# Properties

 Label 5070.2.a.bd.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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ 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 $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.12311 q^{11} +1.00000 q^{12} -5.12311 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +5.12311 q^{21} -3.12311 q^{22} -3.12311 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} -5.12311 q^{31} -1.00000 q^{32} +3.12311 q^{33} +2.00000 q^{34} +5.12311 q^{35} +1.00000 q^{36} -3.12311 q^{37} +6.00000 q^{38} -1.00000 q^{40} +9.12311 q^{41} -5.12311 q^{42} +10.2462 q^{43} +3.12311 q^{44} +1.00000 q^{45} +3.12311 q^{46} -10.2462 q^{47} +1.00000 q^{48} +19.2462 q^{49} -1.00000 q^{50} -2.00000 q^{51} +11.3693 q^{53} -1.00000 q^{54} +3.12311 q^{55} -5.12311 q^{56} -6.00000 q^{57} -2.00000 q^{58} -7.12311 q^{59} +1.00000 q^{60} +10.0000 q^{61} +5.12311 q^{62} +5.12311 q^{63} +1.00000 q^{64} -3.12311 q^{66} +13.1231 q^{67} -2.00000 q^{68} -3.12311 q^{69} -5.12311 q^{70} -6.24621 q^{71} -1.00000 q^{72} +4.87689 q^{73} +3.12311 q^{74} +1.00000 q^{75} -6.00000 q^{76} +16.0000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -9.12311 q^{82} +10.2462 q^{83} +5.12311 q^{84} -2.00000 q^{85} -10.2462 q^{86} +2.00000 q^{87} -3.12311 q^{88} +5.12311 q^{89} -1.00000 q^{90} -3.12311 q^{92} -5.12311 q^{93} +10.2462 q^{94} -6.00000 q^{95} -1.00000 q^{96} -4.87689 q^{97} -19.2462 q^{98} +3.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} - 2q^{14} + 2q^{15} + 2q^{16} - 4q^{17} - 2q^{18} - 12q^{19} + 2q^{20} + 2q^{21} + 2q^{22} + 2q^{23} - 2q^{24} + 2q^{25} + 2q^{27} + 2q^{28} + 4q^{29} - 2q^{30} - 2q^{31} - 2q^{32} - 2q^{33} + 4q^{34} + 2q^{35} + 2q^{36} + 2q^{37} + 12q^{38} - 2q^{40} + 10q^{41} - 2q^{42} + 4q^{43} - 2q^{44} + 2q^{45} - 2q^{46} - 4q^{47} + 2q^{48} + 22q^{49} - 2q^{50} - 4q^{51} - 2q^{53} - 2q^{54} - 2q^{55} - 2q^{56} - 12q^{57} - 4q^{58} - 6q^{59} + 2q^{60} + 20q^{61} + 2q^{62} + 2q^{63} + 2q^{64} + 2q^{66} + 18q^{67} - 4q^{68} + 2q^{69} - 2q^{70} + 4q^{71} - 2q^{72} + 18q^{73} - 2q^{74} + 2q^{75} - 12q^{76} + 32q^{77} + 16q^{79} + 2q^{80} + 2q^{81} - 10q^{82} + 4q^{83} + 2q^{84} - 4q^{85} - 4q^{86} + 4q^{87} + 2q^{88} + 2q^{89} - 2q^{90} + 2q^{92} - 2q^{93} + 4q^{94} - 12q^{95} - 2q^{96} - 18q^{97} - 22q^{98} - 2q^{99} + O(q^{100})$$

## 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
$$7$$ 5.12311 1.93635 0.968176 0.250270i $$-0.0805195\pi$$
0.968176 + 0.250270i $$0.0805195\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 3.12311 0.941652 0.470826 0.882226i $$-0.343956\pi$$
0.470826 + 0.882226i $$0.343956\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −5.12311 −1.36921
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 5.12311 1.11795
$$22$$ −3.12311 −0.665848
$$23$$ −3.12311 −0.651213 −0.325606 0.945505i $$-0.605568\pi$$
−0.325606 + 0.945505i $$0.605568\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 5.12311 0.968176
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −5.12311 −0.920137 −0.460068 0.887883i $$-0.652175\pi$$
−0.460068 + 0.887883i $$0.652175\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 3.12311 0.543663
$$34$$ 2.00000 0.342997
$$35$$ 5.12311 0.865963
$$36$$ 1.00000 0.166667
$$37$$ −3.12311 −0.513435 −0.256718 0.966486i $$-0.582641\pi$$
−0.256718 + 0.966486i $$0.582641\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 9.12311 1.42479 0.712395 0.701779i $$-0.247611\pi$$
0.712395 + 0.701779i $$0.247611\pi$$
$$42$$ −5.12311 −0.790512
$$43$$ 10.2462 1.56253 0.781266 0.624198i $$-0.214574\pi$$
0.781266 + 0.624198i $$0.214574\pi$$
$$44$$ 3.12311 0.470826
$$45$$ 1.00000 0.149071
$$46$$ 3.12311 0.460477
$$47$$ −10.2462 −1.49456 −0.747282 0.664507i $$-0.768641\pi$$
−0.747282 + 0.664507i $$0.768641\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 19.2462 2.74946
$$50$$ −1.00000 −0.141421
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 11.3693 1.56170 0.780848 0.624721i $$-0.214787\pi$$
0.780848 + 0.624721i $$0.214787\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 3.12311 0.421119
$$56$$ −5.12311 −0.684604
$$57$$ −6.00000 −0.794719
$$58$$ −2.00000 −0.262613
$$59$$ −7.12311 −0.927349 −0.463675 0.886006i $$-0.653469\pi$$
−0.463675 + 0.886006i $$0.653469\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 5.12311 0.650635
$$63$$ 5.12311 0.645451
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −3.12311 −0.384428
$$67$$ 13.1231 1.60324 0.801621 0.597832i $$-0.203971\pi$$
0.801621 + 0.597832i $$0.203971\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −3.12311 −0.375978
$$70$$ −5.12311 −0.612328
$$71$$ −6.24621 −0.741289 −0.370644 0.928775i $$-0.620863\pi$$
−0.370644 + 0.928775i $$0.620863\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 4.87689 0.570797 0.285399 0.958409i $$-0.407874\pi$$
0.285399 + 0.958409i $$0.407874\pi$$
$$74$$ 3.12311 0.363054
