Properties

Label 4016.2.a.m.1.13
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 1.13
Character \(\chi\) = 4016.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.201914 q^{3} -1.79819 q^{5} -3.34181 q^{7} -2.95923 q^{9} +O(q^{10})\) \(q+0.201914 q^{3} -1.79819 q^{5} -3.34181 q^{7} -2.95923 q^{9} +5.67355 q^{11} -3.71810 q^{13} -0.363080 q^{15} +3.29744 q^{17} +0.297273 q^{19} -0.674758 q^{21} -8.29149 q^{23} -1.76650 q^{25} -1.20325 q^{27} +7.61220 q^{29} -7.30347 q^{31} +1.14557 q^{33} +6.00922 q^{35} -5.21090 q^{37} -0.750736 q^{39} +7.48959 q^{41} +0.751052 q^{43} +5.32127 q^{45} -7.73856 q^{47} +4.16770 q^{49} +0.665799 q^{51} +8.79352 q^{53} -10.2021 q^{55} +0.0600236 q^{57} +11.6953 q^{59} +3.46801 q^{61} +9.88919 q^{63} +6.68586 q^{65} -14.0862 q^{67} -1.67417 q^{69} +1.05938 q^{71} -13.8342 q^{73} -0.356681 q^{75} -18.9599 q^{77} +0.746282 q^{79} +8.63474 q^{81} +2.81989 q^{83} -5.92943 q^{85} +1.53701 q^{87} +11.6951 q^{89} +12.4252 q^{91} -1.47467 q^{93} -0.534554 q^{95} +4.20944 q^{97} -16.7893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} + O(q^{10}) \) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} - 8q^{11} + 8q^{13} - 7q^{15} + 19q^{17} + 9q^{19} + 9q^{21} - 21q^{23} + 65q^{25} - 5q^{27} + 10q^{29} + 9q^{31} + 34q^{33} - 12q^{35} + 11q^{37} + 9q^{39} + 35q^{41} + 9q^{43} + 29q^{45} - 37q^{47} + 77q^{49} + 17q^{51} + 38q^{53} + 20q^{55} + 51q^{57} - 17q^{59} - 22q^{63} + 41q^{65} - 9q^{67} + 8q^{69} - 13q^{71} + 41q^{73} - 25q^{75} + 36q^{77} + 36q^{79} + 127q^{81} - 29q^{83} + 34q^{85} - 10q^{87} + 36q^{89} + 6q^{91} + 36q^{93} - 25q^{95} + 40q^{97} - 19q^{99} + O(q^{100}) \)

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
\(3\) 0.201914 0.116575 0.0582875 0.998300i \(-0.481436\pi\)
0.0582875 + 0.998300i \(0.481436\pi\)
\(4\) 0 0
\(5\) −1.79819 −0.804176 −0.402088 0.915601i \(-0.631715\pi\)
−0.402088 + 0.915601i \(0.631715\pi\)
\(6\) 0 0
\(7\) −3.34181 −1.26309 −0.631543 0.775341i \(-0.717578\pi\)
−0.631543 + 0.775341i \(0.717578\pi\)
\(8\) 0 0
\(9\) −2.95923 −0.986410
\(10\) 0 0
\(11\) 5.67355 1.71064 0.855319 0.518102i \(-0.173361\pi\)
0.855319 + 0.518102i \(0.173361\pi\)
\(12\) 0 0
\(13\) −3.71810 −1.03122 −0.515608 0.856825i \(-0.672434\pi\)
−0.515608 + 0.856825i \(0.672434\pi\)
\(14\) 0 0
\(15\) −0.363080 −0.0937469
\(16\) 0 0
\(17\) 3.29744 0.799746 0.399873 0.916570i \(-0.369054\pi\)
0.399873 + 0.916570i \(0.369054\pi\)
\(18\) 0 0
\(19\) 0.297273 0.0681991 0.0340996 0.999418i \(-0.489144\pi\)
0.0340996 + 0.999418i \(0.489144\pi\)
\(20\) 0 0
\(21\) −0.674758 −0.147244
\(22\) 0 0
\(23\) −8.29149 −1.72890 −0.864448 0.502723i \(-0.832332\pi\)
−0.864448 + 0.502723i \(0.832332\pi\)
\(24\) 0 0
\(25\) −1.76650 −0.353300
\(26\) 0 0
\(27\) −1.20325 −0.231566
\(28\) 0 0
\(29\) 7.61220 1.41355 0.706775 0.707438i \(-0.250149\pi\)
0.706775 + 0.707438i \(0.250149\pi\)
\(30\) 0 0
\(31\) −7.30347 −1.31174 −0.655871 0.754873i \(-0.727698\pi\)
−0.655871 + 0.754873i \(0.727698\pi\)
\(32\) 0 0
\(33\) 1.14557 0.199418
\(34\) 0 0
\(35\) 6.00922 1.01574
\(36\) 0 0
\(37\) −5.21090 −0.856666 −0.428333 0.903621i \(-0.640899\pi\)
−0.428333 + 0.903621i \(0.640899\pi\)
\(38\) 0 0
\(39\) −0.750736 −0.120214
\(40\) 0 0
\(41\) 7.48959 1.16968 0.584838 0.811150i \(-0.301158\pi\)
0.584838 + 0.811150i \(0.301158\pi\)
\(42\) 0 0
\(43\) 0.751052 0.114534 0.0572672 0.998359i \(-0.481761\pi\)
0.0572672 + 0.998359i \(0.481761\pi\)
\(44\) 0 0
\(45\) 5.32127 0.793248
\(46\) 0 0
\(47\) −7.73856 −1.12879 −0.564393 0.825506i \(-0.690890\pi\)
−0.564393 + 0.825506i \(0.690890\pi\)
\(48\) 0 0
\(49\) 4.16770 0.595385
\(50\) 0 0
\(51\) 0.665799 0.0932305
\(52\) 0 0
\(53\) 8.79352 1.20788 0.603941 0.797029i \(-0.293596\pi\)
0.603941 + 0.797029i \(0.293596\pi\)
\(54\) 0 0
\(55\) −10.2021 −1.37565
\(56\) 0 0
\(57\) 0.0600236 0.00795032
\(58\) 0 0
\(59\) 11.6953 1.52259 0.761297 0.648403i \(-0.224563\pi\)
0.761297 + 0.648403i \(0.224563\pi\)
\(60\) 0 0
\(61\) 3.46801 0.444033 0.222016 0.975043i \(-0.428736\pi\)
0.222016 + 0.975043i \(0.428736\pi\)
\(62\) 0 0
\(63\) 9.88919 1.24592
\(64\) 0 0
\(65\) 6.68586 0.829279
\(66\) 0 0
\(67\) −14.0862 −1.72090 −0.860452 0.509532i \(-0.829819\pi\)
