Properties

Label 1620.3.o.f.701.1
Level $1620$
Weight $3$
Character 1620.701
Analytic conductor $44.142$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 701.1
Root \(-1.40294 + 1.01575i\) of defining polynomial
Character \(\chi\) \(=\) 1620.701
Dual form 1620.3.o.f.1241.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.93649 - 1.11803i) q^{5} +(-2.74342 - 4.75174i) q^{7} +O(q^{10})\) \(q+(-1.93649 - 1.11803i) q^{5} +(-2.74342 - 4.75174i) q^{7} +(7.94472 - 4.58688i) q^{11} +(-5.74342 + 9.94789i) q^{13} +16.9706i q^{17} +26.9737 q^{19} +(-4.27048 - 2.46556i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-17.7749 + 10.2624i) q^{29} +(-10.4868 + 18.1637i) q^{31} +12.2689i q^{35} -62.4605 q^{37} +(35.4531 + 20.4689i) q^{41} +(-0.513167 - 0.888831i) q^{43} +(74.6772 - 43.1149i) q^{47} +(9.44733 - 16.3633i) q^{49} -96.0920i q^{53} -20.5132 q^{55} +(97.3188 + 56.1871i) q^{59} +(33.4605 + 57.9553i) q^{61} +(22.2442 - 12.8427i) q^{65} +(38.0000 - 65.8179i) q^{67} -24.0789i q^{71} -18.9210 q^{73} +(-43.5913 - 25.1675i) q^{77} +(-53.4605 - 92.5963i) q^{79} +(39.1273 - 22.5902i) q^{83} +(18.9737 - 32.8634i) q^{85} -115.928i q^{89} +63.0263 q^{91} +(-52.2343 - 30.1575i) q^{95} +(43.5132 + 75.3670i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) −1.93649 1.11803i −0.387298 0.223607i
\(6\) 0 0
\(7\) −2.74342 4.75174i −0.391917 0.678820i 0.600786 0.799410i \(-0.294855\pi\)
−0.992702 + 0.120591i \(0.961521\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 7.94472 4.58688i 0.722247 0.416989i −0.0933322 0.995635i \(-0.529752\pi\)
0.815579 + 0.578646i \(0.196419\pi\)
\(12\) 0 0
\(13\) −5.74342 + 9.94789i −0.441801 + 0.765222i −0.997823 0.0659454i \(-0.978994\pi\)
0.556022 + 0.831168i \(0.312327\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.9706i 0.998268i 0.866525 + 0.499134i \(0.166349\pi\)
−0.866525 + 0.499134i \(0.833651\pi\)
\(18\) 0 0
\(19\) 26.9737 1.41967 0.709833 0.704370i \(-0.248770\pi\)
0.709833 + 0.704370i \(0.248770\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.27048 2.46556i −0.185673 0.107198i 0.404282 0.914634i \(-0.367521\pi\)
−0.589955 + 0.807436i \(0.700855\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −17.7749 + 10.2624i −0.612929 + 0.353874i −0.774111 0.633050i \(-0.781803\pi\)
0.161182 + 0.986925i \(0.448469\pi\)
\(30\) 0 0
\(31\) −10.4868 + 18.1637i −0.338285 + 0.585927i −0.984110 0.177559i \(-0.943180\pi\)
0.645825 + 0.763485i \(0.276513\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.2689i 0.350541i
\(36\) 0 0
\(37\) −62.4605 −1.68812 −0.844061 0.536247i \(-0.819841\pi\)
−0.844061 + 0.536247i \(0.819841\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 35.4531 + 20.4689i 0.864710 + 0.499240i 0.865587 0.500759i \(-0.166946\pi\)
−0.000876936 1.00000i \(0.500279\pi\)
\(42\) 0 0
\(43\) −0.513167 0.888831i −0.0119341 0.0206705i 0.859997 0.510300i \(-0.170466\pi\)
−0.871931 + 0.489629i \(0.837132\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 74.6772 43.1149i 1.58888 0.917338i 0.595384 0.803442i \(-0.297000\pi\)
0.993493 0.113897i \(-0.0363333\pi\)
\(48\) 0 0
\(49\) 9.44733 16.3633i 0.192803 0.333944i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 96.0920i 1.81306i −0.422144 0.906529i \(-0.638722\pi\)
0.422144 0.906529i \(-0.361278\pi\)
\(54\) 0 0
\(55\) −20.5132 −0.372967
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 97.3188 + 56.1871i 1.64947 + 0.952323i 0.977282 + 0.211944i \(0.0679796\pi\)
0.672190 + 0.740379i \(0.265354\pi\)
\(60\) 0 0
\(61\) 33.4605 + 57.9553i 0.548533 + 0.950087i 0.998375 + 0.0569788i \(0.0181467\pi\)
−0.449843 + 0.893108i \(0.648520\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.2442 12.8427i 0.342218 0.197580i
