Properties

Label 1728.3.q.e.449.2
Level $1728$
Weight $3$
Character 1728.449
Analytic conductor $47.085$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.2
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1728.449
Dual form 1728.3.q.e.1601.2

$q$-expansion

\(f(q)\) \(=\) \(q+(6.39898 + 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +O(q^{10})\) \(q+(6.39898 + 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +(5.29796 - 3.05878i) q^{11} +(8.39898 - 14.5475i) q^{13} +25.1701i q^{17} +17.5959 q^{19} +(-12.3990 - 7.15855i) q^{23} +(14.7980 + 25.6308i) q^{25} +(16.1969 - 9.35131i) q^{29} +(23.3990 - 40.5282i) q^{31} +50.2295i q^{35} +49.5959 q^{37} +(34.5000 + 19.9186i) q^{41} +(-22.0959 - 38.2713i) q^{43} +(-28.8031 + 16.6295i) q^{47} +(1.39388 - 2.41427i) q^{49} -10.1708i q^{53} +45.2020 q^{55} +(-14.2980 - 8.25493i) q^{59} +(10.6010 + 18.3615i) q^{61} +(107.490 - 62.0593i) q^{65} +(-43.4898 + 75.3265i) q^{67} -30.2555i q^{71} -48.7878 q^{73} +(36.0153 + 20.7934i) q^{77} +(-55.7929 - 96.6361i) q^{79} +(-85.0857 + 49.1243i) q^{83} +(-92.9898 + 161.063i) q^{85} +75.5103i q^{89} +114.192 q^{91} +(112.596 + 65.0073i) q^{95} +(70.2980 + 121.760i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} - 6q^{7} + O(q^{10}) \) \( 4q + 6q^{5} - 6q^{7} - 18q^{11} + 14q^{13} - 8q^{19} - 30q^{23} + 20q^{25} + 6q^{29} + 74q^{31} + 120q^{37} + 138q^{41} - 10q^{43} - 174q^{47} - 112q^{49} + 220q^{55} - 18q^{59} + 62q^{61} + 234q^{65} + 22q^{67} + 40q^{73} + 438q^{77} - 86q^{79} - 66q^{83} - 176q^{85} + 300q^{91} + 372q^{95} + 242q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 6.39898 + 3.69445i 1.27980 + 0.738891i 0.976811 0.214105i \(-0.0686834\pi\)
0.302985 + 0.952995i \(0.402017\pi\)
\(6\) 0 0
\(7\) 3.39898 + 5.88721i 0.485568 + 0.841029i 0.999862 0.0165847i \(-0.00527930\pi\)
−0.514294 + 0.857614i \(0.671946\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.29796 3.05878i 0.481633 0.278071i −0.239464 0.970905i \(-0.576972\pi\)
0.721097 + 0.692835i \(0.243638\pi\)
\(12\) 0 0
\(13\) 8.39898 14.5475i 0.646075 1.11904i −0.337977 0.941154i \(-0.609743\pi\)
0.984052 0.177881i \(-0.0569242\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 25.1701i 1.48059i 0.672279 + 0.740297i \(0.265315\pi\)
−0.672279 + 0.740297i \(0.734685\pi\)
\(18\) 0 0
\(19\) 17.5959 0.926101 0.463050 0.886332i \(-0.346755\pi\)
0.463050 + 0.886332i \(0.346755\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12.3990 7.15855i −0.539086 0.311241i 0.205622 0.978631i \(-0.434078\pi\)
−0.744708 + 0.667390i \(0.767411\pi\)
\(24\) 0 0
\(25\) 14.7980 + 25.6308i 0.591918 + 1.02523i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 16.1969 9.35131i 0.558515 0.322459i −0.194034 0.980995i \(-0.562157\pi\)
0.752549 + 0.658536i \(0.228824\pi\)
\(30\) 0 0
\(31\) 23.3990 40.5282i 0.754806 1.30736i −0.190665 0.981655i \(-0.561064\pi\)
0.945471 0.325707i \(-0.105602\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 50.2295i 1.43513i
\(36\) 0 0
\(37\) 49.5959 1.34043 0.670215 0.742167i \(-0.266202\pi\)
0.670215 + 0.742167i \(0.266202\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 34.5000 + 19.9186i 0.841463 + 0.485819i 0.857761 0.514048i \(-0.171855\pi\)
−0.0162980 + 0.999867i \(0.505188\pi\)
\(42\) 0 0
\(43\) −22.0959 38.2713i −0.513859 0.890029i −0.999871 0.0160771i \(-0.994882\pi\)
0.486012 0.873952i \(-0.338451\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −28.8031 + 16.6295i −0.612831 + 0.353818i −0.774073 0.633097i \(-0.781784\pi\)
0.161242 + 0.986915i \(0.448450\pi\)
\(48\) 0 0
\(49\) 1.39388 2.41427i 0.0284465 0.0492707i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.1708i 0.191902i −0.995386 0.0959509i \(-0.969411\pi\)
