Properties

Label 8004.2.a.i.1.13
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(3.01816\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.01816 q^{5} +3.11628 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.01816 q^{5} +3.11628 q^{7} +1.00000 q^{9} +4.01390 q^{11} +2.08194 q^{13} -3.01816 q^{15} +1.51150 q^{17} -3.39944 q^{19} -3.11628 q^{21} +1.00000 q^{23} +4.10928 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.311954 q^{31} -4.01390 q^{33} +9.40544 q^{35} -0.640134 q^{37} -2.08194 q^{39} +4.74044 q^{41} +2.47895 q^{43} +3.01816 q^{45} -8.97233 q^{47} +2.71123 q^{49} -1.51150 q^{51} +5.80006 q^{53} +12.1146 q^{55} +3.39944 q^{57} -11.3298 q^{59} +10.1299 q^{61} +3.11628 q^{63} +6.28362 q^{65} +9.23512 q^{67} -1.00000 q^{69} +10.8687 q^{71} +13.2814 q^{73} -4.10928 q^{75} +12.5084 q^{77} +15.2895 q^{79} +1.00000 q^{81} -7.22584 q^{83} +4.56194 q^{85} -1.00000 q^{87} -8.32140 q^{89} +6.48791 q^{91} +0.311954 q^{93} -10.2601 q^{95} -19.6100 q^{97} +4.01390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.01816 1.34976 0.674881 0.737927i \(-0.264195\pi\)
0.674881 + 0.737927i \(0.264195\pi\)
\(6\) 0 0
\(7\) 3.11628 1.17784 0.588922 0.808190i \(-0.299552\pi\)
0.588922 + 0.808190i \(0.299552\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.01390 1.21023 0.605117 0.796136i \(-0.293126\pi\)
0.605117 + 0.796136i \(0.293126\pi\)
\(12\) 0 0
\(13\) 2.08194 0.577426 0.288713 0.957416i \(-0.406773\pi\)
0.288713 + 0.957416i \(0.406773\pi\)
\(14\) 0 0
\(15\) −3.01816 −0.779285
\(16\) 0 0
\(17\) 1.51150 0.366592 0.183296 0.983058i \(-0.441323\pi\)
0.183296 + 0.983058i \(0.441323\pi\)
\(18\) 0 0
\(19\) −3.39944 −0.779885 −0.389943 0.920839i \(-0.627505\pi\)
−0.389943 + 0.920839i \(0.627505\pi\)
\(20\) 0 0
\(21\) −3.11628 −0.680029
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.10928 0.821857
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.311954 −0.0560287 −0.0280143 0.999608i \(-0.508918\pi\)
−0.0280143 + 0.999608i \(0.508918\pi\)
\(32\) 0 0
\(33\) −4.01390 −0.698729
\(34\) 0 0
\(35\) 9.40544 1.58981
\(36\) 0 0
\(37\) −0.640134 −0.105237 −0.0526187 0.998615i \(-0.516757\pi\)
−0.0526187 + 0.998615i \(0.516757\pi\)
\(38\) 0 0
\(39\) −2.08194 −0.333377
\(40\) 0 0
\(41\) 4.74044 0.740333 0.370166 0.928965i \(-0.379301\pi\)
0.370166 + 0.928965i \(0.379301\pi\)
\(42\) 0 0
\(43\) 2.47895 0.378036 0.189018 0.981974i \(-0.439470\pi\)
0.189018 + 0.981974i \(0.439470\pi\)
\(44\) 0 0
\(45\) 3.01816 0.449921
\(46\) 0 0
\(47\) −8.97233 −1.30875 −0.654374 0.756171i \(-0.727068\pi\)
−0.654374 + 0.756171i \(0.727068\pi\)
\(48\) 0 0
\(49\) 2.71123 0.387318
\(50\) 0 0
\(51\) −1.51150 −0.211652
\(52\) 0 0
\(53\) 5.80006 0.796700 0.398350 0.917233i \(-0.369583\pi\)
0.398350 + 0.917233i \(0.369583\pi\)
\(54\) 0 0
\(55\) 12.1146 1.63353
\(56\) 0 0
\(57\) 3.39944 0.450267
\(58\) 0 0
\(59\) −11.3298 −1.47501 −0.737505 0.675341i \(-0.763996\pi\)
−0.737505 + 0.675341i \(0.763996\pi\)
\(60\) 0 0
\(61\) 10.1299 1.29701 0.648503 0.761212i \(-0.275396\pi\)
0.648503 + 0.761212i \(0.275396\pi\)
\(62\) 0 0
\(63\) 3.11628 0.392615
\(64\) 0 0
\(65\) 6.28362 0.779388
\(66\) 0 0
\(67\) 9.23512 1.12825 0.564125 0.825689i \(-0.309214\pi\)
0.564125 + 0.825689i \(0.309214\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.8687 1.28988 0.644938 0.764235i \(-0.276883\pi\)
0.644938 + 0.764235i \(0.276883\pi\)
\(72\) 0 0
\(73\) 13.2814 1.55447 0.777234 0.629212i \(-0.216622\pi\)
0.777234 + 0.629212i \(0.216622\pi\)
\(74\) 0 0
\(75\) −4.10928 −0.474499
\(76\) 0 0
\(77\) 12.5084 1.42547
\(78\) 0 0
\(79\) 15.2895 1.72021 0.860103 0.510120i \(-0.170399\pi\)
0.860103 + 0.510120i \(0.170399\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.22584 −0.793139 −0.396570 0.918005i \(-0.629799\pi\)
−0.396570 + 0.918005i \(0.629799\pi\)
\(84\) 0 0
\(85\) 4.56194 0.494812
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −8.32140 −0.882067 −0.441033 0.897491i \(-0.645388\pi\)
−0.441033 + 0.897491i \(0.645388\pi\)
\(90\) 0 0
\(91\) 6.48791 0.680118
