# Properties

 Label 4851.2.a.bi.1.3 Level $4851$ Weight $2$ Character 4851.1 Self dual yes Analytic conductor $38.735$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.7354300205$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\chi$$ $$=$$ 4851.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.93543 q^{2} +1.74590 q^{4} +4.18953 q^{5} -0.491797 q^{8} +O(q^{10})$$ $$q+1.93543 q^{2} +1.74590 q^{4} +4.18953 q^{5} -0.491797 q^{8} +8.10856 q^{10} +1.00000 q^{11} +3.17313 q^{13} -4.44364 q^{16} +6.85446 q^{17} +0.318669 q^{19} +7.31450 q^{20} +1.93543 q^{22} +1.87086 q^{23} +12.5522 q^{25} +6.14137 q^{26} +3.17313 q^{29} -9.23353 q^{31} -7.61676 q^{32} +13.2663 q^{34} -7.55220 q^{37} +0.616763 q^{38} -2.06040 q^{40} +9.36266 q^{41} -10.8873 q^{43} +1.74590 q^{44} +3.62093 q^{46} -8.06040 q^{47} +24.2939 q^{50} +5.53996 q^{52} -0.508203 q^{53} +4.18953 q^{55} +6.14137 q^{58} -7.04399 q^{59} +2.00000 q^{61} -17.8709 q^{62} -5.85446 q^{64} +13.2939 q^{65} -2.66492 q^{67} +11.9672 q^{68} +5.01641 q^{71} +4.82687 q^{73} -14.6168 q^{74} +0.556364 q^{76} +5.01641 q^{79} -18.6168 q^{80} +18.1208 q^{82} +3.52461 q^{83} +28.7170 q^{85} -21.0716 q^{86} -0.491797 q^{88} -1.74173 q^{89} +3.26634 q^{92} -15.6004 q^{94} +1.33508 q^{95} +12.2499 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 6 q^{4} + 4 q^{5} - 3 q^{8}+O(q^{10})$$ 3 * q - 2 * q^2 + 6 * q^4 + 4 * q^5 - 3 * q^8 $$3 q - 2 q^{2} + 6 q^{4} + 4 q^{5} - 3 q^{8} + 11 q^{10} + 3 q^{11} + 4 q^{13} - 4 q^{16} + 8 q^{17} + 8 q^{19} - 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} - q^{26} + 4 q^{29} + 2 q^{31} - 8 q^{32} + 4 q^{34} - 13 q^{38} + 18 q^{40} + 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 19 q^{50} + 29 q^{52} + 4 q^{55} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} + 42 q^{68} + 12 q^{71} + 20 q^{73} - 29 q^{74} + 11 q^{76} + 12 q^{79} - 41 q^{80} + 6 q^{82} + 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} + 26 q^{89} - 26 q^{92} - 35 q^{94} + 8 q^{95} + 4 q^{97}+O(q^{100})$$ 3 * q - 2 * q^2 + 6 * q^4 + 4 * q^5 - 3 * q^8 + 11 * q^10 + 3 * q^11 + 4 * q^13 - 4 * q^16 + 8 * q^17 + 8 * q^19 - 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 - q^26 + 4 * q^29 + 2 * q^31 - 8 * q^32 + 4 * q^34 - 13 * q^38 + 18 * q^40 + 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 19 * q^50 + 29 * q^52 + 4 * q^55 - q^58 + 6 * q^61 - 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 + 42 * q^68 + 12 * q^71 + 20 * q^73 - 29 * q^74 + 11 * q^76 + 12 * q^79 - 41 * q^80 + 6 * q^82 + 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 + 26 * q^89 - 26 * q^92 - 35 * q^94 + 8 * q^95 + 4 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.93543 1.36856 0.684279 0.729221i $$-0.260117\pi$$
0.684279 + 0.729221i $$0.260117\pi$$
$$3$$ 0 0
$$4$$ 1.74590 0.872949
$$5$$ 4.18953 1.87362 0.936808 0.349843i $$-0.113765\pi$$
0.936808 + 0.349843i $$0.113765\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −0.491797 −0.173876
$$9$$ 0 0
$$10$$ 8.10856 2.56415
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 3.17313 0.880067 0.440034 0.897981i $$-0.354967\pi$$
0.440034 + 0.897981i $$0.354967\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.44364 −1.11091
$$17$$ 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 $$0.488362\pi$$
$$20$$ 7.31450 1.63557
$$21$$ 0 0
$$22$$ 1.93543 0.412636
$$23$$ 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
$$26$$ 6.14137 1.20442
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.17313 0.589235 0.294617 0.955615i $$-0.404808\pi$$
0.294617 + 0.955615i $$0.404808\pi$$
$$30$$ 0 0
$$31$$ −9.23353 −1.65839 −0.829195 0.558959i $$-0.811201\pi$$
−0.829195 + 0.558959i $$0.811201\pi$$
$$32$$ −7.61676 −1.34647
$$33$$ 0 0
$$34$$ 13.2663 2.27516
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.55220 −1.24157 −0.620787 0.783980i $$-0.713187\pi$$
−0.620787 + 0.783980i $$0.713187\pi$$
$$38$$ 0.616763 0.100052
$$39$$ 0 0
$$40$$ −2.06040 −0.325778
$$41$$ 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 $$-0.811743\pi$$
−0.830147 + 0.557545i $$0.811743\pi$$
$$44$$ 1.74590 0.263204
$$45$$ 0 0
