Properties

Label 177.2.a.d.1.2
Level $177$
Weight $2$
Character 177.1
Self dual yes
Analytic conductor $1.413$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.254102 q^{2} -1.00000 q^{3} -1.93543 q^{4} +1.68133 q^{5} +0.254102 q^{6} +2.74590 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.254102 q^{2} -1.00000 q^{3} -1.93543 q^{4} +1.68133 q^{5} +0.254102 q^{6} +2.74590 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.427229 q^{10} +2.18953 q^{11} +1.93543 q^{12} +4.18953 q^{13} -0.697737 q^{14} -1.68133 q^{15} +3.61676 q^{16} -6.29809 q^{17} -0.254102 q^{18} +4.93543 q^{19} -3.25410 q^{20} -2.74590 q^{21} -0.556364 q^{22} +3.44364 q^{23} -1.00000 q^{24} -2.17313 q^{25} -1.06457 q^{26} -1.00000 q^{27} -5.31450 q^{28} -9.12497 q^{29} +0.427229 q^{30} -0.616763 q^{31} -2.91903 q^{32} -2.18953 q^{33} +1.60036 q^{34} +4.61676 q^{35} -1.93543 q^{36} +1.44364 q^{37} -1.25410 q^{38} -4.18953 q^{39} +1.68133 q^{40} +4.10856 q^{41} +0.697737 q^{42} +8.53579 q^{43} -4.23769 q^{44} +1.68133 q^{45} -0.875034 q^{46} -6.48763 q^{47} -3.61676 q^{48} +0.539958 q^{49} +0.552195 q^{50} +6.29809 q^{51} -8.10856 q^{52} -0.664924 q^{53} +0.254102 q^{54} +3.68133 q^{55} +2.74590 q^{56} -4.93543 q^{57} +2.31867 q^{58} +1.00000 q^{59} +3.25410 q^{60} -10.9958 q^{61} +0.156721 q^{62} +2.74590 q^{63} -6.49180 q^{64} +7.04399 q^{65} +0.556364 q^{66} -9.93126 q^{67} +12.1895 q^{68} -3.44364 q^{69} -1.17313 q^{70} +6.82687 q^{71} +1.00000 q^{72} +13.5040 q^{73} -0.366830 q^{74} +2.17313 q^{75} -9.55220 q^{76} +6.01224 q^{77} +1.06457 q^{78} -1.17313 q^{79} +6.08097 q^{80} +1.00000 q^{81} -1.04399 q^{82} -15.3421 q^{83} +5.31450 q^{84} -10.5892 q^{85} -2.16896 q^{86} +9.12497 q^{87} +2.18953 q^{88} +1.97942 q^{89} -0.427229 q^{90} +11.5040 q^{91} -6.66492 q^{92} +0.616763 q^{93} +1.64852 q^{94} +8.29809 q^{95} +2.91903 q^{96} +3.33508 q^{97} -0.137204 q^{98} +2.18953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + 3q^{8} + 3q^{9} + 5q^{10} - 2q^{11} - 2q^{12} + 4q^{13} + 8q^{14} + 2q^{15} - 4q^{16} + 3q^{17} + 7q^{19} - 9q^{20} - 9q^{21} - 11q^{22} + q^{23} - 3q^{24} - q^{25} - 11q^{26} - 3q^{27} + 9q^{28} - 11q^{29} - 5q^{30} + 13q^{31} - 4q^{32} + 2q^{33} - 7q^{34} - q^{35} + 2q^{36} - 5q^{37} - 3q^{38} - 4q^{39} - 2q^{40} - q^{41} - 8q^{42} + 6q^{43} - 15q^{44} - 2q^{45} - 19q^{46} + 11q^{47} + 4q^{48} + 14q^{49} - 21q^{50} - 3q^{51} - 11q^{52} + 2q^{53} + 4q^{55} + 9q^{56} - 7q^{57} + 14q^{58} + 3q^{59} + 9q^{60} - q^{61} - 2q^{62} + 9q^{63} - 21q^{64} + 11q^{66} + 10q^{67} + 28q^{68} - q^{69} + 2q^{70} + 26q^{71} + 3q^{72} + 7q^{73} - 19q^{74} + q^{75} - 6q^{76} - 17q^{77} + 11q^{78} + 2q^{79} + 23q^{80} + 3q^{81} + 18q^{82} - 3q^{83} - 9q^{84} - 35q^{85} + 31q^{86} + 11q^{87} - 2q^{88} - 23q^{89} + 5q^{90} + q^{91} - 16q^{92} - 13q^{93} + 4q^{94} + 3q^{95} + 4q^{96} + 14q^{97} + 51q^{98} - 2q^{99} + O(q^{100}) \)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.254102 −0.179677 −0.0898385 0.995956i \(-0.528635\pi\)
−0.0898385 + 0.995956i \(0.528635\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.93543 −0.967716
\(5\) 1.68133 0.751914 0.375957 0.926637i \(-0.377314\pi\)
0.375957 + 0.926637i \(0.377314\pi\)
\(6\) 0.254102 0.103737
\(7\) 2.74590 1.03785 0.518926 0.854819i \(-0.326332\pi\)
0.518926 + 0.854819i \(0.326332\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.427229 −0.135102
\(11\) 2.18953 0.660169 0.330085 0.943951i \(-0.392923\pi\)
0.330085 + 0.943951i \(0.392923\pi\)
\(12\) 1.93543 0.558711
\(13\) 4.18953 1.16197 0.580984 0.813915i \(-0.302668\pi\)
0.580984 + 0.813915i \(0.302668\pi\)
\(14\) −0.697737 −0.186478
\(15\) −1.68133 −0.434118
\(16\) 3.61676 0.904191
\(17\) −6.29809 −1.52751 −0.763756 0.645505i \(-0.776647\pi\)
−0.763756 + 0.645505i \(0.776647\pi\)
\(18\) −0.254102 −0.0598923
\(19\) 4.93543 1.13227 0.566133 0.824314i \(-0.308439\pi\)
0.566133 + 0.824314i \(0.308439\pi\)
\(20\) −3.25410 −0.727639
\(21\) −2.74590 −0.599204
\(22\) −0.556364 −0.118617
\(23\) 3.44364 0.718048 0.359024 0.933328i \(-0.383110\pi\)
0.359024 + 0.933328i \(0.383110\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.17313 −0.434625
\(26\) −1.06457 −0.208779
\(27\) −1.00000 −0.192450
\(28\) −5.31450 −1.00435
\(29\) −9.12497 −1.69446 −0.847232 0.531223i \(-0.821733\pi\)
−0.847232 + 0.531223i \(0.821733\pi\)
\(30\) 0.427229 0.0780010
\(31\) −0.616763 −0.110774 −0.0553870 0.998465i \(-0.517639\pi\)
−0.0553870 + 0.998465i \(0.517639\pi\)
\(32\) −2.91903 −0.516016
\(33\) −2.18953 −0.381149
\(34\) 1.60036 0.274459
\(35\) 4.61676 0.780375
\(36\) −1.93543 −0.322572
\(37\) 1.44364 0.237332 0.118666 0.992934i \(-0.462138\pi\)
0.118666 + 0.992934i \(0.462138\pi\)
\(38\) −1.25410 −0.203442
\(39\) −4.18953 −0.670862
\(40\) 1.68133 0.265842
\(41\) 4.10856 0.641649 0.320825 0.947139i \(-0.396040\pi\)
0.320825 + 0.947139i \(0.396040\pi\)
\(42\) 0.697737 0.107663
\(43\) 8.53579 1.30170 0.650848 0.759208i \(-0.274414\pi\)
0.650848 + 0.759208i \(0.274414\pi\)
\(44\) −4.23769 −0.638857
\(45\) 1.68133 0.250638
\(46\) −0.875034 −0.129017
\(47\) −6.48763 −0.946318 −0.473159 0.880977i \(-0.656886\pi\)
−0.473159 + 0.880977i \(0.656886\pi\)
\(48\) −3.61676 −0.522035
\(49\) 0.539958 0.0771368
\(50\) 0.552195 0.0780922
\(51\) 6.29809 0.881910
