Properties

Label 162.3.b.b.161.4
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 161.4
Root \(-0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 162.161
Dual form 162.3.b.b.161.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +1.55291i q^{5} +12.3923 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} +1.55291i q^{5} +12.3923 q^{7} -2.82843i q^{8} -2.19615 q^{10} +14.6969i q^{11} -10.8038 q^{13} +17.5254i q^{14} +4.00000 q^{16} +28.9778i q^{17} +3.60770 q^{19} -3.10583i q^{20} -20.7846 q^{22} -14.6969i q^{23} +22.5885 q^{25} -15.2789i q^{26} -24.7846 q^{28} -28.1456i q^{29} +8.00000 q^{31} +5.65685i q^{32} -40.9808 q^{34} +19.2442i q^{35} +22.5692 q^{37} +5.10205i q^{38} +4.39230 q^{40} -25.1512i q^{41} -53.1769 q^{43} -29.3939i q^{44} +20.7846 q^{46} +16.9706i q^{47} +104.569 q^{49} +31.9449i q^{50} +21.6077 q^{52} -84.5482i q^{53} -22.8231 q^{55} -35.0507i q^{56} +39.8038 q^{58} -91.0645i q^{59} -13.0000 q^{61} +11.3137i q^{62} -8.00000 q^{64} -16.7774i q^{65} -41.1769 q^{67} -57.9555i q^{68} -27.2154 q^{70} -16.3613i q^{71} +71.5885 q^{73} +31.9177i q^{74} -7.21539 q^{76} +182.129i q^{77} -46.7461 q^{79} +6.21166i q^{80} +35.5692 q^{82} -15.3062i q^{83} -45.0000 q^{85} -75.2035i q^{86} +41.5692 q^{88} +78.9756i q^{89} -133.885 q^{91} +29.3939i q^{92} -24.0000 q^{94} +5.60244i q^{95} +91.1384 q^{97} +147.883i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) \) 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
\(5\) 1.55291i 0.310583i 0.987869 + 0.155291i \(0.0496317\pi\)
−0.987869 + 0.155291i \(0.950368\pi\)
\(6\) 0 0
\(7\) 12.3923 1.77033 0.885165 0.465278i \(-0.154046\pi\)
0.885165 + 0.465278i \(0.154046\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) −2.19615 −0.219615
\(11\) 14.6969i 1.33609i 0.744123 + 0.668043i \(0.232868\pi\)
−0.744123 + 0.668043i \(0.767132\pi\)
\(12\) 0 0
\(13\) −10.8038 −0.831065 −0.415533 0.909578i \(-0.636405\pi\)
−0.415533 + 0.909578i \(0.636405\pi\)
\(14\) 17.5254i 1.25181i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 28.9778i 1.70457i 0.523074 + 0.852287i \(0.324785\pi\)
−0.523074 + 0.852287i \(0.675215\pi\)
\(18\) 0 0
\(19\) 3.60770 0.189879 0.0949393 0.995483i \(-0.469734\pi\)
0.0949393 + 0.995483i \(0.469734\pi\)
\(20\) − 3.10583i − 0.155291i
\(21\) 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
\(25\) 22.5885 0.903538
\(26\) − 15.2789i − 0.587652i
\(27\) 0 0
\(28\) −24.7846 −0.885165
\(29\) − 28.1456i − 0.970537i −0.874365 0.485268i \(-0.838722\pi\)
0.874365 0.485268i \(-0.161278\pi\)
\(30\) 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
\(34\) −40.9808 −1.20532
\(35\) 19.2442i 0.549834i
\(36\) 0 0
\(37\) 22.5692 0.609979 0.304989 0.952356i \(-0.401347\pi\)
0.304989 + 0.952356i \(0.401347\pi\)
\(38\) 5.10205i 0.134265i
\(39\) 0 0
\(40\) 4.39230 0.109808
\(41\) − 25.1512i − 0.613445i −0.951799 0.306722i \(-0.900768\pi\)
0.951799 0.306722i \(-0.0992323\pi\)
\(42\) 0 0
\(43\) −53.1769 −1.23667 −0.618336 0.785914i \(-0.712193\pi\)
−0.618336 + 0.785914i \(0.712193\pi\)
\(44\) − 29.3939i − 0.668043i
\(45\) 0 0
\(46\) 20.7846 0.451839
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 104.569 2.13407
\(50\) 31.9449i 0.638898i
\(51\) 0 0
\(52\) 21.6077 0.415533
\(53\) − 84.5482i − 1.59525i −0.603154 0.797625i \(-0.706090\pi\)
0.603154 0.797625i \(-0.293910\pi\)
\(54\) 0 0
\(55\) −22.8231 −0.414965
\(56\) − 35.0507i − 0.625906i
\(57\) 0 0
\(58\) 39.8038 0.686273
\(59\) − 91.0645i − 1.54347i −0.635947 0.771733i \(-0.719390\pi\)
0.635947 0.771733i \(-0.280610\pi\)
\(60\) 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
\(65\) − 16.7774i − 0.258115i
\(66\) 0 0
\(67\) −41.1769 −0.614581 −0.307290 0.951616i \(-0.599422\pi\)
−0.307290 + 0.951616i \(0.599422\pi\)
\(68\) − 57.9555i − 0.852287i
\(69\) 0 0
\(70\) −27.2154 −0.388791
\(71\) − 16.3613i − 0.230442i −0.993340 0.115221i \(-0.963242\pi\)
0.993340 0.115221i \(-0.0367575\pi\)
\(72\) 0 0
\(73\) 71.5885 0.980664 0.490332 0.871536i \(-0.336876\pi\)
0.490332 + 0.871536i \(0.336876\pi\)
\(74\) 31.9177i 0.431320i
\(75\) 0 0
\(76\) −7.21539 −0.0949393
\(77\) 182.129i 2.36531i
\(78\) 0 0
\(79\) −46.7461 −0.591723 −0.295862 0.955231i \(-0.595607\pi\)
−0.295862 + 0.955231i \(0.595607\pi\)
\(80\) 6.21166i 0.0776457i
\(81\) 0 0
\(82\) 35.5692 0.433771
\(83\) − 15.3062i − 0.184411i −0.995740 0.0922057i \(-0.970608\pi\)
0.995740 0.0922057i \(-0.0293917\pi\)
\(84\) 0 0
\(85\) −45.0000 −0.529412
\(86\) − 75.2035i − 0.874459i
\(87\) 0 0
\(88\) 41.5692 0.472377
\(89\) 78.9756i 0.887367i 0.896184 + 0.443683i \(0.146329\pi\)
