Properties

Label 4016.2.a.m.1.1
Level 4016
Weight 2
Character 4016.1
Self dual yes
Analytic conductor 32.068
Analytic rank 0
Dimension 23
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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 1.1
Character \(\chi\) = 4016.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.32587 q^{3} -0.316897 q^{5} -0.268063 q^{7} +8.06143 q^{9} +O(q^{10})\) \(q-3.32587 q^{3} -0.316897 q^{5} -0.268063 q^{7} +8.06143 q^{9} +0.366288 q^{11} -3.69132 q^{13} +1.05396 q^{15} +5.91866 q^{17} -7.61514 q^{19} +0.891542 q^{21} +6.61559 q^{23} -4.89958 q^{25} -16.8337 q^{27} +5.48804 q^{29} +4.15696 q^{31} -1.21823 q^{33} +0.0849482 q^{35} +8.30100 q^{37} +12.2769 q^{39} +3.92223 q^{41} -6.01718 q^{43} -2.55464 q^{45} -1.79035 q^{47} -6.92814 q^{49} -19.6847 q^{51} -10.5522 q^{53} -0.116076 q^{55} +25.3270 q^{57} +3.99709 q^{59} +1.88567 q^{61} -2.16097 q^{63} +1.16977 q^{65} -1.28140 q^{67} -22.0026 q^{69} -8.15695 q^{71} -16.0154 q^{73} +16.2954 q^{75} -0.0981882 q^{77} -7.93927 q^{79} +31.8024 q^{81} +11.7328 q^{83} -1.87560 q^{85} -18.2525 q^{87} -1.39073 q^{89} +0.989507 q^{91} -13.8255 q^{93} +2.41321 q^{95} -9.53422 q^{97} +2.95281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} + O(q^{10}) \) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} - 8q^{11} + 8q^{13} - 7q^{15} + 19q^{17} + 9q^{19} + 9q^{21} - 21q^{23} + 65q^{25} - 5q^{27} + 10q^{29} + 9q^{31} + 34q^{33} - 12q^{35} + 11q^{37} + 9q^{39} + 35q^{41} + 9q^{43} + 29q^{45} - 37q^{47} + 77q^{49} + 17q^{51} + 38q^{53} + 20q^{55} + 51q^{57} - 17q^{59} - 22q^{63} + 41q^{65} - 9q^{67} + 8q^{69} - 13q^{71} + 41q^{73} - 25q^{75} + 36q^{77} + 36q^{79} + 127q^{81} - 29q^{83} + 34q^{85} - 10q^{87} + 36q^{89} + 6q^{91} + 36q^{93} - 25q^{95} + 40q^{97} - 19q^{99} + O(q^{100}) \)

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
\(3\) −3.32587 −1.92019 −0.960097 0.279668i \(-0.909776\pi\)
−0.960097 + 0.279668i \(0.909776\pi\)
\(4\) 0 0
\(5\) −0.316897 −0.141721 −0.0708603 0.997486i \(-0.522574\pi\)
−0.0708603 + 0.997486i \(0.522574\pi\)
\(6\) 0 0
\(7\) −0.268063 −0.101318 −0.0506591 0.998716i \(-0.516132\pi\)
−0.0506591 + 0.998716i \(0.516132\pi\)
\(8\) 0 0
\(9\) 8.06143 2.68714
\(10\) 0 0
\(11\) 0.366288 0.110440 0.0552200 0.998474i \(-0.482414\pi\)
0.0552200 + 0.998474i \(0.482414\pi\)
\(12\) 0 0
\(13\) −3.69132 −1.02379 −0.511895 0.859048i \(-0.671056\pi\)
−0.511895 + 0.859048i \(0.671056\pi\)
\(14\) 0 0
\(15\) 1.05396 0.272131
\(16\) 0 0
\(17\) 5.91866 1.43549 0.717743 0.696308i \(-0.245175\pi\)
0.717743 + 0.696308i \(0.245175\pi\)
\(18\) 0 0
\(19\) −7.61514 −1.74703 −0.873517 0.486794i \(-0.838166\pi\)
−0.873517 + 0.486794i \(0.838166\pi\)
\(20\) 0 0
\(21\) 0.891542 0.194551
\(22\) 0 0
\(23\) 6.61559 1.37945 0.689723 0.724073i \(-0.257732\pi\)
0.689723 + 0.724073i \(0.257732\pi\)
\(24\) 0 0
\(25\) −4.89958 −0.979915
\(26\) 0 0
\(27\) −16.8337 −3.23964
\(28\) 0 0
\(29\) 5.48804 1.01910 0.509552 0.860440i \(-0.329811\pi\)
0.509552 + 0.860440i \(0.329811\pi\)
\(30\) 0 0
\(31\) 4.15696 0.746613 0.373306 0.927708i \(-0.378224\pi\)
0.373306 + 0.927708i \(0.378224\pi\)
\(32\) 0 0
\(33\) −1.21823 −0.212066
\(34\) 0 0
\(35\) 0.0849482 0.0143589
\(36\) 0 0
\(37\) 8.30100 1.36468 0.682338 0.731037i \(-0.260963\pi\)
0.682338 + 0.731037i \(0.260963\pi\)
\(38\) 0 0
\(39\) 12.2769 1.96587
\(40\) 0 0
\(41\) 3.92223 0.612550 0.306275 0.951943i \(-0.400917\pi\)
0.306275 + 0.951943i \(0.400917\pi\)
\(42\) 0 0
\(43\) −6.01718 −0.917611 −0.458805 0.888537i \(-0.651722\pi\)
−0.458805 + 0.888537i \(0.651722\pi\)
\(44\) 0 0
\(45\) −2.55464 −0.380823
\(46\) 0 0
\(47\) −1.79035 −0.261149 −0.130574 0.991439i \(-0.541682\pi\)
−0.130574 + 0.991439i \(0.541682\pi\)
\(48\) 0 0
\(49\) −6.92814 −0.989735
\(50\) 0 0
\(51\) −19.6847 −2.75641
\(52\) 0 0
\(53\) −10.5522 −1.44945 −0.724725 0.689039i \(-0.758033\pi\)
−0.724725 + 0.689039i \(0.758033\pi\)
\(54\) 0 0
\(55\) −0.116076 −0.0156516
\(56\) 0 0
\(57\) 25.3270 3.35464
\(58\) 0 0
\(59\) 3.99709 0.520377 0.260188 0.965558i \(-0.416215\pi\)
0.260188 + 0.965558i \(0.416215\pi\)
\(60\) 0 0
\(61\) 1.88567 0.241436 0.120718 0.992687i \(-0.461480\pi\)
