# Properties

 Label 5070.2.b.w.1351.6 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.6 Root $$1.80194i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.w.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +4.80194i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +4.80194i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.55496i q^{11} -1.00000 q^{12} -4.80194 q^{14} +1.00000i q^{15} +1.00000 q^{16} +3.02715 q^{17} +1.00000i q^{18} +3.89977i q^{19} -1.00000i q^{20} +4.80194i q^{21} -2.55496 q^{22} +1.64310 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -4.80194i q^{28} +5.07606 q^{29} -1.00000 q^{30} -5.44504i q^{31} +1.00000i q^{32} +2.55496i q^{33} +3.02715i q^{34} -4.80194 q^{35} -1.00000 q^{36} +7.51573i q^{37} -3.89977 q^{38} +1.00000 q^{40} +5.89977i q^{41} -4.80194 q^{42} +10.1468 q^{43} -2.55496i q^{44} +1.00000i q^{45} +1.64310i q^{46} +6.42758i q^{47} +1.00000 q^{48} -16.0586 q^{49} -1.00000i q^{50} +3.02715 q^{51} -6.91185 q^{53} +1.00000i q^{54} -2.55496 q^{55} +4.80194 q^{56} +3.89977i q^{57} +5.07606i q^{58} +7.60925i q^{59} -1.00000i q^{60} +9.65279 q^{61} +5.44504 q^{62} +4.80194i q^{63} -1.00000 q^{64} -2.55496 q^{66} -0.929312i q^{67} -3.02715 q^{68} +1.64310 q^{69} -4.80194i q^{70} -14.9487i q^{71} -1.00000i q^{72} +14.1129i q^{73} -7.51573 q^{74} -1.00000 q^{75} -3.89977i q^{76} -12.2687 q^{77} -5.40581 q^{79} +1.00000i q^{80} +1.00000 q^{81} -5.89977 q^{82} -4.33513i q^{83} -4.80194i q^{84} +3.02715i q^{85} +10.1468i q^{86} +5.07606 q^{87} +2.55496 q^{88} -16.8267i q^{89} -1.00000 q^{90} -1.64310 q^{92} -5.44504i q^{93} -6.42758 q^{94} -3.89977 q^{95} +1.00000i q^{96} -17.9855i q^{97} -16.0586i q^{98} +2.55496i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} - 6q^{12} - 20q^{14} + 6q^{16} + 6q^{17} - 16q^{22} + 18q^{23} - 6q^{25} + 6q^{27} - 6q^{30} - 20q^{35} - 6q^{36} + 22q^{38} + 6q^{40} - 20q^{42} + 6q^{43} + 6q^{48} - 34q^{49} + 6q^{51} - 34q^{53} - 16q^{55} + 20q^{56} + 22q^{61} + 32q^{62} - 6q^{64} - 16q^{66} - 6q^{68} + 18q^{69} - 20q^{74} - 6q^{75} - 58q^{77} - 6q^{79} + 6q^{81} + 10q^{82} + 16q^{88} - 6q^{90} - 18q^{92} - 6q^{94} + 22q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$5$$ 1.00000i 0.447214i
$$6$$ 1.00000i 0.408248i
$$7$$ 4.80194i 1.81496i 0.420093 + 0.907481i $$0.361997\pi$$
−0.420093 + 0.907481i $$0.638003\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 2.55496i 0.770349i 0.922844 + 0.385174i $$0.125859\pi$$
−0.922844 + 0.385174i $$0.874141\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −4.80194 −1.28337
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 3.02715 0.734191 0.367096 0.930183i $$-0.380352\pi$$
0.367096 + 0.930183i $$0.380352\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 3.89977i 0.894669i 0.894367 + 0.447335i $$0.147627\pi$$
−0.894367 + 0.447335i $$0.852373\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ 4.80194i 1.04787i
$$22$$ −2.55496 −0.544719
$$23$$ 1.64310 0.342611 0.171305 0.985218i $$-0.445202\pi$$
0.171305 + 0.985218i $$0.445202\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 4.80194i − 0.907481i
$$29$$ 5.07606 0.942601 0.471301 0.881973i $$-0.343785\pi$$
0.471301 + 0.881973i $$0.343785\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 5.44504i − 0.977958i −0.872295 0.488979i $$-0.837369\pi$$
0.872295 0.488979i $$-0.162631\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 2.55496i 0.444761i
$$34$$ 3.02715i 0.519151i
$$35$$ −4.80194 −0.811676
$$36$$ −1.00000 −0.166667
$$37$$ 7.51573i 1.23558i 0.786344 + 0.617789i $$0.211971\pi$$
−0.786344 + 0.617789i $$0.788029\pi$$
$$38$$ −3.89977 −0.632627
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 5.89977i 0.921390i 0.887559 + 0.460695i $$0.152400\pi$$
−0.887559 + 0.460695i $$0.847600\pi$$
$$42$$ −4.80194 −0.740955
$$43$$ 10.1468 1.54737 0.773683 0.633573i $$-0.218413\pi$$
0.773683 + 0.633573i $$0.218413\pi$$
$$44$$ − 2.55496i − 0.385174i
$$45$$ 1.00000i 0.149071i
$$46$$ 1.64310i 0.242262i
$$47$$ 6.42758i 0.937559i 0.883315 + 0.468780i $$0.155306\pi$$
−0.883315 + 0.468780i $$0.844694\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −16.0586 −2.29409
$$50$$ − 1.00000i − 0.141421i
$$51$$ 3.02715 0.423885
$$52$$ 0 0
$$53$$ −6.91185 −0.949416 −0.474708 0.880143i $$-0.657446\pi$$
−0.474708 + 0.880143i $$0.657446\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ −2.55496 −0.344510
$$56$$ 4.80194 0.641686
$$57$$ 3.89977i 0.516537i
$$58$$ 5.07606i 0.666520i
$$59$$ 7.60925i 0.990640i 0.868711 + 0.495320i $$0.164949\pi$$
−0.868711 + 0.495320i $$0.835051\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ 9.65279 1.23591 0.617957 0.786212i $$-0.287961\pi$$
0.617957 + 0.786212i $$0.287961\pi$$
$$62$$ 5.44504 0.691521
$$63$$ 4.80194i 0.604987i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.55496 −0.314494
$$67$$ − 0.929312i − 0.113534i −0.998387 0.0567668i $$-0.981921\pi$$
0.998387 0.0567668i $$-0.0180791\pi$$
$$68$$ −3.02715 −0.367096
$$69$$ 1.64310 0.197806
$$70$$ − 4.80194i − 0.573941i
$$71$$ − 14.9487i − 1.77408i −0.461690 0.887042i $$-0.652757\pi$$
0.461690 0.887042i $$-0.347243\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 14.1129i 1.65179i 0.563824 + 0.825895i $$0.309330\pi$$
