Properties

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

$q$-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 \(-0.950368\pi\)
0.987869 0.155291i \(-0.0496317\pi\)
\(6\) 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 \(-0.767132\pi\)
0.744123 0.668043i \(-0.232868\pi\)
\(12\) 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 \(-0.675215\pi\)
0.523074 0.852287i \(-0.324785\pi\)
\(18\) 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 \(0.103514\pi\)
−0.947587 + 0.319499i \(0.896486\pi\)
\(24\) 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 \(0.161278\pi\)
−0.874365 + 0.485268i \(0.838722\pi\)
\(30\) 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 \(0.0992323\pi\)
−0.951799 + 0.306722i \(0.900768\pi\)
\(42\) 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 \(-0.942216\pi\)
0.983568 0.180538i \(-0.0577838\pi\)
\(48\) 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 \(0.293910\pi\)
−0.603154 + 0.797625i \(0.706090\pi\)
\(54\) 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 \(0.280610\pi\)
−0.635947 + 0.771733i \(0.719390\pi\)
\(60\) 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 \(0.0367575\pi\)
−0.993340 + 0.115221i \(0.963242\pi\)
\(72\) 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 \(0.0293917\pi\)
−0.995740 + 0.0922057i \(0.970608\pi\)
\(84\) 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 \(-0.853671\pi\)
0.896184 0.443683i \(-0.146329\pi\)
\(90\) 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 \(0.144331\pi\)
−0.898950 + 0.438051i \(0.855669\pi\)
\(102\) 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 \(0.0904680\pi\)
−0.959882 + 0.280403i \(0.909532\pi\)
\(108\) 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 \(-0.871775\pi\)
0.919954 0.392025i \(-0.128225\pi\)
\(114\) 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 \(-0.897933\pi\)
0.949029 0.315188i \(-0.102067\pi\)
\(132\) 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 \(0.286682\pi\)
−0.621110 + 0.783723i \(0.713318\pi\)
\(138\) 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 \(-0.967211\pi\)
0.994699 0.102826i \(-0.0327885\pi\)
\(150\) 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 \(0.244995\pi\)
−0.718136 + 0.695902i \(0.755005\pi\)
\(168\) 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 \(0.0198361\pi\)
−0.998059 + 0.0622765i \(0.980164\pi\)
\(174\) 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 \(-0.717341\pi\)
0.630964 0.775812i \(-0.282659\pi\)
\(180\) 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 \(0.130377\pi\)
−0.917283 + 0.398235i \(0.869623\pi\)
\(192\) 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 \(-0.890142\pi\)
0.941031 0.338319i \(-0.109858\pi\)
\(198\) 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 \(0.230073\pi\)
−0.749959 + 0.661484i \(0.769927\pi\)
\(228\) 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 \(0.0802512\pi\)
−0.968387 + 0.249454i \(0.919749\pi\)
\(234\) 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 \(-0.888251\pi\)
0.939005 0.343904i \(-0.111749\pi\)
\(240\) 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 \(-0.710296\pi\)
0.613642 0.789584i \(-0.289704\pi\)
\(252\) 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 \(-0.862655\pi\)
0.908348 0.418216i \(-0.137345\pi\)
\(258\) 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 \(-0.793429\pi\)
0.796711 0.604360i \(-0.206571\pi\)
\(264\) 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 \(-0.873304\pi\)
0.921828 0.387599i \(-0.126696\pi\)
\(270\) 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 \(-0.874878\pi\)
0.923732 0.383039i \(-0.125122\pi\)
\(282\) 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 \(0.197671\pi\)
−0.813296 + 0.581850i \(0.802329\pi\)
\(294\) 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 \(-0.897841\pi\)
0.948939 0.315461i \(-0.102159\pi\)
\(312\) 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 \(0.0241929\pi\)
−0.997113 + 0.0759311i \(0.975807\pi\)
\(318\) 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 \(0.0177920\pi\)
−0.998438 + 0.0558662i \(0.982208\pi\)
\(348\) 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 \(0.000539487\pi\)
−0.999999 + 0.00169485i \(0.999461\pi\)
\(354\) 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 \(0.267326\pi\)
−0.667589 + 0.744530i \(0.732674\pi\)
\(360\) 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 \(-0.807607\pi\)
0.822832 0.568284i \(-0.192393\pi\)
\(384\) 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 \(0.0217192\pi\)
−0.997673 + 0.0681799i \(0.978281\pi\)
\(390\) 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 \(0.286775\pi\)
−0.620882 + 0.783904i \(0.713225\pi\)
\(402\) 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 \(-0.847914\pi\)
0.888012 0.459821i \(-0.152086\pi\)
\(420\) 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 \(0.0398777\pi\)
−0.992163 + 0.124952i \(0.960122\pi\)
\(432\) 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 \(0.398364\pi\)
−0.313902 + 0.949455i \(0.601636\pi\)
\(444\) 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 \(0.140157\pi\)
−0.904616 + 0.426227i \(0.859843\pi\)
\(450\) 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 \(-0.755049\pi\)
0.718232 0.695803i \(-0.244951\pi\)
\(462\) 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 \(-0.744019\pi\)
0.693696 0.720268i \(-0.255981\pi\)
\(468\) 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 \(-0.958142\pi\)
