799\pi$$
0.476794 + 0.879015i $$0.341799\pi$$
$$180$$ 0 0
$$181$$ −3.23353 −0.240346 −0.120173 0.992753i $$-0.538345\pi$$
−0.120173 + 0.992753i $$0.538345\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ −31.6402 −2.32623
$$186$$ 0 0
$$187$$ 6.85446 0.501248
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −20.9753 −1.51772 −0.758858 0.651256i $$-0.774242\pi$$
−0.758858 + 0.651256i $$0.774242\pi$$
$$192$$ 0 0
$$193$$ 0.249933 0.0179905 0.00899527 0.999960i $$-0.497137\pi$$
0.00899527 + 0.999960i $$0.497137\pi$$
$$194$$ 0 0
$$195$$ 13.2939 0.951998
$$196$$ 0 0
$$197$$ 18.4999 1.31806 0.659030 0.752116i $$-0.270967\pi$$
0.659030 + 0.752116i $$0.270967\pi$$
$$198$$ 0 0
$$199$$ −9.87086 −0.699727 −0.349864 0.936801i $$-0.613772\pi$$
−0.349864 + 0.936801i $$0.613772\pi$$
$$200$$ 0 0
$$201$$ −2.66492 −0.187969
$$202$$ 0 0
$$203$$ −3.17313 −0.222710
$$204$$ 0 0
$$205$$ 39.2252 2.73961
$$206$$ 0 0
$$207$$ 1.87086 0.130034
$$208$$ 0 0
$$209$$ 0.318669 0.0220428
$$210$$ 0 0
$$211$$ 4.63734 0.319248 0.159624 0.987178i $$-0.448972\pi$$
0.159624 + 0.987178i $$0.448972\pi$$
$$212$$ 0 0
$$213$$ −5.01641 −0.343719
$$214$$ 0 0
$$215$$ 45.6126 3.11075
$$216$$ 0 0
$$217$$ −9.23353 −0.626813
$$218$$ 0 0
$$219$$ 4.82687 0.326170
$$220$$ 0 0
$$221$$ −21.7501 −1.46307
$$222$$ 0 0
$$223$$ 18.3463 1.22856 0.614278 0.789090i $$-0.289447\pi$$
0.614278 + 0.789090i $$0.289447\pi$$
$$224$$ 0 0
$$225$$ 12.5522 0.836813
$$226$$ 0 0
$$227$$ −0.379068 −0.0251596 −0.0125798 0.999921i $$-0.504004\pi$$
−0.0125798 + 0.999921i $$0.504004\pi$$
$$228$$ 0 0
$$229$$ 24.9424 1.64824 0.824121 0.566413i $$-0.191669\pi$$
0.824121 + 0.566413i $$0.191669\pi$$
$$230$$ 0 0
$$231$$ −1.00000 −0.0657952
$$232$$ 0 0
$$233$$ 23.4506 1.53630 0.768151 0.640268i $$-0.221177\pi$$
0.768151 + 0.640268i $$0.221177\pi$$
$$234$$ 0 0
$$235$$ 33.7693 2.20287
$$236$$ 0 0
$$237$$ 5.01641 0.325851
$$238$$ 0 0
$$239$$ −5.07681 −0.328391 −0.164196 0.986428i $$-0.552503\pi$$
−0.164196 + 0.986428i $$0.552503\pi$$
$$240$$ 0 0
$$241$$ 19.2939 1.24283 0.621415 0.783481i $$-0.286558\pi$$
0.621415 + 0.783481i $$0.286558\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 4.18953 0.267660
$$246$$ 0 0
$$247$$ −1.01118 −0.0643397
$$248$$ 0 0
$$249$$ 3.52461 0.223363
$$250$$ 0 0
$$251$$ −23.8021 −1.50238 −0.751188 0.660088i $$-0.770519\pi$$
−0.751188 + 0.660088i $$0.770519\pi$$
$$252$$ 0 0
$$253$$ 1.87086 0.117620
$$254$$ 0 0
$$255$$ −28.7170 −1.79833
$$256$$ 0 0
$$257$$ 14.9149 0.930363 0.465182 0.885215i $$-0.345989\pi$$
0.465182 + 0.885215i $$0.345989\pi$$
$$258$$ 0 0
$$259$$ −7.55220 −0.469271
$$260$$ 0 0
$$261$$ −3.17313 −0.196412
$$262$$ 0 0
$$263$$ −2.92319 −0.180252 −0.0901259 0.995930i $$-0.528727\pi$$
−0.0901259 + 0.995930i $$0.528727\pi$$
$$264$$ 0 0
$$265$$ 2.12914 0.130792
$$266$$ 0 0
$$267$$ 1.74173 0.106592
$$268$$ 0 0
$$269$$ 11.9672 0.729652 0.364826 0.931076i $$-0.381128\pi$$
0.364826 + 0.931076i $$0.381128\pi$$
$$270$$ 0 0
$$271$$ 20.3187 1.23427 0.617136 0.786857i $$-0.288293\pi$$
0.617136 + 0.786857i $$0.288293\pi$$
$$272$$ 0 0
$$273$$ 3.17313 0.192046
$$274$$ 0 0
$$275$$ 12.5522 0.756926
$$276$$ 0 0
$$277$$ 18.0552 1.08483 0.542415 0.840111i $$-0.317510\pi$$
