Properties

Label 4032.2.v.e.1583.13
Level 4032
Weight 2
Character 4032.1583
Analytic conductor 32.196
Analytic rank 0
Dimension 40
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1583.13
Character \(\chi\) = 4032.1583
Dual form 4032.2.v.e.3599.13

$q$-expansion

\(f(q)\) \(=\) \(q+(0.667815 + 0.667815i) q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+(0.667815 + 0.667815i) q^{5} -1.00000 q^{7} +(-1.57551 + 1.57551i) q^{11} +(-1.83034 - 1.83034i) q^{13} -3.40687i q^{17} +(-3.18485 + 3.18485i) q^{19} -0.793288i q^{23} -4.10805i q^{25} +(-1.73542 + 1.73542i) q^{29} +3.28367i q^{31} +(-0.667815 - 0.667815i) q^{35} +(7.72049 - 7.72049i) q^{37} +7.19799 q^{41} +(5.84265 + 5.84265i) q^{43} +13.0051 q^{47} +1.00000 q^{49} +(3.34052 + 3.34052i) q^{53} -2.10430 q^{55} +(7.41533 - 7.41533i) q^{59} +(1.93050 + 1.93050i) q^{61} -2.44465i q^{65} +(-6.38033 + 6.38033i) q^{67} -3.41542i q^{71} +8.13689i q^{73} +(1.57551 - 1.57551i) q^{77} -0.0502773i q^{79} +(-2.29129 - 2.29129i) q^{83} +(2.27516 - 2.27516i) q^{85} -7.18090 q^{89} +(1.83034 + 1.83034i) q^{91} -4.25379 q^{95} +1.49996 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{7} + O(q^{10}) \) \( 40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 0.667815 + 0.667815i 0.298656 + 0.298656i 0.840487 0.541831i \(-0.182269\pi\)
−0.541831 + 0.840487i \(0.682269\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.57551 + 1.57551i −0.475033 + 0.475033i −0.903539 0.428506i \(-0.859040\pi\)
0.428506 + 0.903539i \(0.359040\pi\)
\(12\) 0 0
\(13\) −1.83034 1.83034i −0.507644 0.507644i 0.406159 0.913803i \(-0.366868\pi\)
−0.913803 + 0.406159i \(0.866868\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.40687i 0.826287i −0.910666 0.413143i \(-0.864431\pi\)
0.910666 0.413143i \(-0.135569\pi\)
\(18\) 0 0
\(19\) −3.18485 + 3.18485i −0.730656 + 0.730656i −0.970750 0.240094i \(-0.922822\pi\)
0.240094 + 0.970750i \(0.422822\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.793288i 0.165412i −0.996574 0.0827060i \(-0.973644\pi\)
0.996574 0.0827060i \(-0.0263562\pi\)
\(24\) 0 0
\(25\) 4.10805i 0.821609i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.73542 + 1.73542i −0.322260 + 0.322260i −0.849634 0.527373i \(-0.823177\pi\)
0.527373 + 0.849634i \(0.323177\pi\)
\(30\) 0 0
\(31\) 3.28367i 0.589765i 0.955534 + 0.294882i \(0.0952805\pi\)
−0.955534 + 0.294882i \(0.904720\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.667815 0.667815i −0.112881 0.112881i
\(36\) 0 0
\(37\) 7.72049 7.72049i 1.26924 1.26924i 0.322760 0.946481i \(-0.395389\pi\)
0.946481 0.322760i \(-0.104611\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.19799 1.12414 0.562069 0.827091i \(-0.310006\pi\)
0.562069 + 0.827091i \(0.310006\pi\)
\(42\) 0 0
\(43\) 5.84265 + 5.84265i 0.890996 + 0.890996i 0.994617 0.103621i \(-0.0330430\pi\)
−0.103621 + 0.994617i \(0.533043\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.0051 1.89699 0.948497 0.316786i \(-0.102604\pi\)
0.948497 + 0.316786i \(0.102604\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.34052 + 3.34052i 0.458856 + 0.458856i 0.898280 0.439424i \(-0.144817\pi\)
−0.439424 + 0.898280i \(0.644817\pi\)
\(54\) 0 0
\(55\) −2.10430 −0.283743
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.41533 7.41533i 0.965394 0.965394i −0.0340273 0.999421i \(-0.510833\pi\)
0.999421 + 0.0340273i \(0.0108333\pi\)
\(60\) 0 0
\(61\) 1.93050 + 1.93050i 0.247175 + 0.247175i 0.819810 0.572635i \(-0.194079\pi\)
