Properties

Label 43.4.a.b.1.5
Level 43
Weight 4
Character 43.1
Self dual yes
Analytic conductor 2.537
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-3.17112\) of \(x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.17112 q^{2} +2.46717 q^{3} +9.39827 q^{4} -7.54340 q^{5} +10.2909 q^{6} +4.58222 q^{7} +5.83236 q^{8} -20.9131 q^{9} +O(q^{10})\) \(q+4.17112 q^{2} +2.46717 q^{3} +9.39827 q^{4} -7.54340 q^{5} +10.2909 q^{6} +4.58222 q^{7} +5.83236 q^{8} -20.9131 q^{9} -31.4645 q^{10} +26.9150 q^{11} +23.1872 q^{12} -15.6529 q^{13} +19.1130 q^{14} -18.6109 q^{15} -50.8587 q^{16} +27.2420 q^{17} -87.2309 q^{18} +38.3104 q^{19} -70.8949 q^{20} +11.3051 q^{21} +112.266 q^{22} +82.5575 q^{23} +14.3894 q^{24} -68.0971 q^{25} -65.2903 q^{26} -118.210 q^{27} +43.0650 q^{28} -34.2852 q^{29} -77.6283 q^{30} +119.055 q^{31} -258.797 q^{32} +66.4040 q^{33} +113.630 q^{34} -34.5655 q^{35} -196.547 q^{36} +378.527 q^{37} +159.797 q^{38} -38.6185 q^{39} -43.9958 q^{40} +385.478 q^{41} +47.1551 q^{42} -43.0000 q^{43} +252.955 q^{44} +157.756 q^{45} +344.358 q^{46} +271.022 q^{47} -125.477 q^{48} -322.003 q^{49} -284.041 q^{50} +67.2108 q^{51} -147.110 q^{52} -329.363 q^{53} -493.068 q^{54} -203.031 q^{55} +26.7252 q^{56} +94.5184 q^{57} -143.008 q^{58} -173.956 q^{59} -174.910 q^{60} +54.5012 q^{61} +496.592 q^{62} -95.8283 q^{63} -672.604 q^{64} +118.076 q^{65} +276.979 q^{66} -906.954 q^{67} +256.028 q^{68} +203.684 q^{69} -144.177 q^{70} -621.376 q^{71} -121.972 q^{72} -1025.87 q^{73} +1578.88 q^{74} -168.007 q^{75} +360.052 q^{76} +123.331 q^{77} -161.082 q^{78} -737.945 q^{79} +383.647 q^{80} +273.009 q^{81} +1607.88 q^{82} +558.465 q^{83} +106.249 q^{84} -205.498 q^{85} -179.358 q^{86} -84.5875 q^{87} +156.978 q^{88} +1631.31 q^{89} +658.018 q^{90} -71.7252 q^{91} +775.898 q^{92} +293.729 q^{93} +1130.47 q^{94} -288.991 q^{95} -638.496 q^{96} -406.607 q^{97} -1343.12 q^{98} -562.875 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 7q^{3} + 22q^{4} + 43q^{5} - 3q^{6} + 8q^{7} + 54q^{8} + 81q^{9} + 57q^{10} - 28q^{11} - 157q^{12} + 56q^{13} - 184q^{14} - 124q^{15} - 54q^{16} + 19q^{17} - 81q^{18} - 75q^{19} + 135q^{20} - 18q^{21} - 504q^{22} + 131q^{23} - 567q^{24} + 105q^{25} + 44q^{26} + 238q^{27} - 404q^{28} + 515q^{29} - 396q^{30} + 237q^{31} + 558q^{32} + 540q^{33} - 107q^{34} + 198q^{35} + 73q^{36} + 269q^{37} + 527q^{38} + 290q^{39} + 613q^{40} + 471q^{41} + 362q^{42} - 258q^{43} - 428q^{44} + 334q^{45} - 67q^{46} + 415q^{47} - 989q^{48} + 350q^{49} + 1335q^{50} - 1241q^{51} - 8q^{52} + 450q^{53} + 402q^{54} - 1732q^{55} - 780q^{56} - 1000q^{57} - 1055q^{58} + 356q^{59} - 2732q^{60} - 1328q^{61} + 1603q^{62} - 2290q^{63} + 466q^{64} - 62q^{65} + 156q^{66} - 632q^{67} + 571q^{68} - 1130q^{69} - 1902q^{70} - 144q^{71} + 567q^{72} + 864q^{73} + 1207q^{74} - 2494q^{75} + 1005q^{76} + 2660q^{77} + 2222q^{78} - 1613q^{79} + 2399q^{80} - 102q^{81} + 1673q^{82} - 682q^{83} + 3758q^{84} + 84q^{85} - 258q^{86} + 449q^{87} - 608q^{88} + 3378q^{89} + 930q^{90} - 3900q^{91} + 3491q^{92} + 1879q^{93} + 3197q^{94} - 79q^{95} - 591q^{96} - 55q^{97} + 2398q^{98} - 1612q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.17112 1.47471 0.737357 0.675503i \(-0.236073\pi\)
0.737357 + 0.675503i \(0.236073\pi\)
\(3\) 2.46717 0.474808 0.237404 0.971411i \(-0.423704\pi\)
0.237404 + 0.971411i \(0.423704\pi\)
\(4\) 9.39827 1.17478
\(5\) −7.54340 −0.674702 −0.337351 0.941379i \(-0.609531\pi\)
−0.337351 + 0.941379i \(0.609531\pi\)
\(6\) 10.2909 0.700206
\(7\) 4.58222 0.247417 0.123708 0.992319i \(-0.460521\pi\)
0.123708 + 0.992319i \(0.460521\pi\)
\(8\) 5.83236 0.257756
\(9\) −20.9131 −0.774558
\(10\) −31.4645 −0.994994
\(11\) 26.9150 0.737744 0.368872 0.929480i \(-0.379744\pi\)
0.368872 + 0.929480i \(0.379744\pi\)
\(12\) 23.1872 0.557796
\(13\) −15.6529 −0.333949 −0.166975 0.985961i \(-0.553400\pi\)
−0.166975 + 0.985961i \(0.553400\pi\)
\(14\) 19.1130 0.364869
\(15\) −18.6109 −0.320354
\(16\) −50.8587 −0.794667
\(17\) 27.2420 0.388657 0.194328 0.980937i \(-0.437747\pi\)
0.194328 + 0.980937i \(0.437747\pi\)
\(18\) −87.2309 −1.14225
\(19\) 38.3104 0.462580 0.231290 0.972885i \(-0.425705\pi\)
0.231290 + 0.972885i \(0.425705\pi\)
\(20\) −70.8949 −0.792629
\(21\) 11.3051 0.117475
\(22\) 112.266 1.08796
\(23\) 82.5575 0.748453 0.374227 0.927337i \(-0.377908\pi\)
0.374227 + 0.927337i \(0.377908\pi\)
\(24\) 14.3894 0.122385
\(25\) −68.0971 −0.544777
\(26\) −65.2903 −0.492480
\(27\) −118.210 −0.842574
\(28\) 43.0650 0.290661
\(29\) −34.2852 −0.219538 −0.109769 0.993957i \(-0.535011\pi\)
−0.109769 + 0.993957i \(0.535011\pi\)
\(30\) −77.6283 −0.472431
\(31\) 119.055 0.689770 0.344885 0.938645i \(-0.387918\pi\)
0.344885 + 0.938645i \(0.387918\pi\)
\(32\) −258.797 −1.42966
\(33\) 66.4040 0.350286
\(34\) 113.630 0.573158
\(35\) −34.5655 −0.166933
\(36\) −196.547 −0.909938
\(37\) 378.527 1.68188 0.840939 0.541129i \(-0.182003\pi\)
0.840939 + 0.541129i \(0.182003\pi\)
\(38\) 159.797 0.682173
\(39\) −38.6185 −0.158562
\(40\) −43.9958 −0.173909
\(41\) 385.478 1.46833 0.734166 0.678970i \(-0.237573\pi\)
0.734166 + 0.678970i \(0.237573\pi\)
\(42\) 47.1551 0.173243
\(43\) −43.0000 −0.152499
\(44\) 252.955 0.866689
\(45\) 157.756 0.522596
\(46\) 344.358 1.10376
