Properties

Label 1680.2.t.j.1009.4
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.4
Root \(-0.854638 - 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.j.1009.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.26180 q^{11} -0.340173i q^{13} +(0.539189 - 2.17009i) q^{15} +5.75872i q^{17} -6.49693 q^{19} +1.00000 q^{21} -8.49693i q^{23} +(4.41855 + 2.34017i) q^{25} -1.00000i q^{27} +2.00000 q^{29} +8.34017 q^{31} +3.26180i q^{33} +(-0.539189 + 2.17009i) q^{35} -6.15676i q^{37} +0.340173 q^{39} +0.340173 q^{41} -8.68035i q^{43} +(2.17009 + 0.539189i) q^{45} -1.00000 q^{49} -5.75872 q^{51} +8.34017i q^{53} +(-7.07838 - 1.75872i) q^{55} -6.49693i q^{57} +6.83710 q^{59} +15.3607 q^{61} +1.00000i q^{63} +(-0.183417 + 0.738205i) q^{65} -14.8371i q^{67} +8.49693 q^{69} +15.9421 q^{71} +1.50307i q^{73} +(-2.34017 + 4.41855i) q^{75} -3.26180i q^{77} +8.68035 q^{79} +1.00000 q^{81} +6.83710i q^{83} +(3.10504 - 12.4969i) q^{85} +2.00000i q^{87} +15.1773 q^{89} -0.340173 q^{91} +8.34017i q^{93} +(14.0989 + 3.50307i) q^{95} -6.49693i q^{97} -3.26180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} + 28 q^{31} - 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} + 16 q^{51} - 36 q^{55} - 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} + 36 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} + 20 q^{91} + 12 q^{95} - 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.17009 0.539189i −0.970492 0.241133i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.26180 0.983468 0.491734 0.870745i \(-0.336363\pi\)
0.491734 + 0.870745i \(0.336363\pi\)
\(12\) 0 0
\(13\) 0.340173i 0.0943470i −0.998887 0.0471735i \(-0.984979\pi\)
0.998887 0.0471735i \(-0.0150214\pi\)
\(14\) 0 0
\(15\) 0.539189 2.17009i 0.139218 0.560314i
\(16\) 0 0
\(17\) 5.75872i 1.39670i 0.715759 + 0.698348i \(0.246081\pi\)
−0.715759 + 0.698348i \(0.753919\pi\)
\(18\) 0 0
\(19\) −6.49693 −1.49050 −0.745249 0.666786i \(-0.767669\pi\)
−0.745249 + 0.666786i \(0.767669\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 8.49693i 1.77173i −0.463941 0.885866i \(-0.653565\pi\)
0.463941 0.885866i \(-0.346435\pi\)
\(24\) 0 0
\(25\) 4.41855 + 2.34017i 0.883710 + 0.468035i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 8.34017 1.49794 0.748970 0.662604i \(-0.230549\pi\)
0.748970 + 0.662604i \(0.230549\pi\)
\(32\) 0 0
\(33\) 3.26180i 0.567806i
\(34\) 0 0
\(35\) −0.539189 + 2.17009i −0.0911396 + 0.366812i
\(36\) 0 0
\(37\) 6.15676i 1.01216i −0.862485 0.506082i \(-0.831093\pi\)
0.862485 0.506082i \(-0.168907\pi\)
\(38\) 0 0
\(39\) 0.340173 0.0544713
\(40\) 0 0
\(41\) 0.340173 0.0531261 0.0265630 0.999647i \(-0.491544\pi\)
0.0265630 + 0.999647i \(0.491544\pi\)
\(42\) 0 0
\(43\) 8.68035i 1.32374i −0.749618 0.661870i \(-0.769763\pi\)
0.749618 0.661870i \(-0.230237\pi\)
\(44\) 0 0
\(45\) 2.17009 + 0.539189i 0.323497 + 0.0803775i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −5.75872 −0.806383
\(52\) 0 0
\(53\) 8.34017i 1.14561i 0.819691 + 0.572805i \(0.194145\pi\)
−0.819691 + 0.572805i \(0.805855\pi\)
\(54\) 0 0
\(55\) −7.07838 1.75872i −0.954448 0.237146i
\(56\) 0 0
\(57\) 6.49693i 0.860539i
\(58\) 0 0
\(59\) 6.83710 0.890115 0.445057 0.895502i \(-0.353183\pi\)
0.445057 + 0.895502i \(0.353183\pi\)
\(60\) 0 0
\(61\) 15.3607 1.96674 0.983368 0.181627i \(-0.0581363\pi\)
0.983368 + 0.181627i \(0.0581363\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) −0.183417 + 0.738205i −0.0227501 + 0.0915630i
\(66\) 0 0
