Properties

Label 4400.2.a.bs.1.2
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +O(q^{10})\) \(q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.00000 q^{13} -6.90833 q^{17} +1.00000 q^{19} +1.60555 q^{21} -7.30278 q^{23} -1.60555 q^{27} +0.908327 q^{29} -10.2111 q^{31} +2.30278 q^{33} -2.39445 q^{37} -11.5139 q^{39} -5.60555 q^{41} +7.21110 q^{43} -3.00000 q^{47} -6.51388 q^{49} -15.9083 q^{51} +1.30278 q^{53} +2.30278 q^{57} +14.2111 q^{59} -7.90833 q^{61} +1.60555 q^{63} -4.00000 q^{67} -16.8167 q^{69} +2.60555 q^{71} +7.90833 q^{73} +0.697224 q^{77} +10.9083 q^{79} -10.6056 q^{81} -3.51388 q^{83} +2.09167 q^{87} +1.69722 q^{89} -3.48612 q^{91} -23.5139 q^{93} -15.3028 q^{97} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + q^{9} + O(q^{10}) \) \( 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51} - q^{53} + q^{57} + 14 q^{59} - 5 q^{61} - 4 q^{63} - 8 q^{67} - 12 q^{69} - 2 q^{71} + 5 q^{73} + 5 q^{77} + 11 q^{79} - 14 q^{81} + 11 q^{83} + 15 q^{87} + 7 q^{89} - 25 q^{91} - 29 q^{93} - 27 q^{97} + q^{99} + O(q^{100}) \)

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
\(3\) 2.30278 1.32951 0.664754 0.747062i \(-0.268536\pi\)
0.664754 + 0.747062i \(0.268536\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.697224 0.263526 0.131763 0.991281i \(-0.457936\pi\)
0.131763 + 0.991281i \(0.457936\pi\)
\(8\) 0 0
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.90833 −1.67552 −0.837758 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 0 0
\(23\) −7.30278 −1.52273 −0.761367 0.648321i \(-0.775471\pi\)
−0.761367 + 0.648321i \(0.775471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.60555 −0.308988
\(28\) 0 0
\(29\) 0.908327 0.168672 0.0843360 0.996437i \(-0.473123\pi\)
0.0843360 + 0.996437i \(0.473123\pi\)
\(30\) 0 0
\(31\) −10.2111 −1.83397 −0.916984 0.398924i \(-0.869384\pi\)
−0.916984 + 0.398924i \(0.869384\pi\)
\(32\) 0 0
\(33\) 2.30278 0.400862
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.39445 −0.393645 −0.196822 0.980439i \(-0.563062\pi\)
−0.196822 + 0.980439i \(0.563062\pi\)
\(38\) 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
\(43\) 7.21110 1.09968 0.549841 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) −15.9083 −2.22761
\(52\) 0 0
\(53\) 1.30278 0.178950 0.0894750 0.995989i \(-0.471481\pi\)
0.0894750 + 0.995989i \(0.471481\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.30278 0.305010
\(58\) 0 0
\(59\) 14.2111 1.85013 0.925064 0.379811i \(-0.124011\pi\)
0.925064 + 0.379811i \(0.124011\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 0 0
\(63\) 1.60555 0.202280
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 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
\(73\) 7.90833 0.925600 0.462800 0.886463i \(-0.346845\pi\)
0.462800 + 0.886463i \(0.346845\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.697224 0.0794561
\(78\) 0 0
\(79\) 10.9083 1.22728 0.613641 0.789585i \(-0.289704\pi\)
0.613641 + 0.789585i \(0.289704\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) −3.51388 −0.385698 −0.192849 0.981228i \(-0.561773\pi\)
−0.192849 + 0.981228i \(0.561773\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.09167 0.224251
\(88\) 0 0
\(89\) 1.69722 0.179905 0.0899527 0.995946i \(-0.471328\pi\)
0.0899527 + 0.995946i \(0.471328\pi\)
\(90\) 0 0
\(91\) −3.48612 −0.365445
\(92\) 0 0
\(93\) −23.5139 −2.43828
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.3028 −1.55376 −0.776881 0.629648i \(-0.783199\pi\)
