Properties

Label 2352.2.h.p.2255.3
Level $2352$
Weight $2$
Character 2352.2255
Analytic conductor $18.781$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2255.3
Root \(-0.667172 - 0.385192i\) of defining polynomial
Character \(\chi\) \(=\) 2352.2255
Dual form 2352.2.h.p.2255.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.70166 + 0.323042i) q^{3} -1.55481i q^{5} +(2.79129 - 1.09941i) q^{9} +O(q^{10})\) \(q+(-1.70166 + 0.323042i) q^{3} -1.55481i q^{5} +(2.79129 - 1.09941i) q^{9} +2.53326 q^{13} +(0.502268 + 2.64575i) q^{15} -4.33991i q^{17} -6.83723i q^{19} -3.80848 q^{23} +2.58258 q^{25} +(-4.39466 + 2.77253i) q^{27} +8.33639i q^{29} +4.89898i q^{31} +1.58258 q^{37} +(-4.31075 + 0.818350i) q^{39} -7.44953i q^{41} +6.20520i q^{43} +(-1.70938 - 4.33991i) q^{45} +5.38601 q^{47} +(1.40197 + 7.38505i) q^{51} -10.5352i q^{53} +(2.20871 + 11.6346i) q^{57} -10.2100 q^{59} -8.19012 q^{61} -3.93874i q^{65} -3.46410i q^{67} +(6.48074 - 1.23030i) q^{69} +14.4391 q^{71} +11.5473 q^{73} +(-4.39466 + 0.834280i) q^{75} -6.92820i q^{79} +(6.58258 - 6.13756i) q^{81} -4.82395 q^{83} -6.74773 q^{85} +(-2.69300 - 14.1857i) q^{87} -4.33991i q^{89} +(-1.58258 - 8.33639i) q^{93} -10.6306 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.70166 + 0.323042i −0.982453 + 0.186508i
\(4\) 0 0
\(5\) 1.55481i 0.695331i −0.937619 0.347665i \(-0.886975\pi\)
0.937619 0.347665i \(-0.113025\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.79129 1.09941i 0.930429 0.366471i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.53326 0.702601 0.351300 0.936263i \(-0.385740\pi\)
0.351300 + 0.936263i \(0.385740\pi\)
\(14\) 0 0
\(15\) 0.502268 + 2.64575i 0.129685 + 0.683130i
\(16\) 0 0
\(17\) 4.33991i 1.05258i −0.850304 0.526292i \(-0.823582\pi\)
0.850304 0.526292i \(-0.176418\pi\)
\(18\) 0 0
\(19\) 6.83723i 1.56857i −0.620402 0.784284i \(-0.713030\pi\)
0.620402 0.784284i \(-0.286970\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.80848 −0.794124 −0.397062 0.917792i \(-0.629970\pi\)
−0.397062 + 0.917792i \(0.629970\pi\)
\(24\) 0 0
\(25\) 2.58258 0.516515
\(26\) 0 0
\(27\) −4.39466 + 2.77253i −0.845753 + 0.533574i
\(28\) 0 0
\(29\) 8.33639i 1.54803i 0.633168 + 0.774015i \(0.281754\pi\)
−0.633168 + 0.774015i \(0.718246\pi\)
\(30\) 0 0
\(31\) 4.89898i 0.879883i 0.898027 + 0.439941i \(0.145001\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.58258 0.260174 0.130087 0.991503i \(-0.458474\pi\)
0.130087 + 0.991503i \(0.458474\pi\)
\(38\) 0 0
\(39\) −4.31075 + 0.818350i −0.690273 + 0.131041i
\(40\) 0 0
\(41\) 7.44953i 1.16342i −0.813396 0.581710i \(-0.802384\pi\)
0.813396 0.581710i \(-0.197616\pi\)
\(42\) 0 0
\(43\) 6.20520i 0.946285i 0.880986 + 0.473142i \(0.156880\pi\)
−0.880986 + 0.473142i \(0.843120\pi\)
\(44\) 0 0
\(45\) −1.70938 4.33991i −0.254819 0.646956i
\(46\) 0 0
\(47\) 5.38601 0.785630 0.392815 0.919617i \(-0.371501\pi\)
0.392815 + 0.919617i \(0.371501\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.40197 + 7.38505i 0.196316 + 1.03411i
\(52\) 0 0
\(53\) 10.5352i 1.44712i −0.690259 0.723562i \(-0.742504\pi\)
0.690259 0.723562i \(-0.257496\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.20871 + 11.6346i 0.292551 + 1.54105i
\(58\) 0 0
\(59\) −10.2100 −1.32922 −0.664611 0.747189i \(-0.731403\pi\)
−0.664611 + 0.747189i \(0.731403\pi\)
\(60\) 0 0
\(61\) −8.19012 −1.04864 −0.524319 0.851522i \(-0.675680\pi\)
