# Properties

 Label 1950.2.f.n.649.3 Level $1950$ Weight $2$ Character 1950.649 Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.3 Root $$-2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.649 Dual form 1950.2.f.n.649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} -4.60555 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} -4.60555 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.00000i q^{12} -3.60555i q^{13} -4.60555 q^{14} +1.00000 q^{16} +4.60555i q^{17} -1.00000 q^{18} -4.60555i q^{19} -4.60555i q^{21} +1.39445i q^{23} +1.00000i q^{24} -3.60555i q^{26} -1.00000i q^{27} -4.60555 q^{28} -4.60555 q^{29} -6.00000i q^{31} +1.00000 q^{32} +4.60555i q^{34} -1.00000 q^{36} -9.21110 q^{37} -4.60555i q^{38} +3.60555 q^{39} -3.21110i q^{41} -4.60555i q^{42} -8.00000i q^{43} +1.39445i q^{46} -9.21110 q^{47} +1.00000i q^{48} +14.2111 q^{49} -4.60555 q^{51} -3.60555i q^{52} -6.00000i q^{53} -1.00000i q^{54} -4.60555 q^{56} +4.60555 q^{57} -4.60555 q^{58} -9.21110i q^{59} -11.2111 q^{61} -6.00000i q^{62} +4.60555 q^{63} +1.00000 q^{64} -3.21110 q^{67} +4.60555i q^{68} -1.39445 q^{69} -9.21110i q^{71} -1.00000 q^{72} +1.39445 q^{73} -9.21110 q^{74} -4.60555i q^{76} +3.60555 q^{78} +14.4222 q^{79} +1.00000 q^{81} -3.21110i q^{82} -2.78890 q^{83} -4.60555i q^{84} -8.00000i q^{86} -4.60555i q^{87} +15.2111i q^{89} +16.6056i q^{91} +1.39445i q^{92} +6.00000 q^{93} -9.21110 q^{94} +1.00000i q^{96} +1.39445 q^{97} +14.2111 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{4} - 4q^{7} + 4q^{8} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{4} - 4q^{7} + 4q^{8} - 4q^{9} - 4q^{14} + 4q^{16} - 4q^{18} - 4q^{28} - 4q^{29} + 4q^{32} - 4q^{36} - 8q^{37} - 8q^{47} + 28q^{49} - 4q^{51} - 4q^{56} + 4q^{57} - 4q^{58} - 16q^{61} + 4q^{63} + 4q^{64} + 16q^{67} - 20q^{69} - 4q^{72} + 20q^{73} - 8q^{74} + 4q^{81} - 40q^{83} + 24q^{93} - 8q^{94} + 20q^{97} + 28q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000i 0.408248i
$$7$$ −4.60555 −1.74073 −0.870367 0.492403i $$-0.836119\pi$$
−0.870367 + 0.492403i $$0.836119\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 3.60555i − 1.00000i
$$14$$ −4.60555 −1.23089
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.60555i 1.11701i 0.829501 + 0.558505i $$0.188625\pi$$
−0.829501 + 0.558505i $$0.811375\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ − 4.60555i − 1.05659i −0.849062 0.528293i $$-0.822832\pi$$
0.849062 0.528293i $$-0.177168\pi$$
$$20$$ 0 0
$$21$$ − 4.60555i − 1.00501i
$$22$$ 0 0
$$23$$ 1.39445i 0.290763i 0.989376 + 0.145381i $$0.0464409\pi$$
−0.989376 + 0.145381i $$0.953559\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ 0 0
$$26$$ − 3.60555i − 0.707107i
$$27$$ − 1.00000i − 0.192450i
$$28$$ −4.60555 −0.870367
$$29$$ −4.60555 −0.855229 −0.427615 0.903961i $$-0.640646\pi$$
−0.427615 + 0.903961i $$0.640646\pi$$
$$30$$ 0 0
$$31$$ − 6.00000i − 1.07763i −0.842424 0.538816i $$-0.818872\pi$$
0.842424 0.538816i $$-0.181128\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.60555i 0.789846i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −9.21110 −1.51430 −0.757148 0.653243i $$-0.773408\pi$$
−0.757148 + 0.653243i $$0.773408\pi$$
$$38$$ − 4.60555i − 0.747119i
$$39$$ 3.60555 0.577350
$$40$$ 0 0
$$41$$ − 3.21110i − 0.501490i −0.968053 0.250745i $$-0.919324\pi$$
0.968053 0.250745i $$-0.0806756\pi$$
$$42$$ − 4.60555i − 0.710652i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.39445i 0.205600i
$$47$$ −9.21110 −1.34358 −0.671789 0.740743i $$-0.734474\pi$$
−0.671789 + 0.740743i $$0.734474\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 14.2111 2.03016
$$50$$ 0 0
$$51$$ −4.60555 −0.644906
$$52$$ − 3.60555i − 0.500000i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0 0
$$56$$ −4.60555 −0.615443
$$57$$ 4.60555 0.610020
$$58$$ −4.60555 −0.604739
$$59$$ − 9.21110i − 1.19918i −0.800306 0.599592i $$-0.795330\pi$$
0.800306 0.599592i $$-0.204670\pi$$
$$60$$ 0 0
$$61$$ −11.2111 −1.43543 −0.717717 0.696335i $$-0.754813\pi$$
−0.717717 + 0.696335i $$0.754813\pi$$
$$62$$ − 6.00000i − 0.762001i
$$63$$ 4.60555 0.580245
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.21110 −0.392299 −0.196149 0.980574i $$-0.562844\pi$$
−0.196149 + 0.980574i $$0.562844\pi$$
$$68$$ 4.60555i 0.558505i
$$69$$ −1.39445 −0.167872
$$70$$ 0 0
$$71$$ − 9.21110i − 1.09316i −0.837408 0.546578i $$-0.815930\pi$$
0.837408 0.546578i $$-0.184070\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 1.39445 0.163208 0.0816039 0.996665i $$-0.473996\pi$$
0.0816039 + 0.996665i $$0.473996\pi$$
$$74$$ −9.21110 −1.07077
$$75$$ 0 0
$$76$$ − 4.60555i − 0.528293i
$$77$$ 0 0
$$78$$ 3.60555 0.408248
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 3.21110i − 0.354607i
