# Properties

 Label 3120.2.a.bc.1.2 Level $3120$ Weight $2$ Character 3120.1 Self dual yes Analytic conductor $24.913$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.9133254306$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3120.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -5.65685 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +8.48528 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.82843 q^{29} -4.00000 q^{31} +5.65685 q^{33} +2.82843 q^{35} -11.6569 q^{37} +1.00000 q^{39} -7.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -0.828427 q^{51} +13.3137 q^{53} -5.65685 q^{55} +2.82843 q^{57} +2.34315 q^{59} +6.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -5.65685 q^{67} -8.48528 q^{69} +5.65685 q^{71} -14.4853 q^{73} -1.00000 q^{75} -16.0000 q^{77} -2.34315 q^{79} +1.00000 q^{81} -6.34315 q^{83} +0.828427 q^{85} +8.82843 q^{87} -15.6569 q^{89} -2.82843 q^{91} +4.00000 q^{93} -2.82843 q^{95} -3.17157 q^{97} -5.65685 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} + 2q^{9} - 2q^{13} - 2q^{15} - 4q^{17} + 2q^{25} - 2q^{27} - 12q^{29} - 8q^{31} - 12q^{37} + 2q^{39} - 4q^{41} - 8q^{43} + 2q^{45} + 16q^{47} + 2q^{49} + 4q^{51} + 4q^{53} + 16q^{59} + 12q^{61} - 2q^{65} - 12q^{73} - 2q^{75} - 32q^{77} - 16q^{79} + 2q^{81} - 24q^{83} - 4q^{85} + 12q^{87} - 20q^{89} + 8q^{93} - 12q^{97} + O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.65685 −1.70561 −0.852803 0.522233i $$-0.825099\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 0.828427 0.200923 0.100462 0.994941i $$-0.467968\pi$$
0.100462 + 0.994941i $$0.467968\pi$$
$$18$$ 0 0
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ −2.82843 −0.617213
$$22$$ 0 0
$$23$$ 8.48528 1.76930 0.884652 0.466252i $$-0.154396\pi$$
0.884652 + 0.466252i $$0.154396\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −8.82843 −1.63940 −0.819699 0.572795i $$-0.805859\pi$$
−0.819699 + 0.572795i $$0.805859\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 5.65685 0.984732
$$34$$ 0 0
$$35$$ 2.82843 0.478091
$$36$$ 0 0
$$37$$ −11.6569 −1.91638 −0.958188 0.286141i $$-0.907627\pi$$
−0.958188 + 0.286141i $$0.907627\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −7.65685 −1.19580 −0.597900 0.801571i $$-0.703998\pi$$
−0.597900 + 0.801571i $$0.703998\pi$$
$$42$$ 0 0
$$43$$ −9.65685 −1.47266 −0.736328 0.676625i $$-0.763442\pi$$
−0.736328 + 0.676625i $$0.763442\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −0.828427 −0.116003
$$52$$ 0 0
$$53$$ 13.3137 1.82878 0.914389 0.404836i $$-0.132671\pi$$
0.914389 + 0.404836i $$0.132671\pi$$
$$54$$ 0 0
$$55$$ −5.65685 −0.762770
$$56$$ 0 0
$$57$$ 2.82843 0.374634
$$58$$ 0 0
$$59$$ 2.34315 0.305052 0.152526 0.988299i $$-0.451259\pi$$
0.152526 + 0.988299i $$0.451259\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 2.82843 0.356348
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −5.65685 −0.691095 −0.345547 0.938401i $$-0.612307\pi$$
−0.345547 + 0.938401i $$0.612307\pi$$
$$68$$ 0 0
$$69$$ −8.48528 −1.02151
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ −14.4853 −1.69537 −0.847687 0.530497i $$-0.822005\pi$$
−0.847687 + 0.530497i $$0.822005\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −16.0000 −1.82337
$$78$$ 0 0
$$79$$ −2.34315 −0.263624 −0.131812 0.991275i $$-0.542080\pi$$
−0.131812 + 0.991275i $$0.542080\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.34315 −0.696251 −0.348125 0.937448i $$-0.613182\pi$$
−0.348125 + 0.937448i $$0.613182\pi$$
$$84$$ 0 0
$$85$$ 0.828427 0.0898555
$$86$$ 0 0
$$87$$ 8.82843 0.946507
$$88$$ 0 0
$$89$$ −15.6569 −1.65962 −0.829812 0.558044i $$-0.811552\pi$$
−0.829812 + 0.558044i $$0.811552\pi$$
$$90$$ 0 0
$$91$$ −2.82843 −0.296500
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ −2.82843 −0.290191
$$96$$ 0 0
$$97$$ −3.17157 −0.322024 −0.161012 0.986952i $$-0.551476\pi$$
−0.161012 + 0.986952i $$0.551476\pi$$
