Properties

Label 4864.2.a.bs.1.6
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.30294\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.458177 q^{3} +3.00397 q^{5} +2.21624 q^{7} -2.79007 q^{9} +O(q^{10})\) \(q-0.458177 q^{3} +3.00397 q^{5} +2.21624 q^{7} -2.79007 q^{9} -5.02382 q^{11} -5.22021 q^{13} -1.37635 q^{15} +7.73025 q^{17} -1.00000 q^{19} -1.01543 q^{21} -6.51613 q^{23} +4.02382 q^{25} +2.65288 q^{27} +2.23562 q^{29} -3.33566 q^{31} +2.30180 q^{33} +6.65751 q^{35} +1.45208 q^{37} +2.39178 q^{39} +3.16461 q^{41} +2.16101 q^{43} -8.38129 q^{45} +9.95256 q^{47} -2.08828 q^{49} -3.54182 q^{51} -6.66810 q^{53} -15.0914 q^{55} +0.458177 q^{57} +3.89126 q^{59} -7.05259 q^{61} -6.18347 q^{63} -15.6813 q^{65} -13.2767 q^{67} +2.98554 q^{69} -5.72840 q^{71} -5.89754 q^{73} -1.84362 q^{75} -11.1340 q^{77} -15.9369 q^{79} +7.15473 q^{81} -8.71734 q^{83} +23.2214 q^{85} -1.02431 q^{87} -3.79938 q^{89} -11.5692 q^{91} +1.52833 q^{93} -3.00397 q^{95} -3.94645 q^{97} +14.0168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{3} + 14q^{9} + O(q^{10}) \) \( 10q - 4q^{3} + 14q^{9} - 20q^{11} + 4q^{17} - 10q^{19} + 10q^{25} - 28q^{27} - 8q^{33} - 36q^{35} - 12q^{41} + 4q^{43} + 26q^{49} - 36q^{51} + 4q^{57} - 52q^{59} - 24q^{65} - 12q^{67} + 12q^{73} + 12q^{75} + 34q^{81} - 16q^{83} - 20q^{89} - 60q^{91} - 28q^{97} - 60q^{99} + O(q^{100}) \)

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
\(3\) −0.458177 −0.264529 −0.132264 0.991214i \(-0.542225\pi\)
−0.132264 + 0.991214i \(0.542225\pi\)
\(4\) 0 0
\(5\) 3.00397 1.34341 0.671707 0.740817i \(-0.265561\pi\)
0.671707 + 0.740817i \(0.265561\pi\)
\(6\) 0 0
\(7\) 2.21624 0.837660 0.418830 0.908065i \(-0.362440\pi\)
0.418830 + 0.908065i \(0.362440\pi\)
\(8\) 0 0
\(9\) −2.79007 −0.930025
\(10\) 0 0
\(11\) −5.02382 −1.51474 −0.757369 0.652987i \(-0.773516\pi\)
−0.757369 + 0.652987i \(0.773516\pi\)
\(12\) 0 0
\(13\) −5.22021 −1.44783 −0.723913 0.689892i \(-0.757658\pi\)
−0.723913 + 0.689892i \(0.757658\pi\)
\(14\) 0 0
\(15\) −1.37635 −0.355372
\(16\) 0 0
\(17\) 7.73025 1.87486 0.937430 0.348174i \(-0.113198\pi\)
0.937430 + 0.348174i \(0.113198\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.01543 −0.221585
\(22\) 0 0
\(23\) −6.51613 −1.35871 −0.679353 0.733811i \(-0.737740\pi\)
−0.679353 + 0.733811i \(0.737740\pi\)
\(24\) 0 0
\(25\) 4.02382 0.804764
\(26\) 0 0
\(27\) 2.65288 0.510547
\(28\) 0 0
\(29\) 2.23562 0.415145 0.207572 0.978220i \(-0.433444\pi\)
0.207572 + 0.978220i \(0.433444\pi\)
\(30\) 0 0
\(31\) −3.33566 −0.599103 −0.299551 0.954080i \(-0.596837\pi\)
−0.299551 + 0.954080i \(0.596837\pi\)
\(32\) 0 0
\(33\) 2.30180 0.400692
\(34\) 0 0
\(35\) 6.65751 1.12533
\(36\) 0 0
\(37\) 1.45208 0.238720 0.119360 0.992851i \(-0.461916\pi\)
0.119360 + 0.992851i \(0.461916\pi\)
\(38\) 0 0
\(39\) 2.39178 0.382991
\(40\) 0 0
\(41\) 3.16461 0.494228 0.247114 0.968986i \(-0.420518\pi\)
0.247114 + 0.968986i \(0.420518\pi\)
\(42\) 0 0
\(43\) 2.16101 0.329551 0.164776 0.986331i \(-0.447310\pi\)
0.164776 + 0.986331i \(0.447310\pi\)
\(44\) 0 0
\(45\) −8.38129 −1.24941
\(46\) 0 0
\(47\) 9.95256 1.45173 0.725865 0.687837i \(-0.241440\pi\)
0.725865 + 0.687837i \(0.241440\pi\)
\(48\) 0 0
\(49\) −2.08828 −0.298325
\(50\) 0 0
\(51\) −3.54182 −0.495954
\(52\) 0 0
\(53\) −6.66810 −0.915935 −0.457967 0.888969i \(-0.651422\pi\)
−0.457967 + 0.888969i \(0.651422\pi\)
\(54\) 0 0
\(55\) −15.0914 −2.03492
\(56\) 0 0
\(57\) 0.458177 0.0606871
\(58\) 0 0
\(59\) 3.89126 0.506599 0.253299 0.967388i \(-0.418484\pi\)
0.253299 + 0.967388i \(0.418484\pi\)
\(60\) 0 0
\(61\) −7.05259 −0.902991 −0.451496 0.892273i \(-0.649109\pi\)
−0.451496 + 0.892273i \(0.649109\pi\)
\(62\) 0 0
\(63\) −6.18347 −0.779044
\(64\) 0 0
\(65\) −15.6813 −1.94503
\(66\) 0 0
\(67\) −13.2767 −1.62201 −0.811004 0.585041i \(-0.801078\pi\)
−0.811004 + 0.585041i \(0.801078\pi\)
\(68\) 0 0
\(69\) 2.98554 0.359417
\(70\) 0 0
\(71\) −5.72840 −0.679836 −0.339918 0.940455i \(-0.610399\pi\)
−0.339918 + 0.940455i \(0.610399\pi\)
\(72\) 0 0
\(73\) −5.89754 −0.690254 −0.345127 0.938556i \(-0.612164\pi\)
−0.345127 + 0.938556i \(0.612164\pi\)
\(74\) 0 0
\(75\) −1.84362 −0.212883
