Properties

Label 1296.4.a.i.1.1
Level $1296$
Weight $4$
Character 1296.1
Self dual yes
Analytic conductor $76.466$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1296.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.3723 q^{5} +5.11684 q^{7} +O(q^{10})\) \(q-10.3723 q^{5} +5.11684 q^{7} +55.9783 q^{11} +37.5842 q^{13} -23.6495 q^{17} -39.0516 q^{19} +71.0733 q^{23} -17.4158 q^{25} +28.3723 q^{29} -12.8832 q^{31} -53.0733 q^{35} -180.103 q^{37} +215.484 q^{41} -61.2337 q^{43} -61.8776 q^{47} -316.818 q^{49} -492.310 q^{53} -580.622 q^{55} +789.630 q^{59} +521.090 q^{61} -389.834 q^{65} -304.429 q^{67} +270.391 q^{71} -925.464 q^{73} +286.432 q^{77} +1289.03 q^{79} +713.834 q^{83} +245.299 q^{85} +404.804 q^{89} +192.313 q^{91} +405.054 q^{95} +75.0273 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 15 q^{5} - 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 15 q^{5} - 7 q^{7} + 66 q^{11} - 11 q^{13} - 99 q^{17} + 77 q^{19} + 33 q^{23} - 121 q^{25} + 51 q^{29} - 43 q^{31} + 3 q^{35} - 50 q^{37} - 132 q^{41} - 88 q^{43} + 399 q^{47} - 513 q^{49} - 54 q^{53} - 627 q^{55} + 798 q^{59} + 439 q^{61} - 165 q^{65} - 988 q^{67} + 1368 q^{71} - 455 q^{73} + 165 q^{77} + 803 q^{79} + 813 q^{83} + 594 q^{85} + 396 q^{89} + 781 q^{91} - 132 q^{95} + 736 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −10.3723 −0.927725 −0.463863 0.885907i \(-0.653537\pi\)
−0.463863 + 0.885907i \(0.653537\pi\)
\(6\) 0 0
\(7\) 5.11684 0.276284 0.138142 0.990412i \(-0.455887\pi\)
0.138142 + 0.990412i \(0.455887\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 55.9783 1.53437 0.767185 0.641425i \(-0.221657\pi\)
0.767185 + 0.641425i \(0.221657\pi\)
\(12\) 0 0
\(13\) 37.5842 0.801845 0.400923 0.916112i \(-0.368690\pi\)
0.400923 + 0.916112i \(0.368690\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −23.6495 −0.337402 −0.168701 0.985667i \(-0.553957\pi\)
−0.168701 + 0.985667i \(0.553957\pi\)
\(18\) 0 0
\(19\) −39.0516 −0.471529 −0.235764 0.971810i \(-0.575759\pi\)
−0.235764 + 0.971810i \(0.575759\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 71.0733 0.644340 0.322170 0.946682i \(-0.395588\pi\)
0.322170 + 0.946682i \(0.395588\pi\)
\(24\) 0 0
\(25\) −17.4158 −0.139326
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 28.3723 0.181676 0.0908379 0.995866i \(-0.471045\pi\)
0.0908379 + 0.995866i \(0.471045\pi\)
\(30\) 0 0
\(31\) −12.8832 −0.0746414 −0.0373207 0.999303i \(-0.511882\pi\)
−0.0373207 + 0.999303i \(0.511882\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −53.0733 −0.256315
\(36\) 0 0
\(37\) −180.103 −0.800237 −0.400119 0.916463i \(-0.631031\pi\)
−0.400119 + 0.916463i \(0.631031\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 215.484 0.820802 0.410401 0.911905i \(-0.365389\pi\)
0.410401 + 0.911905i \(0.365389\pi\)
\(42\) 0 0
\(43\) −61.2337 −0.217164 −0.108582 0.994087i \(-0.534631\pi\)
−0.108582 + 0.994087i \(0.534631\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −61.8776 −0.192038 −0.0960189 0.995380i \(-0.530611\pi\)
−0.0960189 + 0.995380i \(0.530611\pi\)
\(48\) 0 0
\(49\) −316.818 −0.923667
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −492.310 −1.27592 −0.637962 0.770068i \(-0.720222\pi\)
−0.637962 + 0.770068i \(0.720222\pi\)
\(54\) 0 0
\(55\) −580.622 −1.42347
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 789.630 1.74239 0.871196 0.490936i \(-0.163345\pi\)
0.871196 + 0.490936i \(0.163345\pi\)
\(60\) 0 0
\(61\) 521.090 1.09375 0.546874 0.837215i \(-0.315818\pi\)
0.546874 + 0.837215i \(0.315818\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −389.834 −0.743892
\(66\) 0 0
\(67\) −304.429 −0.555104 −0.277552 0.960711i \(-0.589523\pi\)
−0.277552 + 0.960711i \(0.589523\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 270.391 0.451966 0.225983 0.974131i \(-0.427441\pi\)
0.225983 + 0.974131i \(0.427441\pi\)
\(72\) 0 0
\(73\) −925.464 −1.48380 −0.741900 0.670510i \(-0.766075\pi\)
−0.741900 + 0.670510i \(0.766075\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 286.432 0.423921
\(78\) 0 0
