Properties

Label 3312.2.a.t.1.1
Level $3312$
Weight $2$
Character 3312.1
Self dual yes
Analytic conductor $26.446$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3312.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{5} -3.12311 q^{11} +0.438447 q^{13} -5.12311 q^{17} +3.12311 q^{19} -1.00000 q^{23} -1.00000 q^{25} -3.56155 q^{29} +2.43845 q^{31} +8.24621 q^{37} +9.80776 q^{41} +8.00000 q^{43} -0.684658 q^{47} -7.00000 q^{49} -2.00000 q^{53} +6.24621 q^{55} +10.2462 q^{59} -4.24621 q^{61} -0.876894 q^{65} -3.12311 q^{67} +13.5616 q^{71} -14.6847 q^{73} -3.12311 q^{79} +14.2462 q^{83} +10.2462 q^{85} -11.3693 q^{89} -6.24621 q^{95} +11.3693 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + O(q^{10}) \) \( 2 q - 4 q^{5} + 2 q^{11} + 5 q^{13} - 2 q^{17} - 2 q^{19} - 2 q^{23} - 2 q^{25} - 3 q^{29} + 9 q^{31} - q^{41} + 16 q^{43} + 11 q^{47} - 14 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} + 8 q^{61} - 10 q^{65} + 2 q^{67} + 23 q^{71} - 17 q^{73} + 2 q^{79} + 12 q^{83} + 4 q^{85} + 2 q^{89} + 4 q^{95} - 2 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) 0 0
\(13\) 0.438447 0.121603 0.0608017 0.998150i \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.12311 −1.24254 −0.621268 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) 0 0
\(19\) 3.12311 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.56155 −0.661364 −0.330682 0.943742i \(-0.607279\pi\)
−0.330682 + 0.943742i \(0.607279\pi\)
\(30\) 0 0
\(31\) 2.43845 0.437958 0.218979 0.975730i \(-0.429727\pi\)
0.218979 + 0.975730i \(0.429727\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24621 1.35567 0.677834 0.735215i \(-0.262919\pi\)
0.677834 + 0.735215i \(0.262919\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.80776 1.53172 0.765858 0.643010i \(-0.222315\pi\)
0.765858 + 0.643010i \(0.222315\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.684658 −0.0998677 −0.0499338 0.998753i \(-0.515901\pi\)
−0.0499338 + 0.998753i \(0.515901\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 6.24621 0.842239
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.2462 1.33394 0.666972 0.745083i \(-0.267590\pi\)
0.666972 + 0.745083i \(0.267590\pi\)
\(60\) 0 0
\(61\) −4.24621 −0.543672 −0.271836 0.962344i \(-0.587631\pi\)
−0.271836 + 0.962344i \(0.587631\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.876894 −0.108765
\(66\) 0 0
\(67\) −3.12311 −0.381548 −0.190774 0.981634i \(-0.561100\pi\)
−0.190774 + 0.981634i \(0.561100\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.5616 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(72\) 0 0
\(73\) −14.6847 −1.71871 −0.859355 0.511380i \(-0.829134\pi\)
−0.859355 + 0.511380i \(0.829134\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.2462 1.56372 0.781862 0.623451i \(-0.214270\pi\)
0.781862 + 0.623451i \(0.214270\pi\)
\(84\) 0 0
\(85\) 10.2462 1.11136
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.3693 −1.20515 −0.602573 0.798064i \(-0.705858\pi\)
−0.602573 + 0.798064i \(0.705858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.24621 −0.640848
\(96\) 0 0
\(97\) 11.3693 1.15438 0.577190 0.816610i \(-0.304149\pi\)
0.577190 + 0.816610i \(0.304149\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.2462 −1.21854 −0.609272 0.792961i \(-0.708538\pi\)
−0.609272 + 0.792961i \(0.708538\pi\)
\(102\) 0 0
\(103\) 14.2462 1.40372 0.701860 0.712314i \(-0.252353\pi\)
0.701860 + 0.712314i \(0.252353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.1231 1.07531 0.537656 0.843165i \(-0.319310\pi\)
0.537656 + 0.843165i \(0.319310\pi\)
\(108\) 0 0
\(109\) 13.1231 1.25697 0.628483 0.777824i \(-0.283676\pi\)
0.628483 + 0.777824i \(0.283676\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.87689 0.270635 0.135318 0.990802i \(-0.456794\pi\)
0.135318 + 0.990802i \(0.456794\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −0.684658 −0.0607536 −0.0303768 0.999539i \(-0.509671\pi\)
