# Properties

 Label 3072.2.a.i.1.2 Level $3072$ Weight $2$ Character 3072.1 Self dual yes Analytic conductor $24.530$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.5300435009$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.74912$$ of defining polynomial Character $$\chi$$ $$=$$ 3072.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -2.47363 q^{5} -2.55765 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -2.47363 q^{5} -2.55765 q^{7} +1.00000 q^{9} +0.669808 q^{11} -4.08402 q^{13} +2.47363 q^{15} +6.44549 q^{17} +6.44549 q^{19} +2.55765 q^{21} +2.82843 q^{23} +1.11882 q^{25} -1.00000 q^{27} -4.35480 q^{29} +6.55765 q^{31} -0.669808 q^{33} +6.32666 q^{35} +3.85970 q^{37} +4.08402 q^{39} +0.788632 q^{41} -0.550984 q^{43} -2.47363 q^{45} -2.82843 q^{47} -0.458440 q^{49} -6.44549 q^{51} -3.64520 q^{53} -1.65685 q^{55} -6.44549 q^{57} -5.65685 q^{59} -6.20285 q^{61} -2.55765 q^{63} +10.1023 q^{65} -2.99647 q^{67} -2.82843 q^{69} +5.11529 q^{71} -14.7721 q^{73} -1.11882 q^{75} -1.71313 q^{77} +6.32000 q^{79} +1.00000 q^{81} -0.907457 q^{83} -15.9437 q^{85} +4.35480 q^{87} -6.31724 q^{89} +10.4455 q^{91} -6.55765 q^{93} -15.9437 q^{95} +12.6533 q^{97} +0.669808 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 4q^{5} + 4q^{7} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 4q^{5} + 4q^{7} + 4q^{9} - 8q^{13} + 4q^{15} - 4q^{21} + 4q^{25} - 4q^{27} - 12q^{29} + 12q^{31} - 16q^{37} + 8q^{39} - 4q^{45} + 4q^{49} - 20q^{53} + 16q^{55} - 16q^{61} + 4q^{63} - 8q^{65} + 16q^{67} - 8q^{71} - 8q^{73} - 4q^{75} - 24q^{77} + 12q^{79} + 4q^{81} - 24q^{85} + 12q^{87} - 8q^{89} + 16q^{91} - 12q^{93} - 24q^{95} + O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −2.47363 −1.10624 −0.553120 0.833102i $$-0.686563\pi$$
−0.553120 + 0.833102i $$0.686563\pi$$
$$6$$ 0 0
$$7$$ −2.55765 −0.966700 −0.483350 0.875427i $$-0.660580\pi$$
−0.483350 + 0.875427i $$0.660580\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.669808 0.201955 0.100977 0.994889i $$-0.467803\pi$$
0.100977 + 0.994889i $$0.467803\pi$$
$$12$$ 0 0
$$13$$ −4.08402 −1.13270 −0.566352 0.824164i $$-0.691646\pi$$
−0.566352 + 0.824164i $$0.691646\pi$$
$$14$$ 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
$$19$$ 6.44549 1.47870 0.739348 0.673323i $$-0.235134\pi$$
0.739348 + 0.673323i $$0.235134\pi$$
$$20$$ 0 0
$$21$$ 2.55765 0.558124
$$22$$ 0 0
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ 1.11882 0.223765
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −4.35480 −0.808666 −0.404333 0.914612i $$-0.632496\pi$$
−0.404333 + 0.914612i $$0.632496\pi$$
$$30$$ 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
$$35$$ 6.32666 1.06940
$$36$$ 0 0
$$37$$ 3.85970 0.634531 0.317265 0.948337i $$-0.397235\pi$$
0.317265 + 0.948337i $$0.397235\pi$$
$$38$$ 0 0
$$39$$ 4.08402 0.653967
$$40$$ 0 0
$$41$$ 0.788632 0.123164 0.0615818 0.998102i $$-0.480385\pi$$
0.0615818 + 0.998102i $$0.480385\pi$$
$$42$$ 0 0
$$43$$ −0.550984 −0.0840242 −0.0420121 0.999117i $$-0.513377\pi$$
−0.0420121 + 0.999117i $$0.513377\pi$$
$$44$$ 0 0
$$45$$ −2.47363 −0.368746
$$46$$ 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
$$51$$ −6.44549 −0.902549
$$52$$ 0 0
$$53$$ −3.64520 −0.500707 −0.250353 0.968155i $$-0.580547\pi$$
−0.250353 + 0.968155i $$0.580547\pi$$
$$54$$ 0 0
$$55$$ −1.65685 −0.223410
$$56$$ 0 0
$$57$$ −6.44549 −0.853726
$$58$$ 0 0
$$59$$ −5.65685 −0.736460 −0.368230 0.929735i $$-0.620036\pi$$
−0.368230 + 0.929735i $$0.620036\pi$$
$$60$$ 0 0
$$61$$ −6.20285 −0.794193 −0.397097 0.917777i $$-0.629982\pi$$
−0.397097 + 0.917777i $$0.629982\pi$$
$$62$$ 0 0
$$63$$ −2.55765 −0.322233
$$64$$ 0 0
$$65$$ 10.1023 1.25304
$$66$$ 0 0
$$67$$ −2.99647 −0.366077 −0.183039 0.983106i $$-0.558593\pi$$
−0.183039 + 0.983106i $$0.558593\pi$$
$$68$$ 0 0
$$69$$ −2.82843 −0.340503
$$70$$ 0 0
$$71$$ 5.11529 0.607074 0.303537 0.952820i $$-0.401832\pi$$