\(47\) 271.022 0.841120 0.420560 0.907265i \(-0.361834\pi\)
0.420560 + 0.907265i \(0.361834\pi\)
\(48\) −125.477 −0.377314
\(49\) −322.003 −0.938785
\(50\) −284.041 −0.803390
\(51\) 67.2108 0.184537
\(52\) −147.110 −0.392318
\(53\) −329.363 −0.853612 −0.426806 0.904343i \(-0.640361\pi\)
−0.426806 + 0.904343i \(0.640361\pi\)
\(54\) −493.068 −1.24256
\(55\) −203.031 −0.497757
\(56\) 26.7252 0.0637732
\(57\) 94.5184 0.219636
\(58\) −143.008 −0.323756
\(59\) −173.956 −0.383849 −0.191924 0.981410i \(-0.561473\pi\)
−0.191924 + 0.981410i \(0.561473\pi\)
\(60\) −174.910 −0.376346
\(61\) 54.5012 0.114396 0.0571981 0.998363i \(-0.481783\pi\)
0.0571981 + 0.998363i \(0.481783\pi\)
\(62\) 496.592 1.01721
\(63\) −95.8283 −0.191639
\(64\) −672.604 −1.31368
\(65\) 118.076 0.225316
\(66\) 276.979 0.516572
\(67\) −906.954 −1.65376 −0.826881 0.562377i \(-0.809887\pi\)
−0.826881 + 0.562377i \(0.809887\pi\)
\(68\) 256.028 0.456588
\(69\) 203.684 0.355371
\(70\) −144.177 −0.246178
\(71\) −621.376 −1.03864 −0.519322 0.854579i \(-0.673816\pi\)
−0.519322 + 0.854579i \(0.673816\pi\)
\(72\) −121.972 −0.199647
\(73\) −1025.87 −1.64477 −0.822387 0.568928i \(-0.807358\pi\)
−0.822387 + 0.568928i \(0.807358\pi\)
\(74\) 1578.88 2.48029
\(75\) −168.007 −0.258664
\(76\) 360.052 0.543431
\(77\) 123.331 0.182530
\(78\) −161.082 −0.233833
\(79\) −737.945 −1.05095 −0.525476 0.850808i \(-0.676113\pi\)
−0.525476 + 0.850808i \(0.676113\pi\)
\(80\) 383.647 0.536164
\(81\) 273.009 0.374497
\(82\) 1607.88 2.16537
\(83\) 558.465 0.738548 0.369274 0.929321i \(-0.379606\pi\)
0.369274 + 0.929321i \(0.379606\pi\)
\(84\) 106.249 0.138008
\(85\) −205.498 −0.262228
\(86\) −179.358 −0.224892
\(87\) −84.5875 −0.104238
\(88\) 156.978 0.190158
\(89\) 1631.31 1.94291 0.971453 0.237231i \(-0.0762398\pi\)
0.971453 + 0.237231i \(0.0762398\pi\)
\(90\) 658.018 0.770680
\(91\) −71.7252 −0.0826247
\(92\) 775.898 0.879271
\(93\) 293.729 0.327508
\(94\) 1130.47 1.24041
\(95\) −288.991 −0.312104
\(96\) −638.496 −0.678815
\(97\) −406.607 −0.425616 −0.212808 0.977094i \(-0.568261\pi\)
−0.212808 + 0.977094i \(0.568261\pi\)
\(98\) −1343.12 −1.38444
\(99\) −562.875 −0.571425
\(100\) −639.995 −0.639995
\(101\) 1000.43 0.985606 0.492803 0.870141i \(-0.335972\pi\)
0.492803 + 0.870141i \(0.335972\pi\)
\(102\) 280.345 0.272140
\(103\) −1659.81 −1.58783 −0.793913 0.608031i \(-0.791960\pi\)
−0.793913 + 0.608031i \(0.791960\pi\)
\(104\) −91.2935 −0.0860775
\(105\) −85.2792 −0.0792609
\(106\) −1373.81 −1.25883
\(107\) −151.590 −0.136961 −0.0684803 0.997652i \(-0.521815\pi\)
−0.0684803 + 0.997652i \(0.521815\pi\)
\(108\) −1110.97 −0.989842
\(109\) 1092.76 0.960254 0.480127 0.877199i \(-0.340591\pi\)
0.480127 + 0.877199i \(0.340591\pi\)
\(110\) −846.866 −0.734050
\(111\) 933.892 0.798569
\(112\) −233.046 −0.196614
\(113\) −970.442 −0.807889 −0.403945 0.914783i \(-0.632361\pi\)
−0.403945 + 0.914783i \(0.632361\pi\)
\(114\) 394.248 0.323901
\(115\) −622.764 −0.504983
\(116\) −322.222 −0.257910
\(117\) 327.351 0.258663
\(118\) −725.590 −0.566068
\(119\) 124.829 0.0961602
\(120\) −108.545 −0.0825732
\(121\) −606.582 −0.455734
\(122\) 227.331 0.168702
\(123\) 951.042 0.697175
\(124\) 1118.91 0.810331
\(125\) 1456.61 1.04226
\(126\) −399.712 −0.282612
\(127\) 2115.78 1.47831 0.739155 0.673535i \(-0.235225\pi\)
0.739155 + 0.673535i \(0.235225\pi\)
\(128\) −735.139 −0.507638
\(129\) −106.088 −0.0724075
\(130\) 492.511 0.332277
\(131\) −1695.44 −1.13077 −0.565386 0.824826i \(-0.691273\pi\)
−0.565386 + 0.824826i \(0.691273\pi\)
\(132\) 624.083 0.411511
\(133\) 175.547 0.114450
\(134\) −3783.02 −2.43883
\(135\) 891.704 0.568486
\(136\) 158.885 0.100179
\(137\) 1613.51 1.00622 0.503108 0.864224i \(-0.332190\pi\)
0.503108 + 0.864224i \(0.332190\pi\)
\(138\) 849.590 0.524071
\(139\) 2072.45 1.26463 0.632313 0.774713i \(-0.282106\pi\)
0.632313 + 0.774713i \(0.282106\pi\)
\(140\) −324.856 −0.196110
\(141\) 668.658 0.399370
\(142\) −2591.84 −1.53170
\(143\) −421.299 −0.246369
\(144\) 1063.61 0.615515
\(145\) 258.627 0.148123
\(146\) −4279.01 −2.42557
\(147\) −794.438 −0.445742
\(148\) 3557.50 1.97584
\(149\) −811.396 −0.446122 −0.223061 0.974805i \(-0.571605\pi\)
−0.223061 + 0.974805i \(0.571605\pi\)
\(150\) −700.779 −0.381456
\(151\) −944.326 −0.508928 −0.254464 0.967082i \(-0.581899\pi\)
−0.254464 + 0.967082i \(0.581899\pi\)
\(152\) 223.440 0.119233
\(153\) −569.714 −0.301037
\(154\) 514.427 0.269180
\(155\) −898.078 −0.465390
\(156\) −362.947 −0.186276
\(157\) 2138.58 1.08712 0.543558 0.839372i \(-0.317077\pi\)
0.543558 + 0.839372i \(0.317077\pi\)
\(158\) −3078.06 −1.54986
\(159\) −812.595 −0.405302
\(160\) 1952.21 0.964597
\(161\) 378.297 0.185180
\(162\) 1138.75 0.552277
\(163\) −2184.57 −1.04975 −0.524873 0.851180i \(-0.675887\pi\)
−0.524873 + 0.851180i \(0.675887\pi\)
\(164\) 3622.83 1.72497
\(165\) −500.912 −0.236339
\(166\) 2329.43 1.08915
\(167\) 3334.42 1.54506 0.772531 0.634978i \(-0.218991\pi\)
0.772531 + 0.634978i \(0.218991\pi\)
\(168\) 65.9356 0.0302800
\(169\) −1951.99 −0.888478
\(170\) −857.156 −0.386711
\(171\) −801.188 −0.358295
\(172\) −404.126 −0.179153
\(173\) 1169.82 0.514105 0.257052 0.966397i \(-0.417249\pi\)
0.257052 + 0.966397i \(0.417249\pi\)
\(174\) −352.825 −0.153722
\(175\) −312.036 −0.134787