\(92\) 0 0
\(93\) 0.311954 0.0323482
\(94\) 0 0
\(95\) −10.2601 −1.05266
\(96\) 0 0
\(97\) −19.6100 −1.99109 −0.995544 0.0942940i \(-0.969941\pi\)
−0.995544 + 0.0942940i \(0.969941\pi\)
\(98\) 0 0
\(99\) 4.01390 0.403412
\(100\) 0 0
\(101\) −19.8128 −1.97145 −0.985725 0.168366i \(-0.946151\pi\)
−0.985725 + 0.168366i \(0.946151\pi\)
\(102\) 0 0
\(103\) −11.1184 −1.09553 −0.547765 0.836632i \(-0.684521\pi\)
−0.547765 + 0.836632i \(0.684521\pi\)
\(104\) 0 0
\(105\) −9.40544 −0.917877
\(106\) 0 0
\(107\) 12.1490 1.17448 0.587242 0.809412i \(-0.300214\pi\)
0.587242 + 0.809412i \(0.300214\pi\)
\(108\) 0 0
\(109\) 12.5557 1.20262 0.601308 0.799017i \(-0.294646\pi\)
0.601308 + 0.799017i \(0.294646\pi\)
\(110\) 0 0
\(111\) 0.640134 0.0607588
\(112\) 0 0
\(113\) 4.44375 0.418033 0.209017 0.977912i \(-0.432974\pi\)
0.209017 + 0.977912i \(0.432974\pi\)
\(114\) 0 0
\(115\) 3.01816 0.281445
\(116\) 0 0
\(117\) 2.08194 0.192475
\(118\) 0 0
\(119\) 4.71026 0.431789
\(120\) 0 0
\(121\) 5.11135 0.464668
\(122\) 0 0
\(123\) −4.74044 −0.427431
\(124\) 0 0
\(125\) −2.68832 −0.240451
\(126\) 0 0
\(127\) −21.3241 −1.89221 −0.946106 0.323858i \(-0.895020\pi\)
−0.946106 + 0.323858i \(0.895020\pi\)
\(128\) 0 0
\(129\) −2.47895 −0.218259
\(130\) 0 0
\(131\) −9.00172 −0.786484 −0.393242 0.919435i \(-0.628647\pi\)
−0.393242 + 0.919435i \(0.628647\pi\)
\(132\) 0 0
\(133\) −10.5936 −0.918584
\(134\) 0 0
\(135\) −3.01816 −0.259762
\(136\) 0 0
\(137\) 21.8217 1.86435 0.932177 0.362004i \(-0.117907\pi\)
0.932177 + 0.362004i \(0.117907\pi\)
\(138\) 0 0
\(139\) 10.5613 0.895796 0.447898 0.894085i \(-0.352173\pi\)
0.447898 + 0.894085i \(0.352173\pi\)
\(140\) 0 0
\(141\) 8.97233 0.755606
\(142\) 0 0
\(143\) 8.35668 0.698821
\(144\) 0 0
\(145\) 3.01816 0.250644
\(146\) 0 0
\(147\) −2.71123 −0.223618
\(148\) 0 0
\(149\) 5.60764 0.459396 0.229698 0.973262i \(-0.426226\pi\)
0.229698 + 0.973262i \(0.426226\pi\)
\(150\) 0 0
\(151\) −2.14664 −0.174691 −0.0873456 0.996178i \(-0.527838\pi\)
−0.0873456 + 0.996178i \(0.527838\pi\)
\(152\) 0 0
\(153\) 1.51150 0.122197
\(154\) 0 0
\(155\) −0.941528 −0.0756253
\(156\) 0 0
\(157\) −9.18809 −0.733290 −0.366645 0.930361i \(-0.619494\pi\)
−0.366645 + 0.930361i \(0.619494\pi\)
\(158\) 0 0
\(159\) −5.80006 −0.459975
\(160\) 0 0
\(161\) 3.11628 0.245598
\(162\) 0 0
\(163\) −22.6642 −1.77520 −0.887598 0.460619i \(-0.847627\pi\)
−0.887598 + 0.460619i \(0.847627\pi\)
\(164\) 0 0
\(165\) −12.1146 −0.943118
\(166\) 0 0
\(167\) 15.2030 1.17644 0.588220 0.808701i \(-0.299829\pi\)
0.588220 + 0.808701i \(0.299829\pi\)
\(168\) 0 0
\(169\) −8.66553 −0.666579
\(170\) 0 0
\(171\) −3.39944 −0.259962
\(172\) 0 0
\(173\) −0.0577373 −0.00438969 −0.00219484 0.999998i \(-0.500699\pi\)
−0.00219484 + 0.999998i \(0.500699\pi\)
\(174\) 0 0
\(175\) 12.8057 0.968020
\(176\) 0 0
\(177\) 11.3298 0.851598
\(178\) 0 0
\(179\) 19.6721 1.47036 0.735182 0.677869i \(-0.237096\pi\)
0.735182 + 0.677869i \(0.237096\pi\)
\(180\) 0 0
\(181\) −4.55761 −0.338765 −0.169382 0.985550i \(-0.554177\pi\)
−0.169382 + 0.985550i \(0.554177\pi\)
\(182\) 0 0
\(183\) −10.1299 −0.748826
\(184\) 0 0
\(185\) −1.93203 −0.142045
\(186\) 0 0
\(187\) 6.06700 0.443663
\(188\) 0 0
\(189\) −3.11628 −0.226676
\(190\) 0 0
\(191\) 4.68909 0.339291 0.169645 0.985505i \(-0.445738\pi\)
0.169645 + 0.985505i \(0.445738\pi\)
\(192\) 0 0
\(193\) 12.3245 0.887134 0.443567 0.896241i \(-0.353713\pi\)
0.443567 + 0.896241i \(0.353713\pi\)
\(194\) 0 0
\(195\) −6.28362 −0.449980
\(196\) 0 0
\(197\) 8.15404 0.580951 0.290476 0.956882i \(-0.406186\pi\)
0.290476 + 0.956882i \(0.406186\pi\)
\(198\) 0 0
\(199\) −1.42058 −0.100702 −0.0503510 0.998732i \(-0.516034\pi\)
−0.0503510 + 0.998732i \(0.516034\pi\)
\(200\) 0 0
\(201\) −9.23512 −0.651395
\(202\) 0 0
\(203\) 3.11628 0.218720
\(204\) 0 0
\(205\) 14.3074 0.999273
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −13.6450 −0.943844
\(210\) 0 0
\(211\) 3.98242 0.274161 0.137081 0.990560i \(-0.456228\pi\)
0.137081 + 0.990560i \(0.456228\pi\)
