Properties

Label 503.2.a.e.1.3
Level 503
Weight 2
Character 503.1
Self dual yes
Analytic conductor 4.016
Analytic rank 1
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
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 1.3
Root \(-0.489003\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.62786 q^{2} -1.48900 q^{3} +0.649933 q^{4} -1.79865 q^{5} +2.42389 q^{6} +0.552233 q^{7} +2.19772 q^{8} -0.782869 q^{9} +O(q^{10})\) \(q-1.62786 q^{2} -1.48900 q^{3} +0.649933 q^{4} -1.79865 q^{5} +2.42389 q^{6} +0.552233 q^{7} +2.19772 q^{8} -0.782869 q^{9} +2.92795 q^{10} +4.33718 q^{11} -0.967753 q^{12} +2.54873 q^{13} -0.898958 q^{14} +2.67819 q^{15} -4.87745 q^{16} +2.52304 q^{17} +1.27440 q^{18} -5.11893 q^{19} -1.16900 q^{20} -0.822276 q^{21} -7.06033 q^{22} -3.78409 q^{23} -3.27241 q^{24} -1.76486 q^{25} -4.14898 q^{26} +5.63270 q^{27} +0.358914 q^{28} +0.907287 q^{29} -4.35973 q^{30} +0.380820 q^{31} +3.54438 q^{32} -6.45807 q^{33} -4.10717 q^{34} -0.993273 q^{35} -0.508813 q^{36} -5.43266 q^{37} +8.33291 q^{38} -3.79507 q^{39} -3.95293 q^{40} +5.72459 q^{41} +1.33855 q^{42} -9.21553 q^{43} +2.81888 q^{44} +1.40811 q^{45} +6.15997 q^{46} -8.81077 q^{47} +7.26254 q^{48} -6.69504 q^{49} +2.87295 q^{50} -3.75682 q^{51} +1.65651 q^{52} -6.46357 q^{53} -9.16926 q^{54} -7.80106 q^{55} +1.21365 q^{56} +7.62210 q^{57} -1.47694 q^{58} +3.40568 q^{59} +1.74065 q^{60} -1.06109 q^{61} -0.619922 q^{62} -0.432326 q^{63} +3.98515 q^{64} -4.58427 q^{65} +10.5129 q^{66} -0.253929 q^{67} +1.63981 q^{68} +5.63451 q^{69} +1.61691 q^{70} +11.7311 q^{71} -1.72053 q^{72} -16.1271 q^{73} +8.84362 q^{74} +2.62789 q^{75} -3.32696 q^{76} +2.39513 q^{77} +6.17785 q^{78} +7.76447 q^{79} +8.77283 q^{80} -6.03851 q^{81} -9.31884 q^{82} -16.3846 q^{83} -0.534425 q^{84} -4.53807 q^{85} +15.0016 q^{86} -1.35095 q^{87} +9.53191 q^{88} -3.09992 q^{89} -2.29220 q^{90} +1.40749 q^{91} -2.45940 q^{92} -0.567042 q^{93} +14.3427 q^{94} +9.20715 q^{95} -5.27759 q^{96} -4.28605 q^{97} +10.8986 q^{98} -3.39544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.62786 −1.15107 −0.575536 0.817776i \(-0.695207\pi\)
−0.575536 + 0.817776i \(0.695207\pi\)
\(3\) −1.48900 −0.859676 −0.429838 0.902906i \(-0.641429\pi\)
−0.429838 + 0.902906i \(0.641429\pi\)
\(4\) 0.649933 0.324967
\(5\) −1.79865 −0.804380 −0.402190 0.915556i \(-0.631751\pi\)
−0.402190 + 0.915556i \(0.631751\pi\)
\(6\) 2.42389 0.989549
\(7\) 0.552233 0.208724 0.104362 0.994539i \(-0.466720\pi\)
0.104362 + 0.994539i \(0.466720\pi\)
\(8\) 2.19772 0.777012
\(9\) −0.782869 −0.260956
\(10\) 2.92795 0.925900
\(11\) 4.33718 1.30771 0.653854 0.756620i \(-0.273151\pi\)
0.653854 + 0.756620i \(0.273151\pi\)
\(12\) −0.967753 −0.279366
\(13\) 2.54873 0.706891 0.353445 0.935455i \(-0.385010\pi\)
0.353445 + 0.935455i \(0.385010\pi\)
\(14\) −0.898958 −0.240257
\(15\) 2.67819 0.691507
\(16\) −4.87745 −1.21936
\(17\) 2.52304 0.611928 0.305964 0.952043i \(-0.401021\pi\)
0.305964 + 0.952043i \(0.401021\pi\)
\(18\) 1.27440 0.300380
\(19\) −5.11893 −1.17436 −0.587181 0.809455i \(-0.699762\pi\)
−0.587181 + 0.809455i \(0.699762\pi\)
\(20\) −1.16900 −0.261397
\(21\) −0.822276 −0.179435
\(22\) −7.06033 −1.50527
\(23\) −3.78409 −0.789036 −0.394518 0.918888i \(-0.629088\pi\)
−0.394518 + 0.918888i \(0.629088\pi\)
\(24\) −3.27241 −0.667979
\(25\) −1.76486 −0.352972
\(26\) −4.14898 −0.813682
\(27\) 5.63270 1.08401
\(28\) 0.358914 0.0678284
\(29\) 0.907287 0.168479 0.0842395 0.996446i \(-0.473154\pi\)
0.0842395 + 0.996446i \(0.473154\pi\)
\(30\) −4.35973 −0.795974
\(31\) 0.380820 0.0683972 0.0341986 0.999415i \(-0.489112\pi\)
0.0341986 + 0.999415i \(0.489112\pi\)
\(32\) 3.54438 0.626563
\(33\) −6.45807 −1.12421
\(34\) −4.10717 −0.704373
\(35\) −0.993273 −0.167894
\(36\) −0.508813 −0.0848021
\(37\) −5.43266 −0.893124 −0.446562 0.894753i \(-0.647352\pi\)
−0.446562 + 0.894753i \(0.647352\pi\)
\(38\) 8.33291 1.35178
\(39\) −3.79507 −0.607697
\(40\) −3.95293 −0.625013
\(41\) 5.72459 0.894031 0.447015 0.894526i \(-0.352487\pi\)
0.447015 + 0.894526i \(0.352487\pi\)
\(42\) 1.33855 0.206543
\(43\) −9.21553 −1.40536 −0.702678 0.711508i \(-0.748012\pi\)
−0.702678 + 0.711508i \(0.748012\pi\)
\(44\) 2.81888 0.424962
\(45\) 1.40811 0.209908
\(46\) 6.15997 0.908238
\(47\) −8.81077 −1.28518 −0.642592 0.766209i \(-0.722141\pi\)
−0.642592 + 0.766209i \(0.722141\pi\)
\(48\) 7.26254 1.04826
\(49\) −6.69504 −0.956434
\(50\) 2.87295 0.406297
\(51\) −3.75682 −0.526060
\(52\) 1.65651 0.229716
\(53\) −6.46357 −0.887840 −0.443920 0.896066i \(-0.646413\pi\)
−0.443920 + 0.896066i \(0.646413\pi\)
\(54\) −9.16926 −1.24778
\(55\) −7.80106 −1.05190
\(56\) 1.21365 0.162181
\(57\) 7.62210 1.00957
\(58\) −1.47694 −0.193932
\(59\) 3.40568 0.443381 0.221691 0.975117i \(-0.428843\pi\)
0.221691 + 0.975117i \(0.428843\pi\)
\(60\) 1.74065 0.224717
\(61\) −1.06109 −0.135859 −0.0679294 0.997690i \(-0.521639\pi\)
−0.0679294 + 0.997690i \(0.521639\pi\)
\(62\) −0.619922 −0.0787301
\(63\) −0.432326 −0.0544679
\(64\) 3.98515 0.498144
\(65\) −4.58427 −0.568609
\(66\) 10.5129 1.29404
\(67\) −0.253929 −0.0310223 −0.0155112 0.999880i \(-0.504938\pi\)
