Properties

Label 1344.2.k.d.1217.3
Level $1344$
Weight $2$
Character 1344.1217
Analytic conductor $10.732$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.3
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1344.1217
Dual form 1344.2.k.d.1217.4

$q$-expansion

\(f(q)\) \(=\) \(q+(1.22474 - 1.22474i) q^{3} +2.44949 q^{5} +(1.00000 - 2.44949i) q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 - 1.22474i) q^{3} +2.44949 q^{5} +(1.00000 - 2.44949i) q^{7} -3.00000i q^{9} +2.44949i q^{13} +(3.00000 - 3.00000i) q^{15} +4.89898 q^{17} +2.44949i q^{19} +(-1.77526 - 4.22474i) q^{21} -6.00000i q^{23} +1.00000 q^{25} +(-3.67423 - 3.67423i) q^{27} +6.00000i q^{29} +(2.44949 - 6.00000i) q^{35} +2.00000 q^{37} +(3.00000 + 3.00000i) q^{39} -4.89898 q^{41} +4.00000 q^{43} -7.34847i q^{45} -4.89898 q^{47} +(-5.00000 - 4.89898i) q^{49} +(6.00000 - 6.00000i) q^{51} -6.00000i q^{53} +(3.00000 + 3.00000i) q^{57} -12.2474 q^{59} +12.2474i q^{61} +(-7.34847 - 3.00000i) q^{63} +6.00000i q^{65} +8.00000 q^{67} +(-7.34847 - 7.34847i) q^{69} +9.79796i q^{73} +(1.22474 - 1.22474i) q^{75} +10.0000 q^{79} -9.00000 q^{81} +2.44949 q^{83} +12.0000 q^{85} +(7.34847 + 7.34847i) q^{87} +(6.00000 + 2.44949i) q^{91} +6.00000i q^{95} +4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 12q^{15} - 12q^{21} + 4q^{25} + 8q^{37} + 12q^{39} + 16q^{43} - 20q^{49} + 24q^{51} + 12q^{57} + 32q^{67} + 40q^{79} - 36q^{81} + 48q^{85} + 24q^{91} + O(q^{100}) \)

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
\(3\) 1.22474 1.22474i 0.707107 0.707107i
\(4\) 0 0
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) 1.00000 2.44949i 0.377964 0.925820i
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0 0
\(15\) 3.00000 3.00000i 0.774597 0.774597i
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) 2.44949i 0.561951i 0.959715 + 0.280976i \(0.0906580\pi\)
−0.959715 + 0.280976i \(0.909342\pi\)
\(20\) 0 0
\(21\) −1.77526 4.22474i −0.387392 0.921915i
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.67423 3.67423i −0.707107 0.707107i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.44949 6.00000i 0.414039 1.01419i
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 3.00000 + 3.00000i 0.480384 + 0.480384i
\(40\) 0 0
\(41\) −4.89898 −0.765092 −0.382546 0.923936i \(-0.624953\pi\)
−0.382546 + 0.923936i \(0.624953\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 7.34847i 1.09545i
\(46\) 0 0
\(47\) −4.89898 −0.714590 −0.357295 0.933992i \(-0.616301\pi\)
−0.357295 + 0.933992i \(0.616301\pi\)
\(48\) 0 0
\(49\) −5.00000 4.89898i −0.714286 0.699854i
\(50\) 0 0
\(51\) 6.00000 6.00000i 0.840168 0.840168i
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 + 3.00000i 0.397360 + 0.397360i
\(58\) 0 0
\(59\) −12.2474 −1.59448 −0.797241 0.603661i \(-0.793708\pi\)
−0.797241 + 0.603661i \(0.793708\pi\)
\(60\) 0 0
\(61\) 12.2474i 1.56813i 0.620682 + 0.784063i \(0.286856\pi\)
−0.620682 + 0.784063i \(0.713144\pi\)
\(62\) 0 0
\(63\) −7.34847 3.00000i −0.925820 0.377964i
\(64\) 0 0
\(65\) 6.00000i 0.744208i
