Properties

Label 2352.4.a.bl.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -0.100505 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -0.100505 q^{5} +9.00000 q^{9} +43.9411 q^{11} -16.6447 q^{13} +0.301515 q^{15} -121.640 q^{17} +127.113 q^{19} -53.5980 q^{23} -124.990 q^{25} -27.0000 q^{27} +235.681 q^{29} +18.7107 q^{31} -131.823 q^{33} -191.882 q^{37} +49.9340 q^{39} -319.713 q^{41} +218.579 q^{43} -0.904546 q^{45} -401.553 q^{47} +364.919 q^{51} +643.117 q^{53} -4.41631 q^{55} -381.338 q^{57} +11.6123 q^{59} -12.2426 q^{61} +1.67287 q^{65} -669.048 q^{67} +160.794 q^{69} -822.098 q^{71} +515.100 q^{73} +374.970 q^{75} +805.754 q^{79} +81.0000 q^{81} +394.863 q^{83} +12.2254 q^{85} -707.044 q^{87} +673.418 q^{89} -56.1320 q^{93} -12.7755 q^{95} -1091.11 q^{97} +395.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 20q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 20q^{5} + 18q^{9} + 20q^{11} - 104q^{13} + 60q^{15} - 116q^{17} + 192q^{19} - 28q^{23} + 146q^{25} - 54q^{27} + 296q^{29} - 104q^{31} - 60q^{33} - 248q^{37} + 312q^{39} - 20q^{41} + 720q^{43} - 180q^{45} - 96q^{47} + 348q^{51} + 268q^{53} + 472q^{55} - 576q^{57} - 616q^{59} - 16q^{61} + 1740q^{65} + 144q^{67} + 84q^{69} - 988q^{71} - 104q^{73} - 438q^{75} + 944q^{79} + 162q^{81} + 1016q^{83} - 100q^{85} - 888q^{87} + 388q^{89} + 312q^{93} - 1304q^{95} - 488q^{97} + 180q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −0.100505 −0.00898945 −0.00449472 0.999990i \(-0.501431\pi\)
−0.00449472 + 0.999990i \(0.501431\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 43.9411 1.20443 0.602216 0.798333i \(-0.294285\pi\)
0.602216 + 0.798333i \(0.294285\pi\)
\(12\) 0 0
\(13\) −16.6447 −0.355108 −0.177554 0.984111i \(-0.556818\pi\)
−0.177554 + 0.984111i \(0.556818\pi\)
\(14\) 0 0
\(15\) 0.301515 0.00519006
\(16\) 0 0
\(17\) −121.640 −1.73541 −0.867704 0.497081i \(-0.834405\pi\)
−0.867704 + 0.497081i \(0.834405\pi\)
\(18\) 0 0
\(19\) 127.113 1.53482 0.767412 0.641154i \(-0.221544\pi\)
0.767412 + 0.641154i \(0.221544\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −53.5980 −0.485911 −0.242955 0.970037i \(-0.578117\pi\)
−0.242955 + 0.970037i \(0.578117\pi\)
\(24\) 0 0
\(25\) −124.990 −0.999919
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 235.681 1.50913 0.754567 0.656223i \(-0.227847\pi\)
0.754567 + 0.656223i \(0.227847\pi\)
\(30\) 0 0
\(31\) 18.7107 0.108404 0.0542022 0.998530i \(-0.482738\pi\)
0.0542022 + 0.998530i \(0.482738\pi\)
\(32\) 0 0
\(33\) −131.823 −0.695379
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −191.882 −0.852574 −0.426287 0.904588i \(-0.640179\pi\)
−0.426287 + 0.904588i \(0.640179\pi\)
\(38\) 0 0
\(39\) 49.9340 0.205021
\(40\) 0 0
\(41\) −319.713 −1.21782 −0.608912 0.793238i \(-0.708394\pi\)
−0.608912 + 0.793238i \(0.708394\pi\)
\(42\) 0 0
\(43\) 218.579 0.775184 0.387592 0.921831i \(-0.373307\pi\)
0.387592 + 0.921831i \(0.373307\pi\)
\(44\) 0 0
\(45\) −0.904546 −0.00299648
\(46\) 0 0
\(47\) −401.553 −1.24623 −0.623113 0.782132i \(-0.714132\pi\)
−0.623113 + 0.782132i \(0.714132\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 364.919 1.00194
\(52\) 0 0
\(53\) 643.117 1.66677 0.833386 0.552692i \(-0.186399\pi\)
0.833386 + 0.552692i \(0.186399\pi\)
\(54\) 0 0
\(55\) −4.41631 −0.0108272
\(56\) 0 0
\(57\) −381.338 −0.886131
\(58\) 0 0
\(59\) 11.6123 0.0256235 0.0128118 0.999918i \(-0.495922\pi\)
0.0128118 + 0.999918i \(0.495922\pi\)
\(60\) 0 0
\(61\) −12.2426 −0.0256969 −0.0128484 0.999917i \(-0.504090\pi\)
−0.0128484 + 0.999917i \(0.504090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.67287 0.00319222
\(66\) 0 0
\(67\) −669.048 −1.21996 −0.609979 0.792417i \(-0.708822\pi\)
−0.609979 + 0.792417i \(0.708822\pi\)
\(68\) 0 0
\(69\) 160.794 0.280541
\(70\) 0 0
\(71\) −822.098 −1.37416 −0.687078 0.726584i \(-0.741107\pi\)
−0.687078 + 0.726584i \(0.741107\pi\)
\(72\) 0 0
\(73\) 515.100 0.825861 0.412930 0.910763i \(-0.364505\pi\)
0.412930 + 0.910763i \(0.364505\pi\)
\(74\) 0 0
\(75\) 374.970 0.577304
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 805.754 1.14752 0.573762 0.819022i \(-0.305483\pi\)