−0.860452 + 0.509532i \(0.829819\pi\)
\(68\) 0 0
\(69\) −1.67417 −0.201546
\(70\) 0 0
\(71\) 1.05938 0.125726 0.0628629 0.998022i \(-0.479977\pi\)
0.0628629 + 0.998022i \(0.479977\pi\)
\(72\) 0 0
\(73\) −13.8342 −1.61917 −0.809587 0.587000i \(-0.800309\pi\)
−0.809587 + 0.587000i \(0.800309\pi\)
\(74\) 0 0
\(75\) −0.356681 −0.0411860
\(76\) 0 0
\(77\) −18.9599 −2.16068
\(78\) 0 0
\(79\) 0.746282 0.0839633 0.0419816 0.999118i \(-0.486633\pi\)
0.0419816 + 0.999118i \(0.486633\pi\)
\(80\) 0 0
\(81\) 8.63474 0.959415
\(82\) 0 0
\(83\) 2.81989 0.309523 0.154761 0.987952i \(-0.450539\pi\)
0.154761 + 0.987952i \(0.450539\pi\)
\(84\) 0 0
\(85\) −5.92943 −0.643137
\(86\) 0 0
\(87\) 1.53701 0.164785
\(88\) 0 0
\(89\) 11.6951 1.23968 0.619838 0.784730i \(-0.287198\pi\)
0.619838 + 0.784730i \(0.287198\pi\)
\(90\) 0 0
\(91\) 12.4252 1.30251
\(92\) 0 0
\(93\) −1.47467 −0.152916
\(94\) 0 0
\(95\) −0.534554 −0.0548441
\(96\) 0 0
\(97\) 4.20944 0.427404 0.213702 0.976899i \(-0.431448\pi\)
0.213702 + 0.976899i \(0.431448\pi\)
\(98\) 0 0
\(99\) −16.7893 −1.68739
\(100\) 0 0
\(101\) 5.63117 0.560323 0.280161 0.959953i \(-0.409612\pi\)
0.280161 + 0.959953i \(0.409612\pi\)
\(102\) 0 0
\(103\) 19.3367 1.90531 0.952653 0.304060i \(-0.0983425\pi\)
0.952653 + 0.304060i \(0.0983425\pi\)
\(104\) 0 0
\(105\) 1.21335 0.118410
\(106\) 0 0
\(107\) −0.533744 −0.0515990 −0.0257995 0.999667i \(-0.508213\pi\)
−0.0257995 + 0.999667i \(0.508213\pi\)
\(108\) 0 0
\(109\) −0.151594 −0.0145200 −0.00726001 0.999974i \(-0.502311\pi\)
−0.00726001 + 0.999974i \(0.502311\pi\)
\(110\) 0 0
\(111\) −1.05215 −0.0998659
\(112\) 0 0
\(113\) 18.6063 1.75033 0.875164 0.483825i \(-0.160753\pi\)
0.875164 + 0.483825i \(0.160753\pi\)
\(114\) 0 0
\(115\) 14.9097 1.39034
\(116\) 0 0
\(117\) 11.0027 1.01720
\(118\) 0 0
\(119\) −11.0194 −1.01015
\(120\) 0 0
\(121\) 21.1891 1.92628
\(122\) 0 0
\(123\) 1.51225 0.136355
\(124\) 0 0
\(125\) 12.1675 1.08829
\(126\) 0 0
\(127\) 10.9350 0.970328 0.485164 0.874423i \(-0.338760\pi\)
0.485164 + 0.874423i \(0.338760\pi\)
\(128\) 0 0
\(129\) 0.151648 0.0133518
\(130\) 0 0
\(131\) −8.51858 −0.744272 −0.372136 0.928178i \(-0.621374\pi\)
−0.372136 + 0.928178i \(0.621374\pi\)
\(132\) 0 0
\(133\) −0.993430 −0.0861413
\(134\) 0 0
\(135\) 2.16368 0.186220
\(136\) 0 0
\(137\) 2.57273 0.219804 0.109902 0.993942i \(-0.464946\pi\)
0.109902 + 0.993942i \(0.464946\pi\)
\(138\) 0 0
\(139\) 1.48109 0.125624 0.0628120 0.998025i \(-0.479993\pi\)
0.0628120 + 0.998025i \(0.479993\pi\)
\(140\) 0 0
\(141\) −1.56252 −0.131588
\(142\) 0 0
\(143\) −21.0948 −1.76404
\(144\) 0 0
\(145\) −13.6882 −1.13674
\(146\) 0 0
\(147\) 0.841516 0.0694071
\(148\) 0 0
\(149\) 18.8091 1.54090 0.770450 0.637500i \(-0.220031\pi\)
0.770450 + 0.637500i \(0.220031\pi\)
\(150\) 0 0
\(151\) 12.1457 0.988407 0.494203 0.869346i \(-0.335460\pi\)
0.494203 + 0.869346i \(0.335460\pi\)
\(152\) 0 0
\(153\) −9.75788 −0.788878
\(154\) 0 0
\(155\) 13.1330 1.05487
\(156\) 0 0
\(157\) 6.19537 0.494444 0.247222 0.968959i \(-0.420482\pi\)
0.247222 + 0.968959i \(0.420482\pi\)
\(158\) 0 0
\(159\) 1.77553 0.140809
\(160\) 0 0
\(161\) 27.7086 2.18374
\(162\) 0 0
\(163\) −7.57212 −0.593094 −0.296547 0.955018i \(-0.595835\pi\)
−0.296547 + 0.955018i \(0.595835\pi\)
\(164\) 0 0
\(165\) −2.05995 −0.160367
\(166\) 0 0
\(167\) −9.61169 −0.743775 −0.371888 0.928278i \(-0.621289\pi\)
−0.371888 + 0.928278i \(0.621289\pi\)
\(168\) 0 0
\(169\) 0.824263 0.0634049
\(170\) 0 0
\(171\) −0.879700 −0.0672723
\(172\) 0 0
\(173\) 6.48244 0.492851 0.246426 0.969162i \(-0.420744\pi\)
0.246426 + 0.969162i \(0.420744\pi\)
\(174\) 0 0
\(175\) 5.90331 0.446249
\(176\) 0 0
\(177\) 2.36144 0.177496
\(178\) 0 0
\(179\) 20.9179 1.56348 0.781740 0.623605i \(-0.214333\pi\)
0.781740 + 0.623605i \(0.214333\pi\)
\(180\) 0 0
\(181\) 14.2301 1.05771 0.528857 0.848711i \(-0.322621\pi\)
0.528857 + 0.848711i \(0.322621\pi\)
\(182\) 0 0
\(183\) 0.700239 0.0517632
\(184\) 0 0
\(185\) 9.37020 0.688911
\(186\) 0 0
\(187\) 18.7082 1.36808
\(188\) 0 0
\(189\) 4.02104 0.292488
\(190\) 0 0
\(191\) −9.18371 −0.664510 −0.332255 0.943190i \(-0.607809\pi\)
