Properties

Label 180.3.f.f.19.2
Level $180$
Weight $3$
Character 180.19
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{15})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.2
Root \(-1.93649 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 180.19
Dual form 180.3.f.f.19.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.93649 + 0.500000i) q^{2} +(3.50000 - 1.93649i) q^{4} +5.00000i q^{5} +(-5.80948 + 5.50000i) q^{8} +O(q^{10})\) \(q+(-1.93649 + 0.500000i) q^{2} +(3.50000 - 1.93649i) q^{4} +5.00000i q^{5} +(-5.80948 + 5.50000i) q^{8} +(-2.50000 - 9.68246i) q^{10} +(8.50000 - 13.5554i) q^{16} +14.0000i q^{17} +30.9839i q^{19} +(9.68246 + 17.5000i) q^{20} -30.9839 q^{23} -25.0000 q^{25} +61.9677i q^{31} +(-9.68246 + 30.5000i) q^{32} +(-7.00000 - 27.1109i) q^{34} +(-15.4919 - 60.0000i) q^{38} +(-27.5000 - 29.0474i) q^{40} +(60.0000 - 15.4919i) q^{46} +92.9516 q^{47} -49.0000 q^{49} +(48.4123 - 12.5000i) q^{50} -86.0000i q^{53} +118.000 q^{61} +(-30.9839 - 120.000i) q^{62} +(3.50000 - 63.9042i) q^{64} +(27.1109 + 49.0000i) q^{68} +(60.0000 + 108.444i) q^{76} -123.935i q^{79} +(67.7772 + 42.5000i) q^{80} -61.9677 q^{83} -70.0000 q^{85} +(-108.444 + 60.0000i) q^{92} +(-180.000 + 46.4758i) q^{94} -154.919 q^{95} +(94.8881 - 24.5000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{4} + O(q^{10}) \) \( 4q + 14q^{4} - 10q^{10} + 34q^{16} - 100q^{25} - 28q^{34} - 110q^{40} + 240q^{46} - 196q^{49} + 472q^{61} + 14q^{64} + 240q^{76} - 280q^{85} - 720q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93649 + 0.500000i −0.968246 + 0.250000i
\(3\) 0 0
\(4\) 3.50000 1.93649i 0.875000 0.484123i
\(5\) 5.00000i 1.00000i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −5.80948 + 5.50000i −0.726184 + 0.687500i
\(9\) 0 0
\(10\) −2.50000 9.68246i −0.250000 0.968246i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 8.50000 13.5554i 0.531250 0.847215i
\(17\) 14.0000i 0.823529i 0.911290 + 0.411765i \(0.135087\pi\)
−0.911290 + 0.411765i \(0.864913\pi\)
\(18\) 0 0
\(19\) 30.9839i 1.63073i 0.578947 + 0.815365i \(0.303464\pi\)
−0.578947 + 0.815365i \(0.696536\pi\)
\(20\) 9.68246 + 17.5000i 0.484123 + 0.875000i
\(21\) 0 0
\(22\) 0 0
\(23\) −30.9839 −1.34712 −0.673562 0.739130i \(-0.735237\pi\)
−0.673562 + 0.739130i \(0.735237\pi\)
\(24\) 0 0
\(25\) −25.0000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 61.9677i 1.99896i 0.0322581 + 0.999480i \(0.489730\pi\)
−0.0322581 + 0.999480i \(0.510270\pi\)
\(32\) −9.68246 + 30.5000i −0.302577 + 0.953125i
\(33\) 0 0
\(34\) −7.00000 27.1109i −0.205882 0.797379i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −15.4919 60.0000i −0.407682 1.57895i
\(39\) 0 0
\(40\) −27.5000 29.0474i −0.687500 0.726184i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 60.0000 15.4919i 1.30435 0.336781i
\(47\) 92.9516 1.97769 0.988847 0.148936i \(-0.0475849\pi\)
0.988847 + 0.148936i \(0.0475849\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 48.4123 12.5000i 0.968246 0.250000i
\(51\) 0 0
\(52\) 0 0
\(53\) 86.0000i 1.62264i −0.584601 0.811321i \(-0.698749\pi\)
0.584601 0.811321i \(-0.301251\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 118.000 1.93443 0.967213 0.253966i \(-0.0817352\pi\)
0.967213 + 0.253966i \(0.0817352\pi\)
\(62\) −30.9839 120.000i −0.499740 1.93548i
\(63\) 0 0