\(52\) −8.10856 −1.12445
\(53\) −0.664924 −0.0913343 −0.0456672 0.998957i \(-0.514541\pi\)
−0.0456672 + 0.998957i \(0.514541\pi\)
\(54\) 0.254102 0.0345789
\(55\) 3.68133 0.496391
\(56\) 2.74590 0.366936
\(57\) −4.93543 −0.653714
\(58\) 2.31867 0.304456
\(59\) 1.00000 0.130189
\(60\) 3.25410 0.420103
\(61\) −10.9958 −1.40787 −0.703936 0.710263i \(-0.748576\pi\)
−0.703936 + 0.710263i \(0.748576\pi\)
\(62\) 0.156721 0.0199035
\(63\) 2.74590 0.345951
\(64\) −6.49180 −0.811475
\(65\) 7.04399 0.873700
\(66\) 0.556364 0.0684837
\(67\) −9.93126 −1.21330 −0.606648 0.794970i \(-0.707486\pi\)
−0.606648 + 0.794970i \(0.707486\pi\)
\(68\) 12.1895 1.47820
\(69\) −3.44364 −0.414565
\(70\) −1.17313 −0.140216
\(71\) 6.82687 0.810201 0.405100 0.914272i \(-0.367237\pi\)
0.405100 + 0.914272i \(0.367237\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.5040 1.58053 0.790264 0.612767i \(-0.209943\pi\)
0.790264 + 0.612767i \(0.209943\pi\)
\(74\) −0.366830 −0.0426432
\(75\) 2.17313 0.250931
\(76\) −9.55220 −1.09571
\(77\) 6.01224 0.685158
\(78\) 1.06457 0.120539
\(79\) −1.17313 −0.131987 −0.0659936 0.997820i \(-0.521022\pi\)
−0.0659936 + 0.997820i \(0.521022\pi\)
\(80\) 6.08097 0.679874
\(81\) 1.00000 0.111111
\(82\) −1.04399 −0.115290
\(83\) −15.3421 −1.68401 −0.842006 0.539468i \(-0.818626\pi\)
−0.842006 + 0.539468i \(0.818626\pi\)
\(84\) 5.31450 0.579860
\(85\) −10.5892 −1.14856
\(86\) −2.16896 −0.233885
\(87\) 9.12497 0.978299
\(88\) 2.18953 0.233405
\(89\) 1.97942 0.209819 0.104909 0.994482i \(-0.466545\pi\)
0.104909 + 0.994482i \(0.466545\pi\)
\(90\) −0.427229 −0.0450339
\(91\) 11.5040 1.20595
\(92\) −6.66492 −0.694866
\(93\) 0.616763 0.0639553
\(94\) 1.64852 0.170032
\(95\) 8.29809 0.851366
\(96\) 2.91903 0.297922
\(97\) 3.33508 0.338626 0.169313 0.985562i \(-0.445845\pi\)
0.169313 + 0.985562i \(0.445845\pi\)
\(98\) −0.137204 −0.0138597
\(99\) 2.18953 0.220056
\(100\) 4.20594 0.420594
\(101\) −3.49180 −0.347447 −0.173723 0.984794i \(-0.555580\pi\)
−0.173723 + 0.984794i \(0.555580\pi\)
\(102\) −1.60036 −0.158459
\(103\) 8.21712 0.809657 0.404828 0.914393i \(-0.367331\pi\)
0.404828 + 0.914393i \(0.367331\pi\)
\(104\) 4.18953 0.410818
\(105\) −4.61676 −0.450550
\(106\) 0.168958 0.0164107
\(107\) 17.7899 1.71981 0.859907 0.510451i \(-0.170522\pi\)
0.859907 + 0.510451i \(0.170522\pi\)
\(108\) 1.93543 0.186237
\(109\) −18.2981 −1.75264 −0.876320 0.481730i \(-0.840009\pi\)
−0.876320 + 0.481730i \(0.840009\pi\)
\(110\) −0.935432 −0.0891900
\(111\) −1.44364 −0.137024
\(112\) 9.93126 0.938416
\(113\) −7.71414 −0.725686 −0.362843 0.931850i \(-0.618194\pi\)
−0.362843 + 0.931850i \(0.618194\pi\)
\(114\) 1.25410 0.117457
\(115\) 5.78989 0.539910
\(116\) 17.6608 1.63976
\(117\) 4.18953 0.387323
\(118\) −0.254102 −0.0233920
\(119\) −17.2939 −1.58533
\(120\) −1.68133 −0.153484
\(121\) −6.20594 −0.564176
\(122\) 2.79406 0.252962
\(123\) −4.10856 −0.370456
\(124\) 1.19370 0.107198
\(125\) −12.0604 −1.07871
\(126\) −0.697737 −0.0621594
\(127\) 6.18953 0.549232 0.274616 0.961554i \(-0.411449\pi\)
0.274616 + 0.961554i \(0.411449\pi\)
\(128\) 7.48763 0.661819
\(129\) −8.53579 −0.751534
\(130\) −1.78989 −0.156984
\(131\) −15.5798 −1.36121 −0.680606 0.732650i \(-0.738283\pi\)
−0.680606 + 0.732650i \(0.738283\pi\)
\(132\) 4.23769 0.368844
\(133\) 13.5522 1.17512
\(134\) 2.52355 0.218002
\(135\) −1.68133 −0.144706
\(136\) −6.29809 −0.540057
\(137\) 0.664924 0.0568083 0.0284041 0.999597i \(-0.490957\pi\)
0.0284041 + 0.999597i \(0.490957\pi\)
\(138\) 0.875034 0.0744878
\(139\) −18.7857 −1.59338 −0.796692 0.604385i \(-0.793419\pi\)
−0.796692 + 0.604385i \(0.793419\pi\)
\(140\) −8.93543 −0.755182
\(141\) 6.48763 0.546357
\(142\) −1.73472 −0.145574
\(143\) 9.17313 0.767095
\(144\) 3.61676 0.301397
\(145\) −15.3421 −1.27409
\(146\) −3.43140 −0.283985
\(147\) −0.539958 −0.0445349
\(148\) −2.79406 −0.229670
\(149\) −8.04816 −0.659331 −0.329666 0.944098i \(-0.606936\pi\)
−0.329666 + 0.944098i \(0.606936\pi\)
\(150\) −0.552195 −0.0450866
\(151\) −17.8503 −1.45264 −0.726318 0.687359i \(-0.758770\pi\)
−0.726318 + 0.687359i \(0.758770\pi\)
\(152\) 4.93543 0.400316
\(153\) −6.29809 −0.509171
\(154\) −1.52772 −0.123107
\(155\) −1.03698 −0.0832924
\(156\) 8.10856 0.649204
\(157\) 10.4395 0.833160 0.416580 0.909099i \(-0.363229\pi\)
0.416580 + 0.909099i \(0.363229\pi\)
\(158\) 0.298094 0.0237151
\(159\) 0.664924 0.0527319
\(160\) −4.90785 −0.387999
\(161\) 9.45587 0.745227
\(162\) −0.254102 −0.0199641
\(163\) −3.76231 −0.294686 −0.147343 0.989085i \(-0.547072\pi\)
−0.147343 + 0.989085i \(0.547072\pi\)
\(164\) −7.95184 −0.620934
\(165\) −3.68133 −0.286591
\(166\) 3.89845 0.302578
\(167\) 18.1086 1.40128 0.700641 0.713514i \(-0.252897\pi\)
0.700641 + 0.713514i \(0.252897\pi\)
\(168\) −2.74590 −0.211851
\(169\) 4.55220 0.350169
\(170\) 2.69073 0.206369
\(171\) 4.93543 0.377422
\(172\) −16.5204 −1.25967
\(173\) −9.66598 −0.734891 −0.367446 0.930045i \(-0.619768\pi\)
−0.367446 + 0.930045i \(0.619768\pi\)
\(174\) −2.31867 −0.175778
\(175\) −5.96719 −0.451077
\(176\) 7.91903 0.596919
\(177\) −1.00000 −0.0751646
\(178\) −0.502975 −0.0376996
\(179\) −8.95601 −0.669403 −0.334702 0.942324i \(-0.608636\pi\)
−0.334702 + 0.942324i \(0.608636\pi\)
\(180\) −3.25410 −0.242546