\(67\) 14.8371i 1.81264i −0.422592 0.906320i \(-0.638880\pi\)
0.422592 0.906320i \(-0.361120\pi\)
\(68\) 0 0
\(69\) 8.49693 1.02291
\(70\) 0 0
\(71\) 15.9421 1.89198 0.945992 0.324190i \(-0.105092\pi\)
0.945992 + 0.324190i \(0.105092\pi\)
\(72\) 0 0
\(73\) 1.50307i 0.175921i 0.996124 + 0.0879606i \(0.0280350\pi\)
−0.996124 + 0.0879606i \(0.971965\pi\)
\(74\) 0 0
\(75\) −2.34017 + 4.41855i −0.270220 + 0.510210i
\(76\) 0 0
\(77\) 3.26180i 0.371716i
\(78\) 0 0
\(79\) 8.68035 0.976615 0.488308 0.872672i \(-0.337614\pi\)
0.488308 + 0.872672i \(0.337614\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.83710i 0.750469i 0.926930 + 0.375235i \(0.122438\pi\)
−0.926930 + 0.375235i \(0.877562\pi\)
\(84\) 0 0
\(85\) 3.10504 12.4969i 0.336789 1.35548i
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 15.1773 1.60879 0.804394 0.594096i \(-0.202490\pi\)
0.804394 + 0.594096i \(0.202490\pi\)
\(90\) 0 0
\(91\) −0.340173 −0.0356598
\(92\) 0 0
\(93\) 8.34017i 0.864836i
\(94\) 0 0
\(95\) 14.0989 + 3.50307i 1.44652 + 0.359408i
\(96\) 0 0
\(97\) 6.49693i 0.659663i −0.944040 0.329832i \(-0.893008\pi\)
0.944040 0.329832i \(-0.106992\pi\)
\(98\) 0 0
\(99\) −3.26180 −0.327823
\(100\) 0 0
\(101\) −2.18342 −0.217258 −0.108629 0.994082i \(-0.534646\pi\)
−0.108629 + 0.994082i \(0.534646\pi\)
\(102\) 0 0
\(103\) 5.84324i 0.575752i −0.957668 0.287876i \(-0.907051\pi\)
0.957668 0.287876i \(-0.0929491\pi\)
\(104\) 0 0
\(105\) −2.17009 0.539189i −0.211779 0.0526194i
\(106\) 0 0
\(107\) 8.18342i 0.791121i 0.918440 + 0.395560i \(0.129450\pi\)
−0.918440 + 0.395560i \(0.870550\pi\)
\(108\) 0 0
\(109\) −16.8371 −1.61270 −0.806351 0.591437i \(-0.798561\pi\)
−0.806351 + 0.591437i \(0.798561\pi\)
\(110\) 0 0
\(111\) 6.15676 0.584373
\(112\) 0 0
\(113\) 13.0205i 1.22487i 0.790522 + 0.612434i \(0.209809\pi\)
−0.790522 + 0.612434i \(0.790191\pi\)
\(114\) 0 0
\(115\) −4.58145 + 18.4391i −0.427222 + 1.71945i
\(116\) 0 0
\(117\) 0.340173i 0.0314490i
\(118\) 0 0
\(119\) 5.75872 0.527901
\(120\) 0 0
\(121\) −0.360692 −0.0327902
\(122\) 0 0
\(123\) 0.340173i 0.0306724i
\(124\) 0 0
\(125\) −8.32684 7.46081i −0.744775 0.667315i
\(126\) 0 0
\(127\) 1.84324i 0.163562i 0.996650 + 0.0817808i \(0.0260607\pi\)
−0.996650 + 0.0817808i \(0.973939\pi\)
\(128\) 0 0
\(129\) 8.68035 0.764262
\(130\) 0 0
\(131\) 0.313511 0.0273916 0.0136958 0.999906i \(-0.495640\pi\)
0.0136958 + 0.999906i \(0.495640\pi\)
\(132\) 0 0
\(133\) 6.49693i 0.563355i
\(134\) 0 0
\(135\) −0.539189 + 2.17009i −0.0464060 + 0.186771i
\(136\) 0 0
\(137\) 2.18342i 0.186542i −0.995641 0.0932710i \(-0.970268\pi\)
0.995641 0.0932710i \(-0.0297323\pi\)
\(138\) 0 0
\(139\) −1.02052 −0.0865593 −0.0432796 0.999063i \(-0.513781\pi\)
−0.0432796 + 0.999063i \(0.513781\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.10957i 0.0927873i
\(144\) 0 0
\(145\) −4.34017 1.07838i −0.360432 0.0895544i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −14.6803 −1.20266 −0.601330 0.799000i \(-0.705362\pi\)
−0.601330 + 0.799000i \(0.705362\pi\)
\(150\) 0 0
\(151\) −2.15676 −0.175514 −0.0877571 0.996142i \(-0.527970\pi\)
−0.0877571 + 0.996142i \(0.527970\pi\)
\(152\) 0 0
\(153\) 5.75872i 0.465565i
\(154\) 0 0
\(155\) −18.0989 4.49693i −1.45374 0.361202i
\(156\) 0 0
\(157\) 6.18342i 0.493490i −0.969080 0.246745i \(-0.920639\pi\)
0.969080 0.246745i \(-0.0793611\pi\)
\(158\) 0 0
\(159\) −8.34017 −0.661419
\(160\) 0 0
\(161\) −8.49693 −0.669652
\(162\) 0 0
\(163\) 4.00000i 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 1.75872 7.07838i 0.136916 0.551051i