0.542415 + 0.840111i $$0.317510\pi$$
$$278$$ 0 0
$$279$$ −9.23353 −0.552797
$$280$$ 0 0
$$281$$ 27.2939 1.62822 0.814110 0.580711i $$-0.197226\pi$$
0.814110 + 0.580711i $$0.197226\pi$$
$$282$$ 0 0
$$283$$ −29.8901 −1.77678 −0.888391 0.459087i $$-0.848177\pi$$
−0.888391 + 0.459087i $$0.848177\pi$$
$$284$$ 0 0
$$285$$ −1.33508 −0.0790831
$$286$$ 0 0
$$287$$ 9.36266 0.552660
$$288$$ 0 0
$$289$$ 29.9836 1.76374
$$290$$ 0 0
$$291$$ 12.2499 0.718104
$$292$$ 0 0
$$293$$ −4.34625 −0.253911 −0.126955 0.991908i $$-0.540521\pi$$
−0.126955 + 0.991908i $$0.540521\pi$$
$$294$$ 0 0
$$295$$ 29.5110 1.71820
$$296$$ 0 0
$$297$$ −1.00000 −0.0580259
$$298$$ 0 0
$$299$$ −5.93649 −0.343316
$$300$$ 0 0
$$301$$ 10.8873 0.627532
$$302$$ 0 0
$$303$$ −4.88727 −0.280766
$$304$$ 0 0
$$305$$ −8.37907 −0.479784
$$306$$ 0 0
$$307$$ 20.7581 1.18473 0.592365 0.805670i $$-0.298194\pi$$
0.592365 + 0.805670i $$0.298194\pi$$
$$308$$ 0 0
$$309$$ −0.637339 −0.0362569
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 8.12914 0.459486 0.229743 0.973251i $$-0.426211\pi$$
0.229743 + 0.973251i $$0.426211\pi$$
$$314$$ 0 0
$$315$$ 4.18953 0.236054
$$316$$ 0 0
$$317$$ −19.9917 −1.12284 −0.561422 0.827530i $$-0.689745\pi$$
−0.561422 + 0.827530i $$0.689745\pi$$
$$318$$ 0 0
$$319$$ −3.17313 −0.177661
$$320$$ 0 0
$$321$$ 0.956008 0.0533592
$$322$$ 0 0
$$323$$ 2.18431 0.121538
$$324$$ 0 0
$$325$$ −39.8297 −2.20935
$$326$$ 0 0
$$327$$ 7.61259 0.420977
$$328$$ 0 0
$$329$$ 8.06040 0.444384
$$330$$ 0 0
$$331$$ −17.0164 −0.935306 −0.467653 0.883912i $$-0.654900\pi$$
−0.467653 + 0.883912i $$0.654900\pi$$
$$332$$ 0 0
$$333$$ −7.55220 −0.413858
$$334$$ 0 0
$$335$$ 11.1648 0.609998
$$336$$ 0 0
$$337$$ 1.52461 0.0830508 0.0415254 0.999137i $$-0.486778\pi$$
0.0415254 + 0.999137i $$0.486778\pi$$
$$338$$ 0 0
$$339$$ 7.70892 0.418691
$$340$$ 0 0
$$341$$ −9.23353 −0.500023
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −7.83805 −0.421986
$$346$$ 0 0
$$347$$ −17.4506 −0.936800 −0.468400 0.883517i $$-0.655169\pi$$
−0.468400 + 0.883517i $$0.655169\pi$$
$$348$$ 0 0
$$349$$ 6.85969 0.367191 0.183595 0.983002i $$-0.441226\pi$$
0.183595 + 0.983002i $$0.441226\pi$$
$$350$$ 0 0
$$351$$ 3.17313 0.169369
$$352$$ 0 0
$$353$$ −31.9313 −1.69953 −0.849765 0.527162i $$-0.823256\pi$$
−0.849765 + 0.527162i $$0.823256\pi$$
$$354$$ 0 0
$$355$$ 21.0164 1.11544
$$356$$ 0 0
$$357$$ −6.85446 −0.362776
$$358$$ 0 0
$$359$$ 24.4342 1.28959 0.644795 0.764356i $$-0.276943\pi$$
0.644795 + 0.764356i $$0.276943\pi$$
$$360$$ 0 0
$$361$$ −18.8984 −0.994655
$$362$$ 0 0
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −20.2223 −1.05849
$$366$$ 0 0
$$367$$ −17.0716 −0.891129 −0.445565 0.895250i $$-0.646997\pi$$
−0.445565 + 0.895250i $$0.646997\pi$$
$$368$$ 0 0
$$369$$ 9.36266 0.487401
$$370$$ 0 0
$$371$$ 0.508203 0.0263846
$$372$$ 0 0
$$373$$ 3.17836 0.164569 0.0822845 0.996609i $$-0.473778\pi$$
0.0822845 + 0.996609i $$0.473778\pi$$
$$374$$ 0 0
$$375$$ −31.6402 −1.63389
$$376$$ 0 0
$$377$$ 10.0687 0.518566
$$378$$ 0 0
$$379$$ −3.93960 −0.202364 −0.101182 0.994868i $$-0.532262\pi$$
−0.101182 + 0.994868i $$0.532262\pi$$
$$380$$ 0 0
$$381$$ −5.49180 −0.281353
$$382$$ 0 0
$$383$$ 24.7581 1.26508 0.632541 0.774527i $$-0.282012\pi$$