−0.0303768 + 0.999539i \(0.509671\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.31534 0.289663 0.144831 0.989456i \(-0.453736\pi\)
0.144831 + 0.989456i \(0.453736\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.87689 −0.587533 −0.293766 0.955877i \(-0.594909\pi\)
−0.293766 + 0.955877i \(0.594909\pi\)
\(138\) 0 0
\(139\) 3.31534 0.281204 0.140602 0.990066i \(-0.455096\pi\)
0.140602 + 0.990066i \(0.455096\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.36932 −0.114508
\(144\) 0 0
\(145\) 7.12311 0.591542
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.2462 1.65863 0.829317 0.558778i \(-0.188730\pi\)
0.829317 + 0.558778i \(0.188730\pi\)
\(150\) 0 0
\(151\) 10.4384 0.849469 0.424734 0.905318i \(-0.360367\pi\)
0.424734 + 0.905318i \(0.360367\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.87689 −0.391722
\(156\) 0 0
\(157\) 21.1231 1.68581 0.842904 0.538064i \(-0.180844\pi\)
0.842904 + 0.538064i \(0.180844\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −18.9309 −1.48278 −0.741390 0.671074i \(-0.765833\pi\)
−0.741390 + 0.671074i \(0.765833\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.31534 −0.247800 −0.123900 0.992295i \(-0.539540\pi\)
−0.123900 + 0.992295i \(0.539540\pi\)
\(180\) 0 0
\(181\) 3.75379 0.279017 0.139508 0.990221i \(-0.455448\pi\)
0.139508 + 0.990221i \(0.455448\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −16.4924 −1.21255
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 25.3693 1.83566 0.917830 0.396974i \(-0.129940\pi\)
0.917830 + 0.396974i \(0.129940\pi\)
\(192\) 0 0
\(193\) −0.438447 −0.0315601 −0.0157801 0.999875i \(-0.505023\pi\)
−0.0157801 + 0.999875i \(0.505023\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.9309 1.20627 0.603137 0.797637i \(-0.293917\pi\)
0.603137 + 0.797637i \(0.293917\pi\)
\(198\) 0 0
\(199\) −11.1231 −0.788496 −0.394248 0.919004i \(-0.628995\pi\)
−0.394248 + 0.919004i \(0.628995\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −19.6155 −1.37001
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.75379 −0.674684
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.24621 −0.151097
\(222\) 0 0
\(223\) 28.4924 1.90799 0.953997 0.299817i \(-0.0969255\pi\)
0.953997 + 0.299817i \(0.0969255\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.24621 −0.414576 −0.207288 0.978280i \(-0.566464\pi\)
−0.207288 + 0.978280i \(0.566464\pi\)
\(228\) 0 0
\(229\) 14.8769 0.983093 0.491546 0.870851i \(-0.336432\pi\)
0.491546 + 0.870851i \(0.336432\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0540 1.05173 0.525865 0.850568i \(-0.323742\pi\)
0.525865 + 0.850568i \(0.323742\pi\)
\(234\) 0 0
\(235\) 1.36932 0.0893244
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.0540 1.68529 0.842646 0.538468i \(-0.180997\pi\)
0.842646 + 0.538468i \(0.180997\pi\)
\(240\) 0 0
\(241\) 6.49242 0.418214 0.209107 0.977893i \(-0.432944\pi\)
0.209107 + 0.977893i \(0.432944\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.0000 0.894427
\(246\) 0 0
\(247\) 1.36932 0.0871275
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.24621 −0.394257 −0.197129 0.980378i \(-0.563162\pi\)
−0.197129 + 0.980378i \(0.563162\pi\)
\(252\) 0 0
\(253\) 3.12311 0.196348
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.31534 0.331562 0.165781 0.986163i \(-0.446986\pi\)
0.165781 + 0.986163i \(0.446986\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −19.1231 −1.17918 −0.589591 0.807702i \(-0.700711\pi\)
−0.589591 + 0.807702i \(0.700711\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.3153 −0.811851 −0.405925 0.913906i \(-0.633051\pi\)
−0.405925 + 0.913906i \(0.633051\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.12311 0.188330
\(276\) 0 0
\(277\) 17.8078 1.06996 0.534982 0.844863i \(-0.320318\pi\)
0.534982 + 0.844863i \(0.320318\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.12311 −0.305619 −0.152809 0.988256i \(-0.548832\pi\)
−0.152809 + 0.988256i \(0.548832\pi\)
