Properties

Label 4600.2.a.bd.1.2
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(5\)
Coefficient field: 5.5.521397.1
Defining polynomial: \(x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.83957\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} +O(q^{10})\) \(q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} +2.10339 q^{11} -5.35673 q^{13} -1.29567 q^{17} +2.10339 q^{19} -7.30809 q^{21} +1.00000 q^{23} +4.81229 q^{27} +6.03130 q^{29} -8.32489 q^{31} -3.86933 q^{33} -5.10131 q^{37} +9.85407 q^{39} -8.33994 q^{41} -7.78253 q^{43} -11.3964 q^{47} +8.78253 q^{49} +2.38347 q^{51} +0.573664 q^{53} -3.86933 q^{57} -9.17951 q^{59} +13.5149 q^{61} +1.52557 q^{63} +15.8314 q^{67} -1.83957 q^{69} +14.9449 q^{71} -8.36459 q^{73} +8.35619 q^{77} -9.41149 q^{79} -10.0046 q^{81} -1.51206 q^{83} -11.0950 q^{87} +10.5903 q^{89} -21.2808 q^{91} +15.3142 q^{93} +0.337451 q^{97} +0.807724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 5q - 4q^{7} + 3q^{9} - 4q^{13} - 6q^{17} + 5q^{23} - 9q^{27} + 12q^{29} - 18q^{31} - 6q^{33} - 10q^{37} + 9q^{39} - 6q^{41} - 10q^{43} - 22q^{47} + 15q^{49} - 6q^{51} - 10q^{53} - 6q^{57} - q^{59} + 10q^{61} - 8q^{67} + 8q^{71} - 6q^{73} - 27q^{81} + 2q^{83} - 39q^{87} + 14q^{89} - 46q^{91} + 3q^{93} - 6q^{97} - 6q^{99} + O(q^{100}) \)

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
\(3\) −1.83957 −1.06208 −0.531038 0.847348i \(-0.678198\pi\)
−0.531038 + 0.847348i \(0.678198\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.97272 1.50155 0.750774 0.660559i \(-0.229681\pi\)
0.750774 + 0.660559i \(0.229681\pi\)
\(8\) 0 0
\(9\) 0.384010 0.128003
\(10\) 0 0
\(11\) 2.10339 0.634196 0.317098 0.948393i \(-0.397292\pi\)
0.317098 + 0.948393i \(0.397292\pi\)
\(12\) 0 0
\(13\) −5.35673 −1.48569 −0.742845 0.669463i \(-0.766524\pi\)
−0.742845 + 0.669463i \(0.766524\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.29567 −0.314246 −0.157123 0.987579i \(-0.550222\pi\)
−0.157123 + 0.987579i \(0.550222\pi\)
\(18\) 0 0
\(19\) 2.10339 0.482551 0.241276 0.970457i \(-0.422434\pi\)
0.241276 + 0.970457i \(0.422434\pi\)
\(20\) 0 0
\(21\) −7.30809 −1.59476
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.81229 0.926126
\(28\) 0 0
\(29\) 6.03130 1.11998 0.559992 0.828498i \(-0.310804\pi\)
0.559992 + 0.828498i \(0.310804\pi\)
\(30\) 0 0
\(31\) −8.32489 −1.49519 −0.747597 0.664153i \(-0.768793\pi\)
−0.747597 + 0.664153i \(0.768793\pi\)
\(32\) 0 0
\(33\) −3.86933 −0.673564
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.10131 −0.838650 −0.419325 0.907836i \(-0.637733\pi\)
−0.419325 + 0.907836i \(0.637733\pi\)
\(38\) 0 0
\(39\) 9.85407 1.57791
\(40\) 0 0
\(41\) −8.33994 −1.30248 −0.651240 0.758872i \(-0.725751\pi\)
−0.651240 + 0.758872i \(0.725751\pi\)
\(42\) 0 0
\(43\) −7.78253 −1.18682 −0.593412 0.804899i \(-0.702220\pi\)
−0.593412 + 0.804899i \(0.702220\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.3964 −1.66234 −0.831171 0.556018i \(-0.812329\pi\)
−0.831171 + 0.556018i \(0.812329\pi\)
\(48\) 0 0
\(49\) 8.78253 1.25465
\(50\) 0 0
\(51\) 2.38347 0.333752
\(52\) 0 0
\(53\) 0.573664 0.0787988 0.0393994 0.999224i \(-0.487456\pi\)
0.0393994 + 0.999224i \(0.487456\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.86933 −0.512505
\(58\) 0 0
\(59\) −9.17951 −1.19507 −0.597535 0.801843i \(-0.703853\pi\)
−0.597535 + 0.801843i \(0.703853\pi\)
\(60\) 0 0
\(61\) 13.5149 1.73040 0.865201 0.501425i \(-0.167191\pi\)
0.865201 + 0.501425i \(0.167191\pi\)
\(62\) 0 0
\(63\) 1.52557 0.192203
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8314 1.93411 0.967055 0.254569i \(-0.0819337\pi\)
0.967055 + 0.254569i \(0.0819337\pi\)
\(68\) 0 0
\(69\) −1.83957 −0.221458
\(70\) 0 0
\(71\) 14.9449 1.77363 0.886817 0.462121i \(-0.152911\pi\)
0.886817 + 0.462121i \(0.152911\pi\)
\(72\) 0 0
\(73\) −8.36459 −0.979002 −0.489501 0.872003i \(-0.662821\pi\)
−0.489501 + 0.872003i \(0.662821\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.35619 0.952276
\(78\) 0 0
\(79\) −9.41149 −1.05887 −0.529437 0.848349i \(-0.677597\pi\)