\(66\) 0 0
\(67\) 38.0000 65.8179i 0.567164 0.982357i −0.429681 0.902981i \(-0.641374\pi\)
0.996845 0.0793762i \(-0.0252928\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 24.0789i 0.339139i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542377\pi\)
\(72\) 0 0
\(73\) −18.9210 −0.259192 −0.129596 0.991567i \(-0.541368\pi\)
−0.129596 + 0.991567i \(0.541368\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −43.5913 25.1675i −0.566121 0.326850i
\(78\) 0 0
\(79\) −53.4605 92.5963i −0.676715 1.17211i −0.975964 0.217930i \(-0.930070\pi\)
0.299249 0.954175i \(-0.403264\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 39.1273 22.5902i 0.471414 0.272171i −0.245418 0.969417i \(-0.578925\pi\)
0.716831 + 0.697247i \(0.245592\pi\)
\(84\) 0 0
\(85\) 18.9737 32.8634i 0.223220 0.386628i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 115.928i 1.30256i −0.758835 0.651282i \(-0.774231\pi\)
0.758835 0.651282i \(-0.225769\pi\)
\(90\) 0 0
\(91\) 63.0263 0.692597
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −52.2343 30.1575i −0.549835 0.317447i
\(96\) 0 0
\(97\) 43.5132 + 75.3670i 0.448589 + 0.776980i 0.998294 0.0583792i \(-0.0185933\pi\)
−0.549705 + 0.835359i \(0.685260\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 111.919 64.6165i 1.10811 0.639767i 0.169770 0.985484i \(-0.445697\pi\)
0.938339 + 0.345717i \(0.112364\pi\)
\(102\) 0 0
\(103\) −57.1512 + 98.9889i −0.554866 + 0.961057i 0.443047 + 0.896498i \(0.353897\pi\)
−0.997914 + 0.0645588i \(0.979436\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 93.3381i 0.872319i 0.899869 + 0.436159i \(0.143662\pi\)
−0.899869 + 0.436159i \(0.856338\pi\)
\(108\) 0 0
\(109\) 120.868 1.10888 0.554442 0.832222i \(-0.312932\pi\)
0.554442 + 0.832222i \(0.312932\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 95.5301 + 55.1543i 0.845399 + 0.488091i 0.859096 0.511815i \(-0.171027\pi\)
−0.0136967 + 0.999906i \(0.504360\pi\)
\(114\) 0 0
\(115\) 5.51317 + 9.54909i 0.0479406 + 0.0830355i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 80.6396 46.5573i 0.677644 0.391238i
\(120\) 0 0
\(121\) −18.4210 + 31.9061i −0.152240 + 0.263687i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 105.381 0.829776 0.414888 0.909873i \(-0.363821\pi\)
0.414888 + 0.909873i \(0.363821\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 121.749 + 70.2920i 0.929383 + 0.536580i 0.886617 0.462505i \(-0.153049\pi\)
0.0427669 + 0.999085i \(0.486383\pi\)
\(132\) 0 0
\(133\) −74.0000 128.172i −0.556391 0.963697i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 231.187 133.476i 1.68749 0.974274i 0.731064 0.682309i \(-0.239024\pi\)
0.956429 0.291965i \(-0.0943092\pi\)
\(138\) 0 0
\(139\) 7.92100 13.7196i 0.0569856 0.0987019i −0.836125 0.548538i \(-0.815184\pi\)
0.893111 + 0.449837i \(0.148518\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 105.378i 0.736906i
\(144\) 0 0
\(145\) 45.8947 0.316515
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 36.0493 + 20.8131i 0.241942 + 0.139685i 0.616069 0.787692i \(-0.288724\pi\)
−0.374127 + 0.927377i \(0.622058\pi\)
\(150\) 0 0
\(151\) −51.9210 89.9298i −0.343848 0.595562i 0.641296 0.767294i \(-0.278397\pi\)
−0.985144 + 0.171732i \(0.945064\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 40.6153 23.4493i 0.262034 0.151286i
\(156\) 0 0
\(157\) 6.25658 10.8367i 0.0398509 0.0690237i −0.845412 0.534115i \(-0.820645\pi\)
0.885263 + 0.465091i \(0.153978\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 27.0563i 0.168051i
\(162\) 0 0
\(163\) −290.868 −1.78447 −0.892234 0.451573i \(-0.850863\pi\)
