# Properties

 Label 5390.2.a.bs.1.1 Level $5390$ Weight $2$ Character 5390.1 Self dual yes Analytic conductor $43.039$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5390.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.0393666895$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 5390.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +1.00000 q^{11} -2.73205 q^{12} +1.46410 q^{13} -2.73205 q^{15} +1.00000 q^{16} +3.46410 q^{17} +4.46410 q^{18} -6.73205 q^{19} +1.00000 q^{20} +1.00000 q^{22} -8.19615 q^{23} -2.73205 q^{24} +1.00000 q^{25} +1.46410 q^{26} -4.00000 q^{27} -4.73205 q^{29} -2.73205 q^{30} -2.00000 q^{31} +1.00000 q^{32} -2.73205 q^{33} +3.46410 q^{34} +4.46410 q^{36} +0.732051 q^{37} -6.73205 q^{38} -4.00000 q^{39} +1.00000 q^{40} +2.19615 q^{41} +2.00000 q^{43} +1.00000 q^{44} +4.46410 q^{45} -8.19615 q^{46} -6.92820 q^{47} -2.73205 q^{48} +1.00000 q^{50} -9.46410 q^{51} +1.46410 q^{52} -7.26795 q^{53} -4.00000 q^{54} +1.00000 q^{55} +18.3923 q^{57} -4.73205 q^{58} -6.92820 q^{59} -2.73205 q^{60} +4.92820 q^{61} -2.00000 q^{62} +1.00000 q^{64} +1.46410 q^{65} -2.73205 q^{66} -4.00000 q^{67} +3.46410 q^{68} +22.3923 q^{69} +9.46410 q^{71} +4.46410 q^{72} +14.3923 q^{73} +0.732051 q^{74} -2.73205 q^{75} -6.73205 q^{76} -4.00000 q^{78} -12.1962 q^{79} +1.00000 q^{80} -2.46410 q^{81} +2.19615 q^{82} +16.3923 q^{83} +3.46410 q^{85} +2.00000 q^{86} +12.9282 q^{87} +1.00000 q^{88} -3.46410 q^{89} +4.46410 q^{90} -8.19615 q^{92} +5.46410 q^{93} -6.92820 q^{94} -6.73205 q^{95} -2.73205 q^{96} -14.5885 q^{97} +4.46410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{15} + 2 q^{16} + 2 q^{18} - 10 q^{19} + 2 q^{20} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 4 q^{26} - 8 q^{27} - 6 q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{36} - 2 q^{37} - 10 q^{38} - 8 q^{39} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 2 q^{44} + 2 q^{45} - 6 q^{46} - 2 q^{48} + 2 q^{50} - 12 q^{51} - 4 q^{52} - 18 q^{53} - 8 q^{54} + 2 q^{55} + 16 q^{57} - 6 q^{58} - 2 q^{60} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 4 q^{65} - 2 q^{66} - 8 q^{67} + 24 q^{69} + 12 q^{71} + 2 q^{72} + 8 q^{73} - 2 q^{74} - 2 q^{75} - 10 q^{76} - 8 q^{78} - 14 q^{79} + 2 q^{80} + 2 q^{81} - 6 q^{82} + 12 q^{83} + 4 q^{86} + 12 q^{87} + 2 q^{88} + 2 q^{90} - 6 q^{92} + 4 q^{93} - 10 q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{99} + O(q^{100})$$

## 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
$$3$$ −2.73205 −1.57735 −0.788675 0.614810i $$-0.789233\pi$$
−0.788675 + 0.614810i $$0.789233\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −2.73205 −1.11536
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 4.46410 1.48803
$$10$$ 1.00000 0.316228
$$11$$ 1.00000 0.301511
$$12$$ −2.73205 −0.788675
$$13$$ 1.46410 0.406069 0.203034 0.979172i $$-0.434920\pi$$
0.203034 + 0.979172i $$0.434920\pi$$
$$14$$ 0 0
$$15$$ −2.73205 −0.705412
$$16$$ 1.00000 0.250000
$$17$$ 3.46410 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$18$$ 4.46410 1.05220
$$19$$ −6.73205 −1.54444 −0.772219 0.635356i $$-0.780853\pi$$
−0.772219 + 0.635356i $$0.780853\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ −8.19615 −1.70902 −0.854508 0.519438i $$-0.826141\pi$$
−0.854508 + 0.519438i $$0.826141\pi$$
$$24$$ −2.73205 −0.557678
$$25$$ 1.00000 0.200000
$$26$$ 1.46410 0.287134
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ −4.73205 −0.878720 −0.439360 0.898311i $$-0.644795\pi$$
−0.439360 + 0.898311i $$0.644795\pi$$
$$30$$ −2.73205 −0.498802
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.73205 −0.475589
$$34$$ 3.46410 0.594089
$$35$$ 0 0
$$36$$ 4.46410 0.744017
$$37$$ 0.732051 0.120348 0.0601742 0.998188i $$-0.480834\pi$$
0.0601742 + 0.998188i $$0.480834\pi$$
$$38$$ −6.73205 −1.09208
$$39$$ −4.00000 −0.640513
$$40$$ 1.00000 0.158114
$$41$$ 2.19615 0.342981 0.171491 0.985186i $$-0.445142\pi$$
0.171491 + 0.985186i $$0.445142\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 4.46410 0.665469
$$46$$ −8.19615 −1.20846
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ −2.73205 −0.394338
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ −9.46410 −1.32524
$$52$$ 1.46410 0.203034
$$53$$ −7.26795 −0.998330 −0.499165 0.866507i $$-0.666360\pi$$
−0.499165 + 0.866507i $$0.666360\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 18.3923 2.43612
$$58$$ −4.73205 −0.621349
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ −2.73205 −0.352706
$$61$$ 4.92820 0.630992 0.315496 0.948927i $$-0.397829\pi$$
0.315496 + 0.948927i $$0.397829\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.46410 0.181599
$$66$$ −2.73205 −0.336292
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 3.46410 0.420084
$$69$$ 22.3923 2.69572
$$70$$ 0 0
