Properties

Label 1089.3.c.b.604.4
Level $1089$
Weight $3$
Character 1089.604
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 604.4
Root \(0.500000 - 3.07253i\) of defining polynomial
Character \(\chi\) \(=\) 1089.604
Dual form 1089.3.c.b.604.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.31662i q^{2} -7.00000 q^{4} +9.38083 q^{5} +9.89949i q^{7} -9.94987i q^{8} +O(q^{10})\) \(q+3.31662i q^{2} -7.00000 q^{4} +9.38083 q^{5} +9.89949i q^{7} -9.94987i q^{8} +31.1127i q^{10} +12.7279i q^{13} -32.8329 q^{14} +5.00000 q^{16} -4.24264i q^{19} -65.6658 q^{20} +28.1425 q^{23} +63.0000 q^{25} -42.2137 q^{26} -69.2965i q^{28} +13.2665i q^{29} -44.0000 q^{31} -23.2164i q^{32} +92.8655i q^{35} +44.0000 q^{37} +14.0712 q^{38} -93.3381i q^{40} +39.7995i q^{41} -29.6985i q^{43} +93.3381i q^{46} +9.38083 q^{47} -49.0000 q^{49} +208.947i q^{50} -89.0955i q^{52} -65.6658 q^{53} +98.4987 q^{56} -44.0000 q^{58} +18.7617 q^{59} +69.2965i q^{61} -145.931i q^{62} +97.0000 q^{64} +119.398i q^{65} -88.0000 q^{67} -308.000 q^{70} -46.9042 q^{71} -41.0122i q^{73} +145.931i q^{74} +29.6985i q^{76} -12.7279i q^{79} +46.9042 q^{80} -132.000 q^{82} -92.8655i q^{83} +98.4987 q^{86} +112.570 q^{89} -126.000 q^{91} -196.997 q^{92} +31.1127i q^{94} -39.7995i q^{95} +70.0000 q^{97} -162.515i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{4} + 20 q^{16} + 252 q^{25} - 176 q^{31} + 176 q^{37} - 196 q^{49} - 176 q^{58} + 388 q^{64} - 352 q^{67} - 1232 q^{70} - 528 q^{82} - 504 q^{91} + 280 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.31662i 1.65831i 0.559017 + 0.829156i \(0.311179\pi\)
−0.559017 + 0.829156i \(0.688821\pi\)
\(3\) 0 0
\(4\) −7.00000 −1.75000
\(5\) 9.38083 1.87617 0.938083 0.346410i \(-0.112599\pi\)
0.938083 + 0.346410i \(0.112599\pi\)
\(6\) 0 0
\(7\) 9.89949i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) − 9.94987i − 1.24373i
\(9\) 0 0
\(10\) 31.1127i 3.11127i
\(11\) 0 0
\(12\) 0 0
\(13\) 12.7279i 0.979071i 0.871983 + 0.489535i \(0.162834\pi\)
−0.871983 + 0.489535i \(0.837166\pi\)
\(14\) −32.8329 −2.34521
\(15\) 0 0
\(16\) 5.00000 0.312500
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) − 4.24264i − 0.223297i −0.993748 0.111648i \(-0.964387\pi\)
0.993748 0.111648i \(-0.0356131\pi\)
\(20\) −65.6658 −3.28329
\(21\) 0 0
\(22\) 0 0
\(23\) 28.1425 1.22359 0.611793 0.791018i \(-0.290448\pi\)
0.611793 + 0.791018i \(0.290448\pi\)
\(24\) 0 0
\(25\) 63.0000 2.52000
\(26\) −42.2137 −1.62361
\(27\) 0 0
\(28\) − 69.2965i − 2.47487i
\(29\) 13.2665i 0.457465i 0.973489 + 0.228733i \(0.0734582\pi\)
−0.973489 + 0.228733i \(0.926542\pi\)
\(30\) 0 0
\(31\) −44.0000 −1.41935 −0.709677 0.704527i \(-0.751159\pi\)
−0.709677 + 0.704527i \(0.751159\pi\)
\(32\) − 23.2164i − 0.725512i
\(33\) 0 0
\(34\) 0 0
\(35\) 92.8655i 2.65330i
\(36\) 0 0
\(37\) 44.0000 1.18919 0.594595 0.804026i \(-0.297313\pi\)
0.594595 + 0.804026i \(0.297313\pi\)
\(38\) 14.0712 0.370296
\(39\) 0 0
\(40\) − 93.3381i − 2.33345i
\(41\) 39.7995i 0.970719i 0.874315 + 0.485360i \(0.161311\pi\)
−0.874315 + 0.485360i \(0.838689\pi\)
\(42\) 0 0
\(43\) − 29.6985i − 0.690662i −0.938481 0.345331i \(-0.887767\pi\)
0.938481 0.345331i \(-0.112233\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 93.3381i 2.02909i
\(47\) 9.38083 0.199592 0.0997961 0.995008i \(-0.468181\pi\)
0.0997961 + 0.995008i \(0.468181\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 208.947i 4.17895i
\(51\) 0 0
\(52\) − 89.0955i − 1.71337i
\(53\) −65.6658 −1.23898 −0.619489 0.785005i \(-0.712660\pi\)
−0.619489 + 0.785005i \(0.712660\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 98.4987 1.75891
\(57\) 0 0
\(58\) −44.0000 −0.758621
\(59\) 18.7617 0.317994 0.158997 0.987279i \(-0.449174\pi\)
0.158997 + 0.987279i \(0.449174\pi\)
\(60\) 0 0
\(61\) 69.2965i 1.13601i 0.823026 + 0.568004i \(0.192284\pi\)
−0.823026 + 0.568004i \(0.807716\pi\)
\(62\) − 145.931i − 2.35373i
\(63\) 0 0
\(64\) 97.0000 1.51562
\(65\) 119.398i 1.83690i
\(66\) 0 0
\(67\) −88.0000 −1.31343 −0.656716 0.754138i \(-0.728055\pi\)
−0.656716 + 0.754138i \(0.728055\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −308.000 −4.40000
\(71\) −46.9042 −0.660622 −0.330311 0.943872i \(-0.607154\pi\)
−0.330311 + 0.943872i \(0.607154\pi\)
\(72\) 0 0
\(73\) − 41.0122i − 0.561811i −0.959735 0.280905i \(-0.909365\pi\)
0.959735 0.280905i \(-0.0906347\pi\)
\(74\) 145.931i 1.97205i
\(75\) 0 0
