Properties

Label 4012.2.b.b.237.3
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.3
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.44

$q$-expansion

\(f(q)\) \(=\) \(q-3.18403i q^{3} -1.45906i q^{5} -1.23253i q^{7} -7.13805 q^{9} +O(q^{10})\) \(q-3.18403i q^{3} -1.45906i q^{5} -1.23253i q^{7} -7.13805 q^{9} +5.92877i q^{11} -3.31160 q^{13} -4.64569 q^{15} +(-4.08333 + 0.571297i) q^{17} -2.44989 q^{19} -3.92443 q^{21} +4.79101i q^{23} +2.87114 q^{25} +13.1757i q^{27} -2.32008i q^{29} -1.94770i q^{31} +18.8774 q^{33} -1.79834 q^{35} -3.37438i q^{37} +10.5442i q^{39} -8.19338i q^{41} -7.42601 q^{43} +10.4148i q^{45} +2.88961 q^{47} +5.48086 q^{49} +(1.81903 + 13.0015i) q^{51} +1.48779 q^{53} +8.65044 q^{55} +7.80053i q^{57} +1.00000 q^{59} +11.0568i q^{61} +8.79789i q^{63} +4.83182i q^{65} +11.4333 q^{67} +15.2547 q^{69} -8.97142i q^{71} +16.1610i q^{73} -9.14180i q^{75} +7.30742 q^{77} +7.60558i q^{79} +20.5376 q^{81} +9.36739 q^{83} +(0.833557 + 5.95783i) q^{85} -7.38721 q^{87} +4.54564 q^{89} +4.08166i q^{91} -6.20153 q^{93} +3.57454i q^{95} +0.608891i q^{97} -42.3199i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.18403i 1.83830i −0.393907 0.919150i \(-0.628877\pi\)
0.393907 0.919150i \(-0.371123\pi\)
\(4\) 0 0
\(5\) 1.45906i 0.652512i −0.945281 0.326256i \(-0.894213\pi\)
0.945281 0.326256i \(-0.105787\pi\)
\(6\) 0 0
\(7\) 1.23253i 0.465854i −0.972494 0.232927i \(-0.925170\pi\)
0.972494 0.232927i \(-0.0748304\pi\)
\(8\) 0 0
\(9\) −7.13805 −2.37935
\(10\) 0 0
\(11\) 5.92877i 1.78759i 0.448474 + 0.893796i \(0.351968\pi\)
−0.448474 + 0.893796i \(0.648032\pi\)
\(12\) 0 0
\(13\) −3.31160 −0.918472 −0.459236 0.888314i \(-0.651877\pi\)
−0.459236 + 0.888314i \(0.651877\pi\)
\(14\) 0 0
\(15\) −4.64569 −1.19951
\(16\) 0 0
\(17\) −4.08333 + 0.571297i −0.990354 + 0.138560i
\(18\) 0 0
\(19\) −2.44989 −0.562044 −0.281022 0.959701i \(-0.590673\pi\)
−0.281022 + 0.959701i \(0.590673\pi\)
\(20\) 0 0
\(21\) −3.92443 −0.856380
\(22\) 0 0
\(23\) 4.79101i 0.998995i 0.866315 + 0.499497i \(0.166482\pi\)
−0.866315 + 0.499497i \(0.833518\pi\)
\(24\) 0 0
\(25\) 2.87114 0.574228
\(26\) 0 0
\(27\) 13.1757i 2.53566i
\(28\) 0 0
\(29\) 2.32008i 0.430828i −0.976523 0.215414i \(-0.930890\pi\)
0.976523 0.215414i \(-0.0691101\pi\)
\(30\) 0 0
\(31\) 1.94770i 0.349817i −0.984585 0.174908i \(-0.944037\pi\)
0.984585 0.174908i \(-0.0559630\pi\)
\(32\) 0 0
\(33\) 18.8774 3.28613
\(34\) 0 0
\(35\) −1.79834 −0.303976
\(36\) 0 0
\(37\) 3.37438i 0.554745i −0.960763 0.277372i \(-0.910536\pi\)
0.960763 0.277372i \(-0.0894635\pi\)
\(38\) 0 0
\(39\) 10.5442i 1.68843i
\(40\) 0 0
\(41\) 8.19338i 1.27959i −0.768545 0.639795i \(-0.779019\pi\)
0.768545 0.639795i \(-0.220981\pi\)
\(42\) 0 0
\(43\) −7.42601 −1.13246 −0.566228 0.824249i \(-0.691598\pi\)
−0.566228 + 0.824249i \(0.691598\pi\)
\(44\) 0 0
\(45\) 10.4148i 1.55255i
\(46\) 0 0
\(47\) 2.88961 0.421493 0.210746 0.977541i \(-0.432411\pi\)
0.210746 + 0.977541i \(0.432411\pi\)
\(48\) 0 0
\(49\) 5.48086 0.782980
\(50\) 0 0
\(51\) 1.81903 + 13.0015i 0.254715 + 1.82057i
\(52\) 0 0
\(53\) 1.48779 0.204364 0.102182 0.994766i \(-0.467418\pi\)
0.102182 + 0.994766i \(0.467418\pi\)
\(54\) 0 0
\(55\) 8.65044 1.16643
\(56\) 0 0
\(57\) 7.80053i 1.03321i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 11.0568i 1.41568i 0.706374 + 0.707838i \(0.250329\pi\)
