Properties

Label 7800.2.a.bt.1.1
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.30056.1
Defining polynomial: \(x^{4} - 11 x^{2} + 26\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.74983\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.92778 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.92778 q^{7} +1.00000 q^{9} -1.38360 q^{11} +1.00000 q^{13} +0.195329 q^{17} +4.92778 q^{21} -2.19533 q^{23} -1.00000 q^{27} +7.49966 q^{29} +1.38360 q^{33} +6.05088 q^{37} -1.00000 q^{39} -2.11605 q^{41} +3.49966 q^{43} +2.31138 q^{47} +17.2830 q^{49} -0.195329 q^{51} -6.05088 q^{53} +9.43449 q^{59} -3.69498 q^{61} -4.92778 q^{63} -12.6228 q^{67} +2.19533 q^{69} +13.9716 q^{71} +4.73245 q^{73} +6.81809 q^{77} -8.07153 q^{79} +1.00000 q^{81} -13.3997 q^{83} -7.49966 q^{87} +3.02770 q^{89} -4.92778 q^{91} +6.01612 q^{97} -1.38360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 7 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} - 7 q^{7} + 4 q^{9} + q^{11} + 4 q^{13} - 3 q^{17} + 7 q^{21} - 5 q^{23} - 4 q^{27} + 8 q^{29} - q^{33} - 5 q^{37} - 4 q^{39} + 7 q^{41} - 8 q^{43} - 10 q^{47} + 9 q^{49} + 3 q^{51} + 5 q^{53} + 2 q^{59} + 11 q^{61} - 7 q^{63} - 12 q^{67} + 5 q^{69} + 15 q^{71} + 10 q^{73} - 15 q^{77} - q^{79} + 4 q^{81} - 22 q^{83} - 8 q^{87} + 9 q^{89} - 7 q^{91} - q^{97} + q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.92778 −1.86252 −0.931262 0.364349i \(-0.881291\pi\)
−0.931262 + 0.364349i \(0.881291\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.38360 −0.417172 −0.208586 0.978004i \(-0.566886\pi\)
−0.208586 + 0.978004i \(0.566886\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.195329 0.0473741 0.0236871 0.999719i \(-0.492459\pi\)
0.0236871 + 0.999719i \(0.492459\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.92778 1.07533
\(22\) 0 0
\(23\) −2.19533 −0.457758 −0.228879 0.973455i \(-0.573506\pi\)
−0.228879 + 0.973455i \(0.573506\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.49966 1.39265 0.696326 0.717726i \(-0.254817\pi\)
0.696326 + 0.717726i \(0.254817\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 1.38360 0.240854
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.05088 0.994759 0.497379 0.867533i \(-0.334296\pi\)
0.497379 + 0.867533i \(0.334296\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.11605 −0.330472 −0.165236 0.986254i \(-0.552839\pi\)
−0.165236 + 0.986254i \(0.552839\pi\)
\(42\) 0 0
\(43\) 3.49966 0.533692 0.266846 0.963739i \(-0.414018\pi\)
0.266846 + 0.963739i \(0.414018\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.31138 0.337150 0.168575 0.985689i \(-0.446084\pi\)
0.168575 + 0.985689i \(0.446084\pi\)
\(48\) 0 0
\(49\) 17.2830 2.46900
\(50\) 0 0
\(51\) −0.195329 −0.0273515
\(52\) 0 0
\(53\) −6.05088 −0.831153 −0.415576 0.909558i \(-0.636420\pi\)
−0.415576 + 0.909558i \(0.636420\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.43449 1.22827 0.614133 0.789203i \(-0.289506\pi\)
0.614133 + 0.789203i \(0.289506\pi\)
\(60\) 0 0
\(61\) −3.69498 −0.473094 −0.236547 0.971620i \(-0.576016\pi\)
−0.236547 + 0.971620i \(0.576016\pi\)
\(62\) 0 0
\(63\) −4.92778 −0.620842
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.6228 −1.54212 −0.771058 0.636765i \(-0.780272\pi\)
−0.771058 + 0.636765i \(0.780272\pi\)
\(68\) 0 0
\(69\) 2.19533 0.264287
\(70\) 0 0
\(71\) 13.9716 1.65812 0.829062 0.559156i \(-0.188875\pi\)
0.829062 + 0.559156i \(0.188875\pi\)
\(72\) 0 0
\(73\) 4.73245 0.553891 0.276946 0.960886i \(-0.410678\pi\)
0.276946 + 0.960886i \(0.410678\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.81809 0.776993
\(78\) 0 0
\(79\) −8.07153 −0.908119 −0.454059 0.890971i \(-0.650025\pi\)
−0.454059 + 0.890971i \(0.650025\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.3997 −1.47081 −0.735406 0.677627i \(-0.763008\pi\)
−0.735406 + 0.677627i \(0.763008\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.49966 −0.804047