−0.896184 + 0.443683i \(0.853671\pi\)
\(90\) 0 0
\(91\) −133.885 −1.47126
\(92\) 29.3939i 0.319499i
\(93\) 0 0
\(94\) −24.0000 −0.255319
\(95\) 5.60244i 0.0589731i
\(96\) 0 0
\(97\) 91.1384 0.939572 0.469786 0.882780i \(-0.344331\pi\)
0.469786 + 0.882780i \(0.344331\pi\)
\(98\) 147.883i 1.50901i
\(99\) 0 0
\(100\) −45.1769 −0.451769
\(101\) − 88.4862i − 0.876101i −0.898950 0.438051i \(-0.855669\pi\)
0.898950 0.438051i \(-0.144331\pi\)
\(102\) 0 0
\(103\) −152.708 −1.48260 −0.741299 0.671175i \(-0.765790\pi\)
−0.741299 + 0.671175i \(0.765790\pi\)
\(104\) 30.5579i 0.293826i
\(105\) 0 0
\(106\) 119.569 1.12801
\(107\) − 60.0062i − 0.560805i −0.959882 0.280403i \(-0.909532\pi\)
0.959882 0.280403i \(-0.0904680\pi\)
\(108\) 0 0
\(109\) −93.9423 −0.861856 −0.430928 0.902386i \(-0.641814\pi\)
−0.430928 + 0.902386i \(0.641814\pi\)
\(110\) − 32.2767i − 0.293425i
\(111\) 0 0
\(112\) 49.5692 0.442582
\(113\) 88.5977i 0.784051i 0.919954 + 0.392025i \(0.128225\pi\)
−0.919954 + 0.392025i \(0.871775\pi\)
\(114\) 0 0
\(115\) 22.8231 0.198462
\(116\) 56.2911i 0.485268i
\(117\) 0 0
\(118\) 128.785 1.09139
\(119\) 359.101i 3.01766i
\(120\) 0 0
\(121\) −95.0000 −0.785124
\(122\) − 18.3848i − 0.150695i
\(123\) 0 0
\(124\) −16.0000 −0.129032
\(125\) 73.9008i 0.591206i
\(126\) 0 0
\(127\) 78.8231 0.620654 0.310327 0.950630i \(-0.399561\pi\)
0.310327 + 0.950630i \(0.399561\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) 23.7269 0.182515
\(131\) 82.5792i 0.630375i 0.949029 + 0.315188i \(0.102067\pi\)
−0.949029 + 0.315188i \(0.897933\pi\)
\(132\) 0 0
\(133\) 44.7077 0.336148
\(134\) − 58.2330i − 0.434574i
\(135\) 0 0
\(136\) 81.9615 0.602658
\(137\) − 214.740i − 1.56745i −0.621110 0.783723i \(-0.713318\pi\)
0.621110 0.783723i \(-0.286682\pi\)
\(138\) 0 0
\(139\) −60.7846 −0.437299 −0.218650 0.975803i \(-0.570165\pi\)
−0.218650 + 0.975803i \(0.570165\pi\)
\(140\) − 38.4884i − 0.274917i
\(141\) 0 0
\(142\) 23.1384 0.162947
\(143\) − 158.783i − 1.11037i
\(144\) 0 0
\(145\) 43.7077 0.301432
\(146\) 101.241i 0.693434i
\(147\) 0 0
\(148\) −45.1384 −0.304989
\(149\) 30.6422i 0.205652i 0.994699 + 0.102826i \(0.0327885\pi\)
−0.994699 + 0.102826i \(0.967211\pi\)
\(150\) 0 0
\(151\) 32.0000 0.211921 0.105960 0.994370i \(-0.466208\pi\)
0.105960 + 0.994370i \(0.466208\pi\)
\(152\) − 10.2041i − 0.0671323i
\(153\) 0 0
\(154\) −257.569 −1.67253
\(155\) 12.4233i 0.0801504i
\(156\) 0 0
\(157\) 249.708 1.59049 0.795247 0.606285i \(-0.207341\pi\)
0.795247 + 0.606285i \(0.207341\pi\)
\(158\) − 66.1090i − 0.418411i
\(159\) 0 0
\(160\) −8.78461 −0.0549038
\(161\) − 182.129i − 1.13124i
\(162\) 0 0
\(163\) −12.7846 −0.0784332 −0.0392166 0.999231i \(-0.512486\pi\)
−0.0392166 + 0.999231i \(0.512486\pi\)
\(164\) 50.3025i 0.306722i
\(165\) 0 0
\(166\) 21.6462 0.130399
\(167\) − 232.431i − 1.39180i −0.718136 0.695902i \(-0.755005\pi\)
0.718136 0.695902i \(-0.244995\pi\)
\(168\) 0 0
\(169\) −52.2769 −0.309331
\(170\) − 63.6396i − 0.374351i
\(171\) 0 0
\(172\) 106.354 0.618336
\(173\) − 21.5477i − 0.124553i −0.998059 0.0622765i \(-0.980164\pi\)
0.998059 0.0622765i \(-0.0198361\pi\)
\(174\) 0 0
\(175\) 279.923 1.59956
\(176\) 58.7878i 0.334021i
\(177\) 0 0
\(178\) −111.688 −0.627463
\(179\) 277.741i 1.55162i 0.630964 + 0.775812i \(0.282659\pi\)
−0.630964 + 0.775812i \(0.717341\pi\)
\(180\) 0 0
\(181\) 174.277 0.962856 0.481428 0.876486i \(-0.340118\pi\)
0.481428 + 0.876486i \(0.340118\pi\)
\(182\) − 189.341i − 1.04034i
\(183\) 0 0
\(184\) −41.5692 −0.225920
\(185\) 35.0481i 0.189449i
\(186\) 0 0
\(187\) −425.885 −2.27746
\(188\) − 33.9411i − 0.180538i
\(189\) 0 0
\(190\) −7.92305 −0.0417003
\(191\) − 152.126i − 0.796470i −0.917283 0.398235i \(-0.869623\pi\)
0.917283 0.398235i \(-0.130377\pi\)
\(192\) 0 0
\(193\) −55.0000 −0.284974 −0.142487 0.989797i \(-0.545510\pi\)
−0.142487 + 0.989797i \(0.545510\pi\)
\(194\) 128.889i 0.664377i
\(195\) 0 0
\(196\) −209.138 −1.06703
\(197\) 133.298i 0.676638i 0.941031 + 0.338319i \(0.109858\pi\)
−0.941031 + 0.338319i \(0.890142\pi\)
\(198\) 0 0
\(199\) −208.862 −1.04956 −0.524778 0.851239i \(-0.675852\pi\)
−0.524778 + 0.851239i \(0.675852\pi\)
\(200\) − 63.8898i − 0.319449i
\(201\) 0 0
\(202\) 125.138 0.619497
\(203\) − 348.788i − 1.71817i
\(204\) 0 0
\(205\) 39.0577 0.190525
\(206\) − 215.961i − 1.04836i
\(207\) 0 0
\(208\) −43.2154 −0.207766
\(209\) 53.0221i 0.253694i
\(210\) 0 0
\(211\) −87.4538 −0.414473 −0.207236 0.978291i \(-0.566447\pi\)
−0.207236 + 0.978291i \(0.566447\pi\)
\(212\) 169.096i 0.797625i
\(213\) 0 0
\(214\) 84.8616 0.396549
\(215\) − 82.5792i − 0.384089i
\(216\) 0 0