−0.332255 + 0.943190i \(0.607809\pi\)
\(192\) 0 0
\(193\) −2.62839 −0.189195 −0.0945977 0.995516i \(-0.530156\pi\)
−0.0945977 + 0.995516i \(0.530156\pi\)
\(194\) 0 0
\(195\) 1.34997 0.0966732
\(196\) 0 0
\(197\) −1.75695 −0.125178 −0.0625889 0.998039i \(-0.519936\pi\)
−0.0625889 + 0.998039i \(0.519936\pi\)
\(198\) 0 0
\(199\) −17.2691 −1.22417 −0.612087 0.790790i \(-0.709670\pi\)
−0.612087 + 0.790790i \(0.709670\pi\)
\(200\) 0 0
\(201\) −2.84420 −0.200614
\(202\) 0 0
\(203\) −25.4385 −1.78544
\(204\) 0 0
\(205\) −13.4677 −0.940627
\(206\) 0 0
\(207\) 24.5364 1.70540
\(208\) 0 0
\(209\) 1.68659 0.116664
\(210\) 0 0
\(211\) −0.0218577 −0.00150475 −0.000752373 1.00000i \(-0.500239\pi\)
−0.000752373 1.00000i \(0.500239\pi\)
\(212\) 0 0
\(213\) 0.213904 0.0146565
\(214\) 0 0
\(215\) −1.35054 −0.0921058
\(216\) 0 0
\(217\) 24.4068 1.65684
\(218\) 0 0
\(219\) −2.79332 −0.188755
\(220\) 0 0
\(221\) −12.2602 −0.824711
\(222\) 0 0
\(223\) 13.3927 0.896842 0.448421 0.893823i \(-0.351987\pi\)
0.448421 + 0.893823i \(0.351987\pi\)
\(224\) 0 0
\(225\) 5.22749 0.348499
\(226\) 0 0
\(227\) −17.8476 −1.18459 −0.592294 0.805722i \(-0.701778\pi\)
−0.592294 + 0.805722i \(0.701778\pi\)
\(228\) 0 0
\(229\) 5.13644 0.339426 0.169713 0.985494i \(-0.445716\pi\)
0.169713 + 0.985494i \(0.445716\pi\)
\(230\) 0 0
\(231\) −3.82827 −0.251882
\(232\) 0 0
\(233\) −25.5471 −1.67365 −0.836823 0.547474i \(-0.815590\pi\)
−0.836823 + 0.547474i \(0.815590\pi\)
\(234\) 0 0
\(235\) 13.9154 0.907743
\(236\) 0 0
\(237\) 0.150685 0.00978802
\(238\) 0 0
\(239\) 21.8897 1.41593 0.707964 0.706249i \(-0.249614\pi\)
0.707964 + 0.706249i \(0.249614\pi\)
\(240\) 0 0
\(241\) 17.8301 1.14854 0.574269 0.818667i \(-0.305286\pi\)
0.574269 + 0.818667i \(0.305286\pi\)
\(242\) 0 0
\(243\) 5.35323 0.343410
\(244\) 0 0
\(245\) −7.49433 −0.478795
\(246\) 0 0
\(247\) −1.10529 −0.0703280
\(248\) 0 0
\(249\) 0.569374 0.0360826
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) −47.0422 −2.95751
\(254\) 0 0
\(255\) −1.19723 −0.0749737
\(256\) 0 0
\(257\) −23.4810 −1.46471 −0.732353 0.680925i \(-0.761578\pi\)
−0.732353 + 0.680925i \(0.761578\pi\)
\(258\) 0 0
\(259\) 17.4138 1.08204
\(260\) 0 0
\(261\) −22.5263 −1.39434
\(262\) 0 0
\(263\) 26.9639 1.66267 0.831333 0.555775i \(-0.187578\pi\)
0.831333 + 0.555775i \(0.187578\pi\)
\(264\) 0 0
\(265\) −15.8124 −0.971351
\(266\) 0 0
\(267\) 2.36140 0.144515
\(268\) 0 0
\(269\) 12.5841 0.767264 0.383632 0.923486i \(-0.374673\pi\)
0.383632 + 0.923486i \(0.374673\pi\)
\(270\) 0 0
\(271\) −23.7129 −1.44046 −0.720228 0.693737i \(-0.755963\pi\)
−0.720228 + 0.693737i \(0.755963\pi\)
\(272\) 0 0
\(273\) 2.50882 0.151841
\(274\) 0 0
\(275\) −10.0223 −0.604369
\(276\) 0 0
\(277\) −23.1666 −1.39194 −0.695972 0.718069i \(-0.745026\pi\)
−0.695972 + 0.718069i \(0.745026\pi\)
\(278\) 0 0
\(279\) 21.6126 1.29392
\(280\) 0 0
\(281\) −11.1605 −0.665780 −0.332890 0.942966i \(-0.608024\pi\)
−0.332890 + 0.942966i \(0.608024\pi\)
\(282\) 0 0
\(283\) 13.6610 0.812064 0.406032 0.913859i \(-0.366912\pi\)
0.406032 + 0.913859i \(0.366912\pi\)
\(284\) 0 0
\(285\) −0.107934 −0.00639346
\(286\) 0 0
\(287\) −25.0288 −1.47740
\(288\) 0 0
\(289\) −6.12690 −0.360406
\(290\) 0 0
\(291\) 0.849944 0.0498246
\(292\) 0 0
\(293\) −6.77614 −0.395866 −0.197933 0.980216i \(-0.563423\pi\)
−0.197933 + 0.980216i \(0.563423\pi\)
\(294\) 0 0
\(295\) −21.0303 −1.22443
\(296\) 0 0
\(297\) −6.82670 −0.396125
\(298\) 0 0
\(299\) 30.8286 1.78286
\(300\) 0 0
\(301\) −2.50987 −0.144667
\(302\) 0 0
\(303\) 1.13701 0.0653196
\(304\) 0 0
\(305\) −6.23615 −0.357081
\(306\) 0 0
\(307\) −27.0975 −1.54653 −0.773267 0.634081i \(-0.781379\pi\)
−0.773267 + 0.634081i \(0.781379\pi\)
\(308\) 0 0
\(309\) 3.90436 0.222111
\(310\) 0 0
\(311\) 1.85540 0.105210 0.0526051 0.998615i \(-0.483248\pi\)
0.0526051 + 0.998615i \(0.483248\pi\)
\(312\) 0 0
\(313\) 0.494028 0.0279241 0.0139621 0.999903i \(-0.495556\pi\)
0.0139621 + 0.999903i \(0.495556\pi\)
\(314\) 0 0
\(315\) −17.7827 −1.00194
\(316\) 0 0
\(317\) 22.0357 1.23765 0.618824 0.785530i \(-0.287610\pi\)
0.618824 + 0.785530i \(0.287610\pi\)
\(318\) 0 0
\(319\) 43.1882 2.41807
\(320\) 0 0
\(321\) −0.107770 −0.00601515
