Properties

Label 2304.2.k.i.1729.1
Level $2304$
Weight $2$
Character 2304.1729
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1729.1
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1729
Dual form 2304.2.k.i.577.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.44949 - 2.44949i) q^{5} -1.41421i q^{7} +O(q^{10})\) \(q+(-2.44949 - 2.44949i) q^{5} -1.41421i q^{7} +(3.46410 + 3.46410i) q^{11} +(1.00000 - 1.00000i) q^{13} +4.89898 q^{17} +(4.24264 - 4.24264i) q^{19} +6.92820i q^{23} +7.00000i q^{25} +(-2.44949 + 2.44949i) q^{29} +1.41421 q^{31} +(-3.46410 + 3.46410i) q^{35} +(-7.00000 - 7.00000i) q^{37} +4.89898i q^{41} +(-4.24264 - 4.24264i) q^{43} +6.92820 q^{47} +5.00000 q^{49} +(2.44949 + 2.44949i) q^{53} -16.9706i q^{55} +(-6.92820 - 6.92820i) q^{59} +(5.00000 - 5.00000i) q^{61} -4.89898 q^{65} +(5.65685 - 5.65685i) q^{67} -13.8564i q^{71} +12.0000i q^{73} +(4.89898 - 4.89898i) q^{77} +15.5563 q^{79} +(10.3923 - 10.3923i) q^{83} +(-12.0000 - 12.0000i) q^{85} -9.79796i q^{89} +(-1.41421 - 1.41421i) q^{91} -20.7846 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{13} - 56q^{37} + 40q^{49} + 40q^{61} - 96q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 0
\(4\) 0 0
\(5\) −2.44949 2.44949i −1.09545 1.09545i −0.994936 0.100509i \(-0.967953\pi\)
−0.100509 0.994936i \(-0.532047\pi\)
\(6\) 0 0
\(7\) 1.41421i 0.534522i −0.963624 0.267261i \(-0.913881\pi\)
0.963624 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 + 3.46410i 1.04447 + 1.04447i 0.998964 + 0.0455017i \(0.0144886\pi\)
0.0455017 + 0.998964i \(0.485511\pi\)
\(12\) 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
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 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
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820i 1.44463i 0.691564 + 0.722315i \(0.256922\pi\)
−0.691564 + 0.722315i \(0.743078\pi\)
\(24\) 0 0
\(25\) 7.00000i 1.40000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.44949 + 2.44949i −0.454859 + 0.454859i −0.896963 0.442105i \(-0.854232\pi\)
0.442105 + 0.896963i \(0.354232\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 + 3.46410i −0.585540 + 0.585540i
\(36\) 0 0
\(37\) −7.00000 7.00000i −1.15079 1.15079i −0.986394 0.164399i \(-0.947432\pi\)
−0.164399 0.986394i \(-0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.89898i 0.765092i 0.923936 + 0.382546i \(0.124953\pi\)
−0.923936 + 0.382546i \(0.875047\pi\)
\(42\) 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
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.44949 + 2.44949i 0.336463 + 0.336463i 0.855034 0.518571i \(-0.173536\pi\)
−0.518571 + 0.855034i \(0.673536\pi\)
\(54\) 0 0
\(55\) 16.9706i 2.28831i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.92820 6.92820i −0.901975 0.901975i 0.0936317 0.995607i \(-0.470152\pi\)
−0.995607 + 0.0936317i \(0.970152\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 0 0
\(65\) −4.89898 −0.607644
\(66\) 0 0
\(67\) 5.65685 5.65685i 0.691095 0.691095i −0.271378 0.962473i \(-0.587479\pi\)
0.962473 + 0.271378i \(0.0874794\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i −0.569160 0.822226i \(-0.692732\pi\)
0.569160 0.822226i \(-0.307268\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i 0.711934 + 0.702247i \(0.247820\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.89898 4.89898i 0.558291 0.558291i
\(78\) 0 0
\(79\) 15.5563 1.75023 0.875113 0.483919i \(-0.160787\pi\)
