Properties

Label 4900.2.e.r.2549.3
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 980)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.r.2549.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214i q^{3} +2.82843 q^{9} +O(q^{10})\) \(q+0.414214i q^{3} +2.82843 q^{9} -3.82843 q^{11} -3.58579i q^{13} +6.41421i q^{17} -3.65685 q^{19} -0.585786i q^{23} +2.41421i q^{27} -6.65685 q^{29} +4.58579 q^{31} -1.58579i q^{33} -3.41421i q^{37} +1.48528 q^{39} +0.585786 q^{41} -11.6569i q^{43} -8.89949i q^{47} -2.65685 q^{51} +3.75736i q^{53} -1.51472i q^{57} -3.41421 q^{59} +5.17157 q^{61} -11.0711i q^{67} +0.242641 q^{69} +6.48528 q^{71} -5.17157i q^{73} -13.1421 q^{79} +7.48528 q^{81} +8.00000i q^{83} -2.75736i q^{87} -16.9706 q^{89} +1.89949i q^{93} +15.7279i q^{97} -10.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} + 8 q^{19} - 4 q^{29} + 24 q^{31} - 28 q^{39} + 8 q^{41} + 12 q^{51} - 8 q^{59} + 32 q^{61} - 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0.414214i 0.239146i 0.992825 + 0.119573i \(0.0381526\pi\)
−0.992825 + 0.119573i \(0.961847\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −3.82843 −1.15431 −0.577157 0.816633i \(-0.695838\pi\)
−0.577157 + 0.816633i \(0.695838\pi\)
\(12\) 0 0
\(13\) − 3.58579i − 0.994518i −0.867602 0.497259i \(-0.834340\pi\)
0.867602 0.497259i \(-0.165660\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.41421i 1.55568i 0.628465 + 0.777838i \(0.283683\pi\)
−0.628465 + 0.777838i \(0.716317\pi\)
\(18\) 0 0
\(19\) −3.65685 −0.838940 −0.419470 0.907769i \(-0.637784\pi\)
−0.419470 + 0.907769i \(0.637784\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 0.585786i − 0.122145i −0.998133 0.0610725i \(-0.980548\pi\)
0.998133 0.0610725i \(-0.0194521\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.41421i 0.464616i
\(28\) 0 0
\(29\) −6.65685 −1.23615 −0.618073 0.786120i \(-0.712087\pi\)
−0.618073 + 0.786120i \(0.712087\pi\)
\(30\) 0 0
\(31\) 4.58579 0.823632 0.411816 0.911267i \(-0.364895\pi\)
0.411816 + 0.911267i \(0.364895\pi\)
\(32\) 0 0
\(33\) − 1.58579i − 0.276050i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.41421i − 0.561293i −0.959811 0.280647i \(-0.909451\pi\)
0.959811 0.280647i \(-0.0905489\pi\)
\(38\) 0 0
\(39\) 1.48528 0.237835
\(40\) 0 0
\(41\) 0.585786 0.0914845 0.0457422 0.998953i \(-0.485435\pi\)
0.0457422 + 0.998953i \(0.485435\pi\)
\(42\) 0 0
\(43\) − 11.6569i − 1.77765i −0.458243 0.888827i \(-0.651521\pi\)
0.458243 0.888827i \(-0.348479\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.89949i − 1.29812i −0.760735 0.649062i \(-0.775161\pi\)
0.760735 0.649062i \(-0.224839\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.65685 −0.372034
\(52\) 0 0
\(53\) 3.75736i 0.516113i 0.966130 + 0.258056i \(0.0830821\pi\)
−0.966130 + 0.258056i \(0.916918\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.51472i − 0.200629i
\(58\) 0 0
\(59\) −3.41421 −0.444493 −0.222246 0.974991i \(-0.571339\pi\)
−0.222246 + 0.974991i \(0.571339\pi\)
\(60\) 0 0
\(61\) 5.17157 0.662152 0.331076 0.943604i \(-0.392588\pi\)
0.331076 + 0.943604i \(0.392588\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.0711i − 1.35255i −0.736651 0.676273i \(-0.763594\pi\)
0.736651 0.676273i \(-0.236406\pi\)
\(68\) 0 0
\(69\) 0.242641 0.0292105
\(70\) 0 0
\(71\) 6.48528 0.769661 0.384831 0.922987i \(-0.374260\pi\)
0.384831 + 0.922987i \(0.374260\pi\)
\(72\) 0 0
\(73\) − 5.17157i − 0.605287i −0.953104 0.302643i \(-0.902131\pi\)
0.953104 0.302643i \(-0.0978691\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.1421 −1.47861 −0.739303 0.673373i \(-0.764845\pi\)
−0.739303 + 0.673373i \(0.764845\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.75736i − 0.295620i
\(88\) 0 0
\(89\) −16.9706 −1.79888 −0.899438 0.437048i \(-0.856024\pi\)