$$46$$ 3.62093 0.533877
$$47$$ −8.06040 −1.17573 −0.587865 0.808959i $$-0.700031\pi$$
−0.587865 + 0.808959i $$0.700031\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 24.2939 3.43568
$$51$$ 0 0
$$52$$ 5.53996 0.768254
$$53$$ −0.508203 −0.0698071 −0.0349036 0.999391i $$-0.511112\pi$$
−0.0349036 + 0.999391i $$0.511112\pi$$
$$54$$ 0 0
$$55$$ 4.18953 0.564917
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.14137 0.806402
$$59$$ −7.04399 −0.917050 −0.458525 0.888682i $$-0.651622\pi$$
−0.458525 + 0.888682i $$0.651622\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −17.8709 −2.26960
$$63$$ 0 0
$$64$$ −5.85446 −0.731807
$$65$$ 13.2939 1.64891
$$66$$ 0 0
$$67$$ −2.66492 −0.325572 −0.162786 0.986661i $$-0.552048\pi$$
−0.162786 + 0.986661i $$0.552048\pi$$
$$68$$ 11.9672 1.45123
$$69$$ 0 0
$$70$$ 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 $$-0.408846\pi$$
0.282471 + 0.959276i $$0.408846\pi$$
$$74$$ −14.6168 −1.69916
$$75$$ 0 0
$$76$$ 0.556364 0.0638194
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.01641 0.564390 0.282195 0.959357i $$-0.408938\pi$$
0.282195 + 0.959357i $$0.408938\pi$$
$$80$$ −18.6168 −2.08142
$$81$$ 0 0
$$82$$ 18.1208 2.00111
$$83$$ 3.52461 0.386876 0.193438 0.981112i $$-0.438036\pi$$
0.193438 + 0.981112i $$0.438036\pi$$
$$84$$ 0 0
$$85$$ 28.7170 3.11479
$$86$$ −21.0716 −2.27221
$$87$$ 0 0
$$88$$ −0.491797 −0.0524257
$$89$$ −1.74173 −0.184623 −0.0923115 0.995730i $$-0.529426\pi$$
−0.0923115 + 0.995730i $$0.529426\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 3.26634 0.340539
$$93$$ 0 0
$$94$$ −15.6004 −1.60905
$$95$$ 1.33508 0.136976
$$96$$ 0 0
$$97$$ 12.2499 1.24379 0.621896 0.783100i $$-0.286363\pi$$
0.621896 + 0.783100i $$0.286363\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 21.9149 2.19149
$$101$$ 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 $$0.490004\pi$$
$$104$$ −1.56053 −0.153023
$$105$$ 0 0
$$106$$ −0.983593 −0.0955350
$$107$$ −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 $$0.618786\pi$$
$$110$$ 8.10856 0.773121
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 7.70892 0.725194 0.362597 0.931946i $$-0.381890\pi$$
0.362597 + 0.931946i $$0.381890\pi$$
$$114$$ 0 0
$$115$$ 7.83805 0.730902
$$116$$ 5.53996 0.514372
$$117$$ 0 0
$$118$$ −13.6332 −1.25504
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 3.87086 0.350452
$$123$$ 0 0
$$124$$ −16.1208 −1.44769
$$125$$ 31.6402 2.82998
$$126$$ 0 0
$$127$$ −5.49180 −0.487318 −0.243659 0.969861i $$-0.578348\pi$$
−0.243659 + 0.969861i $$0.578348\pi$$
$$128$$ 3.90262 0.344946
$$129$$ 0 0
$$130$$ 25.7295 2.25663
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −5.15778 −0.445564
$$135$$ 0 0
$$136$$ −3.37100 −0.289061
$$137$$ −15.6126 −1.33387 −0.666937 0.745114i $$-0.732395\pi$$
−0.666937 + 0.745114i $$0.732395\pi$$
$$138$$ 0 0
$$139$$ 9.01641 0.764762 0.382381 0.924005i $$-0.375104\pi$$
0.382381 + 0.924005i $$0.375104\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 9.70892 0.814754
$$143$$ 3.17313 0.265350
$$144$$ 0 0
$$145$$ 13.2939 1.10400
$$146$$ 9.34209 0.773157
$$147$$ 0 0
$$148$$ −13.1854 −1.08383
$$149$$ 5.20594 0.426487 0.213244 0.976999i $$-0.431597\pi$$
0.213244 + 0.976999i $$0.431597\pi$$
$$150$$ 0 0
$$151$$ 6.24993 0.508612 0.254306 0.967124i $$-0.418153\pi$$
0.254306 + 0.967124i $$0.418153\pi$$
$$152$$ −0.156721 −0.0127117
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −38.6842 −3.10719
$$156$$ 0 0
$$157$$ 18.1208 1.44620 0.723099 0.690745i $$-0.242717\pi$$
0.723099 + 0.690745i $$0.242717\pi$$
$$158$$ 9.70892 0.772400
$$159$$ 0 0
$$160$$ −31.9107 −2.52276
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −2.66492 −0.208733 −0.104366 0.994539i $$-0.533281\pi$$
−0.104366 + 0.994539i $$0.533281\pi$$
$$164$$ 16.3463 1.27643
$$165$$ 0 0
$$166$$ 6.82164 0.529462
$$167$$ −11.1455 −0.862468 −0.431234 0.902240i $$-0.641922\pi$$
−0.431234 + 0.902240i $$0.641922\pi$$
$$168$$ 0 0
$$169$$ −2.93126 −0.225482
$$170$$ 55.5798 4.26277
$$171$$ 0 0
$$172$$ −19.0081 −1.44935
$$173$$ −24.8461 −1.88902 −0.944508 0.328489i $$-0.893461\pi$$
−0.944508 + 0.328489i $$0.893461\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.44364 −0.334952
