Properties

Label 4600.2.e.w.4049.8
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.8
Root \(1.83957i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.83957i q^{3} +3.97272i q^{7} -0.384010 q^{9} +O(q^{10})\) \(q+1.83957i q^{3} +3.97272i q^{7} -0.384010 q^{9} +2.10339 q^{11} +5.35673i q^{13} -1.29567i q^{17} -2.10339 q^{19} -7.30809 q^{21} -1.00000i q^{23} +4.81229i q^{27} -6.03130 q^{29} -8.32489 q^{31} +3.86933i q^{33} -5.10131i q^{37} -9.85407 q^{39} -8.33994 q^{41} +7.78253i q^{43} -11.3964i q^{47} -8.78253 q^{49} +2.38347 q^{51} -0.573664i q^{53} -3.86933i q^{57} +9.17951 q^{59} +13.5149 q^{61} -1.52557i q^{63} +15.8314i q^{67} +1.83957 q^{69} +14.9449 q^{71} +8.36459i q^{73} +8.35619i q^{77} +9.41149 q^{79} -10.0046 q^{81} +1.51206i q^{83} -11.0950i q^{87} -10.5903 q^{89} -21.2808 q^{91} -15.3142i q^{93} +0.337451i q^{97} -0.807724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} + O(q^{10}) \) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.83957i 1.06208i 0.847348 + 0.531038i \(0.178198\pi\)
−0.847348 + 0.531038i \(0.821802\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.97272i 1.50155i 0.660559 + 0.750774i \(0.270319\pi\)
−0.660559 + 0.750774i \(0.729681\pi\)
\(8\) 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
\(13\) 5.35673i 1.48569i 0.669463 + 0.742845i \(0.266524\pi\)
−0.669463 + 0.742845i \(0.733476\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.29567i − 0.314246i −0.987579 0.157123i \(-0.949778\pi\)
0.987579 0.157123i \(-0.0502218\pi\)
\(18\) 0 0
\(19\) −2.10339 −0.482551 −0.241276 0.970457i \(-0.577566\pi\)
−0.241276 + 0.970457i \(0.577566\pi\)
\(20\) 0 0
\(21\) −7.30809 −1.59476
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.81229i 0.926126i
\(28\) 0 0
\(29\) −6.03130 −1.11998 −0.559992 0.828498i \(-0.689196\pi\)
−0.559992 + 0.828498i \(0.689196\pi\)
\(30\) 0 0
\(31\) −8.32489 −1.49519 −0.747597 0.664153i \(-0.768793\pi\)
−0.747597 + 0.664153i \(0.768793\pi\)
\(32\) 0 0
\(33\) 3.86933i 0.673564i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.10131i − 0.838650i −0.907836 0.419325i \(-0.862267\pi\)
0.907836 0.419325i \(-0.137733\pi\)
\(38\) 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
\(43\) 7.78253i 1.18682i 0.804899 + 0.593412i \(0.202220\pi\)
−0.804899 + 0.593412i \(0.797780\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 11.3964i − 1.66234i −0.556018 0.831171i \(-0.687671\pi\)
0.556018 0.831171i \(-0.312329\pi\)
\(48\) 0 0
\(49\) −8.78253 −1.25465
\(50\) 0 0
\(51\) 2.38347 0.333752
\(52\) 0 0
\(53\) − 0.573664i − 0.0787988i −0.999224 0.0393994i \(-0.987456\pi\)
0.999224 0.0393994i \(-0.0125445\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.86933i − 0.512505i
\(58\) 0 0
\(59\) 9.17951 1.19507 0.597535 0.801843i \(-0.296147\pi\)
0.597535 + 0.801843i \(0.296147\pi\)
\(60\) 0 0
\(61\) 13.5149 1.73040 0.865201 0.501425i \(-0.167191\pi\)
0.865201 + 0.501425i \(0.167191\pi\)
\(62\) 0 0
\(63\) − 1.52557i − 0.192203i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8314i 1.93411i 0.254569 + 0.967055i \(0.418066\pi\)
−0.254569 + 0.967055i \(0.581934\pi\)
\(68\) 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
\(73\) 8.36459i 0.979002i 0.872003 + 0.489501i \(0.162821\pi\)
−0.872003 + 0.489501i \(0.837179\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.35619i 0.952276i
\(78\) 0 0
\(79\) 9.41149 1.05887 0.529437 0.848349i \(-0.322403\pi\)
0.529437 + 0.848349i \(0.322403\pi\)
\(80\) 0 0
\(81\) −10.0046 −1.11162
\(82\) 0 0
\(83\) 1.51206i 0.165970i 0.996551 + 0.0829849i \(0.0264453\pi\)
−0.996551 + 0.0829849i \(0.973555\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 11.0950i − 1.18951i
\(88\) 0 0