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −7.34847 7.34847i −0.884652 0.884652i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 9.79796i 1.14676i 0.819288 + 0.573382i \(0.194369\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 1.22474 1.22474i 0.141421 0.141421i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 2.44949 0.268866 0.134433 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) 7.34847 + 7.34847i 0.787839 + 0.787839i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 6.00000 + 2.44949i 0.628971 + 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000i 0.615587i
\(96\) 0 0
\(97\) 4.89898i 0.497416i 0.968579 + 0.248708i \(0.0800060\pi\)
−0.968579 + 0.248708i \(0.919994\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.34847 −0.731200 −0.365600 0.930772i \(-0.619136\pi\)
−0.365600 + 0.930772i \(0.619136\pi\)
\(102\) 0 0
\(103\) 9.79796i 0.965422i −0.875780 0.482711i \(-0.839652\pi\)
0.875780 0.482711i \(-0.160348\pi\)
\(104\) 0 0
\(105\) −4.34847 10.3485i −0.424367 1.00991i
\(106\) 0 0
\(107\) 12.0000i 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 2.44949 2.44949i 0.232495 0.232495i
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 14.6969i 1.37050i
\(116\) 0 0
\(117\) 7.34847 0.679366
\(118\) 0 0
\(119\) 4.89898 12.0000i 0.449089 1.10004i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −6.00000 + 6.00000i −0.541002 + 0.541002i
\(124\) 0 0
\(125\) −9.79796 −0.876356
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 4.89898 4.89898i 0.431331 0.431331i
\(130\) 0 0
\(131\) 7.34847 0.642039 0.321019 0.947073i \(-0.395975\pi\)
0.321019 + 0.947073i \(0.395975\pi\)
\(132\) 0 0
\(133\) 6.00000 + 2.44949i 0.520266 + 0.212398i
\(134\) 0 0
\(135\) −9.00000 9.00000i −0.774597 0.774597i
\(136\) 0 0
\(137\) 12.0000i 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 2.44949i 0.207763i 0.994590 + 0.103882i \(0.0331263\pi\)
−0.994590 + 0.103882i \(0.966874\pi\)
\(140\) 0 0
\(141\) −6.00000 + 6.00000i −0.505291 + 0.505291i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 14.6969i 1.22051i
\(146\) 0 0
\(147\) −12.1237 + 0.123724i −0.999948 + 0.0102046i
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 14.6969i 1.18818i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.34847i 0.586472i 0.956040 + 0.293236i \(0.0947321\pi\)
−0.956040 + 0.293236i \(0.905268\pi\)
\(158\) 0 0
\(159\) −7.34847 7.34847i −0.582772 0.582772i
\(160\) 0 0
\(161\) −14.6969 6.00000i −1.15828 0.472866i
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.89898 −0.379094 −0.189547 0.981872i \(-0.560702\pi\)
−0.189547 + 0.981872i \(0.560702\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 7.34847 0.561951
\(172\) 0 0
\(173\) 22.0454 1.67608 0.838041 0.545608i \(-0.183701\pi\)
0.838041 + 0.545608i \(0.183701\pi\)
\(174\) 0 0
\(175\) 1.00000 2.44949i 0.0755929 0.185164i
\(176\) 0 0
\(177\) −15.0000 + 15.0000i −1.12747 + 1.12747i
\(178\) 0 0
\(179\) 24.0000i 1.79384i 0.442189 + 0.896922i \(0.354202\pi\)
−0.442189 + 0.896922i \(0.645798\pi\)
\(180\) 0 0
\(181\) 12.2474i 0.910346i −0.890403 0.455173i \(-0.849577\pi\)
0.890403 0.455173i \(-0.150423\pi\)
\(182\) 0 0
\(183\) 15.0000 + 15.0000i 1.10883 + 1.10883i
\(184\) 0 0
\(185\) 4.89898 0.360180
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −12.6742 + 5.32577i −0.921915 + 0.387392i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 7.34847 + 7.34847i 0.526235 + 0.526235i
\(196\) 0 0
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 9.79796i 0.694559i 0.937762 + 0.347279i \(0.112894\pi\)