\(76\) 29.6985i 0.390770i
\(77\) 0 0
\(78\) 0 0
\(79\) − 12.7279i − 0.161113i −0.996750 0.0805565i \(-0.974330\pi\)
0.996750 0.0805565i \(-0.0256697\pi\)
\(80\) 46.9042 0.586302
\(81\) 0 0
\(82\) −132.000 −1.60976
\(83\) − 92.8655i − 1.11886i −0.828877 0.559431i \(-0.811020\pi\)
0.828877 0.559431i \(-0.188980\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 98.4987 1.14533
\(87\) 0 0
\(88\) 0 0
\(89\) 112.570 1.26483 0.632416 0.774629i \(-0.282064\pi\)
0.632416 + 0.774629i \(0.282064\pi\)
\(90\) 0 0
\(91\) −126.000 −1.38462
\(92\) −196.997 −2.14128
\(93\) 0 0
\(94\) 31.1127i 0.330986i
\(95\) − 39.7995i − 0.418942i
\(96\) 0 0
\(97\) 70.0000 0.721649 0.360825 0.932634i \(-0.382495\pi\)
0.360825 + 0.932634i \(0.382495\pi\)
\(98\) − 162.515i − 1.65831i
\(99\) 0 0
\(100\) −441.000 −4.41000
\(101\) 26.5330i 0.262703i 0.991336 + 0.131351i \(0.0419316\pi\)
−0.991336 + 0.131351i \(0.958068\pi\)
\(102\) 0 0
\(103\) 14.0000 0.135922 0.0679612 0.997688i \(-0.478351\pi\)
0.0679612 + 0.997688i \(0.478351\pi\)
\(104\) 126.641 1.21770
\(105\) 0 0
\(106\) − 217.789i − 2.05461i
\(107\) 26.5330i 0.247972i 0.992284 + 0.123986i \(0.0395678\pi\)
−0.992284 + 0.123986i \(0.960432\pi\)
\(108\) 0 0
\(109\) − 63.6396i − 0.583850i −0.956441 0.291925i \(-0.905704\pi\)
0.956441 0.291925i \(-0.0942957\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 49.4975i 0.441942i
\(113\) −131.332 −1.16223 −0.581113 0.813823i \(-0.697383\pi\)
−0.581113 + 0.813823i \(0.697383\pi\)
\(114\) 0 0
\(115\) 264.000 2.29565
\(116\) − 92.8655i − 0.800565i
\(117\) 0 0
\(118\) 62.2254i 0.527334i
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) −229.830 −1.88386
\(123\) 0 0
\(124\) 308.000 2.48387
\(125\) 356.472 2.85177
\(126\) 0 0
\(127\) − 117.380i − 0.924250i −0.886815 0.462125i \(-0.847087\pi\)
0.886815 0.462125i \(-0.152913\pi\)
\(128\) 228.847i 1.78787i
\(129\) 0 0
\(130\) −396.000 −3.04615
\(131\) 198.997i 1.51906i 0.650469 + 0.759532i \(0.274572\pi\)
−0.650469 + 0.759532i \(0.725428\pi\)
\(132\) 0 0
\(133\) 42.0000 0.315789
\(134\) − 291.863i − 2.17808i
\(135\) 0 0
\(136\) 0 0
\(137\) −131.332 −0.958625 −0.479313 0.877644i \(-0.659114\pi\)
−0.479313 + 0.877644i \(0.659114\pi\)
\(138\) 0 0
\(139\) − 227.688i − 1.63805i −0.573761 0.819023i \(-0.694516\pi\)
0.573761 0.819023i \(-0.305484\pi\)
\(140\) − 650.058i − 4.64327i
\(141\) 0 0
\(142\) − 155.563i − 1.09552i
\(143\) 0 0
\(144\) 0 0
\(145\) 124.451i 0.858281i
\(146\) 136.022 0.931658
\(147\) 0 0
\(148\) −308.000 −2.08108
\(149\) − 119.398i − 0.801332i −0.916224 0.400666i \(-0.868779\pi\)
0.916224 0.400666i \(-0.131221\pi\)
\(150\) 0 0
\(151\) − 89.0955i − 0.590036i −0.955492 0.295018i \(-0.904674\pi\)
0.955492 0.295018i \(-0.0953257\pi\)
\(152\) −42.2137 −0.277722
\(153\) 0 0
\(154\) 0 0
\(155\) −412.757 −2.66295
\(156\) 0 0
\(157\) −44.0000 −0.280255 −0.140127 0.990133i \(-0.544751\pi\)
−0.140127 + 0.990133i \(0.544751\pi\)
\(158\) 42.2137 0.267176
\(159\) 0 0
\(160\) − 217.789i − 1.36118i
\(161\) 278.596i 1.73041i
\(162\) 0 0
\(163\) 194.000 1.19018 0.595092 0.803658i \(-0.297116\pi\)
0.595092 + 0.803658i \(0.297116\pi\)
\(164\) − 278.596i − 1.69876i
\(165\) 0 0
\(166\) 308.000 1.85542
\(167\) 238.797i 1.42992i 0.699164 + 0.714961i \(0.253556\pi\)
−0.699164 + 0.714961i \(0.746444\pi\)
\(168\) 0 0
\(169\) 7.00000 0.0414201
\(170\) 0 0
\(171\) 0 0
\(172\) 207.889i 1.20866i
\(173\) 198.997i 1.15027i 0.818057 + 0.575137i \(0.195051\pi\)
−0.818057 + 0.575137i \(0.804949\pi\)
\(174\) 0 0
\(175\) 623.668i 3.56382i
\(176\) 0 0
\(177\) 0 0
\(178\) 373.352i 2.09749i
\(179\) −75.0467 −0.419255 −0.209628 0.977781i \(-0.567225\pi\)
−0.209628 + 0.977781i \(0.567225\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.0552486 −0.0276243 0.999618i \(-0.508794\pi\)
−0.0276243 + 0.999618i \(0.508794\pi\)
\(182\) − 417.895i − 2.29612i
\(183\) 0 0
\(184\) − 280.014i − 1.52182i
\(185\) 412.757 2.23112
\(186\) 0 0
\(187\) 0 0
\(188\) −65.6658 −0.349286
\(189\) 0 0
\(190\) 132.000 0.694737
\(191\) −103.189 −0.540257 −0.270129 0.962824i \(-0.587066\pi\)
−0.270129 + 0.962824i \(0.587066\pi\)
\(192\) 0 0
\(193\) 148.492i 0.769391i 0.923044 + 0.384695i \(0.125694\pi\)
−0.923044 + 0.384695i \(0.874306\pi\)
\(194\) 232.164i 1.19672i
\(195\) 0 0
\(196\) 343.000 1.75000
\(197\) − 344.929i − 1.75091i −0.483301 0.875454i \(-0.660562\pi\)
0.483301 0.875454i \(-0.339438\pi\)
\(198\) 0 0
