Properties

Label 2352.2.h.i.2255.1
Level $2352$
Weight $2$
Character 2352.2255
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2255.1
Root \(1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2352.2255
Dual form 2352.2.h.i.2255.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.65831 - 0.500000i) q^{3} +3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +O(q^{10})\) \(q+(-1.65831 - 0.500000i) q^{3} +3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +3.31662 q^{11} -4.00000 q^{13} +(1.65831 - 5.50000i) q^{15} -3.31662i q^{17} -7.00000i q^{19} +3.31662 q^{23} -6.00000 q^{25} +(-3.31662 - 4.00000i) q^{27} -6.63325i q^{29} -3.00000i q^{31} +(-5.50000 - 1.65831i) q^{33} -1.00000 q^{37} +(6.63325 + 2.00000i) q^{39} +6.63325i q^{41} +2.00000i q^{43} +(-5.50000 + 8.29156i) q^{45} +9.94987 q^{47} +(-1.65831 + 5.50000i) q^{51} +3.31662i q^{53} +11.0000i q^{55} +(-3.50000 + 11.6082i) q^{57} -3.31662 q^{59} +3.00000 q^{61} -13.2665i q^{65} -9.00000i q^{67} +(-5.50000 - 1.65831i) q^{69} +13.2665 q^{71} +7.00000 q^{73} +(9.94987 + 3.00000i) q^{75} -9.00000i q^{79} +(3.50000 + 8.29156i) q^{81} -13.2665 q^{83} +11.0000 q^{85} +(-3.31662 + 11.0000i) q^{87} -3.31662i q^{89} +(-1.50000 + 4.97494i) q^{93} +23.2164 q^{95} +8.00000 q^{97} +(8.29156 + 5.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{9} - 16 q^{13} - 24 q^{25} - 22 q^{33} - 4 q^{37} - 22 q^{45} - 14 q^{57} + 12 q^{61} - 22 q^{69} + 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} + 32 q^{97}+O(q^{100}) \) 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
\(3\) −1.65831 0.500000i −0.957427 0.288675i
\(4\) 0 0
\(5\) 3.31662i 1.48324i 0.670820 + 0.741620i \(0.265942\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.50000 + 1.65831i 0.833333 + 0.552771i
\(10\) 0 0
\(11\) 3.31662 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 1.65831 5.50000i 0.428174 1.42009i
\(16\) 0 0
\(17\) 3.31662i 0.804400i −0.915552 0.402200i \(-0.868246\pi\)
0.915552 0.402200i \(-0.131754\pi\)
\(18\) 0 0
\(19\) 7.00000i 1.60591i −0.596040 0.802955i \(-0.703260\pi\)
0.596040 0.802955i \(-0.296740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662 0.691564 0.345782 0.938315i \(-0.387614\pi\)
0.345782 + 0.938315i \(0.387614\pi\)
\(24\) 0 0
\(25\) −6.00000 −1.20000
\(26\) 0 0
\(27\) −3.31662 4.00000i −0.638285 0.769800i
\(28\) 0 0
\(29\) 6.63325i 1.23176i −0.787839 0.615882i \(-0.788800\pi\)
0.787839 0.615882i \(-0.211200\pi\)
\(30\) 0 0
\(31\) 3.00000i 0.538816i −0.963026 0.269408i \(-0.913172\pi\)
0.963026 0.269408i \(-0.0868280\pi\)
\(32\) 0 0
\(33\) −5.50000 1.65831i −0.957427 0.288675i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 6.63325 + 2.00000i 1.06217 + 0.320256i
\(40\) 0 0
\(41\) 6.63325i 1.03594i 0.855399 + 0.517970i \(0.173312\pi\)
−0.855399 + 0.517970i \(0.826688\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) −5.50000 + 8.29156i −0.819892 + 1.23603i
\(46\) 0 0
\(47\) 9.94987 1.45134 0.725669 0.688044i \(-0.241530\pi\)
0.725669 + 0.688044i \(0.241530\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.65831 + 5.50000i −0.232210 + 0.770154i
\(52\) 0 0
\(53\) 3.31662i 0.455573i 0.973711 + 0.227787i \(0.0731489\pi\)
−0.973711 + 0.227787i \(0.926851\pi\)
\(54\) 0 0
\(55\) 11.0000i 1.48324i
\(56\) 0 0
\(57\) −3.50000 + 11.6082i −0.463586 + 1.53754i
\(58\) 0 0
