Properties

Label 4400.2.b.y.4049.4
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.y.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.30278i q^{3} -0.697224i q^{7} -2.30278 q^{9} +O(q^{10})\) \(q+2.30278i q^{3} -0.697224i q^{7} -2.30278 q^{9} +1.00000 q^{11} -5.00000i q^{13} +6.90833i q^{17} -1.00000 q^{19} +1.60555 q^{21} -7.30278i q^{23} +1.60555i q^{27} -0.908327 q^{29} -10.2111 q^{31} +2.30278i q^{33} +2.39445i q^{37} +11.5139 q^{39} -5.60555 q^{41} +7.21110i q^{43} +3.00000i q^{47} +6.51388 q^{49} -15.9083 q^{51} +1.30278i q^{53} -2.30278i q^{57} -14.2111 q^{59} -7.90833 q^{61} +1.60555i q^{63} +4.00000i q^{67} +16.8167 q^{69} +2.60555 q^{71} +7.90833i q^{73} -0.697224i q^{77} -10.9083 q^{79} -10.6056 q^{81} -3.51388i q^{83} -2.09167i q^{87} -1.69722 q^{89} -3.48612 q^{91} -23.5139i q^{93} +15.3028i q^{97} -2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 2.30278i 1.32951i 0.747062 + 0.664754i \(0.231464\pi\)
−0.747062 + 0.664754i \(0.768536\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.697224i − 0.263526i −0.991281 0.131763i \(-0.957936\pi\)
0.991281 0.131763i \(-0.0420638\pi\)
\(8\) 0 0
\(9\) −2.30278 −0.767592
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 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
\(17\) 6.90833i 1.67552i 0.546042 + 0.837758i \(0.316134\pi\)
−0.546042 + 0.837758i \(0.683866\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 0 0
\(23\) − 7.30278i − 1.52273i −0.648321 0.761367i \(-0.724529\pi\)
0.648321 0.761367i \(-0.275471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.60555i 0.308988i
\(28\) 0 0
\(29\) −0.908327 −0.168672 −0.0843360 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 0 0
\(31\) −10.2111 −1.83397 −0.916984 0.398924i \(-0.869384\pi\)
−0.916984 + 0.398924i \(0.869384\pi\)
\(32\) 0 0
\(33\) 2.30278i 0.400862i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.39445i 0.393645i 0.980439 + 0.196822i \(0.0630623\pi\)
−0.980439 + 0.196822i \(0.936938\pi\)
\(38\) 0 0
\(39\) 11.5139 1.84370
\(40\) 0 0
\(41\) −5.60555 −0.875440 −0.437720 0.899111i \(-0.644214\pi\)
−0.437720 + 0.899111i \(0.644214\pi\)
\(42\) 0 0
\(43\) 7.21110i 1.09968i 0.835269 + 0.549841i \(0.185312\pi\)
−0.835269 + 0.549841i \(0.814688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) 6.51388 0.930554
\(50\) 0 0
\(51\) −15.9083 −2.22761
\(52\) 0 0
\(53\) 1.30278i 0.178950i 0.995989 + 0.0894750i \(0.0285189\pi\)
−0.995989 + 0.0894750i \(0.971481\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.30278i − 0.305010i
\(58\) 0 0
\(59\) −14.2111 −1.85013 −0.925064 0.379811i \(-0.875989\pi\)
−0.925064 + 0.379811i \(0.875989\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 0 0
\(63\) 1.60555i 0.202280i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 16.8167 2.02449
\(70\) 0 0
\(71\) 2.60555 0.309222 0.154611 0.987975i \(-0.450588\pi\)
0.154611 + 0.987975i \(0.450588\pi\)
\(72\) 0 0
\(73\) 7.90833i 0.925600i 0.886463 + 0.462800i \(0.153155\pi\)
−0.886463 + 0.462800i \(0.846845\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.697224i − 0.0794561i
\(78\) 0 0
\(79\) −10.9083 −1.22728 −0.613641 0.789585i \(-0.710296\pi\)
−0.613641 + 0.789585i \(0.710296\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) − 3.51388i − 0.385698i −0.981228 0.192849i \(-0.938227\pi\)
0.981228 0.192849i \(-0.0617728\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.09167i − 0.224251i
\(88\) 0 0
\(89\) −1.69722 −0.179905 −0.0899527 0.995946i \(-0.528672\pi\)
−0.0899527 + 0.995946i \(0.528672\pi\)
\(90\) 0 0
\(91\) −3.48612 −0.365445
\(92\) 0 0
\(93\) − 23.5139i − 2.43828i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.3028i 1.55376i 0.629648 + 0.776881i \(0.283199\pi\)
−0.629648 + 0.776881i \(0.716801\pi\)
\(98\) 0 0
\(99\) −2.30278 −0.231438
\(100\) 0 0
\(101\) −0.513878 −0.0511328 −0.0255664 0.999673i \(-0.508139\pi\)
−0.0255664 + 0.999673i \(0.508139\pi\)
\(102\) 0 0
\(103\) 2.90833i 0.286566i 0.989682 + 0.143283i \(0.0457659\pi\)