\(79\) 1289.03 1.83579 0.917897 0.396818i \(-0.129886\pi\)
0.917897 + 0.396818i \(0.129886\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 713.834 0.944018 0.472009 0.881594i \(-0.343529\pi\)
0.472009 + 0.881594i \(0.343529\pi\)
\(84\) 0 0
\(85\) 245.299 0.313017
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 404.804 0.482125 0.241063 0.970510i \(-0.422504\pi\)
0.241063 + 0.970510i \(0.422504\pi\)
\(90\) 0 0
\(91\) 192.313 0.221537
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 405.054 0.437449
\(96\) 0 0
\(97\) 75.0273 0.0785347 0.0392674 0.999229i \(-0.487498\pi\)
0.0392674 + 0.999229i \(0.487498\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1087.88 1.07176 0.535881 0.844294i \(-0.319980\pi\)
0.535881 + 0.844294i \(0.319980\pi\)
\(102\) 0 0
\(103\) 1091.82 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1029.15 −0.929833 −0.464917 0.885354i \(-0.653916\pi\)
−0.464917 + 0.885354i \(0.653916\pi\)
\(108\) 0 0
\(109\) 1776.52 1.56110 0.780548 0.625096i \(-0.214940\pi\)
0.780548 + 0.625096i \(0.214940\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1615.94 1.34526 0.672631 0.739978i \(-0.265164\pi\)
0.672631 + 0.739978i \(0.265164\pi\)
\(114\) 0 0
\(115\) −737.193 −0.597770
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −121.011 −0.0932187
\(120\) 0 0
\(121\) 1802.56 1.35429
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1477.18 1.05698
\(126\) 0 0
\(127\) 1206.10 0.842711 0.421356 0.906895i \(-0.361554\pi\)
0.421356 + 0.906895i \(0.361554\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1027.86 −0.685528 −0.342764 0.939422i \(-0.611363\pi\)
−0.342764 + 0.939422i \(0.611363\pi\)
\(132\) 0 0
\(133\) −199.821 −0.130276
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1260.91 0.786326 0.393163 0.919469i \(-0.371381\pi\)
0.393163 + 0.919469i \(0.371381\pi\)
\(138\) 0 0
\(139\) −461.832 −0.281813 −0.140907 0.990023i \(-0.545002\pi\)
−0.140907 + 0.990023i \(0.545002\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2103.90 1.23033
\(144\) 0 0
\(145\) −294.285 −0.168545
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1459.32 −0.802365 −0.401182 0.915998i \(-0.631401\pi\)
−0.401182 + 0.915998i \(0.631401\pi\)
\(150\) 0 0
\(151\) −1541.32 −0.830666 −0.415333 0.909669i \(-0.636335\pi\)
−0.415333 + 0.909669i \(0.636335\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 133.628 0.0692467
\(156\) 0 0
\(157\) −3215.57 −1.63459 −0.817295 0.576220i \(-0.804527\pi\)
−0.817295 + 0.576220i \(0.804527\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 363.671 0.178021
\(162\) 0 0
\(163\) −947.587 −0.455342 −0.227671 0.973738i \(-0.573111\pi\)
−0.227671 + 0.973738i \(0.573111\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 685.960 0.317851 0.158926 0.987291i \(-0.449197\pi\)
0.158926 + 0.987291i \(0.449197\pi\)
\(168\) 0 0
\(169\) −784.426 −0.357044
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2212.83 0.972475 0.486237 0.873827i \(-0.338369\pi\)
0.486237 + 0.873827i \(0.338369\pi\)
\(174\) 0 0
\(175\) −89.1138 −0.0384936
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3023.22 1.26238 0.631190 0.775629i \(-0.282567\pi\)
0.631190 + 0.775629i \(0.282567\pi\)
\(180\) 0 0
\(181\) 391.445 0.160751 0.0803753 0.996765i \(-0.474388\pi\)
0.0803753 + 0.996765i \(0.474388\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1868.08 0.742400
\(186\) 0 0
\(187\) −1323.86 −0.517700
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3485.59 1.32046 0.660231 0.751062i \(-0.270458\pi\)
0.660231 + 0.751062i \(0.270458\pi\)
\(192\) 0 0
\(193\) 2215.07 0.826136 0.413068 0.910700i \(-0.364457\pi\)
0.413068 + 0.910700i \(0.364457\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3975.11 1.43764 0.718820 0.695196i \(-0.244682\pi\)
0.718820 + 0.695196i \(0.244682\pi\)
\(198\) 0 0
\(199\) 1555.34 0.554046 0.277023 0.960863i \(-0.410652\pi\)
0.277023 + 0.960863i \(0.410652\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 145.177 0.0501941
\(204\) 0 0
\(205\) −2235.06 −0.761479
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2186.04 −0.723500
\(210\) 0 0
