Properties

Label 4012.2.b.b.237.5
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 237.5
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.42

$q$-expansion

\(f(q)\) \(=\) \(q-2.81557i q^{3} +1.77357i q^{5} -1.01302i q^{7} -4.92746 q^{9} +O(q^{10})\) \(q-2.81557i q^{3} +1.77357i q^{5} -1.01302i q^{7} -4.92746 q^{9} +1.98267i q^{11} +5.96810 q^{13} +4.99361 q^{15} +(-3.47244 - 2.22309i) q^{17} +1.12852 q^{19} -2.85222 q^{21} +3.54324i q^{23} +1.85446 q^{25} +5.42691i q^{27} -3.47464i q^{29} +9.40563i q^{31} +5.58236 q^{33} +1.79665 q^{35} +4.75181i q^{37} -16.8036i q^{39} +10.0497i q^{41} -2.36274 q^{43} -8.73919i q^{45} -5.00336 q^{47} +5.97380 q^{49} +(-6.25928 + 9.77692i) q^{51} +11.5834 q^{53} -3.51640 q^{55} -3.17742i q^{57} +1.00000 q^{59} +7.35908i q^{61} +4.99160i q^{63} +10.5848i q^{65} +9.95061 q^{67} +9.97626 q^{69} +1.41124i q^{71} -13.9379i q^{73} -5.22136i q^{75} +2.00848 q^{77} +1.05486i q^{79} +0.497499 q^{81} -6.70479 q^{83} +(3.94281 - 6.15861i) q^{85} -9.78311 q^{87} -16.1957 q^{89} -6.04578i q^{91} +26.4823 q^{93} +2.00150i q^{95} +10.4135i q^{97} -9.76953i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 2.81557i 1.62557i −0.582562 0.812786i \(-0.697950\pi\)
0.582562 0.812786i \(-0.302050\pi\)
\(4\) 0 0
\(5\) 1.77357i 0.793164i 0.917999 + 0.396582i \(0.129804\pi\)
−0.917999 + 0.396582i \(0.870196\pi\)
\(6\) 0 0
\(7\) 1.01302i 0.382884i −0.981504 0.191442i \(-0.938684\pi\)
0.981504 0.191442i \(-0.0613164\pi\)
\(8\) 0 0
\(9\) −4.92746 −1.64249
\(10\) 0 0
\(11\) 1.98267i 0.597797i 0.954285 + 0.298899i \(0.0966193\pi\)
−0.954285 + 0.298899i \(0.903381\pi\)
\(12\) 0 0
\(13\) 5.96810 1.65525 0.827627 0.561279i \(-0.189691\pi\)
0.827627 + 0.561279i \(0.189691\pi\)
\(14\) 0 0
\(15\) 4.99361 1.28935
\(16\) 0 0
\(17\) −3.47244 2.22309i −0.842191 0.539179i
\(18\) 0 0
\(19\) 1.12852 0.258899 0.129450 0.991586i \(-0.458679\pi\)
0.129450 + 0.991586i \(0.458679\pi\)
\(20\) 0 0
\(21\) −2.85222 −0.622406
\(22\) 0 0
\(23\) 3.54324i 0.738817i 0.929267 + 0.369409i \(0.120440\pi\)
−0.929267 + 0.369409i \(0.879560\pi\)
\(24\) 0 0
\(25\) 1.85446 0.370891
\(26\) 0 0
\(27\) 5.42691i 1.04441i
\(28\) 0 0
\(29\) 3.47464i 0.645224i −0.946531 0.322612i \(-0.895439\pi\)
0.946531 0.322612i \(-0.104561\pi\)
\(30\) 0 0
\(31\) 9.40563i 1.68930i 0.535318 + 0.844651i \(0.320192\pi\)
−0.535318 + 0.844651i \(0.679808\pi\)
\(32\) 0 0
\(33\) 5.58236 0.971763
\(34\) 0 0
\(35\) 1.79665 0.303690
\(36\) 0 0
\(37\) 4.75181i 0.781193i 0.920562 + 0.390597i \(0.127731\pi\)
−0.920562 + 0.390597i \(0.872269\pi\)
\(38\) 0 0
\(39\) 16.8036i 2.69073i
\(40\) 0 0
\(41\) 10.0497i 1.56950i 0.619814 + 0.784749i \(0.287208\pi\)
−0.619814 + 0.784749i \(0.712792\pi\)
\(42\) 0 0
\(43\) −2.36274 −0.360315 −0.180157 0.983638i \(-0.557661\pi\)
−0.180157 + 0.983638i \(0.557661\pi\)
\(44\) 0 0
\(45\) 8.73919i 1.30276i
\(46\) 0 0
\(47\) −5.00336 −0.729815 −0.364908 0.931044i \(-0.618899\pi\)
−0.364908 + 0.931044i \(0.618899\pi\)
\(48\) 0 0
\(49\) 5.97380 0.853400
\(50\) 0 0
\(51\) −6.25928 + 9.77692i −0.876475 + 1.36904i
\(52\) 0 0
\(53\) 11.5834 1.59111 0.795553 0.605884i \(-0.207180\pi\)
0.795553 + 0.605884i \(0.207180\pi\)
\(54\) 0 0
\(55\) −3.51640 −0.474151
\(56\) 0 0
\(57\) 3.17742i 0.420860i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 7.35908i 0.942234i 0.882071 + 0.471117i \(0.156149\pi\)