−0.989682 + 0.143283i \(0.954234\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) −11.5139 −1.10283 −0.551415 0.834231i \(-0.685912\pi\)
−0.551415 + 0.834231i \(0.685912\pi\)
\(110\) 0 0
\(111\) −5.51388 −0.523354
\(112\) 0 0
\(113\) 10.8167i 1.01755i 0.860901 + 0.508773i \(0.169901\pi\)
−0.860901 + 0.508773i \(0.830099\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 11.5139i 1.06446i
\(118\) 0 0
\(119\) 4.81665 0.441542
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 12.9083i − 1.16390i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.11943i − 0.720483i −0.932859 0.360241i \(-0.882694\pi\)
0.932859 0.360241i \(-0.117306\pi\)
\(128\) 0 0
\(129\) −16.6056 −1.46204
\(130\) 0 0
\(131\) 9.90833 0.865695 0.432847 0.901467i \(-0.357509\pi\)
0.432847 + 0.901467i \(0.357509\pi\)
\(132\) 0 0
\(133\) 0.697224i 0.0604570i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.9083i − 1.10283i −0.834230 0.551416i \(-0.814088\pi\)
0.834230 0.551416i \(-0.185912\pi\)
\(138\) 0 0
\(139\) −6.21110 −0.526819 −0.263409 0.964684i \(-0.584847\pi\)
−0.263409 + 0.964684i \(0.584847\pi\)
\(140\) 0 0
\(141\) −6.90833 −0.581786
\(142\) 0 0
\(143\) − 5.00000i − 0.418121i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.0000i 1.23718i
\(148\) 0 0
\(149\) −17.2111 −1.40999 −0.704994 0.709213i \(-0.749050\pi\)
−0.704994 + 0.709213i \(0.749050\pi\)
\(150\) 0 0
\(151\) −0.816654 −0.0664583 −0.0332292 0.999448i \(-0.510579\pi\)
−0.0332292 + 0.999448i \(0.510579\pi\)
\(152\) 0 0
\(153\) − 15.9083i − 1.28611i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.2111i 1.53321i 0.642117 + 0.766606i \(0.278056\pi\)
−0.642117 + 0.766606i \(0.721944\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −5.09167 −0.401280
\(162\) 0 0
\(163\) 9.30278i 0.728650i 0.931272 + 0.364325i \(0.118700\pi\)
−0.931272 + 0.364325i \(0.881300\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 13.4222i − 1.03864i −0.854579 0.519321i \(-0.826185\pi\)
0.854579 0.519321i \(-0.173815\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 2.30278 0.176098
\(172\) 0 0
\(173\) 4.81665i 0.366203i 0.983094 + 0.183102i \(0.0586138\pi\)
−0.983094 + 0.183102i \(0.941386\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 32.7250i − 2.45976i
\(178\) 0 0
\(179\) 12.5139 0.935331 0.467666 0.883905i \(-0.345095\pi\)
0.467666 + 0.883905i \(0.345095\pi\)
\(180\) 0 0
\(181\) −19.9083 −1.47977 −0.739887 0.672731i \(-0.765121\pi\)
−0.739887 + 0.672731i \(0.765121\pi\)
\(182\) 0 0
\(183\) − 18.2111i − 1.34620i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.90833i 0.505187i
\(188\) 0 0
\(189\) 1.11943 0.0814265
\(190\) 0 0
\(191\) −10.3028 −0.745483 −0.372741 0.927935i \(-0.621582\pi\)
−0.372741 + 0.927935i \(0.621582\pi\)
\(192\) 0 0
\(193\) − 13.2111i − 0.950956i −0.879728 0.475478i \(-0.842275\pi\)
0.879728 0.475478i \(-0.157725\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.3028i 0.947784i 0.880583 + 0.473892i \(0.157151\pi\)
−0.880583 + 0.473892i \(0.842849\pi\)
\(198\) 0 0
\(199\) −6.48612 −0.459789 −0.229894 0.973216i \(-0.573838\pi\)
−0.229894 + 0.973216i \(0.573838\pi\)
\(200\) 0 0
\(201\) −9.21110 −0.649701
\(202\) 0 0
\(203\) 0.633308i 0.0444495i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 16.8167i 1.16884i
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 25.2389 1.73751 0.868757 0.495238i \(-0.164919\pi\)
0.868757 + 0.495238i \(0.164919\pi\)
\(212\) 0 0
\(213\) 6.00000i 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.11943i 0.483298i
\(218\) 0 0
\(219\) −18.2111 −1.23059
\(220\) 0 0
\(221\) 34.5416 2.32352
\(222\) 0 0
\(223\) − 22.6333i − 1.51564i −0.652465 0.757819i \(-0.726265\pi\)
0.652465 0.757819i \(-0.273735\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.69722i − 0.112649i −0.998413 0.0563244i \(-0.982062\pi\)
0.998413 0.0563244i \(-0.0179381\pi\)
\(228\) 0 0
\(229\) 18.7250 1.23738 0.618691 0.785635i \(-0.287663\pi\)