$$98$$ 0 0
$$99$$ −5.65685 −0.568535
$$100$$ 0 0
$$101$$ 16.1421 1.60620 0.803101 0.595843i $$-0.203182\pi$$
0.803101 + 0.595843i $$0.203182\pi$$
$$102$$ 0 0
$$103$$ 1.65685 0.163255 0.0816274 0.996663i $$-0.473988\pi$$
0.0816274 + 0.996663i $$0.473988\pi$$
$$104$$ 0 0
$$105$$ −2.82843 −0.276026
$$106$$ 0 0
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ 8.82843 0.845610 0.422805 0.906221i $$-0.361046\pi$$
0.422805 + 0.906221i $$0.361046\pi$$
$$110$$ 0 0
$$111$$ 11.6569 1.10642
$$112$$ 0 0
$$113$$ 6.48528 0.610084 0.305042 0.952339i $$-0.401330\pi$$
0.305042 + 0.952339i $$0.401330\pi$$
$$114$$ 0 0
$$115$$ 8.48528 0.791257
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 21.0000 1.90909
$$122$$ 0 0
$$123$$ 7.65685 0.690395
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −9.65685 −0.856907 −0.428454 0.903564i $$-0.640941\pi$$
−0.428454 + 0.903564i $$0.640941\pi$$
$$128$$ 0 0
$$129$$ 9.65685 0.850239
$$130$$ 0 0
$$131$$ 6.14214 0.536641 0.268320 0.963330i $$-0.413531\pi$$
0.268320 + 0.963330i $$0.413531\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −17.3137 −1.47921 −0.739605 0.673041i $$-0.764988\pi$$
−0.739605 + 0.673041i $$0.764988\pi$$
$$138$$ 0 0
$$139$$ −6.34315 −0.538019 −0.269009 0.963138i $$-0.586696\pi$$
−0.269009 + 0.963138i $$0.586696\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ 5.65685 0.473050
$$144$$ 0 0
$$145$$ −8.82843 −0.733161
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 3.65685 0.299581 0.149791 0.988718i $$-0.452140\pi$$
0.149791 + 0.988718i $$0.452140\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 0.828427 0.0669744
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 5.31371 0.424080 0.212040 0.977261i $$-0.431989\pi$$
0.212040 + 0.977261i $$0.431989\pi$$
$$158$$ 0 0
$$159$$ −13.3137 −1.05585
$$160$$ 0 0
$$161$$ 24.0000 1.89146
$$162$$ 0 0
$$163$$ −11.3137 −0.886158 −0.443079 0.896483i $$-0.646114\pi$$
−0.443079 + 0.896483i $$0.646114\pi$$
$$164$$ 0 0
$$165$$ 5.65685 0.440386
$$166$$ 0 0
$$167$$ −8.97056 −0.694163 −0.347081 0.937835i $$-0.612827\pi$$
−0.347081 + 0.937835i $$0.612827\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.82843 −0.216295
$$172$$ 0 0
$$173$$ −9.31371 −0.708108 −0.354054 0.935225i $$-0.615197\pi$$
−0.354054 + 0.935225i $$0.615197\pi$$
$$174$$ 0 0
$$175$$ 2.82843 0.213809
$$176$$ 0 0
$$177$$ −2.34315 −0.176122
$$178$$ 0 0
$$179$$ −7.51472 −0.561676 −0.280838 0.959755i $$-0.590612\pi$$
−0.280838 + 0.959755i $$0.590612\pi$$
$$180$$ 0 0
$$181$$ −7.65685 −0.569129 −0.284565 0.958657i $$-0.591849\pi$$
−0.284565 + 0.958657i $$0.591849\pi$$
$$182$$ 0 0
$$183$$ −6.00000 −0.443533
$$184$$ 0 0
$$185$$ −11.6569 −0.857029
$$186$$ 0 0
$$187$$ −4.68629 −0.342696
$$188$$ 0 0
$$189$$ −2.82843 −0.205738
$$190$$ 0 0
$$191$$ −11.3137 −0.818631 −0.409316 0.912393i $$-0.634232\pi$$
−0.409316 + 0.912393i $$0.634232\pi$$
$$192$$ 0 0
$$193$$ 2.48528 0.178894 0.0894472 0.995992i $$-0.471490\pi$$
0.0894472 + 0.995992i $$0.471490\pi$$
$$194$$ 0 0
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ −13.3137 −0.948562 −0.474281 0.880373i $$-0.657292\pi$$
−0.474281 + 0.880373i $$0.657292\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ 5.65685 0.399004
$$202$$ 0 0
$$203$$ −24.9706 −1.75259
$$204$$ 0 0
$$205$$ −7.65685 −0.534778
$$206$$ 0 0
$$207$$ 8.48528 0.589768
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −0.686292 −0.0472463 −0.0236231 0.999721i $$-0.507520\pi$$
−0.0236231 + 0.999721i $$0.507520\pi$$
$$212$$ 0 0
$$213$$ −5.65685 −0.387601
$$214$$ 0 0
$$215$$ −9.65685 −0.658592
$$216$$ 0 0
$$217$$ −11.3137 −0.768025
$$218$$ 0 0
$$219$$ 14.4853 0.978825
$$220$$ 0 0
$$221$$ −0.828427 −0.0557260
$$222$$ 0 0
$$223$$ −10.8284 −0.725125 −0.362563 0.931959i $$-0.618098\pi$$
−0.362563 + 0.931959i $$0.618098\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ 4.14214 0.273720 0.136860 0.990590i $$-0.456299\pi$$
