# Properties

 Label 4864.2.a.bs.1.6 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $1$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 2432) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-2.30294$$ of defining polynomial Character $$\chi$$ $$=$$ 4864.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.458177 q^{3} +3.00397 q^{5} +2.21624 q^{7} -2.79007 q^{9} +O(q^{10})$$ $$q-0.458177 q^{3} +3.00397 q^{5} +2.21624 q^{7} -2.79007 q^{9} -5.02382 q^{11} -5.22021 q^{13} -1.37635 q^{15} +7.73025 q^{17} -1.00000 q^{19} -1.01543 q^{21} -6.51613 q^{23} +4.02382 q^{25} +2.65288 q^{27} +2.23562 q^{29} -3.33566 q^{31} +2.30180 q^{33} +6.65751 q^{35} +1.45208 q^{37} +2.39178 q^{39} +3.16461 q^{41} +2.16101 q^{43} -8.38129 q^{45} +9.95256 q^{47} -2.08828 q^{49} -3.54182 q^{51} -6.66810 q^{53} -15.0914 q^{55} +0.458177 q^{57} +3.89126 q^{59} -7.05259 q^{61} -6.18347 q^{63} -15.6813 q^{65} -13.2767 q^{67} +2.98554 q^{69} -5.72840 q^{71} -5.89754 q^{73} -1.84362 q^{75} -11.1340 q^{77} -15.9369 q^{79} +7.15473 q^{81} -8.71734 q^{83} +23.2214 q^{85} -1.02431 q^{87} -3.79938 q^{89} -11.5692 q^{91} +1.52833 q^{93} -3.00397 q^{95} -3.94645 q^{97} +14.0168 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 4q^{3} + 14q^{9} + O(q^{10})$$ $$10q - 4q^{3} + 14q^{9} - 20q^{11} + 4q^{17} - 10q^{19} + 10q^{25} - 28q^{27} - 8q^{33} - 36q^{35} - 12q^{41} + 4q^{43} + 26q^{49} - 36q^{51} + 4q^{57} - 52q^{59} - 24q^{65} - 12q^{67} + 12q^{73} + 12q^{75} + 34q^{81} - 16q^{83} - 20q^{89} - 60q^{91} - 28q^{97} - 60q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −0.458177 −0.264529 −0.132264 0.991214i $$-0.542225\pi$$
−0.132264 + 0.991214i $$0.542225\pi$$
$$4$$ 0 0
$$5$$ 3.00397 1.34341 0.671707 0.740817i $$-0.265561\pi$$
0.671707 + 0.740817i $$0.265561\pi$$
$$6$$ 0 0
$$7$$ 2.21624 0.837660 0.418830 0.908065i $$-0.362440\pi$$
0.418830 + 0.908065i $$0.362440\pi$$
$$8$$ 0 0
$$9$$ −2.79007 −0.930025
$$10$$ 0 0
$$11$$ −5.02382 −1.51474 −0.757369 0.652987i $$-0.773516\pi$$
−0.757369 + 0.652987i $$0.773516\pi$$
$$12$$ 0 0
$$13$$ −5.22021 −1.44783 −0.723913 0.689892i $$-0.757658\pi$$
−0.723913 + 0.689892i $$0.757658\pi$$
$$14$$ 0 0
$$15$$ −1.37635 −0.355372
$$16$$ 0 0
$$17$$ 7.73025 1.87486 0.937430 0.348174i $$-0.113198\pi$$
0.937430 + 0.348174i $$0.113198\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.01543 −0.221585
$$22$$ 0 0
$$23$$ −6.51613 −1.35871 −0.679353 0.733811i $$-0.737740\pi$$
−0.679353 + 0.733811i $$0.737740\pi$$
$$24$$ 0 0
$$25$$ 4.02382 0.804764
$$26$$ 0 0
$$27$$ 2.65288 0.510547
$$28$$ 0 0
$$29$$ 2.23562 0.415145 0.207572 0.978220i $$-0.433444\pi$$
0.207572 + 0.978220i $$0.433444\pi$$
$$30$$ 0 0
$$31$$ −3.33566 −0.599103 −0.299551 0.954080i $$-0.596837\pi$$
−0.299551 + 0.954080i $$0.596837\pi$$
$$32$$ 0 0
$$33$$ 2.30180 0.400692
$$34$$ 0 0
$$35$$ 6.65751 1.12533
$$36$$ 0 0
$$37$$ 1.45208 0.238720 0.119360 0.992851i $$-0.461916\pi$$
0.119360 + 0.992851i $$0.461916\pi$$
$$38$$ 0 0
$$39$$ 2.39178 0.382991
$$40$$ 0 0
$$41$$ 3.16461 0.494228 0.247114 0.968986i $$-0.420518\pi$$
0.247114 + 0.968986i $$0.420518\pi$$
$$42$$ 0 0
$$43$$ 2.16101 0.329551 0.164776 0.986331i $$-0.447310\pi$$
0.164776 + 0.986331i $$0.447310\pi$$
$$44$$ 0 0
$$45$$ −8.38129 −1.24941
$$46$$ 0 0
$$47$$ 9.95256 1.45173 0.725865 0.687837i $$-0.241440\pi$$
0.725865 + 0.687837i $$0.241440\pi$$
$$48$$ 0 0
$$49$$ −2.08828 −0.298325
$$50$$ 0 0
$$51$$ −3.54182 −0.495954
$$52$$ 0 0
$$53$$ −6.66810 −0.915935 −0.457967 0.888969i $$-0.651422\pi$$
−0.457967 + 0.888969i $$0.651422\pi$$
$$54$$ 0 0
$$55$$ −15.0914 −2.03492
$$56$$ 0 0
$$57$$ 0.458177 0.0606871
$$58$$ 0 0
$$59$$ 3.89126 0.506599 0.253299 0.967388i $$-0.418484\pi$$
0.253299 + 0.967388i $$0.418484\pi$$
$$60$$ 0 0
$$61$$ −7.05259 −0.902991 −0.451496 0.892273i $$-0.649109\pi$$
−0.451496 + 0.892273i $$0.649109\pi$$
$$62$$ 0 0
$$63$$ −6.18347 −0.779044
$$64$$ 0 0
$$65$$ −15.6813 −1.94503
$$66$$ 0 0
$$67$$ −13.2767 −1.62201 −0.811004 0.585041i $$-0.801078\pi$$
−0.811004 + 0.585041i $$0.801078\pi$$
$$68$$ 0 0
$$69$$ 2.98554 0.359417
$$70$$ 0 0
$$71$$ −5.72840 −0.679836 −0.339918 0.940455i $$-0.610399\pi$$
−0.339918 + 0.940455i $$0.610399\pi$$
$$72$$ 0 0