−0.892234 + 0.451573i \(0.850863\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 151.240 + 87.3184i 0.905628 + 0.522865i 0.879022 0.476781i \(-0.158197\pi\)
0.0266061 + 0.999646i \(0.491530\pi\)
\(168\) 0 0
\(169\) 18.5263 + 32.0886i 0.109623 + 0.189873i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 37.7413 21.7900i 0.218158 0.125954i −0.386939 0.922105i \(-0.626468\pi\)
0.605097 + 0.796152i \(0.293134\pi\)
\(174\) 0 0
\(175\) 13.7171 23.7587i 0.0783833 0.135764i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 88.2952i 0.493270i −0.969109 0.246635i \(-0.920675\pi\)
0.969109 0.246635i \(-0.0793248\pi\)
\(180\) 0 0
\(181\) −56.8683 −0.314190 −0.157095 0.987584i \(-0.550213\pi\)
−0.157095 + 0.987584i \(0.550213\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 120.954 + 69.8330i 0.653807 + 0.377475i
\(186\) 0 0
\(187\) 77.8420 + 134.826i 0.416267 + 0.720996i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −49.0543 + 28.3215i −0.256829 + 0.148280i −0.622887 0.782312i \(-0.714040\pi\)
0.366058 + 0.930592i \(0.380707\pi\)
\(192\) 0 0
\(193\) −55.0000 + 95.2628i −0.284974 + 0.493590i −0.972603 0.232473i \(-0.925318\pi\)
0.687629 + 0.726062i \(0.258652\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 245.049i 1.24391i −0.783055 0.621953i \(-0.786339\pi\)
0.783055 0.621953i \(-0.213661\pi\)
\(198\) 0 0
\(199\) −169.895 −0.853742 −0.426871 0.904313i \(-0.640384\pi\)
−0.426871 + 0.904313i \(0.640384\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 97.5281 + 56.3078i 0.480434 + 0.277379i
\(204\) 0 0
\(205\) −45.7698 79.2755i −0.223267 0.386710i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 214.298 123.725i 1.02535 0.591986i
\(210\) 0 0
\(211\) 132.789 229.998i 0.629333 1.09004i −0.358352 0.933586i \(-0.616661\pi\)
0.987686 0.156451i \(-0.0500053\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.29495i 0.0106742i
\(216\) 0 0
\(217\) 115.079 0.530318
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −168.821 97.4690i −0.763897 0.441036i
\(222\) 0 0
\(223\) 93.6644 + 162.232i 0.420020 + 0.727496i 0.995941 0.0900097i \(-0.0286898\pi\)
−0.575921 + 0.817505i \(0.695356\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −364.651 + 210.532i −1.60639 + 0.927452i −0.616226 + 0.787569i \(0.711339\pi\)
−0.990168 + 0.139883i \(0.955327\pi\)
\(228\) 0 0
\(229\) 51.0527 88.4258i 0.222937 0.386139i −0.732761 0.680486i \(-0.761769\pi\)
0.955699 + 0.294347i \(0.0951021\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 99.0694i 0.425191i −0.977140 0.212595i \(-0.931808\pi\)
0.977140 0.212595i \(-0.0681916\pi\)
\(234\) 0 0
\(235\) −192.816 −0.820492
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −67.5222 38.9840i −0.282520 0.163113i 0.352044 0.935984i \(-0.385487\pi\)
−0.634564 + 0.772871i \(0.718820\pi\)
\(240\) 0 0
\(241\) 128.974 + 223.389i 0.535160 + 0.926925i 0.999156 + 0.0410873i \(0.0130822\pi\)
−0.463995 + 0.885838i \(0.653584\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −36.5894 + 21.1249i −0.149344 + 0.0862240i
\(246\) 0 0
\(247\) −154.921 + 268.331i −0.627211 + 1.08636i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 274.971i 1.09550i −0.836641 0.547752i \(-0.815484\pi\)
0.836641 0.547752i \(-0.184516\pi\)
\(252\) 0 0
\(253\) −45.2370 −0.178802
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 113.998 + 65.8168i 0.443572 + 0.256096i 0.705112 0.709096i \(-0.250897\pi\)
−0.261540 + 0.965193i \(0.584230\pi\)
\(258\) 0 0
\(259\) 171.355 + 296.796i 0.661603 + 1.14593i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 397.317 229.391i 1.51071 0.872209i 0.510789 0.859706i \(-0.329354\pi\)
0.999922 0.0125026i \(-0.00397982\pi\)
\(264\) 0 0