\(76\) 0 0
\(77\) −11.1340 −1.26884
\(78\) 0 0
\(79\) −15.9369 −1.79304 −0.896522 0.442998i \(-0.853915\pi\)
−0.896522 + 0.442998i \(0.853915\pi\)
\(80\) 0 0
\(81\) 7.15473 0.794970
\(82\) 0 0
\(83\) −8.71734 −0.956852 −0.478426 0.878128i \(-0.658793\pi\)
−0.478426 + 0.878128i \(0.658793\pi\)
\(84\) 0 0
\(85\) 23.2214 2.51871
\(86\) 0 0
\(87\) −1.02431 −0.109818
\(88\) 0 0
\(89\) −3.79938 −0.402734 −0.201367 0.979516i \(-0.564538\pi\)
−0.201367 + 0.979516i \(0.564538\pi\)
\(90\) 0 0
\(91\) −11.5692 −1.21279
\(92\) 0 0
\(93\) 1.52833 0.158480
\(94\) 0 0
\(95\) −3.00397 −0.308201
\(96\) 0 0
\(97\) −3.94645 −0.400701 −0.200351 0.979724i \(-0.564208\pi\)
−0.200351 + 0.979724i \(0.564208\pi\)
\(98\) 0 0
\(99\) 14.0168 1.40874
\(100\) 0 0
\(101\) 6.32025 0.628888 0.314444 0.949276i \(-0.398182\pi\)
0.314444 + 0.949276i \(0.398182\pi\)
\(102\) 0 0
\(103\) 4.71201 0.464288 0.232144 0.972681i \(-0.425426\pi\)
0.232144 + 0.972681i \(0.425426\pi\)
\(104\) 0 0
\(105\) −3.05032 −0.297681
\(106\) 0 0
\(107\) −18.5550 −1.79378 −0.896892 0.442249i \(-0.854181\pi\)
−0.896892 + 0.442249i \(0.854181\pi\)
\(108\) 0 0
\(109\) 9.25995 0.886942 0.443471 0.896289i \(-0.353747\pi\)
0.443471 + 0.896289i \(0.353747\pi\)
\(110\) 0 0
\(111\) −0.665309 −0.0631483
\(112\) 0 0
\(113\) −7.97386 −0.750118 −0.375059 0.927001i \(-0.622378\pi\)
−0.375059 + 0.927001i \(0.622378\pi\)
\(114\) 0 0
\(115\) −19.5742 −1.82531
\(116\) 0 0
\(117\) 14.5648 1.34651
\(118\) 0 0
\(119\) 17.1321 1.57050
\(120\) 0 0
\(121\) 14.2387 1.29443
\(122\) 0 0
\(123\) −1.44995 −0.130738
\(124\) 0 0
\(125\) −2.93242 −0.262284
\(126\) 0 0
\(127\) −8.32912 −0.739090 −0.369545 0.929213i \(-0.620486\pi\)
−0.369545 + 0.929213i \(0.620486\pi\)
\(128\) 0 0
\(129\) −0.990126 −0.0871758
\(130\) 0 0
\(131\) −16.3831 −1.43140 −0.715700 0.698408i \(-0.753892\pi\)
−0.715700 + 0.698408i \(0.753892\pi\)
\(132\) 0 0
\(133\) −2.21624 −0.192172
\(134\) 0 0
\(135\) 7.96916 0.685876
\(136\) 0 0
\(137\) 16.2267 1.38634 0.693172 0.720772i \(-0.256212\pi\)
0.693172 + 0.720772i \(0.256212\pi\)
\(138\) 0 0
\(139\) −4.38904 −0.372273 −0.186137 0.982524i \(-0.559597\pi\)
−0.186137 + 0.982524i \(0.559597\pi\)
\(140\) 0 0
\(141\) −4.56004 −0.384024
\(142\) 0 0
\(143\) 26.2254 2.19308
\(144\) 0 0
\(145\) 6.71574 0.557712
\(146\) 0 0
\(147\) 0.956802 0.0789157
\(148\) 0 0
\(149\) 1.28542 0.105306 0.0526528 0.998613i \(-0.483232\pi\)
0.0526528 + 0.998613i \(0.483232\pi\)
\(150\) 0 0
\(151\) −9.07295 −0.738346 −0.369173 0.929361i \(-0.620359\pi\)
−0.369173 + 0.929361i \(0.620359\pi\)
\(152\) 0 0
\(153\) −21.5680 −1.74367
\(154\) 0 0
\(155\) −10.0202 −0.804844
\(156\) 0 0
\(157\) 3.13656 0.250325 0.125162 0.992136i \(-0.460055\pi\)
0.125162 + 0.992136i \(0.460055\pi\)
\(158\) 0 0
\(159\) 3.05517 0.242291
\(160\) 0 0
\(161\) −14.4413 −1.13813
\(162\) 0 0
\(163\) −14.9640 −1.17207 −0.586035 0.810286i \(-0.699312\pi\)
−0.586035 + 0.810286i \(0.699312\pi\)
\(164\) 0 0
\(165\) 6.91453 0.538295
\(166\) 0 0
\(167\) 21.0799 1.63121 0.815607 0.578606i \(-0.196403\pi\)
0.815607 + 0.578606i \(0.196403\pi\)
\(168\) 0 0
\(169\) 14.2506 1.09620
\(170\) 0 0
\(171\) 2.79007 0.213362
\(172\) 0 0
\(173\) −23.7808 −1.80802 −0.904011 0.427510i \(-0.859391\pi\)
−0.904011 + 0.427510i \(0.859391\pi\)
\(174\) 0 0
\(175\) 8.91775 0.674118
\(176\) 0 0
\(177\) −1.78289 −0.134010
\(178\) 0 0
\(179\) −14.3018 −1.06897 −0.534483 0.845179i \(-0.679494\pi\)
−0.534483 + 0.845179i \(0.679494\pi\)
\(180\) 0 0
\(181\) −13.9014 −1.03328 −0.516640 0.856203i \(-0.672818\pi\)
−0.516640 + 0.856203i \(0.672818\pi\)
\(182\) 0 0
\(183\) 3.23133 0.238867
\(184\) 0 0
\(185\) 4.36199 0.320700
\(186\) 0 0
\(187\) −38.8353 −2.83992
\(188\) 0 0
\(189\) 5.87942 0.427665
\(190\) 0 0
\(191\) 2.17747 0.157557 0.0787783 0.996892i \(-0.474898\pi\)
0.0787783 + 0.996892i \(0.474898\pi\)
\(192\) 0 0
\(193\) 26.6410 1.91766 0.958831 0.283978i \(-0.0916541\pi\)
0.958831 + 0.283978i \(0.0916541\pi\)
\(194\) 0 0
\(195\) 7.18483 0.514516
\(196\) 0 0
\(197\) 20.8500 1.48550 0.742749 0.669570i \(-0.233521\pi\)
0.742749 + 0.669570i \(0.233521\pi\)
\(198\) 0 0
\(199\) −4.24710 −0.301069 −0.150535 0.988605i \(-0.548099\pi\)
