Properties

Label 4864.2.a.bm.1.7
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: 8.8.34309996544.1
Defining polynomial: \(x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.904430\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.06644 q^{3} -1.45635 q^{5} -0.196723 q^{7} +1.27016 q^{9} +O(q^{10})\) \(q+2.06644 q^{3} -1.45635 q^{5} -0.196723 q^{7} +1.27016 q^{9} -3.27016 q^{11} +6.72211 q^{13} -3.00945 q^{15} -3.82843 q^{17} +1.00000 q^{19} -0.406516 q^{21} -6.72211 q^{23} -2.87905 q^{25} -3.57461 q^{27} -4.26907 q^{29} +5.92215 q^{31} -6.75758 q^{33} +0.286497 q^{35} +10.1782 q^{37} +13.8908 q^{39} +7.16502 q^{41} -10.4194 q^{43} -1.84979 q^{45} -3.61269 q^{47} -6.96130 q^{49} -7.91120 q^{51} -8.52508 q^{53} +4.76249 q^{55} +2.06644 q^{57} -0.137287 q^{59} -10.3879 q^{61} -0.249870 q^{63} -9.78973 q^{65} -2.22167 q^{67} -13.8908 q^{69} -8.11856 q^{71} -2.01634 q^{73} -5.94938 q^{75} +0.643316 q^{77} +13.6473 q^{79} -11.1972 q^{81} +1.14921 q^{83} +5.57552 q^{85} -8.82177 q^{87} -6.11706 q^{89} -1.32239 q^{91} +12.2377 q^{93} -1.45635 q^{95} -0.132873 q^{97} -4.15362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100}) \)

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
\(3\) 2.06644 1.19306 0.596529 0.802592i \(-0.296546\pi\)
0.596529 + 0.802592i \(0.296546\pi\)
\(4\) 0 0
\(5\) −1.45635 −0.651299 −0.325649 0.945491i \(-0.605583\pi\)
−0.325649 + 0.945491i \(0.605583\pi\)
\(6\) 0 0
\(7\) −0.196723 −0.0743544 −0.0371772 0.999309i \(-0.511837\pi\)
−0.0371772 + 0.999309i \(0.511837\pi\)
\(8\) 0 0
\(9\) 1.27016 0.423387
\(10\) 0 0
\(11\) −3.27016 −0.985990 −0.492995 0.870032i \(-0.664098\pi\)
−0.492995 + 0.870032i \(0.664098\pi\)
\(12\) 0 0
\(13\) 6.72211 1.86438 0.932189 0.361973i \(-0.117897\pi\)
0.932189 + 0.361973i \(0.117897\pi\)
\(14\) 0 0
\(15\) −3.00945 −0.777037
\(16\) 0 0
\(17\) −3.82843 −0.928530 −0.464265 0.885696i \(-0.653681\pi\)
−0.464265 + 0.885696i \(0.653681\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.406516 −0.0887090
\(22\) 0 0
\(23\) −6.72211 −1.40166 −0.700828 0.713330i \(-0.747186\pi\)
−0.700828 + 0.713330i \(0.747186\pi\)
\(24\) 0 0
\(25\) −2.87905 −0.575810
\(26\) 0 0
\(27\) −3.57461 −0.687933
\(28\) 0 0
\(29\) −4.26907 −0.792747 −0.396374 0.918089i \(-0.629732\pi\)
−0.396374 + 0.918089i \(0.629732\pi\)
\(30\) 0 0
\(31\) 5.92215 1.06365 0.531824 0.846855i \(-0.321507\pi\)
0.531824 + 0.846855i \(0.321507\pi\)
\(32\) 0 0
\(33\) −6.75758 −1.17634
\(34\) 0 0
\(35\) 0.286497 0.0484269
\(36\) 0 0
\(37\) 10.1782 1.67328 0.836639 0.547755i \(-0.184517\pi\)
0.836639 + 0.547755i \(0.184517\pi\)
\(38\) 0 0
\(39\) 13.8908 2.22431
\(40\) 0 0
\(41\) 7.16502 1.11899 0.559494 0.828834i \(-0.310995\pi\)
0.559494 + 0.828834i \(0.310995\pi\)
\(42\) 0 0
\(43\) −10.4194 −1.58894 −0.794470 0.607304i \(-0.792251\pi\)
−0.794470 + 0.607304i \(0.792251\pi\)
\(44\) 0 0
\(45\) −1.84979 −0.275751
\(46\) 0 0
\(47\) −3.61269 −0.526965 −0.263482 0.964664i \(-0.584871\pi\)
−0.263482 + 0.964664i \(0.584871\pi\)
\(48\) 0 0
\(49\) −6.96130 −0.994471
\(50\) 0 0
\(51\) −7.91120 −1.10779
\(52\) 0 0
\(53\) −8.52508 −1.17101 −0.585505 0.810669i \(-0.699104\pi\)
−0.585505 + 0.810669i \(0.699104\pi\)
\(54\) 0 0
\(55\) 4.76249 0.642174
\(56\) 0 0
\(57\) 2.06644 0.273706
\(58\) 0 0
\(59\) −0.137287 −0.0178732 −0.00893660 0.999960i \(-0.502845\pi\)
−0.00893660 + 0.999960i \(0.502845\pi\)
\(60\) 0 0
\(61\) −10.3879 −1.33004 −0.665020 0.746826i \(-0.731577\pi\)
−0.665020 + 0.746826i \(0.731577\pi\)
\(62\) 0 0
\(63\) −0.249870 −0.0314806
\(64\) 0 0
\(65\) −9.78973 −1.21427
\(66\) 0 0
\(67\) −2.22167 −0.271420 −0.135710 0.990749i \(-0.543332\pi\)
−0.135710 + 0.990749i \(0.543332\pi\)
\(68\) 0 0
\(69\) −13.8908 −1.67226
\(70\) 0 0
\(71\) −8.11856 −0.963496 −0.481748 0.876310i \(-0.659998\pi\)
−0.481748 + 0.876310i \(0.659998\pi\)
\(72\) 0 0
\(73\) −2.01634 −0.235994 −0.117997 0.993014i \(-0.537647\pi\)
−0.117997 + 0.993014i \(0.537647\pi\)
\(74\) 0 0
\(75\) −5.94938 −0.686975
\(76\) 0 0
\(77\) 0.643316 0.0733127
\(78\) 0 0
\(79\) 13.6473 1.53544 0.767719 0.640787i \(-0.221392\pi\)
0.767719 + 0.640787i \(0.221392\pi\)
\(80\) 0 0
