Properties

Label 1143.3.b.a.890.5
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.5
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.80

$q$-expansion

\(f(q)\) \(=\) \(q-3.63886i q^{2} -9.24130 q^{4} -3.49336i q^{5} -9.62828 q^{7} +19.0724i q^{8} +O(q^{10})\) \(q-3.63886i q^{2} -9.24130 q^{4} -3.49336i q^{5} -9.62828 q^{7} +19.0724i q^{8} -12.7118 q^{10} +19.0515i q^{11} +20.1962 q^{13} +35.0360i q^{14} +32.4365 q^{16} -0.819801i q^{17} +7.28753 q^{19} +32.2832i q^{20} +69.3258 q^{22} +0.446991i q^{23} +12.7965 q^{25} -73.4911i q^{26} +88.9779 q^{28} -43.3692i q^{29} -12.5475 q^{31} -41.7423i q^{32} -2.98314 q^{34} +33.6350i q^{35} -31.1753 q^{37} -26.5183i q^{38} +66.6266 q^{40} -17.5224i q^{41} +21.4646 q^{43} -176.061i q^{44} +1.62654 q^{46} +61.7080i q^{47} +43.7038 q^{49} -46.5645i q^{50} -186.639 q^{52} +65.6615i q^{53} +66.5537 q^{55} -183.634i q^{56} -157.814 q^{58} +15.3487i q^{59} +59.7978 q^{61} +45.6585i q^{62} -22.1485 q^{64} -70.5525i q^{65} -51.8217 q^{67} +7.57603i q^{68} +122.393 q^{70} +11.6672i q^{71} +123.785 q^{73} +113.443i q^{74} -67.3463 q^{76} -183.433i q^{77} +52.3561 q^{79} -113.312i q^{80} -63.7615 q^{82} -92.4955i q^{83} -2.86386 q^{85} -78.1067i q^{86} -363.357 q^{88} +1.56409i q^{89} -194.455 q^{91} -4.13078i q^{92} +224.547 q^{94} -25.4580i q^{95} -40.8492 q^{97} -159.032i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.63886i − 1.81943i −0.415233 0.909715i \(-0.636300\pi\)
0.415233 0.909715i \(-0.363700\pi\)
\(3\) 0 0
\(4\) −9.24130 −2.31033
\(5\) − 3.49336i − 0.698671i −0.936998 0.349336i \(-0.886407\pi\)
0.936998 0.349336i \(-0.113593\pi\)
\(6\) 0 0
\(7\) −9.62828 −1.37547 −0.687734 0.725962i \(-0.741395\pi\)
−0.687734 + 0.725962i \(0.741395\pi\)
\(8\) 19.0724i 2.38405i
\(9\) 0 0
\(10\) −12.7118 −1.27118
\(11\) 19.0515i 1.73196i 0.500083 + 0.865978i \(0.333303\pi\)
−0.500083 + 0.865978i \(0.666697\pi\)
\(12\) 0 0
\(13\) 20.1962 1.55355 0.776776 0.629777i \(-0.216854\pi\)
0.776776 + 0.629777i \(0.216854\pi\)
\(14\) 35.0360i 2.50257i
\(15\) 0 0
\(16\) 32.4365 2.02728
\(17\) − 0.819801i − 0.0482236i −0.999709 0.0241118i \(-0.992324\pi\)
0.999709 0.0241118i \(-0.00767577\pi\)
\(18\) 0 0
\(19\) 7.28753 0.383554 0.191777 0.981438i \(-0.438575\pi\)
0.191777 + 0.981438i \(0.438575\pi\)
\(20\) 32.2832i 1.61416i
\(21\) 0 0
\(22\) 69.3258 3.15117
\(23\) 0.446991i 0.0194344i 0.999953 + 0.00971719i \(0.00309313\pi\)
−0.999953 + 0.00971719i \(0.996907\pi\)
\(24\) 0 0
\(25\) 12.7965 0.511858
\(26\) − 73.4911i − 2.82658i
\(27\) 0 0
\(28\) 88.9779 3.17778
\(29\) − 43.3692i − 1.49549i −0.663986 0.747745i \(-0.731137\pi\)
0.663986 0.747745i \(-0.268863\pi\)
\(30\) 0 0
\(31\) −12.5475 −0.404757 −0.202379 0.979307i \(-0.564867\pi\)
−0.202379 + 0.979307i \(0.564867\pi\)
\(32\) − 41.7423i − 1.30445i
\(33\) 0 0
\(34\) −2.98314 −0.0877395
\(35\) 33.6350i 0.961001i
\(36\) 0 0
\(37\) −31.1753 −0.842576 −0.421288 0.906927i \(-0.638422\pi\)
−0.421288 + 0.906927i \(0.638422\pi\)
\(38\) − 26.5183i − 0.697850i
\(39\) 0 0
\(40\) 66.6266 1.66566
\(41\) − 17.5224i − 0.427375i −0.976902 0.213688i \(-0.931453\pi\)
0.976902 0.213688i \(-0.0685474\pi\)
\(42\) 0 0
\(43\) 21.4646 0.499177 0.249588 0.968352i \(-0.419705\pi\)
0.249588 + 0.968352i \(0.419705\pi\)
\(44\) − 176.061i − 4.00138i
\(45\) 0 0
\(46\) 1.62654 0.0353595
\(47\) 61.7080i 1.31294i 0.754354 + 0.656468i \(0.227950\pi\)
−0.754354 + 0.656468i \(0.772050\pi\)
\(48\) 0 0
\(49\) 43.7038 0.891915
\(50\) − 46.5645i − 0.931290i
\(51\) 0 0
\(52\) −186.639 −3.58921
\(53\) 65.6615i 1.23890i 0.785038 + 0.619448i \(0.212644\pi\)
−0.785038 + 0.619448i \(0.787356\pi\)
\(54\) 0 0
\(55\) 66.5537 1.21007
\(56\) − 183.634i − 3.27918i
\(57\) 0 0
\(58\) −157.814 −2.72094
\(59\) 15.3487i 0.260147i 0.991504 + 0.130073i \(0.0415213\pi\)
−0.991504 + 0.130073i \(0.958479\pi\)
\(60\) 0 0
\(61\) 59.7978 0.980292 0.490146 0.871640i \(-0.336943\pi\)
0.490146 + 0.871640i \(0.336943\pi\)
\(62\) 45.6585i 0.736427i
\(63\) 0 0
\(64\) −22.1485 −0.346070
\(65\) − 70.5525i − 1.08542i
\(66\) 0 0
\(67\) −51.8217 −0.773458 −0.386729 0.922193i \(-0.626395\pi\)
−0.386729 + 0.922193i \(0.626395\pi\)
\(68\) 7.57603i 0.111412i
\(69\) 0 0
\(70\) 122.393 1.74847
\(71\) 11.6672i 0.164327i 0.996619 + 0.0821635i \(0.0261830\pi\)
−0.996619 + 0.0821635i \(0.973817\pi\)
\(72\) 0 0
\(73\) 123.785 1.69568 0.847842 0.530248i \(-0.177901\pi\)
0.847842 + 0.530248i \(0.177901\pi\)
\(74\) 113.443i 1.53301i
\(75\) 0 0
\(76\) −67.3463 −0.886136
\(77\) − 183.433i − 2.38225i
\(78\) 0 0