$$83$$ −2.78890 −0.306121 −0.153061 0.988217i $$-0.548913\pi$$
−0.153061 + 0.988217i $$0.548913\pi$$
$$84$$ − 4.60555i − 0.502507i
$$85$$ 0 0
$$86$$ − 8.00000i − 0.862662i
$$87$$ − 4.60555i − 0.493767i
$$88$$ 0 0
$$89$$ 15.2111i 1.61237i 0.591661 + 0.806187i $$0.298472\pi$$
−0.591661 + 0.806187i $$0.701528\pi$$
$$90$$ 0 0
$$91$$ 16.6056i 1.74073i
$$92$$ 1.39445i 0.145381i
$$93$$ 6.00000 0.622171
$$94$$ −9.21110 −0.950053
$$95$$ 0 0
$$96$$ 1.00000i 0.102062i
$$97$$ 1.39445 0.141585 0.0707924 0.997491i $$-0.477447\pi$$
0.0707924 + 0.997491i $$0.477447\pi$$
$$98$$ 14.2111 1.43554
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.39445 −0.735775 −0.367888 0.929870i $$-0.619919\pi$$
−0.367888 + 0.929870i $$0.619919\pi$$
$$102$$ −4.60555 −0.456018
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ − 3.60555i − 0.353553i
$$105$$ 0 0
$$106$$ − 6.00000i − 0.582772i
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 1.39445i 0.133564i 0.997768 + 0.0667820i $$0.0212732\pi$$
−0.997768 + 0.0667820i $$0.978727\pi$$
$$110$$ 0 0
$$111$$ − 9.21110i − 0.874279i
$$112$$ −4.60555 −0.435184
$$113$$ 13.8167i 1.29976i 0.760036 + 0.649881i $$0.225181\pi$$
−0.760036 + 0.649881i $$0.774819\pi$$
$$114$$ 4.60555 0.431349
$$115$$ 0 0
$$116$$ −4.60555 −0.427615
$$117$$ 3.60555i 0.333333i
$$118$$ − 9.21110i − 0.847951i
$$119$$ − 21.2111i − 1.94442i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ −11.2111 −1.01501
$$123$$ 3.21110 0.289535
$$124$$ − 6.00000i − 0.538816i
$$125$$ 0 0
$$126$$ 4.60555 0.410295
$$127$$ 1.21110i 0.107468i 0.998555 + 0.0537340i $$0.0171123\pi$$
−0.998555 + 0.0537340i $$0.982888\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 22.6056 1.97506 0.987528 0.157443i $$-0.0503250\pi$$
0.987528 + 0.157443i $$0.0503250\pi$$
$$132$$ 0 0
$$133$$ 21.2111i 1.83924i
$$134$$ −3.21110 −0.277397
$$135$$ 0 0
$$136$$ 4.60555i 0.394923i
$$137$$ 3.21110 0.274343 0.137172 0.990547i $$-0.456199\pi$$
0.137172 + 0.990547i $$0.456199\pi$$
$$138$$ −1.39445 −0.118703
$$139$$ −17.2111 −1.45983 −0.729913 0.683540i $$-0.760440\pi$$
−0.729913 + 0.683540i $$0.760440\pi$$
$$140$$ 0 0
$$141$$ − 9.21110i − 0.775715i
$$142$$ − 9.21110i − 0.772979i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 1.39445 0.115405
$$147$$ 14.2111i 1.17211i
$$148$$ −9.21110 −0.757148
$$149$$ 15.2111i 1.24614i 0.782165 + 0.623071i $$0.214115\pi$$
−0.782165 + 0.623071i $$0.785885\pi$$
$$150$$ 0 0
$$151$$ − 6.00000i − 0.488273i −0.969741 0.244137i $$-0.921495\pi$$
0.969741 0.244137i $$-0.0785045\pi$$
$$152$$ − 4.60555i − 0.373560i
$$153$$ − 4.60555i − 0.372337i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.60555 0.288675
$$157$$ 20.4222i 1.62987i 0.579553 + 0.814935i $$0.303227\pi$$
−0.579553 + 0.814935i $$0.696773\pi$$
$$158$$ 14.4222 1.14737
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ − 6.42221i − 0.506141i
$$162$$ 1.00000 0.0785674
$$163$$ −24.4222 −1.91289 −0.956447 0.291905i $$-0.905711\pi$$
−0.956447 + 0.291905i $$0.905711\pi$$
$$164$$ − 3.21110i − 0.250745i
$$165$$ 0 0
$$166$$ −2.78890 −0.216460
$$167$$ 9.21110 0.712777 0.356388 0.934338i $$-0.384008\pi$$
0.356388 + 0.934338i $$0.384008\pi$$
$$168$$ − 4.60555i − 0.355326i
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 4.60555i 0.352195i
$$172$$ − 8.00000i − 0.609994i
$$173$$ 12.4222i 0.944443i 0.881480 + 0.472221i $$0.156548\pi$$
−0.881480 + 0.472221i $$0.843452\pi$$
$$174$$ − 4.60555i − 0.349146i
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 9.21110 0.692349
$$178$$ 15.2111i 1.14012i
$$179$$ 19.8167 1.48117 0.740583 0.671965i $$-0.234549\pi$$
0.740583 + 0.671965i $$0.234549\pi$$
$$180$$ 0 0
$$181$$ 8.42221 0.626018 0.313009 0.949750i $$-0.398663\pi$$
0.313009 + 0.949750i $$0.398663\pi$$
$$182$$ 16.6056i 1.23089i
$$183$$ − 11.2111i − 0.828749i
$$184$$ 1.39445i 0.102800i
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ 0 0
$$188$$ −9.21110 −0.671789
$$189$$ 4.60555i 0.335005i
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ −7.81665 −0.562655 −0.281328 0.959612i $$-0.590775\pi$$
−0.281328 + 0.959612i $$0.590775\pi$$
$$194$$ 1.39445 0.100116
$$195$$ 0 0
$$196$$ 14.2111 1.01508
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −22.4222 −1.58947 −0.794734 0.606958i $$-0.792390\pi$$
−0.794734 + 0.606958i $$0.792390\pi$$
$$200$$ 0 0
$$201$$ − 3.21110i − 0.226494i
$$202$$ −7.39445 −0.520272
$$203$$ 21.2111 1.48873
$$204$$ −4.60555 −0.322453
$$205$$ 0 0
$$206$$ 4.00000i 0.278693i
$$207$$ − 1.39445i − 0.0969209i
$$208$$ − 3.60555i − 0.250000i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −17.2111 −1.18486 −0.592431 0.805622i $$-0.701832\pi$$
−0.592431 + 0.805622i $$0.701832\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 9.21110 0.631134