−0.882071 + 0.471117i \(0.843851\pi\)
\(62\) 0 0
\(63\) 4.99160i 0.628882i
\(64\) 0 0
\(65\) 10.5848i 1.31289i
\(66\) 0 0
\(67\) 9.95061 1.21566 0.607830 0.794067i \(-0.292040\pi\)
0.607830 + 0.794067i \(0.292040\pi\)
\(68\) 0 0
\(69\) 9.97626 1.20100
\(70\) 0 0
\(71\) 1.41124i 0.167483i 0.996488 + 0.0837415i \(0.0266870\pi\)
−0.996488 + 0.0837415i \(0.973313\pi\)
\(72\) 0 0
\(73\) 13.9379i 1.63131i −0.578536 0.815657i \(-0.696376\pi\)
0.578536 0.815657i \(-0.303624\pi\)
\(74\) 0 0
\(75\) 5.22136i 0.602911i
\(76\) 0 0
\(77\) 2.00848 0.228887
\(78\) 0 0
\(79\) 1.05486i 0.118681i 0.998238 + 0.0593407i \(0.0188999\pi\)
−0.998238 + 0.0593407i \(0.981100\pi\)
\(80\) 0 0
\(81\) 0.497499 0.0552776
\(82\) 0 0
\(83\) −6.70479 −0.735947 −0.367973 0.929836i \(-0.619948\pi\)
−0.367973 + 0.929836i \(0.619948\pi\)
\(84\) 0 0
\(85\) 3.94281 6.15861i 0.427657 0.667995i
\(86\) 0 0
\(87\) −9.78311 −1.04886
\(88\) 0 0
\(89\) −16.1957 −1.71674 −0.858368 0.513034i \(-0.828521\pi\)
−0.858368 + 0.513034i \(0.828521\pi\)
\(90\) 0 0
\(91\) 6.04578i 0.633770i
\(92\) 0 0
\(93\) 26.4823 2.74608
\(94\) 0 0
\(95\) 2.00150i 0.205349i
\(96\) 0 0
\(97\) 10.4135i 1.05733i 0.848830 + 0.528665i \(0.177307\pi\)
−0.848830 + 0.528665i \(0.822693\pi\)
\(98\) 0 0
\(99\) 9.76953i 0.981875i
\(100\) 0 0
\(101\) 19.4250 1.93286 0.966431 0.256925i \(-0.0827095\pi\)
0.966431 + 0.256925i \(0.0827095\pi\)
\(102\) 0 0
\(103\) 8.62785 0.850128 0.425064 0.905163i \(-0.360252\pi\)
0.425064 + 0.905163i \(0.360252\pi\)
\(104\) 0 0
\(105\) 5.05861i 0.493670i
\(106\) 0 0
\(107\) 12.6904i 1.22683i −0.789762 0.613414i \(-0.789796\pi\)
0.789762 0.613414i \(-0.210204\pi\)
\(108\) 0 0
\(109\) 10.4306i 0.999068i −0.866294 0.499534i \(-0.833505\pi\)
0.866294 0.499534i \(-0.166495\pi\)
\(110\) 0 0
\(111\) 13.3791 1.26989
\(112\) 0 0
\(113\) 16.6706i 1.56824i 0.620610 + 0.784119i \(0.286885\pi\)
−0.620610 + 0.784119i \(0.713115\pi\)
\(114\) 0 0
\(115\) −6.28418 −0.586003
\(116\) 0 0
\(117\) −29.4076 −2.71873
\(118\) 0 0
\(119\) −2.25203 + 3.51764i −0.206443 + 0.322461i
\(120\) 0 0
\(121\) 7.06902 0.642638
\(122\) 0 0
\(123\) 28.2956 2.55133
\(124\) 0 0
\(125\) 12.1568i 1.08734i
\(126\) 0 0
\(127\) 19.6325 1.74210 0.871049 0.491196i \(-0.163440\pi\)
0.871049 + 0.491196i \(0.163440\pi\)
\(128\) 0 0
\(129\) 6.65247i 0.585718i
\(130\) 0 0
\(131\) 14.4265i 1.26045i −0.776413 0.630224i \(-0.782963\pi\)
0.776413 0.630224i \(-0.217037\pi\)
\(132\) 0 0
\(133\) 1.14320i 0.0991284i
\(134\) 0 0
\(135\) −9.62500 −0.828388
\(136\) 0 0
\(137\) 10.2949 0.879554 0.439777 0.898107i \(-0.355057\pi\)
0.439777 + 0.898107i \(0.355057\pi\)
\(138\) 0 0
\(139\) 12.6278i 1.07108i 0.844510 + 0.535540i \(0.179892\pi\)
−0.844510 + 0.535540i \(0.820108\pi\)
\(140\) 0 0
\(141\) 14.0873i 1.18637i
\(142\) 0 0
\(143\) 11.8328i 0.989506i
\(144\) 0 0
\(145\) 6.16251 0.511769
\(146\) 0 0
\(147\) 16.8197i 1.38726i
\(148\) 0 0
\(149\) 5.16778 0.423361 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(150\) 0 0
\(151\) 15.5358 1.26428 0.632142 0.774853i \(-0.282176\pi\)
0.632142 + 0.774853i \(0.282176\pi\)
\(152\) 0 0
\(153\) 17.1103 + 10.9542i 1.38329 + 0.885595i
\(154\) 0 0
\(155\) −16.6815 −1.33989
\(156\) 0 0
\(157\) −5.90512 −0.471280 −0.235640 0.971840i \(-0.575719\pi\)
−0.235640 + 0.971840i \(0.575719\pi\)
\(158\) 0 0
\(159\) 32.6140i 2.58646i
\(160\) 0 0
