Properties

Label 4600.2.e.p.4049.6
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.431642176.2
Defining polynomial: \(x^{6} + 19 x^{4} + 97 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.6
Root \(3.07912i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.p.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.07912i q^{3} +2.07912i q^{7} -6.48097 q^{9} +O(q^{10})\) \(q+3.07912i q^{3} +2.07912i q^{7} -6.48097 q^{9} +5.07912 q^{11} -3.48097i q^{13} -6.48097i q^{17} +7.48097 q^{19} -6.40185 q^{21} +1.00000i q^{23} -10.7183i q^{27} +1.56009 q^{29} +0.0791189 q^{31} +15.6392i q^{33} +9.71833i q^{37} +10.7183 q^{39} -0.480973 q^{41} +8.00000i q^{43} -6.96195i q^{47} +2.67726 q^{49} +19.9557 q^{51} +11.7183i q^{53} +23.0348i q^{57} +11.5601 q^{59} +7.88283 q^{61} -13.4747i q^{63} +9.71833i q^{67} -3.07912 q^{69} -9.67726 q^{71} -13.2784i q^{73} +10.5601i q^{77} +12.3165 q^{79} +13.5601 q^{81} -4.59815i q^{83} +4.80371i q^{87} -8.31648 q^{89} +7.23736 q^{91} +0.243616i q^{93} -7.23736i q^{97} -32.9176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 20q^{9} + O(q^{10}) \) \( 6q - 20q^{9} + 14q^{11} + 26q^{19} - 36q^{21} - 26q^{29} - 16q^{31} - 4q^{39} + 16q^{41} + 2q^{49} + 2q^{51} + 34q^{59} + 26q^{61} - 2q^{69} - 44q^{71} + 8q^{79} + 46q^{81} + 16q^{89} - 6q^{91} - 42q^{99} + O(q^{100}) \)

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
\(3\) 3.07912i 1.77773i 0.458169 + 0.888865i \(0.348505\pi\)
−0.458169 + 0.888865i \(0.651495\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.07912i 0.785833i 0.919574 + 0.392917i \(0.128534\pi\)
−0.919574 + 0.392917i \(0.871466\pi\)
\(8\) 0 0
\(9\) −6.48097 −2.16032
\(10\) 0 0
\(11\) 5.07912 1.53141 0.765706 0.643191i \(-0.222390\pi\)
0.765706 + 0.643191i \(0.222390\pi\)
\(12\) 0 0
\(13\) − 3.48097i − 0.965448i −0.875772 0.482724i \(-0.839647\pi\)
0.875772 0.482724i \(-0.160353\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.48097i − 1.57187i −0.618311 0.785933i \(-0.712183\pi\)
0.618311 0.785933i \(-0.287817\pi\)
\(18\) 0 0
\(19\) 7.48097 1.71625 0.858126 0.513438i \(-0.171629\pi\)
0.858126 + 0.513438i \(0.171629\pi\)
\(20\) 0 0
\(21\) −6.40185 −1.39700
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 10.7183i − 2.06274i
\(28\) 0 0
\(29\) 1.56009 0.289702 0.144851 0.989453i \(-0.453730\pi\)
0.144851 + 0.989453i \(0.453730\pi\)
\(30\) 0 0
\(31\) 0.0791189 0.0142102 0.00710508 0.999975i \(-0.497738\pi\)
0.00710508 + 0.999975i \(0.497738\pi\)
\(32\) 0 0
\(33\) 15.6392i 2.72244i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.71833i 1.59768i 0.601541 + 0.798842i \(0.294554\pi\)
−0.601541 + 0.798842i \(0.705446\pi\)
\(38\) 0 0
\(39\) 10.7183 1.71631
\(40\) 0 0
\(41\) −0.480973 −0.0751154 −0.0375577 0.999294i \(-0.511958\pi\)
−0.0375577 + 0.999294i \(0.511958\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.96195i − 1.01550i −0.861503 0.507752i \(-0.830477\pi\)
0.861503 0.507752i \(-0.169523\pi\)
\(48\) 0 0
\(49\) 2.67726 0.382466
\(50\) 0 0
\(51\) 19.9557 2.79435
\(52\) 0 0
\(53\) 11.7183i 1.60964i 0.593521 + 0.804818i \(0.297737\pi\)
−0.593521 + 0.804818i \(0.702263\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 23.0348i 3.05103i
\(58\) 0 0
\(59\) 11.5601 1.50500 0.752498 0.658595i \(-0.228849\pi\)
0.752498 + 0.658595i \(0.228849\pi\)
\(60\) 0 0
\(61\) 7.88283 1.00929 0.504646 0.863326i \(-0.331623\pi\)
0.504646 + 0.863326i \(0.331623\pi\)
\(62\) 0 0
\(63\) − 13.4747i − 1.69765i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.71833i 1.18728i 0.804730 + 0.593641i \(0.202310\pi\)
−0.804730 + 0.593641i \(0.797690\pi\)
\(68\) 0 0
\(69\) −3.07912 −0.370682
\(70\) 0 0
\(71\) −9.67726 −1.14848 −0.574240 0.818687i \(-0.694702\pi\)
−0.574240 + 0.818687i \(0.694702\pi\)
\(72\) 0 0
\(73\) − 13.2784i − 1.55412i −0.629425 0.777061i \(-0.716710\pi\)
0.629425 0.777061i \(-0.283290\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.5601i 1.20343i
\(78\) 0 0
\(79\) 12.3165 1.38571 0.692856 0.721076i \(-0.256352\pi\)
0.692856 + 0.721076i \(0.256352\pi\)
\(80\) 0 0
