Properties

Label 8880.2.a.cg.1.2
Level $8880$
Weight $2$
Character 8880.1
Self dual yes
Analytic conductor $70.907$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.9071569949\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.339102\) of defining polynomial
Character \(\chi\) \(=\) 8880.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.84556 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.84556 q^{7} +1.00000 q^{9} -0.832640 q^{11} +3.21973 q^{13} -1.00000 q^{15} -7.95030 q^{17} -4.20681 q^{19} +1.84556 q^{21} +6.57614 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.03945 q^{29} +4.73057 q^{31} +0.832640 q^{33} -1.84556 q^{35} -1.00000 q^{37} -3.21973 q^{39} +2.73057 q^{41} +5.03945 q^{43} +1.00000 q^{45} -9.11766 q^{47} -3.59390 q^{49} +7.95030 q^{51} +3.95030 q^{53} -0.832640 q^{55} +4.20681 q^{57} +3.69113 q^{59} +5.40878 q^{61} -1.84556 q^{63} +3.21973 q^{65} -11.7959 q^{67} -6.57614 q^{69} +13.7700 q^{71} -3.52860 q^{73} -1.00000 q^{75} +1.53669 q^{77} -4.00000 q^{79} +1.00000 q^{81} +3.55445 q^{83} -7.95030 q^{85} -3.03945 q^{87} -6.55029 q^{89} -5.94222 q^{91} -4.73057 q^{93} -4.20681 q^{95} -10.4918 q^{97} -0.832640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} - 2q^{11} - 3q^{13} - 4q^{15} + 6q^{17} - 3q^{19} + 4q^{21} + q^{23} + 4q^{25} - 4q^{27} - 3q^{29} - 3q^{31} + 2q^{33} - 4q^{35} - 4q^{37} + 3q^{39} - 11q^{41} + 5q^{43} + 4q^{45} + 14q^{49} - 6q^{51} - 22q^{53} - 2q^{55} + 3q^{57} + 8q^{59} - 5q^{61} - 4q^{63} - 3q^{65} - 6q^{67} - q^{69} + 18q^{71} - 5q^{73} - 4q^{75} - 4q^{77} - 16q^{79} + 4q^{81} + q^{83} + 6q^{85} + 3q^{87} - 5q^{89} + 13q^{91} + 3q^{93} - 3q^{95} + 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(7\) −1.84556 −0.697557 −0.348779 0.937205i \(-0.613404\pi\)
−0.348779 + 0.937205i \(0.613404\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.832640 −0.251050 −0.125525 0.992090i \(-0.540062\pi\)
−0.125525 + 0.992090i \(0.540062\pi\)
\(12\) 0 0
\(13\) 3.21973 0.892992 0.446496 0.894786i \(-0.352672\pi\)
0.446496 + 0.894786i \(0.352672\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −7.95030 −1.92823 −0.964116 0.265482i \(-0.914469\pi\)
−0.964116 + 0.265482i \(0.914469\pi\)
\(18\) 0 0
\(19\) −4.20681 −0.965108 −0.482554 0.875866i \(-0.660291\pi\)
−0.482554 + 0.875866i \(0.660291\pi\)
\(20\) 0 0
\(21\) 1.84556 0.402735
\(22\) 0 0
\(23\) 6.57614 1.37122 0.685610 0.727969i \(-0.259536\pi\)
0.685610 + 0.727969i \(0.259536\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.03945 0.564411 0.282206 0.959354i \(-0.408934\pi\)
0.282206 + 0.959354i \(0.408934\pi\)
\(30\) 0 0
\(31\) 4.73057 0.849636 0.424818 0.905279i \(-0.360338\pi\)
0.424818 + 0.905279i \(0.360338\pi\)
\(32\) 0 0
\(33\) 0.832640 0.144944
\(34\) 0 0
\(35\) −1.84556 −0.311957
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −3.21973 −0.515569
\(40\) 0 0
\(41\) 2.73057 0.426444 0.213222 0.977004i \(-0.431604\pi\)
0.213222 + 0.977004i \(0.431604\pi\)
\(42\) 0 0
\(43\) 5.03945 0.768508 0.384254 0.923227i \(-0.374459\pi\)
0.384254 + 0.923227i \(0.374459\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −9.11766 −1.32995 −0.664974 0.746867i \(-0.731557\pi\)
−0.664974 + 0.746867i \(0.731557\pi\)
\(48\) 0 0
\(49\) −3.59390 −0.513414
\(50\) 0 0
\(51\) 7.95030 1.11327
\(52\) 0 0
\(53\) 3.95030 0.542616 0.271308 0.962493i \(-0.412544\pi\)
0.271308 + 0.962493i \(0.412544\pi\)
\(54\) 0 0
\(55\) −0.832640 −0.112273
\(56\) 0 0
\(57\) 4.20681 0.557205
\(58\) 0 0
\(59\) 3.69113 0.480544 0.240272 0.970706i \(-0.422763\pi\)
0.240272 + 0.970706i \(0.422763\pi\)
\(60\) 0 0
\(61\) 5.40878 0.692523 0.346261 0.938138i \(-0.387451\pi\)
0.346261 + 0.938138i \(0.387451\pi\)
\(62\) 0 0
\(63\) −1.84556 −0.232519
\(64\) 0 0
\(65\) 3.21973 0.399358
\(66\) 0 0
\(67\) −11.7959 −1.44109 −0.720547 0.693406i \(-0.756109\pi\)
−0.720547 + 0.693406i \(0.756109\pi\)
\(68\) 0 0
\(69\) −6.57614 −0.791674
\(70\) 0 0
\(71\) 13.7700 1.63420 0.817100 0.576495i \(-0.195580\pi\)
0.817100 + 0.576495i \(0.195580\pi\)
\(72\) 0 0
\(73\) −3.52860 −0.412992 −0.206496 0.978447i \(-0.566206\pi\)