−0.563824 + 0.825895i $$0.690670\pi$$
$$74$$ −7.51573 −0.873686
$$75$$ −1.00000 −0.115470
$$76$$ − 3.89977i − 0.447335i
$$77$$ −12.2687 −1.39815
$$78$$ 0 0
$$79$$ −5.40581 −0.608202 −0.304101 0.952640i $$-0.598356\pi$$
−0.304101 + 0.952640i $$0.598356\pi$$
$$80$$ 1.00000i 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −5.89977 −0.651521
$$83$$ − 4.33513i − 0.475842i −0.971285 0.237921i $$-0.923534\pi$$
0.971285 0.237921i $$-0.0764659\pi$$
$$84$$ − 4.80194i − 0.523934i
$$85$$ 3.02715i 0.328340i
$$86$$ 10.1468i 1.09415i
$$87$$ 5.07606 0.544211
$$88$$ 2.55496 0.272359
$$89$$ − 16.8267i − 1.78363i −0.452404 0.891813i $$-0.649434\pi$$
0.452404 0.891813i $$-0.350566\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −1.64310 −0.171305
$$93$$ − 5.44504i − 0.564625i
$$94$$ −6.42758 −0.662955
$$95$$ −3.89977 −0.400108
$$96$$ 1.00000i 0.102062i
$$97$$ − 17.9855i − 1.82615i −0.407788 0.913077i $$-0.633700\pi$$
0.407788 0.913077i $$-0.366300\pi$$
$$98$$ − 16.0586i − 1.62216i
$$99$$ 2.55496i 0.256783i
$$100$$ 1.00000 0.100000
$$101$$ 15.7453 1.56671 0.783355 0.621574i $$-0.213506\pi$$
0.783355 + 0.621574i $$0.213506\pi$$
$$102$$ 3.02715i 0.299732i
$$103$$ −12.0858 −1.19084 −0.595422 0.803413i $$-0.703015\pi$$
−0.595422 + 0.803413i $$0.703015\pi$$
$$104$$ 0 0
$$105$$ −4.80194 −0.468621
$$106$$ − 6.91185i − 0.671339i
$$107$$ −4.64071 −0.448634 −0.224317 0.974516i $$-0.572015\pi$$
−0.224317 + 0.974516i $$0.572015\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.20344i 0.402616i 0.979528 + 0.201308i $$0.0645193\pi$$
−0.979528 + 0.201308i $$0.935481\pi$$
$$110$$ − 2.55496i − 0.243606i
$$111$$ 7.51573i 0.713361i
$$112$$ 4.80194i 0.453740i
$$113$$ −0.170915 −0.0160783 −0.00803917 0.999968i $$-0.502559\pi$$
−0.00803917 + 0.999968i $$0.502559\pi$$
$$114$$ −3.89977 −0.365247
$$115$$ 1.64310i 0.153220i
$$116$$ −5.07606 −0.471301
$$117$$ 0 0
$$118$$ −7.60925 −0.700488
$$119$$ 14.5362i 1.33253i
$$120$$ 1.00000 0.0912871
$$121$$ 4.47219 0.406563
$$122$$ 9.65279i 0.873923i
$$123$$ 5.89977i 0.531965i
$$124$$ 5.44504i 0.488979i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ −4.80194 −0.427791
$$127$$ −5.58211 −0.495332 −0.247666 0.968846i $$-0.579664\pi$$
−0.247666 + 0.968846i $$0.579664\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 10.1468 0.893372
$$130$$ 0 0
$$131$$ −3.06100 −0.267441 −0.133720 0.991019i $$-0.542692\pi$$
−0.133720 + 0.991019i $$0.542692\pi$$
$$132$$ − 2.55496i − 0.222381i
$$133$$ −18.7265 −1.62379
$$134$$ 0.929312 0.0802804
$$135$$ 1.00000i 0.0860663i
$$136$$ − 3.02715i − 0.259576i
$$137$$ − 1.03385i − 0.0883279i −0.999024 0.0441640i $$-0.985938\pi$$
0.999024 0.0441640i $$-0.0140624\pi$$
$$138$$ 1.64310i 0.139870i
$$139$$ −16.7995 −1.42492 −0.712459 0.701713i $$-0.752419\pi$$
−0.712459 + 0.701713i $$0.752419\pi$$
$$140$$ 4.80194 0.405838
$$141$$ 6.42758i 0.541300i
$$142$$ 14.9487 1.25447
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 5.07606i 0.421544i
$$146$$ −14.1129 −1.16799
$$147$$ −16.0586 −1.32449
$$148$$ − 7.51573i − 0.617789i
$$149$$ − 17.8756i − 1.46443i −0.681075 0.732213i $$-0.738487\pi$$
0.681075 0.732213i $$-0.261513\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ 0.390748i 0.0317986i 0.999874 + 0.0158993i $$0.00506112\pi$$
−0.999874 + 0.0158993i $$0.994939\pi$$
$$152$$ 3.89977 0.316313
$$153$$ 3.02715 0.244730
$$154$$ − 12.2687i − 0.988644i
$$155$$ 5.44504 0.437356
$$156$$ 0 0
$$157$$ 9.50902 0.758903 0.379451 0.925212i $$-0.376113\pi$$
0.379451 + 0.925212i $$0.376113\pi$$
$$158$$ − 5.40581i − 0.430063i
$$159$$ −6.91185 −0.548146
$$160$$ −1.00000 −0.0790569
$$161$$ 7.89008i 0.621826i
$$162$$ 1.00000i 0.0785674i
$$163$$ − 5.84846i − 0.458087i −0.973416 0.229043i $$-0.926440\pi$$
0.973416 0.229043i $$-0.0735598\pi$$
$$164$$ − 5.89977i − 0.460695i
$$165$$ −2.55496 −0.198903
$$166$$ 4.33513 0.336471
$$167$$ − 2.55496i − 0.197709i −0.995102 0.0988543i $$-0.968482\pi$$
0.995102 0.0988543i $$-0.0315178\pi$$
$$168$$ 4.80194 0.370478
$$169$$ 0 0
$$170$$ −3.02715 −0.232172
$$171$$ 3.89977i 0.298223i
$$172$$ −10.1468 −0.773683
$$173$$ 21.5308 1.63696 0.818478 0.574538i $$-0.194818\pi$$
0.818478 + 0.574538i $$0.194818\pi$$
$$174$$ 5.07606i 0.384815i
$$175$$ − 4.80194i − 0.362992i
$$176$$ 2.55496i 0.192587i
$$177$$ 7.60925i 0.571946i
$$178$$ 16.8267 1.26121
$$179$$ −15.5483 −1.16213 −0.581066 0.813857i $$-0.697364\pi$$
−0.581066 + 0.813857i $$0.697364\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ −14.4058 −1.07078 −0.535388 0.844606i $$-0.679835\pi$$
−0.535388 + 0.844606i $$0.679835\pi$$
$$182$$ 0 0
$$183$$ 9.65279 0.713555
$$184$$ − 1.64310i − 0.121131i
$$185$$ −7.51573 −0.552567
$$186$$ 5.44504 0.399250
$$187$$ 7.73423i 0.565583i
$$188$$ − 6.42758i − 0.468780i
$$189$$ 4.80194i 0.349290i
$$190$$ − 3.89977i − 0.282919i
$$191$$ −4.02177 −0.291005 −0.145503 0.989358i $$-0.546480\pi$$
−0.145503 + 0.989358i $$0.546480\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 20.1008i − 1.44689i −0.690383 0.723444i $$-0.742558\pi$$
0.690383 0.723444i $$-0.257442\pi$$
$$194$$ 17.9855 1.29129
$$195$$ 0 0
$$196$$ 16.0586 1.14704
$$197$$ 15.3884i 1.09637i 0.836356 + 0.548187i $$0.184682\pi$$
