Properties

Label 4004.2.m.c.2157.12
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.12
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.11

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-1.08723 q^{3}\) \(-3.34996i q^{5}\) \(-1.00000i q^{7}\) \(-1.81792 q^{9}\) \(+O(q^{10})\) \(q\)\(-1.08723 q^{3}\) \(-3.34996i q^{5}\) \(-1.00000i q^{7}\) \(-1.81792 q^{9}\) \(+1.00000i q^{11}\) \(+(-3.52800 - 0.743786i) q^{13}\) \(+3.64219i q^{15}\) \(+4.31834 q^{17}\) \(+7.13566i q^{19}\) \(+1.08723i q^{21}\) \(+1.49079 q^{23}\) \(-6.22222 q^{25}\) \(+5.23821 q^{27}\) \(+2.80792 q^{29}\) \(+6.91308i q^{31}\) \(-1.08723i q^{33}\) \(-3.34996 q^{35}\) \(-3.56372i q^{37}\) \(+(3.83576 + 0.808669i) q^{39}\) \(+8.88418i q^{41}\) \(-6.06590 q^{43}\) \(+6.08996i q^{45}\) \(+4.61283i q^{47}\) \(-1.00000 q^{49}\) \(-4.69505 q^{51}\) \(+12.9549 q^{53}\) \(+3.34996 q^{55}\) \(-7.75813i q^{57}\) \(-9.89042i q^{59}\) \(+3.97309 q^{61}\) \(+1.81792i q^{63}\) \(+(-2.49165 + 11.8187i) q^{65}\) \(+12.7654i q^{67}\) \(-1.62084 q^{69}\) \(+6.59867i q^{71}\) \(-12.0334i q^{73}\) \(+6.76501 q^{75}\) \(+1.00000 q^{77}\) \(+9.43515 q^{79}\) \(-0.241394 q^{81}\) \(-0.795337i q^{83}\) \(-14.4663i q^{85}\) \(-3.05286 q^{87}\) \(+1.02531i q^{89}\) \(+(-0.743786 + 3.52800i) q^{91}\) \(-7.51614i q^{93}\) \(+23.9041 q^{95}\) \(-12.9734i q^{97}\) \(-1.81792i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.08723 −0.627715 −0.313857 0.949470i \(-0.601621\pi\)
−0.313857 + 0.949470i \(0.601621\pi\)
\(4\) 0 0
\(5\) 3.34996i 1.49815i −0.662487 0.749073i \(-0.730499\pi\)
0.662487 0.749073i \(-0.269501\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.81792 −0.605974
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) −3.52800 0.743786i −0.978491 0.206289i
\(14\) 0 0
\(15\) 3.64219i 0.940409i
\(16\) 0 0
\(17\) 4.31834 1.04735 0.523676 0.851918i \(-0.324560\pi\)
0.523676 + 0.851918i \(0.324560\pi\)
\(18\) 0 0
\(19\) 7.13566i 1.63703i 0.574484 + 0.818516i \(0.305203\pi\)
−0.574484 + 0.818516i \(0.694797\pi\)
\(20\) 0 0
\(21\) 1.08723i 0.237254i
\(22\) 0 0
\(23\) 1.49079 0.310851 0.155426 0.987848i \(-0.450325\pi\)
0.155426 + 0.987848i \(0.450325\pi\)
\(24\) 0 0
\(25\) −6.22222 −1.24444
\(26\) 0 0
\(27\) 5.23821 1.00809
\(28\) 0 0
\(29\) 2.80792 0.521417 0.260708 0.965418i \(-0.416044\pi\)
0.260708 + 0.965418i \(0.416044\pi\)
\(30\) 0 0
\(31\) 6.91308i 1.24163i 0.783959 + 0.620813i \(0.213197\pi\)
−0.783959 + 0.620813i \(0.786803\pi\)
\(32\) 0 0
\(33\) 1.08723i 0.189263i
\(34\) 0 0
\(35\) −3.34996 −0.566246
\(36\) 0 0
\(37\) 3.56372i 0.585873i −0.956132 0.292936i \(-0.905368\pi\)
0.956132 0.292936i \(-0.0946324\pi\)
\(38\) 0 0
\(39\) 3.83576 + 0.808669i 0.614213 + 0.129491i
\(40\) 0 0
\(41\) 8.88418i 1.38748i 0.720227 + 0.693738i \(0.244037\pi\)
−0.720227 + 0.693738i \(0.755963\pi\)
\(42\) 0 0
\(43\) −6.06590 −0.925040 −0.462520 0.886609i \(-0.653055\pi\)
−0.462520 + 0.886609i \(0.653055\pi\)
\(44\) 0 0
\(45\) 6.08996i 0.907838i
\(46\) 0 0
\(47\) 4.61283i 0.672850i 0.941710 + 0.336425i \(0.109218\pi\)
−0.941710 + 0.336425i \(0.890782\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −4.69505 −0.657438
\(52\) 0 0
\(53\) 12.9549 1.77950 0.889749 0.456451i \(-0.150879\pi\)
0.889749 + 0.456451i \(0.150879\pi\)
\(54\) 0 0
\(55\) 3.34996 0.451708
\(56\) 0 0
\(57\) 7.75813i 1.02759i
\(58\) 0 0
\(59\) 9.89042i 1.28762i −0.765184 0.643811i \(-0.777352\pi\)