\(166\) 0 0
\(167\) 1.47641i 0.114248i −0.998367 0.0571240i \(-0.981807\pi\)
0.998367 0.0571240i \(-0.0181930\pi\)
\(168\) 0 0
\(169\) 12.8843 0.991099
\(170\) 0 0
\(171\) 6.49693 0.496833
\(172\) 0 0
\(173\) 1.75872i 0.133713i −0.997763 0.0668566i \(-0.978703\pi\)
0.997763 0.0668566i \(-0.0212970\pi\)
\(174\) 0 0
\(175\) 2.34017 4.41855i 0.176900 0.334011i
\(176\) 0 0
\(177\) 6.83710i 0.513908i
\(178\) 0 0
\(179\) 0.424694 0.0317431 0.0158716 0.999874i \(-0.494948\pi\)
0.0158716 + 0.999874i \(0.494948\pi\)
\(180\) 0 0
\(181\) 10.3668 0.770561 0.385280 0.922800i \(-0.374105\pi\)
0.385280 + 0.922800i \(0.374105\pi\)
\(182\) 0 0
\(183\) 15.3607i 1.13550i
\(184\) 0 0
\(185\) −3.31965 + 13.3607i −0.244066 + 0.982298i
\(186\) 0 0
\(187\) 18.7838i 1.37361i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 21.7321 1.57248 0.786238 0.617923i \(-0.212026\pi\)
0.786238 + 0.617923i \(0.212026\pi\)
\(192\) 0 0
\(193\) 8.36683i 0.602258i −0.953583 0.301129i \(-0.902637\pi\)
0.953583 0.301129i \(-0.0973635\pi\)
\(194\) 0 0
\(195\) −0.738205 0.183417i −0.0528639 0.0131348i
\(196\) 0 0
\(197\) 13.3340i 0.950010i −0.879983 0.475005i \(-0.842446\pi\)
0.879983 0.475005i \(-0.157554\pi\)
\(198\) 0 0
\(199\) 6.49693 0.460555 0.230278 0.973125i \(-0.426037\pi\)
0.230278 + 0.973125i \(0.426037\pi\)
\(200\) 0 0
\(201\) 14.8371 1.04653
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) −0.738205 0.183417i −0.0515585 0.0128104i
\(206\) 0 0
\(207\) 8.49693i 0.590577i
\(208\) 0 0
\(209\) −21.1917 −1.46586
\(210\) 0 0
\(211\) −6.83710 −0.470685 −0.235343 0.971912i \(-0.575621\pi\)
−0.235343 + 0.971912i \(0.575621\pi\)
\(212\) 0 0
\(213\) 15.9421i 1.09234i
\(214\) 0 0
\(215\) −4.68035 + 18.8371i −0.319197 + 1.28468i
\(216\) 0 0
\(217\) 8.34017i 0.566168i
\(218\) 0 0
\(219\) −1.50307 −0.101568
\(220\) 0 0
\(221\) 1.95896 0.131774
\(222\) 0 0
\(223\) 17.3607i 1.16256i −0.813704 0.581279i \(-0.802553\pi\)
0.813704 0.581279i \(-0.197447\pi\)
\(224\) 0 0
\(225\) −4.41855 2.34017i −0.294570 0.156012i
\(226\) 0 0
\(227\) 10.1568i 0.674128i 0.941482 + 0.337064i \(0.109434\pi\)
−0.941482 + 0.337064i \(0.890566\pi\)
\(228\) 0 0
\(229\) −14.9939 −0.990822 −0.495411 0.868659i \(-0.664983\pi\)
−0.495411 + 0.868659i \(0.664983\pi\)
\(230\) 0 0
\(231\) 3.26180 0.214610
\(232\) 0 0
\(233\) 11.5441i 0.756280i −0.925748 0.378140i \(-0.876564\pi\)
0.925748 0.378140i \(-0.123436\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.68035i 0.563849i
\(238\) 0 0
\(239\) −10.8950 −0.704736 −0.352368 0.935861i \(-0.614624\pi\)
−0.352368 + 0.935861i \(0.614624\pi\)
\(240\) 0 0
\(241\) −6.68035 −0.430319 −0.215159 0.976579i \(-0.569027\pi\)
−0.215159 + 0.976579i \(0.569027\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.17009 + 0.539189i 0.138642 + 0.0344475i
\(246\) 0 0
\(247\) 2.21008i 0.140624i
\(248\) 0 0
\(249\) −6.83710 −0.433284
\(250\) 0 0
\(251\) −24.1978 −1.52735 −0.763676 0.645600i \(-0.776607\pi\)
−0.763676 + 0.645600i \(0.776607\pi\)
\(252\) 0 0
\(253\) 27.7152i 1.74244i
\(254\) 0 0
\(255\) 12.4969 + 3.10504i 0.782588 + 0.194445i
\(256\) 0 0
\(257\) 15.2351i 0.950342i −0.879894 0.475171i \(-0.842386\pi\)
0.879894 0.475171i \(-0.157614\pi\)
\(258\) 0 0
\(259\) −6.15676 −0.382562
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 29.1773i 1.79915i −0.436769 0.899574i \(-0.643877\pi\)
0.436769 0.899574i \(-0.356123\pi\)
\(264\) 0 0
\(265\) 4.49693 18.0989i 0.276244 1.11181i
\(266\) 0 0
\(267\) 15.1773i 0.928834i