0.120718 + 0.992687i \(0.461480\pi\)
\(62\) 0 0
\(63\) −2.16097 −0.272256
\(64\) 0 0
\(65\) 1.16977 0.145092
\(66\) 0 0
\(67\) −1.28140 −0.156548 −0.0782739 0.996932i \(-0.524941\pi\)
−0.0782739 + 0.996932i \(0.524941\pi\)
\(68\) 0 0
\(69\) −22.0026 −2.64880
\(70\) 0 0
\(71\) −8.15695 −0.968052 −0.484026 0.875054i \(-0.660826\pi\)
−0.484026 + 0.875054i \(0.660826\pi\)
\(72\) 0 0
\(73\) −16.0154 −1.87446 −0.937230 0.348712i \(-0.886619\pi\)
−0.937230 + 0.348712i \(0.886619\pi\)
\(74\) 0 0
\(75\) 16.2954 1.88163
\(76\) 0 0
\(77\) −0.0981882 −0.0111896
\(78\) 0 0
\(79\) −7.93927 −0.893237 −0.446619 0.894724i \(-0.647372\pi\)
−0.446619 + 0.894724i \(0.647372\pi\)
\(80\) 0 0
\(81\) 31.8024 3.53360
\(82\) 0 0
\(83\) 11.7328 1.28784 0.643919 0.765094i \(-0.277307\pi\)
0.643919 + 0.765094i \(0.277307\pi\)
\(84\) 0 0
\(85\) −1.87560 −0.203438
\(86\) 0 0
\(87\) −18.2525 −1.95688
\(88\) 0 0
\(89\) −1.39073 −0.147417 −0.0737083 0.997280i \(-0.523483\pi\)
−0.0737083 + 0.997280i \(0.523483\pi\)
\(90\) 0 0
\(91\) 0.989507 0.103728
\(92\) 0 0
\(93\) −13.8255 −1.43364
\(94\) 0 0
\(95\) 2.41321 0.247591
\(96\) 0 0
\(97\) −9.53422 −0.968053 −0.484027 0.875053i \(-0.660826\pi\)
−0.484027 + 0.875053i \(0.660826\pi\)
\(98\) 0 0
\(99\) 2.95281 0.296768
\(100\) 0 0
\(101\) −6.51231 −0.647999 −0.323999 0.946057i \(-0.605028\pi\)
−0.323999 + 0.946057i \(0.605028\pi\)
\(102\) 0 0
\(103\) −7.24438 −0.713810 −0.356905 0.934141i \(-0.616168\pi\)
−0.356905 + 0.934141i \(0.616168\pi\)
\(104\) 0 0
\(105\) −0.282527 −0.0275718
\(106\) 0 0
\(107\) −9.89532 −0.956617 −0.478308 0.878192i \(-0.658750\pi\)
−0.478308 + 0.878192i \(0.658750\pi\)
\(108\) 0 0
\(109\) 14.9095 1.42807 0.714034 0.700111i \(-0.246866\pi\)
0.714034 + 0.700111i \(0.246866\pi\)
\(110\) 0 0
\(111\) −27.6081 −2.62044
\(112\) 0 0
\(113\) 15.9835 1.50360 0.751798 0.659393i \(-0.229187\pi\)
0.751798 + 0.659393i \(0.229187\pi\)
\(114\) 0 0
\(115\) −2.09646 −0.195496
\(116\) 0 0
\(117\) −29.7574 −2.75107
\(118\) 0 0
\(119\) −1.58657 −0.145441
\(120\) 0 0
\(121\) −10.8658 −0.987803
\(122\) 0 0
\(123\) −13.0448 −1.17621
\(124\) 0 0
\(125\) 3.13714 0.280595
\(126\) 0 0
\(127\) 9.21799 0.817964 0.408982 0.912542i \(-0.365884\pi\)
0.408982 + 0.912542i \(0.365884\pi\)
\(128\) 0 0
\(129\) 20.0124 1.76199
\(130\) 0 0
\(131\) 21.7817 1.90307 0.951536 0.307538i \(-0.0995050\pi\)
0.951536 + 0.307538i \(0.0995050\pi\)
\(132\) 0 0
\(133\) 2.04134 0.177006
\(134\) 0 0
\(135\) 5.33454 0.459124
\(136\) 0 0
\(137\) 7.50480 0.641178 0.320589 0.947218i \(-0.396119\pi\)
0.320589 + 0.947218i \(0.396119\pi\)
\(138\) 0 0
\(139\) 7.54416 0.639887 0.319944 0.947437i \(-0.396336\pi\)
0.319944 + 0.947437i \(0.396336\pi\)
\(140\) 0 0
\(141\) 5.95447 0.501457
\(142\) 0 0
\(143\) −1.35209 −0.113067
\(144\) 0 0
\(145\) −1.73914 −0.144428
\(146\) 0 0
\(147\) 23.0421 1.90048
\(148\) 0 0
\(149\) −4.41372 −0.361586 −0.180793 0.983521i \(-0.557866\pi\)
−0.180793 + 0.983521i \(0.557866\pi\)
\(150\) 0 0
\(151\) 20.7013 1.68464 0.842322 0.538975i \(-0.181188\pi\)
0.842322 + 0.538975i \(0.181188\pi\)
\(152\) 0 0
\(153\) 47.7129 3.85736
\(154\) 0 0
\(155\) −1.31733 −0.105810
\(156\) 0 0
\(157\) −18.1328 −1.44715 −0.723576 0.690245i \(-0.757503\pi\)
−0.723576 + 0.690245i \(0.757503\pi\)
\(158\) 0 0
\(159\) 35.0951 2.78322
\(160\) 0 0
\(161\) −1.77339 −0.139763
\(162\) 0 0
\(163\) 0.195775 0.0153343 0.00766715 0.999971i \(-0.497559\pi\)
0.00766715 + 0.999971i \(0.497559\pi\)
\(164\) 0 0
\(165\) 0.386052 0.0300541
\(166\) 0 0
\(167\) 1.69068 0.130829 0.0654144 0.997858i \(-0.479163\pi\)
0.0654144 + 0.997858i \(0.479163\pi\)
\(168\) 0 0
\(169\) 0.625880 0.0481446
\(170\) 0 0
\(171\) −61.3889 −4.69453
\(172\) 0 0
\(173\) 2.39382 0.181999 0.0909995 0.995851i \(-0.470994\pi\)
0.0909995 + 0.995851i \(0.470994\pi\)
\(174\) 0 0
\(175\) 1.31339 0.0992832
\(176\) 0 0
\(177\) −13.2938 −0.999224
\(178\) 0 0
\(179\) 22.2730 1.66476 0.832380 0.554205i \(-0.186978\pi\)
0.832380 + 0.554205i \(0.186978\pi\)
\(180\) 0 0
\(181\) 18.2384 1.35565 0.677827 0.735222i \(-0.262922\pi\)
0.677827 + 0.735222i \(0.262922\pi\)
\(182\) 0 0
\(183\) −6.27151 −0.463603
\(184\) 0 0
\(185\) −2.63056 −0.193403
\(186\) 0 0
