Properties

Label 688.6.a.h.1.6
Level 688
Weight 6
Character 688.1
Self dual yes
Analytic conductor 110.344
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.344068031\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(9.86547\) of defining polynomial
Character \(\chi\) \(=\) 688.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.50169 q^{3} -37.8251 q^{5} +124.747 q^{7} -240.745 q^{9} +O(q^{10})\) \(q-1.50169 q^{3} -37.8251 q^{5} +124.747 q^{7} -240.745 q^{9} -590.079 q^{11} +434.774 q^{13} +56.8015 q^{15} +1925.58 q^{17} -654.129 q^{19} -187.332 q^{21} -2805.03 q^{23} -1694.26 q^{25} +726.434 q^{27} -1456.23 q^{29} -4419.85 q^{31} +886.115 q^{33} -4718.58 q^{35} +3753.09 q^{37} -652.894 q^{39} +1972.85 q^{41} -1849.00 q^{43} +9106.20 q^{45} -2204.84 q^{47} -1245.12 q^{49} -2891.62 q^{51} +24984.2 q^{53} +22319.8 q^{55} +982.298 q^{57} +42756.2 q^{59} -21022.8 q^{61} -30032.3 q^{63} -16445.4 q^{65} +25272.1 q^{67} +4212.28 q^{69} -48082.8 q^{71} +58801.2 q^{73} +2544.25 q^{75} -73610.8 q^{77} -92704.7 q^{79} +57410.1 q^{81} +1849.64 q^{83} -72835.2 q^{85} +2186.80 q^{87} -70380.3 q^{89} +54236.8 q^{91} +6637.23 q^{93} +24742.5 q^{95} -88842.1 q^{97} +142059. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 28q^{3} + 138q^{5} - 60q^{7} + 1356q^{9} + O(q^{10}) \) \( 10q - 28q^{3} + 138q^{5} - 60q^{7} + 1356q^{9} - 745q^{11} + 1917q^{13} - 1688q^{15} + 4017q^{17} + 2404q^{19} - 228q^{21} - 1733q^{23} + 7120q^{25} + 2324q^{27} + 6996q^{29} + 4899q^{31} - 15734q^{33} - 7084q^{35} + 1466q^{37} + 26542q^{39} + 10297q^{41} - 18490q^{43} + 73822q^{45} - 48592q^{47} + 29458q^{49} - 92972q^{51} + 127165q^{53} - 106672q^{55} + 34060q^{57} - 99372q^{59} + 17408q^{61} - 2244q^{63} + 54484q^{65} + 2021q^{67} + 1654q^{69} - 11286q^{71} + 49892q^{73} + 44662q^{75} + 98144q^{77} + 91524q^{79} - 26450q^{81} + 105203q^{83} - 87212q^{85} - 181200q^{87} - 62682q^{89} + 295304q^{91} - 238430q^{93} + 305340q^{95} + 108383q^{97} + 270499q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.50169 −0.0963333 −0.0481667 0.998839i \(-0.515338\pi\)
−0.0481667 + 0.998839i \(0.515338\pi\)
\(4\) 0 0
\(5\) −37.8251 −0.676636 −0.338318 0.941032i \(-0.609858\pi\)
−0.338318 + 0.941032i \(0.609858\pi\)
\(6\) 0 0
\(7\) 124.747 0.962246 0.481123 0.876653i \(-0.340229\pi\)
0.481123 + 0.876653i \(0.340229\pi\)
\(8\) 0 0
\(9\) −240.745 −0.990720
\(10\) 0 0
\(11\) −590.079 −1.47038 −0.735188 0.677863i \(-0.762906\pi\)
−0.735188 + 0.677863i \(0.762906\pi\)
\(12\) 0 0
\(13\) 434.774 0.713518 0.356759 0.934196i \(-0.383882\pi\)
0.356759 + 0.934196i \(0.383882\pi\)
\(14\) 0 0
\(15\) 56.8015 0.0651826
\(16\) 0 0
\(17\) 1925.58 1.61599 0.807995 0.589189i \(-0.200553\pi\)
0.807995 + 0.589189i \(0.200553\pi\)
\(18\) 0 0
\(19\) −654.129 −0.415700 −0.207850 0.978161i \(-0.566647\pi\)
−0.207850 + 0.978161i \(0.566647\pi\)
\(20\) 0 0
\(21\) −187.332 −0.0926963
\(22\) 0 0
\(23\) −2805.03 −1.10565 −0.552825 0.833297i \(-0.686450\pi\)
−0.552825 + 0.833297i \(0.686450\pi\)
\(24\) 0 0
\(25\) −1694.26 −0.542163
\(26\) 0 0
\(27\) 726.434 0.191773
\(28\) 0 0
\(29\) −1456.23 −0.321540 −0.160770 0.986992i \(-0.551398\pi\)
−0.160770 + 0.986992i \(0.551398\pi\)
\(30\) 0 0
\(31\) −4419.85 −0.826044 −0.413022 0.910721i \(-0.635527\pi\)
−0.413022 + 0.910721i \(0.635527\pi\)
\(32\) 0 0
\(33\) 886.115 0.141646
\(34\) 0 0
\(35\) −4718.58 −0.651090
\(36\) 0 0
\(37\) 3753.09 0.450697 0.225348 0.974278i \(-0.427648\pi\)
0.225348 + 0.974278i \(0.427648\pi\)
\(38\) 0 0
\(39\) −652.894 −0.0687356
\(40\) 0 0
\(41\) 1972.85 0.183288 0.0916442 0.995792i \(-0.470788\pi\)
0.0916442 + 0.995792i \(0.470788\pi\)
\(42\) 0 0
\(43\) −1849.00 −0.152499
\(44\) 0 0
\(45\) 9106.20 0.670357
\(46\) 0 0
\(47\) −2204.84 −0.145590 −0.0727950 0.997347i \(-0.523192\pi\)
−0.0727950 + 0.997347i \(0.523192\pi\)
\(48\) 0 0
\(49\) −1245.12 −0.0740834
\(50\) 0 0
\(51\) −2891.62 −0.155674
\(52\) 0 0
\(53\) 24984.2 1.22173 0.610865 0.791735i \(-0.290822\pi\)
0.610865 + 0.791735i \(0.290822\pi\)
\(54\) 0 0
\(55\) 22319.8 0.994910
\(56\) 0 0
\(57\) 982.298 0.0400457
\(58\) 0 0
\(59\) 42756.2 1.59907 0.799537 0.600616i \(-0.205078\pi\)
0.799537 + 0.600616i \(0.205078\pi\)
\(60\) 0 0
\(61\) −21022.8 −0.723379 −0.361689 0.932299i \(-0.617800\pi\)
−0.361689 + 0.932299i \(0.617800\pi\)