−0.206496 + 0.978447i \(0.566206\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.53669 0.175122
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.55445 0.390151 0.195076 0.980788i \(-0.437505\pi\)
0.195076 + 0.980788i \(0.437505\pi\)
\(84\) 0 0
\(85\) −7.95030 −0.862331
\(86\) 0 0
\(87\) −3.03945 −0.325863
\(88\) 0 0
\(89\) −6.55029 −0.694329 −0.347165 0.937804i \(-0.612856\pi\)
−0.347165 + 0.937804i \(0.612856\pi\)
\(90\) 0 0
\(91\) −5.94222 −0.622913
\(92\) 0 0
\(93\) −4.73057 −0.490538
\(94\) 0 0
\(95\) −4.20681 −0.431609
\(96\) 0 0
\(97\) −10.4918 −1.06528 −0.532642 0.846341i \(-0.678801\pi\)
−0.532642 + 0.846341i \(0.678801\pi\)
\(98\) 0 0
\(99\) −0.832640 −0.0836835
\(100\) 0 0
\(101\) −3.63067 −0.361265 −0.180633 0.983551i \(-0.557814\pi\)
−0.180633 + 0.983551i \(0.557814\pi\)
\(102\) 0 0
\(103\) 9.11766 0.898390 0.449195 0.893434i \(-0.351711\pi\)
0.449195 + 0.893434i \(0.351711\pi\)
\(104\) 0 0
\(105\) 1.84556 0.180109
\(106\) 0 0
\(107\) −8.91086 −0.861445 −0.430723 0.902484i \(-0.641741\pi\)
−0.430723 + 0.902484i \(0.641741\pi\)
\(108\) 0 0
\(109\) 11.2979 1.08215 0.541073 0.840975i \(-0.318018\pi\)
0.541073 + 0.840975i \(0.318018\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) −11.0394 −1.03850 −0.519252 0.854621i \(-0.673789\pi\)
−0.519252 + 0.854621i \(0.673789\pi\)
\(114\) 0 0
\(115\) 6.57614 0.613228
\(116\) 0 0
\(117\) 3.21973 0.297664
\(118\) 0 0
\(119\) 14.6728 1.34505
\(120\) 0 0
\(121\) −10.3067 −0.936974
\(122\) 0 0
\(123\) −2.73057 −0.246208
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.136678 0.0121282 0.00606410 0.999982i \(-0.498070\pi\)
0.00606410 + 0.999982i \(0.498070\pi\)
\(128\) 0 0
\(129\) −5.03945 −0.443699
\(130\) 0 0
\(131\) −2.07889 −0.181634 −0.0908169 0.995868i \(-0.528948\pi\)
−0.0908169 + 0.995868i \(0.528948\pi\)
\(132\) 0 0
\(133\) 7.76393 0.673218
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 3.06597 0.261943 0.130972 0.991386i \(-0.458190\pi\)
0.130972 + 0.991386i \(0.458190\pi\)
\(138\) 0 0
\(139\) −22.1917 −1.88228 −0.941139 0.338021i \(-0.890243\pi\)
−0.941139 + 0.338021i \(0.890243\pi\)
\(140\) 0 0
\(141\) 9.11766 0.767846
\(142\) 0 0
\(143\) −2.68088 −0.224186
\(144\) 0 0
\(145\) 3.03945 0.252412
\(146\) 0 0
\(147\) 3.59390 0.296420
\(148\) 0 0
\(149\) −14.4999 −1.18788 −0.593940 0.804510i \(-0.702428\pi\)
−0.593940 + 0.804510i \(0.702428\pi\)
\(150\) 0 0
\(151\) −8.24142 −0.670677 −0.335339 0.942098i \(-0.608851\pi\)
−0.335339 + 0.942098i \(0.608851\pi\)
\(152\) 0 0
\(153\) −7.95030 −0.642744
\(154\) 0 0
\(155\) 4.73057 0.379969
\(156\) 0 0
\(157\) −6.11283 −0.487857 −0.243928 0.969793i \(-0.578436\pi\)
−0.243928 + 0.969793i \(0.578436\pi\)
\(158\) 0 0
\(159\) −3.95030 −0.313279
\(160\) 0 0
\(161\) −12.1367 −0.956504
\(162\) 0 0
\(163\) 21.4373 1.67910 0.839549 0.543283i \(-0.182819\pi\)
0.839549 + 0.543283i \(0.182819\pi\)
\(164\) 0 0
\(165\) 0.832640 0.0648210
\(166\) 0 0
\(167\) 22.5815 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(168\) 0 0
\(169\) −2.63334 −0.202565
\(170\) 0 0
\(171\) −4.20681 −0.321703
\(172\) 0 0
\(173\) −6.48916 −0.493361 −0.246681 0.969097i \(-0.579340\pi\)
−0.246681 + 0.969097i \(0.579340\pi\)
\(174\) 0 0
\(175\) −1.84556 −0.139511
\(176\) 0 0
\(177\) −3.69113 −0.277442
\(178\) 0 0
\(179\) −16.2095 −1.21155 −0.605777 0.795635i \(-0.707138\pi\)
−0.605777 + 0.795635i \(0.707138\pi\)
\(180\) 0 0
\(181\) 5.38225 0.400060 0.200030 0.979790i \(-0.435896\pi\)
0.200030 + 0.979790i \(0.435896\pi\)
\(182\) 0 0
\(183\) −5.40878 −0.399828
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 6.61974 0.484083
\(188\) 0 0
\(189\) 1.84556 0.134245
\(190\) 0 0
\(191\) 10.5680 0.764677 0.382339 0.924022i \(-0.375119\pi\)
0.382339 + 0.924022i \(0.375119\pi\)
\(192\) 0 0
\(193\) 3.11766 0.224414 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(194\) 0 0
\(195\) −3.21973 −0.230570
\(196\) 0 0
\(197\) 0.576137 0.0410481 0.0205240 0.999789i \(-0.493467\pi\)
0.0205240 + 0.999789i \(0.493467\pi\)
\(198\) 0 0
\(199\) −6.64359 −0.470952 −0.235476 0.971880i \(-0.575665\pi\)
−0.235476 + 0.971880i \(0.575665\pi\)
