Properties

Label 4016.2.a.m.1.12
Level 4016
Weight 2
Character 4016.1
Self dual yes
Analytic conductor 32.068
Analytic rank 0
Dimension 23
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(23\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) = 4016.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.0812979 q^{3} +4.08000 q^{5} -4.55548 q^{7} -2.99339 q^{9} +O(q^{10})\) \(q+0.0812979 q^{3} +4.08000 q^{5} -4.55548 q^{7} -2.99339 q^{9} -3.53247 q^{11} -1.88643 q^{13} +0.331695 q^{15} +7.25153 q^{17} +6.82679 q^{19} -0.370351 q^{21} +0.510976 q^{23} +11.6464 q^{25} -0.487250 q^{27} -0.891263 q^{29} -4.69490 q^{31} -0.287182 q^{33} -18.5864 q^{35} +5.85500 q^{37} -0.153363 q^{39} -10.5623 q^{41} +5.97649 q^{43} -12.2130 q^{45} +3.59740 q^{47} +13.7524 q^{49} +0.589534 q^{51} -7.13887 q^{53} -14.4125 q^{55} +0.555004 q^{57} +11.2148 q^{59} +5.92985 q^{61} +13.6363 q^{63} -7.69665 q^{65} +8.31848 q^{67} +0.0415413 q^{69} +8.26092 q^{71} -9.74189 q^{73} +0.946827 q^{75} +16.0921 q^{77} -10.0811 q^{79} +8.94056 q^{81} +8.97449 q^{83} +29.5862 q^{85} -0.0724579 q^{87} +3.15098 q^{89} +8.59362 q^{91} -0.381685 q^{93} +27.8533 q^{95} -5.31571 q^{97} +10.5741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} + O(q^{10}) \) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} - 8q^{11} + 8q^{13} - 7q^{15} + 19q^{17} + 9q^{19} + 9q^{21} - 21q^{23} + 65q^{25} - 5q^{27} + 10q^{29} + 9q^{31} + 34q^{33} - 12q^{35} + 11q^{37} + 9q^{39} + 35q^{41} + 9q^{43} + 29q^{45} - 37q^{47} + 77q^{49} + 17q^{51} + 38q^{53} + 20q^{55} + 51q^{57} - 17q^{59} - 22q^{63} + 41q^{65} - 9q^{67} + 8q^{69} - 13q^{71} + 41q^{73} - 25q^{75} + 36q^{77} + 36q^{79} + 127q^{81} - 29q^{83} + 34q^{85} - 10q^{87} + 36q^{89} + 6q^{91} + 36q^{93} - 25q^{95} + 40q^{97} - 19q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.0812979 0.0469374 0.0234687 0.999725i \(-0.492529\pi\)
0.0234687 + 0.999725i \(0.492529\pi\)
\(4\) 0 0
\(5\) 4.08000 1.82463 0.912315 0.409489i \(-0.134293\pi\)
0.912315 + 0.409489i \(0.134293\pi\)
\(6\) 0 0
\(7\) −4.55548 −1.72181 −0.860905 0.508765i \(-0.830102\pi\)
−0.860905 + 0.508765i \(0.830102\pi\)
\(8\) 0 0
\(9\) −2.99339 −0.997797
\(10\) 0 0
\(11\) −3.53247 −1.06508 −0.532540 0.846405i \(-0.678762\pi\)
−0.532540 + 0.846405i \(0.678762\pi\)
\(12\) 0 0
\(13\) −1.88643 −0.523203 −0.261601 0.965176i \(-0.584251\pi\)
−0.261601 + 0.965176i \(0.584251\pi\)
\(14\) 0 0
\(15\) 0.331695 0.0856434
\(16\) 0 0
\(17\) 7.25153 1.75875 0.879377 0.476127i \(-0.157960\pi\)
0.879377 + 0.476127i \(0.157960\pi\)
\(18\) 0 0
\(19\) 6.82679 1.56617 0.783087 0.621912i \(-0.213644\pi\)
0.783087 + 0.621912i \(0.213644\pi\)
\(20\) 0 0
\(21\) −0.370351 −0.0808173
\(22\) 0 0
\(23\) 0.510976 0.106546 0.0532729 0.998580i \(-0.483035\pi\)
0.0532729 + 0.998580i \(0.483035\pi\)
\(24\) 0 0
\(25\) 11.6464 2.32928
\(26\) 0 0
\(27\) −0.487250 −0.0937714
\(28\) 0 0
\(29\) −0.891263 −0.165503 −0.0827517 0.996570i \(-0.526371\pi\)
−0.0827517 + 0.996570i \(0.526371\pi\)
\(30\) 0 0
\(31\) −4.69490 −0.843229 −0.421614 0.906775i \(-0.638536\pi\)
−0.421614 + 0.906775i \(0.638536\pi\)
\(32\) 0 0
\(33\) −0.287182 −0.0499920
\(34\) 0 0
\(35\) −18.5864 −3.14167
\(36\) 0 0
\(37\) 5.85500 0.962557 0.481278 0.876568i \(-0.340173\pi\)
0.481278 + 0.876568i \(0.340173\pi\)
\(38\) 0 0
\(39\) −0.153363 −0.0245578
\(40\) 0 0
\(41\) −10.5623 −1.64955 −0.824775 0.565461i \(-0.808698\pi\)
−0.824775 + 0.565461i \(0.808698\pi\)
\(42\) 0 0
\(43\) 5.97649 0.911407 0.455703 0.890132i \(-0.349388\pi\)
0.455703 + 0.890132i \(0.349388\pi\)
\(44\) 0 0
\(45\) −12.2130 −1.82061
\(46\) 0 0
\(47\) 3.59740 0.524734 0.262367 0.964968i \(-0.415497\pi\)
0.262367 + 0.964968i \(0.415497\pi\)
\(48\) 0 0
\(49\) 13.7524 1.96463
\(50\) 0 0
\(51\) 0.589534 0.0825513
\(52\) 0 0
\(53\) −7.13887 −0.980600 −0.490300 0.871554i \(-0.663113\pi\)
−0.490300 + 0.871554i \(0.663113\pi\)
\(54\) 0 0
\(55\) −14.4125 −1.94338
\(56\) 0 0
\(57\) 0.555004 0.0735121
\(58\) 0 0
\(59\) 11.2148 1.46004 0.730021 0.683425i \(-0.239510\pi\)
0.730021 + 0.683425i \(0.239510\pi\)
\(60\) 0 0
\(61\) 5.92985 0.759240 0.379620 0.925142i \(-0.376055\pi\)
0.379620 + 0.925142i \(0.376055\pi\)
\(62\) 0 0
