Properties

Label 4600.2.e.w.4049.4
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 4049.4
Root \(-1.36629i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.36629i q^{3} -3.28093i q^{7} +1.13327 q^{9} +O(q^{10})\) \(q-1.36629i q^{3} -3.28093i q^{7} +1.13327 q^{9} +3.49709 q^{11} -3.41420i q^{13} -7.46023i q^{17} -3.49709 q^{19} -4.48269 q^{21} -1.00000i q^{23} -5.64722i q^{27} +3.46268 q^{29} +2.01105 q^{31} -4.77803i q^{33} +0.511497i q^{37} -4.66477 q^{39} -7.07954 q^{41} +2.76452i q^{43} -0.889198i q^{47} -3.76452 q^{49} -10.1928 q^{51} +14.2383i q^{53} +4.77803i q^{57} +4.71325 q^{59} +13.4769 q^{61} -3.71817i q^{63} -2.30025i q^{67} -1.36629 q^{69} -10.6214 q^{71} -3.70765i q^{73} -11.4737i q^{77} +7.97978 q^{79} -4.31592 q^{81} -9.42336i q^{83} -4.73101i q^{87} -0.801390 q^{89} -11.2018 q^{91} -2.74766i q^{93} +11.7722i q^{97} +3.96313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} + O(q^{10}) \) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99} + O(q^{100}) \)

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
\(3\) − 1.36629i − 0.788825i −0.918933 0.394413i \(-0.870948\pi\)
0.918933 0.394413i \(-0.129052\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.28093i − 1.24008i −0.784572 0.620038i \(-0.787117\pi\)
0.784572 0.620038i \(-0.212883\pi\)
\(8\) 0 0
\(9\) 1.13327 0.377755
\(10\) 0 0
\(11\) 3.49709 1.05441 0.527207 0.849737i \(-0.323239\pi\)
0.527207 + 0.849737i \(0.323239\pi\)
\(12\) 0 0
\(13\) − 3.41420i − 0.946928i −0.880813 0.473464i \(-0.843003\pi\)
0.880813 0.473464i \(-0.156997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.46023i − 1.80937i −0.426080 0.904685i \(-0.640106\pi\)
0.426080 0.904685i \(-0.359894\pi\)
\(18\) 0 0
\(19\) −3.49709 −0.802288 −0.401144 0.916015i \(-0.631387\pi\)
−0.401144 + 0.916015i \(0.631387\pi\)
\(20\) 0 0
\(21\) −4.48269 −0.978203
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.64722i − 1.08681i
\(28\) 0 0
\(29\) 3.46268 0.643004 0.321502 0.946909i \(-0.395812\pi\)
0.321502 + 0.946909i \(0.395812\pi\)
\(30\) 0 0
\(31\) 2.01105 0.361195 0.180597 0.983557i \(-0.442197\pi\)
0.180597 + 0.983557i \(0.442197\pi\)
\(32\) 0 0
\(33\) − 4.77803i − 0.831748i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.511497i 0.0840896i 0.999116 + 0.0420448i \(0.0133872\pi\)
−0.999116 + 0.0420448i \(0.986613\pi\)
\(38\) 0 0
\(39\) −4.66477 −0.746961
\(40\) 0 0
\(41\) −7.07954 −1.10564 −0.552819 0.833301i \(-0.686448\pi\)
−0.552819 + 0.833301i \(0.686448\pi\)
\(42\) 0 0
\(43\) 2.76452i 0.421586i 0.977531 + 0.210793i \(0.0676046\pi\)
−0.977531 + 0.210793i \(0.932395\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.889198i − 0.129703i −0.997895 0.0648514i \(-0.979343\pi\)
0.997895 0.0648514i \(-0.0206573\pi\)
\(48\) 0 0
\(49\) −3.76452 −0.537789
\(50\) 0 0
\(51\) −10.1928 −1.42728
\(52\) 0 0
\(53\) 14.2383i 1.95577i 0.209132 + 0.977887i \(0.432936\pi\)
−0.209132 + 0.977887i \(0.567064\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.77803i 0.632865i
\(58\) 0 0
\(59\) 4.71325 0.613613 0.306807 0.951772i \(-0.400739\pi\)
0.306807 + 0.951772i \(0.400739\pi\)
\(60\) 0 0
\(61\) 13.4769 1.72554 0.862769 0.505599i \(-0.168728\pi\)
0.862769 + 0.505599i \(0.168728\pi\)
\(62\) 0 0
\(63\) − 3.71817i − 0.468445i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.30025i − 0.281020i −0.990079 0.140510i \(-0.955126\pi\)
0.990079 0.140510i \(-0.0448743\pi\)
\(68\) 0 0
\(69\) −1.36629 −0.164481
\(70\) 0 0
\(71\) −10.6214 −1.26053 −0.630264 0.776381i \(-0.717053\pi\)
−0.630264 + 0.776381i \(0.717053\pi\)
\(72\) 0 0
\(73\) − 3.70765i − 0.433948i −0.976177 0.216974i \(-0.930381\pi\)
0.976177 0.216974i \(-0.0696186\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 11.4737i − 1.30755i
\(78\) 0 0
\(79\) 7.97978 0.897796 0.448898 0.893583i \(-0.351817\pi\)
0.448898 + 0.893583i \(0.351817\pi\)
\(80\) 0 0
\(81\) −4.31592 −0.479546
\(82\) 0 0
\(83\) − 9.42336i − 1.03435i −0.855880 0.517174i \(-0.826984\pi\)
0.855880 0.517174i \(-0.173016\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.73101i − 0.507218i