−0.836356 + 0.548187i $$0.815318\pi$$
$$198$$ −2.55496 −0.181573
$$199$$ −24.7724 −1.75607 −0.878034 0.478598i $$-0.841145\pi$$
−0.878034 + 0.478598i $$0.841145\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ − 0.929312i − 0.0655486i
$$202$$ 15.7453i 1.10783i
$$203$$ 24.3749i 1.71079i
$$204$$ −3.02715 −0.211943
$$205$$ −5.89977 −0.412058
$$206$$ − 12.0858i − 0.842054i
$$207$$ 1.64310 0.114204
$$208$$ 0 0
$$209$$ −9.96376 −0.689207
$$210$$ − 4.80194i − 0.331365i
$$211$$ 6.49396 0.447063 0.223531 0.974697i $$-0.428242\pi$$
0.223531 + 0.974697i $$0.428242\pi$$
$$212$$ 6.91185 0.474708
$$213$$ − 14.9487i − 1.02427i
$$214$$ − 4.64071i − 0.317232i
$$215$$ 10.1468i 0.692003i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 26.1468 1.77496
$$218$$ −4.20344 −0.284693
$$219$$ 14.1129i 0.953661i
$$220$$ 2.55496 0.172255
$$221$$ 0 0
$$222$$ −7.51573 −0.504423
$$223$$ − 20.1250i − 1.34767i −0.738883 0.673834i $$-0.764646\pi$$
0.738883 0.673834i $$-0.235354\pi$$
$$224$$ −4.80194 −0.320843
$$225$$ −1.00000 −0.0666667
$$226$$ − 0.170915i − 0.0113691i
$$227$$ 24.2379i 1.60872i 0.594139 + 0.804362i $$0.297493\pi$$
−0.594139 + 0.804362i $$0.702507\pi$$
$$228$$ − 3.89977i − 0.258269i
$$229$$ − 2.71618i − 0.179491i −0.995965 0.0897453i $$-0.971395\pi$$
0.995965 0.0897453i $$-0.0286053\pi$$
$$230$$ −1.64310 −0.108343
$$231$$ −12.2687 −0.807224
$$232$$ − 5.07606i − 0.333260i
$$233$$ 2.16421 0.141782 0.0708911 0.997484i $$-0.477416\pi$$
0.0708911 + 0.997484i $$0.477416\pi$$
$$234$$ 0 0
$$235$$ −6.42758 −0.419289
$$236$$ − 7.60925i − 0.495320i
$$237$$ −5.40581 −0.351145
$$238$$ −14.5362 −0.942240
$$239$$ − 14.5114i − 0.938666i −0.883021 0.469333i $$-0.844494\pi$$
0.883021 0.469333i $$-0.155506\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ − 18.1142i − 1.16684i −0.812171 0.583420i $$-0.801714\pi$$
0.812171 0.583420i $$-0.198286\pi$$
$$242$$ 4.47219i 0.287483i
$$243$$ 1.00000 0.0641500
$$244$$ −9.65279 −0.617957
$$245$$ − 16.0586i − 1.02595i
$$246$$ −5.89977 −0.376156
$$247$$ 0 0
$$248$$ −5.44504 −0.345761
$$249$$ − 4.33513i − 0.274727i
$$250$$ 1.00000 0.0632456
$$251$$ −8.55257 −0.539833 −0.269917 0.962884i $$-0.586996\pi$$
−0.269917 + 0.962884i $$0.586996\pi$$
$$252$$ − 4.80194i − 0.302494i
$$253$$ 4.19806i 0.263930i
$$254$$ − 5.58211i − 0.350252i
$$255$$ 3.02715i 0.189567i
$$256$$ 1.00000 0.0625000
$$257$$ −1.30260 −0.0812541 −0.0406270 0.999174i $$-0.512936\pi$$
−0.0406270 + 0.999174i $$0.512936\pi$$
$$258$$ 10.1468i 0.631709i
$$259$$ −36.0901 −2.24253
$$260$$ 0 0
$$261$$ 5.07606 0.314200
$$262$$ − 3.06100i − 0.189109i
$$263$$ 8.24027 0.508117 0.254059 0.967189i $$-0.418234\pi$$
0.254059 + 0.967189i $$0.418234\pi$$
$$264$$ 2.55496 0.157247
$$265$$ − 6.91185i − 0.424592i
$$266$$ − 18.7265i − 1.14819i
$$267$$ − 16.8267i − 1.02978i
$$268$$ 0.929312i 0.0567668i
$$269$$ 22.6601 1.38161 0.690805 0.723041i $$-0.257256\pi$$
0.690805 + 0.723041i $$0.257256\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 13.9172i 0.845412i 0.906267 + 0.422706i $$0.138920\pi$$
−0.906267 + 0.422706i $$0.861080\pi$$
$$272$$ 3.02715 0.183548
$$273$$ 0 0
$$274$$ 1.03385 0.0624573
$$275$$ − 2.55496i − 0.154070i
$$276$$ −1.64310 −0.0989032
$$277$$ 24.2610 1.45770 0.728851 0.684673i $$-0.240055\pi$$
0.728851 + 0.684673i $$0.240055\pi$$
$$278$$ − 16.7995i − 1.00757i
$$279$$ − 5.44504i − 0.325986i
$$280$$ 4.80194i 0.286971i
$$281$$ 10.5961i 0.632111i 0.948741 + 0.316055i $$0.102359\pi$$
−0.948741 + 0.316055i $$0.897641\pi$$
$$282$$ −6.42758 −0.382757
$$283$$ −9.86294 −0.586291 −0.293145 0.956068i $$-0.594702\pi$$
−0.293145 + 0.956068i $$0.594702\pi$$
$$284$$ 14.9487i 0.887042i
$$285$$ −3.89977 −0.231003
$$286$$ 0 0
$$287$$ −28.3303 −1.67229
$$288$$ 1.00000i 0.0589256i
$$289$$ −7.83638 −0.460964
$$290$$ −5.07606 −0.298077
$$291$$ − 17.9855i − 1.05433i
$$292$$ − 14.1129i − 0.825895i
$$293$$ 5.68233i 0.331965i 0.986129 + 0.165983i $$0.0530796\pi$$
−0.986129 + 0.165983i $$0.946920\pi$$
$$294$$ − 16.0586i − 0.936557i
$$295$$ −7.60925 −0.443028
$$296$$ 7.51573 0.436843
$$297$$ 2.55496i 0.148254i
$$298$$ 17.8756 1.03551
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 48.7241i 2.80841i
$$302$$ −0.390748 −0.0224850
$$303$$ 15.7453 0.904541
$$304$$ 3.89977i 0.223667i
$$305$$ 9.65279i 0.552717i
$$306$$ 3.02715i 0.173050i
$$307$$ − 11.5961i − 0.661825i −0.943661 0.330912i $$-0.892644\pi$$
0.943661 0.330912i $$-0.107356\pi$$
$$308$$ 12.2687 0.699077
$$309$$ −12.0858 −0.687534
$$310$$ 5.44504i 0.309258i
$$311$$ −4.43429 −0.251445 −0.125723 0.992065i $$-0.540125\pi$$
−0.125723 + 0.992065i $$0.540125\pi$$
$$312$$ 0 0
$$313$$ 11.2881 0.638043 0.319021 0.947748i $$-0.396646\pi$$
0.319021 + 0.947748i $$0.396646\pi$$
$$314$$ 9.50902i 0.536625i
$$315$$ −4.80194 −0.270559
$$316$$ 5.40581 0.304101
$$317$$ 6.31229i 0.354534i 0.984163 + 0.177267i $$0.0567255\pi$$
−0.984163 + 0.177267i $$0.943274\pi$$
$$318$$ − 6.91185i − 0.387598i
$$319$$ 12.9691i 0.726132i
$$320$$ − 1.00000i − 0.0559017i
$$321$$ −4.64071 −0.259019
$$322$$ −7.89008 −0.439697
$$323$$ 11.8052i 0.656858i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 5.84846 0.323916
$$327$$ 4.20344i 0.232451i
$$328$$ 5.89977 0.325760
$$329$$ −30.8649 −1.70163