\(161\) 3.58936 0.282881
\(162\) 0 0
\(163\) 10.5183i 0.823859i −0.911216 0.411929i \(-0.864855\pi\)
0.911216 0.411929i \(-0.135145\pi\)
\(164\) 0 0
\(165\) 9.90069i 0.770767i
\(166\) 0 0
\(167\) 25.0986i 1.94219i −0.238700 0.971093i \(-0.576721\pi\)
0.238700 0.971093i \(-0.423279\pi\)
\(168\) 0 0
\(169\) 22.6182 1.73986
\(170\) 0 0
\(171\) −5.56072 −0.425239
\(172\) 0 0
\(173\) 7.25160i 0.551329i −0.961254 0.275664i \(-0.911102\pi\)
0.961254 0.275664i \(-0.0888978\pi\)
\(174\) 0 0
\(175\) 1.87859i 0.142008i
\(176\) 0 0
\(177\) 2.81557i 0.211632i
\(178\) 0 0
\(179\) −20.5865 −1.53870 −0.769352 0.638825i \(-0.779421\pi\)
−0.769352 + 0.638825i \(0.779421\pi\)
\(180\) 0 0
\(181\) 16.2432i 1.20735i 0.797230 + 0.603675i \(0.206298\pi\)
−0.797230 + 0.603675i \(0.793702\pi\)
\(182\) 0 0
\(183\) 20.7200 1.53167
\(184\) 0 0
\(185\) −8.42766 −0.619614
\(186\) 0 0
\(187\) 4.40766 6.88471i 0.322320 0.503460i
\(188\) 0 0
\(189\) 5.49755 0.399888
\(190\) 0 0
\(191\) −0.717439 −0.0519121 −0.0259560 0.999663i \(-0.508263\pi\)
−0.0259560 + 0.999663i \(0.508263\pi\)
\(192\) 0 0
\(193\) 15.8853i 1.14345i 0.820447 + 0.571723i \(0.193725\pi\)
−0.820447 + 0.571723i \(0.806275\pi\)
\(194\) 0 0
\(195\) 29.8024 2.13419
\(196\) 0 0
\(197\) 10.5858i 0.754205i −0.926172 0.377102i \(-0.876921\pi\)
0.926172 0.377102i \(-0.123079\pi\)
\(198\) 0 0
\(199\) 12.3598i 0.876163i −0.898935 0.438082i \(-0.855658\pi\)
0.898935 0.438082i \(-0.144342\pi\)
\(200\) 0 0
\(201\) 28.0167i 1.97615i
\(202\) 0 0
\(203\) −3.51986 −0.247046
\(204\) 0 0
\(205\) −17.8238 −1.24487
\(206\) 0 0
\(207\) 17.4592i 1.21350i
\(208\) 0 0
\(209\) 2.23747i 0.154769i
\(210\) 0 0
\(211\) 15.7388i 1.08350i −0.840538 0.541752i \(-0.817761\pi\)
0.840538 0.541752i \(-0.182239\pi\)
\(212\) 0 0
\(213\) 3.97344 0.272256
\(214\) 0 0
\(215\) 4.19048i 0.285788i
\(216\) 0 0
\(217\) 9.52805 0.646806
\(218\) 0 0
\(219\) −39.2433 −2.65182
\(220\) 0 0
\(221\) −20.7239 13.2676i −1.39404 0.892478i
\(222\) 0 0
\(223\) −14.7639 −0.988666 −0.494333 0.869273i \(-0.664588\pi\)
−0.494333 + 0.869273i \(0.664588\pi\)
\(224\) 0 0
\(225\) −9.13777 −0.609184
\(226\) 0 0
\(227\) 3.24334i 0.215268i 0.994191 + 0.107634i \(0.0343275\pi\)
−0.994191 + 0.107634i \(0.965673\pi\)
\(228\) 0 0
\(229\) −4.52390 −0.298948 −0.149474 0.988766i \(-0.547758\pi\)
−0.149474 + 0.988766i \(0.547758\pi\)
\(230\) 0 0
\(231\) 5.65501i 0.372073i
\(232\) 0 0
\(233\) 13.1725i 0.862957i −0.902123 0.431479i \(-0.857992\pi\)
0.902123 0.431479i \(-0.142008\pi\)
\(234\) 0 0
\(235\) 8.87380i 0.578863i
\(236\) 0 0
\(237\) 2.97005 0.192925
\(238\) 0 0
\(239\) −8.11549 −0.524947 −0.262474 0.964939i \(-0.584538\pi\)
−0.262474 + 0.964939i \(0.584538\pi\)
\(240\) 0 0
\(241\) 28.9264i 1.86331i 0.363341 + 0.931656i \(0.381636\pi\)
−0.363341 + 0.931656i \(0.618364\pi\)
\(242\) 0 0
\(243\) 14.8800i 0.954552i
\(244\) 0 0
\(245\) 10.5949i 0.676886i
\(246\) 0 0
\(247\) 6.73509 0.428544
\(248\) 0 0
\(249\) 18.8778i 1.19633i
\(250\) 0 0
\(251\) 0.609374 0.0384634 0.0192317 0.999815i \(-0.493878\pi\)
0.0192317 + 0.999815i \(0.493878\pi\)
\(252\) 0 0
\(253\) −7.02508 −0.441663
\(254\) 0 0
\(255\) −17.3400 11.1013i −1.08588 0.695188i
\(256\) 0 0
\(257\) −4.50186 −0.280818 −0.140409 0.990094i \(-0.544842\pi\)
−0.140409 + 0.990094i \(0.544842\pi\)
\(258\) 0 0
\(259\) 4.81366 0.299106
\(260\) 0 0
\(261\) 17.1212i 1.05977i