\(268\) 0 0
\(269\) 6.13009 0.373758 0.186879 0.982383i \(-0.440163\pi\)
0.186879 + 0.982383i \(0.440163\pi\)
\(270\) 0 0
\(271\) −25.7009 −1.56122 −0.780608 0.625021i \(-0.785091\pi\)
−0.780608 + 0.625021i \(0.785091\pi\)
\(272\) 0 0
\(273\) 0.340173i 0.0205882i
\(274\) 0 0
\(275\) 14.4124 + 7.63317i 0.869101 + 0.460297i
\(276\) 0 0
\(277\) 11.2039i 0.673179i −0.941651 0.336590i \(-0.890726\pi\)
0.941651 0.336590i \(-0.109274\pi\)
\(278\) 0 0
\(279\) −8.34017 −0.499313
\(280\) 0 0
\(281\) 24.3545 1.45287 0.726435 0.687235i \(-0.241176\pi\)
0.726435 + 0.687235i \(0.241176\pi\)
\(282\) 0 0
\(283\) 6.15676i 0.365981i 0.983115 + 0.182991i \(0.0585778\pi\)
−0.983115 + 0.182991i \(0.941422\pi\)
\(284\) 0 0
\(285\) −3.50307 + 14.0989i −0.207504 + 0.835147i
\(286\) 0 0
\(287\) 0.340173i 0.0200798i
\(288\) 0 0
\(289\) −16.1629 −0.950759
\(290\) 0 0
\(291\) 6.49693 0.380857
\(292\) 0 0
\(293\) 1.75872i 0.102746i 0.998680 + 0.0513729i \(0.0163597\pi\)
−0.998680 + 0.0513729i \(0.983640\pi\)
\(294\) 0 0
\(295\) −14.8371 3.68649i −0.863849 0.214636i
\(296\) 0 0
\(297\) 3.26180i 0.189269i
\(298\) 0 0
\(299\) −2.89043 −0.167158
\(300\) 0 0
\(301\) −8.68035 −0.500327
\(302\) 0 0
\(303\) 2.18342i 0.125434i
\(304\) 0 0
\(305\) −33.3340 8.28231i −1.90870 0.474244i
\(306\) 0 0
\(307\) 27.2039i 1.55261i −0.630357 0.776305i \(-0.717092\pi\)
0.630357 0.776305i \(-0.282908\pi\)
\(308\) 0 0
\(309\) 5.84324 0.332411
\(310\) 0 0
\(311\) 21.8432 1.23862 0.619308 0.785148i \(-0.287413\pi\)
0.619308 + 0.785148i \(0.287413\pi\)
\(312\) 0 0
\(313\) 25.3874i 1.43498i 0.696570 + 0.717489i \(0.254709\pi\)
−0.696570 + 0.717489i \(0.745291\pi\)
\(314\) 0 0
\(315\) 0.539189 2.17009i 0.0303799 0.122271i
\(316\) 0 0
\(317\) 22.6947i 1.27466i 0.770590 + 0.637331i \(0.219962\pi\)
−0.770590 + 0.637331i \(0.780038\pi\)
\(318\) 0 0
\(319\) 6.52359 0.365251
\(320\) 0 0
\(321\) −8.18342 −0.456754
\(322\) 0 0
\(323\) 37.4140i 2.08177i
\(324\) 0 0
\(325\) 0.796064 1.50307i 0.0441577 0.0833754i
\(326\) 0 0
\(327\) 16.8371i 0.931094i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.47641 −0.301011 −0.150505 0.988609i \(-0.548090\pi\)
−0.150505 + 0.988609i \(0.548090\pi\)
\(332\) 0 0
\(333\) 6.15676i 0.337388i
\(334\) 0 0
\(335\) −8.00000 + 32.1978i −0.437087 + 1.75915i
\(336\) 0 0
\(337\) 13.6742i 0.744881i 0.928056 + 0.372441i \(0.121479\pi\)
−0.928056 + 0.372441i \(0.878521\pi\)
\(338\) 0 0
\(339\) −13.0205 −0.707178
\(340\) 0 0
\(341\) 27.2039 1.47318
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −18.4391 4.58145i −0.992726 0.246657i
\(346\) 0 0
\(347\) 0.183417i 0.00984637i 0.999988 + 0.00492318i \(0.00156710\pi\)
−0.999988 + 0.00492318i \(0.998433\pi\)
\(348\) 0 0
\(349\) 7.67420 0.410791 0.205395 0.978679i \(-0.434152\pi\)
0.205395 + 0.978679i \(0.434152\pi\)
\(350\) 0 0
\(351\) −0.340173 −0.0181571
\(352\) 0 0
\(353\) 19.4329i 1.03431i 0.855892 + 0.517155i \(0.173009\pi\)
−0.855892 + 0.517155i \(0.826991\pi\)
\(354\) 0 0
\(355\) −34.5958 8.59583i −1.83616 0.456219i
\(356\) 0 0
\(357\) 5.75872i 0.304784i
\(358\) 0 0
\(359\) −25.4186 −1.34154 −0.670770 0.741666i \(-0.734036\pi\)
−0.670770 + 0.741666i \(0.734036\pi\)
\(360\) 0 0
\(361\) 23.2101 1.22158
\(362\) 0 0
\(363\) 0.360692i 0.0189314i
\(364\) 0 0
\(365\) 0.810439 3.26180i 0.0424203 0.170730i
\(366\) 0 0
\(367\) 9.36069i 0.488624i 0.969697 + 0.244312i \(0.0785621\pi\)
−0.969697 + 0.244312i \(0.921438\pi\)
\(368\) 0 0
\(369\) −0.340173 −0.0177087
\(370\) 0 0
\(371\) 8.34017 0.433000