0.618691 + 0.785635i \(0.287663\pi\)
\(230\) 0 0
\(231\) 1.60555 0.105638
\(232\) 0 0
\(233\) − 15.9083i − 1.04219i −0.853499 0.521095i \(-0.825524\pi\)
0.853499 0.521095i \(-0.174476\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 25.1194i − 1.63168i
\(238\) 0 0
\(239\) 21.1194 1.36610 0.683051 0.730371i \(-0.260653\pi\)
0.683051 + 0.730371i \(0.260653\pi\)
\(240\) 0 0
\(241\) 21.9361 1.41303 0.706514 0.707699i \(-0.250267\pi\)
0.706514 + 0.707699i \(0.250267\pi\)
\(242\) 0 0
\(243\) − 19.6056i − 1.25770i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000i 0.318142i
\(248\) 0 0
\(249\) 8.09167 0.512789
\(250\) 0 0
\(251\) −6.90833 −0.436050 −0.218025 0.975943i \(-0.569961\pi\)
−0.218025 + 0.975943i \(0.569961\pi\)
\(252\) 0 0
\(253\) − 7.30278i − 0.459122i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 1.66947 0.103736
\(260\) 0 0
\(261\) 2.09167 0.129471
\(262\) 0 0
\(263\) − 22.8167i − 1.40694i −0.710727 0.703468i \(-0.751634\pi\)
0.710727 0.703468i \(-0.248366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 3.90833i − 0.239186i
\(268\) 0 0
\(269\) −8.72498 −0.531971 −0.265986 0.963977i \(-0.585697\pi\)
−0.265986 + 0.963977i \(0.585697\pi\)
\(270\) 0 0
\(271\) 0.211103 0.0128236 0.00641178 0.999979i \(-0.497959\pi\)
0.00641178 + 0.999979i \(0.497959\pi\)
\(272\) 0 0
\(273\) − 8.02776i − 0.485862i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.3944i 0.864879i 0.901663 + 0.432439i \(0.142347\pi\)
−0.901663 + 0.432439i \(0.857653\pi\)
\(278\) 0 0
\(279\) 23.5139 1.40774
\(280\) 0 0
\(281\) −1.18335 −0.0705925 −0.0352963 0.999377i \(-0.511237\pi\)
−0.0352963 + 0.999377i \(0.511237\pi\)
\(282\) 0 0
\(283\) 6.30278i 0.374661i 0.982297 + 0.187331i \(0.0599836\pi\)
−0.982297 + 0.187331i \(0.940016\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.90833i 0.230701i
\(288\) 0 0
\(289\) −30.7250 −1.80735
\(290\) 0 0
\(291\) −35.2389 −2.06574
\(292\) 0 0
\(293\) − 0.788897i − 0.0460879i −0.999734 0.0230439i \(-0.992664\pi\)
0.999734 0.0230439i \(-0.00733576\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.60555i 0.0931635i
\(298\) 0 0
\(299\) −36.5139 −2.11165
\(300\) 0 0
\(301\) 5.02776 0.289795
\(302\) 0 0
\(303\) − 1.18335i − 0.0679815i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.9083i 0.965009i 0.875893 + 0.482505i \(0.160273\pi\)
−0.875893 + 0.482505i \(0.839727\pi\)
\(308\) 0 0
\(309\) −6.69722 −0.380992
\(310\) 0 0
\(311\) −4.81665 −0.273127 −0.136564 0.990631i \(-0.543606\pi\)
−0.136564 + 0.990631i \(0.543606\pi\)
\(312\) 0 0
\(313\) − 0.183346i − 0.0103633i −0.999987 0.00518167i \(-0.998351\pi\)
0.999987 0.00518167i \(-0.00164938\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 0.908327i − 0.0510167i −0.999675 0.0255084i \(-0.991880\pi\)
0.999675 0.0255084i \(-0.00812044\pi\)
\(318\) 0 0
\(319\) −0.908327 −0.0508565
\(320\) 0 0
\(321\) −6.90833 −0.385585
\(322\) 0 0
\(323\) − 6.90833i − 0.384390i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 26.5139i − 1.46622i
\(328\) 0 0
\(329\) 2.09167 0.115318
\(330\) 0 0
\(331\) 21.6056 1.18755 0.593774 0.804632i \(-0.297637\pi\)
0.593774 + 0.804632i \(0.297637\pi\)
\(332\) 0 0
\(333\) − 5.51388i − 0.302159i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 30.8444i − 1.68020i −0.542430 0.840101i \(-0.682496\pi\)
0.542430 0.840101i \(-0.317504\pi\)
\(338\) 0 0
\(339\) −24.9083 −1.35283
\(340\) 0 0
\(341\) −10.2111 −0.552962
\(342\) 0 0
\(343\) − 9.42221i − 0.508751i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.5139i 0.671780i 0.941901 + 0.335890i \(0.109037\pi\)
−0.941901 + 0.335890i \(0.890963\pi\)
\(348\) 0 0
\(349\) 5.18335 0.277458 0.138729 0.990330i \(-0.455698\pi\)
0.138729 + 0.990330i \(0.455698\pi\)
\(350\) 0 0
\(351\) 8.02776 0.428490
\(352\) 0 0
\(353\) − 18.6333i − 0.991751i −0.868394 0.495875i \(-0.834847\pi\)
0.868394 0.495875i \(-0.165153\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 11.0917i 0.587034i
\(358\) 0 0
\(359\) 0.788897 0.0416364 0.0208182 0.999783i \(-0.493373\pi\)
