# Properties

 Label 3072.2.d.f.1537.3 Level $3072$ Weight $2$ Character 3072.1537 Analytic conductor $24.530$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1537.3 Root $$0.500000 + 1.44392i$$ of defining polynomial Character $$\chi$$ $$=$$ 3072.1537 Dual form 3072.2.d.f.1537.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +2.47363i q^{5} +2.55765 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +2.47363i q^{5} +2.55765 q^{7} -1.00000 q^{9} -0.669808i q^{11} -4.08402i q^{13} +2.47363 q^{15} +6.44549 q^{17} +6.44549i q^{19} -2.55765i q^{21} -2.82843 q^{23} -1.11882 q^{25} +1.00000i q^{27} -4.35480i q^{29} +6.55765 q^{31} -0.669808 q^{33} +6.32666i q^{35} -3.85970i q^{37} -4.08402 q^{39} -0.788632 q^{41} +0.550984i q^{43} -2.47363i q^{45} -2.82843 q^{47} -0.458440 q^{49} -6.44549i q^{51} +3.64520i q^{53} +1.65685 q^{55} +6.44549 q^{57} +5.65685i q^{59} -6.20285i q^{61} -2.55765 q^{63} +10.1023 q^{65} -2.99647i q^{67} +2.82843i q^{69} -5.11529 q^{71} +14.7721 q^{73} +1.11882i q^{75} -1.71313i q^{77} +6.32000 q^{79} +1.00000 q^{81} -0.907457i q^{83} +15.9437i q^{85} -4.35480 q^{87} +6.31724 q^{89} -10.4455i q^{91} -6.55765i q^{93} -15.9437 q^{95} +12.6533 q^{97} +0.669808i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} - 8q^{9} + O(q^{10})$$ $$8q - 8q^{7} - 8q^{9} + 8q^{15} - 8q^{25} + 24q^{31} - 16q^{39} + 8q^{49} - 32q^{55} + 8q^{63} - 16q^{65} + 16q^{71} + 16q^{73} + 24q^{79} + 8q^{81} - 24q^{87} + 16q^{89} - 48q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 2.47363i 1.10624i 0.833102 + 0.553120i $$0.186563\pi$$
−0.833102 + 0.553120i $$0.813437\pi$$
$$6$$ 0 0
$$7$$ 2.55765 0.966700 0.483350 0.875427i $$-0.339420\pi$$
0.483350 + 0.875427i $$0.339420\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 0.669808i − 0.201955i −0.994889 0.100977i $$-0.967803\pi$$
0.994889 0.100977i $$-0.0321970\pi$$
$$12$$ 0 0
$$13$$ − 4.08402i − 1.13270i −0.824164 0.566352i $$-0.808354\pi$$
0.824164 0.566352i $$-0.191646\pi$$
$$14$$ 0 0
$$15$$ 2.47363 0.638687
$$16$$ 0 0
$$17$$ 6.44549 1.56326 0.781630 0.623742i $$-0.214389\pi$$
0.781630 + 0.623742i $$0.214389\pi$$
$$18$$ 0 0
$$19$$ 6.44549i 1.47870i 0.673323 + 0.739348i $$0.264866\pi$$
−0.673323 + 0.739348i $$0.735134\pi$$
$$20$$ 0 0
$$21$$ − 2.55765i − 0.558124i
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ −1.11882 −0.223765
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ − 4.35480i − 0.808666i −0.914612 0.404333i $$-0.867504\pi$$
0.914612 0.404333i $$-0.132496\pi$$
$$30$$ 0 0
$$31$$ 6.55765 1.17779 0.588894 0.808210i $$-0.299563\pi$$
0.588894 + 0.808210i $$0.299563\pi$$
$$32$$ 0 0
$$33$$ −0.669808 −0.116599
$$34$$ 0 0
$$35$$ 6.32666i 1.06940i
$$36$$ 0 0
$$37$$ − 3.85970i − 0.634531i −0.948337 0.317265i $$-0.897235\pi$$
0.948337 0.317265i $$-0.102765\pi$$
$$38$$ 0 0
$$39$$ −4.08402 −0.653967
$$40$$ 0 0
$$41$$ −0.788632 −0.123164 −0.0615818 0.998102i $$-0.519615\pi$$
−0.0615818 + 0.998102i $$0.519615\pi$$
$$42$$ 0 0
$$43$$ 0.550984i 0.0840242i 0.999117 + 0.0420121i $$0.0133768\pi$$
−0.999117 + 0.0420121i $$0.986623\pi$$
$$44$$ 0 0
$$45$$ − 2.47363i − 0.368746i
$$46$$ 0 0
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ −0.458440 −0.0654915
$$50$$ 0 0
$$51$$ − 6.44549i − 0.902549i
$$52$$ 0 0
$$53$$ 3.64520i 0.500707i 0.968155 + 0.250353i $$0.0805468\pi$$
−0.968155 + 0.250353i $$0.919453\pi$$
$$54$$ 0 0
$$55$$ 1.65685 0.223410
$$56$$ 0 0
$$57$$ 6.44549 0.853726
$$58$$ 0 0
$$59$$ 5.65685i 0.736460i 0.929735 + 0.368230i $$0.120036\pi$$
−0.929735 + 0.368230i $$0.879964\pi$$
$$60$$ 0 0
$$61$$ − 6.20285i − 0.794193i −0.917777 0.397097i $$-0.870018\pi$$
0.917777 0.397097i $$-0.129982\pi$$
$$62$$ 0 0
$$63$$ −2.55765 −0.322233
$$64$$ 0 0
$$65$$ 10.1023 1.25304
$$66$$ 0 0
$$67$$ − 2.99647i − 0.366077i −0.983106 0.183039i $$-0.941407\pi$$
0.983106 0.183039i $$-0.0585933\pi$$
$$68$$ 0 0
$$69$$ 2.82843i 0.340503i
$$70$$ 0 0
$$71$$ −5.11529 −0.607074 −0.303537 0.952820i $$-0.598168\pi$$
−0.303537 + 0.952820i $$0.598168\pi$$
$$72$$ 0 0
$$73$$ 14.7721 1.72895 0.864475 0.502676i $$-0.167651\pi$$
0.864475 + 0.502676i $$0.167651\pi$$
$$74$$ 0 0
$$75$$ 1.11882i 0.129191i
$$76$$ 0 0
$$77$$ − 1.71313i − 0.195230i
$$78$$ 0 0
$$79$$ 6.32000 0.711055 0.355528 0.934666i $$-0.384301\pi$$
0.355528 + 0.934666i $$0.384301\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 0.907457i − 0.0996063i −0.998759 0.0498032i $$-0.984141\pi$$