\(81\) 13.5601 1.50668
\(82\) 0 0
\(83\) − 4.59815i − 0.504712i −0.967634 0.252356i \(-0.918795\pi\)
0.967634 0.252356i \(-0.0812054\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.80371i 0.515012i
\(88\) 0 0
\(89\) −8.31648 −0.881545 −0.440772 0.897619i \(-0.645295\pi\)
−0.440772 + 0.897619i \(0.645295\pi\)
\(90\) 0 0
\(91\) 7.23736 0.758681
\(92\) 0 0
\(93\) 0.243616i 0.0252618i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.23736i − 0.734842i −0.930055 0.367421i \(-0.880241\pi\)
0.930055 0.367421i \(-0.119759\pi\)
\(98\) 0 0
\(99\) −32.9176 −3.30835
\(100\) 0 0
\(101\) 2.43991 0.242780 0.121390 0.992605i \(-0.461265\pi\)
0.121390 + 0.992605i \(0.461265\pi\)
\(102\) 0 0
\(103\) 7.88283i 0.776718i 0.921508 + 0.388359i \(0.126958\pi\)
−0.921508 + 0.388359i \(0.873042\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.6803i 1.61254i 0.591546 + 0.806272i \(0.298518\pi\)
−0.591546 + 0.806272i \(0.701482\pi\)
\(108\) 0 0
\(109\) −5.32274 −0.509826 −0.254913 0.966964i \(-0.582047\pi\)
−0.254913 + 0.966964i \(0.582047\pi\)
\(110\) 0 0
\(111\) −29.9239 −2.84025
\(112\) 0 0
\(113\) − 2.20556i − 0.207482i −0.994604 0.103741i \(-0.966919\pi\)
0.994604 0.103741i \(-0.0330813\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 22.5601i 2.08568i
\(118\) 0 0
\(119\) 13.4747 1.23522
\(120\) 0 0
\(121\) 14.7974 1.34522
\(122\) 0 0
\(123\) − 1.48097i − 0.133535i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 6.15824i − 0.546455i −0.961949 0.273228i \(-0.911909\pi\)
0.961949 0.273228i \(-0.0880912\pi\)
\(128\) 0 0
\(129\) −24.6330 −2.16881
\(130\) 0 0
\(131\) −7.19629 −0.628743 −0.314371 0.949300i \(-0.601794\pi\)
−0.314371 + 0.949300i \(0.601794\pi\)
\(132\) 0 0
\(133\) 15.5538i 1.34869i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.6012i − 1.07659i −0.842757 0.538295i \(-0.819069\pi\)
0.842757 0.538295i \(-0.180931\pi\)
\(138\) 0 0
\(139\) 8.75638 0.742707 0.371353 0.928492i \(-0.378894\pi\)
0.371353 + 0.928492i \(0.378894\pi\)
\(140\) 0 0
\(141\) 21.4367 1.80529
\(142\) 0 0
\(143\) − 17.6803i − 1.47850i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.24362i 0.679922i
\(148\) 0 0
\(149\) −1.79745 −0.147253 −0.0736264 0.997286i \(-0.523457\pi\)
−0.0736264 + 0.997286i \(0.523457\pi\)
\(150\) 0 0
\(151\) −19.3956 −1.57839 −0.789196 0.614142i \(-0.789502\pi\)
−0.789196 + 0.614142i \(0.789502\pi\)
\(152\) 0 0
\(153\) 42.0030i 3.39574i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.36380i − 0.348269i −0.984722 0.174135i \(-0.944287\pi\)
0.984722 0.174135i \(-0.0557128\pi\)
\(158\) 0 0
\(159\) −36.0821 −2.86150
\(160\) 0 0
\(161\) −2.07912 −0.163858
\(162\) 0 0
\(163\) 5.32274i 0.416909i 0.978032 + 0.208454i \(0.0668433\pi\)
−0.978032 + 0.208454i \(0.933157\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 0.882827 0.0679098
\(170\) 0 0
\(171\) −48.4840 −3.70766
\(172\) 0 0
\(173\) − 9.23736i − 0.702303i −0.936319 0.351152i \(-0.885790\pi\)
0.936319 0.351152i \(-0.114210\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 35.5949i 2.67548i
\(178\) 0 0
\(179\) −2.96195 −0.221386 −0.110693 0.993855i \(-0.535307\pi\)
−0.110693 + 0.993855i \(0.535307\pi\)
\(180\) 0 0
\(181\) 10.8355 0.805397 0.402698 0.915333i \(-0.368072\pi\)
0.402698 + 0.915333i \(0.368072\pi\)
\(182\) 0 0
\(183\) 24.2722i 1.79425i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 32.9176i − 2.40718i
\(188\) 0 0
\(189\) 22.2847 1.62097
\(190\) 0 0
\(191\) 7.76565 0.561903 0.280952 0.959722i \(-0.409350\pi\)
0.280952 + 0.959722i \(0.409350\pi\)
\(192\) 0 0
\(193\) − 2.15824i − 0.155353i −0.996979 0.0776767i \(-0.975250\pi\)
0.996979 0.0776767i \(-0.0247502\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 12.0411i − 0.857890i −0.903331 0.428945i \(-0.858885\pi\)
0.903331 0.428945i \(-0.141115\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −29.9239 −2.11067
\(202\) 0 0
\(203\) 3.24362i 0.227657i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.48097i − 0.450459i
\(208\) 0 0
\(209\) 37.9968 2.62829
\(210\) 0 0
\(211\) 3.40185 0.234193 0.117097 0.993121i \(-0.462641\pi\)