$$330$$ − 2.55496i − 0.140646i
$$331$$ − 10.9554i − 0.602163i −0.953598 0.301081i $$-0.902652\pi$$
0.953598 0.301081i $$-0.0973476\pi$$
$$332$$ 4.33513i 0.237921i
$$333$$ 7.51573i 0.411859i
$$334$$ 2.55496 0.139801
$$335$$ 0.929312 0.0507738
$$336$$ 4.80194i 0.261967i
$$337$$ −26.6668 −1.45263 −0.726316 0.687361i $$-0.758769\pi$$
−0.726316 + 0.687361i $$0.758769\pi$$
$$338$$ 0 0
$$339$$ −0.170915 −0.00928284
$$340$$ − 3.02715i − 0.164170i
$$341$$ 13.9119 0.753369
$$342$$ −3.89977 −0.210876
$$343$$ − 43.4989i − 2.34872i
$$344$$ − 10.1468i − 0.547076i
$$345$$ 1.64310i 0.0884618i
$$346$$ 21.5308i 1.15750i
$$347$$ 25.0694 1.34579 0.672897 0.739736i $$-0.265050\pi$$
0.672897 + 0.739736i $$0.265050\pi$$
$$348$$ −5.07606 −0.272106
$$349$$ − 21.5623i − 1.15420i −0.816673 0.577100i $$-0.804184\pi$$
0.816673 0.577100i $$-0.195816\pi$$
$$350$$ 4.80194 0.256674
$$351$$ 0 0
$$352$$ −2.55496 −0.136180
$$353$$ 28.2959i 1.50604i 0.657998 + 0.753019i $$0.271403\pi$$
−0.657998 + 0.753019i $$0.728597\pi$$
$$354$$ −7.60925 −0.404427
$$355$$ 14.9487 0.793394
$$356$$ 16.8267i 0.891813i
$$357$$ 14.5362i 0.769336i
$$358$$ − 15.5483i − 0.821751i
$$359$$ 34.2717i 1.80879i 0.426693 + 0.904396i $$0.359678\pi$$
−0.426693 + 0.904396i $$0.640322\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 3.79178 0.199567
$$362$$ − 14.4058i − 0.757153i
$$363$$ 4.47219 0.234729
$$364$$ 0 0
$$365$$ −14.1129 −0.738703
$$366$$ 9.65279i 0.504560i
$$367$$ −23.4209 −1.22256 −0.611280 0.791414i $$-0.709345\pi$$
−0.611280 + 0.791414i $$0.709345\pi$$
$$368$$ 1.64310 0.0856527
$$369$$ 5.89977i 0.307130i
$$370$$ − 7.51573i − 0.390724i
$$371$$ − 33.1903i − 1.72315i
$$372$$ 5.44504i 0.282312i
$$373$$ 1.82371 0.0944280 0.0472140 0.998885i $$-0.484966\pi$$
0.0472140 + 0.998885i $$0.484966\pi$$
$$374$$ −7.73423 −0.399928
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 6.42758 0.331477
$$377$$ 0 0
$$378$$ −4.80194 −0.246985
$$379$$ 15.3260i 0.787245i 0.919272 + 0.393623i $$0.128778\pi$$
−0.919272 + 0.393623i $$0.871222\pi$$
$$380$$ 3.89977 0.200054
$$381$$ −5.58211 −0.285980
$$382$$ − 4.02177i − 0.205772i
$$383$$ 29.6722i 1.51618i 0.652152 + 0.758089i $$0.273867\pi$$
−0.652152 + 0.758089i $$0.726133\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ − 12.2687i − 0.625273i
$$386$$ 20.1008 1.02310
$$387$$ 10.1468 0.515788
$$388$$ 17.9855i 0.913077i
$$389$$ −11.8291 −0.599758 −0.299879 0.953977i $$-0.596946\pi$$
−0.299879 + 0.953977i $$0.596946\pi$$
$$390$$ 0 0
$$391$$ 4.97392 0.251542
$$392$$ 16.0586i 0.811082i
$$393$$ −3.06100 −0.154407
$$394$$ −15.3884 −0.775254
$$395$$ − 5.40581i − 0.271996i
$$396$$ − 2.55496i − 0.128391i
$$397$$ 12.1153i 0.608049i 0.952664 + 0.304025i $$0.0983305\pi$$
−0.952664 + 0.304025i $$0.901670\pi$$
$$398$$ − 24.7724i − 1.24173i
$$399$$ −18.7265 −0.937496
$$400$$ −1.00000 −0.0500000
$$401$$ − 10.8291i − 0.540779i −0.962751 0.270389i $$-0.912848\pi$$
0.962751 0.270389i $$-0.0871524\pi$$
$$402$$ 0.929312 0.0463499
$$403$$ 0 0
$$404$$ −15.7453 −0.783355
$$405$$ 1.00000i 0.0496904i
$$406$$ −24.3749 −1.20971
$$407$$ −19.2024 −0.951826
$$408$$ − 3.02715i − 0.149866i
$$409$$ 33.9396i 1.67820i 0.543973 + 0.839102i $$0.316919\pi$$
−0.543973 + 0.839102i $$0.683081\pi$$
$$410$$ − 5.89977i − 0.291369i
$$411$$ − 1.03385i − 0.0509962i
$$412$$ 12.0858 0.595422
$$413$$ −36.5392 −1.79797
$$414$$ 1.64310i 0.0807542i
$$415$$ 4.33513 0.212803
$$416$$ 0 0
$$417$$ −16.7995 −0.822677
$$418$$ − 9.96376i − 0.487343i
$$419$$ 26.8562 1.31201 0.656006 0.754755i $$-0.272244\pi$$
0.656006 + 0.754755i $$0.272244\pi$$
$$420$$ 4.80194 0.234311
$$421$$ − 27.9928i − 1.36429i −0.731219 0.682143i $$-0.761048\pi$$
0.731219 0.682143i $$-0.238952\pi$$
$$422$$ 6.49396i 0.316121i
$$423$$ 6.42758i 0.312520i
$$424$$ 6.91185i 0.335669i
$$425$$ −3.02715 −0.146838
$$426$$ 14.9487 0.724266
$$427$$ 46.3521i 2.24314i
$$428$$ 4.64071 0.224317
$$429$$ 0 0
$$430$$ −10.1468 −0.489320
$$431$$ 33.4088i 1.60925i 0.593787 + 0.804623i $$0.297632\pi$$
−0.593787 + 0.804623i $$0.702368\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 13.6420 0.655595 0.327797 0.944748i $$-0.393694\pi$$
0.327797 + 0.944748i $$0.393694\pi$$
$$434$$ 26.1468i 1.25508i
$$435$$ 5.07606i 0.243379i
$$436$$ − 4.20344i − 0.201308i
$$437$$ 6.40773i 0.306523i
$$438$$ −14.1129 −0.674340
$$439$$ 30.6437 1.46254 0.731272 0.682086i $$-0.238927\pi$$
0.731272 + 0.682086i $$0.238927\pi$$
$$440$$ 2.55496i 0.121803i
$$441$$ −16.0586 −0.764696
$$442$$ 0 0
$$443$$ −19.7409 −0.937920 −0.468960 0.883219i $$-0.655371\pi$$
−0.468960 + 0.883219i $$0.655371\pi$$
$$444$$ − 7.51573i − 0.356681i
$$445$$ 16.8267 0.797662
$$446$$ 20.1250 0.952946
$$447$$ − 17.8756i − 0.845487i
$$448$$ − 4.80194i − 0.226870i
$$449$$ 36.2664i 1.71152i 0.517377 + 0.855758i $$0.326909\pi$$
−0.517377 + 0.855758i $$0.673091\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ −15.0737 −0.709791
$$452$$ 0.170915 0.00803917
$$453$$ 0.390748i 0.0183589i
$$454$$ −24.2379 −1.13754
$$455$$ 0 0
$$456$$ 3.89977 0.182624
$$457$$ − 3.55602i − 0.166344i −0.996535 0.0831719i $$-0.973495\pi$$
0.996535 0.0831719i $$-0.0265050\pi$$
$$458$$ 2.71618 0.126919
$$459$$ 3.02715 0.141295
$$460$$ − 1.64310i − 0.0766101i