0.136860 + 0.990590i $$0.456299\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 0 0
$$233$$ 5.51472 0.361281 0.180641 0.983549i $$-0.442183\pi$$
0.180641 + 0.983549i $$0.442183\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ 2.34315 0.152204
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 5.31371 0.342286 0.171143 0.985246i $$-0.445254\pi$$
0.171143 + 0.985246i $$0.445254\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 2.82843 0.179969
$$248$$ 0 0
$$249$$ 6.34315 0.401981
$$250$$ 0 0
$$251$$ −10.8284 −0.683484 −0.341742 0.939794i $$-0.611017\pi$$
−0.341742 + 0.939794i $$0.611017\pi$$
$$252$$ 0 0
$$253$$ −48.0000 −3.01773
$$254$$ 0 0
$$255$$ −0.828427 −0.0518781
$$256$$ 0 0
$$257$$ −4.82843 −0.301189 −0.150595 0.988596i $$-0.548119\pi$$
−0.150595 + 0.988596i $$0.548119\pi$$
$$258$$ 0 0
$$259$$ −32.9706 −2.04869
$$260$$ 0 0
$$261$$ −8.82843 −0.546466
$$262$$ 0 0
$$263$$ −16.4853 −1.01653 −0.508263 0.861202i $$-0.669712\pi$$
−0.508263 + 0.861202i $$0.669712\pi$$
$$264$$ 0 0
$$265$$ 13.3137 0.817855
$$266$$ 0 0
$$267$$ 15.6569 0.958184
$$268$$ 0 0
$$269$$ −14.4853 −0.883183 −0.441592 0.897216i $$-0.645586\pi$$
−0.441592 + 0.897216i $$0.645586\pi$$
$$270$$ 0 0
$$271$$ −7.31371 −0.444276 −0.222138 0.975015i $$-0.571304\pi$$
−0.222138 + 0.975015i $$0.571304\pi$$
$$272$$ 0 0
$$273$$ 2.82843 0.171184
$$274$$ 0 0
$$275$$ −5.65685 −0.341121
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 8.34315 0.497710 0.248855 0.968541i $$-0.419946\pi$$
0.248855 + 0.968541i $$0.419946\pi$$
$$282$$ 0 0
$$283$$ 17.6569 1.04959 0.524796 0.851228i $$-0.324142\pi$$
0.524796 + 0.851228i $$0.324142\pi$$
$$284$$ 0 0
$$285$$ 2.82843 0.167542
$$286$$ 0 0
$$287$$ −21.6569 −1.27836
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 0 0
$$291$$ 3.17157 0.185921
$$292$$ 0 0
$$293$$ −16.6274 −0.971384 −0.485692 0.874130i $$-0.661432\pi$$
−0.485692 + 0.874130i $$0.661432\pi$$
$$294$$ 0 0
$$295$$ 2.34315 0.136423
$$296$$ 0 0
$$297$$ 5.65685 0.328244
$$298$$ 0 0
$$299$$ −8.48528 −0.490716
$$300$$ 0 0
$$301$$ −27.3137 −1.57434
$$302$$ 0 0
$$303$$ −16.1421 −0.927341
$$304$$ 0 0
$$305$$ 6.00000 0.343559
$$306$$ 0 0
$$307$$ 21.6569 1.23602 0.618011 0.786169i $$-0.287939\pi$$
0.618011 + 0.786169i $$0.287939\pi$$
$$308$$ 0 0
$$309$$ −1.65685 −0.0942551
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −30.9706 −1.75056 −0.875280 0.483617i $$-0.839323\pi$$
−0.875280 + 0.483617i $$0.839323\pi$$
$$314$$ 0 0
$$315$$ 2.82843 0.159364
$$316$$ 0 0
$$317$$ 25.3137 1.42176 0.710880 0.703314i $$-0.248297\pi$$
0.710880 + 0.703314i $$0.248297\pi$$
$$318$$ 0 0
$$319$$ 49.9411 2.79617
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ −2.34315 −0.130376
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −8.82843 −0.488213
$$328$$ 0 0
$$329$$ 22.6274 1.24749
$$330$$ 0 0
$$331$$ 8.48528 0.466393 0.233197 0.972430i $$-0.425081\pi$$
0.233197 + 0.972430i $$0.425081\pi$$
$$332$$ 0 0
$$333$$ −11.6569 −0.638792
$$334$$ 0 0
$$335$$ −5.65685 −0.309067
$$336$$ 0 0
$$337$$ 10.9706 0.597605 0.298802 0.954315i $$-0.403413\pi$$
0.298802 + 0.954315i $$0.403413\pi$$
$$338$$ 0 0
$$339$$ −6.48528 −0.352232
$$340$$ 0 0
$$341$$ 22.6274 1.22534
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ −8.48528 −0.456832
$$346$$ 0 0
$$347$$ 9.65685 0.518407 0.259204 0.965823i $$-0.416540\pi$$
0.259204 + 0.965823i $$0.416540\pi$$
$$348$$ 0 0
$$349$$ 12.1421 0.649954 0.324977 0.945722i $$-0.394644\pi$$
0.324977 + 0.945722i $$0.394644\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ 5.31371 0.282820 0.141410 0.989951i $$-0.454836\pi$$
0.141410 + 0.989951i $$0.454836\pi$$
$$354$$ 0 0
$$355$$ 5.65685 0.300235
$$356$$ 0 0
$$357$$ −2.34315 −0.124012
$$358$$ 0 0
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ −21.0000 −1.10221
$$364$$ 0 0
$$365$$ −14.4853 −0.758194
$$366$$ 0 0
$$367$$ −25.6569 −1.33928 −0.669638 0.742687i $$-0.733551\pi$$