−0.0155112 + 0.999880i \(0.504938\pi\)
\(68\) 1.63981 0.198856
\(69\) 5.63451 0.678316
\(70\) 1.61691 0.193258
\(71\) 11.7311 1.39222 0.696112 0.717933i \(-0.254912\pi\)
0.696112 + 0.717933i \(0.254912\pi\)
\(72\) −1.72053 −0.202766
\(73\) −16.1271 −1.88753 −0.943765 0.330617i \(-0.892743\pi\)
−0.943765 + 0.330617i \(0.892743\pi\)
\(74\) 8.84362 1.02805
\(75\) 2.62789 0.303442
\(76\) −3.32696 −0.381629
\(77\) 2.39513 0.272951
\(78\) 6.17785 0.699504
\(79\) 7.76447 0.873571 0.436785 0.899566i \(-0.356117\pi\)
0.436785 + 0.899566i \(0.356117\pi\)
\(80\) 8.77283 0.980832
\(81\) −6.03851 −0.670945
\(82\) −9.31884 −1.02909
\(83\) −16.3846 −1.79844 −0.899222 0.437492i \(-0.855867\pi\)
−0.899222 + 0.437492i \(0.855867\pi\)
\(84\) −0.534425 −0.0583105
\(85\) −4.53807 −0.492223
\(86\) 15.0016 1.61767
\(87\) −1.35095 −0.144837
\(88\) 9.53191 1.01611
\(89\) −3.09992 −0.328590 −0.164295 0.986411i \(-0.552535\pi\)
−0.164295 + 0.986411i \(0.552535\pi\)
\(90\) −2.29220 −0.241619
\(91\) 1.40749 0.147545
\(92\) −2.45940 −0.256410
\(93\) −0.567042 −0.0587995
\(94\) 14.3427 1.47934
\(95\) 9.20715 0.944634
\(96\) −5.27759 −0.538641
\(97\) −4.28605 −0.435183 −0.217591 0.976040i \(-0.569820\pi\)
−0.217591 + 0.976040i \(0.569820\pi\)
\(98\) 10.8986 1.10092
\(99\) −3.39544 −0.341255
\(100\) −1.14704 −0.114704
\(101\) −3.73446 −0.371592 −0.185796 0.982588i \(-0.559486\pi\)
−0.185796 + 0.982588i \(0.559486\pi\)
\(102\) 6.11559 0.605533
\(103\) 2.83220 0.279065 0.139533 0.990217i \(-0.455440\pi\)
0.139533 + 0.990217i \(0.455440\pi\)
\(104\) 5.60140 0.549263
\(105\) 1.47899 0.144334
\(106\) 10.5218 1.02197
\(107\) 10.0994 0.976343 0.488172 0.872748i \(-0.337664\pi\)
0.488172 + 0.872748i \(0.337664\pi\)
\(108\) 3.66088 0.352269
\(109\) −6.86764 −0.657800 −0.328900 0.944365i \(-0.606678\pi\)
−0.328900 + 0.944365i \(0.606678\pi\)
\(110\) 12.6991 1.21081
\(111\) 8.08925 0.767798
\(112\) −2.69349 −0.254511
\(113\) 2.80558 0.263927 0.131963 0.991255i \(-0.457872\pi\)
0.131963 + 0.991255i \(0.457872\pi\)
\(114\) −12.4077 −1.16209
\(115\) 6.80624 0.634685
\(116\) 0.589676 0.0547501
\(117\) −1.99532 −0.184468
\(118\) −5.54397 −0.510364
\(119\) 1.39331 0.127724
\(120\) 5.88593 0.537309
\(121\) 7.81112 0.710102
\(122\) 1.72731 0.156383
\(123\) −8.52393 −0.768577
\(124\) 0.247507 0.0222268
\(125\) 12.1676 1.08830
\(126\) 0.703767 0.0626965
\(127\) −14.9147 −1.32347 −0.661734 0.749739i \(-0.730179\pi\)
−0.661734 + 0.749739i \(0.730179\pi\)
\(128\) −13.5760 −1.19996
\(129\) 13.7220 1.20815
\(130\) 7.46256 0.654510
\(131\) 13.5207 1.18131 0.590654 0.806925i \(-0.298870\pi\)
0.590654 + 0.806925i \(0.298870\pi\)
\(132\) −4.19732 −0.365330
\(133\) −2.82684 −0.245118
\(134\) 0.413361 0.0357090
\(135\) −10.1313 −0.871960
\(136\) 5.54495 0.475476
\(137\) −12.4479 −1.06350 −0.531750 0.846901i \(-0.678465\pi\)
−0.531750 + 0.846901i \(0.678465\pi\)
\(138\) −9.17221 −0.780790
\(139\) 2.15443 0.182736 0.0913681 0.995817i \(-0.470876\pi\)
0.0913681 + 0.995817i \(0.470876\pi\)
\(140\) −0.645561 −0.0545599
\(141\) 13.1193 1.10484
\(142\) −19.0966 −1.60255
\(143\) 11.0543 0.924407
\(144\) 3.81841 0.318201
\(145\) −1.63189 −0.135521
\(146\) 26.2526 2.17268
\(147\) 9.96894 0.822224
\(148\) −3.53087 −0.290236
\(149\) −8.59253 −0.703927 −0.351964 0.936014i \(-0.614486\pi\)
−0.351964 + 0.936014i \(0.614486\pi\)
\(150\) −4.27783 −0.349284
\(151\) 0.527862 0.0429568 0.0214784 0.999769i \(-0.493163\pi\)
0.0214784 + 0.999769i \(0.493163\pi\)
\(152\) −11.2500 −0.912494
\(153\) −1.97521 −0.159687
\(154\) −3.89894 −0.314186
\(155\) −0.684961 −0.0550174
\(156\) −2.46654 −0.197481
\(157\) 11.1711 0.891554 0.445777 0.895144i \(-0.352927\pi\)
0.445777 + 0.895144i \(0.352927\pi\)
\(158\) −12.6395 −1.00554
\(159\) 9.62428 0.763255
\(160\) −6.37509 −0.503995
\(161\) −2.08970 −0.164691
\(162\) 9.82985 0.772306
\(163\) 20.0109 1.56738 0.783689 0.621153i \(-0.213336\pi\)
0.783689 + 0.621153i \(0.213336\pi\)
\(164\) 3.72060 0.290530
\(165\) 11.6158 0.904289
\(166\) 26.6719 2.07014
\(167\) −11.9520 −0.924877 −0.462439 0.886651i \(-0.653025\pi\)
−0.462439 + 0.886651i \(0.653025\pi\)
\(168\) −1.80713 −0.139423
\(169\) −6.50397 −0.500305
\(170\) 7.38735 0.566584
\(171\) 4.00745 0.306457
\(172\) −5.98948 −0.456694
\(173\) 3.14561 0.239156 0.119578 0.992825i \(-0.461846\pi\)
0.119578 + 0.992825i \(0.461846\pi\)
\(174\) 2.19917 0.166718
\(175\) −0.974615 −0.0736739
\(176\) −21.1544 −1.59457
\(177\) −5.07106 −0.381164
\(178\) 5.04623 0.378231
\(179\) −6.02096 −0.450028 −0.225014 0.974356i \(-0.572243\pi\)
−0.225014 + 0.974356i \(0.572243\pi\)
\(180\) 0.915176 0.0682132
\(181\) −23.0475 −1.71311 −0.856553 0.516059i \(-0.827398\pi\)
−0.856553 + 0.516059i \(0.827398\pi\)
\(182\) −2.29120 −0.169835
\(183\) 1.57997 0.116795
\(184\) −8.31637 −0.613091
\(185\) 9.77145 0.718412
\(186\) 0.923065 0.0676824
\(187\) 10.9429 0.800224
\(188\) −5.72641 −0.417642
\(189\) 3.11056 0.226260
\(190\) −14.9880 −1.08734
\(191\) −17.9774 −1.30080 −0.650398 0.759594i \(-0.725398\pi\)
−0.650398 + 0.759594i \(0.725398\pi\)
\(192\) −5.93391 −0.428243