$$73$$ −5.89754 −0.690254 −0.345127 0.938556i $$-0.612164\pi$$
−0.345127 + 0.938556i $$0.612164\pi$$
$$74$$ 0 0
$$75$$ −1.84362 −0.212883
$$76$$ 0 0
$$77$$ −11.1340 −1.26884
$$78$$ 0 0
$$79$$ −15.9369 −1.79304 −0.896522 0.442998i $$-0.853915\pi$$
−0.896522 + 0.442998i $$0.853915\pi$$
$$80$$ 0 0
$$81$$ 7.15473 0.794970
$$82$$ 0 0
$$83$$ −8.71734 −0.956852 −0.478426 0.878128i $$-0.658793\pi$$
−0.478426 + 0.878128i $$0.658793\pi$$
$$84$$ 0 0
$$85$$ 23.2214 2.51871
$$86$$ 0 0
$$87$$ −1.02431 −0.109818
$$88$$ 0 0
$$89$$ −3.79938 −0.402734 −0.201367 0.979516i $$-0.564538\pi$$
−0.201367 + 0.979516i $$0.564538\pi$$
$$90$$ 0 0
$$91$$ −11.5692 −1.21279
$$92$$ 0 0
$$93$$ 1.52833 0.158480
$$94$$ 0 0
$$95$$ −3.00397 −0.308201
$$96$$ 0 0
$$97$$ −3.94645 −0.400701 −0.200351 0.979724i $$-0.564208\pi$$
−0.200351 + 0.979724i $$0.564208\pi$$
$$98$$ 0 0
$$99$$ 14.0168 1.40874
$$100$$ 0 0
$$101$$ 6.32025 0.628888 0.314444 0.949276i $$-0.398182\pi$$
0.314444 + 0.949276i $$0.398182\pi$$
$$102$$ 0 0
$$103$$ 4.71201 0.464288 0.232144 0.972681i $$-0.425426\pi$$
0.232144 + 0.972681i $$0.425426\pi$$
$$104$$ 0 0
$$105$$ −3.05032 −0.297681
$$106$$ 0 0
$$107$$ −18.5550 −1.79378 −0.896892 0.442249i $$-0.854181\pi$$
−0.896892 + 0.442249i $$0.854181\pi$$
$$108$$ 0 0
$$109$$ 9.25995 0.886942 0.443471 0.896289i $$-0.353747\pi$$
0.443471 + 0.896289i $$0.353747\pi$$
$$110$$ 0 0
$$111$$ −0.665309 −0.0631483
$$112$$ 0 0
$$113$$ −7.97386 −0.750118 −0.375059 0.927001i $$-0.622378\pi$$
−0.375059 + 0.927001i $$0.622378\pi$$
$$114$$ 0 0
$$115$$ −19.5742 −1.82531
$$116$$ 0 0
$$117$$ 14.5648 1.34651
$$118$$ 0 0
$$119$$ 17.1321 1.57050
$$120$$ 0 0
$$121$$ 14.2387 1.29443
$$122$$ 0 0
$$123$$ −1.44995 −0.130738
$$124$$ 0 0
$$125$$ −2.93242 −0.262284
$$126$$ 0 0
$$127$$ −8.32912 −0.739090 −0.369545 0.929213i $$-0.620486\pi$$
−0.369545 + 0.929213i $$0.620486\pi$$
$$128$$ 0 0
$$129$$ −0.990126 −0.0871758
$$130$$ 0 0
$$131$$ −16.3831 −1.43140 −0.715700 0.698408i $$-0.753892\pi$$
−0.715700 + 0.698408i $$0.753892\pi$$
$$132$$ 0 0
$$133$$ −2.21624 −0.192172
$$134$$ 0 0
$$135$$ 7.96916 0.685876
$$136$$ 0 0
$$137$$ 16.2267 1.38634 0.693172 0.720772i $$-0.256212\pi$$
0.693172 + 0.720772i $$0.256212\pi$$
$$138$$ 0 0
$$139$$ −4.38904 −0.372273 −0.186137 0.982524i $$-0.559597\pi$$
−0.186137 + 0.982524i $$0.559597\pi$$
$$140$$ 0 0
$$141$$ −4.56004 −0.384024
$$142$$ 0 0
$$143$$ 26.2254 2.19308
$$144$$ 0 0
$$145$$ 6.71574 0.557712
$$146$$ 0 0
$$147$$ 0.956802 0.0789157
$$148$$ 0 0
$$149$$ 1.28542 0.105306 0.0526528 0.998613i $$-0.483232\pi$$
0.0526528 + 0.998613i $$0.483232\pi$$
$$150$$ 0 0
$$151$$ −9.07295 −0.738346 −0.369173 0.929361i $$-0.620359\pi$$
−0.369173 + 0.929361i $$0.620359\pi$$
$$152$$ 0 0
$$153$$ −21.5680 −1.74367
$$154$$ 0 0
$$155$$ −10.0202 −0.804844
$$156$$ 0 0
$$157$$ 3.13656 0.250325 0.125162 0.992136i $$-0.460055\pi$$
0.125162 + 0.992136i $$0.460055\pi$$
$$158$$ 0 0
$$159$$ 3.05517 0.242291
$$160$$ 0 0
$$161$$ −14.4413 −1.13813
$$162$$ 0 0
$$163$$ −14.9640 −1.17207 −0.586035 0.810286i $$-0.699312\pi$$
−0.586035 + 0.810286i $$0.699312\pi$$
$$164$$ 0 0
$$165$$ 6.91453 0.538295
$$166$$ 0 0
$$167$$ 21.0799 1.63121 0.815607 0.578606i $$-0.196403\pi$$
0.815607 + 0.578606i $$0.196403\pi$$
$$168$$ 0 0
$$169$$ 14.2506 1.09620
$$170$$ 0 0
$$171$$ 2.79007 0.213362
$$172$$ 0 0
$$173$$ −23.7808 −1.80802 −0.904011 0.427510i $$-0.859391\pi$$
−0.904011 + 0.427510i $$0.859391\pi$$
$$174$$ 0 0
$$175$$ 8.91775 0.674118
$$176$$ 0 0
$$177$$ −1.78289 −0.134010
$$178$$ 0 0
$$179$$ −14.3018 −1.06897 −0.534483 0.845179i $$-0.679494\pi$$
−0.534483 + 0.845179i $$0.679494\pi$$
$$180$$ 0 0
$$181$$ −13.9014 −1.03328 −0.516640 0.856203i $$-0.672818\pi$$
−0.516640 + 0.856203i $$0.672818\pi$$
$$182$$ 0 0
$$183$$ 3.23133 0.238867
$$184$$ 0 0
$$185$$ 4.36199 0.320700
$$186$$ 0 0
$$187$$ −38.8353 −2.83992
$$188$$ 0 0
$$189$$ 5.87942 0.427665
$$190$$ 0 0
$$191$$ 2.17747 0.157557 0.0787783 0.996892i $$-0.474898\pi$$
0.0787783 + 0.996892i $$0.474898\pi$$
$$192$$ 0 0
$$193$$ 26.6410 1.91766 0.958831 0.283978i $$-0.0916541\pi$$
0.958831 + 0.283978i $$0.0916541\pi$$
$$194$$ 0 0
$$195$$ 7.18483 0.514516
$$196$$ 0 0
$$197$$ 20.8500 1.48550 0.742749 0.669570i $$-0.233521\pi$$
