Properties

 Label 845.2.a.l.1.1 Level $845$ Weight $2$ Character 845.1 Self dual yes Analytic conductor $6.747$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.74735897080$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4752.1 Defining polynomial: $$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$2.49551$$ of defining polynomial Character $$\chi$$ $$=$$ 845.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-2.49551 q^{2} -2.82684 q^{3} +4.22756 q^{4} +1.00000 q^{5} +7.05440 q^{6} -1.90521 q^{7} -5.55889 q^{8} +4.99102 q^{9} +O(q^{10})$$ $$q-2.49551 q^{2} -2.82684 q^{3} +4.22756 q^{4} +1.00000 q^{5} +7.05440 q^{6} -1.90521 q^{7} -5.55889 q^{8} +4.99102 q^{9} -2.49551 q^{10} -1.06939 q^{11} -11.9506 q^{12} +4.75447 q^{14} -2.82684 q^{15} +5.41713 q^{16} +0.637263 q^{17} -12.4551 q^{18} -5.73205 q^{19} +4.22756 q^{20} +5.38573 q^{21} +2.66867 q^{22} +3.81785 q^{23} +15.7141 q^{24} +1.00000 q^{25} -5.62828 q^{27} -8.05440 q^{28} +9.45512 q^{29} +7.05440 q^{30} +1.46410 q^{31} -2.40072 q^{32} +3.02299 q^{33} -1.59030 q^{34} -1.90521 q^{35} +21.0998 q^{36} -0.757449 q^{37} +14.3044 q^{38} -5.55889 q^{40} -0.267949 q^{41} -13.4401 q^{42} +0.637263 q^{43} -4.52091 q^{44} +4.99102 q^{45} -9.52748 q^{46} +9.44613 q^{47} -15.3134 q^{48} -3.37017 q^{49} -2.49551 q^{50} -1.80144 q^{51} -6.99102 q^{53} +14.0454 q^{54} -1.06939 q^{55} +10.5909 q^{56} +16.2036 q^{57} -23.5953 q^{58} -0.741035 q^{59} -11.9506 q^{60} +4.19856 q^{61} -3.65368 q^{62} -9.50894 q^{63} -4.84325 q^{64} -7.54390 q^{66} -8.09479 q^{67} +2.69407 q^{68} -10.7925 q^{69} +4.75447 q^{70} -9.76488 q^{71} -27.7445 q^{72} +3.71649 q^{73} +1.89022 q^{74} -2.82684 q^{75} -24.2326 q^{76} +2.03741 q^{77} -9.31937 q^{79} +5.41713 q^{80} +0.937188 q^{81} +0.668669 q^{82} -5.11778 q^{83} +22.7685 q^{84} +0.637263 q^{85} -1.59030 q^{86} -26.7281 q^{87} +5.94462 q^{88} +12.5783 q^{89} -12.4551 q^{90} +16.1402 q^{92} -4.13878 q^{93} -23.5729 q^{94} -5.73205 q^{95} +6.78645 q^{96} +4.22155 q^{97} +8.41027 q^{98} -5.33734 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9} + O(q^{10})$$ $$4 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} + 4 q^{6} - 10 q^{7} - 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 20 q^{18} - 16 q^{19} + 2 q^{20} - 4 q^{21} + 12 q^{22} - 10 q^{23} + 24 q^{24} + 4 q^{25} - 2 q^{27} - 8 q^{28} + 8 q^{29} + 4 q^{30} - 8 q^{31} - 4 q^{32} - 18 q^{33} + 4 q^{34} - 10 q^{35} + 20 q^{36} + 2 q^{37} + 8 q^{38} - 6 q^{40} - 8 q^{41} - 4 q^{42} - 2 q^{43} - 12 q^{44} + 4 q^{45} - 16 q^{46} - 8 q^{47} - 28 q^{48} + 12 q^{49} - 2 q^{50} + 4 q^{51} - 12 q^{53} + 16 q^{54} + 12 q^{56} + 14 q^{57} - 22 q^{58} - 12 q^{59} - 10 q^{60} + 28 q^{61} + 4 q^{62} - 4 q^{63} + 4 q^{64} + 6 q^{66} - 30 q^{67} + 14 q^{68} - 16 q^{69} + 2 q^{70} - 4 q^{71} - 12 q^{72} + 8 q^{73} - 10 q^{74} - 2 q^{75} - 20 q^{76} + 18 q^{77} - 8 q^{79} + 2 q^{80} - 8 q^{81} + 4 q^{82} + 12 q^{83} + 28 q^{84} - 2 q^{85} + 4 q^{86} - 22 q^{87} - 18 q^{88} + 12 q^{89} - 20 q^{90} + 22 q^{92} - 8 q^{93} - 32 q^{94} - 16 q^{95} - 4 q^{96} - 2 q^{97} + 24 q^{98} - 24 q^{99} + O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.49551 −1.76459 −0.882295 0.470696i $$-0.844003\pi$$
−0.882295 + 0.470696i $$0.844003\pi$$
$$3$$ −2.82684 −1.63208 −0.816038 0.577998i $$-0.803834\pi$$
−0.816038 + 0.577998i $$0.803834\pi$$
$$4$$ 4.22756 2.11378
$$5$$ 1.00000 0.447214
$$6$$ 7.05440 2.87995
$$7$$ −1.90521 −0.720103 −0.360051 0.932933i $$-0.617241\pi$$
−0.360051 + 0.932933i $$0.617241\pi$$
$$8$$ −5.55889 −1.96536
$$9$$ 4.99102 1.66367
$$10$$ −2.49551 −0.789149
$$11$$ −1.06939 −0.322433 −0.161217 0.986919i $$-0.551542\pi$$
−0.161217 + 0.986919i $$0.551542\pi$$
$$12$$ −11.9506 −3.44985
$$13$$ 0 0
$$14$$ 4.75447 1.27069
$$15$$ −2.82684 −0.729887
$$16$$ 5.41713 1.35428
$$17$$ 0.637263 0.154559 0.0772795 0.997009i $$-0.475377\pi$$
0.0772795 + 0.997009i $$0.475377\pi$$
$$18$$ −12.4551 −2.93570
$$19$$ −5.73205 −1.31502 −0.657511 0.753445i $$-0.728391\pi$$
−0.657511 + 0.753445i $$0.728391\pi$$
$$20$$ 4.22756 0.945311
$$21$$ 5.38573 1.17526
$$22$$ 2.66867 0.568962
$$23$$ 3.81785 0.796078 0.398039 0.917369i $$-0.369691\pi$$
0.398039 + 0.917369i $$0.369691\pi$$
$$24$$ 15.7141 3.20762
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.62828 −1.08316
$$28$$ −8.05440 −1.52214
$$29$$ 9.45512 1.75577 0.877886 0.478870i $$-0.158954\pi$$
0.877886 + 0.478870i $$0.158954\pi$$
$$30$$ 7.05440 1.28795
$$31$$ 1.46410 0.262960 0.131480 0.991319i $$-0.458027\pi$$
0.131480 + 0.991319i $$0.458027\pi$$
$$32$$ −2.40072 −0.424391
$$33$$ 3.02299 0.526235
$$34$$ −1.59030 −0.272733
$$35$$ −1.90521 −0.322040
$$36$$ 21.0998 3.51663
$$37$$ −0.757449 −0.124524 −0.0622619 0.998060i $$-0.519831\pi$$
−0.0622619 + 0.998060i $$0.519831\pi$$
$$38$$ 14.3044 2.32048
$$39$$ 0 0
$$40$$ −5.55889 −0.878938
$$41$$ −0.267949 −0.0418466 −0.0209233 0.999781i $$-0.506661\pi$$
−0.0209233 + 0.999781i $$0.506661\pi$$
$$42$$ −13.4401 −2.07386
$$43$$ 0.637263 0.0971817 0.0485909 0.998819i $$-0.484527\pi$$
0.0485909 + 0.998819i $$0.484527\pi$$
$$44$$ −4.52091 −0.681552
$$45$$ 4.99102 0.744017
$$46$$ −9.52748 −1.40475
$$47$$ 9.44613 1.37786 0.688930 0.724828i $$-0.258081\pi$$
0.688930 + 0.724828i $$0.258081\pi$$
$$48$$ −15.3134 −2.21029
