Properties

Label 4600.2.e.d.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.d.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -5.00000i q^{13} +4.00000i q^{17} -1.00000 q^{19} -6.00000 q^{21} -1.00000i q^{23} +4.00000i q^{27} -9.00000 q^{29} -2.00000 q^{31} +10.0000i q^{33} +2.00000i q^{37} +10.0000 q^{39} +3.00000 q^{41} -7.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000 q^{51} +12.0000i q^{53} -2.00000i q^{57} +6.00000 q^{59} -10.0000 q^{61} -3.00000i q^{63} +8.00000i q^{67} +2.00000 q^{69} +2.00000 q^{71} +1.00000i q^{73} +15.0000i q^{77} +11.0000 q^{79} -11.0000 q^{81} +9.00000i q^{83} -18.0000i q^{87} +14.0000 q^{89} +15.0000 q^{91} -4.00000i q^{93} +16.0000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} + 10 q^{11} - 2 q^{19} - 12 q^{21} - 18 q^{29} - 4 q^{31} + 20 q^{39} + 6 q^{41} - 4 q^{49} - 16 q^{51} + 12 q^{59} - 20 q^{61} + 4 q^{69} + 4 q^{71} + 22 q^{79} - 22 q^{81} + 28 q^{89} + 30 q^{91} - 10 q^{99} + O(q^{100}) \)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 10.0000i 1.74078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 10.0000 1.60128
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) − 7.00000i − 1.06749i −0.845645 0.533745i \(-0.820784\pi\)
0.845645 0.533745i \(-0.179216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 3.00000i − 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.0000i 1.70941i
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 18.0000i − 1.92980i
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 15.0000 1.57243
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) − 9.00000i − 0.886796i −0.896325 0.443398i \(-0.853773\pi\)
0.896325 0.443398i \(-0.146227\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 16.0000i − 1.54678i −0.633932 0.773389i \(-0.718560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 8.00000i 0.752577i 0.926503 + 0.376288i \(0.122800\pi\)
−0.926503 + 0.376288i \(0.877200\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.00000i 0.462250i
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 0 0
\(129\) 14.0000 1.23263
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) − 3.00000i − 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) − 25.0000i − 2.09061i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 4.00000i − 0.329914i
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) − 4.00000i − 0.323381i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 8.00000i − 0.638470i −0.947676 0.319235i \(-0.896574\pi\)
0.947676 0.319235i \(-0.103426\pi\)
\(158\) 0 0
\(159\) −24.0000 −1.90332
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.0000i 1.08335i 0.840587 + 0.541676i \(0.182210\pi\)
−0.840587 + 0.541676i \(0.817790\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 5.00000i 0.380143i 0.981770 + 0.190071i \(0.0608720\pi\)
−0.981770 + 0.190071i \(0.939128\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) − 20.0000i − 1.47844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.0000i 1.46254i
\(188\) 0 0
\(189\) −12.0000 −0.872872
\(190\) 0 0
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.0000i 1.63868i 0.573306 + 0.819341i \(0.305660\pi\)
−0.573306 + 0.819341i \(0.694340\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) − 27.0000i − 1.89503i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 4.00000i 0.274075i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 6.00000i − 0.407307i
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) 30.0000 1.98246 0.991228 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) 0 0
\(231\) −30.0000 −1.97386
\(232\) 0 0
\(233\) − 25.0000i − 1.63780i −0.573933 0.818902i \(-0.694583\pi\)
0.573933 0.818902i \(-0.305417\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 22.0000i 1.42905i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000i 0.318142i
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) − 5.00000i − 0.314347i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 28.0000i 1.71357i
\(268\) 0 0
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 30.0000i 1.81568i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 13.0000i − 0.781094i −0.920583 0.390547i \(-0.872286\pi\)
0.920583 0.390547i \(-0.127714\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.00000i 0.531253i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −32.0000 −1.87587
\(292\) 0 0
\(293\) 28.0000i 1.63578i 0.575376 + 0.817889i \(0.304856\pi\)
−0.575376 + 0.817889i \(0.695144\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 20.0000i 1.16052i
\(298\) 0 0
\(299\) −5.00000 −0.289157
\(300\) 0 0
\(301\) 21.0000 1.21042
\(302\) 0 0
\(303\) − 28.0000i − 1.60856i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) 18.0000 1.02398
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 17.0000i − 0.954815i −0.878682 0.477408i \(-0.841577\pi\)
