Properties

Label 4560.2.d.e.2431.1
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.9821011968.3
Defining polynomial: \(x^{6} + 20 x^{4} + 118 x^{2} + 192\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.1
Root \(-3.24237i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.e.2431.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -3.24237i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -3.24237i q^{7} +1.00000 q^{9} -5.82288i q^{11} -3.24237i q^{13} +1.00000 q^{15} -4.51298 q^{17} +(-3.51298 - 2.58050i) q^{19} +3.24237i q^{21} -5.56750i q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.82288i q^{29} -7.85398 q^{31} +5.82288i q^{33} +3.24237i q^{35} +0.255376i q^{37} +3.24237i q^{39} -0.661868i q^{41} +0.255376i q^{43} -1.00000 q^{45} +0.406492i q^{47} -3.51298 q^{49} +4.51298 q^{51} +5.56750i q^{53} +5.82288i q^{55} +(3.51298 + 2.58050i) q^{57} +9.02595 q^{59} -5.85398 q^{61} -3.24237i q^{63} +3.24237i q^{65} +10.3670 q^{67} +5.56750i q^{69} +9.02595 q^{71} -5.68495 q^{73} -1.00000 q^{75} -18.8799 q^{77} +4.65900 q^{79} +1.00000 q^{81} -6.07825i q^{83} +4.51298 q^{85} +5.82288i q^{87} -3.64886i q^{89} -10.5130 q^{91} +7.85398 q^{93} +(3.51298 + 2.58050i) q^{95} +13.0536i q^{97} -5.82288i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{5} + 6q^{9} + 6q^{15} - 4q^{17} + 2q^{19} + 6q^{25} - 6q^{27} - 12q^{31} - 6q^{45} + 2q^{49} + 4q^{51} - 2q^{57} + 8q^{59} + 4q^{67} + 8q^{71} - 6q^{75} - 32q^{77} + 40q^{79} + 6q^{81} + 4q^{85} - 40q^{91} + 12q^{93} - 2q^{95} + O(q^{100}) \)

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
\(7\) 3.24237i 1.22550i −0.790276 0.612751i \(-0.790063\pi\)
0.790276 0.612751i \(-0.209937\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.82288i 1.75566i −0.478970 0.877832i \(-0.658990\pi\)
0.478970 0.877832i \(-0.341010\pi\)
\(12\) 0 0
\(13\) 3.24237i 0.899272i −0.893212 0.449636i \(-0.851554\pi\)
0.893212 0.449636i \(-0.148446\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.51298 −1.09456 −0.547279 0.836950i \(-0.684336\pi\)
−0.547279 + 0.836950i \(0.684336\pi\)
\(18\) 0 0
\(19\) −3.51298 2.58050i −0.805932 0.592008i
\(20\) 0 0
\(21\) 3.24237i 0.707544i
\(22\) 0 0
\(23\) 5.56750i 1.16090i −0.814295 0.580452i \(-0.802876\pi\)
0.814295 0.580452i \(-0.197124\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.82288i 1.08128i −0.841254 0.540640i \(-0.818182\pi\)
0.841254 0.540640i \(-0.181818\pi\)
\(30\) 0 0
\(31\) −7.85398 −1.41062 −0.705308 0.708901i \(-0.749191\pi\)
−0.705308 + 0.708901i \(0.749191\pi\)
\(32\) 0 0
\(33\) 5.82288i 1.01363i
\(34\) 0 0
\(35\) 3.24237i 0.548061i
\(36\) 0 0
\(37\) 0.255376i 0.0419836i 0.999780 + 0.0209918i \(0.00668239\pi\)
−0.999780 + 0.0209918i \(0.993318\pi\)
\(38\) 0 0
\(39\) 3.24237i 0.519195i
\(40\) 0 0
\(41\) 0.661868i 0.103366i −0.998664 0.0516832i \(-0.983541\pi\)
0.998664 0.0516832i \(-0.0164586\pi\)
\(42\) 0 0
\(43\) 0.255376i 0.0389445i 0.999810 + 0.0194723i \(0.00619860\pi\)
−0.999810 + 0.0194723i \(0.993801\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0.406492i 0.0592930i 0.999560 + 0.0296465i \(0.00943815\pi\)
−0.999560 + 0.0296465i \(0.990562\pi\)
\(48\) 0 0
\(49\) −3.51298 −0.501854
\(50\) 0 0
\(51\) 4.51298 0.631943
\(52\) 0 0
\(53\) 5.56750i 0.764755i 0.924006 + 0.382377i \(0.124895\pi\)
−0.924006 + 0.382377i \(0.875105\pi\)
\(54\) 0 0
\(55\) 5.82288i 0.785156i
\(56\) 0 0
