# Properties

 Label 845.2.c.g.506.8 Level $845$ Weight $2$ Character 845.506 Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 506.8 Root $$0.665665 - 1.24775i$$ of defining polynomial Character $$\chi$$ $$=$$ 845.506 Dual form 845.2.c.g.506.1

## $q$-expansion

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.49551i 1.76459i 0.470696 + 0.882295i $$0.344003\pi$$
−0.470696 + 0.882295i $$0.655997\pi$$
$$3$$ −2.82684 −1.63208 −0.816038 0.577998i $$-0.803834\pi$$
−0.816038 + 0.577998i $$0.803834\pi$$
$$4$$ −4.22756 −2.11378
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 7.05440i − 2.87995i
$$7$$ − 1.90521i − 0.720103i −0.932933 0.360051i $$-0.882759\pi$$
0.932933 0.360051i $$-0.117241\pi$$
$$8$$ − 5.55889i − 1.96536i
$$9$$ 4.99102 1.66367
$$10$$ 2.49551 0.789149
$$11$$ − 1.06939i − 0.322433i −0.986919 0.161217i $$-0.948458\pi$$
0.986919 0.161217i $$-0.0515417\pi$$
$$12$$ 11.9506 3.44985
$$13$$ 0 0
$$14$$ 4.75447 1.27069
$$15$$ 2.82684i 0.729887i
$$16$$ 5.41713 1.35428
$$17$$ −0.637263 −0.154559 −0.0772795 0.997009i $$-0.524623\pi$$
−0.0772795 + 0.997009i $$0.524623\pi$$
$$18$$ 12.4551i 2.93570i
$$19$$ 5.73205i 1.31502i 0.753445 + 0.657511i $$0.228391\pi$$
−0.753445 + 0.657511i $$0.771609\pi$$
$$20$$ 4.22756i 0.945311i
$$21$$ 5.38573i 1.17526i
$$22$$ 2.66867 0.568962
$$23$$ −3.81785 −0.796078 −0.398039 0.917369i $$-0.630309\pi$$
−0.398039 + 0.917369i $$0.630309\pi$$
$$24$$ 15.7141i 3.20762i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −5.62828 −1.08316
$$28$$ 8.05440i 1.52214i
$$29$$ 9.45512 1.75577 0.877886 0.478870i $$-0.158954\pi$$
0.877886 + 0.478870i $$0.158954\pi$$
$$30$$ −7.05440 −1.28795
$$31$$ − 1.46410i − 0.262960i −0.991319 0.131480i $$-0.958027\pi$$
0.991319 0.131480i $$-0.0419730\pi$$
$$32$$ 2.40072i 0.424391i
$$33$$ 3.02299i 0.526235i
$$34$$ − 1.59030i − 0.272733i
$$35$$ −1.90521 −0.322040
$$36$$ −21.0998 −3.51663
$$37$$ − 0.757449i − 0.124524i −0.998060 0.0622619i $$-0.980169\pi$$
0.998060 0.0622619i $$-0.0198314\pi$$
$$38$$ −14.3044 −2.32048
$$39$$ 0 0
$$40$$ −5.55889 −0.878938
$$41$$ 0.267949i 0.0418466i 0.999781 + 0.0209233i $$0.00666058\pi$$
−0.999781 + 0.0209233i $$0.993339\pi$$
$$42$$ −13.4401 −2.07386
$$43$$ −0.637263 −0.0971817 −0.0485909 0.998819i $$-0.515473\pi$$
−0.0485909 + 0.998819i $$0.515473\pi$$
$$44$$ 4.52091i 0.681552i
$$45$$ − 4.99102i − 0.744017i
$$46$$ − 9.52748i − 1.40475i
$$47$$ 9.44613i 1.37786i 0.724828 + 0.688930i $$0.241919\pi$$
−0.724828 + 0.688930i $$0.758081\pi$$
$$48$$ −15.3134 −2.21029
$$49$$ 3.37017 0.481452
$$50$$ − 2.49551i − 0.352918i
$$51$$ 1.80144 0.252252
$$52$$ 0 0
$$53$$ −6.99102 −0.960290 −0.480145 0.877189i $$-0.659416\pi$$
−0.480145 + 0.877189i $$0.659416\pi$$
$$54$$ − 14.0454i − 1.91134i
$$55$$ −1.06939 −0.144196
$$56$$ −10.5909 −1.41526
$$57$$ − 16.2036i − 2.14622i
$$58$$ 23.5953i 3.09822i
$$59$$ − 0.741035i − 0.0964746i −0.998836 0.0482373i $$-0.984640\pi$$
0.998836 0.0482373i $$-0.0153604\pi$$
$$60$$ − 11.9506i − 1.54282i
$$61$$ 4.19856 0.537571 0.268785 0.963200i $$-0.413378\pi$$
0.268785 + 0.963200i $$0.413378\pi$$
$$62$$ 3.65368 0.464017
$$63$$ − 9.50894i − 1.19801i
$$64$$ 4.84325 0.605406
$$65$$ 0 0
$$66$$ −7.54390 −0.928589
$$67$$ 8.09479i 0.988936i 0.869196 + 0.494468i $$0.164637\pi$$
−0.869196 + 0.494468i $$0.835363\pi$$
$$68$$ 2.69407 0.326704
$$69$$ 10.7925 1.29926
$$70$$ − 4.75447i − 0.568268i
$$71$$ 9.76488i 1.15888i 0.815016 + 0.579439i $$0.196728\pi$$
−0.815016 + 0.579439i $$0.803272\pi$$
$$72$$ − 27.7445i − 3.26972i
$$73$$ 3.71649i 0.434982i 0.976062 + 0.217491i $$0.0697873\pi$$
−0.976062 + 0.217491i $$0.930213\pi$$
$$74$$ 1.89022 0.219734
$$75$$ 2.82684 0.326415
$$76$$ − 24.2326i − 2.77967i
$$77$$ −2.03741 −0.232185
$$78$$ 0 0
$$79$$ −9.31937 −1.04851 −0.524255 0.851561i $$-0.675656\pi$$
−0.524255 + 0.851561i $$0.675656\pi$$
$$80$$ − 5.41713i − 0.605654i
$$81$$ 0.937188 0.104132
$$82$$ −0.668669 −0.0738422
$$83$$ 5.11778i 0.561749i 0.959744 + 0.280875i $$0.0906245\pi$$
−0.959744 + 0.280875i $$0.909376\pi$$
$$84$$ − 22.7685i − 2.48424i
$$85$$ 0.637263i 0.0691209i
$$86$$ − 1.59030i − 0.171486i
$$87$$ −26.7281 −2.86555
$$88$$ −5.94462 −0.633698
$$89$$ 12.5783i 1.33330i 0.745371 + 0.666650i $$0.232273\pi$$
−0.745371 + 0.666650i $$0.767727\pi$$
$$90$$ 12.4551 1.31288
$$91$$ 0 0
$$92$$ 16.1402 1.68273
$$93$$ 4.13878i 0.429171i
$$94$$ −23.5729 −2.43136
$$95$$ 5.73205 0.588096
$$96$$ − 6.78645i − 0.692639i
$$97$$ − 4.22155i − 0.428634i −0.976764 0.214317i $$-0.931248\pi$$
0.976764 0.214317i $$-0.0687525\pi$$
$$98$$ 8.41027i 0.849566i
$$99$$ − 5.33734i − 0.536423i
$$100$$ 4.22756 0.422756
$$101$$ 15.2476 1.51719 0.758595 0.651562i $$-0.225886\pi$$
0.758595 + 0.651562i $$0.225886\pi$$
$$102$$ 4.49551i 0.445122i
$$103$$ 13.5269 1.33285 0.666423 0.745574i $$-0.267824\pi$$
0.666423 + 0.745574i $$0.267824\pi$$
$$104$$ 0 0
$$105$$ 5.38573 0.525593
$$106$$ − 17.4461i − 1.69452i
$$107$$ −7.36274 −0.711783 −0.355891 0.934527i $$-0.615823\pi$$
−0.355891 + 0.934527i $$0.615823\pi$$
$$108$$ 23.7939 2.28957
$$109$$ 10.0760i 0.965103i 0.875868 + 0.482551i $$0.160290\pi$$