0.765184 0.643811i \(-0.222648\pi\)
\(60\) 0 0
\(61\) 3.97309 0.508701 0.254351 0.967112i \(-0.418138\pi\)
0.254351 + 0.967112i \(0.418138\pi\)
\(62\) 0 0
\(63\) 1.81792i 0.229037i
\(64\) 0 0
\(65\) −2.49165 + 11.8187i −0.309051 + 1.46592i
\(66\) 0 0
\(67\) 12.7654i 1.55954i 0.626065 + 0.779771i \(0.284664\pi\)
−0.626065 + 0.779771i \(0.715336\pi\)
\(68\) 0 0
\(69\) −1.62084 −0.195126
\(70\) 0 0
\(71\) 6.59867i 0.783118i 0.920153 + 0.391559i \(0.128064\pi\)
−0.920153 + 0.391559i \(0.871936\pi\)
\(72\) 0 0
\(73\) 12.0334i 1.40841i −0.709998 0.704204i \(-0.751304\pi\)
0.709998 0.704204i \(-0.248696\pi\)
\(74\) 0 0
\(75\) 6.76501 0.781156
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 9.43515 1.06154 0.530769 0.847517i \(-0.321903\pi\)
0.530769 + 0.847517i \(0.321903\pi\)
\(80\) 0 0
\(81\) −0.241394 −0.0268215
\(82\) 0 0
\(83\) 0.795337i 0.0872995i −0.999047 0.0436498i \(-0.986101\pi\)
0.999047 0.0436498i \(-0.0138986\pi\)
\(84\) 0 0
\(85\) 14.4663i 1.56909i
\(86\) 0 0
\(87\) −3.05286 −0.327301
\(88\) 0 0
\(89\) 1.02531i 0.108683i 0.998522 + 0.0543415i \(0.0173060\pi\)
−0.998522 + 0.0543415i \(0.982694\pi\)
\(90\) 0 0
\(91\) −0.743786 + 3.52800i −0.0779700 + 0.369835i
\(92\) 0 0
\(93\) 7.51614i 0.779387i
\(94\) 0 0
\(95\) 23.9041 2.45251
\(96\) 0 0
\(97\) 12.9734i 1.31725i −0.752472 0.658625i \(-0.771139\pi\)
0.752472 0.658625i \(-0.228861\pi\)
\(98\) 0 0
\(99\) 1.81792i 0.182708i
\(100\) 0 0
\(101\) 2.21979 0.220877 0.110439 0.993883i \(-0.464774\pi\)
0.110439 + 0.993883i \(0.464774\pi\)
\(102\) 0 0
\(103\) −8.76051 −0.863199 −0.431600 0.902065i \(-0.642051\pi\)
−0.431600 + 0.902065i \(0.642051\pi\)
\(104\) 0 0
\(105\) 3.64219 0.355441
\(106\) 0 0
\(107\) 5.48980 0.530719 0.265360 0.964150i \(-0.414509\pi\)
0.265360 + 0.964150i \(0.414509\pi\)
\(108\) 0 0
\(109\) 4.16288i 0.398731i −0.979925 0.199366i \(-0.936112\pi\)
0.979925 0.199366i \(-0.0638882\pi\)
\(110\) 0 0
\(111\) 3.87460i 0.367761i
\(112\) 0 0
\(113\) −11.4716 −1.07916 −0.539578 0.841936i \(-0.681416\pi\)
−0.539578 + 0.841936i \(0.681416\pi\)
\(114\) 0 0
\(115\) 4.99408i 0.465701i
\(116\) 0 0
\(117\) 6.41363 + 1.35214i 0.592940 + 0.125006i
\(118\) 0 0
\(119\) 4.31834i 0.395862i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 9.65918i 0.870939i
\(124\) 0 0
\(125\) 4.09438i 0.366212i
\(126\) 0 0
\(127\) −4.72215 −0.419023 −0.209511 0.977806i \(-0.567187\pi\)
−0.209511 + 0.977806i \(0.567187\pi\)
\(128\) 0 0
\(129\) 6.59505 0.580662
\(130\) 0 0
\(131\) 1.01648 0.0888100 0.0444050 0.999014i \(-0.485861\pi\)
0.0444050 + 0.999014i \(0.485861\pi\)
\(132\) 0 0
\(133\) 7.13566 0.618740
\(134\) 0 0
\(135\) 17.5478i 1.51027i
\(136\) 0 0
\(137\) 2.51956i 0.215261i −0.994191 0.107630i \(-0.965674\pi\)
0.994191 0.107630i \(-0.0343263\pi\)
\(138\) 0 0
\(139\) 9.39901 0.797213 0.398607 0.917122i \(-0.369494\pi\)
0.398607 + 0.917122i \(0.369494\pi\)
\(140\) 0 0
\(141\) 5.01523i 0.422358i
\(142\) 0 0
\(143\) 0.743786 3.52800i 0.0621985 0.295026i
\(144\) 0 0
\(145\) 9.40640i 0.781159i
\(146\) 0 0
\(147\) 1.08723 0.0896736
\(148\) 0 0
\(149\) 8.40917i 0.688906i 0.938804 + 0.344453i \(0.111936\pi\)
−0.938804 + 0.344453i \(0.888064\pi\)
\(150\) 0 0
\(151\) 4.61043i 0.375192i 0.982246 + 0.187596i \(0.0600695\pi\)
−0.982246 + 0.187596i \(0.939930\pi\)
\(152\) 0 0
\(153\) −7.85041 −0.634668
\(154\) 0 0
\(155\) 23.1585 1.86014
\(156\) 0 0
\(157\) −17.0879 −1.36376 −0.681880 0.731464i \(-0.738838\pi\)