−0.706374 + 0.707838i \(0.749671\pi\)
\(62\) 0 0
\(63\) 8.79789i 1.10843i
\(64\) 0 0
\(65\) 4.83182i 0.599314i
\(66\) 0 0
\(67\) 11.4333 1.39679 0.698397 0.715710i \(-0.253897\pi\)
0.698397 + 0.715710i \(0.253897\pi\)
\(68\) 0 0
\(69\) 15.2547 1.83645
\(70\) 0 0
\(71\) 8.97142i 1.06471i −0.846521 0.532356i \(-0.821307\pi\)
0.846521 0.532356i \(-0.178693\pi\)
\(72\) 0 0
\(73\) 16.1610i 1.89151i 0.324887 + 0.945753i \(0.394674\pi\)
−0.324887 + 0.945753i \(0.605326\pi\)
\(74\) 0 0
\(75\) 9.14180i 1.05560i
\(76\) 0 0
\(77\) 7.30742 0.832758
\(78\) 0 0
\(79\) 7.60558i 0.855694i 0.903851 + 0.427847i \(0.140728\pi\)
−0.903851 + 0.427847i \(0.859272\pi\)
\(80\) 0 0
\(81\) 20.5376 2.28195
\(82\) 0 0
\(83\) 9.36739 1.02820 0.514102 0.857729i \(-0.328125\pi\)
0.514102 + 0.857729i \(0.328125\pi\)
\(84\) 0 0
\(85\) 0.833557 + 5.95783i 0.0904119 + 0.646218i
\(86\) 0 0
\(87\) −7.38721 −0.791992
\(88\) 0 0
\(89\) 4.54564 0.481837 0.240919 0.970545i \(-0.422551\pi\)
0.240919 + 0.970545i \(0.422551\pi\)
\(90\) 0 0
\(91\) 4.08166i 0.427874i
\(92\) 0 0
\(93\) −6.20153 −0.643069
\(94\) 0 0
\(95\) 3.57454i 0.366740i
\(96\) 0 0
\(97\) 0.608891i 0.0618235i 0.999522 + 0.0309118i \(0.00984109\pi\)
−0.999522 + 0.0309118i \(0.990159\pi\)
\(98\) 0 0
\(99\) 42.3199i 4.25331i
\(100\) 0 0
\(101\) −7.06408 −0.702903 −0.351451 0.936206i \(-0.614312\pi\)
−0.351451 + 0.936206i \(0.614312\pi\)
\(102\) 0 0
\(103\) 5.85515 0.576925 0.288463 0.957491i \(-0.406856\pi\)
0.288463 + 0.957491i \(0.406856\pi\)
\(104\) 0 0
\(105\) 5.72598i 0.558798i
\(106\) 0 0
\(107\) 18.6545i 1.80340i −0.432367 0.901698i \(-0.642322\pi\)
0.432367 0.901698i \(-0.357678\pi\)
\(108\) 0 0
\(109\) 20.3443i 1.94863i 0.225186 + 0.974316i \(0.427701\pi\)
−0.225186 + 0.974316i \(0.572299\pi\)
\(110\) 0 0
\(111\) −10.7441 −1.01979
\(112\) 0 0
\(113\) 1.86914i 0.175834i −0.996128 0.0879168i \(-0.971979\pi\)
0.996128 0.0879168i \(-0.0280210\pi\)
\(114\) 0 0
\(115\) 6.99038 0.651856
\(116\) 0 0
\(117\) 23.6383 2.18537
\(118\) 0 0
\(119\) 0.704143 + 5.03285i 0.0645487 + 0.461361i
\(120\) 0 0
\(121\) −24.1503 −2.19549
\(122\) 0 0
\(123\) −26.0880 −2.35227
\(124\) 0 0
\(125\) 11.4845i 1.02720i
\(126\) 0 0
\(127\) −5.86748 −0.520654 −0.260327 0.965520i \(-0.583830\pi\)
−0.260327 + 0.965520i \(0.583830\pi\)
\(128\) 0 0
\(129\) 23.6446i 2.08180i
\(130\) 0 0
\(131\) 21.4508i 1.87417i 0.349107 + 0.937083i \(0.386485\pi\)
−0.349107 + 0.937083i \(0.613515\pi\)
\(132\) 0 0
\(133\) 3.01958i 0.261831i
\(134\) 0 0
\(135\) 19.2241 1.65455
\(136\) 0 0
\(137\) 2.62103 0.223930 0.111965 0.993712i \(-0.464286\pi\)
0.111965 + 0.993712i \(0.464286\pi\)
\(138\) 0 0
\(139\) 0.836202i 0.0709257i −0.999371 0.0354629i \(-0.988709\pi\)
0.999371 0.0354629i \(-0.0112906\pi\)
\(140\) 0 0
\(141\) 9.20060i 0.774831i
\(142\) 0 0
\(143\) 19.6337i 1.64185i
\(144\) 0 0
\(145\) −3.38514 −0.281121
\(146\) 0 0
\(147\) 17.4512i 1.43935i
\(148\) 0 0
\(149\) −17.2552 −1.41361 −0.706803 0.707411i \(-0.749863\pi\)
−0.706803 + 0.707411i \(0.749863\pi\)
\(150\) 0 0
\(151\) −5.67871 −0.462127 −0.231064 0.972939i \(-0.574221\pi\)
−0.231064 + 0.972939i \(0.574221\pi\)
\(152\) 0 0
\(153\) 29.1470 4.07794i 2.35640 0.329682i
\(154\) 0 0
\(155\) −2.84181 −0.228260
\(156\) 0 0
\(157\) −5.89767 −0.470685 −0.235343 0.971912i \(-0.575621\pi\)
−0.235343 + 0.971912i \(0.575621\pi\)
\(158\) 0 0
\(159\) 4.73717i 0.375682i
