Properties

Label 342.3.d.a.37.1
Level $342$
Weight $3$
Character 342.37
Analytic conductor $9.319$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.1
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 342.37
Dual form 342.3.d.a.37.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} +2.82843i q^{8} -1.41421i q^{10} -5.00000 q^{11} -16.9706i q^{13} -7.07107i q^{14} +4.00000 q^{16} +25.0000 q^{17} +19.0000 q^{19} -2.00000 q^{20} +7.07107i q^{22} +10.0000 q^{23} -24.0000 q^{25} -24.0000 q^{26} -10.0000 q^{28} -42.4264i q^{29} -42.4264i q^{31} -5.65685i q^{32} -35.3553i q^{34} +5.00000 q^{35} -25.4558i q^{37} -26.8701i q^{38} +2.82843i q^{40} +42.4264i q^{41} +5.00000 q^{43} +10.0000 q^{44} -14.1421i q^{46} -5.00000 q^{47} -24.0000 q^{49} +33.9411i q^{50} +33.9411i q^{52} +25.4558i q^{53} -5.00000 q^{55} +14.1421i q^{56} -60.0000 q^{58} -84.8528i q^{59} +95.0000 q^{61} -60.0000 q^{62} -8.00000 q^{64} -16.9706i q^{65} +110.309i q^{67} -50.0000 q^{68} -7.07107i q^{70} -25.0000 q^{73} -36.0000 q^{74} -38.0000 q^{76} -25.0000 q^{77} +42.4264i q^{79} +4.00000 q^{80} +60.0000 q^{82} +130.000 q^{83} +25.0000 q^{85} -7.07107i q^{86} -14.1421i q^{88} +127.279i q^{89} -84.8528i q^{91} -20.0000 q^{92} +7.07107i q^{94} +19.0000 q^{95} +16.9706i q^{97} +33.9411i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 2q^{5} + 10q^{7} + O(q^{10}) \) \( 2q - 4q^{4} + 2q^{5} + 10q^{7} - 10q^{11} + 8q^{16} + 50q^{17} + 38q^{19} - 4q^{20} + 20q^{23} - 48q^{25} - 48q^{26} - 20q^{28} + 10q^{35} + 10q^{43} + 20q^{44} - 10q^{47} - 48q^{49} - 10q^{55} - 120q^{58} + 190q^{61} - 120q^{62} - 16q^{64} - 100q^{68} - 50q^{73} - 72q^{74} - 76q^{76} - 50q^{77} + 8q^{80} + 120q^{82} + 260q^{83} + 50q^{85} - 40q^{92} + 38q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 1.00000 0.200000 0.100000 0.994987i \(-0.468116\pi\)
0.100000 + 0.994987i \(0.468116\pi\)
\(6\) 0 0
\(7\) 5.00000 0.714286 0.357143 0.934050i \(-0.383751\pi\)
0.357143 + 0.934050i \(0.383751\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) − 1.41421i − 0.141421i
\(11\) −5.00000 −0.454545 −0.227273 0.973831i \(-0.572981\pi\)
−0.227273 + 0.973831i \(0.572981\pi\)
\(12\) 0 0
\(13\) − 16.9706i − 1.30543i −0.757604 0.652714i \(-0.773630\pi\)
0.757604 0.652714i \(-0.226370\pi\)
\(14\) − 7.07107i − 0.505076i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 25.0000 1.47059 0.735294 0.677748i \(-0.237044\pi\)
0.735294 + 0.677748i \(0.237044\pi\)
\(18\) 0 0
\(19\) 19.0000 1.00000
\(20\) −2.00000 −0.100000
\(21\) 0 0
\(22\) 7.07107i 0.321412i
\(23\) 10.0000 0.434783 0.217391 0.976085i \(-0.430245\pi\)
0.217391 + 0.976085i \(0.430245\pi\)
\(24\) 0 0
\(25\) −24.0000 −0.960000
\(26\) −24.0000 −0.923077
\(27\) 0 0
\(28\) −10.0000 −0.357143
\(29\) − 42.4264i − 1.46298i −0.681852 0.731490i \(-0.738825\pi\)
0.681852 0.731490i \(-0.261175\pi\)
\(30\) 0 0
\(31\) − 42.4264i − 1.36859i −0.729204 0.684297i \(-0.760109\pi\)
0.729204 0.684297i \(-0.239891\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) − 35.3553i − 1.03986i
\(35\) 5.00000 0.142857
\(36\) 0 0
\(37\) − 25.4558i − 0.687996i −0.938970 0.343998i \(-0.888219\pi\)
0.938970 0.343998i \(-0.111781\pi\)
\(38\) − 26.8701i − 0.707107i
\(39\) 0 0
\(40\) 2.82843i 0.0707107i
\(41\) 42.4264i 1.03479i 0.855747 + 0.517395i \(0.173098\pi\)
−0.855747 + 0.517395i \(0.826902\pi\)
\(42\) 0 0
\(43\) 5.00000 0.116279 0.0581395 0.998308i \(-0.481483\pi\)
0.0581395 + 0.998308i \(0.481483\pi\)
\(44\) 10.0000 0.227273
\(45\) 0 0
\(46\) − 14.1421i − 0.307438i
\(47\) −5.00000 −0.106383 −0.0531915 0.998584i \(-0.516939\pi\)
−0.0531915 + 0.998584i \(0.516939\pi\)
\(48\) 0 0
\(49\) −24.0000 −0.489796
\(50\) 33.9411i 0.678823i
\(51\) 0 0
\(52\) 33.9411i 0.652714i
\(53\) 25.4558i 0.480299i 0.970736 + 0.240149i \(0.0771965\pi\)
−0.970736 + 0.240149i \(0.922804\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.0909091
\(56\) 14.1421i 0.252538i
\(57\) 0 0
\(58\) −60.0000 −1.03448
\(59\) − 84.8528i − 1.43818i −0.694915 0.719092i \(-0.744558\pi\)
0.694915 0.719092i \(-0.255442\pi\)
\(60\) 0 0
\(61\) 95.0000 1.55738 0.778689 0.627411i \(-0.215885\pi\)
0.778689 + 0.627411i \(0.215885\pi\)
\(62\) −60.0000 −0.967742
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 16.9706i − 0.261086i
\(66\) 0 0
\(67\) 110.309i 1.64640i 0.567753 + 0.823199i \(0.307813\pi\)
−0.567753 + 0.823199i \(0.692187\pi\)
\(68\) −50.0000 −0.735294
\(69\) 0 0
\(70\) − 7.07107i − 0.101015i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −25.0000 −0.342466 −0.171233 0.985231i \(-0.554775\pi\)
−0.171233 + 0.985231i \(0.554775\pi\)
\(74\) −36.0000 −0.486486
\(75\) 0 0
\(76\) −38.0000 −0.500000
\(77\) −25.0000 −0.324675
\(78\) 0 0
