Properties

Label 1520.2.d.d.609.1
Level $1520$
Weight $2$
Character 1520.609
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.d.609.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421i q^{3} +(2.12132 - 0.707107i) q^{5} -2.41421i q^{7} -2.82843 q^{9} +O(q^{10})\) \(q-2.41421i q^{3} +(2.12132 - 0.707107i) q^{5} -2.41421i q^{7} -2.82843 q^{9} -1.41421 q^{11} +1.82843i q^{13} +(-1.70711 - 5.12132i) q^{15} -1.00000i q^{17} -1.00000 q^{19} -5.82843 q^{21} -5.24264i q^{23} +(4.00000 - 3.00000i) q^{25} -0.414214i q^{27} -3.82843 q^{29} -3.41421 q^{31} +3.41421i q^{33} +(-1.70711 - 5.12132i) q^{35} -5.17157i q^{37} +4.41421 q^{39} -7.07107 q^{41} -2.24264i q^{43} +(-6.00000 + 2.00000i) q^{45} +8.00000i q^{47} +1.17157 q^{49} -2.41421 q^{51} -1.82843i q^{53} +(-3.00000 + 1.00000i) q^{55} +2.41421i q^{57} +14.4142 q^{59} +3.41421 q^{61} +6.82843i q^{63} +(1.29289 + 3.87868i) q^{65} -6.07107i q^{67} -12.6569 q^{69} -3.07107 q^{71} +13.8284i q^{73} +(-7.24264 - 9.65685i) q^{75} +3.41421i q^{77} -0.828427 q^{79} -9.48528 q^{81} +2.48528i q^{83} +(-0.707107 - 2.12132i) q^{85} +9.24264i q^{87} +3.75736 q^{89} +4.41421 q^{91} +8.24264i q^{93} +(-2.12132 + 0.707107i) q^{95} -7.65685i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{15} - 4q^{19} - 12q^{21} + 16q^{25} - 4q^{29} - 8q^{31} - 4q^{35} + 12q^{39} - 24q^{45} + 16q^{49} - 4q^{51} - 12q^{55} + 52q^{59} + 8q^{61} + 8q^{65} - 28q^{69} + 16q^{71} - 12q^{75} + 8q^{79} - 4q^{81} + 32q^{89} + 12q^{91} + 16q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.41421i 1.39385i −0.717146 0.696923i \(-0.754552\pi\)
0.717146 0.696923i \(-0.245448\pi\)
\(4\) 0 0
\(5\) 2.12132 0.707107i 0.948683 0.316228i
\(6\) 0 0
\(7\) 2.41421i 0.912487i −0.889855 0.456243i \(-0.849195\pi\)
0.889855 0.456243i \(-0.150805\pi\)
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 1.82843i 0.507114i 0.967320 + 0.253557i \(0.0816006\pi\)
−0.967320 + 0.253557i \(0.918399\pi\)
\(14\) 0 0
\(15\) −1.70711 5.12132i −0.440773 1.32232i
\(16\) 0 0
\(17\) 1.00000i 0.242536i −0.992620 0.121268i \(-0.961304\pi\)
0.992620 0.121268i \(-0.0386960\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −5.82843 −1.27187
\(22\) 0 0
\(23\) 5.24264i 1.09317i −0.837405 0.546583i \(-0.815928\pi\)
0.837405 0.546583i \(-0.184072\pi\)
\(24\) 0 0
\(25\) 4.00000 3.00000i 0.800000 0.600000i
\(26\) 0 0
\(27\) 0.414214i 0.0797154i
\(28\) 0 0
\(29\) −3.82843 −0.710921 −0.355461 0.934691i \(-0.615676\pi\)
−0.355461 + 0.934691i \(0.615676\pi\)
\(30\) 0 0
\(31\) −3.41421 −0.613211 −0.306605 0.951837i \(-0.599193\pi\)
−0.306605 + 0.951837i \(0.599193\pi\)
\(32\) 0 0
\(33\) 3.41421i 0.594338i
\(34\) 0 0
\(35\) −1.70711 5.12132i −0.288554 0.865661i
\(36\) 0 0
\(37\) 5.17157i 0.850201i −0.905146 0.425101i \(-0.860239\pi\)
0.905146 0.425101i \(-0.139761\pi\)
\(38\) 0 0
\(39\) 4.41421 0.706840
\(40\) 0 0
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 0 0
\(43\) 2.24264i 0.341999i −0.985271 0.171000i \(-0.945300\pi\)
0.985271 0.171000i \(-0.0546997\pi\)
\(44\) 0 0
\(45\) −6.00000 + 2.00000i −0.894427 + 0.298142i
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 1.17157 0.167368
\(50\) 0 0
\(51\) −2.41421 −0.338058
\(52\) 0 0
\(53\) 1.82843i 0.251154i −0.992084 0.125577i \(-0.959922\pi\)
0.992084 0.125577i \(-0.0400782\pi\)
\(54\) 0 0
\(55\) −3.00000 + 1.00000i −0.404520 + 0.134840i
\(56\) 0 0
\(57\) 2.41421i 0.319770i
\(58\) 0 0
\(59\) 14.4142 1.87657 0.938285 0.345862i \(-0.112413\pi\)
0.938285 + 0.345862i \(0.112413\pi\)