−0.529437 + 0.848349i \(0.677597\pi\)
\(80\) 0 0
\(81\) −10.0046 −1.11162
\(82\) 0 0
\(83\) −1.51206 −0.165970 −0.0829849 0.996551i \(-0.526445\pi\)
−0.0829849 + 0.996551i \(0.526445\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −11.0950 −1.18951
\(88\) 0 0
\(89\) 10.5903 1.12256 0.561282 0.827624i \(-0.310308\pi\)
0.561282 + 0.827624i \(0.310308\pi\)
\(90\) 0 0
\(91\) −21.2808 −2.23084
\(92\) 0 0
\(93\) 15.3142 1.58801
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.337451 0.0342630 0.0171315 0.999853i \(-0.494547\pi\)
0.0171315 + 0.999853i \(0.494547\pi\)
\(98\) 0 0
\(99\) 0.807724 0.0811793
\(100\) 0 0
\(101\) −6.19119 −0.616047 −0.308023 0.951379i \(-0.599667\pi\)
−0.308023 + 0.951379i \(0.599667\pi\)
\(102\) 0 0
\(103\) −12.6267 −1.24414 −0.622071 0.782961i \(-0.713708\pi\)
−0.622071 + 0.782961i \(0.713708\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.7940 −1.23684 −0.618419 0.785848i \(-0.712227\pi\)
−0.618419 + 0.785848i \(0.712227\pi\)
\(108\) 0 0
\(109\) 14.3165 1.37127 0.685635 0.727945i \(-0.259524\pi\)
0.685635 + 0.727945i \(0.259524\pi\)
\(110\) 0 0
\(111\) 9.38421 0.890710
\(112\) 0 0
\(113\) −4.59463 −0.432226 −0.216113 0.976368i \(-0.569338\pi\)
−0.216113 + 0.976368i \(0.569338\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.05704 −0.190173
\(118\) 0 0
\(119\) −5.14733 −0.471855
\(120\) 0 0
\(121\) −6.57574 −0.597795
\(122\) 0 0
\(123\) 15.3419 1.38333
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.6623 −1.12360 −0.561801 0.827273i \(-0.689891\pi\)
−0.561801 + 0.827273i \(0.689891\pi\)
\(128\) 0 0
\(129\) 14.3165 1.26050
\(130\) 0 0
\(131\) 4.68370 0.409217 0.204609 0.978844i \(-0.434408\pi\)
0.204609 + 0.978844i \(0.434408\pi\)
\(132\) 0 0
\(133\) 8.35619 0.724574
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.468120 0.0399942 0.0199971 0.999800i \(-0.493634\pi\)
0.0199971 + 0.999800i \(0.493634\pi\)
\(138\) 0 0
\(139\) 0.739205 0.0626985 0.0313493 0.999508i \(-0.490020\pi\)
0.0313493 + 0.999508i \(0.490020\pi\)
\(140\) 0 0
\(141\) 20.9645 1.76553
\(142\) 0 0
\(143\) −11.2673 −0.942220
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −16.1561 −1.33253
\(148\) 0 0
\(149\) 14.5903 1.19528 0.597640 0.801765i \(-0.296105\pi\)
0.597640 + 0.801765i \(0.296105\pi\)
\(150\) 0 0
\(151\) 21.6885 1.76498 0.882491 0.470329i \(-0.155865\pi\)
0.882491 + 0.470329i \(0.155865\pi\)
\(152\) 0 0
\(153\) −0.497550 −0.0402245
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.71253 −0.615527 −0.307763 0.951463i \(-0.599580\pi\)
−0.307763 + 0.951463i \(0.599580\pi\)
\(158\) 0 0
\(159\) −1.05529 −0.0836902
\(160\) 0 0
\(161\) 3.97272 0.313094
\(162\) 0 0
\(163\) 4.69177 0.367488 0.183744 0.982974i \(-0.441178\pi\)
0.183744 + 0.982974i \(0.441178\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.6738 −0.903343 −0.451671 0.892184i \(-0.649172\pi\)
−0.451671 + 0.892184i \(0.649172\pi\)
\(168\) 0 0
\(169\) 15.6946 1.20728
\(170\) 0 0
\(171\) 0.807724 0.0617682
\(172\) 0 0
\(173\) 8.31723 0.632347 0.316174 0.948701i \(-0.397602\pi\)
0.316174 + 0.948701i \(0.397602\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.8863 1.26925
\(178\) 0 0
\(179\) −4.07517 −0.304592 −0.152296 0.988335i \(-0.548667\pi\)
−0.152296 + 0.988335i \(0.548667\pi\)
\(180\) 0 0
\(181\) 2.52349 0.187569 0.0937846 0.995593i \(-0.470103\pi\)
0.0937846 + 0.995593i \(0.470103\pi\)
\(182\) 0 0
\(183\) −24.8615 −1.83782
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.72530 −0.199293
\(188\) 0 0
\(189\) 19.1179 1.39062
\(190\) 0 0
\(191\) −23.9380 −1.73210 −0.866048 0.499961i \(-0.833348\pi\)
−0.866048 + 0.499961i \(0.833348\pi\)
\(192\) 0 0
\(193\) −18.7733 −1.35133 −0.675667 0.737207i \(-0.736144\pi\)
−0.675667 + 0.737207i \(0.736144\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.9052 −1.13320 −0.566600 0.823993i \(-0.691741\pi\)
−0.566600 + 0.823993i \(0.691741\pi\)
\(198\) 0 0
\(199\) 9.87544 0.700052 0.350026 0.936740i \(-0.386173\pi\)
0.350026 + 0.936740i \(0.386173\pi\)
