Properties

Label 108.6.a.b.1.2
Level 108
Weight 6
Character 108.1
Self dual yes
Analytic conductor 17.321
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3214525398\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 108.1

$q$-expansion

\(f(q)\) \(=\) \(q+57.6281 q^{5} -188.884 q^{7} +O(q^{10})\) \(q+57.6281 q^{5} -188.884 q^{7} -704.025 q^{11} +795.537 q^{13} +180.562 q^{17} -661.769 q^{19} -3632.51 q^{23} +196.000 q^{25} -8021.87 q^{29} -3003.88 q^{31} -10885.0 q^{35} -1520.31 q^{37} -3461.57 q^{41} +11815.8 q^{43} +5979.82 q^{47} +18870.3 q^{49} -13819.2 q^{53} -40571.6 q^{55} -22640.2 q^{59} -37404.2 q^{61} +45845.3 q^{65} +70964.7 q^{67} +67767.0 q^{71} -31562.4 q^{73} +132979. q^{77} +62791.6 q^{79} -93431.5 q^{83} +10405.5 q^{85} +68793.1 q^{89} -150265. q^{91} -38136.5 q^{95} +117774. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{7} + O(q^{10}) \) \( 2q - 32q^{7} - 486q^{11} + 208q^{13} - 1944q^{17} - 632q^{19} - 6804q^{23} + 392q^{25} - 11664q^{29} + 3328q^{31} - 19926q^{35} - 9956q^{37} - 13608q^{41} + 4960q^{43} - 18468q^{47} + 26676q^{49} + 11664q^{53} - 53136q^{55} - 1944q^{59} + 8176q^{61} + 79704q^{65} + 90064q^{67} + 44712q^{71} - 121214q^{73} + 167184q^{77} + 28768q^{79} + 15066q^{83} + 132840q^{85} + 178848q^{89} - 242440q^{91} - 39852q^{95} + 88942q^{97} + O(q^{100}) \)

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
\(5\) 57.6281 1.03088 0.515442 0.856925i \(-0.327628\pi\)
0.515442 + 0.856925i \(0.327628\pi\)
\(6\) 0 0
\(7\) −188.884 −1.45697 −0.728485 0.685061i \(-0.759775\pi\)
−0.728485 + 0.685061i \(0.759775\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −704.025 −1.75431 −0.877155 0.480207i \(-0.840561\pi\)
−0.877155 + 0.480207i \(0.840561\pi\)
\(12\) 0 0
\(13\) 795.537 1.30558 0.652788 0.757541i \(-0.273599\pi\)
0.652788 + 0.757541i \(0.273599\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 180.562 0.151532 0.0757661 0.997126i \(-0.475860\pi\)
0.0757661 + 0.997126i \(0.475860\pi\)
\(18\) 0 0
\(19\) −661.769 −0.420554 −0.210277 0.977642i \(-0.567437\pi\)
−0.210277 + 0.977642i \(0.567437\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3632.51 −1.43182 −0.715909 0.698194i \(-0.753987\pi\)
−0.715909 + 0.698194i \(0.753987\pi\)
\(24\) 0 0
\(25\) 196.000 0.0627200
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8021.87 −1.77125 −0.885626 0.464398i \(-0.846271\pi\)
−0.885626 + 0.464398i \(0.846271\pi\)
\(30\) 0 0
\(31\) −3003.88 −0.561407 −0.280704 0.959795i \(-0.590568\pi\)
−0.280704 + 0.959795i \(0.590568\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −10885.0 −1.50197
\(36\) 0 0
\(37\) −1520.31 −0.182570 −0.0912848 0.995825i \(-0.529097\pi\)
−0.0912848 + 0.995825i \(0.529097\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3461.57 −0.321598 −0.160799 0.986987i \(-0.551407\pi\)
−0.160799 + 0.986987i \(0.551407\pi\)
\(42\) 0 0
\(43\) 11815.8 0.974519 0.487260 0.873257i \(-0.337997\pi\)
0.487260 + 0.873257i \(0.337997\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5979.82 0.394861 0.197430 0.980317i \(-0.436740\pi\)
0.197430 + 0.980317i \(0.436740\pi\)
\(48\) 0 0
\(49\) 18870.3 1.12276
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13819.2 −0.675761 −0.337880 0.941189i \(-0.609710\pi\)
−0.337880 + 0.941189i \(0.609710\pi\)
\(54\) 0 0
\(55\) −40571.6 −1.80849
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −22640.2 −0.846739 −0.423370 0.905957i \(-0.639153\pi\)
−0.423370 + 0.905957i \(0.639153\pi\)
\(60\) 0 0
\(61\) −37404.2 −1.28705 −0.643526 0.765424i \(-0.722529\pi\)
−0.643526 + 0.765424i \(0.722529\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 45845.3 1.34590
\(66\) 0 0
\(67\) 70964.7 1.93132 0.965662 0.259802i \(-0.0836573\pi\)
0.965662 + 0.259802i \(0.0836573\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 67767.0 1.59541 0.797705 0.603048i \(-0.206047\pi\)
0.797705 + 0.603048i \(0.206047\pi\)
\(72\) 0 0
\(73\) −31562.4 −0.693208 −0.346604 0.938012i \(-0.612665\pi\)
−0.346604 + 0.938012i \(0.612665\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 132979. 2.55598