−0.572635 + 0.819810i \(0.694079\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.44465i 0.303222i
\(66\) 0 0
\(67\) −6.38033 + 6.38033i −0.779481 + 0.779481i −0.979742 0.200262i \(-0.935821\pi\)
0.200262 + 0.979742i \(0.435821\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.41542i 0.405336i −0.979247 0.202668i \(-0.935039\pi\)
0.979247 0.202668i \(-0.0649612\pi\)
\(72\) 0 0
\(73\) 8.13689i 0.952351i 0.879350 + 0.476176i \(0.157977\pi\)
−0.879350 + 0.476176i \(0.842023\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.57551 1.57551i 0.179546 0.179546i
\(78\) 0 0
\(79\) 0.0502773i 0.00565663i −0.999996 0.00282832i \(-0.999100\pi\)
0.999996 0.00282832i \(-0.000900282\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.29129 2.29129i −0.251502 0.251502i 0.570084 0.821586i \(-0.306911\pi\)
−0.821586 + 0.570084i \(0.806911\pi\)
\(84\) 0 0
\(85\) 2.27516 2.27516i 0.246776 0.246776i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.18090 −0.761174 −0.380587 0.924745i \(-0.624278\pi\)
−0.380587 + 0.924745i \(0.624278\pi\)
\(90\) 0 0
\(91\) 1.83034 + 1.83034i 0.191871 + 0.191871i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.25379 −0.436429
\(96\) 0 0
\(97\) 1.49996 0.152298 0.0761490 0.997096i \(-0.475738\pi\)
0.0761490 + 0.997096i \(0.475738\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.08005 + 6.08005i 0.604988 + 0.604988i 0.941632 0.336644i \(-0.109292\pi\)
−0.336644 + 0.941632i \(0.609292\pi\)
\(102\) 0 0
\(103\) 7.74912 0.763543 0.381772 0.924257i \(-0.375314\pi\)
0.381772 + 0.924257i \(0.375314\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.05853 2.05853i 0.199006 0.199006i −0.600568 0.799574i \(-0.705059\pi\)
0.799574 + 0.600568i \(0.205059\pi\)
\(108\) 0 0
\(109\) 7.13047 + 7.13047i 0.682975 + 0.682975i 0.960670 0.277694i \(-0.0895702\pi\)
−0.277694 + 0.960670i \(0.589570\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.17748i 0.392984i −0.980505 0.196492i \(-0.937045\pi\)
0.980505 0.196492i \(-0.0629550\pi\)
\(114\) 0 0
\(115\) 0.529770 0.529770i 0.0494013 0.0494013i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.40687i 0.312307i
\(120\) 0 0
\(121\) 6.03555i 0.548686i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.08249 6.08249i 0.544035 0.544035i
\(126\) 0 0
\(127\) 5.16063i 0.457932i 0.973434 + 0.228966i \(0.0735345\pi\)
−0.973434 + 0.228966i \(0.926466\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.53587 1.53587i −0.134189 0.134189i 0.636822 0.771011i \(-0.280249\pi\)
−0.771011 + 0.636822i \(0.780249\pi\)
\(132\) 0 0
\(133\) 3.18485 3.18485i 0.276162 0.276162i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.0645 1.71423 0.857113 0.515128i \(-0.172256\pi\)
0.857113 + 0.515128i \(0.172256\pi\)
\(138\) 0 0
\(139\) 3.01947 + 3.01947i 0.256108 + 0.256108i 0.823469 0.567361i \(-0.192036\pi\)
−0.567361 + 0.823469i \(0.692036\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.76742 0.482296
\(144\) 0 0
\(145\) −2.31789 −0.192490
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.11391 + 2.11391i 0.173178 + 0.173178i 0.788374 0.615196i \(-0.210923\pi\)
−0.615196 + 0.788374i \(0.710923\pi\)
\(150\) 0 0
\(151\) −7.08056 −0.576208 −0.288104 0.957599i \(-0.593025\pi\)
−0.288104 + 0.957599i \(0.593025\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.19289 + 2.19289i −0.176137 + 0.176137i
\(156\) 0 0
\(157\) 16.0565 + 16.0565i 1.28144 + 1.28144i 0.939846 + 0.341598i \(0.110968\pi\)
0.341598 + 0.939846i \(0.389032\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.793288i 0.0625199i