\(79\) 52.3561 0.662736 0.331368 0.943502i \(-0.392490\pi\)
0.331368 + 0.943502i \(0.392490\pi\)
\(80\) − 113.312i − 1.41640i
\(81\) 0 0
\(82\) −63.7615 −0.777579
\(83\) − 92.4955i − 1.11440i −0.830377 0.557202i \(-0.811875\pi\)
0.830377 0.557202i \(-0.188125\pi\)
\(84\) 0 0
\(85\) −2.86386 −0.0336925
\(86\) − 78.1067i − 0.908217i
\(87\) 0 0
\(88\) −363.357 −4.12906
\(89\) 1.56409i 0.0175740i 0.999961 + 0.00878700i \(0.00279703\pi\)
−0.999961 + 0.00878700i \(0.997203\pi\)
\(90\) 0 0
\(91\) −194.455 −2.13686
\(92\) − 4.13078i − 0.0448998i
\(93\) 0 0
\(94\) 224.547 2.38880
\(95\) − 25.4580i − 0.267978i
\(96\) 0 0
\(97\) −40.8492 −0.421125 −0.210563 0.977580i \(-0.567530\pi\)
−0.210563 + 0.977580i \(0.567530\pi\)
\(98\) − 159.032i − 1.62278i
\(99\) 0 0
\(100\) −118.256 −1.18256
\(101\) 95.6674i 0.947202i 0.880739 + 0.473601i \(0.157046\pi\)
−0.880739 + 0.473601i \(0.842954\pi\)
\(102\) 0 0
\(103\) −20.6832 −0.200808 −0.100404 0.994947i \(-0.532013\pi\)
−0.100404 + 0.994947i \(0.532013\pi\)
\(104\) 385.189i 3.70374i
\(105\) 0 0
\(106\) 238.933 2.25409
\(107\) 102.522i 0.958152i 0.877773 + 0.479076i \(0.159028\pi\)
−0.877773 + 0.479076i \(0.840972\pi\)
\(108\) 0 0
\(109\) 100.438 0.921450 0.460725 0.887543i \(-0.347589\pi\)
0.460725 + 0.887543i \(0.347589\pi\)
\(110\) − 242.180i − 2.20163i
\(111\) 0 0
\(112\) −312.307 −2.78846
\(113\) − 117.964i − 1.04393i −0.852967 0.521964i \(-0.825199\pi\)
0.852967 0.521964i \(-0.174801\pi\)
\(114\) 0 0
\(115\) 1.56150 0.0135782
\(116\) 400.788i 3.45507i
\(117\) 0 0
\(118\) 55.8517 0.473319
\(119\) 7.89328i 0.0663301i
\(120\) 0 0
\(121\) −241.960 −1.99967
\(122\) − 217.596i − 1.78357i
\(123\) 0 0
\(124\) 115.955 0.935121
\(125\) − 132.037i − 1.05629i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) − 86.3740i − 0.674797i
\(129\) 0 0
\(130\) −256.731 −1.97485
\(131\) − 180.115i − 1.37493i −0.726219 0.687463i \(-0.758724\pi\)
0.726219 0.687463i \(-0.241276\pi\)
\(132\) 0 0
\(133\) −70.1664 −0.527567
\(134\) 188.572i 1.40725i
\(135\) 0 0
\(136\) 15.6355 0.114967
\(137\) − 179.632i − 1.31118i −0.755116 0.655591i \(-0.772419\pi\)
0.755116 0.655591i \(-0.227581\pi\)
\(138\) 0 0
\(139\) 238.348 1.71474 0.857369 0.514703i \(-0.172098\pi\)
0.857369 + 0.514703i \(0.172098\pi\)
\(140\) − 310.831i − 2.22022i
\(141\) 0 0
\(142\) 42.4554 0.298981
\(143\) 384.768i 2.69068i
\(144\) 0 0
\(145\) −151.504 −1.04486
\(146\) − 450.436i − 3.08518i
\(147\) 0 0
\(148\) 288.100 1.94662
\(149\) − 152.663i − 1.02458i −0.858812 0.512292i \(-0.828797\pi\)
0.858812 0.512292i \(-0.171203\pi\)
\(150\) 0 0
\(151\) 298.056 1.97388 0.986939 0.161092i \(-0.0515015\pi\)
0.986939 + 0.161092i \(0.0515015\pi\)
\(152\) 138.991i 0.914411i
\(153\) 0 0
\(154\) −667.488 −4.33434
\(155\) 43.8328i 0.282792i
\(156\) 0 0
\(157\) 90.2523 0.574855 0.287428 0.957802i \(-0.407200\pi\)
0.287428 + 0.957802i \(0.407200\pi\)
\(158\) − 190.517i − 1.20580i
\(159\) 0 0
\(160\) −145.821 −0.911379
\(161\) − 4.30375i − 0.0267314i
\(162\) 0 0
\(163\) 307.712 1.88780 0.943901 0.330229i \(-0.107126\pi\)
0.943901 + 0.330229i \(0.107126\pi\)
\(164\) 161.930i 0.987376i
\(165\) 0 0
\(166\) −336.578 −2.02758
\(167\) 40.6463i 0.243391i 0.992567 + 0.121696i \(0.0388331\pi\)
−0.992567 + 0.121696i \(0.961167\pi\)
\(168\) 0 0
\(169\) 238.886 1.41352
\(170\) 10.4212i 0.0613011i
\(171\) 0 0
\(172\) −198.361 −1.15326
\(173\) − 234.964i − 1.35818i −0.734057 0.679088i \(-0.762376\pi\)
0.734057 0.679088i \(-0.237624\pi\)
\(174\) 0 0
\(175\) −123.208 −0.704045
\(176\) 617.963i 3.51116i
\(177\) 0 0
\(178\) 5.69149 0.0319747
\(179\) 254.540i 1.42201i 0.703186 + 0.711006i \(0.251760\pi\)
−0.703186 + 0.711006i \(0.748240\pi\)
\(180\) 0 0
\(181\) −160.679 −0.887731 −0.443865 0.896094i \(-0.646393\pi\)
−0.443865 + 0.896094i \(0.646393\pi\)
\(182\) 707.593i 3.88787i
\(183\) 0 0
\(184\) −8.52517 −0.0463325
\(185\) 108.906i 0.588684i
\(186\) 0 0
\(187\) 15.6184 0.0835211
\(188\) − 570.262i − 3.03331i
\(189\) 0 0
\(190\) −92.6379 −0.487568
\(191\) 168.426i 0.881814i 0.897553 + 0.440907i \(0.145343\pi\)
−0.897553 + 0.440907i \(0.854657\pi\)
\(192\) 0 0
\(193\) 325.539 1.68673 0.843364 0.537343i \(-0.180572\pi\)
0.843364 + 0.537343i \(0.180572\pi\)
\(194\) 148.644i 0.766208i
\(195\) 0 0
\(196\) −403.880 −2.06061
\(197\) 154.639i 0.784968i 0.919759 + 0.392484i \(0.128384\pi\)
−0.919759 + 0.392484i \(0.871616\pi\)
\(198\) 0 0
\(199\) −310.045 −1.55801 −0.779007 0.627016i \(-0.784276\pi\)
−0.779007 + 0.627016i \(0.784276\pi\)
\(200\) 244.059i 1.22029i
\(201\) 0 0
\(202\) 348.120 1.72337