\(59\) −3.31662 −0.431788 −0.215894 0.976417i \(-0.569267\pi\)
−0.215894 + 0.976417i \(0.569267\pi\)
\(60\) 0 0
\(61\) 3.00000 0.384111 0.192055 0.981384i \(-0.438485\pi\)
0.192055 + 0.981384i \(0.438485\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.2665i 1.64551i
\(66\) 0 0
\(67\) 9.00000i 1.09952i −0.835321 0.549762i \(-0.814718\pi\)
0.835321 0.549762i \(-0.185282\pi\)
\(68\) 0 0
\(69\) −5.50000 1.65831i −0.662122 0.199637i
\(70\) 0 0
\(71\) 13.2665 1.57444 0.787222 0.616670i \(-0.211519\pi\)
0.787222 + 0.616670i \(0.211519\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 9.94987 + 3.00000i 1.14891 + 0.346410i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.00000i 1.01258i −0.862364 0.506290i \(-0.831017\pi\)
0.862364 0.506290i \(-0.168983\pi\)
\(80\) 0 0
\(81\) 3.50000 + 8.29156i 0.388889 + 0.921285i
\(82\) 0 0
\(83\) −13.2665 −1.45619 −0.728094 0.685478i \(-0.759593\pi\)
−0.728094 + 0.685478i \(0.759593\pi\)
\(84\) 0 0
\(85\) 11.0000 1.19312
\(86\) 0 0
\(87\) −3.31662 + 11.0000i −0.355580 + 1.17932i
\(88\) 0 0
\(89\) 3.31662i 0.351562i −0.984429 0.175781i \(-0.943755\pi\)
0.984429 0.175781i \(-0.0562450\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.50000 + 4.97494i −0.155543 + 0.515877i
\(94\) 0 0
\(95\) 23.2164 2.38195
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 8.29156 + 5.50000i 0.833333 + 0.552771i
\(100\) 0 0
\(101\) 3.31662i 0.330017i −0.986292 0.165008i \(-0.947235\pi\)
0.986292 0.165008i \(-0.0527651\pi\)
\(102\) 0 0
\(103\) 11.0000i 1.08386i 0.840423 + 0.541931i \(0.182307\pi\)
−0.840423 + 0.541931i \(0.817693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.31662 −0.320630 −0.160315 0.987066i \(-0.551251\pi\)
−0.160315 + 0.987066i \(0.551251\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 1.65831 + 0.500000i 0.157400 + 0.0474579i
\(112\) 0 0
\(113\) 6.63325i 0.624004i −0.950082 0.312002i \(-0.899000\pi\)
0.950082 0.312002i \(-0.101000\pi\)
\(114\) 0 0
\(115\) 11.0000i 1.02576i
\(116\) 0 0
\(117\) −10.0000 6.63325i −0.924500 0.613244i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 3.31662 11.0000i 0.299050 0.991837i
\(124\) 0 0
\(125\) 3.31662i 0.296648i
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) 1.00000 3.31662i 0.0880451 0.292013i
\(130\) 0 0
\(131\) 16.5831 1.44887 0.724437 0.689341i \(-0.242100\pi\)
0.724437 + 0.689341i \(0.242100\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 13.2665 11.0000i 1.14180 0.946729i
\(136\) 0 0
\(137\) 16.5831i 1.41679i −0.705815 0.708396i \(-0.749419\pi\)
0.705815 0.708396i \(-0.250581\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i −0.905618 0.424094i \(-0.860592\pi\)
0.905618 0.424094i \(-0.139408\pi\)
\(140\) 0 0
\(141\) −16.5000 4.97494i −1.38955 0.418965i
\(142\) 0 0
\(143\) −13.2665 −1.10940
\(144\) 0 0
\(145\) 22.0000 1.82700
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.5831i 1.35854i 0.733887 + 0.679271i \(0.237704\pi\)
−0.733887 + 0.679271i \(0.762296\pi\)
\(150\) 0 0
\(151\) 7.00000i 0.569652i −0.958579 0.284826i \(-0.908064\pi\)
0.958579 0.284826i \(-0.0919358\pi\)
\(152\) 0 0
\(153\) 5.50000 8.29156i 0.444649 0.670333i
\(154\) 0 0
\(155\) 9.94987 0.799193
\(156\) 0 0
\(157\) 3.00000 0.239426 0.119713 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) 0 0
\(159\) 1.65831 5.50000i 0.131513 0.436178i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.00000i 0.548282i 0.961689 + 0.274141i \(0.0883936\pi\)