0.632541 + 0.774527i $$0.282012\pi$$
$$384$$ 0 0
$$385$$ 4.18953 0.213518
$$386$$ 0 0
$$387$$ 10.8873 0.553431
$$388$$ 0 0
$$389$$ 3.65375 0.185252 0.0926261 0.995701i $$-0.470474\pi$$
0.0926261 + 0.995701i $$0.470474\pi$$
$$390$$ 0 0
$$391$$ 12.8238 0.648526
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ 0 0
$$395$$ −21.0164 −1.05745
$$396$$ 0 0
$$397$$ 4.56337 0.229029 0.114515 0.993422i $$-0.463469\pi$$
0.114515 + 0.993422i $$0.463469\pi$$
$$398$$ 0 0
$$399$$ −0.318669 −0.0159534
$$400$$ 0 0
$$401$$ −7.23353 −0.361225 −0.180613 0.983554i $$-0.557808\pi$$
−0.180613 + 0.983554i $$0.557808\pi$$
$$402$$ 0 0
$$403$$ 29.2992 1.45949
$$404$$ 0 0
$$405$$ 4.18953 0.208180
$$406$$ 0 0
$$407$$ −7.55220 −0.374348
$$408$$ 0 0
$$409$$ 5.30749 0.262439 0.131219 0.991353i $$-0.458111\pi$$
0.131219 + 0.991353i $$0.458111\pi$$
$$410$$ 0 0
$$411$$ −15.6126 −0.770112
$$412$$ 0 0
$$413$$ 7.04399 0.346612
$$414$$ 0 0
$$415$$ −14.7665 −0.724858
$$416$$ 0 0
$$417$$ −9.01641 −0.441535
$$418$$ 0 0
$$419$$ 11.4231 0.558053 0.279026 0.960283i $$-0.409988\pi$$
0.279026 + 0.960283i $$0.409988\pi$$
$$420$$ 0 0
$$421$$ −27.4147 −1.33611 −0.668056 0.744111i $$-0.732873\pi$$
−0.668056 + 0.744111i $$0.732873\pi$$
$$422$$ 0 0
$$423$$ 8.06040 0.391910
$$424$$ 0 0
$$425$$ 86.0385 4.17348
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ 0 0
$$429$$ 3.17313 0.153200
$$430$$ 0 0
$$431$$ 2.28586 0.110106 0.0550529 0.998483i $$-0.482467\pi$$
0.0550529 + 0.998483i $$0.482467\pi$$
$$432$$ 0 0
$$433$$ 31.5714 1.51723 0.758613 0.651541i $$-0.225877\pi$$
0.758613 + 0.651541i $$0.225877\pi$$
$$434$$ 0 0
$$435$$ 13.2939 0.637395
$$436$$ 0 0
$$437$$ 0.596187 0.0285195
$$438$$ 0 0
$$439$$ 18.5275 0.884267 0.442133 0.896949i $$-0.354222\pi$$
0.442133 + 0.896949i $$0.354222\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −28.4342 −1.35095 −0.675476 0.737382i $$-0.736062\pi$$
−0.675476 + 0.737382i $$0.736062\pi$$
$$444$$ 0 0
$$445$$ −7.29703 −0.345913
$$446$$ 0 0
$$447$$ 5.20594 0.246233
$$448$$ 0 0
$$449$$ −5.68656 −0.268365 −0.134183 0.990957i $$-0.542841\pi$$
−0.134183 + 0.990957i $$0.542841\pi$$
$$450$$ 0 0
$$451$$ 9.36266 0.440871
$$452$$ 0 0
$$453$$ 6.24993 0.293647
$$454$$ 0 0
$$455$$ −13.2939 −0.623229
$$456$$ 0 0
$$457$$ −13.0081 −0.608492 −0.304246 0.952594i $$-0.598404\pi$$
−0.304246 + 0.952594i $$0.598404\pi$$
$$458$$ 0 0
$$459$$ −6.85446 −0.319939
$$460$$ 0 0
$$461$$ −6.37907 −0.297103 −0.148551 0.988905i $$-0.547461\pi$$
−0.148551 + 0.988905i $$0.547461\pi$$
$$462$$ 0 0
$$463$$ −34.0932 −1.58445 −0.792223 0.610232i $$-0.791076\pi$$
−0.792223 + 0.610232i $$0.791076\pi$$
$$464$$ 0 0
$$465$$ 38.6842 1.79394
$$466$$ 0 0
$$467$$ −18.1484 −0.839807 −0.419903 0.907569i $$-0.637936\pi$$
−0.419903 + 0.907569i $$0.637936\pi$$
$$468$$ 0 0
$$469$$ 2.66492 0.123055
$$470$$ 0 0
$$471$$ 18.1208 0.834962
$$472$$ 0 0
$$473$$ 10.8873 0.500597
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 0.508203 0.0232690
$$478$$ 0 0
$$479$$ 42.7498 1.95329 0.976644 0.214864i $$-0.0689307\pi$$
0.976644 + 0.214864i $$0.0689307\pi$$
$$480$$ 0 0
$$481$$ 23.9641 1.09267
$$482$$ 0 0
$$483$$ −1.87086 −0.0851273
$$484$$ 0 0
$$485$$ −51.3215 −2.33039