\(200\) 0 0
\(201\) −29.1229 −2.05417
\(202\) 0 0
\(203\) 23.9607 1.68171
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.384010 0.0266906
\(208\) 0 0
\(209\) 4.42426 0.306032
\(210\) 0 0
\(211\) −7.06980 −0.486705 −0.243353 0.969938i \(-0.578247\pi\)
−0.243353 + 0.969938i \(0.578247\pi\)
\(212\) 0 0
\(213\) −27.4922 −1.88373
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −33.0725 −2.24511
\(218\) 0 0
\(219\) 15.3872 1.03977
\(220\) 0 0
\(221\) 6.94055 0.466872
\(222\) 0 0
\(223\) −8.58817 −0.575106 −0.287553 0.957765i \(-0.592842\pi\)
−0.287553 + 0.957765i \(0.592842\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.2068 −0.942937 −0.471469 0.881883i \(-0.656276\pi\)
−0.471469 + 0.881883i \(0.656276\pi\)
\(228\) 0 0
\(229\) −7.49721 −0.495429 −0.247715 0.968833i \(-0.579680\pi\)
−0.247715 + 0.968833i \(0.579680\pi\)
\(230\) 0 0
\(231\) −15.3718 −1.01139
\(232\) 0 0
\(233\) 16.9301 1.10912 0.554562 0.832142i \(-0.312886\pi\)
0.554562 + 0.832142i \(0.312886\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 17.3131 1.12460
\(238\) 0 0
\(239\) −17.7850 −1.15042 −0.575208 0.818007i \(-0.695079\pi\)
−0.575208 + 0.818007i \(0.695079\pi\)
\(240\) 0 0
\(241\) −9.55263 −0.615339 −0.307669 0.951493i \(-0.599549\pi\)
−0.307669 + 0.951493i \(0.599549\pi\)
\(242\) 0 0
\(243\) 3.96721 0.254497
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −11.2673 −0.716922
\(248\) 0 0
\(249\) 2.78153 0.176272
\(250\) 0 0
\(251\) −13.2075 −0.833651 −0.416826 0.908986i \(-0.636857\pi\)
−0.416826 + 0.908986i \(0.636857\pi\)
\(252\) 0 0
\(253\) 2.10339 0.132239
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.41506 0.213025 0.106513 0.994311i \(-0.466032\pi\)
0.106513 + 0.994311i \(0.466032\pi\)
\(258\) 0 0
\(259\) −20.2661 −1.25927
\(260\) 0 0
\(261\) 2.31608 0.143362
\(262\) 0 0
\(263\) −18.6589 −1.15056 −0.575279 0.817957i \(-0.695106\pi\)
−0.575279 + 0.817957i \(0.695106\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −19.4815 −1.19225
\(268\) 0 0
\(269\) −24.2224 −1.47687 −0.738434 0.674326i \(-0.764434\pi\)
−0.738434 + 0.674326i \(0.764434\pi\)
\(270\) 0 0
\(271\) −10.1201 −0.614749 −0.307375 0.951589i \(-0.599450\pi\)
−0.307375 + 0.951589i \(0.599450\pi\)
\(272\) 0 0
\(273\) 39.1475 2.36932
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.5298 1.41377 0.706884 0.707329i \(-0.250100\pi\)
0.706884 + 0.707329i \(0.250100\pi\)
\(278\) 0 0
\(279\) −3.19684 −0.191390
\(280\) 0 0
\(281\) 1.77314 0.105776 0.0528882 0.998600i \(-0.483157\pi\)
0.0528882 + 0.998600i \(0.483157\pi\)
\(282\) 0 0
\(283\) −5.96641 −0.354666 −0.177333 0.984151i \(-0.556747\pi\)
−0.177333 + 0.984151i \(0.556747\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −33.1323 −1.95574
\(288\) 0 0
\(289\) −15.3212 −0.901250
\(290\) 0 0
\(291\) −0.620765 −0.0363899
\(292\) 0 0
\(293\) −12.2404 −0.715090 −0.357545 0.933896i \(-0.616386\pi\)
−0.357545 + 0.933896i \(0.616386\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.1221 0.587346
\(298\) 0 0
\(299\) −5.35673 −0.309788
\(300\) 0 0
\(301\) −30.9178 −1.78207
\(302\) 0 0
\(303\) 11.3891 0.654288
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.82713 0.503791 0.251896 0.967754i \(-0.418946\pi\)
0.251896 + 0.967754i \(0.418946\pi\)
\(308\) 0 0
\(309\) 23.2276 1.32137
\(310\) 0 0
\(311\) −13.9202 −0.789345 −0.394672 0.918822i \(-0.629142\pi\)
−0.394672 + 0.918822i \(0.629142\pi\)
\(312\) 0 0
\(313\) −23.5030 −1.32847 −0.664235 0.747523i \(-0.731243\pi\)
−0.664235 + 0.747523i \(0.731243\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.3279 −0.748570 −0.374285 0.927314i \(-0.622112\pi\)
−0.374285 + 0.927314i \(0.622112\pi\)
\(318\) 0 0
\(319\) 12.6862 0.710290
\(320\) 0 0
\(321\) 23.5354 1.31362
\(322\) 0 0
\(323\) −2.72530 −0.151640
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −26.3362 −1.45639
\(328\) 0 0
\(329\) −45.2749 −2.49609
\(330\) 0 0
\(331\) −0.270137 −0.0148481 −0.00742404 0.999972i \(-0.502363\pi\)
−0.00742404 + 0.999972i \(0.502363\pi\)
\(332\) 0 0
\(333\) −1.95896 −0.107350
