Properties

Label 216.4.a.e.1.2
Level $216$
Weight $4$
Character 216.1
Self dual yes
Analytic conductor $12.744$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 216.1

$q$-expansion

\(f(q)\) \(=\) \(q+13.2337 q^{5} -5.23369 q^{7} +O(q^{10})\) \(q+13.2337 q^{5} -5.23369 q^{7} -1.00000 q^{11} +84.9348 q^{13} -40.9348 q^{17} +57.5326 q^{19} -114.935 q^{23} +50.1305 q^{25} +202.467 q^{29} +274.168 q^{31} -69.2610 q^{35} +242.935 q^{37} +328.337 q^{41} +281.272 q^{43} -23.8695 q^{47} -315.609 q^{49} -300.038 q^{53} -13.2337 q^{55} -753.609 q^{59} +495.478 q^{61} +1124.00 q^{65} -409.945 q^{67} -1114.80 q^{71} -287.000 q^{73} +5.23369 q^{77} -1231.48 q^{79} +942.478 q^{83} -541.718 q^{85} +190.206 q^{89} -444.522 q^{91} +761.369 q^{95} -306.870 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} + 24q^{7} + O(q^{10}) \) \( 2q - 8q^{5} + 24q^{7} - 2q^{11} + 32q^{13} + 56q^{17} + 184q^{19} - 92q^{23} + 376q^{25} + 336q^{29} + 376q^{31} - 690q^{35} + 348q^{37} + 312q^{41} + 80q^{43} + 228q^{47} + 196q^{49} - 152q^{53} + 8q^{55} - 680q^{59} - 112q^{61} + 2248q^{65} + 352q^{67} - 1816q^{71} - 574q^{73} - 24q^{77} - 1360q^{79} + 782q^{83} - 2600q^{85} - 240q^{89} - 1992q^{91} - 1924q^{95} - 338q^{97} + O(q^{100}) \)

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
\(5\) 13.2337 1.18366 0.591829 0.806064i \(-0.298406\pi\)
0.591829 + 0.806064i \(0.298406\pi\)
\(6\) 0 0
\(7\) −5.23369 −0.282593 −0.141296 0.989967i \(-0.545127\pi\)
−0.141296 + 0.989967i \(0.545127\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.0274101 −0.0137051 0.999906i \(-0.504363\pi\)
−0.0137051 + 0.999906i \(0.504363\pi\)
\(12\) 0 0
\(13\) 84.9348 1.81205 0.906025 0.423223i \(-0.139101\pi\)
0.906025 + 0.423223i \(0.139101\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −40.9348 −0.584008 −0.292004 0.956417i \(-0.594322\pi\)
−0.292004 + 0.956417i \(0.594322\pi\)
\(18\) 0 0
\(19\) 57.5326 0.694678 0.347339 0.937740i \(-0.387085\pi\)
0.347339 + 0.937740i \(0.387085\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −114.935 −1.04198 −0.520990 0.853563i \(-0.674437\pi\)
−0.520990 + 0.853563i \(0.674437\pi\)
\(24\) 0 0
\(25\) 50.1305 0.401044
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 202.467 1.29646 0.648228 0.761446i \(-0.275510\pi\)
0.648228 + 0.761446i \(0.275510\pi\)
\(30\) 0 0
\(31\) 274.168 1.58846 0.794228 0.607620i \(-0.207876\pi\)
0.794228 + 0.607620i \(0.207876\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −69.2610 −0.334493
\(36\) 0 0
\(37\) 242.935 1.07941 0.539706 0.841854i \(-0.318535\pi\)
0.539706 + 0.841854i \(0.318535\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 328.337 1.25067 0.625337 0.780355i \(-0.284962\pi\)
0.625337 + 0.780355i \(0.284962\pi\)
\(42\) 0 0
\(43\) 281.272 0.997524 0.498762 0.866739i \(-0.333788\pi\)
0.498762 + 0.866739i \(0.333788\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −23.8695 −0.0740793 −0.0370396 0.999314i \(-0.511793\pi\)
−0.0370396 + 0.999314i \(0.511793\pi\)
\(48\) 0 0
\(49\) −315.609 −0.920141
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −300.038 −0.777611 −0.388805 0.921320i \(-0.627112\pi\)
−0.388805 + 0.921320i \(0.627112\pi\)
\(54\) 0 0
\(55\) −13.2337 −0.0324442
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −753.609 −1.66291 −0.831453 0.555595i \(-0.812491\pi\)
−0.831453 + 0.555595i \(0.812491\pi\)
\(60\) 0 0
\(61\) 495.478 1.03999 0.519996 0.854169i \(-0.325934\pi\)
0.519996 + 0.854169i \(0.325934\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1124.00 2.14485
\(66\) 0 0
\(67\) −409.945 −0.747504 −0.373752 0.927529i \(-0.621929\pi\)
−0.373752 + 0.927529i \(0.621929\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1114.80 −1.86342 −0.931711 0.363201i \(-0.881684\pi\)
−0.931711 + 0.363201i \(0.881684\pi\)
\(72\) 0 0
\(73\) −287.000 −0.460148 −0.230074 0.973173i \(-0.573897\pi\)
−0.230074 + 0.973173i \(0.573897\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.23369 0.00774590
\(78\) 0 0
\(79\) −1231.48 −1.75382 −0.876912 0.480651i \(-0.840400\pi\)