0.742749 + 0.669570i $$0.233521\pi$$
$$198$$ 0 0
$$199$$ −4.24710 −0.301069 −0.150535 0.988605i $$-0.548099\pi$$
−0.150535 + 0.988605i $$0.548099\pi$$
$$200$$ 0 0
$$201$$ 6.08308 0.429068
$$202$$ 0 0
$$203$$ 4.95468 0.347750
$$204$$ 0 0
$$205$$ 9.50637 0.663954
$$206$$ 0 0
$$207$$ 18.1805 1.26363
$$208$$ 0 0
$$209$$ 5.02382 0.347505
$$210$$ 0 0
$$211$$ 18.1134 1.24698 0.623488 0.781833i $$-0.285715\pi$$
0.623488 + 0.781833i $$0.285715\pi$$
$$212$$ 0 0
$$213$$ 2.62462 0.179836
$$214$$ 0 0
$$215$$ 6.49161 0.442724
$$216$$ 0 0
$$217$$ −7.39263 −0.501845
$$218$$ 0 0
$$219$$ 2.70212 0.182592
$$220$$ 0 0
$$221$$ −40.3535 −2.71447
$$222$$ 0 0
$$223$$ −3.71952 −0.249078 −0.124539 0.992215i $$-0.539745\pi$$
−0.124539 + 0.992215i $$0.539745\pi$$
$$224$$ 0 0
$$225$$ −11.2267 −0.748450
$$226$$ 0 0
$$227$$ 17.8318 1.18354 0.591769 0.806108i $$-0.298430\pi$$
0.591769 + 0.806108i $$0.298430\pi$$
$$228$$ 0 0
$$229$$ 13.5159 0.893158 0.446579 0.894744i $$-0.352642\pi$$
0.446579 + 0.894744i $$0.352642\pi$$
$$230$$ 0 0
$$231$$ 5.10134 0.335644
$$232$$ 0 0
$$233$$ −18.5061 −1.21238 −0.606189 0.795321i $$-0.707302\pi$$
−0.606189 + 0.795321i $$0.707302\pi$$
$$234$$ 0 0
$$235$$ 29.8972 1.95028
$$236$$ 0 0
$$237$$ 7.30194 0.474312
$$238$$ 0 0
$$239$$ 11.6874 0.755995 0.377997 0.925807i $$-0.376613\pi$$
0.377997 + 0.925807i $$0.376613\pi$$
$$240$$ 0 0
$$241$$ 10.2701 0.661554 0.330777 0.943709i $$-0.392689\pi$$
0.330777 + 0.943709i $$0.392689\pi$$
$$242$$ 0 0
$$243$$ −11.2368 −0.720839
$$244$$ 0 0
$$245$$ −6.27312 −0.400775
$$246$$ 0 0
$$247$$ 5.22021 0.332154
$$248$$ 0 0
$$249$$ 3.99409 0.253115
$$250$$ 0 0
$$251$$ −25.1064 −1.58470 −0.792350 0.610066i $$-0.791143\pi$$
−0.792350 + 0.610066i $$0.791143\pi$$
$$252$$ 0 0
$$253$$ 32.7358 2.05808
$$254$$ 0 0
$$255$$ −10.6395 −0.666273
$$256$$ 0 0
$$257$$ 12.5905 0.785374 0.392687 0.919672i $$-0.371546\pi$$
0.392687 + 0.919672i $$0.371546\pi$$
$$258$$ 0 0
$$259$$ 3.21815 0.199966
$$260$$ 0 0
$$261$$ −6.23755 −0.386095
$$262$$ 0 0
$$263$$ −22.7749 −1.40436 −0.702181 0.711999i $$-0.747790\pi$$
−0.702181 + 0.711999i $$0.747790\pi$$
$$264$$ 0 0
$$265$$ −20.0308 −1.23048
$$266$$ 0 0
$$267$$ 1.74079 0.106535
$$268$$ 0 0
$$269$$ 17.0271 1.03816 0.519081 0.854725i $$-0.326274\pi$$
0.519081 + 0.854725i $$0.326274\pi$$
$$270$$ 0 0
$$271$$ 8.19591 0.497866 0.248933 0.968521i $$-0.419920\pi$$
0.248933 + 0.968521i $$0.419920\pi$$
$$272$$ 0 0
$$273$$ 5.30076 0.320817
$$274$$ 0 0
$$275$$ −20.2149 −1.21901
$$276$$ 0 0
$$277$$ −18.7174 −1.12462 −0.562309 0.826927i $$-0.690087\pi$$
−0.562309 + 0.826927i $$0.690087\pi$$
$$278$$ 0 0
$$279$$ 9.30675 0.557180
$$280$$ 0 0
$$281$$ 0.189116 0.0112817 0.00564087 0.999984i $$-0.498204\pi$$
0.00564087 + 0.999984i $$0.498204\pi$$
$$282$$ 0 0
$$283$$ −10.6357 −0.632226 −0.316113 0.948722i $$-0.602378\pi$$
−0.316113 + 0.948722i $$0.602378\pi$$
$$284$$ 0 0
$$285$$ 1.37635 0.0815279
$$286$$ 0 0
$$287$$ 7.01353 0.413995
$$288$$ 0 0
$$289$$ 42.7567 2.51510
$$290$$ 0 0
$$291$$ 1.80817 0.105997
$$292$$ 0 0
$$293$$ −27.6765 −1.61688 −0.808439 0.588580i $$-0.799687\pi$$
−0.808439 + 0.588580i $$0.799687\pi$$
$$294$$ 0 0
$$295$$ 11.6892 0.680572
$$296$$ 0 0
$$297$$ −13.3276 −0.773345
$$298$$ 0 0
$$299$$ 34.0155 1.96717
$$300$$ 0 0
$$301$$ 4.78932 0.276052
$$302$$ 0 0
$$303$$ −2.89579 −0.166359
$$304$$ 0 0
$$305$$ −21.1857 −1.21309
$$306$$ 0 0
$$307$$ −8.14707 −0.464978 −0.232489 0.972599i $$-0.574687\pi$$
−0.232489 + 0.972599i $$0.574687\pi$$
$$308$$ 0 0
$$309$$ −2.15894 −0.122818
$$310$$ 0 0
$$311$$ 11.1283 0.631030 0.315515 0.948921i $$-0.397823\pi$$
0.315515 + 0.948921i $$0.397823\pi$$
$$312$$ 0 0
$$313$$ −10.2647 −0.580198 −0.290099 0.956997i $$-0.593688\pi$$
−0.290099 + 0.956997i $$0.593688\pi$$
$$314$$ 0 0
$$315$$ −18.5750 −1.04658
$$316$$ 0 0
$$317$$ 7.14866 0.401509 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$318$$ 0 0
$$319$$ −11.2314 −0.628836
$$320$$ 0 0
$$321$$ 8.50150 0.474508
$$322$$ 0 0
$$323$$ −7.73025 −0.430122
$$324$$ 0 0
$$325$$ −21.0052 −1.16516
$$326$$ 0 0
$$327$$ −4.24270 −0.234622
$$328$$ 0 0
$$329$$ 22.0573 1.21606
$$330$$ 0 0
$$331$$ 28.4370 1.56304 0.781519 0.623881i $$-0.214445\pi$$
0.781519 + 0.623881i $$0.214445\pi$$