\(372\) 0 0
\(373\) 27.5174i 1.42480i 0.701774 + 0.712400i \(0.252392\pi\)
−0.701774 + 0.712400i \(0.747608\pi\)
\(374\) 0 0
\(375\) 7.46081 8.32684i 0.385275 0.429996i
\(376\) 0 0
\(377\) 0.680346i 0.0350396i
\(378\) 0 0
\(379\) 9.84324 0.505614 0.252807 0.967517i \(-0.418646\pi\)
0.252807 + 0.967517i \(0.418646\pi\)
\(380\) 0 0
\(381\) −1.84324 −0.0944323
\(382\) 0 0
\(383\) 27.8310i 1.42210i 0.703144 + 0.711048i \(0.251779\pi\)
−0.703144 + 0.711048i \(0.748221\pi\)
\(384\) 0 0
\(385\) −1.75872 + 7.07838i −0.0896329 + 0.360748i
\(386\) 0 0
\(387\) 8.68035i 0.441247i
\(388\) 0 0
\(389\) 3.67420 0.186289 0.0931447 0.995653i \(-0.470308\pi\)
0.0931447 + 0.995653i \(0.470308\pi\)
\(390\) 0 0
\(391\) 48.9315 2.47457
\(392\) 0 0
\(393\) 0.313511i 0.0158145i
\(394\) 0 0
\(395\) −18.8371 4.68035i −0.947797 0.235494i
\(396\) 0 0
\(397\) 22.8638i 1.14750i −0.819031 0.573750i \(-0.805488\pi\)
0.819031 0.573750i \(-0.194512\pi\)
\(398\) 0 0
\(399\) −6.49693 −0.325253
\(400\) 0 0
\(401\) 5.31965 0.265651 0.132825 0.991139i \(-0.457595\pi\)
0.132825 + 0.991139i \(0.457595\pi\)
\(402\) 0 0
\(403\) 2.83710i 0.141326i
\(404\) 0 0
\(405\) −2.17009 0.539189i −0.107832 0.0267925i
\(406\) 0 0
\(407\) 20.0821i 0.995432i
\(408\) 0 0
\(409\) −27.7275 −1.37104 −0.685519 0.728055i \(-0.740425\pi\)
−0.685519 + 0.728055i \(0.740425\pi\)
\(410\) 0 0
\(411\) 2.18342 0.107700
\(412\) 0 0
\(413\) 6.83710i 0.336432i
\(414\) 0 0
\(415\) 3.68649 14.8371i 0.180963 0.728325i
\(416\) 0 0
\(417\) 1.02052i 0.0499750i
\(418\) 0 0
\(419\) −23.5174 −1.14890 −0.574451 0.818539i \(-0.694785\pi\)
−0.574451 + 0.818539i \(0.694785\pi\)
\(420\) 0 0
\(421\) −1.15061 −0.0560774 −0.0280387 0.999607i \(-0.508926\pi\)
−0.0280387 + 0.999607i \(0.508926\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −13.4764 + 25.4452i −0.653702 + 1.23427i
\(426\) 0 0
\(427\) 15.3607i 0.743356i
\(428\) 0 0
\(429\) 1.10957 0.0535708
\(430\) 0 0
\(431\) −17.4186 −0.839022 −0.419511 0.907750i \(-0.637798\pi\)
−0.419511 + 0.907750i \(0.637798\pi\)
\(432\) 0 0
\(433\) 26.0144i 1.25017i 0.780556 + 0.625086i \(0.214936\pi\)
−0.780556 + 0.625086i \(0.785064\pi\)
\(434\) 0 0
\(435\) 1.07838 4.34017i 0.0517043 0.208095i
\(436\) 0 0
\(437\) 55.2039i 2.64076i
\(438\) 0 0
\(439\) 5.13624 0.245139 0.122570 0.992460i \(-0.460887\pi\)
0.122570 + 0.992460i \(0.460887\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.5380i 0.880765i 0.897810 + 0.440383i \(0.145157\pi\)
−0.897810 + 0.440383i \(0.854843\pi\)
\(444\) 0 0
\(445\) −32.9360 8.18342i −1.56132 0.387931i
\(446\) 0 0
\(447\) 14.6803i 0.694357i
\(448\) 0 0
\(449\) 10.3135 0.486725 0.243362 0.969935i \(-0.421750\pi\)
0.243362 + 0.969935i \(0.421750\pi\)
\(450\) 0 0
\(451\) 1.10957 0.0522478
\(452\) 0 0
\(453\) 2.15676i 0.101333i
\(454\) 0 0
\(455\) 0.738205 + 0.183417i 0.0346076 + 0.00859874i
\(456\) 0 0
\(457\) 4.36683i 0.204272i 0.994770 + 0.102136i \(0.0325677\pi\)
−0.994770 + 0.102136i \(0.967432\pi\)
\(458\) 0 0
\(459\) 5.75872 0.268794
\(460\) 0 0
\(461\) −33.7009 −1.56961 −0.784803 0.619745i \(-0.787236\pi\)
−0.784803 + 0.619745i \(0.787236\pi\)
\(462\) 0 0
\(463\) 10.4703i 0.486595i −0.969952 0.243297i \(-0.921771\pi\)
0.969952 0.243297i \(-0.0782291\pi\)
\(464\) 0 0
\(465\) 4.49693 18.0989i 0.208540 0.839316i
\(466\) 0 0
\(467\) 3.51745i 0.162768i 0.996683 + 0.0813840i \(0.0259340\pi\)
−0.996683 + 0.0813840i \(0.974066\pi\)
\(468\) 0 0
\(469\) −14.8371 −0.685114
\(470\) 0 0
\(471\) 6.18342 0.284917
\(472\) 0 0
