Properties

Label 1440.2.bi.d.847.7
Level $1440$
Weight $2$
Character 1440.847
Analytic conductor $11.498$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.bi (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 847.7
Root \(0.512386 + 1.31813i\) of defining polynomial
Character \(\chi\) \(=\) 1440.847
Dual form 1440.2.bi.d.1423.7

$q$-expansion

\(f(q)\) \(=\) \(q+(1.83051 - 1.28422i) q^{5} +(-2.94984 + 2.94984i) q^{7} +O(q^{10})\) \(q+(1.83051 - 1.28422i) q^{5} +(-2.94984 + 2.94984i) q^{7} +1.61148 q^{11} +(-2.50967 - 2.50967i) q^{13} +(-4.59398 - 4.59398i) q^{17} -4.00000i q^{19} +(-1.09259 - 1.09259i) q^{23} +(1.70156 - 4.70156i) q^{25} +4.75362 q^{29} -5.01934i q^{31} +(-1.61148 + 9.18797i) q^{35} +(2.50967 - 2.50967i) q^{37} +9.18797 q^{41} +(7.40312 - 7.40312i) q^{43} +(-7.32206 + 7.32206i) q^{47} -10.4031i q^{49} +(-3.11473 - 3.11473i) q^{53} +(2.94984 - 2.06950i) q^{55} -1.61148i q^{59} +6.78003i q^{61} +(-7.81695 - 1.37102i) q^{65} +(-7.40312 - 7.40312i) q^{67} +(5.00000 - 5.00000i) q^{73} +(-4.75362 + 4.75362i) q^{77} -5.01934 q^{79} +(7.57648 - 7.57648i) q^{83} +(-14.3090 - 2.50967i) q^{85} +2.74204i q^{89} +14.8062 q^{91} +(-5.13688 - 7.32206i) q^{95} +(2.40312 + 2.40312i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{25} + 16q^{43} - 16q^{67} + 80q^{73} + 32q^{91} - 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.83051 1.28422i 0.818631 0.574320i
\(6\) 0 0
\(7\) −2.94984 + 2.94984i −1.11494 + 1.11494i −0.122462 + 0.992473i \(0.539079\pi\)
−0.992473 + 0.122462i \(0.960921\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.61148 0.485880 0.242940 0.970041i \(-0.421888\pi\)
0.242940 + 0.970041i \(0.421888\pi\)
\(12\) 0 0
\(13\) −2.50967 2.50967i −0.696057 0.696057i 0.267501 0.963558i \(-0.413802\pi\)
−0.963558 + 0.267501i \(0.913802\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.59398 4.59398i −1.11420 1.11420i −0.992576 0.121629i \(-0.961188\pi\)
−0.121629 0.992576i \(-0.538812\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.09259 1.09259i −0.227821 0.227821i 0.583961 0.811782i \(-0.301502\pi\)
−0.811782 + 0.583961i \(0.801502\pi\)
\(24\) 0 0
\(25\) 1.70156 4.70156i 0.340312 0.940312i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.75362 0.882725 0.441362 0.897329i \(-0.354495\pi\)
0.441362 + 0.897329i \(0.354495\pi\)
\(30\) 0 0
\(31\) 5.01934i 0.901500i −0.892650 0.450750i \(-0.851157\pi\)
0.892650 0.450750i \(-0.148843\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.61148 + 9.18797i −0.272390 + 1.55305i
\(36\) 0 0
\(37\) 2.50967 2.50967i 0.412587 0.412587i −0.470052 0.882639i \(-0.655765\pi\)
0.882639 + 0.470052i \(0.155765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.18797 1.43492 0.717460 0.696600i \(-0.245305\pi\)
0.717460 + 0.696600i \(0.245305\pi\)
\(42\) 0 0
\(43\) 7.40312 7.40312i 1.12897 1.12897i 0.138620 0.990346i \(-0.455733\pi\)
0.990346 0.138620i \(-0.0442667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.32206 + 7.32206i −1.06803 + 1.06803i −0.0705213 + 0.997510i \(0.522466\pi\)
−0.997510 + 0.0705213i \(0.977534\pi\)
\(48\) 0 0
\(49\) 10.4031i 1.48616i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.11473 3.11473i −0.427841 0.427841i 0.460051 0.887892i \(-0.347831\pi\)
−0.887892 + 0.460051i \(0.847831\pi\)
\(54\) 0 0
\(55\) 2.94984 2.06950i 0.397756 0.279051i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.61148i 0.209797i −0.994483 0.104899i \(-0.966548\pi\)
0.994483 0.104899i \(-0.0334518\pi\)
\(60\) 0 0