0.573762 + 0.819022i \(0.305483\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 394.863 0.522191 0.261095 0.965313i \(-0.415916\pi\)
0.261095 + 0.965313i \(0.415916\pi\)
\(84\) 0 0
\(85\) 12.2254 0.0156004
\(86\) 0 0
\(87\) −707.044 −0.871299
\(88\) 0 0
\(89\) 673.418 0.802047 0.401024 0.916068i \(-0.368655\pi\)
0.401024 + 0.916068i \(0.368655\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −56.1320 −0.0625873
\(94\) 0 0
\(95\) −12.7755 −0.0137972
\(96\) 0 0
\(97\) −1091.11 −1.14212 −0.571061 0.820908i \(-0.693468\pi\)
−0.571061 + 0.820908i \(0.693468\pi\)
\(98\) 0 0
\(99\) 395.470 0.401477
\(100\) 0 0
\(101\) −1370.79 −1.35048 −0.675242 0.737597i \(-0.735961\pi\)
−0.675242 + 0.737597i \(0.735961\pi\)
\(102\) 0 0
\(103\) 1413.96 1.35263 0.676316 0.736611i \(-0.263575\pi\)
0.676316 + 0.736611i \(0.263575\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 343.539 0.310385 0.155192 0.987884i \(-0.450400\pi\)
0.155192 + 0.987884i \(0.450400\pi\)
\(108\) 0 0
\(109\) −317.657 −0.279138 −0.139569 0.990212i \(-0.544572\pi\)
−0.139569 + 0.990212i \(0.544572\pi\)
\(110\) 0 0
\(111\) 575.647 0.492234
\(112\) 0 0
\(113\) 798.373 0.664643 0.332321 0.943166i \(-0.392168\pi\)
0.332321 + 0.943166i \(0.392168\pi\)
\(114\) 0 0
\(115\) 5.38687 0.00436807
\(116\) 0 0
\(117\) −149.802 −0.118369
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 599.823 0.450656
\(122\) 0 0
\(123\) 959.138 0.703110
\(124\) 0 0
\(125\) 25.1253 0.0179782
\(126\) 0 0
\(127\) −1071.40 −0.748593 −0.374297 0.927309i \(-0.622116\pi\)
−0.374297 + 0.927309i \(0.622116\pi\)
\(128\) 0 0
\(129\) −655.736 −0.447553
\(130\) 0 0
\(131\) 2515.02 1.67739 0.838695 0.544601i \(-0.183319\pi\)
0.838695 + 0.544601i \(0.183319\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.71364 0.00173002
\(136\) 0 0
\(137\) 251.064 0.156568 0.0782841 0.996931i \(-0.475056\pi\)
0.0782841 + 0.996931i \(0.475056\pi\)
\(138\) 0 0
\(139\) −886.067 −0.540685 −0.270343 0.962764i \(-0.587137\pi\)
−0.270343 + 0.962764i \(0.587137\pi\)
\(140\) 0 0
\(141\) 1204.66 0.719508
\(142\) 0 0
\(143\) −731.385 −0.427703
\(144\) 0 0
\(145\) −23.6872 −0.0135663
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 582.626 0.320339 0.160170 0.987090i \(-0.448796\pi\)
0.160170 + 0.987090i \(0.448796\pi\)
\(150\) 0 0
\(151\) 2811.76 1.51535 0.757676 0.652631i \(-0.226335\pi\)
0.757676 + 0.652631i \(0.226335\pi\)
\(152\) 0 0
\(153\) −1094.76 −0.578469
\(154\) 0 0
\(155\) −1.88052 −0.000974496 0
\(156\) 0 0
\(157\) −1691.34 −0.859770 −0.429885 0.902884i \(-0.641446\pi\)
−0.429885 + 0.902884i \(0.641446\pi\)
\(158\) 0 0
\(159\) −1929.35 −0.962311
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 40.7232 0.0195686 0.00978432 0.999952i \(-0.496886\pi\)
0.00978432 + 0.999952i \(0.496886\pi\)
\(164\) 0 0
\(165\) 13.2489 0.00625107
\(166\) 0 0
\(167\) −2900.47 −1.34398 −0.671990 0.740560i \(-0.734560\pi\)
−0.671990 + 0.740560i \(0.734560\pi\)
\(168\) 0 0
\(169\) −1919.96 −0.873899
\(170\) 0 0
\(171\) 1144.01 0.511608
\(172\) 0 0
\(173\) −2146.15 −0.943171 −0.471585 0.881820i \(-0.656318\pi\)
−0.471585 + 0.881820i \(0.656318\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −34.8368 −0.0147938
\(178\) 0 0
\(179\) −1203.54 −0.502552 −0.251276 0.967916i \(-0.580850\pi\)
−0.251276 + 0.967916i \(0.580850\pi\)
\(180\) 0 0
\(181\) −2990.47 −1.22807 −0.614033 0.789280i \(-0.710454\pi\)
−0.614033 + 0.789280i \(0.710454\pi\)
\(182\) 0 0
\(183\) 36.7279 0.0148361
\(184\) 0 0
\(185\) 19.2851 0.00766417
\(186\) 0 0
\(187\) −5344.98 −2.09018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2807.41 1.06354 0.531772 0.846887i \(-0.321526\pi\)
0.531772 + 0.846887i \(0.321526\pi\)
\(192\) 0 0
\(193\) 3336.37 1.24434 0.622169 0.782883i \(-0.286252\pi\)
0.622169 + 0.782883i \(0.286252\pi\)
\(194\) 0 0
\(195\) −5.01862 −0.00184303
\(196\) 0 0
\(197\) −4226.65 −1.52861 −0.764305 0.644855i \(-0.776918\pi\)
−0.764305 + 0.644855i \(0.776918\pi\)
\(198\) 0 0
\(199\) −4385.69 −1.56228 −0.781140 0.624356i \(-0.785361\pi\)
−0.781140 + 0.624356i \(0.785361\pi\)
\(200\) 0 0
\(201\) 2007.14 0.704343
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 32.1328 0.0109476
