Properties

Label 4560.2.d.j.2431.11
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.11
Root \(0.687758i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.j.2431.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +3.95229i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +3.95229i q^{7} +1.00000 q^{9} -2.16866i q^{11} +6.98743i q^{13} -1.00000 q^{15} -2.03991 q^{17} +(-2.20175 + 3.76196i) q^{19} -3.95229i q^{21} +8.87858i q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.63276i q^{29} -1.31415 q^{31} +2.16866i q^{33} +3.95229i q^{35} +1.77374i q^{37} -6.98743i q^{39} -4.92629i q^{41} +9.57516i q^{43} +1.00000 q^{45} -9.15609i q^{47} -8.62061 q^{49} +2.03991 q^{51} -5.41448i q^{53} -2.16866i q^{55} +(2.20175 - 3.76196i) q^{57} -12.9348 q^{59} +1.62061 q^{61} +3.95229i q^{63} +6.98743i q^{65} -6.56189 q^{67} -8.87858i q^{69} -5.35406 q^{71} +2.95056 q^{73} -1.00000 q^{75} +8.57117 q^{77} +11.1906 q^{79} +1.00000 q^{81} -4.37136i q^{83} -2.03991 q^{85} +5.63276i q^{87} -6.91831i q^{89} -27.6164 q^{91} +1.31415 q^{93} +(-2.20175 + 3.76196i) q^{95} +13.0393i q^{97} -2.16866i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} - 12q^{15} + 4q^{17} + 12q^{25} - 12q^{27} + 12q^{31} + 12q^{45} - 28q^{49} - 4q^{51} - 52q^{59} - 56q^{61} + 32q^{67} - 8q^{71} + 32q^{73} - 12q^{75} + 24q^{77} + 28q^{79} + 12q^{81} + 4q^{85} - 32q^{91} - 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.95229i 1.49383i 0.664922 + 0.746913i \(0.268465\pi\)
−0.664922 + 0.746913i \(0.731535\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.16866i 0.653875i −0.945046 0.326937i \(-0.893983\pi\)
0.945046 0.326937i \(-0.106017\pi\)
\(12\) 0 0
\(13\) 6.98743i 1.93796i 0.247130 + 0.968982i \(0.420513\pi\)
−0.247130 + 0.968982i \(0.579487\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.03991 −0.494750 −0.247375 0.968920i \(-0.579568\pi\)
−0.247375 + 0.968920i \(0.579568\pi\)
\(18\) 0 0
\(19\) −2.20175 + 3.76196i −0.505115 + 0.863052i
\(20\) 0 0
\(21\) 3.95229i 0.862461i
\(22\) 0 0
\(23\) 8.87858i 1.85131i 0.378366 + 0.925656i \(0.376486\pi\)
−0.378366 + 0.925656i \(0.623514\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.63276i 1.04598i −0.852340 0.522989i \(-0.824817\pi\)
0.852340 0.522989i \(-0.175183\pi\)
\(30\) 0 0
\(31\) −1.31415 −0.236028 −0.118014 0.993012i \(-0.537653\pi\)
−0.118014 + 0.993012i \(0.537653\pi\)
\(32\) 0 0
\(33\) 2.16866i 0.377515i
\(34\) 0 0
\(35\) 3.95229i 0.668059i
\(36\) 0 0
\(37\) 1.77374i 0.291601i 0.989314 + 0.145801i \(0.0465758\pi\)
−0.989314 + 0.145801i \(0.953424\pi\)
\(38\) 0 0
\(39\) 6.98743i 1.11888i
\(40\) 0 0
\(41\) 4.92629i 0.769357i −0.923051 0.384679i \(-0.874312\pi\)
0.923051 0.384679i \(-0.125688\pi\)
\(42\) 0 0
\(43\) 9.57516i 1.46020i 0.683341 + 0.730099i \(0.260526\pi\)
−0.683341 + 0.730099i \(0.739474\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 9.15609i 1.33555i −0.744362 0.667776i \(-0.767246\pi\)
0.744362 0.667776i \(-0.232754\pi\)
\(48\) 0 0
\(49\) −8.62061 −1.23152
\(50\) 0 0
\(51\) 2.03991 0.285644
\(52\) 0 0
\(53\) 5.41448i 0.743736i −0.928286 0.371868i \(-0.878717\pi\)
0.928286 0.371868i \(-0.121283\pi\)
\(54\) 0 0
\(55\) 2.16866i 0.292422i
\(56\) 0 0
\(57\) 2.20175 3.76196i 0.291629 0.498283i
\(58\) 0 0
\(59\) −12.9348 −1.68396 −0.841981 0.539507i \(-0.818611\pi\)
−0.841981 + 0.539507i \(0.818611\pi\)
\(60\) 0 0