\(81\) −11.1972 −1.24413
\(82\) 0 0
\(83\) 1.14921 0.126142 0.0630711 0.998009i \(-0.479911\pi\)
0.0630711 + 0.998009i \(0.479911\pi\)
\(84\) 0 0
\(85\) 5.57552 0.604750
\(86\) 0 0
\(87\) −8.82177 −0.945793
\(88\) 0 0
\(89\) −6.11706 −0.648407 −0.324204 0.945987i \(-0.605096\pi\)
−0.324204 + 0.945987i \(0.605096\pi\)
\(90\) 0 0
\(91\) −1.32239 −0.138625
\(92\) 0 0
\(93\) 12.2377 1.26899
\(94\) 0 0
\(95\) −1.45635 −0.149418
\(96\) 0 0
\(97\) −0.132873 −0.0134912 −0.00674560 0.999977i \(-0.502147\pi\)
−0.00674560 + 0.999977i \(0.502147\pi\)
\(98\) 0 0
\(99\) −4.15362 −0.417455
\(100\) 0 0
\(101\) 7.81876 0.777996 0.388998 0.921239i \(-0.372821\pi\)
0.388998 + 0.921239i \(0.372821\pi\)
\(102\) 0 0
\(103\) 19.6631 1.93746 0.968729 0.248121i \(-0.0798130\pi\)
0.968729 + 0.248121i \(0.0798130\pi\)
\(104\) 0 0
\(105\) 0.592028 0.0577761
\(106\) 0 0
\(107\) −3.18952 −0.308343 −0.154171 0.988044i \(-0.549271\pi\)
−0.154171 + 0.988044i \(0.549271\pi\)
\(108\) 0 0
\(109\) −5.44941 −0.521959 −0.260980 0.965344i \(-0.584046\pi\)
−0.260980 + 0.965344i \(0.584046\pi\)
\(110\) 0 0
\(111\) 21.0325 1.99632
\(112\) 0 0
\(113\) −12.6890 −1.19368 −0.596841 0.802360i \(-0.703578\pi\)
−0.596841 + 0.802360i \(0.703578\pi\)
\(114\) 0 0
\(115\) 9.78973 0.912897
\(116\) 0 0
\(117\) 8.53815 0.789352
\(118\) 0 0
\(119\) 0.753140 0.0690403
\(120\) 0 0
\(121\) −0.306056 −0.0278233
\(122\) 0 0
\(123\) 14.8061 1.33502
\(124\) 0 0
\(125\) 11.4746 1.02632
\(126\) 0 0
\(127\) −8.83484 −0.783965 −0.391983 0.919973i \(-0.628211\pi\)
−0.391983 + 0.919973i \(0.628211\pi\)
\(128\) 0 0
\(129\) −21.5310 −1.89570
\(130\) 0 0
\(131\) −21.1285 −1.84600 −0.923001 0.384798i \(-0.874271\pi\)
−0.923001 + 0.384798i \(0.874271\pi\)
\(132\) 0 0
\(133\) −0.196723 −0.0170581
\(134\) 0 0
\(135\) 5.20587 0.448050
\(136\) 0 0
\(137\) 12.3464 1.05482 0.527411 0.849610i \(-0.323163\pi\)
0.527411 + 0.849610i \(0.323163\pi\)
\(138\) 0 0
\(139\) −9.84638 −0.835159 −0.417579 0.908640i \(-0.637121\pi\)
−0.417579 + 0.908640i \(0.637121\pi\)
\(140\) 0 0
\(141\) −7.46539 −0.628699
\(142\) 0 0
\(143\) −21.9824 −1.83826
\(144\) 0 0
\(145\) 6.21726 0.516315
\(146\) 0 0
\(147\) −14.3851 −1.18646
\(148\) 0 0
\(149\) 1.06290 0.0870763 0.0435381 0.999052i \(-0.486137\pi\)
0.0435381 + 0.999052i \(0.486137\pi\)
\(150\) 0 0
\(151\) −2.91270 −0.237032 −0.118516 0.992952i \(-0.537814\pi\)
−0.118516 + 0.992952i \(0.537814\pi\)
\(152\) 0 0
\(153\) −4.86271 −0.393127
\(154\) 0 0
\(155\) −8.62470 −0.692753
\(156\) 0 0
\(157\) 8.63179 0.688892 0.344446 0.938806i \(-0.388067\pi\)
0.344446 + 0.938806i \(0.388067\pi\)
\(158\) 0 0
\(159\) −17.6165 −1.39708
\(160\) 0 0
\(161\) 1.32239 0.104219
\(162\) 0 0
\(163\) 1.45541 0.113996 0.0569982 0.998374i \(-0.481847\pi\)
0.0569982 + 0.998374i \(0.481847\pi\)
\(164\) 0 0
\(165\) 9.84138 0.766151
\(166\) 0 0
\(167\) −14.5539 −1.12622 −0.563109 0.826383i \(-0.690395\pi\)
−0.563109 + 0.826383i \(0.690395\pi\)
\(168\) 0 0
\(169\) 32.1867 2.47590
\(170\) 0 0
\(171\) 1.27016 0.0971315
\(172\) 0 0
\(173\) −17.7002 −1.34572 −0.672861 0.739769i \(-0.734935\pi\)
−0.672861 + 0.739769i \(0.734935\pi\)
\(174\) 0 0
\(175\) 0.566376 0.0428140
\(176\) 0 0
\(177\) −0.283694 −0.0213238
\(178\) 0 0
\(179\) −11.5725 −0.864967 −0.432483 0.901642i \(-0.642363\pi\)
−0.432483 + 0.901642i \(0.642363\pi\)
\(180\) 0 0
\(181\) −3.95620 −0.294062 −0.147031 0.989132i \(-0.546972\pi\)
−0.147031 + 0.989132i \(0.546972\pi\)
\(182\) 0 0
\(183\) −21.4660 −1.58681
\(184\) 0 0
\(185\) −14.8229 −1.08980
\(186\) 0 0
\(187\) 12.5196 0.915521
\(188\) 0 0
\(189\) 0.703208 0.0511508
\(190\) 0 0
\(191\) −4.02874 −0.291510 −0.145755 0.989321i \(-0.546561\pi\)
−0.145755 + 0.989321i \(0.546561\pi\)
\(192\) 0 0
\(193\) 13.2979 0.957203 0.478602 0.878032i \(-0.341144\pi\)
0.478602 + 0.878032i \(0.341144\pi\)
\(194\) 0 0
\(195\) −20.2298 −1.44869
\(196\) 0 0
\(197\) 6.61228 0.471106 0.235553 0.971862i \(-0.424310\pi\)
0.235553 + 0.971862i \(0.424310\pi\)
\(198\) 0 0
\(199\) −22.3159 −1.58193 −0.790967 0.611859i \(-0.790422\pi\)
−0.790967 + 0.611859i \(0.790422\pi\)
\(200\) 0 0
\(201\) −4.59094 −0.323820
\(202\) 0 0
\(203\) 0.839826 0.0589442