\(162\) 0 0
\(163\) −10.9257 + 10.9257i −0.855764 + 0.855764i −0.990836 0.135072i \(-0.956873\pi\)
0.135072 + 0.990836i \(0.456873\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.8556i 1.14956i −0.818307 0.574782i \(-0.805087\pi\)
0.818307 0.574782i \(-0.194913\pi\)
\(168\) 0 0
\(169\) 6.29974i 0.484595i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.49652 2.49652i 0.189807 0.189807i −0.605806 0.795613i \(-0.707149\pi\)
0.795613 + 0.605806i \(0.207149\pi\)
\(174\) 0 0
\(175\) 4.10805i 0.310539i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.5658 13.5658i −1.01396 1.01396i −0.999901 0.0140578i \(-0.995525\pi\)
−0.0140578 0.999901i \(1.49553\pi\)
\(180\) 0 0
\(181\) −4.13837 + 4.13837i −0.307602 + 0.307602i −0.843979 0.536376i \(-0.819793\pi\)
0.536376 + 0.843979i \(0.319793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.3117 0.758133
\(186\) 0 0
\(187\) 5.36755 + 5.36755i 0.392514 + 0.392514i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.9065 1.58510 0.792549 0.609808i \(-0.208754\pi\)
0.792549 + 0.609808i \(0.208754\pi\)
\(192\) 0 0
\(193\) −21.1429 −1.52190 −0.760950 0.648811i \(-0.775267\pi\)
−0.760950 + 0.648811i \(0.775267\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.95026 + 4.95026i 0.352692 + 0.352692i 0.861110 0.508418i \(-0.169770\pi\)
−0.508418 + 0.861110i \(0.669770\pi\)
\(198\) 0 0
\(199\) −10.6455 −0.754639 −0.377319 0.926083i \(-0.623154\pi\)
−0.377319 + 0.926083i \(0.623154\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.73542 1.73542i 0.121803 0.121803i
\(204\) 0 0
\(205\) 4.80693 + 4.80693i 0.335730 + 0.335730i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0355i 0.694172i
\(210\) 0 0
\(211\) −0.563683 + 0.563683i −0.0388056 + 0.0388056i −0.726243 0.687438i \(-0.758735\pi\)
0.687438 + 0.726243i \(0.258735\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.80362i 0.532202i
\(216\) 0 0
\(217\) 3.28367i 0.222910i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.23571 + 6.23571i −0.419459 + 0.419459i
\(222\) 0 0
\(223\) 21.6604i 1.45049i −0.688493 0.725243i \(-0.741727\pi\)
0.688493 0.725243i \(-0.258273\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.98117 2.98117i −0.197867 0.197867i 0.601218 0.799085i \(-0.294682\pi\)
−0.799085 + 0.601218i \(0.794682\pi\)
\(228\) 0 0
\(229\) −10.1998 + 10.1998i −0.674019 + 0.674019i −0.958640 0.284621i \(-0.908132\pi\)
0.284621 + 0.958640i \(0.408132\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.9408 1.10983 0.554914 0.831907i \(-0.312751\pi\)
0.554914 + 0.831907i \(0.312751\pi\)
\(234\) 0 0
\(235\) 8.68503 + 8.68503i 0.566549 + 0.566549i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.7743 0.761618 0.380809 0.924654i \(-0.375646\pi\)
0.380809 + 0.924654i \(0.375646\pi\)
\(240\) 0 0
\(241\) −1.77496 −0.114335 −0.0571677 0.998365i \(-0.518207\pi\)
−0.0571677 + 0.998365i \(0.518207\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.667815 + 0.667815i 0.0426652 + 0.0426652i
\(246\) 0 0
\(247\) 11.6587 0.741826
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.7495 + 10.7495i −0.678503 + 0.678503i −0.959661 0.281158i \(-0.909281\pi\)
0.281158 + 0.959661i \(0.409281\pi\)
\(252\) 0 0
\(253\) 1.24983 + 1.24983i 0.0785762 + 0.0785762i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.2878i 1.45265i −0.687351 0.726325i \(-0.741227\pi\)
0.687351 0.726325i \(-0.258773\pi\)
\(258\) 0 0
\(259\) −7.72049 + 7.72049i −0.479728 + 0.479728i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.00910i 0.308874i −0.988003 0.154437i \(-0.950644\pi\)
