# Properties

 Label 230.2.a.d.1.1 Level $230$ Weight $2$ Character 230.1 Self dual yes Analytic conductor $1.837$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.11903$$ of defining polynomial Character $$\chi$$ $$=$$ 230.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -3.11903 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.11903 q^{6} +4.50973 q^{7} +1.00000 q^{8} +6.72833 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -3.11903 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.11903 q^{6} +4.50973 q^{7} +1.00000 q^{8} +6.72833 q^{9} -1.00000 q^{10} +4.33763 q^{11} -3.11903 q^{12} -3.72833 q^{13} +4.50973 q^{14} +3.11903 q^{15} +1.00000 q^{16} +1.11903 q^{17} +6.72833 q^{18} +4.50973 q^{19} -1.00000 q^{20} -14.0660 q^{21} +4.33763 q^{22} -1.00000 q^{23} -3.11903 q^{24} +1.00000 q^{25} -3.72833 q^{26} -11.6288 q^{27} +4.50973 q^{28} -8.23805 q^{29} +3.11903 q^{30} +1.72833 q^{31} +1.00000 q^{32} -13.5292 q^{33} +1.11903 q^{34} -4.50973 q^{35} +6.72833 q^{36} -0.781399 q^{37} +4.50973 q^{38} +11.6288 q^{39} -1.00000 q^{40} +3.90043 q^{41} -14.0660 q^{42} +8.00000 q^{43} +4.33763 q^{44} -6.72833 q^{45} -1.00000 q^{46} -11.4567 q^{47} -3.11903 q^{48} +13.3376 q^{49} +1.00000 q^{50} -3.49027 q^{51} -3.72833 q^{52} -6.00000 q^{53} -11.6288 q^{54} -4.33763 q^{55} +4.50973 q^{56} -14.0660 q^{57} -8.23805 q^{58} -2.23805 q^{59} +3.11903 q^{60} +3.55623 q^{61} +1.72833 q^{62} +30.3429 q^{63} +1.00000 q^{64} +3.72833 q^{65} -13.5292 q^{66} +2.43720 q^{67} +1.11903 q^{68} +3.11903 q^{69} -4.50973 q^{70} +7.11903 q^{71} +6.72833 q^{72} -9.45665 q^{73} -0.781399 q^{74} -3.11903 q^{75} +4.50973 q^{76} +19.5615 q^{77} +11.6288 q^{78} -14.9133 q^{79} -1.00000 q^{80} +16.0854 q^{81} +3.90043 q^{82} +2.78140 q^{83} -14.0660 q^{84} -1.11903 q^{85} +8.00000 q^{86} +25.6947 q^{87} +4.33763 q^{88} -7.69471 q^{89} -6.72833 q^{90} -16.8137 q^{91} -1.00000 q^{92} -5.39070 q^{93} -11.4567 q^{94} -4.50973 q^{95} -3.11903 q^{96} -0.642920 q^{97} +13.3376 q^{98} +29.1850 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + q^{3} + 3q^{4} - 3q^{5} + q^{6} + 3q^{7} + 3q^{8} + 10q^{9} + O(q^{10})$$ $$3q + 3q^{2} + q^{3} + 3q^{4} - 3q^{5} + q^{6} + 3q^{7} + 3q^{8} + 10q^{9} - 3q^{10} + 3q^{11} + q^{12} - q^{13} + 3q^{14} - q^{15} + 3q^{16} - 7q^{17} + 10q^{18} + 3q^{19} - 3q^{20} - 22q^{21} + 3q^{22} - 3q^{23} + q^{24} + 3q^{25} - q^{26} - 14q^{27} + 3q^{28} - 4q^{29} - q^{30} - 5q^{31} + 3q^{32} - 9q^{33} - 7q^{34} - 3q^{35} + 10q^{36} - 2q^{37} + 3q^{38} + 14q^{39} - 3q^{40} + q^{41} - 22q^{42} + 24q^{43} + 3q^{44} - 10q^{45} - 3q^{46} - 14q^{47} + q^{48} + 30q^{49} + 3q^{50} - 21q^{51} - q^{52} - 18q^{53} - 14q^{54} - 3q^{55} + 3q^{56} - 22q^{57} - 4q^{58} + 14q^{59} - q^{60} + q^{61} - 5q^{62} + 8q^{63} + 3q^{64} + q^{65} - 9q^{66} + 8q^{67} - 7q^{68} - q^{69} - 3q^{70} + 11q^{71} + 10q^{72} - 8q^{73} - 2q^{74} + q^{75} + 3q^{76} - 24q^{77} + 14q^{78} - 4q^{79} - 3q^{80} + 7q^{81} + q^{82} + 8q^{83} - 22q^{84} + 7q^{85} + 24q^{86} + 36q^{87} + 3q^{88} + 18q^{89} - 10q^{90} + q^{91} - 3q^{92} - 16q^{93} - 14q^{94} - 3q^{95} + q^{96} - 33q^{97} + 30q^{98} + 57q^{99} + O(q^{100})$$

## 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
$$3$$ −3.11903 −1.80077 −0.900385 0.435093i $$-0.856715\pi$$
−0.900385 + 0.435093i $$0.856715\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −3.11903 −1.27334
$$7$$ 4.50973 1.70452 0.852258 0.523122i $$-0.175233\pi$$
0.852258 + 0.523122i $$0.175233\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 6.72833 2.24278
$$10$$ −1.00000 −0.316228
$$11$$ 4.33763 1.30784 0.653922 0.756562i $$-0.273122\pi$$
0.653922 + 0.756562i $$0.273122\pi$$
$$12$$ −3.11903 −0.900385
$$13$$ −3.72833 −1.03405 −0.517026 0.855970i $$-0.672961\pi$$
−0.517026 + 0.855970i $$0.672961\pi$$
$$14$$ 4.50973 1.20527
$$15$$ 3.11903 0.805329
$$16$$ 1.00000 0.250000
$$17$$ 1.11903 0.271404 0.135702 0.990750i $$-0.456671\pi$$
0.135702 + 0.990750i $$0.456671\pi$$
$$18$$ 6.72833 1.58588
$$19$$ 4.50973 1.03460 0.517301 0.855803i $$-0.326937\pi$$
0.517301 + 0.855803i $$0.326937\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −14.0660 −3.06944
$$22$$ 4.33763 0.924785
$$23$$ −1.00000 −0.208514
$$24$$ −3.11903 −0.636669
$$25$$ 1.00000 0.200000
$$26$$ −3.72833 −0.731185
$$27$$ −11.6288 −2.23795
$$28$$ 4.50973 0.852258
$$29$$ −8.23805 −1.52977 −0.764884 0.644168i $$-0.777204\pi$$
−0.764884 + 0.644168i $$0.777204\pi$$
$$30$$ 3.11903 0.569454
$$31$$ 1.72833 0.310417 0.155208 0.987882i $$-0.450395\pi$$
0.155208 + 0.987882i $$0.450395\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −13.5292 −2.35513
$$34$$ 1.11903 0.191911
$$35$$ −4.50973 −0.762283
$$36$$ 6.72833 1.12139
$$37$$ −0.781399 −0.128461 −0.0642306 0.997935i $$-0.520459\pi$$
−0.0642306 + 0.997935i $$0.520459\pi$$
$$38$$ 4.50973 0.731574
$$39$$ 11.6288 1.86209
$$40$$ −1.00000 −0.158114
$$41$$ 3.90043 0.609144 0.304572 0.952489i $$-0.401487\pi$$
0.304572 + 0.952489i $$0.401487\pi$$
$$42$$ −14.0660 −2.17042
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 4.33763 0.653922
$$45$$ −6.72833 −1.00300
$$46$$ −1.00000 −0.147442
$$47$$ −11.4567 −1.67112 −0.835562 0.549396i $$-0.814858\pi$$
−0.835562 + 0.549396i $$0.814858\pi$$
$$48$$ −3.11903 −0.450193
