Properties

Label 4600.2.a.bi.1.1
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.49931\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} +O(q^{10})\) \(q-2.49931 q^{3} +2.92777 q^{7} +3.24657 q^{9} -4.10909 q^{11} -0.0122971 q^{13} +0.155378 q^{17} +4.32471 q^{19} -7.31743 q^{21} -1.00000 q^{23} -0.616262 q^{27} -6.79978 q^{29} +2.20713 q^{31} +10.2699 q^{33} -4.60741 q^{37} +0.0307342 q^{39} -7.67826 q^{41} +8.38997 q^{43} +6.38116 q^{47} +1.57186 q^{49} -0.388339 q^{51} -7.80358 q^{53} -10.8088 q^{57} -14.1275 q^{59} +7.05297 q^{61} +9.50523 q^{63} +7.31363 q^{67} +2.49931 q^{69} -5.84061 q^{71} +0.727674 q^{73} -12.0305 q^{77} +4.81003 q^{79} -8.19948 q^{81} +7.75797 q^{83} +16.9948 q^{87} +6.77560 q^{89} -0.0360030 q^{91} -5.51631 q^{93} +14.8705 q^{97} -13.3405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{3} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 7q + 3q^{3} + 4q^{7} + 2q^{9} - 7q^{11} - 7q^{13} - 7q^{19} - 6q^{21} - 7q^{23} - 11q^{29} - 10q^{31} - 19q^{33} - 19q^{37} - 24q^{39} - 16q^{41} + 6q^{43} + 6q^{47} - 17q^{49} - 7q^{51} - 15q^{53} - 8q^{57} - 11q^{59} + 5q^{61} + 13q^{63} + 9q^{67} - 3q^{69} - 14q^{71} - 10q^{73} - 6q^{77} - 32q^{79} - 5q^{81} + q^{83} + 10q^{87} - 24q^{89} - 7q^{91} - 26q^{93} + 7q^{97} - 61q^{99} + O(q^{100}) \)

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
\(3\) −2.49931 −1.44298 −0.721490 0.692425i \(-0.756542\pi\)
−0.721490 + 0.692425i \(0.756542\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.92777 1.10659 0.553297 0.832984i \(-0.313369\pi\)
0.553297 + 0.832984i \(0.313369\pi\)
\(8\) 0 0
\(9\) 3.24657 1.08219
\(10\) 0 0
\(11\) −4.10909 −1.23894 −0.619468 0.785022i \(-0.712652\pi\)
−0.619468 + 0.785022i \(0.712652\pi\)
\(12\) 0 0
\(13\) −0.0122971 −0.00341059 −0.00170530 0.999999i \(-0.500543\pi\)
−0.00170530 + 0.999999i \(0.500543\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.155378 0.0376847 0.0188424 0.999822i \(-0.494002\pi\)
0.0188424 + 0.999822i \(0.494002\pi\)
\(18\) 0 0
\(19\) 4.32471 0.992158 0.496079 0.868277i \(-0.334773\pi\)
0.496079 + 0.868277i \(0.334773\pi\)
\(20\) 0 0
\(21\) −7.31743 −1.59679
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.616262 −0.118600
\(28\) 0 0
\(29\) −6.79978 −1.26269 −0.631344 0.775503i \(-0.717496\pi\)
−0.631344 + 0.775503i \(0.717496\pi\)
\(30\) 0 0
\(31\) 2.20713 0.396412 0.198206 0.980160i \(-0.436489\pi\)
0.198206 + 0.980160i \(0.436489\pi\)
\(32\) 0 0
\(33\) 10.2699 1.78776
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.60741 −0.757453 −0.378726 0.925509i \(-0.623638\pi\)
−0.378726 + 0.925509i \(0.623638\pi\)
\(38\) 0 0
\(39\) 0.0307342 0.00492142
\(40\) 0 0
\(41\) −7.67826 −1.19914 −0.599571 0.800321i \(-0.704662\pi\)
−0.599571 + 0.800321i \(0.704662\pi\)
\(42\) 0 0
\(43\) 8.38997 1.27946 0.639729 0.768600i \(-0.279046\pi\)
0.639729 + 0.768600i \(0.279046\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.38116 0.930787 0.465394 0.885104i \(-0.345913\pi\)
0.465394 + 0.885104i \(0.345913\pi\)
\(48\) 0 0
\(49\) 1.57186 0.224551
\(50\) 0 0
\(51\) −0.388339 −0.0543783
\(52\) 0 0
\(53\) −7.80358 −1.07190 −0.535952 0.844248i \(-0.680047\pi\)
−0.535952 + 0.844248i \(0.680047\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.8088 −1.43166
\(58\) 0 0
\(59\) −14.1275 −1.83924 −0.919622 0.392805i \(-0.871505\pi\)
−0.919622 + 0.392805i \(0.871505\pi\)
\(60\) 0 0
\(61\) 7.05297 0.903040 0.451520 0.892261i \(-0.350882\pi\)
0.451520 + 0.892261i \(0.350882\pi\)
\(62\) 0 0
\(63\) 9.50523 1.19755
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.31363 0.893502 0.446751 0.894658i \(-0.352581\pi\)
0.446751 + 0.894658i \(0.352581\pi\)
\(68\) 0 0
\(69\) 2.49931 0.300882
\(70\) 0 0
\(71\) −5.84061 −0.693153 −0.346576 0.938022i \(-0.612656\pi\)
−0.346576 + 0.938022i \(0.612656\pi\)
\(72\) 0 0
\(73\) 0.727674 0.0851678 0.0425839 0.999093i \(-0.486441\pi\)
0.0425839 + 0.999093i \(0.486441\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0305 −1.37100
\(78\) 0 0
\(79\) 4.81003 0.541171 0.270586 0.962696i \(-0.412783\pi\)