0.0208182 + 0.999783i \(0.493373\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 2.30278i 0.120864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 20.6972i 1.08039i 0.841541 + 0.540193i \(0.181649\pi\)
−0.841541 + 0.540193i \(0.818351\pi\)
\(368\) 0 0
\(369\) 12.9083 0.671981
\(370\) 0 0
\(371\) 0.908327 0.0471580
\(372\) 0 0
\(373\) − 27.4222i − 1.41987i −0.704268 0.709934i \(-0.748725\pi\)
0.704268 0.709934i \(-0.251275\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.54163i 0.233906i
\(378\) 0 0
\(379\) 3.18335 0.163518 0.0817588 0.996652i \(-0.473946\pi\)
0.0817588 + 0.996652i \(0.473946\pi\)
\(380\) 0 0
\(381\) 18.6972 0.957888
\(382\) 0 0
\(383\) 21.6333i 1.10541i 0.833377 + 0.552705i \(0.186404\pi\)
−0.833377 + 0.552705i \(0.813596\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 16.6056i − 0.844108i
\(388\) 0 0
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 50.4500 2.55136
\(392\) 0 0
\(393\) 22.8167i 1.15095i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 21.6972i 1.08895i 0.838776 + 0.544476i \(0.183272\pi\)
−0.838776 + 0.544476i \(0.816728\pi\)
\(398\) 0 0
\(399\) −1.60555 −0.0803781
\(400\) 0 0
\(401\) −12.7889 −0.638647 −0.319324 0.947646i \(-0.603456\pi\)
−0.319324 + 0.947646i \(0.603456\pi\)
\(402\) 0 0
\(403\) 51.0555i 2.54326i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.39445i 0.118688i
\(408\) 0 0
\(409\) 6.21110 0.307119 0.153560 0.988139i \(-0.450926\pi\)
0.153560 + 0.988139i \(0.450926\pi\)
\(410\) 0 0
\(411\) 29.7250 1.46623
\(412\) 0 0
\(413\) 9.90833i 0.487557i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 14.3028i − 0.700410i
\(418\) 0 0
\(419\) 6.39445 0.312389 0.156195 0.987726i \(-0.450077\pi\)
0.156195 + 0.987726i \(0.450077\pi\)
\(420\) 0 0
\(421\) 0.697224 0.0339806 0.0169903 0.999856i \(-0.494592\pi\)
0.0169903 + 0.999856i \(0.494592\pi\)
\(422\) 0 0
\(423\) − 6.90833i − 0.335894i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.51388i 0.266835i
\(428\) 0 0
\(429\) 11.5139 0.555895
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) − 5.00000i − 0.240285i −0.992757 0.120142i \(-0.961665\pi\)
0.992757 0.120142i \(-0.0383351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.30278i 0.349339i
\(438\) 0 0
\(439\) 24.3028 1.15991 0.579954 0.814649i \(-0.303070\pi\)
0.579954 + 0.814649i \(0.303070\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) − 8.60555i − 0.408862i −0.978881 0.204431i \(-0.934466\pi\)
0.978881 0.204431i \(-0.0655344\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 39.6333i − 1.87459i
\(448\) 0 0
\(449\) −23.4861 −1.10838 −0.554189 0.832391i \(-0.686972\pi\)
−0.554189 + 0.832391i \(0.686972\pi\)
\(450\) 0 0
\(451\) −5.60555 −0.263955
\(452\) 0 0
\(453\) − 1.88057i − 0.0883569i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 20.6972i − 0.968175i −0.875020 0.484088i \(-0.839152\pi\)
0.875020 0.484088i \(-0.160848\pi\)
\(458\) 0 0
\(459\) −11.0917 −0.517715
\(460\) 0 0
\(461\) 32.2111 1.50022 0.750110 0.661313i \(-0.230000\pi\)
0.750110 + 0.661313i \(0.230000\pi\)
\(462\) 0 0
\(463\) 11.7889i 0.547877i 0.961747 + 0.273938i \(0.0883264\pi\)
−0.961747 + 0.273938i \(0.911674\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.6333i 0.862247i 0.902293 + 0.431123i \(0.141883\pi\)
−0.902293 + 0.431123i \(0.858117\pi\)
\(468\) 0 0
\(469\) 2.78890 0.128779
\(470\) 0 0
\(471\) −44.2389 −2.03842
\(472\) 0 0
\(473\) 7.21110i 0.331567i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 3.00000i − 0.137361i
\(478\) 0 0
\(479\) −34.8167 −1.59081 −0.795407 0.606076i \(-0.792743\pi\)
−0.795407 + 0.606076i \(0.792743\pi\)
\(480\) 0 0
\(481\) 11.9722 0.545887
\(482\) 0 0
\(483\) − 11.7250i − 0.533505i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.21110i − 0.190823i −0.995438 0.0954116i \(-0.969583\pi\)
0.995438 0.0954116i \(-0.0304167\pi\)
\(488\) 0 0
\(489\) −21.4222 −0.968746
\(490\) 0 0
\(491\) 9.78890 0.441767 0.220883 0.975300i \(-0.429106\pi\)