\(206\) 0 0
\(207\) −482.382 −0.161970
\(208\) 0 0
\(209\) 5585.48 1.84859
\(210\) 0 0
\(211\) −2291.56 −0.747665 −0.373833 0.927496i \(-0.621957\pi\)
−0.373833 + 0.927496i \(0.621957\pi\)
\(212\) 0 0
\(213\) 2466.29 0.793369
\(214\) 0 0
\(215\) −21.9683 −0.00696848
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1545.30 −0.476811
\(220\) 0 0
\(221\) 2024.65 0.616257
\(222\) 0 0
\(223\) 217.970 0.0654544 0.0327272 0.999464i \(-0.489581\pi\)
0.0327272 + 0.999464i \(0.489581\pi\)
\(224\) 0 0
\(225\) −1124.91 −0.333306
\(226\) 0 0
\(227\) −1835.36 −0.536639 −0.268320 0.963330i \(-0.586468\pi\)
−0.268320 + 0.963330i \(0.586468\pi\)
\(228\) 0 0
\(229\) 2774.01 0.800488 0.400244 0.916409i \(-0.368925\pi\)
0.400244 + 0.916409i \(0.368925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −988.712 −0.277994 −0.138997 0.990293i \(-0.544388\pi\)
−0.138997 + 0.990293i \(0.544388\pi\)
\(234\) 0 0
\(235\) 40.3581 0.0112029
\(236\) 0 0
\(237\) −2417.26 −0.662524
\(238\) 0 0
\(239\) 837.928 0.226783 0.113391 0.993550i \(-0.463829\pi\)
0.113391 + 0.993550i \(0.463829\pi\)
\(240\) 0 0
\(241\) −3454.99 −0.923466 −0.461733 0.887019i \(-0.652772\pi\)
−0.461733 + 0.887019i \(0.652772\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2115.75 −0.545028
\(248\) 0 0
\(249\) −1184.59 −0.301487
\(250\) 0 0
\(251\) −5635.01 −1.41705 −0.708523 0.705688i \(-0.750638\pi\)
−0.708523 + 0.705688i \(0.750638\pi\)
\(252\) 0 0
\(253\) −2355.16 −0.585246
\(254\) 0 0
\(255\) −36.6762 −0.00900687
\(256\) 0 0
\(257\) −2271.16 −0.551248 −0.275624 0.961265i \(-0.588885\pi\)
−0.275624 + 0.961265i \(0.588885\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2121.13 0.503045
\(262\) 0 0
\(263\) −163.867 −0.0384201 −0.0192101 0.999815i \(-0.506115\pi\)
−0.0192101 + 0.999815i \(0.506115\pi\)
\(264\) 0 0
\(265\) −64.6365 −0.0149834
\(266\) 0 0
\(267\) −2020.26 −0.463062
\(268\) 0 0
\(269\) −5167.10 −1.17116 −0.585582 0.810613i \(-0.699134\pi\)
−0.585582 + 0.810613i \(0.699134\pi\)
\(270\) 0 0
\(271\) 1622.27 0.363638 0.181819 0.983332i \(-0.441801\pi\)
0.181819 + 0.983332i \(0.441801\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5492.20 −1.20433
\(276\) 0 0
\(277\) −4612.37 −1.00047 −0.500235 0.865890i \(-0.666753\pi\)
−0.500235 + 0.865890i \(0.666753\pi\)
\(278\) 0 0
\(279\) 168.396 0.0361348
\(280\) 0 0
\(281\) −2125.22 −0.451174 −0.225587 0.974223i \(-0.572430\pi\)
−0.225587 + 0.974223i \(0.572430\pi\)
\(282\) 0 0
\(283\) −2571.54 −0.540149 −0.270075 0.962839i \(-0.587048\pi\)
−0.270075 + 0.962839i \(0.587048\pi\)
\(284\) 0 0
\(285\) 38.3264 0.00796583
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9883.19 2.01164
\(290\) 0 0
\(291\) 3273.34 0.659404
\(292\) 0 0
\(293\) −3324.96 −0.662957 −0.331478 0.943463i \(-0.607547\pi\)
−0.331478 + 0.943463i \(0.607547\pi\)
\(294\) 0 0
\(295\) −1.16709 −0.000230341 0
\(296\) 0 0
\(297\) −1186.41 −0.231793
\(298\) 0 0
\(299\) 892.120 0.172551
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4112.37 0.779702
\(304\) 0 0
\(305\) 1.23045 0.000231001 0
\(306\) 0 0
\(307\) −887.096 −0.164916 −0.0824580 0.996595i \(-0.526277\pi\)
−0.0824580 + 0.996595i \(0.526277\pi\)
\(308\) 0 0
\(309\) −4241.87 −0.780943
\(310\) 0 0
\(311\) −4510.82 −0.822460 −0.411230 0.911532i \(-0.634901\pi\)
−0.411230 + 0.911532i \(0.634901\pi\)
\(312\) 0 0
\(313\) −3715.78 −0.671018 −0.335509 0.942037i \(-0.608908\pi\)
−0.335509 + 0.942037i \(0.608908\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6954.52 1.23219 0.616096 0.787671i \(-0.288713\pi\)
0.616096 + 0.787671i \(0.288713\pi\)
\(318\) 0 0
\(319\) 10356.1 1.81765
\(320\) 0 0
\(321\) −1030.62 −0.179201
\(322\) 0 0
\(323\) −15461.9 −2.66355
\(324\) 0 0
\(325\) 2080.41 0.355079
\(326\) 0 0
\(327\) 952.971 0.161160
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9863.18 1.63785 0.818926 0.573899i \(-0.194570\pi\)
0.818926 + 0.573899i \(0.194570\pi\)
\(332\) 0 0
\(333\) −1726.94 −0.284191
\(334\) 0 0
\(335\) 67.2427 0.0109668
\(336\) 0 0
\(337\) −5945.06 −0.960974 −0.480487 0.877002i \(-0.659540\pi\)
−0.480487 + 0.877002i \(0.659540\pi\)
\(338\) 0 0
\(339\) −2395.12 −0.383732
\(340\) 0 0
