Properties

Label 162.3.b.b.161.2
Level $162$
Weight $3$
Character 162.161
Analytic conductor $4.414$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.41418028264\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.2
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.3.b.b.161.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +5.79555i q^{5} -8.39230 q^{7} +2.82843i q^{8} +O(q^{10})\) \(q-1.41421i q^{2} -2.00000 q^{4} +5.79555i q^{5} -8.39230 q^{7} +2.82843i q^{8} +8.19615 q^{10} +14.6969i q^{11} -21.1962 q^{13} +11.8685i q^{14} +4.00000 q^{16} +7.76457i q^{17} +24.3923 q^{19} -11.5911i q^{20} +20.7846 q^{22} -14.6969i q^{23} -8.58846 q^{25} +29.9759i q^{26} +16.7846 q^{28} +35.4940i q^{29} +8.00000 q^{31} -5.65685i q^{32} +10.9808 q^{34} -48.6381i q^{35} -60.5692 q^{37} -34.4959i q^{38} -16.3923 q^{40} -33.6365i q^{41} +9.17691 q^{43} -29.3939i q^{44} -20.7846 q^{46} -16.9706i q^{47} +21.4308 q^{49} +12.1459i q^{50} +42.3923 q^{52} +25.7605i q^{53} -85.1769 q^{55} -23.7370i q^{56} +50.1962 q^{58} +61.6706i q^{59} -13.0000 q^{61} -11.3137i q^{62} -8.00000 q^{64} -122.843i q^{65} +21.1769 q^{67} -15.5291i q^{68} -68.7846 q^{70} -101.214i q^{71} +40.4115 q^{73} +85.6578i q^{74} -48.7846 q^{76} -123.341i q^{77} +98.7461 q^{79} +23.1822i q^{80} -47.5692 q^{82} +103.488i q^{83} -45.0000 q^{85} -12.9781i q^{86} -41.5692 q^{88} +134.130i q^{89} +177.885 q^{91} +29.3939i q^{92} -24.0000 q^{94} +141.367i q^{95} -75.1384 q^{97} -30.3077i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 8 q^{7} + 12 q^{10} - 64 q^{13} + 16 q^{16} + 56 q^{19} + 28 q^{25} - 16 q^{28} + 32 q^{31} - 60 q^{34} - 76 q^{37} - 24 q^{40} - 88 q^{43} + 252 q^{49} + 128 q^{52} - 216 q^{55} + 180 q^{58} - 52 q^{61} - 32 q^{64} - 40 q^{67} - 192 q^{70} + 224 q^{73} - 112 q^{76} + 104 q^{79} - 24 q^{82} - 180 q^{85} + 88 q^{91} - 96 q^{94} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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
\(5\) 5.79555i 1.15911i 0.814933 + 0.579555i \(0.196774\pi\)
−0.814933 + 0.579555i \(0.803226\pi\)
\(6\) 0 0
\(7\) −8.39230 −1.19890 −0.599450 0.800412i \(-0.704614\pi\)
−0.599450 + 0.800412i \(0.704614\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) 8.19615 0.819615
\(11\) 14.6969i 1.33609i 0.744123 + 0.668043i \(0.232868\pi\)
−0.744123 + 0.668043i \(0.767132\pi\)
\(12\) 0 0
\(13\) −21.1962 −1.63047 −0.815237 0.579128i \(-0.803393\pi\)
−0.815237 + 0.579128i \(0.803393\pi\)
\(14\) 11.8685i 0.847751i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 7.76457i 0.456739i 0.973574 + 0.228370i \(0.0733395\pi\)
−0.973574 + 0.228370i \(0.926660\pi\)
\(18\) 0 0
\(19\) 24.3923 1.28381 0.641903 0.766786i \(-0.278145\pi\)
0.641903 + 0.766786i \(0.278145\pi\)
\(20\) − 11.5911i − 0.579555i
\(21\) 0 0
\(22\) 20.7846 0.944755
\(23\) − 14.6969i − 0.638997i −0.947587 0.319499i \(-0.896486\pi\)
0.947587 0.319499i \(-0.103514\pi\)
\(24\) 0 0
\(25\) −8.58846 −0.343538
\(26\) 29.9759i 1.15292i
\(27\) 0 0
\(28\) 16.7846 0.599450
\(29\) 35.4940i 1.22393i 0.790884 + 0.611966i \(0.209621\pi\)
−0.790884 + 0.611966i \(0.790379\pi\)
\(30\) 0 0
\(31\) 8.00000 0.258065 0.129032 0.991640i \(-0.458813\pi\)
0.129032 + 0.991640i \(0.458813\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 10.9808 0.322964
\(35\) − 48.6381i − 1.38966i
\(36\) 0 0
\(37\) −60.5692 −1.63701 −0.818503 0.574502i \(-0.805196\pi\)
−0.818503 + 0.574502i \(0.805196\pi\)
\(38\) − 34.4959i − 0.907788i
\(39\) 0 0
\(40\) −16.3923 −0.409808
\(41\) − 33.6365i − 0.820403i −0.911995 0.410201i \(-0.865458\pi\)
0.911995 0.410201i \(-0.134542\pi\)
\(42\) 0 0
\(43\) 9.17691 0.213417 0.106708 0.994290i \(-0.465969\pi\)
0.106708 + 0.994290i \(0.465969\pi\)
\(44\) − 29.3939i − 0.668043i
\(45\) 0 0
\(46\) −20.7846 −0.451839
\(47\) − 16.9706i − 0.361076i −0.983568 0.180538i \(-0.942216\pi\)
0.983568 0.180538i \(-0.0577838\pi\)
\(48\) 0 0
\(49\) 21.4308 0.437363
\(50\) 12.1459i 0.242918i
\(51\) 0 0
\(52\) 42.3923 0.815237
\(53\) 25.7605i 0.486046i 0.970020 + 0.243023i \(0.0781391\pi\)
−0.970020 + 0.243023i \(0.921861\pi\)
\(54\) 0 0
\(55\) −85.1769 −1.54867
\(56\) − 23.7370i − 0.423875i
\(57\) 0 0
\(58\) 50.1962 0.865451
\(59\) 61.6706i 1.04526i 0.852558 + 0.522632i \(0.175050\pi\)
−0.852558 + 0.522632i \(0.824950\pi\)
\(60\) 0 0
\(61\) −13.0000 −0.213115 −0.106557 0.994307i \(-0.533983\pi\)
−0.106557 + 0.994307i \(0.533983\pi\)
\(62\) − 11.3137i − 0.182479i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 122.843i − 1.88990i
\(66\) 0 0
\(67\) 21.1769 0.316073 0.158037 0.987433i \(-0.449484\pi\)
0.158037 + 0.987433i \(0.449484\pi\)
\(68\) − 15.5291i − 0.228370i
\(69\) 0 0
\(70\) −68.7846 −0.982637
