Properties

Label 768.2.j.c.193.2
Level $768$
Weight $2$
Character 768.193
Analytic conductor $6.133$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 768.193
Dual form 768.2.j.c.577.2

$q$-expansion

\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +1.41421i q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{3} +1.41421i q^{7} -1.00000i q^{9} +(1.00000 - 1.00000i) q^{13} +6.00000 q^{17} +(4.24264 - 4.24264i) q^{19} +(1.00000 + 1.00000i) q^{21} +8.48528i q^{23} -5.00000i q^{25} +(-0.707107 - 0.707107i) q^{27} +(6.00000 - 6.00000i) q^{29} -1.41421 q^{31} +(5.00000 + 5.00000i) q^{37} -1.41421i q^{39} +6.00000i q^{41} +(-4.24264 - 4.24264i) q^{43} -8.48528 q^{47} +5.00000 q^{49} +(4.24264 - 4.24264i) q^{51} +(-6.00000 - 6.00000i) q^{53} -6.00000i q^{57} +(5.00000 - 5.00000i) q^{61} +1.41421 q^{63} +(2.82843 - 2.82843i) q^{67} +(6.00000 + 6.00000i) q^{69} +8.48528i q^{71} +(-3.53553 - 3.53553i) q^{75} +1.41421 q^{79} -1.00000 q^{81} +(-8.48528 + 8.48528i) q^{83} -8.48528i q^{87} +6.00000i q^{89} +(1.41421 + 1.41421i) q^{91} +(-1.00000 + 1.00000i) q^{93} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{13} + 24 q^{17} + 4 q^{21} + 24 q^{29} + 20 q^{37} + 20 q^{49} - 24 q^{53} + 20 q^{61} + 24 q^{69} - 4 q^{81} - 4 q^{93} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 0.707107 0.707107i 0.408248 0.408248i
\(4\) 0 0
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 1.41421i 0.534522i 0.963624 + 0.267261i \(0.0861187\pi\)
−0.963624 + 0.267261i \(0.913881\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0 0
\(13\) 1.00000 1.00000i 0.277350 0.277350i −0.554700 0.832050i \(-0.687167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.24264 4.24264i 0.973329 0.973329i −0.0263249 0.999653i \(-0.508380\pi\)
0.999653 + 0.0263249i \(0.00838045\pi\)
\(20\) 0 0
\(21\) 1.00000 + 1.00000i 0.218218 + 0.218218i
\(22\) 0 0
\(23\) 8.48528i 1.76930i 0.466252 + 0.884652i \(0.345604\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 6.00000 6.00000i 1.11417 1.11417i 0.121592 0.992580i \(-0.461200\pi\)
0.992580 0.121592i \(-0.0387999\pi\)
\(30\) 0 0
\(31\) −1.41421 −0.254000 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 + 5.00000i 0.821995 + 0.821995i 0.986394 0.164399i \(-0.0525685\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 1.41421i 0.226455i
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) −4.24264 4.24264i −0.646997 0.646997i 0.305269 0.952266i \(-0.401253\pi\)
−0.952266 + 0.305269i \(0.901253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.48528 −1.23771 −0.618853 0.785507i \(-0.712402\pi\)
−0.618853 + 0.785507i \(0.712402\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 4.24264 4.24264i 0.594089 0.594089i
\(52\) 0 0
\(53\) −6.00000 6.00000i −0.824163 0.824163i 0.162539 0.986702i \(-0.448032\pi\)
−0.986702 + 0.162539i \(0.948032\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 5.00000 5.00000i 0.640184 0.640184i −0.310416 0.950601i \(-0.600468\pi\)
0.950601 + 0.310416i \(0.100468\pi\)
\(62\) 0 0
\(63\) 1.41421 0.178174
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.82843 2.82843i 0.345547 0.345547i −0.512901 0.858448i \(-0.671429\pi\)
0.858448 + 0.512901i \(0.171429\pi\)
\(68\) 0 0
\(69\) 6.00000 + 6.00000i 0.722315 + 0.722315i
\(70\) 0 0
\(71\) 8.48528i 1.00702i 0.863990 + 0.503509i \(0.167958\pi\)