\(473\) 28.3135i 1.30186i
\(474\) 0 0
\(475\) −28.7070 15.2039i −1.31717 0.697604i
\(476\) 0 0
\(477\) 8.34017i 0.381870i
\(478\) 0 0
\(479\) −21.8432 −0.998043 −0.499022 0.866590i \(-0.666307\pi\)
−0.499022 + 0.866590i \(0.666307\pi\)
\(480\) 0 0
\(481\) −2.09436 −0.0954947
\(482\) 0 0
\(483\) 8.49693i 0.386624i
\(484\) 0 0
\(485\) −3.50307 + 14.0989i −0.159066 + 0.640198i
\(486\) 0 0
\(487\) 7.88428i 0.357271i −0.983915 0.178635i \(-0.942832\pi\)
0.983915 0.178635i \(-0.0571683\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −19.7731 −0.892347 −0.446174 0.894946i \(-0.647214\pi\)
−0.446174 + 0.894946i \(0.647214\pi\)
\(492\) 0 0
\(493\) 11.5174i 0.518720i
\(494\) 0 0
\(495\) 7.07838 + 1.75872i 0.318149 + 0.0790488i
\(496\) 0 0
\(497\) 15.9421i 0.715103i
\(498\) 0 0
\(499\) 39.5174 1.76904 0.884522 0.466499i \(-0.154485\pi\)
0.884522 + 0.466499i \(0.154485\pi\)
\(500\) 0 0
\(501\) 1.47641 0.0659611
\(502\) 0 0
\(503\) 11.2039i 0.499559i −0.968303 0.249779i \(-0.919642\pi\)
0.968303 0.249779i \(-0.0803581\pi\)
\(504\) 0 0
\(505\) 4.73820 + 1.17727i 0.210847 + 0.0523880i
\(506\) 0 0
\(507\) 12.8843i 0.572211i
\(508\) 0 0
\(509\) 23.5441 1.04357 0.521787 0.853076i \(-0.325266\pi\)
0.521787 + 0.853076i \(0.325266\pi\)
\(510\) 0 0
\(511\) 1.50307 0.0664920
\(512\) 0 0
\(513\) 6.49693i 0.286846i
\(514\) 0 0
\(515\) −3.15061 + 12.6803i −0.138833 + 0.558763i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.75872 0.0771994
\(520\) 0 0
\(521\) 24.6537 1.08010 0.540049 0.841634i \(-0.318406\pi\)
0.540049 + 0.841634i \(0.318406\pi\)
\(522\) 0 0
\(523\) 2.63931i 0.115409i −0.998334 0.0577044i \(-0.981622\pi\)
0.998334 0.0577044i \(-0.0183781\pi\)
\(524\) 0 0
\(525\) 4.41855 + 2.34017i 0.192841 + 0.102134i
\(526\) 0 0
\(527\) 48.0288i 2.09217i
\(528\) 0 0
\(529\) −49.1978 −2.13903
\(530\) 0 0
\(531\) −6.83710 −0.296705
\(532\) 0 0
\(533\) 0.115718i 0.00501229i
\(534\) 0 0
\(535\) 4.41241 17.7587i 0.190765 0.767777i
\(536\) 0 0
\(537\) 0.424694i 0.0183269i
\(538\) 0 0
\(539\) −3.26180 −0.140495
\(540\) 0 0
\(541\) 25.1506 1.08131 0.540655 0.841245i \(-0.318177\pi\)
0.540655 + 0.841245i \(0.318177\pi\)
\(542\) 0 0
\(543\) 10.3668i 0.444883i
\(544\) 0 0
\(545\) 36.5380 + 9.07838i 1.56511 + 0.388875i
\(546\) 0 0
\(547\) 35.1506i 1.50293i 0.659772 + 0.751466i \(0.270653\pi\)
−0.659772 + 0.751466i \(0.729347\pi\)
\(548\) 0 0
\(549\) −15.3607 −0.655578
\(550\) 0 0
\(551\) −12.9939 −0.553557
\(552\) 0 0
\(553\) 8.68035i 0.369126i
\(554\) 0 0
\(555\) −13.3607 3.31965i −0.567130 0.140911i
\(556\) 0 0
\(557\) 26.3812i 1.11781i 0.829232 + 0.558904i \(0.188778\pi\)
−0.829232 + 0.558904i \(0.811222\pi\)
\(558\) 0 0
\(559\) −2.95282 −0.124891
\(560\) 0 0
\(561\) −18.7838 −0.793052
\(562\) 0 0
\(563\) 12.9939i 0.547626i 0.961783 + 0.273813i \(0.0882849\pi\)
−0.961783 + 0.273813i \(0.911715\pi\)
\(564\) 0 0
\(565\) 7.02052 28.2557i 0.295355 1.18872i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 6.36683 0.266912 0.133456 0.991055i \(-0.457393\pi\)
0.133456 + 0.991055i \(0.457393\pi\)
\(570\) 0 0
\(571\) −2.63931 −0.110452 −0.0552258 0.998474i \(-0.517588\pi\)
−0.0552258 + 0.998474i \(0.517588\pi\)
\(572\) 0 0
\(573\) 21.7321i 0.907870i
\(574\) 0 0
\(575\) 19.8843 37.5441i 0.829232 1.56570i
\(576\) 0 0
\(577\) 33.3340i 1.38771i 0.720113 + 0.693857i \(0.244090\pi\)
−0.720113 + 0.693857i \(0.755910\pi\)
\(578\) 0 0
\(579\) 8.36683 0.347714
\(580\) 0 0
\(581\) 6.83710 0.283651
\(582\) 0 0
\(583\) 27.2039i 1.12667i
\(584\) 0 0
