Properties

Label 4212.2.i.b.1405.1
Level $4212$
Weight $2$
Character 4212.1405
Analytic conductor $33.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4212.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.6329893314\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1405.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 4212.1405
Dual form 4212.2.i.b.2809.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.00000 + 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(-2.00000 + 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-2.00000 - 3.46410i) q^{11} +(-0.500000 + 0.866025i) q^{13} -2.00000 q^{17} -2.00000 q^{19} +(-5.50000 - 9.52628i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(5.00000 - 8.66025i) q^{31} -8.00000 q^{35} +10.0000 q^{37} +(4.00000 - 6.92820i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +10.0000 q^{53} +16.0000 q^{55} +(-4.00000 + 6.92820i) q^{59} +(7.00000 + 12.1244i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(-1.00000 + 1.73205i) q^{67} -16.0000 q^{71} -10.0000 q^{73} +(4.00000 - 6.92820i) q^{77} +(8.00000 + 13.8564i) q^{79} +(4.00000 - 6.92820i) q^{85} +4.00000 q^{89} -2.00000 q^{91} +(4.00000 - 6.92820i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - 4q^{5} + 2q^{7} - 4q^{11} - q^{13} - 4q^{17} - 4q^{19} - 11q^{25} - 6q^{29} + 10q^{31} - 16q^{35} + 20q^{37} + 8q^{41} - 4q^{43} - 4q^{47} + 3q^{49} + 20q^{53} + 32q^{55} - 8q^{59} + 14q^{61} - 4q^{65} - 2q^{67} - 32q^{71} - 20q^{73} + 8q^{77} + 16q^{79} + 8q^{85} + 8q^{89} - 4q^{91} + 8q^{95} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times\).

\(n\) \(2107\) \(3485\) \(3889\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.00000 + 3.46410i −0.894427 + 1.54919i −0.0599153 + 0.998203i \(0.519083\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 3.46410i −0.603023 1.04447i −0.992361 0.123371i \(-0.960630\pi\)
0.389338 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −5.50000 9.52628i −1.10000 1.90526i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 5.19615i −0.557086 0.964901i −0.997738 0.0672232i \(-0.978586\pi\)
0.440652 0.897678i \(-0.354747\pi\)
\(30\) 0 0
\(31\) 5.00000 8.66025i 0.898027 1.55543i 0.0680129 0.997684i \(-0.478334\pi\)
0.830014 0.557743i \(-0.188333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 6.92820i 0.624695 1.08200i −0.363905 0.931436i \(-0.618557\pi\)
0.988600 0.150567i \(-0.0481100\pi\)
\(42\) 0 0
\(43\) −2.00000 3.46410i −0.304997 0.528271i 0.672264 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 3.46410i −0.291730 0.505291i 0.682489 0.730896i \(-0.260898\pi\)
−0.974219 + 0.225605i \(0.927564\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 16.0000 2.15744
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 + 6.92820i −0.520756 + 0.901975i 0.478953 + 0.877841i \(0.341016\pi\)
−0.999709 + 0.0241347i \(0.992317\pi\)
\(60\) 0 0
\(61\) 7.00000 + 12.1244i 0.896258 + 1.55236i 0.832240 + 0.554416i \(0.187058\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 3.46410i −0.248069 0.429669i
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 6.92820i 0.455842 0.789542i
\(78\) 0 0
\(79\) 8.00000 + 13.8564i 0.900070 + 1.55897i 0.827401 + 0.561611i \(0.189818\pi\)
0.0726692 + 0.997356i \(0.476848\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 4.00000 6.92820i 0.433861 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 6.92820i 0.410391 0.710819i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.00000 + 8.66025i 0.497519 + 0.861727i 0.999996 0.00286291i \(-0.000911295\pi\)