−0.863990 + 0.503509i \(0.832042\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −3.53553 3.53553i −0.408248 0.408248i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.41421 0.159111 0.0795557 0.996830i \(-0.474650\pi\)
0.0795557 + 0.996830i \(0.474650\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −8.48528 + 8.48528i −0.931381 + 0.931381i −0.997792 0.0664117i \(-0.978845\pi\)
0.0664117 + 0.997792i \(0.478845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.48528i 0.909718i
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 1.41421 + 1.41421i 0.148250 + 0.148250i
\(92\) 0 0
\(93\) −1.00000 + 1.00000i −0.103695 + 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 6.00000i −0.597022 0.597022i 0.342497 0.939519i \(-0.388727\pi\)
−0.939519 + 0.342497i \(0.888727\pi\)
\(102\) 0 0
\(103\) 9.89949i 0.975426i −0.873004 0.487713i \(-0.837831\pi\)
0.873004 0.487713i \(-0.162169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.48528 + 8.48528i 0.820303 + 0.820303i 0.986151 0.165848i \(-0.0530362\pi\)
−0.165848 + 0.986151i \(0.553036\pi\)
\(108\) 0 0
\(109\) −13.0000 + 13.0000i −1.24517 + 1.24517i −0.287348 + 0.957826i \(0.592774\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 7.07107 0.671156
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 1.00000i −0.0924500 0.0924500i
\(118\) 0 0
\(119\) 8.48528i 0.777844i
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 4.24264 + 4.24264i 0.382546 + 0.382546i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.89949 0.878438 0.439219 0.898380i \(-0.355255\pi\)
0.439219 + 0.898380i \(0.355255\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −8.48528 + 8.48528i −0.741362 + 0.741362i −0.972840 0.231478i \(-0.925644\pi\)
0.231478 + 0.972840i \(0.425644\pi\)
\(132\) 0 0
\(133\) 6.00000 + 6.00000i 0.520266 + 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) 2.82843 + 2.82843i 0.239904 + 0.239904i 0.816810 0.576906i \(-0.195740\pi\)
−0.576906 + 0.816810i \(0.695740\pi\)
\(140\) 0 0
\(141\) −6.00000 + 6.00000i −0.505291 + 0.505291i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.53553 3.53553i 0.291606 0.291606i
\(148\) 0 0
\(149\) −12.0000 12.0000i −0.983078 0.983078i 0.0167809 0.999859i \(-0.494658\pi\)
−0.999859 + 0.0167809i \(0.994658\pi\)
\(150\) 0 0
\(151\) 18.3848i 1.49613i −0.663624 0.748066i \(-0.730983\pi\)
0.663624 0.748066i \(-0.269017\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.00000 + 5.00000i −0.399043 + 0.399043i −0.877896 0.478852i \(-0.841053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) −8.48528 −0.672927
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −4.24264 + 4.24264i −0.332309 + 0.332309i −0.853463 0.521154i \(-0.825502\pi\)
0.521154 + 0.853463i \(0.325502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.9706i 1.31322i −0.754230 0.656611i \(-0.771989\pi\)
0.754230 0.656611i \(-0.228011\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) −4.24264 4.24264i −0.324443 0.324443i
\(172\) 0 0
\(173\) −12.0000 + 12.0000i −0.912343 + 0.912343i −0.996456 0.0841131i \(-0.973194\pi\)
0.0841131 + 0.996456i \(0.473194\pi\)
\(174\) 0 0
\(175\) 7.07107 0.534522
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.9706 + 16.9706i −1.26844 + 1.26844i −0.321545 + 0.946894i \(0.604202\pi\)
−0.946894 + 0.321545i \(0.895798\pi\)
\(180\) 0 0
\(181\) −11.0000 11.0000i −0.817624 0.817624i 0.168140 0.985763i \(-0.446224\pi\)
−0.985763 + 0.168140i \(0.946224\pi\)
\(182\) 0 0