−0.876912 + 0.480651i \(0.840400\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 942.478 1.24639 0.623195 0.782066i \(-0.285834\pi\)
0.623195 + 0.782066i \(0.285834\pi\)
\(84\) 0 0
\(85\) −541.718 −0.691265
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 190.206 0.226537 0.113269 0.993564i \(-0.463868\pi\)
0.113269 + 0.993564i \(0.463868\pi\)
\(90\) 0 0
\(91\) −444.522 −0.512072
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 761.369 0.822261
\(96\) 0 0
\(97\) −306.870 −0.321215 −0.160608 0.987018i \(-0.551345\pi\)
−0.160608 + 0.987018i \(0.551345\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 672.897 0.662928 0.331464 0.943468i \(-0.392457\pi\)
0.331464 + 0.943468i \(0.392457\pi\)
\(102\) 0 0
\(103\) 840.152 0.803715 0.401857 0.915702i \(-0.368365\pi\)
0.401857 + 0.915702i \(0.368365\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1658.09 −1.49807 −0.749034 0.662532i \(-0.769482\pi\)
−0.749034 + 0.662532i \(0.769482\pi\)
\(108\) 0 0
\(109\) −1263.76 −1.11052 −0.555258 0.831678i \(-0.687381\pi\)
−0.555258 + 0.831678i \(0.687381\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −474.358 −0.394901 −0.197451 0.980313i \(-0.563266\pi\)
−0.197451 + 0.980313i \(0.563266\pi\)
\(114\) 0 0
\(115\) −1521.01 −1.23335
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 214.240 0.165036
\(120\) 0 0
\(121\) −1330.00 −0.999249
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −990.800 −0.708959
\(126\) 0 0
\(127\) −560.375 −0.391537 −0.195769 0.980650i \(-0.562720\pi\)
−0.195769 + 0.980650i \(0.562720\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −789.304 −0.526426 −0.263213 0.964738i \(-0.584782\pi\)
−0.263213 + 0.964738i \(0.584782\pi\)
\(132\) 0 0
\(133\) −301.108 −0.196311
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1330.47 0.829704 0.414852 0.909889i \(-0.363833\pi\)
0.414852 + 0.909889i \(0.363833\pi\)
\(138\) 0 0
\(139\) 3139.37 1.91567 0.957834 0.287323i \(-0.0927653\pi\)
0.957834 + 0.287323i \(0.0927653\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −84.9348 −0.0496685
\(144\) 0 0
\(145\) 2679.39 1.53456
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2666.24 −1.46595 −0.732977 0.680253i \(-0.761870\pi\)
−0.732977 + 0.680253i \(0.761870\pi\)
\(150\) 0 0
\(151\) 479.407 0.258368 0.129184 0.991621i \(-0.458764\pi\)
0.129184 + 0.991621i \(0.458764\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3628.26 1.88019
\(156\) 0 0
\(157\) −2004.83 −1.01912 −0.509562 0.860434i \(-0.670193\pi\)
−0.509562 + 0.860434i \(0.670193\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 601.533 0.294456
\(162\) 0 0
\(163\) −832.043 −0.399820 −0.199910 0.979814i \(-0.564065\pi\)
−0.199910 + 0.979814i \(0.564065\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2871.65 1.33063 0.665314 0.746563i \(-0.268297\pi\)
0.665314 + 0.746563i \(0.268297\pi\)
\(168\) 0 0
\(169\) 5016.91 2.28353
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3579.44 −1.57306 −0.786531 0.617551i \(-0.788125\pi\)
−0.786531 + 0.617551i \(0.788125\pi\)
\(174\) 0 0
\(175\) −262.367 −0.113332
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2162.09 0.902804 0.451402 0.892321i \(-0.350924\pi\)
0.451402 + 0.892321i \(0.350924\pi\)
\(180\) 0 0
\(181\) 1740.26 0.714655 0.357328 0.933979i \(-0.383688\pi\)
0.357328 + 0.933979i \(0.383688\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3214.92 1.27765
\(186\) 0 0
\(187\) 40.9348 0.0160077
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2537.80 −0.961408 −0.480704 0.876883i \(-0.659619\pi\)
−0.480704 + 0.876883i \(0.659619\pi\)
\(192\) 0 0
\(193\) 190.348 0.0709923 0.0354962 0.999370i \(-0.488699\pi\)
0.0354962 + 0.999370i \(0.488699\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5314.25 −1.92195 −0.960977 0.276629i \(-0.910783\pi\)
−0.960977 + 0.276629i \(0.910783\pi\)
\(198\) 0 0
\(199\) 3787.15 1.34906 0.674532 0.738246i \(-0.264346\pi\)
0.674532 + 0.738246i \(0.264346\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1059.65 −0.366369
