# Properties

 Label 4704.2.a.bw.1.2 Level $4704$ Weight $2$ Character 4704.1 Self dual yes Analytic conductor $37.562$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4704 = 2^{5} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4704.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.5616291108$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ x^4 - 2*x^3 - 3*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.22833$$ of defining polynomial Character $$\chi$$ $$=$$ 4704.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.04244 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.04244 q^{5} +1.00000 q^{9} -3.93089 q^{11} -4.88845 q^{13} +3.04244 q^{15} +5.34511 q^{17} +2.30266 q^{19} +7.93089 q^{23} +4.25646 q^{25} -1.00000 q^{27} +5.55912 q^{29} +0.645810 q^{31} +3.93089 q^{33} +5.65685 q^{37} +4.88845 q^{39} -10.0886 q^{41} -8.91331 q^{43} -3.04244 q^{45} +6.61065 q^{47} -5.34511 q^{51} +1.25646 q^{53} +11.9595 q^{55} -2.30266 q^{57} -3.04621 q^{59} +2.97334 q^{61} +14.8728 q^{65} +13.5186 q^{67} -7.93089 q^{69} +13.5877 q^{71} -4.67067 q^{73} -4.25646 q^{75} +1.05153 q^{79} +1.00000 q^{81} -8.60533 q^{83} -16.2622 q^{85} -5.55912 q^{87} -4.85983 q^{89} -0.645810 q^{93} -7.00572 q^{95} -18.3275 q^{97} -3.93089 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} + 4 q^{11} - 8 q^{13} + 4 q^{15} - 4 q^{17} - 8 q^{19} + 12 q^{23} + 12 q^{25} - 4 q^{27} + 8 q^{31} - 4 q^{33} + 8 q^{39} - 20 q^{41} - 8 q^{43} - 4 q^{45} + 16 q^{47} + 4 q^{51} + 8 q^{55} + 8 q^{57} - 16 q^{61} - 8 q^{65} - 8 q^{67} - 12 q^{69} + 12 q^{71} - 8 q^{73} - 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} - 28 q^{89} - 8 q^{93} + 24 q^{95} - 40 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^9 + 4 * q^11 - 8 * q^13 + 4 * q^15 - 4 * q^17 - 8 * q^19 + 12 * q^23 + 12 * q^25 - 4 * q^27 + 8 * q^31 - 4 * q^33 + 8 * q^39 - 20 * q^41 - 8 * q^43 - 4 * q^45 + 16 * q^47 + 4 * q^51 + 8 * q^55 + 8 * q^57 - 16 * q^61 - 8 * q^65 - 8 * q^67 - 12 * q^69 + 12 * q^71 - 8 * q^73 - 12 * q^75 + 16 * q^79 + 4 * q^81 - 8 * q^85 - 28 * q^89 - 8 * q^93 + 24 * q^95 - 40 * q^97 + 4 * q^99

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −3.04244 −1.36062 −0.680311 0.732924i $$-0.738155\pi$$
−0.680311 + 0.732924i $$0.738155\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.93089 −1.18521 −0.592604 0.805494i $$-0.701900\pi$$
−0.592604 + 0.805494i $$0.701900\pi$$
$$12$$ 0 0
$$13$$ −4.88845 −1.35581 −0.677906 0.735149i $$-0.737112\pi$$
−0.677906 + 0.735149i $$0.737112\pi$$
$$14$$ 0 0
$$15$$ 3.04244 0.785555
$$16$$ 0 0
$$17$$ 5.34511 1.29638 0.648189 0.761479i $$-0.275526\pi$$
0.648189 + 0.761479i $$0.275526\pi$$
$$18$$ 0 0
$$19$$ 2.30266 0.528267 0.264134 0.964486i $$-0.414914\pi$$
0.264134 + 0.964486i $$0.414914\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.93089 1.65371 0.826853 0.562418i $$-0.190129\pi$$
0.826853 + 0.562418i $$0.190129\pi$$
$$24$$ 0 0
$$25$$ 4.25646 0.851292
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.55912 1.03230 0.516152 0.856497i $$-0.327364\pi$$
0.516152 + 0.856497i $$0.327364\pi$$
$$30$$ 0 0
$$31$$ 0.645810 0.115991 0.0579954 0.998317i $$-0.481529\pi$$
0.0579954 + 0.998317i $$0.481529\pi$$
$$32$$ 0 0
$$33$$ 3.93089 0.684281
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.65685 0.929981 0.464991 0.885316i $$-0.346058\pi$$
0.464991 + 0.885316i $$0.346058\pi$$
$$38$$ 0 0
$$39$$ 4.88845 0.782779
$$40$$ 0 0
$$41$$ −10.0886 −1.57558 −0.787791 0.615943i $$-0.788775\pi$$
−0.787791 + 0.615943i $$0.788775\pi$$
$$42$$ 0 0
$$43$$ −8.91331 −1.35927 −0.679634 0.733552i $$-0.737861\pi$$
−0.679634 + 0.733552i $$0.737861\pi$$
$$44$$ 0 0
$$45$$ −3.04244 −0.453541
$$46$$ 0 0
$$47$$ 6.61065 0.964262 0.482131 0.876099i $$-0.339863\pi$$
0.482131 + 0.876099i $$0.339863\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −5.34511 −0.748465
$$52$$ 0 0
$$53$$ 1.25646 0.172588 0.0862939 0.996270i $$-0.472498\pi$$
0.0862939 + 0.996270i $$0.472498\pi$$
$$54$$ 0 0
$$55$$ 11.9595 1.61262
$$56$$ 0 0
$$57$$ −2.30266 −0.304995
$$58$$ 0 0
$$59$$ −3.04621 −0.396582 −0.198291 0.980143i $$-0.563539\pi$$
−0.198291 + 0.980143i $$0.563539\pi$$
$$60$$ 0 0
