Properties

Label 1344.4.p.c.223.21
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.21
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.c.223.22

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +15.6891 q^{5} +(10.8361 + 15.0193i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +15.6891 q^{5} +(10.8361 + 15.0193i) q^{7} -9.00000 q^{9} +41.3237 q^{11} +8.02995 q^{13} -47.0674i q^{15} -14.2211i q^{17} +10.2063i q^{19} +(45.0579 - 32.5083i) q^{21} -203.033i q^{23} +121.149 q^{25} +27.0000i q^{27} -82.6180i q^{29} -156.241 q^{31} -123.971i q^{33} +(170.009 + 235.640i) q^{35} -106.169i q^{37} -24.0898i q^{39} -315.657i q^{41} +154.276 q^{43} -141.202 q^{45} +474.835 q^{47} +(-108.158 + 325.501i) q^{49} -42.6632 q^{51} -266.866i q^{53} +648.334 q^{55} +30.6190 q^{57} +425.632i q^{59} -8.30642 q^{61} +(-97.5248 - 135.174i) q^{63} +125.983 q^{65} +820.303 q^{67} -609.099 q^{69} +109.956i q^{71} +53.8640i q^{73} -363.446i q^{75} +(447.788 + 620.653i) q^{77} +1171.20i q^{79} +81.0000 q^{81} -507.294i q^{83} -223.116i q^{85} -247.854 q^{87} -1061.99i q^{89} +(87.0133 + 120.604i) q^{91} +468.722i q^{93} +160.128i q^{95} +573.319i q^{97} -371.914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 15.6891 1.40328 0.701639 0.712532i \(-0.252452\pi\)
0.701639 + 0.712532i \(0.252452\pi\)
\(6\) 0 0
\(7\) 10.8361 + 15.0193i 0.585094 + 0.810965i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 41.3237 1.13269 0.566344 0.824169i \(-0.308357\pi\)
0.566344 + 0.824169i \(0.308357\pi\)
\(12\) 0 0
\(13\) 8.02995 0.171316 0.0856580 0.996325i \(-0.472701\pi\)
0.0856580 + 0.996325i \(0.472701\pi\)
\(14\) 0 0
\(15\) 47.0674i 0.810183i
\(16\) 0 0
\(17\) 14.2211i 0.202889i −0.994841 0.101444i \(-0.967654\pi\)
0.994841 0.101444i \(-0.0323464\pi\)
\(18\) 0 0
\(19\) 10.2063i 0.123236i 0.998100 + 0.0616182i \(0.0196261\pi\)
−0.998100 + 0.0616182i \(0.980374\pi\)
\(20\) 0 0
\(21\) 45.0579 32.5083i 0.468211 0.337804i
\(22\) 0 0
\(23\) 203.033i 1.84067i −0.391135 0.920333i \(-0.627917\pi\)
0.391135 0.920333i \(-0.372083\pi\)
\(24\) 0 0
\(25\) 121.149 0.969190
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 82.6180i 0.529027i −0.964382 0.264513i \(-0.914789\pi\)
0.964382 0.264513i \(-0.0852113\pi\)
\(30\) 0 0
\(31\) −156.241 −0.905214 −0.452607 0.891710i \(-0.649506\pi\)
−0.452607 + 0.891710i \(0.649506\pi\)
\(32\) 0 0
\(33\) 123.971i 0.653958i
\(34\) 0 0
\(35\) 170.009 + 235.640i 0.821050 + 1.13801i
\(36\) 0 0
\(37\) 106.169i 0.471732i −0.971786 0.235866i \(-0.924207\pi\)
0.971786 0.235866i \(-0.0757927\pi\)
\(38\) 0 0
\(39\) 24.0898i 0.0989093i
\(40\) 0 0
\(41\) 315.657i 1.20238i −0.799108 0.601188i \(-0.794694\pi\)
0.799108 0.601188i \(-0.205306\pi\)
\(42\) 0 0
\(43\) 154.276 0.547137 0.273569 0.961853i \(-0.411796\pi\)
0.273569 + 0.961853i \(0.411796\pi\)
\(44\) 0 0
\(45\) −141.202 −0.467759
\(46\) 0 0
\(47\) 474.835 1.47365 0.736827 0.676081i \(-0.236323\pi\)
0.736827 + 0.676081i \(0.236323\pi\)
\(48\) 0 0
\(49\) −108.158 + 325.501i −0.315330 + 0.948982i
\(50\) 0 0
\(51\) −42.6632 −0.117138
\(52\) 0 0
\(53\) 266.866i 0.691638i −0.938301 0.345819i \(-0.887601\pi\)
0.938301 0.345819i \(-0.112399\pi\)
\(54\) 0 0
\(55\) 648.334 1.58948
\(56\) 0 0
\(57\) 30.6190 0.0711506
\(58\) 0 0
\(59\) 425.632i 0.939197i 0.882880 + 0.469598i \(0.155601\pi\)
−0.882880 + 0.469598i \(0.844399\pi\)
\(60\) 0 0
\(61\) −8.30642 −0.0174349 −0.00871744 0.999962i \(-0.502775\pi\)
−0.00871744 + 0.999962i \(0.502775\pi\)
\(62\) 0 0