\(200\) 0 0
\(201\) 11.7959 0.832016
\(202\) 0 0
\(203\) −5.60949 −0.393709
\(204\) 0 0
\(205\) 2.73057 0.190712
\(206\) 0 0
\(207\) 6.57614 0.457073
\(208\) 0 0
\(209\) 3.50276 0.242291
\(210\) 0 0
\(211\) 8.83531 0.608248 0.304124 0.952632i \(-0.401636\pi\)
0.304124 + 0.952632i \(0.401636\pi\)
\(212\) 0 0
\(213\) −13.7700 −0.943506
\(214\) 0 0
\(215\) 5.03945 0.343687
\(216\) 0 0
\(217\) −8.73057 −0.592670
\(218\) 0 0
\(219\) 3.52860 0.238441
\(220\) 0 0
\(221\) −25.5978 −1.72190
\(222\) 0 0
\(223\) −17.8828 −1.19752 −0.598762 0.800927i \(-0.704340\pi\)
−0.598762 + 0.800927i \(0.704340\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −25.1959 −1.67231 −0.836155 0.548494i \(-0.815201\pi\)
−0.836155 + 0.548494i \(0.815201\pi\)
\(228\) 0 0
\(229\) −5.48699 −0.362591 −0.181295 0.983429i \(-0.558029\pi\)
−0.181295 + 0.983429i \(0.558029\pi\)
\(230\) 0 0
\(231\) −1.53669 −0.101107
\(232\) 0 0
\(233\) −28.5788 −1.87226 −0.936130 0.351654i \(-0.885619\pi\)
−0.936130 + 0.351654i \(0.885619\pi\)
\(234\) 0 0
\(235\) −9.11766 −0.594771
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −1.77811 −0.115016 −0.0575081 0.998345i \(-0.518316\pi\)
−0.0575081 + 0.998345i \(0.518316\pi\)
\(240\) 0 0
\(241\) 22.2612 1.43397 0.716984 0.697090i \(-0.245522\pi\)
0.716984 + 0.697090i \(0.245522\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.59390 −0.229606
\(246\) 0 0
\(247\) −13.5448 −0.861834
\(248\) 0 0
\(249\) −3.55445 −0.225254
\(250\) 0 0
\(251\) 28.6748 1.80994 0.904968 0.425479i \(-0.139894\pi\)
0.904968 + 0.425479i \(0.139894\pi\)
\(252\) 0 0
\(253\) −5.47556 −0.344245
\(254\) 0 0
\(255\) 7.95030 0.497867
\(256\) 0 0
\(257\) 9.19388 0.573499 0.286749 0.958006i \(-0.407425\pi\)
0.286749 + 0.958006i \(0.407425\pi\)
\(258\) 0 0
\(259\) 1.84556 0.114678
\(260\) 0 0
\(261\) 3.03945 0.188137
\(262\) 0 0
\(263\) 8.02652 0.494937 0.247468 0.968896i \(-0.420401\pi\)
0.247468 + 0.968896i \(0.420401\pi\)
\(264\) 0 0
\(265\) 3.95030 0.242665
\(266\) 0 0
\(267\) 6.55029 0.400871
\(268\) 0 0
\(269\) 21.3849 1.30386 0.651931 0.758278i \(-0.273959\pi\)
0.651931 + 0.758278i \(0.273959\pi\)
\(270\) 0 0
\(271\) −2.64359 −0.160587 −0.0802934 0.996771i \(-0.525586\pi\)
−0.0802934 + 0.996771i \(0.525586\pi\)
\(272\) 0 0
\(273\) 5.94222 0.359639
\(274\) 0 0
\(275\) −0.832640 −0.0502101
\(276\) 0 0
\(277\) −14.4456 −0.867949 −0.433975 0.900925i \(-0.642889\pi\)
−0.433975 + 0.900925i \(0.642889\pi\)
\(278\) 0 0
\(279\) 4.73057 0.283212
\(280\) 0 0
\(281\) −11.2197 −0.669313 −0.334656 0.942340i \(-0.608620\pi\)
−0.334656 + 0.942340i \(0.608620\pi\)
\(282\) 0 0
\(283\) −3.86332 −0.229651 −0.114825 0.993386i \(-0.536631\pi\)
−0.114825 + 0.993386i \(0.536631\pi\)
\(284\) 0 0
\(285\) 4.20681 0.249190
\(286\) 0 0
\(287\) −5.03945 −0.297469
\(288\) 0 0
\(289\) 46.2073 2.71808
\(290\) 0 0
\(291\) 10.4918 0.615042
\(292\) 0 0
\(293\) −16.1544 −0.943752 −0.471876 0.881665i \(-0.656423\pi\)
−0.471876 + 0.881665i \(0.656423\pi\)
\(294\) 0 0
\(295\) 3.69113 0.214906
\(296\) 0 0
\(297\) 0.832640 0.0483147
\(298\) 0 0
\(299\) 21.1734 1.22449
\(300\) 0 0
\(301\) −9.30062 −0.536079
\(302\) 0 0
\(303\) 3.63067 0.208577
\(304\) 0 0
\(305\) 5.40878 0.309706
\(306\) 0 0
\(307\) −6.28303 −0.358591 −0.179296 0.983795i \(-0.557382\pi\)
−0.179296 + 0.983795i \(0.557382\pi\)
\(308\) 0 0
\(309\) −9.11766 −0.518686
\(310\) 0 0
\(311\) −20.6312 −1.16989 −0.584943 0.811074i \(-0.698883\pi\)
−0.584943 + 0.811074i \(0.698883\pi\)
\(312\) 0 0
\(313\) −7.32180 −0.413852 −0.206926 0.978357i \(-0.566346\pi\)
−0.206926 + 0.978357i \(0.566346\pi\)
\(314\) 0 0
\(315\) −1.84556 −0.103986
\(316\) 0 0
\(317\) 19.1442 1.07524 0.537622 0.843186i \(-0.319323\pi\)
0.537622 + 0.843186i \(0.319323\pi\)
\(318\) 0 0
\(319\) −2.53077 −0.141696
\(320\) 0 0
\(321\) 8.91086 0.497356
\(322\) 0 0
\(323\) 33.4454 1.86095
\(324\) 0 0
\(325\) 3.21973 0.178598
\(326\) 0 0
\(327\) −11.2979 −0.624778
\(328\) 0 0
\(329\) 16.8272 0.927715
\(330\) 0 0
\(331\) −0.908184 −0.0499183 −0.0249591 0.999688i \(-0.507946\pi\)
−0.0249591 + 0.999688i \(0.507946\pi\)
\(332\) 0 0