\(89\) −10.5903 −1.12256 −0.561282 0.827624i \(-0.689692\pi\)
−0.561282 + 0.827624i \(0.689692\pi\)
\(90\) 0 0
\(91\) −21.2808 −2.23084
\(92\) 0 0
\(93\) − 15.3142i − 1.58801i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.337451i 0.0342630i 0.999853 + 0.0171315i \(0.00545339\pi\)
−0.999853 + 0.0171315i \(0.994547\pi\)
\(98\) 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
\(103\) 12.6267i 1.24414i 0.782961 + 0.622071i \(0.213708\pi\)
−0.782961 + 0.622071i \(0.786292\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.7940i − 1.23684i −0.785848 0.618419i \(-0.787773\pi\)
0.785848 0.618419i \(-0.212227\pi\)
\(108\) 0 0
\(109\) −14.3165 −1.37127 −0.685635 0.727945i \(-0.740476\pi\)
−0.685635 + 0.727945i \(0.740476\pi\)
\(110\) 0 0
\(111\) 9.38421 0.890710
\(112\) 0 0
\(113\) 4.59463i 0.432226i 0.976368 + 0.216113i \(0.0693380\pi\)
−0.976368 + 0.216113i \(0.930662\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.05704i − 0.190173i
\(118\) 0 0
\(119\) 5.14733 0.471855
\(120\) 0 0
\(121\) −6.57574 −0.597795
\(122\) 0 0
\(123\) − 15.3419i − 1.38333i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.6623i − 1.12360i −0.827273 0.561801i \(-0.810109\pi\)
0.827273 0.561801i \(-0.189891\pi\)
\(128\) 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
\(133\) − 8.35619i − 0.724574i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.468120i 0.0399942i 0.999800 + 0.0199971i \(0.00636570\pi\)
−0.999800 + 0.0199971i \(0.993634\pi\)
\(138\) 0 0
\(139\) −0.739205 −0.0626985 −0.0313493 0.999508i \(-0.509980\pi\)
−0.0313493 + 0.999508i \(0.509980\pi\)
\(140\) 0 0
\(141\) 20.9645 1.76553
\(142\) 0 0
\(143\) 11.2673i 0.942220i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 16.1561i − 1.33253i
\(148\) 0 0
\(149\) −14.5903 −1.19528 −0.597640 0.801765i \(-0.703895\pi\)
−0.597640 + 0.801765i \(0.703895\pi\)
\(150\) 0 0
\(151\) 21.6885 1.76498 0.882491 0.470329i \(-0.155865\pi\)
0.882491 + 0.470329i \(0.155865\pi\)
\(152\) 0 0
\(153\) 0.497550i 0.0402245i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 7.71253i − 0.615527i −0.951463 0.307763i \(-0.900420\pi\)
0.951463 0.307763i \(-0.0995805\pi\)
\(158\) 0 0
\(159\) 1.05529 0.0836902
\(160\) 0 0
\(161\) 3.97272 0.313094
\(162\) 0 0
\(163\) − 4.69177i − 0.367488i −0.982974 0.183744i \(-0.941178\pi\)
0.982974 0.183744i \(-0.0588217\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 11.6738i − 0.903343i −0.892184 0.451671i \(-0.850828\pi\)
0.892184 0.451671i \(-0.149172\pi\)
\(168\) 0 0
\(169\) −15.6946 −1.20728
\(170\) 0 0
\(171\) 0.807724 0.0617682
\(172\) 0 0
\(173\) − 8.31723i − 0.632347i −0.948701 0.316174i \(-0.897602\pi\)
0.948701 0.316174i \(-0.102398\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.8863i 1.26925i
\(178\) 0 0
\(179\) 4.07517 0.304592 0.152296 0.988335i \(-0.451333\pi\)
0.152296 + 0.988335i \(0.451333\pi\)
\(180\) 0 0
\(181\) 2.52349 0.187569 0.0937846 0.995593i \(-0.470103\pi\)
0.0937846 + 0.995593i \(0.470103\pi\)
\(182\) 0 0
\(183\) 24.8615i 1.83782i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.72530i − 0.199293i
\(188\) 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
\(193\) 18.7733i 1.35133i 0.737207 + 0.675667i \(0.236144\pi\)
−0.737207 + 0.675667i \(0.763856\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.9052i − 1.13320i −0.823993 0.566600i \(-0.808259\pi\)
0.823993 0.566600i \(-0.191741\pi\)
\(198\) 0 0
\(199\) −9.87544 −0.700052 −0.350026 0.936740i \(-0.613827\pi\)
−0.350026 + 0.936740i \(0.613827\pi\)
\(200\) 0 0
\(201\) −29.1229 −2.05417
\(202\) 0 0
\(203\) − 23.9607i − 1.68171i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.384010i 0.0266906i
