Properties

Label 4006.2.a.g.1.9
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 1.9
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.91121 q^{3} +1.00000 q^{4} -1.39712 q^{5} +1.91121 q^{6} +3.46874 q^{7} -1.00000 q^{8} +0.652717 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.91121 q^{3} +1.00000 q^{4} -1.39712 q^{5} +1.91121 q^{6} +3.46874 q^{7} -1.00000 q^{8} +0.652717 q^{9} +1.39712 q^{10} -5.38542 q^{11} -1.91121 q^{12} +0.707447 q^{13} -3.46874 q^{14} +2.67020 q^{15} +1.00000 q^{16} +1.30288 q^{17} -0.652717 q^{18} +1.25277 q^{19} -1.39712 q^{20} -6.62949 q^{21} +5.38542 q^{22} -1.63099 q^{23} +1.91121 q^{24} -3.04804 q^{25} -0.707447 q^{26} +4.48615 q^{27} +3.46874 q^{28} +9.27361 q^{29} -2.67020 q^{30} +3.30654 q^{31} -1.00000 q^{32} +10.2927 q^{33} -1.30288 q^{34} -4.84626 q^{35} +0.652717 q^{36} -2.58531 q^{37} -1.25277 q^{38} -1.35208 q^{39} +1.39712 q^{40} -5.07679 q^{41} +6.62949 q^{42} +4.23096 q^{43} -5.38542 q^{44} -0.911927 q^{45} +1.63099 q^{46} -9.82173 q^{47} -1.91121 q^{48} +5.03217 q^{49} +3.04804 q^{50} -2.49007 q^{51} +0.707447 q^{52} +2.68330 q^{53} -4.48615 q^{54} +7.52410 q^{55} -3.46874 q^{56} -2.39431 q^{57} -9.27361 q^{58} -4.53284 q^{59} +2.67020 q^{60} -9.89612 q^{61} -3.30654 q^{62} +2.26411 q^{63} +1.00000 q^{64} -0.988392 q^{65} -10.2927 q^{66} -7.12165 q^{67} +1.30288 q^{68} +3.11715 q^{69} +4.84626 q^{70} +15.6360 q^{71} -0.652717 q^{72} +7.20729 q^{73} +2.58531 q^{74} +5.82545 q^{75} +1.25277 q^{76} -18.6806 q^{77} +1.35208 q^{78} +12.9876 q^{79} -1.39712 q^{80} -10.5321 q^{81} +5.07679 q^{82} +1.68735 q^{83} -6.62949 q^{84} -1.82028 q^{85} -4.23096 q^{86} -17.7238 q^{87} +5.38542 q^{88} +14.6065 q^{89} +0.911927 q^{90} +2.45395 q^{91} -1.63099 q^{92} -6.31948 q^{93} +9.82173 q^{94} -1.75028 q^{95} +1.91121 q^{96} -4.62606 q^{97} -5.03217 q^{98} -3.51515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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
\(3\) −1.91121 −1.10344 −0.551718 0.834031i \(-0.686028\pi\)
−0.551718 + 0.834031i \(0.686028\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.39712 −0.624813 −0.312407 0.949949i \(-0.601135\pi\)
−0.312407 + 0.949949i \(0.601135\pi\)
\(6\) 1.91121 0.780248
\(7\) 3.46874 1.31106 0.655531 0.755169i \(-0.272445\pi\)
0.655531 + 0.755169i \(0.272445\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0.652717 0.217572
\(10\) 1.39712 0.441810
\(11\) −5.38542 −1.62376 −0.811882 0.583821i \(-0.801557\pi\)
−0.811882 + 0.583821i \(0.801557\pi\)
\(12\) −1.91121 −0.551718
\(13\) 0.707447 0.196211 0.0981053 0.995176i \(-0.468722\pi\)
0.0981053 + 0.995176i \(0.468722\pi\)
\(14\) −3.46874 −0.927060
\(15\) 2.67020 0.689442
\(16\) 1.00000 0.250000
\(17\) 1.30288 0.315995 0.157997 0.987440i \(-0.449496\pi\)
0.157997 + 0.987440i \(0.449496\pi\)
\(18\) −0.652717 −0.153847
\(19\) 1.25277 0.287406 0.143703 0.989621i \(-0.454099\pi\)
0.143703 + 0.989621i \(0.454099\pi\)
\(20\) −1.39712 −0.312407
\(21\) −6.62949 −1.44667
\(22\) 5.38542 1.14817
\(23\) −1.63099 −0.340084 −0.170042 0.985437i \(-0.554390\pi\)
−0.170042 + 0.985437i \(0.554390\pi\)
\(24\) 1.91121 0.390124
\(25\) −3.04804 −0.609609
\(26\) −0.707447 −0.138742
\(27\) 4.48615 0.863359
\(28\) 3.46874 0.655531
\(29\) 9.27361 1.72207 0.861033 0.508549i \(-0.169818\pi\)
0.861033 + 0.508549i \(0.169818\pi\)
\(30\) −2.67020 −0.487509
\(31\) 3.30654 0.593872 0.296936 0.954897i \(-0.404035\pi\)
0.296936 + 0.954897i \(0.404035\pi\)
\(32\) −1.00000 −0.176777
\(33\) 10.2927 1.79172
\(34\) −1.30288 −0.223442
\(35\) −4.84626 −0.819168
\(36\) 0.652717 0.108786
\(37\) −2.58531 −0.425022 −0.212511 0.977159i \(-0.568164\pi\)
−0.212511 + 0.977159i \(0.568164\pi\)
\(38\) −1.25277 −0.203227
\(39\) −1.35208 −0.216506
\(40\) 1.39712 0.220905
\(41\) −5.07679 −0.792862 −0.396431 0.918065i \(-0.629751\pi\)
−0.396431 + 0.918065i \(0.629751\pi\)
\(42\) 6.62949 1.02295
\(43\) 4.23096 0.645215 0.322607 0.946533i \(-0.395441\pi\)
0.322607 + 0.946533i \(0.395441\pi\)
\(44\) −5.38542 −0.811882
\(45\) −0.911927 −0.135942
\(46\) 1.63099 0.240476
\(47\) −9.82173 −1.43265 −0.716323 0.697769i \(-0.754176\pi\)
−0.716323 + 0.697769i \(0.754176\pi\)
\(48\) −1.91121 −0.275859
\(49\) 5.03217 0.718881
\(50\) 3.04804 0.431058
\(51\) −2.49007 −0.348680
\(52\) 0.707447 0.0981053
\(53\) 2.68330 0.368579 0.184290 0.982872i \(-0.441002\pi\)
0.184290 + 0.982872i \(0.441002\pi\)
\(54\) −4.48615 −0.610487
\(55\) 7.52410 1.01455
\(56\) −3.46874 −0.463530
\(57\) −2.39431 −0.317134
\(58\) −9.27361 −1.21768
\(59\) −4.53284 −0.590125 −0.295062 0.955478i \(-0.595340\pi\)
−0.295062 + 0.955478i \(0.595340\pi\)
\(60\) 2.67020 0.344721
\(61\) −9.89612 −1.26707 −0.633534 0.773715i \(-0.718396\pi\)
−0.633534 + 0.773715i \(0.718396\pi\)
\(62\) −3.30654 −0.419931
\(63\) 2.26411 0.285251
\(64\) 1.00000 0.125000
\(65\) −0.988392 −0.122595
\(66\) −10.2927 −1.26694
\(67\) −7.12165 −0.870049 −0.435024 0.900419i \(-0.643260\pi\)
−0.435024 + 0.900419i \(0.643260\pi\)
\(68\) 1.30288 0.157997
\(69\) 3.11715 0.375261
\(70\) 4.84626 0.579239