\(63\) −97.5248 135.174i −0.195031 0.270322i
\(64\) 0 0
\(65\) 125.983 0.240404
\(66\) 0 0
\(67\) 820.303 1.49576 0.747880 0.663834i \(-0.231072\pi\)
0.747880 + 0.663834i \(0.231072\pi\)
\(68\) 0 0
\(69\) −609.099 −1.06271
\(70\) 0 0
\(71\) 109.956i 0.183795i 0.995768 + 0.0918974i \(0.0292932\pi\)
−0.995768 + 0.0918974i \(0.970707\pi\)
\(72\) 0 0
\(73\) 53.8640i 0.0863604i 0.999067 + 0.0431802i \(0.0137490\pi\)
−0.999067 + 0.0431802i \(0.986251\pi\)
\(74\) 0 0
\(75\) 363.446i 0.559562i
\(76\) 0 0
\(77\) 447.788 + 620.653i 0.662730 + 0.918571i
\(78\) 0 0
\(79\) 1171.20i 1.66798i 0.551782 + 0.833988i \(0.313948\pi\)
−0.551782 + 0.833988i \(0.686052\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 507.294i 0.670876i −0.942062 0.335438i \(-0.891116\pi\)
0.942062 0.335438i \(-0.108884\pi\)
\(84\) 0 0
\(85\) 223.116i 0.284710i
\(86\) 0 0
\(87\) −247.854 −0.305434
\(88\) 0 0
\(89\) 1061.99i 1.26484i −0.774627 0.632418i \(-0.782062\pi\)
0.774627 0.632418i \(-0.217938\pi\)
\(90\) 0 0
\(91\) 87.0133 + 120.604i 0.100236 + 0.138931i
\(92\) 0 0
\(93\) 468.722i 0.522626i
\(94\) 0 0
\(95\) 160.128i 0.172935i
\(96\) 0 0
\(97\) 573.319i 0.600121i 0.953920 + 0.300060i \(0.0970068\pi\)
−0.953920 + 0.300060i \(0.902993\pi\)
\(98\) 0 0
\(99\) −371.914 −0.377563
\(100\) 0 0
\(101\) −1515.77 −1.49331 −0.746657 0.665209i \(-0.768342\pi\)
−0.746657 + 0.665209i \(0.768342\pi\)
\(102\) 0 0
\(103\) 1642.32 1.57109 0.785545 0.618804i \(-0.212382\pi\)
0.785545 + 0.618804i \(0.212382\pi\)
\(104\) 0 0
\(105\) 706.919 510.027i 0.657031 0.474033i
\(106\) 0 0
\(107\) 900.675 0.813752 0.406876 0.913483i \(-0.366618\pi\)
0.406876 + 0.913483i \(0.366618\pi\)
\(108\) 0 0
\(109\) 1603.61i 1.40915i 0.709627 + 0.704577i \(0.248863\pi\)
−0.709627 + 0.704577i \(0.751137\pi\)
\(110\) 0 0
\(111\) −318.507 −0.272355
\(112\) 0 0
\(113\) 821.120 0.683580 0.341790 0.939776i \(-0.388967\pi\)
0.341790 + 0.939776i \(0.388967\pi\)
\(114\) 0 0
\(115\) 3185.41i 2.58297i
\(116\) 0 0
\(117\) −72.2695 −0.0571053
\(118\) 0 0
\(119\) 213.590 154.101i 0.164536 0.118709i
\(120\) 0 0
\(121\) 376.652 0.282984
\(122\) 0 0
\(123\) −946.972 −0.694192
\(124\) 0 0
\(125\) −60.4223 −0.0432347
\(126\) 0 0
\(127\) 2459.21i 1.71826i 0.511754 + 0.859132i \(0.328996\pi\)
−0.511754 + 0.859132i \(0.671004\pi\)
\(128\) 0 0
\(129\) 462.828i 0.315890i
\(130\) 0 0
\(131\) 1027.71i 0.685428i 0.939440 + 0.342714i \(0.111346\pi\)
−0.939440 + 0.342714i \(0.888654\pi\)
\(132\) 0 0
\(133\) −153.292 + 110.597i −0.0999405 + 0.0721049i
\(134\) 0 0
\(135\) 423.606i 0.270061i
\(136\) 0 0
\(137\) 1372.27 0.855771 0.427886 0.903833i \(-0.359259\pi\)
0.427886 + 0.903833i \(0.359259\pi\)
\(138\) 0 0
\(139\) 1161.76i 0.708915i −0.935072 0.354458i \(-0.884666\pi\)
0.935072 0.354458i \(-0.115334\pi\)
\(140\) 0 0
\(141\) 1424.50i 0.850815i
\(142\) 0 0
\(143\) 331.827 0.194048
\(144\) 0 0
\(145\) 1296.20i 0.742372i
\(146\) 0 0
\(147\) 976.503 + 324.474i 0.547895 + 0.182056i
\(148\) 0 0
\(149\) 647.153i 0.355818i −0.984047 0.177909i \(-0.943067\pi\)
0.984047 0.177909i \(-0.0569332\pi\)
\(150\) 0 0
\(151\) 1459.53i 0.786588i −0.919413 0.393294i \(-0.871336\pi\)
0.919413 0.393294i \(-0.128664\pi\)
\(152\) 0 0
\(153\) 127.990i 0.0676297i
\(154\) 0 0
\(155\) −2451.28 −1.27027
\(156\) 0 0
\(157\) −1989.05 −1.01110 −0.505552 0.862796i \(-0.668711\pi\)
−0.505552 + 0.862796i \(0.668711\pi\)
\(158\) 0 0
\(159\) −800.597 −0.399317
\(160\) 0 0
\(161\) 3049.41 2200.09i 1.49272 1.07696i
\(162\) 0 0