\(204\) 0 0
\(205\) −10.4348 −0.728796
\(206\) 0 0
\(207\) −8.53815 −0.593442
\(208\) 0 0
\(209\) −3.27016 −0.226202
\(210\) 0 0
\(211\) −15.2315 −1.04858 −0.524288 0.851541i \(-0.675669\pi\)
−0.524288 + 0.851541i \(0.675669\pi\)
\(212\) 0 0
\(213\) −16.7765 −1.14951
\(214\) 0 0
\(215\) 15.1742 1.03487
\(216\) 0 0
\(217\) −1.16502 −0.0790869
\(218\) 0 0
\(219\) −4.16663 −0.281555
\(220\) 0 0
\(221\) −25.7351 −1.73113
\(222\) 0 0
\(223\) −1.00343 −0.0671947 −0.0335973 0.999435i \(-0.510696\pi\)
−0.0335973 + 0.999435i \(0.510696\pi\)
\(224\) 0 0
\(225\) −3.65685 −0.243790
\(226\) 0 0
\(227\) −8.20586 −0.544642 −0.272321 0.962206i \(-0.587791\pi\)
−0.272321 + 0.962206i \(0.587791\pi\)
\(228\) 0 0
\(229\) −25.4321 −1.68060 −0.840300 0.542122i \(-0.817621\pi\)
−0.840300 + 0.542122i \(0.817621\pi\)
\(230\) 0 0
\(231\) 1.32937 0.0874662
\(232\) 0 0
\(233\) 17.8702 1.17072 0.585359 0.810774i \(-0.300954\pi\)
0.585359 + 0.810774i \(0.300954\pi\)
\(234\) 0 0
\(235\) 5.26133 0.343211
\(236\) 0 0
\(237\) 28.2012 1.83186
\(238\) 0 0
\(239\) 5.17212 0.334556 0.167278 0.985910i \(-0.446502\pi\)
0.167278 + 0.985910i \(0.446502\pi\)
\(240\) 0 0
\(241\) 24.9189 1.60516 0.802582 0.596542i \(-0.203459\pi\)
0.802582 + 0.596542i \(0.203459\pi\)
\(242\) 0 0
\(243\) −12.4144 −0.796386
\(244\) 0 0
\(245\) 10.1381 0.647698
\(246\) 0 0
\(247\) 6.72211 0.427717
\(248\) 0 0
\(249\) 2.37477 0.150495
\(250\) 0 0
\(251\) −13.4673 −0.850051 −0.425025 0.905181i \(-0.639735\pi\)
−0.425025 + 0.905181i \(0.639735\pi\)
\(252\) 0 0
\(253\) 21.9824 1.38202
\(254\) 0 0
\(255\) 11.5215 0.721502
\(256\) 0 0
\(257\) 7.55666 0.471371 0.235686 0.971829i \(-0.424266\pi\)
0.235686 + 0.971829i \(0.424266\pi\)
\(258\) 0 0
\(259\) −2.00228 −0.124415
\(260\) 0 0
\(261\) −5.42241 −0.335639
\(262\) 0 0
\(263\) −14.3441 −0.884498 −0.442249 0.896892i \(-0.645819\pi\)
−0.442249 + 0.896892i \(0.645819\pi\)
\(264\) 0 0
\(265\) 12.4155 0.762677
\(266\) 0 0
\(267\) −12.6405 −0.773587
\(268\) 0 0
\(269\) 22.1161 1.34844 0.674221 0.738530i \(-0.264480\pi\)
0.674221 + 0.738530i \(0.264480\pi\)
\(270\) 0 0
\(271\) −5.51563 −0.335051 −0.167525 0.985868i \(-0.553578\pi\)
−0.167525 + 0.985868i \(0.553578\pi\)
\(272\) 0 0
\(273\) −2.73264 −0.165387
\(274\) 0 0
\(275\) 9.41496 0.567743
\(276\) 0 0
\(277\) −14.4135 −0.866022 −0.433011 0.901389i \(-0.642549\pi\)
−0.433011 + 0.901389i \(0.642549\pi\)
\(278\) 0 0
\(279\) 7.52207 0.450335
\(280\) 0 0
\(281\) 18.4956 1.10335 0.551677 0.834058i \(-0.313988\pi\)
0.551677 + 0.834058i \(0.313988\pi\)
\(282\) 0 0
\(283\) 4.83004 0.287116 0.143558 0.989642i \(-0.454146\pi\)
0.143558 + 0.989642i \(0.454146\pi\)
\(284\) 0 0
\(285\) −3.00945 −0.178264
\(286\) 0 0
\(287\) −1.40953 −0.0832017
\(288\) 0 0
\(289\) −2.34315 −0.137832
\(290\) 0 0
\(291\) −0.274573 −0.0160958
\(292\) 0 0
\(293\) 1.94022 0.113349 0.0566745 0.998393i \(-0.481950\pi\)
0.0566745 + 0.998393i \(0.481950\pi\)
\(294\) 0 0
\(295\) 0.199937 0.0116408
\(296\) 0 0
\(297\) 11.6895 0.678295
\(298\) 0 0
\(299\) −45.1867 −2.61322
\(300\) 0 0
\(301\) 2.04973 0.118145
\(302\) 0 0
\(303\) 16.1570 0.928194
\(304\) 0 0
\(305\) 15.1285 0.866253
\(306\) 0 0
\(307\) 13.7053 0.782205 0.391103 0.920347i \(-0.372094\pi\)
0.391103 + 0.920347i \(0.372094\pi\)
\(308\) 0 0
\(309\) 40.6325 2.31150
\(310\) 0 0
\(311\) 29.8349 1.69178 0.845890 0.533357i \(-0.179070\pi\)
0.845890 + 0.533357i \(0.179070\pi\)
\(312\) 0 0
\(313\) −20.2244 −1.14315 −0.571575 0.820550i \(-0.693667\pi\)
−0.571575 + 0.820550i \(0.693667\pi\)
\(314\) 0 0
\(315\) 0.363897 0.0205033
\(316\) 0 0
\(317\) −13.6037 −0.764057 −0.382029 0.924150i \(-0.624774\pi\)
−0.382029 + 0.924150i \(0.624774\pi\)
\(318\) 0 0
\(319\) 13.9606 0.781641
\(320\) 0 0
\(321\) −6.59094 −0.367871
\(322\) 0 0
\(323\) −3.82843 −0.213019
\(324\) 0 0
\(325\) −19.3533 −1.07353
\(326\) 0 0
\(327\) −11.2609 −0.622727
\(328\) 0 0
\(329\) 0.710700 0.0391821
\(330\) 0 0
\(331\) −6.66501 −0.366342 −0.183171 0.983081i \(-0.558636\pi\)
−0.183171 + 0.983081i \(0.558636\pi\)
\(332\) 0 0
\(333\) 12.9279 0.708443
\(334\) 0 0
\(335\) 3.23553 0.176776
\(336\) 0 0
\(337\) 33.8420 1.84349 0.921745 0.387796i \(-0.126764\pi\)