\(322\) 0 0
\(323\) 0.980240 0.0545420
\(324\) 0 0
\(325\) 6.56803 0.364329
\(326\) 0 0
\(327\) −0.0306088 −0.00169267
\(328\) 0 0
\(329\) 25.8608 1.42575
\(330\) 0 0
\(331\) −34.2083 −1.88026 −0.940129 0.340819i \(-0.889296\pi\)
−0.940129 + 0.340819i \(0.889296\pi\)
\(332\) 0 0
\(333\) 15.4202 0.845024
\(334\) 0 0
\(335\) 25.3297 1.38391
\(336\) 0 0
\(337\) 20.0437 1.09185 0.545925 0.837834i \(-0.316178\pi\)
0.545925 + 0.837834i \(0.316178\pi\)
\(338\) 0 0
\(339\) 3.75686 0.204045
\(340\) 0 0
\(341\) −41.4365 −2.24391
\(342\) 0 0
\(343\) 9.46502 0.511063
\(344\) 0 0
\(345\) 3.01048 0.162079
\(346\) 0 0
\(347\) 12.0886 0.648948 0.324474 0.945895i \(-0.394813\pi\)
0.324474 + 0.945895i \(0.394813\pi\)
\(348\) 0 0
\(349\) −9.53974 −0.510650 −0.255325 0.966855i \(-0.582183\pi\)
−0.255325 + 0.966855i \(0.582183\pi\)
\(350\) 0 0
\(351\) 4.47381 0.238794
\(352\) 0 0
\(353\) 32.6432 1.73742 0.868712 0.495317i \(-0.164948\pi\)
0.868712 + 0.495317i \(0.164948\pi\)
\(354\) 0 0
\(355\) −1.90498 −0.101106
\(356\) 0 0
\(357\) −2.22497 −0.117758
\(358\) 0 0
\(359\) −15.4908 −0.817574 −0.408787 0.912630i \(-0.634048\pi\)
−0.408787 + 0.912630i \(0.634048\pi\)
\(360\) 0 0
\(361\) −18.9116 −0.995349
\(362\) 0 0
\(363\) 4.27838 0.224557
\(364\) 0 0
\(365\) 24.8766 1.30210
\(366\) 0 0
\(367\) 5.91353 0.308684 0.154342 0.988018i \(-0.450674\pi\)
0.154342 + 0.988018i \(0.450674\pi\)
\(368\) 0 0
\(369\) −22.1634 −1.15378
\(370\) 0 0
\(371\) −29.3863 −1.52566
\(372\) 0 0
\(373\) −18.5340 −0.959652 −0.479826 0.877364i \(-0.659300\pi\)
−0.479826 + 0.877364i \(0.659300\pi\)
\(374\) 0 0
\(375\) 2.45678 0.126868
\(376\) 0 0
\(377\) −28.3029 −1.45768
\(378\) 0 0
\(379\) −12.6334 −0.648933 −0.324467 0.945897i \(-0.605185\pi\)
−0.324467 + 0.945897i \(0.605185\pi\)
\(380\) 0 0
\(381\) 2.20794 0.113116
\(382\) 0 0
\(383\) 19.7360 1.00846 0.504231 0.863569i \(-0.331776\pi\)
0.504231 + 0.863569i \(0.331776\pi\)
\(384\) 0 0
\(385\) 34.0936 1.73757
\(386\) 0 0
\(387\) −2.22254 −0.112978
\(388\) 0 0
\(389\) 29.9438 1.51821 0.759106 0.650967i \(-0.225637\pi\)
0.759106 + 0.650967i \(0.225637\pi\)
\(390\) 0 0
\(391\) −27.3407 −1.38268
\(392\) 0 0
\(393\) −1.72002 −0.0867635
\(394\) 0 0
\(395\) −1.34196 −0.0675213
\(396\) 0 0
\(397\) 9.08019 0.455721 0.227861 0.973694i \(-0.426827\pi\)
0.227861 + 0.973694i \(0.426827\pi\)
\(398\) 0 0
\(399\) −0.200587 −0.0100419
\(400\) 0 0
\(401\) −5.18938 −0.259145 −0.129573 0.991570i \(-0.541361\pi\)
−0.129573 + 0.991570i \(0.541361\pi\)
\(402\) 0 0
\(403\) 27.1550 1.35269
\(404\) 0 0
\(405\) −15.5269 −0.771539
\(406\) 0 0
\(407\) −29.5643 −1.46545
\(408\) 0 0
\(409\) −16.7709 −0.829269 −0.414634 0.909988i \(-0.636091\pi\)
−0.414634 + 0.909988i \(0.636091\pi\)
\(410\) 0 0
\(411\) 0.519471 0.0256236
\(412\) 0 0
\(413\) −39.0834 −1.92317
\(414\) 0 0
\(415\) −5.07070 −0.248911
\(416\) 0 0
\(417\) 0.299052 0.0146446
\(418\) 0 0
\(419\) −35.9968 −1.75856 −0.879279 0.476307i \(-0.841975\pi\)
−0.879279 + 0.476307i \(0.841975\pi\)
\(420\) 0 0
\(421\) 12.6445 0.616256 0.308128 0.951345i \(-0.400298\pi\)
0.308128 + 0.951345i \(0.400298\pi\)
\(422\) 0 0
\(423\) 22.9002 1.11345
\(424\) 0 0
\(425\) −5.82493 −0.282551
\(426\) 0 0
\(427\) −11.5894 −0.560852
\(428\) 0 0
\(429\) −4.25933 −0.205643
\(430\) 0 0
\(431\) 8.66730 0.417489 0.208744 0.977970i \(-0.433062\pi\)
0.208744 + 0.977970i \(0.433062\pi\)
\(432\) 0 0
\(433\) 12.3222 0.592168 0.296084 0.955162i \(-0.404319\pi\)
0.296084 + 0.955162i \(0.404319\pi\)
\(434\) 0 0
\(435\) −2.76384 −0.132516
\(436\) 0 0
\(437\) −2.46484 −0.117909
\(438\) 0 0
\(439\) −6.56535 −0.313347 −0.156674 0.987650i \(-0.550077\pi\)
−0.156674 + 0.987650i \(0.550077\pi\)
\(440\) 0 0
\(441\) −12.3332 −0.587294
\(442\) 0 0
\(443\) −7.06939 −0.335877 −0.167938 0.985797i \(-0.553711\pi\)
−0.167938 + 0.985797i \(0.553711\pi\)
\(444\) 0 0
\(445\) −21.0300 −0.996918
\(446\) 0 0
\(447\) 3.79782 0.179631
\(448\) 0 0
\(449\) 17.0058 0.802554 0.401277 0.915957i \(-0.368566\pi\)
0.401277 + 0.915957i \(0.368566\pi\)
\(450\) 0 0
\(451\) 42.4925 2.00089
\(452\) 0 0
\(453\) 2.45239 0.115224
\(454\) 0 0
\(455\) −22.3429 −1.04745
\(456\) 0 0