\(62\) 0 0
\(63\) −30032.3 −0.953316
\(64\) 0 0
\(65\) −16445.4 −0.482792
\(66\) 0 0
\(67\) 25272.1 0.687788 0.343894 0.939008i \(-0.388254\pi\)
0.343894 + 0.939008i \(0.388254\pi\)
\(68\) 0 0
\(69\) 4212.28 0.106511
\(70\) 0 0
\(71\) −48082.8 −1.13199 −0.565997 0.824407i \(-0.691509\pi\)
−0.565997 + 0.824407i \(0.691509\pi\)
\(72\) 0 0
\(73\) 58801.2 1.29145 0.645727 0.763568i \(-0.276554\pi\)
0.645727 + 0.763568i \(0.276554\pi\)
\(74\) 0 0
\(75\) 2544.25 0.0522284
\(76\) 0 0
\(77\) −73610.8 −1.41486
\(78\) 0 0
\(79\) −92704.7 −1.67122 −0.835611 0.549322i \(-0.814886\pi\)
−0.835611 + 0.549322i \(0.814886\pi\)
\(80\) 0 0
\(81\) 57410.1 0.972246
\(82\) 0 0
\(83\) 1849.64 0.0294708 0.0147354 0.999891i \(-0.495309\pi\)
0.0147354 + 0.999891i \(0.495309\pi\)
\(84\) 0 0
\(85\) −72835.2 −1.09344
\(86\) 0 0
\(87\) 2186.80 0.0309750
\(88\) 0 0
\(89\) −70380.3 −0.941838 −0.470919 0.882176i \(-0.656078\pi\)
−0.470919 + 0.882176i \(0.656078\pi\)
\(90\) 0 0
\(91\) 54236.8 0.686579
\(92\) 0 0
\(93\) 6637.23 0.0795756
\(94\) 0 0
\(95\) 24742.5 0.281277
\(96\) 0 0
\(97\) −88842.1 −0.958714 −0.479357 0.877620i \(-0.659130\pi\)
−0.479357 + 0.877620i \(0.659130\pi\)
\(98\) 0 0
\(99\) 142059. 1.45673
\(100\) 0 0
\(101\) 179221. 1.74818 0.874090 0.485763i \(-0.161458\pi\)
0.874090 + 0.485763i \(0.161458\pi\)
\(102\) 0 0
\(103\) −123443. −1.14650 −0.573248 0.819382i \(-0.694317\pi\)
−0.573248 + 0.819382i \(0.694317\pi\)
\(104\) 0 0
\(105\) 7085.84 0.0627217
\(106\) 0 0
\(107\) −98991.7 −0.835871 −0.417936 0.908477i \(-0.637246\pi\)
−0.417936 + 0.908477i \(0.637246\pi\)
\(108\) 0 0
\(109\) 46356.0 0.373715 0.186857 0.982387i \(-0.440170\pi\)
0.186857 + 0.982387i \(0.440170\pi\)
\(110\) 0 0
\(111\) −5635.97 −0.0434171
\(112\) 0 0
\(113\) 185076. 1.36350 0.681750 0.731586i \(-0.261219\pi\)
0.681750 + 0.731586i \(0.261219\pi\)
\(114\) 0 0
\(115\) 106101. 0.748123
\(116\) 0 0
\(117\) −104670. −0.706896
\(118\) 0 0
\(119\) 240211. 1.55498
\(120\) 0 0
\(121\) 187142. 1.16201
\(122\) 0 0
\(123\) −2962.61 −0.0176568
\(124\) 0 0
\(125\) 182289. 1.04348
\(126\) 0 0
\(127\) 237332. 1.30571 0.652856 0.757482i \(-0.273571\pi\)
0.652856 + 0.757482i \(0.273571\pi\)
\(128\) 0 0
\(129\) 2776.62 0.0146907
\(130\) 0 0
\(131\) 283496. 1.44334 0.721671 0.692236i \(-0.243374\pi\)
0.721671 + 0.692236i \(0.243374\pi\)
\(132\) 0 0
\(133\) −81600.9 −0.400005
\(134\) 0 0
\(135\) −27477.5 −0.129760
\(136\) 0 0
\(137\) 330992. 1.50666 0.753331 0.657642i \(-0.228446\pi\)
0.753331 + 0.657642i \(0.228446\pi\)
\(138\) 0 0
\(139\) −105975. −0.465230 −0.232615 0.972569i \(-0.574728\pi\)
−0.232615 + 0.972569i \(0.574728\pi\)
\(140\) 0 0
\(141\) 3310.98 0.0140252
\(142\) 0 0
\(143\) −256551. −1.04914
\(144\) 0 0
\(145\) 55082.0 0.217565
\(146\) 0 0
\(147\) 1869.78 0.00713670
\(148\) 0 0
\(149\) −196700. −0.725836 −0.362918 0.931821i \(-0.618220\pi\)
−0.362918 + 0.931821i \(0.618220\pi\)
\(150\) 0 0
\(151\) 412513. 1.47230 0.736148 0.676820i \(-0.236643\pi\)
0.736148 + 0.676820i \(0.236643\pi\)
\(152\) 0 0
\(153\) −463573. −1.60099
\(154\) 0 0
\(155\) 167181. 0.558931
\(156\) 0 0
\(157\) 410971. 1.33064 0.665322 0.746557i \(-0.268294\pi\)
0.665322 + 0.746557i \(0.268294\pi\)
\(158\) 0 0
\(159\) −37518.4 −0.117693
\(160\) 0 0
\(161\) −349920. −1.06391
\(162\) 0 0
\(163\) 82488.7 0.243179 0.121589 0.992580i \(-0.461201\pi\)
0.121589 + 0.992580i \(0.461201\pi\)
\(164\) 0 0
\(165\) −33517.4 −0.0958430
\(166\) 0 0
\(167\) 380336. 1.05530 0.527650 0.849462i \(-0.323073\pi\)
0.527650 + 0.849462i \(0.323073\pi\)
\(168\) 0 0
\(169\) −182265. −0.490892
\(170\) 0 0
\(171\) 157478. 0.411842
\(172\) 0 0
\(173\) 91482.2 0.232392 0.116196 0.993226i \(-0.462930\pi\)
0.116196 + 0.993226i \(0.462930\pi\)
\(174\) 0 0
\(175\) −211354. −0.521694
\(176\) 0 0
\(177\) −64206.4 −0.154044
\(178\) 0 0
\(179\) 607572. 1.41731 0.708656 0.705554i \(-0.249302\pi\)
0.708656 + 0.705554i \(0.249302\pi\)
\(180\) 0 0
\(181\) 351519. 0.797540 0.398770 0.917051i \(-0.369437\pi\)
0.398770 + 0.917051i \(0.369437\pi\)
\(182\) 0 0
\(183\) 31569.7 0.0696855
\(184\) 0 0
\(185\) −141961. −0.304958
\(186\) 0 0
\(187\) −1.13624e6 −2.37611
\(188\) 0 0
\(189\) 90620.7 0.184532
\(190\) 0 0
\(191\) 123308. 0.244573 0.122286 0.992495i \(-0.460977\pi\)