−0.669638 + 0.742687i $$0.733551\pi$$
$$368$$ 0 0
$$369$$ −7.65685 −0.398600
$$370$$ 0 0
$$371$$ 37.6569 1.95505
$$372$$ 0 0
$$373$$ −2.68629 −0.139091 −0.0695455 0.997579i $$-0.522155\pi$$
−0.0695455 + 0.997579i $$0.522155\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 8.82843 0.454687
$$378$$ 0 0
$$379$$ 7.51472 0.386005 0.193003 0.981198i $$-0.438177\pi$$
0.193003 + 0.981198i $$0.438177\pi$$
$$380$$ 0 0
$$381$$ 9.65685 0.494736
$$382$$ 0 0
$$383$$ 29.6569 1.51539 0.757697 0.652606i $$-0.226324\pi$$
0.757697 + 0.652606i $$0.226324\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ 0 0
$$387$$ −9.65685 −0.490885
$$388$$ 0 0
$$389$$ −6.48528 −0.328817 −0.164408 0.986392i $$-0.552571\pi$$
−0.164408 + 0.986392i $$0.552571\pi$$
$$390$$ 0 0
$$391$$ 7.02944 0.355494
$$392$$ 0 0
$$393$$ −6.14214 −0.309830
$$394$$ 0 0
$$395$$ −2.34315 −0.117896
$$396$$ 0 0
$$397$$ 30.2843 1.51992 0.759962 0.649968i $$-0.225218\pi$$
0.759962 + 0.649968i $$0.225218\pi$$
$$398$$ 0 0
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −26.9706 −1.34685 −0.673423 0.739258i $$-0.735177\pi$$
−0.673423 + 0.739258i $$0.735177\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 65.9411 3.26858
$$408$$ 0 0
$$409$$ −3.65685 −0.180820 −0.0904099 0.995905i $$-0.528818\pi$$
−0.0904099 + 0.995905i $$0.528818\pi$$
$$410$$ 0 0
$$411$$ 17.3137 0.854022
$$412$$ 0 0
$$413$$ 6.62742 0.326114
$$414$$ 0 0
$$415$$ −6.34315 −0.311373
$$416$$ 0 0
$$417$$ 6.34315 0.310625
$$418$$ 0 0
$$419$$ 10.8284 0.529003 0.264502 0.964385i $$-0.414793\pi$$
0.264502 + 0.964385i $$0.414793\pi$$
$$420$$ 0 0
$$421$$ −24.1421 −1.17662 −0.588308 0.808637i $$-0.700206\pi$$
−0.588308 + 0.808637i $$0.700206\pi$$
$$422$$ 0 0
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ 0.828427 0.0401846
$$426$$ 0 0
$$427$$ 16.9706 0.821263
$$428$$ 0 0
$$429$$ −5.65685 −0.273115
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ −22.9706 −1.10389 −0.551947 0.833879i $$-0.686115\pi$$
−0.551947 + 0.833879i $$0.686115\pi$$
$$434$$ 0 0
$$435$$ 8.82843 0.423291
$$436$$ 0 0
$$437$$ −24.0000 −1.14808
$$438$$ 0 0
$$439$$ −22.6274 −1.07995 −0.539974 0.841682i $$-0.681566\pi$$
−0.539974 + 0.841682i $$0.681566\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −30.3431 −1.44165 −0.720823 0.693119i $$-0.756236\pi$$
−0.720823 + 0.693119i $$0.756236\pi$$
$$444$$ 0 0
$$445$$ −15.6569 −0.742206
$$446$$ 0 0
$$447$$ −3.65685 −0.172963
$$448$$ 0 0
$$449$$ 26.2843 1.24043 0.620216 0.784431i $$-0.287045\pi$$
0.620216 + 0.784431i $$0.287045\pi$$
$$450$$ 0 0
$$451$$ 43.3137 2.03956
$$452$$ 0 0
$$453$$ 12.0000 0.563809
$$454$$ 0 0
$$455$$ −2.82843 −0.132599
$$456$$ 0 0
$$457$$ 20.8284 0.974313 0.487156 0.873315i $$-0.338034\pi$$
0.487156 + 0.873315i $$0.338034\pi$$
$$458$$ 0 0
$$459$$ −0.828427 −0.0386677
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 0 0
$$463$$ 3.79899 0.176554 0.0882770 0.996096i $$-0.471864\pi$$
0.0882770 + 0.996096i $$0.471864\pi$$
$$464$$ 0 0
$$465$$ 4.00000 0.185496
$$466$$ 0 0
$$467$$ −7.31371 −0.338438 −0.169219 0.985578i $$-0.554125\pi$$
−0.169219 + 0.985578i $$0.554125\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −5.31371 −0.244843
$$472$$ 0 0
$$473$$ 54.6274 2.51177
$$474$$ 0 0
$$475$$ −2.82843 −0.129777
$$476$$ 0 0
$$477$$ 13.3137 0.609593
$$478$$ 0 0
$$479$$ 11.3137 0.516937 0.258468 0.966020i $$-0.416782\pi$$
0.258468 + 0.966020i $$0.416782\pi$$
$$480$$ 0 0
$$481$$ 11.6569 0.531507
$$482$$ 0 0
$$483$$ −24.0000 −1.09204
$$484$$ 0 0
$$485$$ −3.17157 −0.144014
$$486$$ 0 0
$$487$$ −16.4853 −0.747019 −0.373510 0.927626i $$-0.621846\pi$$
−0.373510 + 0.927626i $$0.621846\pi$$
$$488$$ 0 0
$$489$$ 11.3137 0.511624
$$490$$ 0 0
$$491$$ 38.1421 1.72133 0.860665 0.509171i $$-0.170048\pi$$
0.860665 + 0.509171i $$0.170048\pi$$
$$492$$ 0 0
$$493$$ −7.31371 −0.329393
$$494$$ 0 0
$$495$$ −5.65685 −0.254257
$$496$$ 0 0
