Properties

Label 1216.2.c.i.609.6
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.6
Root \(-1.52274 - 1.63746i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.i.609.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.31342i q^{5} +4.77753 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.31342i q^{5} +4.77753 q^{7} +3.00000 q^{9} -2.27492i q^{11} -6.09095i q^{13} -4.27492 q^{17} +1.00000i q^{19} +3.46410 q^{23} +3.27492 q^{25} -6.09095i q^{29} -2.62685 q^{31} +6.27492i q^{35} -0.837253i q^{37} -10.5498 q^{41} +10.2749i q^{43} +3.94027i q^{45} -4.77753 q^{47} +15.8248 q^{49} -10.3923i q^{53} +2.98793 q^{55} +8.54983i q^{59} +1.31342i q^{61} +14.3326 q^{63} +8.00000 q^{65} +8.54983i q^{67} +14.8087 q^{71} +4.27492 q^{73} -10.8685i q^{77} +4.30136 q^{79} +9.00000 q^{81} -13.0997i q^{83} -5.61478i q^{85} -10.0000 q^{89} -29.0997i q^{91} -1.31342 q^{95} +1.45017 q^{97} -6.82475i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 4q^{17} - 4q^{25} - 24q^{41} + 36q^{49} + 64q^{65} + 4q^{73} + 72q^{81} - 80q^{89} + 72q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 1.31342i 0.587381i 0.955901 + 0.293691i \(0.0948835\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) 4.77753 1.80573 0.902867 0.429919i \(-0.141458\pi\)
0.902867 + 0.429919i \(0.141458\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) − 2.27492i − 0.685913i −0.939351 0.342957i \(-0.888572\pi\)
0.939351 0.342957i \(-0.111428\pi\)
\(12\) 0 0
\(13\) − 6.09095i − 1.68933i −0.535299 0.844663i \(-0.679801\pi\)
0.535299 0.844663i \(-0.320199\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.27492 −1.03682 −0.518410 0.855132i \(-0.673476\pi\)
−0.518410 + 0.855132i \(0.673476\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 3.27492 0.654983
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.09095i − 1.13106i −0.824727 0.565530i \(-0.808671\pi\)
0.824727 0.565530i \(-0.191329\pi\)
\(30\) 0 0
\(31\) −2.62685 −0.471796 −0.235898 0.971778i \(-0.575803\pi\)
−0.235898 + 0.971778i \(0.575803\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.27492i 1.06065i
\(36\) 0 0
\(37\) − 0.837253i − 0.137644i −0.997629 0.0688218i \(-0.978076\pi\)
0.997629 0.0688218i \(-0.0219240\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.5498 −1.64761 −0.823804 0.566875i \(-0.808152\pi\)
−0.823804 + 0.566875i \(0.808152\pi\)
\(42\) 0 0
\(43\) 10.2749i 1.56691i 0.621448 + 0.783455i \(0.286545\pi\)
−0.621448 + 0.783455i \(0.713455\pi\)
\(44\) 0 0
\(45\) 3.94027i 0.587381i
\(46\) 0 0
\(47\) −4.77753 −0.696874 −0.348437 0.937332i \(-0.613287\pi\)
−0.348437 + 0.937332i \(0.613287\pi\)
\(48\) 0 0
\(49\) 15.8248 2.26068
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 10.3923i − 1.42749i −0.700404 0.713746i \(-0.746997\pi\)
0.700404 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) 2.98793 0.402893
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.54983i 1.11309i 0.830816 + 0.556547i \(0.187874\pi\)
−0.830816 + 0.556547i \(0.812126\pi\)
\(60\) 0 0
\(61\) 1.31342i 0.168167i 0.996459 + 0.0840834i \(0.0267962\pi\)
−0.996459 + 0.0840834i \(0.973204\pi\)
\(62\) 0 0
\(63\) 14.3326 1.80573
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 8.54983i 1.04453i 0.852784 + 0.522264i \(0.174913\pi\)
−0.852784 + 0.522264i \(0.825087\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.8087 1.75748 0.878738 0.477305i \(-0.158386\pi\)
0.878738 + 0.477305i \(0.158386\pi\)
\(72\) 0 0
\(73\) 4.27492 0.500341 0.250171 0.968202i \(-0.419513\pi\)
0.250171 + 0.968202i \(0.419513\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.8685i − 1.23858i
\(78\) 0 0
\(79\) 4.30136 0.483940 0.241970 0.970284i \(-0.422206\pi\)
0.241970 + 0.970284i \(0.422206\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 13.0997i − 1.43788i −0.695074 0.718938i \(-0.744629\pi\)
0.695074 0.718938i \(-0.255371\pi\)
\(84\) 0 0