\(79\) 42.4264i 0.537043i 0.963274 + 0.268522i \(0.0865351\pi\)
−0.963274 + 0.268522i \(0.913465\pi\)
\(80\) 4.00000 0.0500000
\(81\) 0 0
\(82\) 60.0000 0.731707
\(83\) 130.000 1.56627 0.783133 0.621855i \(-0.213621\pi\)
0.783133 + 0.621855i \(0.213621\pi\)
\(84\) 0 0
\(85\) 25.0000 0.294118
\(86\) − 7.07107i − 0.0822217i
\(87\) 0 0
\(88\) − 14.1421i − 0.160706i
\(89\) 127.279i 1.43010i 0.699071 + 0.715052i \(0.253597\pi\)
−0.699071 + 0.715052i \(0.746403\pi\)
\(90\) 0 0
\(91\) − 84.8528i − 0.932449i
\(92\) −20.0000 −0.217391
\(93\) 0 0
\(94\) 7.07107i 0.0752241i
\(95\) 19.0000 0.200000
\(96\) 0 0
\(97\) 16.9706i 0.174954i 0.996167 + 0.0874771i \(0.0278805\pi\)
−0.996167 + 0.0874771i \(0.972120\pi\)
\(98\) 33.9411i 0.346338i
\(99\) 0 0
\(100\) 48.0000 0.480000
\(101\) −50.0000 −0.495050 −0.247525 0.968882i \(-0.579617\pi\)
−0.247525 + 0.968882i \(0.579617\pi\)
\(102\) 0 0
\(103\) − 16.9706i − 0.164763i −0.996601 0.0823814i \(-0.973747\pi\)
0.996601 0.0823814i \(-0.0262526\pi\)
\(104\) 48.0000 0.461538
\(105\) 0 0
\(106\) 36.0000 0.339623
\(107\) 101.823i 0.951620i 0.879548 + 0.475810i \(0.157845\pi\)
−0.879548 + 0.475810i \(0.842155\pi\)
\(108\) 0 0
\(109\) 127.279i 1.16770i 0.811862 + 0.583850i \(0.198454\pi\)
−0.811862 + 0.583850i \(0.801546\pi\)
\(110\) 7.07107i 0.0642824i
\(111\) 0 0
\(112\) 20.0000 0.178571
\(113\) − 110.309i − 0.976183i −0.872793 0.488091i \(-0.837693\pi\)
0.872793 0.488091i \(-0.162307\pi\)
\(114\) 0 0
\(115\) 10.0000 0.0869565
\(116\) 84.8528i 0.731490i
\(117\) 0 0
\(118\) −120.000 −1.01695
\(119\) 125.000 1.05042
\(120\) 0 0
\(121\) −96.0000 −0.793388
\(122\) − 134.350i − 1.10123i
\(123\) 0 0
\(124\) 84.8528i 0.684297i
\(125\) −49.0000 −0.392000
\(126\) 0 0
\(127\) 229.103i 1.80396i 0.431780 + 0.901979i \(0.357886\pi\)
−0.431780 + 0.901979i \(0.642114\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) −24.0000 −0.184615
\(131\) 163.000 1.24427 0.622137 0.782908i \(-0.286265\pi\)
0.622137 + 0.782908i \(0.286265\pi\)
\(132\) 0 0
\(133\) 95.0000 0.714286
\(134\) 156.000 1.16418
\(135\) 0 0
\(136\) 70.7107i 0.519931i
\(137\) −95.0000 −0.693431 −0.346715 0.937970i \(-0.612703\pi\)
−0.346715 + 0.937970i \(0.612703\pi\)
\(138\) 0 0
\(139\) 125.000 0.899281 0.449640 0.893210i \(-0.351552\pi\)
0.449640 + 0.893210i \(0.351552\pi\)
\(140\) −10.0000 −0.0714286
\(141\) 0 0
\(142\) 0 0
\(143\) 84.8528i 0.593376i
\(144\) 0 0
\(145\) − 42.4264i − 0.292596i
\(146\) 35.3553i 0.242160i
\(147\) 0 0
\(148\) 50.9117i 0.343998i
\(149\) −215.000 −1.44295 −0.721477 0.692439i \(-0.756536\pi\)
−0.721477 + 0.692439i \(0.756536\pi\)
\(150\) 0 0
\(151\) − 84.8528i − 0.561939i −0.959717 0.280970i \(-0.909344\pi\)
0.959717 0.280970i \(-0.0906560\pi\)
\(152\) 53.7401i 0.353553i
\(153\) 0 0
\(154\) 35.3553i 0.229580i
\(155\) − 42.4264i − 0.273719i
\(156\) 0 0
\(157\) −190.000 −1.21019 −0.605096 0.796153i \(-0.706865\pi\)
−0.605096 + 0.796153i \(0.706865\pi\)
\(158\) 60.0000 0.379747
\(159\) 0 0
\(160\) − 5.65685i − 0.0353553i
\(161\) 50.0000 0.310559
\(162\) 0 0
\(163\) 110.000 0.674847 0.337423 0.941353i \(-0.390445\pi\)
0.337423 + 0.941353i \(0.390445\pi\)
\(164\) − 84.8528i − 0.517395i
\(165\) 0 0
\(166\) − 183.848i − 1.10752i
\(167\) − 59.3970i − 0.355670i −0.984060 0.177835i \(-0.943091\pi\)
0.984060 0.177835i \(-0.0569094\pi\)
\(168\) 0 0
\(169\) −119.000 −0.704142
\(170\) − 35.3553i − 0.207973i
\(171\) 0 0
\(172\) −10.0000 −0.0581395
\(173\) 186.676i 1.07905i 0.841969 + 0.539527i \(0.181397\pi\)
−0.841969 + 0.539527i \(0.818603\pi\)
\(174\) 0 0
\(175\) −120.000 −0.685714
\(176\) −20.0000 −0.113636
\(177\) 0 0
\(178\) 180.000 1.01124
\(179\) 127.279i 0.711057i 0.934665 + 0.355529i \(0.115699\pi\)
−0.934665 + 0.355529i \(0.884301\pi\)
\(180\) 0 0
\(181\) − 254.558i − 1.40640i −0.710992 0.703200i \(-0.751754\pi\)
0.710992 0.703200i \(-0.248246\pi\)
\(182\) −120.000 −0.659341
\(183\) 0 0
\(184\) 28.2843i 0.153719i
\(185\) − 25.4558i − 0.137599i
\(186\) 0 0
\(187\) −125.000 −0.668449
\(188\) 10.0000 0.0531915
\(189\) 0 0
\(190\) − 26.8701i − 0.141421i
\(191\) −293.000 −1.53403 −0.767016 0.641628i \(-0.778259\pi\)
−0.767016 + 0.641628i \(0.778259\pi\)
\(192\) 0 0
\(193\) 59.3970i 0.307756i 0.988090 + 0.153878i \(0.0491763\pi\)
−0.988090 + 0.153878i \(0.950824\pi\)
\(194\) 24.0000 0.123711
\(195\) 0 0
\(196\) 48.0000 0.244898
\(197\) 70.0000 0.355330 0.177665 0.984091i \(-0.443146\pi\)
0.177665 + 0.984091i \(0.443146\pi\)
\(198\) 0 0
\(199\) 173.000 0.869347 0.434673 0.900588i \(-0.356864\pi\)
0.434673 + 0.900588i \(0.356864\pi\)
\(200\) − 67.8823i − 0.339411i
\(201\) 0 0
\(202\) 70.7107i 0.350053i
\(203\) − 212.132i − 1.04499i
\(204\) 0 0
\(205\) 42.4264i 0.206958i