\(78\) 0 0
\(79\) 62791.6 1.13197 0.565984 0.824416i \(-0.308496\pi\)
0.565984 + 0.824416i \(0.308496\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −93431.5 −1.48867 −0.744334 0.667807i \(-0.767233\pi\)
−0.744334 + 0.667807i \(0.767233\pi\)
\(84\) 0 0
\(85\) 10405.5 0.156212
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 68793.1 0.920598 0.460299 0.887764i \(-0.347742\pi\)
0.460299 + 0.887764i \(0.347742\pi\)
\(90\) 0 0
\(91\) −150265. −1.90219
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −38136.5 −0.433542
\(96\) 0 0
\(97\) 117774. 1.27093 0.635463 0.772132i \(-0.280809\pi\)
0.635463 + 0.772132i \(0.280809\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 161886. 1.57908 0.789542 0.613697i \(-0.210318\pi\)
0.789542 + 0.613697i \(0.210318\pi\)
\(102\) 0 0
\(103\) 59324.6 0.550987 0.275494 0.961303i \(-0.411159\pi\)
0.275494 + 0.961303i \(0.411159\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −21124.1 −0.178369 −0.0891845 0.996015i \(-0.528426\pi\)
−0.0891845 + 0.996015i \(0.528426\pi\)
\(108\) 0 0
\(109\) −168898. −1.36162 −0.680812 0.732458i \(-0.738373\pi\)
−0.680812 + 0.732458i \(0.738373\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −61274.0 −0.451419 −0.225710 0.974195i \(-0.572470\pi\)
−0.225710 + 0.974195i \(0.572470\pi\)
\(114\) 0 0
\(115\) −209335. −1.47604
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −34105.4 −0.220778
\(120\) 0 0
\(121\) 334600. 2.07760
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −168793. −0.966226
\(126\) 0 0
\(127\) 23607.7 0.129881 0.0649404 0.997889i \(-0.479314\pi\)
0.0649404 + 0.997889i \(0.479314\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −206849. −1.05311 −0.526557 0.850140i \(-0.676517\pi\)
−0.526557 + 0.850140i \(0.676517\pi\)
\(132\) 0 0
\(133\) 124998. 0.612736
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −191495. −0.871678 −0.435839 0.900025i \(-0.643548\pi\)
−0.435839 + 0.900025i \(0.643548\pi\)
\(138\) 0 0
\(139\) 250418. 1.09933 0.549666 0.835385i \(-0.314755\pi\)
0.549666 + 0.835385i \(0.314755\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −560078. −2.29039
\(144\) 0 0
\(145\) −462285. −1.82595
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 200857. 0.741177 0.370588 0.928797i \(-0.379156\pi\)
0.370588 + 0.928797i \(0.379156\pi\)
\(150\) 0 0
\(151\) −23263.1 −0.0830283 −0.0415141 0.999138i \(-0.513218\pi\)
−0.0415141 + 0.999138i \(0.513218\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −173108. −0.578745
\(156\) 0 0
\(157\) −523985. −1.69656 −0.848281 0.529546i \(-0.822362\pi\)
−0.848281 + 0.529546i \(0.822362\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 686125. 2.08612
\(162\) 0 0
\(163\) 194530. 0.573479 0.286739 0.958009i \(-0.407429\pi\)
0.286739 + 0.958009i \(0.407429\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −462757. −1.28399 −0.641995 0.766709i \(-0.721893\pi\)
−0.641995 + 0.766709i \(0.721893\pi\)
\(168\) 0 0
\(169\) 261587. 0.704529
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 39891.8 0.101337 0.0506685 0.998716i \(-0.483865\pi\)
0.0506685 + 0.998716i \(0.483865\pi\)
\(174\) 0 0
\(175\) −37021.3 −0.0913812
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −221514. −0.516736 −0.258368 0.966047i \(-0.583185\pi\)
−0.258368 + 0.966047i \(0.583185\pi\)
\(180\) 0 0
\(181\) 800588. 1.81640 0.908202 0.418532i \(-0.137455\pi\)
0.908202 + 0.418532i \(0.137455\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −87612.8 −0.188208
\(186\) 0 0
\(187\) −127120. −0.265834
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −70550.6 −0.139932 −0.0699661 0.997549i \(-0.522289\pi\)
−0.0699661 + 0.997549i \(0.522289\pi\)
\(192\) 0 0
\(193\) −458196. −0.885437 −0.442719 0.896661i \(-0.645986\pi\)
−0.442719 + 0.896661i \(0.645986\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 605955. 1.11244 0.556218 0.831037i \(-0.312252\pi\)
0.556218 + 0.831037i \(0.312252\pi\)
\(198\) 0 0
\(199\) −108129. −0.193557 −0.0967783 0.995306i \(-0.530854\pi\)