$$332$$ 0 0
$$333$$ −4.05140 −0.222016
$$334$$ 0 0
$$335$$ −39.8828 −2.17903
$$336$$ 0 0
$$337$$ −18.9197 −1.03062 −0.515311 0.857003i $$-0.672324\pi$$
−0.515311 + 0.857003i $$0.672324\pi$$
$$338$$ 0 0
$$339$$ 3.65344 0.198428
$$340$$ 0 0
$$341$$ 16.7578 0.907484
$$342$$ 0 0
$$343$$ −20.1418 −1.08756
$$344$$ 0 0
$$345$$ 8.96847 0.482846
$$346$$ 0 0
$$347$$ −2.34140 −0.125693 −0.0628466 0.998023i $$-0.520018\pi$$
−0.0628466 + 0.998023i $$0.520018\pi$$
$$348$$ 0 0
$$349$$ −14.8506 −0.794935 −0.397468 0.917616i $$-0.630111\pi$$
−0.397468 + 0.917616i $$0.630111\pi$$
$$350$$ 0 0
$$351$$ −13.8486 −0.739183
$$352$$ 0 0
$$353$$ −16.9690 −0.903168 −0.451584 0.892229i $$-0.649141\pi$$
−0.451584 + 0.892229i $$0.649141\pi$$
$$354$$ 0 0
$$355$$ −17.2079 −0.913302
$$356$$ 0 0
$$357$$ −7.84953 −0.415441
$$358$$ 0 0
$$359$$ 32.2107 1.70002 0.850008 0.526770i $$-0.176597\pi$$
0.850008 + 0.526770i $$0.176597\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −6.52387 −0.342414
$$364$$ 0 0
$$365$$ −17.7160 −0.927298
$$366$$ 0 0
$$367$$ 10.2366 0.534347 0.267173 0.963648i $$-0.413910\pi$$
0.267173 + 0.963648i $$0.413910\pi$$
$$368$$ 0 0
$$369$$ −8.82948 −0.459644
$$370$$ 0 0
$$371$$ −14.7781 −0.767242
$$372$$ 0 0
$$373$$ −30.1908 −1.56322 −0.781610 0.623767i $$-0.785601\pi$$
−0.781610 + 0.623767i $$0.785601\pi$$
$$374$$ 0 0
$$375$$ 1.34357 0.0693815
$$376$$ 0 0
$$377$$ −11.6704 −0.601057
$$378$$ 0 0
$$379$$ 15.9865 0.821173 0.410586 0.911822i $$-0.365324\pi$$
0.410586 + 0.911822i $$0.365324\pi$$
$$380$$ 0 0
$$381$$ 3.81622 0.195511
$$382$$ 0 0
$$383$$ 22.6961 1.15971 0.579857 0.814718i $$-0.303108\pi$$
0.579857 + 0.814718i $$0.303108\pi$$
$$384$$ 0 0
$$385$$ −33.4461 −1.70457
$$386$$ 0 0
$$387$$ −6.02938 −0.306491
$$388$$ 0 0
$$389$$ 12.1956 0.618341 0.309170 0.951007i $$-0.399949\pi$$
0.309170 + 0.951007i $$0.399949\pi$$
$$390$$ 0 0
$$391$$ −50.3713 −2.54738
$$392$$ 0 0
$$393$$ 7.50638 0.378647
$$394$$ 0 0
$$395$$ −47.8740 −2.40880
$$396$$ 0 0
$$397$$ 10.7422 0.539137 0.269568 0.962981i $$-0.413119\pi$$
0.269568 + 0.962981i $$0.413119\pi$$
$$398$$ 0 0
$$399$$ 1.01543 0.0508351
$$400$$ 0 0
$$401$$ −5.65983 −0.282639 −0.141319 0.989964i $$-0.545134\pi$$
−0.141319 + 0.989964i $$0.545134\pi$$
$$402$$ 0 0
$$403$$ 17.4129 0.867396
$$404$$ 0 0
$$405$$ 21.4926 1.06797
$$406$$ 0 0
$$407$$ −7.29497 −0.361598
$$408$$ 0 0
$$409$$ −27.6645 −1.36792 −0.683960 0.729520i $$-0.739744\pi$$
−0.683960 + 0.729520i $$0.739744\pi$$
$$410$$ 0 0
$$411$$ −7.43473 −0.366728
$$412$$ 0 0
$$413$$ 8.62396 0.424357
$$414$$ 0 0
$$415$$ −26.1866 −1.28545
$$416$$ 0 0
$$417$$ 2.01096 0.0984770
$$418$$ 0 0
$$419$$ 6.91763 0.337948 0.168974 0.985620i $$-0.445955\pi$$
0.168974 + 0.985620i $$0.445955\pi$$
$$420$$ 0 0
$$421$$ −3.07015 −0.149630 −0.0748150 0.997197i $$-0.523837\pi$$
−0.0748150 + 0.997197i $$0.523837\pi$$
$$422$$ 0 0
$$423$$ −27.7684 −1.35014
$$424$$ 0 0
$$425$$ 31.1051 1.50882
$$426$$ 0 0
$$427$$ −15.6302 −0.756400
$$428$$ 0 0
$$429$$ −12.0159 −0.580132
$$430$$ 0 0
$$431$$ −13.2015 −0.635893 −0.317946 0.948109i $$-0.602993\pi$$
−0.317946 + 0.948109i $$0.602993\pi$$
$$432$$ 0 0
$$433$$ −20.1544 −0.968559 −0.484279 0.874913i $$-0.660918\pi$$
−0.484279 + 0.874913i $$0.660918\pi$$
$$434$$ 0 0
$$435$$ −3.07700 −0.147531
$$436$$ 0 0
$$437$$ 6.51613 0.311709
$$438$$ 0 0
$$439$$ 1.72144 0.0821601 0.0410800 0.999156i $$-0.486920\pi$$
0.0410800 + 0.999156i $$0.486920\pi$$
$$440$$ 0 0
$$441$$ 5.82645 0.277450
$$442$$ 0 0
$$443$$ −38.0168 −1.80623 −0.903117 0.429395i $$-0.858727\pi$$
−0.903117 + 0.429395i $$0.858727\pi$$
$$444$$ 0 0
$$445$$ −11.4132 −0.541039
$$446$$ 0 0
$$447$$ −0.588950 −0.0278564
$$448$$ 0 0
$$449$$ −27.1861 −1.28299 −0.641496 0.767127i $$-0.721686\pi$$
−0.641496 + 0.767127i $$0.721686\pi$$
$$450$$ 0 0
$$451$$ −15.8984 −0.748626
$$452$$ 0 0
$$453$$ 4.15702 0.195314
$$454$$ 0 0
$$455$$ −34.7536 −1.62927
$$456$$ 0 0
$$457$$ 15.8069 0.739415 0.369708 0.929148i $$-0.379458\pi$$
0.369708 + 0.929148i $$0.379458\pi$$
$$458$$ 0 0
$$459$$ 20.5074 0.957204
$$460$$ 0 0
$$461$$ 29.7741 1.38672 0.693358 0.720593i $$-0.256130\pi$$
0.693358 + 0.720593i $$0.256130\pi$$
$$462$$ 0 0