0.875113 + 0.483919i \(0.160787\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3923 10.3923i 1.14070 1.14070i 0.152382 0.988322i \(-0.451306\pi\)
0.988322 0.152382i \(-0.0486944\pi\)
\(84\) 0 0
\(85\) −12.0000 12.0000i −1.30158 1.30158i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.79796i 1.03858i −0.854598 0.519291i \(-0.826196\pi\)
0.854598 0.519291i \(-0.173804\pi\)
\(90\) 0 0
\(91\) −1.41421 1.41421i −0.148250 0.148250i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.7846 −2.13246
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.34847 + 7.34847i 0.731200 + 0.731200i 0.970858 0.239657i \(-0.0770351\pi\)
−0.239657 + 0.970858i \(0.577035\pi\)
\(102\) 0 0
\(103\) 7.07107i 0.696733i −0.937358 0.348367i \(-0.886736\pi\)
0.937358 0.348367i \(-0.113264\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) −1.00000 + 1.00000i −0.0957826 + 0.0957826i −0.753374 0.657592i \(-0.771575\pi\)
0.657592 + 0.753374i \(0.271575\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.5959 −1.84343 −0.921714 0.387869i \(-0.873211\pi\)
−0.921714 + 0.387869i \(0.873211\pi\)
\(114\) 0 0
\(115\) 16.9706 16.9706i 1.58251 1.58251i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.92820i 0.635107i
\(120\) 0 0
\(121\) 13.0000i 1.18182i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.89898 4.89898i 0.438178 0.438178i
\(126\) 0 0
\(127\) −9.89949 −0.878438 −0.439219 0.898380i \(-0.644745\pi\)
−0.439219 + 0.898380i \(0.644745\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.92820 + 6.92820i −0.605320 + 0.605320i −0.941719 0.336399i \(-0.890791\pi\)
0.336399 + 0.941719i \(0.390791\pi\)
\(132\) 0 0
\(133\) −6.00000 6.00000i −0.520266 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.6969i 1.25564i −0.778357 0.627822i \(-0.783947\pi\)
0.778357 0.627822i \(-0.216053\pi\)
\(138\) 0 0
\(139\) −11.3137 11.3137i −0.959616 0.959616i 0.0395994 0.999216i \(-0.487392\pi\)
−0.999216 + 0.0395994i \(0.987392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.92820 0.579365
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.44949 2.44949i −0.200670 0.200670i 0.599617 0.800287i \(-0.295320\pi\)
−0.800287 + 0.599617i \(0.795320\pi\)
\(150\) 0 0
\(151\) 15.5563i 1.26596i −0.774169 0.632979i \(-0.781832\pi\)
0.774169 0.632979i \(-0.218168\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.46410 3.46410i −0.278243 0.278243i
\(156\) 0 0
\(157\) 7.00000 7.00000i 0.558661 0.558661i −0.370265 0.928926i \(-0.620733\pi\)
0.928926 + 0.370265i \(0.120733\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.79796 0.772187
\(162\) 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
\(167\) 6.92820i 0.536120i 0.963402 + 0.268060i \(0.0863826\pi\)
−0.963402 + 0.268060i \(0.913617\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.2474 + 12.2474i −0.931156 + 0.931156i −0.997778 0.0666220i \(-0.978778\pi\)
0.0666220 + 0.997778i \(0.478778\pi\)
\(174\) 0 0
\(175\) 9.89949 0.748331
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(180\) 0 0
\(181\) 1.00000 + 1.00000i 0.0743294 + 0.0743294i 0.743294 0.668965i \(-0.233262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 34.2929i 2.52126i
\(186\) 0 0
\(187\) 16.9706 + 16.9706i 1.24101 + 1.24101i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.44949 2.44949i −0.174519 0.174519i 0.614443 0.788962i \(-0.289381\pi\)
−0.788962 + 0.614443i \(0.789381\pi\)
\(198\) 0 0