\(57\) 3.51298 + 2.58050i 0.465305 + 0.341796i
\(58\) 0 0
\(59\) 9.02595 1.17508 0.587539 0.809196i \(-0.300097\pi\)
0.587539 + 0.809196i \(0.300097\pi\)
\(60\) 0 0
\(61\) −5.85398 −0.749525 −0.374762 0.927121i \(-0.622276\pi\)
−0.374762 + 0.927121i \(0.622276\pi\)
\(62\) 0 0
\(63\) 3.24237i 0.408500i
\(64\) 0 0
\(65\) 3.24237i 0.402167i
\(66\) 0 0
\(67\) 10.3670 1.26652 0.633262 0.773937i \(-0.281715\pi\)
0.633262 + 0.773937i \(0.281715\pi\)
\(68\) 0 0
\(69\) 5.56750i 0.670248i
\(70\) 0 0
\(71\) 9.02595 1.07118 0.535592 0.844477i \(-0.320089\pi\)
0.535592 + 0.844477i \(0.320089\pi\)
\(72\) 0 0
\(73\) −5.68495 −0.665373 −0.332687 0.943037i \(-0.607955\pi\)
−0.332687 + 0.943037i \(0.607955\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −18.8799 −2.15157
\(78\) 0 0
\(79\) 4.65900 0.524178 0.262089 0.965044i \(-0.415589\pi\)
0.262089 + 0.965044i \(0.415589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.07825i 0.667175i −0.942719 0.333587i \(-0.891741\pi\)
0.942719 0.333587i \(-0.108259\pi\)
\(84\) 0 0
\(85\) 4.51298 0.489501
\(86\) 0 0
\(87\) 5.82288i 0.624278i
\(88\) 0 0
\(89\) 3.64886i 0.386779i −0.981122 0.193389i \(-0.938052\pi\)
0.981122 0.193389i \(-0.0619481\pi\)
\(90\) 0 0
\(91\) −10.5130 −1.10206
\(92\) 0 0
\(93\) 7.85398 0.814419
\(94\) 0 0
\(95\) 3.51298 + 2.58050i 0.360424 + 0.264754i
\(96\) 0 0
\(97\) 13.0536i 1.32540i 0.748887 + 0.662698i \(0.230589\pi\)
−0.748887 + 0.662698i \(0.769411\pi\)
\(98\) 0 0
\(99\) 5.82288i 0.585221i
\(100\) 0 0
\(101\) 13.7080 1.36399 0.681996 0.731356i \(-0.261112\pi\)
0.681996 + 0.731356i \(0.261112\pi\)
\(102\) 0 0
\(103\) −5.02595 −0.495222 −0.247611 0.968860i \(-0.579645\pi\)
−0.247611 + 0.968860i \(0.579645\pi\)
\(104\) 0 0
\(105\) 3.24237i 0.316423i
\(106\) 0 0
\(107\) 17.3929 1.68144 0.840718 0.541474i \(-0.182133\pi\)
0.840718 + 0.541474i \(0.182133\pi\)
\(108\) 0 0
\(109\) 4.65025i 0.445414i −0.974885 0.222707i \(-0.928511\pi\)
0.974885 0.222707i \(-0.0714893\pi\)
\(110\) 0 0
\(111\) 0.255376i 0.0242392i
\(112\) 0 0
\(113\) 11.2393i 1.05730i 0.848840 + 0.528650i \(0.177302\pi\)
−0.848840 + 0.528650i \(0.822698\pi\)
\(114\) 0 0
\(115\) 5.56750i 0.519172i
\(116\) 0 0
\(117\) 3.24237i 0.299757i
\(118\) 0 0
\(119\) 14.6327i 1.34138i
\(120\) 0 0
\(121\) −22.9059 −2.08235
\(122\) 0 0
\(123\) 0.661868i 0.0596787i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.68495 −0.681929 −0.340965 0.940076i \(-0.610754\pi\)
−0.340965 + 0.940076i \(0.610754\pi\)
\(128\) 0 0
\(129\) 0.255376i 0.0224846i
\(130\) 0 0
\(131\) 1.00139i 0.0874919i −0.999043 0.0437460i \(-0.986071\pi\)
0.999043 0.0437460i \(-0.0139292\pi\)
\(132\) 0 0
\(133\) −8.36695 + 11.3904i −0.725507 + 0.987671i
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −3.68495 −0.314827 −0.157413 0.987533i \(-0.550315\pi\)
−0.157413 + 0.987533i \(0.550315\pi\)
\(138\) 0 0
\(139\) 1.32374i 0.112278i −0.998423 0.0561389i \(-0.982121\pi\)
0.998423 0.0561389i \(-0.0178790\pi\)
\(140\) 0 0
\(141\) 0.406492i 0.0342328i
\(142\) 0 0
\(143\) −18.8799 −1.57882
\(144\) 0 0
\(145\) 5.82288i 0.483564i
\(146\) 0 0
\(147\) 3.51298 0.289745
\(148\) 0 0
\(149\) 15.0260 1.23097 0.615487 0.788147i \(-0.288959\pi\)
0.615487 + 0.788147i \(0.288959\pi\)
\(150\) 0 0
\(151\) 22.2209 1.80831 0.904157 0.427201i \(-0.140500\pi\)
0.904157 + 0.427201i \(0.140500\pi\)
\(152\) 0 0
\(153\) −4.51298 −0.364852
