Properties

Label 672.2.j.d.239.2
Level 672
Weight 2
Character 672.239
Analytic conductor 5.366
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.2
Root \(1.19877 - 0.750295i\)
Character \(\chi\) = 672.239
Dual form 672.2.j.d.239.4

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.67298 - 0.448478i) q^{3} +0.896956 q^{5} -1.00000i q^{7} +(2.59774 + 1.50059i) q^{9} +O(q^{10})\) \(q+(-1.67298 - 0.448478i) q^{3} +0.896956 q^{5} -1.00000i q^{7} +(2.59774 + 1.50059i) q^{9} +1.84951i q^{13} +(-1.50059 - 0.402265i) q^{15} -4.12397i q^{17} +0.654037 q^{19} +(-0.448478 + 1.67298i) q^{21} +7.12515 q^{23} -4.19547 q^{25} +(-3.67298 - 3.67549i) q^{27} +7.79627 q^{29} -0.896956i q^{35} -10.6919i q^{37} +(0.829463 - 3.09419i) q^{39} -10.1263i q^{41} +1.19547 q^{43} +(2.33005 + 1.34596i) q^{45} +9.59019 q^{47} -1.00000 q^{49} +(-1.84951 + 6.89932i) q^{51} +8.24793 q^{53} +(-1.09419 - 0.293321i) q^{57} -6.89932i q^{59} +4.54143i q^{61} +(1.50059 - 2.59774i) q^{63} +1.65893i q^{65} +5.49646 q^{67} +(-11.9202 - 3.19547i) q^{69} -4.12397 q^{71} -6.00000 q^{73} +(7.01894 + 1.88158i) q^{75} +13.3839i q^{79} +(4.49646 + 7.79627i) q^{81} -13.3533i q^{83} -3.69901i q^{85} +(-13.0430 - 3.49646i) q^{87} +13.7142i q^{89} +1.84951 q^{91} +0.586642 q^{95} -8.99291 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{9} + O(q^{10}) \) \( 12q + 4q^{9} + 48q^{19} + 4q^{25} - 24q^{27} - 40q^{43} - 12q^{49} - 8q^{51} + 40q^{57} + 40q^{67} - 72q^{73} + 24q^{75} + 28q^{81} + 8q^{91} - 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.67298 0.448478i −0.965896 0.258929i
\(4\) 0 0
\(5\) 0.896956 0.401131 0.200565 0.979680i \(-0.435722\pi\)
0.200565 + 0.979680i \(0.435722\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.59774 + 1.50059i 0.865912 + 0.500197i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.84951i 0.512961i 0.966549 + 0.256480i \(0.0825629\pi\)
−0.966549 + 0.256480i \(0.917437\pi\)
\(14\) 0 0
\(15\) −1.50059 0.402265i −0.387451 0.103864i
\(16\) 0 0
\(17\) 4.12397i 1.00021i −0.865965 0.500104i \(-0.833295\pi\)
0.865965 0.500104i \(-0.166705\pi\)
\(18\) 0 0
\(19\) 0.654037 0.150046 0.0750232 0.997182i \(-0.476097\pi\)
0.0750232 + 0.997182i \(0.476097\pi\)
\(20\) 0 0
\(21\) −0.448478 + 1.67298i −0.0978659 + 0.365075i
\(22\) 0 0
\(23\) 7.12515 1.48570 0.742848 0.669460i \(-0.233475\pi\)
0.742848 + 0.669460i \(0.233475\pi\)
\(24\) 0 0
\(25\) −4.19547 −0.839094
\(26\) 0 0
\(27\) −3.67298 3.67549i −0.706866 0.707348i
\(28\) 0 0
\(29\) 7.79627 1.44773 0.723866 0.689941i \(-0.242363\pi\)
0.723866 + 0.689941i \(0.242363\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.896956i 0.151613i
\(36\) 0 0
\(37\) 10.6919i 1.75774i −0.477060 0.878871i \(-0.658297\pi\)
0.477060 0.878871i \(-0.341703\pi\)
\(38\) 0 0
\(39\) 0.829463 3.09419i 0.132820 0.495467i
\(40\) 0 0
\(41\) 10.1263i 1.58147i −0.612161 0.790733i \(-0.709699\pi\)
0.612161 0.790733i \(-0.290301\pi\)
\(42\) 0 0
\(43\) 1.19547 0.182308 0.0911538 0.995837i \(-0.470945\pi\)
0.0911538 + 0.995837i \(0.470945\pi\)
\(44\) 0 0
\(45\) 2.33005 + 1.34596i 0.347344 + 0.200644i
\(46\) 0 0
\(47\) 9.59019 1.39887 0.699436 0.714695i \(-0.253435\pi\)
0.699436 + 0.714695i \(0.253435\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −1.84951 + 6.89932i −0.258983 + 0.966098i
\(52\) 0 0
\(53\) 8.24793 1.13294 0.566470 0.824082i \(-0.308309\pi\)
0.566470 + 0.824082i \(0.308309\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.09419 0.293321i −0.144929 0.0388513i
\(58\) 0 0
\(59\) 6.89932i 0.898215i −0.893478 0.449107i \(-0.851742\pi\)
0.893478 0.449107i \(-0.148258\pi\)