−0.961689 + 0.274141i \(0.911606\pi\)
\(164\) 0 0
\(165\) 5.50000 18.2414i 0.428174 1.42009i
\(166\) 0 0
\(167\) 6.63325 0.513296 0.256648 0.966505i \(-0.417382\pi\)
0.256648 + 0.966505i \(0.417382\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 11.6082 17.5000i 0.887700 1.33826i
\(172\) 0 0
\(173\) 9.94987i 0.756475i 0.925709 + 0.378237i \(0.123470\pi\)
−0.925709 + 0.378237i \(0.876530\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.50000 + 1.65831i 0.413405 + 0.124646i
\(178\) 0 0
\(179\) 3.31662 0.247896 0.123948 0.992289i \(-0.460444\pi\)
0.123948 + 0.992289i \(0.460444\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) −4.97494 1.50000i −0.367758 0.110883i
\(184\) 0 0
\(185\) 3.31662i 0.243843i
\(186\) 0 0
\(187\) 11.0000i 0.804400i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.94987 0.719948 0.359974 0.932962i \(-0.382786\pi\)
0.359974 + 0.932962i \(0.382786\pi\)
\(192\) 0 0
\(193\) −3.00000 −0.215945 −0.107972 0.994154i \(-0.534436\pi\)
−0.107972 + 0.994154i \(0.534436\pi\)
\(194\) 0 0
\(195\) −6.63325 + 22.0000i −0.475017 + 1.57545i
\(196\) 0 0
\(197\) 19.8997i 1.41780i 0.705310 + 0.708899i \(0.250808\pi\)
−0.705310 + 0.708899i \(0.749192\pi\)
\(198\) 0 0
\(199\) 1.00000i 0.0708881i 0.999372 + 0.0354441i \(0.0112846\pi\)
−0.999372 + 0.0354441i \(0.988715\pi\)
\(200\) 0 0
\(201\) −4.50000 + 14.9248i −0.317406 + 1.05272i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −22.0000 −1.53655
\(206\) 0 0
\(207\) 8.29156 + 5.50000i 0.576303 + 0.382276i
\(208\) 0 0
\(209\) 23.2164i 1.60591i
\(210\) 0 0
\(211\) 26.0000i 1.78991i −0.446153 0.894957i \(-0.647206\pi\)
0.446153 0.894957i \(-0.352794\pi\)
\(212\) 0 0
\(213\) −22.0000 6.63325i −1.50742 0.454503i
\(214\) 0 0
\(215\) −6.63325 −0.452384
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11.6082 3.50000i −0.784409 0.236508i
\(220\) 0 0
\(221\) 13.2665i 0.892401i
\(222\) 0 0
\(223\) 14.0000i 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) −15.0000 9.94987i −1.00000 0.663325i
\(226\) 0 0
\(227\) 3.31662 0.220132 0.110066 0.993924i \(-0.464894\pi\)
0.110066 + 0.993924i \(0.464894\pi\)
\(228\) 0 0
\(229\) 29.0000 1.91637 0.958187 0.286143i \(-0.0923732\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.5831i 1.08640i 0.839604 + 0.543198i \(0.182787\pi\)
−0.839604 + 0.543198i \(0.817213\pi\)
\(234\) 0 0
\(235\) 33.0000i 2.15268i
\(236\) 0 0
\(237\) −4.50000 + 14.9248i −0.292306 + 0.969471i
\(238\) 0 0
\(239\) −6.63325 −0.429069 −0.214535 0.976716i \(-0.568823\pi\)
−0.214535 + 0.976716i \(0.568823\pi\)
\(240\) 0 0
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) 0 0
\(243\) −1.65831 15.5000i −0.106381 0.994325i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 28.0000i 1.78160i
\(248\) 0 0
\(249\) 22.0000 + 6.63325i 1.39419 + 0.420365i
\(250\) 0 0
\(251\) −19.8997 −1.25606 −0.628031 0.778189i \(-0.716139\pi\)
−0.628031 + 0.778189i \(0.716139\pi\)
\(252\) 0 0
\(253\) 11.0000 0.691564
\(254\) 0 0
\(255\) −18.2414 5.50000i −1.14232 0.344423i
\(256\) 0 0
\(257\) 23.2164i 1.44820i −0.689696 0.724099i \(-0.742256\pi\)
0.689696 0.724099i \(-0.257744\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 11.0000 16.5831i 0.680883 1.02647i
\(262\) 0 0
\(263\) −3.31662 −0.204512 −0.102256 0.994758i \(-0.532606\pi\)
