Properties

Label 1134.2.a.h.1.1
Level $1134$
Weight $2$
Character 1134.1
Self dual yes
Analytic conductor $9.055$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1134.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} +6.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} +3.00000 q^{20} +6.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +3.00000 q^{35} +2.00000 q^{37} -7.00000 q^{38} +3.00000 q^{40} +2.00000 q^{43} +6.00000 q^{44} -3.00000 q^{46} +1.00000 q^{49} +4.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +18.0000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +5.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} +3.00000 q^{70} -3.00000 q^{71} +2.00000 q^{73} +2.00000 q^{74} -7.00000 q^{76} +6.00000 q^{77} +5.00000 q^{79} +3.00000 q^{80} -12.0000 q^{83} -18.0000 q^{85} +2.00000 q^{86} +6.00000 q^{88} +2.00000 q^{91} -3.00000 q^{92} -21.0000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −7.00000 −1.13555
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 18.0000 2.42712
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −18.0000 −1.95237
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) 0 0
\(95\) −21.0000 −2.15455
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 18.0000 1.71623
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 5.00000 0.452679
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) 9.00000 0.786334 0.393167 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) −3.00000 −0.251754
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 23.0000 1.87171 0.935857 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) −7.00000 −0.567775
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 5.00000 0.397779
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −18.0000 −1.38054
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −36.0000 −2.63258
\(188\) 0 0
\(189\) 0 0
\(190\) −21.0000 −1.52350
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 0 0
\(193\) 17.0000 1.22369 0.611843 0.790979i \(-0.290428\pi\)
0.611843 + 0.790979i \(0.290428\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −9.00000 −0.633238
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −42.0000 −2.90520
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 18.0000 1.21356
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 25.0000 1.60706
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −14.0000 −0.890799
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 17.0000 1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 9.00000 0.556022
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) −7.00000 −0.429198
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 24.0000 1.44725
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 5.00000 0.299880
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 0 0
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) −3.00000 −0.178017
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −18.0000 −1.05700
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 23.0000 1.32350
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) 15.0000 0.858898
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 6.00000 0.341882
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −13.0000 −0.733632
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) 3.00000 0.167705
\(321\) 0 0
\(322\) −3.00000 −0.167183
\(323\) 42.0000 2.33694
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) 2.00000 0.110770
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −18.0000 −0.976187
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) 0 0
\(358\) −18.0000 −0.951330
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −25.0000 −1.31397
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) −21.0000 −1.07728
\(381\) 0 0
\(382\) 9.00000 0.460480
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 18.0000 0.917365
\(386\) 17.0000 0.865277
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 15.0000 0.754732
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 14.0000 0.701757
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −10.0000 −0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) −36.0000 −1.76717
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −42.0000 −2.05429
\(419\) −15.0000 −0.732798 −0.366399 0.930458i \(-0.619409\pi\)
−0.366399 + 0.930458i \(0.619409\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 21.0000 1.00457
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 18.0000 0.858116
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) 0 0
\(454\) 15.0000 0.703985
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) −1.00000 −0.0467269
\(459\) 0 0