0.270586 + 0.962696i \(0.412783\pi\)
\(80\) 0 0
\(81\) −8.19948 −0.911054
\(82\) 0 0
\(83\) 7.75797 0.851548 0.425774 0.904830i \(-0.360002\pi\)
0.425774 + 0.904830i \(0.360002\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 16.9948 1.82203
\(88\) 0 0
\(89\) 6.77560 0.718212 0.359106 0.933297i \(-0.383082\pi\)
0.359106 + 0.933297i \(0.383082\pi\)
\(90\) 0 0
\(91\) −0.0360030 −0.00377414
\(92\) 0 0
\(93\) −5.51631 −0.572014
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.8705 1.50987 0.754935 0.655799i \(-0.227668\pi\)
0.754935 + 0.655799i \(0.227668\pi\)
\(98\) 0 0
\(99\) −13.3405 −1.34077
\(100\) 0 0
\(101\) −13.0023 −1.29378 −0.646889 0.762584i \(-0.723930\pi\)
−0.646889 + 0.762584i \(0.723930\pi\)
\(102\) 0 0
\(103\) −1.85814 −0.183088 −0.0915442 0.995801i \(-0.529180\pi\)
−0.0915442 + 0.995801i \(0.529180\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.471267 0.0455591 0.0227795 0.999741i \(-0.492748\pi\)
0.0227795 + 0.999741i \(0.492748\pi\)
\(108\) 0 0
\(109\) −1.83022 −0.175303 −0.0876515 0.996151i \(-0.527936\pi\)
−0.0876515 + 0.996151i \(0.527936\pi\)
\(110\) 0 0
\(111\) 11.5154 1.09299
\(112\) 0 0
\(113\) 7.81151 0.734845 0.367422 0.930054i \(-0.380240\pi\)
0.367422 + 0.930054i \(0.380240\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.0399233 −0.00369091
\(118\) 0 0
\(119\) 0.454912 0.0417017
\(120\) 0 0
\(121\) 5.88461 0.534964
\(122\) 0 0
\(123\) 19.1904 1.73034
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.62237 −0.676376 −0.338188 0.941079i \(-0.609814\pi\)
−0.338188 + 0.941079i \(0.609814\pi\)
\(128\) 0 0
\(129\) −20.9692 −1.84623
\(130\) 0 0
\(131\) 19.4575 1.70001 0.850007 0.526772i \(-0.176598\pi\)
0.850007 + 0.526772i \(0.176598\pi\)
\(132\) 0 0
\(133\) 12.6618 1.09792
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.3163 1.39399 0.696997 0.717074i \(-0.254519\pi\)
0.696997 + 0.717074i \(0.254519\pi\)
\(138\) 0 0
\(139\) −19.9289 −1.69035 −0.845175 0.534490i \(-0.820504\pi\)
−0.845175 + 0.534490i \(0.820504\pi\)
\(140\) 0 0
\(141\) −15.9485 −1.34311
\(142\) 0 0
\(143\) 0.0505297 0.00422551
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.92857 −0.324023
\(148\) 0 0
\(149\) −8.04540 −0.659105 −0.329552 0.944137i \(-0.606898\pi\)
−0.329552 + 0.944137i \(0.606898\pi\)
\(150\) 0 0
\(151\) 3.40963 0.277472 0.138736 0.990329i \(-0.455696\pi\)
0.138736 + 0.990329i \(0.455696\pi\)
\(152\) 0 0
\(153\) 0.504446 0.0407821
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −21.1868 −1.69089 −0.845446 0.534061i \(-0.820665\pi\)
−0.845446 + 0.534061i \(0.820665\pi\)
\(158\) 0 0
\(159\) 19.5036 1.54674
\(160\) 0 0
\(161\) −2.92777 −0.230741
\(162\) 0 0
\(163\) −20.9487 −1.64083 −0.820415 0.571769i \(-0.806257\pi\)
−0.820415 + 0.571769i \(0.806257\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.9861 0.850131 0.425065 0.905163i \(-0.360251\pi\)
0.425065 + 0.905163i \(0.360251\pi\)
\(168\) 0 0
\(169\) −12.9998 −0.999988
\(170\) 0 0
\(171\) 14.0405 1.07370
\(172\) 0 0
\(173\) −4.31110 −0.327767 −0.163884 0.986480i \(-0.552402\pi\)
−0.163884 + 0.986480i \(0.552402\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 35.3091 2.65399
\(178\) 0 0
\(179\) −19.0791 −1.42604 −0.713018 0.701146i \(-0.752672\pi\)
−0.713018 + 0.701146i \(0.752672\pi\)
\(180\) 0 0
\(181\) 4.09312 0.304239 0.152120 0.988362i \(-0.451390\pi\)
0.152120 + 0.988362i \(0.451390\pi\)
\(182\) 0 0
\(183\) −17.6276 −1.30307
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.638462 −0.0466890
\(188\) 0 0
\(189\) −1.80428 −0.131242
\(190\) 0 0
\(191\) 6.21032 0.449363 0.224682 0.974432i \(-0.427866\pi\)
0.224682 + 0.974432i \(0.427866\pi\)
\(192\) 0 0
\(193\) −13.2674 −0.955005 −0.477503 0.878630i \(-0.658458\pi\)
−0.477503 + 0.878630i \(0.658458\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.8836 −0.989163 −0.494582 0.869131i \(-0.664679\pi\)
−0.494582 + 0.869131i \(0.664679\pi\)
\(198\) 0 0
\(199\) −22.8004 −1.61628 −0.808139 0.588992i \(-0.799525\pi\)
−0.808139 + 0.588992i \(0.799525\pi\)
\(200\) 0 0
\(201\) −18.2791 −1.28931