−0.937762 + 0.347279i \(0.887106\pi\)
\(200\) 0 0
\(201\) 9.79796 9.79796i 0.691095 0.691095i
\(202\) 0 0
\(203\) 14.6969 + 6.00000i 1.03152 + 0.421117i
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) −18.0000 −1.25109
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.79796 0.668215
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.0000 + 12.0000i 0.810885 + 0.810885i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 14.6969i 0.984180i 0.870544 + 0.492090i \(0.163767\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) 3.00000i 0.200000i
\(226\) 0 0
\(227\) −7.34847 −0.487735 −0.243868 0.969809i \(-0.578416\pi\)
−0.243868 + 0.969809i \(0.578416\pi\)
\(228\) 0 0
\(229\) 22.0454i 1.45680i 0.685151 + 0.728401i \(0.259736\pi\)
−0.685151 + 0.728401i \(0.740264\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000i 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 12.2474 12.2474i 0.795557 0.795557i
\(238\) 0 0
\(239\) 6.00000i 0.388108i 0.980991 + 0.194054i \(0.0621637\pi\)
−0.980991 + 0.194054i \(0.937836\pi\)
\(240\) 0 0
\(241\) 24.4949i 1.57786i 0.614486 + 0.788928i \(0.289363\pi\)
−0.614486 + 0.788928i \(0.710637\pi\)
\(242\) 0 0
\(243\) −11.0227 + 11.0227i −0.707107 + 0.707107i
\(244\) 0 0
\(245\) −12.2474 12.0000i −0.782461 0.766652i
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 3.00000 3.00000i 0.190117 0.190117i
\(250\) 0 0
\(251\) −17.1464 −1.08227 −0.541136 0.840935i \(-0.682006\pi\)
−0.541136 + 0.840935i \(0.682006\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 14.6969 14.6969i 0.920358 0.920358i
\(256\) 0 0
\(257\) 29.3939 1.83354 0.916770 0.399416i \(-0.130787\pi\)
0.916770 + 0.399416i \(0.130787\pi\)
\(258\) 0 0
\(259\) 2.00000 4.89898i 0.124274 0.304408i
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) 14.6969i 0.902826i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.2474 0.746740 0.373370 0.927682i \(-0.378202\pi\)
0.373370 + 0.927682i \(0.378202\pi\)
\(270\) 0 0
\(271\) 24.4949i 1.48796i 0.668202 + 0.743980i \(0.267064\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 0 0
\(273\) 10.3485 4.34847i 0.626318 0.263181i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 22.0454i 1.31046i 0.755428 + 0.655232i \(0.227429\pi\)
−0.755428 + 0.655232i \(0.772571\pi\)
\(284\) 0 0
\(285\) 7.34847 + 7.34847i 0.435286 + 0.435286i
\(286\) 0 0
\(287\) −4.89898 + 12.0000i −0.289178 + 0.708338i
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 6.00000 + 6.00000i 0.351726 + 0.351726i
\(292\) 0 0
\(293\) −2.44949 −0.143101 −0.0715504 0.997437i \(-0.522795\pi\)
−0.0715504 + 0.997437i \(0.522795\pi\)
\(294\) 0 0
\(295\) −30.0000 −1.74667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 14.6969 0.849946
\(300\) 0 0
\(301\) 4.00000 9.79796i 0.230556 0.564745i
\(302\) 0 0
\(303\) −9.00000 + 9.00000i −0.517036 + 0.517036i
\(304\) 0 0
\(305\) 30.0000i 1.71780i
\(306\) 0 0
\(307\) 7.34847i 0.419399i −0.977766 0.209700i \(-0.932751\pi\)
0.977766 0.209700i \(-0.0672486\pi\)
\(308\) 0 0
\(309\) −12.0000 12.0000i −0.682656 0.682656i
\(310\) 0 0
\(311\) −19.5959 −1.11118 −0.555591 0.831456i \(-0.687508\pi\)
−0.555591 + 0.831456i \(0.687508\pi\)
\(312\) 0 0
\(313\) 34.2929i 1.93835i 0.246380 + 0.969173i \(0.420759\pi\)
−0.246380 + 0.969173i \(0.579241\pi\)
\(314\) 0 0
\(315\) −18.0000 7.34847i −1.01419 0.414039i