−0.502477 + 0.864590i \(0.667578\pi\)
\(102\) 0 0
\(103\) 4.00000 6.92820i 0.394132 0.682656i −0.598858 0.800855i \(-0.704379\pi\)
0.992990 + 0.118199i \(0.0377120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.00000 5.19615i 0.282216 0.488813i −0.689714 0.724082i \(-0.742264\pi\)
0.971930 + 0.235269i \(0.0755971\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 3.46410i −0.183340 0.317554i
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 3.46410i 0.174741 0.302660i −0.765331 0.643637i \(-0.777425\pi\)
0.940072 + 0.340977i \(0.110758\pi\)
\(132\) 0 0
\(133\) −2.00000 3.46410i −0.173422 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000 + 6.92820i 0.341743 + 0.591916i 0.984757 0.173939i \(-0.0556494\pi\)
−0.643013 + 0.765855i \(0.722316\pi\)
\(138\) 0 0
\(139\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 24.0000 1.99309
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) −9.00000 15.5885i −0.732410 1.26857i −0.955851 0.293853i \(-0.905062\pi\)
0.223441 0.974717i \(-0.428271\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 20.0000 + 34.6410i 1.60644 + 2.78243i
\(156\) 0 0
\(157\) −1.00000 + 1.73205i −0.0798087 + 0.138233i −0.903167 0.429289i \(-0.858764\pi\)
0.823359 + 0.567521i \(0.192098\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 + 10.3923i −0.464294 + 0.804181i −0.999169 0.0407502i \(-0.987025\pi\)
0.534875 + 0.844931i \(0.320359\pi\)
\(168\) 0 0
\(169\) −0.500000 0.866025i −0.0384615 0.0666173i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.00000 + 1.73205i 0.0760286 + 0.131685i 0.901533 0.432710i \(-0.142443\pi\)
−0.825505 + 0.564396i \(0.809109\pi\)
\(174\) 0 0
\(175\) 11.0000 19.0526i 0.831522 1.44024i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −20.0000 + 34.6410i −1.47043 + 2.54686i
\(186\) 0 0
\(187\) 4.00000 + 6.92820i 0.292509 + 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 6.92820i −0.289430 0.501307i 0.684244 0.729253i \(-0.260132\pi\)
−0.973674 + 0.227946i \(0.926799\pi\)
\(192\) 0 0
\(193\) 7.00000 12.1244i 0.503871 0.872730i −0.496119 0.868255i \(-0.665242\pi\)
0.999990 0.00447566i \(-0.00142465\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000 10.3923i 0.421117 0.729397i
\(204\) 0 0
\(205\) 16.0000 + 27.7128i 1.11749 + 1.93555i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.00000 + 6.92820i 0.276686 + 0.479234i
\(210\) 0 0
\(211\) 6.00000 10.3923i 0.413057 0.715436i −0.582165 0.813070i \(-0.697794\pi\)
0.995222 + 0.0976347i \(0.0311277\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) 20.0000 1.35769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.00000 1.73205i 0.0672673 0.116510i
\(222\) 0 0
\(223\) −7.00000 12.1244i −0.468755 0.811907i 0.530607 0.847618i \(-0.321964\pi\)
−0.999362 + 0.0357107i \(0.988630\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.00000 3.46410i −0.132745 0.229920i 0.791989 0.610535i \(-0.209046\pi\)
−0.924734 + 0.380615i \(0.875712\pi\)
\(228\) 0 0
\(229\) 1.00000 1.73205i 0.0660819 0.114457i −0.831092 0.556136i \(-0.812283\pi\)
0.897173 + 0.441679i \(0.145617\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 + 20.7846i −0.776215 + 1.34444i 0.157893 + 0.987456i \(0.449530\pi\)
−0.934109 + 0.356988i \(0.883804\pi\)
\(240\) 0 0