\(85\) − 5.61478i − 0.609008i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) − 29.0997i − 3.05047i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.31342 −0.134754
\(96\) 0 0
\(97\) 1.45017 0.147242 0.0736210 0.997286i \(-0.476544\pi\)
0.0736210 + 0.997286i \(0.476544\pi\)
\(98\) 0 0
\(99\) − 6.82475i − 0.685913i
\(100\) 0 0
\(101\) 13.8564i 1.37876i 0.724398 + 0.689382i \(0.242118\pi\)
−0.724398 + 0.689382i \(0.757882\pi\)
\(102\) 0 0
\(103\) −2.62685 −0.258831 −0.129416 0.991590i \(-0.541310\pi\)
−0.129416 + 0.991590i \(0.541310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.5498i 1.59993i 0.600045 + 0.799966i \(0.295149\pi\)
−0.600045 + 0.799966i \(0.704851\pi\)
\(108\) 0 0
\(109\) 3.46410i 0.331801i 0.986143 + 0.165900i \(0.0530530\pi\)
−0.986143 + 0.165900i \(0.946947\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.54983 −0.239868 −0.119934 0.992782i \(-0.538268\pi\)
−0.119934 + 0.992782i \(0.538268\pi\)
\(114\) 0 0
\(115\) 4.54983i 0.424274i
\(116\) 0 0
\(117\) − 18.2728i − 1.68933i
\(118\) 0 0
\(119\) −20.4235 −1.87222
\(120\) 0 0
\(121\) 5.82475 0.529523
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.8685i 0.972106i
\(126\) 0 0
\(127\) 17.4356 1.54716 0.773579 0.633699i \(-0.218464\pi\)
0.773579 + 0.633699i \(0.218464\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.27492i − 0.198760i −0.995050 0.0993802i \(-0.968314\pi\)
0.995050 0.0993802i \(-0.0316860\pi\)
\(132\) 0 0
\(133\) 4.77753i 0.414264i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.2749 −1.73220 −0.866102 0.499868i \(-0.833382\pi\)
−0.866102 + 0.499868i \(0.833382\pi\)
\(138\) 0 0
\(139\) 10.2749i 0.871507i 0.900066 + 0.435754i \(0.143518\pi\)
−0.900066 + 0.435754i \(0.856482\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.8564 −1.15873
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.98793i 0.244781i 0.992482 + 0.122390i \(0.0390560\pi\)
−0.992482 + 0.122390i \(0.960944\pi\)
\(150\) 0 0
\(151\) −9.55505 −0.777579 −0.388790 0.921327i \(-0.627107\pi\)
−0.388790 + 0.921327i \(0.627107\pi\)
\(152\) 0 0
\(153\) −12.8248 −1.03682
\(154\) 0 0
\(155\) − 3.45017i − 0.277124i
\(156\) 0 0
\(157\) 5.25370i 0.419291i 0.977778 + 0.209645i \(0.0672309\pi\)
−0.977778 + 0.209645i \(0.932769\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.5498 1.30431
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.67451 −0.129577 −0.0647886 0.997899i \(-0.520637\pi\)
−0.0647886 + 0.997899i \(0.520637\pi\)
\(168\) 0 0
\(169\) −24.0997 −1.85382
\(170\) 0 0
\(171\) 3.00000i 0.229416i
\(172\) 0 0
\(173\) − 8.71780i − 0.662802i −0.943490 0.331401i \(-0.892479\pi\)
0.943490 0.331401i \(-0.107521\pi\)
\(174\) 0 0
\(175\) 15.6460 1.18273
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.09967i 0.680141i 0.940400 + 0.340071i \(0.110451\pi\)
−0.940400 + 0.340071i \(0.889549\pi\)
\(180\) 0 0
\(181\) − 3.46410i − 0.257485i −0.991678 0.128742i \(-0.958906\pi\)
0.991678 0.128742i \(-0.0410940\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.09967 0.0808493
\(186\) 0 0
\(187\) 9.72508i 0.711168i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0312 −0.725834 −0.362917 0.931822i \(-0.618219\pi\)
−0.362917 + 0.931822i \(0.618219\pi\)
\(192\) 0 0
\(193\) −19.0997 −1.37482 −0.687412 0.726268i \(-0.741253\pi\)
−0.687412 + 0.726268i \(0.741253\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.1819i 0.867924i 0.900931 + 0.433962i \(0.142885\pi\)
−0.900931 + 0.433962i \(0.857115\pi\)
\(198\) 0 0
\(199\) −18.6339 −1.32092 −0.660462 0.750859i \(-0.729640\pi\)
−0.660462 + 0.750859i \(0.729640\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 29.0997i − 2.04240i
\(204\) 0 0
\(205\) − 13.8564i − 0.967773i
\(206\) 0 0
\(207\) 10.3923 0.722315
\(208\) 0 0
\(209\) 2.27492 0.157359
\(210\) 0 0
\(211\) − 11.4502i − 0.788262i −0.919054 0.394131i \(-0.871046\pi\)
0.919054 0.394131i \(-0.128954\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.4953 −0.920373