\(163\) −3462.74 −1.66395 −0.831973 0.554817i \(-0.812788\pi\)
−0.831973 + 0.554817i \(0.812788\pi\)
\(164\) 0 0
\(165\) 1945.00i 0.917685i
\(166\) 0 0
\(167\) 826.203 0.382835 0.191418 0.981509i \(-0.438692\pi\)
0.191418 + 0.981509i \(0.438692\pi\)
\(168\) 0 0
\(169\) −2132.52 −0.970651
\(170\) 0 0
\(171\) 91.8570i 0.0410788i
\(172\) 0 0
\(173\) −2511.03 −1.10352 −0.551762 0.834001i \(-0.686044\pi\)
−0.551762 + 0.834001i \(0.686044\pi\)
\(174\) 0 0
\(175\) 1312.78 + 1819.57i 0.567067 + 0.785980i
\(176\) 0 0
\(177\) 1276.90 0.542246
\(178\) 0 0
\(179\) 3839.23 1.60312 0.801558 0.597917i \(-0.204005\pi\)
0.801558 + 0.597917i \(0.204005\pi\)
\(180\) 0 0
\(181\) 641.534 0.263452 0.131726 0.991286i \(-0.457948\pi\)
0.131726 + 0.991286i \(0.457948\pi\)
\(182\) 0 0
\(183\) 24.9193i 0.0100660i
\(184\) 0 0
\(185\) 1665.70i 0.661971i
\(186\) 0 0
\(187\) 587.667i 0.229810i
\(188\) 0 0
\(189\) −405.521 + 292.575i −0.156070 + 0.112601i
\(190\) 0 0
\(191\) 1641.13i 0.621719i 0.950456 + 0.310859i \(0.100617\pi\)
−0.950456 + 0.310859i \(0.899383\pi\)
\(192\) 0 0
\(193\) 2245.24 0.837390 0.418695 0.908127i \(-0.362488\pi\)
0.418695 + 0.908127i \(0.362488\pi\)
\(194\) 0 0
\(195\) 377.949i 0.138797i
\(196\) 0 0
\(197\) 3758.27i 1.35922i −0.733574 0.679609i \(-0.762149\pi\)
0.733574 0.679609i \(-0.237851\pi\)
\(198\) 0 0
\(199\) 4894.52 1.74353 0.871767 0.489921i \(-0.162974\pi\)
0.871767 + 0.489921i \(0.162974\pi\)
\(200\) 0 0
\(201\) 2460.91i 0.863577i
\(202\) 0 0
\(203\) 1240.86 895.256i 0.429022 0.309530i
\(204\) 0 0
\(205\) 4952.39i 1.68727i
\(206\) 0 0
\(207\) 1827.30i 0.613556i
\(208\) 0 0
\(209\) 421.764i 0.139589i
\(210\) 0 0
\(211\) −34.5211 −0.0112632 −0.00563160 0.999984i \(-0.501793\pi\)
−0.00563160 + 0.999984i \(0.501793\pi\)
\(212\) 0 0
\(213\) 329.869 0.106114
\(214\) 0 0
\(215\) 2420.46 0.767786
\(216\) 0 0
\(217\) −1693.04 2346.62i −0.529635 0.734097i
\(218\) 0 0
\(219\) 161.592 0.0498602
\(220\) 0 0
\(221\) 114.194i 0.0347581i
\(222\) 0 0
\(223\) 2469.84 0.741671 0.370835 0.928699i \(-0.379071\pi\)
0.370835 + 0.928699i \(0.379071\pi\)
\(224\) 0 0
\(225\) −1090.34 −0.323063
\(226\) 0 0
\(227\) 3287.81i 0.961320i −0.876907 0.480660i \(-0.840397\pi\)
0.876907 0.480660i \(-0.159603\pi\)
\(228\) 0 0
\(229\) 4047.39 1.16794 0.583972 0.811774i \(-0.301498\pi\)
0.583972 + 0.811774i \(0.301498\pi\)
\(230\) 0 0
\(231\) 1861.96 1343.36i 0.530337 0.382627i
\(232\) 0 0
\(233\) 7.42114 0.00208659 0.00104329 0.999999i \(-0.499668\pi\)
0.00104329 + 0.999999i \(0.499668\pi\)
\(234\) 0 0
\(235\) 7449.74 2.06795
\(236\) 0 0
\(237\) 3513.60 0.963007
\(238\) 0 0
\(239\) 2959.05i 0.800858i 0.916328 + 0.400429i \(0.131139\pi\)
−0.916328 + 0.400429i \(0.868861\pi\)
\(240\) 0 0
\(241\) 5561.26i 1.48644i −0.669047 0.743220i \(-0.733298\pi\)
0.669047 0.743220i \(-0.266702\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −1696.91 + 5106.83i −0.442495 + 1.33169i
\(246\) 0 0
\(247\) 81.9563i 0.0211124i
\(248\) 0 0
\(249\) −1521.88 −0.387331
\(250\) 0 0
\(251\) 129.815i 0.0326449i −0.999867 0.0163224i \(-0.994804\pi\)
0.999867 0.0163224i \(-0.00519583\pi\)
\(252\) 0 0
\(253\) 8390.09i 2.08490i
\(254\) 0 0
\(255\) −669.348 −0.164377
\(256\) 0 0
\(257\) 341.353i 0.0828521i −0.999142 0.0414260i \(-0.986810\pi\)
0.999142 0.0414260i \(-0.0131901\pi\)
\(258\) 0 0
\(259\) 1594.58 1150.46i 0.382558 0.276008i
\(260\) 0 0
\(261\) 743.562i 0.176342i
\(262\) 0 0
\(263\) 477.176i 0.111878i −0.998434 0.0559391i \(-0.982185\pi\)