−0.899438 + 0.437048i \(0.856024\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.89949i 0.196968i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.7279i 1.59693i 0.602042 + 0.798464i \(0.294354\pi\)
−0.602042 + 0.798464i \(0.705646\pi\)
\(98\) 0 0
\(99\) −10.8284 −1.08830
\(100\) 0 0
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) 0 0
\(103\) − 1.58579i − 0.156252i −0.996943 0.0781261i \(-0.975106\pi\)
0.996943 0.0781261i \(-0.0248937\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 16.8284i − 1.62687i −0.581659 0.813433i \(-0.697596\pi\)
0.581659 0.813433i \(-0.302404\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 1.41421 0.134231
\(112\) 0 0
\(113\) 5.07107i 0.477046i 0.971137 + 0.238523i \(0.0766632\pi\)
−0.971137 + 0.238523i \(0.923337\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 10.1421i − 0.937641i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.65685 0.332441
\(122\) 0 0
\(123\) 0.242641i 0.0218782i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 21.8995i − 1.94327i −0.236494 0.971633i \(-0.575998\pi\)
0.236494 0.971633i \(-0.424002\pi\)
\(128\) 0 0
\(129\) 4.82843 0.425119
\(130\) 0 0
\(131\) −11.7574 −1.02725 −0.513623 0.858016i \(-0.671697\pi\)
−0.513623 + 0.858016i \(0.671697\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.9706i 1.10815i 0.832467 + 0.554075i \(0.186928\pi\)
−0.832467 + 0.554075i \(0.813072\pi\)
\(138\) 0 0
\(139\) 13.8995 1.17894 0.589470 0.807790i \(-0.299337\pi\)
0.589470 + 0.807790i \(0.299337\pi\)
\(140\) 0 0
\(141\) 3.68629 0.310442
\(142\) 0 0
\(143\) 13.7279i 1.14799i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.51472 −0.287937 −0.143968 0.989582i \(-0.545986\pi\)
−0.143968 + 0.989582i \(0.545986\pi\)
\(150\) 0 0
\(151\) −11.8284 −0.962584 −0.481292 0.876560i \(-0.659832\pi\)
−0.481292 + 0.876560i \(0.659832\pi\)
\(152\) 0 0
\(153\) 18.1421i 1.46670i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.4853i 0.836817i 0.908259 + 0.418408i \(0.137412\pi\)
−0.908259 + 0.418408i \(0.862588\pi\)
\(158\) 0 0
\(159\) −1.55635 −0.123427
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 21.5563i − 1.68842i −0.536010 0.844212i \(-0.680069\pi\)
0.536010 0.844212i \(-0.319931\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.41421i 0.186817i 0.995628 + 0.0934087i \(0.0297763\pi\)
−0.995628 + 0.0934087i \(0.970224\pi\)
\(168\) 0 0
\(169\) 0.142136 0.0109335
\(170\) 0 0
\(171\) −10.3431 −0.790960
\(172\) 0 0
\(173\) − 18.5563i − 1.41081i −0.708803 0.705407i \(-0.750764\pi\)
0.708803 0.705407i \(-0.249236\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.41421i − 0.106299i
\(178\) 0 0
\(179\) −6.48528 −0.484733 −0.242366 0.970185i \(-0.577924\pi\)
−0.242366 + 0.970185i \(0.577924\pi\)
\(180\) 0 0
\(181\) 1.75736 0.130623 0.0653117 0.997865i \(-0.479196\pi\)
0.0653117 + 0.997865i \(0.479196\pi\)
\(182\) 0 0
\(183\) 2.14214i 0.158351i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 24.5563i − 1.79574i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.6569 1.20525 0.602624 0.798025i \(-0.294122\pi\)
0.602624 + 0.798025i \(0.294122\pi\)
\(192\) 0 0
\(193\) − 5.65685i − 0.407189i −0.979055 0.203595i \(-0.934738\pi\)
0.979055 0.203595i \(-0.0652625\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 27.5563i − 1.96331i −0.190669 0.981654i \(-0.561066\pi\)
0.190669 0.981654i \(-0.438934\pi\)
\(198\) 0 0
\(199\) 26.7279 1.89469 0.947346 0.320212i \(-0.103754\pi\)
0.947346 + 0.320212i \(0.103754\pi\)
\(200\) 0 0
\(201\) 4.58579 0.323456
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.65685i − 0.115159i
\(208\) 0 0
\(209\) 14.0000 0.968400
\(210\) 0 0
\(211\) −24.3137 −1.67382 −0.836912 0.547337i \(-0.815642\pi\)
−0.836912 + 0.547337i \(0.815642\pi\)
\(212\) 0 0
\(213\) 2.68629i 0.184062i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.14214 0.144752
\(220\) 0 0
\(221\) 23.0000 1.54715
\(222\) 0 0
