Properties

Label 4012.2.b.b.237.7
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 237.7
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.40

$q$-expansion

\(f(q)\) \(=\) \(q-2.57223i q^{3} +2.31913i q^{5} +3.71324i q^{7} -3.61638 q^{9} +O(q^{10})\) \(q-2.57223i q^{3} +2.31913i q^{5} +3.71324i q^{7} -3.61638 q^{9} +5.32303i q^{11} -2.64206 q^{13} +5.96535 q^{15} +(3.19542 + 2.60563i) q^{17} -4.86403 q^{19} +9.55131 q^{21} -0.402486i q^{23} -0.378384 q^{25} +1.58548i q^{27} -7.10049i q^{29} -5.18537i q^{31} +13.6921 q^{33} -8.61149 q^{35} -0.811616i q^{37} +6.79599i q^{39} +11.0444i q^{41} -6.22805 q^{43} -8.38687i q^{45} +4.15852 q^{47} -6.78812 q^{49} +(6.70229 - 8.21936i) q^{51} -9.41694 q^{53} -12.3448 q^{55} +12.5114i q^{57} +1.00000 q^{59} -9.06997i q^{61} -13.4285i q^{63} -6.12729i q^{65} +14.3241 q^{67} -1.03529 q^{69} +13.3448i q^{71} -13.8115i q^{73} +0.973292i q^{75} -19.7657 q^{77} +13.1249i q^{79} -6.77093 q^{81} -12.3267 q^{83} +(-6.04281 + 7.41060i) q^{85} -18.2641 q^{87} -14.8859 q^{89} -9.81059i q^{91} -13.3380 q^{93} -11.2803i q^{95} +1.69852i q^{97} -19.2501i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 2.57223i 1.48508i −0.669802 0.742540i \(-0.733621\pi\)
0.669802 0.742540i \(-0.266379\pi\)
\(4\) 0 0
\(5\) 2.31913i 1.03715i 0.855033 + 0.518574i \(0.173537\pi\)
−0.855033 + 0.518574i \(0.826463\pi\)
\(6\) 0 0
\(7\) 3.71324i 1.40347i 0.712437 + 0.701736i \(0.247591\pi\)
−0.712437 + 0.701736i \(0.752409\pi\)
\(8\) 0 0
\(9\) −3.61638 −1.20546
\(10\) 0 0
\(11\) 5.32303i 1.60495i 0.596683 + 0.802477i \(0.296485\pi\)
−0.596683 + 0.802477i \(0.703515\pi\)
\(12\) 0 0
\(13\) −2.64206 −0.732775 −0.366388 0.930462i \(-0.619406\pi\)
−0.366388 + 0.930462i \(0.619406\pi\)
\(14\) 0 0
\(15\) 5.96535 1.54025
\(16\) 0 0
\(17\) 3.19542 + 2.60563i 0.775002 + 0.631958i
\(18\) 0 0
\(19\) −4.86403 −1.11588 −0.557942 0.829880i \(-0.688409\pi\)
−0.557942 + 0.829880i \(0.688409\pi\)
\(20\) 0 0
\(21\) 9.55131 2.08427
\(22\) 0 0
\(23\) 0.402486i 0.0839242i −0.999119 0.0419621i \(-0.986639\pi\)
0.999119 0.0419621i \(-0.0133609\pi\)
\(24\) 0 0
\(25\) −0.378384 −0.0756768
\(26\) 0 0
\(27\) 1.58548i 0.305125i
\(28\) 0 0
\(29\) 7.10049i 1.31853i −0.751911 0.659264i \(-0.770868\pi\)
0.751911 0.659264i \(-0.229132\pi\)
\(30\) 0 0
\(31\) 5.18537i 0.931320i −0.884964 0.465660i \(-0.845817\pi\)
0.884964 0.465660i \(-0.154183\pi\)
\(32\) 0 0
\(33\) 13.6921 2.38348
\(34\) 0 0
\(35\) −8.61149 −1.45561
\(36\) 0 0
\(37\) 0.811616i 0.133429i −0.997772 0.0667144i \(-0.978748\pi\)
0.997772 0.0667144i \(-0.0212516\pi\)
\(38\) 0 0
\(39\) 6.79599i 1.08823i
\(40\) 0 0
\(41\) 11.0444i 1.72485i 0.506185 + 0.862425i \(0.331055\pi\)
−0.506185 + 0.862425i \(0.668945\pi\)
\(42\) 0 0
\(43\) −6.22805 −0.949769 −0.474884 0.880048i \(-0.657510\pi\)
−0.474884 + 0.880048i \(0.657510\pi\)
\(44\) 0 0
\(45\) 8.38687i 1.25024i
\(46\) 0 0
\(47\) 4.15852 0.606582 0.303291 0.952898i \(-0.401915\pi\)
0.303291 + 0.952898i \(0.401915\pi\)
\(48\) 0 0
\(49\) −6.78812 −0.969732
\(50\) 0 0
\(51\) 6.70229 8.21936i 0.938508 1.15094i
\(52\) 0 0
\(53\) −9.41694 −1.29352 −0.646758 0.762695i \(-0.723876\pi\)
−0.646758 + 0.762695i \(0.723876\pi\)
\(54\) 0 0
\(55\) −12.3448 −1.66458
\(56\) 0 0
\(57\) 12.5114i 1.65718i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 9.06997i 1.16129i −0.814157 0.580645i \(-0.802800\pi\)