\(265\) −107.434 + 186.081i −0.405412 + 0.702194i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 301.916i 1.12236i 0.827692 + 0.561182i \(0.189653\pi\)
−0.827692 + 0.561182i \(0.810347\pi\)
\(270\) 0 0
\(271\) 475.842 1.75587 0.877937 0.478776i \(-0.158919\pi\)
0.877937 + 0.478776i \(0.158919\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 39.7236 + 22.9344i 0.144449 + 0.0833979i
\(276\) 0 0
\(277\) −161.020 278.894i −0.581298 1.00684i −0.995326 0.0965736i \(-0.969212\pi\)
0.414028 0.910264i \(-0.364122\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 304.961 176.070i 1.08527 0.626582i 0.152958 0.988233i \(-0.451120\pi\)
0.932314 + 0.361650i \(0.117787\pi\)
\(282\) 0 0
\(283\) 140.816 243.900i 0.497582 0.861837i −0.502414 0.864627i \(-0.667555\pi\)
0.999996 + 0.00279000i \(0.000888086\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 224.618i 0.782642i
\(288\) 0 0
\(289\) 1.00000 0.00346021
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 55.7098 + 32.1640i 0.190136 + 0.109775i 0.592046 0.805904i \(-0.298320\pi\)
−0.401910 + 0.915679i \(0.631654\pi\)
\(294\) 0 0
\(295\) −125.638 217.612i −0.425892 0.737666i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 49.0543 28.3215i 0.164061 0.0947208i
\(300\) 0 0
\(301\) −2.81566 + 4.87687i −0.00935436 + 0.0162022i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 149.640i 0.490623i
\(306\) 0 0
\(307\) 230.158 0.749700 0.374850 0.927085i \(-0.377694\pi\)
0.374850 + 0.927085i \(0.377694\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.34847 + 4.24264i 0.0236285 + 0.0136419i 0.511768 0.859124i \(-0.328991\pi\)
−0.488139 + 0.872766i \(0.662324\pi\)
\(312\) 0 0
\(313\) −302.789 524.446i −0.967378 1.67555i −0.703085 0.711105i \(-0.748195\pi\)
−0.264293 0.964443i \(-0.585138\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 76.1757 43.9800i 0.240302 0.138738i −0.375014 0.927019i \(-0.622362\pi\)
0.615315 + 0.788281i \(0.289029\pi\)
\(318\) 0 0
\(319\) −94.1445 + 163.063i −0.295124 + 0.511169i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 457.758i 1.41721i
\(324\) 0 0
\(325\) −57.4342 −0.176721
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −409.741 236.564i −1.24541 0.719040i
\(330\) 0 0
\(331\) 24.6185 + 42.6405i 0.0743761 + 0.128823i 0.900815 0.434204i \(-0.142970\pi\)
−0.826439 + 0.563027i \(0.809637\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −147.173 + 84.9706i −0.439323 + 0.253644i
\(336\) 0 0
\(337\) 1.35516 2.34721i 0.00402125 0.00696502i −0.864008 0.503478i \(-0.832053\pi\)
0.868029 + 0.496513i \(0.165387\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 192.408i 0.564245i
\(342\) 0 0
\(343\) −372.527 −1.08608
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 64.9437 + 37.4953i 0.187158 + 0.108056i 0.590651 0.806927i \(-0.298871\pi\)
−0.403494 + 0.914982i \(0.632204\pi\)
\(348\) 0 0
\(349\) 75.4605 + 130.701i 0.216219 + 0.374503i 0.953649 0.300921i \(-0.0972941\pi\)
−0.737430 + 0.675424i \(0.763961\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −253.232 + 146.204i −0.717371 + 0.414174i −0.813784 0.581167i \(-0.802596\pi\)
0.0964134 + 0.995341i \(0.469263\pi\)
\(354\) 0 0
\(355\) −26.9210 + 46.6285i −0.0758338 + 0.131348i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 576.776i 1.60662i 0.595563 + 0.803309i \(0.296929\pi\)
−0.595563 + 0.803309i \(0.703071\pi\)
\(360\) 0 0
\(361\) 366.579 1.01545
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 36.6404 + 21.1543i 0.100385 + 0.0579570i
\(366\) 0 0
\(367\) −121.770 210.911i −0.331798 0.574690i 0.651067 0.759021i \(-0.274322\pi\)
−0.982864 + 0.184330i \(0.940988\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −456.604 + 263.620i −1.23074 + 0.710567i