\(60\) 0 0
\(61\) 3.41421 0.437145 0.218573 0.975821i \(-0.429860\pi\)
0.218573 + 0.975821i \(0.429860\pi\)
\(62\) 0 0
\(63\) 6.82843i 0.860301i
\(64\) 0 0
\(65\) 1.29289 + 3.87868i 0.160364 + 0.481091i
\(66\) 0 0
\(67\) 6.07107i 0.741699i −0.928693 0.370849i \(-0.879067\pi\)
0.928693 0.370849i \(-0.120933\pi\)
\(68\) 0 0
\(69\) −12.6569 −1.52371
\(70\) 0 0
\(71\) −3.07107 −0.364469 −0.182234 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) 13.8284i 1.61849i 0.587468 + 0.809247i \(0.300125\pi\)
−0.587468 + 0.809247i \(0.699875\pi\)
\(74\) 0 0
\(75\) −7.24264 9.65685i −0.836308 1.11508i
\(76\) 0 0
\(77\) 3.41421i 0.389086i
\(78\) 0 0
\(79\) −0.828427 −0.0932053 −0.0466027 0.998914i \(-0.514839\pi\)
−0.0466027 + 0.998914i \(0.514839\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 2.48528i 0.272795i 0.990654 + 0.136398i \(0.0435524\pi\)
−0.990654 + 0.136398i \(0.956448\pi\)
\(84\) 0 0
\(85\) −0.707107 2.12132i −0.0766965 0.230089i
\(86\) 0 0
\(87\) 9.24264i 0.990915i
\(88\) 0 0
\(89\) 3.75736 0.398279 0.199140 0.979971i \(-0.436185\pi\)
0.199140 + 0.979971i \(0.436185\pi\)
\(90\) 0 0
\(91\) 4.41421 0.462735
\(92\) 0 0
\(93\) 8.24264i 0.854722i
\(94\) 0 0
\(95\) −2.12132 + 0.707107i −0.217643 + 0.0725476i
\(96\) 0 0
\(97\) 7.65685i 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 15.8995 1.58206 0.791029 0.611778i \(-0.209545\pi\)
0.791029 + 0.611778i \(0.209545\pi\)
\(102\) 0 0
\(103\) 9.89949i 0.975426i 0.873004 + 0.487713i \(0.162169\pi\)
−0.873004 + 0.487713i \(0.837831\pi\)
\(104\) 0 0
\(105\) −12.3640 + 4.12132i −1.20660 + 0.402200i
\(106\) 0 0
\(107\) 2.41421i 0.233391i −0.993168 0.116695i \(-0.962770\pi\)
0.993168 0.116695i \(-0.0372301\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) −12.4853 −1.18505
\(112\) 0 0
\(113\) 13.0711i 1.22962i −0.788674 0.614811i \(-0.789232\pi\)
0.788674 0.614811i \(-0.210768\pi\)
\(114\) 0 0
\(115\) −3.70711 11.1213i −0.345689 1.03707i
\(116\) 0 0
\(117\) 5.17157i 0.478112i
\(118\) 0 0
\(119\) −2.41421 −0.221311
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 17.0711i 1.53925i
\(124\) 0 0
\(125\) 6.36396 9.19239i 0.569210 0.822192i
\(126\) 0 0
\(127\) 11.1716i 0.991317i 0.868518 + 0.495658i \(0.165073\pi\)
−0.868518 + 0.495658i \(0.834927\pi\)
\(128\) 0 0
\(129\) −5.41421 −0.476695
\(130\) 0 0
\(131\) −21.6569 −1.89217 −0.946084 0.323921i \(-0.894999\pi\)
−0.946084 + 0.323921i \(0.894999\pi\)
\(132\) 0 0
\(133\) 2.41421i 0.209339i
\(134\) 0 0
\(135\) −0.292893 0.878680i −0.0252082 0.0756247i
\(136\) 0 0
\(137\) 16.3137i 1.39377i −0.717181 0.696887i \(-0.754568\pi\)
0.717181 0.696887i \(-0.245432\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 19.3137 1.62651
\(142\) 0 0
\(143\) 2.58579i 0.216234i
\(144\) 0 0
\(145\) −8.12132 + 2.70711i −0.674439 + 0.224813i
\(146\) 0 0
\(147\) 2.82843i 0.233285i
\(148\) 0 0
\(149\) 17.6569 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(150\) 0 0
\(151\) −11.1716 −0.909130 −0.454565 0.890714i \(-0.650205\pi\)
−0.454565 + 0.890714i \(0.650205\pi\)
\(152\) 0 0
\(153\) 2.82843i 0.228665i
\(154\) 0 0
\(155\) −7.24264 + 2.41421i −0.581743 + 0.193914i
\(156\) 0 0
\(157\) 0.343146i 0.0273860i −0.999906 0.0136930i \(-0.995641\pi\)
0.999906 0.0136930i \(-0.00435876\pi\)
\(158\) 0 0
\(159\) −4.41421 −0.350070
\(160\) 0 0
\(161\) −12.6569 −0.997500
\(162\) 0 0
\(163\) 2.92893i 0.229412i −0.993400 0.114706i \(-0.963407\pi\)
0.993400 0.114706i \(-0.0365925\pi\)
\(164\) 0 0
\(165\) 2.41421 + 7.24264i 0.187946 + 0.563839i