$$75$$ 1.00000 0.115470
$$76$$ −6.00000 −0.688247
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −9.12311 −1.00748
$$83$$ 10.2462 1.12467 0.562334 0.826910i $$-0.309904\pi$$
0.562334 + 0.826910i $$0.309904\pi$$
$$84$$ 5.12311 0.558977
$$85$$ −2.00000 −0.216930
$$86$$ −10.2462 −1.10488
$$87$$ 2.00000 0.214423
$$88$$ −3.12311 −0.332924
$$89$$ 5.12311 0.543048 0.271524 0.962432i $$-0.412472\pi$$
0.271524 + 0.962432i $$0.412472\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −3.12311 −0.325606
$$93$$ −5.12311 −0.531241
$$94$$ 10.2462 1.05682
$$95$$ −6.00000 −0.615587
$$96$$ −1.00000 −0.102062
$$97$$ −4.87689 −0.495174 −0.247587 0.968866i $$-0.579638\pi$$
−0.247587 + 0.968866i $$0.579638\pi$$
$$98$$ −19.2462 −1.94416
$$99$$ 3.12311 0.313884
$$100$$ 1.00000 0.100000
$$101$$ −4.24621 −0.422514 −0.211257 0.977431i $$-0.567756\pi$$
−0.211257 + 0.977431i $$0.567756\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 4.87689 0.480535 0.240267 0.970707i $$-0.422765\pi$$
0.240267 + 0.970707i $$0.422765\pi$$
$$104$$ 0 0
$$105$$ 5.12311 0.499964
$$106$$ −11.3693 −1.10429
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 11.1231 1.06540 0.532700 0.846304i $$-0.321177\pi$$
0.532700 + 0.846304i $$0.321177\pi$$
$$110$$ −3.12311 −0.297776
$$111$$ −3.12311 −0.296432
$$112$$ 5.12311 0.484088
$$113$$ −4.24621 −0.399450 −0.199725 0.979852i $$-0.564005\pi$$
−0.199725 + 0.979852i $$0.564005\pi$$
$$114$$ 6.00000 0.561951
$$115$$ −3.12311 −0.291231
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 7.12311 0.655735
$$119$$ −10.2462 −0.939269
$$120$$ −1.00000 −0.0912871
$$121$$ −1.24621 −0.113292
$$122$$ −10.0000 −0.905357
$$123$$ 9.12311 0.822603
$$124$$ −5.12311 −0.460068
$$125$$ 1.00000 0.0894427
$$126$$ −5.12311 −0.456403
$$127$$ 4.87689 0.432754 0.216377 0.976310i $$-0.430576\pi$$
0.216377 + 0.976310i $$0.430576\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 10.2462 0.902129
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 3.12311 0.271831
$$133$$ −30.7386 −2.66538
$$134$$ −13.1231 −1.13366
$$135$$ 1.00000 0.0860663
$$136$$ 2.00000 0.171499
$$137$$ −22.4924 −1.92166 −0.960829 0.277143i $$-0.910612\pi$$
−0.960829 + 0.277143i $$0.910612\pi$$
$$138$$ 3.12311 0.265856
$$139$$ 16.4924 1.39887 0.699435 0.714697i $$-0.253435\pi$$
0.699435 + 0.714697i $$0.253435\pi$$
$$140$$ 5.12311 0.432981
$$141$$ −10.2462 −0.862887
$$142$$ 6.24621 0.524170
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 2.00000 0.166091
$$146$$ −4.87689 −0.403615
$$147$$ 19.2462 1.58740
$$148$$ −3.12311 −0.256718
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 11.3693 0.925222 0.462611 0.886561i $$-0.346913\pi$$
0.462611 + 0.886561i $$0.346913\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −2.00000 −0.161690
$$154$$ −16.0000 −1.28932
$$155$$ −5.12311 −0.411498
$$156$$ 0 0
$$157$$ −3.36932 −0.268901 −0.134450 0.990920i $$-0.542927\pi$$
−0.134450 + 0.990920i $$0.542927\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 11.3693 0.901645
$$160$$ −1.00000 −0.0790569
$$161$$ −16.0000 −1.26098
$$162$$ −1.00000 −0.0785674
$$163$$ −1.12311 −0.0879684 −0.0439842 0.999032i $$-0.514005\pi$$
−0.0439842 + 0.999032i $$0.514005\pi$$
$$164$$ 9.12311 0.712395
$$165$$ 3.12311 0.243133
$$166$$ −10.2462 −0.795260
$$167$$ 5.75379 0.445242 0.222621 0.974905i $$-0.428539\pi$$
0.222621 + 0.974905i $$0.428539\pi$$
$$168$$ −5.12311 −0.395256
$$169$$ 0 0
$$170$$ 2.00000 0.153393
$$171$$ −6.00000 −0.458831
$$172$$ 10.2462 0.781266
$$173$$ −14.8769 −1.13107 −0.565535 0.824725i $$-0.691330\pi$$
−0.565535 + 0.824725i $$0.691330\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 5.12311 0.387270
$$176$$ 3.12311 0.235413
$$177$$ −7.12311 −0.535405
$$178$$ −5.12311 −0.383993
$$179$$ 16.4924 1.23270 0.616351 0.787472i $$-0.288610\pi$$
0.616351 + 0.787472i $$0.288610\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 3.75379 0.279017 0.139508 0.990221i $$-0.455448\pi$$
0.139508 + 0.990221i $$0.455448\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 3.12311 0.230238
$$185$$ −3.12311 −0.229615
$$186$$ 5.12311 0.375644
$$187$$ −6.24621 −0.456768
$$188$$ −10.2462 −0.747282
$$189$$ 5.12311 0.372651
$$190$$ 6.00000 0.435286
$$191$$ −16.4924 −1.19335 −0.596675 0.802483i $$-0.703512\pi$$
−0.596675 + 0.802483i $$0.703512\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 16.8769 1.21483 0.607413 0.794386i $$-0.292207\pi$$
0.607413 + 0.794386i $$0.292207\pi$$
$$194$$ 4.87689 0.350141
$$195$$ 0 0
$$196$$ 19.2462 1.37473
$$197$$ −0.246211 −0.0175418 −0.00877091 0.999962i $$-0.502792\pi$$
−0.00877091 + 0.999962i $$0.502792\pi$$
$$198$$ −3.12311 −0.221949
$$199$$ −1.75379 −0.124323 −0.0621614 0.998066i $$-0.519799\pi$$
−0.0621614 + 0.998066i $$0.519799\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 13.1231 0.925633
$$202$$ 4.24621 0.298762
$$203$$ 10.2462 0.719143
$$204$$ −2.00000 −0.140028
$$205$$ 9.12311 0.637185
$$206$$ −4.87689 −0.339789