0.122286 + 0.992495i \(0.460977\pi\)
\(192\) 0 0
\(193\) −163481. −0.315917 −0.157959 0.987446i \(-0.550491\pi\)
−0.157959 + 0.987446i \(0.550491\pi\)
\(194\) 0 0
\(195\) 24695.8 0.0465090
\(196\) 0 0
\(197\) 581124. 1.06685 0.533425 0.845848i \(-0.320905\pi\)
0.533425 + 0.845848i \(0.320905\pi\)
\(198\) 0 0
\(199\) −351526. −0.629253 −0.314627 0.949216i \(-0.601879\pi\)
−0.314627 + 0.949216i \(0.601879\pi\)
\(200\) 0 0
\(201\) −37950.9 −0.0662570
\(202\) 0 0
\(203\) −181660. −0.309400
\(204\) 0 0
\(205\) −74623.4 −0.124020
\(206\) 0 0
\(207\) 675297. 1.09539
\(208\) 0 0
\(209\) 385988. 0.611235
\(210\) 0 0
\(211\) −327480. −0.506383 −0.253192 0.967416i \(-0.581480\pi\)
−0.253192 + 0.967416i \(0.581480\pi\)
\(212\) 0 0
\(213\) 72205.4 0.109049
\(214\) 0 0
\(215\) 69938.6 0.103186
\(216\) 0 0
\(217\) −551364. −0.794857
\(218\) 0 0
\(219\) −88301.0 −0.124410
\(220\) 0 0
\(221\) 837191. 1.15304
\(222\) 0 0
\(223\) −619602. −0.834355 −0.417177 0.908825i \(-0.636981\pi\)
−0.417177 + 0.908825i \(0.636981\pi\)
\(224\) 0 0
\(225\) 407885. 0.537132
\(226\) 0 0
\(227\) 757446. 0.975635 0.487817 0.872946i \(-0.337793\pi\)
0.487817 + 0.872946i \(0.337793\pi\)
\(228\) 0 0
\(229\) 1.21739e6 1.53405 0.767025 0.641617i \(-0.221736\pi\)
0.767025 + 0.641617i \(0.221736\pi\)
\(230\) 0 0
\(231\) 110540. 0.136299
\(232\) 0 0
\(233\) 379715. 0.458213 0.229107 0.973401i \(-0.426420\pi\)
0.229107 + 0.973401i \(0.426420\pi\)
\(234\) 0 0
\(235\) 83398.2 0.0985115
\(236\) 0 0
\(237\) 139214. 0.160994
\(238\) 0 0
\(239\) −281756. −0.319064 −0.159532 0.987193i \(-0.550999\pi\)
−0.159532 + 0.987193i \(0.550999\pi\)
\(240\) 0 0
\(241\) −1.12524e6 −1.24796 −0.623981 0.781439i \(-0.714486\pi\)
−0.623981 + 0.781439i \(0.714486\pi\)
\(242\) 0 0
\(243\) −262736. −0.285432
\(244\) 0 0
\(245\) 47096.8 0.0501275
\(246\) 0 0
\(247\) −284398. −0.296609
\(248\) 0 0
\(249\) −2777.58 −0.00283902
\(250\) 0 0
\(251\) 675567. 0.676837 0.338419 0.940996i \(-0.390108\pi\)
0.338419 + 0.940996i \(0.390108\pi\)
\(252\) 0 0
\(253\) 1.65519e6 1.62572
\(254\) 0 0
\(255\) 109376. 0.105335
\(256\) 0 0
\(257\) 1.85623e6 1.75306 0.876532 0.481343i \(-0.159851\pi\)
0.876532 + 0.481343i \(0.159851\pi\)
\(258\) 0 0
\(259\) 468188. 0.433681
\(260\) 0 0
\(261\) 350580. 0.318556
\(262\) 0 0
\(263\) 1.54160e6 1.37430 0.687149 0.726516i \(-0.258862\pi\)
0.687149 + 0.726516i \(0.258862\pi\)
\(264\) 0 0
\(265\) −945029. −0.826667
\(266\) 0 0
\(267\) 105689. 0.0907304
\(268\) 0 0
\(269\) 1.16775e6 0.983945 0.491973 0.870611i \(-0.336276\pi\)
0.491973 + 0.870611i \(0.336276\pi\)
\(270\) 0 0
\(271\) 1.20256e6 0.994684 0.497342 0.867555i \(-0.334309\pi\)
0.497342 + 0.867555i \(0.334309\pi\)
\(272\) 0 0
\(273\) −81446.8 −0.0661405
\(274\) 0 0
\(275\) 999748. 0.797184
\(276\) 0 0
\(277\) −504463. −0.395030 −0.197515 0.980300i \(-0.563287\pi\)
−0.197515 + 0.980300i \(0.563287\pi\)
\(278\) 0 0
\(279\) 1.06406e6 0.818378
\(280\) 0 0
\(281\) 1.52712e6 1.15374 0.576870 0.816836i \(-0.304274\pi\)
0.576870 + 0.816836i \(0.304274\pi\)
\(282\) 0 0
\(283\) −1.81916e6 −1.35022 −0.675110 0.737717i \(-0.735904\pi\)
−0.675110 + 0.737717i \(0.735904\pi\)
\(284\) 0 0
\(285\) −37155.5 −0.0270964
\(286\) 0 0
\(287\) 246108. 0.176369
\(288\) 0 0
\(289\) 2.28799e6 1.61143
\(290\) 0 0
\(291\) 133413. 0.0923562
\(292\) 0 0
\(293\) −2.35975e6 −1.60582 −0.802911 0.596099i \(-0.796717\pi\)
−0.802911 + 0.596099i \(0.796717\pi\)
\(294\) 0 0
\(295\) −1.61726e6 −1.08199
\(296\) 0 0
\(297\) −428654. −0.281978
\(298\) 0 0
\(299\) −1.21955e6 −0.788901
\(300\) 0 0
\(301\) −230658. −0.146741
\(302\) 0 0
\(303\) −269135. −0.168408
\(304\) 0 0
\(305\) 795189. 0.489464
\(306\) 0 0
\(307\) 2.59790e6 1.57317 0.786587 0.617480i \(-0.211846\pi\)
0.786587 + 0.617480i \(0.211846\pi\)
\(308\) 0 0
\(309\) 185373. 0.110446
\(310\) 0 0
\(311\) 2.21471e6 1.29842 0.649212 0.760607i \(-0.275099\pi\)
0.649212 + 0.760607i \(0.275099\pi\)
\(312\) 0 0
\(313\) −2.43630e6 −1.40563 −0.702813 0.711374i \(-0.748073\pi\)
−0.702813 + 0.711374i \(0.748073\pi\)
\(314\) 0 0
\(315\) 1.13597e6 0.645048
\(316\) 0 0
\(317\) −2.59108e6 −1.44821 −0.724106 0.689689i \(-0.757747\pi\)
−0.724106 + 0.689689i \(0.757747\pi\)
\(318\) 0 0
\(319\) 859290. 0.472784
\(320\) 0 0
\(321\) 148655. 0.0805223
\(322\) 0 0
\(323\) −1.25958e6 −0.671767