\(166\) 0 0
\(167\) 23.2132i 1.79629i 0.439698 + 0.898146i \(0.355086\pi\)
−0.439698 + 0.898146i \(0.644914\pi\)
\(168\) 0 0
\(169\) 9.65685 0.742835
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) 18.8284i 1.43150i 0.698357 + 0.715749i \(0.253915\pi\)
−0.698357 + 0.715749i \(0.746085\pi\)
\(174\) 0 0
\(175\) −7.24264 9.65685i −0.547492 0.729990i
\(176\) 0 0
\(177\) 34.7990i 2.61565i
\(178\) 0 0
\(179\) −22.9706 −1.71690 −0.858450 0.512897i \(-0.828572\pi\)
−0.858450 + 0.512897i \(0.828572\pi\)
\(180\) 0 0
\(181\) 0.485281 0.0360707 0.0180353 0.999837i \(-0.494259\pi\)
0.0180353 + 0.999837i \(0.494259\pi\)
\(182\) 0 0
\(183\) 8.24264i 0.609314i
\(184\) 0 0
\(185\) −3.65685 10.9706i −0.268857 0.806572i
\(186\) 0 0
\(187\) 1.41421i 0.103418i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 11.2426 0.813489 0.406744 0.913542i \(-0.366664\pi\)
0.406744 + 0.913542i \(0.366664\pi\)
\(192\) 0 0
\(193\) 7.65685i 0.551152i 0.961279 + 0.275576i \(0.0888686\pi\)
−0.961279 + 0.275576i \(0.911131\pi\)
\(194\) 0 0
\(195\) 9.36396 3.12132i 0.670567 0.223522i
\(196\) 0 0
\(197\) 20.7279i 1.47680i −0.674361 0.738402i \(-0.735581\pi\)
0.674361 0.738402i \(-0.264419\pi\)
\(198\) 0 0
\(199\) 16.0711 1.13925 0.569624 0.821905i \(-0.307089\pi\)
0.569624 + 0.821905i \(0.307089\pi\)
\(200\) 0 0
\(201\) −14.6569 −1.03381
\(202\) 0 0
\(203\) 9.24264i 0.648706i
\(204\) 0 0
\(205\) −15.0000 + 5.00000i −1.04765 + 0.349215i
\(206\) 0 0
\(207\) 14.8284i 1.03065i
\(208\) 0 0
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) 24.8995 1.71415 0.857076 0.515190i \(-0.172279\pi\)
0.857076 + 0.515190i \(0.172279\pi\)
\(212\) 0 0
\(213\) 7.41421i 0.508014i
\(214\) 0 0
\(215\) −1.58579 4.75736i −0.108150 0.324449i
\(216\) 0 0
\(217\) 8.24264i 0.559547i
\(218\) 0 0
\(219\) 33.3848 2.25593
\(220\) 0 0
\(221\) 1.82843 0.122993
\(222\) 0 0
\(223\) 3.17157i 0.212384i −0.994346 0.106192i \(-0.966134\pi\)
0.994346 0.106192i \(-0.0338659\pi\)
\(224\) 0 0
\(225\) −11.3137 + 8.48528i −0.754247 + 0.565685i
\(226\) 0 0
\(227\) 17.2426i 1.14443i −0.820102 0.572217i \(-0.806083\pi\)
0.820102 0.572217i \(-0.193917\pi\)
\(228\) 0 0
\(229\) 26.9706 1.78226 0.891132 0.453743i \(-0.149912\pi\)
0.891132 + 0.453743i \(0.149912\pi\)
\(230\) 0 0
\(231\) 8.24264 0.542326
\(232\) 0 0
\(233\) 2.34315i 0.153505i −0.997050 0.0767523i \(-0.975545\pi\)
0.997050 0.0767523i \(-0.0244551\pi\)
\(234\) 0 0
\(235\) 5.65685 + 16.9706i 0.369012 + 1.10704i
\(236\) 0 0
\(237\) 2.00000i 0.129914i
\(238\) 0 0
\(239\) 2.27208 0.146969 0.0734843 0.997296i \(-0.476588\pi\)
0.0734843 + 0.997296i \(0.476588\pi\)
\(240\) 0 0
\(241\) 5.65685 0.364390 0.182195 0.983262i \(-0.441680\pi\)
0.182195 + 0.983262i \(0.441680\pi\)
\(242\) 0 0
\(243\) 21.6569i 1.38929i
\(244\) 0 0
\(245\) 2.48528 0.828427i 0.158779 0.0529263i
\(246\) 0 0
\(247\) 1.82843i 0.116340i
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 27.5563 1.73934 0.869671 0.493632i \(-0.164331\pi\)
0.869671 + 0.493632i \(0.164331\pi\)
\(252\) 0 0
\(253\) 7.41421i 0.466128i
\(254\) 0 0
\(255\) −5.12132 + 1.70711i −0.320710 + 0.106903i
\(256\) 0 0
\(257\) 3.27208i 0.204107i −0.994779 0.102053i \(-0.967459\pi\)
0.994779 0.102053i \(-0.0325412\pi\)
\(258\) 0 0
\(259\) −12.4853 −0.775798
\(260\) 0 0
\(261\) 10.8284 0.670263
\(262\) 0 0
\(263\) 4.34315i 0.267810i 0.990994 + 0.133905i \(0.0427517\pi\)
−0.990994 + 0.133905i \(0.957248\pi\)
\(264\) 0 0
\(265\) −1.29289 3.87868i −0.0794218 0.238265i
\(266\) 0 0