\(217\) 99.1384 0.456859
\(218\) − 132.854i − 0.609424i
\(219\) 0 0
\(220\) 45.6462 0.207483
\(221\) − 313.071i − 1.41661i
\(222\) 0 0
\(223\) 222.592 0.998171 0.499086 0.866553i \(-0.333669\pi\)
0.499086 + 0.866553i \(0.333669\pi\)
\(224\) 70.1015i 0.312953i
\(225\) 0 0
\(226\) −125.296 −0.554408
\(227\) − 300.314i − 1.32297i −0.749959 0.661484i \(-0.769927\pi\)
0.749959 0.661484i \(-0.230073\pi\)
\(228\) 0 0
\(229\) 229.942 1.00411 0.502057 0.864834i \(-0.332577\pi\)
0.502057 + 0.864834i \(0.332577\pi\)
\(230\) 32.2767i 0.140334i
\(231\) 0 0
\(232\) −79.6077 −0.343137
\(233\) − 116.246i − 0.498908i −0.968387 0.249454i \(-0.919749\pi\)
0.968387 0.249454i \(-0.0802512\pi\)
\(234\) 0 0
\(235\) −26.3538 −0.112144
\(236\) 182.129i 0.771733i
\(237\) 0 0
\(238\) −507.846 −2.13381
\(239\) 164.386i 0.687807i 0.939005 + 0.343904i \(0.111749\pi\)
−0.939005 + 0.343904i \(0.888251\pi\)
\(240\) 0 0
\(241\) −81.3116 −0.337392 −0.168696 0.985668i \(-0.553956\pi\)
−0.168696 + 0.985668i \(0.553956\pi\)
\(242\) − 134.350i − 0.555166i
\(243\) 0 0
\(244\) 26.0000 0.106557
\(245\) 162.387i 0.662804i
\(246\) 0 0
\(247\) −38.9770 −0.157802
\(248\) − 22.6274i − 0.0912396i
\(249\) 0 0
\(250\) −104.512 −0.418046
\(251\) 396.371i 1.57917i 0.613642 + 0.789584i \(0.289704\pi\)
−0.613642 + 0.789584i \(0.710296\pi\)
\(252\) 0 0
\(253\) 216.000 0.853755
\(254\) 111.473i 0.438869i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 214.963i 0.836432i 0.908348 + 0.418216i \(0.137345\pi\)
−0.908348 + 0.418216i \(0.862655\pi\)
\(258\) 0 0
\(259\) 279.685 1.07986
\(260\) 33.5549i 0.129057i
\(261\) 0 0
\(262\) −116.785 −0.445743
\(263\) 317.893i 1.20872i 0.796711 + 0.604360i \(0.206571\pi\)
−0.796711 + 0.604360i \(0.793429\pi\)
\(264\) 0 0
\(265\) 131.296 0.495457
\(266\) 63.2262i 0.237692i
\(267\) 0 0
\(268\) 82.3538 0.307290
\(269\) 208.528i 0.775199i 0.921828 + 0.387599i \(0.126696\pi\)
−0.921828 + 0.387599i \(0.873304\pi\)
\(270\) 0 0
\(271\) 409.885 1.51249 0.756245 0.654289i \(-0.227032\pi\)
0.756245 + 0.654289i \(0.227032\pi\)
\(272\) 115.911i 0.426144i
\(273\) 0 0
\(274\) 303.688 1.10835
\(275\) 331.981i 1.20720i
\(276\) 0 0
\(277\) 497.415 1.79572 0.897862 0.440278i \(-0.145120\pi\)
0.897862 + 0.440278i \(0.145120\pi\)
\(278\) − 85.9624i − 0.309217i
\(279\) 0 0
\(280\) 54.4308 0.194396
\(281\) 215.268i 0.766077i 0.923732 + 0.383039i \(0.125122\pi\)
−0.923732 + 0.383039i \(0.874878\pi\)
\(282\) 0 0
\(283\) −296.708 −1.04844 −0.524218 0.851584i \(-0.675642\pi\)
−0.524218 + 0.851584i \(0.675642\pi\)
\(284\) 32.7227i 0.115221i
\(285\) 0 0
\(286\) 224.554 0.785153
\(287\) − 311.682i − 1.08600i
\(288\) 0 0
\(289\) −550.711 −1.90558
\(290\) 61.8120i 0.213145i
\(291\) 0 0
\(292\) −143.177 −0.490332
\(293\) − 340.964i − 1.16370i −0.813296 0.581850i \(-0.802329\pi\)
0.813296 0.581850i \(-0.197671\pi\)
\(294\) 0 0
\(295\) 141.415 0.479374
\(296\) − 63.8354i − 0.215660i
\(297\) 0 0
\(298\) −43.3346 −0.145418
\(299\) 158.783i 0.531048i
\(300\) 0 0
\(301\) −658.985 −2.18932
\(302\) 45.2548i 0.149850i
\(303\) 0 0
\(304\) 14.4308 0.0474697
\(305\) − 20.1879i − 0.0661898i
\(306\) 0 0
\(307\) −114.354 −0.372488 −0.186244 0.982504i \(-0.559632\pi\)
−0.186244 + 0.982504i \(0.559632\pi\)
\(308\) − 364.258i − 1.18266i
\(309\) 0 0
\(310\) −17.5692 −0.0566749
\(311\) 196.217i 0.630922i 0.948939 + 0.315461i \(0.102159\pi\)
−0.948939 + 0.315461i \(0.897841\pi\)
\(312\) 0 0
\(313\) −146.723 −0.468764 −0.234382 0.972145i \(-0.575307\pi\)
−0.234382 + 0.972145i \(0.575307\pi\)
\(314\) 353.140i 1.12465i
\(315\) 0 0
\(316\) 93.4923 0.295862
\(317\) − 48.1403i − 0.151862i −0.997113 0.0759311i \(-0.975807\pi\)
0.997113 0.0759311i \(-0.0241929\pi\)
\(318\) 0 0
\(319\) 413.654 1.29672
\(320\) − 12.4233i − 0.0388229i
\(321\) 0 0
\(322\) 257.569 0.799904
\(323\) 104.543i 0.323662i
\(324\) 0 0
\(325\) −244.042 −0.750899
\(326\) − 18.0802i − 0.0554606i
\(327\) 0 0
\(328\) −71.1384 −0.216885
\(329\) 210.304i 0.639223i
\(330\) 0 0
\(331\) −99.4538 −0.300465 −0.150232 0.988651i \(-0.548002\pi\)
−0.150232 + 0.988651i \(0.548002\pi\)
\(332\) 30.6123i 0.0922057i
\(333\) 0 0
\(334\) 328.708 0.984155
\(335\) − 63.9442i − 0.190878i
\(336\) 0 0
\(337\) −425.261 −1.26190 −0.630952 0.775822i \(-0.717335\pi\)
−0.630952 + 0.775822i \(0.717335\pi\)
\(338\) − 73.9307i − 0.218730i
\(339\) 0 0
\(340\) 90.0000 0.264706
\(341\) 117.576i 0.344796i
\(342\) 0 0
\(343\) 688.631 2.00767
\(344\) 150.407i 0.437230i
\(345\) 0 0
\(346\) 30.4730 0.0880723
\(347\) − 38.7711i − 0.111732i −0.998438 0.0558662i \(-0.982208\pi\)
0.998438 0.0558662i \(-0.0177920\pi\)