−0.875868 + 0.482551i $$0.839710\pi$$
$$110$$ − 2.66867i − 0.254448i
$$111$$ 2.14119i 0.203232i
$$112$$ − 10.3208i − 0.975223i
$$113$$ −6.68806 −0.629160 −0.314580 0.949231i $$-0.601864\pi$$
−0.314580 + 0.949231i $$0.601864\pi$$
$$114$$ 40.4362 3.78719
$$115$$ 3.81785i 0.356017i
$$116$$ −39.9721 −3.71131
$$117$$ 0 0
$$118$$ 1.84926 0.170238
$$119$$ 1.21412i 0.111298i
$$120$$ 15.7141 1.43449
$$121$$ 9.85641 0.896037
$$122$$ 10.4775i 0.948592i
$$123$$ − 0.757449i − 0.0682969i
$$124$$ 6.18958i 0.555840i
$$125$$ 1.00000i 0.0894427i
$$126$$ 23.7296 2.11400
$$127$$ −1.48950 −0.132172 −0.0660859 0.997814i $$-0.521051\pi$$
−0.0660859 + 0.997814i $$0.521051\pi$$
$$128$$ 16.8878i 1.49269i
$$129$$ 1.80144 0.158608
$$130$$ 0 0
$$131$$ 4.12676 0.360557 0.180278 0.983616i $$-0.442300\pi$$
0.180278 + 0.983616i $$0.442300\pi$$
$$132$$ − 12.7799i − 1.11234i
$$133$$ 10.9208 0.946951
$$134$$ −20.2006 −1.74507
$$135$$ 5.62828i 0.484405i
$$136$$ 3.54248i 0.303765i
$$137$$ 20.1096i 1.71808i 0.511906 + 0.859041i $$0.328940\pi$$
−0.511906 + 0.859041i $$0.671060\pi$$
$$138$$ 26.9327i 2.29266i
$$139$$ 20.8253 1.76638 0.883189 0.469018i $$-0.155392\pi$$
0.883189 + 0.469018i $$0.155392\pi$$
$$140$$ 8.05440 0.680721
$$141$$ − 26.7027i − 2.24877i
$$142$$ −24.3683 −2.04494
$$143$$ 0 0
$$144$$ 27.0370 2.25308
$$145$$ − 9.45512i − 0.785205i
$$146$$ −9.27453 −0.767565
$$147$$ −9.52691 −0.785767
$$148$$ 3.20216i 0.263216i
$$149$$ 13.3678i 1.09513i 0.836763 + 0.547565i $$0.184445\pi$$
−0.836763 + 0.547565i $$0.815555\pi$$
$$150$$ 7.05440i 0.575989i
$$151$$ − 18.2984i − 1.48910i −0.667567 0.744550i $$-0.732664\pi$$
0.667567 0.744550i $$-0.267336\pi$$
$$152$$ 31.8638 2.58450
$$153$$ −3.18059 −0.257136
$$154$$ − 5.08438i − 0.409711i
$$155$$ −1.46410 −0.117599
$$156$$ 0 0
$$157$$ 2.42229 0.193320 0.0966599 0.995317i $$-0.469184\pi$$
0.0966599 + 0.995317i $$0.469184\pi$$
$$158$$ − 23.2566i − 1.85019i
$$159$$ 19.7625 1.56727
$$160$$ 2.40072 0.189794
$$161$$ 7.27382i 0.573258i
$$162$$ 2.33876i 0.183750i
$$163$$ − 15.9829i − 1.25188i −0.779873 0.625938i $$-0.784716\pi$$
0.779873 0.625938i $$-0.215284\pi$$
$$164$$ − 1.13277i − 0.0884545i
$$165$$ 3.02299 0.235340
$$166$$ −12.7715 −0.991257
$$167$$ 14.3932i 1.11378i 0.830588 + 0.556888i $$0.188005\pi$$
−0.830588 + 0.556888i $$0.811995\pi$$
$$168$$ 29.9387 2.30982
$$169$$ 0 0
$$170$$ −1.59030 −0.121970
$$171$$ 28.6088i 2.18777i
$$172$$ 2.69407 0.205421
$$173$$ 24.3489 1.85122 0.925608 0.378484i $$-0.123555\pi$$
0.925608 + 0.378484i $$0.123555\pi$$
$$174$$ − 66.7001i − 5.05653i
$$175$$ 1.90521i 0.144021i
$$176$$ − 5.79302i − 0.436666i
$$177$$ 2.09479i 0.157454i
$$178$$ −31.3893 −2.35273
$$179$$ −3.78829 −0.283150 −0.141575 0.989928i $$-0.545217\pi$$
−0.141575 + 0.989928i $$0.545217\pi$$
$$180$$ 21.0998i 1.57269i
$$181$$ 8.48794 0.630904 0.315452 0.948942i $$-0.397844\pi$$
0.315452 + 0.948942i $$0.397844\pi$$
$$182$$ 0 0
$$183$$ −11.8687 −0.877356
$$184$$ 21.2230i 1.56458i
$$185$$ −0.757449 −0.0556888
$$186$$ −10.3284 −0.757312
$$187$$ 0.681482i 0.0498349i
$$188$$ − 39.9341i − 2.91249i
$$189$$ 10.7231i 0.779988i
$$190$$ 14.3044i 1.03775i
$$191$$ −5.44310 −0.393849 −0.196924 0.980419i $$-0.563095\pi$$
−0.196924 + 0.980419i $$0.563095\pi$$
$$192$$ −13.6911 −0.988069
$$193$$ − 12.1576i − 0.875123i −0.899188 0.437562i $$-0.855842\pi$$
0.899188 0.437562i $$-0.144158\pi$$
$$194$$ 10.5349 0.756363
$$195$$ 0 0
$$196$$ −14.2476 −1.01768
$$197$$ − 4.37830i − 0.311941i −0.987762 0.155970i $$-0.950150\pi$$
0.987762 0.155970i $$-0.0498505\pi$$
$$198$$ 13.3194 0.946566
$$199$$ −20.8373 −1.47712 −0.738558 0.674189i $$-0.764493\pi$$
−0.738558 + 0.674189i $$0.764493\pi$$
$$200$$ 5.55889i 0.393073i
$$201$$ − 22.8827i − 1.61402i
$$202$$ 38.0504i 2.67722i
$$203$$ − 18.0140i − 1.26434i
$$204$$ −7.61569 −0.533205
$$205$$ 0.267949 0.0187144
$$206$$ 33.7565i 2.35193i
$$207$$ −19.0550 −1.32441
$$208$$ 0 0
$$209$$ 6.12979 0.424007
$$210$$ 13.4401i 0.927457i
$$211$$ −10.6537 −0.733429 −0.366715 0.930333i $$-0.619517\pi$$
−0.366715 + 0.930333i $$0.619517\pi$$
$$212$$ 29.5549 2.02984
$$213$$ − 27.6037i − 1.89138i
$$214$$ − 18.3738i − 1.25600i
$$215$$ 0.637263i 0.0434610i
$$216$$ 31.2870i 2.12881i
$$217$$ −2.78942 −0.189358
$$218$$ −25.1447 −1.70301
$$219$$ − 10.5059i − 0.709924i
$$220$$ 4.52091 0.304799
$$221$$ 0 0
$$222$$ −5.34335 −0.358622
$$223$$ 21.3393i 1.42899i 0.699642 + 0.714494i $$0.253343\pi$$
−0.699642 + 0.714494i $$0.746657\pi$$
$$224$$ 4.57388 0.305605
$$225$$ −4.99102 −0.332734
$$226$$ − 16.6901i − 1.11021i
$$227$$ 15.6857i 1.04109i 0.853833 + 0.520547i $$0.174272\pi$$
−0.853833 + 0.520547i $$0.825728\pi$$
$$228$$ 68.5016i 4.53663i
$$229$$ 7.62085i 0.503600i 0.967779 + 0.251800i $$0.0810225\pi$$
−0.967779 + 0.251800i $$0.918977\pi$$
$$230$$ −9.52748 −0.628224
$$231$$ 5.75944 0.378943
$$232$$ − 52.5599i − 3.45073i
$$233$$ 19.0550 1.24833 0.624166 0.781292i $$-0.285439\pi$$
0.624166 + 0.781292i $$0.285439\pi$$
$$234$$ 0 0
$$235$$ 9.44613 0.616198
$$236$$ 3.13277i 0.203926i
$$237$$ 26.3444 1.71125
$$238$$ −3.02985 −0.196396
$$239$$ 12.7535i 0.824954i 0.910968 + 0.412477i $$0.135336\pi$$