\(193\) −10.4429 −0.751697 −0.375848 0.926681i \(-0.622649\pi\)
−0.375848 + 0.926681i \(0.622649\pi\)
\(194\) 6.97710 0.500927
\(195\) 6.82600 0.488820
\(196\) −4.35133 −0.310809
\(197\) −4.36026 −0.310656 −0.155328 0.987863i \(-0.549643\pi\)
−0.155328 + 0.987863i \(0.549643\pi\)
\(198\) 5.52731 0.392809
\(199\) 14.0205 0.993888 0.496944 0.867783i \(-0.334455\pi\)
0.496944 + 0.867783i \(0.334455\pi\)
\(200\) −3.87868 −0.274264
\(201\) 0.378101 0.0266692
\(202\) 6.07918 0.427729
\(203\) 0.501034 0.0351657
\(204\) −2.44168 −0.170952
\(205\) −10.2965 −0.719141
\(206\) −4.61043 −0.321224
\(207\) 2.96244 0.205904
\(208\) −12.4313 −0.861957
\(209\) −22.2017 −1.53572
\(210\) −2.40758 −0.166139
\(211\) −11.0580 −0.761265 −0.380633 0.924726i \(-0.624294\pi\)
−0.380633 + 0.924726i \(0.624294\pi\)
\(212\) −4.20089 −0.288518
\(213\) −17.4676 −1.19686
\(214\) −16.4404 −1.12384
\(215\) 16.5755 1.13044
\(216\) 12.3791 0.842292
\(217\) 0.210301 0.0142762
\(218\) 11.1796 0.757175
\(219\) 24.0133 1.62267
\(220\) −5.07017 −0.341831
\(221\) 6.43056 0.432566
\(222\) −13.1682 −0.883791
\(223\) −15.5502 −1.04132 −0.520658 0.853765i \(-0.674313\pi\)
−0.520658 + 0.853765i \(0.674313\pi\)
\(224\) 1.95732 0.130779
\(225\) 1.38166 0.0921104
\(226\) −4.56710 −0.303799
\(227\) 1.27577 0.0846759 0.0423379 0.999103i \(-0.486519\pi\)
0.0423379 + 0.999103i \(0.486519\pi\)
\(228\) 4.95386 0.328077
\(229\) −15.5232 −1.02580 −0.512901 0.858448i \(-0.671429\pi\)
−0.512901 + 0.858448i \(0.671429\pi\)
\(230\) −11.0796 −0.730568
\(231\) −3.56636 −0.234649
\(232\) 1.99397 0.130910
\(233\) −27.8010 −1.82130 −0.910651 0.413177i \(-0.864419\pi\)
−0.910651 + 0.413177i \(0.864419\pi\)
\(234\) 3.24811 0.212336
\(235\) 15.8475 1.03378
\(236\) 2.21346 0.144084
\(237\) −11.5613 −0.750988
\(238\) −2.26811 −0.147020
\(239\) 1.20369 0.0778601 0.0389301 0.999242i \(-0.487605\pi\)
0.0389301 + 0.999242i \(0.487605\pi\)
\(240\) −13.0628 −0.843198
\(241\) −25.6025 −1.64920 −0.824602 0.565713i \(-0.808601\pi\)
−0.824602 + 0.565713i \(0.808601\pi\)
\(242\) −12.7154 −0.817379
\(243\) −7.90676 −0.507219
\(244\) −0.689638 −0.0441496
\(245\) 12.0420 0.769337
\(246\) 13.8758 0.884688
\(247\) −13.0468 −0.830146
\(248\) 0.836936 0.0531455
\(249\) 24.3967 1.54608
\(250\) −19.8072 −1.25272
\(251\) 13.2638 0.837205 0.418603 0.908170i \(-0.362520\pi\)
0.418603 + 0.908170i \(0.362520\pi\)
\(252\) −0.280983 −0.0177003
\(253\) −16.4123 −1.03183
\(254\) 24.2791 1.52341
\(255\) 6.75720 0.423152
\(256\) 14.1296 0.883099
\(257\) −16.1006 −1.00433 −0.502163 0.864773i \(-0.667462\pi\)
−0.502163 + 0.864773i \(0.667462\pi\)
\(258\) −22.3374 −1.39067
\(259\) −3.00009 −0.186417
\(260\) −2.97947 −0.184779
\(261\) −0.710288 −0.0439657
\(262\) −22.0098 −1.35977
\(263\) 6.77336 0.417663 0.208832 0.977952i \(-0.433034\pi\)
0.208832 + 0.977952i \(0.433034\pi\)
\(264\) −14.1930 −0.873522
\(265\) 11.6257 0.714161
\(266\) 4.60170 0.282148
\(267\) 4.61579 0.282481
\(268\) −0.165037 −0.0100812
\(269\) 23.4853 1.43192 0.715962 0.698139i \(-0.245988\pi\)
0.715962 + 0.698139i \(0.245988\pi\)
\(270\) 16.4923 1.00369
\(271\) 17.5886 1.06843 0.534216 0.845348i \(-0.320607\pi\)
0.534216 + 0.845348i \(0.320607\pi\)
\(272\) −12.3060 −0.746163
\(273\) −2.09576 −0.126841
\(274\) 20.2635 1.22417
\(275\) −7.65452 −0.461585
\(276\) 3.66206 0.220430
\(277\) −11.3448 −0.681645 −0.340823 0.940128i \(-0.610706\pi\)
−0.340823 + 0.940128i \(0.610706\pi\)
\(278\) −3.50711 −0.210343
\(279\) −0.298132 −0.0178487
\(280\) −2.18294 −0.130455
\(281\) −9.36752 −0.558820 −0.279410 0.960172i \(-0.590139\pi\)
−0.279410 + 0.960172i \(0.590139\pi\)
\(282\) −21.3564 −1.27175
\(283\) 26.8789 1.59778 0.798892 0.601474i \(-0.205420\pi\)
0.798892 + 0.601474i \(0.205420\pi\)
\(284\) 7.62443 0.452426
\(285\) −13.7095 −0.812080
\(286\) −17.9949 −1.06406
\(287\) 3.16131 0.186606
\(288\) −2.77478 −0.163506
\(289\) −10.6342 −0.625544
\(290\) 2.65649 0.155995
\(291\) 6.38195 0.374117
\(292\) −10.4815 −0.613384
\(293\) 17.2365 1.00697 0.503485 0.864004i \(-0.332051\pi\)
0.503485 + 0.864004i \(0.332051\pi\)
\(294\) −16.2280 −0.946439
\(295\) −6.12561 −0.356647
\(296\) −11.9395 −0.693968
\(297\) 24.4301 1.41758
\(298\) 13.9874 0.810271
\(299\) −9.64462 −0.557763
\(300\) 1.70795 0.0986086
\(301\) −5.08912 −0.293332
\(302\) −0.859286 −0.0494464
\(303\) 5.56062 0.319449
\(304\) 24.9673 1.43197
\(305\) 1.90853 0.109282
\(306\) 3.21538 0.183811
\(307\) 29.0001 1.65512 0.827561 0.561376i \(-0.189728\pi\)
0.827561 + 0.561376i \(0.189728\pi\)
\(308\) 1.55668 0.0886998
\(309\) −4.21716 −0.239906
\(310\) 1.11502 0.0633290
\(311\) 27.2312 1.54414 0.772071 0.635536i \(-0.219221\pi\)
0.772071 + 0.635536i \(0.219221\pi\)
\(312\) −8.34051 −0.472188
\(313\) 5.53059 0.312607 0.156304 0.987709i \(-0.450042\pi\)
0.156304 + 0.987709i \(0.450042\pi\)
\(314\) −18.1851 −1.02624
\(315\) 0.777603 0.0438129
\(316\) 5.04639 0.283881
\(317\) 31.2030 1.75253 0.876267 0.481826i \(-0.160026\pi\)
0.876267 + 0.481826i \(0.160026\pi\)
\(318\) −15.6670 −0.878562
\(319\) 3.93507 0.220322
\(320\) −7.16789 −0.400697
\(321\) −15.0380 −0.839339
\(322\) 3.40173 0.189571
