Properties

Label 4006.2.a.g.1.15
Level 4006
Weight 2
Character 4006.1
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.941592 q^{3} +1.00000 q^{4} -0.129713 q^{5} +0.941592 q^{6} +0.350515 q^{7} -1.00000 q^{8} -2.11340 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.941592 q^{3} +1.00000 q^{4} -0.129713 q^{5} +0.941592 q^{6} +0.350515 q^{7} -1.00000 q^{8} -2.11340 q^{9} +0.129713 q^{10} +3.66879 q^{11} -0.941592 q^{12} -0.342580 q^{13} -0.350515 q^{14} +0.122137 q^{15} +1.00000 q^{16} -0.324847 q^{17} +2.11340 q^{18} -0.916569 q^{19} -0.129713 q^{20} -0.330042 q^{21} -3.66879 q^{22} +1.25046 q^{23} +0.941592 q^{24} -4.98317 q^{25} +0.342580 q^{26} +4.81474 q^{27} +0.350515 q^{28} +2.88923 q^{29} -0.122137 q^{30} -8.36762 q^{31} -1.00000 q^{32} -3.45450 q^{33} +0.324847 q^{34} -0.0454664 q^{35} -2.11340 q^{36} +10.3336 q^{37} +0.916569 q^{38} +0.322571 q^{39} +0.129713 q^{40} -1.92216 q^{41} +0.330042 q^{42} -8.00424 q^{43} +3.66879 q^{44} +0.274136 q^{45} -1.25046 q^{46} -2.84720 q^{47} -0.941592 q^{48} -6.87714 q^{49} +4.98317 q^{50} +0.305874 q^{51} -0.342580 q^{52} +7.31739 q^{53} -4.81474 q^{54} -0.475890 q^{55} -0.350515 q^{56} +0.863034 q^{57} -2.88923 q^{58} +3.01002 q^{59} +0.122137 q^{60} +10.1935 q^{61} +8.36762 q^{62} -0.740780 q^{63} +1.00000 q^{64} +0.0444371 q^{65} +3.45450 q^{66} +2.69312 q^{67} -0.324847 q^{68} -1.17742 q^{69} +0.0454664 q^{70} +5.69491 q^{71} +2.11340 q^{72} -10.4653 q^{73} -10.3336 q^{74} +4.69212 q^{75} -0.916569 q^{76} +1.28597 q^{77} -0.322571 q^{78} +1.47461 q^{79} -0.129713 q^{80} +1.80669 q^{81} +1.92216 q^{82} -10.6060 q^{83} -0.330042 q^{84} +0.0421370 q^{85} +8.00424 q^{86} -2.72047 q^{87} -3.66879 q^{88} -17.5017 q^{89} -0.274136 q^{90} -0.120080 q^{91} +1.25046 q^{92} +7.87888 q^{93} +2.84720 q^{94} +0.118891 q^{95} +0.941592 q^{96} -15.5913 q^{97} +6.87714 q^{98} -7.75363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.941592 −0.543629 −0.271814 0.962350i \(-0.587624\pi\)
−0.271814 + 0.962350i \(0.587624\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.129713 −0.0580095 −0.0290047 0.999579i \(-0.509234\pi\)
−0.0290047 + 0.999579i \(0.509234\pi\)
\(6\) 0.941592 0.384403
\(7\) 0.350515 0.132482 0.0662411 0.997804i \(-0.478899\pi\)
0.0662411 + 0.997804i \(0.478899\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.11340 −0.704468
\(10\) 0.129713 0.0410189
\(11\) 3.66879 1.10618 0.553091 0.833121i \(-0.313448\pi\)
0.553091 + 0.833121i \(0.313448\pi\)
\(12\) −0.941592 −0.271814
\(13\) −0.342580 −0.0950146 −0.0475073 0.998871i \(-0.515128\pi\)
−0.0475073 + 0.998871i \(0.515128\pi\)
\(14\) −0.350515 −0.0936791
\(15\) 0.122137 0.0315356
\(16\) 1.00000 0.250000
\(17\) −0.324847 −0.0787871 −0.0393935 0.999224i \(-0.512543\pi\)
−0.0393935 + 0.999224i \(0.512543\pi\)
\(18\) 2.11340 0.498134
\(19\) −0.916569 −0.210275 −0.105138 0.994458i \(-0.533528\pi\)
−0.105138 + 0.994458i \(0.533528\pi\)
\(20\) −0.129713 −0.0290047
\(21\) −0.330042 −0.0720212
\(22\) −3.66879 −0.782189
\(23\) 1.25046 0.260739 0.130369 0.991466i \(-0.458384\pi\)
0.130369 + 0.991466i \(0.458384\pi\)
\(24\) 0.941592 0.192202
\(25\) −4.98317 −0.996635
\(26\) 0.342580 0.0671855
\(27\) 4.81474 0.926598
\(28\) 0.350515 0.0662411
\(29\) 2.88923 0.536516 0.268258 0.963347i \(-0.413552\pi\)
0.268258 + 0.963347i \(0.413552\pi\)
\(30\) −0.122137 −0.0222990
\(31\) −8.36762 −1.50287 −0.751434 0.659808i \(-0.770638\pi\)
−0.751434 + 0.659808i \(0.770638\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.45450 −0.601352
\(34\) 0.324847 0.0557109
\(35\) −0.0454664 −0.00768523
\(36\) −2.11340 −0.352234
\(37\) 10.3336 1.69884 0.849419 0.527719i \(-0.176953\pi\)
0.849419 + 0.527719i \(0.176953\pi\)
\(38\) 0.916569 0.148687
\(39\) 0.322571 0.0516527
\(40\) 0.129713 0.0205094
\(41\) −1.92216 −0.300190 −0.150095 0.988672i \(-0.547958\pi\)
−0.150095 + 0.988672i \(0.547958\pi\)
\(42\) 0.330042 0.0509266
\(43\) −8.00424 −1.22063 −0.610317 0.792157i \(-0.708958\pi\)
−0.610317 + 0.792157i \(0.708958\pi\)
\(44\) 3.66879 0.553091
\(45\) 0.274136 0.0408658
\(46\) −1.25046 −0.184370
\(47\) −2.84720 −0.415306 −0.207653 0.978203i \(-0.566583\pi\)
−0.207653 + 0.978203i \(0.566583\pi\)
\(48\) −0.941592 −0.135907
\(49\) −6.87714 −0.982448
\(50\) 4.98317 0.704727
\(51\) 0.305874 0.0428309
\(52\) −0.342580 −0.0475073
\(53\) 7.31739 1.00512 0.502560 0.864542i \(-0.332391\pi\)
0.502560 + 0.864542i \(0.332391\pi\)
\(54\) −4.81474 −0.655203
\(55\) −0.475890 −0.0641690
\(56\) −0.350515 −0.0468396
\(57\) 0.863034 0.114312
\(58\) −2.88923 −0.379374
\(59\) 3.01002 0.391871 0.195935 0.980617i \(-0.437226\pi\)
0.195935 + 0.980617i \(0.437226\pi\)
\(60\) 0.122137 0.0157678
\(61\) 10.1935 1.30515 0.652574 0.757725i \(-0.273689\pi\)
0.652574 + 0.757725i \(0.273689\pi\)
\(62\) 8.36762 1.06269
\(63\) −0.740780 −0.0933295
\(64\) 1.00000 0.125000
\(65\) 0.0444371 0.00551175
\(66\) 3.45450 0.425220
\(67\) 2.69312 0.329017 0.164508 0.986376i \(-0.447396\pi\)
0.164508 + 0.986376i \(0.447396\pi\)
\(68\) −0.324847 −0.0393935
\(69\) −1.17742 −0.141745
\(70\) 0.0454664 0.00543428
\(71\) 5.69491 0.675862 0.337931 0.941171i \(-0.390273\pi\)
0.337931 + 0.941171i \(0.390273\pi\)