$$486$$ 0 0
$$487$$ −26.7909 −1.21401 −0.607007 0.794697i $$-0.707630\pi$$
−0.607007 + 0.794697i $$0.707630\pi$$
$$488$$ 0 0
$$489$$ −2.66492 −0.120512
$$490$$ 0 0
$$491$$ −31.6813 −1.42976 −0.714879 0.699248i $$-0.753518\pi$$
−0.714879 + 0.699248i $$0.753518\pi$$
$$492$$ 0 0
$$493$$ −21.7501 −0.979574
$$494$$ 0 0
$$495$$ 4.18953 0.188306
$$496$$ 0 0
$$497$$ 5.01641 0.225017
$$498$$ 0 0
$$499$$ 24.1260 1.08003 0.540015 0.841656i $$-0.318419\pi$$
0.540015 + 0.841656i $$0.318419\pi$$
$$500$$ 0 0
$$501$$ −11.1455 −0.497946
$$502$$ 0 0
$$503$$ −30.8873 −1.37720 −0.688598 0.725144i $$-0.741773\pi$$
−0.688598 + 0.725144i $$0.741773\pi$$
$$504$$ 0 0
$$505$$ 20.4754 0.911143
$$506$$ 0 0
$$507$$ 2.93126 0.130182
$$508$$ 0 0
$$509$$ 28.2088 1.25033 0.625166 0.780492i $$-0.285031\pi$$
0.625166 + 0.780492i $$0.285031\pi$$
$$510$$ 0 0
$$511$$ −4.82687 −0.213528
$$512$$ 0 0
$$513$$ −0.318669 −0.0140696
$$514$$ 0 0
$$515$$ 2.67015 0.117661
$$516$$ 0 0
$$517$$ 8.06040 0.354496
$$518$$ 0 0
$$519$$ 24.8461 1.09062
$$520$$ 0 0
$$521$$ −33.7610 −1.47910 −0.739548 0.673104i $$-0.764961\pi$$
−0.739548 + 0.673104i $$0.764961\pi$$
$$522$$ 0 0
$$523$$ −12.8185 −0.560515 −0.280258 0.959925i $$-0.590420\pi$$
−0.280258 + 0.959925i $$0.590420\pi$$
$$524$$ 0 0
$$525$$ −12.5522 −0.547823
$$526$$ 0 0
$$527$$ −63.2908 −2.75699
$$528$$ 0 0
$$529$$ −19.4999 −0.847820
$$530$$ 0 0
$$531$$ 7.04399 0.305683
$$532$$ 0 0
$$533$$ −29.7089 −1.28684
$$534$$ 0 0
$$535$$ −4.00523 −0.173161
$$536$$ 0 0
$$537$$ −12.7581 −0.550554
$$538$$ 0 0
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ −21.8625 −0.939943 −0.469972 0.882681i $$-0.655736\pi$$
−0.469972 + 0.882681i $$0.655736\pi$$
$$542$$ 0 0
$$543$$ 3.23353 0.138764
$$544$$ 0 0
$$545$$ −31.8932 −1.36616
$$546$$ 0 0
$$547$$ −23.4178 −1.00127 −0.500637 0.865657i $$-0.666901\pi$$
−0.500637 + 0.865657i $$0.666901\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −1.01118 −0.0430776
$$552$$ 0 0
$$553$$ −5.01641 −0.213319
$$554$$ 0 0
$$555$$ 31.6402 1.34305
$$556$$ 0 0
$$557$$ −6.91486 −0.292992 −0.146496 0.989211i $$-0.546800\pi$$
−0.146496 + 0.989211i $$0.546800\pi$$
$$558$$ 0 0
$$559$$ −34.5467 −1.46117
$$560$$ 0 0
$$561$$ −6.85446 −0.289395
$$562$$ 0 0
$$563$$ −11.8381 −0.498914 −0.249457 0.968386i $$-0.580252\pi$$
−0.249457 + 0.968386i $$0.580252\pi$$
$$564$$ 0 0
$$565$$ −32.2968 −1.35874
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 15.2252 0.637154 0.318577 0.947897i $$-0.396795\pi$$
0.318577 + 0.947897i $$0.396795\pi$$
$$572$$ 0 0
$$573$$ 20.9753 0.876254
$$574$$ 0 0
$$575$$ 23.4835 0.979328
$$576$$ 0 0
$$577$$ 17.3215 0.721104 0.360552 0.932739i $$-0.382588\pi$$
0.360552 + 0.932739i $$0.382588\pi$$
$$578$$ 0 0
$$579$$ −0.249933 −0.0103868
$$580$$ 0 0
$$581$$ −3.52461 −0.146225
$$582$$ 0 0
$$583$$ 0.508203 0.0210476
$$584$$ 0 0
$$585$$ −13.2939 −0.549636
$$586$$ 0 0
$$587$$ −27.8678 −1.15023 −0.575113 0.818074i $$-0.695042\pi$$
−0.575113 + 0.818074i $$0.695042\pi$$
$$588$$ 0 0
$$589$$ −2.94244 −0.121241
$$590$$ 0 0
$$591$$ −18.4999 −0.760983
$$592$$ 0 0
$$593$$ −24.3463 −0.999781 −0.499890 0.866089i $$-0.666626\pi$$
−0.499890 + 0.866089i $$0.666626\pi$$
$$594$$ 0 0
$$595$$ 28.7170 1.17728