\(199\) 308.000 1.54774 0.773869 0.633345i \(-0.218319\pi\)
0.773869 + 0.633345i \(0.218319\pi\)
\(200\) − 626.842i − 3.13421i
\(201\) 0 0
\(202\) −88.0000 −0.435644
\(203\) −131.332 −0.646954
\(204\) 0 0
\(205\) 373.352i 1.82123i
\(206\) 46.4327i 0.225402i
\(207\) 0 0
\(208\) 63.6396i 0.305960i
\(209\) 0 0
\(210\) 0 0
\(211\) − 168.291i − 0.797590i −0.917040 0.398795i \(-0.869429\pi\)
0.917040 0.398795i \(-0.130571\pi\)
\(212\) 459.661 2.16821
\(213\) 0 0
\(214\) −88.0000 −0.411215
\(215\) − 278.596i − 1.29580i
\(216\) 0 0
\(217\) − 435.578i − 2.00727i
\(218\) 211.069 0.968205
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 126.000 0.565022 0.282511 0.959264i \(-0.408833\pi\)
0.282511 + 0.959264i \(0.408833\pi\)
\(224\) 229.830 1.02603
\(225\) 0 0
\(226\) − 435.578i − 1.92734i
\(227\) − 397.995i − 1.75328i −0.481145 0.876641i \(-0.659779\pi\)
0.481145 0.876641i \(-0.340221\pi\)
\(228\) 0 0
\(229\) −132.000 −0.576419 −0.288210 0.957567i \(-0.593060\pi\)
−0.288210 + 0.957567i \(0.593060\pi\)
\(230\) 875.589i 3.80691i
\(231\) 0 0
\(232\) 132.000 0.568966
\(233\) − 92.8655i − 0.398564i −0.979942 0.199282i \(-0.936139\pi\)
0.979942 0.199282i \(-0.0638610\pi\)
\(234\) 0 0
\(235\) 88.0000 0.374468
\(236\) −131.332 −0.556490
\(237\) 0 0
\(238\) 0 0
\(239\) 106.132i 0.444067i 0.975039 + 0.222033i \(0.0712694\pi\)
−0.975039 + 0.222033i \(0.928731\pi\)
\(240\) 0 0
\(241\) − 156.978i − 0.651360i −0.945480 0.325680i \(-0.894407\pi\)
0.945480 0.325680i \(-0.105593\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 485.075i − 1.98801i
\(245\) −459.661 −1.87617
\(246\) 0 0
\(247\) 54.0000 0.218623
\(248\) 437.794i 1.76530i
\(249\) 0 0
\(250\) 1182.28i 4.72913i
\(251\) 168.855 0.672729 0.336364 0.941732i \(-0.390803\pi\)
0.336364 + 0.941732i \(0.390803\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 389.305 1.53269
\(255\) 0 0
\(256\) −371.000 −1.44922
\(257\) 187.617 0.730026 0.365013 0.931002i \(-0.381065\pi\)
0.365013 + 0.931002i \(0.381065\pi\)
\(258\) 0 0
\(259\) 435.578i 1.68177i
\(260\) − 835.789i − 3.21457i
\(261\) 0 0
\(262\) −660.000 −2.51908
\(263\) − 39.7995i − 0.151329i −0.997133 0.0756644i \(-0.975892\pi\)
0.997133 0.0756644i \(-0.0241078\pi\)
\(264\) 0 0
\(265\) −616.000 −2.32453
\(266\) 139.298i 0.523678i
\(267\) 0 0
\(268\) 616.000 2.29851
\(269\) 328.329 1.22055 0.610277 0.792188i \(-0.291058\pi\)
0.610277 + 0.792188i \(0.291058\pi\)
\(270\) 0 0
\(271\) − 114.551i − 0.422699i −0.977411 0.211349i \(-0.932214\pi\)
0.977411 0.211349i \(-0.0677857\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) − 435.578i − 1.58970i
\(275\) 0 0
\(276\) 0 0
\(277\) 386.080i 1.39379i 0.717172 + 0.696896i \(0.245436\pi\)
−0.717172 + 0.696896i \(0.754564\pi\)
\(278\) 755.157 2.71639
\(279\) 0 0
\(280\) 924.000 3.30000
\(281\) 477.594i 1.69962i 0.527087 + 0.849811i \(0.323284\pi\)
−0.527087 + 0.849811i \(0.676716\pi\)
\(282\) 0 0
\(283\) 57.9828i 0.204886i 0.994739 + 0.102443i \(0.0326659\pi\)
−0.994739 + 0.102443i \(0.967334\pi\)
\(284\) 328.329 1.15609
\(285\) 0 0
\(286\) 0 0
\(287\) −393.995 −1.37280
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) −412.757 −1.42330
\(291\) 0 0
\(292\) 287.085i 0.983169i
\(293\) 371.462i 1.26779i 0.773420 + 0.633894i \(0.218545\pi\)
−0.773420 + 0.633894i \(0.781455\pi\)
\(294\) 0 0
\(295\) 176.000 0.596610
\(296\) − 437.794i − 1.47904i
\(297\) 0 0
\(298\) 396.000 1.32886
\(299\) 358.195i 1.19798i
\(300\) 0 0
\(301\) 294.000 0.976744
\(302\) 295.496 0.978464
\(303\) 0 0
\(304\) − 21.2132i − 0.0697803i
\(305\) 650.058i 2.13134i
\(306\) 0 0
\(307\) − 530.330i − 1.72746i −0.503955 0.863730i \(-0.668122\pi\)
0.503955 0.863730i \(-0.331878\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) − 1368.96i − 4.41600i
\(311\) −459.661 −1.47801 −0.739004 0.673701i \(-0.764704\pi\)
−0.739004 + 0.673701i \(0.764704\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) − 145.931i − 0.464750i
\(315\) 0 0
\(316\) 89.0955i 0.281948i
\(317\) −365.852 −1.15411 −0.577054 0.816706i \(-0.695798\pi\)
−0.577054 + 0.816706i \(0.695798\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 909.941 2.84356
\(321\) 0 0
\(322\) −924.000 −2.86957
\(323\) 0 0
\(324\) 0 0
\(325\) 801.859i 2.46726i
\(326\) 643.425i 1.97370i
\(327\) 0 0
\(328\) 396.000 1.20732
\(329\) 92.8655i 0.282266i
\(330\) 0 0
\(331\) −350.000 −1.05740 −0.528701 0.848808i \(-0.677321\pi\)
−0.528701 + 0.848808i \(0.677321\pi\)