\(457\) 30.9648 1.44847 0.724237 0.689551i \(-0.242192\pi\)
0.724237 + 0.689551i \(0.242192\pi\)
\(458\) 0 0
\(459\) −3.96765 −0.185194
\(460\) 0 0
\(461\) −15.9416 −0.742473 −0.371237 0.928538i \(-0.621066\pi\)
−0.371237 + 0.928538i \(0.621066\pi\)
\(462\) 0 0
\(463\) 18.7896 0.873227 0.436613 0.899649i \(-0.356178\pi\)
0.436613 + 0.899649i \(0.356178\pi\)
\(464\) 0 0
\(465\) 2.65174 0.122972
\(466\) 0 0
\(467\) 33.5619 1.55306 0.776531 0.630079i \(-0.216978\pi\)
0.776531 + 0.630079i \(0.216978\pi\)
\(468\) 0 0
\(469\) 47.0734 2.17365
\(470\) 0 0
\(471\) 1.25093 0.0576399
\(472\) 0 0
\(473\) 4.26113 0.195927
\(474\) 0 0
\(475\) −0.525133 −0.0240948
\(476\) 0 0
\(477\) −26.0221 −1.19147
\(478\) 0 0
\(479\) −2.55330 −0.116663 −0.0583316 0.998297i \(-0.518578\pi\)
−0.0583316 + 0.998297i \(0.518578\pi\)
\(480\) 0 0
\(481\) 19.3746 0.883407
\(482\) 0 0
\(483\) 5.59475 0.254570
\(484\) 0 0
\(485\) −7.56938 −0.343708
\(486\) 0 0
\(487\) −3.10886 −0.140876 −0.0704380 0.997516i \(-0.522440\pi\)
−0.0704380 + 0.997516i \(0.522440\pi\)
\(488\) 0 0
\(489\) −1.52892 −0.0691399
\(490\) 0 0
\(491\) 17.6594 0.796956 0.398478 0.917178i \(-0.369539\pi\)
0.398478 + 0.917178i \(0.369539\pi\)
\(492\) 0 0
\(493\) 25.1008 1.13048
\(494\) 0 0
\(495\) 30.1905 1.35696
\(496\) 0 0
\(497\) −3.54026 −0.158802
\(498\) 0 0
\(499\) −19.2196 −0.860388 −0.430194 0.902736i \(-0.641555\pi\)
−0.430194 + 0.902736i \(0.641555\pi\)
\(500\) 0 0
\(501\) −1.94073 −0.0867056
\(502\) 0 0
\(503\) 0.483311 0.0215498 0.0107749 0.999942i \(-0.496570\pi\)
0.0107749 + 0.999942i \(0.496570\pi\)
\(504\) 0 0
\(505\) −10.1259 −0.450598
\(506\) 0 0
\(507\) 0.166430 0.00739143
\(508\) 0 0
\(509\) −36.2312 −1.60592 −0.802961 0.596032i \(-0.796743\pi\)
−0.802961 + 0.596032i \(0.796743\pi\)
\(510\) 0 0
\(511\) 46.2314 2.04515
\(512\) 0 0
\(513\) −0.357694 −0.0157926
\(514\) 0 0
\(515\) −34.7712 −1.53220
\(516\) 0 0
\(517\) −43.9051 −1.93094
\(518\) 0 0
\(519\) 1.30890 0.0574541
\(520\) 0 0
\(521\) −11.2503 −0.492882 −0.246441 0.969158i \(-0.579261\pi\)
−0.246441 + 0.969158i \(0.579261\pi\)
\(522\) 0 0
\(523\) −2.69662 −0.117915 −0.0589574 0.998261i \(-0.518778\pi\)
−0.0589574 + 0.998261i \(0.518778\pi\)
\(524\) 0 0
\(525\) 1.19196 0.0520214
\(526\) 0 0
\(527\) −24.0827 −1.04906
\(528\) 0 0
\(529\) 45.7488 1.98908
\(530\) 0 0
\(531\) −34.6090 −1.50190
\(532\) 0 0
\(533\) −27.8470 −1.20619
\(534\) 0 0
\(535\) 0.959775 0.0414947
\(536\) 0 0
\(537\) 4.22362 0.182263
\(538\) 0 0
\(539\) 23.6456 1.01849
\(540\) 0 0
\(541\) 4.54198 0.195275 0.0976375 0.995222i \(-0.468871\pi\)
0.0976375 + 0.995222i \(0.468871\pi\)
\(542\) 0 0
\(543\) 2.87325 0.123303
\(544\) 0 0
\(545\) 0.272594 0.0116767
\(546\) 0 0
\(547\) −23.4436 −1.00238 −0.501188 0.865339i \(-0.667103\pi\)
−0.501188 + 0.865339i \(0.667103\pi\)
\(548\) 0 0
\(549\) −10.2626 −0.437999
\(550\) 0 0
\(551\) 2.26290 0.0964029
\(552\) 0 0
\(553\) −2.49393 −0.106053
\(554\) 0 0
\(555\) 1.89197 0.0803098
\(556\) 0 0
\(557\) −13.4645 −0.570511 −0.285255 0.958452i \(-0.592078\pi\)
−0.285255 + 0.958452i \(0.592078\pi\)
\(558\) 0 0
\(559\) −2.79249 −0.118110
\(560\) 0 0
\(561\) 3.77744 0.159484
\(562\) 0 0
\(563\) 3.30829 0.139428 0.0697140 0.997567i \(-0.477791\pi\)
0.0697140 + 0.997567i \(0.477791\pi\)
\(564\) 0 0
\(565\) −33.4576 −1.40757
\(566\) 0 0
\(567\) −28.8557 −1.21182
\(568\) 0 0
\(569\) −22.3573 −0.937268 −0.468634 0.883392i \(-0.655254\pi\)
−0.468634 + 0.883392i \(0.655254\pi\)
\(570\) 0 0
\(571\) 25.1726 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(572\) 0 0
\(573\) −1.85432 −0.0774653
\(574\) 0 0
\(575\) 14.6469 0.610819
\(576\) 0 0
\(577\) −3.46194 −0.144122 −0.0720612 0.997400i \(-0.522958\pi\)
−0.0720612 + 0.997400i \(0.522958\pi\)
\(578\) 0 0
\(579\) −0.530708 −0.0220555
\(580\) 0 0
\(581\) −9.42352 −0.390954
\(582\) 0 0
\(583\) 49.8904 2.06625
\(584\) 0 0
\(585\) −19.7850 −0.818009
\(586\) 0 0
\(587\) 18.6098 0.768107 0.384053 0.923311i \(-0.374528\pi\)
0.384053 + 0.923311i \(0.374528\pi\)
\(588\) 0 0
\(589\) −2.17112 −0.0894596
\(590\) 0 0
\(591\) −0.354754 −0.0145926
\(592\) 0 0
\(593\) −4.00728 −0.164559 −0.0822797 0.996609i \(-0.526220\pi\)
−0.0822797 + 0.996609i \(0.526220\pi\)