0.303537 + 0.952820i $$0.401832\pi$$
$$72$$ 0 0
$$73$$ −14.7721 −1.72895 −0.864475 0.502676i $$-0.832349\pi$$
−0.864475 + 0.502676i $$0.832349\pi$$
$$74$$ 0 0
$$75$$ −1.11882 −0.129191
$$76$$ 0 0
$$77$$ −1.71313 −0.195230
$$78$$ 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
$$83$$ −0.907457 −0.0996063 −0.0498032 0.998759i $$-0.515859\pi$$
−0.0498032 + 0.998759i $$0.515859\pi$$
$$84$$ 0 0
$$85$$ −15.9437 −1.72934
$$86$$ 0 0
$$87$$ 4.35480 0.466884
$$88$$ 0 0
$$89$$ −6.31724 −0.669626 −0.334813 0.942285i $$-0.608673\pi$$
−0.334813 + 0.942285i $$0.608673\pi$$
$$90$$ 0 0
$$91$$ 10.4455 1.09498
$$92$$ 0 0
$$93$$ −6.55765 −0.679996
$$94$$ 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
$$99$$ 0.669808 0.0673182
$$100$$ 0 0
$$101$$ −10.6417 −1.05889 −0.529443 0.848346i $$-0.677599\pi$$
−0.529443 + 0.848346i $$0.677599\pi$$
$$102$$ 0 0
$$103$$ 3.33686 0.328790 0.164395 0.986395i $$-0.447433\pi$$
0.164395 + 0.986395i $$0.447433\pi$$
$$104$$ 0 0
$$105$$ −6.32666 −0.617419
$$106$$ 0 0
$$107$$ −19.8874 −1.92259 −0.961296 0.275518i $$-0.911151\pi$$
−0.961296 + 0.275518i $$0.911151\pi$$
$$108$$ 0 0
$$109$$ −3.91598 −0.375083 −0.187541 0.982257i $$-0.560052\pi$$
−0.187541 + 0.982257i $$0.560052\pi$$
$$110$$ 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
$$115$$ −6.99647 −0.652424
$$116$$ 0 0
$$117$$ −4.08402 −0.377568
$$118$$ 0 0
$$119$$ −16.4853 −1.51120
$$120$$ 0 0
$$121$$ −10.5514 −0.959214
$$122$$ 0 0
$$123$$ −0.788632 −0.0711086
$$124$$ 0 0
$$125$$ 9.60058 0.858702
$$126$$ 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
$$131$$ −5.33962 −0.466524 −0.233262 0.972414i $$-0.574940\pi$$
−0.233262 + 0.972414i $$0.574940\pi$$
$$132$$ 0 0
$$133$$ −16.4853 −1.42946
$$134$$ 0 0
$$135$$ 2.47363 0.212896
$$136$$ 0 0
$$137$$ −5.10587 −0.436224 −0.218112 0.975924i $$-0.569990\pi$$
−0.218112 + 0.975924i $$0.569990\pi$$
$$138$$ 0 0
$$139$$ 16.6533 1.41252 0.706258 0.707954i $$-0.250382\pi$$
0.706258 + 0.707954i $$0.250382\pi$$
$$140$$ 0 0
$$141$$ 2.82843 0.238197
$$142$$ 0 0
$$143$$ −2.73551 −0.228755
$$144$$ 0 0
$$145$$ 10.7721 0.894578
$$146$$ 0 0
$$147$$ 0.458440 0.0378115
$$148$$ 0 0
$$149$$ −11.1832 −0.916166 −0.458083 0.888909i $$-0.651464\pi$$
−0.458083 + 0.888909i $$0.651464\pi$$
$$150$$ 0 0
$$151$$ 14.6506 1.19225 0.596123 0.802893i $$-0.296707\pi$$
0.596123 + 0.802893i $$0.296707\pi$$
$$152$$ 0 0
$$153$$ 6.44549 0.521087
$$154$$ 0 0
$$155$$ −16.2212 −1.30292
$$156$$ 0 0
$$157$$ −4.45754 −0.355750 −0.177875 0.984053i $$-0.556922\pi$$
−0.177875 + 0.984053i $$0.556922\pi$$
$$158$$ 0 0
$$159$$ 3.64520 0.289083
$$160$$ 0 0
$$161$$ −7.23412 −0.570128
$$162$$ 0 0
$$163$$ 7.78510 0.609776 0.304888 0.952388i $$-0.401381\pi$$
0.304888 + 0.952388i $$0.401381\pi$$
$$164$$ 0 0
$$165$$ 1.65685 0.128986
$$166$$ 0 0
$$167$$ −20.1814 −1.56168 −0.780841 0.624730i $$-0.785209\pi$$
−0.780841 + 0.624730i $$0.785209\pi$$
$$168$$ 0 0
$$169$$ 3.67923 0.283018
$$170$$ 0 0
$$171$$ 6.44549 0.492899
$$172$$ 0 0
$$173$$ −6.15639 −0.468061 −0.234031 0.972229i $$-0.575192\pi$$
−0.234031 + 0.972229i $$0.575192\pi$$
$$174$$ 0 0
$$175$$ −2.86156 −0.216313
$$176$$ 0 0
$$177$$ 5.65685 0.425195
$$178$$ 0 0
$$179$$ 18.7855 1.40409 0.702046 0.712131i $$-0.252270\pi$$
0.702046 + 0.712131i $$0.252270\pi$$
$$180$$ 0 0
$$181$$ 8.97499 0.667106 0.333553 0.942731i $$-0.391752\pi$$
0.333553 + 0.942731i $$0.391752\pi$$
$$182$$ 0 0
$$183$$ 6.20285 0.458528
$$184$$ 0 0
$$185$$ −9.54745 −0.701943
$$186$$ 0 0
$$187$$ 4.31724 0.315708
$$188$$ 0 0
$$189$$ 2.55765 0.186041
$$190$$ 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
$$195$$ −10.1023 −0.723444
$$196$$ 0 0
$$197$$ −1.75070 −0.124732 −0.0623659 0.998053i $$-0.519865\pi$$
−0.0623659 + 0.998053i $$0.519865\pi$$
$$198$$ 0 0
$$199$$ −0.993710 −0.0704422 −0.0352211 0.999380i $$-0.511214\pi$$
−0.0352211 + 0.999380i $$0.511214\pi$$
$$200$$ 0 0