0.988003 0.154437i \(-0.0493564\pi\)
\(264\) 0 0
\(265\) 4.46170i 0.274080i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.1632 11.1632i 0.680633 0.680633i −0.279510 0.960143i \(-0.590172\pi\)
0.960143 + 0.279510i \(0.0901718\pi\)
\(270\) 0 0
\(271\) 16.1939i 0.983706i −0.870678 0.491853i \(-0.836320\pi\)
0.870678 0.491853i \(-0.163680\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.47226 + 6.47226i 0.390292 + 0.390292i
\(276\) 0 0
\(277\) 19.0299 19.0299i 1.14339 1.14339i 0.155568 0.987825i \(-0.450279\pi\)
0.987825 0.155568i \(-0.0497207\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.3974 −0.858875 −0.429438 0.903097i \(-0.641288\pi\)
−0.429438 + 0.903097i \(0.641288\pi\)
\(282\) 0 0
\(283\) 21.2690 + 21.2690i 1.26431 + 1.26431i 0.948982 + 0.315331i \(0.102116\pi\)
0.315331 + 0.948982i \(0.397884\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.19799 −0.424884
\(288\) 0 0
\(289\) 5.39325 0.317250
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.9536 + 18.9536i 1.10728 + 1.10728i 0.993507 + 0.113775i \(0.0362942\pi\)
0.113775 + 0.993507i \(0.463706\pi\)
\(294\) 0 0
\(295\) 9.90414 0.576641
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.45198 + 1.45198i −0.0839704 + 0.0839704i
\(300\) 0 0
\(301\) −5.84265 5.84265i −0.336765 0.336765i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.57843i 0.147641i
\(306\) 0 0
\(307\) 7.90363 7.90363i 0.451084 0.451084i −0.444630 0.895714i \(-0.646665\pi\)
0.895714 + 0.444630i \(0.146665\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.8764i 1.41061i −0.708904 0.705305i \(-0.750810\pi\)
0.708904 0.705305i \(-0.249190\pi\)
\(312\) 0 0
\(313\) 28.7289i 1.62385i 0.583759 + 0.811927i \(0.301581\pi\)
−0.583759 + 0.811927i \(0.698419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.354075 0.354075i 0.0198869 0.0198869i −0.697093 0.716980i \(-0.745524\pi\)
0.716980 + 0.697093i \(0.245524\pi\)
\(318\) 0 0
\(319\) 5.46835i 0.306169i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.8504 + 10.8504i 0.603731 + 0.603731i
\(324\) 0 0
\(325\) −7.51910 + 7.51910i −0.417085 + 0.417085i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −13.0051 −0.716996
\(330\) 0 0
\(331\) 11.1492 + 11.1492i 0.612818 + 0.612818i 0.943679 0.330862i \(-0.107339\pi\)
−0.330862 + 0.943679i \(0.607339\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.52176 −0.465593
\(336\) 0 0
\(337\) −32.8646 −1.79025 −0.895124 0.445817i \(-0.852913\pi\)
−0.895124 + 0.445817i \(0.852913\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.17345 5.17345i −0.280158 0.280158i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −11.4329 + 11.4329i −0.613751 + 0.613751i −0.943921 0.330170i \(-0.892894\pi\)
0.330170 + 0.943921i \(0.392894\pi\)
\(348\) 0 0
\(349\) 19.4418 + 19.4418i 1.04070 + 1.04070i 0.999136 + 0.0415602i \(0.0132328\pi\)
0.0415602 + 0.999136i \(0.486767\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.70827i 0.516719i −0.966049 0.258360i \(-0.916818\pi\)
0.966049 0.258360i \(-0.0831819\pi\)
\(354\) 0 0
\(355\) 2.28087 2.28087i 0.121056 0.121056i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.535153i 0.0282443i 0.999900 + 0.0141222i \(0.00449537\pi\)
−0.999900 + 0.0141222i \(0.995505\pi\)
\(360\) 0 0
\(361\) 1.28659i 0.0677152i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.43394 + 5.43394i −0.284426 + 0.284426i
\(366\) 0 0
\(367\) 14.3854i 0.750913i −0.926840 0.375456i \(-0.877486\pi\)