\(323\) −12.9153 −0.718626
\(324\) −3.92463 −0.218035
\(325\) −4.49816 −0.249513
\(326\) −32.5751 −1.80417
\(327\) 10.2259 0.565495
\(328\) 12.5811 0.694673
\(329\) −4.86560 −0.268249
\(330\) −18.9089 −1.04090
\(331\) 16.8025 0.923551 0.461775 0.886997i \(-0.347213\pi\)
0.461775 + 0.886997i \(0.347213\pi\)
\(332\) −10.6489 −0.584434
\(333\) 4.25307 0.233067
\(334\) 19.4563 1.06460
\(335\) 0.456729 0.0249538
\(336\) 4.01061 0.218797
\(337\) 7.57943 0.412878 0.206439 0.978459i \(-0.433813\pi\)
0.206439 + 0.978459i \(0.433813\pi\)
\(338\) 10.5876 0.575887
\(339\) −4.17752 −0.226892
\(340\) −2.94944 −0.159956
\(341\) 1.65168 0.0894437
\(342\) −6.52358 −0.352755
\(343\) −7.56285 −0.408355
\(344\) −20.2532 −1.09198
\(345\) −10.1345 −0.545624
\(346\) −5.12062 −0.275286
\(347\) −28.0855 −1.50771 −0.753853 0.657043i \(-0.771807\pi\)
−0.753853 + 0.657043i \(0.771807\pi\)
\(348\) −0.878030 −0.0470673
\(349\) 11.6230 0.622164 0.311082 0.950383i \(-0.399309\pi\)
0.311082 + 0.950383i \(0.399309\pi\)
\(350\) 1.58654 0.0848040
\(351\) 14.3563 0.766280
\(352\) 15.3726 0.819362
\(353\) −6.16556 −0.328160 −0.164080 0.986447i \(-0.552465\pi\)
−0.164080 + 0.986447i \(0.552465\pi\)
\(354\) 8.25499 0.438748
\(355\) −21.1001 −1.11988
\(356\) −2.01474 −0.106781
\(357\) −2.07464 −0.109802
\(358\) 9.80129 0.518014
\(359\) −10.3896 −0.548344 −0.274172 0.961681i \(-0.588404\pi\)
−0.274172 + 0.961681i \(0.588404\pi\)
\(360\) 3.09463 0.163101
\(361\) 7.20342 0.379127
\(362\) 37.5181 1.97191
\(363\) −11.6308 −0.610458
\(364\) 0.914776 0.0479473
\(365\) 29.0069 1.51829
\(366\) −2.57197 −0.134439
\(367\) 0.571294 0.0298213 0.0149107 0.999889i \(-0.495254\pi\)
0.0149107 + 0.999889i \(0.495254\pi\)
\(368\) 18.4567 0.962122
\(369\) −4.48161 −0.233303
\(370\) −15.9066 −0.826943
\(371\) −3.56940 −0.185314
\(372\) −0.368539 −0.0191079
\(373\) 2.95332 0.152917 0.0764586 0.997073i \(-0.475639\pi\)
0.0764586 + 0.997073i \(0.475639\pi\)
\(374\) −17.8135 −0.921115
\(375\) −18.1176 −0.935590
\(376\) −19.3636 −0.998603
\(377\) 2.31243 0.119096
\(378\) −5.06357 −0.260442
\(379\) −26.2705 −1.34942 −0.674711 0.738082i \(-0.735732\pi\)
−0.674711 + 0.738082i \(0.735732\pi\)
\(380\) 5.98404 0.306975
\(381\) 22.2081 1.13775
\(382\) 29.2646 1.49731
\(383\) 3.44503 0.176033 0.0880164 0.996119i \(-0.471947\pi\)
0.0880164 + 0.996119i \(0.471947\pi\)
\(384\) 20.2148 1.03158
\(385\) −4.30800 −0.219556
\(386\) 16.9996 0.865257
\(387\) 7.21456 0.366736
\(388\) −2.78565 −0.141420
\(389\) −7.95150 −0.403157 −0.201578 0.979472i \(-0.564607\pi\)
−0.201578 + 0.979472i \(0.564607\pi\)
\(390\) −11.1118 −0.562667
\(391\) −9.54742 −0.482834
\(392\) −14.7138 −0.743161
\(393\) −20.1323 −1.01554
\(394\) 7.09790 0.357587
\(395\) −13.9656 −0.702683
\(396\) −2.20681 −0.110896
\(397\) 21.7255 1.09037 0.545186 0.838315i \(-0.316459\pi\)
0.545186 + 0.838315i \(0.316459\pi\)
\(398\) −22.8235 −1.14404
\(399\) 4.20917 0.210722
\(400\) 8.60803 0.430402
\(401\) 14.1072 0.704481 0.352241 0.935909i \(-0.385420\pi\)
0.352241 + 0.935909i \(0.385420\pi\)
\(402\) −0.615496 −0.0306981
\(403\) 0.970607 0.0483494
\(404\) −2.42715 −0.120755
\(405\) 10.8612 0.539695
\(406\) −0.815614 −0.0404782
\(407\) −23.5624 −1.16795
\(408\) −8.25645 −0.408755
\(409\) −10.2905 −0.508834 −0.254417 0.967095i \(-0.581884\pi\)
−0.254417 + 0.967095i \(0.581884\pi\)
\(410\) 16.7613 0.827783
\(411\) 18.5350 0.914266
\(412\) 1.84074 0.0906869
\(413\) 1.88072 0.0925444
\(414\) −4.82245 −0.237010
\(415\) 29.4702 1.44663
\(416\) 9.03366 0.442912
\(417\) −3.20795 −0.157094
\(418\) 36.1413 1.76773
\(419\) 19.0965 0.932925 0.466463 0.884541i \(-0.345528\pi\)
0.466463 + 0.884541i \(0.345528\pi\)
\(420\) 0.961242 0.0469038
\(421\) 26.3025 1.28191 0.640953 0.767580i \(-0.278539\pi\)
0.640953 + 0.767580i \(0.278539\pi\)
\(422\) 18.0009 0.876271
\(423\) 6.89768 0.335377
\(424\) −14.2051 −0.689862
\(425\) −4.45283 −0.215994
\(426\) 28.4349 1.37767
\(427\) −0.585969 −0.0283570
\(428\) 6.56392 0.317279
\(429\) −16.4599 −0.794691
\(430\) −26.9826 −1.30122
\(431\) −16.5246 −0.795963 −0.397982 0.917393i \(-0.630289\pi\)
−0.397982 + 0.917393i \(0.630289\pi\)
\(432\) −27.4733 −1.32181
\(433\) 4.52467 0.217442 0.108721 0.994072i \(-0.465325\pi\)
0.108721 + 0.994072i \(0.465325\pi\)
\(434\) −0.342341 −0.0164329
\(435\) 2.42989 0.116504
\(436\) −4.46351 −0.213763
\(437\) 19.3705 0.926615
\(438\) −39.0902 −1.86780
\(439\) −9.06783 −0.432784 −0.216392 0.976307i \(-0.569429\pi\)
−0.216392 + 0.976307i \(0.569429\pi\)
\(440\) −17.1446 −0.817335
\(441\) 5.24134 0.249588
\(442\) −10.4681 −0.497915
\(443\) −12.6542 −0.601219 −0.300609 0.953747i \(-0.597190\pi\)
−0.300609 + 0.953747i \(0.597190\pi\)
\(444\) 5.25748 0.249509
\(445\) 5.57566 0.264312
\(446\) 25.3135 1.19863
\(447\) 12.7943 0.605150
\(448\) 2.20073 0.103975
\(449\) 34.8336 1.64390 0.821950 0.569560i \(-0.192886\pi\)
0.821950 + 0.569560i \(0.192886\pi\)
\(450\) −2.24915 −0.106026
\(451\) 24.8286 1.16913
\(452\) 1.82344 0.0857674
\(453\) −0.785988 −0.0369290
\(454\) −2.07678 −0.0974680
\(455\) −2.53158 −0.118683
\(456\) 16.7513 0.784449