\(64\) 3.50000 63.9042i 0.0546875 0.998504i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 27.1109 + 49.0000i 0.398689 + 0.720588i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 60.0000 + 108.444i 0.789474 + 1.42689i
\(77\) 0 0
\(78\) 0 0
\(79\) 123.935i 1.56880i −0.620253 0.784402i \(-0.712970\pi\)
0.620253 0.784402i \(-0.287030\pi\)
\(80\) 67.7772 + 42.5000i 0.847215 + 0.531250i
\(81\) 0 0
\(82\) 0 0
\(83\) −61.9677 −0.746599 −0.373300 0.927711i \(-0.621774\pi\)
−0.373300 + 0.927711i \(0.621774\pi\)
\(84\) 0 0
\(85\) −70.0000 −0.823529
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −108.444 + 60.0000i −1.17873 + 0.652174i
\(93\) 0 0
\(94\) −180.000 + 46.4758i −1.91489 + 0.494423i
\(95\) −154.919 −1.63073
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 94.8881 24.5000i 0.968246 0.250000i
\(99\) 0 0
\(100\) −87.5000 + 48.4123i −0.875000 + 0.484123i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 43.0000 + 166.538i 0.405660 + 1.57112i
\(107\) 185.903 1.73741 0.868707 0.495327i \(-0.164952\pi\)
0.868707 + 0.495327i \(0.164952\pi\)
\(108\) 0 0
\(109\) −22.0000 −0.201835 −0.100917 0.994895i \(-0.532178\pi\)
−0.100917 + 0.994895i \(0.532178\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 206.000i 1.82301i 0.411290 + 0.911504i \(0.365078\pi\)
−0.411290 + 0.911504i \(0.634922\pi\)
\(114\) 0 0
\(115\) 154.919i 1.34712i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) −228.506 + 59.0000i −1.87300 + 0.483607i
\(123\) 0 0
\(124\) 120.000 + 216.887i 0.967742 + 1.74909i
\(125\) 125.000i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 25.1744 + 125.500i 0.196675 + 0.980469i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −77.0000 81.3327i −0.566176 0.598034i
\(137\) 226.000i 1.64964i 0.565399 + 0.824818i \(0.308722\pi\)
−0.565399 + 0.824818i \(0.691278\pi\)
\(138\) 0 0
\(139\) 92.9516i 0.668717i −0.942446 0.334358i \(-0.891480\pi\)
0.942446 0.334358i \(-0.108520\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 185.903i 1.23115i −0.788079 0.615574i \(-0.788924\pi\)
0.788079 0.615574i \(-0.211076\pi\)
\(152\) −170.411 180.000i −1.12113 1.18421i
\(153\) 0 0
\(154\) 0 0
\(155\) −309.839 −1.99896
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 61.9677 + 240.000i 0.392201 + 1.51899i
\(159\) 0 0
\(160\) −152.500 48.4123i −0.953125 0.302577i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 120.000 30.9839i 0.722892 0.186650i
\(167\) 216.887 1.29872 0.649362 0.760479i \(-0.275036\pi\)
0.649362 + 0.760479i \(0.275036\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 135.554 35.0000i 0.797379 0.205882i
\(171\) 0 0
\(172\) 0 0
\(173\) 154.000i 0.890173i −0.895487 0.445087i \(-0.853173\pi\)
0.895487 0.445087i \(-0.146827\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −122.000 −0.674033 −0.337017 0.941499i \(-0.609418\pi\)
−0.337017 + 0.941499i \(0.609418\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 180.000 170.411i 0.978261 0.926148i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 325.331 180.000i 1.73048 0.957447i
\(189\) 0 0
\(190\) 300.000 77.4597i 1.57895 0.407682i
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −171.500 + 94.8881i −0.875000 + 0.484123i
\(197\) 374.000i 1.89848i −0.314557 0.949239i \(-0.601856\pi\)
0.314557 0.949239i \(-0.398144\pi\)
\(198\) 0 0
\(199\) 371.806i 1.86837i 0.356784 + 0.934187i \(0.383873\pi\)
−0.356784 + 0.934187i \(0.616127\pi\)