−0.910968 + 0.412477i $$0.864664\pi$$
$$240$$ 15.3134i 0.988473i
$$241$$ − 25.9288i − 1.67022i −0.550081 0.835111i $$-0.685403\pi$$
0.550081 0.835111i $$-0.314597\pi$$
$$242$$ 24.5967i 1.58114i
$$243$$ 14.2356 0.913211
$$244$$ −17.7497 −1.13631
$$245$$ − 3.37017i − 0.215312i
$$246$$ 1.89022 0.120516
$$247$$ 0 0
$$248$$ −8.13878 −0.516813
$$249$$ − 14.4671i − 0.916817i
$$250$$ −2.49551 −0.157830
$$251$$ −7.61186 −0.480457 −0.240228 0.970716i $$-0.577222\pi$$
−0.240228 + 0.970716i $$0.577222\pi$$
$$252$$ 40.1996i 2.53234i
$$253$$ 4.08277i 0.256682i
$$254$$ − 3.71706i − 0.233229i
$$255$$ − 1.80144i − 0.112811i
$$256$$ −32.4572 −2.02857
$$257$$ −0.335783 −0.0209456 −0.0104728 0.999945i $$-0.503334\pi$$
−0.0104728 + 0.999945i $$0.503334\pi$$
$$258$$ 4.49551i 0.279878i
$$259$$ −1.44310 −0.0896700
$$260$$ 0 0
$$261$$ 47.1906 2.92103
$$262$$ 10.2984i 0.636235i
$$263$$ −5.37589 −0.331492 −0.165746 0.986169i $$-0.553003\pi$$
−0.165746 + 0.986169i $$0.553003\pi$$
$$264$$ 16.8045 1.03424
$$265$$ 6.99102i 0.429455i
$$266$$ 27.2529i 1.67098i
$$267$$ − 35.5569i − 2.17605i
$$268$$ − 34.2212i − 2.09039i
$$269$$ −1.31038 −0.0798956 −0.0399478 0.999202i $$-0.512719\pi$$
−0.0399478 + 0.999202i $$0.512719\pi$$
$$270$$ −14.0454 −0.854777
$$271$$ 11.6453i 0.707403i 0.935358 + 0.353701i $$0.115077\pi$$
−0.935358 + 0.353701i $$0.884923\pi$$
$$272$$ −3.45214 −0.209317
$$273$$ 0 0
$$274$$ −50.1838 −3.03171
$$275$$ 1.06939i 0.0644866i
$$276$$ −45.6257 −2.74635
$$277$$ 20.3161 1.22068 0.610338 0.792141i $$-0.291033\pi$$
0.610338 + 0.792141i $$0.291033\pi$$
$$278$$ 51.9697i 3.11693i
$$279$$ − 7.30735i − 0.437480i
$$280$$ 10.5909i 0.632925i
$$281$$ 11.8744i 0.708366i 0.935176 + 0.354183i $$0.115241\pi$$
−0.935176 + 0.354183i $$0.884759\pi$$
$$282$$ 66.6368 3.96816
$$283$$ 22.6521 1.34653 0.673264 0.739402i $$-0.264892\pi$$
0.673264 + 0.739402i $$0.264892\pi$$
$$284$$ − 41.2816i − 2.44961i
$$285$$ −16.2036 −0.959817
$$286$$ 0 0
$$287$$ 0.510500 0.0301339
$$288$$ 11.9820i 0.706048i
$$289$$ −16.5939 −0.976112
$$290$$ 23.5953 1.38556
$$291$$ 11.9336i 0.699562i
$$292$$ − 15.7117i − 0.919456i
$$293$$ − 18.6127i − 1.08737i −0.839290 0.543683i $$-0.817029\pi$$
0.839290 0.543683i $$-0.182971\pi$$
$$294$$ − 23.7745i − 1.38656i
$$295$$ −0.741035 −0.0431448
$$296$$ −4.21058 −0.244735
$$297$$ 6.01882i 0.349247i
$$298$$ −33.3593 −1.93245
$$299$$ 0 0
$$300$$ −11.9506 −0.689970
$$301$$ 1.21412i 0.0699808i
$$302$$ 45.6637 2.62765
$$303$$ −43.1024 −2.47617
$$304$$ 31.0513i 1.78091i
$$305$$ − 4.19856i − 0.240409i
$$306$$ − 7.93719i − 0.453739i
$$307$$ − 3.14776i − 0.179652i −0.995957 0.0898262i $$-0.971369\pi$$
0.995957 0.0898262i $$-0.0286311\pi$$
$$308$$ 8.61329 0.490788
$$309$$ −38.2384 −2.17531
$$310$$ − 3.65368i − 0.207515i
$$311$$ 3.18059 0.180355 0.0901774 0.995926i $$-0.471257\pi$$
0.0901774 + 0.995926i $$0.471257\pi$$
$$312$$ 0 0
$$313$$ 35.3533 1.99829 0.999144 0.0413596i $$-0.0131689\pi$$
0.999144 + 0.0413596i $$0.0131689\pi$$
$$314$$ 6.04484i 0.341130i
$$315$$ −9.50894 −0.535768
$$316$$ 39.3982 2.21632
$$317$$ − 13.6357i − 0.765858i −0.923778 0.382929i $$-0.874915\pi$$
0.923778 0.382929i $$-0.125085\pi$$
$$318$$ 49.3174i 2.76558i
$$319$$ − 10.1112i − 0.566119i
$$320$$ − 4.84325i − 0.270746i
$$321$$ 20.8133 1.16168
$$322$$ −18.1519 −1.01156
$$323$$ − 3.65283i − 0.203249i
$$324$$ −3.96202 −0.220112
$$325$$ 0 0
$$326$$ 39.8854 2.20905
$$327$$ − 28.4831i − 1.57512i
$$328$$ 1.48950 0.0822439
$$329$$ 17.9969 0.992201
$$330$$ 7.54390i 0.415278i
$$331$$ − 28.7959i − 1.58277i −0.611320 0.791383i $$-0.709361\pi$$
0.611320 0.791383i $$-0.290639\pi$$
$$332$$ − 21.6357i − 1.18741i
$$333$$ − 3.78044i − 0.207167i
$$334$$ −35.9182 −1.96536
$$335$$ 8.09479 0.442265
$$336$$ 29.1752i 1.59164i
$$337$$ −11.7493 −0.640026 −0.320013 0.947413i $$-0.603687\pi$$
−0.320013 + 0.947413i $$0.603687\pi$$
$$338$$ 0 0
$$339$$ 18.9061 1.02684
$$340$$ − 2.69407i − 0.146106i
$$341$$ −1.56569 −0.0847871
$$342$$ −71.3934 −3.86051
$$343$$ − 19.7574i − 1.06680i
$$344$$ 3.54248i 0.190997i
$$345$$ − 10.7925i − 0.581046i
$$346$$ 60.7630i 3.26664i
$$347$$ 1.89977 0.101985 0.0509926 0.998699i $$-0.483762\pi$$
0.0509926 + 0.998699i $$0.483762\pi$$
$$348$$ 112.995 6.05714
$$349$$ 10.2691i 0.549692i 0.961488 + 0.274846i $$0.0886268\pi$$
−0.961488 + 0.274846i $$0.911373\pi$$
$$350$$ −4.75447 −0.254137
$$351$$ 0 0
$$352$$ 2.56730 0.136838
$$353$$ 0.800589i 0.0426110i 0.999773 + 0.0213055i $$0.00678227\pi$$
−0.999773 + 0.0213055i $$0.993218\pi$$
$$354$$ −5.22756 −0.277842
$$355$$ 9.76488 0.518266
$$356$$ − 53.1756i − 2.81830i
$$357$$ − 3.43213i − 0.181647i
$$358$$ − 9.45370i − 0.499643i
$$359$$ − 8.13272i − 0.429228i −0.976699 0.214614i $$-0.931151\pi$$
0.976699 0.214614i $$-0.0688494\pi$$
$$360$$ −27.7445 −1.46226
$$361$$ −13.8564 −0.729285
$$362$$ 21.1817i 1.11329i
$$363$$ −27.8625 −1.46240
$$364$$ 0 0
$$365$$ 3.71649 0.194530
$$366$$ − 29.6183i − 1.54817i
$$367$$ −20.5265 −1.07147 −0.535737 0.844385i $$-0.679966\pi$$
−0.535737 + 0.844385i $$0.679966\pi$$
$$368$$ −20.6818 −1.07811
$$369$$ 1.33734i 0.0696191i