\(88\) 0 0
\(89\) 3.02770 0.320936 0.160468 0.987041i \(-0.448700\pi\)
0.160468 + 0.987041i \(0.448700\pi\)
\(90\) 0 0
\(91\) −4.92778 −0.516571
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.01612 0.610845 0.305422 0.952217i \(-0.401202\pi\)
0.305422 + 0.952217i \(0.401202\pi\)
\(98\) 0 0
\(99\) −1.38360 −0.139057
\(100\) 0 0
\(101\) −3.10900 −0.309357 −0.154678 0.987965i \(-0.549434\pi\)
−0.154678 + 0.987965i \(0.549434\pi\)
\(102\) 0 0
\(103\) 16.8690 1.66215 0.831075 0.556161i \(-0.187726\pi\)
0.831075 + 0.556161i \(0.187726\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.1946 −1.66227 −0.831134 0.556072i \(-0.812308\pi\)
−0.831134 + 0.556072i \(0.812308\pi\)
\(108\) 0 0
\(109\) 8.76721 0.839746 0.419873 0.907583i \(-0.362075\pi\)
0.419873 + 0.907583i \(0.362075\pi\)
\(110\) 0 0
\(111\) −6.05088 −0.574324
\(112\) 0 0
\(113\) 7.46490 0.702238 0.351119 0.936331i \(-0.385801\pi\)
0.351119 + 0.936331i \(0.385801\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −0.962536 −0.0882355
\(120\) 0 0
\(121\) −9.08564 −0.825967
\(122\) 0 0
\(123\) 2.11605 0.190798
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.1231 −0.987016 −0.493508 0.869741i \(-0.664286\pi\)
−0.493508 + 0.869741i \(0.664286\pi\)
\(128\) 0 0
\(129\) −3.49966 −0.308127
\(130\) 0 0
\(131\) −1.82080 −0.159084 −0.0795418 0.996832i \(-0.525346\pi\)
−0.0795418 + 0.996832i \(0.525346\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.70204 −0.743465 −0.371733 0.928340i \(-0.621236\pi\)
−0.371733 + 0.928340i \(0.621236\pi\)
\(138\) 0 0
\(139\) 6.81809 0.578303 0.289151 0.957283i \(-0.406627\pi\)
0.289151 + 0.957283i \(0.406627\pi\)
\(140\) 0 0
\(141\) −2.31138 −0.194653
\(142\) 0 0
\(143\) −1.38360 −0.115703
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −17.2830 −1.42548
\(148\) 0 0
\(149\) −7.48537 −0.613225 −0.306613 0.951834i \(-0.599196\pi\)
−0.306613 + 0.951834i \(0.599196\pi\)
\(150\) 0 0
\(151\) 12.8549 1.04611 0.523057 0.852298i \(-0.324791\pi\)
0.523057 + 0.852298i \(0.324791\pi\)
\(152\) 0 0
\(153\) 0.195329 0.0157914
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.7318 −1.41515 −0.707574 0.706639i \(-0.750210\pi\)
−0.707574 + 0.706639i \(0.750210\pi\)
\(158\) 0 0
\(159\) 6.05088 0.479866
\(160\) 0 0
\(161\) 10.8181 0.852585
\(162\) 0 0
\(163\) 18.1385 1.42072 0.710360 0.703838i \(-0.248532\pi\)
0.710360 + 0.703838i \(0.248532\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.06517 −0.314572 −0.157286 0.987553i \(-0.550274\pi\)
−0.157286 + 0.987553i \(0.550274\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −23.2455 −1.76732 −0.883662 0.468125i \(-0.844930\pi\)
−0.883662 + 0.468125i \(0.844930\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −9.43449 −0.709139
\(178\) 0 0
\(179\) 14.1224 1.05556 0.527779 0.849381i \(-0.323025\pi\)
0.527779 + 0.849381i \(0.323025\pi\)
\(180\) 0 0
\(181\) 3.67433 0.273111 0.136556 0.990632i \(-0.456397\pi\)
0.136556 + 0.990632i \(0.456397\pi\)
\(182\) 0 0
\(183\) 3.69498 0.273141
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.270257 −0.0197632
\(188\) 0 0
\(189\) 4.92778 0.358443
\(190\) 0 0
\(191\) −16.8549 −1.21958 −0.609788 0.792565i \(-0.708745\pi\)
−0.609788 + 0.792565i \(0.708745\pi\)
\(192\) 0 0
\(193\) 6.53712 0.470552 0.235276 0.971929i \(-0.424401\pi\)
0.235276 + 0.971929i \(0.424401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.7890 1.83739 0.918695 0.394967i \(-0.129244\pi\)
0.918695 + 0.394967i \(0.129244\pi\)
\(198\) 0 0
\(199\) −16.9646 −1.20259 −0.601293 0.799029i \(-0.705347\pi\)
−0.601293 + 0.799029i \(0.705347\pi\)
\(200\) 0 0
\(201\) 12.6228 0.890341
\(202\) 0 0
\(203\) −36.9566 −2.59385
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.19533 −0.152586
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −26.4577 −1.82142 −0.910710 0.413046i \(-0.864465\pi\)
−0.910710 + 0.413046i \(0.864465\pi\)