\(216\) 0 0
\(217\) −12.5498 −0.851938
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 26.0383i 1.75153i
\(222\) 0 0
\(223\) −17.4356 −1.16757 −0.583787 0.811907i \(-0.698430\pi\)
−0.583787 + 0.811907i \(0.698430\pi\)
\(224\) 0 0
\(225\) 9.82475 0.654983
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) 22.0980i 1.46028i 0.683298 + 0.730140i \(0.260545\pi\)
−0.683298 + 0.730140i \(0.739455\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.3746 1.13825 0.569123 0.822252i \(-0.307283\pi\)
0.569123 + 0.822252i \(0.307283\pi\)
\(234\) 0 0
\(235\) − 6.27492i − 0.409330i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.82518 −0.247431 −0.123715 0.992318i \(-0.539481\pi\)
−0.123715 + 0.992318i \(0.539481\pi\)
\(240\) 0 0
\(241\) 2.54983 0.164249 0.0821246 0.996622i \(-0.473829\pi\)
0.0821246 + 0.996622i \(0.473829\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.7846i 1.32788i
\(246\) 0 0
\(247\) 6.09095 0.387558
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 18.2749i − 1.15350i −0.816920 0.576751i \(-0.804320\pi\)
0.816920 0.576751i \(-0.195680\pi\)
\(252\) 0 0
\(253\) − 7.88054i − 0.495446i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.5498 1.40662 0.703310 0.710883i \(-0.251705\pi\)
0.703310 + 0.710883i \(0.251705\pi\)
\(258\) 0 0
\(259\) − 4.00000i − 0.248548i
\(260\) 0 0
\(261\) − 18.2728i − 1.13106i
\(262\) 0 0
\(263\) 14.3326 0.883785 0.441892 0.897068i \(-0.354307\pi\)
0.441892 + 0.897068i \(0.354307\pi\)
\(264\) 0 0
\(265\) 13.6495 0.838482
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.5742i 1.37637i 0.725534 + 0.688187i \(0.241593\pi\)
−0.725534 + 0.688187i \(0.758407\pi\)
\(270\) 0 0
\(271\) 7.04329 0.427849 0.213925 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 7.45017i − 0.449262i
\(276\) 0 0
\(277\) − 2.98793i − 0.179527i −0.995963 0.0897637i \(-0.971389\pi\)
0.995963 0.0897637i \(-0.0286112\pi\)
\(278\) 0 0
\(279\) −7.88054 −0.471796
\(280\) 0 0
\(281\) −19.0997 −1.13939 −0.569695 0.821856i \(-0.692939\pi\)
−0.569695 + 0.821856i \(0.692939\pi\)
\(282\) 0 0
\(283\) 11.3746i 0.676149i 0.941119 + 0.338074i \(0.109776\pi\)
−0.941119 + 0.338074i \(0.890224\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −50.4021 −2.97514
\(288\) 0 0
\(289\) 1.27492 0.0749951
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.6219i 1.26316i 0.775310 + 0.631581i \(0.217594\pi\)
−0.775310 + 0.631581i \(0.782406\pi\)
\(294\) 0 0
\(295\) −11.2296 −0.653810
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 21.0997i − 1.22023i
\(300\) 0 0
\(301\) 49.0887i 2.82942i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.72508 −0.0987780
\(306\) 0 0
\(307\) − 16.5498i − 0.944549i −0.881452 0.472274i \(-0.843433\pi\)
0.881452 0.472274i \(-0.156567\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.72987 0.324911 0.162455 0.986716i \(-0.448059\pi\)
0.162455 + 0.986716i \(0.448059\pi\)
\(312\) 0 0
\(313\) −15.0997 −0.853484 −0.426742 0.904373i \(-0.640339\pi\)
−0.426742 + 0.904373i \(0.640339\pi\)
\(314\) 0 0
\(315\) 18.8248i 1.06065i
\(316\) 0 0
\(317\) 8.71780i 0.489640i 0.969569 + 0.244820i \(0.0787289\pi\)
−0.969569 + 0.244820i \(0.921271\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 4.27492i − 0.237863i
\(324\) 0 0
\(325\) − 19.9474i − 1.10648i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −22.8248 −1.25837
\(330\) 0 0
\(331\) 24.5498i 1.34938i 0.738101 + 0.674690i \(0.235723\pi\)
−0.738101 + 0.674690i \(0.764277\pi\)
\(332\) 0 0
\(333\) − 2.51176i − 0.137644i
\(334\) 0 0
\(335\) −11.2296 −0.613536
\(336\) 0 0
\(337\) −11.6495 −0.634589 −0.317294 0.948327i \(-0.602774\pi\)
−0.317294 + 0.948327i \(0.602774\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.97586i 0.323611i
\(342\) 0 0
\(343\) 42.1605 2.27645
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 14.8248i − 0.795834i −0.917421 0.397917i \(-0.869733\pi\)