\(60\) 0 0
\(61\) 4.54143i 0.581471i 0.956803 + 0.290735i \(0.0939000\pi\)
−0.956803 + 0.290735i \(0.906100\pi\)
\(62\) 0 0
\(63\) 1.50059 2.59774i 0.189057 0.327284i
\(64\) 0 0
\(65\) 1.65893i 0.205764i
\(66\) 0 0
\(67\) 5.49646 0.671499 0.335750 0.941951i \(-0.391010\pi\)
0.335750 + 0.941951i \(0.391010\pi\)
\(68\) 0 0
\(69\) −11.9202 3.19547i −1.43503 0.384689i
\(70\) 0 0
\(71\) −4.12397 −0.489425 −0.244712 0.969596i \(-0.578694\pi\)
−0.244712 + 0.969596i \(0.578694\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 7.01894 + 1.88158i 0.810478 + 0.217266i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.3839i 1.50580i 0.658134 + 0.752901i \(0.271346\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(80\) 0 0
\(81\) 4.49646 + 7.79627i 0.499606 + 0.866253i
\(82\) 0 0
\(83\) 13.3533i 1.46572i −0.680380 0.732860i \(-0.738185\pi\)
0.680380 0.732860i \(-0.261815\pi\)
\(84\) 0 0
\(85\) 3.69901i 0.401214i
\(86\) 0 0
\(87\) −13.0430 3.49646i −1.39836 0.374859i
\(88\) 0 0
\(89\) 13.7142i 1.45370i 0.686798 + 0.726849i \(0.259016\pi\)
−0.686798 + 0.726849i \(0.740984\pi\)
\(90\) 0 0
\(91\) 1.84951 0.193881
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.586642 0.0601882
\(96\) 0 0
\(97\) −8.99291 −0.913092 −0.456546 0.889700i \(-0.650914\pi\)
−0.456546 + 0.889700i \(0.650914\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.35097 −0.731449 −0.365725 0.930723i \(-0.619179\pi\)
−0.365725 + 0.930723i \(0.619179\pi\)
\(102\) 0 0
\(103\) 13.3839i 1.31875i −0.751814 0.659375i \(-0.770821\pi\)
0.751814 0.659375i \(-0.229179\pi\)
\(104\) 0 0
\(105\) −0.402265 + 1.50059i −0.0392570 + 0.146443i
\(106\) 0 0
\(107\) 1.79391i 0.173424i 0.996233 + 0.0867120i \(0.0276360\pi\)
−0.996233 + 0.0867120i \(0.972364\pi\)
\(108\) 0 0
\(109\) 5.30807i 0.508421i 0.967149 + 0.254211i \(0.0818156\pi\)
−0.967149 + 0.254211i \(0.918184\pi\)
\(110\) 0 0
\(111\) −4.79509 + 17.8874i −0.455130 + 1.69780i
\(112\) 0 0
\(113\) 3.00118i 0.282327i −0.989986 0.141164i \(-0.954916\pi\)
0.989986 0.141164i \(-0.0450844\pi\)
\(114\) 0 0
\(115\) 6.39094 0.595958
\(116\) 0 0
\(117\) −2.77535 + 4.80453i −0.256581 + 0.444179i
\(118\) 0 0
\(119\) −4.12397 −0.378043
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −4.54143 + 16.9412i −0.409487 + 1.52753i
\(124\) 0 0
\(125\) −8.24793 −0.737717
\(126\) 0 0
\(127\) 14.1884i 1.25902i 0.776994 + 0.629508i \(0.216743\pi\)
−0.776994 + 0.629508i \(0.783257\pi\)
\(128\) 0 0
\(129\) −2.00000 0.536142i −0.176090 0.0472047i
\(130\) 0 0
\(131\) 16.9412i 1.48016i 0.672521 + 0.740078i \(0.265211\pi\)
−0.672521 + 0.740078i \(0.734789\pi\)
\(132\) 0 0
\(133\) 0.654037i 0.0567122i
\(134\) 0 0
\(135\) −3.29450 3.29675i −0.283546 0.283739i
\(136\) 0 0
\(137\) 11.8358i 1.01120i 0.862769 + 0.505598i \(0.168728\pi\)
−0.862769 + 0.505598i \(0.831272\pi\)
\(138\) 0 0
\(139\) −11.0450 −0.936823 −0.468411 0.883510i \(-0.655173\pi\)
−0.468411 + 0.883510i \(0.655173\pi\)
\(140\) 0 0
\(141\) −16.0442 4.30099i −1.35117 0.362208i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.99291 0.580730
\(146\) 0 0
\(147\) 1.67298 + 0.448478i 0.137985 + 0.0369898i
\(148\) 0 0
\(149\) −4.66011 −0.381771 −0.190885 0.981612i \(-0.561136\pi\)
−0.190885 + 0.981612i \(0.561136\pi\)
\(150\) 0 0
\(151\) 12.5793i 1.02369i −0.859078 0.511845i \(-0.828962\pi\)
0.859078 0.511845i \(-0.171038\pi\)
\(152\) 0 0
\(153\) 6.18838 10.7130i 0.500301 0.866092i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.5343i 1.55901i 0.626396 + 0.779505i \(0.284529\pi\)
−0.626396 + 0.779505i \(0.715471\pi\)
\(158\) 0 0
\(159\) −13.7986 3.69901i −1.09430 0.293351i