\(316\) 0 0
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −14.6969 14.6969i −0.820303 0.820303i
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 2.44949i 0.135873i
\(326\) 0 0
\(327\) −12.2474 + 12.2474i −0.677285 + 0.677285i
\(328\) 0 0
\(329\) −4.89898 + 12.0000i −0.270089 + 0.661581i
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 19.5959 1.07064
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 7.34847 + 7.34847i 0.399114 + 0.399114i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −17.0000 + 7.34847i −0.917914 + 0.396780i
\(344\) 0 0
\(345\) −18.0000 18.0000i −0.969087 0.969087i
\(346\) 0 0
\(347\) 12.0000i 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 2.44949i 0.131118i −0.997849 0.0655591i \(-0.979117\pi\)
0.997849 0.0655591i \(-0.0208831\pi\)
\(350\) 0 0
\(351\) 9.00000 9.00000i 0.480384 0.480384i
\(352\) 0 0
\(353\) −9.79796 −0.521493 −0.260746 0.965407i \(-0.583969\pi\)
−0.260746 + 0.965407i \(0.583969\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.69694 20.6969i −0.460291 1.09540i
\(358\) 0 0
\(359\) 6.00000i 0.316668i 0.987386 + 0.158334i \(0.0506123\pi\)
−0.987386 + 0.158334i \(0.949388\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 13.4722 13.4722i 0.707107 0.707107i
\(364\) 0 0
\(365\) 24.0000i 1.25622i
\(366\) 0 0
\(367\) 4.89898i 0.255725i −0.991792 0.127862i \(-0.959188\pi\)
0.991792 0.127862i \(-0.0408116\pi\)
\(368\) 0 0
\(369\) 14.6969i 0.765092i
\(370\) 0 0
\(371\) −14.6969 6.00000i −0.763027 0.311504i
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −12.0000 + 12.0000i −0.619677 + 0.619677i
\(376\) 0 0
\(377\) −14.6969 −0.756931
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −9.79796 + 9.79796i −0.501965 + 0.501965i
\(382\) 0 0
\(383\) 34.2929 1.75228 0.876142 0.482054i \(-0.160109\pi\)
0.876142 + 0.482054i \(0.160109\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.0000i 0.609994i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) 29.3939i 1.48651i
\(392\) 0 0
\(393\) 9.00000 9.00000i 0.453990 0.453990i
\(394\) 0 0
\(395\) 24.4949 1.23247
\(396\) 0 0
\(397\) 7.34847i 0.368809i 0.982850 + 0.184405i \(0.0590357\pi\)
−0.982850 + 0.184405i \(0.940964\pi\)
\(398\) 0 0
\(399\) 10.3485 4.34847i 0.518071 0.217696i
\(400\) 0 0
\(401\) 30.0000i 1.49813i 0.662497 + 0.749064i \(0.269497\pi\)
−0.662497 + 0.749064i \(0.730503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −22.0454 −1.09545
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 34.2929i 1.69567i −0.530258 0.847836i \(-0.677905\pi\)
0.530258 0.847836i \(-0.322095\pi\)
\(410\) 0 0
\(411\) −14.6969 14.6969i −0.724947 0.724947i
\(412\) 0 0
\(413\) −12.2474 + 30.0000i −0.602658 + 1.47620i
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 3.00000 + 3.00000i 0.146911 + 0.146911i
\(418\) 0 0
\(419\) 12.2474 0.598327 0.299164 0.954202i \(-0.403292\pi\)
0.299164 + 0.954202i \(0.403292\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 14.6969i 0.714590i
\(424\) 0 0
\(425\) 4.89898 0.237635
\(426\) 0 0
\(427\) 30.0000 + 12.2474i 1.45180 + 0.592696i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000i 1.44505i −0.691345 0.722525i \(-0.742982\pi\)
0.691345 0.722525i \(-0.257018\pi\)
\(432\) 0 0
\(433\) 14.6969i 0.706290i −0.935569 0.353145i \(-0.885112\pi\)
