# Properties

 Label 5070.2.a.bf.1.2 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.60555 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.60555 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +4.60555 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.60555 q^{17} +1.00000 q^{18} -4.60555 q^{19} +1.00000 q^{20} -4.60555 q^{21} +1.39445 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.60555 q^{28} +4.60555 q^{29} -1.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -4.60555 q^{34} +4.60555 q^{35} +1.00000 q^{36} +9.21110 q^{37} -4.60555 q^{38} +1.00000 q^{40} +3.21110 q^{41} -4.60555 q^{42} -8.00000 q^{43} +1.00000 q^{45} +1.39445 q^{46} +9.21110 q^{47} -1.00000 q^{48} +14.2111 q^{49} +1.00000 q^{50} +4.60555 q^{51} +6.00000 q^{53} -1.00000 q^{54} +4.60555 q^{56} +4.60555 q^{57} +4.60555 q^{58} +9.21110 q^{59} -1.00000 q^{60} -11.2111 q^{61} +6.00000 q^{62} +4.60555 q^{63} +1.00000 q^{64} -3.21110 q^{67} -4.60555 q^{68} -1.39445 q^{69} +4.60555 q^{70} +9.21110 q^{71} +1.00000 q^{72} +1.39445 q^{73} +9.21110 q^{74} -1.00000 q^{75} -4.60555 q^{76} -14.4222 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.21110 q^{82} +2.78890 q^{83} -4.60555 q^{84} -4.60555 q^{85} -8.00000 q^{86} -4.60555 q^{87} -15.2111 q^{89} +1.00000 q^{90} +1.39445 q^{92} -6.00000 q^{93} +9.21110 q^{94} -4.60555 q^{95} -1.00000 q^{96} +1.39445 q^{97} +14.2111 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} + 2q^{14} - 2q^{15} + 2q^{16} - 2q^{17} + 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} + 10q^{23} - 2q^{24} + 2q^{25} - 2q^{27} + 2q^{28} + 2q^{29} - 2q^{30} + 12q^{31} + 2q^{32} - 2q^{34} + 2q^{35} + 2q^{36} + 4q^{37} - 2q^{38} + 2q^{40} - 8q^{41} - 2q^{42} - 16q^{43} + 2q^{45} + 10q^{46} + 4q^{47} - 2q^{48} + 14q^{49} + 2q^{50} + 2q^{51} + 12q^{53} - 2q^{54} + 2q^{56} + 2q^{57} + 2q^{58} + 4q^{59} - 2q^{60} - 8q^{61} + 12q^{62} + 2q^{63} + 2q^{64} + 8q^{67} - 2q^{68} - 10q^{69} + 2q^{70} + 4q^{71} + 2q^{72} + 10q^{73} + 4q^{74} - 2q^{75} - 2q^{76} + 2q^{80} + 2q^{81} - 8q^{82} + 20q^{83} - 2q^{84} - 2q^{85} - 16q^{86} - 2q^{87} - 16q^{89} + 2q^{90} + 10q^{92} - 12q^{93} + 4q^{94} - 2q^{95} - 2q^{96} + 10q^{97} + 14q^{98} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 4.60555 1.74073 0.870367 0.492403i $$-0.163881\pi$$
0.870367 + 0.492403i $$0.163881\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 4.60555 1.23089
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −4.60555 −1.11701 −0.558505 0.829501i $$-0.688625\pi$$
−0.558505 + 0.829501i $$0.688625\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.60555 −1.05659 −0.528293 0.849062i $$-0.677168\pi$$
−0.528293 + 0.849062i $$0.677168\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −4.60555 −1.00501
$$22$$ 0 0
$$23$$ 1.39445 0.290763 0.145381 0.989376i $$-0.453559\pi$$
0.145381 + 0.989376i $$0.453559\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 4.60555 0.870367
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −4.60555 −0.789846
$$35$$ 4.60555 0.778480
$$36$$ 1.00000 0.166667
$$37$$ 9.21110 1.51430 0.757148 0.653243i $$-0.226592\pi$$
0.757148 + 0.653243i $$0.226592\pi$$
$$38$$ −4.60555 −0.747119
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 3.21110 0.501490 0.250745 0.968053i $$-0.419324\pi$$
0.250745 + 0.968053i $$0.419324\pi$$
$$42$$ −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
$$45$$ 1.00000 0.149071
$$46$$ 1.39445 0.205600
$$47$$ 9.21110 1.34358 0.671789 0.740743i $$-0.265526\pi$$
0.671789 + 0.740743i $$0.265526\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 14.2111 2.03016
$$50$$ 1.00000 0.141421
$$51$$ 4.60555 0.644906
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 4.60555 0.615443
$$57$$ 4.60555 0.610020
$$58$$ 4.60555 0.604739
$$59$$ 9.21110 1.19918 0.599592 0.800306i $$-0.295330\pi$$
0.599592 + 0.800306i $$0.295330\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −11.2111 −1.43543 −0.717717 0.696335i $$-0.754813\pi$$
−0.717717 + 0.696335i $$0.754813\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 4.60555 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 $$0.562844\pi$$
$$68$$ −4.60555 −0.558505
$$69$$ −1.39445 −0.167872
$$70$$ 4.60555 0.550469
$$71$$ 9.21110 1.09316 0.546578 0.837408i $$-0.315930\pi$$
0.546578 + 0.837408i $$0.315930\pi$$
$$72$$ 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
$$75$$ −1.00000 −0.115470
$$76$$ −4.60555 −0.528293