\(160\) 0 0
\(161\) 7.12515i 0.561540i
\(162\) 0 0
\(163\) −9.19547 −0.720245 −0.360122 0.932905i \(-0.617265\pi\)
−0.360122 + 0.932905i \(0.617265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.8358 −0.915878 −0.457939 0.888984i \(-0.651412\pi\)
−0.457939 + 0.888984i \(0.651412\pi\)
\(168\) 0 0
\(169\) 9.57932 0.736871
\(170\) 0 0
\(171\) 1.69901 + 0.981441i 0.129927 + 0.0750527i
\(172\) 0 0
\(173\) 19.3557 1.47159 0.735793 0.677206i \(-0.236810\pi\)
0.735793 + 0.677206i \(0.236810\pi\)
\(174\) 0 0
\(175\) 4.19547i 0.317148i
\(176\) 0 0
\(177\) −3.09419 + 11.5424i −0.232574 + 0.867582i
\(178\) 0 0
\(179\) 14.7019i 1.09888i 0.835535 + 0.549438i \(0.185158\pi\)
−0.835535 + 0.549438i \(0.814842\pi\)
\(180\) 0 0
\(181\) 4.54143i 0.337562i 0.985654 + 0.168781i \(0.0539831\pi\)
−0.985654 + 0.168781i \(0.946017\pi\)
\(182\) 0 0
\(183\) 2.03673 7.59774i 0.150560 0.561641i
\(184\) 0 0
\(185\) 9.59019i 0.705084i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.67549 + 3.67298i −0.267352 + 0.267170i
\(190\) 0 0
\(191\) 1.70943 0.123690 0.0618449 0.998086i \(-0.480302\pi\)
0.0618449 + 0.998086i \(0.480302\pi\)
\(192\) 0 0
\(193\) 16.1884 1.16527 0.582633 0.812736i \(-0.302023\pi\)
0.582633 + 0.812736i \(0.302023\pi\)
\(194\) 0 0
\(195\) 0.743992 2.77535i 0.0532783 0.198747i
\(196\) 0 0
\(197\) −9.59019 −0.683272 −0.341636 0.939832i \(-0.610981\pi\)
−0.341636 + 0.939832i \(0.610981\pi\)
\(198\) 0 0
\(199\) 14.3909i 1.02015i −0.860131 0.510073i \(-0.829618\pi\)
0.860131 0.510073i \(-0.170382\pi\)
\(200\) 0 0
\(201\) −9.19547 2.46504i −0.648598 0.173870i
\(202\) 0 0
\(203\) 7.79627i 0.547191i
\(204\) 0 0
\(205\) 9.08287i 0.634375i
\(206\) 0 0
\(207\) 18.5092 + 10.6919i 1.28648 + 0.743140i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −15.8874 −1.09373 −0.546867 0.837220i \(-0.684180\pi\)
−0.546867 + 0.837220i \(0.684180\pi\)
\(212\) 0 0
\(213\) 6.89932 + 1.84951i 0.472733 + 0.126726i
\(214\) 0 0
\(215\) 1.07228 0.0731292
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.0379 + 2.69087i 0.678298 + 0.181832i
\(220\) 0 0
\(221\) 7.62730 0.513068
\(222\) 0 0
\(223\) 9.00709i 0.603159i −0.953441 0.301580i \(-0.902486\pi\)
0.953441 0.301580i \(-0.0975139\pi\)
\(224\) 0 0
\(225\) −10.8987 6.29568i −0.726581 0.419712i
\(226\) 0 0
\(227\) 22.9435i 1.52282i −0.648274 0.761408i \(-0.724509\pi\)
0.648274 0.761408i \(-0.275491\pi\)
\(228\) 0 0
\(229\) 10.9324i 0.722432i −0.932482 0.361216i \(-0.882362\pi\)
0.932482 0.361216i \(-0.117638\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.24793i 0.540340i −0.962813 0.270170i \(-0.912920\pi\)
0.962813 0.270170i \(-0.0870799\pi\)
\(234\) 0 0
\(235\) 8.60197 0.561131
\(236\) 0 0
\(237\) 6.00236 22.3909i 0.389895 1.45445i
\(238\) 0 0
\(239\) 4.87958 0.315634 0.157817 0.987468i \(-0.449554\pi\)
0.157817 + 0.987468i \(0.449554\pi\)
\(240\) 0 0
\(241\) −4.99291 −0.321622 −0.160811 0.986985i \(-0.551411\pi\)
−0.160811 + 0.986985i \(0.551411\pi\)
\(242\) 0 0
\(243\) −4.02603 15.0596i −0.258270 0.966073i
\(244\) 0 0
\(245\) −0.896956 −0.0573044
\(246\) 0 0
\(247\) 1.20965i 0.0769679i
\(248\) 0 0
\(249\) −5.98868 + 22.3399i −0.379517 + 1.41573i
\(250\) 0 0
\(251\) 11.1078i 0.701116i 0.936541 + 0.350558i \(0.114008\pi\)
−0.936541 + 0.350558i \(0.885992\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.65893 + 6.18838i −0.103886 + 0.387532i
\(256\) 0 0
\(257\) 11.4686i 0.715390i 0.933838 + 0.357695i \(0.116437\pi\)
−0.933838 + 0.357695i \(0.883563\pi\)
\(258\) 0 0
\(259\) −10.6919 −0.664364
\(260\) 0 0
\(261\) 20.2527 + 11.6990i 1.25361 + 0.724151i
\(262\) 0 0