\(206\) −24.0000 −0.116505
\(207\) 0 0
\(208\) − 67.8823i − 0.326357i
\(209\) −95.0000 −0.454545
\(210\) 0 0
\(211\) 84.8528i 0.402146i 0.979576 + 0.201073i \(0.0644429\pi\)
−0.979576 + 0.201073i \(0.935557\pi\)
\(212\) − 50.9117i − 0.240149i
\(213\) 0 0
\(214\) 144.000 0.672897
\(215\) 5.00000 0.0232558
\(216\) 0 0
\(217\) − 212.132i − 0.977567i
\(218\) 180.000 0.825688
\(219\) 0 0
\(220\) 10.0000 0.0454545
\(221\) − 424.264i − 1.91975i
\(222\) 0 0
\(223\) − 364.867i − 1.63618i −0.575094 0.818088i \(-0.695034\pi\)
0.575094 0.818088i \(-0.304966\pi\)
\(224\) − 28.2843i − 0.126269i
\(225\) 0 0
\(226\) −156.000 −0.690265
\(227\) − 67.8823i − 0.299041i −0.988759 0.149520i \(-0.952227\pi\)
0.988759 0.149520i \(-0.0477730\pi\)
\(228\) 0 0
\(229\) −145.000 −0.633188 −0.316594 0.948561i \(-0.602539\pi\)
−0.316594 + 0.948561i \(0.602539\pi\)
\(230\) − 14.1421i − 0.0614875i
\(231\) 0 0
\(232\) 120.000 0.517241
\(233\) −335.000 −1.43777 −0.718884 0.695130i \(-0.755347\pi\)
−0.718884 + 0.695130i \(0.755347\pi\)
\(234\) 0 0
\(235\) −5.00000 −0.0212766
\(236\) 169.706i 0.719092i
\(237\) 0 0
\(238\) − 176.777i − 0.742759i
\(239\) −197.000 −0.824268 −0.412134 0.911123i \(-0.635216\pi\)
−0.412134 + 0.911123i \(0.635216\pi\)
\(240\) 0 0
\(241\) 296.985i 1.23230i 0.787628 + 0.616151i \(0.211309\pi\)
−0.787628 + 0.616151i \(0.788691\pi\)
\(242\) 135.765i 0.561010i
\(243\) 0 0
\(244\) −190.000 −0.778689
\(245\) −24.0000 −0.0979592
\(246\) 0 0
\(247\) − 322.441i − 1.30543i
\(248\) 120.000 0.483871
\(249\) 0 0
\(250\) 69.2965i 0.277186i
\(251\) −173.000 −0.689243 −0.344622 0.938742i \(-0.611993\pi\)
−0.344622 + 0.938742i \(0.611993\pi\)
\(252\) 0 0
\(253\) −50.0000 −0.197628
\(254\) 324.000 1.27559
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 67.8823i 0.264133i 0.991241 + 0.132067i \(0.0421613\pi\)
−0.991241 + 0.132067i \(0.957839\pi\)
\(258\) 0 0
\(259\) − 127.279i − 0.491426i
\(260\) 33.9411i 0.130543i
\(261\) 0 0
\(262\) − 230.517i − 0.879835i
\(263\) 355.000 1.34981 0.674905 0.737905i \(-0.264185\pi\)
0.674905 + 0.737905i \(0.264185\pi\)
\(264\) 0 0
\(265\) 25.4558i 0.0960598i
\(266\) − 134.350i − 0.505076i
\(267\) 0 0
\(268\) − 220.617i − 0.823199i
\(269\) 381.838i 1.41947i 0.704468 + 0.709735i \(0.251186\pi\)
−0.704468 + 0.709735i \(0.748814\pi\)
\(270\) 0 0
\(271\) 110.000 0.405904 0.202952 0.979189i \(-0.434946\pi\)
0.202952 + 0.979189i \(0.434946\pi\)
\(272\) 100.000 0.367647
\(273\) 0 0
\(274\) 134.350i 0.490330i
\(275\) 120.000 0.436364
\(276\) 0 0
\(277\) −265.000 −0.956679 −0.478339 0.878175i \(-0.658761\pi\)
−0.478339 + 0.878175i \(0.658761\pi\)
\(278\) − 176.777i − 0.635887i
\(279\) 0 0
\(280\) 14.1421i 0.0505076i
\(281\) − 424.264i − 1.50984i −0.655819 0.754918i \(-0.727677\pi\)
0.655819 0.754918i \(-0.272323\pi\)
\(282\) 0 0
\(283\) 125.000 0.441696 0.220848 0.975308i \(-0.429118\pi\)
0.220848 + 0.975308i \(0.429118\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 120.000 0.419580
\(287\) 212.132i 0.739136i
\(288\) 0 0
\(289\) 336.000 1.16263
\(290\) −60.0000 −0.206897
\(291\) 0 0
\(292\) 50.0000 0.171233
\(293\) − 186.676i − 0.637120i −0.947903 0.318560i \(-0.896801\pi\)
0.947903 0.318560i \(-0.103199\pi\)
\(294\) 0 0
\(295\) − 84.8528i − 0.287637i
\(296\) 72.0000 0.243243
\(297\) 0 0
\(298\) 304.056i 1.02032i
\(299\) − 169.706i − 0.567577i
\(300\) 0 0
\(301\) 25.0000 0.0830565
\(302\) −120.000 −0.397351
\(303\) 0 0
\(304\) 76.0000 0.250000
\(305\) 95.0000 0.311475
\(306\) 0 0
\(307\) 280.014i 0.912099i 0.889955 + 0.456049i \(0.150736\pi\)
−0.889955 + 0.456049i \(0.849264\pi\)
\(308\) 50.0000 0.162338
\(309\) 0 0
\(310\) −60.0000 −0.193548
\(311\) 235.000 0.755627 0.377814 0.925882i \(-0.376676\pi\)
0.377814 + 0.925882i \(0.376676\pi\)
\(312\) 0 0
\(313\) −310.000 −0.990415 −0.495208 0.868775i \(-0.664908\pi\)
−0.495208 + 0.868775i \(0.664908\pi\)
\(314\) 268.701i 0.855734i
\(315\) 0 0
\(316\) − 84.8528i − 0.268522i
\(317\) 186.676i 0.588884i 0.955669 + 0.294442i \(0.0951338\pi\)
−0.955669 + 0.294442i \(0.904866\pi\)
\(318\) 0 0
\(319\) 212.132i 0.664991i
\(320\) −8.00000 −0.0250000
\(321\) 0 0
\(322\) − 70.7107i − 0.219598i
\(323\) 475.000 1.47059
\(324\) 0 0
\(325\) 407.294i 1.25321i
\(326\) − 155.563i − 0.477189i
\(327\) 0 0
\(328\) −120.000 −0.365854
\(329\) −25.0000 −0.0759878
\(330\) 0 0
\(331\) 296.985i 0.897235i 0.893724 + 0.448618i \(0.148083\pi\)
−0.893724 + 0.448618i \(0.851917\pi\)
\(332\) −260.000 −0.783133
\(333\) 0 0
\(334\) −84.0000 −0.251497
\(335\) 110.309i 0.329280i
\(336\) 0 0
\(337\) − 526.087i − 1.56109i −0.625099 0.780545i \(-0.714942\pi\)
0.625099 0.780545i \(-0.285058\pi\)
\(338\) 168.291i 0.497904i
\(339\) 0 0
\(340\) −50.0000 −0.147059
