Properties

Label 2800.2.g.s.449.1
Level $2800$
Weight $2$
Character 2800.449
Analytic conductor $22.358$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.s.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +O(q^{10})\) \(q-3.23607i q^{3} -1.00000i q^{7} -7.47214 q^{9} +0.236068 q^{11} -1.23607i q^{13} +2.47214i q^{17} -4.47214 q^{19} -3.23607 q^{21} +6.23607i q^{23} +14.4721i q^{27} -5.00000 q^{29} -3.70820 q^{31} -0.763932i q^{33} +3.00000i q^{37} -4.00000 q^{39} +4.76393 q^{41} +1.76393i q^{43} +2.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} -8.47214i q^{53} +14.4721i q^{57} +11.7082 q^{59} -9.70820 q^{61} +7.47214i q^{63} +4.23607i q^{67} +20.1803 q^{69} -8.70820 q^{71} +8.76393i q^{73} -0.236068i q^{77} -11.1803 q^{79} +24.4164 q^{81} -7.70820i q^{83} +16.1803i q^{87} -17.2361 q^{89} -1.23607 q^{91} +12.0000i q^{93} +5.23607i q^{97} -1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 8q^{11} - 4q^{21} - 20q^{29} + 12q^{31} - 16q^{39} + 28q^{41} - 4q^{49} + 32q^{51} + 20q^{59} - 12q^{61} + 36q^{69} - 8q^{71} + 44q^{81} - 60q^{89} + 4q^{91} - 16q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.23607i − 1.86834i −0.356822 0.934172i \(-0.616140\pi\)
0.356822 0.934172i \(-0.383860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) 0.236068 0.0711772 0.0355886 0.999367i \(-0.488669\pi\)
0.0355886 + 0.999367i \(0.488669\pi\)
\(12\) 0 0
\(13\) − 1.23607i − 0.342824i −0.985199 0.171412i \(-0.945167\pi\)
0.985199 0.171412i \(-0.0548329\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.47214i 0.599581i 0.954005 + 0.299791i \(0.0969168\pi\)
−0.954005 + 0.299791i \(0.903083\pi\)
\(18\) 0 0
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) 6.23607i 1.30031i 0.759802 + 0.650155i \(0.225296\pi\)
−0.759802 + 0.650155i \(0.774704\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.4721i 2.78516i
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −3.70820 −0.666013 −0.333007 0.942925i \(-0.608063\pi\)
−0.333007 + 0.942925i \(0.608063\pi\)
\(32\) 0 0
\(33\) − 0.763932i − 0.132983i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 4.76393 0.744001 0.372001 0.928232i \(-0.378672\pi\)
0.372001 + 0.928232i \(0.378672\pi\)
\(42\) 0 0
\(43\) 1.76393i 0.268997i 0.990914 + 0.134499i \(0.0429424\pi\)
−0.990914 + 0.134499i \(0.957058\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) − 8.47214i − 1.16374i −0.813283 0.581869i \(-0.802322\pi\)
0.813283 0.581869i \(-0.197678\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.4721i 1.91688i
\(58\) 0 0
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) −9.70820 −1.24301 −0.621504 0.783411i \(-0.713478\pi\)
−0.621504 + 0.783411i \(0.713478\pi\)
\(62\) 0 0
\(63\) 7.47214i 0.941401i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.23607i 0.517518i 0.965942 + 0.258759i \(0.0833136\pi\)
−0.965942 + 0.258759i \(0.916686\pi\)
\(68\) 0 0
\(69\) 20.1803 2.42943
\(70\) 0 0
\(71\) −8.70820 −1.03347 −0.516737 0.856144i \(-0.672853\pi\)
−0.516737 + 0.856144i \(0.672853\pi\)
\(72\) 0 0
\(73\) 8.76393i 1.02574i 0.858466 + 0.512870i \(0.171418\pi\)
−0.858466 + 0.512870i \(0.828582\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.236068i − 0.0269024i
\(78\) 0 0
\(79\) −11.1803 −1.25789 −0.628943 0.777451i \(-0.716512\pi\)
−0.628943 + 0.777451i \(0.716512\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) − 7.70820i − 0.846085i −0.906110 0.423043i \(-0.860962\pi\)
0.906110 0.423043i \(-0.139038\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 16.1803i 1.73471i
\(88\) 0 0
\(89\) −17.2361 −1.82702 −0.913510 0.406817i \(-0.866639\pi\)
−0.913510 + 0.406817i \(0.866639\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 0 0
\(93\) 12.0000i 1.24434i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.23607i 0.531642i 0.964022 + 0.265821i \(0.0856430\pi\)
−0.964022 + 0.265821i \(0.914357\pi\)
\(98\) 0 0
\(99\) −1.76393 −0.177282
\(100\) 0 0
\(101\) 4.76393 0.474029 0.237014 0.971506i \(-0.423831\pi\)