0.878682 0.477408i \(-0.158423\pi\)
\(318\) 0 0
\(319\) −45.0000 −2.51952
\(320\) 0 0
\(321\) 32.0000 1.78607
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.0000i 1.10600i
\(328\) 0 0
\(329\) −36.0000 −1.98474
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 32.0000i − 1.74315i −0.490261 0.871576i \(-0.663099\pi\)
0.490261 0.871576i \(-0.336901\pi\)
\(338\) 0 0
\(339\) −16.0000 −0.869001
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 32.0000i − 1.71785i −0.512101 0.858925i \(-0.671133\pi\)
0.512101 0.858925i \(-0.328867\pi\)
\(348\) 0 0
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) 0 0
\(353\) 21.0000i 1.11772i 0.829263 + 0.558859i \(0.188761\pi\)
−0.829263 + 0.558859i \(0.811239\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 24.0000i − 1.27021i
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 28.0000i 1.46962i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.0000i 0.782994i 0.920179 + 0.391497i \(0.128043\pi\)
−0.920179 + 0.391497i \(0.871957\pi\)
\(368\) 0 0
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 45.0000i 2.31762i
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) 0 0
\(383\) − 23.0000i − 1.17525i −0.809135 0.587623i \(-0.800064\pi\)
0.809135 0.587623i \(-0.199936\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.00000i 0.355830i
\(388\) 0 0
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 10.0000i 0.498135i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000i 0.495682i
\(408\) 0 0
\(409\) 35.0000 1.73064 0.865319 0.501221i \(-0.167116\pi\)
0.865319 + 0.501221i \(0.167116\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 18.0000i 0.885722i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 24.0000i − 1.17529i
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) − 12.0000i − 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 30.0000i − 1.45180i
\(428\) 0 0
\(429\) 50.0000 2.41402
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 6.00000i 0.288342i 0.989553 + 0.144171i \(0.0460515\pi\)
−0.989553 + 0.144171i \(0.953949\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.00000i 0.0478365i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 14.0000i 0.665160i 0.943075 + 0.332580i \(0.107919\pi\)
−0.943075 + 0.332580i \(0.892081\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 28.0000i − 1.32435i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 15.0000 0.706322
\(452\) 0 0
\(453\) − 36.0000i − 1.69143i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 0 0
\(459\) −16.0000 −0.746816
\(460\) 0 0
\(461\) 41.0000 1.90956 0.954780 0.297313i \(-0.0960904\pi\)
0.954780 + 0.297313i \(0.0960904\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0000i 0.971764i 0.874024 + 0.485882i \(0.161502\pi\)
−0.874024 + 0.485882i \(0.838498\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 0 0
\(473\) − 35.0000i − 1.60930i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0 0
\(479\) 17.0000 0.776750 0.388375 0.921501i \(-0.373037\pi\)
0.388375 + 0.921501i \(0.373037\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 0 0
\(489\) −32.0000 −1.44709
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) − 36.0000i − 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000i 0.269137i
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) −28.0000 −1.25095
\(502\) 0 0
\(503\) − 5.00000i − 0.222939i −0.993768 0.111469i \(-0.964444\pi\)
0.993768 0.111469i \(-0.0355557\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 24.0000i − 1.06588i
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 60.0000i 2.63880i
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 7.00000i 0.306089i 0.988219 + 0.153044i \(0.0489077\pi\)
−0.988219 + 0.153044i \(0.951092\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 8.00000i − 0.348485i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) − 15.0000i − 0.649722i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 24.0000i − 1.03568i
\(538\) 0 0
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) −23.0000 −0.988847 −0.494424 0.869221i \(-0.664621\pi\)
−0.494424 + 0.869221i \(0.664621\pi\)
\(542\) 0 0
\(543\) − 40.0000i − 1.71656i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 44.0000i − 1.88130i −0.339372 0.940652i \(-0.610215\pi\)
0.339372 0.940652i \(-0.389785\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 33.0000i 1.40330i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 42.0000i − 1.77960i −0.456354 0.889799i \(-0.650845\pi\)
0.456354 0.889799i \(-0.349155\pi\)
\(558\) 0 0
\(559\) −35.0000 −1.48034
\(560\) 0 0
\(561\) −40.0000 −1.68880
\(562\) 0 0
\(563\) − 39.0000i − 1.64365i −0.569737 0.821827i \(-0.692955\pi\)
0.569737 0.821827i \(-0.307045\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 33.0000i − 1.38587i
