Properties

Label 4012.2.b.b.237.8
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 237.8
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.39

$q$-expansion

\(f(q)\) \(=\) \(q-2.34087i q^{3} -2.91532i q^{5} -3.43927i q^{7} -2.47969 q^{9} +O(q^{10})\) \(q-2.34087i q^{3} -2.91532i q^{5} -3.43927i q^{7} -2.47969 q^{9} -1.22244i q^{11} -4.98028 q^{13} -6.82439 q^{15} +(0.473595 + 4.09582i) q^{17} +3.37891 q^{19} -8.05090 q^{21} -0.446807i q^{23} -3.49908 q^{25} -1.21799i q^{27} -2.96447i q^{29} -1.59177i q^{31} -2.86158 q^{33} -10.0266 q^{35} +3.24673i q^{37} +11.6582i q^{39} +1.29097i q^{41} -7.24180 q^{43} +7.22908i q^{45} +3.49672 q^{47} -4.82860 q^{49} +(9.58779 - 1.10863i) q^{51} -11.1260 q^{53} -3.56380 q^{55} -7.90961i q^{57} +1.00000 q^{59} +12.7963i q^{61} +8.52832i q^{63} +14.5191i q^{65} -10.5687 q^{67} -1.04592 q^{69} +0.203780i q^{71} -13.1593i q^{73} +8.19090i q^{75} -4.20431 q^{77} -10.4001i q^{79} -10.2902 q^{81} +7.96025 q^{83} +(11.9406 - 1.38068i) q^{85} -6.93945 q^{87} -8.89649 q^{89} +17.1285i q^{91} -3.72614 q^{93} -9.85061i q^{95} +11.9595i q^{97} +3.03127i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 2.34087i 1.35150i −0.737129 0.675752i \(-0.763819\pi\)
0.737129 0.675752i \(-0.236181\pi\)
\(4\) 0 0
\(5\) 2.91532i 1.30377i −0.758318 0.651885i \(-0.773979\pi\)
0.758318 0.651885i \(-0.226021\pi\)
\(6\) 0 0
\(7\) 3.43927i 1.29992i −0.759967 0.649962i \(-0.774785\pi\)
0.759967 0.649962i \(-0.225215\pi\)
\(8\) 0 0
\(9\) −2.47969 −0.826563
\(10\) 0 0
\(11\) 1.22244i 0.368580i −0.982872 0.184290i \(-0.941001\pi\)
0.982872 0.184290i \(-0.0589985\pi\)
\(12\) 0 0
\(13\) −4.98028 −1.38128 −0.690640 0.723198i \(-0.742671\pi\)
−0.690640 + 0.723198i \(0.742671\pi\)
\(14\) 0 0
\(15\) −6.82439 −1.76205
\(16\) 0 0
\(17\) 0.473595 + 4.09582i 0.114864 + 0.993381i
\(18\) 0 0
\(19\) 3.37891 0.775176 0.387588 0.921833i \(-0.373308\pi\)
0.387588 + 0.921833i \(0.373308\pi\)
\(20\) 0 0
\(21\) −8.05090 −1.75685
\(22\) 0 0
\(23\) 0.446807i 0.0931656i −0.998914 0.0465828i \(-0.985167\pi\)
0.998914 0.0465828i \(-0.0148331\pi\)
\(24\) 0 0
\(25\) −3.49908 −0.699816
\(26\) 0 0
\(27\) 1.21799i 0.234401i
\(28\) 0 0
\(29\) 2.96447i 0.550489i −0.961374 0.275244i \(-0.911241\pi\)
0.961374 0.275244i \(-0.0887587\pi\)
\(30\) 0 0
\(31\) 1.59177i 0.285891i −0.989731 0.142945i \(-0.954343\pi\)
0.989731 0.142945i \(-0.0456573\pi\)
\(32\) 0 0
\(33\) −2.86158 −0.498137
\(34\) 0 0
\(35\) −10.0266 −1.69480
\(36\) 0 0
\(37\) 3.24673i 0.533759i 0.963730 + 0.266880i \(0.0859926\pi\)
−0.963730 + 0.266880i \(0.914007\pi\)
\(38\) 0 0
\(39\) 11.6582i 1.86681i
\(40\) 0 0
\(41\) 1.29097i 0.201615i 0.994906 + 0.100807i \(0.0321426\pi\)
−0.994906 + 0.100807i \(0.967857\pi\)
\(42\) 0 0
\(43\) −7.24180 −1.10436 −0.552182 0.833724i \(-0.686205\pi\)
−0.552182 + 0.833724i \(0.686205\pi\)
\(44\) 0 0
\(45\) 7.22908i 1.07765i
\(46\) 0 0
\(47\) 3.49672 0.510050 0.255025 0.966935i \(-0.417916\pi\)
0.255025 + 0.966935i \(0.417916\pi\)
\(48\) 0 0
\(49\) −4.82860 −0.689800
\(50\) 0 0
\(51\) 9.58779 1.10863i 1.34256 0.155239i
\(52\) 0 0
\(53\) −11.1260 −1.52827 −0.764136 0.645055i \(-0.776834\pi\)
−0.764136 + 0.645055i \(0.776834\pi\)
\(54\) 0 0
\(55\) −3.56380 −0.480543
\(56\) 0 0
\(57\) 7.90961i 1.04765i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 12.7963i 1.63840i 0.573507 + 0.819201i \(0.305583\pi\)