\(61\) 6.78003i 0.868093i 0.900890 + 0.434047i \(0.142915\pi\)
−0.900890 + 0.434047i \(0.857085\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.81695 1.37102i −0.969573 0.170054i
\(66\) 0 0
\(67\) −7.40312 7.40312i −0.904436 0.904436i 0.0913805 0.995816i \(-0.470872\pi\)
−0.995816 + 0.0913805i \(0.970872\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.00000 5.00000i 0.585206 0.585206i −0.351123 0.936329i \(-0.614200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.75362 + 4.75362i −0.541725 + 0.541725i
\(78\) 0 0
\(79\) −5.01934 −0.564720 −0.282360 0.959309i \(-0.591117\pi\)
−0.282360 + 0.959309i \(0.591117\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.57648 7.57648i 0.831627 0.831627i −0.156112 0.987739i \(-0.549896\pi\)
0.987739 + 0.156112i \(0.0498961\pi\)
\(84\) 0 0
\(85\) −14.3090 2.50967i −1.55203 0.272212i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.74204i 0.290655i 0.989384 + 0.145328i \(0.0464236\pi\)
−0.989384 + 0.145328i \(0.953576\pi\)
\(90\) 0 0
\(91\) 14.8062 1.55212
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.13688 7.32206i −0.527032 0.751227i
\(96\) 0 0
\(97\) 2.40312 + 2.40312i 0.244000 + 0.244000i 0.818503 0.574503i \(-0.194804\pi\)
−0.574503 + 0.818503i \(0.694804\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.79790i 0.875424i 0.899115 + 0.437712i \(0.144211\pi\)
−0.899115 + 0.437712i \(0.855789\pi\)
\(102\) 0 0
\(103\) −7.96918 7.96918i −0.785227 0.785227i 0.195481 0.980707i \(-0.437373\pi\)
−0.980707 + 0.195481i \(0.937373\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.61148 + 1.61148i 0.155788 + 0.155788i 0.780697 0.624909i \(-0.214864\pi\)
−0.624909 + 0.780697i \(0.714864\pi\)
\(108\) 0 0
\(109\) −11.7994 −1.13017 −0.565087 0.825031i \(-0.691157\pi\)
−0.565087 + 0.825031i \(0.691157\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.59398 + 4.59398i −0.432166 + 0.432166i −0.889364 0.457199i \(-0.848853\pi\)
0.457199 + 0.889364i \(0.348853\pi\)
\(114\) 0 0
\(115\) −3.40312 0.596876i −0.317343 0.0556590i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 27.1030 2.48453
\(120\) 0 0
\(121\) −8.40312 −0.763920
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.92310 10.7915i −0.261450 0.965217i
\(126\) 0 0
\(127\) −3.83019 + 3.83019i −0.339874 + 0.339874i −0.856320 0.516446i \(-0.827255\pi\)
0.516446 + 0.856320i \(0.327255\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.5415 −1.18313 −0.591563 0.806259i \(-0.701489\pi\)
−0.591563 + 0.806259i \(0.701489\pi\)
\(132\) 0 0
\(133\) 11.7994 + 11.7994i 1.02313 + 1.02313i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0399 + 11.0399i 0.943203 + 0.943203i 0.998472 0.0552681i \(-0.0176013\pi\)
−0.0552681 + 0.998472i \(0.517601\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.04429 4.04429i −0.338200 0.338200i
\(144\) 0 0
\(145\) 8.70156 6.10469i 0.722625 0.506967i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.79790 0.720752 0.360376 0.932807i \(-0.382648\pi\)
0.360376 + 0.932807i \(0.382648\pi\)
\(150\) 0 0
\(151\) 0.880344i 0.0716414i −0.999358 0.0358207i \(-0.988595\pi\)
0.999358 0.0358207i \(-0.0114045\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.44593 9.18797i −0.517750 0.737995i
\(156\) 0 0
\(157\) 9.28970 9.28970i 0.741398 0.741398i −0.231449 0.972847i \(-0.574347\pi\)
0.972847 + 0.231449i \(0.0743465\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.44593 0.508010
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.32206 7.32206i 0.566598 0.566598i −0.364576 0.931174i \(-0.618786\pi\)