\(204\) 0 0
\(205\) 4345.11 1.48037
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −57.5326 −0.0190412
\(210\) 0 0
\(211\) 2442.05 0.796768 0.398384 0.917219i \(-0.369571\pi\)
0.398384 + 0.917219i \(0.369571\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3722.26 1.18073
\(216\) 0 0
\(217\) −1434.91 −0.448886
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3476.78 −1.05825
\(222\) 0 0
\(223\) −2329.87 −0.699639 −0.349820 0.936817i \(-0.613757\pi\)
−0.349820 + 0.936817i \(0.613757\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1326.39 −0.387822 −0.193911 0.981019i \(-0.562117\pi\)
−0.193911 + 0.981019i \(0.562117\pi\)
\(228\) 0 0
\(229\) −1544.13 −0.445585 −0.222793 0.974866i \(-0.571517\pi\)
−0.222793 + 0.974866i \(0.571517\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2266.32 0.637216 0.318608 0.947887i \(-0.396785\pi\)
0.318608 + 0.947887i \(0.396785\pi\)
\(234\) 0 0
\(235\) −315.882 −0.0876844
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 51.6752 0.0139857 0.00699286 0.999976i \(-0.497774\pi\)
0.00699286 + 0.999976i \(0.497774\pi\)
\(240\) 0 0
\(241\) −2323.95 −0.621158 −0.310579 0.950548i \(-0.600523\pi\)
−0.310579 + 0.950548i \(0.600523\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4176.66 −1.08913
\(246\) 0 0
\(247\) 4886.52 1.25879
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6958.52 1.74987 0.874936 0.484239i \(-0.160903\pi\)
0.874936 + 0.484239i \(0.160903\pi\)
\(252\) 0 0
\(253\) 114.935 0.0285608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2974.91 0.722062 0.361031 0.932554i \(-0.382425\pi\)
0.361031 + 0.932554i \(0.382425\pi\)
\(258\) 0 0
\(259\) −1271.44 −0.305034
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 116.847 0.0273958 0.0136979 0.999906i \(-0.495640\pi\)
0.0136979 + 0.999906i \(0.495640\pi\)
\(264\) 0 0
\(265\) −3970.61 −0.920425
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3334.81 0.755863 0.377932 0.925834i \(-0.376635\pi\)
0.377932 + 0.925834i \(0.376635\pi\)
\(270\) 0 0
\(271\) −3120.23 −0.699412 −0.349706 0.936860i \(-0.613718\pi\)
−0.349706 + 0.936860i \(0.613718\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −50.1305 −0.0109927
\(276\) 0 0
\(277\) −2594.39 −0.562750 −0.281375 0.959598i \(-0.590791\pi\)
−0.281375 + 0.959598i \(0.590791\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2283.11 −0.484693 −0.242347 0.970190i \(-0.577917\pi\)
−0.242347 + 0.970190i \(0.577917\pi\)
\(282\) 0 0
\(283\) 1647.60 0.346076 0.173038 0.984915i \(-0.444642\pi\)
0.173038 + 0.984915i \(0.444642\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1718.41 −0.353431
\(288\) 0 0
\(289\) −3237.35 −0.658935
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2933.31 0.584867 0.292434 0.956286i \(-0.405535\pi\)
0.292434 + 0.956286i \(0.405535\pi\)
\(294\) 0 0
\(295\) −9973.02 −1.96831
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9761.95 −1.88812
\(300\) 0 0
\(301\) −1472.09 −0.281893
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6557.00 1.23099
\(306\) 0 0
\(307\) −8216.93 −1.52757 −0.763787 0.645468i \(-0.776662\pi\)
−0.763787 + 0.645468i \(0.776662\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 337.630 0.0615602 0.0307801 0.999526i \(-0.490201\pi\)
0.0307801 + 0.999526i \(0.490201\pi\)
\(312\) 0 0
\(313\) −772.566 −0.139514 −0.0697572 0.997564i \(-0.522222\pi\)
−0.0697572 + 0.997564i \(0.522222\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7773.93 −1.37737 −0.688686 0.725059i \(-0.741812\pi\)
−0.688686 + 0.725059i \(0.741812\pi\)
\(318\) 0 0
\(319\) −202.467 −0.0355360
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2355.08 −0.405698
\(324\) 0 0
\(325\) 4257.82 0.726712
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 124.926 0.0209342
\(330\) 0 0
\(331\) 4261.31 0.707622 0.353811 0.935317i \(-0.384886\pi\)
0.353811 + 0.935317i \(0.384886\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5425.09 −0.884789
\(336\) 0 0