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) −11.7959 −0.644477
\(336\) 0 0
\(337\) −8.11083 −0.441825 −0.220913 0.975294i \(-0.570904\pi\)
−0.220913 + 0.975294i \(0.570904\pi\)
\(338\) 0 0
\(339\) 11.0394 0.599580
\(340\) 0 0
\(341\) −3.93887 −0.213302
\(342\) 0 0
\(343\) 19.5517 1.05569
\(344\) 0 0
\(345\) −6.57614 −0.354047
\(346\) 0 0
\(347\) −9.35641 −0.502278 −0.251139 0.967951i \(-0.580805\pi\)
−0.251139 + 0.967951i \(0.580805\pi\)
\(348\) 0 0
\(349\) −35.4665 −1.89848 −0.949239 0.314556i \(-0.898144\pi\)
−0.949239 + 0.314556i \(0.898144\pi\)
\(350\) 0 0
\(351\) −3.21973 −0.171856
\(352\) 0 0
\(353\) −36.0923 −1.92100 −0.960500 0.278279i \(-0.910236\pi\)
−0.960500 + 0.278279i \(0.910236\pi\)
\(354\) 0 0
\(355\) 13.7700 0.730837
\(356\) 0 0
\(357\) −14.6728 −0.776566
\(358\) 0 0
\(359\) −2.07889 −0.109720 −0.0548599 0.998494i \(-0.517471\pi\)
−0.0548599 + 0.998494i \(0.517471\pi\)
\(360\) 0 0
\(361\) −1.30278 −0.0685674
\(362\) 0 0
\(363\) 10.3067 0.540962
\(364\) 0 0
\(365\) −3.52860 −0.184696
\(366\) 0 0
\(367\) −34.2645 −1.78859 −0.894297 0.447474i \(-0.852324\pi\)
−0.894297 + 0.447474i \(0.852324\pi\)
\(368\) 0 0
\(369\) 2.73057 0.142148
\(370\) 0 0
\(371\) −7.29053 −0.378506
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 9.78620 0.504015
\(378\) 0 0
\(379\) −29.8217 −1.53184 −0.765919 0.642937i \(-0.777716\pi\)
−0.765919 + 0.642937i \(0.777716\pi\)
\(380\) 0 0
\(381\) −0.136678 −0.00700222
\(382\) 0 0
\(383\) 6.78027 0.346456 0.173228 0.984882i \(-0.444580\pi\)
0.173228 + 0.984882i \(0.444580\pi\)
\(384\) 0 0
\(385\) 1.53669 0.0783170
\(386\) 0 0
\(387\) 5.03945 0.256169
\(388\) 0 0
\(389\) 35.1959 1.78450 0.892251 0.451540i \(-0.149125\pi\)
0.892251 + 0.451540i \(0.149125\pi\)
\(390\) 0 0
\(391\) −52.2823 −2.64403
\(392\) 0 0
\(393\) 2.07889 0.104866
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −3.56054 −0.178698 −0.0893492 0.996000i \(-0.528479\pi\)
−0.0893492 + 0.996000i \(0.528479\pi\)
\(398\) 0 0
\(399\) −7.76393 −0.388683
\(400\) 0 0
\(401\) 1.44971 0.0723950 0.0361975 0.999345i \(-0.488475\pi\)
0.0361975 + 0.999345i \(0.488475\pi\)
\(402\) 0 0
\(403\) 15.2312 0.758718
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0.832640 0.0412724
\(408\) 0 0
\(409\) 25.1781 1.24498 0.622489 0.782629i \(-0.286122\pi\)
0.622489 + 0.782629i \(0.286122\pi\)
\(410\) 0 0
\(411\) −3.06597 −0.151233
\(412\) 0 0
\(413\) −6.81221 −0.335207
\(414\) 0 0
\(415\) 3.55445 0.174481
\(416\) 0 0
\(417\) 22.1917 1.08673
\(418\) 0 0
\(419\) −31.6938 −1.54834 −0.774172 0.632976i \(-0.781833\pi\)
−0.774172 + 0.632976i \(0.781833\pi\)
\(420\) 0 0
\(421\) −5.14351 −0.250679 −0.125340 0.992114i \(-0.540002\pi\)
−0.125340 + 0.992114i \(0.540002\pi\)
\(422\) 0 0
\(423\) −9.11766 −0.443316
\(424\) 0 0
\(425\) −7.95030 −0.385646
\(426\) 0 0
\(427\) −9.98224 −0.483075
\(428\) 0 0
\(429\) 2.68088 0.129434
\(430\) 0 0
\(431\) −2.72448 −0.131234 −0.0656168 0.997845i \(-0.520902\pi\)
−0.0656168 + 0.997845i \(0.520902\pi\)
\(432\) 0 0
\(433\) −8.68088 −0.417176 −0.208588 0.978004i \(-0.566887\pi\)
−0.208588 + 0.978004i \(0.566887\pi\)
\(434\) 0 0
\(435\) −3.03945 −0.145730
\(436\) 0 0
\(437\) −27.6645 −1.32337
\(438\) 0 0
\(439\) −7.64752 −0.364996 −0.182498 0.983206i \(-0.558418\pi\)
−0.182498 + 0.983206i \(0.558418\pi\)
\(440\) 0 0
\(441\) −3.59390 −0.171138
\(442\) 0 0
\(443\) −16.7917 −0.797798 −0.398899 0.916995i \(-0.630608\pi\)
−0.398899 + 0.916995i \(0.630608\pi\)
\(444\) 0 0
\(445\) −6.55029 −0.310514
\(446\) 0 0
\(447\) 14.4999 0.685823
\(448\) 0 0
\(449\) 26.1047 1.23196 0.615979 0.787762i \(-0.288760\pi\)
0.615979 + 0.787762i \(0.288760\pi\)
\(450\) 0 0
\(451\) −2.27359 −0.107059
\(452\) 0 0
\(453\) 8.24142 0.387216
\(454\) 0 0
\(455\) −5.94222 −0.278575
\(456\) 0 0
\(457\) −17.6646 −0.826315 −0.413158 0.910660i \(-0.635574\pi\)
−0.413158 + 0.910660i \(0.635574\pi\)
\(458\) 0 0
\(459\) 7.95030 0.371088
\(460\) 0 0
\(461\) −21.2231 −0.988457 −0.494229 0.869332i \(-0.664549\pi\)
−0.494229 + 0.869332i \(0.664549\pi\)
\(462\) 0 0
\(463\) −32.5530 −1.51286 −0.756432 0.654072i \(-0.773059\pi\)