0.117097 + 0.993121i \(0.462641\pi\)
\(212\) 0 0
\(213\) − 29.7974i − 2.04169i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.164498i 0.0111668i
\(218\) 0 0
\(219\) 40.8858 2.76281
\(220\) 0 0
\(221\) −22.5601 −1.51756
\(222\) 0 0
\(223\) 5.19629i 0.347969i 0.984748 + 0.173985i \(0.0556643\pi\)
−0.984748 + 0.173985i \(0.944336\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 5.35453i − 0.355393i −0.984085 0.177696i \(-0.943136\pi\)
0.984085 0.177696i \(-0.0568645\pi\)
\(228\) 0 0
\(229\) 0.803708 0.0531105 0.0265553 0.999647i \(-0.491546\pi\)
0.0265553 + 0.999647i \(0.491546\pi\)
\(230\) 0 0
\(231\) −32.5158 −2.13938
\(232\) 0 0
\(233\) 14.1582i 0.927537i 0.885956 + 0.463768i \(0.153503\pi\)
−0.885956 + 0.463768i \(0.846497\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 37.9239i 2.46342i
\(238\) 0 0
\(239\) −10.9146 −0.706008 −0.353004 0.935622i \(-0.614840\pi\)
−0.353004 + 0.935622i \(0.614840\pi\)
\(240\) 0 0
\(241\) −27.4367 −1.76735 −0.883675 0.468100i \(-0.844939\pi\)
−0.883675 + 0.468100i \(0.844939\pi\)
\(242\) 0 0
\(243\) 9.59815i 0.615721i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 26.0411i − 1.65695i
\(248\) 0 0
\(249\) 14.1582 0.897242
\(250\) 0 0
\(251\) −16.5190 −1.04267 −0.521336 0.853352i \(-0.674566\pi\)
−0.521336 + 0.853352i \(0.674566\pi\)
\(252\) 0 0
\(253\) 5.07912i 0.319321i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 19.7657i − 1.23295i −0.787375 0.616474i \(-0.788561\pi\)
0.787375 0.616474i \(-0.211439\pi\)
\(258\) 0 0
\(259\) −20.2056 −1.25551
\(260\) 0 0
\(261\) −10.1109 −0.625850
\(262\) 0 0
\(263\) 7.44292i 0.458950i 0.973315 + 0.229475i \(0.0737009\pi\)
−0.973315 + 0.229475i \(0.926299\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 25.6074i − 1.56715i
\(268\) 0 0
\(269\) 22.7564 1.38748 0.693741 0.720225i \(-0.255961\pi\)
0.693741 + 0.720225i \(0.255961\pi\)
\(270\) 0 0
\(271\) −19.0411 −1.15666 −0.578331 0.815802i \(-0.696296\pi\)
−0.578331 + 0.815802i \(0.696296\pi\)
\(272\) 0 0
\(273\) 22.2847i 1.34873i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.3165i 1.10053i 0.834990 + 0.550265i \(0.185473\pi\)
−0.834990 + 0.550265i \(0.814527\pi\)
\(278\) 0 0
\(279\) −0.512767 −0.0306986
\(280\) 0 0
\(281\) 26.0821 1.55593 0.777965 0.628308i \(-0.216252\pi\)
0.777965 + 0.628308i \(0.216252\pi\)
\(282\) 0 0
\(283\) − 25.6422i − 1.52427i −0.647417 0.762136i \(-0.724151\pi\)
0.647417 0.762136i \(-0.275849\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.00000i − 0.0590281i
\(288\) 0 0
\(289\) −25.0030 −1.47077
\(290\) 0 0
\(291\) 22.2847 1.30635
\(292\) 0 0
\(293\) 30.8385i 1.80161i 0.434229 + 0.900803i \(0.357021\pi\)
−0.434229 + 0.900803i \(0.642979\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 54.4397i − 3.15891i
\(298\) 0 0
\(299\) 3.48097 0.201310
\(300\) 0 0
\(301\) −16.6330 −0.958707
\(302\) 0 0
\(303\) 7.51277i 0.431597i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.88283i 0.107459i 0.998556 + 0.0537293i \(0.0171108\pi\)
−0.998556 + 0.0537293i \(0.982889\pi\)
\(308\) 0 0
\(309\) −24.2722 −1.38080
\(310\) 0 0
\(311\) 1.60742 0.0911482 0.0455741 0.998961i \(-0.485488\pi\)
0.0455741 + 0.998961i \(0.485488\pi\)
\(312\) 0 0
\(313\) − 1.68654i − 0.0953286i −0.998863 0.0476643i \(-0.984822\pi\)
0.998863 0.0476643i \(-0.0151778\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.6773i 0.712026i 0.934481 + 0.356013i \(0.115864\pi\)
−0.934481 + 0.356013i \(0.884136\pi\)
\(318\) 0 0
\(319\) 7.92389 0.443653
\(320\) 0 0
\(321\) −51.3606 −2.86667
\(322\) 0 0
\(323\) − 48.4840i − 2.69772i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 16.3893i − 0.906332i
\(328\) 0 0
\(329\) 14.4747 0.798017
\(330\) 0 0
\(331\) 33.3893 1.83524 0.917622 0.397454i \(-0.130106\pi\)
0.917622 + 0.397454i \(0.130106\pi\)
\(332\) 0 0
\(333\) − 62.9842i − 3.45151i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 17.6392i 0.960869i 0.877031 + 0.480435i \(0.159521\pi\)
−0.877031 + 0.480435i \(0.840479\pi\)
\(338\) 0 0
\(339\) 6.79119 0.368847
\(340\) 0 0