\(199\) 7.07107i 0.501255i 0.968084 + 0.250627i \(0.0806369\pi\)
−0.968084 + 0.250627i \(0.919363\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.46410 + 3.46410i 0.243132 + 0.243132i
\(204\) 0 0
\(205\) 12.0000 12.0000i 0.838116 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.3939 2.03322
\(210\) 0 0
\(211\) 11.3137 11.3137i 0.778868 0.778868i −0.200770 0.979638i \(-0.564345\pi\)
0.979638 + 0.200770i \(0.0643445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.7846i 1.41750i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.89898 4.89898i 0.329541 0.329541i
\(222\) 0 0
\(223\) −7.07107 −0.473514 −0.236757 0.971569i \(-0.576084\pi\)
−0.236757 + 0.971569i \(0.576084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.46410 3.46410i 0.229920 0.229920i −0.582739 0.812659i \(-0.698019\pi\)
0.812659 + 0.582739i \(0.198019\pi\)
\(228\) 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
\(233\) 9.79796i 0.641886i 0.947099 + 0.320943i \(0.104000\pi\)
−0.947099 + 0.320943i \(0.896000\pi\)
\(234\) 0 0
\(235\) −16.9706 16.9706i −1.10704 1.10704i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −12.2474 12.2474i −0.782461 0.782461i
\(246\) 0 0
\(247\) 8.48528i 0.539906i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205 + 17.3205i 1.09326 + 1.09326i 0.995178 + 0.0980825i \(0.0312709\pi\)
0.0980825 + 0.995178i \(0.468729\pi\)
\(252\) 0 0
\(253\) −24.0000 + 24.0000i −1.50887 + 1.50887i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.79796 0.611180 0.305590 0.952163i \(-0.401146\pi\)
0.305590 + 0.952163i \(0.401146\pi\)
\(258\) 0 0
\(259\) −9.89949 + 9.89949i −0.615125 + 0.615125i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.8564i 0.854423i 0.904152 + 0.427211i \(0.140504\pi\)
−0.904152 + 0.427211i \(0.859496\pi\)
\(264\) 0 0
\(265\) 12.0000i 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.34847 + 7.34847i −0.448044 + 0.448044i −0.894704 0.446660i \(-0.852613\pi\)
0.446660 + 0.894704i \(0.352613\pi\)
\(270\) 0 0
\(271\) 7.07107 0.429537 0.214768 0.976665i \(-0.431100\pi\)
0.214768 + 0.976665i \(0.431100\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.2487 + 24.2487i −1.46225 + 1.46225i
\(276\) 0 0
\(277\) −1.00000 1.00000i −0.0600842 0.0600842i 0.676426 0.736510i \(-0.263528\pi\)
−0.736510 + 0.676426i \(0.763528\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 29.3939i 1.75349i −0.480954 0.876746i \(-0.659710\pi\)
0.480954 0.876746i \(-0.340290\pi\)
\(282\) 0 0
\(283\) 5.65685 + 5.65685i 0.336265 + 0.336265i 0.854960 0.518695i \(-0.173582\pi\)
−0.518695 + 0.854960i \(0.673582\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.92820 0.408959
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −12.2474 12.2474i −0.715504 0.715504i 0.252177 0.967681i \(-0.418853\pi\)
−0.967681 + 0.252177i \(0.918853\pi\)
\(294\) 0 0
\(295\) 33.9411i 1.97613i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 + 6.92820i 0.400668 + 0.400668i
\(300\) 0 0
\(301\) −6.00000 + 6.00000i −0.345834 + 0.345834i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.4949 −1.40257
\(306\) 0 0
\(307\) −5.65685 + 5.65685i −0.322854 + 0.322854i −0.849861 0.527007i \(-0.823314\pi\)
0.527007 + 0.849861i \(0.323314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.8564i 0.785725i −0.919597 0.392862i \(-0.871485\pi\)
0.919597 0.392862i \(-0.128515\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.34847 + 7.34847i −0.412731 + 0.412731i −0.882689 0.469958i \(-0.844269\pi\)
