Properties

Label 4560.2.a.bf.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} -6.24264 q^{11} -0.585786 q^{13} +1.00000 q^{15} +6.82843 q^{17} +1.00000 q^{19} +1.41421 q^{21} +3.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.41421 q^{29} +8.82843 q^{31} +6.24264 q^{33} +1.41421 q^{35} -0.585786 q^{37} +0.585786 q^{39} +8.24264 q^{41} -3.75736 q^{43} -1.00000 q^{45} -3.65685 q^{47} -5.00000 q^{49} -6.82843 q^{51} +8.00000 q^{53} +6.24264 q^{55} -1.00000 q^{57} +4.48528 q^{59} -15.3137 q^{61} -1.41421 q^{63} +0.585786 q^{65} -1.65685 q^{67} -3.65685 q^{69} +5.17157 q^{71} +3.65685 q^{73} -1.00000 q^{75} +8.82843 q^{77} +1.00000 q^{81} -7.17157 q^{83} -6.82843 q^{85} +1.41421 q^{87} -13.8995 q^{89} +0.828427 q^{91} -8.82843 q^{93} -1.00000 q^{95} -18.2426 q^{97} -6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{9} - 4q^{11} - 4q^{13} + 2q^{15} + 8q^{17} + 2q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 12q^{31} + 4q^{33} - 4q^{37} + 4q^{39} + 8q^{41} - 16q^{43} - 2q^{45} + 4q^{47} - 10q^{49} - 8q^{51} + 16q^{53} + 4q^{55} - 2q^{57} - 8q^{59} - 8q^{61} + 4q^{65} + 8q^{67} + 4q^{69} + 16q^{71} - 4q^{73} - 2q^{75} + 12q^{77} + 2q^{81} - 20q^{83} - 8q^{85} - 8q^{89} - 4q^{91} - 12q^{93} - 2q^{95} - 28q^{97} - 4q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.24264 −1.88223 −0.941113 0.338091i \(-0.890219\pi\)
−0.941113 + 0.338091i \(0.890219\pi\)
\(12\) 0 0
\(13\) −0.585786 −0.162468 −0.0812340 0.996695i \(-0.525886\pi\)
−0.0812340 + 0.996695i \(0.525886\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) 0 0
\(23\) 3.65685 0.762507 0.381253 0.924471i \(-0.375493\pi\)
0.381253 + 0.924471i \(0.375493\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.41421 −0.262613 −0.131306 0.991342i \(-0.541917\pi\)
−0.131306 + 0.991342i \(0.541917\pi\)
\(30\) 0 0
\(31\) 8.82843 1.58563 0.792816 0.609461i \(-0.208614\pi\)
0.792816 + 0.609461i \(0.208614\pi\)
\(32\) 0 0
\(33\) 6.24264 1.08670
\(34\) 0 0
\(35\) 1.41421 0.239046
\(36\) 0 0
\(37\) −0.585786 −0.0963027 −0.0481513 0.998840i \(-0.515333\pi\)
−0.0481513 + 0.998840i \(0.515333\pi\)
\(38\) 0 0
\(39\) 0.585786 0.0938009
\(40\) 0 0
\(41\) 8.24264 1.28728 0.643642 0.765327i \(-0.277423\pi\)
0.643642 + 0.765327i \(0.277423\pi\)
\(42\) 0 0
\(43\) −3.75736 −0.572992 −0.286496 0.958081i \(-0.592491\pi\)
−0.286496 + 0.958081i \(0.592491\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −3.65685 −0.533407 −0.266704 0.963779i \(-0.585934\pi\)
−0.266704 + 0.963779i \(0.585934\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −6.82843 −0.956171
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 6.24264 0.841757
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 4.48528 0.583934 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(60\) 0 0
\(61\) −15.3137 −1.96072 −0.980360 0.197218i \(-0.936809\pi\)
−0.980360 + 0.197218i \(0.936809\pi\)
\(62\) 0 0
\(63\) −1.41421 −0.178174
\(64\) 0 0
\(65\) 0.585786 0.0726579
\(66\) 0 0
\(67\) −1.65685 −0.202417 −0.101208 0.994865i \(-0.532271\pi\)
−0.101208 + 0.994865i \(0.532271\pi\)
\(68\) 0 0
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) 5.17157 0.613753 0.306876 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) 0 0
\(73\) 3.65685 0.428002 0.214001 0.976833i \(-0.431350\pi\)
0.214001 + 0.976833i \(0.431350\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.82843 1.00609
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.17157 −0.787182 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(84\) 0 0
\(85\) −6.82843 −0.740647
\(86\) 0 0
\(87\) 1.41421 0.151620
\(88\) 0 0
\(89\) −13.8995 −1.47334 −0.736672 0.676250i \(-0.763604\pi\)
−0.736672 + 0.676250i \(0.763604\pi\)
\(90\) 0 0
\(91\) 0.828427 0.0868428
\(92\) 0 0
\(93\) −8.82843 −0.915465
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −18.2426 −1.85226 −0.926130 0.377205i \(-0.876885\pi\)
