# Properties

 Label 1950.2.b.k.1351.2 Level $1950$ Weight $2$ Character 1950.1351 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.b (of order $$2$$, degree $$1$$, 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 1351.2 Root $$2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1351 Dual form 1950.2.b.k.1351.3

## $q$-expansion

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

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ − 1.00000i − 0.408248i
$$7$$ 4.60555i 1.74073i 0.492403 + 0.870367i $$0.336119\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −3.60555 −1.00000
$$14$$ 4.60555 1.23089
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.60555 −1.11701 −0.558505 0.829501i $$-0.688625\pi$$
−0.558505 + 0.829501i $$0.688625\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ − 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
$$23$$ 1.39445 0.290763 0.145381 0.989376i $$-0.453559\pi$$
0.145381 + 0.989376i $$0.453559\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ 0 0
$$26$$ 3.60555i 0.707107i
$$27$$ 1.00000 0.192450
$$28$$ − 4.60555i − 0.870367i
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ 0 0
$$31$$ 6.00000i 1.07763i 0.842424 + 0.538816i $$0.181128\pi$$
−0.842424 + 0.538816i $$0.818872\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 4.60555i 0.789846i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 9.21110i 1.51430i 0.653243 + 0.757148i $$0.273408\pi$$
−0.653243 + 0.757148i $$0.726592\pi$$
$$38$$ −4.60555 −0.747119
$$39$$ −3.60555 −0.577350
$$40$$ 0 0
$$41$$ 3.21110i 0.501490i 0.968053 + 0.250745i $$0.0806756\pi$$
−0.968053 + 0.250745i $$0.919324\pi$$
$$42$$ 4.60555 0.710652
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ − 1.39445i − 0.205600i
$$47$$ 9.21110i 1.34358i 0.740743 + 0.671789i $$0.234474\pi$$
−0.740743 + 0.671789i $$0.765526\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −14.2111 −2.03016
$$50$$ 0 0
$$51$$ −4.60555 −0.644906
$$52$$ 3.60555 0.500000
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0 0
$$56$$ −4.60555 −0.615443
$$57$$ − 4.60555i − 0.610020i
$$58$$ − 4.60555i − 0.604739i
$$59$$ − 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 $$0.754813\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 4.60555i 0.580245i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.21110i 0.392299i 0.980574 + 0.196149i $$0.0628437\pi$$
−0.980574 + 0.196149i $$0.937156\pi$$
$$68$$ 4.60555 0.558505
$$69$$ 1.39445 0.167872
$$70$$ 0 0
$$71$$ 9.21110i 1.09316i 0.837408 + 0.546578i $$0.184070\pi$$
−0.837408 + 0.546578i $$0.815930\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 1.39445i 0.163208i 0.996665 + 0.0816039i $$0.0260043\pi$$
−0.996665 + 0.0816039i $$0.973996\pi$$
$$74$$ 9.21110 1.07077
$$75$$ 0 0
$$76$$ 4.60555i 0.528293i
$$77$$ 0 0
$$78$$ 3.60555i 0.408248i
$$79$$ −14.4222 −1.62262 −0.811312 0.584613i $$-0.801246\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.21110 0.354607
$$83$$ − 2.78890i − 0.306121i −0.988217 0.153061i $$-0.951087\pi$$
0.988217 0.153061i $$-0.0489130\pi$$
$$84$$ − 4.60555i − 0.502507i
$$85$$ 0 0
$$86$$ 8.00000i 0.862662i
$$87$$ 4.60555 0.493767
$$88$$ 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
$$92$$ −1.39445 −0.145381
$$93$$ 6.00000i 0.622171i
$$94$$ 9.21110 0.950053
$$95$$ 0 0
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 1.39445i − 0.141585i −0.997491 0.0707924i $$-0.977447\pi$$
0.997491 0.0707924i $$-0.0225528\pi$$
$$98$$ 14.2111i 1.43554i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.39445 −0.735775 −0.367888 0.929870i $$-0.619919\pi$$
−0.367888 + 0.929870i $$0.619919\pi$$
$$102$$ 4.60555i 0.456018i
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ − 3.60555i − 0.353553i
$$105$$ 0 0
$$106$$ 6.00000i 0.582772i
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 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
$$112$$ 4.60555i 0.435184i
$$113$$ 13.8167 1.29976 0.649881 0.760036i $$-0.274819\pi$$
0.649881 + 0.760036i $$0.274819\pi$$
$$114$$ −4.60555 −0.431349
$$115$$ 0 0
$$116$$ −4.60555 −0.427615
$$117$$ −3.60555 −0.333333
$$118$$ −9.21110 −0.847951
$$119$$ − 21.2111i − 1.94442i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 11.2111i 1.01501i
$$123$$ 3.21110i 0.289535i
$$124$$ − 6.00000i − 0.538816i