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ −0.485281 −0.0217242 −0.0108621 0.999941i $$-0.503458\pi$$
−0.0108621 + 0.999941i $$0.503458\pi$$
$$500$$ 0 0
$$501$$ 8.97056 0.400775
$$502$$ 0 0
$$503$$ 23.5147 1.04847 0.524235 0.851574i $$-0.324351\pi$$
0.524235 + 0.851574i $$0.324351\pi$$
$$504$$ 0 0
$$505$$ 16.1421 0.718316
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ −37.3137 −1.65390 −0.826951 0.562275i $$-0.809926\pi$$
−0.826951 + 0.562275i $$0.809926\pi$$
$$510$$ 0 0
$$511$$ −40.9706 −1.81243
$$512$$ 0 0
$$513$$ 2.82843 0.124878
$$514$$ 0 0
$$515$$ 1.65685 0.0730097
$$516$$ 0 0
$$517$$ −45.2548 −1.99031
$$518$$ 0 0
$$519$$ 9.31371 0.408826
$$520$$ 0 0
$$521$$ 26.9706 1.18160 0.590801 0.806817i $$-0.298812\pi$$
0.590801 + 0.806817i $$0.298812\pi$$
$$522$$ 0 0
$$523$$ 10.6274 0.464704 0.232352 0.972632i $$-0.425358\pi$$
0.232352 + 0.972632i $$0.425358\pi$$
$$524$$ 0 0
$$525$$ −2.82843 −0.123443
$$526$$ 0 0
$$527$$ −3.31371 −0.144347
$$528$$ 0 0
$$529$$ 49.0000 2.13043
$$530$$ 0 0
$$531$$ 2.34315 0.101684
$$532$$ 0 0
$$533$$ 7.65685 0.331655
$$534$$ 0 0
$$535$$ 4.00000 0.172935
$$536$$ 0 0
$$537$$ 7.51472 0.324284
$$538$$ 0 0
$$539$$ −5.65685 −0.243658
$$540$$ 0 0
$$541$$ 14.4853 0.622771 0.311385 0.950284i $$-0.399207\pi$$
0.311385 + 0.950284i $$0.399207\pi$$
$$542$$ 0 0
$$543$$ 7.65685 0.328587
$$544$$ 0 0
$$545$$ 8.82843 0.378168
$$546$$ 0 0
$$547$$ −0.686292 −0.0293437 −0.0146719 0.999892i $$-0.504670\pi$$
−0.0146719 + 0.999892i $$0.504670\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 24.9706 1.06378
$$552$$ 0 0
$$553$$ −6.62742 −0.281826
$$554$$ 0 0
$$555$$ 11.6569 0.494806
$$556$$ 0 0
$$557$$ 10.6863 0.452793 0.226396 0.974035i $$-0.427306\pi$$
0.226396 + 0.974035i $$0.427306\pi$$
$$558$$ 0 0
$$559$$ 9.65685 0.408441
$$560$$ 0 0
$$561$$ 4.68629 0.197855
$$562$$ 0 0
$$563$$ 30.3431 1.27881 0.639406 0.768870i $$-0.279181\pi$$
0.639406 + 0.768870i $$0.279181\pi$$
$$564$$ 0 0
$$565$$ 6.48528 0.272838
$$566$$ 0 0
$$567$$ 2.82843 0.118783
$$568$$ 0 0
$$569$$ 31.6569 1.32712 0.663562 0.748121i $$-0.269044\pi$$
0.663562 + 0.748121i $$0.269044\pi$$
$$570$$ 0 0
$$571$$ −20.9706 −0.877591 −0.438795 0.898587i $$-0.644595\pi$$
−0.438795 + 0.898587i $$0.644595\pi$$
$$572$$ 0 0
$$573$$ 11.3137 0.472637
$$574$$ 0 0
$$575$$ 8.48528 0.353861
$$576$$ 0 0
$$577$$ −23.4558 −0.976480 −0.488240 0.872710i $$-0.662361\pi$$
−0.488240 + 0.872710i $$0.662361\pi$$
$$578$$ 0 0
$$579$$ −2.48528 −0.103285
$$580$$ 0 0
$$581$$ −17.9411 −0.744323
$$582$$ 0 0
$$583$$ −75.3137 −3.11918
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 0 0
$$587$$ −2.62742 −0.108445 −0.0542226 0.998529i $$-0.517268\pi$$
−0.0542226 + 0.998529i $$0.517268\pi$$
$$588$$ 0 0
$$589$$ 11.3137 0.466173
$$590$$ 0 0
$$591$$ 13.3137 0.547653
$$592$$ 0 0
$$593$$ −0.343146 −0.0140913 −0.00704565 0.999975i $$-0.502243\pi$$
−0.00704565 + 0.999975i $$0.502243\pi$$
$$594$$ 0 0
$$595$$ 2.34315 0.0960596
$$596$$ 0 0
$$597$$ 10.3431 0.423317
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 29.3137 1.19573 0.597866 0.801596i $$-0.296016\pi$$
0.597866 + 0.801596i $$0.296016\pi$$
$$602$$ 0 0
$$603$$ −5.65685 −0.230365
$$604$$ 0 0
$$605$$ 21.0000 0.853771
$$606$$ 0 0
$$607$$ −28.9706 −1.17588 −0.587939 0.808905i $$-0.700061\pi$$
−0.587939 + 0.808905i $$0.700061\pi$$
$$608$$ 0 0
$$609$$ 24.9706 1.01186
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ 22.2843 0.900053 0.450027 0.893015i $$-0.351414\pi$$
0.450027 + 0.893015i $$0.351414\pi$$
$$614$$ 0 0
$$615$$ 7.65685 0.308754
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ 34.8284 1.39987 0.699936 0.714205i $$-0.253212\pi$$
0.699936 + 0.714205i $$0.253212\pi$$
$$620$$ 0 0
$$621$$ −8.48528 −0.340503
$$622$$ 0 0
$$623$$ −44.2843 −1.77421
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −16.0000 −0.638978
$$628$$ 0 0
$$629$$ −9.65685 −0.385044
$$630$$ 0 0