0.995386 0.0959509i \(-0.0305892\pi\)
\(54\) 0 0
\(55\) 45.2020 0.821855
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.2980 8.25493i −0.242338 0.139914i 0.373913 0.927464i \(-0.378016\pi\)
−0.616251 + 0.787550i \(0.711349\pi\)
\(60\) 0 0
\(61\) 10.6010 + 18.3615i 0.173787 + 0.301008i 0.939741 0.341887i \(-0.111066\pi\)
−0.765954 + 0.642896i \(0.777733\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 107.490 62.0593i 1.65369 0.954758i
\(66\) 0 0
\(67\) −43.4898 + 75.3265i −0.649101 + 1.12428i 0.334236 + 0.942489i \(0.391522\pi\)
−0.983338 + 0.181787i \(0.941812\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 30.2555i 0.426134i −0.977038 0.213067i \(-0.931655\pi\)
0.977038 0.213067i \(-0.0683453\pi\)
\(72\) 0 0
\(73\) −48.7878 −0.668325 −0.334163 0.942515i \(-0.608454\pi\)
−0.334163 + 0.942515i \(0.608454\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 36.0153 + 20.7934i 0.467731 + 0.270045i
\(78\) 0 0
\(79\) −55.7929 96.6361i −0.706239 1.22324i −0.966243 0.257634i \(-0.917057\pi\)
0.260004 0.965608i \(-0.416276\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −85.0857 + 49.1243i −1.02513 + 0.591859i −0.915585 0.402123i \(-0.868272\pi\)
−0.109544 + 0.993982i \(0.534939\pi\)
\(84\) 0 0
\(85\) −92.9898 + 161.063i −1.09400 + 1.89486i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 75.5103i 0.848431i 0.905561 + 0.424215i \(0.139450\pi\)
−0.905561 + 0.424215i \(0.860550\pi\)
\(90\) 0 0
\(91\) 114.192 1.25486
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 112.596 + 65.0073i 1.18522 + 0.684287i
\(96\) 0 0
\(97\) 70.2980 + 121.760i 0.724721 + 1.25525i 0.959089 + 0.283106i \(0.0913648\pi\)
−0.234367 + 0.972148i \(0.575302\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 28.1969 16.2795i 0.279178 0.161183i −0.353873 0.935293i \(-0.615136\pi\)
0.633051 + 0.774110i \(0.281802\pi\)
\(102\) 0 0
\(103\) −67.7929 + 117.421i −0.658183 + 1.14001i 0.322903 + 0.946432i \(0.395341\pi\)
−0.981086 + 0.193574i \(0.937992\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 35.3409i 0.330289i −0.986269 0.165144i \(-0.947191\pi\)
0.986269 0.165144i \(-0.0528090\pi\)
\(108\) 0 0
\(109\) −53.5959 −0.491706 −0.245853 0.969307i \(-0.579068\pi\)
−0.245853 + 0.969307i \(0.579068\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −143.076 82.6047i −1.26615 0.731015i −0.291897 0.956450i \(-0.594286\pi\)
−0.974258 + 0.225435i \(0.927620\pi\)
\(114\) 0 0
\(115\) −52.8939 91.6149i −0.459947 0.796651i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −148.182 + 85.5527i −1.24522 + 0.718930i
\(120\) 0 0
\(121\) −41.7878 + 72.3785i −0.345353 + 0.598170i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 33.9588i 0.271670i
\(126\) 0 0
\(127\) 11.9796 0.0943275 0.0471637 0.998887i \(-0.484982\pi\)
0.0471637 + 0.998887i \(0.484982\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.3082 7.68347i −0.101589 0.0586525i 0.448345 0.893861i \(-0.352014\pi\)
−0.549934 + 0.835208i \(0.685347\pi\)
\(132\) 0 0
\(133\) 59.8082 + 103.591i 0.449685 + 0.778878i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 47.7122 27.5467i 0.348265 0.201071i −0.315656 0.948874i \(-0.602225\pi\)
0.663921 + 0.747803i \(0.268891\pi\)
\(138\) 0 0
\(139\) 50.4898 87.4509i 0.363236 0.629143i −0.625255 0.780420i \(-0.715005\pi\)
0.988491 + 0.151277i \(0.0483386\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 102.762i 0.718619i
\(144\) 0 0
\(145\) 138.192 0.953047
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 187.389 + 108.189i 1.25764 + 0.726100i 0.972616 0.232419i \(-0.0746641\pi\)
0.285027 + 0.958519i \(0.407997\pi\)
\(150\) 0 0
\(151\) 76.7929 + 133.009i 0.508562 + 0.880855i 0.999951 + 0.00991488i \(0.00315606\pi\)