−0.776881 + 0.629648i \(0.783199\pi\)
\(98\) 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
\(103\) 2.90833 0.286566 0.143283 0.989682i \(-0.454234\pi\)
0.143283 + 0.989682i \(0.454234\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) 11.5139 1.10283 0.551415 0.834231i \(-0.314088\pi\)
0.551415 + 0.834231i \(0.314088\pi\)
\(110\) 0 0
\(111\) −5.51388 −0.523354
\(112\) 0 0
\(113\) 10.8167 1.01755 0.508773 0.860901i \(-0.330099\pi\)
0.508773 + 0.860901i \(0.330099\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −11.5139 −1.06446
\(118\) 0 0
\(119\) −4.81665 −0.441542
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −12.9083 −1.16390
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.11943 0.720483 0.360241 0.932859i \(-0.382694\pi\)
0.360241 + 0.932859i \(0.382694\pi\)
\(128\) 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
\(133\) 0.697224 0.0604570
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.9083 1.10283 0.551416 0.834230i \(-0.314088\pi\)
0.551416 + 0.834230i \(0.314088\pi\)
\(138\) 0 0
\(139\) 6.21110 0.526819 0.263409 0.964684i \(-0.415153\pi\)
0.263409 + 0.964684i \(0.415153\pi\)
\(140\) 0 0
\(141\) −6.90833 −0.581786
\(142\) 0 0
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −15.0000 −1.23718
\(148\) 0 0
\(149\) 17.2111 1.40999 0.704994 0.709213i \(-0.250950\pi\)
0.704994 + 0.709213i \(0.250950\pi\)
\(150\) 0 0
\(151\) −0.816654 −0.0664583 −0.0332292 0.999448i \(-0.510579\pi\)
−0.0332292 + 0.999448i \(0.510579\pi\)
\(152\) 0 0
\(153\) −15.9083 −1.28611
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −19.2111 −1.53321 −0.766606 0.642117i \(-0.778056\pi\)
−0.766606 + 0.642117i \(0.778056\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −5.09167 −0.401280
\(162\) 0 0
\(163\) 9.30278 0.728650 0.364325 0.931272i \(-0.381300\pi\)
0.364325 + 0.931272i \(0.381300\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.4222 1.03864 0.519321 0.854579i \(-0.326185\pi\)
0.519321 + 0.854579i \(0.326185\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 2.30278 0.176098
\(172\) 0 0
\(173\) 4.81665 0.366203 0.183102 0.983094i \(-0.441386\pi\)
0.183102 + 0.983094i \(0.441386\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 32.7250 2.45976
\(178\) 0 0
\(179\) −12.5139 −0.935331 −0.467666 0.883905i \(-0.654905\pi\)
−0.467666 + 0.883905i \(0.654905\pi\)
\(180\) 0 0
\(181\) −19.9083 −1.47977 −0.739887 0.672731i \(-0.765121\pi\)
−0.739887 + 0.672731i \(0.765121\pi\)
\(182\) 0 0
\(183\) −18.2111 −1.34620
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.90833 −0.505187
\(188\) 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
\(193\) −13.2111 −0.950956 −0.475478 0.879728i \(-0.657725\pi\)
−0.475478 + 0.879728i \(0.657725\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.3028 −0.947784 −0.473892 0.880583i \(-0.657151\pi\)
−0.473892 + 0.880583i \(0.657151\pi\)
\(198\) 0 0
\(199\) 6.48612 0.459789 0.229894 0.973216i \(-0.426162\pi\)
0.229894 + 0.973216i \(0.426162\pi\)
\(200\) 0 0
\(201\) −9.21110 −0.649701
\(202\) 0 0
\(203\) 0.633308 0.0444495
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −16.8167 −1.16884
\(208\) 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
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.11943 −0.483298
\(218\) 0 0
\(219\) 18.2111 1.23059
\(220\) 0 0
\(221\) 34.5416 2.32352
\(222\) 0 0
\(223\) −22.6333 −1.51564 −0.757819 0.652465i \(-0.773735\pi\)
−0.757819 + 0.652465i \(0.773735\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.69722 0.112649 0.0563244 0.998413i \(-0.482062\pi\)