\(63\) 13.6363 1.71802
\(64\) 0 0
\(65\) −7.69665 −0.954652
\(66\) 0 0
\(67\) 8.31848 1.01626 0.508132 0.861279i \(-0.330336\pi\)
0.508132 + 0.861279i \(0.330336\pi\)
\(68\) 0 0
\(69\) 0.0415413 0.00500098
\(70\) 0 0
\(71\) 8.26092 0.980391 0.490196 0.871613i \(-0.336925\pi\)
0.490196 + 0.871613i \(0.336925\pi\)
\(72\) 0 0
\(73\) −9.74189 −1.14020 −0.570101 0.821575i \(-0.693096\pi\)
−0.570101 + 0.821575i \(0.693096\pi\)
\(74\) 0 0
\(75\) 0.946827 0.109330
\(76\) 0 0
\(77\) 16.0921 1.83386
\(78\) 0 0
\(79\) −10.0811 −1.13421 −0.567107 0.823644i \(-0.691937\pi\)
−0.567107 + 0.823644i \(0.691937\pi\)
\(80\) 0 0
\(81\) 8.94056 0.993396
\(82\) 0 0
\(83\) 8.97449 0.985078 0.492539 0.870290i \(-0.336069\pi\)
0.492539 + 0.870290i \(0.336069\pi\)
\(84\) 0 0
\(85\) 29.5862 3.20908
\(86\) 0 0
\(87\) −0.0724579 −0.00776830
\(88\) 0 0
\(89\) 3.15098 0.334003 0.167002 0.985957i \(-0.446591\pi\)
0.167002 + 0.985957i \(0.446591\pi\)
\(90\) 0 0
\(91\) 8.59362 0.900856
\(92\) 0 0
\(93\) −0.381685 −0.0395789
\(94\) 0 0
\(95\) 27.8533 2.85769
\(96\) 0 0
\(97\) −5.31571 −0.539729 −0.269864 0.962898i \(-0.586979\pi\)
−0.269864 + 0.962898i \(0.586979\pi\)
\(98\) 0 0
\(99\) 10.5741 1.06273
\(100\) 0 0
\(101\) 8.10417 0.806395 0.403198 0.915113i \(-0.367899\pi\)
0.403198 + 0.915113i \(0.367899\pi\)
\(102\) 0 0
\(103\) −6.08265 −0.599341 −0.299670 0.954043i \(-0.596877\pi\)
−0.299670 + 0.954043i \(0.596877\pi\)
\(104\) 0 0
\(105\) −1.51103 −0.147462
\(106\) 0 0
\(107\) −4.24257 −0.410145 −0.205072 0.978747i \(-0.565743\pi\)
−0.205072 + 0.978747i \(0.565743\pi\)
\(108\) 0 0
\(109\) 19.1124 1.83064 0.915320 0.402727i \(-0.131938\pi\)
0.915320 + 0.402727i \(0.131938\pi\)
\(110\) 0 0
\(111\) 0.476000 0.0451799
\(112\) 0 0
\(113\) −0.811208 −0.0763121 −0.0381560 0.999272i \(-0.512148\pi\)
−0.0381560 + 0.999272i \(0.512148\pi\)
\(114\) 0 0
\(115\) 2.08478 0.194407
\(116\) 0 0
\(117\) 5.64683 0.522050
\(118\) 0 0
\(119\) −33.0342 −3.02824
\(120\) 0 0
\(121\) 1.47833 0.134393
\(122\) 0 0
\(123\) −0.858691 −0.0774255
\(124\) 0 0
\(125\) 27.1172 2.42544
\(126\) 0 0
\(127\) 20.0145 1.77600 0.887998 0.459848i \(-0.152096\pi\)
0.887998 + 0.459848i \(0.152096\pi\)
\(128\) 0 0
\(129\) 0.485877 0.0427790
\(130\) 0 0
\(131\) 15.7560 1.37661 0.688304 0.725422i \(-0.258355\pi\)
0.688304 + 0.725422i \(0.258355\pi\)
\(132\) 0 0
\(133\) −31.0993 −2.69665
\(134\) 0 0
\(135\) −1.98798 −0.171098
\(136\) 0 0
\(137\) 19.6854 1.68183 0.840917 0.541165i \(-0.182016\pi\)
0.840917 + 0.541165i \(0.182016\pi\)
\(138\) 0 0
\(139\) 8.46512 0.718002 0.359001 0.933337i \(-0.383117\pi\)
0.359001 + 0.933337i \(0.383117\pi\)
\(140\) 0 0
\(141\) 0.292461 0.0246297
\(142\) 0 0
\(143\) 6.66377 0.557252
\(144\) 0 0
\(145\) −3.63635 −0.301983
\(146\) 0 0
\(147\) 1.11804 0.0922146
\(148\) 0 0
\(149\) 3.07600 0.251996 0.125998 0.992031i \(-0.459787\pi\)
0.125998 + 0.992031i \(0.459787\pi\)
\(150\) 0 0
\(151\) −7.28405 −0.592768 −0.296384 0.955069i \(-0.595781\pi\)
−0.296384 + 0.955069i \(0.595781\pi\)
\(152\) 0 0
\(153\) −21.7066 −1.75488
\(154\) 0 0
\(155\) −19.1552 −1.53858
\(156\) 0 0
\(157\) −9.36041 −0.747042 −0.373521 0.927622i \(-0.621850\pi\)
−0.373521 + 0.927622i \(0.621850\pi\)
\(158\) 0 0
\(159\) −0.580376 −0.0460268
\(160\) 0 0
\(161\) −2.32774 −0.183452
\(162\) 0 0
\(163\) −0.991367 −0.0776498 −0.0388249 0.999246i \(-0.512361\pi\)
−0.0388249 + 0.999246i \(0.512361\pi\)
\(164\) 0 0
\(165\) −1.17170 −0.0912170
\(166\) 0 0
\(167\) 7.81810 0.604982 0.302491 0.953152i \(-0.402182\pi\)
0.302491 + 0.953152i \(0.402182\pi\)
\(168\) 0 0
\(169\) −9.44137 −0.726259
\(170\) 0 0
\(171\) −20.4353 −1.56272
\(172\) 0 0
\(173\) 8.88338 0.675391 0.337695 0.941255i \(-0.390353\pi\)
0.337695 + 0.941255i \(0.390353\pi\)
\(174\) 0 0
\(175\) −53.0549 −4.01057
\(176\) 0 0
\(177\) 0.911739 0.0685305
\(178\) 0 0
\(179\) −25.0766 −1.87431 −0.937156 0.348911i \(-0.886552\pi\)
−0.937156 + 0.348911i \(0.886552\pi\)
\(180\) 0 0
\(181\) 13.1745 0.979254 0.489627 0.871932i \(-0.337133\pi\)
0.489627 + 0.871932i \(0.337133\pi\)
\(182\) 0 0
\(183\) 0.482085 0.0356367
\(184\) 0 0
\(185\) 23.8884 1.75631
\(186\) 0 0
\(187\) −25.6158 −1.87321
\(188\) 0 0
