# Properties

 Label 171.2.a.e.1.2 Level $171$ Weight $2$ Character 171.1 Self dual yes Analytic conductor $1.365$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} - x + 1$$ x^4 - x^3 - 6*x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.548230$$ of defining polynomial Character $$\chi$$ $$=$$ 171.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.27582 q^{2} -0.372281 q^{4} -2.15121 q^{5} +3.37228 q^{7} +3.02661 q^{8} +O(q^{10})$$ $$q-1.27582 q^{2} -0.372281 q^{4} -2.15121 q^{5} +3.37228 q^{7} +3.02661 q^{8} +2.74456 q^{10} +4.70285 q^{11} +2.00000 q^{13} -4.30243 q^{14} -3.11684 q^{16} -2.15121 q^{17} +1.00000 q^{19} +0.800857 q^{20} -6.00000 q^{22} +6.85407 q^{23} -0.372281 q^{25} -2.55164 q^{26} -1.25544 q^{28} +6.85407 q^{29} -6.74456 q^{31} -2.07668 q^{32} +2.74456 q^{34} -7.25450 q^{35} -0.744563 q^{37} -1.27582 q^{38} -6.51087 q^{40} +2.55164 q^{41} +6.11684 q^{43} -1.75079 q^{44} -8.74456 q^{46} -9.00528 q^{47} +4.37228 q^{49} +0.474964 q^{50} -0.744563 q^{52} -11.9574 q^{53} -10.1168 q^{55} +10.2066 q^{56} -8.74456 q^{58} -5.10328 q^{59} +12.1168 q^{61} +8.60485 q^{62} +8.88316 q^{64} -4.30243 q^{65} -4.00000 q^{67} +0.800857 q^{68} +9.25544 q^{70} +13.7081 q^{71} +12.1168 q^{73} +0.949929 q^{74} -0.372281 q^{76} +15.8593 q^{77} -4.00000 q^{79} +6.70500 q^{80} -3.25544 q^{82} -1.75079 q^{83} +4.62772 q^{85} -7.80400 q^{86} +14.2337 q^{88} -11.9574 q^{89} +6.74456 q^{91} -2.55164 q^{92} +11.4891 q^{94} -2.15121 q^{95} -15.4891 q^{97} -5.57825 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{4} + 2 q^{7}+O(q^{10})$$ 4 * q + 10 * q^4 + 2 * q^7 $$4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100})$$ 4 * q + 10 * q^4 + 2 * q^7 - 12 * q^10 + 8 * q^13 + 22 * q^16 + 4 * q^19 - 24 * q^22 + 10 * q^25 - 28 * q^28 - 4 * q^31 - 12 * q^34 + 20 * q^37 - 72 * q^40 - 10 * q^43 - 12 * q^46 + 6 * q^49 + 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 - 16 * q^67 + 60 * q^70 + 14 * q^73 + 10 * q^76 - 16 * q^79 - 36 * q^82 + 30 * q^85 - 12 * q^88 + 4 * q^91 - 16 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.27582 −0.902142 −0.451071 0.892488i $$-0.648958\pi$$
−0.451071 + 0.892488i $$0.648958\pi$$
$$3$$ 0 0
$$4$$ −0.372281 −0.186141
$$5$$ −2.15121 −0.962052 −0.481026 0.876706i $$-0.659736\pi$$
−0.481026 + 0.876706i $$0.659736\pi$$
$$6$$ 0 0
$$7$$ 3.37228 1.27460 0.637301 0.770615i $$-0.280051\pi$$
0.637301 + 0.770615i $$0.280051\pi$$
$$8$$ 3.02661 1.07007
$$9$$ 0 0
$$10$$ 2.74456 0.867907
$$11$$ 4.70285 1.41796 0.708982 0.705227i $$-0.249155\pi$$
0.708982 + 0.705227i $$0.249155\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −4.30243 −1.14987
$$15$$ 0 0
$$16$$ −3.11684 −0.779211
$$17$$ −2.15121 −0.521746 −0.260873 0.965373i $$-0.584010\pi$$
−0.260873 + 0.965373i $$0.584010\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0.800857 0.179077
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 6.85407 1.42917 0.714586 0.699548i $$-0.246615\pi$$
0.714586 + 0.699548i $$0.246615\pi$$
$$24$$ 0 0
$$25$$ −0.372281 −0.0744563
$$26$$ −2.55164 −0.500418
$$27$$ 0 0
$$28$$ −1.25544 −0.237255
$$29$$ 6.85407 1.27277 0.636384 0.771372i $$-0.280429\pi$$
0.636384 + 0.771372i $$0.280429\pi$$
$$30$$ 0 0
$$31$$ −6.74456 −1.21136 −0.605680 0.795709i $$-0.707099\pi$$
−0.605680 + 0.795709i $$0.707099\pi$$
$$32$$ −2.07668 −0.367108
$$33$$ 0 0
$$34$$ 2.74456 0.470689
$$35$$ −7.25450 −1.22623
$$36$$ 0 0
$$37$$ −0.744563 −0.122405 −0.0612027 0.998125i $$-0.519494\pi$$
−0.0612027 + 0.998125i $$0.519494\pi$$
$$38$$ −1.27582 −0.206965
$$39$$ 0 0
$$40$$ −6.51087 −1.02946
$$41$$ 2.55164 0.398499 0.199250 0.979949i $$-0.436150\pi$$
0.199250 + 0.979949i $$0.436150\pi$$
$$42$$ 0 0
$$43$$ 6.11684 0.932810 0.466405 0.884571i $$-0.345549\pi$$
0.466405 + 0.884571i $$0.345549\pi$$
$$44$$ −1.75079 −0.263941
$$45$$ 0 0
$$46$$ −8.74456 −1.28932
$$47$$ −9.00528 −1.31356 −0.656778 0.754084i $$-0.728081\pi$$
−0.656778 + 0.754084i $$0.728081\pi$$
$$48$$ 0 0
$$49$$ 4.37228 0.624612
$$50$$ 0.474964 0.0671701
$$51$$ 0 0
$$52$$ −0.744563 −0.103252
$$53$$ −11.9574 −1.64247 −0.821234 0.570591i $$-0.806714\pi$$
−0.821234 + 0.570591i $$0.806714\pi$$
$$54$$ 0 0
$$55$$ −10.1168 −1.36415
$$56$$ 10.2066 1.36391
$$57$$ 0 0
$$58$$ −8.74456 −1.14822
$$59$$ −5.10328 −0.664391 −0.332195 0.943211i $$-0.607789\pi$$
−0.332195 + 0.943211i $$0.607789\pi$$
$$60$$ 0 0
$$61$$ 12.1168 1.55140 0.775701 0.631100i $$-0.217396\pi$$
0.775701 + 0.631100i $$0.217396\pi$$