\(88\) 0 0
\(89\) −0.801390 −0.0849471 −0.0424736 0.999098i \(-0.513524\pi\)
−0.0424736 + 0.999098i \(0.513524\pi\)
\(90\) 0 0
\(91\) −11.2018 −1.17426
\(92\) 0 0
\(93\) − 2.74766i − 0.284919i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.7722i 1.19529i 0.801762 + 0.597644i \(0.203896\pi\)
−0.801762 + 0.597644i \(0.796104\pi\)
\(98\) 0 0
\(99\) 3.96313 0.398310
\(100\) 0 0
\(101\) 11.1559 1.11006 0.555028 0.831831i \(-0.312707\pi\)
0.555028 + 0.831831i \(0.312707\pi\)
\(102\) 0 0
\(103\) − 1.28585i − 0.126698i −0.997991 0.0633491i \(-0.979822\pi\)
0.997991 0.0633491i \(-0.0201782\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 13.8975i − 1.34352i −0.740770 0.671759i \(-0.765539\pi\)
0.740770 0.671759i \(-0.234461\pi\)
\(108\) 0 0
\(109\) 3.77713 0.361783 0.180892 0.983503i \(-0.442102\pi\)
0.180892 + 0.983503i \(0.442102\pi\)
\(110\) 0 0
\(111\) 0.698851 0.0663320
\(112\) 0 0
\(113\) 19.3111i 1.81663i 0.418282 + 0.908317i \(0.362632\pi\)
−0.418282 + 0.908317i \(0.637368\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.86919i − 0.357707i
\(118\) 0 0
\(119\) −24.4765 −2.24376
\(120\) 0 0
\(121\) 1.22966 0.111788
\(122\) 0 0
\(123\) 9.67267i 0.872155i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 13.7612i − 1.22111i −0.791975 0.610553i \(-0.790947\pi\)
0.791975 0.610553i \(-0.209053\pi\)
\(128\) 0 0
\(129\) 3.77713 0.332558
\(130\) 0 0
\(131\) −7.41665 −0.647996 −0.323998 0.946058i \(-0.605027\pi\)
−0.323998 + 0.946058i \(0.605027\pi\)
\(132\) 0 0
\(133\) 11.4737i 0.994899i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.5502i 1.75573i 0.478912 + 0.877863i \(0.341031\pi\)
−0.478912 + 0.877863i \(0.658969\pi\)
\(138\) 0 0
\(139\) 5.49652 0.466209 0.233104 0.972452i \(-0.425112\pi\)
0.233104 + 0.972452i \(0.425112\pi\)
\(140\) 0 0
\(141\) −1.21490 −0.102313
\(142\) 0 0
\(143\) − 11.9398i − 0.998454i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.14341i 0.424221i
\(148\) 0 0
\(149\) −4.80139 −0.393345 −0.196673 0.980469i \(-0.563014\pi\)
−0.196673 + 0.980469i \(0.563014\pi\)
\(150\) 0 0
\(151\) −3.26862 −0.265997 −0.132998 0.991116i \(-0.542461\pi\)
−0.132998 + 0.991116i \(0.542461\pi\)
\(152\) 0 0
\(153\) − 8.45441i − 0.683499i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 21.8392i − 1.74296i −0.490430 0.871480i \(-0.663160\pi\)
0.490430 0.871480i \(-0.336840\pi\)
\(158\) 0 0
\(159\) 19.4535 1.54276
\(160\) 0 0
\(161\) −3.28093 −0.258574
\(162\) 0 0
\(163\) 23.2435i 1.82057i 0.413982 + 0.910285i \(0.364138\pi\)
−0.413982 + 0.910285i \(0.635862\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.13179i 0.319728i 0.987139 + 0.159864i \(0.0511055\pi\)
−0.987139 + 0.159864i \(0.948894\pi\)
\(168\) 0 0
\(169\) 1.34325 0.103327
\(170\) 0 0
\(171\) −3.96313 −0.303068
\(172\) 0 0
\(173\) 5.88548i 0.447465i 0.974651 + 0.223732i \(0.0718241\pi\)
−0.974651 + 0.223732i \(0.928176\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.43965i − 0.484034i
\(178\) 0 0
\(179\) 6.85789 0.512583 0.256292 0.966600i \(-0.417499\pi\)
0.256292 + 0.966600i \(0.417499\pi\)
\(180\) 0 0
\(181\) −2.29042 −0.170246 −0.0851229 0.996370i \(-0.527128\pi\)
−0.0851229 + 0.996370i \(0.527128\pi\)
\(182\) 0 0
\(183\) − 18.4133i − 1.36115i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 26.0891i − 1.90782i
\(188\) 0 0
\(189\) −18.5281 −1.34772
\(190\) 0 0
\(191\) −9.58748 −0.693725 −0.346863 0.937916i \(-0.612753\pi\)
−0.346863 + 0.937916i \(0.612753\pi\)
\(192\) 0 0
\(193\) 13.9410i 1.00350i 0.865013 + 0.501749i \(0.167310\pi\)
−0.865013 + 0.501749i \(0.832690\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.1011i − 1.07591i −0.842974 0.537954i \(-0.819197\pi\)
0.842974 0.537954i \(-0.180803\pi\)
\(198\) 0 0
\(199\) −14.5128 −1.02879 −0.514394 0.857554i \(-0.671983\pi\)
−0.514394 + 0.857554i \(0.671983\pi\)
\(200\) 0 0
\(201\) −3.14280 −0.221676
\(202\) 0 0
\(203\) − 11.3608i − 0.797374i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.13327i − 0.0787674i
\(208\) 0 0
\(209\) −12.2297 −0.845944
\(210\) 0 0
\(211\) −23.4846 −1.61674 −0.808372 0.588672i \(-0.799651\pi\)
