# Properties

 Label 2736.2.k.n.2431.1 Level $2736$ Weight $2$ Character 2736.2431 Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2431.1 Root $$-1.18614 + 1.26217i$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.2431 Dual form 2736.2.k.n.2431.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.37228 q^{5} -2.52434i q^{7} +O(q^{10})$$ $$q-2.37228 q^{5} -2.52434i q^{7} +2.52434i q^{11} -1.58457i q^{13} +0.372281 q^{17} +(-4.00000 - 1.73205i) q^{19} +1.87953i q^{23} +0.627719 q^{25} -3.16915i q^{29} +2.74456 q^{31} +5.98844i q^{35} -1.58457i q^{37} +6.92820i q^{41} +0.644810i q^{43} +0.939764i q^{47} +0.627719 q^{49} +10.0974i q^{53} -5.98844i q^{55} -4.00000 q^{59} -0.372281 q^{61} +3.75906i q^{65} -13.4891 q^{67} -4.00000 q^{71} +13.1168 q^{73} +6.37228 q^{77} +6.74456 q^{79} +3.46410i q^{83} -0.883156 q^{85} +13.2665i q^{89} -4.00000 q^{91} +(9.48913 + 4.10891i) q^{95} +13.2665i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + O(q^{10})$$ $$4q + 2q^{5} - 10q^{17} - 16q^{19} + 14q^{25} - 12q^{31} + 14q^{49} - 16q^{59} + 10q^{61} - 8q^{67} - 16q^{71} + 18q^{73} + 14q^{77} + 4q^{79} - 38q^{85} - 16q^{91} - 8q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −2.37228 −1.06092 −0.530458 0.847711i $$-0.677980\pi$$
−0.530458 + 0.847711i $$0.677980\pi$$
$$6$$ 0 0
$$7$$ 2.52434i 0.954110i −0.878873 0.477055i $$-0.841704\pi$$
0.878873 0.477055i $$-0.158296\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.52434i 0.761116i 0.924757 + 0.380558i $$0.124268\pi$$
−0.924757 + 0.380558i $$0.875732\pi$$
$$12$$ 0 0
$$13$$ 1.58457i 0.439482i −0.975558 0.219741i $$-0.929479\pi$$
0.975558 0.219741i $$-0.0705212\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.372281 0.0902915 0.0451457 0.998980i $$-0.485625\pi$$
0.0451457 + 0.998980i $$0.485625\pi$$
$$18$$ 0 0
$$19$$ −4.00000 1.73205i −0.917663 0.397360i
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.87953i 0.391909i 0.980613 + 0.195954i $$0.0627804\pi$$
−0.980613 + 0.195954i $$0.937220\pi$$
$$24$$ 0 0
$$25$$ 0.627719 0.125544
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.16915i 0.588496i −0.955729 0.294248i $$-0.904931\pi$$
0.955729 0.294248i $$-0.0950692\pi$$
$$30$$ 0 0
$$31$$ 2.74456 0.492938 0.246469 0.969151i $$-0.420730\pi$$
0.246469 + 0.969151i $$0.420730\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 5.98844i 1.01223i
$$36$$ 0 0
$$37$$ 1.58457i 0.260502i −0.991481 0.130251i $$-0.958422\pi$$
0.991481 0.130251i $$-0.0415784\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820i 1.08200i 0.841021 + 0.541002i $$0.181955\pi$$
−0.841021 + 0.541002i $$0.818045\pi$$
$$42$$ 0 0
$$43$$ 0.644810i 0.0983326i 0.998791 + 0.0491663i $$0.0156564\pi$$
−0.998791 + 0.0491663i $$0.984344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0.939764i 0.137079i 0.997648 + 0.0685393i $$0.0218339\pi$$
−0.997648 + 0.0685393i $$0.978166\pi$$
$$48$$ 0 0
$$49$$ 0.627719 0.0896741
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.0974i 1.38698i 0.720467 + 0.693489i $$0.243927\pi$$
−0.720467 + 0.693489i $$0.756073\pi$$
$$54$$ 0 0
$$55$$ 5.98844i 0.807481i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −0.372281 −0.0476657 −0.0238329 0.999716i $$-0.507587\pi$$
−0.0238329 + 0.999716i $$0.507587\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.75906i 0.466253i
$$66$$ 0 0
$$67$$ −13.4891 −1.64796 −0.823979 0.566620i $$-0.808251\pi$$
−0.823979 + 0.566620i $$0.808251\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 13.1168 1.53521 0.767605 0.640923i $$-0.221448\pi$$
0.767605 + 0.640923i $$0.221448\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.37228 0.726189