$$49$$ −3.37017 −0.481452
$$50$$ −2.49551 −0.352918
$$51$$ −1.80144 −0.252252
$$52$$ 0 0
$$53$$ −6.99102 −0.960290 −0.480145 0.877189i $$-0.659416\pi$$
−0.480145 + 0.877189i $$0.659416\pi$$
$$54$$ 14.0454 1.91134
$$55$$ −1.06939 −0.144196
$$56$$ 10.5909 1.41526
$$57$$ 16.2036 2.14622
$$58$$ −23.5953 −3.09822
$$59$$ −0.741035 −0.0964746 −0.0482373 0.998836i $$-0.515360\pi$$
−0.0482373 + 0.998836i $$0.515360\pi$$
$$60$$ −11.9506 −1.54282
$$61$$ 4.19856 0.537571 0.268785 0.963200i $$-0.413378\pi$$
0.268785 + 0.963200i $$0.413378\pi$$
$$62$$ −3.65368 −0.464017
$$63$$ −9.50894 −1.19801
$$64$$ −4.84325 −0.605406
$$65$$ 0 0
$$66$$ −7.54390 −0.928589
$$67$$ −8.09479 −0.988936 −0.494468 0.869196i $$-0.664637\pi$$
−0.494468 + 0.869196i $$0.664637\pi$$
$$68$$ 2.69407 0.326704
$$69$$ −10.7925 −1.29926
$$70$$ 4.75447 0.568268
$$71$$ −9.76488 −1.15888 −0.579439 0.815016i $$-0.696728\pi$$
−0.579439 + 0.815016i $$0.696728\pi$$
$$72$$ −27.7445 −3.26972
$$73$$ 3.71649 0.434982 0.217491 0.976062i $$-0.430213\pi$$
0.217491 + 0.976062i $$0.430213\pi$$
$$74$$ 1.89022 0.219734
$$75$$ −2.82684 −0.326415
$$76$$ −24.2326 −2.77967
$$77$$ 2.03741 0.232185
$$78$$ 0 0
$$79$$ −9.31937 −1.04851 −0.524255 0.851561i $$-0.675656\pi$$
−0.524255 + 0.851561i $$0.675656\pi$$
$$80$$ 5.41713 0.605654
$$81$$ 0.937188 0.104132
$$82$$ 0.668669 0.0738422
$$83$$ −5.11778 −0.561749 −0.280875 0.959744i $$-0.590624\pi$$
−0.280875 + 0.959744i $$0.590624\pi$$
$$84$$ 22.7685 2.48424
$$85$$ 0.637263 0.0691209
$$86$$ −1.59030 −0.171486
$$87$$ −26.7281 −2.86555
$$88$$ 5.94462 0.633698
$$89$$ 12.5783 1.33330 0.666650 0.745371i $$-0.267727\pi$$
0.666650 + 0.745371i $$0.267727\pi$$
$$90$$ −12.4551 −1.31288
$$91$$ 0 0
$$92$$ 16.1402 1.68273
$$93$$ −4.13878 −0.429171
$$94$$ −23.5729 −2.43136
$$95$$ −5.73205 −0.588096
$$96$$ 6.78645 0.692639
$$97$$ 4.22155 0.428634 0.214317 0.976764i $$-0.431248\pi$$
0.214317 + 0.976764i $$0.431248\pi$$
$$98$$ 8.41027 0.849566
$$99$$ −5.33734 −0.536423
$$100$$ 4.22756 0.422756
$$101$$ −15.2476 −1.51719 −0.758595 0.651562i $$-0.774114\pi$$
−0.758595 + 0.651562i $$0.774114\pi$$
$$102$$ 4.49551 0.445122
$$103$$ −13.5269 −1.33285 −0.666423 0.745574i $$-0.732176\pi$$
−0.666423 + 0.745574i $$0.732176\pi$$
$$104$$ 0 0
$$105$$ 5.38573 0.525593
$$106$$ 17.4461 1.69452
$$107$$ −7.36274 −0.711783 −0.355891 0.934527i $$-0.615823\pi$$
−0.355891 + 0.934527i $$0.615823\pi$$
$$108$$ −23.7939 −2.28957
$$109$$ −10.0760 −0.965103 −0.482551 0.875868i $$-0.660290\pi$$
−0.482551 + 0.875868i $$0.660290\pi$$
$$110$$ 2.66867 0.254448
$$111$$ 2.14119 0.203232
$$112$$ −10.3208 −0.975223
$$113$$ −6.68806 −0.629160 −0.314580 0.949231i $$-0.601864\pi$$
−0.314580 + 0.949231i $$0.601864\pi$$
$$114$$ −40.4362 −3.78719
$$115$$ 3.81785 0.356017
$$116$$ 39.9721 3.71131
$$117$$ 0 0
$$118$$ 1.84926 0.170238
$$119$$ −1.21412 −0.111298
$$120$$ 15.7141 1.43449
$$121$$ −9.85641 −0.896037
$$122$$ −10.4775 −0.948592
$$123$$ 0.757449 0.0682969
$$124$$ 6.18958 0.555840
$$125$$ 1.00000 0.0894427
$$126$$ 23.7296 2.11400
$$127$$ 1.48950 0.132172 0.0660859 0.997814i $$-0.478949\pi$$
0.0660859 + 0.997814i $$0.478949\pi$$
$$128$$ 16.8878 1.49269
$$129$$ −1.80144 −0.158608
$$130$$ 0 0
$$131$$ 4.12676 0.360557 0.180278 0.983616i $$-0.442300\pi$$
0.180278 + 0.983616i $$0.442300\pi$$
$$132$$ 12.7799 1.11234
$$133$$ 10.9208 0.946951
$$134$$ 20.2006 1.74507
$$135$$ −5.62828 −0.484405
$$136$$ −3.54248 −0.303765
$$137$$ 20.1096 1.71808 0.859041 0.511906i $$-0.171060\pi$$
0.859041 + 0.511906i $$0.171060\pi$$
$$138$$ 26.9327 2.29266
$$139$$ 20.8253 1.76638 0.883189 0.469018i $$-0.155392\pi$$
0.883189 + 0.469018i $$0.155392\pi$$
$$140$$ −8.05440 −0.680721
$$141$$ −26.7027 −2.24877
$$142$$ 24.3683 2.04494
$$143$$ 0 0
$$144$$ 27.0370 2.25308
$$145$$ 9.45512 0.785205
$$146$$ −9.27453 −0.767565
$$147$$ 9.52691 0.785767
$$148$$ −3.20216 −0.263216
$$149$$ −13.3678 −1.09513 −0.547565 0.836763i $$-0.684445\pi$$
−0.547565 + 0.836763i $$0.684445\pi$$
$$150$$ 7.05440 0.575989
$$151$$ −18.2984 −1.48910 −0.744550 0.667567i $$-0.767336\pi$$
−0.744550 + 0.667567i $$0.767336\pi$$
$$152$$ 31.8638 2.58450
$$153$$ 3.18059 0.257136
$$154$$ −5.08438 −0.409711
$$155$$ 1.46410 0.117599
$$156$$ 0 0
$$157$$ 2.42229 0.193320 0.0966599 0.995317i $$-0.469184\pi$$
0.0966599 + 0.995317i $$0.469184\pi$$
$$158$$ 23.2566 1.85019
$$159$$ 19.7625 1.56727
$$160$$ −2.40072 −0.189794
$$161$$ −7.27382 −0.573258
$$162$$ −2.33876 −0.183750
$$163$$ −15.9829 −1.25188 −0.625938 0.779873i $$-0.715284\pi$$
−0.625938 + 0.779873i $$0.715284\pi$$
$$164$$ −1.13277 −0.0884545
$$165$$ 3.02299 0.235340
$$166$$ 12.7715 0.991257
$$167$$ 14.3932 1.11378 0.556888 0.830588i $$-0.311995\pi$$
0.556888 + 0.830588i $$0.311995\pi$$
$$168$$ −29.9387 −2.30982
$$169$$ 0 0
$$170$$ −1.59030 −0.121970
$$171$$ −28.6088 −2.18777
$$172$$ 2.69407 0.205421
$$173$$ −24.3489 −1.85122 −0.925608 0.378484i $$-0.876445\pi$$
−0.925608 + 0.378484i $$0.876445\pi$$
$$174$$ 66.7001 5.05653
$$175$$ −1.90521 −0.144021
$$176$$ −5.79302 −0.436666
$$177$$ 2.09479 0.157454
$$178$$ −31.3893 −2.35273
$$179$$ 3.78829 0.283150 0.141575 0.989928i $$-0.454783\pi$$
0.141575 + 0.989928i $$0.454783\pi$$
$$180$$ 21.0998 1.57269
$$181$$ −8.48794 −0.630904 −0.315452 0.948942i $$-0.602156\pi$$