0.237014 + 0.971506i \(0.423831\pi\)
\(102\) 0 0
\(103\) 8.47214i 0.834784i 0.908726 + 0.417392i \(0.137056\pi\)
−0.908726 + 0.417392i \(0.862944\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) −8.41641 −0.806146 −0.403073 0.915168i \(-0.632058\pi\)
−0.403073 + 0.915168i \(0.632058\pi\)
\(110\) 0 0
\(111\) 9.70820 0.921462
\(112\) 0 0
\(113\) 14.4164i 1.35618i 0.734978 + 0.678091i \(0.237192\pi\)
−0.734978 + 0.678091i \(0.762808\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.23607i 0.853875i
\(118\) 0 0
\(119\) 2.47214 0.226620
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) 0 0
\(123\) − 15.4164i − 1.39005i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 13.6525i − 1.21146i −0.795670 0.605731i \(-0.792881\pi\)
0.795670 0.605731i \(-0.207119\pi\)
\(128\) 0 0
\(129\) 5.70820 0.502579
\(130\) 0 0
\(131\) 16.9443 1.48043 0.740214 0.672371i \(-0.234724\pi\)
0.740214 + 0.672371i \(0.234724\pi\)
\(132\) 0 0
\(133\) 4.47214i 0.387783i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.9443i − 0.935032i −0.883985 0.467516i \(-0.845149\pi\)
0.883985 0.467516i \(-0.154851\pi\)
\(138\) 0 0
\(139\) 10.6525 0.903531 0.451766 0.892137i \(-0.350794\pi\)
0.451766 + 0.892137i \(0.350794\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 0 0
\(143\) − 0.291796i − 0.0244012i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.23607i 0.266906i
\(148\) 0 0
\(149\) −3.94427 −0.323127 −0.161564 0.986862i \(-0.551654\pi\)
−0.161564 + 0.986862i \(0.551654\pi\)
\(150\) 0 0
\(151\) 20.2361 1.64679 0.823394 0.567470i \(-0.192078\pi\)
0.823394 + 0.567470i \(0.192078\pi\)
\(152\) 0 0
\(153\) − 18.4721i − 1.49338i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.763932i 0.0609684i 0.999535 + 0.0304842i \(0.00970493\pi\)
−0.999535 + 0.0304842i \(0.990295\pi\)
\(158\) 0 0
\(159\) −27.4164 −2.17426
\(160\) 0 0
\(161\) 6.23607 0.491471
\(162\) 0 0
\(163\) − 1.52786i − 0.119672i −0.998208 0.0598358i \(-0.980942\pi\)
0.998208 0.0598358i \(-0.0190577\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 5.23607i − 0.405179i −0.979264 0.202590i \(-0.935064\pi\)
0.979264 0.202590i \(-0.0649357\pi\)
\(168\) 0 0
\(169\) 11.4721 0.882472
\(170\) 0 0
\(171\) 33.4164 2.55542
\(172\) 0 0
\(173\) 11.5279i 0.876447i 0.898866 + 0.438224i \(0.144392\pi\)
−0.898866 + 0.438224i \(0.855608\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 37.8885i − 2.84788i
\(178\) 0 0
\(179\) −23.4164 −1.75022 −0.875112 0.483920i \(-0.839213\pi\)
−0.875112 + 0.483920i \(0.839213\pi\)
\(180\) 0 0
\(181\) 8.18034 0.608040 0.304020 0.952666i \(-0.401671\pi\)
0.304020 + 0.952666i \(0.401671\pi\)
\(182\) 0 0
\(183\) 31.4164i 2.32237i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.583592i 0.0426765i
\(188\) 0 0
\(189\) 14.4721 1.05269
\(190\) 0 0
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) 0 0
\(193\) − 12.4164i − 0.893753i −0.894596 0.446876i \(-0.852536\pi\)
0.894596 0.446876i \(-0.147464\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.47214i − 0.104885i −0.998624 0.0524427i \(-0.983299\pi\)
0.998624 0.0524427i \(-0.0167007\pi\)
\(198\) 0 0
\(199\) 7.23607 0.512951 0.256476 0.966551i \(-0.417439\pi\)
0.256476 + 0.966551i \(0.417439\pi\)
\(200\) 0 0
\(201\) 13.7082 0.966902
\(202\) 0 0
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 46.5967i − 3.23870i
\(208\) 0 0
\(209\) −1.05573 −0.0730262
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 28.1803i 1.93089i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.70820i 0.251729i
\(218\) 0 0
\(219\) 28.3607 1.91644
\(220\) 0 0
\(221\) 3.05573 0.205551
\(222\) 0 0
\(223\) 20.1803i 1.35138i 0.737188 + 0.675688i \(0.236153\pi\)
−0.737188 + 0.675688i \(0.763847\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 21.4164i − 1.42146i −0.703466 0.710728i \(-0.748365\pi\)
0.703466 0.710728i \(-0.251635\pi\)
\(228\) 0 0
\(229\) 4.47214 0.295527 0.147764 0.989023i \(-0.452793\pi\)