\(71\) 15.6360 1.85565 0.927825 0.373015i \(-0.121676\pi\)
0.927825 + 0.373015i \(0.121676\pi\)
\(72\) −0.652717 −0.0769235
\(73\) 7.20729 0.843550 0.421775 0.906701i \(-0.361407\pi\)
0.421775 + 0.906701i \(0.361407\pi\)
\(74\) 2.58531 0.300536
\(75\) 5.82545 0.672665
\(76\) 1.25277 0.143703
\(77\) −18.6806 −2.12885
\(78\) 1.35208 0.153093
\(79\) 12.9876 1.46122 0.730612 0.682793i \(-0.239235\pi\)
0.730612 + 0.682793i \(0.239235\pi\)
\(80\) −1.39712 −0.156203
\(81\) −10.5321 −1.17023
\(82\) 5.07679 0.560638
\(83\) 1.68735 0.185210 0.0926052 0.995703i \(-0.470481\pi\)
0.0926052 + 0.995703i \(0.470481\pi\)
\(84\) −6.62949 −0.723336
\(85\) −1.82028 −0.197438
\(86\) −4.23096 −0.456236
\(87\) −17.7238 −1.90019
\(88\) 5.38542 0.574087
\(89\) 14.6065 1.54829 0.774145 0.633009i \(-0.218180\pi\)
0.774145 + 0.633009i \(0.218180\pi\)
\(90\) 0.911927 0.0961256
\(91\) 2.45395 0.257244
\(92\) −1.63099 −0.170042
\(93\) −6.31948 −0.655300
\(94\) 9.82173 1.01303
\(95\) −1.75028 −0.179575
\(96\) 1.91121 0.195062
\(97\) −4.62606 −0.469705 −0.234853 0.972031i \(-0.575461\pi\)
−0.234853 + 0.972031i \(0.575461\pi\)
\(98\) −5.03217 −0.508326
\(99\) −3.51515 −0.353286
\(100\) −3.04804 −0.304804
\(101\) −18.7019 −1.86091 −0.930456 0.366405i \(-0.880589\pi\)
−0.930456 + 0.366405i \(0.880589\pi\)
\(102\) 2.49007 0.246554
\(103\) 2.87078 0.282866 0.141433 0.989948i \(-0.454829\pi\)
0.141433 + 0.989948i \(0.454829\pi\)
\(104\) −0.707447 −0.0693709
\(105\) 9.26222 0.903900
\(106\) −2.68330 −0.260625
\(107\) −12.9417 −1.25113 −0.625563 0.780174i \(-0.715131\pi\)
−0.625563 + 0.780174i \(0.715131\pi\)
\(108\) 4.48615 0.431680
\(109\) 8.58800 0.822581 0.411290 0.911504i \(-0.365078\pi\)
0.411290 + 0.911504i \(0.365078\pi\)
\(110\) −7.52410 −0.717394
\(111\) 4.94106 0.468985
\(112\) 3.46874 0.327765
\(113\) 6.36794 0.599045 0.299523 0.954089i \(-0.403173\pi\)
0.299523 + 0.954089i \(0.403173\pi\)
\(114\) 2.39431 0.224248
\(115\) 2.27869 0.212489
\(116\) 9.27361 0.861033
\(117\) 0.461763 0.0426900
\(118\) 4.53284 0.417281
\(119\) 4.51935 0.414288
\(120\) −2.67020 −0.243754
\(121\) 18.0027 1.63661
\(122\) 9.89612 0.895953
\(123\) 9.70281 0.874873
\(124\) 3.30654 0.296936
\(125\) 11.2441 1.00570
\(126\) −2.26411 −0.201703
\(127\) 11.5882 1.02829 0.514144 0.857704i \(-0.328110\pi\)
0.514144 + 0.857704i \(0.328110\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.08624 −0.711954
\(130\) 0.988392 0.0866877
\(131\) 12.7769 1.11632 0.558159 0.829734i \(-0.311508\pi\)
0.558159 + 0.829734i \(0.311508\pi\)
\(132\) 10.2927 0.895860
\(133\) 4.34554 0.376806
\(134\) 7.12165 0.615217
\(135\) −6.26770 −0.539438
\(136\) −1.30288 −0.111721
\(137\) −12.7306 −1.08765 −0.543825 0.839199i \(-0.683024\pi\)
−0.543825 + 0.839199i \(0.683024\pi\)
\(138\) −3.11715 −0.265350
\(139\) 10.3838 0.880746 0.440373 0.897815i \(-0.354846\pi\)
0.440373 + 0.897815i \(0.354846\pi\)
\(140\) −4.84626 −0.409584
\(141\) 18.7714 1.58083
\(142\) −15.6360 −1.31214
\(143\) −3.80990 −0.318600
\(144\) 0.652717 0.0543931
\(145\) −12.9564 −1.07597
\(146\) −7.20729 −0.596480
\(147\) −9.61752 −0.793240
\(148\) −2.58531 −0.212511
\(149\) −12.1004 −0.991306 −0.495653 0.868521i \(-0.665071\pi\)
−0.495653 + 0.868521i \(0.665071\pi\)
\(150\) −5.82545 −0.475646
\(151\) 2.15561 0.175421 0.0877105 0.996146i \(-0.472045\pi\)
0.0877105 + 0.996146i \(0.472045\pi\)
\(152\) −1.25277 −0.101613
\(153\) 0.850412 0.0687517
\(154\) 18.6806 1.50533
\(155\) −4.61964 −0.371059
\(156\) −1.35208 −0.108253
\(157\) 6.01394 0.479965 0.239982 0.970777i \(-0.422858\pi\)
0.239982 + 0.970777i \(0.422858\pi\)
\(158\) −12.9876 −1.03324
\(159\) −5.12834 −0.406704
\(160\) 1.39712 0.110452
\(161\) −5.65747 −0.445871
\(162\) 10.5321 0.827481
\(163\) −19.3553 −1.51603 −0.758013 0.652239i \(-0.773830\pi\)
−0.758013 + 0.652239i \(0.773830\pi\)
\(164\) −5.07679 −0.396431
\(165\) −14.3801 −1.11949
\(166\) −1.68735 −0.130964
\(167\) 17.6636 1.36685 0.683425 0.730021i \(-0.260490\pi\)
0.683425 + 0.730021i \(0.260490\pi\)
\(168\) 6.62949 0.511476
\(169\) −12.4995 −0.961501
\(170\) 1.82028 0.139609
\(171\) 0.817706 0.0625316
\(172\) 4.23096 0.322607
\(173\) −18.7071 −1.42228 −0.711139 0.703052i \(-0.751820\pi\)
−0.711139 + 0.703052i \(0.751820\pi\)
\(174\) 17.7238 1.34364
\(175\) −10.5729 −0.799234
\(176\) −5.38542 −0.405941
\(177\) 8.66319 0.651165
\(178\) −14.6065 −1.09481
\(179\) −23.0957 −1.72626 −0.863128 0.504985i \(-0.831498\pi\)
−0.863128 + 0.504985i \(0.831498\pi\)
\(180\) −0.911927 −0.0679710
\(181\) 21.8782 1.62619 0.813096 0.582130i \(-0.197780\pi\)
0.813096 + 0.582130i \(0.197780\pi\)
\(182\) −2.45395 −0.181899
\(183\) 18.9135 1.39813
\(184\) 1.63099 0.120238
\(185\) 3.61199 0.265559
\(186\) 6.31948 0.463367
\(187\) −7.01655 −0.513101
\(188\) −9.82173 −0.716323
\(189\) 15.5613 1.13192
\(190\) 1.75028 0.126979
\(191\) −19.0384 −1.37757 −0.688787 0.724964i \(-0.741856\pi\)
−0.688787 + 0.724964i \(0.741856\pi\)
\(192\) −1.91121 −0.137930
\(193\) 19.8450 1.42847 0.714237 0.699904i \(-0.246774\pi\)
0.714237 + 0.699904i \(0.246774\pi\)
\(194\) 4.62606 0.332132
\(195\) 1.88902 0.135276