\(341\) 822.168 0.130566
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −16.1606 −0.00252191
\(346\) 0 0
\(347\) −1169.57 −0.180939 −0.0904697 0.995899i \(-0.528837\pi\)
−0.0904697 + 0.995899i \(0.528837\pi\)
\(348\) 0 0
\(349\) 9176.66 1.40749 0.703747 0.710451i \(-0.251509\pi\)
0.703747 + 0.710451i \(0.251509\pi\)
\(350\) 0 0
\(351\) 449.406 0.0683405
\(352\) 0 0
\(353\) −10587.2 −1.59632 −0.798158 0.602448i \(-0.794192\pi\)
−0.798158 + 0.602448i \(0.794192\pi\)
\(354\) 0 0
\(355\) 82.6250 0.0123529
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8615.21 −1.26656 −0.633278 0.773924i \(-0.718291\pi\)
−0.633278 + 0.773924i \(0.718291\pi\)
\(360\) 0 0
\(361\) 9298.64 1.35568
\(362\) 0 0
\(363\) −1799.47 −0.260186
\(364\) 0 0
\(365\) −51.7701 −0.00742403
\(366\) 0 0
\(367\) 8297.79 1.18022 0.590110 0.807323i \(-0.299084\pi\)
0.590110 + 0.807323i \(0.299084\pi\)
\(368\) 0 0
\(369\) −2877.41 −0.405941
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5123.86 −0.711269 −0.355634 0.934625i \(-0.615735\pi\)
−0.355634 + 0.934625i \(0.615735\pi\)
\(374\) 0 0
\(375\) −75.3758 −0.0103797
\(376\) 0 0
\(377\) −3922.83 −0.535905
\(378\) 0 0
\(379\) 1502.49 0.203635 0.101817 0.994803i \(-0.467534\pi\)
0.101817 + 0.994803i \(0.467534\pi\)
\(380\) 0 0
\(381\) 3214.20 0.432200
\(382\) 0 0
\(383\) −10872.9 −1.45060 −0.725301 0.688431i \(-0.758300\pi\)
−0.725301 + 0.688431i \(0.758300\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1967.21 0.258395
\(388\) 0 0
\(389\) 4618.99 0.602036 0.301018 0.953618i \(-0.402674\pi\)
0.301018 + 0.953618i \(0.402674\pi\)
\(390\) 0 0
\(391\) 6519.64 0.843254
\(392\) 0 0
\(393\) −7545.05 −0.968442
\(394\) 0 0
\(395\) −80.9824 −0.0103156
\(396\) 0 0
\(397\) −9606.95 −1.21451 −0.607253 0.794508i \(-0.707729\pi\)
−0.607253 + 0.794508i \(0.707729\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10501.0 −1.30772 −0.653862 0.756614i \(-0.726852\pi\)
−0.653862 + 0.756614i \(0.726852\pi\)
\(402\) 0 0
\(403\) −311.433 −0.0384952
\(404\) 0 0
\(405\) −8.14091 −0.000998827 0
\(406\) 0 0
\(407\) −8431.52 −1.02687
\(408\) 0 0
\(409\) 12066.9 1.45885 0.729427 0.684059i \(-0.239787\pi\)
0.729427 + 0.684059i \(0.239787\pi\)
\(410\) 0 0
\(411\) −753.192 −0.0903947
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −39.6857 −0.00469421
\(416\) 0 0
\(417\) 2658.20 0.312165
\(418\) 0 0
\(419\) −6366.31 −0.742278 −0.371139 0.928577i \(-0.621033\pi\)
−0.371139 + 0.928577i \(0.621033\pi\)
\(420\) 0 0
\(421\) −4731.84 −0.547781 −0.273890 0.961761i \(-0.588311\pi\)
−0.273890 + 0.961761i \(0.588311\pi\)
\(422\) 0 0
\(423\) −3613.98 −0.415408
\(424\) 0 0
\(425\) 15203.7 1.73527
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2194.16 0.246934
\(430\) 0 0
\(431\) −3752.78 −0.419409 −0.209704 0.977765i \(-0.567250\pi\)
−0.209704 + 0.977765i \(0.567250\pi\)
\(432\) 0 0
\(433\) −11709.2 −1.29956 −0.649780 0.760122i \(-0.725139\pi\)
−0.649780 + 0.760122i \(0.725139\pi\)
\(434\) 0 0
\(435\) 71.0615 0.00783250
\(436\) 0 0
\(437\) −6812.98 −0.745788
\(438\) 0 0
\(439\) −14924.7 −1.62259 −0.811296 0.584635i \(-0.801238\pi\)
−0.811296 + 0.584635i \(0.801238\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8517.66 0.913513 0.456756 0.889592i \(-0.349011\pi\)
0.456756 + 0.889592i \(0.349011\pi\)
\(444\) 0 0
\(445\) −67.6820 −0.00720996
\(446\) 0 0
\(447\) −1747.88 −0.184948
\(448\) 0 0
\(449\) −5965.73 −0.627038 −0.313519 0.949582i \(-0.601508\pi\)
−0.313519 + 0.949582i \(0.601508\pi\)
\(450\) 0 0
\(451\) −14048.5 −1.46678
\(452\) 0 0
\(453\) −8435.29 −0.874889
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13860.6 −1.41875 −0.709376 0.704830i \(-0.751023\pi\)
−0.709376 + 0.704830i \(0.751023\pi\)
\(458\) 0 0
\(459\) 3284.27 0.333979
\(460\) 0 0
\(461\) 149.312 0.0150850 0.00754249 0.999972i \(-0.497599\pi\)
0.00754249 + 0.999972i \(0.497599\pi\)
\(462\) 0 0
\(463\) −5403.95 −0.542425 −0.271213 0.962519i \(-0.587425\pi\)
−0.271213 + 0.962519i \(0.587425\pi\)
\(464\) 0 0
\(465\) 5.64155 0.000562625 0
\(466\) 0 0
\(467\) 3704.66 0.367090 0.183545 0.983011i \(-0.441243\pi\)
0.183545 + 0.983011i \(0.441243\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5074.03 0.496389