−0.102256 + 0.994758i \(0.532606\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) −1.65831 + 5.50000i −0.101487 + 0.336595i
\(268\) 0 0
\(269\) 16.5831i 1.01109i 0.862800 + 0.505545i \(0.168709\pi\)
−0.862800 + 0.505545i \(0.831291\pi\)
\(270\) 0 0
\(271\) 17.0000i 1.03268i −0.856385 0.516338i \(-0.827295\pi\)
0.856385 0.516338i \(-0.172705\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −19.8997 −1.20000
\(276\) 0 0
\(277\) −13.0000 −0.781094 −0.390547 0.920583i \(-0.627714\pi\)
−0.390547 + 0.920583i \(0.627714\pi\)
\(278\) 0 0
\(279\) 4.97494 7.50000i 0.297842 0.449013i
\(280\) 0 0
\(281\) 6.63325i 0.395706i 0.980232 + 0.197853i \(0.0633969\pi\)
−0.980232 + 0.197853i \(0.936603\pi\)
\(282\) 0 0
\(283\) 15.0000i 0.891657i −0.895118 0.445829i \(-0.852909\pi\)
0.895118 0.445829i \(-0.147091\pi\)
\(284\) 0 0
\(285\) −38.5000 11.6082i −2.28054 0.687610i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.00000 0.352941
\(290\) 0 0
\(291\) −13.2665 4.00000i −0.777696 0.234484i
\(292\) 0 0
\(293\) 13.2665i 0.775037i 0.921862 + 0.387519i \(0.126668\pi\)
−0.921862 + 0.387519i \(0.873332\pi\)
\(294\) 0 0
\(295\) 11.0000i 0.640445i
\(296\) 0 0
\(297\) −11.0000 13.2665i −0.638285 0.769800i
\(298\) 0 0
\(299\) −13.2665 −0.767221
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.65831 + 5.50000i −0.0952676 + 0.315967i
\(304\) 0 0
\(305\) 9.94987i 0.569728i
\(306\) 0 0
\(307\) 6.00000i 0.342438i 0.985233 + 0.171219i \(0.0547706\pi\)
−0.985233 + 0.171219i \(0.945229\pi\)
\(308\) 0 0
\(309\) 5.50000 18.2414i 0.312884 1.03772i
\(310\) 0 0
\(311\) 9.94987 0.564206 0.282103 0.959384i \(-0.408968\pi\)
0.282103 + 0.959384i \(0.408968\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.8496i 1.67652i −0.545269 0.838261i \(-0.683573\pi\)
0.545269 0.838261i \(-0.316427\pi\)
\(318\) 0 0
\(319\) 22.0000i 1.23176i
\(320\) 0 0
\(321\) 5.50000 + 1.65831i 0.306980 + 0.0925580i
\(322\) 0 0
\(323\) −23.2164 −1.29179
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 0 0
\(327\) −8.29156 2.50000i −0.458524 0.138250i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.0000i 0.934405i −0.884150 0.467202i \(-0.845262\pi\)
0.884150 0.467202i \(-0.154738\pi\)
\(332\) 0 0
\(333\) −2.50000 1.65831i −0.136999 0.0908750i
\(334\) 0 0
\(335\) 29.8496 1.63086
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 0 0
\(339\) −3.31662 + 11.0000i −0.180134 + 0.597438i
\(340\) 0 0
\(341\) 9.94987i 0.538816i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.50000 18.2414i 0.296110 0.982086i
\(346\) 0 0
\(347\) −29.8496 −1.60241 −0.801206 0.598389i \(-0.795808\pi\)
−0.801206 + 0.598389i \(0.795808\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 13.2665 + 16.0000i 0.708113 + 0.854017i
\(352\) 0 0
\(353\) 36.4829i 1.94179i 0.239511 + 0.970894i \(0.423013\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 44.0000i 2.33528i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 36.4829 1.92549 0.962746 0.270407i \(-0.0871582\pi\)
0.962746 + 0.270407i \(0.0871582\pi\)
\(360\) 0 0
\(361\) −30.0000 −1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 23.2164i 1.21520i
\(366\) 0 0
\(367\) 1.00000i 0.0521996i −0.999659 0.0260998i \(-0.991691\pi\)
0.999659 0.0260998i \(-0.00830876\pi\)
\(368\) 0 0
\(369\) −11.0000 + 16.5831i −0.572637 + 0.863283i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) −1.65831 + 5.50000i −0.0856349 + 0.284019i