0.931174 + 0.364576i \(0.118786\pi\)
\(168\) 0 0
\(169\) 0.403124i 0.0310096i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.86835 7.86835i −0.598220 0.598220i 0.341619 0.939839i \(-0.389025\pi\)
−0.939839 + 0.341619i \(0.889025\pi\)
\(174\) 0 0
\(175\) 8.84952 + 18.8882i 0.668961 + 1.42781i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 26.4333i 1.97572i −0.155343 0.987861i \(-0.549648\pi\)
0.155343 0.987861i \(-0.450352\pi\)
\(180\) 0 0
\(181\) 11.7994i 0.877040i −0.898721 0.438520i \(-0.855503\pi\)
0.898721 0.438520i \(-0.144497\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.37102 7.81695i 0.100799 0.574713i
\(186\) 0 0
\(187\) −7.40312 7.40312i −0.541370 0.541370i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0145i 1.37584i 0.725787 + 0.687919i \(0.241476\pi\)
−0.725787 + 0.687919i \(0.758524\pi\)
\(192\) 0 0
\(193\) −12.4031 + 12.4031i −0.892796 + 0.892796i −0.994786 0.101989i \(-0.967479\pi\)
0.101989 + 0.994786i \(0.467479\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.34420 + 9.34420i −0.665747 + 0.665747i −0.956729 0.290982i \(-0.906018\pi\)
0.290982 + 0.956729i \(0.406018\pi\)
\(198\) 0 0
\(199\) 20.9577 1.48565 0.742826 0.669485i \(-0.233485\pi\)
0.742826 + 0.669485i \(0.233485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.0224 + 14.0224i −0.984181 + 0.984181i
\(204\) 0 0
\(205\) 16.8187 11.7994i 1.17467 0.824103i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.44593i 0.445874i
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.04429 23.0588i 0.275818 1.57259i
\(216\) 0 0
\(217\) 14.8062 + 14.8062i 1.00511 + 1.00511i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.0588i 1.55110i
\(222\) 0 0
\(223\) −3.83019 3.83019i −0.256488 0.256488i 0.567136 0.823624i \(-0.308051\pi\)
−0.823624 + 0.567136i \(0.808051\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.6339 + 15.6339i 1.03766 + 1.03766i 0.999263 + 0.0383956i \(0.0122247\pi\)
0.0383956 + 0.999263i \(0.487775\pi\)
\(228\) 0 0
\(229\) 6.78003 0.448037 0.224018 0.974585i \(-0.428082\pi\)
0.224018 + 0.974585i \(0.428082\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.0399 + 11.0399i −0.723249 + 0.723249i −0.969266 0.246017i \(-0.920878\pi\)
0.246017 + 0.969266i \(0.420878\pi\)
\(234\) 0 0
\(235\) −4.00000 + 22.8062i −0.260931 + 1.48772i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.08857 −0.523206 −0.261603 0.965176i \(-0.584251\pi\)
−0.261603 + 0.965176i \(0.584251\pi\)
\(240\) 0 0
\(241\) −6.80625 −0.438429 −0.219215 0.975677i \(-0.570349\pi\)
−0.219215 + 0.975677i \(0.570349\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −13.3599 19.0431i −0.853532 1.21662i
\(246\) 0 0
\(247\) −10.0387 + 10.0387i −0.638746 + 0.638746i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.9874 1.26159 0.630797 0.775948i \(-0.282728\pi\)
0.630797 + 0.775948i \(0.282728\pi\)
\(252\) 0 0
\(253\) −1.76069 1.76069i −0.110694 0.110694i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.7820 + 13.7820i 0.859694 + 0.859694i 0.991302 0.131607i \(-0.0420138\pi\)
−0.131607 + 0.991302i \(0.542014\pi\)
\(258\) 0 0
\(259\) 14.8062i 0.920016i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.95170 + 2.95170i 0.182009 + 0.182009i 0.792231 0.610221i \(-0.208920\pi\)
−0.610221 + 0.792231i \(0.708920\pi\)
\(264\) 0 0
\(265\) −9.70156 1.70156i −0.595962 0.104526i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.3051 1.11608 0.558042 0.829813i \(-0.311553\pi\)