\(160\) 0 0
\(161\) 5.90509 0.465386
\(162\) 0 0
\(163\) 7.23741i 0.566878i 0.958990 + 0.283439i \(0.0914753\pi\)
−0.958990 + 0.283439i \(0.908525\pi\)
\(164\) 0 0
\(165\) 27.5433i 2.14424i
\(166\) 0 0
\(167\) 21.9836i 1.70114i 0.525860 + 0.850571i \(0.323744\pi\)
−0.525860 + 0.850571i \(0.676256\pi\)
\(168\) 0 0
\(169\) −2.03332 −0.156409
\(170\) 0 0
\(171\) 17.4874 1.33730
\(172\) 0 0
\(173\) 4.50222i 0.342297i 0.985245 + 0.171149i \(0.0547478\pi\)
−0.985245 + 0.171149i \(0.945252\pi\)
\(174\) 0 0
\(175\) 3.53878i 0.267507i
\(176\) 0 0
\(177\) 3.18403i 0.239326i
\(178\) 0 0
\(179\) 5.15021 0.384945 0.192472 0.981302i \(-0.438349\pi\)
0.192472 + 0.981302i \(0.438349\pi\)
\(180\) 0 0
\(181\) 2.67026i 0.198479i −0.995064 0.0992393i \(-0.968359\pi\)
0.995064 0.0992393i \(-0.0316409\pi\)
\(182\) 0 0
\(183\) 35.2051 2.60244
\(184\) 0 0
\(185\) −4.92343 −0.361977
\(186\) 0 0
\(187\) −3.38709 24.2092i −0.247688 1.77035i
\(188\) 0 0
\(189\) 16.2395 1.18125
\(190\) 0 0
\(191\) −23.9344 −1.73183 −0.865917 0.500188i \(-0.833264\pi\)
−0.865917 + 0.500188i \(0.833264\pi\)
\(192\) 0 0
\(193\) 16.3152i 1.17439i 0.809444 + 0.587197i \(0.199769\pi\)
−0.809444 + 0.587197i \(0.800231\pi\)
\(194\) 0 0
\(195\) 15.3847 1.10172
\(196\) 0 0
\(197\) 26.6339i 1.89759i 0.315897 + 0.948793i \(0.397694\pi\)
−0.315897 + 0.948793i \(0.602306\pi\)
\(198\) 0 0
\(199\) 4.39212i 0.311349i 0.987808 + 0.155674i \(0.0497551\pi\)
−0.987808 + 0.155674i \(0.950245\pi\)
\(200\) 0 0
\(201\) 36.4038i 2.56773i
\(202\) 0 0
\(203\) −2.85958 −0.200703
\(204\) 0 0
\(205\) −11.9546 −0.834948
\(206\) 0 0
\(207\) 34.1985i 2.37696i
\(208\) 0 0
\(209\) 14.5248i 1.00470i
\(210\) 0 0
\(211\) 4.06297i 0.279706i −0.990172 0.139853i \(-0.955337\pi\)
0.990172 0.139853i \(-0.0446630\pi\)
\(212\) 0 0
\(213\) −28.5653 −1.95726
\(214\) 0 0
\(215\) 10.8350i 0.738941i
\(216\) 0 0
\(217\) −2.40061 −0.162964
\(218\) 0 0
\(219\) 51.4572 3.47716
\(220\) 0 0
\(221\) 13.5224 1.89190i 0.909613 0.127263i
\(222\) 0 0
\(223\) 24.1345 1.61617 0.808084 0.589067i \(-0.200505\pi\)
0.808084 + 0.589067i \(0.200505\pi\)
\(224\) 0 0
\(225\) −20.4943 −1.36629
\(226\) 0 0
\(227\) 6.76308i 0.448882i −0.974488 0.224441i \(-0.927944\pi\)
0.974488 0.224441i \(-0.0720555\pi\)
\(228\) 0 0
\(229\) 6.27977 0.414979 0.207489 0.978237i \(-0.433471\pi\)
0.207489 + 0.978237i \(0.433471\pi\)
\(230\) 0 0
\(231\) 23.2670i 1.53086i
\(232\) 0 0
\(233\) 7.30784i 0.478753i −0.970927 0.239376i \(-0.923057\pi\)
0.970927 0.239376i \(-0.0769430\pi\)
\(234\) 0 0
\(235\) 4.21612i 0.275029i
\(236\) 0 0
\(237\) 24.2164 1.57302
\(238\) 0 0
\(239\) −19.6417 −1.27051 −0.635257 0.772301i \(-0.719106\pi\)
−0.635257 + 0.772301i \(0.719106\pi\)
\(240\) 0 0
\(241\) 12.0508i 0.776259i −0.921605 0.388130i \(-0.873121\pi\)
0.921605 0.388130i \(-0.126879\pi\)
\(242\) 0 0
\(243\) 25.8652i 1.65926i
\(244\) 0 0
\(245\) 7.99691i 0.510904i
\(246\) 0 0
\(247\) 8.11306 0.516221
\(248\) 0 0
\(249\) 29.8260i 1.89015i
\(250\) 0 0
\(251\) −6.54963 −0.413409 −0.206705 0.978403i \(-0.566274\pi\)
−0.206705 + 0.978403i \(0.566274\pi\)
\(252\) 0 0
\(253\) −28.4048 −1.78580
\(254\) 0 0
\(255\) 18.9699 2.65407i 1.18794 0.166204i
\(256\) 0 0
\(257\) −13.8588 −0.864486 −0.432243 0.901757i \(-0.642278\pi\)
−0.432243 + 0.901757i \(0.642278\pi\)
\(258\) 0 0
\(259\) −4.15904 −0.258430
\(260\) 0 0
\(261\) 16.5608i 1.02509i