\(472\) 0 0
\(473\) 9604.59 0.933657
\(474\) 0 0
\(475\) −15887.8 −1.53470
\(476\) 0 0
\(477\) 5788.05 0.555591
\(478\) 0 0
\(479\) −10671.8 −1.01796 −0.508982 0.860777i \(-0.669978\pi\)
−0.508982 + 0.860777i \(0.669978\pi\)
\(480\) 0 0
\(481\) 3193.82 0.302756
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 109.662 0.0102670
\(486\) 0 0
\(487\) −5853.92 −0.544695 −0.272348 0.962199i \(-0.587800\pi\)
−0.272348 + 0.962199i \(0.587800\pi\)
\(488\) 0 0
\(489\) −122.170 −0.0112980
\(490\) 0 0
\(491\) −4065.31 −0.373656 −0.186828 0.982393i \(-0.559821\pi\)
−0.186828 + 0.982393i \(0.559821\pi\)
\(492\) 0 0
\(493\) −28668.2 −2.61896
\(494\) 0 0
\(495\) −39.7468 −0.00360906
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4811.19 −0.431620 −0.215810 0.976435i \(-0.569239\pi\)
−0.215810 + 0.976435i \(0.569239\pi\)
\(500\) 0 0
\(501\) 8701.40 0.775947
\(502\) 0 0
\(503\) 17001.2 1.50705 0.753526 0.657418i \(-0.228351\pi\)
0.753526 + 0.657418i \(0.228351\pi\)
\(504\) 0 0
\(505\) 137.771 0.0121401
\(506\) 0 0
\(507\) 5759.87 0.504546
\(508\) 0 0
\(509\) −13797.2 −1.20148 −0.600738 0.799446i \(-0.705126\pi\)
−0.600738 + 0.799446i \(0.705126\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3432.04 −0.295377
\(514\) 0 0
\(515\) −142.110 −0.0121594
\(516\) 0 0
\(517\) −17644.7 −1.50099
\(518\) 0 0
\(519\) 6438.44 0.544540
\(520\) 0 0
\(521\) −3936.61 −0.331029 −0.165515 0.986207i \(-0.552928\pi\)
−0.165515 + 0.986207i \(0.552928\pi\)
\(522\) 0 0
\(523\) 17459.3 1.45973 0.729866 0.683590i \(-0.239582\pi\)
0.729866 + 0.683590i \(0.239582\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2275.96 −0.188126
\(528\) 0 0
\(529\) −9294.26 −0.763891
\(530\) 0 0
\(531\) 104.510 0.00854118
\(532\) 0 0
\(533\) 5321.51 0.432458
\(534\) 0 0
\(535\) −34.5274 −0.00279019
\(536\) 0 0
\(537\) 3610.62 0.290148
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 19101.3 1.51798 0.758992 0.651100i \(-0.225692\pi\)
0.758992 + 0.651100i \(0.225692\pi\)
\(542\) 0 0
\(543\) 8971.41 0.709024
\(544\) 0 0
\(545\) 31.9261 0.00250929
\(546\) 0 0
\(547\) −15413.5 −1.20481 −0.602407 0.798189i \(-0.705792\pi\)
−0.602407 + 0.798189i \(0.705792\pi\)
\(548\) 0 0
\(549\) −110.184 −0.00856563
\(550\) 0 0
\(551\) 29958.1 2.31626
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −57.8554 −0.00442491
\(556\) 0 0
\(557\) −20492.9 −1.55891 −0.779453 0.626460i \(-0.784503\pi\)
−0.779453 + 0.626460i \(0.784503\pi\)
\(558\) 0 0
\(559\) −3638.17 −0.275274
\(560\) 0 0
\(561\) 16034.9 1.20677
\(562\) 0 0
\(563\) −7142.49 −0.534671 −0.267336 0.963603i \(-0.586143\pi\)
−0.267336 + 0.963603i \(0.586143\pi\)
\(564\) 0 0
\(565\) −80.2406 −0.00597477
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4097.91 0.301922 0.150961 0.988540i \(-0.451763\pi\)
0.150961 + 0.988540i \(0.451763\pi\)
\(570\) 0 0
\(571\) 2838.48 0.208033 0.104016 0.994576i \(-0.466831\pi\)
0.104016 + 0.994576i \(0.466831\pi\)
\(572\) 0 0
\(573\) −8422.23 −0.614038
\(574\) 0 0
\(575\) 6699.21 0.485872
\(576\) 0 0
\(577\) 15464.5 1.11576 0.557881 0.829921i \(-0.311615\pi\)
0.557881 + 0.829921i \(0.311615\pi\)
\(578\) 0 0
\(579\) −10009.1 −0.718418
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 28259.3 2.00751
\(584\) 0 0
\(585\) 15.0559 0.00106407
\(586\) 0 0
\(587\) 14003.6 0.984652 0.492326 0.870411i \(-0.336147\pi\)
0.492326 + 0.870411i \(0.336147\pi\)
\(588\) 0 0
\(589\) 2378.36 0.166382
\(590\) 0 0
\(591\) 12679.9 0.882543
\(592\) 0 0
\(593\) −6504.50 −0.450435 −0.225217 0.974309i \(-0.572309\pi\)
−0.225217 + 0.974309i \(0.572309\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13157.1 0.901982
\(598\) 0 0
\(599\) 12616.1 0.860567 0.430284 0.902694i \(-0.358414\pi\)
0.430284 + 0.902694i \(0.358414\pi\)
\(600\) 0 0
\(601\) −8270.87 −0.561358 −0.280679 0.959802i \(-0.590560\pi\)
−0.280679 + 0.959802i \(0.590560\pi\)
\(602\) 0 0
\(603\) −6021.43 −0.406653
\(604\) 0 0
\(605\) −60.2852 −0.00405114
\(606\) 0 0
\(607\) 3811.84 0.254889 0.127445 0.991846i \(-0.459323\pi\)
0.127445 + 0.991846i \(0.459323\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6683.72 0.442544