0.935569 0.353145i \(-0.114888\pi\)
\(434\) 0 0
\(435\) 18.0000 + 18.0000i 0.863034 + 0.863034i
\(436\) 0 0
\(437\) 14.6969 0.703050
\(438\) 0 0
\(439\) 14.6969i 0.701447i −0.936479 0.350723i \(-0.885936\pi\)
0.936479 0.350723i \(-0.114064\pi\)
\(440\) 0 0
\(441\) −14.6969 + 15.0000i −0.699854 + 0.714286i
\(442\) 0 0
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7.34847 + 7.34847i 0.347571 + 0.347571i
\(448\) 0 0
\(449\) 36.0000i 1.69895i −0.527633 0.849473i \(-0.676920\pi\)
0.527633 0.849473i \(-0.323080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 9.79796 9.79796i 0.460348 0.460348i
\(454\) 0 0
\(455\) 14.6969 + 6.00000i 0.689003 + 0.281284i
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 0 0
\(459\) −18.0000 18.0000i −0.840168 0.840168i
\(460\) 0 0
\(461\) −31.8434 −1.48309 −0.741547 0.670901i \(-0.765907\pi\)
−0.741547 + 0.670901i \(0.765907\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.34847 −0.340047 −0.170023 0.985440i \(-0.554384\pi\)
−0.170023 + 0.985440i \(0.554384\pi\)
\(468\) 0 0
\(469\) 8.00000 19.5959i 0.369406 0.904855i
\(470\) 0 0
\(471\) 9.00000 + 9.00000i 0.414698 + 0.414698i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.44949i 0.112390i
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 0 0
\(479\) −24.4949 −1.11920 −0.559600 0.828763i \(-0.689045\pi\)
−0.559600 + 0.828763i \(0.689045\pi\)
\(480\) 0 0
\(481\) 4.89898i 0.223374i
\(482\) 0 0
\(483\) −25.3485 + 10.6515i −1.15340 + 0.484661i
\(484\) 0 0
\(485\) 12.0000i 0.544892i
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) −19.5959 + 19.5959i −0.886158 + 0.886158i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 29.3939i 1.32383i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −6.00000 + 6.00000i −0.268060 + 0.268060i
\(502\) 0 0
\(503\) −39.1918 −1.74748 −0.873739 0.486395i \(-0.838311\pi\)
−0.873739 + 0.486395i \(0.838311\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 8.57321 8.57321i 0.380750 0.380750i
\(508\) 0 0
\(509\) 12.2474 0.542859 0.271429 0.962458i \(-0.412504\pi\)
0.271429 + 0.962458i \(0.412504\pi\)
\(510\) 0 0
\(511\) 24.0000 + 9.79796i 1.06170 + 0.433436i
\(512\) 0 0
\(513\) 9.00000 9.00000i 0.397360 0.397360i
\(514\) 0 0
\(515\) 24.0000i 1.05757i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 27.0000 27.0000i 1.18517 1.18517i
\(520\) 0 0
\(521\) −4.89898 −0.214628 −0.107314 0.994225i \(-0.534225\pi\)
−0.107314 + 0.994225i \(0.534225\pi\)
\(522\) 0 0
\(523\) 2.44949i 0.107109i −0.998565 0.0535544i \(-0.982945\pi\)
0.998565 0.0535544i \(-0.0170550\pi\)
\(524\) 0 0
\(525\) −1.77526 4.22474i −0.0774785 0.184383i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 36.7423i 1.59448i
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 29.3939i 1.27081i
\(536\) 0 0
\(537\) 29.3939 + 29.3939i 1.26844 + 1.26844i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) −15.0000 15.0000i −0.643712 0.643712i
\(544\) 0 0
\(545\) −24.4949 −1.04925
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 36.7423 1.56813
\(550\) 0 0
\(551\) −14.6969 −0.626111
\(552\) 0 0
\(553\) 10.0000 24.4949i 0.425243 1.04163i
\(554\) 0 0
\(555\) 6.00000 6.00000i 0.254686 0.254686i
\(556\) 0 0
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 9.79796i 0.414410i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.0454 −0.929103 −0.464552 0.885546i \(-0.653784\pi\)