$$49$$ 13.3376 1.90538
$$50$$ 1.00000 0.141421
$$51$$ −3.49027 −0.488736
$$52$$ −3.72833 −0.517026
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −11.6288 −1.58247
$$55$$ −4.33763 −0.584886
$$56$$ 4.50973 0.602637
$$57$$ −14.0660 −1.86308
$$58$$ −8.23805 −1.08171
$$59$$ −2.23805 −0.291370 −0.145685 0.989331i $$-0.546539\pi$$
−0.145685 + 0.989331i $$0.546539\pi$$
$$60$$ 3.11903 0.402665
$$61$$ 3.55623 0.455329 0.227664 0.973740i $$-0.426891\pi$$
0.227664 + 0.973740i $$0.426891\pi$$
$$62$$ 1.72833 0.219498
$$63$$ 30.3429 3.82285
$$64$$ 1.00000 0.125000
$$65$$ 3.72833 0.462442
$$66$$ −13.5292 −1.66533
$$67$$ 2.43720 0.297752 0.148876 0.988856i $$-0.452435\pi$$
0.148876 + 0.988856i $$0.452435\pi$$
$$68$$ 1.11903 0.135702
$$69$$ 3.11903 0.375487
$$70$$ −4.50973 −0.539015
$$71$$ 7.11903 0.844873 0.422437 0.906393i $$-0.361175\pi$$
0.422437 + 0.906393i $$0.361175\pi$$
$$72$$ 6.72833 0.792941
$$73$$ −9.45665 −1.10682 −0.553409 0.832910i $$-0.686673\pi$$
−0.553409 + 0.832910i $$0.686673\pi$$
$$74$$ −0.781399 −0.0908357
$$75$$ −3.11903 −0.360154
$$76$$ 4.50973 0.517301
$$77$$ 19.5615 2.22924
$$78$$ 11.6288 1.31670
$$79$$ −14.9133 −1.67788 −0.838939 0.544225i $$-0.816824\pi$$
−0.838939 + 0.544225i $$0.816824\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 16.0854 1.78727
$$82$$ 3.90043 0.430730
$$83$$ 2.78140 0.305298 0.152649 0.988280i $$-0.451220\pi$$
0.152649 + 0.988280i $$0.451220\pi$$
$$84$$ −14.0660 −1.53472
$$85$$ −1.11903 −0.121375
$$86$$ 8.00000 0.862662
$$87$$ 25.6947 2.75476
$$88$$ 4.33763 0.462393
$$89$$ −7.69471 −0.815637 −0.407819 0.913063i $$-0.633710\pi$$
−0.407819 + 0.913063i $$0.633710\pi$$
$$90$$ −6.72833 −0.709228
$$91$$ −16.8137 −1.76256
$$92$$ −1.00000 −0.104257
$$93$$ −5.39070 −0.558989
$$94$$ −11.4567 −1.18166
$$95$$ −4.50973 −0.462688
$$96$$ −3.11903 −0.318334
$$97$$ −0.642920 −0.0652786 −0.0326393 0.999467i $$-0.510391\pi$$
−0.0326393 + 0.999467i $$0.510391\pi$$
$$98$$ 13.3376 1.34730
$$99$$ 29.1850 2.93320
$$100$$ 1.00000 0.100000
$$101$$ −8.23805 −0.819717 −0.409858 0.912149i $$-0.634422\pi$$
−0.409858 + 0.912149i $$0.634422\pi$$
$$102$$ −3.49027 −0.345589
$$103$$ 12.3376 1.21566 0.607831 0.794066i $$-0.292040\pi$$
0.607831 + 0.794066i $$0.292040\pi$$
$$104$$ −3.72833 −0.365593
$$105$$ 14.0660 1.37270
$$106$$ −6.00000 −0.582772
$$107$$ −15.9328 −1.54028 −0.770139 0.637876i $$-0.779813\pi$$
−0.770139 + 0.637876i $$0.779813\pi$$
$$108$$ −11.6288 −1.11898
$$109$$ −1.49027 −0.142742 −0.0713712 0.997450i $$-0.522737\pi$$
−0.0713712 + 0.997450i $$0.522737\pi$$
$$110$$ −4.33763 −0.413577
$$111$$ 2.43720 0.231329
$$112$$ 4.50973 0.426129
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −14.0660 −1.31740
$$115$$ 1.00000 0.0932505
$$116$$ −8.23805 −0.764884
$$117$$ −25.0854 −2.31915
$$118$$ −2.23805 −0.206030
$$119$$ 5.04650 0.462612
$$120$$ 3.11903 0.284727
$$121$$ 7.81502 0.710456
$$122$$ 3.55623 0.321966
$$123$$ −12.1655 −1.09693
$$124$$ 1.72833 0.155208
$$125$$ −1.00000 −0.0894427
$$126$$ 30.3429 2.70316
$$127$$ −0.675256 −0.0599193 −0.0299597 0.999551i $$-0.509538\pi$$
−0.0299597 + 0.999551i $$0.509538\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −24.9522 −2.19692
$$130$$ 3.72833 0.326996
$$131$$ −13.6947 −1.19651 −0.598256 0.801305i $$-0.704139\pi$$
−0.598256 + 0.801305i $$0.704139\pi$$
$$132$$ −13.5292 −1.17756
$$133$$ 20.3376 1.76350
$$134$$ 2.43720 0.210542
$$135$$ 11.6288 1.00084
$$136$$ 1.11903 0.0959557
$$137$$ 7.52918 0.643261 0.321631 0.946865i $$-0.395769\pi$$
0.321631 + 0.946865i $$0.395769\pi$$
$$138$$ 3.11903 0.265509
$$139$$ 4.67526 0.396550 0.198275 0.980146i $$-0.436466\pi$$
0.198275 + 0.980146i $$0.436466\pi$$
$$140$$ −4.50973 −0.381141
$$141$$ 35.7336 3.00931
$$142$$ 7.11903 0.597415
$$143$$ −16.1721 −1.35238
$$144$$ 6.72833 0.560694
$$145$$ 8.23805 0.684133
$$146$$ −9.45665 −0.782638
$$147$$ −41.6004 −3.43114
$$148$$ −0.781399 −0.0642306
$$149$$ 7.52918 0.616814 0.308407 0.951254i $$-0.400204\pi$$
0.308407 + 0.951254i $$0.400204\pi$$
$$150$$ −3.11903 −0.254667
$$151$$ −13.3571 −1.08698 −0.543492 0.839414i $$-0.682898\pi$$
−0.543492 + 0.839414i $$0.682898\pi$$
$$152$$ 4.50973 0.365787
$$153$$ 7.52918 0.608698
$$154$$ 19.5615 1.57631
$$155$$ −1.72833 −0.138823
$$156$$ 11.6288 0.931045
$$157$$ 16.2381 1.29594 0.647969 0.761667i $$-0.275619\pi$$
0.647969 + 0.761667i $$0.275619\pi$$
$$158$$ −14.9133 −1.18644
$$159$$ 18.7142 1.48413
$$160$$ −1.00000 −0.0790569
$$161$$ −4.50973 −0.355416
$$162$$ 16.0854 1.26379
$$163$$ −3.29112 −0.257781 −0.128890 0.991659i $$-0.541142\pi$$
−0.128890 + 0.991659i $$0.541142\pi$$
$$164$$ 3.90043 0.304572
$$165$$ 13.5292 1.05325
$$166$$ 2.78140 0.215878
$$167$$ 22.9133 1.77309 0.886543 0.462647i $$-0.153100\pi$$
0.886543 + 0.462647i $$0.153100\pi$$
$$168$$ −14.0660 −1.08521
$$169$$ 0.900425 0.0692635
$$170$$ −1.11903 −0.0858254
$$171$$ 30.3429 2.32038
$$172$$ 8.00000 0.609994
$$173$$ 0.575681 0.0437683 0.0218841 0.999761i $$-0.493034\pi$$
0.0218841 + 0.999761i $$0.493034\pi$$
$$174$$ 25.6947 1.94791
$$175$$ 4.50973 0.340903
$$176$$ 4.33763 0.326961
$$177$$ 6.98055 0.524690
$$178$$ −7.69471 −0.576743
$$179$$ 5.01945 0.375171 0.187586 0.982248i $$-0.439934\pi$$
0.187586 + 0.982248i $$0.439934\pi$$
$$180$$ −6.72833 −0.501500