0.998434 0.0559391i \(-0.0178153\pi\)
\(264\) 0 0
\(265\) 4186.89i 0.970560i
\(266\) 0 0
\(267\) −3185.96 −0.730254
\(268\) 0 0
\(269\) −147.535 −0.0334399 −0.0167200 0.999860i \(-0.505322\pi\)
−0.0167200 + 0.999860i \(0.505322\pi\)
\(270\) 0 0
\(271\) −6071.49 −1.36095 −0.680474 0.732773i \(-0.738226\pi\)
−0.680474 + 0.732773i \(0.738226\pi\)
\(272\) 0 0
\(273\) 361.812 261.040i 0.0802120 0.0578712i
\(274\) 0 0
\(275\) 5006.32 1.09779
\(276\) 0 0
\(277\) 569.213i 0.123468i 0.998093 + 0.0617341i \(0.0196631\pi\)
−0.998093 + 0.0617341i \(0.980337\pi\)
\(278\) 0 0
\(279\) 1406.17 0.301738
\(280\) 0 0
\(281\) −784.629 −0.166573 −0.0832866 0.996526i \(-0.526542\pi\)
−0.0832866 + 0.996526i \(0.526542\pi\)
\(282\) 0 0
\(283\) 7125.60i 1.49672i 0.663290 + 0.748362i \(0.269160\pi\)
−0.663290 + 0.748362i \(0.730840\pi\)
\(284\) 0 0
\(285\) 480.385 0.0998441
\(286\) 0 0
\(287\) 4740.95 3420.49i 0.975085 0.703503i
\(288\) 0 0
\(289\) 4710.76 0.958836
\(290\) 0 0
\(291\) 1719.96 0.346480
\(292\) 0 0
\(293\) −6069.86 −1.21026 −0.605128 0.796129i \(-0.706878\pi\)
−0.605128 + 0.796129i \(0.706878\pi\)
\(294\) 0 0
\(295\) 6677.80i 1.31795i
\(296\) 0 0
\(297\) 1115.74i 0.217986i
\(298\) 0 0
\(299\) 1630.35i 0.315335i
\(300\) 0 0
\(301\) 1671.75 + 2317.12i 0.320127 + 0.443709i
\(302\) 0 0
\(303\) 4547.31i 0.862165i
\(304\) 0 0
\(305\) −130.321 −0.0244660
\(306\) 0 0
\(307\) 4655.34i 0.865453i −0.901525 0.432727i \(-0.857552\pi\)
0.901525 0.432727i \(-0.142448\pi\)
\(308\) 0 0
\(309\) 4926.95i 0.907070i
\(310\) 0 0
\(311\) −3077.49 −0.561121 −0.280560 0.959836i \(-0.590520\pi\)
−0.280560 + 0.959836i \(0.590520\pi\)
\(312\) 0 0
\(313\) 485.087i 0.0875999i −0.999040 0.0437999i \(-0.986054\pi\)
0.999040 0.0437999i \(-0.0139464\pi\)
\(314\) 0 0
\(315\) −1530.08 2120.76i −0.273683 0.379337i
\(316\) 0 0
\(317\) 8675.14i 1.53705i 0.639821 + 0.768524i \(0.279008\pi\)
−0.639821 + 0.768524i \(0.720992\pi\)
\(318\) 0 0
\(319\) 3414.08i 0.599223i
\(320\) 0 0
\(321\) 2702.02i 0.469820i
\(322\) 0 0
\(323\) 145.145 0.0250033
\(324\) 0 0
\(325\) 972.818 0.166038
\(326\) 0 0
\(327\) 4810.83 0.813576
\(328\) 0 0
\(329\) 5145.35 + 7131.68i 0.862227 + 1.19508i
\(330\) 0 0
\(331\) −10094.7 −1.67630 −0.838152 0.545436i \(-0.816364\pi\)
−0.838152 + 0.545436i \(0.816364\pi\)
\(332\) 0 0
\(333\) 955.522i 0.157244i
\(334\) 0 0
\(335\) 12869.8 2.09897
\(336\) 0 0
\(337\) −2044.06 −0.330406 −0.165203 0.986260i \(-0.552828\pi\)
−0.165203 + 0.986260i \(0.552828\pi\)
\(338\) 0 0
\(339\) 2463.36i 0.394665i
\(340\) 0 0
\(341\) −6456.44 −1.02533
\(342\) 0 0
\(343\) −6060.80 + 1902.70i −0.954089 + 0.299522i
\(344\) 0 0
\(345\) −9556.24 −1.49128
\(346\) 0 0
\(347\) 7094.50 1.09756 0.548779 0.835967i \(-0.315093\pi\)
0.548779 + 0.835967i \(0.315093\pi\)
\(348\) 0 0
\(349\) −6628.07 −1.01660 −0.508299 0.861181i \(-0.669725\pi\)
−0.508299 + 0.861181i \(0.669725\pi\)
\(350\) 0 0
\(351\) 216.809i 0.0329698i
\(352\) 0 0
\(353\) 6785.57i 1.02311i 0.859249 + 0.511557i \(0.170931\pi\)
−0.859249 + 0.511557i \(0.829069\pi\)
\(354\) 0 0
\(355\) 1725.12i 0.257915i
\(356\) 0 0
\(357\) −462.302 640.770i −0.0685368 0.0949949i
\(358\) 0 0
\(359\) 3251.11i 0.477959i 0.971025 + 0.238979i \(0.0768128\pi\)
−0.971025 + 0.238979i \(0.923187\pi\)
\(360\) 0 0
\(361\) 6754.83 0.984813
\(362\) 0 0
\(363\) 1129.95i 0.163381i
\(364\) 0 0
\(365\) 845.080i 0.121188i
\(366\) 0 0
\(367\) −2278.93 −0.324139 −0.162070 0.986779i \(-0.551817\pi\)
−0.162070 + 0.986779i \(0.551817\pi\)
\(368\) 0 0