\(341\) 212.132i 0.622088i
\(342\) 0 0
\(343\) −365.000 −1.06414
\(344\) 14.1421i 0.0411109i
\(345\) 0 0
\(346\) 264.000 0.763006
\(347\) −125.000 −0.360231 −0.180115 0.983646i \(-0.557647\pi\)
−0.180115 + 0.983646i \(0.557647\pi\)
\(348\) 0 0
\(349\) 23.0000 0.0659026 0.0329513 0.999457i \(-0.489509\pi\)
0.0329513 + 0.999457i \(0.489509\pi\)
\(350\) 169.706i 0.484873i
\(351\) 0 0
\(352\) 28.2843i 0.0803530i
\(353\) −410.000 −1.16147 −0.580737 0.814092i \(-0.697235\pi\)
−0.580737 + 0.814092i \(0.697235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 254.558i − 0.715052i
\(357\) 0 0
\(358\) 180.000 0.502793
\(359\) 475.000 1.32312 0.661560 0.749892i \(-0.269895\pi\)
0.661560 + 0.749892i \(0.269895\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) −360.000 −0.994475
\(363\) 0 0
\(364\) 169.706i 0.466224i
\(365\) −25.0000 −0.0684932
\(366\) 0 0
\(367\) 230.000 0.626703 0.313351 0.949637i \(-0.398548\pi\)
0.313351 + 0.949637i \(0.398548\pi\)
\(368\) 40.0000 0.108696
\(369\) 0 0
\(370\) −36.0000 −0.0972973
\(371\) 127.279i 0.343071i
\(372\) 0 0
\(373\) − 67.8823i − 0.181990i −0.995851 0.0909950i \(-0.970995\pi\)
0.995851 0.0909950i \(-0.0290047\pi\)
\(374\) 176.777i 0.472665i
\(375\) 0 0
\(376\) − 14.1421i − 0.0376121i
\(377\) −720.000 −1.90981
\(378\) 0 0
\(379\) − 254.558i − 0.671658i −0.941923 0.335829i \(-0.890984\pi\)
0.941923 0.335829i \(-0.109016\pi\)
\(380\) −38.0000 −0.100000
\(381\) 0 0
\(382\) 414.365i 1.08472i
\(383\) 144.250i 0.376631i 0.982109 + 0.188316i \(0.0603028\pi\)
−0.982109 + 0.188316i \(0.939697\pi\)
\(384\) 0 0
\(385\) −25.0000 −0.0649351
\(386\) 84.0000 0.217617
\(387\) 0 0
\(388\) − 33.9411i − 0.0874771i
\(389\) 553.000 1.42159 0.710797 0.703397i \(-0.248334\pi\)
0.710797 + 0.703397i \(0.248334\pi\)
\(390\) 0 0
\(391\) 250.000 0.639386
\(392\) − 67.8823i − 0.173169i
\(393\) 0 0
\(394\) − 98.9949i − 0.251256i
\(395\) 42.4264i 0.107409i
\(396\) 0 0
\(397\) 335.000 0.843829 0.421914 0.906636i \(-0.361358\pi\)
0.421914 + 0.906636i \(0.361358\pi\)
\(398\) − 244.659i − 0.614721i
\(399\) 0 0
\(400\) −96.0000 −0.240000
\(401\) − 212.132i − 0.529008i −0.964385 0.264504i \(-0.914792\pi\)
0.964385 0.264504i \(-0.0852082\pi\)
\(402\) 0 0
\(403\) −720.000 −1.78660
\(404\) 100.000 0.247525
\(405\) 0 0
\(406\) −300.000 −0.738916
\(407\) 127.279i 0.312725i
\(408\) 0 0
\(409\) 721.249i 1.76344i 0.471769 + 0.881722i \(0.343616\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(410\) 60.0000 0.146341
\(411\) 0 0
\(412\) 33.9411i 0.0823814i
\(413\) − 424.264i − 1.02727i
\(414\) 0 0
\(415\) 130.000 0.313253
\(416\) −96.0000 −0.230769
\(417\) 0 0
\(418\) 134.350i 0.321412i
\(419\) −62.0000 −0.147971 −0.0739857 0.997259i \(-0.523572\pi\)
−0.0739857 + 0.997259i \(0.523572\pi\)
\(420\) 0 0
\(421\) 296.985i 0.705427i 0.935731 + 0.352714i \(0.114741\pi\)
−0.935731 + 0.352714i \(0.885259\pi\)
\(422\) 120.000 0.284360
\(423\) 0 0
\(424\) −72.0000 −0.169811
\(425\) −600.000 −1.41176
\(426\) 0 0
\(427\) 475.000 1.11241
\(428\) − 203.647i − 0.475810i
\(429\) 0 0
\(430\) − 7.07107i − 0.0164443i
\(431\) 509.117i 1.18125i 0.806948 + 0.590623i \(0.201118\pi\)
−0.806948 + 0.590623i \(0.798882\pi\)
\(432\) 0 0
\(433\) − 229.103i − 0.529105i −0.964371 0.264553i \(-0.914776\pi\)
0.964371 0.264553i \(-0.0852243\pi\)
\(434\) −300.000 −0.691244
\(435\) 0 0
\(436\) − 254.558i − 0.583850i
\(437\) 190.000 0.434783
\(438\) 0 0
\(439\) 806.102i 1.83622i 0.396323 + 0.918111i \(0.370286\pi\)
−0.396323 + 0.918111i \(0.629714\pi\)
\(440\) − 14.1421i − 0.0321412i
\(441\) 0 0
\(442\) −600.000 −1.35747
\(443\) −365.000 −0.823928 −0.411964 0.911200i \(-0.635157\pi\)
−0.411964 + 0.911200i \(0.635157\pi\)
\(444\) 0 0
\(445\) 127.279i 0.286021i
\(446\) −516.000 −1.15695
\(447\) 0 0
\(448\) −40.0000 −0.0892857
\(449\) 763.675i 1.70084i 0.526108 + 0.850418i \(0.323651\pi\)
−0.526108 + 0.850418i \(0.676349\pi\)
\(450\) 0 0
\(451\) − 212.132i − 0.470359i
\(452\) 220.617i 0.488091i
\(453\) 0 0
\(454\) −96.0000 −0.211454
\(455\) − 84.8528i − 0.186490i
\(456\) 0 0
\(457\) −265.000 −0.579869 −0.289934 0.957047i \(-0.593633\pi\)
−0.289934 + 0.957047i \(0.593633\pi\)
\(458\) 205.061i 0.447731i
\(459\) 0 0
\(460\) −20.0000 −0.0434783
\(461\) 553.000 1.19957 0.599783 0.800163i \(-0.295254\pi\)
0.599783 + 0.800163i \(0.295254\pi\)
\(462\) 0 0
\(463\) 485.000 1.04752 0.523758 0.851867i \(-0.324530\pi\)
0.523758 + 0.851867i \(0.324530\pi\)
\(464\) − 169.706i − 0.365745i
\(465\) 0 0
\(466\) 473.762i 1.01666i
\(467\) 115.000 0.246253 0.123126 0.992391i \(-0.460708\pi\)
0.123126 + 0.992391i \(0.460708\pi\)
\(468\) 0 0
\(469\) 551.543i 1.17600i
\(470\) 7.07107i 0.0150448i
\(471\) 0 0
\(472\) 240.000 0.508475