\(348\) 0 0
\(349\) −651.969 −1.86811 −0.934053 0.357134i \(-0.883754\pi\)
−0.934053 + 0.357134i \(0.883754\pi\)
\(350\) 395.871i 1.13106i
\(351\) 0 0
\(352\) −83.1384 −0.236189
\(353\) − 1.19656i − 0.00338969i −0.999999 0.00169485i \(-0.999461\pi\)
0.999999 0.00169485i \(-0.000539487\pi\)
\(354\) 0 0
\(355\) 25.4078 0.0715712
\(356\) − 157.951i − 0.443683i
\(357\) 0 0
\(358\) −392.785 −1.09716
\(359\) − 534.573i − 1.48906i −0.667589 0.744530i \(-0.732674\pi\)
0.667589 0.744530i \(-0.267326\pi\)
\(360\) 0 0
\(361\) −347.985 −0.963946
\(362\) 246.465i 0.680842i
\(363\) 0 0
\(364\) 267.769 0.735630
\(365\) 111.171i 0.304577i
\(366\) 0 0
\(367\) 264.708 0.721274 0.360637 0.932706i \(-0.382559\pi\)
0.360637 + 0.932706i \(0.382559\pi\)
\(368\) − 58.7878i − 0.159749i
\(369\) 0 0
\(370\) −49.5654 −0.133961
\(371\) − 1047.75i − 2.82412i
\(372\) 0 0
\(373\) 71.0155 0.190390 0.0951950 0.995459i \(-0.469653\pi\)
0.0951950 + 0.995459i \(0.469653\pi\)
\(374\) − 602.292i − 1.61041i
\(375\) 0 0
\(376\) 48.0000 0.127660
\(377\) 304.080i 0.806579i
\(378\) 0 0
\(379\) −696.785 −1.83848 −0.919241 0.393696i \(-0.871196\pi\)
−0.919241 + 0.393696i \(0.871196\pi\)
\(380\) − 11.2049i − 0.0294865i
\(381\) 0 0
\(382\) 215.138 0.563190
\(383\) 435.306i 1.13657i 0.822832 + 0.568284i \(0.192393\pi\)
−0.822832 + 0.568284i \(0.807607\pi\)
\(384\) 0 0
\(385\) −282.831 −0.734625
\(386\) − 77.7817i − 0.201507i
\(387\) 0 0
\(388\) −182.277 −0.469786
\(389\) − 53.0439i − 0.136360i −0.997673 0.0681799i \(-0.978281\pi\)
0.997673 0.0681799i \(-0.0217192\pi\)
\(390\) 0 0
\(391\) 425.885 1.08922
\(392\) − 295.766i − 0.754506i
\(393\) 0 0
\(394\) −188.512 −0.478456
\(395\) − 72.5927i − 0.183779i
\(396\) 0 0
\(397\) 63.7077 0.160473 0.0802363 0.996776i \(-0.474432\pi\)
0.0802363 + 0.996776i \(0.474432\pi\)
\(398\) − 295.375i − 0.742148i
\(399\) 0 0
\(400\) 90.3538 0.225885
\(401\) − 628.691i − 1.56781i −0.620882 0.783904i \(-0.713225\pi\)
0.620882 0.783904i \(-0.286775\pi\)
\(402\) 0 0
\(403\) −86.4308 −0.214468
\(404\) 176.972i 0.438051i
\(405\) 0 0
\(406\) 493.261 1.21493
\(407\) 331.698i 0.814984i
\(408\) 0 0
\(409\) 535.281 1.30875 0.654377 0.756168i \(-0.272931\pi\)
0.654377 + 0.756168i \(0.272931\pi\)
\(410\) 55.2359i 0.134722i
\(411\) 0 0
\(412\) 305.415 0.741299
\(413\) − 1128.50i − 2.73244i
\(414\) 0 0
\(415\) 23.7691 0.0572750
\(416\) − 61.1158i − 0.146913i
\(417\) 0 0
\(418\) −74.9845 −0.179389
\(419\) 385.330i 0.919641i 0.888012 + 0.459821i \(0.152086\pi\)
−0.888012 + 0.459821i \(0.847914\pi\)
\(420\) 0 0
\(421\) −755.319 −1.79411 −0.897054 0.441922i \(-0.854297\pi\)
−0.897054 + 0.441922i \(0.854297\pi\)
\(422\) − 123.678i − 0.293077i
\(423\) 0 0
\(424\) −239.138 −0.564006
\(425\) 654.563i 1.54015i
\(426\) 0 0
\(427\) −161.100 −0.377283
\(428\) 120.012i 0.280403i
\(429\) 0 0
\(430\) 116.785 0.271592
\(431\) − 107.709i − 0.249904i −0.992163 0.124952i \(-0.960122\pi\)
0.992163 0.124952i \(-0.0398777\pi\)
\(432\) 0 0
\(433\) 655.123 1.51299 0.756493 0.654002i \(-0.226911\pi\)
0.756493 + 0.654002i \(0.226911\pi\)
\(434\) 140.203i 0.323048i
\(435\) 0 0
\(436\) 187.885 0.430928
\(437\) − 53.0221i − 0.121332i
\(438\) 0 0
\(439\) −472.000 −1.07517 −0.537585 0.843209i \(-0.680663\pi\)
−0.537585 + 0.843209i \(0.680663\pi\)
\(440\) 64.5534i 0.146712i
\(441\) 0 0
\(442\) 442.750 1.00170
\(443\) − 841.218i − 1.89891i −0.313902 0.949455i \(-0.601636\pi\)
0.313902 0.949455i \(-0.398364\pi\)
\(444\) 0 0
\(445\) −122.642 −0.275601
\(446\) 314.793i 0.705814i
\(447\) 0 0
\(448\) −99.1384 −0.221291
\(449\) − 382.751i − 0.852453i −0.904616 0.426227i \(-0.859843\pi\)
0.904616 0.426227i \(-0.140157\pi\)
\(450\) 0 0
\(451\) 369.646 0.819615
\(452\) − 177.195i − 0.392025i
\(453\) 0 0
\(454\) 424.708 0.935479
\(455\) − 207.911i − 0.456948i
\(456\) 0 0
\(457\) 323.704 0.708324 0.354162 0.935184i \(-0.384766\pi\)
0.354162 + 0.935184i \(0.384766\pi\)
\(458\) 325.187i 0.710016i
\(459\) 0 0
\(460\) −45.6462 −0.0992308
\(461\) 641.531i 1.39161i 0.718232 + 0.695803i \(0.244951\pi\)
−0.718232 + 0.695803i \(0.755049\pi\)
\(462\) 0 0
\(463\) 129.492 0.279681 0.139840 0.990174i \(-0.455341\pi\)
0.139840 + 0.990174i \(0.455341\pi\)
\(464\) − 112.582i − 0.242634i
\(465\) 0 0
\(466\) 164.396 0.352781
\(467\) 672.730i 1.44054i 0.693696 + 0.720268i \(0.255981\pi\)
−0.693696 + 0.720268i \(0.744019\pi\)
\(468\) 0 0
\(469\) −510.277 −1.08801
\(470\) − 37.2699i − 0.0792977i
\(471\) 0 0
\(472\) −257.569 −0.545697
\(473\) − 781.538i − 1.65230i
\(474\) 0 0
\(475\) 81.4923 0.171563
\(476\) − 718.203i − 1.50883i
\(477\) 0 0