0.0563244 + 0.998413i \(0.482062\pi\)
\(228\) 0 0
\(229\) −18.7250 −1.23738 −0.618691 0.785635i \(-0.712337\pi\)
−0.618691 + 0.785635i \(0.712337\pi\)
\(230\) 0 0
\(231\) 1.60555 0.105638
\(232\) 0 0
\(233\) −15.9083 −1.04219 −0.521095 0.853499i \(-0.674476\pi\)
−0.521095 + 0.853499i \(0.674476\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 25.1194 1.63168
\(238\) 0 0
\(239\) −21.1194 −1.36610 −0.683051 0.730371i \(-0.739347\pi\)
−0.683051 + 0.730371i \(0.739347\pi\)
\(240\) 0 0
\(241\) 21.9361 1.41303 0.706514 0.707699i \(-0.250267\pi\)
0.706514 + 0.707699i \(0.250267\pi\)
\(242\) 0 0
\(243\) −19.6056 −1.25770
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) 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
\(253\) −7.30278 −0.459122
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −1.66947 −0.103736
\(260\) 0 0
\(261\) 2.09167 0.129471
\(262\) 0 0
\(263\) −22.8167 −1.40694 −0.703468 0.710727i \(-0.748366\pi\)
−0.703468 + 0.710727i \(0.748366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.90833 0.239186
\(268\) 0 0
\(269\) 8.72498 0.531971 0.265986 0.963977i \(-0.414303\pi\)
0.265986 + 0.963977i \(0.414303\pi\)
\(270\) 0 0
\(271\) 0.211103 0.0128236 0.00641178 0.999979i \(-0.497959\pi\)
0.00641178 + 0.999979i \(0.497959\pi\)
\(272\) 0 0
\(273\) −8.02776 −0.485862
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.3944 −0.864879 −0.432439 0.901663i \(-0.642347\pi\)
−0.432439 + 0.901663i \(0.642347\pi\)
\(278\) 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
\(283\) 6.30278 0.374661 0.187331 0.982297i \(-0.440016\pi\)
0.187331 + 0.982297i \(0.440016\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.90833 −0.230701
\(288\) 0 0
\(289\) 30.7250 1.80735
\(290\) 0 0
\(291\) −35.2389 −2.06574
\(292\) 0 0
\(293\) −0.788897 −0.0460879 −0.0230439 0.999734i \(-0.507336\pi\)
−0.0230439 + 0.999734i \(0.507336\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.60555 −0.0931635
\(298\) 0 0
\(299\) 36.5139 2.11165
\(300\) 0 0
\(301\) 5.02776 0.289795
\(302\) 0 0
\(303\) −1.18335 −0.0679815
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −16.9083 −0.965009 −0.482505 0.875893i \(-0.660273\pi\)
−0.482505 + 0.875893i \(0.660273\pi\)
\(308\) 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
\(313\) −0.183346 −0.0103633 −0.00518167 0.999987i \(-0.501649\pi\)
−0.00518167 + 0.999987i \(0.501649\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.908327 0.0510167 0.0255084 0.999675i \(-0.491880\pi\)
0.0255084 + 0.999675i \(0.491880\pi\)
\(318\) 0 0
\(319\) 0.908327 0.0508565
\(320\) 0 0
\(321\) −6.90833 −0.385585
\(322\) 0 0
\(323\) −6.90833 −0.384390
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 26.5139 1.46622
\(328\) 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
\(333\) −5.51388 −0.302159
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 30.8444 1.68020 0.840101 0.542430i \(-0.182496\pi\)
0.840101 + 0.542430i \(0.182496\pi\)
\(338\) 0 0
\(339\) 24.9083 1.35283
\(340\) 0 0
\(341\) −10.2111 −0.552962
\(342\) 0 0
\(343\) −9.42221 −0.508751
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.5139 −0.671780 −0.335890 0.941901i \(-0.609037\pi\)
−0.335890 + 0.941901i \(0.609037\pi\)
\(348\) 0 0
\(349\) −5.18335 −0.277458 −0.138729 0.990330i \(-0.544302\pi\)
−0.138729 + 0.990330i \(0.544302\pi\)
\(350\) 0 0
\(351\) 8.02776 0.428490
\(352\) 0 0
\(353\) −18.6333 −0.991751 −0.495875 0.868394i \(-0.665153\pi\)