\(341\) 0.401854 0.0217616
\(342\) 0 0
\(343\) 20.1202i 1.08639i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 19.7121i − 1.05820i −0.848560 0.529100i \(-0.822530\pi\)
0.848560 0.529100i \(-0.177470\pi\)
\(348\) 0 0
\(349\) 16.0348 0.858323 0.429162 0.903228i \(-0.358809\pi\)
0.429162 + 0.903228i \(0.358809\pi\)
\(350\) 0 0
\(351\) −37.3102 −1.99147
\(352\) 0 0
\(353\) − 8.00000i − 0.425797i −0.977074 0.212899i \(-0.931710\pi\)
0.977074 0.212899i \(-0.0682904\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 41.4902i 2.19590i
\(358\) 0 0
\(359\) 13.1202 0.692457 0.346228 0.938150i \(-0.387462\pi\)
0.346228 + 0.938150i \(0.387462\pi\)
\(360\) 0 0
\(361\) 36.9650 1.94552
\(362\) 0 0
\(363\) 45.5631i 2.39144i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.47796i − 0.181548i −0.995872 0.0907741i \(-0.971066\pi\)
0.995872 0.0907741i \(-0.0289341\pi\)
\(368\) 0 0
\(369\) 3.11717 0.162274
\(370\) 0 0
\(371\) −24.3638 −1.26491
\(372\) 0 0
\(373\) 21.5949i 1.11814i 0.829120 + 0.559071i \(0.188842\pi\)
−0.829120 + 0.559071i \(0.811158\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5.43064i − 0.279692i
\(378\) 0 0
\(379\) −7.80070 −0.400695 −0.200347 0.979725i \(-0.564207\pi\)
−0.200347 + 0.979725i \(0.564207\pi\)
\(380\) 0 0
\(381\) 18.9619 0.971450
\(382\) 0 0
\(383\) 32.4274i 1.65696i 0.560017 + 0.828481i \(0.310795\pi\)
−0.560017 + 0.828481i \(0.689205\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 51.8478i − 2.63557i
\(388\) 0 0
\(389\) −23.5538 −1.19423 −0.597113 0.802157i \(-0.703686\pi\)
−0.597113 + 0.802157i \(0.703686\pi\)
\(390\) 0 0
\(391\) 6.48097 0.327757
\(392\) 0 0
\(393\) − 22.1582i − 1.11774i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.370060i 0.0185728i 0.999957 + 0.00928639i \(0.00295599\pi\)
−0.999957 + 0.00928639i \(0.997044\pi\)
\(398\) 0 0
\(399\) −47.8921 −2.39760
\(400\) 0 0
\(401\) 11.3545 0.567018 0.283509 0.958970i \(-0.408501\pi\)
0.283509 + 0.958970i \(0.408501\pi\)
\(402\) 0 0
\(403\) − 0.275411i − 0.0137192i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 49.3606i 2.44671i
\(408\) 0 0
\(409\) −4.88283 −0.241440 −0.120720 0.992687i \(-0.538520\pi\)
−0.120720 + 0.992687i \(0.538520\pi\)
\(410\) 0 0
\(411\) 38.8005 1.91389
\(412\) 0 0
\(413\) 24.0348i 1.18268i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 26.9619i 1.32033i
\(418\) 0 0
\(419\) −1.51277 −0.0739035 −0.0369518 0.999317i \(-0.511765\pi\)
−0.0369518 + 0.999317i \(0.511765\pi\)
\(420\) 0 0
\(421\) 22.2722 1.08548 0.542739 0.839901i \(-0.317387\pi\)
0.542739 + 0.839901i \(0.317387\pi\)
\(422\) 0 0
\(423\) 45.1202i 2.19382i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.3893i 0.793135i
\(428\) 0 0
\(429\) 54.4397 2.62837
\(430\) 0 0
\(431\) −16.8858 −0.813362 −0.406681 0.913570i \(-0.633314\pi\)
−0.406681 + 0.913570i \(0.633314\pi\)
\(432\) 0 0
\(433\) 24.3956i 1.17238i 0.810175 + 0.586189i \(0.199372\pi\)
−0.810175 + 0.586189i \(0.800628\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.48097i 0.357863i
\(438\) 0 0
\(439\) −40.0378 −1.91090 −0.955450 0.295152i \(-0.904630\pi\)
−0.955450 + 0.295152i \(0.904630\pi\)
\(440\) 0 0
\(441\) −17.3513 −0.826251
\(442\) 0 0
\(443\) − 5.22809i − 0.248394i −0.992258 0.124197i \(-0.960365\pi\)
0.992258 0.124197i \(-0.0396355\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 5.53456i − 0.261776i
\(448\) 0 0
\(449\) −4.72459 −0.222967 −0.111484 0.993766i \(-0.535560\pi\)
−0.111484 + 0.993766i \(0.535560\pi\)
\(450\) 0 0
\(451\) −2.44292 −0.115033
\(452\) 0 0
\(453\) − 59.7213i − 2.80595i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 33.2311i − 1.55449i −0.629201 0.777243i \(-0.716618\pi\)
0.629201 0.777243i \(-0.283382\pi\)
\(458\) 0 0
\(459\) −69.4652 −3.24236
\(460\) 0 0
\(461\) −28.9619 −1.34889 −0.674446 0.738324i \(-0.735618\pi\)
−0.674446 + 0.738324i \(0.735618\pi\)
\(462\) 0 0
\(463\) 4.39258i 0.204141i 0.994777 + 0.102070i \(0.0325467\pi\)
−0.994777 + 0.102070i \(0.967453\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17.0729i − 0.790038i −0.918673 0.395019i \(-0.870738\pi\)