\(369\) 2840.92i 0.400792i
\(370\) 0 0
\(371\) 4008.13 2891.78i 0.560894 0.404673i
\(372\) 0 0
\(373\) 711.350i 0.0987461i −0.998780 0.0493730i \(-0.984278\pi\)
0.998780 0.0493730i \(-0.0157223\pi\)
\(374\) 0 0
\(375\) 181.267i 0.0249615i
\(376\) 0 0
\(377\) 663.418i 0.0906307i
\(378\) 0 0
\(379\) −11988.5 −1.62482 −0.812412 0.583084i \(-0.801846\pi\)
−0.812412 + 0.583084i \(0.801846\pi\)
\(380\) 0 0
\(381\) 7377.63 0.992040
\(382\) 0 0
\(383\) −2949.04 −0.393443 −0.196722 0.980459i \(-0.563029\pi\)
−0.196722 + 0.980459i \(0.563029\pi\)
\(384\) 0 0
\(385\) 7025.40 + 9737.51i 0.929994 + 1.28901i
\(386\) 0 0
\(387\) −1388.49 −0.182379
\(388\) 0 0
\(389\) 12522.4i 1.63217i 0.577934 + 0.816083i \(0.303859\pi\)
−0.577934 + 0.816083i \(0.696141\pi\)
\(390\) 0 0
\(391\) −2887.35 −0.373451
\(392\) 0 0
\(393\) 3083.12 0.395732
\(394\) 0 0
\(395\) 18375.1i 2.34064i
\(396\) 0 0
\(397\) −11913.8 −1.50614 −0.753069 0.657942i \(-0.771427\pi\)
−0.753069 + 0.657942i \(0.771427\pi\)
\(398\) 0 0
\(399\) 331.790 + 459.875i 0.0416298 + 0.0577007i
\(400\) 0 0
\(401\) 7252.00 0.903111 0.451555 0.892243i \(-0.350869\pi\)
0.451555 + 0.892243i \(0.350869\pi\)
\(402\) 0 0
\(403\) −1254.60 −0.155078
\(404\) 0 0
\(405\) 1270.82 0.155920
\(406\) 0 0
\(407\) 4387.30i 0.534326i
\(408\) 0 0
\(409\) 13209.0i 1.59693i −0.602043 0.798464i \(-0.705646\pi\)
0.602043 0.798464i \(-0.294354\pi\)
\(410\) 0 0
\(411\) 4116.80i 0.494080i
\(412\) 0 0
\(413\) −6392.70 + 4612.19i −0.761656 + 0.549519i
\(414\) 0 0
\(415\) 7959.00i 0.941426i
\(416\) 0 0
\(417\) −3485.28 −0.409292
\(418\) 0 0
\(419\) 5362.97i 0.625294i 0.949869 + 0.312647i \(0.101216\pi\)
−0.949869 + 0.312647i \(0.898784\pi\)
\(420\) 0 0
\(421\) 3238.08i 0.374856i 0.982278 + 0.187428i \(0.0600151\pi\)
−0.982278 + 0.187428i \(0.939985\pi\)
\(422\) 0 0
\(423\) −4273.51 −0.491218
\(424\) 0 0
\(425\) 1722.86i 0.196638i
\(426\) 0 0
\(427\) −90.0092 124.757i −0.0102011 0.0141391i
\(428\) 0 0
\(429\) 995.482i 0.112033i
\(430\) 0 0
\(431\) 16968.0i 1.89633i −0.317779 0.948165i \(-0.602937\pi\)
0.317779 0.948165i \(-0.397063\pi\)
\(432\) 0 0
\(433\) 129.696i 0.0143944i −0.999974 0.00719722i \(-0.997709\pi\)
0.999974 0.00719722i \(-0.00229096\pi\)
\(434\) 0 0
\(435\) −3888.61 −0.428609
\(436\) 0 0
\(437\) 2072.22 0.226837
\(438\) 0 0
\(439\) −16075.4 −1.74769 −0.873847 0.486201i \(-0.838382\pi\)
−0.873847 + 0.486201i \(0.838382\pi\)
\(440\) 0 0
\(441\) 973.423 2929.51i 0.105110 0.316327i
\(442\) 0 0
\(443\) −6218.21 −0.666899 −0.333449 0.942768i \(-0.608213\pi\)
−0.333449 + 0.942768i \(0.608213\pi\)
\(444\) 0 0
\(445\) 16661.7i 1.77492i
\(446\) 0 0
\(447\) −1941.46 −0.205431
\(448\) 0 0
\(449\) −4008.24 −0.421293 −0.210647 0.977562i \(-0.567557\pi\)
−0.210647 + 0.977562i \(0.567557\pi\)
\(450\) 0 0
\(451\) 13044.1i 1.36192i
\(452\) 0 0
\(453\) −4378.58 −0.454137
\(454\) 0 0
\(455\) 1365.16 + 1892.17i 0.140659 + 0.194959i
\(456\) 0 0
\(457\) 10539.9 1.07886 0.539429 0.842031i \(-0.318640\pi\)
0.539429 + 0.842031i \(0.318640\pi\)
\(458\) 0 0
\(459\) 383.969 0.0390460
\(460\) 0 0
\(461\) 9445.51 0.954276 0.477138 0.878828i \(-0.341674\pi\)
0.477138 + 0.878828i \(0.341674\pi\)
\(462\) 0 0
\(463\) 3097.02i 0.310865i −0.987847 0.155433i \(-0.950323\pi\)
0.987847 0.155433i \(-0.0496771\pi\)
\(464\) 0 0
\(465\) 7353.84i 0.733389i
\(466\) 0 0
\(467\) 397.479i 0.0393857i −0.999806 0.0196929i \(-0.993731\pi\)
0.999806 0.0196929i \(-0.00626884\pi\)
\(468\) 0 0
\(469\) 8888.88 + 12320.4i 0.875160 + 1.21301i