\(208\) 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
\(213\) 27.4922i 1.88373i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 33.0725i − 2.24511i
\(218\) 0 0
\(219\) −15.3872 −1.03977
\(220\) 0 0
\(221\) 6.94055 0.466872
\(222\) 0 0
\(223\) 8.58817i 0.575106i 0.957765 + 0.287553i \(0.0928418\pi\)
−0.957765 + 0.287553i \(0.907158\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 14.2068i − 0.942937i −0.881883 0.471469i \(-0.843724\pi\)
0.881883 0.471469i \(-0.156276\pi\)
\(228\) 0 0
\(229\) 7.49721 0.495429 0.247715 0.968833i \(-0.420320\pi\)
0.247715 + 0.968833i \(0.420320\pi\)
\(230\) 0 0
\(231\) −15.3718 −1.01139
\(232\) 0 0
\(233\) − 16.9301i − 1.10912i −0.832142 0.554562i \(-0.812886\pi\)
0.832142 0.554562i \(-0.187114\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 17.3131i 1.12460i
\(238\) 0 0
\(239\) 17.7850 1.15042 0.575208 0.818007i \(-0.304921\pi\)
0.575208 + 0.818007i \(0.304921\pi\)
\(240\) 0 0
\(241\) −9.55263 −0.615339 −0.307669 0.951493i \(-0.599549\pi\)
−0.307669 + 0.951493i \(0.599549\pi\)
\(242\) 0 0
\(243\) − 3.96721i − 0.254497i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 11.2673i − 0.716922i
\(248\) 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
\(253\) − 2.10339i − 0.132239i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.41506i 0.213025i 0.994311 + 0.106513i \(0.0339685\pi\)
−0.994311 + 0.106513i \(0.966032\pi\)
\(258\) 0 0
\(259\) 20.2661 1.25927
\(260\) 0 0
\(261\) 2.31608 0.143362
\(262\) 0 0
\(263\) 18.6589i 1.15056i 0.817957 + 0.575279i \(0.195106\pi\)
−0.817957 + 0.575279i \(0.804894\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 19.4815i − 1.19225i
\(268\) 0 0
\(269\) 24.2224 1.47687 0.738434 0.674326i \(-0.235566\pi\)
0.738434 + 0.674326i \(0.235566\pi\)
\(270\) 0 0
\(271\) −10.1201 −0.614749 −0.307375 0.951589i \(-0.599450\pi\)
−0.307375 + 0.951589i \(0.599450\pi\)
\(272\) 0 0
\(273\) − 39.1475i − 2.36932i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.5298i 1.41377i 0.707329 + 0.706884i \(0.249900\pi\)
−0.707329 + 0.706884i \(0.750100\pi\)
\(278\) 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
\(283\) 5.96641i 0.354666i 0.984151 + 0.177333i \(0.0567470\pi\)
−0.984151 + 0.177333i \(0.943253\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 33.1323i − 1.95574i
\(288\) 0 0
\(289\) 15.3212 0.901250
\(290\) 0 0
\(291\) −0.620765 −0.0363899
\(292\) 0 0
\(293\) 12.2404i 0.715090i 0.933896 + 0.357545i \(0.116386\pi\)
−0.933896 + 0.357545i \(0.883614\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.1221i 0.587346i
\(298\) 0 0
\(299\) 5.35673 0.309788
\(300\) 0 0
\(301\) −30.9178 −1.78207
\(302\) 0 0
\(303\) − 11.3891i − 0.654288i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.82713i 0.503791i 0.967754 + 0.251896i \(0.0810539\pi\)
−0.967754 + 0.251896i \(0.918946\pi\)
\(308\) 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
\(313\) 23.5030i 1.32847i 0.747523 + 0.664235i \(0.231243\pi\)
−0.747523 + 0.664235i \(0.768757\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 13.3279i − 0.748570i −0.927314 0.374285i \(-0.877888\pi\)
0.927314 0.374285i \(-0.122112\pi\)
\(318\) 0 0
\(319\) −12.6862 −0.710290
\(320\) 0 0
\(321\) 23.5354 1.31362
\(322\) 0 0
\(323\) 2.72530i 0.151640i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 26.3362i − 1.45639i
\(328\) 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
\(333\) 1.95896i 0.107350i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.0205i 1.09058i 0.838246 + 0.545292i \(0.183581\pi\)
−0.838246 + 0.545292i \(0.816419\pi\)
\(338\) 0 0
\(339\) −8.45213 −0.459057
\(340\) 0 0
\(341\) −17.5105 −0.948247
\(342\) 0 0