\(478\) −232.477 −0.486353
\(479\) 125.615i 0.262244i 0.991366 + 0.131122i \(0.0418579\pi\)
−0.991366 + 0.131122i \(0.958142\pi\)
\(480\) 0 0
\(481\) −243.834 −0.506932
\(482\) − 114.992i − 0.238572i
\(483\) 0 0
\(484\) 190.000 0.392562
\(485\) 141.530i 0.291815i
\(486\) 0 0
\(487\) −448.631 −0.921213 −0.460606 0.887604i \(-0.652368\pi\)
−0.460606 + 0.887604i \(0.652368\pi\)
\(488\) 36.7696i 0.0753474i
\(489\) 0 0
\(490\) −229.650 −0.468673
\(491\) 452.157i 0.920890i 0.887688 + 0.460445i \(0.152310\pi\)
−0.887688 + 0.460445i \(0.847690\pi\)
\(492\) 0 0
\(493\) 815.596 1.65435
\(494\) − 55.1218i − 0.111583i
\(495\) 0 0
\(496\) 32.0000 0.0645161
\(497\) − 202.755i − 0.407957i
\(498\) 0 0
\(499\) 126.592 0.253692 0.126846 0.991922i \(-0.459515\pi\)
0.126846 + 0.991922i \(0.459515\pi\)
\(500\) − 147.802i − 0.295603i
\(501\) 0 0
\(502\) −560.554 −1.11664
\(503\) − 296.822i − 0.590103i −0.955481 0.295051i \(-0.904663\pi\)
0.955481 0.295051i \(-0.0953367\pi\)
\(504\) 0 0
\(505\) 137.412 0.272102
\(506\) 305.470i 0.603696i
\(507\) 0 0
\(508\) −157.646 −0.310327
\(509\) 654.846i 1.28653i 0.765642 + 0.643267i \(0.222422\pi\)
−0.765642 + 0.643267i \(0.777578\pi\)
\(510\) 0 0
\(511\) 887.146 1.73610
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) −304.004 −0.591447
\(515\) − 237.142i − 0.460470i
\(516\) 0 0
\(517\) −249.415 −0.482428
\(518\) 395.534i 0.763579i
\(519\) 0 0
\(520\) −47.4538 −0.0912573
\(521\) 690.006i 1.32439i 0.749333 + 0.662193i \(0.230374\pi\)
−0.749333 + 0.662193i \(0.769626\pi\)
\(522\) 0 0
\(523\) 616.238 1.17828 0.589138 0.808032i \(-0.299467\pi\)
0.589138 + 0.808032i \(0.299467\pi\)
\(524\) − 165.158i − 0.315188i
\(525\) 0 0
\(526\) −449.569 −0.854694
\(527\) 231.822i 0.439890i
\(528\) 0 0
\(529\) 313.000 0.591682
\(530\) 185.681i 0.350341i
\(531\) 0 0
\(532\) −89.4153 −0.168074
\(533\) 271.730i 0.509813i
\(534\) 0 0
\(535\) 93.1845 0.174177
\(536\) 116.466i 0.217287i
\(537\) 0 0
\(538\) −294.904 −0.548148
\(539\) 1536.85i 2.85129i
\(540\) 0 0
\(541\) −548.734 −1.01430 −0.507148 0.861859i \(-0.669300\pi\)
−0.507148 + 0.861859i \(0.669300\pi\)
\(542\) 579.664i 1.06949i
\(543\) 0 0
\(544\) −163.923 −0.301329
\(545\) − 145.884i − 0.267678i
\(546\) 0 0
\(547\) −819.769 −1.49866 −0.749332 0.662195i \(-0.769625\pi\)
−0.749332 + 0.662195i \(0.769625\pi\)
\(548\) 429.480i 0.783723i
\(549\) 0 0
\(550\) −469.492 −0.853622
\(551\) − 101.541i − 0.184284i
\(552\) 0 0
\(553\) −579.292 −1.04754
\(554\) 703.451i 1.26977i
\(555\) 0 0
\(556\) 121.569 0.218650
\(557\) 125.808i 0.225867i 0.993603 + 0.112934i \(0.0360247\pi\)
−0.993603 + 0.112934i \(0.963975\pi\)
\(558\) 0 0
\(559\) 574.515 1.02776
\(560\) 76.9767i 0.137458i
\(561\) 0 0
\(562\) −304.435 −0.541698
\(563\) − 508.790i − 0.903713i −0.892091 0.451856i \(-0.850762\pi\)
0.892091 0.451856i \(-0.149238\pi\)
\(564\) 0 0
\(565\) −137.585 −0.243513
\(566\) − 419.608i − 0.741357i
\(567\) 0 0
\(568\) −46.2769 −0.0814734
\(569\) 96.8600i 0.170229i 0.996371 + 0.0851143i \(0.0271255\pi\)
−0.996371 + 0.0851143i \(0.972874\pi\)
\(570\) 0 0
\(571\) −454.200 −0.795446 −0.397723 0.917505i \(-0.630200\pi\)
−0.397723 + 0.917505i \(0.630200\pi\)
\(572\) 317.567i 0.555187i
\(573\) 0 0
\(574\) 440.785 0.767917
\(575\) − 331.981i − 0.577359i
\(576\) 0 0
\(577\) −39.1230 −0.0678041 −0.0339021 0.999425i \(-0.510793\pi\)
−0.0339021 + 0.999425i \(0.510793\pi\)
\(578\) − 778.824i − 1.34745i
\(579\) 0 0
\(580\) −87.4153 −0.150716
\(581\) − 189.679i − 0.326469i
\(582\) 0 0
\(583\) 1242.60 2.13139
\(584\) − 202.483i − 0.346717i
\(585\) 0 0
\(586\) 482.196 0.822860
\(587\) 419.227i 0.714186i 0.934069 + 0.357093i \(0.116232\pi\)
−0.934069 + 0.357093i \(0.883768\pi\)
\(588\) 0 0
\(589\) 28.8616 0.0490010
\(590\) 199.991i 0.338969i
\(591\) 0 0
\(592\) 90.2769 0.152495
\(593\) 329.210i 0.555160i 0.960703 + 0.277580i \(0.0895323\pi\)
−0.960703 + 0.277580i \(0.910468\pi\)
\(594\) 0 0
\(595\) −557.654 −0.937233
\(596\) − 61.2844i − 0.102826i
\(597\) 0 0
\(598\) −224.554 −0.375508
\(599\) − 184.239i − 0.307578i −0.988104 0.153789i \(-0.950852\pi\)
0.988104 0.153789i \(-0.0491476\pi\)
\(600\) 0 0
\(601\) 218.415 0.363420 0.181710 0.983352i \(-0.441837\pi\)
0.181710 + 0.983352i \(0.441837\pi\)
\(602\) − 931.945i − 1.54808i
\(603\) 0 0
\(604\) −64.0000 −0.105960
\(605\) − 147.527i − 0.243846i
\(606\) 0 0
\(607\) 10.2693 0.0169182 0.00845909 0.999964i \(-0.497307\pi\)
0.00845909 + 0.999964i \(0.497307\pi\)
\(608\) 20.4082i 0.0335661i
\(609\) 0 0
\(610\) 28.5500 0.0468032
\(611\) − 183.347i − 0.300078i
\(612\) 0 0
\(613\) −180.585 −0.294592 −0.147296 0.989092i \(-0.547057\pi\)