0.921745 + 0.387796i \(0.126764\pi\)
\(338\) 0 0
\(339\) −26.2210 −1.42413
\(340\) 0 0
\(341\) −19.3664 −1.04875
\(342\) 0 0
\(343\) 2.74651 0.148298
\(344\) 0 0
\(345\) 20.2298 1.08914
\(346\) 0 0
\(347\) −4.69716 −0.252157 −0.126079 0.992020i \(-0.540239\pi\)
−0.126079 + 0.992020i \(0.540239\pi\)
\(348\) 0 0
\(349\) 4.27540 0.228857 0.114428 0.993431i \(-0.463496\pi\)
0.114428 + 0.993431i \(0.463496\pi\)
\(350\) 0 0
\(351\) −24.0289 −1.28257
\(352\) 0 0
\(353\) 4.82401 0.256756 0.128378 0.991725i \(-0.459023\pi\)
0.128378 + 0.991725i \(0.459023\pi\)
\(354\) 0 0
\(355\) 11.8235 0.627524
\(356\) 0 0
\(357\) 1.55632 0.0823690
\(358\) 0 0
\(359\) 5.52237 0.291460 0.145730 0.989324i \(-0.453447\pi\)
0.145730 + 0.989324i \(0.453447\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.632446 −0.0331948
\(364\) 0 0
\(365\) 2.93649 0.153703
\(366\) 0 0
\(367\) 10.1750 0.531133 0.265566 0.964093i \(-0.414441\pi\)
0.265566 + 0.964093i \(0.414441\pi\)
\(368\) 0 0
\(369\) 9.10072 0.473765
\(370\) 0 0
\(371\) 1.67708 0.0870697
\(372\) 0 0
\(373\) 0.640102 0.0331432 0.0165716 0.999863i \(-0.494725\pi\)
0.0165716 + 0.999863i \(0.494725\pi\)
\(374\) 0 0
\(375\) 23.7116 1.22446
\(376\) 0 0
\(377\) −28.6972 −1.47798
\(378\) 0 0
\(379\) −32.9406 −1.69204 −0.846022 0.533149i \(-0.821009\pi\)
−0.846022 + 0.533149i \(0.821009\pi\)
\(380\) 0 0
\(381\) −18.2566 −0.935316
\(382\) 0 0
\(383\) −8.11545 −0.414680 −0.207340 0.978269i \(-0.566481\pi\)
−0.207340 + 0.978269i \(0.566481\pi\)
\(384\) 0 0
\(385\) −0.936892 −0.0477484
\(386\) 0 0
\(387\) −13.2343 −0.672735
\(388\) 0 0
\(389\) 32.5702 1.65138 0.825689 0.564126i \(-0.190787\pi\)
0.825689 + 0.564126i \(0.190787\pi\)
\(390\) 0 0
\(391\) 25.7351 1.30148
\(392\) 0 0
\(393\) −43.6606 −2.20239
\(394\) 0 0
\(395\) −19.8752 −1.00003
\(396\) 0 0
\(397\) −27.6380 −1.38711 −0.693557 0.720402i \(-0.743957\pi\)
−0.693557 + 0.720402i \(0.743957\pi\)
\(398\) 0 0
\(399\) −0.406516 −0.0203512
\(400\) 0 0
\(401\) 37.4219 1.86876 0.934381 0.356275i \(-0.115953\pi\)
0.934381 + 0.356275i \(0.115953\pi\)
\(402\) 0 0
\(403\) 39.8093 1.98304
\(404\) 0 0
\(405\) 16.3070 0.810300
\(406\) 0 0
\(407\) −33.2842 −1.64984
\(408\) 0 0
\(409\) −26.5114 −1.31090 −0.655452 0.755237i \(-0.727522\pi\)
−0.655452 + 0.755237i \(0.727522\pi\)
\(410\) 0 0
\(411\) 25.5130 1.25846
\(412\) 0 0
\(413\) 0.0270075 0.00132895
\(414\) 0 0
\(415\) −1.67365 −0.0821563
\(416\) 0 0
\(417\) −20.3469 −0.996392
\(418\) 0 0
\(419\) 29.3006 1.43143 0.715714 0.698393i \(-0.246101\pi\)
0.715714 + 0.698393i \(0.246101\pi\)
\(420\) 0 0
\(421\) 4.10610 0.200119 0.100060 0.994981i \(-0.468097\pi\)
0.100060 + 0.994981i \(0.468097\pi\)
\(422\) 0 0
\(423\) −4.58869 −0.223110
\(424\) 0 0
\(425\) 11.0222 0.534657
\(426\) 0 0
\(427\) 2.04355 0.0988943
\(428\) 0 0
\(429\) −45.4252 −2.19315
\(430\) 0 0
\(431\) 17.0502 0.821277 0.410639 0.911798i \(-0.365306\pi\)
0.410639 + 0.911798i \(0.365306\pi\)
\(432\) 0 0
\(433\) 16.1356 0.775427 0.387713 0.921780i \(-0.373265\pi\)
0.387713 + 0.921780i \(0.373265\pi\)
\(434\) 0 0
\(435\) 12.8476 0.615994
\(436\) 0 0
\(437\) −6.72211 −0.321562
\(438\) 0 0
\(439\) −37.3232 −1.78134 −0.890670 0.454651i \(-0.849764\pi\)
−0.890670 + 0.454651i \(0.849764\pi\)
\(440\) 0 0
\(441\) −8.84196 −0.421046
\(442\) 0 0
\(443\) −11.2254 −0.533337 −0.266668 0.963788i \(-0.585923\pi\)
−0.266668 + 0.963788i \(0.585923\pi\)
\(444\) 0 0
\(445\) 8.90857 0.422307
\(446\) 0 0
\(447\) 2.19642 0.103887
\(448\) 0 0
\(449\) −15.3791 −0.725783 −0.362891 0.931831i \(-0.618210\pi\)
−0.362891 + 0.931831i \(0.618210\pi\)
\(450\) 0 0
\(451\) −23.4308 −1.10331
\(452\) 0 0
\(453\) −6.01890 −0.282793
\(454\) 0 0
\(455\) 1.92587 0.0902860
\(456\) 0 0
\(457\) −24.8093 −1.16053 −0.580265 0.814428i \(-0.697051\pi\)
−0.580265 + 0.814428i \(0.697051\pi\)
\(458\) 0 0
\(459\) 13.6851 0.638767
\(460\) 0 0
\(461\) 5.74289 0.267473 0.133737 0.991017i \(-0.457302\pi\)
0.133737 + 0.991017i \(0.457302\pi\)
\(462\) 0 0
\(463\) 23.5494 1.09443 0.547217 0.836991i \(-0.315687\pi\)
0.547217 + 0.836991i \(0.315687\pi\)
\(464\) 0 0
\(465\) −17.8224 −0.826494
\(466\) 0 0