$$78$$ 0 0
$$79$$ 6.74456 0.758823 0.379411 0.925228i $$-0.376127\pi$$
0.379411 + 0.925228i $$0.376127\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.46410i 0.380235i 0.981761 + 0.190117i $$0.0608868\pi$$
−0.981761 + 0.190117i $$0.939113\pi$$
$$84$$ 0 0
$$85$$ −0.883156 −0.0957917
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 13.2665i 1.40625i 0.711068 + 0.703123i $$0.248212\pi$$
−0.711068 + 0.703123i $$0.751788\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 9.48913 + 4.10891i 0.973564 + 0.421565i
$$96$$ 0 0
$$97$$ 13.2665i 1.34701i 0.739183 + 0.673504i $$0.235212\pi$$
−0.739183 + 0.673504i $$0.764788\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.25544 0.323928 0.161964 0.986797i $$-0.448217\pi$$
0.161964 + 0.986797i $$0.448217\pi$$
$$102$$ 0 0
$$103$$ 1.25544 0.123702 0.0618510 0.998085i $$-0.480300\pi$$
0.0618510 + 0.998085i $$0.480300\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −13.4891 −1.30404 −0.652021 0.758200i $$-0.726079\pi$$
−0.652021 + 0.758200i $$0.726079\pi$$
$$108$$ 0 0
$$109$$ 11.6819i 1.11893i 0.828855 + 0.559463i $$0.188993\pi$$
−0.828855 + 0.559463i $$0.811007\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 17.0256i 1.60163i 0.598912 + 0.800815i $$0.295600\pi$$
−0.598912 + 0.800815i $$0.704400\pi$$
$$114$$ 0 0
$$115$$ 4.45877i 0.415782i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.939764i 0.0861480i
$$120$$ 0 0
$$121$$ 4.62772 0.420702
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 10.3723 0.927725
$$126$$ 0 0
$$127$$ −1.25544 −0.111402 −0.0557010 0.998447i $$-0.517739\pi$$
−0.0557010 + 0.998447i $$0.517739\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 7.57301i 0.661657i −0.943691 0.330829i $$-0.892672\pi$$
0.943691 0.330829i $$-0.107328\pi$$
$$132$$ 0 0
$$133$$ −4.37228 + 10.0974i −0.379125 + 0.875551i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −13.1168 −1.12065 −0.560324 0.828274i $$-0.689323\pi$$
−0.560324 + 0.828274i $$0.689323\pi$$
$$138$$ 0 0
$$139$$ 17.6704i 1.49878i 0.662128 + 0.749390i $$0.269653\pi$$
−0.662128 + 0.749390i $$0.730347\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 7.51811i 0.624345i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 15.1168 1.23842 0.619210 0.785225i $$-0.287453\pi$$
0.619210 + 0.785225i $$0.287453\pi$$
$$150$$ 0 0
$$151$$ −2.74456 −0.223349 −0.111675 0.993745i $$-0.535621\pi$$
−0.111675 + 0.993745i $$0.535621\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.51087 −0.522966
$$156$$ 0 0
$$157$$ −15.4891 −1.23617 −0.618083 0.786113i $$-0.712091\pi$$
−0.618083 + 0.786113i $$0.712091\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4.74456 0.373924
$$162$$ 0 0
$$163$$ 19.8997i 1.55867i 0.626608 + 0.779334i $$0.284443\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −5.48913 −0.424761 −0.212381 0.977187i $$-0.568122\pi$$
−0.212381 + 0.977187i $$0.568122\pi$$
$$168$$ 0 0
$$169$$ 10.4891 0.806856
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.33830i 0.481892i −0.970539 0.240946i $$-0.922542\pi$$
0.970539 0.240946i $$-0.0774576\pi$$
$$174$$ 0 0
$$175$$ 1.58457i 0.119783i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −21.4891 −1.60617 −0.803086 0.595863i $$-0.796810\pi$$
−0.803086 + 0.595863i $$0.796810\pi$$
$$180$$ 0 0
$$181$$ 25.5383i 1.89825i 0.314901 + 0.949125i $$0.398029\pi$$
−0.314901 + 0.949125i $$0.601971\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 3.75906i 0.276371i
$$186$$ 0 0
$$187$$ 0.939764i 0.0687223i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 26.1831i 1.89455i −0.320428 0.947273i $$-0.603827\pi$$
0.320428 0.947273i $$-0.396173\pi$$