\(262\) 0 0
\(263\) −7.19803 −0.443850 −0.221925 0.975064i \(-0.571234\pi\)
−0.221925 + 0.975064i \(0.571234\pi\)
\(264\) 0 0
\(265\) 20.5440i 1.26201i
\(266\) 0 0
\(267\) 45.6001i 2.79068i
\(268\) 0 0
\(269\) 20.3740i 1.24223i −0.783721 0.621113i \(-0.786681\pi\)
0.783721 0.621113i \(-0.213319\pi\)
\(270\) 0 0
\(271\) −3.58302 −0.217653 −0.108826 0.994061i \(-0.534709\pi\)
−0.108826 + 0.994061i \(0.534709\pi\)
\(272\) 0 0
\(273\) −17.0223 −1.03024
\(274\) 0 0
\(275\) 3.67678i 0.221718i
\(276\) 0 0
\(277\) 3.80453i 0.228592i 0.993447 + 0.114296i \(0.0364613\pi\)
−0.993447 + 0.114296i \(0.963539\pi\)
\(278\) 0 0
\(279\) 46.3459i 2.77466i
\(280\) 0 0
\(281\) −27.4803 −1.63934 −0.819668 0.572839i \(-0.805842\pi\)
−0.819668 + 0.572839i \(0.805842\pi\)
\(282\) 0 0
\(283\) 18.4731i 1.09811i −0.835787 0.549055i \(-0.814988\pi\)
0.835787 0.549055i \(-0.185012\pi\)
\(284\) 0 0
\(285\) 5.63537 0.333811
\(286\) 0 0
\(287\) 10.1805 0.600935
\(288\) 0 0
\(289\) 7.11572 + 15.4391i 0.418571 + 0.908184i
\(290\) 0 0
\(291\) 29.3200 1.71877
\(292\) 0 0
\(293\) 25.0094 1.46106 0.730532 0.682878i \(-0.239272\pi\)
0.730532 + 0.682878i \(0.239272\pi\)
\(294\) 0 0
\(295\) 1.77357i 0.103261i
\(296\) 0 0
\(297\) −10.7598 −0.624346
\(298\) 0 0
\(299\) 21.1464i 1.22293i
\(300\) 0 0
\(301\) 2.39349i 0.137959i
\(302\) 0 0
\(303\) 54.6926i 3.14201i
\(304\) 0 0
\(305\) −13.0518 −0.747346
\(306\) 0 0
\(307\) 28.2474 1.61217 0.806083 0.591802i \(-0.201583\pi\)
0.806083 + 0.591802i \(0.201583\pi\)
\(308\) 0 0
\(309\) 24.2924i 1.38194i
\(310\) 0 0
\(311\) 21.4590i 1.21683i 0.793620 + 0.608414i \(0.208194\pi\)
−0.793620 + 0.608414i \(0.791806\pi\)
\(312\) 0 0
\(313\) 0.372065i 0.0210303i 0.999945 + 0.0105152i \(0.00334714\pi\)
−0.999945 + 0.0105152i \(0.996653\pi\)
\(314\) 0 0
\(315\) −8.85294 −0.498806
\(316\) 0 0
\(317\) 13.6019i 0.763960i −0.924171 0.381980i \(-0.875242\pi\)
0.924171 0.381980i \(-0.124758\pi\)
\(318\) 0 0
\(319\) 6.88906 0.385713
\(320\) 0 0
\(321\) −35.7308 −1.99430
\(322\) 0 0
\(323\) −3.91871 2.50880i −0.218043 0.139593i
\(324\) 0 0
\(325\) 11.0676 0.613919
\(326\) 0 0
\(327\) −29.3681 −1.62406
\(328\) 0 0
\(329\) 5.06848i 0.279435i
\(330\) 0 0
\(331\) −31.6998 −1.74238 −0.871189 0.490949i \(-0.836650\pi\)
−0.871189 + 0.490949i \(0.836650\pi\)
\(332\) 0 0
\(333\) 23.4144i 1.28310i
\(334\) 0 0
\(335\) 17.6481i 0.964218i
\(336\) 0 0
\(337\) 15.4649i 0.842426i 0.906962 + 0.421213i \(0.138396\pi\)
−0.906962 + 0.421213i \(0.861604\pi\)
\(338\) 0 0
\(339\) 46.9373 2.54929
\(340\) 0 0
\(341\) −18.6483 −1.00986
\(342\) 0 0
\(343\) 13.1427i 0.709637i
\(344\) 0 0
\(345\) 17.6936i 0.952591i
\(346\) 0 0
\(347\) 10.1605i 0.545446i −0.962093 0.272723i \(-0.912076\pi\)
0.962093 0.272723i \(-0.0879242\pi\)
\(348\) 0 0
\(349\) 2.33941 0.125226 0.0626128 0.998038i \(-0.480057\pi\)
0.0626128 + 0.998038i \(0.480057\pi\)
\(350\) 0 0
\(351\) 32.3884i 1.72876i
\(352\) 0 0
\(353\) 19.7266 1.04994 0.524970 0.851121i \(-0.324076\pi\)
0.524970 + 0.851121i \(0.324076\pi\)
\(354\) 0 0
\(355\) −2.50292 −0.132841
\(356\) 0 0
\(357\) 9.90418 + 6.34075i 0.524185 + 0.335588i
\(358\) 0 0
\(359\) −8.02116 −0.423341 −0.211670 0.977341i \(-0.567890\pi\)
−0.211670 + 0.977341i \(0.567890\pi\)
\(360\) 0 0
\(361\) −17.7265 −0.932971
\(362\) 0 0
\(363\) 19.9034i 1.04466i
\(364\) 0 0
\(365\) 24.7199 1.29390