\(267\) 9.07107i 0.555140i
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 21.3848 1.29903 0.649516 0.760348i \(-0.274971\pi\)
0.649516 + 0.760348i \(0.274971\pi\)
\(272\) 0 0
\(273\) 10.6569i 0.644982i
\(274\) 0 0
\(275\) −5.65685 + 4.24264i −0.341121 + 0.255841i
\(276\) 0 0
\(277\) 19.3137i 1.16045i −0.814457 0.580224i \(-0.802965\pi\)
0.814457 0.580224i \(-0.197035\pi\)
\(278\) 0 0
\(279\) 9.65685 0.578141
\(280\) 0 0
\(281\) −12.2426 −0.730335 −0.365167 0.930942i \(-0.618988\pi\)
−0.365167 + 0.930942i \(0.618988\pi\)
\(282\) 0 0
\(283\) 26.4853i 1.57439i −0.616706 0.787193i \(-0.711533\pi\)
0.616706 0.787193i \(-0.288467\pi\)
\(284\) 0 0
\(285\) 1.70711 + 5.12132i 0.101120 + 0.303361i
\(286\) 0 0
\(287\) 17.0711i 1.00767i
\(288\) 0 0
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) −18.4853 −1.08363
\(292\) 0 0
\(293\) 28.7990i 1.68245i −0.540681 0.841227i \(-0.681834\pi\)
0.540681 0.841227i \(-0.318166\pi\)
\(294\) 0 0
\(295\) 30.5772 10.1924i 1.78027 0.593424i
\(296\) 0 0
\(297\) 0.585786i 0.0339908i
\(298\) 0 0
\(299\) 9.58579 0.554360
\(300\) 0 0
\(301\) −5.41421 −0.312070
\(302\) 0 0
\(303\) 38.3848i 2.20515i
\(304\) 0 0
\(305\) 7.24264 2.41421i 0.414712 0.138237i
\(306\) 0 0
\(307\) 24.2843i 1.38598i −0.720949 0.692988i \(-0.756294\pi\)
0.720949 0.692988i \(-0.243706\pi\)
\(308\) 0 0
\(309\) 23.8995 1.35959
\(310\) 0 0
\(311\) −14.7574 −0.836813 −0.418407 0.908260i \(-0.637411\pi\)
−0.418407 + 0.908260i \(0.637411\pi\)
\(312\) 0 0
\(313\) 4.65685i 0.263221i −0.991302 0.131610i \(-0.957985\pi\)
0.991302 0.131610i \(-0.0420148\pi\)
\(314\) 0 0
\(315\) 4.82843 + 14.4853i 0.272051 + 0.816153i
\(316\) 0 0
\(317\) 32.7990i 1.84217i 0.389356 + 0.921087i \(0.372698\pi\)
−0.389356 + 0.921087i \(0.627302\pi\)
\(318\) 0 0
\(319\) 5.41421 0.303138
\(320\) 0 0
\(321\) −5.82843 −0.325311
\(322\) 0 0
\(323\) 1.00000i 0.0556415i
\(324\) 0 0
\(325\) 5.48528 + 7.31371i 0.304269 + 0.405692i
\(326\) 0 0
\(327\) 12.0711i 0.667532i
\(328\) 0 0
\(329\) 19.3137 1.06480
\(330\) 0 0
\(331\) 17.3848 0.955554 0.477777 0.878481i \(-0.341443\pi\)
0.477777 + 0.878481i \(0.341443\pi\)
\(332\) 0 0
\(333\) 14.6274i 0.801578i
\(334\) 0 0
\(335\) −4.29289 12.8787i −0.234546 0.703637i
\(336\) 0 0
\(337\) 12.7279i 0.693334i 0.937988 + 0.346667i \(0.112687\pi\)
−0.937988 + 0.346667i \(0.887313\pi\)
\(338\) 0 0
\(339\) −31.5563 −1.71391
\(340\) 0 0
\(341\) 4.82843 0.261474
\(342\) 0 0
\(343\) 19.7279i 1.06521i
\(344\) 0 0
\(345\) −26.8492 + 8.94975i −1.44551 + 0.481838i
\(346\) 0 0
\(347\) 20.8284i 1.11813i 0.829124 + 0.559064i \(0.188840\pi\)
−0.829124 + 0.559064i \(0.811160\pi\)
\(348\) 0 0
\(349\) 16.6274 0.890045 0.445023 0.895519i \(-0.353196\pi\)
0.445023 + 0.895519i \(0.353196\pi\)
\(350\) 0 0
\(351\) 0.757359 0.0404248
\(352\) 0 0
\(353\) 26.1127i 1.38984i −0.719088 0.694919i \(-0.755440\pi\)
0.719088 0.694919i \(-0.244560\pi\)
\(354\) 0 0
\(355\) −6.51472 + 2.17157i −0.345765 + 0.115255i
\(356\) 0 0
\(357\) 5.82843i 0.308473i
\(358\) 0 0
\(359\) 8.07107 0.425975 0.212987 0.977055i \(-0.431681\pi\)
0.212987 + 0.977055i \(0.431681\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.7279i 1.14042i
\(364\) 0 0
\(365\) 9.77817 + 29.3345i 0.511813 + 1.53544i
\(366\) 0 0
\(367\) 17.4558i 0.911188i 0.890188 + 0.455594i \(0.150573\pi\)
−0.890188 + 0.455594i \(0.849427\pi\)
\(368\) 0 0
\(369\) 20.0000 1.04116
\(370\) 0 0
\(371\) −4.41421 −0.229175
\(372\) 0 0
\(373\) 5.97056i 0.309144i 0.987982 + 0.154572i \(0.0493999\pi\)