\(211\) −1747.73 −0.570231 −0.285115 0.958493i \(-0.592032\pi\)
−0.285115 + 0.958493i \(0.592032\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 635.133 0.201468
\(216\) 0 0
\(217\) −65.9211 −0.0206222
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −888.847 −0.270544
\(222\) 0 0
\(223\) 2541.94 0.763323 0.381662 0.924302i \(-0.375352\pi\)
0.381662 + 0.924302i \(0.375352\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2993.26 −0.875197 −0.437598 0.899171i \(-0.644171\pi\)
−0.437598 + 0.899171i \(0.644171\pi\)
\(228\) 0 0
\(229\) −4305.31 −1.24237 −0.621185 0.783664i \(-0.713348\pi\)
−0.621185 + 0.783664i \(0.713348\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5581.34 1.56930 0.784648 0.619942i \(-0.212844\pi\)
0.784648 + 0.619942i \(0.212844\pi\)
\(234\) 0 0
\(235\) 641.812 0.178158
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1409.63 0.381512 0.190756 0.981638i \(-0.438906\pi\)
0.190756 + 0.981638i \(0.438906\pi\)
\(240\) 0 0
\(241\) 626.572 0.167473 0.0837366 0.996488i \(-0.473315\pi\)
0.0837366 + 0.996488i \(0.473315\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3286.12 0.856909
\(246\) 0 0
\(247\) −1467.72 −0.378093
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1705.53 0.428892 0.214446 0.976736i \(-0.431205\pi\)
0.214446 + 0.976736i \(0.431205\pi\)
\(252\) 0 0
\(253\) 3978.56 0.988656
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3597.38 0.873146 0.436573 0.899669i \(-0.356192\pi\)
0.436573 + 0.899669i \(0.356192\pi\)
\(258\) 0 0
\(259\) −921.560 −0.221092
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4137.50 0.970074 0.485037 0.874494i \(-0.338806\pi\)
0.485037 + 0.874494i \(0.338806\pi\)
\(264\) 0 0
\(265\) 5106.37 1.18371
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6090.99 −1.38057 −0.690287 0.723536i \(-0.742516\pi\)
−0.690287 + 0.723536i \(0.742516\pi\)
\(270\) 0 0
\(271\) 3196.62 0.716534 0.358267 0.933619i \(-0.383368\pi\)
0.358267 + 0.933619i \(0.383368\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −974.905 −0.213778
\(276\) 0 0
\(277\) −3119.36 −0.676622 −0.338311 0.941034i \(-0.609856\pi\)
−0.338311 + 0.941034i \(0.609856\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4948.33 −1.05051 −0.525254 0.850946i \(-0.676030\pi\)
−0.525254 + 0.850946i \(0.676030\pi\)
\(282\) 0 0
\(283\) 4544.93 0.954658 0.477329 0.878725i \(-0.341605\pi\)
0.477329 + 0.878725i \(0.341605\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1102.60 0.226774
\(288\) 0 0
\(289\) −4353.70 −0.886160
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6860.11 −1.36782 −0.683911 0.729566i \(-0.739722\pi\)
−0.683911 + 0.729566i \(0.739722\pi\)
\(294\) 0 0
\(295\) −8190.27 −1.61646
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2671.24 0.516661
\(300\) 0 0
\(301\) −313.323 −0.0599988
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5404.89 −1.01470
\(306\) 0 0
\(307\) −6332.25 −1.17720 −0.588600 0.808424i \(-0.700321\pi\)
−0.588600 + 0.808424i \(0.700321\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7077.67 1.29048 0.645238 0.763982i \(-0.276758\pi\)
0.645238 + 0.763982i \(0.276758\pi\)
\(312\) 0 0
\(313\) 1381.30 0.249443 0.124721 0.992192i \(-0.460196\pi\)
0.124721 + 0.992192i \(0.460196\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8174.93 1.44842 0.724211 0.689578i \(-0.242204\pi\)
0.724211 + 0.689578i \(0.242204\pi\)
\(318\) 0 0
\(319\) 1588.23 0.278758
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 923.549 0.159095
\(324\) 0 0
\(325\) −654.559 −0.111718
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −316.618 −0.0530569
\(330\) 0 0
\(331\) 9661.28 1.60433 0.802163 0.597105i \(-0.203682\pi\)
0.802163 + 0.597105i \(0.203682\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3157.63 0.514984
\(336\) 0 0
\(337\) −4956.02 −0.801103 −0.400552 0.916274i \(-0.631181\pi\)
−0.400552 + 0.916274i \(0.631181\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −721.177 −0.114528
\(342\) 0 0
\(343\) −3376.19 −0.531478
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1015.60 −0.157120 −0.0785598 0.996909i \(-0.525032\pi\)
