# Properties

 Label 845.2.a.m.1.4 Level $845$ Weight $2$ Character 845.1 Self dual yes Analytic conductor $6.747$ Analytic rank $0$ 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: $$0$$ 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 1.4 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 $$-0.155997\pi$$
0.882295 + 0.470696i $$0.155997\pi$$
$$3$$ −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 $$-0.382759\pi$$
0.360051 + 0.932933i $$0.382759\pi$$
$$8$$ 5.55889 1.96536
$$9$$ 4.99102 1.66367
$$10$$ −2.49551 −0.789149
$$11$$ 1.06939 0.322433 0.161217 0.986919i $$-0.448458\pi$$
0.161217 + 0.986919i $$0.448458\pi$$
$$12$$ −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 $$-0.271609\pi$$
0.657511 + 0.753445i $$0.271609\pi$$
$$20$$ −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 $$-0.541973\pi$$
−0.131480 + 0.991319i $$0.541973\pi$$
$$32$$ 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 $$-0.480169\pi$$
0.0622619 + 0.998060i $$0.480169\pi$$
$$38$$ 14.3044 2.32048
$$39$$ 0 0
$$40$$ −5.55889 −0.878938
$$41$$ 0.267949 0.0418466 0.0209233 0.999781i $$-0.493339\pi$$
0.0209233 + 0.999781i $$0.493339\pi$$
$$42$$ −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 $$-0.741919\pi$$
−0.688930 + 0.724828i $$0.741919\pi$$
$$48$$ −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 $$-0.484640\pi$$
0.0482373 + 0.998836i $$0.484640\pi$$
$$60$$ 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 $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$68$$ 2.69407 0.326704
$$69$$ −10.7925 −1.29926
$$70$$ −4.75447 −0.568268
$$71$$ 9.76488 1.15888 0.579439 0.815016i $$-0.303272\pi$$
0.579439 + 0.815016i $$0.303272\pi$$
$$72$$ 27.7445 3.26972
$$73$$ −3.71649 −0.434982 −0.217491 0.976062i $$-0.569787\pi$$
−0.217491 + 0.976062i $$0.569787\pi$$
$$74$$ 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 $$-0.409376\pi$$
0.280875 + 0.959744i $$0.409376\pi$$
$$84$$ −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 $$-0.732273\pi$$
−0.666650 + 0.745371i $$0.732273\pi$$
$$90$$ −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 $$-0.568752\pi$$
−0.214317 + 0.976764i $$0.568752\pi$$
$$98$$ −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 $$-0.339710\pi$$
0.482551 + 0.875868i $$0.339710\pi$$
$$110$$ −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 $$-0.828940\pi$$
−0.859041 + 0.511906i $$0.828940\pi$$
$$138$$ −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 $$-0.315555\pi$$
0.547565 + 0.836763i $$0.315555\pi$$
$$150$$ −7.05440 −0.575989
$$151$$ 18.2984 1.48910 0.744550 0.667567i $$-0.232664\pi$$
0.744550 + 0.667567i $$0.232664\pi$$
$$152$$ 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 $$-0.284716\pi$$
0.625938 + 0.779873i $$0.284716\pi$$
$$164$$ 1.13277 0.0884545
$$165$$ 3.02299 0.235340
$$166$$ 12.7715 0.991257
$$167$$ −14.3932 −1.11378 −0.556888 0.830588i $$-0.688005\pi$$
−0.556888 + 0.830588i $$0.688005\pi$$
$$168$$ −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 $$-0.355842\pi$$
0.437562 + 0.899188i $$0.355842\pi$$
$$194$$ −10.5349 −0.756363
$$195$$ 0 0
$$196$$ −14.2476 −1.01768
$$197$$ −4.37830 −0.311941 −0.155970 0.987762i $$-0.549850\pi$$
−0.155970 + 0.987762i $$0.549850\pi$$