$$201$$ 2.99647 0.211355
$$202$$ 0 0
$$203$$ 11.1380 0.781738
$$204$$ 0 0
$$205$$ −1.95078 −0.136248
$$206$$ 0 0
$$207$$ 2.82843 0.196589
$$208$$ 0 0
$$209$$ 4.31724 0.298630
$$210$$ 0 0
$$211$$ −5.97409 −0.411273 −0.205637 0.978628i $$-0.565927\pi$$
−0.205637 + 0.978628i $$0.565927\pi$$
$$212$$ 0 0
$$213$$ −5.11529 −0.350494
$$214$$ 0 0
$$215$$ 1.36293 0.0929509
$$216$$ 0 0
$$217$$ −16.7721 −1.13857
$$218$$ 0 0
$$219$$ 14.7721 0.998209
$$220$$ 0 0
$$221$$ −26.3235 −1.77071
$$222$$ 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
$$227$$ −0.907457 −0.0602300 −0.0301150 0.999546i $$-0.509587\pi$$
−0.0301150 + 0.999546i $$0.509587\pi$$
$$228$$ 0 0
$$229$$ −7.55579 −0.499301 −0.249650 0.968336i $$-0.580316\pi$$
−0.249650 + 0.968336i $$0.580316\pi$$
$$230$$ 0 0
$$231$$ 1.71313 0.112716
$$232$$ 0 0
$$233$$ −23.2271 −1.52166 −0.760828 0.648954i $$-0.775207\pi$$
−0.760828 + 0.648954i $$0.775207\pi$$
$$234$$ 0 0
$$235$$ 6.99647 0.456399
$$236$$ 0 0
$$237$$ −6.32000 −0.410528
$$238$$ 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
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 1.13401 0.0724492
$$246$$ 0 0
$$247$$ −26.3235 −1.67492
$$248$$ 0 0
$$249$$ 0.907457 0.0575077
$$250$$ 0 0
$$251$$ 13.7984 0.870949 0.435475 0.900201i $$-0.356581\pi$$
0.435475 + 0.900201i $$0.356581\pi$$
$$252$$ 0 0
$$253$$ 1.89450 0.119106
$$254$$ 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
$$259$$ −9.87175 −0.613401
$$260$$ 0 0
$$261$$ −4.35480 −0.269555
$$262$$ 0 0
$$263$$ −29.9929 −1.84944 −0.924722 0.380643i $$-0.875703\pi$$
−0.924722 + 0.380643i $$0.875703\pi$$
$$264$$ 0 0
$$265$$ 9.01686 0.553901
$$266$$ 0 0
$$267$$ 6.31724 0.386609
$$268$$ 0 0
$$269$$ −29.1332 −1.77628 −0.888142 0.459569i $$-0.848004\pi$$
−0.888142 + 0.459569i $$0.848004\pi$$
$$270$$ 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
$$275$$ 0.749397 0.0451904
$$276$$ 0 0
$$277$$ 17.1430 1.03003 0.515013 0.857183i $$-0.327787\pi$$
0.515013 + 0.857183i $$0.327787\pi$$
$$278$$ 0 0
$$279$$ 6.55765 0.392596
$$280$$ 0 0
$$281$$ 2.76588 0.164999 0.0824993 0.996591i $$-0.473710\pi$$
0.0824993 + 0.996591i $$0.473710\pi$$
$$282$$ 0 0
$$283$$ −6.34315 −0.377061 −0.188530 0.982067i $$-0.560372\pi$$
−0.188530 + 0.982067i $$0.560372\pi$$
$$284$$ 0 0
$$285$$ 15.9437 0.944425
$$286$$ 0 0
$$287$$ −2.01704 −0.119062
$$288$$ 0 0
$$289$$ 24.5443 1.44378
$$290$$ 0 0
$$291$$ −12.6533 −0.741751
$$292$$ 0 0
$$293$$ 11.6078 0.678133 0.339067 0.940762i $$-0.389889\pi$$
0.339067 + 0.940762i $$0.389889\pi$$
$$294$$ 0 0
$$295$$ 13.9929 0.814700
$$296$$ 0 0
$$297$$ −0.669808 −0.0388662
$$298$$ 0 0
$$299$$ −11.5514 −0.668032
$$300$$ 0 0
$$301$$ 1.40922 0.0812262
$$302$$ 0 0
$$303$$ 10.6417 0.611348
$$304$$ 0 0
$$305$$ 15.3435 0.878567
$$306$$ 0 0
$$307$$ −14.7855 −0.843852 −0.421926 0.906630i $$-0.638646\pi$$
−0.421926 + 0.906630i $$0.638646\pi$$
$$308$$ 0 0
$$309$$ −3.33686 −0.189827
$$310$$ 0 0
$$311$$ −15.0761 −0.854885 −0.427442 0.904043i $$-0.640585\pi$$
−0.427442 + 0.904043i $$0.640585\pi$$
$$312$$ 0 0
$$313$$ 23.0027 1.30019 0.650096 0.759852i $$-0.274729\pi$$
0.650096 + 0.759852i $$0.274729\pi$$
$$314$$ 0 0
$$315$$ 6.32666 0.356467
$$316$$ 0 0
$$317$$ −9.55855 −0.536862 −0.268431 0.963299i $$-0.586505\pi$$
−0.268431 + 0.963299i $$0.586505\pi$$
$$318$$ 0 0
$$319$$ −2.91688 −0.163314
$$320$$ 0 0
$$321$$ 19.8874 1.11001
$$322$$ 0 0
$$323$$ 41.5443 2.31159
$$324$$ 0 0
$$325$$ −4.56930 −0.253459
$$326$$ 0 0
$$327$$ 3.91598 0.216554
$$328$$ 0 0
$$329$$ 7.23412 0.398830
$$330$$ 0 0
$$331$$ −27.8079 −1.52846 −0.764229 0.644945i $$-0.776880\pi$$
−0.764229 + 0.644945i $$0.776880\pi$$
$$332$$ 0 0
$$333$$ 3.85970 0.211510
$$334$$ 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
$$339$$ 2.23765 0.121532
$$340$$ 0 0
$$341$$ 4.39236 0.237860
$$342$$ 0 0
$$343$$ 19.0761 1.03001
$$344$$ 0 0
$$345$$ 6.99647 0.376677
$$346$$ 0 0