\(72\) 2.11340 0.249067
\(73\) −10.4653 −1.22487 −0.612436 0.790520i \(-0.709810\pi\)
−0.612436 + 0.790520i \(0.709810\pi\)
\(74\) −10.3336 −1.20126
\(75\) 4.69212 0.541799
\(76\) −0.916569 −0.105138
\(77\) 1.28597 0.146549
\(78\) −0.322571 −0.0365239
\(79\) 1.47461 0.165907 0.0829535 0.996553i \(-0.473565\pi\)
0.0829535 + 0.996553i \(0.473565\pi\)
\(80\) −0.129713 −0.0145024
\(81\) 1.80669 0.200743
\(82\) 1.92216 0.212267
\(83\) −10.6060 −1.16416 −0.582078 0.813133i \(-0.697760\pi\)
−0.582078 + 0.813133i \(0.697760\pi\)
\(84\) −0.330042 −0.0360106
\(85\) 0.0421370 0.00457040
\(86\) 8.00424 0.863119
\(87\) −2.72047 −0.291665
\(88\) −3.66879 −0.391094
\(89\) −17.5017 −1.85518 −0.927588 0.373604i \(-0.878122\pi\)
−0.927588 + 0.373604i \(0.878122\pi\)
\(90\) −0.274136 −0.0288965
\(91\) −0.120080 −0.0125878
\(92\) 1.25046 0.130369
\(93\) 7.87888 0.817002
\(94\) 2.84720 0.293666
\(95\) 0.118891 0.0121980
\(96\) 0.941592 0.0961009
\(97\) −15.5913 −1.58306 −0.791530 0.611130i \(-0.790715\pi\)
−0.791530 + 0.611130i \(0.790715\pi\)
\(98\) 6.87714 0.694696
\(99\) −7.75363 −0.779270
\(100\) −4.98317 −0.498317
\(101\) 6.78412 0.675046 0.337523 0.941317i \(-0.390411\pi\)
0.337523 + 0.941317i \(0.390411\pi\)
\(102\) −0.305874 −0.0302860
\(103\) 11.3369 1.11706 0.558528 0.829485i \(-0.311366\pi\)
0.558528 + 0.829485i \(0.311366\pi\)
\(104\) 0.342580 0.0335927
\(105\) 0.0428108 0.00417791
\(106\) −7.31739 −0.710728
\(107\) −11.4350 −1.10547 −0.552733 0.833358i \(-0.686415\pi\)
−0.552733 + 0.833358i \(0.686415\pi\)
\(108\) 4.81474 0.463299
\(109\) 18.1642 1.73982 0.869909 0.493212i \(-0.164177\pi\)
0.869909 + 0.493212i \(0.164177\pi\)
\(110\) 0.475890 0.0453743
\(111\) −9.73007 −0.923537
\(112\) 0.350515 0.0331206
\(113\) −1.75797 −0.165376 −0.0826881 0.996575i \(-0.526351\pi\)
−0.0826881 + 0.996575i \(0.526351\pi\)
\(114\) −0.863034 −0.0808306
\(115\) −0.162201 −0.0151253
\(116\) 2.88923 0.268258
\(117\) 0.724010 0.0669348
\(118\) −3.01002 −0.277094
\(119\) −0.113864 −0.0104379
\(120\) −0.122137 −0.0111495
\(121\) 2.46002 0.223638
\(122\) −10.1935 −0.922879
\(123\) 1.80989 0.163192
\(124\) −8.36762 −0.751434
\(125\) 1.29495 0.115824
\(126\) 0.740780 0.0659939
\(127\) 19.0525 1.69064 0.845320 0.534261i \(-0.179410\pi\)
0.845320 + 0.534261i \(0.179410\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.53673 0.663572
\(130\) −0.0444371 −0.00389739
\(131\) −12.1222 −1.05912 −0.529562 0.848271i \(-0.677644\pi\)
−0.529562 + 0.848271i \(0.677644\pi\)
\(132\) −3.45450 −0.300676
\(133\) −0.321271 −0.0278578
\(134\) −2.69312 −0.232650
\(135\) −0.624535 −0.0537514
\(136\) 0.324847 0.0278554
\(137\) −9.76659 −0.834416 −0.417208 0.908811i \(-0.636991\pi\)
−0.417208 + 0.908811i \(0.636991\pi\)
\(138\) 1.17742 0.100229
\(139\) −0.211615 −0.0179490 −0.00897448 0.999960i \(-0.502857\pi\)
−0.00897448 + 0.999960i \(0.502857\pi\)
\(140\) −0.0454664 −0.00384261
\(141\) 2.68090 0.225772
\(142\) −5.69491 −0.477907
\(143\) −1.25685 −0.105103
\(144\) −2.11340 −0.176117
\(145\) −0.374771 −0.0311230
\(146\) 10.4653 0.866116
\(147\) 6.47546 0.534087
\(148\) 10.3336 0.849419
\(149\) −0.581612 −0.0476475 −0.0238238 0.999716i \(-0.507584\pi\)
−0.0238238 + 0.999716i \(0.507584\pi\)
\(150\) −4.69212 −0.383110
\(151\) −2.43880 −0.198466 −0.0992332 0.995064i \(-0.531639\pi\)
−0.0992332 + 0.995064i \(0.531639\pi\)
\(152\) 0.916569 0.0743436
\(153\) 0.686534 0.0555030
\(154\) −1.28597 −0.103626
\(155\) 1.08539 0.0871806
\(156\) 0.322571 0.0258263
\(157\) 20.0875 1.60315 0.801577 0.597891i \(-0.203994\pi\)
0.801577 + 0.597891i \(0.203994\pi\)
\(158\) −1.47461 −0.117314
\(159\) −6.89000 −0.546412
\(160\) 0.129713 0.0102547
\(161\) 0.438304 0.0345432
\(162\) −1.80669 −0.141947
\(163\) 6.78425 0.531384 0.265692 0.964058i \(-0.414400\pi\)
0.265692 + 0.964058i \(0.414400\pi\)
\(164\) −1.92216 −0.150095
\(165\) 0.448095 0.0348841
\(166\) 10.6060 0.823182
\(167\) −18.7317 −1.44950 −0.724752 0.689009i \(-0.758046\pi\)
−0.724752 + 0.689009i \(0.758046\pi\)
\(168\) 0.330042 0.0254633
\(169\) −12.8826 −0.990972
\(170\) −0.0421370 −0.00323176
\(171\) 1.93708 0.148132
\(172\) −8.00424 −0.610317
\(173\) −25.2897 −1.92274 −0.961370 0.275258i \(-0.911237\pi\)
−0.961370 + 0.275258i \(0.911237\pi\)
\(174\) 2.72047 0.206239
\(175\) −1.74668 −0.132036
\(176\) 3.66879 0.276545
\(177\) −2.83421 −0.213032
\(178\) 17.5017 1.31181
\(179\) −14.2700 −1.06659 −0.533296 0.845929i \(-0.679047\pi\)
−0.533296 + 0.845929i \(0.679047\pi\)
\(180\) 0.274136 0.0204329
\(181\) −1.28838 −0.0957645 −0.0478823 0.998853i \(-0.515247\pi\)
−0.0478823 + 0.998853i \(0.515247\pi\)
\(182\) 0.120080 0.00890089
\(183\) −9.59815 −0.709516
\(184\) −1.25046 −0.0921850
\(185\) −1.34041 −0.0985487
\(186\) −7.87888 −0.577708
\(187\) −1.19180 −0.0871528
\(188\) −2.84720 −0.207653
\(189\) 1.68764 0.122758
\(190\) −0.118891 −0.00862526
\(191\) −7.65560 −0.553940 −0.276970 0.960879i \(-0.589330\pi\)
−0.276970 + 0.960879i \(0.589330\pi\)
\(192\) −0.941592 −0.0679536
\(193\) 15.6131 1.12386 0.561930 0.827185i \(-0.310059\pi\)
0.561930 + 0.827185i \(0.310059\pi\)
\(194\) 15.5913 1.11939
\(195\) −0.0418417 −0.00299634