\(262\) 0 0
\(263\) 19.2879 1.18934 0.594672 0.803968i \(-0.297282\pi\)
0.594672 + 0.803968i \(0.297282\pi\)
\(264\) 0 0
\(265\) 2.17078i 0.133350i
\(266\) 0 0
\(267\) 14.4735i 0.885762i
\(268\) 0 0
\(269\) 21.5692i 1.31509i 0.753413 + 0.657547i \(0.228406\pi\)
−0.753413 + 0.657547i \(0.771594\pi\)
\(270\) 0 0
\(271\) −14.1592 −0.860109 −0.430054 0.902803i \(-0.641506\pi\)
−0.430054 + 0.902803i \(0.641506\pi\)
\(272\) 0 0
\(273\) 12.9961 0.786561
\(274\) 0 0
\(275\) 17.0223i 1.02649i
\(276\) 0 0
\(277\) 0.541294i 0.0325232i 0.999868 + 0.0162616i \(0.00517645\pi\)
−0.999868 + 0.0162616i \(0.994824\pi\)
\(278\) 0 0
\(279\) 13.9028i 0.832337i
\(280\) 0 0
\(281\) −4.83498 −0.288431 −0.144215 0.989546i \(-0.546066\pi\)
−0.144215 + 0.989546i \(0.546066\pi\)
\(282\) 0 0
\(283\) 25.8603i 1.53723i 0.639709 + 0.768617i \(0.279055\pi\)
−0.639709 + 0.768617i \(0.720945\pi\)
\(284\) 0 0
\(285\) 11.3814 0.674179
\(286\) 0 0
\(287\) −10.0986 −0.596103
\(288\) 0 0
\(289\) 16.3472 4.66559i 0.961602 0.274446i
\(290\) 0 0
\(291\) 1.93873 0.113650
\(292\) 0 0
\(293\) −23.6848 −1.38368 −0.691840 0.722051i \(-0.743200\pi\)
−0.691840 + 0.722051i \(0.743200\pi\)
\(294\) 0 0
\(295\) 1.45906i 0.0849498i
\(296\) 0 0
\(297\) −78.1155 −4.53272
\(298\) 0 0
\(299\) 15.8659i 0.917549i
\(300\) 0 0
\(301\) 9.15282i 0.527560i
\(302\) 0 0
\(303\) 22.4923i 1.29215i
\(304\) 0 0
\(305\) 16.1325 0.923746
\(306\) 0 0
\(307\) 30.4472 1.73771 0.868855 0.495066i \(-0.164856\pi\)
0.868855 + 0.495066i \(0.164856\pi\)
\(308\) 0 0
\(309\) 18.6430i 1.06056i
\(310\) 0 0
\(311\) 4.55904i 0.258519i 0.991611 + 0.129260i \(0.0412601\pi\)
−0.991611 + 0.129260i \(0.958740\pi\)
\(312\) 0 0
\(313\) 15.6866i 0.886661i 0.896358 + 0.443331i \(0.146203\pi\)
−0.896358 + 0.443331i \(0.853797\pi\)
\(314\) 0 0
\(315\) 12.8367 0.723264
\(316\) 0 0
\(317\) 1.65225i 0.0927993i −0.998923 0.0463997i \(-0.985225\pi\)
0.998923 0.0463997i \(-0.0147748\pi\)
\(318\) 0 0
\(319\) 13.7552 0.770145
\(320\) 0 0
\(321\) −59.3964 −3.31518
\(322\) 0 0
\(323\) 10.0037 1.39961i 0.556622 0.0778766i
\(324\) 0 0
\(325\) −9.50806 −0.527412
\(326\) 0 0
\(327\) 64.7769 3.58217
\(328\) 0 0
\(329\) 3.56154i 0.196354i
\(330\) 0 0
\(331\) −13.1265 −0.721498 −0.360749 0.932663i \(-0.617479\pi\)
−0.360749 + 0.932663i \(0.617479\pi\)
\(332\) 0 0
\(333\) 24.0865i 1.31993i
\(334\) 0 0
\(335\) 16.6818i 0.911425i
\(336\) 0 0
\(337\) 1.09349i 0.0595662i 0.999556 + 0.0297831i \(0.00948165\pi\)
−0.999556 + 0.0297831i \(0.990518\pi\)
\(338\) 0 0
\(339\) −5.95139 −0.323235
\(340\) 0 0
\(341\) 11.5475 0.625330
\(342\) 0 0
\(343\) 15.3831i 0.830609i
\(344\) 0 0
\(345\) 22.2576i 1.19831i
\(346\) 0 0
\(347\) 24.1509i 1.29649i 0.761432 + 0.648244i \(0.224496\pi\)
−0.761432 + 0.648244i \(0.775504\pi\)
\(348\) 0 0
\(349\) 21.8561 1.16993 0.584964 0.811059i \(-0.301108\pi\)
0.584964 + 0.811059i \(0.301108\pi\)
\(350\) 0 0
\(351\) 43.6325i 2.32893i
\(352\) 0 0
\(353\) −20.9809 −1.11670 −0.558351 0.829605i \(-0.688566\pi\)
−0.558351 + 0.829605i \(0.688566\pi\)
\(354\) 0 0
\(355\) −13.0898 −0.694737
\(356\) 0 0
\(357\) 16.0248 2.24201i 0.848120 0.118660i
\(358\) 0 0
\(359\) −10.4184 −0.549861 −0.274931 0.961464i \(-0.588655\pi\)
−0.274931 + 0.961464i \(0.588655\pi\)
\(360\) 0 0
\(361\) −12.9980 −0.684107
\(362\) 0 0
\(363\) 76.8954i 4.03596i
\(364\) 0 0
\(365\) 23.5799 1.23423
\(366\) 0 0