\(154\) 0 0
\(155\) 7.85398 0.630847
\(156\) 0 0
\(157\) −4.36695 −0.348521 −0.174260 0.984700i \(-0.555753\pi\)
−0.174260 + 0.984700i \(0.555753\pi\)
\(158\) 0 0
\(159\) 5.56750i 0.441531i
\(160\) 0 0
\(161\) −18.0519 −1.42269
\(162\) 0 0
\(163\) 6.56889i 0.514515i 0.966343 + 0.257258i \(0.0828189\pi\)
−0.966343 + 0.257258i \(0.917181\pi\)
\(164\) 0 0
\(165\) 5.82288i 0.453310i
\(166\) 0 0
\(167\) −1.48702 −0.115069 −0.0575347 0.998344i \(-0.518324\pi\)
−0.0575347 + 0.998344i \(0.518324\pi\)
\(168\) 0 0
\(169\) 2.48702 0.191310
\(170\) 0 0
\(171\) −3.51298 2.58050i −0.268644 0.197336i
\(172\) 0 0
\(173\) 16.4003i 1.24689i 0.781868 + 0.623445i \(0.214267\pi\)
−0.781868 + 0.623445i \(0.785733\pi\)
\(174\) 0 0
\(175\) 3.24237i 0.245100i
\(176\) 0 0
\(177\) −9.02595 −0.678432
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 24.6152i 1.82964i −0.403866 0.914818i \(-0.632334\pi\)
0.403866 0.914818i \(-0.367666\pi\)
\(182\) 0 0
\(183\) 5.85398 0.432738
\(184\) 0 0
\(185\) 0.255376i 0.0187756i
\(186\) 0 0
\(187\) 26.2785i 1.92167i
\(188\) 0 0
\(189\) 3.24237i 0.235848i
\(190\) 0 0
\(191\) 1.17262i 0.0848479i 0.999100 + 0.0424239i \(0.0135080\pi\)
−0.999100 + 0.0424239i \(0.986492\pi\)
\(192\) 0 0
\(193\) 17.5356i 1.26224i −0.775685 0.631120i \(-0.782596\pi\)
0.775685 0.631120i \(-0.217404\pi\)
\(194\) 0 0
\(195\) 3.24237i 0.232191i
\(196\) 0 0
\(197\) 7.48702 0.533428 0.266714 0.963776i \(-0.414062\pi\)
0.266714 + 0.963776i \(0.414062\pi\)
\(198\) 0 0
\(199\) 0.510752i 0.0362063i 0.999836 + 0.0181031i \(0.00576272\pi\)
−0.999836 + 0.0181031i \(0.994237\pi\)
\(200\) 0 0
\(201\) −10.3670 −0.731228
\(202\) 0 0
\(203\) −18.8799 −1.32511
\(204\) 0 0
\(205\) 0.661868i 0.0462269i
\(206\) 0 0
\(207\) 5.56750i 0.386968i
\(208\) 0 0
\(209\) −15.0260 + 20.4556i −1.03937 + 1.41495i
\(210\) 0 0
\(211\) −18.3670 −1.26443 −0.632217 0.774792i \(-0.717855\pi\)
−0.632217 + 0.774792i \(0.717855\pi\)
\(212\) 0 0
\(213\) −9.02595 −0.618448
\(214\) 0 0
\(215\) 0.255376i 0.0174165i
\(216\) 0 0
\(217\) 25.4655i 1.72871i
\(218\) 0 0
\(219\) 5.68495 0.384153
\(220\) 0 0
\(221\) 14.6327i 0.984305i
\(222\) 0 0
\(223\) −23.0779 −1.54541 −0.772704 0.634767i \(-0.781096\pi\)
−0.772704 + 0.634767i \(0.781096\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 8.85693 0.587855 0.293927 0.955828i \(-0.405038\pi\)
0.293927 + 0.955828i \(0.405038\pi\)
\(228\) 0 0
\(229\) −20.5130 −1.35554 −0.677768 0.735276i \(-0.737053\pi\)
−0.677768 + 0.735276i \(0.737053\pi\)
\(230\) 0 0
\(231\) 18.8799 1.24221
\(232\) 0 0
\(233\) −28.7858 −1.88582 −0.942911 0.333046i \(-0.891924\pi\)
−0.942911 + 0.333046i \(0.891924\pi\)
\(234\) 0 0
\(235\) 0.406492i 0.0265166i
\(236\) 0 0
\(237\) −4.65900 −0.302635
\(238\) 0 0
\(239\) 25.6166i 1.65700i −0.559988 0.828501i \(-0.689194\pi\)
0.559988 0.828501i \(-0.310806\pi\)
\(240\) 0 0
\(241\) 22.4785i 1.44797i 0.689816 + 0.723984i \(0.257691\pi\)
−0.689816 + 0.723984i \(0.742309\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.51298 0.224436
\(246\) 0 0
\(247\) −8.36695 + 11.3904i −0.532376 + 0.724752i
\(248\) 0 0
\(249\) 6.07825i 0.385194i
\(250\) 0 0
\(251\) 15.8054i 0.997626i −0.866710 0.498813i \(-0.833769\pi\)
0.866710 0.498813i \(-0.166231\pi\)
\(252\) 0 0
\(253\) −32.4189 −2.03816
\(254\) 0 0
\(255\) −4.51298 −0.282614
\(256\) 0 0
\(257\) 18.0262i 1.12445i −0.826986 0.562223i \(-0.809946\pi\)