$$214$$ 0 0
$$215$$ 0 0
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 27.6333i 1.87587i
$$218$$ 1.39445i 0.0944440i
$$219$$ 1.39445i 0.0942281i
$$220$$ 0 0
$$221$$ 16.6056 1.11701
$$222$$ − 9.21110i − 0.618209i
$$223$$ −1.81665 −0.121652 −0.0608261 0.998148i $$-0.519373\pi$$
−0.0608261 + 0.998148i $$0.519373\pi$$
$$224$$ −4.60555 −0.307721
$$225$$ 0 0
$$226$$ 13.8167i 0.919070i
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 4.60555 0.305010
$$229$$ − 19.8167i − 1.30952i −0.755836 0.654761i $$-0.772769\pi$$
0.755836 0.654761i $$-0.227231\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −4.60555 −0.302369
$$233$$ 1.81665i 0.119013i 0.998228 + 0.0595065i $$0.0189527\pi$$
−0.998228 + 0.0595065i $$0.981047\pi$$
$$234$$ 3.60555i 0.235702i
$$235$$ 0 0
$$236$$ − 9.21110i − 0.599592i
$$237$$ 14.4222i 0.936823i
$$238$$ − 21.2111i − 1.37491i
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 6.42221i 0.413691i 0.978374 + 0.206845i $$0.0663197\pi$$
−0.978374 + 0.206845i $$0.933680\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 1.00000i 0.0641500i
$$244$$ −11.2111 −0.717717
$$245$$ 0 0
$$246$$ 3.21110 0.204732
$$247$$ −16.6056 −1.05659
$$248$$ − 6.00000i − 0.381000i
$$249$$ − 2.78890i − 0.176739i
$$250$$ 0 0
$$251$$ −13.3944 −0.845450 −0.422725 0.906258i $$-0.638926\pi$$
−0.422725 + 0.906258i $$0.638926\pi$$
$$252$$ 4.60555 0.290122
$$253$$ 0 0
$$254$$ 1.21110i 0.0759913i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 28.6056i − 1.78437i −0.451675 0.892183i $$-0.649173\pi$$
0.451675 0.892183i $$-0.350827\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 42.4222 2.63599
$$260$$ 0 0
$$261$$ 4.60555 0.285076
$$262$$ 22.6056 1.39658
$$263$$ 7.81665i 0.481996i 0.970526 + 0.240998i $$0.0774746\pi$$
−0.970526 + 0.240998i $$0.922525\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 21.2111i 1.30054i
$$267$$ −15.2111 −0.930904
$$268$$ −3.21110 −0.196149
$$269$$ −25.8167 −1.57407 −0.787035 0.616909i $$-0.788385\pi$$
−0.787035 + 0.616909i $$0.788385\pi$$
$$270$$ 0 0
$$271$$ 0.422205i 0.0256471i 0.999918 + 0.0128236i $$0.00408198\pi$$
−0.999918 + 0.0128236i $$0.995918\pi$$
$$272$$ 4.60555i 0.279253i
$$273$$ −16.6056 −1.00501
$$274$$ 3.21110 0.193990
$$275$$ 0 0
$$276$$ −1.39445 −0.0839359
$$277$$ − 16.4222i − 0.986715i −0.869827 0.493357i $$-0.835770\pi$$
0.869827 0.493357i $$-0.164230\pi$$
$$278$$ −17.2111 −1.03225
$$279$$ 6.00000i 0.359211i
$$280$$ 0 0
$$281$$ − 27.2111i − 1.62328i −0.584159 0.811639i $$-0.698576\pi$$
0.584159 0.811639i $$-0.301424\pi$$
$$282$$ − 9.21110i − 0.548513i
$$283$$ − 10.4222i − 0.619536i −0.950812 0.309768i $$-0.899749\pi$$
0.950812 0.309768i $$-0.100251\pi$$
$$284$$ − 9.21110i − 0.546578i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 14.7889i 0.872961i
$$288$$ −1.00000 −0.0589256
$$289$$ −4.21110 −0.247712
$$290$$ 0 0
$$291$$ 1.39445i 0.0817440i
$$292$$ 1.39445 0.0816039
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 14.2111i 0.828808i
$$295$$ 0 0
$$296$$ −9.21110 −0.535384
$$297$$ 0 0
$$298$$ 15.2111i 0.881156i
$$299$$ 5.02776 0.290763
$$300$$ 0 0
$$301$$ 36.8444i 2.12368i
$$302$$ − 6.00000i − 0.345261i
$$303$$ − 7.39445i − 0.424800i
$$304$$ − 4.60555i − 0.264146i
$$305$$ 0 0
$$306$$ − 4.60555i − 0.263282i
$$307$$ −8.78890 −0.501609 −0.250804 0.968038i $$-0.580695\pi$$
−0.250804 + 0.968038i $$0.580695\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 3.60555 0.204124
$$313$$ − 3.57779i − 0.202229i −0.994875 0.101114i $$-0.967759\pi$$
0.994875 0.101114i $$-0.0322408\pi$$
$$314$$ 20.4222i 1.15249i
$$315$$ 0 0
$$316$$ 14.4222 0.811312
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 6.42221i − 0.357895i
$$323$$ 21.2111 1.18022
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −24.4222 −1.35262
$$327$$ −1.39445 −0.0771132
$$328$$ − 3.21110i − 0.177303i
$$329$$ 42.4222 2.33881
$$330$$ 0 0
$$331$$ 16.6056i 0.912724i 0.889794 + 0.456362i $$0.150848\pi$$
−0.889794 + 0.456362i $$0.849152\pi$$
$$332$$ −2.78890 −0.153061
$$333$$ 9.21110 0.504765
$$334$$ 9.21110 0.504009
$$335$$ 0 0
$$336$$ − 4.60555i − 0.251253i
$$337$$ 13.6333i 0.742654i 0.928502 + 0.371327i $$0.121097\pi$$
−0.928502 + 0.371327i $$0.878903\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ −13.8167 −0.750418
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.60555i 0.249040i
$$343$$ −33.2111 −1.79323
$$344$$ − 8.00000i − 0.431331i
$$345$$ 0 0
$$346$$ 12.4222i 0.667822i
$$347$$ − 27.6333i − 1.48343i −0.670713 0.741717i $$-0.734012\pi$$
0.670713 0.741717i $$-0.265988\pi$$
$$348$$ − 4.60555i − 0.246883i
$$349$$ − 7.81665i − 0.418416i −0.977871 0.209208i $$-0.932911\pi$$
0.977871 0.209208i $$-0.0670886\pi$$