\(332\) 650.058i 1.95801i
\(333\) 0 0
\(334\) −792.000 −2.37126
\(335\) −825.513 −2.46422
\(336\) 0 0
\(337\) 414.365i 1.22957i 0.788695 + 0.614784i \(0.210757\pi\)
−0.788695 + 0.614784i \(0.789243\pi\)
\(338\) 23.2164i 0.0686875i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −295.496 −0.859001
\(345\) 0 0
\(346\) −660.000 −1.90751
\(347\) 464.327i 1.33812i 0.743209 + 0.669060i \(0.233303\pi\)
−0.743209 + 0.669060i \(0.766697\pi\)
\(348\) 0 0
\(349\) 284.257i 0.814490i 0.913319 + 0.407245i \(0.133510\pi\)
−0.913319 + 0.407245i \(0.866490\pi\)
\(350\) −2068.47 −5.90992
\(351\) 0 0
\(352\) 0 0
\(353\) 281.425 0.797238 0.398619 0.917117i \(-0.369490\pi\)
0.398619 + 0.917117i \(0.369490\pi\)
\(354\) 0 0
\(355\) −440.000 −1.23944
\(356\) −787.990 −2.21345
\(357\) 0 0
\(358\) − 248.902i − 0.695256i
\(359\) − 650.058i − 1.81075i −0.424615 0.905374i \(-0.639591\pi\)
0.424615 0.905374i \(-0.360409\pi\)
\(360\) 0 0
\(361\) 343.000 0.950139
\(362\) − 33.1662i − 0.0916195i
\(363\) 0 0
\(364\) 882.000 2.42308
\(365\) − 384.728i − 1.05405i
\(366\) 0 0
\(367\) 338.000 0.920981 0.460490 0.887665i \(-0.347674\pi\)
0.460490 + 0.887665i \(0.347674\pi\)
\(368\) 140.712 0.382371
\(369\) 0 0
\(370\) 1368.96i 3.69989i
\(371\) − 650.058i − 1.75218i
\(372\) 0 0
\(373\) 250.316i 0.671088i 0.942025 + 0.335544i \(0.108920\pi\)
−0.942025 + 0.335544i \(0.891080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) − 93.3381i − 0.248240i
\(377\) −168.855 −0.447891
\(378\) 0 0
\(379\) −78.0000 −0.205805 −0.102902 0.994691i \(-0.532813\pi\)
−0.102902 + 0.994691i \(0.532813\pi\)
\(380\) 278.596i 0.733149i
\(381\) 0 0
\(382\) − 342.240i − 0.895915i
\(383\) −347.091 −0.906242 −0.453121 0.891449i \(-0.649689\pi\)
−0.453121 + 0.891449i \(0.649689\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −492.494 −1.27589
\(387\) 0 0
\(388\) −490.000 −1.26289
\(389\) 121.951 0.313498 0.156749 0.987638i \(-0.449899\pi\)
0.156749 + 0.987638i \(0.449899\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 487.544i 1.24373i
\(393\) 0 0
\(394\) 1144.00 2.90355
\(395\) − 119.398i − 0.302275i
\(396\) 0 0
\(397\) 266.000 0.670025 0.335013 0.942214i \(-0.391259\pi\)
0.335013 + 0.942214i \(0.391259\pi\)
\(398\) 1021.52i 2.56663i
\(399\) 0 0
\(400\) 315.000 0.787500
\(401\) 262.663 0.655021 0.327510 0.944848i \(-0.393790\pi\)
0.327510 + 0.944848i \(0.393790\pi\)
\(402\) 0 0
\(403\) − 560.029i − 1.38965i
\(404\) − 185.731i − 0.459730i
\(405\) 0 0
\(406\) − 435.578i − 1.07285i
\(407\) 0 0
\(408\) 0 0
\(409\) 106.066i 0.259330i 0.991558 + 0.129665i \(0.0413902\pi\)
−0.991558 + 0.129665i \(0.958610\pi\)
\(410\) −1238.27 −3.02017
\(411\) 0 0
\(412\) −98.0000 −0.237864
\(413\) 185.731i 0.449712i
\(414\) 0 0
\(415\) − 871.156i − 2.09917i
\(416\) 295.496 0.710327
\(417\) 0 0
\(418\) 0 0
\(419\) 37.5233 0.0895545 0.0447772 0.998997i \(-0.485742\pi\)
0.0447772 + 0.998997i \(0.485742\pi\)
\(420\) 0 0
\(421\) 484.000 1.14964 0.574822 0.818279i \(-0.305071\pi\)
0.574822 + 0.818279i \(0.305071\pi\)
\(422\) 558.159 1.32265
\(423\) 0 0
\(424\) 653.367i 1.54096i
\(425\) 0 0
\(426\) 0 0
\(427\) −686.000 −1.60656
\(428\) − 185.731i − 0.433951i
\(429\) 0 0
\(430\) 924.000 2.14884
\(431\) 411.261i 0.954203i 0.878848 + 0.477101i \(0.158313\pi\)
−0.878848 + 0.477101i \(0.841687\pi\)
\(432\) 0 0
\(433\) 690.000 1.59353 0.796767 0.604287i \(-0.206542\pi\)
0.796767 + 0.604287i \(0.206542\pi\)
\(434\) 1444.65 3.32868
\(435\) 0 0
\(436\) 445.477i 1.02174i
\(437\) − 119.398i − 0.273223i
\(438\) 0 0
\(439\) 499.217i 1.13717i 0.822625 + 0.568585i \(0.192509\pi\)
−0.822625 + 0.568585i \(0.807491\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 525.327 1.18584 0.592919 0.805262i \(-0.297975\pi\)
0.592919 + 0.805262i \(0.297975\pi\)
\(444\) 0 0
\(445\) 1056.00 2.37303
\(446\) 417.895i 0.936984i
\(447\) 0 0
\(448\) 960.251i 2.14342i
\(449\) −262.663 −0.584996 −0.292498 0.956266i \(-0.594487\pi\)
−0.292498 + 0.956266i \(0.594487\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 919.321 2.03390
\(453\) 0 0
\(454\) 1320.00 2.90749
\(455\) −1181.98 −2.59777
\(456\) 0 0
\(457\) − 660.438i − 1.44516i −0.691288 0.722580i \(-0.742956\pi\)
0.691288 0.722580i \(-0.257044\pi\)
\(458\) − 437.794i − 0.955883i
\(459\) 0 0
\(460\) −1848.00 −4.01739
\(461\) 371.462i 0.805774i 0.915250 + 0.402887i \(0.131993\pi\)
−0.915250 + 0.402887i \(0.868007\pi\)
\(462\) 0 0