−0.491389 + 0.870940i \(0.663511\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 299.459 172.893i 1.93199 1.11544i
\(156\) 0 0
\(157\) 40.9847 70.9876i 0.261049 0.452150i −0.705472 0.708738i \(-0.749265\pi\)
0.966521 + 0.256588i \(0.0825983\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 97.3271i 0.604516i
\(162\) 0 0
\(163\) −55.2122 −0.338725 −0.169363 0.985554i \(-0.554171\pi\)
−0.169363 + 0.985554i \(0.554171\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 133.803 + 77.2512i 0.801216 + 0.462582i 0.843896 0.536507i \(-0.180256\pi\)
−0.0426802 + 0.999089i \(0.513590\pi\)
\(168\) 0 0
\(169\) −56.5857 98.0093i −0.334827 0.579937i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.8031 13.1654i 0.131810 0.0761003i −0.432645 0.901564i \(-0.642420\pi\)
0.564455 + 0.825464i \(0.309086\pi\)
\(174\) 0 0
\(175\) −100.596 + 174.237i −0.574834 + 0.995641i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 266.700i 1.48995i −0.667094 0.744973i \(-0.732462\pi\)
0.667094 0.744973i \(-0.267538\pi\)
\(180\) 0 0
\(181\) 58.4041 0.322674 0.161337 0.986899i \(-0.448419\pi\)
0.161337 + 0.986899i \(0.448419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 317.363 + 183.230i 1.71548 + 0.990431i
\(186\) 0 0
\(187\) 76.9898 + 133.350i 0.411710 + 0.713103i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 99.5602 57.4811i 0.521258 0.300948i −0.216191 0.976351i \(-0.569364\pi\)
0.737449 + 0.675403i \(0.236030\pi\)
\(192\) 0 0
\(193\) −108.490 + 187.910i −0.562123 + 0.973626i 0.435188 + 0.900340i \(0.356682\pi\)
−0.997311 + 0.0732863i \(0.976651\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 171.105i 0.868555i −0.900779 0.434278i \(-0.857004\pi\)
0.900779 0.434278i \(-0.142996\pi\)
\(198\) 0 0
\(199\) −62.0000 −0.311558 −0.155779 0.987792i \(-0.549789\pi\)
−0.155779 + 0.987792i \(0.549789\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 110.106 + 63.5698i 0.542395 + 0.313152i
\(204\) 0 0
\(205\) 147.177 + 254.917i 0.717934 + 1.24350i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 93.2225 53.8220i 0.446040 0.257522i
\(210\) 0 0
\(211\) −64.7020 + 112.067i −0.306645 + 0.531124i −0.977626 0.210350i \(-0.932540\pi\)
0.670981 + 0.741474i \(0.265873\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 326.529i 1.51874i
\(216\) 0 0
\(217\) 318.131 1.46604
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 366.161 + 211.403i 1.65684 + 0.956576i
\(222\) 0 0
\(223\) −49.1867 85.1939i −0.220568 0.382036i 0.734412 0.678704i \(-0.237458\pi\)
−0.954981 + 0.296668i \(0.904125\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −383.651 + 221.501i −1.69009 + 0.975775i −0.735659 + 0.677352i \(0.763127\pi\)
−0.954434 + 0.298423i \(0.903539\pi\)
\(228\) 0 0
\(229\) −56.0051 + 97.0037i −0.244564 + 0.423597i −0.962009 0.273018i \(-0.911978\pi\)
0.717445 + 0.696615i \(0.245311\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 80.8526i 0.347007i 0.984833 + 0.173503i \(0.0555088\pi\)
−0.984833 + 0.173503i \(0.944491\pi\)
\(234\) 0 0
\(235\) −245.747 −1.04573
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −37.3888 21.5864i −0.156438 0.0903197i 0.419737 0.907646i \(-0.362122\pi\)
−0.576176 + 0.817326i \(0.695456\pi\)
\(240\) 0 0
\(241\) 140.904 + 244.053i 0.584664 + 1.01267i 0.994917 + 0.100696i \(0.0321071\pi\)
−0.410253 + 0.911972i \(0.634560\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 17.8388 10.2992i 0.0728113 0.0420376i
\(246\) 0 0
\(247\) 147.788 255.976i 0.598331 1.03634i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.5131i 0.0618050i −0.999522 0.0309025i \(-0.990162\pi\)
0.999522 0.0309025i \(-0.00983814\pi\)
\(252\) 0 0
\(253\) −87.5857 −0.346189
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 312.035 + 180.153i 1.21414 + 0.700986i 0.963659 0.267135i \(-0.0860770\pi\)