−0.464552 + 0.885546i \(0.653784\pi\)
\(564\) 0 0
\(565\) 14.6969i 0.618305i
\(566\) 0 0
\(567\) −9.00000 + 22.0454i −0.377964 + 0.925820i
\(568\) 0 0
\(569\) 6.00000i 0.251533i −0.992060 0.125767i \(-0.959861\pi\)
0.992060 0.125767i \(-0.0401390\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000i 0.250217i
\(576\) 0 0
\(577\) 19.5959i 0.815789i −0.913029 0.407894i \(-0.866263\pi\)
0.913029 0.407894i \(-0.133737\pi\)
\(578\) 0 0
\(579\) 4.89898 4.89898i 0.203595 0.203595i
\(580\) 0 0
\(581\) 2.44949 6.00000i 0.101622 0.248922i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 18.0000 0.744208
\(586\) 0 0
\(587\) −7.34847 −0.303304 −0.151652 0.988434i \(-0.548459\pi\)
−0.151652 + 0.988434i \(0.548459\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −22.0454 22.0454i −0.906827 0.906827i
\(592\) 0 0
\(593\) 39.1918 1.60942 0.804708 0.593671i \(-0.202322\pi\)
0.804708 + 0.593671i \(0.202322\pi\)
\(594\) 0 0
\(595\) 12.0000 29.3939i 0.491952 1.20503i
\(596\) 0 0
\(597\) 12.0000 + 12.0000i 0.491127 + 0.491127i
\(598\) 0 0
\(599\) 24.0000i 0.980613i −0.871550 0.490307i \(-0.836885\pi\)
0.871550 0.490307i \(-0.163115\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) 26.9444 1.09545
\(606\) 0 0
\(607\) 4.89898i 0.198843i −0.995045 0.0994217i \(-0.968301\pi\)
0.995045 0.0994217i \(-0.0316993\pi\)
\(608\) 0 0
\(609\) 25.3485 10.6515i 1.02717 0.431622i
\(610\) 0 0
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) −14.6969 + 14.6969i −0.592638 + 0.592638i
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 26.9444i 1.08299i 0.840705 + 0.541493i \(0.182141\pi\)
−0.840705 + 0.541493i \(0.817859\pi\)
\(620\) 0 0
\(621\) −22.0454 + 22.0454i −0.884652 + 0.884652i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.79796 0.390670
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 0 0
\(633\) −9.79796 + 9.79796i −0.389434 + 0.389434i
\(634\) 0 0
\(635\) −19.5959 −0.777640
\(636\) 0 0
\(637\) 12.0000 12.2474i 0.475457 0.485262i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000i 1.18493i −0.805597 0.592464i \(-0.798155\pi\)
0.805597 0.592464i \(-0.201845\pi\)
\(642\) 0 0
\(643\) 22.0454i 0.869386i 0.900579 + 0.434693i \(0.143143\pi\)
−0.900579 + 0.434693i \(0.856857\pi\)
\(644\) 0 0
\(645\) 12.0000 12.0000i 0.472500 0.472500i
\(646\) 0 0
\(647\) 44.0908 1.73339 0.866694 0.498839i \(-0.166240\pi\)
0.866694 + 0.498839i \(0.166240\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000i 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) 29.3939 1.14676
\(658\) 0 0
\(659\) 36.0000i 1.40236i −0.712984 0.701180i \(-0.752657\pi\)
0.712984 0.701180i \(-0.247343\pi\)
\(660\) 0 0
\(661\) 12.2474i 0.476371i −0.971220 0.238185i \(-0.923447\pi\)
0.971220 0.238185i \(-0.0765525\pi\)
\(662\) 0 0
\(663\) 14.6969 + 14.6969i 0.570782 + 0.570782i
\(664\) 0 0
\(665\) 14.6969 + 6.00000i 0.569923 + 0.232670i
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) 18.0000 + 18.0000i 0.695920 + 0.695920i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) −3.67423 3.67423i −0.141421 0.141421i
\(676\) 0 0
\(677\) 7.34847 0.282425 0.141212 0.989979i \(-0.454900\pi\)
0.141212 + 0.989979i \(0.454900\pi\)
\(678\) 0 0
\(679\) 12.0000 + 4.89898i 0.460518 + 0.188006i
\(680\) 0 0