\(463\) 174.000 0.375810 0.187905 0.982187i \(-0.439830\pi\)
0.187905 + 0.982187i \(0.439830\pi\)
\(464\) 66.3325i 0.142958i
\(465\) 0 0
\(466\) 308.000 0.660944
\(467\) 863.036 1.84804 0.924022 0.382339i \(-0.124881\pi\)
0.924022 + 0.382339i \(0.124881\pi\)
\(468\) 0 0
\(469\) − 871.156i − 1.85747i
\(470\) 291.863i 0.620985i
\(471\) 0 0
\(472\) − 186.676i − 0.395500i
\(473\) 0 0
\(474\) 0 0
\(475\) − 267.286i − 0.562708i
\(476\) 0 0
\(477\) 0 0
\(478\) −352.000 −0.736402
\(479\) − 623.525i − 1.30172i −0.759196 0.650862i \(-0.774408\pi\)
0.759196 0.650862i \(-0.225592\pi\)
\(480\) 0 0
\(481\) 560.029i 1.16430i
\(482\) 520.636 1.08016
\(483\) 0 0
\(484\) 0 0
\(485\) 656.658 1.35393
\(486\) 0 0
\(487\) −484.000 −0.993840 −0.496920 0.867796i \(-0.665536\pi\)
−0.496920 + 0.867796i \(0.665536\pi\)
\(488\) 689.491 1.41289
\(489\) 0 0
\(490\) − 1524.52i − 3.11127i
\(491\) 106.132i 0.216155i 0.994142 + 0.108077i \(0.0344694\pi\)
−0.994142 + 0.108077i \(0.965531\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 179.098i 0.362546i
\(495\) 0 0
\(496\) −220.000 −0.443548
\(497\) − 464.327i − 0.934261i
\(498\) 0 0
\(499\) 880.000 1.76353 0.881764 0.471692i \(-0.156356\pi\)
0.881764 + 0.471692i \(0.156356\pi\)
\(500\) −2495.30 −4.99060
\(501\) 0 0
\(502\) 560.029i 1.11559i
\(503\) 504.127i 1.00224i 0.865378 + 0.501120i \(0.167079\pi\)
−0.865378 + 0.501120i \(0.832921\pi\)
\(504\) 0 0
\(505\) 248.902i 0.492874i
\(506\) 0 0
\(507\) 0 0
\(508\) 821.658i 1.61744i
\(509\) 196.997 0.387028 0.193514 0.981097i \(-0.438011\pi\)
0.193514 + 0.981097i \(0.438011\pi\)
\(510\) 0 0
\(511\) 406.000 0.794521
\(512\) − 315.079i − 0.615389i
\(513\) 0 0
\(514\) 622.254i 1.21061i
\(515\) 131.332 0.255013
\(516\) 0 0
\(517\) 0 0
\(518\) −1444.65 −2.78890
\(519\) 0 0
\(520\) 1188.00 2.28462
\(521\) 544.088 1.04432 0.522158 0.852849i \(-0.325127\pi\)
0.522158 + 0.852849i \(0.325127\pi\)
\(522\) 0 0
\(523\) − 411.536i − 0.786876i −0.919351 0.393438i \(-0.871286\pi\)
0.919351 0.393438i \(-0.128714\pi\)
\(524\) − 1392.98i − 2.65836i
\(525\) 0 0
\(526\) 132.000 0.250951
\(527\) 0 0
\(528\) 0 0
\(529\) 263.000 0.497164
\(530\) − 2043.04i − 3.85479i
\(531\) 0 0
\(532\) −294.000 −0.552632
\(533\) −506.565 −0.950403
\(534\) 0 0
\(535\) 248.902i 0.465237i
\(536\) 875.589i 1.63356i
\(537\) 0 0
\(538\) 1088.94i 2.02406i
\(539\) 0 0
\(540\) 0 0
\(541\) 422.850i 0.781608i 0.920474 + 0.390804i \(0.127803\pi\)
−0.920474 + 0.390804i \(0.872197\pi\)
\(542\) 379.924 0.700966
\(543\) 0 0
\(544\) 0 0
\(545\) − 596.992i − 1.09540i
\(546\) 0 0
\(547\) − 999.849i − 1.82788i −0.405852 0.913939i \(-0.633025\pi\)
0.405852 0.913939i \(-0.366975\pi\)
\(548\) 919.321 1.67759
\(549\) 0 0
\(550\) 0 0
\(551\) 56.2850 0.102151
\(552\) 0 0
\(553\) 126.000 0.227848
\(554\) −1280.48 −2.31134
\(555\) 0 0
\(556\) 1593.82i 2.86658i
\(557\) 610.259i 1.09562i 0.836604 + 0.547809i \(0.184538\pi\)
−0.836604 + 0.547809i \(0.815462\pi\)
\(558\) 0 0
\(559\) 378.000 0.676208
\(560\) 464.327i 0.829156i
\(561\) 0 0
\(562\) −1584.00 −2.81851
\(563\) 159.198i 0.282767i 0.989955 + 0.141384i \(0.0451551\pi\)
−0.989955 + 0.141384i \(0.954845\pi\)
\(564\) 0 0
\(565\) −1232.00 −2.18053
\(566\) −192.307 −0.339765
\(567\) 0 0
\(568\) 466.690i 0.821638i
\(569\) − 742.924i − 1.30567i −0.757502 0.652833i \(-0.773580\pi\)
0.757502 0.652833i \(-0.226420\pi\)
\(570\) 0 0
\(571\) 465.276i 0.814845i 0.913240 + 0.407422i \(0.133572\pi\)
−0.913240 + 0.407422i \(0.866428\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 1306.73i − 2.27654i
\(575\) 1772.98 3.08344
\(576\) 0 0
\(577\) 880.000 1.52513 0.762565 0.646912i \(-0.223940\pi\)
0.762565 + 0.646912i \(0.223940\pi\)
\(578\) 958.505i 1.65831i
\(579\) 0 0
\(580\) − 871.156i − 1.50199i
\(581\) 919.321 1.58231
\(582\) 0 0
\(583\) 0 0
\(584\) −408.066 −0.698743
\(585\) 0 0
\(586\) −1232.00 −2.10239
\(587\) 694.182 1.18259 0.591296 0.806455i \(-0.298616\pi\)
0.591296 + 0.806455i \(0.298616\pi\)
\(588\) 0 0
\(589\) 186.676i 0.316938i
\(590\) 583.726i 0.989366i
\(591\) 0 0
\(592\) 220.000 0.371622
\(593\) − 928.655i − 1.56603i −0.622004 0.783014i \(-0.713681\pi\)
0.622004 0.783014i \(-0.286319\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 835.789i 1.40233i
\(597\) 0 0
\(598\) −1188.00 −1.98662
\(599\) 647.277 1.08060 0.540298 0.841474i \(-0.318311\pi\)
0.540298 + 0.841474i \(0.318311\pi\)
\(600\) 0 0