$$62$$ 8.60485 1.09282
$$63$$ 0 0
$$64$$ 8.88316 1.11039
$$65$$ −4.30243 −0.533650
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0.800857 0.0971181
$$69$$ 0 0
$$70$$ 9.25544 1.10624
$$71$$ 13.7081 1.62686 0.813428 0.581665i $$-0.197599\pi$$
0.813428 + 0.581665i $$0.197599\pi$$
$$72$$ 0 0
$$73$$ 12.1168 1.41817 0.709085 0.705123i $$-0.249108\pi$$
0.709085 + 0.705123i $$0.249108\pi$$
$$74$$ 0.949929 0.110427
$$75$$ 0 0
$$76$$ −0.372281 −0.0427036
$$77$$ 15.8593 1.80734
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 6.70500 0.749641
$$81$$ 0 0
$$82$$ −3.25544 −0.359503
$$83$$ −1.75079 −0.192174 −0.0960868 0.995373i $$-0.530633\pi$$
−0.0960868 + 0.995373i $$0.530633\pi$$
$$84$$ 0 0
$$85$$ 4.62772 0.501947
$$86$$ −7.80400 −0.841527
$$87$$ 0 0
$$88$$ 14.2337 1.51732
$$89$$ −11.9574 −1.26748 −0.633738 0.773547i $$-0.718480\pi$$
−0.633738 + 0.773547i $$0.718480\pi$$
$$90$$ 0 0
$$91$$ 6.74456 0.707022
$$92$$ −2.55164 −0.266027
$$93$$ 0 0
$$94$$ 11.4891 1.18501
$$95$$ −2.15121 −0.220710
$$96$$ 0 0
$$97$$ −15.4891 −1.57268 −0.786341 0.617792i $$-0.788027\pi$$
−0.786341 + 0.617792i $$0.788027\pi$$
$$98$$ −5.57825 −0.563488
$$99$$ 0 0
$$100$$ 0.138593 0.0138593
$$101$$ 8.60485 0.856215 0.428107 0.903728i $$-0.359180\pi$$
0.428107 + 0.903728i $$0.359180\pi$$
$$102$$ 0 0
$$103$$ −18.7446 −1.84696 −0.923478 0.383651i $$-0.874667\pi$$
−0.923478 + 0.383651i $$0.874667\pi$$
$$104$$ 6.05321 0.593566
$$105$$ 0 0
$$106$$ 15.2554 1.48174
$$107$$ −9.40571 −0.909284 −0.454642 0.890674i $$-0.650233\pi$$
−0.454642 + 0.890674i $$0.650233\pi$$
$$108$$ 0 0
$$109$$ 16.7446 1.60384 0.801919 0.597433i $$-0.203812\pi$$
0.801919 + 0.597433i $$0.203812\pi$$
$$110$$ 12.9073 1.23066
$$111$$ 0 0
$$112$$ −10.5109 −0.993184
$$113$$ −1.75079 −0.164700 −0.0823500 0.996603i $$-0.526243\pi$$
−0.0823500 + 0.996603i $$0.526243\pi$$
$$114$$ 0 0
$$115$$ −14.7446 −1.37494
$$116$$ −2.55164 −0.236914
$$117$$ 0 0
$$118$$ 6.51087 0.599375
$$119$$ −7.25450 −0.665019
$$120$$ 0 0
$$121$$ 11.1168 1.01062
$$122$$ −15.4589 −1.39958
$$123$$ 0 0
$$124$$ 2.51087 0.225483
$$125$$ 11.5569 1.03368
$$126$$ 0 0
$$127$$ −1.25544 −0.111402 −0.0557010 0.998447i $$-0.517739\pi$$
−0.0557010 + 0.998447i $$0.517739\pi$$
$$128$$ −7.17996 −0.634625
$$129$$ 0 0
$$130$$ 5.48913 0.481428
$$131$$ −3.90200 −0.340919 −0.170460 0.985365i $$-0.554525\pi$$
−0.170460 + 0.985365i $$0.554525\pi$$
$$132$$ 0 0
$$133$$ 3.37228 0.292414
$$134$$ 5.10328 0.440857
$$135$$ 0 0
$$136$$ −6.51087 −0.558303
$$137$$ −16.6602 −1.42338 −0.711689 0.702495i $$-0.752069\pi$$
−0.711689 + 0.702495i $$0.752069\pi$$
$$138$$ 0 0
$$139$$ −14.1168 −1.19738 −0.598688 0.800983i $$-0.704311\pi$$
−0.598688 + 0.800983i $$0.704311\pi$$
$$140$$ 2.70071 0.228252
$$141$$ 0 0
$$142$$ −17.4891 −1.46765
$$143$$ 9.40571 0.786545
$$144$$ 0 0
$$145$$ −14.7446 −1.22447
$$146$$ −15.4589 −1.27939
$$147$$ 0 0
$$148$$ 0.277187 0.0227846
$$149$$ 1.35036 0.110626 0.0553128 0.998469i $$-0.482384\pi$$
0.0553128 + 0.998469i $$0.482384\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 3.02661 0.245490
$$153$$ 0 0
$$154$$ −20.2337 −1.63048
$$155$$ 14.5090 1.16539
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 5.10328 0.405995
$$159$$ 0 0
$$160$$ 4.46738 0.353177
$$161$$ 23.1138 1.82163
$$162$$ 0 0
$$163$$ 1.48913 0.116637 0.0583186 0.998298i $$-0.481426\pi$$
0.0583186 + 0.998298i $$0.481426\pi$$
$$164$$ −0.949929 −0.0741770
$$165$$ 0 0
$$166$$ 2.23369 0.173368
$$167$$ −4.30243 −0.332932 −0.166466 0.986047i $$-0.553236\pi$$
−0.166466 + 0.986047i $$0.553236\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −5.90414 −0.452827
$$171$$ 0 0
$$172$$ −2.27719 −0.173634
$$173$$ −6.85407 −0.521105 −0.260553 0.965460i $$-0.583905\pi$$
−0.260553 + 0.965460i $$0.583905\pi$$
$$174$$ 0 0
$$175$$ −1.25544 −0.0949021
$$176$$ −14.6581 −1.10489
$$177$$ 0 0
$$178$$ 15.2554 1.14344
$$179$$ 9.40571 0.703016 0.351508 0.936185i $$-0.385669\pi$$
0.351508 + 0.936185i $$0.385669\pi$$
$$180$$ 0 0
$$181$$ −3.48913 −0.259345 −0.129672 0.991557i $$-0.541393\pi$$
−0.129672 + 0.991557i $$0.541393\pi$$
$$182$$ −8.60485 −0.637834
$$183$$ 0 0
$$184$$ 20.7446 1.52931
$$185$$ 1.60171 0.117760
$$186$$ 0 0
$$187$$ −10.1168 −0.739817
$$188$$ 3.35250 0.244506
$$189$$ 0 0
$$190$$ 2.74456 0.199112
$$191$$ −8.20442 −0.593651 −0.296826 0.954932i $$-0.595928\pi$$
−0.296826 + 0.954932i $$0.595928\pi$$
$$192$$ 0 0
$$193$$ −18.2337 −1.31249 −0.656245 0.754548i $$-0.727856\pi$$