\(612\) 0 0
\(613\) 11359.3 0.748445 0.374222 0.927339i \(-0.377910\pi\)
0.374222 + 0.927339i \(0.377910\pi\)
\(614\) 0 0
\(615\) −96.3983 −0.00632057
\(616\) 0 0
\(617\) −18272.2 −1.19224 −0.596118 0.802896i \(-0.703291\pi\)
−0.596118 + 0.802896i \(0.703291\pi\)
\(618\) 0 0
\(619\) 29600.2 1.92203 0.961013 0.276503i \(-0.0891756\pi\)
0.961013 + 0.276503i \(0.0891756\pi\)
\(620\) 0 0
\(621\) 1447.15 0.0935136
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15621.2 0.999758
\(626\) 0 0
\(627\) −16756.4 −1.06728
\(628\) 0 0
\(629\) 23340.5 1.47956
\(630\) 0 0
\(631\) −7185.41 −0.453322 −0.226661 0.973974i \(-0.572781\pi\)
−0.226661 + 0.973974i \(0.572781\pi\)
\(632\) 0 0
\(633\) 6874.67 0.431665
\(634\) 0 0
\(635\) 107.681 0.00672944
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −7398.88 −0.458052
\(640\) 0 0
\(641\) −232.982 −0.0143561 −0.00717803 0.999974i \(-0.502285\pi\)
−0.00717803 + 0.999974i \(0.502285\pi\)
\(642\) 0 0
\(643\) −1837.96 −0.112725 −0.0563624 0.998410i \(-0.517950\pi\)
−0.0563624 + 0.998410i \(0.517950\pi\)
\(644\) 0 0
\(645\) 65.9048 0.00402325
\(646\) 0 0
\(647\) −18594.7 −1.12988 −0.564941 0.825131i \(-0.691101\pi\)
−0.564941 + 0.825131i \(0.691101\pi\)
\(648\) 0 0
\(649\) 510.256 0.0308618
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28864.7 −1.72980 −0.864902 0.501940i \(-0.832620\pi\)
−0.864902 + 0.501940i \(0.832620\pi\)
\(654\) 0 0
\(655\) −252.772 −0.0150788
\(656\) 0 0
\(657\) 4635.90 0.275287
\(658\) 0 0
\(659\) 29066.3 1.71815 0.859076 0.511847i \(-0.171039\pi\)
0.859076 + 0.511847i \(0.171039\pi\)
\(660\) 0 0
\(661\) 3979.51 0.234168 0.117084 0.993122i \(-0.462645\pi\)
0.117084 + 0.993122i \(0.462645\pi\)
\(662\) 0 0
\(663\) −6073.95 −0.355796
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12632.0 −0.733305
\(668\) 0 0
\(669\) −653.909 −0.0377901
\(670\) 0 0
\(671\) −537.955 −0.0309501
\(672\) 0 0
\(673\) −184.229 −0.0105520 −0.00527601 0.999986i \(-0.501679\pi\)
−0.00527601 + 0.999986i \(0.501679\pi\)
\(674\) 0 0
\(675\) 3374.73 0.192435
\(676\) 0 0
\(677\) 16683.5 0.947116 0.473558 0.880763i \(-0.342969\pi\)
0.473558 + 0.880763i \(0.342969\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 5506.08 0.309829
\(682\) 0 0
\(683\) −17808.2 −0.997676 −0.498838 0.866695i \(-0.666240\pi\)
−0.498838 + 0.866695i \(0.666240\pi\)
\(684\) 0 0
\(685\) −25.2332 −0.00140746
\(686\) 0 0
\(687\) −8322.03 −0.462162
\(688\) 0 0
\(689\) −10704.5 −0.591883
\(690\) 0 0
\(691\) −20145.7 −1.10908 −0.554542 0.832156i \(-0.687106\pi\)
−0.554542 + 0.832156i \(0.687106\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 89.0542 0.00486046
\(696\) 0 0
\(697\) 38889.7 2.11342
\(698\) 0 0
\(699\) 2966.14 0.160500
\(700\) 0 0
\(701\) −2719.67 −0.146534 −0.0732672 0.997312i \(-0.523343\pi\)
−0.0732672 + 0.997312i \(0.523343\pi\)
\(702\) 0 0
\(703\) −24390.7 −1.30855
\(704\) 0 0
\(705\) −121.074 −0.00646798
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −625.708 −0.0331438 −0.0165719 0.999863i \(-0.505275\pi\)
−0.0165719 + 0.999863i \(0.505275\pi\)
\(710\) 0 0
\(711\) 7251.79 0.382508
\(712\) 0 0
\(713\) −1002.85 −0.0526749
\(714\) 0 0
\(715\) 73.5079 0.00384481
\(716\) 0 0
\(717\) −2513.78 −0.130933
\(718\) 0 0
\(719\) −9577.54 −0.496776 −0.248388 0.968661i \(-0.579901\pi\)
−0.248388 + 0.968661i \(0.579901\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 10365.0 0.533163
\(724\) 0 0
\(725\) −29457.8 −1.50901
\(726\) 0 0
\(727\) −16741.2 −0.854053 −0.427027 0.904239i \(-0.640439\pi\)
−0.427027 + 0.904239i \(0.640439\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −26587.8 −1.34526
\(732\) 0 0
\(733\) 6496.04 0.327335 0.163668 0.986516i \(-0.447668\pi\)
0.163668 + 0.986516i \(0.447668\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −29398.7 −1.46936
\(738\) 0 0
\(739\) −499.280 −0.0248529 −0.0124265 0.999923i \(-0.503956\pi\)
−0.0124265 + 0.999923i \(0.503956\pi\)
\(740\) 0 0
\(741\) 6347.24 0.314672
\(742\) 0 0
\(743\) 6367.30 0.314393 0.157196 0.987567i \(-0.449754\pi\)
0.157196 + 0.987567i \(0.449754\pi\)
\(744\) 0 0
\(745\) −58.5568 −0.00287967
\(746\) 0 0
\(747\) 3553.77 0.174064