$$461$$ − 1.46011i − 0.0680040i −0.999422 0.0340020i $$-0.989175\pi$$
0.999422 0.0340020i $$-0.0108253\pi$$
$$462$$ − 12.2687i − 0.570794i
$$463$$ − 13.7942i − 0.641069i −0.947237 0.320535i $$-0.896137\pi$$
0.947237 0.320535i $$-0.103863\pi$$
$$464$$ 5.07606 0.235650
$$465$$ 5.44504 0.252508
$$466$$ 2.16421i 0.100255i
$$467$$ 1.72886 0.0800020 0.0400010 0.999200i $$-0.487264\pi$$
0.0400010 + 0.999200i $$0.487264\pi$$
$$468$$ 0 0
$$469$$ 4.46250 0.206059
$$470$$ − 6.42758i − 0.296482i
$$471$$ 9.50902 0.438153
$$472$$ 7.60925 0.350244
$$473$$ 25.9245i 1.19201i
$$474$$ − 5.40581i − 0.248297i
$$475$$ − 3.89977i − 0.178934i
$$476$$ − 14.5362i − 0.666264i
$$477$$ −6.91185 −0.316472
$$478$$ 14.5114 0.663737
$$479$$ 14.5496i 0.664787i 0.943141 + 0.332394i $$0.107856\pi$$
−0.943141 + 0.332394i $$0.892144\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 18.1142 0.825080
$$483$$ 7.89008i 0.359011i
$$484$$ −4.47219 −0.203281
$$485$$ 17.9855 0.816681
$$486$$ 1.00000i 0.0453609i
$$487$$ − 30.5526i − 1.38447i −0.721673 0.692234i $$-0.756626\pi$$
0.721673 0.692234i $$-0.243374\pi$$
$$488$$ − 9.65279i − 0.436961i
$$489$$ − 5.84846i − 0.264477i
$$490$$ 16.0586 0.725454
$$491$$ −9.68233 −0.436958 −0.218479 0.975842i $$-0.570109\pi$$
−0.218479 + 0.975842i $$0.570109\pi$$
$$492$$ − 5.89977i − 0.265982i
$$493$$ 15.3660 0.692050
$$494$$ 0 0
$$495$$ −2.55496 −0.114837
$$496$$ − 5.44504i − 0.244490i
$$497$$ 71.7827 3.21989
$$498$$ 4.33513 0.194262
$$499$$ − 18.6025i − 0.832764i −0.909190 0.416382i $$-0.863298\pi$$
0.909190 0.416382i $$-0.136702\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ − 2.55496i − 0.114147i
$$502$$ − 8.55257i − 0.381720i
$$503$$ 41.6644 1.85772 0.928862 0.370426i $$-0.120788\pi$$
0.928862 + 0.370426i $$0.120788\pi$$
$$504$$ 4.80194 0.213895
$$505$$ 15.7453i 0.700654i
$$506$$ −4.19806 −0.186627
$$507$$ 0 0
$$508$$ 5.58211 0.247666
$$509$$ − 41.1648i − 1.82460i −0.409525 0.912299i $$-0.634306\pi$$
0.409525 0.912299i $$-0.365694\pi$$
$$510$$ −3.02715 −0.134044
$$511$$ −67.7693 −2.99794
$$512$$ 1.00000i 0.0441942i
$$513$$ 3.89977i 0.172179i
$$514$$ − 1.30260i − 0.0574553i
$$515$$ − 12.0858i − 0.532562i
$$516$$ −10.1468 −0.446686
$$517$$ −16.4222 −0.722248
$$518$$ − 36.0901i − 1.58571i
$$519$$ 21.5308 0.945097
$$520$$ 0 0
$$521$$ 1.12737 0.0493912 0.0246956 0.999695i $$-0.492138\pi$$
0.0246956 + 0.999695i $$0.492138\pi$$
$$522$$ 5.07606i 0.222173i
$$523$$ −22.7922 −0.996635 −0.498318 0.866994i $$-0.666049\pi$$
−0.498318 + 0.866994i $$0.666049\pi$$
$$524$$ 3.06100 0.133720
$$525$$ − 4.80194i − 0.209574i
$$526$$ 8.24027i 0.359293i
$$527$$ − 16.4829i − 0.718008i
$$528$$ 2.55496i 0.111190i
$$529$$ −20.3002 −0.882618
$$530$$ 6.91185 0.300232
$$531$$ 7.60925i 0.330213i
$$532$$ 18.7265 0.811895
$$533$$ 0 0
$$534$$ 16.8267 0.728162
$$535$$ − 4.64071i − 0.200635i
$$536$$ −0.929312 −0.0401402
$$537$$ −15.5483 −0.670957
$$538$$ 22.6601i 0.976946i
$$539$$ − 41.0291i − 1.76725i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ − 41.1148i − 1.76766i −0.467804 0.883832i $$-0.654955\pi$$
0.467804 0.883832i $$-0.345045\pi$$
$$542$$ −13.9172 −0.597796
$$543$$ −14.4058 −0.618213
$$544$$ 3.02715i 0.129788i
$$545$$ −4.20344 −0.180056
$$546$$ 0 0
$$547$$ 3.96615 0.169580 0.0847901 0.996399i $$-0.472978\pi$$
0.0847901 + 0.996399i $$0.472978\pi$$
$$548$$ 1.03385i 0.0441640i
$$549$$ 9.65279 0.411971
$$550$$ 2.55496 0.108944
$$551$$ 19.7955i 0.843316i
$$552$$ − 1.64310i − 0.0699352i
$$553$$ − 25.9584i − 1.10386i
$$554$$ 24.2610i 1.03075i
$$555$$ −7.51573 −0.319025
$$556$$ 16.7995 0.712459
$$557$$ − 4.25236i − 0.180178i −0.995934 0.0900891i $$-0.971285\pi$$
0.995934 0.0900891i $$-0.0287152\pi$$
$$558$$ 5.44504 0.230507
$$559$$ 0 0
$$560$$ −4.80194 −0.202919
$$561$$ 7.73423i 0.326540i
$$562$$ −10.5961 −0.446970
$$563$$ 32.7700 1.38109 0.690546 0.723289i $$-0.257371\pi$$
0.690546 + 0.723289i $$0.257371\pi$$
$$564$$ − 6.42758i − 0.270650i
$$565$$ − 0.170915i − 0.00719046i
$$566$$ − 9.86294i − 0.414570i
$$567$$ 4.80194i 0.201662i
$$568$$ −14.9487 −0.627233
$$569$$ 20.6189 0.864391 0.432195 0.901780i $$-0.357739\pi$$
0.432195 + 0.901780i $$0.357739\pi$$
$$570$$ − 3.89977i − 0.163343i
$$571$$ −23.1215 −0.967606 −0.483803 0.875177i $$-0.660745\pi$$
−0.483803 + 0.875177i $$0.660745\pi$$
$$572$$ 0 0
$$573$$ −4.02177 −0.168012
$$574$$ − 28.3303i − 1.18249i
$$575$$ −1.64310 −0.0685222
$$576$$ −1.00000 −0.0416667
$$577$$ − 28.0000i − 1.16566i −0.812596 0.582828i $$-0.801946\pi$$
0.812596 0.582828i $$-0.198054\pi$$
$$578$$ − 7.83638i − 0.325950i
$$579$$ − 20.1008i − 0.835362i
$$580$$ − 5.07606i − 0.210772i
$$581$$ 20.8170 0.863635
$$582$$ 17.9855 0.745524
$$583$$ − 17.6595i − 0.731382i
$$584$$ 14.1129 0.583996
$$585$$ 0 0
$$586$$ −5.68233 −0.234735
$$587$$ − 21.1081i − 0.871225i −0.900134 0.435613i $$-0.856532\pi$$
0.900134 0.435613i $$-0.143468\pi$$
$$588$$ 16.0586 0.662246
$$589$$ 21.2344 0.874949
$$590$$ − 7.60925i − 0.313268i
$$591$$ 15.3884i 0.632992i
$$592$$ 7.51573i 0.308895i
$$593$$ − 15.4276i − 0.633535i −0.948503 0.316767i $$-0.897403\pi$$
0.948503 0.316767i $$-0.102597\pi$$
$$594$$ −2.55496 −0.104831
$$595$$ −14.5362 −0.595925
$$596$$ 17.8756i 0.732213i
$$597$$ −24.7724 −1.01387
$$598$$ 0 0