\(282\) 0 0
\(283\) −3.12311 −0.185649 −0.0928247 0.995682i \(-0.529590\pi\)
−0.0928247 + 0.995682i \(0.529590\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.24621 0.543895
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.3693 0.897885 0.448943 0.893561i \(-0.351801\pi\)
0.448943 + 0.893561i \(0.351801\pi\)
\(294\) 0 0
\(295\) −20.4924 −1.19311
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.438447 −0.0253561
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.49242 0.486275
\(306\) 0 0
\(307\) 18.2462 1.04137 0.520683 0.853750i \(-0.325677\pi\)
0.520683 + 0.853750i \(0.325677\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.56155 0.315367 0.157683 0.987490i \(-0.449597\pi\)
0.157683 + 0.987490i \(0.449597\pi\)
\(312\) 0 0
\(313\) −15.3693 −0.868725 −0.434363 0.900738i \(-0.643026\pi\)
−0.434363 + 0.900738i \(0.643026\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.75379 0.210834 0.105417 0.994428i \(-0.466382\pi\)
0.105417 + 0.994428i \(0.466382\pi\)
\(318\) 0 0
\(319\) 11.1231 0.622774
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) −0.438447 −0.0243207
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −31.4233 −1.72718 −0.863590 0.504194i \(-0.831789\pi\)
−0.863590 + 0.504194i \(0.831789\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.24621 0.341267
\(336\) 0 0
\(337\) −21.6155 −1.17747 −0.588736 0.808325i \(-0.700374\pi\)
−0.588736 + 0.808325i \(0.700374\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.61553 −0.412404
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.4924 −0.885360 −0.442680 0.896680i \(-0.645972\pi\)
−0.442680 + 0.896680i \(0.645972\pi\)
\(348\) 0 0
\(349\) −34.6847 −1.85663 −0.928314 0.371798i \(-0.878741\pi\)
−0.928314 + 0.371798i \(0.878741\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.5616 −0.828258 −0.414129 0.910218i \(-0.635914\pi\)
−0.414129 + 0.910218i \(0.635914\pi\)
\(354\) 0 0
\(355\) −27.1231 −1.43954
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −23.6155 −1.24638 −0.623190 0.782071i \(-0.714164\pi\)
−0.623190 + 0.782071i \(0.714164\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 29.3693 1.53726
\(366\) 0 0
\(367\) 14.2462 0.743646 0.371823 0.928304i \(-0.378733\pi\)
0.371823 + 0.928304i \(0.378733\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.4924 −0.543277 −0.271639 0.962399i \(-0.587566\pi\)
−0.271639 + 0.962399i \(0.587566\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.56155 −0.0804241
\(378\) 0 0
\(379\) 12.4924 0.641693 0.320846 0.947131i \(-0.396033\pi\)
0.320846 + 0.947131i \(0.396033\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.75379 0.498395 0.249198 0.968453i \(-0.419833\pi\)
0.249198 + 0.968453i \(0.419833\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.3693 1.18487 0.592436 0.805618i \(-0.298166\pi\)
0.592436 + 0.805618i \(0.298166\pi\)
\(390\) 0 0
\(391\) 5.12311 0.259087
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.24621 0.314281
\(396\) 0 0
\(397\) 16.4384 0.825022 0.412511 0.910953i \(-0.364652\pi\)
0.412511 + 0.910953i \(0.364652\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 28.2462 1.41055 0.705274 0.708935i \(-0.250824\pi\)
0.705274 + 0.708935i \(0.250824\pi\)
\(402\) 0 0
\(403\) 1.06913 0.0532572
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.7538 −1.27657
\(408\) 0 0
\(409\) 26.3002 1.30046 0.650230 0.759737i \(-0.274672\pi\)
0.650230 + 0.759737i \(0.274672\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −28.4924 −1.39864
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.4924 1.00112 0.500560 0.865702i \(-0.333127\pi\)
0.500560 + 0.865702i \(0.333127\pi\)
\(420\) 0 0
\(421\) −18.8769 −0.920004 −0.460002 0.887918i \(-0.652151\pi\)
−0.460002 + 0.887918i \(0.652151\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.12311 0.248507
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.49242 0.216392 0.108196 0.994130i \(-0.465493\pi\)