−0.150535 + 0.988605i \(0.548099\pi\)
\(200\) 0 0
\(201\) 6.08308 0.429068
\(202\) 0 0
\(203\) 4.95468 0.347750
\(204\) 0 0
\(205\) 9.50637 0.663954
\(206\) 0 0
\(207\) 18.1805 1.26363
\(208\) 0 0
\(209\) 5.02382 0.347505
\(210\) 0 0
\(211\) 18.1134 1.24698 0.623488 0.781833i \(-0.285715\pi\)
0.623488 + 0.781833i \(0.285715\pi\)
\(212\) 0 0
\(213\) 2.62462 0.179836
\(214\) 0 0
\(215\) 6.49161 0.442724
\(216\) 0 0
\(217\) −7.39263 −0.501845
\(218\) 0 0
\(219\) 2.70212 0.182592
\(220\) 0 0
\(221\) −40.3535 −2.71447
\(222\) 0 0
\(223\) −3.71952 −0.249078 −0.124539 0.992215i \(-0.539745\pi\)
−0.124539 + 0.992215i \(0.539745\pi\)
\(224\) 0 0
\(225\) −11.2267 −0.748450
\(226\) 0 0
\(227\) 17.8318 1.18354 0.591769 0.806108i \(-0.298430\pi\)
0.591769 + 0.806108i \(0.298430\pi\)
\(228\) 0 0
\(229\) 13.5159 0.893158 0.446579 0.894744i \(-0.352642\pi\)
0.446579 + 0.894744i \(0.352642\pi\)
\(230\) 0 0
\(231\) 5.10134 0.335644
\(232\) 0 0
\(233\) −18.5061 −1.21238 −0.606189 0.795321i \(-0.707302\pi\)
−0.606189 + 0.795321i \(0.707302\pi\)
\(234\) 0 0
\(235\) 29.8972 1.95028
\(236\) 0 0
\(237\) 7.30194 0.474312
\(238\) 0 0
\(239\) 11.6874 0.755995 0.377997 0.925807i \(-0.376613\pi\)
0.377997 + 0.925807i \(0.376613\pi\)
\(240\) 0 0
\(241\) 10.2701 0.661554 0.330777 0.943709i \(-0.392689\pi\)
0.330777 + 0.943709i \(0.392689\pi\)
\(242\) 0 0
\(243\) −11.2368 −0.720839
\(244\) 0 0
\(245\) −6.27312 −0.400775
\(246\) 0 0
\(247\) 5.22021 0.332154
\(248\) 0 0
\(249\) 3.99409 0.253115
\(250\) 0 0
\(251\) −25.1064 −1.58470 −0.792350 0.610066i \(-0.791143\pi\)
−0.792350 + 0.610066i \(0.791143\pi\)
\(252\) 0 0
\(253\) 32.7358 2.05808
\(254\) 0 0
\(255\) −10.6395 −0.666273
\(256\) 0 0
\(257\) 12.5905 0.785374 0.392687 0.919672i \(-0.371546\pi\)
0.392687 + 0.919672i \(0.371546\pi\)
\(258\) 0 0
\(259\) 3.21815 0.199966
\(260\) 0 0
\(261\) −6.23755 −0.386095
\(262\) 0 0
\(263\) −22.7749 −1.40436 −0.702181 0.711999i \(-0.747790\pi\)
−0.702181 + 0.711999i \(0.747790\pi\)
\(264\) 0 0
\(265\) −20.0308 −1.23048
\(266\) 0 0
\(267\) 1.74079 0.106535
\(268\) 0 0
\(269\) 17.0271 1.03816 0.519081 0.854725i \(-0.326274\pi\)
0.519081 + 0.854725i \(0.326274\pi\)
\(270\) 0 0
\(271\) 8.19591 0.497866 0.248933 0.968521i \(-0.419920\pi\)
0.248933 + 0.968521i \(0.419920\pi\)
\(272\) 0 0
\(273\) 5.30076 0.320817
\(274\) 0 0
\(275\) −20.2149 −1.21901
\(276\) 0 0
\(277\) −18.7174 −1.12462 −0.562309 0.826927i \(-0.690087\pi\)
−0.562309 + 0.826927i \(0.690087\pi\)
\(278\) 0 0
\(279\) 9.30675 0.557180
\(280\) 0 0
\(281\) 0.189116 0.0112817 0.00564087 0.999984i \(-0.498204\pi\)
0.00564087 + 0.999984i \(0.498204\pi\)
\(282\) 0 0
\(283\) −10.6357 −0.632226 −0.316113 0.948722i \(-0.602378\pi\)
−0.316113 + 0.948722i \(0.602378\pi\)
\(284\) 0 0
\(285\) 1.37635 0.0815279
\(286\) 0 0
\(287\) 7.01353 0.413995
\(288\) 0 0
\(289\) 42.7567 2.51510
\(290\) 0 0
\(291\) 1.80817 0.105997
\(292\) 0 0
\(293\) −27.6765 −1.61688 −0.808439 0.588580i \(-0.799687\pi\)
−0.808439 + 0.588580i \(0.799687\pi\)
\(294\) 0 0
\(295\) 11.6892 0.680572
\(296\) 0 0
\(297\) −13.3276 −0.773345
\(298\) 0 0
\(299\) 34.0155 1.96717
\(300\) 0 0
\(301\) 4.78932 0.276052
\(302\) 0 0
\(303\) −2.89579 −0.166359
\(304\) 0 0
\(305\) −21.1857 −1.21309
\(306\) 0 0
\(307\) −8.14707 −0.464978 −0.232489 0.972599i \(-0.574687\pi\)
−0.232489 + 0.972599i \(0.574687\pi\)
\(308\) 0 0
\(309\) −2.15894 −0.122818
\(310\) 0 0
\(311\) 11.1283 0.631030 0.315515 0.948921i \(-0.397823\pi\)
0.315515 + 0.948921i \(0.397823\pi\)
\(312\) 0 0
\(313\) −10.2647 −0.580198 −0.290099 0.956997i \(-0.593688\pi\)
−0.290099 + 0.956997i \(0.593688\pi\)
\(314\) 0 0
\(315\) −18.5750 −1.04658
\(316\) 0 0
\(317\) 7.14866 0.401509 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(318\) 0 0
\(319\) −11.2314 −0.628836
\(320\) 0 0
\(321\) 8.50150 0.474508
\(322\) 0 0
\(323\) −7.73025 −0.430122
\(324\) 0 0
\(325\) −21.0052 −1.16516
\(326\) 0 0
\(327\) −4.24270 −0.234622
\(328\) 0 0
\(329\) 22.0573 1.21606
\(330\) 0 0
\(331\) 28.4370 1.56304 0.781519 0.623881i \(-0.214445\pi\)
0.781519 + 0.623881i \(0.214445\pi\)
\(332\) 0 0
\(333\) −4.05140 −0.222016
\(334\) 0 0
\(335\) −39.8828 −2.17903