\(470\) 0 0
\(471\) 5967.15i 0.583761i
\(472\) 0 0
\(473\) 6375.27 0.619736
\(474\) 0 0
\(475\) 1236.48i 0.119440i
\(476\) 0 0
\(477\) 2401.79i 0.230546i
\(478\) 0 0
\(479\) 7591.55 0.724148 0.362074 0.932149i \(-0.382069\pi\)
0.362074 + 0.932149i \(0.382069\pi\)
\(480\) 0 0
\(481\) 852.532i 0.0808152i
\(482\) 0 0
\(483\) −6600.26 9148.24i −0.621785 0.861821i
\(484\) 0 0
\(485\) 8994.87i 0.842136i
\(486\) 0 0
\(487\) 4582.61i 0.426403i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683890\pi\)
\(488\) 0 0
\(489\) 10388.2i 0.960679i
\(490\) 0 0
\(491\) 14345.7 1.31856 0.659280 0.751897i \(-0.270861\pi\)
0.659280 + 0.751897i \(0.270861\pi\)
\(492\) 0 0
\(493\) −1174.91 −0.107334
\(494\) 0 0
\(495\) −5835.00 −0.529826
\(496\) 0 0
\(497\) −1651.47 + 1191.50i −0.149051 + 0.107537i
\(498\) 0 0
\(499\) 5091.08 0.456729 0.228365 0.973576i \(-0.426662\pi\)
0.228365 + 0.973576i \(0.426662\pi\)
\(500\) 0 0
\(501\) 2478.61i 0.221030i
\(502\) 0 0
\(503\) −13924.4 −1.23431 −0.617155 0.786842i \(-0.711715\pi\)
−0.617155 + 0.786842i \(0.711715\pi\)
\(504\) 0 0
\(505\) −23781.1 −2.09554
\(506\) 0 0
\(507\) 6397.56i 0.560406i
\(508\) 0 0
\(509\) 404.309 0.0352076 0.0176038 0.999845i \(-0.494396\pi\)
0.0176038 + 0.999845i \(0.494396\pi\)
\(510\) 0 0
\(511\) −809.000 + 583.676i −0.0700353 + 0.0505290i
\(512\) 0 0
\(513\) −275.571 −0.0237169
\(514\) 0 0
\(515\) 25766.5 2.20468
\(516\) 0 0
\(517\) 19621.9 1.66919
\(518\) 0 0
\(519\) 7533.08i 0.637120i
\(520\) 0 0
\(521\) 14904.9i 1.25335i 0.779279 + 0.626677i \(0.215585\pi\)
−0.779279 + 0.626677i \(0.784415\pi\)
\(522\) 0 0
\(523\) 20930.9i 1.74999i 0.484136 + 0.874993i \(0.339134\pi\)
−0.484136 + 0.874993i \(0.660866\pi\)
\(524\) 0 0
\(525\) 5458.71 3938.34i 0.453786 0.327397i
\(526\) 0 0
\(527\) 2221.91i 0.183658i
\(528\) 0 0
\(529\) −29055.4 −2.38805
\(530\) 0 0
\(531\) 3830.69i 0.313066i
\(532\) 0 0
\(533\) 2534.71i 0.205986i
\(534\) 0 0
\(535\) 14130.8 1.14192
\(536\) 0 0
\(537\) 11517.7i 0.925559i
\(538\) 0 0
\(539\) −4469.50 + 13450.9i −0.357171 + 1.07490i
\(540\) 0 0
\(541\) 903.665i 0.0718144i 0.999355 + 0.0359072i \(0.0114321\pi\)
−0.999355 + 0.0359072i \(0.988568\pi\)
\(542\) 0 0
\(543\) 1924.60i 0.152104i
\(544\) 0 0
\(545\) 25159.2i 1.97744i
\(546\) 0 0
\(547\) −390.629 −0.0305340 −0.0152670 0.999883i \(-0.504860\pi\)
−0.0152670 + 0.999883i \(0.504860\pi\)
\(548\) 0 0
\(549\) 74.7578 0.00581163
\(550\) 0 0
\(551\) 843.226 0.0651954
\(552\) 0 0
\(553\) −17590.6 + 12691.2i −1.35267 + 0.975923i
\(554\) 0 0
\(555\) −4997.10 −0.382189
\(556\) 0 0
\(557\) 9201.82i 0.699989i 0.936752 + 0.349994i \(0.113816\pi\)
−0.936752 + 0.349994i \(0.886184\pi\)
\(558\) 0 0
\(559\) 1238.83 0.0937333
\(560\) 0 0
\(561\) −1763.00 −0.132681
\(562\) 0 0
\(563\) 12105.2i 0.906169i 0.891468 + 0.453084i \(0.149676\pi\)
−0.891468 + 0.453084i \(0.850324\pi\)
\(564\) 0 0
\(565\) 12882.7 0.959252
\(566\) 0 0
\(567\) 877.724 + 1216.56i 0.0650105 + 0.0901073i
\(568\) 0 0
\(569\) −20681.2 −1.52373 −0.761864 0.647737i \(-0.775716\pi\)
−0.761864 + 0.647737i \(0.775716\pi\)
\(570\) 0 0
\(571\) −25231.2 −1.84920 −0.924599 0.380943i \(-0.875600\pi\)
−0.924599 + 0.380943i \(0.875600\pi\)
\(572\) 0 0
\(573\) 4923.40 0.358950
\(574\) 0 0
\(575\) 24597.2i 1.78396i
\(576\) 0 0
\(577\) 6102.77i 0.440314i −0.975464 0.220157i \(-0.929343\pi\)
0.975464 0.220157i \(-0.0706570\pi\)
\(578\) 0 0
\(579\) 6735.73i 0.483467i
\(580\) 0 0
\(581\) 7619.20 5497.09i 0.544058 0.392526i
\(582\) 0 0