\(71\) − 101.214i − 1.42555i −0.701392 0.712776i \(-0.747438\pi\)
0.701392 0.712776i \(-0.252562\pi\)
\(72\) 0 0
\(73\) 40.4115 0.553583 0.276791 0.960930i \(-0.410729\pi\)
0.276791 + 0.960930i \(0.410729\pi\)
\(74\) 85.6578i 1.15754i
\(75\) 0 0
\(76\) −48.7846 −0.641903
\(77\) − 123.341i − 1.60183i
\(78\) 0 0
\(79\) 98.7461 1.24995 0.624976 0.780644i \(-0.285109\pi\)
0.624976 + 0.780644i \(0.285109\pi\)
\(80\) 23.1822i 0.289778i
\(81\) 0 0
\(82\) −47.5692 −0.580112
\(83\) 103.488i 1.24684i 0.781887 + 0.623420i \(0.214257\pi\)
−0.781887 + 0.623420i \(0.785743\pi\)
\(84\) 0 0
\(85\) −45.0000 −0.529412
\(86\) − 12.9781i − 0.150908i
\(87\) 0 0
\(88\) −41.5692 −0.472377
\(89\) 134.130i 1.50708i 0.657403 + 0.753539i \(0.271655\pi\)
−0.657403 + 0.753539i \(0.728345\pi\)
\(90\) 0 0
\(91\) 177.885 1.95478
\(92\) 29.3939i 0.319499i
\(93\) 0 0
\(94\) −24.0000 −0.255319
\(95\) 141.367i 1.48807i
\(96\) 0 0
\(97\) −75.1384 −0.774623 −0.387312 0.921949i \(-0.626596\pi\)
−0.387312 + 0.921949i \(0.626596\pi\)
\(98\) − 30.3077i − 0.309262i
\(99\) 0 0
\(100\) 17.1769 0.171769
\(101\) − 29.0893i − 0.288013i −0.989577 0.144006i \(-0.954001\pi\)
0.989577 0.144006i \(-0.0459986\pi\)
\(102\) 0 0
\(103\) 96.7077 0.938909 0.469455 0.882957i \(-0.344451\pi\)
0.469455 + 0.882957i \(0.344451\pi\)
\(104\) − 59.9518i − 0.576459i
\(105\) 0 0
\(106\) 36.4308 0.343687
\(107\) 177.582i 1.65964i 0.558030 + 0.829821i \(0.311558\pi\)
−0.558030 + 0.829821i \(0.688442\pi\)
\(108\) 0 0
\(109\) 61.9423 0.568278 0.284139 0.958783i \(-0.408292\pi\)
0.284139 + 0.958783i \(0.408292\pi\)
\(110\) 120.458i 1.09508i
\(111\) 0 0
\(112\) −33.5692 −0.299725
\(113\) 109.811i 0.971778i 0.874020 + 0.485889i \(0.161504\pi\)
−0.874020 + 0.485889i \(0.838496\pi\)
\(114\) 0 0
\(115\) 85.1769 0.740669
\(116\) − 70.9881i − 0.611966i
\(117\) 0 0
\(118\) 87.2154 0.739113
\(119\) − 65.1626i − 0.547585i
\(120\) 0 0
\(121\) −95.0000 −0.785124
\(122\) 18.3848i 0.150695i
\(123\) 0 0
\(124\) −16.0000 −0.129032
\(125\) 95.1140i 0.760912i
\(126\) 0 0
\(127\) 141.177 1.11163 0.555815 0.831306i \(-0.312406\pi\)
0.555815 + 0.831306i \(0.312406\pi\)
\(128\) 11.3137i 0.0883883i
\(129\) 0 0
\(130\) −173.727 −1.33636
\(131\) − 53.1853i − 0.405995i −0.979179 0.202997i \(-0.934932\pi\)
0.979179 0.202997i \(-0.0650683\pi\)
\(132\) 0 0
\(133\) −204.708 −1.53916
\(134\) − 29.9487i − 0.223498i
\(135\) 0 0
\(136\) −21.9615 −0.161482
\(137\) 1.63453i 0.0119309i 0.999982 + 0.00596545i \(0.00189887\pi\)
−0.999982 + 0.00596545i \(0.998101\pi\)
\(138\) 0 0
\(139\) −19.2154 −0.138240 −0.0691201 0.997608i \(-0.522019\pi\)
−0.0691201 + 0.997608i \(0.522019\pi\)
\(140\) 97.2761i 0.694829i
\(141\) 0 0
\(142\) −143.138 −1.00802
\(143\) − 311.519i − 2.17845i
\(144\) 0 0
\(145\) −205.708 −1.41867
\(146\) − 57.1506i − 0.391442i
\(147\) 0 0
\(148\) 121.138 0.818503
\(149\) 94.2818i 0.632764i 0.948632 + 0.316382i \(0.102468\pi\)
−0.948632 + 0.316382i \(0.897532\pi\)
\(150\) 0 0
\(151\) 32.0000 0.211921 0.105960 0.994370i \(-0.466208\pi\)
0.105960 + 0.994370i \(0.466208\pi\)
\(152\) 68.9919i 0.453894i
\(153\) 0 0
\(154\) −174.431 −1.13267
\(155\) 46.3644i 0.299125i
\(156\) 0 0
\(157\) 0.292342 0.00186205 0.000931025 1.00000i \(-0.499704\pi\)
0.000931025 1.00000i \(0.499704\pi\)
\(158\) − 139.648i − 0.883849i
\(159\) 0 0
\(160\) 32.7846 0.204904
\(161\) 123.341i 0.766094i
\(162\) 0 0
\(163\) 28.7846 0.176593 0.0882963 0.996094i \(-0.471858\pi\)
0.0882963 + 0.996094i \(0.471858\pi\)
\(164\) 67.2730i 0.410201i
\(165\) 0 0
\(166\) 146.354 0.881650
\(167\) 56.0682i 0.335737i 0.985809 + 0.167869i \(0.0536885\pi\)
−0.985809 + 0.167869i \(0.946312\pi\)
\(168\) 0 0
\(169\) 280.277 1.65844
\(170\) 63.6396i 0.374351i
\(171\) 0 0
\(172\) −18.3538 −0.106708
\(173\) − 220.952i − 1.27718i −0.769548 0.638589i \(-0.779518\pi\)
0.769548 0.638589i \(-0.220482\pi\)
\(174\) 0 0
\(175\) 72.0770 0.411868
\(176\) 58.7878i 0.334021i
\(177\) 0 0
\(178\) 189.688 1.06567
\(179\) − 248.347i − 1.38741i −0.720258 0.693706i \(-0.755977\pi\)
0.720258 0.693706i \(-0.244023\pi\)
\(180\) 0 0
\(181\) −158.277 −0.874458 −0.437229 0.899350i \(-0.644040\pi\)
−0.437229 + 0.899350i \(0.644040\pi\)
\(182\) − 251.567i − 1.38224i
\(183\) 0 0
\(184\) 41.5692 0.225920
\(185\) − 351.032i − 1.89747i
\(186\) 0 0
\(187\) −114.115 −0.610243
\(188\) 33.9411i 0.180538i
\(189\) 0 0
\(190\) 199.923 1.05223
\(191\) 34.5503i 0.180892i 0.995901 + 0.0904459i \(0.0288292\pi\)
−0.995901 + 0.0904459i \(0.971171\pi\)
\(192\) 0 0
\(193\) −55.0000 −0.284974 −0.142487 0.989797i \(-0.545510\pi\)
−0.142487 + 0.989797i \(0.545510\pi\)
\(194\) 106.262i 0.547741i
\(195\) 0 0
\(196\) −42.8616 −0.218681
\(197\) 35.7170i 0.181305i 0.995883 + 0.0906524i \(0.0288952\pi\)