0.926840 0.375456i \(-0.122514\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.34052 3.34052i −0.173431 0.173431i
\(372\) 0 0
\(373\) 12.7345 12.7345i 0.659368 0.659368i −0.295862 0.955231i \(-0.595607\pi\)
0.955231 + 0.295862i \(0.0956070\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.35282 0.327187
\(378\) 0 0
\(379\) −18.4758 18.4758i −0.949037 0.949037i 0.0497262 0.998763i \(-0.484165\pi\)
−0.998763 + 0.0497262i \(0.984165\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 29.0133 1.48251 0.741256 0.671222i \(-0.234230\pi\)
0.741256 + 0.671222i \(0.234230\pi\)
\(384\) 0 0
\(385\) 2.10430 0.107245
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.75723 7.75723i −0.393307 0.393307i 0.482557 0.875864i \(-0.339708\pi\)
−0.875864 + 0.482557i \(0.839708\pi\)
\(390\) 0 0
\(391\) −2.70263 −0.136678
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.0335759 0.0335759i 0.00168939 0.00168939i
\(396\) 0 0
\(397\) 13.0436 + 13.0436i 0.654638 + 0.654638i 0.954106 0.299468i \(-0.0968093\pi\)
−0.299468 + 0.954106i \(0.596809\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.31443i 0.115577i 0.998329 + 0.0577887i \(0.0184050\pi\)
−0.998329 + 0.0577887i \(0.981595\pi\)
\(402\) 0 0
\(403\) 6.01022 6.01022i 0.299390 0.299390i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.3274i 1.20586i
\(408\) 0 0
\(409\) 3.87844i 0.191777i −0.995392 0.0958883i \(-0.969431\pi\)
0.995392 0.0958883i \(-0.0305692\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.41533 + 7.41533i −0.364884 + 0.364884i
\(414\) 0 0
\(415\) 3.06032i 0.150225i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.40127 1.40127i −0.0684563 0.0684563i 0.672050 0.740506i \(-0.265414\pi\)
−0.740506 + 0.672050i \(0.765414\pi\)
\(420\) 0 0
\(421\) 7.61424 7.61424i 0.371095 0.371095i −0.496781 0.867876i \(-0.665485\pi\)
0.867876 + 0.496781i \(0.165485\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −13.9956 −0.678885
\(426\) 0 0
\(427\) −1.93050 1.93050i −0.0934234 0.0934234i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.496260 −0.0239040 −0.0119520 0.999929i \(-0.503805\pi\)
−0.0119520 + 0.999929i \(0.503805\pi\)
\(432\) 0 0
\(433\) −32.9889 −1.58535 −0.792673 0.609647i \(-0.791311\pi\)
−0.792673 + 0.609647i \(0.791311\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.52651 + 2.52651i 0.120859 + 0.120859i
\(438\) 0 0
\(439\) 28.1034 1.34130 0.670651 0.741773i \(-0.266015\pi\)
0.670651 + 0.741773i \(0.266015\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.8047 23.8047i 1.13099 1.13099i 0.140981 0.990012i \(-0.454974\pi\)
0.990012 0.140981i \(-0.0450257\pi\)
\(444\) 0 0
\(445\) −4.79552 4.79552i −0.227329 0.227329i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.3175i 1.28919i −0.764523 0.644597i \(-0.777025\pi\)
0.764523 0.644597i \(-0.222975\pi\)
\(450\) 0 0
\(451\) −11.3405 + 11.3405i −0.534003 + 0.534003i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.44465i 0.114607i
\(456\) 0 0
\(457\) 3.28370i 0.153605i −0.997046 0.0768026i \(-0.975529\pi\)
0.997046 0.0768026i \(-0.0244711\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.3723 + 21.3723i −0.995407 + 0.995407i −0.999989 0.00458260i \(-0.998541\pi\)
0.00458260 + 0.999989i \(0.498541\pi\)
\(462\) 0 0
\(463\) 36.4190i 1.69254i −0.532758 0.846268i \(-0.678844\pi\)
0.532758 0.846268i \(-0.321156\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.4979 + 22.4979i 1.04108 + 1.04108i 0.999119 + 0.0419609i \(0.0133605\pi\)
0.0419609 + 0.999119i \(0.486639\pi\)