0.558042 + 0.829813i \(0.311553\pi\)
\(270\) 0 0
\(271\) 12.6797i 0.770237i −0.922867 0.385119i \(-0.874160\pi\)
0.922867 0.385119i \(-0.125840\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.74204 7.57648i 0.165351 0.456879i
\(276\) 0 0
\(277\) −7.52901 + 7.52901i −0.452374 + 0.452374i −0.896142 0.443768i \(-0.853642\pi\)
0.443768 + 0.896142i \(0.353642\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −27.5639 −1.64432 −0.822162 0.569253i \(-0.807232\pi\)
−0.822162 + 0.569253i \(0.807232\pi\)
\(282\) 0 0
\(283\) −7.40312 + 7.40312i −0.440070 + 0.440070i −0.892035 0.451965i \(-0.850723\pi\)
0.451965 + 0.892035i \(0.350723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −27.1030 + 27.1030i −1.59984 + 1.59984i
\(288\) 0 0
\(289\) 25.2094i 1.48290i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.11473 + 3.11473i 0.181965 + 0.181965i 0.792211 0.610247i \(-0.208930\pi\)
−0.610247 + 0.792211i \(0.708930\pi\)
\(294\) 0 0
\(295\) −2.06950 2.94984i −0.120491 0.171746i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.48408i 0.317152i
\(300\) 0 0
\(301\) 43.6761i 2.51745i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.70704 + 12.4109i 0.498564 + 0.710648i
\(306\) 0 0
\(307\) 20.0000 + 20.0000i 1.14146 + 1.14146i 0.988183 + 0.153277i \(0.0489827\pi\)
0.153277 + 0.988183i \(0.451017\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.08857i 0.458661i 0.973349 + 0.229330i \(0.0736537\pi\)
−0.973349 + 0.229330i \(0.926346\pi\)
\(312\) 0 0
\(313\) −19.8062 + 19.8062i −1.11952 + 1.11952i −0.127703 + 0.991812i \(0.540760\pi\)
−0.991812 + 0.127703i \(0.959240\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.59058 + 4.59058i −0.257833 + 0.257833i −0.824172 0.566339i \(-0.808359\pi\)
0.566339 + 0.824172i \(0.308359\pi\)
\(318\) 0 0
\(319\) 7.66037 0.428898
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −18.3759 + 18.3759i −1.02246 + 1.02246i
\(324\) 0 0
\(325\) −16.0697 + 7.52901i −0.891388 + 0.417634i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 43.1978i 2.38157i
\(330\) 0 0
\(331\) −2.80625 −0.154245 −0.0771227 0.997022i \(-0.524573\pi\)
−0.0771227 + 0.997022i \(0.524573\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −23.0588 4.04429i −1.25983 0.220963i
\(336\) 0 0
\(337\) −2.40312 2.40312i −0.130907 0.130907i 0.638618 0.769524i \(-0.279507\pi\)
−0.769524 + 0.638618i \(0.779507\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.08857i 0.438021i
\(342\) 0 0
\(343\) 10.0387 + 10.0387i 0.542038 + 0.542038i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.74204 2.74204i −0.147200 0.147200i 0.629666 0.776866i \(-0.283192\pi\)
−0.776866 + 0.629666i \(0.783192\pi\)
\(348\) 0 0
\(349\) −6.78003 −0.362926 −0.181463 0.983398i \(-0.558083\pi\)
−0.181463 + 0.983398i \(0.558083\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.7820 13.7820i 0.733539 0.733539i −0.237780 0.971319i \(-0.576420\pi\)
0.971319 + 0.237780i \(0.0764197\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 35.1916 1.85734 0.928671 0.370904i \(-0.120952\pi\)
0.928671 + 0.370904i \(0.120952\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.73147 15.5737i 0.142972 0.815163i
\(366\) 0 0
\(367\) 14.7492 14.7492i 0.769902 0.769902i −0.208187 0.978089i \(-0.566756\pi\)
0.978089 + 0.208187i \(0.0667562\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18.3759 0.954031
\(372\) 0 0
\(373\) 14.3090 + 14.3090i 0.740894 + 0.740894i 0.972750 0.231856i \(-0.0744799\pi\)