\(176\) −1368.86 −0.586260
\(177\) −429.178 −0.182254
\(178\) 6804.40 2.86523
\(179\) 206.597 0.0862669 0.0431334 0.999069i \(-0.486266\pi\)
0.0431334 + 0.999069i \(0.486266\pi\)
\(180\) 1482.63 0.613937
\(181\) 652.242 0.267849 0.133925 0.990992i \(-0.457242\pi\)
0.133925 + 0.990992i \(0.457242\pi\)
\(182\) −299.175 −0.121848
\(183\) 134.464 0.0543162
\(184\) 481.505 0.192919
\(185\) −2855.38 −1.13477
\(186\) 1225.18 0.482981
\(187\) 733.220 0.286729
\(188\) 2547.14 0.988134
\(189\) −541.664 −0.208467
\(190\) −1205.42 −0.460264
\(191\) −1128.21 −0.427404 −0.213702 0.976899i \(-0.568552\pi\)
−0.213702 + 0.976899i \(0.568552\pi\)
\(192\) −1659.43 −0.623745
\(193\) 355.606 0.132627 0.0663136 0.997799i \(-0.478876\pi\)
0.0663136 + 0.997799i \(0.478876\pi\)
\(194\) −1696.01 −0.627662
\(195\) 291.315 0.106982
\(196\) −3026.27 −1.10287
\(197\) 4347.00 1.57214 0.786068 0.618140i \(-0.212114\pi\)
0.786068 + 0.618140i \(0.212114\pi\)
\(198\) −2347.82 −0.842689
\(199\) 430.375 0.153309 0.0766545 0.997058i \(-0.475576\pi\)
0.0766545 + 0.997058i \(0.475576\pi\)
\(200\) −397.167 −0.140420
\(201\) −2237.61 −0.785219
\(202\) 4172.91 1.45349
\(203\) −157.102 −0.0543174
\(204\) 631.665 0.216791
\(205\) −2907.82 −0.990687
\(206\) −6923.28 −2.34159
\(207\) −1726.53 −0.579720
\(208\) 796.087 0.265378
\(209\) 1031.13 0.341265
\(210\) −355.710 −0.116887
\(211\) 103.702 0.0338347 0.0169174 0.999857i \(-0.494615\pi\)
0.0169174 + 0.999857i \(0.494615\pi\)
\(212\) −3095.44 −1.00281
\(213\) −1533.04 −0.493156
\(214\) −632.302 −0.201978
\(215\) 324.366 0.102891
\(216\) −689.442 −0.217179
\(217\) 545.535 0.170661
\(218\) 4558.05 1.41610
\(219\) −2530.99 −0.780951
\(220\) −1908.14 −0.584757
\(221\) −426.418 −0.129792
\(222\) 3895.38 1.17766
\(223\) 4447.94 1.33568 0.667839 0.744306i \(-0.267219\pi\)
0.667839 + 0.744306i \(0.267219\pi\)
\(224\) −1185.86 −0.353723
\(225\) 1424.12 0.421961
\(226\) −4047.83 −1.19141
\(227\) 509.808 0.149062 0.0745311 0.997219i \(-0.476254\pi\)
0.0745311 + 0.997219i \(0.476254\pi\)
\(228\) 888.310 0.258025
\(229\) −4754.86 −1.37210 −0.686049 0.727556i \(-0.740656\pi\)
−0.686049 + 0.727556i \(0.740656\pi\)
\(230\) −2597.63 −0.744706
\(231\) 304.278 0.0866667
\(232\) −199.964 −0.0565873
\(233\) 893.223 0.251146 0.125573 0.992084i \(-0.459923\pi\)
0.125573 + 0.992084i \(0.459923\pi\)
\(234\) 1365.42 0.381454
\(235\) −2044.43 −0.567506
\(236\) −1634.88 −0.450939
\(237\) −1820.64 −0.499000
\(238\) 520.677 0.141809
\(239\) −3883.75 −1.05112 −0.525562 0.850755i \(-0.676145\pi\)
−0.525562 + 0.850755i \(0.676145\pi\)
\(240\) 946.524 0.254575
\(241\) −3118.83 −0.833616 −0.416808 0.908995i \(-0.636851\pi\)
−0.416808 + 0.908995i \(0.636851\pi\)
\(242\) −2530.13 −0.672078
\(243\) 3865.22 1.02039
\(244\) 512.217 0.134391
\(245\) 2429.00 0.633400
\(246\) 3966.91 1.02813
\(247\) −599.670 −0.154478
\(248\) 694.370 0.177793
\(249\) 1377.83 0.350668
\(250\) 6075.70 1.53704
\(251\) 5196.19 1.30669 0.653347 0.757058i \(-0.273364\pi\)
0.653347 + 0.757058i \(0.273364\pi\)
\(252\) −900.620 −0.225134
\(253\) 2222.04 0.552167
\(254\) 8825.19 2.18009
\(255\) −506.998 −0.124508
\(256\) 2314.47 0.565057
\(257\) −2441.32 −0.592550 −0.296275 0.955103i \(-0.595744\pi\)
−0.296275 + 0.955103i \(0.595744\pi\)
\(258\) −442.508 −0.106780
\(259\) 1734.50 0.416125
\(260\) 1109.71 0.264698
\(261\) 717.008 0.170045
\(262\) −7071.89 −1.66757
\(263\) 5831.01 1.36713 0.683566 0.729889i \(-0.260428\pi\)
0.683566 + 0.729889i \(0.260428\pi\)
\(264\) 387.292 0.0902885
\(265\) 2484.52 0.575934
\(266\) 732.227 0.168781
\(267\) 4024.73 0.922507
\(268\) −8523.80 −1.94281
\(269\) −2605.41 −0.590538 −0.295269 0.955414i \(-0.595409\pi\)
−0.295269 + 0.955414i \(0.595409\pi\)
\(270\) 3719.41 0.838355
\(271\) 1793.84 0.402097 0.201049 0.979581i \(-0.435565\pi\)
0.201049 + 0.979581i \(0.435565\pi\)
\(272\) −1385.49 −0.308853
\(273\) −176.958 −0.0392308
\(274\) 6730.16 1.48388
\(275\) −1832.83 −0.401906
\(276\) 1914.27 0.417485
\(277\) −825.071 −0.178967 −0.0894833 0.995988i \(-0.528522\pi\)
−0.0894833 + 0.995988i \(0.528522\pi\)
\(278\) 8644.45 1.86496
\(279\) −2489.80 −0.534267
\(280\) −201.599 −0.0430279
\(281\) −5114.69 −1.08582 −0.542912 0.839789i \(-0.682678\pi\)
−0.542912 + 0.839789i \(0.682678\pi\)
\(282\) 2789.06 0.588957
\(283\) −3703.40 −0.777895 −0.388947 0.921260i \(-0.627161\pi\)
−0.388947 + 0.921260i \(0.627161\pi\)
\(284\) −5839.86 −1.22018
\(285\) −712.990 −0.148189
\(286\) −1757.29 −0.363324
\(287\) 1766.35 0.363290
\(288\) 5412.23 1.10736
\(289\) −4170.87 −0.848946
\(290\) 1078.77 0.218439
\(291\) −1003.17 −0.202085
\(292\) −9641.37 −1.93225
\(293\) 749.193 0.149380 0.0746899 0.997207i \(-0.476203\pi\)
0.0746899 + 0.997207i \(0.476203\pi\)
\(294\) −3313.70 −0.657343
\(295\) 1312.22 0.258984
\(296\) 2207.71 0.433515
\(297\) −3181.62 −0.621603
\(298\) −3384.43 −0.657902
\(299\) −1292.27 −0.249946
\(300\) −1578.98 −0.303874
\(301\) −197.036 −0.0377307
\(302\) −3938.90 −0.750524
\(303\) 2468.23 0.467973
\(304\) −1948.42 −0.367597
\(305\) −411.125 −0.0771834
\(306\) −2376.35 −0.443944
\(307\) −4761.00 −0.885097 −0.442548 0.896745i \(-0.645925\pi\)
−0.442548 + 0.896745i \(0.645925\pi\)
\(308\) 1159.09 0.214433
\(309\) −4095.04 −0.753912
\(310\) −3745.99 −0.686317