\(196\) 5.03217 0.359441
\(197\) −10.3872 −0.740054 −0.370027 0.929021i \(-0.620652\pi\)
−0.370027 + 0.929021i \(0.620652\pi\)
\(198\) 3.51515 0.249811
\(199\) −28.0304 −1.98702 −0.993511 0.113735i \(-0.963719\pi\)
−0.993511 + 0.113735i \(0.963719\pi\)
\(200\) 3.04804 0.215529
\(201\) 13.6110 0.960043
\(202\) 18.7019 1.31586
\(203\) 32.1678 2.25773
\(204\) −2.49007 −0.174340
\(205\) 7.09291 0.495390
\(206\) −2.87078 −0.200017
\(207\) −1.06457 −0.0739929
\(208\) 0.707447 0.0490526
\(209\) −6.74670 −0.466679
\(210\) −9.26222 −0.639154
\(211\) −21.9817 −1.51328 −0.756642 0.653829i \(-0.773161\pi\)
−0.756642 + 0.653829i \(0.773161\pi\)
\(212\) 2.68330 0.184290
\(213\) −29.8836 −2.04759
\(214\) 12.9417 0.884680
\(215\) −5.91117 −0.403139
\(216\) −4.48615 −0.305244
\(217\) 11.4695 0.778602
\(218\) −8.58800 −0.581653
\(219\) −13.7746 −0.930804
\(220\) 7.52410 0.507274
\(221\) 0.921719 0.0620015
\(222\) −4.94106 −0.331622
\(223\) −11.3740 −0.761661 −0.380830 0.924645i \(-0.624362\pi\)
−0.380830 + 0.924645i \(0.624362\pi\)
\(224\) −3.46874 −0.231765
\(225\) −1.98951 −0.132634
\(226\) −6.36794 −0.423589
\(227\) 4.39704 0.291842 0.145921 0.989296i \(-0.453386\pi\)
0.145921 + 0.989296i \(0.453386\pi\)
\(228\) −2.39431 −0.158567
\(229\) 0.737047 0.0487054 0.0243527 0.999703i \(-0.492248\pi\)
0.0243527 + 0.999703i \(0.492248\pi\)
\(230\) −2.27869 −0.150252
\(231\) 35.7026 2.34906
\(232\) −9.27361 −0.608842
\(233\) −29.1104 −1.90709 −0.953543 0.301257i \(-0.902594\pi\)
−0.953543 + 0.301257i \(0.902594\pi\)
\(234\) −0.461763 −0.0301864
\(235\) 13.7222 0.895136
\(236\) −4.53284 −0.295062
\(237\) −24.8221 −1.61237
\(238\) −4.51935 −0.292946
\(239\) −13.9227 −0.900584 −0.450292 0.892881i \(-0.648680\pi\)
−0.450292 + 0.892881i \(0.648680\pi\)
\(240\) 2.67020 0.172360
\(241\) 15.9988 1.03057 0.515287 0.857018i \(-0.327685\pi\)
0.515287 + 0.857018i \(0.327685\pi\)
\(242\) −18.0027 −1.15726
\(243\) 6.67062 0.427921
\(244\) −9.89612 −0.633534
\(245\) −7.03057 −0.449166
\(246\) −9.70281 −0.618628
\(247\) 0.886271 0.0563920
\(248\) −3.30654 −0.209965
\(249\) −3.22487 −0.204368
\(250\) −11.2441 −0.711140
\(251\) −24.5132 −1.54726 −0.773629 0.633639i \(-0.781561\pi\)
−0.773629 + 0.633639i \(0.781561\pi\)
\(252\) 2.26411 0.142625
\(253\) 8.78354 0.552216
\(254\) −11.5882 −0.727110
\(255\) 3.47894 0.217860
\(256\) 1.00000 0.0625000
\(257\) −5.56235 −0.346970 −0.173485 0.984837i \(-0.555503\pi\)
−0.173485 + 0.984837i \(0.555503\pi\)
\(258\) 8.08624 0.503427
\(259\) −8.96776 −0.557230
\(260\) −0.988392 −0.0612975
\(261\) 6.05305 0.374674
\(262\) −12.7769 −0.789357
\(263\) 29.4620 1.81670 0.908351 0.418208i \(-0.137342\pi\)
0.908351 + 0.418208i \(0.137342\pi\)
\(264\) −10.2927 −0.633469
\(265\) −3.74890 −0.230293
\(266\) −4.34554 −0.266442
\(267\) −27.9161 −1.70844
\(268\) −7.12165 −0.435024
\(269\) 0.299339 0.0182510 0.00912551 0.999958i \(-0.497095\pi\)
0.00912551 + 0.999958i \(0.497095\pi\)
\(270\) 6.26770 0.381440
\(271\) 29.9220 1.81763 0.908816 0.417198i \(-0.136988\pi\)
0.908816 + 0.417198i \(0.136988\pi\)
\(272\) 1.30288 0.0789987
\(273\) −4.69001 −0.283853
\(274\) 12.7306 0.769085
\(275\) 16.4150 0.989861
\(276\) 3.11715 0.187631
\(277\) −13.6109 −0.817802 −0.408901 0.912579i \(-0.634088\pi\)
−0.408901 + 0.912579i \(0.634088\pi\)
\(278\) −10.3838 −0.622782
\(279\) 2.15823 0.129210
\(280\) 4.84626 0.289620
\(281\) −20.9957 −1.25250 −0.626248 0.779624i \(-0.715410\pi\)
−0.626248 + 0.779624i \(0.715410\pi\)
\(282\) −18.7714 −1.11782
\(283\) −29.0276 −1.72551 −0.862755 0.505621i \(-0.831263\pi\)
−0.862755 + 0.505621i \(0.831263\pi\)
\(284\) 15.6360 0.927825
\(285\) 3.34515 0.198149
\(286\) 3.80990 0.225284
\(287\) −17.6101 −1.03949
\(288\) −0.652717 −0.0384617
\(289\) −15.3025 −0.900147
\(290\) 12.9564 0.760825
\(291\) 8.84137 0.518290
\(292\) 7.20729 0.421775
\(293\) −7.17957 −0.419435 −0.209717 0.977762i \(-0.567254\pi\)
−0.209717 + 0.977762i \(0.567254\pi\)
\(294\) 9.61752 0.560905
\(295\) 6.33294 0.368718
\(296\) 2.58531 0.150268
\(297\) −24.1598 −1.40189
\(298\) 12.1004 0.700959
\(299\) −1.15384 −0.0667281
\(300\) 5.82545 0.336332
\(301\) 14.6761 0.845916
\(302\) −2.15561 −0.124041
\(303\) 35.7433 2.05340
\(304\) 1.25277 0.0718514
\(305\) 13.8261 0.791681
\(306\) −0.850412 −0.0486148
\(307\) −21.9860 −1.25481 −0.627404 0.778694i \(-0.715883\pi\)
−0.627404 + 0.778694i \(0.715883\pi\)
\(308\) −18.6806 −1.06443
\(309\) −5.48666 −0.312125
\(310\) 4.61964 0.262378
\(311\) −6.43014 −0.364620 −0.182310 0.983241i \(-0.558357\pi\)
−0.182310 + 0.983241i \(0.558357\pi\)
\(312\) 1.35208 0.0765464
\(313\) 7.21113 0.407597 0.203799 0.979013i \(-0.434671\pi\)
0.203799 + 0.979013i \(0.434671\pi\)
\(314\) −6.01394 −0.339386
\(315\) −3.16324 −0.178228
\(316\) 12.9876 0.730612
\(317\) −17.4497 −0.980071 −0.490035 0.871702i \(-0.663016\pi\)
−0.490035 + 0.871702i \(0.663016\pi\)
\(318\) 5.12834 0.287583
\(319\) −49.9422 −2.79623
\(320\) −1.39712 −0.0781016
\(321\) 24.7344 1.38054
\(322\) 5.65747 0.315278
\(323\) 1.63221 0.0908187
\(324\) −10.5321 −0.585117
\(325\) −2.15633 −0.119612