\(212\) 0 0
\(213\) −13.9716 −0.957319
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.73245 −0.319789
\(220\) 0 0
\(221\) 0.195329 0.0131392
\(222\) 0 0
\(223\) −2.02065 −0.135313 −0.0676564 0.997709i \(-0.521552\pi\)
−0.0676564 + 0.997709i \(0.521552\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.7904 −1.18079 −0.590395 0.807115i \(-0.701028\pi\)
−0.590395 + 0.807115i \(0.701028\pi\)
\(228\) 0 0
\(229\) −18.3339 −1.21154 −0.605768 0.795641i \(-0.707134\pi\)
−0.605768 + 0.795641i \(0.707134\pi\)
\(230\) 0 0
\(231\) −6.81809 −0.448597
\(232\) 0 0
\(233\) −23.0361 −1.50914 −0.754572 0.656217i \(-0.772156\pi\)
−0.754572 + 0.656217i \(0.772156\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.07153 0.524302
\(238\) 0 0
\(239\) −0.986402 −0.0638051 −0.0319025 0.999491i \(-0.510157\pi\)
−0.0319025 + 0.999491i \(0.510157\pi\)
\(240\) 0 0
\(241\) 1.52031 0.0979316 0.0489658 0.998800i \(-0.484407\pi\)
0.0489658 + 0.998800i \(0.484407\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 13.3997 0.849173
\(250\) 0 0
\(251\) 11.6763 0.737005 0.368502 0.929627i \(-0.379871\pi\)
0.368502 + 0.929627i \(0.379871\pi\)
\(252\) 0 0
\(253\) 3.03746 0.190964
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.00069 −0.187178 −0.0935889 0.995611i \(-0.529834\pi\)
−0.0935889 + 0.995611i \(0.529834\pi\)
\(258\) 0 0
\(259\) −29.8174 −1.85276
\(260\) 0 0
\(261\) 7.49966 0.464217
\(262\) 0 0
\(263\) 25.8683 1.59511 0.797553 0.603248i \(-0.206127\pi\)
0.797553 + 0.603248i \(0.206127\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.02770 −0.185293
\(268\) 0 0
\(269\) 11.1090 0.677328 0.338664 0.940907i \(-0.390025\pi\)
0.338664 + 0.940907i \(0.390025\pi\)
\(270\) 0 0
\(271\) −13.9859 −0.849582 −0.424791 0.905291i \(-0.639653\pi\)
−0.424791 + 0.905291i \(0.639653\pi\)
\(272\) 0 0
\(273\) 4.92778 0.298243
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0529 0.604020 0.302010 0.953305i \(-0.402342\pi\)
0.302010 + 0.953305i \(0.402342\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −19.6318 −1.17114 −0.585568 0.810623i \(-0.699129\pi\)
−0.585568 + 0.810623i \(0.699129\pi\)
\(282\) 0 0
\(283\) −25.5667 −1.51978 −0.759890 0.650051i \(-0.774747\pi\)
−0.759890 + 0.650051i \(0.774747\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.4274 0.615512
\(288\) 0 0
\(289\) −16.9618 −0.997756
\(290\) 0 0
\(291\) −6.01612 −0.352671
\(292\) 0 0
\(293\) 6.71614 0.392361 0.196181 0.980568i \(-0.437146\pi\)
0.196181 + 0.980568i \(0.437146\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.38360 0.0802848
\(298\) 0 0
\(299\) −2.19533 −0.126959
\(300\) 0 0
\(301\) −17.2455 −0.994015
\(302\) 0 0
\(303\) 3.10900 0.178607
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 14.2689 0.814368 0.407184 0.913346i \(-0.366511\pi\)
0.407184 + 0.913346i \(0.366511\pi\)
\(308\) 0 0
\(309\) −16.8690 −0.959642
\(310\) 0 0
\(311\) −7.15786 −0.405885 −0.202943 0.979191i \(-0.565050\pi\)
−0.202943 + 0.979191i \(0.565050\pi\)
\(312\) 0 0
\(313\) 21.6568 1.22412 0.612058 0.790813i \(-0.290342\pi\)
0.612058 + 0.790813i \(0.290342\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.0232 −1.63010 −0.815052 0.579388i \(-0.803292\pi\)
−0.815052 + 0.579388i \(0.803292\pi\)
\(318\) 0 0
\(319\) −10.3765 −0.580975
\(320\) 0 0
\(321\) 17.1946 0.959711
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −8.76721 −0.484828
\(328\) 0 0
\(329\) −11.3900 −0.627949
\(330\) 0 0
\(331\) 21.8683 1.20199 0.600995 0.799253i \(-0.294771\pi\)
0.600995 + 0.799253i \(0.294771\pi\)
\(332\) 0 0
\(333\) 6.05088 0.331586
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.2991 −1.21471 −0.607355 0.794431i \(-0.707769\pi\)
−0.607355 + 0.794431i \(0.707769\pi\)
\(338\) 0 0
\(339\) −7.46490 −0.405438
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −50.6723 −2.73605
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.9199 1.23040 0.615201 0.788370i \(-0.289075\pi\)