$$125$$ 0 0
$$126$$ 4.60555 0.410295
$$127$$ −1.21110 −0.107468 −0.0537340 0.998555i $$-0.517112\pi$$
−0.0537340 + 0.998555i $$0.517112\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −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
$$133$$ 21.2111 1.83924
$$134$$ 3.21110 0.277397
$$135$$ 0 0
$$136$$ − 4.60555i − 0.394923i
$$137$$ − 3.21110i − 0.274343i −0.990547 0.137172i $$-0.956199\pi$$
0.990547 0.137172i $$-0.0438011\pi$$
$$138$$ − 1.39445i − 0.118703i
$$139$$ 17.2111 1.45983 0.729913 0.683540i $$-0.239560\pi$$
0.729913 + 0.683540i $$0.239560\pi$$
$$140$$ 0 0
$$141$$ 9.21110i 0.775715i
$$142$$ 9.21110 0.772979
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 1.39445 0.115405
$$147$$ −14.2111 −1.17211
$$148$$ − 9.21110i − 0.757148i
$$149$$ 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 $$0.0785045\pi$$
−0.969741 + 0.244137i $$0.921495\pi$$
$$152$$ 4.60555 0.373560
$$153$$ −4.60555 −0.372337
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.60555 0.288675
$$157$$ −20.4222 −1.62987 −0.814935 0.579553i $$-0.803227\pi$$
−0.814935 + 0.579553i $$0.803227\pi$$
$$158$$ 14.4222i 1.14737i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 6.42221i 0.506141i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 24.4222i − 1.91289i −0.291905 0.956447i $$-0.594289\pi$$
0.291905 0.956447i $$-0.405711\pi$$
$$164$$ − 3.21110i − 0.250745i
$$165$$ 0 0
$$166$$ −2.78890 −0.216460
$$167$$ − 9.21110i − 0.712777i −0.934338 0.356388i $$-0.884008\pi$$
0.934338 0.356388i $$-0.115992\pi$$
$$168$$ −4.60555 −0.355326
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ − 4.60555i − 0.352195i
$$172$$ 8.00000 0.609994
$$173$$ 12.4222 0.944443 0.472221 0.881480i $$-0.343452\pi$$
0.472221 + 0.881480i $$0.343452\pi$$
$$174$$ − 4.60555i − 0.349146i
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 9.21110i − 0.692349i
$$178$$ 15.2111 1.14012
$$179$$ −19.8167 −1.48117 −0.740583 0.671965i $$-0.765451\pi$$
−0.740583 + 0.671965i $$0.765451\pi$$
$$180$$ 0 0
$$181$$ 8.42221 0.626018 0.313009 0.949750i $$-0.398663\pi$$
0.313009 + 0.949750i $$0.398663\pi$$
$$182$$ −16.6056 −1.23089
$$183$$ −11.2111 −0.828749
$$184$$ 1.39445i 0.102800i
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ 0 0
$$188$$ − 9.21110i − 0.671789i
$$189$$ 4.60555i 0.335005i
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 7.81665i − 0.562655i −0.959612 0.281328i $$-0.909225\pi$$
0.959612 0.281328i $$-0.0907747\pi$$
$$194$$ −1.39445 −0.100116
$$195$$ 0 0
$$196$$ 14.2111 1.01508
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 22.4222 1.58947 0.794734 0.606958i $$-0.207610\pi$$
0.794734 + 0.606958i $$0.207610\pi$$
$$200$$ 0 0
$$201$$ 3.21110i 0.226494i
$$202$$ 7.39445i 0.520272i
$$203$$ 21.2111i 1.48873i
$$204$$ 4.60555 0.322453
$$205$$ 0 0
$$206$$ − 4.00000i − 0.278693i
$$207$$ 1.39445 0.0969209
$$208$$ −3.60555 −0.250000
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −17.2111 −1.18486 −0.592431 0.805622i $$-0.701832\pi$$
−0.592431 + 0.805622i $$0.701832\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 9.21110i 0.631134i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000i 0.0680414i
$$217$$ −27.6333 −1.87587
$$218$$ 1.39445 0.0944440
$$219$$ 1.39445i 0.0942281i
$$220$$ 0 0
$$221$$ 16.6056 1.11701
$$222$$ 9.21110 0.618209
$$223$$ − 1.81665i − 0.121652i −0.998148 0.0608261i $$-0.980627\pi$$
0.998148 0.0608261i $$-0.0193735\pi$$
$$224$$ 4.60555 0.307721
$$225$$ 0 0
$$226$$ − 13.8167i − 0.919070i
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ 4.60555i 0.305010i
$$229$$ − 19.8167i − 1.30952i −0.755836 0.654761i $$-0.772769\pi$$
0.755836 0.654761i $$-0.227231\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 4.60555i 0.302369i
$$233$$ 1.81665 0.119013 0.0595065 0.998228i $$-0.481047\pi$$
0.0595065 + 0.998228i $$0.481047\pi$$
$$234$$ 3.60555i 0.235702i
$$235$$ 0 0
$$236$$ 9.21110i 0.599592i
$$237$$ −14.4222 −0.936823
$$238$$ −21.2111 −1.37491
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ − 6.42221i − 0.413691i −0.978374 0.206845i $$-0.933680\pi$$
0.978374 0.206845i $$-0.0663197\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000 0.0641500
$$244$$ 11.2111 0.717717
$$245$$ 0 0
$$246$$ 3.21110 0.204732
$$247$$ 16.6056i 1.05659i
$$248$$ −6.00000 −0.381000