\(568\) 0 0
\(569\) 4.00000 0.167689 0.0838444 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) − 2.00000i − 0.0835512i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.0000i 0.957503i 0.877951 + 0.478751i \(0.158910\pi\)
−0.877951 + 0.478751i \(0.841090\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −27.0000 −1.12015
\(582\) 0 0
\(583\) 60.0000i 2.48495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 30.0000i − 1.23823i −0.785299 0.619116i \(-0.787491\pi\)
0.785299 0.619116i \(-0.212509\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) −46.0000 −1.89219
\(592\) 0 0
\(593\) − 17.0000i − 0.698106i −0.937103 0.349053i \(-0.886503\pi\)
0.937103 0.349053i \(-0.113497\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 10.0000i − 0.409273i
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 0 0
\(609\) 54.0000 2.18819
\(610\) 0 0
\(611\) 60.0000 2.42734
\(612\) 0 0
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.0000i 1.28827i 0.764911 + 0.644136i \(0.222783\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 42.0000i 1.68269i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 10.0000i − 0.399362i
\(628\) 0 0
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 10.0000i 0.396214i
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 29.0000i 1.14365i 0.820376 + 0.571824i \(0.193764\pi\)
−0.820376 + 0.571824i \(0.806236\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000i 1.25805i 0.777385 + 0.629025i \(0.216546\pi\)
−0.777385 + 0.629025i \(0.783454\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) − 29.0000i − 1.13486i −0.823422 0.567429i \(-0.807938\pi\)
0.823422 0.567429i \(-0.192062\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.00000i − 0.0390137i
\(658\) 0 0
\(659\) 45.0000 1.75295 0.876476 0.481446i \(-0.159888\pi\)
0.876476 + 0.481446i \(0.159888\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 0 0
\(663\) 40.0000i 1.55347i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −50.0000 −1.93023
\(672\) 0 0
\(673\) 15.0000i 0.578208i 0.957298 + 0.289104i \(0.0933573\pi\)
−0.957298 + 0.289104i \(0.906643\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) −48.0000 −1.84207
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 0 0
\(683\) − 34.0000i − 1.30097i −0.759517 0.650487i \(-0.774565\pi\)
0.759517 0.650487i \(-0.225435\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 60.0000i 2.28914i
\(688\) 0 0
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 0 0
\(693\) − 15.0000i − 0.569803i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 0 0
\(699\) 50.0000 1.89117
\(700\) 0 0
\(701\) 8.00000 0.302156 0.151078 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) 0 0
\(703\) − 2.00000i − 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 42.0000i − 1.57957i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 0 0
\(713\) 2.00000i 0.0749006i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) 27.0000 1.00553
\(722\) 0 0
\(723\) − 8.00000i − 0.297523i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 28.0000 1.03562
\(732\) 0 0
\(733\) − 26.0000i − 0.960332i −0.877178 0.480166i \(-0.840576\pi\)
0.877178 0.480166i \(-0.159424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.0000i 1.47342i
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 0 0
\(741\) −10.0000 −0.367359
\(742\) 0 0
\(743\) 21.0000i 0.770415i 0.922830 + 0.385208i \(0.125870\pi\)
−0.922830 + 0.385208i \(0.874130\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 9.00000i − 0.329293i
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 19.0000 0.693320 0.346660 0.937991i \(-0.387316\pi\)
0.346660 + 0.937991i \(0.387316\pi\)
\(752\) 0 0
\(753\) 40.0000i 1.45768i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 10.0000 0.362977
\(760\) 0 0
\(761\) 1.00000 0.0362500 0.0181250 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) 0 0
\(763\) 30.0000i 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 30.0000i − 1.08324i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 0 0
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 12.0000i − 0.430498i
\(778\) 0 0
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) − 36.0000i − 1.28654i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 25.0000i 0.891154i 0.895244 + 0.445577i \(0.147001\pi\)
−0.895244 + 0.445577i \(0.852999\pi\)
\(788\) 0 0
\(789\) 32.0000 1.13923
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 50.0000i 1.77555i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) 5.00000i 0.176446i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 10.0000i − 0.352017i
\(808\) 0 0
\(809\) 23.0000 0.808637 0.404318 0.914618i \(-0.367509\pi\)
0.404318 + 0.914618i \(0.367509\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.00000i 0.244899i