\(336\) 0 0
\(337\) −18.9197 −1.03062 −0.515311 0.857003i \(-0.672324\pi\)
−0.515311 + 0.857003i \(0.672324\pi\)
\(338\) 0 0
\(339\) 3.65344 0.198428
\(340\) 0 0
\(341\) 16.7578 0.907484
\(342\) 0 0
\(343\) −20.1418 −1.08756
\(344\) 0 0
\(345\) 8.96847 0.482846
\(346\) 0 0
\(347\) −2.34140 −0.125693 −0.0628466 0.998023i \(-0.520018\pi\)
−0.0628466 + 0.998023i \(0.520018\pi\)
\(348\) 0 0
\(349\) −14.8506 −0.794935 −0.397468 0.917616i \(-0.630111\pi\)
−0.397468 + 0.917616i \(0.630111\pi\)
\(350\) 0 0
\(351\) −13.8486 −0.739183
\(352\) 0 0
\(353\) −16.9690 −0.903168 −0.451584 0.892229i \(-0.649141\pi\)
−0.451584 + 0.892229i \(0.649141\pi\)
\(354\) 0 0
\(355\) −17.2079 −0.913302
\(356\) 0 0
\(357\) −7.84953 −0.415441
\(358\) 0 0
\(359\) 32.2107 1.70002 0.850008 0.526770i \(-0.176597\pi\)
0.850008 + 0.526770i \(0.176597\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −6.52387 −0.342414
\(364\) 0 0
\(365\) −17.7160 −0.927298
\(366\) 0 0
\(367\) 10.2366 0.534347 0.267173 0.963648i \(-0.413910\pi\)
0.267173 + 0.963648i \(0.413910\pi\)
\(368\) 0 0
\(369\) −8.82948 −0.459644
\(370\) 0 0
\(371\) −14.7781 −0.767242
\(372\) 0 0
\(373\) −30.1908 −1.56322 −0.781610 0.623767i \(-0.785601\pi\)
−0.781610 + 0.623767i \(0.785601\pi\)
\(374\) 0 0
\(375\) 1.34357 0.0693815
\(376\) 0 0
\(377\) −11.6704 −0.601057
\(378\) 0 0
\(379\) 15.9865 0.821173 0.410586 0.911822i \(-0.365324\pi\)
0.410586 + 0.911822i \(0.365324\pi\)
\(380\) 0 0
\(381\) 3.81622 0.195511
\(382\) 0 0
\(383\) 22.6961 1.15971 0.579857 0.814718i \(-0.303108\pi\)
0.579857 + 0.814718i \(0.303108\pi\)
\(384\) 0 0
\(385\) −33.4461 −1.70457
\(386\) 0 0
\(387\) −6.02938 −0.306491
\(388\) 0 0
\(389\) 12.1956 0.618341 0.309170 0.951007i \(-0.399949\pi\)
0.309170 + 0.951007i \(0.399949\pi\)
\(390\) 0 0
\(391\) −50.3713 −2.54738
\(392\) 0 0
\(393\) 7.50638 0.378647
\(394\) 0 0
\(395\) −47.8740 −2.40880
\(396\) 0 0
\(397\) 10.7422 0.539137 0.269568 0.962981i \(-0.413119\pi\)
0.269568 + 0.962981i \(0.413119\pi\)
\(398\) 0 0
\(399\) 1.01543 0.0508351
\(400\) 0 0
\(401\) −5.65983 −0.282639 −0.141319 0.989964i \(-0.545134\pi\)
−0.141319 + 0.989964i \(0.545134\pi\)
\(402\) 0 0
\(403\) 17.4129 0.867396
\(404\) 0 0
\(405\) 21.4926 1.06797
\(406\) 0 0
\(407\) −7.29497 −0.361598
\(408\) 0 0
\(409\) −27.6645 −1.36792 −0.683960 0.729520i \(-0.739744\pi\)
−0.683960 + 0.729520i \(0.739744\pi\)
\(410\) 0 0
\(411\) −7.43473 −0.366728
\(412\) 0 0
\(413\) 8.62396 0.424357
\(414\) 0 0
\(415\) −26.1866 −1.28545
\(416\) 0 0
\(417\) 2.01096 0.0984770
\(418\) 0 0
\(419\) 6.91763 0.337948 0.168974 0.985620i \(-0.445955\pi\)
0.168974 + 0.985620i \(0.445955\pi\)
\(420\) 0 0
\(421\) −3.07015 −0.149630 −0.0748150 0.997197i \(-0.523837\pi\)
−0.0748150 + 0.997197i \(0.523837\pi\)
\(422\) 0 0
\(423\) −27.7684 −1.35014
\(424\) 0 0
\(425\) 31.1051 1.50882
\(426\) 0 0
\(427\) −15.6302 −0.756400
\(428\) 0 0
\(429\) −12.0159 −0.580132
\(430\) 0 0
\(431\) −13.2015 −0.635893 −0.317946 0.948109i \(-0.602993\pi\)
−0.317946 + 0.948109i \(0.602993\pi\)
\(432\) 0 0
\(433\) −20.1544 −0.968559 −0.484279 0.874913i \(-0.660918\pi\)
−0.484279 + 0.874913i \(0.660918\pi\)
\(434\) 0 0
\(435\) −3.07700 −0.147531
\(436\) 0 0
\(437\) 6.51613 0.311709
\(438\) 0 0
\(439\) 1.72144 0.0821601 0.0410800 0.999156i \(-0.486920\pi\)
0.0410800 + 0.999156i \(0.486920\pi\)
\(440\) 0 0
\(441\) 5.82645 0.277450
\(442\) 0 0
\(443\) −38.0168 −1.80623 −0.903117 0.429395i \(-0.858727\pi\)
−0.903117 + 0.429395i \(0.858727\pi\)
\(444\) 0 0
\(445\) −11.4132 −0.541039
\(446\) 0 0
\(447\) −0.588950 −0.0278564
\(448\) 0 0
\(449\) −27.1861 −1.28299 −0.641496 0.767127i \(-0.721686\pi\)
−0.641496 + 0.767127i \(0.721686\pi\)
\(450\) 0 0
\(451\) −15.8984 −0.748626
\(452\) 0 0
\(453\) 4.15702 0.195314
\(454\) 0 0
\(455\) −34.7536 −1.62927
\(456\) 0 0
\(457\) 15.8069 0.739415 0.369708 0.929148i \(-0.379458\pi\)
0.369708 + 0.929148i \(0.379458\pi\)
\(458\) 0 0
\(459\) 20.5074 0.957204
\(460\) 0 0
\(461\) 29.7741 1.38672 0.693358 0.720593i \(-0.256130\pi\)
0.693358 + 0.720593i \(0.256130\pi\)
\(462\) 0 0
\(463\) 11.8339 0.549967 0.274984 0.961449i \(-0.411328\pi\)
0.274984 + 0.961449i \(0.411328\pi\)