\(203\) 417.571i 2.05700i
\(204\) 0 0
\(205\) −61.2119 −0.298595
\(206\) 75.2632i 0.365355i
\(207\) 0 0
\(208\) 655.092 3.14948
\(209\) 138.838i 0.664299i
\(210\) 0 0
\(211\) 212.697 1.00804 0.504021 0.863691i \(-0.331853\pi\)
0.504021 + 0.863691i \(0.331853\pi\)
\(212\) − 606.798i − 2.86225i
\(213\) 0 0
\(214\) 373.064 1.74329
\(215\) − 74.9835i − 0.348761i
\(216\) 0 0
\(217\) 120.811 0.556731
\(218\) − 365.480i − 1.67651i
\(219\) 0 0
\(220\) −615.043 −2.79565
\(221\) − 16.5569i − 0.0749179i
\(222\) 0 0
\(223\) −259.955 −1.16572 −0.582859 0.812573i \(-0.698066\pi\)
−0.582859 + 0.812573i \(0.698066\pi\)
\(224\) 401.906i 1.79423i
\(225\) 0 0
\(226\) −429.254 −1.89936
\(227\) − 213.321i − 0.939742i −0.882735 0.469871i \(-0.844300\pi\)
0.882735 0.469871i \(-0.155700\pi\)
\(228\) 0 0
\(229\) −72.3915 −0.316120 −0.158060 0.987430i \(-0.550524\pi\)
−0.158060 + 0.987430i \(0.550524\pi\)
\(230\) − 5.68208i − 0.0247047i
\(231\) 0 0
\(232\) 827.153 3.56532
\(233\) 254.852i 1.09378i 0.837203 + 0.546892i \(0.184189\pi\)
−0.837203 + 0.546892i \(0.815811\pi\)
\(234\) 0 0
\(235\) 215.568 0.917311
\(236\) − 141.842i − 0.601024i
\(237\) 0 0
\(238\) 28.7225 0.120683
\(239\) 99.5547i 0.416547i 0.978071 + 0.208273i \(0.0667844\pi\)
−0.978071 + 0.208273i \(0.933216\pi\)
\(240\) 0 0
\(241\) 280.958 1.16580 0.582900 0.812544i \(-0.301918\pi\)
0.582900 + 0.812544i \(0.301918\pi\)
\(242\) 880.458i 3.63826i
\(243\) 0 0
\(244\) −552.610 −2.26479
\(245\) − 152.673i − 0.623155i
\(246\) 0 0
\(247\) 147.180 0.595872
\(248\) − 239.310i − 0.964960i
\(249\) 0 0
\(250\) −480.462 −1.92185
\(251\) 394.884i 1.57324i 0.617436 + 0.786621i \(0.288172\pi\)
−0.617436 + 0.786621i \(0.711828\pi\)
\(252\) 0 0
\(253\) −8.51585 −0.0336595
\(254\) 41.0079i 0.161448i
\(255\) 0 0
\(256\) −402.897 −1.57381
\(257\) − 21.2709i − 0.0827660i −0.999143 0.0413830i \(-0.986824\pi\)
0.999143 0.0413830i \(-0.0131764\pi\)
\(258\) 0 0
\(259\) 300.165 1.15894
\(260\) 651.997i 2.50768i
\(261\) 0 0
\(262\) −655.415 −2.50158
\(263\) − 380.580i − 1.44707i −0.690287 0.723536i \(-0.742516\pi\)
0.690287 0.723536i \(-0.257484\pi\)
\(264\) 0 0
\(265\) 229.379 0.865581
\(266\) 255.326i 0.959872i
\(267\) 0 0
\(268\) 478.900 1.78694
\(269\) 184.288i 0.685087i 0.939502 + 0.342543i \(0.111288\pi\)
−0.939502 + 0.342543i \(0.888712\pi\)
\(270\) 0 0
\(271\) −509.810 −1.88122 −0.940608 0.339494i \(-0.889744\pi\)
−0.940608 + 0.339494i \(0.889744\pi\)
\(272\) − 26.5915i − 0.0977627i
\(273\) 0 0
\(274\) −653.656 −2.38561
\(275\) 243.792i 0.886515i
\(276\) 0 0
\(277\) 78.0363 0.281720 0.140860 0.990030i \(-0.455013\pi\)
0.140860 + 0.990030i \(0.455013\pi\)
\(278\) − 867.317i − 3.11984i
\(279\) 0 0
\(280\) −641.500 −2.29107
\(281\) − 143.781i − 0.511676i −0.966720 0.255838i \(-0.917649\pi\)
0.966720 0.255838i \(-0.0823514\pi\)
\(282\) 0 0
\(283\) 275.281 0.972724 0.486362 0.873757i \(-0.338324\pi\)
0.486362 + 0.873757i \(0.338324\pi\)
\(284\) − 107.820i − 0.379649i
\(285\) 0 0
\(286\) 1400.12 4.89551
\(287\) 168.710i 0.587841i
\(288\) 0 0
\(289\) 288.328 0.997674
\(290\) 551.302i 1.90104i
\(291\) 0 0
\(292\) −1143.93 −3.91758
\(293\) 267.650i 0.913480i 0.889600 + 0.456740i \(0.150983\pi\)
−0.889600 + 0.456740i \(0.849017\pi\)
\(294\) 0 0
\(295\) 53.6184 0.181757
\(296\) − 594.587i − 2.00874i
\(297\) 0 0
\(298\) −555.519 −1.86416
\(299\) 9.02751i 0.0301923i
\(300\) 0 0
\(301\) −206.667 −0.686602
\(302\) − 1084.58i − 3.59133i
\(303\) 0 0
\(304\) 236.382 0.777572
\(305\) − 208.895i − 0.684902i
\(306\) 0 0
\(307\) 151.145 0.492329 0.246164 0.969228i \(-0.420830\pi\)
0.246164 + 0.969228i \(0.420830\pi\)
\(308\) 1695.16i 5.50377i
\(309\) 0 0
\(310\) 159.501 0.514521
\(311\) − 493.265i − 1.58606i −0.609182 0.793030i \(-0.708502\pi\)
0.609182 0.793030i \(-0.291498\pi\)
\(312\) 0 0
\(313\) 99.1298 0.316708 0.158354 0.987382i \(-0.449381\pi\)
0.158354 + 0.987382i \(0.449381\pi\)
\(314\) − 328.415i − 1.04591i
\(315\) 0 0
\(316\) −483.839 −1.53114
\(317\) 195.074i 0.615375i 0.951487 + 0.307687i \(0.0995551\pi\)
−0.951487 + 0.307687i \(0.900445\pi\)
\(318\) 0 0
\(319\) 826.249 2.59012
\(320\) 77.3725i 0.241789i
\(321\) 0 0
\(322\) −15.6608 −0.0486359
\(323\) − 5.97433i − 0.0184964i
\(324\) 0 0
\(325\) 258.439 0.795198
\(326\) − 1119.72i − 3.43472i
\(327\) 0 0
\(328\) 334.193 1.01888
\(329\) − 594.142i − 1.80590i
\(330\) 0 0
\(331\) 116.566 0.352163 0.176082 0.984376i \(-0.443658\pi\)
0.176082 + 0.984376i \(0.443658\pi\)
\(332\) 854.778i 2.57463i
\(333\) 0 0
\(334\) 147.906 0.442833
\(335\) 181.032i 0.540393i