0.147764 + 0.989023i \(0.452793\pi\)
\(230\) 0 0
\(231\) −0.763932 −0.0502630
\(232\) 0 0
\(233\) − 7.94427i − 0.520447i −0.965548 0.260223i \(-0.916204\pi\)
0.965548 0.260223i \(-0.0837962\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 36.1803i 2.35017i
\(238\) 0 0
\(239\) −5.52786 −0.357568 −0.178784 0.983888i \(-0.557216\pi\)
−0.178784 + 0.983888i \(0.557216\pi\)
\(240\) 0 0
\(241\) −3.52786 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(242\) 0 0
\(243\) − 35.5967i − 2.28353i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.52786i 0.351730i
\(248\) 0 0
\(249\) −24.9443 −1.58078
\(250\) 0 0
\(251\) −6.47214 −0.408518 −0.204259 0.978917i \(-0.565478\pi\)
−0.204259 + 0.978917i \(0.565478\pi\)
\(252\) 0 0
\(253\) 1.47214i 0.0925524i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 12.6525i − 0.789240i −0.918844 0.394620i \(-0.870876\pi\)
0.918844 0.394620i \(-0.129124\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) 37.3607 2.31257
\(262\) 0 0
\(263\) 16.2361i 1.00116i 0.865691 + 0.500579i \(0.166880\pi\)
−0.865691 + 0.500579i \(0.833120\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 55.7771i 3.41350i
\(268\) 0 0
\(269\) 11.7082 0.713862 0.356931 0.934131i \(-0.383823\pi\)
0.356931 + 0.934131i \(0.383823\pi\)
\(270\) 0 0
\(271\) −23.7082 −1.44017 −0.720085 0.693885i \(-0.755897\pi\)
−0.720085 + 0.693885i \(0.755897\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 19.8885i − 1.19499i −0.801874 0.597493i \(-0.796163\pi\)
0.801874 0.597493i \(-0.203837\pi\)
\(278\) 0 0
\(279\) 27.7082 1.65885
\(280\) 0 0
\(281\) −15.3607 −0.916341 −0.458171 0.888864i \(-0.651495\pi\)
−0.458171 + 0.888864i \(0.651495\pi\)
\(282\) 0 0
\(283\) 17.4164i 1.03530i 0.855593 + 0.517649i \(0.173193\pi\)
−0.855593 + 0.517649i \(0.826807\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4.76393i − 0.281206i
\(288\) 0 0
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) 16.9443 0.993291
\(292\) 0 0
\(293\) 31.1246i 1.81832i 0.416448 + 0.909160i \(0.363275\pi\)
−0.416448 + 0.909160i \(0.636725\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.41641i 0.198240i
\(298\) 0 0
\(299\) 7.70820 0.445777
\(300\) 0 0
\(301\) 1.76393 0.101671
\(302\) 0 0
\(303\) − 15.4164i − 0.885649i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 4.58359i − 0.261599i −0.991409 0.130800i \(-0.958246\pi\)
0.991409 0.130800i \(-0.0417545\pi\)
\(308\) 0 0
\(309\) 27.4164 1.55966
\(310\) 0 0
\(311\) −24.3607 −1.38137 −0.690684 0.723157i \(-0.742690\pi\)
−0.690684 + 0.723157i \(0.742690\pi\)
\(312\) 0 0
\(313\) − 19.5279i − 1.10378i −0.833917 0.551890i \(-0.813907\pi\)
0.833917 0.551890i \(-0.186093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.3607i 1.42440i 0.701978 + 0.712199i \(0.252301\pi\)
−0.701978 + 0.712199i \(0.747699\pi\)
\(318\) 0 0
\(319\) −1.18034 −0.0660863
\(320\) 0 0
\(321\) −25.8885 −1.44496
\(322\) 0 0
\(323\) − 11.0557i − 0.615157i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 27.2361i 1.50616i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 24.7082 1.35809 0.679043 0.734099i \(-0.262395\pi\)
0.679043 + 0.734099i \(0.262395\pi\)
\(332\) 0 0
\(333\) − 22.4164i − 1.22841i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 16.4721i − 0.897294i −0.893709 0.448647i \(-0.851906\pi\)
0.893709 0.448647i \(-0.148094\pi\)
\(338\) 0 0
\(339\) 46.6525 2.53381
\(340\) 0 0
\(341\) −0.875388 −0.0474049
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 20.2361i − 1.08633i −0.839626 0.543165i \(-0.817226\pi\)
0.839626 0.543165i \(-0.182774\pi\)
\(348\) 0 0
\(349\) −4.47214 −0.239388 −0.119694 0.992811i \(-0.538191\pi\)
−0.119694 + 0.992811i \(0.538191\pi\)
\(350\) 0 0
\(351\) 17.8885 0.954820
\(352\) 0 0
\(353\) 2.18034i 0.116048i 0.998315 + 0.0580239i \(0.0184800\pi\)
−0.998315 + 0.0580239i \(0.981520\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 8.00000i − 0.423405i
\(358\) 0 0
\(359\) −30.1246 −1.58992 −0.794958 0.606664i \(-0.792507\pi\)