$$350$$ 0 0
$$351$$ −3.60555 −0.192450
$$352$$ 0 0
$$353$$ 8.78890 0.467786 0.233893 0.972262i $$-0.424853\pi$$
0.233893 + 0.972262i $$0.424853\pi$$
$$354$$ 9.21110 0.489565
$$355$$ 0 0
$$356$$ 15.2111i 0.806187i
$$357$$ 21.2111 1.12261
$$358$$ 19.8167 1.04734
$$359$$ 15.6333i 0.825094i 0.910936 + 0.412547i $$0.135361\pi$$
−0.910936 + 0.412547i $$0.864639\pi$$
$$360$$ 0 0
$$361$$ −2.21110 −0.116374
$$362$$ 8.42221 0.442661
$$363$$ 11.0000i 0.577350i
$$364$$ 16.6056i 0.870367i
$$365$$ 0 0
$$366$$ − 11.2111i − 0.586014i
$$367$$ − 19.6333i − 1.02485i −0.858732 0.512425i $$-0.828747\pi$$
0.858732 0.512425i $$-0.171253\pi$$
$$368$$ 1.39445i 0.0726907i
$$369$$ 3.21110i 0.167163i
$$370$$ 0 0
$$371$$ 27.6333i 1.43465i
$$372$$ 6.00000 0.311086
$$373$$ 20.4222i 1.05742i 0.848802 + 0.528711i $$0.177324\pi$$
−0.848802 + 0.528711i $$0.822676\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.21110 −0.475026
$$377$$ 16.6056i 0.855229i
$$378$$ 4.60555i 0.236884i
$$379$$ 35.0278i 1.79925i 0.436658 + 0.899627i $$0.356162\pi$$
−0.436658 + 0.899627i $$0.643838\pi$$
$$380$$ 0 0
$$381$$ −1.21110 −0.0620467
$$382$$ −12.0000 −0.613973
$$383$$ 27.6333 1.41200 0.705998 0.708214i $$-0.250499\pi$$
0.705998 + 0.708214i $$0.250499\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ −7.81665 −0.397857
$$387$$ 8.00000i 0.406663i
$$388$$ 1.39445 0.0707924
$$389$$ −4.60555 −0.233511 −0.116755 0.993161i $$-0.537249\pi$$
−0.116755 + 0.993161i $$0.537249\pi$$
$$390$$ 0 0
$$391$$ −6.42221 −0.324785
$$392$$ 14.2111 0.717769
$$393$$ 22.6056i 1.14030i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3.63331 −0.182350 −0.0911752 0.995835i $$-0.529062\pi$$
−0.0911752 + 0.995835i $$0.529062\pi$$
$$398$$ −22.4222 −1.12392
$$399$$ −21.2111 −1.06188
$$400$$ 0 0
$$401$$ − 8.78890i − 0.438897i −0.975624 0.219448i $$-0.929574\pi$$
0.975624 0.219448i $$-0.0704257\pi$$
$$402$$ − 3.21110i − 0.160155i
$$403$$ −21.6333 −1.07763
$$404$$ −7.39445 −0.367888
$$405$$ 0 0
$$406$$ 21.2111 1.05269
$$407$$ 0 0
$$408$$ −4.60555 −0.228009
$$409$$ − 14.7889i − 0.731264i −0.930760 0.365632i $$-0.880853\pi$$
0.930760 0.365632i $$-0.119147\pi$$
$$410$$ 0 0
$$411$$ 3.21110i 0.158392i
$$412$$ 4.00000i 0.197066i
$$413$$ 42.4222i 2.08746i
$$414$$ − 1.39445i − 0.0685334i
$$415$$ 0 0
$$416$$ − 3.60555i − 0.176777i
$$417$$ − 17.2111i − 0.842831i
$$418$$ 0 0
$$419$$ −4.18335 −0.204370 −0.102185 0.994765i $$-0.532583\pi$$
−0.102185 + 0.994765i $$0.532583\pi$$
$$420$$ 0 0
$$421$$ 19.8167i 0.965805i 0.875674 + 0.482902i $$0.160417\pi$$
−0.875674 + 0.482902i $$0.839583\pi$$
$$422$$ −17.2111 −0.837823
$$423$$ 9.21110 0.447859
$$424$$ − 6.00000i − 0.291386i
$$425$$ 0 0
$$426$$ 9.21110 0.446279
$$427$$ 51.6333 2.49871
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 12.0000i − 0.578020i −0.957326 0.289010i $$-0.906674\pi$$
0.957326 0.289010i $$-0.0933260\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 19.2111i − 0.923227i −0.887081 0.461613i $$-0.847271\pi$$
0.887081 0.461613i $$-0.152729\pi$$
$$434$$ 27.6333i 1.32644i
$$435$$ 0 0
$$436$$ 1.39445i 0.0667820i
$$437$$ 6.42221 0.307216
$$438$$ 1.39445i 0.0666293i
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −14.2111 −0.676719
$$442$$ 16.6056 0.789846
$$443$$ − 15.6333i − 0.742761i −0.928481 0.371380i $$-0.878885\pi$$
0.928481 0.371380i $$-0.121115\pi$$
$$444$$ − 9.21110i − 0.437140i
$$445$$ 0 0
$$446$$ −1.81665 −0.0860211
$$447$$ −15.2111 −0.719460
$$448$$ −4.60555 −0.217592
$$449$$ 33.6333i 1.58725i 0.608405 + 0.793627i $$0.291810\pi$$
−0.608405 + 0.793627i $$0.708190\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 13.8167i 0.649881i
$$453$$ 6.00000 0.281905
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ 4.60555 0.215675
$$457$$ 38.2389 1.78874 0.894369 0.447330i $$-0.147625\pi$$
0.894369 + 0.447330i $$0.147625\pi$$
$$458$$ − 19.8167i − 0.925971i
$$459$$ 4.60555 0.214969
$$460$$ 0 0
$$461$$ 33.6333i 1.56646i 0.621733 + 0.783230i $$0.286429\pi$$
−0.621733 + 0.783230i $$0.713571\pi$$
$$462$$ 0 0
$$463$$ 31.3944 1.45902 0.729512 0.683968i $$-0.239747\pi$$
0.729512 + 0.683968i $$0.239747\pi$$
$$464$$ −4.60555 −0.213807
$$465$$ 0 0
$$466$$ 1.81665i 0.0841549i
$$467$$ − 30.4222i − 1.40777i −0.710313 0.703886i $$-0.751447\pi$$
0.710313 0.703886i $$-0.248553\pi$$
$$468$$ 3.60555i 0.166667i
$$469$$ 14.7889 0.682888
$$470$$ 0 0
$$471$$ −20.4222 −0.941006
$$472$$ − 9.21110i − 0.423975i
$$473$$ 0 0
$$474$$ 14.4222i 0.662434i
$$475$$ 0 0
$$476$$ − 21.2111i − 0.972209i
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ 5.57779i 0.254856i 0.991848 + 0.127428i $$0.0406722\pi$$
−0.991848 + 0.127428i $$0.959328\pi$$
$$480$$ 0 0
$$481$$ 33.2111i 1.51430i
$$482$$ 6.42221i 0.292523i