\(223\) − 24.0711i − 1.61192i −0.591971 0.805959i \(-0.701650\pi\)
0.591971 0.805959i \(-0.298350\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.72792i 0.512920i 0.966555 + 0.256460i \(0.0825562\pi\)
−0.966555 + 0.256460i \(0.917444\pi\)
\(228\) 0 0
\(229\) −23.8995 −1.57932 −0.789662 0.613543i \(-0.789744\pi\)
−0.789662 + 0.613543i \(0.789744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.17157i 0.600850i 0.953805 + 0.300425i \(0.0971285\pi\)
−0.953805 + 0.300425i \(0.902872\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 5.44365i − 0.353603i
\(238\) 0 0
\(239\) −14.1716 −0.916683 −0.458341 0.888776i \(-0.651556\pi\)
−0.458341 + 0.888776i \(0.651556\pi\)
\(240\) 0 0
\(241\) −3.55635 −0.229085 −0.114542 0.993418i \(-0.536540\pi\)
−0.114542 + 0.993418i \(0.536540\pi\)
\(242\) 0 0
\(243\) 10.3431i 0.663513i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.1127i 0.834341i
\(248\) 0 0
\(249\) −3.31371 −0.209998
\(250\) 0 0
\(251\) −27.0711 −1.70871 −0.854355 0.519689i \(-0.826048\pi\)
−0.854355 + 0.519689i \(0.826048\pi\)
\(252\) 0 0
\(253\) 2.24264i 0.140994i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 25.7990i − 1.60930i −0.593752 0.804648i \(-0.702354\pi\)
0.593752 0.804648i \(-0.297646\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −18.8284 −1.16545
\(262\) 0 0
\(263\) − 10.0000i − 0.616626i −0.951285 0.308313i \(-0.900236\pi\)
0.951285 0.308313i \(-0.0997645\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 7.02944i − 0.430195i
\(268\) 0 0
\(269\) 16.7279 1.01992 0.509960 0.860198i \(-0.329660\pi\)
0.509960 + 0.860198i \(0.329660\pi\)
\(270\) 0 0
\(271\) 28.9706 1.75984 0.879918 0.475125i \(-0.157597\pi\)
0.879918 + 0.475125i \(0.157597\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 7.41421i − 0.445477i −0.974878 0.222738i \(-0.928500\pi\)
0.974878 0.222738i \(-0.0714996\pi\)
\(278\) 0 0
\(279\) 12.9706 0.776527
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 18.7574i 1.11501i 0.830174 + 0.557505i \(0.188241\pi\)
−0.830174 + 0.557505i \(0.811759\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −24.1421 −1.42013
\(290\) 0 0
\(291\) −6.51472 −0.381900
\(292\) 0 0
\(293\) 14.4142i 0.842087i 0.907040 + 0.421044i \(0.138336\pi\)
−0.907040 + 0.421044i \(0.861664\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 9.24264i − 0.536312i
\(298\) 0 0
\(299\) −2.10051 −0.121475
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.58579i 0.204652i 0.994751 + 0.102326i \(0.0326284\pi\)
−0.994751 + 0.102326i \(0.967372\pi\)
\(308\) 0 0
\(309\) 0.656854 0.0373671
\(310\) 0 0
\(311\) 6.97056 0.395264 0.197632 0.980276i \(-0.436675\pi\)
0.197632 + 0.980276i \(0.436675\pi\)
\(312\) 0 0
\(313\) − 16.5563i − 0.935820i −0.883776 0.467910i \(-0.845007\pi\)
0.883776 0.467910i \(-0.154993\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 14.9706i − 0.840831i −0.907332 0.420415i \(-0.861884\pi\)
0.907332 0.420415i \(-0.138116\pi\)
\(318\) 0 0
\(319\) 25.4853 1.42690
\(320\) 0 0
\(321\) 6.97056 0.389059
\(322\) 0 0
\(323\) − 23.4558i − 1.30512i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3.72792i − 0.206155i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.82843 0.265394 0.132697 0.991157i \(-0.457636\pi\)
0.132697 + 0.991157i \(0.457636\pi\)
\(332\) 0 0
\(333\) − 9.65685i − 0.529192i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 18.7279i − 1.02017i −0.860123 0.510087i \(-0.829613\pi\)
0.860123 0.510087i \(-0.170387\pi\)
\(338\) 0 0
\(339\) −2.10051 −0.114084
\(340\) 0 0
\(341\) −17.5563 −0.950730
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.41421i − 0.398016i −0.979998 0.199008i \(-0.936228\pi\)
0.979998 0.199008i \(-0.0637719\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 8.65685 0.462069
\(352\) 0 0
\(353\) 2.07107i 0.110232i 0.998480 + 0.0551159i \(0.0175528\pi\)