$$207$$ −3.12311 −0.217071
$$208$$ 0 0
$$209$$ −18.7386 −1.29618
$$210$$ −5.12311 −0.353528
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 11.3693 0.780848
$$213$$ −6.24621 −0.427983
$$214$$ 8.00000 0.546869
$$215$$ 10.2462 0.698786
$$216$$ −1.00000 −0.0680414
$$217$$ −26.2462 −1.78171
$$218$$ −11.1231 −0.753352
$$219$$ 4.87689 0.329550
$$220$$ 3.12311 0.210560
$$221$$ 0 0
$$222$$ 3.12311 0.209609
$$223$$ −15.3693 −1.02921 −0.514603 0.857429i $$-0.672061\pi$$
−0.514603 + 0.857429i $$0.672061\pi$$
$$224$$ −5.12311 −0.342302
$$225$$ 1.00000 0.0666667
$$226$$ 4.24621 0.282454
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ −6.00000 −0.397360
$$229$$ −3.12311 −0.206381 −0.103190 0.994662i $$-0.532905\pi$$
−0.103190 + 0.994662i $$0.532905\pi$$
$$230$$ 3.12311 0.205931
$$231$$ 16.0000 1.05272
$$232$$ −2.00000 −0.131306
$$233$$ 24.2462 1.58842 0.794211 0.607642i $$-0.207884\pi$$
0.794211 + 0.607642i $$0.207884\pi$$
$$234$$ 0 0
$$235$$ −10.2462 −0.668389
$$236$$ −7.12311 −0.463675
$$237$$ 8.00000 0.519656
$$238$$ 10.2462 0.664163
$$239$$ −28.4924 −1.84302 −0.921511 0.388353i $$-0.873044\pi$$
−0.921511 + 0.388353i $$0.873044\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −2.24621 −0.144691 −0.0723456 0.997380i $$-0.523048\pi$$
−0.0723456 + 0.997380i $$0.523048\pi$$
$$242$$ 1.24621 0.0801095
$$243$$ 1.00000 0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 19.2462 1.22960
$$246$$ −9.12311 −0.581668
$$247$$ 0 0
$$248$$ 5.12311 0.325318
$$249$$ 10.2462 0.649327
$$250$$ −1.00000 −0.0632456
$$251$$ −9.75379 −0.615654 −0.307827 0.951442i $$-0.599602\pi$$
−0.307827 + 0.951442i $$0.599602\pi$$
$$252$$ 5.12311 0.322725
$$253$$ −9.75379 −0.613215
$$254$$ −4.87689 −0.306004
$$255$$ −2.00000 −0.125245
$$256$$ 1.00000 0.0625000
$$257$$ −4.24621 −0.264871 −0.132436 0.991192i $$-0.542280\pi$$
−0.132436 + 0.991192i $$0.542280\pi$$
$$258$$ −10.2462 −0.637901
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ −4.00000 −0.247121
$$263$$ −2.63068 −0.162215 −0.0811074 0.996705i $$-0.525846\pi$$
−0.0811074 + 0.996705i $$0.525846\pi$$
$$264$$ −3.12311 −0.192214
$$265$$ 11.3693 0.698412
$$266$$ 30.7386 1.88471
$$267$$ 5.12311 0.313529
$$268$$ 13.1231 0.801621
$$269$$ 0.246211 0.0150118 0.00750588 0.999972i $$-0.497611\pi$$
0.00750588 + 0.999972i $$0.497611\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −14.8769 −0.903707 −0.451853 0.892092i $$-0.649237\pi$$
−0.451853 + 0.892092i $$0.649237\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 22.4924 1.35882
$$275$$ 3.12311 0.188330
$$276$$ −3.12311 −0.187989
$$277$$ −27.8617 −1.67405 −0.837025 0.547165i $$-0.815707\pi$$
−0.837025 + 0.547165i $$0.815707\pi$$
$$278$$ −16.4924 −0.989150
$$279$$ −5.12311 −0.306712
$$280$$ −5.12311 −0.306164
$$281$$ −5.12311 −0.305619 −0.152809 0.988256i $$-0.548832\pi$$
−0.152809 + 0.988256i $$0.548832\pi$$
$$282$$ 10.2462 0.610153
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −6.24621 −0.370644
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 46.7386 2.75889
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ −2.00000 −0.117444
$$291$$ −4.87689 −0.285889
$$292$$ 4.87689 0.285399
$$293$$ 20.7386 1.21156 0.605782 0.795631i $$-0.292860\pi$$
0.605782 + 0.795631i $$0.292860\pi$$
$$294$$ −19.2462 −1.12246
$$295$$ −7.12311 −0.414723
$$296$$ 3.12311 0.181527
$$297$$ 3.12311 0.181221
$$298$$ 14.0000 0.810998
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 52.4924 3.02561
$$302$$ −11.3693 −0.654231
$$303$$ −4.24621 −0.243938
$$304$$ −6.00000 −0.344124
$$305$$ 10.0000 0.572598
$$306$$ 2.00000 0.114332
$$307$$ −22.8769 −1.30565 −0.652827 0.757507i $$-0.726417\pi$$
−0.652827 + 0.757507i $$0.726417\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 4.87689 0.277437
$$310$$ 5.12311 0.290973
$$311$$ 24.4924 1.38884 0.694419 0.719571i $$-0.255661\pi$$
0.694419 + 0.719571i $$0.255661\pi$$
$$312$$ 0 0
$$313$$ −0.246211 −0.0139167 −0.00695834 0.999976i $$-0.502215\pi$$
−0.00695834 + 0.999976i $$0.502215\pi$$
$$314$$ 3.36932 0.190142
$$315$$ 5.12311 0.288654
$$316$$ 8.00000 0.450035
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ −11.3693 −0.637560
$$319$$ 6.24621 0.349721
$$320$$ 1.00000 0.0559017
$$321$$ −8.00000 −0.446516
$$322$$ 16.0000 0.891645
$$323$$ 12.0000 0.667698
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 1.12311 0.0622031
$$327$$ 11.1231 0.615109
$$328$$ −9.12311 −0.503739
$$329$$ −52.4924 −2.89400
$$330$$ −3.12311 −0.171921
$$331$$ −24.2462 −1.33269 −0.666346 0.745643i $$-0.732143\pi$$
−0.666346 + 0.745643i $$0.732143\pi$$
$$332$$ 10.2462 0.562334
$$333$$ −3.12311 −0.171145
$$334$$ −5.75379 −0.314833
$$335$$ 13.1231 0.716992
$$336$$ 5.12311 0.279488
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 0 0
$$339$$ −4.24621 −0.230623
$$340$$ −2.00000 −0.108465
$$341$$ −16.0000 −0.866449
$$342$$ 6.00000 0.324443
$$343$$ 62.7386 3.38757
$$344$$ −10.2462 −0.552439