$$347$$ −8.87449 −0.476408 −0.238204 0.971215i $$-0.576559\pi$$
−0.238204 + 0.971215i $$0.576559\pi$$
$$348$$ 0 0
$$349$$ −6.70698 −0.359016 −0.179508 0.983757i $$-0.557451\pi$$
−0.179508 + 0.983757i $$0.557451\pi$$
$$350$$ 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
$$355$$ −12.6533 −0.671569
$$356$$ 0 0
$$357$$ 16.4853 0.872494
$$358$$ 0 0
$$359$$ 32.7917 1.73068 0.865341 0.501184i $$-0.167102\pi$$
0.865341 + 0.501184i $$0.167102\pi$$
$$360$$ 0 0
$$361$$ 22.5443 1.18654
$$362$$ 0 0
$$363$$ 10.5514 0.553803
$$364$$ 0 0
$$365$$ 36.5408 1.91263
$$366$$ 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
$$371$$ 9.32313 0.484033
$$372$$ 0 0
$$373$$ −23.4995 −1.21676 −0.608379 0.793646i $$-0.708180\pi$$
−0.608379 + 0.793646i $$0.708180\pi$$
$$374$$ 0 0
$$375$$ −9.60058 −0.495772
$$376$$ 0 0
$$377$$ 17.7851 0.915979
$$378$$ 0 0
$$379$$ −11.0004 −0.565051 −0.282526 0.959260i $$-0.591172\pi$$
−0.282526 + 0.959260i $$0.591172\pi$$
$$380$$ 0 0
$$381$$ −12.2145 −0.625768
$$382$$ 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
$$387$$ −0.550984 −0.0280081
$$388$$ 0 0
$$389$$ 33.7311 1.71023 0.855116 0.518436i $$-0.173486\pi$$
0.855116 + 0.518436i $$0.173486\pi$$
$$390$$ 0 0
$$391$$ 18.2306 0.921961
$$392$$ 0 0
$$393$$ 5.33962 0.269348
$$394$$ 0 0
$$395$$ −15.6333 −0.786597
$$396$$ 0 0
$$397$$ 14.5201 0.728742 0.364371 0.931254i $$-0.381284\pi$$
0.364371 + 0.931254i $$0.381284\pi$$
$$398$$ 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
$$403$$ −26.7816 −1.33409
$$404$$ 0 0
$$405$$ −2.47363 −0.122915
$$406$$ 0 0
$$407$$ 2.58526 0.128146
$$408$$ 0 0
$$409$$ −11.5702 −0.572110 −0.286055 0.958213i $$-0.592344\pi$$
−0.286055 + 0.958213i $$0.592344\pi$$
$$410$$ 0 0
$$411$$ 5.10587 0.251854
$$412$$ 0 0
$$413$$ 14.4682 0.711935
$$414$$ 0 0
$$415$$ 2.24471 0.110188
$$416$$ 0 0
$$417$$ −16.6533 −0.815517
$$418$$ 0 0
$$419$$ 9.54193 0.466154 0.233077 0.972458i $$-0.425121\pi$$
0.233077 + 0.972458i $$0.425121\pi$$
$$420$$ 0 0
$$421$$ 24.3583 1.18715 0.593576 0.804778i $$-0.297716\pi$$
0.593576 + 0.804778i $$0.297716\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ 7.21137 0.349803
$$426$$ 0 0
$$427$$ 15.8647 0.767746
$$428$$ 0 0
$$429$$ 2.73551 0.132072
$$430$$ 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
$$435$$ −10.7721 −0.516485
$$436$$ 0 0
$$437$$ 18.2306 0.872087
$$438$$ 0 0
$$439$$ 17.7122 0.845356 0.422678 0.906280i $$-0.361090\pi$$
0.422678 + 0.906280i $$0.361090\pi$$
$$440$$ 0 0
$$441$$ −0.458440 −0.0218305
$$442$$ 0 0
$$443$$ −22.1953 −1.05453 −0.527264 0.849701i $$-0.676782\pi$$
−0.527264 + 0.849701i $$0.676782\pi$$
$$444$$ 0 0
$$445$$ 15.6265 0.740766
$$446$$ 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
$$451$$ 0.528232 0.0248735
$$452$$ 0 0
$$453$$ −14.6506 −0.688344
$$454$$ 0 0
$$455$$ −25.8382 −1.21131
$$456$$ 0 0
$$457$$ −17.3396 −0.811113 −0.405557 0.914070i $$-0.632922\pi$$
−0.405557 + 0.914070i $$0.632922\pi$$
$$458$$ 0 0
$$459$$ −6.44549 −0.300850
$$460$$ 0 0
$$461$$ 2.39404 0.111501 0.0557507 0.998445i $$-0.482245\pi$$
0.0557507 + 0.998445i $$0.482245\pi$$
$$462$$ 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
$$467$$ 24.2023 1.11995 0.559975 0.828510i $$-0.310811\pi$$
0.559975 + 0.828510i $$0.310811\pi$$
$$468$$ 0 0
$$469$$ 7.66391 0.353887
$$470$$ 0 0
$$471$$ 4.45754 0.205393
$$472$$ 0 0
$$473$$ −0.369053 −0.0169691
$$474$$ 0 0
$$475$$ 7.21137 0.330880
$$476$$ 0 0
$$477$$ −3.64520 −0.166902
$$478$$ 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
$$483$$ 7.23412 0.329164
$$484$$ 0 0
$$485$$ −31.2996 −1.42124
$$486$$ 0 0
$$487$$ 13.9839 0.633672 0.316836 0.948480i $$-0.397380\pi$$
0.316836 + 0.948480i $$0.397380\pi$$
$$488$$ 0 0
$$489$$ −7.78510 −0.352055
$$490$$ 0 0
$$491$$ 10.2306 0.461700 0.230850 0.972989i $$-0.425849\pi$$
0.230850 + 0.972989i $$0.425849\pi$$
$$492$$ 0 0
$$493$$ −28.0688 −1.26416
$$494$$ 0 0