−0.987982 + 0.154572i \(0.950600\pi\)
\(374\) 0 0
\(375\) −22.1924 15.3640i −1.14601 0.793392i
\(376\) 0 0
\(377\) 7.00000i 0.360518i
\(378\) 0 0
\(379\) 33.3848 1.71486 0.857430 0.514600i \(-0.172060\pi\)
0.857430 + 0.514600i \(0.172060\pi\)
\(380\) 0 0
\(381\) 26.9706 1.38174
\(382\) 0 0
\(383\) 4.72792i 0.241586i −0.992678 0.120793i \(-0.961456\pi\)
0.992678 0.120793i \(-0.0385437\pi\)
\(384\) 0 0
\(385\) 2.41421 + 7.24264i 0.123040 + 0.369119i
\(386\) 0 0
\(387\) 6.34315i 0.322440i
\(388\) 0 0
\(389\) 11.8995 0.603328 0.301664 0.953414i \(-0.402458\pi\)
0.301664 + 0.953414i \(0.402458\pi\)
\(390\) 0 0
\(391\) −5.24264 −0.265132
\(392\) 0 0
\(393\) 52.2843i 2.63739i
\(394\) 0 0
\(395\) −1.75736 + 0.585786i −0.0884223 + 0.0294741i
\(396\) 0 0
\(397\) 22.6274i 1.13564i 0.823154 + 0.567819i \(0.192213\pi\)
−0.823154 + 0.567819i \(0.807787\pi\)
\(398\) 0 0
\(399\) 5.82843 0.291786
\(400\) 0 0
\(401\) −25.8995 −1.29336 −0.646680 0.762762i \(-0.723843\pi\)
−0.646680 + 0.762762i \(0.723843\pi\)
\(402\) 0 0
\(403\) 6.24264i 0.310968i
\(404\) 0 0
\(405\) −20.1213 + 6.70711i −0.999836 + 0.333279i
\(406\) 0 0
\(407\) 7.31371i 0.362527i
\(408\) 0 0
\(409\) −5.41421 −0.267716 −0.133858 0.991001i \(-0.542737\pi\)
−0.133858 + 0.991001i \(0.542737\pi\)
\(410\) 0 0
\(411\) −39.3848 −1.94271
\(412\) 0 0
\(413\) 34.7990i 1.71235i
\(414\) 0 0
\(415\) 1.75736 + 5.27208i 0.0862654 + 0.258796i
\(416\) 0 0
\(417\) 9.65685i 0.472898i
\(418\) 0 0
\(419\) 15.2132 0.743214 0.371607 0.928390i \(-0.378807\pi\)
0.371607 + 0.928390i \(0.378807\pi\)
\(420\) 0 0
\(421\) −15.1421 −0.737983 −0.368991 0.929433i \(-0.620297\pi\)
−0.368991 + 0.929433i \(0.620297\pi\)
\(422\) 0 0
\(423\) 22.6274i 1.10018i
\(424\) 0 0
\(425\) −3.00000 4.00000i −0.145521 0.194029i
\(426\) 0 0
\(427\) 8.24264i 0.398889i
\(428\) 0 0
\(429\) −6.24264 −0.301398
\(430\) 0 0
\(431\) 2.58579 0.124553 0.0622765 0.998059i \(-0.480164\pi\)
0.0622765 + 0.998059i \(0.480164\pi\)
\(432\) 0 0
\(433\) 9.07107i 0.435928i −0.975957 0.217964i \(-0.930059\pi\)
0.975957 0.217964i \(-0.0699415\pi\)
\(434\) 0 0
\(435\) 6.53553 + 19.6066i 0.313355 + 0.940065i
\(436\) 0 0
\(437\) 5.24264i 0.250790i
\(438\) 0 0
\(439\) −5.27208 −0.251623 −0.125811 0.992054i \(-0.540153\pi\)
−0.125811 + 0.992054i \(0.540153\pi\)
\(440\) 0 0
\(441\) −3.31371 −0.157796
\(442\) 0 0
\(443\) 15.5563i 0.739104i 0.929210 + 0.369552i \(0.120489\pi\)
−0.929210 + 0.369552i \(0.879511\pi\)
\(444\) 0 0
\(445\) 7.97056 2.65685i 0.377841 0.125947i
\(446\) 0 0
\(447\) 42.6274i 2.01621i
\(448\) 0 0
\(449\) −31.9411 −1.50739 −0.753697 0.657222i \(-0.771732\pi\)
−0.753697 + 0.657222i \(0.771732\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 26.9706i 1.26719i
\(454\) 0 0
\(455\) 9.36396 3.12132i 0.438989 0.146330i
\(456\) 0 0
\(457\) 6.31371i 0.295343i 0.989036 + 0.147671i \(0.0471778\pi\)
−0.989036 + 0.147671i \(0.952822\pi\)
\(458\) 0 0
\(459\) −0.414214 −0.0193338
\(460\) 0 0
\(461\) −32.7279 −1.52429 −0.762146 0.647406i \(-0.775854\pi\)
−0.762146 + 0.647406i \(0.775854\pi\)
\(462\) 0 0
\(463\) 6.14214i 0.285449i 0.989762 + 0.142725i \(0.0455863\pi\)
−0.989762 + 0.142725i \(0.954414\pi\)
\(464\) 0 0
\(465\) 5.82843 + 17.4853i 0.270287 + 0.810861i
\(466\) 0 0
\(467\) 2.10051i 0.0971998i −0.998818 0.0485999i \(-0.984524\pi\)
0.998818 0.0485999i \(-0.0154759\pi\)
\(468\) 0 0
\(469\) −14.6569 −0.676791
\(470\) 0 0
\(471\) −0.828427 −0.0381719
\(472\) 0 0
\(473\) 3.17157i 0.145829i
\(474\) 0 0
\(475\) −4.00000 + 3.00000i −0.183533 + 0.137649i