$$177$$ 0 0
$$178$$ −3.37100 −0.252667
$$179$$ 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 $$-0.461655\pi$$
0.120173 + 0.992753i $$0.461655\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −0.920085 −0.0678296
$$185$$ −31.6402 −2.32623
$$186$$ 0 0
$$187$$ 6.85446 0.501248
$$188$$ −14.0726 −1.02635
$$189$$ 0 0
$$190$$ 2.58395 0.187459
$$191$$ −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 $$0.497137\pi$$
$$194$$ 23.7089 1.70220
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.4999 −1.31806 −0.659030 0.752116i $$-0.729033\pi$$
−0.659030 + 0.752116i $$0.729033\pi$$
$$198$$ 0 0
$$199$$ −9.87086 −0.699727 −0.349864 0.936801i $$-0.613772\pi$$
−0.349864 + 0.936801i $$0.613772\pi$$
$$200$$ −6.17313 −0.436506
$$201$$ 0 0
$$202$$ 9.45898 0.665532
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 39.2252 2.73961
$$206$$ 1.23353 0.0859438
$$207$$ 0 0
$$208$$ −14.1002 −0.977674
$$209$$ 0.318669 0.0220428
$$210$$ 0 0
$$211$$ −4.63734 −0.319248 −0.159624 0.987178i $$-0.551028\pi$$
−0.159624 + 0.987178i $$0.551028\pi$$
$$212$$ −0.887271 −0.0609381
$$213$$ 0 0
$$214$$ −1.85029 −0.126483
$$215$$ −45.6126 −3.11075
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −14.7337 −0.997889
$$219$$ 0 0
$$220$$ 7.31450 0.493144
$$221$$ 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
$$225$$ 0 0
$$226$$ 14.9201 0.992469
$$227$$ 0.379068 0.0251596 0.0125798 0.999921i $$-0.495996\pi$$
0.0125798 + 0.999921i $$0.495996\pi$$
$$228$$ 0 0
$$229$$ −24.9424 −1.64824 −0.824121 0.566413i $$-0.808331\pi$$
−0.824121 + 0.566413i $$0.808331\pi$$
$$230$$ 15.1700 1.00028
$$231$$ 0 0
$$232$$ −1.56053 −0.102454
$$233$$ −23.4506 −1.53630 −0.768151 0.640268i $$-0.778823\pi$$
−0.768151 + 0.640268i $$0.778823\pi$$
$$234$$ 0 0
$$235$$ −33.7693 −2.20287
$$236$$ −12.2981 −0.800538
$$237$$ 0 0
$$238$$ 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 $$-0.713442\pi$$
−0.621415 + 0.783481i $$0.713442\pi$$
$$242$$ 1.93543 0.124414
$$243$$ 0 0
$$244$$ 3.49180 0.223539
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.01118 0.0643397
$$248$$ 4.54102 0.288355
$$249$$ 0 0
$$250$$ 61.2374 3.87299
$$251$$ 23.8021 1.50238 0.751188 0.660088i $$-0.229481\pi$$
0.751188 + 0.660088i $$0.229481\pi$$
$$252$$ 0 0
$$253$$ 1.87086 0.117620
$$254$$ −10.6290 −0.666923
$$255$$ 0 0
$$256$$ 19.2622 1.20389
$$257$$ 14.9149 0.930363 0.465182 0.885215i $$-0.345989\pi$$
0.465182 + 0.885215i $$0.345989\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 23.2098 1.43941
$$261$$ 0 0
$$262$$ 7.74173 0.478286
$$263$$ −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
$$267$$ 0 0
$$268$$ −4.65269 −0.284208
$$269$$ 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 $$0.288293\pi$$
$$272$$ −30.4587 −1.84683
$$273$$ 0 0
$$274$$ −30.2171 −1.82548
$$275$$ 12.5522 0.756926
$$276$$ 0 0
$$277$$ 18.0552 1.08483 0.542415 0.840111i $$-0.317510\pi$$
0.542415 + 0.840111i $$0.317510\pi$$
$$278$$ 17.4506 1.04662
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −27.2939 −1.62822 −0.814110 0.580711i $$-0.802774\pi$$
−0.814110 + 0.580711i $$0.802774\pi$$
$$282$$ 0 0
$$283$$ −29.8901 −1.77678 −0.888391 0.459087i $$-0.848177\pi$$
−0.888391 + 0.459087i $$0.848177\pi$$
$$284$$ 8.75814 0.519700
$$285$$ 0 0
$$286$$ 6.14137 0.363147
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 29.9836 1.76374
$$290$$ 25.7295 1.51089
$$291$$ 0 0
$$292$$ 8.42723 0.493166
$$293$$ −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
$$296$$ 3.71414 0.215880
$$297$$ 0 0
$$298$$ 10.0757 0.583672
$$299$$ 5.93649 0.343316
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 12.0963 0.696065
$$303$$ 0 0
$$304$$ −1.41605 −0.0812161
$$305$$ 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 0
$$310$$ −74.8706 −4.25236
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −8.12914 −0.459486 −0.229743 0.973251i $$-0.573789\pi$$
−0.229743 + 0.973251i $$0.573789\pi$$
$$314$$ 35.0716 1.97920
$$315$$ 0 0
$$316$$ 8.75814 0.492684
$$317$$ 19.9917 1.12284 0.561422 0.827530i $$-0.310255\pi$$
0.561422 + 0.827530i $$0.310255\pi$$
$$318$$ 0 0
$$319$$ 3.17313 0.177661