\(460\) −9.00000 −0.419627
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −9.00000 −0.416917
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 15.0000 0.686084
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 29.0000 1.31412 0.657058 0.753840i \(-0.271801\pi\)
0.657058 + 0.753840i \(0.271801\pi\)
\(488\) 5.00000 0.226339
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) −14.0000 −0.629890
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −3.00000 −0.134568
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −3.00000 −0.134164
\(501\) 0 0
\(502\) −3.00000 −0.133897
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −27.0000 −1.20148
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) 17.0000 0.754253
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −30.0000 −1.32196
\(516\) 0 0
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) 9.00000 0.393167
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −18.0000 −0.781870
\(531\) 0 0
\(532\) −7.00000 −0.303488
\(533\) 0 0
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 9.00000 0.388018
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −28.0000 −1.20270
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −30.0000 −1.28506
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 24.0000 1.02336
\(551\) 42.0000 1.78926
\(552\) 0 0
\(553\) 5.00000 0.212622
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) 33.0000 1.39078 0.695392 0.718631i \(-0.255231\pi\)
0.695392 + 0.718631i \(0.255231\pi\)
\(564\) 0 0
\(565\) −45.0000 −1.89316
\(566\) −19.0000 −0.798630
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −18.0000 −0.747409
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 3.00000 0.123929
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) −18.0000 −0.737928
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 2.00000 0.0815139
\(603\) 0 0
\(604\) 23.0000 0.935857
\(605\) 75.0000 3.04918
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −7.00000 −0.283887
\(609\) 0 0
\(610\) 15.0000 0.607332
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −25.0000 −1.00892
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −7.00000 −0.281354 −0.140677 0.990056i \(-0.544928\pi\)
−0.140677 + 0.990056i \(0.544928\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 5.00000 0.198889
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 51.0000 2.02387
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 42.0000 1.65247
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 27.0000 1.05498
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 26.0000 1.01052
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −21.0000 −0.814345
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) −37.0000 −1.42625 −0.713123 0.701039i \(-0.752720\pi\)
−0.713123 + 0.701039i \(0.752720\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) −18.0000 −0.690268
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 47.0000 1.78796 0.893982 0.448103i \(-0.147900\pi\)
0.893982 + 0.448103i \(0.147900\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 15.0000 0.568982
\(696\) 0 0
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −9.00000 −0.338480
\(708\) 0 0
\(709\) −52.0000 −1.95290 −0.976450 0.215742i \(-0.930783\pi\)
−0.976450 + 0.215742i \(0.930783\pi\)
\(710\) −9.00000 −0.337764
\(711\) 0 0
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) −18.0000 −0.672692
\(717\) 0 0
\(718\) 3.00000 0.111959
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) −25.0000 −0.929118
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 29.0000 1.07114 0.535570 0.844491i \(-0.320097\pi\)
0.535570 + 0.844491i \(0.320097\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) −36.0000 −1.31629
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 69.0000 2.51117
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 2.00000 0.0726433
\(759\) 0 0
\(760\) −21.0000 −0.761750
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) 18.0000 0.650366
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 18.0000 0.648675
\(771\) 0 0
\(772\) 17.0000 0.611843
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 18.0000 0.643679
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −39.0000 −1.39197
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 15.0000 0.533676
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −3.00000 −0.105934
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) −9.00000 −0.316619
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) −14.0000 −0.489798
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −36.0000 −1.24958
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) −42.0000 −1.45260