\(263\) 12.3719 0.762884 0.381442 0.924393i \(-0.375428\pi\)
0.381442 + 0.924393i \(0.375428\pi\)
\(264\) 0 0
\(265\) 7.39803 0.454457
\(266\) 0 0
\(267\) 6.15049 22.9435i 0.376404 1.40412i
\(268\) 0 0
\(269\) −24.7374 −1.50827 −0.754134 0.656721i \(-0.771943\pi\)
−0.754134 + 0.656721i \(0.771943\pi\)
\(270\) 0 0
\(271\) 20.7819i 1.26241i 0.775616 + 0.631205i \(0.217439\pi\)
−0.775616 + 0.631205i \(0.782561\pi\)
\(272\) 0 0
\(273\) −3.09419 0.829463i −0.187269 0.0502014i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.60906i 0.0966790i −0.998831 0.0483395i \(-0.984607\pi\)
0.998831 0.0483395i \(-0.0153929\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.66011i 0.277999i −0.990292 0.138999i \(-0.955611\pi\)
0.990292 0.138999i \(-0.0443886\pi\)
\(282\) 0 0
\(283\) −13.7369 −0.816574 −0.408287 0.912854i \(-0.633874\pi\)
−0.408287 + 0.912854i \(0.633874\pi\)
\(284\) 0 0
\(285\) −0.981441 0.263096i −0.0581356 0.0155845i
\(286\) 0 0
\(287\) −10.1263 −0.597738
\(288\) 0 0
\(289\) −0.00708757 −0.000416916
\(290\) 0 0
\(291\) 15.0450 + 4.03312i 0.881952 + 0.236426i
\(292\) 0 0
\(293\) 19.3557 1.13077 0.565386 0.824826i \(-0.308727\pi\)
0.565386 + 0.824826i \(0.308727\pi\)
\(294\) 0 0
\(295\) 6.18838i 0.360302i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.1780i 0.762104i
\(300\) 0 0
\(301\) 1.19547i 0.0689058i
\(302\) 0 0
\(303\) 12.2980 + 3.29675i 0.706504 + 0.189393i
\(304\) 0 0
\(305\) 4.07347i 0.233246i
\(306\) 0 0
\(307\) 30.8198 1.75898 0.879489 0.475920i \(-0.157885\pi\)
0.879489 + 0.475920i \(0.157885\pi\)
\(308\) 0 0
\(309\) −6.00236 + 22.3909i −0.341462 + 1.27378i
\(310\) 0 0
\(311\) 6.00236 0.340363 0.170181 0.985413i \(-0.445565\pi\)
0.170181 + 0.985413i \(0.445565\pi\)
\(312\) 0 0
\(313\) −13.7748 −0.778597 −0.389299 0.921112i \(-0.627283\pi\)
−0.389299 + 0.921112i \(0.627283\pi\)
\(314\) 0 0
\(315\) 1.34596 2.33005i 0.0758364 0.131284i
\(316\) 0 0
\(317\) −3.75679 −0.211003 −0.105501 0.994419i \(-0.533645\pi\)
−0.105501 + 0.994419i \(0.533645\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.804530 3.00118i 0.0449045 0.167510i
\(322\) 0 0
\(323\) 2.69722i 0.150078i
\(324\) 0 0
\(325\) 7.75955i 0.430423i
\(326\) 0 0
\(327\) 2.38055 8.88031i 0.131645 0.491082i
\(328\) 0 0
\(329\) 9.59019i 0.528724i
\(330\) 0 0
\(331\) 6.20256 0.340923 0.170462 0.985364i \(-0.445474\pi\)
0.170462 + 0.985364i \(0.445474\pi\)
\(332\) 0 0
\(333\) 16.0442 27.7748i 0.879217 1.52205i
\(334\) 0 0
\(335\) 4.93008 0.269359
\(336\) 0 0
\(337\) −14.5793 −0.794186 −0.397093 0.917778i \(-0.629981\pi\)
−0.397093 + 0.917778i \(0.629981\pi\)
\(338\) 0 0
\(339\) −1.34596 + 5.02092i −0.0731027 + 0.272699i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −10.6919 2.86620i −0.575634 0.154311i
\(346\) 0 0
\(347\) 8.41690i 0.451843i 0.974146 + 0.225921i \(0.0725393\pi\)
−0.974146 + 0.225921i \(0.927461\pi\)
\(348\) 0 0
\(349\) 25.9253i 1.38775i 0.720096 + 0.693874i \(0.244098\pi\)
−0.720096 + 0.693874i \(0.755902\pi\)
\(350\) 0 0
\(351\) 6.79784 6.79321i 0.362842 0.362594i
\(352\) 0 0
\(353\) 13.7142i 0.729931i −0.931021 0.364965i \(-0.881081\pi\)
0.931021 0.364965i \(-0.118919\pi\)
\(354\) 0 0
\(355\) −3.69901 −0.196323
\(356\) 0 0
\(357\) 6.89932 + 1.84951i 0.365151 + 0.0978863i
\(358\) 0 0
\(359\) −26.3055 −1.38835 −0.694176 0.719805i \(-0.744231\pi\)
−0.694176 + 0.719805i \(0.744231\pi\)
\(360\) 0 0
\(361\) −18.5722 −0.977486
\(362\) 0 0
\(363\) −18.4028 4.93326i −0.965896 0.258929i
\(364\) 0 0
\(365\) −5.38173 −0.281693
\(366\) 0 0
\(367\) 19.7748i 1.03224i 0.856518 + 0.516118i \(0.172623\pi\)
−0.856518 + 0.516118i \(0.827377\pi\)