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.0205 1.09058 0.545292 0.838246i \(-0.316419\pi\)
0.545292 + 0.838246i \(0.316419\pi\)
\(338\) 0 0
\(339\) 8.45213 0.459057
\(340\) 0 0
\(341\) −17.5105 −0.948247
\(342\) 0 0
\(343\) 7.08149 0.382364
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.1156 0.811448 0.405724 0.913996i \(-0.367019\pi\)
0.405724 + 0.913996i \(0.367019\pi\)
\(348\) 0 0
\(349\) −14.5673 −0.779771 −0.389886 0.920863i \(-0.627485\pi\)
−0.389886 + 0.920863i \(0.627485\pi\)
\(350\) 0 0
\(351\) −25.7782 −1.37594
\(352\) 0 0
\(353\) −18.5376 −0.986656 −0.493328 0.869843i \(-0.664220\pi\)
−0.493328 + 0.869843i \(0.664220\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 9.46886 0.501145
\(358\) 0 0
\(359\) −24.1834 −1.27635 −0.638175 0.769891i \(-0.720311\pi\)
−0.638175 + 0.769891i \(0.720311\pi\)
\(360\) 0 0
\(361\) −14.5757 −0.767144
\(362\) 0 0
\(363\) 12.0965 0.634903
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −5.79604 −0.302551 −0.151275 0.988492i \(-0.548338\pi\)
−0.151275 + 0.988492i \(0.548338\pi\)
\(368\) 0 0
\(369\) −3.20262 −0.166722
\(370\) 0 0
\(371\) 2.27901 0.118320
\(372\) 0 0
\(373\) 25.5753 1.32424 0.662119 0.749399i \(-0.269657\pi\)
0.662119 + 0.749399i \(0.269657\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −32.3081 −1.66395
\(378\) 0 0
\(379\) −9.82223 −0.504534 −0.252267 0.967658i \(-0.581176\pi\)
−0.252267 + 0.967658i \(0.581176\pi\)
\(380\) 0 0
\(381\) 23.2932 1.19335
\(382\) 0 0
\(383\) 33.9797 1.73628 0.868141 0.496319i \(-0.165315\pi\)
0.868141 + 0.496319i \(0.165315\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.98857 −0.151918
\(388\) 0 0
\(389\) −24.6744 −1.25104 −0.625521 0.780207i \(-0.715114\pi\)
−0.625521 + 0.780207i \(0.715114\pi\)
\(390\) 0 0
\(391\) −1.29567 −0.0655247
\(392\) 0 0
\(393\) −8.61599 −0.434619
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.17340 0.159268 0.0796342 0.996824i \(-0.474625\pi\)
0.0796342 + 0.996824i \(0.474625\pi\)
\(398\) 0 0
\(399\) −15.3718 −0.769552
\(400\) 0 0
\(401\) 4.07720 0.203606 0.101803 0.994805i \(-0.467539\pi\)
0.101803 + 0.994805i \(0.467539\pi\)
\(402\) 0 0
\(403\) 44.5942 2.22140
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.7301 −0.531869
\(408\) 0 0
\(409\) 18.5426 0.916871 0.458435 0.888728i \(-0.348410\pi\)
0.458435 + 0.888728i \(0.348410\pi\)
\(410\) 0 0
\(411\) −0.861138 −0.0424768
\(412\) 0 0
\(413\) −36.4676 −1.79445
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.35982 −0.0665906
\(418\) 0 0
\(419\) −3.22587 −0.157594 −0.0787970 0.996891i \(-0.525108\pi\)
−0.0787970 + 0.996891i \(0.525108\pi\)
\(420\) 0 0
\(421\) 18.0595 0.880167 0.440084 0.897957i \(-0.354949\pi\)
0.440084 + 0.897957i \(0.354949\pi\)
\(422\) 0 0
\(423\) −4.37635 −0.212785
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 53.6909 2.59828
\(428\) 0 0
\(429\) 20.7270 1.00071
\(430\) 0 0
\(431\) −9.62458 −0.463600 −0.231800 0.972763i \(-0.574461\pi\)
−0.231800 + 0.972763i \(0.574461\pi\)
\(432\) 0 0
\(433\) 1.83990 0.0884200 0.0442100 0.999022i \(-0.485923\pi\)
0.0442100 + 0.999022i \(0.485923\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.10339 0.100619
\(438\) 0 0
\(439\) 6.73907 0.321638 0.160819 0.986984i \(-0.448586\pi\)
0.160819 + 0.986984i \(0.448586\pi\)
\(440\) 0 0
\(441\) 3.37258 0.160599
\(442\) 0 0
\(443\) −11.4099 −0.542103 −0.271051 0.962565i \(-0.587371\pi\)
−0.271051 + 0.962565i \(0.587371\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −26.8398 −1.26948
\(448\) 0 0
\(449\) 27.8081 1.31235 0.656173 0.754610i \(-0.272174\pi\)
0.656173 + 0.754610i \(0.272174\pi\)
\(450\) 0 0
\(451\) −17.5422 −0.826028
\(452\) 0 0
\(453\) −39.8974 −1.87454
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −26.6835 −1.24820 −0.624102 0.781343i \(-0.714535\pi\)
−0.624102 + 0.781343i \(0.714535\pi\)
\(458\) 0 0
\(459\) −6.23513 −0.291031
\(460\) 0 0
\(461\) −23.0398 −1.07307 −0.536534 0.843879i \(-0.680267\pi\)
−0.536534 + 0.843879i \(0.680267\pi\)
\(462\) 0 0