−0.808372 + 0.588672i \(0.799651\pi\)
\(212\) 0 0
\(213\) 14.5119i 0.994336i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 6.59811i − 0.447909i
\(218\) 0 0
\(219\) −5.06571 −0.342309
\(220\) 0 0
\(221\) −25.4707 −1.71334
\(222\) 0 0
\(223\) − 8.20720i − 0.549595i −0.961502 0.274797i \(-0.911389\pi\)
0.961502 0.274797i \(-0.0886108\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 16.9942i − 1.12794i −0.825794 0.563972i \(-0.809273\pi\)
0.825794 0.563972i \(-0.190727\pi\)
\(228\) 0 0
\(229\) −19.6818 −1.30061 −0.650306 0.759672i \(-0.725359\pi\)
−0.650306 + 0.759672i \(0.725359\pi\)
\(230\) 0 0
\(231\) −15.6764 −1.03143
\(232\) 0 0
\(233\) 24.8799i 1.62993i 0.579507 + 0.814967i \(0.303245\pi\)
−0.579507 + 0.814967i \(0.696755\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 10.9027i − 0.708204i
\(238\) 0 0
\(239\) 0.0718483 0.00464748 0.00232374 0.999997i \(-0.499260\pi\)
0.00232374 + 0.999997i \(0.499260\pi\)
\(240\) 0 0
\(241\) −8.50658 −0.547957 −0.273979 0.961736i \(-0.588340\pi\)
−0.273979 + 0.961736i \(0.588340\pi\)
\(242\) 0 0
\(243\) − 11.0449i − 0.708530i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 11.9398i 0.759709i
\(248\) 0 0
\(249\) −12.8750 −0.815920
\(250\) 0 0
\(251\) −19.8858 −1.25518 −0.627591 0.778543i \(-0.715959\pi\)
−0.627591 + 0.778543i \(0.715959\pi\)
\(252\) 0 0
\(253\) − 3.49709i − 0.219860i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.4062i 0.711498i 0.934582 + 0.355749i \(0.115774\pi\)
−0.934582 + 0.355749i \(0.884226\pi\)
\(258\) 0 0
\(259\) 1.67819 0.104277
\(260\) 0 0
\(261\) 3.92414 0.242898
\(262\) 0 0
\(263\) − 13.3903i − 0.825679i −0.910804 0.412840i \(-0.864537\pi\)
0.910804 0.412840i \(-0.135463\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.09493i 0.0670084i
\(268\) 0 0
\(269\) −13.8103 −0.842027 −0.421014 0.907054i \(-0.638326\pi\)
−0.421014 + 0.907054i \(0.638326\pi\)
\(270\) 0 0
\(271\) 26.7574 1.62540 0.812699 0.582683i \(-0.197997\pi\)
0.812699 + 0.582683i \(0.197997\pi\)
\(272\) 0 0
\(273\) 15.3048i 0.926288i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.80667i 0.108552i 0.998526 + 0.0542760i \(0.0172851\pi\)
−0.998526 + 0.0542760i \(0.982715\pi\)
\(278\) 0 0
\(279\) 2.27905 0.136443
\(280\) 0 0
\(281\) 27.1348 1.61873 0.809364 0.587308i \(-0.199812\pi\)
0.809364 + 0.587308i \(0.199812\pi\)
\(282\) 0 0
\(283\) 20.9875i 1.24758i 0.781594 + 0.623788i \(0.214407\pi\)
−0.781594 + 0.623788i \(0.785593\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.2275i 1.37108i
\(288\) 0 0
\(289\) −38.6550 −2.27382
\(290\) 0 0
\(291\) 16.0842 0.942873
\(292\) 0 0
\(293\) 0.00671252i 0 0.000392149i 1.00000 0.000196075i \(6.24125e-5\pi\)
−1.00000 0.000196075i \(0.999938\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 19.7489i − 1.14594i
\(298\) 0 0
\(299\) −3.41420 −0.197448
\(300\) 0 0
\(301\) 9.07022 0.522799
\(302\) 0 0
\(303\) − 15.2422i − 0.875641i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.9767i 1.14013i 0.821598 + 0.570067i \(0.193083\pi\)
−0.821598 + 0.570067i \(0.806917\pi\)
\(308\) 0 0
\(309\) −1.75683 −0.0999427
\(310\) 0 0
\(311\) 23.8601 1.35298 0.676491 0.736451i \(-0.263500\pi\)
0.676491 + 0.736451i \(0.263500\pi\)
\(312\) 0 0
\(313\) − 17.4406i − 0.985803i −0.870085 0.492901i \(-0.835936\pi\)
0.870085 0.492901i \(-0.164064\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.35581i − 0.0761499i −0.999275 0.0380750i \(-0.987877\pi\)
0.999275 0.0380750i \(-0.0121226\pi\)
\(318\) 0 0
\(319\) 12.1093 0.677992
\(320\) 0 0
\(321\) −18.9879 −1.05980
\(322\) 0 0
\(323\) 26.0891i 1.45164i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 5.16063i − 0.285384i
\(328\) 0 0
\(329\) −2.91740 −0.160841
\(330\) 0 0
\(331\) 17.4050 0.956667 0.478333 0.878178i \(-0.341241\pi\)
0.478333 + 0.878178i \(0.341241\pi\)
\(332\) 0 0
\(333\) 0.579662i 0.0317653i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.4648i − 1.22373i −0.790961 0.611867i \(-0.790419\pi\)
0.790961 0.611867i \(-0.209581\pi\)
\(338\) 0 0
\(339\) 26.3845 1.43301
\(340\) 0 0
\(341\) 7.03282 0.380849