\(367\) 24.4045i 1.27390i −0.770904 0.636951i \(-0.780195\pi\)
0.770904 0.636951i \(-0.219805\pi\)
\(368\) 0 0
\(369\) 58.4847i 3.04459i
\(370\) 0 0
\(371\) 1.83375i 0.0952037i
\(372\) 0 0
\(373\) 30.5197 1.58025 0.790124 0.612947i \(-0.210016\pi\)
0.790124 + 0.612947i \(0.210016\pi\)
\(374\) 0 0
\(375\) −36.5669 −1.88831
\(376\) 0 0
\(377\) 7.68318i 0.395704i
\(378\) 0 0
\(379\) 5.24374i 0.269353i −0.990890 0.134677i \(-0.957000\pi\)
0.990890 0.134677i \(-0.0429995\pi\)
\(380\) 0 0
\(381\) 18.6822i 0.957119i
\(382\) 0 0
\(383\) −17.4888 −0.893634 −0.446817 0.894625i \(-0.647442\pi\)
−0.446817 + 0.894625i \(0.647442\pi\)
\(384\) 0 0
\(385\) 10.6620i 0.543384i
\(386\) 0 0
\(387\) 53.0072 2.69451
\(388\) 0 0
\(389\) 11.6054 0.588417 0.294209 0.955741i \(-0.404944\pi\)
0.294209 + 0.955741i \(0.404944\pi\)
\(390\) 0 0
\(391\) −2.73709 19.5633i −0.138420 0.989359i
\(392\) 0 0
\(393\) 68.3000 3.44528
\(394\) 0 0
\(395\) 11.0970 0.558351
\(396\) 0 0
\(397\) 39.0760i 1.96117i 0.196102 + 0.980584i \(0.437172\pi\)
−0.196102 + 0.980584i \(0.562828\pi\)
\(398\) 0 0
\(399\) 9.61442 0.481323
\(400\) 0 0
\(401\) 17.1430i 0.856080i 0.903760 + 0.428040i \(0.140796\pi\)
−0.903760 + 0.428040i \(0.859204\pi\)
\(402\) 0 0
\(403\) 6.44999i 0.321297i
\(404\) 0 0
\(405\) 29.9656i 1.48900i
\(406\) 0 0
\(407\) 20.0059 0.991657
\(408\) 0 0
\(409\) 23.3558 1.15487 0.577435 0.816437i \(-0.304054\pi\)
0.577435 + 0.816437i \(0.304054\pi\)
\(410\) 0 0
\(411\) 8.34544i 0.411650i
\(412\) 0 0
\(413\) 1.23253i 0.0606491i
\(414\) 0 0
\(415\) 13.6676i 0.670916i
\(416\) 0 0
\(417\) −2.66249 −0.130383
\(418\) 0 0
\(419\) 10.1123i 0.494020i 0.969013 + 0.247010i \(0.0794481\pi\)
−0.969013 + 0.247010i \(0.920552\pi\)
\(420\) 0 0
\(421\) −7.94430 −0.387182 −0.193591 0.981082i \(-0.562013\pi\)
−0.193591 + 0.981082i \(0.562013\pi\)
\(422\) 0 0
\(423\) −20.6262 −1.00288
\(424\) 0 0
\(425\) −11.7238 + 1.64027i −0.568689 + 0.0795649i
\(426\) 0 0
\(427\) 13.6279 0.659499
\(428\) 0 0
\(429\) −62.5143 −3.01822
\(430\) 0 0
\(431\) 21.7853i 1.04936i −0.851299 0.524681i \(-0.824185\pi\)
0.851299 0.524681i \(-0.175815\pi\)
\(432\) 0 0
\(433\) −18.5810 −0.892944 −0.446472 0.894798i \(-0.647320\pi\)
−0.446472 + 0.894798i \(0.647320\pi\)
\(434\) 0 0
\(435\) 10.7784i 0.516784i
\(436\) 0 0
\(437\) 11.7375i 0.561479i
\(438\) 0 0
\(439\) 34.4288i 1.64320i −0.570067 0.821598i \(-0.693083\pi\)
0.570067 0.821598i \(-0.306917\pi\)
\(440\) 0 0
\(441\) −39.1226 −1.86298
\(442\) 0 0
\(443\) −31.7022 −1.50622 −0.753109 0.657896i \(-0.771447\pi\)
−0.753109 + 0.657896i \(0.771447\pi\)
\(444\) 0 0
\(445\) 6.63237i 0.314405i
\(446\) 0 0
\(447\) 54.9412i 2.59863i
\(448\) 0 0
\(449\) 0.310627i 0.0146594i 0.999973 + 0.00732969i \(0.00233314\pi\)
−0.999973 + 0.00732969i \(0.997667\pi\)
\(450\) 0 0
\(451\) 48.5767 2.28739
\(452\) 0 0
\(453\) 18.0812i 0.849529i
\(454\) 0 0
\(455\) 5.95539 0.279193
\(456\) 0 0
\(457\) −15.6630 −0.732686 −0.366343 0.930480i \(-0.619390\pi\)
−0.366343 + 0.930480i \(0.619390\pi\)
\(458\) 0 0
\(459\) −7.52721 53.8006i −0.351340 2.51120i
\(460\) 0 0
\(461\) 5.35063 0.249204 0.124602 0.992207i \(-0.460235\pi\)
0.124602 + 0.992207i \(0.460235\pi\)
\(462\) 0 0
\(463\) −4.79377 −0.222785 −0.111393 0.993776i \(-0.535531\pi\)
−0.111393 + 0.993776i \(0.535531\pi\)
\(464\) 0 0
\(465\) 9.04841i 0.419610i
\(466\) 0 0