−0.926130 + 0.377205i \(0.876885\pi\)
\(98\) 0 0
\(99\) −6.24264 −0.627409
\(100\) 0 0
\(101\) −8.82843 −0.878461 −0.439231 0.898374i \(-0.644749\pi\)
−0.439231 + 0.898374i \(0.644749\pi\)
\(102\) 0 0
\(103\) −15.3137 −1.50890 −0.754452 0.656355i \(-0.772097\pi\)
−0.754452 + 0.656355i \(0.772097\pi\)
\(104\) 0 0
\(105\) −1.41421 −0.138013
\(106\) 0 0
\(107\) 3.31371 0.320348 0.160174 0.987089i \(-0.448794\pi\)
0.160174 + 0.987089i \(0.448794\pi\)
\(108\) 0 0
\(109\) 10.4853 1.00431 0.502154 0.864778i \(-0.332541\pi\)
0.502154 + 0.864778i \(0.332541\pi\)
\(110\) 0 0
\(111\) 0.585786 0.0556004
\(112\) 0 0
\(113\) −18.1421 −1.70667 −0.853334 0.521364i \(-0.825423\pi\)
−0.853334 + 0.521364i \(0.825423\pi\)
\(114\) 0 0
\(115\) −3.65685 −0.341003
\(116\) 0 0
\(117\) −0.585786 −0.0541560
\(118\) 0 0
\(119\) −9.65685 −0.885242
\(120\) 0 0
\(121\) 27.9706 2.54278
\(122\) 0 0
\(123\) −8.24264 −0.743214
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.31371 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(128\) 0 0
\(129\) 3.75736 0.330817
\(130\) 0 0
\(131\) 11.4142 0.997264 0.498632 0.866814i \(-0.333836\pi\)
0.498632 + 0.866814i \(0.333836\pi\)
\(132\) 0 0
\(133\) −1.41421 −0.122628
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −14.1421 −1.19952 −0.599760 0.800180i \(-0.704737\pi\)
−0.599760 + 0.800180i \(0.704737\pi\)
\(140\) 0 0
\(141\) 3.65685 0.307963
\(142\) 0 0
\(143\) 3.65685 0.305802
\(144\) 0 0
\(145\) 1.41421 0.117444
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −10.4853 −0.853280 −0.426640 0.904422i \(-0.640303\pi\)
−0.426640 + 0.904422i \(0.640303\pi\)
\(152\) 0 0
\(153\) 6.82843 0.552046
\(154\) 0 0
\(155\) −8.82843 −0.709116
\(156\) 0 0
\(157\) −6.48528 −0.517582 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) −5.17157 −0.407577
\(162\) 0 0
\(163\) −2.10051 −0.164524 −0.0822621 0.996611i \(-0.526214\pi\)
−0.0822621 + 0.996611i \(0.526214\pi\)
\(164\) 0 0
\(165\) −6.24264 −0.485989
\(166\) 0 0
\(167\) −5.31371 −0.411187 −0.205594 0.978637i \(-0.565912\pi\)
−0.205594 + 0.978637i \(0.565912\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 19.7990 1.50529 0.752645 0.658427i \(-0.228778\pi\)
0.752645 + 0.658427i \(0.228778\pi\)
\(174\) 0 0
\(175\) −1.41421 −0.106904
\(176\) 0 0
\(177\) −4.48528 −0.337134
\(178\) 0 0
\(179\) 0.485281 0.0362716 0.0181358 0.999836i \(-0.494227\pi\)
0.0181358 + 0.999836i \(0.494227\pi\)
\(180\) 0 0
\(181\) −15.1716 −1.12769 −0.563847 0.825879i \(-0.690679\pi\)
−0.563847 + 0.825879i \(0.690679\pi\)
\(182\) 0 0
\(183\) 15.3137 1.13202
\(184\) 0 0
\(185\) 0.585786 0.0430679
\(186\) 0 0
\(187\) −42.6274 −3.11723
\(188\) 0 0
\(189\) 1.41421 0.102869
\(190\) 0 0
\(191\) −1.75736 −0.127158 −0.0635790 0.997977i \(-0.520251\pi\)
−0.0635790 + 0.997977i \(0.520251\pi\)
\(192\) 0 0
\(193\) −9.07107 −0.652950 −0.326475 0.945206i \(-0.605861\pi\)
−0.326475 + 0.945206i \(0.605861\pi\)
\(194\) 0 0
\(195\) −0.585786 −0.0419490
\(196\) 0 0
\(197\) 1.17157 0.0834711 0.0417356 0.999129i \(-0.486711\pi\)
0.0417356 + 0.999129i \(0.486711\pi\)
\(198\) 0 0
\(199\) 10.1421 0.718957 0.359478 0.933153i \(-0.382955\pi\)
0.359478 + 0.933153i \(0.382955\pi\)
\(200\) 0 0
\(201\) 1.65685 0.116865
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −8.24264 −0.575691
\(206\) 0 0
\(207\) 3.65685 0.254169
\(208\) 0 0
\(209\) −6.24264 −0.431812
\(210\) 0 0
\(211\) 15.3137 1.05424 0.527120 0.849791i \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(212\) 0 0
\(213\) −5.17157 −0.354350
\(214\) 0 0
\(215\) 3.75736 0.256250
\(216\) 0 0
\(217\) −12.4853 −0.847556
\(218\) 0 0
\(219\) −3.65685 −0.247107
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 26.6274 1.78310 0.891552 0.452919i \(-0.149617\pi\)
0.891552 + 0.452919i \(0.149617\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 18.9706 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(228\) 0 0