\(324\) 0 0
\(325\) −736620. −0.386843
\(326\) 0 0
\(327\) −69612.3 −0.0360012
\(328\) 0 0
\(329\) −275047. −0.140093
\(330\) 0 0
\(331\) 2.49362e6 1.25101 0.625503 0.780221i \(-0.284894\pi\)
0.625503 + 0.780221i \(0.284894\pi\)
\(332\) 0 0
\(333\) −903537. −0.446514
\(334\) 0 0
\(335\) −955921. −0.465383
\(336\) 0 0
\(337\) −159199. −0.0763600 −0.0381800 0.999271i \(-0.512156\pi\)
−0.0381800 + 0.999271i \(0.512156\pi\)
\(338\) 0 0
\(339\) −277927. −0.131350
\(340\) 0 0
\(341\) 2.60806e6 1.21460
\(342\) 0 0
\(343\) −2.25195e6 −1.03353
\(344\) 0 0
\(345\) −159330. −0.0720692
\(346\) 0 0
\(347\) −4.35264e6 −1.94057 −0.970285 0.241966i \(-0.922208\pi\)
−0.970285 + 0.241966i \(0.922208\pi\)
\(348\) 0 0
\(349\) −502730. −0.220938 −0.110469 0.993880i \(-0.535235\pi\)
−0.110469 + 0.993880i \(0.535235\pi\)
\(350\) 0 0
\(351\) 315834. 0.136833
\(352\) 0 0
\(353\) −990932. −0.423260 −0.211630 0.977350i \(-0.567877\pi\)
−0.211630 + 0.977350i \(0.567877\pi\)
\(354\) 0 0
\(355\) 1.81874e6 0.765948
\(356\) 0 0
\(357\) −360721. −0.149796
\(358\) 0 0
\(359\) 1.78805e6 0.732223 0.366112 0.930571i \(-0.380689\pi\)
0.366112 + 0.930571i \(0.380689\pi\)
\(360\) 0 0
\(361\) −2.04821e6 −0.827194
\(362\) 0 0
\(363\) −281029. −0.111940
\(364\) 0 0
\(365\) −2.22416e6 −0.873844
\(366\) 0 0
\(367\) 68021.3 0.0263621 0.0131810 0.999913i \(-0.495804\pi\)
0.0131810 + 0.999913i \(0.495804\pi\)
\(368\) 0 0
\(369\) −474954. −0.181588
\(370\) 0 0
\(371\) 3.11671e6 1.17560
\(372\) 0 0
\(373\) −949477. −0.353356 −0.176678 0.984269i \(-0.556535\pi\)
−0.176678 + 0.984269i \(0.556535\pi\)
\(374\) 0 0
\(375\) −273741. −0.100522
\(376\) 0 0
\(377\) −633130. −0.229424
\(378\) 0 0
\(379\) −2.53069e6 −0.904982 −0.452491 0.891769i \(-0.649465\pi\)
−0.452491 + 0.891769i \(0.649465\pi\)
\(380\) 0 0
\(381\) −356399. −0.125784
\(382\) 0 0
\(383\) −500631. −0.174390 −0.0871948 0.996191i \(-0.527790\pi\)
−0.0871948 + 0.996191i \(0.527790\pi\)
\(384\) 0 0
\(385\) 2.78434e6 0.957348
\(386\) 0 0
\(387\) 445137. 0.151083
\(388\) 0 0
\(389\) −4.08546e6 −1.36889 −0.684443 0.729066i \(-0.739955\pi\)
−0.684443 + 0.729066i \(0.739955\pi\)
\(390\) 0 0
\(391\) −5.40130e6 −1.78672
\(392\) 0 0
\(393\) −425723. −0.139042
\(394\) 0 0
\(395\) 3.50657e6 1.13081
\(396\) 0 0
\(397\) 953651. 0.303678 0.151839 0.988405i \(-0.451480\pi\)
0.151839 + 0.988405i \(0.451480\pi\)
\(398\) 0 0
\(399\) 122539. 0.0385338
\(400\) 0 0
\(401\) 4.81115e6 1.49413 0.747064 0.664752i \(-0.231463\pi\)
0.747064 + 0.664752i \(0.231463\pi\)
\(402\) 0 0
\(403\) −1.92163e6 −0.589397
\(404\) 0 0
\(405\) −2.17155e6 −0.657857
\(406\) 0 0
\(407\) −2.21462e6 −0.662694
\(408\) 0 0
\(409\) −1.42551e6 −0.421370 −0.210685 0.977554i \(-0.567569\pi\)
−0.210685 + 0.977554i \(0.567569\pi\)
\(410\) 0 0
\(411\) −497046. −0.145142
\(412\) 0 0
\(413\) 5.33372e6 1.53870
\(414\) 0 0
\(415\) −69962.7 −0.0199410
\(416\) 0 0
\(417\) 159142. 0.0448171
\(418\) 0 0
\(419\) −2.91631e6 −0.811520 −0.405760 0.913980i \(-0.632993\pi\)
−0.405760 + 0.913980i \(0.632993\pi\)
\(420\) 0 0
\(421\) 6.91223e6 1.90070 0.950349 0.311186i \(-0.100726\pi\)
0.950349 + 0.311186i \(0.100726\pi\)
\(422\) 0 0
\(423\) 530803. 0.144239
\(424\) 0 0
\(425\) −3.26243e6 −0.876131
\(426\) 0 0
\(427\) −2.62254e6 −0.696068
\(428\) 0 0
\(429\) 385259. 0.101067
\(430\) 0 0
\(431\) −2.58575e6 −0.670493 −0.335246 0.942131i \(-0.608820\pi\)
−0.335246 + 0.942131i \(0.608820\pi\)
\(432\) 0 0
\(433\) −3.71200e6 −0.951455 −0.475727 0.879593i \(-0.657815\pi\)
−0.475727 + 0.879593i \(0.657815\pi\)
\(434\) 0 0
\(435\) −82716.0 −0.0209588
\(436\) 0 0
\(437\) 1.83485e6 0.459619
\(438\) 0 0
\(439\) 415772. 0.102966 0.0514830 0.998674i \(-0.483605\pi\)
0.0514830 + 0.998674i \(0.483605\pi\)
\(440\) 0 0
\(441\) 299756. 0.0733959
\(442\) 0 0
\(443\) 602736. 0.145921 0.0729605 0.997335i \(-0.476755\pi\)
0.0729605 + 0.997335i \(0.476755\pi\)
\(444\) 0 0
\(445\) 2.66214e6 0.637282
\(446\) 0 0
\(447\) 295382. 0.0699223
\(448\) 0 0
\(449\) 1.84907e6 0.432849 0.216424 0.976299i \(-0.430560\pi\)
0.216424 + 0.976299i \(0.430560\pi\)
\(450\) 0 0
\(451\) −1.16414e6 −0.269503
\(452\) 0 0
\(453\) −619466. −0.141831
\(454\) 0 0
\(455\) −2.05151e6 −0.464564
\(456\) 0 0
\(457\) −7.35472e6 −1.64731 −0.823655 0.567091i \(-0.808069\pi\)