\(467\) 35.4649 1.64112 0.820560 0.571561i \(-0.193662\pi\)
0.820560 + 0.571561i \(0.193662\pi\)
\(468\) 0 0
\(469\) 14.0919i 0.650703i
\(470\) 0 0
\(471\) 18.7784i 0.865261i
\(472\) 0 0
\(473\) 44.0271i 2.02437i
\(474\) 0 0
\(475\) −7.03398 −0.322741
\(476\) 0 0
\(477\) −10.6199 −0.486252
\(478\) 0 0
\(479\) 29.4834i 1.34713i −0.739128 0.673565i \(-0.764762\pi\)
0.739128 0.673565i \(-0.235238\pi\)
\(480\) 0 0
\(481\) 11.1746i 0.509517i
\(482\) 0 0
\(483\) 18.8020i 0.855520i
\(484\) 0 0
\(485\) 0.888409 0.0403406
\(486\) 0 0
\(487\) 30.1177i 1.36476i −0.730996 0.682382i \(-0.760944\pi\)
0.730996 0.682382i \(-0.239056\pi\)
\(488\) 0 0
\(489\) 23.0441 1.04209
\(490\) 0 0
\(491\) −27.5413 −1.24292 −0.621461 0.783445i \(-0.713461\pi\)
−0.621461 + 0.783445i \(0.713461\pi\)
\(492\) 0 0
\(493\) 1.32545 + 9.47367i 0.0596955 + 0.426673i
\(494\) 0 0
\(495\) −61.7473 −2.77533
\(496\) 0 0
\(497\) −11.0576 −0.496000
\(498\) 0 0
\(499\) 20.6410i 0.924016i −0.886876 0.462008i \(-0.847129\pi\)
0.886876 0.462008i \(-0.152871\pi\)
\(500\) 0 0
\(501\) 69.9965 3.12721
\(502\) 0 0
\(503\) 29.7160i 1.32497i 0.749075 + 0.662485i \(0.230498\pi\)
−0.749075 + 0.662485i \(0.769502\pi\)
\(504\) 0 0
\(505\) 10.3069i 0.458652i
\(506\) 0 0
\(507\) 6.47415i 0.287527i
\(508\) 0 0
\(509\) 8.45892 0.374935 0.187467 0.982271i \(-0.439972\pi\)
0.187467 + 0.982271i \(0.439972\pi\)
\(510\) 0 0
\(511\) 19.9190 0.881166
\(512\) 0 0
\(513\) 32.2790i 1.42515i
\(514\) 0 0
\(515\) 8.54302i 0.376451i
\(516\) 0 0
\(517\) 17.1318i 0.753457i
\(518\) 0 0
\(519\) 14.3352 0.629245
\(520\) 0 0
\(521\) 24.2625i 1.06296i 0.847071 + 0.531480i \(0.178364\pi\)
−0.847071 + 0.531480i \(0.821636\pi\)
\(522\) 0 0
\(523\) 8.31208 0.363462 0.181731 0.983348i \(-0.441830\pi\)
0.181731 + 0.983348i \(0.441830\pi\)
\(524\) 0 0
\(525\) −11.2676 −0.491758
\(526\) 0 0
\(527\) 1.11271 + 7.95310i 0.0484706 + 0.346443i
\(528\) 0 0
\(529\) 0.0462149 0.00200935
\(530\) 0 0
\(531\) −7.13805 −0.309765
\(532\) 0 0
\(533\) 27.1332i 1.17527i
\(534\) 0 0
\(535\) −27.2180 −1.17674
\(536\) 0 0
\(537\) 16.3984i 0.707644i
\(538\) 0 0
\(539\) 32.4948i 1.39965i
\(540\) 0 0
\(541\) 18.8560i 0.810685i −0.914165 0.405342i \(-0.867152\pi\)
0.914165 0.405342i \(-0.132848\pi\)
\(542\) 0 0
\(543\) −8.50218 −0.364863
\(544\) 0 0
\(545\) 29.6836 1.27151
\(546\) 0 0
\(547\) 37.5805i 1.60683i 0.595421 + 0.803414i \(0.296985\pi\)
−0.595421 + 0.803414i \(0.703015\pi\)
\(548\) 0 0
\(549\) 78.9239i 3.36839i
\(550\) 0 0
\(551\) 5.68395i 0.242144i
\(552\) 0 0
\(553\) 9.37414 0.398629
\(554\) 0 0
\(555\) 15.6763i 0.665423i
\(556\) 0 0
\(557\) −33.9610 −1.43898 −0.719488 0.694505i \(-0.755623\pi\)
−0.719488 + 0.694505i \(0.755623\pi\)
\(558\) 0 0
\(559\) 24.5920 1.04013
\(560\) 0 0
\(561\) −77.0827 + 10.7846i −3.25443 + 0.455326i
\(562\) 0 0
\(563\) −29.2925 −1.23453 −0.617265 0.786755i \(-0.711759\pi\)
−0.617265 + 0.786755i \(0.711759\pi\)
\(564\) 0 0
\(565\) −2.72718 −0.114734
\(566\) 0 0
\(567\) 25.3133i 1.06306i
\(568\) 0 0
\(569\) −13.5056 −0.566184 −0.283092 0.959093i \(-0.591360\pi\)
−0.283092 + 0.959093i \(0.591360\pi\)
\(570\) 0 0
\(571\) 4.26022i 0.178285i −0.996019 0.0891424i \(-0.971587\pi\)
0.996019 0.0891424i \(-0.0284126\pi\)
\(572\) 0 0
\(573\) 76.2079i 3.18363i
\(574\) 0 0
\(575\) 13.7557i 0.573651i
\(576\) 0 0
\(577\) 9.52454 0.396512 0.198256 0.980150i \(-0.436472\pi\)