$$370$$ − 1.89022i − 0.0982679i
$$371$$ 13.3194i 0.691507i
$$372$$ − 17.4969i − 0.907173i
$$373$$ −17.8058 −0.921951 −0.460976 0.887413i $$-0.652500\pi$$
−0.460976 + 0.887413i $$0.652500\pi$$
$$374$$ −1.70064 −0.0879382
$$375$$ − 2.82684i − 0.145977i
$$376$$ 52.5100 2.70800
$$377$$ 0 0
$$378$$ −26.7595 −1.37636
$$379$$ 2.04555i 0.105073i 0.998619 + 0.0525363i $$0.0167305\pi$$
−0.998619 + 0.0525363i $$0.983269\pi$$
$$380$$ −24.2326 −1.24311
$$381$$ 4.21058 0.215714
$$382$$ − 13.5833i − 0.694982i
$$383$$ − 7.90521i − 0.403937i −0.979392 0.201969i $$-0.935266\pi$$
0.979392 0.201969i $$-0.0647339\pi$$
$$384$$ − 47.7391i − 2.43618i
$$385$$ 2.03741i 0.103836i
$$386$$ 30.3394 1.54423
$$387$$ −3.18059 −0.161679
$$388$$ 17.8469i 0.906037i
$$389$$ 9.21171 0.467052 0.233526 0.972351i $$-0.424974\pi$$
0.233526 + 0.972351i $$0.424974\pi$$
$$390$$ 0 0
$$391$$ 2.43298 0.123041
$$392$$ − 18.7344i − 0.946229i
$$393$$ −11.6657 −0.588456
$$394$$ 10.9261 0.550448
$$395$$ 9.31937i 0.468908i
$$396$$ 22.5639i 1.13388i
$$397$$ 6.35438i 0.318917i 0.987205 + 0.159458i $$0.0509748\pi$$
−0.987205 + 0.159458i $$0.949025\pi$$
$$398$$ − 51.9996i − 2.60651i
$$399$$ −30.8713 −1.54550
$$400$$ −5.41713 −0.270857
$$401$$ − 4.16920i − 0.208200i −0.994567 0.104100i $$-0.966804\pi$$
0.994567 0.104100i $$-0.0331962\pi$$
$$402$$ 57.1038 2.84808
$$403$$ 0 0
$$404$$ −64.4600 −3.20701
$$405$$ − 0.937188i − 0.0465692i
$$406$$ 44.9541 2.23103
$$407$$ −0.810008 −0.0401506
$$408$$ − 10.0140i − 0.495767i
$$409$$ 10.1681i 0.502778i 0.967886 + 0.251389i $$0.0808874\pi$$
−0.967886 + 0.251389i $$0.919113\pi$$
$$410$$ 0.668669i 0.0330232i
$$411$$ − 56.8467i − 2.80404i
$$412$$ −57.1858 −2.81734
$$413$$ −1.41183 −0.0694716
$$414$$ − 47.5518i − 2.33704i
$$415$$ 5.11778 0.251222
$$416$$ 0 0
$$417$$ −58.8697 −2.88286
$$418$$ 15.2969i 0.748198i
$$419$$ 28.5909 1.39676 0.698378 0.715730i $$-0.253906\pi$$
0.698378 + 0.715730i $$0.253906\pi$$
$$420$$ −22.7685 −1.11099
$$421$$ 2.01797i 0.0983498i 0.998790 + 0.0491749i $$0.0156592\pi$$
−0.998790 + 0.0491749i $$0.984341\pi$$
$$422$$ − 26.5863i − 1.29420i
$$423$$ 47.1458i 2.29231i
$$424$$ 38.8623i 1.88732i
$$425$$ 0.637263 0.0309118
$$426$$ 68.8853 3.33750
$$427$$ − 7.99915i − 0.387106i
$$428$$ 31.1264 1.50455
$$429$$ 0 0
$$430$$ −1.59030 −0.0766908
$$431$$ − 20.6123i − 0.992860i −0.868077 0.496430i $$-0.834644\pi$$
0.868077 0.496430i $$-0.165356\pi$$
$$432$$ −30.4891 −1.46691
$$433$$ −29.4356 −1.41458 −0.707292 0.706921i $$-0.750083\pi$$
−0.707292 + 0.706921i $$0.750083\pi$$
$$434$$ − 6.96103i − 0.334140i
$$435$$ 26.7281i 1.28151i
$$436$$ − 42.5967i − 2.04001i
$$437$$ − 21.8841i − 1.04686i
$$438$$ 26.2176 1.25272
$$439$$ −16.9520 −0.809077 −0.404538 0.914521i $$-0.632568\pi$$
−0.404538 + 0.914521i $$0.632568\pi$$
$$440$$ 5.94462i 0.283398i
$$441$$ 16.8205 0.800978
$$442$$ 0 0
$$443$$ −24.1399 −1.14692 −0.573461 0.819233i $$-0.694400\pi$$
−0.573461 + 0.819233i $$0.694400\pi$$
$$444$$ − 9.05199i − 0.429588i
$$445$$ 12.5783 0.596270
$$446$$ −53.2525 −2.52158
$$447$$ − 37.7885i − 1.78733i
$$448$$ − 9.22742i − 0.435955i
$$449$$ 20.8630i 0.984585i 0.870430 + 0.492293i $$0.163841\pi$$
−0.870430 + 0.492293i $$0.836159\pi$$
$$450$$ − 12.4551i − 0.587140i
$$451$$ 0.286542 0.0134927
$$452$$ 28.2742 1.32990
$$453$$ 51.7265i 2.43032i
$$454$$ −39.1437 −1.83710
$$455$$ 0 0
$$456$$ −90.0739 −4.21810
$$457$$ − 30.5659i − 1.42981i −0.699220 0.714906i $$-0.746469\pi$$
0.699220 0.714906i $$-0.253531\pi$$
$$458$$ −19.0179 −0.888648
$$459$$ 3.58669 0.167413
$$460$$ − 16.1402i − 0.752541i
$$461$$ 4.67822i 0.217887i 0.994048 + 0.108943i $$0.0347467\pi$$
−0.994048 + 0.108943i $$0.965253\pi$$
$$462$$ 14.3727i 0.668680i
$$463$$ 14.0011i 0.650688i 0.945596 + 0.325344i $$0.105480\pi$$
−0.945596 + 0.325344i $$0.894520\pi$$
$$464$$ 51.2196 2.37781
$$465$$ 4.13878 0.191931
$$466$$ 47.5518i 2.20280i
$$467$$ 6.98506 0.323230 0.161615 0.986854i $$-0.448330\pi$$
0.161615 + 0.986854i $$0.448330\pi$$
$$468$$ 0 0
$$469$$ 15.4223 0.712135
$$470$$ 23.5729i 1.08734i
$$471$$ −6.84742 −0.315513
$$472$$ −4.11933 −0.189608
$$473$$ 0.681482i 0.0313346i
$$474$$ 65.7425i 3.01965i
$$475$$ − 5.73205i − 0.263005i
$$476$$ − 5.13277i − 0.235260i
$$477$$ −34.8923 −1.59761
$$478$$ −31.8264 −1.45571
$$479$$ 16.2888i 0.744252i 0.928182 + 0.372126i $$0.121371\pi$$
−0.928182 + 0.372126i $$0.878629\pi$$
$$480$$ −6.78645 −0.309758
$$481$$ 0 0
$$482$$ 64.7056 2.94726
$$483$$ − 20.5619i − 0.935600i
$$484$$ −41.6685 −1.89402
$$485$$ −4.22155 −0.191691
$$486$$ 35.5249i 1.61144i
$$487$$ 20.0409i 0.908139i 0.890966 + 0.454069i $$0.150028\pi$$
−0.890966 + 0.454069i $$0.849972\pi$$
$$488$$ − 23.3393i − 1.05652i
$$489$$ 45.1810i 2.04316i
$$490$$ 8.41027 0.379937
$$491$$ 15.7983 0.712969 0.356484 0.934301i $$-0.383975\pi$$
0.356484 + 0.934301i $$0.383975\pi$$
$$492$$ 3.20216i 0.144365i
$$493$$ −6.02540 −0.271370
$$494$$ 0 0
$$495$$ −5.33734 −0.239896
$$496$$ − 7.93123i − 0.356123i
$$497$$ 18.6042 0.834511
$$498$$ 36.1028 1.61781
$$499$$ 1.24651i 0.0558016i 0.999611 + 0.0279008i $$0.00888226\pi$$
−0.999611 + 0.0279008i $$0.991118\pi$$