$$320$$ −24.5275 −1.37113
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 2.18431 0.121538
$$324$$ 0 0
$$325$$ 39.8297 2.20935
$$326$$ −5.15778 −0.285663
$$327$$ 0 0
$$328$$ −4.60453 −0.254242
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 17.0164 0.935306 0.467653 0.883912i $$-0.345100\pi$$
0.467653 + 0.883912i $$0.345100\pi$$
$$332$$ 6.15361 0.337723
$$333$$ 0 0
$$334$$ −21.5714 −1.18034
$$335$$ −11.1648 −0.609998
$$336$$ 0 0
$$337$$ 1.52461 0.0830508 0.0415254 0.999137i $$-0.486778\pi$$
0.0415254 + 0.999137i $$0.486778\pi$$
$$338$$ −5.67326 −0.308585
$$339$$ 0 0
$$340$$ 50.1369 2.71906
$$341$$ −9.23353 −0.500023
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 5.35432 0.288686
$$345$$ 0 0
$$346$$ −48.0880 −2.58523
$$347$$ −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 $$-0.558774\pi$$
−0.183595 + 0.983002i $$0.558774\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −7.61676 −0.405975
$$353$$ −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
$$356$$ −3.04088 −0.161166
$$357$$ 0 0
$$358$$ 24.6925 1.30504
$$359$$ 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
$$362$$ 6.25827 0.328927
$$363$$ 0 0
$$364$$ 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 $$0.646997\pi$$
$$368$$ −8.31344 −0.433368
$$369$$ 0 0
$$370$$ −61.2374 −3.18358
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.17836 0.164569 0.0822845 0.996609i $$-0.473778\pi$$
0.0822845 + 0.996609i $$0.473778\pi$$
$$374$$ 13.2663 0.685986
$$375$$ 0 0
$$376$$ 3.96408 0.204432
$$377$$ 10.0687 0.518566
$$378$$ 0 0
$$379$$ 3.93960 0.202364 0.101182 0.994868i $$-0.467738\pi$$
0.101182 + 0.994868i $$0.467738\pi$$
$$380$$ 2.33091 0.119573
$$381$$ 0 0
$$382$$ −40.5962 −2.07708
$$383$$ −24.7581 −1.26508 −0.632541 0.774527i $$-0.717988\pi$$
−0.632541 + 0.774527i $$0.717988\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0.483728 0.0246211
$$387$$ 0 0
$$388$$ 21.3871 1.08577
$$389$$ −3.65375 −0.185252 −0.0926261 0.995701i $$-0.529526\pi$$
−0.0926261 + 0.995701i $$0.529526\pi$$
$$390$$ 0 0
$$391$$ 12.8238 0.648526
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −35.8052 −1.80384
$$395$$ 21.0164 1.05745
$$396$$ 0 0
$$397$$ −4.56337 −0.229029 −0.114515 0.993422i $$-0.536531\pi$$
−0.114515 + 0.993422i $$0.536531\pi$$
$$398$$ −19.1044 −0.957617
$$399$$ 0 0
$$400$$ −55.7774 −2.78887
$$401$$ 7.23353 0.361225 0.180613 0.983554i $$-0.442192\pi$$
0.180613 + 0.983554i $$0.442192\pi$$
$$402$$ 0 0
$$403$$ −29.2992 −1.45949
$$404$$ 8.53268 0.424517
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7.55220 −0.374348
$$408$$ 0 0
$$409$$ −5.30749 −0.262439 −0.131219 0.991353i $$-0.541889\pi$$
−0.131219 + 0.991353i $$0.541889\pi$$
$$410$$ 75.9177 3.74931
$$411$$ 0 0
$$412$$ 1.11273 0.0548202
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 14.7665 0.724858
$$416$$ −24.1690 −1.18498
$$417$$ 0 0
$$418$$ 0.616763 0.0301669
$$419$$ −11.4231 −0.558053 −0.279026 0.960283i $$-0.590012\pi$$
−0.279026 + 0.960283i $$0.590012\pi$$
$$420$$ 0 0
$$421$$ −27.4147 −1.33611 −0.668056 0.744111i $$-0.732873\pi$$
−0.668056 + 0.744111i $$0.732873\pi$$
$$422$$ −8.97526 −0.436909
$$423$$ 0 0
$$424$$ 0.249933 0.0121378
$$425$$ 86.0385 4.17348
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −1.66909 −0.0806786
$$429$$ 0 0
$$430$$ −88.2801 −4.25724
$$431$$ 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 $$-0.774123\pi$$
−0.758613 + 0.651541i $$0.774123\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −13.2908 −0.636515
$$437$$ 0.596187 0.0285195
$$438$$ 0 0
$$439$$ 18.5275 0.884267 0.442133 0.896949i $$-0.354222\pi$$
0.442133 + 0.896949i $$0.354222\pi$$
$$440$$ −2.06040 −0.0982257
$$441$$ 0 0
$$442$$ 42.0958 2.00229
$$443$$ −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
$$446$$ 35.5079 1.68135
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.68656 0.268365 0.134183 0.990957i $$-0.457159\pi$$
0.134183 + 0.990957i $$0.457159\pi$$
$$450$$ 0 0
$$451$$ 9.36266 0.440871
$$452$$ 13.4590 0.633057
$$453$$ 0 0
$$454$$ 0.733661 0.0344324
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13.0081 −0.608492 −0.304246 0.952594i $$-0.598404\pi$$