\(202\) 0 0
\(203\) −19.9082 −1.39728
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.24657 −0.225652
\(208\) 0 0
\(209\) −17.7706 −1.22922
\(210\) 0 0
\(211\) −0.698944 −0.0481173 −0.0240587 0.999711i \(-0.507659\pi\)
−0.0240587 + 0.999711i \(0.507659\pi\)
\(212\) 0 0
\(213\) 14.5975 1.00021
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.46197 0.438667
\(218\) 0 0
\(219\) −1.81869 −0.122895
\(220\) 0 0
\(221\) −0.00191069 −0.000128527 0
\(222\) 0 0
\(223\) 27.3719 1.83296 0.916480 0.400082i \(-0.131018\pi\)
0.916480 + 0.400082i \(0.131018\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.6070 −0.704011 −0.352005 0.935998i \(-0.614500\pi\)
−0.352005 + 0.935998i \(0.614500\pi\)
\(228\) 0 0
\(229\) 3.11433 0.205801 0.102900 0.994692i \(-0.467188\pi\)
0.102900 + 0.994692i \(0.467188\pi\)
\(230\) 0 0
\(231\) 30.0680 1.97833
\(232\) 0 0
\(233\) −23.5295 −1.54147 −0.770735 0.637156i \(-0.780111\pi\)
−0.770735 + 0.637156i \(0.780111\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −12.0218 −0.780899
\(238\) 0 0
\(239\) 24.4214 1.57969 0.789844 0.613308i \(-0.210162\pi\)
0.789844 + 0.613308i \(0.210162\pi\)
\(240\) 0 0
\(241\) 27.5525 1.77481 0.887407 0.460986i \(-0.152504\pi\)
0.887407 + 0.460986i \(0.152504\pi\)
\(242\) 0 0
\(243\) 22.3419 1.43323
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.0531813 −0.00338385
\(248\) 0 0
\(249\) −19.3896 −1.22877
\(250\) 0 0
\(251\) −25.3692 −1.60129 −0.800645 0.599139i \(-0.795510\pi\)
−0.800645 + 0.599139i \(0.795510\pi\)
\(252\) 0 0
\(253\) 4.10909 0.258336
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.8501 −1.23822 −0.619108 0.785306i \(-0.712506\pi\)
−0.619108 + 0.785306i \(0.712506\pi\)
\(258\) 0 0
\(259\) −13.4894 −0.838193
\(260\) 0 0
\(261\) −22.0760 −1.36647
\(262\) 0 0
\(263\) −22.9087 −1.41261 −0.706306 0.707906i \(-0.749640\pi\)
−0.706306 + 0.707906i \(0.749640\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −16.9344 −1.03637
\(268\) 0 0
\(269\) 1.03048 0.0628295 0.0314148 0.999506i \(-0.489999\pi\)
0.0314148 + 0.999506i \(0.489999\pi\)
\(270\) 0 0
\(271\) −3.79576 −0.230576 −0.115288 0.993332i \(-0.536779\pi\)
−0.115288 + 0.993332i \(0.536779\pi\)
\(272\) 0 0
\(273\) 0.0899829 0.00544601
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.48782 0.0893946 0.0446973 0.999001i \(-0.485768\pi\)
0.0446973 + 0.999001i \(0.485768\pi\)
\(278\) 0 0
\(279\) 7.16560 0.428993
\(280\) 0 0
\(281\) −10.2796 −0.613228 −0.306614 0.951834i \(-0.599196\pi\)
−0.306614 + 0.951834i \(0.599196\pi\)
\(282\) 0 0
\(283\) 4.62427 0.274884 0.137442 0.990510i \(-0.456112\pi\)
0.137442 + 0.990510i \(0.456112\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −22.4802 −1.32697
\(288\) 0 0
\(289\) −16.9759 −0.998580
\(290\) 0 0
\(291\) −37.1661 −2.17871
\(292\) 0 0
\(293\) −16.3067 −0.952650 −0.476325 0.879269i \(-0.658031\pi\)
−0.476325 + 0.879269i \(0.658031\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.53228 0.146938
\(298\) 0 0
\(299\) 0.0122971 0.000711158 0
\(300\) 0 0
\(301\) 24.5639 1.41584
\(302\) 0 0
\(303\) 32.4969 1.86690
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.28659 −0.530014 −0.265007 0.964246i \(-0.585374\pi\)
−0.265007 + 0.964246i \(0.585374\pi\)
\(308\) 0 0
\(309\) 4.64409 0.264193
\(310\) 0 0
\(311\) −24.3713 −1.38197 −0.690984 0.722870i \(-0.742822\pi\)
−0.690984 + 0.722870i \(0.742822\pi\)
\(312\) 0 0
\(313\) −15.0048 −0.848119 −0.424060 0.905634i \(-0.639395\pi\)
−0.424060 + 0.905634i \(0.639395\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.84747 0.440758 0.220379 0.975414i \(-0.429271\pi\)
0.220379 + 0.975414i \(0.429271\pi\)
\(318\) 0 0
\(319\) 27.9409 1.56439
\(320\) 0 0
\(321\) −1.17784 −0.0657408
\(322\) 0 0
\(323\) 0.671966 0.0373892
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.57429 0.252959
\(328\) 0 0
\(329\) 18.6826 1.03000
\(330\) 0 0
\(331\) 9.28156 0.510160 0.255080 0.966920i \(-0.417898\pi\)
0.255080 + 0.966920i \(0.417898\pi\)
\(332\) 0 0
\(333\) −14.9583 −0.819709