\(187\) 2.16793 0.158535
\(188\) 0 0
\(189\) 4.51248 0.328235
\(190\) 0 0
\(191\) 11.0429 0.799033 0.399517 0.916726i \(-0.369178\pi\)
0.399517 + 0.916726i \(0.369178\pi\)
\(192\) 0 0
\(193\) −1.08590 −0.0781646 −0.0390823 0.999236i \(-0.512443\pi\)
−0.0390823 + 0.999236i \(0.512443\pi\)
\(194\) 0 0
\(195\) −3.89050 −0.278605
\(196\) 0 0
\(197\) 23.2852 1.65901 0.829503 0.558503i \(-0.188624\pi\)
0.829503 + 0.558503i \(0.188624\pi\)
\(198\) 0 0
\(199\) 17.8332 1.26416 0.632080 0.774903i \(-0.282201\pi\)
0.632080 + 0.774903i \(0.282201\pi\)
\(200\) 0 0
\(201\) 4.26177 0.300602
\(202\) 0 0
\(203\) −1.47114 −0.103254
\(204\) 0 0
\(205\) −1.24294 −0.0868109
\(206\) 0 0
\(207\) 53.3311 3.70677
\(208\) 0 0
\(209\) −2.78934 −0.192942
\(210\) 0 0
\(211\) 26.6400 1.83397 0.916986 0.398920i \(-0.130615\pi\)
0.916986 + 0.398920i \(0.130615\pi\)
\(212\) 0 0
\(213\) 27.1290 1.85885
\(214\) 0 0
\(215\) 1.90682 0.130044
\(216\) 0 0
\(217\) −1.11433 −0.0756454
\(218\) 0 0
\(219\) 53.2652 3.59933
\(220\) 0 0
\(221\) −21.8477 −1.46963
\(222\) 0 0
\(223\) −21.2507 −1.42305 −0.711525 0.702661i \(-0.751995\pi\)
−0.711525 + 0.702661i \(0.751995\pi\)
\(224\) 0 0
\(225\) −39.4976 −2.63317
\(226\) 0 0
\(227\) −9.08131 −0.602747 −0.301374 0.953506i \(-0.597445\pi\)
−0.301374 + 0.953506i \(0.597445\pi\)
\(228\) 0 0
\(229\) −17.7753 −1.17463 −0.587313 0.809360i \(-0.699814\pi\)
−0.587313 + 0.809360i \(0.699814\pi\)
\(230\) 0 0
\(231\) 0.326561 0.0214862
\(232\) 0 0
\(233\) −8.94497 −0.586004 −0.293002 0.956112i \(-0.594654\pi\)
−0.293002 + 0.956112i \(0.594654\pi\)
\(234\) 0 0
\(235\) 0.567355 0.0370102
\(236\) 0 0
\(237\) 26.4050 1.71519
\(238\) 0 0
\(239\) −8.57703 −0.554802 −0.277401 0.960754i \(-0.589473\pi\)
−0.277401 + 0.960754i \(0.589473\pi\)
\(240\) 0 0
\(241\) −0.371399 −0.0239239 −0.0119620 0.999928i \(-0.503808\pi\)
−0.0119620 + 0.999928i \(0.503808\pi\)
\(242\) 0 0
\(243\) −55.2696 −3.54555
\(244\) 0 0
\(245\) 2.19551 0.140266
\(246\) 0 0
\(247\) 28.1100 1.78859
\(248\) 0 0
\(249\) −39.0217 −2.47290
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 2.42321 0.152346
\(254\) 0 0
\(255\) 6.23802 0.390640
\(256\) 0 0
\(257\) 19.0422 1.18782 0.593910 0.804531i \(-0.297583\pi\)
0.593910 + 0.804531i \(0.297583\pi\)
\(258\) 0 0
\(259\) −2.22519 −0.138267
\(260\) 0 0
\(261\) 44.2414 2.73848
\(262\) 0 0
\(263\) 16.3403 1.00759 0.503795 0.863824i \(-0.331937\pi\)
0.503795 + 0.863824i \(0.331937\pi\)
\(264\) 0 0
\(265\) 3.34394 0.205417
\(266\) 0 0
\(267\) 4.62538 0.283068
\(268\) 0 0
\(269\) 22.5392 1.37424 0.687121 0.726543i \(-0.258874\pi\)
0.687121 + 0.726543i \(0.258874\pi\)
\(270\) 0 0
\(271\) 8.55125 0.519451 0.259726 0.965682i \(-0.416368\pi\)
0.259726 + 0.965682i \(0.416368\pi\)
\(272\) 0 0
\(273\) −3.29097 −0.199179
\(274\) 0 0
\(275\) −1.79466 −0.108222
\(276\) 0 0
\(277\) −7.92242 −0.476012 −0.238006 0.971264i \(-0.576494\pi\)
−0.238006 + 0.971264i \(0.576494\pi\)
\(278\) 0 0
\(279\) 33.5111 2.00625
\(280\) 0 0
\(281\) −5.74555 −0.342751 −0.171375 0.985206i \(-0.554821\pi\)
−0.171375 + 0.985206i \(0.554821\pi\)
\(282\) 0 0
\(283\) −6.77595 −0.402789 −0.201394 0.979510i \(-0.564547\pi\)
−0.201394 + 0.979510i \(0.564547\pi\)
\(284\) 0 0
\(285\) −8.02604 −0.475422
\(286\) 0 0
\(287\) −1.05140 −0.0620624
\(288\) 0 0
\(289\) 18.0305 1.06062
\(290\) 0 0
\(291\) 31.7096 1.85885
\(292\) 0 0
\(293\) −14.2153 −0.830464 −0.415232 0.909716i \(-0.636300\pi\)
−0.415232 + 0.909716i \(0.636300\pi\)
\(294\) 0 0
\(295\) −1.26666 −0.0737481
\(296\) 0 0
\(297\) −6.16597 −0.357786
\(298\) 0 0
\(299\) −24.4203 −1.41226
\(300\) 0 0
\(301\) 1.61298 0.0929706
\(302\) 0 0
\(303\) 21.6591 1.24428
\(304\) 0 0
\(305\) −0.597563 −0.0342164
\(306\) 0 0
\(307\) 23.7876 1.35763 0.678816 0.734308i \(-0.262493\pi\)
0.678816 + 0.734308i \(0.262493\pi\)
\(308\) 0 0
\(309\) 24.0939 1.37065
\(310\) 0 0
\(311\) −9.00045 −0.510369 −0.255184 0.966892i \(-0.582136\pi\)
−0.255184 + 0.966892i \(0.582136\pi\)
\(312\) 0 0
\(313\) 22.8553 1.29186 0.645928 0.763398i \(-0.276471\pi\)
0.645928 + 0.763398i \(0.276471\pi\)
\(314\) 0 0
\(315\) 0.684804 0.0385843
\(316\) 0 0
\(317\) 24.6076 1.38210 0.691051 0.722806i \(-0.257148\pi\)