\(189\) 2.21966 0.161456
\(190\) 0 0
\(191\) 2.38371 0.172479 0.0862396 0.996274i \(-0.472515\pi\)
0.0862396 + 0.996274i \(0.472515\pi\)
\(192\) 0 0
\(193\) 4.28291 0.308291 0.154145 0.988048i \(-0.450738\pi\)
0.154145 + 0.988048i \(0.450738\pi\)
\(194\) 0 0
\(195\) −0.625721 −0.0448088
\(196\) 0 0
\(197\) 14.2113 1.01252 0.506258 0.862382i \(-0.331029\pi\)
0.506258 + 0.862382i \(0.331029\pi\)
\(198\) 0 0
\(199\) −25.4472 −1.80390 −0.901952 0.431837i \(-0.857866\pi\)
−0.901952 + 0.431837i \(0.857866\pi\)
\(200\) 0 0
\(201\) 0.676275 0.0477008
\(202\) 0 0
\(203\) 4.06013 0.284966
\(204\) 0 0
\(205\) −43.0940 −3.00982
\(206\) 0 0
\(207\) −1.52955 −0.106311
\(208\) 0 0
\(209\) −24.1154 −1.66810
\(210\) 0 0
\(211\) 23.9641 1.64976 0.824878 0.565311i \(-0.191244\pi\)
0.824878 + 0.565311i \(0.191244\pi\)
\(212\) 0 0
\(213\) 0.671596 0.0460170
\(214\) 0 0
\(215\) 24.3841 1.66298
\(216\) 0 0
\(217\) 21.3875 1.45188
\(218\) 0 0
\(219\) −0.791996 −0.0535181
\(220\) 0 0
\(221\) −13.6795 −0.920184
\(222\) 0 0
\(223\) 11.8891 0.796156 0.398078 0.917352i \(-0.369677\pi\)
0.398078 + 0.917352i \(0.369677\pi\)
\(224\) 0 0
\(225\) −34.8622 −2.32414
\(226\) 0 0
\(227\) −0.0493739 −0.00327706 −0.00163853 0.999999i \(-0.500522\pi\)
−0.00163853 + 0.999999i \(0.500522\pi\)
\(228\) 0 0
\(229\) 11.0320 0.729013 0.364507 0.931201i \(-0.381238\pi\)
0.364507 + 0.931201i \(0.381238\pi\)
\(230\) 0 0
\(231\) 1.30825 0.0860768
\(232\) 0 0
\(233\) 16.3230 1.06935 0.534677 0.845057i \(-0.320433\pi\)
0.534677 + 0.845057i \(0.320433\pi\)
\(234\) 0 0
\(235\) 14.6774 0.957446
\(236\) 0 0
\(237\) −0.819574 −0.0532370
\(238\) 0 0
\(239\) 5.21570 0.337376 0.168688 0.985670i \(-0.446047\pi\)
0.168688 + 0.985670i \(0.446047\pi\)
\(240\) 0 0
\(241\) −20.0438 −1.29114 −0.645568 0.763703i \(-0.723379\pi\)
−0.645568 + 0.763703i \(0.723379\pi\)
\(242\) 0 0
\(243\) 2.18860 0.140399
\(244\) 0 0
\(245\) 56.1098 3.58473
\(246\) 0 0
\(247\) −12.8783 −0.819426
\(248\) 0 0
\(249\) 0.729608 0.0462370
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) −1.80500 −0.113480
\(254\) 0 0
\(255\) 2.40530 0.150626
\(256\) 0 0
\(257\) 8.26409 0.515499 0.257750 0.966212i \(-0.417019\pi\)
0.257750 + 0.966212i \(0.417019\pi\)
\(258\) 0 0
\(259\) −26.6724 −1.65734
\(260\) 0 0
\(261\) 2.66790 0.165139
\(262\) 0 0
\(263\) −6.14540 −0.378942 −0.189471 0.981886i \(-0.560677\pi\)
−0.189471 + 0.981886i \(0.560677\pi\)
\(264\) 0 0
\(265\) −29.1266 −1.78923
\(266\) 0 0
\(267\) 0.256168 0.0156772
\(268\) 0 0
\(269\) −28.2396 −1.72180 −0.860901 0.508773i \(-0.830099\pi\)
−0.860901 + 0.508773i \(0.830099\pi\)
\(270\) 0 0
\(271\) −9.25843 −0.562409 −0.281205 0.959648i \(-0.590734\pi\)
−0.281205 + 0.959648i \(0.590734\pi\)
\(272\) 0 0
\(273\) 0.698643 0.0422838
\(274\) 0 0
\(275\) −41.1405 −2.48086
\(276\) 0 0
\(277\) −18.8351 −1.13169 −0.565847 0.824510i \(-0.691451\pi\)
−0.565847 + 0.824510i \(0.691451\pi\)
\(278\) 0 0
\(279\) 14.0537 0.841371
\(280\) 0 0
\(281\) 1.15954 0.0691721 0.0345860 0.999402i \(-0.488989\pi\)
0.0345860 + 0.999402i \(0.488989\pi\)
\(282\) 0 0
\(283\) 24.2403 1.44094 0.720468 0.693488i \(-0.243927\pi\)
0.720468 + 0.693488i \(0.243927\pi\)
\(284\) 0 0
\(285\) 2.26442 0.134132
\(286\) 0 0
\(287\) 48.1162 2.84021
\(288\) 0 0
\(289\) 35.5846 2.09321
\(290\) 0 0
\(291\) −0.432156 −0.0253334
\(292\) 0 0
\(293\) 14.1841 0.828641 0.414321 0.910131i \(-0.364019\pi\)
0.414321 + 0.910131i \(0.364019\pi\)
\(294\) 0 0
\(295\) 45.7563 2.66404
\(296\) 0 0
\(297\) 1.72120 0.0998739
\(298\) 0 0
\(299\) −0.963922 −0.0557450
\(300\) 0 0
\(301\) −27.2258 −1.56927
\(302\) 0 0
\(303\) 0.658852 0.0378501
\(304\) 0 0
\(305\) 24.1938 1.38533
\(306\) 0 0
\(307\) −18.1114 −1.03367 −0.516836 0.856085i \(-0.672890\pi\)
−0.516836 + 0.856085i \(0.672890\pi\)
\(308\) 0 0
\(309\) −0.494507 −0.0281315
\(310\) 0 0
\(311\) −21.7357 −1.23252 −0.616261 0.787542i \(-0.711353\pi\)
−0.616261 + 0.787542i \(0.711353\pi\)
\(312\) 0 0
\(313\) −24.3780 −1.37793 −0.688963 0.724796i \(-0.741934\pi\)
−0.688963 + 0.724796i \(0.741934\pi\)
\(314\) 0 0
\(315\) 55.6362 3.13475
\(316\) 0 0
\(317\) 19.3696 1.08791 0.543953 0.839116i \(-0.316927\pi\)
0.543953 + 0.839116i \(0.316927\pi\)
\(318\) 0 0