0.250484 + 0.968121i \(0.419410\pi\)
\(258\) 0 0
\(259\) 168.576 + 291.981i 0.650871 + 1.12734i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 42.4296 24.4967i 0.161329 0.0931435i −0.417162 0.908832i \(-0.636975\pi\)
0.578491 + 0.815689i \(0.303642\pi\)
\(264\) 0 0
\(265\) 37.5755 65.0827i 0.141794 0.245595i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 281.700i 1.04721i 0.851961 + 0.523606i \(0.175413\pi\)
−0.851961 + 0.523606i \(0.824587\pi\)
\(270\) 0 0
\(271\) −89.5959 −0.330612 −0.165306 0.986242i \(-0.552861\pi\)
−0.165306 + 0.986242i \(0.552861\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 156.798 + 90.5273i 0.570174 + 0.329190i
\(276\) 0 0
\(277\) 42.1969 + 73.0872i 0.152336 + 0.263853i 0.932086 0.362238i \(-0.117987\pi\)
−0.779750 + 0.626091i \(0.784654\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.2673 13.4334i 0.0828019 0.0478057i −0.458027 0.888938i \(-0.651444\pi\)
0.540829 + 0.841132i \(0.318110\pi\)
\(282\) 0 0
\(283\) 90.4898 156.733i 0.319752 0.553827i −0.660684 0.750664i \(-0.729734\pi\)
0.980436 + 0.196837i \(0.0630671\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 270.811i 0.943594i
\(288\) 0 0
\(289\) −344.535 −1.19216
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −407.985 235.550i −1.39244 0.803925i −0.398854 0.917014i \(-0.630592\pi\)
−0.993585 + 0.113089i \(0.963925\pi\)
\(294\) 0 0
\(295\) −60.9949 105.646i −0.206762 0.358123i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −208.278 + 120.249i −0.696580 + 0.402171i
\(300\) 0 0
\(301\) 150.207 260.166i 0.499027 0.864340i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 156.660i 0.513639i
\(306\) 0 0
\(307\) −464.747 −1.51383 −0.756917 0.653511i \(-0.773295\pi\)
−0.756917 + 0.653511i \(0.773295\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 218.348 + 126.063i 0.702083 + 0.405348i 0.808123 0.589014i \(-0.200484\pi\)
−0.106039 + 0.994362i \(0.533817\pi\)
\(312\) 0 0
\(313\) −98.1061 169.925i −0.313438 0.542891i 0.665666 0.746250i \(-0.268147\pi\)
−0.979104 + 0.203359i \(0.934814\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 98.9847 57.1488i 0.312255 0.180280i −0.335680 0.941976i \(-0.608966\pi\)
0.647935 + 0.761696i \(0.275633\pi\)
\(318\) 0 0
\(319\) 57.2071 99.0857i 0.179333 0.310613i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 442.891i 1.37118i
\(324\) 0 0
\(325\) 497.151 1.52970
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −195.802 113.046i −0.595143 0.343606i
\(330\) 0 0
\(331\) 27.2980 + 47.2815i 0.0824712 + 0.142844i 0.904311 0.426875i \(-0.140385\pi\)
−0.821840 + 0.569719i \(0.807052\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −556.581 + 321.342i −1.66143 + 0.959230i
\(336\) 0 0
\(337\) 118.884 205.913i 0.352771 0.611016i −0.633963 0.773363i \(-0.718573\pi\)
0.986734 + 0.162347i \(0.0519063\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 286.289i 0.839558i
\(342\) 0 0
\(343\) 352.051 1.02639
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 108.349 + 62.5553i 0.312245 + 0.180275i 0.647931 0.761699i \(-0.275635\pi\)
−0.335686 + 0.941974i \(0.608968\pi\)
\(348\) 0 0
\(349\) −269.985 467.627i −0.773595 1.33991i −0.935581 0.353113i \(-0.885123\pi\)
0.161986 0.986793i \(-0.448210\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −254.490 + 146.930i −0.720934 + 0.416232i −0.815096 0.579325i \(-0.803316\pi\)
0.0941622 + 0.995557i \(0.469983\pi\)
\(354\) 0 0
\(355\) 111.778 193.604i 0.314866 0.545364i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 422.550i 1.17702i −0.808490 0.588509i \(-0.799715\pi\)
0.808490 0.588509i \(-0.200285\pi\)
\(360\) 0 0
\(361\) −51.3837 −0.142337