0.691051 + 0.722806i \(0.257148\pi\)
\(318\) 0 0
\(319\) 2.01020 0.112550
\(320\) 0 0
\(321\) 32.9106 1.83689
\(322\) 0 0
\(323\) −45.0714 −2.50784
\(324\) 0 0
\(325\) 18.0859 1.00323
\(326\) 0 0
\(327\) −49.5870 −2.74217
\(328\) 0 0
\(329\) 0.479925 0.0264591
\(330\) 0 0
\(331\) 17.7024 0.973013 0.486507 0.873677i \(-0.338271\pi\)
0.486507 + 0.873677i \(0.338271\pi\)
\(332\) 0 0
\(333\) 66.9179 3.66708
\(334\) 0 0
\(335\) 0.406071 0.0221860
\(336\) 0 0
\(337\) 32.2381 1.75612 0.878060 0.478551i \(-0.158838\pi\)
0.878060 + 0.478551i \(0.158838\pi\)
\(338\) 0 0
\(339\) −53.1589 −2.88720
\(340\) 0 0
\(341\) 1.52265 0.0824559
\(342\) 0 0
\(343\) 3.73362 0.201596
\(344\) 0 0
\(345\) 6.97256 0.375390
\(346\) 0 0
\(347\) 7.82960 0.420315 0.210157 0.977668i \(-0.432602\pi\)
0.210157 + 0.977668i \(0.432602\pi\)
\(348\) 0 0
\(349\) 3.97080 0.212552 0.106276 0.994337i \(-0.466107\pi\)
0.106276 + 0.994337i \(0.466107\pi\)
\(350\) 0 0
\(351\) 62.1385 3.31671
\(352\) 0 0
\(353\) 0.605256 0.0322145 0.0161073 0.999870i \(-0.494873\pi\)
0.0161073 + 0.999870i \(0.494873\pi\)
\(354\) 0 0
\(355\) 2.58491 0.137193
\(356\) 0 0
\(357\) 5.27674 0.279274
\(358\) 0 0
\(359\) −20.2211 −1.06723 −0.533614 0.845728i \(-0.679167\pi\)
−0.533614 + 0.845728i \(0.679167\pi\)
\(360\) 0 0
\(361\) 38.9904 2.05213
\(362\) 0 0
\(363\) 36.1384 1.89677
\(364\) 0 0
\(365\) 5.07523 0.265650
\(366\) 0 0
\(367\) −22.9860 −1.19986 −0.599931 0.800052i \(-0.704805\pi\)
−0.599931 + 0.800052i \(0.704805\pi\)
\(368\) 0 0
\(369\) 31.6188 1.64601
\(370\) 0 0
\(371\) 2.82864 0.146856
\(372\) 0 0
\(373\) −32.6800 −1.69211 −0.846054 0.533098i \(-0.821028\pi\)
−0.846054 + 0.533098i \(0.821028\pi\)
\(374\) 0 0
\(375\) −10.4337 −0.538796
\(376\) 0 0
\(377\) −20.2581 −1.04335
\(378\) 0 0
\(379\) −2.52675 −0.129790 −0.0648951 0.997892i \(-0.520671\pi\)
−0.0648951 + 0.997892i \(0.520671\pi\)
\(380\) 0 0
\(381\) −30.6579 −1.57065
\(382\) 0 0
\(383\) −26.2117 −1.33935 −0.669677 0.742652i \(-0.733568\pi\)
−0.669677 + 0.742652i \(0.733568\pi\)
\(384\) 0 0
\(385\) 0.0311155 0.00158579
\(386\) 0 0
\(387\) −48.5070 −2.46575
\(388\) 0 0
\(389\) −23.0102 −1.16667 −0.583333 0.812233i \(-0.698252\pi\)
−0.583333 + 0.812233i \(0.698252\pi\)
\(390\) 0 0
\(391\) 39.1554 1.98018
\(392\) 0 0
\(393\) −72.4430 −3.65427
\(394\) 0 0
\(395\) 2.51593 0.126590
\(396\) 0 0
\(397\) −1.25813 −0.0631439 −0.0315719 0.999501i \(-0.510051\pi\)
−0.0315719 + 0.999501i \(0.510051\pi\)
\(398\) 0 0
\(399\) −6.78922 −0.339886
\(400\) 0 0
\(401\) 4.13472 0.206478 0.103239 0.994657i \(-0.467079\pi\)
0.103239 + 0.994657i \(0.467079\pi\)
\(402\) 0 0
\(403\) −15.3447 −0.764374
\(404\) 0 0
\(405\) −10.0781 −0.500783
\(406\) 0 0
\(407\) 3.04056 0.150715
\(408\) 0 0
\(409\) −12.8022 −0.633030 −0.316515 0.948588i \(-0.602513\pi\)
−0.316515 + 0.948588i \(0.602513\pi\)
\(410\) 0 0
\(411\) −24.9600 −1.23119
\(412\) 0 0
\(413\) −1.07147 −0.0527236
\(414\) 0 0
\(415\) −3.71807 −0.182513
\(416\) 0 0
\(417\) −25.0909 −1.22871
\(418\) 0 0
\(419\) 17.3329 0.846767 0.423384 0.905950i \(-0.360842\pi\)
0.423384 + 0.905950i \(0.360842\pi\)
\(420\) 0 0
\(421\) −10.3663 −0.505222 −0.252611 0.967568i \(-0.581289\pi\)
−0.252611 + 0.967568i \(0.581289\pi\)
\(422\) 0 0
\(423\) −14.4328 −0.701745
\(424\) 0 0
\(425\) −28.9989 −1.40665
\(426\) 0 0
\(427\) −0.505478 −0.0244618
\(428\) 0 0
\(429\) 4.49687 0.217111
\(430\) 0 0
\(431\) 3.16300 0.152357 0.0761783 0.997094i \(-0.475728\pi\)
0.0761783 + 0.997094i \(0.475728\pi\)
\(432\) 0 0
\(433\) −18.9740 −0.911832 −0.455916 0.890023i \(-0.650688\pi\)
−0.455916 + 0.890023i \(0.650688\pi\)
\(434\) 0 0
\(435\) 5.78417 0.277330
\(436\) 0 0
\(437\) −50.3787 −2.40994
\(438\) 0 0
\(439\) 29.5408 1.40991 0.704953 0.709254i \(-0.250968\pi\)
0.704953 + 0.709254i \(0.250968\pi\)
\(440\) 0 0
\(441\) −55.8507 −2.65956
\(442\) 0 0
\(443\) −12.3408 −0.586330 −0.293165 0.956062i \(-0.594709\pi\)
−0.293165 + 0.956062i \(0.594709\pi\)
\(444\) 0 0
\(445\) 0.440717 0.0208920
\(446\) 0 0
\(447\) 14.6795 0.694315
\(448\) 0 0
\(449\) 36.3313 1.71458 0.857289 0.514835i \(-0.172147\pi\)
0.857289 + 0.514835i \(0.172147\pi\)
\(450\) 0 0
\(451\) 1.43667 0.0676500
\(452\) 0 0