\(229\) 22.6274 1.49526 0.747631 0.664114i \(-0.231191\pi\)
0.747631 + 0.664114i \(0.231191\pi\)
\(230\) 0 0
\(231\) −8.82843 −0.580868
\(232\) 0 0
\(233\) −11.6569 −0.763666 −0.381833 0.924231i \(-0.624707\pi\)
−0.381833 + 0.924231i \(0.624707\pi\)
\(234\) 0 0
\(235\) 3.65685 0.238547
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.27208 0.0822839 0.0411419 0.999153i \(-0.486900\pi\)
0.0411419 + 0.999153i \(0.486900\pi\)
\(240\) 0 0
\(241\) −8.34315 −0.537429 −0.268715 0.963220i \(-0.586599\pi\)
−0.268715 + 0.963220i \(0.586599\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.00000 0.319438
\(246\) 0 0
\(247\) −0.585786 −0.0372727
\(248\) 0 0
\(249\) 7.17157 0.454480
\(250\) 0 0
\(251\) 10.2426 0.646510 0.323255 0.946312i \(-0.395223\pi\)
0.323255 + 0.946312i \(0.395223\pi\)
\(252\) 0 0
\(253\) −22.8284 −1.43521
\(254\) 0 0
\(255\) 6.82843 0.427613
\(256\) 0 0
\(257\) 12.4853 0.778810 0.389405 0.921067i \(-0.372681\pi\)
0.389405 + 0.921067i \(0.372681\pi\)
\(258\) 0 0
\(259\) 0.828427 0.0514760
\(260\) 0 0
\(261\) −1.41421 −0.0875376
\(262\) 0 0
\(263\) −27.4558 −1.69300 −0.846500 0.532389i \(-0.821294\pi\)
−0.846500 + 0.532389i \(0.821294\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 13.8995 0.850635
\(268\) 0 0
\(269\) 11.0711 0.675015 0.337507 0.941323i \(-0.390416\pi\)
0.337507 + 0.941323i \(0.390416\pi\)
\(270\) 0 0
\(271\) 5.17157 0.314151 0.157075 0.987587i \(-0.449793\pi\)
0.157075 + 0.987587i \(0.449793\pi\)
\(272\) 0 0
\(273\) −0.828427 −0.0501387
\(274\) 0 0
\(275\) −6.24264 −0.376445
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 8.82843 0.528544
\(280\) 0 0
\(281\) −17.4142 −1.03884 −0.519422 0.854518i \(-0.673853\pi\)
−0.519422 + 0.854518i \(0.673853\pi\)
\(282\) 0 0
\(283\) −26.3848 −1.56841 −0.784206 0.620500i \(-0.786929\pi\)
−0.784206 + 0.620500i \(0.786929\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −11.6569 −0.688082
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 18.2426 1.06940
\(292\) 0 0
\(293\) −28.4853 −1.66413 −0.832064 0.554680i \(-0.812841\pi\)
−0.832064 + 0.554680i \(0.812841\pi\)
\(294\) 0 0
\(295\) −4.48528 −0.261143
\(296\) 0 0
\(297\) 6.24264 0.362235
\(298\) 0 0
\(299\) −2.14214 −0.123883
\(300\) 0 0
\(301\) 5.31371 0.306277
\(302\) 0 0
\(303\) 8.82843 0.507180
\(304\) 0 0
\(305\) 15.3137 0.876860
\(306\) 0 0
\(307\) −26.8284 −1.53118 −0.765590 0.643329i \(-0.777553\pi\)
−0.765590 + 0.643329i \(0.777553\pi\)
\(308\) 0 0
\(309\) 15.3137 0.871166
\(310\) 0 0
\(311\) −2.24264 −0.127168 −0.0635842 0.997976i \(-0.520253\pi\)
−0.0635842 + 0.997976i \(0.520253\pi\)
\(312\) 0 0
\(313\) −33.7990 −1.91043 −0.955216 0.295910i \(-0.904377\pi\)
−0.955216 + 0.295910i \(0.904377\pi\)
\(314\) 0 0
\(315\) 1.41421 0.0796819
\(316\) 0 0
\(317\) 18.6274 1.04622 0.523110 0.852265i \(-0.324772\pi\)
0.523110 + 0.852265i \(0.324772\pi\)
\(318\) 0 0
\(319\) 8.82843 0.494297
\(320\) 0 0
\(321\) −3.31371 −0.184953
\(322\) 0 0
\(323\) 6.82843 0.379944
\(324\) 0 0
\(325\) −0.585786 −0.0324936
\(326\) 0 0
\(327\) −10.4853 −0.579837
\(328\) 0 0
\(329\) 5.17157 0.285118
\(330\) 0 0
\(331\) 0.142136 0.00781248 0.00390624 0.999992i \(-0.498757\pi\)
0.00390624 + 0.999992i \(0.498757\pi\)
\(332\) 0 0
\(333\) −0.585786 −0.0321009
\(334\) 0 0
\(335\) 1.65685 0.0905236
\(336\) 0 0
\(337\) −27.4142 −1.49335 −0.746674 0.665191i \(-0.768350\pi\)
−0.746674 + 0.665191i \(0.768350\pi\)
\(338\) 0 0
\(339\) 18.1421 0.985346
\(340\) 0 0
\(341\) −55.1127 −2.98452
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 3.65685 0.196878
\(346\) 0 0
\(347\) −23.4558 −1.25918 −0.629588 0.776929i \(-0.716776\pi\)
−0.629588 + 0.776929i \(0.716776\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 0.585786 0.0312670
\(352\) 0 0
\(353\) −7.65685 −0.407533 −0.203767 0.979019i \(-0.565318\pi\)