\(473\) −25.0000 −0.0528541
\(474\) 0 0
\(475\) −456.000 −0.960000
\(476\) −250.000 −0.525210
\(477\) 0 0
\(478\) 278.600i 0.582845i
\(479\) 490.000 1.02296 0.511482 0.859294i \(-0.329097\pi\)
0.511482 + 0.859294i \(0.329097\pi\)
\(480\) 0 0
\(481\) −432.000 −0.898129
\(482\) 420.000 0.871369
\(483\) 0 0
\(484\) 192.000 0.396694
\(485\) 16.9706i 0.0349909i
\(486\) 0 0
\(487\) − 610.940i − 1.25450i −0.778819 0.627249i \(-0.784181\pi\)
0.778819 0.627249i \(-0.215819\pi\)
\(488\) 268.701i 0.550616i
\(489\) 0 0
\(490\) 33.9411i 0.0692676i
\(491\) 82.0000 0.167006 0.0835031 0.996508i \(-0.473389\pi\)
0.0835031 + 0.996508i \(0.473389\pi\)
\(492\) 0 0
\(493\) − 1060.66i − 2.15144i
\(494\) −456.000 −0.923077
\(495\) 0 0
\(496\) − 169.706i − 0.342148i
\(497\) 0 0
\(498\) 0 0
\(499\) 485.000 0.971944 0.485972 0.873974i \(-0.338466\pi\)
0.485972 + 0.873974i \(0.338466\pi\)
\(500\) 98.0000 0.196000
\(501\) 0 0
\(502\) 244.659i 0.487368i
\(503\) 250.000 0.497018 0.248509 0.968630i \(-0.420059\pi\)
0.248509 + 0.968630i \(0.420059\pi\)
\(504\) 0 0
\(505\) −50.0000 −0.0990099
\(506\) 70.7107i 0.139744i
\(507\) 0 0
\(508\) − 458.205i − 0.901979i
\(509\) 169.706i 0.333410i 0.986007 + 0.166705i \(0.0533127\pi\)
−0.986007 + 0.166705i \(0.946687\pi\)
\(510\) 0 0
\(511\) −125.000 −0.244618
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) 96.0000 0.186770
\(515\) − 16.9706i − 0.0329525i
\(516\) 0 0
\(517\) 25.0000 0.0483559
\(518\) −180.000 −0.347490
\(519\) 0 0
\(520\) 48.0000 0.0923077
\(521\) 127.279i 0.244298i 0.992512 + 0.122149i \(0.0389786\pi\)
−0.992512 + 0.122149i \(0.961021\pi\)
\(522\) 0 0
\(523\) 356.382i 0.681418i 0.940169 + 0.340709i \(0.110667\pi\)
−0.940169 + 0.340709i \(0.889333\pi\)
\(524\) −326.000 −0.622137
\(525\) 0 0
\(526\) − 502.046i − 0.954460i
\(527\) − 1060.66i − 2.01264i
\(528\) 0 0
\(529\) −429.000 −0.810964
\(530\) 36.0000 0.0679245
\(531\) 0 0
\(532\) −190.000 −0.357143
\(533\) 720.000 1.35084
\(534\) 0 0
\(535\) 101.823i 0.190324i
\(536\) −312.000 −0.582090
\(537\) 0 0
\(538\) 540.000 1.00372
\(539\) 120.000 0.222635
\(540\) 0 0
\(541\) −25.0000 −0.0462107 −0.0231054 0.999733i \(-0.507355\pi\)
−0.0231054 + 0.999733i \(0.507355\pi\)
\(542\) − 155.563i − 0.287018i
\(543\) 0 0
\(544\) − 141.421i − 0.259966i
\(545\) 127.279i 0.233540i
\(546\) 0 0
\(547\) 16.9706i 0.0310248i 0.999880 + 0.0155124i \(0.00493795\pi\)
−0.999880 + 0.0155124i \(0.995062\pi\)
\(548\) 190.000 0.346715
\(549\) 0 0
\(550\) − 169.706i − 0.308556i
\(551\) − 806.102i − 1.46298i
\(552\) 0 0
\(553\) 212.132i 0.383602i
\(554\) 374.767i 0.676474i
\(555\) 0 0
\(556\) −250.000 −0.449640
\(557\) 745.000 1.33752 0.668761 0.743477i \(-0.266825\pi\)
0.668761 + 0.743477i \(0.266825\pi\)
\(558\) 0 0
\(559\) − 84.8528i − 0.151794i
\(560\) 20.0000 0.0357143
\(561\) 0 0
\(562\) −600.000 −1.06762
\(563\) − 313.955i − 0.557647i −0.960342 0.278824i \(-0.910056\pi\)
0.960342 0.278824i \(-0.0899445\pi\)
\(564\) 0 0
\(565\) − 110.309i − 0.195237i
\(566\) − 176.777i − 0.312326i
\(567\) 0 0
\(568\) 0 0
\(569\) − 424.264i − 0.745631i −0.927905 0.372816i \(-0.878392\pi\)
0.927905 0.372816i \(-0.121608\pi\)
\(570\) 0 0
\(571\) 1070.00 1.87391 0.936953 0.349456i \(-0.113634\pi\)
0.936953 + 0.349456i \(0.113634\pi\)
\(572\) − 169.706i − 0.296688i
\(573\) 0 0
\(574\) 300.000 0.522648
\(575\) −240.000 −0.417391
\(576\) 0 0
\(577\) −25.0000 −0.0433276 −0.0216638 0.999765i \(-0.506896\pi\)
−0.0216638 + 0.999765i \(0.506896\pi\)
\(578\) − 475.176i − 0.822103i
\(579\) 0 0
\(580\) 84.8528i 0.146298i
\(581\) 650.000 1.11876
\(582\) 0 0
\(583\) − 127.279i − 0.218318i
\(584\) − 70.7107i − 0.121080i
\(585\) 0 0
\(586\) −264.000 −0.450512
\(587\) −725.000 −1.23509 −0.617547 0.786534i \(-0.711873\pi\)
−0.617547 + 0.786534i \(0.711873\pi\)
\(588\) 0 0
\(589\) − 806.102i − 1.36859i
\(590\) −120.000 −0.203390
\(591\) 0 0
\(592\) − 101.823i − 0.171999i
\(593\) −650.000 −1.09612 −0.548061 0.836439i \(-0.684634\pi\)
−0.548061 + 0.836439i \(0.684634\pi\)
\(594\) 0 0
\(595\) 125.000 0.210084
\(596\) 430.000 0.721477
\(597\) 0 0
\(598\) −240.000 −0.401338
\(599\) 296.985i 0.495801i 0.968785 + 0.247901i \(0.0797406\pi\)
−0.968785 + 0.247901i \(0.920259\pi\)
\(600\) 0 0
\(601\) − 848.528i − 1.41186i −0.708281 0.705930i \(-0.750529\pi\)
0.708281 0.705930i \(-0.249471\pi\)
\(602\) − 35.3553i − 0.0587298i
\(603\) 0 0
\(604\) 169.706i 0.280970i
\(605\) −96.0000 −0.158678
\(606\) 0 0
\(607\) − 271.529i − 0.447329i −0.974666 0.223665i \(-0.928198\pi\)
0.974666 0.223665i \(-0.0718021\pi\)
\(608\) − 107.480i − 0.176777i
\(609\) 0 0
\(610\) − 134.350i − 0.220246i
\(611\) 84.8528i 0.138875i
\(612\) 0 0
\(613\) 1055.00 1.72104 0.860522 0.509413i \(-0.170138\pi\)