0.615201 + 0.788370i \(0.289075\pi\)
\(348\) 0 0
\(349\) 6.75310 0.361485 0.180743 0.983530i \(-0.442150\pi\)
0.180743 + 0.983530i \(0.442150\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 3.07859 0.163857 0.0819283 0.996638i \(-0.473892\pi\)
0.0819283 + 0.996638i \(0.473892\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.962536 0.0509428
\(358\) 0 0
\(359\) −13.0091 −0.686592 −0.343296 0.939227i \(-0.611543\pi\)
−0.343296 + 0.939227i \(0.611543\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 9.08564 0.476872
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −27.4041 −1.43048 −0.715241 0.698878i \(-0.753683\pi\)
−0.715241 + 0.698878i \(0.753683\pi\)
\(368\) 0 0
\(369\) −2.11605 −0.110157
\(370\) 0 0
\(371\) 29.8174 1.54804
\(372\) 0 0
\(373\) 19.9573 1.03335 0.516675 0.856181i \(-0.327169\pi\)
0.516675 + 0.856181i \(0.327169\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.49966 0.386252
\(378\) 0 0
\(379\) −29.2042 −1.50012 −0.750060 0.661370i \(-0.769975\pi\)
−0.750060 + 0.661370i \(0.769975\pi\)
\(380\) 0 0
\(381\) 11.1231 0.569854
\(382\) 0 0
\(383\) 15.5764 0.795918 0.397959 0.917403i \(-0.369719\pi\)
0.397959 + 0.917403i \(0.369719\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.49966 0.177897
\(388\) 0 0
\(389\) 20.0529 1.01672 0.508361 0.861144i \(-0.330251\pi\)
0.508361 + 0.861144i \(0.330251\pi\)
\(390\) 0 0
\(391\) −0.428810 −0.0216859
\(392\) 0 0
\(393\) 1.82080 0.0918470
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −30.9057 −1.55112 −0.775558 0.631277i \(-0.782531\pi\)
−0.775558 + 0.631277i \(0.782531\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.9902 −0.598764 −0.299382 0.954133i \(-0.596780\pi\)
−0.299382 + 0.954133i \(0.596780\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.37202 −0.414986
\(408\) 0 0
\(409\) 39.2169 1.93915 0.969577 0.244788i \(-0.0787183\pi\)
0.969577 + 0.244788i \(0.0787183\pi\)
\(410\) 0 0
\(411\) 8.70204 0.429240
\(412\) 0 0
\(413\) −46.4910 −2.28767
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.81809 −0.333883
\(418\) 0 0
\(419\) −17.8621 −0.872621 −0.436310 0.899796i \(-0.643715\pi\)
−0.436310 + 0.899796i \(0.643715\pi\)
\(420\) 0 0
\(421\) 28.8690 1.40699 0.703494 0.710701i \(-0.251622\pi\)
0.703494 + 0.710701i \(0.251622\pi\)
\(422\) 0 0
\(423\) 2.31138 0.112383
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.2081 0.881150
\(428\) 0 0
\(429\) 1.38360 0.0668010
\(430\) 0 0
\(431\) 17.0373 0.820657 0.410329 0.911938i \(-0.365414\pi\)
0.410329 + 0.911938i \(0.365414\pi\)
\(432\) 0 0
\(433\) −29.1554 −1.40112 −0.700558 0.713595i \(-0.747066\pi\)
−0.700558 + 0.713595i \(0.747066\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −23.2087 −1.10769 −0.553847 0.832619i \(-0.686841\pi\)
−0.553847 + 0.832619i \(0.686841\pi\)
\(440\) 0 0
\(441\) 17.2830 0.822999
\(442\) 0 0
\(443\) −14.9484 −0.710221 −0.355111 0.934824i \(-0.615557\pi\)
−0.355111 + 0.934824i \(0.615557\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7.48537 0.354046
\(448\) 0 0
\(449\) −42.2023 −1.99165 −0.995826 0.0912763i \(-0.970905\pi\)
−0.995826 + 0.0912763i \(0.970905\pi\)
\(450\) 0 0
\(451\) 2.92778 0.137864
\(452\) 0 0
\(453\) −12.8549 −0.603974
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.1599 −0.709149 −0.354575 0.935028i \(-0.615374\pi\)
−0.354575 + 0.935028i \(0.615374\pi\)
\(458\) 0 0
\(459\) −0.195329 −0.00911716
\(460\) 0 0
\(461\) −0.848501 −0.0395186 −0.0197593 0.999805i \(-0.506290\pi\)
−0.0197593 + 0.999805i \(0.506290\pi\)
\(462\) 0 0
\(463\) 13.8109 0.641845 0.320922 0.947105i \(-0.396007\pi\)
0.320922 + 0.947105i \(0.396007\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.5165 0.486644 0.243322 0.969946i \(-0.421763\pi\)
0.243322 + 0.969946i \(0.421763\pi\)
\(468\) 0 0
\(469\) 62.2022 2.87223
\(470\) 0 0
\(471\) 17.7318 0.817036
\(472\) 0 0
\(473\) −4.84214 −0.222642