$$495$$ −1.65685 −0.0744701
$$496$$ 0 0
$$497$$ −13.0831 −0.586858
$$498$$ 0 0
$$499$$ −3.66391 −0.164019 −0.0820097 0.996632i $$-0.526134\pi$$
−0.0820097 + 0.996632i $$0.526134\pi$$
$$500$$ 0 0
$$501$$ 20.1814 0.901637
$$502$$ 0 0
$$503$$ −39.6443 −1.76765 −0.883825 0.467817i $$-0.845041\pi$$
−0.883825 + 0.467817i $$0.845041\pi$$
$$504$$ 0 0
$$505$$ 26.3235 1.17138
$$506$$ 0 0
$$507$$ −3.67923 −0.163400
$$508$$ 0 0
$$509$$ 28.6909 1.27170 0.635851 0.771812i $$-0.280649\pi$$
0.635851 + 0.771812i $$0.280649\pi$$
$$510$$ 0 0
$$511$$ 37.7819 1.67137
$$512$$ 0 0
$$513$$ −6.44549 −0.284575
$$514$$ 0 0
$$515$$ −8.25413 −0.363721
$$516$$ 0 0
$$517$$ −1.89450 −0.0833201
$$518$$ 0 0
$$519$$ 6.15639 0.270235
$$520$$ 0 0
$$521$$ 23.1784 1.01546 0.507732 0.861515i $$-0.330484\pi$$
0.507732 + 0.861515i $$0.330484\pi$$
$$522$$ 0 0
$$523$$ −8.18193 −0.357771 −0.178885 0.983870i $$-0.557249\pi$$
−0.178885 + 0.983870i $$0.557249\pi$$
$$524$$ 0 0
$$525$$ 2.86156 0.124889
$$526$$ 0 0
$$527$$ 42.2672 1.84119
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ −5.65685 −0.245487
$$532$$ 0 0
$$533$$ −3.22079 −0.139508
$$534$$ 0 0
$$535$$ 49.1941 2.12685
$$536$$ 0 0
$$537$$ −18.7855 −0.810653
$$538$$ 0 0
$$539$$ −0.307067 −0.0132263
$$540$$ 0 0
$$541$$ 6.43715 0.276755 0.138377 0.990380i $$-0.455811\pi$$
0.138377 + 0.990380i $$0.455811\pi$$
$$542$$ 0 0
$$543$$ −8.97499 −0.385154
$$544$$ 0 0
$$545$$ 9.68667 0.414931
$$546$$ 0 0
$$547$$ −39.2239 −1.67709 −0.838546 0.544830i $$-0.816594\pi$$
−0.838546 + 0.544830i $$0.816594\pi$$
$$548$$ 0 0
$$549$$ −6.20285 −0.264731
$$550$$ 0 0
$$551$$ −28.0688 −1.19577
$$552$$ 0 0
$$553$$ −16.1643 −0.687377
$$554$$ 0 0
$$555$$ 9.54745 0.405267
$$556$$ 0 0
$$557$$ 1.66224 0.0704315 0.0352157 0.999380i $$-0.488788\pi$$
0.0352157 + 0.999380i $$0.488788\pi$$
$$558$$ 0 0
$$559$$ 2.25023 0.0951745
$$560$$ 0 0
$$561$$ −4.31724 −0.182274
$$562$$ 0 0
$$563$$ 40.6368 1.71264 0.856319 0.516447i $$-0.172746\pi$$
0.856319 + 0.516447i $$0.172746\pi$$
$$564$$ 0 0
$$565$$ 5.53511 0.232864
$$566$$ 0 0
$$567$$ −2.55765 −0.107411
$$568$$ 0 0
$$569$$ −27.0004 −1.13191 −0.565957 0.824435i $$-0.691493\pi$$
−0.565957 + 0.824435i $$0.691493\pi$$
$$570$$ 0 0
$$571$$ 20.9706 0.877591 0.438795 0.898587i $$-0.355405\pi$$
0.438795 + 0.898587i $$0.355405\pi$$
$$572$$ 0 0
$$573$$ −5.60058 −0.233967
$$574$$ 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
$$579$$ 19.4514 0.808372
$$580$$ 0 0
$$581$$ 2.32095 0.0962894
$$582$$ 0 0
$$583$$ −2.44158 −0.101120
$$584$$ 0 0
$$585$$ 10.1023 0.417680
$$586$$ 0 0
$$587$$ −44.2047 −1.82452 −0.912261 0.409609i $$-0.865665\pi$$
−0.912261 + 0.409609i $$0.865665\pi$$
$$588$$ 0 0
$$589$$ 42.2672 1.74159
$$590$$ 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
$$595$$ 40.7784 1.67175
$$596$$ 0 0
$$597$$ 0.993710 0.0406698
$$598$$ 0 0
$$599$$ 22.0296 0.900104 0.450052 0.893002i $$-0.351405\pi$$
0.450052 + 0.893002i $$0.351405\pi$$
$$600$$ 0 0
$$601$$ 10.7721 0.439405 0.219703 0.975567i $$-0.429491\pi$$
0.219703 + 0.975567i $$0.429491\pi$$
$$602$$ 0 0
$$603$$ −2.99647 −0.122026
$$604$$ 0 0
$$605$$ 26.1001 1.06112
$$606$$ 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
$$611$$ 11.5514 0.467318
$$612$$ 0 0
$$613$$ 14.8562 0.600035 0.300018 0.953934i $$-0.403007\pi$$
0.300018 + 0.953934i $$0.403007\pi$$
$$614$$ 0 0
$$615$$ 1.95078 0.0786631
$$616$$ 0 0
$$617$$ 22.2235 0.894686 0.447343 0.894363i $$-0.352370\pi$$
0.447343 + 0.894363i $$0.352370\pi$$
$$618$$ 0 0
$$619$$ 16.4612 0.661631 0.330815 0.943696i $$-0.392676\pi$$
0.330815 + 0.943696i $$0.392676\pi$$
$$620$$ 0 0
$$621$$ −2.82843 −0.113501
$$622$$ 0 0
$$623$$ 16.1573 0.647327
$$624$$ 0 0
$$625$$ −29.3424 −1.17369
$$626$$ 0 0
$$627$$ −4.31724 −0.172414
$$628$$ 0 0
$$629$$ 24.8776 0.991937
$$630$$ 0 0
$$631$$ −4.06977 −0.162015 −0.0810075 0.996713i $$-0.525814\pi$$
−0.0810075 + 0.996713i $$0.525814\pi$$