\(372\) 0 0
\(373\) −57.9541 + 100.379i −0.155373 + 0.269114i −0.933195 0.359371i \(-0.882991\pi\)
0.777822 + 0.628485i \(0.216325\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 235.764i 0.625369i
\(378\) 0 0
\(379\) 30.2107 0.0797115 0.0398558 0.999205i \(-0.487310\pi\)
0.0398558 + 0.999205i \(0.487310\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −564.059 325.660i −1.47274 0.850286i −0.473209 0.880950i \(-0.656905\pi\)
−0.999530 + 0.0306636i \(0.990238\pi\)
\(384\) 0 0
\(385\) 56.2762 + 97.4732i 0.146172 + 0.253177i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −178.055 + 102.800i −0.457726 + 0.264268i −0.711087 0.703104i \(-0.751797\pi\)
0.253362 + 0.967372i \(0.418464\pi\)
\(390\) 0 0
\(391\) 41.8420 72.4725i 0.107013 0.185352i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 239.083i 0.605272i
\(396\) 0 0
\(397\) 118.355 0.298124 0.149062 0.988828i \(-0.452375\pi\)
0.149062 + 0.988828i \(0.452375\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −148.065 85.4854i −0.369240 0.213181i 0.303887 0.952708i \(-0.401716\pi\)
−0.673126 + 0.739528i \(0.735049\pi\)
\(402\) 0 0
\(403\) −120.460 208.644i −0.298909 0.517726i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −496.231 + 286.499i −1.21924 + 0.703929i
\(408\) 0 0
\(409\) 167.974 290.939i 0.410694 0.711342i −0.584272 0.811558i \(-0.698620\pi\)
0.994966 + 0.100216i \(0.0319533\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 616.578i 1.49292i
\(414\) 0 0
\(415\) −101.026 −0.243437
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 352.936 + 203.768i 0.842329 + 0.486319i 0.858055 0.513558i \(-0.171673\pi\)
−0.0157264 + 0.999876i \(0.505006\pi\)
\(420\) 0 0
\(421\) 385.605 + 667.887i 0.915926 + 1.58643i 0.805539 + 0.592542i \(0.201876\pi\)
0.110387 + 0.993889i \(0.464791\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −73.4847 + 42.4264i −0.172905 + 0.0998268i
\(426\) 0 0
\(427\) 183.592 317.991i 0.429958 0.744710i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 436.210i 1.01209i 0.862508 + 0.506044i \(0.168893\pi\)
−0.862508 + 0.506044i \(0.831107\pi\)
\(432\) 0 0
\(433\) −838.500 −1.93649 −0.968244 0.250006i \(-0.919568\pi\)
−0.968244 + 0.250006i \(0.919568\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −115.191 66.5053i −0.263594 0.152186i
\(438\) 0 0
\(439\) −25.0000 43.3013i −0.0569476 0.0986362i 0.836146 0.548507i \(-0.184803\pi\)
−0.893094 + 0.449871i \(0.851470\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −426.823 + 246.426i −0.963483 + 0.556267i −0.897243 0.441536i \(-0.854434\pi\)
−0.0662400 + 0.997804i \(0.521100\pi\)
\(444\) 0 0
\(445\) −129.612 + 224.494i −0.291262 + 0.504481i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 483.102i 1.07595i −0.842960 0.537976i \(-0.819189\pi\)
0.842960 0.537976i \(-0.180811\pi\)
\(450\) 0 0
\(451\) 375.553 0.832712
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −122.050 70.4656i −0.268242 0.154869i
\(456\) 0 0
\(457\) 273.460 + 473.647i 0.598382 + 1.03643i 0.993060 + 0.117608i \(0.0375227\pi\)
−0.394678 + 0.918819i \(0.629144\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −112.918 + 65.1932i −0.244941 + 0.141417i −0.617446 0.786613i \(-0.711833\pi\)
0.372504 + 0.928030i \(0.378499\pi\)
\(462\) 0 0
\(463\) −12.3883 + 21.4571i −0.0267565 + 0.0463436i −0.879094 0.476649i \(-0.841851\pi\)
0.852337 + 0.522993i \(0.175185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 671.491i 1.43788i 0.695071 + 0.718941i \(0.255373\pi\)
−0.695071 + 0.718941i \(0.744627\pi\)
\(468\) 0 0
\(469\) −416.999 −0.889124
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.15393 4.70767i −0.0172388 0.00995280i