\(311\) −5995.99 −1.09325 −0.546626 0.837377i \(-0.684088\pi\)
−0.546626 + 0.837377i \(0.684088\pi\)
\(312\) −225.237 −0.0408703
\(313\) 398.261 0.0719203 0.0359602 0.999353i \(-0.488551\pi\)
0.0359602 + 0.999353i \(0.488551\pi\)
\(314\) 8920.27 1.60318
\(315\) 722.871 0.129299
\(316\) −6935.40 −1.23464
\(317\) 4657.19 0.825154 0.412577 0.910923i \(-0.364629\pi\)
0.412577 + 0.910923i \(0.364629\pi\)
\(318\) −3389.43 −0.597704
\(319\) −922.786 −0.161963
\(320\) 5073.72 0.886342
\(321\) −374.000 −0.0650300
\(322\) 1577.92 0.273088
\(323\) 1043.65 0.179785
\(324\) 2565.81 0.439954
\(325\) 1065.92 0.181928
\(326\) −9112.11 −1.54808
\(327\) 2696.03 0.455936
\(328\) 2248.25 0.378472
\(329\) 1241.88 0.208107
\(330\) −2089.37 −0.348533
\(331\) 10013.6 1.66282 0.831412 0.555656i \(-0.187533\pi\)
0.831412 + 0.555656i \(0.187533\pi\)
\(332\) 5248.60 0.867634
\(333\) −7916.16 −1.30271
\(334\) 13908.3 2.27852
\(335\) 6841.52 1.11580
\(336\) −574.964 −0.0933538
\(337\) −3872.11 −0.625897 −0.312949 0.949770i \(-0.601317\pi\)
−0.312949 + 0.949770i \(0.601317\pi\)
\(338\) −8141.97 −1.31025
\(339\) −2394.25 −0.383592
\(340\) −1931.32 −0.308061
\(341\) 3204.36 0.508874
\(342\) −3341.85 −0.528382
\(343\) −3047.19 −0.479688
\(344\) −250.791 −0.0393075
\(345\) −1536.47 −0.239770
\(346\) 4879.48 0.758158
\(347\) 10290.8 1.59204 0.796022 0.605267i \(-0.206934\pi\)
0.796022 + 0.605267i \(0.206934\pi\)
\(348\) −794.976 −0.122457
\(349\) 700.572 0.107452 0.0537260 0.998556i \(-0.482890\pi\)
0.0537260 + 0.998556i \(0.482890\pi\)
\(350\) −1301.54 −0.198772
\(351\) 1850.33 0.281377
\(352\) −6965.52 −1.05473
\(353\) 3608.33 0.544057 0.272028 0.962289i \(-0.412306\pi\)
0.272028 + 0.962289i \(0.412306\pi\)
\(354\) −1790.16 −0.268773
\(355\) 4687.29 0.700776
\(356\) 15331.5 2.28250
\(357\) 307.975 0.0456576
\(358\) 861.741 0.127219
\(359\) −12085.9 −1.77680 −0.888399 0.459072i \(-0.848182\pi\)
−0.888399 + 0.459072i \(0.848182\pi\)
\(360\) 920.087 0.134702
\(361\) −5391.31 −0.786020
\(362\) 2720.58 0.395001
\(363\) −1496.54 −0.216386
\(364\) −674.093 −0.0970661
\(365\) 7738.52 1.10973
\(366\) 560.866 0.0801009
\(367\) 968.974 0.137820 0.0689101 0.997623i \(-0.478048\pi\)
0.0689101 + 0.997623i \(0.478048\pi\)
\(368\) −4198.77 −0.594771
\(369\) −8061.53 −1.13731
\(370\) −11910.2 −1.67346
\(371\) −1509.21 −0.211198
\(372\) 2760.54 0.384751
\(373\) 13006.0 1.80543 0.902713 0.430243i \(-0.141572\pi\)
0.902713 + 0.430243i \(0.141572\pi\)
\(374\) 3058.35 0.422844
\(375\) 3593.71 0.494875
\(376\) 1580.70 0.216804
\(377\) 536.664 0.0733146
\(378\) −2259.35 −0.307429
\(379\) 1901.08 0.257657 0.128828 0.991667i \(-0.458878\pi\)
0.128828 + 0.991667i \(0.458878\pi\)
\(380\) −2716.01 −0.366654
\(381\) 5220.00 0.701913
\(382\) −4705.89 −0.630299
\(383\) −9501.65 −1.26765 −0.633827 0.773475i \(-0.718517\pi\)
−0.633827 + 0.773475i \(0.718517\pi\)
\(384\) −1813.71 −0.241031
\(385\) −930.332 −0.123154
\(386\) 1483.27 0.195587
\(387\) 899.262 0.118119
\(388\) −3821.40 −0.500006
\(389\) 4640.26 0.604808 0.302404 0.953180i \(-0.402211\pi\)
0.302404 + 0.953180i \(0.402211\pi\)
\(390\) 1215.11 0.157768
\(391\) 2249.03 0.290891
\(392\) −1878.04 −0.241978
\(393\) −4182.94 −0.536899
\(394\) 18131.9 2.31845
\(395\) 5566.61 0.709080
\(396\) −5290.05 −0.671301
\(397\) 4633.90 0.585815 0.292908 0.956141i \(-0.405377\pi\)
0.292908 + 0.956141i \(0.405377\pi\)
\(398\) 1795.15 0.226087
\(399\) 433.104 0.0543417
\(400\) 3463.33 0.432916
\(401\) −11801.3 −1.46965 −0.734823 0.678258i \(-0.762735\pi\)
−0.734823 + 0.678258i \(0.762735\pi\)
\(402\) −9333.35 −1.15797
\(403\) −1863.56 −0.230348
\(404\) 9402.29 1.15787
\(405\) −2059.41 −0.252674
\(406\) −655.293 −0.0801026
\(407\) 10188.1 1.24080
\(408\) 391.998 0.0475656
\(409\) 5588.72 0.675658 0.337829 0.941207i \(-0.390307\pi\)
0.337829 + 0.941207i \(0.390307\pi\)
\(410\) −12128.9 −1.46098
\(411\) 3980.81 0.477759
\(412\) −15599.4 −1.86535
\(413\) −797.103 −0.0949706
\(414\) −7201.57 −0.854922
\(415\) −4212.72 −0.498300
\(416\) 4050.93 0.477435
\(417\) 5113.09 0.600454
\(418\) 4300.95 0.503269
\(419\) −6908.94 −0.805547 −0.402773 0.915300i \(-0.631954\pi\)
−0.402773 + 0.915300i \(0.631954\pi\)
\(420\) −801.477 −0.0931144
\(421\) 16499.6 1.91007 0.955036 0.296491i \(-0.0958165\pi\)
0.955036 + 0.296491i \(0.0958165\pi\)
\(422\) 432.553 0.0498966
\(423\) −5667.90 −0.651496
\(424\) −1920.96 −0.220024
\(425\) −1855.10 −0.211731
\(426\) −6394.51 −0.727265
\(427\) 249.737 0.0283035
\(428\) −1424.69 −0.160899
\(429\) −1039.42 −0.116978
\(430\) 1352.97 0.151735
\(431\) −7101.67 −0.793679 −0.396839 0.917888i \(-0.629893\pi\)
−0.396839 + 0.917888i \(0.629893\pi\)
\(432\) 6011.99 0.669565
\(433\) −8935.72 −0.991739 −0.495870 0.868397i \(-0.665151\pi\)
−0.495870 + 0.868397i \(0.665151\pi\)
\(434\) 2275.50 0.251676
\(435\) 638.077 0.0703298
\(436\) 10270.1 1.12809
\(437\) 3162.81 0.346219
\(438\) −10557.1 −1.15168
\(439\) 17789.3 1.93403 0.967015 0.254720i \(-0.0819833\pi\)
0.967015 + 0.254720i \(0.0819833\pi\)
\(440\) −1184.15 −0.128300
\(441\) 6734.07 0.727143
\(442\) −1778.64 −0.191406
\(443\) 2744.62 0.294359 0.147180 0.989110i \(-0.452980\pi\)
0.147180 + 0.989110i \(0.452980\pi\)
\(444\) 8776.97 0.938146
\(445\) −12305.6 −1.31088
\(446\) 18552.9 1.96974