−0.756432 + 0.654072i \(0.773059\pi\)
\(464\) 0 0
\(465\) −4.73057 −0.219375
\(466\) 0 0
\(467\) −15.1170 −0.699531 −0.349765 0.936837i \(-0.613739\pi\)
−0.349765 + 0.936837i \(0.613739\pi\)
\(468\) 0 0
\(469\) 21.7700 1.00525
\(470\) 0 0
\(471\) 6.11283 0.281664
\(472\) 0 0
\(473\) −4.19605 −0.192934
\(474\) 0 0
\(475\) −4.20681 −0.193022
\(476\) 0 0
\(477\) 3.95030 0.180872
\(478\) 0 0
\(479\) −3.44971 −0.157621 −0.0788106 0.996890i \(-0.525112\pi\)
−0.0788106 + 0.996890i \(0.525112\pi\)
\(480\) 0 0
\(481\) −3.21973 −0.146807
\(482\) 0 0
\(483\) 12.1367 0.552238
\(484\) 0 0
\(485\) −10.4918 −0.476409
\(486\) 0 0
\(487\) 15.6619 0.709707 0.354853 0.934922i \(-0.384531\pi\)
0.354853 + 0.934922i \(0.384531\pi\)
\(488\) 0 0
\(489\) −21.4373 −0.969428
\(490\) 0 0
\(491\) −7.89793 −0.356429 −0.178214 0.983992i \(-0.557032\pi\)
−0.178214 + 0.983992i \(0.557032\pi\)
\(492\) 0 0
\(493\) −24.1645 −1.08832
\(494\) 0 0
\(495\) −0.832640 −0.0374244
\(496\) 0 0
\(497\) −25.4134 −1.13995
\(498\) 0 0
\(499\) −5.87209 −0.262871 −0.131435 0.991325i \(-0.541959\pi\)
−0.131435 + 0.991325i \(0.541959\pi\)
\(500\) 0 0
\(501\) −22.5815 −1.00887
\(502\) 0 0
\(503\) −3.33056 −0.148502 −0.0742512 0.997240i \(-0.523657\pi\)
−0.0742512 + 0.997240i \(0.523657\pi\)
\(504\) 0 0
\(505\) −3.63067 −0.161563
\(506\) 0 0
\(507\) 2.63334 0.116951
\(508\) 0 0
\(509\) −30.9508 −1.37187 −0.685935 0.727663i \(-0.740607\pi\)
−0.685935 + 0.727663i \(0.740607\pi\)
\(510\) 0 0
\(511\) 6.51226 0.288085
\(512\) 0 0
\(513\) 4.20681 0.185735
\(514\) 0 0
\(515\) 9.11766 0.401772
\(516\) 0 0
\(517\) 7.59173 0.333884
\(518\) 0 0
\(519\) 6.48916 0.284842
\(520\) 0 0
\(521\) 7.45306 0.326524 0.163262 0.986583i \(-0.447798\pi\)
0.163262 + 0.986583i \(0.447798\pi\)
\(522\) 0 0
\(523\) −31.5917 −1.38141 −0.690705 0.723137i \(-0.742700\pi\)
−0.690705 + 0.723137i \(0.742700\pi\)
\(524\) 0 0
\(525\) 1.84556 0.0805470
\(526\) 0 0
\(527\) −37.6095 −1.63830
\(528\) 0 0
\(529\) 20.2456 0.880242
\(530\) 0 0
\(531\) 3.69113 0.160181
\(532\) 0 0
\(533\) 8.79171 0.380811
\(534\) 0 0
\(535\) −8.91086 −0.385250
\(536\) 0 0
\(537\) 16.2095 0.699491
\(538\) 0 0
\(539\) 2.99242 0.128893
\(540\) 0 0
\(541\) 36.0816 1.55127 0.775634 0.631183i \(-0.217430\pi\)
0.775634 + 0.631183i \(0.217430\pi\)
\(542\) 0 0
\(543\) −5.38225 −0.230975
\(544\) 0 0
\(545\) 11.2979 0.483951
\(546\) 0 0
\(547\) 38.7420 1.65649 0.828244 0.560367i \(-0.189340\pi\)
0.828244 + 0.560367i \(0.189340\pi\)
\(548\) 0 0
\(549\) 5.40878 0.230841
\(550\) 0 0
\(551\) −12.7864 −0.544717
\(552\) 0 0
\(553\) 7.38225 0.313925
\(554\) 0 0
\(555\) 1.00000 0.0424476
\(556\) 0 0
\(557\) −23.3618 −0.989869 −0.494935 0.868930i \(-0.664808\pi\)
−0.494935 + 0.868930i \(0.664808\pi\)
\(558\) 0 0
\(559\) 16.2257 0.686272
\(560\) 0 0
\(561\) −6.61974 −0.279486
\(562\) 0 0
\(563\) −39.0351 −1.64513 −0.822567 0.568668i \(-0.807459\pi\)
−0.822567 + 0.568668i \(0.807459\pi\)
\(564\) 0 0
\(565\) −11.0394 −0.464433
\(566\) 0 0
\(567\) −1.84556 −0.0775064
\(568\) 0 0
\(569\) 35.4809 1.48744 0.743718 0.668493i \(-0.233060\pi\)
0.743718 + 0.668493i \(0.233060\pi\)
\(570\) 0 0
\(571\) −37.0093 −1.54879 −0.774395 0.632702i \(-0.781946\pi\)
−0.774395 + 0.632702i \(0.781946\pi\)
\(572\) 0 0
\(573\) −10.5680 −0.441487
\(574\) 0 0
\(575\) 6.57614 0.274244
\(576\) 0 0
\(577\) −15.7354 −0.655074 −0.327537 0.944838i \(-0.606219\pi\)
−0.327537 + 0.944838i \(0.606219\pi\)
\(578\) 0 0
\(579\) −3.11766 −0.129566
\(580\) 0 0
\(581\) −6.55996 −0.272153
\(582\) 0 0
\(583\) −3.28918 −0.136224
\(584\) 0 0
\(585\) 3.21973 0.133119
\(586\) 0 0
\(587\) −37.3006 −1.53956 −0.769781 0.638309i \(-0.779634\pi\)
−0.769781 + 0.638309i \(0.779634\pi\)
\(588\) 0 0
\(589\) −19.9006 −0.819990
\(590\) 0 0
\(591\) −0.576137 −0.0236991
\(592\) 0 0
\(593\) −17.9829 −0.738470 −0.369235 0.929336i \(-0.620380\pi\)
−0.369235 + 0.929336i \(0.620380\pi\)
\(594\) 0 0
\(595\) 14.6728 0.601526
\(596\) 0 0
\(597\) 6.64359 0.271904
\(598\) 0 0
\(599\) 32.9998 1.34834 0.674168 0.738578i \(-0.264502\pi\)
0.674168 + 0.738578i \(0.264502\pi\)
\(600\) 0 0