−0.147296 + 0.989092i \(0.547057\pi\)
\(614\) − 161.721i − 0.263389i
\(615\) 0 0
\(616\) 515.138 0.836264
\(617\) − 906.209i − 1.46873i −0.678753 0.734367i \(-0.737479\pi\)
0.678753 0.734367i \(-0.262521\pi\)
\(618\) 0 0
\(619\) 761.646 1.23045 0.615223 0.788353i \(-0.289066\pi\)
0.615223 + 0.788353i \(0.289066\pi\)
\(620\) − 24.8466i − 0.0400752i
\(621\) 0 0
\(622\) −277.492 −0.446129
\(623\) 978.690i 1.57093i
\(624\) 0 0
\(625\) 449.950 0.719920
\(626\) − 207.498i − 0.331466i
\(627\) 0 0
\(628\) −499.415 −0.795247
\(629\) 654.006i 1.03975i
\(630\) 0 0
\(631\) 601.108 0.952627 0.476313 0.879276i \(-0.341973\pi\)
0.476313 + 0.879276i \(0.341973\pi\)
\(632\) 132.218i 0.209206i
\(633\) 0 0
\(634\) 68.0807 0.107383
\(635\) 122.405i 0.192765i
\(636\) 0 0
\(637\) −1129.75 −1.77355
\(638\) 584.995i 0.916920i
\(639\) 0 0
\(640\) 17.5692 0.0274519
\(641\) − 577.453i − 0.900863i −0.892811 0.450431i \(-0.851270\pi\)
0.892811 0.450431i \(-0.148730\pi\)
\(642\) 0 0
\(643\) 130.123 0.202369 0.101184 0.994868i \(-0.467737\pi\)
0.101184 + 0.994868i \(0.467737\pi\)
\(644\) 364.258i 0.565618i
\(645\) 0 0
\(646\) −147.846 −0.228864
\(647\) − 985.467i − 1.52313i −0.648087 0.761567i \(-0.724430\pi\)
0.648087 0.761567i \(-0.275570\pi\)
\(648\) 0 0
\(649\) 1338.37 2.06220
\(650\) − 345.128i − 0.530966i
\(651\) 0 0
\(652\) 25.5692 0.0392166
\(653\) 600.323i 0.919330i 0.888092 + 0.459665i \(0.152031\pi\)
−0.888092 + 0.459665i \(0.847969\pi\)
\(654\) 0 0
\(655\) −128.238 −0.195784
\(656\) − 100.605i − 0.153361i
\(657\) 0 0
\(658\) −297.415 −0.451999
\(659\) 15.5889i 0.0236554i 0.999930 + 0.0118277i \(0.00376496\pi\)
−0.999930 + 0.0118277i \(0.996235\pi\)
\(660\) 0 0
\(661\) 407.831 0.616990 0.308495 0.951226i \(-0.400175\pi\)
0.308495 + 0.951226i \(0.400175\pi\)
\(662\) − 140.649i − 0.212461i
\(663\) 0 0
\(664\) −43.2923 −0.0651993
\(665\) 69.4272i 0.104402i
\(666\) 0 0
\(667\) −413.654 −0.620170
\(668\) 464.863i 0.695902i
\(669\) 0 0
\(670\) 90.4308 0.134971
\(671\) − 191.060i − 0.284739i
\(672\) 0 0
\(673\) −474.569 −0.705155 −0.352577 0.935783i \(-0.614695\pi\)
−0.352577 + 0.935783i \(0.614695\pi\)
\(674\) − 601.410i − 0.892300i
\(675\) 0 0
\(676\) 104.554 0.154665
\(677\) − 282.147i − 0.416760i −0.978048 0.208380i \(-0.933181\pi\)
0.978048 0.208380i \(-0.0668191\pi\)
\(678\) 0 0
\(679\) 1129.42 1.66335
\(680\) 127.279i 0.187175i
\(681\) 0 0
\(682\) −166.277 −0.243808
\(683\) − 1085.46i − 1.58926i −0.607095 0.794629i \(-0.707665\pi\)
0.607095 0.794629i \(-0.292335\pi\)
\(684\) 0 0
\(685\) 333.473 0.486822
\(686\) 973.871i 1.41964i
\(687\) 0 0
\(688\) −212.708 −0.309168
\(689\) 913.446i 1.32576i
\(690\) 0 0
\(691\) −1007.78 −1.45843 −0.729216 0.684283i \(-0.760115\pi\)
−0.729216 + 0.684283i \(0.760115\pi\)
\(692\) 43.0954i 0.0622765i
\(693\) 0 0
\(694\) 54.8306 0.0790067
\(695\) − 94.3933i − 0.135818i
\(696\) 0 0
\(697\) 728.827 1.04566
\(698\) − 922.024i − 1.32095i
\(699\) 0 0
\(700\) −559.846 −0.799780
\(701\) − 216.731i − 0.309174i −0.987979 0.154587i \(-0.950595\pi\)
0.987979 0.154587i \(-0.0494047\pi\)
\(702\) 0 0
\(703\) 81.4229 0.115822
\(704\) − 117.576i − 0.167011i
\(705\) 0 0
\(706\) 1.69219 0.00239688
\(707\) − 1096.55i − 1.55099i
\(708\) 0 0
\(709\) 994.496 1.40267 0.701337 0.712830i \(-0.252587\pi\)
0.701337 + 0.712830i \(0.252587\pi\)
\(710\) 35.9320i 0.0506085i
\(711\) 0 0
\(712\) 223.377 0.313732
\(713\) − 117.576i − 0.164903i
\(714\) 0 0
\(715\) 246.577 0.344863
\(716\) − 555.481i − 0.775812i
\(717\) 0 0
\(718\) 756.000 1.05292
\(719\) 340.912i 0.474148i 0.971492 + 0.237074i \(0.0761884\pi\)
−0.971492 + 0.237074i \(0.923812\pi\)
\(720\) 0 0
\(721\) −1892.40 −2.62469
\(722\) − 492.124i − 0.681613i
\(723\) 0 0
\(724\) −348.554 −0.481428
\(725\) − 635.765i − 0.876917i
\(726\) 0 0
\(727\) −971.777 −1.33669 −0.668347 0.743850i \(-0.732998\pi\)
−0.668347 + 0.743850i \(0.732998\pi\)
\(728\) 378.683i 0.520169i
\(729\) 0 0
\(730\) −157.219 −0.215369
\(731\) − 1540.95i − 2.10800i
\(732\) 0 0
\(733\) 229.169 0.312646 0.156323 0.987706i \(-0.450036\pi\)
0.156323 + 0.987706i \(0.450036\pi\)
\(734\) 374.353i 0.510018i
\(735\) 0 0
\(736\) 83.1384 0.112960
\(737\) − 605.175i − 0.821132i
\(738\) 0 0
\(739\) −629.892 −0.852357 −0.426179 0.904639i \(-0.640141\pi\)
−0.426179 + 0.904639i \(0.640141\pi\)
\(740\) − 70.0961i − 0.0947245i
\(741\) 0 0
\(742\) 1481.74 1.99695
\(743\) 188.014i 0.253047i 0.991964 + 0.126524i \(0.0403820\pi\)
−0.991964 + 0.126524i \(0.959618\pi\)
\(744\) 0 0
\(745\) −47.5847 −0.0638721
\(746\) 100.431i 0.134626i
\(747\) 0 0