$$198$$ 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 $$-0.246657\pi$$
0.714494 + 0.699642i $$0.246657\pi$$
$$224$$ 4.57388 0.305605
$$225$$ 4.99102 0.332734
$$226$$ −16.6901 −1.11021
$$227$$ 15.6857 1.04109 0.520547 0.853833i $$-0.325728\pi$$
0.520547 + 0.853833i $$0.325728\pi$$
$$228$$ −68.5016 −4.53663
$$229$$ −7.62085 −0.503600 −0.251800 0.967779i $$-0.581023\pi$$
−0.251800 + 0.967779i $$0.581023\pi$$
$$230$$ −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 $$-0.364664\pi$$
0.412477 + 0.910968i $$0.364664\pi$$
$$240$$ 15.3134 0.988473
$$241$$ 25.9288 1.67022 0.835111 0.550081i $$-0.185403\pi$$
0.835111 + 0.550081i $$0.185403\pi$$
$$242$$ −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 $$-0.615077\pi$$
−0.353701 + 0.935358i $$0.615077\pi$$
$$272$$ 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 $$-0.615241\pi$$
−0.354183 + 0.935176i $$0.615241\pi$$
$$282$$ 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 $$-0.317029\pi$$
0.543683 + 0.839290i $$0.317029\pi$$
$$294$$ 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 $$-0.471369\pi$$
0.0898262 + 0.995957i $$0.471369\pi$$
$$308$$ 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 $$-0.625085\pi$$
−0.382929 + 0.923778i $$0.625085\pi$$
$$318$$ 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 $$-0.790639\pi$$
−0.791383 + 0.611320i $$0.790639\pi$$
$$332$$ 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 $$-0.588627\pi$$
−0.274846 + 0.961488i $$0.588627\pi$$
$$350$$ 4.75447 0.254137
$$351$$ 0 0
$$352$$ 2.56730 0.136838
$$353$$ 0.800589 0.0426110 0.0213055 0.999773i $$-0.493218\pi$$
0.0213055 + 0.999773i $$0.493218\pi$$
$$354$$ −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 $$-0.431151\pi$$
0.214614 + 0.976699i $$0.431151\pi$$
$$360$$ −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 $$-0.483269\pi$$
0.0525363 + 0.998619i $$0.483269\pi$$
$$380$$ −24.2326 −1.24311
$$381$$ −4.21058 −0.215714
$$382$$ −13.5833 −0.694982
$$383$$ −7.90521 −0.403937 −0.201969 0.979392i $$-0.564734\pi$$
−0.201969 + 0.979392i $$0.564734\pi$$
$$384$$ 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 $$-0.550975\pi$$
−0.159458 + 0.987205i $$0.550975\pi$$
$$398$$ 51.9996 2.60651
$$399$$ −30.8713 −1.54550
$$400$$ 5.41713 0.270857
$$401$$ 4.16920 0.208200 0.104100 0.994567i $$-0.466804\pi$$
0.104100 + 0.994567i $$0.466804\pi$$
$$402$$ −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 $$-0.419113\pi$$
0.251389 + 0.967886i $$0.419113\pi$$
$$410$$ −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 $$-0.484341\pi$$
0.0491749 + 0.998790i $$0.484341\pi$$
$$422$$ −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 $$-0.665356\pi$$
−0.496430 + 0.868077i $$0.665356\pi$$
$$432$$ −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 $$-0.663841\pi$$
−0.492293 + 0.870430i $$0.663841\pi$$
$$450$$ 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 $$-0.753531\pi$$
−0.714906 + 0.699220i $$0.753531\pi$$
$$458$$ −19.0179 −0.888648
$$459$$ −3.58669 −0.167413
$$460$$ −16.1402 −0.752541
$$461$$ 4.67822 0.217887 0.108943 0.994048i $$-0.465253\pi$$
0.108943 + 0.994048i $$0.465253\pi$$
$$462$$ −14.3727 −0.668680