\(583\) 11027.9i 0.783410i
\(584\) 0 0
\(585\) −1133.85 −0.0801346
\(586\) 0 0
\(587\) 18315.2i 1.28781i 0.765103 + 0.643907i \(0.222688\pi\)
−0.765103 + 0.643907i \(0.777312\pi\)
\(588\) 0 0
\(589\) 1594.64i 0.111555i
\(590\) 0 0
\(591\) −11274.8 −0.784745
\(592\) 0 0
\(593\) 19068.9i 1.32052i −0.751038 0.660259i \(-0.770447\pi\)
0.751038 0.660259i \(-0.229553\pi\)
\(594\) 0 0
\(595\) 3351.04 2417.71i 0.230890 0.166582i
\(596\) 0 0
\(597\) 14683.6i 1.00663i
\(598\) 0 0
\(599\) 909.641i 0.0620483i 0.999519 + 0.0310241i \(0.00987688\pi\)
−0.999519 + 0.0310241i \(0.990123\pi\)
\(600\) 0 0
\(601\) 16132.6i 1.09494i 0.836824 + 0.547471i \(0.184410\pi\)
−0.836824 + 0.547471i \(0.815590\pi\)
\(602\) 0 0
\(603\) −7382.72 −0.498587
\(604\) 0 0
\(605\) 5909.33 0.397105
\(606\) 0 0
\(607\) −23364.8 −1.56235 −0.781177 0.624309i \(-0.785380\pi\)
−0.781177 + 0.624309i \(0.785380\pi\)
\(608\) 0 0
\(609\) −2685.77 3722.59i −0.178707 0.247696i
\(610\) 0 0
\(611\) 3812.90 0.252461
\(612\) 0 0
\(613\) 25176.1i 1.65882i −0.558643 0.829408i \(-0.688678\pi\)
0.558643 0.829408i \(-0.311322\pi\)
\(614\) 0 0
\(615\) −14857.2 −0.974144
\(616\) 0 0
\(617\) 28030.4 1.82895 0.914474 0.404645i \(-0.132605\pi\)
0.914474 + 0.404645i \(0.132605\pi\)
\(618\) 0 0
\(619\) 24278.8i 1.57649i −0.615361 0.788245i \(-0.710990\pi\)
0.615361 0.788245i \(-0.289010\pi\)
\(620\) 0 0
\(621\) 5481.89 0.354236
\(622\) 0 0
\(623\) 15950.3 11507.8i 1.02574 0.740048i
\(624\) 0 0
\(625\) −16091.6 −1.02986
\(626\) 0 0
\(627\) 1265.29 0.0805915
\(628\) 0 0
\(629\) −1509.84 −0.0957093
\(630\) 0 0
\(631\) 9973.29i 0.629208i 0.949223 + 0.314604i \(0.101872\pi\)
−0.949223 + 0.314604i \(0.898128\pi\)
\(632\) 0 0
\(633\) 103.563i 0.00650281i
\(634\) 0 0
\(635\) 38582.9i 2.41120i
\(636\) 0 0
\(637\) −868.504 + 2613.76i −0.0540210 + 0.162576i
\(638\) 0 0
\(639\) 989.608i 0.0612650i
\(640\) 0 0
\(641\) −17715.9 −1.09163 −0.545815 0.837906i \(-0.683780\pi\)
−0.545815 + 0.837906i \(0.683780\pi\)
\(642\) 0 0
\(643\) 3370.63i 0.206726i −0.994644 0.103363i \(-0.967040\pi\)
0.994644 0.103363i \(-0.0329603\pi\)
\(644\) 0 0
\(645\) 7261.38i 0.443281i
\(646\) 0 0
\(647\) −22938.8 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(648\) 0 0
\(649\) 17588.7i 1.06382i
\(650\) 0 0
\(651\) −7039.87 + 5079.11i −0.423831 + 0.305785i
\(652\) 0 0
\(653\) 21021.4i 1.25977i 0.776688 + 0.629885i \(0.216898\pi\)
−0.776688 + 0.629885i \(0.783102\pi\)
\(654\) 0 0
\(655\) 16123.8i 0.961847i
\(656\) 0 0
\(657\) 484.776i 0.0287868i
\(658\) 0 0
\(659\) 21351.4 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(660\) 0 0
\(661\) −5778.37 −0.340019 −0.170010 0.985442i \(-0.554380\pi\)
−0.170010 + 0.985442i \(0.554380\pi\)
\(662\) 0 0
\(663\) −342.583 −0.0200676
\(664\) 0 0
\(665\) −2405.02 + 1735.17i −0.140244 + 0.101183i
\(666\) 0 0
\(667\) −16774.2 −0.973762
\(668\) 0 0
\(669\) 7409.51i 0.428204i
\(670\) 0 0
\(671\) −343.252 −0.0197483
\(672\) 0 0
\(673\) 9562.36 0.547700 0.273850 0.961772i \(-0.411703\pi\)
0.273850 + 0.961772i \(0.411703\pi\)
\(674\) 0 0
\(675\) 3271.02i 0.186521i
\(676\) 0 0
\(677\) −25060.1 −1.42266 −0.711329 0.702860i \(-0.751906\pi\)
−0.711329 + 0.702860i \(0.751906\pi\)
\(678\) 0 0
\(679\) −8610.84 + 6212.54i −0.486677 + 0.351127i
\(680\) 0 0
\(681\) −9863.43 −0.555018
\(682\) 0 0
\(683\) −16236.9 −0.909644 −0.454822 0.890582i \(-0.650297\pi\)
−0.454822 + 0.890582i \(0.650297\pi\)
\(684\) 0 0
\(685\) 21529.7 1.20089
\(686\) 0 0
\(687\) 12142.2i 0.674313i
\(688\) 0 0