\(183\) 7.07107i 0.522708i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 1.00000i 0.0727393 0.0727393i
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 + 6.00000i 0.427482 + 0.427482i 0.887770 0.460288i \(-0.152254\pi\)
−0.460288 + 0.887770i \(0.652254\pi\)
\(198\) 0 0
\(199\) 9.89949i 0.701757i 0.936421 + 0.350878i \(0.114117\pi\)
−0.936421 + 0.350878i \(0.885883\pi\)
\(200\) 0 0
\(201\) 4.00000i 0.282138i
\(202\) 0 0
\(203\) 8.48528 + 8.48528i 0.595550 + 0.595550i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.48528 0.589768
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −2.82843 + 2.82843i −0.194717 + 0.194717i −0.797731 0.603014i \(-0.793966\pi\)
0.603014 + 0.797731i \(0.293966\pi\)
\(212\) 0 0
\(213\) 6.00000 + 6.00000i 0.411113 + 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 6.00000i 0.403604 0.403604i
\(222\) 0 0
\(223\) −26.8701 −1.79935 −0.899676 0.436558i \(-0.856197\pi\)
−0.899676 + 0.436558i \(0.856197\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 8.48528 8.48528i 0.563188 0.563188i −0.367024 0.930212i \(-0.619623\pi\)
0.930212 + 0.367024i \(0.119623\pi\)
\(228\) 0 0
\(229\) 1.00000 + 1.00000i 0.0660819 + 0.0660819i 0.739375 0.673293i \(-0.235121\pi\)
−0.673293 + 0.739375i \(0.735121\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00000 1.00000i 0.0649570 0.0649570i
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −0.707107 + 0.707107i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.48528i 0.539906i
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) −16.9706 16.9706i −1.07117 1.07117i −0.997265 0.0739073i \(-0.976453\pi\)
−0.0739073 0.997265i \(-0.523547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −7.07107 + 7.07107i −0.439375 + 0.439375i
\(260\) 0 0
\(261\) −6.00000 6.00000i −0.371391 0.371391i
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.24264 + 4.24264i 0.259645 + 0.259645i
\(268\) 0 0
\(269\) 6.00000 6.00000i 0.365826 0.365826i −0.500126 0.865953i \(-0.666713\pi\)
0.865953 + 0.500126i \(0.166713\pi\)
\(270\) 0 0
\(271\) 26.8701 1.63224 0.816120 0.577883i \(-0.196121\pi\)
0.816120 + 0.577883i \(0.196121\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.0000 + 11.0000i 0.660926 + 0.660926i 0.955598 0.294672i \(-0.0952105\pi\)
−0.294672 + 0.955598i \(0.595211\pi\)
\(278\) 0 0
\(279\) 1.41421i 0.0846668i
\(280\) 0 0
\(281\) 18.0000i 1.07379i −0.843649 0.536895i \(-0.819597\pi\)
0.843649 0.536895i \(-0.180403\pi\)
\(282\) 0 0
\(283\) 2.82843 + 2.82843i 0.168133 + 0.168133i 0.786158 0.618026i \(-0.212067\pi\)
−0.618026 + 0.786158i \(0.712067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.48528 −0.500870
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −8.48528 + 8.48528i −0.497416 + 0.497416i
\(292\) 0 0
\(293\) 6.00000 + 6.00000i 0.350524 + 0.350524i 0.860304 0.509781i \(-0.170273\pi\)
−0.509781 + 0.860304i \(0.670273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.48528 + 8.48528i 0.490716 + 0.490716i
\(300\) 0 0
\(301\) 6.00000 6.00000i 0.345834 0.345834i
\(302\) 0 0
\(303\) −8.48528 −0.487467
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 14.1421 14.1421i 0.807134 0.807134i −0.177065 0.984199i \(-0.556660\pi\)
0.984199 + 0.177065i \(0.0566602\pi\)
\(308\) 0 0
\(309\) −7.00000 7.00000i −0.398216 0.398216i
\(310\) 0 0
\(311\) 16.9706i 0.962312i 0.876635 + 0.481156i \(0.159783\pi\)