\(337\) −4728.91 −0.764393 −0.382196 0.924081i \(-0.624832\pi\)
−0.382196 + 0.924081i \(0.624832\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −274.168 −0.0435397
\(342\) 0 0
\(343\) 3446.95 0.542618
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2584.04 0.399765 0.199883 0.979820i \(-0.435944\pi\)
0.199883 + 0.979820i \(0.435944\pi\)
\(348\) 0 0
\(349\) 4122.46 0.632292 0.316146 0.948711i \(-0.397611\pi\)
0.316146 + 0.948711i \(0.397611\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9524.92 1.43615 0.718074 0.695967i \(-0.245024\pi\)
0.718074 + 0.695967i \(0.245024\pi\)
\(354\) 0 0
\(355\) −14753.0 −2.20565
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3048.39 0.448156 0.224078 0.974571i \(-0.428063\pi\)
0.224078 + 0.974571i \(0.428063\pi\)
\(360\) 0 0
\(361\) −3549.00 −0.517422
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3798.07 −0.544658
\(366\) 0 0
\(367\) −5237.90 −0.745003 −0.372501 0.928032i \(-0.621500\pi\)
−0.372501 + 0.928032i \(0.621500\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1570.30 0.219747
\(372\) 0 0
\(373\) 2871.50 0.398608 0.199304 0.979938i \(-0.436132\pi\)
0.199304 + 0.979938i \(0.436132\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17196.5 2.34925
\(378\) 0 0
\(379\) 1649.20 0.223518 0.111759 0.993735i \(-0.464351\pi\)
0.111759 + 0.993735i \(0.464351\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8182.76 −1.09170 −0.545848 0.837884i \(-0.683792\pi\)
−0.545848 + 0.837884i \(0.683792\pi\)
\(384\) 0 0
\(385\) 69.2610 0.00916849
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1380.58 0.179944 0.0899719 0.995944i \(-0.471322\pi\)
0.0899719 + 0.995944i \(0.471322\pi\)
\(390\) 0 0
\(391\) 4704.83 0.608525
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16297.0 −2.07593
\(396\) 0 0
\(397\) −5772.41 −0.729746 −0.364873 0.931057i \(-0.618888\pi\)
−0.364873 + 0.931057i \(0.618888\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9378.85 1.16797 0.583987 0.811763i \(-0.301492\pi\)
0.583987 + 0.811763i \(0.301492\pi\)
\(402\) 0 0
\(403\) 23286.4 2.87836
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −242.935 −0.0295868
\(408\) 0 0
\(409\) 7073.09 0.855114 0.427557 0.903988i \(-0.359374\pi\)
0.427557 + 0.903988i \(0.359374\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3944.15 0.469925
\(414\) 0 0
\(415\) 12472.5 1.47530
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9435.65 1.10015 0.550073 0.835116i \(-0.314600\pi\)
0.550073 + 0.835116i \(0.314600\pi\)
\(420\) 0 0
\(421\) −2620.13 −0.303319 −0.151659 0.988433i \(-0.548462\pi\)
−0.151659 + 0.988433i \(0.548462\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2052.08 −0.234213
\(426\) 0 0
\(427\) −2593.18 −0.293894
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3485.65 −0.389555 −0.194777 0.980847i \(-0.562398\pi\)
−0.194777 + 0.980847i \(0.562398\pi\)
\(432\) 0 0
\(433\) 9818.69 1.08974 0.544869 0.838521i \(-0.316579\pi\)
0.544869 + 0.838521i \(0.316579\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6612.50 −0.723841
\(438\) 0 0
\(439\) −1235.00 −0.134268 −0.0671338 0.997744i \(-0.521385\pi\)
−0.0671338 + 0.997744i \(0.521385\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10744.3 −1.15232 −0.576162 0.817336i \(-0.695450\pi\)
−0.576162 + 0.817336i \(0.695450\pi\)
\(444\) 0 0
\(445\) 2517.13 0.268143
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16166.3 −1.69919 −0.849595 0.527435i \(-0.823154\pi\)
−0.849595 + 0.527435i \(0.823154\pi\)
\(450\) 0 0
\(451\) −328.337 −0.0342811
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5882.67 −0.606118
\(456\) 0 0
\(457\) −2122.26 −0.217232 −0.108616 0.994084i \(-0.534642\pi\)
−0.108616 + 0.994084i \(0.534642\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3473.41 0.350917 0.175459 0.984487i \(-0.443859\pi\)
0.175459 + 0.984487i \(0.443859\pi\)
\(462\) 0 0
\(463\) 3575.70 0.358913 0.179457 0.983766i \(-0.442566\pi\)
0.179457 + 0.983766i \(0.442566\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9406.95 0.932124 0.466062 0.884752i \(-0.345672\pi\)