0.918673 0.395019i \(-0.129262\pi\)
\(468\) 0 0
\(469\) −20.2056 −0.933006
\(470\) 0 0
\(471\) 13.4367 0.619129
\(472\) 0 0
\(473\) 40.6330i 1.86831i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 75.9462i − 3.47734i
\(478\) 0 0
\(479\) −27.0441 −1.23568 −0.617838 0.786306i \(-0.711991\pi\)
−0.617838 + 0.786306i \(0.711991\pi\)
\(480\) 0 0
\(481\) 33.8292 1.54248
\(482\) 0 0
\(483\) − 6.40185i − 0.291294i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 28.0821i 1.27252i 0.771474 + 0.636261i \(0.219520\pi\)
−0.771474 + 0.636261i \(0.780480\pi\)
\(488\) 0 0
\(489\) −16.3893 −0.741151
\(490\) 0 0
\(491\) −7.48398 −0.337747 −0.168874 0.985638i \(-0.554013\pi\)
−0.168874 + 0.985638i \(0.554013\pi\)
\(492\) 0 0
\(493\) − 10.1109i − 0.455373i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 20.1202i − 0.902514i
\(498\) 0 0
\(499\) −21.7183 −0.972246 −0.486123 0.873890i \(-0.661589\pi\)
−0.486123 + 0.873890i \(0.661589\pi\)
\(500\) 0 0
\(501\) −24.6330 −1.10052
\(502\) 0 0
\(503\) − 7.27541i − 0.324395i −0.986758 0.162197i \(-0.948142\pi\)
0.986758 0.162197i \(-0.0518581\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.71833i 0.120725i
\(508\) 0 0
\(509\) −22.7151 −1.00683 −0.503414 0.864045i \(-0.667923\pi\)
−0.503414 + 0.864045i \(0.667923\pi\)
\(510\) 0 0
\(511\) 27.6074 1.22128
\(512\) 0 0
\(513\) − 80.1835i − 3.54019i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 35.3606i − 1.55516i
\(518\) 0 0
\(519\) 28.4429 1.24851
\(520\) 0 0
\(521\) −16.5568 −0.725368 −0.362684 0.931912i \(-0.618140\pi\)
−0.362684 + 0.931912i \(0.618140\pi\)
\(522\) 0 0
\(523\) 24.8037i 1.08459i 0.840188 + 0.542295i \(0.182445\pi\)
−0.840188 + 0.542295i \(0.817555\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 0.512767i − 0.0223365i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −74.9206 −3.25128
\(532\) 0 0
\(533\) 1.67425i 0.0725200i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.12018i − 0.393565i
\(538\) 0 0
\(539\) 13.5981 0.585714
\(540\) 0 0
\(541\) −21.8292 −0.938512 −0.469256 0.883062i \(-0.655478\pi\)
−0.469256 + 0.883062i \(0.655478\pi\)
\(542\) 0 0
\(543\) 33.3638i 1.43178i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 29.0759i 1.24319i 0.783337 + 0.621597i \(0.213516\pi\)
−0.783337 + 0.621597i \(0.786484\pi\)
\(548\) 0 0
\(549\) −51.0884 −2.18040
\(550\) 0 0
\(551\) 11.6710 0.497202
\(552\) 0 0
\(553\) 25.6074i 1.08894i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0348i 0.933645i 0.884351 + 0.466822i \(0.154601\pi\)
−0.884351 + 0.466822i \(0.845399\pi\)
\(558\) 0 0
\(559\) 27.8478 1.17784
\(560\) 0 0
\(561\) 101.357 4.27931
\(562\) 0 0
\(563\) 1.40185i 0.0590811i 0.999564 + 0.0295406i \(0.00940442\pi\)
−0.999564 + 0.0295406i \(0.990596\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 28.1930i 1.18400i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −18.3575 −0.768239 −0.384120 0.923283i \(-0.625495\pi\)
−0.384120 + 0.923283i \(0.625495\pi\)
\(572\) 0 0
\(573\) 23.9114i 0.998912i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.5128i 0.479283i 0.970861 + 0.239641i \(0.0770299\pi\)
−0.970861 + 0.239641i \(0.922970\pi\)
\(578\) 0 0
\(579\) 6.64547 0.276176
\(580\) 0 0
\(581\) 9.56009 0.396619
\(582\) 0 0
\(583\) 59.5188i 2.46502i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 40.1232i − 1.65606i −0.560683 0.828031i \(-0.689461\pi\)
0.560683 0.828031i \(-0.310539\pi\)
\(588\) 0 0
\(589\) 0.591886 0.0243882
\(590\) 0 0
\(591\) 37.0759 1.52510
\(592\) 0 0
\(593\) 18.7912i 0.771662i 0.922570 + 0.385831i \(0.126085\pi\)
−0.922570 + 0.385831i \(0.873915\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 43.1077i 1.76428i
\(598\) 0 0
\(599\) 27.4777 1.12271 0.561355 0.827575i \(-0.310280\pi\)
0.561355 + 0.827575i \(0.310280\pi\)
\(600\) 0 0
\(601\) −19.5283 −0.796576 −0.398288 0.917260i \(-0.630396\pi\)
−0.398288 + 0.917260i \(0.630396\pi\)
\(602\) 0 0
\(603\) − 62.9842i − 2.56492i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 45.9239i − 1.86399i −0.362467 0.931997i \(-0.618065\pi\)