\(594\) 0 0
\(595\) 19.8150 0.812337
\(596\) 0 0
\(597\) −3.48687 −0.142708
\(598\) 0 0
\(599\) 46.1066 1.88386 0.941932 0.335805i \(-0.109008\pi\)
0.941932 + 0.335805i \(0.109008\pi\)
\(600\) 0 0
\(601\) −15.4501 −0.630222 −0.315111 0.949055i \(-0.602042\pi\)
−0.315111 + 0.949055i \(0.602042\pi\)
\(602\) 0 0
\(603\) 41.6843 1.69752
\(604\) 0 0
\(605\) −38.1021 −1.54907
\(606\) 0 0
\(607\) −36.9285 −1.49888 −0.749441 0.662071i \(-0.769678\pi\)
−0.749441 + 0.662071i \(0.769678\pi\)
\(608\) 0 0
\(609\) −5.13640 −0.208137
\(610\) 0 0
\(611\) 28.7727 1.16402
\(612\) 0 0
\(613\) 30.6685 1.23869 0.619345 0.785119i \(-0.287398\pi\)
0.619345 + 0.785119i \(0.287398\pi\)
\(614\) 0 0
\(615\) −2.71932 −0.109654
\(616\) 0 0
\(617\) 30.4887 1.22743 0.613714 0.789528i \(-0.289675\pi\)
0.613714 + 0.789528i \(0.289675\pi\)
\(618\) 0 0
\(619\) 11.4511 0.460258 0.230129 0.973160i \(-0.426085\pi\)
0.230129 + 0.973160i \(0.426085\pi\)
\(620\) 0 0
\(621\) 9.97675 0.400353
\(622\) 0 0
\(623\) −39.0827 −1.56582
\(624\) 0 0
\(625\) −13.0470 −0.521879
\(626\) 0 0
\(627\) 0.340546 0.0136001
\(628\) 0 0
\(629\) −17.1826 −0.685116
\(630\) 0 0
\(631\) 15.2408 0.606727 0.303363 0.952875i \(-0.401890\pi\)
0.303363 + 0.952875i \(0.401890\pi\)
\(632\) 0 0
\(633\) −0.00441337 −0.000175416 0
\(634\) 0 0
\(635\) −19.6633 −0.780315
\(636\) 0 0
\(637\) −15.4959 −0.613970
\(638\) 0 0
\(639\) −3.13496 −0.124017
\(640\) 0 0
\(641\) 44.4574 1.75596 0.877981 0.478695i \(-0.158890\pi\)
0.877981 + 0.478695i \(0.158890\pi\)
\(642\) 0 0
\(643\) −23.7397 −0.936204 −0.468102 0.883675i \(-0.655062\pi\)
−0.468102 + 0.883675i \(0.655062\pi\)
\(644\) 0 0
\(645\) −0.272692 −0.0107372
\(646\) 0 0
\(647\) −4.14223 −0.162848 −0.0814239 0.996680i \(-0.525947\pi\)
−0.0814239 + 0.996680i \(0.525947\pi\)
\(648\) 0 0
\(649\) 66.3536 2.60461
\(650\) 0 0
\(651\) 4.92807 0.193146
\(652\) 0 0
\(653\) 21.5920 0.844959 0.422479 0.906373i \(-0.361160\pi\)
0.422479 + 0.906373i \(0.361160\pi\)
\(654\) 0 0
\(655\) 15.3180 0.598526
\(656\) 0 0
\(657\) 40.9387 1.59717
\(658\) 0 0
\(659\) −0.00140660 −5.47932e−5 0 −2.73966e−5 1.00000i \(-0.500009\pi\)
−2.73966e−5 1.00000i \(0.500009\pi\)
\(660\) 0 0
\(661\) −40.7801 −1.58616 −0.793081 0.609116i \(-0.791525\pi\)
−0.793081 + 0.609116i \(0.791525\pi\)
\(662\) 0 0
\(663\) −2.47551 −0.0961407
\(664\) 0 0
\(665\) 1.78638 0.0692728
\(666\) 0 0
\(667\) −63.1165 −2.44388
\(668\) 0 0
\(669\) 2.70417 0.104549
\(670\) 0 0
\(671\) 19.6759 0.759580
\(672\) 0 0
\(673\) 15.3615 0.592141 0.296071 0.955166i \(-0.404324\pi\)
0.296071 + 0.955166i \(0.404324\pi\)
\(674\) 0 0
\(675\) 2.12555 0.0818123
\(676\) 0 0
\(677\) −27.9057 −1.07250 −0.536251 0.844059i \(-0.680160\pi\)
−0.536251 + 0.844059i \(0.680160\pi\)
\(678\) 0 0
\(679\) −14.0671 −0.539848
\(680\) 0 0
\(681\) −3.60369 −0.138094
\(682\) 0 0
\(683\) −17.6691 −0.676091 −0.338045 0.941130i \(-0.609766\pi\)
−0.338045 + 0.941130i \(0.609766\pi\)
\(684\) 0 0
\(685\) −4.62627 −0.176761
\(686\) 0 0
\(687\) 1.03712 0.0395686
\(688\) 0 0
\(689\) −32.6952 −1.24559
\(690\) 0 0
\(691\) 21.6744 0.824532 0.412266 0.911063i \(-0.364737\pi\)
0.412266 + 0.911063i \(0.364737\pi\)
\(692\) 0 0
\(693\) 56.1068 2.13132
\(694\) 0 0
\(695\) −2.66328 −0.101024
\(696\) 0 0
\(697\) 24.6965 0.935445
\(698\) 0 0
\(699\) −5.15831 −0.195105
\(700\) 0 0
\(701\) 15.6555 0.591299 0.295649 0.955296i \(-0.404464\pi\)
0.295649 + 0.955296i \(0.404464\pi\)
\(702\) 0 0
\(703\) −1.54906 −0.0584239
\(704\) 0 0
\(705\) 2.80972 0.105820
\(706\) 0 0
\(707\) −18.8183 −0.707735
\(708\) 0 0
\(709\) 25.4117 0.954356 0.477178 0.878807i \(-0.341660\pi\)
0.477178 + 0.878807i \(0.341660\pi\)
\(710\) 0 0
\(711\) −2.20842 −0.0828222
\(712\) 0 0
\(713\) 60.5566 2.26786
\(714\) 0 0
\(715\) 37.9325 1.41860
\(716\) 0 0
\(717\) 4.41984 0.165062
\(718\) 0 0
\(719\) 10.0811 0.375960 0.187980 0.982173i \(-0.439806\pi\)
0.187980 + 0.982173i \(0.439806\pi\)
\(720\) 0 0
\(721\) −64.6197 −2.40656
\(722\) 0 0
\(723\) 3.60015 0.133891
\(724\) 0 0
\(725\) −13.4470 −0.499408
\(726\) 0 0
\(727\) 41.3278 1.53276 0.766381 0.642386i \(-0.222056\pi\)
0.766381 + 0.642386i \(0.222056\pi\)
\(728\) 0 0