\(467\) 26.2375 1.21413 0.607063 0.794653i \(-0.292347\pi\)
0.607063 + 0.794653i \(0.292347\pi\)
\(468\) 0 0
\(469\) 0.437054 0.0201813
\(470\) 0 0
\(471\) 17.8371 0.821888
\(472\) 0 0
\(473\) 34.0730 1.56668
\(474\) 0 0
\(475\) −2.87905 −0.132100
\(476\) 0 0
\(477\) −10.8282 −0.495790
\(478\) 0 0
\(479\) 19.5262 0.892176 0.446088 0.894989i \(-0.352817\pi\)
0.446088 + 0.894989i \(0.352817\pi\)
\(480\) 0 0
\(481\) 68.4186 3.11962
\(482\) 0 0
\(483\) 2.73264 0.124340
\(484\) 0 0
\(485\) 0.193509 0.00878680
\(486\) 0 0
\(487\) 27.5206 1.24708 0.623539 0.781792i \(-0.285694\pi\)
0.623539 + 0.781792i \(0.285694\pi\)
\(488\) 0 0
\(489\) 3.00751 0.136004
\(490\) 0 0
\(491\) 0.870365 0.0392790 0.0196395 0.999807i \(-0.493748\pi\)
0.0196395 + 0.999807i \(0.493748\pi\)
\(492\) 0 0
\(493\) 16.3438 0.736090
\(494\) 0 0
\(495\) 6.04912 0.271888
\(496\) 0 0
\(497\) 1.59711 0.0716402
\(498\) 0 0
\(499\) −1.37036 −0.0613456 −0.0306728 0.999529i \(-0.509765\pi\)
−0.0306728 + 0.999529i \(0.509765\pi\)
\(500\) 0 0
\(501\) −30.0748 −1.34364
\(502\) 0 0
\(503\) 22.2354 0.991430 0.495715 0.868485i \(-0.334906\pi\)
0.495715 + 0.868485i \(0.334906\pi\)
\(504\) 0 0
\(505\) −11.3868 −0.506708
\(506\) 0 0
\(507\) 66.5118 2.95389
\(508\) 0 0
\(509\) 19.0391 0.843895 0.421947 0.906620i \(-0.361347\pi\)
0.421947 + 0.906620i \(0.361347\pi\)
\(510\) 0 0
\(511\) 0.396660 0.0175472
\(512\) 0 0
\(513\) −3.57461 −0.157823
\(514\) 0 0
\(515\) −28.6362 −1.26186
\(516\) 0 0
\(517\) 11.8141 0.519582
\(518\) 0 0
\(519\) −36.5764 −1.60552
\(520\) 0 0
\(521\) 17.8425 0.781694 0.390847 0.920456i \(-0.372182\pi\)
0.390847 + 0.920456i \(0.372182\pi\)
\(522\) 0 0
\(523\) −39.9928 −1.74876 −0.874382 0.485239i \(-0.838733\pi\)
−0.874382 + 0.485239i \(0.838733\pi\)
\(524\) 0 0
\(525\) 1.17038 0.0510796
\(526\) 0 0
\(527\) −22.6725 −0.987630
\(528\) 0 0
\(529\) 22.1867 0.964641
\(530\) 0 0
\(531\) −0.174376 −0.00756727
\(532\) 0 0
\(533\) 48.1641 2.08622
\(534\) 0 0
\(535\) 4.64505 0.200823
\(536\) 0 0
\(537\) −23.9138 −1.03196
\(538\) 0 0
\(539\) 22.7646 0.980539
\(540\) 0 0
\(541\) 25.3449 1.08966 0.544830 0.838546i \(-0.316594\pi\)
0.544830 + 0.838546i \(0.316594\pi\)
\(542\) 0 0
\(543\) −8.17524 −0.350833
\(544\) 0 0
\(545\) 7.93624 0.339951
\(546\) 0 0
\(547\) −22.4541 −0.960068 −0.480034 0.877250i \(-0.659376\pi\)
−0.480034 + 0.877250i \(0.659376\pi\)
\(548\) 0 0
\(549\) −13.1943 −0.563121
\(550\) 0 0
\(551\) −4.26907 −0.181869
\(552\) 0 0
\(553\) −2.68473 −0.114166
\(554\) 0 0
\(555\) −30.6306 −1.30020
\(556\) 0 0
\(557\) 26.8511 1.13772 0.568860 0.822434i \(-0.307385\pi\)
0.568860 + 0.822434i \(0.307385\pi\)
\(558\) 0 0
\(559\) −70.0401 −2.96238
\(560\) 0 0
\(561\) 25.8709 1.09227
\(562\) 0 0
\(563\) 37.0560 1.56172 0.780862 0.624703i \(-0.214780\pi\)
0.780862 + 0.624703i \(0.214780\pi\)
\(564\) 0 0
\(565\) 18.4796 0.777443
\(566\) 0 0
\(567\) 2.20274 0.0925065
\(568\) 0 0
\(569\) 2.20600 0.0924804 0.0462402 0.998930i \(-0.485276\pi\)
0.0462402 + 0.998930i \(0.485276\pi\)
\(570\) 0 0
\(571\) 20.4444 0.855571 0.427786 0.903880i \(-0.359294\pi\)
0.427786 + 0.903880i \(0.359294\pi\)
\(572\) 0 0
\(573\) −8.32514 −0.347788
\(574\) 0 0
\(575\) 19.3533 0.807088
\(576\) 0 0
\(577\) −41.8850 −1.74369 −0.871847 0.489779i \(-0.837077\pi\)
−0.871847 + 0.489779i \(0.837077\pi\)
\(578\) 0 0
\(579\) 27.4793 1.14200
\(580\) 0 0
\(581\) −0.226076 −0.00937923
\(582\) 0 0
\(583\) 27.8784 1.15460
\(584\) 0 0
\(585\) −12.4345 −0.514104
\(586\) 0 0
\(587\) −33.4301 −1.37981 −0.689904 0.723901i \(-0.742347\pi\)
−0.689904 + 0.723901i \(0.742347\pi\)
\(588\) 0 0
\(589\) 5.92215 0.244018
\(590\) 0 0
\(591\) 13.6639 0.562056
\(592\) 0 0
\(593\) 4.83769 0.198660 0.0993301 0.995055i \(-0.468330\pi\)
0.0993301 + 0.995055i \(0.468330\pi\)
\(594\) 0 0
\(595\) −1.09683 −0.0449658
\(596\) 0 0
\(597\) −46.1144 −1.88734
\(598\) 0 0
\(599\) −4.64816 −0.189919 −0.0949594 0.995481i \(-0.530272\pi\)
−0.0949594 + 0.995481i \(0.530272\pi\)
\(600\) 0 0
\(601\) −12.8661 −0.524819 −0.262409 0.964957i \(-0.584517\pi\)
−0.262409 + 0.964957i \(0.584517\pi\)
\(602\) 0 0
\(603\) −2.82188 −0.114916
\(604\) 0 0
\(605\) 0.445725 0.0181213
\(606\) 0 0