0.469958 + 0.882689i \(0.344269\pi\)
\(318\) 0 0
\(319\) −16.9706 −0.950169
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.7846 20.7846i 1.15649 1.15649i
\(324\) 0 0
\(325\) 7.00000 + 7.00000i 0.388290 + 0.388290i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.79796i 0.540179i
\(330\) 0 0
\(331\) 11.3137 + 11.3137i 0.621858 + 0.621858i 0.946006 0.324149i \(-0.105078\pi\)
−0.324149 + 0.946006i \(0.605078\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.7128 −1.51411
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.89898 + 4.89898i 0.265295 + 0.265295i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.46410 + 3.46410i 0.185963 + 0.185963i 0.793948 0.607985i \(-0.208022\pi\)
−0.607985 + 0.793948i \(0.708022\pi\)
\(348\) 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
\(351\) 0 0
\(352\) 0 0
\(353\) −9.79796 −0.521493 −0.260746 0.965407i \(-0.583969\pi\)
−0.260746 + 0.965407i \(0.583969\pi\)
\(354\) 0 0
\(355\) −33.9411 + 33.9411i −1.80141 + 1.80141i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 17.0000i 0.894737i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 29.3939 29.3939i 1.53855 1.53855i
\(366\) 0 0
\(367\) 32.5269 1.69789 0.848945 0.528480i \(-0.177238\pi\)
0.848945 + 0.528480i \(0.177238\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.46410 3.46410i 0.179847 0.179847i
\(372\) 0 0
\(373\) 19.0000 + 19.0000i 0.983783 + 0.983783i 0.999871 0.0160879i \(-0.00512115\pi\)
−0.0160879 + 0.999871i \(0.505121\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.89898i 0.252310i
\(378\) 0 0
\(379\) 4.24264 + 4.24264i 0.217930 + 0.217930i 0.807626 0.589696i \(-0.200752\pi\)
−0.589696 + 0.807626i \(0.700752\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.92820 0.354015 0.177007 0.984210i \(-0.443358\pi\)
0.177007 + 0.984210i \(0.443358\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.1464 + 17.1464i 0.869358 + 0.869358i 0.992401 0.123043i \(-0.0392653\pi\)
−0.123043 + 0.992401i \(0.539265\pi\)
\(390\) 0 0
\(391\) 33.9411i 1.71648i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −38.1051 38.1051i −1.91728 1.91728i
\(396\) 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
\(399\) 0 0
\(400\) 0 0
\(401\) 14.6969 0.733930 0.366965 0.930235i \(-0.380397\pi\)
0.366965 + 0.930235i \(0.380397\pi\)
\(402\) 0 0
\(403\) 1.41421 1.41421i 0.0704470 0.0704470i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 48.4974i 2.40393i
\(408\) 0 0
\(409\) 24.0000i 1.18672i −0.804936 0.593362i \(-0.797800\pi\)
0.804936 0.593362i \(-0.202200\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.79796 + 9.79796i −0.482126 + 0.482126i
\(414\) 0 0
\(415\) −50.9117 −2.49916
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923 10.3923i 0.507697 0.507697i −0.406122 0.913819i \(-0.633119\pi\)
0.913819 + 0.406122i \(0.133119\pi\)
\(420\) 0 0
\(421\) 25.0000 + 25.0000i 1.21843 + 1.21843i 0.968183 + 0.250242i \(0.0805102\pi\)
0.250242 + 0.968183i \(0.419490\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 34.2929i 1.66345i
\(426\) 0 0
\(427\) −7.07107 7.07107i −0.342193 0.342193i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 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
\(437\) 29.3939 + 29.3939i 1.40610 + 1.40610i
\(438\) 0 0
\(439\) 26.8701i 1.28244i 0.767358 + 0.641219i \(0.221571\pi\)