$$71$$ 9.46410 1.12318 0.561591 0.827415i $$-0.310189\pi$$
0.561591 + 0.827415i $$0.310189\pi$$
$$72$$ 4.46410 0.526099
$$73$$ 14.3923 1.68449 0.842246 0.539093i $$-0.181233\pi$$
0.842246 + 0.539093i $$0.181233\pi$$
$$74$$ 0.732051 0.0850992
$$75$$ −2.73205 −0.315470
$$76$$ −6.73205 −0.772219
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ −12.1962 −1.37217 −0.686087 0.727519i $$-0.740673\pi$$
−0.686087 + 0.727519i $$0.740673\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −2.46410 −0.273789
$$82$$ 2.19615 0.242524
$$83$$ 16.3923 1.79929 0.899645 0.436623i $$-0.143826\pi$$
0.899645 + 0.436623i $$0.143826\pi$$
$$84$$ 0 0
$$85$$ 3.46410 0.375735
$$86$$ 2.00000 0.215666
$$87$$ 12.9282 1.38605
$$88$$ 1.00000 0.106600
$$89$$ −3.46410 −0.367194 −0.183597 0.983002i $$-0.558774\pi$$
−0.183597 + 0.983002i $$0.558774\pi$$
$$90$$ 4.46410 0.470558
$$91$$ 0 0
$$92$$ −8.19615 −0.854508
$$93$$ 5.46410 0.566601
$$94$$ −6.92820 −0.714590
$$95$$ −6.73205 −0.690694
$$96$$ −2.73205 −0.278839
$$97$$ −14.5885 −1.48123 −0.740617 0.671928i $$-0.765467\pi$$
−0.740617 + 0.671928i $$0.765467\pi$$
$$98$$ 0 0
$$99$$ 4.46410 0.448659
$$100$$ 1.00000 0.100000
$$101$$ 7.85641 0.781742 0.390871 0.920446i $$-0.372174\pi$$
0.390871 + 0.920446i $$0.372174\pi$$
$$102$$ −9.46410 −0.937086
$$103$$ −12.3923 −1.22105 −0.610525 0.791997i $$-0.709042\pi$$
−0.610525 + 0.791997i $$0.709042\pi$$
$$104$$ 1.46410 0.143567
$$105$$ 0 0
$$106$$ −7.26795 −0.705926
$$107$$ 7.85641 0.759507 0.379754 0.925088i $$-0.376009\pi$$
0.379754 + 0.925088i $$0.376009\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ −15.6603 −1.49998 −0.749990 0.661449i $$-0.769942\pi$$
−0.749990 + 0.661449i $$0.769942\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 7.85641 0.739069 0.369534 0.929217i $$-0.379517\pi$$
0.369534 + 0.929217i $$0.379517\pi$$
$$114$$ 18.3923 1.72260
$$115$$ −8.19615 −0.764295
$$116$$ −4.73205 −0.439360
$$117$$ 6.53590 0.604244
$$118$$ −6.92820 −0.637793
$$119$$ 0 0
$$120$$ −2.73205 −0.249401
$$121$$ 1.00000 0.0909091
$$122$$ 4.92820 0.446179
$$123$$ −6.00000 −0.541002
$$124$$ −2.00000 −0.179605
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 1.07180 0.0951066 0.0475533 0.998869i $$-0.484858\pi$$
0.0475533 + 0.998869i $$0.484858\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −5.46410 −0.481087
$$130$$ 1.46410 0.128410
$$131$$ −5.66025 −0.494539 −0.247269 0.968947i $$-0.579533\pi$$
−0.247269 + 0.968947i $$0.579533\pi$$
$$132$$ −2.73205 −0.237795
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ −4.00000 −0.344265
$$136$$ 3.46410 0.297044
$$137$$ −0.928203 −0.0793018 −0.0396509 0.999214i $$-0.512625\pi$$
−0.0396509 + 0.999214i $$0.512625\pi$$
$$138$$ 22.3923 1.90616
$$139$$ −13.6603 −1.15865 −0.579324 0.815097i $$-0.696683\pi$$
−0.579324 + 0.815097i $$0.696683\pi$$
$$140$$ 0 0
$$141$$ 18.9282 1.59404
$$142$$ 9.46410 0.794210
$$143$$ 1.46410 0.122434
$$144$$ 4.46410 0.372008
$$145$$ −4.73205 −0.392975
$$146$$ 14.3923 1.19112
$$147$$ 0 0
$$148$$ 0.732051 0.0601742
$$149$$ −7.26795 −0.595414 −0.297707 0.954657i $$-0.596222\pi$$
−0.297707 + 0.954657i $$0.596222\pi$$
$$150$$ −2.73205 −0.223071
$$151$$ 11.1244 0.905287 0.452644 0.891692i $$-0.350481\pi$$
0.452644 + 0.891692i $$0.350481\pi$$
$$152$$ −6.73205 −0.546041
$$153$$ 15.4641 1.25020
$$154$$ 0 0
$$155$$ −2.00000 −0.160644
$$156$$ −4.00000 −0.320256
$$157$$ −4.53590 −0.362004 −0.181002 0.983483i $$-0.557934\pi$$
−0.181002 + 0.983483i $$0.557934\pi$$
$$158$$ −12.1962 −0.970274
$$159$$ 19.8564 1.57472
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −2.46410 −0.193598
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.19615 0.171491
$$165$$ −2.73205 −0.212690
$$166$$ 16.3923 1.27229
$$167$$ −13.8564 −1.07224 −0.536120 0.844141i $$-0.680111\pi$$
−0.536120 + 0.844141i $$0.680111\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ 3.46410 0.265684
$$171$$ −30.0526 −2.29818
$$172$$ 2.00000 0.152499
$$173$$ −0.928203 −0.0705700 −0.0352850 0.999377i $$-0.511234\pi$$
−0.0352850 + 0.999377i $$0.511234\pi$$
$$174$$ 12.9282 0.980085
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 18.9282 1.42273
$$178$$ −3.46410 −0.259645
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 4.46410 0.332734
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −13.4641 −0.995295
$$184$$ −8.19615 −0.604228
$$185$$ 0.732051 0.0538214
$$186$$ 5.46410 0.400647
$$187$$ 3.46410 0.253320
$$188$$ −6.92820 −0.505291
$$189$$ 0 0
$$190$$ −6.73205 −0.488394
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −2.73205 −0.197169
$$193$$ 12.3923 0.892018 0.446009 0.895029i $$-0.352845\pi$$
0.446009 + 0.895029i $$0.352845\pi$$
$$194$$ −14.5885 −1.04739
$$195$$ −4.00000 −0.286446
$$196$$ 0 0
$$197$$ −24.2487 −1.72765 −0.863825 0.503793i $$-0.831938\pi$$
−0.863825 + 0.503793i $$0.831938\pi$$