−0.573507 + 0.819201i \(0.694417\pi\)
\(62\) 0 0
\(63\) 8.52832i 1.07447i
\(64\) 0 0
\(65\) 14.5191i 1.80087i
\(66\) 0 0
\(67\) −10.5687 −1.29117 −0.645586 0.763688i \(-0.723387\pi\)
−0.645586 + 0.763688i \(0.723387\pi\)
\(68\) 0 0
\(69\) −1.04592 −0.125914
\(70\) 0 0
\(71\) 0.203780i 0.0241842i 0.999927 + 0.0120921i \(0.00384914\pi\)
−0.999927 + 0.0120921i \(0.996151\pi\)
\(72\) 0 0
\(73\) 13.1593i 1.54018i −0.637937 0.770088i \(-0.720212\pi\)
0.637937 0.770088i \(-0.279788\pi\)
\(74\) 0 0
\(75\) 8.19090i 0.945804i
\(76\) 0 0
\(77\) −4.20431 −0.479125
\(78\) 0 0
\(79\) 10.4001i 1.17010i −0.810996 0.585051i \(-0.801074\pi\)
0.810996 0.585051i \(-0.198926\pi\)
\(80\) 0 0
\(81\) −10.2902 −1.14336
\(82\) 0 0
\(83\) 7.96025 0.873751 0.436875 0.899522i \(-0.356085\pi\)
0.436875 + 0.899522i \(0.356085\pi\)
\(84\) 0 0
\(85\) 11.9406 1.38068i 1.29514 0.149756i
\(86\) 0 0
\(87\) −6.93945 −0.743987
\(88\) 0 0
\(89\) −8.89649 −0.943026 −0.471513 0.881859i \(-0.656292\pi\)
−0.471513 + 0.881859i \(0.656292\pi\)
\(90\) 0 0
\(91\) 17.1285i 1.79556i
\(92\) 0 0
\(93\) −3.72614 −0.386382
\(94\) 0 0
\(95\) 9.85061i 1.01065i
\(96\) 0 0
\(97\) 11.9595i 1.21430i 0.794586 + 0.607152i \(0.207688\pi\)
−0.794586 + 0.607152i \(0.792312\pi\)
\(98\) 0 0
\(99\) 3.03127i 0.304654i
\(100\) 0 0
\(101\) −4.07128 −0.405108 −0.202554 0.979271i \(-0.564924\pi\)
−0.202554 + 0.979271i \(0.564924\pi\)
\(102\) 0 0
\(103\) 18.0364 1.77718 0.888592 0.458699i \(-0.151684\pi\)
0.888592 + 0.458699i \(0.151684\pi\)
\(104\) 0 0
\(105\) 23.4709i 2.29053i
\(106\) 0 0
\(107\) 14.2888i 1.38135i 0.723164 + 0.690677i \(0.242687\pi\)
−0.723164 + 0.690677i \(0.757313\pi\)
\(108\) 0 0
\(109\) 15.5752i 1.49183i −0.666040 0.745916i \(-0.732012\pi\)
0.666040 0.745916i \(-0.267988\pi\)
\(110\) 0 0
\(111\) 7.60018 0.721378
\(112\) 0 0
\(113\) 0.667011i 0.0627471i 0.999508 + 0.0313736i \(0.00998816\pi\)
−0.999508 + 0.0313736i \(0.990012\pi\)
\(114\) 0 0
\(115\) −1.30258 −0.121467
\(116\) 0 0
\(117\) 12.3495 1.14171
\(118\) 0 0
\(119\) 14.0866 1.62882i 1.29132 0.149314i
\(120\) 0 0
\(121\) 9.50564 0.864149
\(122\) 0 0
\(123\) 3.02199 0.272483
\(124\) 0 0
\(125\) 4.37566i 0.391371i
\(126\) 0 0
\(127\) −6.95451 −0.617113 −0.308556 0.951206i \(-0.599846\pi\)
−0.308556 + 0.951206i \(0.599846\pi\)
\(128\) 0 0
\(129\) 16.9521i 1.49255i
\(130\) 0 0
\(131\) 7.68462i 0.671408i −0.941967 0.335704i \(-0.891026\pi\)
0.941967 0.335704i \(-0.108974\pi\)
\(132\) 0 0
\(133\) 11.6210i 1.00767i
\(134\) 0 0
\(135\) −3.55081 −0.305605
\(136\) 0 0
\(137\) 19.9196 1.70185 0.850924 0.525290i \(-0.176043\pi\)
0.850924 + 0.525290i \(0.176043\pi\)
\(138\) 0 0
\(139\) 8.29999i 0.703996i −0.936001 0.351998i \(-0.885502\pi\)
0.936001 0.351998i \(-0.114498\pi\)
\(140\) 0 0
\(141\) 8.18539i 0.689334i
\(142\) 0 0
\(143\) 6.08810i 0.509112i
\(144\) 0 0
\(145\) −8.64238 −0.717710
\(146\) 0 0
\(147\) 11.3031i 0.932268i
\(148\) 0 0
\(149\) 16.5265 1.35391 0.676953 0.736026i \(-0.263300\pi\)
0.676953 + 0.736026i \(0.263300\pi\)
\(150\) 0 0
\(151\) 21.0461 1.71271 0.856353 0.516391i \(-0.172725\pi\)
0.856353 + 0.516391i \(0.172725\pi\)
\(152\) 0 0
\(153\) −1.17437 10.1563i −0.0949420 0.821092i
\(154\) 0 0
\(155\) −4.64052 −0.372736
\(156\) 0 0
\(157\) −9.33570 −0.745070 −0.372535 0.928018i \(-0.621511\pi\)
−0.372535 + 0.928018i \(0.621511\pi\)
\(158\) 0 0
\(159\) 26.0445i 2.06547i