\(607\) −14.4246 −0.585477 −0.292739 0.956193i \(-0.594567\pi\)
−0.292739 + 0.956193i \(0.594567\pi\)
\(608\) 0 0
\(609\) 1.73545 0.0703238
\(610\) 0 0
\(611\) −24.2849 −0.982461
\(612\) 0 0
\(613\) −37.2502 −1.50452 −0.752261 0.658865i \(-0.771037\pi\)
−0.752261 + 0.658865i \(0.771037\pi\)
\(614\) 0 0
\(615\) −21.5628 −0.869495
\(616\) 0 0
\(617\) 28.6841 1.15478 0.577388 0.816470i \(-0.304072\pi\)
0.577388 + 0.816470i \(0.304072\pi\)
\(618\) 0 0
\(619\) −4.26470 −0.171413 −0.0857063 0.996320i \(-0.527315\pi\)
−0.0857063 + 0.996320i \(0.527315\pi\)
\(620\) 0 0
\(621\) 24.0289 0.964246
\(622\) 0 0
\(623\) 1.20337 0.0482119
\(624\) 0 0
\(625\) −2.31581 −0.0926324
\(626\) 0 0
\(627\) −6.75758 −0.269872
\(628\) 0 0
\(629\) −38.9663 −1.55369
\(630\) 0 0
\(631\) 9.34444 0.371996 0.185998 0.982550i \(-0.440448\pi\)
0.185998 + 0.982550i \(0.440448\pi\)
\(632\) 0 0
\(633\) −31.4748 −1.25101
\(634\) 0 0
\(635\) 12.8666 0.510596
\(636\) 0 0
\(637\) −46.7946 −1.85407
\(638\) 0 0
\(639\) −10.3119 −0.407931
\(640\) 0 0
\(641\) 33.0359 1.30484 0.652420 0.757858i \(-0.273754\pi\)
0.652420 + 0.757858i \(0.273754\pi\)
\(642\) 0 0
\(643\) −3.53485 −0.139401 −0.0697005 0.997568i \(-0.522204\pi\)
−0.0697005 + 0.997568i \(0.522204\pi\)
\(644\) 0 0
\(645\) 31.3566 1.23466
\(646\) 0 0
\(647\) −36.0925 −1.41894 −0.709471 0.704734i \(-0.751066\pi\)
−0.709471 + 0.704734i \(0.751066\pi\)
\(648\) 0 0
\(649\) 0.448949 0.0176228
\(650\) 0 0
\(651\) −2.40745 −0.0943553
\(652\) 0 0
\(653\) −19.8193 −0.775588 −0.387794 0.921746i \(-0.626763\pi\)
−0.387794 + 0.921746i \(0.626763\pi\)
\(654\) 0 0
\(655\) 30.7704 1.20230
\(656\) 0 0
\(657\) −2.56107 −0.0999169
\(658\) 0 0
\(659\) −40.8393 −1.59087 −0.795437 0.606036i \(-0.792759\pi\)
−0.795437 + 0.606036i \(0.792759\pi\)
\(660\) 0 0
\(661\) 26.8218 1.04325 0.521623 0.853176i \(-0.325327\pi\)
0.521623 + 0.853176i \(0.325327\pi\)
\(662\) 0 0
\(663\) −53.1799 −2.06534
\(664\) 0 0
\(665\) 0.286497 0.0111099
\(666\) 0 0
\(667\) 28.6972 1.11116
\(668\) 0 0
\(669\) −2.07352 −0.0801671
\(670\) 0 0
\(671\) 33.9702 1.31141
\(672\) 0 0
\(673\) 28.6633 1.10489 0.552445 0.833549i \(-0.313695\pi\)
0.552445 + 0.833549i \(0.313695\pi\)
\(674\) 0 0
\(675\) 10.2915 0.396119
\(676\) 0 0
\(677\) −7.30155 −0.280621 −0.140311 0.990108i \(-0.544810\pi\)
−0.140311 + 0.990108i \(0.544810\pi\)
\(678\) 0 0
\(679\) 0.0261392 0.00100313
\(680\) 0 0
\(681\) −16.9569 −0.649789
\(682\) 0 0
\(683\) −25.3780 −0.971063 −0.485531 0.874219i \(-0.661374\pi\)
−0.485531 + 0.874219i \(0.661374\pi\)
\(684\) 0 0
\(685\) −17.9806 −0.687005
\(686\) 0 0
\(687\) −52.5538 −2.00505
\(688\) 0 0
\(689\) −57.3065 −2.18320
\(690\) 0 0
\(691\) 32.3136 1.22927 0.614633 0.788813i \(-0.289304\pi\)
0.614633 + 0.788813i \(0.289304\pi\)
\(692\) 0 0
\(693\) 0.817114 0.0310396
\(694\) 0 0
\(695\) 14.3397 0.543938
\(696\) 0 0
\(697\) −27.4308 −1.03901
\(698\) 0 0
\(699\) 36.9277 1.39673
\(700\) 0 0
\(701\) 12.8573 0.485612 0.242806 0.970075i \(-0.421932\pi\)
0.242806 + 0.970075i \(0.421932\pi\)
\(702\) 0 0
\(703\) 10.1782 0.383876
\(704\) 0 0
\(705\) 10.8722 0.409471
\(706\) 0 0
\(707\) −1.53813 −0.0578474
\(708\) 0 0
\(709\) 41.1583 1.54573 0.772867 0.634568i \(-0.218822\pi\)
0.772867 + 0.634568i \(0.218822\pi\)
\(710\) 0 0
\(711\) 17.3342 0.650083
\(712\) 0 0
\(713\) −39.8093 −1.49087
\(714\) 0 0
\(715\) 32.0140 1.19725
\(716\) 0 0
\(717\) 10.6878 0.399145
\(718\) 0 0
\(719\) −36.3230 −1.35462 −0.677309 0.735698i \(-0.736854\pi\)
−0.677309 + 0.735698i \(0.736854\pi\)
\(720\) 0 0
\(721\) −3.86818 −0.144058
\(722\) 0 0
\(723\) 51.4932 1.91505
\(724\) 0 0
\(725\) 12.2909 0.456472
\(726\) 0 0
\(727\) −1.34652 −0.0499398 −0.0249699 0.999688i \(-0.507949\pi\)
−0.0249699 + 0.999688i \(0.507949\pi\)
\(728\) 0 0
\(729\) 7.93789 0.293996
\(730\) 0 0
\(731\) 39.8898 1.47538
\(732\) 0 0
\(733\) −9.03789 −0.333822 −0.166911 0.985972i \(-0.553379\pi\)
−0.166911 + 0.985972i \(0.553379\pi\)
\(734\) 0 0
\(735\) 20.9497 0.772741
\(736\) 0 0
\(737\) 7.26522 0.267618
\(738\) 0 0
\(739\) −36.7004 −1.35005 −0.675023 0.737796i \(-0.735866\pi\)
−0.675023 + 0.737796i \(0.735866\pi\)
\(740\) 0 0
\(741\) 13.8908 0.510292