0.814157 0.580645i \(-0.197200\pi\)
\(62\) 0 0
\(63\) 13.4285i 1.69183i
\(64\) 0 0
\(65\) 6.12729i 0.759997i
\(66\) 0 0
\(67\) 14.3241 1.74997 0.874985 0.484150i \(-0.160871\pi\)
0.874985 + 0.484150i \(0.160871\pi\)
\(68\) 0 0
\(69\) −1.03529 −0.124634
\(70\) 0 0
\(71\) 13.3448i 1.58374i 0.610689 + 0.791871i \(0.290893\pi\)
−0.610689 + 0.791871i \(0.709107\pi\)
\(72\) 0 0
\(73\) 13.8115i 1.61651i −0.588831 0.808256i \(-0.700412\pi\)
0.588831 0.808256i \(-0.299588\pi\)
\(74\) 0 0
\(75\) 0.973292i 0.112386i
\(76\) 0 0
\(77\) −19.7657 −2.25251
\(78\) 0 0
\(79\) 13.1249i 1.47667i 0.674434 + 0.738335i \(0.264388\pi\)
−0.674434 + 0.738335i \(0.735612\pi\)
\(80\) 0 0
\(81\) −6.77093 −0.752326
\(82\) 0 0
\(83\) −12.3267 −1.35304 −0.676518 0.736426i \(-0.736512\pi\)
−0.676518 + 0.736426i \(0.736512\pi\)
\(84\) 0 0
\(85\) −6.04281 + 7.41060i −0.655434 + 0.803793i
\(86\) 0 0
\(87\) −18.2641 −1.95812
\(88\) 0 0
\(89\) −14.8859 −1.57790 −0.788950 0.614457i \(-0.789375\pi\)
−0.788950 + 0.614457i \(0.789375\pi\)
\(90\) 0 0
\(91\) 9.81059i 1.02843i
\(92\) 0 0
\(93\) −13.3380 −1.38308
\(94\) 0 0
\(95\) 11.2803i 1.15734i
\(96\) 0 0
\(97\) 1.69852i 0.172458i 0.996275 + 0.0862292i \(0.0274817\pi\)
−0.996275 + 0.0862292i \(0.972518\pi\)
\(98\) 0 0
\(99\) 19.2501i 1.93471i
\(100\) 0 0
\(101\) −1.48096 −0.147361 −0.0736807 0.997282i \(-0.523475\pi\)
−0.0736807 + 0.997282i \(0.523475\pi\)
\(102\) 0 0
\(103\) −11.1542 −1.09905 −0.549527 0.835476i \(-0.685192\pi\)
−0.549527 + 0.835476i \(0.685192\pi\)
\(104\) 0 0
\(105\) 22.1508i 2.16169i
\(106\) 0 0
\(107\) 7.98392i 0.771834i 0.922533 + 0.385917i \(0.126115\pi\)
−0.922533 + 0.385917i \(0.873885\pi\)
\(108\) 0 0
\(109\) 2.78131i 0.266402i 0.991089 + 0.133201i \(0.0425255\pi\)
−0.991089 + 0.133201i \(0.957474\pi\)
\(110\) 0 0
\(111\) −2.08767 −0.198152
\(112\) 0 0
\(113\) 6.31139i 0.593725i −0.954920 0.296863i \(-0.904060\pi\)
0.954920 0.296863i \(-0.0959404\pi\)
\(114\) 0 0
\(115\) 0.933420 0.0870418
\(116\) 0 0
\(117\) 9.55469 0.883332
\(118\) 0 0
\(119\) −9.67532 + 11.8653i −0.886935 + 1.08769i
\(120\) 0 0
\(121\) −17.3347 −1.57588
\(122\) 0 0
\(123\) 28.4088 2.56154
\(124\) 0 0
\(125\) 10.7181i 0.958660i
\(126\) 0 0
\(127\) −3.22852 −0.286485 −0.143243 0.989688i \(-0.545753\pi\)
−0.143243 + 0.989688i \(0.545753\pi\)
\(128\) 0 0
\(129\) 16.0200i 1.41048i
\(130\) 0 0
\(131\) 6.43250i 0.562010i −0.959706 0.281005i \(-0.909332\pi\)
0.959706 0.281005i \(-0.0906677\pi\)
\(132\) 0 0
\(133\) 18.0613i 1.56611i
\(134\) 0 0
\(135\) −3.67693 −0.316460
\(136\) 0 0
\(137\) −13.4755 −1.15129 −0.575644 0.817701i \(-0.695248\pi\)
−0.575644 + 0.817701i \(0.695248\pi\)
\(138\) 0 0
\(139\) 21.7545i 1.84520i −0.385764 0.922598i \(-0.626062\pi\)
0.385764 0.922598i \(-0.373938\pi\)
\(140\) 0 0
\(141\) 10.6967i 0.900823i
\(142\) 0 0
\(143\) 14.0638i 1.17607i
\(144\) 0 0
\(145\) 16.4670 1.36751
\(146\) 0 0
\(147\) 17.4606i 1.44013i
\(148\) 0 0
\(149\) 2.38227 0.195163 0.0975815 0.995228i \(-0.468889\pi\)
0.0975815 + 0.995228i \(0.468889\pi\)
\(150\) 0 0
\(151\) 12.6698 1.03106 0.515528 0.856873i \(-0.327596\pi\)
0.515528 + 0.856873i \(0.327596\pi\)
\(152\) 0 0
\(153\) −11.5558 9.42295i −0.934235 0.761801i
\(154\) 0 0
\(155\) 12.0256 0.965917
\(156\) 0 0
\(157\) −22.5994 −1.80363 −0.901815 0.432122i \(-0.857765\pi\)
−0.901815 + 0.432122i \(0.857765\pi\)
\(158\) 0 0