\(601\) − 165.463i − 0.275313i −0.990480 0.137656i \(-0.956043\pi\)
0.990480 0.137656i \(-0.0439570\pi\)
\(602\) 975.088i 1.61975i
\(603\) 0 0
\(604\) 623.668i 1.03256i
\(605\) 0 0
\(606\) 0 0
\(607\) 173.948i 0.286570i 0.989681 + 0.143285i \(0.0457666\pi\)
−0.989681 + 0.143285i \(0.954233\pi\)
\(608\) −98.4987 −0.162004
\(609\) 0 0
\(610\) −2156.00 −3.53443
\(611\) 119.398i 0.195415i
\(612\) 0 0
\(613\) − 89.0955i − 0.145343i −0.997356 0.0726717i \(-0.976847\pi\)
0.997356 0.0726717i \(-0.0231525\pi\)
\(614\) 1758.91 2.86467
\(615\) 0 0
\(616\) 0 0
\(617\) 375.233 0.608158 0.304079 0.952647i \(-0.401651\pi\)
0.304079 + 0.952647i \(0.401651\pi\)
\(618\) 0 0
\(619\) 490.000 0.791599 0.395800 0.918337i \(-0.370467\pi\)
0.395800 + 0.918337i \(0.370467\pi\)
\(620\) 2889.30 4.66016
\(621\) 0 0
\(622\) − 1524.52i − 2.45100i
\(623\) 1114.39i 1.78874i
\(624\) 0 0
\(625\) 1769.00 2.83040
\(626\) 0 0
\(627\) 0 0
\(628\) 308.000 0.490446
\(629\) 0 0
\(630\) 0 0
\(631\) −308.000 −0.488114 −0.244057 0.969761i \(-0.578478\pi\)
−0.244057 + 0.969761i \(0.578478\pi\)
\(632\) −126.641 −0.200382
\(633\) 0 0
\(634\) − 1213.40i − 1.91387i
\(635\) − 1101.12i − 1.73405i
\(636\) 0 0
\(637\) − 623.668i − 0.979071i
\(638\) 0 0
\(639\) 0 0
\(640\) 2146.78i 3.35434i
\(641\) 1257.03 1.96105 0.980524 0.196401i \(-0.0629253\pi\)
0.980524 + 0.196401i \(0.0629253\pi\)
\(642\) 0 0
\(643\) −714.000 −1.11042 −0.555210 0.831710i \(-0.687362\pi\)
−0.555210 + 0.831710i \(0.687362\pi\)
\(644\) − 1950.18i − 3.02822i
\(645\) 0 0
\(646\) 0 0
\(647\) 46.9042 0.0724948 0.0362474 0.999343i \(-0.488460\pi\)
0.0362474 + 0.999343i \(0.488460\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −2659.47 −4.09149
\(651\) 0 0
\(652\) −1358.00 −2.08282
\(653\) −853.656 −1.30728 −0.653641 0.756804i \(-0.726760\pi\)
−0.653641 + 0.756804i \(0.726760\pi\)
\(654\) 0 0
\(655\) 1866.76i 2.85002i
\(656\) 198.997i 0.303350i
\(657\) 0 0
\(658\) −308.000 −0.468085
\(659\) 1034.79i 1.57024i 0.619345 + 0.785119i \(0.287398\pi\)
−0.619345 + 0.785119i \(0.712602\pi\)
\(660\) 0 0
\(661\) −308.000 −0.465961 −0.232980 0.972481i \(-0.574848\pi\)
−0.232980 + 0.972481i \(0.574848\pi\)
\(662\) − 1160.82i − 1.75350i
\(663\) 0 0
\(664\) −924.000 −1.39157
\(665\) 393.995 0.592474
\(666\) 0 0
\(667\) 373.352i 0.559749i
\(668\) − 1671.58i − 2.50236i
\(669\) 0 0
\(670\) − 2737.92i − 4.08644i
\(671\) 0 0
\(672\) 0 0
\(673\) − 852.771i − 1.26712i −0.773694 0.633559i \(-0.781593\pi\)
0.773694 0.633559i \(-0.218407\pi\)
\(674\) −1374.29 −2.03901
\(675\) 0 0
\(676\) −49.0000 −0.0724852
\(677\) 371.462i 0.548688i 0.961632 + 0.274344i \(0.0884607\pi\)
−0.961632 + 0.274344i \(0.911539\pi\)
\(678\) 0 0
\(679\) 692.965i 1.02057i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −262.663 −0.384573 −0.192286 0.981339i \(-0.561590\pi\)
−0.192286 + 0.981339i \(0.561590\pi\)
\(684\) 0 0
\(685\) −1232.00 −1.79854
\(686\) 0 0
\(687\) 0 0
\(688\) − 148.492i − 0.215832i
\(689\) − 835.789i − 1.21305i
\(690\) 0 0
\(691\) −1126.00 −1.62952 −0.814761 0.579797i \(-0.803132\pi\)
−0.814761 + 0.579797i \(0.803132\pi\)
\(692\) − 1392.98i − 2.01298i
\(693\) 0 0
\(694\) −1540.00 −2.21902
\(695\) − 2135.91i − 3.07325i
\(696\) 0 0
\(697\) 0 0
\(698\) −942.774 −1.35068
\(699\) 0 0
\(700\) − 4365.68i − 6.23668i
\(701\) − 1114.39i − 1.58971i −0.606800 0.794854i \(-0.707547\pi\)
0.606800 0.794854i \(-0.292453\pi\)
\(702\) 0 0
\(703\) − 186.676i − 0.265542i
\(704\) 0 0
\(705\) 0 0
\(706\) 933.381i 1.32207i
\(707\) −262.663 −0.371518
\(708\) 0 0
\(709\) −44.0000 −0.0620592 −0.0310296 0.999518i \(-0.509879\pi\)
−0.0310296 + 0.999518i \(0.509879\pi\)
\(710\) − 1459.31i − 2.05537i
\(711\) 0 0
\(712\) − 1120.06i − 1.57311i
\(713\) −1238.27 −1.73670
\(714\) 0 0
\(715\) 0 0
\(716\) 525.327 0.733696
\(717\) 0 0
\(718\) 2156.00 3.00279
\(719\) −778.609 −1.08291 −0.541453 0.840731i \(-0.682125\pi\)
−0.541453 + 0.840731i \(0.682125\pi\)
\(720\) 0 0
\(721\) 138.593i 0.192223i
\(722\) 1137.60i 1.57563i
\(723\) 0 0
\(724\) 70.0000 0.0966851
\(725\) 835.789i 1.15281i
\(726\) 0 0
\(727\) −1012.00 −1.39202 −0.696011 0.718031i \(-0.745044\pi\)
−0.696011 + 0.718031i \(0.745044\pi\)
\(728\) 1253.68i 1.72209i
\(729\) 0 0
\(730\) 1276.00 1.74795
\(731\) 0 0
\(732\) 0 0
\(733\) 425.678i 0.580734i 0.956915 + 0.290367i \(0.0937774\pi\)
−0.956915 + 0.290367i \(0.906223\pi\)
\(734\) 1121.02i 1.52727i
\(735\) 0 0
\(736\) − 653.367i − 0.887726i