\(447\) −2001.85 −0.211822
\(448\) −3082.02 −0.325026
\(449\) −12051.9 −1.26673 −0.633366 0.773853i \(-0.718327\pi\)
−0.633366 + 0.773853i \(0.718327\pi\)
\(450\) 5940.17 0.622272
\(451\) 10375.2 1.08325
\(452\) −9120.47 −0.949095
\(453\) −2329.81 −0.241643
\(454\) 2126.47 0.219824
\(455\) 541.052 0.0557471
\(456\) 551.265 0.0566126
\(457\) 130.425 0.0133502 0.00667509 0.999978i \(-0.497875\pi\)
0.00667509 + 0.999978i \(0.497875\pi\)
\(458\) −19833.1 −2.02345
\(459\) −3220.28 −0.327472
\(460\) −5852.91 −0.593246
\(461\) −981.122 −0.0991223 −0.0495612 0.998771i \(-0.515782\pi\)
−0.0495612 + 0.998771i \(0.515782\pi\)
\(462\) 1269.18 0.127809
\(463\) 6359.89 0.638378 0.319189 0.947691i \(-0.396589\pi\)
0.319189 + 0.947691i \(0.396589\pi\)
\(464\) 1743.70 0.174460
\(465\) −2215.71 −0.220970
\(466\) 3725.75 0.370369
\(467\) −14159.9 −1.40309 −0.701546 0.712624i \(-0.747506\pi\)
−0.701546 + 0.712624i \(0.747506\pi\)
\(468\) 3076.53 0.303873
\(469\) −4155.86 −0.409168
\(470\) −8527.57 −0.836909
\(471\) 5276.24 0.516170
\(472\) −1014.57 −0.0989395
\(473\) −1157.35 −0.112505
\(474\) −7594.10 −0.735883
\(475\) −2608.83 −0.252003
\(476\) 1173.18 0.112967
\(477\) 6887.98 0.661172
\(478\) −16199.6 −1.55011
\(479\) −20583.3 −1.96342 −0.981709 0.190390i \(-0.939025\pi\)
−0.981709 + 0.190390i \(0.939025\pi\)
\(480\) 4816.43 0.457998
\(481\) −5925.06 −0.561662
\(482\) −13009.0 −1.22935
\(483\) 933.324 0.0879248
\(484\) −5700.82 −0.535389
\(485\) 3067.20 0.287164
\(486\) 16122.3 1.50478
\(487\) −17672.3 −1.64437 −0.822186 0.569218i \(-0.807246\pi\)
−0.822186 + 0.569218i \(0.807246\pi\)
\(488\) 317.871 0.0294863
\(489\) −5389.71 −0.498428
\(490\) 10131.7 0.934085
\(491\) −3484.50 −0.320271 −0.160136 0.987095i \(-0.551193\pi\)
−0.160136 + 0.987095i \(0.551193\pi\)
\(492\) 8938.15 0.819030
\(493\) −933.998 −0.0853249
\(494\) −2501.30 −0.227811
\(495\) 4245.99 0.385542
\(496\) −6054.97 −0.548137
\(497\) −2847.28 −0.256978
\(498\) 5747.09 0.517135
\(499\) 1393.45 0.125009 0.0625046 0.998045i \(-0.480091\pi\)
0.0625046 + 0.998045i \(0.480091\pi\)
\(500\) 13689.6 1.22444
\(501\) 8226.59 0.733607
\(502\) 21673.9 1.92700
\(503\) 3191.15 0.282875 0.141438 0.989947i \(-0.454828\pi\)
0.141438 + 0.989947i \(0.454828\pi\)
\(504\) −558.905 −0.0493960
\(505\) −7546.63 −0.664991
\(506\) 9268.39 0.814289
\(507\) −4815.89 −0.421856
\(508\) 19884.7 1.73670
\(509\) 10831.8 0.943247 0.471624 0.881800i \(-0.343668\pi\)
0.471624 + 0.881800i \(0.343668\pi\)
\(510\) −2114.75 −0.183613
\(511\) −4700.75 −0.406945
\(512\) 15535.1 1.34094
\(513\) −4528.67 −0.389757
\(514\) −10183.0 −0.873842
\(515\) 12520.6 1.07131
\(516\) −997.048 −0.0850631
\(517\) 7294.56 0.620531
\(518\) 7234.80 0.613666
\(519\) 2886.16 0.244101
\(520\) 688.664 0.0580767
\(521\) 14277.5 1.20059 0.600296 0.799778i \(-0.295049\pi\)
0.600296 + 0.799778i \(0.295049\pi\)
\(522\) 2990.73 0.250768
\(523\) 4181.28 0.349588 0.174794 0.984605i \(-0.444074\pi\)
0.174794 + 0.984605i \(0.444074\pi\)
\(524\) −15934.2 −1.32841
\(525\) −769.847 −0.0639978
\(526\) 24321.9 2.01613
\(527\) 3243.29 0.268084
\(528\) −3377.22 −0.278361
\(529\) −5351.26 −0.439817
\(530\) 10363.2 0.849339
\(531\) 3637.94 0.297313
\(532\) 1649.84 0.134454
\(533\) −6033.87 −0.490348
\(534\) 16787.6 1.36043
\(535\) 1143.51 0.0924077
\(536\) −5289.68 −0.426267
\(537\) 509.710 0.0409602
\(538\) −10867.5 −0.870875
\(539\) −8666.72 −0.692583
\(540\) 8380.48 0.667849
\(541\) 5123.26 0.407146 0.203573 0.979060i \(-0.434745\pi\)
0.203573 + 0.979060i \(0.434745\pi\)
\(542\) 7482.35 0.592979
\(543\) 1609.19 0.127177
\(544\) −7050.15 −0.555648
\(545\) −8243.14 −0.647885
\(546\) −738.116 −0.0578543
\(547\) −15435.6 −1.20654 −0.603271 0.797537i \(-0.706136\pi\)
−0.603271 + 0.797537i \(0.706136\pi\)
\(548\) 15164.2 1.18209
\(549\) −1139.79 −0.0886064
\(550\) −7644.98 −0.592696
\(551\) −1313.48 −0.101554
\(552\) 1187.96 0.0915992
\(553\) −3381.43 −0.260023
\(554\) −3441.47 −0.263925
\(555\) −7044.73 −0.538796
\(556\) 19477.5 1.48566
\(557\) −978.643 −0.0744460 −0.0372230 0.999307i \(-0.511851\pi\)
−0.0372230 + 0.999307i \(0.511851\pi\)
\(558\) −10385.3 −0.787891
\(559\) 673.076 0.0509268
\(560\) 1757.96 0.132656
\(561\) 1808.98 0.136141
\(562\) −21334.0 −1.60128
\(563\) −24362.4 −1.82371 −0.911857 0.410507i \(-0.865352\pi\)
−0.911857 + 0.410507i \(0.865352\pi\)
\(564\) 6284.23 0.469174
\(565\) 7320.43 0.545085
\(566\) −15447.3 −1.14717
\(567\) 1250.99 0.0926569
\(568\) −3624.09 −0.267717
\(569\) −12733.1 −0.938135 −0.469067 0.883162i \(-0.655410\pi\)
−0.469067 + 0.883162i \(0.655410\pi\)
\(570\) −2973.97 −0.218537
\(571\) −6363.95 −0.466415 −0.233208 0.972427i \(-0.574922\pi\)
−0.233208 + 0.972427i \(0.574922\pi\)
\(572\) −3959.48 −0.289430
\(573\) −2783.48 −0.202935
\(574\) 7367.65 0.535749
\(575\) −5621.93 −0.407740
\(576\) 14066.2 1.01752
\(577\) −11981.3 −0.864449 −0.432225 0.901766i \(-0.642271\pi\)
−0.432225 + 0.901766i \(0.642271\pi\)
\(578\) −17397.2 −1.25195
\(579\) 877.340 0.0629724
\(580\) 2430.65 0.174012
\(581\) 2559.01 0.182729
\(582\) −4184.35 −0.298018
\(583\) −8864.80 −0.629747
\(584\) −5983.22 −0.423951
\(585\) −2469.34 −0.174521
\(586\) 3124.97 0.220293
\(587\) 17927.4 1.26055 0.630276 0.776371i \(-0.282942\pi\)