0.108196 + 0.994130i \(0.465493\pi\)
\(432\) 0 0
\(433\) −24.7386 −1.18886 −0.594431 0.804146i \(-0.702623\pi\)
−0.594431 + 0.804146i \(0.702623\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.12311 −0.149398
\(438\) 0 0
\(439\) 26.0540 1.24349 0.621744 0.783220i \(-0.286424\pi\)
0.621744 + 0.783220i \(0.286424\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −34.9309 −1.65962 −0.829808 0.558049i \(-0.811550\pi\)
−0.829808 + 0.558049i \(0.811550\pi\)
\(444\) 0 0
\(445\) 22.7386 1.07791
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.24621 −0.389163 −0.194581 0.980886i \(-0.562335\pi\)
−0.194581 + 0.980886i \(0.562335\pi\)
\(450\) 0 0
\(451\) −30.6307 −1.44234
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.3693 0.906058 0.453029 0.891496i \(-0.350343\pi\)
0.453029 + 0.891496i \(0.350343\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.80776 −0.456793 −0.228397 0.973568i \(-0.573348\pi\)
−0.228397 + 0.973568i \(0.573348\pi\)
\(462\) 0 0
\(463\) 20.4924 0.952364 0.476182 0.879347i \(-0.342020\pi\)
0.476182 + 0.879347i \(0.342020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.36932 0.433560 0.216780 0.976220i \(-0.430445\pi\)
0.216780 + 0.976220i \(0.430445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −24.9848 −1.14880
\(474\) 0 0
\(475\) −3.12311 −0.143298
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.24621 −0.285397 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(480\) 0 0
\(481\) 3.61553 0.164854
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.7386 −1.03251
\(486\) 0 0
\(487\) −24.6847 −1.11857 −0.559284 0.828976i \(-0.688924\pi\)
−0.559284 + 0.828976i \(0.688924\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.80776 −0.352359 −0.176180 0.984358i \(-0.556374\pi\)
−0.176180 + 0.984358i \(0.556374\pi\)
\(492\) 0 0
\(493\) 18.2462 0.821768
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0540 −0.629142 −0.314571 0.949234i \(-0.601861\pi\)
−0.314571 + 0.949234i \(0.601861\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.3693 −0.774460 −0.387230 0.921983i \(-0.626568\pi\)
−0.387230 + 0.921983i \(0.626568\pi\)
\(504\) 0 0
\(505\) 24.4924 1.08990
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −38.3002 −1.69763 −0.848813 0.528693i \(-0.822682\pi\)
−0.848813 + 0.528693i \(0.822682\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −28.4924 −1.25553
\(516\) 0 0
\(517\) 2.13826 0.0940406
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.6155 0.946993 0.473497 0.880796i \(-0.342992\pi\)
0.473497 + 0.880796i \(0.342992\pi\)
\(522\) 0 0
\(523\) −22.2462 −0.972759 −0.486379 0.873748i \(-0.661683\pi\)
−0.486379 + 0.873748i \(0.661683\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.4924 −0.544178
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.30019 0.186262
\(534\) 0 0
\(535\) −22.2462 −0.961788
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 21.8617 0.941652
\(540\) 0 0
\(541\) 7.06913 0.303926 0.151963 0.988386i \(-0.451441\pi\)
0.151963 + 0.988386i \(0.451441\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −26.2462 −1.12426
\(546\) 0 0
\(547\) 26.5464 1.13504 0.567521 0.823359i \(-0.307903\pi\)
0.567521 + 0.823359i \(0.307903\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.1231 −0.473860
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.12311 0.386558 0.193279 0.981144i \(-0.438088\pi\)
0.193279 + 0.981144i \(0.438088\pi\)
\(558\) 0 0
\(559\) 3.50758 0.148355
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.2462 0.937566 0.468783 0.883313i \(-0.344693\pi\)
0.468783 + 0.883313i \(0.344693\pi\)
\(564\) 0 0
\(565\) −5.75379 −0.242064
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.24621 −0.345699 −0.172850 0.984948i \(-0.555297\pi\)
−0.172850 + 0.984948i \(0.555297\pi\)
\(570\) 0 0
\(571\) 44.4924 1.86195 0.930975 0.365083i \(-0.118959\pi\)
0.930975 + 0.365083i \(0.118959\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −12.9309 −0.538319 −0.269160 0.963096i \(-0.586746\pi\)