−0.315452 + 0.948942i $$0.602156\pi$$
$$182$$ 0 0
$$183$$ −11.8687 −0.877356
$$184$$ −21.2230 −1.56458
$$185$$ −0.757449 −0.0556888
$$186$$ 10.3284 0.757312
$$187$$ −0.681482 −0.0498349
$$188$$ 39.9341 2.91249
$$189$$ 10.7231 0.779988
$$190$$ 14.3044 1.03775
$$191$$ −5.44310 −0.393849 −0.196924 0.980419i $$-0.563095\pi$$
−0.196924 + 0.980419i $$0.563095\pi$$
$$192$$ 13.6911 0.988069
$$193$$ −12.1576 −0.875123 −0.437562 0.899188i $$-0.644158\pi$$
−0.437562 + 0.899188i $$0.644158\pi$$
$$194$$ −10.5349 −0.756363
$$195$$ 0 0
$$196$$ −14.2476 −1.01768
$$197$$ 4.37830 0.311941 0.155970 0.987762i $$-0.450150\pi$$
0.155970 + 0.987762i $$0.450150\pi$$
$$198$$ 13.3194 0.946566
$$199$$ 20.8373 1.47712 0.738558 0.674189i $$-0.235507\pi$$
0.738558 + 0.674189i $$0.235507\pi$$
$$200$$ −5.55889 −0.393073
$$201$$ 22.8827 1.61402
$$202$$ 38.0504 2.67722
$$203$$ −18.0140 −1.26434
$$204$$ −7.61569 −0.533205
$$205$$ −0.267949 −0.0187144
$$206$$ 33.7565 2.35193
$$207$$ 19.0550 1.32441
$$208$$ 0 0
$$209$$ 6.12979 0.424007
$$210$$ −13.4401 −0.927457
$$211$$ −10.6537 −0.733429 −0.366715 0.930333i $$-0.619517\pi$$
−0.366715 + 0.930333i $$0.619517\pi$$
$$212$$ −29.5549 −2.02984
$$213$$ 27.6037 1.89138
$$214$$ 18.3738 1.25600
$$215$$ 0.637263 0.0434610
$$216$$ 31.2870 2.12881
$$217$$ −2.78942 −0.189358
$$218$$ 25.1447 1.70301
$$219$$ −10.5059 −0.709924
$$220$$ −4.52091 −0.304799
$$221$$ 0 0
$$222$$ −5.34335 −0.358622
$$223$$ −21.3393 −1.42899 −0.714494 0.699642i $$-0.753343\pi$$
−0.714494 + 0.699642i $$0.753343\pi$$
$$224$$ 4.57388 0.305605
$$225$$ 4.99102 0.332734
$$226$$ 16.6901 1.11021
$$227$$ −15.6857 −1.04109 −0.520547 0.853833i $$-0.674272\pi$$
−0.520547 + 0.853833i $$0.674272\pi$$
$$228$$ 68.5016 4.53663
$$229$$ 7.62085 0.503600 0.251800 0.967779i $$-0.418977\pi$$
0.251800 + 0.967779i $$0.418977\pi$$
$$230$$ −9.52748 −0.628224
$$231$$ −5.75944 −0.378943
$$232$$ −52.5599 −3.45073
$$233$$ −19.0550 −1.24833 −0.624166 0.781292i $$-0.714561\pi$$
−0.624166 + 0.781292i $$0.714561\pi$$
$$234$$ 0 0
$$235$$ 9.44613 0.616198
$$236$$ −3.13277 −0.203926
$$237$$ 26.3444 1.71125
$$238$$ 3.02985 0.196396
$$239$$ −12.7535 −0.824954 −0.412477 0.910968i $$-0.635336\pi$$
−0.412477 + 0.910968i $$0.635336\pi$$
$$240$$ −15.3134 −0.988473
$$241$$ −25.9288 −1.67022 −0.835111 0.550081i $$-0.814597\pi$$
−0.835111 + 0.550081i $$0.814597\pi$$
$$242$$ 24.5967 1.58114
$$243$$ 14.2356 0.913211
$$244$$ 17.7497 1.13631
$$245$$ −3.37017 −0.215312
$$246$$ −1.89022 −0.120516
$$247$$ 0 0
$$248$$ −8.13878 −0.516813
$$249$$ 14.4671 0.916817
$$250$$ −2.49551 −0.157830
$$251$$ 7.61186 0.480457 0.240228 0.970716i $$-0.422778\pi$$
0.240228 + 0.970716i $$0.422778\pi$$
$$252$$ −40.1996 −2.53234
$$253$$ −4.08277 −0.256682
$$254$$ −3.71706 −0.233229
$$255$$ −1.80144 −0.112811
$$256$$ −32.4572 −2.02857
$$257$$ 0.335783 0.0209456 0.0104728 0.999945i $$-0.496666\pi$$
0.0104728 + 0.999945i $$0.496666\pi$$
$$258$$ 4.49551 0.279878
$$259$$ 1.44310 0.0896700
$$260$$ 0 0
$$261$$ 47.1906 2.92103
$$262$$ −10.2984 −0.636235
$$263$$ −5.37589 −0.331492 −0.165746 0.986169i $$-0.553003\pi$$
−0.165746 + 0.986169i $$0.553003\pi$$
$$264$$ −16.8045 −1.03424
$$265$$ −6.99102 −0.429455
$$266$$ −27.2529 −1.67098
$$267$$ −35.5569 −2.17605
$$268$$ −34.2212 −2.09039
$$269$$ −1.31038 −0.0798956 −0.0399478 0.999202i $$-0.512719\pi$$
−0.0399478 + 0.999202i $$0.512719\pi$$
$$270$$ 14.0454 0.854777
$$271$$ 11.6453 0.707403 0.353701 0.935358i $$-0.384923\pi$$
0.353701 + 0.935358i $$0.384923\pi$$
$$272$$ 3.45214 0.209317
$$273$$ 0 0
$$274$$ −50.1838 −3.03171
$$275$$ −1.06939 −0.0644866
$$276$$ −45.6257 −2.74635
$$277$$ −20.3161 −1.22068 −0.610338 0.792141i $$-0.708967\pi$$
−0.610338 + 0.792141i $$0.708967\pi$$
$$278$$ −51.9697 −3.11693
$$279$$ 7.30735 0.437480
$$280$$ 10.5909 0.632925
$$281$$ 11.8744 0.708366 0.354183 0.935176i $$-0.384759\pi$$
0.354183 + 0.935176i $$0.384759\pi$$
$$282$$ 66.6368 3.96816
$$283$$ −22.6521 −1.34653 −0.673264 0.739402i $$-0.735108\pi$$
−0.673264 + 0.739402i $$0.735108\pi$$
$$284$$ −41.2816 −2.44961
$$285$$ 16.2036 0.959817
$$286$$ 0 0
$$287$$ 0.510500 0.0301339
$$288$$ −11.9820 −0.706048
$$289$$ −16.5939 −0.976112
$$290$$ −23.5953 −1.38556
$$291$$ −11.9336 −0.699562
$$292$$ 15.7117 0.919456
$$293$$ −18.6127 −1.08737 −0.543683 0.839290i $$-0.682971\pi$$
−0.543683 + 0.839290i $$0.682971\pi$$
$$294$$ −23.7745 −1.38656
$$295$$ −0.741035 −0.0431448
$$296$$ 4.21058 0.244735
$$297$$ 6.01882 0.349247
$$298$$ 33.3593 1.93245
$$299$$ 0 0
$$300$$ −11.9506 −0.689970
$$301$$ −1.21412 −0.0699808
$$302$$ 45.6637 2.62765
$$303$$ 43.1024 2.47617
$$304$$ −31.0513 −1.78091
$$305$$ 4.19856 0.240409
$$306$$ −7.93719 −0.453739
$$307$$ −3.14776 −0.179652 −0.0898262 0.995957i $$-0.528631\pi$$
−0.0898262 + 0.995957i $$0.528631\pi$$
$$308$$ 8.61329 0.490788
$$309$$ 38.2384 2.17531
$$310$$ −3.65368 −0.207515
$$311$$ −3.18059 −0.180355 −0.0901774 0.995926i $$-0.528743\pi$$
−0.0901774 + 0.995926i $$0.528743\pi$$
$$312$$ 0 0
$$313$$ 35.3533 1.99829 0.999144 0.0413596i $$-0.0131689\pi$$
0.999144 + 0.0413596i $$0.0131689\pi$$
$$314$$ −6.04484 −0.341130
$$315$$ −9.50894 −0.535768
$$316$$ −39.3982 −2.21632
$$317$$ 13.6357 0.765858 0.382929 0.923778i $$-0.374915\pi$$
0.382929 + 0.923778i $$0.374915\pi$$
$$318$$ −49.3174 −2.76558