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −26.3760 −1.43679 −0.718397 0.695634i \(-0.755124\pi\)
−0.718397 + 0.695634i \(0.755124\pi\)
\(338\) 0 0
\(339\) −19.5234 −1.06037
\(340\) 0 0
\(341\) −9.06928 −0.491129
\(342\) 0 0
\(343\) −15.8924 −0.858107
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.3316 −0.984093 −0.492047 0.870569i \(-0.663751\pi\)
−0.492047 + 0.870569i \(0.663751\pi\)
\(348\) 0 0
\(349\) −32.2759 −1.72769 −0.863845 0.503758i \(-0.831950\pi\)
−0.863845 + 0.503758i \(0.831950\pi\)
\(350\) 0 0
\(351\) 0.00757822 0.000404495 0
\(352\) 0 0
\(353\) −29.5653 −1.57360 −0.786800 0.617208i \(-0.788264\pi\)
−0.786800 + 0.617208i \(0.788264\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.13697 −0.0601747
\(358\) 0 0
\(359\) 33.4169 1.76368 0.881838 0.471552i \(-0.156306\pi\)
0.881838 + 0.471552i \(0.156306\pi\)
\(360\) 0 0
\(361\) −0.296842 −0.0156232
\(362\) 0 0
\(363\) −14.7075 −0.771943
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −25.4764 −1.32986 −0.664928 0.746908i \(-0.731538\pi\)
−0.664928 + 0.746908i \(0.731538\pi\)
\(368\) 0 0
\(369\) −24.9280 −1.29770
\(370\) 0 0
\(371\) −22.8471 −1.18616
\(372\) 0 0
\(373\) −3.28172 −0.169921 −0.0849605 0.996384i \(-0.527076\pi\)
−0.0849605 + 0.996384i \(0.527076\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.0836174 0.00430651
\(378\) 0 0
\(379\) 5.66762 0.291126 0.145563 0.989349i \(-0.453501\pi\)
0.145563 + 0.989349i \(0.453501\pi\)
\(380\) 0 0
\(381\) 19.0507 0.975997
\(382\) 0 0
\(383\) 34.4343 1.75951 0.879755 0.475428i \(-0.157707\pi\)
0.879755 + 0.475428i \(0.157707\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 27.2387 1.38462
\(388\) 0 0
\(389\) −14.3737 −0.728773 −0.364387 0.931248i \(-0.618721\pi\)
−0.364387 + 0.931248i \(0.618721\pi\)
\(390\) 0 0
\(391\) −0.155378 −0.00785781
\(392\) 0 0
\(393\) −48.6305 −2.45309
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.6149 −0.733500 −0.366750 0.930319i \(-0.619530\pi\)
−0.366750 + 0.930319i \(0.619530\pi\)
\(398\) 0 0
\(399\) −31.6458 −1.58427
\(400\) 0 0
\(401\) −3.65553 −0.182548 −0.0912742 0.995826i \(-0.529094\pi\)
−0.0912742 + 0.995826i \(0.529094\pi\)
\(402\) 0 0
\(403\) −0.0271412 −0.00135200
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 18.9322 0.938436
\(408\) 0 0
\(409\) 4.40926 0.218024 0.109012 0.994040i \(-0.465231\pi\)
0.109012 + 0.994040i \(0.465231\pi\)
\(410\) 0 0
\(411\) −40.7795 −2.01150
\(412\) 0 0
\(413\) −41.3621 −2.03530
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 49.8087 2.43914
\(418\) 0 0
\(419\) 32.5906 1.59216 0.796078 0.605195i \(-0.206905\pi\)
0.796078 + 0.605195i \(0.206905\pi\)
\(420\) 0 0
\(421\) 4.23954 0.206622 0.103311 0.994649i \(-0.467056\pi\)
0.103311 + 0.994649i \(0.467056\pi\)
\(422\) 0 0
\(423\) 20.7169 1.00729
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.6495 0.999299
\(428\) 0 0
\(429\) −0.126290 −0.00609732
\(430\) 0 0
\(431\) 1.81123 0.0872436 0.0436218 0.999048i \(-0.486110\pi\)
0.0436218 + 0.999048i \(0.486110\pi\)
\(432\) 0 0
\(433\) −1.76263 −0.0847068 −0.0423534 0.999103i \(-0.513486\pi\)
−0.0423534 + 0.999103i \(0.513486\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.32471 −0.206879
\(438\) 0 0
\(439\) −26.7848 −1.27837 −0.639184 0.769054i \(-0.720728\pi\)
−0.639184 + 0.769054i \(0.720728\pi\)
\(440\) 0 0
\(441\) 5.10316 0.243008
\(442\) 0 0
\(443\) 16.6631 0.791686 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 20.1080 0.951075
\(448\) 0 0
\(449\) −34.1236 −1.61039 −0.805196 0.593008i \(-0.797940\pi\)
−0.805196 + 0.593008i \(0.797940\pi\)
\(450\) 0 0
\(451\) 31.5507 1.48566
\(452\) 0 0
\(453\) −8.52174 −0.400386
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.5980 −1.15065 −0.575324 0.817926i \(-0.695124\pi\)
−0.575324 + 0.817926i \(0.695124\pi\)
\(458\) 0 0
\(459\) −0.0957537 −0.00446940
\(460\) 0 0
\(461\) −10.7297 −0.499733 −0.249866 0.968280i \(-0.580387\pi\)
−0.249866 + 0.968280i \(0.580387\pi\)
\(462\) 0 0
\(463\) −22.3124 −1.03695 −0.518473 0.855094i \(-0.673499\pi\)
−0.518473 + 0.855094i \(0.673499\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.0218 −0.926497 −0.463248 0.886229i \(-0.653316\pi\)