0.220883 + 0.975300i \(0.429106\pi\)
\(492\) 0 0
\(493\) − 6.27502i − 0.282613i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.81665i − 0.0814881i
\(498\) 0 0
\(499\) −3.48612 −0.156060 −0.0780301 0.996951i \(-0.524863\pi\)
−0.0780301 + 0.996951i \(0.524863\pi\)
\(500\) 0 0
\(501\) 30.9083 1.38088
\(502\) 0 0
\(503\) 9.39445i 0.418878i 0.977822 + 0.209439i \(0.0671637\pi\)
−0.977822 + 0.209439i \(0.932836\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 27.6333i − 1.22724i
\(508\) 0 0
\(509\) 22.6972 1.00604 0.503018 0.864276i \(-0.332223\pi\)
0.503018 + 0.864276i \(0.332223\pi\)
\(510\) 0 0
\(511\) 5.51388 0.243920
\(512\) 0 0
\(513\) − 1.60555i − 0.0708868i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.00000i 0.131940i
\(518\) 0 0
\(519\) −11.0917 −0.486870
\(520\) 0 0
\(521\) −41.4500 −1.81596 −0.907978 0.419018i \(-0.862374\pi\)
−0.907978 + 0.419018i \(0.862374\pi\)
\(522\) 0 0
\(523\) − 32.4222i − 1.41772i −0.705347 0.708862i \(-0.749209\pi\)
0.705347 0.708862i \(-0.250791\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 70.5416i − 3.07284i
\(528\) 0 0
\(529\) −30.3305 −1.31872
\(530\) 0 0
\(531\) 32.7250 1.42014
\(532\) 0 0
\(533\) 28.0278i 1.21402i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 28.8167i 1.24353i
\(538\) 0 0
\(539\) 6.51388 0.280573
\(540\) 0 0
\(541\) 25.7250 1.10600 0.553002 0.833180i \(-0.313482\pi\)
0.553002 + 0.833180i \(0.313482\pi\)
\(542\) 0 0
\(543\) − 45.8444i − 1.96737i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.11943i 0.304405i 0.988349 + 0.152202i \(0.0486366\pi\)
−0.988349 + 0.152202i \(0.951363\pi\)
\(548\) 0 0
\(549\) 18.2111 0.777231
\(550\) 0 0
\(551\) 0.908327 0.0386960
\(552\) 0 0
\(553\) 7.60555i 0.323421i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.4222i 0.822945i 0.911422 + 0.411473i \(0.134985\pi\)
−0.911422 + 0.411473i \(0.865015\pi\)
\(558\) 0 0
\(559\) 36.0555 1.52499
\(560\) 0 0
\(561\) −15.9083 −0.671650
\(562\) 0 0
\(563\) 8.09167i 0.341023i 0.985356 + 0.170512i \(0.0545421\pi\)
−0.985356 + 0.170512i \(0.945458\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.39445i 0.310538i
\(568\) 0 0
\(569\) 46.1472 1.93459 0.967295 0.253653i \(-0.0816321\pi\)
0.967295 + 0.253653i \(0.0816321\pi\)
\(570\) 0 0
\(571\) −22.3305 −0.934504 −0.467252 0.884124i \(-0.654756\pi\)
−0.467252 + 0.884124i \(0.654756\pi\)
\(572\) 0 0
\(573\) − 23.7250i − 0.991125i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 44.3583i 1.84666i 0.384008 + 0.923330i \(0.374544\pi\)
−0.384008 + 0.923330i \(0.625456\pi\)
\(578\) 0 0
\(579\) 30.4222 1.26430
\(580\) 0 0
\(581\) −2.44996 −0.101642
\(582\) 0 0
\(583\) 1.30278i 0.0539555i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 16.5416i − 0.682746i −0.939928 0.341373i \(-0.889108\pi\)
0.939928 0.341373i \(-0.110892\pi\)
\(588\) 0 0
\(589\) 10.2111 0.420741
\(590\) 0 0
\(591\) −30.6333 −1.26009
\(592\) 0 0
\(593\) 6.39445i 0.262589i 0.991343 + 0.131294i \(0.0419133\pi\)
−0.991343 + 0.131294i \(0.958087\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 14.9361i − 0.611293i
\(598\) 0 0
\(599\) −24.9083 −1.01773 −0.508863 0.860847i \(-0.669934\pi\)
−0.508863 + 0.860847i \(0.669934\pi\)
\(600\) 0 0
\(601\) −1.90833 −0.0778423 −0.0389211 0.999242i \(-0.512392\pi\)
−0.0389211 + 0.999242i \(0.512392\pi\)
\(602\) 0 0
\(603\) − 9.21110i − 0.375105i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.21110i − 0.292690i −0.989234 0.146345i \(-0.953249\pi\)
0.989234 0.146345i \(-0.0467509\pi\)
\(608\) 0 0
\(609\) −1.45837 −0.0590959
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) 15.8806i 0.641410i 0.947179 + 0.320705i \(0.103920\pi\)
−0.947179 + 0.320705i \(0.896080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.39445i 0.136655i 0.997663 + 0.0683277i \(0.0217663\pi\)
−0.997663 + 0.0683277i \(0.978234\pi\)
\(618\) 0 0
\(619\) −11.4222 −0.459097 −0.229549 0.973297i \(-0.573725\pi\)
−0.229549 + 0.973297i \(0.573725\pi\)
\(620\) 0 0