$$596$$ 0 0
$$597$$ 9.87086 0.403988
$$598$$ 0 0
$$599$$ −21.0164 −0.858707 −0.429354 0.903136i $$-0.641259\pi$$
−0.429354 + 0.903136i $$0.641259\pi$$
$$600$$ 0 0
$$601$$ 40.9477 1.67029 0.835145 0.550030i $$-0.185384\pi$$
0.835145 + 0.550030i $$0.185384\pi$$
$$602$$ 0 0
$$603$$ 2.66492 0.108524
$$604$$ 0 0
$$605$$ 4.18953 0.170329
$$606$$ 0 0
$$607$$ 14.0276 0.569362 0.284681 0.958622i $$-0.408112\pi$$
0.284681 + 0.958622i $$0.408112\pi$$
$$608$$ 0 0
$$609$$ 3.17313 0.128582
$$610$$ 0 0
$$611$$ −25.5767 −1.03472
$$612$$ 0 0
$$613$$ 18.5306 0.748442 0.374221 0.927339i $$-0.377910\pi$$
0.374221 + 0.927339i $$0.377910\pi$$
$$614$$ 0 0
$$615$$ −39.2252 −1.58171
$$616$$ 0 0
$$617$$ 2.43424 0.0979987 0.0489994 0.998799i $$-0.484397\pi$$
0.0489994 + 0.998799i $$0.484397\pi$$
$$618$$ 0 0
$$619$$ 30.4259 1.22292 0.611460 0.791275i $$-0.290582\pi$$
0.611460 + 0.791275i $$0.290582\pi$$
$$620$$ 0 0
$$621$$ −1.87086 −0.0750752
$$622$$ 0 0
$$623$$ −1.74173 −0.0697809
$$624$$ 0 0
$$625$$ 69.7966 2.79187
$$626$$ 0 0
$$627$$ −0.318669 −0.0127264
$$628$$ 0 0
$$629$$ −51.7662 −2.06405
$$630$$ 0 0
$$631$$ −34.9836 −1.39267 −0.696337 0.717715i $$-0.745188\pi$$
−0.696337 + 0.717715i $$0.745188\pi$$
$$632$$ 0 0
$$633$$ −4.63734 −0.184318
$$634$$ 0 0
$$635$$ 23.0081 0.913047
$$636$$ 0 0
$$637$$ −3.17313 −0.125724
$$638$$ 0 0
$$639$$ 5.01641 0.198446
$$640$$ 0 0
$$641$$ 8.56337 0.338233 0.169116 0.985596i $$-0.445909\pi$$
0.169116 + 0.985596i $$0.445909\pi$$
$$642$$ 0 0
$$643$$ 5.11273 0.201626 0.100813 0.994905i $$-0.467856\pi$$
0.100813 + 0.994905i $$0.467856\pi$$
$$644$$ 0 0
$$645$$ −45.6126 −1.79599
$$646$$ 0 0
$$647$$ −25.7693 −1.01310 −0.506548 0.862212i $$-0.669079\pi$$
−0.506548 + 0.862212i $$0.669079\pi$$
$$648$$ 0 0
$$649$$ 7.04399 0.276501
$$650$$ 0 0
$$651$$ 9.23353 0.361890
$$652$$ 0 0
$$653$$ −47.8953 −1.87429 −0.937145 0.348941i $$-0.886541\pi$$
−0.937145 + 0.348941i $$0.886541\pi$$
$$654$$ 0 0
$$655$$ −16.7581 −0.654795
$$656$$ 0 0
$$657$$ −4.82687 −0.188314
$$658$$ 0 0
$$659$$ −8.95601 −0.348877 −0.174438 0.984668i $$-0.555811\pi$$
−0.174438 + 0.984668i $$0.555811\pi$$
$$660$$ 0 0
$$661$$ −10.7993 −0.420044 −0.210022 0.977697i $$-0.567353\pi$$
−0.210022 + 0.977697i $$0.567353\pi$$
$$662$$ 0 0
$$663$$ 21.7501 0.844703
$$664$$ 0 0
$$665$$ 1.33508 0.0517720
$$666$$ 0 0
$$667$$ −5.93649 −0.229862
$$668$$ 0 0
$$669$$ −18.3463 −0.709307
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 0 0
$$673$$ 32.2499 1.24314 0.621572 0.783357i $$-0.286494\pi$$
0.621572 + 0.783357i $$0.286494\pi$$
$$674$$ 0 0
$$675$$ −12.5522 −0.483134
$$676$$ 0 0
$$677$$ −27.7089 −1.06494 −0.532470 0.846449i $$-0.678736\pi$$
−0.532470 + 0.846449i $$0.678736\pi$$
$$678$$ 0 0
$$679$$ −12.2499 −0.470109
$$680$$ 0 0
$$681$$ 0.379068 0.0145259
$$682$$ 0 0
$$683$$ −22.3156 −0.853881 −0.426941 0.904280i $$-0.640409\pi$$
−0.426941 + 0.904280i $$0.640409\pi$$
$$684$$ 0 0
$$685$$ 65.4095 2.49917
$$686$$ 0 0
$$687$$ −24.9424 −0.951614
$$688$$ 0 0
$$689$$ −1.61259 −0.0614349
$$690$$ 0 0
$$691$$ −25.1372 −0.956264 −0.478132 0.878288i $$-0.658686\pi$$
−0.478132 + 0.878288i $$0.658686\pi$$
$$692$$ 0 0
$$693$$ 1.00000 0.0379869
$$694$$ 0 0
$$695$$ 37.7745 1.43287