\(729\) −24.8233 −0.919382
\(730\) 0 0
\(731\) 2.47655 0.0915984
\(732\) 0 0
\(733\) −7.71188 −0.284845 −0.142422 0.989806i \(-0.545489\pi\)
−0.142422 + 0.989806i \(0.545489\pi\)
\(734\) 0 0
\(735\) −1.51321 −0.0558155
\(736\) 0 0
\(737\) −79.9187 −2.94384
\(738\) 0 0
\(739\) −8.81195 −0.324153 −0.162076 0.986778i \(-0.551819\pi\)
−0.162076 + 0.986778i \(0.551819\pi\)
\(740\) 0 0
\(741\) −0.223174 −0.00819849
\(742\) 0 0
\(743\) −10.2517 −0.376100 −0.188050 0.982159i \(-0.560217\pi\)
−0.188050 + 0.982159i \(0.560217\pi\)
\(744\) 0 0
\(745\) −33.8224 −1.23916
\(746\) 0 0
\(747\) −8.34469 −0.305316
\(748\) 0 0
\(749\) 1.78367 0.0651739
\(750\) 0 0
\(751\) −14.9259 −0.544653 −0.272326 0.962205i \(-0.587793\pi\)
−0.272326 + 0.962205i \(0.587793\pi\)
\(752\) 0 0
\(753\) 0.201914 0.00735815
\(754\) 0 0
\(755\) −21.8404 −0.794853
\(756\) 0 0
\(757\) −22.3520 −0.812396 −0.406198 0.913785i \(-0.633146\pi\)
−0.406198 + 0.913785i \(0.633146\pi\)
\(758\) 0 0
\(759\) −9.49846 −0.344772
\(760\) 0 0
\(761\) −0.735230 −0.0266521 −0.0133260 0.999911i \(-0.504242\pi\)
−0.0133260 + 0.999911i \(0.504242\pi\)
\(762\) 0 0
\(763\) 0.506597 0.0183400
\(764\) 0 0
\(765\) 17.5466 0.634397
\(766\) 0 0
\(767\) −43.4842 −1.57012
\(768\) 0 0
\(769\) 9.13226 0.329318 0.164659 0.986351i \(-0.447348\pi\)
0.164659 + 0.986351i \(0.447348\pi\)
\(770\) 0 0
\(771\) −4.74115 −0.170748
\(772\) 0 0
\(773\) 20.2905 0.729797 0.364899 0.931047i \(-0.381104\pi\)
0.364899 + 0.931047i \(0.381104\pi\)
\(774\) 0 0
\(775\) 12.9016 0.463439
\(776\) 0 0
\(777\) 3.51609 0.126139
\(778\) 0 0
\(779\) 2.22645 0.0797710
\(780\) 0 0
\(781\) 6.01046 0.215071
\(782\) 0 0
\(783\) −9.15940 −0.327330
\(784\) 0 0
\(785\) −11.1405 −0.397620
\(786\) 0 0
\(787\) 43.6653 1.55650 0.778250 0.627954i \(-0.216108\pi\)
0.778250 + 0.627954i \(0.216108\pi\)
\(788\) 0 0
\(789\) 5.44439 0.193825
\(790\) 0 0
\(791\) −62.1786 −2.21082
\(792\) 0 0
\(793\) −12.8944 −0.457894
\(794\) 0 0
\(795\) −3.19275 −0.113235
\(796\) 0 0
\(797\) −43.9528 −1.55689 −0.778443 0.627715i \(-0.783990\pi\)
−0.778443 + 0.627715i \(0.783990\pi\)
\(798\) 0 0
\(799\) −25.5174 −0.902742
\(800\) 0 0
\(801\) −34.6084 −1.22283
\(802\) 0 0
\(803\) −78.4891 −2.76982
\(804\) 0 0
\(805\) −49.8254 −1.75611
\(806\) 0 0
\(807\) 2.54090 0.0894439
\(808\) 0 0
\(809\) −23.2769 −0.818371 −0.409185 0.912451i \(-0.634187\pi\)
−0.409185 + 0.912451i \(0.634187\pi\)
\(810\) 0 0
\(811\) −15.8360 −0.556076 −0.278038 0.960570i \(-0.589684\pi\)
−0.278038 + 0.960570i \(0.589684\pi\)
\(812\) 0 0
\(813\) −4.78796 −0.167921
\(814\) 0 0
\(815\) 13.6161 0.476952
\(816\) 0 0
\(817\) 0.223268 0.00781114
\(818\) 0 0
\(819\) −36.7690 −1.28481
\(820\) 0 0
\(821\) 54.6129 1.90600 0.953001 0.302967i \(-0.0979770\pi\)
0.953001 + 0.302967i \(0.0979770\pi\)
\(822\) 0 0
\(823\) 9.89849 0.345040 0.172520 0.985006i \(-0.444809\pi\)
0.172520 + 0.985006i \(0.444809\pi\)
\(824\) 0 0
\(825\) −2.02365 −0.0704543
\(826\) 0 0
\(827\) −43.8078 −1.52335 −0.761673 0.647962i \(-0.775622\pi\)
−0.761673 + 0.647962i \(0.775622\pi\)
\(828\) 0 0
\(829\) 29.4488 1.02280 0.511400 0.859343i \(-0.329127\pi\)
0.511400 + 0.859343i \(0.329127\pi\)
\(830\) 0 0
\(831\) −4.67765 −0.162266
\(832\) 0 0
\(833\) 13.7427 0.476157
\(834\) 0 0
\(835\) 17.2837 0.598126
\(836\) 0 0
\(837\) 8.78791 0.303755
\(838\) 0 0
\(839\) −21.9325 −0.757194 −0.378597 0.925562i \(-0.623593\pi\)
−0.378597 + 0.925562i \(0.623593\pi\)
\(840\) 0 0
\(841\) 28.9456 0.998126
\(842\) 0 0
\(843\) −2.25346 −0.0776133
\(844\) 0 0
\(845\) −1.48218 −0.0509887
\(846\) 0 0
\(847\) −70.8100 −2.43306
\(848\) 0 0
\(849\) 2.75835 0.0946664
\(850\) 0 0
\(851\) 43.2061 1.48109
\(852\) 0 0
\(853\) 28.6166 0.979813 0.489907 0.871775i \(-0.337031\pi\)
0.489907 + 0.871775i \(0.337031\pi\)
\(854\) 0 0
\(855\) 1.58187 0.0540988
\(856\) 0 0
\(857\) 48.8480 1.66862 0.834308 0.551299i \(-0.185868\pi\)
0.834308 + 0.551299i \(0.185868\pi\)
\(858\) 0 0
\(859\) 50.7577 1.73183 0.865914 0.500192i \(-0.166737\pi\)
0.865914 + 0.500192i \(0.166737\pi\)
\(860\) 0 0
\(861\) −5.05366 −0.172228
\(862\) 0 0
\(863\) −45.3849 −1.54492 −0.772460 0.635064i \(-0.780974\pi\)
−0.772460 + 0.635064i \(0.780974\pi\)