0.860522 + 0.509413i \(0.170138\pi\)
\(614\) 396.000 0.644951
\(615\) 0 0
\(616\) − 70.7107i − 0.114790i
\(617\) 505.000 0.818476 0.409238 0.912428i \(-0.365794\pi\)
0.409238 + 0.912428i \(0.365794\pi\)
\(618\) 0 0
\(619\) −130.000 −0.210016 −0.105008 0.994471i \(-0.533487\pi\)
−0.105008 + 0.994471i \(0.533487\pi\)
\(620\) 84.8528i 0.136859i
\(621\) 0 0
\(622\) − 332.340i − 0.534309i
\(623\) 636.396i 1.02150i
\(624\) 0 0
\(625\) 551.000 0.881600
\(626\) 438.406i 0.700329i
\(627\) 0 0
\(628\) 380.000 0.605096
\(629\) − 636.396i − 1.01176i
\(630\) 0 0
\(631\) −475.000 −0.752773 −0.376387 0.926463i \(-0.622834\pi\)
−0.376387 + 0.926463i \(0.622834\pi\)
\(632\) −120.000 −0.189873
\(633\) 0 0
\(634\) 264.000 0.416404
\(635\) 229.103i 0.360791i
\(636\) 0 0
\(637\) 407.294i 0.639393i
\(638\) 300.000 0.470219
\(639\) 0 0
\(640\) 11.3137i 0.0176777i
\(641\) 848.528i 1.32376i 0.749611 + 0.661878i \(0.230240\pi\)
−0.749611 + 0.661878i \(0.769760\pi\)
\(642\) 0 0
\(643\) −955.000 −1.48523 −0.742613 0.669721i \(-0.766414\pi\)
−0.742613 + 0.669721i \(0.766414\pi\)
\(644\) −100.000 −0.155280
\(645\) 0 0
\(646\) − 671.751i − 1.03986i
\(647\) −965.000 −1.49150 −0.745750 0.666226i \(-0.767908\pi\)
−0.745750 + 0.666226i \(0.767908\pi\)
\(648\) 0 0
\(649\) 424.264i 0.653720i
\(650\) 576.000 0.886154
\(651\) 0 0
\(652\) −220.000 −0.337423
\(653\) −935.000 −1.43185 −0.715926 0.698176i \(-0.753995\pi\)
−0.715926 + 0.698176i \(0.753995\pi\)
\(654\) 0 0
\(655\) 163.000 0.248855
\(656\) 169.706i 0.258698i
\(657\) 0 0
\(658\) 35.3553i 0.0537315i
\(659\) 84.8528i 0.128760i 0.997925 + 0.0643800i \(0.0205070\pi\)
−0.997925 + 0.0643800i \(0.979493\pi\)
\(660\) 0 0
\(661\) − 678.823i − 1.02696i −0.858101 0.513481i \(-0.828356\pi\)
0.858101 0.513481i \(-0.171644\pi\)
\(662\) 420.000 0.634441
\(663\) 0 0
\(664\) 367.696i 0.553758i
\(665\) 95.0000 0.142857
\(666\) 0 0
\(667\) − 424.264i − 0.636078i
\(668\) 118.794i 0.177835i
\(669\) 0 0
\(670\) 156.000 0.232836
\(671\) −475.000 −0.707899
\(672\) 0 0
\(673\) − 186.676i − 0.277379i −0.990336 0.138690i \(-0.955711\pi\)
0.990336 0.138690i \(-0.0442890\pi\)
\(674\) −744.000 −1.10386
\(675\) 0 0
\(676\) 238.000 0.352071
\(677\) − 907.925i − 1.34110i −0.741864 0.670550i \(-0.766058\pi\)
0.741864 0.670550i \(-0.233942\pi\)
\(678\) 0 0
\(679\) 84.8528i 0.124967i
\(680\) 70.7107i 0.103986i
\(681\) 0 0
\(682\) 300.000 0.439883
\(683\) − 1120.06i − 1.63991i −0.572429 0.819954i \(-0.693999\pi\)
0.572429 0.819954i \(-0.306001\pi\)
\(684\) 0 0
\(685\) −95.0000 −0.138686
\(686\) 516.188i 0.752461i
\(687\) 0 0
\(688\) 20.0000 0.0290698
\(689\) 432.000 0.626996
\(690\) 0 0
\(691\) −715.000 −1.03473 −0.517366 0.855764i \(-0.673087\pi\)
−0.517366 + 0.855764i \(0.673087\pi\)
\(692\) − 373.352i − 0.539527i
\(693\) 0 0
\(694\) 176.777i 0.254721i
\(695\) 125.000 0.179856
\(696\) 0 0
\(697\) 1060.66i 1.52175i
\(698\) − 32.5269i − 0.0466002i
\(699\) 0 0
\(700\) 240.000 0.342857
\(701\) 430.000 0.613409 0.306705 0.951805i \(-0.400774\pi\)
0.306705 + 0.951805i \(0.400774\pi\)
\(702\) 0 0
\(703\) − 483.661i − 0.687996i
\(704\) 40.0000 0.0568182
\(705\) 0 0
\(706\) 579.828i 0.821285i
\(707\) −250.000 −0.353607
\(708\) 0 0
\(709\) −382.000 −0.538787 −0.269394 0.963030i \(-0.586823\pi\)
−0.269394 + 0.963030i \(0.586823\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −360.000 −0.505618
\(713\) − 424.264i − 0.595041i
\(714\) 0 0
\(715\) 84.8528i 0.118675i
\(716\) − 254.558i − 0.355529i
\(717\) 0 0
\(718\) − 671.751i − 0.935587i
\(719\) 115.000 0.159944 0.0799722 0.996797i \(-0.474517\pi\)
0.0799722 + 0.996797i \(0.474517\pi\)
\(720\) 0 0
\(721\) − 84.8528i − 0.117688i
\(722\) − 510.531i − 0.707107i
\(723\) 0 0
\(724\) 509.117i 0.703200i
\(725\) 1018.23i 1.40446i
\(726\) 0 0
\(727\) −1075.00 −1.47868 −0.739340 0.673333i \(-0.764862\pi\)
−0.739340 + 0.673333i \(0.764862\pi\)
\(728\) 240.000 0.329670
\(729\) 0 0
\(730\) 35.3553i 0.0484320i
\(731\) 125.000 0.170999
\(732\) 0 0
\(733\) 530.000 0.723056 0.361528 0.932361i \(-0.382255\pi\)
0.361528 + 0.932361i \(0.382255\pi\)
\(734\) − 325.269i − 0.443146i
\(735\) 0 0
\(736\) − 56.5685i − 0.0768594i
\(737\) − 551.543i − 0.748363i
\(738\) 0 0
\(739\) −547.000 −0.740189 −0.370095 0.928994i \(-0.620675\pi\)
−0.370095 + 0.928994i \(0.620675\pi\)
\(740\) 50.9117i 0.0687996i
\(741\) 0 0
\(742\) 180.000 0.242588
\(743\) 958.837i 1.29049i 0.763974 + 0.645247i \(0.223245\pi\)
−0.763974 + 0.645247i \(0.776755\pi\)
\(744\) 0 0
\(745\) −215.000 −0.288591
\(746\) −96.0000 −0.128686
\(747\) 0 0
\(748\) 250.000 0.334225
\(749\) 509.117i 0.679729i
\(750\) 0 0
\(751\) − 169.706i − 0.225973i −0.993597 0.112986i \(-0.963958\pi\)