\(366\) 0 0
\(367\) 23.3745i 1.22014i 0.792349 + 0.610069i \(0.208858\pi\)
−0.792349 + 0.610069i \(0.791142\pi\)
\(368\) 0 0
\(369\) 49.5195i 2.57788i
\(370\) 0 0
\(371\) 11.7342i 0.609209i
\(372\) 0 0
\(373\) −6.07516 −0.314560 −0.157280 0.987554i \(-0.550273\pi\)
−0.157280 + 0.987554i \(0.550273\pi\)
\(374\) 0 0
\(375\) 34.2285 1.76755
\(376\) 0 0
\(377\) 20.7370i 1.06801i
\(378\) 0 0
\(379\) 0.180751i 0.00928458i −0.999989 0.00464229i \(-0.998522\pi\)
0.999989 0.00464229i \(-0.00147769\pi\)
\(380\) 0 0
\(381\) 55.2766i 2.83191i
\(382\) 0 0
\(383\) −19.5919 −1.00110 −0.500549 0.865708i \(-0.666868\pi\)
−0.500549 + 0.865708i \(0.666868\pi\)
\(384\) 0 0
\(385\) 3.56217i 0.181545i
\(386\) 0 0
\(387\) 11.6423 0.591812
\(388\) 0 0
\(389\) 26.1382 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(390\) 0 0
\(391\) 7.87696 12.3037i 0.398355 0.622225i
\(392\) 0 0
\(393\) −40.6188 −2.04895
\(394\) 0 0
\(395\) −1.87087 −0.0941339
\(396\) 0 0
\(397\) 9.85946i 0.494832i 0.968909 + 0.247416i \(0.0795815\pi\)
−0.968909 + 0.247416i \(0.920419\pi\)
\(398\) 0 0
\(399\) −3.21878 −0.161140
\(400\) 0 0
\(401\) 8.62190i 0.430557i −0.976553 0.215279i \(-0.930934\pi\)
0.976553 0.215279i \(-0.0690660\pi\)
\(402\) 0 0
\(403\) 56.1337i 2.79622i
\(404\) 0 0
\(405\) 0.882348i 0.0438442i
\(406\) 0 0
\(407\) −9.42128 −0.466995
\(408\) 0 0
\(409\) 21.4765 1.06194 0.530972 0.847389i \(-0.321827\pi\)
0.530972 + 0.847389i \(0.321827\pi\)
\(410\) 0 0
\(411\) 28.9861i 1.42978i
\(412\) 0 0
\(413\) 1.01302i 0.0498472i
\(414\) 0 0
\(415\) 11.8914i 0.583726i
\(416\) 0 0
\(417\) 35.5547 1.74112
\(418\) 0 0
\(419\) 27.4699i 1.34199i 0.741461 + 0.670996i \(0.234133\pi\)
−0.741461 + 0.670996i \(0.765867\pi\)
\(420\) 0 0
\(421\) 12.3256 0.600712 0.300356 0.953827i \(-0.402895\pi\)
0.300356 + 0.953827i \(0.402895\pi\)
\(422\) 0 0
\(423\) 24.6539 1.19871
\(424\) 0 0
\(425\) −6.43949 4.12263i −0.312361 0.199977i
\(426\) 0 0
\(427\) 7.45487 0.360766
\(428\) 0 0
\(429\) 33.3161 1.60851
\(430\) 0 0
\(431\) 33.6273i 1.61977i −0.586587 0.809886i \(-0.699529\pi\)
0.586587 0.809886i \(-0.300471\pi\)
\(432\) 0 0
\(433\) 20.2399 0.972665 0.486333 0.873774i \(-0.338334\pi\)
0.486333 + 0.873774i \(0.338334\pi\)
\(434\) 0 0
\(435\) 17.3510i 0.831917i
\(436\) 0 0
\(437\) 3.99860i 0.191279i
\(438\) 0 0
\(439\) 33.9267i 1.61923i −0.586959 0.809616i \(-0.699675\pi\)
0.586959 0.809616i \(-0.300325\pi\)
\(440\) 0 0
\(441\) −29.4357 −1.40170
\(442\) 0 0
\(443\) 16.4757 0.782782 0.391391 0.920224i \(-0.371994\pi\)
0.391391 + 0.920224i \(0.371994\pi\)
\(444\) 0 0
\(445\) 28.7241i 1.36165i
\(446\) 0 0
\(447\) 14.5503i 0.688205i
\(448\) 0 0
\(449\) 20.0182i 0.944720i 0.881406 + 0.472360i \(0.156598\pi\)
−0.881406 + 0.472360i \(0.843402\pi\)
\(450\) 0 0
\(451\) −19.9252 −0.938242
\(452\) 0 0
\(453\) 43.7421i 2.05518i
\(454\) 0 0
\(455\) 10.7226 0.502683
\(456\) 0 0
\(457\) −36.5676 −1.71056 −0.855280 0.518166i \(-0.826615\pi\)
−0.855280 + 0.518166i \(0.826615\pi\)
\(458\) 0 0
\(459\) 12.0645 18.8447i 0.563124 0.879593i
\(460\) 0 0
\(461\) 38.8086 1.80750 0.903748 0.428065i \(-0.140805\pi\)
0.903748 + 0.428065i \(0.140805\pi\)
\(462\) 0 0
\(463\) −16.6910 −0.775697 −0.387848 0.921723i \(-0.626782\pi\)
−0.387848 + 0.921723i \(0.626782\pi\)
\(464\) 0 0
\(465\) 46.9681i 2.17809i
\(466\) 0 0
\(467\) 34.8224 1.61139 0.805694 0.592332i \(-0.201793\pi\)