\(326\) 19.3553 1.07199
\(327\) −16.4135 −0.907666
\(328\) 5.07679 0.280319
\(329\) −34.0690 −1.87829
\(330\) 14.3801 0.791599
\(331\) 6.76232 0.371691 0.185845 0.982579i \(-0.440498\pi\)
0.185845 + 0.982579i \(0.440498\pi\)
\(332\) 1.68735 0.0926052
\(333\) −1.68747 −0.0924730
\(334\) −17.6636 −0.966508
\(335\) 9.94984 0.543618
\(336\) −6.62949 −0.361668
\(337\) 3.88491 0.211625 0.105812 0.994386i \(-0.466256\pi\)
0.105812 + 0.994386i \(0.466256\pi\)
\(338\) 12.4995 0.679884
\(339\) −12.1705 −0.661009
\(340\) −1.82028 −0.0987188
\(341\) −17.8071 −0.964308
\(342\) −0.817706 −0.0442165
\(343\) −6.82590 −0.368564
\(344\) −4.23096 −0.228118
\(345\) −4.35505 −0.234468
\(346\) 18.7071 1.00570
\(347\) 22.6570 1.21629 0.608146 0.793825i \(-0.291913\pi\)
0.608146 + 0.793825i \(0.291913\pi\)
\(348\) −17.7238 −0.950095
\(349\) −14.3862 −0.770074 −0.385037 0.922901i \(-0.625811\pi\)
−0.385037 + 0.922901i \(0.625811\pi\)
\(350\) 10.5729 0.565144
\(351\) 3.17371 0.169400
\(352\) 5.38542 0.287044
\(353\) −31.0634 −1.65334 −0.826668 0.562689i \(-0.809767\pi\)
−0.826668 + 0.562689i \(0.809767\pi\)
\(354\) −8.66319 −0.460444
\(355\) −21.8454 −1.15943
\(356\) 14.6065 0.774145
\(357\) −8.63743 −0.457141
\(358\) 23.0957 1.22065
\(359\) −7.74628 −0.408833 −0.204417 0.978884i \(-0.565530\pi\)
−0.204417 + 0.978884i \(0.565530\pi\)
\(360\) 0.911927 0.0480628
\(361\) −17.4306 −0.917398
\(362\) −21.8782 −1.14989
\(363\) −34.4069 −1.80590
\(364\) 2.45395 0.128622
\(365\) −10.0695 −0.527061
\(366\) −18.9135 −0.988627
\(367\) 3.81909 0.199355 0.0996774 0.995020i \(-0.468219\pi\)
0.0996774 + 0.995020i \(0.468219\pi\)
\(368\) −1.63099 −0.0850210
\(369\) −3.31371 −0.172505
\(370\) −3.61199 −0.187779
\(371\) 9.30767 0.483230
\(372\) −6.31948 −0.327650
\(373\) 26.7451 1.38481 0.692403 0.721510i \(-0.256552\pi\)
0.692403 + 0.721510i \(0.256552\pi\)
\(374\) 7.01655 0.362817
\(375\) −21.4899 −1.10973
\(376\) 9.82173 0.506517
\(377\) 6.56059 0.337888
\(378\) −15.5613 −0.800386
\(379\) 0.580027 0.0297940 0.0148970 0.999889i \(-0.495258\pi\)
0.0148970 + 0.999889i \(0.495258\pi\)
\(380\) −1.75028 −0.0897874
\(381\) −22.1475 −1.13465
\(382\) 19.0384 0.974091
\(383\) 12.4377 0.635536 0.317768 0.948168i \(-0.397067\pi\)
0.317768 + 0.948168i \(0.397067\pi\)
\(384\) 1.91121 0.0975309
\(385\) 26.0991 1.33014
\(386\) −19.8450 −1.01008
\(387\) 2.76162 0.140381
\(388\) −4.62606 −0.234853
\(389\) 1.38667 0.0703070 0.0351535 0.999382i \(-0.488808\pi\)
0.0351535 + 0.999382i \(0.488808\pi\)
\(390\) −1.88902 −0.0956544
\(391\) −2.12498 −0.107465
\(392\) −5.03217 −0.254163
\(393\) −24.4192 −1.23179
\(394\) 10.3872 0.523297
\(395\) −18.1454 −0.912992
\(396\) −3.51515 −0.176643
\(397\) 15.5049 0.778167 0.389084 0.921202i \(-0.372792\pi\)
0.389084 + 0.921202i \(0.372792\pi\)
\(398\) 28.0304 1.40504
\(399\) −8.30524 −0.415782
\(400\) −3.04804 −0.152402
\(401\) −6.49569 −0.324379 −0.162190 0.986760i \(-0.551856\pi\)
−0.162190 + 0.986760i \(0.551856\pi\)
\(402\) −13.6110 −0.678853
\(403\) 2.33920 0.116524
\(404\) −18.7019 −0.930456
\(405\) 14.7147 0.731178
\(406\) −32.1678 −1.59646
\(407\) 13.9230 0.690135
\(408\) 2.49007 0.123277
\(409\) −22.7868 −1.12674 −0.563368 0.826206i \(-0.690495\pi\)
−0.563368 + 0.826206i \(0.690495\pi\)
\(410\) −7.09291 −0.350294
\(411\) 24.3309 1.20015
\(412\) 2.87078 0.141433
\(413\) −15.7232 −0.773690
\(414\) 1.06457 0.0523209
\(415\) −2.35743 −0.115722
\(416\) −0.707447 −0.0346855
\(417\) −19.8457 −0.971848
\(418\) 6.74670 0.329992
\(419\) −6.04978 −0.295551 −0.147776 0.989021i \(-0.547211\pi\)
−0.147776 + 0.989021i \(0.547211\pi\)
\(420\) 9.26222 0.451950
\(421\) −33.3435 −1.62506 −0.812531 0.582917i \(-0.801911\pi\)
−0.812531 + 0.582917i \(0.801911\pi\)
\(422\) 21.9817 1.07005
\(423\) −6.41081 −0.311704
\(424\) −2.68330 −0.130313
\(425\) −3.97123 −0.192633
\(426\) 29.8836 1.44787
\(427\) −34.3271 −1.66120
\(428\) −12.9417 −0.625563
\(429\) 7.28151 0.351555
\(430\) 5.91117 0.285062
\(431\) −21.0016 −1.01161 −0.505807 0.862647i \(-0.668805\pi\)
−0.505807 + 0.862647i \(0.668805\pi\)
\(432\) 4.48615 0.215840
\(433\) −16.0873 −0.773106 −0.386553 0.922267i \(-0.626334\pi\)
−0.386553 + 0.922267i \(0.626334\pi\)
\(434\) −11.4695 −0.550555
\(435\) 24.7624 1.18726
\(436\) 8.58800 0.411290
\(437\) −2.04325 −0.0977421
\(438\) 13.7746 0.658178
\(439\) 9.20885 0.439515 0.219757 0.975555i \(-0.429473\pi\)
0.219757 + 0.975555i \(0.429473\pi\)
\(440\) −7.52410 −0.358697
\(441\) 3.28458 0.156409
\(442\) −0.921719 −0.0438417
\(443\) 14.3123 0.679997 0.339999 0.940426i \(-0.389573\pi\)
0.339999 + 0.940426i \(0.389573\pi\)
\(444\) 4.94106 0.234492
\(445\) −20.4071 −0.967391
\(446\) 11.3740 0.538576
\(447\) 23.1265 1.09384
\(448\) 3.46874 0.163883
\(449\) −5.34142 −0.252077 −0.126039 0.992025i \(-0.540226\pi\)
−0.126039 + 0.992025i \(0.540226\pi\)
\(450\) 1.98951 0.0937864
\(451\) 27.3406 1.28742
\(452\) 6.36794 0.299523
\(453\) −4.11982 −0.193566
\(454\) −4.39704 −0.206363
\(455\) −3.42848 −0.160729
\(456\) 2.39431 0.112124
\(457\) −35.0989 −1.64186 −0.820928 0.571031i \(-0.806543\pi\)