−0.998480 + 0.0551159i \(0.982447\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) −5.62742 −0.296180
\(362\) 0 0
\(363\) 1.51472i 0.0795021i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.7279i 1.02979i 0.857254 + 0.514895i \(0.172169\pi\)
−0.857254 + 0.514895i \(0.827831\pi\)
\(368\) 0 0
\(369\) 1.65685 0.0862524
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 16.4853i − 0.853576i −0.904352 0.426788i \(-0.859645\pi\)
0.904352 0.426788i \(-0.140355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.8701i 1.22937i
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 0 0
\(381\) 9.07107 0.464725
\(382\) 0 0
\(383\) 3.51472i 0.179594i 0.995960 + 0.0897969i \(0.0286218\pi\)
−0.995960 + 0.0897969i \(0.971378\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 32.9706i − 1.67599i
\(388\) 0 0
\(389\) −13.1421 −0.666333 −0.333166 0.942868i \(-0.608117\pi\)
−0.333166 + 0.942868i \(0.608117\pi\)
\(390\) 0 0
\(391\) 3.75736 0.190018
\(392\) 0 0
\(393\) − 4.87006i − 0.245662i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.4142i 0.723429i 0.932289 + 0.361714i \(0.117808\pi\)
−0.932289 + 0.361714i \(0.882192\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.514719 0.0257038 0.0128519 0.999917i \(-0.495909\pi\)
0.0128519 + 0.999917i \(0.495909\pi\)
\(402\) 0 0
\(403\) − 16.4437i − 0.819117i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.0711i 0.647909i
\(408\) 0 0
\(409\) −32.8284 −1.62326 −0.811631 0.584171i \(-0.801420\pi\)
−0.811631 + 0.584171i \(0.801420\pi\)
\(410\) 0 0
\(411\) −5.37258 −0.265010
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.75736i 0.281939i
\(418\) 0 0
\(419\) 9.55635 0.466858 0.233429 0.972374i \(-0.425005\pi\)
0.233429 + 0.972374i \(0.425005\pi\)
\(420\) 0 0
\(421\) 20.3137 0.990030 0.495015 0.868885i \(-0.335163\pi\)
0.495015 + 0.868885i \(0.335163\pi\)
\(422\) 0 0
\(423\) − 25.1716i − 1.22388i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −5.68629 −0.274537
\(430\) 0 0
\(431\) −9.82843 −0.473419 −0.236709 0.971581i \(-0.576069\pi\)
−0.236709 + 0.971581i \(0.576069\pi\)
\(432\) 0 0
\(433\) 22.2843i 1.07091i 0.844563 + 0.535457i \(0.179861\pi\)
−0.844563 + 0.535457i \(0.820139\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.14214i 0.102472i
\(438\) 0 0
\(439\) −24.2426 −1.15704 −0.578519 0.815669i \(-0.696369\pi\)
−0.578519 + 0.815669i \(0.696369\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.4558i 1.68456i 0.539042 + 0.842279i \(0.318786\pi\)
−0.539042 + 0.842279i \(0.681214\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1.45584i − 0.0688591i
\(448\) 0 0
\(449\) −23.8284 −1.12453 −0.562267 0.826956i \(-0.690070\pi\)
−0.562267 + 0.826956i \(0.690070\pi\)
\(450\) 0 0
\(451\) −2.24264 −0.105602
\(452\) 0 0
\(453\) − 4.89949i − 0.230198i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 15.0711i − 0.704995i −0.935813 0.352497i \(-0.885333\pi\)
0.935813 0.352497i \(-0.114667\pi\)
\(458\) 0 0
\(459\) −15.4853 −0.722791
\(460\) 0 0
\(461\) 22.3431 1.04062 0.520312 0.853976i \(-0.325816\pi\)
0.520312 + 0.853976i \(0.325816\pi\)
\(462\) 0 0
\(463\) 13.4558i 0.625346i 0.949861 + 0.312673i \(0.101224\pi\)
−0.949861 + 0.312673i \(0.898776\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.899495i 0.0416237i 0.999783 + 0.0208118i \(0.00662509\pi\)
−0.999783 + 0.0208118i \(0.993375\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.34315 −0.200122
\(472\) 0 0
\(473\) 44.6274i 2.05197i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.6274i 0.486596i
\(478\) 0 0
\(479\) −9.41421 −0.430146 −0.215073 0.976598i \(-0.568999\pi\)
−0.215073 + 0.976598i \(0.568999\pi\)
\(480\) 0 0
\(481\) −12.2426 −0.558216
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.41421i − 0.335970i −0.985790 0.167985i \(-0.946274\pi\)
0.985790 0.167985i \(-0.0537260\pi\)
\(488\) 0 0
\(489\) 8.92893 0.403780
\(490\) 0 0