\(319\) 3.14836 0.176274
\(320\) 0 0
\(321\) −0.344912 −0.0192511
\(322\) 0 0
\(323\) 49.5047 2.75451
\(324\) 0 0
\(325\) −21.9701 −1.21868
\(326\) 0 0
\(327\) 1.55380 0.0859255
\(328\) 0 0
\(329\) −16.3879 −0.903493
\(330\) 0 0
\(331\) −5.55908 −0.305555 −0.152777 0.988261i \(-0.548822\pi\)
−0.152777 + 0.988261i \(0.548822\pi\)
\(332\) 0 0
\(333\) −17.5263 −0.960436
\(334\) 0 0
\(335\) 33.9394 1.85431
\(336\) 0 0
\(337\) −30.5103 −1.66200 −0.831001 0.556271i \(-0.812232\pi\)
−0.831001 + 0.556271i \(0.812232\pi\)
\(338\) 0 0
\(339\) −0.0659496 −0.00358189
\(340\) 0 0
\(341\) 16.5846 0.898105
\(342\) 0 0
\(343\) −30.7605 −1.66091
\(344\) 0 0
\(345\) 0.169488 0.00912494
\(346\) 0 0
\(347\) −10.1759 −0.546269 −0.273134 0.961976i \(-0.588060\pi\)
−0.273134 + 0.961976i \(0.588060\pi\)
\(348\) 0 0
\(349\) −4.86972 −0.260670 −0.130335 0.991470i \(-0.541605\pi\)
−0.130335 + 0.991470i \(0.541605\pi\)
\(350\) 0 0
\(351\) 0.919165 0.0490614
\(352\) 0 0
\(353\) −11.6941 −0.622415 −0.311208 0.950342i \(-0.600733\pi\)
−0.311208 + 0.950342i \(0.600733\pi\)
\(354\) 0 0
\(355\) 33.7045 1.78885
\(356\) 0 0
\(357\) −2.68561 −0.142138
\(358\) 0 0
\(359\) 6.69853 0.353535 0.176767 0.984253i \(-0.443436\pi\)
0.176767 + 0.984253i \(0.443436\pi\)
\(360\) 0 0
\(361\) 27.6051 1.45290
\(362\) 0 0
\(363\) 0.120185 0.00630808
\(364\) 0 0
\(365\) −39.7469 −2.08045
\(366\) 0 0
\(367\) −8.72221 −0.455296 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(368\) 0 0
\(369\) 31.6170 1.64592
\(370\) 0 0
\(371\) 32.5210 1.68841
\(372\) 0 0
\(373\) −18.8557 −0.976309 −0.488154 0.872757i \(-0.662330\pi\)
−0.488154 + 0.872757i \(0.662330\pi\)
\(374\) 0 0
\(375\) 2.20457 0.113844
\(376\) 0 0
\(377\) 1.68131 0.0865918
\(378\) 0 0
\(379\) 34.7023 1.78254 0.891268 0.453476i \(-0.149816\pi\)
0.891268 + 0.453476i \(0.149816\pi\)
\(380\) 0 0
\(381\) 1.62713 0.0833606
\(382\) 0 0
\(383\) −12.9708 −0.662774 −0.331387 0.943495i \(-0.607517\pi\)
−0.331387 + 0.943495i \(0.607517\pi\)
\(384\) 0 0
\(385\) 65.6557 3.34612
\(386\) 0 0
\(387\) −17.8900 −0.909399
\(388\) 0 0
\(389\) −16.5551 −0.839378 −0.419689 0.907668i \(-0.637861\pi\)
−0.419689 + 0.907668i \(0.637861\pi\)
\(390\) 0 0
\(391\) 3.70535 0.187388
\(392\) 0 0
\(393\) 1.28093 0.0646144
\(394\) 0 0
\(395\) −41.1309 −2.06952
\(396\) 0 0
\(397\) −37.4152 −1.87781 −0.938907 0.344172i \(-0.888160\pi\)
−0.938907 + 0.344172i \(0.888160\pi\)
\(398\) 0 0
\(399\) −2.52831 −0.126574
\(400\) 0 0
\(401\) −19.8151 −0.989518 −0.494759 0.869030i \(-0.664744\pi\)
−0.494759 + 0.869030i \(0.664744\pi\)
\(402\) 0 0
\(403\) 8.85662 0.441179
\(404\) 0 0
\(405\) 36.4775 1.81258
\(406\) 0 0
\(407\) −20.6826 −1.02520
\(408\) 0 0
\(409\) 29.3248 1.45002 0.725010 0.688739i \(-0.241835\pi\)
0.725010 + 0.688739i \(0.241835\pi\)
\(410\) 0 0
\(411\) 1.60038 0.0789408
\(412\) 0 0
\(413\) −51.0888 −2.51391
\(414\) 0 0
\(415\) 36.6159 1.79740
\(416\) 0 0
\(417\) 0.688197 0.0337012
\(418\) 0 0
\(419\) 26.5237 1.29577 0.647884 0.761739i \(-0.275654\pi\)
0.647884 + 0.761739i \(0.275654\pi\)
\(420\) 0 0
\(421\) −8.62688 −0.420448 −0.210224 0.977653i \(-0.567419\pi\)
−0.210224 + 0.977653i \(0.567419\pi\)
\(422\) 0 0
\(423\) −10.7684 −0.523578
\(424\) 0 0
\(425\) 84.4540 4.09662
\(426\) 0 0
\(427\) −27.0133 −1.30727
\(428\) 0 0
\(429\) 0.541750 0.0261560
\(430\) 0 0
\(431\) 7.33946 0.353529 0.176765 0.984253i \(-0.443437\pi\)
0.176765 + 0.984253i \(0.443437\pi\)
\(432\) 0 0
\(433\) 37.8792 1.82036 0.910180 0.414213i \(-0.135943\pi\)
0.910180 + 0.414213i \(0.135943\pi\)
\(434\) 0 0
\(435\) −0.295628 −0.0141743
\(436\) 0 0
\(437\) 3.48833 0.166869
\(438\) 0 0
\(439\) −11.5455 −0.551035 −0.275518 0.961296i \(-0.588849\pi\)
−0.275518 + 0.961296i \(0.588849\pi\)
\(440\) 0 0
\(441\) −41.1664 −1.96030
\(442\) 0 0
\(443\) −5.58800 −0.265494 −0.132747 0.991150i \(-0.542380\pi\)
−0.132747 + 0.991150i \(0.542380\pi\)
\(444\) 0 0
\(445\) 12.8560 0.609432
\(446\) 0 0
\(447\) 0.250072 0.0118280
\(448\) 0 0
\(449\) 19.4504 0.917922 0.458961 0.888456i \(-0.348222\pi\)
0.458961 + 0.888456i \(0.348222\pi\)
\(450\) 0 0
\(451\) 37.3109 1.75690
\(452\) 0 0
\(453\) −0.592178 −0.0278230
\(454\) 0 0
\(455\) 35.0619 1.64373