−0.524319 + 0.851522i \(0.675680\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.93874i 0.488540i
\(66\) 0 0
\(67\) 3.46410i 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 6.48074 1.23030i 0.780189 0.148111i
\(70\) 0 0
\(71\) 14.4391 1.71360 0.856800 0.515648i \(-0.172449\pi\)
0.856800 + 0.515648i \(0.172449\pi\)
\(72\) 0 0
\(73\) 11.5473 1.35151 0.675753 0.737128i \(-0.263819\pi\)
0.675753 + 0.737128i \(0.263819\pi\)
\(74\) 0 0
\(75\) −4.39466 + 0.834280i −0.507452 + 0.0963344i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.92820i 0.779484i −0.920924 0.389742i \(-0.872564\pi\)
0.920924 0.389742i \(-0.127436\pi\)
\(80\) 0 0
\(81\) 6.58258 6.13756i 0.731397 0.681952i
\(82\) 0 0
\(83\) −4.82395 −0.529497 −0.264748 0.964318i \(-0.585289\pi\)
−0.264748 + 0.964318i \(0.585289\pi\)
\(84\) 0 0
\(85\) −6.74773 −0.731894
\(86\) 0 0
\(87\) −2.69300 14.1857i −0.288720 1.52087i
\(88\) 0 0
\(89\) 4.33991i 0.460030i −0.973187 0.230015i \(-0.926122\pi\)
0.973187 0.230015i \(-0.0738775\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.58258 8.33639i −0.164105 0.864444i
\(94\) 0 0
\(95\) −10.6306 −1.09067
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.5635i 1.94664i −0.229455 0.973319i \(-0.573694\pi\)
0.229455 0.973319i \(-0.426306\pi\)
\(102\) 0 0
\(103\) 3.87650i 0.381963i 0.981594 + 0.190982i \(0.0611671\pi\)
−0.981594 + 0.190982i \(0.938833\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.4391 −1.39588 −0.697938 0.716158i \(-0.745899\pi\)
−0.697938 + 0.716158i \(0.745899\pi\)
\(108\) 0 0
\(109\) −13.5826 −1.30097 −0.650487 0.759517i \(-0.725435\pi\)
−0.650487 + 0.759517i \(0.725435\pi\)
\(110\) 0 0
\(111\) −2.69300 + 0.511238i −0.255609 + 0.0485246i
\(112\) 0 0
\(113\) 14.4740i 1.36160i 0.732471 + 0.680798i \(0.238367\pi\)
−0.732471 + 0.680798i \(0.761633\pi\)
\(114\) 0 0
\(115\) 5.92146i 0.552179i
\(116\) 0 0
\(117\) 7.07107 2.78511i 0.653720 0.257483i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 2.40651 + 12.6766i 0.216988 + 1.14301i
\(124\) 0 0
\(125\) 11.7894i 1.05448i
\(126\) 0 0
\(127\) 10.3923i 0.922168i −0.887357 0.461084i \(-0.847461\pi\)
0.887357 0.461084i \(-0.152539\pi\)
\(128\) 0 0
\(129\) −2.00454 10.5591i −0.176490 0.929681i
\(130\) 0 0
\(131\) −15.5960 −1.36263 −0.681313 0.731992i \(-0.738591\pi\)
−0.681313 + 0.731992i \(0.738591\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.31075 + 6.83285i 0.371010 + 0.588078i
\(136\) 0 0
\(137\) 1.73991i 0.148650i −0.997234 0.0743251i \(-0.976320\pi\)
0.997234 0.0743251i \(-0.0236803\pi\)
\(138\) 0 0
\(139\) 11.7362i 0.995452i −0.867334 0.497726i \(-0.834168\pi\)
0.867334 0.497726i \(-0.165832\pi\)
\(140\) 0 0
\(141\) −9.16515 + 1.73991i −0.771845 + 0.146527i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 12.9615 1.07639
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.5352i 0.863079i −0.902094 0.431540i \(-0.857971\pi\)
0.902094 0.431540i \(-0.142029\pi\)
\(150\) 0 0
\(151\) 15.8745i 1.29185i −0.763401 0.645925i \(-0.776472\pi\)
0.763401 0.645925i \(-0.223528\pi\)
\(152\) 0 0
\(153\) −4.77136 12.1139i −0.385742 0.979355i
\(154\) 0 0
\(155\) 7.61697 0.611809
\(156\) 0 0
\(157\) 8.19012 0.653643 0.326821 0.945086i \(-0.394022\pi\)
0.326821 + 0.945086i \(0.394022\pi\)
\(158\) 0 0
\(159\) 3.40332 + 17.9274i 0.269901 + 1.42173i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0616i 1.57135i −0.618642 0.785673i \(-0.712317\pi\)