\(453\) −68.8497 −3.23484
\(454\) 0 0
\(455\) −0.313571 −0.0147005
\(456\) 0 0
\(457\) 12.9686 0.606646 0.303323 0.952888i \(-0.401904\pi\)
0.303323 + 0.952888i \(0.401904\pi\)
\(458\) 0 0
\(459\) −99.6328 −4.65046
\(460\) 0 0
\(461\) −6.19845 −0.288690 −0.144345 0.989527i \(-0.546108\pi\)
−0.144345 + 0.989527i \(0.546108\pi\)
\(462\) 0 0
\(463\) 35.9488 1.67068 0.835340 0.549733i \(-0.185271\pi\)
0.835340 + 0.549733i \(0.185271\pi\)
\(464\) 0 0
\(465\) 4.38127 0.203176
\(466\) 0 0
\(467\) 12.3324 0.570677 0.285339 0.958427i \(-0.407894\pi\)
0.285339 + 0.958427i \(0.407894\pi\)
\(468\) 0 0
\(469\) 0.343495 0.0158611
\(470\) 0 0
\(471\) 60.3072 2.77881
\(472\) 0 0
\(473\) −2.20402 −0.101341
\(474\) 0 0
\(475\) 37.3110 1.71194
\(476\) 0 0
\(477\) −85.0654 −3.89488
\(478\) 0 0
\(479\) −20.4813 −0.935813 −0.467906 0.883778i \(-0.654992\pi\)
−0.467906 + 0.883778i \(0.654992\pi\)
\(480\) 0 0
\(481\) −30.6417 −1.39714
\(482\) 0 0
\(483\) 5.89808 0.268372
\(484\) 0 0
\(485\) 3.02136 0.137193
\(486\) 0 0
\(487\) 13.4600 0.609930 0.304965 0.952364i \(-0.401355\pi\)
0.304965 + 0.952364i \(0.401355\pi\)
\(488\) 0 0
\(489\) −0.651124 −0.0294448
\(490\) 0 0
\(491\) −2.63215 −0.118787 −0.0593937 0.998235i \(-0.518917\pi\)
−0.0593937 + 0.998235i \(0.518917\pi\)
\(492\) 0 0
\(493\) 32.4818 1.46291
\(494\) 0 0
\(495\) −0.935735 −0.0420581
\(496\) 0 0
\(497\) 2.18657 0.0980813
\(498\) 0 0
\(499\) 43.1846 1.93321 0.966604 0.256274i \(-0.0824950\pi\)
0.966604 + 0.256274i \(0.0824950\pi\)
\(500\) 0 0
\(501\) −5.62299 −0.251217
\(502\) 0 0
\(503\) 4.19813 0.187185 0.0935926 0.995611i \(-0.470165\pi\)
0.0935926 + 0.995611i \(0.470165\pi\)
\(504\) 0 0
\(505\) 2.06373 0.0918348
\(506\) 0 0
\(507\) −2.08160 −0.0924470
\(508\) 0 0
\(509\) 36.4290 1.61469 0.807344 0.590081i \(-0.200904\pi\)
0.807344 + 0.590081i \(0.200904\pi\)
\(510\) 0 0
\(511\) 4.29313 0.189917
\(512\) 0 0
\(513\) 128.191 5.65976
\(514\) 0 0
\(515\) 2.29572 0.101162
\(516\) 0 0
\(517\) −0.655783 −0.0288413
\(518\) 0 0
\(519\) −7.96155 −0.349473
\(520\) 0 0
\(521\) −4.13667 −0.181231 −0.0906155 0.995886i \(-0.528883\pi\)
−0.0906155 + 0.995886i \(0.528883\pi\)
\(522\) 0 0
\(523\) −9.67212 −0.422932 −0.211466 0.977385i \(-0.567824\pi\)
−0.211466 + 0.977385i \(0.567824\pi\)
\(524\) 0 0
\(525\) −4.36818 −0.190643
\(526\) 0 0
\(527\) 24.6036 1.07175
\(528\) 0 0
\(529\) 20.7661 0.902873
\(530\) 0 0
\(531\) 32.2223 1.39833
\(532\) 0 0
\(533\) −14.4782 −0.627122
\(534\) 0 0
\(535\) 3.13580 0.135572
\(536\) 0 0
\(537\) −74.0771 −3.19666
\(538\) 0 0
\(539\) −2.53770 −0.109306
\(540\) 0 0
\(541\) 15.6611 0.673324 0.336662 0.941626i \(-0.390702\pi\)
0.336662 + 0.941626i \(0.390702\pi\)
\(542\) 0 0
\(543\) −60.6588 −2.60312
\(544\) 0 0
\(545\) −4.72476 −0.202387
\(546\) 0 0
\(547\) 10.8854 0.465424 0.232712 0.972546i \(-0.425240\pi\)
0.232712 + 0.972546i \(0.425240\pi\)
\(548\) 0 0
\(549\) 15.2012 0.648772
\(550\) 0 0
\(551\) −41.7922 −1.78041
\(552\) 0 0
\(553\) 2.12822 0.0905012
\(554\) 0 0
\(555\) 8.74891 0.371371
\(556\) 0 0
\(557\) 2.76645 0.117218 0.0586091 0.998281i \(-0.481333\pi\)
0.0586091 + 0.998281i \(0.481333\pi\)
\(558\) 0 0
\(559\) 22.2114 0.939440
\(560\) 0 0
\(561\) −7.21027 −0.304418
\(562\) 0 0
\(563\) 6.67356 0.281257 0.140629 0.990062i \(-0.455088\pi\)
0.140629 + 0.990062i \(0.455088\pi\)
\(564\) 0 0
\(565\) −5.06511 −0.213091
\(566\) 0 0
\(567\) −8.52503 −0.358017
\(568\) 0 0
\(569\) 6.27226 0.262947 0.131473 0.991320i \(-0.458029\pi\)
0.131473 + 0.991320i \(0.458029\pi\)
\(570\) 0 0
\(571\) 33.7487 1.41234 0.706169 0.708043i \(-0.250422\pi\)
0.706169 + 0.708043i \(0.250422\pi\)
\(572\) 0 0
\(573\) −36.7272 −1.53430
\(574\) 0 0
\(575\) −32.4136 −1.35174
\(576\) 0 0
\(577\) 10.1858 0.424041 0.212021 0.977265i \(-0.431996\pi\)
0.212021 + 0.977265i \(0.431996\pi\)
\(578\) 0 0
\(579\) 3.61156 0.150091
\(580\) 0 0
\(581\) −3.14511 −0.130481
\(582\) 0 0
\(583\) −3.86513 −0.160077
\(584\) 0 0
\(585\) 9.43001 0.389883
\(586\) 0 0
\(587\) −2.99433 −0.123589 −0.0617946 0.998089i \(-0.519682\pi\)
−0.0617946 + 0.998089i \(0.519682\pi\)
\(588\) 0 0
\(589\) −31.6559 −1.30436
\(590\) 0 0
\(591\) −77.4438 −3.18561
\(592\) 0 0