−0.304246 + 0.952594i $$0.598404\pi$$
$$458$$ −48.2744 −2.25571
$$459$$ 0 0
$$460$$ 13.6844 0.638040
$$461$$ −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 $$-0.208924\pi$$
0.792223 + 0.610232i $$0.208924\pi$$
$$464$$ −14.1002 −0.654586
$$465$$ 0 0
$$466$$ −45.3871 −2.10252
$$467$$ 18.1484 0.839807 0.419903 0.907569i $$-0.362064\pi$$
0.419903 + 0.907569i $$0.362064\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −65.3582 −3.01475
$$471$$ 0 0
$$472$$ 3.46421 0.159453
$$473$$ −10.8873 −0.500597
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −9.82581 −0.449422
$$479$$ −42.7498 −1.95329 −0.976644 0.214864i $$-0.931069\pi$$
−0.976644 + 0.214864i $$0.931069\pi$$
$$480$$ 0 0
$$481$$ −23.9641 −1.09267
$$482$$ −37.3421 −1.70089
$$483$$ 0 0
$$484$$ 1.74590 0.0793590
$$485$$ 51.3215 2.33039
$$486$$ 0 0
$$487$$ 26.7909 1.21401 0.607007 0.794697i $$-0.292370\pi$$
0.607007 + 0.794697i $$0.292370\pi$$
$$488$$ −0.983593 −0.0445252
$$489$$ 0 0
$$490$$ 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
$$494$$ 1.95707 0.0880526
$$495$$ 0 0
$$496$$ 41.0304 1.84232
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −24.1260 −1.08003 −0.540015 0.841656i $$-0.681581\pi$$
−0.540015 + 0.841656i $$0.681581\pi$$
$$500$$ 55.2405 2.47043
$$501$$ 0 0
$$502$$ 46.0674 2.05609
$$503$$ 30.8873 1.37720 0.688598 0.725144i $$-0.258227\pi$$
0.688598 + 0.725144i $$0.258227\pi$$
$$504$$ 0 0
$$505$$ 20.4754 0.911143
$$506$$ 3.62093 0.160970
$$507$$ 0 0
$$508$$ −9.58812 −0.425404
$$509$$ 28.2088 1.25033 0.625166 0.780492i $$-0.285031\pi$$
0.625166 + 0.780492i $$0.285031\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 29.4754 1.30264
$$513$$ 0 0
$$514$$ 28.8667 1.27326
$$515$$ 2.67015 0.117661
$$516$$ 0 0
$$517$$ −8.06040 −0.354496
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6.53791 −0.286706
$$521$$ −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 $$0.590420\pi$$
$$524$$ 6.98359 0.305080
$$525$$ 0 0
$$526$$ −5.65765 −0.246685
$$527$$ −63.2908 −2.75699
$$528$$ 0 0
$$529$$ −19.4999 −0.847820
$$530$$ −4.12080 −0.178996
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 29.7089 1.28684
$$534$$ 0 0
$$535$$ −4.00523 −0.173161
$$536$$ 1.31060 0.0566093
$$537$$ 0 0
$$538$$ 23.1617 0.998571
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −21.8625 −0.939943 −0.469972 0.882681i $$-0.655736\pi$$
−0.469972 + 0.882681i $$0.655736\pi$$
$$542$$ 39.3254 1.68917
$$543$$ 0 0
$$544$$ −52.2088 −2.23843
$$545$$ −31.8932 −1.36616
$$546$$ 0 0
$$547$$ 23.4178 1.00127 0.500637 0.865657i $$-0.333099\pi$$
0.500637 + 0.865657i $$0.333099\pi$$
$$548$$ −27.2580 −1.16440
$$549$$ 0 0
$$550$$ 24.2939 1.03590
$$551$$ 1.01118 0.0430776
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 34.9446 1.48465
$$555$$ 0 0
$$556$$ 15.7417 0.667598
$$557$$ 6.91486 0.292992 0.146496 0.989211i $$-0.453200\pi$$
0.146496 + 0.989211i $$0.453200\pi$$
$$558$$ 0 0
$$559$$ −34.5467 −1.46117
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −52.8255 −2.22831
$$563$$ 11.8381 0.498914 0.249457 0.968386i $$-0.419748\pi$$
0.249457 + 0.968386i $$0.419748\pi$$
$$564$$ 0 0
$$565$$ 32.2968 1.35874
$$566$$ −57.8503 −2.43163
$$567$$ 0 0
$$568$$ −2.46705 −0.103515
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −15.2252 −0.637154 −0.318577 0.947897i $$-0.603205\pi$$
−0.318577 + 0.947897i $$0.603205\pi$$
$$572$$ 5.53996 0.231637
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 23.4835 0.979328
$$576$$ 0 0
$$577$$ −17.3215 −0.721104 −0.360552 0.932739i $$-0.617412\pi$$
−0.360552 + 0.932739i $$0.617412\pi$$
$$578$$ 58.0312 2.41378
$$579$$ 0 0
$$580$$ 23.2098 0.963736
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −0.508203 −0.0210476
$$584$$ −2.37384 −0.0982302
$$585$$ 0 0
$$586$$ −8.41188 −0.347492
$$587$$ 27.8678 1.15023 0.575113 0.818074i $$-0.304958\pi$$
0.575113 + 0.818074i $$0.304958\pi$$
$$588$$ 0 0
$$589$$ −2.94244 −0.121241
$$590$$ −57.1166 −2.35145
$$591$$ 0 0
$$592$$ 33.5592 1.37927
$$593$$ −24.3463 −0.999781 −0.499890 0.866089i $$-0.666626\pi$$
−0.499890 + 0.866089i $$0.666626\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 9.08904 0.372302