−0.823655 + 0.567091i \(0.808069\pi\)
\(458\) 0 0
\(459\) 1.39881e6 0.309903
\(460\) 0 0
\(461\) −3.51032e6 −0.769298 −0.384649 0.923063i \(-0.625678\pi\)
−0.384649 + 0.923063i \(0.625678\pi\)
\(462\) 0 0
\(463\) 6.86607e6 1.48852 0.744262 0.667888i \(-0.232801\pi\)
0.744262 + 0.667888i \(0.232801\pi\)
\(464\) 0 0
\(465\) −251054. −0.0538437
\(466\) 0 0
\(467\) −5.66490e6 −1.20199 −0.600993 0.799254i \(-0.705228\pi\)
−0.600993 + 0.799254i \(0.705228\pi\)
\(468\) 0 0
\(469\) 3.15263e6 0.661821
\(470\) 0 0
\(471\) −617150. −0.128185
\(472\) 0 0
\(473\) 1.09106e6 0.224230
\(474\) 0 0
\(475\) 1.10827e6 0.225377
\(476\) 0 0
\(477\) −6.01481e6 −1.21039
\(478\) 0 0
\(479\) −4.55961e6 −0.908006 −0.454003 0.891000i \(-0.650004\pi\)
−0.454003 + 0.891000i \(0.650004\pi\)
\(480\) 0 0
\(481\) 1.63174e6 0.321580
\(482\) 0 0
\(483\) 525470. 0.102490
\(484\) 0 0
\(485\) 3.36046e6 0.648701
\(486\) 0 0
\(487\) 9.17550e6 1.75310 0.876552 0.481308i \(-0.159838\pi\)
0.876552 + 0.481308i \(0.159838\pi\)
\(488\) 0 0
\(489\) −123872. −0.0234262
\(490\) 0 0
\(491\) −3.41449e6 −0.639179 −0.319589 0.947556i \(-0.603545\pi\)
−0.319589 + 0.947556i \(0.603545\pi\)
\(492\) 0 0
\(493\) −2.80408e6 −0.519605
\(494\) 0 0
\(495\) −5.37338e6 −0.985677
\(496\) 0 0
\(497\) −5.99820e6 −1.08926
\(498\) 0 0
\(499\) 1.99364e6 0.358423 0.179211 0.983811i \(-0.442645\pi\)
0.179211 + 0.983811i \(0.442645\pi\)
\(500\) 0 0
\(501\) −571146. −0.101661
\(502\) 0 0
\(503\) −9.15685e6 −1.61371 −0.806856 0.590748i \(-0.798833\pi\)
−0.806856 + 0.590748i \(0.798833\pi\)
\(504\) 0 0
\(505\) −6.77907e6 −1.18288
\(506\) 0 0
\(507\) 273705. 0.0472893
\(508\) 0 0
\(509\) 6.56697e6 1.12349 0.561746 0.827309i \(-0.310130\pi\)
0.561746 + 0.827309i \(0.310130\pi\)
\(510\) 0 0
\(511\) 7.33529e6 1.24270
\(512\) 0 0
\(513\) −475182. −0.0797199
\(514\) 0 0
\(515\) 4.66924e6 0.775761
\(516\) 0 0
\(517\) 1.30103e6 0.214072
\(518\) 0 0
\(519\) −137378. −0.0223871
\(520\) 0 0
\(521\) 6.45805e6 1.04234 0.521168 0.853454i \(-0.325497\pi\)
0.521168 + 0.853454i \(0.325497\pi\)
\(522\) 0 0
\(523\) −184405. −0.0294794 −0.0147397 0.999891i \(-0.504692\pi\)
−0.0147397 + 0.999891i \(0.504692\pi\)
\(524\) 0 0
\(525\) 317388. 0.0502566
\(526\) 0 0
\(527\) −8.51076e6 −1.33488
\(528\) 0 0
\(529\) 1.43185e6 0.222463
\(530\) 0 0
\(531\) −1.02933e7 −1.58424
\(532\) 0 0
\(533\) 857745. 0.130780
\(534\) 0 0
\(535\) 3.74437e6 0.565581
\(536\) 0 0
\(537\) −912384. −0.136534
\(538\) 0 0
\(539\) 734719. 0.108931
\(540\) 0 0
\(541\) −7.10369e6 −1.04350 −0.521748 0.853100i \(-0.674720\pi\)
−0.521748 + 0.853100i \(0.674720\pi\)
\(542\) 0 0
\(543\) −527872. −0.0768297
\(544\) 0 0
\(545\) −1.75342e6 −0.252869
\(546\) 0 0
\(547\) 9.75210e6 1.39357 0.696787 0.717279i \(-0.254612\pi\)
0.696787 + 0.717279i \(0.254612\pi\)
\(548\) 0 0
\(549\) 5.06113e6 0.716666
\(550\) 0 0
\(551\) 952562. 0.133664
\(552\) 0 0
\(553\) −1.15647e7 −1.60813
\(554\) 0 0
\(555\) 213181. 0.0293776
\(556\) 0 0
\(557\) −2.28586e6 −0.312185 −0.156092 0.987742i \(-0.549890\pi\)
−0.156092 + 0.987742i \(0.549890\pi\)
\(558\) 0 0
\(559\) −803896. −0.108810
\(560\) 0 0
\(561\) 1.70628e6 0.228899
\(562\) 0 0
\(563\) −2.31099e6 −0.307275 −0.153638 0.988127i \(-0.549099\pi\)
−0.153638 + 0.988127i \(0.549099\pi\)
\(564\) 0 0
\(565\) −7.00054e6 −0.922593
\(566\) 0 0
\(567\) 7.16176e6 0.935539
\(568\) 0 0
\(569\) 1.17226e7 1.51790 0.758949 0.651151i \(-0.225713\pi\)
0.758949 + 0.651151i \(0.225713\pi\)
\(570\) 0 0
\(571\) 5.10057e6 0.654679 0.327340 0.944907i \(-0.393848\pi\)
0.327340 + 0.944907i \(0.393848\pi\)
\(572\) 0 0
\(573\) −185170. −0.0235605
\(574\) 0 0
\(575\) 4.75245e6 0.599443
\(576\) 0 0
\(577\) 599429. 0.0749546 0.0374773 0.999297i \(-0.488068\pi\)
0.0374773 + 0.999297i \(0.488068\pi\)
\(578\) 0 0
\(579\) 245497. 0.0304334
\(580\) 0 0
\(581\) 230737. 0.0283581
\(582\) 0 0
\(583\) −1.47426e7 −1.79640
\(584\) 0 0
\(585\) 3.95914e6 0.478312
\(586\) 0 0
\(587\) −4.32755e6 −0.518379 −0.259189 0.965826i \(-0.583455\pi\)
−0.259189 + 0.965826i \(0.583455\pi\)
\(588\) 0 0
\(589\) 2.89115e6 0.343386
\(590\) 0 0
\(591\) −872667. −0.102773
\(592\) 0 0
\(593\) −1.59099e7 −1.85794 −0.928968 0.370160i \(-0.879303\pi\)
−0.928968 + 0.370160i \(0.879303\pi\)
\(594\) 0 0
\(595\) −9.08599e6 −1.05216
\(596\) 0 0