\(200\) 145.237 137.500i 0.726184 0.687500i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 216.887i 1.02790i 0.857820 + 0.513950i \(0.171818\pi\)
−0.857820 + 0.513950i \(0.828182\pi\)
\(212\) −166.538 301.000i −0.785558 1.41981i
\(213\) 0 0
\(214\) −360.000 + 92.9516i −1.68224 + 0.434353i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 42.6028 11.0000i 0.195426 0.0504587i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −103.000 398.917i −0.455752 1.76512i
\(227\) 433.774 1.91090 0.955450 0.295154i \(-0.0953710\pi\)
0.955450 + 0.295154i \(0.0953710\pi\)
\(228\) 0 0
\(229\) −218.000 −0.951965 −0.475983 0.879455i \(-0.657907\pi\)
−0.475983 + 0.879455i \(0.657907\pi\)
\(230\) 77.4597 + 300.000i 0.336781 + 1.30435i
\(231\) 0 0
\(232\) 0 0
\(233\) 34.0000i 0.145923i 0.997335 + 0.0729614i \(0.0232450\pi\)
−0.997335 + 0.0729614i \(0.976755\pi\)
\(234\) 0 0
\(235\) 464.758i 1.97769i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −478.000 −1.98340 −0.991701 0.128564i \(-0.958963\pi\)
−0.991701 + 0.128564i \(0.958963\pi\)
\(242\) −234.315 + 60.5000i −0.968246 + 0.250000i
\(243\) 0 0
\(244\) 413.000 228.506i 1.69262 0.936500i
\(245\) 245.000i 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) −340.823 360.000i −1.37428 1.45161i
\(249\) 0 0
\(250\) 62.5000 + 242.061i 0.250000 + 0.968246i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −111.500 230.443i −0.435547 0.900166i
\(257\) 466.000i 1.81323i 0.421959 + 0.906615i \(0.361343\pi\)
−0.421959 + 0.906615i \(0.638657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −278.855 −1.06028 −0.530142 0.847909i \(-0.677861\pi\)
−0.530142 + 0.847909i \(0.677861\pi\)
\(264\) 0 0
\(265\) 430.000 1.62264
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 247.871i 0.914653i 0.889299 + 0.457326i \(0.151193\pi\)
−0.889299 + 0.457326i \(0.848807\pi\)
\(272\) 189.776 + 119.000i 0.697707 + 0.437500i
\(273\) 0 0
\(274\) −113.000 437.647i −0.412409 1.59725i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 46.4758 + 180.000i 0.167179 + 0.647482i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 93.0000 0.321799
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 394.000i 1.34471i −0.740229 0.672355i \(-0.765283\pi\)
0.740229 0.672355i \(-0.234717\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 92.9516 + 360.000i 0.307787 + 1.19205i
\(303\) 0 0
\(304\) 420.000 + 263.363i 1.38158 + 0.866325i
\(305\) 590.000i 1.93443i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 600.000 154.919i 1.93548 0.499740i
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −240.000 433.774i −0.759494 1.37270i
\(317\) 134.000i 0.422713i 0.977409 + 0.211356i \(0.0677881\pi\)
−0.977409 + 0.211356i \(0.932212\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 319.521 + 17.5000i 0.998504 + 0.0546875i
\(321\) 0 0
\(322\) 0 0
\(323\) −433.774 −1.34295
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 650.661i 1.96574i −0.184290 0.982872i \(-0.558999\pi\)
0.184290 0.982872i \(-0.441001\pi\)
\(332\) −216.887 + 120.000i −0.653274 + 0.361446i
\(333\) 0 0
\(334\) −420.000 + 108.444i −1.25749 + 0.324681i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −327.267 + 84.5000i −0.968246 + 0.250000i
\(339\) 0 0
\(340\) −245.000 + 135.554i −0.720588 + 0.398689i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 77.0000 + 298.220i 0.222543 + 0.861907i