−0.495875 + 0.868394i \(0.665153\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −11.0917 −0.587034
\(358\) 0 0
\(359\) −0.788897 −0.0416364 −0.0208182 0.999783i \(-0.506627\pi\)
−0.0208182 + 0.999783i \(0.506627\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 2.30278 0.120864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −20.6972 −1.08039 −0.540193 0.841541i \(-0.681649\pi\)
−0.540193 + 0.841541i \(0.681649\pi\)
\(368\) 0 0
\(369\) −12.9083 −0.671981
\(370\) 0 0
\(371\) 0.908327 0.0471580
\(372\) 0 0
\(373\) −27.4222 −1.41987 −0.709934 0.704268i \(-0.751275\pi\)
−0.709934 + 0.704268i \(0.751275\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.54163 −0.233906
\(378\) 0 0
\(379\) −3.18335 −0.163518 −0.0817588 0.996652i \(-0.526054\pi\)
−0.0817588 + 0.996652i \(0.526054\pi\)
\(380\) 0 0
\(381\) 18.6972 0.957888
\(382\) 0 0
\(383\) 21.6333 1.10541 0.552705 0.833377i \(-0.313596\pi\)
0.552705 + 0.833377i \(0.313596\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.6056 0.844108
\(388\) 0 0
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 50.4500 2.55136
\(392\) 0 0
\(393\) 22.8167 1.15095
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −21.6972 −1.08895 −0.544476 0.838776i \(-0.683272\pi\)
−0.544476 + 0.838776i \(0.683272\pi\)
\(398\) 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
\(403\) 51.0555 2.54326
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.39445 −0.118688
\(408\) 0 0
\(409\) −6.21110 −0.307119 −0.153560 0.988139i \(-0.549074\pi\)
−0.153560 + 0.988139i \(0.549074\pi\)
\(410\) 0 0
\(411\) 29.7250 1.46623
\(412\) 0 0
\(413\) 9.90833 0.487557
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.3028 0.700410
\(418\) 0 0
\(419\) −6.39445 −0.312389 −0.156195 0.987726i \(-0.549923\pi\)
−0.156195 + 0.987726i \(0.549923\pi\)
\(420\) 0 0
\(421\) 0.697224 0.0339806 0.0169903 0.999856i \(-0.494592\pi\)
0.0169903 + 0.999856i \(0.494592\pi\)
\(422\) 0 0
\(423\) −6.90833 −0.335894
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.51388 −0.266835
\(428\) 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
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.30278 −0.349339
\(438\) 0 0
\(439\) −24.3028 −1.15991 −0.579954 0.814649i \(-0.696930\pi\)
−0.579954 + 0.814649i \(0.696930\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) −8.60555 −0.408862 −0.204431 0.978881i \(-0.565534\pi\)
−0.204431 + 0.978881i \(0.565534\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 39.6333 1.87459
\(448\) 0 0
\(449\) 23.4861 1.10838 0.554189 0.832391i \(-0.313028\pi\)
0.554189 + 0.832391i \(0.313028\pi\)
\(450\) 0 0
\(451\) −5.60555 −0.263955
\(452\) 0 0
\(453\) −1.88057 −0.0883569
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.6972 0.968175 0.484088 0.875020i \(-0.339152\pi\)
0.484088 + 0.875020i \(0.339152\pi\)
\(458\) 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
\(463\) 11.7889 0.547877 0.273938 0.961747i \(-0.411674\pi\)
0.273938 + 0.961747i \(0.411674\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.6333 −0.862247 −0.431123 0.902293i \(-0.641883\pi\)
−0.431123 + 0.902293i \(0.641883\pi\)
\(468\) 0 0
\(469\) −2.78890 −0.128779
\(470\) 0 0
\(471\) −44.2389 −2.03842
\(472\) 0 0
\(473\) 7.21110 0.331567
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 34.8167 1.59081 0.795407 0.606076i \(-0.207257\pi\)
0.795407 + 0.606076i \(0.207257\pi\)
\(480\) 0 0
\(481\) 11.9722 0.545887
\(482\) 0 0
\(483\) −11.7250 −0.533505