\(742\) 0 0
\(743\) 36.7973 1.34996 0.674981 0.737835i \(-0.264152\pi\)
0.674981 + 0.737835i \(0.264152\pi\)
\(744\) 0 0
\(745\) −1.54795 −0.0567127
\(746\) 0 0
\(747\) 1.45968 0.0534069
\(748\) 0 0
\(749\) 0.627453 0.0229266
\(750\) 0 0
\(751\) 42.2062 1.54013 0.770064 0.637967i \(-0.220224\pi\)
0.770064 + 0.637967i \(0.220224\pi\)
\(752\) 0 0
\(753\) −27.8294 −1.01416
\(754\) 0 0
\(755\) 4.24190 0.154378
\(756\) 0 0
\(757\) 8.89431 0.323269 0.161635 0.986851i \(-0.448323\pi\)
0.161635 + 0.986851i \(0.448323\pi\)
\(758\) 0 0
\(759\) 45.4252 1.64883
\(760\) 0 0
\(761\) 36.3702 1.31842 0.659210 0.751959i \(-0.270891\pi\)
0.659210 + 0.751959i \(0.270891\pi\)
\(762\) 0 0
\(763\) 1.07203 0.0388099
\(764\) 0 0
\(765\) 7.08180 0.256043
\(766\) 0 0
\(767\) −0.922856 −0.0333224
\(768\) 0 0
\(769\) 8.11654 0.292690 0.146345 0.989234i \(-0.453249\pi\)
0.146345 + 0.989234i \(0.453249\pi\)
\(770\) 0 0
\(771\) 15.6154 0.562373
\(772\) 0 0
\(773\) −41.0053 −1.47486 −0.737429 0.675424i \(-0.763961\pi\)
−0.737429 + 0.675424i \(0.763961\pi\)
\(774\) 0 0
\(775\) −17.0502 −0.612460
\(776\) 0 0
\(777\) −4.13758 −0.148435
\(778\) 0 0
\(779\) 7.16502 0.256714
\(780\) 0 0
\(781\) 26.5490 0.949998
\(782\) 0 0
\(783\) 15.2603 0.545357
\(784\) 0 0
\(785\) −12.5709 −0.448674
\(786\) 0 0
\(787\) −27.1644 −0.968305 −0.484153 0.874984i \(-0.660872\pi\)
−0.484153 + 0.874984i \(0.660872\pi\)
\(788\) 0 0
\(789\) −29.6413 −1.05526
\(790\) 0 0
\(791\) 2.49622 0.0887554
\(792\) 0 0
\(793\) −69.8289 −2.47970
\(794\) 0 0
\(795\) 25.6558 0.909918
\(796\) 0 0
\(797\) −4.55623 −0.161390 −0.0806949 0.996739i \(-0.525714\pi\)
−0.0806949 + 0.996739i \(0.525714\pi\)
\(798\) 0 0
\(799\) 13.8309 0.489303
\(800\) 0 0
\(801\) −7.76964 −0.274527
\(802\) 0 0
\(803\) 6.59375 0.232688
\(804\) 0 0
\(805\) −1.92587 −0.0678779
\(806\) 0 0
\(807\) 45.7015 1.60877
\(808\) 0 0
\(809\) −22.5386 −0.792414 −0.396207 0.918161i \(-0.629674\pi\)
−0.396207 + 0.918161i \(0.629674\pi\)
\(810\) 0 0
\(811\) 36.0887 1.26725 0.633623 0.773642i \(-0.281567\pi\)
0.633623 + 0.773642i \(0.281567\pi\)
\(812\) 0 0
\(813\) −11.3977 −0.399735
\(814\) 0 0
\(815\) −2.11958 −0.0742457
\(816\) 0 0
\(817\) −10.4194 −0.364528
\(818\) 0 0
\(819\) −1.67965 −0.0586918
\(820\) 0 0
\(821\) −14.0875 −0.491658 −0.245829 0.969313i \(-0.579060\pi\)
−0.245829 + 0.969313i \(0.579060\pi\)
\(822\) 0 0
\(823\) −37.0424 −1.29122 −0.645608 0.763669i \(-0.723396\pi\)
−0.645608 + 0.763669i \(0.723396\pi\)
\(824\) 0 0
\(825\) 19.4554 0.677350
\(826\) 0 0
\(827\) −5.03700 −0.175154 −0.0875768 0.996158i \(-0.527912\pi\)
−0.0875768 + 0.996158i \(0.527912\pi\)
\(828\) 0 0
\(829\) 28.9611 1.00586 0.502930 0.864327i \(-0.332255\pi\)
0.502930 + 0.864327i \(0.332255\pi\)
\(830\) 0 0
\(831\) −29.7845 −1.03321
\(832\) 0 0
\(833\) 26.6508 0.923397
\(834\) 0 0
\(835\) 21.1956 0.733504
\(836\) 0 0
\(837\) −21.1693 −0.731719
\(838\) 0 0
\(839\) 36.5197 1.26080 0.630400 0.776270i \(-0.282891\pi\)
0.630400 + 0.776270i \(0.282891\pi\)
\(840\) 0 0
\(841\) −10.7750 −0.371552
\(842\) 0 0
\(843\) 38.2200 1.31637
\(844\) 0 0
\(845\) −46.8751 −1.61255
\(846\) 0 0
\(847\) 0.0602084 0.00206878
\(848\) 0 0
\(849\) 9.98097 0.342546
\(850\) 0 0
\(851\) −68.4186 −2.34536
\(852\) 0 0
\(853\) 5.19322 0.177812 0.0889062 0.996040i \(-0.471663\pi\)
0.0889062 + 0.996040i \(0.471663\pi\)
\(854\) 0 0
\(855\) −1.84979 −0.0632616
\(856\) 0 0
\(857\) 21.1808 0.723524 0.361762 0.932271i \(-0.382175\pi\)
0.361762 + 0.932271i \(0.382175\pi\)
\(858\) 0 0
\(859\) 13.3430 0.455257 0.227629 0.973748i \(-0.426903\pi\)
0.227629 + 0.973748i \(0.426903\pi\)
\(860\) 0 0
\(861\) −2.91270 −0.0992644
\(862\) 0 0
\(863\) 36.8361 1.25392 0.626958 0.779053i \(-0.284300\pi\)
0.626958 + 0.779053i \(0.284300\pi\)
\(864\) 0 0
\(865\) 25.7777 0.876467
\(866\) 0 0
\(867\) −4.84196 −0.164442
\(868\) 0 0
\(869\) −44.6287 −1.51393
\(870\) 0 0
\(871\) −14.9343 −0.506030
\(872\) 0 0
\(873\) −0.168770 −0.00571199
\(874\) 0 0
\(875\) −2.25733 −0.0763116
\(876\) 0 0
\(877\) −15.1343 −0.511047 −0.255524 0.966803i \(-0.582248\pi\)
−0.255524 + 0.966803i \(0.582248\pi\)
\(878\) 0 0
\(879\) 4.00935 0.135232
\(880\) 0 0