−0.767358 + 0.641219i \(0.778429\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.3923 10.3923i −0.493753 0.493753i 0.415733 0.909487i \(-0.363525\pi\)
−0.909487 + 0.415733i \(0.863525\pi\)
\(444\) 0 0
\(445\) −24.0000 + 24.0000i −1.13771 + 1.13771i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −34.2929 −1.61838 −0.809190 0.587547i \(-0.800094\pi\)
−0.809190 + 0.587547i \(0.800094\pi\)
\(450\) 0 0
\(451\) −16.9706 + 16.9706i −0.799113 + 0.799113i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.92820i 0.324799i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.44949 2.44949i 0.114084 0.114084i −0.647760 0.761844i \(-0.724294\pi\)
0.761844 + 0.647760i \(0.224294\pi\)
\(462\) 0 0
\(463\) −24.0416 −1.11731 −0.558655 0.829400i \(-0.688682\pi\)
−0.558655 + 0.829400i \(0.688682\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.46410 + 3.46410i −0.160300 + 0.160300i −0.782699 0.622400i \(-0.786158\pi\)
0.622400 + 0.782699i \(0.286158\pi\)
\(468\) 0 0
\(469\) −8.00000 8.00000i −0.369406 0.369406i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 29.3939i 1.35153i
\(474\) 0 0
\(475\) 29.6985 + 29.6985i 1.36266 + 1.36266i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.7128 1.26623 0.633115 0.774057i \(-0.281776\pi\)
0.633115 + 0.774057i \(0.281776\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 35.3553i 1.60210i 0.598595 + 0.801052i \(0.295726\pi\)
−0.598595 + 0.801052i \(0.704274\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20.7846 20.7846i −0.937996 0.937996i 0.0601906 0.998187i \(-0.480829\pi\)
−0.998187 + 0.0601906i \(0.980829\pi\)
\(492\) 0 0
\(493\) −12.0000 + 12.0000i −0.540453 + 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.5959 −0.878997
\(498\) 0 0
\(499\) 5.65685 5.65685i 0.253236 0.253236i −0.569060 0.822296i \(-0.692693\pi\)
0.822296 + 0.569060i \(0.192693\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 36.0000i 1.60198i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.34847 + 7.34847i −0.325715 + 0.325715i −0.850955 0.525239i \(-0.823976\pi\)
0.525239 + 0.850955i \(0.323976\pi\)
\(510\) 0 0
\(511\) 16.9706 0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.3205 + 17.3205i −0.763233 + 0.763233i
\(516\) 0 0
\(517\) 24.0000 + 24.0000i 1.05552 + 1.05552i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.6969i 0.643885i −0.946759 0.321942i \(-0.895664\pi\)
0.946759 0.321942i \(-0.104336\pi\)
\(522\) 0 0
\(523\) −21.2132 21.2132i −0.927589 0.927589i 0.0699611 0.997550i \(-0.477712\pi\)
−0.997550 + 0.0699611i \(0.977712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.92820 0.301797
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.89898 + 4.89898i 0.212198 + 0.212198i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.3205 + 17.3205i 0.746047 + 0.746047i
\(540\) 0 0
\(541\) 11.0000 11.0000i 0.472927 0.472927i −0.429934 0.902861i \(-0.641463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.89898 0.209849
\(546\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) 20.7846i 0.885454i
\(552\) 0 0
\(553\) 22.0000i 0.935535i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.34847 + 7.34847i −0.311365 + 0.311365i −0.845438 0.534073i \(-0.820661\pi\)
0.534073 + 0.845438i \(0.320661\pi\)
\(558\) 0 0
\(559\) −8.48528 −0.358889
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.46410 3.46410i 0.145994 0.145994i −0.630332 0.776326i \(-0.717081\pi\)
0.776326 + 0.630332i \(0.217081\pi\)
\(564\) 0 0