$$61$$ 2.97334 0.380697 0.190348 0.981717i $$-0.439038\pi$$
0.190348 + 0.981717i $$0.439038\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 14.8728 1.84475
$$66$$ 0 0
$$67$$ 13.5186 1.65156 0.825782 0.563989i $$-0.190734\pi$$
0.825782 + 0.563989i $$0.190734\pi$$
$$68$$ 0 0
$$69$$ −7.93089 −0.954767
$$70$$ 0 0
$$71$$ 13.5877 1.61257 0.806284 0.591528i $$-0.201475\pi$$
0.806284 + 0.591528i $$0.201475\pi$$
$$72$$ 0 0
$$73$$ −4.67067 −0.546661 −0.273330 0.961920i $$-0.588125\pi$$
−0.273330 + 0.961920i $$0.588125\pi$$
$$74$$ 0 0
$$75$$ −4.25646 −0.491494
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.05153 0.118306 0.0591530 0.998249i $$-0.481160\pi$$
0.0591530 + 0.998249i $$0.481160\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −8.60533 −0.944557 −0.472279 0.881449i $$-0.656568\pi$$
−0.472279 + 0.881449i $$0.656568\pi$$
$$84$$ 0 0
$$85$$ −16.2622 −1.76388
$$86$$ 0 0
$$87$$ −5.55912 −0.596001
$$88$$ 0 0
$$89$$ −4.85983 −0.515140 −0.257570 0.966260i $$-0.582922\pi$$
−0.257570 + 0.966260i $$0.582922\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −0.645810 −0.0669673
$$94$$ 0 0
$$95$$ −7.00572 −0.718772
$$96$$ 0 0
$$97$$ −18.3275 −1.86088 −0.930439 0.366446i $$-0.880574\pi$$
−0.930439 + 0.366446i $$0.880574\pi$$
$$98$$ 0 0
$$99$$ −3.93089 −0.395070
$$100$$ 0 0
$$101$$ −5.87087 −0.584173 −0.292087 0.956392i $$-0.594350\pi$$
−0.292087 + 0.956392i $$0.594350\pi$$
$$102$$ 0 0
$$103$$ −15.0778 −1.48566 −0.742828 0.669482i $$-0.766516\pi$$
−0.742828 + 0.669482i $$0.766516\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −0.238878 −0.0230932 −0.0115466 0.999933i $$-0.503675\pi$$
−0.0115466 + 0.999933i $$0.503675\pi$$
$$108$$ 0 0
$$109$$ −12.9133 −1.23687 −0.618436 0.785836i $$-0.712233\pi$$
−0.618436 + 0.785836i $$0.712233\pi$$
$$110$$ 0 0
$$111$$ −5.65685 −0.536925
$$112$$ 0 0
$$113$$ −13.3137 −1.25245 −0.626224 0.779643i $$-0.715401\pi$$
−0.626224 + 0.779643i $$0.715401\pi$$
$$114$$ 0 0
$$115$$ −24.1293 −2.25007
$$116$$ 0 0
$$117$$ −4.88845 −0.451937
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 4.45192 0.404720
$$122$$ 0 0
$$123$$ 10.0886 0.909663
$$124$$ 0 0
$$125$$ 2.26218 0.202336
$$126$$ 0 0
$$127$$ −3.69202 −0.327613 −0.163807 0.986492i $$-0.552377\pi$$
−0.163807 + 0.986492i $$0.552377\pi$$
$$128$$ 0 0
$$129$$ 8.91331 0.784773
$$130$$ 0 0
$$131$$ 18.4320 1.61041 0.805204 0.592998i $$-0.202056\pi$$
0.805204 + 0.592998i $$0.202056\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.04244 0.261852
$$136$$ 0 0
$$137$$ −4.26750 −0.364597 −0.182299 0.983243i $$-0.558354\pi$$
−0.182299 + 0.983243i $$0.558354\pi$$
$$138$$ 0 0
$$139$$ 7.31371 0.620341 0.310170 0.950681i $$-0.399614\pi$$
0.310170 + 0.950681i $$0.399614\pi$$
$$140$$ 0 0
$$141$$ −6.61065 −0.556717
$$142$$ 0 0
$$143$$ 19.2160 1.60692
$$144$$ 0 0
$$145$$ −16.9133 −1.40457
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 14.0572 1.15161 0.575807 0.817585i $$-0.304688\pi$$
0.575807 + 0.817585i $$0.304688\pi$$
$$150$$ 0 0
$$151$$ −22.5702 −1.83673 −0.918367 0.395730i $$-0.870492\pi$$
−0.918367 + 0.395730i $$0.870492\pi$$
$$152$$ 0 0
$$153$$ 5.34511 0.432126
$$154$$ 0 0
$$155$$ −1.96484 −0.157820
$$156$$ 0 0
$$157$$ 12.3147 0.982818 0.491409 0.870929i $$-0.336482\pi$$
0.491409 + 0.870929i $$0.336482\pi$$
$$158$$ 0 0
$$159$$ −1.25646 −0.0996436
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3.45192 −0.270375 −0.135188 0.990820i $$-0.543164\pi$$
−0.135188 + 0.990820i $$0.543164\pi$$
$$164$$ 0 0
$$165$$ −11.9595 −0.931047
$$166$$ 0 0
$$167$$ 12.5076 0.967867 0.483933 0.875105i $$-0.339208\pi$$
0.483933 + 0.875105i $$0.339208\pi$$
$$168$$ 0 0
$$169$$ 10.8969 0.838227
$$170$$ 0 0
$$171$$ 2.30266 0.176089
$$172$$ 0 0
$$173$$ −19.0997 −1.45212 −0.726061 0.687630i $$-0.758651\pi$$
−0.726061 + 0.687630i $$0.758651\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.04621 0.228967
$$178$$ 0 0
$$179$$ 11.6908 0.873811 0.436906 0.899507i $$-0.356074\pi$$
0.436906 + 0.899507i $$0.356074\pi$$
$$180$$ 0 0
$$181$$ 11.3204 0.841439 0.420720 0.907191i $$-0.361778\pi$$
0.420720 + 0.907191i $$0.361778\pi$$
$$182$$ 0 0
$$183$$ −2.97334 −0.219795
$$184$$ 0 0
$$185$$ −17.2107 −1.26535