−0.203767 + 0.979019i \(0.565318\pi\)
\(354\) 0 0
\(355\) −5.17157 −0.274479
\(356\) 0 0
\(357\) 9.65685 0.511095
\(358\) 0 0
\(359\) 28.8701 1.52370 0.761852 0.647752i \(-0.224290\pi\)
0.761852 + 0.647752i \(0.224290\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −27.9706 −1.46807
\(364\) 0 0
\(365\) −3.65685 −0.191408
\(366\) 0 0
\(367\) −22.3848 −1.16848 −0.584238 0.811582i \(-0.698606\pi\)
−0.584238 + 0.811582i \(0.698606\pi\)
\(368\) 0 0
\(369\) 8.24264 0.429095
\(370\) 0 0
\(371\) −11.3137 −0.587378
\(372\) 0 0
\(373\) 3.41421 0.176781 0.0883906 0.996086i \(-0.471828\pi\)
0.0883906 + 0.996086i \(0.471828\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0.828427 0.0426662
\(378\) 0 0
\(379\) −8.82843 −0.453486 −0.226743 0.973955i \(-0.572808\pi\)
−0.226743 + 0.973955i \(0.572808\pi\)
\(380\) 0 0
\(381\) −3.31371 −0.169766
\(382\) 0 0
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 0 0
\(385\) −8.82843 −0.449938
\(386\) 0 0
\(387\) −3.75736 −0.190997
\(388\) 0 0
\(389\) −2.97056 −0.150614 −0.0753068 0.997160i \(-0.523994\pi\)
−0.0753068 + 0.997160i \(0.523994\pi\)
\(390\) 0 0
\(391\) 24.9706 1.26282
\(392\) 0 0
\(393\) −11.4142 −0.575771
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.6274 −0.834506 −0.417253 0.908790i \(-0.637007\pi\)
−0.417253 + 0.908790i \(0.637007\pi\)
\(398\) 0 0
\(399\) 1.41421 0.0707992
\(400\) 0 0
\(401\) −16.2426 −0.811119 −0.405559 0.914069i \(-0.632923\pi\)
−0.405559 + 0.914069i \(0.632923\pi\)
\(402\) 0 0
\(403\) −5.17157 −0.257614
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 3.65685 0.181264
\(408\) 0 0
\(409\) −7.17157 −0.354611 −0.177306 0.984156i \(-0.556738\pi\)
−0.177306 + 0.984156i \(0.556738\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) −6.34315 −0.312126
\(414\) 0 0
\(415\) 7.17157 0.352039
\(416\) 0 0
\(417\) 14.1421 0.692543
\(418\) 0 0
\(419\) −0.585786 −0.0286175 −0.0143088 0.999898i \(-0.504555\pi\)
−0.0143088 + 0.999898i \(0.504555\pi\)
\(420\) 0 0
\(421\) −13.3137 −0.648870 −0.324435 0.945908i \(-0.605174\pi\)
−0.324435 + 0.945908i \(0.605174\pi\)
\(422\) 0 0
\(423\) −3.65685 −0.177802
\(424\) 0 0
\(425\) 6.82843 0.331227
\(426\) 0 0
\(427\) 21.6569 1.04805
\(428\) 0 0
\(429\) −3.65685 −0.176555
\(430\) 0 0
\(431\) −31.1127 −1.49865 −0.749323 0.662205i \(-0.769621\pi\)
−0.749323 + 0.662205i \(0.769621\pi\)
\(432\) 0 0
\(433\) −25.0711 −1.20484 −0.602419 0.798180i \(-0.705796\pi\)
−0.602419 + 0.798180i \(0.705796\pi\)
\(434\) 0 0
\(435\) −1.41421 −0.0678064
\(436\) 0 0
\(437\) 3.65685 0.174931
\(438\) 0 0
\(439\) 10.3431 0.493651 0.246826 0.969060i \(-0.420612\pi\)
0.246826 + 0.969060i \(0.420612\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) 13.8995 0.658899
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 26.8701 1.26808 0.634038 0.773302i \(-0.281396\pi\)
0.634038 + 0.773302i \(0.281396\pi\)
\(450\) 0 0
\(451\) −51.4558 −2.42296
\(452\) 0 0
\(453\) 10.4853 0.492641
\(454\) 0 0
\(455\) −0.828427 −0.0388373
\(456\) 0 0
\(457\) 27.1716 1.27103 0.635516 0.772087i \(-0.280787\pi\)
0.635516 + 0.772087i \(0.280787\pi\)
\(458\) 0 0
\(459\) −6.82843 −0.318724
\(460\) 0 0
\(461\) 8.34315 0.388579 0.194290 0.980944i \(-0.437760\pi\)
0.194290 + 0.980944i \(0.437760\pi\)
\(462\) 0 0
\(463\) −15.7574 −0.732307 −0.366153 0.930555i \(-0.619325\pi\)
−0.366153 + 0.930555i \(0.619325\pi\)
\(464\) 0 0
\(465\) 8.82843 0.409409
\(466\) 0 0
\(467\) −24.3431 −1.12647 −0.563233 0.826298i \(-0.690443\pi\)
−0.563233 + 0.826298i \(0.690443\pi\)
\(468\) 0 0
\(469\) 2.34315 0.108196
\(470\) 0 0
\(471\) 6.48528 0.298826
\(472\) 0 0
\(473\) 23.4558 1.07850
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 0 0
\(479\) −2.92893 −0.133826 −0.0669132 0.997759i \(-0.521315\pi\)
−0.0669132 + 0.997759i \(0.521315\pi\)
\(480\) 0 0
\(481\) 0.343146 0.0156461
\(482\) 0 0
\(483\) 5.17157 0.235315