$$192$$ 0 0
$$193$$ 13.8564i 0.997406i −0.866773 0.498703i $$-0.833810\pi$$
0.866773 0.498703i $$-0.166190\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.7446 1.19300 0.596500 0.802613i $$-0.296557\pi$$
0.596500 + 0.802613i $$0.296557\pi$$
$$198$$ 0 0
$$199$$ 18.2603i 1.29444i 0.762305 + 0.647218i $$0.224068\pi$$
−0.762305 + 0.647218i $$0.775932\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ 16.4356i 1.14792i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.37228 10.0974i 0.302437 0.698448i
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1.52967i 0.104323i
$$216$$ 0 0
$$217$$ 6.92820i 0.470317i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.589907i 0.0396815i
$$222$$ 0 0
$$223$$ 10.7446 0.719509 0.359755 0.933047i $$-0.382860\pi$$
0.359755 + 0.933047i $$0.382860\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −6.88316 −0.454852 −0.227426 0.973795i $$-0.573031\pi$$
−0.227426 + 0.973795i $$0.573031\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.1168 −0.859313 −0.429657 0.902992i $$-0.641365\pi$$
−0.429657 + 0.902992i $$0.641365\pi$$
$$234$$ 0 0
$$235$$ 2.22938i 0.145429i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.57835i 0.425518i −0.977105 0.212759i $$-0.931755\pi$$
0.977105 0.212759i $$-0.0682449\pi$$
$$240$$ 0 0
$$241$$ 3.75906i 0.242142i 0.992644 + 0.121071i $$0.0386329\pi$$
−0.992644 + 0.121071i $$0.961367\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1.48913 −0.0951367
$$246$$ 0 0
$$247$$ −2.74456 + 6.33830i −0.174632 + 0.403296i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 9.45254i 0.596639i 0.954466 + 0.298320i $$0.0964261\pi$$
−0.954466 + 0.298320i $$0.903574\pi$$
$$252$$ 0 0
$$253$$ −4.74456 −0.298288
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 17.0256i 1.06202i −0.847364 0.531012i $$-0.821812\pi$$
0.847364 0.531012i $$-0.178188\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 14.7962i 0.912371i 0.889885 + 0.456185i $$0.150785\pi$$
−0.889885 + 0.456185i $$0.849215\pi$$
$$264$$ 0 0
$$265$$ 23.9538i 1.47147i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 23.9538i 1.46049i 0.683187 + 0.730243i $$0.260593\pi$$
−0.683187 + 0.730243i $$0.739407\pi$$
$$270$$ 0 0
$$271$$ 0.294954i 0.0179172i 0.999960 + 0.00895858i $$0.00285164\pi$$
−0.999960 + 0.00895858i $$0.997148\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.58457i 0.0955534i
$$276$$ 0 0
$$277$$ −8.37228 −0.503042 −0.251521 0.967852i $$-0.580931\pi$$
−0.251521 + 0.967852i $$0.580931\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3.16915i 0.189056i −0.995522 0.0945278i $$-0.969866\pi$$
0.995522 0.0945278i $$-0.0301341\pi$$
$$282$$ 0 0
$$283$$ 29.0573i 1.72728i −0.504110 0.863640i $$-0.668179\pi$$
0.504110 0.863640i $$-0.331821\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 17.4891 1.03235
$$288$$ 0 0
$$289$$ −16.8614 −0.991847
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 13.8564i 0.809500i 0.914427 + 0.404750i $$0.132641\pi$$
−0.914427 + 0.404750i $$0.867359\pi$$
$$294$$ 0 0
$$295$$ 9.48913 0.552478
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.97825 0.172237
$$300$$ 0 0
$$301$$ 1.62772 0.0938201
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0.883156 0.0505694
$$306$$ 0 0
$$307$$ −5.48913 −0.313281 −0.156640 0.987656i $$-0.550066\pi$$
−0.156640 + 0.987656i $$0.550066\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.22938i 0.126417i −0.998000 0.0632084i $$-0.979867\pi$$
0.998000 0.0632084i $$-0.0201333\pi$$
$$312$$ 0 0
$$313$$ −3.48913 −0.197217 −0.0986085 0.995126i $$-0.531439\pi$$