−0.794958 + 0.606664i \(0.792507\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 35.4164i 1.85888i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 37.1246i 1.93789i 0.247278 + 0.968944i \(0.420464\pi\)
−0.247278 + 0.968944i \(0.579536\pi\)
\(368\) 0 0
\(369\) −35.5967 −1.85309
\(370\) 0 0
\(371\) −8.47214 −0.439851
\(372\) 0 0
\(373\) 37.8328i 1.95891i 0.201665 + 0.979454i \(0.435365\pi\)
−0.201665 + 0.979454i \(0.564635\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.18034i 0.318304i
\(378\) 0 0
\(379\) −11.1803 −0.574295 −0.287148 0.957886i \(-0.592707\pi\)
−0.287148 + 0.957886i \(0.592707\pi\)
\(380\) 0 0
\(381\) −44.1803 −2.26343
\(382\) 0 0
\(383\) − 33.2361i − 1.69828i −0.528165 0.849142i \(-0.677120\pi\)
0.528165 0.849142i \(-0.322880\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 13.1803i − 0.669994i
\(388\) 0 0
\(389\) −2.88854 −0.146455 −0.0732275 0.997315i \(-0.523330\pi\)
−0.0732275 + 0.997315i \(0.523330\pi\)
\(390\) 0 0
\(391\) −15.4164 −0.779641
\(392\) 0 0
\(393\) − 54.8328i − 2.76595i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.05573i 0.454494i 0.973837 + 0.227247i \(0.0729725\pi\)
−0.973837 + 0.227247i \(0.927028\pi\)
\(398\) 0 0
\(399\) 14.4721 0.724513
\(400\) 0 0
\(401\) 2.52786 0.126236 0.0631178 0.998006i \(-0.479896\pi\)
0.0631178 + 0.998006i \(0.479896\pi\)
\(402\) 0 0
\(403\) 4.58359i 0.228325i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.708204i 0.0351044i
\(408\) 0 0
\(409\) −24.4721 −1.21007 −0.605035 0.796199i \(-0.706841\pi\)
−0.605035 + 0.796199i \(0.706841\pi\)
\(410\) 0 0
\(411\) −35.4164 −1.74696
\(412\) 0 0
\(413\) − 11.7082i − 0.576123i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 34.4721i − 1.68811i
\(418\) 0 0
\(419\) −26.1803 −1.27899 −0.639497 0.768794i \(-0.720857\pi\)
−0.639497 + 0.768794i \(0.720857\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 0 0
\(423\) − 14.9443i − 0.726615i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.70820i 0.469813i
\(428\) 0 0
\(429\) −0.944272 −0.0455899
\(430\) 0 0
\(431\) −17.5279 −0.844288 −0.422144 0.906529i \(-0.638722\pi\)
−0.422144 + 0.906529i \(0.638722\pi\)
\(432\) 0 0
\(433\) 28.3607i 1.36293i 0.731852 + 0.681464i \(0.238656\pi\)
−0.731852 + 0.681464i \(0.761344\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 27.8885i − 1.33409i
\(438\) 0 0
\(439\) −8.29180 −0.395746 −0.197873 0.980228i \(-0.563403\pi\)
−0.197873 + 0.980228i \(0.563403\pi\)
\(440\) 0 0
\(441\) 7.47214 0.355816
\(442\) 0 0
\(443\) − 19.4164i − 0.922501i −0.887270 0.461251i \(-0.847401\pi\)
0.887270 0.461251i \(-0.152599\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.7639i 0.603713i
\(448\) 0 0
\(449\) −20.5279 −0.968770 −0.484385 0.874855i \(-0.660957\pi\)
−0.484385 + 0.874855i \(0.660957\pi\)
\(450\) 0 0
\(451\) 1.12461 0.0529559
\(452\) 0 0
\(453\) − 65.4853i − 3.07677i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 12.5279i − 0.586029i −0.956108 0.293014i \(-0.905342\pi\)
0.956108 0.293014i \(-0.0946584\pi\)
\(458\) 0 0
\(459\) −35.7771 −1.66993
\(460\) 0 0
\(461\) −14.1803 −0.660444 −0.330222 0.943903i \(-0.607124\pi\)
−0.330222 + 0.943903i \(0.607124\pi\)
\(462\) 0 0
\(463\) − 13.8885i − 0.645455i −0.946492 0.322728i \(-0.895400\pi\)
0.946492 0.322728i \(-0.104600\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.94427i − 0.321343i −0.987008 0.160671i \(-0.948634\pi\)
0.987008 0.160671i \(-0.0513659\pi\)
\(468\) 0 0
\(469\) 4.23607 0.195603
\(470\) 0 0
\(471\) 2.47214 0.113910
\(472\) 0 0
\(473\) 0.416408i 0.0191465i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 63.3050i 2.89853i
\(478\) 0 0
\(479\) −26.1803 −1.19621 −0.598105 0.801418i \(-0.704079\pi\)
−0.598105 + 0.801418i \(0.704079\pi\)
\(480\) 0 0
\(481\) 3.70820 0.169080
\(482\) 0 0
\(483\) − 20.1803i − 0.918237i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.76393i − 0.261189i −0.991436 0.130594i \(-0.958311\pi\)