−0.0785598 + 0.996909i \(0.525032\pi\)
\(348\) 0 0
\(349\) 12158.6 1.86485 0.932426 0.361360i \(-0.117687\pi\)
0.932426 + 0.361360i \(0.117687\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4236.08 −0.638708 −0.319354 0.947635i \(-0.603466\pi\)
−0.319354 + 0.947635i \(0.603466\pi\)
\(354\) 0 0
\(355\) −2804.58 −0.419300
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −517.939 −0.0761443 −0.0380721 0.999275i \(-0.512122\pi\)
−0.0380721 + 0.999275i \(0.512122\pi\)
\(360\) 0 0
\(361\) −5333.97 −0.777660
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9599.18 1.37656
\(366\) 0 0
\(367\) 4616.29 0.656590 0.328295 0.944575i \(-0.393526\pi\)
0.328295 + 0.944575i \(0.393526\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2519.07 −0.352517
\(372\) 0 0
\(373\) 4765.42 0.661512 0.330756 0.943716i \(-0.392696\pi\)
0.330756 + 0.943716i \(0.392696\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1066.35 0.145676
\(378\) 0 0
\(379\) 2000.33 0.271108 0.135554 0.990770i \(-0.456719\pi\)
0.135554 + 0.990770i \(0.456719\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 990.294 0.132119 0.0660596 0.997816i \(-0.478957\pi\)
0.0660596 + 0.997816i \(0.478957\pi\)
\(384\) 0 0
\(385\) −2970.95 −0.393283
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −404.411 −0.0527106 −0.0263553 0.999653i \(-0.508390\pi\)
−0.0263553 + 0.999653i \(0.508390\pi\)
\(390\) 0 0
\(391\) −1680.85 −0.217402
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13370.2 −1.70311
\(396\) 0 0
\(397\) 2919.61 0.369096 0.184548 0.982824i \(-0.440918\pi\)
0.184548 + 0.982824i \(0.440918\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10186.2 1.26852 0.634258 0.773121i \(-0.281306\pi\)
0.634258 + 0.773121i \(0.281306\pi\)
\(402\) 0 0
\(403\) −484.203 −0.0598508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10081.9 −1.22786
\(408\) 0 0
\(409\) 6914.24 0.835910 0.417955 0.908468i \(-0.362747\pi\)
0.417955 + 0.908468i \(0.362747\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4040.41 0.481394
\(414\) 0 0
\(415\) −7404.09 −0.875789
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5120.31 0.597002 0.298501 0.954409i \(-0.403513\pi\)
0.298501 + 0.954409i \(0.403513\pi\)
\(420\) 0 0
\(421\) 1866.49 0.216074 0.108037 0.994147i \(-0.465543\pi\)
0.108037 + 0.994147i \(0.465543\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 411.874 0.0470090
\(426\) 0 0
\(427\) 2666.33 0.302185
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4090.64 −0.457168 −0.228584 0.973524i \(-0.573410\pi\)
−0.228584 + 0.973524i \(0.573410\pi\)
\(432\) 0 0
\(433\) 633.052 0.0702599 0.0351299 0.999383i \(-0.488815\pi\)
0.0351299 + 0.999383i \(0.488815\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2775.53 −0.303825
\(438\) 0 0
\(439\) 11306.5 1.22923 0.614614 0.788828i \(-0.289312\pi\)
0.614614 + 0.788828i \(0.289312\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8281.30 −0.888163 −0.444082 0.895986i \(-0.646470\pi\)
−0.444082 + 0.895986i \(0.646470\pi\)
\(444\) 0 0
\(445\) −4198.74 −0.447280
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6888.40 −0.724017 −0.362008 0.932175i \(-0.617909\pi\)
−0.362008 + 0.932175i \(0.617909\pi\)
\(450\) 0 0
\(451\) 12062.4 1.25941
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1994.72 −0.205525
\(456\) 0 0
\(457\) 4283.60 0.438465 0.219233 0.975673i \(-0.429645\pi\)
0.219233 + 0.975673i \(0.429645\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −13778.3 −1.39202 −0.696009 0.718033i \(-0.745042\pi\)
−0.696009 + 0.718033i \(0.745042\pi\)
\(462\) 0 0
\(463\) −5734.53 −0.575608 −0.287804 0.957689i \(-0.592925\pi\)
−0.287804 + 0.957689i \(0.592925\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8950.97 −0.886941 −0.443470 0.896289i \(-0.646253\pi\)
−0.443470 + 0.896289i \(0.646253\pi\)
\(468\) 0 0
\(469\) −1557.72 −0.153366
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3427.75 −0.333210
\(474\) 0 0
\(475\) 680.114 0.0656964
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9681.01 0.923459 0.461729 0.887021i \(-0.347229\pi\)