−0.463248 + 0.886229i \(0.653316\pi\)
\(468\) 0 0
\(469\) 21.4127 0.988744
\(470\) 0 0
\(471\) 52.9525 2.43992
\(472\) 0 0
\(473\) −34.4751 −1.58517
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −25.3349 −1.16001
\(478\) 0 0
\(479\) −7.61905 −0.348123 −0.174062 0.984735i \(-0.555689\pi\)
−0.174062 + 0.984735i \(0.555689\pi\)
\(480\) 0 0
\(481\) 0.0566576 0.00258336
\(482\) 0 0
\(483\) 7.31743 0.332954
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 21.5163 0.974997 0.487499 0.873124i \(-0.337909\pi\)
0.487499 + 0.873124i \(0.337909\pi\)
\(488\) 0 0
\(489\) 52.3574 2.36768
\(490\) 0 0
\(491\) −10.3705 −0.468015 −0.234007 0.972235i \(-0.575184\pi\)
−0.234007 + 0.972235i \(0.575184\pi\)
\(492\) 0 0
\(493\) −1.05654 −0.0475841
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −17.1000 −0.767039
\(498\) 0 0
\(499\) −16.2850 −0.729018 −0.364509 0.931200i \(-0.618763\pi\)
−0.364509 + 0.931200i \(0.618763\pi\)
\(500\) 0 0
\(501\) −27.4577 −1.22672
\(502\) 0 0
\(503\) 16.9326 0.754986 0.377493 0.926012i \(-0.376786\pi\)
0.377493 + 0.926012i \(0.376786\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 32.4907 1.44296
\(508\) 0 0
\(509\) 39.0075 1.72898 0.864488 0.502654i \(-0.167643\pi\)
0.864488 + 0.502654i \(0.167643\pi\)
\(510\) 0 0
\(511\) 2.13046 0.0942462
\(512\) 0 0
\(513\) −2.66516 −0.117670
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −26.2207 −1.15319
\(518\) 0 0
\(519\) 10.7748 0.472961
\(520\) 0 0
\(521\) 33.1380 1.45180 0.725902 0.687798i \(-0.241423\pi\)
0.725902 + 0.687798i \(0.241423\pi\)
\(522\) 0 0
\(523\) 12.4056 0.542460 0.271230 0.962515i \(-0.412570\pi\)
0.271230 + 0.962515i \(0.412570\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.342939 0.0149387
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −45.8660 −1.99041
\(532\) 0 0
\(533\) 0.0944201 0.00408979
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 47.6846 2.05774
\(538\) 0 0
\(539\) −6.45891 −0.278205
\(540\) 0 0
\(541\) −8.53472 −0.366936 −0.183468 0.983026i \(-0.558732\pi\)
−0.183468 + 0.983026i \(0.558732\pi\)
\(542\) 0 0
\(543\) −10.2300 −0.439011
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.45828 0.276136 0.138068 0.990423i \(-0.455911\pi\)
0.138068 + 0.990423i \(0.455911\pi\)
\(548\) 0 0
\(549\) 22.8980 0.977262
\(550\) 0 0
\(551\) −29.4071 −1.25279
\(552\) 0 0
\(553\) 14.0827 0.598857
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.08413 −0.173050 −0.0865251 0.996250i \(-0.527576\pi\)
−0.0865251 + 0.996250i \(0.527576\pi\)
\(558\) 0 0
\(559\) −0.103172 −0.00436371
\(560\) 0 0
\(561\) 1.59572 0.0673713
\(562\) 0 0
\(563\) −15.4494 −0.651114 −0.325557 0.945522i \(-0.605552\pi\)
−0.325557 + 0.945522i \(0.605552\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −24.0062 −1.00817
\(568\) 0 0
\(569\) 42.2633 1.77177 0.885886 0.463903i \(-0.153552\pi\)
0.885886 + 0.463903i \(0.153552\pi\)
\(570\) 0 0
\(571\) −2.74497 −0.114873 −0.0574367 0.998349i \(-0.518293\pi\)
−0.0574367 + 0.998349i \(0.518293\pi\)
\(572\) 0 0
\(573\) −15.5215 −0.648422
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.93871 0.372123 0.186062 0.982538i \(-0.440428\pi\)
0.186062 + 0.982538i \(0.440428\pi\)
\(578\) 0 0
\(579\) 33.1593 1.37805
\(580\) 0 0
\(581\) 22.7136 0.942318
\(582\) 0 0
\(583\) 32.0656 1.32802
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.90821 0.243858 0.121929 0.992539i \(-0.461092\pi\)
0.121929 + 0.992539i \(0.461092\pi\)
\(588\) 0 0
\(589\) 9.54520 0.393303
\(590\) 0 0
\(591\) 34.6994 1.42734
\(592\) 0 0
\(593\) 27.8823 1.14499 0.572494 0.819909i \(-0.305976\pi\)
0.572494 + 0.819909i \(0.305976\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 56.9854 2.33226
\(598\) 0 0
\(599\) 10.5644 0.431648 0.215824 0.976432i \(-0.430756\pi\)
0.215824 + 0.976432i \(0.430756\pi\)
\(600\) 0 0
\(601\) 35.8910 1.46403 0.732013 0.681291i \(-0.238581\pi\)
0.732013 + 0.681291i \(0.238581\pi\)
\(602\) 0 0
\(603\) 23.7442 0.966940