\(212\) 0 0
\(213\) −10.8687 −0.744710
\(214\) 0 0
\(215\) 7.48186 0.510258
\(216\) 0 0
\(217\) −0.972138 −0.0659930
\(218\) 0 0
\(219\) −13.2814 −0.897473
\(220\) 0 0
\(221\) 3.14685 0.211680
\(222\) 0 0
\(223\) −7.29555 −0.488546 −0.244273 0.969706i \(-0.578549\pi\)
−0.244273 + 0.969706i \(0.578549\pi\)
\(224\) 0 0
\(225\) 4.10928 0.273952
\(226\) 0 0
\(227\) −19.4143 −1.28857 −0.644285 0.764786i \(-0.722845\pi\)
−0.644285 + 0.764786i \(0.722845\pi\)
\(228\) 0 0
\(229\) −16.7203 −1.10491 −0.552453 0.833544i \(-0.686308\pi\)
−0.552453 + 0.833544i \(0.686308\pi\)
\(230\) 0 0
\(231\) −12.5084 −0.822995
\(232\) 0 0
\(233\) −0.394749 −0.0258608 −0.0129304 0.999916i \(-0.504116\pi\)
−0.0129304 + 0.999916i \(0.504116\pi\)
\(234\) 0 0
\(235\) −27.0799 −1.76650
\(236\) 0 0
\(237\) −15.2895 −0.993162
\(238\) 0 0
\(239\) 4.88353 0.315889 0.157945 0.987448i \(-0.449513\pi\)
0.157945 + 0.987448i \(0.449513\pi\)
\(240\) 0 0
\(241\) 1.59740 0.102898 0.0514488 0.998676i \(-0.483616\pi\)
0.0514488 + 0.998676i \(0.483616\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 8.18291 0.522787
\(246\) 0 0
\(247\) −7.07743 −0.450326
\(248\) 0 0
\(249\) 7.22584 0.457919
\(250\) 0 0
\(251\) −30.2217 −1.90758 −0.953789 0.300476i \(-0.902854\pi\)
−0.953789 + 0.300476i \(0.902854\pi\)
\(252\) 0 0
\(253\) 4.01390 0.252351
\(254\) 0 0
\(255\) −4.56194 −0.285680
\(256\) 0 0
\(257\) −19.9318 −1.24331 −0.621656 0.783290i \(-0.713540\pi\)
−0.621656 + 0.783290i \(0.713540\pi\)
\(258\) 0 0
\(259\) −1.99484 −0.123953
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −21.1395 −1.30352 −0.651759 0.758426i \(-0.725969\pi\)
−0.651759 + 0.758426i \(0.725969\pi\)
\(264\) 0 0
\(265\) 17.5055 1.07536
\(266\) 0 0
\(267\) 8.32140 0.509261
\(268\) 0 0
\(269\) −22.7862 −1.38930 −0.694650 0.719348i \(-0.744441\pi\)
−0.694650 + 0.719348i \(0.744441\pi\)
\(270\) 0 0
\(271\) 16.0246 0.973428 0.486714 0.873561i \(-0.338195\pi\)
0.486714 + 0.873561i \(0.338195\pi\)
\(272\) 0 0
\(273\) −6.48791 −0.392666
\(274\) 0 0
\(275\) 16.4942 0.994640
\(276\) 0 0
\(277\) 1.48741 0.0893699 0.0446850 0.999001i \(-0.485772\pi\)
0.0446850 + 0.999001i \(0.485772\pi\)
\(278\) 0 0
\(279\) −0.311954 −0.0186762
\(280\) 0 0
\(281\) 9.54063 0.569146 0.284573 0.958654i \(-0.408148\pi\)
0.284573 + 0.958654i \(0.408148\pi\)
\(282\) 0 0
\(283\) −18.6641 −1.10946 −0.554732 0.832029i \(-0.687179\pi\)
−0.554732 + 0.832029i \(0.687179\pi\)
\(284\) 0 0
\(285\) 10.2601 0.607753
\(286\) 0 0
\(287\) 14.7726 0.871997
\(288\) 0 0
\(289\) −14.7154 −0.865610
\(290\) 0 0
\(291\) 19.6100 1.14956
\(292\) 0 0
\(293\) 24.4969 1.43112 0.715562 0.698550i \(-0.246171\pi\)
0.715562 + 0.698550i \(0.246171\pi\)
\(294\) 0 0
\(295\) −34.1951 −1.99091
\(296\) 0 0
\(297\) −4.01390 −0.232910
\(298\) 0 0
\(299\) 2.08194 0.120402
\(300\) 0 0
\(301\) 7.72510 0.445267
\(302\) 0 0
\(303\) 19.8128 1.13822
\(304\) 0 0
\(305\) 30.5738 1.75065
\(306\) 0 0
\(307\) 28.1002 1.60376 0.801881 0.597483i \(-0.203833\pi\)
0.801881 + 0.597483i \(0.203833\pi\)
\(308\) 0 0
\(309\) 11.1184 0.632504
\(310\) 0 0
\(311\) −22.7191 −1.28828 −0.644140 0.764907i \(-0.722785\pi\)
−0.644140 + 0.764907i \(0.722785\pi\)
\(312\) 0 0
\(313\) 21.1963 1.19809 0.599044 0.800716i \(-0.295547\pi\)
0.599044 + 0.800716i \(0.295547\pi\)
\(314\) 0 0
\(315\) 9.40544 0.529937
\(316\) 0 0
\(317\) 28.0936 1.57790 0.788948 0.614460i \(-0.210626\pi\)
0.788948 + 0.614460i \(0.210626\pi\)
\(318\) 0 0
\(319\) 4.01390 0.224735
\(320\) 0 0
\(321\) −12.1490 −0.678088
\(322\) 0 0
\(323\) −5.13825 −0.285900
\(324\) 0 0
\(325\) 8.55528 0.474562
\(326\) 0 0
\(327\) −12.5557 −0.694331
\(328\) 0 0
\(329\) −27.9603 −1.54150
\(330\) 0 0
\(331\) −13.6408 −0.749765 −0.374882 0.927072i \(-0.622317\pi\)
−0.374882 + 0.927072i \(0.622317\pi\)
\(332\) 0 0
\(333\) −0.640134 −0.0350791
\(334\) 0 0
\(335\) 27.8731 1.52287
\(336\) 0 0
\(337\) 31.4994 1.71588 0.857942 0.513747i \(-0.171743\pi\)
0.857942 + 0.513747i \(0.171743\pi\)
\(338\) 0 0
\(339\) −4.44375 −0.241352
\(340\) 0 0
\(341\) −1.25215 −0.0678078