0.461729 + 0.887021i \(0.347229\pi\)
\(480\) 0 0
\(481\) −6769.04 −0.641666
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −778.204 −0.0728586
\(486\) 0 0
\(487\) −8704.66 −0.809950 −0.404975 0.914328i \(-0.632720\pi\)
−0.404975 + 0.914328i \(0.632720\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15595.7 −1.43345 −0.716725 0.697356i \(-0.754360\pi\)
−0.716725 + 0.697356i \(0.754360\pi\)
\(492\) 0 0
\(493\) −670.989 −0.0612979
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1383.55 0.124871
\(498\) 0 0
\(499\) 9696.28 0.869870 0.434935 0.900462i \(-0.356771\pi\)
0.434935 + 0.900462i \(0.356771\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20949.7 1.85706 0.928532 0.371253i \(-0.121072\pi\)
0.928532 + 0.371253i \(0.121072\pi\)
\(504\) 0 0
\(505\) −11283.8 −0.994300
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11274.7 −0.981816 −0.490908 0.871211i \(-0.663335\pi\)
−0.490908 + 0.871211i \(0.663335\pi\)
\(510\) 0 0
\(511\) −4735.46 −0.409950
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11324.6 −0.968977
\(516\) 0 0
\(517\) −3463.80 −0.294657
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −8675.49 −0.729520 −0.364760 0.931102i \(-0.618849\pi\)
−0.364760 + 0.931102i \(0.618849\pi\)
\(522\) 0 0
\(523\) 4226.14 0.353339 0.176670 0.984270i \(-0.443468\pi\)
0.176670 + 0.984270i \(0.443468\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 304.680 0.0251842
\(528\) 0 0
\(529\) −7115.58 −0.584826
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8098.78 0.658156
\(534\) 0 0
\(535\) 10674.7 0.862630
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17734.9 −1.41725
\(540\) 0 0
\(541\) 13357.8 1.06154 0.530771 0.847515i \(-0.321902\pi\)
0.530771 + 0.847515i \(0.321902\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −18426.5 −1.44827
\(546\) 0 0
\(547\) −21671.1 −1.69395 −0.846974 0.531634i \(-0.821578\pi\)
−0.846974 + 0.531634i \(0.821578\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1107.98 −0.0856654
\(552\) 0 0
\(553\) 6595.79 0.507200
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7477.63 −0.568828 −0.284414 0.958702i \(-0.591799\pi\)
−0.284414 + 0.958702i \(0.591799\pi\)
\(558\) 0 0
\(559\) −2301.42 −0.174132
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23304.7 −1.74454 −0.872269 0.489026i \(-0.837352\pi\)
−0.872269 + 0.489026i \(0.837352\pi\)
\(564\) 0 0
\(565\) −16761.0 −1.24803
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14649.1 1.07930 0.539650 0.841890i \(-0.318557\pi\)
0.539650 + 0.841890i \(0.318557\pi\)
\(570\) 0 0
\(571\) −23164.0 −1.69769 −0.848846 0.528640i \(-0.822702\pi\)
−0.848846 + 0.528640i \(0.822702\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1237.80 −0.0897735
\(576\) 0 0
\(577\) 7865.97 0.567529 0.283765 0.958894i \(-0.408417\pi\)
0.283765 + 0.958894i \(0.408417\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3652.58 0.260817
\(582\) 0 0
\(583\) −27558.6 −1.95774
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −956.182 −0.0672332 −0.0336166 0.999435i \(-0.510703\pi\)
−0.0336166 + 0.999435i \(0.510703\pi\)
\(588\) 0 0
\(589\) 503.108 0.0351956
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16966.0 1.17489 0.587444 0.809265i \(-0.300134\pi\)
0.587444 + 0.809265i \(0.300134\pi\)
\(594\) 0 0
\(595\) 1255.16 0.0864813
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6191.41 −0.422327 −0.211164 0.977451i \(-0.567725\pi\)
−0.211164 + 0.977451i \(0.567725\pi\)
\(600\) 0 0
\(601\) −2718.54 −0.184512 −0.0922559 0.995735i \(-0.529408\pi\)
−0.0922559 + 0.995735i \(0.529408\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −18696.7 −1.25641
\(606\) 0 0
\(607\) −16825.0 −1.12505 −0.562524 0.826781i \(-0.690170\pi\)
−0.562524 + 0.826781i \(0.690170\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2325.62 −0.153985
\(612\) 0 0
\(613\) −20175.1 −1.32930 −0.664652 0.747153i \(-0.731420\pi\)
−0.664652 + 0.747153i \(0.731420\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11310.6 −0.738004 −0.369002 0.929429i \(-0.620301\pi\)