−0.656245 + 0.754548i $$0.727856\pi$$
$$194$$ 19.7613 1.41878
$$195$$ 0 0
$$196$$ −1.62772 −0.116266
$$197$$ −3.50157 −0.249477 −0.124738 0.992190i $$-0.539809\pi$$
−0.124738 + 0.992190i $$0.539809\pi$$
$$198$$ 0 0
$$199$$ 18.1168 1.28427 0.642135 0.766592i $$-0.278049\pi$$
0.642135 + 0.766592i $$0.278049\pi$$
$$200$$ −1.12675 −0.0796732
$$201$$ 0 0
$$202$$ −10.9783 −0.772427
$$203$$ 23.1138 1.62227
$$204$$ 0 0
$$205$$ −5.48913 −0.383377
$$206$$ 23.9147 1.66622
$$207$$ 0 0
$$208$$ −6.23369 −0.432228
$$209$$ 4.70285 0.325303
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 4.45150 0.305730
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −13.1586 −0.897412
$$216$$ 0 0
$$217$$ −22.7446 −1.54400
$$218$$ −21.3631 −1.44689
$$219$$ 0 0
$$220$$ 3.76631 0.253925
$$221$$ −4.30243 −0.289413
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −7.00314 −0.467917
$$225$$ 0 0
$$226$$ 2.23369 0.148583
$$227$$ 12.9073 0.856686 0.428343 0.903616i $$-0.359097\pi$$
0.428343 + 0.903616i $$0.359097\pi$$
$$228$$ 0 0
$$229$$ 21.3723 1.41232 0.706160 0.708052i $$-0.250426\pi$$
0.706160 + 0.708052i $$0.250426\pi$$
$$230$$ 18.8114 1.24039
$$231$$ 0 0
$$232$$ 20.7446 1.36195
$$233$$ 7.25450 0.475258 0.237629 0.971356i $$-0.423630\pi$$
0.237629 + 0.971356i $$0.423630\pi$$
$$234$$ 0 0
$$235$$ 19.3723 1.26371
$$236$$ 1.89986 0.123670
$$237$$ 0 0
$$238$$ 9.25544 0.599941
$$239$$ −27.0158 −1.74751 −0.873755 0.486367i $$-0.838322\pi$$
−0.873755 + 0.486367i $$0.838322\pi$$
$$240$$ 0 0
$$241$$ 10.2337 0.659210 0.329605 0.944119i $$-0.393084\pi$$
0.329605 + 0.944119i $$0.393084\pi$$
$$242$$ −14.1831 −0.911724
$$243$$ 0 0
$$244$$ −4.51087 −0.288779
$$245$$ −9.40571 −0.600909
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ −20.4131 −1.29624
$$249$$ 0 0
$$250$$ −14.7446 −0.932528
$$251$$ −14.9094 −0.941074 −0.470537 0.882380i $$-0.655940\pi$$
−0.470537 + 0.882380i $$0.655940\pi$$
$$252$$ 0 0
$$253$$ 32.2337 2.02651
$$254$$ 1.60171 0.100500
$$255$$ 0 0
$$256$$ −8.60597 −0.537873
$$257$$ −2.55164 −0.159167 −0.0795835 0.996828i $$-0.525359\pi$$
−0.0795835 + 0.996828i $$0.525359\pi$$
$$258$$ 0 0
$$259$$ −2.51087 −0.156018
$$260$$ 1.60171 0.0993340
$$261$$ 0 0
$$262$$ 4.97825 0.307557
$$263$$ −9.80614 −0.604672 −0.302336 0.953201i $$-0.597767\pi$$
−0.302336 + 0.953201i $$0.597767\pi$$
$$264$$ 0 0
$$265$$ 25.7228 1.58014
$$266$$ −4.30243 −0.263799
$$267$$ 0 0
$$268$$ 1.48913 0.0909628
$$269$$ −25.6655 −1.56485 −0.782426 0.622743i $$-0.786018\pi$$
−0.782426 + 0.622743i $$0.786018\pi$$
$$270$$ 0 0
$$271$$ 2.51087 0.152525 0.0762624 0.997088i $$-0.475701\pi$$
0.0762624 + 0.997088i $$0.475701\pi$$
$$272$$ 6.70500 0.406550
$$273$$ 0 0
$$274$$ 21.2554 1.28409
$$275$$ −1.75079 −0.105576
$$276$$ 0 0
$$277$$ −8.11684 −0.487694 −0.243847 0.969814i $$-0.578409\pi$$
−0.243847 + 0.969814i $$0.578409\pi$$
$$278$$ 18.0106 1.08020
$$279$$ 0 0
$$280$$ −21.9565 −1.31215
$$281$$ 10.3556 0.617766 0.308883 0.951100i $$-0.400045\pi$$
0.308883 + 0.951100i $$0.400045\pi$$
$$282$$ 0 0
$$283$$ 0.627719 0.0373140 0.0186570 0.999826i $$-0.494061\pi$$
0.0186570 + 0.999826i $$0.494061\pi$$
$$284$$ −5.10328 −0.302824
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 8.60485 0.507928
$$288$$ 0 0
$$289$$ −12.3723 −0.727781
$$290$$ 18.8114 1.10464
$$291$$ 0 0
$$292$$ −4.51087 −0.263979
$$293$$ 26.4663 1.54618 0.773090 0.634296i $$-0.218710\pi$$
0.773090 + 0.634296i $$0.218710\pi$$
$$294$$ 0 0
$$295$$ 10.9783 0.639178
$$296$$ −2.25350 −0.130982
$$297$$ 0 0
$$298$$ −1.72281 −0.0997999
$$299$$ 13.7081 0.792762
$$300$$ 0 0
$$301$$ 20.6277 1.18896
$$302$$ 5.10328 0.293661
$$303$$ 0 0
$$304$$ −3.11684 −0.178763
$$305$$ −26.0659 −1.49253
$$306$$ 0 0
$$307$$ 2.51087 0.143303 0.0716516 0.997430i $$-0.477173\pi$$
0.0716516 + 0.997430i $$0.477173\pi$$
$$308$$ −5.90414 −0.336420
$$309$$ 0 0
$$310$$ −18.5109 −1.05135
$$311$$ 27.0158 1.53193 0.765964 0.642883i $$-0.222262\pi$$
0.765964 + 0.642883i $$0.222262\pi$$
$$312$$ 0 0
$$313$$ −15.4891 −0.875497 −0.437749 0.899097i $$-0.644224\pi$$
−0.437749 + 0.899097i $$0.644224\pi$$
$$314$$ −2.55164 −0.143997
$$315$$ 0 0
$$316$$ 1.48913 0.0837698
$$317$$ 35.0712 1.96979 0.984897 0.173139i $$-0.0553911\pi$$
0.984897 + 0.173139i $$0.0553911\pi$$
$$318$$ 0 0
$$319$$ 32.2337 1.80474
$$320$$ −19.1096 −1.06826
$$321$$ 0 0
$$322$$ −29.4891 −1.64336
$$323$$ −2.15121 −0.119697
$$324$$ 0 0
$$325$$ −0.744563 −0.0413009
$$326$$ −1.89986 −0.105223
$$327$$ 0 0