$$632$$ 0 0
$$633$$ 5.97409 0.237449
$$634$$ 0 0
$$635$$ −30.2141 −1.19901
$$636$$ 0 0
$$637$$ 1.87228 0.0741824
$$638$$ 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
$$643$$ −10.4266 −0.411186 −0.205593 0.978638i $$-0.565912\pi$$
−0.205593 + 0.978638i $$0.565912\pi$$
$$644$$ 0 0
$$645$$ −1.36293 −0.0536652
$$646$$ 0 0
$$647$$ 11.6132 0.456560 0.228280 0.973595i $$-0.426690\pi$$
0.228280 + 0.973595i $$0.426690\pi$$
$$648$$ 0 0
$$649$$ −3.78901 −0.148731
$$650$$ 0 0
$$651$$ 16.7721 0.657352
$$652$$ 0 0
$$653$$ 2.73012 0.106838 0.0534190 0.998572i $$-0.482988\pi$$
0.0534190 + 0.998572i $$0.482988\pi$$
$$654$$ 0 0
$$655$$ 13.2082 0.516088
$$656$$ 0 0
$$657$$ −14.7721 −0.576316
$$658$$ 0 0
$$659$$ −31.5514 −1.22907 −0.614533 0.788891i $$-0.710656\pi$$
−0.614533 + 0.788891i $$0.710656\pi$$
$$660$$ 0 0
$$661$$ 15.1368 0.588752 0.294376 0.955690i $$-0.404888\pi$$
0.294376 + 0.955690i $$0.404888\pi$$
$$662$$ 0 0
$$663$$ 26.3235 1.02232
$$664$$ 0 0
$$665$$ 40.7784 1.58132
$$666$$ 0 0
$$667$$ −12.3172 −0.476925
$$668$$ 0 0
$$669$$ 23.7659 0.918841
$$670$$ 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
$$675$$ −1.11882 −0.0430636
$$676$$ 0 0
$$677$$ −37.9357 −1.45799 −0.728994 0.684520i $$-0.760012\pi$$
−0.728994 + 0.684520i $$0.760012\pi$$
$$678$$ 0 0
$$679$$ −32.3627 −1.24197
$$680$$ 0 0
$$681$$ 0.907457 0.0347738
$$682$$ 0 0
$$683$$ 18.2471 0.698205 0.349102 0.937085i $$-0.386487\pi$$
0.349102 + 0.937085i $$0.386487\pi$$
$$684$$ 0 0
$$685$$ 12.6300 0.482568
$$686$$ 0 0
$$687$$ 7.55579 0.288271
$$688$$ 0 0
$$689$$ 14.8871 0.567152
$$690$$ 0 0
$$691$$ 30.2533 1.15089 0.575446 0.817840i $$-0.304829\pi$$
0.575446 + 0.817840i $$0.304829\pi$$
$$692$$ 0 0
$$693$$ −1.71313 −0.0650765
$$694$$ 0 0
$$695$$ −41.1941 −1.56258
$$696$$ 0 0
$$697$$ 5.08312 0.192537
$$698$$ 0 0
$$699$$ 23.2271 0.878528
$$700$$ 0 0
$$701$$ 20.0875 0.758696 0.379348 0.925254i $$-0.376148\pi$$
0.379348 + 0.925254i $$0.376148\pi$$
$$702$$ 0 0
$$703$$ 24.8776 0.938278
$$704$$ 0 0
$$705$$ −6.99647 −0.263502
$$706$$ 0 0
$$707$$ 27.2176 1.02362
$$708$$ 0 0
$$709$$ 41.7864 1.56932 0.784660 0.619926i $$-0.212837\pi$$
0.784660 + 0.619926i $$0.212837\pi$$
$$710$$ 0 0
$$711$$ 6.32000 0.237018
$$712$$ 0 0
$$713$$ 18.5478 0.694622
$$714$$ 0 0
$$715$$ 6.76663 0.253058
$$716$$ 0 0
$$717$$ 26.9213 1.00540
$$718$$ 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
$$723$$ −10.3494 −0.384899
$$724$$ 0 0
$$725$$ −4.87226 −0.180951
$$726$$ 0 0
$$727$$ −20.4843 −0.759722 −0.379861 0.925044i $$-0.624028\pi$$
−0.379861 + 0.925044i $$0.624028\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −3.55136 −0.131352
$$732$$ 0 0
$$733$$ 48.0777 1.77579 0.887895 0.460045i $$-0.152167\pi$$
0.887895 + 0.460045i $$0.152167\pi$$
$$734$$ 0 0
$$735$$ −1.13401 −0.0418286
$$736$$ 0 0
$$737$$ −2.00706 −0.0739310
$$738$$ 0 0
$$739$$ 21.4459 0.788899 0.394449 0.918918i $$-0.370935\pi$$
0.394449 + 0.918918i $$0.370935\pi$$
$$740$$ 0 0
$$741$$ 26.3235 0.967018
$$742$$ 0 0
$$743$$ 2.17431 0.0797677 0.0398839 0.999204i $$-0.487301\pi$$
0.0398839 + 0.999204i $$0.487301\pi$$
$$744$$ 0 0
$$745$$ 27.6631 1.01350
$$746$$ 0 0
$$747$$ −0.907457 −0.0332021
$$748$$ 0 0
$$749$$ 50.8651 1.85857
$$750$$ 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
$$755$$ −36.2400 −1.31891
$$756$$ 0 0
$$757$$ −21.6791 −0.787939 −0.393970 0.919123i $$-0.628899\pi$$
−0.393970 + 0.919123i $$0.628899\pi$$
$$758$$ 0 0
$$759$$ −1.89450 −0.0687661
$$760$$ 0 0
$$761$$ 4.29449 0.155675 0.0778375 0.996966i $$-0.475198\pi$$
0.0778375 + 0.996966i $$0.475198\pi$$
$$762$$ 0 0
$$763$$ 10.0157 0.362592
$$764$$ 0 0
$$765$$ −15.9437 −0.576446
$$766$$ 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
$$771$$ −16.9965 −0.612113
$$772$$ 0 0
$$773$$ 49.5300 1.78147 0.890736 0.454521i $$-0.150190\pi$$
0.890736 + 0.454521i $$0.150190\pi$$
$$774$$ 0 0
$$775$$ 7.33686 0.263548