$$631$$ 33.6569 1.33986 0.669929 0.742425i $$-0.266324\pi$$
0.669929 + 0.742425i $$0.266324\pi$$
$$632$$ 0 0
$$633$$ 0.686292 0.0272776
$$634$$ 0 0
$$635$$ −9.65685 −0.383221
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 5.65685 0.223782
$$640$$ 0 0
$$641$$ −4.62742 −0.182772 −0.0913860 0.995816i $$-0.529130\pi$$
−0.0913860 + 0.995816i $$0.529130\pi$$
$$642$$ 0 0
$$643$$ −39.5980 −1.56159 −0.780796 0.624786i $$-0.785186\pi$$
−0.780796 + 0.624786i $$0.785186\pi$$
$$644$$ 0 0
$$645$$ 9.65685 0.380238
$$646$$ 0 0
$$647$$ −8.48528 −0.333591 −0.166795 0.985992i $$-0.553342\pi$$
−0.166795 + 0.985992i $$0.553342\pi$$
$$648$$ 0 0
$$649$$ −13.2548 −0.520298
$$650$$ 0 0
$$651$$ 11.3137 0.443419
$$652$$ 0 0
$$653$$ −42.2843 −1.65471 −0.827356 0.561678i $$-0.810156\pi$$
−0.827356 + 0.561678i $$0.810156\pi$$
$$654$$ 0 0
$$655$$ 6.14214 0.239993
$$656$$ 0 0
$$657$$ −14.4853 −0.565125
$$658$$ 0 0
$$659$$ 7.51472 0.292732 0.146366 0.989231i $$-0.453242\pi$$
0.146366 + 0.989231i $$0.453242\pi$$
$$660$$ 0 0
$$661$$ −8.14214 −0.316692 −0.158346 0.987384i $$-0.550616\pi$$
−0.158346 + 0.987384i $$0.550616\pi$$
$$662$$ 0 0
$$663$$ 0.828427 0.0321734
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ −74.9117 −2.90059
$$668$$ 0 0
$$669$$ 10.8284 0.418651
$$670$$ 0 0
$$671$$ −33.9411 −1.31028
$$672$$ 0 0
$$673$$ 32.6274 1.25769 0.628847 0.777529i $$-0.283527\pi$$
0.628847 + 0.777529i $$0.283527\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 12.3431 0.474386 0.237193 0.971463i $$-0.423773\pi$$
0.237193 + 0.971463i $$0.423773\pi$$
$$678$$ 0 0
$$679$$ −8.97056 −0.344259
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ −33.6569 −1.28784 −0.643922 0.765091i $$-0.722694\pi$$
−0.643922 + 0.765091i $$0.722694\pi$$
$$684$$ 0 0
$$685$$ −17.3137 −0.661523
$$686$$ 0 0
$$687$$ −4.14214 −0.158032
$$688$$ 0 0
$$689$$ −13.3137 −0.507212
$$690$$ 0 0
$$691$$ 27.7990 1.05752 0.528762 0.848770i $$-0.322657\pi$$
0.528762 + 0.848770i $$0.322657\pi$$
$$692$$ 0 0
$$693$$ −16.0000 −0.607790
$$694$$ 0 0
$$695$$ −6.34315 −0.240609
$$696$$ 0 0
$$697$$ −6.34315 −0.240264
$$698$$ 0 0
$$699$$ −5.51472 −0.208586
$$700$$ 0 0
$$701$$ 0.142136 0.00536839 0.00268419 0.999996i $$-0.499146\pi$$
0.00268419 + 0.999996i $$0.499146\pi$$
$$702$$ 0 0
$$703$$ 32.9706 1.24351
$$704$$ 0 0
$$705$$ −8.00000 −0.301297
$$706$$ 0 0
$$707$$ 45.6569 1.71710
$$708$$ 0 0
$$709$$ −7.17157 −0.269334 −0.134667 0.990891i $$-0.542996\pi$$
−0.134667 + 0.990891i $$0.542996\pi$$
$$710$$ 0 0
$$711$$ −2.34315 −0.0878748
$$712$$ 0 0
$$713$$ −33.9411 −1.27111
$$714$$ 0 0
$$715$$ 5.65685 0.211554
$$716$$ 0 0
$$717$$ 16.0000 0.597531
$$718$$ 0 0
$$719$$ 29.6569 1.10601 0.553007 0.833177i $$-0.313480\pi$$
0.553007 + 0.833177i $$0.313480\pi$$
$$720$$ 0 0
$$721$$ 4.68629 0.174527
$$722$$ 0 0
$$723$$ −5.31371 −0.197619
$$724$$ 0 0
$$725$$ −8.82843 −0.327880
$$726$$ 0 0
$$727$$ 45.9411 1.70386 0.851931 0.523654i $$-0.175432\pi$$
0.851931 + 0.523654i $$0.175432\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −0.343146 −0.0126744 −0.00633719 0.999980i $$-0.502017\pi$$
−0.00633719 + 0.999980i $$0.502017\pi$$
$$734$$ 0 0
$$735$$ −1.00000 −0.0368856
$$736$$ 0 0
$$737$$ 32.0000 1.17874
$$738$$ 0 0
$$739$$ 14.1421 0.520227 0.260113 0.965578i $$-0.416240\pi$$
0.260113 + 0.965578i $$0.416240\pi$$
$$740$$ 0 0
$$741$$ −2.82843 −0.103905
$$742$$ 0 0
$$743$$ 36.2843 1.33114 0.665570 0.746335i $$-0.268188\pi$$
0.665570 + 0.746335i $$0.268188\pi$$
$$744$$ 0 0
$$745$$ 3.65685 0.133977
$$746$$ 0 0
$$747$$ −6.34315 −0.232084
$$748$$ 0 0
$$749$$ 11.3137 0.413394
$$750$$ 0 0
$$751$$ −11.3137 −0.412843 −0.206422 0.978463i $$-0.566182\pi$$
−0.206422 + 0.978463i $$0.566182\pi$$
$$752$$ 0 0
$$753$$ 10.8284 0.394610
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ 19.9411 0.724773 0.362386 0.932028i $$-0.381962\pi$$
0.362386 + 0.932028i $$0.381962\pi$$
$$758$$ 0 0
$$759$$ 48.0000 1.74229