$$463$$ 11.8339 0.549967 0.274984 0.961449i $$-0.411328\pi$$
0.274984 + 0.961449i $$0.411328\pi$$
$$464$$ 0 0
$$465$$ 4.59104 0.212904
$$466$$ 0 0
$$467$$ −30.6052 −1.41624 −0.708121 0.706091i $$-0.750457\pi$$
−0.708121 + 0.706091i $$0.750457\pi$$
$$468$$ 0 0
$$469$$ −29.4244 −1.35869
$$470$$ 0 0
$$471$$ −1.43710 −0.0662181
$$472$$ 0 0
$$473$$ −10.8565 −0.499184
$$474$$ 0 0
$$475$$ −4.02382 −0.184625
$$476$$ 0 0
$$477$$ 18.6045 0.851842
$$478$$ 0 0
$$479$$ 12.3908 0.566150 0.283075 0.959098i $$-0.408645\pi$$
0.283075 + 0.959098i $$0.408645\pi$$
$$480$$ 0 0
$$481$$ −7.58015 −0.345625
$$482$$ 0 0
$$483$$ 6.61668 0.301069
$$484$$ 0 0
$$485$$ −11.8550 −0.538308
$$486$$ 0 0
$$487$$ 7.61617 0.345122 0.172561 0.984999i $$-0.444796\pi$$
0.172561 + 0.984999i $$0.444796\pi$$
$$488$$ 0 0
$$489$$ 6.85616 0.310046
$$490$$ 0 0
$$491$$ 3.09291 0.139581 0.0697906 0.997562i $$-0.477767\pi$$
0.0697906 + 0.997562i $$0.477767\pi$$
$$492$$ 0 0
$$493$$ 17.2819 0.778338
$$494$$ 0 0
$$495$$ 42.1061 1.89253
$$496$$ 0 0
$$497$$ −12.6955 −0.569472
$$498$$ 0 0
$$499$$ 6.55042 0.293237 0.146618 0.989193i $$-0.453161\pi$$
0.146618 + 0.989193i $$0.453161\pi$$
$$500$$ 0 0
$$501$$ −9.65835 −0.431503
$$502$$ 0 0
$$503$$ 34.9332 1.55760 0.778798 0.627275i $$-0.215830\pi$$
0.778798 + 0.627275i $$0.215830\pi$$
$$504$$ 0 0
$$505$$ 18.9858 0.844858
$$506$$ 0 0
$$507$$ −6.52928 −0.289976
$$508$$ 0 0
$$509$$ −10.6461 −0.471882 −0.235941 0.971767i $$-0.575817\pi$$
−0.235941 + 0.971767i $$0.575817\pi$$
$$510$$ 0 0
$$511$$ −13.0704 −0.578199
$$512$$ 0 0
$$513$$ −2.65288 −0.117128
$$514$$ 0 0
$$515$$ 14.1547 0.623732
$$516$$ 0 0
$$517$$ −49.9999 −2.19899
$$518$$ 0 0
$$519$$ 10.8958 0.478274
$$520$$ 0 0
$$521$$ −33.7087 −1.47681 −0.738403 0.674359i $$-0.764420\pi$$
−0.738403 + 0.674359i $$0.764420\pi$$
$$522$$ 0 0
$$523$$ 3.61224 0.157952 0.0789761 0.996877i $$-0.474835\pi$$
0.0789761 + 0.996877i $$0.474835\pi$$
$$524$$ 0 0
$$525$$ −4.08591 −0.178324
$$526$$ 0 0
$$527$$ −25.7855 −1.12323
$$528$$ 0 0
$$529$$ 19.4599 0.846084
$$530$$ 0 0
$$531$$ −10.8569 −0.471149
$$532$$ 0 0
$$533$$ −16.5199 −0.715556
$$534$$ 0 0
$$535$$ −55.7388 −2.40980
$$536$$ 0 0
$$537$$ 6.55276 0.282772
$$538$$ 0 0
$$539$$ 10.4911 0.451885
$$540$$ 0 0
$$541$$ 45.1835 1.94259 0.971295 0.237876i $$-0.0764513\pi$$
0.971295 + 0.237876i $$0.0764513\pi$$
$$542$$ 0 0
$$543$$ 6.36929 0.273332
$$544$$ 0 0
$$545$$ 27.8166 1.19153
$$546$$ 0 0
$$547$$ 25.9716 1.11047 0.555233 0.831695i $$-0.312629\pi$$
0.555233 + 0.831695i $$0.312629\pi$$
$$548$$ 0 0
$$549$$ 19.6772 0.839804
$$550$$ 0 0
$$551$$ −2.23562 −0.0952408
$$552$$ 0 0
$$553$$ −35.3201 −1.50196
$$554$$ 0 0
$$555$$ −1.99857 −0.0848344
$$556$$ 0 0
$$557$$ 27.5515 1.16739 0.583697 0.811972i $$-0.301606\pi$$
0.583697 + 0.811972i $$0.301606\pi$$
$$558$$ 0 0
$$559$$ −11.2809 −0.477132
$$560$$ 0 0
$$561$$ 17.7935 0.751241
$$562$$ 0 0
$$563$$ 14.8531 0.625985 0.312992 0.949756i $$-0.398669\pi$$
0.312992 + 0.949756i $$0.398669\pi$$
$$564$$ 0 0
$$565$$ −23.9532 −1.00772
$$566$$ 0 0
$$567$$ 15.8566 0.665915
$$568$$ 0 0
$$569$$ −4.86281 −0.203859 −0.101930 0.994792i $$-0.532502\pi$$
−0.101930 + 0.994792i $$0.532502\pi$$
$$570$$ 0 0
$$571$$ −21.6960 −0.907948 −0.453974 0.891015i $$-0.649994\pi$$
−0.453974 + 0.891015i $$0.649994\pi$$
$$572$$ 0 0
$$573$$ −0.997669 −0.0416782
$$574$$ 0 0
$$575$$ −26.2197 −1.09344
$$576$$ 0 0
$$577$$ 28.5083 1.18682 0.593409 0.804901i $$-0.297782\pi$$
0.593409 + 0.804901i $$0.297782\pi$$
$$578$$ 0 0
$$579$$ −12.2063 −0.507277
$$580$$ 0 0
$$581$$ −19.3197 −0.801517
$$582$$ 0 0
$$583$$ 33.4993 1.38740
$$584$$ 0 0
$$585$$ 43.7521 1.80893
$$586$$ 0 0
$$587$$ 22.5133 0.929224 0.464612 0.885514i $$-0.346194\pi$$
0.464612 + 0.885514i $$0.346194\pi$$
$$588$$ 0 0
$$589$$ 3.33566 0.137444
$$590$$ 0 0
$$591$$ −9.55298 −0.392957
$$592$$ 0 0
$$593$$ −11.0209 −0.452574 −0.226287 0.974061i $$-0.572659\pi$$
−0.226287 + 0.974061i $$0.572659\pi$$
$$594$$ 0 0
$$595$$ 51.4642 2.10983
$$596$$ 0 0
$$597$$ 1.94593 0.0796414
$$598$$ 0 0
$$599$$ −18.7308 −0.765319 −0.382659 0.923890i $$-0.624992\pi$$
−0.382659 + 0.923890i $$0.624992\pi$$
$$600$$ 0 0
$$601$$ 14.8859 0.607209 0.303604 0.952798i $$-0.401810\pi$$
0.303604 + 0.952798i $$0.401810\pi$$