\(342\) 0 0
\(343\) −13.3650 −0.721644
\(344\) 0 0
\(345\) −3.01816 −0.162492
\(346\) 0 0
\(347\) 9.35775 0.502351 0.251175 0.967942i \(-0.419183\pi\)
0.251175 + 0.967942i \(0.419183\pi\)
\(348\) 0 0
\(349\) 21.8298 1.16852 0.584262 0.811565i \(-0.301384\pi\)
0.584262 + 0.811565i \(0.301384\pi\)
\(350\) 0 0
\(351\) −2.08194 −0.111126
\(352\) 0 0
\(353\) −14.5936 −0.776741 −0.388371 0.921503i \(-0.626962\pi\)
−0.388371 + 0.921503i \(0.626962\pi\)
\(354\) 0 0
\(355\) 32.8034 1.74102
\(356\) 0 0
\(357\) −4.71026 −0.249293
\(358\) 0 0
\(359\) −27.7744 −1.46588 −0.732939 0.680294i \(-0.761852\pi\)
−0.732939 + 0.680294i \(0.761852\pi\)
\(360\) 0 0
\(361\) −7.44380 −0.391779
\(362\) 0 0
\(363\) −5.11135 −0.268276
\(364\) 0 0
\(365\) 40.0853 2.09816
\(366\) 0 0
\(367\) −18.1887 −0.949444 −0.474722 0.880136i \(-0.657451\pi\)
−0.474722 + 0.880136i \(0.657451\pi\)
\(368\) 0 0
\(369\) 4.74044 0.246778
\(370\) 0 0
\(371\) 18.0746 0.938389
\(372\) 0 0
\(373\) −25.6001 −1.32552 −0.662760 0.748832i \(-0.730615\pi\)
−0.662760 + 0.748832i \(0.730615\pi\)
\(374\) 0 0
\(375\) 2.68832 0.138824
\(376\) 0 0
\(377\) 2.08194 0.107225
\(378\) 0 0
\(379\) −20.1736 −1.03625 −0.518125 0.855305i \(-0.673370\pi\)
−0.518125 + 0.855305i \(0.673370\pi\)
\(380\) 0 0
\(381\) 21.3241 1.09247
\(382\) 0 0
\(383\) −10.4984 −0.536443 −0.268222 0.963357i \(-0.586436\pi\)
−0.268222 + 0.963357i \(0.586436\pi\)
\(384\) 0 0
\(385\) 37.7525 1.92404
\(386\) 0 0
\(387\) 2.47895 0.126012
\(388\) 0 0
\(389\) 3.11093 0.157730 0.0788652 0.996885i \(-0.474870\pi\)
0.0788652 + 0.996885i \(0.474870\pi\)
\(390\) 0 0
\(391\) 1.51150 0.0764398
\(392\) 0 0
\(393\) 9.00172 0.454077
\(394\) 0 0
\(395\) 46.1462 2.32187
\(396\) 0 0
\(397\) 28.9655 1.45374 0.726869 0.686776i \(-0.240975\pi\)
0.726869 + 0.686776i \(0.240975\pi\)
\(398\) 0 0
\(399\) 10.5936 0.530344
\(400\) 0 0
\(401\) −2.06390 −0.103066 −0.0515330 0.998671i \(-0.516411\pi\)
−0.0515330 + 0.998671i \(0.516411\pi\)
\(402\) 0 0
\(403\) −0.649470 −0.0323524
\(404\) 0 0
\(405\) 3.01816 0.149974
\(406\) 0 0
\(407\) −2.56943 −0.127362
\(408\) 0 0
\(409\) −32.6300 −1.61345 −0.806724 0.590928i \(-0.798762\pi\)
−0.806724 + 0.590928i \(0.798762\pi\)
\(410\) 0 0
\(411\) −21.8217 −1.07638
\(412\) 0 0
\(413\) −35.3068 −1.73733
\(414\) 0 0
\(415\) −21.8087 −1.07055
\(416\) 0 0
\(417\) −10.5613 −0.517188
\(418\) 0 0
\(419\) −12.9519 −0.632740 −0.316370 0.948636i \(-0.602464\pi\)
−0.316370 + 0.948636i \(0.602464\pi\)
\(420\) 0 0
\(421\) 9.36719 0.456529 0.228265 0.973599i \(-0.426695\pi\)
0.228265 + 0.973599i \(0.426695\pi\)
\(422\) 0 0
\(423\) −8.97233 −0.436250
\(424\) 0 0
\(425\) 6.21118 0.301286
\(426\) 0 0
\(427\) 31.5678 1.52767
\(428\) 0 0
\(429\) −8.35668 −0.403465
\(430\) 0 0
\(431\) 15.6930 0.755907 0.377954 0.925825i \(-0.376628\pi\)
0.377954 + 0.925825i \(0.376628\pi\)
\(432\) 0 0
\(433\) 15.8098 0.759772 0.379886 0.925033i \(-0.375963\pi\)
0.379886 + 0.925033i \(0.375963\pi\)
\(434\) 0 0
\(435\) −3.01816 −0.144710
\(436\) 0 0
\(437\) −3.39944 −0.162617
\(438\) 0 0
\(439\) −14.6868 −0.700964 −0.350482 0.936569i \(-0.613982\pi\)
−0.350482 + 0.936569i \(0.613982\pi\)
\(440\) 0 0
\(441\) 2.71123 0.129106
\(442\) 0 0
\(443\) −4.79849 −0.227983 −0.113991 0.993482i \(-0.536364\pi\)
−0.113991 + 0.993482i \(0.536364\pi\)
\(444\) 0 0
\(445\) −25.1153 −1.19058
\(446\) 0 0
\(447\) −5.60764 −0.265232
\(448\) 0 0
\(449\) 2.59321 0.122381 0.0611905 0.998126i \(-0.480510\pi\)
0.0611905 + 0.998126i \(0.480510\pi\)
\(450\) 0 0
\(451\) 19.0276 0.895976
\(452\) 0 0
\(453\) 2.14664 0.100858
\(454\) 0 0
\(455\) 19.5816 0.917997
\(456\) 0 0
\(457\) 29.1359 1.36292 0.681460 0.731855i \(-0.261345\pi\)
0.681460 + 0.731855i \(0.261345\pi\)
\(458\) 0 0
\(459\) −1.51150 −0.0705507
\(460\) 0 0
\(461\) 15.8875 0.739954 0.369977 0.929041i \(-0.379366\pi\)
0.369977 + 0.929041i \(0.379366\pi\)
\(462\) 0 0
\(463\) 28.1147 1.30660 0.653301 0.757099i \(-0.273384\pi\)
0.653301 + 0.757099i \(0.273384\pi\)
\(464\) 0 0
\(465\) 0.941528 0.0436623