0.998759 0.0498032i $$-0.0158594\pi$$
$$84$$ 0 0
$$85$$ 15.9437i 1.72934i
$$86$$ 0 0
$$87$$ −4.35480 −0.466884
$$88$$ 0 0
$$89$$ 6.31724 0.669626 0.334813 0.942285i $$-0.391327\pi$$
0.334813 + 0.942285i $$0.391327\pi$$
$$90$$ 0 0
$$91$$ − 10.4455i − 1.09498i
$$92$$ 0 0
$$93$$ − 6.55765i − 0.679996i
$$94$$ 0 0
$$95$$ −15.9437 −1.63579
$$96$$ 0 0
$$97$$ 12.6533 1.28475 0.642375 0.766390i $$-0.277949\pi$$
0.642375 + 0.766390i $$0.277949\pi$$
$$98$$ 0 0
$$99$$ 0.669808i 0.0673182i
$$100$$ 0 0
$$101$$ 10.6417i 1.05889i 0.848346 + 0.529443i $$0.177599\pi$$
−0.848346 + 0.529443i $$0.822401\pi$$
$$102$$ 0 0
$$103$$ −3.33686 −0.328790 −0.164395 0.986395i $$-0.552567\pi$$
−0.164395 + 0.986395i $$0.552567\pi$$
$$104$$ 0 0
$$105$$ 6.32666 0.617419
$$106$$ 0 0
$$107$$ 19.8874i 1.92259i 0.275518 + 0.961296i $$0.411151\pi$$
−0.275518 + 0.961296i $$0.588849\pi$$
$$108$$ 0 0
$$109$$ − 3.91598i − 0.375083i −0.982257 0.187541i $$-0.939948\pi$$
0.982257 0.187541i $$-0.0600519\pi$$
$$110$$ 0 0
$$111$$ −3.85970 −0.366347
$$112$$ 0 0
$$113$$ −2.23765 −0.210500 −0.105250 0.994446i $$-0.533564\pi$$
−0.105250 + 0.994446i $$0.533564\pi$$
$$114$$ 0 0
$$115$$ − 6.99647i − 0.652424i
$$116$$ 0 0
$$117$$ 4.08402i 0.377568i
$$118$$ 0 0
$$119$$ 16.4853 1.51120
$$120$$ 0 0
$$121$$ 10.5514 0.959214
$$122$$ 0 0
$$123$$ 0.788632i 0.0711086i
$$124$$ 0 0
$$125$$ 9.60058i 0.858702i
$$126$$ 0 0
$$127$$ 12.2145 1.08386 0.541931 0.840423i $$-0.317693\pi$$
0.541931 + 0.840423i $$0.317693\pi$$
$$128$$ 0 0
$$129$$ 0.550984 0.0485114
$$130$$ 0 0
$$131$$ − 5.33962i − 0.466524i −0.972414 0.233262i $$-0.925060\pi$$
0.972414 0.233262i $$-0.0749400\pi$$
$$132$$ 0 0
$$133$$ 16.4853i 1.42946i
$$134$$ 0 0
$$135$$ −2.47363 −0.212896
$$136$$ 0 0
$$137$$ 5.10587 0.436224 0.218112 0.975924i $$-0.430010\pi$$
0.218112 + 0.975924i $$0.430010\pi$$
$$138$$ 0 0
$$139$$ − 16.6533i − 1.41252i −0.707954 0.706258i $$-0.750382\pi$$
0.707954 0.706258i $$-0.249618\pi$$
$$140$$ 0 0
$$141$$ 2.82843i 0.238197i
$$142$$ 0 0
$$143$$ −2.73551 −0.228755
$$144$$ 0 0
$$145$$ 10.7721 0.894578
$$146$$ 0 0
$$147$$ 0.458440i 0.0378115i
$$148$$ 0 0
$$149$$ 11.1832i 0.916166i 0.888909 + 0.458083i $$0.151464\pi$$
−0.888909 + 0.458083i $$0.848536\pi$$
$$150$$ 0 0
$$151$$ −14.6506 −1.19225 −0.596123 0.802893i $$-0.703293\pi$$
−0.596123 + 0.802893i $$0.703293\pi$$
$$152$$ 0 0
$$153$$ −6.44549 −0.521087
$$154$$ 0 0
$$155$$ 16.2212i 1.30292i
$$156$$ 0 0
$$157$$ − 4.45754i − 0.355750i −0.984053 0.177875i $$-0.943078\pi$$
0.984053 0.177875i $$-0.0569223\pi$$
$$158$$ 0 0
$$159$$ 3.64520 0.289083
$$160$$ 0 0
$$161$$ −7.23412 −0.570128
$$162$$ 0 0
$$163$$ 7.78510i 0.609776i 0.952388 + 0.304888i $$0.0986191\pi$$
−0.952388 + 0.304888i $$0.901381\pi$$
$$164$$ 0 0
$$165$$ − 1.65685i − 0.128986i
$$166$$ 0 0
$$167$$ 20.1814 1.56168 0.780841 0.624730i $$-0.214791\pi$$
0.780841 + 0.624730i $$0.214791\pi$$
$$168$$ 0 0
$$169$$ −3.67923 −0.283018
$$170$$ 0 0
$$171$$ − 6.44549i − 0.492899i
$$172$$ 0 0
$$173$$ − 6.15639i − 0.468061i −0.972229 0.234031i $$-0.924808\pi$$
0.972229 0.234031i $$-0.0751916\pi$$
$$174$$ 0 0
$$175$$ −2.86156 −0.216313
$$176$$ 0 0
$$177$$ 5.65685 0.425195
$$178$$ 0 0
$$179$$ 18.7855i 1.40409i 0.712131 + 0.702046i $$0.247730\pi$$
−0.712131 + 0.702046i $$0.752270\pi$$
$$180$$ 0 0
$$181$$ − 8.97499i − 0.667106i −0.942731 0.333553i $$-0.891752\pi$$
0.942731 0.333553i $$-0.108248\pi$$
$$182$$ 0 0
$$183$$ −6.20285 −0.458528
$$184$$ 0 0
$$185$$ 9.54745 0.701943
$$186$$ 0 0
$$187$$ − 4.31724i − 0.315708i
$$188$$ 0 0
$$189$$ 2.55765i 0.186041i
$$190$$ 0 0
$$191$$ 5.60058 0.405243 0.202622 0.979257i $$-0.435054\pi$$
0.202622 + 0.979257i $$0.435054\pi$$
$$192$$ 0 0
$$193$$ −19.4514 −1.40014 −0.700071 0.714074i $$-0.746848\pi$$
−0.700071 + 0.714074i $$0.746848\pi$$
$$194$$ 0 0
$$195$$ − 10.1023i − 0.723444i
$$196$$ 0 0
$$197$$ 1.75070i 0.124732i 0.998053 + 0.0623659i $$0.0198646\pi$$
−0.998053 + 0.0623659i $$0.980135\pi$$
$$198$$ 0 0
$$199$$ 0.993710 0.0704422 0.0352211 0.999380i $$-0.488786\pi$$
0.0352211 + 0.999380i $$0.488786\pi$$
$$200$$ 0 0
$$201$$ −2.99647 −0.211355
$$202$$ 0 0
$$203$$ − 11.1380i − 0.781738i
$$204$$ 0 0
$$205$$ − 1.95078i − 0.136248i
$$206$$ 0 0
$$207$$ 2.82843 0.196589
$$208$$ 0 0
$$209$$ 4.31724 0.298630
$$210$$ 0 0
$$211$$ − 5.97409i − 0.411273i −0.978628 0.205637i $$-0.934073\pi$$
0.978628 0.205637i $$-0.0659265\pi$$
$$212$$ 0 0
$$213$$ 5.11529i 0.350494i
$$214$$ 0 0
$$215$$ −1.36293 −0.0929509