$$249$$ − 2.78890i − 0.176739i
$$250$$ 0 0
$$251$$ −13.3944 −0.845450 −0.422725 0.906258i $$-0.638926\pi$$
−0.422725 + 0.906258i $$0.638926\pi$$
$$252$$ − 4.60555i − 0.290122i
$$253$$ 0 0
$$254$$ 1.21110i 0.0759913i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 28.6056 1.78437 0.892183 0.451675i $$-0.149173\pi$$
0.892183 + 0.451675i $$0.149173\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −42.4222 −2.63599
$$260$$ 0 0
$$261$$ 4.60555 0.285076
$$262$$ − 22.6056i − 1.39658i
$$263$$ 7.81665 0.481996 0.240998 0.970526i $$-0.422525\pi$$
0.240998 + 0.970526i $$0.422525\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ − 21.2111i − 1.30054i
$$267$$ 15.2111i 0.930904i
$$268$$ − 3.21110i − 0.196149i
$$269$$ 25.8167 1.57407 0.787035 0.616909i $$-0.211615\pi$$
0.787035 + 0.616909i $$0.211615\pi$$
$$270$$ 0 0
$$271$$ − 0.422205i − 0.0256471i −0.999918 0.0128236i $$-0.995918\pi$$
0.999918 0.0128236i $$-0.00408198\pi$$
$$272$$ −4.60555 −0.279253
$$273$$ − 16.6056i − 1.00501i
$$274$$ −3.21110 −0.193990
$$275$$ 0 0
$$276$$ −1.39445 −0.0839359
$$277$$ 16.4222 0.986715 0.493357 0.869827i $$-0.335770\pi$$
0.493357 + 0.869827i $$0.335770\pi$$
$$278$$ − 17.2111i − 1.03225i
$$279$$ 6.00000i 0.359211i
$$280$$ 0 0
$$281$$ 27.2111i 1.62328i 0.584159 + 0.811639i $$0.301424\pi$$
−0.584159 + 0.811639i $$0.698576\pi$$
$$282$$ 9.21110 0.548513
$$283$$ −10.4222 −0.619536 −0.309768 0.950812i $$-0.600251\pi$$
−0.309768 + 0.950812i $$0.600251\pi$$
$$284$$ − 9.21110i − 0.546578i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −14.7889 −0.872961
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 4.21110 0.247712
$$290$$ 0 0
$$291$$ − 1.39445i − 0.0817440i
$$292$$ − 1.39445i − 0.0816039i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ 14.2111i 0.828808i
$$295$$ 0 0
$$296$$ −9.21110 −0.535384
$$297$$ 0 0
$$298$$ 15.2111 0.881156
$$299$$ −5.02776 −0.290763
$$300$$ 0 0
$$301$$ − 36.8444i − 2.12368i
$$302$$ 6.00000 0.345261
$$303$$ −7.39445 −0.424800
$$304$$ − 4.60555i − 0.264146i
$$305$$ 0 0
$$306$$ 4.60555i 0.263282i
$$307$$ 8.78890i 0.501609i 0.968038 + 0.250804i $$0.0806951\pi$$
−0.968038 + 0.250804i $$0.919305\pi$$
$$308$$ 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 $$0.389495\pi$$
$$312$$ − 3.60555i − 0.204124i
$$313$$ −3.57779 −0.202229 −0.101114 0.994875i $$-0.532241\pi$$
−0.101114 + 0.994875i $$0.532241\pi$$
$$314$$ 20.4222i 1.15249i
$$315$$ 0 0
$$316$$ 14.4222 0.811312
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 6.42221 0.357895
$$323$$ 21.2111i 1.18022i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −24.4222 −1.35262
$$327$$ 1.39445i 0.0771132i
$$328$$ −3.21110 −0.177303
$$329$$ −42.4222 −2.33881
$$330$$ 0 0
$$331$$ − 16.6056i − 0.912724i −0.889794 0.456362i $$-0.849152\pi$$
0.889794 0.456362i $$-0.150848\pi$$
$$332$$ 2.78890i 0.153061i
$$333$$ 9.21110i 0.504765i
$$334$$ −9.21110 −0.504009
$$335$$ 0 0
$$336$$ 4.60555i 0.251253i
$$337$$ −13.6333 −0.742654 −0.371327 0.928502i $$-0.621097\pi$$
−0.371327 + 0.928502i $$0.621097\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ 13.8167 0.750418
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −4.60555 −0.249040
$$343$$ − 33.2111i − 1.79323i
$$344$$ − 8.00000i − 0.431331i
$$345$$ 0 0
$$346$$ − 12.4222i − 0.667822i
$$347$$ 27.6333 1.48343 0.741717 0.670713i $$-0.234012\pi$$
0.741717 + 0.670713i $$0.234012\pi$$
$$348$$ −4.60555 −0.246883
$$349$$ − 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
$$353$$ 8.78890i 0.467786i 0.972262 + 0.233893i $$0.0751465\pi$$
−0.972262 + 0.233893i $$0.924853\pi$$
$$354$$ −9.21110 −0.489565
$$355$$ 0 0
$$356$$ − 15.2111i − 0.806187i
$$357$$ − 21.2111i − 1.12261i
$$358$$ 19.8167i 1.04734i
$$359$$ 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
$$362$$ − 8.42221i − 0.442661i
$$363$$ 11.0000 0.577350
$$364$$ 16.6056i 0.870367i
$$365$$ 0 0
$$366$$ 11.2111i 0.586014i
$$367$$ 19.6333 1.02485 0.512425 0.858732i $$-0.328747\pi$$
0.512425 + 0.858732i $$0.328747\pi$$
$$368$$ 1.39445 0.0726907
$$369$$ 3.21110i 0.167163i
$$370$$ 0 0
$$371$$ − 27.6333i − 1.43465i
$$372$$ − 6.00000i − 0.311086i
$$373$$ 20.4222 1.05742 0.528711 0.848802i $$-0.322676\pi$$
0.528711 + 0.848802i $$0.322676\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.21110 −0.475026