$$198$$ 4.46410 0.317250
$$199$$ −2.92820 −0.207575 −0.103787 0.994600i $$-0.533096\pi$$
−0.103787 + 0.994600i $$0.533096\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 10.9282 0.770816
$$202$$ 7.85641 0.552775
$$203$$ 0 0
$$204$$ −9.46410 −0.662620
$$205$$ 2.19615 0.153386
$$206$$ −12.3923 −0.863413
$$207$$ −36.5885 −2.54307
$$208$$ 1.46410 0.101517
$$209$$ −6.73205 −0.465666
$$210$$ 0 0
$$211$$ 26.9282 1.85381 0.926907 0.375291i $$-0.122457\pi$$
0.926907 + 0.375291i $$0.122457\pi$$
$$212$$ −7.26795 −0.499165
$$213$$ −25.8564 −1.77165
$$214$$ 7.85641 0.537053
$$215$$ 2.00000 0.136399
$$216$$ −4.00000 −0.272166
$$217$$ 0 0
$$218$$ −15.6603 −1.06065
$$219$$ −39.3205 −2.65703
$$220$$ 1.00000 0.0674200
$$221$$ 5.07180 0.341166
$$222$$ −2.00000 −0.134231
$$223$$ 25.4641 1.70520 0.852601 0.522562i $$-0.175024\pi$$
0.852601 + 0.522562i $$0.175024\pi$$
$$224$$ 0 0
$$225$$ 4.46410 0.297607
$$226$$ 7.85641 0.522600
$$227$$ −6.92820 −0.459841 −0.229920 0.973209i $$-0.573847\pi$$
−0.229920 + 0.973209i $$0.573847\pi$$
$$228$$ 18.3923 1.21806
$$229$$ −24.3923 −1.61189 −0.805944 0.591991i $$-0.798342\pi$$
−0.805944 + 0.591991i $$0.798342\pi$$
$$230$$ −8.19615 −0.540438
$$231$$ 0 0
$$232$$ −4.73205 −0.310674
$$233$$ −7.85641 −0.514690 −0.257345 0.966320i $$-0.582848\pi$$
−0.257345 + 0.966320i $$0.582848\pi$$
$$234$$ 6.53590 0.427265
$$235$$ −6.92820 −0.451946
$$236$$ −6.92820 −0.450988
$$237$$ 33.3205 2.16440
$$238$$ 0 0
$$239$$ 1.26795 0.0820168 0.0410084 0.999159i $$-0.486943\pi$$
0.0410084 + 0.999159i $$0.486943\pi$$
$$240$$ −2.73205 −0.176353
$$241$$ −3.26795 −0.210507 −0.105254 0.994445i $$-0.533565\pi$$
−0.105254 + 0.994445i $$0.533565\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 18.7321 1.20166
$$244$$ 4.92820 0.315496
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ −9.85641 −0.627148
$$248$$ −2.00000 −0.127000
$$249$$ −44.7846 −2.83811
$$250$$ 1.00000 0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −8.19615 −0.515288
$$254$$ 1.07180 0.0672505
$$255$$ −9.46410 −0.592665
$$256$$ 1.00000 0.0625000
$$257$$ 23.6603 1.47589 0.737943 0.674863i $$-0.235797\pi$$
0.737943 + 0.674863i $$0.235797\pi$$
$$258$$ −5.46410 −0.340180
$$259$$ 0 0
$$260$$ 1.46410 0.0907997
$$261$$ −21.1244 −1.30756
$$262$$ −5.66025 −0.349692
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ −2.73205 −0.168146
$$265$$ −7.26795 −0.446467
$$266$$ 0 0
$$267$$ 9.46410 0.579194
$$268$$ −4.00000 −0.244339
$$269$$ −28.3923 −1.73111 −0.865555 0.500814i $$-0.833034\pi$$
−0.865555 + 0.500814i $$0.833034\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ −0.392305 −0.0238308 −0.0119154 0.999929i $$-0.503793\pi$$
−0.0119154 + 0.999929i $$0.503793\pi$$
$$272$$ 3.46410 0.210042
$$273$$ 0 0
$$274$$ −0.928203 −0.0560748
$$275$$ 1.00000 0.0603023
$$276$$ 22.3923 1.34786
$$277$$ 7.07180 0.424903 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$278$$ −13.6603 −0.819288
$$279$$ −8.92820 −0.534518
$$280$$ 0 0
$$281$$ 22.3923 1.33581 0.667906 0.744245i $$-0.267191\pi$$
0.667906 + 0.744245i $$0.267191\pi$$
$$282$$ 18.9282 1.12716
$$283$$ 31.7128 1.88513 0.942566 0.334021i $$-0.108406\pi$$
0.942566 + 0.334021i $$0.108406\pi$$
$$284$$ 9.46410 0.561591
$$285$$ 18.3923 1.08947
$$286$$ 1.46410 0.0865741
$$287$$ 0 0
$$288$$ 4.46410 0.263050
$$289$$ −5.00000 −0.294118
$$290$$ −4.73205 −0.277876
$$291$$ 39.8564 2.33642
$$292$$ 14.3923 0.842246
$$293$$ −21.4641 −1.25395 −0.626973 0.779041i $$-0.715706\pi$$
−0.626973 + 0.779041i $$0.715706\pi$$
$$294$$ 0 0
$$295$$ −6.92820 −0.403376
$$296$$ 0.732051 0.0425496
$$297$$ −4.00000 −0.232104
$$298$$ −7.26795 −0.421021
$$299$$ −12.0000 −0.693978
$$300$$ −2.73205 −0.157735
$$301$$ 0 0
$$302$$ 11.1244 0.640135
$$303$$ −21.4641 −1.23308
$$304$$ −6.73205 −0.386110
$$305$$ 4.92820 0.282188
$$306$$ 15.4641 0.884024
$$307$$ −24.3923 −1.39214 −0.696071 0.717973i $$-0.745070\pi$$
−0.696071 + 0.717973i $$0.745070\pi$$
$$308$$ 0 0
$$309$$ 33.8564 1.92602
$$310$$ −2.00000 −0.113592
$$311$$ −24.9282 −1.41355 −0.706774 0.707439i $$-0.749850\pi$$
−0.706774 + 0.707439i $$0.749850\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −22.1962 −1.25460 −0.627300 0.778777i $$-0.715840\pi$$
−0.627300 + 0.778777i $$0.715840\pi$$
$$314$$ −4.53590 −0.255976
$$315$$ 0 0
$$316$$ −12.1962 −0.686087
$$317$$ 30.5885 1.71802 0.859009 0.511960i $$-0.171080\pi$$
0.859009 + 0.511960i $$0.171080\pi$$
$$318$$ 19.8564 1.11349
$$319$$ −4.73205 −0.264944
$$320$$ 1.00000 0.0559017
$$321$$ −21.4641 −1.19801
$$322$$ 0 0
$$323$$ −23.3205 −1.29759
$$324$$ −2.46410 −0.136895
$$325$$ 1.46410 0.0812137
$$326$$ −4.00000 −0.221540
$$327$$ 42.7846 2.36599
$$328$$ 2.19615 0.121262
$$329$$ 0 0
$$330$$ −2.73205 −0.150394
$$331$$ −18.7846 −1.03250 −0.516248 0.856439i $$-0.672672\pi$$
−0.516248 + 0.856439i $$0.672672\pi$$