\(585\) 0.183417 0.738205i 0.00758338 0.0305210i
\(586\) 0 0
\(587\) 17.3074i 0.714352i 0.934037 + 0.357176i \(0.116260\pi\)
−0.934037 + 0.357176i \(0.883740\pi\)
\(588\) 0 0
\(589\) −54.1855 −2.23267
\(590\) 0 0
\(591\) 13.3340 0.548489
\(592\) 0 0
\(593\) 47.1194i 1.93496i −0.252943 0.967481i \(-0.581398\pi\)
0.252943 0.967481i \(-0.418602\pi\)
\(594\) 0 0
\(595\) −12.4969 3.10504i −0.512324 0.127294i
\(596\) 0 0
\(597\) 6.49693i 0.265902i
\(598\) 0 0
\(599\) −13.1050 −0.535457 −0.267729 0.963494i \(-0.586273\pi\)
−0.267729 + 0.963494i \(0.586273\pi\)
\(600\) 0 0
\(601\) 29.0349 1.18436 0.592179 0.805806i \(-0.298268\pi\)
0.592179 + 0.805806i \(0.298268\pi\)
\(602\) 0 0
\(603\) 14.8371i 0.604213i
\(604\) 0 0
\(605\) 0.782733 + 0.194481i 0.0318226 + 0.00790678i
\(606\) 0 0
\(607\) 25.9877i 1.05481i −0.849614 0.527404i \(-0.823165\pi\)
0.849614 0.527404i \(-0.176835\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 20.1978i 0.815781i −0.913031 0.407891i \(-0.866264\pi\)
0.913031 0.407891i \(-0.133736\pi\)
\(614\) 0 0
\(615\) 0.183417 0.738205i 0.00739611 0.0297673i
\(616\) 0 0
\(617\) 47.8576i 1.92668i 0.268292 + 0.963338i \(0.413541\pi\)
−0.268292 + 0.963338i \(0.586459\pi\)
\(618\) 0 0
\(619\) −18.3812 −0.738803 −0.369402 0.929270i \(-0.620437\pi\)
−0.369402 + 0.929270i \(0.620437\pi\)
\(620\) 0 0
\(621\) −8.49693 −0.340970
\(622\) 0 0
\(623\) 15.1773i 0.608065i
\(624\) 0 0
\(625\) 14.0472 + 20.6803i 0.561887 + 0.827214i
\(626\) 0 0
\(627\) 21.1917i 0.846313i
\(628\) 0 0
\(629\) 35.4551 1.41369
\(630\) 0 0
\(631\) 7.94668 0.316352 0.158176 0.987411i \(-0.449439\pi\)
0.158176 + 0.987411i \(0.449439\pi\)
\(632\) 0 0
\(633\) 6.83710i 0.271750i
\(634\) 0 0
\(635\) 0.993857 4.00000i 0.0394400 0.158735i
\(636\) 0 0
\(637\) 0.340173i 0.0134781i
\(638\) 0 0
\(639\) −15.9421 −0.630661
\(640\) 0 0
\(641\) 15.7275 0.621200 0.310600 0.950541i \(-0.399470\pi\)
0.310600 + 0.950541i \(0.399470\pi\)
\(642\) 0 0
\(643\) 44.5646i 1.75746i 0.477322 + 0.878729i \(0.341608\pi\)
−0.477322 + 0.878729i \(0.658392\pi\)
\(644\) 0 0
\(645\) −18.8371 4.68035i −0.741710 0.184288i
\(646\) 0 0
\(647\) 5.16290i 0.202974i 0.994837 + 0.101487i \(0.0323601\pi\)
−0.994837 + 0.101487i \(0.967640\pi\)
\(648\) 0 0
\(649\) 22.3012 0.875400
\(650\) 0 0
\(651\) 8.34017 0.326877
\(652\) 0 0
\(653\) 7.49079i 0.293137i 0.989201 + 0.146569i \(0.0468229\pi\)
−0.989201 + 0.146569i \(0.953177\pi\)
\(654\) 0 0
\(655\) −0.680346 0.169042i −0.0265833 0.00660500i
\(656\) 0 0
\(657\) 1.50307i 0.0586404i
\(658\) 0 0
\(659\) −22.7259 −0.885276 −0.442638 0.896700i \(-0.645957\pi\)
−0.442638 + 0.896700i \(0.645957\pi\)
\(660\) 0 0
\(661\) −12.3258 −0.479418 −0.239709 0.970845i \(-0.577052\pi\)
−0.239709 + 0.970845i \(0.577052\pi\)
\(662\) 0 0
\(663\) 1.95896i 0.0760798i
\(664\) 0 0
\(665\) 3.50307 14.0989i 0.135843 0.546732i
\(666\) 0 0
\(667\) 16.9939i 0.658005i
\(668\) 0 0
\(669\) 17.3607 0.671203
\(670\) 0 0
\(671\) 50.1034 1.93422
\(672\) 0 0
\(673\) 25.6742i 0.989668i −0.868988 0.494834i \(-0.835229\pi\)
0.868988 0.494834i \(-0.164771\pi\)
\(674\) 0 0
\(675\) 2.34017 4.41855i 0.0900733 0.170070i
\(676\) 0 0
\(677\) 29.2762i 1.12517i −0.826738 0.562587i \(-0.809806\pi\)
0.826738 0.562587i \(-0.190194\pi\)
\(678\) 0 0
\(679\) −6.49693 −0.249329
\(680\) 0 0
\(681\) −10.1568 −0.389208
\(682\) 0 0
\(683\) 25.5441i 0.977418i −0.872447 0.488709i \(-0.837468\pi\)
0.872447 0.488709i \(-0.162532\pi\)
\(684\) 0 0
\(685\) −1.17727 + 4.73820i −0.0449813 + 0.181037i