0.826986 0.562223i \(-0.190054\pi\)
\(258\) 0 0
\(259\) 0.828025 0.0514510
\(260\) 0 0
\(261\) 5.82288i 0.360427i
\(262\) 0 0
\(263\) 18.0262i 1.11155i −0.831334 0.555773i \(-0.812423\pi\)
0.831334 0.555773i \(-0.187577\pi\)
\(264\) 0 0
\(265\) 5.56750i 0.342009i
\(266\) 0 0
\(267\) 3.64886i 0.223307i
\(268\) 0 0
\(269\) 3.34663i 0.204048i −0.994782 0.102024i \(-0.967468\pi\)
0.994782 0.102024i \(-0.0325318\pi\)
\(270\) 0 0
\(271\) 3.83727i 0.233098i 0.993185 + 0.116549i \(0.0371831\pi\)
−0.993185 + 0.116549i \(0.962817\pi\)
\(272\) 0 0
\(273\) 10.5130 0.636274
\(274\) 0 0
\(275\) 5.82288i 0.351133i
\(276\) 0 0
\(277\) 21.7080 1.30430 0.652152 0.758088i \(-0.273866\pi\)
0.652152 + 0.758088i \(0.273866\pi\)
\(278\) 0 0
\(279\) −7.85398 −0.470205
\(280\) 0 0
\(281\) 17.8081i 1.06235i 0.847264 + 0.531173i \(0.178248\pi\)
−0.847264 + 0.531173i \(0.821752\pi\)
\(282\) 0 0
\(283\) 14.8881i 0.885007i 0.896767 + 0.442504i \(0.145910\pi\)
−0.896767 + 0.442504i \(0.854090\pi\)
\(284\) 0 0
\(285\) −3.51298 2.58050i −0.208091 0.152856i
\(286\) 0 0
\(287\) −2.14602 −0.126676
\(288\) 0 0
\(289\) 3.36695 0.198056
\(290\) 0 0
\(291\) 13.0536i 0.765218i
\(292\) 0 0
\(293\) 17.2133i 1.00561i 0.864400 + 0.502804i \(0.167698\pi\)
−0.864400 + 0.502804i \(0.832302\pi\)
\(294\) 0 0
\(295\) −9.02595 −0.525511
\(296\) 0 0
\(297\) 5.82288i 0.337878i
\(298\) 0 0
\(299\) −18.0519 −1.04397
\(300\) 0 0
\(301\) 0.828025 0.0477265
\(302\) 0 0
\(303\) −13.7080 −0.787501
\(304\) 0 0
\(305\) 5.85398 0.335198
\(306\) 0 0
\(307\) 6.97405 0.398030 0.199015 0.979996i \(-0.436226\pi\)
0.199015 + 0.979996i \(0.436226\pi\)
\(308\) 0 0
\(309\) 5.02595 0.285916
\(310\) 0 0
\(311\) 9.83138i 0.557486i −0.960366 0.278743i \(-0.910082\pi\)
0.960366 0.278743i \(-0.0899178\pi\)
\(312\) 0 0
\(313\) −14.7109 −0.831509 −0.415755 0.909477i \(-0.636483\pi\)
−0.415755 + 0.909477i \(0.636483\pi\)
\(314\) 0 0
\(315\) 3.24237i 0.182687i
\(316\) 0 0
\(317\) 25.5325i 1.43405i −0.697049 0.717024i \(-0.745504\pi\)
0.697049 0.717024i \(-0.254496\pi\)
\(318\) 0 0
\(319\) −33.9059 −1.89836
\(320\) 0 0
\(321\) −17.3929 −0.970777
\(322\) 0 0
\(323\) 15.8540 + 11.6458i 0.882139 + 0.647987i
\(324\) 0 0
\(325\) 3.24237i 0.179854i
\(326\) 0 0
\(327\) 4.65025i 0.257160i
\(328\) 0 0
\(329\) 1.31800 0.0726636
\(330\) 0 0
\(331\) −3.53893 −0.194517 −0.0972585 0.995259i \(-0.531007\pi\)
−0.0972585 + 0.995259i \(0.531007\pi\)
\(332\) 0 0
\(333\) 0.255376i 0.0139945i
\(334\) 0 0
\(335\) −10.3670 −0.566407
\(336\) 0 0
\(337\) 29.8231i 1.62457i 0.583262 + 0.812284i \(0.301776\pi\)
−0.583262 + 0.812284i \(0.698224\pi\)
\(338\) 0 0
\(339\) 11.2393i 0.610433i
\(340\) 0 0
\(341\) 45.7327i 2.47657i
\(342\) 0 0
\(343\) 11.3062i 0.610479i
\(344\) 0 0
\(345\) 5.56750i 0.299744i
\(346\) 0 0
\(347\) 3.56472i 0.191364i 0.995412 + 0.0956820i \(0.0305032\pi\)
−0.995412 + 0.0956820i \(0.969497\pi\)
\(348\) 0 0
\(349\) 28.4189 1.52123 0.760613 0.649205i \(-0.224898\pi\)
0.760613 + 0.649205i \(0.224898\pi\)
\(350\) 0 0
\(351\) 3.24237i 0.173065i
\(352\) 0 0
\(353\) −12.0519 −0.641458 −0.320729 0.947171i \(-0.603928\pi\)
−0.320729 + 0.947171i \(0.603928\pi\)
\(354\) 0 0
\(355\) −9.02595 −0.479048
\(356\) 0 0
\(357\) 14.6327i 0.774447i
\(358\) 0 0
\(359\) 35.0884i 1.85189i 0.377654 + 0.925947i \(0.376731\pi\)
−0.377654 + 0.925947i \(0.623269\pi\)
\(360\) 0 0
\(361\) 5.68200 + 18.1305i 0.299053 + 0.954237i