$$345$$ −3.12311 −0.168142
$$346$$ 14.8769 0.799787
$$347$$ 2.24621 0.120583 0.0602915 0.998181i $$-0.480797\pi$$
0.0602915 + 0.998181i $$0.480797\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 5.36932 0.287413 0.143706 0.989620i $$-0.454098\pi$$
0.143706 + 0.989620i $$0.454098\pi$$
$$350$$ −5.12311 −0.273842
$$351$$ 0 0
$$352$$ −3.12311 −0.166462
$$353$$ 4.24621 0.226003 0.113002 0.993595i $$-0.463954\pi$$
0.113002 + 0.993595i $$0.463954\pi$$
$$354$$ 7.12311 0.378589
$$355$$ −6.24621 −0.331514
$$356$$ 5.12311 0.271524
$$357$$ −10.2462 −0.542287
$$358$$ −16.4924 −0.871652
$$359$$ −34.2462 −1.80745 −0.903723 0.428118i $$-0.859177\pi$$
−0.903723 + 0.428118i $$0.859177\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 17.0000 0.894737
$$362$$ −3.75379 −0.197295
$$363$$ −1.24621 −0.0654091
$$364$$ 0 0
$$365$$ 4.87689 0.255268
$$366$$ −10.0000 −0.522708
$$367$$ 33.3693 1.74186 0.870932 0.491403i $$-0.163516\pi$$
0.870932 + 0.491403i $$0.163516\pi$$
$$368$$ −3.12311 −0.162803
$$369$$ 9.12311 0.474930
$$370$$ 3.12311 0.162363
$$371$$ 58.2462 3.02399
$$372$$ −5.12311 −0.265621
$$373$$ −1.12311 −0.0581522 −0.0290761 0.999577i $$-0.509257\pi$$
−0.0290761 + 0.999577i $$0.509257\pi$$
$$374$$ 6.24621 0.322984
$$375$$ 1.00000 0.0516398
$$376$$ 10.2462 0.528408
$$377$$ 0 0
$$378$$ −5.12311 −0.263504
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 4.87689 0.249851
$$382$$ 16.4924 0.843826
$$383$$ 18.2462 0.932338 0.466169 0.884696i $$-0.345634\pi$$
0.466169 + 0.884696i $$0.345634\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 16.0000 0.815436
$$386$$ −16.8769 −0.859011
$$387$$ 10.2462 0.520844
$$388$$ −4.87689 −0.247587
$$389$$ −16.2462 −0.823716 −0.411858 0.911248i $$-0.635120\pi$$
−0.411858 + 0.911248i $$0.635120\pi$$
$$390$$ 0 0
$$391$$ 6.24621 0.315884
$$392$$ −19.2462 −0.972080
$$393$$ 4.00000 0.201773
$$394$$ 0.246211 0.0124039
$$395$$ 8.00000 0.402524
$$396$$ 3.12311 0.156942
$$397$$ −9.36932 −0.470233 −0.235116 0.971967i $$-0.575547\pi$$
−0.235116 + 0.971967i $$0.575547\pi$$
$$398$$ 1.75379 0.0879095
$$399$$ −30.7386 −1.53886
$$400$$ 1.00000 0.0500000
$$401$$ −23.3693 −1.16701 −0.583504 0.812110i $$-0.698319\pi$$
−0.583504 + 0.812110i $$0.698319\pi$$
$$402$$ −13.1231 −0.654521
$$403$$ 0 0
$$404$$ −4.24621 −0.211257
$$405$$ 1.00000 0.0496904
$$406$$ −10.2462 −0.508511
$$407$$ −9.75379 −0.483477
$$408$$ 2.00000 0.0990148
$$409$$ −24.4924 −1.21107 −0.605536 0.795818i $$-0.707041\pi$$
−0.605536 + 0.795818i $$0.707041\pi$$
$$410$$ −9.12311 −0.450558
$$411$$ −22.4924 −1.10947
$$412$$ 4.87689 0.240267
$$413$$ −36.4924 −1.79567
$$414$$ 3.12311 0.153492
$$415$$ 10.2462 0.502967
$$416$$ 0 0
$$417$$ 16.4924 0.807637
$$418$$ 18.7386 0.916537
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 5.12311 0.249982
$$421$$ −25.3693 −1.23642 −0.618212 0.786011i $$-0.712143\pi$$
−0.618212 + 0.786011i $$0.712143\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ −10.2462 −0.498188
$$424$$ −11.3693 −0.552143
$$425$$ −2.00000 −0.0970143
$$426$$ 6.24621 0.302630
$$427$$ 51.2311 2.47924
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ −10.2462 −0.494116
$$431$$ 0.492423 0.0237192 0.0118596 0.999930i $$-0.496225\pi$$
0.0118596 + 0.999930i $$0.496225\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ 26.2462 1.25986
$$435$$ 2.00000 0.0958927
$$436$$ 11.1231 0.532700
$$437$$ 18.7386 0.896390
$$438$$ −4.87689 −0.233027
$$439$$ 3.50758 0.167408 0.0837038 0.996491i $$-0.473325\pi$$
0.0837038 + 0.996491i $$0.473325\pi$$
$$440$$ −3.12311 −0.148888
$$441$$ 19.2462 0.916486
$$442$$ 0 0
$$443$$ −36.4924 −1.73381 −0.866904 0.498476i $$-0.833893\pi$$
−0.866904 + 0.498476i $$0.833893\pi$$
$$444$$ −3.12311 −0.148216
$$445$$ 5.12311 0.242858
$$446$$ 15.3693 0.727758
$$447$$ −14.0000 −0.662177
$$448$$ 5.12311 0.242044
$$449$$ 37.1231 1.75195 0.875974 0.482359i $$-0.160220\pi$$
0.875974 + 0.482359i $$0.160220\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 28.4924 1.34166
$$452$$ −4.24621 −0.199725
$$453$$ 11.3693 0.534177
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ 6.63068 0.310170 0.155085 0.987901i $$-0.450435\pi$$
0.155085 + 0.987901i $$0.450435\pi$$
$$458$$ 3.12311 0.145933
$$459$$ −2.00000 −0.0933520
$$460$$ −3.12311 −0.145616
$$461$$ −14.4924 −0.674979 −0.337490 0.941329i $$-0.609578\pi$$
−0.337490 + 0.941329i $$0.609578\pi$$
$$462$$ −16.0000 −0.744387
$$463$$ −35.8617 −1.66664 −0.833318 0.552794i $$-0.813562\pi$$
−0.833318 + 0.552794i $$0.813562\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ −5.12311 −0.237578
$$466$$ −24.2462 −1.12318
$$467$$ 5.75379 0.266254 0.133127 0.991099i $$-0.457498\pi$$
0.133127 + 0.991099i $$0.457498\pi$$
$$468$$ 0 0
$$469$$ 67.2311 3.10444
$$470$$ 10.2462 0.472622
$$471$$ −3.36932 −0.155250
$$472$$ 7.12311 0.327868
$$473$$ 32.0000 1.47136
$$474$$ −8.00000 −0.367452
$$475$$ −6.00000 −0.275299
$$476$$ −10.2462 −0.469634
$$477$$ 11.3693 0.520565