\(464\) 0 0
\(465\) 4.59104 0.212904
\(466\) 0 0
\(467\) −30.6052 −1.41624 −0.708121 0.706091i \(-0.750457\pi\)
−0.708121 + 0.706091i \(0.750457\pi\)
\(468\) 0 0
\(469\) −29.4244 −1.35869
\(470\) 0 0
\(471\) −1.43710 −0.0662181
\(472\) 0 0
\(473\) −10.8565 −0.499184
\(474\) 0 0
\(475\) −4.02382 −0.184625
\(476\) 0 0
\(477\) 18.6045 0.851842
\(478\) 0 0
\(479\) 12.3908 0.566150 0.283075 0.959098i \(-0.408645\pi\)
0.283075 + 0.959098i \(0.408645\pi\)
\(480\) 0 0
\(481\) −7.58015 −0.345625
\(482\) 0 0
\(483\) 6.61668 0.301069
\(484\) 0 0
\(485\) −11.8550 −0.538308
\(486\) 0 0
\(487\) 7.61617 0.345122 0.172561 0.984999i \(-0.444796\pi\)
0.172561 + 0.984999i \(0.444796\pi\)
\(488\) 0 0
\(489\) 6.85616 0.310046
\(490\) 0 0
\(491\) 3.09291 0.139581 0.0697906 0.997562i \(-0.477767\pi\)
0.0697906 + 0.997562i \(0.477767\pi\)
\(492\) 0 0
\(493\) 17.2819 0.778338
\(494\) 0 0
\(495\) 42.1061 1.89253
\(496\) 0 0
\(497\) −12.6955 −0.569472
\(498\) 0 0
\(499\) 6.55042 0.293237 0.146618 0.989193i \(-0.453161\pi\)
0.146618 + 0.989193i \(0.453161\pi\)
\(500\) 0 0
\(501\) −9.65835 −0.431503
\(502\) 0 0
\(503\) 34.9332 1.55760 0.778798 0.627275i \(-0.215830\pi\)
0.778798 + 0.627275i \(0.215830\pi\)
\(504\) 0 0
\(505\) 18.9858 0.844858
\(506\) 0 0
\(507\) −6.52928 −0.289976
\(508\) 0 0
\(509\) −10.6461 −0.471882 −0.235941 0.971767i \(-0.575817\pi\)
−0.235941 + 0.971767i \(0.575817\pi\)
\(510\) 0 0
\(511\) −13.0704 −0.578199
\(512\) 0 0
\(513\) −2.65288 −0.117128
\(514\) 0 0
\(515\) 14.1547 0.623732
\(516\) 0 0
\(517\) −49.9999 −2.19899
\(518\) 0 0
\(519\) 10.8958 0.478274
\(520\) 0 0
\(521\) −33.7087 −1.47681 −0.738403 0.674359i \(-0.764420\pi\)
−0.738403 + 0.674359i \(0.764420\pi\)
\(522\) 0 0
\(523\) 3.61224 0.157952 0.0789761 0.996877i \(-0.474835\pi\)
0.0789761 + 0.996877i \(0.474835\pi\)
\(524\) 0 0
\(525\) −4.08591 −0.178324
\(526\) 0 0
\(527\) −25.7855 −1.12323
\(528\) 0 0
\(529\) 19.4599 0.846084
\(530\) 0 0
\(531\) −10.8569 −0.471149
\(532\) 0 0
\(533\) −16.5199 −0.715556
\(534\) 0 0
\(535\) −55.7388 −2.40980
\(536\) 0 0
\(537\) 6.55276 0.282772
\(538\) 0 0
\(539\) 10.4911 0.451885
\(540\) 0 0
\(541\) 45.1835 1.94259 0.971295 0.237876i \(-0.0764513\pi\)
0.971295 + 0.237876i \(0.0764513\pi\)
\(542\) 0 0
\(543\) 6.36929 0.273332
\(544\) 0 0
\(545\) 27.8166 1.19153
\(546\) 0 0
\(547\) 25.9716 1.11047 0.555233 0.831695i \(-0.312629\pi\)
0.555233 + 0.831695i \(0.312629\pi\)
\(548\) 0 0
\(549\) 19.6772 0.839804
\(550\) 0 0
\(551\) −2.23562 −0.0952408
\(552\) 0 0
\(553\) −35.3201 −1.50196
\(554\) 0 0
\(555\) −1.99857 −0.0848344
\(556\) 0 0
\(557\) 27.5515 1.16739 0.583697 0.811972i \(-0.301606\pi\)
0.583697 + 0.811972i \(0.301606\pi\)
\(558\) 0 0
\(559\) −11.2809 −0.477132
\(560\) 0 0
\(561\) 17.7935 0.751241
\(562\) 0 0
\(563\) 14.8531 0.625985 0.312992 0.949756i \(-0.398669\pi\)
0.312992 + 0.949756i \(0.398669\pi\)
\(564\) 0 0
\(565\) −23.9532 −1.00772
\(566\) 0 0
\(567\) 15.8566 0.665915
\(568\) 0 0
\(569\) −4.86281 −0.203859 −0.101930 0.994792i \(-0.532502\pi\)
−0.101930 + 0.994792i \(0.532502\pi\)
\(570\) 0 0
\(571\) −21.6960 −0.907948 −0.453974 0.891015i \(-0.649994\pi\)
−0.453974 + 0.891015i \(0.649994\pi\)
\(572\) 0 0
\(573\) −0.997669 −0.0416782
\(574\) 0 0
\(575\) −26.2197 −1.09344
\(576\) 0 0
\(577\) 28.5083 1.18682 0.593409 0.804901i \(-0.297782\pi\)
0.593409 + 0.804901i \(0.297782\pi\)
\(578\) 0 0
\(579\) −12.2063 −0.507277
\(580\) 0 0
\(581\) −19.3197 −0.801517
\(582\) 0 0
\(583\) 33.4993 1.38740
\(584\) 0 0
\(585\) 43.7521 1.80893
\(586\) 0 0
\(587\) 22.5133 0.929224 0.464612 0.885514i \(-0.346194\pi\)
0.464612 + 0.885514i \(0.346194\pi\)
\(588\) 0 0
\(589\) 3.33566 0.137444
\(590\) 0 0
\(591\) −9.55298 −0.392957
\(592\) 0 0
\(593\) −11.0209 −0.452574 −0.226287 0.974061i \(-0.572659\pi\)
−0.226287 + 0.974061i \(0.572659\pi\)
\(594\) 0 0
\(595\) 51.4642 2.10983
\(596\) 0 0
\(597\) 1.94593 0.0796414
\(598\) 0 0
\(599\) −18.7308 −0.765319 −0.382659 0.923890i \(-0.624992\pi\)
−0.382659 + 0.923890i \(0.624992\pi\)
\(600\) 0 0
\(601\) 14.8859 0.607209 0.303604 0.952798i \(-0.401810\pi\)
0.303604 + 0.952798i \(0.401810\pi\)
\(602\) 0 0
\(603\) 37.0430 1.50851
\(604\) 0 0