\(362\) 0 0
\(363\) 22.9059 1.20225
\(364\) 0 0
\(365\) 5.68495 0.297564
\(366\) 0 0
\(367\) 32.1683i 1.67917i −0.543225 0.839587i \(-0.682797\pi\)
0.543225 0.839587i \(-0.317203\pi\)
\(368\) 0 0
\(369\) 0.661868i 0.0344555i
\(370\) 0 0
\(371\) 18.0519 0.937208
\(372\) 0 0
\(373\) 2.42939i 0.125789i 0.998020 + 0.0628945i \(0.0200332\pi\)
−0.998020 + 0.0628945i \(0.979967\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −18.8799 −0.972366
\(378\) 0 0
\(379\) −3.53893 −0.181783 −0.0908913 0.995861i \(-0.528972\pi\)
−0.0908913 + 0.995861i \(0.528972\pi\)
\(380\) 0 0
\(381\) 7.68495 0.393712
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 18.8799 0.962210
\(386\) 0 0
\(387\) 0.255376i 0.0129815i
\(388\) 0 0
\(389\) 21.0779 1.06869 0.534345 0.845267i \(-0.320558\pi\)
0.534345 + 0.845267i \(0.320558\pi\)
\(390\) 0 0
\(391\) 25.1260i 1.27068i
\(392\) 0 0
\(393\) 1.00139i 0.0505135i
\(394\) 0 0
\(395\) −4.65900 −0.234420
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 8.36695 11.3904i 0.418872 0.570232i
\(400\) 0 0
\(401\) 14.1421i 0.706223i −0.935581 0.353112i \(-0.885124\pi\)
0.935581 0.353112i \(-0.114876\pi\)
\(402\) 0 0
\(403\) 25.4655i 1.26853i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 1.48702 0.0737090
\(408\) 0 0
\(409\) 18.6412i 0.921750i −0.887465 0.460875i \(-0.847536\pi\)
0.887465 0.460875i \(-0.152464\pi\)
\(410\) 0 0
\(411\) 3.68495 0.181765
\(412\) 0 0
\(413\) 29.2655i 1.44006i
\(414\) 0 0
\(415\) 6.07825i 0.298370i
\(416\) 0 0
\(417\) 1.32374i 0.0648237i
\(418\) 0 0
\(419\) 6.63586i 0.324183i −0.986776 0.162091i \(-0.948176\pi\)
0.986776 0.162091i \(-0.0518240\pi\)
\(420\) 0 0
\(421\) 12.1565i 0.592472i −0.955115 0.296236i \(-0.904269\pi\)
0.955115 0.296236i \(-0.0957314\pi\)
\(422\) 0 0
\(423\) 0.406492i 0.0197643i
\(424\) 0 0
\(425\) −4.51298 −0.218911
\(426\) 0 0
\(427\) 18.9808i 0.918544i
\(428\) 0 0
\(429\) 18.8799 0.911532
\(430\) 0 0
\(431\) 9.02595 0.434765 0.217382 0.976087i \(-0.430248\pi\)
0.217382 + 0.976087i \(0.430248\pi\)
\(432\) 0 0
\(433\) 14.8881i 0.715478i 0.933822 + 0.357739i \(0.116452\pi\)
−0.933822 + 0.357739i \(0.883548\pi\)
\(434\) 0 0
\(435\) 5.82288i 0.279186i
\(436\) 0 0
\(437\) −14.3670 + 19.5585i −0.687265 + 0.935610i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.51298 −0.167285
\(442\) 0 0
\(443\) 34.0200i 1.61634i 0.588950 + 0.808170i \(0.299542\pi\)
−0.588950 + 0.808170i \(0.700458\pi\)
\(444\) 0 0
\(445\) 3.64886i 0.172973i
\(446\) 0 0
\(447\) −15.0260 −0.710703
\(448\) 0 0
\(449\) 30.6064i 1.44441i −0.691681 0.722203i \(-0.743130\pi\)
0.691681 0.722203i \(-0.256870\pi\)
\(450\) 0 0
\(451\) −3.85398 −0.181477
\(452\) 0 0
\(453\) −22.2209 −1.04403
\(454\) 0 0
\(455\) 10.5130 0.492856
\(456\) 0 0
\(457\) 12.3670 0.578502 0.289251 0.957253i \(-0.406594\pi\)
0.289251 + 0.957253i \(0.406594\pi\)
\(458\) 0 0
\(459\) 4.51298 0.210648
\(460\) 0 0
\(461\) 33.4159 1.55633 0.778167 0.628057i \(-0.216150\pi\)
0.778167 + 0.628057i \(0.216150\pi\)
\(462\) 0 0
\(463\) 17.0621i 0.792945i 0.918047 + 0.396472i \(0.129766\pi\)
−0.918047 + 0.396472i \(0.870234\pi\)
\(464\) 0 0
\(465\) −7.85398 −0.364219
\(466\) 0 0
\(467\) 27.0245i 1.25055i −0.780406 0.625273i \(-0.784988\pi\)
0.780406 0.625273i \(-0.215012\pi\)
\(468\) 0 0
\(469\) 33.6135i 1.55213i
\(470\) 0 0
\(471\) 4.36695 0.201219
\(472\) 0 0
\(473\) 1.48702 0.0683734