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.05088 −0.277051
\(478\) 0 0
\(479\) −6.41905 −0.293294 −0.146647 0.989189i \(-0.546848\pi\)
−0.146647 + 0.989189i \(0.546848\pi\)
\(480\) 0 0
\(481\) 6.05088 0.275897
\(482\) 0 0
\(483\) −10.8181 −0.492240
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.5492 1.06711 0.533557 0.845764i \(-0.320855\pi\)
0.533557 + 0.845764i \(0.320855\pi\)
\(488\) 0 0
\(489\) −18.1385 −0.820253
\(490\) 0 0
\(491\) 3.99346 0.180222 0.0901111 0.995932i \(-0.471278\pi\)
0.0901111 + 0.995932i \(0.471278\pi\)
\(492\) 0 0
\(493\) 1.46490 0.0659756
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −68.8490 −3.08830
\(498\) 0 0
\(499\) −5.30231 −0.237364 −0.118682 0.992932i \(-0.537867\pi\)
−0.118682 + 0.992932i \(0.537867\pi\)
\(500\) 0 0
\(501\) 4.06517 0.181618
\(502\) 0 0
\(503\) 34.8408 1.55347 0.776736 0.629826i \(-0.216874\pi\)
0.776736 + 0.629826i \(0.216874\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 42.2526 1.87281 0.936406 0.350918i \(-0.114130\pi\)
0.936406 + 0.350918i \(0.114130\pi\)
\(510\) 0 0
\(511\) −23.3205 −1.03164
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.19803 −0.140649
\(518\) 0 0
\(519\) 23.2455 1.02037
\(520\) 0 0
\(521\) −2.65098 −0.116141 −0.0580707 0.998312i \(-0.518495\pi\)
−0.0580707 + 0.998312i \(0.518495\pi\)
\(522\) 0 0
\(523\) −31.6014 −1.38183 −0.690917 0.722934i \(-0.742793\pi\)
−0.690917 + 0.722934i \(0.742793\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −18.1805 −0.790458
\(530\) 0 0
\(531\) 9.43449 0.409422
\(532\) 0 0
\(533\) −2.11605 −0.0916564
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −14.1224 −0.609427
\(538\) 0 0
\(539\) −23.9128 −1.03000
\(540\) 0 0
\(541\) 1.54852 0.0665761 0.0332881 0.999446i \(-0.489402\pi\)
0.0332881 + 0.999446i \(0.489402\pi\)
\(542\) 0 0
\(543\) −3.67433 −0.157681
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −33.2376 −1.42114 −0.710569 0.703628i \(-0.751562\pi\)
−0.710569 + 0.703628i \(0.751562\pi\)
\(548\) 0 0
\(549\) −3.69498 −0.157698
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 39.7747 1.69139
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.64594 0.0697407 0.0348703 0.999392i \(-0.488898\pi\)
0.0348703 + 0.999392i \(0.488898\pi\)
\(558\) 0 0
\(559\) 3.49966 0.148020
\(560\) 0 0
\(561\) 0.270257 0.0114103
\(562\) 0 0
\(563\) −18.7486 −0.790158 −0.395079 0.918647i \(-0.629283\pi\)
−0.395079 + 0.918647i \(0.629283\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −4.92778 −0.206947
\(568\) 0 0
\(569\) −41.6543 −1.74624 −0.873120 0.487505i \(-0.837907\pi\)
−0.873120 + 0.487505i \(0.837907\pi\)
\(570\) 0 0
\(571\) −27.3184 −1.14324 −0.571620 0.820518i \(-0.693685\pi\)
−0.571620 + 0.820518i \(0.693685\pi\)
\(572\) 0 0
\(573\) 16.8549 0.704122
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −24.0715 −1.00211 −0.501056 0.865415i \(-0.667055\pi\)
−0.501056 + 0.865415i \(0.667055\pi\)
\(578\) 0 0
\(579\) −6.53712 −0.271673
\(580\) 0 0
\(581\) 66.0309 2.73942
\(582\) 0 0
\(583\) 8.37202 0.346734
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.6872 0.977677 0.488839 0.872374i \(-0.337421\pi\)
0.488839 + 0.872374i \(0.337421\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −25.7890 −1.06082
\(592\) 0 0
\(593\) 38.8632 1.59592 0.797961 0.602709i \(-0.205912\pi\)
0.797961 + 0.602709i \(0.205912\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.9646 0.694313
\(598\) 0 0
\(599\) 18.4783 0.755003 0.377502 0.926009i \(-0.376783\pi\)
0.377502 + 0.926009i \(0.376783\pi\)
\(600\) 0 0
\(601\) −2.08702 −0.0851313 −0.0425656 0.999094i \(-0.513553\pi\)
−0.0425656 + 0.999094i \(0.513553\pi\)
\(602\) 0 0
\(603\) −12.6228 −0.514039
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.59120 −0.145762 −0.0728812 0.997341i \(-0.523219\pi\)
−0.0728812 + 0.997341i \(0.523219\pi\)
\(608\) 0 0
\(609\) 36.9566 1.49756
\(610\) 0 0