−0.876635 + 0.481156i \(0.840217\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 + 6.00000i −0.336994 + 0.336994i −0.855235 0.518241i \(-0.826587\pi\)
0.518241 + 0.855235i \(0.326587\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 25.4558 25.4558i 1.41640 1.41640i
\(324\) 0 0
\(325\) −5.00000 5.00000i −0.277350 0.277350i
\(326\) 0 0
\(327\) 18.3848i 1.01668i
\(328\) 0 0
\(329\) 12.0000i 0.661581i
\(330\) 0 0
\(331\) −19.7990 19.7990i −1.08825 1.08825i −0.995709 0.0925421i \(-0.970501\pi\)
−0.0925421 0.995709i \(-0.529499\pi\)
\(332\) 0 0
\(333\) 5.00000 5.00000i 0.273998 0.273998i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.0000 1.30736 0.653682 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) 0 0
\(339\) 4.24264 4.24264i 0.230429 0.230429i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.48528 8.48528i −0.455514 0.455514i 0.441666 0.897180i \(-0.354388\pi\)
−0.897180 + 0.441666i \(0.854388\pi\)
\(348\) 0 0
\(349\) 19.0000 19.0000i 1.01705 1.01705i 0.0171945 0.999852i \(-0.494527\pi\)
0.999852 0.0171945i \(-0.00547346\pi\)
\(350\) 0 0
\(351\) −1.41421 −0.0754851
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6.00000 + 6.00000i 0.317554 + 0.317554i
\(358\) 0 0
\(359\) 25.4558i 1.34351i −0.740774 0.671754i \(-0.765541\pi\)
0.740774 0.671754i \(-0.234459\pi\)
\(360\) 0 0
\(361\) 17.0000i 0.894737i
\(362\) 0 0
\(363\) −7.77817 7.77817i −0.408248 0.408248i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −32.5269 −1.69789 −0.848945 0.528480i \(-0.822762\pi\)
−0.848945 + 0.528480i \(0.822762\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 8.48528 8.48528i 0.440534 0.440534i
\(372\) 0 0
\(373\) −5.00000 5.00000i −0.258890 0.258890i 0.565712 0.824603i \(-0.308601\pi\)
−0.824603 + 0.565712i \(0.808601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 21.2132 + 21.2132i 1.08965 + 1.08965i 0.995564 + 0.0940849i \(0.0299925\pi\)
0.0940849 + 0.995564i \(0.470007\pi\)
\(380\) 0 0
\(381\) 7.00000 7.00000i 0.358621 0.358621i
\(382\) 0 0
\(383\) 33.9411 1.73431 0.867155 0.498038i \(-0.165946\pi\)
0.867155 + 0.498038i \(0.165946\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.24264 + 4.24264i −0.215666 + 0.215666i
\(388\) 0 0
\(389\) 12.0000 + 12.0000i 0.608424 + 0.608424i 0.942534 0.334110i \(-0.108436\pi\)
−0.334110 + 0.942534i \(0.608436\pi\)
\(390\) 0 0
\(391\) 50.9117i 2.57471i
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.00000 + 7.00000i −0.351320 + 0.351320i −0.860601 0.509281i \(-0.829912\pi\)
0.509281 + 0.860601i \(0.329912\pi\)
\(398\) 0 0
\(399\) 8.48528 0.424795
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −1.41421 + 1.41421i −0.0704470 + 0.0704470i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 12.7279 + 12.7279i 0.627822 + 0.627822i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 25.4558 25.4558i 1.24360 1.24360i 0.285102 0.958497i \(-0.407972\pi\)
0.958497 0.285102i \(-0.0920276\pi\)
\(420\) 0 0
\(421\) −11.0000 11.0000i −0.536107 0.536107i 0.386276 0.922383i \(-0.373761\pi\)
−0.922383 + 0.386276i \(0.873761\pi\)
\(422\) 0 0
\(423\) 8.48528i 0.412568i
\(424\) 0 0
\(425\) 30.0000i 1.45521i
\(426\) 0 0
\(427\) 7.07107 + 7.07107i 0.342193 + 0.342193i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.48528 −0.408722 −0.204361 0.978896i \(-0.565512\pi\)
−0.204361 + 0.978896i \(0.565512\pi\)
\(432\) 0 0
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 36.0000 + 36.0000i 1.72211 + 1.72211i
\(438\) 0 0