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −29.0640 −1.17967 −0.589835 0.807524i \(-0.700807\pi\)
−0.589835 + 0.807524i \(0.700807\pi\)
\(608\) 0 0
\(609\) 49.7569 2.01625
\(610\) 0 0
\(611\) −0.0784695 −0.00317454
\(612\) 0 0
\(613\) 24.4282 0.986644 0.493322 0.869847i \(-0.335782\pi\)
0.493322 + 0.869847i \(0.335782\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29.5614 1.19010 0.595049 0.803689i \(-0.297133\pi\)
0.595049 + 0.803689i \(0.297133\pi\)
\(618\) 0 0
\(619\) −30.5880 −1.22944 −0.614718 0.788747i \(-0.710730\pi\)
−0.614718 + 0.788747i \(0.710730\pi\)
\(620\) 0 0
\(621\) 0.616262 0.0247298
\(622\) 0 0
\(623\) 19.8374 0.794769
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 44.4144 1.77374
\(628\) 0 0
\(629\) −0.715890 −0.0285444
\(630\) 0 0
\(631\) 10.5713 0.420835 0.210418 0.977612i \(-0.432518\pi\)
0.210418 + 0.977612i \(0.432518\pi\)
\(632\) 0 0
\(633\) 1.74688 0.0694323
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.0193293 −0.000765853 0
\(638\) 0 0
\(639\) −18.9620 −0.750123
\(640\) 0 0
\(641\) −20.4738 −0.808666 −0.404333 0.914612i \(-0.632496\pi\)
−0.404333 + 0.914612i \(0.632496\pi\)
\(642\) 0 0
\(643\) 18.4989 0.729525 0.364763 0.931101i \(-0.381150\pi\)
0.364763 + 0.931101i \(0.381150\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.6328 0.732531 0.366266 0.930510i \(-0.380636\pi\)
0.366266 + 0.930510i \(0.380636\pi\)
\(648\) 0 0
\(649\) 58.0511 2.27871
\(650\) 0 0
\(651\) −16.1505 −0.632988
\(652\) 0 0
\(653\) 30.5290 1.19469 0.597347 0.801983i \(-0.296222\pi\)
0.597347 + 0.801983i \(0.296222\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.36245 0.0921678
\(658\) 0 0
\(659\) −26.6231 −1.03709 −0.518545 0.855051i \(-0.673526\pi\)
−0.518545 + 0.855051i \(0.673526\pi\)
\(660\) 0 0
\(661\) −13.0529 −0.507700 −0.253850 0.967244i \(-0.581697\pi\)
−0.253850 + 0.967244i \(0.581697\pi\)
\(662\) 0 0
\(663\) 0.00477543 0.000185462 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.79978 0.263289
\(668\) 0 0
\(669\) −68.4110 −2.64492
\(670\) 0 0
\(671\) −28.9813 −1.11881
\(672\) 0 0
\(673\) 28.9120 1.11448 0.557238 0.830353i \(-0.311861\pi\)
0.557238 + 0.830353i \(0.311861\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.4824 −0.479738 −0.239869 0.970805i \(-0.577104\pi\)
−0.239869 + 0.970805i \(0.577104\pi\)
\(678\) 0 0
\(679\) 43.5375 1.67081
\(680\) 0 0
\(681\) 26.5102 1.01587
\(682\) 0 0
\(683\) 40.0241 1.53148 0.765740 0.643150i \(-0.222373\pi\)
0.765740 + 0.643150i \(0.222373\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −7.78369 −0.296966
\(688\) 0 0
\(689\) 0.0959612 0.00365583
\(690\) 0 0
\(691\) −38.9938 −1.48340 −0.741698 0.670734i \(-0.765979\pi\)
−0.741698 + 0.670734i \(0.765979\pi\)
\(692\) 0 0
\(693\) −39.0578 −1.48368
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.19303 −0.0451894
\(698\) 0 0
\(699\) 58.8077 2.22431
\(700\) 0 0
\(701\) 21.9042 0.827309 0.413655 0.910434i \(-0.364252\pi\)
0.413655 + 0.910434i \(0.364252\pi\)
\(702\) 0 0
\(703\) −19.9257 −0.751513
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −38.0678 −1.43169
\(708\) 0 0
\(709\) 32.1360 1.20689 0.603447 0.797403i \(-0.293793\pi\)
0.603447 + 0.797403i \(0.293793\pi\)
\(710\) 0 0
\(711\) 15.6161 0.585650
\(712\) 0 0
\(713\) −2.20713 −0.0826576
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −61.0367 −2.27946
\(718\) 0 0
\(719\) −24.4018 −0.910033 −0.455017 0.890483i \(-0.650367\pi\)
−0.455017 + 0.890483i \(0.650367\pi\)
\(720\) 0 0
\(721\) −5.44023 −0.202605
\(722\) 0 0
\(723\) −68.8625 −2.56102
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.8573 −1.29279 −0.646393 0.763005i \(-0.723723\pi\)
−0.646393 + 0.763005i \(0.723723\pi\)
\(728\) 0 0
\(729\) −31.2409 −1.15707
\(730\) 0 0
\(731\) 1.30362 0.0482161
\(732\) 0 0
\(733\) 30.7343 1.13520 0.567598 0.823306i \(-0.307873\pi\)
0.567598 + 0.823306i \(0.307873\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.0523 −1.10699
\(738\) 0 0
\(739\) −38.7573 −1.42571 −0.712855 0.701312i \(-0.752598\pi\)
−0.712855 + 0.701312i \(0.752598\pi\)
\(740\) 0 0
\(741\) 0.132917 0.00488282
\(742\) 0 0