\(347\) −371.806 −1.07149 −0.535744 0.844380i \(-0.679969\pi\)
−0.535744 + 0.844380i \(0.679969\pi\)
\(348\) 0 0
\(349\) 458.000 1.31232 0.656160 0.754621i \(-0.272179\pi\)
0.656160 + 0.754621i \(0.272179\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 274.000i 0.776204i 0.921616 + 0.388102i \(0.126869\pi\)
−0.921616 + 0.388102i \(0.873131\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −599.000 −1.65928
\(362\) 236.252 61.0000i 0.652630 0.168508i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −263.363 + 420.000i −0.715660 + 1.14130i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −540.000 + 511.234i −1.43617 + 1.35966i
\(377\) 0 0
\(378\) 0 0
\(379\) 154.919i 0.408758i 0.978892 + 0.204379i \(0.0655175\pi\)
−0.978892 + 0.204379i \(0.934482\pi\)
\(380\) −542.218 + 300.000i −1.42689 + 0.789474i
\(381\) 0 0
\(382\) 0 0
\(383\) 340.823 0.889876 0.444938 0.895561i \(-0.353226\pi\)
0.444938 + 0.895561i \(0.353226\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 433.774i 1.10940i
\(392\) 284.664 269.500i 0.726184 0.687500i
\(393\) 0 0
\(394\) 187.000 + 724.248i 0.474619 + 1.83819i
\(395\) 619.677 1.56880
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −185.903 720.000i −0.467093 1.80905i
\(399\) 0 0
\(400\) −212.500 + 338.886i −0.531250 + 0.847215i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −142.000 −0.347188 −0.173594 0.984817i \(-0.555538\pi\)
−0.173594 + 0.984817i \(0.555538\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 309.839i 0.746599i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 602.000 1.42993 0.714964 0.699161i \(-0.246443\pi\)
0.714964 + 0.699161i \(0.246443\pi\)
\(422\) −108.444 420.000i −0.256975 0.995261i
\(423\) 0 0
\(424\) 473.000 + 499.615i 1.11557 + 1.17834i
\(425\) 350.000i 0.823529i
\(426\) 0 0
\(427\) 0 0
\(428\) 650.661 360.000i 1.52024 0.841121i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −77.0000 + 42.6028i −0.176606 + 0.0977129i
\(437\) 960.000i 2.19680i
\(438\) 0 0
\(439\) 619.677i 1.41157i −0.708428 0.705783i \(-0.750595\pi\)
0.708428 0.705783i \(-0.249405\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 681.645 1.53870 0.769351 0.638826i \(-0.220580\pi\)
0.769351 + 0.638826i \(0.220580\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 398.917 + 721.000i 0.882560 + 1.59513i
\(453\) 0 0
\(454\) −840.000 + 216.887i −1.85022 + 0.477725i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 422.155 109.000i 0.921736 0.237991i
\(459\) 0 0
\(460\) −300.000 542.218i −0.652174 1.17873i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −17.0000 65.8407i −0.0364807 0.141289i
\(467\) −867.548 −1.85771 −0.928853 0.370450i \(-0.879204\pi\)
−0.928853 + 0.370450i \(0.879204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −232.379 900.000i −0.494423 1.91489i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 774.597i 1.63073i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 925.643 239.000i 1.92042 0.495851i
\(483\) 0 0
\(484\) 423.500 234.315i 0.875000 0.484123i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −685.518 + 649.000i −1.40475 + 1.32992i
\(489\) 0 0
\(490\) 122.500 + 474.440i 0.250000 + 0.968246i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 840.000 + 526.726i 1.69355 + 1.06195i
\(497\) 0 0
\(498\) 0 0
\(499\) 340.823i 0.683011i −0.939880 0.341506i \(-0.889063\pi\)
0.939880 0.341506i \(-0.110937\pi\)