\(466\) 0 0
\(467\) 9.49080 0.439182 0.219591 0.975592i \(-0.429528\pi\)
0.219591 + 0.975592i \(0.429528\pi\)
\(468\) 0 0
\(469\) 28.7793 1.32890
\(470\) 0 0
\(471\) 9.18809 0.423365
\(472\) 0 0
\(473\) 9.95023 0.457512
\(474\) 0 0
\(475\) −13.9693 −0.640954
\(476\) 0 0
\(477\) 5.80006 0.265567
\(478\) 0 0
\(479\) 18.4200 0.841633 0.420816 0.907146i \(-0.361744\pi\)
0.420816 + 0.907146i \(0.361744\pi\)
\(480\) 0 0
\(481\) −1.33272 −0.0607668
\(482\) 0 0
\(483\) −3.11628 −0.141796
\(484\) 0 0
\(485\) −59.1859 −2.68750
\(486\) 0 0
\(487\) −40.7305 −1.84568 −0.922838 0.385189i \(-0.874136\pi\)
−0.922838 + 0.385189i \(0.874136\pi\)
\(488\) 0 0
\(489\) 22.6642 1.02491
\(490\) 0 0
\(491\) −12.0643 −0.544455 −0.272227 0.962233i \(-0.587760\pi\)
−0.272227 + 0.962233i \(0.587760\pi\)
\(492\) 0 0
\(493\) 1.51150 0.0680745
\(494\) 0 0
\(495\) 12.1146 0.544510
\(496\) 0 0
\(497\) 33.8699 1.51927
\(498\) 0 0
\(499\) 37.1644 1.66371 0.831855 0.554994i \(-0.187279\pi\)
0.831855 + 0.554994i \(0.187279\pi\)
\(500\) 0 0
\(501\) −15.2030 −0.679218
\(502\) 0 0
\(503\) 37.1622 1.65698 0.828491 0.560003i \(-0.189200\pi\)
0.828491 + 0.560003i \(0.189200\pi\)
\(504\) 0 0
\(505\) −59.7982 −2.66099
\(506\) 0 0
\(507\) 8.66553 0.384850
\(508\) 0 0
\(509\) 0.747383 0.0331272 0.0165636 0.999863i \(-0.494727\pi\)
0.0165636 + 0.999863i \(0.494727\pi\)
\(510\) 0 0
\(511\) 41.3886 1.83092
\(512\) 0 0
\(513\) 3.39944 0.150089
\(514\) 0 0
\(515\) −33.5571 −1.47870
\(516\) 0 0
\(517\) −36.0140 −1.58389
\(518\) 0 0
\(519\) 0.0577373 0.00253439
\(520\) 0 0
\(521\) 33.4372 1.46491 0.732455 0.680815i \(-0.238374\pi\)
0.732455 + 0.680815i \(0.238374\pi\)
\(522\) 0 0
\(523\) 5.12177 0.223960 0.111980 0.993710i \(-0.464281\pi\)
0.111980 + 0.993710i \(0.464281\pi\)
\(524\) 0 0
\(525\) −12.8057 −0.558886
\(526\) 0 0
\(527\) −0.471518 −0.0205397
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.3298 −0.491670
\(532\) 0 0
\(533\) 9.86931 0.427487
\(534\) 0 0
\(535\) 36.6675 1.58527
\(536\) 0 0
\(537\) −19.6721 −0.848916
\(538\) 0 0
\(539\) 10.8826 0.468746
\(540\) 0 0
\(541\) −23.8365 −1.02481 −0.512405 0.858744i \(-0.671245\pi\)
−0.512405 + 0.858744i \(0.671245\pi\)
\(542\) 0 0
\(543\) 4.55761 0.195586
\(544\) 0 0
\(545\) 37.8951 1.62325
\(546\) 0 0
\(547\) −3.60333 −0.154067 −0.0770336 0.997029i \(-0.524545\pi\)
−0.0770336 + 0.997029i \(0.524545\pi\)
\(548\) 0 0
\(549\) 10.1299 0.432335
\(550\) 0 0
\(551\) −3.39944 −0.144821
\(552\) 0 0
\(553\) 47.6465 2.02614
\(554\) 0 0
\(555\) 1.93203 0.0820100
\(556\) 0 0
\(557\) 36.9165 1.56420 0.782102 0.623151i \(-0.214148\pi\)
0.782102 + 0.623151i \(0.214148\pi\)
\(558\) 0 0
\(559\) 5.16102 0.218288
\(560\) 0 0
\(561\) −6.06700 −0.256149
\(562\) 0 0
\(563\) 32.7021 1.37823 0.689115 0.724652i \(-0.258000\pi\)
0.689115 + 0.724652i \(0.258000\pi\)
\(564\) 0 0
\(565\) 13.4120 0.564245
\(566\) 0 0
\(567\) 3.11628 0.130872
\(568\) 0 0
\(569\) 38.3521 1.60780 0.803901 0.594763i \(-0.202754\pi\)
0.803901 + 0.594763i \(0.202754\pi\)
\(570\) 0 0
\(571\) −9.73555 −0.407420 −0.203710 0.979031i \(-0.565300\pi\)
−0.203710 + 0.979031i \(0.565300\pi\)
\(572\) 0 0
\(573\) −4.68909 −0.195890
\(574\) 0 0
\(575\) 4.10928 0.171369
\(576\) 0 0
\(577\) −43.0538 −1.79235 −0.896176 0.443698i \(-0.853666\pi\)
−0.896176 + 0.443698i \(0.853666\pi\)
\(578\) 0 0
\(579\) −12.3245 −0.512187
\(580\) 0 0
\(581\) −22.5178 −0.934195
\(582\) 0 0
\(583\) 23.2808 0.964194
\(584\) 0 0
\(585\) 6.28362 0.259796
\(586\) 0 0
\(587\) −25.5360 −1.05398 −0.526992 0.849870i \(-0.676680\pi\)
−0.526992 + 0.849870i \(0.676680\pi\)
\(588\) 0 0
\(589\) 1.06047 0.0436959
\(590\) 0 0
\(591\) −8.15404 −0.335412
\(592\) 0 0
\(593\) −17.7807 −0.730165 −0.365083 0.930975i \(-0.618959\pi\)
−0.365083 + 0.930975i \(0.618959\pi\)
\(594\) 0 0
\(595\) 14.2163 0.582812
\(596\) 0 0
\(597\) 1.42058 0.0581404
\(598\) 0 0
\(599\) 26.4509 1.08075 0.540377 0.841423i \(-0.318281\pi\)
0.540377 + 0.841423i \(0.318281\pi\)
\(600\) 0 0
\(601\) 31.8168 1.29783 0.648916 0.760860i \(-0.275223\pi\)