\(463\) 23.3998 1.08748 0.543740 0.839253i \(-0.317008\pi\)
0.543740 + 0.839253i \(0.317008\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.29380 −0.198693 −0.0993467 0.995053i \(-0.531675\pi\)
−0.0993467 + 0.995053i \(0.531675\pi\)
\(468\) 0 0
\(469\) 62.8936 2.90416
\(470\) 0 0
\(471\) 14.1877 0.653735
\(472\) 0 0
\(473\) −16.3697 −0.752680
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.220293 0.0100865
\(478\) 0 0
\(479\) −26.6694 −1.21856 −0.609278 0.792957i \(-0.708541\pi\)
−0.609278 + 0.792957i \(0.708541\pi\)
\(480\) 0 0
\(481\) 27.3264 1.24597
\(482\) 0 0
\(483\) −7.30809 −0.332530
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.67114 0.392927 0.196463 0.980511i \(-0.437054\pi\)
0.196463 + 0.980511i \(0.437054\pi\)
\(488\) 0 0
\(489\) −8.63083 −0.390300
\(490\) 0 0
\(491\) −25.0900 −1.13229 −0.566147 0.824304i \(-0.691566\pi\)
−0.566147 + 0.824304i \(0.691566\pi\)
\(492\) 0 0
\(493\) −7.81456 −0.351950
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 59.3720 2.66320
\(498\) 0 0
\(499\) −34.9323 −1.56379 −0.781893 0.623412i \(-0.785746\pi\)
−0.781893 + 0.623412i \(0.785746\pi\)
\(500\) 0 0
\(501\) 21.4747 0.959418
\(502\) 0 0
\(503\) −25.7782 −1.14939 −0.574695 0.818367i \(-0.694879\pi\)
−0.574695 + 0.818367i \(0.694879\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −28.8713 −1.28222
\(508\) 0 0
\(509\) 4.21667 0.186900 0.0934502 0.995624i \(-0.470210\pi\)
0.0934502 + 0.995624i \(0.470210\pi\)
\(510\) 0 0
\(511\) −33.2302 −1.47002
\(512\) 0 0
\(513\) 10.1221 0.446903
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −23.9712 −1.05425
\(518\) 0 0
\(519\) −15.3001 −0.671600
\(520\) 0 0
\(521\) 29.3112 1.28415 0.642073 0.766644i \(-0.278075\pi\)
0.642073 + 0.766644i \(0.278075\pi\)
\(522\) 0 0
\(523\) 31.1549 1.36231 0.681154 0.732140i \(-0.261478\pi\)
0.681154 + 0.732140i \(0.261478\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.7863 0.469858
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.52502 −0.152973
\(532\) 0 0
\(533\) 44.6748 1.93508
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.49655 0.323500
\(538\) 0 0
\(539\) 18.4731 0.795692
\(540\) 0 0
\(541\) 15.7393 0.676683 0.338342 0.941023i \(-0.390134\pi\)
0.338342 + 0.941023i \(0.390134\pi\)
\(542\) 0 0
\(543\) −4.64212 −0.199213
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 39.2935 1.68007 0.840034 0.542534i \(-0.182535\pi\)
0.840034 + 0.542534i \(0.182535\pi\)
\(548\) 0 0
\(549\) 5.18985 0.221497
\(550\) 0 0
\(551\) 12.6862 0.540450
\(552\) 0 0
\(553\) −37.3892 −1.58995
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −38.1005 −1.61437 −0.807186 0.590297i \(-0.799010\pi\)
−0.807186 + 0.590297i \(0.799010\pi\)
\(558\) 0 0
\(559\) 41.6889 1.76325
\(560\) 0 0
\(561\) 5.01337 0.211665
\(562\) 0 0
\(563\) −32.6868 −1.37758 −0.688792 0.724959i \(-0.741859\pi\)
−0.688792 + 0.724959i \(0.741859\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −39.7454 −1.66915
\(568\) 0 0
\(569\) 21.0256 0.881438 0.440719 0.897645i \(-0.354724\pi\)
0.440719 + 0.897645i \(0.354724\pi\)
\(570\) 0 0
\(571\) −7.43876 −0.311303 −0.155651 0.987812i \(-0.549748\pi\)
−0.155651 + 0.987812i \(0.549748\pi\)
\(572\) 0 0
\(573\) 44.0357 1.83962
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.4148 1.26618 0.633091 0.774077i \(-0.281786\pi\)
0.633091 + 0.774077i \(0.281786\pi\)
\(578\) 0 0
\(579\) 34.5348 1.43522
\(580\) 0 0
\(581\) −6.00698 −0.249212
\(582\) 0 0
\(583\) 1.20664 0.0499739
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.7544 1.39319 0.696596 0.717464i \(-0.254697\pi\)
0.696596 + 0.717464i \(0.254697\pi\)
\(588\) 0 0
\(589\) −17.5105 −0.721508
\(590\) 0 0
\(591\) 29.2587 1.20354
\(592\) 0 0
\(593\) −27.3980 −1.12510 −0.562550 0.826763i \(-0.690180\pi\)
−0.562550 + 0.826763i \(0.690180\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.1666 −0.743507
\(598\) 0 0
\(599\) 37.2927 1.52374 0.761869 0.647731i \(-0.224282\pi\)
0.761869 + 0.647731i \(0.224282\pi\)
\(600\) 0 0
\(601\) −2.26039 −0.0922032 −0.0461016 0.998937i \(-0.514680\pi\)