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.21110 0.190823 0.0954116 0.995438i \(-0.469583\pi\)
0.0954116 + 0.995438i \(0.469583\pi\)
\(488\) 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
\(493\) −6.27502 −0.282613
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.81665 0.0814881
\(498\) 0 0
\(499\) 3.48612 0.156060 0.0780301 0.996951i \(-0.475137\pi\)
0.0780301 + 0.996951i \(0.475137\pi\)
\(500\) 0 0
\(501\) 30.9083 1.38088
\(502\) 0 0
\(503\) 9.39445 0.418878 0.209439 0.977822i \(-0.432836\pi\)
0.209439 + 0.977822i \(0.432836\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 27.6333 1.22724
\(508\) 0 0
\(509\) −22.6972 −1.00604 −0.503018 0.864276i \(-0.667777\pi\)
−0.503018 + 0.864276i \(0.667777\pi\)
\(510\) 0 0
\(511\) 5.51388 0.243920
\(512\) 0 0
\(513\) −1.60555 −0.0708868
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.00000 −0.131940
\(518\) 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
\(523\) −32.4222 −1.41772 −0.708862 0.705347i \(-0.750791\pi\)
−0.708862 + 0.705347i \(0.750791\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 70.5416 3.07284
\(528\) 0 0
\(529\) 30.3305 1.31872
\(530\) 0 0
\(531\) 32.7250 1.42014
\(532\) 0 0
\(533\) 28.0278 1.21402
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −28.8167 −1.24353
\(538\) 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
\(543\) −45.8444 −1.96737
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7.11943 −0.304405 −0.152202 0.988349i \(-0.548637\pi\)
−0.152202 + 0.988349i \(0.548637\pi\)
\(548\) 0 0
\(549\) −18.2111 −0.777231
\(550\) 0 0
\(551\) 0.908327 0.0386960
\(552\) 0 0
\(553\) 7.60555 0.323421
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.4222 −0.822945 −0.411473 0.911422i \(-0.634985\pi\)
−0.411473 + 0.911422i \(0.634985\pi\)
\(558\) 0 0
\(559\) −36.0555 −1.52499
\(560\) 0 0
\(561\) −15.9083 −0.671650
\(562\) 0 0
\(563\) 8.09167 0.341023 0.170512 0.985356i \(-0.445458\pi\)
0.170512 + 0.985356i \(0.445458\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.39445 −0.310538
\(568\) 0 0
\(569\) −46.1472 −1.93459 −0.967295 0.253653i \(-0.918368\pi\)
−0.967295 + 0.253653i \(0.918368\pi\)
\(570\) 0 0
\(571\) −22.3305 −0.934504 −0.467252 0.884124i \(-0.654756\pi\)
−0.467252 + 0.884124i \(0.654756\pi\)
\(572\) 0 0
\(573\) −23.7250 −0.991125
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −44.3583 −1.84666 −0.923330 0.384008i \(-0.874544\pi\)
−0.923330 + 0.384008i \(0.874544\pi\)
\(578\) 0 0
\(579\) −30.4222 −1.26430
\(580\) 0 0
\(581\) −2.44996 −0.101642
\(582\) 0 0
\(583\) 1.30278 0.0539555
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.5416 0.682746 0.341373 0.939928i \(-0.389108\pi\)
0.341373 + 0.939928i \(0.389108\pi\)
\(588\) 0 0
\(589\) −10.2111 −0.420741
\(590\) 0 0
\(591\) −30.6333 −1.26009
\(592\) 0 0
\(593\) 6.39445 0.262589 0.131294 0.991343i \(-0.458087\pi\)
0.131294 + 0.991343i \(0.458087\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.9361 0.611293
\(598\) 0 0
\(599\) 24.9083 1.01773 0.508863 0.860847i \(-0.330066\pi\)
0.508863 + 0.860847i \(0.330066\pi\)
\(600\) 0 0
\(601\) −1.90833 −0.0778423 −0.0389211 0.999242i \(-0.512392\pi\)
−0.0389211 + 0.999242i \(0.512392\pi\)
\(602\) 0 0
\(603\) −9.21110 −0.375105
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.21110 0.292690 0.146345 0.989234i \(-0.453249\pi\)
0.146345 + 0.989234i \(0.453249\pi\)
\(608\) 0 0
\(609\) 1.45837 0.0590959
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) 15.8806 0.641410 0.320705 0.947179i \(-0.396080\pi\)