$$478$$ 28.4924 1.30321
$$479$$ −20.4924 −0.936323 −0.468161 0.883643i $$-0.655083\pi$$
−0.468161 + 0.883643i $$0.655083\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 2.24621 0.102312
$$483$$ −16.0000 −0.728025
$$484$$ −1.24621 −0.0566460
$$485$$ −4.87689 −0.221448
$$486$$ −1.00000 −0.0453609
$$487$$ −7.36932 −0.333936 −0.166968 0.985962i $$-0.553398\pi$$
−0.166968 + 0.985962i $$0.553398\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ −1.12311 −0.0507886
$$490$$ −19.2462 −0.869455
$$491$$ 10.7386 0.484628 0.242314 0.970198i $$-0.422094\pi$$
0.242314 + 0.970198i $$0.422094\pi$$
$$492$$ 9.12311 0.411301
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 3.12311 0.140373
$$496$$ −5.12311 −0.230034
$$497$$ −32.0000 −1.43540
$$498$$ −10.2462 −0.459144
$$499$$ 1.50758 0.0674884 0.0337442 0.999431i $$-0.489257\pi$$
0.0337442 + 0.999431i $$0.489257\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 5.75379 0.257060
$$502$$ 9.75379 0.435333
$$503$$ 10.6307 0.473999 0.236999 0.971510i $$-0.423836\pi$$
0.236999 + 0.971510i $$0.423836\pi$$
$$504$$ −5.12311 −0.228201
$$505$$ −4.24621 −0.188954
$$506$$ 9.75379 0.433609
$$507$$ 0 0
$$508$$ 4.87689 0.216377
$$509$$ 15.7538 0.698274 0.349137 0.937072i $$-0.386475\pi$$
0.349137 + 0.937072i $$0.386475\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 24.9848 1.10526
$$512$$ −1.00000 −0.0441942
$$513$$ −6.00000 −0.264906
$$514$$ 4.24621 0.187292
$$515$$ 4.87689 0.214902
$$516$$ 10.2462 0.451064
$$517$$ −32.0000 −1.40736
$$518$$ 16.0000 0.703000
$$519$$ −14.8769 −0.653023
$$520$$ 0 0
$$521$$ −16.2462 −0.711759 −0.355880 0.934532i $$-0.615819\pi$$
−0.355880 + 0.934532i $$0.615819\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ 22.7386 0.994291 0.497146 0.867667i $$-0.334382\pi$$
0.497146 + 0.867667i $$0.334382\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 5.12311 0.223591
$$526$$ 2.63068 0.114703
$$527$$ 10.2462 0.446332
$$528$$ 3.12311 0.135916
$$529$$ −13.2462 −0.575922
$$530$$ −11.3693 −0.493852
$$531$$ −7.12311 −0.309116
$$532$$ −30.7386 −1.33269
$$533$$ 0 0
$$534$$ −5.12311 −0.221698
$$535$$ −8.00000 −0.345870
$$536$$ −13.1231 −0.566832
$$537$$ 16.4924 0.711701
$$538$$ −0.246211 −0.0106149
$$539$$ 60.1080 2.58903
$$540$$ 1.00000 0.0430331
$$541$$ −19.1231 −0.822167 −0.411083 0.911598i $$-0.634849\pi$$
−0.411083 + 0.911598i $$0.634849\pi$$
$$542$$ 14.8769 0.639017
$$543$$ 3.75379 0.161090
$$544$$ 2.00000 0.0857493
$$545$$ 11.1231 0.476461
$$546$$ 0 0
$$547$$ −44.9848 −1.92341 −0.961707 0.274081i $$-0.911626\pi$$
−0.961707 + 0.274081i $$0.911626\pi$$
$$548$$ −22.4924 −0.960829
$$549$$ 10.0000 0.426790
$$550$$ −3.12311 −0.133170
$$551$$ −12.0000 −0.511217
$$552$$ 3.12311 0.132928
$$553$$ 40.9848 1.74285
$$554$$ 27.8617 1.18373
$$555$$ −3.12311 −0.132568
$$556$$ 16.4924 0.699435
$$557$$ −28.2462 −1.19683 −0.598415 0.801186i $$-0.704203\pi$$
−0.598415 + 0.801186i $$0.704203\pi$$
$$558$$ 5.12311 0.216878
$$559$$ 0 0
$$560$$ 5.12311 0.216491
$$561$$ −6.24621 −0.263715
$$562$$ 5.12311 0.216105
$$563$$ −32.9848 −1.39015 −0.695073 0.718939i $$-0.744628\pi$$
−0.695073 + 0.718939i $$0.744628\pi$$
$$564$$ −10.2462 −0.431443
$$565$$ −4.24621 −0.178639
$$566$$ −4.00000 −0.168133
$$567$$ 5.12311 0.215150
$$568$$ 6.24621 0.262085
$$569$$ −36.7386 −1.54016 −0.770082 0.637945i $$-0.779785\pi$$
−0.770082 + 0.637945i $$0.779785\pi$$
$$570$$ 6.00000 0.251312
$$571$$ −24.4924 −1.02498 −0.512488 0.858694i $$-0.671276\pi$$
−0.512488 + 0.858694i $$0.671276\pi$$
$$572$$ 0 0
$$573$$ −16.4924 −0.688981
$$574$$ −46.7386 −1.95083
$$575$$ −3.12311 −0.130243
$$576$$ 1.00000 0.0416667
$$577$$ 2.63068 0.109517 0.0547584 0.998500i $$-0.482561\pi$$
0.0547584 + 0.998500i $$0.482561\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 16.8769 0.701380
$$580$$ 2.00000 0.0830455
$$581$$ 52.4924 2.17775
$$582$$ 4.87689 0.202154
$$583$$ 35.5076 1.47057
$$584$$ −4.87689 −0.201807
$$585$$ 0 0
$$586$$ −20.7386 −0.856705
$$587$$ 16.4924 0.680715 0.340358 0.940296i $$-0.389452\pi$$
0.340358 + 0.940296i $$0.389452\pi$$
$$588$$ 19.2462 0.793700
$$589$$ 30.7386 1.26656
$$590$$ 7.12311 0.293254
$$591$$ −0.246211 −0.0101278
$$592$$ −3.12311 −0.128359
$$593$$ 38.4924 1.58069 0.790347 0.612659i $$-0.209900\pi$$
0.790347 + 0.612659i $$0.209900\pi$$
$$594$$ −3.12311 −0.128143
$$595$$ −10.2462 −0.420054
$$596$$ −14.0000 −0.573462
$$597$$ −1.75379 −0.0717778
$$598$$ 0 0
$$599$$ −3.50758 −0.143316 −0.0716579 0.997429i $$-0.522829\pi$$
−0.0716579 + 0.997429i $$0.522829\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ −52.4924 −2.13943
$$603$$ 13.1231 0.534414
$$604$$ 11.3693 0.462611
$$605$$ −1.24621 −0.0506657
$$606$$ 4.24621 0.172491
$$607$$ −9.36932 −0.380289 −0.190144 0.981756i $$-0.560896\pi$$
−0.190144 + 0.981756i $$0.560896\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 10.2462 0.415197
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ 14.6307 0.590928 0.295464 0.955354i $$-0.404526\pi$$