$$319$$ −10.1112 −0.566119
$$320$$ −4.84325 −0.270746
$$321$$ 20.8133 1.16168
$$322$$ 18.1519 1.01156
$$323$$ −3.65283 −0.203249
$$324$$ 3.96202 0.220112
$$325$$ 0 0
$$326$$ 39.8854 2.20905
$$327$$ 28.4831 1.57512
$$328$$ 1.48950 0.0822439
$$329$$ −17.9969 −0.992201
$$330$$ −7.54390 −0.415278
$$331$$ 28.7959 1.58277 0.791383 0.611320i $$-0.209361\pi$$
0.791383 + 0.611320i $$0.209361\pi$$
$$332$$ −21.6357 −1.18741
$$333$$ −3.78044 −0.207167
$$334$$ −35.9182 −1.96536
$$335$$ −8.09479 −0.442265
$$336$$ 29.1752 1.59164
$$337$$ 11.7493 0.640026 0.320013 0.947413i $$-0.396313\pi$$
0.320013 + 0.947413i $$0.396313\pi$$
$$338$$ 0 0
$$339$$ 18.9061 1.02684
$$340$$ 2.69407 0.146106
$$341$$ −1.56569 −0.0847871
$$342$$ 71.3934 3.86051
$$343$$ 19.7574 1.06680
$$344$$ −3.54248 −0.190997
$$345$$ −10.7925 −0.581046
$$346$$ 60.7630 3.26664
$$347$$ 1.89977 0.101985 0.0509926 0.998699i $$-0.483762\pi$$
0.0509926 + 0.998699i $$0.483762\pi$$
$$348$$ −112.995 −6.05714
$$349$$ 10.2691 0.549692 0.274846 0.961488i $$-0.411373\pi$$
0.274846 + 0.961488i $$0.411373\pi$$
$$350$$ 4.75447 0.254137
$$351$$ 0 0
$$352$$ 2.56730 0.136838
$$353$$ −0.800589 −0.0426110 −0.0213055 0.999773i $$-0.506782\pi$$
−0.0213055 + 0.999773i $$0.506782\pi$$
$$354$$ −5.22756 −0.277842
$$355$$ −9.76488 −0.518266
$$356$$ 53.1756 2.81830
$$357$$ 3.43213 0.181647
$$358$$ −9.45370 −0.499643
$$359$$ −8.13272 −0.429228 −0.214614 0.976699i $$-0.568849\pi$$
−0.214614 + 0.976699i $$0.568849\pi$$
$$360$$ −27.7445 −1.46226
$$361$$ 13.8564 0.729285
$$362$$ 21.1817 1.11329
$$363$$ 27.8625 1.46240
$$364$$ 0 0
$$365$$ 3.71649 0.194530
$$366$$ 29.6183 1.54817
$$367$$ −20.5265 −1.07147 −0.535737 0.844385i $$-0.679966\pi$$
−0.535737 + 0.844385i $$0.679966\pi$$
$$368$$ 20.6818 1.07811
$$369$$ −1.33734 −0.0696191
$$370$$ 1.89022 0.0982679
$$371$$ 13.3194 0.691507
$$372$$ −17.4969 −0.907173
$$373$$ −17.8058 −0.921951 −0.460976 0.887413i $$-0.652500\pi$$
−0.460976 + 0.887413i $$0.652500\pi$$
$$374$$ 1.70064 0.0879382
$$375$$ −2.82684 −0.145977
$$376$$ −52.5100 −2.70800
$$377$$ 0 0
$$378$$ −26.7595 −1.37636
$$379$$ −2.04555 −0.105073 −0.0525363 0.998619i $$-0.516731\pi$$
−0.0525363 + 0.998619i $$0.516731\pi$$
$$380$$ −24.2326 −1.24311
$$381$$ −4.21058 −0.215714
$$382$$ 13.5833 0.694982
$$383$$ 7.90521 0.403937 0.201969 0.979392i $$-0.435266\pi$$
0.201969 + 0.979392i $$0.435266\pi$$
$$384$$ −47.7391 −2.43618
$$385$$ 2.03741 0.103836
$$386$$ 30.3394 1.54423
$$387$$ 3.18059 0.161679
$$388$$ 17.8469 0.906037
$$389$$ −9.21171 −0.467052 −0.233526 0.972351i $$-0.575026\pi$$
−0.233526 + 0.972351i $$0.575026\pi$$
$$390$$ 0 0
$$391$$ 2.43298 0.123041
$$392$$ 18.7344 0.946229
$$393$$ −11.6657 −0.588456
$$394$$ −10.9261 −0.550448
$$395$$ −9.31937 −0.468908
$$396$$ −22.5639 −1.13388
$$397$$ 6.35438 0.318917 0.159458 0.987205i $$-0.449025\pi$$
0.159458 + 0.987205i $$0.449025\pi$$
$$398$$ −51.9996 −2.60651
$$399$$ −30.8713 −1.54550
$$400$$ 5.41713 0.270857
$$401$$ −4.16920 −0.208200 −0.104100 0.994567i $$-0.533196\pi$$
−0.104100 + 0.994567i $$0.533196\pi$$
$$402$$ −57.1038 −2.84808
$$403$$ 0 0
$$404$$ −64.4600 −3.20701
$$405$$ 0.937188 0.0465692
$$406$$ 44.9541 2.23103
$$407$$ 0.810008 0.0401506
$$408$$ 10.0140 0.495767
$$409$$ −10.1681 −0.502778 −0.251389 0.967886i $$-0.580887\pi$$
−0.251389 + 0.967886i $$0.580887\pi$$
$$410$$ 0.668669 0.0330232
$$411$$ −56.8467 −2.80404
$$412$$ −57.1858 −2.81734
$$413$$ 1.41183 0.0694716
$$414$$ −47.5518 −2.33704
$$415$$ −5.11778 −0.251222
$$416$$ 0 0
$$417$$ −58.8697 −2.88286
$$418$$ −15.2969 −0.748198
$$419$$ 28.5909 1.39676 0.698378 0.715730i $$-0.253906\pi$$
0.698378 + 0.715730i $$0.253906\pi$$
$$420$$ 22.7685 1.11099
$$421$$ −2.01797 −0.0983498 −0.0491749 0.998790i $$-0.515659\pi$$
−0.0491749 + 0.998790i $$0.515659\pi$$
$$422$$ 26.5863 1.29420
$$423$$ 47.1458 2.29231
$$424$$ 38.8623 1.88732
$$425$$ 0.637263 0.0309118
$$426$$ −68.8853 −3.33750
$$427$$ −7.99915 −0.387106
$$428$$ −31.1264 −1.50455
$$429$$ 0 0
$$430$$ −1.59030 −0.0766908
$$431$$ 20.6123 0.992860 0.496430 0.868077i $$-0.334644\pi$$
0.496430 + 0.868077i $$0.334644\pi$$
$$432$$ −30.4891 −1.46691
$$433$$ 29.4356 1.41458 0.707292 0.706921i $$-0.249917\pi$$
0.707292 + 0.706921i $$0.249917\pi$$
$$434$$ 6.96103 0.334140
$$435$$ −26.7281 −1.28151
$$436$$ −42.5967 −2.04001
$$437$$ −21.8841 −1.04686
$$438$$ 26.2176 1.25272
$$439$$ 16.9520 0.809077 0.404538 0.914521i $$-0.367432\pi$$
0.404538 + 0.914521i $$0.367432\pi$$
$$440$$ 5.94462 0.283398
$$441$$ −16.8205 −0.800978
$$442$$ 0 0
$$443$$ −24.1399 −1.14692 −0.573461 0.819233i $$-0.694400\pi$$
−0.573461 + 0.819233i $$0.694400\pi$$
$$444$$ 9.05199 0.429588
$$445$$ 12.5783 0.596270
$$446$$ 53.2525 2.52158
$$447$$ 37.7885 1.78733
$$448$$ 9.22742 0.435955
$$449$$ 20.8630 0.984585 0.492293 0.870430i $$-0.336159\pi$$
0.492293 + 0.870430i $$0.336159\pi$$
$$450$$ −12.4551 −0.587140
$$451$$ 0.286542 0.0134927
$$452$$ −28.2742 −1.32990
$$453$$ 51.7265 2.43032
$$454$$ 39.1437 1.83710
$$455$$ 0 0
$$456$$ −90.0739 −4.21810
$$457$$ 30.5659 1.42981 0.714906 0.699220i $$-0.246469\pi$$
0.714906 + 0.699220i $$0.246469\pi$$
$$458$$ −19.0179 −0.888648
$$459$$ −3.58669 −0.167413
$$460$$ 16.1402 0.752541
$$461$$ −4.67822 −0.217887 −0.108943 0.994048i $$-0.534747\pi$$
−0.108943 + 0.994048i $$0.534747\pi$$