$$186$$ 0 0
$$187$$ −21.0110 −1.53648
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.9015 1.22295 0.611473 0.791265i $$-0.290577\pi$$
0.611473 + 0.791265i $$0.290577\pi$$
$$192$$ 0 0
$$193$$ −23.6884 −1.70513 −0.852565 0.522622i $$-0.824954\pi$$
−0.852565 + 0.522622i $$0.824954\pi$$
$$194$$ 0 0
$$195$$ −14.8728 −1.06507
$$196$$ 0 0
$$197$$ −11.0831 −0.789637 −0.394819 0.918759i $$-0.629193\pi$$
−0.394819 + 0.918759i $$0.629193\pi$$
$$198$$ 0 0
$$199$$ 6.51292 0.461688 0.230844 0.972991i $$-0.425851\pi$$
0.230844 + 0.972991i $$0.425851\pi$$
$$200$$ 0 0
$$201$$ −13.5186 −0.953531
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 30.6941 2.14377
$$206$$ 0 0
$$207$$ 7.93089 0.551235
$$208$$ 0 0
$$209$$ −9.05153 −0.626107
$$210$$ 0 0
$$211$$ −3.83023 −0.263684 −0.131842 0.991271i $$-0.542089\pi$$
−0.131842 + 0.991271i $$0.542089\pi$$
$$212$$ 0 0
$$213$$ −13.5877 −0.931017
$$214$$ 0 0
$$215$$ 27.1182 1.84945
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 4.67067 0.315615
$$220$$ 0 0
$$221$$ −26.1293 −1.75765
$$222$$ 0 0
$$223$$ −12.4099 −0.831026 −0.415513 0.909587i $$-0.636398\pi$$
−0.415513 + 0.909587i $$0.636398\pi$$
$$224$$ 0 0
$$225$$ 4.25646 0.283764
$$226$$ 0 0
$$227$$ 17.4782 1.16007 0.580033 0.814593i $$-0.303040\pi$$
0.580033 + 0.814593i $$0.303040\pi$$
$$228$$ 0 0
$$229$$ −21.1783 −1.39950 −0.699750 0.714388i $$-0.746705\pi$$
−0.699750 + 0.714388i $$0.746705\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5.75459 0.376995 0.188498 0.982074i $$-0.439638\pi$$
0.188498 + 0.982074i $$0.439638\pi$$
$$234$$ 0 0
$$235$$ −20.1125 −1.31200
$$236$$ 0 0
$$237$$ −1.05153 −0.0683040
$$238$$ 0 0
$$239$$ 7.93089 0.513007 0.256503 0.966543i $$-0.417430\pi$$
0.256503 + 0.966543i $$0.417430\pi$$
$$240$$ 0 0
$$241$$ −20.9605 −1.35018 −0.675092 0.737734i $$-0.735896\pi$$
−0.675092 + 0.737734i $$0.735896\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −11.2565 −0.716231
$$248$$ 0 0
$$249$$ 8.60533 0.545340
$$250$$ 0 0
$$251$$ 7.45607 0.470623 0.235311 0.971920i $$-0.424389\pi$$
0.235311 + 0.971920i $$0.424389\pi$$
$$252$$ 0 0
$$253$$ −31.1755 −1.95999
$$254$$ 0 0
$$255$$ 16.2622 1.01838
$$256$$ 0 0
$$257$$ 12.3433 0.769954 0.384977 0.922926i $$-0.374209\pi$$
0.384977 + 0.922926i $$0.374209\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 5.55912 0.344101
$$262$$ 0 0
$$263$$ −23.4144 −1.44379 −0.721896 0.692002i $$-0.756729\pi$$
−0.721896 + 0.692002i $$0.756729\pi$$
$$264$$ 0 0
$$265$$ −3.82270 −0.234827
$$266$$ 0 0
$$267$$ 4.85983 0.297416
$$268$$ 0 0
$$269$$ −5.81362 −0.354463 −0.177231 0.984169i $$-0.556714\pi$$
−0.177231 + 0.984169i $$0.556714\pi$$
$$270$$ 0 0
$$271$$ 24.3694 1.48033 0.740167 0.672423i $$-0.234746\pi$$
0.740167 + 0.672423i $$0.234746\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −16.7317 −1.00896
$$276$$ 0 0
$$277$$ 15.7587 0.946851 0.473425 0.880834i $$-0.343017\pi$$
0.473425 + 0.880834i $$0.343017\pi$$
$$278$$ 0 0
$$279$$ 0.645810 0.0386636
$$280$$ 0 0
$$281$$ −30.7698 −1.83557 −0.917786 0.397076i $$-0.870025\pi$$
−0.917786 + 0.397076i $$0.870025\pi$$
$$282$$ 0 0
$$283$$ −22.9080 −1.36174 −0.680869 0.732405i $$-0.738398\pi$$
−0.680869 + 0.732405i $$0.738398\pi$$
$$284$$ 0 0
$$285$$ 7.00572 0.414983
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 11.5702 0.680598
$$290$$ 0 0
$$291$$ 18.3275 1.07438
$$292$$ 0 0
$$293$$ −19.3323 −1.12940 −0.564701 0.825295i $$-0.691009\pi$$
−0.564701 + 0.825295i $$0.691009\pi$$
$$294$$ 0 0
$$295$$ 9.26791 0.539598
$$296$$ 0 0
$$297$$ 3.93089 0.228094
$$298$$ 0 0
$$299$$ −38.7698 −2.24211
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 5.87087 0.337273
$$304$$ 0 0
$$305$$ −9.04621 −0.517984
$$306$$ 0 0
$$307$$ 26.8377 1.53171 0.765853 0.643015i $$-0.222317\pi$$
0.765853 + 0.643015i $$0.222317\pi$$
$$308$$ 0 0
$$309$$ 15.0778 0.857744
$$310$$ 0 0
$$311$$ 3.49240 0.198036 0.0990180 0.995086i $$-0.468430\pi$$
0.0990180 + 0.995086i $$0.468430\pi$$
$$312$$ 0 0
$$313$$ 2.75556 0.155753 0.0778767 0.996963i $$-0.475186\pi$$
0.0778767 + 0.996963i $$0.475186\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7.15341 −0.401775 −0.200888 0.979614i $$-0.564383\pi$$