$$328$$ 7.72281 0.426421
$$329$$ −30.3683 −1.67426
$$330$$ 0 0
$$331$$ −26.9783 −1.48286 −0.741429 0.671031i $$-0.765852\pi$$
−0.741429 + 0.671031i $$0.765852\pi$$
$$332$$ 0.651785 0.0357713
$$333$$ 0 0
$$334$$ 5.48913 0.300352
$$335$$ 8.60485 0.470133
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 11.4824 0.624560
$$339$$ 0 0
$$340$$ −1.72281 −0.0934327
$$341$$ −31.7187 −1.71766
$$342$$ 0 0
$$343$$ −8.86141 −0.478471
$$344$$ 18.5133 0.998169
$$345$$ 0 0
$$346$$ 8.74456 0.470111
$$347$$ 3.10114 0.166478 0.0832390 0.996530i $$-0.473474\pi$$
0.0832390 + 0.996530i $$0.473474\pi$$
$$348$$ 0 0
$$349$$ −10.8614 −0.581398 −0.290699 0.956815i $$-0.593888\pi$$
−0.290699 + 0.956815i $$0.593888\pi$$
$$350$$ 1.60171 0.0856152
$$351$$ 0 0
$$352$$ −9.76631 −0.520546
$$353$$ 22.3130 1.18760 0.593800 0.804612i $$-0.297627\pi$$
0.593800 + 0.804612i $$0.297627\pi$$
$$354$$ 0 0
$$355$$ −29.4891 −1.56512
$$356$$ 4.45150 0.235929
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 27.8167 1.46811 0.734055 0.679090i $$-0.237626\pi$$
0.734055 + 0.679090i $$0.237626\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 4.45150 0.233966
$$363$$ 0 0
$$364$$ −2.51087 −0.131606
$$365$$ −26.0659 −1.36435
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −21.3631 −1.11363
$$369$$ 0 0
$$370$$ −2.04350 −0.106236
$$371$$ −40.3236 −2.09349
$$372$$ 0 0
$$373$$ 10.2337 0.529880 0.264940 0.964265i $$-0.414648\pi$$
0.264940 + 0.964265i $$0.414648\pi$$
$$374$$ 12.9073 0.667420
$$375$$ 0 0
$$376$$ −27.2554 −1.40559
$$377$$ 13.7081 0.706005
$$378$$ 0 0
$$379$$ 17.2554 0.886352 0.443176 0.896435i $$-0.353852\pi$$
0.443176 + 0.896435i $$0.353852\pi$$
$$380$$ 0.800857 0.0410831
$$381$$ 0 0
$$382$$ 10.4674 0.535558
$$383$$ 0.800857 0.0409219 0.0204609 0.999791i $$-0.493487\pi$$
0.0204609 + 0.999791i $$0.493487\pi$$
$$384$$ 0 0
$$385$$ −34.1168 −1.73876
$$386$$ 23.2629 1.18405
$$387$$ 0 0
$$388$$ 5.76631 0.292740
$$389$$ −16.6602 −0.844706 −0.422353 0.906431i $$-0.638796\pi$$
−0.422353 + 0.906431i $$0.638796\pi$$
$$390$$ 0 0
$$391$$ −14.7446 −0.745665
$$392$$ 13.2332 0.668376
$$393$$ 0 0
$$394$$ 4.46738 0.225063
$$395$$ 8.60485 0.432957
$$396$$ 0 0
$$397$$ 0.116844 0.00586423 0.00293212 0.999996i $$-0.499067\pi$$
0.00293212 + 0.999996i $$0.499067\pi$$
$$398$$ −23.1138 −1.15859
$$399$$ 0 0
$$400$$ 1.16034 0.0580171
$$401$$ −16.2598 −0.811975 −0.405987 0.913879i $$-0.633072\pi$$
−0.405987 + 0.913879i $$0.633072\pi$$
$$402$$ 0 0
$$403$$ −13.4891 −0.671941
$$404$$ −3.20343 −0.159376
$$405$$ 0 0
$$406$$ −29.4891 −1.46352
$$407$$ −3.50157 −0.173566
$$408$$ 0 0
$$409$$ −15.4891 −0.765888 −0.382944 0.923772i $$-0.625090\pi$$
−0.382944 + 0.923772i $$0.625090\pi$$
$$410$$ 7.00314 0.345860
$$411$$ 0 0
$$412$$ 6.97825 0.343794
$$413$$ −17.2097 −0.846834
$$414$$ 0 0
$$415$$ 3.76631 0.184881
$$416$$ −4.15335 −0.203635
$$417$$ 0 0
$$418$$ −6.00000 −0.293470
$$419$$ 1.75079 0.0855314 0.0427657 0.999085i $$-0.486383\pi$$
0.0427657 + 0.999085i $$0.486383\pi$$
$$420$$ 0 0
$$421$$ 36.9783 1.80221 0.901105 0.433601i $$-0.142757\pi$$
0.901105 + 0.433601i $$0.142757\pi$$
$$422$$ 5.10328 0.248424
$$423$$ 0 0
$$424$$ −36.1902 −1.75755
$$425$$ 0.800857 0.0388472
$$426$$ 0 0
$$427$$ 40.8614 1.97742
$$428$$ 3.50157 0.169255
$$429$$ 0 0
$$430$$ 16.7881 0.809592
$$431$$ −7.80400 −0.375905 −0.187953 0.982178i $$-0.560185\pi$$
−0.187953 + 0.982178i $$0.560185\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 29.0180 1.39291
$$435$$ 0 0
$$436$$ −6.23369 −0.298540
$$437$$ 6.85407 0.327875
$$438$$ 0 0
$$439$$ 16.2337 0.774792 0.387396 0.921913i $$-0.373375\pi$$
0.387396 + 0.921913i $$0.373375\pi$$
$$440$$ −30.6197 −1.45974
$$441$$ 0 0
$$442$$ 5.48913 0.261091
$$443$$ −12.5069 −0.594218 −0.297109 0.954843i $$-0.596023\pi$$
−0.297109 + 0.954843i $$0.596023\pi$$
$$444$$ 0 0
$$445$$ 25.7228 1.21938
$$446$$ 5.10328 0.241647
$$447$$ 0 0
$$448$$ 29.9565 1.41531
$$449$$ 33.4695 1.57952 0.789761 0.613414i $$-0.210204\pi$$
0.789761 + 0.613414i $$0.210204\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 0.651785 0.0306574
$$453$$ 0 0
$$454$$ −16.4674 −0.772852
$$455$$ −14.5090 −0.680192
$$456$$ 0 0
$$457$$ −2.62772 −0.122919 −0.0614597 0.998110i $$-0.519576\pi$$
−0.0614597 + 0.998110i $$0.519576\pi$$
$$458$$ −27.2672 −1.27411
$$459$$ 0 0
$$460$$ 5.48913 0.255932
$$461$$ −5.65278 −0.263276 −0.131638 0.991298i $$-0.542024\pi$$
−0.131638 + 0.991298i $$0.542024\pi$$
$$462$$ 0 0