\(748\) 851.769 1.13873
\(749\) − 743.615i − 0.992810i
\(750\) 0 0
\(751\) −1386.67 −1.84643 −0.923215 0.384283i \(-0.874449\pi\)
−0.923215 + 0.384283i \(0.874449\pi\)
\(752\) 67.8823i 0.0902690i
\(753\) 0 0
\(754\) −430.035 −0.570338
\(755\) 49.6933i 0.0658189i
\(756\) 0 0
\(757\) 1222.12 1.61443 0.807215 0.590258i \(-0.200974\pi\)
0.807215 + 0.590258i \(0.200974\pi\)
\(758\) − 985.402i − 1.30000i
\(759\) 0 0
\(760\) 15.8461 0.0208501
\(761\) 565.683i 0.743341i 0.928365 + 0.371671i \(0.121215\pi\)
−0.928365 + 0.371671i \(0.878785\pi\)
\(762\) 0 0
\(763\) −1164.16 −1.52577
\(764\) 304.252i 0.398235i
\(765\) 0 0
\(766\) −615.615 −0.803675
\(767\) 983.847i 1.28272i
\(768\) 0 0
\(769\) −598.815 −0.778693 −0.389347 0.921091i \(-0.627299\pi\)
−0.389347 + 0.921091i \(0.627299\pi\)
\(770\) − 399.983i − 0.519458i
\(771\) 0 0
\(772\) 110.000 0.142487
\(773\) − 446.970i − 0.578228i −0.957295 0.289114i \(-0.906639\pi\)
0.957295 0.289114i \(-0.0933607\pi\)
\(774\) 0 0
\(775\) 180.708 0.233171
\(776\) − 257.778i − 0.332189i
\(777\) 0 0
\(778\) 75.0155 0.0964209
\(779\) − 90.7380i − 0.116480i
\(780\) 0 0
\(781\) 240.462 0.307890
\(782\) 602.292i 0.770194i
\(783\) 0 0
\(784\) 418.277 0.533516
\(785\) 387.775i 0.493980i
\(786\) 0 0
\(787\) −955.300 −1.21385 −0.606925 0.794759i \(-0.707597\pi\)
−0.606925 + 0.794759i \(0.707597\pi\)
\(788\) − 266.596i − 0.338319i
\(789\) 0 0
\(790\) 102.662 0.129951
\(791\) 1097.93i 1.38803i
\(792\) 0 0
\(793\) 140.450 0.177112
\(794\) 90.0962i 0.113471i
\(795\) 0 0
\(796\) 417.723 0.524778
\(797\) − 221.866i − 0.278376i −0.990266 0.139188i \(-0.955551\pi\)
0.990266 0.139188i \(-0.0444492\pi\)
\(798\) 0 0
\(799\) −491.769 −0.615481
\(800\) 127.780i 0.159725i
\(801\) 0 0
\(802\) 889.104 1.10861
\(803\) 1052.13i 1.31025i
\(804\) 0 0
\(805\) 282.831 0.351342
\(806\) − 122.232i − 0.151652i
\(807\) 0 0
\(808\) −250.277 −0.309749
\(809\) − 1180.17i − 1.45881i −0.684084 0.729403i \(-0.739798\pi\)
0.684084 0.729403i \(-0.260202\pi\)
\(810\) 0 0
\(811\) −627.307 −0.773499 −0.386749 0.922185i \(-0.626402\pi\)
−0.386749 + 0.922185i \(0.626402\pi\)
\(812\) 697.577i 0.859085i
\(813\) 0 0
\(814\) −469.092 −0.576281
\(815\) − 19.8534i − 0.0243600i
\(816\) 0 0
\(817\) −191.846 −0.234818
\(818\) 757.001i 0.925429i
\(819\) 0 0
\(820\) −78.1154 −0.0952627
\(821\) − 390.880i − 0.476103i −0.971253 0.238051i \(-0.923491\pi\)
0.971253 0.238051i \(-0.0765087\pi\)
\(822\) 0 0
\(823\) 326.200 0.396355 0.198177 0.980166i \(-0.436498\pi\)
0.198177 + 0.980166i \(0.436498\pi\)
\(824\) 431.922i 0.524178i
\(825\) 0 0
\(826\) 1595.94 1.93213
\(827\) 1103.70i 1.33458i 0.744799 + 0.667289i \(0.232545\pi\)
−0.744799 + 0.667289i \(0.767455\pi\)
\(828\) 0 0
\(829\) −441.569 −0.532653 −0.266326 0.963883i \(-0.585810\pi\)
−0.266326 + 0.963883i \(0.585810\pi\)
\(830\) 33.6146i 0.0404996i
\(831\) 0 0
\(832\) 86.4308 0.103883
\(833\) 3030.18i 3.63768i
\(834\) 0 0
\(835\) 360.946 0.432271
\(836\) − 106.044i − 0.126847i
\(837\) 0 0
\(838\) −544.939 −0.650285
\(839\) − 503.960i − 0.600668i −0.953834 0.300334i \(-0.902902\pi\)
0.953834 0.300334i \(-0.0970981\pi\)
\(840\) 0 0
\(841\) 48.8269 0.0580581
\(842\) − 1068.18i − 1.26863i
\(843\) 0 0
\(844\) 174.908 0.207236
\(845\) − 81.1815i − 0.0960728i
\(846\) 0 0
\(847\) −1177.27 −1.38993
\(848\) − 338.193i − 0.398812i
\(849\) 0 0
\(850\) −925.692 −1.08905
\(851\) − 331.698i − 0.389775i
\(852\) 0 0
\(853\) 944.831 1.10766 0.553828 0.832631i \(-0.313166\pi\)
0.553828 + 0.832631i \(0.313166\pi\)
\(854\) − 227.830i − 0.266780i
\(855\) 0 0
\(856\) −169.723 −0.198275
\(857\) − 741.377i − 0.865084i −0.901614 0.432542i \(-0.857617\pi\)
0.901614 0.432542i \(-0.142383\pi\)
\(858\) 0 0
\(859\) −1361.26 −1.58470 −0.792352 0.610064i \(-0.791144\pi\)
−0.792352 + 0.610064i \(0.791144\pi\)
\(860\) 165.158i 0.192045i
\(861\) 0 0
\(862\) 152.323 0.176709
\(863\) − 805.003i − 0.932796i −0.884575 0.466398i \(-0.845551\pi\)
0.884575 0.466398i \(-0.154449\pi\)
\(864\) 0 0
\(865\) 33.4617 0.0386841
\(866\) 926.484i 1.06984i
\(867\) 0 0
\(868\) −198.277 −0.228430
\(869\) − 687.025i − 0.790593i
\(870\) 0 0
\(871\) 444.869 0.510757
\(872\) 265.709i 0.304712i
\(873\) 0 0
\(874\) 74.9845 0.0857947
\(875\) 915.801i 1.04663i
\(876\) 0 0
\(877\) 42.6922 0.0486798 0.0243399 0.999704i \(-0.492252\pi\)
0.0243399 + 0.999704i \(0.492252\pi\)
\(878\) − 667.509i − 0.760261i
\(879\) 0 0
\(880\) −91.2923 −0.103741
\(881\) − 264.567i − 0.300303i −0.988663 0.150151i \(-0.952024\pi\)
0.988663 0.150151i \(-0.0479761\pi\)
\(882\) 0 0
\(883\) −393.338 −0.445457 −0.222728 0.974881i \(-0.571496\pi\)