\(611\) 2.31138 0.0935084
\(612\) 0 0
\(613\) −24.7895 −1.00124 −0.500620 0.865667i \(-0.666894\pi\)
−0.500620 + 0.865667i \(0.666894\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.7574 −0.594112 −0.297056 0.954860i \(-0.596005\pi\)
−0.297056 + 0.954860i \(0.596005\pi\)
\(618\) 0 0
\(619\) 25.4081 1.02124 0.510619 0.859807i \(-0.329416\pi\)
0.510619 + 0.859807i \(0.329416\pi\)
\(620\) 0 0
\(621\) 2.19533 0.0880955
\(622\) 0 0
\(623\) −14.9199 −0.597751
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.18191 0.0471258
\(630\) 0 0
\(631\) 9.46490 0.376792 0.188396 0.982093i \(-0.439671\pi\)
0.188396 + 0.982093i \(0.439671\pi\)
\(632\) 0 0
\(633\) 26.4577 1.05160
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.2830 0.684777
\(638\) 0 0
\(639\) 13.9716 0.552708
\(640\) 0 0
\(641\) 7.89686 0.311907 0.155954 0.987764i \(-0.450155\pi\)
0.155954 + 0.987764i \(0.450155\pi\)
\(642\) 0 0
\(643\) 16.1204 0.635727 0.317863 0.948137i \(-0.397035\pi\)
0.317863 + 0.948137i \(0.397035\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.99616 0.117791 0.0588956 0.998264i \(-0.481242\pi\)
0.0588956 + 0.998264i \(0.481242\pi\)
\(648\) 0 0
\(649\) −13.0536 −0.512398
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.5330 0.842653 0.421326 0.906909i \(-0.361565\pi\)
0.421326 + 0.906909i \(0.361565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.73245 0.184630
\(658\) 0 0
\(659\) 17.4429 0.679478 0.339739 0.940520i \(-0.389661\pi\)
0.339739 + 0.940520i \(0.389661\pi\)
\(660\) 0 0
\(661\) −22.4642 −0.873756 −0.436878 0.899521i \(-0.643916\pi\)
−0.436878 + 0.899521i \(0.643916\pi\)
\(662\) 0 0
\(663\) −0.195329 −0.00758593
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16.4642 −0.637497
\(668\) 0 0
\(669\) 2.02065 0.0781229
\(670\) 0 0
\(671\) 5.11239 0.197362
\(672\) 0 0
\(673\) −46.8122 −1.80448 −0.902239 0.431237i \(-0.858077\pi\)
−0.902239 + 0.431237i \(0.858077\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −31.2682 −1.20173 −0.600867 0.799349i \(-0.705178\pi\)
−0.600867 + 0.799349i \(0.705178\pi\)
\(678\) 0 0
\(679\) −29.6461 −1.13771
\(680\) 0 0
\(681\) 17.7904 0.681729
\(682\) 0 0
\(683\) 44.1797 1.69049 0.845244 0.534381i \(-0.179455\pi\)
0.845244 + 0.534381i \(0.179455\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.3339 0.699481
\(688\) 0 0
\(689\) −6.05088 −0.230520
\(690\) 0 0
\(691\) −25.8360 −0.982849 −0.491425 0.870920i \(-0.663524\pi\)
−0.491425 + 0.870920i \(0.663524\pi\)
\(692\) 0 0
\(693\) 6.81809 0.258998
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.413325 −0.0156558
\(698\) 0 0
\(699\) 23.0361 0.871305
\(700\) 0 0
\(701\) 15.5887 0.588777 0.294388 0.955686i \(-0.404884\pi\)
0.294388 + 0.955686i \(0.404884\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.3205 0.576185
\(708\) 0 0
\(709\) −4.05541 −0.152304 −0.0761521 0.997096i \(-0.524263\pi\)
−0.0761521 + 0.997096i \(0.524263\pi\)
\(710\) 0 0
\(711\) −8.07153 −0.302706
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.986402 0.0368379
\(718\) 0 0
\(719\) 13.4081 0.500038 0.250019 0.968241i \(-0.419563\pi\)
0.250019 + 0.968241i \(0.419563\pi\)
\(720\) 0 0
\(721\) −83.1265 −3.09579
\(722\) 0 0
\(723\) −1.52031 −0.0565408
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −12.9787 −0.481352 −0.240676 0.970606i \(-0.577369\pi\)
−0.240676 + 0.970606i \(0.577369\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.683583 0.0252832
\(732\) 0 0
\(733\) −49.2410 −1.81876 −0.909379 0.415969i \(-0.863443\pi\)
−0.909379 + 0.415969i \(0.863443\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.4649 0.643328
\(738\) 0 0
\(739\) −11.3359 −0.416999 −0.208500 0.978022i \(-0.566858\pi\)
−0.208500 + 0.978022i \(0.566858\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.07790 −0.296349 −0.148175 0.988961i \(-0.547340\pi\)
−0.148175 + 0.988961i \(0.547340\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.3997 −0.490270