\(605\) 42.7727 1.73896
\(606\) 0 0
\(607\) 37.9701 1.54116 0.770579 0.637345i \(-0.219967\pi\)
0.770579 + 0.637345i \(0.219967\pi\)
\(608\) 0 0
\(609\) −2.27012 −0.0919900
\(610\) 0 0
\(611\) −51.9544 −2.10185
\(612\) 0 0
\(613\) 28.5379 1.15264 0.576318 0.817226i \(-0.304489\pi\)
0.576318 + 0.817226i \(0.304489\pi\)
\(614\) 0 0
\(615\) −4.35560 −0.175635
\(616\) 0 0
\(617\) 3.91835 0.157747 0.0788733 0.996885i \(-0.474868\pi\)
0.0788733 + 0.996885i \(0.474868\pi\)
\(618\) 0 0
\(619\) 5.37059 0.215862 0.107931 0.994158i \(-0.465577\pi\)
0.107931 + 0.994158i \(0.465577\pi\)
\(620\) 0 0
\(621\) −17.2865 −0.693684
\(622\) 0 0
\(623\) −8.42035 −0.337354
\(624\) 0 0
\(625\) −28.9280 −1.15712
\(626\) 0 0
\(627\) −2.30180 −0.0919250
\(628\) 0 0
\(629\) 11.2249 0.447567
\(630\) 0 0
\(631\) 18.6483 0.742377 0.371189 0.928557i \(-0.378950\pi\)
0.371189 + 0.928557i \(0.378950\pi\)
\(632\) 0 0
\(633\) −8.29913 −0.329861
\(634\) 0 0
\(635\) −25.0204 −0.992905
\(636\) 0 0
\(637\) 10.9012 0.431923
\(638\) 0 0
\(639\) 15.9827 0.632264
\(640\) 0 0
\(641\) 42.2857 1.67018 0.835092 0.550111i \(-0.185415\pi\)
0.835092 + 0.550111i \(0.185415\pi\)
\(642\) 0 0
\(643\) 18.4426 0.727305 0.363652 0.931535i \(-0.381530\pi\)
0.363652 + 0.931535i \(0.381530\pi\)
\(644\) 0 0
\(645\) −2.97431 −0.117113
\(646\) 0 0
\(647\) 2.19012 0.0861024 0.0430512 0.999073i \(-0.486292\pi\)
0.0430512 + 0.999073i \(0.486292\pi\)
\(648\) 0 0
\(649\) −19.5490 −0.767364
\(650\) 0 0
\(651\) 3.38714 0.132752
\(652\) 0 0
\(653\) 43.8208 1.71484 0.857420 0.514617i \(-0.172066\pi\)
0.857420 + 0.514617i \(0.172066\pi\)
\(654\) 0 0
\(655\) −49.2144 −1.92296
\(656\) 0 0
\(657\) 16.4546 0.641954
\(658\) 0 0
\(659\) 11.0297 0.429657 0.214829 0.976652i \(-0.431081\pi\)
0.214829 + 0.976652i \(0.431081\pi\)
\(660\) 0 0
\(661\) 7.49137 0.291381 0.145690 0.989330i \(-0.453460\pi\)
0.145690 + 0.989330i \(0.453460\pi\)
\(662\) 0 0
\(663\) 18.4890 0.718055
\(664\) 0 0
\(665\) −6.65751 −0.258167
\(666\) 0 0
\(667\) −14.5676 −0.564060
\(668\) 0 0
\(669\) 1.70420 0.0658882
\(670\) 0 0
\(671\) 35.4309 1.36780
\(672\) 0 0
\(673\) −34.6195 −1.33448 −0.667242 0.744841i \(-0.732525\pi\)
−0.667242 + 0.744841i \(0.732525\pi\)
\(674\) 0 0
\(675\) 10.6747 0.410870
\(676\) 0 0
\(677\) −29.0201 −1.11533 −0.557666 0.830066i \(-0.688303\pi\)
−0.557666 + 0.830066i \(0.688303\pi\)
\(678\) 0 0
\(679\) −8.74629 −0.335652
\(680\) 0 0
\(681\) −8.17012 −0.313080
\(682\) 0 0
\(683\) 13.3799 0.511966 0.255983 0.966681i \(-0.417601\pi\)
0.255983 + 0.966681i \(0.417601\pi\)
\(684\) 0 0
\(685\) 48.7446 1.86244
\(686\) 0 0
\(687\) −6.19269 −0.236266
\(688\) 0 0
\(689\) 34.8089 1.32611
\(690\) 0 0
\(691\) 40.7415 1.54988 0.774940 0.632035i \(-0.217780\pi\)
0.774940 + 0.632035i \(0.217780\pi\)
\(692\) 0 0
\(693\) 31.0646 1.18005
\(694\) 0 0
\(695\) −13.1845 −0.500118
\(696\) 0 0
\(697\) 24.4632 0.926609
\(698\) 0 0
\(699\) 8.47909 0.320709
\(700\) 0 0
\(701\) −43.7760 −1.65340 −0.826698 0.562645i \(-0.809784\pi\)
−0.826698 + 0.562645i \(0.809784\pi\)
\(702\) 0 0
\(703\) −1.45208 −0.0547662
\(704\) 0 0
\(705\) −13.6982 −0.515904
\(706\) 0 0
\(707\) 14.0072 0.526795
\(708\) 0 0
\(709\) 26.0155 0.977031 0.488516 0.872555i \(-0.337538\pi\)
0.488516 + 0.872555i \(0.337538\pi\)
\(710\) 0 0
\(711\) 44.4652 1.66758
\(712\) 0 0
\(713\) 21.7356 0.814005
\(714\) 0 0
\(715\) 78.7801 2.94621
\(716\) 0 0
\(717\) −5.35490 −0.199982
\(718\) 0 0
\(719\) −31.3051 −1.16748 −0.583741 0.811940i \(-0.698412\pi\)
−0.583741 + 0.811940i \(0.698412\pi\)
\(720\) 0 0
\(721\) 10.4430 0.388916
\(722\) 0 0
\(723\) −4.70552 −0.175000
\(724\) 0 0
\(725\) 8.99574 0.334093
\(726\) 0 0
\(727\) 7.31403 0.271262 0.135631 0.990759i \(-0.456694\pi\)
0.135631 + 0.990759i \(0.456694\pi\)
\(728\) 0 0
\(729\) −16.3158 −0.604287
\(730\) 0 0
\(731\) 16.7051 0.617862
\(732\) 0 0
\(733\) −13.0935 −0.483620 −0.241810 0.970324i \(-0.577741\pi\)
−0.241810 + 0.970324i \(0.577741\pi\)
\(734\) 0 0
\(735\) 2.87420 0.106016
\(736\) 0 0
\(737\) 66.6997 2.45692
\(738\) 0 0
\(739\) 10.5782 0.389124 0.194562 0.980890i \(-0.437672\pi\)
0.194562 + 0.980890i \(0.437672\pi\)
\(740\) 0 0
\(741\) −2.39178 −0.0878642