\(457\) −14.3718 −0.672283 −0.336141 0.941812i \(-0.609122\pi\)
−0.336141 + 0.941812i \(0.609122\pi\)
\(458\) 25.2696 1.18077
\(459\) 14.2116 0.663339
\(460\) 4.42360 0.206252
\(461\) −20.9323 −0.974914 −0.487457 0.873147i \(-0.662075\pi\)
−0.487457 + 0.873147i \(0.662075\pi\)
\(462\) 5.80554 0.270098
\(463\) 19.4204 0.902541 0.451271 0.892387i \(-0.350971\pi\)
0.451271 + 0.892387i \(0.350971\pi\)
\(464\) −4.42525 −0.205437
\(465\) 1.01991 0.0472971
\(466\) 45.2561 2.09645
\(467\) 22.0847 1.02196 0.510980 0.859593i \(-0.329283\pi\)
0.510980 + 0.859593i \(0.329283\pi\)
\(468\) −1.29683 −0.0599459
\(469\) −0.140228 −0.00647512
\(470\) −25.7975 −1.18995
\(471\) −16.6339 −0.766448
\(472\) 7.48473 0.344512
\(473\) −39.9694 −1.83780
\(474\) 18.8202 0.864442
\(475\) 9.03420 0.414518
\(476\) 0.905557 0.0415061
\(477\) 5.06013 0.231688
\(478\) −1.95944 −0.0896226
\(479\) 25.4118 1.16109 0.580547 0.814227i \(-0.302839\pi\)
0.580547 + 0.814227i \(0.302839\pi\)
\(480\) 9.49252 0.433272
\(481\) −13.8464 −0.631342
\(482\) 41.6774 1.89835
\(483\) 3.11156 0.141581
\(484\) 5.07671 0.230760
\(485\) 7.70911 0.350053
\(486\) 12.8711 0.583845
\(487\) −36.7671 −1.66608 −0.833039 0.553214i \(-0.813401\pi\)
−0.833039 + 0.553214i \(0.813401\pi\)
\(488\) −2.33198 −0.105564
\(489\) −29.7964 −1.34744
\(490\) −19.6027 −0.885562
\(491\) −12.6572 −0.571212 −0.285606 0.958347i \(-0.592195\pi\)
−0.285606 + 0.958347i \(0.592195\pi\)
\(492\) −5.53999 −0.249762
\(493\) 2.28913 0.103097
\(494\) 21.2383 0.955558
\(495\) 6.10721 0.274499
\(496\) −1.85743 −0.0834011
\(497\) 6.47829 0.290591
\(498\) −39.7145 −1.77965
\(499\) 2.39500 0.107215 0.0536074 0.998562i \(-0.482928\pi\)
0.0536074 + 0.998562i \(0.482928\pi\)
\(500\) 7.90814 0.353663
\(501\) 17.7966 0.795095
\(502\) −21.5917 −0.963683
\(503\) −1.00000 −0.0445878
\(504\) −0.950132 −0.0423222
\(505\) 6.71697 0.298901
\(506\) 26.7169 1.18771
\(507\) 9.68443 0.430101
\(508\) −9.69357 −0.430083
\(509\) 6.46329 0.286480 0.143240 0.989688i \(-0.454248\pi\)
0.143240 + 0.989688i \(0.454248\pi\)
\(510\) −10.9998 −0.487079
\(511\) −8.90589 −0.393973
\(512\) 4.15104 0.183452
\(513\) −28.8334 −1.27303
\(514\) 26.2095 1.15605
\(515\) −5.09414 −0.224475
\(516\) 8.91836 0.392609
\(517\) −38.2139 −1.68065
\(518\) 4.88374 0.214579
\(519\) −4.68382 −0.205597
\(520\) −10.0750 −0.441816
\(521\) 34.3036 1.50287 0.751434 0.659808i \(-0.229363\pi\)
0.751434 + 0.659808i \(0.229363\pi\)
\(522\) 1.15625 0.0506077
\(523\) 30.3662 1.32782 0.663910 0.747812i \(-0.268896\pi\)
0.663910 + 0.747812i \(0.268896\pi\)
\(524\) 8.78754 0.383885
\(525\) 1.45120 0.0633357
\(526\) −11.0261 −0.480761
\(527\) 0.960825 0.0418542
\(528\) 31.4990 1.37082
\(529\) −8.68070 −0.377422
\(530\) −18.9250 −0.822051
\(531\) −2.66620 −0.115703
\(532\) −1.83726 −0.0796552
\(533\) 14.5904 0.631982
\(534\) −7.51386 −0.325157
\(535\) −18.1652 −0.785351
\(536\) −0.558065 −0.0241047
\(537\) 8.96523 0.386878
\(538\) −38.2308 −1.64825
\(539\) −29.0376 −1.25074
\(540\) −6.58464 −0.283358
\(541\) 39.1371 1.68263 0.841317 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(542\) −28.6318 −1.22984
\(543\) 34.3178 1.47272
\(544\) 8.94262 0.383412
\(545\) 12.3525 0.529121
\(546\) 3.41161 0.146003
\(547\) 24.0487 1.02825 0.514125 0.857715i \(-0.328117\pi\)
0.514125 + 0.857715i \(0.328117\pi\)
\(548\) −8.09034 −0.345602
\(549\) 0.830696 0.0354532
\(550\) 12.4605 0.531318
\(551\) −4.64434 −0.197855
\(552\) 12.3831 0.527060
\(553\) 4.28779 0.182335
\(554\) 18.4678 0.784623
\(555\) −14.5497 −0.617602
\(556\) 1.40023 0.0593832
\(557\) −17.2987 −0.732971 −0.366486 0.930424i \(-0.619439\pi\)
−0.366486 + 0.930424i \(0.619439\pi\)
\(558\) 0.485318 0.0205451
\(559\) −23.4879 −0.993433
\(560\) 4.84464 0.204723
\(561\) −16.2940 −0.687934
\(562\) 15.2490 0.643242
\(563\) 5.63944 0.237674 0.118837 0.992914i \(-0.462083\pi\)
0.118837 + 0.992914i \(0.462083\pi\)
\(564\) 8.52665 0.359037
\(565\) −5.04626 −0.212298
\(566\) −43.7551 −1.83917
\(567\) −3.33466 −0.140043
\(568\) 25.7817 1.08177
\(569\) −12.0350 −0.504535 −0.252268 0.967658i \(-0.581176\pi\)
−0.252268 + 0.967658i \(0.581176\pi\)
\(570\) 22.3171 0.934762
\(571\) 29.9404 1.25297 0.626484 0.779434i \(-0.284493\pi\)
0.626484 + 0.779434i \(0.284493\pi\)
\(572\) 7.18456 0.300402
\(573\) 26.7683 1.11826
\(574\) −5.14617 −0.214797
\(575\) 6.67839 0.278508
\(576\) −3.11985 −0.129994
\(577\) 17.6080 0.733032 0.366516 0.930412i \(-0.380551\pi\)
0.366516 + 0.930412i \(0.380551\pi\)
\(578\) 17.3111 0.720046
\(579\) 15.5495 0.646216
\(580\) −1.06062 −0.0440399
\(581\) −9.04811 −0.375379
\(582\) −10.3889 −0.430635
\(583\) −28.0337 −1.16104
\(584\) −35.4428 −1.46663
\(585\) 3.58889 0.148382
\(586\) −28.0587 −1.15909
\(587\) −21.2268 −0.876125 −0.438062 0.898945i \(-0.644335\pi\)
−0.438062 + 0.898945i \(0.644335\pi\)
\(588\) 6.47914 0.267195
\(589\) −1.94939 −0.0803231
\(590\) 9.97165 0.410526
\(591\) 6.49244 0.267063
\(592\) 26.4976 1.08904
\(593\) 1.78783 0.0734175 0.0367087 0.999326i \(-0.488313\pi\)
0.0367087 + 0.999326i \(0.488313\pi\)
\(594\) −39.7687 −1.63173
\(595\) −2.50607 −0.102739
\(596\) −5.58457 −0.228753