−0.0986085 + 0.995126i $$0.531439\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 23.9538i 1.34538i −0.739926 0.672689i $$-0.765139\pi$$
0.739926 0.672689i $$-0.234861\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.48913 0.644810i −0.0828571 0.0358782i
$$324$$ 0 0
$$325$$ 0.994667i 0.0551742i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 2.37228 0.130788
$$330$$ 0 0
$$331$$ 13.4891 0.741429 0.370715 0.928747i $$-0.379113\pi$$
0.370715 + 0.928747i $$0.379113\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 32.0000 1.74835
$$336$$ 0 0
$$337$$ 20.7846i 1.13221i −0.824333 0.566105i $$-0.808450\pi$$
0.824333 0.566105i $$-0.191550\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.92820i 0.375183i
$$342$$ 0 0
$$343$$ 19.2549i 1.03967i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 16.3807i 0.879364i 0.898153 + 0.439682i $$0.144909\pi$$
−0.898153 + 0.439682i $$0.855091\pi$$
$$348$$ 0 0
$$349$$ 14.6060 0.781840 0.390920 0.920425i $$-0.372157\pi$$
0.390920 + 0.920425i $$0.372157\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0.510875 0.0271911 0.0135956 0.999908i $$-0.495672\pi$$
0.0135956 + 0.999908i $$0.495672\pi$$
$$354$$ 0 0
$$355$$ 9.48913 0.503630
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 29.3523i 1.54915i −0.632479 0.774577i $$-0.717963\pi$$
0.632479 0.774577i $$-0.282037\pi$$
$$360$$ 0 0
$$361$$ 13.0000 + 13.8564i 0.684211 + 0.729285i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −31.1168 −1.62873
$$366$$ 0 0
$$367$$ 12.9715i 0.677109i 0.940947 + 0.338555i $$0.109938\pi$$
−0.940947 + 0.338555i $$0.890062\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 25.4891 1.32333
$$372$$ 0 0
$$373$$ 8.51278i 0.440775i 0.975412 + 0.220387i $$0.0707322\pi$$
−0.975412 + 0.220387i $$0.929268\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5.02175 −0.258633
$$378$$ 0 0
$$379$$ 18.9783 0.974847 0.487424 0.873166i $$-0.337937\pi$$
0.487424 + 0.873166i $$0.337937\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −14.9783 −0.765353 −0.382676 0.923882i $$-0.624998\pi$$
−0.382676 + 0.923882i $$0.624998\pi$$
$$384$$ 0 0
$$385$$ −15.1168 −0.770426
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 11.1168 0.563646 0.281823 0.959466i $$-0.409061\pi$$
0.281823 + 0.959466i $$0.409061\pi$$
$$390$$ 0 0
$$391$$ 0.699713i 0.0353860i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ 29.1168 1.46133 0.730666 0.682735i $$-0.239210\pi$$
0.730666 + 0.682735i $$0.239210\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.6873i 0.533696i −0.963739 0.266848i $$-0.914018\pi$$
0.963739 0.266848i $$-0.0859822\pi$$
$$402$$ 0 0
$$403$$ 4.34896i 0.216637i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ 6.92820i 0.342578i 0.985221 + 0.171289i $$0.0547931\pi$$
−0.985221 + 0.171289i $$0.945207\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 10.0974i 0.496858i
$$414$$ 0 0
$$415$$ 8.21782i 0.403397i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 13.5615i 0.662520i −0.943539 0.331260i $$-0.892526\pi$$
0.943539 0.331260i $$-0.107474\pi$$
$$420$$ 0 0
$$421$$ 26.1282i 1.27341i −0.771106 0.636706i $$-0.780296\pi$$
0.771106 0.636706i $$-0.219704\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0.233688 0.0113355
$$426$$ 0 0
$$427$$ 0.939764i 0.0454784i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 41.4891 1.99846 0.999230 0.0392245i $$-0.0124888\pi$$
0.999230 + 0.0392245i $$0.0124888\pi$$
$$432$$ 0 0
$$433$$ 23.3639i 1.12279i 0.827546 + 0.561397i $$0.189736\pi$$
−0.827546 + 0.561397i $$0.810264\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.25544 7.51811i 0.155729 0.359640i
$$438$$ 0 0
$$439$$ −17.7228 −0.845864 −0.422932 0.906161i $$-0.638999\pi$$