\(439\) 7.07107i 0.337484i 0.985660 + 0.168742i \(0.0539704\pi\)
−0.985660 + 0.168742i \(0.946030\pi\)
\(440\) 0 0
\(441\) 5.00000i 0.238095i
\(442\) 0 0
\(443\) −8.48528 8.48528i −0.403148 0.403148i 0.476193 0.879341i \(-0.342016\pi\)
−0.879341 + 0.476193i \(0.842016\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −16.9706 −0.802680
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −13.0000 13.0000i −0.610793 0.610793i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) 0 0
\(459\) −4.24264 4.24264i −0.198030 0.198030i
\(460\) 0 0
\(461\) 24.0000 24.0000i 1.11779 1.11779i 0.125726 0.992065i \(-0.459874\pi\)
0.992065 0.125726i \(-0.0401262\pi\)
\(462\) 0 0
\(463\) 7.07107 0.328620 0.164310 0.986409i \(-0.447460\pi\)
0.164310 + 0.986409i \(0.447460\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.48528 8.48528i 0.392652 0.392652i −0.482980 0.875632i \(-0.660445\pi\)
0.875632 + 0.482980i \(0.160445\pi\)
\(468\) 0 0
\(469\) 4.00000 + 4.00000i 0.184703 + 0.184703i
\(470\) 0 0
\(471\) 7.07107i 0.325818i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −21.2132 21.2132i −0.973329 0.973329i
\(476\) 0 0
\(477\) −6.00000 + 6.00000i −0.274721 + 0.274721i
\(478\) 0 0
\(479\) −8.48528 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) −8.48528 + 8.48528i −0.386094 + 0.386094i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18.3848i 0.833094i −0.909114 0.416547i \(-0.863240\pi\)
0.909114 0.416547i \(-0.136760\pi\)
\(488\) 0 0
\(489\) 6.00000i 0.271329i
\(490\) 0 0
\(491\) −16.9706 16.9706i −0.765871 0.765871i 0.211506 0.977377i \(-0.432163\pi\)
−0.977377 + 0.211506i \(0.932163\pi\)
\(492\) 0 0
\(493\) 36.0000 36.0000i 1.62136 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 2.82843 2.82843i 0.126618 0.126618i −0.640958 0.767576i \(-0.721463\pi\)
0.767576 + 0.640958i \(0.221463\pi\)
\(500\) 0 0
\(501\) −12.0000 12.0000i −0.536120 0.536120i
\(502\) 0 0
\(503\) 8.48528i 0.378340i 0.981944 + 0.189170i \(0.0605797\pi\)
−0.981944 + 0.189170i \(0.939420\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.77817 + 7.77817i 0.345441 + 0.345441i
\(508\) 0 0
\(509\) −18.0000 + 18.0000i −0.797836 + 0.797836i −0.982754 0.184918i \(-0.940798\pi\)
0.184918 + 0.982754i \(0.440798\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.00000 −0.264906
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 16.9706i 0.744925i
\(520\) 0 0
\(521\) 30.0000i 1.31432i −0.753749 0.657162i \(-0.771757\pi\)
0.753749 0.657162i \(-0.228243\pi\)
\(522\) 0 0
\(523\) 12.7279 + 12.7279i 0.556553 + 0.556553i 0.928324 0.371771i \(-0.121249\pi\)
−0.371771 + 0.928324i \(0.621249\pi\)
\(524\) 0 0
\(525\) 5.00000 5.00000i 0.218218 0.218218i
\(526\) 0 0
\(527\) −8.48528 −0.369625
\(528\) 0 0
\(529\) −49.0000 −2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 + 6.00000i 0.259889 + 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 24.0000i 1.03568i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −25.0000 + 25.0000i −1.07483 + 1.07483i −0.0778705 + 0.996963i \(0.524812\pi\)
−0.996963 + 0.0778705i \(0.975188\pi\)
\(542\) 0 0
\(543\) −15.5563 −0.667587
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12.7279 + 12.7279i −0.544207 + 0.544207i −0.924759 0.380553i \(-0.875734\pi\)
0.380553 + 0.924759i \(0.375734\pi\)
\(548\) 0 0
\(549\) −5.00000 5.00000i −0.213395 0.213395i
\(550\) 0 0
\(551\) 50.9117i 2.16891i