$$463$$ −14.0011 −0.650688 −0.325344 0.945596i $$-0.605480\pi$$
−0.325344 + 0.945596i $$0.605480\pi$$
$$464$$ 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 $$-0.621371\pi$$
−0.372126 + 0.928182i $$0.621371\pi$$
$$480$$ 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 $$-0.349972\pi$$
0.454069 + 0.890966i $$0.349972\pi$$
$$488$$ 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 $$-0.491118\pi$$
0.0279008 + 0.999611i $$0.491118\pi$$
$$500$$ −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 $$-0.306904\pi$$
0.570101 + 0.821575i $$0.306904\pi$$
$$510$$ 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 $$-0.610949\pi$$
−0.341541 + 0.939867i $$0.610949\pi$$
$$542$$ −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 $$-0.446815\pi$$
0.166309 + 0.986074i $$0.446815\pi$$
$$558$$ −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 $$-0.433417\pi$$
0.207655 + 0.978202i $$0.433417\pi$$
$$578$$ −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 $$-0.665372\pi$$
−0.496472 + 0.868053i $$0.665372\pi$$
$$588$$ 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 $$-0.493855\pi$$
0.0193051 + 0.999814i $$0.493855\pi$$
$$594$$ −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 $$-0.502221\pi$$
−0.00697673 + 0.999976i $$0.502221\pi$$
$$614$$ 7.85527 0.317013
$$615$$ 0.757449 0.0305433
$$616$$ 11.3258 0.456328
$$617$$ 38.6850 1.55740 0.778700 0.627397i $$-0.215879\pi$$
0.778700 + 0.627397i $$0.215879\pi$$
$$618$$ 95.4242 3.83852
$$619$$ 14.8971 0.598764 0.299382 0.954133i $$-0.403219\pi$$
0.299382 + 0.954133i $$0.403219\pi$$
$$620$$ 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 $$-0.781344\pi$$
−0.773198 + 0.634165i $$0.781344\pi$$
$$632$$ −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 $$-0.442549\pi$$
0.179509 + 0.983756i $$0.442549\pi$$
$$644$$ 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 $$-0.462601\pi$$
0.117223 + 0.993106i $$0.462601\pi$$
$$662$$ −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 $$-0.662209\pi$$
−0.487823 + 0.872943i $$0.662209\pi$$
$$684$$ 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 $$-0.540043\pi$$
−0.125468 + 0.992098i $$0.540043\pi$$
$$692$$ −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 $$-0.434181\pi$$
0.205307 + 0.978698i $$0.434181\pi$$
$$710$$ −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 $$-0.558070\pi$$
−0.181421 + 0.983406i $$0.558070\pi$$
$$734$$ −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 $$-0.141510\pi$$
0.902797 + 0.430068i $$0.141510\pi$$
$$740$$ −3.20216 −0.117714
$$741$$ 0 0
$$742$$ −33.2386 −1.22023
$$743$$ 40.8375 1.49818 0.749091 0.662467i $$-0.230490\pi$$
0.749091 + 0.662467i $$0.230490\pi$$
$$744$$ 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 $$-0.566065\pi$$
−0.206061 + 0.978539i $$0.566065\pi$$
$$762$$ −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 $$-0.623957\pi$$
−0.379654 + 0.925129i $$0.623957\pi$$
$$770$$ −5.08438 −0.183228
$$771$$ −0.949203 −0.0341847
$$772$$ 51.3970 1.84982
$$773$$ 14.0829 0.506526 0.253263 0.967397i $$-0.418496\pi$$
0.253263 + 0.967397i $$0.418496\pi$$
$$774$$ 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 $$-0.299682\pi$$
0.588593 + 0.808429i $$0.299682\pi$$