\(196\) −6.87714 −0.491224
\(197\) −2.44421 −0.174142 −0.0870712 0.996202i \(-0.527751\pi\)
−0.0870712 + 0.996202i \(0.527751\pi\)
\(198\) 7.75363 0.551027
\(199\) −27.3011 −1.93532 −0.967661 0.252255i \(-0.918828\pi\)
−0.967661 + 0.252255i \(0.918828\pi\)
\(200\) 4.98317 0.352364
\(201\) −2.53582 −0.178863
\(202\) −6.78412 −0.477329
\(203\) 1.01272 0.0710788
\(204\) 0.305874 0.0214155
\(205\) 0.249329 0.0174139
\(206\) −11.3369 −0.789878
\(207\) −2.64272 −0.183682
\(208\) −0.342580 −0.0237537
\(209\) −3.36270 −0.232603
\(210\) −0.0428108 −0.00295423
\(211\) 4.19124 0.288537 0.144269 0.989539i \(-0.453917\pi\)
0.144269 + 0.989539i \(0.453917\pi\)
\(212\) 7.31739 0.502560
\(213\) −5.36229 −0.367418
\(214\) 11.4350 0.781683
\(215\) 1.03825 0.0708084
\(216\) −4.81474 −0.327602
\(217\) −2.93298 −0.199103
\(218\) −18.1642 −1.23024
\(219\) 9.85406 0.665876
\(220\) −0.475890 −0.0320845
\(221\) 0.111286 0.00748592
\(222\) 9.73007 0.653039
\(223\) 4.63209 0.310188 0.155094 0.987900i \(-0.450432\pi\)
0.155094 + 0.987900i \(0.450432\pi\)
\(224\) −0.350515 −0.0234198
\(225\) 10.5315 0.702097
\(226\) 1.75797 0.116939
\(227\) 10.4114 0.691027 0.345513 0.938414i \(-0.387705\pi\)
0.345513 + 0.938414i \(0.387705\pi\)
\(228\) 0.863034 0.0571558
\(229\) −8.95876 −0.592012 −0.296006 0.955186i \(-0.595655\pi\)
−0.296006 + 0.955186i \(0.595655\pi\)
\(230\) 0.162201 0.0106952
\(231\) −1.21086 −0.0796685
\(232\) −2.88923 −0.189687
\(233\) 21.0993 1.38226 0.691132 0.722729i \(-0.257113\pi\)
0.691132 + 0.722729i \(0.257113\pi\)
\(234\) −0.724010 −0.0473300
\(235\) 0.369319 0.0240917
\(236\) 3.01002 0.195935
\(237\) −1.38849 −0.0901918
\(238\) 0.113864 0.00738070
\(239\) 4.13040 0.267174 0.133587 0.991037i \(-0.457350\pi\)
0.133587 + 0.991037i \(0.457350\pi\)
\(240\) 0.122137 0.00788390
\(241\) −13.0441 −0.840242 −0.420121 0.907468i \(-0.638012\pi\)
−0.420121 + 0.907468i \(0.638012\pi\)
\(242\) −2.46002 −0.158136
\(243\) −16.1454 −1.03573
\(244\) 10.1935 0.652574
\(245\) 0.892055 0.0569913
\(246\) −1.80989 −0.115394
\(247\) 0.313998 0.0199792
\(248\) 8.36762 0.531344
\(249\) 9.98649 0.632868
\(250\) −1.29495 −0.0818998
\(251\) 20.2778 1.27992 0.639962 0.768406i \(-0.278950\pi\)
0.639962 + 0.768406i \(0.278950\pi\)
\(252\) −0.740780 −0.0466648
\(253\) 4.58767 0.288424
\(254\) −19.0525 −1.19546
\(255\) −0.0396759 −0.00248460
\(256\) 1.00000 0.0625000
\(257\) −0.336181 −0.0209704 −0.0104852 0.999945i \(-0.503338\pi\)
−0.0104852 + 0.999945i \(0.503338\pi\)
\(258\) −7.53673 −0.469216
\(259\) 3.62209 0.225066
\(260\) 0.0444371 0.00275587
\(261\) −6.10610 −0.377958
\(262\) 12.1222 0.748914
\(263\) −24.0852 −1.48516 −0.742578 0.669759i \(-0.766397\pi\)
−0.742578 + 0.669759i \(0.766397\pi\)
\(264\) 3.45450 0.212610
\(265\) −0.949161 −0.0583065
\(266\) 0.321271 0.0196984
\(267\) 16.4795 1.00853
\(268\) 2.69312 0.164508
\(269\) −8.89648 −0.542428 −0.271214 0.962519i \(-0.587425\pi\)
−0.271214 + 0.962519i \(0.587425\pi\)
\(270\) 0.624535 0.0380080
\(271\) −15.9456 −0.968624 −0.484312 0.874895i \(-0.660930\pi\)
−0.484312 + 0.874895i \(0.660930\pi\)
\(272\) −0.324847 −0.0196968
\(273\) 0.113066 0.00684306
\(274\) 9.76659 0.590021
\(275\) −18.2822 −1.10246
\(276\) −1.17742 −0.0708725
\(277\) −24.6530 −1.48125 −0.740626 0.671917i \(-0.765471\pi\)
−0.740626 + 0.671917i \(0.765471\pi\)
\(278\) 0.211615 0.0126918
\(279\) 17.6842 1.05872
\(280\) 0.0454664 0.00271714
\(281\) −12.2409 −0.730233 −0.365116 0.930962i \(-0.618971\pi\)
−0.365116 + 0.930962i \(0.618971\pi\)
\(282\) −2.68090 −0.159645
\(283\) 21.5413 1.28050 0.640249 0.768168i \(-0.278831\pi\)
0.640249 + 0.768168i \(0.278831\pi\)
\(284\) 5.69491 0.337931
\(285\) −0.111947 −0.00663116
\(286\) 1.25685 0.0743193
\(287\) −0.673745 −0.0397699
\(288\) 2.11340 0.124534
\(289\) −16.8945 −0.993793
\(290\) 0.374771 0.0220073
\(291\) 14.6807 0.860597
\(292\) −10.4653 −0.612436
\(293\) −22.8468 −1.33473 −0.667363 0.744733i \(-0.732577\pi\)
−0.667363 + 0.744733i \(0.732577\pi\)
\(294\) −6.47546 −0.377657
\(295\) −0.390439 −0.0227322
\(296\) −10.3336 −0.600630
\(297\) 17.6643 1.02499
\(298\) 0.581612 0.0336919
\(299\) −0.428382 −0.0247740
\(300\) 4.69212 0.270900
\(301\) −2.80561 −0.161712
\(302\) 2.43880 0.140337
\(303\) −6.38788 −0.366974
\(304\) −0.916569 −0.0525688
\(305\) −1.32223 −0.0757109
\(306\) −0.686534 −0.0392465
\(307\) −9.56090 −0.545670 −0.272835 0.962061i \(-0.587961\pi\)
−0.272835 + 0.962061i \(0.587961\pi\)
\(308\) 1.28597 0.0732747
\(309\) −10.6747 −0.607264
\(310\) −1.08539 −0.0616460
\(311\) −10.2196 −0.579501 −0.289750 0.957102i \(-0.593572\pi\)
−0.289750 + 0.957102i \(0.593572\pi\)
\(312\) −0.322571 −0.0182620
\(313\) −10.7607 −0.608229 −0.304114 0.952636i \(-0.598360\pi\)
−0.304114 + 0.952636i \(0.598360\pi\)
\(314\) −20.0875 −1.13360
\(315\) 0.0960889 0.00541400
\(316\) 1.47461 0.0829535
\(317\) 14.7775 0.829985 0.414993 0.909825i \(-0.363784\pi\)
0.414993 + 0.909825i \(0.363784\pi\)
\(318\) 6.89000 0.386372
\(319\) 10.6000 0.593484
\(320\) −0.129713 −0.00725118
\(321\) 10.7671 0.600963
\(322\) −0.438304 −0.0244258
\(323\) 0.297745 0.0165670
\(324\) 1.80669 0.100372
\(325\) 1.70714 0.0946949
\(326\) −6.78425 −0.375745