0.618642 0.785673i \(-0.287683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −6.58258 −0.506352
\(170\) 0 0
\(171\) −7.51695 19.0847i −0.574836 1.45944i
\(172\) 0 0
\(173\) 1.55481i 0.118210i −0.998252 0.0591049i \(-0.981175\pi\)
0.998252 0.0591049i \(-0.0188246\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 17.3739 3.29824i 1.30590 0.247911i
\(178\) 0 0
\(179\) 14.4391 1.07923 0.539613 0.841913i \(-0.318571\pi\)
0.539613 + 0.841913i \(0.318571\pi\)
\(180\) 0 0
\(181\) 10.4282 0.775123 0.387562 0.921844i \(-0.373317\pi\)
0.387562 + 0.921844i \(0.373317\pi\)
\(182\) 0 0
\(183\) 13.9368 2.64575i 1.03024 0.195580i
\(184\) 0 0
\(185\) 2.46060i 0.180907i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.0560 1.59592 0.797959 0.602712i \(-0.205913\pi\)
0.797959 + 0.602712i \(0.205913\pi\)
\(192\) 0 0
\(193\) 8.74773 0.629675 0.314838 0.949146i \(-0.398050\pi\)
0.314838 + 0.949146i \(0.398050\pi\)
\(194\) 0 0
\(195\) 1.27238 + 6.70239i 0.0911168 + 0.479968i
\(196\) 0 0
\(197\) 10.5352i 0.750604i 0.926903 + 0.375302i \(0.122461\pi\)
−0.926903 + 0.375302i \(0.877539\pi\)
\(198\) 0 0
\(199\) 9.79796i 0.694559i 0.937762 + 0.347279i \(0.112894\pi\)
−0.937762 + 0.347279i \(0.887106\pi\)
\(200\) 0 0
\(201\) 1.11905 + 5.89472i 0.0789317 + 0.415782i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −11.5826 −0.808962
\(206\) 0 0
\(207\) −10.6306 + 4.18710i −0.738876 + 0.291024i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.8745i 1.09285i −0.837509 0.546423i \(-0.815989\pi\)
0.837509 0.546423i \(-0.184011\pi\)
\(212\) 0 0
\(213\) −24.5704 + 4.66442i −1.68353 + 0.319601i
\(214\) 0 0
\(215\) 9.64789 0.657981
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −19.6495 + 3.73025i −1.32779 + 0.252067i
\(220\) 0 0
\(221\) 10.9941i 0.739546i
\(222\) 0 0
\(223\) 23.4724i 1.57183i −0.618335 0.785915i \(-0.712192\pi\)
0.618335 0.785915i \(-0.287808\pi\)
\(224\) 0 0
\(225\) 7.20871 2.83932i 0.480581 0.189288i
\(226\) 0 0
\(227\) −4.82395 −0.320177 −0.160088 0.987103i \(-0.551178\pi\)
−0.160088 + 0.987103i \(0.551178\pi\)
\(228\) 0 0
\(229\) −8.78044 −0.580228 −0.290114 0.956992i \(-0.593693\pi\)
−0.290114 + 0.956992i \(0.593693\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.73991i 0.113985i 0.998375 + 0.0569925i \(0.0181511\pi\)
−0.998375 + 0.0569925i \(0.981849\pi\)
\(234\) 0 0
\(235\) 8.37420i 0.546273i
\(236\) 0 0
\(237\) 2.23810 + 11.7894i 0.145380 + 0.765806i
\(238\) 0 0
\(239\) −3.80848 −0.246350 −0.123175 0.992385i \(-0.539308\pi\)
−0.123175 + 0.992385i \(0.539308\pi\)
\(240\) 0 0
\(241\) −3.06199 −0.197240 −0.0986199 0.995125i \(-0.531443\pi\)
−0.0986199 + 0.995125i \(0.531443\pi\)
\(242\) 0 0
\(243\) −9.21861 + 12.5705i −0.591374 + 0.806397i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.3205i 1.10208i
\(248\) 0 0
\(249\) 8.20871 1.55834i 0.520206 0.0987556i
\(250\) 0 0
\(251\) −0.562063 −0.0354771 −0.0177385 0.999843i \(-0.505647\pi\)
−0.0177385 + 0.999843i \(0.505647\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 11.4823 2.17980i 0.719052 0.136504i
\(256\) 0 0
\(257\) 13.6688i 0.852633i 0.904574 + 0.426317i \(0.140189\pi\)
−0.904574 + 0.426317i \(0.859811\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 9.16515 + 23.2693i 0.567309 + 1.44033i
\(262\) 0 0
\(263\) −29.6730 −1.82971 −0.914857 0.403777i \(-0.867697\pi\)
−0.914857 + 0.403777i \(0.867697\pi\)
\(264\) 0 0
\(265\) −16.3802 −1.00623
\(266\) 0 0