\(476\) 0 0
\(477\) 5.17157i 0.236790i
\(478\) 0 0
\(479\) −17.4558 −0.797578 −0.398789 0.917043i \(-0.630569\pi\)
−0.398789 + 0.917043i \(0.630569\pi\)
\(480\) 0 0
\(481\) 9.45584 0.431149
\(482\) 0 0
\(483\) 30.5563i 1.39036i
\(484\) 0 0
\(485\) −5.41421 16.2426i −0.245847 0.737540i
\(486\) 0 0
\(487\) 12.8284i 0.581312i 0.956828 + 0.290656i \(0.0938734\pi\)
−0.956828 + 0.290656i \(0.906127\pi\)
\(488\) 0 0
\(489\) −7.07107 −0.319765
\(490\) 0 0
\(491\) −26.2426 −1.18431 −0.592157 0.805823i \(-0.701723\pi\)
−0.592157 + 0.805823i \(0.701723\pi\)
\(492\) 0 0
\(493\) 3.82843i 0.172424i
\(494\) 0 0
\(495\) 8.48528 2.82843i 0.381385 0.127128i
\(496\) 0 0
\(497\) 7.41421i 0.332573i
\(498\) 0 0
\(499\) 33.0711 1.48046 0.740232 0.672351i \(-0.234716\pi\)
0.740232 + 0.672351i \(0.234716\pi\)
\(500\) 0 0
\(501\) 56.0416 2.50376
\(502\) 0 0
\(503\) 13.1005i 0.584123i −0.956400 0.292061i \(-0.905659\pi\)
0.956400 0.292061i \(-0.0943411\pi\)
\(504\) 0 0
\(505\) 33.7279 11.2426i 1.50087 0.500291i
\(506\) 0 0
\(507\) 23.3137i 1.03540i
\(508\) 0 0
\(509\) 1.65685 0.0734388 0.0367194 0.999326i \(-0.488309\pi\)
0.0367194 + 0.999326i \(0.488309\pi\)
\(510\) 0 0
\(511\) 33.3848 1.47686
\(512\) 0 0
\(513\) 0.414214i 0.0182880i
\(514\) 0 0
\(515\) 7.00000 + 21.0000i 0.308457 + 0.925371i
\(516\) 0 0
\(517\) 11.3137i 0.497576i
\(518\) 0 0
\(519\) 45.4558 1.99529
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 22.2132i 0.971316i −0.874149 0.485658i \(-0.838580\pi\)
0.874149 0.485658i \(-0.161420\pi\)
\(524\) 0 0
\(525\) −23.3137 + 17.4853i −1.01749 + 0.763120i
\(526\) 0 0
\(527\) 3.41421i 0.148725i
\(528\) 0 0
\(529\) −4.48528 −0.195012
\(530\) 0 0
\(531\) −40.7696 −1.76925
\(532\) 0 0
\(533\) 12.9289i 0.560014i
\(534\) 0 0
\(535\) −1.70711 5.12132i −0.0738047 0.221414i
\(536\) 0 0
\(537\) 55.4558i 2.39310i
\(538\) 0 0
\(539\) −1.65685 −0.0713658
\(540\) 0 0
\(541\) −18.0416 −0.775670 −0.387835 0.921729i \(-0.626777\pi\)
−0.387835 + 0.921729i \(0.626777\pi\)
\(542\) 0 0
\(543\) 1.17157i 0.0502770i
\(544\) 0 0
\(545\) 10.6066 3.53553i 0.454337 0.151446i
\(546\) 0 0
\(547\) 43.9411i 1.87879i 0.342841 + 0.939393i \(0.388611\pi\)
−0.342841 + 0.939393i \(0.611389\pi\)
\(548\) 0 0
\(549\) −9.65685 −0.412144
\(550\) 0 0
\(551\) 3.82843 0.163096
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 0 0
\(555\) −26.4853 + 8.82843i −1.12424 + 0.374746i
\(556\) 0 0
\(557\) 20.6863i 0.876506i 0.898852 + 0.438253i \(0.144403\pi\)
−0.898852 + 0.438253i \(0.855597\pi\)
\(558\) 0 0
\(559\) 4.10051 0.173433
\(560\) 0 0
\(561\) 3.41421 0.144148
\(562\) 0 0
\(563\) 11.0294i 0.464835i 0.972616 + 0.232418i \(0.0746636\pi\)
−0.972616 + 0.232418i \(0.925336\pi\)
\(564\) 0 0
\(565\) −9.24264 27.7279i −0.388841 1.16652i
\(566\) 0 0
\(567\) 22.8995i 0.961688i
\(568\) 0 0
\(569\) −24.9706 −1.04682 −0.523410 0.852081i \(-0.675340\pi\)
−0.523410 + 0.852081i \(0.675340\pi\)
\(570\) 0 0
\(571\) 11.6985 0.489566 0.244783 0.969578i \(-0.421283\pi\)
0.244783 + 0.969578i \(0.421283\pi\)
\(572\) 0 0
\(573\) 27.1421i 1.13388i
\(574\) 0 0
\(575\) −15.7279 20.9706i −0.655900 0.874533i
\(576\) 0 0
\(577\) 25.0000i 1.04076i 0.853934 + 0.520382i \(0.174210\pi\)
−0.853934 + 0.520382i \(0.825790\pi\)
\(578\) 0 0
\(579\) 18.4853 0.768222
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 2.58579i 0.107092i
\(584\) 0 0
\(585\) −3.65685 10.9706i −0.151192 0.453577i
\(586\) 0 0
\(587\) 4.38478i 0.180979i 0.995897 + 0.0904895i \(0.0288432\pi\)
−0.995897 + 0.0904895i \(0.971157\pi\)