0.320705 + 0.947179i \(0.396080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.39445 −0.136655 −0.0683277 0.997663i \(-0.521766\pi\)
−0.0683277 + 0.997663i \(0.521766\pi\)
\(618\) 0 0
\(619\) 11.4222 0.459097 0.229549 0.973297i \(-0.426275\pi\)
0.229549 + 0.973297i \(0.426275\pi\)
\(620\) 0 0
\(621\) 11.7250 0.470507
\(622\) 0 0
\(623\) 1.18335 0.0474098
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.30278 0.0919640
\(628\) 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
\(633\) 58.1194 2.31004
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 32.5694 1.29045
\(638\) 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
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.2389 −1.30675 −0.653377 0.757033i \(-0.726648\pi\)
−0.653377 + 0.757033i \(0.726648\pi\)
\(648\) 0 0
\(649\) 14.2111 0.557835
\(650\) 0 0
\(651\) −16.3944 −0.642549
\(652\) 0 0
\(653\) −6.11943 −0.239472 −0.119736 0.992806i \(-0.538205\pi\)
−0.119736 + 0.992806i \(0.538205\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.2111 0.710483
\(658\) 0 0
\(659\) 30.9083 1.20402 0.602009 0.798489i \(-0.294367\pi\)
0.602009 + 0.798489i \(0.294367\pi\)
\(660\) 0 0
\(661\) −8.81665 −0.342928 −0.171464 0.985190i \(-0.554850\pi\)
−0.171464 + 0.985190i \(0.554850\pi\)
\(662\) 0 0
\(663\) 79.5416 3.08914
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.63331 −0.256843
\(668\) 0 0
\(669\) −52.1194 −2.01505
\(670\) 0 0
\(671\) −7.90833 −0.305298
\(672\) 0 0
\(673\) −30.0278 −1.15748 −0.578742 0.815510i \(-0.696456\pi\)
−0.578742 + 0.815510i \(0.696456\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.2389 −0.931575 −0.465788 0.884897i \(-0.654229\pi\)
−0.465788 + 0.884897i \(0.654229\pi\)
\(678\) 0 0
\(679\) −10.6695 −0.409457
\(680\) 0 0
\(681\) 3.90833 0.149767
\(682\) 0 0
\(683\) 47.8444 1.83072 0.915358 0.402642i \(-0.131908\pi\)
0.915358 + 0.402642i \(0.131908\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −43.1194 −1.64511
\(688\) 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
\(693\) 1.60555 0.0609898
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 38.7250 1.46681
\(698\) 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
\(703\) −2.39445 −0.0903083
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.358288 −0.0134748
\(708\) 0 0
\(709\) −31.6333 −1.18801 −0.594007 0.804460i \(-0.702455\pi\)
−0.594007 + 0.804460i \(0.702455\pi\)
\(710\) 0 0
\(711\) 25.1194 0.942052
\(712\) 0 0
\(713\) 74.5694 2.79265
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −48.6333 −1.81624
\(718\) 0 0
\(719\) −7.18335 −0.267894 −0.133947 0.990989i \(-0.542765\pi\)
−0.133947 + 0.990989i \(0.542765\pi\)
\(720\) 0 0
\(721\) 2.02776 0.0755176
\(722\) 0 0
\(723\) 50.5139 1.87863
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −39.3305 −1.45869 −0.729344 0.684147i \(-0.760175\pi\)
−0.729344 + 0.684147i \(0.760175\pi\)
\(728\) 0 0
\(729\) −13.3305 −0.493723
\(730\) 0 0
\(731\) −49.8167 −1.84254
\(732\) 0 0
\(733\) −19.6056 −0.724148 −0.362074 0.932149i \(-0.617931\pi\)
−0.362074 + 0.932149i \(0.617931\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −35.1194 −1.29189 −0.645945 0.763384i \(-0.723536\pi\)
−0.645945 + 0.763384i \(0.723536\pi\)
\(740\) 0 0
\(741\) −11.5139 −0.422973
\(742\) 0 0
\(743\) 40.6972 1.49304 0.746518 0.665365i \(-0.231724\pi\)
0.746518 + 0.665365i \(0.231724\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8.09167 −0.296059
\(748\) 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