\(327\) −17.1033 −0.945815
\(328\) 1.92216 0.106133
\(329\) −0.997985 −0.0550207
\(330\) −0.448095 −0.0246668
\(331\) 4.76669 0.262001 0.131000 0.991382i \(-0.458181\pi\)
0.131000 + 0.991382i \(0.458181\pi\)
\(332\) −10.6060 −0.582078
\(333\) −21.8391 −1.19678
\(334\) 18.7317 1.02495
\(335\) −0.349333 −0.0190861
\(336\) −0.330042 −0.0180053
\(337\) −0.980382 −0.0534048 −0.0267024 0.999643i \(-0.508501\pi\)
−0.0267024 + 0.999643i \(0.508501\pi\)
\(338\) 12.8826 0.700723
\(339\) 1.65529 0.0899033
\(340\) 0.0421370 0.00228520
\(341\) −30.6990 −1.66245
\(342\) −1.93708 −0.104745
\(343\) −4.86415 −0.262639
\(344\) 8.00424 0.431560
\(345\) 0.152727 0.00822255
\(346\) 25.2897 1.35958
\(347\) −23.2784 −1.24965 −0.624826 0.780764i \(-0.714830\pi\)
−0.624826 + 0.780764i \(0.714830\pi\)
\(348\) −2.72047 −0.145833
\(349\) 5.88858 0.315208 0.157604 0.987502i \(-0.449623\pi\)
0.157604 + 0.987502i \(0.449623\pi\)
\(350\) 1.74668 0.0933639
\(351\) −1.64943 −0.0880403
\(352\) −3.66879 −0.195547
\(353\) −17.0303 −0.906434 −0.453217 0.891400i \(-0.649724\pi\)
−0.453217 + 0.891400i \(0.649724\pi\)
\(354\) 2.83421 0.150636
\(355\) −0.738705 −0.0392064
\(356\) −17.5017 −0.927588
\(357\) 0.107213 0.00567434
\(358\) 14.2700 0.754195
\(359\) −15.5151 −0.818855 −0.409428 0.912343i \(-0.634272\pi\)
−0.409428 + 0.912343i \(0.634272\pi\)
\(360\) −0.274136 −0.0144482
\(361\) −18.1599 −0.955784
\(362\) 1.28838 0.0677158
\(363\) −2.31633 −0.121576
\(364\) −0.120080 −0.00629388
\(365\) 1.35749 0.0710542
\(366\) 9.59815 0.501703
\(367\) 26.9568 1.40714 0.703568 0.710628i \(-0.251589\pi\)
0.703568 + 0.710628i \(0.251589\pi\)
\(368\) 1.25046 0.0651846
\(369\) 4.06229 0.211474
\(370\) 1.34041 0.0696845
\(371\) 2.56486 0.133161
\(372\) 7.87888 0.408501
\(373\) −5.36389 −0.277732 −0.138866 0.990311i \(-0.544346\pi\)
−0.138866 + 0.990311i \(0.544346\pi\)
\(374\) 1.19180 0.0616263
\(375\) −1.21931 −0.0629651
\(376\) 2.84720 0.146833
\(377\) −0.989791 −0.0509768
\(378\) −1.68764 −0.0868028
\(379\) −13.8038 −0.709056 −0.354528 0.935045i \(-0.615358\pi\)
−0.354528 + 0.935045i \(0.615358\pi\)
\(380\) 0.118891 0.00609898
\(381\) −17.9397 −0.919080
\(382\) 7.65560 0.391694
\(383\) −23.4129 −1.19634 −0.598171 0.801369i \(-0.704105\pi\)
−0.598171 + 0.801369i \(0.704105\pi\)
\(384\) 0.941592 0.0480504
\(385\) −0.166807 −0.00850126
\(386\) −15.6131 −0.794688
\(387\) 16.9162 0.859898
\(388\) −15.5913 −0.791530
\(389\) −23.9313 −1.21336 −0.606682 0.794945i \(-0.707500\pi\)
−0.606682 + 0.794945i \(0.707500\pi\)
\(390\) 0.0418417 0.00211874
\(391\) −0.406208 −0.0205428
\(392\) 6.87714 0.347348
\(393\) 11.4142 0.575770
\(394\) 2.44421 0.123137
\(395\) −0.191277 −0.00962418
\(396\) −7.75363 −0.389635
\(397\) −4.43559 −0.222616 −0.111308 0.993786i \(-0.535504\pi\)
−0.111308 + 0.993786i \(0.535504\pi\)
\(398\) 27.3011 1.36848
\(399\) 0.302507 0.0151443
\(400\) −4.98317 −0.249159
\(401\) −6.81431 −0.340290 −0.170145 0.985419i \(-0.554424\pi\)
−0.170145 + 0.985419i \(0.554424\pi\)
\(402\) 2.53582 0.126475
\(403\) 2.86658 0.142794
\(404\) 6.78412 0.337523
\(405\) −0.234351 −0.0116450
\(406\) −1.01272 −0.0502603
\(407\) 37.9119 1.87922
\(408\) −0.305874 −0.0151430
\(409\) −24.3559 −1.20432 −0.602161 0.798375i \(-0.705693\pi\)
−0.602161 + 0.798375i \(0.705693\pi\)
\(410\) −0.249329 −0.0123135
\(411\) 9.19615 0.453612
\(412\) 11.3369 0.558528
\(413\) 1.05506 0.0519159
\(414\) 2.64272 0.129883
\(415\) 1.37573 0.0675320
\(416\) 0.342580 0.0167964
\(417\) 0.199255 0.00975757
\(418\) 3.36270 0.164475
\(419\) 29.0451 1.41894 0.709472 0.704734i \(-0.248933\pi\)
0.709472 + 0.704734i \(0.248933\pi\)
\(420\) 0.0428108 0.00208895
\(421\) −14.3006 −0.696968 −0.348484 0.937315i \(-0.613303\pi\)
−0.348484 + 0.937315i \(0.613303\pi\)
\(422\) −4.19124 −0.204027
\(423\) 6.01727 0.292570
\(424\) −7.31739 −0.355364
\(425\) 1.61877 0.0785219
\(426\) 5.36229 0.259804
\(427\) 3.57299 0.172909
\(428\) −11.4350 −0.552733
\(429\) 1.18344 0.0571372
\(430\) −1.03825 −0.0500691
\(431\) −22.7210 −1.09443 −0.547217 0.836991i \(-0.684313\pi\)
−0.547217 + 0.836991i \(0.684313\pi\)
\(432\) 4.81474 0.231649
\(433\) −35.0246 −1.68317 −0.841587 0.540121i \(-0.818378\pi\)
−0.841587 + 0.540121i \(0.818378\pi\)
\(434\) 2.93298 0.140787
\(435\) 0.352881 0.0169194
\(436\) 18.1642 0.869909
\(437\) −1.14613 −0.0548269
\(438\) −9.85406 −0.470845
\(439\) 3.41021 0.162761 0.0813803 0.996683i \(-0.474067\pi\)
0.0813803 + 0.996683i \(0.474067\pi\)
\(440\) 0.475890 0.0226872
\(441\) 14.5342 0.692103
\(442\) −0.111286 −0.00529335
\(443\) −2.49849 −0.118707 −0.0593535 0.998237i \(-0.518904\pi\)
−0.0593535 + 0.998237i \(0.518904\pi\)
\(444\) −9.73007 −0.461768
\(445\) 2.27020 0.107618
\(446\) −4.63209 −0.219336
\(447\) 0.547642 0.0259026
\(448\) 0.350515 0.0165603
\(449\) −16.2151 −0.765237 −0.382619 0.923906i \(-0.624978\pi\)
−0.382619 + 0.923906i \(0.624978\pi\)
\(450\) −10.5315 −0.496458
\(451\) −7.05199 −0.332065
\(452\) −1.75797 −0.0826881
\(453\) 2.29635 0.107892
\(454\) −10.4114 −0.488630
\(455\) 0.0155759 0.000730209 0
\(456\) −0.863034 −0.0404153
\(457\) 13.4972 0.631372 0.315686 0.948864i \(-0.397765\pi\)