\(241\) 1.00000 + 1.73205i 0.0644157 + 0.111571i 0.896435 0.443176i \(-0.146148\pi\)
−0.832019 + 0.554747i \(0.812815\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.00000 + 10.3923i 0.383326 + 0.663940i
\(246\) 0 0
\(247\) 1.00000 1.73205i 0.0636285 0.110208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.0000 25.9808i 0.935674 1.62064i 0.162247 0.986750i \(-0.448126\pi\)
0.773427 0.633885i \(-0.218541\pi\)
\(258\) 0 0
\(259\) 10.0000 + 17.3205i 0.621370 + 1.07624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.00000 13.8564i −0.493301 0.854423i 0.506669 0.862141i \(-0.330877\pi\)
−0.999970 + 0.00771799i \(0.997543\pi\)
\(264\) 0 0
\(265\) −20.0000 + 34.6410i −1.22859 + 2.12798i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.0000 + 38.1051i −1.32665 + 2.29783i
\(276\) 0 0
\(277\) −11.0000 19.0526i −0.660926 1.14476i −0.980373 0.197153i \(-0.936830\pi\)
0.319447 0.947604i \(-0.396503\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 6.92820i −0.238620 0.413302i 0.721699 0.692207i \(-0.243362\pi\)
−0.960319 + 0.278906i \(0.910028\pi\)
\(282\) 0 0
\(283\) 8.00000 13.8564i 0.475551 0.823678i −0.524057 0.851683i \(-0.675582\pi\)
0.999608 + 0.0280052i \(0.00891551\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.0000 0.944450
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.00000 6.92820i 0.233682 0.404750i −0.725206 0.688531i \(-0.758256\pi\)
0.958889 + 0.283782i \(0.0915890\pi\)
\(294\) 0 0
\(295\) −16.0000 27.7128i −0.931556 1.61350i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 6.92820i 0.230556 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −56.0000 −3.20655
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 20.7846i 0.680458 1.17859i −0.294384 0.955687i \(-0.595114\pi\)
0.974841 0.222900i \(-0.0715523\pi\)
\(312\) 0 0
\(313\) 3.00000 + 5.19615i 0.169570 + 0.293704i 0.938269 0.345907i \(-0.112429\pi\)
−0.768699 + 0.639611i \(0.779095\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 10.3923i −0.336994 0.583690i 0.646872 0.762598i \(-0.276077\pi\)
−0.983866 + 0.178908i \(0.942743\pi\)
\(318\) 0 0
\(319\) −12.0000 + 20.7846i −0.671871 + 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 6.92820i 0.220527 0.381964i
\(330\) 0 0
\(331\) −1.00000 1.73205i −0.0549650 0.0952021i 0.837234 0.546845i \(-0.184171\pi\)
−0.892199 + 0.451643i \(0.850838\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 6.92820i −0.218543 0.378528i
\(336\) 0 0
\(337\) 15.0000 25.9808i 0.817102 1.41526i −0.0907066 0.995878i \(-0.528913\pi\)
0.907809 0.419385i \(-0.137754\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −40.0000 −2.16612
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 10.3923i 0.322097 0.557888i −0.658824 0.752297i \(-0.728946\pi\)
0.980921 + 0.194409i \(0.0622790\pi\)
\(348\) 0 0
\(349\) −9.00000 15.5885i −0.481759 0.834431i 0.518022 0.855367i \(-0.326669\pi\)
−0.999781 + 0.0209364i \(0.993335\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.00000 13.8564i −0.425797 0.737502i 0.570697 0.821160i \(-0.306673\pi\)
−0.996495 + 0.0836583i \(0.973340\pi\)
\(354\) 0 0
\(355\) 32.0000 55.4256i 1.69838 2.94169i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 20.0000 34.6410i 1.04685 1.81319i
\(366\) 0 0
\(367\) −2.00000 3.46410i −0.104399 0.180825i 0.809093 0.587680i \(-0.199959\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.0000 + 17.3205i 0.519174 + 0.899236i
\(372\) 0 0