$$483$$ 6.42221 0.292220
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000i 0.0453609i
$$487$$ −0.972244 −0.0440566 −0.0220283 0.999757i $$-0.507012\pi$$
−0.0220283 + 0.999757i $$0.507012\pi$$
$$488$$ −11.2111 −0.507503
$$489$$ − 24.4222i − 1.10441i
$$490$$ 0 0
$$491$$ 7.81665 0.352761 0.176380 0.984322i $$-0.443561\pi$$
0.176380 + 0.984322i $$0.443561\pi$$
$$492$$ 3.21110 0.144768
$$493$$ − 21.2111i − 0.955300i
$$494$$ −16.6056 −0.747119
$$495$$ 0 0
$$496$$ − 6.00000i − 0.269408i
$$497$$ 42.4222i 1.90290i
$$498$$ − 2.78890i − 0.124973i
$$499$$ 23.0278i 1.03086i 0.856930 + 0.515432i $$0.172369\pi$$
−0.856930 + 0.515432i $$0.827631\pi$$
$$500$$ 0 0
$$501$$ 9.21110i 0.411522i
$$502$$ −13.3944 −0.597824
$$503$$ − 23.4500i − 1.04558i −0.852461 0.522791i $$-0.824891\pi$$
0.852461 0.522791i $$-0.175109\pi$$
$$504$$ 4.60555 0.205148
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 13.0000i − 0.577350i
$$508$$ 1.21110i 0.0537340i
$$509$$ − 33.6333i − 1.49077i −0.666634 0.745385i $$-0.732266\pi$$
0.666634 0.745385i $$-0.267734\pi$$
$$510$$ 0 0
$$511$$ −6.42221 −0.284102
$$512$$ 1.00000 0.0441942
$$513$$ −4.60555 −0.203340
$$514$$ − 28.6056i − 1.26174i
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 42.4222 1.86392
$$519$$ −12.4222 −0.545274
$$520$$ 0 0
$$521$$ 21.6333 0.947772 0.473886 0.880586i $$-0.342851\pi$$
0.473886 + 0.880586i $$0.342851\pi$$
$$522$$ 4.60555 0.201580
$$523$$ − 32.8444i − 1.43619i −0.695947 0.718093i $$-0.745015\pi$$
0.695947 0.718093i $$-0.254985\pi$$
$$524$$ 22.6056 0.987528
$$525$$ 0 0
$$526$$ 7.81665i 0.340822i
$$527$$ 27.6333 1.20373
$$528$$ 0 0
$$529$$ 21.0555 0.915457
$$530$$ 0 0
$$531$$ 9.21110i 0.399728i
$$532$$ 21.2111i 0.919618i
$$533$$ −11.5778 −0.501490
$$534$$ −15.2111 −0.658249
$$535$$ 0 0
$$536$$ −3.21110 −0.138699
$$537$$ 19.8167i 0.855152i
$$538$$ −25.8167 −1.11303
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 6.97224i − 0.299760i −0.988704 0.149880i $$-0.952111\pi$$
0.988704 0.149880i $$-0.0478888\pi$$
$$542$$ 0.422205i 0.0181353i
$$543$$ 8.42221i 0.361431i
$$544$$ 4.60555i 0.197461i
$$545$$ 0 0
$$546$$ −16.6056 −0.710652
$$547$$ 14.4222i 0.616649i 0.951281 + 0.308324i $$0.0997682\pi$$
−0.951281 + 0.308324i $$0.900232\pi$$
$$548$$ 3.21110 0.137172
$$549$$ 11.2111 0.478478
$$550$$ 0 0
$$551$$ 21.2111i 0.903623i
$$552$$ −1.39445 −0.0593517
$$553$$ −66.4222 −2.82456
$$554$$ − 16.4222i − 0.697713i
$$555$$ 0 0
$$556$$ −17.2111 −0.729913
$$557$$ 11.5778 0.490567 0.245283 0.969451i $$-0.421119\pi$$
0.245283 + 0.969451i $$0.421119\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ −28.8444 −1.21999
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 27.2111i − 1.14783i
$$563$$ − 34.0555i − 1.43527i −0.696420 0.717634i $$-0.745225\pi$$
0.696420 0.717634i $$-0.254775\pi$$
$$564$$ − 9.21110i − 0.387857i
$$565$$ 0 0
$$566$$ − 10.4222i − 0.438078i
$$567$$ −4.60555 −0.193415
$$568$$ − 9.21110i − 0.386489i
$$569$$ −33.6333 −1.40998 −0.704991 0.709216i $$-0.749049\pi$$
−0.704991 + 0.709216i $$0.749049\pi$$
$$570$$ 0 0
$$571$$ −30.0555 −1.25778 −0.628892 0.777493i $$-0.716491\pi$$
−0.628892 + 0.777493i $$0.716491\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ 14.7889i 0.617277i
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ −37.3944 −1.55675 −0.778376 0.627799i $$-0.783956\pi$$
−0.778376 + 0.627799i $$0.783956\pi$$
$$578$$ −4.21110 −0.175159
$$579$$ − 7.81665i − 0.324849i
$$580$$ 0 0
$$581$$ 12.8444 0.532876
$$582$$ 1.39445i 0.0578018i
$$583$$ 0 0
$$584$$ 1.39445 0.0577027
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −6.42221 −0.265073 −0.132536 0.991178i $$-0.542312\pi$$
−0.132536 + 0.991178i $$0.542312\pi$$
$$588$$ 14.2111i 0.586056i
$$589$$ −27.6333 −1.13861
$$590$$ 0 0
$$591$$ − 6.00000i − 0.246807i
$$592$$ −9.21110 −0.378574
$$593$$ 24.4222 1.00290 0.501450 0.865187i $$-0.332800\pi$$
0.501450 + 0.865187i $$0.332800\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 15.2111i 0.623071i
$$597$$ − 22.4222i − 0.917680i
$$598$$ 5.02776 0.205600
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ 36.8444i 1.50167i
$$603$$ 3.21110 0.130766
$$604$$ − 6.00000i − 0.244137i
$$605$$ 0 0
$$606$$ − 7.39445i − 0.300379i
$$607$$ 17.2111i 0.698577i 0.937015 + 0.349289i $$0.113577\pi$$
−0.937015 + 0.349289i $$0.886423\pi$$
$$608$$ − 4.60555i − 0.186780i
$$609$$ 21.2111i 0.859517i
$$610$$ 0 0
$$611$$ 33.2111i 1.34358i
$$612$$ − 4.60555i − 0.186168i
$$613$$ 33.2111 1.34138 0.670692 0.741736i $$-0.265997\pi$$
0.670692 + 0.741736i $$0.265997\pi$$
$$614$$ −8.78890 −0.354691
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.4222 0.500099 0.250050 0.968233i $$-0.419553\pi$$