\(491\) 17.2843 0.780028 0.390014 0.920809i \(-0.372470\pi\)
0.390014 + 0.920809i \(0.372470\pi\)
\(492\) 0 0
\(493\) − 42.6985i − 1.92304i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 13.6274 0.610047 0.305023 0.952345i \(-0.401336\pi\)
0.305023 + 0.952345i \(0.401336\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) − 39.0416i − 1.74078i −0.492363 0.870390i \(-0.663867\pi\)
0.492363 0.870390i \(-0.336133\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.0588745i 0.00261471i
\(508\) 0 0
\(509\) 35.0711 1.55450 0.777249 0.629193i \(-0.216615\pi\)
0.777249 + 0.629193i \(0.216615\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) − 8.82843i − 0.389785i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 34.0711i 1.49844i
\(518\) 0 0
\(519\) 7.68629 0.337391
\(520\) 0 0
\(521\) 6.68629 0.292932 0.146466 0.989216i \(-0.453210\pi\)
0.146466 + 0.989216i \(0.453210\pi\)
\(522\) 0 0
\(523\) − 1.85786i − 0.0812387i −0.999175 0.0406194i \(-0.987067\pi\)
0.999175 0.0406194i \(-0.0129331\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.4142i 1.28130i
\(528\) 0 0
\(529\) 22.6569 0.985081
\(530\) 0 0
\(531\) −9.65685 −0.419072
\(532\) 0 0
\(533\) − 2.10051i − 0.0909830i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2.68629i − 0.115922i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15.9706 0.686628 0.343314 0.939221i \(-0.388450\pi\)
0.343314 + 0.939221i \(0.388450\pi\)
\(542\) 0 0
\(543\) 0.727922i 0.0312381i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.51472i 0.321306i 0.987011 + 0.160653i \(0.0513600\pi\)
−0.987011 + 0.160653i \(0.948640\pi\)
\(548\) 0 0
\(549\) 14.6274 0.624283
\(550\) 0 0
\(551\) 24.3431 1.03705
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 7.79899i − 0.330454i −0.986256 0.165227i \(-0.947164\pi\)
0.986256 0.165227i \(-0.0528356\pi\)
\(558\) 0 0
\(559\) −41.7990 −1.76791
\(560\) 0 0
\(561\) 10.1716 0.429444
\(562\) 0 0
\(563\) 20.6274i 0.869342i 0.900589 + 0.434671i \(0.143135\pi\)
−0.900589 + 0.434671i \(0.856865\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.7990 1.33308 0.666542 0.745468i \(-0.267774\pi\)
0.666542 + 0.745468i \(0.267774\pi\)
\(570\) 0 0
\(571\) 4.82843 0.202063 0.101032 0.994883i \(-0.467786\pi\)
0.101032 + 0.994883i \(0.467786\pi\)
\(572\) 0 0
\(573\) 6.89949i 0.288231i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 14.0711i − 0.585786i −0.956145 0.292893i \(-0.905382\pi\)
0.956145 0.292893i \(-0.0946180\pi\)
\(578\) 0 0
\(579\) 2.34315 0.0973778
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 14.3848i − 0.595757i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 30.8284i − 1.27243i −0.771514 0.636213i \(-0.780500\pi\)
0.771514 0.636213i \(-0.219500\pi\)
\(588\) 0 0
\(589\) −16.7696 −0.690977
\(590\) 0 0
\(591\) 11.4142 0.469518
\(592\) 0 0
\(593\) − 17.7279i − 0.727999i −0.931399 0.363999i \(-0.881411\pi\)
0.931399 0.363999i \(-0.118589\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0711i 0.453109i
\(598\) 0 0
\(599\) 12.7990 0.522953 0.261476 0.965210i \(-0.415791\pi\)
0.261476 + 0.965210i \(0.415791\pi\)
\(600\) 0 0
\(601\) 39.6569 1.61764 0.808818 0.588058i \(-0.200108\pi\)
0.808818 + 0.588058i \(0.200108\pi\)
\(602\) 0 0
\(603\) − 31.3137i − 1.27519i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13.9289i 0.565358i 0.959215 + 0.282679i \(0.0912231\pi\)
−0.959215 + 0.282679i \(0.908777\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.9117 −1.29101
\(612\) 0 0
\(613\) 41.3137i 1.66864i 0.551277 + 0.834322i \(0.314141\pi\)
−0.551277 + 0.834322i \(0.685859\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.8701i 1.64537i 0.568500 + 0.822683i \(0.307524\pi\)
−0.568500 + 0.822683i \(0.692476\pi\)
\(618\) 0 0
\(619\) −10.8701 −0.436905 −0.218452 0.975848i \(-0.570101\pi\)
−0.218452 + 0.975848i \(0.570101\pi\)
\(620\) 0 0