\(61\) 1.62061 0.207497 0.103749 0.994604i \(-0.466916\pi\)
0.103749 + 0.994604i \(0.466916\pi\)
\(62\) 0 0
\(63\) 3.95229i 0.497942i
\(64\) 0 0
\(65\) 6.98743i 0.866684i
\(66\) 0 0
\(67\) −6.56189 −0.801662 −0.400831 0.916152i \(-0.631279\pi\)
−0.400831 + 0.916152i \(0.631279\pi\)
\(68\) 0 0
\(69\) 8.87858i 1.06886i
\(70\) 0 0
\(71\) −5.35406 −0.635410 −0.317705 0.948190i \(-0.602912\pi\)
−0.317705 + 0.948190i \(0.602912\pi\)
\(72\) 0 0
\(73\) 2.95056 0.345337 0.172669 0.984980i \(-0.444761\pi\)
0.172669 + 0.984980i \(0.444761\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.57117 0.976775
\(78\) 0 0
\(79\) 11.1906 1.25904 0.629521 0.776984i \(-0.283251\pi\)
0.629521 + 0.776984i \(0.283251\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.37136i 0.479819i −0.970795 0.239910i \(-0.922882\pi\)
0.970795 0.239910i \(-0.0771178\pi\)
\(84\) 0 0
\(85\) −2.03991 −0.221259
\(86\) 0 0
\(87\) 5.63276i 0.603895i
\(88\) 0 0
\(89\) 6.91831i 0.733340i −0.930351 0.366670i \(-0.880498\pi\)
0.930351 0.366670i \(-0.119502\pi\)
\(90\) 0 0
\(91\) −27.6164 −2.89498
\(92\) 0 0
\(93\) 1.31415 0.136271
\(94\) 0 0
\(95\) −2.20175 + 3.76196i −0.225894 + 0.385968i
\(96\) 0 0
\(97\) 13.0393i 1.32394i 0.749532 + 0.661968i \(0.230279\pi\)
−0.749532 + 0.661968i \(0.769721\pi\)
\(98\) 0 0
\(99\) 2.16866i 0.217958i
\(100\) 0 0
\(101\) 6.88415 0.684998 0.342499 0.939518i \(-0.388727\pi\)
0.342499 + 0.939518i \(0.388727\pi\)
\(102\) 0 0
\(103\) 6.56189 0.646562 0.323281 0.946303i \(-0.395214\pi\)
0.323281 + 0.946303i \(0.395214\pi\)
\(104\) 0 0
\(105\) 3.95229i 0.385704i
\(106\) 0 0
\(107\) −9.20783 −0.890155 −0.445077 0.895492i \(-0.646824\pi\)
−0.445077 + 0.895492i \(0.646824\pi\)
\(108\) 0 0
\(109\) 17.5873i 1.68455i −0.539045 0.842277i \(-0.681215\pi\)
0.539045 0.842277i \(-0.318785\pi\)
\(110\) 0 0
\(111\) 1.77374i 0.168356i
\(112\) 0 0
\(113\) 0.870836i 0.0819213i 0.999161 + 0.0409607i \(0.0130418\pi\)
−0.999161 + 0.0409607i \(0.986958\pi\)
\(114\) 0 0
\(115\) 8.87858i 0.827932i
\(116\) 0 0
\(117\) 6.98743i 0.645988i
\(118\) 0 0
\(119\) 8.06230i 0.739070i
\(120\) 0 0
\(121\) 6.29692 0.572447
\(122\) 0 0
\(123\) 4.92629i 0.444189i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.61133 0.497925 0.248962 0.968513i \(-0.419910\pi\)
0.248962 + 0.968513i \(0.419910\pi\)
\(128\) 0 0
\(129\) 9.57516i 0.843046i
\(130\) 0 0
\(131\) 13.1751i 1.15112i −0.817761 0.575558i \(-0.804785\pi\)
0.817761 0.575558i \(-0.195215\pi\)
\(132\) 0 0
\(133\) −14.8683 8.70195i −1.28925 0.754554i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 13.9976 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(138\) 0 0
\(139\) 21.0062i 1.78173i 0.454273 + 0.890863i \(0.349899\pi\)
−0.454273 + 0.890863i \(0.650101\pi\)
\(140\) 0 0
\(141\) 9.15609i 0.771082i
\(142\) 0 0
\(143\) 15.1533 1.26719
\(144\) 0 0
\(145\) 5.63276i 0.467775i
\(146\) 0 0
\(147\) 8.62061 0.711016
\(148\) 0 0
\(149\) 5.98236 0.490094 0.245047 0.969511i \(-0.421197\pi\)
0.245047 + 0.969511i \(0.421197\pi\)
\(150\) 0 0
\(151\) 12.2317 0.995400 0.497700 0.867349i \(-0.334178\pi\)
0.497700 + 0.867349i \(0.334178\pi\)
\(152\) 0 0
\(153\) −2.03991 −0.164917
\(154\) 0 0
\(155\) −1.31415 −0.105555
\(156\) 0 0
\(157\) −6.13042 −0.489261 −0.244630 0.969616i \(-0.578667\pi\)
−0.244630 + 0.969616i \(0.578667\pi\)
\(158\) 0 0
\(159\) 5.41448i 0.429396i
\(160\) 0 0
\(161\) −35.0907 −2.76554
\(162\) 0 0