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 + 12.0000i −0.508456 + 0.508456i −0.914052 0.405596i \(-0.867064\pi\)
0.405596 + 0.914052i \(0.367064\pi\)
\(558\) 0 0
\(559\) −8.48528 −0.358889
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.41421i 0.0593914i
\(568\) 0 0
\(569\) 18.0000i 0.754599i 0.926091 + 0.377300i \(0.123147\pi\)
−0.926091 + 0.377300i \(0.876853\pi\)
\(570\) 0 0
\(571\) −2.82843 2.82843i −0.118366 0.118366i 0.645443 0.763809i \(-0.276673\pi\)
−0.763809 + 0.645443i \(0.776673\pi\)
\(572\) 0 0
\(573\) 12.0000 12.0000i 0.501307 0.501307i
\(574\) 0 0
\(575\) 42.4264 1.76930
\(576\) 0 0
\(577\) 36.0000 1.49870 0.749350 0.662174i \(-0.230366\pi\)
0.749350 + 0.662174i \(0.230366\pi\)
\(578\) 0 0
\(579\) −9.89949 + 9.89949i −0.411409 + 0.411409i
\(580\) 0 0
\(581\) −12.0000 12.0000i −0.497844 0.497844i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) −6.00000 + 6.00000i −0.247226 + 0.247226i
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000 + 7.00000i 0.286491 + 0.286491i
\(598\) 0 0
\(599\) 25.4558i 1.04010i −0.854137 0.520049i \(-0.825914\pi\)
0.854137 0.520049i \(-0.174086\pi\)
\(600\) 0 0
\(601\) 12.0000i 0.489490i 0.969587 + 0.244745i \(0.0787043\pi\)
−0.969587 + 0.244745i \(0.921296\pi\)
\(602\) 0 0
\(603\) −2.82843 2.82843i −0.115182 0.115182i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −24.0416 −0.975820 −0.487910 0.872894i \(-0.662241\pi\)
−0.487910 + 0.872894i \(0.662241\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −8.48528 + 8.48528i −0.343278 + 0.343278i
\(612\) 0 0
\(613\) −19.0000 19.0000i −0.767403 0.767403i 0.210246 0.977649i \(-0.432574\pi\)
−0.977649 + 0.210246i \(0.932574\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) −19.7990 19.7990i −0.795789 0.795789i 0.186640 0.982428i \(-0.440240\pi\)
−0.982428 + 0.186640i \(0.940240\pi\)
\(620\) 0 0
\(621\) 6.00000 6.00000i 0.240772 0.240772i
\(622\) 0 0
\(623\) −8.48528 −0.339956
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 30.0000 + 30.0000i 1.19618 + 1.19618i
\(630\) 0 0
\(631\) 18.3848i 0.731886i 0.930637 + 0.365943i \(0.119254\pi\)
−0.930637 + 0.365943i \(0.880746\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.00000 5.00000i 0.198107 0.198107i
\(638\) 0 0
\(639\) 8.48528 0.335673
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) −12.7279 + 12.7279i −0.501940 + 0.501940i −0.912040 0.410100i \(-0.865494\pi\)
0.410100 + 0.912040i \(0.365494\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.48528i 0.333591i −0.985992 0.166795i \(-0.946658\pi\)
0.985992 0.166795i \(-0.0533419\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.41421 1.41421i −0.0554274 0.0554274i
\(652\) 0 0
\(653\) −12.0000 + 12.0000i −0.469596 + 0.469596i −0.901784 0.432187i \(-0.857742\pi\)
0.432187 + 0.901784i \(0.357742\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0 0
\(661\) −7.00000 7.00000i −0.272268 0.272268i 0.557744 0.830013i \(-0.311667\pi\)
−0.830013 + 0.557744i \(0.811667\pi\)
\(662\) 0 0
\(663\) 8.48528i 0.329541i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 50.9117 + 50.9117i 1.97131 + 1.97131i
\(668\) 0 0
\(669\) −19.0000 + 19.0000i −0.734582 + 0.734582i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) −3.53553 + 3.53553i −0.136083 + 0.136083i
\(676\) 0 0
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 12.0000i 0.459841i
\(682\) 0 0
\(683\) 25.4558 + 25.4558i 0.974041 + 0.974041i 0.999671 0.0256307i \(-0.00815939\pi\)