\(601\) 33.0334 1.34746 0.673729 0.738978i \(-0.264691\pi\)
0.673729 + 0.738978i \(0.264691\pi\)
\(602\) 0 0
\(603\) −11.7959 −0.480365
\(604\) 0 0
\(605\) −10.3067 −0.419027
\(606\) 0 0
\(607\) −27.6102 −1.12066 −0.560331 0.828269i \(-0.689326\pi\)
−0.560331 + 0.828269i \(0.689326\pi\)
\(608\) 0 0
\(609\) 5.60949 0.227308
\(610\) 0 0
\(611\) −29.3564 −1.18763
\(612\) 0 0
\(613\) 1.62999 0.0658348 0.0329174 0.999458i \(-0.489520\pi\)
0.0329174 + 0.999458i \(0.489520\pi\)
\(614\) 0 0
\(615\) −2.73057 −0.110107
\(616\) 0 0
\(617\) 28.6143 1.15197 0.575985 0.817460i \(-0.304619\pi\)
0.575985 + 0.817460i \(0.304619\pi\)
\(618\) 0 0
\(619\) −19.6830 −0.791128 −0.395564 0.918438i \(-0.629451\pi\)
−0.395564 + 0.918438i \(0.629451\pi\)
\(620\) 0 0
\(621\) −6.57614 −0.263891
\(622\) 0 0
\(623\) 12.0890 0.484335
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.50276 −0.139887
\(628\) 0 0
\(629\) 7.95030 0.316999
\(630\) 0 0
\(631\) 5.76193 0.229379 0.114689 0.993401i \(-0.463413\pi\)
0.114689 + 0.993401i \(0.463413\pi\)
\(632\) 0 0
\(633\) −8.83531 −0.351172
\(634\) 0 0
\(635\) 0.136678 0.00542390
\(636\) 0 0
\(637\) −11.5714 −0.458474
\(638\) 0 0
\(639\) 13.7700 0.544734
\(640\) 0 0
\(641\) 20.8967 0.825369 0.412685 0.910874i \(-0.364591\pi\)
0.412685 + 0.910874i \(0.364591\pi\)
\(642\) 0 0
\(643\) −30.6197 −1.20752 −0.603762 0.797164i \(-0.706332\pi\)
−0.603762 + 0.797164i \(0.706332\pi\)
\(644\) 0 0
\(645\) −5.03945 −0.198428
\(646\) 0 0
\(647\) 10.6278 0.417823 0.208912 0.977935i \(-0.433008\pi\)
0.208912 + 0.977935i \(0.433008\pi\)
\(648\) 0 0
\(649\) −3.07338 −0.120641
\(650\) 0 0
\(651\) 8.73057 0.342178
\(652\) 0 0
\(653\) −5.90061 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(654\) 0 0
\(655\) −2.07889 −0.0812291
\(656\) 0 0
\(657\) −3.52860 −0.137664
\(658\) 0 0
\(659\) −19.8305 −0.772486 −0.386243 0.922397i \(-0.626227\pi\)
−0.386243 + 0.922397i \(0.626227\pi\)
\(660\) 0 0
\(661\) −9.32379 −0.362653 −0.181327 0.983423i \(-0.558039\pi\)
−0.181327 + 0.983423i \(0.558039\pi\)
\(662\) 0 0
\(663\) 25.5978 0.994137
\(664\) 0 0
\(665\) 7.76393 0.301072
\(666\) 0 0
\(667\) 19.9878 0.773931
\(668\) 0 0
\(669\) 17.8828 0.691391
\(670\) 0 0
\(671\) −4.50357 −0.173858
\(672\) 0 0
\(673\) 33.0113 1.27249 0.636245 0.771487i \(-0.280487\pi\)
0.636245 + 0.771487i \(0.280487\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 26.1245 1.00405 0.502023 0.864854i \(-0.332589\pi\)
0.502023 + 0.864854i \(0.332589\pi\)
\(678\) 0 0
\(679\) 19.3633 0.743097
\(680\) 0 0
\(681\) 25.1959 0.965508
\(682\) 0 0
\(683\) 39.3957 1.50743 0.753717 0.657199i \(-0.228259\pi\)
0.753717 + 0.657199i \(0.228259\pi\)
\(684\) 0 0
\(685\) 3.06597 0.117145
\(686\) 0 0
\(687\) 5.48699 0.209342
\(688\) 0 0
\(689\) 12.7189 0.484552
\(690\) 0 0
\(691\) 9.45306 0.359611 0.179806 0.983702i \(-0.442453\pi\)
0.179806 + 0.983702i \(0.442453\pi\)
\(692\) 0 0
\(693\) 1.53669 0.0583740
\(694\) 0 0
\(695\) −22.1917 −0.841780
\(696\) 0 0
\(697\) −21.7089 −0.822283
\(698\) 0 0
\(699\) 28.5788 1.08095
\(700\) 0 0
\(701\) 37.2842 1.40821 0.704103 0.710098i \(-0.251349\pi\)
0.704103 + 0.710098i \(0.251349\pi\)
\(702\) 0 0
\(703\) 4.20681 0.158663
\(704\) 0 0
\(705\) 9.11766 0.343391
\(706\) 0 0
\(707\) 6.70063 0.252003
\(708\) 0 0
\(709\) −19.8110 −0.744016 −0.372008 0.928230i \(-0.621331\pi\)
−0.372008 + 0.928230i \(0.621331\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 31.1089 1.16504
\(714\) 0 0
\(715\) −2.68088 −0.100259
\(716\) 0 0
\(717\) 1.77811 0.0664046
\(718\) 0 0
\(719\) 19.2136 0.716548 0.358274 0.933617i \(-0.383365\pi\)
0.358274 + 0.933617i \(0.383365\pi\)
\(720\) 0 0
\(721\) −16.8272 −0.626679
\(722\) 0 0
\(723\) −22.2612 −0.827902
\(724\) 0 0
\(725\) 3.03945 0.112882
\(726\) 0 0
\(727\) 45.0183 1.66964 0.834818 0.550527i \(-0.185573\pi\)
0.834818 + 0.550527i \(0.185573\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −40.0651 −1.48186
\(732\) 0 0
\(733\) −20.7262 −0.765541 −0.382771 0.923843i \(-0.625030\pi\)
−0.382771 + 0.923843i \(0.625030\pi\)
\(734\) 0 0
\(735\) 3.59390 0.132563
\(736\) 0 0
\(737\) 9.82171 0.361787