0.805694 + 0.592332i \(0.201793\pi\)
\(468\) 0 0
\(469\) 10.0801i 0.465457i
\(470\) 0 0
\(471\) 16.6263i 0.766100i
\(472\) 0 0
\(473\) 4.68453i 0.215395i
\(474\) 0 0
\(475\) 2.09278 0.0960235
\(476\) 0 0
\(477\) −57.0769 −2.61337
\(478\) 0 0
\(479\) 29.0259i 1.32623i 0.748519 + 0.663114i \(0.230765\pi\)
−0.748519 + 0.663114i \(0.769235\pi\)
\(480\) 0 0
\(481\) 28.3593i 1.29307i
\(482\) 0 0
\(483\) 10.1061i 0.459844i
\(484\) 0 0
\(485\) −18.4690 −0.838636
\(486\) 0 0
\(487\) 14.7029i 0.666250i −0.942883 0.333125i \(-0.891897\pi\)
0.942883 0.333125i \(-0.108103\pi\)
\(488\) 0 0
\(489\) −29.6151 −1.33924
\(490\) 0 0
\(491\) −19.6682 −0.887613 −0.443807 0.896123i \(-0.646372\pi\)
−0.443807 + 0.896123i \(0.646372\pi\)
\(492\) 0 0
\(493\) −7.72445 + 12.0655i −0.347892 + 0.543402i
\(494\) 0 0
\(495\) 17.3269 0.778787
\(496\) 0 0
\(497\) 1.42960 0.0641265
\(498\) 0 0
\(499\) 11.1454i 0.498936i 0.968383 + 0.249468i \(0.0802558\pi\)
−0.968383 + 0.249468i \(0.919744\pi\)
\(500\) 0 0
\(501\) −70.6669 −3.15717
\(502\) 0 0
\(503\) 13.7333i 0.612338i −0.951977 0.306169i \(-0.900953\pi\)
0.951977 0.306169i \(-0.0990472\pi\)
\(504\) 0 0
\(505\) 34.4516i 1.53308i
\(506\) 0 0
\(507\) 63.6833i 2.82827i
\(508\) 0 0
\(509\) 22.5571 0.999826 0.499913 0.866076i \(-0.333365\pi\)
0.499913 + 0.866076i \(0.333365\pi\)
\(510\) 0 0
\(511\) −14.1194 −0.624604
\(512\) 0 0
\(513\) 6.12436i 0.270397i
\(514\) 0 0
\(515\) 15.3021i 0.674290i
\(516\) 0 0
\(517\) 9.92001i 0.436282i
\(518\) 0 0
\(519\) −20.4174 −0.896225
\(520\) 0 0
\(521\) 6.46984i 0.283449i −0.989906 0.141724i \(-0.954735\pi\)
0.989906 0.141724i \(-0.0452647\pi\)
\(522\) 0 0
\(523\) 36.0518 1.57643 0.788217 0.615397i \(-0.211004\pi\)
0.788217 + 0.615397i \(0.211004\pi\)
\(524\) 0 0
\(525\) −5.28932 −0.230845
\(526\) 0 0
\(527\) 20.9096 32.6605i 0.910836 1.42271i
\(528\) 0 0
\(529\) 10.4454 0.454149
\(530\) 0 0
\(531\) −4.92746 −0.213834
\(532\) 0 0
\(533\) 59.9775i 2.59792i
\(534\) 0 0
\(535\) 22.5073 0.973075
\(536\) 0 0
\(537\) 57.9627i 2.50128i
\(538\) 0 0
\(539\) 11.8441i 0.510160i
\(540\) 0 0
\(541\) 25.2692i 1.08641i 0.839601 + 0.543203i \(0.182789\pi\)
−0.839601 + 0.543203i \(0.817211\pi\)
\(542\) 0 0
\(543\) 45.7340 1.96264
\(544\) 0 0
\(545\) 18.4993 0.792424
\(546\) 0 0
\(547\) 23.7132i 1.01390i 0.861975 + 0.506951i \(0.169228\pi\)
−0.861975 + 0.506951i \(0.830772\pi\)
\(548\) 0 0
\(549\) 36.2616i 1.54761i
\(550\) 0 0
\(551\) 3.92119i 0.167048i
\(552\) 0 0
\(553\) 1.06859 0.0454412
\(554\) 0 0
\(555\) 23.7287i 1.00723i
\(556\) 0 0
\(557\) −14.1225 −0.598391 −0.299195 0.954192i \(-0.596718\pi\)
−0.299195 + 0.954192i \(0.596718\pi\)
\(558\) 0 0
\(559\) −14.1011 −0.596412
\(560\) 0 0
\(561\) −19.3844 12.4101i −0.818410 0.523955i
\(562\) 0 0
\(563\) 3.11376 0.131229 0.0656146 0.997845i \(-0.479099\pi\)
0.0656146 + 0.997845i \(0.479099\pi\)
\(564\) 0 0
\(565\) −29.5664 −1.24387
\(566\) 0 0
\(567\) 0.503974i 0.0211649i
\(568\) 0 0
\(569\) 39.6305 1.66140 0.830699 0.556723i \(-0.187941\pi\)
0.830699 + 0.556723i \(0.187941\pi\)
\(570\) 0 0
\(571\) 10.7523i 0.449972i −0.974362 0.224986i \(-0.927766\pi\)
0.974362 0.224986i \(-0.0722336\pi\)
\(572\) 0 0
\(573\) 2.02000i 0.0843869i
\(574\) 0 0
\(575\) 6.57079i 0.274021i
\(576\) 0 0
\(577\) 42.7902 1.78138 0.890689 0.454614i \(-0.150223\pi\)
0.890689 + 0.454614i \(0.150223\pi\)