$$462$$ 14.3727 0.668680
$$463$$ 14.0011 0.650688 0.325344 0.945596i $$-0.394520\pi$$
0.325344 + 0.945596i $$0.394520\pi$$
$$464$$ 51.2196 2.37781
$$465$$ −4.13878 −0.191931
$$466$$ 47.5518 2.20280
$$467$$ −6.98506 −0.323230 −0.161615 0.986854i $$-0.551670\pi$$
−0.161615 + 0.986854i $$0.551670\pi$$
$$468$$ 0 0
$$469$$ 15.4223 0.712135
$$470$$ −23.5729 −1.08734
$$471$$ −6.84742 −0.315513
$$472$$ 4.11933 0.189608
$$473$$ −0.681482 −0.0313346
$$474$$ −65.7425 −3.01965
$$475$$ −5.73205 −0.263005
$$476$$ −5.13277 −0.235260
$$477$$ −34.8923 −1.59761
$$478$$ 31.8264 1.45571
$$479$$ 16.2888 0.744252 0.372126 0.928182i $$-0.378629\pi$$
0.372126 + 0.928182i $$0.378629\pi$$
$$480$$ 6.78645 0.309758
$$481$$ 0 0
$$482$$ 64.7056 2.94726
$$483$$ 20.5619 0.935600
$$484$$ −41.6685 −1.89402
$$485$$ 4.22155 0.191691
$$486$$ −35.5249 −1.61144
$$487$$ −20.0409 −0.908139 −0.454069 0.890966i $$-0.650028\pi$$
−0.454069 + 0.890966i $$0.650028\pi$$
$$488$$ −23.3393 −1.05652
$$489$$ 45.1810 2.04316
$$490$$ 8.41027 0.379937
$$491$$ −15.7983 −0.712969 −0.356484 0.934301i $$-0.616025\pi$$
−0.356484 + 0.934301i $$0.616025\pi$$
$$492$$ 3.20216 0.144365
$$493$$ 6.02540 0.271370
$$494$$ 0 0
$$495$$ −5.33734 −0.239896
$$496$$ 7.93123 0.356123
$$497$$ 18.6042 0.834511
$$498$$ −36.1028 −1.61781
$$499$$ −1.24651 −0.0558016 −0.0279008 0.999611i $$-0.508882\pi$$
−0.0279008 + 0.999611i $$0.508882\pi$$
$$500$$ 4.22756 0.189062
$$501$$ −40.6871 −1.81777
$$502$$ −18.9955 −0.847809
$$503$$ −7.65345 −0.341250 −0.170625 0.985336i $$-0.554579\pi$$
−0.170625 + 0.985336i $$0.554579\pi$$
$$504$$ 52.8592 2.35453
$$505$$ −15.2476 −0.678508
$$506$$ 10.1886 0.452938
$$507$$ 0 0
$$508$$ 6.29695 0.279382
$$509$$ −25.7241 −1.14020 −0.570101 0.821575i $$-0.693096\pi$$
−0.570101 + 0.821575i $$0.693096\pi$$
$$510$$ 4.49551 0.199064
$$511$$ −7.08070 −0.313232
$$512$$ 47.2215 2.08691
$$513$$ 32.2616 1.42438
$$514$$ −0.837948 −0.0369603
$$515$$ −13.5269 −0.596067
$$516$$ −7.61569 −0.335262
$$517$$ −10.1016 −0.444268
$$518$$ −3.60127 −0.158231
$$519$$ 68.8305 3.02132
$$520$$ 0 0
$$521$$ −30.1519 −1.32098 −0.660490 0.750835i $$-0.729651\pi$$
−0.660490 + 0.750835i $$0.729651\pi$$
$$522$$ −117.765 −5.15442
$$523$$ 3.93752 0.172176 0.0860880 0.996288i $$-0.472563\pi$$
0.0860880 + 0.996288i $$0.472563\pi$$
$$524$$ 17.4461 0.762138
$$525$$ 5.38573 0.235052
$$526$$ 13.4156 0.584947
$$527$$ 0.933018 0.0406429
$$528$$ 16.3759 0.712672
$$529$$ −8.42399 −0.366261
$$530$$ 17.4461 0.757812
$$531$$ −3.69852 −0.160502
$$532$$ 46.1682 2.00165
$$533$$ 0 0
$$534$$ 88.7326 3.83983
$$535$$ −7.36274 −0.318319
$$536$$ 44.9980 1.94362
$$537$$ −10.7089 −0.462122
$$538$$ 3.27007 0.140983
$$539$$ 3.60402 0.155236
$$540$$ −23.7939 −1.02393
$$541$$ 15.8881 0.683083 0.341541 0.939867i $$-0.389051\pi$$
0.341541 + 0.939867i $$0.389051\pi$$
$$542$$ −29.0610 −1.24828
$$543$$ 23.9940 1.02968
$$544$$ −1.52989 −0.0655935
$$545$$ −10.0760 −0.431607
$$546$$ 0 0
$$547$$ −6.56107 −0.280531 −0.140266 0.990114i $$-0.544796\pi$$
−0.140266 + 0.990114i $$0.544796\pi$$
$$548$$ 85.0147 3.63165
$$549$$ 20.9551 0.894341
$$550$$ 2.66867 0.113792
$$551$$ −54.1972 −2.30888
$$552$$ 59.9941 2.55352
$$553$$ 17.7554 0.755035
$$554$$ 50.6990 2.15399
$$555$$ 2.14119 0.0908883
$$556$$ 88.0401 3.73373
$$557$$ −7.85006 −0.332618 −0.166309 0.986074i $$-0.553185\pi$$
−0.166309 + 0.986074i $$0.553185\pi$$
$$558$$ −18.2356 −0.771973
$$559$$ 0 0
$$560$$ −10.3208 −0.436133
$$561$$ 1.92644 0.0813344
$$562$$ −29.6326 −1.24998
$$563$$ −15.5595 −0.655755 −0.327878 0.944720i $$-0.606333\pi$$
−0.327878 + 0.944720i $$0.606333\pi$$
$$564$$ −112.887 −4.75341
$$565$$ −6.68806 −0.281369
$$566$$ 56.5285 2.37607
$$567$$ −1.78554 −0.0749857
$$568$$ 54.2819 2.27762
$$569$$ 3.47915 0.145853 0.0729267 0.997337i $$-0.476766\pi$$
0.0729267 + 0.997337i $$0.476766\pi$$
$$570$$ −40.4362 −1.69368
$$571$$ −21.5118 −0.900240 −0.450120 0.892968i $$-0.648619\pi$$
−0.450120 + 0.892968i $$0.648619\pi$$
$$572$$ 0 0
$$573$$ 15.3868 0.642791
$$574$$ −1.27396 −0.0531739
$$575$$ 3.81785 0.159216
$$576$$ −24.1727 −1.00720
$$577$$ −9.97608 −0.415310 −0.207655 0.978202i $$-0.566583\pi$$
−0.207655 + 0.978202i $$0.566583\pi$$
$$578$$ 41.4102 1.72244
$$579$$ 34.3676 1.42827
$$580$$ 39.9721 1.65975
$$581$$ 9.75045 0.404517
$$582$$ 29.7805 1.23444
$$583$$ 7.47612 0.309629
$$584$$ −20.6595 −0.854898
$$585$$ 0 0
$$586$$ 46.4482 1.91876
$$587$$ 24.0571 0.992945 0.496472 0.868053i $$-0.334628\pi$$
0.496472 + 0.868053i $$0.334628\pi$$
$$588$$ 40.2756 1.66094
$$589$$ −8.39230 −0.345799
$$590$$ 1.84926 0.0761328
$$591$$ −12.3767 −0.509111
$$592$$ −4.10320 −0.168641
$$593$$ −0.940219 −0.0386102 −0.0193051 0.999814i $$-0.506145\pi$$
−0.0193051 + 0.999814i $$0.506145\pi$$
$$594$$ −15.0200 −0.616279
$$595$$ −1.21412 −0.0497741
$$596$$ −56.5130 −2.31486
$$597$$ −58.9037 −2.41077
$$598$$ 0 0
$$599$$ −11.4270 −0.466896 −0.233448 0.972369i $$-0.575001\pi$$
−0.233448 + 0.972369i $$0.575001\pi$$
$$600$$ 15.7141 0.641525
$$601$$ −36.0431 −1.47023 −0.735114 0.677944i $$-0.762871\pi$$
−0.735114 + 0.677944i $$0.762871\pi$$
$$602$$ 3.02985 0.123487
$$603$$ −40.4012 −1.64526
$$604$$ −77.3574 −3.14763
$$605$$ −9.85641 −0.400720
$$606$$ −107.562 −4.36942
$$607$$ 39.8907 1.61911 0.809557 0.587041i $$-0.199707\pi$$