\(343\) − 7.08149i − 0.382364i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.1156i 0.811448i 0.913996 + 0.405724i \(0.132981\pi\)
−0.913996 + 0.405724i \(0.867019\pi\)
\(348\) 0 0
\(349\) 14.5673 0.779771 0.389886 0.920863i \(-0.372515\pi\)
0.389886 + 0.920863i \(0.372515\pi\)
\(350\) 0 0
\(351\) −25.7782 −1.37594
\(352\) 0 0
\(353\) 18.5376i 0.986656i 0.869843 + 0.493328i \(0.164220\pi\)
−0.869843 + 0.493328i \(0.835780\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 9.46886i 0.501145i
\(358\) 0 0
\(359\) 24.1834 1.27635 0.638175 0.769891i \(-0.279689\pi\)
0.638175 + 0.769891i \(0.279689\pi\)
\(360\) 0 0
\(361\) −14.5757 −0.767144
\(362\) 0 0
\(363\) − 12.0965i − 0.634903i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 5.79604i − 0.302551i −0.988492 0.151275i \(-0.951662\pi\)
0.988492 0.151275i \(-0.0483380\pi\)
\(368\) 0 0
\(369\) 3.20262 0.166722
\(370\) 0 0
\(371\) 2.27901 0.118320
\(372\) 0 0
\(373\) − 25.5753i − 1.32424i −0.749399 0.662119i \(-0.769657\pi\)
0.749399 0.662119i \(-0.230343\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 32.3081i − 1.66395i
\(378\) 0 0
\(379\) 9.82223 0.504534 0.252267 0.967658i \(-0.418824\pi\)
0.252267 + 0.967658i \(0.418824\pi\)
\(380\) 0 0
\(381\) 23.2932 1.19335
\(382\) 0 0
\(383\) − 33.9797i − 1.73628i −0.496319 0.868141i \(-0.665315\pi\)
0.496319 0.868141i \(-0.334685\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 2.98857i − 0.151918i
\(388\) 0 0
\(389\) 24.6744 1.25104 0.625521 0.780207i \(-0.284886\pi\)
0.625521 + 0.780207i \(0.284886\pi\)
\(390\) 0 0
\(391\) −1.29567 −0.0655247
\(392\) 0 0
\(393\) 8.61599i 0.434619i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.17340i 0.159268i 0.996824 + 0.0796342i \(0.0253752\pi\)
−0.996824 + 0.0796342i \(0.974625\pi\)
\(398\) 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
\(403\) − 44.5942i − 2.22140i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 10.7301i − 0.531869i
\(408\) 0 0
\(409\) −18.5426 −0.916871 −0.458435 0.888728i \(-0.651590\pi\)
−0.458435 + 0.888728i \(0.651590\pi\)
\(410\) 0 0
\(411\) −0.861138 −0.0424768
\(412\) 0 0
\(413\) 36.4676i 1.79445i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 1.35982i − 0.0665906i
\(418\) 0 0
\(419\) 3.22587 0.157594 0.0787970 0.996891i \(-0.474892\pi\)
0.0787970 + 0.996891i \(0.474892\pi\)
\(420\) 0 0
\(421\) 18.0595 0.880167 0.440084 0.897957i \(-0.354949\pi\)
0.440084 + 0.897957i \(0.354949\pi\)
\(422\) 0 0
\(423\) 4.37635i 0.212785i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 53.6909i 2.59828i
\(428\) 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
\(433\) − 1.83990i − 0.0884200i −0.999022 0.0442100i \(-0.985923\pi\)
0.999022 0.0442100i \(-0.0140771\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.10339i 0.100619i
\(438\) 0 0
\(439\) −6.73907 −0.321638 −0.160819 0.986984i \(-0.551414\pi\)
−0.160819 + 0.986984i \(0.551414\pi\)
\(440\) 0 0
\(441\) 3.37258 0.160599
\(442\) 0 0
\(443\) 11.4099i 0.542103i 0.962565 + 0.271051i \(0.0873714\pi\)
−0.962565 + 0.271051i \(0.912629\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 26.8398i − 1.26948i
\(448\) 0 0
\(449\) −27.8081 −1.31235 −0.656173 0.754610i \(-0.727826\pi\)
−0.656173 + 0.754610i \(0.727826\pi\)
\(450\) 0 0
\(451\) −17.5422 −0.826028
\(452\) 0 0
\(453\) 39.8974i 1.87454i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 26.6835i − 1.24820i −0.781343 0.624102i \(-0.785465\pi\)
0.781343 0.624102i \(-0.214535\pi\)
\(458\) 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
\(463\) − 23.3998i − 1.08748i −0.839253 0.543740i \(-0.817008\pi\)
0.839253 0.543740i \(-0.182992\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4.29380i − 0.198693i −0.995053 0.0993467i \(-0.968325\pi\)