−0.422932 + 0.906161i $$0.638999\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3.81396i 0.181207i −0.995887 0.0906033i $$-0.971120\pi$$
0.995887 0.0906033i $$-0.0288795\pi$$
$$444$$ 0 0
$$445$$ 31.4719i 1.49191i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.8820i 1.45741i −0.684828 0.728705i $$-0.740123\pi$$
0.684828 0.728705i $$-0.259877\pi$$
$$450$$ 0 0
$$451$$ −17.4891 −0.823531
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 9.48913 0.444857
$$456$$ 0 0
$$457$$ 9.11684 0.426468 0.213234 0.977001i $$-0.431600\pi$$
0.213234 + 0.977001i $$0.431600\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −38.3723 −1.78718 −0.893588 0.448889i $$-0.851820\pi$$
−0.893588 + 0.448889i $$0.851820\pi$$
$$462$$ 0 0
$$463$$ 29.6472i 1.37782i −0.724845 0.688912i $$-0.758089\pi$$
0.724845 0.688912i $$-0.241911\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.0318i 0.556764i 0.960470 + 0.278382i $$0.0897982\pi$$
−0.960470 + 0.278382i $$0.910202\pi$$
$$468$$ 0 0
$$469$$ 34.0511i 1.57233i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −1.62772 −0.0748426
$$474$$ 0 0
$$475$$ −2.51087 1.08724i −0.115207 0.0498860i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 11.3870i 0.520284i 0.965570 + 0.260142i $$0.0837694\pi$$
−0.965570 + 0.260142i $$0.916231\pi$$
$$480$$ 0 0
$$481$$ −2.51087 −0.114486
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 31.4719i 1.42906i
$$486$$ 0 0
$$487$$ −40.2337 −1.82316 −0.911581 0.411120i $$-0.865138\pi$$
−0.911581 + 0.411120i $$0.865138\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 34.3461i 1.55002i 0.631951 + 0.775008i $$0.282254\pi$$
−0.631951 + 0.775008i $$0.717746\pi$$
$$492$$ 0 0
$$493$$ 1.17981i 0.0531362i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10.0974i 0.452928i
$$498$$ 0 0
$$499$$ 14.5012i 0.649164i 0.945858 + 0.324582i $$0.105224\pi$$
−0.945858 + 0.324582i $$0.894776\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 35.9306i 1.60207i 0.598619 + 0.801034i $$0.295716\pi$$
−0.598619 + 0.801034i $$0.704284\pi$$
$$504$$ 0 0
$$505$$ −7.72281 −0.343661
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 13.2665i 0.588027i −0.955801 0.294014i $$-0.905009\pi$$
0.955801 0.294014i $$-0.0949911\pi$$
$$510$$ 0 0
$$511$$ 33.1113i 1.46476i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −2.97825 −0.131237
$$516$$ 0 0
$$517$$ −2.37228 −0.104333
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 17.0256i 0.745903i 0.927851 + 0.372952i $$0.121654\pi$$
−0.927851 + 0.372952i $$0.878346\pi$$
$$522$$ 0 0
$$523$$ −32.0000 −1.39926 −0.699631 0.714504i $$-0.746652\pi$$
−0.699631 + 0.714504i $$0.746652\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.02175 0.0445081
$$528$$ 0 0
$$529$$ 19.4674 0.846408
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 10.9783 0.475521
$$534$$ 0 0
$$535$$ 32.0000 1.38348
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1.58457i 0.0682524i
$$540$$ 0 0
$$541$$ 13.1168 0.563937 0.281969 0.959424i $$-0.409013\pi$$
0.281969 + 0.959424i $$0.409013\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 27.7128i 1.18709i
$$546$$ 0 0
$$547$$ 6.51087 0.278385 0.139192 0.990265i $$-0.455549\pi$$
0.139192 + 0.990265i $$0.455549\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5.48913 + 12.6766i −0.233845 + 0.540041i
$$552$$ 0 0
$$553$$ 17.0256i 0.724000i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −39.8614 −1.68898 −0.844491 0.535570i $$-0.820097\pi$$
−0.844491 + 0.535570i $$0.820097\pi$$
$$558$$ 0 0
$$559$$ 1.02175 0.0432154
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 22.9783 0.968418 0.484209 0.874952i $$-0.339107\pi$$