$$599$$ 9.16985 0.374670 0.187335 0.982296i $$-0.440015\pi$$
0.187335 + 0.982296i $$0.440015\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ −25.8864 −1.05593 −0.527963 0.849267i $$-0.677044\pi$$
−0.527963 + 0.849267i $$0.677044\pi$$
$$602$$ −48.7241 −1.98584
$$603$$ − 0.929312i − 0.0378445i
$$604$$ − 0.390748i − 0.0158993i
$$605$$ 4.47219i 0.181820i
$$606$$ 15.7453i 0.639607i
$$607$$ −11.6093 −0.471205 −0.235603 0.971850i $$-0.575706\pi$$
−0.235603 + 0.971850i $$0.575706\pi$$
$$608$$ −3.89977 −0.158157
$$609$$ 24.3749i 0.987723i
$$610$$ −9.65279 −0.390830
$$611$$ 0 0
$$612$$ −3.02715 −0.122365
$$613$$ − 16.5985i − 0.670407i −0.942146 0.335204i $$-0.891195\pi$$
0.942146 0.335204i $$-0.108805\pi$$
$$614$$ 11.5961 0.467981
$$615$$ −5.89977 −0.237902
$$616$$ 12.2687i 0.494322i
$$617$$ 33.4373i 1.34613i 0.739582 + 0.673067i $$0.235023\pi$$
−0.739582 + 0.673067i $$0.764977\pi$$
$$618$$ − 12.0858i − 0.486160i
$$619$$ 25.4198i 1.02171i 0.859667 + 0.510854i $$0.170671\pi$$
−0.859667 + 0.510854i $$0.829329\pi$$
$$620$$ −5.44504 −0.218678
$$621$$ 1.64310 0.0659355
$$622$$ − 4.43429i − 0.177799i
$$623$$ 80.8007 3.23721
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 11.2881i 0.451164i
$$627$$ −9.96376 −0.397914
$$628$$ −9.50902 −0.379451
$$629$$ 22.7512i 0.907150i
$$630$$ − 4.80194i − 0.191314i
$$631$$ 15.1879i 0.604621i 0.953210 + 0.302310i $$0.0977579\pi$$
−0.953210 + 0.302310i $$0.902242\pi$$
$$632$$ 5.40581i 0.215032i
$$633$$ 6.49396 0.258112
$$634$$ −6.31229 −0.250693
$$635$$ − 5.58211i − 0.221519i
$$636$$ 6.91185 0.274073
$$637$$ 0 0
$$638$$ −12.9691 −0.513453
$$639$$ − 14.9487i − 0.591361i
$$640$$ 1.00000 0.0395285
$$641$$ −16.8009 −0.663595 −0.331797 0.943351i $$-0.607655\pi$$
−0.331797 + 0.943351i $$0.607655\pi$$
$$642$$ − 4.64071i − 0.183154i
$$643$$ 35.2597i 1.39050i 0.718766 + 0.695252i $$0.244707\pi$$
−0.718766 + 0.695252i $$0.755293\pi$$
$$644$$ − 7.89008i − 0.310913i
$$645$$ 10.1468i 0.399528i
$$646$$ −11.8052 −0.464469
$$647$$ 26.7985 1.05356 0.526778 0.850003i $$-0.323400\pi$$
0.526778 + 0.850003i $$0.323400\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −19.4413 −0.763139
$$650$$ 0 0
$$651$$ 26.1468 1.02477
$$652$$ 5.84846i 0.229043i
$$653$$ −10.8358 −0.424037 −0.212019 0.977266i $$-0.568004\pi$$
−0.212019 + 0.977266i $$0.568004\pi$$
$$654$$ −4.20344 −0.164367
$$655$$ − 3.06100i − 0.119603i
$$656$$ 5.89977i 0.230347i
$$657$$ 14.1129i 0.550597i
$$658$$ − 30.8649i − 1.20324i
$$659$$ 49.3870 1.92385 0.961923 0.273322i $$-0.0881223\pi$$
0.961923 + 0.273322i $$0.0881223\pi$$
$$660$$ 2.55496 0.0994516
$$661$$ 2.18167i 0.0848571i 0.999100 + 0.0424285i $$0.0135095\pi$$
−0.999100 + 0.0424285i $$0.986491\pi$$
$$662$$ 10.9554 0.425793
$$663$$ 0 0
$$664$$ −4.33513 −0.168236
$$665$$ − 18.7265i − 0.726181i
$$666$$ −7.51573 −0.291229
$$667$$ 8.34050 0.322946
$$668$$ 2.55496i 0.0988543i
$$669$$ − 20.1250i − 0.778077i
$$670$$ 0.929312i 0.0359025i
$$671$$ 24.6625i 0.952085i
$$672$$ −4.80194 −0.185239
$$673$$ 24.2319 0.934072 0.467036 0.884238i $$-0.345322\pi$$
0.467036 + 0.884238i $$0.345322\pi$$
$$674$$ − 26.6668i − 1.02717i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 24.0422 0.924017 0.462009 0.886875i $$-0.347129\pi$$
0.462009 + 0.886875i $$0.347129\pi$$
$$678$$ − 0.170915i − 0.00656396i
$$679$$ 86.3654 3.31440
$$680$$ 3.02715 0.116086
$$681$$ 24.2379i 0.928798i
$$682$$ 13.9119i 0.532712i
$$683$$ − 24.3183i − 0.930512i −0.885176 0.465256i $$-0.845962\pi$$
0.885176 0.465256i $$-0.154038\pi$$
$$684$$ − 3.89977i − 0.149112i
$$685$$ 1.03385 0.0395014
$$686$$ 43.4989 1.66079
$$687$$ − 2.71618i − 0.103629i
$$688$$ 10.1468 0.386841
$$689$$ 0 0
$$690$$ −1.64310 −0.0625519
$$691$$ − 18.2054i − 0.692564i −0.938130 0.346282i $$-0.887444\pi$$
0.938130 0.346282i $$-0.112556\pi$$
$$692$$ −21.5308 −0.818478
$$693$$ −12.2687 −0.466051
$$694$$ 25.0694i 0.951620i
$$695$$ − 16.7995i − 0.637243i
$$696$$ − 5.07606i − 0.192408i
$$697$$ 17.8595i 0.676476i
$$698$$ 21.5623 0.816143
$$699$$ 2.16421 0.0818580
$$700$$ 4.80194i 0.181496i
$$701$$ 16.0441 0.605978 0.302989 0.952994i $$-0.402015\pi$$
0.302989 + 0.952994i $$0.402015\pi$$
$$702$$ 0 0
$$703$$ −29.3096 −1.10543
$$704$$ − 2.55496i − 0.0962936i
$$705$$ −6.42758 −0.242077
$$706$$ −28.2959 −1.06493
$$707$$ 75.6077i 2.84352i
$$708$$ − 7.60925i − 0.285973i
$$709$$ − 20.8025i − 0.781255i −0.920549 0.390628i $$-0.872258\pi$$
0.920549 0.390628i $$-0.127742\pi$$
$$710$$ 14.9487i 0.561014i
$$711$$ −5.40581 −0.202734
$$712$$ −16.8267 −0.630607
$$713$$ − 8.94677i − 0.335059i
$$714$$ −14.5362 −0.544003
$$715$$ 0 0
$$716$$ 15.5483 0.581066
$$717$$ − 14.5114i − 0.541939i
$$718$$ −34.2717 −1.27901
$$719$$ 40.3424 1.50452 0.752259 0.658867i $$-0.228964\pi$$
0.752259 + 0.658867i $$0.228964\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ − 58.0350i − 2.16134i
$$722$$ 3.79178i 0.141115i
$$723$$ − 18.1142i − 0.673675i
$$724$$ 14.4058 0.535388
$$725$$ −5.07606 −0.188520
$$726$$ 4.47219i 0.165978i
$$727$$ 47.2073 1.75082 0.875410 0.483380i $$-0.160591\pi$$
0.875410 + 0.483380i $$0.160591\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 14.1129i − 0.522342i
$$731$$ 30.7157 1.13606
$$732$$ −9.65279 −0.356777