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −496.098 −0.0241050 −0.0120525 0.999927i \(-0.503837\pi\)
−0.0120525 + 0.999927i \(0.503837\pi\)
\(752\) 0 0
\(753\) 16905.0 0.818132
\(754\) 0 0
\(755\) −282.597 −0.0136222
\(756\) 0 0
\(757\) 13025.9 0.625408 0.312704 0.949851i \(-0.398765\pi\)
0.312704 + 0.949851i \(0.398765\pi\)
\(758\) 0 0
\(759\) 7065.47 0.337892
\(760\) 0 0
\(761\) 25474.6 1.21347 0.606736 0.794904i \(-0.292479\pi\)
0.606736 + 0.794904i \(0.292479\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 110.029 0.00520012
\(766\) 0 0
\(767\) −193.282 −0.00909911
\(768\) 0 0
\(769\) 29054.0 1.36244 0.681218 0.732080i \(-0.261450\pi\)
0.681218 + 0.732080i \(0.261450\pi\)
\(770\) 0 0
\(771\) 6813.47 0.318263
\(772\) 0 0
\(773\) −1897.35 −0.0882834 −0.0441417 0.999025i \(-0.514055\pi\)
−0.0441417 + 0.999025i \(0.514055\pi\)
\(774\) 0 0
\(775\) −2338.65 −0.108396
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −40639.6 −1.86914
\(780\) 0 0
\(781\) −36123.9 −1.65508
\(782\) 0 0
\(783\) −6363.39 −0.290433
\(784\) 0 0
\(785\) 169.989 0.00772886
\(786\) 0 0
\(787\) 42650.3 1.93179 0.965895 0.258936i \(-0.0833718\pi\)
0.965895 + 0.258936i \(0.0833718\pi\)
\(788\) 0 0
\(789\) 491.602 0.0221819
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 203.775 0.00912516
\(794\) 0 0
\(795\) 193.910 0.00865064
\(796\) 0 0
\(797\) −36822.8 −1.63655 −0.818275 0.574828i \(-0.805069\pi\)
−0.818275 + 0.574828i \(0.805069\pi\)
\(798\) 0 0
\(799\) 48844.8 2.16271
\(800\) 0 0
\(801\) 6060.77 0.267349
\(802\) 0 0
\(803\) 22634.1 0.994693
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15501.3 0.676172
\(808\) 0 0
\(809\) 5081.94 0.220855 0.110427 0.993884i \(-0.464778\pi\)
0.110427 + 0.993884i \(0.464778\pi\)
\(810\) 0 0
\(811\) 11873.4 0.514097 0.257048 0.966399i \(-0.417250\pi\)
0.257048 + 0.966399i \(0.417250\pi\)
\(812\) 0 0
\(813\) −4866.82 −0.209947
\(814\) 0 0
\(815\) −4.09289 −0.000175911 0
\(816\) 0 0
\(817\) 27784.1 1.18977
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16969.4 0.721361 0.360681 0.932689i \(-0.382544\pi\)
0.360681 + 0.932689i \(0.382544\pi\)
\(822\) 0 0
\(823\) −3995.94 −0.169246 −0.0846231 0.996413i \(-0.526969\pi\)
−0.0846231 + 0.996413i \(0.526969\pi\)
\(824\) 0 0
\(825\) 16476.6 0.695323
\(826\) 0 0
\(827\) 13589.7 0.571417 0.285708 0.958317i \(-0.407771\pi\)
0.285708 + 0.958317i \(0.407771\pi\)
\(828\) 0 0
\(829\) 30646.0 1.28393 0.641966 0.766733i \(-0.278119\pi\)
0.641966 + 0.766733i \(0.278119\pi\)
\(830\) 0 0
\(831\) 13837.1 0.577622
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 291.511 0.0120816
\(836\) 0 0
\(837\) −505.188 −0.0208624
\(838\) 0 0
\(839\) −7497.57 −0.308516 −0.154258 0.988031i \(-0.549299\pi\)
−0.154258 + 0.988031i \(0.549299\pi\)
\(840\) 0 0
\(841\) 31156.6 1.27749
\(842\) 0 0
\(843\) 6375.65 0.260485
\(844\) 0 0
\(845\) 192.965 0.00785586
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7714.62 0.311855
\(850\) 0 0
\(851\) 10284.5 0.414275
\(852\) 0 0
\(853\) 10347.6 0.415352 0.207676 0.978198i \(-0.433410\pi\)
0.207676 + 0.978198i \(0.433410\pi\)
\(854\) 0 0
\(855\) −114.979 −0.00459907
\(856\) 0 0
\(857\) −1550.00 −0.0617817 −0.0308909 0.999523i \(-0.509834\pi\)
−0.0308909 + 0.999523i \(0.509834\pi\)
\(858\) 0 0
\(859\) 15187.5 0.603250 0.301625 0.953427i \(-0.402471\pi\)
0.301625 + 0.953427i \(0.402471\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −41550.2 −1.63892 −0.819459 0.573138i \(-0.805726\pi\)
−0.819459 + 0.573138i \(0.805726\pi\)
\(864\) 0 0
\(865\) 215.699 0.00847858
\(866\) 0 0
\(867\) −29649.6 −1.16142
\(868\) 0 0
\(869\) 35405.8 1.38212
\(870\) 0 0
\(871\) 11136.1 0.433216
\(872\) 0 0
\(873\) −9820.03 −0.380707
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27837.4 1.07184 0.535919 0.844269i \(-0.319965\pi\)
0.535919 + 0.844269i \(0.319965\pi\)
\(878\) 0 0
\(879\) 9974.89 0.382758
\(880\) 0 0
\(881\) 2587.85 0.0989635 0.0494817 0.998775i \(-0.484243\pi\)
0.0494817 + 0.998775i \(0.484243\pi\)
\(882\) 0 0
\(883\) 16382.0 0.624346 0.312173 0.950025i \(-0.398943\pi\)
0.312173 + 0.950025i \(0.398943\pi\)
\(884\) 0 0
\(885\) 3.50127 0.000132988 0
\(886\) 0 0