−0.0256307 + 0.999671i \(0.508159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.41421 0.0539556
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −29.6985 + 29.6985i −1.12978 + 1.12978i −0.139572 + 0.990212i \(0.544573\pi\)
−0.990212 + 0.139572i \(0.955427\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) 4.24264 + 4.24264i 0.160471 + 0.160471i
\(700\) 0 0
\(701\) −18.0000 + 18.0000i −0.679851 + 0.679851i −0.959966 0.280116i \(-0.909627\pi\)
0.280116 + 0.959966i \(0.409627\pi\)
\(702\) 0 0
\(703\) 42.4264 1.60014
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.48528 8.48528i 0.319122 0.319122i
\(708\) 0 0
\(709\) −1.00000 1.00000i −0.0375558 0.0375558i 0.688080 0.725635i \(-0.258454\pi\)
−0.725635 + 0.688080i \(0.758454\pi\)
\(710\) 0 0
\(711\) 1.41421i 0.0530372i
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 + 12.0000i −0.448148 + 0.448148i
\(718\) 0 0
\(719\) 8.48528 0.316448 0.158224 0.987403i \(-0.449423\pi\)
0.158224 + 0.987403i \(0.449423\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) 0 0
\(723\) −7.07107 + 7.07107i −0.262976 + 0.262976i
\(724\) 0 0
\(725\) −30.0000 30.0000i −1.11417 1.11417i
\(726\) 0 0
\(727\) 32.5269i 1.20636i −0.797606 0.603178i \(-0.793901\pi\)
0.797606 0.603178i \(-0.206099\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −25.4558 25.4558i −0.941518 0.941518i
\(732\) 0 0
\(733\) −35.0000 + 35.0000i −1.29275 + 1.29275i −0.359678 + 0.933076i \(0.617113\pi\)
−0.933076 + 0.359678i \(0.882887\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −14.1421 + 14.1421i −0.520227 + 0.520227i −0.917640 0.397413i \(-0.869908\pi\)
0.397413 + 0.917640i \(0.369908\pi\)
\(740\) 0 0
\(741\) −6.00000 6.00000i −0.220416 0.220416i
\(742\) 0 0
\(743\) 16.9706i 0.622590i −0.950313 0.311295i \(-0.899237\pi\)
0.950313 0.311295i \(-0.100763\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.48528 + 8.48528i 0.310460 + 0.310460i
\(748\) 0 0
\(749\) −12.0000 + 12.0000i −0.438470 + 0.438470i
\(750\) 0 0
\(751\) 49.4975 1.80619 0.903094 0.429442i \(-0.141290\pi\)
0.903094 + 0.429442i \(0.141290\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000 + 1.00000i 0.0363456 + 0.0363456i 0.725046 0.688700i \(-0.241818\pi\)
−0.688700 + 0.725046i \(0.741818\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000i 0.217500i −0.994069 0.108750i \(-0.965315\pi\)
0.994069 0.108750i \(-0.0346848\pi\)
\(762\) 0 0
\(763\) −18.3848 18.3848i −0.665574 0.665574i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −4.24264 + 4.24264i −0.152795 + 0.152795i
\(772\) 0 0
\(773\) −18.0000 18.0000i −0.647415 0.647415i 0.304953 0.952368i \(-0.401359\pi\)
−0.952368 + 0.304953i \(0.901359\pi\)
\(774\) 0 0
\(775\) 7.07107i 0.254000i
\(776\) 0 0
\(777\) 10.0000i 0.358748i
\(778\) 0 0
\(779\) 25.4558 + 25.4558i 0.912050 + 0.912050i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.48528 −0.303239
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.7279 12.7279i 0.453701 0.453701i −0.442880 0.896581i \(-0.646043\pi\)
0.896581 + 0.442880i \(0.146043\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.48528i 0.301702i
\(792\) 0 0
\(793\) 10.0000i 0.355110i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 12.0000i 0.425062 0.425062i −0.461880 0.886942i \(-0.652825\pi\)
0.886942 + 0.461880i \(0.152825\pi\)
\(798\) 0 0
\(799\) −50.9117 −1.80113
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0