$$216$$ 0 0
$$217$$ 16.7721 1.13857
$$218$$ 0 0
$$219$$ − 14.7721i − 0.998209i
$$220$$ 0 0
$$221$$ − 26.3235i − 1.77071i
$$222$$ 0 0
$$223$$ −23.7659 −1.59148 −0.795740 0.605639i $$-0.792918\pi$$
−0.795740 + 0.605639i $$0.792918\pi$$
$$224$$ 0 0
$$225$$ 1.11882 0.0745883
$$226$$ 0 0
$$227$$ − 0.907457i − 0.0602300i −0.999546 0.0301150i $$-0.990413\pi$$
0.999546 0.0301150i $$-0.00958735\pi$$
$$228$$ 0 0
$$229$$ 7.55579i 0.499301i 0.968336 + 0.249650i $$0.0803157\pi$$
−0.968336 + 0.249650i $$0.919684\pi$$
$$230$$ 0 0
$$231$$ −1.71313 −0.112716
$$232$$ 0 0
$$233$$ 23.2271 1.52166 0.760828 0.648954i $$-0.224793\pi$$
0.760828 + 0.648954i $$0.224793\pi$$
$$234$$ 0 0
$$235$$ − 6.99647i − 0.456399i
$$236$$ 0 0
$$237$$ − 6.32000i − 0.410528i
$$238$$ 0 0
$$239$$ −26.9213 −1.74140 −0.870698 0.491817i $$-0.836333\pi$$
−0.870698 + 0.491817i $$0.836333\pi$$
$$240$$ 0 0
$$241$$ 10.3494 0.666664 0.333332 0.942809i $$-0.391827\pi$$
0.333332 + 0.942809i $$0.391827\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ − 1.13401i − 0.0724492i
$$246$$ 0 0
$$247$$ 26.3235 1.67492
$$248$$ 0 0
$$249$$ −0.907457 −0.0575077
$$250$$ 0 0
$$251$$ − 13.7984i − 0.870949i −0.900201 0.435475i $$-0.856581\pi$$
0.900201 0.435475i $$-0.143419\pi$$
$$252$$ 0 0
$$253$$ 1.89450i 0.119106i
$$254$$ 0 0
$$255$$ 15.9437 0.998435
$$256$$ 0 0
$$257$$ 16.9965 1.06021 0.530105 0.847932i $$-0.322152\pi$$
0.530105 + 0.847932i $$0.322152\pi$$
$$258$$ 0 0
$$259$$ − 9.87175i − 0.613401i
$$260$$ 0 0
$$261$$ 4.35480i 0.269555i
$$262$$ 0 0
$$263$$ 29.9929 1.84944 0.924722 0.380643i $$-0.124297\pi$$
0.924722 + 0.380643i $$0.124297\pi$$
$$264$$ 0 0
$$265$$ −9.01686 −0.553901
$$266$$ 0 0
$$267$$ − 6.31724i − 0.386609i
$$268$$ 0 0
$$269$$ − 29.1332i − 1.77628i −0.459569 0.888142i $$-0.651996\pi$$
0.459569 0.888142i $$-0.348004\pi$$
$$270$$ 0 0
$$271$$ 26.6506 1.61891 0.809453 0.587184i $$-0.199764\pi$$
0.809453 + 0.587184i $$0.199764\pi$$
$$272$$ 0 0
$$273$$ −10.4455 −0.632190
$$274$$ 0 0
$$275$$ 0.749397i 0.0451904i
$$276$$ 0 0
$$277$$ − 17.1430i − 1.03003i −0.857183 0.515013i $$-0.827787\pi$$
0.857183 0.515013i $$-0.172213\pi$$
$$278$$ 0 0
$$279$$ −6.55765 −0.392596
$$280$$ 0 0
$$281$$ −2.76588 −0.164999 −0.0824993 0.996591i $$-0.526290\pi$$
−0.0824993 + 0.996591i $$0.526290\pi$$
$$282$$ 0 0
$$283$$ 6.34315i 0.377061i 0.982067 + 0.188530i $$0.0603724\pi$$
−0.982067 + 0.188530i $$0.939628\pi$$
$$284$$ 0 0
$$285$$ 15.9437i 0.944425i
$$286$$ 0 0
$$287$$ −2.01704 −0.119062
$$288$$ 0 0
$$289$$ 24.5443 1.44378
$$290$$ 0 0
$$291$$ − 12.6533i − 0.741751i
$$292$$ 0 0
$$293$$ − 11.6078i − 0.678133i −0.940762 0.339067i $$-0.889889\pi$$
0.940762 0.339067i $$-0.110111\pi$$
$$294$$ 0 0
$$295$$ −13.9929 −0.814700
$$296$$ 0 0
$$297$$ 0.669808 0.0388662
$$298$$ 0 0
$$299$$ 11.5514i 0.668032i
$$300$$ 0 0
$$301$$ 1.40922i 0.0812262i
$$302$$ 0 0
$$303$$ 10.6417 0.611348
$$304$$ 0 0
$$305$$ 15.3435 0.878567
$$306$$ 0 0
$$307$$ − 14.7855i − 0.843852i −0.906630 0.421926i $$-0.861354\pi$$
0.906630 0.421926i $$-0.138646\pi$$
$$308$$ 0 0
$$309$$ 3.33686i 0.189827i
$$310$$ 0 0
$$311$$ 15.0761 0.854885 0.427442 0.904043i $$-0.359415\pi$$
0.427442 + 0.904043i $$0.359415\pi$$
$$312$$ 0 0
$$313$$ −23.0027 −1.30019 −0.650096 0.759852i $$-0.725271\pi$$
−0.650096 + 0.759852i $$0.725271\pi$$
$$314$$ 0 0
$$315$$ − 6.32666i − 0.356467i
$$316$$ 0 0
$$317$$ − 9.55855i − 0.536862i −0.963299 0.268431i $$-0.913495\pi$$
0.963299 0.268431i $$-0.0865051\pi$$
$$318$$ 0 0
$$319$$ −2.91688 −0.163314
$$320$$ 0 0
$$321$$ 19.8874 1.11001
$$322$$ 0 0
$$323$$ 41.5443i 2.31159i
$$324$$ 0 0
$$325$$ 4.56930i 0.253459i
$$326$$ 0 0
$$327$$ −3.91598 −0.216554
$$328$$ 0 0
$$329$$ −7.23412 −0.398830
$$330$$ 0 0
$$331$$ 27.8079i 1.52846i 0.644945 + 0.764229i $$0.276880\pi$$
−0.644945 + 0.764229i $$0.723120\pi$$
$$332$$ 0 0
$$333$$ 3.85970i 0.211510i
$$334$$ 0 0
$$335$$ 7.41215 0.404969
$$336$$ 0 0
$$337$$ −3.00980 −0.163954 −0.0819771 0.996634i $$-0.526123\pi$$
−0.0819771 + 0.996634i $$0.526123\pi$$
$$338$$ 0 0
$$339$$ 2.23765i 0.121532i
$$340$$ 0 0
$$341$$ − 4.39236i − 0.237860i
$$342$$ 0 0
$$343$$ −19.0761 −1.03001
$$344$$ 0 0
$$345$$ −6.99647 −0.376677
$$346$$ 0 0
$$347$$ 8.87449i 0.476408i 0.971215 + 0.238204i $$0.0765586\pi$$
−0.971215 + 0.238204i $$0.923441\pi$$
$$348$$ 0 0
$$349$$ − 6.70698i − 0.359016i −0.983757 0.179508i $$-0.942549\pi$$
0.983757 0.179508i $$-0.0574506\pi$$
$$350$$ 0 0
$$351$$ 4.08402 0.217989