−0.369002 + 0.929429i \(0.620301\pi\)
\(618\) 0 0
\(619\) 17059.9 1.10775 0.553873 0.832601i \(-0.313149\pi\)
0.553873 + 0.832601i \(0.313149\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2071.32 0.133203
\(624\) 0 0
\(625\) −13144.7 −0.841262
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4259.34 0.270002
\(630\) 0 0
\(631\) 13186.3 0.831916 0.415958 0.909384i \(-0.363446\pi\)
0.415958 + 0.909384i \(0.363446\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12510.0 −0.781805
\(636\) 0 0
\(637\) −11907.4 −0.740638
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −16362.0 −1.00820 −0.504102 0.863644i \(-0.668177\pi\)
−0.504102 + 0.863644i \(0.668177\pi\)
\(642\) 0 0
\(643\) 28044.9 1.72004 0.860019 0.510262i \(-0.170452\pi\)
0.860019 + 0.510262i \(0.170452\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21247.7 1.29109 0.645543 0.763724i \(-0.276631\pi\)
0.645543 + 0.763724i \(0.276631\pi\)
\(648\) 0 0
\(649\) 44202.1 2.67347
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1259.86 −0.0755007 −0.0377504 0.999287i \(-0.512019\pi\)
−0.0377504 + 0.999287i \(0.512019\pi\)
\(654\) 0 0
\(655\) 10661.2 0.635981
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12046.7 −0.712098 −0.356049 0.934467i \(-0.615876\pi\)
−0.356049 + 0.934467i \(0.615876\pi\)
\(660\) 0 0
\(661\) 13108.1 0.771324 0.385662 0.922640i \(-0.373973\pi\)
0.385662 + 0.922640i \(0.373973\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2072.60 0.120860
\(666\) 0 0
\(667\) 2016.51 0.117061
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 29169.7 1.67822
\(672\) 0 0
\(673\) 2743.65 0.157147 0.0785734 0.996908i \(-0.474963\pi\)
0.0785734 + 0.996908i \(0.474963\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25004.0 −1.41947 −0.709735 0.704468i \(-0.751186\pi\)
−0.709735 + 0.704468i \(0.751186\pi\)
\(678\) 0 0
\(679\) 383.903 0.0216979
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4846.23 −0.271502 −0.135751 0.990743i \(-0.543345\pi\)
−0.135751 + 0.990743i \(0.543345\pi\)
\(684\) 0 0
\(685\) −13078.5 −0.729494
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −18503.1 −1.02309
\(690\) 0 0
\(691\) −3484.58 −0.191837 −0.0959187 0.995389i \(-0.530579\pi\)
−0.0959187 + 0.995389i \(0.530579\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4790.25 0.261445
\(696\) 0 0
\(697\) −5096.07 −0.276940
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15701.4 −0.845981 −0.422991 0.906134i \(-0.639020\pi\)
−0.422991 + 0.906134i \(0.639020\pi\)
\(702\) 0 0
\(703\) 7033.32 0.377335
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5566.50 0.296110
\(708\) 0 0
\(709\) 15643.4 0.828634 0.414317 0.910133i \(-0.364020\pi\)
0.414317 + 0.910133i \(0.364020\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −915.649 −0.0480944
\(714\) 0 0
\(715\) −21822.2 −1.14141
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6964.13 −0.361222 −0.180611 0.983555i \(-0.557807\pi\)
−0.180611 + 0.983555i \(0.557807\pi\)
\(720\) 0 0
\(721\) 5586.66 0.288569
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −494.125 −0.0253122
\(726\) 0 0
\(727\) 14207.2 0.724782 0.362391 0.932026i \(-0.381961\pi\)
0.362391 + 0.932026i \(0.381961\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1448.14 0.0732716
\(732\) 0 0
\(733\) 26530.5 1.33687 0.668437 0.743769i \(-0.266964\pi\)
0.668437 + 0.743769i \(0.266964\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −17041.4 −0.851735
\(738\) 0 0
\(739\) 5683.47 0.282909 0.141455 0.989945i \(-0.454822\pi\)
0.141455 + 0.989945i \(0.454822\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15568.6 −0.768715 −0.384358 0.923184i \(-0.625577\pi\)
−0.384358 + 0.923184i \(0.625577\pi\)
\(744\) 0 0
\(745\) 15136.5 0.744374
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5266.02 −0.256898
\(750\) 0 0
\(751\) 8261.64 0.401427 0.200713 0.979650i \(-0.435674\pi\)
0.200713 + 0.979650i \(0.435674\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15987.0 0.770630
\(756\) 0 0
\(757\) −13381.5 −0.642481 −0.321240 0.946998i \(-0.604100\pi\)