\(181\) −0.487628 −0.0362451 −0.0181225 0.999836i \(-0.505769\pi\)
−0.0181225 + 0.999836i \(0.505769\pi\)
\(182\) −2.92319 −0.216682
\(183\) 10.9958 0.812835
\(184\) 3.44364 0.253868
\(185\) 2.42723 0.178453
\(186\) −0.156721 −0.0114913
\(187\) −13.7899 −1.00842
\(188\) 12.5564 0.915767
\(189\) −2.74590 −0.199735
\(190\) −2.10856 −0.152971
\(191\) 5.07681 0.367345 0.183672 0.982988i \(-0.441201\pi\)
0.183672 + 0.982988i \(0.441201\pi\)
\(192\) 6.49180 0.468505
\(193\) 9.36266 0.673939 0.336970 0.941516i \(-0.390598\pi\)
0.336970 + 0.941516i \(0.390598\pi\)
\(194\) −0.847448 −0.0608433
\(195\) −7.04399 −0.504431
\(196\) −1.04505 −0.0746465
\(197\) −16.2499 −1.15776 −0.578880 0.815413i \(-0.696510\pi\)
−0.578880 + 0.815413i \(0.696510\pi\)
\(198\) −0.556364 −0.0395391
\(199\) 6.64958 0.471376 0.235688 0.971829i \(-0.424266\pi\)
0.235688 + 0.971829i \(0.424266\pi\)
\(200\) −2.17313 −0.153663
\(201\) 9.93126 0.700497
\(202\) 0.887271 0.0624282
\(203\) −25.0562 −1.75860
\(204\) −12.1895 −0.853438
\(205\) 6.90785 0.482465
\(206\) −2.08798 −0.145477
\(207\) 3.44364 0.239349
\(208\) 15.1526 1.05064
\(209\) 10.8063 0.747487
\(210\) 1.17313 0.0809535
\(211\) 13.1526 0.905459 0.452729 0.891648i \(-0.350450\pi\)
0.452729 + 0.891648i \(0.350450\pi\)
\(212\) 1.28692 0.0883857
\(213\) −6.82687 −0.467770
\(214\) −4.52044 −0.309011
\(215\) 14.3515 0.978763
\(216\) −1.00000 −0.0680414
\(217\) −1.69357 −0.114967
\(218\) 4.64958 0.314909
\(219\) −13.5040 −0.912518
\(220\) −7.12497 −0.480365
\(221\) −26.3861 −1.77492
\(222\) 0.366830 0.0246200
\(223\) −9.70892 −0.650157 −0.325079 0.945687i \(-0.605391\pi\)
−0.325079 + 0.945687i \(0.605391\pi\)
\(224\) −8.01535 −0.535548
\(225\) −2.17313 −0.144875
\(226\) 1.96018 0.130389
\(227\) 6.16896 0.409448 0.204724 0.978820i \(-0.434370\pi\)
0.204724 + 0.978820i \(0.434370\pi\)
\(228\) 9.55220 0.632610
\(229\) 25.8175 1.70607 0.853033 0.521856i \(-0.174760\pi\)
0.853033 + 0.521856i \(0.174760\pi\)
\(230\) −1.47122 −0.0970094
\(231\) −6.01224 −0.395576
\(232\) −9.12497 −0.599083
\(233\) 12.2499 0.802520 0.401260 0.915964i \(-0.368572\pi\)
0.401260 + 0.915964i \(0.368572\pi\)
\(234\) −1.06457 −0.0695930
\(235\) −10.9078 −0.711549
\(236\) −1.93543 −0.125986
\(237\) 1.17313 0.0762028
\(238\) 4.39442 0.284848
\(239\) −22.9477 −1.48436 −0.742181 0.670200i \(-0.766208\pi\)
−0.742181 + 0.670200i \(0.766208\pi\)
\(240\) −6.08097 −0.392525
\(241\) 4.67015 0.300831 0.150415 0.988623i \(-0.451939\pi\)
0.150415 + 0.988623i \(0.451939\pi\)
\(242\) 1.57694 0.101370
\(243\) −1.00000 −0.0641500
\(244\) 21.2817 1.36242
\(245\) 0.907847 0.0580002
\(246\) 1.04399 0.0665625
\(247\) 20.6772 1.31566
\(248\) −0.616763 −0.0391645
\(249\) 15.3421 0.972265
\(250\) 3.06457 0.193820
\(251\) 22.4067 1.41430 0.707148 0.707066i \(-0.249982\pi\)
0.707148 + 0.707066i \(0.249982\pi\)
\(252\) −5.31450 −0.334782
\(253\) 7.53996 0.474033
\(254\) −1.57277 −0.0986844
\(255\) 10.5892 0.663120
\(256\) 11.0810 0.692561
\(257\) 31.1372 1.94229 0.971143 0.238499i \(-0.0766554\pi\)
0.971143 + 0.238499i \(0.0766554\pi\)
\(258\) 2.16896 0.135033
\(259\) 3.96408 0.246316
\(260\) −13.6332 −0.845493
\(261\) −9.12497 −0.564821
\(262\) 3.95885 0.244578
\(263\) 24.9547 1.53877 0.769386 0.638784i \(-0.220562\pi\)
0.769386 + 0.638784i \(0.220562\pi\)
\(264\) −2.18953 −0.134757
\(265\) −1.11796 −0.0686755
\(266\) −3.44364 −0.211143
\(267\) −1.97942 −0.121139
\(268\) 19.2213 1.17413
\(269\) 5.99477 0.365508 0.182754 0.983159i \(-0.441499\pi\)
0.182754 + 0.983159i \(0.441499\pi\)
\(270\) 0.427229 0.0260003
\(271\) −1.97241 −0.119816 −0.0599078 0.998204i \(-0.519081\pi\)
−0.0599078 + 0.998204i \(0.519081\pi\)
\(272\) −22.7787 −1.38116
\(273\) −11.5040 −0.696256
\(274\) −0.168958 −0.0102071
\(275\) −4.75814 −0.286926
\(276\) 6.66492 0.401181
\(277\) −2.53579 −0.152361 −0.0761804 0.997094i \(-0.524272\pi\)
−0.0761804 + 0.997094i \(0.524272\pi\)
\(278\) 4.77348 0.286295
\(279\) −0.616763 −0.0369246
\(280\) 4.61676 0.275904
\(281\) −18.8545 −1.12476 −0.562381 0.826878i \(-0.690115\pi\)
−0.562381 + 0.826878i \(0.690115\pi\)
\(282\) −1.64852 −0.0981678
\(283\) −5.78989 −0.344173 −0.172087 0.985082i \(-0.555051\pi\)
−0.172087 + 0.985082i \(0.555051\pi\)
\(284\) −13.2130 −0.784044
\(285\) −8.29809 −0.491537
\(286\) −2.33091 −0.137829
\(287\) 11.2817 0.665937
\(288\) −2.91903 −0.172005
\(289\) 22.6660 1.33329
\(290\) 3.89845 0.228925
\(291\) −3.33508 −0.195506
\(292\) −26.1361 −1.52950
\(293\) 7.73472 0.451867 0.225934 0.974143i \(-0.427457\pi\)
0.225934 + 0.974143i \(0.427457\pi\)
\(294\) 0.137204 0.00800191
\(295\) 1.68133 0.0978909
\(296\) 1.44364 0.0839096
\(297\) −2.18953 −0.127050
\(298\) 2.04505 0.118467
\(299\) 14.4272 0.834348
\(300\) −4.20594 −0.242830
\(301\) 23.4384 1.35097
\(302\) 4.53579 0.261005
\(303\) 3.49180 0.200598
\(304\) 17.8503 1.02378
\(305\) −18.4876 −1.05860
\(306\) 1.60036 0.0914863
\(307\) 16.4999 0.941697 0.470849 0.882214i \(-0.343948\pi\)
0.470849 + 0.882214i \(0.343948\pi\)
\(308\) −11.6363 −0.663039
\(309\) −8.21712 −0.467456
\(310\) 0.263499 0.0149657
\(311\) 24.9354 1.41396 0.706979 0.707234i \(-0.250057\pi\)
0.706979 + 0.707234i \(0.250057\pi\)
\(312\) −4.18953 −0.237186
\(313\) −24.1690 −1.36611 −0.683055 0.730367i \(-0.739349\pi\)