$$352$$ 0 0
$$353$$ −8.75882 −0.466185 −0.233093 0.972455i $$-0.574884\pi$$
−0.233093 + 0.972455i $$0.574884\pi$$
$$354$$ 0 0
$$355$$ − 12.6533i − 0.671569i
$$356$$ 0 0
$$357$$ − 16.4853i − 0.872494i
$$358$$ 0 0
$$359$$ −32.7917 −1.73068 −0.865341 0.501184i $$-0.832898\pi$$
−0.865341 + 0.501184i $$0.832898\pi$$
$$360$$ 0 0
$$361$$ −22.5443 −1.18654
$$362$$ 0 0
$$363$$ − 10.5514i − 0.553803i
$$364$$ 0 0
$$365$$ 36.5408i 1.91263i
$$366$$ 0 0
$$367$$ −20.6435 −1.07758 −0.538791 0.842439i $$-0.681119\pi$$
−0.538791 + 0.842439i $$0.681119\pi$$
$$368$$ 0 0
$$369$$ 0.788632 0.0410546
$$370$$ 0 0
$$371$$ 9.32313i 0.484033i
$$372$$ 0 0
$$373$$ 23.4995i 1.21676i 0.793646 + 0.608379i $$0.208180\pi$$
−0.793646 + 0.608379i $$0.791820\pi$$
$$374$$ 0 0
$$375$$ 9.60058 0.495772
$$376$$ 0 0
$$377$$ −17.7851 −0.915979
$$378$$ 0 0
$$379$$ 11.0004i 0.565051i 0.959260 + 0.282526i $$0.0911722\pi$$
−0.959260 + 0.282526i $$0.908828\pi$$
$$380$$ 0 0
$$381$$ − 12.2145i − 0.625768i
$$382$$ 0 0
$$383$$ −17.2037 −0.879070 −0.439535 0.898225i $$-0.644857\pi$$
−0.439535 + 0.898225i $$0.644857\pi$$
$$384$$ 0 0
$$385$$ 4.23765 0.215971
$$386$$ 0 0
$$387$$ − 0.550984i − 0.0280081i
$$388$$ 0 0
$$389$$ − 33.7311i − 1.71023i −0.518436 0.855116i $$-0.673486\pi$$
0.518436 0.855116i $$-0.326514\pi$$
$$390$$ 0 0
$$391$$ −18.2306 −0.921961
$$392$$ 0 0
$$393$$ −5.33962 −0.269348
$$394$$ 0 0
$$395$$ 15.6333i 0.786597i
$$396$$ 0 0
$$397$$ 14.5201i 0.728742i 0.931254 + 0.364371i $$0.118716\pi$$
−0.931254 + 0.364371i $$0.881284\pi$$
$$398$$ 0 0
$$399$$ 16.4853 0.825296
$$400$$ 0 0
$$401$$ −32.2274 −1.60936 −0.804681 0.593708i $$-0.797663\pi$$
−0.804681 + 0.593708i $$0.797663\pi$$
$$402$$ 0 0
$$403$$ − 26.7816i − 1.33409i
$$404$$ 0 0
$$405$$ 2.47363i 0.122915i
$$406$$ 0 0
$$407$$ −2.58526 −0.128146
$$408$$ 0 0
$$409$$ 11.5702 0.572110 0.286055 0.958213i $$-0.407656\pi$$
0.286055 + 0.958213i $$0.407656\pi$$
$$410$$ 0 0
$$411$$ − 5.10587i − 0.251854i
$$412$$ 0 0
$$413$$ 14.4682i 0.711935i
$$414$$ 0 0
$$415$$ 2.24471 0.110188
$$416$$ 0 0
$$417$$ −16.6533 −0.815517
$$418$$ 0 0
$$419$$ 9.54193i 0.466154i 0.972458 + 0.233077i $$0.0748794\pi$$
−0.972458 + 0.233077i $$0.925121\pi$$
$$420$$ 0 0
$$421$$ − 24.3583i − 1.18715i −0.804778 0.593576i $$-0.797716\pi$$
0.804778 0.593576i $$-0.202284\pi$$
$$422$$ 0 0
$$423$$ 2.82843 0.137523
$$424$$ 0 0
$$425$$ −7.21137 −0.349803
$$426$$ 0 0
$$427$$ − 15.8647i − 0.767746i
$$428$$ 0 0
$$429$$ 2.73551i 0.132072i
$$430$$ 0 0
$$431$$ −40.7088 −1.96087 −0.980437 0.196832i $$-0.936935\pi$$
−0.980437 + 0.196832i $$0.936935\pi$$
$$432$$ 0 0
$$433$$ −7.31371 −0.351474 −0.175737 0.984437i $$-0.556231\pi$$
−0.175737 + 0.984437i $$0.556231\pi$$
$$434$$ 0 0
$$435$$ − 10.7721i − 0.516485i
$$436$$ 0 0
$$437$$ − 18.2306i − 0.872087i
$$438$$ 0 0
$$439$$ −17.7122 −0.845356 −0.422678 0.906280i $$-0.638910\pi$$
−0.422678 + 0.906280i $$0.638910\pi$$
$$440$$ 0 0
$$441$$ 0.458440 0.0218305
$$442$$ 0 0
$$443$$ 22.1953i 1.05453i 0.849701 + 0.527264i $$0.176782\pi$$
−0.849701 + 0.527264i $$0.823218\pi$$
$$444$$ 0 0
$$445$$ 15.6265i 0.740766i
$$446$$ 0 0
$$447$$ 11.1832 0.528949
$$448$$ 0 0
$$449$$ −28.3400 −1.33745 −0.668723 0.743511i $$-0.733159\pi$$
−0.668723 + 0.743511i $$0.733159\pi$$
$$450$$ 0 0
$$451$$ 0.528232i 0.0248735i
$$452$$ 0 0
$$453$$ 14.6506i 0.688344i
$$454$$ 0 0
$$455$$ 25.8382 1.21131
$$456$$ 0 0
$$457$$ 17.3396 0.811113 0.405557 0.914070i $$-0.367078\pi$$
0.405557 + 0.914070i $$0.367078\pi$$
$$458$$ 0 0
$$459$$ 6.44549i 0.300850i
$$460$$ 0 0
$$461$$ 2.39404i 0.111501i 0.998445 + 0.0557507i $$0.0177552\pi$$
−0.998445 + 0.0557507i $$0.982245\pi$$
$$462$$ 0 0
$$463$$ −2.70238 −0.125590 −0.0627951 0.998026i $$-0.520001\pi$$
−0.0627951 + 0.998026i $$0.520001\pi$$
$$464$$ 0 0
$$465$$ 16.2212 0.752239
$$466$$ 0 0
$$467$$ 24.2023i 1.11995i 0.828510 + 0.559975i $$0.189189\pi$$
−0.828510 + 0.559975i $$0.810811\pi$$
$$468$$ 0 0
$$469$$ − 7.66391i − 0.353887i
$$470$$ 0 0
$$471$$ −4.45754 −0.205393
$$472$$ 0 0
$$473$$ 0.369053 0.0169691
$$474$$ 0 0
$$475$$ − 7.21137i − 0.330880i
$$476$$ 0 0
$$477$$ − 3.64520i − 0.166902i
$$478$$ 0 0
$$479$$ −22.2251 −1.01549 −0.507745 0.861508i $$-0.669521\pi$$
−0.507745 + 0.861508i $$0.669521\pi$$
$$480$$ 0 0
$$481$$ −15.7631 −0.718735
$$482$$ 0 0
$$483$$ 7.23412i 0.329164i
$$484$$ 0 0
$$485$$ 31.2996i 1.42124i
$$486$$ 0 0
$$487$$ −13.9839 −0.633672 −0.316836 0.948480i $$-0.602620\pi$$