0.198256 + 0.980150i \(0.436472\pi\)
\(578\) 0 0
\(579\) 51.9481 2.15889
\(580\) 0 0
\(581\) 11.5456i 0.478993i
\(582\) 0 0
\(583\) 8.82076i 0.365319i
\(584\) 0 0
\(585\) 34.4898i 1.42598i
\(586\) 0 0
\(587\) 38.4819 1.58832 0.794160 0.607709i \(-0.207911\pi\)
0.794160 + 0.607709i \(0.207911\pi\)
\(588\) 0 0
\(589\) 4.77165i 0.196612i
\(590\) 0 0
\(591\) 84.8031 3.48833
\(592\) 0 0
\(593\) −38.7928 −1.59303 −0.796515 0.604618i \(-0.793326\pi\)
−0.796515 + 0.604618i \(0.793326\pi\)
\(594\) 0 0
\(595\) 7.34324 1.02739i 0.301043 0.0421188i
\(596\) 0 0
\(597\) 13.9846 0.572353
\(598\) 0 0
\(599\) 7.53260 0.307774 0.153887 0.988088i \(-0.450821\pi\)
0.153887 + 0.988088i \(0.450821\pi\)
\(600\) 0 0
\(601\) 37.8971i 1.54585i 0.634494 + 0.772927i \(0.281208\pi\)
−0.634494 + 0.772927i \(0.718792\pi\)
\(602\) 0 0
\(603\) −81.6111 −3.32346
\(604\) 0 0
\(605\) 35.2368i 1.43258i
\(606\) 0 0
\(607\) 19.1958i 0.779133i 0.920998 + 0.389567i \(0.127375\pi\)
−0.920998 + 0.389567i \(0.872625\pi\)
\(608\) 0 0
\(609\) 9.10499i 0.368953i
\(610\) 0 0
\(611\) −9.56922 −0.387129
\(612\) 0 0
\(613\) −28.0662 −1.13358 −0.566792 0.823861i \(-0.691816\pi\)
−0.566792 + 0.823861i \(0.691816\pi\)
\(614\) 0 0
\(615\) 38.0639i 1.53489i
\(616\) 0 0
\(617\) 20.2211i 0.814072i −0.913412 0.407036i \(-0.866562\pi\)
0.913412 0.407036i \(-0.133438\pi\)
\(618\) 0 0
\(619\) 40.1443i 1.61354i −0.590869 0.806768i \(-0.701215\pi\)
0.590869 0.806768i \(-0.298785\pi\)
\(620\) 0 0
\(621\) −63.1248 −2.53311
\(622\) 0 0
\(623\) 5.60266i 0.224466i
\(624\) 0 0
\(625\) −2.40085 −0.0960339
\(626\) 0 0
\(627\) −46.2476 −1.84695
\(628\) 0 0
\(629\) 1.92777 + 13.7787i 0.0768653 + 0.549394i
\(630\) 0 0
\(631\) 26.1841 1.04237 0.521186 0.853443i \(-0.325490\pi\)
0.521186 + 0.853443i \(0.325490\pi\)
\(632\) 0 0
\(633\) −12.9366 −0.514184
\(634\) 0 0
\(635\) 8.56101i 0.339733i
\(636\) 0 0
\(637\) −18.1504 −0.719145
\(638\) 0 0
\(639\) 64.0384i 2.53332i
\(640\) 0 0
\(641\) 15.0919i 0.596096i 0.954551 + 0.298048i \(0.0963355\pi\)
−0.954551 + 0.298048i \(0.903665\pi\)
\(642\) 0 0
\(643\) 8.11859i 0.320166i −0.987104 0.160083i \(-0.948824\pi\)
0.987104 0.160083i \(-0.0511762\pi\)
\(644\) 0 0
\(645\) 34.4990 1.35840
\(646\) 0 0
\(647\) 8.93033 0.351088 0.175544 0.984472i \(-0.443832\pi\)
0.175544 + 0.984472i \(0.443832\pi\)
\(648\) 0 0
\(649\) 5.92877i 0.232725i
\(650\) 0 0
\(651\) 7.64360i 0.299576i
\(652\) 0 0
\(653\) 23.8172i 0.932040i 0.884774 + 0.466020i \(0.154312\pi\)
−0.884774 + 0.466020i \(0.845688\pi\)
\(654\) 0 0
\(655\) 31.2980 1.22292
\(656\) 0 0
\(657\) 115.358i 4.50055i
\(658\) 0 0
\(659\) 7.17063 0.279328 0.139664 0.990199i \(-0.455398\pi\)
0.139664 + 0.990199i \(0.455398\pi\)
\(660\) 0 0
\(661\) 0.249180 0.00969197 0.00484599 0.999988i \(-0.498457\pi\)
0.00484599 + 0.999988i \(0.498457\pi\)
\(662\) 0 0
\(663\) −6.02388 43.0556i −0.233948 1.67214i
\(664\) 0 0
\(665\) 4.40575 0.170848
\(666\) 0 0
\(667\) 11.1155 0.430395
\(668\) 0 0
\(669\) 76.8451i 2.97100i
\(670\) 0 0
\(671\) −65.5532 −2.53065
\(672\) 0 0
\(673\) 6.92563i 0.266964i 0.991051 + 0.133482i \(0.0426158\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(674\) 0 0
\(675\) 37.8292i 1.45605i
\(676\) 0 0
\(677\) 8.31848i 0.319705i −0.987141 0.159853i \(-0.948898\pi\)
0.987141 0.159853i \(-0.0511019\pi\)
\(678\) 0 0
\(679\) 0.750479 0.0288008
\(680\) 0 0
\(681\) −21.5339 −0.825179