\(342\) 0 0
\(343\) − 10.6154i − 0.573177i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.97296i − 0.159597i −0.996811 0.0797984i \(-0.974572\pi\)
0.996811 0.0797984i \(-0.0254277\pi\)
\(348\) 0 0
\(349\) 4.36975 0.233908 0.116954 0.993137i \(-0.462687\pi\)
0.116954 + 0.993137i \(0.462687\pi\)
\(350\) 0 0
\(351\) −19.2807 −1.02913
\(352\) 0 0
\(353\) 31.4419i 1.67348i 0.547598 + 0.836742i \(0.315542\pi\)
−0.547598 + 0.836742i \(0.684458\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 33.4419i 1.76993i
\(358\) 0 0
\(359\) −27.7912 −1.46676 −0.733381 0.679818i \(-0.762059\pi\)
−0.733381 + 0.679818i \(0.762059\pi\)
\(360\) 0 0
\(361\) −6.77034 −0.356333
\(362\) 0 0
\(363\) − 1.68007i − 0.0881809i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 13.9061i − 0.725890i −0.931811 0.362945i \(-0.881771\pi\)
0.931811 0.362945i \(-0.118229\pi\)
\(368\) 0 0
\(369\) −8.02299 −0.417660
\(370\) 0 0
\(371\) 46.7148 2.42531
\(372\) 0 0
\(373\) − 26.2799i − 1.36072i −0.732876 0.680362i \(-0.761823\pi\)
0.732876 0.680362i \(-0.238177\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 11.8223i − 0.608879i
\(378\) 0 0
\(379\) 3.06792 0.157588 0.0787942 0.996891i \(-0.474893\pi\)
0.0787942 + 0.996891i \(0.474893\pi\)
\(380\) 0 0
\(381\) −18.8017 −0.963239
\(382\) 0 0
\(383\) 28.8872i 1.47607i 0.674763 + 0.738034i \(0.264246\pi\)
−0.674763 + 0.738034i \(0.735754\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.13294i 0.159256i
\(388\) 0 0
\(389\) −24.4697 −1.24066 −0.620331 0.784340i \(-0.713002\pi\)
−0.620331 + 0.784340i \(0.713002\pi\)
\(390\) 0 0
\(391\) −7.46023 −0.377280
\(392\) 0 0
\(393\) 10.1333i 0.511156i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.44828i 0.424007i 0.977269 + 0.212004i \(0.0679988\pi\)
−0.977269 + 0.212004i \(0.932001\pi\)
\(398\) 0 0
\(399\) 15.6764 0.784801
\(400\) 0 0
\(401\) 20.3352 1.01549 0.507746 0.861507i \(-0.330479\pi\)
0.507746 + 0.861507i \(0.330479\pi\)
\(402\) 0 0
\(403\) − 6.86611i − 0.342025i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.78875i 0.0886652i
\(408\) 0 0
\(409\) −6.05655 −0.299477 −0.149738 0.988726i \(-0.547843\pi\)
−0.149738 + 0.988726i \(0.547843\pi\)
\(410\) 0 0
\(411\) 28.0775 1.38496
\(412\) 0 0
\(413\) − 15.4639i − 0.760927i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 7.50982i − 0.367757i
\(418\) 0 0
\(419\) −7.02434 −0.343162 −0.171581 0.985170i \(-0.554887\pi\)
−0.171581 + 0.985170i \(0.554887\pi\)
\(420\) 0 0
\(421\) 7.17652 0.349762 0.174881 0.984590i \(-0.444046\pi\)
0.174881 + 0.984590i \(0.444046\pi\)
\(422\) 0 0
\(423\) − 1.00770i − 0.0489959i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 44.2167i − 2.13980i
\(428\) 0 0
\(429\) −16.3131 −0.787605
\(430\) 0 0
\(431\) 11.2944 0.544034 0.272017 0.962292i \(-0.412309\pi\)
0.272017 + 0.962292i \(0.412309\pi\)
\(432\) 0 0
\(433\) − 22.2266i − 1.06814i −0.845439 0.534072i \(-0.820661\pi\)
0.845439 0.534072i \(-0.179339\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.49709i 0.167289i
\(438\) 0 0
\(439\) 30.2573 1.44410 0.722052 0.691839i \(-0.243199\pi\)
0.722052 + 0.691839i \(0.243199\pi\)
\(440\) 0 0
\(441\) −4.26620 −0.203153
\(442\) 0 0
\(443\) 14.0307i 0.666620i 0.942817 + 0.333310i \(0.108166\pi\)
−0.942817 + 0.333310i \(0.891834\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.56007i 0.310281i
\(448\) 0 0
\(449\) −8.70110 −0.410630 −0.205315 0.978696i \(-0.565822\pi\)
−0.205315 + 0.978696i \(0.565822\pi\)
\(450\) 0 0
\(451\) −24.7578 −1.16580
\(452\) 0 0
\(453\) 4.46587i 0.209825i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 33.8379i 1.58287i 0.611255 + 0.791434i \(0.290665\pi\)
−0.611255 + 0.791434i \(0.709335\pi\)
\(458\) 0 0
\(459\) −42.1295 −1.96644
\(460\) 0 0
\(461\) 39.6512 1.84674 0.923370 0.383912i \(-0.125423\pi\)
0.923370 + 0.383912i \(0.125423\pi\)
\(462\) 0 0
\(463\) − 34.8490i − 1.61957i −0.586727 0.809784i \(-0.699584\pi\)
0.586727 0.809784i \(-0.300416\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 13.9442i − 0.645258i −0.946525 0.322629i \(-0.895433\pi\)