\(368\) 0 0
\(369\) 15.1955 26.3055i 0.791045 1.36941i
\(370\) 0 0
\(371\) 8.24793i 0.428211i
\(372\) 0 0
\(373\) 3.77479i 0.195451i 0.995213 + 0.0977257i \(0.0311568\pi\)
−0.995213 + 0.0977257i \(0.968843\pi\)
\(374\) 0 0
\(375\) 13.7986 + 3.69901i 0.712558 + 0.191016i
\(376\) 0 0
\(377\) 14.4193i 0.742630i
\(378\) 0 0
\(379\) 8.59350 0.441418 0.220709 0.975340i \(-0.429163\pi\)
0.220709 + 0.975340i \(0.429163\pi\)
\(380\) 0 0
\(381\) 6.36318 23.7369i 0.325995 1.21608i
\(382\) 0 0
\(383\) 17.8381 0.911485 0.455743 0.890112i \(-0.349374\pi\)
0.455743 + 0.890112i \(0.349374\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.10552 + 1.79391i 0.157862 + 0.0911896i
\(388\) 0 0
\(389\) 7.79627 0.395287 0.197643 0.980274i \(-0.436671\pi\)
0.197643 + 0.980274i \(0.436671\pi\)
\(390\) 0 0
\(391\) 29.3839i 1.48601i
\(392\) 0 0
\(393\) 7.59774 28.3422i 0.383255 1.42968i
\(394\) 0 0
\(395\) 12.0047i 0.604023i
\(396\) 0 0
\(397\) 22.2263i 1.11550i −0.830007 0.557752i \(-0.811664\pi\)
0.830007 0.557752i \(-0.188336\pi\)
\(398\) 0 0
\(399\) −0.293321 + 1.09419i −0.0146844 + 0.0547781i
\(400\) 0 0
\(401\) 5.24675i 0.262010i −0.991382 0.131005i \(-0.958180\pi\)
0.991382 0.131005i \(-0.0418204\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.03312 + 6.99291i 0.200407 + 0.347481i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 28.1657 1.39271 0.696353 0.717699i \(-0.254805\pi\)
0.696353 + 0.717699i \(0.254805\pi\)
\(410\) 0 0
\(411\) 5.30807 19.8010i 0.261828 0.976711i
\(412\) 0 0
\(413\) −6.89932 −0.339493
\(414\) 0 0
\(415\) 11.9774i 0.587945i
\(416\) 0 0
\(417\) 18.4780 + 4.95343i 0.904874 + 0.242570i
\(418\) 0 0
\(419\) 2.23921i 0.109393i −0.998503 0.0546963i \(-0.982581\pi\)
0.998503 0.0546963i \(-0.0174191\pi\)
\(420\) 0 0
\(421\) 6.99291i 0.340814i −0.985374 0.170407i \(-0.945492\pi\)
0.985374 0.170407i \(-0.0545082\pi\)
\(422\) 0 0
\(423\) 24.9128 + 14.3909i 1.21130 + 0.699711i
\(424\) 0 0
\(425\) 17.3020i 0.839269i
\(426\) 0 0
\(427\) 4.54143 0.219775
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.80966 −0.472515 −0.236257 0.971691i \(-0.575921\pi\)
−0.236257 + 0.971691i \(0.575921\pi\)
\(432\) 0 0
\(433\) −23.3839 −1.12376 −0.561878 0.827220i \(-0.689921\pi\)
−0.561878 + 0.827220i \(0.689921\pi\)
\(434\) 0 0
\(435\) −11.6990 3.13617i −0.560925 0.150368i
\(436\) 0 0
\(437\) 4.66011 0.222923
\(438\) 0 0
\(439\) 20.3768i 0.972530i 0.873811 + 0.486265i \(0.161641\pi\)
−0.873811 + 0.486265i \(0.838359\pi\)
\(440\) 0 0
\(441\) −2.59774 1.50059i −0.123702 0.0714567i
\(442\) 0 0
\(443\) 36.5668i 1.73734i −0.495389 0.868671i \(-0.664974\pi\)
0.495389 0.868671i \(-0.335026\pi\)
\(444\) 0 0
\(445\) 12.3010i 0.583123i
\(446\) 0 0
\(447\) 7.79627 + 2.08995i 0.368751 + 0.0988515i
\(448\) 0 0
\(449\) 19.1804i 0.905178i 0.891719 + 0.452589i \(0.149499\pi\)
−0.891719 + 0.452589i \(0.850501\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −5.64155 + 21.0450i −0.265063 + 0.988779i
\(454\) 0 0
\(455\) 1.65893 0.0777717
\(456\) 0 0
\(457\) −20.5935 −0.963323 −0.481662 0.876357i \(-0.659967\pi\)
−0.481662 + 0.876357i \(0.659967\pi\)
\(458\) 0 0
\(459\) −15.1576 + 15.1472i −0.707495 + 0.707013i
\(460\) 0 0
\(461\) −2.69087 −0.125326 −0.0626631 0.998035i \(-0.519959\pi\)
−0.0626631 + 0.998035i \(0.519959\pi\)
\(462\) 0 0
\(463\) 4.78188i 0.222233i 0.993807 + 0.111116i \(0.0354427\pi\)
−0.993807 + 0.111116i \(0.964557\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.8341i 1.10291i 0.834204 + 0.551456i \(0.185927\pi\)
−0.834204 + 0.551456i \(0.814073\pi\)
\(468\) 0 0
\(469\) 5.49646i 0.253803i
\(470\) 0 0
