# Properties

 Label 171.2.a.e.1.3 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 1.3 Root $$-1.82405$$ 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 $$-0.351042\pi$$
0.451071 + 0.892488i $$0.351042\pi$$
$$3$$ 0 0
$$4$$ −0.372281 −0.186141
$$5$$ 2.15121 0.962052 0.481026 0.876706i $$-0.340264\pi$$
0.481026 + 0.876706i $$0.340264\pi$$
$$6$$ 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 $$-0.750845\pi$$
−0.708982 + 0.705227i $$0.750845\pi$$
$$12$$ 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 $$-0.415990\pi$$
0.260873 + 0.965373i $$0.415990\pi$$
$$18$$ 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 $$-0.753385\pi$$
−0.714586 + 0.699548i $$0.753385\pi$$
$$24$$ 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 $$-0.719571\pi$$
−0.636384 + 0.771372i $$0.719571\pi$$
$$30$$ 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 $$-0.563850\pi$$
−0.199250 + 0.979949i $$0.563850\pi$$
$$42$$ 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 $$-0.271919\pi$$
0.656778 + 0.754084i $$0.271919\pi$$
$$48$$ 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 $$-0.193286\pi$$
0.821234 + 0.570591i $$0.193286\pi$$
$$54$$ 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 $$-0.392211\pi$$
0.332195 + 0.943211i $$0.392211\pi$$
$$60$$ 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 $$-0.802401\pi$$
−0.813428 + 0.581665i $$0.802401\pi$$
$$72$$ 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 $$-0.469367\pi$$
0.0960868 + 0.995373i $$0.469367\pi$$
$$84$$ 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 $$-0.281520\pi$$
0.633738 + 0.773547i $$0.281520\pi$$
$$90$$ 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 $$-0.640820\pi$$
−0.428107 + 0.903728i $$0.640820\pi$$
$$102$$ 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 $$-0.349767\pi$$
0.454642 + 0.890674i $$0.349767\pi$$
$$108$$ 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 $$-0.473757\pi$$
0.0823500 + 0.996603i $$0.473757\pi$$
$$114$$ 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 $$-0.445475\pi$$
0.170460 + 0.985365i $$0.445475\pi$$
$$132$$ 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 $$-0.247931\pi$$
0.711689 + 0.702495i $$0.247931\pi$$
$$138$$ 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 $$-0.517616\pi$$
−0.0553128 + 0.998469i $$0.517616\pi$$
$$150$$ 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 $$-0.446764\pi$$
0.166466 + 0.986047i $$0.446764\pi$$
$$168$$ 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 $$-0.416095\pi$$
0.260553 + 0.965460i $$0.416095\pi$$
$$174$$ 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 $$-0.614331\pi$$
−0.351508 + 0.936185i $$0.614331\pi$$
$$180$$ 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 $$-0.404072\pi$$
0.296826 + 0.954932i $$0.404072\pi$$
$$192$$ 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 $$-0.460191\pi$$
0.124738 + 0.992190i $$0.460191\pi$$
$$198$$ 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 $$-0.640903\pi$$
−0.428343 + 0.903616i $$0.640903\pi$$
$$228$$ 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 $$-0.576370\pi$$
−0.237629 + 0.971356i $$0.576370\pi$$
$$234$$ 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 $$-0.161678\pi$$
0.873755 + 0.486367i $$0.161678\pi$$
$$240$$ 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 $$-0.344060\pi$$
0.470537 + 0.882380i $$0.344060\pi$$
$$252$$ 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 $$-0.474641\pi$$
0.0795835 + 0.996828i $$0.474641\pi$$
$$258$$ 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 $$-0.402233\pi$$
0.302336 + 0.953201i $$0.402233\pi$$
$$264$$ 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 $$-0.213982\pi$$
0.782426 + 0.622743i $$0.213982\pi$$
$$270$$ 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 $$-0.599955\pi$$
−0.308883 + 0.951100i $$0.599955\pi$$
$$282$$ 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 $$-0.781290\pi$$
−0.773090 + 0.634296i $$0.781290\pi$$
$$294$$ 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 $$-0.777738\pi$$
−0.765964 + 0.642883i $$0.777738\pi$$
$$312$$ 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 $$-0.944609\pi$$
−0.984897 + 0.173139i $$0.944609\pi$$
$$318$$ 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 $$-0.526526\pi$$
−0.0832390 + 0.996530i $$0.526526\pi$$
$$348$$ 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 $$-0.702373\pi$$
−0.593800 + 0.804612i $$0.702373\pi$$
$$354$$ 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 $$-0.762374\pi$$
−0.734055 + 0.679090i $$0.762374\pi$$
$$360$$ 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 $$-0.506513\pi$$
−0.0204609 + 0.999791i $$0.506513\pi$$
$$384$$ 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 $$-0.361204\pi$$
0.422353 + 0.906431i $$0.361204\pi$$
$$390$$ 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 $$-0.366928\pi$$
0.405987 + 0.913879i $$0.366928\pi$$
$$402$$ 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 $$-0.513617\pi$$
−0.0427657 + 0.999085i $$0.513617\pi$$
$$420$$ 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 $$-0.439815\pi$$
0.187953 + 0.982178i $$0.439815\pi$$
$$432$$ 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 $$-0.403977\pi$$
0.297109 + 0.954843i $$0.403977\pi$$
$$444$$ 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 $$-0.789796\pi$$