\(743\) 43.0371 1.57888 0.789438 0.613830i \(-0.210372\pi\)
0.789438 + 0.613830i \(0.210372\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 25.1868 0.921538
\(748\) 0 0
\(749\) 1.37976 0.0504154
\(750\) 0 0
\(751\) −35.3761 −1.29089 −0.645447 0.763805i \(-0.723329\pi\)
−0.645447 + 0.763805i \(0.723329\pi\)
\(752\) 0 0
\(753\) 63.4056 2.31063
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −51.2605 −1.86309 −0.931547 0.363621i \(-0.881540\pi\)
−0.931547 + 0.363621i \(0.881540\pi\)
\(758\) 0 0
\(759\) −10.2699 −0.372774
\(760\) 0 0
\(761\) −13.4331 −0.486950 −0.243475 0.969907i \(-0.578287\pi\)
−0.243475 + 0.969907i \(0.578287\pi\)
\(762\) 0 0
\(763\) −5.35846 −0.193989
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.173727 0.00627291
\(768\) 0 0
\(769\) −24.7779 −0.893512 −0.446756 0.894656i \(-0.647421\pi\)
−0.446756 + 0.894656i \(0.647421\pi\)
\(770\) 0 0
\(771\) 49.6117 1.78672
\(772\) 0 0
\(773\) 11.9946 0.431417 0.215708 0.976458i \(-0.430794\pi\)
0.215708 + 0.976458i \(0.430794\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 33.7144 1.20950
\(778\) 0 0
\(779\) −33.2063 −1.18974
\(780\) 0 0
\(781\) 23.9996 0.858772
\(782\) 0 0
\(783\) 4.19045 0.149755
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.4110 0.371114 0.185557 0.982634i \(-0.440591\pi\)
0.185557 + 0.982634i \(0.440591\pi\)
\(788\) 0 0
\(789\) 57.2561 2.03837
\(790\) 0 0
\(791\) 22.8703 0.813175
\(792\) 0 0
\(793\) −0.0867308 −0.00307990
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.0848 0.994814 0.497407 0.867517i \(-0.334286\pi\)
0.497407 + 0.867517i \(0.334286\pi\)
\(798\) 0 0
\(799\) 0.991492 0.0350765
\(800\) 0 0
\(801\) 21.9975 0.777243
\(802\) 0 0
\(803\) −2.99008 −0.105517
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2.57550 −0.0906618
\(808\) 0 0
\(809\) −11.1238 −0.391092 −0.195546 0.980695i \(-0.562648\pi\)
−0.195546 + 0.980695i \(0.562648\pi\)
\(810\) 0 0
\(811\) −28.7572 −1.00980 −0.504900 0.863178i \(-0.668471\pi\)
−0.504900 + 0.863178i \(0.668471\pi\)
\(812\) 0 0
\(813\) 9.48679 0.332716
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 36.2842 1.26942
\(818\) 0 0
\(819\) −0.116886 −0.00408434
\(820\) 0 0
\(821\) 37.0808 1.29413 0.647064 0.762435i \(-0.275996\pi\)
0.647064 + 0.762435i \(0.275996\pi\)
\(822\) 0 0
\(823\) −10.9501 −0.381698 −0.190849 0.981619i \(-0.561124\pi\)
−0.190849 + 0.981619i \(0.561124\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.9973 0.486732 0.243366 0.969934i \(-0.421748\pi\)
0.243366 + 0.969934i \(0.421748\pi\)
\(828\) 0 0
\(829\) 16.0622 0.557864 0.278932 0.960311i \(-0.410020\pi\)
0.278932 + 0.960311i \(0.410020\pi\)
\(830\) 0 0
\(831\) −3.71853 −0.128995
\(832\) 0 0
\(833\) 0.244233 0.00846216
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.36017 −0.0470144
\(838\) 0 0
\(839\) −9.49187 −0.327696 −0.163848 0.986486i \(-0.552391\pi\)
−0.163848 + 0.986486i \(0.552391\pi\)
\(840\) 0 0
\(841\) 17.2371 0.594381
\(842\) 0 0
\(843\) 25.6919 0.884876
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 17.2288 0.591988
\(848\) 0 0
\(849\) −11.5575 −0.396653
\(850\) 0 0
\(851\) 4.60741 0.157940
\(852\) 0 0
\(853\) −3.45965 −0.118456 −0.0592280 0.998244i \(-0.518864\pi\)
−0.0592280 + 0.998244i \(0.518864\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.729973 −0.0249354 −0.0124677 0.999922i \(-0.503969\pi\)
−0.0124677 + 0.999922i \(0.503969\pi\)
\(858\) 0 0
\(859\) 0.357241 0.0121889 0.00609446 0.999981i \(-0.498060\pi\)
0.00609446 + 0.999981i \(0.498060\pi\)
\(860\) 0 0
\(861\) 56.1851 1.91478
\(862\) 0 0
\(863\) −16.8358 −0.573098 −0.286549 0.958066i \(-0.592508\pi\)
−0.286549 + 0.958066i \(0.592508\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 42.4280 1.44093
\(868\) 0 0
\(869\) −19.7649 −0.670477
\(870\) 0 0
\(871\) −0.0899362 −0.00304737
\(872\) 0 0
\(873\) 48.2782 1.63397
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.06147 0.272217 0.136108 0.990694i \(-0.456540\pi\)
0.136108 + 0.990694i \(0.456540\pi\)
\(878\) 0 0
\(879\) 40.7557 1.37466
\(880\) 0 0