\(163\) 9.47515i 0.742151i −0.928603 0.371076i \(-0.878989\pi\)
0.928603 0.371076i \(-0.121011\pi\)
\(164\) 0 0
\(165\) 2.16866i 0.168830i
\(166\) 0 0
\(167\) −21.7817 −1.68551 −0.842757 0.538294i \(-0.819069\pi\)
−0.842757 + 0.538294i \(0.819069\pi\)
\(168\) 0 0
\(169\) −35.8242 −2.75571
\(170\) 0 0
\(171\) −2.20175 + 3.76196i −0.168372 + 0.287684i
\(172\) 0 0
\(173\) 22.4728i 1.70857i −0.519802 0.854287i \(-0.673994\pi\)
0.519802 0.854287i \(-0.326006\pi\)
\(174\) 0 0
\(175\) 3.95229i 0.298765i
\(176\) 0 0
\(177\) 12.9348 0.972236
\(178\) 0 0
\(179\) 3.11079 0.232512 0.116256 0.993219i \(-0.462911\pi\)
0.116256 + 0.993219i \(0.462911\pi\)
\(180\) 0 0
\(181\) 2.28171i 0.169598i 0.996398 + 0.0847992i \(0.0270249\pi\)
−0.996398 + 0.0847992i \(0.972975\pi\)
\(182\) 0 0
\(183\) −1.62061 −0.119799
\(184\) 0 0
\(185\) 1.77374i 0.130408i
\(186\) 0 0
\(187\) 4.42386i 0.323505i
\(188\) 0 0
\(189\) 3.95229i 0.287487i
\(190\) 0 0
\(191\) 25.4170i 1.83911i 0.392960 + 0.919555i \(0.371451\pi\)
−0.392960 + 0.919555i \(0.628549\pi\)
\(192\) 0 0
\(193\) 12.2011i 0.878256i −0.898425 0.439128i \(-0.855287\pi\)
0.898425 0.439128i \(-0.144713\pi\)
\(194\) 0 0
\(195\) 6.98743i 0.500380i
\(196\) 0 0
\(197\) −23.2941 −1.65964 −0.829818 0.558034i \(-0.811556\pi\)
−0.829818 + 0.558034i \(0.811556\pi\)
\(198\) 0 0
\(199\) 5.68961i 0.403326i −0.979455 0.201663i \(-0.935365\pi\)
0.979455 0.201663i \(-0.0646345\pi\)
\(200\) 0 0
\(201\) 6.56189 0.462840
\(202\) 0 0
\(203\) 22.2623 1.56251
\(204\) 0 0
\(205\) 4.92629i 0.344067i
\(206\) 0 0
\(207\) 8.87858i 0.617104i
\(208\) 0 0
\(209\) 8.15839 + 4.77484i 0.564328 + 0.330282i
\(210\) 0 0
\(211\) 20.4764 1.40965 0.704827 0.709379i \(-0.251025\pi\)
0.704827 + 0.709379i \(0.251025\pi\)
\(212\) 0 0
\(213\) 5.35406 0.366854
\(214\) 0 0
\(215\) 9.57516i 0.653021i
\(216\) 0 0
\(217\) 5.19391i 0.352585i
\(218\) 0 0
\(219\) −2.95056 −0.199381
\(220\) 0 0
\(221\) 14.2537i 0.958808i
\(222\) 0 0
\(223\) −24.2530 −1.62410 −0.812051 0.583586i \(-0.801649\pi\)
−0.812051 + 0.583586i \(0.801649\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −11.0735 −0.734976 −0.367488 0.930028i \(-0.619782\pi\)
−0.367488 + 0.930028i \(0.619782\pi\)
\(228\) 0 0
\(229\) −20.6524 −1.36475 −0.682375 0.731003i \(-0.739053\pi\)
−0.682375 + 0.731003i \(0.739053\pi\)
\(230\) 0 0
\(231\) −8.57117 −0.563941
\(232\) 0 0
\(233\) −3.98277 −0.260920 −0.130460 0.991454i \(-0.541645\pi\)
−0.130460 + 0.991454i \(0.541645\pi\)
\(234\) 0 0
\(235\) 9.15609i 0.597277i
\(236\) 0 0
\(237\) −11.1906 −0.726908
\(238\) 0 0
\(239\) 13.8555i 0.896241i 0.893973 + 0.448120i \(0.147906\pi\)
−0.893973 + 0.448120i \(0.852094\pi\)
\(240\) 0 0
\(241\) 11.1943i 0.721091i 0.932742 + 0.360545i \(0.117409\pi\)
−0.932742 + 0.360545i \(0.882591\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −8.62061 −0.550750
\(246\) 0 0
\(247\) −26.2864 15.3846i −1.67256 0.978896i
\(248\) 0 0
\(249\) 4.37136i 0.277024i
\(250\) 0 0
\(251\) 17.3655i 1.09610i 0.836446 + 0.548050i \(0.184630\pi\)
−0.836446 + 0.548050i \(0.815370\pi\)
\(252\) 0 0
\(253\) 19.2546 1.21053
\(254\) 0 0
\(255\) 2.03991 0.127744
\(256\) 0 0
\(257\) 1.35467i 0.0845021i 0.999107 + 0.0422510i \(0.0134529\pi\)
−0.999107 + 0.0422510i \(0.986547\pi\)
\(258\) 0 0
\(259\) −7.01035 −0.435602
\(260\) 0 0
\(261\) 5.63276i 0.348659i
\(262\) 0 0
\(263\) 5.19286i 0.320205i 0.987100 + 0.160103i \(0.0511826\pi\)