−0.269160 + 0.963096i \(0.586746\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.24621 0.258692
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.0540 0.580070 0.290035 0.957016i \(-0.406333\pi\)
0.290035 + 0.957016i \(0.406333\pi\)
\(588\) 0 0
\(589\) 7.61553 0.313792
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.73863 −0.194592 −0.0972962 0.995255i \(-0.531019\pi\)
−0.0972962 + 0.995255i \(0.531019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −18.1922 −0.742077 −0.371038 0.928618i \(-0.620998\pi\)
−0.371038 + 0.928618i \(0.620998\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.49242 0.101331
\(606\) 0 0
\(607\) −34.7386 −1.41000 −0.704999 0.709208i \(-0.749052\pi\)
−0.704999 + 0.709208i \(0.749052\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.300187 −0.0121442
\(612\) 0 0
\(613\) 46.1080 1.86228 0.931141 0.364659i \(-0.118814\pi\)
0.931141 + 0.364659i \(0.118814\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.9848 1.24740 0.623701 0.781663i \(-0.285628\pi\)
0.623701 + 0.781663i \(0.285628\pi\)
\(618\) 0 0
\(619\) 4.49242 0.180566 0.0902829 0.995916i \(-0.471223\pi\)
0.0902829 + 0.995916i \(0.471223\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −42.2462 −1.68447
\(630\) 0 0
\(631\) −42.7386 −1.70140 −0.850699 0.525653i \(-0.823821\pi\)
−0.850699 + 0.525653i \(0.823821\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.36932 0.0543397
\(636\) 0 0
\(637\) −3.06913 −0.121603
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.3693 −0.449061 −0.224531 0.974467i \(-0.572085\pi\)
−0.224531 + 0.974467i \(0.572085\pi\)
\(642\) 0 0
\(643\) −1.36932 −0.0540006 −0.0270003 0.999635i \(-0.508596\pi\)
−0.0270003 + 0.999635i \(0.508596\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.3002 0.955339 0.477669 0.878540i \(-0.341482\pi\)
0.477669 + 0.878540i \(0.341482\pi\)
\(648\) 0 0
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.0540 0.471709 0.235854 0.971788i \(-0.424211\pi\)
0.235854 + 0.971788i \(0.424211\pi\)
\(654\) 0 0
\(655\) −6.63068 −0.259082
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14.2462 −0.554954 −0.277477 0.960732i \(-0.589498\pi\)
−0.277477 + 0.960732i \(0.589498\pi\)
\(660\) 0 0
\(661\) 16.2462 0.631904 0.315952 0.948775i \(-0.397676\pi\)
0.315952 + 0.948775i \(0.397676\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.56155 0.137904
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 13.2614 0.511949
\(672\) 0 0
\(673\) −40.0540 −1.54397 −0.771984 0.635642i \(-0.780735\pi\)
−0.771984 + 0.635642i \(0.780735\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26.9848 −1.03711 −0.518556 0.855044i \(-0.673530\pi\)
−0.518556 + 0.855044i \(0.673530\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28.6847 1.09759 0.548794 0.835958i \(-0.315087\pi\)
0.548794 + 0.835958i \(0.315087\pi\)
\(684\) 0 0
\(685\) 13.7538 0.525505
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.876894 −0.0334070
\(690\) 0 0
\(691\) 44.9848 1.71130 0.855652 0.517551i \(-0.173156\pi\)
0.855652 + 0.517551i \(0.173156\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.63068 −0.251516
\(696\) 0 0
\(697\) −50.2462 −1.90321
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27.8617 1.05232 0.526162 0.850385i \(-0.323631\pi\)
0.526162 + 0.850385i \(0.323631\pi\)
\(702\) 0 0
\(703\) 25.7538 0.971323
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −12.6307 −0.474355 −0.237178 0.971466i \(-0.576222\pi\)
−0.237178 + 0.971466i \(0.576222\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.43845 −0.0913206
\(714\) 0 0
\(715\) 2.73863 0.102419
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28.4924 −1.06259 −0.531294 0.847187i \(-0.678294\pi\)
−0.531294 + 0.847187i \(0.678294\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.56155 0.132273
\(726\) 0 0
\(727\) 14.6307 0.542622 0.271311 0.962492i \(-0.412543\pi\)