\(753\) −15.9083 −0.579732
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −49.0555 −1.78295 −0.891476 0.453067i \(-0.850330\pi\)
−0.891476 + 0.453067i \(0.850330\pi\)
\(758\) 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
\(763\) 8.02776 0.290624
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −71.0555 −2.56567
\(768\) 0 0
\(769\) −5.18335 −0.186916 −0.0934581 0.995623i \(-0.529792\pi\)
−0.0934581 + 0.995623i \(0.529792\pi\)
\(770\) 0 0
\(771\) 41.4500 1.49278
\(772\) 0 0
\(773\) −3.11943 −0.112198 −0.0560990 0.998425i \(-0.517866\pi\)
−0.0560990 + 0.998425i \(0.517866\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3.84441 −0.137917
\(778\) 0 0
\(779\) −5.60555 −0.200840
\(780\) 0 0
\(781\) 2.60555 0.0932340
\(782\) 0 0
\(783\) −1.45837 −0.0521177
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.2111 0.363986 0.181993 0.983300i \(-0.441745\pi\)
0.181993 + 0.983300i \(0.441745\pi\)
\(788\) 0 0
\(789\) −52.5416 −1.87053
\(790\) 0 0
\(791\) 7.54163 0.268150
\(792\) 0 0
\(793\) 39.5416 1.40416
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.51388 −0.124468 −0.0622340 0.998062i \(-0.519822\pi\)
−0.0622340 + 0.998062i \(0.519822\pi\)
\(798\) 0 0
\(799\) 20.7250 0.733197
\(800\) 0 0
\(801\) 3.90833 0.138094
\(802\) 0 0
\(803\) 7.90833 0.279079
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20.0917 0.707260
\(808\) 0 0
\(809\) 39.6333 1.39343 0.696716 0.717347i \(-0.254644\pi\)
0.696716 + 0.717347i \(0.254644\pi\)
\(810\) 0 0
\(811\) −38.8722 −1.36499 −0.682493 0.730892i \(-0.739104\pi\)
−0.682493 + 0.730892i \(0.739104\pi\)
\(812\) 0 0
\(813\) 0.486122 0.0170490
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.21110 0.252285
\(818\) 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
\(823\) 18.4222 0.642158 0.321079 0.947052i \(-0.395955\pi\)
0.321079 + 0.947052i \(0.395955\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.8167 0.480452 0.240226 0.970717i \(-0.422778\pi\)
0.240226 + 0.970717i \(0.422778\pi\)
\(828\) 0 0
\(829\) 29.7527 1.03336 0.516678 0.856180i \(-0.327169\pi\)
0.516678 + 0.856180i \(0.327169\pi\)
\(830\) 0 0
\(831\) −33.1472 −1.14986
\(832\) 0 0
\(833\) 45.0000 1.55916
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 16.3944 0.566675
\(838\) 0 0
\(839\) 9.11943 0.314838 0.157419 0.987532i \(-0.449683\pi\)
0.157419 + 0.987532i \(0.449683\pi\)
\(840\) 0 0
\(841\) −28.1749 −0.971550
\(842\) 0 0
\(843\) −2.72498 −0.0938533
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.697224 0.0239569
\(848\) 0 0
\(849\) 14.5139 0.498115
\(850\) 0 0
\(851\) 17.4861 0.599417
\(852\) 0 0
\(853\) 12.7250 0.435695 0.217848 0.975983i \(-0.430096\pi\)
0.217848 + 0.975983i \(0.430096\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −41.3944 −1.41236 −0.706180 0.708032i \(-0.749583\pi\)
−0.706180 + 0.708032i \(0.749583\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) −12.3944 −0.421912 −0.210956 0.977496i \(-0.567658\pi\)
−0.210956 + 0.977496i \(0.567658\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 70.7527 2.40289
\(868\) 0 0
\(869\) 10.9083 0.370040
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) −35.2389 −1.19265
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 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
\(883\) 52.4500 1.76508 0.882541 0.470236i \(-0.155831\pi\)
0.882541 + 0.470236i \(0.155831\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.23886 −0.108750 −0.0543751 0.998521i \(-0.517317\pi\)