\(159\) 24.2226i 1.92097i
\(160\) 0 0
\(161\) 1.49453 0.117785
\(162\) 0 0
\(163\) 8.29301i 0.649559i 0.945790 + 0.324779i \(0.105290\pi\)
−0.945790 + 0.324779i \(0.894710\pi\)
\(164\) 0 0
\(165\) 31.7538i 2.47203i
\(166\) 0 0
\(167\) 11.5763i 0.895804i 0.894083 + 0.447902i \(0.147829\pi\)
−0.894083 + 0.447902i \(0.852171\pi\)
\(168\) 0 0
\(169\) −6.01952 −0.463040
\(170\) 0 0
\(171\) 17.5902 1.34515
\(172\) 0 0
\(173\) 12.8513i 0.977067i 0.872545 + 0.488534i \(0.162468\pi\)
−0.872545 + 0.488534i \(0.837532\pi\)
\(174\) 0 0
\(175\) 1.40503i 0.106210i
\(176\) 0 0
\(177\) 2.57223i 0.193341i
\(178\) 0 0
\(179\) 5.73192 0.428424 0.214212 0.976787i \(-0.431282\pi\)
0.214212 + 0.976787i \(0.431282\pi\)
\(180\) 0 0
\(181\) 0.416007i 0.0309215i 0.999880 + 0.0154608i \(0.00492151\pi\)
−0.999880 + 0.0154608i \(0.995078\pi\)
\(182\) 0 0
\(183\) −23.3301 −1.72461
\(184\) 0 0
\(185\) 1.88225 0.138386
\(186\) 0 0
\(187\) −13.8699 + 17.0093i −1.01426 + 1.24384i
\(188\) 0 0
\(189\) −5.88725 −0.428234
\(190\) 0 0
\(191\) −2.35405 −0.170333 −0.0851665 0.996367i \(-0.527142\pi\)
−0.0851665 + 0.996367i \(0.527142\pi\)
\(192\) 0 0
\(193\) 10.0371i 0.722487i 0.932471 + 0.361244i \(0.117648\pi\)
−0.932471 + 0.361244i \(0.882352\pi\)
\(194\) 0 0
\(195\) −15.7608 −1.12866
\(196\) 0 0
\(197\) 23.2730i 1.65813i −0.559151 0.829066i \(-0.688873\pi\)
0.559151 0.829066i \(-0.311127\pi\)
\(198\) 0 0
\(199\) 6.09988i 0.432409i 0.976348 + 0.216205i \(0.0693678\pi\)
−0.976348 + 0.216205i \(0.930632\pi\)
\(200\) 0 0
\(201\) 36.8450i 2.59884i
\(202\) 0 0
\(203\) 26.3658 1.85052
\(204\) 0 0
\(205\) −25.6135 −1.78893
\(206\) 0 0
\(207\) 1.45554i 0.101167i
\(208\) 0 0
\(209\) 25.8914i 1.79094i
\(210\) 0 0
\(211\) 17.7961i 1.22513i −0.790419 0.612566i \(-0.790137\pi\)
0.790419 0.612566i \(-0.209863\pi\)
\(212\) 0 0
\(213\) 34.3260 2.35198
\(214\) 0 0
\(215\) 14.4437i 0.985051i
\(216\) 0 0
\(217\) 19.2545 1.30708
\(218\) 0 0
\(219\) −35.5264 −2.40065
\(220\) 0 0
\(221\) −8.44248 6.88423i −0.567903 0.463083i
\(222\) 0 0
\(223\) 1.96657 0.131691 0.0658455 0.997830i \(-0.479026\pi\)
0.0658455 + 0.997830i \(0.479026\pi\)
\(224\) 0 0
\(225\) 1.36838 0.0912254
\(226\) 0 0
\(227\) 12.5249i 0.831304i 0.909524 + 0.415652i \(0.136447\pi\)
−0.909524 + 0.415652i \(0.863553\pi\)
\(228\) 0 0
\(229\) 19.3439 1.27828 0.639141 0.769089i \(-0.279290\pi\)
0.639141 + 0.769089i \(0.279290\pi\)
\(230\) 0 0
\(231\) 50.8419i 3.34515i
\(232\) 0 0
\(233\) 20.5770i 1.34804i −0.738712 0.674021i \(-0.764566\pi\)
0.738712 0.674021i \(-0.235434\pi\)
\(234\) 0 0
\(235\) 9.64416i 0.629116i
\(236\) 0 0
\(237\) 33.7604 2.19297
\(238\) 0 0
\(239\) 8.11145 0.524686 0.262343 0.964975i \(-0.415505\pi\)
0.262343 + 0.964975i \(0.415505\pi\)
\(240\) 0 0
\(241\) 26.9970i 1.73903i −0.493905 0.869516i \(-0.664431\pi\)
0.493905 0.869516i \(-0.335569\pi\)
\(242\) 0 0
\(243\) 22.1728i 1.42239i
\(244\) 0 0
\(245\) 15.7426i 1.00576i
\(246\) 0 0
\(247\) 12.8510 0.817692
\(248\) 0 0
\(249\) 31.7073i 2.00937i
\(250\) 0 0
\(251\) 14.2903 0.901996 0.450998 0.892525i \(-0.351068\pi\)
0.450998 + 0.892525i \(0.351068\pi\)
\(252\) 0 0
\(253\) 2.14245 0.134694
\(254\) 0 0
\(255\) 19.0618 + 15.5435i 1.19370 + 0.973372i
\(256\) 0 0
\(257\) 23.4765 1.46442 0.732210 0.681078i \(-0.238489\pi\)
0.732210 + 0.681078i \(0.238489\pi\)
\(258\) 0 0
\(259\) 3.01372 0.187264
\(260\) 0 0