−0.681880 + 0.731464i \(0.738838\pi\)
\(158\) 0 0
\(159\) −14.0850 −1.11702
\(160\) 0 0
\(161\) 1.49079i 0.117491i
\(162\) 0 0
\(163\) 22.8787i 1.79200i −0.444057 0.895999i \(-0.646461\pi\)
0.444057 0.895999i \(-0.353539\pi\)
\(164\) 0 0
\(165\) −3.64219 −0.283544
\(166\) 0 0
\(167\) 17.2131i 1.33199i −0.745957 0.665994i \(-0.768008\pi\)
0.745957 0.665994i \(-0.231992\pi\)
\(168\) 0 0
\(169\) 11.8936 + 5.24815i 0.914890 + 0.403704i
\(170\) 0 0
\(171\) 12.9721i 0.991999i
\(172\) 0 0
\(173\) −11.9899 −0.911575 −0.455788 0.890089i \(-0.650642\pi\)
−0.455788 + 0.890089i \(0.650642\pi\)
\(174\) 0 0
\(175\) 6.22222i 0.470355i
\(176\) 0 0
\(177\) 10.7532i 0.808260i
\(178\) 0 0
\(179\) 19.8321 1.48232 0.741162 0.671327i \(-0.234275\pi\)
0.741162 + 0.671327i \(0.234275\pi\)
\(180\) 0 0
\(181\) −19.6548 −1.46093 −0.730464 0.682951i \(-0.760696\pi\)
−0.730464 + 0.682951i \(0.760696\pi\)
\(182\) 0 0
\(183\) −4.31967 −0.319319
\(184\) 0 0
\(185\) −11.9383 −0.877723
\(186\) 0 0
\(187\) 4.31834i 0.315788i
\(188\) 0 0
\(189\) 5.23821i 0.381024i
\(190\) 0 0
\(191\) 13.7595 0.995601 0.497800 0.867292i \(-0.334141\pi\)
0.497800 + 0.867292i \(0.334141\pi\)
\(192\) 0 0
\(193\) 16.3099i 1.17401i 0.809582 + 0.587007i \(0.199694\pi\)
−0.809582 + 0.587007i \(0.800306\pi\)
\(194\) 0 0
\(195\) 2.70901 12.8496i 0.193996 0.920182i
\(196\) 0 0
\(197\) 18.3513i 1.30748i 0.756719 + 0.653740i \(0.226801\pi\)
−0.756719 + 0.653740i \(0.773199\pi\)
\(198\) 0 0
\(199\) 20.4478 1.44950 0.724752 0.689010i \(-0.241954\pi\)
0.724752 + 0.689010i \(0.241954\pi\)
\(200\) 0 0
\(201\) 13.8790i 0.978948i
\(202\) 0 0
\(203\) 2.80792i 0.197077i
\(204\) 0 0
\(205\) 29.7616 2.07864
\(206\) 0 0
\(207\) −2.71014 −0.188368
\(208\) 0 0
\(209\) −7.13566 −0.493584
\(210\) 0 0
\(211\) 13.8352 0.952454 0.476227 0.879322i \(-0.342004\pi\)
0.476227 + 0.879322i \(0.342004\pi\)
\(212\) 0 0
\(213\) 7.17430i 0.491575i
\(214\) 0 0
\(215\) 20.3205i 1.38585i
\(216\) 0 0
\(217\) 6.91308 0.469291
\(218\) 0 0
\(219\) 13.0832i 0.884079i
\(220\) 0 0
\(221\) −15.2351 3.21192i −1.02482 0.216057i
\(222\) 0 0
\(223\) 11.0590i 0.740568i 0.928919 + 0.370284i \(0.120740\pi\)
−0.928919 + 0.370284i \(0.879260\pi\)
\(224\) 0 0
\(225\) 11.3115 0.754100
\(226\) 0 0
\(227\) 16.2606i 1.07925i −0.841904 0.539627i \(-0.818565\pi\)
0.841904 0.539627i \(-0.181435\pi\)
\(228\) 0 0
\(229\) 17.6808i 1.16838i 0.811617 + 0.584189i \(0.198587\pi\)
−0.811617 + 0.584189i \(0.801413\pi\)
\(230\) 0 0
\(231\) −1.08723 −0.0715348
\(232\) 0 0
\(233\) 17.8756 1.17107 0.585535 0.810647i \(-0.300885\pi\)
0.585535 + 0.810647i \(0.300885\pi\)
\(234\) 0 0
\(235\) 15.4528 1.00803
\(236\) 0 0
\(237\) −10.2582 −0.666343
\(238\) 0 0
\(239\) 10.3242i 0.667819i 0.942605 + 0.333909i \(0.108368\pi\)
−0.942605 + 0.333909i \(0.891632\pi\)
\(240\) 0 0
\(241\) 0.774848i 0.0499124i 0.999689 + 0.0249562i \(0.00794462\pi\)
−0.999689 + 0.0249562i \(0.992055\pi\)
\(242\) 0 0
\(243\) −15.4522 −0.991258
\(244\) 0 0
\(245\) 3.34996i 0.214021i
\(246\) 0 0
\(247\) 5.30740 25.1746i 0.337702 1.60182i
\(248\) 0 0
\(249\) 0.864717i 0.0547992i
\(250\) 0 0
\(251\) −5.64264 −0.356161 −0.178080 0.984016i \(-0.556989\pi\)
−0.178080 + 0.984016i \(0.556989\pi\)
\(252\) 0 0
\(253\) 1.49079i 0.0937251i
\(254\) 0 0
\(255\) 15.7282i 0.984939i
\(256\) 0 0
\(257\) −11.7966 −0.735853 −0.367926 0.929855i \(-0.619932\pi\)
−0.367926 + 0.929855i \(0.619932\pi\)