\(484\) 0 0
\(485\) 18.2426 0.828356
\(486\) 0 0
\(487\) −3.51472 −0.159267 −0.0796336 0.996824i \(-0.525375\pi\)
−0.0796336 + 0.996824i \(0.525375\pi\)
\(488\) 0 0
\(489\) 2.10051 0.0949881
\(490\) 0 0
\(491\) 10.2426 0.462244 0.231122 0.972925i \(-0.425760\pi\)
0.231122 + 0.972925i \(0.425760\pi\)
\(492\) 0 0
\(493\) −9.65685 −0.434923
\(494\) 0 0
\(495\) 6.24264 0.280586
\(496\) 0 0
\(497\) −7.31371 −0.328065
\(498\) 0 0
\(499\) 10.8284 0.484747 0.242373 0.970183i \(-0.422074\pi\)
0.242373 + 0.970183i \(0.422074\pi\)
\(500\) 0 0
\(501\) 5.31371 0.237399
\(502\) 0 0
\(503\) 0.828427 0.0369377 0.0184689 0.999829i \(-0.494121\pi\)
0.0184689 + 0.999829i \(0.494121\pi\)
\(504\) 0 0
\(505\) 8.82843 0.392860
\(506\) 0 0
\(507\) 12.6569 0.562111
\(508\) 0 0
\(509\) 24.7279 1.09605 0.548023 0.836463i \(-0.315381\pi\)
0.548023 + 0.836463i \(0.315381\pi\)
\(510\) 0 0
\(511\) −5.17157 −0.228777
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 15.3137 0.674803
\(516\) 0 0
\(517\) 22.8284 1.00399
\(518\) 0 0
\(519\) −19.7990 −0.869079
\(520\) 0 0
\(521\) 16.2426 0.711603 0.355802 0.934562i \(-0.384208\pi\)
0.355802 + 0.934562i \(0.384208\pi\)
\(522\) 0 0
\(523\) 32.4853 1.42048 0.710241 0.703959i \(-0.248586\pi\)
0.710241 + 0.703959i \(0.248586\pi\)
\(524\) 0 0
\(525\) 1.41421 0.0617213
\(526\) 0 0
\(527\) 60.2843 2.62602
\(528\) 0 0
\(529\) −9.62742 −0.418583
\(530\) 0 0
\(531\) 4.48528 0.194645
\(532\) 0 0
\(533\) −4.82843 −0.209142
\(534\) 0 0
\(535\) −3.31371 −0.143264
\(536\) 0 0
\(537\) −0.485281 −0.0209414
\(538\) 0 0
\(539\) 31.2132 1.34445
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 15.1716 0.651075
\(544\) 0 0
\(545\) −10.4853 −0.449140
\(546\) 0 0
\(547\) −7.51472 −0.321306 −0.160653 0.987011i \(-0.551360\pi\)
−0.160653 + 0.987011i \(0.551360\pi\)
\(548\) 0 0
\(549\) −15.3137 −0.653573
\(550\) 0 0
\(551\) −1.41421 −0.0602475
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.585786 −0.0248652
\(556\) 0 0
\(557\) −2.68629 −0.113822 −0.0569109 0.998379i \(-0.518125\pi\)
−0.0569109 + 0.998379i \(0.518125\pi\)
\(558\) 0 0
\(559\) 2.20101 0.0930928
\(560\) 0 0
\(561\) 42.6274 1.79973
\(562\) 0 0
\(563\) −26.2843 −1.10775 −0.553875 0.832600i \(-0.686851\pi\)
−0.553875 + 0.832600i \(0.686851\pi\)
\(564\) 0 0
\(565\) 18.1421 0.763245
\(566\) 0 0
\(567\) −1.41421 −0.0593914
\(568\) 0 0
\(569\) 16.9289 0.709698 0.354849 0.934924i \(-0.384532\pi\)
0.354849 + 0.934924i \(0.384532\pi\)
\(570\) 0 0
\(571\) −14.8284 −0.620550 −0.310275 0.950647i \(-0.600421\pi\)
−0.310275 + 0.950647i \(0.600421\pi\)
\(572\) 0 0
\(573\) 1.75736 0.0734147
\(574\) 0 0
\(575\) 3.65685 0.152501
\(576\) 0 0
\(577\) −31.4558 −1.30952 −0.654762 0.755835i \(-0.727231\pi\)
−0.654762 + 0.755835i \(0.727231\pi\)
\(578\) 0 0
\(579\) 9.07107 0.376981
\(580\) 0 0
\(581\) 10.1421 0.420767
\(582\) 0 0
\(583\) −49.9411 −2.06835
\(584\) 0 0
\(585\) 0.585786 0.0242193
\(586\) 0 0
\(587\) 23.6569 0.976423 0.488211 0.872725i \(-0.337649\pi\)
0.488211 + 0.872725i \(0.337649\pi\)
\(588\) 0 0
\(589\) 8.82843 0.363769
\(590\) 0 0
\(591\) −1.17157 −0.0481921
\(592\) 0 0
\(593\) −8.62742 −0.354286 −0.177143 0.984185i \(-0.556685\pi\)
−0.177143 + 0.984185i \(0.556685\pi\)
\(594\) 0 0
\(595\) 9.65685 0.395892
\(596\) 0 0
\(597\) −10.1421 −0.415090
\(598\) 0 0
\(599\) 36.9706 1.51058 0.755288 0.655393i \(-0.227497\pi\)
0.755288 + 0.655393i \(0.227497\pi\)
\(600\) 0 0
\(601\) −20.1421 −0.821615 −0.410807 0.911722i \(-0.634753\pi\)
−0.410807 + 0.911722i \(0.634753\pi\)
\(602\) 0 0
\(603\) −1.65685 −0.0674723
\(604\) 0 0
\(605\) −27.9706 −1.13717
\(606\) 0 0
\(607\) 0.485281 0.0196970 0.00984848 0.999952i \(-0.496865\pi\)
0.00984848 + 0.999952i \(0.496865\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 2.14214 0.0866615
\(612\) 0 0
\(613\) 6.48528 0.261938 0.130969 0.991386i \(-0.458191\pi\)