\(261\) 25.6781i 1.58943i
\(262\) 0 0
\(263\) −1.24249 −0.0766152 −0.0383076 0.999266i \(-0.512197\pi\)
−0.0383076 + 0.999266i \(0.512197\pi\)
\(264\) 0 0
\(265\) 21.8391i 1.34157i
\(266\) 0 0
\(267\) 38.2900i 2.34331i
\(268\) 0 0
\(269\) 1.76424i 0.107567i −0.998553 0.0537837i \(-0.982872\pi\)
0.998553 0.0537837i \(-0.0171281\pi\)
\(270\) 0 0
\(271\) −3.91306 −0.237702 −0.118851 0.992912i \(-0.537921\pi\)
−0.118851 + 0.992912i \(0.537921\pi\)
\(272\) 0 0
\(273\) −25.2351 −1.52730
\(274\) 0 0
\(275\) 2.01415i 0.121458i
\(276\) 0 0
\(277\) 17.7866i 1.06869i 0.845266 + 0.534346i \(0.179442\pi\)
−0.845266 + 0.534346i \(0.820558\pi\)
\(278\) 0 0
\(279\) 18.7523i 1.12267i
\(280\) 0 0
\(281\) 23.3743 1.39440 0.697198 0.716878i \(-0.254430\pi\)
0.697198 + 0.716878i \(0.254430\pi\)
\(282\) 0 0
\(283\) 2.29742i 0.136568i −0.997666 0.0682838i \(-0.978248\pi\)
0.997666 0.0682838i \(-0.0217523\pi\)
\(284\) 0 0
\(285\) −29.0156 −1.71874
\(286\) 0 0
\(287\) −41.0106 −2.42078
\(288\) 0 0
\(289\) 3.42138 + 16.6522i 0.201258 + 0.979538i
\(290\) 0 0
\(291\) 4.36898 0.256114
\(292\) 0 0
\(293\) −6.28286 −0.367048 −0.183524 0.983015i \(-0.558751\pi\)
−0.183524 + 0.983015i \(0.558751\pi\)
\(294\) 0 0
\(295\) 2.31913i 0.135025i
\(296\) 0 0
\(297\) −8.43954 −0.489712
\(298\) 0 0
\(299\) 1.06339i 0.0614976i
\(300\) 0 0
\(301\) 23.1262i 1.33297i
\(302\) 0 0
\(303\) 3.80938i 0.218843i
\(304\) 0 0
\(305\) 21.0345 1.20443
\(306\) 0 0
\(307\) −34.9114 −1.99250 −0.996249 0.0865310i \(-0.972422\pi\)
−0.996249 + 0.0865310i \(0.972422\pi\)
\(308\) 0 0
\(309\) 28.6911i 1.63218i
\(310\) 0 0
\(311\) 16.2144i 0.919435i 0.888065 + 0.459717i \(0.152049\pi\)
−0.888065 + 0.459717i \(0.847951\pi\)
\(312\) 0 0
\(313\) 4.92509i 0.278382i 0.990266 + 0.139191i \(0.0444503\pi\)
−0.990266 + 0.139191i \(0.955550\pi\)
\(314\) 0 0
\(315\) 31.1424 1.75468
\(316\) 0 0
\(317\) 9.85029i 0.553247i −0.960978 0.276624i \(-0.910784\pi\)
0.960978 0.276624i \(-0.0892156\pi\)
\(318\) 0 0
\(319\) 37.7961 2.11618
\(320\) 0 0
\(321\) 20.5365 1.14624
\(322\) 0 0
\(323\) −15.5426 12.6739i −0.864813 0.705192i
\(324\) 0 0
\(325\) 0.999713 0.0554541
\(326\) 0 0
\(327\) 7.15419 0.395627
\(328\) 0 0
\(329\) 15.4416i 0.851321i
\(330\) 0 0
\(331\) −23.1819 −1.27419 −0.637097 0.770784i \(-0.719865\pi\)
−0.637097 + 0.770784i \(0.719865\pi\)
\(332\) 0 0
\(333\) 2.93511i 0.160843i
\(334\) 0 0
\(335\) 33.2196i 1.81498i
\(336\) 0 0
\(337\) 0.371583i 0.0202414i −0.999949 0.0101207i \(-0.996778\pi\)
0.999949 0.0101207i \(-0.00322158\pi\)
\(338\) 0 0
\(339\) −16.2344 −0.881729
\(340\) 0 0
\(341\) 27.6019 1.49473
\(342\) 0 0
\(343\) 0.786747i 0.0424804i
\(344\) 0 0
\(345\) 2.40097i 0.129264i
\(346\) 0 0
\(347\) 4.37180i 0.234690i −0.993091 0.117345i \(-0.962562\pi\)
0.993091 0.117345i \(-0.0374384\pi\)
\(348\) 0 0
\(349\) −4.53122 −0.242551 −0.121275 0.992619i \(-0.538698\pi\)
−0.121275 + 0.992619i \(0.538698\pi\)
\(350\) 0 0
\(351\) 4.18892i 0.223588i
\(352\) 0 0
\(353\) −30.0520 −1.59951 −0.799754 0.600328i \(-0.795037\pi\)
−0.799754 + 0.600328i \(0.795037\pi\)
\(354\) 0 0
\(355\) −30.9485 −1.64257
\(356\) 0 0
\(357\) 30.5204 + 24.8872i 1.61531 + 1.31717i
\(358\) 0 0
\(359\) −7.97010 −0.420646 −0.210323 0.977632i \(-0.567451\pi\)
−0.210323 + 0.977632i \(0.567451\pi\)
\(360\) 0 0
\(361\) 4.65875 0.245197
\(362\) 0 0
\(363\) 44.5888i 2.34030i