0.993597 0.112986i \(-0.0360417\pi\)
\(752\) −20.0000 −0.0265957
\(753\) 0 0
\(754\) 1018.23i 1.35044i
\(755\) − 84.8528i − 0.112388i
\(756\) 0 0
\(757\) 1055.00 1.39366 0.696830 0.717237i \(-0.254593\pi\)
0.696830 + 0.717237i \(0.254593\pi\)
\(758\) −360.000 −0.474934
\(759\) 0 0
\(760\) 53.7401i 0.0707107i
\(761\) −215.000 −0.282523 −0.141261 0.989972i \(-0.545116\pi\)
−0.141261 + 0.989972i \(0.545116\pi\)
\(762\) 0 0
\(763\) 636.396i 0.834071i
\(764\) 586.000 0.767016
\(765\) 0 0
\(766\) 204.000 0.266319
\(767\) −1440.00 −1.87744
\(768\) 0 0
\(769\) −145.000 −0.188557 −0.0942783 0.995546i \(-0.530054\pi\)
−0.0942783 + 0.995546i \(0.530054\pi\)
\(770\) 35.3553i 0.0459160i
\(771\) 0 0
\(772\) − 118.794i − 0.153878i
\(773\) − 407.294i − 0.526900i −0.964673 0.263450i \(-0.915140\pi\)
0.964673 0.263450i \(-0.0848604\pi\)
\(774\) 0 0
\(775\) 1018.23i 1.31385i
\(776\) −48.0000 −0.0618557
\(777\) 0 0
\(778\) − 782.060i − 1.00522i
\(779\) 806.102i 1.03479i
\(780\) 0 0
\(781\) 0 0
\(782\) − 353.553i − 0.452114i
\(783\) 0 0
\(784\) −96.0000 −0.122449
\(785\) −190.000 −0.242038
\(786\) 0 0
\(787\) 186.676i 0.237200i 0.992942 + 0.118600i \(0.0378406\pi\)
−0.992942 + 0.118600i \(0.962159\pi\)
\(788\) −140.000 −0.177665
\(789\) 0 0
\(790\) 60.0000 0.0759494
\(791\) − 551.543i − 0.697273i
\(792\) 0 0
\(793\) − 1612.20i − 2.03304i
\(794\) − 473.762i − 0.596677i
\(795\) 0 0
\(796\) −346.000 −0.434673
\(797\) 704.278i 0.883662i 0.897098 + 0.441831i \(0.145671\pi\)
−0.897098 + 0.441831i \(0.854329\pi\)
\(798\) 0 0
\(799\) −125.000 −0.156446
\(800\) 135.765i 0.169706i
\(801\) 0 0
\(802\) −300.000 −0.374065
\(803\) 125.000 0.155666
\(804\) 0 0
\(805\) 50.0000 0.0621118
\(806\) 1018.23i 1.26332i
\(807\) 0 0
\(808\) − 141.421i − 0.175026i
\(809\) 457.000 0.564895 0.282447 0.959283i \(-0.408854\pi\)
0.282447 + 0.959283i \(0.408854\pi\)
\(810\) 0 0
\(811\) 509.117i 0.627764i 0.949462 + 0.313882i \(0.101630\pi\)
−0.949462 + 0.313882i \(0.898370\pi\)
\(812\) 424.264i 0.522493i
\(813\) 0 0
\(814\) 180.000 0.221130
\(815\) 110.000 0.134969
\(816\) 0 0
\(817\) 95.0000 0.116279
\(818\) 1020.00 1.24694
\(819\) 0 0
\(820\) − 84.8528i − 0.103479i
\(821\) −167.000 −0.203410 −0.101705 0.994815i \(-0.532430\pi\)
−0.101705 + 0.994815i \(0.532430\pi\)
\(822\) 0 0
\(823\) −1315.00 −1.59781 −0.798906 0.601455i \(-0.794588\pi\)
−0.798906 + 0.601455i \(0.794588\pi\)
\(824\) 48.0000 0.0582524
\(825\) 0 0
\(826\) −600.000 −0.726392
\(827\) 534.573i 0.646400i 0.946331 + 0.323200i \(0.104759\pi\)
−0.946331 + 0.323200i \(0.895241\pi\)
\(828\) 0 0
\(829\) 763.675i 0.921201i 0.887608 + 0.460600i \(0.152366\pi\)
−0.887608 + 0.460600i \(0.847634\pi\)
\(830\) − 183.848i − 0.221503i
\(831\) 0 0
\(832\) 135.765i 0.163178i
\(833\) −600.000 −0.720288
\(834\) 0 0
\(835\) − 59.3970i − 0.0711341i
\(836\) 190.000 0.227273
\(837\) 0 0
\(838\) 87.6812i 0.104632i
\(839\) 339.411i 0.404543i 0.979330 + 0.202271i \(0.0648323\pi\)
−0.979330 + 0.202271i \(0.935168\pi\)
\(840\) 0 0
\(841\) −959.000 −1.14031
\(842\) 420.000 0.498812
\(843\) 0 0
\(844\) − 169.706i − 0.201073i
\(845\) −119.000 −0.140828
\(846\) 0 0
\(847\) −480.000 −0.566706
\(848\) 101.823i 0.120075i
\(849\) 0 0
\(850\) 848.528i 0.998268i
\(851\) − 254.558i − 0.299129i
\(852\) 0 0
\(853\) 770.000 0.902696 0.451348 0.892348i \(-0.350943\pi\)
0.451348 + 0.892348i \(0.350943\pi\)
\(854\) − 671.751i − 0.786594i
\(855\) 0 0
\(856\) −288.000 −0.336449
\(857\) 1255.82i 1.46537i 0.680568 + 0.732685i \(0.261733\pi\)
−0.680568 + 0.732685i \(0.738267\pi\)
\(858\) 0 0
\(859\) 557.000 0.648428 0.324214 0.945984i \(-0.394900\pi\)
0.324214 + 0.945984i \(0.394900\pi\)
\(860\) −10.0000 −0.0116279
\(861\) 0 0
\(862\) 720.000 0.835267
\(863\) − 992.778i − 1.15038i −0.818020 0.575190i \(-0.804928\pi\)
0.818020 0.575190i \(-0.195072\pi\)
\(864\) 0 0
\(865\) 186.676i 0.215811i
\(866\) −324.000 −0.374134
\(867\) 0 0
\(868\) 424.264i 0.488783i
\(869\) − 212.132i − 0.244111i
\(870\) 0 0
\(871\) 1872.00 2.14925
\(872\) −360.000 −0.412844
\(873\) 0 0
\(874\) − 268.701i − 0.307438i
\(875\) −245.000 −0.280000
\(876\) 0 0
\(877\) 186.676i 0.212858i 0.994320 + 0.106429i \(0.0339416\pi\)
−0.994320 + 0.106429i \(0.966058\pi\)
\(878\) 1140.00 1.29841
\(879\) 0 0
\(880\) −20.0000 −0.0227273
\(881\) 25.0000 0.0283768 0.0141884 0.999899i \(-0.495484\pi\)
0.0141884 + 0.999899i \(0.495484\pi\)
\(882\) 0 0
\(883\) 965.000 1.09287 0.546433 0.837503i \(-0.315985\pi\)
0.546433 + 0.837503i \(0.315985\pi\)
\(884\) 848.528i 0.959873i
\(885\) 0 0
\(886\) 516.188i 0.582605i
\(887\) 780.646i 0.880097i 0.897974 + 0.440048i \(0.145039\pi\)
−0.897974 + 0.440048i \(0.854961\pi\)
\(888\) 0 0
\(889\) 1145.51i 1.28854i