−0.200888 + 0.979614i $$0.564383\pi$$
$$318$$ 0 0
$$319$$ −21.8523 −1.22349
$$320$$ 0 0
$$321$$ 0.238878 0.0133329
$$322$$ 0 0
$$323$$ 12.3080 0.684835
$$324$$ 0 0
$$325$$ −20.8075 −1.15419
$$326$$ 0 0
$$327$$ 12.9133 0.714108
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −31.7236 −1.74369 −0.871843 0.489786i $$-0.837075\pi$$
−0.871843 + 0.489786i $$0.837075\pi$$
$$332$$ 0 0
$$333$$ 5.65685 0.309994
$$334$$ 0 0
$$335$$ −41.1297 −2.24716
$$336$$ 0 0
$$337$$ 12.6053 0.686656 0.343328 0.939216i $$-0.388446\pi$$
0.343328 + 0.939216i $$0.388446\pi$$
$$338$$ 0 0
$$339$$ 13.3137 0.723101
$$340$$ 0 0
$$341$$ −2.53861 −0.137473
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 24.1293 1.29908
$$346$$ 0 0
$$347$$ 31.1064 1.66988 0.834939 0.550342i $$-0.185503\pi$$
0.834939 + 0.550342i $$0.185503\pi$$
$$348$$ 0 0
$$349$$ −15.7741 −0.844370 −0.422185 0.906510i $$-0.638737\pi$$
−0.422185 + 0.906510i $$0.638737\pi$$
$$350$$ 0 0
$$351$$ 4.88845 0.260926
$$352$$ 0 0
$$353$$ 18.3785 0.978187 0.489094 0.872231i $$-0.337328\pi$$
0.489094 + 0.872231i $$0.337328\pi$$
$$354$$ 0 0
$$355$$ −41.3399 −2.19410
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −17.7575 −0.937206 −0.468603 0.883409i $$-0.655243\pi$$
−0.468603 + 0.883409i $$0.655243\pi$$
$$360$$ 0 0
$$361$$ −13.6977 −0.720934
$$362$$ 0 0
$$363$$ −4.45192 −0.233665
$$364$$ 0 0
$$365$$ 14.2103 0.743799
$$366$$ 0 0
$$367$$ −20.9449 −1.09331 −0.546657 0.837357i $$-0.684100\pi$$
−0.546657 + 0.837357i $$0.684100\pi$$
$$368$$ 0 0
$$369$$ −10.0886 −0.525194
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −20.2365 −1.04781 −0.523903 0.851778i $$-0.675525\pi$$
−0.523903 + 0.851778i $$0.675525\pi$$
$$374$$ 0 0
$$375$$ −2.26218 −0.116819
$$376$$ 0 0
$$377$$ −27.1755 −1.39961
$$378$$ 0 0
$$379$$ −36.5244 −1.87613 −0.938065 0.346459i $$-0.887384\pi$$
−0.938065 + 0.346459i $$0.887384\pi$$
$$380$$ 0 0
$$381$$ 3.69202 0.189148
$$382$$ 0 0
$$383$$ 2.97417 0.151973 0.0759864 0.997109i $$-0.475789\pi$$
0.0759864 + 0.997109i $$0.475789\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.91331 −0.453089
$$388$$ 0 0
$$389$$ −2.16445 −0.109742 −0.0548710 0.998493i $$-0.517475\pi$$
−0.0548710 + 0.998493i $$0.517475\pi$$
$$390$$ 0 0
$$391$$ 42.3915 2.14383
$$392$$ 0 0
$$393$$ −18.4320 −0.929769
$$394$$ 0 0
$$395$$ −3.19921 −0.160970
$$396$$ 0 0
$$397$$ 12.0249 0.603511 0.301755 0.953385i $$-0.402427\pi$$
0.301755 + 0.953385i $$0.402427\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −21.5591 −1.07661 −0.538306 0.842750i $$-0.680935\pi$$
−0.538306 + 0.842750i $$0.680935\pi$$
$$402$$ 0 0
$$403$$ −3.15701 −0.157262
$$404$$ 0 0
$$405$$ −3.04244 −0.151180
$$406$$ 0 0
$$407$$ −22.2365 −1.10222
$$408$$ 0 0
$$409$$ 24.5143 1.21215 0.606077 0.795406i $$-0.292742\pi$$
0.606077 + 0.795406i $$0.292742\pi$$
$$410$$ 0 0
$$411$$ 4.26750 0.210500
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 26.1812 1.28519
$$416$$ 0 0
$$417$$ −7.31371 −0.358154
$$418$$ 0 0
$$419$$ −35.3048 −1.72475 −0.862376 0.506269i $$-0.831024\pi$$
−0.862376 + 0.506269i $$0.831024\pi$$
$$420$$ 0 0
$$421$$ 2.47776 0.120758 0.0603792 0.998176i $$-0.480769\pi$$
0.0603792 + 0.998176i $$0.480769\pi$$
$$422$$ 0 0
$$423$$ 6.61065 0.321421
$$424$$ 0 0
$$425$$ 22.7512 1.10360
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −19.2160 −0.927756
$$430$$ 0 0
$$431$$ 20.3408 0.979780 0.489890 0.871784i $$-0.337037\pi$$
0.489890 + 0.871784i $$0.337037\pi$$
$$432$$ 0 0
$$433$$ −15.7964 −0.759129 −0.379564 0.925165i $$-0.623926\pi$$
−0.379564 + 0.925165i $$0.623926\pi$$
$$434$$ 0 0
$$435$$ 16.9133 0.810931
$$436$$ 0 0
$$437$$ 18.2622 0.873599
$$438$$ 0 0
$$439$$ 12.6863 0.605484 0.302742 0.953073i $$-0.402098\pi$$
0.302742 + 0.953073i $$0.402098\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 25.9273 1.23184 0.615921 0.787808i $$-0.288784\pi$$
0.615921 + 0.787808i $$0.288784\pi$$
$$444$$ 0 0
$$445$$ 14.7857 0.700911
$$446$$ 0 0
$$447$$ −14.0572 −0.664885
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 39.6574 1.86739
$$452$$ 0 0
$$453$$ 22.5702 1.06044
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 29.7236 1.39041 0.695205 0.718811i $$-0.255314\pi$$