\(474\) 0 0
\(475\) 67.4342 + 116.799i 0.141967 + 0.245893i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −466.144 + 269.128i −0.973161 + 0.561855i −0.900198 0.435480i \(-0.856579\pi\)
−0.0729624 + 0.997335i \(0.523245\pi\)
\(480\) 0 0
\(481\) 358.737 621.350i 0.745814 1.29179i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 194.597i 0.401231i
\(486\) 0 0
\(487\) 608.250 1.24897 0.624486 0.781036i \(-0.285308\pi\)
0.624486 + 0.781036i \(0.285308\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 124.328 + 71.7806i 0.253213 + 0.146193i 0.621235 0.783625i \(-0.286631\pi\)
−0.368021 + 0.929817i \(0.619965\pi\)
\(492\) 0 0
\(493\) −174.158 301.651i −0.353262 0.611867i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −114.416 + 66.0584i −0.230214 + 0.132914i
\(498\) 0 0
\(499\) −308.302 + 533.996i −0.617841 + 1.07013i 0.372038 + 0.928217i \(0.378659\pi\)
−0.989879 + 0.141914i \(0.954674\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 374.729i 0.744989i 0.928034 + 0.372494i \(0.121497\pi\)
−0.928034 + 0.372494i \(0.878503\pi\)
\(504\) 0 0
\(505\) −288.974 −0.572225
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −149.854 86.5182i −0.294408 0.169977i 0.345520 0.938411i \(-0.387703\pi\)
−0.639928 + 0.768435i \(0.721036\pi\)
\(510\) 0 0
\(511\) 51.9082 + 89.9076i 0.101582 + 0.175944i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 221.346 127.794i 0.429798 0.248144i
\(516\) 0 0
\(517\) 395.526 685.071i 0.765041 1.32509i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 455.116i 0.873543i −0.899572 0.436772i \(-0.856122\pi\)
0.899572 0.436772i \(-0.143878\pi\)
\(522\) 0 0
\(523\) −295.395 −0.564809 −0.282404 0.959295i \(-0.591132\pi\)
−0.282404 + 0.959295i \(0.591132\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −308.249 177.967i −0.584912 0.337699i
\(528\) 0 0
\(529\) −252.342 437.069i −0.477017 0.826218i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −407.244 + 235.122i −0.764060 + 0.441130i
\(534\) 0 0
\(535\) 104.355 180.748i 0.195056 0.337848i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 173.335i 0.321587i
\(540\) 0 0
\(541\) 906.394 1.67541 0.837703 0.546127i \(-0.183898\pi\)
0.837703 + 0.546127i \(0.183898\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −234.061 135.135i −0.429469 0.247954i
\(546\) 0 0
\(547\) 336.460 + 582.767i 0.615101 + 1.06539i 0.990367 + 0.138470i \(0.0442183\pi\)
−0.375265 + 0.926917i \(0.622448\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −479.455 + 276.813i −0.870154 + 0.502384i
\(552\) 0 0
\(553\) −293.329 + 508.060i −0.530432 + 0.918735i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 870.336i 1.56254i 0.624192 + 0.781271i \(0.285428\pi\)
−0.624192 + 0.781271i \(0.714572\pi\)
\(558\) 0 0
\(559\) 11.7893 0.0210900
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20.4659 11.8160i −0.0363515 0.0209875i 0.481714 0.876328i \(-0.340014\pi\)
−0.518066 + 0.855341i \(0.673348\pi\)
\(564\) 0 0
\(565\) −123.329 213.612i −0.218281 0.378074i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −255.714 + 147.636i −0.449409 + 0.259466i −0.707581 0.706633i \(-0.750213\pi\)
0.258172 + 0.966099i \(0.416880\pi\)
\(570\) 0 0
\(571\) −446.960 + 774.158i −0.782767 + 1.35579i 0.147556 + 0.989054i \(0.452859\pi\)
−0.930324 + 0.366739i \(0.880474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.6556i 0.0428794i
\(576\) 0 0
\(577\) −264.763 −0.458861 −0.229431 0.973325i \(-0.573686\pi\)
−0.229431 + 0.973325i \(0.573686\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −214.685 123.949i −0.369510 0.213337i
\(582\) 0 0