−0.222728 + 0.974881i \(0.571496\pi\)
\(884\) 626.143i 0.708306i
\(885\) 0 0
\(886\) 1189.66 1.34273
\(887\) 1548.05i 1.74527i 0.488375 + 0.872634i \(0.337590\pi\)
−0.488375 + 0.872634i \(0.662410\pi\)
\(888\) 0 0
\(889\) 976.800 1.09876
\(890\) − 173.443i − 0.194879i
\(891\) 0 0
\(892\) −445.184 −0.499086
\(893\) 61.2246i 0.0685606i
\(894\) 0 0
\(895\) −431.307 −0.481908
\(896\) − 140.203i − 0.156476i
\(897\) 0 0
\(898\) 541.292 0.602775
\(899\) − 225.165i − 0.250461i
\(900\) 0 0
\(901\) 2450.02 2.71922
\(902\) 522.759i 0.579555i
\(903\) 0 0
\(904\) 250.592 0.277204
\(905\) 270.637i 0.299046i
\(906\) 0 0
\(907\) 1377.49 1.51873 0.759367 0.650662i \(-0.225509\pi\)
0.759367 + 0.650662i \(0.225509\pi\)
\(908\) 600.627i 0.661484i
\(909\) 0 0
\(910\) 294.031 0.323111
\(911\) 1406.08i 1.54344i 0.635961 + 0.771721i \(0.280604\pi\)
−0.635961 + 0.771721i \(0.719396\pi\)
\(912\) 0 0
\(913\) 224.954 0.246389
\(914\) 457.786i 0.500860i
\(915\) 0 0
\(916\) −459.885 −0.502057
\(917\) 1023.35i 1.11597i
\(918\) 0 0
\(919\) 479.992 0.522299 0.261149 0.965298i \(-0.415899\pi\)
0.261149 + 0.965298i \(0.415899\pi\)
\(920\) − 64.5534i − 0.0701668i
\(921\) 0 0
\(922\) −907.261 −0.984015
\(923\) 176.765i 0.191512i
\(924\) 0 0
\(925\) 509.804 0.551139
\(926\) 183.130i 0.197764i
\(927\) 0 0
\(928\) 159.215 0.171568
\(929\) 678.733i 0.730606i 0.930889 + 0.365303i \(0.119035\pi\)
−0.930889 + 0.365303i \(0.880965\pi\)
\(930\) 0 0
\(931\) 377.254 0.405214
\(932\) 232.491i 0.249454i
\(933\) 0 0
\(934\) −951.384 −1.01861
\(935\) − 661.362i − 0.707339i
\(936\) 0 0
\(937\) −666.600 −0.711420 −0.355710 0.934596i \(-0.615761\pi\)
−0.355710 + 0.934596i \(0.615761\pi\)
\(938\) − 721.640i − 0.769340i
\(939\) 0 0
\(940\) 52.7077 0.0560720
\(941\) 1197.96i 1.27307i 0.771249 + 0.636533i \(0.219632\pi\)
−0.771249 + 0.636533i \(0.780368\pi\)
\(942\) 0 0
\(943\) −369.646 −0.391990
\(944\) − 364.258i − 0.385866i
\(945\) 0 0
\(946\) 1105.26 1.16835
\(947\) 454.713i 0.480162i 0.970753 + 0.240081i \(0.0771740\pi\)
−0.970753 + 0.240081i \(0.922826\pi\)
\(948\) 0 0
\(949\) −773.431 −0.814996
\(950\) 115.247i 0.121313i
\(951\) 0 0
\(952\) 1015.69 1.06690
\(953\) 1060.16i 1.11245i 0.831033 + 0.556224i \(0.187750\pi\)
−0.831033 + 0.556224i \(0.812250\pi\)
\(954\) 0 0
\(955\) 236.238 0.247370
\(956\) − 328.772i − 0.343904i
\(957\) 0 0
\(958\) −177.646 −0.185434
\(959\) − 2661.13i − 2.77490i
\(960\) 0 0
\(961\) −897.000 −0.933403
\(962\) − 344.834i − 0.358455i
\(963\) 0 0
\(964\) 162.623 0.168696
\(965\) − 85.4103i − 0.0885081i
\(966\) 0 0
\(967\) −1655.31 −1.71180 −0.855902 0.517138i \(-0.826998\pi\)
−0.855902 + 0.517138i \(0.826998\pi\)
\(968\) 268.701i 0.277583i
\(969\) 0 0
\(970\) −200.154 −0.206344
\(971\) − 797.780i − 0.821606i −0.911724 0.410803i \(-0.865248\pi\)
0.911724 0.410803i \(-0.134752\pi\)
\(972\) 0 0
\(973\) −753.261 −0.774164
\(974\) − 634.460i − 0.651396i
\(975\) 0 0
\(976\) −52.0000 −0.0532787
\(977\) − 312.432i − 0.319787i −0.987134 0.159894i \(-0.948885\pi\)
0.987134 0.159894i \(-0.0511152\pi\)
\(978\) 0 0
\(979\) −1160.70 −1.18560
\(980\) − 324.774i − 0.331402i
\(981\) 0 0
\(982\) −639.446 −0.651167
\(983\) 370.143i 0.376544i 0.982117 + 0.188272i \(0.0602887\pi\)
−0.982117 + 0.188272i \(0.939711\pi\)
\(984\) 0 0
\(985\) −207.000 −0.210152
\(986\) 1153.43i 1.16980i
\(987\) 0 0
\(988\) 77.9540 0.0789008
\(989\) 781.538i 0.790230i
\(990\) 0 0
\(991\) −1324.48 −1.33651 −0.668256 0.743931i \(-0.732959\pi\)
−0.668256 + 0.743931i \(0.732959\pi\)
\(992\) 45.2548i 0.0456198i
\(993\) 0 0
\(994\) 286.739 0.288469
\(995\) − 324.344i − 0.325974i
\(996\) 0 0
\(997\) 879.246 0.881892 0.440946 0.897534i \(-0.354643\pi\)
0.440946 + 0.897534i \(0.354643\pi\)
\(998\) 179.028i 0.179387i
\(999\) 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.161.4 yes 4
3.2 odd 2 inner 162.3.b.b.161.1 4
4.3 odd 2 1296.3.e.d.161.3 4
9.2 odd 6 162.3.d.c.53.2 8
9.4 even 3 162.3.d.c.107.2 8
9.5 odd 6 162.3.d.c.107.3 8
9.7 even 3 162.3.d.c.53.3 8
12.11 even 2 1296.3.e.d.161.2 4
36.7 odd 6 1296.3.q.o.1025.2 8
36.11 even 6 1296.3.q.o.1025.3 8
36.23 even 6 1296.3.q.o.593.2 8
36.31 odd 6 1296.3.q.o.593.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.3.b.b.161.1 4 3.2 odd 2 inner
162.3.b.b.161.4 yes 4 1.1 even 1 trivial
162.3.d.c.53.2 8 9.2 odd 6
162.3.d.c.53.3 8 9.7 even 3
162.3.d.c.107.2 8 9.4 even 3
162.3.d.c.107.3 8 9.5 odd 6
1296.3.e.d.161.2 4 12.11 even 2
1296.3.e.d.161.3 4 4.3 odd 2
1296.3.q.o.593.2 8 36.23 even 6
1296.3.q.o.593.3 8 36.31 odd 6
1296.3.q.o.1025.2 8 36.7 odd 6
1296.3.q.o.1025.3 8 36.11 even 6