−0.0461016 + 0.998937i \(0.514680\pi\)
\(602\) 0 0
\(603\) 6.07941 0.247573
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 23.4782 0.952951 0.476476 0.879188i \(-0.341914\pi\)
0.476476 + 0.879188i \(0.341914\pi\)
\(608\) 0 0
\(609\) −44.0773 −1.78610
\(610\) 0 0
\(611\) 61.0477 2.46972
\(612\) 0 0
\(613\) 25.0418 1.01143 0.505714 0.862701i \(-0.331229\pi\)
0.505714 + 0.862701i \(0.331229\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.2705 0.735541 0.367771 0.929916i \(-0.380121\pi\)
0.367771 + 0.929916i \(0.380121\pi\)
\(618\) 0 0
\(619\) 7.17743 0.288485 0.144243 0.989542i \(-0.453925\pi\)
0.144243 + 0.989542i \(0.453925\pi\)
\(620\) 0 0
\(621\) 4.81229 0.193111
\(622\) 0 0
\(623\) 42.0721 1.68558
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −8.13872 −0.325029
\(628\) 0 0
\(629\) 6.60960 0.263542
\(630\) 0 0
\(631\) −28.7621 −1.14500 −0.572500 0.819904i \(-0.694026\pi\)
−0.572500 + 0.819904i \(0.694026\pi\)
\(632\) 0 0
\(633\) 13.0054 0.516917
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −47.0457 −1.86402
\(638\) 0 0
\(639\) 5.73900 0.227031
\(640\) 0 0
\(641\) −40.7383 −1.60907 −0.804533 0.593907i \(-0.797585\pi\)
−0.804533 + 0.593907i \(0.797585\pi\)
\(642\) 0 0
\(643\) 8.24043 0.324971 0.162485 0.986711i \(-0.448049\pi\)
0.162485 + 0.986711i \(0.448049\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.1134 1.30182 0.650911 0.759154i \(-0.274387\pi\)
0.650911 + 0.759154i \(0.274387\pi\)
\(648\) 0 0
\(649\) −19.3081 −0.757909
\(650\) 0 0
\(651\) 60.8391 2.38447
\(652\) 0 0
\(653\) −14.5452 −0.569199 −0.284599 0.958647i \(-0.591861\pi\)
−0.284599 + 0.958647i \(0.591861\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.21209 −0.125316
\(658\) 0 0
\(659\) −24.4498 −0.952427 −0.476214 0.879330i \(-0.657991\pi\)
−0.476214 + 0.879330i \(0.657991\pi\)
\(660\) 0 0
\(661\) 1.43400 0.0557763 0.0278882 0.999611i \(-0.491122\pi\)
0.0278882 + 0.999611i \(0.491122\pi\)
\(662\) 0 0
\(663\) −12.7676 −0.495853
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.03130 0.233533
\(668\) 0 0
\(669\) 15.7985 0.610806
\(670\) 0 0
\(671\) 28.4271 1.09742
\(672\) 0 0
\(673\) 3.39873 0.131011 0.0655056 0.997852i \(-0.479134\pi\)
0.0655056 + 0.997852i \(0.479134\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.7751 0.606287 0.303144 0.952945i \(-0.401964\pi\)
0.303144 + 0.952945i \(0.401964\pi\)
\(678\) 0 0
\(679\) 1.34060 0.0514475
\(680\) 0 0
\(681\) 26.1343 1.00147
\(682\) 0 0
\(683\) 4.99979 0.191312 0.0956559 0.995414i \(-0.469505\pi\)
0.0956559 + 0.995414i \(0.469505\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.7916 0.526183
\(688\) 0 0
\(689\) −3.07296 −0.117071
\(690\) 0 0
\(691\) 40.6864 1.54779 0.773893 0.633317i \(-0.218307\pi\)
0.773893 + 0.633317i \(0.218307\pi\)
\(692\) 0 0
\(693\) 3.20886 0.121895
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.8058 0.409298
\(698\) 0 0
\(699\) −31.1440 −1.17797
\(700\) 0 0
\(701\) −1.30372 −0.0492407 −0.0246204 0.999697i \(-0.507838\pi\)
−0.0246204 + 0.999697i \(0.507838\pi\)
\(702\) 0 0
\(703\) −10.7301 −0.404692
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −24.5959 −0.925024
\(708\) 0 0
\(709\) 52.5228 1.97253 0.986267 0.165157i \(-0.0528131\pi\)
0.986267 + 0.165157i \(0.0528131\pi\)
\(710\) 0 0
\(711\) −3.61411 −0.135540
\(712\) 0 0
\(713\) −8.32489 −0.311770
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 32.7167 1.22183
\(718\) 0 0
\(719\) −24.2329 −0.903735 −0.451868 0.892085i \(-0.649242\pi\)
−0.451868 + 0.892085i \(0.649242\pi\)
\(720\) 0 0
\(721\) −50.1622 −1.86814
\(722\) 0 0
\(723\) 17.5727 0.653536
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −37.0943 −1.37575 −0.687875 0.725829i \(-0.741456\pi\)
−0.687875 + 0.725829i \(0.741456\pi\)
\(728\) 0 0
\(729\) 22.7158 0.841324
\(730\) 0 0
\(731\) 10.0836 0.372954
\(732\) 0 0
\(733\) −46.3065 −1.71037 −0.855184 0.518325i \(-0.826556\pi\)
−0.855184 + 0.518325i \(0.826556\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.2996 1.22660
\(738\) 0 0