\(258\) 0 0
\(259\) −3.56372 −0.221439
\(260\) 0 0
\(261\) −5.10457 −0.315965
\(262\) 0 0
\(263\) 3.17206 0.195598 0.0977988 0.995206i \(-0.468820\pi\)
0.0977988 + 0.995206i \(0.468820\pi\)
\(264\) 0 0
\(265\) 43.3985i 2.66595i
\(266\) 0 0
\(267\) 1.11476i 0.0682220i
\(268\) 0 0
\(269\) 10.9042 0.664839 0.332420 0.943132i \(-0.392135\pi\)
0.332420 + 0.943132i \(0.392135\pi\)
\(270\) 0 0
\(271\) 11.9374i 0.725147i 0.931955 + 0.362574i \(0.118102\pi\)
−0.931955 + 0.362574i \(0.881898\pi\)
\(272\) 0 0
\(273\) 0.808669 3.83576i 0.0489429 0.232151i
\(274\) 0 0
\(275\) 6.22222i 0.375214i
\(276\) 0 0
\(277\) 25.8415 1.55267 0.776334 0.630322i \(-0.217077\pi\)
0.776334 + 0.630322i \(0.217077\pi\)
\(278\) 0 0
\(279\) 12.5674i 0.752393i
\(280\) 0 0
\(281\) 8.45300i 0.504264i −0.967693 0.252132i \(-0.918868\pi\)
0.967693 0.252132i \(-0.0811317\pi\)
\(282\) 0 0
\(283\) 29.9613 1.78102 0.890508 0.454968i \(-0.150349\pi\)
0.890508 + 0.454968i \(0.150349\pi\)
\(284\) 0 0
\(285\) −25.9894 −1.53948
\(286\) 0 0
\(287\) 8.88418 0.524417
\(288\) 0 0
\(289\) 1.64806 0.0969449
\(290\) 0 0
\(291\) 14.1051i 0.826857i
\(292\) 0 0
\(293\) 4.86174i 0.284026i 0.989865 + 0.142013i \(0.0453574\pi\)
−0.989865 + 0.142013i \(0.954643\pi\)
\(294\) 0 0
\(295\) −33.1325 −1.92905
\(296\) 0 0
\(297\) 5.23821i 0.303952i
\(298\) 0 0
\(299\) −5.25950 1.10883i −0.304165 0.0641252i
\(300\) 0 0
\(301\) 6.06590i 0.349632i
\(302\) 0 0
\(303\) −2.41343 −0.138648
\(304\) 0 0
\(305\) 13.3097i 0.762109i
\(306\) 0 0
\(307\) 33.5920i 1.91719i −0.284765 0.958597i \(-0.591916\pi\)
0.284765 0.958597i \(-0.408084\pi\)
\(308\) 0 0
\(309\) 9.52473 0.541843
\(310\) 0 0
\(311\) −0.887922 −0.0503494 −0.0251747 0.999683i \(-0.508014\pi\)
−0.0251747 + 0.999683i \(0.508014\pi\)
\(312\) 0 0
\(313\) 25.4993 1.44131 0.720654 0.693295i \(-0.243842\pi\)
0.720654 + 0.693295i \(0.243842\pi\)
\(314\) 0 0
\(315\) 6.08996 0.343130
\(316\) 0 0
\(317\) 16.0285i 0.900252i −0.892965 0.450126i \(-0.851379\pi\)
0.892965 0.450126i \(-0.148621\pi\)
\(318\) 0 0
\(319\) 2.80792i 0.157213i
\(320\) 0 0
\(321\) −5.96870 −0.333140
\(322\) 0 0
\(323\) 30.8142i 1.71455i
\(324\) 0 0
\(325\) 21.9520 + 4.62800i 1.21768 + 0.256715i
\(326\) 0 0
\(327\) 4.52602i 0.250290i
\(328\) 0 0
\(329\) 4.61283 0.254314
\(330\) 0 0
\(331\) 26.4189i 1.45211i 0.687634 + 0.726057i \(0.258649\pi\)
−0.687634 + 0.726057i \(0.741351\pi\)
\(332\) 0 0
\(333\) 6.47857i 0.355024i
\(334\) 0 0
\(335\) 42.7635 2.33642
\(336\) 0 0
\(337\) 5.97919 0.325707 0.162854 0.986650i \(-0.447930\pi\)
0.162854 + 0.986650i \(0.447930\pi\)
\(338\) 0 0
\(339\) 12.4723 0.677402
\(340\) 0 0
\(341\) −6.91308 −0.374364
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 5.42974i 0.292327i
\(346\) 0 0
\(347\) 15.6757 0.841517 0.420758 0.907173i \(-0.361764\pi\)
0.420758 + 0.907173i \(0.361764\pi\)
\(348\) 0 0
\(349\) 12.5860i 0.673711i 0.941556 + 0.336855i \(0.109363\pi\)
−0.941556 + 0.336855i \(0.890637\pi\)
\(350\) 0 0
\(351\) −18.4804 3.89611i −0.986411 0.207959i
\(352\) 0 0
\(353\) 19.0812i 1.01559i 0.861478 + 0.507794i \(0.169539\pi\)
−0.861478 + 0.507794i \(0.830461\pi\)
\(354\) 0 0
\(355\) 22.1053 1.17323
\(356\) 0 0
\(357\) 4.69505i 0.248488i
\(358\) 0 0
\(359\) 18.2818i 0.964879i 0.875929 + 0.482439i \(0.160249\pi\)
−0.875929 + 0.482439i \(0.839751\pi\)
\(360\) 0 0
\(361\) −31.9176 −1.67987
\(362\) 0 0
\(363\) 1.08723 0.0570650
\(364\) 0 0