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −312.192 180.244i −0.855320 0.493819i
\(366\) 0 0
\(367\) 131.358 + 227.519i 0.357924 + 0.619943i 0.987614 0.156904i \(-0.0501513\pi\)
−0.629690 + 0.776847i \(0.716818\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 59.8775 34.5703i 0.161395 0.0931814i
\(372\) 0 0
\(373\) 60.9847 105.629i 0.163498 0.283187i −0.772623 0.634865i \(-0.781056\pi\)
0.936121 + 0.351679i \(0.114389\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 314.166i 0.833331i
\(378\) 0 0
\(379\) 641.151 1.69169 0.845846 0.533428i \(-0.179096\pi\)
0.845846 + 0.533428i \(0.179096\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −640.681 369.897i −1.67280 0.965789i −0.966063 0.258308i \(-0.916835\pi\)
−0.706733 0.707481i \(-0.749832\pi\)
\(384\) 0 0
\(385\) 153.641 + 266.114i 0.399067 + 0.691204i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 75.7520 43.7355i 0.194735 0.112430i −0.399462 0.916750i \(-0.630803\pi\)
0.594197 + 0.804319i \(0.297470\pi\)
\(390\) 0 0
\(391\) 180.182 312.084i 0.460823 0.798168i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 824.496i 2.08733i
\(396\) 0 0
\(397\) −483.090 −1.21685 −0.608425 0.793611i \(-0.708199\pi\)
−0.608425 + 0.793611i \(0.708199\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −317.682 183.414i −0.792224 0.457390i 0.0485212 0.998822i \(-0.484549\pi\)
−0.840745 + 0.541432i \(0.817882\pi\)
\(402\) 0 0
\(403\) −393.055 680.791i −0.975323 1.68931i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 262.757 151.703i 0.645595 0.372734i
\(408\) 0 0
\(409\) −267.641 + 463.567i −0.654379 + 1.13342i 0.327671 + 0.944792i \(0.393736\pi\)
−0.982049 + 0.188625i \(0.939597\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 112.233i 0.271751i
\(414\) 0 0
\(415\) −725.949 −1.74927
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 605.620 + 349.655i 1.44539 + 0.834499i 0.998202 0.0599386i \(-0.0190905\pi\)
0.447193 + 0.894438i \(0.352424\pi\)
\(420\) 0 0
\(421\) −180.772 313.107i −0.429388 0.743722i 0.567431 0.823421i \(-0.307937\pi\)
−0.996819 + 0.0796989i \(0.974604\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −645.131 + 372.466i −1.51795 + 0.876391i
\(426\) 0 0
\(427\) −72.0653 + 124.821i −0.168771 + 0.292320i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 463.747i 1.07598i 0.842952 + 0.537989i \(0.180816\pi\)
−0.842952 + 0.537989i \(0.819184\pi\)
\(432\) 0 0
\(433\) −689.514 −1.59241 −0.796206 0.605026i \(-0.793163\pi\)
−0.796206 + 0.605026i \(0.793163\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −218.171 125.961i −0.499248 0.288241i
\(438\) 0 0
\(439\) 310.772 + 538.274i 0.707910 + 1.22614i 0.965631 + 0.259917i \(0.0836951\pi\)
−0.257721 + 0.966219i \(0.582972\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −698.843 + 403.477i −1.57752 + 0.910784i −0.582320 + 0.812960i \(0.697855\pi\)
−0.995204 + 0.0978236i \(0.968812\pi\)
\(444\) 0 0
\(445\) −278.969 + 483.189i −0.626897 + 1.08582i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 317.554i 0.707248i −0.935388 0.353624i \(-0.884949\pi\)
0.935388 0.353624i \(-0.115051\pi\)
\(450\) 0 0
\(451\) 243.706 0.540368
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 730.711 + 421.876i 1.60596 + 0.927201i
\(456\) 0 0
\(457\) −285.843 495.094i −0.625477 1.08336i −0.988448 0.151557i \(-0.951571\pi\)
0.362972 0.931800i \(-0.381762\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −478.550 + 276.291i −1.03807 + 0.599330i −0.919286 0.393591i \(-0.871233\pi\)
−0.118784 + 0.992920i \(0.537899\pi\)
\(462\) 0 0
\(463\) 60.1663 104.211i 0.129949 0.225078i −0.793708 0.608299i \(-0.791852\pi\)
0.923657 + 0.383221i \(0.125185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 880.440i 1.88531i −0.333767 0.942656i \(-0.608320\pi\)