\(373\) 9.00000 15.5885i 0.466002 0.807140i −0.533244 0.845962i \(-0.679027\pi\)
0.999246 + 0.0388219i \(0.0123605\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.00000 10.3923i 0.306586 0.531022i −0.671027 0.741433i \(-0.734147\pi\)
0.977613 + 0.210411i \(0.0674801\pi\)
\(384\) 0 0
\(385\) 16.0000 + 27.7128i 0.815436 + 1.41238i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.0000 22.5167i −0.659126 1.14164i −0.980842 0.194804i \(-0.937593\pi\)
0.321716 0.946836i \(-0.395740\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −64.0000 −3.22019
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 + 10.3923i −0.299626 + 0.518967i −0.976050 0.217545i \(-0.930195\pi\)
0.676425 + 0.736512i \(0.263528\pi\)
\(402\) 0 0
\(403\) 5.00000 + 8.66025i 0.249068 + 0.431398i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0000 34.6410i −0.991363 1.71709i
\(408\) 0 0
\(409\) −7.00000 + 12.1244i −0.346128 + 0.599511i −0.985558 0.169338i \(-0.945837\pi\)
0.639430 + 0.768849i \(0.279170\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.0000 + 17.3205i −0.488532 + 0.846162i −0.999913 0.0131919i \(-0.995801\pi\)
0.511381 + 0.859354i \(0.329134\pi\)
\(420\) 0 0
\(421\) 1.00000 + 1.73205i 0.0487370 + 0.0844150i 0.889365 0.457198i \(-0.151147\pi\)
−0.840628 + 0.541613i \(0.817814\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 11.0000 + 19.0526i 0.533578 + 0.924185i
\(426\) 0 0
\(427\) −14.0000 + 24.2487i −0.677507 + 1.17348i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 0 0
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 13.8564i −0.381819 0.661330i 0.609503 0.792784i \(-0.291369\pi\)
−0.991322 + 0.131453i \(0.958036\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.00000 3.46410i −0.0950229 0.164584i 0.814595 0.580030i \(-0.196959\pi\)
−0.909618 + 0.415445i \(0.863626\pi\)
\(444\) 0 0
\(445\) −8.00000 + 13.8564i −0.379236 + 0.656857i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 6.92820i 0.187523 0.324799i
\(456\) 0 0
\(457\) 9.00000 + 15.5885i 0.421002 + 0.729197i 0.996038 0.0889312i \(-0.0283451\pi\)
−0.575036 + 0.818128i \(0.695012\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 + 10.3923i 0.279448 + 0.484018i 0.971248 0.238071i \(-0.0765153\pi\)
−0.691800 + 0.722089i \(0.743182\pi\)
\(462\) 0 0
\(463\) −11.0000 + 19.0526i −0.511213 + 0.885448i 0.488702 + 0.872451i \(0.337470\pi\)
−0.999916 + 0.0129968i \(0.995863\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 + 13.8564i −0.367840 + 0.637118i
\(474\) 0 0
\(475\) 11.0000 + 19.0526i 0.504715 + 0.874191i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.00000 13.8564i −0.365529 0.633115i 0.623332 0.781958i \(-0.285779\pi\)
−0.988861 + 0.148842i \(0.952445\pi\)
\(480\) 0 0
\(481\) −5.00000 + 8.66025i −0.227980 + 0.394874i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.00000 10.3923i 0.270776 0.468998i −0.698285 0.715820i \(-0.746053\pi\)
0.969061 + 0.246822i \(0.0793863\pi\)
\(492\) 0 0
\(493\) 6.00000 + 10.3923i 0.270226 + 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.0000 27.7128i −0.717698 1.24309i
\(498\) 0 0
\(499\) 3.00000 5.19615i 0.134298 0.232612i −0.791031 0.611776i \(-0.790455\pi\)
0.925329 + 0.379165i \(0.123789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −40.0000 −1.77998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 10.3923i 0.265945 0.460631i −0.701866 0.712309i \(-0.747649\pi\)