\(689\) 2142.92i 0.118489i
\(690\) 0 0
\(691\) 15848.6i 0.872516i −0.899822 0.436258i \(-0.856304\pi\)
0.899822 0.436258i \(-0.143696\pi\)
\(692\) 0 0
\(693\) −4030.09 5585.88i −0.220910 0.306190i
\(694\) 0 0
\(695\) 18227.0i 0.994805i
\(696\) 0 0
\(697\) −4488.98 −0.243949
\(698\) 0 0
\(699\) 22.2634i 0.00120469i
\(700\) 0 0
\(701\) 21638.9i 1.16589i 0.812511 + 0.582946i \(0.198100\pi\)
−0.812511 + 0.582946i \(0.801900\pi\)
\(702\) 0 0
\(703\) 1083.60 0.0581346
\(704\) 0 0
\(705\) 22349.2i 1.19393i
\(706\) 0 0
\(707\) −16425.0 22765.8i −0.873729 1.21103i
\(708\) 0 0
\(709\) 15659.1i 0.829462i 0.909944 + 0.414731i \(0.136124\pi\)
−0.909944 + 0.414731i \(0.863876\pi\)
\(710\) 0 0
\(711\) 10540.8i 0.555992i
\(712\) 0 0
\(713\) 31722.0i 1.66620i
\(714\) 0 0
\(715\) 5206.08 0.272303
\(716\) 0 0
\(717\) 8877.15 0.462376
\(718\) 0 0
\(719\) −26401.4 −1.36941 −0.684705 0.728820i \(-0.740069\pi\)
−0.684705 + 0.728820i \(0.740069\pi\)
\(720\) 0 0
\(721\) 17796.3 + 24666.4i 0.919236 + 1.27410i
\(722\) 0 0
\(723\) −16683.8 −0.858197
\(724\) 0 0
\(725\) 10009.1i 0.512728i
\(726\) 0 0
\(727\) 12097.7 0.617164 0.308582 0.951198i \(-0.400146\pi\)
0.308582 + 0.951198i \(0.400146\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 2193.97i 0.111008i
\(732\) 0 0
\(733\) 32395.5 1.63241 0.816205 0.577762i \(-0.196074\pi\)
0.816205 + 0.577762i \(0.196074\pi\)
\(734\) 0 0
\(735\) 15320.5 + 5090.72i 0.768849 + 0.255475i
\(736\) 0 0
\(737\) 33898.0 1.69423
\(738\) 0 0
\(739\) 3055.66 0.152103 0.0760516 0.997104i \(-0.475769\pi\)
0.0760516 + 0.997104i \(0.475769\pi\)
\(740\) 0 0
\(741\) 245.869 0.0121892
\(742\) 0 0
\(743\) 31219.4i 1.54149i −0.637143 0.770746i \(-0.719884\pi\)
0.637143 0.770746i \(-0.280116\pi\)
\(744\) 0 0
\(745\) 10153.3i 0.499311i
\(746\) 0 0
\(747\) 4565.65i 0.223625i
\(748\) 0 0
\(749\) 9759.79 + 13527.5i 0.476122 + 0.659925i
\(750\) 0 0
\(751\) 19939.2i 0.968830i 0.874838 + 0.484415i \(0.160968\pi\)
−0.874838 + 0.484415i \(0.839032\pi\)
\(752\) 0 0
\(753\) −389.446 −0.0188475
\(754\) 0 0
\(755\) 22898.7i 1.10380i
\(756\) 0 0
\(757\) 27284.6i 1.31001i 0.755626 + 0.655004i \(0.227333\pi\)
−0.755626 + 0.655004i \(0.772667\pi\)
\(758\) 0 0
\(759\) −25170.3 −1.20372
\(760\) 0 0
\(761\) 13346.7i 0.635765i −0.948130 0.317883i \(-0.897028\pi\)
0.948130 0.317883i \(-0.102972\pi\)
\(762\) 0 0
\(763\) −24085.1 + 17376.9i −1.14278 + 0.824488i
\(764\) 0 0
\(765\) 2008.04i 0.0949032i
\(766\) 0 0
\(767\) 3417.81i 0.160899i
\(768\) 0 0
\(769\) 7158.77i 0.335698i 0.985813 + 0.167849i \(0.0536821\pi\)
−0.985813 + 0.167849i \(0.946318\pi\)
\(770\) 0 0
\(771\) −1024.06 −0.0478347
\(772\) 0 0
\(773\) −32460.8 −1.51039 −0.755196 0.655499i \(-0.772458\pi\)
−0.755196 + 0.655499i \(0.772458\pi\)
\(774\) 0 0
\(775\) −18928.4 −0.877325
\(776\) 0 0
\(777\) −3451.37 4783.75i −0.159353 0.220870i
\(778\) 0 0
\(779\) 3221.70 0.148176
\(780\) 0 0
\(781\) 4543.81i 0.208182i
\(782\) 0 0
\(783\) 2230.69 0.101811
\(784\) 0 0
\(785\) −31206.4 −1.41886
\(786\) 0 0
\(787\) 4441.50i 0.201172i −0.994928 0.100586i \(-0.967928\pi\)
0.994928 0.100586i \(-0.0320717\pi\)
\(788\) 0 0
\(789\) −1431.53 −0.0645929
\(790\) 0 0
\(791\) 8897.74 + 12332.6i 0.399958 + 0.554359i
\(792\) 0 0
\(793\) −66.7001 −0.00298687
\(794\) 0 0
\(795\) −12560.7 −0.560353
\(796\) 0 0
\(797\) −24575.5 −1.09223 −0.546115 0.837710i \(-0.683894\pi\)
−0.546115 + 0.837710i \(0.683894\pi\)
\(798\) 0 0
\(799\) 6752.65i 0.298988i
\(800\) 0 0
\(801\) 9557.89i 0.421612i