$$463$$ 3.37228 0.156723 0.0783616 0.996925i $$-0.475031\pi$$
0.0783616 + 0.996925i $$0.475031\pi$$
$$464$$ −21.3631 −0.991755
$$465$$ 0 0
$$466$$ −9.25544 −0.428750
$$467$$ −30.5174 −1.41218 −0.706089 0.708123i $$-0.749542\pi$$
−0.706089 + 0.708123i $$0.749542\pi$$
$$468$$ 0 0
$$469$$ −13.4891 −0.622870
$$470$$ −24.7156 −1.14004
$$471$$ 0 0
$$472$$ −15.4456 −0.710943
$$473$$ 28.7666 1.32269
$$474$$ 0 0
$$475$$ −0.372281 −0.0170814
$$476$$ 2.70071 0.123787
$$477$$ 0 0
$$478$$ 34.4674 1.57650
$$479$$ −8.45578 −0.386355 −0.193177 0.981164i $$-0.561879\pi$$
−0.193177 + 0.981164i $$0.561879\pi$$
$$480$$ 0 0
$$481$$ −1.48913 −0.0678983
$$482$$ −13.0564 −0.594701
$$483$$ 0 0
$$484$$ −4.13859 −0.188118
$$485$$ 33.3204 1.51300
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 36.6729 1.66010
$$489$$ 0 0
$$490$$ 12.0000 0.542105
$$491$$ −6.85407 −0.309320 −0.154660 0.987968i $$-0.549428\pi$$
−0.154660 + 0.987968i $$0.549428\pi$$
$$492$$ 0 0
$$493$$ −14.7446 −0.664062
$$494$$ −2.55164 −0.114804
$$495$$ 0 0
$$496$$ 21.0217 0.943904
$$497$$ 46.2277 2.07360
$$498$$ 0 0
$$499$$ 6.11684 0.273828 0.136914 0.990583i $$-0.456282\pi$$
0.136914 + 0.990583i $$0.456282\pi$$
$$500$$ −4.30243 −0.192410
$$501$$ 0 0
$$502$$ 19.0217 0.848982
$$503$$ 22.1639 0.988240 0.494120 0.869394i $$-0.335490\pi$$
0.494120 + 0.869394i $$0.335490\pi$$
$$504$$ 0 0
$$505$$ −18.5109 −0.823723
$$506$$ −41.1244 −1.82820
$$507$$ 0 0
$$508$$ 0.467376 0.0207365
$$509$$ 10.3556 0.459006 0.229503 0.973308i $$-0.426290\pi$$
0.229503 + 0.973308i $$0.426290\pi$$
$$510$$ 0 0
$$511$$ 40.8614 1.80760
$$512$$ 25.3396 1.11986
$$513$$ 0 0
$$514$$ 3.25544 0.143591
$$515$$ 40.3236 1.77687
$$516$$ 0 0
$$517$$ −42.3505 −1.86257
$$518$$ 3.20343 0.140750
$$519$$ 0 0
$$520$$ −13.0217 −0.571041
$$521$$ −7.65492 −0.335368 −0.167684 0.985841i $$-0.553629\pi$$
−0.167684 + 0.985841i $$0.553629\pi$$
$$522$$ 0 0
$$523$$ 16.2337 0.709850 0.354925 0.934895i $$-0.384506\pi$$
0.354925 + 0.934895i $$0.384506\pi$$
$$524$$ 1.45264 0.0634589
$$525$$ 0 0
$$526$$ 12.5109 0.545500
$$527$$ 14.5090 0.632022
$$528$$ 0 0
$$529$$ 23.9783 1.04253
$$530$$ −32.8177 −1.42551
$$531$$ 0 0
$$532$$ −1.25544 −0.0544301
$$533$$ 5.10328 0.221048
$$534$$ 0 0
$$535$$ 20.2337 0.874779
$$536$$ −12.1064 −0.522918
$$537$$ 0 0
$$538$$ 32.7446 1.41172
$$539$$ 20.5622 0.885677
$$540$$ 0 0
$$541$$ 3.88316 0.166950 0.0834750 0.996510i $$-0.473398\pi$$
0.0834750 + 0.996510i $$0.473398\pi$$
$$542$$ −3.20343 −0.137599
$$543$$ 0 0
$$544$$ 4.46738 0.191537
$$545$$ −36.0211 −1.54298
$$546$$ 0 0
$$547$$ −12.2337 −0.523075 −0.261537 0.965193i $$-0.584229\pi$$
−0.261537 + 0.965193i $$0.584229\pi$$
$$548$$ 6.20228 0.264948
$$549$$ 0 0
$$550$$ 2.23369 0.0952448
$$551$$ 6.85407 0.291993
$$552$$ 0 0
$$553$$ −13.4891 −0.573616
$$554$$ 10.3556 0.439969
$$555$$ 0 0
$$556$$ 5.25544 0.222880
$$557$$ −1.35036 −0.0572165 −0.0286082 0.999591i $$-0.509108\pi$$
−0.0286082 + 0.999591i $$0.509108\pi$$
$$558$$ 0 0
$$559$$ 12.2337 0.517430
$$560$$ 22.6111 0.955495
$$561$$ 0 0
$$562$$ −13.2119 −0.557312
$$563$$ −25.8146 −1.08795 −0.543977 0.839100i $$-0.683082\pi$$
−0.543977 + 0.839100i $$0.683082\pi$$
$$564$$ 0 0
$$565$$ 3.76631 0.158450
$$566$$ −0.800857 −0.0336625
$$567$$ 0 0
$$568$$ 41.4891 1.74085
$$569$$ 20.5622 0.862012 0.431006 0.902349i $$-0.358159\pi$$
0.431006 + 0.902349i $$0.358159\pi$$
$$570$$ 0 0
$$571$$ 25.4891 1.06669 0.533343 0.845899i $$-0.320935\pi$$
0.533343 + 0.845899i $$0.320935\pi$$
$$572$$ −3.50157 −0.146408
$$573$$ 0 0
$$574$$ −10.9783 −0.458223
$$575$$ −2.55164 −0.106411
$$576$$ 0 0
$$577$$ −23.8832 −0.994269 −0.497134 0.867674i $$-0.665614\pi$$
−0.497134 + 0.867674i $$0.665614\pi$$
$$578$$ 15.7848 0.656562
$$579$$ 0 0
$$580$$ 5.48913 0.227924
$$581$$ −5.90414 −0.244945
$$582$$ 0 0
$$583$$ −56.2337 −2.32896
$$584$$ 36.6729 1.51754
$$585$$ 0 0
$$586$$ −33.7663 −1.39487
$$587$$ −32.1191 −1.32570 −0.662849 0.748753i $$-0.730653\pi$$
−0.662849 + 0.748753i $$0.730653\pi$$
$$588$$ 0 0
$$589$$ −6.74456 −0.277905
$$590$$ −14.0063 −0.576629
$$591$$ 0 0
$$592$$ 2.32069 0.0953796
$$593$$ 27.4163 1.12585 0.562926 0.826508i $$-0.309676\pi$$
0.562926 + 0.826508i $$0.309676\pi$$
$$594$$ 0 0
$$595$$ 15.6060 0.639782
$$596$$ −0.502713 −0.0205919
$$597$$ 0 0
$$598$$ −17.4891 −0.715184
$$599$$ 8.60485 0.351585 0.175792 0.984427i $$-0.443751\pi$$
0.175792 + 0.984427i $$0.443751\pi$$
$$600$$ 0 0
$$601$$ −36.7446 −1.49884 −0.749421 0.662094i $$-0.769668\pi$$