0.809557 + 0.587041i $$0.199707\pi$$
$$608$$ 13.7610 0.558084
$$609$$ 50.9227 2.06349
$$610$$ −10.4775 −0.424223
$$611$$ 0 0
$$612$$ 13.4461 0.543528
$$613$$ 0.345472 0.0139535 0.00697673 0.999976i $$-0.497779\pi$$
0.00697673 + 0.999976i $$0.497779\pi$$
$$614$$ 7.85527 0.317013
$$615$$ 0.757449 0.0305433
$$616$$ −11.3258 −0.456328
$$617$$ −38.6850 −1.55740 −0.778700 0.627397i $$-0.784121\pi$$
−0.778700 + 0.627397i $$0.784121\pi$$
$$618$$ −95.4242 −3.83852
$$619$$ −14.8971 −0.598764 −0.299382 0.954133i $$-0.596781\pi$$
−0.299382 + 0.954133i $$0.596781\pi$$
$$620$$ 6.18958 0.248579
$$621$$ −21.4879 −0.862281
$$622$$ 7.93719 0.318252
$$623$$ −23.9644 −0.960113
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −88.2245 −3.52616
$$627$$ −17.3279 −0.692011
$$628$$ 10.2404 0.408635
$$629$$ −0.482694 −0.0192463
$$630$$ 23.7296 0.945412
$$631$$ 38.8450 1.54640 0.773198 0.634165i $$-0.218656\pi$$
0.773198 + 0.634165i $$0.218656\pi$$
$$632$$ 51.8053 2.06071
$$633$$ 30.1162 1.19701
$$634$$ −34.0280 −1.35143
$$635$$ 1.48950 0.0591090
$$636$$ 83.5470 3.31285
$$637$$ 0 0
$$638$$ 25.2326 0.998967
$$639$$ −48.7367 −1.92799
$$640$$ 16.8878 0.667549
$$641$$ 37.1816 1.46859 0.734293 0.678832i $$-0.237514\pi$$
0.734293 + 0.678832i $$0.237514\pi$$
$$642$$ −51.9397 −2.04990
$$643$$ −9.10377 −0.359018 −0.179509 0.983756i $$-0.557451\pi$$
−0.179509 + 0.983756i $$0.557451\pi$$
$$644$$ −30.7505 −1.21174
$$645$$ −1.80144 −0.0709316
$$646$$ 9.11565 0.358651
$$647$$ −19.1224 −0.751778 −0.375889 0.926665i $$-0.622663\pi$$
−0.375889 + 0.926665i $$0.622663\pi$$
$$648$$ −5.20972 −0.204657
$$649$$ 0.792455 0.0311066
$$650$$ 0 0
$$651$$ 7.88525 0.309047
$$652$$ −67.5686 −2.64619
$$653$$ 34.6324 1.35527 0.677636 0.735397i $$-0.263004\pi$$
0.677636 + 0.735397i $$0.263004\pi$$
$$654$$ −71.0799 −2.77944
$$655$$ 4.12676 0.161246
$$656$$ −1.45152 −0.0566722
$$657$$ 18.5491 0.723667
$$658$$ 44.9114 1.75083
$$659$$ −6.69852 −0.260937 −0.130469 0.991452i $$-0.541648\pi$$
−0.130469 + 0.991452i $$0.541648\pi$$
$$660$$ 12.7799 0.497456
$$661$$ −6.02758 −0.234446 −0.117223 0.993106i $$-0.537399\pi$$
−0.117223 + 0.993106i $$0.537399\pi$$
$$662$$ −71.8604 −2.79294
$$663$$ 0 0
$$664$$ 28.4492 1.10404
$$665$$ 10.9208 0.423489
$$666$$ 9.43412 0.365565
$$667$$ 36.0983 1.39773
$$668$$ 60.8479 2.35428
$$669$$ 60.3228 2.33222
$$670$$ 20.2006 0.780417
$$671$$ −4.48990 −0.173330
$$672$$ −12.9296 −0.498771
$$673$$ 23.3568 0.900338 0.450169 0.892943i $$-0.351364\pi$$
0.450169 + 0.892943i $$0.351364\pi$$
$$674$$ −29.3205 −1.12938
$$675$$ −5.62828 −0.216633
$$676$$ 0 0
$$677$$ −45.4042 −1.74503 −0.872513 0.488590i $$-0.837511\pi$$
−0.872513 + 0.488590i $$0.837511\pi$$
$$678$$ −47.1802 −1.81195
$$679$$ −8.04295 −0.308660
$$680$$ −3.54248 −0.135848
$$681$$ 44.3408 1.69914
$$682$$ 3.90720 0.149615
$$683$$ 25.4978 0.975645 0.487823 0.872943i $$-0.337791\pi$$
0.487823 + 0.872943i $$0.337791\pi$$
$$684$$ −120.945 −4.62445
$$685$$ 20.1096 0.768350
$$686$$ −49.3047 −1.88246
$$687$$ −21.5429 −0.821913
$$688$$ 3.45214 0.131612
$$689$$ 0 0
$$690$$ 26.9327 1.02531
$$691$$ 6.59630 0.250935 0.125468 0.992098i $$-0.459957\pi$$
0.125468 + 0.992098i $$0.459957\pi$$
$$692$$ −102.937 −3.91306
$$693$$ 10.1688 0.386279
$$694$$ −4.74090 −0.179962
$$695$$ 20.8253 0.789948
$$696$$ 148.578 5.63185
$$697$$ −0.170754 −0.00646778
$$698$$ −25.6266 −0.969981
$$699$$ 53.8653 2.03737
$$700$$ −8.05440 −0.304428
$$701$$ 29.2474 1.10466 0.552329 0.833626i $$-0.313739\pi$$
0.552329 + 0.833626i $$0.313739\pi$$
$$702$$ 0 0
$$703$$ 4.34174 0.163752
$$704$$ 5.17932 0.195203
$$705$$ −26.7027 −1.00568
$$706$$ 1.99787 0.0751910
$$707$$ 29.0499 1.09253
$$708$$ 8.85584 0.332823
$$709$$ −10.9335 −0.410614 −0.205307 0.978698i $$-0.565819\pi$$
−0.205307 + 0.978698i $$0.565819\pi$$
$$710$$ 24.3683 0.914527
$$711$$ −46.5131 −1.74438
$$712$$ −69.9216 −2.62042
$$713$$ 5.58973 0.209337
$$714$$ −8.56490 −0.320533
$$715$$ 0 0
$$716$$ 16.0152 0.598516
$$717$$ 36.0520 1.34639
$$718$$ 20.2953 0.757412
$$719$$ 16.0598 0.598929 0.299464 0.954107i $$-0.403192\pi$$
0.299464 + 0.954107i $$0.403192\pi$$
$$720$$ 27.0370 1.00761
$$721$$ 25.7716 0.959786
$$722$$ −34.5788 −1.28689
$$723$$ 73.2966 2.72593
$$724$$ −35.8833 −1.33359
$$725$$ 9.45512 0.351154
$$726$$ −69.5310 −2.58054
$$727$$ 51.3754 1.90541 0.952704 0.303900i $$-0.0982889\pi$$
0.952704 + 0.303900i $$0.0982889\pi$$
$$728$$ 0 0
$$729$$ −43.0532 −1.59456
$$730$$ −9.27453 −0.343266
$$731$$ 0.406104 0.0150203
$$732$$ −50.1754 −1.85454
$$733$$ 9.82358 0.362842 0.181421 0.983406i $$-0.441930\pi$$
0.181421 + 0.983406i $$0.441930\pi$$
$$734$$ 51.2240 1.89071
$$735$$ 9.52691 0.351406
$$736$$ −9.16560 −0.337848
$$737$$ 8.65648 0.318866
$$738$$ 3.33734 0.122849
$$739$$ −49.0842 −1.80559 −0.902797 0.430068i $$-0.858490\pi$$
−0.902797 + 0.430068i $$0.858490\pi$$
$$740$$ −3.20216 −0.117714
$$741$$ 0 0
$$742$$ −33.2386 −1.22023
$$743$$ −40.8375 −1.49818 −0.749091 0.662467i $$-0.769510\pi$$
−0.749091 + 0.662467i $$0.769510\pi$$
$$744$$ 23.0070 0.843478
$$745$$ −13.3678 −0.489757
$$746$$ 44.4346 1.62687
$$747$$ −25.5429 −0.934566
$$748$$ −2.88101 −0.105340
$$749$$ 14.0276 0.512557
$$750$$ 7.05440 0.257590
$$751$$ −2.72680 −0.0995024 −0.0497512 0.998762i $$-0.515843\pi$$