\(583\) −440.763 763.424i −0.756026 1.30947i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 685.793 395.943i 1.16830 0.674519i 0.215022 0.976609i \(-0.431018\pi\)
0.953279 + 0.302090i \(0.0976845\pi\)
\(588\) 0 0
\(589\) −282.868 + 489.942i −0.480252 + 0.831821i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.82388i 0.00307567i −0.999999 0.00153784i \(-0.999510\pi\)
0.999999 0.00153784i \(-0.000489509\pi\)
\(594\) 0 0
\(595\) −208.211 −0.349934
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −656.624 379.102i −1.09620 0.632891i −0.160980 0.986958i \(-0.551465\pi\)
−0.935220 + 0.354067i \(0.884799\pi\)
\(600\) 0 0
\(601\) 141.789 + 245.586i 0.235922 + 0.408629i 0.959540 0.281571i \(-0.0908556\pi\)
−0.723618 + 0.690201i \(0.757522\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 71.3442 41.1906i 0.117924 0.0680836i
\(606\) 0 0
\(607\) −83.9142 + 145.344i −0.138244 + 0.239446i −0.926832 0.375476i \(-0.877479\pi\)
0.788588 + 0.614922i \(0.210813\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 990.507i 1.62112i
\(612\) 0 0
\(613\) −395.698 −0.645510 −0.322755 0.946483i \(-0.604609\pi\)
−0.322755 + 0.946483i \(0.604609\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −740.616 427.595i −1.20035 0.693022i −0.239717 0.970843i \(-0.577055\pi\)
−0.960633 + 0.277821i \(0.910388\pi\)
\(618\) 0 0
\(619\) −154.079 266.873i −0.248916 0.431135i 0.714309 0.699830i \(-0.246741\pi\)
−0.963225 + 0.268695i \(0.913408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −550.861 + 318.040i −0.884206 + 0.510497i
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1059.99i 1.68520i
\(630\) 0 0
\(631\) 344.974 0.546709 0.273355 0.961913i \(-0.411867\pi\)
0.273355 + 0.961913i \(0.411867\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −204.070 117.820i −0.321371 0.185543i
\(636\) 0 0
\(637\) 108.520 + 187.962i 0.170361 + 0.295074i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 175.492 101.321i 0.273779 0.158066i −0.356825 0.934171i \(-0.616141\pi\)
0.630604 + 0.776105i \(0.282807\pi\)
\(642\) 0 0
\(643\) −362.302 + 627.526i −0.563456 + 0.975935i 0.433735 + 0.901040i \(0.357195\pi\)
−0.997191 + 0.0748947i \(0.976138\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 125.679i 0.194249i 0.995272 + 0.0971243i \(0.0309644\pi\)
−0.995272 + 0.0971243i \(0.969036\pi\)
\(648\) 0 0
\(649\) 1030.89 1.58843
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −477.376 275.613i −0.731050 0.422072i 0.0877559 0.996142i \(-0.472030\pi\)
−0.818806 + 0.574070i \(0.805364\pi\)
\(654\) 0 0
\(655\) −157.178 272.240i −0.239966 0.415633i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −694.349 + 400.883i −1.05364 + 0.608320i −0.923666 0.383198i \(-0.874823\pi\)
−0.129974 + 0.991517i \(0.541490\pi\)
\(660\) 0 0
\(661\) 264.540 458.196i 0.400211 0.693186i −0.593540 0.804804i \(-0.702270\pi\)
0.993751 + 0.111619i \(0.0356035\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 330.938i 0.497651i
\(666\) 0 0
\(667\) 101.210 0.151739
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 531.668 + 306.959i 0.792352 + 0.457465i
\(672\) 0 0
\(673\) −206.698 358.011i −0.307129 0.531962i 0.670604 0.741815i \(-0.266035\pi\)
−0.977733 + 0.209853i \(0.932701\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 311.714 179.968i 0.460434 0.265832i −0.251793 0.967781i \(-0.581020\pi\)
0.712227 + 0.701950i \(0.247687\pi\)
\(678\) 0 0
\(679\) 238.749 413.526i 0.351619 0.609022i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.5484i 0.0315496i −0.999876 0.0157748i \(-0.994979\pi\)
0.999876 0.0157748i \(-0.00502148\pi\)
\(684\) 0 0
\(685\) −596.921 −0.871418
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 955.913 + 551.897i 1.38739 + 0.801011i