\(267\) 1.40197 + 7.38505i 0.0857994 + 0.451958i
\(268\) 0 0
\(269\) 1.55481i 0.0947982i 0.998876 + 0.0473991i \(0.0150933\pi\)
−0.998876 + 0.0473991i \(0.984907\pi\)
\(270\) 0 0
\(271\) 19.5959i 1.19037i −0.803590 0.595184i \(-0.797079\pi\)
0.803590 0.595184i \(-0.202921\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 5.38601 + 13.6745i 0.322452 + 0.818669i
\(280\) 0 0
\(281\) 14.9329i 0.890821i −0.895326 0.445410i \(-0.853058\pi\)
0.895326 0.445410i \(-0.146942\pi\)
\(282\) 0 0
\(283\) 26.4331i 1.57129i 0.618679 + 0.785644i \(0.287668\pi\)
−0.618679 + 0.785644i \(0.712332\pi\)
\(284\) 0 0
\(285\) 18.0896 3.43412i 1.07154 0.203420i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.83485 −0.107932
\(290\) 0 0
\(291\) 16.8456 3.19795i 0.987505 0.187467i
\(292\) 0 0
\(293\) 13.3442i 0.779579i 0.920904 + 0.389790i \(0.127452\pi\)
−0.920904 + 0.389790i \(0.872548\pi\)
\(294\) 0 0
\(295\) 15.8745i 0.924250i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.64789 −0.557952
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.31982 + 33.2904i 0.363064 + 1.91248i
\(304\) 0 0
\(305\) 12.7341i 0.729150i
\(306\) 0 0
\(307\) 11.7362i 0.669821i 0.942250 + 0.334910i \(0.108706\pi\)
−0.942250 + 0.334910i \(0.891294\pi\)
\(308\) 0 0
\(309\) −1.25227 6.59649i −0.0712393 0.375261i
\(310\) 0 0
\(311\) 15.0339 0.852494 0.426247 0.904607i \(-0.359835\pi\)
0.426247 + 0.904607i \(0.359835\pi\)
\(312\) 0 0
\(313\) −5.89041 −0.332946 −0.166473 0.986046i \(-0.553238\pi\)
−0.166473 + 0.986046i \(0.553238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.65775i 0.149274i −0.997211 0.0746371i \(-0.976220\pi\)
0.997211 0.0746371i \(-0.0237798\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 24.5704 4.66442i 1.37138 0.260343i
\(322\) 0 0
\(323\) −29.6730 −1.65105
\(324\) 0 0
\(325\) 6.54234 0.362904
\(326\) 0 0
\(327\) 23.1129 4.38774i 1.27815 0.242643i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.1334i 0.721877i −0.932590 0.360939i \(-0.882456\pi\)
0.932590 0.360939i \(-0.117544\pi\)
\(332\) 0 0
\(333\) 4.41742 1.73991i 0.242073 0.0953463i
\(334\) 0 0
\(335\) −5.38601 −0.294269
\(336\) 0 0
\(337\) −1.25227 −0.0682157 −0.0341078 0.999418i \(-0.510859\pi\)
−0.0341078 + 0.999418i \(0.510859\pi\)
\(338\) 0 0
\(339\) −4.67569 24.6297i −0.253949 1.33770i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.91288 10.0763i −0.102986 0.542490i
\(346\) 0 0
\(347\) 36.4951 1.95916 0.979579 0.201058i \(-0.0644381\pi\)
0.979579 + 0.201058i \(0.0644381\pi\)
\(348\) 0 0
\(349\) −4.77136 −0.255405 −0.127703 0.991813i \(-0.540760\pi\)
−0.127703 + 0.991813i \(0.540760\pi\)
\(350\) 0 0
\(351\) −11.1328 + 7.02355i −0.594227 + 0.374890i
\(352\) 0 0
\(353\) 27.9188i 1.48597i 0.669309 + 0.742984i \(0.266590\pi\)
−0.669309 + 0.742984i \(0.733410\pi\)
\(354\) 0 0
\(355\) 22.4499i 1.19152i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.80848 0.201004 0.100502 0.994937i \(-0.467955\pi\)
0.100502 + 0.994937i \(0.467955\pi\)
\(360\) 0 0
\(361\) −27.7477 −1.46041
\(362\) 0 0
\(363\) 18.7183 3.55346i 0.982453 0.186508i
\(364\) 0 0
\(365\) 17.9538i 0.939743i
\(366\) 0 0
\(367\) 23.4724i 1.22525i 0.790374 + 0.612625i \(0.209886\pi\)
−0.790374 + 0.612625i \(0.790114\pi\)
\(368\) 0 0
\(369\) −8.19012 20.7938i −0.426361 1.08248i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 3.80848 + 20.0616i 0.196669 + 1.03598i
\(376\) 0 0
\(377\) 21.1183i 1.08765i
\(378\) 0 0