$$500$$ − 4.22756i − 0.189062i
$$501$$ − 40.6871i − 1.81777i
$$502$$ − 18.9955i − 0.847809i
$$503$$ −7.65345 −0.341250 −0.170625 0.985336i $$-0.554579\pi$$
−0.170625 + 0.985336i $$0.554579\pi$$
$$504$$ −52.8592 −2.35453
$$505$$ − 15.2476i − 0.678508i
$$506$$ −10.1886 −0.452938
$$507$$ 0 0
$$508$$ 6.29695 0.279382
$$509$$ 25.7241i 1.14020i 0.821575 + 0.570101i $$0.193096\pi$$
−0.821575 + 0.570101i $$0.806904\pi$$
$$510$$ 4.49551 0.199064
$$511$$ 7.08070 0.313232
$$512$$ − 47.2215i − 2.08691i
$$513$$ − 32.2616i − 1.42438i
$$514$$ − 0.837948i − 0.0369603i
$$515$$ − 13.5269i − 0.596067i
$$516$$ −7.61569 −0.335262
$$517$$ 10.1016 0.444268
$$518$$ − 3.60127i − 0.158231i
$$519$$ −68.8305 −3.02132
$$520$$ 0 0
$$521$$ −30.1519 −1.32098 −0.660490 0.750835i $$-0.729651\pi$$
−0.660490 + 0.750835i $$0.729651\pi$$
$$522$$ 117.765i 5.15442i
$$523$$ 3.93752 0.172176 0.0860880 0.996288i $$-0.472563\pi$$
0.0860880 + 0.996288i $$0.472563\pi$$
$$524$$ −17.4461 −0.762138
$$525$$ − 5.38573i − 0.235052i
$$526$$ − 13.4156i − 0.584947i
$$527$$ 0.933018i 0.0406429i
$$528$$ 16.3759i 0.712672i
$$529$$ −8.42399 −0.366261
$$530$$ −17.4461 −0.757812
$$531$$ − 3.69852i − 0.160502i
$$532$$ −46.1682 −2.00165
$$533$$ 0 0
$$534$$ 88.7326 3.83983
$$535$$ 7.36274i 0.318319i
$$536$$ 44.9980 1.94362
$$537$$ 10.7089 0.462122
$$538$$ − 3.27007i − 0.140983i
$$539$$ − 3.60402i − 0.155236i
$$540$$ − 23.7939i − 1.02393i
$$541$$ 15.8881i 0.683083i 0.939867 + 0.341541i $$0.110949\pi$$
−0.939867 + 0.341541i $$0.889051\pi$$
$$542$$ −29.0610 −1.24828
$$543$$ −23.9940 −1.02968
$$544$$ − 1.52989i − 0.0655935i
$$545$$ 10.0760 0.431607
$$546$$ 0 0
$$547$$ −6.56107 −0.280531 −0.140266 0.990114i $$-0.544796\pi$$
−0.140266 + 0.990114i $$0.544796\pi$$
$$548$$ − 85.0147i − 3.63165i
$$549$$ 20.9551 0.894341
$$550$$ −2.66867 −0.113792
$$551$$ 54.1972i 2.30888i
$$552$$ − 59.9941i − 2.55352i
$$553$$ 17.7554i 0.755035i
$$554$$ 50.6990i 2.15399i
$$555$$ 2.14119 0.0908883
$$556$$ −88.0401 −3.73373
$$557$$ − 7.85006i − 0.332618i −0.986074 0.166309i $$-0.946815\pi$$
0.986074 0.166309i $$-0.0531849\pi$$
$$558$$ 18.2356 0.771973
$$559$$ 0 0
$$560$$ −10.3208 −0.436133
$$561$$ − 1.92644i − 0.0813344i
$$562$$ −29.6326 −1.24998
$$563$$ 15.5595 0.655755 0.327878 0.944720i $$-0.393667\pi$$
0.327878 + 0.944720i $$0.393667\pi$$
$$564$$ 112.887i 4.75341i
$$565$$ 6.68806i 0.281369i
$$566$$ 56.5285i 2.37607i
$$567$$ − 1.78554i − 0.0749857i
$$568$$ 54.2819 2.27762
$$569$$ −3.47915 −0.145853 −0.0729267 0.997337i $$-0.523234\pi$$
−0.0729267 + 0.997337i $$0.523234\pi$$
$$570$$ − 40.4362i − 1.69368i
$$571$$ 21.5118 0.900240 0.450120 0.892968i $$-0.351381\pi$$
0.450120 + 0.892968i $$0.351381\pi$$
$$572$$ 0 0
$$573$$ 15.3868 0.642791
$$574$$ 1.27396i 0.0531739i
$$575$$ 3.81785 0.159216
$$576$$ 24.1727 1.00720
$$577$$ 9.97608i 0.415310i 0.978202 + 0.207655i $$0.0665831\pi$$
−0.978202 + 0.207655i $$0.933417\pi$$
$$578$$ − 41.4102i − 1.72244i
$$579$$ 34.3676i 1.42827i
$$580$$ 39.9721i 1.65975i
$$581$$ 9.75045 0.404517
$$582$$ −29.7805 −1.23444
$$583$$ 7.47612i 0.309629i
$$584$$ 20.6595 0.854898
$$585$$ 0 0
$$586$$ 46.4482 1.91876
$$587$$ − 24.0571i − 0.992945i −0.868053 0.496472i $$-0.834628\pi$$
0.868053 0.496472i $$-0.165372\pi$$
$$588$$ 40.2756 1.66094
$$589$$ 8.39230 0.345799
$$590$$ − 1.84926i − 0.0761328i
$$591$$ 12.3767i 0.509111i
$$592$$ − 4.10320i − 0.168641i
$$593$$ − 0.940219i − 0.0386102i −0.999814 0.0193051i $$-0.993855\pi$$
0.999814 0.0193051i $$-0.00614538\pi$$
$$594$$ −15.0200 −0.616279
$$595$$ 1.21412 0.0497741
$$596$$ − 56.5130i − 2.31486i
$$597$$ 58.9037 2.41077
$$598$$ 0 0
$$599$$ −11.4270 −0.466896 −0.233448 0.972369i $$-0.575001\pi$$
−0.233448 + 0.972369i $$0.575001\pi$$
$$600$$ − 15.7141i − 0.641525i
$$601$$ −36.0431 −1.47023 −0.735114 0.677944i $$-0.762871\pi$$
−0.735114 + 0.677944i $$0.762871\pi$$
$$602$$ −3.02985 −0.123487
$$603$$ 40.4012i 1.64526i
$$604$$ 77.3574i 3.14763i
$$605$$ − 9.85641i − 0.400720i
$$606$$ − 107.562i − 4.36942i
$$607$$ 39.8907 1.61911 0.809557 0.587041i $$-0.199707\pi$$
0.809557 + 0.587041i $$0.199707\pi$$
$$608$$ −13.7610 −0.558084
$$609$$ 50.9227i 2.06349i
$$610$$ 10.4775 0.424223
$$611$$ 0 0
$$612$$ 13.4461 0.543528
$$613$$ − 0.345472i − 0.0139535i −0.999976 0.00697673i $$-0.997779\pi$$
0.999976 0.00697673i $$-0.00222078\pi$$
$$614$$ 7.85527 0.317013
$$615$$ −0.757449 −0.0305433
$$616$$ 11.3258i 0.456328i
$$617$$ 38.6850i 1.55740i 0.627397 + 0.778700i $$0.284121\pi$$
−0.627397 + 0.778700i $$0.715879\pi$$
$$618$$ − 95.4242i − 3.83852i
$$619$$ − 14.8971i − 0.598764i −0.954133 0.299382i $$-0.903219\pi$$
0.954133 0.299382i $$-0.0967805\pi$$
$$620$$ 6.18958 0.248579
$$621$$ 21.4879 0.862281
$$622$$ 7.93719i 0.318252i
$$623$$ 23.9644 0.960113
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 88.2245i 3.52616i
$$627$$ −17.3279 −0.692011
$$628$$ −10.2404 −0.408635
$$629$$ 0.482694i 0.0192463i
$$630$$ − 23.7296i − 0.945412i
$$631$$ 38.8450i 1.54640i 0.634165 + 0.773198i $$0.281344\pi$$
−0.634165 + 0.773198i $$0.718656\pi$$
$$632$$ 51.8053i 2.06071i
$$633$$ 30.1162 1.19701
$$634$$ 34.0280 1.35143