\(864\) 0 0
\(865\) −11.6567 −0.396339
\(866\) 0 0
\(867\) −1.23711 −0.0420143
\(868\) 0 0
\(869\) 4.23406 0.143631
\(870\) 0 0
\(871\) 52.3739 1.77462
\(872\) 0 0
\(873\) −12.4567 −0.421596
\(874\) 0 0
\(875\) −40.6614 −1.37461
\(876\) 0 0
\(877\) 24.2677 0.819462 0.409731 0.912206i \(-0.365623\pi\)
0.409731 + 0.912206i \(0.365623\pi\)
\(878\) 0 0
\(879\) −1.36820 −0.0461481
\(880\) 0 0
\(881\) −48.7992 −1.64409 −0.822044 0.569424i \(-0.807166\pi\)
−0.822044 + 0.569424i \(0.807166\pi\)
\(882\) 0 0
\(883\) −27.5261 −0.926327 −0.463164 0.886273i \(-0.653286\pi\)
−0.463164 + 0.886273i \(0.653286\pi\)
\(884\) 0 0
\(885\) −4.24632 −0.142738
\(886\) 0 0
\(887\) −51.0022 −1.71249 −0.856244 0.516572i \(-0.827208\pi\)
−0.856244 + 0.516572i \(0.827208\pi\)
\(888\) 0 0
\(889\) −36.5428 −1.22561
\(890\) 0 0
\(891\) 48.9896 1.64121
\(892\) 0 0
\(893\) −2.30047 −0.0769822
\(894\) 0 0
\(895\) −37.6145 −1.25731
\(896\) 0 0
\(897\) 6.22472 0.207837
\(898\) 0 0
\(899\) −55.5955 −1.85421
\(900\) 0 0
\(901\) 28.9961 0.966000
\(902\) 0 0
\(903\) −0.506778 −0.0168645
\(904\) 0 0
\(905\) −25.5884 −0.850588
\(906\) 0 0
\(907\) 6.17559 0.205057 0.102528 0.994730i \(-0.467307\pi\)
0.102528 + 0.994730i \(0.467307\pi\)
\(908\) 0 0
\(909\) −16.6639 −0.552708
\(910\) 0 0
\(911\) 17.6971 0.586331 0.293165 0.956062i \(-0.405291\pi\)
0.293165 + 0.956062i \(0.405291\pi\)
\(912\) 0 0
\(913\) 15.9987 0.529481
\(914\) 0 0
\(915\) −1.25917 −0.0416267
\(916\) 0 0
\(917\) 28.4675 0.940079
\(918\) 0 0
\(919\) −44.6548 −1.47303 −0.736514 0.676423i \(-0.763529\pi\)
−0.736514 + 0.676423i \(0.763529\pi\)
\(920\) 0 0
\(921\) −5.47135 −0.180287
\(922\) 0 0
\(923\) −3.93889 −0.129650
\(924\) 0 0
\(925\) 9.20506 0.302660
\(926\) 0 0
\(927\) −57.2219 −1.87941
\(928\) 0 0
\(929\) −27.5676 −0.904462 −0.452231 0.891901i \(-0.649372\pi\)
−0.452231 + 0.891901i \(0.649372\pi\)
\(930\) 0 0
\(931\) 1.23894 0.0406048
\(932\) 0 0
\(933\) 0.374631 0.0122649
\(934\) 0 0
\(935\) −33.6409 −1.10017
\(936\) 0 0
\(937\) −1.22196 −0.0399196 −0.0199598 0.999801i \(-0.506354\pi\)
−0.0199598 + 0.999801i \(0.506354\pi\)
\(938\) 0 0
\(939\) 0.0997511 0.00325526
\(940\) 0 0
\(941\) −42.9102 −1.39883 −0.699416 0.714715i \(-0.746556\pi\)
−0.699416 + 0.714715i \(0.746556\pi\)
\(942\) 0 0
\(943\) −62.0998 −2.02225
\(944\) 0 0
\(945\) −7.23060 −0.235212
\(946\) 0 0
\(947\) −25.6450 −0.833352 −0.416676 0.909055i \(-0.636805\pi\)
−0.416676 + 0.909055i \(0.636805\pi\)
\(948\) 0 0
\(949\) 51.4370 1.66972
\(950\) 0 0
\(951\) 4.44931 0.144279
\(952\) 0 0
\(953\) 37.8088 1.22475 0.612374 0.790568i \(-0.290215\pi\)
0.612374 + 0.790568i \(0.290215\pi\)
\(954\) 0 0
\(955\) 16.5141 0.534383
\(956\) 0 0
\(957\) 8.72029 0.281887
\(958\) 0 0
\(959\) −8.59759 −0.277631
\(960\) 0 0
\(961\) 22.3406 0.720665
\(962\) 0 0
\(963\) 1.57947 0.0508978
\(964\) 0 0
\(965\) 4.72635 0.152146
\(966\) 0 0
\(967\) −4.36445 −0.140351 −0.0701757 0.997535i \(-0.522356\pi\)
−0.0701757 + 0.997535i \(0.522356\pi\)
\(968\) 0 0
\(969\) 0.197924 0.00635824
\(970\) 0 0
\(971\) 3.58064 0.114908 0.0574541 0.998348i \(-0.481702\pi\)
0.0574541 + 0.998348i \(0.481702\pi\)
\(972\) 0 0
\(973\) −4.94951 −0.158674
\(974\) 0 0
\(975\) 1.32618 0.0424716
\(976\) 0 0
\(977\) 29.3579 0.939244 0.469622 0.882868i \(-0.344390\pi\)
0.469622 + 0.882868i \(0.344390\pi\)
\(978\) 0 0
\(979\) 66.3526 2.12064
\(980\) 0 0
\(981\) 0.448600 0.0143227
\(982\) 0 0
\(983\) 17.1708 0.547663 0.273832 0.961778i \(-0.411709\pi\)
0.273832 + 0.961778i \(0.411709\pi\)
\(984\) 0 0
\(985\) 3.15934 0.100665
\(986\) 0 0
\(987\) 5.22166 0.166207
\(988\) 0 0
\(989\) −6.22734 −0.198018
\(990\) 0 0
\(991\) −22.9166 −0.727971 −0.363985 0.931405i \(-0.618584\pi\)
−0.363985 + 0.931405i \(0.618584\pi\)
\(992\) 0 0
\(993\) −6.90713 −0.219191
\(994\) 0 0
\(995\) 31.0532 0.984452
\(996\) 0 0
\(997\) −32.3122 −1.02334 −0.511669 0.859183i \(-0.670973\pi\)
−0.511669 + 0.859183i \(0.670973\pi\)
\(998\) 0 0
\(999\) 6.27002 0.198375
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.1.13 23
4.3 odd 2 2008.2.a.d.1.11 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.d.1.11 23 4.3 odd 2
4016.2.a.m.1.13 23 1.1 even 1 trivial