\(593\) −0.634339 −0.0260492 −0.0130246 0.999915i \(-0.504146\pi\)
−0.0130246 + 0.999915i \(0.504146\pi\)
\(594\) 0 0
\(595\) 0.502780 0.0206120
\(596\) 0 0
\(597\) −59.3109 −2.42743
\(598\) 0 0
\(599\) 27.3229 1.11638 0.558192 0.829712i \(-0.311495\pi\)
0.558192 + 0.829712i \(0.311495\pi\)
\(600\) 0 0
\(601\) 37.4715 1.52849 0.764247 0.644924i \(-0.223111\pi\)
0.764247 + 0.644924i \(0.223111\pi\)
\(602\) 0 0
\(603\) −10.3299 −0.420666
\(604\) 0 0
\(605\) 3.44335 0.139992
\(606\) 0 0
\(607\) −11.8099 −0.479348 −0.239674 0.970853i \(-0.577041\pi\)
−0.239674 + 0.970853i \(0.577041\pi\)
\(608\) 0 0
\(609\) 4.89282 0.198267
\(610\) 0 0
\(611\) 6.60875 0.267362
\(612\) 0 0
\(613\) −41.7932 −1.68801 −0.844005 0.536335i \(-0.819808\pi\)
−0.844005 + 0.536335i \(0.819808\pi\)
\(614\) 0 0
\(615\) 4.13387 0.166694
\(616\) 0 0
\(617\) 0.825343 0.0332270 0.0166135 0.999862i \(-0.494712\pi\)
0.0166135 + 0.999862i \(0.494712\pi\)
\(618\) 0 0
\(619\) −32.1673 −1.29291 −0.646456 0.762952i \(-0.723749\pi\)
−0.646456 + 0.762952i \(0.723749\pi\)
\(620\) 0 0
\(621\) −111.365 −4.46891
\(622\) 0 0
\(623\) 0.372802 0.0149360
\(624\) 0 0
\(625\) 23.5037 0.940149
\(626\) 0 0
\(627\) 9.27697 0.370487
\(628\) 0 0
\(629\) 49.1308 1.95897
\(630\) 0 0
\(631\) 4.55806 0.181453 0.0907267 0.995876i \(-0.471081\pi\)
0.0907267 + 0.995876i \(0.471081\pi\)
\(632\) 0 0
\(633\) −88.6012 −3.52158
\(634\) 0 0
\(635\) −2.92115 −0.115922
\(636\) 0 0
\(637\) 25.5740 1.01328
\(638\) 0 0
\(639\) −65.7567 −2.60129
\(640\) 0 0
\(641\) −5.26084 −0.207791 −0.103895 0.994588i \(-0.533131\pi\)
−0.103895 + 0.994588i \(0.533131\pi\)
\(642\) 0 0
\(643\) −7.88916 −0.311118 −0.155559 0.987827i \(-0.549718\pi\)
−0.155559 + 0.987827i \(0.549718\pi\)
\(644\) 0 0
\(645\) −6.34185 −0.249710
\(646\) 0 0
\(647\) 13.2551 0.521113 0.260557 0.965459i \(-0.416094\pi\)
0.260557 + 0.965459i \(0.416094\pi\)
\(648\) 0 0
\(649\) 1.46409 0.0574704
\(650\) 0 0
\(651\) 3.70611 0.145254
\(652\) 0 0
\(653\) 1.98919 0.0778431 0.0389215 0.999242i \(-0.487608\pi\)
0.0389215 + 0.999242i \(0.487608\pi\)
\(654\) 0 0
\(655\) −6.90254 −0.269704
\(656\) 0 0
\(657\) −129.107 −5.03694
\(658\) 0 0
\(659\) 27.2300 1.06073 0.530366 0.847769i \(-0.322055\pi\)
0.530366 + 0.847769i \(0.322055\pi\)
\(660\) 0 0
\(661\) 0.551019 0.0214322 0.0107161 0.999943i \(-0.496589\pi\)
0.0107161 + 0.999943i \(0.496589\pi\)
\(662\) 0 0
\(663\) 72.6627 2.82198
\(664\) 0 0
\(665\) −0.646893 −0.0250854
\(666\) 0 0
\(667\) 36.3066 1.40580
\(668\) 0 0
\(669\) 70.6770 2.73253
\(670\) 0 0
\(671\) 0.690699 0.0266641
\(672\) 0 0
\(673\) 11.0122 0.424488 0.212244 0.977217i \(-0.431923\pi\)
0.212244 + 0.977217i \(0.431923\pi\)
\(674\) 0 0
\(675\) 82.4779 3.17457
\(676\) 0 0
\(677\) 43.3014 1.66421 0.832105 0.554618i \(-0.187136\pi\)
0.832105 + 0.554618i \(0.187136\pi\)
\(678\) 0 0
\(679\) 2.55577 0.0980814
\(680\) 0 0
\(681\) 30.2033 1.15739
\(682\) 0 0
\(683\) −45.7493 −1.75055 −0.875274 0.483627i \(-0.839319\pi\)
−0.875274 + 0.483627i \(0.839319\pi\)
\(684\) 0 0
\(685\) −2.37825 −0.0908682
\(686\) 0 0
\(687\) 59.1184 2.25551
\(688\) 0 0
\(689\) 38.9514 1.48393
\(690\) 0 0
\(691\) −12.2254 −0.465076 −0.232538 0.972587i \(-0.574703\pi\)
−0.232538 + 0.972587i \(0.574703\pi\)
\(692\) 0 0
\(693\) −0.791537 −0.0300680
\(694\) 0 0
\(695\) −2.39072 −0.0906852
\(696\) 0 0
\(697\) 23.2144 0.879307
\(698\) 0 0
\(699\) 29.7498 1.12524
\(700\) 0 0
\(701\) −28.4854 −1.07588 −0.537939 0.842984i \(-0.680797\pi\)
−0.537939 + 0.842984i \(0.680797\pi\)
\(702\) 0 0
\(703\) −63.2133 −2.38414
\(704\) 0 0
\(705\) −1.88695 −0.0710667
\(706\) 0 0
\(707\) 1.74571 0.0656541
\(708\) 0 0
\(709\) −11.5494 −0.433747 −0.216874 0.976200i \(-0.569586\pi\)
−0.216874 + 0.976200i \(0.569586\pi\)
\(710\) 0 0
\(711\) −64.0018 −2.40026
\(712\) 0 0
\(713\) 27.5008 1.02991
\(714\) 0 0
\(715\) 0.428472 0.0160240
\(716\) 0 0
\(717\) 28.5261 1.06533
\(718\) 0 0
\(719\) 43.1550 1.60941 0.804704 0.593676i \(-0.202324\pi\)
0.804704 + 0.593676i \(0.202324\pi\)
\(720\) 0 0
\(721\) 1.94195 0.0723219
\(722\) 0 0
\(723\) 1.23523 0.0459385
\(724\) 0 0
\(725\) −26.8891 −0.998635
\(726\) 0 0
\(727\) 18.8853 0.700416 0.350208 0.936672i \(-0.386111\pi\)