$$602$$ 0 0
$$603$$ 37.0430 1.50851
$$604$$ 0 0
$$605$$ 42.7727 1.73896
$$606$$ 0 0
$$607$$ 37.9701 1.54116 0.770579 0.637345i $$-0.219967\pi$$
0.770579 + 0.637345i $$0.219967\pi$$
$$608$$ 0 0
$$609$$ −2.27012 −0.0919900
$$610$$ 0 0
$$611$$ −51.9544 −2.10185
$$612$$ 0 0
$$613$$ 28.5379 1.15264 0.576318 0.817226i $$-0.304489\pi$$
0.576318 + 0.817226i $$0.304489\pi$$
$$614$$ 0 0
$$615$$ −4.35560 −0.175635
$$616$$ 0 0
$$617$$ 3.91835 0.157747 0.0788733 0.996885i $$-0.474868\pi$$
0.0788733 + 0.996885i $$0.474868\pi$$
$$618$$ 0 0
$$619$$ 5.37059 0.215862 0.107931 0.994158i $$-0.465577\pi$$
0.107931 + 0.994158i $$0.465577\pi$$
$$620$$ 0 0
$$621$$ −17.2865 −0.693684
$$622$$ 0 0
$$623$$ −8.42035 −0.337354
$$624$$ 0 0
$$625$$ −28.9280 −1.15712
$$626$$ 0 0
$$627$$ −2.30180 −0.0919250
$$628$$ 0 0
$$629$$ 11.2249 0.447567
$$630$$ 0 0
$$631$$ 18.6483 0.742377 0.371189 0.928557i $$-0.378950\pi$$
0.371189 + 0.928557i $$0.378950\pi$$
$$632$$ 0 0
$$633$$ −8.29913 −0.329861
$$634$$ 0 0
$$635$$ −25.0204 −0.992905
$$636$$ 0 0
$$637$$ 10.9012 0.431923
$$638$$ 0 0
$$639$$ 15.9827 0.632264
$$640$$ 0 0
$$641$$ 42.2857 1.67018 0.835092 0.550111i $$-0.185415\pi$$
0.835092 + 0.550111i $$0.185415\pi$$
$$642$$ 0 0
$$643$$ 18.4426 0.727305 0.363652 0.931535i $$-0.381530\pi$$
0.363652 + 0.931535i $$0.381530\pi$$
$$644$$ 0 0
$$645$$ −2.97431 −0.117113
$$646$$ 0 0
$$647$$ 2.19012 0.0861024 0.0430512 0.999073i $$-0.486292\pi$$
0.0430512 + 0.999073i $$0.486292\pi$$
$$648$$ 0 0
$$649$$ −19.5490 −0.767364
$$650$$ 0 0
$$651$$ 3.38714 0.132752
$$652$$ 0 0
$$653$$ 43.8208 1.71484 0.857420 0.514617i $$-0.172066\pi$$
0.857420 + 0.514617i $$0.172066\pi$$
$$654$$ 0 0
$$655$$ −49.2144 −1.92296
$$656$$ 0 0
$$657$$ 16.4546 0.641954
$$658$$ 0 0
$$659$$ 11.0297 0.429657 0.214829 0.976652i $$-0.431081\pi$$
0.214829 + 0.976652i $$0.431081\pi$$
$$660$$ 0 0
$$661$$ 7.49137 0.291381 0.145690 0.989330i $$-0.453460\pi$$
0.145690 + 0.989330i $$0.453460\pi$$
$$662$$ 0 0
$$663$$ 18.4890 0.718055
$$664$$ 0 0
$$665$$ −6.65751 −0.258167
$$666$$ 0 0
$$667$$ −14.5676 −0.564060
$$668$$ 0 0
$$669$$ 1.70420 0.0658882
$$670$$ 0 0
$$671$$ 35.4309 1.36780
$$672$$ 0 0
$$673$$ −34.6195 −1.33448 −0.667242 0.744841i $$-0.732525\pi$$
−0.667242 + 0.744841i $$0.732525\pi$$
$$674$$ 0 0
$$675$$ 10.6747 0.410870
$$676$$ 0 0
$$677$$ −29.0201 −1.11533 −0.557666 0.830066i $$-0.688303\pi$$
−0.557666 + 0.830066i $$0.688303\pi$$
$$678$$ 0 0
$$679$$ −8.74629 −0.335652
$$680$$ 0 0
$$681$$ −8.17012 −0.313080
$$682$$ 0 0
$$683$$ 13.3799 0.511966 0.255983 0.966681i $$-0.417601\pi$$
0.255983 + 0.966681i $$0.417601\pi$$
$$684$$ 0 0
$$685$$ 48.7446 1.86244
$$686$$ 0 0
$$687$$ −6.19269 −0.236266
$$688$$ 0 0
$$689$$ 34.8089 1.32611
$$690$$ 0 0
$$691$$ 40.7415 1.54988 0.774940 0.632035i $$-0.217780\pi$$
0.774940 + 0.632035i $$0.217780\pi$$
$$692$$ 0 0
$$693$$ 31.0646 1.18005
$$694$$ 0 0
$$695$$ −13.1845 −0.500118
$$696$$ 0 0
$$697$$ 24.4632 0.926609
$$698$$ 0 0
$$699$$ 8.47909 0.320709
$$700$$ 0 0
$$701$$ −43.7760 −1.65340 −0.826698 0.562645i $$-0.809784\pi$$
−0.826698 + 0.562645i $$0.809784\pi$$
$$702$$ 0 0
$$703$$ −1.45208 −0.0547662
$$704$$ 0 0
$$705$$ −13.6982 −0.515904
$$706$$ 0 0
$$707$$ 14.0072 0.526795
$$708$$ 0 0
$$709$$ 26.0155 0.977031 0.488516 0.872555i $$-0.337538\pi$$
0.488516 + 0.872555i $$0.337538\pi$$
$$710$$ 0 0
$$711$$ 44.4652 1.66758
$$712$$ 0 0
$$713$$ 21.7356 0.814005
$$714$$ 0 0
$$715$$ 78.7801 2.94621
$$716$$ 0 0
$$717$$ −5.35490 −0.199982
$$718$$ 0 0
$$719$$ −31.3051 −1.16748 −0.583741 0.811940i $$-0.698412\pi$$
−0.583741 + 0.811940i $$0.698412\pi$$
$$720$$ 0 0
$$721$$ 10.4430 0.388916
$$722$$ 0 0
$$723$$ −4.70552 −0.175000
$$724$$ 0 0
$$725$$ 8.99574 0.334093
$$726$$ 0 0
$$727$$ 7.31403 0.271262 0.135631 0.990759i $$-0.456694\pi$$
0.135631 + 0.990759i $$0.456694\pi$$
$$728$$ 0 0
$$729$$ −16.3158 −0.604287
$$730$$ 0 0
$$731$$ 16.7051 0.617862
$$732$$ 0 0
$$733$$ −13.0935 −0.483620 −0.241810 0.970324i $$-0.577741\pi$$
−0.241810 + 0.970324i $$0.577741\pi$$
$$734$$ 0 0
$$735$$ 2.87420 0.106016
$$736$$ 0 0
$$737$$ 66.6997 2.45692
$$738$$ 0 0
$$739$$ 10.5782 0.389124 0.194562 0.980890i $$-0.437672\pi$$
0.194562 + 0.980890i $$0.437672\pi$$
$$740$$ 0 0