$$635$$ 1.48950i 0.0591090i
$$636$$ −83.5470 −3.31285
$$637$$ 0 0
$$638$$ 25.2326 0.998967
$$639$$ 48.7367i 1.92799i
$$640$$ 16.8878 0.667549
$$641$$ −37.1816 −1.46859 −0.734293 0.678832i $$-0.762486\pi$$
−0.734293 + 0.678832i $$0.762486\pi$$
$$642$$ 51.9397i 2.04990i
$$643$$ 9.10377i 0.359018i 0.983756 + 0.179509i $$0.0574509\pi$$
−0.983756 + 0.179509i $$0.942549\pi$$
$$644$$ − 30.7505i − 1.21174i
$$645$$ − 1.80144i − 0.0709316i
$$646$$ 9.11565 0.358651
$$647$$ 19.1224 0.751778 0.375889 0.926665i $$-0.377337\pi$$
0.375889 + 0.926665i $$0.377337\pi$$
$$648$$ − 5.20972i − 0.204657i
$$649$$ −0.792455 −0.0311066
$$650$$ 0 0
$$651$$ 7.88525 0.309047
$$652$$ 67.5686i 2.64619i
$$653$$ 34.6324 1.35527 0.677636 0.735397i $$-0.263004\pi$$
0.677636 + 0.735397i $$0.263004\pi$$
$$654$$ 71.0799 2.77944
$$655$$ − 4.12676i − 0.161246i
$$656$$ 1.45152i 0.0566722i
$$657$$ 18.5491i 0.723667i
$$658$$ 44.9114i 1.75083i
$$659$$ −6.69852 −0.260937 −0.130469 0.991452i $$-0.541648\pi$$
−0.130469 + 0.991452i $$0.541648\pi$$
$$660$$ −12.7799 −0.497456
$$661$$ − 6.02758i − 0.234446i −0.993106 0.117223i $$-0.962601\pi$$
0.993106 0.117223i $$-0.0373992\pi$$
$$662$$ 71.8604 2.79294
$$663$$ 0 0
$$664$$ 28.4492 1.10404
$$665$$ − 10.9208i − 0.423489i
$$666$$ 9.43412 0.365565
$$667$$ −36.0983 −1.39773
$$668$$ − 60.8479i − 2.35428i
$$669$$ − 60.3228i − 2.33222i
$$670$$ 20.2006i 0.780417i
$$671$$ − 4.48990i − 0.173330i
$$672$$ −12.9296 −0.498771
$$673$$ −23.3568 −0.900338 −0.450169 0.892943i $$-0.648636\pi$$
−0.450169 + 0.892943i $$0.648636\pi$$
$$674$$ − 29.3205i − 1.12938i
$$675$$ 5.62828 0.216633
$$676$$ 0 0
$$677$$ −45.4042 −1.74503 −0.872513 0.488590i $$-0.837511\pi$$
−0.872513 + 0.488590i $$0.837511\pi$$
$$678$$ 47.1802i 1.81195i
$$679$$ −8.04295 −0.308660
$$680$$ 3.54248 0.135848
$$681$$ − 44.3408i − 1.69914i
$$682$$ − 3.90720i − 0.149615i
$$683$$ 25.4978i 0.975645i 0.872943 + 0.487823i $$0.162209\pi$$
−0.872943 + 0.487823i $$0.837791\pi$$
$$684$$ − 120.945i − 4.62445i
$$685$$ 20.1096 0.768350
$$686$$ 49.3047 1.88246
$$687$$ − 21.5429i − 0.821913i
$$688$$ −3.45214 −0.131612
$$689$$ 0 0
$$690$$ 26.9327 1.02531
$$691$$ − 6.59630i − 0.250935i −0.992098 0.125468i $$-0.959957\pi$$
0.992098 0.125468i $$-0.0400431\pi$$
$$692$$ −102.937 −3.91306
$$693$$ −10.1688 −0.386279
$$694$$ 4.74090i 0.179962i
$$695$$ − 20.8253i − 0.789948i
$$696$$ 148.578i 5.63185i
$$697$$ − 0.170754i − 0.00646778i
$$698$$ −25.6266 −0.969981
$$699$$ −53.8653 −2.03737
$$700$$ − 8.05440i − 0.304428i
$$701$$ −29.2474 −1.10466 −0.552329 0.833626i $$-0.686261\pi$$
−0.552329 + 0.833626i $$0.686261\pi$$
$$702$$ 0 0
$$703$$ 4.34174 0.163752
$$704$$ − 5.17932i − 0.195203i
$$705$$ −26.7027 −1.00568
$$706$$ −1.99787 −0.0751910
$$707$$ − 29.0499i − 1.09253i
$$708$$ − 8.85584i − 0.332823i
$$709$$ − 10.9335i − 0.410614i −0.978698 0.205307i $$-0.934181\pi$$
0.978698 0.205307i $$-0.0658194\pi$$
$$710$$ 24.3683i 0.914527i
$$711$$ −46.5131 −1.74438
$$712$$ 69.9216 2.62042
$$713$$ 5.58973i 0.209337i
$$714$$ 8.56490 0.320533
$$715$$ 0 0
$$716$$ 16.0152 0.598516
$$717$$ − 36.0520i − 1.34639i
$$718$$ 20.2953 0.757412
$$719$$ −16.0598 −0.598929 −0.299464 0.954107i $$-0.596808\pi$$
−0.299464 + 0.954107i $$0.596808\pi$$
$$720$$ − 27.0370i − 1.00761i
$$721$$ − 25.7716i − 0.959786i
$$722$$ − 34.5788i − 1.28689i
$$723$$ 73.2966i 2.72593i
$$724$$ −35.8833 −1.33359
$$725$$ −9.45512 −0.351154
$$726$$ − 69.5310i − 2.58054i
$$727$$ −51.3754 −1.90541 −0.952704 0.303900i $$-0.901711\pi$$
−0.952704 + 0.303900i $$0.901711\pi$$
$$728$$ 0 0
$$729$$ −43.0532 −1.59456
$$730$$ 9.27453i 0.343266i
$$731$$ 0.406104 0.0150203
$$732$$ 50.1754 1.85454
$$733$$ − 9.82358i − 0.362842i −0.983406 0.181421i $$-0.941930\pi$$
0.983406 0.181421i $$-0.0580697\pi$$
$$734$$ − 51.2240i − 1.89071i
$$735$$ 9.52691i 0.351406i
$$736$$ − 9.16560i − 0.337848i
$$737$$ 8.65648 0.318866
$$738$$ −3.33734 −0.122849
$$739$$ − 49.0842i − 1.80559i −0.430068 0.902797i $$-0.641510\pi$$
0.430068 0.902797i $$-0.358490\pi$$
$$740$$ 3.20216 0.117714
$$741$$ 0 0
$$742$$ −33.2386 −1.22023
$$743$$ 40.8375i 1.49818i 0.662467 + 0.749091i $$0.269510\pi$$
−0.662467 + 0.749091i $$0.730490\pi$$
$$744$$ 23.0070 0.843478
$$745$$ 13.3678 0.489757
$$746$$ − 44.4346i − 1.62687i
$$747$$ 25.5429i 0.934566i
$$748$$ − 2.88101i − 0.105340i
$$749$$ 14.0276i 0.512557i
$$750$$ 7.05440 0.257590
$$751$$ 2.72680 0.0995024 0.0497512 0.998762i $$-0.484157\pi$$
0.0497512 + 0.998762i $$0.484157\pi$$
$$752$$ 51.1710i 1.86601i
$$753$$ 21.5175 0.784142
$$754$$ 0 0
$$755$$ −18.2984 −0.665946
$$756$$ − 45.3324i − 1.64872i
$$757$$ 14.8060 0.538134 0.269067 0.963121i $$-0.413285\pi$$
0.269067 + 0.963121i $$0.413285\pi$$
$$758$$ −5.10468 −0.185410
$$759$$ − 11.5413i − 0.418924i
$$760$$ − 31.8638i − 1.15582i
$$761$$ 11.3689i 0.412122i 0.978539 + 0.206061i $$0.0660646\pi$$
−0.978539 + 0.206061i $$0.933935\pi$$
$$762$$ 10.5075i 0.380647i
$$763$$ 19.1969 0.694973
$$764$$ 23.0110 0.832510
$$765$$ 3.18059i 0.114994i
$$766$$ 19.7275 0.712784
$$767$$ 0 0
$$768$$ 91.7512 3.31078
$$769$$ − 21.0562i − 0.759307i −0.925129 0.379654i $$-0.876043\pi$$