\(690\) 0 0
\(691\) 192.460 + 333.351i 0.278525 + 0.482419i 0.971018 0.239005i \(-0.0768213\pi\)
−0.692494 + 0.721424i \(0.743488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −30.6779 + 17.7119i −0.0441409 + 0.0254847i
\(696\) 0 0
\(697\) −347.368 + 601.659i −0.498376 + 0.863212i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 439.057i 0.626330i −0.949699 0.313165i \(-0.898611\pi\)
0.949699 0.313165i \(-0.101389\pi\)
\(702\) 0 0
\(703\) −1684.79 −2.39657
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −614.081 354.540i −0.868573 0.501471i
\(708\) 0 0
\(709\) 63.2107 + 109.484i 0.0891547 + 0.154420i 0.907154 0.420799i \(-0.138250\pi\)
−0.817999 + 0.575219i \(0.804917\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 89.5676 51.7119i 0.125621 0.0725272i
\(714\) 0 0
\(715\) 117.816 204.063i 0.164777 0.285402i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 571.715i 0.795153i 0.917569 + 0.397576i \(0.130149\pi\)
−0.917569 + 0.397576i \(0.869851\pi\)
\(720\) 0 0
\(721\) 627.159 0.869846
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −88.8746 51.3118i −0.122586 0.0707749i
\(726\) 0 0
\(727\) 420.519 + 728.361i 0.578431 + 1.00187i 0.995660 + 0.0930701i \(0.0296681\pi\)
−0.417229 + 0.908802i \(0.636999\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.0840 8.70873i 0.0206347 0.0119135i
\(732\) 0 0
\(733\) −247.375 + 428.466i −0.337483 + 0.584537i −0.983959 0.178397i \(-0.942909\pi\)
0.646476 + 0.762934i \(0.276242\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 697.206i 0.946006i
\(738\) 0 0
\(739\) 1263.97 1.71038 0.855191 0.518312i \(-0.173439\pi\)
0.855191 + 0.518312i \(0.173439\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 828.217 + 478.171i 1.11469 + 0.643568i 0.940041 0.341062i \(-0.110787\pi\)
0.174652 + 0.984630i \(0.444120\pi\)
\(744\) 0 0
\(745\) −46.5395 80.6088i −0.0624691 0.108200i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 443.518 256.065i 0.592147 0.341876i
\(750\) 0 0
\(751\) 560.828 971.383i 0.746776 1.29345i −0.202585 0.979265i \(-0.564934\pi\)
0.949361 0.314189i \(-0.101732\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 232.198i 0.307547i
\(756\) 0 0
\(757\) −466.433 −0.616160 −0.308080 0.951360i \(-0.599687\pi\)
−0.308080 + 0.951360i \(0.599687\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1059.50 611.703i −1.39225 0.803814i −0.398684 0.917089i \(-0.630533\pi\)
−0.993564 + 0.113274i \(0.963866\pi\)
\(762\) 0 0
\(763\) −331.592 574.334i −0.434590 0.752732i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1117.89 + 645.411i −1.45748 + 0.841475i
\(768\) 0 0
\(769\) 645.315 1117.72i 0.839162 1.45347i −0.0514349 0.998676i \(-0.516379\pi\)
0.890597 0.454794i \(-0.150287\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 329.455i 0.426204i −0.977030 0.213102i \(-0.931643\pi\)
0.977030 0.213102i \(-0.0683566\pi\)
\(774\) 0 0
\(775\) −104.868 −0.135314
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 956.300 + 552.120i 1.22760 + 0.708755i
\(780\) 0 0
\(781\) −110.447 191.300i −0.141417 0.244942i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −24.2316 + 13.9901i −0.0308683 + 0.0178218i
\(786\) 0 0
\(787\) 753.065 1304.35i 0.956881 1.65737i 0.226878 0.973923i \(-0.427148\pi\)
0.730003 0.683444i \(-0.239518\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 605.245i 0.765165i
\(792\) 0 0
\(793\) −768.710 −0.969370
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 217.264 + 125.437i 0.272602 + 0.157387i 0.630069 0.776539i \(-0.283026\pi\)
−0.357468 + 0.933926i \(0.616360\pi\)
\(798\) 0 0
\(799\) 731.684 + 1267.31i 0.915750 + 1.58612i
\(800\) 0 0