\(160\) 0 0
\(161\) −1.53669 −0.121108
\(162\) 0 0
\(163\) 8.27257i 0.647957i 0.946064 + 0.323979i \(0.105021\pi\)
−0.946064 + 0.323979i \(0.894979\pi\)
\(164\) 0 0
\(165\) 8.34241i 0.649456i
\(166\) 0 0
\(167\) 20.2252i 1.56508i −0.622603 0.782538i \(-0.713925\pi\)
0.622603 0.782538i \(-0.286075\pi\)
\(168\) 0 0
\(169\) 11.8032 0.907936
\(170\) 0 0
\(171\) −8.37865 −0.640731
\(172\) 0 0
\(173\) 20.2456i 1.53925i −0.638498 0.769623i \(-0.720444\pi\)
0.638498 0.769623i \(-0.279556\pi\)
\(174\) 0 0
\(175\) 12.0343i 0.909707i
\(176\) 0 0
\(177\) 2.34087i 0.175951i
\(178\) 0 0
\(179\) 0.893467 0.0667808 0.0333904 0.999442i \(-0.489370\pi\)
0.0333904 + 0.999442i \(0.489370\pi\)
\(180\) 0 0
\(181\) 9.58982i 0.712806i 0.934332 + 0.356403i \(0.115997\pi\)
−0.934332 + 0.356403i \(0.884003\pi\)
\(182\) 0 0
\(183\) 29.9546 2.21431
\(184\) 0 0
\(185\) 9.46525 0.695899
\(186\) 0 0
\(187\) 5.00689 0.578942i 0.366140 0.0423364i
\(188\) 0 0
\(189\) −4.18898 −0.304704
\(190\) 0 0
\(191\) −14.4637 −1.04655 −0.523277 0.852163i \(-0.675291\pi\)
−0.523277 + 0.852163i \(0.675291\pi\)
\(192\) 0 0
\(193\) 26.2572i 1.89003i −0.327024 0.945016i \(-0.606046\pi\)
0.327024 0.945016i \(-0.393954\pi\)
\(194\) 0 0
\(195\) 33.9874 2.43389
\(196\) 0 0
\(197\) 8.98094i 0.639865i 0.947440 + 0.319933i \(0.103660\pi\)
−0.947440 + 0.319933i \(0.896340\pi\)
\(198\) 0 0
\(199\) 2.21702i 0.157160i −0.996908 0.0785801i \(-0.974961\pi\)
0.996908 0.0785801i \(-0.0250386\pi\)
\(200\) 0 0
\(201\) 24.7400i 1.74502i
\(202\) 0 0
\(203\) −10.1956 −0.715593
\(204\) 0 0
\(205\) 3.76358 0.262859
\(206\) 0 0
\(207\) 1.10794i 0.0770072i
\(208\) 0 0
\(209\) 4.13052i 0.285714i
\(210\) 0 0
\(211\) 7.70593i 0.530498i 0.964180 + 0.265249i \(0.0854542\pi\)
−0.964180 + 0.265249i \(0.914546\pi\)
\(212\) 0 0
\(213\) 0.477023 0.0326851
\(214\) 0 0
\(215\) 21.1121i 1.43984i
\(216\) 0 0
\(217\) −5.47454 −0.371636
\(218\) 0 0
\(219\) −30.8042 −2.08155
\(220\) 0 0
\(221\) −2.35864 20.3983i −0.158659 1.37214i
\(222\) 0 0
\(223\) −16.7478 −1.12152 −0.560758 0.827980i \(-0.689490\pi\)
−0.560758 + 0.827980i \(0.689490\pi\)
\(224\) 0 0
\(225\) 8.67662 0.578442
\(226\) 0 0
\(227\) 12.1224i 0.804591i 0.915510 + 0.402296i \(0.131788\pi\)
−0.915510 + 0.402296i \(0.868212\pi\)
\(228\) 0 0
\(229\) −15.6583 −1.03473 −0.517364 0.855766i \(-0.673087\pi\)
−0.517364 + 0.855766i \(0.673087\pi\)
\(230\) 0 0
\(231\) 9.84175i 0.647540i
\(232\) 0 0
\(233\) 2.53784i 0.166259i 0.996539 + 0.0831297i \(0.0264916\pi\)
−0.996539 + 0.0831297i \(0.973508\pi\)
\(234\) 0 0
\(235\) 10.1941i 0.664987i
\(236\) 0 0
\(237\) −24.3453 −1.58140
\(238\) 0 0
\(239\) 16.2543 1.05140 0.525702 0.850669i \(-0.323803\pi\)
0.525702 + 0.850669i \(0.323803\pi\)
\(240\) 0 0
\(241\) 3.83534i 0.247056i 0.992341 + 0.123528i \(0.0394208\pi\)
−0.992341 + 0.123528i \(0.960579\pi\)
\(242\) 0 0
\(243\) 20.4341i 1.31085i
\(244\) 0 0
\(245\) 14.0769i 0.899341i
\(246\) 0 0
\(247\) −16.8279 −1.07074
\(248\) 0 0
\(249\) 18.6339i 1.18088i
\(250\) 0 0
\(251\) −17.2399 −1.08817 −0.544087 0.839029i \(-0.683124\pi\)
−0.544087 + 0.839029i \(0.683124\pi\)
\(252\) 0 0
\(253\) −0.546195 −0.0343390
\(254\) 0 0
\(255\) −3.23200 27.9514i −0.202396 1.75039i
\(256\) 0 0
\(257\) 18.8519 1.17595 0.587974 0.808880i \(-0.299926\pi\)
0.587974 + 0.808880i \(0.299926\pi\)
\(258\) 0 0
\(259\) 11.1664 0.693846
\(260\) 0 0