−0.683055 + 0.730367i \(0.739349\pi\)
\(314\) −2.65269 −0.149700
\(315\) 4.61676 0.260125
\(316\) 2.27051 0.127726
\(317\) −18.1812 −1.02116 −0.510579 0.859831i \(-0.670569\pi\)
−0.510579 + 0.859831i \(0.670569\pi\)
\(318\) −0.168958 −0.00947471
\(319\) −19.9794 −1.11863
\(320\) −10.9149 −0.610159
\(321\) −17.7899 −0.992935
\(322\) −2.40275 −0.133900
\(323\) −31.0838 −1.72955
\(324\) −1.93543 −0.107524
\(325\) −9.10439 −0.505021
\(326\) 0.956008 0.0529484
\(327\) 18.2981 1.01189
\(328\) 4.10856 0.226857
\(329\) −17.8144 −0.982138
\(330\) 0.935432 0.0514939
\(331\) −14.6290 −0.804083 −0.402041 0.915622i \(-0.631699\pi\)
−0.402041 + 0.915622i \(0.631699\pi\)
\(332\) 29.6936 1.62965
\(333\) 1.44364 0.0791107
\(334\) −4.60142 −0.251778
\(335\) −16.6977 −0.912295
\(336\) −9.93126 −0.541795
\(337\) −19.1526 −1.04331 −0.521653 0.853158i \(-0.674684\pi\)
−0.521653 + 0.853158i \(0.674684\pi\)
\(338\) −1.15672 −0.0629173
\(339\) 7.71414 0.418975
\(340\) 20.4946 1.11148
\(341\) −1.35042 −0.0731295
\(342\) −1.25410 −0.0678141
\(343\) −17.7386 −0.957795
\(344\) 8.53579 0.460219
\(345\) −5.78989 −0.311717
\(346\) 2.45614 0.132043
\(347\) 29.4301 1.57989 0.789944 0.613178i \(-0.210109\pi\)
0.789944 + 0.613178i \(0.210109\pi\)
\(348\) −17.6608 −0.946716
\(349\) 28.9065 1.54733 0.773665 0.633595i \(-0.218421\pi\)
0.773665 + 0.633595i \(0.218421\pi\)
\(350\) 1.51627 0.0810482
\(351\) −4.18953 −0.223621
\(352\) −6.39131 −0.340658
\(353\) 0.0481609 0.00256335 0.00128167 0.999999i \(-0.499592\pi\)
0.00128167 + 0.999999i \(0.499592\pi\)
\(354\) 0.254102 0.0135054
\(355\) 11.4782 0.609201
\(356\) −3.83104 −0.203045
\(357\) 17.2939 0.915292
\(358\) 2.27574 0.120276
\(359\) 11.7417 0.619705 0.309852 0.950785i \(-0.399720\pi\)
0.309852 + 0.950785i \(0.399720\pi\)
\(360\) 1.68133 0.0886139
\(361\) 5.35849 0.282026
\(362\) 0.123907 0.00651241
\(363\) 6.20594 0.325727
\(364\) −22.2653 −1.16702
\(365\) 22.7047 1.18842
\(366\) −2.79406 −0.146048
\(367\) −7.04399 −0.367693 −0.183847 0.982955i \(-0.558855\pi\)
−0.183847 + 0.982955i \(0.558855\pi\)
\(368\) 12.4548 0.649252
\(369\) 4.10856 0.213883
\(370\) −0.616763 −0.0320640
\(371\) −1.82581 −0.0947915
\(372\) −1.19370 −0.0618906
\(373\) 31.5316 1.63265 0.816323 0.577596i \(-0.196009\pi\)
0.816323 + 0.577596i \(0.196009\pi\)
\(374\) 3.50403 0.181189
\(375\) 12.0604 0.622796
\(376\) −6.48763 −0.334574
\(377\) −38.2294 −1.96891
\(378\) 0.697737 0.0358877
\(379\) −10.5030 −0.539502 −0.269751 0.962930i \(-0.586941\pi\)
−0.269751 + 0.962930i \(0.586941\pi\)
\(380\) −16.0604 −0.823881
\(381\) −6.18953 −0.317099
\(382\) −1.29002 −0.0660034
\(383\) 23.3023 1.19069 0.595345 0.803470i \(-0.297015\pi\)
0.595345 + 0.803470i \(0.297015\pi\)
\(384\) −7.48763 −0.382101
\(385\) 10.1086 0.515180
\(386\) −2.37907 −0.121091
\(387\) 8.53579 0.433899
\(388\) −6.45481 −0.327694
\(389\) −33.4835 −1.69768 −0.848839 0.528651i \(-0.822698\pi\)
−0.848839 + 0.528651i \(0.822698\pi\)
\(390\) 1.78989 0.0906346
\(391\) −21.6883 −1.09683
\(392\) 0.539958 0.0272720
\(393\) 15.5798 0.785896
\(394\) 4.12914 0.208023
\(395\) −1.97241 −0.0992430
\(396\) −4.23769 −0.212952
\(397\) −10.6977 −0.536904 −0.268452 0.963293i \(-0.586512\pi\)
−0.268452 + 0.963293i \(0.586512\pi\)
\(398\) −1.68967 −0.0846954
\(399\) −13.5522 −0.678458
\(400\) −7.85969 −0.392984
\(401\) −23.3145 −1.16427 −0.582135 0.813092i \(-0.697783\pi\)
−0.582135 + 0.813092i \(0.697783\pi\)
\(402\) −2.52355 −0.125863
\(403\) −2.58395 −0.128716
\(404\) 6.75814 0.336230
\(405\) 1.68133 0.0835460
\(406\) 6.36683 0.315980
\(407\) 3.16089 0.156679
\(408\) 6.29809 0.311802
\(409\) −8.65970 −0.428194 −0.214097 0.976812i \(-0.568681\pi\)
−0.214097 + 0.976812i \(0.568681\pi\)
\(410\) −1.75530 −0.0866879
\(411\) −0.664924 −0.0327983
\(412\) −15.9037 −0.783518
\(413\) 2.74590 0.135117
\(414\) −0.875034 −0.0430056
\(415\) −25.7951 −1.26623
\(416\) −12.2294 −0.599594
\(417\) 18.7857 0.919941
\(418\) −2.74590 −0.134306
\(419\) 4.63734 0.226549 0.113274 0.993564i \(-0.463866\pi\)
0.113274 + 0.993564i \(0.463866\pi\)
\(420\) 8.93543 0.436004
\(421\) 1.68133 0.0819430 0.0409715 0.999160i \(-0.486955\pi\)
0.0409715 + 0.999160i \(0.486955\pi\)
\(422\) −3.34209 −0.162690
\(423\) −6.48763 −0.315439
\(424\) −0.664924 −0.0322916
\(425\) 13.6866 0.663896
\(426\) 1.73472 0.0840475
\(427\) −30.1934 −1.46116
\(428\) −34.4311 −1.66429
\(429\) −9.17313 −0.442883
\(430\) −3.64674 −0.175861
\(431\) 11.5040 0.554130 0.277065 0.960851i \(-0.410638\pi\)
0.277065 + 0.960851i \(0.410638\pi\)
\(432\) −3.61676 −0.174012
\(433\) 24.5069 1.17773 0.588863 0.808233i \(-0.299576\pi\)
0.588863 + 0.808233i \(0.299576\pi\)
\(434\) 0.430339 0.0206569
\(435\) 15.3421 0.735597
\(436\) 35.4147 1.69606
\(437\) 16.9958 0.813021
\(438\) 3.43140 0.163959
\(439\) −5.63840 −0.269106 −0.134553 0.990906i \(-0.542960\pi\)
−0.134553 + 0.990906i \(0.542960\pi\)
\(440\) 3.68133 0.175501
\(441\) 0.539958 0.0257123
\(442\) 6.70475 0.318912
\(443\) −38.0070 −1.80577 −0.902884 0.429885i \(-0.858554\pi\)
−0.902884 + 0.429885i \(0.858554\pi\)
\(444\) 2.79406 0.132600
\(445\) 3.32807 0.157765
\(446\) 2.46705 0.116818
\(447\) 8.04816 0.380665
\(448\) −17.8258 −0.842191
\(449\) −11.0357 −0.520805 −0.260402 0.965500i \(-0.583855\pi\)