0.362467 0.931997i \(-0.381935\pi\)
\(608\) 0 0
\(609\) −9.98748 −0.404713
\(610\) 0 0
\(611\) −24.2343 −0.980417
\(612\) 0 0
\(613\) 12.7091i 0.513314i 0.966503 + 0.256657i \(0.0826211\pi\)
−0.966503 + 0.256657i \(0.917379\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 48.3102i − 1.94490i −0.233120 0.972448i \(-0.574893\pi\)
0.233120 0.972448i \(-0.425107\pi\)
\(618\) 0 0
\(619\) −0.677265 −0.0272216 −0.0136108 0.999907i \(-0.504333\pi\)
−0.0136108 + 0.999907i \(0.504333\pi\)
\(620\) 0 0
\(621\) 10.7183 0.430112
\(622\) 0 0
\(623\) − 17.2909i − 0.692747i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 116.997i 4.67239i
\(628\) 0 0
\(629\) 62.9842 2.51135
\(630\) 0 0
\(631\) 44.3986 1.76748 0.883740 0.467978i \(-0.155017\pi\)
0.883740 + 0.467978i \(0.155017\pi\)
\(632\) 0 0
\(633\) 10.4747i 0.416332i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 9.31949i − 0.369251i
\(638\) 0 0
\(639\) 62.7181 2.48109
\(640\) 0 0
\(641\) 16.8798 0.666713 0.333356 0.942801i \(-0.391819\pi\)
0.333356 + 0.942801i \(0.391819\pi\)
\(642\) 0 0
\(643\) − 26.7564i − 1.05517i −0.849503 0.527584i \(-0.823098\pi\)
0.849503 0.527584i \(-0.176902\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 32.5568i − 1.27994i −0.768399 0.639971i \(-0.778946\pi\)
0.768399 0.639971i \(-0.221054\pi\)
\(648\) 0 0
\(649\) 58.7151 2.30477
\(650\) 0 0
\(651\) −0.506507 −0.0198516
\(652\) 0 0
\(653\) − 9.87356i − 0.386382i −0.981161 0.193191i \(-0.938116\pi\)
0.981161 0.193191i \(-0.0618837\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 86.0571i 3.35741i
\(658\) 0 0
\(659\) 13.5128 0.526383 0.263191 0.964744i \(-0.415225\pi\)
0.263191 + 0.964744i \(0.415225\pi\)
\(660\) 0 0
\(661\) −26.7687 −1.04118 −0.520590 0.853807i \(-0.674288\pi\)
−0.520590 + 0.853807i \(0.674288\pi\)
\(662\) 0 0
\(663\) − 69.4652i − 2.69780i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.56009i 0.0604070i
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 40.0378 1.54564
\(672\) 0 0
\(673\) − 0.803708i − 0.0309807i −0.999880 0.0154903i \(-0.995069\pi\)
0.999880 0.0154903i \(-0.00493092\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 33.6422i 1.29298i 0.762924 + 0.646488i \(0.223763\pi\)
−0.762924 + 0.646488i \(0.776237\pi\)
\(678\) 0 0
\(679\) 15.0473 0.577463
\(680\) 0 0
\(681\) 16.4872 0.631792
\(682\) 0 0
\(683\) − 11.8736i − 0.454329i −0.973856 0.227165i \(-0.927054\pi\)
0.973856 0.227165i \(-0.0729455\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.47471i 0.0944162i
\(688\) 0 0
\(689\) 40.7912 1.55402
\(690\) 0 0
\(691\) 27.9114 1.06180 0.530899 0.847435i \(-0.321854\pi\)
0.530899 + 0.847435i \(0.321854\pi\)
\(692\) 0 0
\(693\) − 68.4397i − 2.59981i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.11717i 0.118071i
\(698\) 0 0
\(699\) −43.5949 −1.64891
\(700\) 0 0
\(701\) −26.3668 −0.995861 −0.497930 0.867217i \(-0.665906\pi\)
−0.497930 + 0.867217i \(0.665906\pi\)
\(702\) 0 0
\(703\) 72.7026i 2.74203i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.07286i 0.190785i
\(708\) 0 0
\(709\) −11.8007 −0.443184 −0.221592 0.975139i \(-0.571125\pi\)
−0.221592 + 0.975139i \(0.571125\pi\)
\(710\) 0 0
\(711\) −79.8227 −2.99359
\(712\) 0 0
\(713\) 0.0791189i 0.00296302i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 33.6074i − 1.25509i
\(718\) 0 0
\(719\) 12.9589 0.483287 0.241643 0.970365i \(-0.422314\pi\)
0.241643 + 0.970365i \(0.422314\pi\)
\(720\) 0 0
\(721\) −16.3893 −0.610371
\(722\) 0 0
\(723\) − 84.4807i − 3.14187i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 19.0318i 0.705850i 0.935652 + 0.352925i \(0.114813\pi\)
−0.935652 + 0.352925i \(0.885187\pi\)
\(728\) 0 0
\(729\) 11.1264 0.412090
\(730\) 0 0
\(731\) 51.8478 1.91766
\(732\) 0 0
\(733\) − 30.5220i − 1.12736i −0.825994 0.563679i \(-0.809386\pi\)
0.825994 0.563679i \(-0.190614\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 49.3606i 1.81822i
\(738\) 0 0
\(739\) 6.18702 0.227593 0.113797 0.993504i \(-0.463699\pi\)
0.113797 + 0.993504i \(0.463699\pi\)
\(740\) 0 0