$$181$$ −11.5292 −0.856957 −0.428479 0.903552i $$-0.640950\pi$$
−0.428479 + 0.903552i $$0.640950\pi$$
$$182$$ −16.8137 −1.24632
$$183$$ −11.0920 −0.819942
$$184$$ −1.00000 −0.0737210
$$185$$ 0.781399 0.0574496
$$186$$ −5.39070 −0.395265
$$187$$ 4.85392 0.354954
$$188$$ −11.4567 −0.835562
$$189$$ −52.4425 −3.81463
$$190$$ −4.50973 −0.327170
$$191$$ −18.7142 −1.35411 −0.677055 0.735933i $$-0.736744\pi$$
−0.677055 + 0.735933i $$0.736744\pi$$
$$192$$ −3.11903 −0.225096
$$193$$ 23.4956 1.69125 0.845624 0.533780i $$-0.179229\pi$$
0.845624 + 0.533780i $$0.179229\pi$$
$$194$$ −0.642920 −0.0461590
$$195$$ −11.6288 −0.832752
$$196$$ 13.3376 0.952688
$$197$$ −18.1385 −1.29231 −0.646157 0.763205i $$-0.723625\pi$$
−0.646157 + 0.763205i $$0.723625\pi$$
$$198$$ 29.1850 2.07409
$$199$$ −23.2575 −1.64868 −0.824340 0.566094i $$-0.808454\pi$$
−0.824340 + 0.566094i $$0.808454\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −7.60170 −0.536183
$$202$$ −8.23805 −0.579627
$$203$$ −37.1514 −2.60751
$$204$$ −3.49027 −0.244368
$$205$$ −3.90043 −0.272418
$$206$$ 12.3376 0.859603
$$207$$ −6.72833 −0.467651
$$208$$ −3.72833 −0.258513
$$209$$ 19.5615 1.35310
$$210$$ 14.0660 0.970643
$$211$$ 4.34420 0.299067 0.149533 0.988757i $$-0.452223\pi$$
0.149533 + 0.988757i $$0.452223\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −22.2044 −1.52142
$$214$$ −15.9328 −1.08914
$$215$$ −8.00000 −0.545595
$$216$$ −11.6288 −0.791236
$$217$$ 7.79428 0.529110
$$218$$ −1.49027 −0.100934
$$219$$ 29.4956 1.99313
$$220$$ −4.33763 −0.292443
$$221$$ −4.17210 −0.280646
$$222$$ 2.43720 0.163574
$$223$$ 12.4761 0.835462 0.417731 0.908571i $$-0.362825\pi$$
0.417731 + 0.908571i $$0.362825\pi$$
$$224$$ 4.50973 0.301319
$$225$$ 6.72833 0.448555
$$226$$ −6.00000 −0.399114
$$227$$ 15.9328 1.05749 0.528747 0.848779i $$-0.322662\pi$$
0.528747 + 0.848779i $$0.322662\pi$$
$$228$$ −14.0660 −0.931541
$$229$$ −3.56280 −0.235436 −0.117718 0.993047i $$-0.537558\pi$$
−0.117718 + 0.993047i $$0.537558\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ −61.0129 −4.01435
$$232$$ −8.23805 −0.540855
$$233$$ −27.4956 −1.80129 −0.900647 0.434552i $$-0.856907\pi$$
−0.900647 + 0.434552i $$0.856907\pi$$
$$234$$ −25.0854 −1.63988
$$235$$ 11.4567 0.747350
$$236$$ −2.23805 −0.145685
$$237$$ 46.5150 3.02147
$$238$$ 5.04650 0.327116
$$239$$ 10.0389 0.649363 0.324681 0.945823i $$-0.394743\pi$$
0.324681 + 0.945823i $$0.394743\pi$$
$$240$$ 3.11903 0.201332
$$241$$ −23.6947 −1.52631 −0.763155 0.646215i $$-0.776351\pi$$
−0.763155 + 0.646215i $$0.776351\pi$$
$$242$$ 7.81502 0.502368
$$243$$ −15.2846 −0.980505
$$244$$ 3.55623 0.227664
$$245$$ −13.3376 −0.852110
$$246$$ −12.1655 −0.775646
$$247$$ −16.8137 −1.06983
$$248$$ 1.72833 0.109749
$$249$$ −8.67526 −0.549772
$$250$$ −1.00000 −0.0632456
$$251$$ 12.4425 0.785363 0.392681 0.919675i $$-0.371548\pi$$
0.392681 + 0.919675i $$0.371548\pi$$
$$252$$ 30.3429 1.91142
$$253$$ −4.33763 −0.272704
$$254$$ −0.675256 −0.0423693
$$255$$ 3.49027 0.218569
$$256$$ 1.00000 0.0625000
$$257$$ 5.45665 0.340377 0.170188 0.985412i $$-0.445562\pi$$
0.170188 + 0.985412i $$0.445562\pi$$
$$258$$ −24.9522 −1.55346
$$259$$ −3.52389 −0.218964
$$260$$ 3.72833 0.231221
$$261$$ −55.4283 −3.43093
$$262$$ −13.6947 −0.846062
$$263$$ 0.138479 0.00853895 0.00426948 0.999991i $$-0.498641\pi$$
0.00426948 + 0.999991i $$0.498641\pi$$
$$264$$ −13.5292 −0.832663
$$265$$ 6.00000 0.368577
$$266$$ 20.3376 1.24698
$$267$$ 24.0000 1.46878
$$268$$ 2.43720 0.148876
$$269$$ 14.6753 0.894766 0.447383 0.894342i $$-0.352356\pi$$
0.447383 + 0.894342i $$0.352356\pi$$
$$270$$ 11.6288 0.707703
$$271$$ −8.31058 −0.504832 −0.252416 0.967619i $$-0.581225\pi$$
−0.252416 + 0.967619i $$0.581225\pi$$
$$272$$ 1.11903 0.0678510
$$273$$ 52.4425 3.17396
$$274$$ 7.52918 0.454854
$$275$$ 4.33763 0.261569
$$276$$ 3.11903 0.187743
$$277$$ 12.9133 0.775886 0.387943 0.921683i $$-0.373186\pi$$
0.387943 + 0.921683i $$0.373186\pi$$
$$278$$ 4.67526 0.280403
$$279$$ 11.6288 0.696195
$$280$$ −4.50973 −0.269508
$$281$$ 2.67526 0.159592 0.0797962 0.996811i $$-0.474573\pi$$
0.0797962 + 0.996811i $$0.474573\pi$$
$$282$$ 35.7336 2.12791
$$283$$ 0.742495 0.0441367 0.0220684 0.999756i $$-0.492975\pi$$
0.0220684 + 0.999756i $$0.492975\pi$$
$$284$$ 7.11903 0.422437
$$285$$ 14.0660 0.833195
$$286$$ −16.1721 −0.956276
$$287$$ 17.5898 1.03830
$$288$$ 6.72833 0.396470
$$289$$ −15.7478 −0.926340
$$290$$ 8.23805 0.483755
$$291$$ 2.00528 0.117552
$$292$$ −9.45665 −0.553409
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −41.6004 −2.42619
$$295$$ 2.23805 0.130305
$$296$$ −0.781399 −0.0454179
$$297$$ −50.4412 −2.92690
$$298$$ 7.52918 0.436154
$$299$$ 3.72833 0.215615
$$300$$ −3.11903 −0.180077
$$301$$ 36.0778 2.07949
$$302$$ −13.3571 −0.768614
$$303$$ 25.6947 1.47612
$$304$$ 4.50973 0.258651
$$305$$ −3.55623 −0.203629
$$306$$ 7.52918 0.430414
$$307$$ 30.5084 1.74121 0.870604 0.491984i $$-0.163728\pi$$
0.870604 + 0.491984i $$0.163728\pi$$
$$308$$ 19.5615 1.11462
$$309$$ −38.4814 −2.18913
$$310$$ −1.72833 −0.0981624
$$311$$ 5.56280 0.315437 0.157719 0.987484i $$-0.449586\pi$$
0.157719 + 0.987484i $$0.449586\pi$$
$$312$$ 11.6288 0.658348
$$313$$ 4.07252 0.230193 0.115096 0.993354i $$-0.463282\pi$$
0.115096 + 0.993354i $$0.463282\pi$$