0.484209 + 0.874952i $$0.339107\pi$$
$$564$$ 0 0
$$565$$ 40.3894i 1.69920i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0.589907i 0.0247302i 0.999924 + 0.0123651i $$0.00393603\pi$$
−0.999924 + 0.0123651i $$0.996064\pi$$
$$570$$ 0 0
$$571$$ 23.6588i 0.990090i −0.868867 0.495045i $$-0.835152\pi$$
0.868867 0.495045i $$-0.164848\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.17981i 0.0492017i
$$576$$ 0 0
$$577$$ 2.60597 0.108488 0.0542440 0.998528i $$-0.482725\pi$$
0.0542440 + 0.998528i $$0.482725\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 8.74456 0.362786
$$582$$ 0 0
$$583$$ −25.4891 −1.05565
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.5986i 1.01529i −0.861566 0.507646i $$-0.830516\pi$$
0.861566 0.507646i $$-0.169484\pi$$
$$588$$ 0 0
$$589$$ −10.9783 4.75372i −0.452351 0.195874i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0.510875 0.0209791 0.0104896 0.999945i $$-0.496661\pi$$
0.0104896 + 0.999945i $$0.496661\pi$$
$$594$$ 0 0
$$595$$ 2.22938i 0.0913958i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.4674 −0.999710 −0.499855 0.866109i $$-0.666613\pi$$
−0.499855 + 0.866109i $$0.666613\pi$$
$$600$$ 0 0
$$601$$ 33.4612i 1.36491i 0.730927 + 0.682455i $$0.239088\pi$$
−0.730927 + 0.682455i $$0.760912\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −10.9783 −0.446329
$$606$$ 0 0
$$607$$ 29.2554 1.18744 0.593721 0.804671i $$-0.297658\pi$$
0.593721 + 0.804671i $$0.297658\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.48913 0.0602436
$$612$$ 0 0
$$613$$ −0.372281 −0.0150363 −0.00751815 0.999972i $$-0.502393\pi$$
−0.00751815 + 0.999972i $$0.502393\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −44.0951 −1.77520 −0.887601 0.460613i $$-0.847629\pi$$
−0.887601 + 0.460613i $$0.847629\pi$$
$$618$$ 0 0
$$619$$ 10.3923i 0.417702i 0.977947 + 0.208851i $$0.0669724\pi$$
−0.977947 + 0.208851i $$0.933028\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 33.4891 1.34171
$$624$$ 0 0
$$625$$ −27.7446 −1.10978
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0.589907i 0.0235211i
$$630$$ 0 0
$$631$$ 20.2496i 0.806124i 0.915173 + 0.403062i $$0.132054\pi$$
−0.915173 + 0.403062i $$0.867946\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2.97825 0.118188
$$636$$ 0 0
$$637$$ 0.994667i 0.0394101i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 13.2665i 0.523995i 0.965069 + 0.261998i $$0.0843813\pi$$
−0.965069 + 0.261998i $$0.915619\pi$$
$$642$$ 0 0
$$643$$ 2.52434i 0.0995502i −0.998760 0.0497751i $$-0.984150\pi$$
0.998760 0.0497751i $$-0.0158505\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 13.5065i 0.530997i −0.964111 0.265499i $$-0.914463\pi$$
0.964111 0.265499i $$-0.0855366\pi$$
$$648$$ 0 0
$$649$$ 10.0974i 0.396356i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −4.88316 −0.191093 −0.0955463 0.995425i $$-0.530460\pi$$
−0.0955463 + 0.995425i $$0.530460\pi$$
$$654$$ 0 0
$$655$$ 17.9653i 0.701963i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −13.4891 −0.525462 −0.262731 0.964869i $$-0.584623\pi$$
−0.262731 + 0.964869i $$0.584623\pi$$
$$660$$ 0 0
$$661$$ 12.8617i 0.500264i −0.968212 0.250132i $$-0.919526\pi$$
0.968212 0.250132i $$-0.0804740\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10.3723 23.9538i 0.402220 0.928887i
$$666$$ 0 0
$$667$$ 5.95650 0.230637
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.939764i 0.0362792i
$$672$$ 0 0
$$673$$ 40.9793i 1.57964i −0.613341 0.789818i $$-0.710175\pi$$
0.613341 0.789818i $$-0.289825\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 47.9075i 1.84124i −0.390465 0.920618i $$-0.627686\pi$$