0.917421 0.397917i \(-0.130267\pi\)
\(348\) 0 0
\(349\) − 35.2323i − 1.88594i −0.332877 0.942970i \(-0.608019\pi\)
0.332877 0.942970i \(-0.391981\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 19.4502i 1.03231i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.476171 0.0251313 0.0125657 0.999921i \(-0.496000\pi\)
0.0125657 + 0.999921i \(0.496000\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.61478i 0.293891i
\(366\) 0 0
\(367\) −1.78959 −0.0934161 −0.0467080 0.998909i \(-0.514873\pi\)
−0.0467080 + 0.998909i \(0.514873\pi\)
\(368\) 0 0
\(369\) −31.6495 −1.64761
\(370\) 0 0
\(371\) − 49.6495i − 2.57767i
\(372\) 0 0
\(373\) − 25.2011i − 1.30486i −0.757849 0.652431i \(-0.773749\pi\)
0.757849 0.652431i \(-0.226251\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −37.0997 −1.91073
\(378\) 0 0
\(379\) − 21.6495i − 1.11206i −0.831162 0.556030i \(-0.812324\pi\)
0.831162 0.556030i \(-0.187676\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.97586 −0.305352 −0.152676 0.988276i \(-0.548789\pi\)
−0.152676 + 0.988276i \(0.548789\pi\)
\(384\) 0 0
\(385\) 14.2749 0.727517
\(386\) 0 0
\(387\) 30.8248i 1.56691i
\(388\) 0 0
\(389\) − 17.7967i − 0.902327i −0.892441 0.451164i \(-0.851009\pi\)
0.892441 0.451164i \(-0.148991\pi\)
\(390\) 0 0
\(391\) −14.8087 −0.748911
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.64950i 0.284257i
\(396\) 0 0
\(397\) − 24.0027i − 1.20466i −0.798247 0.602331i \(-0.794239\pi\)
0.798247 0.602331i \(-0.205761\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.90033 0.244711 0.122355 0.992486i \(-0.460955\pi\)
0.122355 + 0.992486i \(0.460955\pi\)
\(402\) 0 0
\(403\) 16.0000i 0.797017i
\(404\) 0 0
\(405\) 11.8208i 0.587381i
\(406\) 0 0
\(407\) −1.90468 −0.0944116
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 40.8471i 2.00995i
\(414\) 0 0
\(415\) 17.2054 0.844581
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000i 0.586238i 0.956076 + 0.293119i \(0.0946933\pi\)
−0.956076 + 0.293119i \(0.905307\pi\)
\(420\) 0 0
\(421\) − 24.2487i − 1.18181i −0.806741 0.590905i \(-0.798771\pi\)
0.806741 0.590905i \(-0.201229\pi\)
\(422\) 0 0
\(423\) −14.3326 −0.696874
\(424\) 0 0
\(425\) −14.0000 −0.679100
\(426\) 0 0
\(427\) 6.27492i 0.303665i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.1101 0.920501 0.460251 0.887789i \(-0.347760\pi\)
0.460251 + 0.887789i \(0.347760\pi\)
\(432\) 0 0
\(433\) 27.6495 1.32875 0.664375 0.747399i \(-0.268698\pi\)
0.664375 + 0.747399i \(0.268698\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410i 0.165710i
\(438\) 0 0
\(439\) 3.57919 0.170825 0.0854127 0.996346i \(-0.472779\pi\)
0.0854127 + 0.996346i \(0.472779\pi\)
\(440\) 0 0
\(441\) 47.4743 2.26068
\(442\) 0 0
\(443\) 27.3746i 1.30061i 0.759675 + 0.650303i \(0.225358\pi\)
−0.759675 + 0.650303i \(0.774642\pi\)
\(444\) 0 0
\(445\) − 13.1342i − 0.622623i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.45017 −0.257209 −0.128605 0.991696i \(-0.541050\pi\)
−0.128605 + 0.991696i \(0.541050\pi\)
\(450\) 0 0
\(451\) 24.0000i 1.13012i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 38.2202 1.79179
\(456\) 0 0
\(457\) −7.72508 −0.361364 −0.180682 0.983542i \(-0.557831\pi\)
−0.180682 + 0.983542i \(0.557831\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 35.2323i − 1.64093i −0.571696 0.820465i \(-0.693714\pi\)
0.571696 0.820465i \(-0.306286\pi\)
\(462\) 0 0
\(463\) −33.4427 −1.55421 −0.777107 0.629369i \(-0.783313\pi\)
−0.777107 + 0.629369i \(0.783313\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 30.8248i − 1.42640i −0.700961 0.713200i \(-0.747245\pi\)
0.700961 0.713200i \(-0.252755\pi\)
\(468\) 0 0
\(469\) 40.8471i 1.88614i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.3746 1.07476
\(474\) 0 0
\(475\) 3.27492i 0.150264i
\(476\) 0 0
\(477\) − 31.1769i − 1.42749i
\(478\) 0 0
\(479\) −17.3205 −0.791394 −0.395697 0.918381i \(-0.629497\pi\)