0.995053 0.0993467i \(-0.0316753\pi\)
\(468\) 0 0
\(469\) −62.8936 −2.90416
\(470\) 0 0
\(471\) 14.1877 0.653735
\(472\) 0 0
\(473\) 16.3697i 0.752680i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.220293i 0.0100865i
\(478\) 0 0
\(479\) 26.6694 1.21856 0.609278 0.792957i \(-0.291459\pi\)
0.609278 + 0.792957i \(0.291459\pi\)
\(480\) 0 0
\(481\) 27.3264 1.24597
\(482\) 0 0
\(483\) 7.30809i 0.332530i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.67114i 0.392927i 0.980511 + 0.196463i \(0.0629457\pi\)
−0.980511 + 0.196463i \(0.937054\pi\)
\(488\) 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
\(493\) 7.81456i 0.351950i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 59.3720i 2.66320i
\(498\) 0 0
\(499\) 34.9323 1.56379 0.781893 0.623412i \(-0.214254\pi\)
0.781893 + 0.623412i \(0.214254\pi\)
\(500\) 0 0
\(501\) 21.4747 0.959418
\(502\) 0 0
\(503\) 25.7782i 1.14939i 0.818367 + 0.574695i \(0.194879\pi\)
−0.818367 + 0.574695i \(0.805121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 28.8713i − 1.28222i
\(508\) 0 0
\(509\) −4.21667 −0.186900 −0.0934502 0.995624i \(-0.529790\pi\)
−0.0934502 + 0.995624i \(0.529790\pi\)
\(510\) 0 0
\(511\) −33.2302 −1.47002
\(512\) 0 0
\(513\) − 10.1221i − 0.446903i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 23.9712i − 1.05425i
\(518\) 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
\(523\) − 31.1549i − 1.36231i −0.732140 0.681154i \(-0.761478\pi\)
0.732140 0.681154i \(-0.238522\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.7863i 0.469858i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −3.52502 −0.152973
\(532\) 0 0
\(533\) − 44.6748i − 1.93508i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.49655i 0.323500i
\(538\) 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
\(543\) 4.64212i 0.199213i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 39.2935i 1.68007i 0.542534 + 0.840034i \(0.317465\pi\)
−0.542534 + 0.840034i \(0.682535\pi\)
\(548\) 0 0
\(549\) −5.18985 −0.221497
\(550\) 0 0
\(551\) 12.6862 0.540450
\(552\) 0 0
\(553\) 37.3892i 1.58995i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 38.1005i − 1.61437i −0.590297 0.807186i \(-0.700990\pi\)
0.590297 0.807186i \(-0.299010\pi\)
\(558\) 0 0
\(559\) −41.6889 −1.76325
\(560\) 0 0
\(561\) 5.01337 0.211665
\(562\) 0 0
\(563\) 32.6868i 1.37758i 0.724959 + 0.688792i \(0.241859\pi\)
−0.724959 + 0.688792i \(0.758141\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 39.7454i − 1.66915i
\(568\) 0 0
\(569\) −21.0256 −0.881438 −0.440719 0.897645i \(-0.645276\pi\)
−0.440719 + 0.897645i \(0.645276\pi\)
\(570\) 0 0
\(571\) −7.43876 −0.311303 −0.155651 0.987812i \(-0.549748\pi\)
−0.155651 + 0.987812i \(0.549748\pi\)
\(572\) 0 0
\(573\) − 44.0357i − 1.83962i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.4148i 1.26618i 0.774077 + 0.633091i \(0.218214\pi\)
−0.774077 + 0.633091i \(0.781786\pi\)
\(578\) 0 0
\(579\) −34.5348 −1.43522
\(580\) 0 0
\(581\) −6.00698 −0.249212
\(582\) 0 0
\(583\) − 1.20664i − 0.0499739i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.7544i 1.39319i 0.717464 + 0.696596i \(0.245303\pi\)
−0.717464 + 0.696596i \(0.754697\pi\)
\(588\) 0 0
\(589\) 17.5105 0.721508
\(590\) 0 0
\(591\) 29.2587 1.20354
\(592\) 0 0
\(593\) 27.3980i 1.12510i 0.826763 + 0.562550i \(0.190180\pi\)
−0.826763 + 0.562550i \(0.809820\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 18.1666i − 0.743507i
\(598\) 0 0
\(599\) −37.2927 −1.52374 −0.761869 0.647731i \(-0.775718\pi\)
−0.761869 + 0.647731i \(0.775718\pi\)
\(600\) 0 0
\(601\) −2.26039 −0.0922032 −0.0461016 0.998937i \(-0.514680\pi\)
−0.0461016 + 0.998937i \(0.514680\pi\)