0.946525 0.322629i \(-0.104567\pi\)
\(468\) 0 0
\(469\) −7.54697 −0.348486
\(470\) 0 0
\(471\) −29.8386 −1.37489
\(472\) 0 0
\(473\) 9.66780i 0.444526i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 16.1357i 0.738804i
\(478\) 0 0
\(479\) 32.4103 1.48086 0.740432 0.672131i \(-0.234621\pi\)
0.740432 + 0.672131i \(0.234621\pi\)
\(480\) 0 0
\(481\) 1.74635 0.0796268
\(482\) 0 0
\(483\) 4.48269i 0.203969i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 25.1999i − 1.14192i −0.820979 0.570959i \(-0.806572\pi\)
0.820979 0.570959i \(-0.193428\pi\)
\(488\) 0 0
\(489\) 31.7572 1.43611
\(490\) 0 0
\(491\) −32.9247 −1.48587 −0.742936 0.669363i \(-0.766567\pi\)
−0.742936 + 0.669363i \(0.766567\pi\)
\(492\) 0 0
\(493\) − 25.8324i − 1.16343i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34.8481i 1.56315i
\(498\) 0 0
\(499\) −12.4047 −0.555309 −0.277654 0.960681i \(-0.589557\pi\)
−0.277654 + 0.960681i \(0.589557\pi\)
\(500\) 0 0
\(501\) 5.64521 0.252209
\(502\) 0 0
\(503\) − 3.74517i − 0.166989i −0.996508 0.0834945i \(-0.973392\pi\)
0.996508 0.0834945i \(-0.0266081\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1.83526i − 0.0815069i
\(508\) 0 0
\(509\) 5.84783 0.259201 0.129600 0.991566i \(-0.458631\pi\)
0.129600 + 0.991566i \(0.458631\pi\)
\(510\) 0 0
\(511\) −12.1646 −0.538128
\(512\) 0 0
\(513\) 19.7489i 0.871933i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.10961i − 0.136760i
\(518\) 0 0
\(519\) 8.04124 0.352971
\(520\) 0 0
\(521\) 18.3808 0.805278 0.402639 0.915359i \(-0.368093\pi\)
0.402639 + 0.915359i \(0.368093\pi\)
\(522\) 0 0
\(523\) 23.4541i 1.02558i 0.858515 + 0.512789i \(0.171388\pi\)
−0.858515 + 0.512789i \(0.828612\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 15.0029i − 0.653535i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 5.34137 0.231796
\(532\) 0 0
\(533\) 24.1710i 1.04696i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.36984i − 0.404338i
\(538\) 0 0
\(539\) −13.1649 −0.567052
\(540\) 0 0
\(541\) 3.03474 0.130474 0.0652368 0.997870i \(-0.479220\pi\)
0.0652368 + 0.997870i \(0.479220\pi\)
\(542\) 0 0
\(543\) 3.12937i 0.134294i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 16.4383i − 0.702851i −0.936216 0.351426i \(-0.885697\pi\)
0.936216 0.351426i \(-0.114303\pi\)
\(548\) 0 0
\(549\) 15.2729 0.651830
\(550\) 0 0
\(551\) −12.1093 −0.515875
\(552\) 0 0
\(553\) − 26.1811i − 1.11334i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.72395i 0.284903i 0.989802 + 0.142451i \(0.0454985\pi\)
−0.989802 + 0.142451i \(0.954502\pi\)
\(558\) 0 0
\(559\) 9.43863 0.399212
\(560\) 0 0
\(561\) −35.6452 −1.50494
\(562\) 0 0
\(563\) − 42.0044i − 1.77027i −0.465331 0.885137i \(-0.654065\pi\)
0.465331 0.885137i \(-0.345935\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 14.1602i 0.594674i
\(568\) 0 0
\(569\) 39.1152 1.63980 0.819898 0.572510i \(-0.194030\pi\)
0.819898 + 0.572510i \(0.194030\pi\)
\(570\) 0 0
\(571\) −13.2607 −0.554944 −0.277472 0.960734i \(-0.589497\pi\)
−0.277472 + 0.960734i \(0.589497\pi\)
\(572\) 0 0
\(573\) 13.0992i 0.547228i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 9.31588i − 0.387825i −0.981019 0.193913i \(-0.937882\pi\)
0.981019 0.193913i \(-0.0621178\pi\)
\(578\) 0 0
\(579\) 19.0474 0.791584
\(580\) 0 0
\(581\) −30.9174 −1.28267
\(582\) 0 0
\(583\) 49.7925i 2.06220i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.80336i − 0.322079i −0.986948 0.161040i \(-0.948515\pi\)
0.986948 0.161040i \(-0.0514847\pi\)
\(588\) 0 0
\(589\) −7.03282 −0.289782
\(590\) 0 0
\(591\) −20.6324 −0.848704
\(592\) 0 0
\(593\) 12.8383i 0.527204i 0.964632 + 0.263602i \(0.0849106\pi\)
−0.964632 + 0.263602i \(0.915089\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19.8287i 0.811534i
\(598\) 0 0
\(599\) 10.4957 0.428843 0.214422 0.976741i \(-0.431213\pi\)
0.214422 + 0.976741i \(0.431213\pi\)
\(600\) 0 0
\(601\) −28.1978 −1.15021 −0.575106 0.818079i \(-0.695039\pi\)
−0.575106 + 0.818079i \(0.695039\pi\)
\(602\) 0 0
\(603\) − 2.60679i − 0.106157i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 46.5662i 1.89007i 0.326976 + 0.945033i \(0.393970\pi\)