−0.316836 + 0.948480i $$0.602620\pi$$
$$488$$ 0 0
$$489$$ 7.78510 0.352055
$$490$$ 0 0
$$491$$ − 10.2306i − 0.461700i −0.972989 0.230850i $$-0.925849\pi$$
0.972989 0.230850i $$-0.0741507\pi$$
$$492$$ 0 0
$$493$$ − 28.0688i − 1.26416i
$$494$$ 0 0
$$495$$ −1.65685 −0.0744701
$$496$$ 0 0
$$497$$ −13.0831 −0.586858
$$498$$ 0 0
$$499$$ − 3.66391i − 0.164019i −0.996632 0.0820097i $$-0.973866\pi$$
0.996632 0.0820097i $$-0.0261338\pi$$
$$500$$ 0 0
$$501$$ − 20.1814i − 0.901637i
$$502$$ 0 0
$$503$$ 39.6443 1.76765 0.883825 0.467817i $$-0.154959\pi$$
0.883825 + 0.467817i $$0.154959\pi$$
$$504$$ 0 0
$$505$$ −26.3235 −1.17138
$$506$$ 0 0
$$507$$ 3.67923i 0.163400i
$$508$$ 0 0
$$509$$ 28.6909i 1.27170i 0.771812 + 0.635851i $$0.219351\pi$$
−0.771812 + 0.635851i $$0.780649\pi$$
$$510$$ 0 0
$$511$$ 37.7819 1.67137
$$512$$ 0 0
$$513$$ −6.44549 −0.284575
$$514$$ 0 0
$$515$$ − 8.25413i − 0.363721i
$$516$$ 0 0
$$517$$ 1.89450i 0.0833201i
$$518$$ 0 0
$$519$$ −6.15639 −0.270235
$$520$$ 0 0
$$521$$ −23.1784 −1.01546 −0.507732 0.861515i $$-0.669516\pi$$
−0.507732 + 0.861515i $$0.669516\pi$$
$$522$$ 0 0
$$523$$ 8.18193i 0.357771i 0.983870 + 0.178885i $$0.0572491\pi$$
−0.983870 + 0.178885i $$0.942751\pi$$
$$524$$ 0 0
$$525$$ 2.86156i 0.124889i
$$526$$ 0 0
$$527$$ 42.2672 1.84119
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ − 5.65685i − 0.245487i
$$532$$ 0 0
$$533$$ 3.22079i 0.139508i
$$534$$ 0 0
$$535$$ −49.1941 −2.12685
$$536$$ 0 0
$$537$$ 18.7855 0.810653
$$538$$ 0 0
$$539$$ 0.307067i 0.0132263i
$$540$$ 0 0
$$541$$ 6.43715i 0.276755i 0.990380 + 0.138377i $$0.0441887\pi$$
−0.990380 + 0.138377i $$0.955811\pi$$
$$542$$ 0 0
$$543$$ −8.97499 −0.385154
$$544$$ 0 0
$$545$$ 9.68667 0.414931
$$546$$ 0 0
$$547$$ − 39.2239i − 1.67709i −0.544830 0.838546i $$-0.683406\pi$$
0.544830 0.838546i $$-0.316594\pi$$
$$548$$ 0 0
$$549$$ 6.20285i 0.264731i
$$550$$ 0 0
$$551$$ 28.0688 1.19577
$$552$$ 0 0
$$553$$ 16.1643 0.687377
$$554$$ 0 0
$$555$$ − 9.54745i − 0.405267i
$$556$$ 0 0
$$557$$ 1.66224i 0.0704315i 0.999380 + 0.0352157i $$0.0112118\pi$$
−0.999380 + 0.0352157i $$0.988788\pi$$
$$558$$ 0 0
$$559$$ 2.25023 0.0951745
$$560$$ 0 0
$$561$$ −4.31724 −0.182274
$$562$$ 0 0
$$563$$ 40.6368i 1.71264i 0.516447 + 0.856319i $$0.327254\pi$$
−0.516447 + 0.856319i $$0.672746\pi$$
$$564$$ 0 0
$$565$$ − 5.53511i − 0.232864i
$$566$$ 0 0
$$567$$ 2.55765 0.107411
$$568$$ 0 0
$$569$$ 27.0004 1.13191 0.565957 0.824435i $$-0.308507\pi$$
0.565957 + 0.824435i $$0.308507\pi$$
$$570$$ 0 0
$$571$$ − 20.9706i − 0.877591i −0.898587 0.438795i $$-0.855405\pi$$
0.898587 0.438795i $$-0.144595\pi$$
$$572$$ 0 0
$$573$$ − 5.60058i − 0.233967i
$$574$$ 0 0
$$575$$ 3.16451 0.131969
$$576$$ 0 0
$$577$$ −37.6372 −1.56686 −0.783429 0.621481i $$-0.786531\pi$$
−0.783429 + 0.621481i $$0.786531\pi$$
$$578$$ 0 0
$$579$$ 19.4514i 0.808372i
$$580$$ 0 0
$$581$$ − 2.32095i − 0.0962894i
$$582$$ 0 0
$$583$$ 2.44158 0.101120
$$584$$ 0 0
$$585$$ −10.1023 −0.417680
$$586$$ 0 0
$$587$$ 44.2047i 1.82452i 0.409609 + 0.912261i $$0.365665\pi$$
−0.409609 + 0.912261i $$0.634335\pi$$
$$588$$ 0 0
$$589$$ 42.2672i 1.74159i
$$590$$ 0 0
$$591$$ 1.75070 0.0720140
$$592$$ 0 0
$$593$$ 3.59611 0.147675 0.0738373 0.997270i $$-0.476475\pi$$
0.0738373 + 0.997270i $$0.476475\pi$$
$$594$$ 0 0
$$595$$ 40.7784i 1.67175i
$$596$$ 0 0
$$597$$ − 0.993710i − 0.0406698i
$$598$$ 0 0
$$599$$ −22.0296 −0.900104 −0.450052 0.893002i $$-0.648595\pi$$
−0.450052 + 0.893002i $$0.648595\pi$$
$$600$$ 0 0
$$601$$ −10.7721 −0.439405 −0.219703 0.975567i $$-0.570509\pi$$
−0.219703 + 0.975567i $$0.570509\pi$$
$$602$$ 0 0
$$603$$ 2.99647i 0.122026i
$$604$$ 0 0
$$605$$ 26.1001i 1.06112i
$$606$$ 0 0
$$607$$ 5.47453 0.222204 0.111102 0.993809i $$-0.464562\pi$$
0.111102 + 0.993809i $$0.464562\pi$$
$$608$$ 0 0
$$609$$ −11.1380 −0.451336
$$610$$ 0 0
$$611$$ 11.5514i 0.467318i
$$612$$ 0 0
$$613$$ − 14.8562i − 0.600035i −0.953934 0.300018i $$-0.903007\pi$$
0.953934 0.300018i $$-0.0969925\pi$$
$$614$$ 0 0
$$615$$ −1.95078 −0.0786631
$$616$$ 0 0
$$617$$ −22.2235 −0.894686 −0.447343 0.894363i $$-0.647630\pi$$
−0.447343 + 0.894363i $$0.647630\pi$$
$$618$$ 0 0
$$619$$ − 16.4612i − 0.661631i −0.943696 0.330815i $$-0.892676\pi$$
0.943696 0.330815i $$-0.107324\pi$$
$$620$$ 0 0
$$621$$ − 2.82843i − 0.113501i
$$622$$ 0 0
$$623$$ 16.1573 0.647327
$$624$$ 0 0
$$625$$ −29.3424 −1.17369
$$626$$ 0 0
$$627$$ − 4.31724i − 0.172414i
$$628$$ 0 0
$$629$$ − 24.8776i − 0.991937i