\(474\) 0 0
\(475\) −3.51298 2.58050i −0.161186 0.118402i
\(476\) 0 0
\(477\) 5.56750i 0.254918i
\(478\) 0 0
\(479\) 21.7794i 0.995124i −0.867428 0.497562i \(-0.834229\pi\)
0.867428 0.497562i \(-0.165771\pi\)
\(480\) 0 0
\(481\) 0.828025 0.0377547
\(482\) 0 0
\(483\) 18.0519 0.821390
\(484\) 0 0
\(485\) 13.0536i 0.592735i
\(486\) 0 0
\(487\) −15.3929 −0.697519 −0.348760 0.937212i \(-0.613397\pi\)
−0.348760 + 0.937212i \(0.613397\pi\)
\(488\) 0 0
\(489\) 6.56889i 0.297055i
\(490\) 0 0
\(491\) 17.4686i 0.788348i −0.919036 0.394174i \(-0.871031\pi\)
0.919036 0.394174i \(-0.128969\pi\)
\(492\) 0 0
\(493\) 26.2785i 1.18352i
\(494\) 0 0
\(495\) 5.82288i 0.261719i
\(496\) 0 0
\(497\) 29.2655i 1.31274i
\(498\) 0 0
\(499\) 5.67176i 0.253903i −0.991909 0.126951i \(-0.959481\pi\)
0.991909 0.126951i \(-0.0405192\pi\)
\(500\) 0 0
\(501\) 1.48702 0.0664353
\(502\) 0 0
\(503\) 18.0262i 0.803750i 0.915695 + 0.401875i \(0.131641\pi\)
−0.915695 + 0.401875i \(0.868359\pi\)
\(504\) 0 0
\(505\) −13.7080 −0.609996
\(506\) 0 0
\(507\) −2.48702 −0.110453
\(508\) 0 0
\(509\) 14.3104i 0.634297i −0.948376 0.317149i \(-0.897275\pi\)
0.948376 0.317149i \(-0.102725\pi\)
\(510\) 0 0
\(511\) 18.4327i 0.815416i
\(512\) 0 0
\(513\) 3.51298 + 2.58050i 0.155102 + 0.113932i
\(514\) 0 0
\(515\) 5.02595 0.221470
\(516\) 0 0
\(517\) 2.36695 0.104098
\(518\) 0 0
\(519\) 16.4003i 0.719892i
\(520\) 0 0
\(521\) 6.29634i 0.275848i −0.990443 0.137924i \(-0.955957\pi\)
0.990443 0.137924i \(-0.0440429\pi\)
\(522\) 0 0
\(523\) −34.7628 −1.52007 −0.760036 0.649881i \(-0.774819\pi\)
−0.760036 + 0.649881i \(0.774819\pi\)
\(524\) 0 0
\(525\) 3.24237i 0.141509i
\(526\) 0 0
\(527\) 35.4448 1.54400
\(528\) 0 0
\(529\) −7.99705 −0.347698
\(530\) 0 0
\(531\) 9.02595 0.391693
\(532\) 0 0
\(533\) −2.14602 −0.0929546
\(534\) 0 0
\(535\) −17.3929 −0.751961
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.4556i 0.881086i
\(540\) 0 0
\(541\) −35.4159 −1.52265 −0.761324 0.648371i \(-0.775451\pi\)
−0.761324 + 0.648371i \(0.775451\pi\)
\(542\) 0 0
\(543\) 24.6152i 1.05634i
\(544\) 0 0
\(545\) 4.65025i 0.199195i
\(546\) 0 0
\(547\) −15.7080 −0.671624 −0.335812 0.941929i \(-0.609011\pi\)
−0.335812 + 0.941929i \(0.609011\pi\)
\(548\) 0 0
\(549\) −5.85398 −0.249842
\(550\) 0 0
\(551\) −15.0260 + 20.4556i −0.640127 + 0.871439i
\(552\) 0 0
\(553\) 15.1062i 0.642381i
\(554\) 0 0
\(555\) 0.255376i 0.0108401i
\(556\) 0 0
\(557\) −6.99705 −0.296475 −0.148237 0.988952i \(-0.547360\pi\)
−0.148237 + 0.988952i \(0.547360\pi\)
\(558\) 0 0
\(559\) 0.828025 0.0350217
\(560\) 0 0
\(561\) 26.2785i 1.10948i
\(562\) 0 0
\(563\) −9.19498 −0.387522 −0.193761 0.981049i \(-0.562069\pi\)
−0.193761 + 0.981049i \(0.562069\pi\)
\(564\) 0 0
\(565\) 11.2393i 0.472839i
\(566\) 0 0
\(567\) 3.24237i 0.136167i
\(568\) 0 0
\(569\) 12.6099i 0.528632i 0.964436 + 0.264316i \(0.0851463\pi\)
−0.964436 + 0.264316i \(0.914854\pi\)
\(570\) 0 0
\(571\) 32.4237i 1.35689i −0.734651 0.678445i \(-0.762654\pi\)
0.734651 0.678445i \(-0.237346\pi\)
\(572\) 0 0
\(573\) 1.17262i 0.0489869i
\(574\) 0 0
\(575\) 5.56750i 0.232181i
\(576\) 0 0
\(577\) 21.3929 0.890598 0.445299 0.895382i \(-0.353097\pi\)
0.445299 + 0.895382i \(0.353097\pi\)
\(578\) 0 0
\(579\) 17.5356i 0.728755i
\(580\) 0 0
\(581\) −19.7080 −0.817624
\(582\) 0 0
\(583\) 32.4189 1.34265
\(584\) 0 0
\(585\) 3.24237i 0.134056i
\(586\) 0 0