−0.749421 + 0.662094i $$0.769668\pi$$
$$602$$ −26.3173 −1.07261
$$603$$ 0 0
$$604$$ 1.48913 0.0605916
$$605$$ −23.9147 −0.972271
$$606$$ 0 0
$$607$$ 17.2554 0.700377 0.350188 0.936679i $$-0.386118\pi$$
0.350188 + 0.936679i $$0.386118\pi$$
$$608$$ −2.07668 −0.0842204
$$609$$ 0 0
$$610$$ 33.2554 1.34647
$$611$$ −18.0106 −0.728629
$$612$$ 0 0
$$613$$ −37.6060 −1.51889 −0.759445 0.650571i $$-0.774530\pi$$
−0.759445 + 0.650571i $$0.774530\pi$$
$$614$$ −3.20343 −0.129280
$$615$$ 0 0
$$616$$ 48.0000 1.93398
$$617$$ 2.15121 0.0866046 0.0433023 0.999062i $$-0.486212\pi$$
0.0433023 + 0.999062i $$0.486212\pi$$
$$618$$ 0 0
$$619$$ −38.9783 −1.56667 −0.783334 0.621601i $$-0.786483\pi$$
−0.783334 + 0.621601i $$0.786483\pi$$
$$620$$ −5.40143 −0.216927
$$621$$ 0 0
$$622$$ −34.4674 −1.38202
$$623$$ −40.3236 −1.61553
$$624$$ 0 0
$$625$$ −23.0000 −0.920000
$$626$$ 19.7613 0.789822
$$627$$ 0 0
$$628$$ −0.744563 −0.0297113
$$629$$ 1.60171 0.0638645
$$630$$ 0 0
$$631$$ −2.11684 −0.0842702 −0.0421351 0.999112i $$-0.513416\pi$$
−0.0421351 + 0.999112i $$0.513416\pi$$
$$632$$ −12.1064 −0.481568
$$633$$ 0 0
$$634$$ −44.7446 −1.77703
$$635$$ 2.70071 0.107175
$$636$$ 0 0
$$637$$ 8.74456 0.346472
$$638$$ −41.1244 −1.62813
$$639$$ 0 0
$$640$$ 15.4456 0.610542
$$641$$ −10.3556 −0.409023 −0.204512 0.978864i $$-0.565561\pi$$
−0.204512 + 0.978864i $$0.565561\pi$$
$$642$$ 0 0
$$643$$ −17.8832 −0.705243 −0.352621 0.935766i $$-0.614710\pi$$
−0.352621 + 0.935766i $$0.614710\pi$$
$$644$$ −8.60485 −0.339079
$$645$$ 0 0
$$646$$ 2.74456 0.107983
$$647$$ −17.6101 −0.692326 −0.346163 0.938174i $$-0.612516\pi$$
−0.346163 + 0.938174i $$0.612516\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0.949929 0.0372593
$$651$$ 0 0
$$652$$ −0.554374 −0.0217109
$$653$$ −2.95207 −0.115523 −0.0577617 0.998330i $$-0.518396\pi$$
−0.0577617 + 0.998330i $$0.518396\pi$$
$$654$$ 0 0
$$655$$ 8.39403 0.327982
$$656$$ −7.95307 −0.310515
$$657$$ 0 0
$$658$$ 38.7446 1.51042
$$659$$ −41.9253 −1.63318 −0.816588 0.577221i $$-0.804137\pi$$
−0.816588 + 0.577221i $$0.804137\pi$$
$$660$$ 0 0
$$661$$ 4.74456 0.184542 0.0922710 0.995734i $$-0.470587\pi$$
0.0922710 + 0.995734i $$0.470587\pi$$
$$662$$ 34.4194 1.33775
$$663$$ 0 0
$$664$$ −5.29894 −0.205639
$$665$$ −7.25450 −0.281317
$$666$$ 0 0
$$667$$ 46.9783 1.81901
$$668$$ 1.60171 0.0619721
$$669$$ 0 0
$$670$$ −10.9783 −0.424127
$$671$$ 56.9838 2.19983
$$672$$ 0 0
$$673$$ −36.7446 −1.41640 −0.708199 0.706012i $$-0.750492\pi$$
−0.708199 + 0.706012i $$0.750492\pi$$
$$674$$ −17.8615 −0.687999
$$675$$ 0 0
$$676$$ 3.35053 0.128867
$$677$$ −13.5591 −0.521117 −0.260559 0.965458i $$-0.583907\pi$$
−0.260559 + 0.965458i $$0.583907\pi$$
$$678$$ 0 0
$$679$$ −52.2337 −2.00454
$$680$$ 14.0063 0.537116
$$681$$ 0 0
$$682$$ 40.4674 1.54958
$$683$$ 35.2203 1.34767 0.673833 0.738884i $$-0.264647\pi$$
0.673833 + 0.738884i $$0.264647\pi$$
$$684$$ 0 0
$$685$$ 35.8397 1.36936
$$686$$ 11.3056 0.431649
$$687$$ 0 0
$$688$$ −19.0652 −0.726856
$$689$$ −23.9147 −0.911078
$$690$$ 0 0
$$691$$ 9.88316 0.375973 0.187986 0.982172i $$-0.439804\pi$$
0.187986 + 0.982172i $$0.439804\pi$$
$$692$$ 2.55164 0.0969989
$$693$$ 0 0
$$694$$ −3.95650 −0.150187
$$695$$ 30.3683 1.15194
$$696$$ 0 0
$$697$$ −5.48913 −0.207915
$$698$$ 13.8572 0.524503
$$699$$ 0 0
$$700$$ 0.467376 0.0176651
$$701$$ 29.0180 1.09599 0.547997 0.836480i $$-0.315390\pi$$
0.547997 + 0.836480i $$0.315390\pi$$
$$702$$ 0 0
$$703$$ −0.744563 −0.0280817
$$704$$ 41.7762 1.57450
$$705$$ 0 0
$$706$$ −28.4674 −1.07138
$$707$$ 29.0180 1.09133
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 37.6228 1.41196
$$711$$ 0 0
$$712$$ −36.1902 −1.35628
$$713$$ −46.2277 −1.73124
$$714$$ 0 0
$$715$$ −20.2337 −0.756697
$$716$$ −3.50157 −0.130860
$$717$$ 0 0
$$718$$ −35.4891 −1.32444
$$719$$ −7.40357 −0.276107 −0.138053 0.990425i $$-0.544085\pi$$
−0.138053 + 0.990425i $$0.544085\pi$$
$$720$$ 0 0
$$721$$ −63.2119 −2.35414
$$722$$ −1.27582 −0.0474811
$$723$$ 0 0
$$724$$ 1.29894 0.0482746
$$725$$ −2.55164 −0.0947656
$$726$$ 0 0
$$727$$ 14.3505 0.532232 0.266116 0.963941i $$-0.414260\pi$$
0.266116 + 0.963941i $$0.414260\pi$$
$$728$$ 20.4131 0.756561
$$729$$ 0 0
$$730$$ 33.2554 1.23084
$$731$$ −13.1586 −0.486690
$$732$$ 0 0
$$733$$ −50.4674 −1.86406 −0.932028 0.362387i $$-0.881962\pi$$
−0.932028 + 0.362387i $$0.881962\pi$$
$$734$$ −10.2066 −0.376731
$$735$$ 0 0
$$736$$ −14.2337 −0.524661
$$737$$ −18.8114 −0.692928
$$738$$ 0 0
$$739$$ 9.88316 0.363558 0.181779 0.983339i $$-0.441814\pi$$