\(597\) −20.8766 −0.854422
\(598\) 15.7001 0.642025
\(599\) −4.15199 −0.169646 −0.0848228 0.996396i \(-0.527032\pi\)
−0.0848228 + 0.996396i \(0.527032\pi\)
\(600\) 5.77536 0.235778
\(601\) −4.87221 −0.198742 −0.0993708 0.995050i \(-0.531683\pi\)
−0.0993708 + 0.995050i \(0.531683\pi\)
\(602\) 8.28438 0.337646
\(603\) 0.198793 0.00809548
\(604\) 0.343075 0.0139595
\(605\) −14.0495 −0.571192
\(606\) −9.05191 −0.367709
\(607\) −22.0579 −0.895303 −0.447652 0.894208i \(-0.647739\pi\)
−0.447652 + 0.894208i \(0.647739\pi\)
\(608\) −18.1434 −0.735812
\(609\) −0.746041 −0.0302311
\(610\) −3.10682 −0.125792
\(611\) −22.4563 −0.908484
\(612\) −1.28376 −0.0518928
\(613\) −19.0369 −0.768895 −0.384447 0.923147i \(-0.625608\pi\)
−0.384447 + 0.923147i \(0.625608\pi\)
\(614\) −47.2081 −1.90516
\(615\) 15.3316 0.618228
\(616\) 5.26383 0.212086
\(617\) −22.0697 −0.888494 −0.444247 0.895904i \(-0.646529\pi\)
−0.444247 + 0.895904i \(0.646529\pi\)
\(618\) 6.86495 0.276149
\(619\) −5.69372 −0.228850 −0.114425 0.993432i \(-0.536503\pi\)
−0.114425 + 0.993432i \(0.536503\pi\)
\(620\) −0.445179 −0.0178788
\(621\) −21.3146 −0.855327
\(622\) −44.3287 −1.77742
\(623\) −1.71187 −0.0685848
\(624\) 18.5103 0.741004
\(625\) −13.0609 −0.522438
\(626\) −9.00303 −0.359833
\(627\) 33.0584 1.32023
\(628\) 7.26050 0.289725
\(629\) −13.7069 −0.546528
\(630\) −1.26583 −0.0504318
\(631\) 37.6327 1.49813 0.749067 0.662494i \(-0.230502\pi\)
0.749067 + 0.662494i \(0.230502\pi\)
\(632\) 17.0641 0.678775
\(633\) 16.4654 0.654442
\(634\) −50.7941 −2.01729
\(635\) 26.8263 1.06457
\(636\) 6.25514 0.248032
\(637\) −17.0639 −0.676095
\(638\) −6.40575 −0.253606
\(639\) −9.18391 −0.363310
\(640\) 24.4185 0.965226
\(641\) 6.63550 0.262086 0.131043 0.991377i \(-0.458167\pi\)
0.131043 + 0.991377i \(0.458167\pi\)
\(642\) 24.4798 0.966140
\(643\) −29.4970 −1.16325 −0.581625 0.813457i \(-0.697583\pi\)
−0.581625 + 0.813457i \(0.697583\pi\)
\(644\) −1.35816 −0.0535191
\(645\) −24.6810 −0.971813
\(646\) 21.0243 0.827190
\(647\) −45.6726 −1.79558 −0.897788 0.440427i \(-0.854827\pi\)
−0.897788 + 0.440427i \(0.854827\pi\)
\(648\) −13.2710 −0.521333
\(649\) 14.7710 0.579813
\(650\) 7.32238 0.287207
\(651\) −0.313139 −0.0122729
\(652\) 13.0058 0.509346
\(653\) −31.6691 −1.23931 −0.619655 0.784875i \(-0.712727\pi\)
−0.619655 + 0.784875i \(0.712727\pi\)
\(654\) −16.6464 −0.650926
\(655\) −24.3190 −0.950220
\(656\) −27.9214 −1.09015
\(657\) 12.6254 0.492563
\(658\) 7.92052 0.308774
\(659\) 9.73186 0.379099 0.189550 0.981871i \(-0.439297\pi\)
0.189550 + 0.981871i \(0.439297\pi\)
\(660\) 7.54950 0.293864
\(661\) −14.5544 −0.566100 −0.283050 0.959105i \(-0.591346\pi\)
−0.283050 + 0.959105i \(0.591346\pi\)
\(662\) −27.3522 −1.06307
\(663\) −9.57513 −0.371867
\(664\) −36.0088 −1.39741
\(665\) 5.08449 0.197168
\(666\) −6.92340 −0.268276
\(667\) −3.43325 −0.132936
\(668\) −7.76803 −0.300554
\(669\) 23.1542 0.895195
\(670\) −0.743491 −0.0287236
\(671\) −4.60214 −0.177664
\(672\) −2.91446 −0.112428
\(673\) −46.1914 −1.78055 −0.890273 0.455427i \(-0.849486\pi\)
−0.890273 + 0.455427i \(0.849486\pi\)
\(674\) −12.3383 −0.475252
\(675\) −9.94095 −0.382627
\(676\) −4.22715 −0.162583
\(677\) −10.6087 −0.407725 −0.203863 0.978999i \(-0.565350\pi\)
−0.203863 + 0.978999i \(0.565350\pi\)
\(678\) 6.80042 0.261169
\(679\) −2.36690 −0.0908333
\(680\) −9.97342 −0.382463
\(681\) −1.89963 −0.0727939
\(682\) −2.68871 −0.102956
\(683\) −35.9985 −1.37745 −0.688723 0.725025i \(-0.741828\pi\)
−0.688723 + 0.725025i \(0.741828\pi\)
\(684\) 2.60458 0.0995885
\(685\) 22.3895 0.855458
\(686\) 12.3113 0.470046
\(687\) 23.1141 0.881858
\(688\) 44.9483 1.71364
\(689\) −16.4739 −0.627606
\(690\) 16.4976 0.628052
\(691\) 24.7115 0.940070 0.470035 0.882648i \(-0.344241\pi\)
0.470035 + 0.882648i \(0.344241\pi\)
\(692\) 2.04444 0.0777178
\(693\) −1.87508 −0.0712282
\(694\) 45.7192 1.73548
\(695\) −3.87506 −0.146989
\(696\) −2.96902 −0.112540
\(697\) 14.4434 0.547083
\(698\) −18.9206 −0.716155
\(699\) 41.3957 1.56573
\(700\) −0.633434 −0.0239416
\(701\) 28.8712 1.09045 0.545225 0.838290i \(-0.316444\pi\)
0.545225 + 0.838290i \(0.316444\pi\)
\(702\) −23.3700 −0.882043
\(703\) 27.8094 1.04885
\(704\) 17.2843 0.651428
\(705\) −23.5970 −0.888713
\(706\) 10.0367 0.377735
\(707\) −2.06229 −0.0775603
\(708\) −3.29585 −0.123866
\(709\) −31.6656 −1.18923 −0.594614 0.804012i \(-0.702695\pi\)
−0.594614 + 0.804012i \(0.702695\pi\)
\(710\) 34.3481 1.28906
\(711\) −6.07856 −0.227964
\(712\) −6.81275 −0.255319
\(713\) −1.44105 −0.0539679
\(714\) 3.37723 0.126389
\(715\) −19.8828 −0.743575
\(716\) −3.91322 −0.146244
\(717\) −1.79230 −0.0669345
\(718\) 16.9129 0.631183
\(719\) 30.1524 1.12449 0.562247 0.826969i \(-0.309937\pi\)
0.562247 + 0.826969i \(0.309937\pi\)
\(720\) −6.86798 −0.255954
\(721\) 1.56403 0.0582477
\(722\) −11.7262 −0.436403
\(723\) 38.1222 1.41778
\(724\) −14.9793 −0.556702
\(725\) −1.60124 −0.0594685
\(726\) 18.9333 0.702681
\(727\) 15.1905 0.563384 0.281692 0.959505i \(-0.409104\pi\)
0.281692 + 0.959505i \(0.409104\pi\)
\(728\) 3.09328 0.114644
\(729\) 29.8887 1.10699