\(837\) 0 0
\(838\) −15.0000 −0.518166
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −27.0000 −0.928828
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −24.0000 −0.823193
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 35.0000 1.19838 0.599189 0.800608i \(-0.295490\pi\)
0.599189 + 0.800608i \(0.295490\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) −9.00000 −0.306364 −0.153182 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) 21.0000 0.710336
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 18.0000 0.606780
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −18.0000 −0.604722
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 17.0000 0.570162
\(890\) 0 0
\(891\) 0 0
\(892\) −28.0000 −0.937509
\(893\) 0 0
\(894\) 0 0
\(895\) −54.0000 −1.80502
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −33.0000 −1.10122
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) −15.0000 −0.498893
\(905\) −75.0000 −2.49308
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 15.0000 0.497792
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) 29.0000 0.959235
\(915\) 0 0
\(916\) −1.00000 −0.0330409
\(917\) 9.00000 0.297206
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) −9.00000 −0.296721
\(921\) 0 0
\(922\) 33.0000 1.08680
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −13.0000 −0.427207
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) −9.00000 −0.294805
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) −108.000 −3.53198
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) −21.0000 −0.684580 −0.342290 0.939594i \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −28.0000 −0.908440
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 27.0000 0.873699
\(956\) 15.0000 0.485135
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) 8.00000 0.257663
\(965\) 51.0000 1.64175
\(966\) 0 0
\(967\) 17.0000 0.546683 0.273342 0.961917i \(-0.411871\pi\)
0.273342 + 0.961917i \(0.411871\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) 29.0000 0.929220
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) −18.0000 −0.574403
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) −54.0000 −1.72058
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) −14.0000 −0.445399
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) −3.00000 −0.0951542
\(995\) 42.0000 1.33149
\(996\) 0 0
\(997\) −55.0000 −1.74187 −0.870934 0.491400i \(-0.836485\pi\)
−0.870934 + 0.491400i \(0.836485\pi\)
\(998\) 32.0000 1.01294
\(999\) 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 1134.2.a.h.1.1 1
3.2 odd 2 1134.2.a.a.1.1 1
4.3 odd 2 9072.2.a.w.1.1 1
7.6 odd 2 7938.2.a.u.1.1 1
9.2 odd 6 126.2.f.a.85.1 yes 2
9.4 even 3 378.2.f.a.127.1 2
9.5 odd 6 126.2.f.a.43.1 2
9.7 even 3 378.2.f.a.253.1 2
12.11 even 2 9072.2.a.c.1.1 1
21.20 even 2 7938.2.a.l.1.1 1
36.7 odd 6 3024.2.r.a.1009.1 2
36.11 even 6 1008.2.r.d.337.1 2
36.23 even 6 1008.2.r.d.673.1 2
36.31 odd 6 3024.2.r.a.2017.1 2
63.2 odd 6 882.2.h.j.67.1 2
63.4 even 3 2646.2.h.e.667.1 2
63.5 even 6 882.2.e.d.655.1 2
63.11 odd 6 882.2.e.b.373.1 2
63.13 odd 6 2646.2.f.c.883.1 2
63.16 even 3 2646.2.h.e.361.1 2
63.20 even 6 882.2.f.h.589.1 2
63.23 odd 6 882.2.e.b.655.1 2
63.25 even 3 2646.2.e.f.1549.1 2
63.31 odd 6 2646.2.h.a.667.1 2
63.32 odd 6 882.2.h.j.79.1 2
63.34 odd 6 2646.2.f.c.1765.1 2
63.38 even 6 882.2.e.d.373.1 2
63.40 odd 6 2646.2.e.j.2125.1 2
63.41 even 6 882.2.f.h.295.1 2
63.47 even 6 882.2.h.f.67.1 2
63.52 odd 6 2646.2.e.j.1549.1 2
63.58 even 3 2646.2.e.f.2125.1 2
63.59 even 6 882.2.h.f.79.1 2
63.61 odd 6 2646.2.h.a.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.a.43.1 2 9.5 odd 6
126.2.f.a.85.1 yes 2 9.2 odd 6
378.2.f.a.127.1 2 9.4 even 3
378.2.f.a.253.1 2 9.7 even 3
882.2.e.b.373.1 2 63.11 odd 6
882.2.e.b.655.1 2 63.23 odd 6
882.2.e.d.373.1 2 63.38 even 6
882.2.e.d.655.1 2 63.5 even 6
882.2.f.h.295.1 2 63.41 even 6
882.2.f.h.589.1 2 63.20 even 6
882.2.h.f.67.1 2 63.47 even 6
882.2.h.f.79.1 2 63.59 even 6
882.2.h.j.67.1 2 63.2 odd 6
882.2.h.j.79.1 2 63.32 odd 6
1008.2.r.d.337.1 2 36.11 even 6
1008.2.r.d.673.1 2 36.23 even 6
1134.2.a.a.1.1 1 3.2 odd 2
1134.2.a.h.1.1 1 1.1 even 1 trivial
2646.2.e.f.1549.1 2 63.25 even 3
2646.2.e.f.2125.1 2 63.58 even 3
2646.2.e.j.1549.1 2 63.52 odd 6
2646.2.e.j.2125.1 2 63.40 odd 6
2646.2.f.c.883.1 2 63.13 odd 6
2646.2.f.c.1765.1 2 63.34 odd 6
2646.2.h.a.361.1 2 63.61 odd 6
2646.2.h.a.667.1 2 63.31 odd 6
2646.2.h.e.361.1 2 63.16 even 3
2646.2.h.e.667.1 2 63.4 even 3
3024.2.r.a.1009.1 2 36.7 odd 6
3024.2.r.a.2017.1 2 36.31 odd 6
7938.2.a.l.1.1 1 21.20 even 2
7938.2.a.u.1.1 1 7.6 odd 2
9072.2.a.c.1.1 1 12.11 even 2
9072.2.a.w.1.1 1 4.3 odd 2