0.648916 + 0.760860i \(0.275223\pi\)
\(602\) 0 0
\(603\) 9.23512 0.376083
\(604\) 0 0
\(605\) 15.4269 0.627192
\(606\) 0 0
\(607\) −36.9352 −1.49916 −0.749578 0.661917i \(-0.769743\pi\)
−0.749578 + 0.661917i \(0.769743\pi\)
\(608\) 0 0
\(609\) −3.11628 −0.126278
\(610\) 0 0
\(611\) −18.6798 −0.755706
\(612\) 0 0
\(613\) 14.9193 0.602587 0.301293 0.953531i \(-0.402582\pi\)
0.301293 + 0.953531i \(0.402582\pi\)
\(614\) 0 0
\(615\) −14.3074 −0.576930
\(616\) 0 0
\(617\) 7.63942 0.307552 0.153776 0.988106i \(-0.450857\pi\)
0.153776 + 0.988106i \(0.450857\pi\)
\(618\) 0 0
\(619\) 5.64294 0.226809 0.113404 0.993549i \(-0.463824\pi\)
0.113404 + 0.993549i \(0.463824\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −25.9318 −1.03894
\(624\) 0 0
\(625\) −28.6602 −1.14641
\(626\) 0 0
\(627\) 13.6450 0.544929
\(628\) 0 0
\(629\) −0.967561 −0.0385792
\(630\) 0 0
\(631\) −28.4240 −1.13154 −0.565772 0.824562i \(-0.691422\pi\)
−0.565772 + 0.824562i \(0.691422\pi\)
\(632\) 0 0
\(633\) −3.98242 −0.158287
\(634\) 0 0
\(635\) −64.3597 −2.55403
\(636\) 0 0
\(637\) 5.64461 0.223647
\(638\) 0 0
\(639\) 10.8687 0.429959
\(640\) 0 0
\(641\) 46.3130 1.82925 0.914626 0.404301i \(-0.132485\pi\)
0.914626 + 0.404301i \(0.132485\pi\)
\(642\) 0 0
\(643\) −27.0406 −1.06638 −0.533188 0.845997i \(-0.679006\pi\)
−0.533188 + 0.845997i \(0.679006\pi\)
\(644\) 0 0
\(645\) −7.48186 −0.294598
\(646\) 0 0
\(647\) 27.8531 1.09502 0.547510 0.836799i \(-0.315576\pi\)
0.547510 + 0.836799i \(0.315576\pi\)
\(648\) 0 0
\(649\) −45.4765 −1.78511
\(650\) 0 0
\(651\) 0.972138 0.0381011
\(652\) 0 0
\(653\) −16.8048 −0.657622 −0.328811 0.944396i \(-0.606648\pi\)
−0.328811 + 0.944396i \(0.606648\pi\)
\(654\) 0 0
\(655\) −27.1686 −1.06157
\(656\) 0 0
\(657\) 13.2814 0.518156
\(658\) 0 0
\(659\) 7.66296 0.298506 0.149253 0.988799i \(-0.452313\pi\)
0.149253 + 0.988799i \(0.452313\pi\)
\(660\) 0 0
\(661\) 21.2426 0.826240 0.413120 0.910677i \(-0.364439\pi\)
0.413120 + 0.910677i \(0.364439\pi\)
\(662\) 0 0
\(663\) −3.14685 −0.122213
\(664\) 0 0
\(665\) −31.9732 −1.23987
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 7.29555 0.282062
\(670\) 0 0
\(671\) 40.6605 1.56968
\(672\) 0 0
\(673\) −26.9404 −1.03848 −0.519238 0.854630i \(-0.673784\pi\)
−0.519238 + 0.854630i \(0.673784\pi\)
\(674\) 0 0
\(675\) −4.10928 −0.158166
\(676\) 0 0
\(677\) 15.3473 0.589845 0.294923 0.955521i \(-0.404706\pi\)
0.294923 + 0.955521i \(0.404706\pi\)
\(678\) 0 0
\(679\) −61.1102 −2.34519
\(680\) 0 0
\(681\) 19.4143 0.743956
\(682\) 0 0
\(683\) −22.9220 −0.877088 −0.438544 0.898710i \(-0.644506\pi\)
−0.438544 + 0.898710i \(0.644506\pi\)
\(684\) 0 0
\(685\) 65.8614 2.51643
\(686\) 0 0
\(687\) 16.7203 0.637918
\(688\) 0 0
\(689\) 12.0754 0.460035
\(690\) 0 0
\(691\) 8.94873 0.340426 0.170213 0.985407i \(-0.445554\pi\)
0.170213 + 0.985407i \(0.445554\pi\)
\(692\) 0 0
\(693\) 12.5084 0.475156
\(694\) 0 0
\(695\) 31.8756 1.20911
\(696\) 0 0
\(697\) 7.16517 0.271400
\(698\) 0 0
\(699\) 0.394749 0.0149308
\(700\) 0 0
\(701\) −0.809113 −0.0305598 −0.0152799 0.999883i \(-0.504864\pi\)
−0.0152799 + 0.999883i \(0.504864\pi\)
\(702\) 0 0
\(703\) 2.17610 0.0820731
\(704\) 0 0
\(705\) 27.0799 1.01989
\(706\) 0 0
\(707\) −61.7424 −2.32206
\(708\) 0 0
\(709\) 26.8125 1.00696 0.503482 0.864006i \(-0.332052\pi\)
0.503482 + 0.864006i \(0.332052\pi\)
\(710\) 0 0
\(711\) 15.2895 0.573402
\(712\) 0 0
\(713\) −0.311954 −0.0116828
\(714\) 0 0
\(715\) 25.2218 0.943242
\(716\) 0 0
\(717\) −4.88353 −0.182379
\(718\) 0 0
\(719\) −1.28642 −0.0479752 −0.0239876 0.999712i \(-0.507636\pi\)
−0.0239876 + 0.999712i \(0.507636\pi\)
\(720\) 0 0
\(721\) −34.6481 −1.29036
\(722\) 0 0
\(723\) −1.59740 −0.0594079
\(724\) 0 0
\(725\) 4.10928 0.152615
\(726\) 0 0
\(727\) −34.4980 −1.27946 −0.639730 0.768600i \(-0.720954\pi\)
−0.639730 + 0.768600i \(0.720954\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.74692 0.138585
\(732\) 0 0
\(733\) −17.9446 −0.662800 −0.331400 0.943490i \(-0.607521\pi\)
−0.331400 + 0.943490i \(0.607521\pi\)