$$332$$ 16.3923 0.899645
$$333$$ 3.26795 0.179083
$$334$$ −13.8564 −0.758189
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 22.7846 1.24116 0.620578 0.784144i $$-0.286898\pi$$
0.620578 + 0.784144i $$0.286898\pi$$
$$338$$ −10.8564 −0.590511
$$339$$ −21.4641 −1.16577
$$340$$ 3.46410 0.187867
$$341$$ −2.00000 −0.108306
$$342$$ −30.0526 −1.62506
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 22.3923 1.20556
$$346$$ −0.928203 −0.0499005
$$347$$ −23.0718 −1.23856 −0.619279 0.785171i $$-0.712575\pi$$
−0.619279 + 0.785171i $$0.712575\pi$$
$$348$$ 12.9282 0.693024
$$349$$ 5.60770 0.300173 0.150087 0.988673i $$-0.452045\pi$$
0.150087 + 0.988673i $$0.452045\pi$$
$$350$$ 0 0
$$351$$ −5.85641 −0.312592
$$352$$ 1.00000 0.0533002
$$353$$ −14.1962 −0.755585 −0.377792 0.925890i $$-0.623317\pi$$
−0.377792 + 0.925890i $$0.623317\pi$$
$$354$$ 18.9282 1.00602
$$355$$ 9.46410 0.502302
$$356$$ −3.46410 −0.183597
$$357$$ 0 0
$$358$$ −6.00000 −0.317110
$$359$$ −34.0526 −1.79723 −0.898613 0.438743i $$-0.855424\pi$$
−0.898613 + 0.438743i $$0.855424\pi$$
$$360$$ 4.46410 0.235279
$$361$$ 26.3205 1.38529
$$362$$ −14.0000 −0.735824
$$363$$ −2.73205 −0.143395
$$364$$ 0 0
$$365$$ 14.3923 0.753328
$$366$$ −13.4641 −0.703780
$$367$$ −12.3923 −0.646873 −0.323437 0.946250i $$-0.604838\pi$$
−0.323437 + 0.946250i $$0.604838\pi$$
$$368$$ −8.19615 −0.427254
$$369$$ 9.80385 0.510368
$$370$$ 0.732051 0.0380575
$$371$$ 0 0
$$372$$ 5.46410 0.283300
$$373$$ 18.3923 0.952317 0.476159 0.879359i $$-0.342029\pi$$
0.476159 + 0.879359i $$0.342029\pi$$
$$374$$ 3.46410 0.179124
$$375$$ −2.73205 −0.141082
$$376$$ −6.92820 −0.357295
$$377$$ −6.92820 −0.356821
$$378$$ 0 0
$$379$$ 6.14359 0.315575 0.157788 0.987473i $$-0.449564\pi$$
0.157788 + 0.987473i $$0.449564\pi$$
$$380$$ −6.73205 −0.345347
$$381$$ −2.92820 −0.150016
$$382$$ −12.0000 −0.613973
$$383$$ 33.4641 1.70994 0.854968 0.518681i $$-0.173577\pi$$
0.854968 + 0.518681i $$0.173577\pi$$
$$384$$ −2.73205 −0.139419
$$385$$ 0 0
$$386$$ 12.3923 0.630752
$$387$$ 8.92820 0.453846
$$388$$ −14.5885 −0.740617
$$389$$ −1.60770 −0.0815134 −0.0407567 0.999169i $$-0.512977\pi$$
−0.0407567 + 0.999169i $$0.512977\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ −28.3923 −1.43586
$$392$$ 0 0
$$393$$ 15.4641 0.780061
$$394$$ −24.2487 −1.22163
$$395$$ −12.1962 −0.613655
$$396$$ 4.46410 0.224330
$$397$$ −30.3923 −1.52535 −0.762673 0.646784i $$-0.776113\pi$$
−0.762673 + 0.646784i $$0.776113\pi$$
$$398$$ −2.92820 −0.146778
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 2.53590 0.126637 0.0633184 0.997993i $$-0.479832\pi$$
0.0633184 + 0.997993i $$0.479832\pi$$
$$402$$ 10.9282 0.545049
$$403$$ −2.92820 −0.145864
$$404$$ 7.85641 0.390871
$$405$$ −2.46410 −0.122442
$$406$$ 0 0
$$407$$ 0.732051 0.0362864
$$408$$ −9.46410 −0.468543
$$409$$ −10.8756 −0.537766 −0.268883 0.963173i $$-0.586654\pi$$
−0.268883 + 0.963173i $$0.586654\pi$$
$$410$$ 2.19615 0.108460
$$411$$ 2.53590 0.125087
$$412$$ −12.3923 −0.610525
$$413$$ 0 0
$$414$$ −36.5885 −1.79822
$$415$$ 16.3923 0.804667
$$416$$ 1.46410 0.0717835
$$417$$ 37.3205 1.82759
$$418$$ −6.73205 −0.329275
$$419$$ −30.9282 −1.51094 −0.755471 0.655182i $$-0.772592\pi$$
−0.755471 + 0.655182i $$0.772592\pi$$
$$420$$ 0 0
$$421$$ −35.8564 −1.74753 −0.873767 0.486344i $$-0.838330\pi$$
−0.873767 + 0.486344i $$0.838330\pi$$
$$422$$ 26.9282 1.31084
$$423$$ −30.9282 −1.50378
$$424$$ −7.26795 −0.352963
$$425$$ 3.46410 0.168034
$$426$$ −25.8564 −1.25275
$$427$$ 0 0
$$428$$ 7.85641 0.379754
$$429$$ −4.00000 −0.193122
$$430$$ 2.00000 0.0964486
$$431$$ 3.12436 0.150495 0.0752475 0.997165i $$-0.476025\pi$$
0.0752475 + 0.997165i $$0.476025\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 18.1962 0.874451 0.437226 0.899352i $$-0.355961\pi$$
0.437226 + 0.899352i $$0.355961\pi$$
$$434$$ 0 0
$$435$$ 12.9282 0.619860
$$436$$ −15.6603 −0.749990
$$437$$ 55.1769 2.63947
$$438$$ −39.3205 −1.87881
$$439$$ −26.2487 −1.25278 −0.626391 0.779509i $$-0.715469\pi$$
−0.626391 + 0.779509i $$0.715469\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 0 0
$$442$$ 5.07180 0.241241
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −3.46410 −0.164214
$$446$$ 25.4641 1.20576
$$447$$ 19.8564 0.939176
$$448$$ 0 0
$$449$$ 40.3923 1.90623 0.953115 0.302607i $$-0.0978570\pi$$
0.953115 + 0.302607i $$0.0978570\pi$$
$$450$$ 4.46410 0.210440
$$451$$ 2.19615 0.103413
$$452$$ 7.85641 0.369534
$$453$$ −30.3923 −1.42796
$$454$$ −6.92820 −0.325157
$$455$$ 0 0
$$456$$ 18.3923 0.861299
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ −24.3923 −1.13978
$$459$$ −13.8564 −0.646762
$$460$$ −8.19615 −0.382148
$$461$$ −22.3923 −1.04291 −0.521457 0.853278i $$-0.674611\pi$$
−0.521457 + 0.853278i $$0.674611\pi$$
$$462$$ 0 0
$$463$$ 27.5167 1.27881 0.639404 0.768871i $$-0.279181\pi$$