0.466062 + 0.884752i \(0.345672\pi\)
\(468\) 0 0
\(469\) 2145.53 0.211239
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −281.272 −0.0273422
\(474\) 0 0
\(475\) 2884.14 0.278597
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18524.0 1.76698 0.883492 0.468447i \(-0.155186\pi\)
0.883492 + 0.468447i \(0.155186\pi\)
\(480\) 0 0
\(481\) 20633.6 1.95595
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4061.02 −0.380209
\(486\) 0 0
\(487\) −825.955 −0.0768533 −0.0384267 0.999261i \(-0.512235\pi\)
−0.0384267 + 0.999261i \(0.512235\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8918.35 −0.819714 −0.409857 0.912150i \(-0.634421\pi\)
−0.409857 + 0.912150i \(0.634421\pi\)
\(492\) 0 0
\(493\) −8287.95 −0.757141
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5834.54 0.526589
\(498\) 0 0
\(499\) 18214.4 1.63404 0.817022 0.576606i \(-0.195623\pi\)
0.817022 + 0.576606i \(0.195623\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2264.42 −0.200726 −0.100363 0.994951i \(-0.532000\pi\)
−0.100363 + 0.994951i \(0.532000\pi\)
\(504\) 0 0
\(505\) 8904.91 0.784679
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14235.2 1.23961 0.619806 0.784755i \(-0.287211\pi\)
0.619806 + 0.784755i \(0.287211\pi\)
\(510\) 0 0
\(511\) 1502.07 0.130034
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11118.3 0.951323
\(516\) 0 0
\(517\) 23.8695 0.00203052
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −11343.3 −0.953856 −0.476928 0.878942i \(-0.658250\pi\)
−0.476928 + 0.878942i \(0.658250\pi\)
\(522\) 0 0
\(523\) −20464.1 −1.71096 −0.855482 0.517832i \(-0.826739\pi\)
−0.855482 + 0.517832i \(0.826739\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11223.0 −0.927670
\(528\) 0 0
\(529\) 1043.00 0.0857234
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 27887.2 2.26628
\(534\) 0 0
\(535\) −21942.6 −1.77320
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 315.609 0.0252212
\(540\) 0 0
\(541\) −1993.69 −0.158439 −0.0792196 0.996857i \(-0.525243\pi\)
−0.0792196 + 0.996857i \(0.525243\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16724.2 −1.31447
\(546\) 0 0
\(547\) −2724.02 −0.212926 −0.106463 0.994317i \(-0.533953\pi\)
−0.106463 + 0.994317i \(0.533953\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11648.5 0.900621
\(552\) 0 0
\(553\) 6445.17 0.495618
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15046.5 −1.14460 −0.572299 0.820045i \(-0.693948\pi\)
−0.572299 + 0.820045i \(0.693948\pi\)
\(558\) 0 0
\(559\) 23889.7 1.80756
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19608.6 1.46786 0.733928 0.679228i \(-0.237685\pi\)
0.733928 + 0.679228i \(0.237685\pi\)
\(564\) 0 0
\(565\) −6277.51 −0.467428
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13968.5 −1.02916 −0.514580 0.857443i \(-0.672052\pi\)
−0.514580 + 0.857443i \(0.672052\pi\)
\(570\) 0 0
\(571\) −23614.0 −1.73068 −0.865339 0.501188i \(-0.832897\pi\)
−0.865339 + 0.501188i \(0.832897\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5761.74 −0.417880
\(576\) 0 0
\(577\) −23002.4 −1.65962 −0.829811 0.558044i \(-0.811552\pi\)
−0.829811 + 0.558044i \(0.811552\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4932.64 −0.352221
\(582\) 0 0
\(583\) 300.038 0.0213144
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8100.44 0.569576 0.284788 0.958591i \(-0.408077\pi\)
0.284788 + 0.958591i \(0.408077\pi\)
\(588\) 0 0
\(589\) 15773.6 1.10347
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22256.2 −1.54124 −0.770618 0.637298i \(-0.780052\pi\)
−0.770618 + 0.637298i \(0.780052\pi\)
\(594\) 0 0
\(595\) 2835.18 0.195346
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8386.30 0.572045 0.286023 0.958223i \(-0.407667\pi\)
0.286023 + 0.958223i \(0.407667\pi\)
\(600\) 0 0
\(601\) −9814.21 −0.666107 −0.333053 0.942908i \(-0.608079\pi\)
−0.333053 + 0.942908i \(0.608079\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −17600.8 −1.18277
\(606\) 0 0
\(607\) −24077.7 −1.61002 −0.805011 0.593260i \(-0.797841\pi\)