\(365\) −40.3115 −2.11000
\(366\) 0 0
\(367\) 33.7377 1.76109 0.880547 0.473959i \(-0.157175\pi\)
0.880547 + 0.473959i \(0.157175\pi\)
\(368\) 0 0
\(369\) 16.1507i 0.840774i
\(370\) 0 0
\(371\) 12.9549i 0.672587i
\(372\) 0 0
\(373\) 31.7608 1.64451 0.822256 0.569118i \(-0.192715\pi\)
0.822256 + 0.569118i \(0.192715\pi\)
\(374\) 0 0
\(375\) 4.45155i 0.229877i
\(376\) 0 0
\(377\) −9.90633 2.08849i −0.510202 0.107563i
\(378\) 0 0
\(379\) 7.77706i 0.399481i 0.979849 + 0.199740i \(0.0640099\pi\)
−0.979849 + 0.199740i \(0.935990\pi\)
\(380\) 0 0
\(381\) 5.13408 0.263027
\(382\) 0 0
\(383\) 20.9064i 1.06827i 0.845400 + 0.534134i \(0.179362\pi\)
−0.845400 + 0.534134i \(0.820638\pi\)
\(384\) 0 0
\(385\) 3.34996i 0.170730i
\(386\) 0 0
\(387\) 11.0273 0.560550
\(388\) 0 0
\(389\) 28.0133 1.42033 0.710166 0.704034i \(-0.248620\pi\)
0.710166 + 0.704034i \(0.248620\pi\)
\(390\) 0 0
\(391\) 6.43774 0.325570
\(392\) 0 0
\(393\) −1.10515 −0.0557474
\(394\) 0 0
\(395\) 31.6074i 1.59034i
\(396\) 0 0
\(397\) 16.5116i 0.828694i 0.910119 + 0.414347i \(0.135990\pi\)
−0.910119 + 0.414347i \(0.864010\pi\)
\(398\) 0 0
\(399\) −7.75813 −0.388392
\(400\) 0 0
\(401\) 22.1648i 1.10686i 0.832896 + 0.553430i \(0.186681\pi\)
−0.832896 + 0.553430i \(0.813319\pi\)
\(402\) 0 0
\(403\) 5.14185 24.3894i 0.256134 1.21492i
\(404\) 0 0
\(405\) 0.808658i 0.0401826i
\(406\) 0 0
\(407\) 3.56372 0.176647
\(408\) 0 0
\(409\) 0.356646i 0.0176350i −0.999961 0.00881751i \(-0.997193\pi\)
0.999961 0.00881751i \(-0.00280674\pi\)
\(410\) 0 0
\(411\) 2.73935i 0.135122i
\(412\) 0 0
\(413\) −9.89042 −0.486676
\(414\) 0 0
\(415\) −2.66434 −0.130788
\(416\) 0 0
\(417\) −10.2189 −0.500423
\(418\) 0 0
\(419\) 33.5854 1.64076 0.820378 0.571821i \(-0.193763\pi\)
0.820378 + 0.571821i \(0.193763\pi\)
\(420\) 0 0
\(421\) 1.02335i 0.0498752i 0.999689 + 0.0249376i \(0.00793870\pi\)
−0.999689 + 0.0249376i \(0.992061\pi\)
\(422\) 0 0
\(423\) 8.38576i 0.407730i
\(424\) 0 0
\(425\) −26.8697 −1.30337
\(426\) 0 0
\(427\) 3.97309i 0.192271i
\(428\) 0 0
\(429\) −0.808669 + 3.83576i −0.0390429 + 0.185192i
\(430\) 0 0
\(431\) 17.8561i 0.860100i 0.902805 + 0.430050i \(0.141504\pi\)
−0.902805 + 0.430050i \(0.858496\pi\)
\(432\) 0 0
\(433\) −17.2041 −0.826775 −0.413388 0.910555i \(-0.635655\pi\)
−0.413388 + 0.910555i \(0.635655\pi\)
\(434\) 0 0
\(435\) 10.2270i 0.490345i
\(436\) 0 0
\(437\) 10.6378i 0.508873i
\(438\) 0 0
\(439\) −15.3419 −0.732229 −0.366114 0.930570i \(-0.619312\pi\)
−0.366114 + 0.930570i \(0.619312\pi\)
\(440\) 0 0
\(441\) 1.81792 0.0865677
\(442\) 0 0
\(443\) −35.0634 −1.66591 −0.832955 0.553340i \(-0.813353\pi\)
−0.832955 + 0.553340i \(0.813353\pi\)
\(444\) 0 0
\(445\) 3.43476 0.162823
\(446\) 0 0
\(447\) 9.14273i 0.432436i
\(448\) 0 0
\(449\) 6.70661i 0.316504i −0.987399 0.158252i \(-0.949414\pi\)
0.987399 0.158252i \(-0.0505859\pi\)
\(450\) 0 0
\(451\) −8.88418 −0.418340
\(452\) 0 0
\(453\) 5.01262i 0.235513i
\(454\) 0 0
\(455\) 11.8187 + 2.49165i 0.554067 + 0.116810i
\(456\) 0 0
\(457\) 37.6775i 1.76248i 0.472668 + 0.881240i \(0.343291\pi\)
−0.472668 + 0.881240i \(0.656709\pi\)
\(458\) 0 0
\(459\) 22.6204 1.05583
\(460\) 0 0
\(461\) 40.6991i 1.89555i −0.318947 0.947773i \(-0.603329\pi\)
0.318947 0.947773i \(-0.396671\pi\)
\(462\) 0 0
\(463\) 2.98402i 0.138679i −0.997593 0.0693396i \(-0.977911\pi\)
0.997593 0.0693396i \(-0.0220892\pi\)
\(464\) 0 0
\(465\) −25.1787 −1.16764