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.4222 −1.62262 −0.811312 0.584613i $$-0.801246\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 3.21110 0.354607
$$83$$ 2.78890 0.306121 0.153061 0.988217i $$-0.451087\pi$$
0.153061 + 0.988217i $$0.451087\pi$$
$$84$$ −4.60555 −0.502507
$$85$$ −4.60555 −0.499542
$$86$$ −8.00000 −0.862662
$$87$$ −4.60555 −0.493767
$$88$$ 0 0
$$89$$ −15.2111 −1.61237 −0.806187 0.591661i $$-0.798472\pi$$
−0.806187 + 0.591661i $$0.798472\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 1.39445 0.145381
$$93$$ −6.00000 −0.622171
$$94$$ 9.21110 0.950053
$$95$$ −4.60555 −0.472520
$$96$$ −1.00000 −0.102062
$$97$$ 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
$$100$$ 1.00000 0.100000
$$101$$ 7.39445 0.735775 0.367888 0.929870i $$-0.380081\pi$$
0.367888 + 0.929870i $$0.380081\pi$$
$$102$$ 4.60555 0.456018
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ −4.60555 −0.449456
$$106$$ 6.00000 0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 1.39445 0.133564 0.0667820 0.997768i $$-0.478727\pi$$
0.0667820 + 0.997768i $$0.478727\pi$$
$$110$$ 0 0
$$111$$ −9.21110 −0.874279
$$112$$ 4.60555 0.435184
$$113$$ −13.8167 −1.29976 −0.649881 0.760036i $$-0.725181\pi$$
−0.649881 + 0.760036i $$0.725181\pi$$
$$114$$ 4.60555 0.431349
$$115$$ 1.39445 0.130033
$$116$$ 4.60555 0.427615
$$117$$ 0 0
$$118$$ 9.21110 0.847951
$$119$$ −21.2111 −1.94442
$$120$$ −1.00000 −0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ −11.2111 −1.01501
$$123$$ −3.21110 −0.289535
$$124$$ 6.00000 0.538816
$$125$$ 1.00000 0.0894427
$$126$$ 4.60555 0.410295
$$127$$ −1.21110 −0.107468 −0.0537340 0.998555i $$-0.517112\pi$$
−0.0537340 + 0.998555i $$0.517112\pi$$
$$128$$ 1.00000 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
$$133$$ −21.2111 −1.83924
$$134$$ −3.21110 −0.277397
$$135$$ −1.00000 −0.0860663
$$136$$ −4.60555 −0.394923
$$137$$ −3.21110 −0.274343 −0.137172 0.990547i $$-0.543801\pi$$
−0.137172 + 0.990547i $$0.543801\pi$$
$$138$$ −1.39445 −0.118703
$$139$$ 17.2111 1.45983 0.729913 0.683540i $$-0.239560\pi$$
0.729913 + 0.683540i $$0.239560\pi$$
$$140$$ 4.60555 0.389240
$$141$$ −9.21110 −0.775715
$$142$$ 9.21110 0.772979
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 4.60555 0.382470
$$146$$ 1.39445 0.115405
$$147$$ −14.2111 −1.17211
$$148$$ 9.21110 0.757148
$$149$$ 15.2111 1.24614 0.623071 0.782165i $$-0.285885\pi$$
0.623071 + 0.782165i $$0.285885\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −6.00000 −0.488273 −0.244137 0.969741i $$-0.578505\pi$$
−0.244137 + 0.969741i $$0.578505\pi$$
$$152$$ −4.60555 −0.373560
$$153$$ −4.60555 −0.372337
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 20.4222 1.62987 0.814935 0.579553i $$-0.196773\pi$$
0.814935 + 0.579553i $$0.196773\pi$$
$$158$$ −14.4222 −1.14737
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 6.42221 0.506141
$$162$$ 1.00000 0.0785674
$$163$$ −24.4222 −1.91289 −0.956447 0.291905i $$-0.905711\pi$$
−0.956447 + 0.291905i $$0.905711\pi$$
$$164$$ 3.21110 0.250745
$$165$$ 0 0
$$166$$ 2.78890 0.216460
$$167$$ −9.21110 −0.712777 −0.356388 0.934338i $$-0.615992\pi$$
−0.356388 + 0.934338i $$0.615992\pi$$
$$168$$ −4.60555 −0.355326
$$169$$ 0 0
$$170$$ −4.60555 −0.353230
$$171$$ −4.60555 −0.352195
$$172$$ −8.00000 −0.609994
$$173$$ 12.4222 0.944443 0.472221 0.881480i $$-0.343452\pi$$
0.472221 + 0.881480i $$0.343452\pi$$
$$174$$ −4.60555 −0.349146
$$175$$ 4.60555 0.348147
$$176$$ 0 0
$$177$$ −9.21110 −0.692349
$$178$$ −15.2111 −1.14012
$$179$$ 19.8167 1.48117 0.740583 0.671965i $$-0.234549\pi$$
0.740583 + 0.671965i $$0.234549\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −8.42221 −0.626018 −0.313009 0.949750i $$-0.601337\pi$$
−0.313009 + 0.949750i $$0.601337\pi$$
$$182$$ 0 0
$$183$$ 11.2111 0.828749
$$184$$ 1.39445 0.102800
$$185$$ 9.21110 0.677214
$$186$$ −6.00000 −0.439941
$$187$$ 0 0
$$188$$ 9.21110 0.671789
$$189$$ −4.60555 −0.335005
$$190$$ −4.60555 −0.334122
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −7.81665 −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 $$0.792390\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 3.21110 0.226494
$$202$$ 7.39445 0.520272
$$203$$ 21.2111 1.48873
$$204$$ 4.60555 0.322453
$$205$$ 3.21110 0.224273
$$206$$ 4.00000 0.278693
$$207$$ 1.39445 0.0969209
$$208$$ 0 0
$$209$$ 0 0
$$210$$ −4.60555 −0.317813
$$211$$ −17.2111 −1.18486 −0.592431 0.805622i $$-0.701832\pi$$