0.991436 0.130594i \(-0.0416885\pi\)
\(488\) 0 0
\(489\) −4.94427 −0.223588
\(490\) 0 0
\(491\) 5.76393 0.260123 0.130061 0.991506i \(-0.458483\pi\)
0.130061 + 0.991506i \(0.458483\pi\)
\(492\) 0 0
\(493\) − 12.3607i − 0.556697i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.70820i 0.390616i
\(498\) 0 0
\(499\) 11.0557 0.494922 0.247461 0.968898i \(-0.420404\pi\)
0.247461 + 0.968898i \(0.420404\pi\)
\(500\) 0 0
\(501\) −16.9443 −0.757014
\(502\) 0 0
\(503\) − 8.11146i − 0.361672i −0.983513 0.180836i \(-0.942120\pi\)
0.983513 0.180836i \(-0.0578803\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 37.1246i − 1.64876i
\(508\) 0 0
\(509\) −40.6525 −1.80189 −0.900945 0.433934i \(-0.857125\pi\)
−0.900945 + 0.433934i \(0.857125\pi\)
\(510\) 0 0
\(511\) 8.76393 0.387694
\(512\) 0 0
\(513\) − 64.7214i − 2.85752i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.472136i 0.0207645i
\(518\) 0 0
\(519\) 37.3050 1.63751
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 16.3607i 0.715403i 0.933836 + 0.357701i \(0.116439\pi\)
−0.933836 + 0.357701i \(0.883561\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 9.16718i − 0.399329i
\(528\) 0 0
\(529\) −15.8885 −0.690806
\(530\) 0 0
\(531\) −87.4853 −3.79654
\(532\) 0 0
\(533\) − 5.88854i − 0.255061i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 75.7771i 3.27002i
\(538\) 0 0
\(539\) −0.236068 −0.0101682
\(540\) 0 0
\(541\) 15.9443 0.685498 0.342749 0.939427i \(-0.388642\pi\)
0.342749 + 0.939427i \(0.388642\pi\)
\(542\) 0 0
\(543\) − 26.4721i − 1.13603i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.76393i 0.417476i 0.977972 + 0.208738i \(0.0669355\pi\)
−0.977972 + 0.208738i \(0.933064\pi\)
\(548\) 0 0
\(549\) 72.5410 3.09598
\(550\) 0 0
\(551\) 22.3607 0.952597
\(552\) 0 0
\(553\) 11.1803i 0.475436i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 9.11146i − 0.386065i −0.981192 0.193032i \(-0.938168\pi\)
0.981192 0.193032i \(-0.0618322\pi\)
\(558\) 0 0
\(559\) 2.18034 0.0922186
\(560\) 0 0
\(561\) 1.88854 0.0797344
\(562\) 0 0
\(563\) 17.4164i 0.734014i 0.930218 + 0.367007i \(0.119618\pi\)
−0.930218 + 0.367007i \(0.880382\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 24.4164i − 1.02539i
\(568\) 0 0
\(569\) 3.94427 0.165352 0.0826762 0.996576i \(-0.473653\pi\)
0.0826762 + 0.996576i \(0.473653\pi\)
\(570\) 0 0
\(571\) −36.5967 −1.53153 −0.765763 0.643123i \(-0.777638\pi\)
−0.765763 + 0.643123i \(0.777638\pi\)
\(572\) 0 0
\(573\) 20.9443i 0.874960i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) −40.1803 −1.66984
\(580\) 0 0
\(581\) −7.70820 −0.319790
\(582\) 0 0
\(583\) − 2.00000i − 0.0828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.7639i 1.02212i 0.859546 + 0.511058i \(0.170746\pi\)
−0.859546 + 0.511058i \(0.829254\pi\)
\(588\) 0 0
\(589\) 16.5836 0.683315
\(590\) 0 0
\(591\) −4.76393 −0.195962
\(592\) 0 0
\(593\) 37.3050i 1.53193i 0.642882 + 0.765965i \(0.277739\pi\)
−0.642882 + 0.765965i \(0.722261\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 23.4164i − 0.958370i
\(598\) 0 0
\(599\) −11.1803 −0.456816 −0.228408 0.973565i \(-0.573352\pi\)
−0.228408 + 0.973565i \(0.573352\pi\)
\(600\) 0 0
\(601\) −36.9443 −1.50699 −0.753494 0.657455i \(-0.771633\pi\)
−0.753494 + 0.657455i \(0.771633\pi\)
\(602\) 0 0
\(603\) − 31.6525i − 1.28899i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.12461i 0.289179i 0.989492 + 0.144590i \(0.0461862\pi\)
−0.989492 + 0.144590i \(0.953814\pi\)
\(608\) 0 0
\(609\) 16.1803 0.655660
\(610\) 0 0
\(611\) 2.47214 0.100012
\(612\) 0 0
\(613\) 44.4164i 1.79396i 0.442069 + 0.896981i \(0.354245\pi\)
−0.442069 + 0.896981i \(0.645755\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5.94427i − 0.239307i −0.992816 0.119654i \(-0.961822\pi\)
0.992816 0.119654i \(-0.0381784\pi\)
\(618\) 0 0
\(619\) −11.7082 −0.470592 −0.235296 0.971924i \(-0.575606\pi\)
−0.235296 + 0.971924i \(0.575606\pi\)
\(620\) 0 0
\(621\) −90.2492 −3.62158
\(622\) 0 0