$$377$$ −16.6056 −0.855229
$$378$$ 4.60555 0.236884
$$379$$ 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
$$382$$ 12.0000i 0.613973i
$$383$$ 27.6333i 1.41200i 0.708214 + 0.705998i $$0.249501\pi$$
−0.708214 + 0.705998i $$0.750499\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ −7.81665 −0.397857
$$387$$ −8.00000 −0.406663
$$388$$ 1.39445i 0.0707924i
$$389$$ 4.60555 0.233511 0.116755 0.993161i $$-0.462751\pi$$
0.116755 + 0.993161i $$0.462751\pi$$
$$390$$ 0 0
$$391$$ −6.42221 −0.324785
$$392$$ − 14.2111i − 0.717769i
$$393$$ 22.6056 1.14030
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 3.63331i 0.182350i 0.995835 + 0.0911752i $$0.0290623\pi$$
−0.995835 + 0.0911752i $$0.970938\pi$$
$$398$$ − 22.4222i − 1.12392i
$$399$$ 21.2111 1.06188
$$400$$ 0 0
$$401$$ 8.78890i 0.438897i 0.975624 + 0.219448i $$0.0704257\pi$$
−0.975624 + 0.219448i $$0.929574\pi$$
$$402$$ 3.21110 0.160155
$$403$$ − 21.6333i − 1.07763i
$$404$$ 7.39445 0.367888
$$405$$ 0 0
$$406$$ 21.2111 1.05269
$$407$$ 0 0
$$408$$ − 4.60555i − 0.228009i
$$409$$ − 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
$$412$$ −4.00000 −0.197066
$$413$$ 42.4222 2.08746
$$414$$ − 1.39445i − 0.0685334i
$$415$$ 0 0
$$416$$ 3.60555i 0.176777i
$$417$$ 17.2111 0.842831
$$418$$ 0 0
$$419$$ 4.18335 0.204370 0.102185 0.994765i $$-0.467417\pi$$
0.102185 + 0.994765i $$0.467417\pi$$
$$420$$ 0 0
$$421$$ − 19.8167i − 0.965805i −0.875674 0.482902i $$-0.839583\pi$$
0.875674 0.482902i $$-0.160417\pi$$
$$422$$ 17.2111i 0.837823i
$$423$$ 9.21110i 0.447859i
$$424$$ − 6.00000i − 0.291386i
$$425$$ 0 0
$$426$$ 9.21110 0.446279
$$427$$ − 51.6333i − 2.49871i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −19.2111 −0.923227 −0.461613 0.887081i $$-0.652729\pi$$
−0.461613 + 0.887081i $$0.652729\pi$$
$$434$$ 27.6333i 1.32644i
$$435$$ 0 0
$$436$$ − 1.39445i − 0.0667820i
$$437$$ − 6.42221i − 0.307216i
$$438$$ 1.39445 0.0666293
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −14.2111 −0.676719
$$442$$ − 16.6056i − 0.789846i
$$443$$ −15.6333 −0.742761 −0.371380 0.928481i $$-0.621115\pi$$
−0.371380 + 0.928481i $$0.621115\pi$$
$$444$$ − 9.21110i − 0.437140i
$$445$$ 0 0
$$446$$ −1.81665 −0.0860211
$$447$$ 15.2111i 0.719460i
$$448$$ − 4.60555i − 0.217592i
$$449$$ 33.6333i 1.58725i 0.608405 + 0.793627i $$0.291810\pi$$
−0.608405 + 0.793627i $$0.708190\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −13.8167 −0.649881
$$453$$ 6.00000i 0.281905i
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ 4.60555 0.215675
$$457$$ − 38.2389i − 1.78874i −0.447330 0.894369i $$-0.647625\pi$$
0.447330 0.894369i $$-0.352375\pi$$
$$458$$ −19.8167 −0.925971
$$459$$ −4.60555 −0.214969
$$460$$ 0 0
$$461$$ − 33.6333i − 1.56646i −0.621733 0.783230i $$-0.713571\pi$$
0.621733 0.783230i $$-0.286429\pi$$
$$462$$ 0 0
$$463$$ 31.3944i 1.45902i 0.683968 + 0.729512i $$0.260253\pi$$
−0.683968 + 0.729512i $$0.739747\pi$$
$$464$$ 4.60555 0.213807
$$465$$ 0 0
$$466$$ − 1.81665i − 0.0841549i
$$467$$ 30.4222 1.40777 0.703886 0.710313i $$-0.251447\pi$$
0.703886 + 0.710313i $$0.251447\pi$$
$$468$$ 3.60555 0.166667
$$469$$ −14.7889 −0.682888
$$470$$ 0 0
$$471$$ −20.4222 −0.941006
$$472$$ 9.21110 0.423975
$$473$$ 0 0
$$474$$ 14.4222i 0.662434i
$$475$$ 0 0
$$476$$ 21.2111i 0.972209i
$$477$$ −6.00000 −0.274721
$$478$$ 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
$$482$$ −6.42221 −0.292523
$$483$$ 6.42221i 0.292220i
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 0.972244i 0.0440566i 0.999757 + 0.0220283i $$0.00701239\pi$$
−0.999757 + 0.0220283i $$0.992988\pi$$
$$488$$ − 11.2111i − 0.507503i
$$489$$ − 24.4222i − 1.10441i
$$490$$ 0 0
$$491$$ 7.81665 0.352761 0.176380 0.984322i $$-0.443561\pi$$
0.176380 + 0.984322i $$0.443561\pi$$
$$492$$ − 3.21110i − 0.144768i
$$493$$ −21.2111 −0.955300
$$494$$ 16.6056 0.747119
$$495$$ 0 0
$$496$$ 6.00000i 0.269408i
$$497$$ −42.4222 −1.90290
$$498$$ −2.78890 −0.124973
$$499$$ 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
$$502$$ 13.3944i 0.597824i
$$503$$ −23.4500 −1.04558 −0.522791 0.852461i $$-0.675109\pi$$
−0.522791 + 0.852461i $$0.675109\pi$$
$$504$$ −4.60555 −0.205148
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 1.21110 0.0537340
$$509$$ − 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