\(336\) 0 0
\(337\) −193.848 −0.575216 −0.287608 0.957748i \(-0.592860\pi\)
−0.287608 + 0.957748i \(0.592860\pi\)
\(338\) − 869.271i − 2.57181i
\(339\) 0 0
\(340\) 26.4658 0.0778405
\(341\) − 239.048i − 0.701021i
\(342\) 0 0
\(343\) 50.9930 0.148668
\(344\) 409.381i 1.19006i
\(345\) 0 0
\(346\) −855.003 −2.47111
\(347\) 365.092i 1.05214i 0.850442 + 0.526069i \(0.176335\pi\)
−0.850442 + 0.526069i \(0.823665\pi\)
\(348\) 0 0
\(349\) 619.739 1.77576 0.887879 0.460078i \(-0.152179\pi\)
0.887879 + 0.460078i \(0.152179\pi\)
\(350\) 448.336i 1.28096i
\(351\) 0 0
\(352\) 795.253 2.25924
\(353\) − 553.745i − 1.56868i −0.620330 0.784341i \(-0.713001\pi\)
0.620330 0.784341i \(-0.286999\pi\)
\(354\) 0 0
\(355\) 40.7577 0.114811
\(356\) − 14.4542i − 0.0406017i
\(357\) 0 0
\(358\) 926.235 2.58725
\(359\) − 97.3629i − 0.271206i −0.990763 0.135603i \(-0.956703\pi\)
0.990763 0.135603i \(-0.0432972\pi\)
\(360\) 0 0
\(361\) −307.892 −0.852886
\(362\) 584.689i 1.61516i
\(363\) 0 0
\(364\) 1797.01 4.93685
\(365\) − 432.425i − 1.18473i
\(366\) 0 0
\(367\) −491.906 −1.34034 −0.670172 0.742206i \(-0.733780\pi\)
−0.670172 + 0.742206i \(0.733780\pi\)
\(368\) 14.4988i 0.0393989i
\(369\) 0 0
\(370\) 396.295 1.07107
\(371\) − 632.208i − 1.70406i
\(372\) 0 0
\(373\) 379.909 1.01852 0.509261 0.860612i \(-0.329919\pi\)
0.509261 + 0.860612i \(0.329919\pi\)
\(374\) − 56.8333i − 0.151961i
\(375\) 0 0
\(376\) −1176.92 −3.13010
\(377\) − 875.892i − 2.32332i
\(378\) 0 0
\(379\) 646.240 1.70512 0.852559 0.522631i \(-0.175049\pi\)
0.852559 + 0.522631i \(0.175049\pi\)
\(380\) 235.265i 0.619118i
\(381\) 0 0
\(382\) 612.880 1.60440
\(383\) − 363.219i − 0.948354i −0.880430 0.474177i \(-0.842746\pi\)
0.880430 0.474177i \(-0.157254\pi\)
\(384\) 0 0
\(385\) −640.798 −1.66441
\(386\) − 1184.59i − 3.06888i
\(387\) 0 0
\(388\) 377.499 0.972937
\(389\) 662.872i 1.70404i 0.523509 + 0.852020i \(0.324623\pi\)
−0.523509 + 0.852020i \(0.675377\pi\)
\(390\) 0 0
\(391\) 0.366444 0.000937196 0
\(392\) 833.535i 2.12637i
\(393\) 0 0
\(394\) 562.708 1.42819
\(395\) − 182.899i − 0.463035i
\(396\) 0 0
\(397\) 24.6407 0.0620674 0.0310337 0.999518i \(-0.490120\pi\)
0.0310337 + 0.999518i \(0.490120\pi\)
\(398\) 1128.21i 2.83470i
\(399\) 0 0
\(400\) 415.072 1.03768
\(401\) 273.425i 0.681858i 0.940089 + 0.340929i \(0.110742\pi\)
−0.940089 + 0.340929i \(0.889258\pi\)
\(402\) 0 0
\(403\) −253.411 −0.628811
\(404\) − 884.092i − 2.18835i
\(405\) 0 0
\(406\) 1519.48 3.74257
\(407\) − 593.936i − 1.45930i
\(408\) 0 0
\(409\) 601.798 1.47139 0.735695 0.677313i \(-0.236856\pi\)
0.735695 + 0.677313i \(0.236856\pi\)
\(410\) 222.742i 0.543272i
\(411\) 0 0
\(412\) 191.140 0.463931
\(413\) − 147.781i − 0.357824i
\(414\) 0 0
\(415\) −323.120 −0.778602
\(416\) − 843.034i − 2.02653i
\(417\) 0 0
\(418\) 505.214 1.20865
\(419\) 242.412i 0.578550i 0.957246 + 0.289275i \(0.0934142\pi\)
−0.957246 + 0.289275i \(0.906586\pi\)
\(420\) 0 0
\(421\) 76.6518 0.182071 0.0910354 0.995848i \(-0.470982\pi\)
0.0910354 + 0.995848i \(0.470982\pi\)
\(422\) − 773.975i − 1.83406i
\(423\) 0 0
\(424\) −1252.32 −2.95359
\(425\) − 10.4906i − 0.0246836i
\(426\) 0 0
\(427\) −575.750 −1.34836
\(428\) − 947.440i − 2.21364i
\(429\) 0 0
\(430\) −272.855 −0.634545
\(431\) 818.889i 1.89997i 0.312290 + 0.949987i \(0.398904\pi\)
−0.312290 + 0.949987i \(0.601096\pi\)
\(432\) 0 0
\(433\) 15.8770 0.0366674 0.0183337 0.999832i \(-0.494164\pi\)
0.0183337 + 0.999832i \(0.494164\pi\)
\(434\) − 439.613i − 1.01293i
\(435\) 0 0
\(436\) −928.178 −2.12885
\(437\) 3.25746i 0.00745414i
\(438\) 0 0
\(439\) −617.332 −1.40622 −0.703111 0.711080i \(-0.748206\pi\)
−0.703111 + 0.711080i \(0.748206\pi\)
\(440\) 1269.34i 2.88486i
\(441\) 0 0
\(442\) −60.2481 −0.136308
\(443\) 548.280i 1.23765i 0.785528 + 0.618826i \(0.212392\pi\)
−0.785528 + 0.618826i \(0.787608\pi\)
\(444\) 0 0
\(445\) 5.46391 0.0122785
\(446\) 945.941i 2.12094i
\(447\) 0 0
\(448\) 213.252 0.476008
\(449\) − 209.223i − 0.465976i −0.972480 0.232988i \(-0.925150\pi\)
0.972480 0.232988i \(-0.0748503\pi\)
\(450\) 0 0
\(451\) 333.828 0.740195
\(452\) 1090.14i 2.41181i
\(453\) 0 0
\(454\) −776.247 −1.70980
\(455\) 679.299i 1.49296i
\(456\) 0 0
\(457\) −505.510 −1.10615 −0.553074 0.833132i \(-0.686545\pi\)
−0.553074 + 0.833132i \(0.686545\pi\)
\(458\) 263.422i 0.575158i
\(459\) 0 0
\(460\) −14.4303 −0.0313702
\(461\) 839.959i 1.82204i 0.412366 + 0.911018i \(0.364702\pi\)
−0.412366 + 0.911018i \(0.635298\pi\)
\(462\) 0 0
\(463\) −287.564 −0.621089 −0.310545 0.950559i \(-0.600511\pi\)