$$760$$ 0 0
$$761$$ 27.6569 1.00256 0.501280 0.865285i $$-0.332863\pi$$
0.501280 + 0.865285i $$0.332863\pi$$
$$762$$ 0 0
$$763$$ 24.9706 0.903995
$$764$$ 0 0
$$765$$ 0.828427 0.0299518
$$766$$ 0 0
$$767$$ −2.34315 −0.0846061
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 4.82843 0.173892
$$772$$ 0 0
$$773$$ −53.3137 −1.91756 −0.958780 0.284148i $$-0.908289\pi$$
−0.958780 + 0.284148i $$0.908289\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 32.9706 1.18281
$$778$$ 0 0
$$779$$ 21.6569 0.775937
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ 8.82843 0.315502
$$784$$ 0 0
$$785$$ 5.31371 0.189654
$$786$$ 0 0
$$787$$ 24.0000 0.855508 0.427754 0.903895i $$-0.359305\pi$$
0.427754 + 0.903895i $$0.359305\pi$$
$$788$$ 0 0
$$789$$ 16.4853 0.586892
$$790$$ 0 0
$$791$$ 18.3431 0.652207
$$792$$ 0 0
$$793$$ −6.00000 −0.213066
$$794$$ 0 0
$$795$$ −13.3137 −0.472189
$$796$$ 0 0
$$797$$ 16.6274 0.588973 0.294487 0.955656i $$-0.404851\pi$$
0.294487 + 0.955656i $$0.404851\pi$$
$$798$$ 0 0
$$799$$ 6.62742 0.234461
$$800$$ 0 0
$$801$$ −15.6569 −0.553208
$$802$$ 0 0
$$803$$ 81.9411 2.89164
$$804$$ 0 0
$$805$$ 24.0000 0.845889
$$806$$ 0 0
$$807$$ 14.4853 0.509906
$$808$$ 0 0
$$809$$ 13.3137 0.468085 0.234043 0.972226i $$-0.424804\pi$$
0.234043 + 0.972226i $$0.424804\pi$$
$$810$$ 0 0
$$811$$ 1.85786 0.0652384 0.0326192 0.999468i $$-0.489615\pi$$
0.0326192 + 0.999468i $$0.489615\pi$$
$$812$$ 0 0
$$813$$ 7.31371 0.256503
$$814$$ 0 0
$$815$$ −11.3137 −0.396302
$$816$$ 0 0
$$817$$ 27.3137 0.955586
$$818$$ 0 0
$$819$$ −2.82843 −0.0988332
$$820$$ 0 0
$$821$$ 34.2843 1.19653 0.598265 0.801299i $$-0.295857\pi$$
0.598265 + 0.801299i $$0.295857\pi$$
$$822$$ 0 0
$$823$$ 52.9706 1.84644 0.923219 0.384275i $$-0.125548\pi$$
0.923219 + 0.384275i $$0.125548\pi$$
$$824$$ 0 0
$$825$$ 5.65685 0.196946
$$826$$ 0 0
$$827$$ −9.65685 −0.335802 −0.167901 0.985804i $$-0.553699\pi$$
−0.167901 + 0.985804i $$0.553699\pi$$
$$828$$ 0 0
$$829$$ −53.3137 −1.85166 −0.925831 0.377938i $$-0.876633\pi$$
−0.925831 + 0.377938i $$0.876633\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ 0.828427 0.0287033
$$834$$ 0 0
$$835$$ −8.97056 −0.310439
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 48.9411 1.68763
$$842$$ 0 0
$$843$$ −8.34315 −0.287353
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 59.3970 2.04090
$$848$$ 0 0
$$849$$ −17.6569 −0.605982
$$850$$ 0 0
$$851$$ −98.9117 −3.39065
$$852$$ 0 0
$$853$$ 38.2843 1.31083 0.655414 0.755270i $$-0.272494\pi$$
0.655414 + 0.755270i $$0.272494\pi$$
$$854$$ 0 0
$$855$$ −2.82843 −0.0967302
$$856$$ 0 0
$$857$$ −20.8284 −0.711486 −0.355743 0.934584i $$-0.615772\pi$$
−0.355743 + 0.934584i $$0.615772\pi$$
$$858$$ 0 0
$$859$$ −37.9411 −1.29453 −0.647267 0.762263i $$-0.724088\pi$$
−0.647267 + 0.762263i $$0.724088\pi$$
$$860$$ 0 0
$$861$$ 21.6569 0.738064
$$862$$ 0 0
$$863$$ 28.2843 0.962808 0.481404 0.876499i $$-0.340127\pi$$
0.481404 + 0.876499i $$0.340127\pi$$
$$864$$ 0 0
$$865$$ −9.31371 −0.316676
$$866$$ 0 0
$$867$$ 16.3137 0.554043
$$868$$ 0 0
$$869$$ 13.2548 0.449639
$$870$$ 0 0
$$871$$ 5.65685 0.191675
$$872$$ 0 0
$$873$$ −3.17157 −0.107341
$$874$$ 0 0
$$875$$ 2.82843 0.0956183
$$876$$ 0 0
$$877$$ −51.2548 −1.73075 −0.865376 0.501122i $$-0.832921\pi$$
−0.865376 + 0.501122i $$0.832921\pi$$
$$878$$ 0 0
$$879$$ 16.6274 0.560829
$$880$$ 0 0
$$881$$ −10.2843 −0.346486 −0.173243 0.984879i $$-0.555425\pi$$
−0.173243 + 0.984879i $$0.555425\pi$$
$$882$$ 0 0
$$883$$ −31.3137 −1.05379 −0.526895 0.849930i $$-0.676644\pi$$
−0.526895 + 0.849930i $$0.676644\pi$$
$$884$$ 0 0
$$885$$ −2.34315 −0.0787640
$$886$$ 0 0
$$887$$ 40.4853 1.35936 0.679681 0.733508i $$-0.262118\pi$$
0.679681 + 0.733508i $$0.262118\pi$$
$$888$$ 0 0
$$889$$ −27.3137 −0.916072
$$890$$ 0 0
$$891$$ −5.65685 −0.189512
$$892$$ 0 0
$$893$$ −22.6274 −0.757198
$$894$$ 0 0
$$895$$ −7.51472 −0.251189
$$896$$ 0 0
$$897$$ 8.48528 0.283315