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 4.60555i − 0.203340i
$$514$$ − 28.6056i − 1.26174i
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 42.4222i 1.86392i
$$519$$ 12.4222 0.545274
$$520$$ 0 0
$$521$$ 21.6333 0.947772 0.473886 0.880586i $$-0.342851\pi$$
0.473886 + 0.880586i $$0.342851\pi$$
$$522$$ − 4.60555i − 0.201580i
$$523$$ −32.8444 −1.43619 −0.718093 0.695947i $$-0.754985\pi$$
−0.718093 + 0.695947i $$0.754985\pi$$
$$524$$ −22.6056 −0.987528
$$525$$ 0 0
$$526$$ − 7.81665i − 0.340822i
$$527$$ − 27.6333i − 1.20373i
$$528$$ 0 0
$$529$$ −21.0555 −0.915457
$$530$$ 0 0
$$531$$ − 9.21110i − 0.399728i
$$532$$ −21.2111 −0.919618
$$533$$ − 11.5778i − 0.501490i
$$534$$ 15.2111 0.658249
$$535$$ 0 0
$$536$$ −3.21110 −0.138699
$$537$$ −19.8167 −0.855152
$$538$$ − 25.8167i − 1.11303i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6.97224i 0.299760i 0.988704 + 0.149880i $$0.0478888\pi$$
−0.988704 + 0.149880i $$0.952111\pi$$
$$542$$ −0.422205 −0.0181353
$$543$$ 8.42221 0.361431
$$544$$ 4.60555i 0.197461i
$$545$$ 0 0
$$546$$ −16.6056 −0.710652
$$547$$ −14.4222 −0.616649 −0.308324 0.951281i $$-0.599768\pi$$
−0.308324 + 0.951281i $$0.599768\pi$$
$$548$$ 3.21110i 0.137172i
$$549$$ −11.2111 −0.478478
$$550$$ 0 0
$$551$$ − 21.2111i − 0.903623i
$$552$$ 1.39445i 0.0593517i
$$553$$ − 66.4222i − 2.82456i
$$554$$ − 16.4222i − 0.697713i
$$555$$ 0 0
$$556$$ −17.2111 −0.729913
$$557$$ − 11.5778i − 0.490567i −0.969451 0.245283i $$-0.921119\pi$$
0.969451 0.245283i $$-0.0788810\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 28.8444 1.21999
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 27.2111 1.14783
$$563$$ −34.0555 −1.43527 −0.717634 0.696420i $$-0.754775\pi$$
−0.717634 + 0.696420i $$0.754775\pi$$
$$564$$ − 9.21110i − 0.387857i
$$565$$ 0 0
$$566$$ 10.4222i 0.438078i
$$567$$ 4.60555i 0.193415i
$$568$$ −9.21110 −0.386489
$$569$$ 33.6333 1.40998 0.704991 0.709216i $$-0.250951\pi$$
0.704991 + 0.709216i $$0.250951\pi$$
$$570$$ 0 0
$$571$$ −30.0555 −1.25778 −0.628892 0.777493i $$-0.716491\pi$$
−0.628892 + 0.777493i $$0.716491\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ 14.7889i 0.617277i
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ 37.3944i 1.55675i 0.627799 + 0.778376i $$0.283956\pi$$
−0.627799 + 0.778376i $$0.716044\pi$$
$$578$$ − 4.21110i − 0.175159i
$$579$$ − 7.81665i − 0.324849i
$$580$$ 0 0
$$581$$ 12.8444 0.532876
$$582$$ −1.39445 −0.0578018
$$583$$ 0 0
$$584$$ −1.39445 −0.0577027
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 6.42221i 0.265073i 0.991178 + 0.132536i $$0.0423121\pi$$
−0.991178 + 0.132536i $$0.957688\pi$$
$$588$$ 14.2111 0.586056
$$589$$ 27.6333 1.13861
$$590$$ 0 0
$$591$$ 6.00000i 0.246807i
$$592$$ 9.21110i 0.378574i
$$593$$ 24.4222i 1.00290i 0.865187 + 0.501450i $$0.167200\pi$$
−0.865187 + 0.501450i $$0.832800\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ − 15.2111i − 0.623071i
$$597$$ 22.4222 0.917680
$$598$$ 5.02776i 0.205600i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ −36.8444 −1.50167
$$603$$ 3.21110i 0.130766i
$$604$$ − 6.00000i − 0.244137i
$$605$$ 0 0
$$606$$ 7.39445i 0.300379i
$$607$$ −17.2111 −0.698577 −0.349289 0.937015i $$-0.613577\pi$$
−0.349289 + 0.937015i $$0.613577\pi$$
$$608$$ −4.60555 −0.186780
$$609$$ 21.2111i 0.859517i
$$610$$ 0 0
$$611$$ − 33.2111i − 1.34358i
$$612$$ 4.60555 0.186168
$$613$$ 33.2111i 1.34138i 0.741736 + 0.670692i $$0.234003\pi$$
−0.741736 + 0.670692i $$0.765997\pi$$
$$614$$ 8.78890 0.354691
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 12.4222i − 0.500099i −0.968233 0.250050i $$-0.919553\pi$$
0.968233 0.250050i $$-0.0804469\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ 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
$$622$$ − 12.0000i − 0.481156i
$$623$$ −70.0555 −2.80671
$$624$$ −3.60555 −0.144338
$$625$$ 0 0
$$626$$ 3.57779i 0.142997i
$$627$$ 0 0
$$628$$ 20.4222 0.814935
$$629$$ − 42.4222i − 1.69148i
$$630$$ 0 0
$$631$$ 3.21110i 0.127832i 0.997955 + 0.0639160i $$0.0203590\pi$$
−0.997955 + 0.0639160i $$0.979641\pi$$
$$632$$ − 14.4222i − 0.573685i
$$633$$ −17.2111 −0.684080
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 51.2389 2.03016
$$638$$ 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