0.630276 + 0.776371i \(0.282942\pi\)
\(588\) −7466.34 −0.523651
\(589\) 4561.04 0.319074
\(590\) 5473.42 0.381927
\(591\) 10724.8 0.746462
\(592\) −19251.4 −1.33653
\(593\) 19927.7 1.37999 0.689993 0.723816i \(-0.257613\pi\)
0.689993 + 0.723816i \(0.257613\pi\)
\(594\) −13270.9 −0.916688
\(595\) −941.636 −0.0648795
\(596\) −7625.72 −0.524097
\(597\) 1061.81 0.0727923
\(598\) −5390.20 −0.368598
\(599\) −4155.72 −0.283469 −0.141735 0.989905i \(-0.545268\pi\)
−0.141735 + 0.989905i \(0.545268\pi\)
\(600\) −979.879 −0.0666723
\(601\) −4104.20 −0.278559 −0.139279 0.990253i \(-0.544479\pi\)
−0.139279 + 0.990253i \(0.544479\pi\)
\(602\) −821.860 −0.0556420
\(603\) 18967.2 1.28093
\(604\) −8875.03 −0.597880
\(605\) 4575.69 0.307485
\(606\) 10295.3 0.690127
\(607\) 5870.16 0.392525 0.196262 0.980551i \(-0.437120\pi\)
0.196262 + 0.980551i \(0.437120\pi\)
\(608\) −9914.61 −0.661333
\(609\) −387.599 −0.0257903
\(610\) −1714.85 −0.113823
\(611\) −4242.29 −0.280891
\(612\) −5354.33 −0.353653
\(613\) 29802.4 1.96363 0.981817 0.189830i \(-0.0607936\pi\)
0.981817 + 0.189830i \(0.0607936\pi\)
\(614\) −19858.7 −1.30527
\(615\) −7174.09 −0.470386
\(616\) 719.308 0.0470483
\(617\) 27394.8 1.78748 0.893738 0.448588i \(-0.148073\pi\)
0.893738 + 0.448588i \(0.148073\pi\)
\(618\) −17080.9 −1.11181
\(619\) −25129.8 −1.63174 −0.815872 0.578232i \(-0.803743\pi\)
−0.815872 + 0.578232i \(0.803743\pi\)
\(620\) −8440.38 −0.546732
\(621\) −9759.11 −0.630627
\(622\) −25010.0 −1.61223
\(623\) 7475.03 0.480708
\(624\) 1964.08 0.126004
\(625\) −2475.65 −0.158442
\(626\) 1661.20 0.106062
\(627\) 2543.96 0.162035
\(628\) 20098.9 1.27713
\(629\) 10311.9 0.653673
\(630\) 3015.19 0.190679
\(631\) −1190.63 −0.0751160 −0.0375580 0.999294i \(-0.511958\pi\)
−0.0375580 + 0.999294i \(0.511958\pi\)
\(632\) −4303.96 −0.270890
\(633\) 255.850 0.0160650
\(634\) 19425.7 1.21687
\(635\) −15960.2 −0.997420
\(636\) −7636.98 −0.476142
\(637\) 5040.29 0.313507
\(638\) −3849.06 −0.238849
\(639\) 12994.9 0.804490
\(640\) 5545.45 0.342505
\(641\) 28828.0 1.77635 0.888174 0.459508i \(-0.151974\pi\)
0.888174 + 0.459508i \(0.151974\pi\)
\(642\) −1560.00 −0.0959007
\(643\) 16853.2 1.03363 0.516816 0.856096i \(-0.327117\pi\)
0.516816 + 0.856096i \(0.327117\pi\)
\(644\) 3555.34 0.217546
\(645\) 800.268 0.0488535
\(646\) 4353.21 0.265131
\(647\) −20234.3 −1.22951 −0.614754 0.788719i \(-0.710745\pi\)
−0.614754 + 0.788719i \(0.710745\pi\)
\(648\) 1592.28 0.0965291
\(649\) −4682.02 −0.283182
\(650\) 4446.08 0.268292
\(651\) 1345.93 0.0810310
\(652\) −20531.2 −1.23323
\(653\) 24965.3 1.49612 0.748062 0.663629i \(-0.230985\pi\)
0.748062 + 0.663629i \(0.230985\pi\)
\(654\) 11245.5 0.672375
\(655\) 12789.4 0.762935
\(656\) −19604.9 −1.16683
\(657\) 21454.0 1.27397
\(658\) 5180.05 0.306899
\(659\) 9033.00 0.533954 0.266977 0.963703i \(-0.413975\pi\)
0.266977 + 0.963703i \(0.413975\pi\)
\(660\) −4707.71 −0.277647
\(661\) −19710.5 −1.15983 −0.579915 0.814677i \(-0.696914\pi\)
−0.579915 + 0.814677i \(0.696914\pi\)
\(662\) 41767.8 2.45219
\(663\) −1052.05 −0.0616261
\(664\) 3257.17 0.190365
\(665\) −1324.22 −0.0772196
\(666\) −33019.3 −1.92113
\(667\) −2830.50 −0.164314
\(668\) 31337.8 1.81511
\(669\) 10973.8 0.634190
\(670\) 28536.8 1.64548
\(671\) 1466.90 0.0843951
\(672\) −2925.73 −0.167950
\(673\) 19293.1 1.10504 0.552521 0.833499i \(-0.313666\pi\)
0.552521 + 0.833499i \(0.313666\pi\)
\(674\) −16151.1 −0.923020
\(675\) 8049.74 0.459014
\(676\) −18345.3 −1.04377
\(677\) −16029.1 −0.909968 −0.454984 0.890500i \(-0.650355\pi\)
−0.454984 + 0.890500i \(0.650355\pi\)
\(678\) −9986.70 −0.565689
\(679\) −1863.16 −0.105304
\(680\) −1198.54 −0.0675908
\(681\) 1257.78 0.0707759
\(682\) 13365.8 0.750443
\(683\) −4293.48 −0.240535 −0.120267 0.992742i \(-0.538375\pi\)
−0.120267 + 0.992742i \(0.538375\pi\)
\(684\) −7529.78 −0.420919
\(685\) −12171.4 −0.678896
\(686\) −12710.2 −0.707403
\(687\) −11731.1 −0.651482
\(688\) 2186.92 0.121186
\(689\) 5155.49 0.285063
\(690\) −6408.80 −0.353592
\(691\) −19631.7 −1.08079 −0.540393 0.841413i \(-0.681724\pi\)
−0.540393 + 0.841413i \(0.681724\pi\)
\(692\) 10994.3 0.603962
\(693\) −2579.22 −0.141380
\(694\) 42924.2 2.34781
\(695\) −15633.3 −0.853246
\(696\) −493.345 −0.0268681
\(697\) 10501.2 0.570677
\(698\) 2922.17 0.158461
\(699\) 2203.74 0.119246
\(700\) −2932.60 −0.158345
\(701\) −13399.7 −0.721967 −0.360983 0.932572i \(-0.617559\pi\)
−0.360983 + 0.932572i \(0.617559\pi\)
\(702\) 7717.95 0.414951
\(703\) 14501.5 0.778003
\(704\) −18103.1 −0.969158
\(705\) −5043.96 −0.269456
\(706\) 15050.8 0.802329
\(707\) 4584.18 0.243855
\(708\) −4033.53 −0.214109
\(709\) −510.833 −0.0270589 −0.0135294 0.999908i \(-0.504307\pi\)
−0.0135294 + 0.999908i \(0.504307\pi\)
\(710\) 19551.3 1.03344
\(711\) 15432.7 0.814023
\(712\) 9514.40 0.500796
\(713\) 9828.87 0.516261
\(714\) 1284.60 0.0673319
\(715\) 3178.03 0.166226
\(716\) 1941.65 0.101345
\(717\) −9581.88 −0.499082
\(718\) −50411.9 −2.62027
\(719\) −29685.4 −1.53975 −0.769875 0.638195i \(-0.779681\pi\)
−0.769875 + 0.638195i \(0.779681\pi\)
\(720\) −8023.24 −0.415290
\(721\) −7605.63 −0.392855
\(722\) −22487.8 −1.15916
\(723\) −7694.69 −0.395807
\(724\) 6129.94 0.314665
\(725\) 2334.72 0.119599
\(726\) −6242.26 −0.319108