$$776$$ 0 0
$$777$$ 9.87175 0.354147
$$778$$ 0 0
$$779$$ 5.08312 0.182122
$$780$$ 0 0
$$781$$ 3.42627 0.122601
$$782$$ 0 0
$$783$$ 4.35480 0.155628
$$784$$ 0 0
$$785$$ 11.0263 0.393545
$$786$$ 0 0
$$787$$ −34.0953 −1.21537 −0.607683 0.794180i $$-0.707901\pi$$
−0.607683 + 0.794180i $$0.707901\pi$$
$$788$$ 0 0
$$789$$ 29.9929 1.06778
$$790$$ 0 0
$$791$$ 5.72312 0.203491
$$792$$ 0 0
$$793$$ 25.3326 0.899585
$$794$$ 0 0
$$795$$ −9.01686 −0.319795
$$796$$ 0 0
$$797$$ −40.6901 −1.44132 −0.720659 0.693290i $$-0.756160\pi$$
−0.720659 + 0.693290i $$0.756160\pi$$
$$798$$ 0 0
$$799$$ −18.2306 −0.644952
$$800$$ 0 0
$$801$$ −6.31724 −0.223209
$$802$$ 0 0
$$803$$ −9.89450 −0.349169
$$804$$ 0 0
$$805$$ 17.8945 0.630698
$$806$$ 0 0
$$807$$ 29.1332 1.02554
$$808$$ 0 0
$$809$$ −10.9926 −0.386478 −0.193239 0.981152i $$-0.561899\pi$$
−0.193239 + 0.981152i $$0.561899\pi$$
$$810$$ 0 0
$$811$$ −21.2498 −0.746182 −0.373091 0.927795i $$-0.621702\pi$$
−0.373091 + 0.927795i $$0.621702\pi$$
$$812$$ 0 0
$$813$$ −26.6506 −0.934676
$$814$$ 0 0
$$815$$ −19.2574 −0.674558
$$816$$ 0 0
$$817$$ −3.55136 −0.124246
$$818$$ 0 0
$$819$$ 10.4455 0.364995
$$820$$ 0 0
$$821$$ −30.0572 −1.04900 −0.524501 0.851410i $$-0.675748\pi$$
−0.524501 + 0.851410i $$0.675748\pi$$
$$822$$ 0 0
$$823$$ −55.0851 −1.92015 −0.960073 0.279751i $$-0.909748\pi$$
−0.960073 + 0.279751i $$0.909748\pi$$
$$824$$ 0 0
$$825$$ −0.749397 −0.0260907
$$826$$ 0 0
$$827$$ 34.5478 1.20135 0.600673 0.799495i $$-0.294899\pi$$
0.600673 + 0.799495i $$0.294899\pi$$
$$828$$ 0 0
$$829$$ −31.0046 −1.07683 −0.538417 0.842679i $$-0.680977\pi$$
−0.538417 + 0.842679i $$0.680977\pi$$
$$830$$ 0 0
$$831$$ −17.1430 −0.594685
$$832$$ 0 0
$$833$$ −2.95487 −0.102380
$$834$$ 0 0
$$835$$ 49.9212 1.72759
$$836$$ 0 0
$$837$$ −6.55765 −0.226665
$$838$$ 0 0
$$839$$ 5.14195 0.177520 0.0887599 0.996053i $$-0.471710\pi$$
0.0887599 + 0.996053i $$0.471710\pi$$
$$840$$ 0 0
$$841$$ −10.0357 −0.346059
$$842$$ 0 0
$$843$$ −2.76588 −0.0952620
$$844$$ 0 0
$$845$$ −9.10104 −0.313085
$$846$$ 0 0
$$847$$ 26.9867 0.927272
$$848$$ 0 0
$$849$$ 6.34315 0.217696
$$850$$ 0 0
$$851$$ 10.9169 0.374226
$$852$$ 0 0
$$853$$ 18.5060 0.633632 0.316816 0.948487i $$-0.397386\pi$$
0.316816 + 0.948487i $$0.397386\pi$$
$$854$$ 0 0
$$855$$ −15.9437 −0.545264
$$856$$ 0 0
$$857$$ 22.8878 0.781833 0.390916 0.920426i $$-0.372158\pi$$
0.390916 + 0.920426i $$0.372158\pi$$
$$858$$ 0 0
$$859$$ 35.5286 1.21222 0.606110 0.795381i $$-0.292729\pi$$
0.606110 + 0.795381i $$0.292729\pi$$
$$860$$ 0 0
$$861$$ 2.01704 0.0687407
$$862$$ 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
$$867$$ −24.5443 −0.833568
$$868$$ 0 0
$$869$$ 4.23319 0.143601
$$870$$ 0 0
$$871$$ 12.2376 0.414657
$$872$$ 0 0
$$873$$ 12.6533 0.428250
$$874$$ 0 0
$$875$$ −24.5549 −0.830107
$$876$$ 0 0
$$877$$ −21.5773 −0.728614 −0.364307 0.931279i $$-0.618694\pi$$
−0.364307 + 0.931279i $$0.618694\pi$$
$$878$$ 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
$$883$$ 0.0834930 0.00280976 0.00140488 0.999999i $$-0.499553\pi$$
0.00140488 + 0.999999i $$0.499553\pi$$
$$884$$ 0 0
$$885$$ −13.9929 −0.470368
$$886$$ 0 0
$$887$$ 30.8043 1.03431 0.517154 0.855892i $$-0.326991\pi$$
0.517154 + 0.855892i $$0.326991\pi$$
$$888$$ 0 0
$$889$$ −31.2404 −1.04777
$$890$$ 0 0
$$891$$ 0.669808 0.0224394
$$892$$ 0 0
$$893$$ −18.2306 −0.610063
$$894$$ 0 0
$$895$$ −46.4682 −1.55326
$$896$$ 0 0
$$897$$ 11.5514 0.385689
$$898$$ 0 0
$$899$$ −28.5573 −0.952438
$$900$$ 0 0
$$901$$ −23.4951 −0.782735
$$902$$ 0 0
$$903$$ −1.40922 −0.0468960
$$904$$ 0 0
$$905$$ −22.2008 −0.737979
$$906$$ 0 0
$$907$$ 49.5215 1.64434 0.822168 0.569245i $$-0.192764\pi$$
0.822168 + 0.569245i $$0.192764\pi$$
$$908$$ 0 0
$$909$$ −10.6417 −0.352962
$$910$$ 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
$$915$$ −15.3435 −0.507241
$$916$$ 0 0
$$917$$ 13.6569 0.450989
$$918$$ 0 0