$$643$$ − 9.63331i − 0.379901i −0.981794 0.189950i $$-0.939167\pi$$
0.981794 0.189950i $$-0.0608327\pi$$
$$644$$ − 6.42221i − 0.253070i
$$645$$ 0 0
$$646$$ 21.2111 0.834540
$$647$$ 34.6056 1.36048 0.680242 0.732987i $$-0.261875\pi$$
0.680242 + 0.732987i $$0.261875\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −27.6333 −1.08303
$$652$$ 24.4222i 0.956447i
$$653$$ −39.2111 −1.53445 −0.767225 0.641379i $$-0.778363\pi$$
−0.767225 + 0.641379i $$0.778363\pi$$
$$654$$ 1.39445 0.0545273
$$655$$ 0 0
$$656$$ 3.21110i 0.125372i
$$657$$ 1.39445i 0.0544026i
$$658$$ 42.4222i 1.65379i
$$659$$ −26.2389 −1.02212 −0.511060 0.859545i $$-0.670747\pi$$
−0.511060 + 0.859545i $$0.670747\pi$$
$$660$$ 0 0
$$661$$ 50.2389i 1.95407i 0.213090 + 0.977033i $$0.431647\pi$$
−0.213090 + 0.977033i $$0.568353\pi$$
$$662$$ −16.6056 −0.645393
$$663$$ 16.6056 0.644906
$$664$$ 2.78890 0.108230
$$665$$ 0 0
$$666$$ 9.21110 0.356923
$$667$$ 6.42221 0.248669
$$668$$ 9.21110i 0.356388i
$$669$$ − 1.81665i − 0.0702359i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 4.60555 0.177663
$$673$$ 37.6333 1.45066 0.725329 0.688403i $$-0.241688\pi$$
0.725329 + 0.688403i $$0.241688\pi$$
$$674$$ 13.6333i 0.525135i
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 28.0555 1.07826 0.539130 0.842222i $$-0.318753\pi$$
0.539130 + 0.842222i $$0.318753\pi$$
$$678$$ − 13.8167i − 0.530625i
$$679$$ 6.42221 0.246462
$$680$$ 0 0
$$681$$ − 24.0000i − 0.919682i
$$682$$ 0 0
$$683$$ − 9.21110i − 0.352453i −0.984350 0.176227i $$-0.943611\pi$$
0.984350 0.176227i $$-0.0563891\pi$$
$$684$$ 4.60555i 0.176098i
$$685$$ 0 0
$$686$$ −33.2111 −1.26801
$$687$$ − 19.8167i − 0.756053i
$$688$$ −8.00000 −0.304997
$$689$$ 21.6333 0.824163
$$690$$ 0 0
$$691$$ 20.2389i 0.769922i 0.922933 + 0.384961i $$0.125785\pi$$
−0.922933 + 0.384961i $$0.874215\pi$$
$$692$$ −12.4222 −0.472221
$$693$$ 0 0
$$694$$ − 27.6333i − 1.04895i
$$695$$ 0 0
$$696$$ 4.60555i 0.174573i
$$697$$ − 14.7889i − 0.560169i
$$698$$ −7.81665 −0.295865
$$699$$ 1.81665 0.0687122
$$700$$ 0 0
$$701$$ 47.0278 1.77621 0.888107 0.459637i $$-0.152020\pi$$
0.888107 + 0.459637i $$0.152020\pi$$
$$702$$ 3.60555i 0.136083i
$$703$$ 42.4222 1.59998
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 8.78890 0.330775
$$707$$ − 34.0555i − 1.28079i
$$708$$ 9.21110i 0.346174i
$$709$$ − 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
$$712$$ −15.2111 −0.570060
$$713$$ 8.36669i 0.313335i
$$714$$ −21.2111 −0.793806
$$715$$ 0 0
$$716$$ 19.8167 0.740583
$$717$$ 0 0
$$718$$ 15.6333 0.583430
$$719$$ −51.6333 −1.92560 −0.962799 0.270220i $$-0.912904\pi$$
−0.962799 + 0.270220i $$0.912904\pi$$
$$720$$ 0 0
$$721$$ 18.4222i 0.686079i
$$722$$ 2.21110i 0.0822887i
$$723$$ − 6.42221i − 0.238844i
$$724$$ −8.42221 −0.313009
$$725$$ 0 0
$$726$$ − 11.0000i − 0.408248i
$$727$$ 14.4222 0.534890 0.267445 0.963573i $$-0.413821\pi$$
0.267445 + 0.963573i $$0.413821\pi$$
$$728$$ 16.6056 0.615443
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 36.8444 1.36274
$$732$$ 11.2111 0.414374
$$733$$ 34.0555i 1.25787i 0.777458 + 0.628935i $$0.216509\pi$$
−0.777458 + 0.628935i $$0.783491\pi$$
$$734$$ − 19.6333i − 0.724679i
$$735$$ 0 0
$$736$$ − 1.39445i − 0.0514001i
$$737$$ 0 0
$$738$$ 3.21110 0.118202
$$739$$ 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
$$742$$ −27.6333 −1.01445
$$743$$ − 36.8444i − 1.35169i −0.737044 0.675845i $$-0.763779\pi$$
0.737044 0.675845i $$-0.236221\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0 0
$$746$$ − 20.4222i − 0.747710i
$$747$$ − 2.78890i − 0.102040i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.4222 −0.380312 −0.190156 0.981754i $$-0.560899\pi$$
−0.190156 + 0.981754i $$0.560899\pi$$
$$752$$ 9.21110i 0.335894i
$$753$$ −13.3944 −0.488121
$$754$$ 16.6056i 0.604739i
$$755$$ 0 0
$$756$$ − 4.60555i − 0.167502i
$$757$$ −12.7889 −0.464820 −0.232410 0.972618i $$-0.574661\pi$$
−0.232410 + 0.972618i $$0.574661\pi$$
$$758$$ 35.0278 1.27227
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.6333i 1.21921i 0.792707 + 0.609603i $$0.208671\pi$$
−0.792707 + 0.609603i $$0.791329\pi$$
$$762$$ 1.21110i 0.0438736i
$$763$$ −6.42221 −0.232499
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 27.6333 0.998432
$$767$$ 33.2111i 1.19918i
$$768$$ 1.00000 0.0360844