\(738\) 0 0
\(739\) −24.2176 −0.890858 −0.445429 0.895317i \(-0.646949\pi\)
−0.445429 + 0.895317i \(0.646949\pi\)
\(740\) 0 0
\(741\) 13.5448 0.497580
\(742\) 0 0
\(743\) −13.9530 −0.511885 −0.255943 0.966692i \(-0.582386\pi\)
−0.255943 + 0.966692i \(0.582386\pi\)
\(744\) 0 0
\(745\) −14.4999 −0.531236
\(746\) 0 0
\(747\) 3.55445 0.130050
\(748\) 0 0
\(749\) 16.4456 0.600907
\(750\) 0 0
\(751\) −41.6176 −1.51865 −0.759324 0.650713i \(-0.774470\pi\)
−0.759324 + 0.650713i \(0.774470\pi\)
\(752\) 0 0
\(753\) −28.6748 −1.04497
\(754\) 0 0
\(755\) −8.24142 −0.299936
\(756\) 0 0
\(757\) −45.4034 −1.65021 −0.825107 0.564977i \(-0.808885\pi\)
−0.825107 + 0.564977i \(0.808885\pi\)
\(758\) 0 0
\(759\) 5.47556 0.198750
\(760\) 0 0
\(761\) 18.9347 0.686383 0.343191 0.939266i \(-0.388492\pi\)
0.343191 + 0.939266i \(0.388492\pi\)
\(762\) 0 0
\(763\) −20.8511 −0.754860
\(764\) 0 0
\(765\) −7.95030 −0.287444
\(766\) 0 0
\(767\) 11.8844 0.429122
\(768\) 0 0
\(769\) 31.0884 1.12108 0.560538 0.828129i \(-0.310594\pi\)
0.560538 + 0.828129i \(0.310594\pi\)
\(770\) 0 0
\(771\) −9.19388 −0.331110
\(772\) 0 0
\(773\) 34.3542 1.23564 0.617818 0.786321i \(-0.288017\pi\)
0.617818 + 0.786321i \(0.288017\pi\)
\(774\) 0 0
\(775\) 4.73057 0.169927
\(776\) 0 0
\(777\) −1.84556 −0.0662092
\(778\) 0 0
\(779\) −11.4870 −0.411564
\(780\) 0 0
\(781\) −11.4655 −0.410267
\(782\) 0 0
\(783\) −3.03945 −0.108621
\(784\) 0 0
\(785\) −6.11283 −0.218176
\(786\) 0 0
\(787\) −36.0053 −1.28345 −0.641726 0.766934i \(-0.721781\pi\)
−0.641726 + 0.766934i \(0.721781\pi\)
\(788\) 0 0
\(789\) −8.02652 −0.285752
\(790\) 0 0
\(791\) 20.3740 0.724416
\(792\) 0 0
\(793\) 17.4148 0.618418
\(794\) 0 0
\(795\) −3.95030 −0.140103
\(796\) 0 0
\(797\) 43.2107 1.53060 0.765300 0.643674i \(-0.222591\pi\)
0.765300 + 0.643674i \(0.222591\pi\)
\(798\) 0 0
\(799\) 72.4882 2.56445
\(800\) 0 0
\(801\) −6.55029 −0.231443
\(802\) 0 0
\(803\) 2.93806 0.103682
\(804\) 0 0
\(805\) −12.1367 −0.427762
\(806\) 0 0
\(807\) −21.3849 −0.752785
\(808\) 0 0
\(809\) −32.2209 −1.13283 −0.566414 0.824121i \(-0.691669\pi\)
−0.566414 + 0.824121i \(0.691669\pi\)
\(810\) 0 0
\(811\) −8.87892 −0.311781 −0.155890 0.987774i \(-0.549825\pi\)
−0.155890 + 0.987774i \(0.549825\pi\)
\(812\) 0 0
\(813\) 2.64359 0.0927148
\(814\) 0 0
\(815\) 21.4373 0.750916
\(816\) 0 0
\(817\) −21.2000 −0.741693
\(818\) 0 0
\(819\) −5.94222 −0.207638
\(820\) 0 0
\(821\) 2.31155 0.0806735 0.0403368 0.999186i \(-0.487157\pi\)
0.0403368 + 0.999186i \(0.487157\pi\)
\(822\) 0 0
\(823\) −13.6280 −0.475042 −0.237521 0.971382i \(-0.576335\pi\)
−0.237521 + 0.971382i \(0.576335\pi\)
\(824\) 0 0
\(825\) 0.832640 0.0289888
\(826\) 0 0
\(827\) 17.8298 0.620003 0.310001 0.950736i \(-0.399670\pi\)
0.310001 + 0.950736i \(0.399670\pi\)
\(828\) 0 0
\(829\) −29.7388 −1.03287 −0.516435 0.856326i \(-0.672741\pi\)
−0.516435 + 0.856326i \(0.672741\pi\)
\(830\) 0 0
\(831\) 14.4456 0.501111
\(832\) 0 0
\(833\) 28.5726 0.989980
\(834\) 0 0
\(835\) 22.5815 0.781464
\(836\) 0 0
\(837\) −4.73057 −0.163513
\(838\) 0 0
\(839\) 10.4556 0.360969 0.180484 0.983578i \(-0.442234\pi\)
0.180484 + 0.983578i \(0.442234\pi\)
\(840\) 0 0
\(841\) −19.7618 −0.681440
\(842\) 0 0
\(843\) 11.2197 0.386428
\(844\) 0 0
\(845\) −2.63334 −0.0905897
\(846\) 0 0
\(847\) 19.0217 0.653593
\(848\) 0 0
\(849\) 3.86332 0.132589
\(850\) 0 0
\(851\) −6.57614 −0.225427
\(852\) 0 0
\(853\) −15.6333 −0.535275 −0.267638 0.963520i \(-0.586243\pi\)
−0.267638 + 0.963520i \(0.586243\pi\)
\(854\) 0 0
\(855\) −4.20681 −0.143870
\(856\) 0 0
\(857\) −14.4945 −0.495123 −0.247561 0.968872i \(-0.579629\pi\)
−0.247561 + 0.968872i \(0.579629\pi\)
\(858\) 0 0
\(859\) −3.38493 −0.115492 −0.0577461 0.998331i \(-0.518391\pi\)
−0.0577461 + 0.998331i \(0.518391\pi\)
\(860\) 0 0
\(861\) 5.03945 0.171744
\(862\) 0 0
\(863\) 56.3924 1.91962 0.959810 0.280649i \(-0.0905498\pi\)
0.959810 + 0.280649i \(0.0905498\pi\)
\(864\) 0 0
\(865\) −6.48916 −0.220638
\(866\) 0 0
\(867\) −46.2073 −1.56928
\(868\) 0 0
\(869\) 3.33056 0.112982
\(870\) 0 0
\(871\) −37.9795 −1.28689
\(872\) 0 0
\(873\) −10.4918 −0.355095