\(881\) −37.7548 −1.27199 −0.635996 0.771693i \(-0.719410\pi\)
−0.635996 + 0.771693i \(0.719410\pi\)
\(882\) 0 0
\(883\) 35.2103 1.18492 0.592460 0.805600i \(-0.298157\pi\)
0.592460 + 0.805600i \(0.298157\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.4225 −0.719297 −0.359649 0.933088i \(-0.617103\pi\)
−0.359649 + 0.933088i \(0.617103\pi\)
\(888\) 0 0
\(889\) −22.3166 −0.748474
\(890\) 0 0
\(891\) 33.6924 1.12874
\(892\) 0 0
\(893\) 27.5967 0.923488
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −0.0307342 −0.00102619
\(898\) 0 0
\(899\) −15.0080 −0.500545
\(900\) 0 0
\(901\) −1.21251 −0.0403944
\(902\) 0 0
\(903\) −61.3930 −2.04303
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 38.7517 1.28673 0.643364 0.765560i \(-0.277538\pi\)
0.643364 + 0.765560i \(0.277538\pi\)
\(908\) 0 0
\(909\) −42.2129 −1.40012
\(910\) 0 0
\(911\) −47.0069 −1.55741 −0.778704 0.627391i \(-0.784123\pi\)
−0.778704 + 0.627391i \(0.784123\pi\)
\(912\) 0 0
\(913\) −31.8782 −1.05501
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 56.9673 1.88123
\(918\) 0 0
\(919\) 12.2360 0.403629 0.201814 0.979424i \(-0.435316\pi\)
0.201814 + 0.979424i \(0.435316\pi\)
\(920\) 0 0
\(921\) 23.2101 0.764799
\(922\) 0 0
\(923\) 0.0718223 0.00236406
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6.03260 −0.198137
\(928\) 0 0
\(929\) −13.0143 −0.426984 −0.213492 0.976945i \(-0.568484\pi\)
−0.213492 + 0.976945i \(0.568484\pi\)
\(930\) 0 0
\(931\) 6.79785 0.222790
\(932\) 0 0
\(933\) 60.9114 1.99415
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −38.6856 −1.26380 −0.631902 0.775048i \(-0.717726\pi\)
−0.631902 + 0.775048i \(0.717726\pi\)
\(938\) 0 0
\(939\) 37.5016 1.22382
\(940\) 0 0
\(941\) −43.8241 −1.42862 −0.714312 0.699828i \(-0.753260\pi\)
−0.714312 + 0.699828i \(0.753260\pi\)
\(942\) 0 0
\(943\) 7.67826 0.250039
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −43.3854 −1.40984 −0.704918 0.709289i \(-0.749016\pi\)
−0.704918 + 0.709289i \(0.749016\pi\)
\(948\) 0 0
\(949\) −0.00894825 −0.000290473 0
\(950\) 0 0
\(951\) −19.6133 −0.636005
\(952\) 0 0
\(953\) 2.80245 0.0907802 0.0453901 0.998969i \(-0.485547\pi\)
0.0453901 + 0.998969i \(0.485547\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −69.8331 −2.25738
\(958\) 0 0
\(959\) 47.7704 1.54259
\(960\) 0 0
\(961\) −26.1286 −0.842858
\(962\) 0 0
\(963\) 1.53000 0.0493036
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 30.0015 0.964783 0.482391 0.875956i \(-0.339768\pi\)
0.482391 + 0.875956i \(0.339768\pi\)
\(968\) 0 0
\(969\) −1.67945 −0.0539518
\(970\) 0 0
\(971\) 53.6632 1.72213 0.861067 0.508492i \(-0.169797\pi\)
0.861067 + 0.508492i \(0.169797\pi\)
\(972\) 0 0
\(973\) −58.3474 −1.87053
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.4590 1.48636 0.743178 0.669094i \(-0.233318\pi\)
0.743178 + 0.669094i \(0.233318\pi\)
\(978\) 0 0
\(979\) −27.8415 −0.889819
\(980\) 0 0
\(981\) −5.94193 −0.189711
\(982\) 0 0
\(983\) −50.8346 −1.62137 −0.810686 0.585481i \(-0.800906\pi\)
−0.810686 + 0.585481i \(0.800906\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −46.6936 −1.48627
\(988\) 0 0
\(989\) −8.38997 −0.266786
\(990\) 0 0
\(991\) 4.30720 0.136823 0.0684114 0.997657i \(-0.478207\pi\)
0.0684114 + 0.997657i \(0.478207\pi\)
\(992\) 0 0
\(993\) −23.1975 −0.736151
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −13.2358 −0.419181 −0.209590 0.977789i \(-0.567213\pi\)
−0.209590 + 0.977789i \(0.567213\pi\)
\(998\) 0 0
\(999\) 2.83937 0.0898337
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 4600.2.a.bi.1.1 7
4.3 odd 2 9200.2.a.cz.1.7 7
5.2 odd 4 920.2.e.b.369.13 yes 14
5.3 odd 4 920.2.e.b.369.2 14
5.4 even 2 4600.2.a.bh.1.7 7
20.3 even 4 1840.2.e.g.369.13 14
20.7 even 4 1840.2.e.g.369.2 14
20.19 odd 2 9200.2.a.dc.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.e.b.369.2 14 5.3 odd 4
920.2.e.b.369.13 yes 14 5.2 odd 4
1840.2.e.g.369.2 14 20.7 even 4
1840.2.e.g.369.13 14 20.3 even 4
4600.2.a.bh.1.7 7 5.4 even 2
4600.2.a.bi.1.1 7 1.1 even 1 trivial
9200.2.a.cz.1.7 7 4.3 odd 2
9200.2.a.dc.1.1 7 20.19 odd 2