\(468\) 0 0
\(469\) 6.38033 6.38033i 0.294616 0.294616i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −18.4103 −0.846505
\(474\) 0 0
\(475\) 13.0835 + 13.0835i 0.600313 + 0.600313i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.8888 1.13720 0.568598 0.822616i \(-0.307486\pi\)
0.568598 + 0.822616i \(0.307486\pi\)
\(480\) 0 0
\(481\) −28.2622 −1.28864
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00170 + 1.00170i 0.0454847 + 0.0454847i
\(486\) 0 0
\(487\) −1.96454 −0.0890217 −0.0445108 0.999009i \(-0.514173\pi\)
−0.0445108 + 0.999009i \(0.514173\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.24360 2.24360i 0.101252 0.101252i −0.654666 0.755918i \(-0.727191\pi\)
0.755918 + 0.654666i \(0.227191\pi\)
\(492\) 0 0
\(493\) 5.91236 + 5.91236i 0.266279 + 0.266279i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.41542i 0.153203i
\(498\) 0 0
\(499\) −25.8623 + 25.8623i −1.15776 + 1.15776i −0.172800 + 0.984957i \(0.555282\pi\)
−0.984957 + 0.172800i \(0.944718\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 32.7702i 1.46115i 0.682833 + 0.730575i \(0.260748\pi\)
−0.682833 + 0.730575i \(0.739252\pi\)
\(504\) 0 0
\(505\) 8.12070i 0.361366i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.28132 + 1.28132i −0.0567935 + 0.0567935i −0.734933 0.678140i \(-0.762786\pi\)
0.678140 + 0.734933i \(0.262786\pi\)
\(510\) 0 0
\(511\) 8.13689i 0.359955i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.17498 + 5.17498i 0.228037 + 0.228037i
\(516\) 0 0
\(517\) −20.4897 + 20.4897i −0.901136 + 0.901136i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.9473 1.13677 0.568386 0.822762i \(-0.307568\pi\)
0.568386 + 0.822762i \(0.307568\pi\)
\(522\) 0 0
\(523\) −10.8853 10.8853i −0.475981 0.475981i 0.427862 0.903844i \(-0.359267\pi\)
−0.903844 + 0.427862i \(0.859267\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.1870 0.487315
\(528\) 0 0
\(529\) 22.3707 0.972639
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −13.1747 13.1747i −0.570661 0.570661i
\(534\) 0 0
\(535\) 2.74943 0.118868
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.57551 + 1.57551i −0.0678619 + 0.0678619i
\(540\) 0 0
\(541\) −11.4352 11.4352i −0.491637 0.491637i 0.417185 0.908822i \(-0.363017\pi\)
−0.908822 + 0.417185i \(0.863017\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.52368i 0.407949i
\(546\) 0 0
\(547\) −0.00768172 + 0.00768172i −0.000328447 + 0.000328447i −0.707271 0.706943i \(-0.750074\pi\)
0.706943 + 0.707271i \(0.250074\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.0541i 0.470922i
\(552\) 0 0
\(553\) 0.0502773i 0.00213801i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.247153 0.247153i 0.0104722 0.0104722i −0.701851 0.712324i \(-0.747643\pi\)
0.712324 + 0.701851i \(0.247643\pi\)
\(558\) 0 0
\(559\) 21.3880i 0.904617i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.42038 + 4.42038i 0.186297 + 0.186297i 0.794093 0.607796i \(-0.207946\pi\)
−0.607796 + 0.794093i \(0.707946\pi\)
\(564\) 0 0
\(565\) 2.78978 2.78978i 0.117367 0.117367i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.4012 −1.02295 −0.511476 0.859297i \(-0.670901\pi\)
−0.511476 + 0.859297i \(0.670901\pi\)
\(570\) 0 0
\(571\) −8.33750 8.33750i −0.348914 0.348914i 0.510791 0.859705i \(-0.329353\pi\)
−0.859705 + 0.510791i \(0.829353\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.25886 −0.135904
\(576\) 0 0
\(577\) 2.92478 0.121760 0.0608800 0.998145i \(-0.480609\pi\)