$$898$$ 0 0
$$899$$ 35.3137 1.17778
$$900$$ 0 0
$$901$$ 11.0294 0.367444
$$902$$ 0 0
$$903$$ 27.3137 0.908943
$$904$$ 0 0
$$905$$ −7.65685 −0.254522
$$906$$ 0 0
$$907$$ −8.28427 −0.275075 −0.137537 0.990497i $$-0.543919\pi$$
−0.137537 + 0.990497i $$0.543919\pi$$
$$908$$ 0 0
$$909$$ 16.1421 0.535401
$$910$$ 0 0
$$911$$ −24.9706 −0.827312 −0.413656 0.910433i $$-0.635748\pi$$
−0.413656 + 0.910433i $$0.635748\pi$$
$$912$$ 0 0
$$913$$ 35.8823 1.18753
$$914$$ 0 0
$$915$$ −6.00000 −0.198354
$$916$$ 0 0
$$917$$ 17.3726 0.573693
$$918$$ 0 0
$$919$$ −41.9411 −1.38351 −0.691755 0.722132i $$-0.743162\pi$$
−0.691755 + 0.722132i $$0.743162\pi$$
$$920$$ 0 0
$$921$$ −21.6569 −0.713618
$$922$$ 0 0
$$923$$ −5.65685 −0.186198
$$924$$ 0 0
$$925$$ −11.6569 −0.383275
$$926$$ 0 0
$$927$$ 1.65685 0.0544182
$$928$$ 0 0
$$929$$ −33.5980 −1.10231 −0.551157 0.834402i $$-0.685813\pi$$
−0.551157 + 0.834402i $$0.685813\pi$$
$$930$$ 0 0
$$931$$ −2.82843 −0.0926980
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ −4.68629 −0.153258
$$936$$ 0 0
$$937$$ 16.6274 0.543194 0.271597 0.962411i $$-0.412448\pi$$
0.271597 + 0.962411i $$0.412448\pi$$
$$938$$ 0 0
$$939$$ 30.9706 1.01069
$$940$$ 0 0
$$941$$ 54.9706 1.79199 0.895995 0.444065i $$-0.146464\pi$$
0.895995 + 0.444065i $$0.146464\pi$$
$$942$$ 0 0
$$943$$ −64.9706 −2.11573
$$944$$ 0 0
$$945$$ −2.82843 −0.0920087
$$946$$ 0 0
$$947$$ −30.3431 −0.986020 −0.493010 0.870024i $$-0.664103\pi$$
−0.493010 + 0.870024i $$0.664103\pi$$
$$948$$ 0 0
$$949$$ 14.4853 0.470212
$$950$$ 0 0
$$951$$ −25.3137 −0.820853
$$952$$ 0 0
$$953$$ −27.8579 −0.902405 −0.451202 0.892422i $$-0.649005\pi$$
−0.451202 + 0.892422i $$0.649005\pi$$
$$954$$ 0 0
$$955$$ −11.3137 −0.366103
$$956$$ 0 0
$$957$$ −49.9411 −1.61437
$$958$$ 0 0
$$959$$ −48.9706 −1.58134
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 4.00000 0.128898
$$964$$ 0 0
$$965$$ 2.48528 0.0800040
$$966$$ 0 0
$$967$$ 7.51472 0.241657 0.120829 0.992673i $$-0.461445\pi$$
0.120829 + 0.992673i $$0.461445\pi$$
$$968$$ 0 0
$$969$$ 2.34315 0.0752727
$$970$$ 0 0
$$971$$ −15.5147 −0.497891 −0.248946 0.968517i $$-0.580084\pi$$
−0.248946 + 0.968517i $$0.580084\pi$$
$$972$$ 0 0
$$973$$ −17.9411 −0.575166
$$974$$ 0 0
$$975$$ 1.00000 0.0320256
$$976$$ 0 0
$$977$$ −8.34315 −0.266921 −0.133460 0.991054i $$-0.542609\pi$$
−0.133460 + 0.991054i $$0.542609\pi$$
$$978$$ 0 0
$$979$$ 88.5685 2.83066
$$980$$ 0 0
$$981$$ 8.82843 0.281870
$$982$$ 0 0
$$983$$ 2.34315 0.0747347 0.0373674 0.999302i $$-0.488103\pi$$
0.0373674 + 0.999302i $$0.488103\pi$$
$$984$$ 0 0
$$985$$ −13.3137 −0.424210
$$986$$ 0 0
$$987$$ −22.6274 −0.720239
$$988$$ 0 0
$$989$$ −81.9411 −2.60558
$$990$$ 0 0
$$991$$ 42.9117 1.36313 0.681567 0.731755i $$-0.261299\pi$$
0.681567 + 0.731755i $$0.261299\pi$$
$$992$$ 0 0
$$993$$ −8.48528 −0.269272
$$994$$ 0 0
$$995$$ −10.3431 −0.327900
$$996$$ 0 0
$$997$$ 61.3137 1.94182 0.970912 0.239435i $$-0.0769623\pi$$
0.970912 + 0.239435i $$0.0769623\pi$$
$$998$$ 0 0
$$999$$ 11.6569 0.368807
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 3120.2.a.bc.1.2 2
3.2 odd 2 9360.2.a.ch.1.2 2
4.3 odd 2 390.2.a.h.1.1 2
12.11 even 2 1170.2.a.o.1.1 2
20.3 even 4 1950.2.e.o.1249.2 4
20.7 even 4 1950.2.e.o.1249.3 4
20.19 odd 2 1950.2.a.bd.1.2 2
52.31 even 4 5070.2.b.q.1351.1 4
52.47 even 4 5070.2.b.q.1351.4 4
52.51 odd 2 5070.2.a.bc.1.2 2
60.23 odd 4 5850.2.e.bk.5149.4 4
60.47 odd 4 5850.2.e.bk.5149.1 4
60.59 even 2 5850.2.a.cl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 4.3 odd 2
1170.2.a.o.1.1 2 12.11 even 2
1950.2.a.bd.1.2 2 20.19 odd 2
1950.2.e.o.1249.2 4 20.3 even 4
1950.2.e.o.1249.3 4 20.7 even 4
3120.2.a.bc.1.2 2 1.1 even 1 trivial
5070.2.a.bc.1.2 2 52.51 odd 2
5070.2.b.q.1351.1 4 52.31 even 4
5070.2.b.q.1351.4 4 52.47 even 4
5850.2.a.cl.1.2 2 60.59 even 2
5850.2.e.bk.5149.1 4 60.47 odd 4
5850.2.e.bk.5149.4 4 60.23 odd 4
9360.2.a.ch.1.2 2 3.2 odd 2