$$314$$ 16.2381 0.916366
$$315$$ −30.3429 −1.70963
$$316$$ −14.9133 −0.838939
$$317$$ −6.16553 −0.346291 −0.173145 0.984896i $$-0.555393\pi$$
−0.173145 + 0.984896i $$0.555393\pi$$
$$318$$ 18.7142 1.04944
$$319$$ −35.7336 −2.00070
$$320$$ −1.00000 −0.0559017
$$321$$ 49.6947 2.77369
$$322$$ −4.50973 −0.251317
$$323$$ 5.04650 0.280795
$$324$$ 16.0854 0.893634
$$325$$ −3.72833 −0.206810
$$326$$ −3.29112 −0.182279
$$327$$ 4.64820 0.257046
$$328$$ 3.90043 0.215365
$$329$$ −51.6664 −2.84846
$$330$$ 13.5292 0.744757
$$331$$ 27.5886 1.51640 0.758202 0.652019i $$-0.226078\pi$$
0.758202 + 0.652019i $$0.226078\pi$$
$$332$$ 2.78140 0.152649
$$333$$ −5.25751 −0.288110
$$334$$ 22.9133 1.25376
$$335$$ −2.43720 −0.133159
$$336$$ −14.0660 −0.767361
$$337$$ −17.4230 −0.949093 −0.474547 0.880230i $$-0.657388\pi$$
−0.474547 + 0.880230i $$0.657388\pi$$
$$338$$ 0.900425 0.0489767
$$339$$ 18.7142 1.01641
$$340$$ −1.11903 −0.0606877
$$341$$ 7.49684 0.405977
$$342$$ 30.3429 1.64076
$$343$$ 28.5810 1.54323
$$344$$ 8.00000 0.431331
$$345$$ −3.11903 −0.167923
$$346$$ 0.575681 0.0309488
$$347$$ 4.88097 0.262024 0.131012 0.991381i $$-0.458177\pi$$
0.131012 + 0.991381i $$0.458177\pi$$
$$348$$ 25.6947 1.37738
$$349$$ 24.0389 1.28677 0.643387 0.765542i $$-0.277529\pi$$
0.643387 + 0.765542i $$0.277529\pi$$
$$350$$ 4.50973 0.241055
$$351$$ 43.3558 2.31416
$$352$$ 4.33763 0.231196
$$353$$ −14.3442 −0.763464 −0.381732 0.924273i $$-0.624672\pi$$
−0.381732 + 0.924273i $$0.624672\pi$$
$$354$$ 6.98055 0.371012
$$355$$ −7.11903 −0.377839
$$356$$ −7.69471 −0.407819
$$357$$ −15.7402 −0.833059
$$358$$ 5.01945 0.265286
$$359$$ 26.7814 1.41347 0.706734 0.707479i $$-0.250168\pi$$
0.706734 + 0.707479i $$0.250168\pi$$
$$360$$ −6.72833 −0.354614
$$361$$ 1.33763 0.0704015
$$362$$ −11.5292 −0.605960
$$363$$ −24.3752 −1.27937
$$364$$ −16.8137 −0.881279
$$365$$ 9.45665 0.494984
$$366$$ −11.0920 −0.579787
$$367$$ −20.4761 −1.06884 −0.534422 0.845218i $$-0.679471\pi$$
−0.534422 + 0.845218i $$0.679471\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 26.2433 1.36617
$$370$$ 0.781399 0.0406230
$$371$$ −27.0584 −1.40480
$$372$$ −5.39070 −0.279495
$$373$$ −3.89386 −0.201616 −0.100808 0.994906i $$-0.532143\pi$$
−0.100808 + 0.994906i $$0.532143\pi$$
$$374$$ 4.85392 0.250990
$$375$$ 3.11903 0.161066
$$376$$ −11.4567 −0.590832
$$377$$ 30.7142 1.58186
$$378$$ −52.4425 −2.69735
$$379$$ −30.3765 −1.56034 −0.780169 0.625569i $$-0.784867\pi$$
−0.780169 + 0.625569i $$0.784867\pi$$
$$380$$ −4.50973 −0.231344
$$381$$ 2.10614 0.107901
$$382$$ −18.7142 −0.957500
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −3.11903 −0.159167
$$385$$ −19.5615 −0.996947
$$386$$ 23.4956 1.19589
$$387$$ 53.8266 2.73616
$$388$$ −0.642920 −0.0326393
$$389$$ −18.6818 −0.947206 −0.473603 0.880738i $$-0.657047\pi$$
−0.473603 + 0.880738i $$0.657047\pi$$
$$390$$ −11.6288 −0.588845
$$391$$ −1.11903 −0.0565916
$$392$$ 13.3376 0.673652
$$393$$ 42.7142 2.15464
$$394$$ −18.1385 −0.913803
$$395$$ 14.9133 0.750370
$$396$$ 29.1850 1.46660
$$397$$ −28.5757 −1.43417 −0.717086 0.696985i $$-0.754525\pi$$
−0.717086 + 0.696985i $$0.754525\pi$$
$$398$$ −23.2575 −1.16579
$$399$$ −63.4336 −3.17565
$$400$$ 1.00000 0.0500000
$$401$$ 12.1061 0.604552 0.302276 0.953220i $$-0.402254\pi$$
0.302276 + 0.953220i $$0.402254\pi$$
$$402$$ −7.60170 −0.379138
$$403$$ −6.44377 −0.320987
$$404$$ −8.23805 −0.409858
$$405$$ −16.0854 −0.799290
$$406$$ −37.1514 −1.84379
$$407$$ −3.38942 −0.168007
$$408$$ −3.49027 −0.172794
$$409$$ 25.2911 1.25057 0.625283 0.780398i $$-0.284984\pi$$
0.625283 + 0.780398i $$0.284984\pi$$
$$410$$ −3.90043 −0.192628
$$411$$ −23.4837 −1.15837
$$412$$ 12.3376 0.607831
$$413$$ −10.0930 −0.496644
$$414$$ −6.72833 −0.330679
$$415$$ −2.78140 −0.136533
$$416$$ −3.72833 −0.182796
$$417$$ −14.5822 −0.714096
$$418$$ 19.5615 0.956785
$$419$$ −17.3505 −0.847628 −0.423814 0.905749i $$-0.639309\pi$$
−0.423814 + 0.905749i $$0.639309\pi$$
$$420$$ 14.0660 0.686348
$$421$$ 21.4230 1.04409 0.522047 0.852916i $$-0.325168\pi$$
0.522047 + 0.852916i $$0.325168\pi$$
$$422$$ 4.34420 0.211472
$$423$$ −77.0841 −3.74796
$$424$$ −6.00000 −0.291386
$$425$$ 1.11903 0.0542808
$$426$$ −22.2044 −1.07581
$$427$$ 16.0376 0.776115
$$428$$ −15.9328 −0.770139
$$429$$ 50.4412 2.43532
$$430$$ −8.00000 −0.385794
$$431$$ 22.5822 1.08775 0.543874 0.839167i $$-0.316957\pi$$
0.543874 + 0.839167i $$0.316957\pi$$
$$432$$ −11.6288 −0.559489
$$433$$ 1.01417 0.0487378 0.0243689 0.999703i $$-0.492242\pi$$
0.0243689 + 0.999703i $$0.492242\pi$$
$$434$$ 7.79428 0.374138
$$435$$ −25.6947 −1.23197
$$436$$ −1.49027 −0.0713712
$$437$$ −4.50973 −0.215729
$$438$$ 29.4956 1.40935
$$439$$ −26.7478 −1.27660 −0.638301 0.769787i $$-0.720362\pi$$
−0.638301 + 0.769787i $$0.720362\pi$$
$$440$$ −4.33763 −0.206788
$$441$$ 89.7399 4.27333
$$442$$ −4.17210 −0.198446
$$443$$ 10.2044 0.484827 0.242414 0.970173i $$-0.422061\pi$$
0.242414 + 0.970173i $$0.422061\pi$$
$$444$$ 2.43720 0.115665
$$445$$ 7.69471 0.364764
$$446$$ 12.4761 0.590761
$$447$$ −23.4837 −1.11074
$$448$$ 4.50973 0.213065
$$449$$ 38.7867 1.83046 0.915228 0.402936i $$-0.132010\pi$$
0.915228 + 0.402936i $$0.132010\pi$$
$$450$$ 6.72833 0.317176
$$451$$ 16.9186 0.796665
$$452$$ −6.00000 −0.282216