\(376\) 0 0
\(377\) 26.5330i 1.36652i
\(378\) 0 0
\(379\) 24.0000i 1.23280i 0.787434 + 0.616399i \(0.211409\pi\)
−0.787434 + 0.616399i \(0.788591\pi\)
\(380\) 0 0
\(381\) 1.00000 3.31662i 0.0512316 0.169916i
\(382\) 0 0
\(383\) −16.5831 −0.847358 −0.423679 0.905812i \(-0.639262\pi\)
−0.423679 + 0.905812i \(0.639262\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.31662 + 5.00000i −0.168594 + 0.254164i
\(388\) 0 0
\(389\) 3.31662i 0.168160i −0.996459 0.0840798i \(-0.973205\pi\)
0.996459 0.0840798i \(-0.0267951\pi\)
\(390\) 0 0
\(391\) 11.0000i 0.556294i
\(392\) 0 0
\(393\) −27.5000 8.29156i −1.38719 0.418254i
\(394\) 0 0
\(395\) 29.8496 1.50190
\(396\) 0 0
\(397\) 13.0000 0.652451 0.326226 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.5831i 0.828122i 0.910249 + 0.414061i \(0.135890\pi\)
−0.910249 + 0.414061i \(0.864110\pi\)
\(402\) 0 0
\(403\) 12.0000i 0.597763i
\(404\) 0 0
\(405\) −27.5000 + 11.6082i −1.36649 + 0.576815i
\(406\) 0 0
\(407\) −3.31662 −0.164399
\(408\) 0 0
\(409\) −15.0000 −0.741702 −0.370851 0.928692i \(-0.620934\pi\)
−0.370851 + 0.928692i \(0.620934\pi\)
\(410\) 0 0
\(411\) −8.29156 + 27.5000i −0.408993 + 1.35647i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 44.0000i 2.15988i
\(416\) 0 0
\(417\) −5.00000 + 16.5831i −0.244851 + 0.812079i
\(418\) 0 0
\(419\) 26.5330 1.29622 0.648111 0.761546i \(-0.275559\pi\)
0.648111 + 0.761546i \(0.275559\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 24.8747 + 16.5000i 1.20945 + 0.802257i
\(424\) 0 0
\(425\) 19.8997i 0.965280i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 22.0000 + 6.63325i 1.06217 + 0.320256i
\(430\) 0 0
\(431\) −23.2164 −1.11829 −0.559147 0.829069i \(-0.688871\pi\)
−0.559147 + 0.829069i \(0.688871\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) −36.4829 11.0000i −1.74922 0.527410i
\(436\) 0 0
\(437\) 23.2164i 1.11059i
\(438\) 0 0
\(439\) 9.00000i 0.429547i 0.976664 + 0.214773i \(0.0689013\pi\)
−0.976664 + 0.214773i \(0.931099\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −29.8496 −1.41820 −0.709099 0.705109i \(-0.750898\pi\)
−0.709099 + 0.705109i \(0.750898\pi\)
\(444\) 0 0
\(445\) 11.0000 0.521450
\(446\) 0 0
\(447\) 8.29156 27.5000i 0.392177 1.30071i
\(448\) 0 0
\(449\) 19.8997i 0.939127i −0.882899 0.469564i \(-0.844411\pi\)
0.882899 0.469564i \(-0.155589\pi\)
\(450\) 0 0
\(451\) 22.0000i 1.03594i
\(452\) 0 0
\(453\) −3.50000 + 11.6082i −0.164444 + 0.545400i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 0 0
\(459\) −13.2665 + 11.0000i −0.619227 + 0.513436i
\(460\) 0 0
\(461\) 33.1662i 1.54471i −0.635193 0.772353i \(-0.719080\pi\)
0.635193 0.772353i \(-0.280920\pi\)
\(462\) 0 0
\(463\) 38.0000i 1.76601i 0.469364 + 0.883005i \(0.344483\pi\)
−0.469364 + 0.883005i \(0.655517\pi\)
\(464\) 0 0
\(465\) −16.5000 4.97494i −0.765169 0.230707i
\(466\) 0 0
\(467\) 3.31662 0.153475 0.0767375 0.997051i \(-0.475550\pi\)
0.0767375 + 0.997051i \(0.475550\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.97494 1.50000i −0.229233 0.0691164i
\(472\) 0 0
\(473\) 6.63325i 0.304997i
\(474\) 0 0
\(475\) 42.0000i 1.92709i
\(476\) 0 0
\(477\) −5.50000 + 8.29156i −0.251828 + 0.379645i
\(478\) 0 0
\(479\) −16.5831 −0.757702 −0.378851 0.925458i \(-0.623681\pi\)