$$741$$ −2.39178 −0.0878642
$$742$$ 0 0
$$743$$ 23.6180 0.866460 0.433230 0.901283i $$-0.357374\pi$$
0.433230 + 0.901283i $$0.357374\pi$$
$$744$$ 0 0
$$745$$ 3.86136 0.141469
$$746$$ 0 0
$$747$$ 24.3220 0.889896
$$748$$ 0 0
$$749$$ −41.1225 −1.50258
$$750$$ 0 0
$$751$$ −24.5994 −0.897644 −0.448822 0.893621i $$-0.648156\pi$$
−0.448822 + 0.893621i $$0.648156\pi$$
$$752$$ 0 0
$$753$$ 11.5032 0.419199
$$754$$ 0 0
$$755$$ −27.2548 −0.991905
$$756$$ 0 0
$$757$$ 3.03483 0.110303 0.0551514 0.998478i $$-0.482436\pi$$
0.0551514 + 0.998478i $$0.482436\pi$$
$$758$$ 0 0
$$759$$ −14.9988 −0.544423
$$760$$ 0 0
$$761$$ −8.15330 −0.295557 −0.147778 0.989020i $$-0.547212\pi$$
−0.147778 + 0.989020i $$0.547212\pi$$
$$762$$ 0 0
$$763$$ 20.5223 0.742956
$$764$$ 0 0
$$765$$ −64.7894 −2.34247
$$766$$ 0 0
$$767$$ −20.3132 −0.733466
$$768$$ 0 0
$$769$$ −14.2744 −0.514747 −0.257374 0.966312i $$-0.582857\pi$$
−0.257374 + 0.966312i $$0.582857\pi$$
$$770$$ 0 0
$$771$$ −5.76868 −0.207754
$$772$$ 0 0
$$773$$ −45.2404 −1.62718 −0.813591 0.581437i $$-0.802491\pi$$
−0.813591 + 0.581437i $$0.802491\pi$$
$$774$$ 0 0
$$775$$ −13.4221 −0.482136
$$776$$ 0 0
$$777$$ −1.47448 −0.0528968
$$778$$ 0 0
$$779$$ −3.16461 −0.113384
$$780$$ 0 0
$$781$$ 28.7784 1.02977
$$782$$ 0 0
$$783$$ 5.93084 0.211951
$$784$$ 0 0
$$785$$ 9.42212 0.336290
$$786$$ 0 0
$$787$$ −50.1656 −1.78821 −0.894106 0.447856i $$-0.852188\pi$$
−0.894106 + 0.447856i $$0.852188\pi$$
$$788$$ 0 0
$$789$$ 10.4349 0.371494
$$790$$ 0 0
$$791$$ −17.6720 −0.628344
$$792$$ 0 0
$$793$$ 36.8160 1.30737
$$794$$ 0 0
$$795$$ 9.17764 0.325497
$$796$$ 0 0
$$797$$ −21.7396 −0.770056 −0.385028 0.922905i $$-0.625808\pi$$
−0.385028 + 0.922905i $$0.625808\pi$$
$$798$$ 0 0
$$799$$ 76.9357 2.72179
$$800$$ 0 0
$$801$$ 10.6006 0.374552
$$802$$ 0 0
$$803$$ 29.6281 1.04555
$$804$$ 0 0
$$805$$ −43.3812 −1.52899
$$806$$ 0 0
$$807$$ −7.80144 −0.274624
$$808$$ 0 0
$$809$$ −9.48695 −0.333543 −0.166772 0.985996i $$-0.553334\pi$$
−0.166772 + 0.985996i $$0.553334\pi$$
$$810$$ 0 0
$$811$$ 22.6529 0.795450 0.397725 0.917505i $$-0.369800\pi$$
0.397725 + 0.917505i $$0.369800\pi$$
$$812$$ 0 0
$$813$$ −3.75518 −0.131700
$$814$$ 0 0
$$815$$ −44.9513 −1.57458
$$816$$ 0 0
$$817$$ −2.16101 −0.0756042
$$818$$ 0 0
$$819$$ 32.2790 1.12792
$$820$$ 0 0
$$821$$ 26.4552 0.923292 0.461646 0.887064i $$-0.347259\pi$$
0.461646 + 0.887064i $$0.347259\pi$$
$$822$$ 0 0
$$823$$ −35.1129 −1.22396 −0.611980 0.790873i $$-0.709627\pi$$
−0.611980 + 0.790873i $$0.709627\pi$$
$$824$$ 0 0
$$825$$ 9.26202 0.322462
$$826$$ 0 0
$$827$$ −27.4003 −0.952803 −0.476402 0.879228i $$-0.658059\pi$$
−0.476402 + 0.879228i $$0.658059\pi$$
$$828$$ 0 0
$$829$$ 25.2213 0.875972 0.437986 0.898982i $$-0.355692\pi$$
0.437986 + 0.898982i $$0.355692\pi$$
$$830$$ 0 0
$$831$$ 8.57588 0.297494
$$832$$ 0 0
$$833$$ −16.1429 −0.559319
$$834$$ 0 0
$$835$$ 63.3234 2.19140
$$836$$ 0 0
$$837$$ −8.84912 −0.305870
$$838$$ 0 0
$$839$$ −36.8869 −1.27348 −0.636738 0.771080i $$-0.719717\pi$$
−0.636738 + 0.771080i $$0.719717\pi$$
$$840$$ 0 0
$$841$$ −24.0020 −0.827655
$$842$$ 0 0
$$843$$ −0.0866488 −0.00298434
$$844$$ 0 0
$$845$$ 42.8082 1.47265
$$846$$ 0 0
$$847$$ 31.5565 1.08429
$$848$$ 0 0
$$849$$ 4.87303 0.167242
$$850$$ 0 0
$$851$$ −9.46193 −0.324351
$$852$$ 0 0
$$853$$ 3.36746 0.115300 0.0576499 0.998337i $$-0.481639\pi$$
0.0576499 + 0.998337i $$0.481639\pi$$
$$854$$ 0 0
$$855$$ 8.38129 0.286634
$$856$$ 0 0
$$857$$ −16.0416 −0.547970 −0.273985 0.961734i $$-0.588342\pi$$
−0.273985 + 0.961734i $$0.588342\pi$$
$$858$$ 0 0
$$859$$ −14.7310 −0.502614 −0.251307 0.967907i $$-0.580860\pi$$
−0.251307 + 0.967907i $$0.580860\pi$$
$$860$$ 0 0
$$861$$ −3.21344 −0.109514
$$862$$ 0 0
$$863$$ 41.3034 1.40598 0.702991 0.711198i $$-0.251847\pi$$
0.702991 + 0.711198i $$0.251847\pi$$
$$864$$ 0 0
$$865$$ −71.4368 −2.42892
$$866$$ 0 0
$$867$$ −19.5901 −0.665316
$$868$$ 0 0
$$869$$ 80.0642 2.71599
$$870$$ 0 0
$$871$$ 69.3071 2.34838
$$872$$ 0 0
$$873$$ 11.0109 0.372662
$$874$$ 0 0
$$875$$ −6.49895 −0.219704
$$876$$ 0 0
$$877$$ −3.84438 −0.129815 −0.0649077 0.997891i $$-0.520675\pi$$
−0.0649077 + 0.997891i $$0.520675\pi$$
$$878$$ 0 0
$$879$$ 12.6807 0.427711
$$880$$ 0 0