$$453$$ 41.6611 1.95741
$$454$$ 15.9328 0.747762
$$455$$ 16.8137 0.788240
$$456$$ −14.0660 −0.658699
$$457$$ 34.9522 1.63500 0.817498 0.575932i $$-0.195361\pi$$
0.817498 + 0.575932i $$0.195361\pi$$
$$458$$ −3.56280 −0.166479
$$459$$ −13.0129 −0.607389
$$460$$ 1.00000 0.0466252
$$461$$ 16.3700 0.762425 0.381213 0.924487i $$-0.375507\pi$$
0.381213 + 0.924487i $$0.375507\pi$$
$$462$$ −61.0129 −2.83858
$$463$$ 29.2186 1.35790 0.678952 0.734183i $$-0.262435\pi$$
0.678952 + 0.734183i $$0.262435\pi$$
$$464$$ −8.23805 −0.382442
$$465$$ 5.39070 0.249988
$$466$$ −27.4956 −1.27371
$$467$$ 24.2770 1.12340 0.561702 0.827340i $$-0.310147\pi$$
0.561702 + 0.827340i $$0.310147\pi$$
$$468$$ −25.0854 −1.15957
$$469$$ 10.9911 0.507523
$$470$$ 11.4567 0.528456
$$471$$ −50.6469 −2.33369
$$472$$ −2.23805 −0.103015
$$473$$ 34.7010 1.59555
$$474$$ 46.5150 2.13651
$$475$$ 4.50973 0.206920
$$476$$ 5.04650 0.231306
$$477$$ −40.3700 −1.84841
$$478$$ 10.0389 0.459169
$$479$$ 24.6080 1.12437 0.562185 0.827012i $$-0.309961\pi$$
0.562185 + 0.827012i $$0.309961\pi$$
$$480$$ 3.11903 0.142363
$$481$$ 2.91331 0.132835
$$482$$ −23.6947 −1.07926
$$483$$ 14.0660 0.640023
$$484$$ 7.81502 0.355228
$$485$$ 0.642920 0.0291935
$$486$$ −15.2846 −0.693322
$$487$$ −30.2381 −1.37022 −0.685108 0.728441i $$-0.740245\pi$$
−0.685108 + 0.728441i $$0.740245\pi$$
$$488$$ 3.55623 0.160983
$$489$$ 10.2651 0.464204
$$490$$ −13.3376 −0.602533
$$491$$ 12.3311 0.556493 0.278246 0.960510i $$-0.410247\pi$$
0.278246 + 0.960510i $$0.410247\pi$$
$$492$$ −12.1655 −0.548464
$$493$$ −9.21860 −0.415185
$$494$$ −16.8137 −0.756486
$$495$$ −29.1850 −1.31177
$$496$$ 1.72833 0.0776042
$$497$$ 32.1049 1.44010
$$498$$ −8.67526 −0.388748
$$499$$ −26.9133 −1.20481 −0.602403 0.798192i $$-0.705790\pi$$
−0.602403 + 0.798192i $$0.705790\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −71.4672 −3.19292
$$502$$ 12.4425 0.555335
$$503$$ 20.5097 0.914483 0.457242 0.889342i $$-0.348837\pi$$
0.457242 + 0.889342i $$0.348837\pi$$
$$504$$ 30.3429 1.35158
$$505$$ 8.23805 0.366589
$$506$$ −4.33763 −0.192831
$$507$$ −2.80845 −0.124728
$$508$$ −0.675256 −0.0299597
$$509$$ −36.7142 −1.62733 −0.813663 0.581336i $$-0.802530\pi$$
−0.813663 + 0.581336i $$0.802530\pi$$
$$510$$ 3.49027 0.154552
$$511$$ −42.6469 −1.88659
$$512$$ 1.00000 0.0441942
$$513$$ −52.4425 −2.31539
$$514$$ 5.45665 0.240683
$$515$$ −12.3376 −0.543661
$$516$$ −24.9522 −1.09846
$$517$$ −49.6947 −2.18557
$$518$$ −3.52389 −0.154831
$$519$$ −1.79557 −0.0788166
$$520$$ 3.72833 0.163498
$$521$$ −4.91331 −0.215256 −0.107628 0.994191i $$-0.534326\pi$$
−0.107628 + 0.994191i $$0.534326\pi$$
$$522$$ −55.4283 −2.42603
$$523$$ −0.344196 −0.0150506 −0.00752531 0.999972i $$-0.502395\pi$$
−0.00752531 + 0.999972i $$0.502395\pi$$
$$524$$ −13.6947 −0.598256
$$525$$ −14.0660 −0.613889
$$526$$ 0.138479 0.00603795
$$527$$ 1.93404 0.0842483
$$528$$ −13.5292 −0.588782
$$529$$ 1.00000 0.0434783
$$530$$ 6.00000 0.260623
$$531$$ −15.0584 −0.653477
$$532$$ 20.3376 0.881748
$$533$$ −14.5421 −0.629887
$$534$$ 24.0000 1.03858
$$535$$ 15.9328 0.688833
$$536$$ 2.43720 0.105271
$$537$$ −15.6558 −0.675598
$$538$$ 14.6753 0.632695
$$539$$ 57.8537 2.49193
$$540$$ 11.6288 0.500422
$$541$$ −6.13191 −0.263631 −0.131816 0.991274i $$-0.542081\pi$$
−0.131816 + 0.991274i $$0.542081\pi$$
$$542$$ −8.31058 −0.356970
$$543$$ 35.9598 1.54318
$$544$$ 1.11903 0.0479779
$$545$$ 1.49027 0.0638363
$$546$$ 52.4425 2.24433
$$547$$ −9.18498 −0.392721 −0.196361 0.980532i $$-0.562912\pi$$
−0.196361 + 0.980532i $$0.562912\pi$$
$$548$$ 7.52918 0.321631
$$549$$ 23.9275 1.02120
$$550$$ 4.33763 0.184957
$$551$$ −37.1514 −1.58270
$$552$$ 3.11903 0.132755
$$553$$ −67.2549 −2.85997
$$554$$ 12.9133 0.548634
$$555$$ −2.43720 −0.103454
$$556$$ 4.67526 0.198275
$$557$$ −4.30529 −0.182421 −0.0912105 0.995832i $$-0.529074\pi$$
−0.0912105 + 0.995832i $$0.529074\pi$$
$$558$$ 11.6288 0.492284
$$559$$ −29.8266 −1.26153
$$560$$ −4.50973 −0.190571
$$561$$ −15.1395 −0.639191
$$562$$ 2.67526 0.112849
$$563$$ 11.1256 0.468888 0.234444 0.972130i $$-0.424673\pi$$
0.234444 + 0.972130i $$0.424673\pi$$
$$564$$ 35.7336 1.50466
$$565$$ 6.00000 0.252422
$$566$$ 0.742495 0.0312094
$$567$$ 72.5408 3.04643
$$568$$ 7.11903 0.298708
$$569$$ −16.0389 −0.672386 −0.336193 0.941793i $$-0.609139\pi$$
−0.336193 + 0.941793i $$0.609139\pi$$
$$570$$ 14.0660 0.589158
$$571$$ 17.9004 0.749109 0.374555 0.927205i $$-0.377796\pi$$
0.374555 + 0.927205i $$0.377796\pi$$
$$572$$ −16.1721 −0.676189
$$573$$ 58.3700 2.43844
$$574$$ 17.5898 0.734186
$$575$$ −1.00000 −0.0417029
$$576$$ 6.72833 0.280347
$$577$$ −9.12559 −0.379903 −0.189952 0.981793i $$-0.560833\pi$$
−0.189952 + 0.981793i $$0.560833\pi$$
$$578$$ −15.7478 −0.655021
$$579$$ −73.2833 −3.04555
$$580$$ 8.23805 0.342067
$$581$$ 12.5433 0.520386
$$582$$ 2.00528 0.0831217
$$583$$ −26.0258 −1.07788
$$584$$ −9.45665 −0.391319
$$585$$ 25.0854 1.03715
$$586$$ −6.00000 −0.247858
$$587$$ −33.6340 −1.38823 −0.694113 0.719866i $$-0.744203\pi$$
−0.694113 + 0.719866i $$0.744203\pi$$
$$588$$ −41.6004 −1.71557
$$589$$ 7.79428 0.321158
$$590$$ 2.23805 0.0921392
$$591$$ 56.5744 2.32716
$$592$$ −0.781399 −0.0321153
$$593$$ −17.4567 −0.716859 −0.358429 0.933557i $$-0.616688\pi$$