−0.592431 + 0.805622i $$0.701832\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −9.21110 −0.631134
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ −1.00000 −0.0680414
$$217$$ 27.6333 1.87587
$$218$$ 1.39445 0.0944440
$$219$$ −1.39445 −0.0942281
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −9.21110 −0.618209
$$223$$ 1.81665 0.121652 0.0608261 0.998148i $$-0.480627\pi$$
0.0608261 + 0.998148i $$0.480627\pi$$
$$224$$ 4.60555 0.307721
$$225$$ 1.00000 0.0666667
$$226$$ −13.8167 −0.919070
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 4.60555 0.305010
$$229$$ 19.8167 1.30952 0.654761 0.755836i $$-0.272769\pi$$
0.654761 + 0.755836i $$0.272769\pi$$
$$230$$ 1.39445 0.0919472
$$231$$ 0 0
$$232$$ 4.60555 0.302369
$$233$$ 1.81665 0.119013 0.0595065 0.998228i $$-0.481047\pi$$
0.0595065 + 0.998228i $$0.481047\pi$$
$$234$$ 0 0
$$235$$ 9.21110 0.600866
$$236$$ 9.21110 0.599592
$$237$$ 14.4222 0.936823
$$238$$ −21.2111 −1.37491
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 6.42221 0.413691 0.206845 0.978374i $$-0.433680\pi$$
0.206845 + 0.978374i $$0.433680\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ −11.2111 −0.717717
$$245$$ 14.2111 0.907914
$$246$$ −3.21110 −0.204732
$$247$$ 0 0
$$248$$ 6.00000 0.381000
$$249$$ −2.78890 −0.176739
$$250$$ 1.00000 0.0632456
$$251$$ 13.3944 0.845450 0.422725 0.906258i $$-0.361074\pi$$
0.422725 + 0.906258i $$0.361074\pi$$
$$252$$ 4.60555 0.290122
$$253$$ 0 0
$$254$$ −1.21110 −0.0759913
$$255$$ 4.60555 0.288411
$$256$$ 1.00000 0.0625000
$$257$$ 28.6056 1.78437 0.892183 0.451675i $$-0.149173\pi$$
0.892183 + 0.451675i $$0.149173\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 42.4222 2.63599
$$260$$ 0 0
$$261$$ 4.60555 0.285076
$$262$$ 22.6056 1.39658
$$263$$ −7.81665 −0.481996 −0.240998 0.970526i $$-0.577475\pi$$
−0.240998 + 0.970526i $$0.577475\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ −21.2111 −1.30054
$$267$$ 15.2111 0.930904
$$268$$ −3.21110 −0.196149
$$269$$ 25.8167 1.57407 0.787035 0.616909i $$-0.211615\pi$$
0.787035 + 0.616909i $$0.211615\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 0.422205 0.0256471 0.0128236 0.999918i $$-0.495918\pi$$
0.0128236 + 0.999918i $$0.495918\pi$$
$$272$$ −4.60555 −0.279253
$$273$$ 0 0
$$274$$ −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 $$0.335770\pi$$
$$278$$ 17.2111 1.03225
$$279$$ 6.00000 0.359211
$$280$$ 4.60555 0.275234
$$281$$ −27.2111 −1.62328 −0.811639 0.584159i $$-0.801424\pi$$
−0.811639 + 0.584159i $$0.801424\pi$$
$$282$$ −9.21110 −0.548513
$$283$$ −10.4222 −0.619536 −0.309768 0.950812i $$-0.600251\pi$$
−0.309768 + 0.950812i $$0.600251\pi$$
$$284$$ 9.21110 0.546578
$$285$$ 4.60555 0.272809
$$286$$ 0 0
$$287$$ 14.7889 0.872961
$$288$$ 1.00000 0.0589256
$$289$$ 4.21110 0.247712
$$290$$ 4.60555 0.270447
$$291$$ −1.39445 −0.0817440
$$292$$ 1.39445 0.0816039
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ −14.2111 −0.828808
$$295$$ 9.21110 0.536291
$$296$$ 9.21110 0.535384
$$297$$ 0 0
$$298$$ 15.2111 0.881156
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ −36.8444 −2.12368
$$302$$ −6.00000 −0.345261
$$303$$ −7.39445 −0.424800
$$304$$ −4.60555 −0.264146
$$305$$ −11.2111 −0.641946
$$306$$ −4.60555 −0.263282
$$307$$ 8.78890 0.501609 0.250804 0.968038i $$-0.419305\pi$$
0.250804 + 0.968038i $$0.419305\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 6.00000 0.340777
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 3.57779 0.202229 0.101114 0.994875i $$-0.467759\pi$$
0.101114 + 0.994875i $$0.467759\pi$$
$$314$$ 20.4222 1.15249
$$315$$ 4.60555 0.259493
$$316$$ −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
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 6.42221 0.357895
$$323$$ 21.2111 1.18022
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −24.4222 −1.35262
$$327$$ −1.39445 −0.0771132
$$328$$ 3.21110 0.177303
$$329$$ 42.4222 2.33881
$$330$$ 0 0
$$331$$ −16.6056 −0.912724 −0.456362 0.889794i $$-0.650848\pi$$
−0.456362 + 0.889794i $$0.650848\pi$$
$$332$$ 2.78890 0.153061
$$333$$ 9.21110 0.504765
$$334$$ −9.21110 −0.504009
$$335$$ −3.21110 −0.175441
$$336$$ −4.60555 −0.251253
$$337$$ −13.6333 −0.742654 −0.371327 0.928502i $$-0.621097\pi$$
−0.371327 + 0.928502i $$0.621097\pi$$
$$338$$ 0 0
$$339$$ 13.8167 0.750418
$$340$$ −4.60555 −0.249771
$$341$$ 0 0
$$342$$ −4.60555 −0.249040
$$343$$ 33.2111 1.79323
$$344$$ −8.00000 −0.431331