−0.321240 + 0.946998i \(0.604100\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5449.84 −0.259601 −0.129801 0.991540i \(-0.541434\pi\)
−0.129801 + 0.991540i \(0.541434\pi\)
\(762\) 0 0
\(763\) 9090.15 0.431305
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29677.6 1.39713
\(768\) 0 0
\(769\) −19364.0 −0.908039 −0.454020 0.890992i \(-0.650010\pi\)
−0.454020 + 0.890992i \(0.650010\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1865.54 0.0868033 0.0434017 0.999058i \(-0.486180\pi\)
0.0434017 + 0.999058i \(0.486180\pi\)
\(774\) 0 0
\(775\) 224.370 0.0103995
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8414.98 −0.387032
\(780\) 0 0
\(781\) 15136.0 0.693483
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 33352.8 1.51645
\(786\) 0 0
\(787\) 19207.3 0.869970 0.434985 0.900438i \(-0.356754\pi\)
0.434985 + 0.900438i \(0.356754\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8268.50 0.371674
\(792\) 0 0
\(793\) 19584.7 0.877017
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −186.074 −0.00826988 −0.00413494 0.999991i \(-0.501316\pi\)
−0.00413494 + 0.999991i \(0.501316\pi\)
\(798\) 0 0
\(799\) 1463.37 0.0647940
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −51805.9 −2.27670
\(804\) 0 0
\(805\) −3772.10 −0.165154
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5903.09 −0.256541 −0.128270 0.991739i \(-0.540943\pi\)
−0.128270 + 0.991739i \(0.540943\pi\)
\(810\) 0 0
\(811\) −23111.0 −1.00066 −0.500331 0.865834i \(-0.666788\pi\)
−0.500331 + 0.865834i \(0.666788\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9828.64 0.422432
\(816\) 0 0
\(817\) 2391.27 0.102399
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9644.29 0.409973 0.204987 0.978765i \(-0.434285\pi\)
0.204987 + 0.978765i \(0.434285\pi\)
\(822\) 0 0
\(823\) 33573.4 1.42199 0.710994 0.703198i \(-0.248245\pi\)
0.710994 + 0.703198i \(0.248245\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25916.1 −1.08971 −0.544855 0.838530i \(-0.683415\pi\)
−0.544855 + 0.838530i \(0.683415\pi\)
\(828\) 0 0
\(829\) −28650.6 −1.20033 −0.600166 0.799876i \(-0.704899\pi\)
−0.600166 + 0.799876i \(0.704899\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7492.58 0.311647
\(834\) 0 0
\(835\) −7114.97 −0.294878
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 712.960 0.0293374 0.0146687 0.999892i \(-0.495331\pi\)
0.0146687 + 0.999892i \(0.495331\pi\)
\(840\) 0 0
\(841\) −23584.0 −0.966994
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8136.29 0.331239
\(846\) 0 0
\(847\) 9223.44 0.374169
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −12800.5 −0.515625
\(852\) 0 0
\(853\) −30367.2 −1.21894 −0.609469 0.792810i \(-0.708617\pi\)
−0.609469 + 0.792810i \(0.708617\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9080.70 −0.361950 −0.180975 0.983488i \(-0.557925\pi\)
−0.180975 + 0.983488i \(0.557925\pi\)
\(858\) 0 0
\(859\) −26160.2 −1.03909 −0.519543 0.854444i \(-0.673898\pi\)
−0.519543 + 0.854444i \(0.673898\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −40102.0 −1.58180 −0.790898 0.611949i \(-0.790386\pi\)
−0.790898 + 0.611949i \(0.790386\pi\)
\(864\) 0 0
\(865\) −22952.1 −0.902189
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 72157.9 2.81679
\(870\) 0 0
\(871\) −11441.7 −0.445108
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 7558.48 0.292027
\(876\) 0 0
\(877\) 25252.2 0.972299 0.486149 0.873876i \(-0.338401\pi\)
0.486149 + 0.873876i \(0.338401\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2049.26 0.0783670 0.0391835 0.999232i \(-0.487524\pi\)
0.0391835 + 0.999232i \(0.487524\pi\)
\(882\) 0 0
\(883\) −39413.4 −1.50211 −0.751057 0.660237i \(-0.770456\pi\)
−0.751057 + 0.660237i \(0.770456\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36968.5 1.39941 0.699707 0.714430i \(-0.253314\pi\)
0.699707 + 0.714430i \(0.253314\pi\)
\(888\) 0 0
\(889\) 6171.44 0.232827
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2416.42 0.0905514
\(894\) 0 0
\(895\) −31357.7 −1.17114
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −365.525 −0.0135605
\(900\) 0 0