$$919$$ 20.1161 0.663568 0.331784 0.943355i $$-0.392350\pi$$
0.331784 + 0.943355i $$0.392350\pi$$
$$920$$ 0 0
$$921$$ 14.7855 0.487198
$$922$$ 0 0
$$923$$ −20.8910 −0.687635
$$924$$ 0 0
$$925$$ 4.31833 0.141986
$$926$$ 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
$$931$$ −2.95487 −0.0968420
$$932$$ 0 0
$$933$$ 15.0761 0.493568
$$934$$ 0 0
$$935$$ −10.6792 −0.349248
$$936$$ 0 0
$$937$$ 33.5780 1.09695 0.548473 0.836168i $$-0.315209\pi$$
0.548473 + 0.836168i $$0.315209\pi$$
$$938$$ 0 0
$$939$$ −23.0027 −0.750666
$$940$$ 0 0
$$941$$ 11.9991 0.391159 0.195579 0.980688i $$-0.437341\pi$$
0.195579 + 0.980688i $$0.437341\pi$$
$$942$$ 0 0
$$943$$ 2.23059 0.0726380
$$944$$ 0 0
$$945$$ −6.32666 −0.205806
$$946$$ 0 0
$$947$$ −25.2537 −0.820635 −0.410318 0.911943i $$-0.634582\pi$$
−0.410318 + 0.911943i $$0.634582\pi$$
$$948$$ 0 0
$$949$$ 60.3298 1.95839
$$950$$ 0 0
$$951$$ 9.55855 0.309957
$$952$$ 0 0
$$953$$ 3.86469 0.125190 0.0625948 0.998039i $$-0.480062\pi$$
0.0625948 + 0.998039i $$0.480062\pi$$
$$954$$ 0 0
$$955$$ −13.8537 −0.448296
$$956$$ 0 0
$$957$$ 2.91688 0.0942894
$$958$$ 0 0
$$959$$ 13.0590 0.421698
$$960$$ 0 0
$$961$$ 12.0027 0.387185
$$962$$ 0 0
$$963$$ −19.8874 −0.640864
$$964$$ 0 0
$$965$$ 48.1154 1.54889
$$966$$ 0 0
$$967$$ −37.8714 −1.21786 −0.608930 0.793224i $$-0.708401\pi$$
−0.608930 + 0.793224i $$0.708401\pi$$
$$968$$ 0 0
$$969$$ −41.5443 −1.33460
$$970$$ 0 0
$$971$$ 3.74587 0.120211 0.0601053 0.998192i $$-0.480856\pi$$
0.0601053 + 0.998192i $$0.480856\pi$$
$$972$$ 0 0
$$973$$ −42.5933 −1.36548
$$974$$ 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
$$979$$ −4.23134 −0.135234
$$980$$ 0 0
$$981$$ −3.91598 −0.125028
$$982$$ 0 0
$$983$$ 22.3557 0.713035 0.356518 0.934289i $$-0.383964\pi$$
0.356518 + 0.934289i $$0.383964\pi$$
$$984$$ 0 0
$$985$$ 4.33057 0.137983
$$986$$ 0 0
$$987$$ −7.23412 −0.230265
$$988$$ 0 0
$$989$$ −1.55842 −0.0495548
$$990$$ 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
$$995$$ 2.45807 0.0779260
$$996$$ 0 0
$$997$$ −6.06146 −0.191968 −0.0959841 0.995383i $$-0.530600\pi$$
−0.0959841 + 0.995383i $$0.530600\pi$$
$$998$$ 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.a.i.1.2 4
3.2 odd 2 9216.2.a.bo.1.3 4
4.3 odd 2 3072.2.a.o.1.2 4
8.3 odd 2 3072.2.a.n.1.3 4
8.5 even 2 3072.2.a.t.1.3 4
12.11 even 2 9216.2.a.bn.1.3 4
16.3 odd 4 3072.2.d.i.1537.7 8
16.5 even 4 3072.2.d.f.1537.6 8
16.11 odd 4 3072.2.d.i.1537.2 8
16.13 even 4 3072.2.d.f.1537.3 8
24.5 odd 2 9216.2.a.y.1.2 4
24.11 even 2 9216.2.a.x.1.2 4
32.3 odd 8 192.2.j.a.145.4 8
32.5 even 8 384.2.j.b.97.3 8
32.11 odd 8 192.2.j.a.49.4 8
32.13 even 8 384.2.j.b.289.3 8
32.19 odd 8 384.2.j.a.289.1 8
32.21 even 8 48.2.j.a.37.2 yes 8
32.27 odd 8 384.2.j.a.97.1 8
32.29 even 8 48.2.j.a.13.2 8
96.5 odd 8 1152.2.k.c.865.4 8
96.11 even 8 576.2.k.b.433.1 8
96.29 odd 8 144.2.k.b.109.3 8
96.35 even 8 576.2.k.b.145.1 8
96.53 odd 8 144.2.k.b.37.3 8
96.59 even 8 1152.2.k.f.865.4 8
96.77 odd 8 1152.2.k.c.289.4 8
96.83 even 8 1152.2.k.f.289.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.2 8 32.29 even 8
48.2.j.a.37.2 yes 8 32.21 even 8
144.2.k.b.37.3 8 96.53 odd 8
144.2.k.b.109.3 8 96.29 odd 8
192.2.j.a.49.4 8 32.11 odd 8
192.2.j.a.145.4 8 32.3 odd 8
384.2.j.a.97.1 8 32.27 odd 8
384.2.j.a.289.1 8 32.19 odd 8
384.2.j.b.97.3 8 32.5 even 8
384.2.j.b.289.3 8 32.13 even 8
576.2.k.b.145.1 8 96.35 even 8
576.2.k.b.433.1 8 96.11 even 8
1152.2.k.c.289.4 8 96.77 odd 8
1152.2.k.c.865.4 8 96.5 odd 8
1152.2.k.f.289.4 8 96.83 even 8
1152.2.k.f.865.4 8 96.59 even 8
3072.2.a.i.1.2 4 1.1 even 1 trivial
3072.2.a.n.1.3 4 8.3 odd 2
3072.2.a.o.1.2 4 4.3 odd 2
3072.2.a.t.1.3 4 8.5 even 2
3072.2.d.f.1537.3 8 16.13 even 4
3072.2.d.f.1537.6 8 16.5 even 4
3072.2.d.i.1537.2 8 16.11 odd 4
3072.2.d.i.1537.7 8 16.3 odd 4
9216.2.a.x.1.2 4 24.11 even 2
9216.2.a.y.1.2 4 24.5 odd 2
9216.2.a.bn.1.3 4 12.11 even 2
9216.2.a.bo.1.3 4 3.2 odd 2