−0.358429 + 0.933557i $$0.616688\pi$$
$$594$$ −50.4412 −2.06963
$$595$$ −5.04650 −0.206886
$$596$$ 7.52918 0.308407
$$597$$ 72.5408 2.96890
$$598$$ 3.72833 0.152463
$$599$$ −11.5951 −0.473764 −0.236882 0.971538i $$-0.576126\pi$$
−0.236882 + 0.971538i $$0.576126\pi$$
$$600$$ −3.11903 −0.127334
$$601$$ −31.6611 −1.29148 −0.645741 0.763556i $$-0.723452\pi$$
−0.645741 + 0.763556i $$0.723452\pi$$
$$602$$ 36.0778 1.47042
$$603$$ 16.3983 0.667790
$$604$$ −13.3571 −0.543492
$$605$$ −7.81502 −0.317726
$$606$$ 25.6947 1.04378
$$607$$ −36.0778 −1.46435 −0.732177 0.681115i $$-0.761495\pi$$
−0.732177 + 0.681115i $$0.761495\pi$$
$$608$$ 4.50973 0.182894
$$609$$ 115.876 4.69554
$$610$$ −3.55623 −0.143988
$$611$$ 42.7142 1.72803
$$612$$ 7.52918 0.304349
$$613$$ −32.0389 −1.29404 −0.647020 0.762473i $$-0.723985\pi$$
−0.647020 + 0.762473i $$0.723985\pi$$
$$614$$ 30.5084 1.23122
$$615$$ 12.1655 0.490562
$$616$$ 19.5615 0.788156
$$617$$ −13.3960 −0.539302 −0.269651 0.962958i $$-0.586908\pi$$
−0.269651 + 0.962958i $$0.586908\pi$$
$$618$$ −38.4814 −1.54795
$$619$$ 37.1309 1.49242 0.746208 0.665713i $$-0.231872\pi$$
0.746208 + 0.665713i $$0.231872\pi$$
$$620$$ −1.72833 −0.0694113
$$621$$ 11.6288 0.466646
$$622$$ 5.56280 0.223048
$$623$$ −34.7010 −1.39027
$$624$$ 11.6288 0.465523
$$625$$ 1.00000 0.0400000
$$626$$ 4.07252 0.162771
$$627$$ −61.0129 −2.43662
$$628$$ 16.2381 0.647969
$$629$$ −0.874406 −0.0348648
$$630$$ −30.3429 −1.20889
$$631$$ 11.1125 0.442380 0.221190 0.975231i $$-0.429006\pi$$
0.221190 + 0.975231i $$0.429006\pi$$
$$632$$ −14.9133 −0.593220
$$633$$ −13.5497 −0.538551
$$634$$ −6.16553 −0.244864
$$635$$ 0.675256 0.0267967
$$636$$ 18.7142 0.742065
$$637$$ −49.7270 −1.97026
$$638$$ −35.7336 −1.41471
$$639$$ 47.8991 1.89486
$$640$$ −1.00000 −0.0395285
$$641$$ 12.3831 0.489103 0.244552 0.969636i $$-0.421359\pi$$
0.244552 + 0.969636i $$0.421359\pi$$
$$642$$ 49.6947 1.96129
$$643$$ −37.4956 −1.47868 −0.739340 0.673332i $$-0.764862\pi$$
−0.739340 + 0.673332i $$0.764862\pi$$
$$644$$ −4.50973 −0.177708
$$645$$ 24.9522 0.982492
$$646$$ 5.04650 0.198552
$$647$$ 14.5691 0.572771 0.286385 0.958114i $$-0.407546\pi$$
0.286385 + 0.958114i $$0.407546\pi$$
$$648$$ 16.0854 0.631894
$$649$$ −9.70784 −0.381066
$$650$$ −3.72833 −0.146237
$$651$$ −24.3106 −0.952807
$$652$$ −3.29112 −0.128890
$$653$$ −4.41672 −0.172840 −0.0864198 0.996259i $$-0.527543\pi$$
−0.0864198 + 0.996259i $$0.527543\pi$$
$$654$$ 4.64820 0.181759
$$655$$ 13.6947 0.535097
$$656$$ 3.90043 0.152286
$$657$$ −63.6275 −2.48234
$$658$$ −51.6664 −2.01416
$$659$$ −31.8655 −1.24130 −0.620652 0.784086i $$-0.713132\pi$$
−0.620652 + 0.784086i $$0.713132\pi$$
$$660$$ 13.5292 0.526623
$$661$$ 33.1190 1.28818 0.644090 0.764949i $$-0.277236\pi$$
0.644090 + 0.764949i $$0.277236\pi$$
$$662$$ 27.5886 1.07226
$$663$$ 13.0129 0.505379
$$664$$ 2.78140 0.107939
$$665$$ −20.3376 −0.788659
$$666$$ −5.25751 −0.203724
$$667$$ 8.23805 0.318979
$$668$$ 22.9133 0.886543
$$669$$ −38.9133 −1.50448
$$670$$ −2.43720 −0.0941574
$$671$$ 15.4256 0.595499
$$672$$ −14.0660 −0.542606
$$673$$ 19.3505 0.745907 0.372954 0.927850i $$-0.378345\pi$$
0.372954 + 0.927850i $$0.378345\pi$$
$$674$$ −17.4230 −0.671110
$$675$$ −11.6288 −0.447591
$$676$$ 0.900425 0.0346317
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 18.7142 0.718713
$$679$$ −2.89939 −0.111268
$$680$$ −1.11903 −0.0429127
$$681$$ −49.6947 −1.90431
$$682$$ 7.49684 0.287069
$$683$$ 38.3495 1.46740 0.733701 0.679472i $$-0.237791\pi$$
0.733701 + 0.679472i $$0.237791\pi$$
$$684$$ 30.3429 1.16019
$$685$$ −7.52918 −0.287675
$$686$$ 28.5810 1.09123
$$687$$ 11.1125 0.423967
$$688$$ 8.00000 0.304997
$$689$$ 22.3700 0.852228
$$690$$ −3.11903 −0.118739
$$691$$ −21.0195 −0.799618 −0.399809 0.916599i $$-0.630923\pi$$
−0.399809 + 0.916599i $$0.630923\pi$$
$$692$$ 0.575681 0.0218841
$$693$$ 131.616 4.99969
$$694$$ 4.88097 0.185279
$$695$$ −4.67526 −0.177343
$$696$$ 25.6947 0.973955
$$697$$ 4.36468 0.165324
$$698$$ 24.0389 0.909886
$$699$$ 85.7594 3.24372
$$700$$ 4.50973 0.170452
$$701$$ 19.3169 0.729589 0.364794 0.931088i $$-0.381139\pi$$
0.364794 + 0.931088i $$0.381139\pi$$
$$702$$ 43.3558 1.63636
$$703$$ −3.52389 −0.132906
$$704$$ 4.33763 0.163481
$$705$$ −35.7336 −1.34581
$$706$$ −14.3442 −0.539851
$$707$$ −37.1514 −1.39722
$$708$$ 6.98055 0.262345
$$709$$ 12.2315 0.459363 0.229682 0.973266i $$-0.426232\pi$$
0.229682 + 0.973266i $$0.426232\pi$$
$$710$$ −7.11903 −0.267172
$$711$$ −100.342 −3.76311
$$712$$ −7.69471 −0.288371
$$713$$ −1.72833 −0.0647264
$$714$$ −15.7402 −0.589061
$$715$$ 16.1721 0.604802
$$716$$ 5.01945 0.187586
$$717$$ −31.3116 −1.16935
$$718$$ 26.7814 0.999473
$$719$$ −40.6416 −1.51568 −0.757839 0.652442i $$-0.773745\pi$$
−0.757839 + 0.652442i $$0.773745\pi$$
$$720$$ −6.72833 −0.250750
$$721$$ 55.6393 2.07212
$$722$$ 1.33763 0.0497814
$$723$$ 73.9044 2.74854
$$724$$ −11.5292 −0.428479
$$725$$ −8.23805 −0.305954
$$726$$ −24.3752 −0.904650
$$727$$ −23.4501 −0.869716 −0.434858 0.900499i $$-0.643201\pi$$
−0.434858 + 0.900499i $$0.643201\pi$$
$$728$$ −16.8137 −0.623158
$$729$$ −0.583281 −0.0216030
$$730$$ 9.45665 0.350006
$$731$$ 8.95221 0.331110
$$732$$ −11.0920 −0.409971
$$733$$ −12.5150 −0.462252 −0.231126 0.972924i $$-0.574241\pi$$
−0.231126 + 0.972924i $$0.574241\pi$$