\(578\) 0 0
\(579\) 44.7262 1.85876
\(580\) 0 0
\(581\) 6.79206i 0.281782i
\(582\) 0 0
\(583\) 22.9661i 0.951159i
\(584\) 0 0
\(585\) 52.1564i 2.15640i
\(586\) 0 0
\(587\) −21.5944 −0.891296 −0.445648 0.895208i \(-0.647027\pi\)
−0.445648 + 0.895208i \(0.647027\pi\)
\(588\) 0 0
\(589\) 10.6144i 0.437359i
\(590\) 0 0
\(591\) −29.8050 −1.22601
\(592\) 0 0
\(593\) −25.2159 −1.03549 −0.517747 0.855534i \(-0.673229\pi\)
−0.517747 + 0.855534i \(0.673229\pi\)
\(594\) 0 0
\(595\) −6.23877 3.99412i −0.255765 0.163743i
\(596\) 0 0
\(597\) −34.7999 −1.42427
\(598\) 0 0
\(599\) −21.3818 −0.873635 −0.436817 0.899550i \(-0.643894\pi\)
−0.436817 + 0.899550i \(0.643894\pi\)
\(600\) 0 0
\(601\) 25.1080i 1.02418i −0.858932 0.512089i \(-0.828872\pi\)
0.858932 0.512089i \(-0.171128\pi\)
\(602\) 0 0
\(603\) −49.0313 −1.99671
\(604\) 0 0
\(605\) 12.5374i 0.509717i
\(606\) 0 0
\(607\) 6.83691i 0.277502i 0.990327 + 0.138751i \(0.0443087\pi\)
−0.990327 + 0.138751i \(0.955691\pi\)
\(608\) 0 0
\(609\) 9.91044i 0.401591i
\(610\) 0 0
\(611\) −29.8606 −1.20803
\(612\) 0 0
\(613\) 18.3054 0.739346 0.369673 0.929162i \(-0.379470\pi\)
0.369673 + 0.929162i \(0.379470\pi\)
\(614\) 0 0
\(615\) 50.1843i 2.02362i
\(616\) 0 0
\(617\) 6.06759i 0.244272i 0.992513 + 0.122136i \(0.0389744\pi\)
−0.992513 + 0.122136i \(0.961026\pi\)
\(618\) 0 0
\(619\) 1.30249i 0.0523513i 0.999657 + 0.0261757i \(0.00833292\pi\)
−0.999657 + 0.0261757i \(0.991667\pi\)
\(620\) 0 0
\(621\) −19.2289 −0.771628
\(622\) 0 0
\(623\) 16.4065i 0.657311i
\(624\) 0 0
\(625\) −12.2887 −0.491548
\(626\) 0 0
\(627\) 6.29978 0.251589
\(628\) 0 0
\(629\) 10.5637 16.5004i 0.421203 0.657914i
\(630\) 0 0
\(631\) −44.7455 −1.78129 −0.890644 0.454700i \(-0.849746\pi\)
−0.890644 + 0.454700i \(0.849746\pi\)
\(632\) 0 0
\(633\) −44.3138 −1.76132
\(634\) 0 0
\(635\) 34.8195i 1.38177i
\(636\) 0 0
\(637\) 35.6522 1.41259
\(638\) 0 0
\(639\) 6.95381i 0.275089i
\(640\) 0 0
\(641\) 18.9106i 0.746922i 0.927646 + 0.373461i \(0.121829\pi\)
−0.927646 + 0.373461i \(0.878171\pi\)
\(642\) 0 0
\(643\) 3.31783i 0.130842i 0.997858 + 0.0654211i \(0.0208391\pi\)
−0.997858 + 0.0654211i \(0.979161\pi\)
\(644\) 0 0
\(645\) −11.7986 −0.464570
\(646\) 0 0
\(647\) 8.23766 0.323856 0.161928 0.986803i \(-0.448229\pi\)
0.161928 + 0.986803i \(0.448229\pi\)
\(648\) 0 0
\(649\) 1.98267i 0.0778266i
\(650\) 0 0
\(651\) 26.8269i 1.05143i
\(652\) 0 0
\(653\) 41.8275i 1.63684i 0.574622 + 0.818419i \(0.305149\pi\)
−0.574622 + 0.818419i \(0.694851\pi\)
\(654\) 0 0
\(655\) 25.5864 0.999741
\(656\) 0 0
\(657\) 68.6787i 2.67941i
\(658\) 0 0
\(659\) −26.8807 −1.04712 −0.523561 0.851988i \(-0.675397\pi\)
−0.523561 + 0.851988i \(0.675397\pi\)
\(660\) 0 0
\(661\) −46.5871 −1.81203 −0.906014 0.423247i \(-0.860890\pi\)
−0.906014 + 0.423247i \(0.860890\pi\)
\(662\) 0 0
\(663\) −37.3560 + 58.3497i −1.45079 + 2.26611i
\(664\) 0 0
\(665\) 2.02755 0.0786250
\(666\) 0 0
\(667\) 12.3115 0.476703
\(668\) 0 0
\(669\) 41.5690i 1.60715i
\(670\) 0 0
\(671\) −14.5906 −0.563265
\(672\) 0 0
\(673\) 50.8104i 1.95860i −0.202417 0.979299i \(-0.564880\pi\)
0.202417 0.979299i \(-0.435120\pi\)
\(674\) 0 0
\(675\) 10.0640i 0.387363i
\(676\) 0 0
\(677\) 40.6993i 1.56420i 0.623151 + 0.782102i \(0.285852\pi\)
−0.623151 + 0.782102i \(0.714148\pi\)
\(678\) 0 0
\(679\) 10.5490 0.404835
\(680\) 0 0
\(681\) 9.13187 0.349934
\(682\) 0 0