\(874\) 0 0
\(875\) −1.84556 −0.0623914
\(876\) 0 0
\(877\) −32.1755 −1.08649 −0.543245 0.839574i \(-0.682805\pi\)
−0.543245 + 0.839574i \(0.682805\pi\)
\(878\) 0 0
\(879\) 16.1544 0.544876
\(880\) 0 0
\(881\) 14.6097 0.492212 0.246106 0.969243i \(-0.420849\pi\)
0.246106 + 0.969243i \(0.420849\pi\)
\(882\) 0 0
\(883\) −18.5203 −0.623259 −0.311630 0.950204i \(-0.600875\pi\)
−0.311630 + 0.950204i \(0.600875\pi\)
\(884\) 0 0
\(885\) −3.69113 −0.124076
\(886\) 0 0
\(887\) 28.5611 0.958986 0.479493 0.877546i \(-0.340821\pi\)
0.479493 + 0.877546i \(0.340821\pi\)
\(888\) 0 0
\(889\) −0.252248 −0.00846012
\(890\) 0 0
\(891\) −0.832640 −0.0278945
\(892\) 0 0
\(893\) 38.3562 1.28354
\(894\) 0 0
\(895\) −16.2095 −0.541823
\(896\) 0 0
\(897\) −21.1734 −0.706959
\(898\) 0 0
\(899\) 14.3783 0.479544
\(900\) 0 0
\(901\) −31.4061 −1.04629
\(902\) 0 0
\(903\) 9.30062 0.309505
\(904\) 0 0
\(905\) 5.38225 0.178912
\(906\) 0 0
\(907\) 25.8428 0.858097 0.429048 0.903282i \(-0.358849\pi\)
0.429048 + 0.903282i \(0.358849\pi\)
\(908\) 0 0
\(909\) −3.63067 −0.120422
\(910\) 0 0
\(911\) −24.9998 −0.828281 −0.414141 0.910213i \(-0.635918\pi\)
−0.414141 + 0.910213i \(0.635918\pi\)
\(912\) 0 0
\(913\) −2.95958 −0.0979477
\(914\) 0 0
\(915\) −5.40878 −0.178809
\(916\) 0 0
\(917\) 3.83673 0.126700
\(918\) 0 0
\(919\) 34.3931 1.13452 0.567262 0.823537i \(-0.308003\pi\)
0.567262 + 0.823537i \(0.308003\pi\)
\(920\) 0 0
\(921\) 6.28303 0.207033
\(922\) 0 0
\(923\) 44.3357 1.45933
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 9.11766 0.299463
\(928\) 0 0
\(929\) 55.6310 1.82519 0.912597 0.408860i \(-0.134074\pi\)
0.912597 + 0.408860i \(0.134074\pi\)
\(930\) 0 0
\(931\) 15.1188 0.495499
\(932\) 0 0
\(933\) 20.6312 0.675434
\(934\) 0 0
\(935\) 6.61974 0.216489
\(936\) 0 0
\(937\) −53.9576 −1.76272 −0.881360 0.472446i \(-0.843371\pi\)
−0.881360 + 0.472446i \(0.843371\pi\)
\(938\) 0 0
\(939\) 7.32180 0.238938
\(940\) 0 0
\(941\) −50.1092 −1.63351 −0.816756 0.576983i \(-0.804230\pi\)
−0.816756 + 0.576983i \(0.804230\pi\)
\(942\) 0 0
\(943\) 17.9566 0.584748
\(944\) 0 0
\(945\) 1.84556 0.0600362
\(946\) 0 0
\(947\) −39.3795 −1.27966 −0.639831 0.768516i \(-0.720996\pi\)
−0.639831 + 0.768516i \(0.720996\pi\)
\(948\) 0 0
\(949\) −11.3611 −0.368798
\(950\) 0 0
\(951\) −19.1442 −0.620793
\(952\) 0 0
\(953\) −20.5096 −0.664371 −0.332185 0.943214i \(-0.607786\pi\)
−0.332185 + 0.943214i \(0.607786\pi\)
\(954\) 0 0
\(955\) 10.5680 0.341974
\(956\) 0 0
\(957\) 2.53077 0.0818080
\(958\) 0 0
\(959\) −5.65844 −0.182721
\(960\) 0 0
\(961\) −8.62168 −0.278119
\(962\) 0 0
\(963\) −8.91086 −0.287148
\(964\) 0 0
\(965\) 3.11766 0.100361
\(966\) 0 0
\(967\) −18.6265 −0.598988 −0.299494 0.954098i \(-0.596818\pi\)
−0.299494 + 0.954098i \(0.596818\pi\)
\(968\) 0 0
\(969\) −33.4454 −1.07442
\(970\) 0 0
\(971\) 3.56656 0.114456 0.0572282 0.998361i \(-0.481774\pi\)
0.0572282 + 0.998361i \(0.481774\pi\)
\(972\) 0 0
\(973\) 40.9562 1.31300
\(974\) 0 0
\(975\) −3.21973 −0.103114
\(976\) 0 0
\(977\) −43.6998 −1.39808 −0.699041 0.715082i \(-0.746389\pi\)
−0.699041 + 0.715082i \(0.746389\pi\)
\(978\) 0 0
\(979\) 5.45404 0.174312
\(980\) 0 0
\(981\) 11.2979 0.360716
\(982\) 0 0
\(983\) −53.1096 −1.69393 −0.846966 0.531647i \(-0.821573\pi\)
−0.846966 + 0.531647i \(0.821573\pi\)
\(984\) 0 0
\(985\) 0.576137 0.0183572
\(986\) 0 0
\(987\) −16.8272 −0.535616
\(988\) 0 0
\(989\) 33.1401 1.05379
\(990\) 0 0
\(991\) 9.14283 0.290432 0.145216 0.989400i \(-0.453612\pi\)
0.145216 + 0.989400i \(0.453612\pi\)
\(992\) 0 0
\(993\) 0.908184 0.0288203
\(994\) 0 0
\(995\) −6.64359 −0.210616
\(996\) 0 0
\(997\) −15.6280 −0.494944 −0.247472 0.968895i \(-0.579600\pi\)
−0.247472 + 0.968895i \(0.579600\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
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 8880.2.a.cg.1.2 4
4.3 odd 2 1110.2.a.s.1.3 4
12.11 even 2 3330.2.a.bj.1.3 4
20.19 odd 2 5550.2.a.cj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.3 4 4.3 odd 2
3330.2.a.bj.1.3 4 12.11 even 2
5550.2.a.cj.1.2 4 20.19 odd 2
8880.2.a.cg.1.2 4 1.1 even 1 trivial