\(621\) 1.41421 0.0567504
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 5.79899i 0.231589i
\(628\) 0 0
\(629\) 21.8995 0.873190
\(630\) 0 0
\(631\) −39.4853 −1.57188 −0.785942 0.618300i \(-0.787822\pi\)
−0.785942 + 0.618300i \(0.787822\pi\)
\(632\) 0 0
\(633\) − 10.0711i − 0.400289i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 18.3431 0.725644
\(640\) 0 0
\(641\) 39.3137 1.55280 0.776399 0.630242i \(-0.217044\pi\)
0.776399 + 0.630242i \(0.217044\pi\)
\(642\) 0 0
\(643\) − 4.21320i − 0.166153i −0.996543 0.0830763i \(-0.973525\pi\)
0.996543 0.0830763i \(-0.0264745\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.1127i 1.38042i 0.723608 + 0.690211i \(0.242482\pi\)
−0.723608 + 0.690211i \(0.757518\pi\)
\(648\) 0 0
\(649\) 13.0711 0.513084
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.6863i 1.04432i 0.852849 + 0.522158i \(0.174873\pi\)
−0.852849 + 0.522158i \(0.825127\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 14.6274i − 0.570670i
\(658\) 0 0
\(659\) 20.6569 0.804677 0.402338 0.915491i \(-0.368198\pi\)
0.402338 + 0.915491i \(0.368198\pi\)
\(660\) 0 0
\(661\) −34.1421 −1.32798 −0.663988 0.747744i \(-0.731137\pi\)
−0.663988 + 0.747744i \(0.731137\pi\)
\(662\) 0 0
\(663\) 9.52691i 0.369995i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.89949i 0.150989i
\(668\) 0 0
\(669\) 9.97056 0.385484
\(670\) 0 0
\(671\) −19.7990 −0.764332
\(672\) 0 0
\(673\) 27.5147i 1.06061i 0.847806 + 0.530307i \(0.177923\pi\)
−0.847806 + 0.530307i \(0.822077\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 10.2721i − 0.394788i −0.980324 0.197394i \(-0.936752\pi\)
0.980324 0.197394i \(-0.0632478\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.20101 −0.122663
\(682\) 0 0
\(683\) − 35.1127i − 1.34355i −0.740755 0.671775i \(-0.765532\pi\)
0.740755 0.671775i \(-0.234468\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 9.89949i − 0.377689i
\(688\) 0 0
\(689\) 13.4731 0.513284
\(690\) 0 0
\(691\) 3.51472 0.133706 0.0668531 0.997763i \(-0.478704\pi\)
0.0668531 + 0.997763i \(0.478704\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.75736i 0.142320i
\(698\) 0 0
\(699\) −3.79899 −0.143691
\(700\) 0 0
\(701\) 0.514719 0.0194407 0.00972033 0.999953i \(-0.496906\pi\)
0.00972033 + 0.999953i \(0.496906\pi\)
\(702\) 0 0
\(703\) 12.4853i 0.470891i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.62742 −0.136231 −0.0681153 0.997677i \(-0.521699\pi\)
−0.0681153 + 0.997677i \(0.521699\pi\)
\(710\) 0 0
\(711\) −37.1716 −1.39404
\(712\) 0 0
\(713\) − 2.68629i − 0.100602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 5.87006i − 0.219221i
\(718\) 0 0
\(719\) −15.7574 −0.587650 −0.293825 0.955859i \(-0.594928\pi\)
−0.293825 + 0.955859i \(0.594928\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 1.47309i − 0.0547847i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 18.0000i − 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) 0 0
\(729\) 18.1716 0.673021
\(730\) 0 0
\(731\) 74.7696 2.76545
\(732\) 0 0
\(733\) 30.6985i 1.13387i 0.823761 + 0.566937i \(0.191872\pi\)
−0.823761 + 0.566937i \(0.808128\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.3848i 1.56126i
\(738\) 0 0
\(739\) −20.6569 −0.759875 −0.379937 0.925012i \(-0.624054\pi\)
−0.379937 + 0.925012i \(0.624054\pi\)
\(740\) 0 0
\(741\) −5.43146 −0.199530
\(742\) 0 0
\(743\) 8.92893i 0.327571i 0.986496 + 0.163785i \(0.0523705\pi\)
−0.986496 + 0.163785i \(0.947630\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22.6274i 0.827894i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.14214 0.0416771 0.0208386 0.999783i \(-0.493366\pi\)
0.0208386 + 0.999783i \(0.493366\pi\)
\(752\) 0 0
\(753\) − 11.2132i − 0.408632i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 37.1716i 1.35102i 0.737349 + 0.675512i \(0.236077\pi\)
−0.737349 + 0.675512i \(0.763923\pi\)
\(758\) 0 0