\(602\) 0 0
\(603\) − 6.07941i − 0.247573i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 23.4782i 0.952951i 0.879188 + 0.476476i \(0.158086\pi\)
−0.879188 + 0.476476i \(0.841914\pi\)
\(608\) 0 0
\(609\) 44.0773 1.78610
\(610\) 0 0
\(611\) 61.0477 2.46972
\(612\) 0 0
\(613\) − 25.0418i − 1.01143i −0.862701 0.505714i \(-0.831229\pi\)
0.862701 0.505714i \(-0.168771\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.2705i 0.735541i 0.929916 + 0.367771i \(0.119879\pi\)
−0.929916 + 0.367771i \(0.880121\pi\)
\(618\) 0 0
\(619\) −7.17743 −0.288485 −0.144243 0.989542i \(-0.546075\pi\)
−0.144243 + 0.989542i \(0.546075\pi\)
\(620\) 0 0
\(621\) 4.81229 0.193111
\(622\) 0 0
\(623\) − 42.0721i − 1.68558i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 8.13872i − 0.325029i
\(628\) 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
\(633\) − 13.0054i − 0.516917i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 47.0457i − 1.86402i
\(638\) 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
\(643\) − 8.24043i − 0.324971i −0.986711 0.162485i \(-0.948049\pi\)
0.986711 0.162485i \(-0.0519510\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.1134i 1.30182i 0.759154 + 0.650911i \(0.225613\pi\)
−0.759154 + 0.650911i \(0.774387\pi\)
\(648\) 0 0
\(649\) 19.3081 0.757909
\(650\) 0 0
\(651\) 60.8391 2.38447
\(652\) 0 0
\(653\) 14.5452i 0.569199i 0.958647 + 0.284599i \(0.0918605\pi\)
−0.958647 + 0.284599i \(0.908139\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 3.21209i − 0.125316i
\(658\) 0 0
\(659\) 24.4498 0.952427 0.476214 0.879330i \(-0.342009\pi\)
0.476214 + 0.879330i \(0.342009\pi\)
\(660\) 0 0
\(661\) 1.43400 0.0557763 0.0278882 0.999611i \(-0.491122\pi\)
0.0278882 + 0.999611i \(0.491122\pi\)
\(662\) 0 0
\(663\) 12.7676i 0.495853i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.03130i 0.233533i
\(668\) 0 0
\(669\) −15.7985 −0.610806
\(670\) 0 0
\(671\) 28.4271 1.09742
\(672\) 0 0
\(673\) − 3.39873i − 0.131011i −0.997852 0.0655056i \(-0.979134\pi\)
0.997852 0.0655056i \(-0.0208660\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.7751i 0.606287i 0.952945 + 0.303144i \(0.0980362\pi\)
−0.952945 + 0.303144i \(0.901964\pi\)
\(678\) 0 0
\(679\) −1.34060 −0.0514475
\(680\) 0 0
\(681\) 26.1343 1.00147
\(682\) 0 0
\(683\) − 4.99979i − 0.191312i −0.995414 0.0956559i \(-0.969505\pi\)
0.995414 0.0956559i \(-0.0304948\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.7916i 0.526183i
\(688\) 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
\(693\) − 3.20886i − 0.121895i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.8058i 0.409298i
\(698\) 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
\(703\) 10.7301i 0.404692i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 24.5959i − 0.925024i
\(708\) 0 0
\(709\) −52.5228 −1.97253 −0.986267 0.165157i \(-0.947187\pi\)
−0.986267 + 0.165157i \(0.947187\pi\)
\(710\) 0 0
\(711\) −3.61411 −0.135540
\(712\) 0 0
\(713\) 8.32489i 0.311770i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 32.7167i 1.22183i
\(718\) 0 0
\(719\) 24.2329 0.903735 0.451868 0.892085i \(-0.350758\pi\)
0.451868 + 0.892085i \(0.350758\pi\)
\(720\) 0 0
\(721\) −50.1622 −1.86814
\(722\) 0 0
\(723\) − 17.5727i − 0.653536i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 37.0943i − 1.37575i −0.725829 0.687875i \(-0.758544\pi\)
0.725829 0.687875i \(-0.241456\pi\)
\(728\) 0 0
\(729\) −22.7158 −0.841324
\(730\) 0 0
\(731\) 10.0836 0.372954
\(732\) 0 0
\(733\) 46.3065i 1.71037i 0.518325 + 0.855184i \(0.326556\pi\)
−0.518325 + 0.855184i \(0.673444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.2996i 1.22660i
\(738\) 0 0
\(739\) 32.3516 1.19007 0.595037 0.803698i \(-0.297137\pi\)