0.925129 0.379654i $$-0.123957\pi$$
$$770$$ −5.08438 −0.183228
$$771$$ 0.949203 0.0341847
$$772$$ 51.3970i 1.84982i
$$773$$ 14.0829i 0.506526i 0.967397 + 0.253263i $$0.0815038\pi$$
−0.967397 + 0.253263i $$0.918496\pi$$
$$774$$ − 7.93719i − 0.285296i
$$775$$ 1.46410i 0.0525921i
$$776$$ −23.4671 −0.842421
$$777$$ 4.07941 0.146348
$$778$$ 22.9879i 0.824156i
$$779$$ −1.53590 −0.0550293
$$780$$ 0 0
$$781$$ 10.4425 0.373660
$$782$$ 6.07151i 0.217117i
$$783$$ −53.2160 −1.90179
$$784$$ 18.2566 0.652023
$$785$$ − 2.42229i − 0.0864552i
$$786$$ − 29.1118i − 1.03838i
$$787$$ − 33.0242i − 1.17719i −0.808429 0.588593i $$-0.799682\pi$$
0.808429 0.588593i $$-0.200318\pi$$
$$788$$ 18.5095i 0.659374i
$$789$$ 15.1968 0.541020
$$790$$ −23.2566 −0.827431
$$791$$ 12.7422i 0.453060i
$$792$$ −29.6697 −1.05427
$$793$$ 0 0
$$794$$ −15.8574 −0.562758
$$795$$ − 19.7625i − 0.700903i
$$796$$ 88.0909 3.12230
$$797$$ 16.9416 0.600102 0.300051 0.953923i $$-0.402996\pi$$
0.300051 + 0.953923i $$0.402996\pi$$
$$798$$ − 77.0395i − 2.72717i
$$799$$ − 6.01967i − 0.212961i
$$800$$ − 2.40072i − 0.0848783i
$$801$$ 62.7787i 2.21817i
$$802$$ 10.4043 0.367387
$$803$$ 3.97437 0.140253
$$804$$ 96.7378i 3.41168i
$$805$$ 7.27382 0.256369
$$806$$ 0 0
$$807$$ 3.70425 0.130396
$$808$$ − 84.7596i − 2.98183i
$$809$$ −51.7635 −1.81991 −0.909954 0.414708i $$-0.863884\pi$$
−0.909954 + 0.414708i $$0.863884\pi$$
$$810$$ 2.33876 0.0821756
$$811$$ 22.6699i 0.796047i 0.917375 + 0.398023i $$0.130304\pi$$
−0.917375 + 0.398023i $$0.869696\pi$$
$$812$$ 76.1553i 2.67253i
$$813$$ − 32.9194i − 1.15453i
$$814$$ − 2.02138i − 0.0708494i
$$815$$ −15.9829 −0.559856
$$816$$ 9.75864 0.341621
$$817$$ − 3.65283i − 0.127796i
$$818$$ −25.3745 −0.887198
$$819$$ 0 0
$$820$$ −1.13277 −0.0395581
$$821$$ 28.6631i 1.00035i 0.865925 + 0.500174i $$0.166731\pi$$
−0.865925 + 0.500174i $$0.833269\pi$$
$$822$$ 141.861 4.94799
$$823$$ 25.8327 0.900472 0.450236 0.892910i $$-0.351340\pi$$
0.450236 + 0.892910i $$0.351340\pi$$
$$824$$ − 75.1946i − 2.61953i
$$825$$ − 3.02299i − 0.105247i
$$826$$ − 3.52323i − 0.122589i
$$827$$ 16.0820i 0.559227i 0.960113 + 0.279613i $$0.0902063\pi$$
−0.960113 + 0.279613i $$0.909794\pi$$
$$828$$ 80.5560 2.79951
$$829$$ 22.5818 0.784298 0.392149 0.919902i $$-0.371732\pi$$
0.392149 + 0.919902i $$0.371732\pi$$
$$830$$ 12.7715i 0.443304i
$$831$$ −57.4304 −1.99224
$$832$$ 0 0
$$833$$ −2.14768 −0.0744128
$$834$$ − 146.910i − 5.08707i
$$835$$ 14.3932 0.498096
$$836$$ −25.9141 −0.896257
$$837$$ 8.24037i 0.284829i
$$838$$ 71.3487i 2.46470i
$$839$$ 17.8440i 0.616042i 0.951380 + 0.308021i $$0.0996667\pi$$
−0.951380 + 0.308021i $$0.900333\pi$$
$$840$$ − 29.9387i − 1.03298i
$$841$$ 60.3992 2.08273
$$842$$ −5.03586 −0.173547
$$843$$ − 33.5669i − 1.15611i
$$844$$ 45.0390 1.55031
$$845$$ 0 0
$$846$$ −117.653 −4.04498
$$847$$ − 18.7785i − 0.645239i
$$848$$ −37.8713 −1.30050
$$849$$ −64.0339 −2.19764
$$850$$ 1.59030i 0.0545467i
$$851$$ 2.89183i 0.0991306i
$$852$$ 116.696i 3.99795i
$$853$$ − 19.7936i − 0.677720i −0.940837 0.338860i $$-0.889959\pi$$
0.940837 0.338860i $$-0.110041\pi$$
$$854$$ 19.9619 0.683083
$$855$$ 28.6088 0.978399
$$856$$ 40.9286i 1.39891i
$$857$$ 11.7302 0.400696 0.200348 0.979725i $$-0.435793\pi$$
0.200348 + 0.979725i $$0.435793\pi$$
$$858$$ 0 0
$$859$$ 5.37452 0.183376 0.0916882 0.995788i $$-0.470774\pi$$
0.0916882 + 0.995788i $$0.470774\pi$$
$$860$$ − 2.69407i − 0.0918669i
$$861$$ −1.44310 −0.0491808
$$862$$ 51.4382 1.75199
$$863$$ − 25.3234i − 0.862017i −0.902348 0.431008i $$-0.858158\pi$$
0.902348 0.431008i $$-0.141842\pi$$
$$864$$ − 13.5119i − 0.459685i
$$865$$ − 24.3489i − 0.827889i
$$866$$ − 73.4567i − 2.49616i
$$867$$ 46.9083 1.59309
$$868$$ 11.7925 0.400262
$$869$$ 9.96603i 0.338075i
$$870$$ −66.7001 −2.26135
$$871$$ 0 0
$$872$$ 56.0112 1.89678
$$873$$ − 21.0698i − 0.713106i
$$874$$ 54.6120 1.84728
$$875$$ 1.90521 0.0644079
$$876$$ 44.4144i 1.50062i
$$877$$ 20.6915i 0.698703i 0.936992 + 0.349352i $$0.113598\pi$$
−0.936992 + 0.349352i $$0.886402\pi$$
$$878$$ − 42.3040i − 1.42769i
$$879$$ 52.6151i 1.77466i
$$880$$ −5.79302 −0.195283
$$881$$ −48.3993 −1.63061 −0.815307 0.579029i $$-0.803432\pi$$
−0.815307 + 0.579029i $$0.803432\pi$$
$$882$$ 41.9758i 1.41340i
$$883$$ −45.8550 −1.54314 −0.771572 0.636142i $$-0.780529\pi$$
−0.771572 + 0.636142i $$0.780529\pi$$
$$884$$ 0 0
$$885$$ 2.09479 0.0704155
$$886$$ − 60.2413i − 2.02385i
$$887$$ −1.08234 −0.0363413 −0.0181707 0.999835i $$-0.505784\pi$$
−0.0181707 + 0.999835i $$0.505784\pi$$
$$888$$ 11.9026 0.399426
$$889$$ 2.83781i 0.0951772i
$$890$$ 31.3893i 1.05217i
$$891$$ − 1.00222i − 0.0335756i
$$892$$ − 90.2133i − 3.02056i
$$893$$ −54.1457 −1.81192
$$894$$ 94.3015 3.15391
$$895$$ 3.78829i 0.126628i
$$896$$ 32.1749 1.07489
$$897$$ 0 0
$$898$$ −52.0637 −1.73739
$$899$$ − 13.8433i − 0.461698i
$$900$$ 21.0998 0.703327
$$901$$ 4.45512 0.148421
$$902$$ 0.715068i 0.0238092i
$$903$$ − 3.43213i − 0.114214i
$$904$$ 37.1782i 1.23653i
$$905$$ − 8.48794i − 0.282149i
$$906$$ −129.084 −4.28853
$$907$$ 45.5307 1.51182 0.755910 0.654675i $$-0.227195\pi$$