$$696$$ 0 0
$$697$$ 64.1760 2.43084
$$698$$ 0 0
$$699$$ −23.4506 −0.886985
$$700$$ 0 0
$$701$$ −19.2747 −0.727995 −0.363997 0.931400i $$-0.618588\pi$$
−0.363997 + 0.931400i $$0.618588\pi$$
$$702$$ 0 0
$$703$$ −2.40665 −0.0907686
$$704$$ 0 0
$$705$$ −33.7693 −1.27183
$$706$$ 0 0
$$707$$ 4.88727 0.183805
$$708$$ 0 0
$$709$$ 9.76098 0.366581 0.183291 0.983059i $$-0.441325\pi$$
0.183291 + 0.983059i $$0.441325\pi$$
$$710$$ 0 0
$$711$$ −5.01641 −0.188130
$$712$$ 0 0
$$713$$ −17.2747 −0.646942
$$714$$ 0 0
$$715$$ −13.2939 −0.497165
$$716$$ 0 0
$$717$$ 5.07681 0.189597
$$718$$ 0 0
$$719$$ −14.0276 −0.523141 −0.261570 0.965184i $$-0.584240\pi$$
−0.261570 + 0.965184i $$0.584240\pi$$
$$720$$ 0 0
$$721$$ 0.637339 0.0237357
$$722$$ 0 0
$$723$$ −19.2939 −0.717549
$$724$$ 0 0
$$725$$ −39.8297 −1.47924
$$726$$ 0 0
$$727$$ 34.3051 1.27231 0.636153 0.771563i $$-0.280525\pi$$
0.636153 + 0.771563i $$0.280525\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 74.6263 2.76016
$$732$$ 0 0
$$733$$ −10.7581 −0.397361 −0.198680 0.980064i $$-0.563666\pi$$
−0.198680 + 0.980064i $$0.563666\pi$$
$$734$$ 0 0
$$735$$ −4.18953 −0.154533
$$736$$ 0 0
$$737$$ 2.66492 0.0981637
$$738$$ 0 0
$$739$$ 6.38741 0.234965 0.117482 0.993075i $$-0.462518\pi$$
0.117482 + 0.993075i $$0.462518\pi$$
$$740$$ 0 0
$$741$$ 1.01118 0.0371466
$$742$$ 0 0
$$743$$ 23.1096 0.847810 0.423905 0.905707i $$-0.360659\pi$$
0.423905 + 0.905707i $$0.360659\pi$$
$$744$$ 0 0
$$745$$ −21.8105 −0.799074
$$746$$ 0 0
$$747$$ −3.52461 −0.128959
$$748$$ 0 0
$$749$$ −0.956008 −0.0349318
$$750$$ 0 0
$$751$$ −31.8678 −1.16287 −0.581435 0.813593i $$-0.697509\pi$$
−0.581435 + 0.813593i $$0.697509\pi$$
$$752$$ 0 0
$$753$$ 23.8021 0.867398
$$754$$ 0 0
$$755$$ −26.1843 −0.952944
$$756$$ 0 0
$$757$$ −48.3103 −1.75587 −0.877934 0.478781i $$-0.841079\pi$$
−0.877934 + 0.478781i $$0.841079\pi$$
$$758$$ 0 0
$$759$$ −1.87086 −0.0679081
$$760$$ 0 0
$$761$$ −17.8625 −0.647516 −0.323758 0.946140i $$-0.604946\pi$$
−0.323758 + 0.946140i $$0.604946\pi$$
$$762$$ 0 0
$$763$$ −7.61259 −0.275594
$$764$$ 0 0
$$765$$ 28.7170 1.03826
$$766$$ 0 0
$$767$$ −22.3515 −0.807065
$$768$$ 0 0
$$769$$ −24.8820 −0.897269 −0.448635 0.893715i $$-0.648090\pi$$
−0.448635 + 0.893715i $$0.648090\pi$$
$$770$$ 0 0
$$771$$ −14.9149 −0.537145
$$772$$ 0 0
$$773$$ −34.9700 −1.25778 −0.628892 0.777492i $$-0.716491\pi$$
−0.628892 + 0.777492i $$0.716491\pi$$
$$774$$ 0 0
$$775$$ −115.901 −4.16329
$$776$$ 0 0
$$777$$ 7.55220 0.270933
$$778$$ 0 0
$$779$$ 2.98359 0.106898
$$780$$ 0 0
$$781$$ 5.01641 0.179501
$$782$$ 0 0
$$783$$ 3.17313 0.113398
$$784$$ 0 0
$$785$$ −75.9177 −2.70962
$$786$$ 0 0
$$787$$ 15.1648 0.540566 0.270283 0.962781i $$-0.412883\pi$$
0.270283 + 0.962781i $$0.412883\pi$$
$$788$$ 0 0
$$789$$ 2.92319 0.104068
$$790$$ 0 0
$$791$$ −7.70892 −0.274097
$$792$$ 0 0
$$793$$ 6.34625 0.225362
$$794$$ 0 0
$$795$$ −2.12914 −0.0755126
$$796$$ 0 0
$$797$$ 8.24470 0.292042 0.146021 0.989281i $$-0.453353\pi$$
0.146021 + 0.989281i $$0.453353\pi$$
$$798$$ 0 0
$$799$$ 55.2497 1.95459
$$800$$ 0 0
$$801$$ −1.74173 −0.0615410
$$802$$ 0 0
$$803$$ −4.82687 −0.170337
$$804$$ 0 0
$$805$$ 7.83805 0.276255
$$806$$ 0 0
$$807$$ −11.9672 −0.421265