\(588\) 0 0
\(589\) 3.41421 0.140680
\(590\) 0 0
\(591\) −50.0416 −2.05844
\(592\) 0 0
\(593\) 42.0000i 1.72473i −0.506284 0.862367i \(-0.668981\pi\)
0.506284 0.862367i \(-0.331019\pi\)
\(594\) 0 0
\(595\) −5.12132 + 1.70711i −0.209954 + 0.0699846i
\(596\) 0 0
\(597\) 38.7990i 1.58794i
\(598\) 0 0
\(599\) −2.44365 −0.0998449 −0.0499224 0.998753i \(-0.515897\pi\)
−0.0499224 + 0.998753i \(0.515897\pi\)
\(600\) 0 0
\(601\) 31.5563 1.28721 0.643605 0.765358i \(-0.277438\pi\)
0.643605 + 0.765358i \(0.277438\pi\)
\(602\) 0 0
\(603\) 17.1716i 0.699281i
\(604\) 0 0
\(605\) −19.0919 + 6.36396i −0.776195 + 0.258732i
\(606\) 0 0
\(607\) 13.5147i 0.548546i 0.961652 + 0.274273i \(0.0884371\pi\)
−0.961652 + 0.274273i \(0.911563\pi\)
\(608\) 0 0
\(609\) 22.3137 0.904197
\(610\) 0 0
\(611\) −14.6274 −0.591762
\(612\) 0 0
\(613\) 22.7279i 0.917972i 0.888443 + 0.458986i \(0.151787\pi\)
−0.888443 + 0.458986i \(0.848213\pi\)
\(614\) 0 0
\(615\) 12.0711 + 36.2132i 0.486752 + 1.46026i
\(616\) 0 0
\(617\) 7.79899i 0.313976i −0.987601 0.156988i \(-0.949822\pi\)
0.987601 0.156988i \(-0.0501783\pi\)
\(618\) 0 0
\(619\) −30.3848 −1.22127 −0.610634 0.791913i \(-0.709085\pi\)
−0.610634 + 0.791913i \(0.709085\pi\)
\(620\) 0 0
\(621\) −2.17157 −0.0871422
\(622\) 0 0
\(623\) 9.07107i 0.363425i
\(624\) 0 0
\(625\) 7.00000 24.0000i 0.280000 0.960000i
\(626\) 0 0
\(627\) 3.41421i 0.136351i
\(628\) 0 0
\(629\) −5.17157 −0.206204
\(630\) 0 0
\(631\) 38.9706 1.55139 0.775697 0.631106i \(-0.217399\pi\)
0.775697 + 0.631106i \(0.217399\pi\)
\(632\) 0 0
\(633\) 60.1127i 2.38927i
\(634\) 0 0
\(635\) 7.89949 + 23.6985i 0.313482 + 0.940446i
\(636\) 0 0
\(637\) 2.14214i 0.0848745i
\(638\) 0 0
\(639\) 8.68629 0.343624
\(640\) 0 0
\(641\) −39.2132 −1.54883 −0.774414 0.632679i \(-0.781955\pi\)
−0.774414 + 0.632679i \(0.781955\pi\)
\(642\) 0 0
\(643\) 31.1716i 1.22929i −0.788805 0.614643i \(-0.789300\pi\)
0.788805 0.614643i \(-0.210700\pi\)
\(644\) 0 0
\(645\) −11.4853 + 3.82843i −0.452233 + 0.150744i
\(646\) 0 0
\(647\) 12.0711i 0.474563i 0.971441 + 0.237281i \(0.0762563\pi\)
−0.971441 + 0.237281i \(0.923744\pi\)
\(648\) 0 0
\(649\) −20.3848 −0.800172
\(650\) 0 0
\(651\) 19.8995 0.779923
\(652\) 0 0
\(653\) 32.2843i 1.26338i −0.775221 0.631691i \(-0.782361\pi\)
0.775221 0.631691i \(-0.217639\pi\)
\(654\) 0 0
\(655\) −45.9411 + 15.3137i −1.79507 + 0.598356i
\(656\) 0 0
\(657\) 39.1127i 1.52593i
\(658\) 0 0
\(659\) −8.89949 −0.346675 −0.173338 0.984862i \(-0.555455\pi\)
−0.173338 + 0.984862i \(0.555455\pi\)
\(660\) 0 0
\(661\) 43.4853 1.69138 0.845691 0.533673i \(-0.179189\pi\)
0.845691 + 0.533673i \(0.179189\pi\)
\(662\) 0 0
\(663\) 4.41421i 0.171434i
\(664\) 0 0
\(665\) 1.70711 + 5.12132i 0.0661988 + 0.198596i
\(666\) 0 0
\(667\) 20.0711i 0.777155i
\(668\) 0 0
\(669\) −7.65685 −0.296031
\(670\) 0 0
\(671\) −4.82843 −0.186399
\(672\) 0 0
\(673\) 28.0000i 1.07932i 0.841883 + 0.539660i \(0.181447\pi\)
−0.841883 + 0.539660i \(0.818553\pi\)
\(674\) 0 0
\(675\) −1.24264 1.65685i −0.0478293 0.0637723i
\(676\) 0 0
\(677\) 21.3431i 0.820284i 0.912022 + 0.410142i \(0.134521\pi\)
−0.912022 + 0.410142i \(0.865479\pi\)
\(678\) 0 0
\(679\) −18.4853 −0.709400
\(680\) 0 0
\(681\) −41.6274 −1.59517
\(682\) 0 0
\(683\) 23.3137i 0.892074i −0.895014 0.446037i \(-0.852835\pi\)
0.895014 0.446037i \(-0.147165\pi\)
\(684\) 0 0
\(685\) −11.5355 34.6066i −0.440750 1.32225i
\(686\) 0 0
\(687\) 65.1127i 2.48420i
\(688\) 0 0
\(689\) 3.34315 0.127364
\(690\) 0 0