\(748\) 0 0
\(749\) 84.7314 3.09602
\(750\) 0 0
\(751\) 18.2403 0.665598 0.332799 0.942998i \(-0.392007\pi\)
0.332799 + 0.942998i \(0.392007\pi\)
\(752\) 0 0
\(753\) −11.6763 −0.425510
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.71111 0.207574 0.103787 0.994600i \(-0.466904\pi\)
0.103787 + 0.994600i \(0.466904\pi\)
\(758\) 0 0
\(759\) −3.03746 −0.110253
\(760\) 0 0
\(761\) 12.6543 0.458718 0.229359 0.973342i \(-0.426337\pi\)
0.229359 + 0.973342i \(0.426337\pi\)
\(762\) 0 0
\(763\) −43.2028 −1.56405
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.43449 0.340660
\(768\) 0 0
\(769\) −21.4222 −0.772505 −0.386252 0.922393i \(-0.626231\pi\)
−0.386252 + 0.922393i \(0.626231\pi\)
\(770\) 0 0
\(771\) 3.00069 0.108067
\(772\) 0 0
\(773\) 36.0547 1.29680 0.648399 0.761300i \(-0.275439\pi\)
0.648399 + 0.761300i \(0.275439\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 29.8174 1.06969
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −19.3312 −0.691723
\(782\) 0 0
\(783\) −7.49966 −0.268016
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −40.8394 −1.45577 −0.727883 0.685701i \(-0.759496\pi\)
−0.727883 + 0.685701i \(0.759496\pi\)
\(788\) 0 0
\(789\) −25.8683 −0.920935
\(790\) 0 0
\(791\) −36.7853 −1.30794
\(792\) 0 0
\(793\) −3.69498 −0.131213
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.4691 −0.902161 −0.451080 0.892483i \(-0.648961\pi\)
−0.451080 + 0.892483i \(0.648961\pi\)
\(798\) 0 0
\(799\) 0.451479 0.0159722
\(800\) 0 0
\(801\) 3.02770 0.106979
\(802\) 0 0
\(803\) −6.54783 −0.231068
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −11.1090 −0.391055
\(808\) 0 0
\(809\) 15.7252 0.552869 0.276435 0.961033i \(-0.410847\pi\)
0.276435 + 0.961033i \(0.410847\pi\)
\(810\) 0 0
\(811\) −49.6815 −1.74455 −0.872277 0.489012i \(-0.837357\pi\)
−0.872277 + 0.489012i \(0.837357\pi\)
\(812\) 0 0
\(813\) 13.9859 0.490507
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −4.92778 −0.172190
\(820\) 0 0
\(821\) 23.9936 0.837384 0.418692 0.908128i \(-0.362489\pi\)
0.418692 + 0.908128i \(0.362489\pi\)
\(822\) 0 0
\(823\) −23.0457 −0.803321 −0.401661 0.915789i \(-0.631567\pi\)
−0.401661 + 0.915789i \(0.631567\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 49.1195 1.70805 0.854027 0.520229i \(-0.174153\pi\)
0.854027 + 0.520229i \(0.174153\pi\)
\(828\) 0 0
\(829\) −5.54739 −0.192669 −0.0963344 0.995349i \(-0.530712\pi\)
−0.0963344 + 0.995349i \(0.530712\pi\)
\(830\) 0 0
\(831\) −10.0529 −0.348731
\(832\) 0 0
\(833\) 3.37586 0.116967
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.7106 −0.991200 −0.495600 0.868551i \(-0.665052\pi\)
−0.495600 + 0.868551i \(0.665052\pi\)
\(840\) 0 0
\(841\) 27.2448 0.939477
\(842\) 0 0
\(843\) 19.6318 0.676156
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 44.7720 1.53838
\(848\) 0 0
\(849\) 25.5667 0.877446
\(850\) 0 0
\(851\) −13.2837 −0.455359
\(852\) 0 0
\(853\) −27.7606 −0.950505 −0.475253 0.879849i \(-0.657643\pi\)
−0.475253 + 0.879849i \(0.657643\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.8869 −1.22587 −0.612937 0.790132i \(-0.710012\pi\)
−0.612937 + 0.790132i \(0.710012\pi\)
\(858\) 0 0
\(859\) −26.6107 −0.907944 −0.453972 0.891016i \(-0.649993\pi\)
−0.453972 + 0.891016i \(0.649993\pi\)
\(860\) 0 0
\(861\) −10.4274 −0.355366
\(862\) 0 0
\(863\) −20.0366 −0.682054 −0.341027 0.940054i \(-0.610775\pi\)
−0.341027 + 0.940054i \(0.610775\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.9618 0.576055
\(868\) 0 0
\(869\) 11.1678 0.378842
\(870\) 0 0
\(871\) −12.6228 −0.427706
\(872\) 0 0
\(873\) 6.01612 0.203615
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.15855 0.0728892 0.0364446 0.999336i \(-0.488397\pi\)
0.0364446 + 0.999336i \(0.488397\pi\)
\(878\) 0 0
\(879\) −6.71614 −0.226530
\(880\) 0 0
\(881\) −14.8421 −0.500044 −0.250022 0.968240i \(-0.580438\pi\)
−0.250022 + 0.968240i \(0.580438\pi\)