\(623\) 17.2361i 0.690548i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.41641i 0.136438i
\(628\) 0 0
\(629\) −7.41641 −0.295712
\(630\) 0 0
\(631\) −27.6525 −1.10083 −0.550414 0.834892i \(-0.685530\pi\)
−0.550414 + 0.834892i \(0.685530\pi\)
\(632\) 0 0
\(633\) 38.8328i 1.54347i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.23607i 0.0489748i
\(638\) 0 0
\(639\) 65.0689 2.57409
\(640\) 0 0
\(641\) 43.8328 1.73129 0.865646 0.500656i \(-0.166908\pi\)
0.865646 + 0.500656i \(0.166908\pi\)
\(642\) 0 0
\(643\) 18.4721i 0.728470i 0.931307 + 0.364235i \(0.118669\pi\)
−0.931307 + 0.364235i \(0.881331\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.8885i 0.781899i 0.920412 + 0.390950i \(0.127853\pi\)
−0.920412 + 0.390950i \(0.872147\pi\)
\(648\) 0 0
\(649\) 2.76393 0.108494
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) − 25.0557i − 0.980506i −0.871580 0.490253i \(-0.836904\pi\)
0.871580 0.490253i \(-0.163096\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 65.4853i − 2.55482i
\(658\) 0 0
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) −42.7214 −1.66167 −0.830834 0.556520i \(-0.812136\pi\)
−0.830834 + 0.556520i \(0.812136\pi\)
\(662\) 0 0
\(663\) − 9.88854i − 0.384039i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 31.1803i − 1.20731i
\(668\) 0 0
\(669\) 65.3050 2.52484
\(670\) 0 0
\(671\) −2.29180 −0.0884738
\(672\) 0 0
\(673\) − 19.5279i − 0.752744i −0.926469 0.376372i \(-0.877171\pi\)
0.926469 0.376372i \(-0.122829\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 14.3607i − 0.551926i −0.961168 0.275963i \(-0.911003\pi\)
0.961168 0.275963i \(-0.0889967\pi\)
\(678\) 0 0
\(679\) 5.23607 0.200942
\(680\) 0 0
\(681\) −69.3050 −2.65577
\(682\) 0 0
\(683\) 14.1246i 0.540463i 0.962795 + 0.270232i \(0.0871003\pi\)
−0.962795 + 0.270232i \(0.912900\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 14.4721i − 0.552146i
\(688\) 0 0
\(689\) −10.4721 −0.398957
\(690\) 0 0
\(691\) 4.18034 0.159028 0.0795138 0.996834i \(-0.474663\pi\)
0.0795138 + 0.996834i \(0.474663\pi\)
\(692\) 0 0
\(693\) 1.76393i 0.0670062i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 11.7771i 0.446089i
\(698\) 0 0
\(699\) −25.7082 −0.972374
\(700\) 0 0
\(701\) −29.0557 −1.09742 −0.548710 0.836013i \(-0.684881\pi\)
−0.548710 + 0.836013i \(0.684881\pi\)
\(702\) 0 0
\(703\) − 13.4164i − 0.506009i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 4.76393i − 0.179166i
\(708\) 0 0
\(709\) 12.1115 0.454855 0.227428 0.973795i \(-0.426968\pi\)
0.227428 + 0.973795i \(0.426968\pi\)
\(710\) 0 0
\(711\) 83.5410 3.13303
\(712\) 0 0
\(713\) − 23.1246i − 0.866024i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 17.8885i 0.668060i
\(718\) 0 0
\(719\) 16.1803 0.603425 0.301712 0.953399i \(-0.402442\pi\)
0.301712 + 0.953399i \(0.402442\pi\)
\(720\) 0 0
\(721\) 8.47214 0.315519
\(722\) 0 0
\(723\) 11.4164i 0.424581i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.05573i 0.113331i 0.998393 + 0.0566653i \(0.0180468\pi\)
−0.998393 + 0.0566653i \(0.981953\pi\)
\(728\) 0 0
\(729\) −41.9443 −1.55349
\(730\) 0 0
\(731\) −4.36068 −0.161286
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000i 0.0368355i
\(738\) 0 0
\(739\) 25.6525 0.943642 0.471821 0.881694i \(-0.343597\pi\)
0.471821 + 0.881694i \(0.343597\pi\)
\(740\) 0 0
\(741\) 17.8885 0.657152
\(742\) 0 0
\(743\) − 10.4721i − 0.384185i −0.981377 0.192093i \(-0.938473\pi\)
0.981377 0.192093i \(-0.0615274\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 57.5967i 2.10735i
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −3.05573 −0.111505 −0.0557526 0.998445i \(-0.517756\pi\)
−0.0557526 + 0.998445i \(0.517756\pi\)
\(752\) 0 0
\(753\) 20.9443i 0.763252i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19.5836i 0.711778i 0.934528 + 0.355889i \(0.115822\pi\)
−0.934528 + 0.355889i \(0.884178\pi\)
\(758\) 0 0
\(759\) 4.76393 0.172920
\(760\) 0 0
\(761\) 27.7771 1.00692 0.503459 0.864019i \(-0.332060\pi\)