\(682\) 0 0
\(683\) 32.6455i 1.24914i 0.780967 + 0.624572i \(0.214726\pi\)
−0.780967 + 0.624572i \(0.785274\pi\)
\(684\) 0 0
\(685\) 3.82425i 0.146117i
\(686\) 0 0
\(687\) 19.9950i 0.762856i
\(688\) 0 0
\(689\) −4.92696 −0.187702
\(690\) 0 0
\(691\) 25.2226i 0.959513i 0.877402 + 0.479756i \(0.159275\pi\)
−0.877402 + 0.479756i \(0.840725\pi\)
\(692\) 0 0
\(693\) −52.1607 −1.98142
\(694\) 0 0
\(695\) −1.22007 −0.0462799
\(696\) 0 0
\(697\) 4.68085 + 33.4563i 0.177300 + 1.26725i
\(698\) 0 0
\(699\) −23.2684 −0.880091
\(700\) 0 0
\(701\) −33.1163 −1.25079 −0.625393 0.780310i \(-0.715061\pi\)
−0.625393 + 0.780310i \(0.715061\pi\)
\(702\) 0 0
\(703\) 8.26686i 0.311791i
\(704\) 0 0
\(705\) −13.4242 −0.505586
\(706\) 0 0
\(707\) 8.70673i 0.327450i
\(708\) 0 0
\(709\) 29.2206i 1.09740i 0.836019 + 0.548701i \(0.184877\pi\)
−0.836019 + 0.548701i \(0.815123\pi\)
\(710\) 0 0
\(711\) 54.2890i 2.03599i
\(712\) 0 0
\(713\) 9.33144 0.349465
\(714\) 0 0
\(715\) −28.6468 −1.07133
\(716\) 0 0
\(717\) 62.5396i 2.33559i
\(718\) 0 0
\(719\) 3.76832i 0.140535i −0.997528 0.0702673i \(-0.977615\pi\)
0.997528 0.0702673i \(-0.0223852\pi\)
\(720\) 0 0
\(721\) 7.21668i 0.268763i
\(722\) 0 0
\(723\) −38.3701 −1.42700
\(724\) 0 0
\(725\) 6.66128i 0.247394i
\(726\) 0 0
\(727\) −14.8098 −0.549265 −0.274632 0.961549i \(-0.588556\pi\)
−0.274632 + 0.961549i \(0.588556\pi\)
\(728\) 0 0
\(729\) −20.7430 −0.768260
\(730\) 0 0
\(731\) 30.3229 4.24246i 1.12153 0.156913i
\(732\) 0 0
\(733\) −48.5122 −1.79184 −0.895920 0.444215i \(-0.853483\pi\)
−0.895920 + 0.444215i \(0.853483\pi\)
\(734\) 0 0
\(735\) −25.4624 −0.939194
\(736\) 0 0
\(737\) 67.7852i 2.49690i
\(738\) 0 0
\(739\) −20.0660 −0.738141 −0.369070 0.929401i \(-0.620324\pi\)
−0.369070 + 0.929401i \(0.620324\pi\)
\(740\) 0 0
\(741\) 25.8322i 0.948970i
\(742\) 0 0
\(743\) 23.6271i 0.866795i −0.901203 0.433398i \(-0.857315\pi\)
0.901203 0.433398i \(-0.142685\pi\)
\(744\) 0 0
\(745\) 25.1765i 0.922394i
\(746\) 0 0
\(747\) −66.8648 −2.44646
\(748\) 0 0
\(749\) −22.9923 −0.840120
\(750\) 0 0
\(751\) 10.6929i 0.390188i −0.980785 0.195094i \(-0.937499\pi\)
0.980785 0.195094i \(-0.0625012\pi\)
\(752\) 0 0
\(753\) 20.8542i 0.759970i
\(754\) 0 0
\(755\) 8.28559i 0.301544i
\(756\) 0 0
\(757\) −32.7900 −1.19177 −0.595886 0.803069i \(-0.703199\pi\)
−0.595886 + 0.803069i \(0.703199\pi\)
\(758\) 0 0
\(759\) 90.4418i 3.28283i
\(760\) 0 0
\(761\) −35.3405 −1.28109 −0.640547 0.767919i \(-0.721292\pi\)
−0.640547 + 0.767919i \(0.721292\pi\)
\(762\) 0 0
\(763\) 25.0751 0.907779
\(764\) 0 0
\(765\) −5.94997 42.5273i −0.215121 1.53758i
\(766\) 0 0
\(767\) −3.31160 −0.119575
\(768\) 0 0
\(769\) 45.1200 1.62707 0.813534 0.581518i \(-0.197541\pi\)
0.813534 + 0.581518i \(0.197541\pi\)
\(770\) 0 0
\(771\) 44.1267i 1.58919i
\(772\) 0 0
\(773\) 31.2692 1.12468 0.562338 0.826908i \(-0.309902\pi\)
0.562338 + 0.826908i \(0.309902\pi\)
\(774\) 0 0
\(775\) 5.59212i 0.200875i
\(776\) 0 0
\(777\) 13.2425i 0.475072i
\(778\) 0 0
\(779\) 20.0729i 0.719186i
\(780\) 0 0
\(781\) 53.1895 1.90327
\(782\) 0 0
\(783\) 30.5686 1.09243
\(784\) 0 0
\(785\) 8.60506i 0.307128i
\(786\) 0 0
\(787\) 8.20289i 0.292402i 0.989255 + 0.146201i \(0.0467046\pi\)
−0.989255 + 0.146201i \(0.953295\pi\)
\(788\) 0 0
\(789\) 61.4133i 2.18637i
\(790\) 0 0
\(791\) −2.30378 −0.0819128
\(792\) 0 0
\(793\) 36.6156i 1.30026i