−0.805011 + 0.593260i \(0.797841\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2027.35 −0.134235
\(612\) 0 0
\(613\) −18679.6 −1.23077 −0.615386 0.788226i \(-0.711000\pi\)
−0.615386 + 0.788226i \(0.711000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1650.88 0.107718 0.0538589 0.998549i \(-0.482848\pi\)
0.0538589 + 0.998549i \(0.482848\pi\)
\(618\) 0 0
\(619\) 25032.9 1.62545 0.812727 0.582645i \(-0.197982\pi\)
0.812727 + 0.582645i \(0.197982\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −995.481 −0.0640178
\(624\) 0 0
\(625\) −19378.2 −1.24021
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9944.47 −0.630385
\(630\) 0 0
\(631\) 1666.38 0.105131 0.0525653 0.998617i \(-0.483260\pi\)
0.0525653 + 0.998617i \(0.483260\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7415.83 −0.463446
\(636\) 0 0
\(637\) −26806.1 −1.66734
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24026.5 1.48049 0.740243 0.672339i \(-0.234710\pi\)
0.740243 + 0.672339i \(0.234710\pi\)
\(642\) 0 0
\(643\) −12772.5 −0.783356 −0.391678 0.920102i \(-0.628105\pi\)
−0.391678 + 0.920102i \(0.628105\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19420.6 1.18007 0.590034 0.807378i \(-0.299114\pi\)
0.590034 + 0.807378i \(0.299114\pi\)
\(648\) 0 0
\(649\) 753.609 0.0455805
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −811.744 −0.0486462 −0.0243231 0.999704i \(-0.507743\pi\)
−0.0243231 + 0.999704i \(0.507743\pi\)
\(654\) 0 0
\(655\) −10445.4 −0.623108
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12165.8 0.719138 0.359569 0.933119i \(-0.382924\pi\)
0.359569 + 0.933119i \(0.382924\pi\)
\(660\) 0 0
\(661\) −3004.33 −0.176785 −0.0883925 0.996086i \(-0.528173\pi\)
−0.0883925 + 0.996086i \(0.528173\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3984.77 −0.232365
\(666\) 0 0
\(667\) −23270.5 −1.35088
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −495.478 −0.0285063
\(672\) 0 0
\(673\) −1272.83 −0.0729032 −0.0364516 0.999335i \(-0.511605\pi\)
−0.0364516 + 0.999335i \(0.511605\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11039.9 0.626731 0.313366 0.949633i \(-0.398544\pi\)
0.313366 + 0.949633i \(0.398544\pi\)
\(678\) 0 0
\(679\) 1606.06 0.0907730
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6103.65 −0.341947 −0.170974 0.985276i \(-0.554691\pi\)
−0.170974 + 0.985276i \(0.554691\pi\)
\(684\) 0 0
\(685\) 17607.0 0.982085
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −25483.6 −1.40907
\(690\) 0 0
\(691\) 30759.2 1.69340 0.846698 0.532074i \(-0.178587\pi\)
0.846698 + 0.532074i \(0.178587\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 41545.4 2.26749
\(696\) 0 0
\(697\) −13440.4 −0.730403
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30619.5 1.64976 0.824880 0.565308i \(-0.191243\pi\)
0.824880 + 0.565308i \(0.191243\pi\)
\(702\) 0 0
\(703\) 13976.7 0.749844
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3521.73 −0.187339
\(708\) 0 0
\(709\) 28651.0 1.51764 0.758822 0.651298i \(-0.225775\pi\)
0.758822 + 0.651298i \(0.225775\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −31511.5 −1.65514
\(714\) 0 0
\(715\) −1124.00 −0.0587905
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24899.6 −1.29151 −0.645756 0.763544i \(-0.723458\pi\)
−0.645756 + 0.763544i \(0.723458\pi\)
\(720\) 0 0
\(721\) −4397.09 −0.227124
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10149.8 0.519936
\(726\) 0 0
\(727\) −34318.0 −1.75074 −0.875369 0.483456i \(-0.839381\pi\)
−0.875369 + 0.483456i \(0.839381\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11513.8 −0.582562
\(732\) 0 0
\(733\) 27751.9 1.39842 0.699209 0.714917i \(-0.253536\pi\)
0.699209 + 0.714917i \(0.253536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 409.945 0.0204892
\(738\) 0 0
\(739\) −24701.0 −1.22955 −0.614777 0.788701i \(-0.710754\pi\)
−0.614777 + 0.788701i \(0.710754\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23186.8 1.14488 0.572438 0.819948i \(-0.305998\pi\)