\(887\) −22980.2 −0.869896 −0.434948 0.900456i \(-0.643233\pi\)
−0.434948 + 0.900456i \(0.643233\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3559.23 0.133826
\(892\) 0 0
\(893\) −51042.5 −1.91274
\(894\) 0 0
\(895\) 120.962 0.00451766
\(896\) 0 0
\(897\) −2676.36 −0.0996222
\(898\) 0 0
\(899\) 4409.76 0.163597
\(900\) 0 0
\(901\) −78228.5 −2.89253
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 300.557 0.0110396
\(906\) 0 0
\(907\) 22715.4 0.831590 0.415795 0.909458i \(-0.363503\pi\)
0.415795 + 0.909458i \(0.363503\pi\)
\(908\) 0 0
\(909\) −12337.1 −0.450161
\(910\) 0 0
\(911\) −34922.3 −1.27006 −0.635031 0.772487i \(-0.719013\pi\)
−0.635031 + 0.772487i \(0.719013\pi\)
\(912\) 0 0
\(913\) 17350.7 0.628943
\(914\) 0 0
\(915\) −3.69134 −0.000133368 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −11702.6 −0.420059 −0.210030 0.977695i \(-0.567356\pi\)
−0.210030 + 0.977695i \(0.567356\pi\)
\(920\) 0 0
\(921\) 2661.29 0.0952144
\(922\) 0 0
\(923\) 13683.5 0.487973
\(924\) 0 0
\(925\) 23983.3 0.852505
\(926\) 0 0
\(927\) 12725.6 0.450878
\(928\) 0 0
\(929\) 53096.5 1.87518 0.937588 0.347747i \(-0.113053\pi\)
0.937588 + 0.347747i \(0.113053\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13532.5 0.474848
\(934\) 0 0
\(935\) 537.198 0.0187896
\(936\) 0 0
\(937\) 39020.6 1.36046 0.680229 0.733000i \(-0.261880\pi\)
0.680229 + 0.733000i \(0.261880\pi\)
\(938\) 0 0
\(939\) 11147.4 0.387412
\(940\) 0 0
\(941\) −30743.8 −1.06506 −0.532528 0.846412i \(-0.678758\pi\)
−0.532528 + 0.846412i \(0.678758\pi\)
\(942\) 0 0
\(943\) 17136.0 0.591754
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16300.3 −0.559332 −0.279666 0.960097i \(-0.590224\pi\)
−0.279666 + 0.960097i \(0.590224\pi\)
\(948\) 0 0
\(949\) −8573.66 −0.293269
\(950\) 0 0
\(951\) −20863.6 −0.711406
\(952\) 0 0
\(953\) −11512.7 −0.391325 −0.195663 0.980671i \(-0.562686\pi\)
−0.195663 + 0.980671i \(0.562686\pi\)
\(954\) 0 0
\(955\) −282.159 −0.00956068
\(956\) 0 0
\(957\) −31068.3 −1.04942
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29440.9 −0.988248
\(962\) 0 0
\(963\) 3091.85 0.103462
\(964\) 0 0
\(965\) −335.322 −0.0111859
\(966\) 0 0
\(967\) 18178.4 0.604528 0.302264 0.953224i \(-0.402258\pi\)
0.302264 + 0.953224i \(0.402258\pi\)
\(968\) 0 0
\(969\) 46385.8 1.53780
\(970\) 0 0
\(971\) −28276.7 −0.934543 −0.467271 0.884114i \(-0.654763\pi\)
−0.467271 + 0.884114i \(0.654763\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −6241.24 −0.205005
\(976\) 0 0
\(977\) −13947.1 −0.456710 −0.228355 0.973578i \(-0.573335\pi\)
−0.228355 + 0.973578i \(0.573335\pi\)
\(978\) 0 0
\(979\) 29590.8 0.966011
\(980\) 0 0
\(981\) −2858.91 −0.0930459
\(982\) 0 0
\(983\) 26576.8 0.862327 0.431164 0.902274i \(-0.358103\pi\)
0.431164 + 0.902274i \(0.358103\pi\)
\(984\) 0 0
\(985\) 424.799 0.0137414
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11715.4 −0.376671
\(990\) 0 0
\(991\) −16249.9 −0.520884 −0.260442 0.965490i \(-0.583868\pi\)
−0.260442 + 0.965490i \(0.583868\pi\)
\(992\) 0 0
\(993\) −29589.5 −0.945615
\(994\) 0 0
\(995\) 440.784 0.0140440
\(996\) 0 0
\(997\) 18814.1 0.597642 0.298821 0.954309i \(-0.403407\pi\)
0.298821 + 0.954309i \(0.403407\pi\)
\(998\) 0 0
\(999\) 5180.82 0.164078
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 2352.4.a.bl.1.2 2
4.3 odd 2 147.4.a.k.1.1 yes 2
7.6 odd 2 2352.4.a.cf.1.1 2
12.11 even 2 441.4.a.o.1.2 2
28.3 even 6 147.4.e.k.79.2 4
28.11 odd 6 147.4.e.j.79.2 4
28.19 even 6 147.4.e.k.67.2 4
28.23 odd 6 147.4.e.j.67.2 4
28.27 even 2 147.4.a.j.1.1 2
84.11 even 6 441.4.e.u.226.1 4
84.23 even 6 441.4.e.u.361.1 4
84.47 odd 6 441.4.e.v.361.1 4
84.59 odd 6 441.4.e.v.226.1 4
84.83 odd 2 441.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.1 2 28.27 even 2
147.4.a.k.1.1 yes 2 4.3 odd 2
147.4.e.j.67.2 4 28.23 odd 6
147.4.e.j.79.2 4 28.11 odd 6
147.4.e.k.67.2 4 28.19 even 6
147.4.e.k.79.2 4 28.3 even 6
441.4.a.n.1.2 2 84.83 odd 2
441.4.a.o.1.2 2 12.11 even 2
441.4.e.u.226.1 4 84.11 even 6
441.4.e.u.361.1 4 84.23 even 6
441.4.e.v.226.1 4 84.59 odd 6
441.4.e.v.361.1 4 84.47 odd 6
2352.4.a.bl.1.2 2 1.1 even 1 trivial
2352.4.a.cf.1.1 2 7.6 odd 2