0.695205 + 0.718811i $$0.255314\pi$$
$$458$$ 0 0
$$459$$ −5.34511 −0.249488
$$460$$ 0 0
$$461$$ −12.6314 −0.588303 −0.294152 0.955759i $$-0.595037\pi$$
−0.294152 + 0.955759i $$0.595037\pi$$
$$462$$ 0 0
$$463$$ −33.6884 −1.56563 −0.782817 0.622252i $$-0.786218\pi$$
−0.782817 + 0.622252i $$0.786218\pi$$
$$464$$ 0 0
$$465$$ 1.96484 0.0911172
$$466$$ 0 0
$$467$$ −34.0941 −1.57769 −0.788844 0.614593i $$-0.789320\pi$$
−0.788844 + 0.614593i $$0.789320\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −12.3147 −0.567431
$$472$$ 0 0
$$473$$ 35.0373 1.61102
$$474$$ 0 0
$$475$$ 9.80119 0.449710
$$476$$ 0 0
$$477$$ 1.25646 0.0575293
$$478$$ 0 0
$$479$$ 0.0977317 0.00446547 0.00223274 0.999998i $$-0.499289\pi$$
0.00223274 + 0.999998i $$0.499289\pi$$
$$480$$ 0 0
$$481$$ −27.6533 −1.26088
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 55.7605 2.53195
$$486$$ 0 0
$$487$$ −38.5702 −1.74778 −0.873891 0.486123i $$-0.838411\pi$$
−0.873891 + 0.486123i $$0.838411\pi$$
$$488$$ 0 0
$$489$$ 3.45192 0.156101
$$490$$ 0 0
$$491$$ −19.0748 −0.860835 −0.430418 0.902630i $$-0.641634\pi$$
−0.430418 + 0.902630i $$0.641634\pi$$
$$492$$ 0 0
$$493$$ 29.7141 1.33826
$$494$$ 0 0
$$495$$ 11.9595 0.537540
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −8.05725 −0.360692 −0.180346 0.983603i $$-0.557722\pi$$
−0.180346 + 0.983603i $$0.557722\pi$$
$$500$$ 0 0
$$501$$ −12.5076 −0.558798
$$502$$ 0 0
$$503$$ −4.94487 −0.220481 −0.110240 0.993905i $$-0.535162\pi$$
−0.110240 + 0.993905i $$0.535162\pi$$
$$504$$ 0 0
$$505$$ 17.8618 0.794839
$$506$$ 0 0
$$507$$ −10.8969 −0.483950
$$508$$ 0 0
$$509$$ −37.2161 −1.64958 −0.824788 0.565442i $$-0.808706\pi$$
−0.824788 + 0.565442i $$0.808706\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2.30266 −0.101665
$$514$$ 0 0
$$515$$ 45.8732 2.02142
$$516$$ 0 0
$$517$$ −25.9858 −1.14285
$$518$$ 0 0
$$519$$ 19.0997 0.838383
$$520$$ 0 0
$$521$$ −13.1050 −0.574141 −0.287071 0.957909i $$-0.592681\pi$$
−0.287071 + 0.957909i $$0.592681\pi$$
$$522$$ 0 0
$$523$$ −19.5795 −0.856151 −0.428076 0.903743i $$-0.640808\pi$$
−0.428076 + 0.903743i $$0.640808\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 3.45192 0.150368
$$528$$ 0 0
$$529$$ 39.8991 1.73474
$$530$$ 0 0
$$531$$ −3.04621 −0.132194
$$532$$ 0 0
$$533$$ 49.3179 2.13619
$$534$$ 0 0
$$535$$ 0.726773 0.0314211
$$536$$ 0 0
$$537$$ −11.6908 −0.504495
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −11.9648 −0.514409 −0.257204 0.966357i $$-0.582801\pi$$
−0.257204 + 0.966357i $$0.582801\pi$$
$$542$$ 0 0
$$543$$ −11.3204 −0.485805
$$544$$ 0 0
$$545$$ 39.2880 1.68291
$$546$$ 0 0
$$547$$ 39.8827 1.70526 0.852631 0.522514i $$-0.175006\pi$$
0.852631 + 0.522514i $$0.175006\pi$$
$$548$$ 0 0
$$549$$ 2.97334 0.126899
$$550$$ 0 0
$$551$$ 12.8008 0.545332
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 17.2107 0.730552
$$556$$ 0 0
$$557$$ 28.9800 1.22792 0.613962 0.789336i $$-0.289575\pi$$
0.613962 + 0.789336i $$0.289575\pi$$
$$558$$ 0 0
$$559$$ 43.5723 1.84291
$$560$$ 0 0
$$561$$ 21.0110 0.887087
$$562$$ 0 0
$$563$$ −27.3857 −1.15417 −0.577086 0.816683i $$-0.695810\pi$$
−0.577086 + 0.816683i $$0.695810\pi$$
$$564$$ 0 0
$$565$$ 40.5062 1.70411
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.37056 −0.267068 −0.133534 0.991044i $$-0.542632\pi$$
−0.133534 + 0.991044i $$0.542632\pi$$
$$570$$ 0 0
$$571$$ −14.9485 −0.625574 −0.312787 0.949823i $$-0.601263\pi$$
−0.312787 + 0.949823i $$0.601263\pi$$
$$572$$ 0 0
$$573$$ −16.9015 −0.706068
$$574$$ 0 0
$$575$$ 33.7575 1.40779
$$576$$ 0 0
$$577$$ 6.21500 0.258734 0.129367 0.991597i $$-0.458705\pi$$
0.129367 + 0.991597i $$0.458705\pi$$
$$578$$ 0 0
$$579$$ 23.6884 0.984457
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.93900 −0.204553
$$584$$ 0 0
$$585$$ 14.8728 0.614916
$$586$$ 0 0
$$587$$ −27.0462 −1.11632 −0.558158 0.829735i $$-0.688492\pi$$
−0.558158 + 0.829735i $$0.688492\pi$$
$$588$$ 0 0
$$589$$ 1.48708 0.0612742
$$590$$ 0 0
$$591$$ 11.0831 0.455897
$$592$$ 0 0
$$593$$ 15.8837 0.652266 0.326133 0.945324i $$-0.394254\pi$$
0.326133 + 0.945324i $$0.394254\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6.51292 −0.266556
$$598$$ 0 0