\(739\) −32.3516 −1.19007 −0.595037 0.803698i \(-0.702863\pi\)
−0.595037 + 0.803698i \(0.702863\pi\)
\(740\) 0 0
\(741\) 20.7270 0.761425
\(742\) 0 0
\(743\) 1.73343 0.0635935 0.0317967 0.999494i \(-0.489877\pi\)
0.0317967 + 0.999494i \(0.489877\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.580645 −0.0212447
\(748\) 0 0
\(749\) −50.8268 −1.85717
\(750\) 0 0
\(751\) 42.9983 1.56903 0.784515 0.620110i \(-0.212912\pi\)
0.784515 + 0.620110i \(0.212912\pi\)
\(752\) 0 0
\(753\) 24.2961 0.885400
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −48.3219 −1.75629 −0.878144 0.478397i \(-0.841218\pi\)
−0.878144 + 0.478397i \(0.841218\pi\)
\(758\) 0 0
\(759\) −3.86933 −0.140448
\(760\) 0 0
\(761\) 19.2528 0.697915 0.348958 0.937139i \(-0.386536\pi\)
0.348958 + 0.937139i \(0.386536\pi\)
\(762\) 0 0
\(763\) 56.8754 2.05903
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 49.1722 1.77550
\(768\) 0 0
\(769\) 9.05816 0.326645 0.163323 0.986573i \(-0.447779\pi\)
0.163323 + 0.986573i \(0.447779\pi\)
\(770\) 0 0
\(771\) −6.28223 −0.226249
\(772\) 0 0
\(773\) −29.1276 −1.04765 −0.523824 0.851827i \(-0.675495\pi\)
−0.523824 + 0.851827i \(0.675495\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 37.2809 1.33744
\(778\) 0 0
\(779\) −17.5422 −0.628513
\(780\) 0 0
\(781\) 31.4350 1.12483
\(782\) 0 0
\(783\) 29.0244 1.03725
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 23.8771 0.851128 0.425564 0.904928i \(-0.360076\pi\)
0.425564 + 0.904928i \(0.360076\pi\)
\(788\) 0 0
\(789\) 34.3243 1.22198
\(790\) 0 0
\(791\) −18.2532 −0.649008
\(792\) 0 0
\(793\) −72.3956 −2.57084
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.8896 0.704526 0.352263 0.935901i \(-0.385412\pi\)
0.352263 + 0.935901i \(0.385412\pi\)
\(798\) 0 0
\(799\) 14.7660 0.522383
\(800\) 0 0
\(801\) 4.06677 0.143692
\(802\) 0 0
\(803\) −17.5940 −0.620879
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 44.5588 1.56854
\(808\) 0 0
\(809\) 26.3598 0.926760 0.463380 0.886160i \(-0.346637\pi\)
0.463380 + 0.886160i \(0.346637\pi\)
\(810\) 0 0
\(811\) −14.2482 −0.500323 −0.250161 0.968204i \(-0.580484\pi\)
−0.250161 + 0.968204i \(0.580484\pi\)
\(812\) 0 0
\(813\) 18.6165 0.652910
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −16.3697 −0.572703
\(818\) 0 0
\(819\) −8.17205 −0.285555
\(820\) 0 0
\(821\) 5.56520 0.194227 0.0971134 0.995273i \(-0.469039\pi\)
0.0971134 + 0.995273i \(0.469039\pi\)
\(822\) 0 0
\(823\) 34.1312 1.18974 0.594869 0.803823i \(-0.297204\pi\)
0.594869 + 0.803823i \(0.297204\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.5085 0.817472 0.408736 0.912653i \(-0.365970\pi\)
0.408736 + 0.912653i \(0.365970\pi\)
\(828\) 0 0
\(829\) −31.7442 −1.10252 −0.551260 0.834333i \(-0.685853\pi\)
−0.551260 + 0.834333i \(0.685853\pi\)
\(830\) 0 0
\(831\) −43.2846 −1.50153
\(832\) 0 0
\(833\) −11.3792 −0.394267
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −40.0618 −1.38474
\(838\) 0 0
\(839\) 24.5409 0.847247 0.423623 0.905838i \(-0.360758\pi\)
0.423623 + 0.905838i \(0.360758\pi\)
\(840\) 0 0
\(841\) 7.37661 0.254366
\(842\) 0 0
\(843\) −3.26180 −0.112343
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −26.1236 −0.897618
\(848\) 0 0
\(849\) 10.9756 0.376682
\(850\) 0 0
\(851\) −5.10131 −0.174871
\(852\) 0 0
\(853\) 23.7499 0.813183 0.406591 0.913610i \(-0.366717\pi\)
0.406591 + 0.913610i \(0.366717\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.0643 1.47105 0.735524 0.677499i \(-0.236936\pi\)
0.735524 + 0.677499i \(0.236936\pi\)
\(858\) 0 0
\(859\) −25.0591 −0.855006 −0.427503 0.904014i \(-0.640607\pi\)
−0.427503 + 0.904014i \(0.640607\pi\)
\(860\) 0 0
\(861\) 60.9490 2.07714
\(862\) 0 0
\(863\) −46.0967 −1.56915 −0.784575 0.620033i \(-0.787119\pi\)
−0.784575 + 0.620033i \(0.787119\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 28.1845 0.957195
\(868\) 0 0
\(869\) −19.7960 −0.671535
\(870\) 0 0
\(871\) −84.8044 −2.87349
\(872\) 0 0
\(873\) 0.129585 0.00438578
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 35.1644 1.18742 0.593709 0.804680i \(-0.297663\pi\)