−0.0967783 + 0.995306i \(0.530854\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.51521e6 2.58066
\(204\) 0 0
\(205\) −199484. −0.331530
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 465902. 0.737783
\(210\) 0 0
\(211\) 274922. 0.425113 0.212556 0.977149i \(-0.431821\pi\)
0.212556 + 0.977149i \(0.431821\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 680920. 1.00462
\(216\) 0 0
\(217\) 567385. 0.817954
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 143644. 0.197837
\(222\) 0 0
\(223\) −503846. −0.678478 −0.339239 0.940700i \(-0.610170\pi\)
−0.339239 + 0.940700i \(0.610170\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 389725. 0.501989 0.250994 0.967989i \(-0.419242\pi\)
0.250994 + 0.967989i \(0.419242\pi\)
\(228\) 0 0
\(229\) −931505. −1.17381 −0.586903 0.809657i \(-0.699653\pi\)
−0.586903 + 0.809657i \(0.699653\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −967962. −1.16807 −0.584034 0.811729i \(-0.698527\pi\)
−0.584034 + 0.811729i \(0.698527\pi\)
\(234\) 0 0
\(235\) 344606. 0.407055
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −335896. −0.380373 −0.190187 0.981748i \(-0.560909\pi\)
−0.190187 + 0.981748i \(0.560909\pi\)
\(240\) 0 0
\(241\) −547142. −0.606817 −0.303408 0.952861i \(-0.598125\pi\)
−0.303408 + 0.952861i \(0.598125\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.08746e6 1.15744
\(246\) 0 0
\(247\) −526462. −0.549066
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −36479.6 −0.0365482 −0.0182741 0.999833i \(-0.505817\pi\)
−0.0182741 + 0.999833i \(0.505817\pi\)
\(252\) 0 0
\(253\) 2.55738e6 2.51185
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.19968e6 −1.13301 −0.566505 0.824058i \(-0.691705\pi\)
−0.566505 + 0.824058i \(0.691705\pi\)
\(258\) 0 0
\(259\) 287163. 0.265999
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 878508. 0.783171 0.391585 0.920142i \(-0.371927\pi\)
0.391585 + 0.920142i \(0.371927\pi\)
\(264\) 0 0
\(265\) −796374. −0.696630
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −434159. −0.365820 −0.182910 0.983130i \(-0.558552\pi\)
−0.182910 + 0.983130i \(0.558552\pi\)
\(270\) 0 0
\(271\) 799583. 0.661364 0.330682 0.943742i \(-0.392721\pi\)
0.330682 + 0.943742i \(0.392721\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −137989. −0.110030
\(276\) 0 0
\(277\) 1.63042e6 1.27674 0.638368 0.769731i \(-0.279610\pi\)
0.638368 + 0.769731i \(0.279610\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.53698e6 −1.91669 −0.958344 0.285617i \(-0.907802\pi\)
−0.958344 + 0.285617i \(0.907802\pi\)
\(282\) 0 0
\(283\) −2.07324e6 −1.53880 −0.769402 0.638765i \(-0.779445\pi\)
−0.769402 + 0.638765i \(0.779445\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 653836. 0.468559
\(288\) 0 0
\(289\) −1.38725e6 −0.977038
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 666768. 0.453739 0.226869 0.973925i \(-0.427151\pi\)
0.226869 + 0.973925i \(0.427151\pi\)
\(294\) 0 0
\(295\) −1.30471e6 −0.872889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.88980e6 −1.86935
\(300\) 0 0
\(301\) −2.23181e6 −1.41985
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.15554e6 −1.32680
\(306\) 0 0
\(307\) 811069. 0.491148 0.245574 0.969378i \(-0.421024\pi\)
0.245574 + 0.969378i \(0.421024\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −985037. −0.577500 −0.288750 0.957405i \(-0.593240\pi\)
−0.288750 + 0.957405i \(0.593240\pi\)
\(312\) 0 0
\(313\) 783184. 0.451859 0.225929 0.974144i \(-0.427458\pi\)
0.225929 + 0.974144i \(0.427458\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.66519e6 −1.48963 −0.744817 0.667268i \(-0.767463\pi\)
−0.744817 + 0.667268i \(0.767463\pi\)
\(318\) 0 0
\(319\) 5.64760e6 3.10733
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −119491. −0.0637275
\(324\) 0 0
\(325\) 155925. 0.0818857
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.12950e6 −0.575300
\(330\) 0 0
\(331\) −2.34153e6 −1.17471 −0.587353 0.809331i \(-0.699830\pi\)
−0.587353 + 0.809331i \(0.699830\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.08956e6 1.99097