$$788$$ −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 $$-0.369696\pi$$
0.398023 + 0.917375i $$0.369696\pi$$
$$812$$ 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 $$-0.333269\pi$$
0.500174 + 0.865925i $$0.333269\pi$$
$$822$$ 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 $$-0.590206\pi$$
−0.279613 + 0.960113i $$0.590206\pi$$
$$828$$ 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 $$-0.599667\pi$$
−0.308021 + 0.951380i $$0.599667\pi$$
$$840$$ 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 $$-0.389959\pi$$
0.338860 + 0.940837i $$0.389959\pi$$
$$854$$ 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 $$-0.641842\pi$$
−0.431008 + 0.902348i $$0.641842\pi$$
$$864$$ −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 $$-0.386402\pi$$
0.349352 + 0.936992i $$0.386402\pi$$
$$878$$ 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 $$-0.580331\pi$$
−0.249698 + 0.968324i $$0.580331\pi$$
$$930$$ −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 $$-0.500258\pi$$
−0.000809396 1.00000i $$0.500258\pi$$
$$942$$ −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 $$-0.598040\pi$$
−0.303154 + 0.952942i $$0.598040\pi$$
$$948$$ 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 $$-0.326943\pi$$
0.517285 + 0.855813i $$0.326943\pi$$
$$968$$ −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 $$-0.419073\pi$$
0.251509 + 0.967855i $$0.419073\pi$$
$$978$$ −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 $$-0.289338\pi$$
0.614548 + 0.788880i $$0.289338\pi$$
$$984$$ −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.m.1.4 4
3.2 odd 2 7605.2.a.cf.1.1 4
5.4 even 2 4225.2.a.bi.1.1 4
13.2 odd 12 845.2.m.g.316.4 8
13.3 even 3 845.2.e.m.191.1 8
13.4 even 6 845.2.e.n.146.4 8
13.5 odd 4 845.2.c.g.506.1 8
13.6 odd 12 65.2.m.a.36.1 8
13.7 odd 12 845.2.m.g.361.4 8
13.8 odd 4 845.2.c.g.506.8 8
13.9 even 3 845.2.e.m.146.1 8
13.10 even 6 845.2.e.n.191.4 8
13.11 odd 12 65.2.m.a.56.1 yes 8
13.12 even 2 845.2.a.l.1.1 4
39.11 even 12 585.2.bu.c.316.4 8
39.32 even 12 585.2.bu.c.361.4 8
39.38 odd 2 7605.2.a.cj.1.4 4
52.11 even 12 1040.2.da.b.641.1 8
52.19 even 12 1040.2.da.b.881.1 8
65.19 odd 12 325.2.n.d.101.4 8
65.24 odd 12 325.2.n.d.251.4 8
65.32 even 12 325.2.m.b.49.1 8
65.37 even 12 325.2.m.c.199.4 8
65.58 even 12 325.2.m.c.49.4 8
65.63 even 12 325.2.m.b.199.1 8
65.64 even 2 4225.2.a.bl.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.1 8 13.6 odd 12
65.2.m.a.56.1 yes 8 13.11 odd 12
325.2.m.b.49.1 8 65.32 even 12
325.2.m.b.199.1 8 65.63 even 12
325.2.m.c.49.4 8 65.58 even 12
325.2.m.c.199.4 8 65.37 even 12
325.2.n.d.101.4 8 65.19 odd 12
325.2.n.d.251.4 8 65.24 odd 12
585.2.bu.c.316.4 8 39.11 even 12
585.2.bu.c.361.4 8 39.32 even 12
845.2.a.l.1.1 4 13.12 even 2
845.2.a.m.1.4 4 1.1 even 1 trivial
845.2.c.g.506.1 8 13.5 odd 4
845.2.c.g.506.8 8 13.8 odd 4
845.2.e.m.146.1 8 13.9 even 3
845.2.e.m.191.1 8 13.3 even 3
845.2.e.n.146.4 8 13.4 even 6
845.2.e.n.191.4 8 13.10 even 6
845.2.m.g.316.4 8 13.2 odd 12
845.2.m.g.361.4 8 13.7 odd 12
1040.2.da.b.641.1 8 52.11 even 12
1040.2.da.b.881.1 8 52.19 even 12
4225.2.a.bi.1.1 4 5.4 even 2
4225.2.a.bl.1.4 4 65.64 even 2
7605.2.a.cf.1.1 4 3.2 odd 2
7605.2.a.cj.1.4 4 39.38 odd 2