−0.326976 + 0.945033i \(0.606030\pi\)
\(608\) 0 0
\(609\) −15.5221 −0.628989
\(610\) 0 0
\(611\) −3.03590 −0.122819
\(612\) 0 0
\(613\) − 30.3120i − 1.22429i −0.790746 0.612145i \(-0.790307\pi\)
0.790746 0.612145i \(-0.209693\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.1879i 0.973768i 0.873467 + 0.486884i \(0.161866\pi\)
−0.873467 + 0.486884i \(0.838134\pi\)
\(618\) 0 0
\(619\) 4.29534 0.172644 0.0863221 0.996267i \(-0.472489\pi\)
0.0863221 + 0.996267i \(0.472489\pi\)
\(620\) 0 0
\(621\) −5.64722 −0.226615
\(622\) 0 0
\(623\) 2.62931i 0.105341i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 16.7092i 0.667301i
\(628\) 0 0
\(629\) 3.81588 0.152149
\(630\) 0 0
\(631\) 12.3857 0.493066 0.246533 0.969134i \(-0.420709\pi\)
0.246533 + 0.969134i \(0.420709\pi\)
\(632\) 0 0
\(633\) 32.0866i 1.27533i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 12.8528i 0.509248i
\(638\) 0 0
\(639\) −12.0369 −0.476171
\(640\) 0 0
\(641\) −5.21653 −0.206041 −0.103020 0.994679i \(-0.532851\pi\)
−0.103020 + 0.994679i \(0.532851\pi\)
\(642\) 0 0
\(643\) − 12.5639i − 0.495471i −0.968828 0.247735i \(-0.920314\pi\)
0.968828 0.247735i \(-0.0796863\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 22.7423i − 0.894094i −0.894510 0.447047i \(-0.852476\pi\)
0.894510 0.447047i \(-0.147524\pi\)
\(648\) 0 0
\(649\) 16.4827 0.647002
\(650\) 0 0
\(651\) −9.01490 −0.353322
\(652\) 0 0
\(653\) − 9.36997i − 0.366675i −0.983050 0.183338i \(-0.941310\pi\)
0.983050 0.183338i \(-0.0586902\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 4.20175i − 0.163926i
\(658\) 0 0
\(659\) 15.4086 0.600235 0.300117 0.953902i \(-0.402974\pi\)
0.300117 + 0.953902i \(0.402974\pi\)
\(660\) 0 0
\(661\) 43.1386 1.67790 0.838948 0.544211i \(-0.183171\pi\)
0.838948 + 0.544211i \(0.183171\pi\)
\(662\) 0 0
\(663\) 34.8002i 1.35153i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.46268i − 0.134076i
\(668\) 0 0
\(669\) −11.2134 −0.433534
\(670\) 0 0
\(671\) 47.1299 1.81943
\(672\) 0 0
\(673\) 12.6356i 0.487066i 0.969893 + 0.243533i \(0.0783065\pi\)
−0.969893 + 0.243533i \(0.921694\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.9139i 0.419454i 0.977760 + 0.209727i \(0.0672575\pi\)
−0.977760 + 0.209727i \(0.932742\pi\)
\(678\) 0 0
\(679\) 38.6238 1.48225
\(680\) 0 0
\(681\) −23.2189 −0.889750
\(682\) 0 0
\(683\) − 17.5334i − 0.670896i −0.942059 0.335448i \(-0.891112\pi\)
0.942059 0.335448i \(-0.108888\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 26.8910i 1.02596i
\(688\) 0 0
\(689\) 48.6122 1.85198
\(690\) 0 0
\(691\) −52.2318 −1.98699 −0.993496 0.113871i \(-0.963675\pi\)
−0.993496 + 0.113871i \(0.963675\pi\)
\(692\) 0 0
\(693\) − 13.0028i − 0.493935i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 52.8150i 2.00051i
\(698\) 0 0
\(699\) 33.9930 1.28573
\(700\) 0 0
\(701\) 26.0270 0.983026 0.491513 0.870870i \(-0.336444\pi\)
0.491513 + 0.870870i \(0.336444\pi\)
\(702\) 0 0
\(703\) − 1.78875i − 0.0674641i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 36.6019i − 1.37655i
\(708\) 0 0
\(709\) 34.7971 1.30683 0.653415 0.757000i \(-0.273336\pi\)
0.653415 + 0.757000i \(0.273336\pi\)
\(710\) 0 0
\(711\) 9.04321 0.339147
\(712\) 0 0
\(713\) − 2.01105i − 0.0753143i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 0.0981652i − 0.00366605i
\(718\) 0 0
\(719\) −4.40110 −0.164133 −0.0820667 0.996627i \(-0.526152\pi\)
−0.0820667 + 0.996627i \(0.526152\pi\)
\(720\) 0 0
\(721\) −4.21878 −0.157115
\(722\) 0 0
\(723\) 11.6224i 0.432242i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 0.744921i − 0.0276276i −0.999905 0.0138138i \(-0.995603\pi\)
0.999905 0.0138138i \(-0.00439721\pi\)
\(728\) 0 0
\(729\) −28.0382 −1.03845
\(730\) 0 0
\(731\) 20.6240 0.762805
\(732\) 0 0
\(733\) − 7.35631i − 0.271711i −0.990729 0.135856i \(-0.956622\pi\)
0.990729 0.135856i \(-0.0433784\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8.04419i − 0.296311i
\(738\) 0 0
\(739\) −13.2496 −0.487394 −0.243697 0.969851i \(-0.578360\pi\)
−0.243697 + 0.969851i \(0.578360\pi\)