−0.260402 + 0.965500i \(0.583855\pi\)
\(450\) 0.552195 0.0260307
\(451\) 8.99583 0.423597
\(452\) 14.9302 0.702258
\(453\) 17.8503 0.838680
\(454\) −1.56754 −0.0735684
\(455\) 19.3421 0.906771
\(456\) −4.93543 −0.231123
\(457\) −32.3913 −1.51520 −0.757601 0.652718i \(-0.773629\pi\)
−0.757601 + 0.652718i \(0.773629\pi\)
\(458\) −6.56026 −0.306541
\(459\) 6.29809 0.293970
\(460\) −11.2059 −0.522480
\(461\) 34.9599 1.62825 0.814123 0.580693i \(-0.197218\pi\)
0.814123 + 0.580693i \(0.197218\pi\)
\(462\) 1.52772 0.0710760
\(463\) −22.8063 −1.05990 −0.529949 0.848029i \(-0.677789\pi\)
−0.529949 + 0.848029i \(0.677789\pi\)
\(464\) −33.0028 −1.53212
\(465\) 1.03698 0.0480889
\(466\) −3.11273 −0.144194
\(467\) −36.2416 −1.67706 −0.838530 0.544855i \(-0.816585\pi\)
−0.838530 + 0.544855i \(0.816585\pi\)
\(468\) −8.10856 −0.374818
\(469\) −27.2702 −1.25922
\(470\) 2.77170 0.127849
\(471\) −10.4395 −0.481025
\(472\) 1.00000 0.0460287
\(473\) 18.6894 0.859340
\(474\) −0.298094 −0.0136919
\(475\) −10.7253 −0.492112
\(476\) 33.4712 1.53415
\(477\) −0.664924 −0.0304448
\(478\) 5.83104 0.266706
\(479\) 17.0562 0.779319 0.389660 0.920959i \(-0.372593\pi\)
0.389660 + 0.920959i \(0.372593\pi\)
\(480\) 4.90785 0.224012
\(481\) 6.04816 0.275772
\(482\) −1.18669 −0.0540524
\(483\) −9.45587 −0.430257
\(484\) 12.0112 0.545963
\(485\) 5.60737 0.254617
\(486\) 0.254102 0.0115263
\(487\) 15.3473 0.695453 0.347727 0.937596i \(-0.386954\pi\)
0.347727 + 0.937596i \(0.386954\pi\)
\(488\) −10.9958 −0.497758
\(489\) 3.76231 0.170137
\(490\) −0.230685 −0.0104213
\(491\) 18.0122 0.812881 0.406440 0.913677i \(-0.366770\pi\)
0.406440 + 0.913677i \(0.366770\pi\)
\(492\) 7.95184 0.358497
\(493\) 57.4699 2.58831
\(494\) −5.25410 −0.236393
\(495\) 3.68133 0.165464
\(496\) −2.23069 −0.100161
\(497\) 18.7459 0.840868
\(498\) −3.89845 −0.174694
\(499\) −21.6074 −0.967279 −0.483639 0.875267i \(-0.660685\pi\)
−0.483639 + 0.875267i \(0.660685\pi\)
\(500\) 23.3421 1.04389
\(501\) −18.1086 −0.809031
\(502\) −5.69357 −0.254116
\(503\) 7.76231 0.346104 0.173052 0.984913i \(-0.444637\pi\)
0.173052 + 0.984913i \(0.444637\pi\)
\(504\) 2.74590 0.122312
\(505\) −5.87086 −0.261250
\(506\) −1.91592 −0.0851729
\(507\) −4.55220 −0.202170
\(508\) −11.9794 −0.531501
\(509\) 40.7704 1.80712 0.903558 0.428467i \(-0.140946\pi\)
0.903558 + 0.428467i \(0.140946\pi\)
\(510\) −2.69073 −0.119147
\(511\) 37.0807 1.64035
\(512\) −17.7909 −0.786256
\(513\) −4.93543 −0.217905
\(514\) −7.91202 −0.348984
\(515\) 13.8157 0.608792
\(516\) 16.5204 0.727272
\(517\) −14.2049 −0.624730
\(518\) −1.00728 −0.0442573
\(519\) 9.66598 0.424290
\(520\) 7.04399 0.308899
\(521\) 41.0427 1.79811 0.899056 0.437834i \(-0.144254\pi\)
0.899056 + 0.437834i \(0.144254\pi\)
\(522\) 2.31867 0.101485
\(523\) −22.5686 −0.986856 −0.493428 0.869787i \(-0.664256\pi\)
−0.493428 + 0.869787i \(0.664256\pi\)
\(524\) 30.1536 1.31727
\(525\) 5.96719 0.260429
\(526\) −6.34103 −0.276482
\(527\) 3.88443 0.169208
\(528\) −7.91903 −0.344631
\(529\) −11.1414 −0.484408
\(530\) 0.284075 0.0123394
\(531\) 1.00000 0.0433963
\(532\) −26.2294 −1.13719
\(533\) 17.2130 0.745576
\(534\) 0.502975 0.0217659
\(535\) 29.9107 1.29315
\(536\) −9.93126 −0.428965
\(537\) 8.95601 0.386480
\(538\) −1.52328 −0.0656733
\(539\) 1.18226 0.0509233
\(540\) 3.25410 0.140034
\(541\) 35.0357 1.50630 0.753150 0.657849i \(-0.228533\pi\)
0.753150 + 0.657849i \(0.228533\pi\)
\(542\) 0.501194 0.0215281
\(543\) 0.487628 0.0209261
\(544\) 18.3843 0.788220
\(545\) −30.7651 −1.31783
\(546\) 2.92319 0.125101
\(547\) 3.70191 0.158282 0.0791410 0.996863i \(-0.474782\pi\)
0.0791410 + 0.996863i \(0.474782\pi\)
\(548\) −1.28692 −0.0549743
\(549\) −10.9958 −0.469291
\(550\) 1.20905 0.0515541
\(551\) −45.0357 −1.91858
\(552\) −3.44364 −0.146571
\(553\) −3.22129 −0.136983
\(554\) 0.644348 0.0273757
\(555\) −2.42723 −0.103030
\(556\) 36.3585 1.54194
\(557\) −16.7253 −0.708675 −0.354337 0.935118i \(-0.615294\pi\)
−0.354337 + 0.935118i \(0.615294\pi\)
\(558\) 0.156721 0.00663451
\(559\) 35.7610 1.51253
\(560\) 16.6977 0.705608
\(561\) 13.7899 0.582210
\(562\) 4.79095 0.202094
\(563\) 24.9149 1.05004 0.525018 0.851091i \(-0.324059\pi\)
0.525018 + 0.851091i \(0.324059\pi\)
\(564\) −12.5564 −0.528718
\(565\) −12.9700 −0.545653
\(566\) 1.47122 0.0618400
\(567\) 2.74590 0.115317
\(568\) 6.82687 0.286449
\(569\) 42.4548 1.77980 0.889899 0.456157i \(-0.150775\pi\)
0.889899 + 0.456157i \(0.150775\pi\)
\(570\) 2.10856 0.0883178
\(571\) −32.6977 −1.36836 −0.684179 0.729314i \(-0.739839\pi\)
−0.684179 + 0.729314i \(0.739839\pi\)
\(572\) −17.7540 −0.742331
\(573\) −5.07681 −0.212087
\(574\) −2.86670 −0.119654
\(575\) −7.48346 −0.312082
\(576\) −6.49180 −0.270492
\(577\) −13.2716 −0.552503 −0.276251 0.961085i \(-0.589092\pi\)
−0.276251 + 0.961085i \(0.589092\pi\)
\(578\) −5.75946 −0.239562
\(579\) −9.36266 −0.389099
\(580\) 29.6936 1.23296
\(581\) −42.1278 −1.74776
\(582\) 0.847448 0.0351279
\(583\) −1.45587 −0.0602961
\(584\) 13.5040 0.558801
\(585\) 7.04399 0.291233
\(586\) −1.96541 −0.0811901
\(587\) −14.4067 −0.594626 −0.297313 0.954780i \(-0.596091\pi\)
−0.297313 + 0.954780i \(0.596091\pi\)
\(588\) 1.04505 0.0430972
\(589\) −3.04399 −0.125426
\(590\) −0.427229 −0.0175887