−0.987100 + 0.160103i \(0.948817\pi\)
\(264\) 0 0
\(265\) 5.41448i 0.332609i
\(266\) 0 0
\(267\) 6.91831i 0.423394i
\(268\) 0 0
\(269\) 0.931395i 0.0567882i −0.999597 0.0283941i \(-0.990961\pi\)
0.999597 0.0283941i \(-0.00903933\pi\)
\(270\) 0 0
\(271\) 26.4790i 1.60848i −0.594302 0.804242i \(-0.702572\pi\)
0.594302 0.804242i \(-0.297428\pi\)
\(272\) 0 0
\(273\) 27.6164 1.67142
\(274\) 0 0
\(275\) 2.16866i 0.130775i
\(276\) 0 0
\(277\) 6.62871 0.398281 0.199140 0.979971i \(-0.436185\pi\)
0.199140 + 0.979971i \(0.436185\pi\)
\(278\) 0 0
\(279\) −1.31415 −0.0786762
\(280\) 0 0
\(281\) 8.67106i 0.517272i 0.965975 + 0.258636i \(0.0832730\pi\)
−0.965975 + 0.258636i \(0.916727\pi\)
\(282\) 0 0
\(283\) 26.2409i 1.55986i −0.625866 0.779930i \(-0.715254\pi\)
0.625866 0.779930i \(-0.284746\pi\)
\(284\) 0 0
\(285\) 2.20175 3.76196i 0.130420 0.222839i
\(286\) 0 0
\(287\) 19.4701 1.14929
\(288\) 0 0
\(289\) −12.8388 −0.755223
\(290\) 0 0
\(291\) 13.0393i 0.764375i
\(292\) 0 0
\(293\) 24.0239i 1.40349i 0.712426 + 0.701747i \(0.247596\pi\)
−0.712426 + 0.701747i \(0.752404\pi\)
\(294\) 0 0
\(295\) −12.9348 −0.753091
\(296\) 0 0
\(297\) 2.16866i 0.125838i
\(298\) 0 0
\(299\) −62.0385 −3.58778
\(300\) 0 0
\(301\) −37.8438 −2.18128
\(302\) 0 0
\(303\) −6.88415 −0.395484
\(304\) 0 0
\(305\) 1.62061 0.0927956
\(306\) 0 0
\(307\) −15.6908 −0.895523 −0.447761 0.894153i \(-0.647779\pi\)
−0.447761 + 0.894153i \(0.647779\pi\)
\(308\) 0 0
\(309\) −6.56189 −0.373293
\(310\) 0 0
\(311\) 5.71614i 0.324133i 0.986780 + 0.162066i \(0.0518159\pi\)
−0.986780 + 0.162066i \(0.948184\pi\)
\(312\) 0 0
\(313\) 2.66076 0.150395 0.0751976 0.997169i \(-0.476041\pi\)
0.0751976 + 0.997169i \(0.476041\pi\)
\(314\) 0 0
\(315\) 3.95229i 0.222686i
\(316\) 0 0
\(317\) 21.7787i 1.22321i 0.791163 + 0.611606i \(0.209476\pi\)
−0.791163 + 0.611606i \(0.790524\pi\)
\(318\) 0 0
\(319\) −12.2155 −0.683938
\(320\) 0 0
\(321\) 9.20783 0.513931
\(322\) 0 0
\(323\) 4.49136 7.67403i 0.249906 0.426995i
\(324\) 0 0
\(325\) 6.98743i 0.387593i
\(326\) 0 0
\(327\) 17.5873i 0.972578i
\(328\) 0 0
\(329\) 36.1875 1.99508
\(330\) 0 0
\(331\) −18.6834 −1.02693 −0.513467 0.858109i \(-0.671639\pi\)
−0.513467 + 0.858109i \(0.671639\pi\)
\(332\) 0 0
\(333\) 1.77374i 0.0972005i
\(334\) 0 0
\(335\) −6.56189 −0.358514
\(336\) 0 0
\(337\) 21.2652i 1.15839i 0.815189 + 0.579195i \(0.196633\pi\)
−0.815189 + 0.579195i \(0.803367\pi\)
\(338\) 0 0
\(339\) 0.870836i 0.0472973i
\(340\) 0 0
\(341\) 2.84994i 0.154333i
\(342\) 0 0
\(343\) 6.40510i 0.345843i
\(344\) 0 0
\(345\) 8.87858i 0.478007i
\(346\) 0 0
\(347\) 18.6435i 1.00084i 0.865784 + 0.500418i \(0.166820\pi\)
−0.865784 + 0.500418i \(0.833180\pi\)
\(348\) 0 0
\(349\) 33.9270 1.81607 0.908036 0.418891i \(-0.137581\pi\)
0.908036 + 0.418891i \(0.137581\pi\)
\(350\) 0 0
\(351\) 6.98743i 0.372961i
\(352\) 0 0
\(353\) −1.05835 −0.0563303 −0.0281652 0.999603i \(-0.508966\pi\)
−0.0281652 + 0.999603i \(0.508966\pi\)
\(354\) 0 0
\(355\) −5.35406 −0.284164
\(356\) 0 0
\(357\) 8.06230i 0.426702i
\(358\) 0 0
\(359\) 27.4331i 1.44786i 0.689871 + 0.723932i \(0.257667\pi\)
−0.689871 + 0.723932i \(0.742333\pi\)
\(360\) 0 0
\(361\) −9.30462 16.5657i −0.489717 0.871882i
\(362\) 0 0
\(363\) −6.29692 −0.330503
\(364\) 0 0
\(365\) 2.95056 0.154439
\(366\) 0 0
\(367\) 2.09954i 0.109595i −0.998497 0.0547975i \(-0.982549\pi\)
0.998497 0.0547975i \(-0.0174513\pi\)