0.390465 0.920618i $$-0.372314\pi$$
$$678$$ 0 0
$$679$$ 33.4891 1.28519
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −38.9783 −1.49146 −0.745731 0.666248i $$-0.767899\pi$$
−0.745731 + 0.666248i $$0.767899\pi$$
$$684$$ 0 0
$$685$$ 31.1168 1.18891
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 16.0000 0.609551
$$690$$ 0 0
$$691$$ 1.93443i 0.0735892i −0.999323 0.0367946i $$-0.988285\pi$$
0.999323 0.0367946i $$-0.0117147\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 41.9191i 1.59008i
$$696$$ 0 0
$$697$$ 2.57924i 0.0976957i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19.7228 0.744920 0.372460 0.928048i $$-0.378514\pi$$
0.372460 + 0.928048i $$0.378514\pi$$
$$702$$ 0 0
$$703$$ −2.74456 + 6.33830i −0.103513 + 0.239053i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.21782i 0.309063i
$$708$$ 0 0
$$709$$ −47.9565 −1.80104 −0.900522 0.434810i $$-0.856815\pi$$
−0.900522 + 0.434810i $$0.856815\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 5.15848i 0.193187i
$$714$$ 0 0
$$715$$ −9.48913 −0.354873
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 29.3523i 1.09466i −0.836918 0.547328i $$-0.815645\pi$$
0.836918 0.547328i $$-0.184355\pi$$
$$720$$ 0 0
$$721$$ 3.16915i 0.118025i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.98933i 0.0738820i
$$726$$ 0 0
$$727$$ 8.16292i 0.302746i 0.988477 + 0.151373i $$0.0483695\pi$$
−0.988477 + 0.151373i $$0.951631\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0.240051i 0.00887860i
$$732$$ 0 0
$$733$$ 24.9783 0.922593 0.461296 0.887246i $$-0.347384\pi$$
0.461296 + 0.887246i $$0.347384\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.0511i 1.25429i
$$738$$ 0 0
$$739$$ 5.10358i 0.187738i −0.995585 0.0938691i $$-0.970076\pi$$
0.995585 0.0938691i $$-0.0299235\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 10.9783 0.402753 0.201376 0.979514i $$-0.435459\pi$$
0.201376 + 0.979514i $$0.435459\pi$$
$$744$$ 0 0
$$745$$ −35.8614 −1.31386
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 34.0511i 1.24420i
$$750$$ 0 0
$$751$$ 41.7228 1.52249 0.761244 0.648466i $$-0.224589\pi$$
0.761244 + 0.648466i $$0.224589\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 6.51087 0.236955
$$756$$ 0 0
$$757$$ −17.8614 −0.649184 −0.324592 0.945854i $$-0.605227\pi$$
−0.324592 + 0.945854i $$0.605227\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 21.8614 0.792475 0.396238 0.918148i $$-0.370316\pi$$
0.396238 + 0.918148i $$0.370316\pi$$
$$762$$ 0 0
$$763$$ 29.4891 1.06758
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 6.33830i 0.228863i
$$768$$ 0 0
$$769$$ −37.8614 −1.36532 −0.682659 0.730737i $$-0.739176\pi$$
−0.682659 + 0.730737i $$0.739176\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 39.7995i 1.43149i −0.698363 0.715744i $$-0.746088\pi$$
0.698363 0.715744i $$-0.253912\pi$$
$$774$$ 0 0
$$775$$ 1.72281 0.0618853
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.0000 27.7128i 0.429945 0.992915i
$$780$$ 0 0
$$781$$ 10.0974i 0.361312i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 36.7446 1.31147
$$786$$ 0 0
$$787$$ 34.9783 1.24684 0.623420 0.781887i $$-0.285743\pi$$
0.623420 + 0.781887i $$0.285743\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 42.9783 1.52813
$$792$$ 0 0
$$793$$ 0.589907i 0.0209482i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3.16915i 0.112257i −0.998424 0.0561285i $$-0.982124\pi$$
0.998424 0.0561285i $$-0.0178756\pi$$
$$798$$ 0 0
$$799$$ 0.349857i 0.0123770i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 33.1113i 1.16847i
$$804$$ 0 0
$$805$$ −11.2554 −0.396702
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 14.8832