\(741\) 80.1835 2.94562
\(742\) 0 0
\(743\) 16.2722i 0.596968i 0.954415 + 0.298484i \(0.0964809\pi\)
−0.954415 + 0.298484i \(0.903519\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 29.8005i 1.09034i
\(748\) 0 0
\(749\) −34.6803 −1.26719
\(750\) 0 0
\(751\) −17.5949 −0.642047 −0.321023 0.947071i \(-0.604027\pi\)
−0.321023 + 0.947071i \(0.604027\pi\)
\(752\) 0 0
\(753\) − 50.8640i − 1.85359i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 41.2496i − 1.49924i −0.661866 0.749622i \(-0.730235\pi\)
0.661866 0.749622i \(-0.269765\pi\)
\(758\) 0 0
\(759\) −15.6392 −0.567667
\(760\) 0 0
\(761\) 26.8067 0.971743 0.485871 0.874030i \(-0.338502\pi\)
0.485871 + 0.874030i \(0.338502\pi\)
\(762\) 0 0
\(763\) − 11.0666i − 0.400638i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 40.2404i − 1.45300i
\(768\) 0 0
\(769\) 26.8798 0.969311 0.484655 0.874705i \(-0.338945\pi\)
0.484655 + 0.874705i \(0.338945\pi\)
\(770\) 0 0
\(771\) 60.8608 2.19185
\(772\) 0 0
\(773\) 41.0441i 1.47625i 0.674662 + 0.738126i \(0.264289\pi\)
−0.674662 + 0.738126i \(0.735711\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 62.2153i − 2.23196i
\(778\) 0 0
\(779\) −3.59815 −0.128917
\(780\) 0 0
\(781\) −49.1520 −1.75880
\(782\) 0 0
\(783\) − 16.7216i − 0.597580i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 21.4840i 0.765821i 0.923785 + 0.382911i \(0.125078\pi\)
−0.923785 + 0.382911i \(0.874922\pi\)
\(788\) 0 0
\(789\) −22.9176 −0.815889
\(790\) 0 0
\(791\) 4.58563 0.163046
\(792\) 0 0
\(793\) − 27.4399i − 0.974420i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 38.2692i − 1.35556i −0.735263 0.677781i \(-0.762942\pi\)
0.735263 0.677781i \(-0.237058\pi\)
\(798\) 0 0
\(799\) −45.1202 −1.59624
\(800\) 0 0
\(801\) 53.8989 1.90442
\(802\) 0 0
\(803\) − 67.4427i − 2.38000i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 70.0696i 2.46657i
\(808\) 0 0
\(809\) 27.2026 0.956391 0.478195 0.878253i \(-0.341291\pi\)
0.478195 + 0.878253i \(0.341291\pi\)
\(810\) 0 0
\(811\) 38.5095 1.35225 0.676126 0.736786i \(-0.263657\pi\)
0.676126 + 0.736786i \(0.263657\pi\)
\(812\) 0 0
\(813\) − 58.6297i − 2.05623i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 59.8478i 2.09381i
\(818\) 0 0
\(819\) −46.9051 −1.63900
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 40.1457i 1.39939i 0.714441 + 0.699696i \(0.246681\pi\)
−0.714441 + 0.699696i \(0.753319\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 0.851033i − 0.0295933i −0.999891 0.0147967i \(-0.995290\pi\)
0.999891 0.0147967i \(-0.00471009\pi\)
\(828\) 0 0
\(829\) 2.83851 0.0985856 0.0492928 0.998784i \(-0.484303\pi\)
0.0492928 + 0.998784i \(0.484303\pi\)
\(830\) 0 0
\(831\) −56.3986 −1.95645
\(832\) 0 0
\(833\) − 17.3513i − 0.601186i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 0.848022i − 0.0293119i
\(838\) 0 0
\(839\) 10.2343 0.353329 0.176664 0.984271i \(-0.443469\pi\)
0.176664 + 0.984271i \(0.443469\pi\)
\(840\) 0 0
\(841\) −26.5661 −0.916073
\(842\) 0 0
\(843\) 80.3100i 2.76602i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 30.7657i 1.05712i
\(848\) 0 0
\(849\) 78.9554 2.70974
\(850\) 0 0
\(851\) −9.71833 −0.333140
\(852\) 0 0
\(853\) 17.7213i 0.606767i 0.952869 + 0.303384i \(0.0981163\pi\)
−0.952869 + 0.303384i \(0.901884\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 17.6835i − 0.604058i −0.953299 0.302029i \(-0.902336\pi\)
0.953299 0.302029i \(-0.0976639\pi\)
\(858\) 0 0
\(859\) 26.6042 0.907722 0.453861 0.891072i \(-0.350046\pi\)
0.453861 + 0.891072i \(0.350046\pi\)
\(860\) 0 0
\(861\) 3.07912 0.104936
\(862\) 0 0
\(863\) − 53.5949i − 1.82439i −0.409755 0.912196i \(-0.634386\pi\)
0.409755 0.912196i \(-0.365614\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 76.9872i − 2.61462i
\(868\) 0 0
\(869\) 62.5568 2.12210
\(870\) 0 0
\(871\) 33.8292 1.14626
\(872\) 0 0
\(873\) 46.9051i 1.58750i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 25.9650i − 0.876774i −0.898786 0.438387i \(-0.855550\pi\)
0.898786 0.438387i \(-0.144450\pi\)
\(878\) 0 0
\(879\) −94.9554 −3.20277
\(880\) 0 0