\(364\) 0 0
\(365\) 32.0307 1.67656
\(366\) 0 0
\(367\) 37.8479i 1.97565i 0.155581 + 0.987823i \(0.450275\pi\)
−0.155581 + 0.987823i \(0.549725\pi\)
\(368\) 0 0
\(369\) 39.9409i 2.07924i
\(370\) 0 0
\(371\) 34.9673i 1.81541i
\(372\) 0 0
\(373\) −1.12201 −0.0580957 −0.0290478 0.999578i \(-0.509248\pi\)
−0.0290478 + 0.999578i \(0.509248\pi\)
\(374\) 0 0
\(375\) 27.5696 1.42369
\(376\) 0 0
\(377\) 18.7599i 0.966185i
\(378\) 0 0
\(379\) 1.75720i 0.0902615i 0.998981 + 0.0451307i \(0.0143704\pi\)
−0.998981 + 0.0451307i \(0.985630\pi\)
\(380\) 0 0
\(381\) 8.30452i 0.425453i
\(382\) 0 0
\(383\) 21.7868 1.11325 0.556626 0.830763i \(-0.312096\pi\)
0.556626 + 0.830763i \(0.312096\pi\)
\(384\) 0 0
\(385\) 45.8392i 2.33618i
\(386\) 0 0
\(387\) 22.5230 1.14491
\(388\) 0 0
\(389\) 13.0693 0.662637 0.331319 0.943519i \(-0.392506\pi\)
0.331319 + 0.943519i \(0.392506\pi\)
\(390\) 0 0
\(391\) 1.04873 1.28611i 0.0530366 0.0650415i
\(392\) 0 0
\(393\) −16.5459 −0.834629
\(394\) 0 0
\(395\) −30.4385 −1.53153
\(396\) 0 0
\(397\) 8.77840i 0.440575i 0.975435 + 0.220288i \(0.0706996\pi\)
−0.975435 + 0.220288i \(0.929300\pi\)
\(398\) 0 0
\(399\) −46.4578 −2.32580
\(400\) 0 0
\(401\) 21.2537i 1.06136i −0.847573 0.530680i \(-0.821937\pi\)
0.847573 0.530680i \(-0.178063\pi\)
\(402\) 0 0
\(403\) 13.7001i 0.682448i
\(404\) 0 0
\(405\) 15.7027i 0.780273i
\(406\) 0 0
\(407\) 4.32026 0.214147
\(408\) 0 0
\(409\) −32.1489 −1.58966 −0.794830 0.606833i \(-0.792440\pi\)
−0.794830 + 0.606833i \(0.792440\pi\)
\(410\) 0 0
\(411\) 34.6621i 1.70975i
\(412\) 0 0
\(413\) 3.71324i 0.182716i
\(414\) 0 0
\(415\) 28.5874i 1.40330i
\(416\) 0 0
\(417\) −55.9577 −2.74026
\(418\) 0 0
\(419\) 35.0045i 1.71008i 0.518562 + 0.855040i \(0.326467\pi\)
−0.518562 + 0.855040i \(0.673533\pi\)
\(420\) 0 0
\(421\) −6.96601 −0.339503 −0.169751 0.985487i \(-0.554296\pi\)
−0.169751 + 0.985487i \(0.554296\pi\)
\(422\) 0 0
\(423\) −15.0388 −0.731211
\(424\) 0 0
\(425\) −1.20909 0.985929i −0.0586497 0.0478246i
\(426\) 0 0
\(427\) 33.6789 1.62984
\(428\) 0 0
\(429\) −36.1753 −1.74656
\(430\) 0 0
\(431\) 33.6442i 1.62059i 0.586025 + 0.810293i \(0.300692\pi\)
−0.586025 + 0.810293i \(0.699308\pi\)
\(432\) 0 0
\(433\) −4.10948 −0.197489 −0.0987445 0.995113i \(-0.531483\pi\)
−0.0987445 + 0.995113i \(0.531483\pi\)
\(434\) 0 0
\(435\) 42.3569i 2.03086i
\(436\) 0 0
\(437\) 1.95770i 0.0936497i
\(438\) 0 0
\(439\) 6.20204i 0.296007i 0.988987 + 0.148004i \(0.0472848\pi\)
−0.988987 + 0.148004i \(0.952715\pi\)
\(440\) 0 0
\(441\) 24.5484 1.16897
\(442\) 0 0
\(443\) 25.2223 1.19835 0.599175 0.800618i \(-0.295495\pi\)
0.599175 + 0.800618i \(0.295495\pi\)
\(444\) 0 0
\(445\) 34.5224i 1.63652i
\(446\) 0 0
\(447\) 6.12775i 0.289832i
\(448\) 0 0
\(449\) 17.1235i 0.808110i 0.914735 + 0.404055i \(0.132400\pi\)
−0.914735 + 0.404055i \(0.867600\pi\)
\(450\) 0 0
\(451\) −58.7898 −2.76831
\(452\) 0 0
\(453\) 32.5897i 1.53120i
\(454\) 0 0
\(455\) 22.7521 1.06663
\(456\) 0 0
\(457\) 15.2448 0.713121 0.356561 0.934272i \(-0.383949\pi\)
0.356561 + 0.934272i \(0.383949\pi\)
\(458\) 0 0
\(459\) −4.13117 + 5.06626i −0.192826 + 0.236473i
\(460\) 0 0
\(461\) −21.7920 −1.01496 −0.507478 0.861665i \(-0.669422\pi\)
−0.507478 + 0.861665i \(0.669422\pi\)
\(462\) 0 0
\(463\) 8.97857 0.417269 0.208635 0.977994i \(-0.433098\pi\)
0.208635 + 0.977994i \(0.433098\pi\)
\(464\) 0 0
\(465\) 30.9326i 1.43446i