\(261\) 7.35096i 0.455013i
\(262\) 0 0
\(263\) −19.0906 −1.17717 −0.588587 0.808434i \(-0.700316\pi\)
−0.588587 + 0.808434i \(0.700316\pi\)
\(264\) 0 0
\(265\) 32.4358i 1.99252i
\(266\) 0 0
\(267\) 20.8255i 1.27450i
\(268\) 0 0
\(269\) 1.57492i 0.0960246i 0.998847 + 0.0480123i \(0.0152887\pi\)
−0.998847 + 0.0480123i \(0.984711\pi\)
\(270\) 0 0
\(271\) 27.1833 1.65127 0.825634 0.564206i \(-0.190818\pi\)
0.825634 + 0.564206i \(0.190818\pi\)
\(272\) 0 0
\(273\) 40.0957 2.42670
\(274\) 0 0
\(275\) 4.27742i 0.257938i
\(276\) 0 0
\(277\) 2.94937i 0.177211i −0.996067 0.0886053i \(-0.971759\pi\)
0.996067 0.0886053i \(-0.0282410\pi\)
\(278\) 0 0
\(279\) 3.94710i 0.236307i
\(280\) 0 0
\(281\) 17.6640 1.05375 0.526873 0.849944i \(-0.323364\pi\)
0.526873 + 0.849944i \(0.323364\pi\)
\(282\) 0 0
\(283\) 18.3900i 1.09317i 0.837403 + 0.546587i \(0.184073\pi\)
−0.837403 + 0.546587i \(0.815927\pi\)
\(284\) 0 0
\(285\) −23.0590 −1.36590
\(286\) 0 0
\(287\) 4.43998 0.262084
\(288\) 0 0
\(289\) −16.5514 + 3.87952i −0.973613 + 0.228207i
\(290\) 0 0
\(291\) 27.9957 1.64114
\(292\) 0 0
\(293\) −23.6641 −1.38247 −0.691235 0.722630i \(-0.742933\pi\)
−0.691235 + 0.722630i \(0.742933\pi\)
\(294\) 0 0
\(295\) 2.91532i 0.169736i
\(296\) 0 0
\(297\) −1.48892 −0.0863956
\(298\) 0 0
\(299\) 2.22522i 0.128688i
\(300\) 0 0
\(301\) 24.9065i 1.43559i
\(302\) 0 0
\(303\) 9.53035i 0.547505i
\(304\) 0 0
\(305\) 37.3054 2.13610
\(306\) 0 0
\(307\) 11.3394 0.647171 0.323586 0.946199i \(-0.395112\pi\)
0.323586 + 0.946199i \(0.395112\pi\)
\(308\) 0 0
\(309\) 42.2210i 2.40187i
\(310\) 0 0
\(311\) 17.0112i 0.964616i −0.876002 0.482308i \(-0.839799\pi\)
0.876002 0.482308i \(-0.160201\pi\)
\(312\) 0 0
\(313\) 2.37527i 0.134258i −0.997744 0.0671291i \(-0.978616\pi\)
0.997744 0.0671291i \(-0.0213839\pi\)
\(314\) 0 0
\(315\) 24.8628 1.40086
\(316\) 0 0
\(317\) 11.7504i 0.659969i −0.943986 0.329985i \(-0.892956\pi\)
0.943986 0.329985i \(-0.107044\pi\)
\(318\) 0 0
\(319\) −3.62389 −0.202899
\(320\) 0 0
\(321\) 33.4483 1.86690
\(322\) 0 0
\(323\) 1.60024 + 13.8394i 0.0890395 + 0.770045i
\(324\) 0 0
\(325\) 17.4264 0.966642
\(326\) 0 0
\(327\) −36.4595 −2.01622
\(328\) 0 0
\(329\) 12.0262i 0.663025i
\(330\) 0 0
\(331\) −4.77937 −0.262698 −0.131349 0.991336i \(-0.541931\pi\)
−0.131349 + 0.991336i \(0.541931\pi\)
\(332\) 0 0
\(333\) 8.05088i 0.441185i
\(334\) 0 0
\(335\) 30.8111i 1.68339i
\(336\) 0 0
\(337\) 14.3814i 0.783406i 0.920092 + 0.391703i \(0.128114\pi\)
−0.920092 + 0.391703i \(0.871886\pi\)
\(338\) 0 0
\(339\) 1.56139 0.0848030
\(340\) 0 0
\(341\) −1.94585 −0.105374
\(342\) 0 0
\(343\) 7.46803i 0.403236i
\(344\) 0 0
\(345\) 3.04918i 0.164163i
\(346\) 0 0
\(347\) 16.1054i 0.864585i 0.901733 + 0.432293i \(0.142295\pi\)
−0.901733 + 0.432293i \(0.857705\pi\)
\(348\) 0 0
\(349\) 10.3672 0.554945 0.277473 0.960734i \(-0.410503\pi\)
0.277473 + 0.960734i \(0.410503\pi\)
\(350\) 0 0
\(351\) 6.06591i 0.323774i
\(352\) 0 0
\(353\) −17.2322 −0.917177 −0.458588 0.888649i \(-0.651645\pi\)
−0.458588 + 0.888649i \(0.651645\pi\)
\(354\) 0 0
\(355\) 0.594084 0.0315307
\(356\) 0 0
\(357\) −3.81287 32.9750i −0.201798 1.74522i
\(358\) 0 0
\(359\) −19.6027 −1.03459 −0.517297 0.855806i \(-0.673062\pi\)
−0.517297 + 0.855806i \(0.673062\pi\)
\(360\) 0 0
\(361\) −7.58295 −0.399102
\(362\) 0 0
\(363\) 22.2515i 1.16790i
\(364\) 0 0
\(365\) −38.3635 −2.00804
\(366\) 0 0