0.295464 + 0.955354i $$0.404526\pi$$
$$614$$ 22.8769 0.923236
$$615$$ 9.12311 0.367879
$$616$$ −16.0000 −0.644658
$$617$$ 8.73863 0.351804 0.175902 0.984408i $$-0.443716\pi$$
0.175902 + 0.984408i $$0.443716\pi$$
$$618$$ −4.87689 −0.196177
$$619$$ 26.9848 1.08461 0.542306 0.840181i $$-0.317551\pi$$
0.542306 + 0.840181i $$0.317551\pi$$
$$620$$ −5.12311 −0.205749
$$621$$ −3.12311 −0.125326
$$622$$ −24.4924 −0.982057
$$623$$ 26.2462 1.05153
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0.246211 0.00984058
$$627$$ −18.7386 −0.748349
$$628$$ −3.36932 −0.134450
$$629$$ 6.24621 0.249053
$$630$$ −5.12311 −0.204109
$$631$$ −5.61553 −0.223551 −0.111775 0.993734i $$-0.535654\pi$$
−0.111775 + 0.993734i $$0.535654\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 4.00000 0.158986
$$634$$ −6.00000 −0.238290
$$635$$ 4.87689 0.193534
$$636$$ 11.3693 0.450823
$$637$$ 0 0
$$638$$ −6.24621 −0.247290
$$639$$ −6.24621 −0.247096
$$640$$ −1.00000 −0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 8.00000 0.315735
$$643$$ −7.36932 −0.290617 −0.145309 0.989386i $$-0.546418\pi$$
−0.145309 + 0.989386i $$0.546418\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 10.2462 0.403444
$$646$$ −12.0000 −0.472134
$$647$$ 11.6155 0.456654 0.228327 0.973585i $$-0.426675\pi$$
0.228327 + 0.973585i $$0.426675\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −22.2462 −0.873240
$$650$$ 0 0
$$651$$ −26.2462 −1.02867
$$652$$ −1.12311 −0.0439842
$$653$$ 43.8617 1.71644 0.858221 0.513280i $$-0.171570\pi$$
0.858221 + 0.513280i $$0.171570\pi$$
$$654$$ −11.1231 −0.434948
$$655$$ 4.00000 0.156293
$$656$$ 9.12311 0.356197
$$657$$ 4.87689 0.190266
$$658$$ 52.4924 2.04637
$$659$$ −38.2462 −1.48986 −0.744930 0.667142i $$-0.767517\pi$$
−0.744930 + 0.667142i $$0.767517\pi$$
$$660$$ 3.12311 0.121567
$$661$$ −0.876894 −0.0341072 −0.0170536 0.999855i $$-0.505429\pi$$
−0.0170536 + 0.999855i $$0.505429\pi$$
$$662$$ 24.2462 0.942356
$$663$$ 0 0
$$664$$ −10.2462 −0.397630
$$665$$ −30.7386 −1.19199
$$666$$ 3.12311 0.121018
$$667$$ −6.24621 −0.241854
$$668$$ 5.75379 0.222621
$$669$$ −15.3693 −0.594212
$$670$$ −13.1231 −0.506990
$$671$$ 31.2311 1.20566
$$672$$ −5.12311 −0.197628
$$673$$ 38.9848 1.50276 0.751378 0.659872i $$-0.229390\pi$$
0.751378 + 0.659872i $$0.229390\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 21.1231 0.811827 0.405913 0.913912i $$-0.366953\pi$$
0.405913 + 0.913912i $$0.366953\pi$$
$$678$$ 4.24621 0.163075
$$679$$ −24.9848 −0.958830
$$680$$ 2.00000 0.0766965
$$681$$ 8.00000 0.306561
$$682$$ 16.0000 0.612672
$$683$$ −48.9848 −1.87435 −0.937177 0.348856i $$-0.886570\pi$$
−0.937177 + 0.348856i $$0.886570\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ −22.4924 −0.859391
$$686$$ −62.7386 −2.39537
$$687$$ −3.12311 −0.119154
$$688$$ 10.2462 0.390633
$$689$$ 0 0
$$690$$ 3.12311 0.118895
$$691$$ −20.7386 −0.788935 −0.394467 0.918910i $$-0.629071\pi$$
−0.394467 + 0.918910i $$0.629071\pi$$
$$692$$ −14.8769 −0.565535
$$693$$ 16.0000 0.607790
$$694$$ −2.24621 −0.0852650
$$695$$ 16.4924 0.625593
$$696$$ −2.00000 −0.0758098
$$697$$ −18.2462 −0.691125
$$698$$ −5.36932 −0.203232
$$699$$ 24.2462 0.917076
$$700$$ 5.12311 0.193635
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 0 0
$$703$$ 18.7386 0.706741
$$704$$ 3.12311 0.117706
$$705$$ −10.2462 −0.385895
$$706$$ −4.24621 −0.159808
$$707$$ −21.7538 −0.818135
$$708$$ −7.12311 −0.267703
$$709$$ −39.6155 −1.48779 −0.743896 0.668295i $$-0.767024\pi$$
−0.743896 + 0.668295i $$0.767024\pi$$
$$710$$ 6.24621 0.234416
$$711$$ 8.00000 0.300023
$$712$$ −5.12311 −0.191997
$$713$$ 16.0000 0.599205
$$714$$ 10.2462 0.383455
$$715$$ 0 0
$$716$$ 16.4924 0.616351
$$717$$ −28.4924 −1.06407
$$718$$ 34.2462 1.27806
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 24.9848 0.930484
$$722$$ −17.0000 −0.632674
$$723$$ −2.24621 −0.0835375
$$724$$ 3.75379 0.139508
$$725$$ 2.00000 0.0742781
$$726$$ 1.24621 0.0462512
$$727$$ 37.8617 1.40421 0.702107 0.712071i $$-0.252243\pi$$
0.702107 + 0.712071i $$0.252243\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −4.87689 −0.180502
$$731$$ −20.4924 −0.757940
$$732$$ 10.0000 0.369611
$$733$$ 10.6307 0.392653 0.196327 0.980539i $$-0.437099\pi$$
0.196327 + 0.980539i $$0.437099\pi$$
$$734$$ −33.3693 −1.23168
$$735$$ 19.2462 0.709907
$$736$$ 3.12311 0.115119
$$737$$ 40.9848 1.50970
$$738$$ −9.12311 −0.335826
$$739$$ −45.2311 −1.66385 −0.831926 0.554887i $$-0.812761\pi$$
−0.831926 + 0.554887i $$0.812761\pi$$
$$740$$ −3.12311 −0.114808
$$741$$ 0 0
$$742$$ −58.2462 −2.13829
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 5.12311 0.187822
$$745$$ −14.0000 −0.512920
$$746$$ 1.12311 0.0411198
$$747$$ 10.2462 0.374889
$$748$$ −6.24621 −0.228384
$$749$$ −40.9848 −1.49755
$$750$$ −1.00000 −0.0365148
$$751$$ 9.75379 0.355921 0.177960 0.984038i $$-0.443050\pi$$
0.177960 + 0.984038i $$0.443050\pi$$