\(890\) 180.000 0.202247
\(891\) 0 0
\(892\) 729.734i 0.818088i
\(893\) −95.0000 −0.106383
\(894\) 0 0
\(895\) 127.279i 0.142211i
\(896\) 56.5685i 0.0631345i
\(897\) 0 0
\(898\) 1080.00 1.20267
\(899\) −1800.00 −2.00222
\(900\) 0 0
\(901\) 636.396i 0.706322i
\(902\) −300.000 −0.332594
\(903\) 0 0
\(904\) 312.000 0.345133
\(905\) − 254.558i − 0.281280i
\(906\) 0 0
\(907\) − 313.955i − 0.346147i −0.984909 0.173074i \(-0.944630\pi\)
0.984909 0.173074i \(-0.0553698\pi\)
\(908\) 135.765i 0.149520i
\(909\) 0 0
\(910\) −120.000 −0.131868
\(911\) − 933.381i − 1.02457i −0.858816 0.512284i \(-0.828800\pi\)
0.858816 0.512284i \(-0.171200\pi\)
\(912\) 0 0
\(913\) −650.000 −0.711939
\(914\) 374.767i 0.410029i
\(915\) 0 0
\(916\) 290.000 0.316594
\(917\) 815.000 0.888768
\(918\) 0 0
\(919\) −538.000 −0.585419 −0.292709 0.956201i \(-0.594557\pi\)
−0.292709 + 0.956201i \(0.594557\pi\)
\(920\) 28.2843i 0.0307438i
\(921\) 0 0
\(922\) − 782.060i − 0.848221i
\(923\) 0 0
\(924\) 0 0
\(925\) 610.940i 0.660476i
\(926\) − 685.894i − 0.740706i
\(927\) 0 0
\(928\) −240.000 −0.258621
\(929\) 742.000 0.798708 0.399354 0.916797i \(-0.369234\pi\)
0.399354 + 0.916797i \(0.369234\pi\)
\(930\) 0 0
\(931\) −456.000 −0.489796
\(932\) 670.000 0.718884
\(933\) 0 0
\(934\) − 162.635i − 0.174127i
\(935\) −125.000 −0.133690
\(936\) 0 0
\(937\) 335.000 0.357524 0.178762 0.983892i \(-0.442791\pi\)
0.178762 + 0.983892i \(0.442791\pi\)
\(938\) 780.000 0.831557
\(939\) 0 0
\(940\) 10.0000 0.0106383
\(941\) 424.264i 0.450865i 0.974259 + 0.225433i \(0.0723795\pi\)
−0.974259 + 0.225433i \(0.927620\pi\)
\(942\) 0 0
\(943\) 424.264i 0.449909i
\(944\) − 339.411i − 0.359546i
\(945\) 0 0
\(946\) 35.3553i 0.0373735i
\(947\) 1210.00 1.27772 0.638860 0.769323i \(-0.279406\pi\)
0.638860 + 0.769323i \(0.279406\pi\)
\(948\) 0 0
\(949\) 424.264i 0.447064i
\(950\) 644.881i 0.678823i
\(951\) 0 0
\(952\) 353.553i 0.371380i
\(953\) − 992.778i − 1.04174i −0.853636 0.520870i \(-0.825608\pi\)
0.853636 0.520870i \(-0.174392\pi\)
\(954\) 0 0
\(955\) −293.000 −0.306806
\(956\) 394.000 0.412134
\(957\) 0 0
\(958\) − 692.965i − 0.723345i
\(959\) −475.000 −0.495308
\(960\) 0 0
\(961\) −839.000 −0.873049
\(962\) 610.940i 0.635073i
\(963\) 0 0
\(964\) − 593.970i − 0.616151i
\(965\) 59.3970i 0.0615513i
\(966\) 0 0
\(967\) 350.000 0.361944 0.180972 0.983488i \(-0.442076\pi\)
0.180972 + 0.983488i \(0.442076\pi\)
\(968\) − 271.529i − 0.280505i
\(969\) 0 0
\(970\) 24.0000 0.0247423
\(971\) − 254.558i − 0.262161i −0.991372 0.131081i \(-0.958155\pi\)
0.991372 0.131081i \(-0.0418447\pi\)
\(972\) 0 0
\(973\) 625.000 0.642343
\(974\) −864.000 −0.887064
\(975\) 0 0
\(976\) 380.000 0.389344
\(977\) 398.808i 0.408197i 0.978950 + 0.204098i \(0.0654262\pi\)
−0.978950 + 0.204098i \(0.934574\pi\)
\(978\) 0 0
\(979\) − 636.396i − 0.650047i
\(980\) 48.0000 0.0489796
\(981\) 0 0
\(982\) − 115.966i − 0.118091i
\(983\) 695.793i 0.707826i 0.935278 + 0.353913i \(0.115149\pi\)
−0.935278 + 0.353913i \(0.884851\pi\)
\(984\) 0 0
\(985\) 70.0000 0.0710660
\(986\) −1500.00 −1.52130
\(987\) 0 0
\(988\) 644.881i 0.652714i
\(989\) 50.0000 0.0505561
\(990\) 0 0
\(991\) 381.838i 0.385305i 0.981267 + 0.192653i \(0.0617091\pi\)
−0.981267 + 0.192653i \(0.938291\pi\)
\(992\) −240.000 −0.241935
\(993\) 0 0
\(994\) 0 0
\(995\) 173.000 0.173869
\(996\) 0 0
\(997\) −265.000 −0.265797 −0.132899 0.991130i \(-0.542428\pi\)
−0.132899 + 0.991130i \(0.542428\pi\)
\(998\) − 685.894i − 0.687268i
\(999\) 0 0
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 342.3.d.a.37.1 2
3.2 odd 2 38.3.b.a.37.2 yes 2
4.3 odd 2 2736.3.o.h.721.1 2
12.11 even 2 304.3.e.c.113.1 2
15.2 even 4 950.3.d.a.949.2 4
15.8 even 4 950.3.d.a.949.3 4
15.14 odd 2 950.3.c.a.151.1 2
19.18 odd 2 inner 342.3.d.a.37.2 2
24.5 odd 2 1216.3.e.j.1025.1 2
24.11 even 2 1216.3.e.i.1025.2 2
57.56 even 2 38.3.b.a.37.1 2
76.75 even 2 2736.3.o.h.721.2 2
228.227 odd 2 304.3.e.c.113.2 2
285.113 odd 4 950.3.d.a.949.1 4
285.227 odd 4 950.3.d.a.949.4 4
285.284 even 2 950.3.c.a.151.2 2
456.227 odd 2 1216.3.e.i.1025.1 2
456.341 even 2 1216.3.e.j.1025.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 57.56 even 2
38.3.b.a.37.2 yes 2 3.2 odd 2
304.3.e.c.113.1 2 12.11 even 2
304.3.e.c.113.2 2 228.227 odd 2
342.3.d.a.37.1 2 1.1 even 1 trivial
342.3.d.a.37.2 2 19.18 odd 2 inner
950.3.c.a.151.1 2 15.14 odd 2
950.3.c.a.151.2 2 285.284 even 2
950.3.d.a.949.1 4 285.113 odd 4
950.3.d.a.949.2 4 15.2 even 4
950.3.d.a.949.3 4 15.8 even 4
950.3.d.a.949.4 4 285.227 odd 4
1216.3.e.i.1025.1 2 456.227 odd 2
1216.3.e.i.1025.2 2 24.11 even 2
1216.3.e.j.1025.1 2 24.5 odd 2
1216.3.e.j.1025.2 2 456.341 even 2
2736.3.o.h.721.1 2 4.3 odd 2
2736.3.o.h.721.2 2 76.75 even 2