\(734\) 0 0
\(735\) −8.18291 −0.301831
\(736\) 0 0
\(737\) 37.0688 1.36545
\(738\) 0 0
\(739\) −47.9158 −1.76261 −0.881306 0.472546i \(-0.843335\pi\)
−0.881306 + 0.472546i \(0.843335\pi\)
\(740\) 0 0
\(741\) 7.07743 0.259996
\(742\) 0 0
\(743\) 28.1651 1.03328 0.516638 0.856204i \(-0.327183\pi\)
0.516638 + 0.856204i \(0.327183\pi\)
\(744\) 0 0
\(745\) 16.9247 0.620075
\(746\) 0 0
\(747\) −7.22584 −0.264380
\(748\) 0 0
\(749\) 37.8596 1.38336
\(750\) 0 0
\(751\) −10.7976 −0.394009 −0.197005 0.980403i \(-0.563121\pi\)
−0.197005 + 0.980403i \(0.563121\pi\)
\(752\) 0 0
\(753\) 30.2217 1.10134
\(754\) 0 0
\(755\) −6.47890 −0.235791
\(756\) 0 0
\(757\) 21.1049 0.767072 0.383536 0.923526i \(-0.374706\pi\)
0.383536 + 0.923526i \(0.374706\pi\)
\(758\) 0 0
\(759\) −4.01390 −0.145695
\(760\) 0 0
\(761\) 49.3580 1.78923 0.894613 0.446841i \(-0.147451\pi\)
0.894613 + 0.446841i \(0.147451\pi\)
\(762\) 0 0
\(763\) 39.1271 1.41650
\(764\) 0 0
\(765\) 4.56194 0.164937
\(766\) 0 0
\(767\) −23.5879 −0.851709
\(768\) 0 0
\(769\) −38.1872 −1.37707 −0.688533 0.725205i \(-0.741745\pi\)
−0.688533 + 0.725205i \(0.741745\pi\)
\(770\) 0 0
\(771\) 19.9318 0.717826
\(772\) 0 0
\(773\) 25.7530 0.926272 0.463136 0.886287i \(-0.346724\pi\)
0.463136 + 0.886287i \(0.346724\pi\)
\(774\) 0 0
\(775\) −1.28191 −0.0460475
\(776\) 0 0
\(777\) 1.99484 0.0715645
\(778\) 0 0
\(779\) −16.1149 −0.577374
\(780\) 0 0
\(781\) 43.6258 1.56105
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) −27.7311 −0.989766
\(786\) 0 0
\(787\) 0.966498 0.0344519 0.0172260 0.999852i \(-0.494517\pi\)
0.0172260 + 0.999852i \(0.494517\pi\)
\(788\) 0 0
\(789\) 21.1395 0.752587
\(790\) 0 0
\(791\) 13.8480 0.492378
\(792\) 0 0
\(793\) 21.0899 0.748925
\(794\) 0 0
\(795\) −17.5055 −0.620857
\(796\) 0 0
\(797\) 26.2327 0.929210 0.464605 0.885518i \(-0.346196\pi\)
0.464605 + 0.885518i \(0.346196\pi\)
\(798\) 0 0
\(799\) −13.5617 −0.479777
\(800\) 0 0
\(801\) −8.32140 −0.294022
\(802\) 0 0
\(803\) 53.3101 1.88127
\(804\) 0 0
\(805\) 9.40544 0.331498
\(806\) 0 0
\(807\) 22.7862 0.802113
\(808\) 0 0
\(809\) −29.2417 −1.02808 −0.514041 0.857765i \(-0.671852\pi\)
−0.514041 + 0.857765i \(0.671852\pi\)
\(810\) 0 0
\(811\) 6.61433 0.232261 0.116130 0.993234i \(-0.462951\pi\)
0.116130 + 0.993234i \(0.462951\pi\)
\(812\) 0 0
\(813\) −16.0246 −0.562009
\(814\) 0 0
\(815\) −68.4041 −2.39609
\(816\) 0 0
\(817\) −8.42703 −0.294825
\(818\) 0 0
\(819\) 6.48791 0.226706
\(820\) 0 0
\(821\) 25.7326 0.898075 0.449038 0.893513i \(-0.351767\pi\)
0.449038 + 0.893513i \(0.351767\pi\)
\(822\) 0 0
\(823\) 36.3098 1.26568 0.632841 0.774282i \(-0.281889\pi\)
0.632841 + 0.774282i \(0.281889\pi\)
\(824\) 0 0
\(825\) −16.4942 −0.574256
\(826\) 0 0
\(827\) −48.8534 −1.69880 −0.849400 0.527749i \(-0.823036\pi\)
−0.849400 + 0.527749i \(0.823036\pi\)
\(828\) 0 0
\(829\) −52.1550 −1.81142 −0.905709 0.423900i \(-0.860661\pi\)
−0.905709 + 0.423900i \(0.860661\pi\)
\(830\) 0 0
\(831\) −1.48741 −0.0515978
\(832\) 0 0
\(833\) 4.09801 0.141988
\(834\) 0 0
\(835\) 45.8850 1.58791
\(836\) 0 0
\(837\) 0.311954 0.0107827
\(838\) 0 0
\(839\) −27.9594 −0.965264 −0.482632 0.875823i \(-0.660319\pi\)
−0.482632 + 0.875823i \(0.660319\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −9.54063 −0.328597
\(844\) 0 0
\(845\) −26.1539 −0.899723
\(846\) 0 0
\(847\) 15.9284 0.547307
\(848\) 0 0
\(849\) 18.6641 0.640550
\(850\) 0 0
\(851\) −0.640134 −0.0219435
\(852\) 0 0
\(853\) −51.1221 −1.75039 −0.875193 0.483774i \(-0.839266\pi\)
−0.875193 + 0.483774i \(0.839266\pi\)
\(854\) 0 0
\(855\) −10.2601 −0.350886
\(856\) 0 0
\(857\) −32.9031 −1.12395 −0.561975 0.827154i \(-0.689958\pi\)
−0.561975 + 0.827154i \(0.689958\pi\)
\(858\) 0 0
\(859\) −21.1992 −0.723308 −0.361654 0.932312i \(-0.617788\pi\)
−0.361654 + 0.932312i \(0.617788\pi\)
\(860\) 0 0
\(861\) −14.7726 −0.503448
\(862\) 0 0
\(863\) −48.3542 −1.64599 −0.822997 0.568045i \(-0.807700\pi\)
−0.822997 + 0.568045i \(0.807700\pi\)
\(864\) 0 0
\(865\) −0.174260 −0.00592503
\(866\) 0 0