0.572438 + 0.819948i \(0.305998\pi\)
\(744\) 0 0
\(745\) −35284.2 −1.73519
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8677.91 0.423343
\(750\) 0 0
\(751\) 24909.9 1.21035 0.605177 0.796091i \(-0.293102\pi\)
0.605177 + 0.796091i \(0.293102\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6344.32 0.305819
\(756\) 0 0
\(757\) 2659.35 0.127682 0.0638412 0.997960i \(-0.479665\pi\)
0.0638412 + 0.997960i \(0.479665\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −25415.3 −1.21065 −0.605324 0.795980i \(-0.706956\pi\)
−0.605324 + 0.795980i \(0.706956\pi\)
\(762\) 0 0
\(763\) 6614.13 0.313824
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −64007.6 −3.01327
\(768\) 0 0
\(769\) 30053.1 1.40929 0.704645 0.709560i \(-0.251106\pi\)
0.704645 + 0.709560i \(0.251106\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6066.30 0.282263 0.141132 0.989991i \(-0.454926\pi\)
0.141132 + 0.989991i \(0.454926\pi\)
\(774\) 0 0
\(775\) 13744.2 0.637040
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18890.1 0.868816
\(780\) 0 0
\(781\) 1114.80 0.0510766
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −26531.2 −1.20629
\(786\) 0 0
\(787\) −5496.18 −0.248942 −0.124471 0.992223i \(-0.539723\pi\)
−0.124471 + 0.992223i \(0.539723\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2482.64 0.111596
\(792\) 0 0
\(793\) 42083.3 1.88452
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 27808.0 1.23590 0.617949 0.786218i \(-0.287964\pi\)
0.617949 + 0.786218i \(0.287964\pi\)
\(798\) 0 0
\(799\) 977.092 0.0432629
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 287.000 0.0126127
\(804\) 0 0
\(805\) 7960.50 0.348535
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6720.58 0.292068 0.146034 0.989280i \(-0.453349\pi\)
0.146034 + 0.989280i \(0.453349\pi\)
\(810\) 0 0
\(811\) 9923.66 0.429675 0.214838 0.976650i \(-0.431078\pi\)
0.214838 + 0.976650i \(0.431078\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11011.0 −0.473249
\(816\) 0 0
\(817\) 16182.3 0.692958
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19118.8 0.812728 0.406364 0.913711i \(-0.366797\pi\)
0.406364 + 0.913711i \(0.366797\pi\)
\(822\) 0 0
\(823\) 28343.3 1.20047 0.600234 0.799825i \(-0.295074\pi\)
0.600234 + 0.799825i \(0.295074\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25909.2 −1.08942 −0.544711 0.838624i \(-0.683361\pi\)
−0.544711 + 0.838624i \(0.683361\pi\)
\(828\) 0 0
\(829\) −3137.00 −0.131426 −0.0657132 0.997839i \(-0.520932\pi\)
−0.0657132 + 0.997839i \(0.520932\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12919.4 0.537370
\(834\) 0 0
\(835\) 38002.5 1.57501
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8110.63 −0.333743 −0.166871 0.985979i \(-0.553366\pi\)
−0.166871 + 0.985979i \(0.553366\pi\)
\(840\) 0 0
\(841\) 16604.0 0.680800
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 66392.2 2.70291
\(846\) 0 0
\(847\) 6960.80 0.282380
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27921.6 −1.12473
\(852\) 0 0
\(853\) 20317.3 0.815536 0.407768 0.913085i \(-0.366307\pi\)
0.407768 + 0.913085i \(0.366307\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2938.63 −0.117132 −0.0585658 0.998284i \(-0.518653\pi\)
−0.0585658 + 0.998284i \(0.518653\pi\)
\(858\) 0 0
\(859\) −10710.3 −0.425412 −0.212706 0.977116i \(-0.568228\pi\)
−0.212706 + 0.977116i \(0.568228\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12400.3 0.489122 0.244561 0.969634i \(-0.421356\pi\)
0.244561 + 0.969634i \(0.421356\pi\)
\(864\) 0 0
\(865\) −47369.2 −1.86197
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1231.48 0.0480725
\(870\) 0 0
\(871\) −34818.6 −1.35452
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5185.54 0.200346
\(876\) 0 0
\(877\) −30790.2 −1.18553 −0.592765 0.805375i \(-0.701964\pi\)
−0.592765 + 0.805375i \(0.701964\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −20320.3 −0.777082 −0.388541 0.921431i \(-0.627021\pi\)
−0.388541 + 0.921431i \(0.627021\pi\)
\(882\) 0 0
\(883\) 27680.1 1.05494 0.527468 0.849575i \(-0.323141\pi\)