$$599$$ 5.24820 0.214436 0.107218 0.994236i $$-0.465806\pi$$
0.107218 + 0.994236i $$0.465806\pi$$
$$600$$ 0 0
$$601$$ 33.3830 1.36172 0.680860 0.732414i $$-0.261606\pi$$
0.680860 + 0.732414i $$0.261606\pi$$
$$602$$ 0 0
$$603$$ 13.5186 0.550522
$$604$$ 0 0
$$605$$ −13.5447 −0.550671
$$606$$ 0 0
$$607$$ 18.9228 0.768052 0.384026 0.923322i $$-0.374537\pi$$
0.384026 + 0.923322i $$0.374537\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −32.3158 −1.30736
$$612$$ 0 0
$$613$$ 12.4928 0.504580 0.252290 0.967652i $$-0.418816\pi$$
0.252290 + 0.967652i $$0.418816\pi$$
$$614$$ 0 0
$$615$$ −30.6941 −1.23771
$$616$$ 0 0
$$617$$ 34.4230 1.38582 0.692910 0.721025i $$-0.256329\pi$$
0.692910 + 0.721025i $$0.256329\pi$$
$$618$$ 0 0
$$619$$ −22.6381 −0.909900 −0.454950 0.890517i $$-0.650343\pi$$
−0.454950 + 0.890517i $$0.650343\pi$$
$$620$$ 0 0
$$621$$ −7.93089 −0.318256
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −28.1649 −1.12659
$$626$$ 0 0
$$627$$ 9.05153 0.361483
$$628$$ 0 0
$$629$$ 30.2365 1.20561
$$630$$ 0 0
$$631$$ −25.3489 −1.00912 −0.504561 0.863376i $$-0.668346\pi$$
−0.504561 + 0.863376i $$0.668346\pi$$
$$632$$ 0 0
$$633$$ 3.83023 0.152238
$$634$$ 0 0
$$635$$ 11.2327 0.445758
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 13.5877 0.537523
$$640$$ 0 0
$$641$$ −13.2196 −0.522142 −0.261071 0.965320i $$-0.584076\pi$$
−0.261071 + 0.965320i $$0.584076\pi$$
$$642$$ 0 0
$$643$$ 23.7194 0.935403 0.467701 0.883887i $$-0.345082\pi$$
0.467701 + 0.883887i $$0.345082\pi$$
$$644$$ 0 0
$$645$$ −27.1182 −1.06778
$$646$$ 0 0
$$647$$ 15.8213 0.622000 0.311000 0.950410i $$-0.399336\pi$$
0.311000 + 0.950410i $$0.399336\pi$$
$$648$$ 0 0
$$649$$ 11.9743 0.470033
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 19.4561 0.761375 0.380687 0.924704i $$-0.375687\pi$$
0.380687 + 0.924704i $$0.375687\pi$$
$$654$$ 0 0
$$655$$ −56.0782 −2.19116
$$656$$ 0 0
$$657$$ −4.67067 −0.182220
$$658$$ 0 0
$$659$$ 10.5820 0.412217 0.206109 0.978529i $$-0.433920\pi$$
0.206109 + 0.978529i $$0.433920\pi$$
$$660$$ 0 0
$$661$$ −9.95814 −0.387327 −0.193663 0.981068i $$-0.562037\pi$$
−0.193663 + 0.981068i $$0.562037\pi$$
$$662$$ 0 0
$$663$$ 26.1293 1.01478
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 44.0888 1.70713
$$668$$ 0 0
$$669$$ 12.4099 0.479793
$$670$$ 0 0
$$671$$ −11.6879 −0.451205
$$672$$ 0 0
$$673$$ −21.0724 −0.812283 −0.406141 0.913810i $$-0.633126\pi$$
−0.406141 + 0.913810i $$0.633126\pi$$
$$674$$ 0 0
$$675$$ −4.25646 −0.163831
$$676$$ 0 0
$$677$$ 4.66021 0.179107 0.0895533 0.995982i $$-0.471456\pi$$
0.0895533 + 0.995982i $$0.471456\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −17.4782 −0.669765
$$682$$ 0 0
$$683$$ 9.24820 0.353873 0.176936 0.984222i $$-0.443381\pi$$
0.176936 + 0.984222i $$0.443381\pi$$
$$684$$ 0 0
$$685$$ 12.9836 0.496079
$$686$$ 0 0
$$687$$ 21.1783 0.808001
$$688$$ 0 0
$$689$$ −6.14214 −0.233997
$$690$$ 0 0
$$691$$ 10.0924 0.383933 0.191967 0.981401i $$-0.438513\pi$$
0.191967 + 0.981401i $$0.438513\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −22.2515 −0.844049
$$696$$ 0 0
$$697$$ −53.9249 −2.04255
$$698$$ 0 0
$$699$$ −5.75459 −0.217658
$$700$$ 0 0
$$701$$ 17.0683 0.644661 0.322330 0.946627i $$-0.395534\pi$$
0.322330 + 0.946627i $$0.395534\pi$$
$$702$$ 0 0
$$703$$ 13.0258 0.491279
$$704$$ 0 0
$$705$$ 20.1125 0.757481
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −27.2012 −1.02156 −0.510781 0.859711i $$-0.670644\pi$$
−0.510781 + 0.859711i $$0.670644\pi$$
$$710$$ 0 0
$$711$$ 1.05153 0.0394353
$$712$$ 0 0
$$713$$ 5.12185 0.191815
$$714$$ 0 0
$$715$$ −58.4635 −2.18641
$$716$$ 0 0
$$717$$ −7.93089 −0.296185
$$718$$ 0 0
$$719$$ −24.5350 −0.915001 −0.457501 0.889209i $$-0.651255\pi$$
−0.457501 + 0.889209i $$0.651255\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 20.9605 0.779529
$$724$$ 0 0
$$725$$ 23.6622 0.878791
$$726$$ 0 0
$$727$$ 40.7603 1.51172 0.755858 0.654735i $$-0.227220\pi$$
0.755858 + 0.654735i $$0.227220\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −47.6426 −1.76213
$$732$$ 0 0
$$733$$ 28.1318 1.03907 0.519537 0.854448i $$-0.326105\pi$$
0.519537 + 0.854448i $$0.326105\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −53.1403 −1.95745