\(456\) 0 0
\(457\) 22.5002 1.05251 0.526257 0.850325i \(-0.323595\pi\)
0.526257 + 0.850325i \(0.323595\pi\)
\(458\) 0 0
\(459\) −3.53331 −0.164921
\(460\) 0 0
\(461\) −12.2605 −0.571027 −0.285514 0.958375i \(-0.592164\pi\)
−0.285514 + 0.958375i \(0.592164\pi\)
\(462\) 0 0
\(463\) −6.40785 −0.297798 −0.148899 0.988852i \(-0.547573\pi\)
−0.148899 + 0.988852i \(0.547573\pi\)
\(464\) 0 0
\(465\) −1.55728 −0.0722169
\(466\) 0 0
\(467\) −1.93134 −0.0893718 −0.0446859 0.999001i \(-0.514229\pi\)
−0.0446859 + 0.999001i \(0.514229\pi\)
\(468\) 0 0
\(469\) −37.8947 −1.74981
\(470\) 0 0
\(471\) −0.760982 −0.0350642
\(472\) 0 0
\(473\) −21.1118 −0.970720
\(474\) 0 0
\(475\) 79.5075 3.64805
\(476\) 0 0
\(477\) 21.3694 0.978439
\(478\) 0 0
\(479\) 32.7239 1.49519 0.747596 0.664154i \(-0.231208\pi\)
0.747596 + 0.664154i \(0.231208\pi\)
\(480\) 0 0
\(481\) −11.0451 −0.503612
\(482\) 0 0
\(483\) −0.189240 −0.00861074
\(484\) 0 0
\(485\) −21.6881 −0.984805
\(486\) 0 0
\(487\) −19.0302 −0.862339 −0.431170 0.902271i \(-0.641899\pi\)
−0.431170 + 0.902271i \(0.641899\pi\)
\(488\) 0 0
\(489\) −0.0805961 −0.00364468
\(490\) 0 0
\(491\) −35.1413 −1.58591 −0.792953 0.609283i \(-0.791457\pi\)
−0.792953 + 0.609283i \(0.791457\pi\)
\(492\) 0 0
\(493\) −6.46302 −0.291080
\(494\) 0 0
\(495\) 43.1421 1.93909
\(496\) 0 0
\(497\) −37.6325 −1.68805
\(498\) 0 0
\(499\) 11.3015 0.505924 0.252962 0.967476i \(-0.418595\pi\)
0.252962 + 0.967476i \(0.418595\pi\)
\(500\) 0 0
\(501\) 0.635595 0.0283963
\(502\) 0 0
\(503\) −31.7607 −1.41614 −0.708069 0.706143i \(-0.750434\pi\)
−0.708069 + 0.706143i \(0.750434\pi\)
\(504\) 0 0
\(505\) 33.0650 1.47137
\(506\) 0 0
\(507\) −0.767564 −0.0340887
\(508\) 0 0
\(509\) −29.8144 −1.32150 −0.660751 0.750605i \(-0.729762\pi\)
−0.660751 + 0.750605i \(0.729762\pi\)
\(510\) 0 0
\(511\) 44.3790 1.96321
\(512\) 0 0
\(513\) −3.32636 −0.146862
\(514\) 0 0
\(515\) −24.8172 −1.09358
\(516\) 0 0
\(517\) −12.7077 −0.558883
\(518\) 0 0
\(519\) 0.722200 0.0317011
\(520\) 0 0
\(521\) 16.5762 0.726218 0.363109 0.931747i \(-0.381715\pi\)
0.363109 + 0.931747i \(0.381715\pi\)
\(522\) 0 0
\(523\) 21.3611 0.934056 0.467028 0.884243i \(-0.345325\pi\)
0.467028 + 0.884243i \(0.345325\pi\)
\(524\) 0 0
\(525\) −4.31325 −0.188246
\(526\) 0 0
\(527\) −34.0452 −1.48303
\(528\) 0 0
\(529\) −22.7389 −0.988648
\(530\) 0 0
\(531\) −33.5702 −1.45682
\(532\) 0 0
\(533\) 19.9250 0.863049
\(534\) 0 0
\(535\) −17.3097 −0.748363
\(536\) 0 0
\(537\) −2.03867 −0.0879753
\(538\) 0 0
\(539\) −48.5800 −2.09249
\(540\) 0 0
\(541\) −8.90934 −0.383042 −0.191521 0.981488i \(-0.561342\pi\)
−0.191521 + 0.981488i \(0.561342\pi\)
\(542\) 0 0
\(543\) 1.07106 0.0459636
\(544\) 0 0
\(545\) 77.9787 3.34024
\(546\) 0 0
\(547\) 15.5948 0.666785 0.333392 0.942788i \(-0.391807\pi\)
0.333392 + 0.942788i \(0.391807\pi\)
\(548\) 0 0
\(549\) −17.7504 −0.757567
\(550\) 0 0
\(551\) −6.08447 −0.259207
\(552\) 0 0
\(553\) 45.9243 1.95290
\(554\) 0 0
\(555\) 1.94208 0.0824366
\(556\) 0 0
\(557\) −8.97903 −0.380454 −0.190227 0.981740i \(-0.560922\pi\)
−0.190227 + 0.981740i \(0.560922\pi\)
\(558\) 0 0
\(559\) −11.2743 −0.476850
\(560\) 0 0
\(561\) −2.08251 −0.0879236
\(562\) 0 0
\(563\) 7.47683 0.315111 0.157555 0.987510i \(-0.449639\pi\)
0.157555 + 0.987510i \(0.449639\pi\)
\(564\) 0 0
\(565\) −3.30973 −0.139241
\(566\) 0 0
\(567\) −40.7286 −1.71044
\(568\) 0 0
\(569\) −22.0584 −0.924737 −0.462369 0.886688i \(-0.653000\pi\)
−0.462369 + 0.886688i \(0.653000\pi\)
\(570\) 0 0
\(571\) −34.0514 −1.42500 −0.712502 0.701670i \(-0.752438\pi\)
−0.712502 + 0.701670i \(0.752438\pi\)
\(572\) 0 0
\(573\) 0.193791 0.00809572
\(574\) 0 0
\(575\) 5.95102 0.248175
\(576\) 0 0
\(577\) −30.4513 −1.26771 −0.633853 0.773454i \(-0.718527\pi\)
−0.633853 + 0.773454i \(0.718527\pi\)
\(578\) 0 0
\(579\) 0.348192 0.0144704
\(580\) 0 0
\(581\) −40.8831 −1.69612
\(582\) 0 0
\(583\) 25.2178 1.04442
\(584\) 0 0
\(585\) 23.0391 0.952548
\(586\) 0 0
\(587\) 4.57228 0.188718 0.0943591 0.995538i \(-0.469920\pi\)
0.0943591 + 0.995538i \(0.469920\pi\)
\(588\) 0 0
\(589\) −32.0511 −1.32064
\(590\) 0 0
\(591\) 1.15535 0.0475248
\(592\) 0 0
\(593\) 33.4875 1.37517 0.687583 0.726105i \(-0.258672\pi\)