\(379\) 6.20520i 0.318740i −0.987219 0.159370i \(-0.949054\pi\)
0.987219 0.159370i \(-0.0509463\pi\)
\(380\) 0 0
\(381\) 3.35715 + 17.6842i 0.171992 + 0.905987i
\(382\) 0 0
\(383\) 15.0339 0.768196 0.384098 0.923292i \(-0.374512\pi\)
0.384098 + 0.923292i \(0.374512\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.82209 + 17.3205i 0.346786 + 0.880451i
\(388\) 0 0
\(389\) 38.2022i 1.93693i 0.249157 + 0.968463i \(0.419846\pi\)
−0.249157 + 0.968463i \(0.580154\pi\)
\(390\) 0 0
\(391\) 16.5285i 0.835882i
\(392\) 0 0
\(393\) 26.5390 5.03815i 1.33872 0.254141i
\(394\) 0 0
\(395\) −10.7720 −0.541999
\(396\) 0 0
\(397\) −38.1222 −1.91330 −0.956648 0.291246i \(-0.905930\pi\)
−0.956648 + 0.291246i \(0.905930\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.5294i 1.07513i 0.843224 + 0.537563i \(0.180655\pi\)
−0.843224 + 0.537563i \(0.819345\pi\)
\(402\) 0 0
\(403\) 12.4104i 0.618206i
\(404\) 0 0
\(405\) −9.54273 10.2346i −0.474182 0.508563i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 17.7944 0.879879 0.439939 0.898027i \(-0.355000\pi\)
0.439939 + 0.898027i \(0.355000\pi\)
\(410\) 0 0
\(411\) 0.562063 + 2.96073i 0.0277245 + 0.146042i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.50030i 0.368175i
\(416\) 0 0
\(417\) 3.79129 + 19.9710i 0.185660 + 0.977986i
\(418\) 0 0
\(419\) −15.5960 −0.761913 −0.380956 0.924593i \(-0.624405\pi\)
−0.380956 + 0.924593i \(0.624405\pi\)
\(420\) 0 0
\(421\) 35.4955 1.72994 0.864971 0.501821i \(-0.167337\pi\)
0.864971 + 0.501821i \(0.167337\pi\)
\(422\) 0 0
\(423\) 15.0339 5.92146i 0.730973 0.287911i
\(424\) 0 0
\(425\) 11.2082i 0.543675i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.4527 −0.840665 −0.420333 0.907370i \(-0.638087\pi\)
−0.420333 + 0.907370i \(0.638087\pi\)
\(432\) 0 0
\(433\) −0.823886 −0.0395935 −0.0197967 0.999804i \(-0.506302\pi\)
−0.0197967 + 0.999804i \(0.506302\pi\)
\(434\) 0 0
\(435\) −22.0560 + 4.18710i −1.05751 + 0.200756i
\(436\) 0 0
\(437\) 26.0395i 1.24564i
\(438\) 0 0
\(439\) 14.6969i 0.701447i −0.936479 0.350723i \(-0.885936\pi\)
0.936479 0.350723i \(-0.114064\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.82209 −0.324127 −0.162064 0.986780i \(-0.551815\pi\)
−0.162064 + 0.986780i \(0.551815\pi\)
\(444\) 0 0
\(445\) −6.74773 −0.319873
\(446\) 0 0
\(447\) 3.40332 + 17.9274i 0.160971 + 0.847935i
\(448\) 0 0
\(449\) 28.9479i 1.36614i −0.730355 0.683068i \(-0.760645\pi\)
0.730355 0.683068i \(-0.239355\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.12813 + 27.0130i 0.240941 + 1.26918i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.7477 −1.15765 −0.578825 0.815452i \(-0.696489\pi\)
−0.578825 + 0.815452i \(0.696489\pi\)
\(458\) 0 0
\(459\) 12.0325 + 19.0725i 0.561631 + 0.890226i
\(460\) 0 0
\(461\) 10.2346i 0.476674i 0.971182 + 0.238337i \(0.0766023\pi\)
−0.971182 + 0.238337i \(0.923398\pi\)
\(462\) 0 0
\(463\) 26.2668i 1.22072i 0.792123 + 0.610361i \(0.208976\pi\)
−0.792123 + 0.610361i \(0.791024\pi\)
\(464\) 0 0
\(465\) −12.9615 + 2.46060i −0.601074 + 0.114108i
\(466\) 0 0
\(467\) −25.2439 −1.16815 −0.584073 0.811701i \(-0.698542\pi\)
−0.584073 + 0.811701i \(0.698542\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −13.9368 + 2.64575i −0.642173 + 0.121910i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 17.6577i 0.810189i
\(476\) 0 0
\(477\) −11.5826 29.4068i −0.530330 1.34645i
\(478\) 0 0
\(479\) 31.1919 1.42520 0.712598 0.701573i \(-0.247518\pi\)