0.527468 + 0.849575i \(0.323141\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 41124.8 1.55675 0.778374 0.627801i \(-0.216045\pi\)
0.778374 + 0.627801i \(0.216045\pi\)
\(888\) 0 0
\(889\) 2932.83 0.110646
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1373.28 −0.0514613
\(894\) 0 0
\(895\) 28612.4 1.06861
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 55510.2 2.05936
\(900\) 0 0
\(901\) 12282.0 0.454131
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 23030.1 0.845907
\(906\) 0 0
\(907\) 26971.3 0.987397 0.493698 0.869633i \(-0.335645\pi\)
0.493698 + 0.869633i \(0.335645\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −46282.0 −1.68319 −0.841597 0.540105i \(-0.818384\pi\)
−0.841597 + 0.540105i \(0.818384\pi\)
\(912\) 0 0
\(913\) −942.478 −0.0341637
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4130.97 0.148764
\(918\) 0 0
\(919\) 36864.3 1.32322 0.661611 0.749847i \(-0.269873\pi\)
0.661611 + 0.749847i \(0.269873\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −94685.6 −3.37661
\(924\) 0 0
\(925\) 12178.4 0.432891
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −34278.6 −1.21060 −0.605299 0.795998i \(-0.706946\pi\)
−0.605299 + 0.795998i \(0.706946\pi\)
\(930\) 0 0
\(931\) −18157.8 −0.639202
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 541.718 0.0189477
\(936\) 0 0
\(937\) 450.515 0.0157072 0.00785362 0.999969i \(-0.497500\pi\)
0.00785362 + 0.999969i \(0.497500\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7098.87 0.245926 0.122963 0.992411i \(-0.460760\pi\)
0.122963 + 0.992411i \(0.460760\pi\)
\(942\) 0 0
\(943\) −37737.3 −1.30318
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9833.35 −0.337425 −0.168712 0.985665i \(-0.553961\pi\)
−0.168712 + 0.985665i \(0.553961\pi\)
\(948\) 0 0
\(949\) −24376.3 −0.833812
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41260.1 1.40246 0.701231 0.712934i \(-0.252634\pi\)
0.701231 + 0.712934i \(0.252634\pi\)
\(954\) 0 0
\(955\) −33584.5 −1.13798
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6963.25 −0.234468
\(960\) 0 0
\(961\) 45377.3 1.52319
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2519.00 0.0840305
\(966\) 0 0
\(967\) −22642.8 −0.752992 −0.376496 0.926418i \(-0.622871\pi\)
−0.376496 + 0.926418i \(0.622871\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3166.82 0.104663 0.0523316 0.998630i \(-0.483335\pi\)
0.0523316 + 0.998630i \(0.483335\pi\)
\(972\) 0 0
\(973\) −16430.5 −0.541353
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20643.0 0.675976 0.337988 0.941150i \(-0.390254\pi\)
0.337988 + 0.941150i \(0.390254\pi\)
\(978\) 0 0
\(979\) −190.206 −0.00620942
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −59289.5 −1.92374 −0.961872 0.273498i \(-0.911819\pi\)
−0.961872 + 0.273498i \(0.911819\pi\)
\(984\) 0 0
\(985\) −70327.2 −2.27493
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32327.9 −1.03940
\(990\) 0 0
\(991\) −14404.0 −0.461712 −0.230856 0.972988i \(-0.574153\pi\)
−0.230856 + 0.972988i \(0.574153\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 50117.9 1.59683
\(996\) 0 0
\(997\) −48894.9 −1.55318 −0.776588 0.630009i \(-0.783051\pi\)
−0.776588 + 0.630009i \(0.783051\pi\)
\(998\) 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 216.4.a.e.1.2 2
3.2 odd 2 216.4.a.h.1.1 yes 2
4.3 odd 2 432.4.a.o.1.2 2
8.3 odd 2 1728.4.a.bs.1.1 2
8.5 even 2 1728.4.a.bt.1.1 2
9.2 odd 6 648.4.i.m.433.2 4
9.4 even 3 648.4.i.s.217.1 4
9.5 odd 6 648.4.i.m.217.2 4
9.7 even 3 648.4.i.s.433.1 4
12.11 even 2 432.4.a.s.1.1 2
24.5 odd 2 1728.4.a.bh.1.2 2
24.11 even 2 1728.4.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.e.1.2 2 1.1 even 1 trivial
216.4.a.h.1.1 yes 2 3.2 odd 2
432.4.a.o.1.2 2 4.3 odd 2
432.4.a.s.1.1 2 12.11 even 2
648.4.i.m.217.2 4 9.5 odd 6
648.4.i.m.433.2 4 9.2 odd 6
648.4.i.s.217.1 4 9.4 even 3
648.4.i.s.433.1 4 9.7 even 3
1728.4.a.bg.1.2 2 24.11 even 2
1728.4.a.bh.1.2 2 24.5 odd 2
1728.4.a.bs.1.1 2 8.3 odd 2
1728.4.a.bt.1.1 2 8.5 even 2