−0.789761 + 0.613414i $$0.789796\pi$$
$$450$$ 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 $$-0.457976\pi$$
0.131638 + 0.991298i $$0.457976\pi$$
$$462$$ 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 $$-0.250458\pi$$
0.706089 + 0.708123i $$0.250458\pi$$
$$468$$ 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 $$-0.438121\pi$$
0.193177 + 0.981164i $$0.438121\pi$$
$$480$$ 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 $$-0.450572\pi$$
0.154660 + 0.987968i $$0.450572\pi$$
$$492$$ 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 $$-0.664510\pi$$
−0.494120 + 0.869394i $$0.664510\pi$$
$$504$$ 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 $$-0.573710\pi$$
−0.229503 + 0.973308i $$0.573710\pi$$
$$510$$ 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 $$-0.446371\pi$$
0.167684 + 0.985841i $$0.446371\pi$$
$$522$$ 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 $$-0.490892\pi$$
0.0286082 + 0.999591i $$0.490892\pi$$
$$558$$ 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 $$-0.316918\pi$$
0.543977 + 0.839100i $$0.316918\pi$$
$$564$$ 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 $$-0.641841\pi$$
−0.431006 + 0.902349i $$0.641841\pi$$
$$570$$ 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 $$-0.269347\pi$$
0.662849 + 0.748753i $$0.269347\pi$$
$$588$$ 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 $$-0.690324\pi$$
−0.562926 + 0.826508i $$0.690324\pi$$
$$594$$ 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 $$-0.556249\pi$$
−0.175792 + 0.984427i $$0.556249\pi$$
$$600$$ 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 $$-0.513788\pi$$
−0.0433023 + 0.999062i $$0.513788\pi$$
$$618$$ 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 $$-0.434439\pi$$
0.204512 + 0.978864i $$0.434439\pi$$
$$642$$ 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 $$-0.387484\pi$$
0.346163 + 0.938174i $$0.387484\pi$$
$$648$$ 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 $$-0.481604\pi$$
0.0577617 + 0.998330i $$0.481604\pi$$
$$654$$ 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 $$-0.195863\pi$$
0.816588 + 0.577221i $$0.195863\pi$$
$$660$$ 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 $$-0.416093\pi$$
0.260559 + 0.965458i $$0.416093\pi$$
$$678$$ 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 $$-0.735353\pi$$
−0.673833 + 0.738884i $$0.735353\pi$$
$$684$$ 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 $$-0.684610\pi$$
−0.547997 + 0.836480i $$0.684610\pi$$
$$702$$ 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 $$-0.455915\pi$$
0.138053 + 0.990425i $$0.455915\pi$$
$$720$$ 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 $$-0.213312\pi$$
0.783735 + 0.621096i $$0.213312\pi$$
$$744$$ 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 $$-0.651409\pi$$
−0.457929 + 0.888989i $$0.651409\pi$$
$$762$$ 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 $$-0.485388\pi$$
0.0458881 + 0.998947i $$0.485388\pi$$
$$774$$ 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 $$-0.436688\pi$$
0.197592 + 0.980284i $$0.436688\pi$$
$$798$$ 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 $$-0.708622\pi$$
−0.609480 + 0.792802i $$0.708622\pi$$
$$810$$ 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 $$-0.598555\pi$$
−0.304698 + 0.952449i $$0.598555\pi$$
$$822$$ 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 $$-0.766534\pi$$
−0.742866 + 0.669440i $$0.766534\pi$$
$$828$$ 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 $$-0.543011\pi$$
−0.134712 + 0.990885i $$0.543011\pi$$
$$840$$ 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 $$-0.244776\pi$$
0.718616 + 0.695407i $$0.244776\pi$$
$$858$$ 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 $$-0.376003\pi$$
0.379771 + 0.925080i $$0.376003\pi$$
$$864$$ 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 $$-0.360063\pi$$
0.425601 + 0.904911i $$0.360063\pi$$
$$882$$ 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 $$-0.445187\pi$$
0.171351 + 0.985210i $$0.445187\pi$$
$$888$$ 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 $$-0.634273\pi$$
−0.409432 + 0.912341i $$0.634273\pi$$
$$912$$ 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 $$-0.226010\pi$$
0.758341 + 0.651858i $$0.226010\pi$$
$$930$$ 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 $$-0.553987\pi$$
−0.168792 + 0.985652i $$0.553987\pi$$
$$942$$ 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 $$-0.562237\pi$$
−0.194281 + 0.980946i $$0.562237\pi$$
$$948$$ 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 $$-0.631936\pi$$
−0.402722 + 0.915322i $$0.631936\pi$$
$$954$$ 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 $$-0.313756\pi$$
0.552286 + 0.833655i $$0.313756\pi$$
$$972$$ 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 $$-0.587986\pi$$
−0.272909 + 0.962040i $$0.587986\pi$$
$$978$$ 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 $$-0.284881\pi$$
0.625534 + 0.780197i $$0.284881\pi$$
$$984$$ 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.1.3 yes 4
3.2 odd 2 inner 171.2.a.e.1.2 4
4.3 odd 2 2736.2.a.bf.1.3 4
5.4 even 2 4275.2.a.bp.1.2 4
7.6 odd 2 8379.2.a.bw.1.3 4
12.11 even 2 2736.2.a.bf.1.2 4
15.14 odd 2 4275.2.a.bp.1.3 4
19.18 odd 2 3249.2.a.bf.1.2 4
21.20 even 2 8379.2.a.bw.1.2 4
57.56 even 2 3249.2.a.bf.1.3 4

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