−0.995883 + 0.0906524i \(0.971105\pi\)
\(198\) 0 0
\(199\) −375.138 −1.88512 −0.942559 0.334040i \(-0.891588\pi\)
−0.942559 + 0.334040i \(0.891588\pi\)
\(200\) − 24.2918i − 0.121459i
\(201\) 0 0
\(202\) −41.1384 −0.203656
\(203\) − 297.877i − 1.46737i
\(204\) 0 0
\(205\) 194.942 0.950938
\(206\) − 136.765i − 0.663909i
\(207\) 0 0
\(208\) −84.7846 −0.407618
\(209\) 358.492i 1.71527i
\(210\) 0 0
\(211\) 307.454 1.45713 0.728563 0.684978i \(-0.240188\pi\)
0.728563 + 0.684978i \(0.240188\pi\)
\(212\) − 51.5209i − 0.243023i
\(213\) 0 0
\(214\) 251.138 1.17354
\(215\) 53.1853i 0.247374i
\(216\) 0 0
\(217\) −67.1384 −0.309394
\(218\) − 87.5996i − 0.401833i
\(219\) 0 0
\(220\) 170.354 0.774336
\(221\) − 164.579i − 0.744702i
\(222\) 0 0
\(223\) −338.592 −1.51835 −0.759175 0.650886i \(-0.774398\pi\)
−0.759175 + 0.650886i \(0.774398\pi\)
\(224\) 47.4740i 0.211938i
\(225\) 0 0
\(226\) 155.296 0.687151
\(227\) 123.950i 0.546037i 0.962009 + 0.273019i \(0.0880220\pi\)
−0.962009 + 0.273019i \(0.911978\pi\)
\(228\) 0 0
\(229\) 74.0577 0.323396 0.161698 0.986840i \(-0.448303\pi\)
0.161698 + 0.986840i \(0.448303\pi\)
\(230\) − 120.458i − 0.523732i
\(231\) 0 0
\(232\) −100.392 −0.432725
\(233\) − 273.223i − 1.17263i −0.810082 0.586316i \(-0.800578\pi\)
0.810082 0.586316i \(-0.199422\pi\)
\(234\) 0 0
\(235\) 98.3538 0.418527
\(236\) − 123.341i − 0.522632i
\(237\) 0 0
\(238\) −92.1539 −0.387201
\(239\) 452.885i 1.89492i 0.319877 + 0.947459i \(0.396359\pi\)
−0.319877 + 0.947459i \(0.603641\pi\)
\(240\) 0 0
\(241\) −382.688 −1.58792 −0.793959 0.607971i \(-0.791984\pi\)
−0.793959 + 0.607971i \(0.791984\pi\)
\(242\) 134.350i 0.555166i
\(243\) 0 0
\(244\) 26.0000 0.106557
\(245\) 124.203i 0.506952i
\(246\) 0 0
\(247\) −517.023 −2.09321
\(248\) 22.6274i 0.0912396i
\(249\) 0 0
\(250\) 134.512 0.538046
\(251\) 73.9307i 0.294544i 0.989096 + 0.147272i \(0.0470493\pi\)
−0.989096 + 0.147272i \(0.952951\pi\)
\(252\) 0 0
\(253\) 216.000 0.853755
\(254\) − 199.654i − 0.786041i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 159.809i 0.621824i 0.950439 + 0.310912i \(0.100634\pi\)
−0.950439 + 0.310912i \(0.899366\pi\)
\(258\) 0 0
\(259\) 508.315 1.96261
\(260\) 245.687i 0.944950i
\(261\) 0 0
\(262\) −75.2154 −0.287082
\(263\) − 259.106i − 0.985193i −0.870258 0.492596i \(-0.836048\pi\)
0.870258 0.492596i \(-0.163952\pi\)
\(264\) 0 0
\(265\) −149.296 −0.563382
\(266\) 289.500i 1.08835i
\(267\) 0 0
\(268\) −42.3538 −0.158037
\(269\) − 24.8168i − 0.0922556i −0.998936 0.0461278i \(-0.985312\pi\)
0.998936 0.0461278i \(-0.0146881\pi\)
\(270\) 0 0
\(271\) 98.1154 0.362050 0.181025 0.983479i \(-0.442059\pi\)
0.181025 + 0.983479i \(0.442059\pi\)
\(272\) 31.0583i 0.114185i
\(273\) 0 0
\(274\) 2.31158 0.00843642
\(275\) − 126.224i − 0.458996i
\(276\) 0 0
\(277\) −1.41532 −0.00510945 −0.00255472 0.999997i \(-0.500813\pi\)
−0.00255472 + 0.999997i \(0.500813\pi\)
\(278\) 27.1747i 0.0977506i
\(279\) 0 0
\(280\) 137.569 0.491319
\(281\) 100.716i 0.358421i 0.983811 + 0.179211i \(0.0573544\pi\)
−0.983811 + 0.179211i \(0.942646\pi\)
\(282\) 0 0
\(283\) −47.2923 −0.167111 −0.0835554 0.996503i \(-0.526628\pi\)
−0.0835554 + 0.996503i \(0.526628\pi\)
\(284\) 202.428i 0.712776i
\(285\) 0 0
\(286\) −440.554 −1.54040
\(287\) 282.288i 0.983582i
\(288\) 0 0
\(289\) 228.711 0.791389
\(290\) 290.915i 1.00315i
\(291\) 0 0
\(292\) −80.8231 −0.276791
\(293\) 333.616i 1.13862i 0.822123 + 0.569310i \(0.192790\pi\)
−0.822123 + 0.569310i \(0.807210\pi\)
\(294\) 0 0
\(295\) −357.415 −1.21158
\(296\) − 171.316i − 0.578769i
\(297\) 0 0
\(298\) 133.335 0.447432
\(299\) 311.519i 1.04187i
\(300\) 0 0
\(301\) −77.0155 −0.255865
\(302\) − 45.2548i − 0.149850i
\(303\) 0 0
\(304\) 97.5692 0.320951
\(305\) − 75.3422i − 0.247024i
\(306\) 0 0
\(307\) 10.3538 0.0337258 0.0168629 0.999858i \(-0.494632\pi\)
0.0168629 + 0.999858i \(0.494632\pi\)
\(308\) 246.682i 0.800917i
\(309\) 0 0
\(310\) 65.5692 0.211514
\(311\) 9.54047i 0.0306768i 0.999882 + 0.0153384i \(0.00488255\pi\)
−0.999882 + 0.0153384i \(0.995117\pi\)
\(312\) 0 0
\(313\) −479.277 −1.53124 −0.765618 0.643295i \(-0.777567\pi\)
−0.765618 + 0.643295i \(0.777567\pi\)
\(314\) − 0.413434i − 0.00131667i
\(315\) 0 0
\(316\) −197.492 −0.624976
\(317\) − 179.662i − 0.566758i −0.959008 0.283379i \(-0.908545\pi\)
0.959008 0.283379i \(-0.0914554\pi\)
\(318\) 0 0
\(319\) −521.654 −1.63528
\(320\) − 46.3644i − 0.144889i
\(321\) 0 0
\(322\) 174.431 0.541710
\(323\) 189.396i 0.586365i
\(324\) 0 0
\(325\) 182.042 0.560130
\(326\) − 40.7076i − 0.124870i
\(327\) 0 0
\(328\) 95.1384 0.290056
\(329\) 142.422i 0.432894i
\(330\) 0 0
\(331\) 295.454 0.892610 0.446305 0.894881i \(-0.352740\pi\)