$$752$$ −10.2462 −0.373641
$$753$$ −9.75379 −0.355448
$$754$$ 0 0
$$755$$ 11.3693 0.413772
$$756$$ 5.12311 0.186326
$$757$$ 5.12311 0.186202 0.0931012 0.995657i $$-0.470322\pi$$
0.0931012 + 0.995657i $$0.470322\pi$$
$$758$$ 6.00000 0.217930
$$759$$ −9.75379 −0.354040
$$760$$ 6.00000 0.217643
$$761$$ −5.12311 −0.185712 −0.0928562 0.995680i $$-0.529600\pi$$
−0.0928562 + 0.995680i $$0.529600\pi$$
$$762$$ −4.87689 −0.176671
$$763$$ 56.9848 2.06299
$$764$$ −16.4924 −0.596675
$$765$$ −2.00000 −0.0723102
$$766$$ −18.2462 −0.659262
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 32.9848 1.18946 0.594732 0.803924i $$-0.297258\pi$$
0.594732 + 0.803924i $$0.297258\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ −4.24621 −0.152924
$$772$$ 16.8769 0.607413
$$773$$ 0.246211 0.00885560 0.00442780 0.999990i $$-0.498591\pi$$
0.00442780 + 0.999990i $$0.498591\pi$$
$$774$$ −10.2462 −0.368292
$$775$$ −5.12311 −0.184027
$$776$$ 4.87689 0.175070
$$777$$ −16.0000 −0.573997
$$778$$ 16.2462 0.582455
$$779$$ −54.7386 −1.96122
$$780$$ 0 0
$$781$$ −19.5076 −0.698036
$$782$$ −6.24621 −0.223364
$$783$$ 2.00000 0.0714742
$$784$$ 19.2462 0.687365
$$785$$ −3.36932 −0.120256
$$786$$ −4.00000 −0.142675
$$787$$ 53.6155 1.91119 0.955594 0.294688i $$-0.0952157\pi$$
0.955594 + 0.294688i $$0.0952157\pi$$
$$788$$ −0.246211 −0.00877091
$$789$$ −2.63068 −0.0936548
$$790$$ −8.00000 −0.284627
$$791$$ −21.7538 −0.773476
$$792$$ −3.12311 −0.110975
$$793$$ 0 0
$$794$$ 9.36932 0.332505
$$795$$ 11.3693 0.403228
$$796$$ −1.75379 −0.0621614
$$797$$ 31.8617 1.12860 0.564300 0.825570i $$-0.309146\pi$$
0.564300 + 0.825570i $$0.309146\pi$$
$$798$$ 30.7386 1.08814
$$799$$ 20.4924 0.724970
$$800$$ −1.00000 −0.0353553
$$801$$ 5.12311 0.181016
$$802$$ 23.3693 0.825199
$$803$$ 15.2311 0.537492
$$804$$ 13.1231 0.462816
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ 0.246211 0.00866705
$$808$$ 4.24621 0.149381
$$809$$ −46.4924 −1.63459 −0.817293 0.576222i $$-0.804526\pi$$
−0.817293 + 0.576222i $$0.804526\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −44.2462 −1.55369 −0.776847 0.629689i $$-0.783182\pi$$
−0.776847 + 0.629689i $$0.783182\pi$$
$$812$$ 10.2462 0.359572
$$813$$ −14.8769 −0.521755
$$814$$ 9.75379 0.341870
$$815$$ −1.12311 −0.0393407
$$816$$ −2.00000 −0.0700140
$$817$$ −61.4773 −2.15082
$$818$$ 24.4924 0.856357
$$819$$ 0 0
$$820$$ 9.12311 0.318593
$$821$$ 27.7538 0.968614 0.484307 0.874898i $$-0.339072\pi$$
0.484307 + 0.874898i $$0.339072\pi$$
$$822$$ 22.4924 0.784513
$$823$$ 51.1231 1.78204 0.891020 0.453965i $$-0.149991\pi$$
0.891020 + 0.453965i $$0.149991\pi$$
$$824$$ −4.87689 −0.169895
$$825$$ 3.12311 0.108733
$$826$$ 36.4924 1.26973
$$827$$ −50.7386 −1.76436 −0.882178 0.470917i $$-0.843923\pi$$
−0.882178 + 0.470917i $$0.843923\pi$$
$$828$$ −3.12311 −0.108535
$$829$$ 7.75379 0.269300 0.134650 0.990893i $$-0.457009\pi$$
0.134650 + 0.990893i $$0.457009\pi$$
$$830$$ −10.2462 −0.355651
$$831$$ −27.8617 −0.966513
$$832$$ 0 0
$$833$$ −38.4924 −1.33368
$$834$$ −16.4924 −0.571086
$$835$$ 5.75379 0.199118
$$836$$ −18.7386 −0.648089
$$837$$ −5.12311 −0.177080
$$838$$ 28.0000 0.967244
$$839$$ −2.73863 −0.0945481 −0.0472741 0.998882i $$-0.515053\pi$$
−0.0472741 + 0.998882i $$0.515053\pi$$
$$840$$ −5.12311 −0.176764
$$841$$ −25.0000 −0.862069
$$842$$ 25.3693 0.874284
$$843$$ −5.12311 −0.176449
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 10.2462 0.352272
$$847$$ −6.38447 −0.219373
$$848$$ 11.3693 0.390424
$$849$$ 4.00000 0.137280
$$850$$ 2.00000 0.0685994
$$851$$ 9.75379 0.334356
$$852$$ −6.24621 −0.213992
$$853$$ 21.8617 0.748532 0.374266 0.927321i $$-0.377895\pi$$
0.374266 + 0.927321i $$0.377895\pi$$
$$854$$ −51.2311 −1.75309
$$855$$ −6.00000 −0.205196
$$856$$ 8.00000 0.273434
$$857$$ −2.49242 −0.0851395 −0.0425698 0.999093i $$-0.513554\pi$$
−0.0425698 + 0.999093i $$0.513554\pi$$
$$858$$ 0 0
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ 10.2462 0.349393
$$861$$ 46.7386 1.59285
$$862$$ −0.492423 −0.0167720
$$863$$ 10.2462 0.348785 0.174393 0.984676i $$-0.444204\pi$$
0.174393 + 0.984676i $$0.444204\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −14.8769 −0.505830
$$866$$ −18.0000 −0.611665
$$867$$ −13.0000 −0.441503
$$868$$ −26.2462 −0.890854
$$869$$ 24.9848 0.847553
$$870$$ −2.00000 −0.0678064
$$871$$ 0 0
$$872$$ −11.1231 −0.376676
$$873$$ −4.87689 −0.165058
$$874$$ −18.7386 −0.633844
$$875$$ 5.12311 0.173193
$$876$$ 4.87689 0.164775
$$877$$ −27.1231 −0.915882 −0.457941 0.888983i $$-0.651413\pi$$
−0.457941 + 0.888983i $$0.651413\pi$$
$$878$$ −3.50758 −0.118375
$$879$$ 20.7386 0.699497
$$880$$ 3.12311 0.105280
$$881$$ −11.7538 −0.395995 −0.197998 0.980203i $$-0.563444\pi$$
−0.197998 + 0.980203i $$0.563444\pi$$
$$882$$ −19.2462 −0.648054
$$883$$ 26.2462 0.883255 0.441628 0.897198i $$-0.354401\pi$$
0.441628 + 0.897198i $$0.354401\pi$$
$$884$$ 0 0
$$885$$ −7.12311 −0.239441
$$886$$ 36.4924 1.22599