\(587\) 37.6890i 1.55559i −0.628517 0.777795i \(-0.716338\pi\)
0.628517 0.777795i \(-0.283662\pi\)
\(588\) 0 0
\(589\) 27.5908 + 20.2672i 1.13686 + 0.835096i
\(590\) 0 0
\(591\) −7.48702 −0.307975
\(592\) 0 0
\(593\) −32.4189 −1.33128 −0.665641 0.746272i \(-0.731842\pi\)
−0.665641 + 0.746272i \(0.731842\pi\)
\(594\) 0 0
\(595\) 14.6327i 0.599884i
\(596\) 0 0
\(597\) 0.510752i 0.0209037i
\(598\) 0 0
\(599\) −4.29205 −0.175368 −0.0876841 0.996148i \(-0.527947\pi\)
−0.0876841 + 0.996148i \(0.527947\pi\)
\(600\) 0 0
\(601\) 40.0983i 1.63564i 0.575473 + 0.817821i \(0.304818\pi\)
−0.575473 + 0.817821i \(0.695182\pi\)
\(602\) 0 0
\(603\) 10.3670 0.422175
\(604\) 0 0
\(605\) 22.9059 0.931256
\(606\) 0 0
\(607\) −25.7369 −1.04463 −0.522313 0.852754i \(-0.674931\pi\)
−0.522313 + 0.852754i \(0.674931\pi\)
\(608\) 0 0
\(609\) 18.8799 0.765053
\(610\) 0 0
\(611\) 1.31800 0.0533205
\(612\) 0 0
\(613\) 32.0749 1.29549 0.647747 0.761856i \(-0.275712\pi\)
0.647747 + 0.761856i \(0.275712\pi\)
\(614\) 0 0
\(615\) 0.661868i 0.0266891i
\(616\) 0 0
\(617\) 13.5389 0.545057 0.272528 0.962148i \(-0.412140\pi\)
0.272528 + 0.962148i \(0.412140\pi\)
\(618\) 0 0
\(619\) 21.2887i 0.855666i −0.903858 0.427833i \(-0.859277\pi\)
0.903858 0.427833i \(-0.140723\pi\)
\(620\) 0 0
\(621\) 5.56750i 0.223416i
\(622\) 0 0
\(623\) −11.8310 −0.473998
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.0260 20.4556i 0.600079 0.816919i
\(628\) 0 0
\(629\) 1.15251i 0.0459535i
\(630\) 0 0
\(631\) 40.4005i 1.60832i 0.594414 + 0.804159i \(0.297384\pi\)
−0.594414 + 0.804159i \(0.702616\pi\)
\(632\) 0 0
\(633\) 18.3670 0.730021
\(634\) 0 0
\(635\) 7.68495 0.304968
\(636\) 0 0
\(637\) 11.3904i 0.451303i
\(638\) 0 0
\(639\) 9.02595 0.357061
\(640\) 0 0
\(641\) 18.4528i 0.728843i 0.931234 + 0.364422i \(0.118733\pi\)
−0.931234 + 0.364422i \(0.881267\pi\)
\(642\) 0 0
\(643\) 12.7141i 0.501396i 0.968065 + 0.250698i \(0.0806600\pi\)
−0.968065 + 0.250698i \(0.919340\pi\)
\(644\) 0 0
\(645\) 0.255376i 0.0100554i
\(646\) 0 0
\(647\) 8.72573i 0.343044i 0.985180 + 0.171522i \(0.0548684\pi\)
−0.985180 + 0.171522i \(0.945132\pi\)
\(648\) 0 0
\(649\) 52.5570i 2.06304i
\(650\) 0 0
\(651\) 25.4655i 0.998072i
\(652\) 0 0
\(653\) −11.8829 −0.465013 −0.232506 0.972595i \(-0.574693\pi\)
−0.232506 + 0.972595i \(0.574693\pi\)
\(654\) 0 0
\(655\) 1.00139i 0.0391276i
\(656\) 0 0
\(657\) −5.68495 −0.221791
\(658\) 0 0
\(659\) −13.3180 −0.518796 −0.259398 0.965771i \(-0.583524\pi\)
−0.259398 + 0.965771i \(0.583524\pi\)
\(660\) 0 0
\(661\) 15.7853i 0.613975i −0.951714 0.306988i \(-0.900679\pi\)
0.951714 0.306988i \(-0.0993210\pi\)
\(662\) 0 0
\(663\) 14.6327i 0.568289i
\(664\) 0 0
\(665\) 8.36695 11.3904i 0.324457 0.441700i
\(666\) 0 0
\(667\) −32.4189 −1.25526
\(668\) 0 0
\(669\) 23.0779 0.892241
\(670\) 0 0
\(671\) 34.0870i 1.31591i
\(672\) 0 0
\(673\) 45.1378i 1.73994i 0.493109 + 0.869968i \(0.335861\pi\)
−0.493109 + 0.869968i \(0.664139\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 4.54599i 0.174717i 0.996177 + 0.0873584i \(0.0278425\pi\)
−0.996177 + 0.0873584i \(0.972157\pi\)
\(678\) 0 0
\(679\) 42.3247 1.62427
\(680\) 0 0
\(681\) −8.85693 −0.339398
\(682\) 0 0
\(683\) 26.7569 1.02382 0.511912 0.859038i \(-0.328937\pi\)
0.511912 + 0.859038i \(0.328937\pi\)
\(684\) 0 0
\(685\) 3.68495 0.140795
\(686\) 0 0
\(687\) 20.5130 0.782619
\(688\) 0 0
\(689\) 18.0519 0.687723