0.446305 + 0.894881i \(0.352740\pi\)
\(332\) − 206.976i − 0.623420i
\(333\) 0 0
\(334\) 79.2923 0.237402
\(335\) 122.732i 0.366364i
\(336\) 0 0
\(337\) 489.261 1.45181 0.725907 0.687793i \(-0.241420\pi\)
0.725907 + 0.687793i \(0.241420\pi\)
\(338\) − 396.371i − 1.17270i
\(339\) 0 0
\(340\) 90.0000 0.264706
\(341\) 117.576i 0.344796i
\(342\) 0 0
\(343\) 231.369 0.674546
\(344\) 25.9562i 0.0754542i
\(345\) 0 0
\(346\) −312.473 −0.903101
\(347\) − 666.682i − 1.92127i −0.277807 0.960637i \(-0.589608\pi\)
0.277807 0.960637i \(-0.410392\pi\)
\(348\) 0 0
\(349\) 511.969 1.46696 0.733480 0.679711i \(-0.237895\pi\)
0.733480 + 0.679711i \(0.237895\pi\)
\(350\) − 101.932i − 0.291235i
\(351\) 0 0
\(352\) 83.1384 0.236189
\(353\) − 586.681i − 1.66199i −0.556283 0.830993i \(-0.687773\pi\)
0.556283 0.830993i \(-0.312227\pi\)
\(354\) 0 0
\(355\) 586.592 1.65237
\(356\) − 268.260i − 0.753539i
\(357\) 0 0
\(358\) −351.215 −0.981049
\(359\) 534.573i 1.48906i 0.667589 + 0.744530i \(0.267326\pi\)
−0.667589 + 0.744530i \(0.732674\pi\)
\(360\) 0 0
\(361\) 233.985 0.648157
\(362\) 223.837i 0.618335i
\(363\) 0 0
\(364\) −355.769 −0.977388
\(365\) 234.207i 0.641664i
\(366\) 0 0
\(367\) 15.2923 0.0416685 0.0208343 0.999783i \(-0.493368\pi\)
0.0208343 + 0.999783i \(0.493368\pi\)
\(368\) − 58.7878i − 0.159749i
\(369\) 0 0
\(370\) −496.435 −1.34172
\(371\) − 216.190i − 0.582721i
\(372\) 0 0
\(373\) 652.985 1.75063 0.875314 0.483554i \(-0.160654\pi\)
0.875314 + 0.483554i \(0.160654\pi\)
\(374\) 161.384i 0.431507i
\(375\) 0 0
\(376\) 48.0000 0.127660
\(377\) − 752.337i − 1.99559i
\(378\) 0 0
\(379\) −655.215 −1.72880 −0.864400 0.502804i \(-0.832302\pi\)
−0.864400 + 0.502804i \(0.832302\pi\)
\(380\) − 282.734i − 0.744037i
\(381\) 0 0
\(382\) 48.8616 0.127910
\(383\) 299.541i 0.782092i 0.920371 + 0.391046i \(0.127887\pi\)
−0.920371 + 0.391046i \(0.872113\pi\)
\(384\) 0 0
\(385\) 714.831 1.85670
\(386\) 77.7817i 0.201507i
\(387\) 0 0
\(388\) 150.277 0.387312
\(389\) 464.558i 1.19424i 0.802153 + 0.597119i \(0.203688\pi\)
−0.802153 + 0.597119i \(0.796312\pi\)
\(390\) 0 0
\(391\) 114.115 0.291855
\(392\) 60.6154i 0.154631i
\(393\) 0 0
\(394\) 50.5115 0.128202
\(395\) 572.289i 1.44883i
\(396\) 0 0
\(397\) −185.708 −0.467777 −0.233889 0.972263i \(-0.575145\pi\)
−0.233889 + 0.972263i \(0.575145\pi\)
\(398\) 530.526i 1.33298i
\(399\) 0 0
\(400\) −34.3538 −0.0858846
\(401\) 62.8592i 0.156756i 0.996924 + 0.0783780i \(0.0249741\pi\)
−0.996924 + 0.0783780i \(0.975026\pi\)
\(402\) 0 0
\(403\) −169.569 −0.420767
\(404\) 58.1785i 0.144006i
\(405\) 0 0
\(406\) −421.261 −1.03759
\(407\) − 890.182i − 2.18718i
\(408\) 0 0
\(409\) −327.281 −0.800197 −0.400099 0.916472i \(-0.631024\pi\)
−0.400099 + 0.916472i \(0.631024\pi\)
\(410\) − 275.690i − 0.672415i
\(411\) 0 0
\(412\) −193.415 −0.469455
\(413\) − 517.558i − 1.25317i
\(414\) 0 0
\(415\) −599.769 −1.44523
\(416\) 119.904i 0.288230i
\(417\) 0 0
\(418\) 506.985 1.21288
\(419\) − 649.875i − 1.55101i −0.631339 0.775507i \(-0.717494\pi\)
0.631339 0.775507i \(-0.282506\pi\)
\(420\) 0 0
\(421\) 3.31913 0.00788391 0.00394196 0.999992i \(-0.498745\pi\)
0.00394196 + 0.999992i \(0.498745\pi\)
\(422\) − 434.805i − 1.03034i
\(423\) 0 0
\(424\) −72.8616 −0.171843
\(425\) − 66.6857i − 0.156908i
\(426\) 0 0
\(427\) 109.100 0.255503
\(428\) − 355.163i − 0.829821i
\(429\) 0 0
\(430\) 75.2154 0.174920
\(431\) − 803.502i − 1.86427i −0.362107 0.932136i \(-0.617943\pi\)
0.362107 0.932136i \(-0.382057\pi\)
\(432\) 0 0
\(433\) −93.1230 −0.215065 −0.107532 0.994202i \(-0.534295\pi\)
−0.107532 + 0.994202i \(0.534295\pi\)
\(434\) 94.9481i 0.218774i
\(435\) 0 0
\(436\) −123.885 −0.284139
\(437\) − 358.492i − 0.820348i
\(438\) 0 0
\(439\) −472.000 −1.07517 −0.537585 0.843209i \(-0.680663\pi\)
−0.537585 + 0.843209i \(0.680663\pi\)
\(440\) − 240.917i − 0.547538i
\(441\) 0 0
\(442\) −232.750 −0.526584
\(443\) − 569.689i − 1.28598i −0.765875 0.642989i \(-0.777694\pi\)
0.765875 0.642989i \(-0.222306\pi\)
\(444\) 0 0
\(445\) −777.358 −1.74687
\(446\) 478.842i 1.07364i
\(447\) 0 0
\(448\) 67.1384 0.149863
\(449\) 559.115i 1.24524i 0.782523 + 0.622622i \(0.213933\pi\)
−0.782523 + 0.622622i \(0.786067\pi\)
\(450\) 0 0
\(451\) 494.354 1.09613
\(452\) − 219.622i − 0.485889i
\(453\) 0 0
\(454\) 175.292 0.386106
\(455\) 1030.94i 2.26580i
\(456\) 0 0
\(457\) 604.296 1.32231 0.661155 0.750249i \(-0.270066\pi\)
0.661155 + 0.750249i \(0.270066\pi\)
\(458\) − 104.733i − 0.228676i
\(459\) 0 0
\(460\) −170.354 −0.370334
\(461\) 5.13459i 0.0111379i 0.999984 + 0.00556897i \(0.00177267\pi\)
−0.999984 + 0.00556897i \(0.998227\pi\)
\(462\) 0 0
\(463\) −161.492 −0.348795 −0.174398 0.984675i \(-0.555798\pi\)