\(466\) 0 0
\(467\) −26.4650 −1.22465 −0.612327 0.790605i \(-0.709766\pi\)
−0.612327 + 0.790605i \(0.709766\pi\)
\(468\) 0 0
\(469\) 12.7654 0.589451
\(470\) 0 0
\(471\) 18.5785 0.856053
\(472\) 0 0
\(473\) 6.06590i 0.278910i
\(474\) 0 0
\(475\) 44.3996i 2.03719i
\(476\) 0 0
\(477\) −23.5511 −1.07833
\(478\) 0 0
\(479\) 11.4762i 0.524361i 0.965019 + 0.262180i \(0.0844416\pi\)
−0.965019 + 0.262180i \(0.915558\pi\)
\(480\) 0 0
\(481\) −2.65065 + 12.5728i −0.120859 + 0.573271i
\(482\) 0 0
\(483\) 1.62084i 0.0737506i
\(484\) 0 0
\(485\) −43.4603 −1.97343
\(486\) 0 0
\(487\) 18.7141i 0.848018i 0.905658 + 0.424009i \(0.139378\pi\)
−0.905658 + 0.424009i \(0.860622\pi\)
\(488\) 0 0
\(489\) 24.8745i 1.12486i
\(490\) 0 0
\(491\) −9.55959 −0.431418 −0.215709 0.976458i \(-0.569206\pi\)
−0.215709 + 0.976458i \(0.569206\pi\)
\(492\) 0 0
\(493\) 12.1255 0.546107
\(494\) 0 0
\(495\) −6.08996 −0.273723
\(496\) 0 0
\(497\) 6.59867 0.295991
\(498\) 0 0
\(499\) 18.2878i 0.818672i 0.912384 + 0.409336i \(0.134240\pi\)
−0.912384 + 0.409336i \(0.865760\pi\)
\(500\) 0 0
\(501\) 18.7147i 0.836109i
\(502\) 0 0
\(503\) −22.0398 −0.982705 −0.491352 0.870961i \(-0.663497\pi\)
−0.491352 + 0.870961i \(0.663497\pi\)
\(504\) 0 0
\(505\) 7.43620i 0.330907i
\(506\) 0 0
\(507\) −12.9311 5.70597i −0.574290 0.253411i
\(508\) 0 0
\(509\) 23.5200i 1.04250i 0.853403 + 0.521252i \(0.174535\pi\)
−0.853403 + 0.521252i \(0.825465\pi\)
\(510\) 0 0
\(511\) −12.0334 −0.532328
\(512\) 0 0
\(513\) 37.3781i 1.65028i
\(514\) 0 0
\(515\) 29.3474i 1.29320i
\(516\) 0 0
\(517\) −4.61283 −0.202872
\(518\) 0 0
\(519\) 13.0358 0.572209
\(520\) 0 0
\(521\) −7.16958 −0.314105 −0.157052 0.987590i \(-0.550199\pi\)
−0.157052 + 0.987590i \(0.550199\pi\)
\(522\) 0 0
\(523\) 43.9276 1.92082 0.960410 0.278590i \(-0.0898670\pi\)
0.960410 + 0.278590i \(0.0898670\pi\)
\(524\) 0 0
\(525\) 6.76501i 0.295249i
\(526\) 0 0
\(527\) 29.8530i 1.30042i
\(528\) 0 0
\(529\) −20.7775 −0.903372
\(530\) 0 0
\(531\) 17.9800i 0.780266i
\(532\) 0 0
\(533\) 6.60793 31.3434i 0.286221 1.35763i
\(534\) 0 0
\(535\) 18.3906i 0.795095i
\(536\) 0 0
\(537\) −21.5622 −0.930476
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 5.02702i 0.216129i 0.994144 + 0.108064i \(0.0344652\pi\)
−0.994144 + 0.108064i \(0.965535\pi\)
\(542\) 0 0
\(543\) 21.3694 0.917047
\(544\) 0 0
\(545\) −13.9455 −0.597358
\(546\) 0 0
\(547\) 29.2289 1.24974 0.624868 0.780730i \(-0.285153\pi\)
0.624868 + 0.780730i \(0.285153\pi\)
\(548\) 0 0
\(549\) −7.22276 −0.308260
\(550\) 0 0
\(551\) 20.0363i 0.853576i
\(552\) 0 0
\(553\) 9.43515i 0.401223i
\(554\) 0 0
\(555\) 12.9798 0.550960
\(556\) 0 0
\(557\) 17.8653i 0.756979i 0.925606 + 0.378489i \(0.123556\pi\)
−0.925606 + 0.378489i \(0.876444\pi\)
\(558\) 0 0
\(559\) 21.4005 + 4.51173i 0.905144 + 0.190826i
\(560\) 0 0
\(561\) 4.69505i 0.198225i
\(562\) 0 0
\(563\) −0.379493 −0.0159937 −0.00799685 0.999968i \(-0.502546\pi\)
−0.00799685 + 0.999968i \(0.502546\pi\)
\(564\) 0 0
\(565\) 38.4293i 1.61673i
\(566\) 0 0
\(567\) 0.241394i 0.0101376i
\(568\) 0 0
\(569\) 35.5588 1.49070 0.745352 0.666672i \(-0.232282\pi\)
0.745352 + 0.666672i \(0.232282\pi\)
\(570\) 0 0
\(571\) −41.3720 −1.73136 −0.865682 0.500595i \(-0.833115\pi\)
−0.865682 + 0.500595i \(0.833115\pi\)
\(572\) 0 0
\(573\) −14.9598 −0.624954
\(574\) 0 0
\(575\) −9.27602 −0.386837
\(576\) 0 0
\(577\) 37.7093i 1.56986i −0.619585 0.784929i \(-0.712699\pi\)