0.991366 0.131122i \(-0.0418579\pi\)
\(480\) 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 \(-0.847690\pi\)
0.887688 0.460445i \(-0.152310\pi\)
\(492\) 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 \(0.0953367\pi\)
−0.955481 + 0.295051i \(0.904663\pi\)
\(504\) 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 \(-0.777578\pi\)
0.765642 0.643267i \(-0.222422\pi\)
\(510\) 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 \(-0.769626\pi\)
0.749333 0.662193i \(-0.230374\pi\)
\(522\) 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 \(-0.963975\pi\)
0.993603 0.112934i \(-0.0360247\pi\)
\(558\) 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 \(0.149238\pi\)
−0.892091 + 0.451856i \(0.850762\pi\)
\(564\) 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 \(-0.972874\pi\)
0.996371 0.0851143i \(-0.0271255\pi\)
\(570\) 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 \(-0.883768\pi\)
0.934069 0.357093i \(-0.116232\pi\)
\(588\) 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 \(-0.910468\pi\)
0.960703 0.277580i \(-0.0895323\pi\)
\(594\) 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 \(0.0491476\pi\)
−0.988104 + 0.153789i \(0.950852\pi\)
\(600\) 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 \(0.262521\pi\)
−0.678753 + 0.734367i \(0.737479\pi\)
\(618\) 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 \(0.148730\pi\)
−0.892811 + 0.450431i \(0.851270\pi\)
\(642\) 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 \(0.275570\pi\)
−0.648087 + 0.761567i \(0.724430\pi\)
\(648\) 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 \(-0.847969\pi\)
0.888092 0.459665i \(-0.152031\pi\)
\(654\) 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 \(-0.996235\pi\)
0.999930 0.0118277i \(-0.00376496\pi\)
\(660\) 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 \(0.0668191\pi\)
−0.978048 + 0.208380i \(0.933181\pi\)
\(678\) 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 \(0.292335\pi\)
−0.607095 + 0.794629i \(0.707665\pi\)
\(684\) 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 \(0.0494047\pi\)
−0.987979 + 0.154587i \(0.950595\pi\)
\(702\) 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 \(-0.923812\pi\)
0.971492 0.237074i \(-0.0761884\pi\)
\(720\) 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 \(-0.959618\pi\)
0.991964 0.126524i \(-0.0403820\pi\)
\(744\) 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 \(-0.878785\pi\)
0.928365 0.371671i \(-0.121215\pi\)
\(762\) 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 \(0.0933607\pi\)
−0.957295 + 0.289114i \(0.906639\pi\)
\(774\) 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 \(0.0444492\pi\)
−0.990266 + 0.139188i \(0.955551\pi\)
\(798\) 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 \(0.260202\pi\)
−0.684084 + 0.729403i \(0.739798\pi\)
\(810\) 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 \(0.0765087\pi\)
−0.971253 + 0.238051i \(0.923491\pi\)
\(822\) 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 \(-0.767455\pi\)
0.744799 0.667289i \(-0.232545\pi\)
\(828\) 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 \(0.0970981\pi\)
−0.953834 + 0.300334i \(0.902902\pi\)
\(840\) 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 \(0.142383\pi\)
−0.901614 + 0.432542i \(0.857617\pi\)
\(858\) 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 \(0.154449\pi\)
−0.884575 + 0.466398i \(0.845551\pi\)
\(864\) 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 \(0.0479761\pi\)
−0.988663 + 0.150151i \(0.952024\pi\)
\(882\) 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 \(-0.662410\pi\)
0.488375 0.872634i \(-0.337590\pi\)
\(888\) 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 \(-0.719396\pi\)
0.635961 0.771721i \(-0.280604\pi\)
\(912\) 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 \(-0.880965\pi\)
0.930889 0.365303i \(-0.119035\pi\)
\(930\) 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 \(-0.780368\pi\)
0.771249 0.636533i \(-0.219632\pi\)
\(942\) 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 \(-0.922826\pi\)
0.970753 0.240081i \(-0.0771740\pi\)
\(948\) 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 \(-0.812250\pi\)
0.831033 0.556224i \(-0.187750\pi\)
\(954\) 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 \(0.134752\pi\)
−0.911724 + 0.410803i \(0.865248\pi\)
\(972\) 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 \(0.0511152\pi\)
−0.987134 + 0.159894i \(0.948885\pi\)
\(978\) 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 \(-0.939711\pi\)
0.982117 0.188272i \(-0.0602887\pi\)
\(984\) 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.161.1 4
3.2 odd 2 inner 162.3.b.b.161.4 yes 4
4.3 odd 2 1296.3.e.d.161.2 4
9.2 odd 6 162.3.d.c.53.3 8
9.4 even 3 162.3.d.c.107.3 8
9.5 odd 6 162.3.d.c.107.2 8
9.7 even 3 162.3.d.c.53.2 8
12.11 even 2 1296.3.e.d.161.3 4
36.7 odd 6 1296.3.q.o.1025.3 8
36.11 even 6 1296.3.q.o.1025.2 8
36.23 even 6 1296.3.q.o.593.3 8
36.31 odd 6 1296.3.q.o.593.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.3.b.b.161.1 4 1.1 even 1 trivial
162.3.b.b.161.4 yes 4 3.2 odd 2 inner
162.3.d.c.53.2 8 9.7 even 3
162.3.d.c.53.3 8 9.2 odd 6
162.3.d.c.107.2 8 9.5 odd 6
162.3.d.c.107.3 8 9.4 even 3
1296.3.e.d.161.2 4 4.3 odd 2
1296.3.e.d.161.3 4 12.11 even 2
1296.3.q.o.593.2 8 36.31 odd 6
1296.3.q.o.593.3 8 36.23 even 6
1296.3.q.o.1025.2 8 36.11 even 6
1296.3.q.o.1025.3 8 36.7 odd 6