\(466\) 0 0
\(467\) 26.6699 1.23413 0.617067 0.786911i \(-0.288321\pi\)
0.617067 + 0.786911i \(0.288321\pi\)
\(468\) 0 0
\(469\) 53.1889i 2.45603i
\(470\) 0 0
\(471\) 58.1310i 2.67853i
\(472\) 0 0
\(473\) 33.1521i 1.52434i
\(474\) 0 0
\(475\) 1.84047 0.0844466
\(476\) 0 0
\(477\) 34.0552 1.55928
\(478\) 0 0
\(479\) 36.6325i 1.67378i −0.547369 0.836891i \(-0.684371\pi\)
0.547369 0.836891i \(-0.315629\pi\)
\(480\) 0 0
\(481\) 2.14434i 0.0977734i
\(482\) 0 0
\(483\) 3.84427i 0.174920i
\(484\) 0 0
\(485\) −3.93909 −0.178865
\(486\) 0 0
\(487\) 17.5363i 0.794646i 0.917679 + 0.397323i \(0.130061\pi\)
−0.917679 + 0.397323i \(0.869939\pi\)
\(488\) 0 0
\(489\) 21.3316 0.964646
\(490\) 0 0
\(491\) −29.1764 −1.31671 −0.658355 0.752707i \(-0.728748\pi\)
−0.658355 + 0.752707i \(0.728748\pi\)
\(492\) 0 0
\(493\) 18.5013 22.6890i 0.833255 1.02186i
\(494\) 0 0
\(495\) 44.6436 2.00658
\(496\) 0 0
\(497\) −49.5526 −2.22274
\(498\) 0 0
\(499\) 24.9940i 1.11889i 0.828869 + 0.559443i \(0.188985\pi\)
−0.828869 + 0.559443i \(0.811015\pi\)
\(500\) 0 0
\(501\) 29.7770 1.33034
\(502\) 0 0
\(503\) 5.93744i 0.264737i 0.991201 + 0.132369i \(0.0422583\pi\)
−0.991201 + 0.132369i \(0.957742\pi\)
\(504\) 0 0
\(505\) 3.43455i 0.152836i
\(506\) 0 0
\(507\) 15.4836i 0.687651i
\(508\) 0 0
\(509\) 11.7369 0.520229 0.260115 0.965578i \(-0.416240\pi\)
0.260115 + 0.965578i \(0.416240\pi\)
\(510\) 0 0
\(511\) 51.2853 2.26873
\(512\) 0 0
\(513\) 7.71180i 0.340484i
\(514\) 0 0
\(515\) 25.8680i 1.13988i
\(516\) 0 0
\(517\) 22.1359i 0.973537i
\(518\) 0 0
\(519\) 33.0566 1.45102
\(520\) 0 0
\(521\) 25.6193i 1.12240i −0.827680 0.561200i \(-0.810340\pi\)
0.827680 0.561200i \(-0.189660\pi\)
\(522\) 0 0
\(523\) −19.7900 −0.865358 −0.432679 0.901548i \(-0.642432\pi\)
−0.432679 + 0.901548i \(0.642432\pi\)
\(524\) 0 0
\(525\) −3.61406 −0.157731
\(526\) 0 0
\(527\) 13.5112 16.5694i 0.588555 0.721775i
\(528\) 0 0
\(529\) 22.8380 0.992957
\(530\) 0 0
\(531\) −3.61638 −0.156938
\(532\) 0 0
\(533\) 29.1800i 1.26393i
\(534\) 0 0
\(535\) −18.5158 −0.800507
\(536\) 0 0
\(537\) 14.7438i 0.636243i
\(538\) 0 0
\(539\) 36.1334i 1.55638i
\(540\) 0 0
\(541\) 8.91469i 0.383272i 0.981466 + 0.191636i \(0.0613794\pi\)
−0.981466 + 0.191636i \(0.938621\pi\)
\(542\) 0 0
\(543\) 1.07007 0.0459209
\(544\) 0 0
\(545\) −6.45024 −0.276298
\(546\) 0 0
\(547\) 14.1018i 0.602951i 0.953474 + 0.301475i \(0.0974791\pi\)
−0.953474 + 0.301475i \(0.902521\pi\)
\(548\) 0 0
\(549\) 32.8005i 1.39989i
\(550\) 0 0
\(551\) 34.5370i 1.47132i
\(552\) 0 0
\(553\) −48.7360 −2.07246
\(554\) 0 0
\(555\) 4.84158i 0.205513i
\(556\) 0 0
\(557\) 18.1269 0.768063 0.384031 0.923320i \(-0.374535\pi\)
0.384031 + 0.923320i \(0.374535\pi\)
\(558\) 0 0
\(559\) 16.4549 0.695967
\(560\) 0 0
\(561\) 43.7519 + 35.6765i 1.84721 + 1.50626i
\(562\) 0 0
\(563\) 27.3726 1.15362 0.576808 0.816879i \(-0.304298\pi\)
0.576808 + 0.816879i \(0.304298\pi\)
\(564\) 0 0
\(565\) 14.6370 0.615781
\(566\) 0 0
\(567\) 25.1421i 1.05587i
\(568\) 0 0
\(569\) −11.5482 −0.484124 −0.242062 0.970261i \(-0.577824\pi\)
−0.242062 + 0.970261i \(0.577824\pi\)
\(570\) 0 0
\(571\) 25.3692i 1.06167i −0.847476 0.530834i \(-0.821879\pi\)
0.847476 0.530834i \(-0.178121\pi\)
\(572\) 0 0
\(573\) 6.05516i 0.252958i
\(574\) 0 0
\(575\) 0.152294i 0.00635112i
\(576\) 0 0
\(577\) −9.37840 −0.390428 −0.195214 0.980761i \(-0.562540\pi\)