0.687583 + 0.726105i \(0.258672\pi\)
\(594\) 0 0
\(595\) −134.779 −5.52542
\(596\) 0 0
\(597\) −2.06880 −0.0846705
\(598\) 0 0
\(599\) 4.13268 0.168857 0.0844283 0.996430i \(-0.473094\pi\)
0.0844283 + 0.996430i \(0.473094\pi\)
\(600\) 0 0
\(601\) 18.1982 0.742321 0.371161 0.928569i \(-0.378960\pi\)
0.371161 + 0.928569i \(0.378960\pi\)
\(602\) 0 0
\(603\) −24.9005 −1.01402
\(604\) 0 0
\(605\) 6.03157 0.245218
\(606\) 0 0
\(607\) 35.6100 1.44537 0.722683 0.691180i \(-0.242909\pi\)
0.722683 + 0.691180i \(0.242909\pi\)
\(608\) 0 0
\(609\) 0.330081 0.0133755
\(610\) 0 0
\(611\) −6.78625 −0.274542
\(612\) 0 0
\(613\) −4.36247 −0.176198 −0.0880992 0.996112i \(-0.528079\pi\)
−0.0880992 + 0.996112i \(0.528079\pi\)
\(614\) 0 0
\(615\) −3.50346 −0.141273
\(616\) 0 0
\(617\) −37.3669 −1.50433 −0.752167 0.658972i \(-0.770991\pi\)
−0.752167 + 0.658972i \(0.770991\pi\)
\(618\) 0 0
\(619\) 15.3067 0.615227 0.307614 0.951511i \(-0.400470\pi\)
0.307614 + 0.951511i \(0.400470\pi\)
\(620\) 0 0
\(621\) −0.248973 −0.00999094
\(622\) 0 0
\(623\) −14.3542 −0.575090
\(624\) 0 0
\(625\) 52.4063 2.09625
\(626\) 0 0
\(627\) −1.96053 −0.0782962
\(628\) 0 0
\(629\) 42.4577 1.69290
\(630\) 0 0
\(631\) −3.68147 −0.146557 −0.0732786 0.997312i \(-0.523346\pi\)
−0.0732786 + 0.997312i \(0.523346\pi\)
\(632\) 0 0
\(633\) 1.94823 0.0774352
\(634\) 0 0
\(635\) 81.6589 3.24054
\(636\) 0 0
\(637\) −25.9430 −1.02790
\(638\) 0 0
\(639\) −24.7282 −0.978231
\(640\) 0 0
\(641\) −9.93361 −0.392354 −0.196177 0.980568i \(-0.562853\pi\)
−0.196177 + 0.980568i \(0.562853\pi\)
\(642\) 0 0
\(643\) −30.8635 −1.21714 −0.608569 0.793501i \(-0.708256\pi\)
−0.608569 + 0.793501i \(0.708256\pi\)
\(644\) 0 0
\(645\) 1.98238 0.0780559
\(646\) 0 0
\(647\) −9.87949 −0.388403 −0.194201 0.980962i \(-0.562212\pi\)
−0.194201 + 0.980962i \(0.562212\pi\)
\(648\) 0 0
\(649\) −39.6159 −1.55506
\(650\) 0 0
\(651\) 1.73876 0.0681474
\(652\) 0 0
\(653\) 27.9126 1.09230 0.546152 0.837686i \(-0.316092\pi\)
0.546152 + 0.837686i \(0.316092\pi\)
\(654\) 0 0
\(655\) 64.2845 2.51180
\(656\) 0 0
\(657\) 29.1613 1.13769
\(658\) 0 0
\(659\) 23.0433 0.897641 0.448820 0.893622i \(-0.351844\pi\)
0.448820 + 0.893622i \(0.351844\pi\)
\(660\) 0 0
\(661\) 40.7362 1.58445 0.792227 0.610227i \(-0.208922\pi\)
0.792227 + 0.610227i \(0.208922\pi\)
\(662\) 0 0
\(663\) −1.11212 −0.0431910
\(664\) 0 0
\(665\) −126.885 −4.92040
\(666\) 0 0
\(667\) −0.455414 −0.0176337
\(668\) 0 0
\(669\) 0.966563 0.0373695
\(670\) 0 0
\(671\) −20.9470 −0.808651
\(672\) 0 0
\(673\) −17.3600 −0.669177 −0.334589 0.942364i \(-0.608597\pi\)
−0.334589 + 0.942364i \(0.608597\pi\)
\(674\) 0 0
\(675\) −5.67470 −0.218419
\(676\) 0 0
\(677\) −42.2063 −1.62212 −0.811059 0.584964i \(-0.801109\pi\)
−0.811059 + 0.584964i \(0.801109\pi\)
\(678\) 0 0
\(679\) 24.2156 0.929310
\(680\) 0 0
\(681\) −0.00401400 −0.000153817 0
\(682\) 0 0
\(683\) −5.97105 −0.228476 −0.114238 0.993453i \(-0.536443\pi\)
−0.114238 + 0.993453i \(0.536443\pi\)
\(684\) 0 0
\(685\) 80.3162 3.06872
\(686\) 0 0
\(687\) 0.896876 0.0342180
\(688\) 0 0
\(689\) 13.4670 0.513052
\(690\) 0 0
\(691\) −18.8137 −0.715706 −0.357853 0.933778i \(-0.616491\pi\)
−0.357853 + 0.933778i \(0.616491\pi\)
\(692\) 0 0
\(693\) −48.1699 −1.82982
\(694\) 0 0
\(695\) 34.5377 1.31009
\(696\) 0 0
\(697\) −76.5926 −2.90115
\(698\) 0 0
\(699\) 1.32702 0.0501927
\(700\) 0 0
\(701\) −9.36923 −0.353871 −0.176936 0.984222i \(-0.556618\pi\)
−0.176936 + 0.984222i \(0.556618\pi\)
\(702\) 0 0
\(703\) 39.9709 1.50753
\(704\) 0 0
\(705\) 1.19324 0.0449400
\(706\) 0 0
\(707\) −36.9184 −1.38846
\(708\) 0 0
\(709\) −27.3166 −1.02590 −0.512948 0.858420i \(-0.671447\pi\)
−0.512948 + 0.858420i \(0.671447\pi\)
\(710\) 0 0
\(711\) 30.1767 1.13172
\(712\) 0 0
\(713\) −2.39898 −0.0898424
\(714\) 0 0
\(715\) 27.1882 1.01678
\(716\) 0 0
\(717\) 0.424026 0.0158355
\(718\) 0 0
\(719\) 23.3970 0.872562 0.436281 0.899810i \(-0.356295\pi\)
0.436281 + 0.899810i \(0.356295\pi\)
\(720\) 0 0
\(721\) 27.7094 1.03195
\(722\) 0 0
\(723\) −1.62952 −0.0606026
\(724\) 0 0
\(725\) −10.3800 −0.385503
\(726\) 0 0
\(727\) −48.2868 −1.79086 −0.895429 0.445205i \(-0.853131\pi\)
−0.895429 + 0.445205i \(0.853131\pi\)