0.503459 + 0.864019i \(0.332060\pi\)
\(762\) 0 0
\(763\) 8.41641i 0.304694i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 14.4721i − 0.522559i
\(768\) 0 0
\(769\) 43.0132 1.55109 0.775547 0.631290i \(-0.217474\pi\)
0.775547 + 0.631290i \(0.217474\pi\)
\(770\) 0 0
\(771\) −40.9443 −1.47457
\(772\) 0 0
\(773\) − 50.1803i − 1.80486i −0.430835 0.902431i \(-0.641781\pi\)
0.430835 0.902431i \(-0.358219\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.70820i − 0.348280i
\(778\) 0 0
\(779\) −21.3050 −0.763329
\(780\) 0 0
\(781\) −2.05573 −0.0735597
\(782\) 0 0
\(783\) − 72.3607i − 2.58596i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 40.7639i − 1.45308i −0.687126 0.726539i \(-0.741128\pi\)
0.687126 0.726539i \(-0.258872\pi\)
\(788\) 0 0
\(789\) 52.5410 1.87051
\(790\) 0 0
\(791\) 14.4164 0.512588
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 35.4164i − 1.25451i −0.778813 0.627257i \(-0.784178\pi\)
0.778813 0.627257i \(-0.215822\pi\)
\(798\) 0 0
\(799\) −4.94427 −0.174916
\(800\) 0 0
\(801\) 128.790 4.55058
\(802\) 0 0
\(803\) 2.06888i 0.0730093i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 37.8885i − 1.33374i
\(808\) 0 0
\(809\) −29.4721 −1.03619 −0.518093 0.855325i \(-0.673358\pi\)
−0.518093 + 0.855325i \(0.673358\pi\)
\(810\) 0 0
\(811\) 42.7214 1.50015 0.750075 0.661353i \(-0.230017\pi\)
0.750075 + 0.661353i \(0.230017\pi\)
\(812\) 0 0
\(813\) 76.7214i 2.69074i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.88854i − 0.275985i
\(818\) 0 0
\(819\) 9.23607 0.322734
\(820\) 0 0
\(821\) 28.8328 1.00627 0.503136 0.864207i \(-0.332179\pi\)
0.503136 + 0.864207i \(0.332179\pi\)
\(822\) 0 0
\(823\) − 31.6525i − 1.10334i −0.834064 0.551668i \(-0.813992\pi\)
0.834064 0.551668i \(-0.186008\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 41.5410i − 1.44452i −0.691620 0.722261i \(-0.743103\pi\)
0.691620 0.722261i \(-0.256897\pi\)
\(828\) 0 0
\(829\) 7.63932 0.265325 0.132662 0.991161i \(-0.457647\pi\)
0.132662 + 0.991161i \(0.457647\pi\)
\(830\) 0 0
\(831\) −64.3607 −2.23265
\(832\) 0 0
\(833\) − 2.47214i − 0.0856544i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 53.6656i − 1.85496i
\(838\) 0 0
\(839\) −30.6525 −1.05824 −0.529120 0.848547i \(-0.677478\pi\)
−0.529120 + 0.848547i \(0.677478\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 49.7082i 1.71204i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.9443i 0.376050i
\(848\) 0 0
\(849\) 56.3607 1.93429
\(850\) 0 0
\(851\) −18.7082 −0.641309
\(852\) 0 0
\(853\) − 27.4164i − 0.938720i −0.883007 0.469360i \(-0.844485\pi\)
0.883007 0.469360i \(-0.155515\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 15.8197i − 0.540389i −0.962806 0.270195i \(-0.912912\pi\)
0.962806 0.270195i \(-0.0870881\pi\)
\(858\) 0 0
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 0 0
\(861\) −15.4164 −0.525390
\(862\) 0 0
\(863\) 18.3475i 0.624557i 0.949991 + 0.312278i \(0.101092\pi\)
−0.949991 + 0.312278i \(0.898908\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 35.2361i − 1.19668i
\(868\) 0 0
\(869\) −2.63932 −0.0895328
\(870\) 0 0
\(871\) 5.23607 0.177417
\(872\) 0 0
\(873\) − 39.1246i − 1.32417i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 30.3607i 1.02521i 0.858625 + 0.512604i \(0.171319\pi\)
−0.858625 + 0.512604i \(0.828681\pi\)
\(878\) 0 0
\(879\) 100.721 3.39725
\(880\) 0 0
\(881\) 5.81966 0.196069 0.0980347 0.995183i \(-0.468744\pi\)
0.0980347 + 0.995183i \(0.468744\pi\)
\(882\) 0 0
\(883\) − 1.40325i − 0.0472232i −0.999721 0.0236116i \(-0.992483\pi\)
0.999721 0.0236116i \(-0.00751650\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.3475i 0.716780i 0.933572 + 0.358390i \(0.116674\pi\)
−0.933572 + 0.358390i \(0.883326\pi\)
\(888\) 0 0
\(889\) −13.6525 −0.457889
\(890\) 0 0
\(891\) 5.76393 0.193099
\(892\) 0 0
\(893\) − 8.94427i − 0.299309i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 24.9443i − 0.832865i
\(898\) 0 0
\(899\) 18.5410 0.618378