\(368\) 0 0
\(369\) 4.92629i 0.256452i
\(370\) 0 0
\(371\) 21.3996 1.11101
\(372\) 0 0
\(373\) 7.00721i 0.362820i −0.983408 0.181410i \(-0.941934\pi\)
0.983408 0.181410i \(-0.0580661\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 39.3585 2.02707
\(378\) 0 0
\(379\) −13.9907 −0.718655 −0.359328 0.933212i \(-0.616994\pi\)
−0.359328 + 0.933212i \(0.616994\pi\)
\(380\) 0 0
\(381\) −5.61133 −0.287477
\(382\) 0 0
\(383\) −7.72187 −0.394569 −0.197284 0.980346i \(-0.563212\pi\)
−0.197284 + 0.980346i \(0.563212\pi\)
\(384\) 0 0
\(385\) 8.57117 0.436827
\(386\) 0 0
\(387\) 9.57516i 0.486733i
\(388\) 0 0
\(389\) −10.8868 −0.551983 −0.275991 0.961160i \(-0.589006\pi\)
−0.275991 + 0.961160i \(0.589006\pi\)
\(390\) 0 0
\(391\) 18.1115i 0.915936i
\(392\) 0 0
\(393\) 13.1751i 0.664597i
\(394\) 0 0
\(395\) 11.1906 0.563060
\(396\) 0 0
\(397\) −23.7551 −1.19224 −0.596118 0.802897i \(-0.703291\pi\)
−0.596118 + 0.802897i \(0.703291\pi\)
\(398\) 0 0
\(399\) 14.8683 + 8.70195i 0.744348 + 0.435642i
\(400\) 0 0
\(401\) 26.6810i 1.33238i 0.745780 + 0.666192i \(0.232077\pi\)
−0.745780 + 0.666192i \(0.767923\pi\)
\(402\) 0 0
\(403\) 9.18254i 0.457415i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 3.84664 0.190671
\(408\) 0 0
\(409\) 24.1730i 1.19528i 0.801765 + 0.597640i \(0.203895\pi\)
−0.801765 + 0.597640i \(0.796105\pi\)
\(410\) 0 0
\(411\) −13.9976 −0.690450
\(412\) 0 0
\(413\) 51.1219i 2.51555i
\(414\) 0 0
\(415\) 4.37136i 0.214582i
\(416\) 0 0
\(417\) 21.0062i 1.02868i
\(418\) 0 0
\(419\) 2.72367i 0.133060i −0.997784 0.0665300i \(-0.978807\pi\)
0.997784 0.0665300i \(-0.0211928\pi\)
\(420\) 0 0
\(421\) 31.1453i 1.51793i 0.651131 + 0.758966i \(0.274295\pi\)
−0.651131 + 0.758966i \(0.725705\pi\)
\(422\) 0 0
\(423\) 9.15609i 0.445184i
\(424\) 0 0
\(425\) −2.03991 −0.0989500
\(426\) 0 0
\(427\) 6.40510i 0.309965i
\(428\) 0 0
\(429\) −15.1533 −0.731611
\(430\) 0 0
\(431\) 15.7900 0.760578 0.380289 0.924868i \(-0.375824\pi\)
0.380289 + 0.924868i \(0.375824\pi\)
\(432\) 0 0
\(433\) 37.1696i 1.78626i 0.449801 + 0.893129i \(0.351495\pi\)
−0.449801 + 0.893129i \(0.648505\pi\)
\(434\) 0 0
\(435\) 5.63276i 0.270070i
\(436\) 0 0
\(437\) −33.4008 19.5484i −1.59778 0.935127i
\(438\) 0 0
\(439\) 6.94673 0.331549 0.165775 0.986164i \(-0.446988\pi\)
0.165775 + 0.986164i \(0.446988\pi\)
\(440\) 0 0
\(441\) −8.62061 −0.410505
\(442\) 0 0
\(443\) 17.3896i 0.826206i −0.910684 0.413103i \(-0.864445\pi\)
0.910684 0.413103i \(-0.135555\pi\)
\(444\) 0 0
\(445\) 6.91831i 0.327959i
\(446\) 0 0
\(447\) −5.98236 −0.282956
\(448\) 0 0
\(449\) 20.9370i 0.988079i −0.869439 0.494040i \(-0.835520\pi\)
0.869439 0.494040i \(-0.164480\pi\)
\(450\) 0 0
\(451\) −10.6834 −0.503064
\(452\) 0 0
\(453\) −12.2317 −0.574694
\(454\) 0 0
\(455\) −27.6164 −1.29468
\(456\) 0 0
\(457\) 2.09321 0.0979162 0.0489581 0.998801i \(-0.484410\pi\)
0.0489581 + 0.998801i \(0.484410\pi\)
\(458\) 0 0
\(459\) 2.03991 0.0952146
\(460\) 0 0
\(461\) −15.2212 −0.708923 −0.354462 0.935071i \(-0.615336\pi\)
−0.354462 + 0.935071i \(0.615336\pi\)
\(462\) 0 0
\(463\) 23.1211i 1.07453i 0.843415 + 0.537263i \(0.180542\pi\)
−0.843415 + 0.537263i \(0.819458\pi\)
\(464\) 0 0
\(465\) 1.31415 0.0609423
\(466\) 0 0
\(467\) 23.8388i 1.10313i −0.834133 0.551563i \(-0.814032\pi\)
0.834133 0.551563i \(-0.185968\pi\)
\(468\) 0 0
\(469\) 25.9345i 1.19754i
\(470\) 0 0
\(471\) 6.13042 0.282475