0.315686 + 0.948864i \(0.397765\pi\)
\(458\) 8.95876 0.418615
\(459\) −1.56406 −0.0730039
\(460\) −0.162201 −0.00756265
\(461\) −37.6469 −1.75339 −0.876696 0.481046i \(-0.840257\pi\)
−0.876696 + 0.481046i \(0.840257\pi\)
\(462\) 1.21086 0.0563341
\(463\) 31.4858 1.46327 0.731635 0.681697i \(-0.238758\pi\)
0.731635 + 0.681697i \(0.238758\pi\)
\(464\) 2.88923 0.134129
\(465\) −1.02199 −0.0473939
\(466\) −21.0993 −0.977408
\(467\) 6.12073 0.283233 0.141617 0.989922i \(-0.454770\pi\)
0.141617 + 0.989922i \(0.454770\pi\)
\(468\) 0.724010 0.0334674
\(469\) 0.943979 0.0435889
\(470\) −0.369319 −0.0170354
\(471\) −18.9142 −0.871521
\(472\) −3.01002 −0.138547
\(473\) −29.3659 −1.35024
\(474\) 1.38849 0.0637753
\(475\) 4.56742 0.209568
\(476\) −0.113864 −0.00521895
\(477\) −15.4646 −0.708075
\(478\) −4.13040 −0.188920
\(479\) 17.8974 0.817752 0.408876 0.912590i \(-0.365921\pi\)
0.408876 + 0.912590i \(0.365921\pi\)
\(480\) −0.122137 −0.00557476
\(481\) −3.54010 −0.161414
\(482\) 13.0441 0.594141
\(483\) −0.412704 −0.0187787
\(484\) 2.46002 0.111819
\(485\) 2.02240 0.0918325
\(486\) 16.1454 0.732370
\(487\) −0.111862 −0.00506897 −0.00253448 0.999997i \(-0.500807\pi\)
−0.00253448 + 0.999997i \(0.500807\pi\)
\(488\) −10.1935 −0.461439
\(489\) −6.38800 −0.288875
\(490\) −0.892055 −0.0402989
\(491\) −23.4744 −1.05939 −0.529693 0.848189i \(-0.677693\pi\)
−0.529693 + 0.848189i \(0.677693\pi\)
\(492\) 1.80989 0.0815960
\(493\) −0.938558 −0.0422705
\(494\) −0.313998 −0.0141274
\(495\) 1.00575 0.0452050
\(496\) −8.36762 −0.375717
\(497\) 1.99615 0.0895397
\(498\) −9.98649 −0.447505
\(499\) −5.14322 −0.230242 −0.115121 0.993351i \(-0.536726\pi\)
−0.115121 + 0.993351i \(0.536726\pi\)
\(500\) 1.29495 0.0579119
\(501\) 17.6377 0.787992
\(502\) −20.2778 −0.905044
\(503\) −9.95795 −0.444003 −0.222002 0.975046i \(-0.571259\pi\)
−0.222002 + 0.975046i \(0.571259\pi\)
\(504\) 0.740780 0.0329970
\(505\) −0.879990 −0.0391590
\(506\) −4.58767 −0.203947
\(507\) 12.1302 0.538721
\(508\) 19.0525 0.845320
\(509\) −16.0954 −0.713416 −0.356708 0.934216i \(-0.616101\pi\)
−0.356708 + 0.934216i \(0.616101\pi\)
\(510\) 0.0396759 0.00175688
\(511\) −3.66825 −0.162274
\(512\) −1.00000 −0.0441942
\(513\) −4.41304 −0.194841
\(514\) 0.336181 0.0148283
\(515\) −1.47054 −0.0647999
\(516\) 7.53673 0.331786
\(517\) −10.4458 −0.459404
\(518\) −3.62209 −0.159146
\(519\) 23.8126 1.04526
\(520\) −0.0444371 −0.00194870
\(521\) 12.6857 0.555769 0.277884 0.960615i \(-0.410367\pi\)
0.277884 + 0.960615i \(0.410367\pi\)
\(522\) 6.10610 0.267257
\(523\) 29.4170 1.28632 0.643159 0.765733i \(-0.277624\pi\)
0.643159 + 0.765733i \(0.277624\pi\)
\(524\) −12.1222 −0.529562
\(525\) 1.64466 0.0717788
\(526\) 24.0852 1.05016
\(527\) 2.71820 0.118407
\(528\) −3.45450 −0.150338
\(529\) −21.4364 −0.932015
\(530\) 0.949161 0.0412289
\(531\) −6.36138 −0.276060
\(532\) −0.321271 −0.0139289
\(533\) 0.658492 0.0285225
\(534\) −16.4795 −0.713136
\(535\) 1.48327 0.0641276
\(536\) −2.69312 −0.116325
\(537\) 13.4366 0.579830
\(538\) 8.89648 0.383555
\(539\) −25.2308 −1.08677
\(540\) −0.624535 −0.0268757
\(541\) 8.95639 0.385065 0.192533 0.981291i \(-0.438330\pi\)
0.192533 + 0.981291i \(0.438330\pi\)
\(542\) 15.9456 0.684921
\(543\) 1.21313 0.0520603
\(544\) 0.324847 0.0139277
\(545\) −2.35614 −0.100926
\(546\) −0.113066 −0.00483878
\(547\) 40.1406 1.71629 0.858145 0.513407i \(-0.171617\pi\)
0.858145 + 0.513407i \(0.171617\pi\)
\(548\) −9.76659 −0.417208
\(549\) −21.5430 −0.919435
\(550\) 18.2822 0.779556
\(551\) −2.64817 −0.112816
\(552\) 1.17742 0.0501144
\(553\) 0.516875 0.0219797
\(554\) 24.6530 1.04740
\(555\) 1.26212 0.0535739
\(556\) −0.211615 −0.00897448
\(557\) 26.0020 1.10174 0.550870 0.834591i \(-0.314296\pi\)
0.550870 + 0.834591i \(0.314296\pi\)
\(558\) −17.6842 −0.748630
\(559\) 2.74209 0.115978
\(560\) −0.0454664 −0.00192131
\(561\) 1.12219 0.0473788
\(562\) 12.2409 0.516352
\(563\) −16.9325 −0.713621 −0.356811 0.934177i \(-0.616136\pi\)
−0.356811 + 0.934177i \(0.616136\pi\)
\(564\) 2.68090 0.112886
\(565\) 0.228032 0.00959339
\(566\) −21.5413 −0.905449
\(567\) 0.633271 0.0265949
\(568\) −5.69491 −0.238953
\(569\) 4.85638 0.203590 0.101795 0.994805i \(-0.467541\pi\)
0.101795 + 0.994805i \(0.467541\pi\)
\(570\) 0.111947 0.00468894
\(571\) −28.5545 −1.19497 −0.597485 0.801880i \(-0.703833\pi\)
−0.597485 + 0.801880i \(0.703833\pi\)
\(572\) −1.25685 −0.0525517
\(573\) 7.20845 0.301137
\(574\) 0.673745 0.0281216
\(575\) −6.23125 −0.259861
\(576\) −2.11340 −0.0880585
\(577\) −24.5989 −1.02406 −0.512032 0.858967i \(-0.671107\pi\)
−0.512032 + 0.858967i \(0.671107\pi\)
\(578\) 16.8945 0.702717
\(579\) −14.7012 −0.610962
\(580\) −0.374771 −0.0155615
\(581\) −3.71755 −0.154230
\(582\) −14.6807 −0.608534
\(583\) 26.8460 1.11185
\(584\) 10.4653 0.433058
\(585\) −0.0939136 −0.00388285
\(586\) 22.8468 0.943794
\(587\) −14.0595 −0.580296 −0.290148 0.956982i \(-0.593705\pi\)
−0.290148 + 0.956982i \(0.593705\pi\)
\(588\) 6.47546 0.267044
\(589\) 7.66950 0.316016
\(590\) 0.390439 0.0160741
\(591\) 2.30145 0.0946688
\(592\) 10.3336 0.424710
\(593\) −12.9160 −0.530395 −0.265197 0.964194i \(-0.585437\pi\)
−0.265197 + 0.964194i \(0.585437\pi\)