0.271311 + 0.962492i \(0.412543\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −40.9848 −1.51588
\(732\) 0 0
\(733\) 24.6307 0.909755 0.454878 0.890554i \(-0.349683\pi\)
0.454878 + 0.890554i \(0.349683\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.75379 0.359285
\(738\) 0 0
\(739\) −12.6847 −0.466613 −0.233306 0.972403i \(-0.574954\pi\)
−0.233306 + 0.972403i \(0.574954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.7538 0.651323 0.325662 0.945486i \(-0.394413\pi\)
0.325662 + 0.945486i \(0.394413\pi\)
\(744\) 0 0
\(745\) −40.4924 −1.48353
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 24.9848 0.911710 0.455855 0.890054i \(-0.349334\pi\)
0.455855 + 0.890054i \(0.349334\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −20.8769 −0.759788
\(756\) 0 0
\(757\) −34.4924 −1.25365 −0.626824 0.779161i \(-0.715646\pi\)
−0.626824 + 0.779161i \(0.715646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.6847 1.11232 0.556159 0.831076i \(-0.312275\pi\)
0.556159 + 0.831076i \(0.312275\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.49242 0.162212
\(768\) 0 0
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −27.3693 −0.984406 −0.492203 0.870480i \(-0.663808\pi\)
−0.492203 + 0.870480i \(0.663808\pi\)
\(774\) 0 0
\(775\) −2.43845 −0.0875916
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.6307 1.09746
\(780\) 0 0
\(781\) −42.3542 −1.51555
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −42.2462 −1.50783
\(786\) 0 0
\(787\) 9.75379 0.347685 0.173843 0.984773i \(-0.444382\pi\)
0.173843 + 0.984773i \(0.444382\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.86174 −0.0661123
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.24621 0.150409 0.0752043 0.997168i \(-0.476039\pi\)
0.0752043 + 0.997168i \(0.476039\pi\)
\(798\) 0 0
\(799\) 3.50758 0.124089
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 45.8617 1.61843
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −5.06913 −0.178001 −0.0890006 0.996032i \(-0.528367\pi\)
−0.0890006 + 0.996032i \(0.528367\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 37.8617 1.32624
\(816\) 0 0
\(817\) 24.9848 0.874109
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −24.7386 −0.863384 −0.431692 0.902021i \(-0.642083\pi\)
−0.431692 + 0.902021i \(0.642083\pi\)
\(822\) 0 0
\(823\) −53.5616 −1.86704 −0.933519 0.358527i \(-0.883279\pi\)
−0.933519 + 0.358527i \(0.883279\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.4924 −0.990779 −0.495389 0.868671i \(-0.664975\pi\)
−0.495389 + 0.868671i \(0.664975\pi\)
\(828\) 0 0
\(829\) −3.75379 −0.130374 −0.0651872 0.997873i \(-0.520764\pi\)
−0.0651872 + 0.997873i \(0.520764\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 35.8617 1.24254
\(834\) 0 0
\(835\) 32.0000 1.10741
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 33.7538 1.16531 0.582655 0.812720i \(-0.302014\pi\)
0.582655 + 0.812720i \(0.302014\pi\)
\(840\) 0 0
\(841\) −16.3153 −0.562598
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.6155 0.881201
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.24621 −0.282676
\(852\) 0 0
\(853\) 30.9848 1.06090 0.530450 0.847716i \(-0.322023\pi\)
0.530450 + 0.847716i \(0.322023\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.4384 −0.424889 −0.212445 0.977173i \(-0.568143\pi\)
−0.212445 + 0.977173i \(0.568143\pi\)
\(858\) 0 0
\(859\) −54.0540 −1.84430 −0.922149 0.386835i \(-0.873568\pi\)
−0.922149 + 0.386835i \(0.873568\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 37.1771 1.26552 0.632761 0.774347i \(-0.281921\pi\)
0.632761 + 0.774347i \(0.281921\pi\)
\(864\) 0 0
\(865\) −20.0000 −0.680020
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.75379 0.330875
\(870\) 0 0
\(871\) −1.36932 −0.0463975
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.9848 −1.18135 −0.590677 0.806908i \(-0.701139\pi\)