−0.231856 + 0.972750i \(0.574480\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −11.9300 11.9300i −0.614427 0.614427i
\(378\) 0 0
\(379\) 18.8062i 0.966012i 0.875617 + 0.483006i \(0.160455\pi\)
−0.875617 + 0.483006i \(0.839545\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −26.0105 26.0105i −1.32907 1.32907i −0.906180 0.422892i \(-0.861015\pi\)
−0.422892 0.906180i \(-0.638985\pi\)
\(384\) 0 0
\(385\) −2.59688 + 14.8062i −0.132349 + 0.754596i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 37.3196 1.89218 0.946090 0.323905i \(-0.104996\pi\)
0.946090 + 0.323905i \(0.104996\pi\)
\(390\) 0 0
\(391\) 10.0387i 0.507678i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.18797 + 6.44593i −0.462297 + 0.324330i
\(396\) 0 0
\(397\) 19.3284 19.3284i 0.970063 0.970063i −0.0295016 0.999565i \(-0.509392\pi\)
0.999565 + 0.0295016i \(0.00939202\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.8919 0.643789 0.321894 0.946776i \(-0.395680\pi\)
0.321894 + 0.946776i \(0.395680\pi\)
\(402\) 0 0
\(403\) −12.5969 + 12.5969i −0.627495 + 0.627495i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.04429 4.04429i 0.200468 0.200468i
\(408\) 0 0
\(409\) 3.40312i 0.168274i 0.996454 + 0.0841368i \(0.0268133\pi\)
−0.996454 + 0.0841368i \(0.973187\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.75362 + 4.75362i 0.233910 + 0.233910i
\(414\) 0 0
\(415\) 4.13899 23.5987i 0.203175 1.15842i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.9174i 1.55927i 0.626235 + 0.779634i \(0.284595\pi\)
−0.626235 + 0.779634i \(0.715405\pi\)
\(420\) 0 0
\(421\) 30.3788i 1.48057i −0.672293 0.740285i \(-0.734691\pi\)
0.672293 0.740285i \(-0.265309\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −29.4158 + 13.7820i −1.42688 + 0.668523i
\(426\) 0 0
\(427\) −20.0000 20.0000i −0.967868 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.1030i 1.30551i 0.757570 + 0.652754i \(0.226386\pi\)
−0.757570 + 0.652754i \(0.773614\pi\)
\(432\) 0 0
\(433\) 12.4031 12.4031i 0.596056 0.596056i −0.343205 0.939261i \(-0.611512\pi\)
0.939261 + 0.343205i \(0.111512\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.37036 + 4.37036i −0.209063 + 0.209063i
\(438\) 0 0
\(439\) −10.9190 −0.521136 −0.260568 0.965455i \(-0.583910\pi\)
−0.260568 + 0.965455i \(0.583910\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.6339 15.6339i 0.742789 0.742789i −0.230325 0.973114i \(-0.573979\pi\)
0.973114 + 0.230325i \(0.0739789\pi\)
\(444\) 0 0
\(445\) 3.52138 + 5.01934i 0.166929 + 0.237939i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.44593i 0.304202i −0.988365 0.152101i \(-0.951396\pi\)
0.988365 0.152101i \(-0.0486039\pi\)
\(450\) 0 0
\(451\) 14.8062 0.697199
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 27.1030 19.0145i 1.27061 0.891412i
\(456\) 0 0
\(457\) −9.80625 9.80625i −0.458717 0.458717i 0.439517 0.898234i \(-0.355150\pi\)
−0.898234 + 0.439517i \(0.855150\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.3051i 0.852555i −0.904592 0.426278i \(-0.859825\pi\)
0.904592 0.426278i \(-0.140175\pi\)
\(462\) 0 0
\(463\) 14.7492 + 14.7492i 0.685454 + 0.685454i 0.961224 0.275770i \(-0.0889328\pi\)
−0.275770 + 0.961224i \(0.588933\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.7994 10.7994i −0.499739 0.499739i 0.411618 0.911357i \(-0.364964\pi\)
−0.911357 + 0.411618i \(0.864964\pi\)
\(468\) 0 0
\(469\) 43.6761 2.01677
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.9300 11.9300i 0.548542 0.548542i
\(474\) 0 0