0.639404 + 0.768871i $$0.279181\pi$$
$$464$$ −4.73205 −0.219680
$$465$$ 5.46410 0.253392
$$466$$ −7.85641 −0.363941
$$467$$ 34.0526 1.57576 0.787882 0.615826i $$-0.211178\pi$$
0.787882 + 0.615826i $$0.211178\pi$$
$$468$$ 6.53590 0.302122
$$469$$ 0 0
$$470$$ −6.92820 −0.319574
$$471$$ 12.3923 0.571007
$$472$$ −6.92820 −0.318896
$$473$$ 2.00000 0.0919601
$$474$$ 33.3205 1.53046
$$475$$ −6.73205 −0.308888
$$476$$ 0 0
$$477$$ −32.4449 −1.48555
$$478$$ 1.26795 0.0579946
$$479$$ 32.7846 1.49797 0.748984 0.662589i $$-0.230542\pi$$
0.748984 + 0.662589i $$0.230542\pi$$
$$480$$ −2.73205 −0.124700
$$481$$ 1.07180 0.0488697
$$482$$ −3.26795 −0.148851
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −14.5885 −0.662428
$$486$$ 18.7321 0.849703
$$487$$ 4.19615 0.190146 0.0950729 0.995470i $$-0.469692\pi$$
0.0950729 + 0.995470i $$0.469692\pi$$
$$488$$ 4.92820 0.223089
$$489$$ 10.9282 0.494190
$$490$$ 0 0
$$491$$ −27.7128 −1.25066 −0.625331 0.780360i $$-0.715036\pi$$
−0.625331 + 0.780360i $$0.715036\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −16.3923 −0.738272
$$494$$ −9.85641 −0.443461
$$495$$ 4.46410 0.200646
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ −44.7846 −2.00685
$$499$$ 12.1436 0.543622 0.271811 0.962351i $$-0.412377\pi$$
0.271811 + 0.962351i $$0.412377\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 37.8564 1.69130
$$502$$ −12.0000 −0.535586
$$503$$ −8.78461 −0.391686 −0.195843 0.980635i $$-0.562744\pi$$
−0.195843 + 0.980635i $$0.562744\pi$$
$$504$$ 0 0
$$505$$ 7.85641 0.349605
$$506$$ −8.19615 −0.364363
$$507$$ 29.6603 1.31726
$$508$$ 1.07180 0.0475533
$$509$$ −11.0718 −0.490749 −0.245374 0.969428i $$-0.578911\pi$$
−0.245374 + 0.969428i $$0.578911\pi$$
$$510$$ −9.46410 −0.419077
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 26.9282 1.18891
$$514$$ 23.6603 1.04361
$$515$$ −12.3923 −0.546070
$$516$$ −5.46410 −0.240544
$$517$$ −6.92820 −0.304702
$$518$$ 0 0
$$519$$ 2.53590 0.111314
$$520$$ 1.46410 0.0642051
$$521$$ −10.3923 −0.455295 −0.227648 0.973744i $$-0.573103\pi$$
−0.227648 + 0.973744i $$0.573103\pi$$
$$522$$ −21.1244 −0.924588
$$523$$ 29.1769 1.27582 0.637909 0.770112i $$-0.279800\pi$$
0.637909 + 0.770112i $$0.279800\pi$$
$$524$$ −5.66025 −0.247269
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ −6.92820 −0.301797
$$528$$ −2.73205 −0.118897
$$529$$ 44.1769 1.92074
$$530$$ −7.26795 −0.315700
$$531$$ −30.9282 −1.34217
$$532$$ 0 0
$$533$$ 3.21539 0.139274
$$534$$ 9.46410 0.409552
$$535$$ 7.85641 0.339662
$$536$$ −4.00000 −0.172774
$$537$$ 16.3923 0.707380
$$538$$ −28.3923 −1.22408
$$539$$ 0 0
$$540$$ −4.00000 −0.172133
$$541$$ −20.7321 −0.891340 −0.445670 0.895197i $$-0.647035\pi$$
−0.445670 + 0.895197i $$0.647035\pi$$
$$542$$ −0.392305 −0.0168509
$$543$$ 38.2487 1.64141
$$544$$ 3.46410 0.148522
$$545$$ −15.6603 −0.670812
$$546$$ 0 0
$$547$$ 28.7846 1.23074 0.615371 0.788238i $$-0.289006\pi$$
0.615371 + 0.788238i $$0.289006\pi$$
$$548$$ −0.928203 −0.0396509
$$549$$ 22.0000 0.938937
$$550$$ 1.00000 0.0426401
$$551$$ 31.8564 1.35713
$$552$$ 22.3923 0.953080
$$553$$ 0 0
$$554$$ 7.07180 0.300452
$$555$$ −2.00000 −0.0848953
$$556$$ −13.6603 −0.579324
$$557$$ −25.6077 −1.08503 −0.542516 0.840045i $$-0.682528\pi$$
−0.542516 + 0.840045i $$0.682528\pi$$
$$558$$ −8.92820 −0.377961
$$559$$ 2.92820 0.123850
$$560$$ 0 0
$$561$$ −9.46410 −0.399575
$$562$$ 22.3923 0.944562
$$563$$ −5.07180 −0.213751 −0.106875 0.994272i $$-0.534085\pi$$
−0.106875 + 0.994272i $$0.534085\pi$$
$$564$$ 18.9282 0.797021
$$565$$ 7.85641 0.330522
$$566$$ 31.7128 1.33299
$$567$$ 0 0
$$568$$ 9.46410 0.397105
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 18.3923 0.770369
$$571$$ 3.60770 0.150977 0.0754887 0.997147i $$-0.475948\pi$$
0.0754887 + 0.997147i $$0.475948\pi$$
$$572$$ 1.46410 0.0612172
$$573$$ 32.7846 1.36960
$$574$$ 0 0
$$575$$ −8.19615 −0.341803
$$576$$ 4.46410 0.186004
$$577$$ 34.5885 1.43994 0.719968 0.694007i $$-0.244156\pi$$
0.719968 + 0.694007i $$0.244156\pi$$
$$578$$ −5.00000 −0.207973
$$579$$ −33.8564 −1.40702
$$580$$ −4.73205 −0.196488
$$581$$ 0 0
$$582$$ 39.8564 1.65210
$$583$$ −7.26795 −0.301008
$$584$$ 14.3923 0.595558
$$585$$ 6.53590 0.270226
$$586$$ −21.4641 −0.886674
$$587$$ 11.4115 0.471005 0.235502 0.971874i $$-0.424326\pi$$
0.235502 + 0.971874i $$0.424326\pi$$
$$588$$ 0 0
$$589$$ 13.4641 0.554779
$$590$$ −6.92820 −0.285230
$$591$$ 66.2487 2.72511
$$592$$ 0.732051 0.0300871
$$593$$ −0.248711 −0.0102133 −0.00510667 0.999987i $$-0.501626\pi$$
−0.00510667 + 0.999987i $$0.501626\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −7.26795 −0.297707
$$597$$ 8.00000 0.327418
$$598$$ −12.0000 −0.490716
$$599$$ 37.1769 1.51901 0.759504 0.650503i $$-0.225442\pi$$
0.759504 + 0.650503i $$0.225442\pi$$
$$600$$ −2.73205 −0.111536
$$601$$ 5.51666 0.225029 0.112515 0.993650i $$-0.464109\pi$$