\(591\) 16.2499 0.668433
\(592\) 5.22129 0.214594
\(593\) −1.85863 −0.0763247 −0.0381623 0.999272i \(-0.512150\pi\)
−0.0381623 + 0.999272i \(0.512150\pi\)
\(594\) 0.556364 0.0228279
\(595\) −29.0768 −1.19203
\(596\) 15.5767 0.638045
\(597\) −6.64958 −0.272149
\(598\) −3.66598 −0.149913
\(599\) −20.2171 −0.826049 −0.413025 0.910720i \(-0.635528\pi\)
−0.413025 + 0.910720i \(0.635528\pi\)
\(600\) 2.17313 0.0887175
\(601\) −29.6608 −1.20989 −0.604944 0.796268i \(-0.706804\pi\)
−0.604944 + 0.796268i \(0.706804\pi\)
\(602\) −5.95574 −0.242738
\(603\) −9.93126 −0.404432
\(604\) 34.5480 1.40574
\(605\) −10.4342 −0.424212
\(606\) −0.887271 −0.0360429
\(607\) 8.06040 0.327161 0.163581 0.986530i \(-0.447696\pi\)
0.163581 + 0.986530i \(0.447696\pi\)
\(608\) −14.4067 −0.584267
\(609\) 25.0562 1.01533
\(610\) 4.69774 0.190206
\(611\) −27.1801 −1.09959
\(612\) 12.1895 0.492733
\(613\) −35.7886 −1.44549 −0.722743 0.691117i \(-0.757119\pi\)
−0.722743 + 0.691117i \(0.757119\pi\)
\(614\) −4.19264 −0.169201
\(615\) −6.90785 −0.278551
\(616\) 6.01224 0.242240
\(617\) 21.7159 0.874250 0.437125 0.899401i \(-0.355997\pi\)
0.437125 + 0.899401i \(0.355997\pi\)
\(618\) 2.08798 0.0839910
\(619\) −15.3679 −0.617688 −0.308844 0.951113i \(-0.599942\pi\)
−0.308844 + 0.951113i \(0.599942\pi\)
\(620\) 2.00701 0.0806034
\(621\) −3.44364 −0.138188
\(622\) −6.33614 −0.254056
\(623\) 5.43530 0.217761
\(624\) −15.1526 −0.606588
\(625\) −9.41188 −0.376475
\(626\) 6.14137 0.245459
\(627\) −10.8063 −0.431562
\(628\) −20.2049 −0.806263
\(629\) −9.09215 −0.362528
\(630\) −1.17313 −0.0467385
\(631\) −24.5358 −0.976754 −0.488377 0.872633i \(-0.662411\pi\)
−0.488377 + 0.872633i \(0.662411\pi\)
\(632\) −1.17313 −0.0466645
\(633\) −13.1526 −0.522767
\(634\) 4.61987 0.183479
\(635\) 10.4067 0.412975
\(636\) −1.28692 −0.0510295
\(637\) 2.26217 0.0896305
\(638\) 5.07681 0.200993
\(639\) 6.82687 0.270067
\(640\) 12.5892 0.497631
\(641\) −16.0521 −0.634018 −0.317009 0.948422i \(-0.602679\pi\)
−0.317009 + 0.948422i \(0.602679\pi\)
\(642\) 4.52044 0.178408
\(643\) −9.47645 −0.373715 −0.186857 0.982387i \(-0.559830\pi\)
−0.186857 + 0.982387i \(0.559830\pi\)
\(644\) −18.3012 −0.721168
\(645\) −14.3515 −0.565089
\(646\) 7.89845 0.310760
\(647\) 41.1924 1.61944 0.809720 0.586817i \(-0.199619\pi\)
0.809720 + 0.586817i \(0.199619\pi\)
\(648\) 1.00000 0.0392837
\(649\) 2.18953 0.0859467
\(650\) 2.31344 0.0907406
\(651\) 1.69357 0.0663762
\(652\) 7.28169 0.285173
\(653\) 14.0757 0.550827 0.275413 0.961326i \(-0.411185\pi\)
0.275413 + 0.961326i \(0.411185\pi\)
\(654\) −4.64958 −0.181813
\(655\) −26.1948 −1.02351
\(656\) 14.8597 0.580173
\(657\) 13.5040 0.526843
\(658\) 4.52666 0.176468
\(659\) −20.6220 −0.803319 −0.401659 0.915789i \(-0.631566\pi\)
−0.401659 + 0.915789i \(0.631566\pi\)
\(660\) 7.12497 0.277339
\(661\) −30.1414 −1.17236 −0.586182 0.810180i \(-0.699370\pi\)
−0.586182 + 0.810180i \(0.699370\pi\)
\(662\) 3.71725 0.144475
\(663\) 26.3861 1.02475
\(664\) −15.3421 −0.595388
\(665\) 22.7857 0.883592
\(666\) −0.366830 −0.0142144
\(667\) −31.4231 −1.21671
\(668\) −35.0479 −1.35604
\(669\) 9.70892 0.375368
\(670\) 4.24292 0.163918
\(671\) −24.0757 −0.929434
\(672\) 8.01535 0.309199
\(673\) 3.27468 0.126230 0.0631148 0.998006i \(-0.479897\pi\)
0.0631148 + 0.998006i \(0.479897\pi\)
\(674\) 4.86670 0.187458
\(675\) 2.17313 0.0836437
\(676\) −8.81047 −0.338864
\(677\) 19.6126 0.753773 0.376887 0.926259i \(-0.376995\pi\)
0.376887 + 0.926259i \(0.376995\pi\)
\(678\) −1.96018 −0.0752801
\(679\) 9.15778 0.351443
\(680\) −10.5892 −0.406076
\(681\) −6.16896 −0.236395
\(682\) 0.343145 0.0131397
\(683\) 12.1812 0.466101 0.233050 0.972465i \(-0.425129\pi\)
0.233050 + 0.972465i \(0.425129\pi\)
\(684\) −9.55220 −0.365237
\(685\) 1.11796 0.0427149
\(686\) 4.50741 0.172094
\(687\) −25.8175 −0.984998
\(688\) 30.8719 1.17698
\(689\) −2.78572 −0.106128
\(690\) 1.47122 0.0560084
\(691\) −13.8052 −0.525176 −0.262588 0.964908i \(-0.584576\pi\)
−0.262588 + 0.964908i \(0.584576\pi\)
\(692\) 18.7079 0.711166
\(693\) 6.01224 0.228386
\(694\) −7.47823 −0.283870
\(695\) −31.5850 −1.19809
\(696\) 9.12497 0.345881
\(697\) −25.8761 −0.980127
\(698\) −7.34520 −0.278020
\(699\) −12.2499 −0.463335
\(700\) 11.5491 0.436514
\(701\) −9.06980 −0.342561 −0.171281 0.985222i \(-0.554791\pi\)
−0.171281 + 0.985222i \(0.554791\pi\)
\(702\) 1.06457 0.0401795
\(703\) 7.12497 0.268723
\(704\) −14.2140 −0.535711
\(705\) 10.9078 0.410813
\(706\) −0.0122378 −0.000460575 0
\(707\) −9.58812 −0.360598
\(708\) 1.93543 0.0727380
\(709\) 26.2171 0.984605 0.492302 0.870424i \(-0.336155\pi\)
0.492302 + 0.870424i \(0.336155\pi\)
\(710\) −2.91664 −0.109459
\(711\) −1.17313 −0.0439957
\(712\) 1.97942 0.0741821
\(713\) −2.12391 −0.0795409
\(714\) −4.39442 −0.164457
\(715\) 15.4231 0.576790
\(716\) 17.3337 0.647793
\(717\) 22.9477 0.856996
\(718\) −2.98359 −0.111347
\(719\) −37.5972 −1.40214 −0.701070 0.713092i \(-0.747294\pi\)
−0.701070 + 0.713092i \(0.747294\pi\)
\(720\) 6.08097 0.226625
\(721\) 22.5634 0.840304
\(722\) −1.36160 −0.0506736
\(723\) −4.67015 −0.173685
\(724\) 0.943770 0.0350749
\(725\) 19.8297 0.736457
\(726\) −1.57694 −0.0585257
\(727\) 42.3463 1.57054 0.785268 0.619156i \(-0.212525\pi\)
0.785268 + 0.619156i \(0.212525\pi\)