\(867\) 14.7154 0.499760
\(868\) 0 0
\(869\) 61.3706 2.08185
\(870\) 0 0
\(871\) 19.2270 0.651481
\(872\) 0 0
\(873\) −19.6100 −0.663696
\(874\) 0 0
\(875\) −8.37757 −0.283214
\(876\) 0 0
\(877\) 14.0704 0.475123 0.237562 0.971372i \(-0.423652\pi\)
0.237562 + 0.971372i \(0.423652\pi\)
\(878\) 0 0
\(879\) −24.4969 −0.826259
\(880\) 0 0
\(881\) 32.7327 1.10279 0.551396 0.834244i \(-0.314096\pi\)
0.551396 + 0.834244i \(0.314096\pi\)
\(882\) 0 0
\(883\) −40.1331 −1.35059 −0.675294 0.737549i \(-0.735983\pi\)
−0.675294 + 0.737549i \(0.735983\pi\)
\(884\) 0 0
\(885\) 34.1951 1.14945
\(886\) 0 0
\(887\) −1.74123 −0.0584648 −0.0292324 0.999573i \(-0.509306\pi\)
−0.0292324 + 0.999573i \(0.509306\pi\)
\(888\) 0 0
\(889\) −66.4521 −2.22873
\(890\) 0 0
\(891\) 4.01390 0.134471
\(892\) 0 0
\(893\) 30.5009 1.02067
\(894\) 0 0
\(895\) 59.3736 1.98464
\(896\) 0 0
\(897\) −2.08194 −0.0695139
\(898\) 0 0
\(899\) −0.311954 −0.0104043
\(900\) 0 0
\(901\) 8.76679 0.292064
\(902\) 0 0
\(903\) −7.72510 −0.257075
\(904\) 0 0
\(905\) −13.7556 −0.457252
\(906\) 0 0
\(907\) −7.92339 −0.263092 −0.131546 0.991310i \(-0.541994\pi\)
−0.131546 + 0.991310i \(0.541994\pi\)
\(908\) 0 0
\(909\) −19.8128 −0.657150
\(910\) 0 0
\(911\) −11.9492 −0.395894 −0.197947 0.980213i \(-0.563427\pi\)
−0.197947 + 0.980213i \(0.563427\pi\)
\(912\) 0 0
\(913\) −29.0038 −0.959885
\(914\) 0 0
\(915\) −30.5738 −1.01074
\(916\) 0 0
\(917\) −28.0519 −0.926356
\(918\) 0 0
\(919\) 35.3329 1.16553 0.582763 0.812642i \(-0.301972\pi\)
0.582763 + 0.812642i \(0.301972\pi\)
\(920\) 0 0
\(921\) −28.1002 −0.925933
\(922\) 0 0
\(923\) 22.6279 0.744808
\(924\) 0 0
\(925\) −2.63049 −0.0864901
\(926\) 0 0
\(927\) −11.1184 −0.365176
\(928\) 0 0
\(929\) 27.3999 0.898961 0.449480 0.893290i \(-0.351609\pi\)
0.449480 + 0.893290i \(0.351609\pi\)
\(930\) 0 0
\(931\) −9.21665 −0.302064
\(932\) 0 0
\(933\) 22.7191 0.743789
\(934\) 0 0
\(935\) 18.3112 0.598839
\(936\) 0 0
\(937\) 22.7656 0.743718 0.371859 0.928289i \(-0.378720\pi\)
0.371859 + 0.928289i \(0.378720\pi\)
\(938\) 0 0
\(939\) −21.1963 −0.691717
\(940\) 0 0
\(941\) 41.6999 1.35938 0.679688 0.733501i \(-0.262115\pi\)
0.679688 + 0.733501i \(0.262115\pi\)
\(942\) 0 0
\(943\) 4.74044 0.154370
\(944\) 0 0
\(945\) −9.40544 −0.305959
\(946\) 0 0
\(947\) −44.4725 −1.44516 −0.722581 0.691286i \(-0.757044\pi\)
−0.722581 + 0.691286i \(0.757044\pi\)
\(948\) 0 0
\(949\) 27.6510 0.897590
\(950\) 0 0
\(951\) −28.0936 −0.910999
\(952\) 0 0
\(953\) 16.8109 0.544560 0.272280 0.962218i \(-0.412222\pi\)
0.272280 + 0.962218i \(0.412222\pi\)
\(954\) 0 0
\(955\) 14.1524 0.457962
\(956\) 0 0
\(957\) −4.01390 −0.129751
\(958\) 0 0
\(959\) 68.0026 2.19592
\(960\) 0 0
\(961\) −30.9027 −0.996861
\(962\) 0 0
\(963\) 12.1490 0.391495
\(964\) 0 0
\(965\) 37.1972 1.19742
\(966\) 0 0
\(967\) −19.1942 −0.617244 −0.308622 0.951185i \(-0.599868\pi\)
−0.308622 + 0.951185i \(0.599868\pi\)
\(968\) 0 0
\(969\) 5.13825 0.165064
\(970\) 0 0
\(971\) −14.4563 −0.463925 −0.231962 0.972725i \(-0.574515\pi\)
−0.231962 + 0.972725i \(0.574515\pi\)
\(972\) 0 0
\(973\) 32.9120 1.05511
\(974\) 0 0
\(975\) −8.55528 −0.273988
\(976\) 0 0
\(977\) −25.9966 −0.831705 −0.415852 0.909432i \(-0.636517\pi\)
−0.415852 + 0.909432i \(0.636517\pi\)
\(978\) 0 0
\(979\) −33.4012 −1.06751
\(980\) 0 0
\(981\) 12.5557 0.400872
\(982\) 0 0
\(983\) 47.6020 1.51827 0.759135 0.650934i \(-0.225622\pi\)
0.759135 + 0.650934i \(0.225622\pi\)
\(984\) 0 0
\(985\) 24.6102 0.784146
\(986\) 0 0
\(987\) 27.9603 0.889987
\(988\) 0 0
\(989\) 2.47895 0.0788259
\(990\) 0 0
\(991\) 19.6633 0.624626 0.312313 0.949979i \(-0.398896\pi\)
0.312313 + 0.949979i \(0.398896\pi\)
\(992\) 0 0
\(993\) 13.6408 0.432877
\(994\) 0 0
\(995\) −4.28753 −0.135924
\(996\) 0 0
\(997\) 16.2571 0.514868 0.257434 0.966296i \(-0.417123\pi\)
0.257434 + 0.966296i \(0.417123\pi\)
\(998\) 0 0
\(999\) 0.640134 0.0202529
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 8004.2.a.i.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.13 16 1.1 even 1 trivial