\(367\) 5.25915i 0.274525i 0.990535 + 0.137263i \(0.0438304\pi\)
−0.990535 + 0.137263i \(0.956170\pi\)
\(368\) 0 0
\(369\) 3.20119i 0.166647i
\(370\) 0 0
\(371\) 38.2653i 1.98664i
\(372\) 0 0
\(373\) −3.75747 −0.194554 −0.0972772 0.995257i \(-0.531013\pi\)
−0.0972772 + 0.995257i \(0.531013\pi\)
\(374\) 0 0
\(375\) −10.2429 −0.528939
\(376\) 0 0
\(377\) 14.7639i 0.760379i
\(378\) 0 0
\(379\) 26.7616i 1.37465i −0.726349 0.687326i \(-0.758784\pi\)
0.726349 0.687326i \(-0.241216\pi\)
\(380\) 0 0
\(381\) 16.2796i 0.834030i
\(382\) 0 0
\(383\) 13.4615 0.687853 0.343926 0.938997i \(-0.388243\pi\)
0.343926 + 0.938997i \(0.388243\pi\)
\(384\) 0 0
\(385\) 12.2569i 0.624669i
\(386\) 0 0
\(387\) 17.9574 0.912826
\(388\) 0 0
\(389\) −13.3494 −0.676839 −0.338420 0.940995i \(-0.609892\pi\)
−0.338420 + 0.940995i \(0.609892\pi\)
\(390\) 0 0
\(391\) 1.83004 0.211605i 0.0925490 0.0107013i
\(392\) 0 0
\(393\) −17.9887 −0.907411
\(394\) 0 0
\(395\) −30.3196 −1.52554
\(396\) 0 0
\(397\) 0.760906i 0.0381888i −0.999818 0.0190944i \(-0.993922\pi\)
0.999818 0.0190944i \(-0.00607831\pi\)
\(398\) 0 0
\(399\) −27.2033 −1.36187
\(400\) 0 0
\(401\) 28.2339i 1.40993i −0.709241 0.704966i \(-0.750962\pi\)
0.709241 0.704966i \(-0.249038\pi\)
\(402\) 0 0
\(403\) 7.92747i 0.394895i
\(404\) 0 0
\(405\) 29.9992i 1.49067i
\(406\) 0 0
\(407\) 3.96894 0.196733
\(408\) 0 0
\(409\) 0.346970 0.0171565 0.00857827 0.999963i \(-0.497269\pi\)
0.00857827 + 0.999963i \(0.497269\pi\)
\(410\) 0 0
\(411\) 46.6293i 2.30005i
\(412\) 0 0
\(413\) 3.43927i 0.169236i
\(414\) 0 0
\(415\) 23.2067i 1.13917i
\(416\) 0 0
\(417\) −19.4292 −0.951453
\(418\) 0 0
\(419\) 35.4146i 1.73012i 0.501671 + 0.865059i \(0.332719\pi\)
−0.501671 + 0.865059i \(0.667281\pi\)
\(420\) 0 0
\(421\) 2.57944 0.125714 0.0628572 0.998023i \(-0.479979\pi\)
0.0628572 + 0.998023i \(0.479979\pi\)
\(422\) 0 0
\(423\) −8.67078 −0.421588
\(424\) 0 0
\(425\) −1.65715 14.3316i −0.0803834 0.695184i
\(426\) 0 0
\(427\) 44.0101 2.12980
\(428\) 0 0
\(429\) 14.2515 0.688067
\(430\) 0 0
\(431\) 23.1050i 1.11293i −0.830872 0.556464i \(-0.812158\pi\)
0.830872 0.556464i \(-0.187842\pi\)
\(432\) 0 0
\(433\) −33.8999 −1.62913 −0.814563 0.580075i \(-0.803023\pi\)
−0.814563 + 0.580075i \(0.803023\pi\)
\(434\) 0 0
\(435\) 20.2307i 0.969988i
\(436\) 0 0
\(437\) 1.50972i 0.0722198i
\(438\) 0 0
\(439\) 3.45855i 0.165067i 0.996588 + 0.0825337i \(0.0263012\pi\)
−0.996588 + 0.0825337i \(0.973699\pi\)
\(440\) 0 0
\(441\) 11.9734 0.570163
\(442\) 0 0
\(443\) 14.5483 0.691210 0.345605 0.938380i \(-0.387674\pi\)
0.345605 + 0.938380i \(0.387674\pi\)
\(444\) 0 0
\(445\) 25.9361i 1.22949i
\(446\) 0 0
\(447\) 38.6865i 1.82981i
\(448\) 0 0
\(449\) 11.4645i 0.541045i −0.962714 0.270523i \(-0.912803\pi\)
0.962714 0.270523i \(-0.0871965\pi\)
\(450\) 0 0
\(451\) 1.57813 0.0743112
\(452\) 0 0
\(453\) 49.2662i 2.31473i
\(454\) 0 0
\(455\) 49.9351 2.34100
\(456\) 0 0
\(457\) −26.4125 −1.23553 −0.617763 0.786364i \(-0.711961\pi\)
−0.617763 + 0.786364i \(0.711961\pi\)
\(458\) 0 0
\(459\) 4.98864 0.576832i 0.232850 0.0269242i
\(460\) 0 0
\(461\) −9.35500 −0.435706 −0.217853 0.975982i \(-0.569905\pi\)
−0.217853 + 0.975982i \(0.569905\pi\)
\(462\) 0 0
\(463\) 10.5912 0.492216 0.246108 0.969242i \(-0.420848\pi\)
0.246108 + 0.969242i \(0.420848\pi\)
\(464\) 0 0
\(465\) 10.8629i 0.503754i
\(466\) 0 0