−0.195214 + 0.980761i \(0.562540\pi\)
\(578\) 0 0
\(579\) 25.8178 1.07295
\(580\) 0 0
\(581\) 45.7721i 1.89895i
\(582\) 0 0
\(583\) 50.1266i 2.07603i
\(584\) 0 0
\(585\) 22.1586i 0.916146i
\(586\) 0 0
\(587\) −5.11578 −0.211151 −0.105576 0.994411i \(-0.533668\pi\)
−0.105576 + 0.994411i \(0.533668\pi\)
\(588\) 0 0
\(589\) 25.2218i 1.03924i
\(590\) 0 0
\(591\) −59.8636 −2.46246
\(592\) 0 0
\(593\) 19.0784 0.783455 0.391727 0.920081i \(-0.371878\pi\)
0.391727 + 0.920081i \(0.371878\pi\)
\(594\) 0 0
\(595\) −27.5173 22.4384i −1.12810 0.919884i
\(596\) 0 0
\(597\) 15.6903 0.642162
\(598\) 0 0
\(599\) −37.0353 −1.51322 −0.756611 0.653865i \(-0.773146\pi\)
−0.756611 + 0.653865i \(0.773146\pi\)
\(600\) 0 0
\(601\) 14.6431i 0.597306i 0.954362 + 0.298653i \(0.0965373\pi\)
−0.954362 + 0.298653i \(0.903463\pi\)
\(602\) 0 0
\(603\) −51.8015 −2.10952
\(604\) 0 0
\(605\) 40.2014i 1.63442i
\(606\) 0 0
\(607\) 2.18191i 0.0885611i −0.999019 0.0442805i \(-0.985900\pi\)
0.999019 0.0442805i \(-0.0140995\pi\)
\(608\) 0 0
\(609\) 67.8190i 2.74816i
\(610\) 0 0
\(611\) −10.9871 −0.444489
\(612\) 0 0
\(613\) −3.18512 −0.128646 −0.0643229 0.997929i \(-0.520489\pi\)
−0.0643229 + 0.997929i \(0.520489\pi\)
\(614\) 0 0
\(615\) 65.8839i 2.65670i
\(616\) 0 0
\(617\) 25.9128i 1.04321i −0.853187 0.521605i \(-0.825334\pi\)
0.853187 0.521605i \(-0.174666\pi\)
\(618\) 0 0
\(619\) 28.8384i 1.15911i −0.814932 0.579557i \(-0.803226\pi\)
0.814932 0.579557i \(-0.196774\pi\)
\(620\) 0 0
\(621\) 0.638132 0.0256074
\(622\) 0 0
\(623\) 55.2748i 2.21454i
\(624\) 0 0
\(625\) −26.7487 −1.06995
\(626\) 0 0
\(627\) −66.5986 −2.65969
\(628\) 0 0
\(629\) 2.11477 2.59345i 0.0843215 0.103408i
\(630\) 0 0
\(631\) 14.9710 0.595984 0.297992 0.954568i \(-0.403683\pi\)
0.297992 + 0.954568i \(0.403683\pi\)
\(632\) 0 0
\(633\) −45.7756 −1.81942
\(634\) 0 0
\(635\) 7.48738i 0.297128i
\(636\) 0 0
\(637\) 17.9346 0.710596
\(638\) 0 0
\(639\) 48.2600i 1.90914i
\(640\) 0 0
\(641\) 38.3957i 1.51654i 0.651942 + 0.758269i \(0.273955\pi\)
−0.651942 + 0.758269i \(0.726045\pi\)
\(642\) 0 0
\(643\) 44.4993i 1.75488i −0.479684 0.877441i \(-0.659249\pi\)
0.479684 0.877441i \(-0.340751\pi\)
\(644\) 0 0
\(645\) −37.1525 −1.46288
\(646\) 0 0
\(647\) −6.85386 −0.269453 −0.134726 0.990883i \(-0.543016\pi\)
−0.134726 + 0.990883i \(0.543016\pi\)
\(648\) 0 0
\(649\) 5.32303i 0.208947i
\(650\) 0 0
\(651\) 49.5271i 1.94112i
\(652\) 0 0
\(653\) 38.3482i 1.50068i −0.661052 0.750340i \(-0.729890\pi\)
0.661052 0.750340i \(-0.270110\pi\)
\(654\) 0 0
\(655\) 14.9178 0.582888
\(656\) 0 0
\(657\) 49.9476i 1.94864i
\(658\) 0 0
\(659\) 19.6120 0.763975 0.381987 0.924168i \(-0.375240\pi\)
0.381987 + 0.924168i \(0.375240\pi\)
\(660\) 0 0
\(661\) −30.0416 −1.16848 −0.584242 0.811579i \(-0.698608\pi\)
−0.584242 + 0.811579i \(0.698608\pi\)
\(662\) 0 0
\(663\) −17.7078 + 21.7160i −0.687716 + 0.843381i
\(664\) 0 0
\(665\) 41.8865 1.62429
\(666\) 0 0
\(667\) −2.85785 −0.110656
\(668\) 0 0
\(669\) 5.05847i 0.195572i
\(670\) 0 0
\(671\) 48.2797 1.86382
\(672\) 0 0
\(673\) 20.2512i 0.780626i 0.920682 + 0.390313i \(0.127633\pi\)
−0.920682 + 0.390313i \(0.872367\pi\)
\(674\) 0 0
\(675\) 0.599919i 0.0230909i
\(676\) 0 0
\(677\) 18.7329i 0.719964i −0.932959 0.359982i \(-0.882783\pi\)
0.932959 0.359982i \(-0.117217\pi\)
\(678\) 0 0
\(679\) −6.30700 −0.242040