\(597\) 527883. 0.0606181
\(598\) 0 0
\(599\) −73202.1 −0.00833598 −0.00416799 0.999991i \(-0.501327\pi\)
−0.00416799 + 0.999991i \(0.501327\pi\)
\(600\) 0 0
\(601\) −3.77328e6 −0.426121 −0.213060 0.977039i \(-0.568343\pi\)
−0.213060 + 0.977039i \(0.568343\pi\)
\(602\) 0 0
\(603\) −6.08414e6 −0.681406
\(604\) 0 0
\(605\) −7.07868e6 −0.786256
\(606\) 0 0
\(607\) −3.39314e6 −0.373792 −0.186896 0.982380i \(-0.559843\pi\)
−0.186896 + 0.982380i \(0.559843\pi\)
\(608\) 0 0
\(609\) 272797. 0.0298055
\(610\) 0 0
\(611\) −958605. −0.103881
\(612\) 0 0
\(613\) 5.93491e6 0.637915 0.318957 0.947769i \(-0.396667\pi\)
0.318957 + 0.947769i \(0.396667\pi\)
\(614\) 0 0
\(615\) 112061. 0.0119472
\(616\) 0 0
\(617\) −1.07871e7 −1.14076 −0.570379 0.821382i \(-0.693204\pi\)
−0.570379 + 0.821382i \(0.693204\pi\)
\(618\) 0 0
\(619\) −9.97743e6 −1.04663 −0.523313 0.852140i \(-0.675304\pi\)
−0.523313 + 0.852140i \(0.675304\pi\)
\(620\) 0 0
\(621\) −2.03767e6 −0.212034
\(622\) 0 0
\(623\) −8.77976e6 −0.906280
\(624\) 0 0
\(625\) −1.60054e6 −0.163895
\(626\) 0 0
\(627\) −579634. −0.0588823
\(628\) 0 0
\(629\) 7.22687e6 0.728322
\(630\) 0 0
\(631\) 3.42994e6 0.342936 0.171468 0.985190i \(-0.445149\pi\)
0.171468 + 0.985190i \(0.445149\pi\)
\(632\) 0 0
\(633\) 491774. 0.0487816
\(634\) 0 0
\(635\) −8.97711e6 −0.883492
\(636\) 0 0
\(637\) −541345. −0.0528598
\(638\) 0 0
\(639\) 1.15757e7 1.12149
\(640\) 0 0
\(641\) −1.41725e7 −1.36239 −0.681193 0.732104i \(-0.738538\pi\)
−0.681193 + 0.732104i \(0.738538\pi\)
\(642\) 0 0
\(643\) −4.26802e6 −0.407098 −0.203549 0.979065i \(-0.565248\pi\)
−0.203549 + 0.979065i \(0.565248\pi\)
\(644\) 0 0
\(645\) −105026. −0.00994026
\(646\) 0 0
\(647\) −1.26038e7 −1.18370 −0.591849 0.806049i \(-0.701602\pi\)
−0.591849 + 0.806049i \(0.701602\pi\)
\(648\) 0 0
\(649\) −2.52295e7 −2.35124
\(650\) 0 0
\(651\) 827977. 0.0765712
\(652\) 0 0
\(653\) 1.67724e7 1.53926 0.769631 0.638489i \(-0.220440\pi\)
0.769631 + 0.638489i \(0.220440\pi\)
\(654\) 0 0
\(655\) −1.07233e7 −0.976618
\(656\) 0 0
\(657\) −1.41561e7 −1.27947
\(658\) 0 0
\(659\) 1.72113e7 1.54383 0.771915 0.635725i \(-0.219299\pi\)
0.771915 + 0.635725i \(0.219299\pi\)
\(660\) 0 0
\(661\) 5.67723e6 0.505397 0.252699 0.967545i \(-0.418682\pi\)
0.252699 + 0.967545i \(0.418682\pi\)
\(662\) 0 0
\(663\) −1.25720e6 −0.111076
\(664\) 0 0
\(665\) 3.08656e6 0.270658
\(666\) 0 0
\(667\) 4.08476e6 0.355510
\(668\) 0 0
\(669\) 930449. 0.0803762
\(670\) 0 0
\(671\) 1.24051e7 1.06364
\(672\) 0 0
\(673\) 4.92989e6 0.419565 0.209783 0.977748i \(-0.432724\pi\)
0.209783 + 0.977748i \(0.432724\pi\)
\(674\) 0 0
\(675\) −1.23077e6 −0.103972
\(676\) 0 0
\(677\) −1.25498e7 −1.05236 −0.526180 0.850373i \(-0.676376\pi\)
−0.526180 + 0.850373i \(0.676376\pi\)
\(678\) 0 0
\(679\) −1.10828e7 −0.922519
\(680\) 0 0
\(681\) −1.13745e6 −0.0939861
\(682\) 0 0
\(683\) 7.08503e6 0.581152 0.290576 0.956852i \(-0.406153\pi\)
0.290576 + 0.956852i \(0.406153\pi\)
\(684\) 0 0
\(685\) −1.25198e7 −1.01946
\(686\) 0 0
\(687\) −1.82813e6 −0.147780
\(688\) 0 0
\(689\) 1.08625e7 0.871726
\(690\) 0 0
\(691\) 9.40053e6 0.748958 0.374479 0.927235i \(-0.377822\pi\)
0.374479 + 0.927235i \(0.377822\pi\)
\(692\) 0 0
\(693\) 1.77214e7 1.40173
\(694\) 0 0
\(695\) 4.00853e6 0.314791
\(696\) 0 0
\(697\) 3.79888e6 0.296192
\(698\) 0 0
\(699\) −570213. −0.0441412
\(700\) 0 0
\(701\) 2.14131e7 1.64583 0.822914 0.568167i \(-0.192347\pi\)
0.822914 + 0.568167i \(0.192347\pi\)
\(702\) 0 0
\(703\) −2.45501e6 −0.187355
\(704\) 0 0
\(705\) −125238. −0.00948995
\(706\) 0 0
\(707\) 2.23574e7 1.68218
\(708\) 0 0
\(709\) 1.70494e7 1.27377 0.636887 0.770957i \(-0.280222\pi\)
0.636887 + 0.770957i \(0.280222\pi\)
\(710\) 0 0
\(711\) 2.23182e7 1.65571
\(712\) 0 0
\(713\) 1.23978e7 0.913316
\(714\) 0 0
\(715\) 9.70406e6 0.709886
\(716\) 0 0
\(717\) 423110. 0.0307365
\(718\) 0 0
\(719\) 1.59376e7 1.14974 0.574872 0.818243i \(-0.305052\pi\)
0.574872 + 0.818243i \(0.305052\pi\)
\(720\) 0 0
\(721\) −1.53991e7 −1.10321
\(722\) 0 0
\(723\) 1.68976e6 0.120220
\(724\) 0 0
\(725\) 2.46723e6 0.174327
\(726\) 0 0
\(727\) −8.26202e6 −0.579763 −0.289881 0.957063i \(-0.593616\pi\)
−0.289881 + 0.957063i \(0.593616\pi\)
\(728\) 0 0
\(729\) −1.35561e7 −0.944749
\(730\) 0 0
\(731\) −3.56039e6 −0.246436
\(732\) 0 0