0.619585 0.784929i \(-0.287301\pi\)
\(578\) 0 0
\(579\) 17.7327i 0.736946i
\(580\) 0 0
\(581\) −0.795337 −0.0329961
\(582\) 0 0
\(583\) 12.9549i 0.536539i
\(584\) 0 0
\(585\) 4.52963 21.4854i 0.187277 0.888311i
\(586\) 0 0
\(587\) 32.3211i 1.33404i −0.745042 0.667018i \(-0.767571\pi\)
0.745042 0.667018i \(-0.232429\pi\)
\(588\) 0 0
\(589\) −49.3294 −2.03258
\(590\) 0 0
\(591\) 19.9522i 0.820724i
\(592\) 0 0
\(593\) 31.2125i 1.28174i 0.767647 + 0.640872i \(0.221427\pi\)
−0.767647 + 0.640872i \(0.778573\pi\)
\(594\) 0 0
\(595\) −14.4663 −0.593059
\(596\) 0 0
\(597\) −22.2315 −0.909875
\(598\) 0 0
\(599\) −28.7899 −1.17633 −0.588163 0.808743i \(-0.700149\pi\)
−0.588163 + 0.808743i \(0.700149\pi\)
\(600\) 0 0
\(601\) −2.90195 −0.118373 −0.0591866 0.998247i \(-0.518851\pi\)
−0.0591866 + 0.998247i \(0.518851\pi\)
\(602\) 0 0
\(603\) 23.2065i 0.945042i
\(604\) 0 0
\(605\) 3.34996i 0.136195i
\(606\) 0 0
\(607\) −6.59972 −0.267875 −0.133937 0.990990i \(-0.542762\pi\)
−0.133937 + 0.990990i \(0.542762\pi\)
\(608\) 0 0
\(609\) 3.05286i 0.123708i
\(610\) 0 0
\(611\) 3.43096 16.2741i 0.138802 0.658378i
\(612\) 0 0
\(613\) 32.0862i 1.29595i −0.761662 0.647975i \(-0.775616\pi\)
0.761662 0.647975i \(-0.224384\pi\)
\(614\) 0 0
\(615\) −32.3579 −1.30479
\(616\) 0 0
\(617\) 42.3420i 1.70462i −0.523034 0.852312i \(-0.675200\pi\)
0.523034 0.852312i \(-0.324800\pi\)
\(618\) 0 0
\(619\) 18.0020i 0.723562i −0.932263 0.361781i \(-0.882169\pi\)
0.932263 0.361781i \(-0.117831\pi\)
\(620\) 0 0
\(621\) 7.80907 0.313367
\(622\) 0 0
\(623\) 1.02531 0.0410783
\(624\) 0 0
\(625\) −17.3951 −0.695804
\(626\) 0 0
\(627\) 7.75813 0.309830
\(628\) 0 0
\(629\) 15.3894i 0.613614i
\(630\) 0 0
\(631\) 27.0601i 1.07725i −0.842547 0.538623i \(-0.818945\pi\)
0.842547 0.538623i \(-0.181055\pi\)
\(632\) 0 0
\(633\) −15.0421 −0.597870
\(634\) 0 0
\(635\) 15.8190i 0.627758i
\(636\) 0 0
\(637\) 3.52800 + 0.743786i 0.139784 + 0.0294699i
\(638\) 0 0
\(639\) 11.9959i 0.474549i
\(640\) 0 0
\(641\) −27.9694 −1.10472 −0.552362 0.833604i \(-0.686273\pi\)
−0.552362 + 0.833604i \(0.686273\pi\)
\(642\) 0 0
\(643\) 8.99767i 0.354834i 0.984136 + 0.177417i \(0.0567741\pi\)
−0.984136 + 0.177417i \(0.943226\pi\)
\(644\) 0 0
\(645\) 22.0931i 0.869916i
\(646\) 0 0
\(647\) −38.2166 −1.50245 −0.751225 0.660046i \(-0.770537\pi\)
−0.751225 + 0.660046i \(0.770537\pi\)
\(648\) 0 0
\(649\) 9.89042 0.388233
\(650\) 0 0
\(651\) −7.51614 −0.294581
\(652\) 0 0
\(653\) 36.7247 1.43715 0.718574 0.695451i \(-0.244795\pi\)
0.718574 + 0.695451i \(0.244795\pi\)
\(654\) 0 0
\(655\) 3.40516i 0.133050i
\(656\) 0 0
\(657\) 21.8759i 0.853459i
\(658\) 0 0
\(659\) 16.5319 0.643992 0.321996 0.946741i \(-0.395646\pi\)
0.321996 + 0.946741i \(0.395646\pi\)
\(660\) 0 0
\(661\) 14.2119i 0.552778i 0.961046 + 0.276389i \(0.0891378\pi\)
−0.961046 + 0.276389i \(0.910862\pi\)
\(662\) 0 0
\(663\) 16.5641 + 3.49211i 0.643297 + 0.135622i
\(664\) 0 0
\(665\) 23.9041i 0.926963i
\(666\) 0 0
\(667\) 4.18601 0.162083
\(668\) 0 0
\(669\) 12.0238i 0.464866i
\(670\) 0 0
\(671\) 3.97309i 0.153379i
\(672\) 0 0
\(673\) 11.9327 0.459973 0.229987 0.973194i \(-0.426132\pi\)
0.229987 + 0.973194i \(0.426132\pi\)
\(674\) 0 0
\(675\) −32.5933 −1.25452
\(676\) 0 0
\(677\) 16.7278 0.642902 0.321451 0.946926i \(-0.395830\pi\)
0.321451 + 0.946926i \(0.395830\pi\)
\(678\) 0 0
\(679\) −12.9734 −0.497873
\(680\) 0 0
\(681\) 17.6791i 0.677464i