\(594\) −17.6643 −0.724774
\(595\) 0.0147696 0.000605497 0
\(596\) −0.581612 −0.0238238
\(597\) 25.7065 1.05210
\(598\) 0.428382 0.0175178
\(599\) 25.1142 1.02614 0.513070 0.858347i \(-0.328508\pi\)
0.513070 + 0.858347i \(0.328508\pi\)
\(600\) −4.69212 −0.191555
\(601\) 25.2372 1.02944 0.514722 0.857357i \(-0.327895\pi\)
0.514722 + 0.857357i \(0.327895\pi\)
\(602\) 2.80561 0.114348
\(603\) −5.69165 −0.231782
\(604\) −2.43880 −0.0992332
\(605\) −0.319096 −0.0129731
\(606\) 6.38788 0.259490
\(607\) 47.4054 1.92413 0.962063 0.272827i \(-0.0879587\pi\)
0.962063 + 0.272827i \(0.0879587\pi\)
\(608\) 0.916569 0.0371718
\(609\) −0.953567 −0.0386405
\(610\) 1.32223 0.0535357
\(611\) 0.975392 0.0394602
\(612\) 0.686534 0.0277515
\(613\) −18.9950 −0.767200 −0.383600 0.923499i \(-0.625316\pi\)
−0.383600 + 0.923499i \(0.625316\pi\)
\(614\) 9.56090 0.385847
\(615\) −0.234766 −0.00946668
\(616\) −1.28597 −0.0518131
\(617\) 44.5007 1.79153 0.895766 0.444525i \(-0.146628\pi\)
0.895766 + 0.444525i \(0.146628\pi\)
\(618\) 10.6747 0.429400
\(619\) −23.2897 −0.936091 −0.468045 0.883704i \(-0.655042\pi\)
−0.468045 + 0.883704i \(0.655042\pi\)
\(620\) 1.08539 0.0435903
\(621\) 6.02063 0.241600
\(622\) 10.2196 0.409769
\(623\) −6.13461 −0.245778
\(624\) 0.322571 0.0129132
\(625\) 24.7479 0.989916
\(626\) 10.7607 0.430083
\(627\) 3.16629 0.126449
\(628\) 20.0875 0.801577
\(629\) −3.35685 −0.133846
\(630\) −0.0960889 −0.00382827
\(631\) −19.2637 −0.766874 −0.383437 0.923567i \(-0.625260\pi\)
−0.383437 + 0.923567i \(0.625260\pi\)
\(632\) −1.47461 −0.0586570
\(633\) −3.94644 −0.156857
\(634\) −14.7775 −0.586888
\(635\) −2.47136 −0.0980731
\(636\) −6.89000 −0.273206
\(637\) 2.35597 0.0933470
\(638\) −10.6000 −0.419657
\(639\) −12.0357 −0.476123
\(640\) 0.129713 0.00512736
\(641\) 3.98859 0.157540 0.0787699 0.996893i \(-0.474901\pi\)
0.0787699 + 0.996893i \(0.474901\pi\)
\(642\) −10.7671 −0.424945
\(643\) 39.6773 1.56472 0.782360 0.622827i \(-0.214016\pi\)
0.782360 + 0.622827i \(0.214016\pi\)
\(644\) 0.438304 0.0172716
\(645\) −0.977613 −0.0384935
\(646\) −0.297745 −0.0117146
\(647\) 39.8399 1.56627 0.783135 0.621852i \(-0.213619\pi\)
0.783135 + 0.621852i \(0.213619\pi\)
\(648\) −1.80669 −0.0709734
\(649\) 11.0431 0.433480
\(650\) −1.70714 −0.0669594
\(651\) 2.76167 0.108238
\(652\) 6.78425 0.265692
\(653\) −14.6957 −0.575088 −0.287544 0.957768i \(-0.592839\pi\)
−0.287544 + 0.957768i \(0.592839\pi\)
\(654\) 17.1033 0.668792
\(655\) 1.57241 0.0614392
\(656\) −1.92216 −0.0750476
\(657\) 22.1174 0.862884
\(658\) 0.997985 0.0389055
\(659\) −18.1529 −0.707137 −0.353569 0.935409i \(-0.615032\pi\)
−0.353569 + 0.935409i \(0.615032\pi\)
\(660\) 0.448095 0.0174421
\(661\) −8.74244 −0.340042 −0.170021 0.985440i \(-0.554383\pi\)
−0.170021 + 0.985440i \(0.554383\pi\)
\(662\) −4.76669 −0.185263
\(663\) −0.104786 −0.00406956
\(664\) 10.6060 0.411591
\(665\) 0.0416731 0.00161601
\(666\) 21.8391 0.846249
\(667\) 3.61286 0.139890
\(668\) −18.7317 −0.724752
\(669\) −4.36154 −0.168627
\(670\) 0.349333 0.0134959
\(671\) 37.3979 1.44373
\(672\) 0.330042 0.0127317
\(673\) 6.40280 0.246810 0.123405 0.992356i \(-0.460619\pi\)
0.123405 + 0.992356i \(0.460619\pi\)
\(674\) 0.980382 0.0377629
\(675\) −23.9927 −0.923479
\(676\) −12.8826 −0.495486
\(677\) −16.2493 −0.624510 −0.312255 0.949998i \(-0.601084\pi\)
−0.312255 + 0.949998i \(0.601084\pi\)
\(678\) −1.65529 −0.0635712
\(679\) −5.46500 −0.209727
\(680\) −0.0421370 −0.00161588
\(681\) −9.80326 −0.375662
\(682\) 30.6990 1.17553
\(683\) 19.7004 0.753815 0.376908 0.926251i \(-0.376987\pi\)
0.376908 + 0.926251i \(0.376987\pi\)
\(684\) 1.93708 0.0740661
\(685\) 1.26686 0.0484040
\(686\) 4.86415 0.185714
\(687\) 8.43550 0.321834
\(688\) −8.00424 −0.305159
\(689\) −2.50679 −0.0955012
\(690\) −0.152727 −0.00581422
\(691\) −16.9280 −0.643971 −0.321986 0.946745i \(-0.604350\pi\)
−0.321986 + 0.946745i \(0.604350\pi\)
\(692\) −25.2897 −0.961370
\(693\) −2.71777 −0.103239
\(694\) 23.2784 0.883637
\(695\) 0.0274493 0.00104121
\(696\) 2.72047 0.103119
\(697\) 0.624407 0.0236511
\(698\) −5.88858 −0.222886
\(699\) −19.8670 −0.751438
\(700\) −1.74668 −0.0660182
\(701\) 14.6088 0.551768 0.275884 0.961191i \(-0.411029\pi\)
0.275884 + 0.961191i \(0.411029\pi\)
\(702\) 1.64943 0.0622539
\(703\) −9.47148 −0.357224
\(704\) 3.66879 0.138273
\(705\) −0.347748 −0.0130969
\(706\) 17.0303 0.640945
\(707\) 2.37794 0.0894316
\(708\) −2.83421 −0.106516
\(709\) −24.6702 −0.926507 −0.463254 0.886226i \(-0.653318\pi\)
−0.463254 + 0.886226i \(0.653318\pi\)
\(710\) 0.738705 0.0277231
\(711\) −3.11646 −0.116876
\(712\) 17.5017 0.655904
\(713\) −10.4634 −0.391856
\(714\) −0.107213 −0.00401236
\(715\) 0.163030 0.00609699
\(716\) −14.2700 −0.533296
\(717\) −3.88916 −0.145243
\(718\) 15.5151 0.579018
\(719\) 13.8705 0.517283 0.258642 0.965973i \(-0.416725\pi\)
0.258642 + 0.965973i \(0.416725\pi\)
\(720\) 0.274136 0.0102165
\(721\) 3.97375 0.147990
\(722\) 18.1599 0.675842
\(723\) 12.2822 0.456780
\(724\) −1.28838 −0.0478823
\(725\) −14.3975 −0.534710
\(726\) 2.31633 0.0859672
\(727\) −7.88316 −0.292370 −0.146185 0.989257i \(-0.546699\pi\)
−0.146185 + 0.989257i \(0.546699\pi\)