$$808$$ 0 0
$$809$$ 29.8433 1.04923 0.524617 0.851338i $$-0.324209\pi$$
0.524617 + 0.851338i $$0.324209\pi$$
$$810$$ 0 0
$$811$$ 16.6321 0.584032 0.292016 0.956413i $$-0.405674\pi$$
0.292016 + 0.956413i $$0.405674\pi$$
$$812$$ 0 0
$$813$$ −20.3187 −0.712607
$$814$$ 0 0
$$815$$ 11.1648 0.391086
$$816$$ 0 0
$$817$$ 3.46944 0.121380
$$818$$ 0 0
$$819$$ −3.17313 −0.110878
$$820$$ 0 0
$$821$$ 30.3327 1.05862 0.529309 0.848429i $$-0.322451\pi$$
0.529309 + 0.848429i $$0.322451\pi$$
$$822$$ 0 0
$$823$$ 18.4067 0.641616 0.320808 0.947144i $$-0.396046\pi$$
0.320808 + 0.947144i $$0.396046\pi$$
$$824$$ 0 0
$$825$$ −12.5522 −0.437011
$$826$$ 0 0
$$827$$ −45.4559 −1.58066 −0.790328 0.612684i $$-0.790090\pi$$
−0.790328 + 0.612684i $$0.790090\pi$$
$$828$$ 0 0
$$829$$ −20.3463 −0.706655 −0.353327 0.935500i $$-0.614950\pi$$
−0.353327 + 0.935500i $$0.614950\pi$$
$$830$$ 0 0
$$831$$ −18.0552 −0.626327
$$832$$ 0 0
$$833$$ 6.85446 0.237493
$$834$$ 0 0
$$835$$ 46.6946 1.61593
$$836$$ 0 0
$$837$$ 9.23353 0.319157
$$838$$ 0 0
$$839$$ −16.1812 −0.558637 −0.279318 0.960199i $$-0.590109\pi$$
−0.279318 + 0.960199i $$0.590109\pi$$
$$840$$ 0 0
$$841$$ −18.9313 −0.652802
$$842$$ 0 0
$$843$$ −27.2939 −0.940053
$$844$$ 0 0
$$845$$ −12.2806 −0.422466
$$846$$ 0 0
$$847$$ 1.00000 0.0343604
$$848$$ 0 0
$$849$$ 29.8901 1.02583
$$850$$ 0 0
$$851$$ −14.1291 −0.484341
$$852$$ 0 0
$$853$$ −20.9836 −0.718465 −0.359232 0.933248i $$-0.616961\pi$$
−0.359232 + 0.933248i $$0.616961\pi$$
$$854$$ 0 0
$$855$$ 1.33508 0.0456586
$$856$$ 0 0
$$857$$ 4.79095 0.163656 0.0818279 0.996646i $$-0.473924\pi$$
0.0818279 + 0.996646i $$0.473924\pi$$
$$858$$ 0 0
$$859$$ 9.96719 0.340076 0.170038 0.985438i $$-0.445611\pi$$
0.170038 + 0.985438i $$0.445611\pi$$
$$860$$ 0 0
$$861$$ −9.36266 −0.319079
$$862$$ 0 0
$$863$$ −25.5470 −0.869629 −0.434814 0.900520i $$-0.643186\pi$$
−0.434814 + 0.900520i $$0.643186\pi$$
$$864$$ 0 0
$$865$$ −104.094 −3.53929
$$866$$ 0 0
$$867$$ −29.9836 −1.01830
$$868$$ 0 0
$$869$$ −5.01641 −0.170170
$$870$$ 0 0
$$871$$ −8.45614 −0.286525
$$872$$ 0 0
$$873$$ −12.2499 −0.414597
$$874$$ 0 0
$$875$$ 31.6402 1.06963
$$876$$ 0 0
$$877$$ −50.9893 −1.72179 −0.860893 0.508787i $$-0.830094\pi$$
−0.860893 + 0.508787i $$0.830094\pi$$
$$878$$ 0 0
$$879$$ 4.34625 0.146596
$$880$$ 0 0
$$881$$ 32.1895 1.08449 0.542246 0.840219i $$-0.317574\pi$$
0.542246 + 0.840219i $$0.317574\pi$$
$$882$$ 0 0
$$883$$ −36.6154 −1.23221 −0.616104 0.787665i $$-0.711290\pi$$
−0.616104 + 0.787665i $$0.711290\pi$$
$$884$$ 0 0
$$885$$ −29.5110 −0.992003
$$886$$ 0 0
$$887$$ 48.2004 1.61841 0.809206 0.587525i $$-0.199897\pi$$
0.809206 + 0.587525i $$0.199897\pi$$
$$888$$ 0 0
$$889$$ 5.49180 0.184189
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ 2.56860 0.0859550
$$894$$ 0 0
$$895$$ 53.4506 1.78666
$$896$$ 0 0
$$897$$ 5.93649 0.198214
$$898$$ 0 0
$$899$$ 29.2992 0.977181
$$900$$ 0 0
$$901$$ 3.48346 0.116051
$$902$$ 0 0
$$903$$ −10.8873 −0.362306
$$904$$ 0 0
$$905$$ −13.5470 −0.450316
$$906$$ 0 0
$$907$$ −22.9013 −0.760425 −0.380212 0.924899i $$-0.624149\pi$$
−0.380212 + 0.924899i $$0.624149\pi$$
$$908$$ 0 0
$$909$$ 4.88727 0.162101
$$910$$ 0 0
$$911$$ 36.0552 1.19456 0.597281 0.802032i $$-0.296248\pi$$