\(682\) 0 0
\(683\) 5.48026i 0.209696i 0.994488 + 0.104848i \(0.0334356\pi\)
−0.994488 + 0.104848i \(0.966564\pi\)
\(684\) 0 0
\(685\) −8.44043 −0.322492
\(686\) 0 0
\(687\) 19.2231i 0.733409i
\(688\) 0 0
\(689\) −45.7050 9.63570i −1.74122 0.367091i
\(690\) 0 0
\(691\) 41.8095i 1.59051i −0.606275 0.795255i \(-0.707337\pi\)
0.606275 0.795255i \(-0.292663\pi\)
\(692\) 0 0
\(693\) −1.81792 −0.0690571
\(694\) 0 0
\(695\) 31.4863i 1.19434i
\(696\) 0 0
\(697\) 38.3649i 1.45317i
\(698\) 0 0
\(699\) −19.4350 −0.735098
\(700\) 0 0
\(701\) −0.566554 −0.0213984 −0.0106992 0.999943i \(-0.503406\pi\)
−0.0106992 + 0.999943i \(0.503406\pi\)
\(702\) 0 0
\(703\) 25.4295 0.959092
\(704\) 0 0
\(705\) −16.8008 −0.632754
\(706\) 0 0
\(707\) 2.21979i 0.0834838i
\(708\) 0 0
\(709\) 8.88377i 0.333637i −0.985988 0.166819i \(-0.946651\pi\)
0.985988 0.166819i \(-0.0533494\pi\)
\(710\) 0 0
\(711\) −17.1524 −0.643264
\(712\) 0 0
\(713\) 10.3059i 0.385961i
\(714\) 0 0
\(715\) −11.8187 2.49165i −0.441992 0.0931825i
\(716\) 0 0
\(717\) 11.2249i 0.419200i
\(718\) 0 0
\(719\) 43.4041 1.61870 0.809350 0.587326i \(-0.199819\pi\)
0.809350 + 0.587326i \(0.199819\pi\)
\(720\) 0 0
\(721\) 8.76051i 0.326259i
\(722\) 0 0
\(723\) 0.842442i 0.0313307i
\(724\) 0 0
\(725\) −17.4715 −0.648874
\(726\) 0 0
\(727\) 15.0438 0.557943 0.278972 0.960299i \(-0.410006\pi\)
0.278972 + 0.960299i \(0.410006\pi\)
\(728\) 0 0
\(729\) 17.5243 0.649049
\(730\) 0 0
\(731\) −26.1946 −0.968842
\(732\) 0 0
\(733\) 5.47061i 0.202062i −0.994883 0.101031i \(-0.967786\pi\)
0.994883 0.101031i \(-0.0322141\pi\)
\(734\) 0 0
\(735\) 3.64219i 0.134344i
\(736\) 0 0
\(737\) −12.7654 −0.470220
\(738\) 0 0
\(739\) 14.8229i 0.545268i −0.962118 0.272634i \(-0.912105\pi\)
0.962118 0.272634i \(-0.0878947\pi\)
\(740\) 0 0
\(741\) −5.77039 + 27.3707i −0.211980 + 1.00549i
\(742\) 0 0
\(743\) 34.1196i 1.25173i 0.779933 + 0.625863i \(0.215253\pi\)
−0.779933 + 0.625863i \(0.784747\pi\)
\(744\) 0 0
\(745\) 28.1704 1.03208
\(746\) 0 0
\(747\) 1.44586i 0.0529013i
\(748\) 0 0
\(749\) 5.48980i 0.200593i
\(750\) 0 0
\(751\) 34.1659 1.24673 0.623366 0.781930i \(-0.285765\pi\)
0.623366 + 0.781930i \(0.285765\pi\)
\(752\) 0 0
\(753\) 6.13487 0.223567
\(754\) 0 0
\(755\) 15.4448 0.562092
\(756\) 0 0
\(757\) 34.0261 1.23670 0.618350 0.785903i \(-0.287801\pi\)
0.618350 + 0.785903i \(0.287801\pi\)
\(758\) 0 0
\(759\) 1.62084i 0.0588327i
\(760\) 0 0
\(761\) 19.9598i 0.723544i 0.932267 + 0.361772i \(0.117828\pi\)
−0.932267 + 0.361772i \(0.882172\pi\)
\(762\) 0 0
\(763\) −4.16288 −0.150706
\(764\) 0 0
\(765\) 26.2985i 0.950825i
\(766\) 0 0
\(767\) −7.35635 + 34.8934i −0.265623 + 1.25993i
\(768\) 0 0
\(769\) 53.3918i 1.92536i −0.270649 0.962678i \(-0.587238\pi\)
0.270649 0.962678i \(-0.412762\pi\)
\(770\) 0 0
\(771\) 12.8257 0.461906
\(772\) 0 0
\(773\) 9.47357i 0.340741i 0.985380 + 0.170370i \(0.0544964\pi\)
−0.985380 + 0.170370i \(0.945504\pi\)
\(774\) 0 0
\(775\) 43.0147i 1.54513i
\(776\) 0 0
\(777\) 3.87460 0.139001
\(778\) 0 0
\(779\) −63.3945 −2.27134
\(780\) 0 0
\(781\) −6.59867 −0.236119
\(782\) 0 0
\(783\) 14.7085 0.525637
\(784\) 0 0
\(785\) 57.2437i 2.04311i
\(786\) 0 0
\(787\) 43.0640i 1.53507i 0.641009 + 0.767533i \(0.278516\pi\)
−0.641009 + 0.767533i \(0.721484\pi\)
\(788\) 0 0
\(789\) −3.44877 −0.122779
\(790\) 0 0
\(791\) 11.4716i 0.407882i
\(792\) 0 0
\(793\) −14.0170 2.95513i −0.497760 0.104940i
\(794\) 0 0