0.181779 + 0.983339i $$0.441814\pi$$
$$740$$ −0.596288 −0.0219200
$$741$$ 0 0
$$742$$ 51.4456 1.88863
$$743$$ −42.7261 −1.56747 −0.783735 0.621096i $$-0.786688\pi$$
−0.783735 + 0.621096i $$0.786688\pi$$
$$744$$ 0 0
$$745$$ −2.90491 −0.106428
$$746$$ −13.0564 −0.478027
$$747$$ 0 0
$$748$$ 3.76631 0.137710
$$749$$ −31.7187 −1.15898
$$750$$ 0 0
$$751$$ −44.4674 −1.62264 −0.811319 0.584604i $$-0.801250\pi$$
−0.811319 + 0.584604i $$0.801250\pi$$
$$752$$ 28.0681 1.02354
$$753$$ 0 0
$$754$$ −17.4891 −0.636916
$$755$$ 8.60485 0.313163
$$756$$ 0 0
$$757$$ 12.1168 0.440394 0.220197 0.975455i $$-0.429330\pi$$
0.220197 + 0.975455i $$0.429330\pi$$
$$758$$ −22.0148 −0.799615
$$759$$ 0 0
$$760$$ −6.51087 −0.236174
$$761$$ 25.2651 0.915858 0.457929 0.888989i $$-0.348591\pi$$
0.457929 + 0.888989i $$0.348591\pi$$
$$762$$ 0 0
$$763$$ 56.4674 2.04426
$$764$$ 3.05435 0.110503
$$765$$ 0 0
$$766$$ −1.02175 −0.0369173
$$767$$ −10.2066 −0.368538
$$768$$ 0 0
$$769$$ 24.1168 0.869676 0.434838 0.900509i $$-0.356806\pi$$
0.434838 + 0.900509i $$0.356806\pi$$
$$770$$ 43.5270 1.56860
$$771$$ 0 0
$$772$$ 6.78806 0.244308
$$773$$ −2.55164 −0.0917762 −0.0458881 0.998947i $$-0.514612\pi$$
−0.0458881 + 0.998947i $$0.514612\pi$$
$$774$$ 0 0
$$775$$ 2.51087 0.0901933
$$776$$ −46.8795 −1.68288
$$777$$ 0 0
$$778$$ 21.2554 0.762044
$$779$$ 2.55164 0.0914220
$$780$$ 0 0
$$781$$ 64.4674 2.30682
$$782$$ 18.8114 0.672695
$$783$$ 0 0
$$784$$ −13.6277 −0.486704
$$785$$ −4.30243 −0.153560
$$786$$ 0 0
$$787$$ 41.9565 1.49559 0.747794 0.663931i $$-0.231113\pi$$
0.747794 + 0.663931i $$0.231113\pi$$
$$788$$ 1.30357 0.0464377
$$789$$ 0 0
$$790$$ −10.9783 −0.390589
$$791$$ −5.90414 −0.209927
$$792$$ 0 0
$$793$$ 24.2337 0.860563
$$794$$ −0.149072 −0.00529037
$$795$$ 0 0
$$796$$ −6.74456 −0.239055
$$797$$ −11.1565 −0.395183 −0.197592 0.980284i $$-0.563312\pi$$
−0.197592 + 0.980284i $$0.563312\pi$$
$$798$$ 0 0
$$799$$ 19.3723 0.685342
$$800$$ 0.773108 0.0273335
$$801$$ 0 0
$$802$$ 20.7446 0.732516
$$803$$ 56.9838 2.01091
$$804$$ 0 0
$$805$$ −49.7228 −1.75250
$$806$$ 17.2097 0.606186
$$807$$ 0 0
$$808$$ 26.0435 0.916207
$$809$$ 34.6708 1.21896 0.609480 0.792802i $$-0.291378\pi$$
0.609480 + 0.792802i $$0.291378\pi$$
$$810$$ 0 0
$$811$$ −25.2554 −0.886838 −0.443419 0.896314i $$-0.646235\pi$$
−0.443419 + 0.896314i $$0.646235\pi$$
$$812$$ −8.60485 −0.301971
$$813$$ 0 0
$$814$$ 4.46738 0.156581
$$815$$ −3.20343 −0.112211
$$816$$ 0 0
$$817$$ 6.11684 0.214001
$$818$$ 19.7613 0.690939
$$819$$ 0 0
$$820$$ 2.04350 0.0713621
$$821$$ 17.4611 0.609395 0.304698 0.952449i $$-0.401445\pi$$
0.304698 + 0.952449i $$0.401445\pi$$
$$822$$ 0 0
$$823$$ −31.6060 −1.10171 −0.550857 0.834599i $$-0.685699\pi$$
−0.550857 + 0.834599i $$0.685699\pi$$
$$824$$ −56.7324 −1.97637
$$825$$ 0 0
$$826$$ 21.9565 0.763964
$$827$$ 42.7261 1.48573 0.742866 0.669440i $$-0.233466\pi$$
0.742866 + 0.669440i $$0.233466\pi$$
$$828$$ 0 0
$$829$$ −12.7446 −0.442637 −0.221318 0.975202i $$-0.571036\pi$$
−0.221318 + 0.975202i $$0.571036\pi$$
$$830$$ −4.80514 −0.166789
$$831$$ 0 0
$$832$$ 17.7663 0.615936
$$833$$ −9.40571 −0.325889
$$834$$ 0 0
$$835$$ 9.25544 0.320298
$$836$$ −1.75079 −0.0605522
$$837$$ 0 0
$$838$$ −2.23369 −0.0771615
$$839$$ 7.80400 0.269424 0.134712 0.990885i $$-0.456989\pi$$
0.134712 + 0.990885i $$0.456989\pi$$
$$840$$ 0 0
$$841$$ 17.9783 0.619940
$$842$$ −47.1776 −1.62585
$$843$$ 0 0
$$844$$ 1.48913 0.0512578
$$845$$ 19.3609 0.666036
$$846$$ 0 0
$$847$$ 37.4891 1.28814
$$848$$ 37.2692 1.27983
$$849$$ 0 0
$$850$$ −1.02175 −0.0350457
$$851$$ −5.10328 −0.174938
$$852$$ 0 0
$$853$$ 30.4674 1.04318 0.521592 0.853195i $$-0.325339\pi$$
0.521592 + 0.853195i $$0.325339\pi$$
$$854$$ −52.1318 −1.78391
$$855$$ 0 0
$$856$$ −28.4674 −0.972995
$$857$$ −42.0743 −1.43723 −0.718616 0.695407i $$-0.755224\pi$$
−0.718616 + 0.695407i $$0.755224\pi$$
$$858$$ 0 0
$$859$$ −14.1168 −0.481661 −0.240830 0.970567i $$-0.577420\pi$$
−0.240830 + 0.970567i $$0.577420\pi$$
$$860$$ 4.89871 0.167045
$$861$$ 0 0
$$862$$ 9.95650 0.339120
$$863$$ −22.3130 −0.759543 −0.379771 0.925080i $$-0.623997\pi$$
−0.379771 + 0.925080i $$0.623997\pi$$
$$864$$ 0 0
$$865$$ 14.7446 0.501330
$$866$$ 28.0681 0.953791
$$867$$ 0 0
$$868$$ 8.46738 0.287401
$$869$$ −18.8114 −0.638134
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 50.6792 1.71621
$$873$$ 0 0
$$874$$ −8.74456 −0.295789
$$875$$ 38.9732 1.31753