−0.378851 + 0.925458i \(0.623681\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 26.5330i 1.20480i
\(486\) 0 0
\(487\) 17.0000i 0.770344i −0.922845 0.385172i \(-0.874142\pi\)
0.922845 0.385172i \(-0.125858\pi\)
\(488\) 0 0
\(489\) 3.50000 11.6082i 0.158275 0.524940i
\(490\) 0 0
\(491\) −6.63325 −0.299354 −0.149677 0.988735i \(-0.547823\pi\)
−0.149677 + 0.988735i \(0.547823\pi\)
\(492\) 0 0
\(493\) −22.0000 −0.990830
\(494\) 0 0
\(495\) −18.2414 + 27.5000i −0.819892 + 1.23603i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 35.0000i 1.56682i 0.621508 + 0.783408i \(0.286520\pi\)
−0.621508 + 0.783408i \(0.713480\pi\)
\(500\) 0 0
\(501\) −11.0000 3.31662i −0.491444 0.148176i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 11.0000 0.489494
\(506\) 0 0
\(507\) −4.97494 1.50000i −0.220945 0.0666173i
\(508\) 0 0
\(509\) 16.5831i 0.735034i 0.930017 + 0.367517i \(0.119792\pi\)
−0.930017 + 0.367517i \(0.880208\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −28.0000 + 23.2164i −1.23623 + 1.02503i
\(514\) 0 0
\(515\) −36.4829 −1.60763
\(516\) 0 0
\(517\) 33.0000 1.45134
\(518\) 0 0
\(519\) 4.97494 16.5000i 0.218376 0.724270i
\(520\) 0 0
\(521\) 23.2164i 1.01713i −0.861024 0.508564i \(-0.830177\pi\)
0.861024 0.508564i \(-0.169823\pi\)
\(522\) 0 0
\(523\) 19.0000i 0.830812i −0.909636 0.415406i \(-0.863640\pi\)
0.909636 0.415406i \(-0.136360\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.94987 −0.433423
\(528\) 0 0
\(529\) −12.0000 −0.521739
\(530\) 0 0
\(531\) −8.29156 5.50000i −0.359823 0.238680i
\(532\) 0 0
\(533\) 26.5330i 1.14927i
\(534\) 0 0
\(535\) 11.0000i 0.475571i
\(536\) 0 0
\(537\) −5.50000 1.65831i −0.237343 0.0715615i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 37.0000 1.59075 0.795377 0.606115i \(-0.207273\pi\)
0.795377 + 0.606115i \(0.207273\pi\)
\(542\) 0 0
\(543\) −29.8496 9.00000i −1.28097 0.386227i
\(544\) 0 0
\(545\) 16.5831i 0.710343i
\(546\) 0 0
\(547\) 40.0000i 1.71028i 0.518400 + 0.855138i \(0.326528\pi\)
−0.518400 + 0.855138i \(0.673472\pi\)
\(548\) 0 0
\(549\) 7.50000 + 4.97494i 0.320092 + 0.212325i
\(550\) 0 0
\(551\) −46.4327 −1.97810
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.65831 + 5.50000i −0.0703914 + 0.233462i
\(556\) 0 0
\(557\) 3.31662i 0.140530i −0.997528 0.0702650i \(-0.977616\pi\)
0.997528 0.0702650i \(-0.0223845\pi\)
\(558\) 0 0
\(559\) 8.00000i 0.338364i
\(560\) 0 0
\(561\) −5.50000 + 18.2414i −0.232210 + 0.770154i
\(562\) 0 0
\(563\) 9.94987 0.419337 0.209669 0.977773i \(-0.432761\pi\)
0.209669 + 0.977773i \(0.432761\pi\)
\(564\) 0 0
\(565\) 22.0000 0.925547
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 29.8496i 1.25136i −0.780079 0.625681i \(-0.784821\pi\)
0.780079 0.625681i \(-0.215179\pi\)
\(570\) 0 0
\(571\) 5.00000i 0.209243i 0.994512 + 0.104622i \(0.0333632\pi\)
−0.994512 + 0.104622i \(0.966637\pi\)
\(572\) 0 0
\(573\) −16.5000 4.97494i −0.689297 0.207831i
\(574\) 0 0
\(575\) −19.8997 −0.829877
\(576\) 0 0
\(577\) 35.0000 1.45707 0.728535 0.685009i \(-0.240202\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 0 0
\(579\) 4.97494 + 1.50000i 0.206751 + 0.0623379i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.0000i 0.455573i
\(584\) 0 0
\(585\) 22.0000 33.1662i 0.909588 1.37126i
\(586\) 0 0
\(587\) −26.5330 −1.09513 −0.547567 0.836762i \(-0.684446\pi\)
−0.547567 + 0.836762i \(0.684446\pi\)
\(588\) 0 0