0.967811 + 0.251679i \(0.0809826\pi\)
\(510\) 0 0
\(511\) −10.0000 17.3205i −0.442374 0.766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.0000 + 27.7128i 0.705044 + 1.22117i
\(516\) 0 0
\(517\) −8.00000 + 13.8564i −0.351840 + 0.609404i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.0000 + 17.3205i −0.435607 + 0.754493i
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.00000 + 6.92820i 0.173259 + 0.300094i
\(534\) 0 0
\(535\) 24.0000 41.5692i 1.03761 1.79719i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.00000 6.92820i 0.171341 0.296772i
\(546\) 0 0
\(547\) −16.0000 27.7128i −0.684111 1.18491i −0.973715 0.227768i \(-0.926857\pi\)
0.289605 0.957146i \(-0.406476\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 + 10.3923i 0.255609 + 0.442727i
\(552\) 0 0
\(553\) −16.0000 + 27.7128i −0.680389 + 1.17847i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.0000 24.2487i 0.590030 1.02196i −0.404198 0.914671i \(-0.632449\pi\)
0.994228 0.107290i \(-0.0342173\pi\)
\(564\) 0 0
\(565\) 12.0000 + 20.7846i 0.504844 + 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 + 5.19615i 0.125767 + 0.217834i 0.922032 0.387113i \(-0.126528\pi\)
−0.796266 + 0.604947i \(0.793194\pi\)
\(570\) 0 0
\(571\) 4.00000 6.92820i 0.167395 0.289936i −0.770108 0.637913i \(-0.779798\pi\)
0.937503 + 0.347977i \(0.113131\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.0000 34.6410i −0.828315 1.43468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.00000 6.92820i −0.165098 0.285958i 0.771592 0.636117i \(-0.219461\pi\)
−0.936690 + 0.350160i \(0.886127\pi\)
\(588\) 0 0
\(589\) −10.0000 + 17.3205i −0.412043 + 0.713679i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 20.7846i 0.490307 0.849236i −0.509631 0.860393i \(-0.670218\pi\)
0.999938 + 0.0111569i \(0.00355143\pi\)
\(600\) 0 0
\(601\) 11.0000 + 19.0526i 0.448699 + 0.777170i 0.998302 0.0582563i \(-0.0185541\pi\)
−0.549602 + 0.835426i \(0.685221\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.0000 17.3205i −0.406558 0.704179i
\(606\) 0 0
\(607\) 2.00000 3.46410i 0.0811775 0.140604i −0.822578 0.568652i \(-0.807465\pi\)
0.903756 + 0.428048i \(0.140799\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0000 + 41.5692i −0.966204 + 1.67351i −0.259858 + 0.965647i \(0.583676\pi\)
−0.706346 + 0.707867i \(0.749658\pi\)
\(618\) 0 0
\(619\) −7.00000 12.1244i −0.281354 0.487319i 0.690365 0.723462i \(-0.257450\pi\)
−0.971718 + 0.236143i \(0.924117\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.00000 + 6.92820i 0.160257 + 0.277573i
\(624\) 0 0
\(625\) −20.5000 + 35.5070i −0.820000 + 1.42028i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −30.0000 −1.19428 −0.597141 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.0000 + 41.5692i −0.952411 + 1.64962i
\(636\) 0 0
\(637\) 1.50000 + 2.59808i 0.0594322 + 0.102940i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.0000 + 22.5167i 0.513469 + 0.889355i 0.999878 + 0.0156233i \(0.00497325\pi\)
−0.486409 + 0.873731i \(0.661693\pi\)
\(642\) 0 0
\(643\) −13.0000 + 22.5167i −0.512670 + 0.887970i 0.487222 + 0.873278i \(0.338010\pi\)
−0.999892 + 0.0146923i \(0.995323\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.0000 + 39.8372i −0.900060 + 1.55895i −0.0726446 + 0.997358i \(0.523144\pi\)
−0.827415 + 0.561591i \(0.810189\pi\)
\(654\) 0 0
\(655\) 8.00000 + 13.8564i 0.312586 + 0.541415i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.00000 10.3923i −0.233727 0.404827i 0.725175 0.688565i \(-0.241759\pi\)