$$738$$ 0 0
$$739$$ 12.6626 0.465800 0.232900 0.972501i $$-0.425178\pi$$
0.232900 + 0.972501i $$0.425178\pi$$
$$740$$ 0 0
$$741$$ 11.2565 0.413516
$$742$$ 0 0
$$743$$ 13.3477 0.489678 0.244839 0.969564i $$-0.421265\pi$$
0.244839 + 0.969564i $$0.421265\pi$$
$$744$$ 0 0
$$745$$ −42.7684 −1.56691
$$746$$ 0 0
$$747$$ −8.60533 −0.314852
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.2843 1.03211 0.516054 0.856556i $$-0.327400\pi$$
0.516054 + 0.856556i $$0.327400\pi$$
$$752$$ 0 0
$$753$$ −7.45607 −0.271714
$$754$$ 0 0
$$755$$ 68.6684 2.49910
$$756$$ 0 0
$$757$$ 9.71410 0.353065 0.176533 0.984295i $$-0.443512\pi$$
0.176533 + 0.984295i $$0.443512\pi$$
$$758$$ 0 0
$$759$$ 31.1755 1.13160
$$760$$ 0 0
$$761$$ 16.5569 0.600188 0.300094 0.953910i $$-0.402982\pi$$
0.300094 + 0.953910i $$0.402982\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −16.2622 −0.587960
$$766$$ 0 0
$$767$$ 14.8912 0.537691
$$768$$ 0 0
$$769$$ 12.3884 0.446736 0.223368 0.974734i $$-0.428295\pi$$
0.223368 + 0.974734i $$0.428295\pi$$
$$770$$ 0 0
$$771$$ −12.3433 −0.444533
$$772$$ 0 0
$$773$$ 55.4211 1.99336 0.996679 0.0814351i $$-0.0259503\pi$$
0.996679 + 0.0814351i $$0.0259503\pi$$
$$774$$ 0 0
$$775$$ 2.74886 0.0987421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −23.2308 −0.832329
$$780$$ 0 0
$$781$$ −53.4120 −1.91123
$$782$$ 0 0
$$783$$ −5.55912 −0.198667
$$784$$ 0 0
$$785$$ −37.4667 −1.33724
$$786$$ 0 0
$$787$$ −13.4061 −0.477877 −0.238938 0.971035i $$-0.576799\pi$$
−0.238938 + 0.971035i $$0.576799\pi$$
$$788$$ 0 0
$$789$$ 23.4144 0.833574
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −14.5350 −0.516153
$$794$$ 0 0
$$795$$ 3.82270 0.135577
$$796$$ 0 0
$$797$$ 26.4837 0.938102 0.469051 0.883171i $$-0.344596\pi$$
0.469051 + 0.883171i $$0.344596\pi$$
$$798$$ 0 0
$$799$$ 35.3346 1.25005
$$800$$ 0 0
$$801$$ −4.85983 −0.171713
$$802$$ 0 0
$$803$$ 18.3599 0.647907
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 5.81362 0.204649
$$808$$ 0 0
$$809$$ −36.5760 −1.28594 −0.642972 0.765889i $$-0.722299\pi$$
−0.642972 + 0.765889i $$0.722299\pi$$
$$810$$ 0 0
$$811$$ −4.67565 −0.164184 −0.0820921 0.996625i $$-0.526160\pi$$
−0.0820921 + 0.996625i $$0.526160\pi$$
$$812$$ 0 0
$$813$$ −24.3694 −0.854672
$$814$$ 0 0
$$815$$ 10.5023 0.367879
$$816$$ 0 0
$$817$$ −20.5244 −0.718057
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42.9212 −1.49796 −0.748979 0.662593i $$-0.769456\pi$$
−0.748979 + 0.662593i $$0.769456\pi$$
$$822$$ 0 0
$$823$$ 12.0888 0.421389 0.210695 0.977552i $$-0.432427\pi$$
0.210695 + 0.977552i $$0.432427\pi$$
$$824$$ 0 0
$$825$$ 16.7317 0.582522
$$826$$ 0 0
$$827$$ 3.23676 0.112553 0.0562765 0.998415i $$-0.482077\pi$$
0.0562765 + 0.998415i $$0.482077\pi$$
$$828$$ 0 0
$$829$$ 2.06002 0.0715476 0.0357738 0.999360i $$-0.488610\pi$$
0.0357738 + 0.999360i $$0.488610\pi$$
$$830$$ 0 0
$$831$$ −15.7587 −0.546664
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −38.0536 −1.31690
$$836$$ 0 0
$$837$$ −0.645810 −0.0223224
$$838$$ 0 0
$$839$$ 5.99468 0.206959 0.103480 0.994632i $$-0.467002\pi$$
0.103480 + 0.994632i $$0.467002\pi$$
$$840$$ 0 0
$$841$$ 1.90384 0.0656498
$$842$$ 0 0
$$843$$ 30.7698 1.05977
$$844$$ 0 0
$$845$$ −33.1533 −1.14051
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 22.9080 0.786200
$$850$$ 0 0
$$851$$ 44.8639 1.53791
$$852$$ 0 0
$$853$$ 40.3040 1.37998 0.689992 0.723817i $$-0.257614\pi$$
0.689992 + 0.723817i $$0.257614\pi$$
$$854$$ 0 0
$$855$$ −7.00572 −0.239591
$$856$$ 0 0
$$857$$ −24.1735 −0.825752 −0.412876 0.910787i $$-0.635476\pi$$
−0.412876 + 0.910787i $$0.635476\pi$$
$$858$$ 0 0
$$859$$ 22.6316 0.772179 0.386090 0.922461i $$-0.373826\pi$$
0.386090 + 0.922461i $$0.373826\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 48.9015 1.66462 0.832312 0.554307i $$-0.187017\pi$$
0.832312 + 0.554307i $$0.187017\pi$$
$$864$$ 0 0
$$865$$ 58.1097 1.97579
$$866$$ 0 0
$$867$$ −11.5702 −0.392943
$$868$$ 0 0
$$869$$ −4.13344 −0.140217
$$870$$ 0 0
$$871$$ −66.0852 −2.23921
$$872$$ 0 0
$$873$$ −18.3275 −0.620293
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −32.1461 −1.08550 −0.542748 0.839896i $$-0.682616\pi$$