$$769$$ − 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
$$772$$ 7.81665i 0.281328i
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ 8.00000i 0.287554i
$$775$$ 0 0
$$776$$ 1.39445 0.0500578
$$777$$ −42.4222 −1.52189
$$778$$ − 4.60555i − 0.165117i
$$779$$ 14.7889 0.529867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 6.42221i 0.229658i
$$783$$ 4.60555 0.164589
$$784$$ −14.2111 −0.507539
$$785$$ 0 0
$$786$$ − 22.6056i − 0.806313i
$$787$$ − 49.2666i − 1.75617i −0.478509 0.878083i $$-0.658823\pi$$
0.478509 0.878083i $$-0.341177\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 7.81665 0.278280
$$790$$ 0 0
$$791$$ 63.6333i 2.26254i
$$792$$ 0 0
$$793$$ 40.4222 1.43543
$$794$$ 3.63331 0.128941
$$795$$ 0 0
$$796$$ −22.4222 −0.794734
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ − 21.2111i − 0.750865i
$$799$$ − 42.4222i − 1.50079i
$$800$$ 0 0
$$801$$ 15.2111i 0.537458i
$$802$$ 8.78890 0.310347
$$803$$ 0 0
$$804$$ − 3.21110i − 0.113247i
$$805$$ 0 0
$$806$$ −21.6333 −0.762001
$$807$$ 25.8167 0.908789
$$808$$ − 7.39445i − 0.260136i
$$809$$ −6.84441 −0.240637 −0.120318 0.992735i $$-0.538392\pi$$
−0.120318 + 0.992735i $$0.538392\pi$$
$$810$$ 0 0
$$811$$ 32.2389i 1.13206i 0.824385 + 0.566030i $$0.191521\pi$$
−0.824385 + 0.566030i $$0.808479\pi$$
$$812$$ − 21.2111i − 0.744364i
$$813$$ − 0.422205i − 0.0148074i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −4.60555 −0.161227
$$817$$ 36.8444i 1.28902i
$$818$$ −14.7889 −0.517082
$$819$$ − 16.6056i − 0.580245i
$$820$$ 0 0
$$821$$ − 3.21110i − 0.112068i −0.998429 0.0560341i $$-0.982154\pi$$
0.998429 0.0560341i $$-0.0178456\pi$$
$$822$$ −3.21110 −0.112000
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 4.00000i 0.139347i
$$825$$ 0 0
$$826$$ − 42.4222i − 1.47606i
$$827$$ 27.6333i 0.960904i 0.877021 + 0.480452i $$0.159527\pi$$
−0.877021 + 0.480452i $$0.840473\pi$$
$$828$$ −1.39445 −0.0484604
$$829$$ 46.8444 1.62697 0.813487 0.581583i $$-0.197567\pi$$
0.813487 + 0.581583i $$0.197567\pi$$
$$830$$ 0 0
$$831$$ 16.4222 0.569680
$$832$$ 3.60555 0.125000
$$833$$ 65.4500 2.26771
$$834$$ − 17.2111i − 0.595972i
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 6.00000i 0.207390i
$$838$$ − 4.18335i − 0.144511i
$$839$$ − 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
$$842$$ −19.8167 −0.682927
$$843$$ 27.2111i 0.937200i
$$844$$ 17.2111 0.592431
$$845$$ 0 0
$$846$$ 9.21110 0.316684
$$847$$ 50.6611i 1.74073i
$$848$$ −6.00000 −0.206041
$$849$$ −10.4222 −0.357689
$$850$$ 0 0
$$851$$ 12.8444i 0.440301i
$$852$$ − 9.21110i − 0.315567i
$$853$$ 14.7889i 0.506362i 0.967419 + 0.253181i $$0.0814769\pi$$
−0.967419 + 0.253181i $$0.918523\pi$$
$$854$$ −51.6333 −1.76686
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 23.0278 0.786613 0.393307 0.919407i $$-0.371331\pi$$
0.393307 + 0.919407i $$0.371331\pi$$
$$858$$ 0 0
$$859$$ 25.2111 0.860192 0.430096 0.902783i $$-0.358480\pi$$
0.430096 + 0.902783i $$0.358480\pi$$
$$860$$ 0 0
$$861$$ −14.7889 −0.504004
$$862$$ 12.0000 0.408722
$$863$$ − 51.6333i − 1.75762i −0.477173 0.878809i $$-0.658339\pi$$
0.477173 0.878809i $$-0.341661\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 0 0
$$866$$ 19.2111i 0.652820i
$$867$$ 4.21110 0.143017
$$868$$ 27.6333 0.937936
$$869$$ 0 0
$$870$$ 0 0
$$871$$ − 11.5778i − 0.392299i
$$872$$ −1.39445 −0.0472220
$$873$$ − 1.39445i − 0.0471949i
$$874$$ −6.42221 −0.217234
$$875$$ 0 0
$$876$$ − 1.39445i − 0.0471141i
$$877$$ 24.8444i 0.838936i 0.907770 + 0.419468i $$0.137783\pi$$
−0.907770 + 0.419468i $$0.862217\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ − 18.0000i − 0.607125i
$$880$$ 0 0
$$881$$ 39.2111 1.32106 0.660528 0.750802i $$-0.270333\pi$$
0.660528 + 0.750802i $$0.270333\pi$$
$$882$$ 14.2111i 0.478513i
$$883$$ −9.57779 −0.322318 −0.161159 0.986928i $$-0.551523\pi$$
−0.161159 + 0.986928i $$0.551523\pi$$
$$884$$ −16.6056 −0.558505
$$885$$ 0 0
$$886$$ 15.6333i 0.525211i
$$887$$ −6.97224 −0.234105 −0.117053 0.993126i $$-0.537345\pi$$
−0.117053 + 0.993126i $$0.537345\pi$$
$$888$$ −9.21110 −0.309104
$$889$$ − 5.57779i − 0.187073i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 1.81665i 0.0608261i
$$893$$ 42.4222 1.41960
$$894$$ 15.2111 0.508735
$$895$$ 0 0
$$896$$ −4.60555 −0.153861
$$897$$ −5.02776 −0.167872
$$898$$ 33.6333 1.12236
$$899$$ 27.6333i 0.921622i