$$887$$ −27.1231 −0.910705 −0.455352 0.890311i $$-0.650487\pi$$
−0.455352 + 0.890311i $$0.650487\pi$$
$$888$$ 3.12311 0.104805
$$889$$ 24.9848 0.837965
$$890$$ −5.12311 −0.171727
$$891$$ 3.12311 0.104628
$$892$$ −15.3693 −0.514603
$$893$$ 61.4773 2.05726
$$894$$ 14.0000 0.468230
$$895$$ 16.4924 0.551281
$$896$$ −5.12311 −0.171151
$$897$$ 0 0
$$898$$ −37.1231 −1.23881
$$899$$ −10.2462 −0.341730
$$900$$ 1.00000 0.0333333
$$901$$ −22.7386 −0.757534
$$902$$ −28.4924 −0.948694
$$903$$ 52.4924 1.74684
$$904$$ 4.24621 0.141227
$$905$$ 3.75379 0.124780
$$906$$ −11.3693 −0.377720
$$907$$ −42.2462 −1.40276 −0.701381 0.712786i $$-0.747433\pi$$
−0.701381 + 0.712786i $$0.747433\pi$$
$$908$$ 8.00000 0.265489
$$909$$ −4.24621 −0.140838
$$910$$ 0 0
$$911$$ 4.00000 0.132526 0.0662630 0.997802i $$-0.478892\pi$$
0.0662630 + 0.997802i $$0.478892\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ 32.0000 1.05905
$$914$$ −6.63068 −0.219324
$$915$$ 10.0000 0.330590
$$916$$ −3.12311 −0.103190
$$917$$ 20.4924 0.676719
$$918$$ 2.00000 0.0660098
$$919$$ −38.2462 −1.26163 −0.630813 0.775935i $$-0.717279\pi$$
−0.630813 + 0.775935i $$0.717279\pi$$
$$920$$ 3.12311 0.102966
$$921$$ −22.8769 −0.753819
$$922$$ 14.4924 0.477283
$$923$$ 0 0
$$924$$ 16.0000 0.526361
$$925$$ −3.12311 −0.102687
$$926$$ 35.8617 1.17849
$$927$$ 4.87689 0.160178
$$928$$ −2.00000 −0.0656532
$$929$$ −46.1080 −1.51275 −0.756376 0.654137i $$-0.773032\pi$$
−0.756376 + 0.654137i $$0.773032\pi$$
$$930$$ 5.12311 0.167993
$$931$$ −115.477 −3.78461
$$932$$ 24.2462 0.794211
$$933$$ 24.4924 0.801846
$$934$$ −5.75379 −0.188270
$$935$$ −6.24621 −0.204273
$$936$$ 0 0
$$937$$ −3.75379 −0.122631 −0.0613155 0.998118i $$-0.519530\pi$$
−0.0613155 + 0.998118i $$0.519530\pi$$
$$938$$ −67.2311 −2.19517
$$939$$ −0.246211 −0.00803480
$$940$$ −10.2462 −0.334195
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 3.36932 0.109778
$$943$$ −28.4924 −0.927841
$$944$$ −7.12311 −0.231837
$$945$$ 5.12311 0.166655
$$946$$ −32.0000 −1.04041
$$947$$ 24.4924 0.795897 0.397948 0.917408i $$-0.369722\pi$$
0.397948 + 0.917408i $$0.369722\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ 6.00000 0.194666
$$951$$ 6.00000 0.194563
$$952$$ 10.2462 0.332082
$$953$$ 42.9848 1.39242 0.696208 0.717840i $$-0.254869\pi$$
0.696208 + 0.717840i $$0.254869\pi$$
$$954$$ −11.3693 −0.368095
$$955$$ −16.4924 −0.533682
$$956$$ −28.4924 −0.921511
$$957$$ 6.24621 0.201911
$$958$$ 20.4924 0.662080
$$959$$ −115.231 −3.72100
$$960$$ 1.00000 0.0322749
$$961$$ −4.75379 −0.153348
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ −2.24621 −0.0723456
$$965$$ 16.8769 0.543286
$$966$$ 16.0000 0.514792
$$967$$ −6.38447 −0.205311 −0.102655 0.994717i $$-0.532734\pi$$
−0.102655 + 0.994717i $$0.532734\pi$$
$$968$$ 1.24621 0.0400547
$$969$$ 12.0000 0.385496
$$970$$ 4.87689 0.156588
$$971$$ 54.2462 1.74084 0.870422 0.492307i $$-0.163846\pi$$
0.870422 + 0.492307i $$0.163846\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 84.4924 2.70870
$$974$$ 7.36932 0.236128
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 44.7386 1.43132 0.715658 0.698451i $$-0.246127\pi$$
0.715658 + 0.698451i $$0.246127\pi$$
$$978$$ 1.12311 0.0359130
$$979$$ 16.0000 0.511362
$$980$$ 19.2462 0.614798
$$981$$ 11.1231 0.355133
$$982$$ −10.7386 −0.342684
$$983$$ 27.5076 0.877355 0.438678 0.898644i $$-0.355447\pi$$
0.438678 + 0.898644i $$0.355447\pi$$
$$984$$ −9.12311 −0.290834
$$985$$ −0.246211 −0.00784494
$$986$$ 4.00000 0.127386
$$987$$ −52.4924 −1.67085
$$988$$ 0 0
$$989$$ −32.0000 −1.01754
$$990$$ −3.12311 −0.0992588
$$991$$ 18.7386 0.595252 0.297626 0.954682i $$-0.403805\pi$$
0.297626 + 0.954682i $$0.403805\pi$$
$$992$$ 5.12311 0.162659
$$993$$ −24.2462 −0.769430
$$994$$ 32.0000 1.01498
$$995$$ −1.75379 −0.0555988
$$996$$ 10.2462 0.324664
$$997$$ −52.3542 −1.65807 −0.829036 0.559195i $$-0.811110\pi$$
−0.829036 + 0.559195i $$0.811110\pi$$
$$998$$ −1.50758 −0.0477215
$$999$$ −3.12311 −0.0988107
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.bd.1.2 2
13.5 odd 4 390.2.b.d.181.4 yes 4
13.8 odd 4 390.2.b.d.181.1 4
13.12 even 2 5070.2.a.bh.1.1 2
39.5 even 4 1170.2.b.f.181.2 4
39.8 even 4 1170.2.b.f.181.3 4
52.31 even 4 3120.2.g.o.961.1 4
52.47 even 4 3120.2.g.o.961.4 4
65.8 even 4 1950.2.f.l.649.3 4
65.18 even 4 1950.2.f.o.649.4 4
65.34 odd 4 1950.2.b.h.1351.4 4
65.44 odd 4 1950.2.b.h.1351.1 4
65.47 even 4 1950.2.f.o.649.2 4
65.57 even 4 1950.2.f.l.649.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.d.181.1 4 13.8 odd 4
390.2.b.d.181.4 yes 4 13.5 odd 4
1170.2.b.f.181.2 4 39.5 even 4
1170.2.b.f.181.3 4 39.8 even 4
1950.2.b.h.1351.1 4 65.44 odd 4
1950.2.b.h.1351.4 4 65.34 odd 4
1950.2.f.l.649.1 4 65.57 even 4
1950.2.f.l.649.3 4 65.8 even 4
1950.2.f.o.649.2 4 65.47 even 4
1950.2.f.o.649.4 4 65.18 even 4
3120.2.g.o.961.1 4 52.31 even 4
3120.2.g.o.961.4 4 52.47 even 4
5070.2.a.bd.1.2 2 1.1 even 1 trivial
5070.2.a.bh.1.1 2 13.12 even 2