\(737\) 0 0
\(738\) 0 0
\(739\) − 292.742i − 0.396133i −0.980189 0.198066i \(-0.936534\pi\)
0.980189 0.198066i \(-0.0634662\pi\)
\(740\) −2889.30 −3.90445
\(741\) 0 0
\(742\) 2156.00 2.90566
\(743\) − 1286.85i − 1.73197i −0.500074 0.865983i \(-0.666694\pi\)
0.500074 0.865983i \(-0.333306\pi\)
\(744\) 0 0
\(745\) − 1120.06i − 1.50343i
\(746\) −830.204 −1.11287
\(747\) 0 0
\(748\) 0 0
\(749\) −262.663 −0.350685
\(750\) 0 0
\(751\) 572.000 0.761651 0.380826 0.924647i \(-0.375640\pi\)
0.380826 + 0.924647i \(0.375640\pi\)
\(752\) 46.9042 0.0623725
\(753\) 0 0
\(754\) − 560.029i − 0.742743i
\(755\) − 835.789i − 1.10701i
\(756\) 0 0
\(757\) −748.000 −0.988111 −0.494055 0.869430i \(-0.664486\pi\)
−0.494055 + 0.869430i \(0.664486\pi\)
\(758\) − 258.697i − 0.341289i
\(759\) 0 0
\(760\) −396.000 −0.521053
\(761\) − 451.061i − 0.592721i −0.955076 0.296361i \(-0.904227\pi\)
0.955076 0.296361i \(-0.0957730\pi\)
\(762\) 0 0
\(763\) 630.000 0.825688
\(764\) 722.324 0.945450
\(765\) 0 0
\(766\) − 1151.17i − 1.50283i
\(767\) 238.797i 0.311339i
\(768\) 0 0
\(769\) − 779.232i − 1.01331i −0.862150 0.506653i \(-0.830883\pi\)
0.862150 0.506653i \(-0.169117\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1039.45i − 1.34643i
\(773\) 872.417 1.12861 0.564306 0.825566i \(-0.309144\pi\)
0.564306 + 0.825566i \(0.309144\pi\)
\(774\) 0 0
\(775\) −2772.00 −3.57677
\(776\) − 696.491i − 0.897540i
\(777\) 0 0
\(778\) 404.465i 0.519878i
\(779\) 168.855 0.216759
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −245.000 −0.312500
\(785\) −412.757 −0.525805
\(786\) 0 0
\(787\) 335.169i 0.425881i 0.977065 + 0.212941i \(0.0683041\pi\)
−0.977065 + 0.212941i \(0.931696\pi\)
\(788\) 2414.50i 3.06409i
\(789\) 0 0
\(790\) 396.000 0.501266
\(791\) − 1300.12i − 1.64364i
\(792\) 0 0
\(793\) −882.000 −1.11223
\(794\) 882.222i 1.11111i
\(795\) 0 0
\(796\) −2156.00 −2.70854
\(797\) 290.806 0.364876 0.182438 0.983217i \(-0.441601\pi\)
0.182438 + 0.983217i \(0.441601\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) − 1462.63i − 1.82829i
\(801\) 0 0
\(802\) 871.156i 1.08623i
\(803\) 0 0
\(804\) 0 0
\(805\) 2613.47i 3.24654i
\(806\) 1857.40 2.30447
\(807\) 0 0
\(808\) 264.000 0.326733
\(809\) − 331.662i − 0.409966i −0.978766 0.204983i \(-0.934286\pi\)
0.978766 0.204983i \(-0.0657139\pi\)
\(810\) 0 0
\(811\) 626.497i 0.772499i 0.922394 + 0.386249i \(0.126230\pi\)
−0.922394 + 0.386249i \(0.873770\pi\)
\(812\) 919.321 1.13217
\(813\) 0 0
\(814\) 0 0
\(815\) 1819.88 2.23298
\(816\) 0 0
\(817\) −126.000 −0.154223
\(818\) −351.781 −0.430050
\(819\) 0 0
\(820\) − 2613.47i − 3.18715i
\(821\) − 159.198i − 0.193907i −0.995289 0.0969537i \(-0.969090\pi\)
0.995289 0.0969537i \(-0.0309099\pi\)
\(822\) 0 0
\(823\) −18.0000 −0.0218712 −0.0109356 0.999940i \(-0.503481\pi\)
−0.0109356 + 0.999940i \(0.503481\pi\)
\(824\) − 139.298i − 0.169051i
\(825\) 0 0
\(826\) −616.000 −0.745763
\(827\) 198.997i 0.240626i 0.992736 + 0.120313i \(0.0383898\pi\)
−0.992736 + 0.120313i \(0.961610\pi\)
\(828\) 0 0
\(829\) −602.000 −0.726176 −0.363088 0.931755i \(-0.618278\pi\)
−0.363088 + 0.931755i \(0.618278\pi\)
\(830\) 2889.30 3.48108
\(831\) 0 0
\(832\) 1234.61i 1.48390i
\(833\) 0 0
\(834\) 0 0
\(835\) 2240.11i 2.68277i
\(836\) 0 0
\(837\) 0 0
\(838\) 124.451i 0.148509i
\(839\) −1153.84 −1.37526 −0.687629 0.726062i \(-0.741349\pi\)
−0.687629 + 0.726062i \(0.741349\pi\)
\(840\) 0 0
\(841\) 665.000 0.790725
\(842\) 1605.25i 1.90647i
\(843\) 0 0
\(844\) 1178.04i 1.39578i
\(845\) 65.6658 0.0777110
\(846\) 0 0
\(847\) 0 0
\(848\) −328.329 −0.387181
\(849\) 0 0
\(850\) 0 0
\(851\) 1238.27 1.45508
\(852\) 0 0
\(853\) − 629.325i − 0.737778i −0.929473 0.368889i \(-0.879738\pi\)
0.929473 0.368889i \(-0.120262\pi\)
\(854\) − 2275.20i − 2.66417i
\(855\) 0 0
\(856\) 264.000 0.308411
\(857\) − 92.8655i − 0.108361i −0.998531 0.0541806i \(-0.982745\pi\)
0.998531 0.0541806i \(-0.0172547\pi\)
\(858\) 0 0
\(859\) 526.000 0.612340 0.306170 0.951977i \(-0.400952\pi\)
0.306170 + 0.951977i \(0.400952\pi\)
\(860\) 1950.18i 2.26765i
\(861\) 0 0
\(862\) −1364.00 −1.58237
\(863\) −1604.12 −1.85877 −0.929387 0.369106i \(-0.879664\pi\)
−0.929387 + 0.369106i \(0.879664\pi\)
\(864\) 0 0
\(865\) 1866.76i 2.15811i
\(866\) 2288.47i 2.64258i
\(867\) 0 0
\(868\) 3049.04i 3.51272i
\(869\) 0 0
\(870\) 0 0
\(871\) − 1120.06i − 1.28594i
\(872\) −633.206 −0.726154
\(873\) 0 0
\(874\) 396.000 0.453089