−0.310545 + 0.950559i \(0.600511\pi\)
\(464\) − 1406.74i − 3.03178i
\(465\) 0 0
\(466\) 927.369 1.99006
\(467\) − 729.628i − 1.56237i −0.624298 0.781187i \(-0.714615\pi\)
0.624298 0.781187i \(-0.285385\pi\)
\(468\) 0 0
\(469\) 498.954 1.06387
\(470\) − 784.422i − 1.66898i
\(471\) 0 0
\(472\) −292.735 −0.620202
\(473\) 408.933i 0.864552i
\(474\) 0 0
\(475\) 93.2546 0.196325
\(476\) − 72.9442i − 0.153244i
\(477\) 0 0
\(478\) 362.266 0.757878
\(479\) − 202.711i − 0.423196i −0.977357 0.211598i \(-0.932133\pi\)
0.977357 0.211598i \(-0.0678667\pi\)
\(480\) 0 0
\(481\) −629.622 −1.30899
\(482\) − 1022.37i − 2.12109i
\(483\) 0 0
\(484\) 2236.02 4.61989
\(485\) 142.701i 0.294228i
\(486\) 0 0
\(487\) 706.928 1.45160 0.725799 0.687907i \(-0.241470\pi\)
0.725799 + 0.687907i \(0.241470\pi\)
\(488\) 1140.49i 2.33706i
\(489\) 0 0
\(490\) −555.556 −1.13379
\(491\) 246.901i 0.502854i 0.967876 + 0.251427i \(0.0808999\pi\)
−0.967876 + 0.251427i \(0.919100\pi\)
\(492\) 0 0
\(493\) −35.5541 −0.0721179
\(494\) − 535.569i − 1.08415i
\(495\) 0 0
\(496\) −406.996 −0.820556
\(497\) − 112.335i − 0.226027i
\(498\) 0 0
\(499\) 480.346 0.962618 0.481309 0.876551i \(-0.340162\pi\)
0.481309 + 0.876551i \(0.340162\pi\)
\(500\) 1220.19i 2.44038i
\(501\) 0 0
\(502\) 1436.93 2.86240
\(503\) 955.396i 1.89940i 0.313167 + 0.949698i \(0.398610\pi\)
−0.313167 + 0.949698i \(0.601390\pi\)
\(504\) 0 0
\(505\) 334.201 0.661783
\(506\) 30.9880i 0.0612411i
\(507\) 0 0
\(508\) 104.144 0.205008
\(509\) 666.159i 1.30876i 0.756166 + 0.654380i \(0.227070\pi\)
−0.756166 + 0.654380i \(0.772930\pi\)
\(510\) 0 0
\(511\) −1191.84 −2.33236
\(512\) 1120.59i 2.18865i
\(513\) 0 0
\(514\) −77.4017 −0.150587
\(515\) 72.2537i 0.140299i
\(516\) 0 0
\(517\) −1175.63 −2.27395
\(518\) − 1092.26i − 2.10860i
\(519\) 0 0
\(520\) 1345.60 2.58770
\(521\) 742.086i 1.42435i 0.702002 + 0.712175i \(0.252290\pi\)
−0.702002 + 0.712175i \(0.747710\pi\)
\(522\) 0 0
\(523\) −171.187 −0.327317 −0.163659 0.986517i \(-0.552330\pi\)
−0.163659 + 0.986517i \(0.552330\pi\)
\(524\) 1664.50i 3.17653i
\(525\) 0 0
\(526\) −1384.88 −2.63285
\(527\) 10.2864i 0.0195189i
\(528\) 0 0
\(529\) 528.800 0.999622
\(530\) − 834.678i − 1.57486i
\(531\) 0 0
\(532\) 648.429 1.21885
\(533\) − 353.885i − 0.663949i
\(534\) 0 0
\(535\) 358.147 0.669434
\(536\) − 988.362i − 1.84396i
\(537\) 0 0
\(538\) 670.599 1.24647
\(539\) 832.624i 1.54476i
\(540\) 0 0
\(541\) −853.931 −1.57843 −0.789215 0.614117i \(-0.789512\pi\)
−0.789215 + 0.614117i \(0.789512\pi\)
\(542\) 1855.13i 3.42274i
\(543\) 0 0
\(544\) −34.2204 −0.0629051
\(545\) − 350.866i − 0.643791i
\(546\) 0 0
\(547\) 304.847 0.557306 0.278653 0.960392i \(-0.410112\pi\)
0.278653 + 0.960392i \(0.410112\pi\)
\(548\) 1660.03i 3.02926i
\(549\) 0 0
\(550\) 887.124 1.61295
\(551\) − 316.055i − 0.573602i
\(552\) 0 0
\(553\) −504.100 −0.911573
\(554\) − 283.963i − 0.512569i
\(555\) 0 0
\(556\) −2202.65 −3.96160
\(557\) − 376.762i − 0.676413i −0.941072 0.338207i \(-0.890180\pi\)
0.941072 0.338207i \(-0.109820\pi\)
\(558\) 0 0
\(559\) 433.503 0.775497
\(560\) 1091.00i 1.94822i
\(561\) 0 0
\(562\) −523.199 −0.930959
\(563\) − 343.827i − 0.610705i −0.952239 0.305352i \(-0.901226\pi\)
0.952239 0.305352i \(-0.0987742\pi\)
\(564\) 0 0
\(565\) −412.090 −0.729363
\(566\) − 1001.71i − 1.76980i
\(567\) 0 0
\(568\) −222.521 −0.391763
\(569\) − 601.227i − 1.05664i −0.849046 0.528319i \(-0.822823\pi\)
0.849046 0.528319i \(-0.177177\pi\)
\(570\) 0 0
\(571\) 595.727 1.04330 0.521652 0.853158i \(-0.325316\pi\)
0.521652 + 0.853158i \(0.325316\pi\)
\(572\) − 3555.75i − 6.21635i
\(573\) 0 0
\(574\) 613.914 1.06954
\(575\) 5.71990i 0.00994765i
\(576\) 0 0
\(577\) 68.3783 0.118507 0.0592533 0.998243i \(-0.481128\pi\)
0.0592533 + 0.998243i \(0.481128\pi\)
\(578\) − 1049.18i − 1.81520i
\(579\) 0 0
\(580\) 1400.10 2.41396
\(581\) 890.572i 1.53283i
\(582\) 0 0
\(583\) −1250.95 −2.14571
\(584\) 2360.87i 4.04259i
\(585\) 0 0
\(586\) 973.939 1.66201
\(587\) − 651.244i − 1.10944i −0.832036 0.554722i \(-0.812825\pi\)
0.832036 0.554722i \(-0.187175\pi\)
\(588\) 0 0
\(589\) −91.4401 −0.155246
\(590\) − 195.110i − 0.330695i
\(591\) 0 0
\(592\) −1011.22 −1.70814
\(593\) − 12.2860i − 0.0207183i −0.999946 0.0103592i \(-0.996703\pi\)
0.999946 0.0103592i \(-0.00329748\pi\)
\(594\) 0 0
\(595\) 27.5740 0.0463429
\(596\) 1410.80i 2.36712i
\(597\) 0 0
\(598\) 32.8498 0.0549328
\(599\) − 10.2076i − 0.0170410i −0.999964 0.00852051i \(-0.997288\pi\)
0.999964 0.00852051i \(-0.00271220\pi\)
\(600\) 0 0