\(728\) 0.120080 0.00445044
\(729\) 9.78231 0.362308
\(730\) −1.35749 −0.0502429
\(731\) 2.60016 0.0961703
\(732\) −9.59815 −0.354758
\(733\) 37.1659 1.37275 0.686377 0.727246i \(-0.259200\pi\)
0.686377 + 0.727246i \(0.259200\pi\)
\(734\) −26.9568 −0.994995
\(735\) −0.839952 −0.0309821
\(736\) −1.25046 −0.0460925
\(737\) 9.88048 0.363952
\(738\) −4.06229 −0.149535
\(739\) 32.2183 1.18517 0.592585 0.805508i \(-0.298108\pi\)
0.592585 + 0.805508i \(0.298108\pi\)
\(740\) −1.34041 −0.0492744
\(741\) −0.295658 −0.0108613
\(742\) −2.56486 −0.0941588
\(743\) −37.0544 −1.35939 −0.679697 0.733493i \(-0.737889\pi\)
−0.679697 + 0.733493i \(0.737889\pi\)
\(744\) −7.87888 −0.288854
\(745\) 0.0754427 0.00276401
\(746\) 5.36389 0.196386
\(747\) 22.4147 0.820110
\(748\) −1.19180 −0.0435764
\(749\) −4.00815 −0.146455
\(750\) 1.21931 0.0445230
\(751\) 49.6468 1.81164 0.905818 0.423667i \(-0.139257\pi\)
0.905818 + 0.423667i \(0.139257\pi\)
\(752\) −2.84720 −0.103827
\(753\) −19.0934 −0.695804
\(754\) 0.989791 0.0360461
\(755\) 0.316344 0.0115129
\(756\) 1.68764 0.0613789
\(757\) −48.3117 −1.75592 −0.877958 0.478737i \(-0.841095\pi\)
−0.877958 + 0.478737i \(0.841095\pi\)
\(758\) 13.8038 0.501378
\(759\) −4.31971 −0.156796
\(760\) −0.118891 −0.00431263
\(761\) 4.36062 0.158072 0.0790362 0.996872i \(-0.474816\pi\)
0.0790362 + 0.996872i \(0.474816\pi\)
\(762\) 17.9397 0.649888
\(763\) 6.36684 0.230495
\(764\) −7.65560 −0.276970
\(765\) −0.0890524 −0.00321970
\(766\) 23.4129 0.845941
\(767\) −1.03117 −0.0372334
\(768\) −0.941592 −0.0339768
\(769\) −15.4316 −0.556478 −0.278239 0.960512i \(-0.589751\pi\)
−0.278239 + 0.960512i \(0.589751\pi\)
\(770\) 0.166807 0.00601130
\(771\) 0.316546 0.0114001
\(772\) 15.6131 0.561930
\(773\) 34.7219 1.24886 0.624430 0.781081i \(-0.285331\pi\)
0.624430 + 0.781081i \(0.285331\pi\)
\(774\) −16.9162 −0.608040
\(775\) 41.6973 1.49781
\(776\) 15.5913 0.559696
\(777\) −3.41054 −0.122352
\(778\) 23.9313 0.857978
\(779\) 1.76179 0.0631226
\(780\) −0.0418417 −0.00149817
\(781\) 20.8934 0.747626
\(782\) 0.406208 0.0145260
\(783\) 13.9109 0.497134
\(784\) −6.87714 −0.245612
\(785\) −2.60561 −0.0929982
\(786\) −11.4142 −0.407131
\(787\) 13.5974 0.484696 0.242348 0.970189i \(-0.422082\pi\)
0.242348 + 0.970189i \(0.422082\pi\)
\(788\) −2.44421 −0.0870712
\(789\) 22.6784 0.807374
\(790\) 0.191277 0.00680532
\(791\) −0.616197 −0.0219094
\(792\) 7.75363 0.275513
\(793\) −3.49210 −0.124008
\(794\) 4.43559 0.157413
\(795\) 0.893723 0.0316971
\(796\) −27.3011 −0.967661
\(797\) 46.3988 1.64353 0.821765 0.569826i \(-0.192989\pi\)
0.821765 + 0.569826i \(0.192989\pi\)
\(798\) −0.302507 −0.0107086
\(799\) 0.924904 0.0327208
\(800\) 4.98317 0.176182
\(801\) 36.9882 1.30691
\(802\) 6.81431 0.240622
\(803\) −38.3951 −1.35493
\(804\) −2.53582 −0.0894315
\(805\) −0.0568538 −0.00200383
\(806\) −2.86658 −0.100971
\(807\) 8.37686 0.294879
\(808\) −6.78412 −0.238665
\(809\) −26.6972 −0.938625 −0.469313 0.883032i \(-0.655498\pi\)
−0.469313 + 0.883032i \(0.655498\pi\)
\(810\) 0.234351 0.00823426
\(811\) −6.42122 −0.225480 −0.112740 0.993625i \(-0.535963\pi\)
−0.112740 + 0.993625i \(0.535963\pi\)
\(812\) 1.01272 0.0355394
\(813\) 15.0142 0.526572
\(814\) −37.9119 −1.32881
\(815\) −0.880007 −0.0308253
\(816\) 0.305874 0.0107077
\(817\) 7.33644 0.256669
\(818\) 24.3559 0.851584
\(819\) 0.253776 0.00886767
\(820\) 0.249329 0.00870694
\(821\) 10.7889 0.376536 0.188268 0.982118i \(-0.439713\pi\)
0.188268 + 0.982118i \(0.439713\pi\)
\(822\) −9.19615 −0.320752
\(823\) −27.6366 −0.963351 −0.481676 0.876350i \(-0.659972\pi\)
−0.481676 + 0.876350i \(0.659972\pi\)
\(824\) −11.3369 −0.394939
\(825\) 17.2144 0.599328
\(826\) −1.05506 −0.0367101
\(827\) 43.2078 1.50248 0.751242 0.660027i \(-0.229455\pi\)
0.751242 + 0.660027i \(0.229455\pi\)
\(828\) −2.64272 −0.0918410
\(829\) −31.9777 −1.11063 −0.555316 0.831639i \(-0.687403\pi\)
−0.555316 + 0.831639i \(0.687403\pi\)
\(830\) −1.37573 −0.0477524
\(831\) 23.2130 0.805251
\(832\) −0.342580 −0.0118768
\(833\) 2.23402 0.0774042
\(834\) −0.199255 −0.00689964
\(835\) 2.42975 0.0840850
\(836\) −3.36270 −0.116301
\(837\) −40.2879 −1.39255
\(838\) −29.0451 −1.00334
\(839\) 6.33484 0.218703 0.109351 0.994003i \(-0.465123\pi\)
0.109351 + 0.994003i \(0.465123\pi\)
\(840\) −0.0428108 −0.00147711
\(841\) −20.6524 −0.712151
\(842\) 14.3006 0.492831
\(843\) 11.5260 0.396975
\(844\) 4.19124 0.144269
\(845\) 1.67105 0.0574858
\(846\) −6.01727 −0.206878
\(847\) 0.862273 0.0296281
\(848\) 7.31739 0.251280
\(849\) −20.2831 −0.696115
\(850\) −1.61877 −0.0555234
\(851\) 12.9218 0.442953
\(852\) −5.36229 −0.183709
\(853\) 28.2288 0.966534 0.483267 0.875473i \(-0.339450\pi\)
0.483267 + 0.875473i \(0.339450\pi\)
\(854\) −3.57299 −0.122265
\(855\) −0.251265 −0.00859307
\(856\) 11.4350 0.390842
\(857\) −31.8090 −1.08657 −0.543287 0.839547i \(-0.682821\pi\)
−0.543287 + 0.839547i \(0.682821\pi\)
\(858\) −1.18344 −0.0404021
\(859\) 41.2690 1.40808 0.704041 0.710160i \(-0.251377\pi\)
0.704041 + 0.710160i \(0.251377\pi\)
\(860\) 1.03825 0.0354042
\(861\) 0.634393 0.0216201
\(862\) 22.7210 0.773881
\(863\) −40.4998 −1.37863 −0.689314 0.724463i \(-0.742088\pi\)