\(467\) 2.63760 0.122053 0.0610267 0.998136i \(-0.480563\pi\)
0.0610267 + 0.998136i \(0.480563\pi\)
\(468\) 0 0
\(469\) 36.3486i 1.67842i
\(470\) 0 0
\(471\) 21.8537i 1.00696i
\(472\) 0 0
\(473\) 8.85267i 0.407046i
\(474\) 0 0
\(475\) −11.8231 −0.542480
\(476\) 0 0
\(477\) 27.5890 1.26321
\(478\) 0 0
\(479\) 19.9390i 0.911034i −0.890227 0.455517i \(-0.849454\pi\)
0.890227 0.455517i \(-0.150546\pi\)
\(480\) 0 0
\(481\) 16.1696i 0.737271i
\(482\) 0 0
\(483\) 3.59720i 0.163678i
\(484\) 0 0
\(485\) 34.8657 1.58317
\(486\) 0 0
\(487\) 2.36250i 0.107055i 0.998566 + 0.0535276i \(0.0170465\pi\)
−0.998566 + 0.0535276i \(0.982954\pi\)
\(488\) 0 0
\(489\) 19.3650 0.875717
\(490\) 0 0
\(491\) −30.9178 −1.39530 −0.697649 0.716439i \(-0.745771\pi\)
−0.697649 + 0.716439i \(0.745771\pi\)
\(492\) 0 0
\(493\) 12.1419 1.40396i 0.546845 0.0632311i
\(494\) 0 0
\(495\) 8.83712 0.397199
\(496\) 0 0
\(497\) 0.700855 0.0314377
\(498\) 0 0
\(499\) 33.2392i 1.48799i 0.668184 + 0.743996i \(0.267072\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(500\) 0 0
\(501\) −47.3447 −2.11521
\(502\) 0 0
\(503\) 17.4723i 0.779053i 0.921015 + 0.389527i \(0.127361\pi\)
−0.921015 + 0.389527i \(0.872639\pi\)
\(504\) 0 0
\(505\) 11.8691i 0.528167i
\(506\) 0 0
\(507\) 27.6297i 1.22708i
\(508\) 0 0
\(509\) −5.67789 −0.251668 −0.125834 0.992051i \(-0.540161\pi\)
−0.125834 + 0.992051i \(0.540161\pi\)
\(510\) 0 0
\(511\) −45.2583 −2.00211
\(512\) 0 0
\(513\) 4.11547i 0.181702i
\(514\) 0 0
\(515\) 52.5820i 2.31704i
\(516\) 0 0
\(517\) 4.27454i 0.187994i
\(518\) 0 0
\(519\) −47.3925 −2.08030
\(520\) 0 0
\(521\) 42.4200i 1.85845i −0.369508 0.929227i \(-0.620474\pi\)
0.369508 0.929227i \(-0.379526\pi\)
\(522\) 0 0
\(523\) 0.0969511 0.00423938 0.00211969 0.999998i \(-0.499325\pi\)
0.00211969 + 0.999998i \(0.499325\pi\)
\(524\) 0 0
\(525\) 28.1708 1.22947
\(526\) 0 0
\(527\) 6.51961 0.753856i 0.283999 0.0328385i
\(528\) 0 0
\(529\) 22.8004 0.991320
\(530\) 0 0
\(531\) −2.47969 −0.107609
\(532\) 0 0
\(533\) 6.42937i 0.278487i
\(534\) 0 0
\(535\) 41.6565 1.80097
\(536\) 0 0
\(537\) 2.09149i 0.0902546i
\(538\) 0 0
\(539\) 5.90268i 0.254246i
\(540\) 0 0
\(541\) 14.2023i 0.610603i −0.952256 0.305302i \(-0.901243\pi\)
0.952256 0.305302i \(-0.0987573\pi\)
\(542\) 0 0
\(543\) 22.4486 0.963360
\(544\) 0 0
\(545\) −45.4066 −1.94500
\(546\) 0 0
\(547\) 17.5292i 0.749494i 0.927127 + 0.374747i \(0.122271\pi\)
−0.927127 + 0.374747i \(0.877729\pi\)
\(548\) 0 0
\(549\) 31.7309i 1.35424i
\(550\) 0 0
\(551\) 10.0167i 0.426725i
\(552\) 0 0
\(553\) −35.7688 −1.52104
\(554\) 0 0
\(555\) 22.1570i 0.940510i
\(556\) 0 0
\(557\) 2.87630 0.121873 0.0609363 0.998142i \(-0.480591\pi\)
0.0609363 + 0.998142i \(0.480591\pi\)
\(558\) 0 0
\(559\) 36.0662 1.52544
\(560\) 0 0
\(561\) −1.35523 11.7205i −0.0572179 0.494840i
\(562\) 0 0
\(563\) −41.7131 −1.75800 −0.878998 0.476825i \(-0.841787\pi\)
−0.878998 + 0.476825i \(0.841787\pi\)
\(564\) 0 0
\(565\) 1.94455 0.0818078
\(566\) 0 0
\(567\) 35.3909i 1.48628i
\(568\) 0 0
\(569\) 21.2573 0.891150 0.445575 0.895245i \(-0.352999\pi\)
0.445575 + 0.895245i \(0.352999\pi\)
\(570\) 0 0
\(571\) 34.3672i 1.43822i 0.694894 + 0.719112i \(0.255451\pi\)
−0.694894 + 0.719112i \(0.744549\pi\)
\(572\) 0 0
\(573\) 33.8576i 1.41442i
\(574\) 0 0
\(575\) 1.56341i 0.0651988i
\(576\) 0 0
\(577\) 16.4478 0.684730 0.342365 0.939567i \(-0.388772\pi\)