\(621\) 11.7250 0.470507
\(622\) 0 0
\(623\) 1.18335i 0.0474098i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 2.30278i − 0.0919640i
\(628\) 0 0
\(629\) −16.5416 −0.659558
\(630\) 0 0
\(631\) −6.93608 −0.276121 −0.138061 0.990424i \(-0.544087\pi\)
−0.138061 + 0.990424i \(0.544087\pi\)
\(632\) 0 0
\(633\) 58.1194i 2.31004i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 32.5694i − 1.29045i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 27.7889 1.09760 0.548798 0.835955i \(-0.315086\pi\)
0.548798 + 0.835955i \(0.315086\pi\)
\(642\) 0 0
\(643\) − 22.0000i − 0.867595i −0.901010 0.433798i \(-0.857173\pi\)
0.901010 0.433798i \(-0.142827\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.2389i 1.30675i 0.757033 + 0.653377i \(0.226648\pi\)
−0.757033 + 0.653377i \(0.773352\pi\)
\(648\) 0 0
\(649\) −14.2111 −0.557835
\(650\) 0 0
\(651\) −16.3944 −0.642549
\(652\) 0 0
\(653\) − 6.11943i − 0.239472i −0.992806 0.119736i \(-0.961795\pi\)
0.992806 0.119736i \(-0.0382048\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 18.2111i − 0.710483i
\(658\) 0 0
\(659\) −30.9083 −1.20402 −0.602009 0.798489i \(-0.705633\pi\)
−0.602009 + 0.798489i \(0.705633\pi\)
\(660\) 0 0
\(661\) −8.81665 −0.342928 −0.171464 0.985190i \(-0.554850\pi\)
−0.171464 + 0.985190i \(0.554850\pi\)
\(662\) 0 0
\(663\) 79.5416i 3.08914i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.63331i 0.256843i
\(668\) 0 0
\(669\) 52.1194 2.01505
\(670\) 0 0
\(671\) −7.90833 −0.305298
\(672\) 0 0
\(673\) − 30.0278i − 1.15748i −0.815510 0.578742i \(-0.803544\pi\)
0.815510 0.578742i \(-0.196456\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.2389i 0.931575i 0.884897 + 0.465788i \(0.154229\pi\)
−0.884897 + 0.465788i \(0.845771\pi\)
\(678\) 0 0
\(679\) 10.6695 0.409457
\(680\) 0 0
\(681\) 3.90833 0.149767
\(682\) 0 0
\(683\) 47.8444i 1.83072i 0.402642 + 0.915358i \(0.368092\pi\)
−0.402642 + 0.915358i \(0.631908\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 43.1194i 1.64511i
\(688\) 0 0
\(689\) 6.51388 0.248159
\(690\) 0 0
\(691\) −27.5416 −1.04773 −0.523867 0.851800i \(-0.675511\pi\)
−0.523867 + 0.851800i \(0.675511\pi\)
\(692\) 0 0
\(693\) 1.60555i 0.0609898i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 38.7250i − 1.46681i
\(698\) 0 0
\(699\) 36.6333 1.38560
\(700\) 0 0
\(701\) −41.2111 −1.55652 −0.778261 0.627941i \(-0.783898\pi\)
−0.778261 + 0.627941i \(0.783898\pi\)
\(702\) 0 0
\(703\) − 2.39445i − 0.0903083i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.358288i 0.0134748i
\(708\) 0 0
\(709\) 31.6333 1.18801 0.594007 0.804460i \(-0.297545\pi\)
0.594007 + 0.804460i \(0.297545\pi\)
\(710\) 0 0
\(711\) 25.1194 0.942052
\(712\) 0 0
\(713\) 74.5694i 2.79265i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 48.6333i 1.81624i
\(718\) 0 0
\(719\) 7.18335 0.267894 0.133947 0.990989i \(-0.457235\pi\)
0.133947 + 0.990989i \(0.457235\pi\)
\(720\) 0 0
\(721\) 2.02776 0.0755176
\(722\) 0 0
\(723\) 50.5139i 1.87863i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 39.3305i 1.45869i 0.684147 + 0.729344i \(0.260175\pi\)
−0.684147 + 0.729344i \(0.739825\pi\)
\(728\) 0 0
\(729\) 13.3305 0.493723
\(730\) 0 0
\(731\) −49.8167 −1.84254
\(732\) 0 0
\(733\) − 19.6056i − 0.724148i −0.932149 0.362074i \(-0.882069\pi\)
0.932149 0.362074i \(-0.117931\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000i 0.147342i
\(738\) 0 0
\(739\) 35.1194 1.29189 0.645945 0.763384i \(-0.276464\pi\)
0.645945 + 0.763384i \(0.276464\pi\)
\(740\) 0 0
\(741\) −11.5139 −0.422973
\(742\) 0 0
\(743\) 40.6972i 1.49304i 0.665365 + 0.746518i \(0.268276\pi\)
−0.665365 + 0.746518i \(0.731724\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.09167i 0.296059i
\(748\) 0 0
\(749\) 2.09167 0.0764281
\(750\) 0 0
\(751\) 45.3305 1.65413 0.827067 0.562103i \(-0.190008\pi\)
0.827067 + 0.562103i \(0.190008\pi\)
\(752\) 0 0
\(753\) − 15.9083i − 0.579732i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 49.0555i 1.78295i 0.453067 + 0.891476i \(0.350330\pi\)