\(759\) −0.928932 −0.0337181
\(760\) 0 0
\(761\) −18.8701 −0.684039 −0.342020 0.939693i \(-0.611111\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.2426i 0.442056i
\(768\) 0 0
\(769\) −28.1421 −1.01483 −0.507416 0.861701i \(-0.669399\pi\)
−0.507416 + 0.861701i \(0.669399\pi\)
\(770\) 0 0
\(771\) 10.6863 0.384857
\(772\) 0 0
\(773\) 43.7279i 1.57278i 0.617728 + 0.786392i \(0.288053\pi\)
−0.617728 + 0.786392i \(0.711947\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.14214 −0.0767500
\(780\) 0 0
\(781\) −24.8284 −0.888431
\(782\) 0 0
\(783\) − 16.0711i − 0.574333i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 43.7279i − 1.55873i −0.626569 0.779366i \(-0.715541\pi\)
0.626569 0.779366i \(-0.284459\pi\)
\(788\) 0 0
\(789\) 4.14214 0.147464
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 18.5442i − 0.658522i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 35.3848i − 1.25339i −0.779263 0.626697i \(-0.784407\pi\)
0.779263 0.626697i \(-0.215593\pi\)
\(798\) 0 0
\(799\) 57.0833 2.01946
\(800\) 0 0
\(801\) −48.0000 −1.69600
\(802\) 0 0
\(803\) 19.7990i 0.698691i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.92893i 0.243910i
\(808\) 0 0
\(809\) −6.02944 −0.211984 −0.105992 0.994367i \(-0.533802\pi\)
−0.105992 + 0.994367i \(0.533802\pi\)
\(810\) 0 0
\(811\) −19.5563 −0.686716 −0.343358 0.939205i \(-0.611564\pi\)
−0.343358 + 0.939205i \(0.611564\pi\)
\(812\) 0 0
\(813\) 12.0000i 0.420858i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 42.6274i 1.49134i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 47.8284 1.66922 0.834612 0.550839i \(-0.185692\pi\)
0.834612 + 0.550839i \(0.185692\pi\)
\(822\) 0 0
\(823\) − 36.5269i − 1.27325i −0.771174 0.636624i \(-0.780330\pi\)
0.771174 0.636624i \(-0.219670\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 10.0416i − 0.349182i −0.984641 0.174591i \(-0.944140\pi\)
0.984641 0.174591i \(-0.0558603\pi\)
\(828\) 0 0
\(829\) −22.7279 −0.789373 −0.394687 0.918816i \(-0.629147\pi\)
−0.394687 + 0.918816i \(0.629147\pi\)
\(830\) 0 0
\(831\) 3.07107 0.106534
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 11.0711i 0.382672i
\(838\) 0 0
\(839\) −22.3848 −0.772808 −0.386404 0.922330i \(-0.626283\pi\)
−0.386404 + 0.922330i \(0.626283\pi\)
\(840\) 0 0
\(841\) 15.3137 0.528059
\(842\) 0 0
\(843\) − 0.414214i − 0.0142663i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7.76955 −0.266650
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 30.2843i 1.03691i 0.855104 + 0.518457i \(0.173493\pi\)
−0.855104 + 0.518457i \(0.826507\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 20.8284i − 0.711486i −0.934584 0.355743i \(-0.884228\pi\)
0.934584 0.355743i \(-0.115772\pi\)
\(858\) 0 0
\(859\) 45.4558 1.55093 0.775467 0.631388i \(-0.217515\pi\)
0.775467 + 0.631388i \(0.217515\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19.7574i 0.672548i 0.941764 + 0.336274i \(0.109167\pi\)
−0.941764 + 0.336274i \(0.890833\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 10.0000i − 0.339618i
\(868\) 0 0
\(869\) 50.3137 1.70678
\(870\) 0 0
\(871\) −39.6985 −1.34513
\(872\) 0 0
\(873\) 44.4853i 1.50560i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.6863i 0.360850i 0.983589 + 0.180425i \(0.0577474\pi\)
−0.983589 + 0.180425i \(0.942253\pi\)
\(878\) 0 0
\(879\) −5.97056 −0.201382
\(880\) 0 0
\(881\) −56.2843 −1.89627 −0.948133 0.317875i \(-0.897031\pi\)
−0.948133 + 0.317875i \(0.897031\pi\)
\(882\) 0 0
\(883\) 7.11270i 0.239361i 0.992812 + 0.119681i \(0.0381871\pi\)
−0.992812 + 0.119681i \(0.961813\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 57.5980i 1.93395i 0.254870 + 0.966975i \(0.417967\pi\)
−0.254870 + 0.966975i \(0.582033\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −28.6569 −0.960041
\(892\) 0 0
\(893\) 32.5442i 1.08905i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 0.870058i − 0.0290504i