$$733$$ − 41.5424i − 1.53440i −0.641406 0.767202i $$-0.721648\pi$$
0.641406 0.767202i $$-0.278352\pi$$
$$734$$ − 23.4209i − 0.864480i
$$735$$ − 16.0586i − 0.592331i
$$736$$ 1.64310i 0.0605656i
$$737$$ 2.37435 0.0874605
$$738$$ −5.89977 −0.217174
$$739$$ 26.3811i 0.970443i 0.874391 + 0.485221i $$0.161261\pi$$
−0.874391 + 0.485221i $$0.838739\pi$$
$$740$$ 7.51573 0.276284
$$741$$ 0 0
$$742$$ 33.1903 1.21845
$$743$$ 20.2881i 0.744299i 0.928173 + 0.372150i $$0.121379\pi$$
−0.928173 + 0.372150i $$0.878621\pi$$
$$744$$ −5.44504 −0.199625
$$745$$ 17.8756 0.654912
$$746$$ 1.82371i 0.0667707i
$$747$$ − 4.33513i − 0.158614i
$$748$$ − 7.73423i − 0.282792i
$$749$$ − 22.2844i − 0.814254i
$$750$$ 1.00000 0.0365148
$$751$$ 21.1280 0.770970 0.385485 0.922714i $$-0.374034\pi$$
0.385485 + 0.922714i $$0.374034\pi$$
$$752$$ 6.42758i 0.234390i
$$753$$ −8.55257 −0.311673
$$754$$ 0 0
$$755$$ −0.390748 −0.0142208
$$756$$ − 4.80194i − 0.174645i
$$757$$ −17.3274 −0.629773 −0.314887 0.949129i $$-0.601967\pi$$
−0.314887 + 0.949129i $$0.601967\pi$$
$$758$$ −15.3260 −0.556666
$$759$$ 4.19806i 0.152380i
$$760$$ 3.89977i 0.141460i
$$761$$ 35.4282i 1.28427i 0.766591 + 0.642135i $$0.221951\pi$$
−0.766591 + 0.642135i $$0.778049\pi$$
$$762$$ − 5.58211i − 0.202218i
$$763$$ −20.1847 −0.730733
$$764$$ 4.02177 0.145503
$$765$$ 3.02715i 0.109447i
$$766$$ −29.6722 −1.07210
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 49.5062i − 1.78524i −0.450812 0.892619i $$-0.648866\pi$$
0.450812 0.892619i $$-0.351134\pi$$
$$770$$ 12.2687 0.442135
$$771$$ −1.30260 −0.0469121
$$772$$ 20.1008i 0.723444i
$$773$$ − 27.9235i − 1.00434i −0.864770 0.502169i $$-0.832536\pi$$
0.864770 0.502169i $$-0.167464\pi$$
$$774$$ 10.1468i 0.364717i
$$775$$ 5.44504i 0.195592i
$$776$$ −17.9855 −0.645643
$$777$$ −36.0901 −1.29472
$$778$$ − 11.8291i − 0.424093i
$$779$$ −23.0078 −0.824339
$$780$$ 0 0
$$781$$ 38.1933 1.36666
$$782$$ 4.97392i 0.177867i
$$783$$ 5.07606 0.181404
$$784$$ −16.0586 −0.573522
$$785$$ 9.50902i 0.339392i
$$786$$ − 3.06100i − 0.109182i
$$787$$ 28.7778i 1.02582i 0.858443 + 0.512908i $$0.171432\pi$$
−0.858443 + 0.512908i $$0.828568\pi$$
$$788$$ − 15.3884i − 0.548187i
$$789$$ 8.24027 0.293362
$$790$$ 5.40581 0.192330
$$791$$ − 0.820724i − 0.0291816i
$$792$$ 2.55496 0.0907865
$$793$$ 0 0
$$794$$ −12.1153 −0.429956
$$795$$ − 6.91185i − 0.245138i
$$796$$ 24.7724 0.878034
$$797$$ 14.6752 0.519821 0.259910 0.965633i $$-0.416307\pi$$
0.259910 + 0.965633i $$0.416307\pi$$
$$798$$ − 18.7265i − 0.662910i
$$799$$ 19.4572i 0.688348i
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 16.8267i − 0.594542i
$$802$$ 10.8291 0.382388
$$803$$ −36.0579 −1.27245
$$804$$ 0.929312i 0.0327743i
$$805$$ −7.89008 −0.278089
$$806$$ 0 0
$$807$$ 22.6601 0.797673
$$808$$ − 15.7453i − 0.553916i
$$809$$ −25.4946 −0.896341 −0.448170 0.893948i $$-0.647924\pi$$
−0.448170 + 0.893948i $$0.647924\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 8.99867i 0.315986i 0.987440 + 0.157993i $$0.0505024\pi$$
−0.987440 + 0.157993i $$0.949498\pi$$
$$812$$ − 24.3749i − 0.855393i
$$813$$ 13.9172i 0.488099i
$$814$$ − 19.2024i − 0.673043i
$$815$$ 5.84846 0.204863
$$816$$ 3.02715 0.105971
$$817$$ 39.5700i 1.38438i
$$818$$ −33.9396 −1.18667
$$819$$ 0 0
$$820$$ 5.89977 0.206029
$$821$$ 37.1105i 1.29517i 0.761995 + 0.647583i $$0.224220\pi$$
−0.761995 + 0.647583i $$0.775780\pi$$
$$822$$ 1.03385 0.0360597
$$823$$ 11.1903 0.390069 0.195035 0.980796i $$-0.437518\pi$$
0.195035 + 0.980796i $$0.437518\pi$$
$$824$$ 12.0858i 0.421027i
$$825$$ − 2.55496i − 0.0889522i
$$826$$ − 36.5392i − 1.27136i
$$827$$ − 27.7808i − 0.966032i −0.875612 0.483016i $$-0.839541\pi$$
0.875612 0.483016i $$-0.160459\pi$$
$$828$$ −1.64310 −0.0571018
$$829$$ 26.5633 0.922582 0.461291 0.887249i $$-0.347386\pi$$
0.461291 + 0.887249i $$0.347386\pi$$
$$830$$ 4.33513i 0.150474i
$$831$$ 24.2610 0.841604
$$832$$ 0 0
$$833$$ −48.6118 −1.68430
$$834$$ − 16.7995i − 0.581721i
$$835$$ 2.55496 0.0884180
$$836$$ 9.96376 0.344604
$$837$$ − 5.44504i − 0.188208i
$$838$$ 26.8562i 0.927733i
$$839$$ − 3.92884i − 0.135639i −0.997698 0.0678193i $$-0.978396\pi$$
0.997698 0.0678193i $$-0.0216041\pi$$
$$840$$ 4.80194i 0.165683i
$$841$$ −3.23357 −0.111502
$$842$$ 27.9928 0.964696
$$843$$ 10.5961i 0.364949i
$$844$$ −6.49396 −0.223531
$$845$$ 0 0
$$846$$ −6.42758 −0.220985
$$847$$ 21.4752i 0.737896i
$$848$$ −6.91185 −0.237354
$$849$$ −9.86294 −0.338495
$$850$$ − 3.02715i − 0.103830i
$$851$$ 12.3491i 0.423323i
$$852$$ 14.9487i 0.512134i
$$853$$ − 48.2097i − 1.65067i −0.564645 0.825334i $$-0.690987\pi$$
0.564645 0.825334i $$-0.309013\pi$$
$$854$$ −46.3521 −1.58614
$$855$$ −3.89977 −0.133369
$$856$$ 4.64071i 0.158616i
$$857$$ −41.2976 −1.41070 −0.705349 0.708860i $$-0.749210\pi$$
−0.705349 + 0.708860i $$0.749210\pi$$
$$858$$ 0 0
$$859$$ −42.1041 −1.43657 −0.718286 0.695748i $$-0.755073\pi$$
−0.718286 + 0.695748i $$0.755073\pi$$
$$860$$ − 10.1468i − 0.346001i
$$861$$ −28.3303 −0.965495
$$862$$ −33.4088 −1.13791
$$863$$ − 45.0646i − 1.53402i −0.641638 0.767008i $$-0.721745\pi$$
0.641638 0.767008i $$-0.278255\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ 21.5308i 0.732069i
$$866$$ 13.6420i 0.463575i
$$867$$ −7.83638 −0.266137