\(472\) 0 0
\(473\) 20.7652 0.954787
\(474\) 0 0
\(475\) −2.20175 + 3.76196i −0.101023 + 0.172610i
\(476\) 0 0
\(477\) 5.41448i 0.247912i
\(478\) 0 0
\(479\) 3.04636i 0.139192i 0.997575 + 0.0695959i \(0.0221710\pi\)
−0.997575 + 0.0695959i \(0.977829\pi\)
\(480\) 0 0
\(481\) −12.3939 −0.565113
\(482\) 0 0
\(483\) 35.0907 1.59668
\(484\) 0 0
\(485\) 13.0393i 0.592082i
\(486\) 0 0
\(487\) 31.7706 1.43967 0.719833 0.694148i \(-0.244219\pi\)
0.719833 + 0.694148i \(0.244219\pi\)
\(488\) 0 0
\(489\) 9.47515i 0.428481i
\(490\) 0 0
\(491\) 19.9060i 0.898347i 0.893445 + 0.449174i \(0.148282\pi\)
−0.893445 + 0.449174i \(0.851718\pi\)
\(492\) 0 0
\(493\) 11.4903i 0.517497i
\(494\) 0 0
\(495\) 2.16866i 0.0974739i
\(496\) 0 0
\(497\) 21.1608i 0.949191i
\(498\) 0 0
\(499\) 9.90847i 0.443564i −0.975096 0.221782i \(-0.928813\pi\)
0.975096 0.221782i \(-0.0711873\pi\)
\(500\) 0 0
\(501\) 21.7817 0.973132
\(502\) 0 0
\(503\) 1.97119i 0.0878911i 0.999034 + 0.0439456i \(0.0139928\pi\)
−0.999034 + 0.0439456i \(0.986007\pi\)
\(504\) 0 0
\(505\) 6.88415 0.306341
\(506\) 0 0
\(507\) 35.8242 1.59101
\(508\) 0 0
\(509\) 23.7299i 1.05181i 0.850544 + 0.525905i \(0.176273\pi\)
−0.850544 + 0.525905i \(0.823727\pi\)
\(510\) 0 0
\(511\) 11.6615i 0.515874i
\(512\) 0 0
\(513\) 2.20175 3.76196i 0.0972095 0.166094i
\(514\) 0 0
\(515\) 6.56189 0.289151
\(516\) 0 0
\(517\) −19.8564 −0.873285
\(518\) 0 0
\(519\) 22.4728i 0.986445i
\(520\) 0 0
\(521\) 33.2733i 1.45773i −0.684657 0.728865i \(-0.740048\pi\)
0.684657 0.728865i \(-0.259952\pi\)
\(522\) 0 0
\(523\) 40.3328 1.76363 0.881816 0.471594i \(-0.156321\pi\)
0.881816 + 0.471594i \(0.156321\pi\)
\(524\) 0 0
\(525\) 3.95229i 0.172492i
\(526\) 0 0
\(527\) 2.68074 0.116775
\(528\) 0 0
\(529\) −55.8292 −2.42736
\(530\) 0 0
\(531\) −12.9348 −0.561321
\(532\) 0 0
\(533\) 34.4221 1.49099
\(534\) 0 0
\(535\) −9.20783 −0.398089
\(536\) 0 0
\(537\) −3.11079 −0.134241
\(538\) 0 0
\(539\) 18.6951i 0.805257i
\(540\) 0 0
\(541\) 18.3966 0.790932 0.395466 0.918481i \(-0.370583\pi\)
0.395466 + 0.918481i \(0.370583\pi\)
\(542\) 0 0
\(543\) 2.28171i 0.0979177i
\(544\) 0 0
\(545\) 17.5873i 0.753355i
\(546\) 0 0
\(547\) −8.64717 −0.369726 −0.184863 0.982764i \(-0.559184\pi\)
−0.184863 + 0.982764i \(0.559184\pi\)
\(548\) 0 0
\(549\) 1.62061 0.0691658
\(550\) 0 0
\(551\) 21.1902 + 12.4019i 0.902732 + 0.528339i
\(552\) 0 0
\(553\) 44.2285i 1.88079i
\(554\) 0 0
\(555\) 1.77374i 0.0752912i
\(556\) 0 0
\(557\) −28.6277 −1.21300 −0.606498 0.795085i \(-0.707426\pi\)
−0.606498 + 0.795085i \(0.707426\pi\)
\(558\) 0 0
\(559\) −66.9058 −2.82981
\(560\) 0 0
\(561\) 4.42386i 0.186775i
\(562\) 0 0
\(563\) −9.69899 −0.408764 −0.204382 0.978891i \(-0.565518\pi\)
−0.204382 + 0.978891i \(0.565518\pi\)
\(564\) 0 0
\(565\) 0.870836i 0.0366363i
\(566\) 0 0
\(567\) 3.95229i 0.165981i
\(568\) 0 0
\(569\) 16.1831i 0.678431i −0.940709 0.339215i \(-0.889838\pi\)
0.940709 0.339215i \(-0.110162\pi\)
\(570\) 0 0
\(571\) 5.53056i 0.231447i −0.993281 0.115723i \(-0.963081\pi\)
0.993281 0.115723i \(-0.0369186\pi\)
\(572\) 0 0
\(573\) 25.4170i 1.06181i
\(574\) 0 0
\(575\) 8.87858i 0.370262i
\(576\) 0 0
\(577\) −10.8363 −0.451121 −0.225560 0.974229i \(-0.572421\pi\)
−0.225560 + 0.974229i \(0.572421\pi\)
\(578\) 0 0
\(579\) 12.2011i 0.507061i
\(580\) 0 0
\(581\) 17.2769 0.716766
\(582\) 0 0
\(583\) −11.7422 −0.486310
\(584\) 0 0
\(585\) 6.98743i 0.288895i