\(475\) −18.8062 6.80625i −0.862890 0.312292i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.08857 −0.369576 −0.184788 0.982778i \(-0.559160\pi\)
−0.184788 + 0.982778i \(0.559160\pi\)
\(480\) 0 0
\(481\) −12.5969 −0.574368
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.48509 + 1.31281i 0.339880 + 0.0596118i
\(486\) 0 0
\(487\) 2.06950 2.06950i 0.0937778 0.0937778i −0.658662 0.752439i \(-0.728877\pi\)
0.752439 + 0.658662i \(0.228877\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.2804 0.509076 0.254538 0.967063i \(-0.418077\pi\)
0.254538 + 0.967063i \(0.418077\pi\)
\(492\) 0 0
\(493\) −21.8380 21.8380i −0.983536 0.983536i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18.8062i 0.841883i −0.907088 0.420942i \(-0.861700\pi\)
0.907088 0.420942i \(-0.138300\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.13688 + 5.13688i 0.229042 + 0.229042i 0.812292 0.583250i \(-0.198219\pi\)
−0.583250 + 0.812292i \(0.698219\pi\)
\(504\) 0 0
\(505\) 11.2984 + 16.1047i 0.502774 + 0.716649i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.8422 0.569220 0.284610 0.958643i \(-0.408136\pi\)
0.284610 + 0.958643i \(0.408136\pi\)
\(510\) 0 0
\(511\) 29.4984i 1.30493i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.8219 4.35352i −1.09378 0.191839i
\(516\) 0 0
\(517\) −11.7994 + 11.7994i −0.518935 + 0.518935i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.44593 0.282401 0.141201 0.989981i \(-0.454904\pi\)
0.141201 + 0.989981i \(0.454904\pi\)
\(522\) 0 0
\(523\) 5.19375 5.19375i 0.227107 0.227107i −0.584376 0.811483i \(-0.698661\pi\)
0.811483 + 0.584376i \(0.198661\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −23.0588 + 23.0588i −1.00446 + 1.00446i
\(528\) 0 0
\(529\) 20.6125i 0.896196i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −23.0588 23.0588i −0.998786 0.998786i
\(534\) 0 0
\(535\) 5.01934 + 0.880344i 0.217005 + 0.0380606i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 16.7645i 0.722096i
\(540\) 0 0
\(541\) 8.27799i 0.355898i −0.984040 0.177949i \(-0.943054\pi\)
0.984040 0.177949i \(-0.0569463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −21.5989 + 15.1530i −0.925195 + 0.649082i
\(546\) 0 0
\(547\) 22.2094 + 22.2094i 0.949604 + 0.949604i 0.998790 0.0491855i \(-0.0156625\pi\)
−0.0491855 + 0.998790i \(0.515663\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.0145i 0.810044i
\(552\) 0 0
\(553\) 14.8062 14.8062i 0.629626 0.629626i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.7146 + 13.7146i −0.581104 + 0.581104i −0.935207 0.354102i \(-0.884786\pi\)
0.354102 + 0.935207i \(0.384786\pi\)
\(558\) 0 0
\(559\) −37.1588 −1.57165
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.74204 2.74204i 0.115563 0.115563i −0.646960 0.762524i \(-0.723960\pi\)
0.762524 + 0.646960i \(0.223960\pi\)
\(564\) 0 0
\(565\) −2.50967 + 14.3090i −0.105583 + 0.601986i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 36.7519i 1.54072i 0.637610 + 0.770359i \(0.279923\pi\)
−0.637610 + 0.770359i \(0.720077\pi\)
\(570\) 0 0
\(571\) −17.6125 −0.737060 −0.368530 0.929616i \(-0.620139\pi\)
−0.368530 + 0.929616i \(0.620139\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.99599 + 3.27777i −0.291753 + 0.136692i
\(576\) 0 0
\(577\) 12.4031 + 12.4031i 0.516349 + 0.516349i 0.916465 0.400116i \(-0.131030\pi\)
−0.400116 + 0.916465i \(0.631030\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 44.6989i 1.85442i
\(582\) 0 0