0.712598 + 0.701573i \(0.247518\pi\)
\(480\) 0 0
\(481\) 4.00908 0.182798
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.3918i 0.698906i
\(486\) 0 0
\(487\) 36.6591i 1.66118i −0.556882 0.830592i \(-0.688002\pi\)
0.556882 0.830592i \(-0.311998\pi\)
\(488\) 0 0
\(489\) 6.48074 + 34.1380i 0.293069 + 1.54377i
\(490\) 0 0
\(491\) −22.0560 −0.995374 −0.497687 0.867357i \(-0.665817\pi\)
−0.497687 + 0.867357i \(0.665817\pi\)
\(492\) 0 0
\(493\) 36.1792 1.62943
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.94630i 0.400492i 0.979746 + 0.200246i \(0.0641741\pi\)
−0.979746 + 0.200246i \(0.935826\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.1919 1.39078 0.695390 0.718633i \(-0.255232\pi\)
0.695390 + 0.718633i \(0.255232\pi\)
\(504\) 0 0
\(505\) −30.4174 −1.35356
\(506\) 0 0
\(507\) 11.2013 2.12645i 0.497467 0.0944389i
\(508\) 0 0
\(509\) 0.905793i 0.0401486i −0.999798 0.0200743i \(-0.993610\pi\)
0.999798 0.0200743i \(-0.00639027\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 18.9564 + 30.0473i 0.836947 + 1.32662i
\(514\) 0 0
\(515\) 6.02721 0.265591
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0.502268 + 2.64575i 0.0220471 + 0.116136i
\(520\) 0 0
\(521\) 27.9188i 1.22314i 0.791189 + 0.611572i \(0.209463\pi\)
−0.791189 + 0.611572i \(0.790537\pi\)
\(522\) 0 0
\(523\) 0.915775i 0.0400440i 0.999800 + 0.0200220i \(0.00637363\pi\)
−0.999800 + 0.0200220i \(0.993626\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.2611 0.926150
\(528\) 0 0
\(529\) −8.49545 −0.369368
\(530\) 0 0
\(531\) −28.4989 + 11.2250i −1.23675 + 0.487122i
\(532\) 0 0
\(533\) 18.8716i 0.817420i
\(534\) 0 0
\(535\) 22.4499i 0.970596i
\(536\) 0 0
\(537\) −24.5704 + 4.66442i −1.06029 + 0.201285i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23.4955 −1.01015 −0.505074 0.863076i \(-0.668535\pi\)
−0.505074 + 0.863076i \(0.668535\pi\)
\(542\) 0 0
\(543\) −17.7453 + 3.36875i −0.761523 + 0.144567i
\(544\) 0 0
\(545\) 21.1183i 0.904608i
\(546\) 0 0
\(547\) 11.6874i 0.499717i −0.968282 0.249859i \(-0.919616\pi\)
0.968282 0.249859i \(-0.0803842\pi\)
\(548\) 0 0
\(549\) −22.8610 + 9.00433i −0.975683 + 0.384296i
\(550\) 0 0
\(551\) 56.9978 2.42819
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.794877 + 4.18710i 0.0337406 + 0.177733i
\(556\) 0 0
\(557\) 2.65775i 0.112613i −0.998414 0.0563063i \(-0.982068\pi\)
0.998414 0.0563063i \(-0.0179323\pi\)
\(558\) 0 0
\(559\) 15.7194i 0.664860i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −0.562063 −0.0236881 −0.0118441 0.999930i \(-0.503770\pi\)
−0.0118441 + 0.999930i \(0.503770\pi\)
\(564\) 0 0
\(565\) 22.5042 0.946759
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.85658i 0.203598i −0.994805 0.101799i \(-0.967540\pi\)
0.994805 0.101799i \(-0.0324599\pi\)
\(570\) 0 0
\(571\) 0.723000i 0.0302566i −0.999886 0.0151283i \(-0.995184\pi\)
0.999886 0.0151283i \(-0.00481567\pi\)
\(572\) 0 0
\(573\) −37.5318 + 7.12502i −1.56791 + 0.297652i
\(574\) 0 0
\(575\) −9.83570 −0.410177
\(576\) 0 0
\(577\) −43.7174 −1.81998 −0.909990 0.414631i \(-0.863911\pi\)
−0.909990 + 0.414631i \(0.863911\pi\)
\(578\) 0 0
\(579\) −14.8856 + 2.82588i −0.618627 + 0.117440i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.33030 10.9941i −0.179036 0.454552i
\(586\) 0 0
\(587\) 36.0159 1.48653 0.743267 0.668995i \(-0.233275\pi\)
0.743267 + 0.668995i \(0.233275\pi\)
\(588\) 0 0
\(589\) 33.4955 1.38016
\(590\) 0 0