$$630$$ 0 0
$$631$$ 4.06977 0.162015 0.0810075 0.996713i $$-0.474186\pi$$
0.0810075 + 0.996713i $$0.474186\pi$$
$$632$$ 0 0
$$633$$ −5.97409 −0.237449
$$634$$ 0 0
$$635$$ 30.2141i 1.19901i
$$636$$ 0 0
$$637$$ 1.87228i 0.0741824i
$$638$$ 0 0
$$639$$ 5.11529 0.202358
$$640$$ 0 0
$$641$$ −8.41958 −0.332553 −0.166277 0.986079i $$-0.553174\pi$$
−0.166277 + 0.986079i $$0.553174\pi$$
$$642$$ 0 0
$$643$$ − 10.4266i − 0.411186i −0.978638 0.205593i $$-0.934088\pi$$
0.978638 0.205593i $$-0.0659124\pi$$
$$644$$ 0 0
$$645$$ 1.36293i 0.0536652i
$$646$$ 0 0
$$647$$ −11.6132 −0.456560 −0.228280 0.973595i $$-0.573310\pi$$
−0.228280 + 0.973595i $$0.573310\pi$$
$$648$$ 0 0
$$649$$ 3.78901 0.148731
$$650$$ 0 0
$$651$$ − 16.7721i − 0.657352i
$$652$$ 0 0
$$653$$ 2.73012i 0.106838i 0.998572 + 0.0534190i $$0.0170119\pi$$
−0.998572 + 0.0534190i $$0.982988\pi$$
$$654$$ 0 0
$$655$$ 13.2082 0.516088
$$656$$ 0 0
$$657$$ −14.7721 −0.576316
$$658$$ 0 0
$$659$$ − 31.5514i − 1.22907i −0.788891 0.614533i $$-0.789344\pi$$
0.788891 0.614533i $$-0.210656\pi$$
$$660$$ 0 0
$$661$$ − 15.1368i − 0.588752i −0.955690 0.294376i $$-0.904888\pi$$
0.955690 0.294376i $$-0.0951118\pi$$
$$662$$ 0 0
$$663$$ −26.3235 −1.02232
$$664$$ 0 0
$$665$$ −40.7784 −1.58132
$$666$$ 0 0
$$667$$ 12.3172i 0.476925i
$$668$$ 0 0
$$669$$ 23.7659i 0.918841i
$$670$$ 0 0
$$671$$ −4.15472 −0.160391
$$672$$ 0 0
$$673$$ −20.6345 −0.795401 −0.397700 0.917515i $$-0.630192\pi$$
−0.397700 + 0.917515i $$0.630192\pi$$
$$674$$ 0 0
$$675$$ − 1.11882i − 0.0430636i
$$676$$ 0 0
$$677$$ 37.9357i 1.45799i 0.684520 + 0.728994i $$0.260012\pi$$
−0.684520 + 0.728994i $$0.739988\pi$$
$$678$$ 0 0
$$679$$ 32.3627 1.24197
$$680$$ 0 0
$$681$$ −0.907457 −0.0347738
$$682$$ 0 0
$$683$$ − 18.2471i − 0.698205i −0.937085 0.349102i $$-0.886487\pi$$
0.937085 0.349102i $$-0.113513\pi$$
$$684$$ 0 0
$$685$$ 12.6300i 0.482568i
$$686$$ 0 0
$$687$$ 7.55579 0.288271
$$688$$ 0 0
$$689$$ 14.8871 0.567152
$$690$$ 0 0
$$691$$ 30.2533i 1.15089i 0.817840 + 0.575446i $$0.195171\pi$$
−0.817840 + 0.575446i $$0.804829\pi$$
$$692$$ 0 0
$$693$$ 1.71313i 0.0650765i
$$694$$ 0 0
$$695$$ 41.1941 1.56258
$$696$$ 0 0
$$697$$ −5.08312 −0.192537
$$698$$ 0 0
$$699$$ − 23.2271i − 0.878528i
$$700$$ 0 0
$$701$$ 20.0875i 0.758696i 0.925254 + 0.379348i $$0.123852\pi$$
−0.925254 + 0.379348i $$0.876148\pi$$
$$702$$ 0 0
$$703$$ 24.8776 0.938278
$$704$$ 0 0
$$705$$ −6.99647 −0.263502
$$706$$ 0 0
$$707$$ 27.2176i 1.02362i
$$708$$ 0 0
$$709$$ − 41.7864i − 1.56932i −0.619926 0.784660i $$-0.712837\pi$$
0.619926 0.784660i $$-0.287163\pi$$
$$710$$ 0 0
$$711$$ −6.32000 −0.237018
$$712$$ 0 0
$$713$$ −18.5478 −0.694622
$$714$$ 0 0
$$715$$ − 6.76663i − 0.253058i
$$716$$ 0 0
$$717$$ 26.9213i 1.00540i
$$718$$ 0 0
$$719$$ −28.3683 −1.05796 −0.528979 0.848635i $$-0.677425\pi$$
−0.528979 + 0.848635i $$0.677425\pi$$
$$720$$ 0 0
$$721$$ −8.53450 −0.317841
$$722$$ 0 0
$$723$$ − 10.3494i − 0.384899i
$$724$$ 0 0
$$725$$ 4.87226i 0.180951i
$$726$$ 0 0
$$727$$ 20.4843 0.759722 0.379861 0.925044i $$-0.375972\pi$$
0.379861 + 0.925044i $$0.375972\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 3.55136i 0.131352i
$$732$$ 0 0
$$733$$ 48.0777i 1.77579i 0.460045 + 0.887895i $$0.347833\pi$$
−0.460045 + 0.887895i $$0.652167\pi$$
$$734$$ 0 0
$$735$$ −1.13401 −0.0418286
$$736$$ 0 0
$$737$$ −2.00706 −0.0739310
$$738$$ 0 0
$$739$$ 21.4459i 0.788899i 0.918918 + 0.394449i $$0.129065\pi$$
−0.918918 + 0.394449i $$0.870935\pi$$
$$740$$ 0 0
$$741$$ − 26.3235i − 0.967018i
$$742$$ 0 0
$$743$$ −2.17431 −0.0797677 −0.0398839 0.999204i $$-0.512699\pi$$
−0.0398839 + 0.999204i $$0.512699\pi$$
$$744$$ 0 0
$$745$$ −27.6631 −1.01350
$$746$$ 0 0
$$747$$ 0.907457i 0.0332021i
$$748$$ 0 0
$$749$$ 50.8651i 1.85857i
$$750$$ 0 0
$$751$$ −29.8980 −1.09099 −0.545497 0.838113i $$-0.683659\pi$$
−0.545497 + 0.838113i $$0.683659\pi$$
$$752$$ 0 0
$$753$$ −13.7984 −0.502843
$$754$$ 0 0
$$755$$ − 36.2400i − 1.31891i
$$756$$ 0 0
$$757$$ 21.6791i 0.787939i 0.919123 + 0.393970i $$0.128899\pi$$
−0.919123 + 0.393970i $$0.871101\pi$$
$$758$$ 0 0
$$759$$ 1.89450 0.0687661
$$760$$ 0 0
$$761$$ −4.29449 −0.155675 −0.0778375 0.996966i $$-0.524802\pi$$
−0.0778375 + 0.996966i $$0.524802\pi$$
$$762$$ 0 0
$$763$$ − 10.0157i − 0.362592i
$$764$$ 0 0
$$765$$ − 15.9437i − 0.576446i
$$766$$ 0 0
$$767$$ 23.1027 0.834191
$$768$$ 0 0
$$769$$ 33.8819 1.22181 0.610907 0.791703i $$-0.290805\pi$$
0.610907 + 0.791703i $$0.290805\pi$$