\(901\) 11642.9 0.430499
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4060.17 −0.149132
\(906\) 0 0
\(907\) −2710.62 −0.0992334 −0.0496167 0.998768i \(-0.515800\pi\)
−0.0496167 + 0.998768i \(0.515800\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 22996.6 0.836345 0.418172 0.908368i \(-0.362671\pi\)
0.418172 + 0.908368i \(0.362671\pi\)
\(912\) 0 0
\(913\) 39959.2 1.44847
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5259.38 −0.189400
\(918\) 0 0
\(919\) 39103.8 1.40361 0.701804 0.712370i \(-0.252378\pi\)
0.701804 + 0.712370i \(0.252378\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10162.5 0.362407
\(924\) 0 0
\(925\) 3136.64 0.111494
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35954.6 1.26979 0.634894 0.772600i \(-0.281044\pi\)
0.634894 + 0.772600i \(0.281044\pi\)
\(930\) 0 0
\(931\) 12372.2 0.435536
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 13731.4 0.480283
\(936\) 0 0
\(937\) −7263.94 −0.253258 −0.126629 0.991950i \(-0.540416\pi\)
−0.126629 + 0.991950i \(0.540416\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7478.91 −0.259092 −0.129546 0.991573i \(-0.541352\pi\)
−0.129546 + 0.991573i \(0.541352\pi\)
\(942\) 0 0
\(943\) 15315.1 0.528875
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13491.4 0.462947 0.231473 0.972841i \(-0.425645\pi\)
0.231473 + 0.972841i \(0.425645\pi\)
\(948\) 0 0
\(949\) −34782.9 −1.18978
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13981.6 0.475246 0.237623 0.971357i \(-0.423632\pi\)
0.237623 + 0.971357i \(0.423632\pi\)
\(954\) 0 0
\(955\) −36153.5 −1.22503
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6451.87 0.217249
\(960\) 0 0
\(961\) −29625.0 −0.994429
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −22975.3 −0.766427
\(966\) 0 0
\(967\) 9081.47 0.302007 0.151003 0.988533i \(-0.451750\pi\)
0.151003 + 0.988533i \(0.451750\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −9709.13 −0.320887 −0.160443 0.987045i \(-0.551292\pi\)
−0.160443 + 0.987045i \(0.551292\pi\)
\(972\) 0 0
\(973\) −2363.12 −0.0778604
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10854.9 −0.355455 −0.177727 0.984080i \(-0.556875\pi\)
−0.177727 + 0.984080i \(0.556875\pi\)
\(978\) 0 0
\(979\) 22660.2 0.739759
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7510.10 −0.243678 −0.121839 0.992550i \(-0.538879\pi\)
−0.121839 + 0.992550i \(0.538879\pi\)
\(984\) 0 0
\(985\) −41231.0 −1.33373
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4352.08 −0.139927
\(990\) 0 0
\(991\) −46125.6 −1.47854 −0.739268 0.673412i \(-0.764828\pi\)
−0.739268 + 0.673412i \(0.764828\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16132.4 −0.514003
\(996\) 0 0
\(997\) 45350.1 1.44057 0.720287 0.693677i \(-0.244010\pi\)
0.720287 + 0.693677i \(0.244010\pi\)
\(998\) 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 1296.4.a.i.1.1 2
3.2 odd 2 1296.4.a.u.1.2 2
4.3 odd 2 81.4.a.a.1.2 2
9.2 odd 6 144.4.i.c.49.2 4
9.4 even 3 432.4.i.c.289.2 4
9.5 odd 6 144.4.i.c.97.2 4
9.7 even 3 432.4.i.c.145.2 4
12.11 even 2 81.4.a.d.1.1 2
20.19 odd 2 2025.4.a.n.1.1 2
36.7 odd 6 27.4.c.a.10.1 4
36.11 even 6 9.4.c.a.4.2 4
36.23 even 6 9.4.c.a.7.2 yes 4
36.31 odd 6 27.4.c.a.19.1 4
60.59 even 2 2025.4.a.g.1.2 2
180.23 odd 12 225.4.k.b.124.2 8
180.47 odd 12 225.4.k.b.49.2 8
180.59 even 6 225.4.e.b.151.1 4
180.83 odd 12 225.4.k.b.49.3 8
180.119 even 6 225.4.e.b.76.1 4
180.167 odd 12 225.4.k.b.124.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.4.c.a.4.2 4 36.11 even 6
9.4.c.a.7.2 yes 4 36.23 even 6
27.4.c.a.10.1 4 36.7 odd 6
27.4.c.a.19.1 4 36.31 odd 6
81.4.a.a.1.2 2 4.3 odd 2
81.4.a.d.1.1 2 12.11 even 2
144.4.i.c.49.2 4 9.2 odd 6
144.4.i.c.97.2 4 9.5 odd 6
225.4.e.b.76.1 4 180.119 even 6
225.4.e.b.151.1 4 180.59 even 6
225.4.k.b.49.2 8 180.47 odd 12
225.4.k.b.49.3 8 180.83 odd 12
225.4.k.b.124.2 8 180.23 odd 12
225.4.k.b.124.3 8 180.167 odd 12
432.4.i.c.145.2 4 9.7 even 3
432.4.i.c.289.2 4 9.4 even 3
1296.4.a.i.1.1 2 1.1 even 1 trivial
1296.4.a.u.1.2 2 3.2 odd 2
2025.4.a.g.1.2 2 60.59 even 2
2025.4.a.n.1.1 2 20.19 odd 2