\(742\) 0 0
\(743\) 23.6180 0.866460 0.433230 0.901283i \(-0.357374\pi\)
0.433230 + 0.901283i \(0.357374\pi\)
\(744\) 0 0
\(745\) 3.86136 0.141469
\(746\) 0 0
\(747\) 24.3220 0.889896
\(748\) 0 0
\(749\) −41.1225 −1.50258
\(750\) 0 0
\(751\) −24.5994 −0.897644 −0.448822 0.893621i \(-0.648156\pi\)
−0.448822 + 0.893621i \(0.648156\pi\)
\(752\) 0 0
\(753\) 11.5032 0.419199
\(754\) 0 0
\(755\) −27.2548 −0.991905
\(756\) 0 0
\(757\) 3.03483 0.110303 0.0551514 0.998478i \(-0.482436\pi\)
0.0551514 + 0.998478i \(0.482436\pi\)
\(758\) 0 0
\(759\) −14.9988 −0.544423
\(760\) 0 0
\(761\) −8.15330 −0.295557 −0.147778 0.989020i \(-0.547212\pi\)
−0.147778 + 0.989020i \(0.547212\pi\)
\(762\) 0 0
\(763\) 20.5223 0.742956
\(764\) 0 0
\(765\) −64.7894 −2.34247
\(766\) 0 0
\(767\) −20.3132 −0.733466
\(768\) 0 0
\(769\) −14.2744 −0.514747 −0.257374 0.966312i \(-0.582857\pi\)
−0.257374 + 0.966312i \(0.582857\pi\)
\(770\) 0 0
\(771\) −5.76868 −0.207754
\(772\) 0 0
\(773\) −45.2404 −1.62718 −0.813591 0.581437i \(-0.802491\pi\)
−0.813591 + 0.581437i \(0.802491\pi\)
\(774\) 0 0
\(775\) −13.4221 −0.482136
\(776\) 0 0
\(777\) −1.47448 −0.0528968
\(778\) 0 0
\(779\) −3.16461 −0.113384
\(780\) 0 0
\(781\) 28.7784 1.02977
\(782\) 0 0
\(783\) 5.93084 0.211951
\(784\) 0 0
\(785\) 9.42212 0.336290
\(786\) 0 0
\(787\) −50.1656 −1.78821 −0.894106 0.447856i \(-0.852188\pi\)
−0.894106 + 0.447856i \(0.852188\pi\)
\(788\) 0 0
\(789\) 10.4349 0.371494
\(790\) 0 0
\(791\) −17.6720 −0.628344
\(792\) 0 0
\(793\) 36.8160 1.30737
\(794\) 0 0
\(795\) 9.17764 0.325497
\(796\) 0 0
\(797\) −21.7396 −0.770056 −0.385028 0.922905i \(-0.625808\pi\)
−0.385028 + 0.922905i \(0.625808\pi\)
\(798\) 0 0
\(799\) 76.9357 2.72179
\(800\) 0 0
\(801\) 10.6006 0.374552
\(802\) 0 0
\(803\) 29.6281 1.04555
\(804\) 0 0
\(805\) −43.3812 −1.52899
\(806\) 0 0
\(807\) −7.80144 −0.274624
\(808\) 0 0
\(809\) −9.48695 −0.333543 −0.166772 0.985996i \(-0.553334\pi\)
−0.166772 + 0.985996i \(0.553334\pi\)
\(810\) 0 0
\(811\) 22.6529 0.795450 0.397725 0.917505i \(-0.369800\pi\)
0.397725 + 0.917505i \(0.369800\pi\)
\(812\) 0 0
\(813\) −3.75518 −0.131700
\(814\) 0 0
\(815\) −44.9513 −1.57458
\(816\) 0 0
\(817\) −2.16101 −0.0756042
\(818\) 0 0
\(819\) 32.2790 1.12792
\(820\) 0 0
\(821\) 26.4552 0.923292 0.461646 0.887064i \(-0.347259\pi\)
0.461646 + 0.887064i \(0.347259\pi\)
\(822\) 0 0
\(823\) −35.1129 −1.22396 −0.611980 0.790873i \(-0.709627\pi\)
−0.611980 + 0.790873i \(0.709627\pi\)
\(824\) 0 0
\(825\) 9.26202 0.322462
\(826\) 0 0
\(827\) −27.4003 −0.952803 −0.476402 0.879228i \(-0.658059\pi\)
−0.476402 + 0.879228i \(0.658059\pi\)
\(828\) 0 0
\(829\) 25.2213 0.875972 0.437986 0.898982i \(-0.355692\pi\)
0.437986 + 0.898982i \(0.355692\pi\)
\(830\) 0 0
\(831\) 8.57588 0.297494
\(832\) 0 0
\(833\) −16.1429 −0.559319
\(834\) 0 0
\(835\) 63.3234 2.19140
\(836\) 0 0
\(837\) −8.84912 −0.305870
\(838\) 0 0
\(839\) −36.8869 −1.27348 −0.636738 0.771080i \(-0.719717\pi\)
−0.636738 + 0.771080i \(0.719717\pi\)
\(840\) 0 0
\(841\) −24.0020 −0.827655
\(842\) 0 0
\(843\) −0.0866488 −0.00298434
\(844\) 0 0
\(845\) 42.8082 1.47265
\(846\) 0 0
\(847\) 31.5565 1.08429
\(848\) 0 0
\(849\) 4.87303 0.167242
\(850\) 0 0
\(851\) −9.46193 −0.324351
\(852\) 0 0
\(853\) 3.36746 0.115300 0.0576499 0.998337i \(-0.481639\pi\)
0.0576499 + 0.998337i \(0.481639\pi\)
\(854\) 0 0
\(855\) 8.38129 0.286634
\(856\) 0 0
\(857\) −16.0416 −0.547970 −0.273985 0.961734i \(-0.588342\pi\)
−0.273985 + 0.961734i \(0.588342\pi\)
\(858\) 0 0
\(859\) −14.7310 −0.502614 −0.251307 0.967907i \(-0.580860\pi\)
−0.251307 + 0.967907i \(0.580860\pi\)
\(860\) 0 0
\(861\) −3.21344 −0.109514
\(862\) 0 0
\(863\) 41.3034 1.40598 0.702991 0.711198i \(-0.251847\pi\)
0.702991 + 0.711198i \(0.251847\pi\)
\(864\) 0 0
\(865\) −71.4368 −2.42892
\(866\) 0 0
\(867\) −19.5901 −0.665316
\(868\) 0 0
\(869\) 80.0642 2.71599
\(870\) 0 0
\(871\) 69.3071 2.34838
\(872\) 0 0
\(873\) 11.0109 0.372662
\(874\) 0 0
\(875\) −6.49895 −0.219704
\(876\) 0 0
\(877\) −3.84438 −0.129815 −0.0649077 0.997891i \(-0.520675\pi\)
−0.0649077 + 0.997891i \(0.520675\pi\)
\(878\) 0 0
\(879\) 12.6807 0.427711
\(880\) 0 0