\(589\) −21.0000 −0.865290
\(590\) 0 0
\(591\) 9.94987 33.0000i 0.409283 1.35744i
\(592\) 0 0
\(593\) 3.31662i 0.136197i 0.997679 + 0.0680987i \(0.0216933\pi\)
−0.997679 + 0.0680987i \(0.978307\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.500000 1.65831i 0.0204636 0.0678702i
\(598\) 0 0
\(599\) 9.94987 0.406541 0.203270 0.979123i \(-0.434843\pi\)
0.203270 + 0.979123i \(0.434843\pi\)
\(600\) 0 0
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 0 0
\(603\) 14.9248 22.5000i 0.607785 0.916271i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.00000i 0.284121i −0.989858 0.142061i \(-0.954627\pi\)
0.989858 0.142061i \(-0.0453728\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −39.7995 −1.61012
\(612\) 0 0
\(613\) 1.00000 0.0403896 0.0201948 0.999796i \(-0.493571\pi\)
0.0201948 + 0.999796i \(0.493571\pi\)
\(614\) 0 0
\(615\) 36.4829 + 11.0000i 1.47113 + 0.443563i
\(616\) 0 0
\(617\) 6.63325i 0.267045i 0.991046 + 0.133522i \(0.0426288\pi\)
−0.991046 + 0.133522i \(0.957371\pi\)
\(618\) 0 0
\(619\) 25.0000i 1.00483i −0.864625 0.502417i \(-0.832444\pi\)
0.864625 0.502417i \(-0.167556\pi\)
\(620\) 0 0
\(621\) −11.0000 13.2665i −0.441415 0.532366i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −11.6082 + 38.5000i −0.463586 + 1.53754i
\(628\) 0 0
\(629\) 3.31662i 0.132242i
\(630\) 0 0
\(631\) 18.0000i 0.716569i −0.933613 0.358284i \(-0.883362\pi\)
0.933613 0.358284i \(-0.116638\pi\)
\(632\) 0 0
\(633\) −13.0000 + 43.1161i −0.516704 + 1.71371i
\(634\) 0 0
\(635\) −6.63325 −0.263232
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 33.1662 + 22.0000i 1.31204 + 0.870307i
\(640\) 0 0
\(641\) 36.4829i 1.44099i −0.693462 0.720493i \(-0.743915\pi\)
0.693462 0.720493i \(-0.256085\pi\)
\(642\) 0 0
\(643\) 2.00000i 0.0788723i −0.999222 0.0394362i \(-0.987444\pi\)
0.999222 0.0394362i \(-0.0125562\pi\)
\(644\) 0 0
\(645\) 11.0000 + 3.31662i 0.433125 + 0.130592i
\(646\) 0 0
\(647\) −3.31662 −0.130390 −0.0651950 0.997873i \(-0.520767\pi\)
−0.0651950 + 0.997873i \(0.520767\pi\)
\(648\) 0 0
\(649\) −11.0000 −0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.31662i 0.129790i −0.997892 0.0648948i \(-0.979329\pi\)
0.997892 0.0648948i \(-0.0206712\pi\)
\(654\) 0 0
\(655\) 55.0000i 2.14903i
\(656\) 0 0
\(657\) 17.5000 + 11.6082i 0.682740 + 0.452878i
\(658\) 0 0
\(659\) −26.5330 −1.03358 −0.516789 0.856113i \(-0.672873\pi\)
−0.516789 + 0.856113i \(0.672873\pi\)
\(660\) 0 0
\(661\) 39.0000 1.51692 0.758462 0.651717i \(-0.225951\pi\)
0.758462 + 0.651717i \(0.225951\pi\)
\(662\) 0 0
\(663\) 6.63325 22.0000i 0.257614 0.854409i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22.0000i 0.851843i
\(668\) 0 0
\(669\) −7.00000 + 23.2164i −0.270636 + 0.897597i
\(670\) 0 0
\(671\) 9.94987 0.384111
\(672\) 0 0
\(673\) 40.0000 1.54189 0.770943 0.636904i \(-0.219785\pi\)
0.770943 + 0.636904i \(0.219785\pi\)
\(674\) 0 0
\(675\) 19.8997 + 24.0000i 0.765942 + 0.923760i
\(676\) 0 0
\(677\) 9.94987i 0.382405i −0.981551 0.191202i \(-0.938761\pi\)
0.981551 0.191202i \(-0.0612387\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −5.50000 1.65831i −0.210760 0.0635467i
\(682\) 0 0
\(683\) −23.2164 −0.888350 −0.444175 0.895940i \(-0.646503\pi\)
−0.444175 + 0.895940i \(0.646503\pi\)
\(684\) 0 0
\(685\) 55.0000 2.10144
\(686\) 0 0
\(687\) −48.0911 14.5000i −1.83479 0.553210i
\(688\) 0 0