\(471\) 8.76072 32.6806i 0.403673 1.50584i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.74399 −0.125903
\(476\) 0 0
\(477\) 21.4259 + 12.3768i 0.981026 + 0.566693i
\(478\) 0 0
\(479\) −15.5925 −0.712442 −0.356221 0.934402i \(-0.615935\pi\)
−0.356221 + 0.934402i \(0.615935\pi\)
\(480\) 0 0
\(481\) 19.7748 0.901653
\(482\) 0 0
\(483\) −3.19547 + 11.9202i −0.145399 + 0.542390i
\(484\) 0 0
\(485\) −8.06624 −0.366269
\(486\) 0 0
\(487\) 16.2026i 0.734208i −0.930180 0.367104i \(-0.880349\pi\)
0.930180 0.367104i \(-0.119651\pi\)
\(488\) 0 0
\(489\) 15.3839 + 4.12397i 0.695682 + 0.186492i
\(490\) 0 0
\(491\) 6.62299i 0.298891i −0.988770 0.149446i \(-0.952251\pi\)
0.988770 0.149446i \(-0.0477489\pi\)
\(492\) 0 0
\(493\) 32.1516i 1.44803i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.12397i 0.184985i
\(498\) 0 0
\(499\) −21.2713 −0.952232 −0.476116 0.879383i \(-0.657956\pi\)
−0.476116 + 0.879383i \(0.657956\pi\)
\(500\) 0 0
\(501\) 19.8010 + 5.30807i 0.884643 + 0.237147i
\(502\) 0 0
\(503\) 1.07228 0.0478108 0.0239054 0.999714i \(-0.492390\pi\)
0.0239054 + 0.999714i \(0.492390\pi\)
\(504\) 0 0
\(505\) −6.59350 −0.293407
\(506\) 0 0
\(507\) −16.0260 4.29611i −0.711741 0.190797i
\(508\) 0 0
\(509\) 1.78755 0.0792320 0.0396160 0.999215i \(-0.487387\pi\)
0.0396160 + 0.999215i \(0.487387\pi\)
\(510\) 0 0
\(511\) 6.00000i 0.265424i
\(512\) 0 0
\(513\) −2.40226 2.40390i −0.106063 0.106135i
\(514\) 0 0
\(515\) 12.0047i 0.528991i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −32.3817 8.68060i −1.42140 0.381036i
\(520\) 0 0
\(521\) 8.95304i 0.392240i 0.980580 + 0.196120i \(0.0628342\pi\)
−0.980580 + 0.196120i \(0.937166\pi\)
\(522\) 0 0
\(523\) −1.25601 −0.0549214 −0.0274607 0.999623i \(-0.508742\pi\)
−0.0274607 + 0.999623i \(0.508742\pi\)
\(524\) 0 0
\(525\) 1.88158 7.01894i 0.0821187 0.306332i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 27.7677 1.20729
\(530\) 0 0
\(531\) 10.3531 17.9226i 0.449284 0.777775i
\(532\) 0 0
\(533\) 18.7287 0.811231
\(534\) 0 0
\(535\) 1.60906i 0.0695657i
\(536\) 0 0
\(537\) 6.59350 24.5961i 0.284530 1.06140i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 44.3768i 1.90791i 0.299956 + 0.953953i \(0.403028\pi\)
−0.299956 + 0.953953i \(0.596972\pi\)
\(542\) 0 0
\(543\) 2.03673 7.59774i 0.0874046 0.326050i
\(544\) 0 0
\(545\) 4.76111i 0.203943i
\(546\) 0 0
\(547\) 6.80453 0.290941 0.145470 0.989363i \(-0.453530\pi\)
0.145470 + 0.989363i \(0.453530\pi\)
\(548\) 0 0
\(549\) −6.81483 + 11.7974i −0.290850 + 0.503503i
\(550\) 0 0
\(551\) 5.09905 0.217227
\(552\) 0 0
\(553\) 13.3839 0.569139
\(554\) 0 0
\(555\) −4.30099 + 16.0442i −0.182567 + 0.681039i
\(556\) 0 0
\(557\) −46.6087 −1.97487 −0.987436 0.158017i \(-0.949490\pi\)
−0.987436 + 0.158017i \(0.949490\pi\)
\(558\) 0 0
\(559\) 2.21103i 0.0935166i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.89932i 0.290772i 0.989375 + 0.145386i \(0.0464423\pi\)
−0.989375 + 0.145386i \(0.953558\pi\)
\(564\) 0 0
\(565\) 2.69193i 0.113250i
\(566\) 0 0
\(567\) 7.79627 4.49646i 0.327413 0.188833i
\(568\) 0 0
\(569\) 6.75798i 0.283309i −0.989916 0.141655i \(-0.954758\pi\)
0.989916 0.141655i \(-0.0452422\pi\)
\(570\) 0 0
\(571\) −17.1955 −0.719608 −0.359804 0.933028i \(-0.617156\pi\)
−0.359804 + 0.933028i \(0.617156\pi\)
\(572\) 0 0
\(573\) −2.85984 0.766640i −0.119471 0.0320268i
\(574\) 0 0
\(575\) −29.8933 −1.24664
\(576\) 0 0
\(577\) −27.1586 −1.13063 −0.565315 0.824875i \(-0.691245\pi\)
−0.565315 + 0.824875i \(0.691245\pi\)
\(578\) 0 0
\(579\) −27.0829 7.26013i −1.12553 0.301721i
\(580\) 0 0
\(581\) −13.3533 −0.553990