0.112515 + 0.993650i $$0.464109\pi$$
$$602$$ 0 0
$$603$$ −17.8564 −0.727169
$$604$$ 11.1244 0.452644
$$605$$ 1.00000 0.0406558
$$606$$ −21.4641 −0.871920
$$607$$ −20.9282 −0.849450 −0.424725 0.905323i $$-0.639629\pi$$
−0.424725 + 0.905323i $$0.639629\pi$$
$$608$$ −6.73205 −0.273021
$$609$$ 0 0
$$610$$ 4.92820 0.199537
$$611$$ −10.1436 −0.410366
$$612$$ 15.4641 0.625099
$$613$$ −35.1769 −1.42078 −0.710391 0.703807i $$-0.751482\pi$$
−0.710391 + 0.703807i $$0.751482\pi$$
$$614$$ −24.3923 −0.984393
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ 17.3205 0.697297 0.348649 0.937253i $$-0.386641\pi$$
0.348649 + 0.937253i $$0.386641\pi$$
$$618$$ 33.8564 1.36190
$$619$$ −28.7846 −1.15695 −0.578476 0.815700i $$-0.696352\pi$$
−0.578476 + 0.815700i $$0.696352\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ 32.7846 1.31560
$$622$$ −24.9282 −0.999530
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −22.1962 −0.887137
$$627$$ 18.3923 0.734518
$$628$$ −4.53590 −0.181002
$$629$$ 2.53590 0.101113
$$630$$ 0 0
$$631$$ −46.9282 −1.86818 −0.934091 0.357035i $$-0.883788\pi$$
−0.934091 + 0.357035i $$0.883788\pi$$
$$632$$ −12.1962 −0.485137
$$633$$ −73.5692 −2.92411
$$634$$ 30.5885 1.21482
$$635$$ 1.07180 0.0425330
$$636$$ 19.8564 0.787358
$$637$$ 0 0
$$638$$ −4.73205 −0.187344
$$639$$ 42.2487 1.67133
$$640$$ 1.00000 0.0395285
$$641$$ −35.5692 −1.40490 −0.702450 0.711733i $$-0.747910\pi$$
−0.702450 + 0.711733i $$0.747910\pi$$
$$642$$ −21.4641 −0.847121
$$643$$ −33.2679 −1.31196 −0.655980 0.754778i $$-0.727744\pi$$
−0.655980 + 0.754778i $$0.727744\pi$$
$$644$$ 0 0
$$645$$ −5.46410 −0.215149
$$646$$ −23.3205 −0.917533
$$647$$ 26.5359 1.04323 0.521617 0.853180i $$-0.325329\pi$$
0.521617 + 0.853180i $$0.325329\pi$$
$$648$$ −2.46410 −0.0967991
$$649$$ −6.92820 −0.271956
$$650$$ 1.46410 0.0574268
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 11.6603 0.456301 0.228151 0.973626i $$-0.426732\pi$$
0.228151 + 0.973626i $$0.426732\pi$$
$$654$$ 42.7846 1.67301
$$655$$ −5.66025 −0.221164
$$656$$ 2.19615 0.0857453
$$657$$ 64.2487 2.50658
$$658$$ 0 0
$$659$$ 30.2487 1.17832 0.589161 0.808015i $$-0.299458\pi$$
0.589161 + 0.808015i $$0.299458\pi$$
$$660$$ −2.73205 −0.106345
$$661$$ 46.0000 1.78919 0.894596 0.446875i $$-0.147463\pi$$
0.894596 + 0.446875i $$0.147463\pi$$
$$662$$ −18.7846 −0.730085
$$663$$ −13.8564 −0.538138
$$664$$ 16.3923 0.636145
$$665$$ 0 0
$$666$$ 3.26795 0.126630
$$667$$ 38.7846 1.50175
$$668$$ −13.8564 −0.536120
$$669$$ −69.5692 −2.68970
$$670$$ −4.00000 −0.154533
$$671$$ 4.92820 0.190251
$$672$$ 0 0
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 22.7846 0.877630
$$675$$ −4.00000 −0.153960
$$676$$ −10.8564 −0.417554
$$677$$ 4.14359 0.159251 0.0796256 0.996825i $$-0.474628\pi$$
0.0796256 + 0.996825i $$0.474628\pi$$
$$678$$ −21.4641 −0.824324
$$679$$ 0 0
$$680$$ 3.46410 0.132842
$$681$$ 18.9282 0.725330
$$682$$ −2.00000 −0.0765840
$$683$$ 6.24871 0.239100 0.119550 0.992828i $$-0.461855\pi$$
0.119550 + 0.992828i $$0.461855\pi$$
$$684$$ −30.0526 −1.14909
$$685$$ −0.928203 −0.0354648
$$686$$ 0 0
$$687$$ 66.6410 2.54251
$$688$$ 2.00000 0.0762493
$$689$$ −10.6410 −0.405390
$$690$$ 22.3923 0.852460
$$691$$ 13.4641 0.512199 0.256099 0.966650i $$-0.417563\pi$$
0.256099 + 0.966650i $$0.417563\pi$$
$$692$$ −0.928203 −0.0352850
$$693$$ 0 0
$$694$$ −23.0718 −0.875793
$$695$$ −13.6603 −0.518163
$$696$$ 12.9282 0.490042
$$697$$ 7.60770 0.288162
$$698$$ 5.60770 0.212254
$$699$$ 21.4641 0.811847
$$700$$ 0 0
$$701$$ 41.9090 1.58288 0.791440 0.611247i $$-0.209332\pi$$
0.791440 + 0.611247i $$0.209332\pi$$
$$702$$ −5.85641 −0.221036
$$703$$ −4.92820 −0.185871
$$704$$ 1.00000 0.0376889
$$705$$ 18.9282 0.712877
$$706$$ −14.1962 −0.534279
$$707$$ 0 0
$$708$$ 18.9282 0.711365
$$709$$ 25.3205 0.950932 0.475466 0.879734i $$-0.342280\pi$$
0.475466 + 0.879734i $$0.342280\pi$$
$$710$$ 9.46410 0.355181
$$711$$ −54.4449 −2.04184
$$712$$ −3.46410 −0.129823
$$713$$ 16.3923 0.613897
$$714$$ 0 0
$$715$$ 1.46410 0.0547543
$$716$$ −6.00000 −0.224231
$$717$$ −3.46410 −0.129369
$$718$$ −34.0526 −1.27083
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 4.46410 0.166367
$$721$$ 0 0
$$722$$ 26.3205 0.979548
$$723$$ 8.92820 0.332043
$$724$$ −14.0000 −0.520306
$$725$$ −4.73205 −0.175744
$$726$$ −2.73205 −0.101396
$$727$$ 4.00000 0.148352 0.0741759 0.997245i $$-0.476367\pi$$
0.0741759 + 0.997245i $$0.476367\pi$$
$$728$$ 0 0
$$729$$ −43.7846 −1.62165
$$730$$ 14.3923 0.532683
$$731$$ 6.92820 0.256249
$$732$$ −13.4641 −0.497648
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ −12.3923 −0.457408
$$735$$ 0 0
$$736$$ −8.19615 −0.302114
$$737$$ −4.00000 −0.147342
$$738$$ 9.80385 0.360885
$$739$$ 11.7128 0.430863 0.215431 0.976519i $$-0.430884\pi$$
0.215431 + 0.976519i $$0.430884\pi$$