0.333767 0.942656i \(-0.391680\pi\)
\(468\) 0 0
\(469\) −591.284 −1.26073
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −234.127 135.173i −0.494982 0.285778i
\(474\) 0 0
\(475\) 260.384 + 450.998i 0.548176 + 0.949469i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −593.793 + 342.826i −1.23965 + 0.715713i −0.969023 0.246971i \(-0.920565\pi\)
−0.270628 + 0.962684i \(0.587231\pi\)
\(480\) 0 0
\(481\) 416.555 721.495i 0.866019 1.49999i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1038.85i 2.14196i
\(486\) 0 0
\(487\) 391.131 0.803143 0.401571 0.915828i \(-0.368464\pi\)
0.401571 + 0.915828i \(0.368464\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 166.469 + 96.1111i 0.339042 + 0.195746i 0.659848 0.751399i \(-0.270621\pi\)
−0.320807 + 0.947145i \(0.603954\pi\)
\(492\) 0 0
\(493\) 235.373 + 407.679i 0.477431 + 0.826935i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 178.120 102.838i 0.358391 0.206917i
\(498\) 0 0
\(499\) 304.692 527.742i 0.610605 1.05760i −0.380534 0.924767i \(-0.624260\pi\)
0.991139 0.132832i \(-0.0424070\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 232.130i 0.461491i −0.973014 0.230746i \(-0.925883\pi\)
0.973014 0.230746i \(-0.0741165\pi\)
\(504\) 0 0
\(505\) 240.576 0.476387
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −223.136 128.827i −0.438381 0.253099i 0.264530 0.964377i \(-0.414783\pi\)
−0.702910 + 0.711278i \(0.748117\pi\)
\(510\) 0 0
\(511\) −165.829 287.224i −0.324518 0.562081i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −867.610 + 500.915i −1.68468 + 0.972650i
\(516\) 0 0
\(517\) −101.732 + 176.204i −0.196773 + 0.340821i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 484.088i 0.929152i −0.885533 0.464576i \(-0.846207\pi\)
0.885533 0.464576i \(-0.153793\pi\)
\(522\) 0 0
\(523\) 644.384 1.23209 0.616046 0.787711i \(-0.288734\pi\)
0.616046 + 0.787711i \(0.288734\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1020.10 + 588.955i 1.93567 + 1.11756i
\(528\) 0 0
\(529\) −162.010 280.610i −0.306257 0.530454i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 579.530 334.592i 1.08730 0.627752i
\(534\) 0 0
\(535\) 130.565 226.146i 0.244047 0.422702i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.0542i 0.0316405i
\(540\) 0 0
\(541\) 332.302 0.614237 0.307118 0.951671i \(-0.400635\pi\)
0.307118 + 0.951671i \(0.400635\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −342.959 198.008i −0.629283 0.363317i
\(546\) 0 0
\(547\) −157.329 272.501i −0.287621 0.498174i 0.685621 0.727959i \(-0.259531\pi\)
−0.973241 + 0.229785i \(0.926198\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 285.000 164.545i 0.517241 0.298629i
\(552\) 0 0
\(553\) 379.278 656.928i 0.685855 1.18793i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 664.080i 1.19224i −0.802894 0.596122i \(-0.796707\pi\)
0.802894 0.596122i \(-0.203293\pi\)
\(558\) 0 0
\(559\) −742.333 −1.32797
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −211.024 121.835i −0.374821 0.216403i 0.300741 0.953706i \(-0.402766\pi\)
−0.675563 + 0.737302i \(0.736099\pi\)
\(564\) 0 0
\(565\) −610.358 1057.17i −1.08028 1.87110i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 502.155 289.919i 0.882522 0.509524i 0.0110330 0.999939i \(-0.496488\pi\)
0.871489 + 0.490415i \(0.163155\pi\)
\(570\) 0 0
\(571\) −356.843 + 618.070i −0.624944 + 1.08243i 0.363608 + 0.931552i \(0.381545\pi\)
−0.988552 + 0.150882i \(0.951789\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 423.728i 0.736918i
\(576\) 0 0
\(577\) 829.433 1.43749 0.718746 0.695273i \(-0.244717\pi\)
0.718746 + 0.695273i \(0.244717\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −578.409 333.945i −0.995541 0.574776i