0.350208 + 0.936672i \(0.386111\pi\)
\(728\) 0 0
\(729\) 88.4125 3.27454
\(730\) 0 0
\(731\) −35.6136 −1.31722
\(732\) 0 0
\(733\) 34.9088 1.28939 0.644693 0.764442i \(-0.276985\pi\)
0.644693 + 0.764442i \(0.276985\pi\)
\(734\) 0 0
\(735\) −7.30197 −0.269337
\(736\) 0 0
\(737\) −0.469361 −0.0172891
\(738\) 0 0
\(739\) −5.98919 −0.220316 −0.110158 0.993914i \(-0.535136\pi\)
−0.110158 + 0.993914i \(0.535136\pi\)
\(740\) 0 0
\(741\) −93.4902 −3.43445
\(742\) 0 0
\(743\) 1.68049 0.0616512 0.0308256 0.999525i \(-0.490186\pi\)
0.0308256 + 0.999525i \(0.490186\pi\)
\(744\) 0 0
\(745\) 1.39869 0.0512442
\(746\) 0 0
\(747\) 94.5828 3.46060
\(748\) 0 0
\(749\) 2.65257 0.0969227
\(750\) 0 0
\(751\) −19.3262 −0.705225 −0.352612 0.935770i \(-0.614707\pi\)
−0.352612 + 0.935770i \(0.614707\pi\)
\(752\) 0 0
\(753\) −3.32587 −0.121202
\(754\) 0 0
\(755\) −6.56016 −0.238749
\(756\) 0 0
\(757\) −32.7816 −1.19147 −0.595734 0.803182i \(-0.703139\pi\)
−0.595734 + 0.803182i \(0.703139\pi\)
\(758\) 0 0
\(759\) −8.05930 −0.292534
\(760\) 0 0
\(761\) −5.10313 −0.184988 −0.0924942 0.995713i \(-0.529484\pi\)
−0.0924942 + 0.995713i \(0.529484\pi\)
\(762\) 0 0
\(763\) −3.99667 −0.144689
\(764\) 0 0
\(765\) −15.1201 −0.546667
\(766\) 0 0
\(767\) −14.7546 −0.532756
\(768\) 0 0
\(769\) −15.7298 −0.567232 −0.283616 0.958938i \(-0.591534\pi\)
−0.283616 + 0.958938i \(0.591534\pi\)
\(770\) 0 0
\(771\) −63.3320 −2.28084
\(772\) 0 0
\(773\) −15.2536 −0.548635 −0.274317 0.961639i \(-0.588452\pi\)
−0.274317 + 0.961639i \(0.588452\pi\)
\(774\) 0 0
\(775\) −20.3674 −0.731617
\(776\) 0 0
\(777\) 7.40070 0.265498
\(778\) 0 0
\(779\) −29.8684 −1.07015
\(780\) 0 0
\(781\) −2.98779 −0.106912
\(782\) 0 0
\(783\) −92.3839 −3.30153
\(784\) 0 0
\(785\) 5.74621 0.205091
\(786\) 0 0
\(787\) −47.6696 −1.69924 −0.849619 0.527397i \(-0.823168\pi\)
−0.849619 + 0.527397i \(0.823168\pi\)
\(788\) 0 0
\(789\) −54.3459 −1.93477
\(790\) 0 0
\(791\) −4.28457 −0.152342
\(792\) 0 0
\(793\) −6.96063 −0.247179
\(794\) 0 0
\(795\) −11.1215 −0.394440
\(796\) 0 0
\(797\) 20.5952 0.729518 0.364759 0.931102i \(-0.381151\pi\)
0.364759 + 0.931102i \(0.381151\pi\)
\(798\) 0 0
\(799\) −10.5965 −0.374876
\(800\) 0 0
\(801\) −11.2112 −0.396130
\(802\) 0 0
\(803\) −5.86625 −0.207015
\(804\) 0 0
\(805\) 0.561983 0.0198073
\(806\) 0 0
\(807\) −74.9626 −2.63881
\(808\) 0 0
\(809\) 4.55199 0.160040 0.0800198 0.996793i \(-0.474502\pi\)
0.0800198 + 0.996793i \(0.474502\pi\)
\(810\) 0 0
\(811\) 5.19296 0.182349 0.0911747 0.995835i \(-0.470938\pi\)
0.0911747 + 0.995835i \(0.470938\pi\)
\(812\) 0 0
\(813\) −28.4404 −0.997447
\(814\) 0 0
\(815\) −0.0620406 −0.00217319
\(816\) 0 0
\(817\) 45.8216 1.60310
\(818\) 0 0
\(819\) 7.97684 0.278733
\(820\) 0 0
\(821\) 27.6078 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(822\) 0 0
\(823\) 15.6633 0.545988 0.272994 0.962016i \(-0.411986\pi\)
0.272994 + 0.962016i \(0.411986\pi\)
\(824\) 0 0
\(825\) 5.96880 0.207807
\(826\) 0 0
\(827\) −37.3658 −1.29933 −0.649667 0.760219i \(-0.725092\pi\)
−0.649667 + 0.760219i \(0.725092\pi\)
\(828\) 0 0
\(829\) 35.4986 1.23292 0.616459 0.787387i \(-0.288566\pi\)
0.616459 + 0.787387i \(0.288566\pi\)
\(830\) 0 0
\(831\) 26.3490 0.914036
\(832\) 0 0
\(833\) −41.0053 −1.42075
\(834\) 0 0
\(835\) −0.535771 −0.0185411
\(836\) 0 0
\(837\) −69.9769 −2.41876
\(838\) 0 0
\(839\) −13.2255 −0.456594 −0.228297 0.973592i \(-0.573316\pi\)
−0.228297 + 0.973592i \(0.573316\pi\)
\(840\) 0 0
\(841\) 1.11858 0.0385718
\(842\) 0 0
\(843\) 19.1090 0.658148
\(844\) 0 0
\(845\) −0.198339 −0.00682308
\(846\) 0 0
\(847\) 2.91272 0.100082
\(848\) 0 0
\(849\) 22.5360 0.773432
\(850\) 0 0
\(851\) 54.9161 1.88250
\(852\) 0 0
\(853\) −2.27716 −0.0779686 −0.0389843 0.999240i \(-0.512412\pi\)
−0.0389843 + 0.999240i \(0.512412\pi\)
\(854\) 0 0
\(855\) 19.4540 0.665311
\(856\) 0 0
\(857\) −20.4240 −0.697670 −0.348835 0.937184i \(-0.613423\pi\)
−0.348835 + 0.937184i \(0.613423\pi\)
\(858\) 0 0
\(859\) 0.815502 0.0278246 0.0139123 0.999903i \(-0.495571\pi\)
0.0139123 + 0.999903i \(0.495571\pi\)
\(860\) 0 0
\(861\) 3.49684 0.119172
\(862\) 0 0
\(863\) 19.3454 0.658524 0.329262 0.944239i \(-0.393200\pi\)