\(881\) −51.9345 −1.74972 −0.874859 0.484377i \(-0.839046\pi\)
−0.874859 + 0.484377i \(0.839046\pi\)
\(882\) 0 0
\(883\) 24.5238 0.825293 0.412646 0.910891i \(-0.364605\pi\)
0.412646 + 0.910891i \(0.364605\pi\)
\(884\) 0 0
\(885\) 0.413157 0.0138881
\(886\) 0 0
\(887\) −45.2252 −1.51851 −0.759257 0.650791i \(-0.774438\pi\)
−0.759257 + 0.650791i \(0.774438\pi\)
\(888\) 0 0
\(889\) 1.73802 0.0582912
\(890\) 0 0
\(891\) 36.6165 1.22670
\(892\) 0 0
\(893\) −3.61269 −0.120894
\(894\) 0 0
\(895\) 16.8535 0.563352
\(896\) 0 0
\(897\) −93.3755 −3.11772
\(898\) 0 0
\(899\) −25.2821 −0.843205
\(900\) 0 0
\(901\) 32.6376 1.08732
\(902\) 0 0
\(903\) 4.23564 0.140953
\(904\) 0 0
\(905\) 5.76161 0.191522
\(906\) 0 0
\(907\) 5.25715 0.174561 0.0872804 0.996184i \(-0.472182\pi\)
0.0872804 + 0.996184i \(0.472182\pi\)
\(908\) 0 0
\(909\) 9.93108 0.329393
\(910\) 0 0
\(911\) −8.95774 −0.296783 −0.148392 0.988929i \(-0.547410\pi\)
−0.148392 + 0.988929i \(0.547410\pi\)
\(912\) 0 0
\(913\) −3.75810 −0.124375
\(914\) 0 0
\(915\) 31.2620 1.03349
\(916\) 0 0
\(917\) 4.15646 0.137258
\(918\) 0 0
\(919\) −29.1737 −0.962351 −0.481175 0.876624i \(-0.659790\pi\)
−0.481175 + 0.876624i \(0.659790\pi\)
\(920\) 0 0
\(921\) 28.3212 0.933216
\(922\) 0 0
\(923\) −54.5739 −1.79632
\(924\) 0 0
\(925\) −29.3034 −0.963490
\(926\) 0 0
\(927\) 24.9752 0.820294
\(928\) 0 0
\(929\) −39.2930 −1.28916 −0.644580 0.764537i \(-0.722968\pi\)
−0.644580 + 0.764537i \(0.722968\pi\)
\(930\) 0 0
\(931\) −6.96130 −0.228147
\(932\) 0 0
\(933\) 61.6519 2.01839
\(934\) 0 0
\(935\) −18.2328 −0.596278
\(936\) 0 0
\(937\) −12.4973 −0.408271 −0.204135 0.978943i \(-0.565438\pi\)
−0.204135 + 0.978943i \(0.565438\pi\)
\(938\) 0 0
\(939\) −41.7924 −1.36384
\(940\) 0 0
\(941\) 23.1052 0.753207 0.376603 0.926375i \(-0.377092\pi\)
0.376603 + 0.926375i \(0.377092\pi\)
\(942\) 0 0
\(943\) −48.1641 −1.56844
\(944\) 0 0
\(945\) −1.02411 −0.0333145
\(946\) 0 0
\(947\) 3.97265 0.129094 0.0645468 0.997915i \(-0.479440\pi\)
0.0645468 + 0.997915i \(0.479440\pi\)
\(948\) 0 0
\(949\) −13.5540 −0.439983
\(950\) 0 0
\(951\) −28.1111 −0.911565
\(952\) 0 0
\(953\) 10.4951 0.339968 0.169984 0.985447i \(-0.445628\pi\)
0.169984 + 0.985447i \(0.445628\pi\)
\(954\) 0 0
\(955\) 5.86725 0.189860
\(956\) 0 0
\(957\) 28.8486 0.932543
\(958\) 0 0
\(959\) −2.42882 −0.0784307
\(960\) 0 0
\(961\) 4.07181 0.131349
\(962\) 0 0
\(963\) −4.05120 −0.130548
\(964\) 0 0
\(965\) −19.3664 −0.623425
\(966\) 0 0
\(967\) 36.6501 1.17859 0.589294 0.807919i \(-0.299406\pi\)
0.589294 + 0.807919i \(0.299406\pi\)
\(968\) 0 0
\(969\) −7.91120 −0.254144
\(970\) 0 0
\(971\) 45.3202 1.45439 0.727197 0.686429i \(-0.240823\pi\)
0.727197 + 0.686429i \(0.240823\pi\)
\(972\) 0 0
\(973\) 1.93701 0.0620977
\(974\) 0 0
\(975\) −39.9923 −1.28078
\(976\) 0 0
\(977\) 22.1618 0.709018 0.354509 0.935053i \(-0.384648\pi\)
0.354509 + 0.935053i \(0.384648\pi\)
\(978\) 0 0
\(979\) 20.0038 0.639323
\(980\) 0 0
\(981\) −6.92162 −0.220990
\(982\) 0 0
\(983\) 42.6385 1.35996 0.679978 0.733233i \(-0.261989\pi\)
0.679978 + 0.733233i \(0.261989\pi\)
\(984\) 0 0
\(985\) −9.62978 −0.306830
\(986\) 0 0
\(987\) 1.46862 0.0467465
\(988\) 0 0
\(989\) 70.0401 2.22715
\(990\) 0 0
\(991\) −3.20296 −0.101745 −0.0508727 0.998705i \(-0.516200\pi\)
−0.0508727 + 0.998705i \(0.516200\pi\)
\(992\) 0 0
\(993\) −13.7728 −0.437068
\(994\) 0 0
\(995\) 32.4997 1.03031
\(996\) 0 0
\(997\) 13.5267 0.428395 0.214198 0.976790i \(-0.431286\pi\)
0.214198 + 0.976790i \(0.431286\pi\)
\(998\) 0 0
\(999\) −36.3829 −1.15110
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.bm.1.7 8
4.3 odd 2 4864.2.a.br.1.1 8
8.3 odd 2 inner 4864.2.a.bm.1.8 8
8.5 even 2 4864.2.a.br.1.2 8
16.3 odd 4 2432.2.c.i.1217.4 yes 16
16.5 even 4 2432.2.c.i.1217.3 16
16.11 odd 4 2432.2.c.i.1217.13 yes 16
16.13 even 4 2432.2.c.i.1217.14 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.i.1217.3 16 16.5 even 4
2432.2.c.i.1217.4 yes 16 16.3 odd 4
2432.2.c.i.1217.13 yes 16 16.11 odd 4
2432.2.c.i.1217.14 yes 16 16.13 even 4
4864.2.a.bm.1.7 8 1.1 even 1 trivial
4864.2.a.bm.1.8 8 8.3 odd 2 inner
4864.2.a.br.1.1 8 4.3 odd 2
4864.2.a.br.1.2 8 8.5 even 2