−0.395697 + 0.918381i \(0.629497\pi\)
\(480\) 0 0
\(481\) −5.09967 −0.232525
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.90468i 0.0864872i
\(486\) 0 0
\(487\) 12.9041 0.584739 0.292370 0.956305i \(-0.405556\pi\)
0.292370 + 0.956305i \(0.405556\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000i 0.541552i 0.962642 + 0.270776i \(0.0872803\pi\)
−0.962642 + 0.270776i \(0.912720\pi\)
\(492\) 0 0
\(493\) 26.0383i 1.17271i
\(494\) 0 0
\(495\) 8.96379 0.402893
\(496\) 0 0
\(497\) 70.7492 3.17353
\(498\) 0 0
\(499\) − 34.2749i − 1.53436i −0.641434 0.767178i \(-0.721660\pi\)
0.641434 0.767178i \(-0.278340\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.81312 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(504\) 0 0
\(505\) −18.1993 −0.809860
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 13.0192i − 0.577064i −0.957470 0.288532i \(-0.906833\pi\)
0.957470 0.288532i \(-0.0931671\pi\)
\(510\) 0 0
\(511\) 20.4235 0.903484
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 3.45017i − 0.152032i
\(516\) 0 0
\(517\) 10.8685i 0.477995i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.09967 0.135799 0.0678995 0.997692i \(-0.478370\pi\)
0.0678995 + 0.997692i \(0.478370\pi\)
\(522\) 0 0
\(523\) − 20.5498i − 0.898582i −0.893386 0.449291i \(-0.851677\pi\)
0.893386 0.449291i \(-0.148323\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.2296 0.489167
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 25.6495i 1.11309i
\(532\) 0 0
\(533\) 64.2585i 2.78335i
\(534\) 0 0
\(535\) −21.7370 −0.939770
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 36.0000i − 1.55063i
\(540\) 0 0
\(541\) − 7.28929i − 0.313391i −0.987647 0.156695i \(-0.949916\pi\)
0.987647 0.156695i \(-0.0500841\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.54983 −0.194893
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 0 0
\(549\) 3.94027i 0.168167i
\(550\) 0 0
\(551\) 6.09095 0.259483
\(552\) 0 0
\(553\) 20.5498 0.873868
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 16.8443i − 0.713717i −0.934158 0.356859i \(-0.883848\pi\)
0.934158 0.356859i \(-0.116152\pi\)
\(558\) 0 0
\(559\) 62.5840 2.64702
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) − 3.34901i − 0.140894i
\(566\) 0 0
\(567\) 42.9977 1.80573
\(568\) 0 0
\(569\) −34.5498 −1.44840 −0.724202 0.689588i \(-0.757792\pi\)
−0.724202 + 0.689588i \(0.757792\pi\)
\(570\) 0 0
\(571\) 10.9003i 0.456165i 0.973642 + 0.228082i \(0.0732455\pi\)
−0.973642 + 0.228082i \(0.926754\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.3446 0.473104
\(576\) 0 0
\(577\) 15.7251 0.654644 0.327322 0.944913i \(-0.393854\pi\)
0.327322 + 0.944913i \(0.393854\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 62.5840i − 2.59642i
\(582\) 0 0
\(583\) −23.6416 −0.979136
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) 29.7251i 1.22689i 0.789739 + 0.613443i \(0.210216\pi\)
−0.789739 + 0.613443i \(0.789784\pi\)
\(588\) 0 0
\(589\) − 2.62685i − 0.108237i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11.0997 −0.455809 −0.227904 0.973684i \(-0.573187\pi\)
−0.227904 + 0.973684i \(0.573187\pi\)
\(594\) 0 0
\(595\) − 26.8248i − 1.09971i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −35.5934 −1.45431 −0.727153 0.686476i \(-0.759157\pi\)
−0.727153 + 0.686476i \(0.759157\pi\)
\(600\) 0 0
\(601\) −19.0997 −0.779092 −0.389546 0.921007i \(-0.627368\pi\)
−0.389546 + 0.921007i \(0.627368\pi\)
\(602\) 0 0
\(603\) 25.6495i 1.04453i
\(604\) 0 0
\(605\) 7.65037i 0.311032i
\(606\) 0 0
\(607\) 46.8229 1.90048 0.950242 0.311513i \(-0.100836\pi\)
0.950242 + 0.311513i \(0.100836\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29.0997i 1.17725i
\(612\) 0 0
\(613\) − 16.8443i − 0.680336i −0.940365 0.340168i \(-0.889516\pi\)
0.940365 0.340168i \(-0.110484\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.2749 0.816237 0.408119 0.912929i \(-0.366185\pi\)