$$881$$ −4.57677 −0.154195 −0.0770976 0.997024i $$-0.524565\pi$$
−0.0770976 + 0.997024i $$0.524565\pi$$
$$882$$ 0 0
$$883$$ 2.34140 0.0787945 0.0393972 0.999224i $$-0.487456\pi$$
0.0393972 + 0.999224i $$0.487456\pi$$
$$884$$ 0 0
$$885$$ −5.35573 −0.180031
$$886$$ 0 0
$$887$$ 48.1958 1.61826 0.809129 0.587631i $$-0.199939\pi$$
0.809129 + 0.587631i $$0.199939\pi$$
$$888$$ 0 0
$$889$$ −18.4593 −0.619106
$$890$$ 0 0
$$891$$ −35.9441 −1.20417
$$892$$ 0 0
$$893$$ −9.95256 −0.333050
$$894$$ 0 0
$$895$$ −42.9621 −1.43607
$$896$$ 0 0
$$897$$ −15.5851 −0.520373
$$898$$ 0 0
$$899$$ −7.45729 −0.248714
$$900$$ 0 0
$$901$$ −51.5461 −1.71725
$$902$$ 0 0
$$903$$ −2.19436 −0.0730237
$$904$$ 0 0
$$905$$ −41.7593 −1.38812
$$906$$ 0 0
$$907$$ 4.08133 0.135518 0.0677592 0.997702i $$-0.478415\pi$$
0.0677592 + 0.997702i $$0.478415\pi$$
$$908$$ 0 0
$$909$$ −17.6340 −0.584881
$$910$$ 0 0
$$911$$ 54.1174 1.79299 0.896495 0.443054i $$-0.146105\pi$$
0.896495 + 0.443054i $$0.146105\pi$$
$$912$$ 0 0
$$913$$ 43.7943 1.44938
$$914$$ 0 0
$$915$$ 9.70682 0.320898
$$916$$ 0 0
$$917$$ −36.3089 −1.19903
$$918$$ 0 0
$$919$$ 32.1252 1.05971 0.529857 0.848087i $$-0.322246\pi$$
0.529857 + 0.848087i $$0.322246\pi$$
$$920$$ 0 0
$$921$$ 3.73280 0.123000
$$922$$ 0 0
$$923$$ 29.9034 0.984284
$$924$$ 0 0
$$925$$ 5.84290 0.192113
$$926$$ 0 0
$$927$$ −13.1469 −0.431800
$$928$$ 0 0
$$929$$ −10.9339 −0.358730 −0.179365 0.983783i $$-0.557404\pi$$
−0.179365 + 0.983783i $$0.557404\pi$$
$$930$$ 0 0
$$931$$ 2.08828 0.0684406
$$932$$ 0 0
$$933$$ −5.09875 −0.166926
$$934$$ 0 0
$$935$$ −116.660 −3.81519
$$936$$ 0 0
$$937$$ −46.5172 −1.51965 −0.759825 0.650127i $$-0.774716\pi$$
−0.759825 + 0.650127i $$0.774716\pi$$
$$938$$ 0 0
$$939$$ 4.70307 0.153479
$$940$$ 0 0
$$941$$ −39.9678 −1.30291 −0.651457 0.758686i $$-0.725842\pi$$
−0.651457 + 0.758686i $$0.725842\pi$$
$$942$$ 0 0
$$943$$ −20.6210 −0.671511
$$944$$ 0 0
$$945$$ 17.6616 0.574531
$$946$$ 0 0
$$947$$ −33.7011 −1.09514 −0.547569 0.836761i $$-0.684447\pi$$
−0.547569 + 0.836761i $$0.684447\pi$$
$$948$$ 0 0
$$949$$ 30.7864 0.999368
$$950$$ 0 0
$$951$$ −3.27535 −0.106211
$$952$$ 0 0
$$953$$ −30.1472 −0.976564 −0.488282 0.872686i $$-0.662376\pi$$
−0.488282 + 0.872686i $$0.662376\pi$$
$$954$$ 0 0
$$955$$ 6.54106 0.211664
$$956$$ 0 0
$$957$$ 5.14596 0.166345
$$958$$ 0 0
$$959$$ 35.9624 1.16129
$$960$$ 0 0
$$961$$ −19.8733 −0.641076
$$962$$ 0 0
$$963$$ 51.7700 1.66826
$$964$$ 0 0
$$965$$ 80.0287 2.57622
$$966$$ 0 0
$$967$$ 35.1010 1.12877 0.564385 0.825511i $$-0.309113\pi$$
0.564385 + 0.825511i $$0.309113\pi$$
$$968$$ 0 0
$$969$$ 3.54182 0.113780
$$970$$ 0 0
$$971$$ 38.1884 1.22552 0.612762 0.790268i $$-0.290059\pi$$
0.612762 + 0.790268i $$0.290059\pi$$
$$972$$ 0 0
$$973$$ −9.72716 −0.311839
$$974$$ 0 0
$$975$$ 9.62409 0.308217
$$976$$ 0 0
$$977$$ 22.6596 0.724946 0.362473 0.931994i $$-0.381933\pi$$
0.362473 + 0.931994i $$0.381933\pi$$
$$978$$ 0 0
$$979$$ 19.0874 0.610036
$$980$$ 0 0
$$981$$ −25.8359 −0.824878
$$982$$ 0 0
$$983$$ 1.47180 0.0469430 0.0234715 0.999725i $$-0.492528\pi$$
0.0234715 + 0.999725i $$0.492528\pi$$
$$984$$ 0 0
$$985$$ 62.6326 1.99564
$$986$$ 0 0
$$987$$ −10.1061 −0.321682
$$988$$ 0 0
$$989$$ −14.0814 −0.447763
$$990$$ 0 0
$$991$$ −30.6228 −0.972767 −0.486383 0.873746i $$-0.661684\pi$$
−0.486383 + 0.873746i $$0.661684\pi$$
$$992$$ 0 0
$$993$$ −13.0292 −0.413469
$$994$$ 0 0
$$995$$ −12.7582 −0.404461
$$996$$ 0 0
$$997$$ −7.61232 −0.241085 −0.120542 0.992708i $$-0.538463\pi$$
−0.120542 + 0.992708i $$0.538463\pi$$
$$998$$ 0 0
$$999$$ 3.85219 0.121878
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 4864.2.a.bs.1.6 10
4.3 odd 2 4864.2.a.bt.1.6 10
8.3 odd 2 inner 4864.2.a.bs.1.5 10
8.5 even 2 4864.2.a.bt.1.5 10
16.3 odd 4 2432.2.c.j.1217.11 yes 20
16.5 even 4 2432.2.c.j.1217.12 yes 20
16.11 odd 4 2432.2.c.j.1217.10 yes 20
16.13 even 4 2432.2.c.j.1217.9 20

By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.9 20 16.13 even 4
2432.2.c.j.1217.10 yes 20 16.11 odd 4
2432.2.c.j.1217.11 yes 20 16.3 odd 4
2432.2.c.j.1217.12 yes 20 16.5 even 4
4864.2.a.bs.1.5 10 8.3 odd 2 inner
4864.2.a.bs.1.6 10 1.1 even 1 trivial
4864.2.a.bt.1.5 10 8.5 even 2
4864.2.a.bt.1.6 10 4.3 odd 2