\(583\) −5.01934 5.01934i −0.207880 0.207880i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.61148 1.61148i −0.0665130 0.0665130i 0.673068 0.739581i \(-0.264976\pi\)
−0.739581 + 0.673068i \(0.764976\pi\)
\(588\) 0 0
\(589\) −20.0774 −0.827273
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.59398 4.59398i 0.188652 0.188652i −0.606461 0.795113i \(-0.707411\pi\)
0.795113 + 0.606461i \(0.207411\pi\)
\(594\) 0 0
\(595\) 49.6125 34.8062i 2.03391 1.42692i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.1030 1.10740 0.553700 0.832716i \(-0.313215\pi\)
0.553700 + 0.832716i \(0.313215\pi\)
\(600\) 0 0
\(601\) 20.2094 0.824358 0.412179 0.911103i \(-0.364768\pi\)
0.412179 + 0.911103i \(0.364768\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.3820 + 10.7915i −0.625369 + 0.438735i
\(606\) 0 0
\(607\) −18.8882 + 18.8882i −0.766648 + 0.766648i −0.977515 0.210867i \(-0.932371\pi\)
0.210867 + 0.977515i \(0.432371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 36.7519 1.48682
\(612\) 0 0
\(613\) −19.3284 19.3284i −0.780666 0.780666i 0.199278 0.979943i \(-0.436140\pi\)
−0.979943 + 0.199278i \(0.936140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.33602 + 7.33602i 0.295337 + 0.295337i 0.839184 0.543847i \(-0.183033\pi\)
−0.543847 + 0.839184i \(0.683033\pi\)
\(618\) 0 0
\(619\) 10.8062i 0.434340i −0.976134 0.217170i \(-0.930317\pi\)
0.976134 0.217170i \(-0.0696826\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.08857 8.08857i −0.324062 0.324062i
\(624\) 0 0
\(625\) −19.2094 16.0000i −0.768375 0.640000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −23.0588 −0.919413
\(630\) 0 0
\(631\) 25.0967i 0.999083i −0.866290 0.499542i \(-0.833502\pi\)
0.866290 0.499542i \(-0.166498\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.09241 + 11.9300i −0.0830348 + 0.473428i
\(636\) 0 0
\(637\) −26.1084 + 26.1084i −1.03445 + 1.03445i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.74204 0.108304 0.0541520 0.998533i \(-0.482754\pi\)
0.0541520 + 0.998533i \(0.482754\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.7810 19.7810i 0.777671 0.777671i −0.201763 0.979434i \(-0.564667\pi\)
0.979434 + 0.201763i \(0.0646672\pi\)
\(648\) 0 0
\(649\) 2.59688i 0.101936i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.1904 15.1904i −0.594447 0.594447i 0.344383 0.938829i \(-0.388088\pi\)
−0.938829 + 0.344383i \(0.888088\pi\)
\(654\) 0 0
\(655\) −24.7879 + 17.3902i −0.968543 + 0.679493i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.83445i 0.188323i −0.995557 0.0941617i \(-0.969983\pi\)
0.995557 0.0941617i \(-0.0300171\pi\)
\(660\) 0 0
\(661\) 26.8574i 1.04463i −0.852752 0.522315i \(-0.825068\pi\)
0.852752 0.522315i \(-0.174932\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 36.7519 + 6.44593i 1.42518 + 0.249962i
\(666\) 0 0
\(667\) −5.19375 5.19375i −0.201103 0.201103i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.9259i 0.421789i
\(672\) 0 0
\(673\) 15.0000 15.0000i 0.578208 0.578208i −0.356202 0.934409i \(-0.615928\pi\)
0.934409 + 0.356202i \(0.115928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.1421 + 18.1421i −0.697258 + 0.697258i −0.963818 0.266560i \(-0.914113\pi\)
0.266560 + 0.963818i \(0.414113\pi\)
\(678\) 0 0
\(679\) −14.1777 −0.544089
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.13056 1.13056i 0.0432595 0.0432595i −0.685146 0.728406i \(-0.740262\pi\)
0.728406 + 0.685146i \(0.240262\pi\)
\(684\) 0 0
\(685\) 34.3864 + 6.03105i 1.31384 + 0.230434i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.6339i 0.595604i