$$734$$ −20.4761 −0.755787
$$735$$ 41.6004 1.53445
$$736$$ −1.00000 −0.0368605
$$737$$ 10.5717 0.389413
$$738$$ 26.2433 0.966031
$$739$$ −21.3505 −0.785391 −0.392696 0.919668i $$-0.628457\pi$$
−0.392696 + 0.919668i $$0.628457\pi$$
$$740$$ 0.781399 0.0287248
$$741$$ 52.4425 1.92652
$$742$$ −27.0584 −0.993343
$$743$$ 24.9858 0.916641 0.458321 0.888787i $$-0.348451\pi$$
0.458321 + 0.888787i $$0.348451\pi$$
$$744$$ −5.39070 −0.197633
$$745$$ −7.52918 −0.275848
$$746$$ −3.89386 −0.142564
$$747$$ 18.7142 0.684715
$$748$$ 4.85392 0.177477
$$749$$ −71.8524 −2.62543
$$750$$ 3.11903 0.113891
$$751$$ −33.6275 −1.22708 −0.613542 0.789662i $$-0.710256\pi$$
−0.613542 + 0.789662i $$0.710256\pi$$
$$752$$ −11.4567 −0.417781
$$753$$ −38.8085 −1.41426
$$754$$ 30.7142 1.11854
$$755$$ 13.3571 0.486114
$$756$$ −52.4425 −1.90731
$$757$$ −37.1230 −1.34926 −0.674630 0.738156i $$-0.735697\pi$$
−0.674630 + 0.738156i $$0.735697\pi$$
$$758$$ −30.3765 −1.10333
$$759$$ 13.5292 0.491078
$$760$$ −4.50973 −0.163585
$$761$$ −3.87337 −0.140410 −0.0702048 0.997533i $$-0.522365\pi$$
−0.0702048 + 0.997533i $$0.522365\pi$$
$$762$$ 2.10614 0.0762975
$$763$$ −6.72073 −0.243307
$$764$$ −18.7142 −0.677055
$$765$$ −7.52918 −0.272218
$$766$$ 0 0
$$767$$ 8.34420 0.301291
$$768$$ −3.11903 −0.112548
$$769$$ 23.1645 0.835333 0.417667 0.908600i $$-0.362848\pi$$
0.417667 + 0.908600i $$0.362848\pi$$
$$770$$ −19.5615 −0.704948
$$771$$ −17.0195 −0.612941
$$772$$ 23.4956 0.845624
$$773$$ −8.78140 −0.315845 −0.157922 0.987452i $$-0.550480\pi$$
−0.157922 + 0.987452i $$0.550480\pi$$
$$774$$ 53.8266 1.93476
$$775$$ 1.72833 0.0620834
$$776$$ −0.642920 −0.0230795
$$777$$ 10.9911 0.394304
$$778$$ −18.6818 −0.669776
$$779$$ 17.5898 0.630222
$$780$$ −11.6288 −0.416376
$$781$$ 30.8797 1.10496
$$782$$ −1.11903 −0.0400163
$$783$$ 95.7983 3.42355
$$784$$ 13.3376 0.476344
$$785$$ −16.2381 −0.579561
$$786$$ 42.7142 1.52356
$$787$$ 49.6275 1.76903 0.884514 0.466513i $$-0.154490\pi$$
0.884514 + 0.466513i $$0.154490\pi$$
$$788$$ −18.1385 −0.646157
$$789$$ −0.431918 −0.0153767
$$790$$ 14.9133 0.530592
$$791$$ −27.0584 −0.962084
$$792$$ 29.1850 1.03704
$$793$$ −13.2588 −0.470833
$$794$$ −28.5757 −1.01411
$$795$$ −18.7142 −0.663723
$$796$$ −23.2575 −0.824340
$$797$$ 18.3311 0.649319 0.324660 0.945831i $$-0.394750\pi$$
0.324660 + 0.945831i $$0.394750\pi$$
$$798$$ −63.4336 −2.24553
$$799$$ −12.8203 −0.453550
$$800$$ 1.00000 0.0353553
$$801$$ −51.7725 −1.82929
$$802$$ 12.1061 0.427483
$$803$$ −41.0195 −1.44755
$$804$$ −7.60170 −0.268091
$$805$$ 4.50973 0.158947
$$806$$ −6.44377 −0.226972
$$807$$ −45.7725 −1.61127
$$808$$ −8.23805 −0.289814
$$809$$ −1.93933 −0.0681832 −0.0340916 0.999419i $$-0.510854\pi$$
−0.0340916 + 0.999419i $$0.510854\pi$$
$$810$$ −16.0854 −0.565184
$$811$$ −5.41775 −0.190243 −0.0951215 0.995466i $$-0.530324\pi$$
−0.0951215 + 0.995466i $$0.530324\pi$$
$$812$$ −37.1514 −1.30376
$$813$$ 25.9209 0.909086
$$814$$ −3.38942 −0.118799
$$815$$ 3.29112 0.115283
$$816$$ −3.49027 −0.122184
$$817$$ 36.0778 1.26220
$$818$$ 25.2911 0.884283
$$819$$ −113.128 −3.95302
$$820$$ −3.90043 −0.136209
$$821$$ 37.8655 1.32152 0.660758 0.750599i $$-0.270235\pi$$
0.660758 + 0.750599i $$0.270235\pi$$
$$822$$ −23.4837 −0.819088
$$823$$ 43.7336 1.52446 0.762229 0.647308i $$-0.224105\pi$$
0.762229 + 0.647308i $$0.224105\pi$$
$$824$$ 12.3376 0.429802
$$825$$ −13.5292 −0.471026
$$826$$ −10.0930 −0.351181
$$827$$ 28.1991 0.980581 0.490290 0.871559i $$-0.336891\pi$$
0.490290 + 0.871559i $$0.336891\pi$$
$$828$$ −6.72833 −0.233826
$$829$$ 1.12559 0.0390935 0.0195468 0.999809i $$-0.493778\pi$$
0.0195468 + 0.999809i $$0.493778\pi$$
$$830$$ −2.78140 −0.0965438
$$831$$ −40.2770 −1.39719
$$832$$ −3.72833 −0.129256
$$833$$ 14.9252 0.517126
$$834$$ −14.5822 −0.504942
$$835$$ −22.9133 −0.792948
$$836$$ 19.5615 0.676549
$$837$$ −20.0983 −0.694699
$$838$$ −17.3505 −0.599364
$$839$$ 39.9328 1.37863 0.689316 0.724461i $$-0.257911\pi$$
0.689316 + 0.724461i $$0.257911\pi$$
$$840$$ 14.0660 0.485322
$$841$$ 38.8655 1.34019
$$842$$ 21.4230 0.738287
$$843$$ −8.34420 −0.287389
$$844$$ 4.34420 0.149533
$$845$$ −0.900425 −0.0309756
$$846$$ −77.0841 −2.65021
$$847$$ 35.2436 1.21098
$$848$$ −6.00000 −0.206041
$$849$$ −2.31586 −0.0794801
$$850$$ 1.11903 0.0383823
$$851$$ 0.781399 0.0267860
$$852$$ −22.2044 −0.760711
$$853$$ 47.9921 1.64322 0.821610 0.570050i $$-0.193076\pi$$
0.821610 + 0.570050i $$0.193076\pi$$
$$854$$ 16.0376 0.548796
$$855$$ −30.3429 −1.03771
$$856$$ −15.9328 −0.544571
$$857$$ −43.4283 −1.48348 −0.741742 0.670686i $$-0.766000\pi$$
−0.741742 + 0.670686i $$0.766000\pi$$
$$858$$ 50.4412 1.72203
$$859$$ 32.5433 1.11036 0.555182 0.831729i $$-0.312648\pi$$
0.555182 + 0.831729i $$0.312648\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ −54.8632 −1.86973
$$862$$ 22.5822 0.769154
$$863$$ 46.1036 1.56938 0.784692 0.619886i $$-0.212821\pi$$
0.784692 + 0.619886i $$0.212821\pi$$
$$864$$ −11.6288 −0.395618
$$865$$ −0.575681 −0.0195738
$$866$$ 1.01417 0.0344628
$$867$$ 49.1177 1.66813
$$868$$ 7.79428 0.264555
$$869$$ −64.6884 −2.19440
$$870$$ −25.6947 −0.871132
$$871$$ −9.08669 −0.307891
$$872$$ −1.49027 −0.0504670
$$873$$ −4.32578 −0.146405
$$874$$ −4.50973 −0.152544
$$875$$ −4.50973 −0.152457