$$597$$ 0 0
$$598$$ 11.4897 0.469848
$$599$$ −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 $$-0.814616\pi$$
−0.835145 + 0.550030i $$0.814616\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 10.9117 0.443993
$$605$$ 4.18953 0.170329
$$606$$ 0 0
$$607$$ 14.0276 0.569362 0.284681 0.958622i $$-0.408112\pi$$
0.284681 + 0.958622i $$0.408112\pi$$
$$608$$ −2.42723 −0.0984371
$$609$$ 0 0
$$610$$ 16.2171 0.656612
$$611$$ −25.5767 −1.03472
$$612$$ 0 0
$$613$$ 18.5306 0.748442 0.374221 0.927339i $$-0.377910\pi$$
0.374221 + 0.927339i $$0.377910\pi$$
$$614$$ 40.1760 1.62137
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.43424 −0.0979987 −0.0489994 0.998799i $$-0.515603\pi$$
−0.0489994 + 0.998799i $$0.515603\pi$$
$$618$$ 0 0
$$619$$ 30.4259 1.22292 0.611460 0.791275i $$-0.290582\pi$$
0.611460 + 0.791275i $$0.290582\pi$$
$$620$$ −67.5386 −2.71242
$$621$$ 0 0
$$622$$ 15.4835 0.620830
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 69.7966 2.79187
$$626$$ −15.7334 −0.628833
$$627$$ 0 0
$$628$$ 31.6371 1.26246
$$629$$ −51.7662 −2.06405
$$630$$ 0 0
$$631$$ 34.9836 1.39267 0.696337 0.717715i $$-0.254812\pi$$
0.696337 + 0.717715i $$0.254812\pi$$
$$632$$ −2.46705 −0.0981341
$$633$$ 0 0
$$634$$ 38.6925 1.53668
$$635$$ −23.0081 −0.913047
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 6.14137 0.243139
$$639$$ 0 0
$$640$$ 16.3502 0.646297
$$641$$ −8.56337 −0.338233 −0.169116 0.985596i $$-0.554091\pi$$
−0.169116 + 0.985596i $$0.554091\pi$$
$$642$$ 0 0
$$643$$ 5.11273 0.201626 0.100813 0.994905i $$-0.467856\pi$$
0.100813 + 0.994905i $$0.467856\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 4.22758 0.166332
$$647$$ 25.7693 1.01310 0.506548 0.862212i $$-0.330921\pi$$
0.506548 + 0.862212i $$0.330921\pi$$
$$648$$ 0 0
$$649$$ −7.04399 −0.276501
$$650$$ 77.0877 3.02363
$$651$$ 0 0
$$652$$ −4.65269 −0.182213
$$653$$ 47.8953 1.87429 0.937145 0.348941i $$-0.113459\pi$$
0.937145 + 0.348941i $$0.113459\pi$$
$$654$$ 0 0
$$655$$ 16.7581 0.654795
$$656$$ −41.6043 −1.62437
$$657$$ 0 0
$$658$$ 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 $$-0.432647\pi$$
0.210022 + 0.977697i $$0.432647\pi$$
$$662$$ 32.9341 1.28002
$$663$$ 0 0
$$664$$ −1.73339 −0.0672686
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.93649 0.229862
$$668$$ −19.4590 −0.752891
$$669$$ 0 0
$$670$$ −21.6087 −0.834817
$$671$$ 2.00000 0.0772091
$$672$$ 0 0
$$673$$ 32.2499 1.24314 0.621572 0.783357i $$-0.286494\pi$$
0.621572 + 0.783357i $$0.286494\pi$$
$$674$$ 2.95078 0.113660
$$675$$ 0 0
$$676$$ −5.11769 −0.196834
$$677$$ −27.7089 −1.06494 −0.532470 0.846449i $$-0.678736\pi$$
−0.532470 + 0.846449i $$0.678736\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −14.1229 −0.541589
$$681$$ 0 0
$$682$$ −17.8709 −0.684311
$$683$$ −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
$$687$$ 0 0
$$688$$ 48.3791 1.84443
$$689$$ −1.61259 −0.0614349
$$690$$ 0 0
$$691$$ −25.1372 −0.956264 −0.478132 0.878288i $$-0.658686\pi$$
−0.478132 + 0.878288i $$0.658686\pi$$
$$692$$ −43.3788 −1.64901
$$693$$ 0 0
$$694$$ −33.7745 −1.28206
$$695$$ 37.7745 1.43287
$$696$$ 0 0
$$697$$ 64.1760 2.43084
$$698$$ −13.2765 −0.502521
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19.2747 0.727995 0.363997 0.931400i $$-0.381412\pi$$
0.363997 + 0.931400i $$0.381412\pi$$
$$702$$ 0 0
$$703$$ −2.40665 −0.0907686
$$704$$ −5.85446 −0.220648
$$705$$ 0 0
$$706$$ −61.8008 −2.32590
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 9.76098 0.366581 0.183291 0.983059i $$-0.441325\pi$$
0.183291 + 0.983059i $$0.441325\pi$$
$$710$$ 40.6758 1.52654
$$711$$ 0 0
$$712$$ 0.856577 0.0321016
$$713$$ −17.2747 −0.646942
$$714$$ 0 0
$$715$$ 13.2939 0.497165
$$716$$ 22.2744 0.832434
$$717$$ 0 0
$$718$$ 47.2908 1.76488
$$719$$ 14.0276 0.523141 0.261570 0.965184i $$-0.415760\pi$$
0.261570 + 0.965184i $$0.415760\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −36.5767 −1.36124
$$723$$ 0 0
$$724$$ 5.64541 0.209810
$$725$$ 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
$$729$$ 0 0
$$730$$ 39.1390 1.44860
$$731$$ −74.6263 −2.76016
$$732$$ 0 0
$$733$$ 10.7581 0.397361 0.198680 0.980064i $$-0.436334\pi$$