0.250050 + 0.968233i $$0.419553\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ 25.8167i 1.03766i 0.854878 + 0.518829i $$0.173632\pi$$
−0.854878 + 0.518829i $$0.826368\pi$$
$$620$$ 0 0
$$621$$ 1.39445 0.0559573
$$622$$ 12.0000 0.481156
$$623$$ − 70.0555i − 2.80671i
$$624$$ 3.60555 0.144338
$$625$$ 0 0
$$626$$ − 3.57779i − 0.142997i
$$627$$ 0 0
$$628$$ 20.4222i 0.814935i
$$629$$ − 42.4222i − 1.69148i
$$630$$ 0 0
$$631$$ − 3.21110i − 0.127832i −0.997955 0.0639160i $$-0.979641\pi$$
0.997955 0.0639160i $$-0.0203590\pi$$
$$632$$ 14.4222 0.573685
$$633$$ − 17.2111i − 0.684080i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 51.2389i − 2.03016i
$$638$$ 0 0
$$639$$ 9.21110i 0.364386i
$$640$$ 0 0
$$641$$ 0.422205 0.0166761 0.00833805 0.999965i $$-0.497346\pi$$
0.00833805 + 0.999965i $$0.497346\pi$$
$$642$$ 0 0
$$643$$ −9.63331 −0.379901 −0.189950 0.981794i $$-0.560833\pi$$
−0.189950 + 0.981794i $$0.560833\pi$$
$$644$$ − 6.42221i − 0.253070i
$$645$$ 0 0
$$646$$ 21.2111 0.834540
$$647$$ − 34.6056i − 1.36048i −0.732987 0.680242i $$-0.761875\pi$$
0.732987 0.680242i $$-0.238125\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −27.6333 −1.08303
$$652$$ −24.4222 −0.956447
$$653$$ − 39.2111i − 1.53445i −0.641379 0.767225i $$-0.721637\pi$$
0.641379 0.767225i $$-0.278363\pi$$
$$654$$ −1.39445 −0.0545273
$$655$$ 0 0
$$656$$ − 3.21110i − 0.125372i
$$657$$ −1.39445 −0.0544026
$$658$$ 42.4222 1.65379
$$659$$ 26.2389 1.02212 0.511060 0.859545i $$-0.329253\pi$$
0.511060 + 0.859545i $$0.329253\pi$$
$$660$$ 0 0
$$661$$ − 50.2389i − 1.95407i −0.213090 0.977033i $$-0.568353\pi$$
0.213090 0.977033i $$-0.431647\pi$$
$$662$$ 16.6056i 0.645393i
$$663$$ 16.6056i 0.644906i
$$664$$ −2.78890 −0.108230
$$665$$ 0 0
$$666$$ 9.21110 0.356923
$$667$$ − 6.42221i − 0.248669i
$$668$$ 9.21110 0.356388
$$669$$ − 1.81665i − 0.0702359i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.60555i − 0.177663i
$$673$$ 37.6333i 1.45066i 0.688403 + 0.725329i $$0.258312\pi$$
−0.688403 + 0.725329i $$0.741688\pi$$
$$674$$ 13.6333i 0.525135i
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ − 28.0555i − 1.07826i −0.842222 0.539130i $$-0.818753\pi$$
0.842222 0.539130i $$-0.181247\pi$$
$$678$$ −13.8167 −0.530625
$$679$$ −6.42221 −0.246462
$$680$$ 0 0
$$681$$ 24.0000i 0.919682i
$$682$$ 0 0
$$683$$ −9.21110 −0.352453 −0.176227 0.984350i $$-0.556389\pi$$
−0.176227 + 0.984350i $$0.556389\pi$$
$$684$$ 4.60555i 0.176098i
$$685$$ 0 0
$$686$$ −33.2111 −1.26801
$$687$$ 19.8167 0.756053
$$688$$ − 8.00000i − 0.304997i
$$689$$ −21.6333 −0.824163
$$690$$ 0 0
$$691$$ − 20.2389i − 0.769922i −0.922933 0.384961i $$-0.874215\pi$$
0.922933 0.384961i $$-0.125785\pi$$
$$692$$ 12.4222i 0.472221i
$$693$$ 0 0
$$694$$ − 27.6333i − 1.04895i
$$695$$ 0 0
$$696$$ − 4.60555i − 0.174573i
$$697$$ 14.7889 0.560169
$$698$$ − 7.81665i − 0.295865i
$$699$$ −1.81665 −0.0687122
$$700$$ 0 0
$$701$$ 47.0278 1.77621 0.888107 0.459637i $$-0.152020\pi$$
0.888107 + 0.459637i $$0.152020\pi$$
$$702$$ −3.60555 −0.136083
$$703$$ 42.4222i 1.59998i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 8.78890 0.330775
$$707$$ 34.0555 1.28079
$$708$$ 9.21110 0.346174
$$709$$ − 1.39445i − 0.0523696i −0.999657 0.0261848i $$-0.991664\pi$$
0.999657 0.0261848i $$-0.00833584\pi$$
$$710$$ 0 0
$$711$$ −14.4222 −0.540875
$$712$$ 15.2111i 0.570060i
$$713$$ 8.36669 0.313335
$$714$$ 21.2111 0.793806
$$715$$ 0 0
$$716$$ 19.8167 0.740583
$$717$$ 0 0
$$718$$ 15.6333i 0.583430i
$$719$$ 51.6333 1.92560 0.962799 0.270220i $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 0 0
$$721$$ − 18.4222i − 0.686079i
$$722$$ −2.21110 −0.0822887
$$723$$ −6.42221 −0.238844
$$724$$ 8.42221 0.313009
$$725$$ 0 0
$$726$$ 11.0000i 0.408248i
$$727$$ − 14.4222i − 0.534890i −0.963573 0.267445i $$-0.913821\pi$$
0.963573 0.267445i $$-0.0861794\pi$$
$$728$$ 16.6056i 0.615443i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 36.8444 1.36274
$$732$$ − 11.2111i − 0.414374i
$$733$$ 34.0555 1.25787 0.628935 0.777458i $$-0.283491\pi$$
0.628935 + 0.777458i $$0.283491\pi$$
$$734$$ − 19.6333i − 0.724679i
$$735$$ 0 0
$$736$$ 1.39445i 0.0514001i
$$737$$ 0 0
$$738$$ 3.21110i 0.118202i
$$739$$ 20.2389i 0.744498i 0.928133 + 0.372249i $$0.121413\pi$$
−0.928133 + 0.372249i $$0.878587\pi$$
$$740$$ 0 0
$$741$$ − 16.6056i − 0.610020i
$$742$$ 27.6333i 1.01445i
$$743$$ −36.8444 −1.35169 −0.675845 0.737044i $$-0.736221\pi$$
−0.675845 + 0.737044i $$0.736221\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ 20.4222i 0.747710i
$$747$$ 2.78890 0.102040
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.4222 −0.380312 −0.190156 0.981754i $$-0.560899\pi$$
−0.190156 + 0.981754i $$0.560899\pi$$
$$752$$ −9.21110 −0.335894
$$753$$ − 13.3944i − 0.488121i
$$754$$ 16.6056i 0.604739i