0.0608800 + 0.998145i \(0.480609\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.29129 + 2.29129i 0.0950589 + 0.0950589i
\(582\) 0 0
\(583\) −10.5260 −0.435944
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.6434 + 16.6434i −0.686945 + 0.686945i −0.961556 0.274611i \(-0.911451\pi\)
0.274611 + 0.961556i \(0.411451\pi\)
\(588\) 0 0
\(589\) −10.4580 10.4580i −0.430915 0.430915i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.7065i 0.686054i −0.939326 0.343027i \(-0.888548\pi\)
0.939326 0.343027i \(-0.111452\pi\)
\(594\) 0 0
\(595\) −2.27516 + 2.27516i −0.0932724 + 0.0932724i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 33.1559i 1.35471i −0.735655 0.677356i \(-0.763126\pi\)
0.735655 0.677356i \(-0.236874\pi\)
\(600\) 0 0
\(601\) 37.5909i 1.53336i −0.642027 0.766682i \(-0.721906\pi\)
0.642027 0.766682i \(-0.278094\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.03063 + 4.03063i −0.163869 + 0.163869i
\(606\) 0 0
\(607\) 0.720055i 0.0292261i 0.999893 + 0.0146131i \(0.00465165\pi\)
−0.999893 + 0.0146131i \(0.995348\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.8038 23.8038i −0.962998 0.962998i
\(612\) 0 0
\(613\) −11.6327 + 11.6327i −0.469839 + 0.469839i −0.901862 0.432024i \(-0.857800\pi\)
0.432024 + 0.901862i \(0.357800\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −47.3378 −1.90575 −0.952873 0.303369i \(-0.901889\pi\)
−0.952873 + 0.303369i \(0.901889\pi\)
\(618\) 0 0
\(619\) 10.9927 + 10.9927i 0.441832 + 0.441832i 0.892628 0.450795i \(-0.148859\pi\)
−0.450795 + 0.892628i \(0.648859\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.18090 0.287697
\(624\) 0 0
\(625\) −12.4163 −0.496651
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −26.3027 26.3027i −1.04876 1.04876i
\(630\) 0 0
\(631\) 1.33698 0.0532242 0.0266121 0.999646i \(-0.491528\pi\)
0.0266121 + 0.999646i \(0.491528\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.44635 + 3.44635i −0.136764 + 0.136764i
\(636\) 0 0
\(637\) −1.83034 1.83034i −0.0725206 0.0725206i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.9298i 1.06367i 0.846850 + 0.531833i \(0.178496\pi\)
−0.846850 + 0.531833i \(0.821504\pi\)
\(642\) 0 0
\(643\) 29.8299 29.8299i 1.17638 1.17638i 0.195716 0.980661i \(-0.437297\pi\)
0.980661 0.195716i \(-0.0627030\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.0946i 1.14383i 0.820314 + 0.571914i \(0.193799\pi\)
−0.820314 + 0.571914i \(0.806201\pi\)
\(648\) 0 0
\(649\) 23.3658i 0.917189i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.8462 24.8462i 0.972308 0.972308i −0.0273185 0.999627i \(-0.508697\pi\)
0.999627 + 0.0273185i \(0.00869683\pi\)
\(654\) 0 0
\(655\) 2.05135i 0.0801530i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22.2096 22.2096i −0.865164 0.865164i 0.126769 0.991932i \(-0.459539\pi\)
−0.991932 + 0.126769i \(0.959539\pi\)
\(660\) 0 0
\(661\) 18.7200 18.7200i 0.728122 0.728122i −0.242124 0.970245i \(-0.577844\pi\)
0.970245 + 0.242124i \(0.0778440\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.25379 0.164955
\(666\) 0 0
\(667\) 1.37669 + 1.37669i 0.0533057 + 0.0533057i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.08303 −0.234833
\(672\) 0 0
\(673\) 16.1228 0.621488 0.310744 0.950494i \(-0.399422\pi\)
0.310744 + 0.950494i \(0.399422\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.0537 12.0537i −0.463261 0.463261i 0.436462 0.899723i \(-0.356231\pi\)
−0.899723 + 0.436462i \(0.856231\pi\)
\(678\) 0 0
\(679\) −1.49996 −0.0575632
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −32.7416 + 32.7416i −1.25282 + 1.25282i −0.298372 + 0.954450i \(0.596444\pi\)