$$345$$ −1.39445 −0.0750746
$$346$$ 12.4222 0.667822
$$347$$ −27.6333 −1.48343 −0.741717 0.670713i $$-0.765988\pi$$
−0.741717 + 0.670713i $$0.765988\pi$$
$$348$$ −4.60555 −0.246883
$$349$$ 7.81665 0.418416 0.209208 0.977871i $$-0.432911\pi$$
0.209208 + 0.977871i $$0.432911\pi$$
$$350$$ 4.60555 0.246177
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −8.78890 −0.467786 −0.233893 0.972262i $$-0.575147\pi$$
−0.233893 + 0.972262i $$0.575147\pi$$
$$354$$ −9.21110 −0.489565
$$355$$ 9.21110 0.488875
$$356$$ −15.2111 −0.806187
$$357$$ 21.2111 1.12261
$$358$$ 19.8167 1.04734
$$359$$ −15.6333 −0.825094 −0.412547 0.910936i $$-0.635361\pi$$
−0.412547 + 0.910936i $$0.635361\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 2.21110 0.116374
$$362$$ −8.42221 −0.442661
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 1.39445 0.0729888
$$366$$ 11.2111 0.586014
$$367$$ −19.6333 −1.02485 −0.512425 0.858732i $$-0.671253\pi$$
−0.512425 + 0.858732i $$0.671253\pi$$
$$368$$ 1.39445 0.0726907
$$369$$ 3.21110 0.167163
$$370$$ 9.21110 0.478862
$$371$$ 27.6333 1.43465
$$372$$ −6.00000 −0.311086
$$373$$ −20.4222 −1.05742 −0.528711 0.848802i $$-0.677324\pi$$
−0.528711 + 0.848802i $$0.677324\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 9.21110 0.475026
$$377$$ 0 0
$$378$$ −4.60555 −0.236884
$$379$$ 35.0278 1.79925 0.899627 0.436658i $$-0.143838\pi$$
0.899627 + 0.436658i $$0.143838\pi$$
$$380$$ −4.60555 −0.236260
$$381$$ 1.21110 0.0620467
$$382$$ −12.0000 −0.613973
$$383$$ −27.6333 −1.41200 −0.705998 0.708214i $$-0.749501\pi$$
−0.705998 + 0.708214i $$0.749501\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −7.81665 −0.397857
$$387$$ −8.00000 −0.406663
$$388$$ 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
$$393$$ −22.6056 −1.14030
$$394$$ −6.00000 −0.302276
$$395$$ −14.4222 −0.725660
$$396$$ 0 0
$$397$$ 3.63331 0.182350 0.0911752 0.995835i $$-0.470938\pi$$
0.0911752 + 0.995835i $$0.470938\pi$$
$$398$$ −22.4222 −1.12392
$$399$$ 21.2111 1.06188
$$400$$ 1.00000 0.0500000
$$401$$ −8.78890 −0.438897 −0.219448 0.975624i $$-0.570426\pi$$
−0.219448 + 0.975624i $$0.570426\pi$$
$$402$$ 3.21110 0.160155
$$403$$ 0 0
$$404$$ 7.39445 0.367888
$$405$$ 1.00000 0.0496904
$$406$$ 21.2111 1.05269
$$407$$ 0 0
$$408$$ 4.60555 0.228009
$$409$$ −14.7889 −0.731264 −0.365632 0.930760i $$-0.619147\pi$$
−0.365632 + 0.930760i $$0.619147\pi$$
$$410$$ 3.21110 0.158585
$$411$$ 3.21110 0.158392
$$412$$ 4.00000 0.197066
$$413$$ 42.4222 2.08746
$$414$$ 1.39445 0.0685334
$$415$$ 2.78890 0.136902
$$416$$ 0 0
$$417$$ −17.2111 −0.842831
$$418$$ 0 0
$$419$$ 4.18335 0.204370 0.102185 0.994765i $$-0.467417\pi$$
0.102185 + 0.994765i $$0.467417\pi$$
$$420$$ −4.60555 −0.224728
$$421$$ −19.8167 −0.965805 −0.482902 0.875674i $$-0.660417\pi$$
−0.482902 + 0.875674i $$0.660417\pi$$
$$422$$ −17.2111 −0.837823
$$423$$ 9.21110 0.447859
$$424$$ 6.00000 0.291386
$$425$$ −4.60555 −0.223402
$$426$$ −9.21110 −0.446279
$$427$$ −51.6333 −2.49871
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −19.2111 −0.923227 −0.461613 0.887081i $$-0.652729\pi$$
−0.461613 + 0.887081i $$0.652729\pi$$
$$434$$ 27.6333 1.32644
$$435$$ −4.60555 −0.220819
$$436$$ 1.39445 0.0667820
$$437$$ −6.42221 −0.307216
$$438$$ −1.39445 −0.0666293
$$439$$ −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
$$442$$ 0 0
$$443$$ 15.6333 0.742761 0.371380 0.928481i $$-0.378885\pi$$
0.371380 + 0.928481i $$0.378885\pi$$
$$444$$ −9.21110 −0.437140
$$445$$ −15.2111 −0.721075
$$446$$ 1.81665 0.0860211
$$447$$ −15.2111 −0.719460
$$448$$ 4.60555 0.217592
$$449$$ −33.6333 −1.58725 −0.793627 0.608405i $$-0.791810\pi$$
−0.793627 + 0.608405i $$0.791810\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −13.8167 −0.649881
$$453$$ 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 $$0.147625\pi$$
$$458$$ 19.8167 0.925971
$$459$$ 4.60555 0.214969
$$460$$ 1.39445 0.0650165
$$461$$ −33.6333 −1.56646 −0.783230 0.621733i $$-0.786429\pi$$
−0.783230 + 0.621733i $$0.786429\pi$$
$$462$$ 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
$$465$$ −6.00000 −0.278243
$$466$$ 1.81665 0.0841549
$$467$$ 30.4222 1.40777 0.703886 0.710313i $$-0.251447\pi$$
0.703886 + 0.710313i $$0.251447\pi$$
$$468$$ 0 0
$$469$$ −14.7889 −0.682888
$$470$$ 9.21110 0.424876
$$471$$ −20.4222 −0.941006
$$472$$ 9.21110 0.423975
$$473$$ 0 0
$$474$$ 14.4222 0.662434
$$475$$ −4.60555 −0.211317
$$476$$ −21.2111 −0.972209