−0.453067 + 0.891476i \(0.649670\pi\)
\(758\) 0 0
\(759\) 16.8167 0.610406
\(760\) 0 0
\(761\) −13.5778 −0.492195 −0.246097 0.969245i \(-0.579148\pi\)
−0.246097 + 0.969245i \(0.579148\pi\)
\(762\) 0 0
\(763\) 8.02776i 0.290624i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 71.0555i 2.56567i
\(768\) 0 0
\(769\) 5.18335 0.186916 0.0934581 0.995623i \(-0.470208\pi\)
0.0934581 + 0.995623i \(0.470208\pi\)
\(770\) 0 0
\(771\) 41.4500 1.49278
\(772\) 0 0
\(773\) − 3.11943i − 0.112198i −0.998425 0.0560990i \(-0.982134\pi\)
0.998425 0.0560990i \(-0.0178663\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.84441i 0.137917i
\(778\) 0 0
\(779\) 5.60555 0.200840
\(780\) 0 0
\(781\) 2.60555 0.0932340
\(782\) 0 0
\(783\) − 1.45837i − 0.0521177i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 10.2111i − 0.363986i −0.983300 0.181993i \(-0.941745\pi\)
0.983300 0.181993i \(-0.0582549\pi\)
\(788\) 0 0
\(789\) 52.5416 1.87053
\(790\) 0 0
\(791\) 7.54163 0.268150
\(792\) 0 0
\(793\) 39.5416i 1.40416i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.51388i 0.124468i 0.998062 + 0.0622340i \(0.0198225\pi\)
−0.998062 + 0.0622340i \(0.980178\pi\)
\(798\) 0 0
\(799\) −20.7250 −0.733197
\(800\) 0 0
\(801\) 3.90833 0.138094
\(802\) 0 0
\(803\) 7.90833i 0.279079i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 20.0917i − 0.707260i
\(808\) 0 0
\(809\) −39.6333 −1.39343 −0.696716 0.717347i \(-0.745356\pi\)
−0.696716 + 0.717347i \(0.745356\pi\)
\(810\) 0 0
\(811\) −38.8722 −1.36499 −0.682493 0.730892i \(-0.739104\pi\)
−0.682493 + 0.730892i \(0.739104\pi\)
\(812\) 0 0
\(813\) 0.486122i 0.0170490i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.21110i − 0.252285i
\(818\) 0 0
\(819\) 8.02776 0.280513
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) 18.4222i 0.642158i 0.947052 + 0.321079i \(0.104045\pi\)
−0.947052 + 0.321079i \(0.895955\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 13.8167i − 0.480452i −0.970717 0.240226i \(-0.922778\pi\)
0.970717 0.240226i \(-0.0772216\pi\)
\(828\) 0 0
\(829\) −29.7527 −1.03336 −0.516678 0.856180i \(-0.672831\pi\)
−0.516678 + 0.856180i \(0.672831\pi\)
\(830\) 0 0
\(831\) −33.1472 −1.14986
\(832\) 0 0
\(833\) 45.0000i 1.55916i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 16.3944i − 0.566675i
\(838\) 0 0
\(839\) −9.11943 −0.314838 −0.157419 0.987532i \(-0.550317\pi\)
−0.157419 + 0.987532i \(0.550317\pi\)
\(840\) 0 0
\(841\) −28.1749 −0.971550
\(842\) 0 0
\(843\) − 2.72498i − 0.0938533i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 0.697224i − 0.0239569i
\(848\) 0 0
\(849\) −14.5139 −0.498115
\(850\) 0 0
\(851\) 17.4861 0.599417
\(852\) 0 0
\(853\) 12.7250i 0.435695i 0.975983 + 0.217848i \(0.0699035\pi\)
−0.975983 + 0.217848i \(0.930096\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 3.00000i − 0.102478i −0.998686 0.0512390i \(-0.983683\pi\)
0.998686 0.0512390i \(-0.0163170\pi\)
\(858\) 0 0
\(859\) 41.3944 1.41236 0.706180 0.708032i \(-0.250417\pi\)
0.706180 + 0.708032i \(0.250417\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) − 12.3944i − 0.421912i −0.977496 0.210956i \(-0.932342\pi\)
0.977496 0.210956i \(-0.0676577\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 70.7527i − 2.40289i
\(868\) 0 0
\(869\) −10.9083 −0.370040
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) − 35.2389i − 1.19265i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 20.0000i 0.675352i 0.941262 + 0.337676i \(0.109641\pi\)
−0.941262 + 0.337676i \(0.890359\pi\)
\(878\) 0 0
\(879\) 1.81665 0.0612742
\(880\) 0 0
\(881\) 19.5416 0.658374 0.329187 0.944265i \(-0.393225\pi\)
0.329187 + 0.944265i \(0.393225\pi\)
\(882\) 0 0
\(883\) 52.4500i 1.76508i 0.470236 + 0.882541i \(0.344169\pi\)
−0.470236 + 0.882541i \(0.655831\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.23886i 0.108750i 0.998521 + 0.0543751i \(0.0173167\pi\)
−0.998521 + 0.0543751i \(0.982683\pi\)
\(888\) 0 0