0.342365 + 0.939567i \(0.388772\pi\)
\(578\) 0 0
\(579\) −61.4647 −2.55439
\(580\) 0 0
\(581\) 27.3775i 1.13581i
\(582\) 0 0
\(583\) 13.6009i 0.563290i
\(584\) 0 0
\(585\) 36.0028i 1.48853i
\(586\) 0 0
\(587\) −19.3375 −0.798143 −0.399071 0.916920i \(-0.630667\pi\)
−0.399071 + 0.916920i \(0.630667\pi\)
\(588\) 0 0
\(589\) 5.37846i 0.221616i
\(590\) 0 0
\(591\) 21.0232 0.864781
\(592\) 0 0
\(593\) −29.0165 −1.19156 −0.595782 0.803146i \(-0.703158\pi\)
−0.595782 + 0.803146i \(0.703158\pi\)
\(594\) 0 0
\(595\) −4.74854 41.0670i −0.194671 1.68358i
\(596\) 0 0
\(597\) −5.18975 −0.212403
\(598\) 0 0
\(599\) 11.2839 0.461046 0.230523 0.973067i \(-0.425956\pi\)
0.230523 + 0.973067i \(0.425956\pi\)
\(600\) 0 0
\(601\) 11.9053i 0.485628i 0.970073 + 0.242814i \(0.0780705\pi\)
−0.970073 + 0.242814i \(0.921930\pi\)
\(602\) 0 0
\(603\) 26.2071 1.06723
\(604\) 0 0
\(605\) 27.7120i 1.12665i
\(606\) 0 0
\(607\) 21.2225i 0.861393i 0.902497 + 0.430696i \(0.141732\pi\)
−0.902497 + 0.430696i \(0.858268\pi\)
\(608\) 0 0
\(609\) 23.8667i 0.967126i
\(610\) 0 0
\(611\) −17.4147 −0.704521
\(612\) 0 0
\(613\) 6.73020 0.271830 0.135915 0.990720i \(-0.456603\pi\)
0.135915 + 0.990720i \(0.456603\pi\)
\(614\) 0 0
\(615\) 8.81005i 0.355256i
\(616\) 0 0
\(617\) 36.3550i 1.46360i −0.681521 0.731798i \(-0.738681\pi\)
0.681521 0.731798i \(-0.261319\pi\)
\(618\) 0 0
\(619\) 24.5215i 0.985604i 0.870141 + 0.492802i \(0.164027\pi\)
−0.870141 + 0.492802i \(0.835973\pi\)
\(620\) 0 0
\(621\) −0.544204 −0.0218382
\(622\) 0 0
\(623\) 30.5975i 1.22586i
\(624\) 0 0
\(625\) −30.2518 −1.21007
\(626\) 0 0
\(627\) −9.66903 −0.386144
\(628\) 0 0
\(629\) −13.2980 + 1.53764i −0.530226 + 0.0613095i
\(630\) 0 0
\(631\) 46.8028 1.86319 0.931594 0.363500i \(-0.118418\pi\)
0.931594 + 0.363500i \(0.118418\pi\)
\(632\) 0 0
\(633\) 18.0386 0.716970
\(634\) 0 0
\(635\) 20.2746i 0.804573i
\(636\) 0 0
\(637\) 24.0478 0.952808
\(638\) 0 0
\(639\) 0.505311i 0.0199898i
\(640\) 0 0
\(641\) 24.5811i 0.970894i −0.874266 0.485447i \(-0.838657\pi\)
0.874266 0.485447i \(-0.161343\pi\)
\(642\) 0 0
\(643\) 14.0089i 0.552458i 0.961092 + 0.276229i \(0.0890848\pi\)
−0.961092 + 0.276229i \(0.910915\pi\)
\(644\) 0 0
\(645\) 49.4208 1.94594
\(646\) 0 0
\(647\) −0.524439 −0.0206178 −0.0103089 0.999947i \(-0.503281\pi\)
−0.0103089 + 0.999947i \(0.503281\pi\)
\(648\) 0 0
\(649\) 1.22244i 0.0479850i
\(650\) 0 0
\(651\) 12.8152i 0.502267i
\(652\) 0 0
\(653\) 42.7492i 1.67290i −0.548040 0.836452i \(-0.684626\pi\)
0.548040 0.836452i \(-0.315374\pi\)
\(654\) 0 0
\(655\) −22.4031 −0.875362
\(656\) 0 0
\(657\) 32.6309i 1.27305i
\(658\) 0 0
\(659\) 17.5822 0.684906 0.342453 0.939535i \(-0.388742\pi\)
0.342453 + 0.939535i \(0.388742\pi\)
\(660\) 0 0
\(661\) −29.1880 −1.13528 −0.567642 0.823276i \(-0.692144\pi\)
−0.567642 + 0.823276i \(0.692144\pi\)
\(662\) 0 0
\(663\) −47.7498 + 5.52127i −1.85445 + 0.214428i
\(664\) 0 0
\(665\) −33.8789 −1.31377
\(666\) 0 0
\(667\) −1.32455 −0.0512866
\(668\) 0 0
\(669\) 39.2045i 1.51573i
\(670\) 0 0
\(671\) 15.6427 0.603882
\(672\) 0 0
\(673\) 40.5799i 1.56424i 0.623128 + 0.782120i \(0.285861\pi\)
−0.623128 + 0.782120i \(0.714139\pi\)
\(674\) 0 0
\(675\) 4.26183i 0.164038i
\(676\) 0 0
\(677\) 46.8767i 1.80162i −0.434217 0.900808i \(-0.642975\pi\)
0.434217 0.900808i \(-0.357025\pi\)
\(678\) 0 0
\(679\) 41.1320 1.57850
\(680\) 0 0
\(681\) 28.3770 1.08741