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ −5.57779 −0.254856 −0.127428 0.991848i $$-0.540672\pi$$
−0.127428 + 0.991848i $$0.540672\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 6.42221 0.292523
$$483$$ −6.42221 −0.292220
$$484$$ −11.0000 −0.500000
$$485$$ 1.39445 0.0633187
$$486$$ −1.00000 −0.0453609
$$487$$ −0.972244 −0.0440566 −0.0220283 0.999757i $$-0.507012\pi$$
−0.0220283 + 0.999757i $$0.507012\pi$$
$$488$$ −11.2111 −0.507503
$$489$$ 24.4222 1.10441
$$490$$ 14.2111 0.641992
$$491$$ −7.81665 −0.352761 −0.176380 0.984322i $$-0.556439\pi$$
−0.176380 + 0.984322i $$0.556439\pi$$
$$492$$ −3.21110 −0.144768
$$493$$ −21.2111 −0.955300
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ 42.4222 1.90290
$$498$$ −2.78890 −0.124973
$$499$$ 23.0278 1.03086 0.515432 0.856930i $$-0.327631\pi$$
0.515432 + 0.856930i $$0.327631\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 9.21110 0.411522
$$502$$ 13.3944 0.597824
$$503$$ 23.4500 1.04558 0.522791 0.852461i $$-0.324891\pi$$
0.522791 + 0.852461i $$0.324891\pi$$
$$504$$ 4.60555 0.205148
$$505$$ 7.39445 0.329049
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −1.21110 −0.0537340
$$509$$ −33.6333 −1.49077 −0.745385 0.666634i $$-0.767734\pi$$
−0.745385 + 0.666634i $$0.767734\pi$$
$$510$$ 4.60555 0.203937
$$511$$ 6.42221 0.284102
$$512$$ 1.00000 0.0441942
$$513$$ 4.60555 0.203340
$$514$$ 28.6056 1.26174
$$515$$ 4.00000 0.176261
$$516$$ 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
$$523$$ 32.8444 1.43619 0.718093 0.695947i $$-0.245015\pi$$
0.718093 + 0.695947i $$0.245015\pi$$
$$524$$ 22.6056 0.987528
$$525$$ −4.60555 −0.201003
$$526$$ −7.81665 −0.340822
$$527$$ −27.6333 −1.20373
$$528$$ 0 0
$$529$$ −21.0555 −0.915457
$$530$$ 6.00000 0.260623
$$531$$ 9.21110 0.399728
$$532$$ −21.2111 −0.919618
$$533$$ 0 0
$$534$$ 15.2111 0.658249
$$535$$ 0 0
$$536$$ −3.21110 −0.138699
$$537$$ −19.8167 −0.855152
$$538$$ 25.8167 1.11303
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ −6.97224 −0.299760 −0.149880 0.988704i $$-0.547889\pi$$
−0.149880 + 0.988704i $$0.547889\pi$$
$$542$$ 0.422205 0.0181353
$$543$$ 8.42221 0.361431
$$544$$ −4.60555 −0.197461
$$545$$ 1.39445 0.0597316
$$546$$ 0 0
$$547$$ 14.4222 0.616649 0.308324 0.951281i $$-0.400232\pi$$
0.308324 + 0.951281i $$0.400232\pi$$
$$548$$ −3.21110 −0.137172
$$549$$ −11.2111 −0.478478
$$550$$ 0 0
$$551$$ −21.2111 −0.903623
$$552$$ −1.39445 −0.0593517
$$553$$ −66.4222 −2.82456
$$554$$ 16.4222 0.697713
$$555$$ −9.21110 −0.390990
$$556$$ 17.2111 0.729913
$$557$$ −11.5778 −0.490567 −0.245283 0.969451i $$-0.578881\pi$$
−0.245283 + 0.969451i $$0.578881\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 0 0
$$560$$ 4.60555 0.194620
$$561$$ 0 0
$$562$$ −27.2111 −1.14783
$$563$$ −34.0555 −1.43527 −0.717634 0.696420i $$-0.754775\pi$$
−0.717634 + 0.696420i $$0.754775\pi$$
$$564$$ −9.21110 −0.387857
$$565$$ −13.8167 −0.581271
$$566$$ −10.4222 −0.438078
$$567$$ 4.60555 0.193415
$$568$$ 9.21110 0.386489
$$569$$ −33.6333 −1.40998 −0.704991 0.709216i $$-0.749049\pi$$
−0.704991 + 0.709216i $$0.749049\pi$$
$$570$$ 4.60555 0.192905
$$571$$ 30.0555 1.25778 0.628892 0.777493i $$-0.283509\pi$$
0.628892 + 0.777493i $$0.283509\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ 14.7889 0.617277
$$575$$ 1.39445 0.0581525
$$576$$ 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
$$579$$ 7.81665 0.324849
$$580$$ 4.60555 0.191235
$$581$$ 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
$$587$$ −6.42221 −0.265073 −0.132536 0.991178i $$-0.542312\pi$$
−0.132536 + 0.991178i $$0.542312\pi$$
$$588$$ −14.2111 −0.586056
$$589$$ −27.6333 −1.13861
$$590$$ 9.21110 0.379215
$$591$$ 6.00000 0.246807
$$592$$ 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
$$595$$ −21.2111 −0.869570
$$596$$ 15.2111 0.623071
$$597$$ 22.4222 0.917680
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ −36.8444 −1.50167
$$603$$ −3.21110 −0.130766
$$604$$ −6.00000 −0.244137
$$605$$ −11.0000 −0.447214
$$606$$ −7.39445 −0.300379
$$607$$ 17.2111 0.698577 0.349289 0.937015i $$-0.386423\pi$$
0.349289 + 0.937015i $$0.386423\pi$$
$$608$$ −4.60555 −0.186780
$$609$$ −21.2111 −0.859517
$$610$$ −11.2111 −0.453924
$$611$$ 0 0
$$612$$ −4.60555 −0.186168
$$613$$ −33.2111 −1.34138 −0.670692 0.741736i $$-0.734003\pi$$
−0.670692 + 0.741736i $$0.734003\pi$$
$$614$$ 8.78890 0.354691
$$615$$ −3.21110 −0.129484
$$616$$ 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