−0.958902 + 0.283738i \(0.908425\pi\)
\(660\) 0 0
\(661\) −5.00000 + 8.66025i −0.194477 + 0.336845i −0.946729 0.322031i \(-0.895634\pi\)
0.752252 + 0.658876i \(0.228968\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.0000 48.4974i 1.08093 1.87222i
\(672\) 0 0
\(673\) −11.0000 19.0526i −0.424019 0.734422i 0.572309 0.820038i \(-0.306048\pi\)
−0.996328 + 0.0856156i \(0.972714\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.0000 + 32.9090i 0.730229 + 1.26479i 0.956785 + 0.290796i \(0.0939201\pi\)
−0.226556 + 0.973998i \(0.572747\pi\)
\(678\) 0 0
\(679\) −2.00000 + 3.46410i −0.0767530 + 0.132940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −32.0000 −1.22266
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.00000 + 8.66025i −0.190485 + 0.329929i
\(690\) 0 0
\(691\) −5.00000 8.66025i −0.190209 0.329452i 0.755110 0.655598i \(-0.227583\pi\)
−0.945319 + 0.326146i \(0.894250\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.00000 + 13.8564i −0.303022 + 0.524849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.0000 + 17.3205i −0.376089 + 0.651405i
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 + 13.8564i −0.299183 + 0.518200i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −33.0000 + 57.1577i −1.22559 + 2.12278i
\(726\) 0 0
\(727\) −22.0000 38.1051i −0.815935 1.41324i −0.908655 0.417548i \(-0.862889\pi\)
0.0927199 0.995692i \(-0.470444\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.00000 + 6.92820i 0.147945 + 0.256249i
\(732\) 0 0
\(733\) −11.0000 + 19.0526i −0.406294 + 0.703722i −0.994471 0.105010i \(-0.966513\pi\)
0.588177 + 0.808732i \(0.299846\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 41.5692i 0.880475 1.52503i 0.0296605 0.999560i \(-0.490557\pi\)
0.850814 0.525467i \(-0.176109\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 20.7846i −0.438470 0.759453i
\(750\) 0 0
\(751\) −4.00000 + 6.92820i −0.145962 + 0.252814i −0.929731 0.368238i \(-0.879961\pi\)
0.783769 + 0.621052i \(0.213294\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 72.0000 2.62035
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.0000 34.6410i 0.724999 1.25574i −0.233975 0.972243i \(-0.575173\pi\)
0.958974 0.283493i \(-0.0914933\pi\)
\(762\) 0 0
\(763\) −2.00000 3.46410i −0.0724049 0.125409i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.00000 6.92820i −0.144432 0.250163i
\(768\) 0 0
\(769\) 11.0000 19.0526i 0.396670 0.687053i −0.596643 0.802507i \(-0.703499\pi\)
0.993313 + 0.115454i \(0.0368323\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.0000 1.72644 0.863220 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) −110.000 −3.95132
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.00000 + 13.8564i −0.286630 + 0.496457i
\(780\) 0 0
\(781\) 32.0000 + 55.4256i 1.14505 + 1.98328i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 6.92820i −0.142766 0.247278i
\(786\) 0 0
\(787\) 25.0000 43.3013i 0.891154 1.54352i 0.0526599 0.998613i \(-0.483230\pi\)
0.838494 0.544911i \(-0.183437\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.0000 + 22.5167i −0.460484 + 0.797581i −0.998985 0.0450436i \(-0.985657\pi\)
0.538501 + 0.842625i \(0.318991\pi\)
\(798\) 0 0
\(799\) 4.00000 + 6.92820i 0.141510 + 0.245102i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.0000 + 34.6410i 0.705785 + 1.22245i
\(804\) 0 0