$$755$$ 0 0
$$756$$ 4.60555i 0.167502i
$$757$$ 12.7889i 0.464820i 0.972618 + 0.232410i $$0.0746612\pi$$
−0.972618 + 0.232410i $$0.925339\pi$$
$$758$$ 35.0278i 1.27227i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 33.6333i − 1.21921i −0.792707 0.609603i $$-0.791329\pi$$
0.792707 0.609603i $$-0.208671\pi$$
$$762$$ −1.21110 −0.0438736
$$763$$ − 6.42221i − 0.232499i
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 27.6333 0.998432
$$767$$ −33.2111 −1.19918
$$768$$ 1.00000i 0.0360844i
$$769$$ − 12.8444i − 0.463181i −0.972813 0.231591i $$-0.925607\pi$$
0.972813 0.231591i $$-0.0743930\pi$$
$$770$$ 0 0
$$771$$ 28.6056 1.03020
$$772$$ −7.81665 −0.281328
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ 8.00000i 0.287554i
$$775$$ 0 0
$$776$$ 1.39445 0.0500578
$$777$$ 42.4222i 1.52189i
$$778$$ −4.60555 −0.165117
$$779$$ −14.7889 −0.529867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −6.42221 −0.229658
$$783$$ 4.60555i 0.164589i
$$784$$ 14.2111 0.507539
$$785$$ 0 0
$$786$$ 22.6056i 0.806313i
$$787$$ 49.2666 1.75617 0.878083 0.478509i $$-0.158823\pi$$
0.878083 + 0.478509i $$0.158823\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −7.81665 −0.278280
$$790$$ 0 0
$$791$$ − 63.6333i − 2.26254i
$$792$$ 0 0
$$793$$ 40.4222i 1.43543i
$$794$$ −3.63331 −0.128941
$$795$$ 0 0
$$796$$ −22.4222 −0.794734
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ −21.2111 −0.750865
$$799$$ − 42.4222i − 1.50079i
$$800$$ 0 0
$$801$$ − 15.2111i − 0.537458i
$$802$$ − 8.78890i − 0.310347i
$$803$$ 0 0
$$804$$ − 3.21110i − 0.113247i
$$805$$ 0 0
$$806$$ −21.6333 −0.762001
$$807$$ − 25.8167i − 0.908789i
$$808$$ −7.39445 −0.260136
$$809$$ 6.84441 0.240637 0.120318 0.992735i $$-0.461608\pi$$
0.120318 + 0.992735i $$0.461608\pi$$
$$810$$ 0 0
$$811$$ − 32.2389i − 1.13206i −0.824385 0.566030i $$-0.808479\pi$$
0.824385 0.566030i $$-0.191521\pi$$
$$812$$ 21.2111 0.744364
$$813$$ −0.422205 −0.0148074
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −4.60555 −0.161227
$$817$$ −36.8444 −1.28902
$$818$$ − 14.7889i − 0.517082i
$$819$$ − 16.6056i − 0.580245i
$$820$$ 0 0
$$821$$ 3.21110i 0.112068i 0.998429 + 0.0560341i $$0.0178456\pi$$
−0.998429 + 0.0560341i $$0.982154\pi$$
$$822$$ 3.21110i 0.112000i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 4.00000i 0.139347i
$$825$$ 0 0
$$826$$ 42.4222i 1.47606i
$$827$$ −27.6333 −0.960904 −0.480452 0.877021i $$-0.659527\pi$$
−0.480452 + 0.877021i $$0.659527\pi$$
$$828$$ − 1.39445i − 0.0484604i
$$829$$ −46.8444 −1.62697 −0.813487 0.581583i $$-0.802433\pi$$
−0.813487 + 0.581583i $$0.802433\pi$$
$$830$$ 0 0
$$831$$ 16.4222 0.569680
$$832$$ − 3.60555i − 0.125000i
$$833$$ 65.4500i 2.26771i
$$834$$ − 17.2111i − 0.595972i
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −6.00000 −0.207390
$$838$$ −4.18335 −0.144511
$$839$$ − 18.4222i − 0.636005i −0.948090 0.318003i $$-0.896988\pi$$
0.948090 0.318003i $$-0.103012\pi$$
$$840$$ 0 0
$$841$$ −7.78890 −0.268583
$$842$$ 19.8167i 0.682927i
$$843$$ 27.2111 0.937200
$$844$$ −17.2111 −0.592431
$$845$$ 0 0
$$846$$ 9.21110 0.316684
$$847$$ −50.6611 −1.74073
$$848$$ − 6.00000i − 0.206041i
$$849$$ 10.4222 0.357689
$$850$$ 0 0
$$851$$ − 12.8444i − 0.440301i
$$852$$ 9.21110 0.315567
$$853$$ 14.7889 0.506362 0.253181 0.967419i $$-0.418523\pi$$
0.253181 + 0.967419i $$0.418523\pi$$
$$854$$ 51.6333 1.76686
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 23.0278i − 0.786613i −0.919407 0.393307i $$-0.871331\pi$$
0.919407 0.393307i $$-0.128669\pi$$
$$858$$ 0 0
$$859$$ −25.2111 −0.860192 −0.430096 0.902783i $$-0.641520\pi$$
−0.430096 + 0.902783i $$0.641520\pi$$
$$860$$ 0 0
$$861$$ −14.7889 −0.504004
$$862$$ − 12.0000i − 0.408722i
$$863$$ −51.6333 −1.75762 −0.878809 0.477173i $$-0.841661\pi$$
−0.878809 + 0.477173i $$0.841661\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 0 0
$$866$$ − 19.2111i − 0.652820i
$$867$$ − 4.21110i − 0.143017i
$$868$$ 27.6333i 0.937936i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 11.5778i 0.392299i
$$872$$ 1.39445i 0.0472220i
$$873$$ −1.39445 −0.0471949
$$874$$ 6.42221 0.217234
$$875$$ 0 0
$$876$$ 1.39445i 0.0471141i
$$877$$ −24.8444 −0.838936 −0.419468 0.907770i $$-0.637783\pi$$
−0.419468 + 0.907770i $$0.637783\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ − 18.0000i − 0.607125i
$$880$$ 0 0
$$881$$ 39.2111 1.32106 0.660528 0.750802i $$-0.270333\pi$$
0.660528 + 0.750802i $$0.270333\pi$$
$$882$$ −14.2111 −0.478513
$$883$$ − 9.57779i − 0.322318i −0.986928 0.161159i $$-0.948477\pi$$
0.986928 0.161159i $$-0.0515233\pi$$
$$884$$ 16.6056 0.558505
$$885$$ 0 0
$$886$$ − 15.6333i − 0.525211i
$$887$$ 6.97224i 0.234105i 0.993126 + 0.117053i $$0.0373446\pi$$
−0.993126 + 0.117053i $$0.962655\pi$$