$$619$$ −25.8167 −1.03766 −0.518829 0.854878i $$-0.673632\pi$$
−0.518829 + 0.854878i $$0.673632\pi$$
$$620$$ 6.00000 0.240966
$$621$$ −1.39445 −0.0559573
$$622$$ −12.0000 −0.481156
$$623$$ −70.0555 −2.80671
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 3.57779 0.142997
$$627$$ 0 0
$$628$$ 20.4222 0.814935
$$629$$ −42.4222 −1.69148
$$630$$ 4.60555 0.183490
$$631$$ −3.21110 −0.127832 −0.0639160 0.997955i $$-0.520359\pi$$
−0.0639160 + 0.997955i $$0.520359\pi$$
$$632$$ −14.4222 −0.573685
$$633$$ 17.2111 0.684080
$$634$$ −18.0000 −0.714871
$$635$$ −1.21110 −0.0480611
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 9.21110 0.364386
$$640$$ 1.00000 0.0395285
$$641$$ −0.422205 −0.0166761 −0.00833805 0.999965i $$-0.502654\pi$$
−0.00833805 + 0.999965i $$0.502654\pi$$
$$642$$ 0 0
$$643$$ 9.63331 0.379901 0.189950 0.981794i $$-0.439167\pi$$
0.189950 + 0.981794i $$0.439167\pi$$
$$644$$ 6.42221 0.253070
$$645$$ 8.00000 0.315000
$$646$$ 21.2111 0.834540
$$647$$ 34.6056 1.36048 0.680242 0.732987i $$-0.261875\pi$$
0.680242 + 0.732987i $$0.261875\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −27.6333 −1.08303
$$652$$ −24.4222 −0.956447
$$653$$ 39.2111 1.53445 0.767225 0.641379i $$-0.221637\pi$$
0.767225 + 0.641379i $$0.221637\pi$$
$$654$$ −1.39445 −0.0545273
$$655$$ 22.6056 0.883272
$$656$$ 3.21110 0.125372
$$657$$ 1.39445 0.0544026
$$658$$ 42.4222 1.65379
$$659$$ −26.2389 −1.02212 −0.511060 0.859545i $$-0.670747\pi$$
−0.511060 + 0.859545i $$0.670747\pi$$
$$660$$ 0 0
$$661$$ −50.2389 −1.95407 −0.977033 0.213090i $$-0.931647\pi$$
−0.977033 + 0.213090i $$0.931647\pi$$
$$662$$ −16.6056 −0.645393
$$663$$ 0 0
$$664$$ 2.78890 0.108230
$$665$$ −21.2111 −0.822531
$$666$$ 9.21110 0.356923
$$667$$ 6.42221 0.248669
$$668$$ −9.21110 −0.356388
$$669$$ −1.81665 −0.0702359
$$670$$ −3.21110 −0.124056
$$671$$ 0 0
$$672$$ −4.60555 −0.177663
$$673$$ 37.6333 1.45066 0.725329 0.688403i $$-0.241688\pi$$
0.725329 + 0.688403i $$0.241688\pi$$
$$674$$ −13.6333 −0.525135
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −28.0555 −1.07826 −0.539130 0.842222i $$-0.681247\pi$$
−0.539130 + 0.842222i $$0.681247\pi$$
$$678$$ 13.8167 0.530625
$$679$$ 6.42221 0.246462
$$680$$ −4.60555 −0.176615
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ −9.21110 −0.352453 −0.176227 0.984350i $$-0.556389\pi$$
−0.176227 + 0.984350i $$0.556389\pi$$
$$684$$ −4.60555 −0.176098
$$685$$ −3.21110 −0.122690
$$686$$ 33.2111 1.26801
$$687$$ −19.8167 −0.756053
$$688$$ −8.00000 −0.304997
$$689$$ 0 0
$$690$$ −1.39445 −0.0530858
$$691$$ 20.2389 0.769922 0.384961 0.922933i $$-0.374215\pi$$
0.384961 + 0.922933i $$0.374215\pi$$
$$692$$ 12.4222 0.472221
$$693$$ 0 0
$$694$$ −27.6333 −1.04895
$$695$$ 17.2111 0.652854
$$696$$ −4.60555 −0.174573
$$697$$ −14.7889 −0.560169
$$698$$ 7.81665 0.295865
$$699$$ −1.81665 −0.0687122
$$700$$ 4.60555 0.174073
$$701$$ −47.0278 −1.77621 −0.888107 0.459637i $$-0.847980\pi$$
−0.888107 + 0.459637i $$0.847980\pi$$
$$702$$ 0 0
$$703$$ −42.4222 −1.59998
$$704$$ 0 0
$$705$$ −9.21110 −0.346910
$$706$$ −8.78890 −0.330775
$$707$$ 34.0555 1.28079
$$708$$ −9.21110 −0.346174
$$709$$ 1.39445 0.0523696 0.0261848 0.999657i $$-0.491664\pi$$
0.0261848 + 0.999657i $$0.491664\pi$$
$$710$$ 9.21110 0.345687
$$711$$ −14.4222 −0.540875
$$712$$ −15.2111 −0.570060
$$713$$ 8.36669 0.313335
$$714$$ 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 $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 18.4222 0.686079
$$722$$ 2.21110 0.0822887
$$723$$ −6.42221 −0.238844
$$724$$ −8.42221 −0.313009
$$725$$ 4.60555 0.171046
$$726$$ 11.0000 0.408248
$$727$$ 14.4222 0.534890 0.267445 0.963573i $$-0.413821\pi$$
0.267445 + 0.963573i $$0.413821\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 1.39445 0.0516109
$$731$$ 36.8444 1.36274
$$732$$ 11.2111 0.414374
$$733$$ −34.0555 −1.25787 −0.628935 0.777458i $$-0.716509\pi$$
−0.628935 + 0.777458i $$0.716509\pi$$
$$734$$ −19.6333 −0.724679
$$735$$ −14.2111 −0.524184
$$736$$ 1.39445 0.0514001
$$737$$ 0 0
$$738$$ 3.21110 0.118202
$$739$$ −20.2389 −0.744498 −0.372249 0.928133i $$-0.621413\pi$$
−0.372249 + 0.928133i $$0.621413\pi$$
$$740$$ 9.21110 0.338607
$$741$$ 0 0
$$742$$ 27.6333 1.01445
$$743$$ 36.8444 1.35169 0.675845 0.737044i $$-0.263779\pi$$
0.675845 + 0.737044i $$0.263779\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 15.2111 0.557292
$$746$$ −20.4222 −0.747710
$$747$$ 2.78890 0.102040