\(336\) 0 0
\(337\) −710742. −0.340908 −0.170454 0.985366i \(-0.554523\pi\)
−0.170454 + 0.985366i \(0.554523\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.11480e6 0.984882
\(342\) 0 0
\(343\) −389725. −0.178864
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 702187. 0.313061 0.156531 0.987673i \(-0.449969\pi\)
0.156531 + 0.987673i \(0.449969\pi\)
\(348\) 0 0
\(349\) −138699. −0.0609553 −0.0304776 0.999535i \(-0.509703\pi\)
−0.0304776 + 0.999535i \(0.509703\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 55094.1 0.0235325 0.0117663 0.999931i \(-0.496255\pi\)
0.0117663 + 0.999931i \(0.496255\pi\)
\(354\) 0 0
\(355\) 3.90528e6 1.64468
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.94311e6 0.795722 0.397861 0.917446i \(-0.369753\pi\)
0.397861 + 0.917446i \(0.369753\pi\)
\(360\) 0 0
\(361\) −2.03816e6 −0.823134
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.81888e6 −0.714616
\(366\) 0 0
\(367\) −1.49593e6 −0.579758 −0.289879 0.957063i \(-0.593615\pi\)
−0.289879 + 0.957063i \(0.593615\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.61023e6 0.984564
\(372\) 0 0
\(373\) −1.71696e6 −0.638982 −0.319491 0.947589i \(-0.603512\pi\)
−0.319491 + 0.947589i \(0.603512\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.38170e6 −2.31251
\(378\) 0 0
\(379\) −3.54794e6 −1.26876 −0.634378 0.773023i \(-0.718744\pi\)
−0.634378 + 0.773023i \(0.718744\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.75302e6 0.958987 0.479493 0.877545i \(-0.340820\pi\)
0.479493 + 0.877545i \(0.340820\pi\)
\(384\) 0 0
\(385\) 7.66335e6 2.63492
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.95371e6 0.989679 0.494839 0.868984i \(-0.335227\pi\)
0.494839 + 0.868984i \(0.335227\pi\)
\(390\) 0 0
\(391\) −655895. −0.216966
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.61856e6 1.16693
\(396\) 0 0
\(397\) −2.21949e6 −0.706767 −0.353383 0.935479i \(-0.614969\pi\)
−0.353383 + 0.935479i \(0.614969\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.43822e6 1.37831 0.689157 0.724612i \(-0.257981\pi\)
0.689157 + 0.724612i \(0.257981\pi\)
\(402\) 0 0
\(403\) −2.38970e6 −0.732960
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.07034e6 0.320284
\(408\) 0 0
\(409\) −331601. −0.0980184 −0.0490092 0.998798i \(-0.515606\pi\)
−0.0490092 + 0.998798i \(0.515606\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.27637e6 1.23367
\(414\) 0 0
\(415\) −5.38428e6 −1.53464
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.49486e6 0.694243 0.347121 0.937820i \(-0.387159\pi\)
0.347121 + 0.937820i \(0.387159\pi\)
\(420\) 0 0
\(421\) 4.55155e6 1.25157 0.625784 0.779997i \(-0.284779\pi\)
0.625784 + 0.779997i \(0.284779\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 35390.2 0.00950410
\(426\) 0 0
\(427\) 7.06508e6 1.87520
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 614314. 0.159293 0.0796466 0.996823i \(-0.474621\pi\)
0.0796466 + 0.996823i \(0.474621\pi\)
\(432\) 0 0
\(433\) 2.42759e6 0.622236 0.311118 0.950371i \(-0.399297\pi\)
0.311118 + 0.950371i \(0.399297\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.40388e6 0.602157
\(438\) 0 0
\(439\) −2.31800e6 −0.574054 −0.287027 0.957923i \(-0.592667\pi\)
−0.287027 + 0.957923i \(0.592667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.23648e6 −1.26774 −0.633870 0.773439i \(-0.718535\pi\)
−0.633870 + 0.773439i \(0.718535\pi\)
\(444\) 0 0
\(445\) 3.96442e6 0.949029
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −985716. −0.230747 −0.115373 0.993322i \(-0.536806\pi\)
−0.115373 + 0.993322i \(0.536806\pi\)
\(450\) 0 0
\(451\) 2.43703e6 0.564183
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.65946e6 −1.96093
\(456\) 0 0
\(457\) −5.09496e6 −1.14117 −0.570585 0.821238i \(-0.693284\pi\)
−0.570585 + 0.821238i \(0.693284\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.89512e6 1.94939 0.974697 0.223531i \(-0.0717584\pi\)
0.974697 + 0.223531i \(0.0717584\pi\)
\(462\) 0 0
\(463\) 3.26267e6 0.707327 0.353664 0.935373i \(-0.384936\pi\)