\(500\) −242.061 437.500i −0.484123 0.875000i
\(501\) 0 0
\(502\) 0 0
\(503\) 154.919 0.307991 0.153995 0.988072i \(-0.450786\pi\)
0.153995 + 0.988072i \(0.450786\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 331.140 + 390.500i 0.646758 + 0.762695i
\(513\) 0 0
\(514\) −233.000 902.405i −0.453307 1.75565i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 540.000 139.427i 1.02662 0.265071i
\(527\) −867.548 −1.64620
\(528\) 0 0
\(529\) 431.000 0.814745
\(530\) −832.691 + 215.000i −1.57112 + 0.405660i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 929.516i 1.73741i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1078.00 1.99261 0.996303 0.0859072i \(-0.0273789\pi\)
0.996303 + 0.0859072i \(0.0273789\pi\)
\(542\) −123.935 480.000i −0.228663 0.885609i
\(543\) 0 0
\(544\) −427.000 135.554i −0.784926 0.249181i
\(545\) 110.000i 0.201835i
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 437.647 + 791.000i 0.798626 + 1.44343i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −180.000 325.331i −0.323741 0.585127i
\(557\) 614.000i 1.10233i 0.834395 + 0.551167i \(0.185817\pi\)
−0.834395 + 0.551167i \(0.814183\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1115.42 1.98121 0.990603 0.136767i \(-0.0436713\pi\)
0.990603 + 0.136767i \(0.0436713\pi\)
\(564\) 0 0
\(565\) −1030.00 −1.82301
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 1084.44i 1.89919i 0.313485 + 0.949593i \(0.398503\pi\)
−0.313485 + 0.949593i \(0.601497\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 774.597 1.34712
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −180.094 + 46.5000i −0.311581 + 0.0804498i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 197.000 + 762.978i 0.336177 + 1.30201i
\(587\) −805.581 −1.37237 −0.686184 0.727428i \(-0.740716\pi\)
−0.686184 + 0.727428i \(0.740716\pi\)
\(588\) 0 0
\(589\) −1920.00 −3.25976
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1166.00i 1.96627i 0.182873 + 0.983137i \(0.441460\pi\)
−0.182873 + 0.983137i \(0.558540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −242.000 −0.402662 −0.201331 0.979523i \(-0.564527\pi\)
−0.201331 + 0.979523i \(0.564527\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −360.000 650.661i −0.596026 1.07725i
\(605\) 605.000i 1.00000i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −945.008 300.000i −1.55429 0.493421i
\(609\) 0 0
\(610\) −295.000 1142.53i −0.483607 1.87300i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1186.00i 1.92220i −0.276193 0.961102i \(-0.589073\pi\)
0.276193 0.961102i \(-0.410927\pi\)
\(618\) 0 0
\(619\) 1022.47i 1.65181i 0.563813 + 0.825903i \(0.309334\pi\)
−0.563813 + 0.825903i \(0.690666\pi\)
\(620\) −1084.44 + 600.000i −1.74909 + 0.967742i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1239.35i 1.96411i 0.188590 + 0.982056i \(0.439608\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(632\) 681.645 + 720.000i 1.07855 + 1.13924i
\(633\) 0 0
\(634\) −67.0000 259.490i −0.105678 0.409290i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −627.500 + 125.872i −0.980469 + 0.196675i
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 840.000 216.887i 1.30031 0.335738i
\(647\) −1084.44 −1.67610 −0.838049 0.545595i \(-0.816304\pi\)
−0.838049 + 0.545595i \(0.816304\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1114.00i 1.70597i 0.521933 + 0.852986i \(0.325211\pi\)