0.329262 + 0.944239i \(0.393200\pi\)
\(864\) 0 0
\(865\) −0.758595 −0.0257930
\(866\) 0 0
\(867\) −59.9672 −2.03659
\(868\) 0 0
\(869\) −2.90806 −0.0986491
\(870\) 0 0
\(871\) 4.73006 0.160272
\(872\) 0 0
\(873\) −76.8594 −2.60130
\(874\) 0 0
\(875\) −0.840951 −0.0284293
\(876\) 0 0
\(877\) 55.3799 1.87005 0.935023 0.354588i \(-0.115379\pi\)
0.935023 + 0.354588i \(0.115379\pi\)
\(878\) 0 0
\(879\) 47.2781 1.59465
\(880\) 0 0
\(881\) 11.2131 0.377779 0.188889 0.981998i \(-0.439511\pi\)
0.188889 + 0.981998i \(0.439511\pi\)
\(882\) 0 0
\(883\) −21.0828 −0.709492 −0.354746 0.934963i \(-0.615433\pi\)
−0.354746 + 0.934963i \(0.615433\pi\)
\(884\) 0 0
\(885\) 4.21277 0.141611
\(886\) 0 0
\(887\) −14.1584 −0.475391 −0.237696 0.971340i \(-0.576392\pi\)
−0.237696 + 0.971340i \(0.576392\pi\)
\(888\) 0 0
\(889\) −2.47100 −0.0828746
\(890\) 0 0
\(891\) 11.6488 0.390250
\(892\) 0 0
\(893\) 13.6337 0.456236
\(894\) 0 0
\(895\) −7.05824 −0.235931
\(896\) 0 0
\(897\) 81.2188 2.71182
\(898\) 0 0
\(899\) 22.8136 0.760875
\(900\) 0 0
\(901\) −62.4546 −2.08066
\(902\) 0 0
\(903\) −5.36457 −0.178522
\(904\) 0 0
\(905\) −5.77971 −0.192124
\(906\) 0 0
\(907\) 15.9425 0.529363 0.264682 0.964336i \(-0.414733\pi\)
0.264682 + 0.964336i \(0.414733\pi\)
\(908\) 0 0
\(909\) −52.4985 −1.74127
\(910\) 0 0
\(911\) −3.68657 −0.122142 −0.0610708 0.998133i \(-0.519452\pi\)
−0.0610708 + 0.998133i \(0.519452\pi\)
\(912\) 0 0
\(913\) 4.29757 0.142229
\(914\) 0 0
\(915\) 1.98742 0.0657021
\(916\) 0 0
\(917\) −5.83885 −0.192816
\(918\) 0 0
\(919\) −31.4633 −1.03788 −0.518939 0.854811i \(-0.673673\pi\)
−0.518939 + 0.854811i \(0.673673\pi\)
\(920\) 0 0
\(921\) −79.1146 −2.60692
\(922\) 0 0
\(923\) 30.1100 0.991082
\(924\) 0 0
\(925\) −40.6714 −1.33727
\(926\) 0 0
\(927\) −58.4000 −1.91811
\(928\) 0 0
\(929\) 15.5846 0.511315 0.255657 0.966767i \(-0.417708\pi\)
0.255657 + 0.966767i \(0.417708\pi\)
\(930\) 0 0
\(931\) 52.7588 1.72910
\(932\) 0 0
\(933\) 29.9344 0.980007
\(934\) 0 0
\(935\) −0.687011 −0.0224677
\(936\) 0 0
\(937\) −59.0343 −1.92857 −0.964283 0.264874i \(-0.914670\pi\)
−0.964283 + 0.264874i \(0.914670\pi\)
\(938\) 0 0
\(939\) −76.0137 −2.48061
\(940\) 0 0
\(941\) −26.6618 −0.869149 −0.434575 0.900636i \(-0.643101\pi\)
−0.434575 + 0.900636i \(0.643101\pi\)
\(942\) 0 0
\(943\) 25.9479 0.844980
\(944\) 0 0
\(945\) −1.42999 −0.0465176
\(946\) 0 0
\(947\) −31.3405 −1.01843 −0.509215 0.860640i \(-0.670064\pi\)
−0.509215 + 0.860640i \(0.670064\pi\)
\(948\) 0 0
\(949\) 59.1180 1.91905
\(950\) 0 0
\(951\) −81.8419 −2.65391
\(952\) 0 0
\(953\) 47.3940 1.53524 0.767621 0.640904i \(-0.221440\pi\)
0.767621 + 0.640904i \(0.221440\pi\)
\(954\) 0 0
\(955\) −3.49945 −0.113239
\(956\) 0 0
\(957\) −6.68568 −0.216117
\(958\) 0 0
\(959\) −2.01176 −0.0649630
\(960\) 0 0
\(961\) −13.7197 −0.442570
\(962\) 0 0
\(963\) −79.7704 −2.57057
\(964\) 0 0
\(965\) 0.344118 0.0110775
\(966\) 0 0
\(967\) 45.8553 1.47461 0.737303 0.675562i \(-0.236099\pi\)
0.737303 + 0.675562i \(0.236099\pi\)
\(968\) 0 0
\(969\) 149.902 4.81554
\(970\) 0 0
\(971\) 8.54398 0.274189 0.137095 0.990558i \(-0.456224\pi\)
0.137095 + 0.990558i \(0.456224\pi\)
\(972\) 0 0
\(973\) −2.02231 −0.0648322
\(974\) 0 0
\(975\) −60.1515 −1.92639
\(976\) 0 0
\(977\) 50.5432 1.61702 0.808510 0.588482i \(-0.200274\pi\)
0.808510 + 0.588482i \(0.200274\pi\)
\(978\) 0 0
\(979\) −0.509406 −0.0162807
\(980\) 0 0
\(981\) 120.192 3.83742
\(982\) 0 0
\(983\) 40.1504 1.28060 0.640299 0.768125i \(-0.278810\pi\)
0.640299 + 0.768125i \(0.278810\pi\)
\(984\) 0 0
\(985\) −7.37902 −0.235115
\(986\) 0 0
\(987\) −1.59617 −0.0508067
\(988\) 0 0
\(989\) −39.8072 −1.26580
\(990\) 0 0
\(991\) −2.67733 −0.0850481 −0.0425241 0.999095i \(-0.513540\pi\)
−0.0425241 + 0.999095i \(0.513540\pi\)
\(992\) 0 0
\(993\) −58.8760 −1.86837
\(994\) 0 0
\(995\) −5.65128 −0.179157
\(996\) 0 0
\(997\) −38.1297 −1.20758 −0.603790 0.797143i \(-0.706343\pi\)
−0.603790 + 0.797143i \(0.706343\pi\)
\(998\) 0 0
\(999\) −139.736 −4.42106
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.1.1 23
4.3 odd 2 2008.2.a.d.1.23 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.d.1.23 23 4.3 odd 2
4016.2.a.m.1.1 23 1.1 even 1 trivial