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.48937 + 4.30945i −0.102923 + 0.178174i
\(586\) 0 0
\(587\) 26.5313i 1.09507i 0.836784 + 0.547533i \(0.184433\pi\)
−0.836784 + 0.547533i \(0.815567\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 16.0442 + 4.30099i 0.659970 + 0.176919i
\(592\) 0 0
\(593\) 27.9644i 1.14836i 0.818728 + 0.574181i \(0.194679\pi\)
−0.818728 + 0.574181i \(0.805321\pi\)
\(594\) 0 0
\(595\) −3.69901 −0.151645
\(596\) 0 0
\(597\) −6.45402 + 24.0758i −0.264145 + 0.985356i
\(598\) 0 0
\(599\) −35.0391 −1.43166 −0.715829 0.698275i \(-0.753951\pi\)
−0.715829 + 0.698275i \(0.753951\pi\)
\(600\) 0 0
\(601\) 6.37677 0.260114 0.130057 0.991507i \(-0.458484\pi\)
0.130057 + 0.991507i \(0.458484\pi\)
\(602\) 0 0
\(603\) 14.2783 + 8.24793i 0.581459 + 0.335882i
\(604\) 0 0
\(605\) 9.86651 0.401131
\(606\) 0 0
\(607\) 8.60197i 0.349143i 0.984644 + 0.174572i \(0.0558541\pi\)
−0.984644 + 0.174572i \(0.944146\pi\)
\(608\) 0 0
\(609\) −3.49646 + 13.0430i −0.141684 + 0.528530i
\(610\) 0 0
\(611\) 17.7371i 0.717567i
\(612\) 0 0
\(613\) 19.2939i 0.779273i −0.920969 0.389637i \(-0.872601\pi\)
0.920969 0.389637i \(-0.127399\pi\)
\(614\) 0 0
\(615\) −4.07347 + 15.1955i −0.164258 + 0.612741i
\(616\) 0 0
\(617\) 10.0758i 0.405638i 0.979216 + 0.202819i \(0.0650102\pi\)
−0.979216 + 0.202819i \(0.934990\pi\)
\(618\) 0 0
\(619\) 16.7298 0.672428 0.336214 0.941786i \(-0.390853\pi\)
0.336214 + 0.941786i \(0.390853\pi\)
\(620\) 0 0
\(621\) −26.1705 26.1884i −1.05019 1.05090i
\(622\) 0 0
\(623\) 13.7142 0.549446
\(624\) 0 0
\(625\) 13.5793 0.543173
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −44.0931 −1.75811
\(630\) 0 0
\(631\) 20.7819i 0.827314i 0.910433 + 0.413657i \(0.135749\pi\)
−0.910433 + 0.413657i \(0.864251\pi\)
\(632\) 0 0
\(633\) 26.5793 + 7.12515i 1.05643 + 0.283199i
\(634\) 0 0
\(635\) 12.7264i 0.505030i
\(636\) 0 0
\(637\) 1.84951i 0.0732801i
\(638\) 0 0
\(639\) −10.7130 6.18838i −0.423799 0.244809i
\(640\) 0 0
\(641\) 37.6051i 1.48531i −0.669672 0.742657i \(-0.733565\pi\)
0.669672 0.742657i \(-0.266435\pi\)
\(642\) 0 0
\(643\) 25.7369 1.01496 0.507482 0.861662i \(-0.330576\pi\)
0.507482 + 0.861662i \(0.330576\pi\)
\(644\) 0 0
\(645\) −1.79391 0.480896i −0.0706352 0.0189352i
\(646\) 0 0
\(647\) −41.8476 −1.64520 −0.822599 0.568622i \(-0.807477\pi\)
−0.822599 + 0.568622i \(0.807477\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.0418 −0.392968 −0.196484 0.980507i \(-0.562952\pi\)
−0.196484 + 0.980507i \(0.562952\pi\)
\(654\) 0 0
\(655\) 15.1955i 0.593736i
\(656\) 0 0
\(657\) −15.5864 9.00354i −0.608084 0.351262i
\(658\) 0 0
\(659\) 48.5843i 1.89257i 0.323327 + 0.946287i \(0.395199\pi\)
−0.323327 + 0.946287i \(0.604801\pi\)
\(660\) 0 0
\(661\) 0.164668i 0.00640485i −0.999995 0.00320242i \(-0.998981\pi\)
0.999995 0.00320242i \(-0.00101936\pi\)
\(662\) 0 0
\(663\) −12.7603 3.42068i −0.495570 0.132848i
\(664\) 0 0
\(665\) 0.586642i 0.0227490i
\(666\) 0 0
\(667\) 55.5496 2.15089
\(668\) 0 0
\(669\) −4.03948 + 15.0687i −0.156175 + 0.582589i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 7.00709 0.270103 0.135052 0.990839i \(-0.456880\pi\)
0.135052 + 0.990839i \(0.456880\pi\)
\(674\) 0 0
\(675\) 15.4099 + 15.4204i 0.593127 + 0.593531i
\(676\) 0 0
\(677\) −6.27869 −0.241310 −0.120655 0.992695i \(-0.538499\pi\)
−0.120655 + 0.992695i \(0.538499\pi\)
\(678\) 0 0
\(679\) 8.99291i 0.345116i
\(680\) 0 0
\(681\) −10.2897 + 38.3841i −0.394301 + 1.47088i
\(682\) 0 0
\(683\) 30.2945i 1.15919i −0.814906 0.579593i \(-0.803211\pi\)
0.814906 0.579593i \(-0.196789\pi\)
\(684\) 0 0
\(685\) 10.6161i 0.405622i