$$740$$ 0.732051 0.0269107
$$741$$ 26.9282 0.989232
$$742$$ 0 0
$$743$$ −32.7846 −1.20275 −0.601375 0.798967i $$-0.705380\pi$$
−0.601375 + 0.798967i $$0.705380\pi$$
$$744$$ 5.46410 0.200324
$$745$$ −7.26795 −0.266277
$$746$$ 18.3923 0.673390
$$747$$ 73.1769 2.67740
$$748$$ 3.46410 0.126660
$$749$$ 0 0
$$750$$ −2.73205 −0.0997604
$$751$$ −11.6077 −0.423571 −0.211785 0.977316i $$-0.567928\pi$$
−0.211785 + 0.977316i $$0.567928\pi$$
$$752$$ −6.92820 −0.252646
$$753$$ 32.7846 1.19474
$$754$$ −6.92820 −0.252310
$$755$$ 11.1244 0.404857
$$756$$ 0 0
$$757$$ −43.3731 −1.57642 −0.788210 0.615406i $$-0.788992\pi$$
−0.788210 + 0.615406i $$0.788992\pi$$
$$758$$ 6.14359 0.223145
$$759$$ 22.3923 0.812789
$$760$$ −6.73205 −0.244197
$$761$$ 12.3397 0.447315 0.223658 0.974668i $$-0.428200\pi$$
0.223658 + 0.974668i $$0.428200\pi$$
$$762$$ −2.92820 −0.106078
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 15.4641 0.559106
$$766$$ 33.4641 1.20911
$$767$$ −10.1436 −0.366264
$$768$$ −2.73205 −0.0985844
$$769$$ 18.1962 0.656170 0.328085 0.944648i $$-0.393597\pi$$
0.328085 + 0.944648i $$0.393597\pi$$
$$770$$ 0 0
$$771$$ −64.6410 −2.32799
$$772$$ 12.3923 0.446009
$$773$$ 0.928203 0.0333851 0.0166926 0.999861i $$-0.494686\pi$$
0.0166926 + 0.999861i $$0.494686\pi$$
$$774$$ 8.92820 0.320918
$$775$$ −2.00000 −0.0718421
$$776$$ −14.5885 −0.523695
$$777$$ 0 0
$$778$$ −1.60770 −0.0576387
$$779$$ −14.7846 −0.529714
$$780$$ −4.00000 −0.143223
$$781$$ 9.46410 0.338652
$$782$$ −28.3923 −1.01531
$$783$$ 18.9282 0.676439
$$784$$ 0 0
$$785$$ −4.53590 −0.161893
$$786$$ 15.4641 0.551586
$$787$$ −18.1436 −0.646749 −0.323375 0.946271i $$-0.604817\pi$$
−0.323375 + 0.946271i $$0.604817\pi$$
$$788$$ −24.2487 −0.863825
$$789$$ 65.5692 2.33433
$$790$$ −12.1962 −0.433920
$$791$$ 0 0
$$792$$ 4.46410 0.158625
$$793$$ 7.21539 0.256226
$$794$$ −30.3923 −1.07858
$$795$$ 19.8564 0.704234
$$796$$ −2.92820 −0.103787
$$797$$ 25.6077 0.907071 0.453536 0.891238i $$-0.350163\pi$$
0.453536 + 0.891238i $$0.350163\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 1.00000 0.0353553
$$801$$ −15.4641 −0.546397
$$802$$ 2.53590 0.0895457
$$803$$ 14.3923 0.507893
$$804$$ 10.9282 0.385408
$$805$$ 0 0
$$806$$ −2.92820 −0.103142
$$807$$ 77.5692 2.73057
$$808$$ 7.85641 0.276387
$$809$$ −31.1769 −1.09612 −0.548061 0.836438i $$-0.684634\pi$$
−0.548061 + 0.836438i $$0.684634\pi$$
$$810$$ −2.46410 −0.0865797
$$811$$ 43.1244 1.51430 0.757150 0.653241i $$-0.226591\pi$$
0.757150 + 0.653241i $$0.226591\pi$$
$$812$$ 0 0
$$813$$ 1.07180 0.0375896
$$814$$ 0.732051 0.0256584
$$815$$ −4.00000 −0.140114
$$816$$ −9.46410 −0.331310
$$817$$ −13.4641 −0.471049
$$818$$ −10.8756 −0.380258
$$819$$ 0 0
$$820$$ 2.19615 0.0766930
$$821$$ 17.9090 0.625027 0.312514 0.949913i $$-0.398829\pi$$
0.312514 + 0.949913i $$0.398829\pi$$
$$822$$ 2.53590 0.0884496
$$823$$ −0.875644 −0.0305230 −0.0152615 0.999884i $$-0.504858\pi$$
−0.0152615 + 0.999884i $$0.504858\pi$$
$$824$$ −12.3923 −0.431706
$$825$$ −2.73205 −0.0951178
$$826$$ 0 0
$$827$$ −25.8564 −0.899115 −0.449558 0.893251i $$-0.648418\pi$$
−0.449558 + 0.893251i $$0.648418\pi$$
$$828$$ −36.5885 −1.27154
$$829$$ −2.24871 −0.0781010 −0.0390505 0.999237i $$-0.512433\pi$$
−0.0390505 + 0.999237i $$0.512433\pi$$
$$830$$ 16.3923 0.568985
$$831$$ −19.3205 −0.670221
$$832$$ 1.46410 0.0507586
$$833$$ 0 0
$$834$$ 37.3205 1.29230
$$835$$ −13.8564 −0.479521
$$836$$ −6.73205 −0.232833
$$837$$ 8.00000 0.276520
$$838$$ −30.9282 −1.06840
$$839$$ 31.8564 1.09981 0.549903 0.835229i $$-0.314665\pi$$
0.549903 + 0.835229i $$0.314665\pi$$
$$840$$ 0 0
$$841$$ −6.60770 −0.227852
$$842$$ −35.8564 −1.23569
$$843$$ −61.1769 −2.10704
$$844$$ 26.9282 0.926907
$$845$$ −10.8564 −0.373472
$$846$$ −30.9282 −1.06333
$$847$$ 0 0
$$848$$ −7.26795 −0.249582
$$849$$ −86.6410 −2.97351
$$850$$ 3.46410 0.118818
$$851$$ −6.00000 −0.205677
$$852$$ −25.8564 −0.885826
$$853$$ 54.7846 1.87579 0.937895 0.346920i $$-0.112773\pi$$
0.937895 + 0.346920i $$0.112773\pi$$
$$854$$ 0 0
$$855$$ −30.0526 −1.02778
$$856$$ 7.85641 0.268526
$$857$$ 27.4641 0.938156 0.469078 0.883157i $$-0.344586\pi$$
0.469078 + 0.883157i $$0.344586\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 12.7846 0.436205 0.218103 0.975926i $$-0.430013\pi$$
0.218103 + 0.975926i $$0.430013\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ 3.12436 0.106416
$$863$$ −7.51666 −0.255870 −0.127935 0.991783i $$-0.540835\pi$$
−0.127935 + 0.991783i $$0.540835\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ −0.928203 −0.0315599
$$866$$ 18.1962 0.618330
$$867$$ 13.6603 0.463927
$$868$$ 0 0
$$869$$ −12.1962 −0.413726
$$870$$ 12.9282 0.438307
$$871$$ −5.85641 −0.198437
$$872$$ −15.6603 −0.530323
$$873$$ −65.1244 −2.20413
$$874$$ 55.1769 1.86639
$$875$$ 0 0
$$876$$ −39.3205 −1.32852