\(740\) 0 0
\(741\) 16.3131 0.599278
\(742\) 0 0
\(743\) − 28.8314i − 1.05772i −0.848708 0.528861i \(-0.822619\pi\)
0.848708 0.528861i \(-0.177381\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 10.6792i − 0.390730i
\(748\) 0 0
\(749\) −45.5966 −1.66607
\(750\) 0 0
\(751\) −24.3962 −0.890228 −0.445114 0.895474i \(-0.646837\pi\)
−0.445114 + 0.895474i \(0.646837\pi\)
\(752\) 0 0
\(753\) 27.1697i 0.990120i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.49825i 0.345220i 0.984990 + 0.172610i \(0.0552200\pi\)
−0.984990 + 0.172610i \(0.944780\pi\)
\(758\) 0 0
\(759\) −4.77803 −0.173431
\(760\) 0 0
\(761\) −8.43957 −0.305934 −0.152967 0.988231i \(-0.548883\pi\)
−0.152967 + 0.988231i \(0.548883\pi\)
\(762\) 0 0
\(763\) − 12.3925i − 0.448639i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 16.0920i − 0.581048i
\(768\) 0 0
\(769\) −8.85911 −0.319468 −0.159734 0.987160i \(-0.551064\pi\)
−0.159734 + 0.987160i \(0.551064\pi\)
\(770\) 0 0
\(771\) 15.5841 0.561247
\(772\) 0 0
\(773\) − 32.7351i − 1.17740i −0.808352 0.588699i \(-0.799640\pi\)
0.808352 0.588699i \(-0.200360\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.29288i − 0.0822567i
\(778\) 0 0
\(779\) 24.7578 0.887041
\(780\) 0 0
\(781\) −37.1440 −1.32912
\(782\) 0 0
\(783\) − 19.5545i − 0.698822i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.736862i 0.0262663i 0.999914 + 0.0131331i \(0.00418053\pi\)
−0.999914 + 0.0131331i \(0.995819\pi\)
\(788\) 0 0
\(789\) −18.2949 −0.651316
\(790\) 0 0
\(791\) 63.3584 2.25276
\(792\) 0 0
\(793\) − 46.0127i − 1.63396i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 41.7353i − 1.47834i −0.673519 0.739170i \(-0.735218\pi\)
0.673519 0.739170i \(-0.264782\pi\)
\(798\) 0 0
\(799\) −6.63362 −0.234681
\(800\) 0 0
\(801\) −0.908187 −0.0320892
\(802\) 0 0
\(803\) − 12.9660i − 0.457560i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.8688i 0.664212i
\(808\) 0 0
\(809\) −21.3182 −0.749506 −0.374753 0.927125i \(-0.622273\pi\)
−0.374753 + 0.927125i \(0.622273\pi\)
\(810\) 0 0
\(811\) 24.3126 0.853732 0.426866 0.904315i \(-0.359618\pi\)
0.426866 + 0.904315i \(0.359618\pi\)
\(812\) 0 0
\(813\) − 36.5583i − 1.28216i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 9.66780i − 0.338233i
\(818\) 0 0
\(819\) −12.6946 −0.443584
\(820\) 0 0
\(821\) 49.3157 1.72113 0.860566 0.509340i \(-0.170110\pi\)
0.860566 + 0.509340i \(0.170110\pi\)
\(822\) 0 0
\(823\) − 41.5427i − 1.44809i −0.689755 0.724043i \(-0.742282\pi\)
0.689755 0.724043i \(-0.257718\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 0.306510i − 0.0106584i −0.999986 0.00532919i \(-0.998304\pi\)
0.999986 0.00532919i \(-0.00169634\pi\)
\(828\) 0 0
\(829\) −17.5423 −0.609268 −0.304634 0.952469i \(-0.598534\pi\)
−0.304634 + 0.952469i \(0.598534\pi\)
\(830\) 0 0
\(831\) 2.46842 0.0856285
\(832\) 0 0
\(833\) 28.0842i 0.973060i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 11.3568i − 0.392549i
\(838\) 0 0
\(839\) 44.1480 1.52416 0.762078 0.647485i \(-0.224179\pi\)
0.762078 + 0.647485i \(0.224179\pi\)
\(840\) 0 0
\(841\) −17.0098 −0.586546
\(842\) 0 0
\(843\) − 37.0739i − 1.27689i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.03445i − 0.138625i
\(848\) 0 0
\(849\) 28.6749 0.984119
\(850\) 0 0
\(851\) 0.511497 0.0175339
\(852\) 0 0
\(853\) 41.2098i 1.41100i 0.708711 + 0.705499i \(0.249277\pi\)
−0.708711 + 0.705499i \(0.750723\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.3245i 0.660114i 0.943961 + 0.330057i \(0.107068\pi\)
−0.943961 + 0.330057i \(0.892932\pi\)
\(858\) 0 0
\(859\) 16.2174 0.553330 0.276665 0.960966i \(-0.410771\pi\)
0.276665 + 0.960966i \(0.410771\pi\)
\(860\) 0 0
\(861\) 31.7354 1.08154
\(862\) 0 0
\(863\) 56.9036i 1.93702i 0.248974 + 0.968510i \(0.419907\pi\)
−0.248974 + 0.968510i \(0.580093\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 52.8137i 1.79365i
\(868\) 0 0
\(869\) 27.9061 0.946648
\(870\) 0 0
\(871\) −7.85351 −0.266106
\(872\) 0 0
\(873\) 13.3410i 0.451526i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 33.4569i 1.12976i 0.825173 + 0.564880i \(0.191077\pi\)