\(565\) 48.0000 + 48.0000i 2.01938 + 2.01938i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.6969i 0.616128i 0.951366 + 0.308064i \(0.0996810\pi\)
−0.951366 + 0.308064i \(0.900319\pi\)
\(570\) 0 0
\(571\) 28.2843 + 28.2843i 1.18366 + 1.18366i 0.978789 + 0.204871i \(0.0656775\pi\)
0.204871 + 0.978789i \(0.434323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −48.4974 −2.02248
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.6969 14.6969i −0.609732 0.609732i
\(582\) 0 0
\(583\) 16.9706i 0.702849i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.7128 + 27.7128i 1.14383 + 1.14383i 0.987743 + 0.156087i \(0.0498880\pi\)
0.156087 + 0.987743i \(0.450112\pi\)
\(588\) 0 0
\(589\) 6.00000 6.00000i 0.247226 0.247226i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.1918 1.60942 0.804708 0.593671i \(-0.202322\pi\)
0.804708 + 0.593671i \(0.202322\pi\)
\(594\) 0 0
\(595\) −16.9706 + 16.9706i −0.695725 + 0.695725i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.92820i 0.283079i −0.989933 0.141539i \(-0.954795\pi\)
0.989933 0.141539i \(-0.0452052\pi\)
\(600\) 0 0
\(601\) 12.0000i 0.489490i 0.969587 + 0.244745i \(0.0787043\pi\)
−0.969587 + 0.244745i \(0.921296\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.8434 31.8434i 1.29462 1.29462i
\(606\) 0 0
\(607\) −9.89949 −0.401808 −0.200904 0.979611i \(-0.564388\pi\)
−0.200904 + 0.979611i \(0.564388\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.92820 6.92820i 0.280285 0.280285i
\(612\) 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
\(617\) 19.5959i 0.788902i −0.918917 0.394451i \(-0.870935\pi\)
0.918917 0.394451i \(-0.129065\pi\)
\(618\) 0 0
\(619\) 11.3137 + 11.3137i 0.454736 + 0.454736i 0.896923 0.442187i \(-0.145797\pi\)
−0.442187 + 0.896923i \(0.645797\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −34.2929 34.2929i −1.36735 1.36735i
\(630\) 0 0
\(631\) 1.41421i 0.0562990i −0.999604 0.0281495i \(-0.991039\pi\)
0.999604 0.0281495i \(-0.00896144\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.2487 + 24.2487i 0.962281 + 0.962281i
\(636\) 0 0
\(637\) 5.00000 5.00000i 0.198107 0.198107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.6969 0.580494 0.290247 0.956952i \(-0.406263\pi\)
0.290247 + 0.956952i \(0.406263\pi\)
\(642\) 0 0
\(643\) −29.6985 + 29.6985i −1.17119 + 1.17119i −0.189269 + 0.981925i \(0.560612\pi\)
−0.981925 + 0.189269i \(0.939388\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.92820i 0.272376i 0.990683 + 0.136188i \(0.0434851\pi\)
−0.990683 + 0.136188i \(0.956515\pi\)
\(648\) 0 0
\(649\) 48.0000i 1.88416i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −26.9444 + 26.9444i −1.05442 + 1.05442i −0.0559837 + 0.998432i \(0.517829\pi\)
−0.998432 + 0.0559837i \(0.982171\pi\)
\(654\) 0 0
\(655\) 33.9411 1.32619
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.92820 6.92820i 0.269884 0.269884i −0.559169 0.829054i \(-0.688880\pi\)
0.829054 + 0.559169i \(0.188880\pi\)
\(660\) 0 0
\(661\) −19.0000 19.0000i −0.739014 0.739014i 0.233373 0.972387i \(-0.425024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 29.3939i 1.13985i
\(666\) 0 0
\(667\) −16.9706 16.9706i −0.657103 0.657103i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 34.6410 1.33730
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26.9444 26.9444i −1.03556 1.03556i −0.999344 0.0362128i \(-0.988471\pi\)