−0.820928 + 0.571031i \(0.806543\pi\)
\(458\) −0.737047 −0.0344400
\(459\) 5.84491 0.272817
\(460\) 2.27869 0.106245
\(461\) −26.7673 −1.24668 −0.623339 0.781952i \(-0.714224\pi\)
−0.623339 + 0.781952i \(0.714224\pi\)
\(462\) −35.7026 −1.66103
\(463\) 27.9389 1.29843 0.649216 0.760604i \(-0.275097\pi\)
0.649216 + 0.760604i \(0.275097\pi\)
\(464\) 9.27361 0.430516
\(465\) 8.82910 0.409440
\(466\) 29.1104 1.34851
\(467\) 14.7050 0.680467 0.340234 0.940341i \(-0.389494\pi\)
0.340234 + 0.940341i \(0.389494\pi\)
\(468\) 0.461763 0.0213450
\(469\) −24.7032 −1.14069
\(470\) −13.7222 −0.632957
\(471\) −11.4939 −0.529611
\(472\) 4.53284 0.208641
\(473\) −22.7855 −1.04768
\(474\) 24.8221 1.14012
\(475\) −3.81850 −0.175205
\(476\) 4.51935 0.207144
\(477\) 1.75144 0.0801927
\(478\) 13.9227 0.636809
\(479\) 10.0084 0.457296 0.228648 0.973509i \(-0.426570\pi\)
0.228648 + 0.973509i \(0.426570\pi\)
\(480\) −2.67020 −0.121877
\(481\) −1.82897 −0.0833938
\(482\) −15.9988 −0.728726
\(483\) 10.8126 0.491991
\(484\) 18.0027 0.818305
\(485\) 6.46318 0.293478
\(486\) −6.67062 −0.302586
\(487\) 33.7144 1.52775 0.763873 0.645366i \(-0.223295\pi\)
0.763873 + 0.645366i \(0.223295\pi\)
\(488\) 9.89612 0.447976
\(489\) 36.9921 1.67284
\(490\) 7.03057 0.317609
\(491\) −5.61119 −0.253230 −0.126615 0.991952i \(-0.540411\pi\)
−0.126615 + 0.991952i \(0.540411\pi\)
\(492\) 9.70281 0.437436
\(493\) 12.0824 0.544164
\(494\) −0.886271 −0.0398752
\(495\) 4.91111 0.220738
\(496\) 3.30654 0.148468
\(497\) 54.2372 2.43287
\(498\) 3.22487 0.144510
\(499\) −16.7330 −0.749072 −0.374536 0.927212i \(-0.622198\pi\)
−0.374536 + 0.927212i \(0.622198\pi\)
\(500\) 11.2441 0.502852
\(501\) −33.7588 −1.50823
\(502\) 24.5132 1.09408
\(503\) 20.8072 0.927746 0.463873 0.885902i \(-0.346459\pi\)
0.463873 + 0.885902i \(0.346459\pi\)
\(504\) −2.26411 −0.100851
\(505\) 26.1289 1.16272
\(506\) −8.78354 −0.390476
\(507\) 23.8892 1.06096
\(508\) 11.5882 0.514144
\(509\) −17.7056 −0.784786 −0.392393 0.919798i \(-0.628353\pi\)
−0.392393 + 0.919798i \(0.628353\pi\)
\(510\) −3.47894 −0.154050
\(511\) 25.0002 1.10595
\(512\) −1.00000 −0.0441942
\(513\) 5.62012 0.248134
\(514\) 5.56235 0.245345
\(515\) −4.01084 −0.176739
\(516\) −8.08624 −0.355977
\(517\) 52.8941 2.32628
\(518\) 8.96776 0.394021
\(519\) 35.7532 1.56939
\(520\) 0.988392 0.0433439
\(521\) −2.08100 −0.0911702 −0.0455851 0.998960i \(-0.514515\pi\)
−0.0455851 + 0.998960i \(0.514515\pi\)
\(522\) −6.05305 −0.264935
\(523\) 25.0270 1.09436 0.547178 0.837016i \(-0.315702\pi\)
0.547178 + 0.837016i \(0.315702\pi\)
\(524\) 12.7769 0.558159
\(525\) 20.2070 0.881904
\(526\) −29.4620 −1.28460
\(527\) 4.30802 0.187660
\(528\) 10.2927 0.447930
\(529\) −20.3399 −0.884343
\(530\) 3.74890 0.162842
\(531\) −2.95866 −0.128395
\(532\) 4.34554 0.188403
\(533\) −3.59156 −0.155568
\(534\) 27.9161 1.20805
\(535\) 18.0812 0.781720
\(536\) 7.12165 0.307609
\(537\) 44.1408 1.90481
\(538\) −0.299339 −0.0129054
\(539\) −27.1003 −1.16729
\(540\) −6.26770 −0.269719
\(541\) −39.4401 −1.69566 −0.847831 0.530266i \(-0.822092\pi\)
−0.847831 + 0.530266i \(0.822092\pi\)
\(542\) −29.9220 −1.28526
\(543\) −41.8137 −1.79440
\(544\) −1.30288 −0.0558605
\(545\) −11.9985 −0.513959
\(546\) 4.69001 0.200714
\(547\) 7.09834 0.303503 0.151752 0.988419i \(-0.451509\pi\)
0.151752 + 0.988419i \(0.451509\pi\)
\(548\) −12.7306 −0.543825
\(549\) −6.45937 −0.275679
\(550\) −16.4150 −0.699937
\(551\) 11.6177 0.494932
\(552\) −3.11715 −0.132675
\(553\) 45.0508 1.91575
\(554\) 13.6109 0.578273
\(555\) −6.90327 −0.293028
\(556\) 10.3838 0.440373
\(557\) 40.0389 1.69650 0.848251 0.529594i \(-0.177656\pi\)
0.848251 + 0.529594i \(0.177656\pi\)
\(558\) −2.15823 −0.0913653
\(559\) 2.99318 0.126598
\(560\) −4.84626 −0.204792
\(561\) 13.4101 0.566174
\(562\) 20.9957 0.885648
\(563\) −3.48487 −0.146870 −0.0734348 0.997300i \(-0.523396\pi\)
−0.0734348 + 0.997300i \(0.523396\pi\)
\(564\) 18.7714 0.790417
\(565\) −8.89681 −0.374291
\(566\) 29.0276 1.22012
\(567\) −36.5332 −1.53425
\(568\) −15.6360 −0.656072
\(569\) −28.4361 −1.19210 −0.596052 0.802946i \(-0.703265\pi\)
−0.596052 + 0.802946i \(0.703265\pi\)
\(570\) −3.34515 −0.140113
\(571\) 7.61716 0.318768 0.159384 0.987217i \(-0.449049\pi\)
0.159384 + 0.987217i \(0.449049\pi\)
\(572\) −3.80990 −0.159300
\(573\) 36.3864 1.52006
\(574\) 17.6101 0.735030
\(575\) 4.97132 0.207318
\(576\) 0.652717 0.0271966
\(577\) −24.5812 −1.02333 −0.511664 0.859185i \(-0.670971\pi\)
−0.511664 + 0.859185i \(0.670971\pi\)
\(578\) 15.3025 0.636500
\(579\) −37.9279 −1.57623
\(580\) −12.9564 −0.537985
\(581\) 5.85297 0.242822
\(582\) −8.84137 −0.366487
\(583\) −14.4507 −0.598486
\(584\) −7.20729 −0.298240
\(585\) −0.645140 −0.0266733
\(586\) 7.17957 0.296585
\(587\) 34.1297 1.40869 0.704343 0.709860i \(-0.251242\pi\)
0.704343 + 0.709860i \(0.251242\pi\)
\(588\) −9.61752 −0.396620
\(589\) 4.14234 0.170682
\(590\) −6.33294 −0.260723
\(591\) 19.8520 0.816603
\(592\) −2.58531 −0.106255
\(593\) −30.8247 −1.26582 −0.632909 0.774226i \(-0.718139\pi\)
−0.632909 + 0.774226i \(0.718139\pi\)
\(594\) 24.1598 0.991287