\(601\) −370.191 −0.615959 −0.307980 0.951393i \(-0.599653\pi\)
−0.307980 + 0.951393i \(0.599653\pi\)
\(602\) 752.033i 1.24922i
\(603\) 0 0
\(604\) −2754.42 −4.56030
\(605\) 845.252i 1.39711i
\(606\) 0 0
\(607\) 166.851 0.274878 0.137439 0.990510i \(-0.456113\pi\)
0.137439 + 0.990510i \(0.456113\pi\)
\(608\) − 304.198i − 0.500326i
\(609\) 0 0
\(610\) −760.140 −1.24613
\(611\) 1246.27i 2.03971i
\(612\) 0 0
\(613\) 6.09767 0.00994726 0.00497363 0.999988i \(-0.498417\pi\)
0.00497363 + 0.999988i \(0.498417\pi\)
\(614\) − 549.995i − 0.895758i
\(615\) 0 0
\(616\) 3498.51 5.67939
\(617\) 304.059i 0.492803i 0.969168 + 0.246401i \(0.0792482\pi\)
−0.969168 + 0.246401i \(0.920752\pi\)
\(618\) 0 0
\(619\) −446.526 −0.721367 −0.360683 0.932688i \(-0.617457\pi\)
−0.360683 + 0.932688i \(0.617457\pi\)
\(620\) − 405.072i − 0.653342i
\(621\) 0 0
\(622\) −1794.92 −2.88573
\(623\) − 15.0595i − 0.0241725i
\(624\) 0 0
\(625\) −141.339 −0.226143
\(626\) − 360.719i − 0.576229i
\(627\) 0 0
\(628\) −834.049 −1.32810
\(629\) 25.5576i 0.0406320i
\(630\) 0 0
\(631\) 786.982 1.24720 0.623599 0.781745i \(-0.285670\pi\)
0.623599 + 0.781745i \(0.285670\pi\)
\(632\) 998.556i 1.57999i
\(633\) 0 0
\(634\) 709.846 1.11963
\(635\) 39.3681i 0.0619971i
\(636\) 0 0
\(637\) 882.650 1.38564
\(638\) − 3006.60i − 4.71254i
\(639\) 0 0
\(640\) −301.735 −0.471461
\(641\) 596.401i 0.930423i 0.885199 + 0.465212i \(0.154022\pi\)
−0.885199 + 0.465212i \(0.845978\pi\)
\(642\) 0 0
\(643\) 773.061 1.20227 0.601136 0.799146i \(-0.294715\pi\)
0.601136 + 0.799146i \(0.294715\pi\)
\(644\) 39.7723i 0.0617582i
\(645\) 0 0
\(646\) −21.7397 −0.0336529
\(647\) − 112.273i − 0.173529i −0.996229 0.0867644i \(-0.972347\pi\)
0.996229 0.0867644i \(-0.0276527\pi\)
\(648\) 0 0
\(649\) −292.415 −0.450563
\(650\) − 940.425i − 1.44681i
\(651\) 0 0
\(652\) −2843.66 −4.36144
\(653\) 1069.49i 1.63781i 0.573929 + 0.818905i \(0.305419\pi\)
−0.573929 + 0.818905i \(0.694581\pi\)
\(654\) 0 0
\(655\) −629.207 −0.960622
\(656\) − 568.364i − 0.866409i
\(657\) 0 0
\(658\) −2162.00 −3.28571
\(659\) − 830.211i − 1.25980i −0.776675 0.629902i \(-0.783095\pi\)
0.776675 0.629902i \(-0.216905\pi\)
\(660\) 0 0
\(661\) −696.017 −1.05298 −0.526488 0.850183i \(-0.676492\pi\)
−0.526488 + 0.850183i \(0.676492\pi\)
\(662\) − 424.168i − 0.640737i
\(663\) 0 0
\(664\) 1764.11 2.65679
\(665\) 245.116i 0.368596i
\(666\) 0 0
\(667\) 19.3856 0.0290639
\(668\) − 375.625i − 0.562313i
\(669\) 0 0
\(670\) 658.749 0.983207
\(671\) 1139.24i 1.69782i
\(672\) 0 0
\(673\) −122.055 −0.181359 −0.0906796 0.995880i \(-0.528904\pi\)
−0.0906796 + 0.995880i \(0.528904\pi\)
\(674\) 705.386i 1.04657i
\(675\) 0 0
\(676\) −2207.61 −3.26570
\(677\) 10.1764i 0.0150317i 0.999972 + 0.00751584i \(0.00239239\pi\)
−0.999972 + 0.00751584i \(0.997608\pi\)
\(678\) 0 0
\(679\) 393.307 0.579245
\(680\) − 54.6206i − 0.0803243i
\(681\) 0 0
\(682\) −869.863 −1.27546
\(683\) − 356.039i − 0.521287i −0.965435 0.260643i \(-0.916065\pi\)
0.965435 0.260643i \(-0.0839347\pi\)
\(684\) 0 0
\(685\) −627.519 −0.916086
\(686\) − 185.557i − 0.270491i
\(687\) 0 0
\(688\) 696.236 1.01197
\(689\) 1326.11i 1.92469i
\(690\) 0 0
\(691\) 707.950 1.02453 0.512265 0.858828i \(-0.328807\pi\)
0.512265 + 0.858828i \(0.328807\pi\)
\(692\) 2171.38i 3.13783i
\(693\) 0 0
\(694\) 1328.52 1.91429
\(695\) − 832.636i − 1.19804i
\(696\) 0 0
\(697\) −14.3649 −0.0206096
\(698\) − 2255.14i − 3.23087i
\(699\) 0 0
\(700\) 1138.60 1.62657
\(701\) 578.940i 0.825877i 0.910759 + 0.412939i \(0.135498\pi\)
−0.910759 + 0.412939i \(0.864502\pi\)
\(702\) 0 0
\(703\) −227.191 −0.323174
\(704\) − 421.962i − 0.599377i
\(705\) 0 0
\(706\) −2015.00 −2.85411
\(707\) − 921.113i − 1.30285i
\(708\) 0 0
\(709\) −115.048 −0.162269 −0.0811343 0.996703i \(-0.525854\pi\)
−0.0811343 + 0.996703i \(0.525854\pi\)
\(710\) − 148.312i − 0.208890i
\(711\) 0 0
\(712\) −29.8308 −0.0418972
\(713\) − 5.60861i − 0.00786621i
\(714\) 0 0
\(715\) 1344.13 1.87990
\(716\) − 2352.28i − 3.28531i
\(717\) 0 0
\(718\) −354.290 −0.493440
\(719\) 626.313i 0.871089i 0.900167 + 0.435545i \(0.143444\pi\)
−0.900167 + 0.435545i \(0.856556\pi\)
\(720\) 0 0
\(721\) 199.143 0.276205
\(722\) 1120.38i 1.55177i
\(723\) 0 0
\(724\) 1484.89 2.05095
\(725\) − 554.972i − 0.765479i
\(726\) 0 0
\(727\) −773.461 −1.06391 −0.531954 0.846773i \(-0.678542\pi\)
−0.531954 + 0.846773i \(0.678542\pi\)
\(728\) − 3708.71i − 5.09438i
\(729\) 0 0
\(730\) −1573.53 −2.15553
\(731\) − 17.5967i − 0.0240721i
\(732\) 0 0
\(733\) 148.562 0.202677 0.101339 0.994852i \(-0.467687\pi\)