$$876$$ 29.4956 0.996563
$$877$$ −24.0996 −0.813785 −0.406892 0.913476i $$-0.633388\pi$$
−0.406892 + 0.913476i $$0.633388\pi$$
$$878$$ −26.7478 −0.902694
$$879$$ 18.7142 0.631213
$$880$$ −4.33763 −0.146221
$$881$$ 2.34420 0.0789780 0.0394890 0.999220i $$-0.487427\pi$$
0.0394890 + 0.999220i $$0.487427\pi$$
$$882$$ 89.7399 3.02170
$$883$$ −41.0505 −1.38146 −0.690730 0.723113i $$-0.742711\pi$$
−0.690730 + 0.723113i $$0.742711\pi$$
$$884$$ −4.17210 −0.140323
$$885$$ −6.98055 −0.234649
$$886$$ 10.2044 0.342825
$$887$$ 54.7788 1.83929 0.919647 0.392747i $$-0.128475\pi$$
0.919647 + 0.392747i $$0.128475\pi$$
$$888$$ 2.43720 0.0817872
$$889$$ −3.04522 −0.102133
$$890$$ 7.69471 0.257927
$$891$$ 69.7725 2.33747
$$892$$ 12.4761 0.417731
$$893$$ −51.6664 −1.72895
$$894$$ −23.4837 −0.785413
$$895$$ −5.01945 −0.167782
$$896$$ 4.50973 0.150659
$$897$$ −11.6288 −0.388273
$$898$$ 38.7867 1.29433
$$899$$ −14.2381 −0.474866
$$900$$ 6.72833 0.224278
$$901$$ −6.71416 −0.223681
$$902$$ 16.9186 0.563328
$$903$$ −112.528 −3.74469
$$904$$ −6.00000 −0.199557
$$905$$ 11.5292 0.383243
$$906$$ 41.6611 1.38410
$$907$$ −10.1061 −0.335569 −0.167784 0.985824i $$-0.553661\pi$$
−0.167784 + 0.985824i $$0.553661\pi$$
$$908$$ 15.9328 0.528747
$$909$$ −55.4283 −1.83844
$$910$$ 16.8137 0.557370
$$911$$ −25.4178 −0.842128 −0.421064 0.907031i $$-0.638343\pi$$
−0.421064 + 0.907031i $$0.638343\pi$$
$$912$$ −14.0660 −0.465770
$$913$$ 12.0647 0.399282
$$914$$ 34.9522 1.15612
$$915$$ 11.0920 0.366689
$$916$$ −3.56280 −0.117718
$$917$$ −61.7594 −2.03947
$$918$$ −13.0129 −0.429489
$$919$$ 23.6017 0.778548 0.389274 0.921122i $$-0.372726\pi$$
0.389274 + 0.921122i $$0.372726\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ −95.1566 −3.13552
$$922$$ 16.3700 0.539116
$$923$$ −26.5421 −0.873643
$$924$$ −61.0129 −2.00718
$$925$$ −0.781399 −0.0256922
$$926$$ 29.2186 0.960183
$$927$$ 83.0116 2.72646
$$928$$ −8.23805 −0.270427
$$929$$ −7.08669 −0.232507 −0.116253 0.993220i $$-0.537088\pi$$
−0.116253 + 0.993220i $$0.537088\pi$$
$$930$$ 5.39070 0.176768
$$931$$ 60.1490 1.97131
$$932$$ −27.4956 −0.900647
$$933$$ −17.3505 −0.568030
$$934$$ 24.2770 0.794366
$$935$$ −4.85392 −0.158740
$$936$$ −25.0854 −0.819942
$$937$$ 27.3169 0.892404 0.446202 0.894932i $$-0.352776\pi$$
0.446202 + 0.894932i $$0.352776\pi$$
$$938$$ 10.9911 0.358873
$$939$$ −12.7023 −0.414524
$$940$$ 11.4567 0.373675
$$941$$ 55.8979 1.82222 0.911109 0.412165i $$-0.135227\pi$$
0.911109 + 0.412165i $$0.135227\pi$$
$$942$$ −50.6469 −1.65017
$$943$$ −3.90043 −0.127015
$$944$$ −2.23805 −0.0728424
$$945$$ 52.4425 1.70595
$$946$$ 34.7010 1.12823
$$947$$ 37.5939 1.22164 0.610818 0.791771i $$-0.290841\pi$$
0.610818 + 0.791771i $$0.290841\pi$$
$$948$$ 46.5150 1.51074
$$949$$ 35.2575 1.14451
$$950$$ 4.50973 0.146315
$$951$$ 19.2305 0.623590
$$952$$ 5.04650 0.163558
$$953$$ −29.3828 −0.951804 −0.475902 0.879498i $$-0.657878\pi$$
−0.475902 + 0.879498i $$0.657878\pi$$
$$954$$ −40.3700 −1.30703
$$955$$ 18.7142 0.605576
$$956$$ 10.0389 0.324681
$$957$$ 111.454 3.60280
$$958$$ 24.6080 0.795049
$$959$$ 33.9545 1.09645
$$960$$ 3.11903 0.100666
$$961$$ −28.0129 −0.903641
$$962$$ 2.91331 0.0939289
$$963$$ −107.201 −3.45450
$$964$$ −23.6947 −0.763155
$$965$$ −23.4956 −0.756349
$$966$$ 14.0660 0.452565
$$967$$ −49.2292 −1.58310 −0.791552 0.611102i $$-0.790726\pi$$
−0.791552 + 0.611102i $$0.790726\pi$$
$$968$$ 7.81502 0.251184
$$969$$ −15.7402 −0.505647
$$970$$ 0.642920 0.0206429
$$971$$ 9.62347 0.308832 0.154416 0.988006i $$-0.450650\pi$$
0.154416 + 0.988006i $$0.450650\pi$$
$$972$$ −15.2846 −0.490252
$$973$$ 21.0841 0.675926
$$974$$ −30.2381 −0.968890
$$975$$ 11.6288 0.372418
$$976$$ 3.55623 0.113832
$$977$$ 18.9858 0.607411 0.303705 0.952766i $$-0.401776\pi$$
0.303705 + 0.952766i $$0.401776\pi$$
$$978$$ 10.2651 0.328242
$$979$$ −33.3768 −1.06673
$$980$$ −13.3376 −0.426055
$$981$$ −10.0271 −0.320139
$$982$$ 12.3311 0.393500
$$983$$ 4.33763 0.138349 0.0691744 0.997605i $$-0.477963\pi$$
0.0691744 + 0.997605i $$0.477963\pi$$
$$984$$ −12.1655 −0.387823
$$985$$ 18.1385 0.577940
$$986$$ −9.21860 −0.293580
$$987$$ 161.149 5.12942
$$988$$ −16.8137 −0.534916
$$989$$ −8.00000 −0.254385
$$990$$ −29.1850 −0.927560
$$991$$ 1.96766 0.0625049 0.0312524 0.999512i $$-0.490050\pi$$
0.0312524 + 0.999512i $$0.490050\pi$$
$$992$$ 1.72833 0.0548744
$$993$$ −86.0495 −2.73070
$$994$$ 32.1049 1.01830
$$995$$ 23.2575 0.737312
$$996$$ −8.67526 −0.274886
$$997$$ 3.96110 0.125449 0.0627246 0.998031i $$-0.480021\pi$$
0.0627246 + 0.998031i $$0.480021\pi$$
$$998$$ −26.9133 −0.851926
$$999$$ 9.08669 0.287490
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 230.2.a.d.1.1 3
3.2 odd 2 2070.2.a.z.1.3 3
4.3 odd 2 1840.2.a.r.1.3 3
5.2 odd 4 1150.2.b.j.599.6 6
5.3 odd 4 1150.2.b.j.599.1 6
5.4 even 2 1150.2.a.q.1.3 3
8.3 odd 2 7360.2.a.ce.1.1 3
8.5 even 2 7360.2.a.bz.1.3 3
20.19 odd 2 9200.2.a.cf.1.1 3
23.22 odd 2 5290.2.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 1.1 even 1 trivial
1150.2.a.q.1.3 3 5.4 even 2
1150.2.b.j.599.1 6 5.3 odd 4
1150.2.b.j.599.6 6 5.2 odd 4
1840.2.a.r.1.3 3 4.3 odd 2
2070.2.a.z.1.3 3 3.2 odd 2
5290.2.a.r.1.1 3 23.22 odd 2
7360.2.a.bz.1.3 3 8.5 even 2
7360.2.a.ce.1.1 3 8.3 odd 2
9200.2.a.cf.1.1 3 20.19 odd 2