\(586\) 0 0
\(587\) 44.2720i 1.82730i −0.406500 0.913651i \(-0.633251\pi\)
0.406500 0.913651i \(-0.366749\pi\)
\(588\) 0 0
\(589\) 2.89343 4.94378i 0.119222 0.203705i
\(590\) 0 0
\(591\) 23.2941 0.958191
\(592\) 0 0
\(593\) 13.1477 0.539912 0.269956 0.962873i \(-0.412991\pi\)
0.269956 + 0.962873i \(0.412991\pi\)
\(594\) 0 0
\(595\) 8.06230i 0.330522i
\(596\) 0 0
\(597\) 5.68961i 0.232860i
\(598\) 0 0
\(599\) −16.0279 −0.654883 −0.327441 0.944872i \(-0.606186\pi\)
−0.327441 + 0.944872i \(0.606186\pi\)
\(600\) 0 0
\(601\) 7.74553i 0.315947i 0.987443 + 0.157973i \(0.0504960\pi\)
−0.987443 + 0.157973i \(0.949504\pi\)
\(602\) 0 0
\(603\) −6.56189 −0.267221
\(604\) 0 0
\(605\) 6.29692 0.256006
\(606\) 0 0
\(607\) 4.88150 0.198134 0.0990670 0.995081i \(-0.468414\pi\)
0.0990670 + 0.995081i \(0.468414\pi\)
\(608\) 0 0
\(609\) −22.2623 −0.902114
\(610\) 0 0
\(611\) 63.9775 2.58825
\(612\) 0 0
\(613\) −19.2515 −0.777563 −0.388781 0.921330i \(-0.627104\pi\)
−0.388781 + 0.921330i \(0.627104\pi\)
\(614\) 0 0
\(615\) 4.92629i 0.198647i
\(616\) 0 0
\(617\) −19.1822 −0.772248 −0.386124 0.922447i \(-0.626186\pi\)
−0.386124 + 0.922447i \(0.626186\pi\)
\(618\) 0 0
\(619\) 11.0144i 0.442705i −0.975194 0.221353i \(-0.928953\pi\)
0.975194 0.221353i \(-0.0710472\pi\)
\(620\) 0 0
\(621\) 8.87858i 0.356285i
\(622\) 0 0
\(623\) 27.3432 1.09548
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.15839 4.77484i −0.325815 0.190689i
\(628\) 0 0
\(629\) 3.61827i 0.144270i
\(630\) 0 0
\(631\) 14.9494i 0.595127i −0.954702 0.297563i \(-0.903826\pi\)
0.954702 0.297563i \(-0.0961740\pi\)
\(632\) 0 0
\(633\) −20.4764 −0.813864
\(634\) 0 0
\(635\) 5.61133 0.222679
\(636\) 0 0
\(637\) 60.2359i 2.38663i
\(638\) 0 0
\(639\) −5.35406 −0.211803
\(640\) 0 0
\(641\) 42.1693i 1.66559i 0.553584 + 0.832794i \(0.313260\pi\)
−0.553584 + 0.832794i \(0.686740\pi\)
\(642\) 0 0
\(643\) 16.9247i 0.667446i −0.942671 0.333723i \(-0.891695\pi\)
0.942671 0.333723i \(-0.108305\pi\)
\(644\) 0 0
\(645\) 9.57516i 0.377022i
\(646\) 0 0
\(647\) 5.87195i 0.230850i −0.993316 0.115425i \(-0.963177\pi\)
0.993316 0.115425i \(-0.0368230\pi\)
\(648\) 0 0
\(649\) 28.0511i 1.10110i
\(650\) 0 0
\(651\) 5.19391i 0.203565i
\(652\) 0 0
\(653\) 34.4871 1.34959 0.674793 0.738007i \(-0.264233\pi\)
0.674793 + 0.738007i \(0.264233\pi\)
\(654\) 0 0
\(655\) 13.1751i 0.514795i
\(656\) 0 0
\(657\) 2.95056 0.115112
\(658\) 0 0
\(659\) 5.72034 0.222833 0.111416 0.993774i \(-0.464461\pi\)
0.111416 + 0.993774i \(0.464461\pi\)
\(660\) 0 0
\(661\) 23.5610i 0.916415i −0.888845 0.458208i \(-0.848492\pi\)
0.888845 0.458208i \(-0.151508\pi\)
\(662\) 0 0
\(663\) 14.2537i 0.553568i
\(664\) 0 0
\(665\) −14.8683 8.70195i −0.576570 0.337447i
\(666\) 0 0
\(667\) 50.0109 1.93643
\(668\) 0 0
\(669\) 24.2530 0.937676
\(670\) 0 0
\(671\) 3.51454i 0.135677i
\(672\) 0 0
\(673\) 34.9200i 1.34607i 0.739611 + 0.673034i \(0.235009\pi\)
−0.739611 + 0.673034i \(0.764991\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 39.6191i 1.52269i −0.648349 0.761343i \(-0.724540\pi\)
0.648349 0.761343i \(-0.275460\pi\)
\(678\) 0 0
\(679\) −51.5350 −1.97773
\(680\) 0 0
\(681\) 11.0735 0.424339
\(682\) 0 0
\(683\) 11.9690 0.457980 0.228990 0.973429i \(-0.426458\pi\)
0.228990 + 0.973429i \(0.426458\pi\)
\(684\) 0 0
\(685\) 13.9976 0.534821
\(686\) 0 0
\(687\) 20.6524 0.787938
\(688\) 0 0
\(689\) 37.8333 1.44133
\(690\) 0 0