\(689\) 13.2665i 0.505413i
\(690\) 0 0
\(691\) 33.0000i 1.25538i −0.778464 0.627690i \(-0.784001\pi\)
0.778464 0.627690i \(-0.215999\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.1662 1.25807
\(696\) 0 0
\(697\) 22.0000 0.833309
\(698\) 0 0
\(699\) 8.29156 27.5000i 0.313616 1.04015i
\(700\) 0 0
\(701\) 13.2665i 0.501069i −0.968108 0.250534i \(-0.919394\pi\)
0.968108 0.250534i \(-0.0806063\pi\)
\(702\) 0 0
\(703\) 7.00000i 0.264010i
\(704\) 0 0
\(705\) 16.5000 54.7243i 0.621426 2.06104i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −33.0000 −1.23934 −0.619671 0.784862i \(-0.712734\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(710\) 0 0
\(711\) 14.9248 22.5000i 0.559724 0.843816i
\(712\) 0 0
\(713\) 9.94987i 0.372626i
\(714\) 0 0
\(715\) 44.0000i 1.64551i
\(716\) 0 0
\(717\) 11.0000 + 3.31662i 0.410803 + 0.123862i
\(718\) 0 0
\(719\) −23.2164 −0.865825 −0.432912 0.901436i \(-0.642514\pi\)
−0.432912 + 0.901436i \(0.642514\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −31.5079 9.50000i −1.17179 0.353309i
\(724\) 0 0
\(725\) 39.7995i 1.47812i
\(726\) 0 0
\(727\) 10.0000i 0.370879i −0.982656 0.185440i \(-0.940629\pi\)
0.982656 0.185440i \(-0.0593710\pi\)
\(728\) 0 0
\(729\) −5.00000 + 26.5330i −0.185185 + 0.982704i
\(730\) 0 0
\(731\) 6.63325 0.245340
\(732\) 0 0
\(733\) 7.00000 0.258551 0.129275 0.991609i \(-0.458735\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 29.8496i 1.09952i
\(738\) 0 0
\(739\) 31.0000i 1.14035i −0.821522 0.570177i \(-0.806875\pi\)
0.821522 0.570177i \(-0.193125\pi\)
\(740\) 0 0
\(741\) 14.0000 46.4327i 0.514303 1.70575i
\(742\) 0 0
\(743\) 13.2665 0.486701 0.243350 0.969938i \(-0.421754\pi\)
0.243350 + 0.969938i \(0.421754\pi\)
\(744\) 0 0
\(745\) −55.0000 −2.01504
\(746\) 0 0
\(747\) −33.1662 22.0000i −1.21349 0.804938i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.00000i 0.182453i 0.995830 + 0.0912263i \(0.0290787\pi\)
−0.995830 + 0.0912263i \(0.970921\pi\)
\(752\) 0 0
\(753\) 33.0000 + 9.94987i 1.20259 + 0.362594i
\(754\) 0 0
\(755\) 23.2164 0.844930
\(756\) 0 0
\(757\) −48.0000 −1.74459 −0.872295 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) 0 0
\(759\) −18.2414 5.50000i −0.662122 0.199637i
\(760\) 0 0
\(761\) 23.2164i 0.841593i −0.907155 0.420796i \(-0.861751\pi\)
0.907155 0.420796i \(-0.138249\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 27.5000 + 18.2414i 0.994265 + 0.659521i
\(766\) 0 0
\(767\) 13.2665 0.479026
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −11.6082 + 38.5000i −0.418059 + 1.38654i
\(772\) 0 0
\(773\) 16.5831i 0.596454i −0.954495 0.298227i \(-0.903605\pi\)
0.954495 0.298227i \(-0.0963952\pi\)
\(774\) 0 0
\(775\) 18.0000i 0.646579i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 46.4327 1.66363
\(780\) 0 0
\(781\) 44.0000 1.57444
\(782\) 0 0
\(783\) −26.5330 + 22.0000i −0.948212 + 0.786216i
\(784\) 0 0
\(785\) 9.94987i 0.355126i
\(786\) 0 0
\(787\) 45.0000i 1.60408i −0.597272 0.802038i \(-0.703749\pi\)
0.597272 0.802038i \(-0.296251\pi\)
\(788\) 0 0
\(789\) 5.50000 + 1.65831i 0.195805 + 0.0590375i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 18.2414 + 5.50000i 0.646957 + 0.195065i
\(796\) 0 0
\(797\) 6.63325i 0.234962i 0.993075 + 0.117481i \(0.0374819\pi\)
−0.993075 + 0.117481i \(0.962518\pi\)
\(798\) 0 0
\(799\) 33.0000i 1.16746i
\(800\) 0 0
\(801\)