−0.954450 + 0.298372i \(0.903556\pi\)
\(684\) 0 0
\(685\) 13.3994 + 13.3994i 0.511964 + 0.511964i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.2285i 0.465871i
\(690\) 0 0
\(691\) 21.3170 21.3170i 0.810936 0.810936i −0.173838 0.984774i \(-0.555617\pi\)
0.984774 + 0.173838i \(0.0556169\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.03290i 0.152977i
\(696\) 0 0
\(697\) 24.5226i 0.928860i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −23.9475 + 23.9475i −0.904483 + 0.904483i −0.995820 0.0913369i \(-0.970886\pi\)
0.0913369 + 0.995820i \(0.470886\pi\)
\(702\) 0 0
\(703\) 49.1773i 1.85476i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.08005 6.08005i −0.228664 0.228664i
\(708\) 0 0
\(709\) 20.3189 20.3189i 0.763092 0.763092i −0.213788 0.976880i \(-0.568580\pi\)
0.976880 + 0.213788i \(0.0685801\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.60490 0.0975542
\(714\) 0 0
\(715\) 3.85157 + 3.85157i 0.144041 + 0.144041i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5.54623 −0.206839 −0.103420 0.994638i \(-0.532978\pi\)
−0.103420 + 0.994638i \(0.532978\pi\)
\(720\) 0 0
\(721\) −7.74912 −0.288592
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.12920 + 7.12920i 0.264772 + 0.264772i
\(726\) 0 0
\(727\) −35.2516 −1.30741 −0.653704 0.756750i \(-0.726786\pi\)
−0.653704 + 0.756750i \(0.726786\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19.9051 19.9051i 0.736218 0.736218i
\(732\) 0 0
\(733\) −25.7407 25.7407i −0.950753 0.950753i 0.0480899 0.998843i \(-0.484687\pi\)
−0.998843 + 0.0480899i \(0.984687\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.1045i 0.740559i
\(738\) 0 0
\(739\) −8.42027 + 8.42027i −0.309745 + 0.309745i −0.844810 0.535066i \(-0.820287\pi\)
0.535066 + 0.844810i \(0.320287\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.6054i 1.37961i 0.723996 + 0.689805i \(0.242304\pi\)
−0.723996 + 0.689805i \(0.757696\pi\)
\(744\) 0 0
\(745\) 2.82341i 0.103442i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.05853 + 2.05853i −0.0752170 + 0.0752170i
\(750\) 0 0
\(751\) 33.8331i 1.23459i 0.786732 + 0.617294i \(0.211771\pi\)
−0.786732 + 0.617294i \(0.788229\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.72850 4.72850i −0.172088 0.172088i
\(756\) 0 0
\(757\) −3.38072 + 3.38072i −0.122874 + 0.122874i −0.765870 0.642996i \(-0.777691\pi\)
0.642996 + 0.765870i \(0.277691\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16.7200 −0.606100 −0.303050 0.952975i \(-0.598005\pi\)
−0.303050 + 0.952975i \(0.598005\pi\)
\(762\) 0 0
\(763\) −7.13047 7.13047i −0.258140 0.258140i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −27.1451 −0.980152
\(768\) 0 0
\(769\) 13.6721 0.493030 0.246515 0.969139i \(-0.420715\pi\)
0.246515 + 0.969139i \(0.420715\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 29.3858 + 29.3858i 1.05694 + 1.05694i 0.998278 + 0.0586568i \(0.0186818\pi\)
0.0586568 + 0.998278i \(0.481318\pi\)
\(774\) 0 0
\(775\) 13.4895 0.484556
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −22.9245 + 22.9245i −0.821357 + 0.821357i
\(780\) 0 0
\(781\) 5.38103 + 5.38103i 0.192548 + 0.192548i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 21.4455i 0.765422i
\(786\) 0 0
\(787\) −31.3525 + 31.3525i −1.11760 + 1.11760i −0.125504 + 0.992093i \(0.540055\pi\)
−0.992093 + 0.125504i \(0.959945\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.17748i 0.148534i
\(792\) 0 0
\(793\) 7.06693i 0.250954i
\(794\) 0 0