\(898\) 0 0
\(899\) −30.5269 −1.01813
\(900\) 0 0
\(901\) −24.1005 −0.802904
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.5269i 0.747994i 0.927430 + 0.373997i \(0.122013\pi\)
−0.927430 + 0.373997i \(0.877987\pi\)
\(908\) 0 0
\(909\) −13.6569 −0.452969
\(910\) 0 0
\(911\) −55.5980 −1.84204 −0.921022 0.389511i \(-0.872644\pi\)
−0.921022 + 0.389511i \(0.872644\pi\)
\(912\) 0 0
\(913\) − 30.6274i − 1.01362i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 18.5147 0.610744 0.305372 0.952233i \(-0.401219\pi\)
0.305372 + 0.952233i \(0.401219\pi\)
\(920\) 0 0
\(921\) −1.48528 −0.0489417
\(922\) 0 0
\(923\) − 23.2548i − 0.765442i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.48528i − 0.147316i
\(928\) 0 0
\(929\) 21.2721 0.697914 0.348957 0.937139i \(-0.386536\pi\)
0.348957 + 0.937139i \(0.386536\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.88730i 0.0945260i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 7.72792i − 0.252460i −0.992001 0.126230i \(-0.959712\pi\)
0.992001 0.126230i \(-0.0402878\pi\)
\(938\) 0 0
\(939\) 6.85786 0.223798
\(940\) 0 0
\(941\) −4.00000 −0.130396 −0.0651981 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) − 0.343146i − 0.0111744i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 32.0416i − 1.04121i −0.853797 0.520607i \(-0.825706\pi\)
0.853797 0.520607i \(-0.174294\pi\)
\(948\) 0 0
\(949\) −18.5442 −0.601969
\(950\) 0 0
\(951\) 6.20101 0.201082
\(952\) 0 0
\(953\) 38.8701i 1.25912i 0.776950 + 0.629562i \(0.216766\pi\)
−0.776950 + 0.629562i \(0.783234\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.5563i 0.341238i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 0 0
\(963\) − 47.5980i − 1.53382i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 7.45584i − 0.239764i −0.992788 0.119882i \(-0.961748\pi\)
0.992788 0.119882i \(-0.0382516\pi\)
\(968\) 0 0
\(969\) 9.71573 0.312114
\(970\) 0 0
\(971\) −22.5269 −0.722923 −0.361462 0.932387i \(-0.617722\pi\)
−0.361462 + 0.932387i \(0.617722\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 8.14214i − 0.260490i −0.991482 0.130245i \(-0.958424\pi\)
0.991482 0.130245i \(-0.0415764\pi\)
\(978\) 0 0
\(979\) 64.9706 2.07647
\(980\) 0 0
\(981\) −25.4558 −0.812743
\(982\) 0 0
\(983\) 42.0122i 1.33998i 0.742370 + 0.669990i \(0.233702\pi\)
−0.742370 + 0.669990i \(0.766298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.82843 −0.217131
\(990\) 0 0
\(991\) 25.3137 0.804116 0.402058 0.915614i \(-0.368295\pi\)
0.402058 + 0.915614i \(0.368295\pi\)
\(992\) 0 0
\(993\) 2.00000i 0.0634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 50.8406i − 1.61014i −0.593181 0.805069i \(-0.702128\pi\)
0.593181 0.805069i \(-0.297872\pi\)
\(998\) 0 0
\(999\) 8.24264 0.260786
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 4900.2.e.r.2549.3 4
5.2 odd 4 4900.2.a.x.1.2 2
5.3 odd 4 980.2.a.k.1.1 yes 2
5.4 even 2 inner 4900.2.e.r.2549.2 4
7.6 odd 2 4900.2.e.q.2549.2 4
15.8 even 4 8820.2.a.bl.1.2 2
20.3 even 4 3920.2.a.bo.1.2 2
35.3 even 12 980.2.i.l.961.1 4
35.13 even 4 980.2.a.j.1.2 2
35.18 odd 12 980.2.i.k.961.2 4
35.23 odd 12 980.2.i.k.361.2 4
35.27 even 4 4900.2.a.z.1.1 2
35.33 even 12 980.2.i.l.361.1 4
35.34 odd 2 4900.2.e.q.2549.3 4
105.83 odd 4 8820.2.a.bg.1.2 2
140.83 odd 4 3920.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.j.1.2 2 35.13 even 4
980.2.a.k.1.1 yes 2 5.3 odd 4
980.2.i.k.361.2 4 35.23 odd 12
980.2.i.k.961.2 4 35.18 odd 12
980.2.i.l.361.1 4 35.33 even 12
980.2.i.l.961.1 4 35.3 even 12
3920.2.a.bo.1.2 2 20.3 even 4
3920.2.a.bx.1.1 2 140.83 odd 4
4900.2.a.x.1.2 2 5.2 odd 4
4900.2.a.z.1.1 2 35.27 even 4
4900.2.e.q.2549.2 4 7.6 odd 2
4900.2.e.q.2549.3 4 35.34 odd 2
4900.2.e.r.2549.2 4 5.4 even 2 inner
4900.2.e.r.2549.3 4 1.1 even 1 trivial
8820.2.a.bg.1.2 2 105.83 odd 4
8820.2.a.bl.1.2 2 15.8 even 4