\(582\) 0 0
\(583\) −31.1102 53.8844i −0.0533623 0.0924261i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 777.480 448.878i 1.32450 0.764699i 0.340054 0.940406i \(-0.389555\pi\)
0.984442 + 0.175707i \(0.0562212\pi\)
\(588\) 0 0
\(589\) 411.727 713.131i 0.699026 1.21075i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 378.065i 0.637547i 0.947831 + 0.318774i \(0.103271\pi\)
−0.947831 + 0.318774i \(0.896729\pi\)
\(594\) 0 0
\(595\) −1264.28 −2.12484
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −822.438 474.835i −1.37302 0.792712i −0.381711 0.924282i \(-0.624665\pi\)
−0.991307 + 0.131569i \(0.957998\pi\)
\(600\) 0 0
\(601\) −252.308 437.011i −0.419814 0.727139i 0.576107 0.817375i \(-0.304571\pi\)
−0.995920 + 0.0902356i \(0.971238\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −534.798 + 308.766i −0.883964 + 0.510357i
\(606\) 0 0
\(607\) −429.954 + 744.702i −0.708326 + 1.22686i 0.257151 + 0.966371i \(0.417216\pi\)
−0.965478 + 0.260486i \(0.916117\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 558.682i 0.914373i
\(612\) 0 0
\(613\) 655.253 1.06893 0.534464 0.845191i \(-0.320513\pi\)
0.534464 + 0.845191i \(0.320513\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 147.227 + 85.0013i 0.238617 + 0.137765i 0.614541 0.788885i \(-0.289341\pi\)
−0.375924 + 0.926650i \(0.622675\pi\)
\(618\) 0 0
\(619\) 270.531 + 468.573i 0.437045 + 0.756983i 0.997460 0.0712282i \(-0.0226918\pi\)
−0.560415 + 0.828212i \(0.689359\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −444.545 + 256.658i −0.713555 + 0.411971i
\(624\) 0 0
\(625\) 244.490 423.469i 0.391184 0.677550i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1248.33i 1.98463i
\(630\) 0 0
\(631\) 260.788 0.413293 0.206646 0.978416i \(-0.433745\pi\)
0.206646 + 0.978416i \(0.433745\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 76.6571 + 44.2580i 0.120720 + 0.0696977i
\(636\) 0 0
\(637\) −23.4143 40.5547i −0.0367571 0.0636652i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −585.418 + 337.991i −0.913289 + 0.527288i −0.881488 0.472206i \(-0.843458\pi\)
−0.0318012 + 0.999494i \(0.510124\pi\)
\(642\) 0 0
\(643\) −378.318 + 655.267i −0.588364 + 1.01908i 0.406082 + 0.913837i \(0.366895\pi\)
−0.994447 + 0.105241i \(0.966439\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 554.770i 0.857450i −0.903435 0.428725i \(-0.858963\pi\)
0.903435 0.428725i \(-0.141037\pi\)
\(648\) 0 0
\(649\) −101.000 −0.155624
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 174.499 + 100.747i 0.267227 + 0.154283i 0.627627 0.778514i \(-0.284026\pi\)
−0.360400 + 0.932798i \(0.617360\pi\)
\(654\) 0 0
\(655\) −56.7724 98.3328i −0.0866755 0.150126i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −89.7122 + 51.7954i −0.136134 + 0.0785969i −0.566520 0.824048i \(-0.691711\pi\)
0.430386 + 0.902645i \(0.358377\pi\)
\(660\) 0 0
\(661\) 109.207 189.152i 0.165215 0.286161i −0.771517 0.636209i \(-0.780502\pi\)
0.936732 + 0.350048i \(0.113835\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 883.834i 1.32907i
\(666\) 0 0
\(667\) −267.767 −0.401450
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 112.328 + 64.8523i 0.167403 + 0.0966503i
\(672\) 0 0
\(673\) −394.429 683.170i −0.586075 1.01511i −0.994740 0.102428i \(-0.967339\pi\)
0.408665 0.912684i \(-0.365994\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −634.550 + 366.358i −0.937297 + 0.541149i −0.889112 0.457690i \(-0.848677\pi\)
−0.0481850 + 0.998838i \(0.515344\pi\)
\(678\) 0 0
\(679\) −477.883 + 827.717i −0.703804 + 1.21902i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1353.33i 1.98145i −0.135883 0.990725i \(-0.543387\pi\)
0.135883 0.990725i \(-0.456613\pi\)
\(684\) 0 0
\(685\) 407.080 0.594277