\(727\) −10575.9 −0.539529 −0.269765 0.962926i \(-0.586946\pi\)
−0.269765 + 0.962926i \(0.586946\pi\)
\(728\) −418.327 −0.0212970
\(729\) 2164.94 0.109990
\(730\) 32278.3 1.63654
\(731\) −1171.41 −0.0592696
\(732\) 1263.73 0.0638098
\(733\) −4428.93 −0.223173 −0.111587 0.993755i \(-0.535593\pi\)
−0.111587 + 0.993755i \(0.535593\pi\)
\(734\) 4041.71 0.203246
\(735\) 5992.76 0.300743
\(736\) −21365.6 −1.07004
\(737\) −24410.7 −1.22005
\(738\) −33625.6 −1.67720
\(739\) 20131.6 1.00210 0.501052 0.865417i \(-0.332947\pi\)
0.501052 + 0.865417i \(0.332947\pi\)
\(740\) −26835.7 −1.33311
\(741\) −1479.49 −0.0733474
\(742\) −6295.11 −0.311457
\(743\) −7362.63 −0.363538 −0.181769 0.983341i \(-0.558182\pi\)
−0.181769 + 0.983341i \(0.558182\pi\)
\(744\) 1713.13 0.0844173
\(745\) 6120.69 0.300999
\(746\) 54249.5 2.66249
\(747\) −11679.2 −0.572048
\(748\) 6891.00 0.336845
\(749\) −694.621 −0.0338864
\(750\) 14989.8 0.729800
\(751\) −7494.93 −0.364173 −0.182087 0.983283i \(-0.558285\pi\)
−0.182087 + 0.983283i \(0.558285\pi\)
\(752\) −13783.8 −0.668410
\(753\) 12819.9 0.620429
\(754\) 2238.49 0.108118
\(755\) 7123.43 0.343375
\(756\) −5090.70 −0.244903
\(757\) 26060.4 1.25123 0.625615 0.780132i \(-0.284848\pi\)
0.625615 + 0.780132i \(0.284848\pi\)
\(758\) 7929.64 0.379971
\(759\) 5482.15 0.262173
\(760\) −1685.50 −0.0804467
\(761\) −23064.0 −1.09865 −0.549323 0.835610i \(-0.685114\pi\)
−0.549323 + 0.835610i \(0.685114\pi\)
\(762\) 21773.3 1.03512
\(763\) 5007.28 0.237583
\(764\) −10603.2 −0.502108
\(765\) 4297.58 0.203110
\(766\) −39632.6 −1.86943
\(767\) 2722.91 0.128186
\(768\) 5710.21 0.268293
\(769\) 19455.7 0.912341 0.456171 0.889892i \(-0.349221\pi\)
0.456171 + 0.889892i \(0.349221\pi\)
\(770\) −3880.53 −0.181616
\(771\) −6023.16 −0.281347
\(772\) 3342.08 0.155808
\(773\) −14025.4 −0.652599 −0.326300 0.945266i \(-0.605802\pi\)
−0.326300 + 0.945266i \(0.605802\pi\)
\(774\) 3750.93 0.174192
\(775\) −8107.28 −0.375771
\(776\) −2371.48 −0.109705
\(777\) 4279.30 0.197579
\(778\) 19355.1 0.891920
\(779\) 14767.8 0.679220
\(780\) 2737.85 0.125681
\(781\) −16724.3 −0.766253
\(782\) 9381.00 0.428982
\(783\) 4052.85 0.184977
\(784\) 16376.7 0.746021
\(785\) −16132.1 −0.733479
\(786\) −17447.6 −0.791774
\(787\) −41219.6 −1.86699 −0.933495 0.358589i \(-0.883258\pi\)
−0.933495 + 0.358589i \(0.883258\pi\)
\(788\) 40854.3 1.84692
\(789\) 14386.1 0.649124
\(790\) 23219.0 1.04569
\(791\) −4446.78 −0.199885
\(792\) −3282.89 −0.147288
\(793\) −853.104 −0.0382025
\(794\) 19328.6 0.863910
\(795\) 6129.73 0.273458
\(796\) 4044.78 0.180105
\(797\) 10289.5 0.457305 0.228653 0.973508i \(-0.426568\pi\)
0.228653 + 0.973508i \(0.426568\pi\)
\(798\) 1806.53 0.0801385
\(799\) 7383.19 0.326907
\(800\) 17623.3 0.778847
\(801\) −34115.7 −1.50489
\(802\) −49224.6 −2.16731
\(803\) −27611.2 −1.21342
\(804\) −21029.7 −0.922462
\(805\) −2853.65 −0.124941
\(806\) −7773.12 −0.339698
\(807\) −6428.00 −0.280392
\(808\) 5834.85 0.254046
\(809\) −20063.0 −0.871912 −0.435956 0.899968i \(-0.643590\pi\)
−0.435956 + 0.899968i \(0.643590\pi\)
\(810\) −8590.07 −0.372623
\(811\) 26523.6 1.14842 0.574210 0.818708i \(-0.305309\pi\)
0.574210 + 0.818708i \(0.305309\pi\)
\(812\) −1476.49 −0.0638112
\(813\) 4425.73 0.190919
\(814\) 42495.7 1.82982
\(815\) 16479.1 0.708267
\(816\) −3418.25 −0.146646
\(817\) −1647.35 −0.0705427
\(818\) 23311.2 0.996403
\(819\) 1499.99 0.0639976
\(820\) −27328.5 −1.16384
\(821\) −9204.45 −0.391276 −0.195638 0.980676i \(-0.562678\pi\)
−0.195638 + 0.980676i \(0.562678\pi\)
\(822\) 16604.5 0.704558
\(823\) 15569.2 0.659427 0.329713 0.944081i \(-0.393048\pi\)
0.329713 + 0.944081i \(0.393048\pi\)
\(824\) −9680.62 −0.409272
\(825\) −4521.92 −0.190828
\(826\) −3324.82 −0.140055
\(827\) −11786.2 −0.495582 −0.247791 0.968814i \(-0.579705\pi\)
−0.247791 + 0.968814i \(0.579705\pi\)
\(828\) −16226.4 −0.681046
\(829\) −33661.5 −1.41027 −0.705133 0.709075i \(-0.749113\pi\)
−0.705133 + 0.709075i \(0.749113\pi\)
\(830\) −17571.8 −0.734850
\(831\) −2035.59 −0.0849747
\(832\) 10528.2 0.438702
\(833\) −8772.02 −0.364865
\(834\) 21327.3 0.885498
\(835\) −25152.9 −1.04246
\(836\) 9690.79 0.400913
\(837\) −14073.4 −0.581182
\(838\) −28818.1 −1.18795
\(839\) −2686.31 −0.110538 −0.0552691 0.998471i \(-0.517602\pi\)
−0.0552691 + 0.998471i \(0.517602\pi\)
\(840\) −497.379 −0.0204300
\(841\) −23213.5 −0.951803
\(842\) 68821.8 2.81681
\(843\) −12618.8 −0.515558
\(844\) 974.618 0.0397485
\(845\) 14724.6 0.599458
\(846\) −23641.5 −0.960771
\(847\) −2779.49 −0.112756
\(848\) 16750.9 0.678337
\(849\) −9136.93 −0.369350
\(850\) −7737.86 −0.312243
\(851\) 31250.3 1.25881
\(852\) −14407.9 −0.579352
\(853\) 7920.78 0.317940 0.158970 0.987283i \(-0.449183\pi\)
0.158970 + 0.987283i \(0.449183\pi\)
\(854\) 1041.68 0.0417396
\(855\) 6043.68 0.241742
\(856\) −884.129 −0.0353025
\(857\) 36678.1 1.46196 0.730980 0.682398i \(-0.239063\pi\)
0.730980 + 0.682398i \(0.239063\pi\)
\(858\) −4335.54 −0.172509
\(859\) 31205.9 1.23950 0.619750 0.784799i \(-0.287234\pi\)
0.619750 + 0.784799i \(0.287234\pi\)
\(860\) 3048.48 0.120875
\(861\) 4357.88 0.172493
\(862\) −29622.0 −1.17045
\(863\) 38340.4 1.51231 0.756155 0.654392i \(-0.227076\pi\)
0.756155 + 0.654392i \(0.227076\pi\)
\(864\) 30592.3 1.20460
\(865\) −8824.46 −0.346868