\(818\) 0 0
\(819\) −15.0000 −0.524142
\(820\) 0 0
\(821\) 51.0000 1.77991 0.889956 0.456046i \(-0.150735\pi\)
0.889956 + 0.456046i \(0.150735\pi\)
\(822\) 0 0
\(823\) 12.0000i 0.418294i 0.977884 + 0.209147i \(0.0670687\pi\)
−0.977884 + 0.209147i \(0.932931\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 7.00000i − 0.243414i −0.992566 0.121707i \(-0.961163\pi\)
0.992566 0.121707i \(-0.0388368\pi\)
\(828\) 0 0
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) − 8.00000i − 0.277184i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 8.00000i − 0.276520i
\(838\) 0 0
\(839\) −15.0000 −0.517858 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) − 16.0000i − 0.551069i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 42.0000i 1.44314i
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.0000i 1.29806i 0.760765 + 0.649028i \(0.224824\pi\)
−0.760765 + 0.649028i \(0.775176\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) 36.0000i 1.22545i 0.790295 + 0.612727i \(0.209928\pi\)
−0.790295 + 0.612727i \(0.790072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.00000i 0.0679236i
\(868\) 0 0
\(869\) 55.0000 1.86575
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) − 16.0000i − 0.541518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 0 0
\(879\) −56.0000 −1.88883
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 16.0000i 0.538443i 0.963078 + 0.269221i \(0.0867663\pi\)
−0.963078 + 0.269221i \(0.913234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 0 0
\(889\) −30.0000 −1.00617
\(890\) 0 0
\(891\) −55.0000 −1.84257
\(892\) 0 0
\(893\) − 12.0000i − 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 10.0000i − 0.333890i
\(898\) 0 0
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) −48.0000 −1.59911
\(902\) 0 0
\(903\) 42.0000i 1.39767i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 15.0000i − 0.498067i −0.968495 0.249033i \(-0.919887\pi\)
0.968495 0.249033i \(-0.0801129\pi\)
\(908\) 0 0
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 5.00000 0.165657 0.0828287 0.996564i \(-0.473605\pi\)
0.0828287 + 0.996564i \(0.473605\pi\)
\(912\) 0 0
\(913\) 45.0000i 1.48928i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.0000i 0.594412i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) − 10.0000i − 0.329154i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.00000i 0.295599i
\(928\) 0 0
\(929\) 13.0000 0.426516 0.213258 0.976996i \(-0.431592\pi\)
0.213258 + 0.976996i \(0.431592\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 36.0000i 1.17859i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44.0000i 1.43742i 0.695311 + 0.718709i \(0.255266\pi\)
−0.695311 + 0.718709i \(0.744734\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) − 3.00000i − 0.0976934i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 18.0000i − 0.584921i −0.956278 0.292461i \(-0.905526\pi\)
0.956278 0.292461i \(-0.0944741\pi\)
\(948\) 0 0
\(949\) 5.00000 0.162307
\(950\) 0 0
\(951\) 34.0000 1.10253
\(952\) 0 0
\(953\) − 2.00000i − 0.0647864i −0.999475 0.0323932i \(-0.989687\pi\)
0.999475 0.0323932i \(-0.0103129\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 90.0000i − 2.90929i
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 16.0000i 0.515593i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 12.0000i − 0.385894i −0.981209 0.192947i \(-0.938195\pi\)
0.981209 0.192947i \(-0.0618045\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −41.0000 −1.31575 −0.657876 0.753126i \(-0.728545\pi\)
−0.657876 + 0.753126i \(0.728545\pi\)
\(972\) 0 0
\(973\) − 36.0000i − 1.15411i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.0000i 0.639857i 0.947442 + 0.319928i \(0.103659\pi\)
−0.947442 + 0.319928i \(0.896341\pi\)
\(978\) 0 0
\(979\) 70.0000 2.23721
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) − 1.00000i − 0.0318950i −0.999873 0.0159475i \(-0.994924\pi\)
0.999873 0.0159475i \(-0.00507647\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 72.0000i − 2.29179i
\(988\) 0 0
\(989\) −7.00000 −0.222587
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 52.0000i 1.65017i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 5.00000i − 0.158352i −0.996861 0.0791758i \(-0.974771\pi\)
0.996861 0.0791758i \(-0.0252288\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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.d.4049.2 2
5.2 odd 4 4600.2.a.m.1.1 yes 1
5.3 odd 4 4600.2.a.d.1.1 1
5.4 even 2 inner 4600.2.e.d.4049.1 2
20.3 even 4 9200.2.a.bc.1.1 1
20.7 even 4 9200.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.d.1.1 1 5.3 odd 4
4600.2.a.m.1.1 yes 1 5.2 odd 4
4600.2.e.d.4049.1 2 5.4 even 2 inner
4600.2.e.d.4049.2 2 1.1 even 1 trivial
9200.2.a.j.1.1 1 20.7 even 4
9200.2.a.bc.1.1 1 20.3 even 4