\(728\) 0 0
\(729\) −26.6437 −0.986806
\(730\) 0 0
\(731\) 43.3387 1.60294
\(732\) 0 0
\(733\) 24.0508 0.888337 0.444169 0.895943i \(-0.353499\pi\)
0.444169 + 0.895943i \(0.353499\pi\)
\(734\) 0 0
\(735\) 4.56161 0.168258
\(736\) 0 0
\(737\) −29.3848 −1.08240
\(738\) 0 0
\(739\) −26.9486 −0.991321 −0.495660 0.868516i \(-0.665074\pi\)
−0.495660 + 0.868516i \(0.665074\pi\)
\(740\) 0 0
\(741\) −1.04698 −0.0384617
\(742\) 0 0
\(743\) 5.23899 0.192200 0.0960999 0.995372i \(-0.469363\pi\)
0.0960999 + 0.995372i \(0.469363\pi\)
\(744\) 0 0
\(745\) 12.5501 0.459799
\(746\) 0 0
\(747\) −26.8642 −0.982908
\(748\) 0 0
\(749\) 19.3269 0.706191
\(750\) 0 0
\(751\) 8.14855 0.297345 0.148672 0.988886i \(-0.452500\pi\)
0.148672 + 0.988886i \(0.452500\pi\)
\(752\) 0 0
\(753\) 0.0812979 0.00296266
\(754\) 0 0
\(755\) −29.7189 −1.08158
\(756\) 0 0
\(757\) 1.25379 0.0455698 0.0227849 0.999740i \(-0.492747\pi\)
0.0227849 + 0.999740i \(0.492747\pi\)
\(758\) 0 0
\(759\) −0.146743 −0.00532644
\(760\) 0 0
\(761\) 28.0354 1.01628 0.508141 0.861274i \(-0.330333\pi\)
0.508141 + 0.861274i \(0.330333\pi\)
\(762\) 0 0
\(763\) −87.0664 −3.15202
\(764\) 0 0
\(765\) −88.5631 −3.20201
\(766\) 0 0
\(767\) −21.1560 −0.763898
\(768\) 0 0
\(769\) 33.2123 1.19767 0.598834 0.800873i \(-0.295631\pi\)
0.598834 + 0.800873i \(0.295631\pi\)
\(770\) 0 0
\(771\) 0.671853 0.0241962
\(772\) 0 0
\(773\) −0.493202 −0.0177392 −0.00886962 0.999961i \(-0.502823\pi\)
−0.00886962 + 0.999961i \(0.502823\pi\)
\(774\) 0 0
\(775\) −54.6786 −1.96411
\(776\) 0 0
\(777\) −2.16841 −0.0777912
\(778\) 0 0
\(779\) −72.1065 −2.58348
\(780\) 0 0
\(781\) −29.1814 −1.04419
\(782\) 0 0
\(783\) 0.434268 0.0155195
\(784\) 0 0
\(785\) −38.1904 −1.36308
\(786\) 0 0
\(787\) 22.7424 0.810679 0.405339 0.914166i \(-0.367153\pi\)
0.405339 + 0.914166i \(0.367153\pi\)
\(788\) 0 0
\(789\) −0.499608 −0.0177865
\(790\) 0 0
\(791\) 3.69545 0.131395
\(792\) 0 0
\(793\) −11.1863 −0.397236
\(794\) 0 0
\(795\) −2.36793 −0.0839819
\(796\) 0 0
\(797\) −31.3453 −1.11031 −0.555154 0.831748i \(-0.687341\pi\)
−0.555154 + 0.831748i \(0.687341\pi\)
\(798\) 0 0
\(799\) 26.0866 0.922878
\(800\) 0 0
\(801\) −9.43211 −0.333267
\(802\) 0 0
\(803\) 34.4129 1.21441
\(804\) 0 0
\(805\) −9.49718 −0.334731
\(806\) 0 0
\(807\) −2.29582 −0.0808168
\(808\) 0 0
\(809\) 20.6486 0.725967 0.362983 0.931796i \(-0.381758\pi\)
0.362983 + 0.931796i \(0.381758\pi\)
\(810\) 0 0
\(811\) 6.41871 0.225391 0.112696 0.993630i \(-0.464051\pi\)
0.112696 + 0.993630i \(0.464051\pi\)
\(812\) 0 0
\(813\) −0.752691 −0.0263980
\(814\) 0 0
\(815\) −4.04477 −0.141682
\(816\) 0 0
\(817\) 40.8003 1.42742
\(818\) 0 0
\(819\) −25.7241 −0.898871
\(820\) 0 0
\(821\) 11.5844 0.404297 0.202149 0.979355i \(-0.435208\pi\)
0.202149 + 0.979355i \(0.435208\pi\)
\(822\) 0 0
\(823\) 11.5089 0.401173 0.200587 0.979676i \(-0.435715\pi\)
0.200587 + 0.979676i \(0.435715\pi\)
\(824\) 0 0
\(825\) −3.34464 −0.116445
\(826\) 0 0
\(827\) −5.58063 −0.194058 −0.0970288 0.995282i \(-0.530934\pi\)
−0.0970288 + 0.995282i \(0.530934\pi\)
\(828\) 0 0
\(829\) 17.3422 0.602320 0.301160 0.953574i \(-0.402626\pi\)
0.301160 + 0.953574i \(0.402626\pi\)
\(830\) 0 0
\(831\) −1.53126 −0.0531188
\(832\) 0 0
\(833\) 99.7260 3.45530
\(834\) 0 0
\(835\) 31.8978 1.10387
\(836\) 0 0
\(837\) 2.28759 0.0790707
\(838\) 0 0
\(839\) 14.0923 0.486522 0.243261 0.969961i \(-0.421783\pi\)
0.243261 + 0.969961i \(0.421783\pi\)
\(840\) 0 0
\(841\) −28.2056 −0.972609
\(842\) 0 0
\(843\) 0.0942678 0.00324676
\(844\) 0 0
\(845\) −38.5208 −1.32515
\(846\) 0 0
\(847\) −6.73450 −0.231400
\(848\) 0 0
\(849\) 1.97069 0.0676338
\(850\) 0 0
\(851\) 2.99176 0.102556
\(852\) 0 0
\(853\) −23.0877 −0.790508 −0.395254 0.918572i \(-0.629343\pi\)
−0.395254 + 0.918572i \(0.629343\pi\)
\(854\) 0 0
\(855\) −83.3758 −2.85139
\(856\) 0 0
\(857\) 25.4684 0.869983 0.434992 0.900434i \(-0.356751\pi\)
0.434992 + 0.900434i \(0.356751\pi\)
\(858\) 0 0
\(859\) −15.9645 −0.544702 −0.272351 0.962198i \(-0.587801\pi\)
−0.272351 + 0.962198i \(0.587801\pi\)
\(860\) 0 0
\(861\) 3.91175 0.133312
\(862\) 0 0
\(863\) −34.8405 −1.18598 −0.592992 0.805208i \(-0.702054\pi\)