\(682\) 0 0
\(683\) 40.3634i 1.54446i −0.635341 0.772231i \(-0.719141\pi\)
0.635341 0.772231i \(-0.280859\pi\)
\(684\) 0 0
\(685\) 58.0720i 2.21882i
\(686\) 0 0
\(687\) 36.6540i 1.39844i
\(688\) 0 0
\(689\) 55.4105 2.11097
\(690\) 0 0
\(691\) 41.0163i 1.56033i 0.625572 + 0.780166i \(0.284866\pi\)
−0.625572 + 0.780166i \(0.715134\pi\)
\(692\) 0 0
\(693\) 10.4254 0.396027
\(694\) 0 0
\(695\) −24.1971 −0.917848
\(696\) 0 0
\(697\) −5.28756 + 0.611395i −0.200281 + 0.0231582i
\(698\) 0 0
\(699\) 5.94076 0.224700
\(700\) 0 0
\(701\) −12.4313 −0.469523 −0.234761 0.972053i \(-0.575431\pi\)
−0.234761 + 0.972053i \(0.575431\pi\)
\(702\) 0 0
\(703\) 10.9704i 0.413757i
\(704\) 0 0
\(705\) −23.8630 −0.898733
\(706\) 0 0
\(707\) 14.0023i 0.526609i
\(708\) 0 0
\(709\) 3.82521i 0.143659i −0.997417 0.0718293i \(-0.977116\pi\)
0.997417 0.0718293i \(-0.0228837\pi\)
\(710\) 0 0
\(711\) 25.7890i 0.967163i
\(712\) 0 0
\(713\) −0.711215 −0.0266352
\(714\) 0 0
\(715\) 17.7487 0.663765
\(716\) 0 0
\(717\) 38.0493i 1.42098i
\(718\) 0 0
\(719\) 14.1491i 0.527671i 0.964568 + 0.263835i \(0.0849876\pi\)
−0.964568 + 0.263835i \(0.915012\pi\)
\(720\) 0 0
\(721\) 62.0323i 2.31020i
\(722\) 0 0
\(723\) 8.97804 0.333897
\(724\) 0 0
\(725\) 10.3729i 0.385241i
\(726\) 0 0
\(727\) −36.9837 −1.37165 −0.685825 0.727766i \(-0.740559\pi\)
−0.685825 + 0.727766i \(0.740559\pi\)
\(728\) 0 0
\(729\) 16.9631 0.628262
\(730\) 0 0
\(731\) −3.42968 29.6611i −0.126851 1.09705i
\(732\) 0 0
\(733\) −8.78570 −0.324507 −0.162254 0.986749i \(-0.551876\pi\)
−0.162254 + 0.986749i \(0.551876\pi\)
\(734\) 0 0
\(735\) 32.9523 1.21546
\(736\) 0 0
\(737\) 12.9196i 0.475900i
\(738\) 0 0
\(739\) −32.3626 −1.19048 −0.595239 0.803549i \(-0.702943\pi\)
−0.595239 + 0.803549i \(0.702943\pi\)
\(740\) 0 0
\(741\) 39.3920i 1.44710i
\(742\) 0 0
\(743\) 8.55664i 0.313913i 0.987606 + 0.156956i \(0.0501682\pi\)
−0.987606 + 0.156956i \(0.949832\pi\)
\(744\) 0 0
\(745\) 48.1801i 1.76518i
\(746\) 0 0
\(747\) −19.7389 −0.722210
\(748\) 0 0
\(749\) 49.1432 1.79565
\(750\) 0 0
\(751\) 14.5093i 0.529451i −0.964324 0.264725i \(-0.914719\pi\)
0.964324 0.264725i \(-0.0852813\pi\)
\(752\) 0 0
\(753\) 40.3565i 1.47067i
\(754\) 0 0
\(755\) 61.3560i 2.23297i
\(756\) 0 0
\(757\) −14.3336 −0.520963 −0.260482 0.965479i \(-0.583881\pi\)
−0.260482 + 0.965479i \(0.583881\pi\)
\(758\) 0 0
\(759\) 1.27857i 0.0464093i
\(760\) 0 0
\(761\) 13.8935 0.503638 0.251819 0.967774i \(-0.418971\pi\)
0.251819 + 0.967774i \(0.418971\pi\)
\(762\) 0 0
\(763\) −53.5673 −1.93927
\(764\) 0 0
\(765\) −29.6090 + 3.42366i −1.07051 + 0.123783i
\(766\) 0 0
\(767\) −4.98028 −0.179827
\(768\) 0 0
\(769\) −4.68199 −0.168837 −0.0844185 0.996430i \(-0.526903\pi\)
−0.0844185 + 0.996430i \(0.526903\pi\)
\(770\) 0 0
\(771\) 44.1298i 1.58930i
\(772\) 0 0
\(773\) −51.7340 −1.86074 −0.930371 0.366619i \(-0.880515\pi\)
−0.930371 + 0.366619i \(0.880515\pi\)
\(774\) 0 0
\(775\) 5.56974i 0.200071i
\(776\) 0 0
\(777\) 26.1391i 0.937735i
\(778\) 0 0
\(779\) 4.36206i 0.156287i
\(780\) 0 0
\(781\) 0.249109 0.00891383
\(782\) 0 0
\(783\) −3.61068 −0.129035
\(784\) 0 0
\(785\) 27.2165i 0.971399i
\(786\) 0 0
\(787\) 8.90248i 0.317339i −0.987332 0.158670i \(-0.949280\pi\)
0.987332 0.158670i \(-0.0507205\pi\)
\(788\) 0 0
\(789\) 44.6886i 1.59095i
\(790\) 0 0
\(791\) 2.29403 0.0815665
\(792\) 0 0
\(793\) 63.7292i 2.26309i