\(881\) 29.9239 1.00816 0.504081 0.863657i \(-0.331831\pi\)
0.504081 + 0.863657i \(0.331831\pi\)
\(882\) 0 0
\(883\) − 4.75939i − 0.160166i −0.996788 0.0800832i \(-0.974481\pi\)
0.996788 0.0800832i \(-0.0255186\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 38.8037i − 1.30290i −0.758691 0.651451i \(-0.774161\pi\)
0.758691 0.651451i \(-0.225839\pi\)
\(888\) 0 0
\(889\) 12.8037 0.429423
\(890\) 0 0
\(891\) 68.8733 2.30734
\(892\) 0 0
\(893\) − 52.0821i − 1.74286i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 10.7183i 0.357875i
\(898\) 0 0
\(899\) 0.123433 0.00411671
\(900\) 0 0
\(901\) 75.9462 2.53013
\(902\) 0 0
\(903\) − 51.2148i − 1.70432i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 51.0729i 1.69585i 0.530119 + 0.847923i \(0.322147\pi\)
−0.530119 + 0.847923i \(0.677853\pi\)
\(908\) 0 0
\(909\) −15.8130 −0.524483
\(910\) 0 0
\(911\) −30.4933 −1.01029 −0.505143 0.863035i \(-0.668560\pi\)
−0.505143 + 0.863035i \(0.668560\pi\)
\(912\) 0 0
\(913\) − 23.3545i − 0.772922i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 14.9619i − 0.494087i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −5.79745 −0.191032
\(922\) 0 0
\(923\) 33.6863i 1.10880i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 51.0884i − 1.67796i
\(928\) 0 0
\(929\) 17.3257 0.568439 0.284220 0.958759i \(-0.408266\pi\)
0.284220 + 0.958759i \(0.408266\pi\)
\(930\) 0 0
\(931\) 20.0285 0.656409
\(932\) 0 0
\(933\) 4.94943i 0.162037i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 47.5631i 1.55382i 0.629612 + 0.776909i \(0.283214\pi\)
−0.629612 + 0.776909i \(0.716786\pi\)
\(938\) 0 0
\(939\) 5.19304 0.169469
\(940\) 0 0
\(941\) 14.2754 0.465365 0.232683 0.972553i \(-0.425250\pi\)
0.232683 + 0.972553i \(0.425250\pi\)
\(942\) 0 0
\(943\) − 0.480973i − 0.0156626i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 36.1900i − 1.17602i −0.808854 0.588009i \(-0.799912\pi\)
0.808854 0.588009i \(-0.200088\pi\)
\(948\) 0 0
\(949\) −46.2218 −1.50042
\(950\) 0 0
\(951\) −39.0348 −1.26579
\(952\) 0 0
\(953\) 37.7942i 1.22427i 0.790752 + 0.612137i \(0.209690\pi\)
−0.790752 + 0.612137i \(0.790310\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 24.3986i 0.788695i
\(958\) 0 0
\(959\) 26.1993 0.846020
\(960\) 0 0
\(961\) −30.9937 −0.999798
\(962\) 0 0
\(963\) − 108.104i − 3.48362i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 50.6390i − 1.62844i −0.580557 0.814220i \(-0.697165\pi\)
0.580557 0.814220i \(-0.302835\pi\)
\(968\) 0 0
\(969\) 149.288 4.79582
\(970\) 0 0
\(971\) 2.82623 0.0906981 0.0453490 0.998971i \(-0.485560\pi\)
0.0453490 + 0.998971i \(0.485560\pi\)
\(972\) 0 0
\(973\) 18.2056i 0.583644i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 25.8448i − 0.826848i −0.910539 0.413424i \(-0.864333\pi\)
0.910539 0.413424i \(-0.135667\pi\)
\(978\) 0 0
\(979\) −42.2404 −1.35001
\(980\) 0 0
\(981\) 34.4965 1.10139
\(982\) 0 0
\(983\) − 41.3668i − 1.31940i −0.751531 0.659698i \(-0.770684\pi\)
0.751531 0.659698i \(-0.229316\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 44.5694i 1.41866i
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 54.4745 1.73044 0.865219 0.501394i \(-0.167179\pi\)
0.865219 + 0.501394i \(0.167179\pi\)
\(992\) 0 0
\(993\) 102.810i 3.26257i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 15.5128i 0.491294i 0.969359 + 0.245647i \(0.0790004\pi\)
−0.969359 + 0.245647i \(0.921000\pi\)
\(998\) 0 0
\(999\) 104.164 3.29561
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.p.4049.6 6
5.2 odd 4 920.2.a.h.1.3 3
5.3 odd 4 4600.2.a.x.1.1 3
5.4 even 2 inner 4600.2.e.p.4049.1 6
15.2 even 4 8280.2.a.bj.1.1 3
20.3 even 4 9200.2.a.ce.1.3 3
20.7 even 4 1840.2.a.s.1.1 3
40.27 even 4 7360.2.a.cc.1.3 3
40.37 odd 4 7360.2.a.by.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.3 3 5.2 odd 4
1840.2.a.s.1.1 3 20.7 even 4
4600.2.a.x.1.1 3 5.3 odd 4
4600.2.e.p.4049.1 6 5.4 even 2 inner
4600.2.e.p.4049.6 6 1.1 even 1 trivial
7360.2.a.by.1.1 3 40.37 odd 4
7360.2.a.cc.1.3 3 40.27 even 4
8280.2.a.bj.1.1 3 15.2 even 4
9200.2.a.ce.1.3 3 20.3 even 4