\(882\) 0 0
\(883\) 13.0742 0.439983 0.219992 0.975502i \(-0.429397\pi\)
0.219992 + 0.975502i \(0.429397\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −39.4254 −1.32377 −0.661887 0.749604i \(-0.730244\pi\)
−0.661887 + 0.749604i \(0.730244\pi\)
\(888\) 0 0
\(889\) 54.8122 1.83834
\(890\) 0 0
\(891\) −1.38360 −0.0463525
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.19533 0.0732999
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −1.18191 −0.0393751
\(902\) 0 0
\(903\) 17.2455 0.573895
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 49.8397 1.65490 0.827450 0.561539i \(-0.189791\pi\)
0.827450 + 0.561539i \(0.189791\pi\)
\(908\) 0 0
\(909\) −3.10900 −0.103119
\(910\) 0 0
\(911\) 8.92438 0.295678 0.147839 0.989011i \(-0.452768\pi\)
0.147839 + 0.989011i \(0.452768\pi\)
\(912\) 0 0
\(913\) 18.5399 0.613581
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.97247 0.296297
\(918\) 0 0
\(919\) −21.9876 −0.725302 −0.362651 0.931925i \(-0.618128\pi\)
−0.362651 + 0.931925i \(0.618128\pi\)
\(920\) 0 0
\(921\) −14.2689 −0.470176
\(922\) 0 0
\(923\) 13.9716 0.459881
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.8690 0.554050
\(928\) 0 0
\(929\) 4.14289 0.135924 0.0679619 0.997688i \(-0.478350\pi\)
0.0679619 + 0.997688i \(0.478350\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 7.15786 0.234338
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 59.2235 1.93475 0.967374 0.253354i \(-0.0815338\pi\)
0.967374 + 0.253354i \(0.0815338\pi\)
\(938\) 0 0
\(939\) −21.6568 −0.706744
\(940\) 0 0
\(941\) 11.7615 0.383415 0.191707 0.981452i \(-0.438598\pi\)
0.191707 + 0.981452i \(0.438598\pi\)
\(942\) 0 0
\(943\) 4.64543 0.151276
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.51629 −0.276742 −0.138371 0.990380i \(-0.544187\pi\)
−0.138371 + 0.990380i \(0.544187\pi\)
\(948\) 0 0
\(949\) 4.73245 0.153622
\(950\) 0 0
\(951\) 29.0232 0.941141
\(952\) 0 0
\(953\) 35.4872 1.14954 0.574772 0.818314i \(-0.305091\pi\)
0.574772 + 0.818314i \(0.305091\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.3765 0.335426
\(958\) 0 0
\(959\) 42.8817 1.38472
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −17.1946 −0.554090
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.51445 −0.273806 −0.136903 0.990584i \(-0.543715\pi\)
−0.136903 + 0.990584i \(0.543715\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −50.5013 −1.62066 −0.810331 0.585972i \(-0.800713\pi\)
−0.810331 + 0.585972i \(0.800713\pi\)
\(972\) 0 0
\(973\) −33.5980 −1.07710
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39.0540 −1.24945 −0.624725 0.780845i \(-0.714789\pi\)
−0.624725 + 0.780845i \(0.714789\pi\)
\(978\) 0 0
\(979\) −4.18914 −0.133886
\(980\) 0 0
\(981\) 8.76721 0.279915
\(982\) 0 0
\(983\) 54.9882 1.75385 0.876925 0.480627i \(-0.159591\pi\)
0.876925 + 0.480627i \(0.159591\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 11.3900 0.362547
\(988\) 0 0
\(989\) −7.68289 −0.244302
\(990\) 0 0
\(991\) −15.0052 −0.476656 −0.238328 0.971185i \(-0.576599\pi\)
−0.238328 + 0.971185i \(0.576599\pi\)
\(992\) 0 0
\(993\) −21.8683 −0.693969
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.4113 0.709773 0.354887 0.934909i \(-0.384519\pi\)
0.354887 + 0.934909i \(0.384519\pi\)
\(998\) 0 0
\(999\) −6.05088 −0.191441
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 7800.2.a.bt.1.1 4
5.2 odd 4 1560.2.l.d.1249.6 yes 8
5.3 odd 4 1560.2.l.d.1249.2 8
5.4 even 2 7800.2.a.by.1.4 4
15.2 even 4 4680.2.l.g.2809.6 8
15.8 even 4 4680.2.l.g.2809.5 8
20.3 even 4 3120.2.l.n.1249.6 8
20.7 even 4 3120.2.l.n.1249.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.d.1249.2 8 5.3 odd 4
1560.2.l.d.1249.6 yes 8 5.2 odd 4
3120.2.l.n.1249.2 8 20.7 even 4
3120.2.l.n.1249.6 8 20.3 even 4
4680.2.l.g.2809.5 8 15.8 even 4
4680.2.l.g.2809.6 8 15.2 even 4
7800.2.a.bt.1.1 4 1.1 even 1 trivial
7800.2.a.by.1.4 4 5.4 even 2