\(733\) 6.20168e6 0.426334 0.213167 0.977016i \(-0.431622\pi\)
0.213167 + 0.977016i \(0.431622\pi\)
\(734\) 0 0
\(735\) −70724.7 −0.00482895
\(736\) 0 0
\(737\) −1.49126e7 −1.01131
\(738\) 0 0
\(739\) 1.19003e7 0.801578 0.400789 0.916170i \(-0.368736\pi\)
0.400789 + 0.916170i \(0.368736\pi\)
\(740\) 0 0
\(741\) 427077. 0.0285734
\(742\) 0 0
\(743\) −7.41722e6 −0.492912 −0.246456 0.969154i \(-0.579266\pi\)
−0.246456 + 0.969154i \(0.579266\pi\)
\(744\) 0 0
\(745\) 7.44020e6 0.491127
\(746\) 0 0
\(747\) −445291. −0.0291973
\(748\) 0 0
\(749\) −1.23489e7 −0.804314
\(750\) 0 0
\(751\) −1.46691e7 −0.949084 −0.474542 0.880233i \(-0.657386\pi\)
−0.474542 + 0.880233i \(0.657386\pi\)
\(752\) 0 0
\(753\) −1.01449e6 −0.0652020
\(754\) 0 0
\(755\) −1.56033e7 −0.996209
\(756\) 0 0
\(757\) 2.41775e6 0.153346 0.0766729 0.997056i \(-0.475570\pi\)
0.0766729 + 0.997056i \(0.475570\pi\)
\(758\) 0 0
\(759\) −2.48558e6 −0.156611
\(760\) 0 0
\(761\) 2.32422e7 1.45484 0.727422 0.686191i \(-0.240718\pi\)
0.727422 + 0.686191i \(0.240718\pi\)
\(762\) 0 0
\(763\) 5.78279e6 0.359605
\(764\) 0 0
\(765\) 1.75347e7 1.08329
\(766\) 0 0
\(767\) 1.85893e7 1.14097
\(768\) 0 0
\(769\) −2.55432e7 −1.55761 −0.778806 0.627265i \(-0.784174\pi\)
−0.778806 + 0.627265i \(0.784174\pi\)
\(770\) 0 0
\(771\) −2.78747e6 −0.168879
\(772\) 0 0
\(773\) 1.72159e7 1.03629 0.518146 0.855292i \(-0.326622\pi\)
0.518146 + 0.855292i \(0.326622\pi\)
\(774\) 0 0
\(775\) 7.48838e6 0.447851
\(776\) 0 0
\(777\) −703072. −0.0417780
\(778\) 0 0
\(779\) −1.29050e6 −0.0761930
\(780\) 0 0
\(781\) 2.83727e7 1.66446
\(782\) 0 0
\(783\) −1.05785e6 −0.0616625
\(784\) 0 0
\(785\) −1.55450e7 −0.900362
\(786\) 0 0
\(787\) 1.48812e7 0.856449 0.428225 0.903672i \(-0.359139\pi\)
0.428225 + 0.903672i \(0.359139\pi\)
\(788\) 0 0
\(789\) −2.31500e6 −0.132391
\(790\) 0 0
\(791\) 2.30878e7 1.31202
\(792\) 0 0
\(793\) −9.14016e6 −0.516144
\(794\) 0 0
\(795\) 1.41914e6 0.0796356
\(796\) 0 0
\(797\) 5.81135e6 0.324064 0.162032 0.986785i \(-0.448195\pi\)
0.162032 + 0.986785i \(0.448195\pi\)
\(798\) 0 0
\(799\) −4.24559e6 −0.235272
\(800\) 0 0
\(801\) 1.69437e7 0.933098
\(802\) 0 0
\(803\) −3.46973e7 −1.89892
\(804\) 0 0
\(805\) 1.32358e7 0.719878
\(806\) 0 0
\(807\) −1.75360e6 −0.0947867
\(808\) 0 0
\(809\) −2.41530e7 −1.29748 −0.648740 0.761010i \(-0.724704\pi\)
−0.648740 + 0.761010i \(0.724704\pi\)
\(810\) 0 0
\(811\) −2.14467e7 −1.14501 −0.572505 0.819901i \(-0.694028\pi\)
−0.572505 + 0.819901i \(0.694028\pi\)
\(812\) 0 0
\(813\) −1.80588e6 −0.0958212
\(814\) 0 0
\(815\) −3.12014e6 −0.164543
\(816\) 0 0
\(817\) 1.20949e6 0.0633936
\(818\) 0 0
\(819\) −1.30572e7 −0.680208
\(820\) 0 0
\(821\) −1.06313e7 −0.550465 −0.275232 0.961378i \(-0.588755\pi\)
−0.275232 + 0.961378i \(0.588755\pi\)
\(822\) 0 0
\(823\) 871518. 0.0448515 0.0224257 0.999749i \(-0.492861\pi\)
0.0224257 + 0.999749i \(0.492861\pi\)
\(824\) 0 0
\(825\) −1.50131e6 −0.0767954
\(826\) 0 0
\(827\) 1.70354e7 0.866141 0.433071 0.901360i \(-0.357430\pi\)
0.433071 + 0.901360i \(0.357430\pi\)
\(828\) 0 0
\(829\) 5.65731e6 0.285906 0.142953 0.989729i \(-0.454340\pi\)
0.142953 + 0.989729i \(0.454340\pi\)
\(830\) 0 0
\(831\) 757546. 0.0380546
\(832\) 0 0
\(833\) −2.39758e6 −0.119718
\(834\) 0 0
\(835\) −1.43862e7 −0.714054
\(836\) 0 0
\(837\) −3.21073e6 −0.158413
\(838\) 0 0
\(839\) −3.42452e7 −1.67956 −0.839779 0.542929i \(-0.817315\pi\)
−0.839779 + 0.542929i \(0.817315\pi\)
\(840\) 0 0
\(841\) −1.83905e7 −0.896612
\(842\) 0 0
\(843\) −2.29326e6 −0.111144
\(844\) 0 0
\(845\) 6.89419e6 0.332155
\(846\) 0 0
\(847\) 2.33455e7 1.11814
\(848\) 0 0
\(849\) 2.73181e6 0.130071
\(850\) 0 0
\(851\) −1.05275e7 −0.498313
\(852\) 0 0
\(853\) −2.21968e7 −1.04452 −0.522261 0.852785i \(-0.674911\pi\)
−0.522261 + 0.852785i \(0.674911\pi\)
\(854\) 0 0
\(855\) −5.95664e6 −0.278667
\(856\) 0 0
\(857\) 3.15803e7 1.46881 0.734403 0.678714i \(-0.237462\pi\)
0.734403 + 0.678714i \(0.237462\pi\)
\(858\) 0 0
\(859\) 1.22037e6 0.0564297 0.0282148 0.999602i \(-0.491018\pi\)
0.0282148 + 0.999602i \(0.491018\pi\)
\(860\) 0 0
\(861\) −369578. −0.0169902
\(862\) 0 0
\(863\) 2.43543e7 1.11314 0.556570 0.830801i \(-0.312117\pi\)
0.556570 + 0.830801i \(0.312117\pi\)
\(864\) 0 0
\(865\) −3.46032e6 −0.157245
\(866\) 0 0
\(867\) −3.43585e6 −0.155234