$$770$$ 0 0
$$771$$ − 16.9965i − 0.612113i
$$772$$ 0 0
$$773$$ − 49.5300i − 1.78147i −0.454521 0.890736i $$-0.650190\pi$$
0.454521 0.890736i $$-0.349810\pi$$
$$774$$ 0 0
$$775$$ −7.33686 −0.263548
$$776$$ 0 0
$$777$$ −9.87175 −0.354147
$$778$$ 0 0
$$779$$ − 5.08312i − 0.182122i
$$780$$ 0 0
$$781$$ 3.42627i 0.122601i
$$782$$ 0 0
$$783$$ 4.35480 0.155628
$$784$$ 0 0
$$785$$ 11.0263 0.393545
$$786$$ 0 0
$$787$$ − 34.0953i − 1.21537i −0.794180 0.607683i $$-0.792099\pi$$
0.794180 0.607683i $$-0.207901\pi$$
$$788$$ 0 0
$$789$$ − 29.9929i − 1.06778i
$$790$$ 0 0
$$791$$ −5.72312 −0.203491
$$792$$ 0 0
$$793$$ −25.3326 −0.899585
$$794$$ 0 0
$$795$$ 9.01686i 0.319795i
$$796$$ 0 0
$$797$$ − 40.6901i − 1.44132i −0.693290 0.720659i $$-0.743840\pi$$
0.693290 0.720659i $$-0.256160\pi$$
$$798$$ 0 0
$$799$$ −18.2306 −0.644952
$$800$$ 0 0
$$801$$ −6.31724 −0.223209
$$802$$ 0 0
$$803$$ − 9.89450i − 0.349169i
$$804$$ 0 0
$$805$$ − 17.8945i − 0.630698i
$$806$$ 0 0
$$807$$ −29.1332 −1.02554
$$808$$ 0 0
$$809$$ 10.9926 0.386478 0.193239 0.981152i $$-0.438101\pi$$
0.193239 + 0.981152i $$0.438101\pi$$
$$810$$ 0 0
$$811$$ 21.2498i 0.746182i 0.927795 + 0.373091i $$0.121702\pi$$
−0.927795 + 0.373091i $$0.878298\pi$$
$$812$$ 0 0
$$813$$ − 26.6506i − 0.934676i
$$814$$ 0 0
$$815$$ −19.2574 −0.674558
$$816$$ 0 0
$$817$$ −3.55136 −0.124246
$$818$$ 0 0
$$819$$ 10.4455i 0.364995i
$$820$$ 0 0
$$821$$ 30.0572i 1.04900i 0.851410 + 0.524501i $$0.175748\pi$$
−0.851410 + 0.524501i $$0.824252\pi$$
$$822$$ 0 0
$$823$$ 55.0851 1.92015 0.960073 0.279751i $$-0.0902518\pi$$
0.960073 + 0.279751i $$0.0902518\pi$$
$$824$$ 0 0
$$825$$ 0.749397 0.0260907
$$826$$ 0 0
$$827$$ − 34.5478i − 1.20135i −0.799495 0.600673i $$-0.794899\pi$$
0.799495 0.600673i $$-0.205101\pi$$
$$828$$ 0 0
$$829$$ − 31.0046i − 1.07683i −0.842679 0.538417i $$-0.819023\pi$$
0.842679 0.538417i $$-0.180977\pi$$
$$830$$ 0 0
$$831$$ −17.1430 −0.594685
$$832$$ 0 0
$$833$$ −2.95487 −0.102380
$$834$$ 0 0
$$835$$ 49.9212i 1.72759i
$$836$$ 0 0
$$837$$ 6.55765i 0.226665i
$$838$$ 0 0
$$839$$ −5.14195 −0.177520 −0.0887599 0.996053i $$-0.528290\pi$$
−0.0887599 + 0.996053i $$0.528290\pi$$
$$840$$ 0 0
$$841$$ 10.0357 0.346059
$$842$$ 0 0
$$843$$ 2.76588i 0.0952620i
$$844$$ 0 0
$$845$$ − 9.10104i − 0.313085i
$$846$$ 0 0
$$847$$ 26.9867 0.927272
$$848$$ 0 0
$$849$$ 6.34315 0.217696
$$850$$ 0 0
$$851$$ 10.9169i 0.374226i
$$852$$ 0 0
$$853$$ − 18.5060i − 0.633632i −0.948487 0.316816i $$-0.897386\pi$$
0.948487 0.316816i $$-0.102614\pi$$
$$854$$ 0 0
$$855$$ 15.9437 0.545264
$$856$$ 0 0
$$857$$ −22.8878 −0.781833 −0.390916 0.920426i $$-0.627842\pi$$
−0.390916 + 0.920426i $$0.627842\pi$$
$$858$$ 0 0
$$859$$ − 35.5286i − 1.21222i −0.795381 0.606110i $$-0.792729\pi$$
0.795381 0.606110i $$-0.207271\pi$$
$$860$$ 0 0
$$861$$ 2.01704i 0.0687407i
$$862$$ 0 0
$$863$$ −43.9296 −1.49538 −0.747691 0.664047i $$-0.768837\pi$$
−0.747691 + 0.664047i $$0.768837\pi$$
$$864$$ 0 0
$$865$$ 15.2286 0.517788
$$866$$ 0 0
$$867$$ − 24.5443i − 0.833568i
$$868$$ 0 0
$$869$$ − 4.23319i − 0.143601i
$$870$$ 0 0
$$871$$ −12.2376 −0.414657
$$872$$ 0 0
$$873$$ −12.6533 −0.428250
$$874$$ 0 0
$$875$$ 24.5549i 0.830107i
$$876$$ 0 0
$$877$$ − 21.5773i − 0.728614i −0.931279 0.364307i $$-0.881306\pi$$
0.931279 0.364307i $$-0.118694\pi$$
$$878$$ 0 0
$$879$$ −11.6078 −0.391520
$$880$$ 0 0
$$881$$ −21.6686 −0.730035 −0.365018 0.931001i $$-0.618937\pi$$
−0.365018 + 0.931001i $$0.618937\pi$$
$$882$$ 0 0
$$883$$ 0.0834930i 0.00280976i 0.999999 + 0.00140488i $$0.000447188\pi$$
−0.999999 + 0.00140488i $$0.999553\pi$$
$$884$$ 0 0
$$885$$ 13.9929i 0.470368i
$$886$$ 0 0
$$887$$ −30.8043 −1.03431 −0.517154 0.855892i $$-0.673009\pi$$
−0.517154 + 0.855892i $$0.673009\pi$$
$$888$$ 0 0
$$889$$ 31.2404 1.04777
$$890$$ 0 0
$$891$$ − 0.669808i − 0.0224394i
$$892$$ 0 0
$$893$$ − 18.2306i − 0.610063i
$$894$$ 0 0
$$895$$ −46.4682 −1.55326
$$896$$ 0 0
$$897$$ 11.5514 0.385689
$$898$$ 0 0
$$899$$ − 28.5573i − 0.952438i
$$900$$ 0 0
$$901$$ 23.4951i 0.782735i
$$902$$ 0 0
$$903$$ 1.40922 0.0468960
$$904$$ 0 0
$$905$$ 22.2008 0.737979
$$906$$ 0 0
$$907$$ − 49.5215i − 1.64434i −0.569245 0.822168i $$-0.692764\pi$$
0.569245 0.822168i $$-0.307236\pi$$
$$908$$ 0 0
$$909$$ − 10.6417i − 0.352962i
$$910$$ 0 0
$$911$$ −0.0829331 −0.00274770 −0.00137385 0.999999i $$-0.500437\pi$$
−0.00137385 + 0.999999i $$0.500437\pi$$
$$912$$ 0 0
$$913$$ −0.607822 −0.0201160