0.408119 + 0.912929i \(0.366185\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −47.7753 −1.91408
\(624\) 0 0
\(625\) 2.09967 0.0839868
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.57919i 0.142712i
\(630\) 0 0
\(631\) 9.07888 0.361425 0.180712 0.983536i \(-0.442160\pi\)
0.180712 + 0.983536i \(0.442160\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.9003i 0.908772i
\(636\) 0 0
\(637\) − 96.3878i − 3.81902i
\(638\) 0 0
\(639\) 44.4262 1.75748
\(640\) 0 0
\(641\) 2.54983 0.100712 0.0503562 0.998731i \(-0.483964\pi\)
0.0503562 + 0.998731i \(0.483964\pi\)
\(642\) 0 0
\(643\) − 7.92442i − 0.312509i −0.987717 0.156254i \(-0.950058\pi\)
0.987717 0.156254i \(-0.0499419\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.82518 0.150384 0.0751918 0.997169i \(-0.476043\pi\)
0.0751918 + 0.997169i \(0.476043\pi\)
\(648\) 0 0
\(649\) 19.4502 0.763486
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.98793i − 0.116927i −0.998290 0.0584634i \(-0.981380\pi\)
0.998290 0.0584634i \(-0.0186201\pi\)
\(654\) 0 0
\(655\) 2.98793 0.116748
\(656\) 0 0
\(657\) 12.8248 0.500341
\(658\) 0 0
\(659\) 28.5498i 1.11214i 0.831134 + 0.556072i \(0.187692\pi\)
−0.831134 + 0.556072i \(0.812308\pi\)
\(660\) 0 0
\(661\) − 0.837253i − 0.0325654i −0.999867 0.0162827i \(-0.994817\pi\)
0.999867 0.0162827i \(-0.00518317\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.27492 −0.243331
\(666\) 0 0
\(667\) − 21.0997i − 0.816982i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.98793 0.115348
\(672\) 0 0
\(673\) −16.9003 −0.651460 −0.325730 0.945463i \(-0.605610\pi\)
−0.325730 + 0.945463i \(0.605610\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 44.3112i − 1.70302i −0.524342 0.851508i \(-0.675688\pi\)
0.524342 0.851508i \(-0.324312\pi\)
\(678\) 0 0
\(679\) 6.92820 0.265880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.1993i 1.46166i 0.682561 + 0.730829i \(0.260866\pi\)
−0.682561 + 0.730829i \(0.739134\pi\)
\(684\) 0 0
\(685\) − 26.6296i − 1.01746i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −63.2990 −2.41150
\(690\) 0 0
\(691\) 44.4743i 1.69188i 0.533278 + 0.845940i \(0.320960\pi\)
−0.533278 + 0.845940i \(0.679040\pi\)
\(692\) 0 0
\(693\) − 32.6054i − 1.23858i
\(694\) 0 0
\(695\) −13.4953 −0.511907
\(696\) 0 0
\(697\) 45.0997 1.70827
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 48.7276i 1.84042i 0.391430 + 0.920208i \(0.371981\pi\)
−0.391430 + 0.920208i \(0.628019\pi\)
\(702\) 0 0
\(703\) 0.837253 0.0315776
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 66.1993i 2.48968i
\(708\) 0 0
\(709\) 3.34901i 0.125775i 0.998021 + 0.0628874i \(0.0200309\pi\)
−0.998021 + 0.0628874i \(0.979969\pi\)
\(710\) 0 0
\(711\) 12.9041 0.483940
\(712\) 0 0
\(713\) −9.09967 −0.340785
\(714\) 0 0
\(715\) − 18.1993i − 0.680617i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.4279 0.463482 0.231741 0.972777i \(-0.425558\pi\)
0.231741 + 0.972777i \(0.425558\pi\)
\(720\) 0 0
\(721\) −12.5498 −0.467380
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 19.9474i − 0.740826i
\(726\) 0 0
\(727\) 19.5863 0.726415 0.363207 0.931708i \(-0.381682\pi\)
0.363207 + 0.931708i \(0.381682\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) − 43.9244i − 1.62460i
\(732\) 0 0
\(733\) − 1.67451i − 0.0618493i −0.999522 0.0309247i \(-0.990155\pi\)
0.999522 0.0309247i \(-0.00984520\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.4502 0.716456
\(738\) 0 0
\(739\) 14.8248i 0.545337i 0.962108 + 0.272669i \(0.0879063\pi\)
−0.962108 + 0.272669i \(0.912094\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −26.7605 −0.981746 −0.490873 0.871231i \(-0.663322\pi\)
−0.490873 + 0.871231i \(0.663322\pi\)
\(744\) 0 0
\(745\) −3.92442 −0.143780
\(746\) 0 0
\(747\) − 39.2990i − 1.43788i
\(748\) 0 0
\(749\) 79.0673i 2.88905i
\(750\) 0 0
\(751\) −53.7511 −1.96141 −0.980703 0.195503i \(-0.937366\pi\)