−0.592992 + 0.805208i \(0.702054\pi\)
\(864\) 0 0
\(865\) 36.2442 1.23234
\(866\) 0 0
\(867\) 2.89296 0.0982499
\(868\) 0 0
\(869\) 35.6112 1.20803
\(870\) 0 0
\(871\) −15.6923 −0.531712
\(872\) 0 0
\(873\) 15.9120 0.538540
\(874\) 0 0
\(875\) −123.532 −4.17615
\(876\) 0 0
\(877\) −37.9157 −1.28032 −0.640161 0.768241i \(-0.721132\pi\)
−0.640161 + 0.768241i \(0.721132\pi\)
\(878\) 0 0
\(879\) 1.15313 0.0388942
\(880\) 0 0
\(881\) 36.6026 1.23317 0.616586 0.787287i \(-0.288515\pi\)
0.616586 + 0.787287i \(0.288515\pi\)
\(882\) 0 0
\(883\) 44.5586 1.49952 0.749758 0.661712i \(-0.230170\pi\)
0.749758 + 0.661712i \(0.230170\pi\)
\(884\) 0 0
\(885\) 3.71989 0.125043
\(886\) 0 0
\(887\) 7.66736 0.257445 0.128722 0.991681i \(-0.458912\pi\)
0.128722 + 0.991681i \(0.458912\pi\)
\(888\) 0 0
\(889\) −91.1755 −3.05793
\(890\) 0 0
\(891\) −31.5822 −1.05804
\(892\) 0 0
\(893\) 24.5587 0.821825
\(894\) 0 0
\(895\) −102.312 −3.41993
\(896\) 0 0
\(897\) −0.0783648 −0.00261653
\(898\) 0 0
\(899\) 4.18439 0.139557
\(900\) 0 0
\(901\) −51.7677 −1.72463
\(902\) 0 0
\(903\) −2.21340 −0.0736574
\(904\) 0 0
\(905\) 53.7520 1.78678
\(906\) 0 0
\(907\) 7.06462 0.234577 0.117288 0.993098i \(-0.462580\pi\)
0.117288 + 0.993098i \(0.462580\pi\)
\(908\) 0 0
\(909\) −24.2590 −0.804619
\(910\) 0 0
\(911\) 7.80712 0.258661 0.129331 0.991602i \(-0.458717\pi\)
0.129331 + 0.991602i \(0.458717\pi\)
\(912\) 0 0
\(913\) −31.7021 −1.04919
\(914\) 0 0
\(915\) 1.96691 0.0650239
\(916\) 0 0
\(917\) −71.7762 −2.37026
\(918\) 0 0
\(919\) 50.5634 1.66793 0.833967 0.551815i \(-0.186064\pi\)
0.833967 + 0.551815i \(0.186064\pi\)
\(920\) 0 0
\(921\) −1.47242 −0.0485178
\(922\) 0 0
\(923\) −15.5837 −0.512943
\(924\) 0 0
\(925\) 68.1896 2.24206
\(926\) 0 0
\(927\) 18.2077 0.598021
\(928\) 0 0
\(929\) −22.5151 −0.738696 −0.369348 0.929291i \(-0.620419\pi\)
−0.369348 + 0.929291i \(0.620419\pi\)
\(930\) 0 0
\(931\) 93.8849 3.07695
\(932\) 0 0
\(933\) −1.76707 −0.0578513
\(934\) 0 0
\(935\) −104.512 −3.41792
\(936\) 0 0
\(937\) 34.7253 1.13443 0.567213 0.823571i \(-0.308022\pi\)
0.567213 + 0.823571i \(0.308022\pi\)
\(938\) 0 0
\(939\) −1.98188 −0.0646763
\(940\) 0 0
\(941\) −13.6640 −0.445432 −0.222716 0.974883i \(-0.571492\pi\)
−0.222716 + 0.974883i \(0.571492\pi\)
\(942\) 0 0
\(943\) −5.39706 −0.175753
\(944\) 0 0
\(945\) 9.05621 0.294598
\(946\) 0 0
\(947\) −14.6968 −0.477581 −0.238790 0.971071i \(-0.576751\pi\)
−0.238790 + 0.971071i \(0.576751\pi\)
\(948\) 0 0
\(949\) 18.3774 0.596557
\(950\) 0 0
\(951\) 1.57471 0.0510635
\(952\) 0 0
\(953\) 0.324327 0.0105060 0.00525299 0.999986i \(-0.498328\pi\)
0.00525299 + 0.999986i \(0.498328\pi\)
\(954\) 0 0
\(955\) 9.72553 0.314711
\(956\) 0 0
\(957\) 0.255955 0.00827385
\(958\) 0 0
\(959\) −89.6763 −2.89580
\(960\) 0 0
\(961\) −8.95793 −0.288966
\(962\) 0 0
\(963\) 12.6997 0.409241
\(964\) 0 0
\(965\) 17.4743 0.562517
\(966\) 0 0
\(967\) −1.81111 −0.0582413 −0.0291206 0.999576i \(-0.509271\pi\)
−0.0291206 + 0.999576i \(0.509271\pi\)
\(968\) 0 0
\(969\) 4.02463 0.129290
\(970\) 0 0
\(971\) 25.3189 0.812521 0.406260 0.913757i \(-0.366833\pi\)
0.406260 + 0.913757i \(0.366833\pi\)
\(972\) 0 0
\(973\) −38.5627 −1.23626
\(974\) 0 0
\(975\) −1.78613 −0.0572018
\(976\) 0 0
\(977\) 33.0716 1.05805 0.529027 0.848605i \(-0.322557\pi\)
0.529027 + 0.848605i \(0.322557\pi\)
\(978\) 0 0
\(979\) −11.1307 −0.355740
\(980\) 0 0
\(981\) −57.2110 −1.82661
\(982\) 0 0
\(983\) −20.1354 −0.642221 −0.321110 0.947042i \(-0.604056\pi\)
−0.321110 + 0.947042i \(0.604056\pi\)
\(984\) 0 0
\(985\) 57.9822 1.84747
\(986\) 0 0
\(987\) −1.33230 −0.0424076
\(988\) 0 0
\(989\) 3.05384 0.0971065
\(990\) 0 0
\(991\) 2.73796 0.0869742 0.0434871 0.999054i \(-0.486153\pi\)
0.0434871 + 0.999054i \(0.486153\pi\)
\(992\) 0 0
\(993\) −0.451942 −0.0143419
\(994\) 0 0
\(995\) −103.824 −3.29146
\(996\) 0 0
\(997\) −31.8558 −1.00888 −0.504442 0.863446i \(-0.668302\pi\)
−0.504442 + 0.863446i \(0.668302\pi\)
\(998\) 0 0
\(999\) −2.85285 −0.0902603
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 4016.2.a.m.1.12 23
4.3 odd 2 2008.2.a.d.1.12 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.d.1.12 23 4.3 odd 2
4016.2.a.m.1.12 23 1.1 even 1 trivial