−0.174398 + 0.984675i \(0.555798\pi\)
\(464\) 141.976i 0.305983i
\(465\) 0 0
\(466\) −386.396 −0.829176
\(467\) 503.025i 1.07714i 0.842581 + 0.538570i \(0.181035\pi\)
−0.842581 + 0.538570i \(0.818965\pi\)
\(468\) 0 0
\(469\) −177.723 −0.378941
\(470\) − 139.093i − 0.295943i
\(471\) 0 0
\(472\) −174.431 −0.369557
\(473\) 134.873i 0.285143i
\(474\) 0 0
\(475\) −209.492 −0.441036
\(476\) 130.325i 0.273793i
\(477\) 0 0
\(478\) 640.477 1.33991
\(479\) − 213.796i − 0.446339i −0.974780 0.223170i \(-0.928360\pi\)
0.974780 0.223170i \(-0.0716404\pi\)
\(480\) 0 0
\(481\) 1283.83 2.66909
\(482\) 541.203i 1.12283i
\(483\) 0 0
\(484\) 190.000 0.392562
\(485\) − 435.469i − 0.897874i
\(486\) 0 0
\(487\) 8.63071 0.0177222 0.00886110 0.999961i \(-0.497179\pi\)
0.00886110 + 0.999961i \(0.497179\pi\)
\(488\) − 36.7696i − 0.0753474i
\(489\) 0 0
\(490\) 175.650 0.358469
\(491\) − 922.459i − 1.87873i −0.342912 0.939367i \(-0.611413\pi\)
0.342912 0.939367i \(-0.388587\pi\)
\(492\) 0 0
\(493\) −275.596 −0.559018
\(494\) 731.181i 1.48012i
\(495\) 0 0
\(496\) 32.0000 0.0645161
\(497\) 849.420i 1.70909i
\(498\) 0 0
\(499\) −434.592 −0.870926 −0.435463 0.900207i \(-0.643415\pi\)
−0.435463 + 0.900207i \(0.643415\pi\)
\(500\) − 190.228i − 0.380456i
\(501\) 0 0
\(502\) 104.554 0.208274
\(503\) − 144.087i − 0.286454i −0.989690 0.143227i \(-0.954252\pi\)
0.989690 0.143227i \(-0.0457480\pi\)
\(504\) 0 0
\(505\) 168.588 0.333839
\(506\) − 305.470i − 0.603696i
\(507\) 0 0
\(508\) −282.354 −0.555815
\(509\) 697.272i 1.36989i 0.728596 + 0.684943i \(0.240173\pi\)
−0.728596 + 0.684943i \(0.759827\pi\)
\(510\) 0 0
\(511\) −339.146 −0.663691
\(512\) − 22.6274i − 0.0441942i
\(513\) 0 0
\(514\) 226.004 0.439696
\(515\) 560.475i 1.08830i
\(516\) 0 0
\(517\) 249.415 0.482428
\(518\) − 718.866i − 1.38777i
\(519\) 0 0
\(520\) 347.454 0.668180
\(521\) 426.962i 0.819504i 0.912197 + 0.409752i \(0.134385\pi\)
−0.912197 + 0.409752i \(0.865615\pi\)
\(522\) 0 0
\(523\) 179.762 0.343712 0.171856 0.985122i \(-0.445024\pi\)
0.171856 + 0.985122i \(0.445024\pi\)
\(524\) 106.371i 0.202997i
\(525\) 0 0
\(526\) −366.431 −0.696636
\(527\) 62.1166i 0.117868i
\(528\) 0 0
\(529\) 313.000 0.591682
\(530\) 211.137i 0.398371i
\(531\) 0 0
\(532\) 409.415 0.769578
\(533\) 712.965i 1.33764i
\(534\) 0 0
\(535\) −1029.18 −1.92371
\(536\) 59.8974i 0.111749i
\(537\) 0 0
\(538\) −35.0962 −0.0652346
\(539\) 314.967i 0.584354i
\(540\) 0 0
\(541\) 708.734 1.31005 0.655023 0.755609i \(-0.272659\pi\)
0.655023 + 0.755609i \(0.272659\pi\)
\(542\) − 138.756i − 0.256008i
\(543\) 0 0
\(544\) 43.9230 0.0807409
\(545\) 358.990i 0.658697i
\(546\) 0 0
\(547\) −196.231 −0.358740 −0.179370 0.983782i \(-0.557406\pi\)
−0.179370 + 0.983782i \(0.557406\pi\)
\(548\) − 3.26907i − 0.00596545i
\(549\) 0 0
\(550\) −178.508 −0.324560
\(551\) 865.781i 1.57129i
\(552\) 0 0
\(553\) −828.708 −1.49857
\(554\) 2.00156i 0.00361292i
\(555\) 0 0
\(556\) 38.4308 0.0691201
\(557\) − 353.610i − 0.634848i −0.948284 0.317424i \(-0.897182\pi\)
0.948284 0.317424i \(-0.102818\pi\)
\(558\) 0 0
\(559\) −194.515 −0.347970
\(560\) − 194.552i − 0.347415i
\(561\) 0 0
\(562\) 142.435 0.253442
\(563\) − 373.026i − 0.662568i −0.943531 0.331284i \(-0.892518\pi\)
0.943531 0.331284i \(-0.107482\pi\)
\(564\) 0 0
\(565\) −636.415 −1.12640
\(566\) 66.8815i 0.118165i
\(567\) 0 0
\(568\) 286.277 0.504009
\(569\) − 60.1177i − 0.105655i −0.998604 0.0528275i \(-0.983177\pi\)
0.998604 0.0528275i \(-0.0168233\pi\)
\(570\) 0 0
\(571\) 86.1999 0.150963 0.0754815 0.997147i \(-0.475951\pi\)
0.0754815 + 0.997147i \(0.475951\pi\)
\(572\) 623.037i 1.08923i
\(573\) 0 0
\(574\) 399.215 0.695497
\(575\) 126.224i 0.219520i
\(576\) 0 0
\(577\) 709.123 1.22898 0.614491 0.788924i \(-0.289361\pi\)
0.614491 + 0.788924i \(0.289361\pi\)
\(578\) − 323.447i − 0.559597i
\(579\) 0 0
\(580\) 411.415 0.709337
\(581\) − 868.501i − 1.49484i
\(582\) 0 0
\(583\) −378.600 −0.649399
\(584\) 114.301i 0.195721i
\(585\) 0 0
\(586\) 471.804 0.805126
\(587\) 962.285i 1.63933i 0.572845 + 0.819664i \(0.305840\pi\)
−0.572845 + 0.819664i \(0.694160\pi\)
\(588\) 0 0
\(589\) 195.138 0.331305
\(590\) 505.462i 0.856715i
\(591\) 0 0
\(592\) −242.277 −0.409251
\(593\) 104.350i 0.175969i 0.996122 + 0.0879847i \(0.0280427\pi\)
−0.996122 + 0.0879847i \(0.971957\pi\)
\(594\) 0 0
\(595\) 377.654 0.634712
\(596\) − 188.564i − 0.316382i
\(597\) 0 0
\(598\) 440.554 0.736712
\(599\) − 286.063i − 0.477567i −0.971073 0.238784i \(-0.923251\pi\)
0.971073 0.238784i \(-0.0767486\pi\)
\(600\) 0 0
\(601\) −280.415 −0.466581 −0.233291 0.972407i \(-0.574949\pi\)
−0.233291 + 0.972407i \(0.574949\pi\)
\(602\) 108.916i 0.180924i
\(603\) 0 0
\(604\) −64.0000 −0.105960