0.130969 + 0.991386i \(0.458191\pi\)
\(614\) 0 0
\(615\) 8.24264 0.332375
\(616\) 0 0
\(617\) 11.5147 0.463565 0.231783 0.972768i \(-0.425544\pi\)
0.231783 + 0.972768i \(0.425544\pi\)
\(618\) 0 0
\(619\) 3.51472 0.141268 0.0706342 0.997502i \(-0.477498\pi\)
0.0706342 + 0.997502i \(0.477498\pi\)
\(620\) 0 0
\(621\) −3.65685 −0.146745
\(622\) 0 0
\(623\) 19.6569 0.787535
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.24264 0.249307
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −15.3137 −0.608665
\(634\) 0 0
\(635\) −3.31371 −0.131501
\(636\) 0 0
\(637\) 2.92893 0.116049
\(638\) 0 0
\(639\) 5.17157 0.204584
\(640\) 0 0
\(641\) 39.3553 1.55444 0.777221 0.629227i \(-0.216629\pi\)
0.777221 + 0.629227i \(0.216629\pi\)
\(642\) 0 0
\(643\) −5.61522 −0.221443 −0.110721 0.993851i \(-0.535316\pi\)
−0.110721 + 0.993851i \(0.535316\pi\)
\(644\) 0 0
\(645\) −3.75736 −0.147946
\(646\) 0 0
\(647\) 7.85786 0.308925 0.154462 0.987999i \(-0.450635\pi\)
0.154462 + 0.987999i \(0.450635\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 12.4853 0.489337
\(652\) 0 0
\(653\) 37.1716 1.45464 0.727318 0.686301i \(-0.240767\pi\)
0.727318 + 0.686301i \(0.240767\pi\)
\(654\) 0 0
\(655\) −11.4142 −0.445990
\(656\) 0 0
\(657\) 3.65685 0.142667
\(658\) 0 0
\(659\) −26.6274 −1.03726 −0.518628 0.855000i \(-0.673557\pi\)
−0.518628 + 0.855000i \(0.673557\pi\)
\(660\) 0 0
\(661\) −43.4558 −1.69024 −0.845118 0.534579i \(-0.820470\pi\)
−0.845118 + 0.534579i \(0.820470\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 1.41421 0.0548408
\(666\) 0 0
\(667\) −5.17157 −0.200244
\(668\) 0 0
\(669\) −26.6274 −1.02948
\(670\) 0 0
\(671\) 95.5980 3.69052
\(672\) 0 0
\(673\) 32.1838 1.24059 0.620297 0.784367i \(-0.287012\pi\)
0.620297 + 0.784367i \(0.287012\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −16.9706 −0.652232 −0.326116 0.945330i \(-0.605740\pi\)
−0.326116 + 0.945330i \(0.605740\pi\)
\(678\) 0 0
\(679\) 25.7990 0.990074
\(680\) 0 0
\(681\) −18.9706 −0.726954
\(682\) 0 0
\(683\) −29.6569 −1.13479 −0.567394 0.823446i \(-0.692048\pi\)
−0.567394 + 0.823446i \(0.692048\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) −22.6274 −0.863290
\(688\) 0 0
\(689\) −4.68629 −0.178533
\(690\) 0 0
\(691\) −41.1716 −1.56624 −0.783120 0.621870i \(-0.786373\pi\)
−0.783120 + 0.621870i \(0.786373\pi\)
\(692\) 0 0
\(693\) 8.82843 0.335364
\(694\) 0 0
\(695\) 14.1421 0.536442
\(696\) 0 0
\(697\) 56.2843 2.13192
\(698\) 0 0
\(699\) 11.6569 0.440903
\(700\) 0 0
\(701\) 29.3137 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(702\) 0 0
\(703\) −0.585786 −0.0220934
\(704\) 0 0
\(705\) −3.65685 −0.137725
\(706\) 0 0
\(707\) 12.4853 0.469557
\(708\) 0 0
\(709\) −4.97056 −0.186673 −0.0933367 0.995635i \(-0.529753\pi\)
−0.0933367 + 0.995635i \(0.529753\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.2843 1.20906
\(714\) 0 0
\(715\) −3.65685 −0.136759
\(716\) 0 0
\(717\) −1.27208 −0.0475066
\(718\) 0 0
\(719\) −13.0711 −0.487469 −0.243734 0.969842i \(-0.578372\pi\)
−0.243734 + 0.969842i \(0.578372\pi\)
\(720\) 0 0
\(721\) 21.6569 0.806543
\(722\) 0 0
\(723\) 8.34315 0.310285
\(724\) 0 0
\(725\) −1.41421 −0.0525226
\(726\) 0 0
\(727\) 23.3553 0.866202 0.433101 0.901345i \(-0.357419\pi\)
0.433101 + 0.901345i \(0.357419\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −25.6569 −0.948953
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) −5.00000 −0.184428
\(736\) 0 0
\(737\) 10.3431 0.380995
\(738\) 0 0
\(739\) −25.6569 −0.943803 −0.471901 0.881651i \(-0.656432\pi\)
−0.471901 + 0.881651i \(0.656432\pi\)
\(740\) 0 0
\(741\) 0.585786 0.0215194
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −7.17157 −0.262394
\(748\) 0 0
\(749\) −4.68629 −0.171233
\(750\) 0 0
\(751\) −4.14214 −0.151149 −0.0755743 0.997140i \(-0.524079\pi\)
−0.0755743 + 0.997140i \(0.524079\pi\)
\(752\) 0 0
\(753\) −10.2426 −0.373263