\(881\) −4.57677 −0.154195 −0.0770976 0.997024i \(-0.524565\pi\)
−0.0770976 + 0.997024i \(0.524565\pi\)
\(882\) 0 0
\(883\) 2.34140 0.0787945 0.0393972 0.999224i \(-0.487456\pi\)
0.0393972 + 0.999224i \(0.487456\pi\)
\(884\) 0 0
\(885\) −5.35573 −0.180031
\(886\) 0 0
\(887\) 48.1958 1.61826 0.809129 0.587631i \(-0.199939\pi\)
0.809129 + 0.587631i \(0.199939\pi\)
\(888\) 0 0
\(889\) −18.4593 −0.619106
\(890\) 0 0
\(891\) −35.9441 −1.20417
\(892\) 0 0
\(893\) −9.95256 −0.333050
\(894\) 0 0
\(895\) −42.9621 −1.43607
\(896\) 0 0
\(897\) −15.5851 −0.520373
\(898\) 0 0
\(899\) −7.45729 −0.248714
\(900\) 0 0
\(901\) −51.5461 −1.71725
\(902\) 0 0
\(903\) −2.19436 −0.0730237
\(904\) 0 0
\(905\) −41.7593 −1.38812
\(906\) 0 0
\(907\) 4.08133 0.135518 0.0677592 0.997702i \(-0.478415\pi\)
0.0677592 + 0.997702i \(0.478415\pi\)
\(908\) 0 0
\(909\) −17.6340 −0.584881
\(910\) 0 0
\(911\) 54.1174 1.79299 0.896495 0.443054i \(-0.146105\pi\)
0.896495 + 0.443054i \(0.146105\pi\)
\(912\) 0 0
\(913\) 43.7943 1.44938
\(914\) 0 0
\(915\) 9.70682 0.320898
\(916\) 0 0
\(917\) −36.3089 −1.19903
\(918\) 0 0
\(919\) 32.1252 1.05971 0.529857 0.848087i \(-0.322246\pi\)
0.529857 + 0.848087i \(0.322246\pi\)
\(920\) 0 0
\(921\) 3.73280 0.123000
\(922\) 0 0
\(923\) 29.9034 0.984284
\(924\) 0 0
\(925\) 5.84290 0.192113
\(926\) 0 0
\(927\) −13.1469 −0.431800
\(928\) 0 0
\(929\) −10.9339 −0.358730 −0.179365 0.983783i \(-0.557404\pi\)
−0.179365 + 0.983783i \(0.557404\pi\)
\(930\) 0 0
\(931\) 2.08828 0.0684406
\(932\) 0 0
\(933\) −5.09875 −0.166926
\(934\) 0 0
\(935\) −116.660 −3.81519
\(936\) 0 0
\(937\) −46.5172 −1.51965 −0.759825 0.650127i \(-0.774716\pi\)
−0.759825 + 0.650127i \(0.774716\pi\)
\(938\) 0 0
\(939\) 4.70307 0.153479
\(940\) 0 0
\(941\) −39.9678 −1.30291 −0.651457 0.758686i \(-0.725842\pi\)
−0.651457 + 0.758686i \(0.725842\pi\)
\(942\) 0 0
\(943\) −20.6210 −0.671511
\(944\) 0 0
\(945\) 17.6616 0.574531
\(946\) 0 0
\(947\) −33.7011 −1.09514 −0.547569 0.836761i \(-0.684447\pi\)
−0.547569 + 0.836761i \(0.684447\pi\)
\(948\) 0 0
\(949\) 30.7864 0.999368
\(950\) 0 0
\(951\) −3.27535 −0.106211
\(952\) 0 0
\(953\) −30.1472 −0.976564 −0.488282 0.872686i \(-0.662376\pi\)
−0.488282 + 0.872686i \(0.662376\pi\)
\(954\) 0 0
\(955\) 6.54106 0.211664
\(956\) 0 0
\(957\) 5.14596 0.166345
\(958\) 0 0
\(959\) 35.9624 1.16129
\(960\) 0 0
\(961\) −19.8733 −0.641076
\(962\) 0 0
\(963\) 51.7700 1.66826
\(964\) 0 0
\(965\) 80.0287 2.57622
\(966\) 0 0
\(967\) 35.1010 1.12877 0.564385 0.825511i \(-0.309113\pi\)
0.564385 + 0.825511i \(0.309113\pi\)
\(968\) 0 0
\(969\) 3.54182 0.113780
\(970\) 0 0
\(971\) 38.1884 1.22552 0.612762 0.790268i \(-0.290059\pi\)
0.612762 + 0.790268i \(0.290059\pi\)
\(972\) 0 0
\(973\) −9.72716 −0.311839
\(974\) 0 0
\(975\) 9.62409 0.308217
\(976\) 0 0
\(977\) 22.6596 0.724946 0.362473 0.931994i \(-0.381933\pi\)
0.362473 + 0.931994i \(0.381933\pi\)
\(978\) 0 0
\(979\) 19.0874 0.610036
\(980\) 0 0
\(981\) −25.8359 −0.824878
\(982\) 0 0
\(983\) 1.47180 0.0469430 0.0234715 0.999725i \(-0.492528\pi\)
0.0234715 + 0.999725i \(0.492528\pi\)
\(984\) 0 0
\(985\) 62.6326 1.99564
\(986\) 0 0
\(987\) −10.1061 −0.321682
\(988\) 0 0
\(989\) −14.0814 −0.447763
\(990\) 0 0
\(991\) −30.6228 −0.972767 −0.486383 0.873746i \(-0.661684\pi\)
−0.486383 + 0.873746i \(0.661684\pi\)
\(992\) 0 0
\(993\) −13.0292 −0.413469
\(994\) 0 0
\(995\) −12.7582 −0.404461
\(996\) 0 0
\(997\) −7.61232 −0.241085 −0.120542 0.992708i \(-0.538463\pi\)
−0.120542 + 0.992708i \(0.538463\pi\)
\(998\) 0 0
\(999\) 3.85219 0.121878
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 4864.2.a.bs.1.6 10
4.3 odd 2 4864.2.a.bt.1.6 10
8.3 odd 2 inner 4864.2.a.bs.1.5 10
8.5 even 2 4864.2.a.bt.1.5 10
16.3 odd 4 2432.2.c.j.1217.11 yes 20
16.5 even 4 2432.2.c.j.1217.12 yes 20
16.11 odd 4 2432.2.c.j.1217.10 yes 20
16.13 even 4 2432.2.c.j.1217.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.9 20 16.13 even 4
2432.2.c.j.1217.10 yes 20 16.11 odd 4
2432.2.c.j.1217.11 yes 20 16.3 odd 4
2432.2.c.j.1217.12 yes 20 16.5 even 4
4864.2.a.bs.1.5 10 8.3 odd 2 inner
4864.2.a.bs.1.6 10 1.1 even 1 trivial
4864.2.a.bt.1.5 10 8.5 even 2
4864.2.a.bt.1.6 10 4.3 odd 2