\(591\) −3.40332 17.9274i −0.139994 0.737433i
\(592\) 0 0
\(593\) 28.5678i 1.17314i 0.809899 + 0.586570i \(0.199522\pi\)
−0.809899 + 0.586570i \(0.800478\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.16515 16.6728i −0.129541 0.682372i
\(598\) 0 0
\(599\) 29.6730 1.21241 0.606203 0.795310i \(-0.292692\pi\)
0.606203 + 0.795310i \(0.292692\pi\)
\(600\) 0 0
\(601\) 0.823886 0.0336070 0.0168035 0.999859i \(-0.494651\pi\)
0.0168035 + 0.999859i \(0.494651\pi\)
\(602\) 0 0
\(603\) −3.80848 9.66930i −0.155093 0.393765i
\(604\) 0 0
\(605\) 17.1029i 0.695331i
\(606\) 0 0
\(607\) 18.5734i 0.753873i 0.926239 + 0.376936i \(0.123022\pi\)
−0.926239 + 0.376936i \(0.876978\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.6442 0.551984
\(612\) 0 0
\(613\) 38.2432 1.54463 0.772314 0.635241i \(-0.219099\pi\)
0.772314 + 0.635241i \(0.219099\pi\)
\(614\) 0 0
\(615\) 19.7096 3.74166i 0.794768 0.150878i
\(616\) 0 0
\(617\) 21.5294i 0.866740i −0.901216 0.433370i \(-0.857324\pi\)
0.901216 0.433370i \(-0.142676\pi\)
\(618\) 0 0
\(619\) 7.85971i 0.315908i 0.987446 + 0.157954i \(0.0504898\pi\)
−0.987446 + 0.157954i \(0.949510\pi\)
\(620\) 0 0
\(621\) 16.7370 10.5591i 0.671633 0.423724i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.41742 −0.216697
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.86824i 0.273855i
\(630\) 0 0
\(631\) 40.1232i 1.59728i −0.601809 0.798640i \(-0.705553\pi\)
0.601809 0.798640i \(-0.294447\pi\)
\(632\) 0 0
\(633\) 5.12813 + 27.0130i 0.203825 + 1.07367i
\(634\) 0 0
\(635\) −16.1580 −0.641212
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 40.3036 15.8745i 1.59438 0.627986i
\(640\) 0 0
\(641\) 4.85658i 0.191823i −0.995390 0.0959117i \(-0.969423\pi\)
0.995390 0.0959117i \(-0.0305766\pi\)
\(642\) 0 0
\(643\) 39.0851i 1.54137i −0.637218 0.770684i \(-0.719915\pi\)
0.637218 0.770684i \(-0.280085\pi\)
\(644\) 0 0
\(645\) −16.4174 + 3.11667i −0.646435 + 0.122719i
\(646\) 0 0
\(647\) 25.8059 1.01454 0.507268 0.861789i \(-0.330656\pi\)
0.507268 + 0.861789i \(0.330656\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.0092i 0.978685i 0.872092 + 0.489342i \(0.162763\pi\)
−0.872092 + 0.489342i \(0.837237\pi\)
\(654\) 0 0
\(655\) 24.2487i 0.947476i
\(656\) 0 0
\(657\) 32.2317 12.6952i 1.25748 0.495288i
\(658\) 0 0
\(659\) −13.6442 −0.531502 −0.265751 0.964042i \(-0.585620\pi\)
−0.265751 + 0.964042i \(0.585620\pi\)
\(660\) 0 0
\(661\) −9.24756 −0.359689 −0.179844 0.983695i \(-0.557559\pi\)
−0.179844 + 0.983695i \(0.557559\pi\)
\(662\) 0 0
\(663\) 3.55157 + 18.7083i 0.137932 + 0.726570i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 31.7490i 1.22933i
\(668\) 0 0
\(669\) 7.58258 + 39.9421i 0.293159 + 1.54425i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −15.1652 −0.584574 −0.292287 0.956331i \(-0.594416\pi\)
−0.292287 + 0.956331i \(0.594416\pi\)
\(674\) 0 0
\(675\) −11.3496 + 7.16027i −0.436844 + 0.275599i
\(676\) 0 0
\(677\) 0.905793i 0.0348124i 0.999849 + 0.0174062i \(0.00554085\pi\)
−0.999849 + 0.0174062i \(0.994459\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 8.20871 1.55834i 0.314559 0.0597156i
\(682\) 0 0
\(683\) 22.0560 0.843950 0.421975 0.906607i \(-0.361337\pi\)
0.421975 + 0.906607i \(0.361337\pi\)
\(684\) 0 0
\(685\) −2.70522 −0.103361
\(686\) 0 0
\(687\) 14.9413 2.83645i 0.570047 0.108217i
\(688\) 0 0
\(689\) 26.6885i 1.01675i
\(690\) 0 0
\(691\) 12.7587i 0.485363i 0.970106 + 0.242682i \(0.0780270\pi\)