\(754\) 0 0
\(755\) 10.4853 0.381598
\(756\) 0 0
\(757\) 52.4264 1.90547 0.952735 0.303802i \(-0.0982562\pi\)
0.952735 + 0.303802i \(0.0982562\pi\)
\(758\) 0 0
\(759\) 22.8284 0.828619
\(760\) 0 0
\(761\) −23.9411 −0.867865 −0.433933 0.900945i \(-0.642874\pi\)
−0.433933 + 0.900945i \(0.642874\pi\)
\(762\) 0 0
\(763\) −14.8284 −0.536825
\(764\) 0 0
\(765\) −6.82843 −0.246882
\(766\) 0 0
\(767\) −2.62742 −0.0948705
\(768\) 0 0
\(769\) 9.02944 0.325610 0.162805 0.986658i \(-0.447946\pi\)
0.162805 + 0.986658i \(0.447946\pi\)
\(770\) 0 0
\(771\) −12.4853 −0.449646
\(772\) 0 0
\(773\) −14.3431 −0.515887 −0.257944 0.966160i \(-0.583045\pi\)
−0.257944 + 0.966160i \(0.583045\pi\)
\(774\) 0 0
\(775\) 8.82843 0.317126
\(776\) 0 0
\(777\) −0.828427 −0.0297197
\(778\) 0 0
\(779\) 8.24264 0.295323
\(780\) 0 0
\(781\) −32.2843 −1.15522
\(782\) 0 0
\(783\) 1.41421 0.0505399
\(784\) 0 0
\(785\) 6.48528 0.231470
\(786\) 0 0
\(787\) 37.1716 1.32502 0.662512 0.749052i \(-0.269490\pi\)
0.662512 + 0.749052i \(0.269490\pi\)
\(788\) 0 0
\(789\) 27.4558 0.977454
\(790\) 0 0
\(791\) 25.6569 0.912253
\(792\) 0 0
\(793\) 8.97056 0.318554
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) 14.1421 0.500940 0.250470 0.968124i \(-0.419415\pi\)
0.250470 + 0.968124i \(0.419415\pi\)
\(798\) 0 0
\(799\) −24.9706 −0.883395
\(800\) 0 0
\(801\) −13.8995 −0.491115
\(802\) 0 0
\(803\) −22.8284 −0.805598
\(804\) 0 0
\(805\) 5.17157 0.182274
\(806\) 0 0
\(807\) −11.0711 −0.389720
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −31.3137 −1.09957 −0.549787 0.835305i \(-0.685291\pi\)
−0.549787 + 0.835305i \(0.685291\pi\)
\(812\) 0 0
\(813\) −5.17157 −0.181375
\(814\) 0 0
\(815\) 2.10051 0.0735775
\(816\) 0 0
\(817\) −3.75736 −0.131453
\(818\) 0 0
\(819\) 0.828427 0.0289476
\(820\) 0 0
\(821\) −2.48528 −0.0867369 −0.0433685 0.999059i \(-0.513809\pi\)
−0.0433685 + 0.999059i \(0.513809\pi\)
\(822\) 0 0
\(823\) −57.0122 −1.98732 −0.993660 0.112426i \(-0.964138\pi\)
−0.993660 + 0.112426i \(0.964138\pi\)
\(824\) 0 0
\(825\) 6.24264 0.217341
\(826\) 0 0
\(827\) 25.3137 0.880244 0.440122 0.897938i \(-0.354935\pi\)
0.440122 + 0.897938i \(0.354935\pi\)
\(828\) 0 0
\(829\) 29.5147 1.02509 0.512544 0.858661i \(-0.328703\pi\)
0.512544 + 0.858661i \(0.328703\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −34.1421 −1.18295
\(834\) 0 0
\(835\) 5.31371 0.183888
\(836\) 0 0
\(837\) −8.82843 −0.305155
\(838\) 0 0
\(839\) 38.1421 1.31681 0.658406 0.752663i \(-0.271231\pi\)
0.658406 + 0.752663i \(0.271231\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) 0 0
\(843\) 17.4142 0.599777
\(844\) 0 0
\(845\) 12.6569 0.435409
\(846\) 0 0
\(847\) −39.5563 −1.35917
\(848\) 0 0
\(849\) 26.3848 0.905523
\(850\) 0 0
\(851\) −2.14214 −0.0734315
\(852\) 0 0
\(853\) −11.1716 −0.382507 −0.191254 0.981541i \(-0.561255\pi\)
−0.191254 + 0.981541i \(0.561255\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 35.3137 1.20629 0.603147 0.797630i \(-0.293913\pi\)
0.603147 + 0.797630i \(0.293913\pi\)
\(858\) 0 0
\(859\) −17.9411 −0.612143 −0.306072 0.952008i \(-0.599015\pi\)
−0.306072 + 0.952008i \(0.599015\pi\)
\(860\) 0 0
\(861\) 11.6569 0.397265
\(862\) 0 0
\(863\) −31.3137 −1.06593 −0.532966 0.846137i \(-0.678922\pi\)
−0.532966 + 0.846137i \(0.678922\pi\)
\(864\) 0 0
\(865\) −19.7990 −0.673186
\(866\) 0 0
\(867\) −29.6274 −1.00620
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0.970563 0.0328863
\(872\) 0 0
\(873\) −18.2426 −0.617420
\(874\) 0 0
\(875\) 1.41421 0.0478091
\(876\) 0 0
\(877\) −49.0711 −1.65701 −0.828506 0.559980i \(-0.810809\pi\)
−0.828506 + 0.559980i \(0.810809\pi\)
\(878\) 0 0
\(879\) 28.4853 0.960785
\(880\) 0 0
\(881\) 33.7990 1.13872 0.569358 0.822089i \(-0.307192\pi\)
0.569358 + 0.822089i \(0.307192\pi\)
\(882\) 0 0
\(883\) −36.0416 −1.21290 −0.606449 0.795123i \(-0.707406\pi\)