0.198680 + 0.980064i $$0.436334\pi$$
$$734$$ −33.0409 −1.21956
$$735$$ 0 0
$$736$$ −14.2499 −0.525259
$$737$$ −2.66492 −0.0981637
$$738$$ 0 0
$$739$$ −6.38741 −0.234965 −0.117482 0.993075i $$-0.537482\pi$$
−0.117482 + 0.993075i $$0.537482\pi$$
$$740$$ −55.2405 −2.03068
$$741$$ 0 0
$$742$$ 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
$$746$$ 6.15149 0.225222
$$747$$ 0 0
$$748$$ 11.9672 0.437564
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 31.8678 1.16287 0.581435 0.813593i $$-0.302491\pi$$
0.581435 + 0.813593i $$0.302491\pi$$
$$752$$ 35.8175 1.30613
$$753$$ 0 0
$$754$$ 19.4874 0.709688
$$755$$ 26.1843 0.952944
$$756$$ 0 0
$$757$$ −48.3103 −1.75587 −0.877934 0.478781i $$-0.841079\pi$$
−0.877934 + 0.478781i $$0.841079\pi$$
$$758$$ 7.62483 0.276946
$$759$$ 0 0
$$760$$ −0.656586 −0.0238169
$$761$$ −17.8625 −0.647516 −0.323758 0.946140i $$-0.604946\pi$$
−0.323758 + 0.946140i $$0.604946\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −36.6207 −1.32489
$$765$$ 0 0
$$766$$ −47.9177 −1.73134
$$767$$ −22.3515 −0.807065
$$768$$ 0 0
$$769$$ 24.8820 0.897269 0.448635 0.893715i $$-0.351910\pi$$
0.448635 + 0.893715i $$0.351910\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0.436357 0.0157048
$$773$$ −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
$$776$$ −6.02448 −0.216266
$$777$$ 0 0
$$778$$ −7.07158 −0.253528
$$779$$ 2.98359 0.106898
$$780$$ 0 0
$$781$$ 5.01641 0.179501
$$782$$ 24.8195 0.887544
$$783$$ 0 0
$$784$$ 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 $$0.412883\pi$$
$$788$$ −32.2989 −1.15060
$$789$$ 0 0
$$790$$ 40.6758 1.44718
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6.34625 0.225362
$$794$$ −8.83210 −0.313440
$$795$$ 0 0
$$796$$ −17.2335 −0.610826
$$797$$ 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
$$800$$ −95.6071 −3.38022
$$801$$ 0 0
$$802$$ 14.0000 0.494357
$$803$$ 4.82687 0.170337
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −56.7065 −1.99740
$$807$$ 0 0
$$808$$ −2.40354 −0.0845564
$$809$$ −29.8433 −1.04923 −0.524617 0.851338i $$-0.675791\pi$$
−0.524617 + 0.851338i $$0.675791\pi$$
$$810$$ 0 0
$$811$$ 16.6321 0.584032 0.292016 0.956413i $$-0.405674\pi$$
0.292016 + 0.956413i $$0.405674\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −14.6168 −0.512317
$$815$$ −11.1648 −0.391086
$$816$$ 0 0
$$817$$ −3.46944 −0.121380
$$818$$ −10.2723 −0.359162
$$819$$ 0 0
$$820$$ 68.4832 2.39154
$$821$$ −30.3327 −1.05862 −0.529309 0.848429i $$-0.677549\pi$$
−0.529309 + 0.848429i $$0.677549\pi$$
$$822$$ 0 0
$$823$$ −18.4067 −0.641616 −0.320808 0.947144i $$-0.603954\pi$$
−0.320808 + 0.947144i $$0.603954\pi$$
$$824$$ −0.313441 −0.0109192
$$825$$ 0 0
$$826$$ 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 $$-0.385050\pi$$
0.353327 + 0.935500i $$0.385050\pi$$
$$830$$ 28.5795 0.992009
$$831$$ 0 0
$$832$$ −18.5769 −0.644040
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −46.6946 −1.61593
$$836$$ 0.556364 0.0192423
$$837$$ 0 0
$$838$$ −22.1086 −0.763728
$$839$$ 16.1812 0.558637 0.279318 0.960199i $$-0.409891\pi$$
0.279318 + 0.960199i $$0.409891\pi$$
$$840$$ 0 0
$$841$$ −18.9313 −0.652802
$$842$$ −53.0593 −1.82855
$$843$$ 0 0
$$844$$ −8.09632 −0.278687
$$845$$ −12.2806 −0.422466
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.25827 0.0775493
$$849$$ 0 0
$$850$$ 166.522 5.71165
$$851$$ −14.1291 −0.484341
$$852$$ 0 0
$$853$$ 20.9836 0.718465 0.359232 0.933248i $$-0.383039\pi$$
0.359232 + 0.933248i $$0.383039\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0.470162 0.0160698
$$857$$ 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 $$0.445611\pi$$
$$860$$ −79.6350 −2.71553
$$861$$ 0 0
$$862$$ 4.42412 0.150686
$$863$$ −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
$$866$$ −61.1044 −2.07641
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 5.01641 0.170170
$$870$$ 0 0
$$871$$ −8.45614 −0.286525
$$872$$ 3.74385 0.126783
$$873$$ 0 0
$$874$$ 1.15388 0.0390306
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −50.9893 −1.72179 −0.860893 0.508787i $$-0.830094\pi$$
−0.860893 + 0.508787i $$0.830094\pi$$
$$878$$ 35.8586 1.21017