0.101339 + 0.994852i \(0.467687\pi\)
\(734\) 1789.98i 2.43866i
\(735\) 0 0
\(736\) 18.6584 0.0253511
\(737\) − 987.281i − 1.33959i
\(738\) 0 0
\(739\) −1320.96 −1.78749 −0.893745 0.448575i \(-0.851932\pi\)
−0.893745 + 0.448575i \(0.851932\pi\)
\(740\) − 1006.44i − 1.36005i
\(741\) 0 0
\(742\) −2300.51 −3.10042
\(743\) − 674.905i − 0.908352i −0.890912 0.454176i \(-0.849934\pi\)
0.890912 0.454176i \(-0.150066\pi\)
\(744\) 0 0
\(745\) −533.306 −0.715847
\(746\) − 1382.44i − 1.85313i
\(747\) 0 0
\(748\) −144.335 −0.192961
\(749\) − 987.114i − 1.31791i
\(750\) 0 0
\(751\) −128.638 −0.171289 −0.0856446 0.996326i \(-0.527295\pi\)
−0.0856446 + 0.996326i \(0.527295\pi\)
\(752\) 2001.59i 2.66169i
\(753\) 0 0
\(754\) −3187.25 −4.22712
\(755\) − 1041.22i − 1.37909i
\(756\) 0 0
\(757\) −877.612 −1.15933 −0.579665 0.814855i \(-0.696816\pi\)
−0.579665 + 0.814855i \(0.696816\pi\)
\(758\) − 2351.58i − 3.10234i
\(759\) 0 0
\(760\) 485.543 0.638873
\(761\) − 379.587i − 0.498800i −0.968401 0.249400i \(-0.919767\pi\)
0.968401 0.249400i \(-0.0802334\pi\)
\(762\) 0 0
\(763\) −967.046 −1.26743
\(764\) − 1556.48i − 2.03728i
\(765\) 0 0
\(766\) −1321.70 −1.72546
\(767\) 309.984i 0.404152i
\(768\) 0 0
\(769\) −37.2784 −0.0484764 −0.0242382 0.999706i \(-0.507716\pi\)
−0.0242382 + 0.999706i \(0.507716\pi\)
\(770\) 2331.77i 3.02828i
\(771\) 0 0
\(772\) −3008.40 −3.89689
\(773\) 90.6078i 0.117216i 0.998281 + 0.0586079i \(0.0186662\pi\)
−0.998281 + 0.0586079i \(0.981334\pi\)
\(774\) 0 0
\(775\) −160.563 −0.207178
\(776\) − 779.090i − 1.00398i
\(777\) 0 0
\(778\) 2412.10 3.10038
\(779\) − 127.695i − 0.163922i
\(780\) 0 0
\(781\) −222.278 −0.284607
\(782\) − 1.33344i − 0.00170516i
\(783\) 0 0
\(784\) 1417.60 1.80816
\(785\) − 315.283i − 0.401635i
\(786\) 0 0
\(787\) −605.048 −0.768804 −0.384402 0.923166i \(-0.625592\pi\)
−0.384402 + 0.923166i \(0.625592\pi\)
\(788\) − 1429.06i − 1.81353i
\(789\) 0 0
\(790\) −665.543 −0.842459
\(791\) 1135.79i 1.43589i
\(792\) 0 0
\(793\) 1207.69 1.52294
\(794\) − 89.6642i − 0.112927i
\(795\) 0 0
\(796\) 2865.22 3.59952
\(797\) − 91.0829i − 0.114282i −0.998366 0.0571411i \(-0.981802\pi\)
0.998366 0.0571411i \(-0.0181985\pi\)
\(798\) 0 0
\(799\) 50.5883 0.0633145
\(800\) − 534.153i − 0.667692i
\(801\) 0 0
\(802\) 994.956 1.24059
\(803\) 2358.29i 2.93685i
\(804\) 0 0
\(805\) −15.0346 −0.0186765
\(806\) 922.127i 1.14408i
\(807\) 0 0
\(808\) −1824.60 −2.25817
\(809\) 936.735i 1.15789i 0.815366 + 0.578947i \(0.196536\pi\)
−0.815366 + 0.578947i \(0.803464\pi\)
\(810\) 0 0
\(811\) −275.363 −0.339536 −0.169768 0.985484i \(-0.554302\pi\)
−0.169768 + 0.985484i \(0.554302\pi\)
\(812\) − 3858.90i − 4.75234i
\(813\) 0 0
\(814\) −2161.25 −2.65510
\(815\) − 1074.95i − 1.31895i
\(816\) 0 0
\(817\) 156.424 0.191461
\(818\) − 2189.86i − 2.67709i
\(819\) 0 0
\(820\) 565.678 0.689851
\(821\) − 243.447i − 0.296525i −0.988948 0.148263i \(-0.952632\pi\)
0.988948 0.148263i \(-0.0473681\pi\)
\(822\) 0 0
\(823\) −554.519 −0.673777 −0.336889 0.941545i \(-0.609375\pi\)
−0.336889 + 0.941545i \(0.609375\pi\)
\(824\) − 394.477i − 0.478734i
\(825\) 0 0
\(826\) −537.756 −0.651036
\(827\) 1064.18i 1.28680i 0.765530 + 0.643400i \(0.222477\pi\)
−0.765530 + 0.643400i \(0.777523\pi\)
\(828\) 0 0
\(829\) 1090.98 1.31602 0.658012 0.753007i \(-0.271397\pi\)
0.658012 + 0.753007i \(0.271397\pi\)
\(830\) 1175.79i 1.41661i
\(831\) 0 0
\(832\) −447.314 −0.537637
\(833\) − 35.8285i − 0.0430113i
\(834\) 0 0
\(835\) 141.992 0.170050
\(836\) − 1283.05i − 1.53475i
\(837\) 0 0
\(838\) 882.105 1.05263
\(839\) 978.978i 1.16684i 0.812171 + 0.583419i \(0.198286\pi\)
−0.812171 + 0.583419i \(0.801714\pi\)
\(840\) 0 0
\(841\) −1039.89 −1.23649
\(842\) − 278.925i − 0.331265i
\(843\) 0 0
\(844\) −1965.60 −2.32891
\(845\) − 834.513i − 0.987589i
\(846\) 0 0
\(847\) 2329.66 2.75048
\(848\) 2129.83i 2.51159i
\(849\) 0 0
\(850\) −38.1736 −0.0449102
\(851\) − 13.9351i − 0.0163749i
\(852\) 0 0
\(853\) 301.741 0.353741 0.176870 0.984234i \(-0.443403\pi\)
0.176870 + 0.984234i \(0.443403\pi\)
\(854\) 2095.08i 2.45325i
\(855\) 0 0
\(856\) −1955.34 −2.28428
\(857\) 1467.38i 1.71222i 0.516790 + 0.856112i \(0.327127\pi\)
−0.516790 + 0.856112i \(0.672873\pi\)
\(858\) 0 0
\(859\) 12.3456 0.0143720 0.00718600 0.999974i \(-0.497713\pi\)
0.00718600 + 0.999974i \(0.497713\pi\)
\(860\) 692.945i 0.805750i
\(861\) 0 0
\(862\) 2979.82 3.45687
\(863\) 10.8046i 0.0125198i 0.999980 + 0.00625990i \(0.00199260\pi\)
−0.999980 + 0.00625990i \(0.998007\pi\)
\(864\) 0 0
\(865\) −820.815 −0.948919