−0.0497512 + 0.998762i $$0.515843\pi$$
$$752$$ 51.1710 1.86601
$$753$$ −21.5175 −0.784142
$$754$$ 0 0
$$755$$ −18.2984 −0.665946
$$756$$ 45.3324 1.64872
$$757$$ 14.8060 0.538134 0.269067 0.963121i $$-0.413285\pi$$
0.269067 + 0.963121i $$0.413285\pi$$
$$758$$ 5.10468 0.185410
$$759$$ 11.5413 0.418924
$$760$$ 31.8638 1.15582
$$761$$ 11.3689 0.412122 0.206061 0.978539i $$-0.433935\pi$$
0.206061 + 0.978539i $$0.433935\pi$$
$$762$$ 10.5075 0.380647
$$763$$ 19.1969 0.694973
$$764$$ −23.0110 −0.832510
$$765$$ 3.18059 0.114994
$$766$$ −19.7275 −0.712784
$$767$$ 0 0
$$768$$ 91.7512 3.31078
$$769$$ 21.0562 0.759307 0.379654 0.925129i $$-0.376043\pi$$
0.379654 + 0.925129i $$0.376043\pi$$
$$770$$ −5.08438 −0.183228
$$771$$ −0.949203 −0.0341847
$$772$$ −51.3970 −1.84982
$$773$$ −14.0829 −0.506526 −0.253263 0.967397i $$-0.581504\pi$$
−0.253263 + 0.967397i $$0.581504\pi$$
$$774$$ −7.93719 −0.285296
$$775$$ 1.46410 0.0525921
$$776$$ −23.4671 −0.842421
$$777$$ −4.07941 −0.146348
$$778$$ 22.9879 0.824156
$$779$$ 1.53590 0.0550293
$$780$$ 0 0
$$781$$ 10.4425 0.373660
$$782$$ −6.07151 −0.217117
$$783$$ −53.2160 −1.90179
$$784$$ −18.2566 −0.652023
$$785$$ 2.42229 0.0864552
$$786$$ 29.1118 1.03838
$$787$$ −33.0242 −1.17719 −0.588593 0.808429i $$-0.700318\pi$$
−0.588593 + 0.808429i $$0.700318\pi$$
$$788$$ 18.5095 0.659374
$$789$$ 15.1968 0.541020
$$790$$ 23.2566 0.827431
$$791$$ 12.7422 0.453060
$$792$$ 29.6697 1.05427
$$793$$ 0 0
$$794$$ −15.8574 −0.562758
$$795$$ 19.7625 0.700903
$$796$$ 88.0909 3.12230
$$797$$ −16.9416 −0.600102 −0.300051 0.953923i $$-0.597004\pi$$
−0.300051 + 0.953923i $$0.597004\pi$$
$$798$$ 77.0395 2.72717
$$799$$ 6.01967 0.212961
$$800$$ −2.40072 −0.0848783
$$801$$ 62.7787 2.21817
$$802$$ 10.4043 0.367387
$$803$$ −3.97437 −0.140253
$$804$$ 96.7378 3.41168
$$805$$ −7.27382 −0.256369
$$806$$ 0 0
$$807$$ 3.70425 0.130396
$$808$$ 84.7596 2.98183
$$809$$ −51.7635 −1.81991 −0.909954 0.414708i $$-0.863884\pi$$
−0.909954 + 0.414708i $$0.863884\pi$$
$$810$$ −2.33876 −0.0821756
$$811$$ −22.6699 −0.796047 −0.398023 0.917375i $$-0.630304\pi$$
−0.398023 + 0.917375i $$0.630304\pi$$
$$812$$ −76.1553 −2.67253
$$813$$ −32.9194 −1.15453
$$814$$ −2.02138 −0.0708494
$$815$$ −15.9829 −0.559856
$$816$$ −9.75864 −0.341621
$$817$$ −3.65283 −0.127796
$$818$$ 25.3745 0.887198
$$819$$ 0 0
$$820$$ −1.13277 −0.0395581
$$821$$ −28.6631 −1.00035 −0.500174 0.865925i $$-0.666731\pi$$
−0.500174 + 0.865925i $$0.666731\pi$$
$$822$$ 141.861 4.94799
$$823$$ −25.8327 −0.900472 −0.450236 0.892910i $$-0.648660\pi$$
−0.450236 + 0.892910i $$0.648660\pi$$
$$824$$ 75.1946 2.61953
$$825$$ 3.02299 0.105247
$$826$$ −3.52323 −0.122589
$$827$$ 16.0820 0.559227 0.279613 0.960113i $$-0.409794\pi$$
0.279613 + 0.960113i $$0.409794\pi$$
$$828$$ 80.5560 2.79951
$$829$$ −22.5818 −0.784298 −0.392149 0.919902i $$-0.628268\pi$$
−0.392149 + 0.919902i $$0.628268\pi$$
$$830$$ 12.7715 0.443304
$$831$$ 57.4304 1.99224
$$832$$ 0 0
$$833$$ −2.14768 −0.0744128
$$834$$ 146.910 5.08707
$$835$$ 14.3932 0.498096
$$836$$ 25.9141 0.896257
$$837$$ −8.24037 −0.284829
$$838$$ −71.3487 −2.46470
$$839$$ 17.8440 0.616042 0.308021 0.951380i $$-0.400333\pi$$
0.308021 + 0.951380i $$0.400333\pi$$
$$840$$ −29.9387 −1.03298
$$841$$ 60.3992 2.08273
$$842$$ 5.03586 0.173547
$$843$$ −33.5669 −1.15611
$$844$$ −45.0390 −1.55031
$$845$$ 0 0
$$846$$ −117.653 −4.04498
$$847$$ 18.7785 0.645239
$$848$$ −37.8713 −1.30050
$$849$$ 64.0339 2.19764
$$850$$ −1.59030 −0.0545467
$$851$$ −2.89183 −0.0991306
$$852$$ 116.696 3.99795
$$853$$ −19.7936 −0.677720 −0.338860 0.940837i $$-0.610041\pi$$
−0.338860 + 0.940837i $$0.610041\pi$$
$$854$$ 19.9619 0.683083
$$855$$ −28.6088 −0.978399
$$856$$ 40.9286 1.39891
$$857$$ −11.7302 −0.400696 −0.200348 0.979725i $$-0.564207\pi$$
−0.200348 + 0.979725i $$0.564207\pi$$
$$858$$ 0 0
$$859$$ 5.37452 0.183376 0.0916882 0.995788i $$-0.470774\pi$$
0.0916882 + 0.995788i $$0.470774\pi$$
$$860$$ 2.69407 0.0918669
$$861$$ −1.44310 −0.0491808
$$862$$ −51.4382 −1.75199
$$863$$ 25.3234 0.862017 0.431008 0.902348i $$-0.358158\pi$$
0.431008 + 0.902348i $$0.358158\pi$$
$$864$$ 13.5119 0.459685
$$865$$ −24.3489 −0.827889
$$866$$ −73.4567 −2.49616
$$867$$ 46.9083 1.59309
$$868$$ −11.7925 −0.400262
$$869$$ 9.96603 0.338075
$$870$$ 66.7001 2.26135
$$871$$ 0 0
$$872$$ 56.0112 1.89678
$$873$$ 21.0698 0.713106
$$874$$ 54.6120 1.84728
$$875$$ −1.90521 −0.0644079
$$876$$ −44.4144 −1.50062
$$877$$ −20.6915 −0.698703 −0.349352 0.936992i $$-0.613598\pi$$
−0.349352 + 0.936992i $$0.613598\pi$$
$$878$$ −42.3040 −1.42769
$$879$$ 52.6151 1.77466
$$880$$ −5.79302 −0.195283
$$881$$ 48.3993 1.63061 0.815307 0.579029i $$-0.196568\pi$$
0.815307 + 0.579029i $$0.196568\pi$$
$$882$$ 41.9758 1.41340
$$883$$ 45.8550 1.54314 0.771572 0.636142i $$-0.219471\pi$$
0.771572 + 0.636142i $$0.219471\pi$$
$$884$$ 0 0
$$885$$ 2.09479 0.0704155
$$886$$ 60.2413 2.02385
$$887$$ −1.08234 −0.0363413 −0.0181707 0.999835i $$-0.505784\pi$$
−0.0181707 + 0.999835i $$0.505784\pi$$
$$888$$ −11.9026 −0.399426
$$889$$ −2.83781 −0.0951772
$$890$$ −31.3893 −1.05217
$$891$$ −1.00222 −0.0335756
$$892$$ −90.2133 −3.02056
$$893$$ −54.1457 −1.81192
$$894$$ −94.3015 −3.15391
$$895$$ 3.78829 0.126628
$$896$$ −32.1749 −1.07489
$$897$$ 0 0
$$898$$ −52.0637 −1.73739