\(686\) 0 0
\(687\) 14.9939i 0.572051i
\(688\) 0 0
\(689\) 2.83710 0.108085
\(690\) 0 0
\(691\) −26.3812 −1.00359 −0.501794 0.864987i \(-0.667327\pi\)
−0.501794 + 0.864987i \(0.667327\pi\)
\(692\) 0 0
\(693\) 3.26180i 0.123905i
\(694\) 0 0
\(695\) 2.21461 + 0.550252i 0.0840051 + 0.0208723i
\(696\) 0 0
\(697\) 1.95896i 0.0742010i
\(698\) 0 0
\(699\) 11.5441 0.436638
\(700\) 0 0
\(701\) −23.6742 −0.894162 −0.447081 0.894493i \(-0.647536\pi\)
−0.447081 + 0.894493i \(0.647536\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.18342i 0.0821159i
\(708\) 0 0
\(709\) 40.7214 1.52932 0.764662 0.644432i \(-0.222906\pi\)
0.764662 + 0.644432i \(0.222906\pi\)
\(710\) 0 0
\(711\) −8.68035 −0.325538
\(712\) 0 0
\(713\) 70.8659i 2.65395i
\(714\) 0 0
\(715\) −0.598270 + 2.40787i −0.0223740 + 0.0900493i
\(716\) 0 0
\(717\) 10.8950i 0.406880i
\(718\) 0 0
\(719\) 22.3545 0.833684 0.416842 0.908979i \(-0.363137\pi\)
0.416842 + 0.908979i \(0.363137\pi\)
\(720\) 0 0
\(721\) −5.84324 −0.217614
\(722\) 0 0
\(723\) 6.68035i 0.248445i
\(724\) 0 0
\(725\) 8.83710 + 4.68035i 0.328202 + 0.173824i
\(726\) 0 0
\(727\) 5.10957i 0.189504i −0.995501 0.0947518i \(-0.969794\pi\)
0.995501 0.0947518i \(-0.0302058\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 49.9877 1.84886
\(732\) 0 0
\(733\) 41.6475i 1.53829i 0.639076 + 0.769144i \(0.279317\pi\)
−0.639076 + 0.769144i \(0.720683\pi\)
\(734\) 0 0
\(735\) −0.539189 + 2.17009i −0.0198883 + 0.0800448i
\(736\) 0 0
\(737\) 48.3956i 1.78267i
\(738\) 0 0
\(739\) 32.3135 1.18867 0.594336 0.804217i \(-0.297415\pi\)
0.594336 + 0.804217i \(0.297415\pi\)
\(740\) 0 0
\(741\) −2.21008 −0.0811893
\(742\) 0 0
\(743\) 30.1711i 1.10687i −0.832892 0.553436i \(-0.813316\pi\)
0.832892 0.553436i \(-0.186684\pi\)
\(744\) 0 0
\(745\) 31.8576 + 7.91548i 1.16717 + 0.290001i
\(746\) 0 0
\(747\) 6.83710i 0.250156i
\(748\) 0 0
\(749\) 8.18342 0.299016
\(750\) 0 0
\(751\) 46.1855 1.68533 0.842667 0.538436i \(-0.180985\pi\)
0.842667 + 0.538436i \(0.180985\pi\)
\(752\) 0 0
\(753\) 24.1978i 0.881817i
\(754\) 0 0
\(755\) 4.68035 + 1.16290i 0.170335 + 0.0423222i
\(756\) 0 0
\(757\) 50.5523i 1.83736i −0.395007 0.918678i \(-0.629258\pi\)
0.395007 0.918678i \(-0.370742\pi\)
\(758\) 0 0
\(759\) 27.7152 1.00600
\(760\) 0 0
\(761\) −12.1711 −0.441203 −0.220602 0.975364i \(-0.570802\pi\)
−0.220602 + 0.975364i \(0.570802\pi\)
\(762\) 0 0
\(763\) 16.8371i 0.609544i
\(764\) 0 0
\(765\) −3.10504 + 12.4969i −0.112263 + 0.451827i
\(766\) 0 0
\(767\) 2.32580i 0.0839797i
\(768\) 0 0
\(769\) −7.36069 −0.265433 −0.132717 0.991154i \(-0.542370\pi\)
−0.132717 + 0.991154i \(0.542370\pi\)
\(770\) 0 0
\(771\) 15.2351 0.548680
\(772\) 0 0
\(773\) 10.5548i 0.379629i −0.981820 0.189815i \(-0.939211\pi\)
0.981820 0.189815i \(-0.0607887\pi\)
\(774\) 0 0
\(775\) 36.8515 + 19.5174i 1.32374 + 0.701087i
\(776\) 0 0
\(777\) 6.15676i 0.220872i
\(778\) 0 0
\(779\) −2.21008 −0.0791843
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) −3.33403 + 13.4186i −0.118997 + 0.478929i
\(786\) 0 0
\(787\) 9.04718i 0.322497i 0.986914 + 0.161249i \(0.0515521\pi\)
−0.986914 + 0.161249i \(0.948448\pi\)
\(788\) 0 0
\(789\) 29.1773 1.03874
\(790\) 0 0
\(791\) 13.0205 0.462956
\(792\) 0 0
\(793\) 5.22529i 0.185556i
\(794\) 0 0
\(795\) 18.0989 + 4.49693i 0.641902 + 0.159490i
\(796\) 0 0
\(797\) 16.9627i 0.600848i −0.953806 0.300424i \(-0.902872\pi\)
0.953806 0.300424i \(-0.0971282\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −15.1773 −0.536263
\(802\) 0 0