−0.689314 + 0.724463i \(0.742088\pi\)
\(864\) −4.81474 −0.163801
\(865\) 3.28041 0.111537
\(866\) 35.0246 1.19018
\(867\) 15.9077 0.540254
\(868\) −2.93298 −0.0995517
\(869\) 5.41005 0.183523
\(870\) −0.352881 −0.0119638
\(871\) −0.922609 −0.0312614
\(872\) −18.1642 −0.615119
\(873\) 32.9508 1.11522
\(874\) 1.14613 0.0387685
\(875\) 0.453899 0.0153446
\(876\) 9.85406 0.332938
\(877\) 32.1167 1.08450 0.542252 0.840216i \(-0.317572\pi\)
0.542252 + 0.840216i \(0.317572\pi\)
\(878\) −3.41021 −0.115089
\(879\) 21.5124 0.725595
\(880\) −0.475890 −0.0160423
\(881\) 14.5979 0.491815 0.245908 0.969293i \(-0.420914\pi\)
0.245908 + 0.969293i \(0.420914\pi\)
\(882\) −14.5342 −0.489391
\(883\) 13.3780 0.450206 0.225103 0.974335i \(-0.427728\pi\)
0.225103 + 0.974335i \(0.427728\pi\)
\(884\) 0.111286 0.00374296
\(885\) 0.367634 0.0123579
\(886\) 2.49849 0.0839385
\(887\) 45.1662 1.51653 0.758266 0.651945i \(-0.226047\pi\)
0.758266 + 0.651945i \(0.226047\pi\)
\(888\) 9.73007 0.326520
\(889\) 6.67820 0.223980
\(890\) −2.27020 −0.0760973
\(891\) 6.62836 0.222058
\(892\) 4.63209 0.155094
\(893\) 2.60965 0.0873286
\(894\) −0.547642 −0.0183159
\(895\) 1.85101 0.0618725
\(896\) −0.350515 −0.0117099
\(897\) 0.403361 0.0134678
\(898\) 16.2151 0.541105
\(899\) −24.1759 −0.806313
\(900\) 10.5315 0.351049
\(901\) −2.37703 −0.0791905
\(902\) 7.05199 0.234805
\(903\) 2.64174 0.0879115
\(904\) 1.75797 0.0584693
\(905\) 0.167120 0.00555525
\(906\) −2.29635 −0.0762911
\(907\) 10.5519 0.350370 0.175185 0.984536i \(-0.443948\pi\)
0.175185 + 0.984536i \(0.443948\pi\)
\(908\) 10.4114 0.345513
\(909\) −14.3376 −0.475548
\(910\) −0.0155759 −0.000516336 0
\(911\) 36.4848 1.20880 0.604398 0.796683i \(-0.293414\pi\)
0.604398 + 0.796683i \(0.293414\pi\)
\(912\) 0.863034 0.0285779
\(913\) −38.9110 −1.28777
\(914\) −13.4972 −0.446447
\(915\) 1.24501 0.0411586
\(916\) −8.95876 −0.296006
\(917\) −4.24903 −0.140315
\(918\) 1.56406 0.0516216
\(919\) 19.5171 0.643810 0.321905 0.946772i \(-0.395677\pi\)
0.321905 + 0.946772i \(0.395677\pi\)
\(920\) 0.162201 0.00534760
\(921\) 9.00247 0.296642
\(922\) 37.6469 1.23983
\(923\) −1.95096 −0.0642168
\(924\) −1.21086 −0.0398342
\(925\) −51.4943 −1.69312
\(926\) −31.4858 −1.03469
\(927\) −23.9594 −0.786931
\(928\) −2.88923 −0.0948435
\(929\) 8.12380 0.266533 0.133267 0.991080i \(-0.457453\pi\)
0.133267 + 0.991080i \(0.457453\pi\)
\(930\) 1.02199 0.0335125
\(931\) 6.30337 0.206585
\(932\) 21.0993 0.691132
\(933\) 9.62271 0.315033
\(934\) −6.12073 −0.200276
\(935\) 0.154592 0.00505569
\(936\) −0.724010 −0.0236650
\(937\) 45.0973 1.47326 0.736632 0.676294i \(-0.236415\pi\)
0.736632 + 0.676294i \(0.236415\pi\)
\(938\) −0.943979 −0.0308220
\(939\) 10.1322 0.330651
\(940\) 0.369319 0.0120458
\(941\) −31.8845 −1.03940 −0.519702 0.854348i \(-0.673957\pi\)
−0.519702 + 0.854348i \(0.673957\pi\)
\(942\) 18.9142 0.616258
\(943\) −2.40358 −0.0782712
\(944\) 3.01002 0.0979677
\(945\) −0.218909 −0.00712111
\(946\) 29.3659 0.954767
\(947\) 59.1408 1.92182 0.960909 0.276866i \(-0.0892957\pi\)
0.960909 + 0.276866i \(0.0892957\pi\)
\(948\) −1.38849 −0.0450959
\(949\) 3.58521 0.116381
\(950\) −4.56742 −0.148187
\(951\) −13.9144 −0.451204
\(952\) 0.113864 0.00369035
\(953\) 23.1786 0.750830 0.375415 0.926857i \(-0.377500\pi\)
0.375415 + 0.926857i \(0.377500\pi\)
\(954\) 15.4646 0.500685
\(955\) 0.993032 0.0321337
\(956\) 4.13040 0.133587
\(957\) −9.98084 −0.322635
\(958\) −17.8974 −0.578238
\(959\) −3.42334 −0.110545
\(960\) 0.122137 0.00394195
\(961\) 39.0170 1.25861
\(962\) 3.54010 0.114137
\(963\) 24.1669 0.778766
\(964\) −13.0441 −0.420121
\(965\) −2.02523 −0.0651945
\(966\) 0.412704 0.0132785
\(967\) 30.2130 0.971586 0.485793 0.874074i \(-0.338531\pi\)
0.485793 + 0.874074i \(0.338531\pi\)
\(968\) −2.46002 −0.0790679
\(969\) −0.280354 −0.00900628
\(970\) −2.02240 −0.0649354
\(971\) −13.9789 −0.448603 −0.224302 0.974520i \(-0.572010\pi\)
−0.224302 + 0.974520i \(0.572010\pi\)
\(972\) −16.1454 −0.517864
\(973\) −0.0741743 −0.00237792
\(974\) 0.111862 0.00358430
\(975\) −1.60743 −0.0514788
\(976\) 10.1935 0.326287
\(977\) 30.2608 0.968129 0.484064 0.875032i \(-0.339160\pi\)
0.484064 + 0.875032i \(0.339160\pi\)
\(978\) 6.38800 0.204266
\(979\) −64.2101 −2.05216
\(980\) 0.892055 0.0284957
\(981\) −38.3884 −1.22565
\(982\) 23.4744 0.749099
\(983\) 28.7492 0.916956 0.458478 0.888706i \(-0.348395\pi\)
0.458478 + 0.888706i \(0.348395\pi\)
\(984\) −1.80989 −0.0576971
\(985\) 0.317046 0.0101019
\(986\) 0.938558 0.0298898
\(987\) 0.939695 0.0299108
\(988\) 0.313998 0.00998962
\(989\) −10.0090 −0.318267
\(990\) −1.00575 −0.0319648
\(991\) 37.4492 1.18961 0.594806 0.803869i \(-0.297229\pi\)
0.594806 + 0.803869i \(0.297229\pi\)
\(992\) 8.36762 0.265672
\(993\) −4.48828 −0.142431
\(994\) −1.99615 −0.0633141
\(995\) 3.54131 0.112267
\(996\) 9.98649 0.316434
\(997\) −53.8516 −1.70550 −0.852748 0.522322i \(-0.825066\pi\)
−0.852748 + 0.522322i \(0.825066\pi\)
\(998\) 5.14322 0.162806
\(999\) 49.7538 1.57414
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 4006.2.a.g.1.15 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.15 40 1.1 even 1 trivial