\(605\) − 550.578i − 0.910046i
\(606\) 0 0
\(607\) 737.731 1.21537 0.607686 0.794177i \(-0.292098\pi\)
0.607686 + 0.794177i \(0.292098\pi\)
\(608\) − 137.984i − 0.226947i
\(609\) 0 0
\(610\) −106.550 −0.174672
\(611\) 359.711i 0.588724i
\(612\) 0 0
\(613\) −679.415 −1.10834 −0.554172 0.832402i \(-0.686965\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(614\) − 14.6425i − 0.0238478i
\(615\) 0 0
\(616\) 348.862 0.566334
\(617\) 472.649i 0.766044i 0.923739 + 0.383022i \(0.125117\pi\)
−0.923739 + 0.383022i \(0.874883\pi\)
\(618\) 0 0
\(619\) 886.354 1.43191 0.715956 0.698145i \(-0.245991\pi\)
0.715956 + 0.698145i \(0.245991\pi\)
\(620\) − 92.7289i − 0.149563i
\(621\) 0 0
\(622\) 13.4923 0.0216917
\(623\) − 1125.66i − 1.80684i
\(624\) 0 0
\(625\) −765.950 −1.22552
\(626\) 677.800i 1.08275i
\(627\) 0 0
\(628\) −0.584684 −0.000931025 0
\(629\) − 470.294i − 0.747685i
\(630\) 0 0
\(631\) −729.108 −1.15548 −0.577740 0.816221i \(-0.696065\pi\)
−0.577740 + 0.816221i \(0.696065\pi\)
\(632\) 279.296i 0.441924i
\(633\) 0 0
\(634\) −254.081 −0.400758
\(635\) 818.199i 1.28850i
\(636\) 0 0
\(637\) −454.250 −0.713108
\(638\) 737.730i 1.15632i
\(639\) 0 0
\(640\) −65.5692 −0.102452
\(641\) − 870.195i − 1.35756i −0.734342 0.678780i \(-0.762509\pi\)
0.734342 0.678780i \(-0.237491\pi\)
\(642\) 0 0
\(643\) −618.123 −0.961311 −0.480656 0.876910i \(-0.659601\pi\)
−0.480656 + 0.876910i \(0.659601\pi\)
\(644\) − 246.682i − 0.383047i
\(645\) 0 0
\(646\) 267.846 0.414622
\(647\) − 425.439i − 0.657556i −0.944407 0.328778i \(-0.893363\pi\)
0.944407 0.328778i \(-0.106637\pi\)
\(648\) 0 0
\(649\) −906.369 −1.39656
\(650\) − 257.447i − 0.396072i
\(651\) 0 0
\(652\) −57.5692 −0.0882963
\(653\) − 188.808i − 0.289140i −0.989495 0.144570i \(-0.953820\pi\)
0.989495 0.144570i \(-0.0461799\pi\)
\(654\) 0 0
\(655\) 308.238 0.470593
\(656\) − 134.546i − 0.205101i
\(657\) 0 0
\(658\) 201.415 0.306102
\(659\) 660.470i 1.00223i 0.865380 + 0.501116i \(0.167077\pi\)
−0.865380 + 0.501116i \(0.832923\pi\)
\(660\) 0 0
\(661\) −589.831 −0.892331 −0.446165 0.894951i \(-0.647211\pi\)
−0.446165 + 0.894951i \(0.647211\pi\)
\(662\) − 417.835i − 0.631170i
\(663\) 0 0
\(664\) −292.708 −0.440825
\(665\) − 1186.39i − 1.78405i
\(666\) 0 0
\(667\) 521.654 0.782090
\(668\) − 112.136i − 0.167869i
\(669\) 0 0
\(670\) 173.569 0.259059
\(671\) − 191.060i − 0.284739i
\(672\) 0 0
\(673\) −391.431 −0.581621 −0.290810 0.956781i \(-0.593925\pi\)
−0.290810 + 0.956781i \(0.593925\pi\)
\(674\) − 691.920i − 1.02659i
\(675\) 0 0
\(676\) −560.554 −0.829222
\(677\) 693.661i 1.02461i 0.858804 + 0.512305i \(0.171208\pi\)
−0.858804 + 0.512305i \(0.828792\pi\)
\(678\) 0 0
\(679\) 630.585 0.928696
\(680\) − 127.279i − 0.187175i
\(681\) 0 0
\(682\) 166.277 0.243808
\(683\) − 678.170i − 0.992928i −0.868057 0.496464i \(-0.834632\pi\)
0.868057 0.496464i \(-0.165368\pi\)
\(684\) 0 0
\(685\) −9.47303 −0.0138292
\(686\) − 327.206i − 0.476976i
\(687\) 0 0
\(688\) 36.7077 0.0533542
\(689\) − 546.022i − 0.792485i
\(690\) 0 0
\(691\) 675.777 0.977969 0.488985 0.872292i \(-0.337367\pi\)
0.488985 + 0.872292i \(0.337367\pi\)
\(692\) 441.904i 0.638589i
\(693\) 0 0
\(694\) −942.831 −1.35855
\(695\) − 111.364i − 0.160236i
\(696\) 0 0
\(697\) 261.173 0.374710
\(698\) − 724.034i − 1.03730i
\(699\) 0 0
\(700\) −144.154 −0.205934
\(701\) 797.260i 1.13732i 0.822573 + 0.568659i \(0.192538\pi\)
−0.822573 + 0.568659i \(0.807462\pi\)
\(702\) 0 0
\(703\) −1477.42 −2.10160
\(704\) − 117.576i − 0.167011i
\(705\) 0 0
\(706\) −829.692 −1.17520
\(707\) 244.126i 0.345298i
\(708\) 0 0
\(709\) 173.504 0.244716 0.122358 0.992486i \(-0.460954\pi\)
0.122358 + 0.992486i \(0.460954\pi\)
\(710\) − 829.567i − 1.16840i
\(711\) 0 0
\(712\) −379.377 −0.532833
\(713\) − 117.576i − 0.164903i
\(714\) 0 0
\(715\) 1805.42 2.52507
\(716\) 496.694i 0.693706i
\(717\) 0 0
\(718\) 756.000 1.05292
\(719\) 188.177i 0.261721i 0.991401 + 0.130860i \(0.0417740\pi\)
−0.991401 + 0.130860i \(0.958226\pi\)
\(720\) 0 0
\(721\) −811.600 −1.12566
\(722\) − 330.904i − 0.458316i
\(723\) 0 0
\(724\) 316.554 0.437229
\(725\) − 304.839i − 0.420468i
\(726\) 0 0
\(727\) 711.777 0.979060 0.489530 0.871986i \(-0.337168\pi\)
0.489530 + 0.871986i \(0.337168\pi\)
\(728\) 503.134i 0.691118i
\(729\) 0 0
\(730\) 331.219 0.453725
\(731\) 71.2548i 0.0974758i
\(732\) 0 0
\(733\) 1226.83 1.67371 0.836856 0.547423i \(-0.184391\pi\)
0.836856 + 0.547423i \(0.184391\pi\)
\(734\) − 21.6266i − 0.0294641i
\(735\) 0 0
\(736\) −83.1384 −0.112960
\(737\) 311.236i 0.422301i
\(738\) 0 0
\(739\) 741.892 1.00391 0.501957 0.864893i \(-0.332614\pi\)
0.501957 + 0.864893i \(0.332614\pi\)
\(740\) 702.064i 0.948736i
\(741\) 0 0
\(742\) −305.738 −0.412046