−0.980703 + 0.195503i \(0.937366\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 12.5498i − 0.456735i
\(756\) 0 0
\(757\) − 1.31342i − 0.0477372i −0.999715 0.0238686i \(-0.992402\pi\)
0.999715 0.0238686i \(-0.00759833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.27492 −0.154966 −0.0774828 0.996994i \(-0.524688\pi\)
−0.0774828 + 0.996994i \(0.524688\pi\)
\(762\) 0 0
\(763\) 16.5498i 0.599144i
\(764\) 0 0
\(765\) − 16.8443i − 0.609008i
\(766\) 0 0
\(767\) 52.0766 1.88038
\(768\) 0 0
\(769\) 8.82475 0.318229 0.159114 0.987260i \(-0.449136\pi\)
0.159114 + 0.987260i \(0.449136\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 0.115088i − 0.00413942i −0.999998 0.00206971i \(-0.999341\pi\)
0.999998 0.00206971i \(-0.000658809\pi\)
\(774\) 0 0
\(775\) −8.60271 −0.309018
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 10.5498i − 0.377987i
\(780\) 0 0
\(781\) − 33.6887i − 1.20548i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.90033 −0.246283
\(786\) 0 0
\(787\) − 46.1993i − 1.64683i −0.567441 0.823414i \(-0.692066\pi\)
0.567441 0.823414i \(-0.307934\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.1819 −0.433138
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.2487i 0.858933i 0.903083 + 0.429467i \(0.141298\pi\)
−0.903083 + 0.429467i \(0.858702\pi\)
\(798\) 0 0
\(799\) 20.4235 0.722532
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) − 9.72508i − 0.343191i
\(804\) 0 0
\(805\) 21.7370i 0.766127i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.8248 0.591527 0.295763 0.955261i \(-0.404426\pi\)
0.295763 + 0.955261i \(0.404426\pi\)
\(810\) 0 0
\(811\) 43.2990i 1.52043i 0.649669 + 0.760217i \(0.274907\pi\)
−0.649669 + 0.760217i \(0.725093\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.25370 −0.184029
\(816\) 0 0
\(817\) −10.2749 −0.359474
\(818\) 0 0
\(819\) − 87.2990i − 3.05047i
\(820\) 0 0
\(821\) − 21.3759i − 0.746023i −0.927827 0.373011i \(-0.878325\pi\)
0.927827 0.373011i \(-0.121675\pi\)
\(822\) 0 0
\(823\) 43.9501 1.53200 0.766002 0.642839i \(-0.222243\pi\)
0.766002 + 0.642839i \(0.222243\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 2.35050i − 0.0817348i −0.999165 0.0408674i \(-0.986988\pi\)
0.999165 0.0408674i \(-0.0130121\pi\)
\(828\) 0 0
\(829\) 35.7084i 1.24021i 0.784521 + 0.620103i \(0.212909\pi\)
−0.784521 + 0.620103i \(0.787091\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −67.6495 −2.34392
\(834\) 0 0
\(835\) − 2.19934i − 0.0761112i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −38.9424 −1.34444 −0.672220 0.740351i \(-0.734659\pi\)
−0.672220 + 0.740351i \(0.734659\pi\)
\(840\) 0 0
\(841\) −8.09967 −0.279299
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 31.6531i − 1.08890i
\(846\) 0 0
\(847\) 27.8279 0.956178
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2.90033i − 0.0994221i
\(852\) 0 0
\(853\) 15.5309i 0.531768i 0.964005 + 0.265884i \(0.0856639\pi\)
−0.964005 + 0.265884i \(0.914336\pi\)
\(854\) 0 0
\(855\) −3.94027 −0.134754
\(856\) 0 0
\(857\) 44.1993 1.50982 0.754910 0.655828i \(-0.227680\pi\)
0.754910 + 0.655828i \(0.227680\pi\)
\(858\) 0 0
\(859\) 12.4743i 0.425616i 0.977094 + 0.212808i \(0.0682609\pi\)
−0.977094 + 0.212808i \(0.931739\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.9430 0.951190 0.475595 0.879664i \(-0.342233\pi\)
0.475595 + 0.879664i \(0.342233\pi\)
\(864\) 0 0
\(865\) 11.4502 0.389317
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 9.78523i − 0.331941i
\(870\) 0 0
\(871\) 52.0766 1.76455
\(872\) 0 0
\(873\) 4.35050 0.147242
\(874\) 0 0
\(875\) 51.9244i 1.75537i
\(876\) 0 0
\(877\) 38.3353i 1.29449i 0.762282 + 0.647245i \(0.224079\pi\)
−0.762282 + 0.647245i \(0.775921\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −20.8248 −0.701604 −0.350802 0.936450i \(-0.614091\pi\)
−0.350802 + 0.936450i \(0.614091\pi\)
\(882\) 0 0
\(883\) − 51.3746i − 1.72889i −0.502725 0.864446i \(-0.667669\pi\)