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ 10.4222 0.380312 0.190156 0.981754i $$-0.439101\pi$$
0.190156 + 0.981754i $$0.439101\pi$$
$$752$$ 9.21110 0.335894
$$753$$ −13.3944 −0.488121
$$754$$ 0 0
$$755$$ −6.00000 −0.218362
$$756$$ −4.60555 −0.167502
$$757$$ 12.7889 0.464820 0.232410 0.972618i $$-0.425339\pi$$
0.232410 + 0.972618i $$0.425339\pi$$
$$758$$ 35.0278 1.27227
$$759$$ 0 0
$$760$$ −4.60555 −0.167061
$$761$$ −33.6333 −1.21921 −0.609603 0.792707i $$-0.708671\pi$$
−0.609603 + 0.792707i $$0.708671\pi$$
$$762$$ 1.21110 0.0438736
$$763$$ 6.42221 0.232499
$$764$$ −12.0000 −0.434145
$$765$$ −4.60555 −0.166514
$$766$$ −27.6333 −0.998432
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −12.8444 −0.463181 −0.231591 0.972813i $$-0.574393\pi$$
−0.231591 + 0.972813i $$0.574393\pi$$
$$770$$ 0 0
$$771$$ −28.6056 −1.03020
$$772$$ −7.81665 −0.281328
$$773$$ 30.0000 1.07903 0.539513 0.841978i $$-0.318609\pi$$
0.539513 + 0.841978i $$0.318609\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 6.00000 0.215526
$$776$$ 1.39445 0.0500578
$$777$$ −42.4222 −1.52189
$$778$$ −4.60555 −0.165117
$$779$$ −14.7889 −0.529867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −6.42221 −0.229658
$$783$$ −4.60555 −0.164589
$$784$$ 14.2111 0.507539
$$785$$ 20.4222 0.728900
$$786$$ −22.6056 −0.806313
$$787$$ −49.2666 −1.75617 −0.878083 0.478509i $$-0.841177\pi$$
−0.878083 + 0.478509i $$0.841177\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 7.81665 0.278280
$$790$$ −14.4222 −0.513119
$$791$$ −63.6333 −2.26254
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 3.63331 0.128941
$$795$$ −6.00000 −0.212798
$$796$$ −22.4222 −0.794734
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 21.2111 0.750865
$$799$$ −42.4222 −1.50079
$$800$$ 1.00000 0.0353553
$$801$$ −15.2111 −0.537458
$$802$$ −8.78890 −0.310347
$$803$$ 0 0
$$804$$ 3.21110 0.113247
$$805$$ 6.42221 0.226353
$$806$$ 0 0
$$807$$ −25.8167 −0.908789
$$808$$ 7.39445 0.260136
$$809$$ −6.84441 −0.240637 −0.120318 0.992735i $$-0.538392\pi$$
−0.120318 + 0.992735i $$0.538392\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 32.2389 1.13206 0.566030 0.824385i $$-0.308479\pi$$
0.566030 + 0.824385i $$0.308479\pi$$
$$812$$ 21.2111 0.744364
$$813$$ −0.422205 −0.0148074
$$814$$ 0 0
$$815$$ −24.4222 −0.855473
$$816$$ 4.60555 0.161227
$$817$$ 36.8444 1.28902
$$818$$ −14.7889 −0.517082
$$819$$ 0 0
$$820$$ 3.21110 0.112137
$$821$$ −3.21110 −0.112068 −0.0560341 0.998429i $$-0.517846\pi$$
−0.0560341 + 0.998429i $$0.517846\pi$$
$$822$$ 3.21110 0.112000
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 42.4222 1.47606
$$827$$ 27.6333 0.960904 0.480452 0.877021i $$-0.340473\pi$$
0.480452 + 0.877021i $$0.340473\pi$$
$$828$$ 1.39445 0.0484604
$$829$$ −46.8444 −1.62697 −0.813487 0.581583i $$-0.802433\pi$$
−0.813487 + 0.581583i $$0.802433\pi$$
$$830$$ 2.78890 0.0968040
$$831$$ −16.4222 −0.569680
$$832$$ 0 0
$$833$$ −65.4500 −2.26771
$$834$$ −17.2111 −0.595972
$$835$$ −9.21110 −0.318763
$$836$$ 0 0
$$837$$ −6.00000 −0.207390
$$838$$ 4.18335 0.144511
$$839$$ 18.4222 0.636005 0.318003 0.948090i $$-0.396988\pi$$
0.318003 + 0.948090i $$0.396988\pi$$
$$840$$ −4.60555 −0.158907
$$841$$ −7.78890 −0.268583
$$842$$ −19.8167 −0.682927
$$843$$ 27.2111 0.937200
$$844$$ −17.2111 −0.592431
$$845$$ 0 0
$$846$$ 9.21110 0.316684
$$847$$ −50.6611 −1.74073
$$848$$ 6.00000 0.206041
$$849$$ 10.4222 0.357689
$$850$$ −4.60555 −0.157969
$$851$$ 12.8444 0.440301
$$852$$ −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
$$855$$ −4.60555 −0.157507
$$856$$ 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 $$0.358480\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ −14.7889 −0.504004
$$862$$ 12.0000 0.408722
$$863$$ 51.6333 1.75762 0.878809 0.477173i $$-0.158339\pi$$
0.878809 + 0.477173i $$0.158339\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 12.4222 0.422368
$$866$$ −19.2111 −0.652820
$$867$$ −4.21110 −0.143017
$$868$$ 27.6333 0.937936
$$869$$ 0 0
$$870$$ −4.60555 −0.156143
$$871$$ 0 0
$$872$$ 1.39445 0.0472220
$$873$$ 1.39445 0.0471949
$$874$$ −6.42221 −0.217234
$$875$$ 4.60555 0.155696
$$876$$ −1.39445 −0.0471141
$$877$$ −24.8444 −0.838936 −0.419468 0.907770i $$-0.637783\pi$$
−0.419468 + 0.907770i $$0.637783\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ −39.2111 −1.32106 −0.660528 0.750802i $$-0.729667\pi$$
−0.660528 + 0.750802i $$0.729667\pi$$
$$882$$ 14.2111 0.478513