0.755910 + 0.654675i $$0.227195\pi$$
$$908$$ − 66.3120i − 2.20064i
$$909$$ 76.1009 2.52411
$$910$$ 0 0
$$911$$ 39.7417 1.31670 0.658350 0.752712i $$-0.271255\pi$$
0.658350 + 0.752712i $$0.271255\pi$$
$$912$$ − 87.7770i − 2.90659i
$$913$$ 5.47290 0.181126
$$914$$ 76.2774 2.52303
$$915$$ 11.8687i 0.392365i
$$916$$ − 32.2176i − 1.06450i
$$917$$ − 7.86236i − 0.259638i
$$918$$ 8.95062i 0.295415i
$$919$$ 46.9938 1.55018 0.775091 0.631850i $$-0.217704\pi$$
0.775091 + 0.631850i $$0.217704\pi$$
$$920$$ 21.2230 0.699702
$$921$$ 8.89822i 0.293206i
$$922$$ −11.6745 −0.384481
$$923$$ 0 0
$$924$$ −24.3484 −0.801002
$$925$$ 0.757449i 0.0249048i
$$926$$ −34.9399 −1.14820
$$927$$ 67.5130 2.21742
$$928$$ 22.6991i 0.745134i
$$929$$ − 15.2213i − 0.499395i −0.968324 0.249698i $$-0.919669\pi$$
0.968324 0.249698i $$-0.0803312\pi$$
$$930$$ 10.3284i 0.338680i
$$931$$ 19.3180i 0.633121i
$$932$$ −80.5560 −2.63870
$$933$$ −8.99102 −0.294353
$$934$$ 17.4313i 0.570369i
$$935$$ 0.681482 0.0222869
$$936$$ 0 0
$$937$$ 6.07285 0.198392 0.0991958 0.995068i $$-0.468373\pi$$
0.0991958 + 0.995068i $$0.468373\pi$$
$$938$$ 38.4864i 1.25663i
$$939$$ −99.9382 −3.26136
$$940$$ −39.9341 −1.30251
$$941$$ − 0.0496576i − 0.00161879i −1.00000 0.000809396i $$-0.999742\pi$$
1.00000 0.000809396i $$-0.000257639\pi$$
$$942$$ − 17.0878i − 0.556750i
$$943$$ − 1.02299i − 0.0333132i
$$944$$ − 4.01429i − 0.130654i
$$945$$ 10.7231 0.348821
$$946$$ −1.70064 −0.0552927
$$947$$ 18.6581i 0.606308i 0.952942 + 0.303154i $$0.0980396\pi$$
−0.952942 + 0.303154i $$0.901960\pi$$
$$948$$ −111.372 −3.61720
$$949$$ 0 0
$$950$$ 14.3044 0.464095
$$951$$ 38.5459i 1.24994i
$$952$$ 6.74917 0.218742
$$953$$ 1.52953 0.0495463 0.0247731 0.999693i $$-0.492114\pi$$
0.0247731 + 0.999693i $$0.492114\pi$$
$$954$$ − 87.0739i − 2.81912i
$$955$$ 5.44310i 0.176135i
$$956$$ − 53.9161i − 1.74377i
$$957$$ 28.5827i 0.923948i
$$958$$ −40.6487 −1.31330
$$959$$ 38.3131 1.23720
$$960$$ 13.6911i 0.441878i
$$961$$ 28.8564 0.930852
$$962$$ 0 0
$$963$$ −36.7475 −1.18417
$$964$$ 109.616i 3.53048i
$$965$$ −12.1576 −0.391367
$$966$$ 51.3124 1.65095
$$967$$ 32.1716i 1.03457i 0.855813 + 0.517285i $$0.173057\pi$$
−0.855813 + 0.517285i $$0.826943\pi$$
$$968$$ − 54.7907i − 1.76104i
$$969$$ 10.3259i 0.331717i
$$970$$ − 10.5349i − 0.338256i
$$971$$ −17.2541 −0.553710 −0.276855 0.960912i $$-0.589292\pi$$
−0.276855 + 0.960912i $$0.589292\pi$$
$$972$$ −60.1816 −1.93033
$$973$$ − 39.6766i − 1.27197i
$$974$$ −50.0122 −1.60249
$$975$$ 0 0
$$976$$ 22.7442 0.728023
$$977$$ 15.7228i 0.503018i 0.967855 + 0.251509i $$0.0809268\pi$$
−0.967855 + 0.251509i $$0.919073\pi$$
$$978$$ −112.750 −3.60533
$$979$$ 13.4511 0.429900
$$980$$ 14.2476i 0.455122i
$$981$$ 50.2893i 1.60561i
$$982$$ 39.4248i 1.25810i
$$983$$ − 38.5356i − 1.22910i −0.788880 0.614548i $$-0.789338\pi$$
0.788880 0.614548i $$-0.210662\pi$$
$$984$$ −4.21058 −0.134228
$$985$$ −4.37830 −0.139504
$$986$$ − 15.0364i − 0.478857i
$$987$$ −50.8743 −1.61935
$$988$$ 0 0
$$989$$ 2.43298 0.0773642
$$990$$ − 13.3194i − 0.423317i
$$991$$ 8.59143 0.272916 0.136458 0.990646i $$-0.456428\pi$$
0.136458 + 0.990646i $$0.456428\pi$$
$$992$$ 3.51490 0.111598
$$993$$ 81.4014i 2.58320i
$$994$$ 46.4268i 1.47257i
$$995$$ 20.8373i 0.660587i
$$996$$ 61.1606i 1.93795i
$$997$$ −20.5374 −0.650425 −0.325213 0.945641i $$-0.605436\pi$$
−0.325213 + 0.945641i $$0.605436\pi$$
$$998$$ −3.11069 −0.0984670
$$999$$ 4.26313i 0.134880i
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.c.g.506.8 8
13.2 odd 12 845.2.e.m.191.1 8
13.3 even 3 65.2.m.a.56.1 yes 8
13.4 even 6 65.2.m.a.36.1 8
13.5 odd 4 845.2.a.m.1.4 4
13.6 odd 12 845.2.e.m.146.1 8
13.7 odd 12 845.2.e.n.146.4 8
13.8 odd 4 845.2.a.l.1.1 4
13.9 even 3 845.2.m.g.361.4 8
13.10 even 6 845.2.m.g.316.4 8
13.11 odd 12 845.2.e.n.191.4 8
13.12 even 2 inner 845.2.c.g.506.1 8
39.5 even 4 7605.2.a.cf.1.1 4
39.8 even 4 7605.2.a.cj.1.4 4
39.17 odd 6 585.2.bu.c.361.4 8
39.29 odd 6 585.2.bu.c.316.4 8
52.3 odd 6 1040.2.da.b.641.1 8
52.43 odd 6 1040.2.da.b.881.1 8
65.3 odd 12 325.2.m.b.199.1 8
65.4 even 6 325.2.n.d.101.4 8
65.17 odd 12 325.2.m.b.49.1 8
65.29 even 6 325.2.n.d.251.4 8
65.34 odd 4 4225.2.a.bl.1.4 4
65.42 odd 12 325.2.m.c.199.4 8
65.43 odd 12 325.2.m.c.49.4 8
65.44 odd 4 4225.2.a.bi.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.1 8 13.4 even 6
65.2.m.a.56.1 yes 8 13.3 even 3
325.2.m.b.49.1 8 65.17 odd 12
325.2.m.b.199.1 8 65.3 odd 12
325.2.m.c.49.4 8 65.43 odd 12
325.2.m.c.199.4 8 65.42 odd 12
325.2.n.d.101.4 8 65.4 even 6
325.2.n.d.251.4 8 65.29 even 6
585.2.bu.c.316.4 8 39.29 odd 6
585.2.bu.c.361.4 8 39.17 odd 6
845.2.a.l.1.1 4 13.8 odd 4
845.2.a.m.1.4 4 13.5 odd 4
845.2.c.g.506.1 8 13.12 even 2 inner
845.2.c.g.506.8 8 1.1 even 1 trivial
845.2.e.m.146.1 8 13.6 odd 12
845.2.e.m.191.1 8 13.2 odd 12
845.2.e.n.146.4 8 13.7 odd 12
845.2.e.n.191.4 8 13.11 odd 12
845.2.m.g.316.4 8 13.10 even 6
845.2.m.g.361.4 8 13.9 even 3
1040.2.da.b.641.1 8 52.3 odd 6
1040.2.da.b.881.1 8 52.43 odd 6
4225.2.a.bi.1.1 4 65.44 odd 4
4225.2.a.bl.1.4 4 65.34 odd 4
7605.2.a.cf.1.1 4 39.5 even 4
7605.2.a.cj.1.4 4 39.8 even 4