\(680\) 0 0
\(681\) 32.2169 1.23455
\(682\) 0 0
\(683\) 31.0343i 1.18750i −0.804651 0.593748i \(-0.797648\pi\)
0.804651 0.593748i \(-0.202352\pi\)
\(684\) 0 0
\(685\) 31.2514i 1.19406i
\(686\) 0 0
\(687\) 49.7571i 1.89835i
\(688\) 0 0
\(689\) 24.8801 0.947856
\(690\) 0 0
\(691\) 41.1096i 1.56388i 0.623353 + 0.781941i \(0.285770\pi\)
−0.623353 + 0.781941i \(0.714230\pi\)
\(692\) 0 0
\(693\) 71.4802 2.71531
\(694\) 0 0
\(695\) 50.4517 1.91374
\(696\) 0 0
\(697\) −28.7777 + 35.2916i −1.09003 + 1.33676i
\(698\) 0 0
\(699\) −52.9287 −2.00195
\(700\) 0 0
\(701\) 16.9073 0.638581 0.319290 0.947657i \(-0.396556\pi\)
0.319290 + 0.947657i \(0.396556\pi\)
\(702\) 0 0
\(703\) 3.94772i 0.148891i
\(704\) 0 0
\(705\) 24.8070 0.934287
\(706\) 0 0
\(707\) 5.49916i 0.206817i
\(708\) 0 0
\(709\) 48.7791i 1.83194i 0.401249 + 0.915969i \(0.368576\pi\)
−0.401249 + 0.915969i \(0.631424\pi\)
\(710\) 0 0
\(711\) 47.4647i 1.78007i
\(712\) 0 0
\(713\) −2.08704 −0.0781602
\(714\) 0 0
\(715\) 32.6158 1.21976
\(716\) 0 0
\(717\) 20.8645i 0.779201i
\(718\) 0 0
\(719\) 39.1320i 1.45938i −0.683780 0.729688i \(-0.739665\pi\)
0.683780 0.729688i \(-0.260335\pi\)
\(720\) 0 0
\(721\) 41.4181i 1.54249i
\(722\) 0 0
\(723\) −69.4427 −2.58260
\(724\) 0 0
\(725\) 2.68671i 0.0997820i
\(726\) 0 0
\(727\) 6.08016 0.225501 0.112750 0.993623i \(-0.464034\pi\)
0.112750 + 0.993623i \(0.464034\pi\)
\(728\) 0 0
\(729\) 36.7209 1.36003
\(730\) 0 0
\(731\) −19.9012 16.2280i −0.736073 0.600214i
\(732\) 0 0
\(733\) 17.8747 0.660216 0.330108 0.943943i \(-0.392915\pi\)
0.330108 + 0.943943i \(0.392915\pi\)
\(734\) 0 0
\(735\) −40.4936 −1.49363
\(736\) 0 0
\(737\) 76.2478i 2.80862i
\(738\) 0 0
\(739\) −4.94281 −0.181824 −0.0909120 0.995859i \(-0.528978\pi\)
−0.0909120 + 0.995859i \(0.528978\pi\)
\(740\) 0 0
\(741\) 33.0559i 1.21434i
\(742\) 0 0
\(743\) 20.5661i 0.754497i 0.926112 + 0.377249i \(0.123130\pi\)
−0.926112 + 0.377249i \(0.876870\pi\)
\(744\) 0 0
\(745\) 5.52480i 0.202413i
\(746\) 0 0
\(747\) 44.5782 1.63103
\(748\) 0 0
\(749\) −29.6462 −1.08325
\(750\) 0 0
\(751\) 37.8162i 1.37993i 0.723841 + 0.689967i \(0.242375\pi\)
−0.723841 + 0.689967i \(0.757625\pi\)
\(752\) 0 0
\(753\) 36.7580i 1.33953i
\(754\) 0 0
\(755\) 29.3830i 1.06936i
\(756\) 0 0
\(757\) −16.1887 −0.588387 −0.294193 0.955746i \(-0.595051\pi\)
−0.294193 + 0.955746i \(0.595051\pi\)
\(758\) 0 0
\(759\) 5.51087i 0.200032i
\(760\) 0 0
\(761\) 2.06693 0.0749261 0.0374631 0.999298i \(-0.488072\pi\)
0.0374631 + 0.999298i \(0.488072\pi\)
\(762\) 0 0
\(763\) −10.3277 −0.373887
\(764\) 0 0
\(765\) 21.8531 26.7996i 0.790100 0.968940i
\(766\) 0 0
\(767\) −2.64206 −0.0953992
\(768\) 0 0
\(769\) 13.7996 0.497625 0.248812 0.968552i \(-0.419960\pi\)
0.248812 + 0.968552i \(0.419960\pi\)
\(770\) 0 0
\(771\) 60.3869i 2.17478i
\(772\) 0 0
\(773\) 48.7429 1.75316 0.876580 0.481257i \(-0.159820\pi\)
0.876580 + 0.481257i \(0.159820\pi\)
\(774\) 0 0
\(775\) 1.96206i 0.0704793i
\(776\) 0 0
\(777\) 7.75200i 0.278101i
\(778\) 0 0
\(779\) 53.7204i 1.92473i
\(780\) 0 0
\(781\) −71.0350 −2.54183
\(782\) 0 0
\(783\) 11.2577 0.402316
\(784\) 0 0
\(785\) 52.4111i 1.87063i
\(786\) 0 0
\(787\) 36.0214i 1.28402i 0.766695 + 0.642012i \(0.221900\pi\)
−0.766695 + 0.642012i \(0.778100\pi\)
\(788\) 0 0
\(789\) 3.19597i 0.113780i
\(790\) 0 0
\(791\) 23.4357 0.833277
\(792\) 0 0
\(793\) 23.9634i 0.850965i