\(743\) 781.984i 1.05247i 0.850340 + 0.526234i \(0.176396\pi\)
−0.850340 + 0.526234i \(0.823604\pi\)
\(744\) 0 0
\(745\) −546.415 −0.733443
\(746\) − 923.460i − 1.23788i
\(747\) 0 0
\(748\) 228.231 0.305121
\(749\) − 1490.32i − 1.98975i
\(750\) 0 0
\(751\) −1033.33 −1.37594 −0.687970 0.725739i \(-0.741498\pi\)
−0.687970 + 0.725739i \(0.741498\pi\)
\(752\) − 67.8823i − 0.0902690i
\(753\) 0 0
\(754\) −1063.97 −1.41109
\(755\) 185.458i 0.245639i
\(756\) 0 0
\(757\) 473.877 0.625993 0.312997 0.949754i \(-0.398667\pi\)
0.312997 + 0.949754i \(0.398667\pi\)
\(758\) 926.614i 1.22245i
\(759\) 0 0
\(760\) −399.846 −0.526113
\(761\) − 940.455i − 1.23581i −0.786251 0.617907i \(-0.787981\pi\)
0.786251 0.617907i \(-0.212019\pi\)
\(762\) 0 0
\(763\) −519.839 −0.681309
\(764\) − 69.1007i − 0.0904459i
\(765\) 0 0
\(766\) 423.615 0.553023
\(767\) − 1307.18i − 1.70428i
\(768\) 0 0
\(769\) 980.815 1.27544 0.637721 0.770267i \(-0.279877\pi\)
0.637721 + 0.770267i \(0.279877\pi\)
\(770\) − 1010.92i − 1.31289i
\(771\) 0 0
\(772\) 110.000 0.142487
\(773\) 1042.20i 1.34825i 0.738618 + 0.674124i \(0.235479\pi\)
−0.738618 + 0.674124i \(0.764521\pi\)
\(774\) 0 0
\(775\) −68.7077 −0.0886550
\(776\) − 212.524i − 0.273871i
\(777\) 0 0
\(778\) 656.985 0.844453
\(779\) − 820.472i − 1.05324i
\(780\) 0 0
\(781\) 1487.54 1.90466
\(782\) − 161.384i − 0.206373i
\(783\) 0 0
\(784\) 85.7231 0.109341
\(785\) 1.69428i 0.00215832i
\(786\) 0 0
\(787\) −144.700 −0.183863 −0.0919315 0.995765i \(-0.529304\pi\)
−0.0919315 + 0.995765i \(0.529304\pi\)
\(788\) − 71.4341i − 0.0906524i
\(789\) 0 0
\(790\) 809.338 1.02448
\(791\) − 921.567i − 1.16507i
\(792\) 0 0
\(793\) 275.550 0.347478
\(794\) 262.630i 0.330769i
\(795\) 0 0
\(796\) 750.277 0.942559
\(797\) 155.729i 0.195394i 0.995216 + 0.0976972i \(0.0311477\pi\)
−0.995216 + 0.0976972i \(0.968852\pi\)
\(798\) 0 0
\(799\) 131.769 0.164918
\(800\) 48.5837i 0.0607296i
\(801\) 0 0
\(802\) 88.8963 0.110843
\(803\) 593.926i 0.739634i
\(804\) 0 0
\(805\) −714.831 −0.887988
\(806\) 239.807i 0.297527i
\(807\) 0 0
\(808\) 82.2769 0.101828
\(809\) 1216.92i 1.50422i 0.659035 + 0.752112i \(0.270965\pi\)
−0.659035 + 0.752112i \(0.729035\pi\)
\(810\) 0 0
\(811\) 1243.31 1.53305 0.766527 0.642212i \(-0.221983\pi\)
0.766527 + 0.642212i \(0.221983\pi\)
\(812\) 595.754i 0.733687i
\(813\) 0 0
\(814\) −1258.91 −1.54657
\(815\) 166.823i 0.204691i
\(816\) 0 0
\(817\) 223.846 0.273985
\(818\) 462.845i 0.565825i
\(819\) 0 0
\(820\) −389.885 −0.475469
\(821\) − 13.2854i − 0.0161820i −0.999967 0.00809098i \(-0.997425\pi\)
0.999967 0.00809098i \(-0.00257547\pi\)
\(822\) 0 0
\(823\) −214.200 −0.260267 −0.130134 0.991496i \(-0.541541\pi\)
−0.130134 + 0.991496i \(0.541541\pi\)
\(824\) 273.531i 0.331955i
\(825\) 0 0
\(826\) −731.938 −0.886124
\(827\) − 1221.27i − 1.47675i −0.674391 0.738374i \(-0.735594\pi\)
0.674391 0.738374i \(-0.264406\pi\)
\(828\) 0 0
\(829\) −358.431 −0.432365 −0.216183 0.976353i \(-0.569361\pi\)
−0.216183 + 0.976353i \(0.569361\pi\)
\(830\) 848.202i 1.02193i
\(831\) 0 0
\(832\) 169.569 0.203809
\(833\) 166.401i 0.199761i
\(834\) 0 0
\(835\) −324.946 −0.389157
\(836\) − 716.984i − 0.857637i
\(837\) 0 0
\(838\) −919.061 −1.09673
\(839\) 327.597i 0.390461i 0.980757 + 0.195231i \(0.0625456\pi\)
−0.980757 + 0.195231i \(0.937454\pi\)
\(840\) 0 0
\(841\) −418.827 −0.498011
\(842\) − 4.69395i − 0.00557477i
\(843\) 0 0
\(844\) −614.908 −0.728563
\(845\) 1624.36i 1.92232i
\(846\) 0 0
\(847\) 797.269 0.941286
\(848\) 103.042i 0.121512i
\(849\) 0 0
\(850\) −94.3078 −0.110950
\(851\) 890.182i 1.04604i
\(852\) 0 0
\(853\) −52.8306 −0.0619351 −0.0309675 0.999520i \(-0.509859\pi\)
−0.0309675 + 0.999520i \(0.509859\pi\)
\(854\) − 154.291i − 0.180668i
\(855\) 0 0
\(856\) −502.277 −0.586772
\(857\) 1248.42i 1.45673i 0.685187 + 0.728367i \(0.259720\pi\)
−0.685187 + 0.728367i \(0.740280\pi\)
\(858\) 0 0
\(859\) −446.739 −0.520068 −0.260034 0.965599i \(-0.583734\pi\)
−0.260034 + 0.965599i \(0.583734\pi\)
\(860\) − 106.371i − 0.123687i
\(861\) 0 0
\(862\) −1136.32 −1.31824
\(863\) − 635.297i − 0.736150i −0.929796 0.368075i \(-0.880017\pi\)
0.929796 0.368075i \(-0.119983\pi\)
\(864\) 0 0
\(865\) 1280.54 1.48039
\(866\) 131.696i 0.152074i
\(867\) 0 0
\(868\) 134.277 0.154697
\(869\) 1451.27i 1.67004i
\(870\) 0 0
\(871\) −448.869 −0.515349
\(872\) 175.199i 0.200917i
\(873\) 0 0
\(874\) −506.985 −0.580074
\(875\) − 798.226i − 0.912258i
\(876\) 0 0
\(877\) −788.692 −0.899307 −0.449653 0.893203i \(-0.648453\pi\)
−0.449653 + 0.893203i \(0.648453\pi\)
\(878\) 667.509i 0.760261i
\(879\) 0 0
\(880\) −340.708 −0.387168
\(881\) 558.506i 0.633945i 0.948435 + 0.316972i \(0.102666\pi\)
−0.948435 + 0.316972i \(0.897334\pi\)