\(690\) 0 0
\(691\) 11.4372i 0.435093i −0.976050 0.217546i \(-0.930195\pi\)
0.976050 0.217546i \(-0.0698053\pi\)
\(692\) 0 0
\(693\) −18.8799 −0.717189
\(694\) 0 0
\(695\) 1.32374i 0.0502122i
\(696\) 0 0
\(697\) 2.98700i 0.113141i
\(698\) 0 0
\(699\) 28.7858 1.08878
\(700\) 0 0
\(701\) 21.0779 0.796100 0.398050 0.917364i \(-0.369687\pi\)
0.398050 + 0.917364i \(0.369687\pi\)
\(702\) 0 0
\(703\) 0.658999 0.897131i 0.0248546 0.0338359i
\(704\) 0 0
\(705\) 0.406492i 0.0153094i
\(706\) 0 0
\(707\) 44.4463i 1.67157i
\(708\) 0 0
\(709\) 37.9578 1.42553 0.712767 0.701401i \(-0.247442\pi\)
0.712767 + 0.701401i \(0.247442\pi\)
\(710\) 0 0
\(711\) 4.65900 0.174726
\(712\) 0 0
\(713\) 43.7270i 1.63759i
\(714\) 0 0
\(715\) 18.8799 0.706069
\(716\) 0 0
\(717\) 25.6166i 0.956671i
\(718\) 0 0
\(719\) 17.9421i 0.669127i −0.942373 0.334563i \(-0.891411\pi\)
0.942373 0.334563i \(-0.108589\pi\)
\(720\) 0 0
\(721\) 16.2960i 0.606895i
\(722\) 0 0
\(723\) 22.4785i 0.835985i
\(724\) 0 0
\(725\) 5.82288i 0.216256i
\(726\) 0 0
\(727\) 7.89263i 0.292721i −0.989231 0.146361i \(-0.953244\pi\)
0.989231 0.146361i \(-0.0467560\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.15251i 0.0426270i
\(732\) 0 0
\(733\) 25.6850 0.948695 0.474348 0.880338i \(-0.342684\pi\)
0.474348 + 0.880338i \(0.342684\pi\)
\(734\) 0 0
\(735\) −3.51298 −0.129578
\(736\) 0 0
\(737\) 60.3655i 2.22359i
\(738\) 0 0
\(739\) 16.5045i 0.607129i 0.952811 + 0.303564i \(0.0981768\pi\)
−0.952811 + 0.303564i \(0.901823\pi\)
\(740\) 0 0
\(741\) 8.36695 11.3904i 0.307368 0.418436i
\(742\) 0 0
\(743\) −34.6820 −1.27236 −0.636180 0.771541i \(-0.719486\pi\)
−0.636180 + 0.771541i \(0.719486\pi\)
\(744\) 0 0
\(745\) −15.0260 −0.550508
\(746\) 0 0
\(747\) 6.07825i 0.222392i
\(748\) 0 0
\(749\) 56.3943i 2.06060i
\(750\) 0 0
\(751\) 40.2728 1.46958 0.734788 0.678297i \(-0.237282\pi\)
0.734788 + 0.678297i \(0.237282\pi\)
\(752\) 0 0
\(753\) 15.8054i 0.575980i
\(754\) 0 0
\(755\) −22.2209 −0.808702
\(756\) 0 0
\(757\) −29.7080 −1.07975 −0.539877 0.841744i \(-0.681529\pi\)
−0.539877 + 0.841744i \(0.681529\pi\)
\(758\) 0 0
\(759\) 32.4189 1.17673
\(760\) 0 0
\(761\) −52.4478 −1.90123 −0.950615 0.310373i \(-0.899546\pi\)
−0.950615 + 0.310373i \(0.899546\pi\)
\(762\) 0 0
\(763\) −15.0779 −0.545855
\(764\) 0 0
\(765\) 4.51298 0.163167
\(766\) 0 0
\(767\) 29.2655i 1.05672i
\(768\) 0 0
\(769\) 38.7858 1.39865 0.699326 0.714803i \(-0.253484\pi\)
0.699326 + 0.714803i \(0.253484\pi\)
\(770\) 0 0
\(771\) 18.0262i 0.649199i
\(772\) 0 0
\(773\) 44.4760i 1.59969i −0.600207 0.799845i \(-0.704915\pi\)
0.600207 0.799845i \(-0.295085\pi\)
\(774\) 0 0
\(775\) −7.85398 −0.282123
\(776\) 0 0
\(777\) −0.828025 −0.0297052
\(778\) 0 0
\(779\) −1.70795 + 2.32513i −0.0611938 + 0.0833063i
\(780\) 0 0
\(781\) 52.5570i 1.88064i
\(782\) 0 0
\(783\) 5.82288i 0.208093i
\(784\) 0 0
\(785\) 4.36695 0.155863
\(786\) 0 0
\(787\) 13.3410 0.475555 0.237778 0.971320i \(-0.423581\pi\)
0.237778 + 0.971320i \(0.423581\pi\)
\(788\) 0 0
\(789\) 18.0262i 0.641751i
\(790\) 0 0
\(791\) 36.4419 1.29572
\(792\) 0 0
\(793\) 18.9808i 0.674027i
\(794\) 0 0
\(795\) 5.56750i 0.197459i
\(796\) 0 0
\(797\) 10.5602i 0.374062i 0.982354 + 0.187031i \(0.0598865\pi\)
−0.982354 + 0.187031i \(0.940114\pi\)
\(798\) 0 0
\(799\) 1.83449i 0.0648996i
\(800\) 0 0
\(801\) 3.64886i 0.128926i
\(802\) 0 0
\(803\) 33.1028i 1.16817i
\(804\)