0.597281 + 0.802032i $$0.296248\pi$$
$$912$$ 0 0
$$913$$ −3.52461 −0.116648
$$914$$ 0 0
$$915$$ 8.37907 0.277003
$$916$$ 0 0
$$917$$ −4.00000 −0.132092
$$918$$ 0 0
$$919$$ 28.3379 0.934782 0.467391 0.884051i $$-0.345194\pi$$
0.467391 + 0.884051i $$0.345194\pi$$
$$920$$ 0 0
$$921$$ −20.7581 −0.684004
$$922$$ 0 0
$$923$$ −15.9177 −0.523937
$$924$$ 0 0
$$925$$ −94.7966 −3.11689
$$926$$ 0 0
$$927$$ 0.637339 0.0209329
$$928$$ 0 0
$$929$$ 5.08514 0.166838 0.0834191 0.996515i $$-0.473416\pi$$
0.0834191 + 0.996515i $$0.473416\pi$$
$$930$$ 0 0
$$931$$ 0.318669 0.0104440
$$932$$ 0 0
$$933$$ 8.00000 0.261908
$$934$$ 0 0
$$935$$ 28.7170 0.939146
$$936$$ 0 0
$$937$$ 32.2088 1.05222 0.526108 0.850418i $$-0.323651\pi$$
0.526108 + 0.850418i $$0.323651\pi$$
$$938$$ 0 0
$$939$$ −8.12914 −0.265284
$$940$$ 0 0
$$941$$ 32.7805 1.06861 0.534307 0.845291i $$-0.320573\pi$$
0.534307 + 0.845291i $$0.320573\pi$$
$$942$$ 0 0
$$943$$ 17.5163 0.570408
$$944$$ 0 0
$$945$$ −4.18953 −0.136286
$$946$$ 0 0
$$947$$ 27.3627 0.889167 0.444584 0.895737i $$-0.353352\pi$$
0.444584 + 0.895737i $$0.353352\pi$$
$$948$$ 0 0
$$949$$ 15.3163 0.497188
$$950$$ 0 0
$$951$$ 19.9917 0.648274
$$952$$ 0 0
$$953$$ −24.6894 −0.799768 −0.399884 0.916566i $$-0.630950\pi$$
−0.399884 + 0.916566i $$0.630950\pi$$
$$954$$ 0 0
$$955$$ −87.8765 −2.84362
$$956$$ 0 0
$$957$$ 3.17313 0.102573
$$958$$ 0 0
$$959$$ 15.6126 0.504157
$$960$$ 0 0
$$961$$ 54.2580 1.75026
$$962$$ 0 0
$$963$$ −0.956008 −0.0308069
$$964$$ 0 0
$$965$$ 1.04710 0.0337074
$$966$$ 0 0
$$967$$ −21.3298 −0.685922 −0.342961 0.939350i $$-0.611430\pi$$
−0.342961 + 0.939350i $$0.611430\pi$$
$$968$$ 0 0
$$969$$ −2.18431 −0.0701700
$$970$$ 0 0
$$971$$ −28.4946 −0.914436 −0.457218 0.889355i $$-0.651154\pi$$
−0.457218 + 0.889355i $$0.651154\pi$$
$$972$$ 0 0
$$973$$ 9.01641 0.289053
$$974$$ 0 0
$$975$$ 39.8297 1.27557
$$976$$ 0 0
$$977$$ −19.8157 −0.633960 −0.316980 0.948432i $$-0.602669\pi$$
−0.316980 + 0.948432i $$0.602669\pi$$
$$978$$ 0 0
$$979$$ −1.74173 −0.0556659
$$980$$ 0 0
$$981$$ −7.61259 −0.243051
$$982$$ 0 0
$$983$$ 53.7745 1.71514 0.857571 0.514366i $$-0.171973\pi$$
0.857571 + 0.514366i $$0.171973\pi$$
$$984$$ 0 0
$$985$$ 77.5058 2.46954
$$986$$ 0 0
$$987$$ −8.06040 −0.256565
$$988$$ 0 0
$$989$$ 20.3686 0.647684
$$990$$ 0 0
$$991$$ −49.7693 −1.58097 −0.790487 0.612479i $$-0.790173\pi$$
−0.790487 + 0.612479i $$0.790173\pi$$
$$992$$ 0 0
$$993$$ 17.0164 0.539999
$$994$$ 0 0
$$995$$ −41.3543 −1.31102
$$996$$ 0 0
$$997$$ 0.659696 0.0208928 0.0104464 0.999945i $$-0.496675\pi$$
0.0104464 + 0.999945i $$0.496675\pi$$
$$998$$ 0 0
$$999$$ 7.55220 0.238941
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 3696.2.a.bo.1.3 3
4.3 odd 2 231.2.a.e.1.1 3
12.11 even 2 693.2.a.l.1.3 3
20.19 odd 2 5775.2.a.bp.1.3 3
28.27 even 2 1617.2.a.t.1.1 3
44.43 even 2 2541.2.a.bg.1.3 3
84.83 odd 2 4851.2.a.bi.1.3 3
132.131 odd 2 7623.2.a.cd.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.1 3 4.3 odd 2
693.2.a.l.1.3 3 12.11 even 2
1617.2.a.t.1.1 3 28.27 even 2
2541.2.a.bg.1.3 3 44.43 even 2
3696.2.a.bo.1.3 3 1.1 even 1 trivial
4851.2.a.bi.1.3 3 84.83 odd 2
5775.2.a.bp.1.3 3 20.19 odd 2
7623.2.a.cd.1.1 3 132.131 odd 2