0.353664 + 0.935373i \(0.384936\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.63482e6 −0.346879 −0.173439 0.984845i \(-0.555488\pi\)
−0.173439 + 0.984845i \(0.555488\pi\)
\(468\) 0 0
\(469\) −1.34041e7 −2.81388
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.31859e6 −1.70961
\(474\) 0 0
\(475\) −129707. −0.0263772
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.57113e6 −1.70687 −0.853433 0.521203i \(-0.825483\pi\)
−0.853433 + 0.521203i \(0.825483\pi\)
\(480\) 0 0
\(481\) −1.20947e6 −0.238359
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.78709e6 1.31018
\(486\) 0 0
\(487\) −5.12742e6 −0.979662 −0.489831 0.871817i \(-0.662942\pi\)
−0.489831 + 0.871817i \(0.662942\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.33990e6 1.37400 0.687000 0.726658i \(-0.258927\pi\)
0.687000 + 0.726658i \(0.258927\pi\)
\(492\) 0 0
\(493\) −1.44845e6 −0.268402
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.28001e7 −2.32446
\(498\) 0 0
\(499\) 6.08886e6 1.09467 0.547336 0.836913i \(-0.315642\pi\)
0.547336 + 0.836913i \(0.315642\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.11508e6 −1.60635 −0.803175 0.595743i \(-0.796858\pi\)
−0.803175 + 0.595743i \(0.796858\pi\)
\(504\) 0 0
\(505\) 9.32917e6 1.62785
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.18164e6 0.544323 0.272162 0.962252i \(-0.412261\pi\)
0.272162 + 0.962252i \(0.412261\pi\)
\(510\) 0 0
\(511\) 5.96165e6 1.00998
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.41876e6 0.568003
\(516\) 0 0
\(517\) −4.20994e6 −0.692708
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27154.5 0.00438276 0.00219138 0.999998i \(-0.499302\pi\)
0.00219138 + 0.999998i \(0.499302\pi\)
\(522\) 0 0
\(523\) −3.93041e6 −0.628324 −0.314162 0.949369i \(-0.601724\pi\)
−0.314162 + 0.949369i \(0.601724\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −542387. −0.0850713
\(528\) 0 0
\(529\) 6.75880e6 1.05010
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.75381e6 −0.419871
\(534\) 0 0
\(535\) −1.21734e6 −0.183878
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.32852e7 −1.96968
\(540\) 0 0
\(541\) 8.38180e6 1.23124 0.615622 0.788042i \(-0.288905\pi\)
0.615622 + 0.788042i \(0.288905\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.73325e6 −1.40367
\(546\) 0 0
\(547\) −7.50548e6 −1.07253 −0.536266 0.844049i \(-0.680166\pi\)
−0.536266 + 0.844049i \(0.680166\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.30862e6 0.744908
\(552\) 0 0
\(553\) −1.18604e7 −1.64924
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.03110e6 −0.140820 −0.0704099 0.997518i \(-0.522431\pi\)
−0.0704099 + 0.997518i \(0.522431\pi\)
\(558\) 0 0
\(559\) 9.39988e6 1.27231
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.31962e7 −1.75461 −0.877303 0.479937i \(-0.840659\pi\)
−0.877303 + 0.479937i \(0.840659\pi\)
\(564\) 0 0
\(565\) −3.53110e6 −0.465360
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.49155e6 0.452103 0.226052 0.974115i \(-0.427418\pi\)
0.226052 + 0.974115i \(0.427418\pi\)
\(570\) 0 0
\(571\) −9.49455e6 −1.21866 −0.609332 0.792915i \(-0.708562\pi\)
−0.609332 + 0.792915i \(0.708562\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −711972. −0.0898036
\(576\) 0 0
\(577\) 2.60369e6 0.325574 0.162787 0.986661i \(-0.447952\pi\)
0.162787 + 0.986661i \(0.447952\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.76477e7 2.16895
\(582\) 0 0
\(583\) 9.72905e6 1.18549
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.04099e6 0.603838 0.301919 0.953334i \(-0.402373\pi\)
0.301919 + 0.953334i \(0.402373\pi\)
\(588\) 0 0
\(589\) 1.98787e6 0.236102
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.35192e7 1.57875 0.789374 0.613912i \(-0.210405\pi\)
0.789374 + 0.613912i \(0.210405\pi\)
\(594\) 0 0
\(595\) −1.96543e6 −0.227596
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −815784. −0.0928984 −0.0464492 0.998921i \(-0.514791\pi\)
−0.0464492 + 0.998921i \(0.514791\pi\)
\(600\) 0 0
\(601\) −6.53058e6 −0.737507 −0.368753 0.929527i \(-0.620215\pi\)