$$868$$ −26.1468 −0.887479
$$869$$ − 13.8116i − 0.468527i
$$870$$ −5.07606 −0.172095
$$871$$ 0 0
$$872$$ 4.20344 0.142346
$$873$$ − 17.9855i − 0.608718i
$$874$$ −6.40773 −0.216745
$$875$$ 4.80194 0.162335
$$876$$ − 14.1129i − 0.476831i
$$877$$ 33.4935i 1.13099i 0.824750 + 0.565497i $$0.191316\pi$$
−0.824750 + 0.565497i $$0.808684\pi$$
$$878$$ 30.6437i 1.03417i
$$879$$ 5.68233i 0.191660i
$$880$$ −2.55496 −0.0861276
$$881$$ −46.0431 −1.55123 −0.775615 0.631206i $$-0.782560\pi$$
−0.775615 + 0.631206i $$0.782560\pi$$
$$882$$ − 16.0586i − 0.540721i
$$883$$ 41.8629 1.40880 0.704400 0.709803i $$-0.251216\pi$$
0.704400 + 0.709803i $$0.251216\pi$$
$$884$$ 0 0
$$885$$ −7.60925 −0.255782
$$886$$ − 19.7409i − 0.663210i
$$887$$ 23.8866 0.802034 0.401017 0.916071i $$-0.368657\pi$$
0.401017 + 0.916071i $$0.368657\pi$$
$$888$$ 7.51573 0.252211
$$889$$ − 26.8049i − 0.899008i
$$890$$ 16.8267i 0.564032i
$$891$$ 2.55496i 0.0855943i
$$892$$ 20.1250i 0.673834i
$$893$$ −25.0661 −0.838805
$$894$$ 17.8756 0.597850
$$895$$ − 15.5483i − 0.519721i
$$896$$ 4.80194 0.160421
$$897$$ 0 0
$$898$$ −36.2664 −1.21022
$$899$$ − 27.6394i − 0.921825i
$$900$$ 1.00000 0.0333333
$$901$$ −20.9232 −0.697053
$$902$$ − 15.0737i − 0.501898i
$$903$$ 48.7241i 1.62144i
$$904$$ 0.170915i 0.00568455i
$$905$$ − 14.4058i − 0.478865i
$$906$$ −0.390748 −0.0129817
$$907$$ 20.4136 0.677822 0.338911 0.940818i $$-0.389941\pi$$
0.338911 + 0.940818i $$0.389941\pi$$
$$908$$ − 24.2379i − 0.804362i
$$909$$ 15.7453 0.522237
$$910$$ 0 0
$$911$$ 23.0108 0.762380 0.381190 0.924497i $$-0.375514\pi$$
0.381190 + 0.924497i $$0.375514\pi$$
$$912$$ 3.89977i 0.129134i
$$913$$ 11.0761 0.366564
$$914$$ 3.55602 0.117623
$$915$$ 9.65279i 0.319111i
$$916$$ 2.71618i 0.0897453i
$$917$$ − 14.6987i − 0.485395i
$$918$$ 3.02715i 0.0999107i
$$919$$ −38.3327 −1.26448 −0.632240 0.774773i $$-0.717864\pi$$
−0.632240 + 0.774773i $$0.717864\pi$$
$$920$$ 1.64310 0.0541715
$$921$$ − 11.5961i − 0.382105i
$$922$$ 1.46011 0.0480861
$$923$$ 0 0
$$924$$ 12.2687 0.403612
$$925$$ − 7.51573i − 0.247116i
$$926$$ 13.7942 0.453304
$$927$$ −12.0858 −0.396948
$$928$$ 5.07606i 0.166630i
$$929$$ − 15.3515i − 0.503667i −0.967771 0.251834i $$-0.918966\pi$$
0.967771 0.251834i $$-0.0810335\pi$$
$$930$$ 5.44504i 0.178550i
$$931$$ − 62.6249i − 2.05245i
$$932$$ −2.16421 −0.0708911
$$933$$ −4.43429 −0.145172
$$934$$ 1.72886i 0.0565699i
$$935$$ −7.73423 −0.252936
$$936$$ 0 0
$$937$$ 51.3629 1.67795 0.838976 0.544169i $$-0.183155\pi$$
0.838976 + 0.544169i $$0.183155\pi$$
$$938$$ 4.46250i 0.145706i
$$939$$ 11.2881 0.368374
$$940$$ 6.42758 0.209645
$$941$$ 17.2413i 0.562052i 0.959700 + 0.281026i $$0.0906747\pi$$
−0.959700 + 0.281026i $$0.909325\pi$$
$$942$$ 9.50902i 0.309821i
$$943$$ 9.69394i 0.315678i
$$944$$ 7.60925i 0.247660i
$$945$$ −4.80194 −0.156207
$$946$$ −25.9245 −0.842879
$$947$$ 5.54586i 0.180216i 0.995932 + 0.0901081i $$0.0287213\pi$$
−0.995932 + 0.0901081i $$0.971279\pi$$
$$948$$ 5.40581 0.175573
$$949$$ 0 0
$$950$$ 3.89977 0.126525
$$951$$ 6.31229i 0.204690i
$$952$$ 14.5362 0.471120
$$953$$ 50.1584 1.62479 0.812394 0.583108i $$-0.198164\pi$$
0.812394 + 0.583108i $$0.198164\pi$$
$$954$$ − 6.91185i − 0.223780i
$$955$$ − 4.02177i − 0.130141i
$$956$$ 14.5114i 0.469333i
$$957$$ 12.9691i 0.419233i
$$958$$ −14.5496 −0.470076
$$959$$ 4.96449 0.160312
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 1.35152 0.0435974
$$962$$ 0 0
$$963$$ −4.64071 −0.149545
$$964$$ 18.1142i 0.583420i
$$965$$ 20.1008 0.647068
$$966$$ −7.89008 −0.253859
$$967$$ 13.6890i 0.440210i 0.975476 + 0.220105i $$0.0706400\pi$$
−0.975476 + 0.220105i $$0.929360\pi$$
$$968$$ − 4.47219i − 0.143742i
$$969$$ 11.8052i 0.379237i
$$970$$ 17.9855i 0.577480i
$$971$$ −50.6075 −1.62407 −0.812035 0.583608i $$-0.801640\pi$$
−0.812035 + 0.583608i $$0.801640\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ − 80.6704i − 2.58617i
$$974$$ 30.5526 0.978967
$$975$$ 0 0
$$976$$ 9.65279 0.308978
$$977$$ 11.2959i 0.361388i 0.983539 + 0.180694i $$0.0578343\pi$$
−0.983539 + 0.180694i $$0.942166\pi$$
$$978$$ 5.84846 0.187013
$$979$$ 42.9915 1.37401
$$980$$ 16.0586i 0.512973i
$$981$$ 4.20344i 0.134205i
$$982$$ − 9.68233i − 0.308976i
$$983$$ − 36.8194i − 1.17436i −0.809458 0.587178i $$-0.800239\pi$$
0.809458 0.587178i $$-0.199761\pi$$
$$984$$ 5.89977 0.188078
$$985$$ −15.3884 −0.490314
$$986$$ 15.3660i 0.489353i
$$987$$ −30.8649 −0.982439
$$988$$ 0 0
$$989$$ 16.6722 0.530144
$$990$$ − 2.55496i − 0.0812019i
$$991$$ 10.8254 0.343879 0.171940 0.985108i $$-0.444997\pi$$
0.171940 + 0.985108i $$0.444997\pi$$
$$992$$ 5.44504 0.172880
$$993$$ − 10.9554i − 0.347659i
$$994$$ 71.7827i 2.27681i
$$995$$ − 24.7724i − 0.785338i
$$996$$ 4.33513i 0.137364i
$$997$$ −4.30319 −0.136283 −0.0681417 0.997676i $$-0.521707\pi$$
−0.0681417 + 0.997676i $$0.521707\pi$$
$$998$$ 18.6025 0.588853
$$999$$ 7.51573i 0.237787i
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.b.w.1351.6 6
13.5 odd 4 5070.2.a.bx.1.1 yes 3
13.8 odd 4 5070.2.a.bo.1.3 3
13.12 even 2 inner 5070.2.b.w.1351.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.3 3 13.8 odd 4
5070.2.a.bx.1.1 yes 3 13.5 odd 4
5070.2.b.w.1351.1 6 13.12 even 2 inner
5070.2.b.w.1351.6 6 1.1 even 1 trivial