$$883$$ −9.57779 −0.322318 −0.161159 0.986928i $$-0.551523\pi$$
−0.161159 + 0.986928i $$0.551523\pi$$
$$884$$ 0 0
$$885$$ −9.21110 −0.309628
$$886$$ 15.6333 0.525211
$$887$$ 6.97224 0.234105 0.117053 0.993126i $$-0.462655\pi$$
0.117053 + 0.993126i $$0.462655\pi$$
$$888$$ −9.21110 −0.309104
$$889$$ −5.57779 −0.187073
$$890$$ −15.2111 −0.509877
$$891$$ 0 0
$$892$$ 1.81665 0.0608261
$$893$$ −42.4222 −1.41960
$$894$$ −15.2111 −0.508735
$$895$$ 19.8167 0.662398
$$896$$ 4.60555 0.153861
$$897$$ 0 0
$$898$$ −33.6333 −1.12236
$$899$$ 27.6333 0.921622
$$900$$ 1.00000 0.0333333
$$901$$ −27.6333 −0.920599
$$902$$ 0 0
$$903$$ 36.8444 1.22611
$$904$$ −13.8167 −0.459535
$$905$$ −8.42221 −0.279964
$$906$$ 6.00000 0.199337
$$907$$ −21.5778 −0.716479 −0.358239 0.933630i $$-0.616623\pi$$
−0.358239 + 0.933630i $$0.616623\pi$$
$$908$$ 24.0000 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
$$915$$ 11.2111 0.370628
$$916$$ 19.8167 0.654761
$$917$$ 104.111 3.43805
$$918$$ 4.60555 0.152006
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 1.39445 0.0459736
$$921$$ −8.78890 −0.289604
$$922$$ −33.6333 −1.10765
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 9.21110 0.302859
$$926$$ 31.3944 1.03169
$$927$$ 4.00000 0.131377
$$928$$ 4.60555 0.151185
$$929$$ 39.2111 1.28647 0.643237 0.765667i $$-0.277591\pi$$
0.643237 + 0.765667i $$0.277591\pi$$
$$930$$ −6.00000 −0.196748
$$931$$ −65.4500 −2.14504
$$932$$ 1.81665 0.0595065
$$933$$ 12.0000 0.392862
$$934$$ 30.4222 0.995445
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.3667 −0.338665 −0.169333 0.985559i $$-0.554161\pi$$
−0.169333 + 0.985559i $$0.554161\pi$$
$$938$$ −14.7889 −0.482875
$$939$$ −3.57779 −0.116757
$$940$$ 9.21110 0.300433
$$941$$ −54.0000 −1.76035 −0.880175 0.474650i $$-0.842575\pi$$
−0.880175 + 0.474650i $$0.842575\pi$$
$$942$$ −20.4222 −0.665391
$$943$$ 4.47772 0.145815
$$944$$ 9.21110 0.299796
$$945$$ −4.60555 −0.149819
$$946$$ 0 0
$$947$$ 15.6333 0.508014 0.254007 0.967202i $$-0.418251\pi$$
0.254007 + 0.967202i $$0.418251\pi$$
$$948$$ 14.4222 0.468411
$$949$$ 0 0
$$950$$ −4.60555 −0.149424
$$951$$ 18.0000 0.583690
$$952$$ −21.2111 −0.687456
$$953$$ 20.2389 0.655601 0.327800 0.944747i $$-0.393693\pi$$
0.327800 + 0.944747i $$0.393693\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −12.0000 −0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −5.57779 −0.180210
$$959$$ −14.7889 −0.477558
$$960$$ −1.00000 −0.0322749
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 6.42221 0.206845
$$965$$ −7.81665 −0.251627
$$966$$ −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
$$969$$ −21.2111 −0.681399
$$970$$ 1.39445 0.0447731
$$971$$ −53.0278 −1.70174 −0.850871 0.525375i $$-0.823925\pi$$
−0.850871 + 0.525375i $$0.823925\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 79.2666 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 $$0.402532\pi$$
$$978$$ 24.4222 0.780936
$$979$$ 0 0
$$980$$ 14.2111 0.453957
$$981$$ 1.39445 0.0445213
$$982$$ −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
$$985$$ −6.00000 −0.191176
$$986$$ −21.2111 −0.675499
$$987$$ −42.4222 −1.35031
$$988$$ 0 0
$$989$$ −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
$$993$$ 16.6056 0.526961
$$994$$ 42.4222 1.34555
$$995$$ −22.4222 −0.710832
$$996$$ −2.78890 −0.0883696
$$997$$ −16.4222 −0.520096 −0.260048 0.965596i $$-0.583738\pi$$
−0.260048 + 0.965596i $$0.583738\pi$$
$$998$$ 23.0278 0.728931
$$999$$ −9.21110 −0.291426
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 5070.2.a.bf.1.2 2
13.5 odd 4 390.2.b.c.181.2 4
13.8 odd 4 390.2.b.c.181.3 yes 4
13.12 even 2 5070.2.a.z.1.1 2
39.5 even 4 1170.2.b.d.181.4 4
39.8 even 4 1170.2.b.d.181.1 4
52.31 even 4 3120.2.g.q.961.1 4
52.47 even 4 3120.2.g.q.961.4 4
65.8 even 4 1950.2.f.n.649.1 4
65.18 even 4 1950.2.f.m.649.2 4
65.34 odd 4 1950.2.b.k.1351.2 4
65.44 odd 4 1950.2.b.k.1351.3 4
65.47 even 4 1950.2.f.m.649.4 4
65.57 even 4 1950.2.f.n.649.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 13.5 odd 4
390.2.b.c.181.3 yes 4 13.8 odd 4
1170.2.b.d.181.1 4 39.8 even 4
1170.2.b.d.181.4 4 39.5 even 4
1950.2.b.k.1351.2 4 65.34 odd 4
1950.2.b.k.1351.3 4 65.44 odd 4
1950.2.f.m.649.2 4 65.18 even 4
1950.2.f.m.649.4 4 65.47 even 4
1950.2.f.n.649.1 4 65.8 even 4
1950.2.f.n.649.3 4 65.57 even 4
3120.2.g.q.961.1 4 52.31 even 4
3120.2.g.q.961.4 4 52.47 even 4
5070.2.a.z.1.1 2 13.12 even 2
5070.2.a.bf.1.2 2 1.1 even 1 trivial