0.502725 0.864446i \(-0.332331\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.55505 0.320827 0.160414 0.987050i \(-0.448717\pi\)
0.160414 + 0.987050i \(0.448717\pi\)
\(888\) 0 0
\(889\) 83.2990 2.79376
\(890\) 0 0
\(891\) − 20.4743i − 0.685913i
\(892\) 0 0
\(893\) − 4.77753i − 0.159874i
\(894\) 0 0
\(895\) −11.9517 −0.399502
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.0000i 0.533630i
\(900\) 0 0
\(901\) 44.4262i 1.48005i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.54983 0.151242
\(906\) 0 0
\(907\) 21.6495i 0.718860i 0.933172 + 0.359430i \(0.117029\pi\)
−0.933172 + 0.359430i \(0.882971\pi\)
\(908\) 0 0
\(909\) 41.5692i 1.37876i
\(910\) 0 0
\(911\) −44.4262 −1.47191 −0.735954 0.677032i \(-0.763266\pi\)
−0.735954 + 0.677032i \(0.763266\pi\)
\(912\) 0 0
\(913\) −29.8007 −0.986258
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 10.8685i − 0.358909i
\(918\) 0 0
\(919\) −34.7561 −1.14650 −0.573249 0.819381i \(-0.694317\pi\)
−0.573249 + 0.819381i \(0.694317\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 90.1993i − 2.96895i
\(924\) 0 0
\(925\) − 2.74194i − 0.0901543i
\(926\) 0 0
\(927\) −7.88054 −0.258831
\(928\) 0 0
\(929\) 23.0997 0.757876 0.378938 0.925422i \(-0.376289\pi\)
0.378938 + 0.925422i \(0.376289\pi\)
\(930\) 0 0
\(931\) 15.8248i 0.518635i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.7732 −0.417727
\(936\) 0 0
\(937\) 37.9244 1.23894 0.619468 0.785022i \(-0.287348\pi\)
0.619468 + 0.785022i \(0.287348\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 31.1769i − 1.01634i −0.861257 0.508169i \(-0.830322\pi\)
0.861257 0.508169i \(-0.169678\pi\)
\(942\) 0 0
\(943\) −36.5457 −1.19009
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.1993i 0.981347i 0.871344 + 0.490673i \(0.163249\pi\)
−0.871344 + 0.490673i \(0.836751\pi\)
\(948\) 0 0
\(949\) − 26.0383i − 0.845239i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 45.2990 1.46738 0.733689 0.679485i \(-0.237797\pi\)
0.733689 + 0.679485i \(0.237797\pi\)
\(954\) 0 0
\(955\) − 13.1752i − 0.426341i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −96.8639 −3.12790
\(960\) 0 0
\(961\) −24.0997 −0.777409
\(962\) 0 0
\(963\) 49.6495i 1.59993i
\(964\) 0 0
\(965\) − 25.0860i − 0.807546i
\(966\) 0 0
\(967\) 20.6695 0.664687 0.332344 0.943158i \(-0.392161\pi\)
0.332344 + 0.943158i \(0.392161\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 44.0000i 1.41203i 0.708198 + 0.706014i \(0.249508\pi\)
−0.708198 + 0.706014i \(0.750492\pi\)
\(972\) 0 0
\(973\) 49.0887i 1.57371i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.1993 1.03015 0.515074 0.857146i \(-0.327764\pi\)
0.515074 + 0.857146i \(0.327764\pi\)
\(978\) 0 0
\(979\) 22.7492i 0.727067i
\(980\) 0 0
\(981\) 10.3923i 0.331801i
\(982\) 0 0
\(983\) 29.6175 0.944651 0.472326 0.881424i \(-0.343415\pi\)
0.472326 + 0.881424i \(0.343415\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 35.5934i 1.13180i
\(990\) 0 0
\(991\) −40.1249 −1.27461 −0.637305 0.770612i \(-0.719951\pi\)
−0.637305 + 0.770612i \(0.719951\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 24.4743i − 0.775886i
\(996\) 0 0
\(997\) − 11.5906i − 0.367079i −0.983012 0.183540i \(-0.941244\pi\)
0.983012 0.183540i \(-0.0587556\pi\)
\(998\) 0 0
\(999\) 0 0
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 1216.2.c.i.609.6 yes 8
4.3 odd 2 inner 1216.2.c.i.609.5 yes 8
8.3 odd 2 inner 1216.2.c.i.609.3 8
8.5 even 2 inner 1216.2.c.i.609.4 yes 8
16.3 odd 4 4864.2.a.bj.1.3 4
16.5 even 4 4864.2.a.bj.1.2 4
16.11 odd 4 4864.2.a.bk.1.2 4
16.13 even 4 4864.2.a.bk.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.i.609.3 8 8.3 odd 2 inner
1216.2.c.i.609.4 yes 8 8.5 even 2 inner
1216.2.c.i.609.5 yes 8 4.3 odd 2 inner
1216.2.c.i.609.6 yes 8 1.1 even 1 trivial
4864.2.a.bj.1.2 4 16.5 even 4
4864.2.a.bj.1.3 4 16.3 odd 4
4864.2.a.bk.1.2 4 16.11 odd 4
4864.2.a.bk.1.3 4 16.13 even 4