\(889\) −5.66106 −0.189866
\(890\) 0 0
\(891\) −10.6056 −0.355299
\(892\) 0 0
\(893\) − 3.00000i − 0.100391i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 84.0833i − 2.80746i
\(898\) 0 0
\(899\) 9.27502 0.309339
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 11.5778i 0.385285i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) 1.18335 0.0392491
\(910\) 0 0
\(911\) 24.7889 0.821293 0.410646 0.911795i \(-0.365303\pi\)
0.410646 + 0.911795i \(0.365303\pi\)
\(912\) 0 0
\(913\) − 3.51388i − 0.116292i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6.90833i − 0.228133i
\(918\) 0 0
\(919\) 26.7889 0.883684 0.441842 0.897093i \(-0.354325\pi\)
0.441842 + 0.897093i \(0.354325\pi\)
\(920\) 0 0
\(921\) −38.9361 −1.28299
\(922\) 0 0
\(923\) − 13.0278i − 0.428814i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 6.69722i − 0.219966i
\(928\) 0 0
\(929\) −53.6056 −1.75874 −0.879371 0.476138i \(-0.842036\pi\)
−0.879371 + 0.476138i \(0.842036\pi\)
\(930\) 0 0
\(931\) −6.51388 −0.213484
\(932\) 0 0
\(933\) − 11.0917i − 0.363125i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 9.21110i − 0.300914i −0.988617 0.150457i \(-0.951926\pi\)
0.988617 0.150457i \(-0.0480744\pi\)
\(938\) 0 0
\(939\) 0.422205 0.0137781
\(940\) 0 0
\(941\) 59.6056 1.94309 0.971543 0.236864i \(-0.0761197\pi\)
0.971543 + 0.236864i \(0.0761197\pi\)
\(942\) 0 0
\(943\) 40.9361i 1.33306i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.63331i − 0.215554i −0.994175 0.107777i \(-0.965627\pi\)
0.994175 0.107777i \(-0.0343732\pi\)
\(948\) 0 0
\(949\) 39.5416 1.28358
\(950\) 0 0
\(951\) 2.09167 0.0678271
\(952\) 0 0
\(953\) − 37.2666i − 1.20718i −0.797293 0.603592i \(-0.793736\pi\)
0.797293 0.603592i \(-0.206264\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 2.09167i − 0.0676142i
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) 73.2666 2.36344
\(962\) 0 0
\(963\) − 6.90833i − 0.222618i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 14.9083i − 0.479419i −0.970845 0.239710i \(-0.922948\pi\)
0.970845 0.239710i \(-0.0770523\pi\)
\(968\) 0 0
\(969\) 15.9083 0.511049
\(970\) 0 0
\(971\) −45.3583 −1.45562 −0.727808 0.685781i \(-0.759461\pi\)
−0.727808 + 0.685781i \(0.759461\pi\)
\(972\) 0 0
\(973\) 4.33053i 0.138830i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 52.0278i − 1.66452i −0.554389 0.832258i \(-0.687048\pi\)
0.554389 0.832258i \(-0.312952\pi\)
\(978\) 0 0
\(979\) −1.69722 −0.0542435
\(980\) 0 0
\(981\) 26.5139 0.846523
\(982\) 0 0
\(983\) 8.84441i 0.282093i 0.990003 + 0.141046i \(0.0450467\pi\)
−0.990003 + 0.141046i \(0.954953\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.81665i 0.153316i
\(988\) 0 0
\(989\) 52.6611 1.67452
\(990\) 0 0
\(991\) 16.9083 0.537111 0.268555 0.963264i \(-0.413454\pi\)
0.268555 + 0.963264i \(0.413454\pi\)
\(992\) 0 0
\(993\) 49.7527i 1.57886i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.7250i 1.47979i 0.672720 + 0.739897i \(0.265126\pi\)
−0.672720 + 0.739897i \(0.734874\pi\)
\(998\) 0 0
\(999\) −3.84441 −0.121632
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 4400.2.b.y.4049.4 4
4.3 odd 2 275.2.b.c.199.2 4
5.2 odd 4 4400.2.a.bs.1.2 2
5.3 odd 4 4400.2.a.bh.1.1 2
5.4 even 2 inner 4400.2.b.y.4049.1 4
12.11 even 2 2475.2.c.k.199.3 4
20.3 even 4 275.2.a.f.1.1 yes 2
20.7 even 4 275.2.a.e.1.2 2
20.19 odd 2 275.2.b.c.199.3 4
60.23 odd 4 2475.2.a.o.1.2 2
60.47 odd 4 2475.2.a.t.1.1 2
60.59 even 2 2475.2.c.k.199.2 4
220.43 odd 4 3025.2.a.h.1.2 2
220.87 odd 4 3025.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 20.7 even 4
275.2.a.f.1.1 yes 2 20.3 even 4
275.2.b.c.199.2 4 4.3 odd 2
275.2.b.c.199.3 4 20.19 odd 2
2475.2.a.o.1.2 2 60.23 odd 4
2475.2.a.t.1.1 2 60.47 odd 4
2475.2.c.k.199.2 4 60.59 even 2
2475.2.c.k.199.3 4 12.11 even 2
3025.2.a.h.1.2 2 220.43 odd 4
3025.2.a.n.1.1 2 220.87 odd 4
4400.2.a.bh.1.1 2 5.3 odd 4
4400.2.a.bs.1.2 2 5.2 odd 4
4400.2.b.y.4049.1 4 5.4 even 2 inner
4400.2.b.y.4049.4 4 1.1 even 1 trivial