−0.521933 + 0.852986i \(0.674789\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 838.000 1.26778 0.633888 0.773425i \(-0.281458\pi\)
0.633888 + 0.773425i \(0.281458\pi\)
\(662\) 325.331 + 1260.00i 0.491436 + 1.90332i
\(663\) 0 0
\(664\) 360.000 340.823i 0.542169 0.513287i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 759.105 420.000i 1.13638 0.628743i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 591.500 327.267i 0.875000 0.484123i
\(677\) 374.000i 0.552437i 0.961095 + 0.276219i \(0.0890814\pi\)
−0.961095 + 0.276219i \(0.910919\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 406.663 385.000i 0.598034 0.566176i
\(681\) 0 0
\(682\) 0 0
\(683\) −1363.29 −1.99603 −0.998016 0.0629575i \(-0.979947\pi\)
−0.998016 + 0.0629575i \(0.979947\pi\)
\(684\) 0 0
\(685\) −1130.00 −1.64964
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 402.790i 0.582909i 0.956585 + 0.291455i \(0.0941392\pi\)
−0.956585 + 0.291455i \(0.905861\pi\)
\(692\) −298.220 539.000i −0.430953 0.778902i
\(693\) 0 0
\(694\) 720.000 185.903i 1.03746 0.267872i
\(695\) 464.758 0.668717
\(696\) 0 0
\(697\) 0 0
\(698\) −886.913 + 229.000i −1.27065 + 0.328080i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −137.000 530.599i −0.194051 0.751556i
\(707\) 0 0
\(708\) 0 0
\(709\) 742.000 1.04654 0.523272 0.852166i \(-0.324711\pi\)
0.523272 + 0.852166i \(0.324711\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1920.00i 2.69285i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1159.96 299.500i 1.60659 0.414820i
\(723\) 0 0
\(724\) −427.000 + 236.252i −0.589779 + 0.326315i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 300.000 945.008i 0.407609 1.28398i
\(737\) 0 0
\(738\) 0 0
\(739\) 216.887i 0.293487i −0.989175 0.146744i \(-0.953121\pi\)
0.989175 0.146744i \(-0.0468792\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1394.27 1.87655 0.938273 0.345895i \(-0.112425\pi\)
0.938273 + 0.345895i \(0.112425\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 433.774i 0.577595i −0.957390 0.288798i \(-0.906745\pi\)
0.957390 0.288798i \(-0.0932555\pi\)
\(752\) 790.089 1260.00i 1.05065 1.67553i
\(753\) 0 0
\(754\) 0 0
\(755\) 929.516 1.23115
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −77.4597 300.000i −0.102190 0.395778i
\(759\) 0 0
\(760\) 900.000 852.056i 1.18421 1.12113i
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −660.000 + 170.411i −0.861619 + 0.222469i
\(767\) 0 0
\(768\) 0 0
\(769\) 578.000 0.751625 0.375813 0.926696i \(-0.377364\pi\)
0.375813 + 0.926696i \(0.377364\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1526.00i 1.97413i −0.160330 0.987063i \(-0.551256\pi\)
0.160330 0.987063i \(-0.448744\pi\)
\(774\) 0 0
\(775\) 1549.19i 1.99896i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 216.887 + 840.000i 0.277349 + 1.07417i
\(783\) 0 0
\(784\) −416.500 + 664.217i −0.531250 + 0.847215i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −724.248 1309.00i −0.919096 1.66117i
\(789\) 0 0
\(790\) −1200.00 + 309.839i −1.51899 + 0.392201i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 720.000 + 1301.32i 0.904523 + 1.63483i
\(797\) 826.000i 1.03639i 0.855264 + 0.518193i \(0.173395\pi\)
−0.855264 + 0.518193i \(0.826605\pi\)
\(798\) 0 0
\(799\) 1301.32i 1.62869i
\(800\) 242.061 762.500i 0.302577 0.953125i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\)