\(595\) −6.31410 −0.258853
\(596\) −12.1004 −0.495653
\(597\) 53.5719 2.19255
\(598\) 1.15384 0.0471839
\(599\) 40.4684 1.65349 0.826746 0.562575i \(-0.190189\pi\)
0.826746 + 0.562575i \(0.190189\pi\)
\(600\) −5.82545 −0.237823
\(601\) −34.6124 −1.41187 −0.705934 0.708277i \(-0.749473\pi\)
−0.705934 + 0.708277i \(0.749473\pi\)
\(602\) −14.6761 −0.598153
\(603\) −4.64843 −0.189299
\(604\) 2.15561 0.0877105
\(605\) −25.1520 −1.02258
\(606\) −35.7433 −1.45197
\(607\) −11.4552 −0.464951 −0.232475 0.972602i \(-0.574683\pi\)
−0.232475 + 0.972602i \(0.574683\pi\)
\(608\) −1.25277 −0.0508066
\(609\) −61.4793 −2.49127
\(610\) −13.8261 −0.559803
\(611\) −6.94835 −0.281100
\(612\) 0.850412 0.0343759
\(613\) 22.3551 0.902912 0.451456 0.892293i \(-0.350905\pi\)
0.451456 + 0.892293i \(0.350905\pi\)
\(614\) 21.9860 0.887284
\(615\) −13.5560 −0.546632
\(616\) 18.6806 0.752664
\(617\) 25.8788 1.04184 0.520921 0.853605i \(-0.325589\pi\)
0.520921 + 0.853605i \(0.325589\pi\)
\(618\) 5.48666 0.220706
\(619\) −1.20842 −0.0485705 −0.0242853 0.999705i \(-0.507731\pi\)
−0.0242853 + 0.999705i \(0.507731\pi\)
\(620\) −4.61964 −0.185529
\(621\) −7.31684 −0.293615
\(622\) 6.43014 0.257825
\(623\) 50.6663 2.02990
\(624\) −1.35208 −0.0541265
\(625\) −0.469214 −0.0187686
\(626\) −7.21113 −0.288215
\(627\) 12.8944 0.514951
\(628\) 6.01394 0.239982
\(629\) −3.36834 −0.134305
\(630\) 3.16324 0.126027
\(631\) 4.80816 0.191410 0.0957049 0.995410i \(-0.469489\pi\)
0.0957049 + 0.995410i \(0.469489\pi\)
\(632\) −12.9876 −0.516621
\(633\) 42.0116 1.66981
\(634\) 17.4497 0.693015
\(635\) −16.1902 −0.642488
\(636\) −5.12834 −0.203352
\(637\) 3.55999 0.141052
\(638\) 49.9422 1.97723
\(639\) 10.2059 0.403739
\(640\) 1.39712 0.0552262
\(641\) 35.8358 1.41543 0.707715 0.706498i \(-0.249726\pi\)
0.707715 + 0.706498i \(0.249726\pi\)
\(642\) −24.7344 −0.976188
\(643\) −8.58870 −0.338705 −0.169353 0.985556i \(-0.554168\pi\)
−0.169353 + 0.985556i \(0.554168\pi\)
\(644\) −5.65747 −0.222936
\(645\) 11.2975 0.444838
\(646\) −1.63221 −0.0642185
\(647\) −7.05683 −0.277433 −0.138716 0.990332i \(-0.544298\pi\)
−0.138716 + 0.990332i \(0.544298\pi\)
\(648\) 10.5321 0.413740
\(649\) 24.4112 0.958224
\(650\) 2.15633 0.0845782
\(651\) −21.9207 −0.859138
\(652\) −19.3553 −0.758013
\(653\) −15.4677 −0.605296 −0.302648 0.953102i \(-0.597871\pi\)
−0.302648 + 0.953102i \(0.597871\pi\)
\(654\) 16.4135 0.641817
\(655\) −17.8509 −0.697490
\(656\) −5.07679 −0.198215
\(657\) 4.70433 0.183533
\(658\) 34.0690 1.32815
\(659\) −9.78593 −0.381206 −0.190603 0.981667i \(-0.561044\pi\)
−0.190603 + 0.981667i \(0.561044\pi\)
\(660\) −14.3801 −0.559745
\(661\) −13.9810 −0.543799 −0.271899 0.962326i \(-0.587652\pi\)
−0.271899 + 0.962326i \(0.587652\pi\)
\(662\) −6.76232 −0.262825
\(663\) −1.76160 −0.0684147
\(664\) −1.68735 −0.0654818
\(665\) −6.07127 −0.235434
\(666\) 1.68747 0.0653883
\(667\) −15.1251 −0.585647
\(668\) 17.6636 0.683425
\(669\) 21.7381 0.840445
\(670\) −9.94984 −0.384396
\(671\) 53.2947 2.05742
\(672\) 6.62949 0.255738
\(673\) 18.4026 0.709369 0.354684 0.934986i \(-0.384588\pi\)
0.354684 + 0.934986i \(0.384588\pi\)
\(674\) −3.88491 −0.149641
\(675\) −13.6740 −0.526311
\(676\) −12.4995 −0.480751
\(677\) 21.9176 0.842362 0.421181 0.906977i \(-0.361616\pi\)
0.421181 + 0.906977i \(0.361616\pi\)
\(678\) 12.1705 0.467404
\(679\) −16.0466 −0.615813
\(680\) 1.82028 0.0698047
\(681\) −8.40366 −0.322029
\(682\) 17.8071 0.681868
\(683\) −44.9468 −1.71984 −0.859920 0.510429i \(-0.829487\pi\)
−0.859920 + 0.510429i \(0.829487\pi\)
\(684\) 0.817706 0.0312658
\(685\) 17.7863 0.679578
\(686\) 6.82590 0.260614
\(687\) −1.40865 −0.0537434
\(688\) 4.23096 0.161304
\(689\) 1.89829 0.0723192
\(690\) 4.35505 0.165794
\(691\) −40.4328 −1.53814 −0.769068 0.639166i \(-0.779279\pi\)
−0.769068 + 0.639166i \(0.779279\pi\)
\(692\) −18.7071 −0.711139
\(693\) −12.1932 −0.463180
\(694\) −22.6570 −0.860048
\(695\) −14.5075 −0.550302
\(696\) 17.7238 0.671819
\(697\) −6.61445 −0.250540
\(698\) 14.3862 0.544525
\(699\) 55.6360 2.10435
\(700\) −10.5729 −0.399617
\(701\) −13.6798 −0.516681 −0.258340 0.966054i \(-0.583176\pi\)
−0.258340 + 0.966054i \(0.583176\pi\)
\(702\) −3.17371 −0.119784
\(703\) −3.23880 −0.122154
\(704\) −5.38542 −0.202971
\(705\) −26.2259 −0.987726
\(706\) 31.0634 1.16909
\(707\) −64.8721 −2.43977
\(708\) 8.66319 0.325583
\(709\) −9.89624 −0.371661 −0.185830 0.982582i \(-0.559498\pi\)
−0.185830 + 0.982582i \(0.559498\pi\)
\(710\) 21.8454 0.819844
\(711\) 8.47726 0.317922
\(712\) −14.6065 −0.547403
\(713\) −5.39292 −0.201966
\(714\) 8.63743 0.323248
\(715\) 5.32290 0.199065
\(716\) −23.0957 −0.863128
\(717\) 26.6092 0.993738
\(718\) 7.74628 0.289089
\(719\) 16.1669 0.602925 0.301462 0.953478i \(-0.402525\pi\)
0.301462 + 0.953478i \(0.402525\pi\)
\(720\) −0.911927 −0.0339855
\(721\) 9.95799 0.370855
\(722\) 17.4306 0.648698
\(723\) −30.5771 −1.13717
\(724\) 21.8782 0.813096
\(725\) −28.2664 −1.04979
\(726\) 34.4069 1.27696
\(727\) 10.9898 0.407588 0.203794 0.979014i \(-0.434673\pi\)
0.203794 + 0.979014i \(0.434673\pi\)