\(900\) 0 0
\(901\) 20.9443 0.697755
\(902\) 0 0
\(903\) − 5.70820i − 0.189957i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 34.8328i − 1.15660i −0.815823 0.578302i \(-0.803715\pi\)
0.815823 0.578302i \(-0.196285\pi\)
\(908\) 0 0
\(909\) −35.5967 −1.18067
\(910\) 0 0
\(911\) −0.819660 −0.0271566 −0.0135783 0.999908i \(-0.504322\pi\)
−0.0135783 + 0.999908i \(0.504322\pi\)
\(912\) 0 0
\(913\) − 1.81966i − 0.0602220i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 16.9443i − 0.559549i
\(918\) 0 0
\(919\) −27.7639 −0.915848 −0.457924 0.888991i \(-0.651407\pi\)
−0.457924 + 0.888991i \(0.651407\pi\)
\(920\) 0 0
\(921\) −14.8328 −0.488758
\(922\) 0 0
\(923\) 10.7639i 0.354299i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 63.3050i − 2.07921i
\(928\) 0 0
\(929\) −38.2918 −1.25631 −0.628157 0.778087i \(-0.716190\pi\)
−0.628157 + 0.778087i \(0.716190\pi\)
\(930\) 0 0
\(931\) 4.47214 0.146568
\(932\) 0 0
\(933\) 78.8328i 2.58087i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.2361i 1.15111i 0.817762 + 0.575556i \(0.195214\pi\)
−0.817762 + 0.575556i \(0.804786\pi\)
\(938\) 0 0
\(939\) −63.1935 −2.06224
\(940\) 0 0
\(941\) −5.23607 −0.170691 −0.0853455 0.996351i \(-0.527199\pi\)
−0.0853455 + 0.996351i \(0.527199\pi\)
\(942\) 0 0
\(943\) 29.7082i 0.967432i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 34.8328i − 1.13191i −0.824435 0.565957i \(-0.808507\pi\)
0.824435 0.565957i \(-0.191493\pi\)
\(948\) 0 0
\(949\) 10.8328 0.351648
\(950\) 0 0
\(951\) 82.0689 2.66127
\(952\) 0 0
\(953\) − 3.47214i − 0.112474i −0.998417 0.0562368i \(-0.982090\pi\)
0.998417 0.0562368i \(-0.0179102\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.81966i 0.123472i
\(958\) 0 0
\(959\) −10.9443 −0.353409
\(960\) 0 0
\(961\) −17.2492 −0.556427
\(962\) 0 0
\(963\) 59.7771i 1.92629i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.1115i 0.453794i 0.973919 + 0.226897i \(0.0728580\pi\)
−0.973919 + 0.226897i \(0.927142\pi\)
\(968\) 0 0
\(969\) −35.7771 −1.14933
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) − 10.6525i − 0.341503i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 11.4721i − 0.367026i −0.983017 0.183513i \(-0.941253\pi\)
0.983017 0.183513i \(-0.0587470\pi\)
\(978\) 0 0
\(979\) −4.06888 −0.130042
\(980\) 0 0
\(981\) 62.8885 2.00788
\(982\) 0 0
\(983\) − 34.5410i − 1.10169i −0.834608 0.550844i \(-0.814306\pi\)
0.834608 0.550844i \(-0.185694\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 6.47214i − 0.206010i
\(988\) 0 0
\(989\) −11.0000 −0.349780
\(990\) 0 0
\(991\) −13.1803 −0.418687 −0.209344 0.977842i \(-0.567133\pi\)
−0.209344 + 0.977842i \(0.567133\pi\)
\(992\) 0 0
\(993\) − 79.9574i − 2.53737i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 45.4164i − 1.43835i −0.694828 0.719176i \(-0.744519\pi\)
0.694828 0.719176i \(-0.255481\pi\)
\(998\) 0 0
\(999\) −43.4164 −1.37363
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 2800.2.g.s.449.1 4
4.3 odd 2 175.2.b.c.99.2 4
5.2 odd 4 2800.2.a.bh.1.1 2
5.3 odd 4 2800.2.a.bp.1.2 2
5.4 even 2 inner 2800.2.g.s.449.4 4
12.11 even 2 1575.2.d.k.1324.3 4
20.3 even 4 175.2.a.e.1.1 yes 2
20.7 even 4 175.2.a.d.1.2 2
20.19 odd 2 175.2.b.c.99.3 4
28.27 even 2 1225.2.b.k.99.2 4
60.23 odd 4 1575.2.a.n.1.2 2
60.47 odd 4 1575.2.a.s.1.1 2
60.59 even 2 1575.2.d.k.1324.2 4
140.27 odd 4 1225.2.a.n.1.2 2
140.83 odd 4 1225.2.a.u.1.1 2
140.139 even 2 1225.2.b.k.99.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 20.7 even 4
175.2.a.e.1.1 yes 2 20.3 even 4
175.2.b.c.99.2 4 4.3 odd 2
175.2.b.c.99.3 4 20.19 odd 2
1225.2.a.n.1.2 2 140.27 odd 4
1225.2.a.u.1.1 2 140.83 odd 4
1225.2.b.k.99.2 4 28.27 even 2
1225.2.b.k.99.3 4 140.139 even 2
1575.2.a.n.1.2 2 60.23 odd 4
1575.2.a.s.1.1 2 60.47 odd 4
1575.2.d.k.1324.2 4 60.59 even 2
1575.2.d.k.1324.3 4 12.11 even 2
2800.2.a.bh.1.1 2 5.2 odd 4
2800.2.a.bp.1.2 2 5.3 odd 4
2800.2.g.s.449.1 4 1.1 even 1 trivial
2800.2.g.s.449.4 4 5.4 even 2 inner