\(683\) 20.6882i 0.791614i 0.918334 + 0.395807i \(0.129535\pi\)
−0.918334 + 0.395807i \(0.870465\pi\)
\(684\) 0 0
\(685\) 18.2587i 0.697630i
\(686\) 0 0
\(687\) 12.7374i 0.485962i
\(688\) 0 0
\(689\) 69.1310 2.63368
\(690\) 0 0
\(691\) 22.4370i 0.853545i 0.904359 + 0.426772i \(0.140349\pi\)
−0.904359 + 0.426772i \(0.859651\pi\)
\(692\) 0 0
\(693\) −9.89669 −0.375944
\(694\) 0 0
\(695\) −22.3963 −0.849542
\(696\) 0 0
\(697\) 22.3414 34.8970i 0.846240 1.32182i
\(698\) 0 0
\(699\) −37.0881 −1.40280
\(700\) 0 0
\(701\) −6.48705 −0.245013 −0.122506 0.992468i \(-0.539093\pi\)
−0.122506 + 0.992468i \(0.539093\pi\)
\(702\) 0 0
\(703\) 5.36250i 0.202250i
\(704\) 0 0
\(705\) −24.9849 −0.940984
\(706\) 0 0
\(707\) 19.6779i 0.740062i
\(708\) 0 0
\(709\) 43.5525i 1.63565i 0.575469 + 0.817824i \(0.304820\pi\)
−0.575469 + 0.817824i \(0.695180\pi\)
\(710\) 0 0
\(711\) 5.19780i 0.194933i
\(712\) 0 0
\(713\) −33.3264 −1.24808
\(714\) 0 0
\(715\) −20.9862 −0.784840
\(716\) 0 0
\(717\) 22.8498i 0.853340i
\(718\) 0 0
\(719\) 38.1879i 1.42417i 0.702094 + 0.712084i \(0.252249\pi\)
−0.702094 + 0.712084i \(0.747751\pi\)
\(720\) 0 0
\(721\) 8.74015i 0.325500i
\(722\) 0 0
\(723\) 81.4444 3.02895
\(724\) 0 0
\(725\) 6.44357i 0.239308i
\(726\) 0 0
\(727\) 38.8269 1.44001 0.720006 0.693968i \(-0.244139\pi\)
0.720006 + 0.693968i \(0.244139\pi\)
\(728\) 0 0
\(729\) 43.3883 1.60697
\(730\) 0 0
\(731\) 8.20448 + 5.25259i 0.303454 + 0.194274i
\(732\) 0 0
\(733\) −22.1228 −0.817125 −0.408562 0.912730i \(-0.633970\pi\)
−0.408562 + 0.912730i \(0.633970\pi\)
\(734\) 0 0
\(735\) 29.8308 1.10033
\(736\) 0 0
\(737\) 19.7288i 0.726719i
\(738\) 0 0
\(739\) −42.5717 −1.56603 −0.783013 0.622006i \(-0.786318\pi\)
−0.783013 + 0.622006i \(0.786318\pi\)
\(740\) 0 0
\(741\) 18.9632i 0.696629i
\(742\) 0 0
\(743\) 6.67140i 0.244750i 0.992484 + 0.122375i \(0.0390510\pi\)
−0.992484 + 0.122375i \(0.960949\pi\)
\(744\) 0 0
\(745\) 9.16541i 0.335795i
\(746\) 0 0
\(747\) 33.0376 1.20878
\(748\) 0 0
\(749\) −12.8556 −0.469732
\(750\) 0 0
\(751\) 11.7081i 0.427233i 0.976918 + 0.213617i \(0.0685243\pi\)
−0.976918 + 0.213617i \(0.931476\pi\)
\(752\) 0 0
\(753\) 1.71574i 0.0625250i
\(754\) 0 0
\(755\) 27.5537i 1.00278i
\(756\) 0 0
\(757\) 20.0015 0.726965 0.363483 0.931601i \(-0.381588\pi\)
0.363483 + 0.931601i \(0.381588\pi\)
\(758\) 0 0
\(759\) 19.7796i 0.717955i
\(760\) 0 0
\(761\) −14.6525 −0.531153 −0.265576 0.964090i \(-0.585562\pi\)
−0.265576 + 0.964090i \(0.585562\pi\)
\(762\) 0 0
\(763\) −10.5663 −0.382527
\(764\) 0 0
\(765\) −19.4280 + 30.3463i −0.702422 + 1.09717i
\(766\) 0 0
\(767\) 5.96810 0.215496
\(768\) 0 0
\(769\) −40.0552 −1.44443 −0.722214 0.691670i \(-0.756875\pi\)
−0.722214 + 0.691670i \(0.756875\pi\)
\(770\) 0 0
\(771\) 12.6753i 0.456490i
\(772\) 0 0
\(773\) −39.8881 −1.43467 −0.717337 0.696726i \(-0.754639\pi\)
−0.717337 + 0.696726i \(0.754639\pi\)
\(774\) 0 0
\(775\) 17.4423i 0.626547i
\(776\) 0 0
\(777\) 13.5532i 0.486219i
\(778\) 0 0
\(779\) 11.3412i 0.406342i
\(780\) 0 0
\(781\) −2.79802 −0.100121
\(782\) 0 0
\(783\) 18.8566 0.673879
\(784\) 0 0
\(785\) 10.4731i 0.373802i
\(786\) 0 0
\(787\) 45.1506i 1.60945i −0.593651 0.804723i \(-0.702314\pi\)
0.593651 0.804723i \(-0.297686\pi\)
\(788\) 0 0
\(789\) 20.2666i 0.721510i
\(790\) 0 0
\(791\) 16.8876 0.600453
\(792\) 0 0
\(793\) 43.9197i 1.55964i
\(794\)