\(728\) −2.45395 −0.0909495
\(729\) 18.8474 0.698051
\(730\) 10.0695 0.372688
\(731\) 5.51243 0.203884
\(732\) 18.9135 0.699065
\(733\) −46.6820 −1.72424 −0.862120 0.506704i \(-0.830864\pi\)
−0.862120 + 0.506704i \(0.830864\pi\)
\(734\) −3.81909 −0.140965
\(735\) 13.4369 0.495627
\(736\) 1.63099 0.0601190
\(737\) 38.3531 1.41275
\(738\) 3.31371 0.121979
\(739\) −8.70226 −0.320118 −0.160059 0.987107i \(-0.551168\pi\)
−0.160059 + 0.987107i \(0.551168\pi\)
\(740\) 3.61199 0.132780
\(741\) −1.69385 −0.0622250
\(742\) −9.30767 −0.341695
\(743\) −32.6614 −1.19823 −0.599115 0.800663i \(-0.704481\pi\)
−0.599115 + 0.800663i \(0.704481\pi\)
\(744\) 6.31948 0.231683
\(745\) 16.9058 0.619381
\(746\) −26.7451 −0.979206
\(747\) 1.10136 0.0402967
\(748\) −7.01655 −0.256550
\(749\) −44.8916 −1.64030
\(750\) 21.4899 0.784698
\(751\) −12.5661 −0.458544 −0.229272 0.973362i \(-0.573634\pi\)
−0.229272 + 0.973362i \(0.573634\pi\)
\(752\) −9.82173 −0.358161
\(753\) 46.8498 1.70730
\(754\) −6.56059 −0.238923
\(755\) −3.01165 −0.109605
\(756\) 15.5613 0.565958
\(757\) −27.4350 −0.997140 −0.498570 0.866849i \(-0.666141\pi\)
−0.498570 + 0.866849i \(0.666141\pi\)
\(758\) −0.580027 −0.0210675
\(759\) −16.7872 −0.609336
\(760\) 1.75028 0.0634893
\(761\) 2.93358 0.106342 0.0531712 0.998585i \(-0.483067\pi\)
0.0531712 + 0.998585i \(0.483067\pi\)
\(762\) 22.1475 0.802320
\(763\) 29.7895 1.07845
\(764\) −19.0384 −0.688787
\(765\) −1.18813 −0.0429570
\(766\) −12.4377 −0.449392
\(767\) −3.20674 −0.115789
\(768\) −1.91121 −0.0689648
\(769\) −45.4125 −1.63762 −0.818808 0.574067i \(-0.805365\pi\)
−0.818808 + 0.574067i \(0.805365\pi\)
\(770\) −26.0991 −0.940548
\(771\) 10.6308 0.382859
\(772\) 19.8450 0.714237
\(773\) 16.9877 0.611004 0.305502 0.952191i \(-0.401176\pi\)
0.305502 + 0.952191i \(0.401176\pi\)
\(774\) −2.76162 −0.0992643
\(775\) −10.0785 −0.362029
\(776\) 4.62606 0.166066
\(777\) 17.1393 0.614867
\(778\) −1.38667 −0.0497146
\(779\) −6.36006 −0.227873
\(780\) 1.88902 0.0676379
\(781\) −84.2063 −3.01314
\(782\) 2.12498 0.0759891
\(783\) 41.6028 1.48676
\(784\) 5.03217 0.179720
\(785\) −8.40223 −0.299888
\(786\) 24.4192 0.871005
\(787\) −46.6616 −1.66331 −0.831653 0.555295i \(-0.812605\pi\)
−0.831653 + 0.555295i \(0.812605\pi\)
\(788\) −10.3872 −0.370027
\(789\) −56.3080 −2.00462
\(790\) 18.1454 0.645583
\(791\) 22.0887 0.785385
\(792\) 3.51515 0.124906
\(793\) −7.00098 −0.248612
\(794\) −15.5049 −0.550247
\(795\) 7.16493 0.254114
\(796\) −28.0304 −0.993511
\(797\) 23.4976 0.832329 0.416165 0.909289i \(-0.363374\pi\)
0.416165 + 0.909289i \(0.363374\pi\)
\(798\) 8.30524 0.294002
\(799\) −12.7965 −0.452709
\(800\) 3.04804 0.107765
\(801\) 9.53394 0.336865
\(802\) 6.49569 0.229371
\(803\) −38.8143 −1.36973
\(804\) 13.6110 0.480022
\(805\) 7.90419 0.278586
\(806\) −2.33920 −0.0823948
\(807\) −0.572099 −0.0201388
\(808\) 18.7019 0.657931
\(809\) 12.5746 0.442098 0.221049 0.975263i \(-0.429052\pi\)
0.221049 + 0.975263i \(0.429052\pi\)
\(810\) −14.7147 −0.517021
\(811\) −12.4741 −0.438025 −0.219013 0.975722i \(-0.570284\pi\)
−0.219013 + 0.975722i \(0.570284\pi\)
\(812\) 32.1678 1.12887
\(813\) −57.1872 −2.00564
\(814\) −13.9230 −0.487999
\(815\) 27.0418 0.947233
\(816\) −2.49007 −0.0871700
\(817\) 5.30043 0.185438
\(818\) 22.7868 0.796723
\(819\) 1.60174 0.0559692
\(820\) 7.09291 0.247695
\(821\) 26.2145 0.914893 0.457446 0.889237i \(-0.348764\pi\)
0.457446 + 0.889237i \(0.348764\pi\)
\(822\) −24.3309 −0.848636
\(823\) −6.03310 −0.210301 −0.105150 0.994456i \(-0.533532\pi\)
−0.105150 + 0.994456i \(0.533532\pi\)
\(824\) −2.87078 −0.100008
\(825\) −31.3725 −1.09225
\(826\) 15.7232 0.547081
\(827\) 30.4989 1.06055 0.530276 0.847825i \(-0.322088\pi\)
0.530276 + 0.847825i \(0.322088\pi\)
\(828\) −1.06457 −0.0369965
\(829\) −39.5946 −1.37518 −0.687589 0.726100i \(-0.741331\pi\)
−0.687589 + 0.726100i \(0.741331\pi\)
\(830\) 2.35743 0.0818277
\(831\) 26.0133 0.902393
\(832\) 0.707447 0.0245263
\(833\) 6.55631 0.227163
\(834\) 19.8457 0.687200
\(835\) −24.6782 −0.854025
\(836\) −6.74670 −0.233340
\(837\) 14.8336 0.512725
\(838\) 6.04978 0.208986
\(839\) 37.2213 1.28502 0.642511 0.766276i \(-0.277893\pi\)
0.642511 + 0.766276i \(0.277893\pi\)
\(840\) −9.26222 −0.319577
\(841\) 56.9998 1.96551
\(842\) 33.3435 1.14909
\(843\) 40.1271 1.38205
\(844\) −21.9817 −0.756642
\(845\) 17.4634 0.600759
\(846\) 6.41081 0.220408
\(847\) 62.4467 2.14570
\(848\) 2.68330 0.0921449
\(849\) 55.4778 1.90399
\(850\) 3.97123 0.136212
\(851\) 4.21660 0.144543
\(852\) −29.8836 −1.02380
\(853\) −38.3368 −1.31263 −0.656314 0.754488i \(-0.727885\pi\)
−0.656314 + 0.754488i \(0.727885\pi\)
\(854\) 34.3271 1.17465
\(855\) −1.14244 −0.0390705
\(856\) 12.9417 0.442340
\(857\) −19.0032 −0.649138 −0.324569 0.945862i \(-0.605219\pi\)
−0.324569 + 0.945862i \(0.605219\pi\)
\(858\) −7.28151 −0.248587
\(859\) −38.9152 −1.32777 −0.663885 0.747835i \(-0.731093\pi\)
−0.663885 + 0.747835i \(0.731093\pi\)
\(860\) −5.91117 −0.201569
\(861\) 33.6565 1.14701
\(862\) 21.0016 0.715319
\(863\) 43.3681 1.47627 0.738133 0.674655i \(-0.235708\pi\)