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −147.959 85.4243i −0.214745 0.123983i
\(690\) 0 0
\(691\) −368.257 637.840i −0.532934 0.923068i −0.999260 0.0384555i \(-0.987756\pi\)
0.466327 0.884613i \(-0.345577\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 646.166 373.064i 0.929736 0.536783i
\(696\) 0 0
\(697\) −501.353 + 868.369i −0.719301 + 1.24587i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1068.34i 1.52403i −0.647561 0.762014i \(-0.724211\pi\)
0.647561 0.762014i \(-0.275789\pi\)
\(702\) 0 0
\(703\) 872.686 1.24137
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 191.682 + 110.667i 0.271120 + 0.156531i
\(708\) 0 0
\(709\) 136.944 + 237.194i 0.193151 + 0.334547i 0.946293 0.323311i \(-0.104796\pi\)
−0.753142 + 0.657858i \(0.771463\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −580.247 + 335.006i −0.813811 + 0.469854i
\(714\) 0 0
\(715\) 379.651 657.575i 0.530980 0.919685i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 654.423i 0.910185i 0.890444 + 0.455092i \(0.150394\pi\)
−0.890444 + 0.455092i \(0.849606\pi\)
\(720\) 0 0
\(721\) −921.706 −1.27837
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 479.363 + 276.761i 0.661191 + 0.381739i
\(726\) 0 0
\(727\) −583.166 1010.07i −0.802155 1.38937i −0.918195 0.396128i \(-0.870354\pi\)
0.116041 0.993244i \(-0.462980\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 963.292 556.157i 1.31777 0.760816i
\(732\) 0 0
\(733\) 439.146 760.623i 0.599108 1.03768i −0.393845 0.919177i \(-0.628855\pi\)
0.992953 0.118508i \(-0.0378112\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 532.103i 0.721984i
\(738\) 0 0
\(739\) 593.151 0.802640 0.401320 0.915938i \(-0.368552\pi\)
0.401320 + 0.915938i \(0.368552\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 47.0561 + 27.1679i 0.0633326 + 0.0365651i 0.531332 0.847164i \(-0.321692\pi\)
−0.467999 + 0.883729i \(0.655025\pi\)
\(744\) 0 0
\(745\) 799.398 + 1384.60i 1.07302 + 1.85852i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 208.059 120.123i 0.277783 0.160378i
\(750\) 0 0
\(751\) −455.570 + 789.071i −0.606618 + 1.05069i 0.385175 + 0.922844i \(0.374141\pi\)
−0.991793 + 0.127850i \(0.959192\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1134.83i 1.50309i
\(756\) 0 0
\(757\) −1272.22 −1.68061 −0.840304 0.542115i \(-0.817624\pi\)
−0.840304 + 0.542115i \(0.817624\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −399.480 230.640i −0.524940 0.303074i 0.214013 0.976831i \(-0.431346\pi\)
−0.738954 + 0.673756i \(0.764680\pi\)
\(762\) 0 0
\(763\) −182.171 315.530i −0.238757 0.413539i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −240.177 + 138.666i −0.313138 + 0.180790i
\(768\) 0 0
\(769\) −269.439 + 466.682i −0.350376 + 0.606868i −0.986315 0.164871i \(-0.947279\pi\)
0.635940 + 0.771739i \(0.280613\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1036.29i 1.34061i −0.742087 0.670304i \(-0.766164\pi\)
0.742087 0.670304i \(-0.233836\pi\)
\(774\) 0 0
\(775\) 1385.03 1.78713
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 607.059 + 350.486i 0.779280 + 0.449918i
\(780\) 0 0
\(781\) −92.5449 160.292i −0.118495 0.205240i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 524.520 302.832i 0.668179 0.385773i
\(786\) 0 0
\(787\) −706.096 + 1222.99i −0.897199 + 1.55399i −0.0661406 + 0.997810i \(0.521069\pi\)
−0.831059 + 0.556185i \(0.812265\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1123.09i 1.41983i
\(792\) 0 0
\(793\) 356.151 0.449119
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 60.4602 + 34.9067i 0.0758597 + 0.0437976i 0.537450 0.843296i \(-0.319388\pi\)
−0.461590 + 0.887093i \(0.652721\pi\)
\(798\) 0 0
\(799\) −418.565 724.976i −0.523861 0.907355i
\(800\) 0 0
\(801\) 0 0