$$879$$ 0 0
$$880$$ −18.6168 −0.627571
$$881$$ 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 $$-0.288710\pi$$
0.616104 + 0.787665i $$0.288710\pi$$
$$884$$ 37.9734 1.27718
$$885$$ 0 0
$$886$$ −55.0325 −1.84885
$$887$$ −48.2004 −1.61841 −0.809206 0.587525i $$-0.800103\pi$$
−0.809206 + 0.587525i $$0.800103\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −14.1229 −0.473401
$$891$$ 0 0
$$892$$ 32.0307 1.07247
$$893$$ −2.56860 −0.0859550
$$894$$ 0 0
$$895$$ 53.4506 1.78666
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 11.0060 0.367273
$$899$$ −29.2992 −0.977181
$$900$$ 0 0
$$901$$ −3.48346 −0.116051
$$902$$ 18.1208 0.603357
$$903$$ 0 0
$$904$$ −3.79122 −0.126094
$$905$$ 13.5470 0.450316
$$906$$ 0 0
$$907$$ 22.9013 0.760425 0.380212 0.924899i $$-0.375851\pi$$
0.380212 + 0.924899i $$0.375851\pi$$
$$908$$ 0.661814 0.0219631
$$909$$ 0 0
$$910$$ 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
$$914$$ −25.1762 −0.832756
$$915$$ 0 0
$$916$$ −43.5470 −1.43883
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −28.3379 −0.934782 −0.467391 0.884051i $$-0.654806\pi$$
−0.467391 + 0.884051i $$0.654806\pi$$
$$920$$ −3.85473 −0.127087
$$921$$ 0 0
$$922$$ −12.3463 −0.406602
$$923$$ 15.9177 0.523937
$$924$$ 0 0
$$925$$ −94.7966 −3.11689
$$926$$ 65.9851 2.16841
$$927$$ 0 0
$$928$$ −24.1690 −0.793385
$$929$$ 5.08514 0.166838 0.0834191 0.996515i $$-0.473416\pi$$
0.0834191 + 0.996515i $$0.473416\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −40.9424 −1.34111
$$933$$ 0 0
$$934$$ 35.1250 1.14932
$$935$$ 28.7170 0.939146
$$936$$ 0 0
$$937$$ −32.2088 −1.05222 −0.526108 0.850418i $$-0.676349\pi$$
−0.526108 + 0.850418i $$0.676349\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −58.9578 −1.92299
$$941$$ 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
$$944$$ 31.3009 1.01876
$$945$$ 0 0
$$946$$ −21.0716 −0.685096
$$947$$ 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
$$950$$ 7.74173 0.251175
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 24.6894 0.799768 0.399884 0.916566i $$-0.369050\pi$$
0.399884 + 0.916566i $$0.369050\pi$$
$$954$$ 0 0
$$955$$ −87.8765 −2.84362
$$956$$ −8.86359 −0.286669
$$957$$ 0 0
$$958$$ −82.7393 −2.67319
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 54.2580 1.75026
$$962$$ −46.3808 −1.49538
$$963$$ 0 0
$$964$$ −33.6852 −1.08493
$$965$$ 1.04710 0.0337074
$$966$$ 0 0
$$967$$ 21.3298 0.685922 0.342961 0.939350i $$-0.388570\pi$$
0.342961 + 0.939350i $$0.388570\pi$$
$$968$$ −0.491797 −0.0158069
$$969$$ 0 0
$$970$$ 99.3293 3.18927
$$971$$ 28.4946 0.914436 0.457218 0.889355i $$-0.348846\pi$$
0.457218 + 0.889355i $$0.348846\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 51.8521 1.66145
$$975$$ 0 0
$$976$$ −8.88727 −0.284475
$$977$$ 19.8157 0.633960 0.316980 0.948432i $$-0.397331\pi$$
0.316980 + 0.948432i $$0.397331\pi$$
$$978$$ 0 0
$$979$$ −1.74173 −0.0556659
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −61.3171 −1.95671
$$983$$ −53.7745 −1.71514 −0.857571 0.514366i $$-0.828027\pi$$
−0.857571 + 0.514366i $$0.828027\pi$$
$$984$$ 0 0
$$985$$ −77.5058 −2.46954
$$986$$ 42.0958 1.34060
$$987$$ 0 0
$$988$$ 1.76541 0.0561653
$$989$$ −20.3686 −0.647684
$$990$$ 0 0
$$991$$ 49.7693 1.58097 0.790487 0.612479i $$-0.209827\pi$$
0.790487 + 0.612479i $$0.209827\pi$$
$$992$$ 70.3296 2.23297
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −41.3543 −1.31102
$$996$$ 0 0
$$997$$ −0.659696 −0.0208928 −0.0104464 0.999945i $$-0.503325\pi$$
−0.0104464 + 0.999945i $$0.503325\pi$$
$$998$$ −46.6943 −1.47808
$$999$$ 0 0
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 4851.2.a.bi.1.3 3
3.2 odd 2 1617.2.a.t.1.1 3
7.6 odd 2 693.2.a.l.1.3 3
21.20 even 2 231.2.a.e.1.1 3
77.76 even 2 7623.2.a.cd.1.1 3
84.83 odd 2 3696.2.a.bo.1.3 3
105.104 even 2 5775.2.a.bp.1.3 3
231.230 odd 2 2541.2.a.bg.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.1 3 21.20 even 2
693.2.a.l.1.3 3 7.6 odd 2
1617.2.a.t.1.1 3 3.2 odd 2
2541.2.a.bg.1.3 3 231.230 odd 2
3696.2.a.bo.1.3 3 84.83 odd 2
4851.2.a.bi.1.3 3 1.1 even 1 trivial
5775.2.a.bp.1.3 3 105.104 even 2
7623.2.a.cd.1.1 3 77.76 even 2