\(868\) 0 0
\(869\) 5.47031e7 2.45732
\(870\) 0 0
\(871\) 1.09877e7 0.490749
\(872\) 0 0
\(873\) 2.13883e7 0.949817
\(874\) 0 0
\(875\) 2.27401e7 1.00409
\(876\) 0 0
\(877\) 2.52016e6 0.110644 0.0553222 0.998469i \(-0.482381\pi\)
0.0553222 + 0.998469i \(0.482381\pi\)
\(878\) 0 0
\(879\) 3.54361e6 0.154694
\(880\) 0 0
\(881\) 1.04311e6 0.0452785 0.0226393 0.999744i \(-0.492793\pi\)
0.0226393 + 0.999744i \(0.492793\pi\)
\(882\) 0 0
\(883\) 3.57753e7 1.54412 0.772061 0.635548i \(-0.219226\pi\)
0.772061 + 0.635548i \(0.219226\pi\)
\(884\) 0 0
\(885\) 2.42862e6 0.104232
\(886\) 0 0
\(887\) 1.66616e7 0.711061 0.355531 0.934665i \(-0.384300\pi\)
0.355531 + 0.934665i \(0.384300\pi\)
\(888\) 0 0
\(889\) 2.96065e7 1.25642
\(890\) 0 0
\(891\) −3.38765e7 −1.42957
\(892\) 0 0
\(893\) 1.44225e6 0.0605218
\(894\) 0 0
\(895\) −2.29815e7 −0.959004
\(896\) 0 0
\(897\) 1.83139e6 0.0759975
\(898\) 0 0
\(899\) 6.43631e6 0.265606
\(900\) 0 0
\(901\) 4.81090e7 1.97430
\(902\) 0 0
\(903\) 346376. 0.0141361
\(904\) 0 0
\(905\) −1.32963e7 −0.539645
\(906\) 0 0
\(907\) −3.59470e7 −1.45092 −0.725462 0.688262i \(-0.758374\pi\)
−0.725462 + 0.688262i \(0.758374\pi\)
\(908\) 0 0
\(909\) −4.31466e7 −1.73196
\(910\) 0 0
\(911\) −7.56404e6 −0.301966 −0.150983 0.988536i \(-0.548244\pi\)
−0.150983 + 0.988536i \(0.548244\pi\)
\(912\) 0 0
\(913\) −1.09143e6 −0.0433331
\(914\) 0 0
\(915\) −1.19413e6 −0.0471517
\(916\) 0 0
\(917\) 3.53654e7 1.38885
\(918\) 0 0
\(919\) 2.22167e7 0.867741 0.433870 0.900975i \(-0.357148\pi\)
0.433870 + 0.900975i \(0.357148\pi\)
\(920\) 0 0
\(921\) −3.90124e6 −0.151549
\(922\) 0 0
\(923\) −2.09051e7 −0.807698
\(924\) 0 0
\(925\) −6.35871e6 −0.244351
\(926\) 0 0
\(927\) 2.97182e7 1.13586
\(928\) 0 0
\(929\) 9.18809e6 0.349290 0.174645 0.984631i \(-0.444122\pi\)
0.174645 + 0.984631i \(0.444122\pi\)
\(930\) 0 0
\(931\) 814470. 0.0307965
\(932\) 0 0
\(933\) −3.32581e6 −0.125082
\(934\) 0 0
\(935\) 4.29785e7 1.60777
\(936\) 0 0
\(937\) 4.06654e7 1.51313 0.756565 0.653919i \(-0.226876\pi\)
0.756565 + 0.653919i \(0.226876\pi\)
\(938\) 0 0
\(939\) 3.65856e6 0.135409
\(940\) 0 0
\(941\) −2.60105e7 −0.957580 −0.478790 0.877930i \(-0.658924\pi\)
−0.478790 + 0.877930i \(0.658924\pi\)
\(942\) 0 0
\(943\) −5.53391e6 −0.202653
\(944\) 0 0
\(945\) −3.42774e6 −0.124861
\(946\) 0 0
\(947\) 2.03967e7 0.739067 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(948\) 0 0
\(949\) 2.55652e7 0.921475
\(950\) 0 0
\(951\) 3.89099e6 0.139511
\(952\) 0 0
\(953\) −4.87462e6 −0.173864 −0.0869318 0.996214i \(-0.527706\pi\)
−0.0869318 + 0.996214i \(0.527706\pi\)
\(954\) 0 0
\(955\) −4.66414e6 −0.165487
\(956\) 0 0
\(957\) −1.29039e6 −0.0455449
\(958\) 0 0
\(959\) 4.12903e7 1.44978
\(960\) 0 0
\(961\) −9.09409e6 −0.317651
\(962\) 0 0
\(963\) 2.38318e7 0.828114
\(964\) 0 0
\(965\) 6.18368e6 0.213761
\(966\) 0 0
\(967\) 1.26016e7 0.433371 0.216686 0.976241i \(-0.430475\pi\)
0.216686 + 0.976241i \(0.430475\pi\)
\(968\) 0 0
\(969\) 1.89149e6 0.0647135
\(970\) 0 0
\(971\) 1.39978e6 0.0476443 0.0238221 0.999716i \(-0.492416\pi\)
0.0238221 + 0.999716i \(0.492416\pi\)
\(972\) 0 0
\(973\) −1.32201e7 −0.447665
\(974\) 0 0
\(975\) 1.10617e6 0.0372659
\(976\) 0 0
\(977\) 1.43010e6 0.0479325 0.0239662 0.999713i \(-0.492371\pi\)
0.0239662 + 0.999713i \(0.492371\pi\)
\(978\) 0 0
\(979\) 4.15300e7 1.38486
\(980\) 0 0
\(981\) −1.11600e7 −0.370247
\(982\) 0 0
\(983\) −4.26255e7 −1.40697 −0.703487 0.710708i \(-0.748375\pi\)
−0.703487 + 0.710708i \(0.748375\pi\)
\(984\) 0 0
\(985\) −2.19811e7 −0.721869
\(986\) 0 0
\(987\) 413035. 0.0134957
\(988\) 0 0
\(989\) 5.18650e6 0.168610
\(990\) 0 0
\(991\) −1.88273e7 −0.608980 −0.304490 0.952516i \(-0.598486\pi\)
−0.304490 + 0.952516i \(0.598486\pi\)
\(992\) 0 0
\(993\) −3.74463e6 −0.120514
\(994\) 0 0
\(995\) 1.32965e7 0.425775
\(996\) 0 0
\(997\) 2.10574e7 0.670915 0.335458 0.942055i \(-0.391109\pi\)
0.335458 + 0.942055i \(0.391109\pi\)
\(998\) 0 0
\(999\) 2.72637e6 0.0864314
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 688.6.a.h.1.6 10
4.3 odd 2 43.6.a.b.1.2 10
12.11 even 2 387.6.a.e.1.9 10
20.19 odd 2 1075.6.a.b.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.2 10 4.3 odd 2
387.6.a.e.1.9 10 12.11 even 2
688.6.a.h.1.6 10 1.1 even 1 trivial
1075.6.a.b.1.9 10 20.19 odd 2