\(866\) − 57.7742i − 0.0667138i
\(867\) 0 0
\(868\) −1116.45 −1.28623
\(869\) 997.463i 1.14783i
\(870\) 0 0
\(871\) −1046.60 −1.20161
\(872\) 1915.59i 2.19678i
\(873\) 0 0
\(874\) 11.8534 0.0135623
\(875\) 1271.28i 1.45290i
\(876\) 0 0
\(877\) 366.912 0.418372 0.209186 0.977876i \(-0.432919\pi\)
0.209186 + 0.977876i \(0.432919\pi\)
\(878\) 2246.38i 2.55852i
\(879\) 0 0
\(880\) 2158.77 2.45314
\(881\) − 308.365i − 0.350017i −0.984567 0.175008i \(-0.944005\pi\)
0.984567 0.175008i \(-0.0559952\pi\)
\(882\) 0 0
\(883\) −497.722 −0.563672 −0.281836 0.959463i \(-0.590943\pi\)
−0.281836 + 0.959463i \(0.590943\pi\)
\(884\) 153.007i 0.173085i
\(885\) 0 0
\(886\) 1995.11 2.25182
\(887\) − 1001.96i − 1.12961i −0.825225 0.564805i \(-0.808952\pi\)
0.825225 0.564805i \(-0.191048\pi\)
\(888\) 0 0
\(889\) 108.505 0.122053
\(890\) − 19.8824i − 0.0223398i
\(891\) 0 0
\(892\) 2402.32 2.69319
\(893\) 449.699i 0.503582i
\(894\) 0 0
\(895\) 889.199 0.993519
\(896\) 831.633i 0.928162i
\(897\) 0 0
\(898\) −761.334 −0.847811
\(899\) 544.174i 0.605310i
\(900\) 0 0
\(901\) 53.8294 0.0597440
\(902\) − 1214.75i − 1.34673i
\(903\) 0 0
\(904\) 2249.85 2.48877
\(905\) 561.310i 0.620232i
\(906\) 0 0
\(907\) 968.351 1.06764 0.533821 0.845598i \(-0.320756\pi\)
0.533821 + 0.845598i \(0.320756\pi\)
\(908\) 1971.37i 2.17111i
\(909\) 0 0
\(910\) 2471.87 2.71635
\(911\) − 620.552i − 0.681176i −0.940213 0.340588i \(-0.889374\pi\)
0.940213 0.340588i \(-0.110626\pi\)
\(912\) 0 0
\(913\) 1762.18 1.93010
\(914\) 1839.48i 2.01256i
\(915\) 0 0
\(916\) 668.992 0.730340
\(917\) 1734.20i 1.89117i
\(918\) 0 0
\(919\) 153.844 0.167403 0.0837016 0.996491i \(-0.473326\pi\)
0.0837016 + 0.996491i \(0.473326\pi\)
\(920\) 29.7815i 0.0323712i
\(921\) 0 0
\(922\) 3056.49 3.31507
\(923\) 235.633i 0.255290i
\(924\) 0 0
\(925\) −398.933 −0.431279
\(926\) 1046.41i 1.13003i
\(927\) 0 0
\(928\) −1810.33 −1.95079
\(929\) − 1440.34i − 1.55042i −0.631705 0.775209i \(-0.717644\pi\)
0.631705 0.775209i \(-0.282356\pi\)
\(930\) 0 0
\(931\) 318.493 0.342098
\(932\) − 2355.16i − 2.52700i
\(933\) 0 0
\(934\) −2655.01 −2.84263
\(935\) − 54.5608i − 0.0583538i
\(936\) 0 0
\(937\) −1174.61 −1.25359 −0.626795 0.779184i \(-0.715634\pi\)
−0.626795 + 0.779184i \(0.715634\pi\)
\(938\) − 1815.62i − 1.93563i
\(939\) 0 0
\(940\) −1992.13 −2.11929
\(941\) − 1621.47i − 1.72313i −0.507644 0.861567i \(-0.669484\pi\)
0.507644 0.861567i \(-0.330516\pi\)
\(942\) 0 0
\(943\) 7.83234 0.00830577
\(944\) 497.857i 0.527390i
\(945\) 0 0
\(946\) 1488.05 1.57299
\(947\) − 642.612i − 0.678576i −0.940682 0.339288i \(-0.889814\pi\)
0.940682 0.339288i \(-0.110186\pi\)
\(948\) 0 0
\(949\) 2499.98 2.63433
\(950\) − 339.340i − 0.357200i
\(951\) 0 0
\(952\) −150.543 −0.158134
\(953\) − 447.223i − 0.469279i −0.972082 0.234639i \(-0.924609\pi\)
0.972082 0.234639i \(-0.0753909\pi\)
\(954\) 0 0
\(955\) 588.374 0.616098
\(956\) − 920.015i − 0.962359i
\(957\) 0 0
\(958\) −737.636 −0.769975
\(959\) 1729.55i 1.80349i
\(960\) 0 0
\(961\) −803.561 −0.836172
\(962\) 2291.11i 2.38161i
\(963\) 0 0
\(964\) −2596.42 −2.69338
\(965\) − 1137.22i − 1.17847i
\(966\) 0 0
\(967\) −1084.47 −1.12148 −0.560742 0.827991i \(-0.689484\pi\)
−0.560742 + 0.827991i \(0.689484\pi\)
\(968\) − 4614.75i − 4.76730i
\(969\) 0 0
\(970\) 519.268 0.535328
\(971\) 202.150i 0.208187i 0.994567 + 0.104094i \(0.0331942\pi\)
−0.994567 + 0.104094i \(0.966806\pi\)
\(972\) 0 0
\(973\) −2294.89 −2.35857
\(974\) − 2572.41i − 2.64108i
\(975\) 0 0
\(976\) 1939.63 1.98733
\(977\) 443.613i 0.454057i 0.973888 + 0.227028i \(0.0729010\pi\)
−0.973888 + 0.227028i \(0.927099\pi\)
\(978\) 0 0
\(979\) −29.7982 −0.0304374
\(980\) 1410.90i 1.43969i
\(981\) 0 0
\(982\) 898.440 0.914908
\(983\) 350.462i 0.356522i 0.983983 + 0.178261i \(0.0570472\pi\)
−0.983983 + 0.178261i \(0.942953\pi\)
\(984\) 0 0
\(985\) 540.208 0.548435
\(986\) 129.377i 0.131213i
\(987\) 0 0
\(988\) −1360.14 −1.37666
\(989\) 9.59448i 0.00970119i
\(990\) 0 0
\(991\) −889.003 −0.897076 −0.448538 0.893764i \(-0.648055\pi\)
−0.448538 + 0.893764i \(0.648055\pi\)
\(992\) 523.760i 0.527984i
\(993\) 0 0
\(994\) −408.772 −0.411240
\(995\) 1083.10i 1.08854i
\(996\) 0 0
\(997\) 1156.35 1.15983 0.579913 0.814678i \(-0.303086\pi\)
0.579913 + 0.814678i \(0.303086\pi\)
\(998\) − 1747.91i − 1.75142i
\(999\) 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 1143.3.b.a.890.5 84
3.2 odd 2 inner 1143.3.b.a.890.80 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.5 84 1.1 even 1 trivial
1143.3.b.a.890.80 yes 84 3.2 odd 2 inner