−0.606449 + 0.795123i \(0.707406\pi\)
\(884\) 0 0
\(885\) 4.48528 0.150771
\(886\) 0 0
\(887\) 41.9411 1.40825 0.704123 0.710078i \(-0.251341\pi\)
0.704123 + 0.710078i \(0.251341\pi\)
\(888\) 0 0
\(889\) −4.68629 −0.157173
\(890\) 0 0
\(891\) −6.24264 −0.209136
\(892\) 0 0
\(893\) −3.65685 −0.122372
\(894\) 0 0
\(895\) −0.485281 −0.0162212
\(896\) 0 0
\(897\) 2.14214 0.0715238
\(898\) 0 0
\(899\) −12.4853 −0.416407
\(900\) 0 0
\(901\) 54.6274 1.81990
\(902\) 0 0
\(903\) −5.31371 −0.176829
\(904\) 0 0
\(905\) 15.1716 0.504320
\(906\) 0 0
\(907\) −10.1421 −0.336764 −0.168382 0.985722i \(-0.553854\pi\)
−0.168382 + 0.985722i \(0.553854\pi\)
\(908\) 0 0
\(909\) −8.82843 −0.292820
\(910\) 0 0
\(911\) −31.3137 −1.03747 −0.518735 0.854935i \(-0.673597\pi\)
−0.518735 + 0.854935i \(0.673597\pi\)
\(912\) 0 0
\(913\) 44.7696 1.48166
\(914\) 0 0
\(915\) −15.3137 −0.506256
\(916\) 0 0
\(917\) −16.1421 −0.533060
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 26.8284 0.884027
\(922\) 0 0
\(923\) −3.02944 −0.0997151
\(924\) 0 0
\(925\) −0.585786 −0.0192605
\(926\) 0 0
\(927\) −15.3137 −0.502968
\(928\) 0 0
\(929\) −29.1127 −0.955157 −0.477578 0.878589i \(-0.658485\pi\)
−0.477578 + 0.878589i \(0.658485\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) 2.24264 0.0734208
\(934\) 0 0
\(935\) 42.6274 1.39407
\(936\) 0 0
\(937\) 9.79899 0.320119 0.160060 0.987107i \(-0.448831\pi\)
0.160060 + 0.987107i \(0.448831\pi\)
\(938\) 0 0
\(939\) 33.7990 1.10299
\(940\) 0 0
\(941\) −30.8701 −1.00634 −0.503168 0.864189i \(-0.667832\pi\)
−0.503168 + 0.864189i \(0.667832\pi\)
\(942\) 0 0
\(943\) 30.1421 0.981563
\(944\) 0 0
\(945\) −1.41421 −0.0460044
\(946\) 0 0
\(947\) 52.8284 1.71669 0.858347 0.513070i \(-0.171492\pi\)
0.858347 + 0.513070i \(0.171492\pi\)
\(948\) 0 0
\(949\) −2.14214 −0.0695367
\(950\) 0 0
\(951\) −18.6274 −0.604035
\(952\) 0 0
\(953\) −2.54416 −0.0824133 −0.0412066 0.999151i \(-0.513120\pi\)
−0.0412066 + 0.999151i \(0.513120\pi\)
\(954\) 0 0
\(955\) 1.75736 0.0568668
\(956\) 0 0
\(957\) −8.82843 −0.285383
\(958\) 0 0
\(959\) −14.1421 −0.456673
\(960\) 0 0
\(961\) 46.9411 1.51423
\(962\) 0 0
\(963\) 3.31371 0.106783
\(964\) 0 0
\(965\) 9.07107 0.292008
\(966\) 0 0
\(967\) 40.0416 1.28765 0.643826 0.765172i \(-0.277346\pi\)
0.643826 + 0.765172i \(0.277346\pi\)
\(968\) 0 0
\(969\) −6.82843 −0.219361
\(970\) 0 0
\(971\) −33.6569 −1.08010 −0.540050 0.841633i \(-0.681595\pi\)
−0.540050 + 0.841633i \(0.681595\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 0 0
\(975\) 0.585786 0.0187602
\(976\) 0 0
\(977\) −24.2843 −0.776923 −0.388461 0.921465i \(-0.626993\pi\)
−0.388461 + 0.921465i \(0.626993\pi\)
\(978\) 0 0
\(979\) 86.7696 2.77317
\(980\) 0 0
\(981\) 10.4853 0.334769
\(982\) 0 0
\(983\) −20.6274 −0.657912 −0.328956 0.944345i \(-0.606697\pi\)
−0.328956 + 0.944345i \(0.606697\pi\)
\(984\) 0 0
\(985\) −1.17157 −0.0373294
\(986\) 0 0
\(987\) −5.17157 −0.164613
\(988\) 0 0
\(989\) −13.7401 −0.436910
\(990\) 0 0
\(991\) −13.6569 −0.433824 −0.216912 0.976191i \(-0.569599\pi\)
−0.216912 + 0.976191i \(0.569599\pi\)
\(992\) 0 0
\(993\) −0.142136 −0.00451054
\(994\) 0 0
\(995\) −10.1421 −0.321527
\(996\) 0 0
\(997\) 58.0833 1.83952 0.919758 0.392487i \(-0.128385\pi\)
0.919758 + 0.392487i \(0.128385\pi\)
\(998\) 0 0
\(999\) 0.585786 0.0185335
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 4560.2.a.bf.1.1 2
4.3 odd 2 285.2.a.g.1.1 2
12.11 even 2 855.2.a.d.1.2 2
20.3 even 4 1425.2.c.l.799.3 4
20.7 even 4 1425.2.c.l.799.2 4
20.19 odd 2 1425.2.a.k.1.2 2
60.59 even 2 4275.2.a.y.1.1 2
76.75 even 2 5415.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.1 2 4.3 odd 2
855.2.a.d.1.2 2 12.11 even 2
1425.2.a.k.1.2 2 20.19 odd 2
1425.2.c.l.799.2 4 20.7 even 4
1425.2.c.l.799.3 4 20.3 even 4
4275.2.a.y.1.1 2 60.59 even 2
4560.2.a.bf.1.1 2 1.1 even 1 trivial
5415.2.a.n.1.2 2 76.75 even 2