\(866\) −37272.0 −1.46253
\(867\) −10290.3 −0.403086
\(868\) 5127.09 0.200489
\(869\) −19861.8 −0.775334
\(870\) 2661.50 0.103716
\(871\) 14196.5 0.552273
\(872\) 6373.38 0.247511
\(873\) 8503.40 0.329664
\(874\) 13192.5 0.510575
\(875\) 6674.51 0.257874
\(876\) −23786.9 −0.917449
\(877\) −38409.6 −1.47890 −0.739452 0.673210i \(-0.764915\pi\)
−0.739452 + 0.673210i \(0.764915\pi\)
\(878\) 74201.5 2.85214
\(879\) 1848.39 0.0709267
\(880\) 10325.9 0.395551
\(881\) 41203.4 1.57568 0.787842 0.615877i \(-0.211198\pi\)
0.787842 + 0.615877i \(0.211198\pi\)
\(882\) 28088.6 1.07233
\(883\) 30194.6 1.15077 0.575384 0.817883i \(-0.304852\pi\)
0.575384 + 0.817883i \(0.304852\pi\)
\(884\) −4007.59 −0.152477
\(885\) 3237.47 0.122967
\(886\) 11448.2 0.434096
\(887\) −13570.8 −0.513712 −0.256856 0.966450i \(-0.582687\pi\)
−0.256856 + 0.966450i \(0.582687\pi\)
\(888\) 5446.80 0.205836
\(889\) 9694.99 0.365759
\(890\) −51328.3 −1.93318
\(891\) 7348.03 0.276283
\(892\) 41803.0 1.56913
\(893\) 10383.0 0.389085
\(894\) −8349.98 −0.312377
\(895\) −1558.44 −0.0582045
\(896\) −3368.57 −0.125598
\(897\) −3188.25 −0.118676
\(898\) −50269.8 −1.86807
\(899\) −4081.82 −0.151431
\(900\) 13384.2 0.495713
\(901\) −8972.51 −0.331762
\(902\) 43276.1 1.59749
\(903\) −486.121 −0.0179148
\(904\) −5659.96 −0.208239
\(905\) −4920.12 −0.180719
\(906\) −9717.94 −0.356354
\(907\) 21966.6 0.804176 0.402088 0.915601i \(-0.368285\pi\)
0.402088 + 0.915601i \(0.368285\pi\)
\(908\) 4791.31 0.175116
\(909\) −20922.0 −0.763409
\(910\) 2256.79 0.0822110
\(911\) 27091.0 0.985252 0.492626 0.870241i \(-0.336037\pi\)
0.492626 + 0.870241i \(0.336037\pi\)
\(912\) −4807.08 −0.174538
\(913\) 15031.1 0.544859
\(914\) 544.020 0.0196877
\(915\) −1014.32 −0.0366473
\(916\) −44687.5 −1.61192
\(917\) −7768.88 −0.279772
\(918\) −13432.2 −0.482928
\(919\) −24533.7 −0.880622 −0.440311 0.897845i \(-0.645132\pi\)
−0.440311 + 0.897845i \(0.645132\pi\)
\(920\) −3632.19 −0.130163
\(921\) −11746.2 −0.420251
\(922\) −4092.38 −0.146177
\(923\) 9726.35 0.346855
\(924\) 2859.69 0.101815
\(925\) −25776.6 −0.916248
\(926\) 26527.9 0.941426
\(927\) 34711.8 1.22986
\(928\) 8872.89 0.313865
\(929\) 9220.40 0.325631 0.162816 0.986657i \(-0.447942\pi\)
0.162816 + 0.986657i \(0.447942\pi\)
\(930\) −9242.02 −0.325868
\(931\) −12336.1 −0.434263
\(932\) 8394.76 0.295042
\(933\) −14793.1 −0.519084
\(934\) −59062.9 −2.06916
\(935\) −5530.97 −0.193457
\(936\) 1909.23 0.0666720
\(937\) 22526.6 0.785392 0.392696 0.919668i \(-0.371542\pi\)
0.392696 + 0.919668i \(0.371542\pi\)
\(938\) −17334.6 −0.603407
\(939\) 982.580 0.0341483
\(940\) −19214.1 −0.666697
\(941\) −776.782 −0.0269100 −0.0134550 0.999909i \(-0.504283\pi\)
−0.0134550 + 0.999909i \(0.504283\pi\)
\(942\) 22007.8 0.761204
\(943\) 31824.1 1.09898
\(944\) 8847.15 0.305032
\(945\) 4085.99 0.140653
\(946\) −4827.43 −0.165913
\(947\) −45677.6 −1.56739 −0.783697 0.621143i \(-0.786669\pi\)
−0.783697 + 0.621143i \(0.786669\pi\)
\(948\) −17110.8 −0.586217
\(949\) 16057.8 0.549271
\(950\) −10881.7 −0.371632
\(951\) 11490.1 0.391790
\(952\) 728.048 0.0247859
\(953\) −41584.5 −1.41349 −0.706743 0.707470i \(-0.749836\pi\)
−0.706743 + 0.707470i \(0.749836\pi\)
\(954\) 28730.6 0.975040
\(955\) 8510.52 0.288371
\(956\) −36500.5 −1.23484
\(957\) −2276.67 −0.0769012
\(958\) −85855.7 −2.89548
\(959\) 7393.47 0.248955
\(960\) 12517.7 0.420842
\(961\) −15617.0 −0.524217
\(962\) −24714.2 −0.828292
\(963\) 3170.22 0.106084
\(964\) −29311.6 −0.979318
\(965\) −2682.48 −0.0894839
\(966\) 3893.01 0.129664
\(967\) 5961.61 0.198255 0.0991274 0.995075i \(-0.468395\pi\)
0.0991274 + 0.995075i \(0.468395\pi\)
\(968\) −3537.80 −0.117468
\(969\) 2574.87 0.0853631
\(970\) 12793.7 0.423485
\(971\) −25246.9 −0.834410 −0.417205 0.908812i \(-0.636990\pi\)
−0.417205 + 0.908812i \(0.636990\pi\)
\(972\) 36326.4 1.19874
\(973\) 9496.43 0.312890
\(974\) −73713.5 −2.42498
\(975\) 2629.81 0.0863807
\(976\) −2771.86 −0.0909068
\(977\) 44505.1 1.45736 0.728682 0.684852i \(-0.240133\pi\)
0.728682 + 0.684852i \(0.240133\pi\)
\(978\) −22481.1 −0.735039
\(979\) 43906.8 1.43337
\(980\) 22828.4 0.744109
\(981\) −22853.0 −0.743772
\(982\) −14534.3 −0.472308
\(983\) 14970.7 0.485750 0.242875 0.970058i \(-0.421910\pi\)
0.242875 + 0.970058i \(0.421910\pi\)
\(984\) 5546.82 0.179701
\(985\) −32791.1 −1.06072
\(986\) −3895.82 −0.125830
\(987\) 3063.94 0.0988109
\(988\) −5635.86 −0.181478
\(989\) −3549.97 −0.114138
\(990\) 17710.6 0.568564
\(991\) −10815.1 −0.346674 −0.173337 0.984863i \(-0.555455\pi\)
−0.173337 + 0.984863i \(0.555455\pi\)
\(992\) −30811.0 −0.986139
\(993\) 24705.2 0.789522
\(994\) −11876.4 −0.378969
\(995\) −3246.49 −0.103438
\(996\) 12949.2 0.411959
\(997\) −39556.1 −1.25652 −0.628262 0.778002i \(-0.716233\pi\)
−0.628262 + 0.778002i \(0.716233\pi\)
\(998\) 5812.27 0.184353
\(999\) −44745.6 −1.41711
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 43.4.a.b.1.5 6
3.2 odd 2 387.4.a.h.1.2 6
4.3 odd 2 688.4.a.i.1.3 6
5.4 even 2 1075.4.a.b.1.2 6
7.6 odd 2 2107.4.a.c.1.5 6
43.42 odd 2 1849.4.a.c.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.a.b.1.5 6 1.1 even 1 trivial
387.4.a.h.1.2 6 3.2 odd 2
688.4.a.i.1.3 6 4.3 odd 2
1075.4.a.b.1.2 6 5.4 even 2
1849.4.a.c.1.2 6 43.42 odd 2
2107.4.a.c.1.5 6 7.6 odd 2