\(691\) −7.45584 −0.283634 −0.141817 0.989893i \(-0.545294\pi\)
−0.141817 + 0.989893i \(0.545294\pi\)
\(692\) 0 0
\(693\) 9.65685i 0.366834i
\(694\) 0 0
\(695\) −8.48528 + 2.82843i −0.321865 + 0.107288i
\(696\) 0 0
\(697\) 7.07107i 0.267836i
\(698\) 0 0
\(699\) −5.65685 −0.213962
\(700\) 0 0
\(701\) 32.9706 1.24528 0.622640 0.782508i \(-0.286060\pi\)
0.622640 + 0.782508i \(0.286060\pi\)
\(702\) 0 0
\(703\) 5.17157i 0.195050i
\(704\) 0 0
\(705\) 40.9706 13.6569i 1.54304 0.514347i
\(706\) 0 0
\(707\) 38.3848i 1.44361i
\(708\) 0 0
\(709\) −10.6863 −0.401332 −0.200666 0.979660i \(-0.564311\pi\)
−0.200666 + 0.979660i \(0.564311\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) 17.8995i 0.670341i
\(714\) 0 0
\(715\) −1.82843 5.48528i −0.0683793 0.205138i
\(716\) 0 0
\(717\) 5.48528i 0.204852i
\(718\) 0 0
\(719\) 30.4142 1.13426 0.567129 0.823629i \(-0.308054\pi\)
0.567129 + 0.823629i \(0.308054\pi\)
\(720\) 0 0
\(721\) 23.8995 0.890064
\(722\) 0 0
\(723\) 13.6569i 0.507904i
\(724\) 0 0
\(725\) −15.3137 + 11.4853i −0.568737 + 0.426553i
\(726\) 0 0
\(727\) 1.78680i 0.0662686i −0.999451 0.0331343i \(-0.989451\pi\)
0.999451 0.0331343i \(-0.0105489\pi\)
\(728\) 0 0
\(729\) 23.8284 0.882534
\(730\) 0 0
\(731\) −2.24264 −0.0829471
\(732\) 0 0
\(733\) 10.6274i 0.392533i −0.980551 0.196266i \(-0.937118\pi\)
0.980551 0.196266i \(-0.0628817\pi\)
\(734\) 0 0
\(735\) −2.00000 6.00000i −0.0737711 0.221313i
\(736\) 0 0
\(737\) 8.58579i 0.316262i
\(738\) 0 0
\(739\) 7.41421 0.272736 0.136368 0.990658i \(-0.456457\pi\)
0.136368 + 0.990658i \(0.456457\pi\)
\(740\) 0 0
\(741\) −4.41421 −0.162160
\(742\) 0 0
\(743\) 14.2426i 0.522512i −0.965270 0.261256i \(-0.915863\pi\)
0.965270 0.261256i \(-0.0841366\pi\)
\(744\) 0 0
\(745\) 37.4558 12.4853i 1.37228 0.457425i
\(746\) 0 0
\(747\) 7.02944i 0.257194i
\(748\) 0 0
\(749\) −5.82843 −0.212966
\(750\) 0 0
\(751\) −29.7574 −1.08586 −0.542931 0.839777i \(-0.682685\pi\)
−0.542931 + 0.839777i \(0.682685\pi\)
\(752\) 0 0
\(753\) 66.5269i 2.42438i
\(754\) 0 0
\(755\) −23.6985 + 7.89949i −0.862476 + 0.287492i
\(756\) 0 0
\(757\) 16.3431i 0.594002i 0.954877 + 0.297001i \(0.0959864\pi\)
−0.954877 + 0.297001i \(0.904014\pi\)
\(758\) 0 0
\(759\) 17.8995 0.649711
\(760\) 0 0
\(761\) −4.02944 −0.146067 −0.0730335 0.997329i \(-0.523268\pi\)
−0.0730335 + 0.997329i \(0.523268\pi\)
\(762\) 0 0
\(763\) 12.0711i 0.437002i
\(764\) 0 0
\(765\) 2.00000 + 6.00000i 0.0723102 + 0.216930i
\(766\) 0 0
\(767\) 26.3553i 0.951636i
\(768\) 0 0
\(769\) 30.7990 1.11064 0.555320 0.831637i \(-0.312596\pi\)
0.555320 + 0.831637i \(0.312596\pi\)
\(770\) 0 0
\(771\) −7.89949 −0.284493
\(772\) 0 0
\(773\) 9.00000i 0.323708i 0.986815 + 0.161854i \(0.0517473\pi\)
−0.986815 + 0.161854i \(0.948253\pi\)
\(774\) 0 0
\(775\) −13.6569 + 10.2426i −0.490569 + 0.367927i
\(776\) 0 0
\(777\) 30.1421i 1.08134i
\(778\) 0 0
\(779\) 7.07107 0.253347
\(780\) 0 0
\(781\) 4.34315 0.155410
\(782\) 0 0
\(783\) 1.58579i 0.0566714i
\(784\) 0 0
\(785\) −0.242641 0.727922i −0.00866022 0.0259807i
\(786\) 0 0
\(787\) 22.7574i 0.811212i −0.914048 0.405606i \(-0.867060\pi\)
0.914048 0.405606i \(-0.132940\pi\)
\(788\) 0 0
\(789\) 10.4853 0.373286
\(790\) 0 0
\(791\) −31.5563 −1.12201
\(792\) 0 0
\(793\) 6.24264i 0.221683i
\(794\) 0 0
\(795\) −9.36396 + 3.12132i −0.332105 + 0.110702i
\(796\) 0 0
\(797\) 35.1421i 1.24480i −0.782700 0.622399i \(-0.786158\pi\)
0.782700 0.622399i \(-0.213842\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −10.6274 −0.375501
\(802\) 0 0
\(803\)