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −20.7113 −0.698972
$$879$$ 0 0
$$880$$ 31.5326 1.06296
$$881$$ −25.2651 −0.851201 −0.425601 0.904911i $$-0.639937\pi$$
−0.425601 + 0.904911i $$0.639937\pi$$
$$882$$ 0 0
$$883$$ −14.1168 −0.475070 −0.237535 0.971379i $$-0.576339\pi$$
−0.237535 + 0.971379i $$0.576339\pi$$
$$884$$ 1.60171 0.0538714
$$885$$ 0 0
$$886$$ 15.9565 0.536069
$$887$$ −10.2066 −0.342703 −0.171351 0.985210i $$-0.554813\pi$$
−0.171351 + 0.985210i $$0.554813\pi$$
$$888$$ 0 0
$$889$$ −4.23369 −0.141993
$$890$$ −32.8177 −1.10005
$$891$$ 0 0
$$892$$ 1.48913 0.0498596
$$893$$ −9.00528 −0.301350
$$894$$ 0 0
$$895$$ −20.2337 −0.676338
$$896$$ −24.2128 −0.808894
$$897$$ 0 0
$$898$$ −42.7011 −1.42495
$$899$$ −46.2277 −1.54178
$$900$$ 0 0
$$901$$ 25.7228 0.856951
$$902$$ −15.3098 −0.509762
$$903$$ 0 0
$$904$$ −5.29894 −0.176240
$$905$$ 7.50585 0.249503
$$906$$ 0 0
$$907$$ 34.7446 1.15367 0.576837 0.816859i $$-0.304287\pi$$
0.576837 + 0.816859i $$0.304287\pi$$
$$908$$ −4.80514 −0.159464
$$909$$ 0 0
$$910$$ 18.5109 0.613630
$$911$$ 24.7156 0.818863 0.409432 0.912341i $$-0.365727\pi$$
0.409432 + 0.912341i $$0.365727\pi$$
$$912$$ 0 0
$$913$$ −8.23369 −0.272495
$$914$$ 3.35250 0.110891
$$915$$ 0 0
$$916$$ −7.95650 −0.262890
$$917$$ −13.1586 −0.434536
$$918$$ 0 0
$$919$$ −26.9783 −0.889930 −0.444965 0.895548i $$-0.646784\pi$$
−0.444965 + 0.895548i $$0.646784\pi$$
$$920$$ −44.6260 −1.47127
$$921$$ 0 0
$$922$$ 7.21194 0.237513
$$923$$ 27.4163 0.902418
$$924$$ 0 0
$$925$$ 0.277187 0.00911384
$$926$$ −4.30243 −0.141387
$$927$$ 0 0
$$928$$ −14.2337 −0.467244
$$929$$ −46.2277 −1.51668 −0.758341 0.651858i $$-0.773990\pi$$
−0.758341 + 0.651858i $$0.773990\pi$$
$$930$$ 0 0
$$931$$ 4.37228 0.143296
$$932$$ −2.70071 −0.0884648
$$933$$ 0 0
$$934$$ 38.9348 1.27398
$$935$$ 21.7635 0.711742
$$936$$ 0 0
$$937$$ 12.1168 0.395840 0.197920 0.980218i $$-0.436581\pi$$
0.197920 + 0.980218i $$0.436581\pi$$
$$938$$ 17.2097 0.561917
$$939$$ 0 0
$$940$$ −7.21194 −0.235227
$$941$$ 10.3556 0.337584 0.168792 0.985652i $$-0.446013\pi$$
0.168792 + 0.985652i $$0.446013\pi$$
$$942$$ 0 0
$$943$$ 17.4891 0.569524
$$944$$ 15.9061 0.517701
$$945$$ 0 0
$$946$$ −36.7011 −1.19325
$$947$$ 11.9574 0.388562 0.194281 0.980946i $$-0.437763\pi$$
0.194281 + 0.980946i $$0.437763\pi$$
$$948$$ 0 0
$$949$$ 24.2337 0.786659
$$950$$ 0.474964 0.0154099
$$951$$ 0 0
$$952$$ −21.9565 −0.711614
$$953$$ 24.8646 0.805444 0.402722 0.915322i $$-0.368064\pi$$
0.402722 + 0.915322i $$0.368064\pi$$
$$954$$ 0 0
$$955$$ 17.6495 0.571123
$$956$$ 10.0575 0.325283
$$957$$ 0 0
$$958$$ 10.7881 0.348546
$$959$$ −56.1829 −1.81424
$$960$$ 0 0
$$961$$ 14.4891 0.467391
$$962$$ 1.89986 0.0612538
$$963$$ 0 0
$$964$$ −3.80981 −0.122706
$$965$$ 39.2246 1.26268
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 33.6463 1.08143
$$969$$ 0 0
$$970$$ −42.5109 −1.36494
$$971$$ −34.4194 −1.10457 −0.552286 0.833655i $$-0.686244\pi$$
−0.552286 + 0.833655i $$0.686244\pi$$
$$972$$ 0 0
$$973$$ −47.6060 −1.52618
$$974$$ −10.2066 −0.327039
$$975$$ 0 0
$$976$$ −37.7663 −1.20887
$$977$$ 17.0606 0.545818 0.272909 0.962040i $$-0.412014\pi$$
0.272909 + 0.962040i $$0.412014\pi$$
$$978$$ 0 0
$$979$$ −56.2337 −1.79724
$$980$$ 3.50157 0.111854
$$981$$ 0 0
$$982$$ 8.74456 0.279050
$$983$$ −39.2246 −1.25107 −0.625534 0.780197i $$-0.715119\pi$$
−0.625534 + 0.780197i $$0.715119\pi$$
$$984$$ 0 0
$$985$$ 7.53262 0.240009
$$986$$ 18.8114 0.599078
$$987$$ 0 0
$$988$$ −0.744563 −0.0236877
$$989$$ 41.9253 1.33315
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 14.0063 0.444700
$$993$$ 0 0
$$994$$ −58.9783 −1.87068
$$995$$ −38.9732 −1.23553
$$996$$ 0 0
$$997$$ 32.3505 1.02455 0.512276 0.858821i $$-0.328803\pi$$
0.512276 + 0.858821i $$0.328803\pi$$
$$998$$ −7.80400 −0.247031
$$999$$ 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 171.2.a.e.1.2 4
3.2 odd 2 inner 171.2.a.e.1.3 yes 4
4.3 odd 2 2736.2.a.bf.1.2 4
5.4 even 2 4275.2.a.bp.1.3 4
7.6 odd 2 8379.2.a.bw.1.2 4
12.11 even 2 2736.2.a.bf.1.3 4
15.14 odd 2 4275.2.a.bp.1.2 4
19.18 odd 2 3249.2.a.bf.1.3 4
21.20 even 2 8379.2.a.bw.1.3 4
57.56 even 2 3249.2.a.bf.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.2 4 1.1 even 1 trivial
171.2.a.e.1.3 yes 4 3.2 odd 2 inner
2736.2.a.bf.1.2 4 4.3 odd 2
2736.2.a.bf.1.3 4 12.11 even 2
3249.2.a.bf.1.2 4 57.56 even 2
3249.2.a.bf.1.3 4 19.18 odd 2
4275.2.a.bp.1.2 4 15.14 odd 2
4275.2.a.bp.1.3 4 5.4 even 2
8379.2.a.bw.1.2 4 7.6 odd 2
8379.2.a.bw.1.3 4 21.20 even 2