$$914$$ 0 0
$$915$$ − 15.3435i − 0.507241i
$$916$$ 0 0
$$917$$ − 13.6569i − 0.450989i
$$918$$ 0 0
$$919$$ −20.1161 −0.663568 −0.331784 0.943355i $$-0.607650\pi$$
−0.331784 + 0.943355i $$0.607650\pi$$
$$920$$ 0 0
$$921$$ −14.7855 −0.487198
$$922$$ 0 0
$$923$$ 20.8910i 0.687635i
$$924$$ 0 0
$$925$$ 4.31833i 0.141986i
$$926$$ 0 0
$$927$$ 3.33686 0.109597
$$928$$ 0 0
$$929$$ 8.55098 0.280549 0.140274 0.990113i $$-0.455202\pi$$
0.140274 + 0.990113i $$0.455202\pi$$
$$930$$ 0 0
$$931$$ − 2.95487i − 0.0968420i
$$932$$ 0 0
$$933$$ − 15.0761i − 0.493568i
$$934$$ 0 0
$$935$$ 10.6792 0.349248
$$936$$ 0 0
$$937$$ −33.5780 −1.09695 −0.548473 0.836168i $$-0.684791\pi$$
−0.548473 + 0.836168i $$0.684791\pi$$
$$938$$ 0 0
$$939$$ 23.0027i 0.750666i
$$940$$ 0 0
$$941$$ 11.9991i 0.391159i 0.980688 + 0.195579i $$0.0626587\pi$$
−0.980688 + 0.195579i $$0.937341\pi$$
$$942$$ 0 0
$$943$$ 2.23059 0.0726380
$$944$$ 0 0
$$945$$ −6.32666 −0.205806
$$946$$ 0 0
$$947$$ − 25.2537i − 0.820635i −0.911943 0.410318i $$-0.865418\pi$$
0.911943 0.410318i $$-0.134582\pi$$
$$948$$ 0 0
$$949$$ − 60.3298i − 1.95839i
$$950$$ 0 0
$$951$$ −9.55855 −0.309957
$$952$$ 0 0
$$953$$ −3.86469 −0.125190 −0.0625948 0.998039i $$-0.519938\pi$$
−0.0625948 + 0.998039i $$0.519938\pi$$
$$954$$ 0 0
$$955$$ 13.8537i 0.448296i
$$956$$ 0 0
$$957$$ 2.91688i 0.0942894i
$$958$$ 0 0
$$959$$ 13.0590 0.421698
$$960$$ 0 0
$$961$$ 12.0027 0.387185
$$962$$ 0 0
$$963$$ − 19.8874i − 0.640864i
$$964$$ 0 0
$$965$$ − 48.1154i − 1.54889i
$$966$$ 0 0
$$967$$ 37.8714 1.21786 0.608930 0.793224i $$-0.291599\pi$$
0.608930 + 0.793224i $$0.291599\pi$$
$$968$$ 0 0
$$969$$ 41.5443 1.33460
$$970$$ 0 0
$$971$$ − 3.74587i − 0.120211i −0.998192 0.0601053i $$-0.980856\pi$$
0.998192 0.0601053i $$-0.0191437\pi$$
$$972$$ 0 0
$$973$$ − 42.5933i − 1.36548i
$$974$$ 0 0
$$975$$ 4.56930 0.146335
$$976$$ 0 0
$$977$$ −17.6530 −0.564768 −0.282384 0.959301i $$-0.591125\pi$$
−0.282384 + 0.959301i $$0.591125\pi$$
$$978$$ 0 0
$$979$$ − 4.23134i − 0.135234i
$$980$$ 0 0
$$981$$ 3.91598i 0.125028i
$$982$$ 0 0
$$983$$ −22.3557 −0.713035 −0.356518 0.934289i $$-0.616036\pi$$
−0.356518 + 0.934289i $$0.616036\pi$$
$$984$$ 0 0
$$985$$ −4.33057 −0.137983
$$986$$ 0 0
$$987$$ 7.23412i 0.230265i
$$988$$ 0 0
$$989$$ − 1.55842i − 0.0495548i
$$990$$ 0 0
$$991$$ 17.8769 0.567878 0.283939 0.958842i $$-0.408359\pi$$
0.283939 + 0.958842i $$0.408359\pi$$
$$992$$ 0 0
$$993$$ 27.8079 0.882456
$$994$$ 0 0
$$995$$ 2.45807i 0.0779260i
$$996$$ 0 0
$$997$$ 6.06146i 0.191968i 0.995383 + 0.0959841i $$0.0305998\pi$$
−0.995383 + 0.0959841i $$0.969400\pi$$
$$998$$ 0 0
$$999$$ 3.85970 0.122116
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 3072.2.d.f.1537.3 8
4.3 odd 2 3072.2.d.i.1537.7 8
8.3 odd 2 3072.2.d.i.1537.2 8
8.5 even 2 inner 3072.2.d.f.1537.6 8
16.3 odd 4 3072.2.a.n.1.3 4
16.5 even 4 3072.2.a.i.1.2 4
16.11 odd 4 3072.2.a.o.1.2 4
16.13 even 4 3072.2.a.t.1.3 4
32.3 odd 8 384.2.j.a.97.1 8
32.5 even 8 384.2.j.b.289.3 8
32.11 odd 8 192.2.j.a.145.4 8
32.13 even 8 48.2.j.a.37.2 yes 8
32.19 odd 8 192.2.j.a.49.4 8
32.21 even 8 48.2.j.a.13.2 8
32.27 odd 8 384.2.j.a.289.1 8
32.29 even 8 384.2.j.b.97.3 8
48.5 odd 4 9216.2.a.bo.1.3 4
48.11 even 4 9216.2.a.bn.1.3 4
48.29 odd 4 9216.2.a.y.1.2 4
48.35 even 4 9216.2.a.x.1.2 4
96.5 odd 8 1152.2.k.c.289.4 8
96.11 even 8 576.2.k.b.145.1 8
96.29 odd 8 1152.2.k.c.865.4 8
96.35 even 8 1152.2.k.f.865.4 8
96.53 odd 8 144.2.k.b.109.3 8
96.59 even 8 1152.2.k.f.289.4 8
96.77 odd 8 144.2.k.b.37.3 8
96.83 even 8 576.2.k.b.433.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.2 8 32.21 even 8
48.2.j.a.37.2 yes 8 32.13 even 8
144.2.k.b.37.3 8 96.77 odd 8
144.2.k.b.109.3 8 96.53 odd 8
192.2.j.a.49.4 8 32.19 odd 8
192.2.j.a.145.4 8 32.11 odd 8
384.2.j.a.97.1 8 32.3 odd 8
384.2.j.a.289.1 8 32.27 odd 8
384.2.j.b.97.3 8 32.29 even 8
384.2.j.b.289.3 8 32.5 even 8
576.2.k.b.145.1 8 96.11 even 8
576.2.k.b.433.1 8 96.83 even 8
1152.2.k.c.289.4 8 96.5 odd 8
1152.2.k.c.865.4 8 96.29 odd 8
1152.2.k.f.289.4 8 96.59 even 8
1152.2.k.f.865.4 8 96.35 even 8
3072.2.a.i.1.2 4 16.5 even 4
3072.2.a.n.1.3 4 16.3 odd 4
3072.2.a.o.1.2 4 16.11 odd 4
3072.2.a.t.1.3 4 16.13 even 4
3072.2.d.f.1537.3 8 1.1 even 1 trivial
3072.2.d.f.1537.6 8 8.5 even 2 inner
3072.2.d.i.1537.2 8 8.3 odd 2
3072.2.d.i.1537.7 8 4.3 odd 2
9216.2.a.x.1.2 4 48.35 even 4
9216.2.a.y.1.2 4 48.29 odd 4
9216.2.a.bn.1.3 4 48.11 even 4
9216.2.a.bo.1.3 4 48.5 odd 4