$$877$$ −21.3205 −0.719942 −0.359971 0.932963i $$-0.617213\pi$$
−0.359971 + 0.932963i $$0.617213\pi$$
$$878$$ −26.2487 −0.885851
$$879$$ 58.6410 1.97791
$$880$$ 1.00000 0.0337100
$$881$$ −11.0718 −0.373018 −0.186509 0.982453i $$-0.559717\pi$$
−0.186509 + 0.982453i $$0.559717\pi$$
$$882$$ 0 0
$$883$$ −35.6077 −1.19829 −0.599147 0.800639i $$-0.704494\pi$$
−0.599147 + 0.800639i $$0.704494\pi$$
$$884$$ 5.07180 0.170583
$$885$$ 18.9282 0.636265
$$886$$ 0 0
$$887$$ −13.8564 −0.465253 −0.232626 0.972566i $$-0.574732\pi$$
−0.232626 + 0.972566i $$0.574732\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ −3.46410 −0.116117
$$891$$ −2.46410 −0.0825505
$$892$$ 25.4641 0.852601
$$893$$ 46.6410 1.56078
$$894$$ 19.8564 0.664098
$$895$$ −6.00000 −0.200558
$$896$$ 0 0
$$897$$ 32.7846 1.09465
$$898$$ 40.3923 1.34791
$$899$$ 9.46410 0.315645
$$900$$ 4.46410 0.148803
$$901$$ −25.1769 −0.838765
$$902$$ 2.19615 0.0731239
$$903$$ 0 0
$$904$$ 7.85641 0.261300
$$905$$ −14.0000 −0.465376
$$906$$ −30.3923 −1.00972
$$907$$ −31.0333 −1.03044 −0.515222 0.857057i $$-0.672291\pi$$
−0.515222 + 0.857057i $$0.672291\pi$$
$$908$$ −6.92820 −0.229920
$$909$$ 35.0718 1.16326
$$910$$ 0 0
$$911$$ 9.46410 0.313560 0.156780 0.987634i $$-0.449889\pi$$
0.156780 + 0.987634i $$0.449889\pi$$
$$912$$ 18.3923 0.609030
$$913$$ 16.3923 0.542506
$$914$$ 2.00000 0.0661541
$$915$$ −13.4641 −0.445109
$$916$$ −24.3923 −0.805944
$$917$$ 0 0
$$918$$ −13.8564 −0.457330
$$919$$ −25.3731 −0.836980 −0.418490 0.908221i $$-0.637441\pi$$
−0.418490 + 0.908221i $$0.637441\pi$$
$$920$$ −8.19615 −0.270219
$$921$$ 66.6410 2.19590
$$922$$ −22.3923 −0.737451
$$923$$ 13.8564 0.456089
$$924$$ 0 0
$$925$$ 0.732051 0.0240697
$$926$$ 27.5167 0.904254
$$927$$ −55.3205 −1.81696
$$928$$ −4.73205 −0.155337
$$929$$ 25.6077 0.840161 0.420081 0.907487i $$-0.362002\pi$$
0.420081 + 0.907487i $$0.362002\pi$$
$$930$$ 5.46410 0.179175
$$931$$ 0 0
$$932$$ −7.85641 −0.257345
$$933$$ 68.1051 2.22966
$$934$$ 34.0526 1.11423
$$935$$ 3.46410 0.113288
$$936$$ 6.53590 0.213633
$$937$$ 1.21539 0.0397051 0.0198525 0.999803i $$-0.493680\pi$$
0.0198525 + 0.999803i $$0.493680\pi$$
$$938$$ 0 0
$$939$$ 60.6410 1.97894
$$940$$ −6.92820 −0.225973
$$941$$ 14.7846 0.481965 0.240982 0.970530i $$-0.422530\pi$$
0.240982 + 0.970530i $$0.422530\pi$$
$$942$$ 12.3923 0.403763
$$943$$ −18.0000 −0.586161
$$944$$ −6.92820 −0.225494
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ 47.3205 1.53771 0.768855 0.639423i $$-0.220827\pi$$
0.768855 + 0.639423i $$0.220827\pi$$
$$948$$ 33.3205 1.08220
$$949$$ 21.0718 0.684019
$$950$$ −6.73205 −0.218417
$$951$$ −83.5692 −2.70992
$$952$$ 0 0
$$953$$ 26.5359 0.859582 0.429791 0.902928i $$-0.358587\pi$$
0.429791 + 0.902928i $$0.358587\pi$$
$$954$$ −32.4449 −1.05044
$$955$$ −12.0000 −0.388311
$$956$$ 1.26795 0.0410084
$$957$$ 12.9282 0.417909
$$958$$ 32.7846 1.05922
$$959$$ 0 0
$$960$$ −2.73205 −0.0881766
$$961$$ −27.0000 −0.870968
$$962$$ 1.07180 0.0345561
$$963$$ 35.0718 1.13017
$$964$$ −3.26795 −0.105254
$$965$$ 12.3923 0.398922
$$966$$ 0 0
$$967$$ 26.9282 0.865953 0.432976 0.901405i $$-0.357463\pi$$
0.432976 + 0.901405i $$0.357463\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 63.7128 2.04675
$$970$$ −14.5885 −0.468407
$$971$$ −42.9282 −1.37763 −0.688816 0.724936i $$-0.741869\pi$$
−0.688816 + 0.724936i $$0.741869\pi$$
$$972$$ 18.7321 0.600831
$$973$$ 0 0
$$974$$ 4.19615 0.134453
$$975$$ −4.00000 −0.128103
$$976$$ 4.92820 0.157748
$$977$$ −33.7128 −1.07857 −0.539284 0.842124i $$-0.681305\pi$$
−0.539284 + 0.842124i $$0.681305\pi$$
$$978$$ 10.9282 0.349445
$$979$$ −3.46410 −0.110713
$$980$$ 0 0
$$981$$ −69.9090 −2.23202
$$982$$ −27.7128 −0.884351
$$983$$ 37.1769 1.18576 0.592880 0.805291i $$-0.297991\pi$$
0.592880 + 0.805291i $$0.297991\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −24.2487 −0.772628
$$986$$ −16.3923 −0.522037
$$987$$ 0 0
$$988$$ −9.85641 −0.313574
$$989$$ −16.3923 −0.521245
$$990$$ 4.46410 0.141878
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 51.3205 1.62861
$$994$$ 0 0
$$995$$ −2.92820 −0.0928303
$$996$$ −44.7846 −1.41905
$$997$$ −17.7128 −0.560970 −0.280485 0.959858i $$-0.590495\pi$$
−0.280485 + 0.959858i $$0.590495\pi$$
$$998$$ 12.1436 0.384399
$$999$$ −2.92820 −0.0926443
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 5390.2.a.bs.1.1 2
7.6 odd 2 770.2.a.j.1.2 2
21.20 even 2 6930.2.a.bv.1.1 2
28.27 even 2 6160.2.a.t.1.1 2
35.13 even 4 3850.2.c.x.1849.2 4
35.27 even 4 3850.2.c.x.1849.3 4
35.34 odd 2 3850.2.a.bd.1.1 2
77.76 even 2 8470.2.a.br.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 7.6 odd 2
3850.2.a.bd.1.1 2 35.34 odd 2
3850.2.c.x.1849.2 4 35.13 even 4
3850.2.c.x.1849.3 4 35.27 even 4
5390.2.a.bs.1.1 2 1.1 even 1 trivial
6160.2.a.t.1.1 2 28.27 even 2
6930.2.a.bv.1.1 2 21.20 even 2
8470.2.a.br.1.2 2 77.76 even 2