\(730\) −47.2193 −1.74766
\(731\) −23.2512 −0.859977
\(732\) 1.02687 0.0379543
\(733\) 39.8270 1.47104 0.735522 0.677501i \(-0.236937\pi\)
0.735522 + 0.677501i \(0.236937\pi\)
\(734\) −0.929988 −0.0343265
\(735\) −17.9306 −0.661381
\(736\) −13.4122 −0.494381
\(737\) −1.10134 −0.0405682
\(738\) 7.29544 0.268549
\(739\) −26.4367 −0.972489 −0.486245 0.873823i \(-0.661634\pi\)
−0.486245 + 0.873823i \(0.661634\pi\)
\(740\) 6.35079 0.233460
\(741\) 19.4267 0.713657
\(742\) 5.81048 0.213310
\(743\) −26.5745 −0.974923 −0.487462 0.873144i \(-0.662077\pi\)
−0.487462 + 0.873144i \(0.662077\pi\)
\(744\) −1.24620 −0.0456879
\(745\) 15.4549 0.566225
\(746\) −4.80760 −0.176019
\(747\) 12.8270 0.469316
\(748\) 7.11215 0.260046
\(749\) 5.57720 0.203787
\(750\) 29.4930 1.07693
\(751\) −30.1977 −1.10193 −0.550964 0.834529i \(-0.685740\pi\)
−0.550964 + 0.834529i \(0.685740\pi\)
\(752\) 42.9741 1.56711
\(753\) −19.7499 −0.719725
\(754\) −3.76432 −0.137088
\(755\) −0.949439 −0.0345536
\(756\) 2.02166 0.0735270
\(757\) 31.3606 1.13982 0.569911 0.821706i \(-0.306978\pi\)
0.569911 + 0.821706i \(0.306978\pi\)
\(758\) 42.7647 1.55328
\(759\) 24.4379 0.887040
\(760\) 20.2348 0.733992
\(761\) 40.3426 1.46242 0.731210 0.682153i \(-0.238956\pi\)
0.731210 + 0.682153i \(0.238956\pi\)
\(762\) −36.1517 −1.30964
\(763\) −3.79253 −0.137299
\(764\) −11.6841 −0.422715
\(765\) 3.55272 0.128449
\(766\) −5.60803 −0.202626
\(767\) 8.68015 0.313422
\(768\) −21.0390 −0.759180
\(769\) −20.4304 −0.736739 −0.368369 0.929680i \(-0.620084\pi\)
−0.368369 + 0.929680i \(0.620084\pi\)
\(770\) 7.01283 0.252725
\(771\) 23.9738 0.863395
\(772\) −6.78719 −0.244276
\(773\) 45.6859 1.64321 0.821605 0.570058i \(-0.193079\pi\)
0.821605 + 0.570058i \(0.193079\pi\)
\(774\) −11.7443 −0.422140
\(775\) −0.672094 −0.0241423
\(776\) −9.41956 −0.338142
\(777\) 4.46715 0.160258
\(778\) 12.9439 0.464063
\(779\) −29.3038 −1.04992
\(780\) 4.43644 0.158850
\(781\) 50.8798 1.82062
\(782\) 15.5419 0.555776
\(783\) 5.11048 0.182634
\(784\) 32.6547 1.16624
\(785\) −20.0930 −0.717149
\(786\) 32.7727 1.16896
\(787\) 15.1038 0.538392 0.269196 0.963085i \(-0.413242\pi\)
0.269196 + 0.963085i \(0.413242\pi\)
\(788\) −2.83388 −0.100953
\(789\) −10.0856 −0.359055
\(790\) 22.7340 0.808839
\(791\) 1.54933 0.0550880
\(792\) −7.46224 −0.265159
\(793\) −2.70444 −0.0960373
\(794\) −35.3661 −1.25510
\(795\) −17.3107 −0.613947
\(796\) 9.11240 0.322980
\(797\) 13.0697 0.462954 0.231477 0.972840i \(-0.425644\pi\)
0.231477 + 0.972840i \(0.425644\pi\)
\(798\) −6.85195 −0.242556
\(799\) −22.2300 −0.786440
\(800\) −6.25533 −0.221159
\(801\) 2.42683 0.0857478
\(802\) −22.9646 −0.810909
\(803\) −69.9460 −2.46834
\(804\) 0.245740 0.00866659
\(805\) 3.75863 0.132474
\(806\) −1.58001 −0.0556536
\(807\) −34.9697 −1.23099
\(808\) −8.20729 −0.288732
\(809\) −19.0282 −0.668996 −0.334498 0.942396i \(-0.608567\pi\)
−0.334498 + 0.942396i \(0.608567\pi\)
\(810\) −17.6805 −0.621228
\(811\) 21.0749 0.740042 0.370021 0.929023i \(-0.379351\pi\)
0.370021 + 0.929023i \(0.379351\pi\)
\(812\) 0.325638 0.0114277
\(813\) −26.1895 −0.918506
\(814\) 38.3564 1.34439
\(815\) −35.9927 −1.26077
\(816\) 18.3237 0.641459
\(817\) 47.1736 1.65040
\(818\) 16.7516 0.585705
\(819\) −1.10188 −0.0385029
\(820\) −6.69206 −0.233697
\(821\) 10.5704 0.368909 0.184454 0.982841i \(-0.440948\pi\)
0.184454 + 0.982841i \(0.440948\pi\)
\(822\) −30.1725 −1.05239
\(823\) 40.5293 1.41276 0.706381 0.707832i \(-0.250327\pi\)
0.706381 + 0.707832i \(0.250327\pi\)
\(824\) 6.22439 0.216837
\(825\) 11.3976 0.396814
\(826\) −3.06156 −0.106525
\(827\) −47.1057 −1.63803 −0.819013 0.573775i \(-0.805478\pi\)
−0.819013 + 0.573775i \(0.805478\pi\)
\(828\) 1.92539 0.0669120
\(829\) −9.69487 −0.336717 −0.168358 0.985726i \(-0.553847\pi\)
−0.168358 + 0.985726i \(0.553847\pi\)
\(830\) −47.9733 −1.66518
\(831\) 16.8925 0.585994
\(832\) 10.1571 0.352134
\(833\) −16.8919 −0.585269
\(834\) 5.22210 0.180827
\(835\) 21.4975 0.743953
\(836\) −14.4296 −0.499059
\(837\) 2.14504 0.0741436
\(838\) −31.0865 −1.07386
\(839\) 29.2517 1.00988 0.504940 0.863154i \(-0.331515\pi\)
0.504940 + 0.863154i \(0.331515\pi\)
\(840\) 3.25040 0.112149
\(841\) −28.1768 −0.971615
\(842\) −42.8168 −1.47557
\(843\) 13.9483 0.480404
\(844\) −7.18697 −0.247386
\(845\) 11.6984 0.402436
\(846\) −11.2285 −0.386043
\(847\) 4.31356 0.148216
\(848\) 31.5258 1.08260
\(849\) −40.0228 −1.37358
\(850\) 7.24858 0.248624
\(851\) 20.5577 0.704708
\(852\) −11.3528 −0.388940
\(853\) −31.0610 −1.06351 −0.531754 0.846899i \(-0.678467\pi\)
−0.531754 + 0.846899i \(0.678467\pi\)
\(854\) 0.953876 0.0326410
\(855\) −7.20800 −0.246508
\(856\) 22.1956 0.758630
\(857\) −27.3567 −0.934486 −0.467243 0.884129i \(-0.654753\pi\)
−0.467243 + 0.884129i \(0.654753\pi\)
\(858\) 26.7944 0.914747
\(859\) −37.0124 −1.26285 −0.631424 0.775438i \(-0.717529\pi\)
−0.631424 + 0.775438i \(0.717529\pi\)
\(860\) 10.7730 0.367355
\(861\) −4.70719 −0.160421
\(862\) 26.8998 0.916211
\(863\) −37.6253 −1.28078 −0.640389 0.768051i \(-0.721227\pi\)
−0.640389 + 0.768051i \(0.721227\pi\)
\(864\) 19.9644 0.679203
\(865\) −5.65785 −0.192373