0.595037 + 0.803698i \(0.297137\pi\)
\(740\) 0 0
\(741\) 20.7270 0.761425
\(742\) 0 0
\(743\) − 1.73343i − 0.0635935i −0.999494 0.0317967i \(-0.989877\pi\)
0.999494 0.0317967i \(-0.0101229\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 0.580645i − 0.0212447i
\(748\) 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
\(753\) − 24.2961i − 0.885400i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 48.3219i − 1.75629i −0.478397 0.878144i \(-0.658782\pi\)
0.478397 0.878144i \(-0.341218\pi\)
\(758\) 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
\(763\) − 56.8754i − 2.05903i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 49.1722i 1.77550i
\(768\) 0 0
\(769\) −9.05816 −0.326645 −0.163323 0.986573i \(-0.552221\pi\)
−0.163323 + 0.986573i \(0.552221\pi\)
\(770\) 0 0
\(771\) −6.28223 −0.226249
\(772\) 0 0
\(773\) 29.1276i 1.04765i 0.851827 + 0.523824i \(0.175495\pi\)
−0.851827 + 0.523824i \(0.824505\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 37.2809i 1.33744i
\(778\) 0 0
\(779\) 17.5422 0.628513
\(780\) 0 0
\(781\) 31.4350 1.12483
\(782\) 0 0
\(783\) − 29.0244i − 1.03725i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 23.8771i 0.851128i 0.904928 + 0.425564i \(0.139924\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(788\) 0 0
\(789\) −34.3243 −1.22198
\(790\) 0 0
\(791\) −18.2532 −0.649008
\(792\) 0 0
\(793\) 72.3956i 2.57084i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.8896i 0.704526i 0.935901 + 0.352263i \(0.114588\pi\)
−0.935901 + 0.352263i \(0.885412\pi\)
\(798\) 0 0
\(799\) −14.7660 −0.522383
\(800\) 0 0
\(801\) 4.06677 0.143692
\(802\) 0 0
\(803\) 17.5940i 0.620879i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 44.5588i 1.56854i
\(808\) 0 0
\(809\) −26.3598 −0.926760 −0.463380 0.886160i \(-0.653363\pi\)
−0.463380 + 0.886160i \(0.653363\pi\)
\(810\) 0 0
\(811\) −14.2482 −0.500323 −0.250161 0.968204i \(-0.580484\pi\)
−0.250161 + 0.968204i \(0.580484\pi\)
\(812\) 0 0
\(813\) − 18.6165i − 0.652910i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.3697i − 0.572703i
\(818\) 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
\(823\) − 34.1312i − 1.18974i −0.803823 0.594869i \(-0.797204\pi\)
0.803823 0.594869i \(-0.202796\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.5085i 0.817472i 0.912653 + 0.408736i \(0.134030\pi\)
−0.912653 + 0.408736i \(0.865970\pi\)
\(828\) 0 0
\(829\) 31.7442 1.10252 0.551260 0.834333i \(-0.314147\pi\)
0.551260 + 0.834333i \(0.314147\pi\)
\(830\) 0 0
\(831\) −43.2846 −1.50153
\(832\) 0 0
\(833\) 11.3792i 0.394267i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 40.0618i − 1.38474i
\(838\) 0 0
\(839\) −24.5409 −0.847247 −0.423623 0.905838i \(-0.639242\pi\)
−0.423623 + 0.905838i \(0.639242\pi\)
\(840\) 0 0
\(841\) 7.37661 0.254366
\(842\) 0 0
\(843\) 3.26180i 0.112343i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 26.1236i − 0.897618i
\(848\) 0 0
\(849\) −10.9756 −0.376682
\(850\) 0 0
\(851\) −5.10131 −0.174871
\(852\) 0 0
\(853\) − 23.7499i − 0.813183i −0.913610 0.406591i \(-0.866717\pi\)
0.913610 0.406591i \(-0.133283\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 43.0643i 1.47105i 0.677499 + 0.735524i \(0.263064\pi\)
−0.677499 + 0.735524i \(0.736936\pi\)
\(858\) 0 0
\(859\) 25.0591 0.855006 0.427503 0.904014i \(-0.359393\pi\)
0.427503 + 0.904014i \(0.359393\pi\)
\(860\) 0 0
\(861\) 60.9490 2.07714
\(862\) 0 0
\(863\) 46.0967i 1.56915i 0.620033 + 0.784575i \(0.287119\pi\)
−0.620033 + 0.784575i \(0.712881\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 28.1845i 0.957195i
\(868\) 0 0
\(869\) 19.7960 0.671535
\(870\) 0 0
\(871\) −84.8044 −2.87349