−0.542748 + 0.839896i $$0.682616\pi$$
$$878$$ 0 0
$$879$$ 19.3323 0.652061
$$880$$ 0 0
$$881$$ −1.94291 −0.0654583 −0.0327291 0.999464i $$-0.510420\pi$$
−0.0327291 + 0.999464i $$0.510420\pi$$
$$882$$ 0 0
$$883$$ −48.8770 −1.64484 −0.822421 0.568880i $$-0.807377\pi$$
−0.822421 + 0.568880i $$0.807377\pi$$
$$884$$ 0 0
$$885$$ −9.26791 −0.311537
$$886$$ 0 0
$$887$$ 5.59546 0.187877 0.0939385 0.995578i $$-0.470054\pi$$
0.0939385 + 0.995578i $$0.470054\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3.93089 −0.131690
$$892$$ 0 0
$$893$$ 15.2221 0.509388
$$894$$ 0 0
$$895$$ −35.5686 −1.18893
$$896$$ 0 0
$$897$$ 38.7698 1.29449
$$898$$ 0 0
$$899$$ 3.59014 0.119738
$$900$$ 0 0
$$901$$ 6.71591 0.223739
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −34.4417 −1.14488
$$906$$ 0 0
$$907$$ −51.0945 −1.69657 −0.848283 0.529543i $$-0.822363\pi$$
−0.848283 + 0.529543i $$0.822363\pi$$
$$908$$ 0 0
$$909$$ −5.87087 −0.194724
$$910$$ 0 0
$$911$$ −22.6965 −0.751969 −0.375985 0.926626i $$-0.622695\pi$$
−0.375985 + 0.926626i $$0.622695\pi$$
$$912$$ 0 0
$$913$$ 33.8266 1.11950
$$914$$ 0 0
$$915$$ 9.04621 0.299058
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 12.9836 0.428291 0.214145 0.976802i $$-0.431303\pi$$
0.214145 + 0.976802i $$0.431303\pi$$
$$920$$ 0 0
$$921$$ −26.8377 −0.884331
$$922$$ 0 0
$$923$$ −66.4230 −2.18634
$$924$$ 0 0
$$925$$ 24.0782 0.791685
$$926$$ 0 0
$$927$$ −15.0778 −0.495219
$$928$$ 0 0
$$929$$ 1.84856 0.0606491 0.0303246 0.999540i $$-0.490346\pi$$
0.0303246 + 0.999540i $$0.490346\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −3.49240 −0.114336
$$934$$ 0 0
$$935$$ 63.9249 2.09057
$$936$$ 0 0
$$937$$ 53.0177 1.73201 0.866007 0.500032i $$-0.166678\pi$$
0.866007 + 0.500032i $$0.166678\pi$$
$$938$$ 0 0
$$939$$ −2.75556 −0.0899242
$$940$$ 0 0
$$941$$ 10.9615 0.357334 0.178667 0.983910i $$-0.442822\pi$$
0.178667 + 0.983910i $$0.442822\pi$$
$$942$$ 0 0
$$943$$ −80.0120 −2.60555
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.61718 0.280021 0.140010 0.990150i $$-0.455286\pi$$
0.140010 + 0.990150i $$0.455286\pi$$
$$948$$ 0 0
$$949$$ 22.8323 0.741169
$$950$$ 0 0
$$951$$ 7.15341 0.231965
$$952$$ 0 0
$$953$$ 6.40986 0.207636 0.103818 0.994596i $$-0.466894\pi$$
0.103818 + 0.994596i $$0.466894\pi$$
$$954$$ 0 0
$$955$$ −51.4217 −1.66397
$$956$$ 0 0
$$957$$ 21.8523 0.706385
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.5829 −0.986546
$$962$$ 0 0
$$963$$ −0.238878 −0.00769774
$$964$$ 0 0
$$965$$ 72.0706 2.32004
$$966$$ 0 0
$$967$$ 43.2749 1.39163 0.695814 0.718222i $$-0.255044\pi$$
0.695814 + 0.718222i $$0.255044\pi$$
$$968$$ 0 0
$$969$$ −12.3080 −0.395389
$$970$$ 0 0
$$971$$ −21.8266 −0.700450 −0.350225 0.936666i $$-0.613895\pi$$
−0.350225 + 0.936666i $$0.613895\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 20.8075 0.666373
$$976$$ 0 0
$$977$$ −58.5040 −1.87171 −0.935854 0.352387i $$-0.885370\pi$$
−0.935854 + 0.352387i $$0.885370\pi$$
$$978$$ 0 0
$$979$$ 19.1035 0.610549
$$980$$ 0 0
$$981$$ −12.9133 −0.412290
$$982$$ 0 0
$$983$$ 6.17337 0.196900 0.0984500 0.995142i $$-0.468612\pi$$
0.0984500 + 0.995142i $$0.468612\pi$$
$$984$$ 0 0
$$985$$ 33.7197 1.07440
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −70.6905 −2.24783
$$990$$ 0 0
$$991$$ −5.11942 −0.162624 −0.0813118 0.996689i $$-0.525911\pi$$
−0.0813118 + 0.996689i $$0.525911\pi$$
$$992$$ 0 0
$$993$$ 31.7236 1.00672
$$994$$ 0 0
$$995$$ −19.8152 −0.628183
$$996$$ 0 0
$$997$$ −49.7448 −1.57543 −0.787717 0.616037i $$-0.788737\pi$$
−0.787717 + 0.616037i $$0.788737\pi$$
$$998$$ 0 0
$$999$$ −5.65685 −0.178975
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 4704.2.a.bw.1.2 4
4.3 odd 2 4704.2.a.by.1.2 yes 4
7.6 odd 2 4704.2.a.bz.1.3 yes 4
8.3 odd 2 9408.2.a.el.1.3 4
8.5 even 2 9408.2.a.en.1.3 4
28.27 even 2 4704.2.a.bx.1.3 yes 4
56.13 odd 2 9408.2.a.ek.1.2 4
56.27 even 2 9408.2.a.em.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.2 4 1.1 even 1 trivial
4704.2.a.bx.1.3 yes 4 28.27 even 2
4704.2.a.by.1.2 yes 4 4.3 odd 2
4704.2.a.bz.1.3 yes 4 7.6 odd 2
9408.2.a.ek.1.2 4 56.13 odd 2
9408.2.a.el.1.3 4 8.3 odd 2
9408.2.a.em.1.2 4 56.27 even 2
9408.2.a.en.1.3 4 8.5 even 2