−0.368753 + 0.929527i \(0.620215\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.92824e7 2.14177
\(606\) 0 0
\(607\) −1.10096e6 −0.121284 −0.0606418 0.998160i \(-0.519315\pi\)
−0.0606418 + 0.998160i \(0.519315\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.75717e6 0.515520
\(612\) 0 0
\(613\) 6.09992e6 0.655651 0.327825 0.944738i \(-0.393684\pi\)
0.327825 + 0.944738i \(0.393684\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 565825. 0.0598369 0.0299184 0.999552i \(-0.490475\pi\)
0.0299184 + 0.999552i \(0.490475\pi\)
\(618\) 0 0
\(619\) −1.37509e7 −1.44246 −0.721232 0.692693i \(-0.756424\pi\)
−0.721232 + 0.692693i \(0.756424\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.29939e7 −1.34128
\(624\) 0 0
\(625\) −1.03397e7 −1.05879
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −274511. −0.0276652
\(630\) 0 0
\(631\) 420973. 0.0420902 0.0210451 0.999779i \(-0.493301\pi\)
0.0210451 + 0.999779i \(0.493301\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.36047e6 0.133892
\(636\) 0 0
\(637\) 1.50120e7 1.46585
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −314421. −0.0302251 −0.0151125 0.999886i \(-0.504811\pi\)
−0.0151125 + 0.999886i \(0.504811\pi\)
\(642\) 0 0
\(643\) 1.38039e7 1.31667 0.658333 0.752727i \(-0.271262\pi\)
0.658333 + 0.752727i \(0.271262\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.43225e6 0.322343 0.161172 0.986926i \(-0.448473\pi\)
0.161172 + 0.986926i \(0.448473\pi\)
\(648\) 0 0
\(649\) 1.59392e7 1.48544
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.74961e7 1.60568 0.802839 0.596196i \(-0.203322\pi\)
0.802839 + 0.596196i \(0.203322\pi\)
\(654\) 0 0
\(655\) −1.19203e7 −1.08564
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.20687e7 −1.08254 −0.541272 0.840848i \(-0.682057\pi\)
−0.541272 + 0.840848i \(0.682057\pi\)
\(660\) 0 0
\(661\) −181872. −0.0161906 −0.00809529 0.999967i \(-0.502577\pi\)
−0.00809529 + 0.999967i \(0.502577\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.20339e6 0.631659
\(666\) 0 0
\(667\) 2.91395e7 2.53611
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.63335e7 2.25789
\(672\) 0 0
\(673\) −1.06383e7 −0.905388 −0.452694 0.891666i \(-0.649537\pi\)
−0.452694 + 0.891666i \(0.649537\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.67911e6 −0.727786 −0.363893 0.931441i \(-0.618553\pi\)
−0.363893 + 0.931441i \(0.618553\pi\)
\(678\) 0 0
\(679\) −2.22457e7 −1.85170
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.67179e7 1.37129 0.685647 0.727934i \(-0.259519\pi\)
0.685647 + 0.727934i \(0.259519\pi\)
\(684\) 0 0
\(685\) −1.10355e7 −0.898598
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.09937e7 −0.882257
\(690\) 0 0
\(691\) −3.85551e6 −0.307175 −0.153588 0.988135i \(-0.549083\pi\)
−0.153588 + 0.988135i \(0.549083\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.44311e7 1.13328
\(696\) 0 0
\(697\) −625029. −0.0487325
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.86020e6 0.142976 0.0714882 0.997441i \(-0.477225\pi\)
0.0714882 + 0.997441i \(0.477225\pi\)
\(702\) 0 0
\(703\) 1.00610e6 0.0767805
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.05777e7 −2.30068
\(708\) 0 0
\(709\) 1.55608e7 1.16256 0.581281 0.813703i \(-0.302552\pi\)
0.581281 + 0.813703i \(0.302552\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.09116e7 0.803832
\(714\) 0 0
\(715\) −3.22763e7 −2.36112
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.35632e6 0.314266 0.157133 0.987577i \(-0.449775\pi\)
0.157133 + 0.987577i \(0.449775\pi\)
\(720\) 0 0
\(721\) −1.12055e7 −0.802772
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.57229e6 −0.111093
\(726\) 0 0
\(727\) 2.96155e6 0.207818 0.103909 0.994587i \(-0.466865\pi\)
0.103909 + 0.994587i \(0.466865\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.13348e6 0.147671
\(732\) 0 0
\(733\) −960851. −0.0660536 −0.0330268 0.999454i \(-0.510515\pi\)
−0.0330268 + 0.999454i \(0.510515\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.99609e7 −3.38814
\(738\) 0 0
\(739\) 1.36123e7 0.916898 0.458449 0.888721i \(-0.348405\pi\)