0.593709 + 0.804680i \(0.297663\pi\)
\(878\) 0 0
\(879\) 22.5170 0.759480
\(880\) 0 0
\(881\) −38.8350 −1.30838 −0.654192 0.756329i \(-0.726991\pi\)
−0.654192 + 0.756329i \(0.726991\pi\)
\(882\) 0 0
\(883\) −10.5954 −0.356562 −0.178281 0.983980i \(-0.557054\pi\)
−0.178281 + 0.983980i \(0.557054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.8733 0.902316 0.451158 0.892444i \(-0.351011\pi\)
0.451158 + 0.892444i \(0.351011\pi\)
\(888\) 0 0
\(889\) −50.3040 −1.68714
\(890\) 0 0
\(891\) −21.0435 −0.704984
\(892\) 0 0
\(893\) −23.9712 −0.802165
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.85407 0.329018
\(898\) 0 0
\(899\) −50.2099 −1.67459
\(900\) 0 0
\(901\) −0.743278 −0.0247622
\(902\) 0 0
\(903\) 56.8754 1.89270
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −58.3210 −1.93652 −0.968259 0.249949i \(-0.919586\pi\)
−0.968259 + 0.249949i \(0.919586\pi\)
\(908\) 0 0
\(909\) −2.37748 −0.0788561
\(910\) 0 0
\(911\) 40.3287 1.33615 0.668074 0.744095i \(-0.267119\pi\)
0.668074 + 0.744095i \(0.267119\pi\)
\(912\) 0 0
\(913\) −3.18045 −0.105257
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.6071 0.614459
\(918\) 0 0
\(919\) −37.6333 −1.24141 −0.620704 0.784045i \(-0.713153\pi\)
−0.620704 + 0.784045i \(0.713153\pi\)
\(920\) 0 0
\(921\) −16.2381 −0.535064
\(922\) 0 0
\(923\) −80.0559 −2.63507
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.84877 −0.159254
\(928\) 0 0
\(929\) 30.7190 1.00786 0.503929 0.863745i \(-0.331887\pi\)
0.503929 + 0.863745i \(0.331887\pi\)
\(930\) 0 0
\(931\) 18.4731 0.605431
\(932\) 0 0
\(933\) 25.6072 0.838343
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 40.5843 1.32583 0.662915 0.748694i \(-0.269319\pi\)
0.662915 + 0.748694i \(0.269319\pi\)
\(938\) 0 0
\(939\) 43.2355 1.41094
\(940\) 0 0
\(941\) 36.5104 1.19020 0.595102 0.803650i \(-0.297112\pi\)
0.595102 + 0.803650i \(0.297112\pi\)
\(942\) 0 0
\(943\) −8.33994 −0.271586
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.9681 −1.13631 −0.568155 0.822921i \(-0.692343\pi\)
−0.568155 + 0.822921i \(0.692343\pi\)
\(948\) 0 0
\(949\) 44.8069 1.45449
\(950\) 0 0
\(951\) 24.5176 0.795038
\(952\) 0 0
\(953\) −35.7164 −1.15697 −0.578484 0.815694i \(-0.696355\pi\)
−0.578484 + 0.815694i \(0.696355\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −23.3371 −0.754382
\(958\) 0 0
\(959\) 1.85971 0.0600532
\(960\) 0 0
\(961\) 38.3038 1.23561
\(962\) 0 0
\(963\) −4.91301 −0.158320
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 43.0422 1.38414 0.692072 0.721829i \(-0.256698\pi\)
0.692072 + 0.721829i \(0.256698\pi\)
\(968\) 0 0
\(969\) 5.01337 0.161053
\(970\) 0 0
\(971\) −9.76196 −0.313276 −0.156638 0.987656i \(-0.550066\pi\)
−0.156638 + 0.987656i \(0.550066\pi\)
\(972\) 0 0
\(973\) 2.93666 0.0941449
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −24.0469 −0.769329 −0.384665 0.923056i \(-0.625683\pi\)
−0.384665 + 0.923056i \(0.625683\pi\)
\(978\) 0 0
\(979\) 22.2754 0.711926
\(980\) 0 0
\(981\) 5.49768 0.175527
\(982\) 0 0
\(983\) 19.2418 0.613717 0.306859 0.951755i \(-0.400722\pi\)
0.306859 + 0.951755i \(0.400722\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 83.2862 2.65103
\(988\) 0 0
\(989\) −7.78253 −0.247470
\(990\) 0 0
\(991\) −38.1944 −1.21328 −0.606642 0.794975i \(-0.707484\pi\)
−0.606642 + 0.794975i \(0.707484\pi\)
\(992\) 0 0
\(993\) 0.496936 0.0157698
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 41.5981 1.31742 0.658712 0.752395i \(-0.271101\pi\)
0.658712 + 0.752395i \(0.271101\pi\)
\(998\) 0 0
\(999\) −24.5490 −0.776696
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.bd.1.2 5
4.3 odd 2 9200.2.a.cv.1.4 5
5.2 odd 4 4600.2.e.w.4049.8 10
5.3 odd 4 4600.2.e.w.4049.3 10
5.4 even 2 4600.2.a.bf.1.4 yes 5
20.19 odd 2 9200.2.a.ct.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.2 5 1.1 even 1 trivial
4600.2.a.bf.1.4 yes 5 5.4 even 2
4600.2.e.w.4049.3 10 5.3 odd 4
4600.2.e.w.4049.8 10 5.2 odd 4
9200.2.a.ct.1.2 5 20.19 odd 2
9200.2.a.cv.1.4 5 4.3 odd 2