\(866\) −7.36554 −0.250291
\(867\) 15.8344 0.537765
\(868\) 0.136682 0.00463928
\(869\) 33.6759 1.14238
\(870\) −3.95553 −0.134105
\(871\) −0.647196 −0.0219294
\(872\) −15.0932 −0.511119
\(873\) 3.35542 0.113564
\(874\) −31.5324 −1.06660
\(875\) 6.71935 0.227156
\(876\) 15.6070 0.527312
\(877\) 35.6727 1.20458 0.602291 0.798277i \(-0.294255\pi\)
0.602291 + 0.798277i \(0.294255\pi\)
\(878\) 14.7612 0.498165
\(879\) −25.6653 −0.865668
\(880\) 38.0493 1.28264
\(881\) −30.3930 −1.02397 −0.511984 0.858995i \(-0.671089\pi\)
−0.511984 + 0.858995i \(0.671089\pi\)
\(882\) −8.53218 −0.287293
\(883\) 23.6017 0.794260 0.397130 0.917762i \(-0.370006\pi\)
0.397130 + 0.917762i \(0.370006\pi\)
\(884\) 4.17944 0.140570
\(885\) 9.12106 0.306601
\(886\) 20.5993 0.692046
\(887\) −18.1804 −0.610437 −0.305219 0.952282i \(-0.598730\pi\)
−0.305219 + 0.952282i \(0.598730\pi\)
\(888\) 17.7779 0.596588
\(889\) −8.23639 −0.276240
\(890\) −9.07640 −0.304242
\(891\) −26.1901 −0.877401
\(892\) −10.1066 −0.338393
\(893\) 45.1017 1.50927
\(894\) −20.8273 −0.696571
\(895\) 10.8296 0.361993
\(896\) −7.49713 −0.250461
\(897\) 14.3609 0.479495
\(898\) −56.7043 −1.89225
\(899\) 0.345513 0.0115235
\(900\) 0.897985 0.0299328
\(901\) −16.3079 −0.543294
\(902\) −40.4175 −1.34576
\(903\) 7.57771 0.252170
\(904\) 6.16589 0.205074
\(905\) 41.4543 1.37799
\(906\) 1.27948 0.0425079
\(907\) −26.8615 −0.891921 −0.445961 0.895053i \(-0.647138\pi\)
−0.445961 + 0.895053i \(0.647138\pi\)
\(908\) 0.829166 0.0275168
\(909\) 2.92359 0.0969694
\(910\) 4.12107 0.136612
\(911\) 42.7019 1.41478 0.707388 0.706825i \(-0.249873\pi\)
0.707388 + 0.706825i \(0.249873\pi\)
\(912\) −37.1764 −1.23103
\(913\) −71.0630 −2.35184
\(914\) 23.3952 0.773846
\(915\) −2.84181 −0.0939473
\(916\) −10.0891 −0.333352
\(917\) 7.46656 0.246568
\(918\) −23.1345 −0.763551
\(919\) −7.59767 −0.250624 −0.125312 0.992117i \(-0.539993\pi\)
−0.125312 + 0.992117i \(0.539993\pi\)
\(920\) 14.9582 0.493158
\(921\) −43.1812 −1.42287
\(922\) 34.0749 1.12220
\(923\) 29.8994 0.984151
\(924\) −2.31790 −0.0762532
\(925\) 9.58790 0.315248
\(926\) −31.6137 −1.03889
\(927\) −2.21724 −0.0728239
\(928\) 3.21577 0.105563
\(929\) −34.5765 −1.13442 −0.567210 0.823574i \(-0.691977\pi\)
−0.567210 + 0.823574i \(0.691977\pi\)
\(930\) −1.66027 −0.0544424
\(931\) 34.2714 1.12320
\(932\) −18.0688 −0.591862
\(933\) −40.5474 −1.32746
\(934\) −35.9509 −1.17635
\(935\) −19.6824 −0.643684
\(936\) −4.38517 −0.143334
\(937\) 40.4353 1.32096 0.660482 0.750842i \(-0.270352\pi\)
0.660482 + 0.750842i \(0.270352\pi\)
\(938\) 0.228271 0.00745333
\(939\) −8.23506 −0.268741
\(940\) 10.2998 0.335943
\(941\) −22.3193 −0.727587 −0.363794 0.931480i \(-0.618519\pi\)
−0.363794 + 0.931480i \(0.618519\pi\)
\(942\) 27.0776 0.882237
\(943\) −21.6623 −0.705423
\(944\) −16.6110 −0.540643
\(945\) −5.59481 −0.181999
\(946\) 65.0647 2.11543
\(947\) −40.0277 −1.30073 −0.650363 0.759623i \(-0.725383\pi\)
−0.650363 + 0.759623i \(0.725383\pi\)
\(948\) −7.51409 −0.244046
\(949\) −41.1036 −1.33428
\(950\) −14.7064 −0.477140
\(951\) −46.4613 −1.50661
\(952\) 3.06210 0.0992433
\(953\) −6.33099 −0.205081 −0.102540 0.994729i \(-0.532697\pi\)
−0.102540 + 0.994729i \(0.532697\pi\)
\(954\) −8.23720 −0.266689
\(955\) 32.3349 1.04633
\(956\) 0.782317 0.0253019
\(957\) −5.85933 −0.189405
\(958\) −41.3669 −1.33650
\(959\) −6.87416 −0.221978
\(960\) 10.6730 0.344470
\(961\) −30.8550 −0.995322
\(962\) 22.5400 0.726720
\(963\) −7.90649 −0.254783
\(964\) −16.6399 −0.535936
\(965\) 18.7831 0.604650
\(966\) −5.06519 −0.162970
\(967\) −12.8388 −0.412867 −0.206433 0.978461i \(-0.566186\pi\)
−0.206433 + 0.978461i \(0.566186\pi\)
\(968\) 17.1667 0.551758
\(969\) 19.2309 0.617785
\(970\) −12.5494 −0.402936
\(971\) 22.1505 0.710842 0.355421 0.934706i \(-0.384338\pi\)
0.355421 + 0.934706i \(0.384338\pi\)
\(972\) −5.13886 −0.164829
\(973\) 1.18975 0.0381415
\(974\) 59.8518 1.91778
\(975\) 6.69778 0.214500
\(976\) 5.17542 0.165661
\(977\) 41.0297 1.31266 0.656328 0.754475i \(-0.272109\pi\)
0.656328 + 0.754475i \(0.272109\pi\)
\(978\) 48.5044 1.55100
\(979\) −13.4449 −0.429701
\(980\) 7.82651 0.250009
\(981\) 5.37646 0.171657
\(982\) 20.6042 0.657506
\(983\) 43.9691 1.40240 0.701199 0.712966i \(-0.252649\pi\)
0.701199 + 0.712966i \(0.252649\pi\)
\(984\) −18.7332 −0.597194
\(985\) 7.84258 0.249885
\(986\) −3.72638 −0.118672
\(987\) 7.24489 0.230607
\(988\) −8.47953 −0.269770
\(989\) 34.8724 1.10888
\(990\) −9.94170 −0.315968
\(991\) 20.6965 0.657446 0.328723 0.944426i \(-0.393382\pi\)
0.328723 + 0.944426i \(0.393382\pi\)
\(992\) 1.34977 0.0428552
\(993\) −25.0190 −0.793955
\(994\) −10.5458 −0.334491
\(995\) −25.2180 −0.799464
\(996\) 15.8562 0.502424
\(997\) 52.9018 1.67542 0.837709 0.546117i \(-0.183895\pi\)
0.837709 + 0.546117i \(0.183895\pi\)
\(998\) −3.89873 −0.123412
\(999\) −30.6006 −0.968160
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 503.2.a.e.1.3 10
3.2 odd 2 4527.2.a.k.1.8 10
4.3 odd 2 8048.2.a.p.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.3 10 1.1 even 1 trivial
4527.2.a.k.1.8 10 3.2 odd 2
8048.2.a.p.1.6 10 4.3 odd 2