0.458449 + 0.888721i \(0.348405\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.73716e6 −0.381263 −0.190632 0.981662i \(-0.561054\pi\)
−0.190632 + 0.981662i \(0.561054\pi\)
\(744\) 0 0
\(745\) 1.15750e7 0.764067
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.99002e6 0.259878
\(750\) 0 0
\(751\) 2.44092e7 1.57926 0.789631 0.613582i \(-0.210272\pi\)
0.789631 + 0.613582i \(0.210272\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.34061e6 −0.0855924
\(756\) 0 0
\(757\) 5.16979e6 0.327894 0.163947 0.986469i \(-0.447577\pi\)
0.163947 + 0.986469i \(0.447577\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.08433e7 −0.678732 −0.339366 0.940654i \(-0.610212\pi\)
−0.339366 + 0.940654i \(0.610212\pi\)
\(762\) 0 0
\(763\) 3.19021e7 1.98385
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.80111e7 −1.10548
\(768\) 0 0
\(769\) −9.27810e6 −0.565774 −0.282887 0.959153i \(-0.591292\pi\)
−0.282887 + 0.959153i \(0.591292\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.33311e7 −0.802450 −0.401225 0.915980i \(-0.631415\pi\)
−0.401225 + 0.915980i \(0.631415\pi\)
\(774\) 0 0
\(775\) −588760. −0.0352115
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.29076e6 0.135249
\(780\) 0 0
\(781\) −4.77096e7 −2.79884
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.01963e7 −1.74896
\(786\) 0 0
\(787\) 6.35818e6 0.365928 0.182964 0.983120i \(-0.441431\pi\)
0.182964 + 0.983120i \(0.441431\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.15737e7 0.657704
\(792\) 0 0
\(793\) −2.97565e7 −1.68035
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.59152e6 −0.256042 −0.128021 0.991771i \(-0.540862\pi\)
−0.128021 + 0.991771i \(0.540862\pi\)
\(798\) 0 0
\(799\) 1.07973e6 0.0598341
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.22207e7 1.21610
\(804\) 0 0
\(805\) 3.95401e7 2.15054
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.91492e6 −0.478901 −0.239451 0.970909i \(-0.576967\pi\)
−0.239451 + 0.970909i \(0.576967\pi\)
\(810\) 0 0
\(811\) −3.29527e6 −0.175930 −0.0879648 0.996124i \(-0.528036\pi\)
−0.0879648 + 0.996124i \(0.528036\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.12104e7 0.591190
\(816\) 0 0
\(817\) −7.81930e6 −0.409838
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.41824e6 0.228766 0.114383 0.993437i \(-0.463511\pi\)
0.114383 + 0.993437i \(0.463511\pi\)
\(822\) 0 0
\(823\) −5.58259e6 −0.287300 −0.143650 0.989629i \(-0.545884\pi\)
−0.143650 + 0.989629i \(0.545884\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.18571e7 −1.11130 −0.555648 0.831418i \(-0.687530\pi\)
−0.555648 + 0.831418i \(0.687530\pi\)
\(828\) 0 0
\(829\) 1.89089e7 0.955608 0.477804 0.878466i \(-0.341433\pi\)
0.477804 + 0.878466i \(0.341433\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.40727e6 0.170135
\(834\) 0 0
\(835\) −2.66678e7 −1.32364
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.05498e7 −0.517413 −0.258707 0.965956i \(-0.583296\pi\)
−0.258707 + 0.965956i \(0.583296\pi\)
\(840\) 0 0
\(841\) 4.38392e7 2.13734
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.50748e7 0.726287
\(846\) 0 0
\(847\) −6.32007e7 −3.02701
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.52256e6 0.261406
\(852\) 0 0
\(853\) −641534. −0.0301889 −0.0150945 0.999886i \(-0.504805\pi\)
−0.0150945 + 0.999886i \(0.504805\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.21711e7 −1.49628 −0.748141 0.663540i \(-0.769053\pi\)
−0.748141 + 0.663540i \(0.769053\pi\)
\(858\) 0 0
\(859\) 1.44739e6 0.0669273 0.0334637 0.999440i \(-0.489346\pi\)
0.0334637 + 0.999440i \(0.489346\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.16338e6 −0.418821 −0.209411 0.977828i \(-0.567155\pi\)
−0.209411 + 0.977828i \(0.567155\pi\)
\(864\) 0 0
\(865\) 2.29889e6 0.104467
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.42069e7 −1.98582
\(870\) 0 0
\(871\) 5.64550e7 2.52149
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.18823e7 1.40776
\(876\) 0 0
\(877\) −2.15236e7 −0.944967 −0.472483 0.881340i \(-0.656642\pi\)