$$899$$ 13.8433 0.461698
$$900$$ 21.0998 0.703327
$$901$$ −4.45512 −0.148421
$$902$$ −0.715068 −0.0238092
$$903$$ 3.43213 0.114214
$$904$$ 37.1782 1.23653
$$905$$ −8.48794 −0.282149
$$906$$ −129.084 −4.28853
$$907$$ −45.5307 −1.51182 −0.755910 0.654675i $$-0.772805\pi$$
−0.755910 + 0.654675i $$0.772805\pi$$
$$908$$ −66.3120 −2.20064
$$909$$ −76.1009 −2.52411
$$910$$ 0 0
$$911$$ 39.7417 1.31670 0.658350 0.752712i $$-0.271255\pi$$
0.658350 + 0.752712i $$0.271255\pi$$
$$912$$ 87.7770 2.90659
$$913$$ 5.47290 0.181126
$$914$$ −76.2774 −2.52303
$$915$$ −11.8687 −0.392365
$$916$$ 32.2176 1.06450
$$917$$ −7.86236 −0.259638
$$918$$ 8.95062 0.295415
$$919$$ 46.9938 1.55018 0.775091 0.631850i $$-0.217704\pi$$
0.775091 + 0.631850i $$0.217704\pi$$
$$920$$ −21.2230 −0.699702
$$921$$ 8.89822 0.293206
$$922$$ 11.6745 0.384481
$$923$$ 0 0
$$924$$ −24.3484 −0.801002
$$925$$ −0.757449 −0.0249048
$$926$$ −34.9399 −1.14820
$$927$$ −67.5130 −2.21742
$$928$$ −22.6991 −0.745134
$$929$$ 15.2213 0.499395 0.249698 0.968324i $$-0.419669\pi$$
0.249698 + 0.968324i $$0.419669\pi$$
$$930$$ 10.3284 0.338680
$$931$$ 19.3180 0.633121
$$932$$ −80.5560 −2.63870
$$933$$ 8.99102 0.294353
$$934$$ 17.4313 0.570369
$$935$$ −0.681482 −0.0222869
$$936$$ 0 0
$$937$$ 6.07285 0.198392 0.0991958 0.995068i $$-0.468373\pi$$
0.0991958 + 0.995068i $$0.468373\pi$$
$$938$$ −38.4864 −1.25663
$$939$$ −99.9382 −3.26136
$$940$$ 39.9341 1.30251
$$941$$ 0.0496576 0.00161879 0.000809396 1.00000i $$-0.499742\pi$$
0.000809396 1.00000i $$0.499742\pi$$
$$942$$ 17.0878 0.556750
$$943$$ −1.02299 −0.0333132
$$944$$ −4.01429 −0.130654
$$945$$ 10.7231 0.348821
$$946$$ 1.70064 0.0552927
$$947$$ 18.6581 0.606308 0.303154 0.952942i $$-0.401960\pi$$
0.303154 + 0.952942i $$0.401960\pi$$
$$948$$ 111.372 3.61720
$$949$$ 0 0
$$950$$ 14.3044 0.464095
$$951$$ −38.5459 −1.24994
$$952$$ 6.74917 0.218742
$$953$$ −1.52953 −0.0495463 −0.0247731 0.999693i $$-0.507886\pi$$
−0.0247731 + 0.999693i $$0.507886\pi$$
$$954$$ 87.0739 2.81912
$$955$$ −5.44310 −0.176135
$$956$$ −53.9161 −1.74377
$$957$$ 28.5827 0.923948
$$958$$ −40.6487 −1.31330
$$959$$ −38.3131 −1.23720
$$960$$ 13.6911 0.441878
$$961$$ −28.8564 −0.930852
$$962$$ 0 0
$$963$$ −36.7475 −1.18417
$$964$$ −109.616 −3.53048
$$965$$ −12.1576 −0.391367
$$966$$ −51.3124 −1.65095
$$967$$ −32.1716 −1.03457 −0.517285 0.855813i $$-0.673057\pi$$
−0.517285 + 0.855813i $$0.673057\pi$$
$$968$$ 54.7907 1.76104
$$969$$ 10.3259 0.331717
$$970$$ −10.5349 −0.338256
$$971$$ −17.2541 −0.553710 −0.276855 0.960912i $$-0.589292\pi$$
−0.276855 + 0.960912i $$0.589292\pi$$
$$972$$ 60.1816 1.93033
$$973$$ −39.6766 −1.27197
$$974$$ 50.0122 1.60249
$$975$$ 0 0
$$976$$ 22.7442 0.728023
$$977$$ −15.7228 −0.503018 −0.251509 0.967855i $$-0.580927\pi$$
−0.251509 + 0.967855i $$0.580927\pi$$
$$978$$ −112.750 −3.60533
$$979$$ −13.4511 −0.429900
$$980$$ −14.2476 −0.455122
$$981$$ −50.2893 −1.60561
$$982$$ 39.4248 1.25810
$$983$$ −38.5356 −1.22910 −0.614548 0.788880i $$-0.710662\pi$$
−0.614548 + 0.788880i $$0.710662\pi$$
$$984$$ −4.21058 −0.134228
$$985$$ 4.37830 0.139504
$$986$$ −15.0364 −0.478857
$$987$$ 50.8743 1.61935
$$988$$ 0 0
$$989$$ 2.43298 0.0773642
$$990$$ 13.3194 0.423317
$$991$$ 8.59143 0.272916 0.136458 0.990646i $$-0.456428\pi$$
0.136458 + 0.990646i $$0.456428\pi$$
$$992$$ −3.51490 −0.111598
$$993$$ −81.4014 −2.58320
$$994$$ −46.4268 −1.47257
$$995$$ 20.8373 0.660587
$$996$$ 61.1606 1.93795
$$997$$ −20.5374 −0.650425 −0.325213 0.945641i $$-0.605436\pi$$
−0.325213 + 0.945641i $$0.605436\pi$$
$$998$$ 3.11069 0.0984670
$$999$$ 4.26313 0.134880
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 845.2.a.l.1.1 4
3.2 odd 2 7605.2.a.cj.1.4 4
5.4 even 2 4225.2.a.bl.1.4 4
13.2 odd 12 65.2.m.a.56.1 yes 8
13.3 even 3 845.2.e.n.191.4 8
13.4 even 6 845.2.e.m.146.1 8
13.5 odd 4 845.2.c.g.506.8 8
13.6 odd 12 845.2.m.g.361.4 8
13.7 odd 12 65.2.m.a.36.1 8
13.8 odd 4 845.2.c.g.506.1 8
13.9 even 3 845.2.e.n.146.4 8
13.10 even 6 845.2.e.m.191.1 8
13.11 odd 12 845.2.m.g.316.4 8
13.12 even 2 845.2.a.m.1.4 4
39.2 even 12 585.2.bu.c.316.4 8
39.20 even 12 585.2.bu.c.361.4 8
39.38 odd 2 7605.2.a.cf.1.1 4
52.7 even 12 1040.2.da.b.881.1 8
52.15 even 12 1040.2.da.b.641.1 8
65.2 even 12 325.2.m.c.199.4 8
65.7 even 12 325.2.m.b.49.1 8
65.28 even 12 325.2.m.b.199.1 8
65.33 even 12 325.2.m.c.49.4 8
65.54 odd 12 325.2.n.d.251.4 8
65.59 odd 12 325.2.n.d.101.4 8
65.64 even 2 4225.2.a.bi.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.1 8 13.7 odd 12
65.2.m.a.56.1 yes 8 13.2 odd 12
325.2.m.b.49.1 8 65.7 even 12
325.2.m.b.199.1 8 65.28 even 12
325.2.m.c.49.4 8 65.33 even 12
325.2.m.c.199.4 8 65.2 even 12
325.2.n.d.101.4 8 65.59 odd 12
325.2.n.d.251.4 8 65.54 odd 12
585.2.bu.c.316.4 8 39.2 even 12
585.2.bu.c.361.4 8 39.20 even 12
845.2.a.l.1.1 4 1.1 even 1 trivial
845.2.a.m.1.4 4 13.12 even 2
845.2.c.g.506.1 8 13.8 odd 4
845.2.c.g.506.8 8 13.5 odd 4
845.2.e.m.146.1 8 13.4 even 6
845.2.e.m.191.1 8 13.10 even 6
845.2.e.n.146.4 8 13.9 even 3
845.2.e.n.191.4 8 13.3 even 3
845.2.m.g.316.4 8 13.11 odd 12
845.2.m.g.361.4 8 13.6 odd 12
1040.2.da.b.641.1 8 52.15 even 12
1040.2.da.b.881.1 8 52.7 even 12
4225.2.a.bi.1.1 4 65.64 even 2
4225.2.a.bl.1.4 4 5.4 even 2
7605.2.a.cf.1.1 4 39.38 odd 2
7605.2.a.cj.1.4 4 3.2 odd 2