$$888$$ − 9.21110i − 0.309104i
$$889$$ − 5.57779i − 0.187073i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −1.81665 −0.0608261
$$893$$ 42.4222i 1.41960i
$$894$$ −15.2111 −0.508735
$$895$$ 0 0
$$896$$ −4.60555 −0.153861
$$897$$ 5.02776i 0.167872i
$$898$$ 33.6333i 1.12236i
$$899$$ 27.6333i 0.921622i
$$900$$ 0 0
$$901$$ 27.6333 0.920599
$$902$$ 0 0
$$903$$ −36.8444 −1.22611
$$904$$ 13.8167i 0.459535i
$$905$$ 0 0
$$906$$ 6.00000 0.199337
$$907$$ 21.5778i 0.716479i 0.933630 + 0.358239i $$0.116623\pi$$
−0.933630 + 0.358239i $$0.883377\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 7.39445 0.245258
$$910$$ 0 0
$$911$$ −27.6333 −0.915532 −0.457766 0.889073i $$-0.651350\pi$$
−0.457766 + 0.889073i $$0.651350\pi$$
$$912$$ 4.60555 0.152505
$$913$$ 0 0
$$914$$ 38.2389 1.26483
$$915$$ 0 0
$$916$$ − 19.8167i − 0.654761i
$$917$$ −104.111 −3.43805
$$918$$ 4.60555 0.152006
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ − 8.78890i − 0.289604i
$$922$$ 33.6333i 1.10765i
$$923$$ −33.2111 −1.09316
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 31.3944 1.03169
$$927$$ − 4.00000i − 0.131377i
$$928$$ −4.60555 −0.151185
$$929$$ 39.2111i 1.28647i 0.765667 + 0.643237i $$0.222409\pi$$
−0.765667 + 0.643237i $$0.777591\pi$$
$$930$$ 0 0
$$931$$ − 65.4500i − 2.14504i
$$932$$ 1.81665i 0.0595065i
$$933$$ 12.0000i 0.392862i
$$934$$ − 30.4222i − 0.995445i
$$935$$ 0 0
$$936$$ 3.60555i 0.117851i
$$937$$ − 10.3667i − 0.338665i −0.985559 0.169333i $$-0.945839\pi$$
0.985559 0.169333i $$-0.0541612\pi$$
$$938$$ 14.7889 0.482875
$$939$$ 3.57779 0.116757
$$940$$ 0 0
$$941$$ 54.0000i 1.76035i 0.474650 + 0.880175i $$0.342575\pi$$
−0.474650 + 0.880175i $$0.657425\pi$$
$$942$$ −20.4222 −0.665391
$$943$$ 4.47772 0.145815
$$944$$ − 9.21110i − 0.299796i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −15.6333 −0.508014 −0.254007 0.967202i $$-0.581749\pi$$
−0.254007 + 0.967202i $$0.581749\pi$$
$$948$$ 14.4222i 0.468411i
$$949$$ − 5.02776i − 0.163208i
$$950$$ 0 0
$$951$$ − 18.0000i − 0.583690i
$$952$$ − 21.2111i − 0.687456i
$$953$$ 20.2389i 0.655601i 0.944747 + 0.327800i $$0.106307\pi$$
−0.944747 + 0.327800i $$0.893693\pi$$
$$954$$ 6.00000i 0.194257i
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 5.57779i 0.180210i
$$959$$ −14.7889 −0.477558
$$960$$ 0 0
$$961$$ −5.00000 −0.161290
$$962$$ 33.2111i 1.07077i
$$963$$ 0 0
$$964$$ 6.42221i 0.206845i
$$965$$ 0 0
$$966$$ 6.42221 0.206631
$$967$$ −8.23886 −0.264944 −0.132472 0.991187i $$-0.542291\pi$$
−0.132472 + 0.991187i $$0.542291\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 21.2111i 0.681399i
$$970$$ 0 0
$$971$$ −53.0278 −1.70174 −0.850871 0.525375i $$-0.823925\pi$$
−0.850871 + 0.525375i $$0.823925\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 79.2666 2.54117
$$974$$ −0.972244 −0.0311527
$$975$$ 0 0
$$976$$ −11.2111 −0.358859
$$977$$ 18.8444 0.602886 0.301443 0.953484i $$-0.402532\pi$$
0.301443 + 0.953484i $$0.402532\pi$$
$$978$$ − 24.4222i − 0.780936i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ − 1.39445i − 0.0445213i
$$982$$ 7.81665 0.249439
$$983$$ 42.4222 1.35306 0.676529 0.736416i $$-0.263483\pi$$
0.676529 + 0.736416i $$0.263483\pi$$
$$984$$ 3.21110 0.102366
$$985$$ 0 0
$$986$$ − 21.2111i − 0.675499i
$$987$$ 42.4222i 1.35031i
$$988$$ −16.6056 −0.528293
$$989$$ 11.1556 0.354727
$$990$$ 0 0
$$991$$ −22.4222 −0.712265 −0.356132 0.934436i $$-0.615905\pi$$
−0.356132 + 0.934436i $$0.615905\pi$$
$$992$$ − 6.00000i − 0.190500i
$$993$$ −16.6056 −0.526961
$$994$$ 42.4222i 1.34555i
$$995$$ 0 0
$$996$$ − 2.78890i − 0.0883696i
$$997$$ − 16.4222i − 0.520096i −0.965596 0.260048i $$-0.916262\pi$$
0.965596 0.260048i $$-0.0837385\pi$$
$$998$$ 23.0278i 0.728931i
$$999$$ 9.21110i 0.291426i
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 1950.2.f.n.649.3 4
5.2 odd 4 1950.2.b.k.1351.3 4
5.3 odd 4 390.2.b.c.181.2 4
5.4 even 2 1950.2.f.m.649.2 4
13.12 even 2 1950.2.f.m.649.4 4
15.8 even 4 1170.2.b.d.181.4 4
20.3 even 4 3120.2.g.q.961.1 4
65.8 even 4 5070.2.a.bf.1.2 2
65.12 odd 4 1950.2.b.k.1351.2 4
65.18 even 4 5070.2.a.z.1.1 2
65.38 odd 4 390.2.b.c.181.3 yes 4
65.64 even 2 inner 1950.2.f.n.649.1 4
195.38 even 4 1170.2.b.d.181.1 4
260.103 even 4 3120.2.g.q.961.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 5.3 odd 4
390.2.b.c.181.3 yes 4 65.38 odd 4
1170.2.b.d.181.1 4 195.38 even 4
1170.2.b.d.181.4 4 15.8 even 4
1950.2.b.k.1351.2 4 65.12 odd 4
1950.2.b.k.1351.3 4 5.2 odd 4
1950.2.f.m.649.2 4 5.4 even 2
1950.2.f.m.649.4 4 13.12 even 2
1950.2.f.n.649.1 4 65.64 even 2 inner
1950.2.f.n.649.3 4 1.1 even 1 trivial
3120.2.g.q.961.1 4 20.3 even 4
3120.2.g.q.961.4 4 260.103 even 4
5070.2.a.z.1.1 2 65.18 even 4
5070.2.a.bf.1.2 2 65.8 even 4