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.13688 + 7.32206i 0.194853 + 0.277741i
\(696\) 0 0
\(697\) −42.2094 42.2094i −1.59879 1.59879i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.33496i 0.125960i 0.998015 + 0.0629798i \(0.0200604\pi\)
−0.998015 + 0.0629798i \(0.979940\pi\)
\(702\) 0 0
\(703\) −10.0387 10.0387i −0.378616 0.378616i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.9524 25.9524i −0.976041 0.976041i
\(708\) 0 0
\(709\) −26.8574 −1.00865 −0.504325 0.863514i \(-0.668259\pi\)
−0.504325 + 0.863514i \(0.668259\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.48408 + 5.48408i −0.205380 + 0.205380i
\(714\) 0 0
\(715\) −12.5969 2.20937i −0.471096 0.0826259i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.08857 −0.301653 −0.150826 0.988560i \(-0.548193\pi\)
−0.150826 + 0.988560i \(0.548193\pi\)
\(720\) 0 0
\(721\) 47.0156 1.75095
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.08857 22.3494i 0.300402 0.830037i
\(726\) 0 0
\(727\) 35.7069 35.7069i 1.32430 1.32430i 0.414034 0.910261i \(-0.364119\pi\)
0.910261 0.414034i \(-0.135881\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −68.0197 −2.51580
\(732\) 0 0
\(733\) −2.50967 2.50967i −0.0926967 0.0926967i 0.659238 0.751935i \(-0.270879\pi\)
−0.751935 + 0.659238i \(0.770879\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.9300 11.9300i −0.439447 0.439447i
\(738\) 0 0
\(739\) 48.4187i 1.78111i −0.454873 0.890556i \(-0.650315\pi\)
0.454873 0.890556i \(-0.349685\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.18116 + 9.18116i 0.336824 + 0.336824i 0.855171 0.518346i \(-0.173452\pi\)
−0.518346 + 0.855171i \(0.673452\pi\)
\(744\) 0 0
\(745\) 16.1047 11.2984i 0.590030 0.413943i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.50723 −0.347387
\(750\) 0 0
\(751\) 15.0580i 0.549475i −0.961519 0.274737i \(-0.911409\pi\)
0.961519 0.274737i \(-0.0885909\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.13056 1.61148i −0.0411451 0.0586479i
\(756\) 0 0
\(757\) −11.0504 + 11.0504i −0.401633 + 0.401633i −0.878808 0.477175i \(-0.841661\pi\)
0.477175 + 0.878808i \(0.341661\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.1978 1.56592 0.782960 0.622073i \(-0.213709\pi\)
0.782960 + 0.622073i \(0.213709\pi\)
\(762\) 0 0
\(763\) 34.8062 34.8062i 1.26007 1.26007i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.04429 + 4.04429i −0.146031 + 0.146031i
\(768\) 0 0
\(769\) 1.40312i 0.0505980i 0.999680 + 0.0252990i \(0.00805377\pi\)
−0.999680 + 0.0252990i \(0.991946\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.4642 + 25.4642i 0.915882 + 0.915882i 0.996727 0.0808446i \(-0.0257617\pi\)
−0.0808446 + 0.996727i \(0.525762\pi\)
\(774\) 0 0
\(775\) −23.5987 8.54071i −0.847691 0.306792i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 36.7519i 1.31677i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.07491 28.9349i 0.181131 1.03273i
\(786\) 0 0
\(787\) 20.0000 + 20.0000i 0.712923 + 0.712923i 0.967146 0.254223i \(-0.0818196\pi\)
−0.254223 + 0.967146i \(0.581820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.1030i 0.963673i
\(792\) 0 0
\(793\) 17.0156 17.0156i 0.604242 0.604242i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.00924 6.00924i 0.212858 0.212858i −0.592622 0.805481i \(-0.701907\pi\)
0.805481 + 0.592622i \(0.201907\pi\)
\(798\) 0 0
\(799\) 67.2748 2.38001
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.05741 8.05741i 0.284340 0.284340i
\(804\) 0 0
\(805\) 11.7994 8.27799i 0.415873 0.291761i