−0.825173 + 0.564880i \(0.808923\pi\)
\(878\) 0 0
\(879\) 0.00917121 0.000309337 0
\(880\) 0 0
\(881\) −5.99700 −0.202044 −0.101022 0.994884i \(-0.532211\pi\)
−0.101022 + 0.994884i \(0.532211\pi\)
\(882\) 0 0
\(883\) − 51.3210i − 1.72709i −0.504273 0.863544i \(-0.668239\pi\)
0.504273 0.863544i \(-0.331761\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.1121i 0.910336i 0.890405 + 0.455168i \(0.150421\pi\)
−0.890405 + 0.455168i \(0.849579\pi\)
\(888\) 0 0
\(889\) −45.1495 −1.51426
\(890\) 0 0
\(891\) −15.0932 −0.505640
\(892\) 0 0
\(893\) 3.10961i 0.104059i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.66477i 0.155752i
\(898\) 0 0
\(899\) 6.96362 0.232250
\(900\) 0 0
\(901\) 106.221 3.53872
\(902\) 0 0
\(903\) − 12.3925i − 0.412397i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.6194i 1.34875i 0.738391 + 0.674373i \(0.235586\pi\)
−0.738391 + 0.674373i \(0.764414\pi\)
\(908\) 0 0
\(909\) 12.6426 0.419329
\(910\) 0 0
\(911\) 32.2475 1.06841 0.534203 0.845356i \(-0.320612\pi\)
0.534203 + 0.845356i \(0.320612\pi\)
\(912\) 0 0
\(913\) − 32.9544i − 1.09063i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.3335i 0.803565i
\(918\) 0 0
\(919\) 30.4042 1.00294 0.501470 0.865175i \(-0.332793\pi\)
0.501470 + 0.865175i \(0.332793\pi\)
\(920\) 0 0
\(921\) 27.2939 0.899366
\(922\) 0 0
\(923\) 36.2636i 1.19363i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 1.45720i − 0.0478609i
\(928\) 0 0
\(929\) 25.2834 0.829520 0.414760 0.909931i \(-0.363866\pi\)
0.414760 + 0.909931i \(0.363866\pi\)
\(930\) 0 0
\(931\) 13.1649 0.431462
\(932\) 0 0
\(933\) − 32.5997i − 1.06727i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.1769i 0.724487i 0.932084 + 0.362243i \(0.117989\pi\)
−0.932084 + 0.362243i \(0.882011\pi\)
\(938\) 0 0
\(939\) −23.8289 −0.777626
\(940\) 0 0
\(941\) 4.23228 0.137968 0.0689842 0.997618i \(-0.478024\pi\)
0.0689842 + 0.997618i \(0.478024\pi\)
\(942\) 0 0
\(943\) 7.07954i 0.230542i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 45.2374i − 1.47002i −0.678058 0.735009i \(-0.737178\pi\)
0.678058 0.735009i \(-0.262822\pi\)
\(948\) 0 0
\(949\) −12.6587 −0.410917
\(950\) 0 0
\(951\) −1.85242 −0.0600690
\(952\) 0 0
\(953\) − 43.7170i − 1.41613i −0.706146 0.708066i \(-0.749568\pi\)
0.706146 0.708066i \(-0.250432\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 16.5448i − 0.534817i
\(958\) 0 0
\(959\) 67.4240 2.17723
\(960\) 0 0
\(961\) −26.9557 −0.869538
\(962\) 0 0
\(963\) − 15.7495i − 0.507521i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 5.58482i 0.179596i 0.995960 + 0.0897979i \(0.0286221\pi\)
−0.995960 + 0.0897979i \(0.971378\pi\)
\(968\) 0 0
\(969\) 35.6452 1.14509
\(970\) 0 0
\(971\) 39.1206 1.25544 0.627720 0.778439i \(-0.283988\pi\)
0.627720 + 0.778439i \(0.283988\pi\)
\(972\) 0 0
\(973\) − 18.0337i − 0.578135i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.3050i 1.67339i 0.547672 + 0.836693i \(0.315514\pi\)
−0.547672 + 0.836693i \(0.684486\pi\)
\(978\) 0 0
\(979\) −2.80253 −0.0895694
\(980\) 0 0
\(981\) 4.28049 0.136665
\(982\) 0 0
\(983\) 22.4395i 0.715710i 0.933777 + 0.357855i \(0.116492\pi\)
−0.933777 + 0.357855i \(0.883508\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.98600i 0.126876i
\(988\) 0 0
\(989\) 2.76452 0.0879067
\(990\) 0 0
\(991\) −0.851383 −0.0270451 −0.0135225 0.999909i \(-0.504304\pi\)
−0.0135225 + 0.999909i \(0.504304\pi\)
\(992\) 0 0
\(993\) − 23.7802i − 0.754643i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 12.1784i − 0.385693i −0.981229 0.192846i \(-0.938228\pi\)
0.981229 0.192846i \(-0.0617719\pi\)
\(998\) 0 0
\(999\) 2.88853 0.0913892
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.4049.4 10
5.2 odd 4 4600.2.a.bf.1.2 yes 5
5.3 odd 4 4600.2.a.bd.1.4 5
5.4 even 2 inner 4600.2.e.w.4049.7 10
20.3 even 4 9200.2.a.cv.1.2 5
20.7 even 4 9200.2.a.ct.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.4 5 5.3 odd 4
4600.2.a.bf.1.2 yes 5 5.2 odd 4
4600.2.e.w.4049.4 10 1.1 even 1 trivial
4600.2.e.w.4049.7 10 5.4 even 2 inner
9200.2.a.ct.1.4 5 20.7 even 4
9200.2.a.cv.1.2 5 20.3 even 4