$$900$$ 0 0
$$901$$ 27.6333 0.920599
$$902$$ 0 0
$$903$$ − 36.8444i − 1.22611i
$$904$$ 13.8167i 0.459535i
$$905$$ 0 0
$$906$$ 6.00000 0.199337
$$907$$ −21.5778 −0.716479 −0.358239 0.933630i $$-0.616623\pi$$
−0.358239 + 0.933630i $$0.616623\pi$$
$$908$$ 24.0000i 0.796468i
$$909$$ −7.39445 −0.245258
$$910$$ 0 0
$$911$$ −27.6333 −0.915532 −0.457766 0.889073i $$-0.651350\pi$$
−0.457766 + 0.889073i $$0.651350\pi$$
$$912$$ − 4.60555i − 0.152505i
$$913$$ 0 0
$$914$$ −38.2389 −1.26483
$$915$$ 0 0
$$916$$ 19.8167i 0.654761i
$$917$$ 104.111i 3.43805i
$$918$$ 4.60555i 0.152006i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 8.78890i 0.289604i
$$922$$ −33.6333 −1.10765
$$923$$ − 33.2111i − 1.09316i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 31.3944 1.03169
$$927$$ 4.00000 0.131377
$$928$$ − 4.60555i − 0.151185i
$$929$$ 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
$$932$$ −1.81665 −0.0595065
$$933$$ 12.0000 0.392862
$$934$$ − 30.4222i − 0.995445i
$$935$$ 0 0
$$936$$ − 3.60555i − 0.117851i
$$937$$ 10.3667 0.338665 0.169333 0.985559i $$-0.445839\pi$$
0.169333 + 0.985559i $$0.445839\pi$$
$$938$$ 14.7889i 0.482875i
$$939$$ −3.57779 −0.116757
$$940$$ 0 0
$$941$$ − 54.0000i − 1.76035i −0.474650 0.880175i $$-0.657425\pi$$
0.474650 0.880175i $$-0.342575\pi$$
$$942$$ 20.4222i 0.665391i
$$943$$ 4.47772i 0.145815i
$$944$$ − 9.21110i − 0.299796i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 15.6333i 0.508014i 0.967202 + 0.254007i $$0.0817487\pi$$
−0.967202 + 0.254007i $$0.918251\pi$$
$$948$$ 14.4222 0.468411
$$949$$ − 5.02776i − 0.163208i
$$950$$ 0 0
$$951$$ 18.0000i 0.583690i
$$952$$ 21.2111 0.687456
$$953$$ 20.2389 0.655601 0.327800 0.944747i $$-0.393693\pi$$
0.327800 + 0.944747i $$0.393693\pi$$
$$954$$ 6.00000i 0.194257i
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 5.57779 0.180210
$$959$$ 14.7889 0.477558
$$960$$ 0 0
$$961$$ −5.00000 −0.161290
$$962$$ −33.2111 −1.07077
$$963$$ 0 0
$$964$$ 6.42221i 0.206845i
$$965$$ 0 0
$$966$$ 6.42221 0.206631
$$967$$ 8.23886i 0.264944i 0.991187 + 0.132472i $$0.0422914\pi$$
−0.991187 + 0.132472i $$0.957709\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 21.2111i 0.681399i
$$970$$ 0 0
$$971$$ −53.0278 −1.70174 −0.850871 0.525375i $$-0.823925\pi$$
−0.850871 + 0.525375i $$0.823925\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 79.2666i 2.54117i
$$974$$ 0.972244 0.0311527
$$975$$ 0 0
$$976$$ −11.2111 −0.358859
$$977$$ − 18.8444i − 0.602886i −0.953484 0.301443i $$-0.902532\pi$$
0.953484 0.301443i $$-0.0974683\pi$$
$$978$$ −24.4222 −0.780936
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1.39445i 0.0445213i
$$982$$ − 7.81665i − 0.249439i
$$983$$ 42.4222i 1.35306i 0.736416 + 0.676529i $$0.236517\pi$$
−0.736416 + 0.676529i $$0.763483\pi$$
$$984$$ −3.21110 −0.102366
$$985$$ 0 0
$$986$$ 21.2111i 0.675499i
$$987$$ −42.4222 −1.35031
$$988$$ − 16.6056i − 0.528293i
$$989$$ −11.1556 −0.354727
$$990$$ 0 0
$$991$$ −22.4222 −0.712265 −0.356132 0.934436i $$-0.615905\pi$$
−0.356132 + 0.934436i $$0.615905\pi$$
$$992$$ 6.00000 0.190500
$$993$$ − 16.6056i − 0.526961i
$$994$$ 42.4222i 1.34555i
$$995$$ 0 0
$$996$$ 2.78890i 0.0883696i
$$997$$ 16.4222 0.520096 0.260048 0.965596i $$-0.416262\pi$$
0.260048 + 0.965596i $$0.416262\pi$$
$$998$$ 23.0278 0.728931
$$999$$ 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.b.k.1351.2 4
5.2 odd 4 1950.2.f.n.649.1 4
5.3 odd 4 1950.2.f.m.649.4 4
5.4 even 2 390.2.b.c.181.3 yes 4
13.12 even 2 inner 1950.2.b.k.1351.3 4
15.14 odd 2 1170.2.b.d.181.1 4
20.19 odd 2 3120.2.g.q.961.4 4
65.12 odd 4 1950.2.f.m.649.2 4
65.34 odd 4 5070.2.a.z.1.1 2
65.38 odd 4 1950.2.f.n.649.3 4
65.44 odd 4 5070.2.a.bf.1.2 2
65.64 even 2 390.2.b.c.181.2 4
195.194 odd 2 1170.2.b.d.181.4 4
260.259 odd 2 3120.2.g.q.961.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 65.64 even 2
390.2.b.c.181.3 yes 4 5.4 even 2
1170.2.b.d.181.1 4 15.14 odd 2
1170.2.b.d.181.4 4 195.194 odd 2
1950.2.b.k.1351.2 4 1.1 even 1 trivial
1950.2.b.k.1351.3 4 13.12 even 2 inner
1950.2.f.m.649.2 4 65.12 odd 4
1950.2.f.m.649.4 4 5.3 odd 4
1950.2.f.n.649.1 4 5.2 odd 4
1950.2.f.n.649.3 4 65.38 odd 4
3120.2.g.q.961.1 4 260.259 odd 2
3120.2.g.q.961.4 4 20.19 odd 2
5070.2.a.z.1.1 2 65.34 odd 4
5070.2.a.bf.1.2 2 65.44 odd 4