−0.0543751 + 0.998521i \(0.517317\pi\)
\(888\) 0 0
\(889\) 5.66106 0.189866
\(890\) 0 0
\(891\) −10.6056 −0.355299
\(892\) 0 0
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 84.0833 2.80746
\(898\) 0 0
\(899\) −9.27502 −0.309339
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 11.5778 0.385285
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 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
\(913\) −3.51388 −0.116292
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.90833 0.228133
\(918\) 0 0
\(919\) −26.7889 −0.883684 −0.441842 0.897093i \(-0.645675\pi\)
−0.441842 + 0.897093i \(0.645675\pi\)
\(920\) 0 0
\(921\) −38.9361 −1.28299
\(922\) 0 0
\(923\) −13.0278 −0.428814
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.69722 0.219966
\(928\) 0 0
\(929\) 53.6056 1.75874 0.879371 0.476138i \(-0.157964\pi\)
0.879371 + 0.476138i \(0.157964\pi\)
\(930\) 0 0
\(931\) −6.51388 −0.213484
\(932\) 0 0
\(933\) −11.0917 −0.363125
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 9.21110 0.300914 0.150457 0.988617i \(-0.451926\pi\)
0.150457 + 0.988617i \(0.451926\pi\)
\(938\) 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
\(943\) 40.9361 1.33306
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.63331 0.215554 0.107777 0.994175i \(-0.465627\pi\)
0.107777 + 0.994175i \(0.465627\pi\)
\(948\) 0 0
\(949\) −39.5416 −1.28358
\(950\) 0 0
\(951\) 2.09167 0.0678271
\(952\) 0 0
\(953\) −37.2666 −1.20718 −0.603592 0.797293i \(-0.706264\pi\)
−0.603592 + 0.797293i \(0.706264\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.09167 0.0676142
\(958\) 0 0
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) 73.2666 2.36344
\(962\) 0 0
\(963\) −6.90833 −0.222618
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.9083 0.479419 0.239710 0.970845i \(-0.422948\pi\)
0.239710 + 0.970845i \(0.422948\pi\)
\(968\) 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
\(973\) 4.33053 0.138830
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.0278 1.66452 0.832258 0.554389i \(-0.187048\pi\)
0.832258 + 0.554389i \(0.187048\pi\)
\(978\) 0 0
\(979\) 1.69722 0.0542435
\(980\) 0 0
\(981\) 26.5139 0.846523
\(982\) 0 0
\(983\) 8.84441 0.282093 0.141046 0.990003i \(-0.454953\pi\)
0.141046 + 0.990003i \(0.454953\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.81665 −0.153316
\(988\) 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
\(993\) 49.7527 1.57886
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −46.7250 −1.47979 −0.739897 0.672720i \(-0.765126\pi\)
−0.739897 + 0.672720i \(0.765126\pi\)
\(998\) 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.a.bs.1.2 2
4.3 odd 2 275.2.a.e.1.2 2
5.2 odd 4 4400.2.b.y.4049.1 4
5.3 odd 4 4400.2.b.y.4049.4 4
5.4 even 2 4400.2.a.bh.1.1 2
12.11 even 2 2475.2.a.t.1.1 2
20.3 even 4 275.2.b.c.199.2 4
20.7 even 4 275.2.b.c.199.3 4
20.19 odd 2 275.2.a.f.1.1 yes 2
44.43 even 2 3025.2.a.n.1.1 2
60.23 odd 4 2475.2.c.k.199.3 4
60.47 odd 4 2475.2.c.k.199.2 4
60.59 even 2 2475.2.a.o.1.2 2
220.219 even 2 3025.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 4.3 odd 2
275.2.a.f.1.1 yes 2 20.19 odd 2
275.2.b.c.199.2 4 20.3 even 4
275.2.b.c.199.3 4 20.7 even 4
2475.2.a.o.1.2 2 60.59 even 2
2475.2.a.t.1.1 2 12.11 even 2
2475.2.c.k.199.2 4 60.47 odd 4
2475.2.c.k.199.3 4 60.23 odd 4
3025.2.a.h.1.2 2 220.219 even 2
3025.2.a.n.1.1 2 44.43 even 2
4400.2.a.bh.1.1 2 5.4 even 2
4400.2.a.bs.1.2 2 1.1 even 1 trivial
4400.2.b.y.4049.1 4 5.2 odd 4
4400.2.b.y.4049.4 4 5.3 odd 4