\(691\) 12.6282i 0.480399i −0.970724 0.240200i \(-0.922787\pi\)
0.970724 0.240200i \(-0.0772129\pi\)
\(692\) 0 0
\(693\) 8.57117 0.325592
\(694\) 0 0
\(695\) 21.0062i 0.796812i
\(696\) 0 0
\(697\) 10.0492i 0.380639i
\(698\) 0 0
\(699\) 3.98277 0.150642
\(700\) 0 0
\(701\) −8.33677 −0.314875 −0.157438 0.987529i \(-0.550323\pi\)
−0.157438 + 0.987529i \(0.550323\pi\)
\(702\) 0 0
\(703\) −6.67274 3.90533i −0.251667 0.147292i
\(704\) 0 0
\(705\) 9.15609i 0.344838i
\(706\) 0 0
\(707\) 27.2082i 1.02327i
\(708\) 0 0
\(709\) −9.89450 −0.371596 −0.185798 0.982588i \(-0.559487\pi\)
−0.185798 + 0.982588i \(0.559487\pi\)
\(710\) 0 0
\(711\) 11.1906 0.419680
\(712\) 0 0
\(713\) 11.6678i 0.436962i
\(714\) 0 0
\(715\) 15.1533 0.566703
\(716\) 0 0
\(717\) 13.8555i 0.517445i
\(718\) 0 0
\(719\) 3.30971i 0.123431i 0.998094 + 0.0617157i \(0.0196572\pi\)
−0.998094 + 0.0617157i \(0.980343\pi\)
\(720\) 0 0
\(721\) 25.9345i 0.965851i
\(722\) 0 0
\(723\) 11.1943i 0.416322i
\(724\) 0 0
\(725\) 5.63276i 0.209195i
\(726\) 0 0
\(727\) 39.2801i 1.45682i −0.685143 0.728409i \(-0.740260\pi\)
0.685143 0.728409i \(-0.259740\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 19.5324i 0.722433i
\(732\) 0 0
\(733\) −1.89079 −0.0698379 −0.0349189 0.999390i \(-0.511117\pi\)
−0.0349189 + 0.999390i \(0.511117\pi\)
\(734\) 0 0
\(735\) 8.62061 0.317976
\(736\) 0 0
\(737\) 14.2305i 0.524187i
\(738\) 0 0
\(739\) 28.4982i 1.04832i −0.851619 0.524161i \(-0.824379\pi\)
0.851619 0.524161i \(-0.175621\pi\)
\(740\) 0 0
\(741\) 26.2864 + 15.3846i 0.965655 + 0.565166i
\(742\) 0 0
\(743\) −4.36639 −0.160187 −0.0800936 0.996787i \(-0.525522\pi\)
−0.0800936 + 0.996787i \(0.525522\pi\)
\(744\) 0 0
\(745\) 5.98236 0.219177
\(746\) 0 0
\(747\) 4.37136i 0.159940i
\(748\) 0 0
\(749\) 36.3920i 1.32974i
\(750\) 0 0
\(751\) 2.16726 0.0790845 0.0395422 0.999218i \(-0.487410\pi\)
0.0395422 + 0.999218i \(0.487410\pi\)
\(752\) 0 0
\(753\) 17.3655i 0.632834i
\(754\) 0 0
\(755\) 12.2317 0.445156
\(756\) 0 0
\(757\) 42.5274 1.54568 0.772842 0.634599i \(-0.218835\pi\)
0.772842 + 0.634599i \(0.218835\pi\)
\(758\) 0 0
\(759\) −19.2546 −0.698898
\(760\) 0 0
\(761\) 18.7893 0.681113 0.340557 0.940224i \(-0.389384\pi\)
0.340557 + 0.940224i \(0.389384\pi\)
\(762\) 0 0
\(763\) 69.5100 2.51643
\(764\) 0 0
\(765\) −2.03991 −0.0737529
\(766\) 0 0
\(767\) 90.3807i 3.26346i
\(768\) 0 0
\(769\) 44.0559 1.58870 0.794349 0.607462i \(-0.207812\pi\)
0.794349 + 0.607462i \(0.207812\pi\)
\(770\) 0 0
\(771\) 1.35467i 0.0487873i
\(772\) 0 0
\(773\) 31.9855i 1.15044i 0.817999 + 0.575220i \(0.195084\pi\)
−0.817999 + 0.575220i \(0.804916\pi\)
\(774\) 0 0
\(775\) −1.31415 −0.0472057
\(776\) 0 0
\(777\) 7.01035 0.251495
\(778\) 0 0
\(779\) 18.5325 + 10.8464i 0.663995 + 0.388614i
\(780\) 0 0
\(781\) 11.6111i 0.415478i
\(782\) 0 0
\(783\) 5.63276i 0.201298i
\(784\) 0 0
\(785\) −6.13042 −0.218804
\(786\) 0 0
\(787\) 3.82963 0.136512 0.0682558 0.997668i \(-0.478257\pi\)
0.0682558 + 0.997668i \(0.478257\pi\)
\(788\) 0 0
\(789\) 5.19286i 0.184871i
\(790\) 0 0
\(791\) −3.44180 −0.122376
\(792\) 0 0
\(793\) 11.3239i 0.402122i
\(794\) 0 0
\(795\) 5.41448i 0.192032i
\(796\) 0 0
\(797\) 24.2446i 0.858788i −0.903117 0.429394i \(-0.858727\pi\)
0.903117 0.429394i \(-0.141273\pi\)
\(798\) 0 0
\(799\) 18.6776i 0.660765i
\(800\) 0 0
\(801\) 6.91831i 0.244447i
\(802\) 0 0
\(803\) 6.39876i 0.225807i
\(804\) 0