\(681\) −9.00000 + 9.00000i −0.344881 + 0.344881i
\(682\) 0 0
\(683\) 24.0000i 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 29.3939i 1.12308i
\(686\) 0 0
\(687\) 27.0000 + 27.0000i 1.03011 + 1.03011i
\(688\) 0 0
\(689\) 14.6969 0.559909
\(690\) 0 0
\(691\) 36.7423i 1.39774i −0.715246 0.698872i \(-0.753686\pi\)
0.715246 0.698872i \(-0.246314\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.00000i 0.227593i
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) −29.3939 29.3939i −1.11178 1.11178i
\(700\) 0 0
\(701\) 30.0000i 1.13308i −0.824033 0.566542i \(-0.808281\pi\)
0.824033 0.566542i \(-0.191719\pi\)
\(702\) 0 0
\(703\) 4.89898i 0.184769i
\(704\) 0 0
\(705\) −14.6969 + 14.6969i −0.553519 + 0.553519i
\(706\) 0 0
\(707\) −7.34847 + 18.0000i −0.276368 + 0.676960i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 30.0000i 1.12509i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.34847 + 7.34847i 0.274434 + 0.274434i
\(718\) 0 0
\(719\) −24.4949 −0.913506 −0.456753 0.889594i \(-0.650988\pi\)
−0.456753 + 0.889594i \(0.650988\pi\)
\(720\) 0 0
\(721\) −24.0000 9.79796i −0.893807 0.364895i
\(722\) 0 0
\(723\) 30.0000 + 30.0000i 1.11571 + 1.11571i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 29.3939i 1.09016i −0.838385 0.545079i \(-0.816500\pi\)
0.838385 0.545079i \(-0.183500\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 19.5959 0.724781
\(732\) 0 0
\(733\) 22.0454i 0.814266i −0.913369 0.407133i \(-0.866529\pi\)
0.913369 0.407133i \(-0.133471\pi\)
\(734\) 0 0
\(735\) −29.6969 + 0.303062i −1.09539 + 0.0111786i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −7.34847 + 7.34847i −0.269953 + 0.269953i
\(742\) 0 0
\(743\) 6.00000i 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) 14.6969i 0.538454i
\(746\) 0 0
\(747\) 7.34847i 0.268866i
\(748\) 0 0
\(749\) −29.3939 12.0000i −1.07403 0.438470i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −21.0000 + 21.0000i −0.765283 + 0.765283i
\(754\) 0 0
\(755\) 19.5959 0.713168
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.89898 −0.177588 −0.0887939 0.996050i \(-0.528301\pi\)
−0.0887939 + 0.996050i \(0.528301\pi\)
\(762\) 0 0
\(763\) −10.0000 + 24.4949i −0.362024 + 0.886775i
\(764\) 0 0
\(765\) 36.0000i 1.30158i
\(766\) 0 0
\(767\) 30.0000i 1.08324i
\(768\) 0 0
\(769\) 34.2929i 1.23663i −0.785930 0.618316i \(-0.787815\pi\)
0.785930 0.618316i \(-0.212185\pi\)
\(770\) 0 0
\(771\) 36.0000 36.0000i 1.29651 1.29651i
\(772\) 0 0
\(773\) −26.9444 −0.969122 −0.484561 0.874757i \(-0.661021\pi\)
−0.484561 + 0.874757i \(0.661021\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3.55051 8.44949i −0.127374 0.303124i
\(778\) 0 0
\(779\) 12.0000i 0.429945i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 22.0454 22.0454i 0.787839 0.787839i
\(784\) 0 0
\(785\) 18.0000i 0.642448i
\(786\) 0 0
\(787\) 31.8434i 1.13509i −0.823341 0.567547i \(-0.807893\pi\)
0.823341 0.567547i \(-0.192107\pi\)
\(788\) 0 0
\(789\) 29.3939 + 29.3939i 1.04645 + 1.04645i
\(790\) 0 0
\(791\) 14.6969 + 6.00000i 0.522563 + 0.213335i
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 0 0
\(795\) −18.0000 18.0000i −0.638394 0.638394i
\(796\) 0 0
\(797\) 7.34847 0.260296 0.130148 0.991495i \(-0.458455\pi\)
0.130148 + 0.991495i \(0.458455\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0