0.738133 + 0.674655i \(0.235708\pi\)
\(864\) −4.48615 −0.152622
\(865\) 26.1362 0.888658
\(866\) 16.0873 0.546668
\(867\) 29.2463 0.993256
\(868\) 11.4695 0.389301
\(869\) −69.9439 −2.37268
\(870\) −24.7624 −0.839522
\(871\) −5.03819 −0.170713
\(872\) −8.58800 −0.290826
\(873\) −3.01951 −0.102195
\(874\) 2.04325 0.0691141
\(875\) 39.0029 1.31854
\(876\) −13.7746 −0.465402
\(877\) −25.1145 −0.848055 −0.424028 0.905649i \(-0.639384\pi\)
−0.424028 + 0.905649i \(0.639384\pi\)
\(878\) −9.20885 −0.310784
\(879\) 13.7216 0.462820
\(880\) 7.52410 0.253637
\(881\) −6.81968 −0.229761 −0.114880 0.993379i \(-0.536648\pi\)
−0.114880 + 0.993379i \(0.536648\pi\)
\(882\) −3.28458 −0.110598
\(883\) 33.9324 1.14192 0.570958 0.820979i \(-0.306572\pi\)
0.570958 + 0.820979i \(0.306572\pi\)
\(884\) 0.921719 0.0310008
\(885\) −12.1036 −0.406857
\(886\) −14.3123 −0.480831
\(887\) −16.2336 −0.545073 −0.272536 0.962145i \(-0.587862\pi\)
−0.272536 + 0.962145i \(0.587862\pi\)
\(888\) −4.94106 −0.165811
\(889\) 40.1966 1.34815
\(890\) 20.4071 0.684049
\(891\) 56.7198 1.90019
\(892\) −11.3740 −0.380830
\(893\) −12.3044 −0.411751
\(894\) −23.1265 −0.773464
\(895\) 32.2676 1.07859
\(896\) −3.46874 −0.115883
\(897\) 2.20522 0.0736302
\(898\) 5.34142 0.178246
\(899\) 30.6635 1.02269
\(900\) −1.98951 −0.0663170
\(901\) 3.49602 0.116469
\(902\) −27.3406 −0.910344
\(903\) −28.0491 −0.933415
\(904\) −6.36794 −0.211795
\(905\) −30.5665 −1.01607
\(906\) 4.11982 0.136872
\(907\) −49.3429 −1.63841 −0.819203 0.573504i \(-0.805584\pi\)
−0.819203 + 0.573504i \(0.805584\pi\)
\(908\) 4.39704 0.145921
\(909\) −12.2071 −0.404883
\(910\) 3.42848 0.113653
\(911\) 42.7550 1.41654 0.708268 0.705944i \(-0.249477\pi\)
0.708268 + 0.705944i \(0.249477\pi\)
\(912\) −2.39431 −0.0792835
\(913\) −9.08706 −0.300738
\(914\) 35.0989 1.16097
\(915\) −26.4246 −0.873570
\(916\) 0.737047 0.0243527
\(917\) 44.3196 1.46356
\(918\) −5.84491 −0.192911
\(919\) 31.5375 1.04033 0.520164 0.854067i \(-0.325871\pi\)
0.520164 + 0.854067i \(0.325871\pi\)
\(920\) −2.27869 −0.0751262
\(921\) 42.0199 1.38460
\(922\) 26.7673 0.881534
\(923\) 11.0616 0.364098
\(924\) 35.7026 1.17453
\(925\) 7.88013 0.259097
\(926\) −27.9389 −0.918131
\(927\) 1.87381 0.0615439
\(928\) −9.27361 −0.304421
\(929\) 23.1958 0.761030 0.380515 0.924775i \(-0.375747\pi\)
0.380515 + 0.924775i \(0.375747\pi\)
\(930\) −8.82910 −0.289518
\(931\) 6.30416 0.206611
\(932\) −29.1104 −0.953543
\(933\) 12.2893 0.402335
\(934\) −14.7050 −0.481163
\(935\) 9.80299 0.320592
\(936\) −0.461763 −0.0150932
\(937\) −17.5749 −0.574148 −0.287074 0.957908i \(-0.592683\pi\)
−0.287074 + 0.957908i \(0.592683\pi\)
\(938\) 24.7032 0.806587
\(939\) −13.7820 −0.449758
\(940\) 13.7222 0.447568
\(941\) 5.79800 0.189009 0.0945047 0.995524i \(-0.469873\pi\)
0.0945047 + 0.995524i \(0.469873\pi\)
\(942\) 11.4939 0.374491
\(943\) 8.28018 0.269640
\(944\) −4.53284 −0.147531
\(945\) −21.7410 −0.707236
\(946\) 22.7855 0.740819
\(947\) −44.2098 −1.43662 −0.718312 0.695721i \(-0.755085\pi\)
−0.718312 + 0.695721i \(0.755085\pi\)
\(948\) −24.8221 −0.806184
\(949\) 5.09878 0.165513
\(950\) 3.81850 0.123889
\(951\) 33.3499 1.08145
\(952\) −4.51935 −0.146473
\(953\) −23.0960 −0.748153 −0.374077 0.927398i \(-0.622040\pi\)
−0.374077 + 0.927398i \(0.622040\pi\)
\(954\) −1.75144 −0.0567048
\(955\) 26.5991 0.860726
\(956\) −13.9227 −0.450292
\(957\) 95.4500 3.08546
\(958\) −10.0084 −0.323357
\(959\) −44.1592 −1.42598
\(960\) 2.67020 0.0861802
\(961\) −20.0668 −0.647316
\(962\) 1.82897 0.0589683
\(963\) −8.44730 −0.272211
\(964\) 15.9988 0.515287
\(965\) −27.7259 −0.892529
\(966\) −10.8126 −0.347890
\(967\) 14.4469 0.464581 0.232291 0.972646i \(-0.425378\pi\)
0.232291 + 0.972646i \(0.425378\pi\)
\(968\) −18.0027 −0.578629
\(969\) −3.11950 −0.100213
\(970\) −6.46318 −0.207520
\(971\) 32.0479 1.02847 0.514233 0.857650i \(-0.328077\pi\)
0.514233 + 0.857650i \(0.328077\pi\)
\(972\) 6.67062 0.213960
\(973\) 36.0189 1.15471
\(974\) −33.7144 −1.08028
\(975\) 4.12120 0.131984
\(976\) −9.89612 −0.316767
\(977\) 58.6860 1.87753 0.938766 0.344555i \(-0.111970\pi\)
0.938766 + 0.344555i \(0.111970\pi\)
\(978\) −36.9921 −1.18288
\(979\) −78.6623 −2.51406
\(980\) −7.03057 −0.224583
\(981\) 5.60553 0.178971
\(982\) 5.61119 0.179060
\(983\) 11.6451 0.371421 0.185711 0.982604i \(-0.440541\pi\)
0.185711 + 0.982604i \(0.440541\pi\)
\(984\) −9.70281 −0.309314
\(985\) 14.5121 0.462395
\(986\) −12.0824 −0.384782
\(987\) 65.1130 2.07257
\(988\) 0.886271 0.0281960
\(989\) −6.90063 −0.219427
\(990\) −4.91111 −0.156085
\(991\) −7.52094 −0.238910 −0.119455 0.992840i \(-0.538115\pi\)
−0.119455 + 0.992840i \(0.538115\pi\)
\(992\) −3.30654 −0.104983
\(993\) −12.9242 −0.410137
\(994\) −54.2372 −1.72030
\(995\) 39.1620 1.24152
\(996\) −3.22487 −0.102184
\(997\) 29.6352 0.938555 0.469277 0.883051i \(-0.344514\pi\)
0.469277 + 0.883051i \(0.344514\pi\)
\(998\) 16.7330 0.529674
\(999\) −11.5981 −0.366946
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.1.9 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.9 40 1.1 even 1 trivial