−0.970106 + 0.242682i \(0.921973\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.2475 −0.692169
\(696\) 0 0
\(697\) −32.3303 −1.22460
\(698\) 0 0
\(699\) −0.562063 2.96073i −0.0212592 0.111985i
\(700\) 0 0
\(701\) 4.85658i 0.183431i −0.995785 0.0917153i \(-0.970765\pi\)
0.995785 0.0917153i \(-0.0292350\pi\)
\(702\) 0 0
\(703\) 10.8204i 0.408100i
\(704\) 0 0
\(705\) 2.70522 + 14.2500i 0.101884 + 0.536688i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 47.0780 1.76805 0.884026 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(710\) 0 0
\(711\) −7.61697 19.3386i −0.285659 0.725255i
\(712\) 0 0
\(713\) 18.6577i 0.698736i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.48074 1.23030i 0.242028 0.0459464i
\(718\) 0 0
\(719\) −25.8059 −0.962398 −0.481199 0.876611i \(-0.659799\pi\)
−0.481199 + 0.876611i \(0.659799\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.21046 0.989150i 0.193779 0.0367869i
\(724\) 0 0
\(725\) 21.5294i 0.799581i
\(726\) 0 0
\(727\) 20.6184i 0.764694i −0.924019 0.382347i \(-0.875116\pi\)
0.924019 0.382347i \(-0.124884\pi\)
\(728\) 0 0
\(729\) 11.6261 24.3687i 0.430598 0.902544i
\(730\) 0 0
\(731\) 26.9300 0.996044
\(732\) 0 0
\(733\) 1.94294 0.0717640 0.0358820 0.999356i \(-0.488576\pi\)
0.0358820 + 0.999356i \(0.488576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 40.8462i 1.50255i −0.659988 0.751276i \(-0.729439\pi\)
0.659988 0.751276i \(-0.270561\pi\)
\(740\) 0 0
\(741\) 5.59525 + 29.4736i 0.205547 + 1.08274i
\(742\) 0 0
\(743\) 25.0696 0.919716 0.459858 0.887993i \(-0.347900\pi\)
0.459858 + 0.887993i \(0.347900\pi\)
\(744\) 0 0
\(745\) −16.3802 −0.600125
\(746\) 0 0
\(747\) −13.4650 + 5.30352i −0.492659 + 0.194046i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 22.8027i 0.832083i −0.909346 0.416041i \(-0.863417\pi\)
0.909346 0.416041i \(-0.136583\pi\)
\(752\) 0 0
\(753\) 0.956439 0.181570i 0.0348546 0.00661677i
\(754\) 0 0
\(755\) −24.6818 −0.898262
\(756\) 0 0
\(757\) −4.74773 −0.172559 −0.0862795 0.996271i \(-0.527498\pi\)
−0.0862795 + 0.996271i \(0.527498\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.6688i 0.495492i −0.968825 0.247746i \(-0.920310\pi\)
0.968825 0.247746i \(-0.0796898\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −18.8348 + 7.41855i −0.680975 + 0.268218i
\(766\) 0 0
\(767\) −25.8645 −0.933913
\(768\) 0 0
\(769\) −11.0801 −0.399560 −0.199780 0.979841i \(-0.564023\pi\)
−0.199780 + 0.979841i \(0.564023\pi\)
\(770\) 0 0
\(771\) −4.41558 23.2596i −0.159023 0.837673i
\(772\) 0 0
\(773\) 1.55481i 0.0559225i −0.999609 0.0279613i \(-0.991098\pi\)
0.999609 0.0279613i \(-0.00890150\pi\)
\(774\) 0 0
\(775\) 12.6520i 0.454473i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −50.9341 −1.82490
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −23.1129 36.6356i −0.825988 1.30925i
\(784\) 0 0
\(785\) 12.7341i 0.454498i
\(786\) 0 0
\(787\) 29.2872i 1.04398i −0.852953 0.521988i \(-0.825191\pi\)
0.852953 0.521988i \(-0.174809\pi\)
\(788\) 0 0
\(789\) 50.4933 9.58562i 1.79761 0.341257i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20.7477 −0.736773
\(794\) 0 0
\(795\) 27.8736 5.29150i 0.988574 0.187670i
\(796\) 0 0
\(797\) 7.77403i 0.275370i −0.990476 0.137685i \(-0.956034\pi\)
0.990476 0.137685i \(-0.0439662\pi\)
\(798\) 0 0
\(799\) 23.3748i 0.826941i
\(800\) 0 0
\(801\) −4.77136 12.1139i −0.168588 0.428025i
\(802\) 0 0
\(803\) 0 0