−0.0362128 0.999344i \(-0.511529\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.3205 + 17.3205i 0.662751 + 0.662751i 0.956028 0.293277i \(-0.0947457\pi\)
−0.293277 + 0.956028i \(0.594746\pi\)
\(684\) 0 0
\(685\) −36.0000 + 36.0000i −1.37549 + 1.37549i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.89898 0.186636
\(690\) 0 0
\(691\) 4.24264 4.24264i 0.161398 0.161398i −0.621788 0.783186i \(-0.713593\pi\)
0.783186 + 0.621788i \(0.213593\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.4256i 2.10241i
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.44949 + 2.44949i −0.0925160 + 0.0925160i −0.751850 0.659334i \(-0.770838\pi\)
0.659334 + 0.751850i \(0.270838\pi\)
\(702\) 0 0
\(703\) −59.3970 −2.24020
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.3923 10.3923i 0.390843 0.390843i
\(708\) 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
\(711\) 0 0
\(712\) 0 0
\(713\) 9.79796i 0.366936i
\(714\) 0 0
\(715\) −16.9706 16.9706i −0.634663 0.634663i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.5692 1.55027 0.775135 0.631795i \(-0.217682\pi\)
0.775135 + 0.631795i \(0.217682\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −17.1464 17.1464i −0.636802 0.636802i
\(726\) 0 0
\(727\) 15.5563i 0.576953i 0.957487 + 0.288477i \(0.0931487\pi\)
−0.957487 + 0.288477i \(0.906851\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20.7846 20.7846i −0.768747 0.768747i
\(732\) 0 0
\(733\) −11.0000 + 11.0000i −0.406294 + 0.406294i −0.880444 0.474150i \(-0.842755\pi\)
0.474150 + 0.880444i \(0.342755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.1918 1.44365
\(738\) 0 0
\(739\) 5.65685 5.65685i 0.208091 0.208091i −0.595365 0.803456i \(-0.702992\pi\)
0.803456 + 0.595365i \(0.202992\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.6410i 1.27086i −0.772160 0.635428i \(-0.780824\pi\)
0.772160 0.635428i \(-0.219176\pi\)
\(744\) 0 0
\(745\) 12.0000i 0.439646i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.3848 0.670870 0.335435 0.942063i \(-0.391117\pi\)
0.335435 + 0.942063i \(0.391117\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −38.1051 + 38.1051i −1.38679 + 1.38679i
\(756\) 0 0
\(757\) 25.0000 + 25.0000i 0.908640 + 0.908640i 0.996163 0.0875221i \(-0.0278948\pi\)
−0.0875221 + 0.996163i \(0.527895\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.89898i 0.177588i 0.996050 + 0.0887939i \(0.0283013\pi\)
−0.996050 + 0.0887939i \(0.971699\pi\)
\(762\) 0 0
\(763\) 1.41421 + 1.41421i 0.0511980 + 0.0511980i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.8564 −0.500326
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 26.9444 + 26.9444i 0.969122 + 0.969122i 0.999537 0.0304151i \(-0.00968292\pi\)
−0.0304151 + 0.999537i \(0.509683\pi\)
\(774\) 0 0
\(775\) 9.89949i 0.355600i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.7846 + 20.7846i 0.744686 + 0.744686i
\(780\) 0 0
\(781\) 48.0000 48.0000i 1.71758 1.71758i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −34.2929 −1.22396
\(786\) 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
\(791\) 27.7128i 0.985354i
\(792\) 0 0
\(793\) 10.0000i 0.355110i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.7423 + 36.7423i −1.30148 + 1.30148i −0.374087 + 0.927394i \(0.622044\pi\)
−0.927394 + 0.374087i \(0.877956\pi\)
\(798\) 0 0
\(799\) 33.9411 1.20075
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −41.5692 + 41.5692i −1.46695 + 1.46695i
\(804\) 0 0
\(805\) −24.0000 24.0000i −0.845889 0.845889i
\(806\) 0 0
\(807\)