−0.590677 + 0.806908i \(0.701139\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −13.1231 −0.442129 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(882\) 0 0
\(883\) 28.9848 0.975418 0.487709 0.873006i \(-0.337833\pi\)
0.487709 + 0.873006i \(0.337833\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.31534 0.245625 0.122813 0.992430i \(-0.460809\pi\)
0.122813 + 0.992430i \(0.460809\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.13826 −0.0715542
\(894\) 0 0
\(895\) 6.63068 0.221639
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.68466 −0.289650
\(900\) 0 0
\(901\) 10.2462 0.341351
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.50758 −0.249560
\(906\) 0 0
\(907\) −51.1231 −1.69751 −0.848757 0.528782i \(-0.822649\pi\)
−0.848757 + 0.528782i \(0.822649\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.8769 −0.956734 −0.478367 0.878160i \(-0.658771\pi\)
−0.478367 + 0.878160i \(0.658771\pi\)
\(912\) 0 0
\(913\) −44.4924 −1.47248
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −54.2462 −1.78942 −0.894709 0.446650i \(-0.852617\pi\)
−0.894709 + 0.446650i \(0.852617\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.94602 0.195716
\(924\) 0 0
\(925\) −8.24621 −0.271134
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.1922 −0.465632 −0.232816 0.972521i \(-0.574794\pi\)
−0.232816 + 0.972521i \(0.574794\pi\)
\(930\) 0 0
\(931\) −21.8617 −0.716490
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −32.0000 −1.04651
\(936\) 0 0
\(937\) 32.2462 1.05344 0.526719 0.850040i \(-0.323422\pi\)
0.526719 + 0.850040i \(0.323422\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30.8769 −1.00656 −0.503279 0.864124i \(-0.667873\pi\)
−0.503279 + 0.864124i \(0.667873\pi\)
\(942\) 0 0
\(943\) −9.80776 −0.319385
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.56155 0.0507436 0.0253718 0.999678i \(-0.491923\pi\)
0.0253718 + 0.999678i \(0.491923\pi\)
\(948\) 0 0
\(949\) −6.43845 −0.209001
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −27.7538 −0.899033 −0.449517 0.893272i \(-0.648404\pi\)
−0.449517 + 0.893272i \(0.648404\pi\)
\(954\) 0 0
\(955\) −50.7386 −1.64186
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25.0540 −0.808193
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.876894 0.0282282
\(966\) 0 0
\(967\) 2.05398 0.0660514 0.0330257 0.999455i \(-0.489486\pi\)
0.0330257 + 0.999455i \(0.489486\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6.63068 0.212789 0.106394 0.994324i \(-0.466069\pi\)
0.106394 + 0.994324i \(0.466069\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.2311 −0.807213 −0.403607 0.914933i \(-0.632244\pi\)
−0.403607 + 0.914933i \(0.632244\pi\)
\(978\) 0 0
\(979\) 35.5076 1.13483
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −52.1080 −1.66199 −0.830993 0.556283i \(-0.812227\pi\)
−0.830993 + 0.556283i \(0.812227\pi\)
\(984\) 0 0
\(985\) −33.8617 −1.07892
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 56.9848 1.81018 0.905092 0.425217i \(-0.139802\pi\)
0.905092 + 0.425217i \(0.139802\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.2462 0.705252
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 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 3312.2.a.t.1.1 2
3.2 odd 2 368.2.a.i.1.1 2
4.3 odd 2 1656.2.a.j.1.2 2
12.11 even 2 184.2.a.e.1.2 2
15.14 odd 2 9200.2.a.br.1.2 2
24.5 odd 2 1472.2.a.p.1.2 2
24.11 even 2 1472.2.a.u.1.1 2
60.23 odd 4 4600.2.e.o.4049.3 4
60.47 odd 4 4600.2.e.o.4049.2 4
60.59 even 2 4600.2.a.s.1.1 2
69.68 even 2 8464.2.a.bd.1.1 2
84.83 odd 2 9016.2.a.w.1.1 2
276.275 odd 2 4232.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.2 2 12.11 even 2
368.2.a.i.1.1 2 3.2 odd 2
1472.2.a.p.1.2 2 24.5 odd 2
1472.2.a.u.1.1 2 24.11 even 2
1656.2.a.j.1.2 2 4.3 odd 2
3312.2.a.t.1.1 2 1.1 even 1 trivial
4232.2.a.o.1.2 2 276.275 odd 2
4600.2.a.s.1.1 2 60.59 even 2
4600.2.e.o.4049.2 4 60.47 odd 4
4600.2.e.o.4049.3 4 60.23 odd 4
8464.2.a.bd.1.1 2 69.68 even 2
9016.2.a.w.1.1 2 84.83 odd 2
9200.2.a.br.1.2 2 15.14 odd 2