−0.472483 + 0.881340i \(0.656642\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.58447e7 1.98998 0.994991 0.0999625i \(-0.0318723\pi\)
0.994991 + 0.0999625i \(0.0318723\pi\)
\(882\) 0 0
\(883\) −3.10719e7 −1.34111 −0.670557 0.741858i \(-0.733945\pi\)
−0.670557 + 0.741858i \(0.733945\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.08214e7 −0.461820 −0.230910 0.972975i \(-0.574170\pi\)
−0.230910 + 0.972975i \(0.574170\pi\)
\(888\) 0 0
\(889\) −4.45913e6 −0.189232
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.95726e6 −0.166060
\(894\) 0 0
\(895\) −1.27654e7 −0.532694
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.40967e7 0.994394
\(900\) 0 0
\(901\) −2.49523e6 −0.102399
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.61364e7 1.87250
\(906\) 0 0
\(907\) 3.71785e7 1.50063 0.750316 0.661080i \(-0.229901\pi\)
0.750316 + 0.661080i \(0.229901\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.76509e7 1.10386 0.551929 0.833891i \(-0.313892\pi\)
0.551929 + 0.833891i \(0.313892\pi\)
\(912\) 0 0
\(913\) 6.57781e7 2.61159
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.90705e7 1.53436
\(918\) 0 0
\(919\) 2.24727e6 0.0877743 0.0438872 0.999036i \(-0.486026\pi\)
0.0438872 + 0.999036i \(0.486026\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.39112e7 2.08293
\(924\) 0 0
\(925\) −297981. −0.0114508
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −791805. −0.0301009 −0.0150504 0.999887i \(-0.504791\pi\)
−0.0150504 + 0.999887i \(0.504791\pi\)
\(930\) 0 0
\(931\) −1.24878e7 −0.472184
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.32571e6 −0.274044
\(936\) 0 0
\(937\) −9.61458e6 −0.357751 −0.178876 0.983872i \(-0.557246\pi\)
−0.178876 + 0.983872i \(0.557246\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.50396e7 −0.553683 −0.276842 0.960916i \(-0.589288\pi\)
−0.276842 + 0.960916i \(0.589288\pi\)
\(942\) 0 0
\(943\) 1.25742e7 0.460470
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.55153e7 0.562194 0.281097 0.959679i \(-0.409302\pi\)
0.281097 + 0.959679i \(0.409302\pi\)
\(948\) 0 0
\(949\) −2.51091e7 −0.905035
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.16599e7 −1.84256 −0.921280 0.388899i \(-0.872855\pi\)
−0.921280 + 0.388899i \(0.872855\pi\)
\(954\) 0 0
\(955\) −4.06570e6 −0.144254
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.61704e7 1.27001
\(960\) 0 0
\(961\) −1.96059e7 −0.684822
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.64050e7 −0.912782
\(966\) 0 0
\(967\) 6.19595e6 0.213079 0.106540 0.994308i \(-0.466023\pi\)
0.106540 + 0.994308i \(0.466023\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.67480e7 1.59116 0.795582 0.605846i \(-0.207165\pi\)
0.795582 + 0.605846i \(0.207165\pi\)
\(972\) 0 0
\(973\) −4.73001e7 −1.60169
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.63907e6 −0.155487 −0.0777435 0.996973i \(-0.524772\pi\)
−0.0777435 + 0.996973i \(0.524772\pi\)
\(978\) 0 0
\(979\) −4.84321e7 −1.61501
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.75449e7 −0.909197 −0.454598 0.890697i \(-0.650217\pi\)
−0.454598 + 0.890697i \(0.650217\pi\)
\(984\) 0 0
\(985\) 3.49200e7 1.14679
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.29209e7 −1.39533
\(990\) 0 0
\(991\) 1.83035e7 0.592038 0.296019 0.955182i \(-0.404341\pi\)
0.296019 + 0.955182i \(0.404341\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.23125e6 −0.199534
\(996\) 0 0
\(997\) −4.44890e7 −1.41747 −0.708736 0.705473i \(-0.750734\pi\)
−0.708736 + 0.705473i \(0.750734\pi\)
\(998\) 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 108.6.a.b.1.2 2
3.2 odd 2 108.6.a.c.1.1 yes 2
4.3 odd 2 432.6.a.r.1.2 2
9.2 odd 6 324.6.e.f.109.2 4
9.4 even 3 324.6.e.g.217.1 4
9.5 odd 6 324.6.e.f.217.2 4
9.7 even 3 324.6.e.g.109.1 4
12.11 even 2 432.6.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.6.a.b.1.2 2 1.1 even 1 trivial
108.6.a.c.1.1 yes 2 3.2 odd 2
324.6.e.f.109.2 4 9.2 odd 6
324.6.e.f.217.2 4 9.5 odd 6
324.6.e.g.109.1 4 9.7 even 3
324.6.e.g.217.1 4 9.4 even 3
432.6.a.q.1.1 2 12.11 even 2
432.6.a.r.1.2 2 4.3 odd 2