\(872\) 0 0
\(873\) − 0.129585i − 0.00438578i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 35.1644i 1.18742i 0.804680 + 0.593709i \(0.202337\pi\)
−0.804680 + 0.593709i \(0.797663\pi\)
\(878\) 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
\(883\) 10.5954i 0.356562i 0.983980 + 0.178281i \(0.0570536\pi\)
−0.983980 + 0.178281i \(0.942946\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.8733i 0.902316i 0.892444 + 0.451158i \(0.148989\pi\)
−0.892444 + 0.451158i \(0.851011\pi\)
\(888\) 0 0
\(889\) 50.3040 1.68714
\(890\) 0 0
\(891\) −21.0435 −0.704984
\(892\) 0 0
\(893\) 23.9712i 0.802165i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.85407i 0.329018i
\(898\) 0 0
\(899\) 50.2099 1.67459
\(900\) 0 0
\(901\) −0.743278 −0.0247622
\(902\) 0 0
\(903\) − 56.8754i − 1.89270i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 58.3210i − 1.93652i −0.249949 0.968259i \(-0.580414\pi\)
0.249949 0.968259i \(-0.419586\pi\)
\(908\) 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
\(913\) 3.18045i 0.105257i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.6071i 0.614459i
\(918\) 0 0
\(919\) 37.6333 1.24141 0.620704 0.784045i \(-0.286847\pi\)
0.620704 + 0.784045i \(0.286847\pi\)
\(920\) 0 0
\(921\) −16.2381 −0.535064
\(922\) 0 0
\(923\) 80.0559i 2.63507i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.84877i − 0.159254i
\(928\) 0 0
\(929\) −30.7190 −1.00786 −0.503929 0.863745i \(-0.668113\pi\)
−0.503929 + 0.863745i \(0.668113\pi\)
\(930\) 0 0
\(931\) 18.4731 0.605431
\(932\) 0 0
\(933\) − 25.6072i − 0.838343i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 40.5843i 1.32583i 0.748694 + 0.662915i \(0.230681\pi\)
−0.748694 + 0.662915i \(0.769319\pi\)
\(938\) 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
\(943\) 8.33994i 0.271586i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 34.9681i − 1.13631i −0.822921 0.568155i \(-0.807657\pi\)
0.822921 0.568155i \(-0.192343\pi\)
\(948\) 0 0
\(949\) −44.8069 −1.45449
\(950\) 0 0
\(951\) 24.5176 0.795038
\(952\) 0 0
\(953\) 35.7164i 1.15697i 0.815694 + 0.578484i \(0.196355\pi\)
−0.815694 + 0.578484i \(0.803645\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 23.3371i − 0.754382i
\(958\) 0 0
\(959\) −1.85971 −0.0600532
\(960\) 0 0
\(961\) 38.3038 1.23561
\(962\) 0 0
\(963\) 4.91301i 0.158320i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 43.0422i 1.38414i 0.721829 + 0.692072i \(0.243302\pi\)
−0.721829 + 0.692072i \(0.756698\pi\)
\(968\) 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
\(973\) − 2.93666i − 0.0941449i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 24.0469i − 0.769329i −0.923056 0.384665i \(-0.874317\pi\)
0.923056 0.384665i \(-0.125683\pi\)
\(978\) 0 0
\(979\) −22.2754 −0.711926
\(980\) 0 0
\(981\) 5.49768 0.175527
\(982\) 0 0
\(983\) − 19.2418i − 0.613717i −0.951755 0.306859i \(-0.900722\pi\)
0.951755 0.306859i \(-0.0992778\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 83.2862i 2.65103i
\(988\) 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
\(993\) − 0.496936i − 0.0157698i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 41.5981i 1.31742i 0.752395 + 0.658712i \(0.228899\pi\)
−0.752395 + 0.658712i \(0.771101\pi\)
\(998\) 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.e.w.4049.8 10
5.2 odd 4 4600.2.a.bf.1.4 yes 5
5.3 odd 4 4600.2.a.bd.1.2 5
5.4 even 2 inner 4600.2.e.w.4049.3 10
20.3 even 4 9200.2.a.cv.1.4 5
20.7 even 4 9200.2.a.ct.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.2 5 5.3 odd 4
4600.2.a.bf.1.4 yes 5 5.2 odd 4
4600.2.e.w.4049.3 10 5.4 even 2 inner
4600.2.e.w.4049.8 10 1.1 even 1 trivial
9200.2.a.ct.1.2 5 20.7 even 4
9200.2.a.cv.1.4 5 20.3 even 4