\(728\) 11.5040 0.426368
\(729\) 1.00000 0.0370370
\(730\) −5.76931 −0.213532
\(731\) −53.7592 −1.98836
\(732\) −21.2817 −0.786594
\(733\) −5.96408 −0.220288 −0.110144 0.993916i \(-0.535131\pi\)
−0.110144 + 0.993916i \(0.535131\pi\)
\(734\) 1.78989 0.0660661
\(735\) −0.907847 −0.0334864
\(736\) −10.0521 −0.370524
\(737\) −21.7448 −0.800981
\(738\) −1.04399 −0.0384299
\(739\) 39.9435 1.46935 0.734673 0.678422i \(-0.237336\pi\)
0.734673 + 0.678422i \(0.237336\pi\)
\(740\) −4.69774 −0.172692
\(741\) −20.6772 −0.759595
\(742\) 0.463942 0.0170319
\(743\) 22.7407 0.834274 0.417137 0.908844i \(-0.363033\pi\)
0.417137 + 0.908844i \(0.363033\pi\)
\(744\) 0.616763 0.0226116
\(745\) −13.5316 −0.495760
\(746\) −8.01224 −0.293349
\(747\) −15.3421 −0.561338
\(748\) 26.6894 0.975861
\(749\) 48.8492 1.78491
\(750\) −3.06457 −0.111902
\(751\) 17.8052 0.649722 0.324861 0.945762i \(-0.394682\pi\)
0.324861 + 0.945762i \(0.394682\pi\)
\(752\) −23.4642 −0.855652
\(753\) −22.4067 −0.816544
\(754\) 9.71414 0.353768
\(755\) −30.0122 −1.09226
\(756\) 5.31450 0.193287
\(757\) 27.4353 0.997153 0.498576 0.866846i \(-0.333856\pi\)
0.498576 + 0.866846i \(0.333856\pi\)
\(758\) 2.66882 0.0969360
\(759\) −7.53996 −0.273683
\(760\) 8.29809 0.301003
\(761\) 10.1812 0.369068 0.184534 0.982826i \(-0.440922\pi\)
0.184534 + 0.982826i \(0.440922\pi\)
\(762\) 1.57277 0.0569755
\(763\) −50.2447 −1.81898
\(764\) −9.82581 −0.355485
\(765\) −10.5892 −0.382853
\(766\) −5.92114 −0.213940
\(767\) 4.18953 0.151275
\(768\) −11.0810 −0.399850
\(769\) 38.0398 1.37175 0.685876 0.727719i \(-0.259419\pi\)
0.685876 + 0.727719i \(0.259419\pi\)
\(770\) −2.56860 −0.0925660
\(771\) −31.1372 −1.12138
\(772\) −18.1208 −0.652182
\(773\) −44.1676 −1.58860 −0.794300 0.607526i \(-0.792162\pi\)
−0.794300 + 0.607526i \(0.792162\pi\)
\(774\) −2.16896 −0.0779616
\(775\) 1.34030 0.0481452
\(776\) 3.33508 0.119722
\(777\) −3.96408 −0.142210
\(778\) 8.50820 0.305034
\(779\) 20.2775 0.726517
\(780\) 13.6332 0.488146
\(781\) 14.9477 0.534870
\(782\) 5.51104 0.197075
\(783\) 9.12497 0.326100
\(784\) 1.95290 0.0697464
\(785\) 17.5522 0.626465
\(786\) −3.95885 −0.141207
\(787\) −16.3379 −0.582384 −0.291192 0.956665i \(-0.594052\pi\)
−0.291192 + 0.956665i \(0.594052\pi\)
\(788\) 31.4506 1.12038
\(789\) −24.9547 −0.888410
\(790\) 0.501194 0.0178317
\(791\) −21.1823 −0.753154
\(792\) 2.18953 0.0778017
\(793\) −46.0674 −1.63590
\(794\) 2.71831 0.0964693
\(795\) 1.11796 0.0396498
\(796\) −12.8698 −0.456158
\(797\) −32.9424 −1.16688 −0.583441 0.812156i \(-0.698294\pi\)
−0.583441 + 0.812156i \(0.698294\pi\)
\(798\) 3.44364 0.121903
\(799\) 40.8597 1.44551
\(800\) 6.34341 0.224274
\(801\) 1.97942 0.0699395
\(802\) 5.92425 0.209193
\(803\) 29.5675 1.04342
\(804\) −19.2213 −0.677883
\(805\) 15.8984 0.560347
\(806\) 0.656586 0.0231273
\(807\) −5.99477 −0.211026
\(808\) −3.49180 −0.122841
\(809\) −51.9659 −1.82702 −0.913511 0.406814i \(-0.866640\pi\)
−0.913511 + 0.406814i \(0.866640\pi\)
\(810\) −0.427229 −0.0150113
\(811\) 26.9424 0.946077 0.473039 0.881042i \(-0.343157\pi\)
0.473039 + 0.881042i \(0.343157\pi\)
\(812\) 48.4946 1.70183
\(813\) 1.97241 0.0691756
\(814\) −0.803187 −0.0281517
\(815\) −6.32568 −0.221579
\(816\) 22.7787 0.797414
\(817\) 42.1278 1.47387
\(818\) 2.20044 0.0769367
\(819\) 11.5040 0.401983
\(820\) −13.3697 −0.466889
\(821\) −27.8914 −0.973418 −0.486709 0.873564i \(-0.661803\pi\)
−0.486709 + 0.873564i \(0.661803\pi\)
\(822\) 0.168958 0.00589310
\(823\) 18.3103 0.638258 0.319129 0.947711i \(-0.396610\pi\)
0.319129 + 0.947711i \(0.396610\pi\)
\(824\) 8.21712 0.286257
\(825\) 4.75814 0.165657
\(826\) −0.697737 −0.0242774
\(827\) −4.37384 −0.152093 −0.0760467 0.997104i \(-0.524230\pi\)
−0.0760467 + 0.997104i \(0.524230\pi\)
\(828\) −6.66492 −0.231622
\(829\) 30.5204 1.06002 0.530009 0.847992i \(-0.322188\pi\)
0.530009 + 0.847992i \(0.322188\pi\)
\(830\) 6.55458 0.227513
\(831\) 2.53579 0.0879655
\(832\) −27.1976 −0.942907
\(833\) −3.40070 −0.117827
\(834\) −4.77348 −0.165292
\(835\) 30.4465 1.05364
\(836\) −20.9149 −0.723355
\(837\) 0.616763 0.0213184
\(838\) −1.17836 −0.0407056
\(839\) −0.859686 −0.0296797 −0.0148398 0.999890i \(-0.504724\pi\)
−0.0148398 + 0.999890i \(0.504724\pi\)
\(840\) −4.61676 −0.159293
\(841\) 54.2650 1.87121
\(842\) −0.427229 −0.0147233
\(843\) 18.8545 0.649382
\(844\) −25.4559 −0.876227
\(845\) 7.65375 0.263297
\(846\) 1.64852 0.0566772
\(847\) −17.0409 −0.585532
\(848\) −2.40487 −0.0825836
\(849\) 5.78989 0.198709
\(850\) −3.47778 −0.119287
\(851\) 4.97136 0.170416
\(852\) 13.2130 0.452668
\(853\) 50.4741 1.72820 0.864099 0.503321i \(-0.167889\pi\)
0.864099 + 0.503321i \(0.167889\pi\)
\(854\) 7.67220 0.262537
\(855\) 8.29809 0.283789
\(856\) 17.7899 0.608046
\(857\) 38.4946 1.31495 0.657476 0.753476i \(-0.271624\pi\)
0.657476 + 0.753476i \(0.271624\pi\)
\(858\) 2.33091 0.0795759
\(859\) −27.1044 −0.924790 −0.462395 0.886674i \(-0.653010\pi\)
−0.462395 + 0.886674i \(0.653010\pi\)
\(860\) −27.7763 −0.947165
\(861\) −11.2817 −0.384479
\(862\) −2.92319 −0.0995644
\(863\) 10.6443 0.362338 0.181169 0.983452i \(-0.442012\pi\)
0.181169 + 0.983452i \(0.442012\pi\)
\(864\) 2.91903 0.0993073
\(865\) −16.2517 −0.552575
\(866\) −6.22724 −0.211610