\(686\) 0 0
\(687\) −4.90293 + 18.2897i −0.187058 + 0.697794i
\(688\) 0 0
\(689\) 15.2546i 0.581154i
\(690\) 0 0
\(691\) 15.6469 0.595238 0.297619 0.954685i \(-0.403807\pi\)
0.297619 + 0.954685i \(0.403807\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9.90686 −0.375788
\(696\) 0 0
\(697\) −41.7606 −1.58180
\(698\) 0 0
\(699\) −3.69901 + 13.7986i −0.139910 + 0.521912i
\(700\) 0 0
\(701\) 6.45402 0.243765 0.121882 0.992545i \(-0.461107\pi\)
0.121882 + 0.992545i \(0.461107\pi\)
\(702\) 0 0
\(703\) 6.99291i 0.263743i
\(704\) 0 0
\(705\) −14.3909 3.85779i −0.541994 0.145293i
\(706\) 0 0
\(707\) 7.35097i 0.276462i
\(708\) 0 0
\(709\) 18.0900i 0.679383i −0.940537 0.339691i \(-0.889677\pi\)
0.940537 0.339691i \(-0.110323\pi\)
\(710\) 0 0
\(711\) −20.0837 + 34.7677i −0.753197 + 1.30389i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.16344 2.18838i −0.304869 0.0817267i
\(718\) 0 0
\(719\) −5.83339 −0.217549 −0.108774 0.994066i \(-0.534693\pi\)
−0.108774 + 0.994066i \(0.534693\pi\)
\(720\) 0 0
\(721\) −13.3839 −0.498441
\(722\) 0 0
\(723\) 8.35305 + 2.23921i 0.310653 + 0.0832771i
\(724\) 0 0
\(725\) −32.7090 −1.21478
\(726\) 0 0
\(727\) 4.22521i 0.156704i 0.996926 + 0.0783521i \(0.0249658\pi\)
−0.996926 + 0.0783521i \(0.975034\pi\)
\(728\) 0 0
\(729\) −0.0184116 + 27.0000i −0.000681912 + 1.00000i
\(730\) 0 0
\(731\) 4.93008i 0.182346i
\(732\) 0 0
\(733\) 43.6101i 1.61078i 0.592747 + 0.805388i \(0.298043\pi\)
−0.592747 + 0.805388i \(0.701957\pi\)
\(734\) 0 0
\(735\) 1.50059 + 0.402265i 0.0553501 + 0.0148378i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.97735 0.219880 0.109940 0.993938i \(-0.464934\pi\)
0.109940 + 0.993938i \(0.464934\pi\)
\(740\) 0 0
\(741\) 0.542499 2.02371i 0.0199292 0.0743430i
\(742\) 0 0
\(743\) 49.9770 1.83348 0.916740 0.399485i \(-0.130811\pi\)
0.916740 + 0.399485i \(0.130811\pi\)
\(744\) 0 0
\(745\) −4.17991 −0.153140
\(746\) 0 0
\(747\) 20.0379 34.6884i 0.733148 1.26918i
\(748\) 0 0
\(749\) 1.79391 0.0655481
\(750\) 0 0
\(751\) 31.7974i 1.16031i −0.814508 0.580153i \(-0.802993\pi\)
0.814508 0.580153i \(-0.197007\pi\)
\(752\) 0 0
\(753\) 4.98159 18.5831i 0.181539 0.677206i
\(754\) 0 0
\(755\) 11.2831i 0.410634i
\(756\) 0 0
\(757\) 34.0900i 1.23902i −0.784989 0.619510i \(-0.787331\pi\)
0.784989 0.619510i \(-0.212669\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.806113i 0.0292216i 0.999893 + 0.0146108i \(0.00465092\pi\)
−0.999893 + 0.0146108i \(0.995349\pi\)
\(762\) 0 0
\(763\) 5.30807 0.192165
\(764\) 0 0
\(765\) 5.55071 9.60906i 0.200686 0.347416i
\(766\) 0 0
\(767\) 12.7603 0.460749
\(768\) 0 0
\(769\) −17.7748 −0.640975 −0.320488 0.947253i \(-0.603847\pi\)
−0.320488 + 0.947253i \(0.603847\pi\)
\(770\) 0 0
\(771\) 5.14341 19.1867i 0.185235 0.690993i
\(772\) 0 0
\(773\) 17.3928 0.625576 0.312788 0.949823i \(-0.398737\pi\)
0.312788 + 0.949823i \(0.398737\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 17.8874 + 4.79509i 0.641707 + 0.172023i
\(778\) 0 0
\(779\) 6.62299i 0.237293i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −28.6356 28.6551i −1.02335 1.02405i
\(784\) 0 0
\(785\) 17.5214i 0.625367i
\(786\) 0 0
\(787\) −19.0450 −0.678880 −0.339440 0.940628i \(-0.610238\pi\)
−0.339440 + 0.940628i \(0.610238\pi\)
\(788\) 0 0
\(789\) −20.6980 5.54852i −0.736867 0.197533i
\(790\) 0 0
\(791\) −3.00118 −0.106710
\(792\) 0 0
\(793\) −8.39941 −0.298272
\(794\) 0 0
\(795\) −12.3768 3.31785i −0.438959 0.117672i
\(796\) 0 0
\(797\) 41.2333 1.46056 0.730279 0.683149i \(-0.239390\pi\)
0.730279 + 0.683149i \(0.239390\pi\)
\(798\) 0 0
\(799\) 39.5496i 1.39916i
\(800\) 0 0