\(882\) 0 0
\(883\) 313.338 0.354857 0.177428 0.984134i \(-0.443222\pi\)
0.177428 + 0.984134i \(0.443222\pi\)
\(884\) 329.158i 0.372351i
\(885\) 0 0
\(886\) −805.661 −0.909324
\(887\) 597.701i 0.673845i 0.941532 + 0.336923i \(0.109386\pi\)
−0.941532 + 0.336923i \(0.890614\pi\)
\(888\) 0 0
\(889\) −1184.80 −1.33273
\(890\) 1099.35i 1.23522i
\(891\) 0 0
\(892\) 677.184 0.759175
\(893\) − 413.951i − 0.463551i
\(894\) 0 0
\(895\) 1439.31 1.60816
\(896\) − 94.9481i − 0.105969i
\(897\) 0 0
\(898\) 790.708 0.880521
\(899\) 283.952i 0.315854i
\(900\) 0 0
\(901\) −200.019 −0.221997
\(902\) − 699.122i − 0.775080i
\(903\) 0 0
\(904\) −310.592 −0.343575
\(905\) − 917.302i − 1.01359i
\(906\) 0 0
\(907\) 1086.51 1.19791 0.598957 0.800781i \(-0.295582\pi\)
0.598957 + 0.800781i \(0.295582\pi\)
\(908\) − 247.901i − 0.273019i
\(909\) 0 0
\(910\) 1457.97 1.60216
\(911\) 710.283i 0.779674i 0.920884 + 0.389837i \(0.127469\pi\)
−0.920884 + 0.389837i \(0.872531\pi\)
\(912\) 0 0
\(913\) −1520.95 −1.66589
\(914\) − 854.604i − 0.935015i
\(915\) 0 0
\(916\) −148.115 −0.161698
\(917\) 446.347i 0.486747i
\(918\) 0 0
\(919\) 1540.01 1.67574 0.837871 0.545868i \(-0.183800\pi\)
0.837871 + 0.545868i \(0.183800\pi\)
\(920\) 240.917i 0.261866i
\(921\) 0 0
\(922\) 7.26141 0.00787572
\(923\) 2145.35i 2.32432i
\(924\) 0 0
\(925\) 520.196 0.562374
\(926\) 228.385i 0.246636i
\(927\) 0 0
\(928\) 200.785 0.216363
\(929\) − 1582.59i − 1.70355i −0.523911 0.851773i \(-0.675527\pi\)
0.523911 0.851773i \(-0.324473\pi\)
\(930\) 0 0
\(931\) 522.746 0.561489
\(932\) 546.447i 0.586316i
\(933\) 0 0
\(934\) 711.384 0.761654
\(935\) − 661.362i − 0.707339i
\(936\) 0 0
\(937\) −1747.40 −1.86489 −0.932444 0.361315i \(-0.882328\pi\)
−0.932444 + 0.361315i \(0.882328\pi\)
\(938\) 251.338i 0.267951i
\(939\) 0 0
\(940\) −196.708 −0.209263
\(941\) − 367.579i − 0.390626i −0.980741 0.195313i \(-0.937428\pi\)
0.980741 0.195313i \(-0.0625722\pi\)
\(942\) 0 0
\(943\) −494.354 −0.524235
\(944\) 246.682i 0.261316i
\(945\) 0 0
\(946\) 190.739 0.201626
\(947\) − 190.168i − 0.200811i −0.994947 0.100406i \(-0.967986\pi\)
0.994947 0.100406i \(-0.0320140\pi\)
\(948\) 0 0
\(949\) −856.569 −0.902602
\(950\) 296.267i 0.311860i
\(951\) 0 0
\(952\) 184.308 0.193601
\(953\) − 861.754i − 0.904254i −0.891954 0.452127i \(-0.850665\pi\)
0.891954 0.452127i \(-0.149335\pi\)
\(954\) 0 0
\(955\) −200.238 −0.209674
\(956\) − 905.771i − 0.947459i
\(957\) 0 0
\(958\) −302.354 −0.315609
\(959\) − 13.7175i − 0.0143040i
\(960\) 0 0
\(961\) −897.000 −0.933403
\(962\) − 1815.62i − 1.88733i
\(963\) 0 0
\(964\) 765.377 0.793959
\(965\) − 318.756i − 0.330317i
\(966\) 0 0
\(967\) 1275.31 1.31884 0.659418 0.751776i \(-0.270803\pi\)
0.659418 + 0.751776i \(0.270803\pi\)
\(968\) − 268.701i − 0.277583i
\(969\) 0 0
\(970\) −615.846 −0.634893
\(971\) 1238.69i 1.27568i 0.770168 + 0.637841i \(0.220172\pi\)
−0.770168 + 0.637841i \(0.779828\pi\)
\(972\) 0 0
\(973\) 161.261 0.165736
\(974\) − 12.2057i − 0.0125315i
\(975\) 0 0
\(976\) −52.0000 −0.0532787
\(977\) 18.4936i 0.0189290i 0.999955 + 0.00946448i \(0.00301268\pi\)
−0.999955 + 0.00946448i \(0.996987\pi\)
\(978\) 0 0
\(979\) −1971.30 −2.01359
\(980\) − 248.407i − 0.253476i
\(981\) 0 0
\(982\) −1304.55 −1.32847
\(983\) 658.643i 0.670033i 0.942212 + 0.335017i \(0.108742\pi\)
−0.942212 + 0.335017i \(0.891258\pi\)
\(984\) 0 0
\(985\) −207.000 −0.210152
\(986\) 389.752i 0.395286i
\(987\) 0 0
\(988\) 1034.05 1.04661
\(989\) − 134.873i − 0.136373i
\(990\) 0 0
\(991\) 608.484 0.614010 0.307005 0.951708i \(-0.400673\pi\)
0.307005 + 0.951708i \(0.400673\pi\)
\(992\) − 45.2548i − 0.0456198i
\(993\) 0 0
\(994\) 1201.26 1.20851
\(995\) − 2174.14i − 2.18506i
\(996\) 0 0
\(997\) −617.246 −0.619103 −0.309552 0.950883i \(-0.600179\pi\)
−0.309552 + 0.950883i \(0.600179\pi\)
\(998\) 614.606i 0.615838i
\(999\) 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 162.3.b.b.161.2 4
3.2 odd 2 inner 162.3.b.b.161.3 yes 4
4.3 odd 2 1296.3.e.d.161.4 4
9.2 odd 6 162.3.d.c.53.4 8
9.4 even 3 162.3.d.c.107.4 8
9.5 odd 6 162.3.d.c.107.1 8
9.7 even 3 162.3.d.c.53.1 8
12.11 even 2 1296.3.e.d.161.1 4
36.7 odd 6 1296.3.q.o.1025.1 8
36.11 even 6 1296.3.q.o.1025.4 8
36.23 even 6 1296.3.q.o.593.1 8
36.31 odd 6 1296.3.q.o.593.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.3.b.b.161.2 4 1.1 even 1 trivial
162.3.b.b.161.3 yes 4 3.2 odd 2 inner
162.3.d.c.53.1 8 9.7 even 3
162.3.d.c.53.4 8 9.2 odd 6
162.3.d.c.107.1 8 9.5 odd 6
162.3.d.c.107.4 8 9.4 even 3
1296.3.e.d.161.1 4 12.11 even 2
1296.3.e.d.161.4 4 4.3 odd 2
1296.3.q.o.593.1 8 36.23 even 6
1296.3.q.o.593.4 8 36.31 odd 6
1296.3.q.o.1025.1 8 36.7 odd 6
1296.3.q.o.1025.4 8 36.11 even 6