\(875\) 3528.89i 4.03302i
\(876\) 0 0
\(877\) − 346.482i − 0.395077i −0.980295 0.197538i \(-0.936705\pi\)
0.980295 0.197538i \(-0.0632947\pi\)
\(878\) −1655.72 −1.88578
\(879\) 0 0
\(880\) 0 0
\(881\) −1538.46 −1.74626 −0.873131 0.487486i \(-0.837914\pi\)
−0.873131 + 0.487486i \(0.837914\pi\)
\(882\) 0 0
\(883\) 1408.00 1.59456 0.797282 0.603607i \(-0.206270\pi\)
0.797282 + 0.603607i \(0.206270\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1742.31i 1.96649i
\(887\) 278.596i 0.314088i 0.987592 + 0.157044i \(0.0501965\pi\)
−0.987592 + 0.157044i \(0.949803\pi\)
\(888\) 0 0
\(889\) 1162.00 1.30709
\(890\) 3502.36i 3.93523i
\(891\) 0 0
\(892\) −882.000 −0.988789
\(893\) − 39.7995i − 0.0445683i
\(894\) 0 0
\(895\) −704.000 −0.786592
\(896\) −2265.47 −2.52843
\(897\) 0 0
\(898\) − 871.156i − 0.970106i
\(899\) − 583.726i − 0.649306i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 1306.73i 1.44550i
\(905\) −93.8083 −0.103656
\(906\) 0 0
\(907\) −714.000 −0.787211 −0.393605 0.919280i \(-0.628772\pi\)
−0.393605 + 0.919280i \(0.628772\pi\)
\(908\) 2785.96i 3.06824i
\(909\) 0 0
\(910\) − 3920.20i − 4.30791i
\(911\) −1510.31 −1.65786 −0.828932 0.559350i \(-0.811051\pi\)
−0.828932 + 0.559350i \(0.811051\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2190.42 2.39653
\(915\) 0 0
\(916\) 924.000 1.00873
\(917\) −1969.97 −2.14828
\(918\) 0 0
\(919\) 504.874i 0.549373i 0.961534 + 0.274687i \(0.0885742\pi\)
−0.961534 + 0.274687i \(0.911426\pi\)
\(920\) − 2626.77i − 2.85518i
\(921\) 0 0
\(922\) −1232.00 −1.33623
\(923\) − 596.992i − 0.646796i
\(924\) 0 0
\(925\) 2772.00 2.99676
\(926\) 577.093i 0.623210i
\(927\) 0 0
\(928\) 308.000 0.331897
\(929\) 206.378 0.222151 0.111076 0.993812i \(-0.464570\pi\)
0.111076 + 0.993812i \(0.464570\pi\)
\(930\) 0 0
\(931\) 207.889i 0.223297i
\(932\) 650.058i 0.697488i
\(933\) 0 0
\(934\) 2862.37i 3.06463i
\(935\) 0 0
\(936\) 0 0
\(937\) 168.291i 0.179607i 0.995960 + 0.0898033i \(0.0286238\pi\)
−0.995960 + 0.0898033i \(0.971376\pi\)
\(938\) 2889.30 3.08027
\(939\) 0 0
\(940\) −616.000 −0.655319
\(941\) 530.660i 0.563932i 0.959424 + 0.281966i \(0.0909865\pi\)
−0.959424 + 0.281966i \(0.909013\pi\)
\(942\) 0 0
\(943\) 1120.06i 1.18776i
\(944\) 93.8083 0.0993732
\(945\) 0 0
\(946\) 0 0
\(947\) 694.182 0.733032 0.366516 0.930412i \(-0.380550\pi\)
0.366516 + 0.930412i \(0.380550\pi\)
\(948\) 0 0
\(949\) 522.000 0.550053
\(950\) 886.489 0.933146
\(951\) 0 0
\(952\) 0 0
\(953\) 172.464i 0.180970i 0.995898 + 0.0904850i \(0.0288417\pi\)
−0.995898 + 0.0904850i \(0.971158\pi\)
\(954\) 0 0
\(955\) −968.000 −1.01361
\(956\) − 742.924i − 0.777117i
\(957\) 0 0
\(958\) 2068.00 2.15866
\(959\) − 1300.12i − 1.35570i
\(960\) 0 0
\(961\) 975.000 1.01457
\(962\) −1857.40 −1.93077
\(963\) 0 0
\(964\) 1098.84i 1.13988i
\(965\) 1392.98i 1.44351i
\(966\) 0 0
\(967\) − 683.065i − 0.706376i −0.935552 0.353188i \(-0.885098\pi\)
0.935552 0.353188i \(-0.114902\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 2177.89i 2.24525i
\(971\) 56.2850 0.0579660 0.0289830 0.999580i \(-0.490773\pi\)
0.0289830 + 0.999580i \(0.490773\pi\)
\(972\) 0 0
\(973\) 2254.00 2.31655
\(974\) − 1605.25i − 1.64810i
\(975\) 0 0
\(976\) 346.482i 0.355002i
\(977\) 337.710 0.345660 0.172830 0.984952i \(-0.444709\pi\)
0.172830 + 0.984952i \(0.444709\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3217.63 3.28329
\(981\) 0 0
\(982\) −352.000 −0.358452
\(983\) −459.661 −0.467610 −0.233805 0.972283i \(-0.575118\pi\)
−0.233805 + 0.972283i \(0.575118\pi\)
\(984\) 0 0
\(985\) − 3235.72i − 3.28500i
\(986\) 0 0
\(987\) 0 0
\(988\) −378.000 −0.382591
\(989\) − 835.789i − 0.845085i
\(990\) 0 0
\(991\) −1452.00 −1.46519 −0.732593 0.680667i \(-0.761690\pi\)
−0.732593 + 0.680667i \(0.761690\pi\)
\(992\) 1021.52i 1.02976i
\(993\) 0 0
\(994\) 1540.00 1.54930
\(995\) 2889.30 2.90382
\(996\) 0 0
\(997\) − 1752.21i − 1.75748i −0.477298 0.878742i \(-0.658384\pi\)
0.477298 0.878742i \(-0.341616\pi\)
\(998\) 2918.63i 2.92448i
\(999\) 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 1089.3.c.b.604.4 yes 4
3.2 odd 2 inner 1089.3.c.b.604.1 4
11.10 odd 2 inner 1089.3.c.b.604.2 yes 4
33.32 even 2 inner 1089.3.c.b.604.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.c.b.604.1 4 3.2 odd 2 inner
1089.3.c.b.604.2 yes 4 11.10 odd 2 inner
1089.3.c.b.604.3 yes 4 33.32 even 2 inner
1089.3.c.b.604.4 yes 4 1.1 even 1 trivial