\(867\) −22.6660 −0.769777
\(868\) 3.27779 0.111255
\(869\) −2.56860 −0.0871339
\(870\) −3.89845 −0.132170
\(871\) −41.6074 −1.40981
\(872\) −18.2981 −0.619652
\(873\) 3.33508 0.112875
\(874\) −4.31867 −0.146081
\(875\) −33.1166 −1.11955
\(876\) 26.1361 0.883058
\(877\) 40.0328 1.35181 0.675906 0.736988i \(-0.263753\pi\)
0.675906 + 0.736988i \(0.263753\pi\)
\(878\) 1.43273 0.0483522
\(879\) −7.73472 −0.260886
\(880\) 13.3145 0.448832
\(881\) 29.9588 1.00934 0.504670 0.863313i \(-0.331614\pi\)
0.504670 + 0.863313i \(0.331614\pi\)
\(882\) −0.137204 −0.00461990
\(883\) −12.3239 −0.414732 −0.207366 0.978263i \(-0.566489\pi\)
−0.207366 + 0.978263i \(0.566489\pi\)
\(884\) 51.0685 1.71762
\(885\) −1.68133 −0.0565173
\(886\) 9.65765 0.324455
\(887\) 25.4559 0.854725 0.427362 0.904080i \(-0.359443\pi\)
0.427362 + 0.904080i \(0.359443\pi\)
\(888\) −1.44364 −0.0484452
\(889\) 16.9958 0.570022
\(890\) −0.845667 −0.0283468
\(891\) 2.18953 0.0733521
\(892\) 18.7909 0.629168
\(893\) −32.0192 −1.07148
\(894\) −2.04505 −0.0683967
\(895\) −15.0580 −0.503334
\(896\) 20.5603 0.686870
\(897\) −14.4272 −0.481711
\(898\) 2.80418 0.0935766
\(899\) 5.62794 0.187702
\(900\) 4.20594 0.140198
\(901\) 4.18775 0.139514
\(902\) −2.28586 −0.0761107
\(903\) −23.4384 −0.779981
\(904\) −7.71414 −0.256569
\(905\) −0.819863 −0.0272532
\(906\) −4.53579 −0.150691
\(907\) 38.4863 1.27792 0.638958 0.769241i \(-0.279366\pi\)
0.638958 + 0.769241i \(0.279366\pi\)
\(908\) −11.9396 −0.396230
\(909\) −3.49180 −0.115816
\(910\) −4.91486 −0.162926
\(911\) 21.2234 0.703163 0.351581 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(912\) −17.8503 −0.591082
\(913\) −33.5920 −1.11173
\(914\) 8.23069 0.272247
\(915\) 18.4876 0.611182
\(916\) −49.9680 −1.65099
\(917\) −42.7805 −1.41274
\(918\) −1.60036 −0.0528196
\(919\) 46.3051 1.52746 0.763732 0.645533i \(-0.223365\pi\)
0.763732 + 0.645533i \(0.223365\pi\)
\(920\) 5.78989 0.190887
\(921\) −16.4999 −0.543689
\(922\) −8.88337 −0.292558
\(923\) 28.6014 0.941427
\(924\) 11.6363 0.382805
\(925\) −3.13720 −0.103151
\(926\) 5.79512 0.190439
\(927\) 8.21712 0.269886
\(928\) 26.6360 0.874370
\(929\) 31.9313 1.04763 0.523815 0.851832i \(-0.324508\pi\)
0.523815 + 0.851832i \(0.324508\pi\)
\(930\) −0.263499 −0.00864047
\(931\) 2.66492 0.0873394
\(932\) −23.7089 −0.776611
\(933\) −24.9354 −0.816349
\(934\) 9.20905 0.301329
\(935\) −23.1854 −0.758243
\(936\) 4.18953 0.136939
\(937\) −33.6813 −1.10032 −0.550161 0.835059i \(-0.685433\pi\)
−0.550161 + 0.835059i \(0.685433\pi\)
\(938\) 6.92941 0.226253
\(939\) 24.1690 0.788724
\(940\) 21.1114 0.688578
\(941\) −22.6977 −0.739925 −0.369963 0.929047i \(-0.620629\pi\)
−0.369963 + 0.929047i \(0.620629\pi\)
\(942\) 2.65269 0.0864292
\(943\) 14.1484 0.460735
\(944\) 3.61676 0.117716
\(945\) −4.61676 −0.150183
\(946\) −4.74901 −0.154404
\(947\) 22.5533 0.732882 0.366441 0.930441i \(-0.380576\pi\)
0.366441 + 0.930441i \(0.380576\pi\)
\(948\) −2.27051 −0.0737427
\(949\) 56.5756 1.83652
\(950\) 2.72532 0.0884211
\(951\) 18.1812 0.589566
\(952\) −17.2939 −0.560499
\(953\) −10.3791 −0.336211 −0.168105 0.985769i \(-0.553765\pi\)
−0.168105 + 0.985769i \(0.553765\pi\)
\(954\) 0.168958 0.00547023
\(955\) 8.53579 0.276212
\(956\) 44.4137 1.43644
\(957\) 19.9794 0.645843
\(958\) −4.33402 −0.140026
\(959\) 1.82581 0.0589586
\(960\) 10.9149 0.352275
\(961\) −30.6196 −0.987729
\(962\) −1.53685 −0.0495500
\(963\) 17.7899 0.573271
\(964\) −9.03876 −0.291119
\(965\) 15.7417 0.506744
\(966\) 2.40275 0.0773073
\(967\) 32.1913 1.03520 0.517601 0.855622i \(-0.326825\pi\)
0.517601 + 0.855622i \(0.326825\pi\)
\(968\) −6.20594 −0.199466
\(969\) 31.0838 0.998556
\(970\) −1.42484 −0.0457489
\(971\) −33.1114 −1.06260 −0.531298 0.847185i \(-0.678295\pi\)
−0.531298 + 0.847185i \(0.678295\pi\)
\(972\) 1.93543 0.0620790
\(973\) −51.5837 −1.65370
\(974\) −3.89978 −0.124957
\(975\) 9.10439 0.291574
\(976\) −39.7693 −1.27298
\(977\) 12.2307 0.391294 0.195647 0.980674i \(-0.437319\pi\)
0.195647 + 0.980674i \(0.437319\pi\)
\(978\) −0.956008 −0.0305698
\(979\) 4.33402 0.138516
\(980\) −1.75708 −0.0561278
\(981\) −18.2981 −0.584213
\(982\) −4.57694 −0.146056
\(983\) −7.26111 −0.231593 −0.115797 0.993273i \(-0.536942\pi\)
−0.115797 + 0.993273i \(0.536942\pi\)
\(984\) −4.10856 −0.130976
\(985\) −27.3215 −0.870536
\(986\) −14.6032 −0.465061
\(987\) 17.8144 0.567038
\(988\) −40.0192 −1.27318
\(989\) 29.3941 0.934679
\(990\) −0.935432 −0.0297300
\(991\) −20.1054 −0.638671 −0.319335 0.947642i \(-0.603460\pi\)
−0.319335 + 0.947642i \(0.603460\pi\)
\(992\) 1.80035 0.0571611
\(993\) 14.6290 0.464237
\(994\) −4.76336 −0.151085
\(995\) 11.1801 0.354434
\(996\) −29.6936 −0.940877
\(997\) −42.3173 −1.34020 −0.670102 0.742269i \(-0.733750\pi\)
−0.670102 + 0.742269i \(0.733750\pi\)
\(998\) 5.49047 0.173798
\(999\) −1.44364 −0.0456746
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.2.a.d.1.2 3
3.2 odd 2 531.2.a.d.1.2 3
4.3 odd 2 2832.2.a.t.1.3 3
5.4 even 2 4425.2.a.w.1.2 3
7.6 odd 2 8673.2.a.s.1.2 3
12.11 even 2 8496.2.a.bl.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.2.a.d.1.2 3 1.1 even 1 trivial
531.2.a.d.1.2 3 3.2 odd 2
2832.2.a.t.1.3 3 4.3 odd 2
4425.2.a.w.1.2 3 5.4 even 2
8496.2.a.bl.1.1 3 12.11 even 2
8673.2.a.s.1.2 3 7.6 odd 2