Properties

Label 1900.2.a.j.1.3
Level $1900$
Weight $2$
Character 1900.1
Self dual yes
Analytic conductor $15.172$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.1715763840\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.874032\) of defining polynomial
Character \(\chi\) \(=\) 1900.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.874032 q^{3} +2.82843 q^{7} -2.23607 q^{9} +O(q^{10})\) \(q+0.874032 q^{3} +2.82843 q^{7} -2.23607 q^{9} -0.763932 q^{11} -5.45052 q^{13} -7.40492 q^{17} -1.00000 q^{19} +2.47214 q^{21} -1.08036 q^{23} -4.57649 q^{27} -4.47214 q^{29} -4.00000 q^{31} -0.667701 q^{33} -2.62210 q^{37} -4.76393 q^{39} -6.00000 q^{41} +8.48528 q^{43} +8.48528 q^{47} +1.00000 q^{49} -6.47214 q^{51} +2.62210 q^{53} -0.874032 q^{57} -1.52786 q^{59} -11.7082 q^{61} -6.32456 q^{63} +11.1074 q^{67} -0.944272 q^{69} -10.4721 q^{71} +5.24419 q^{73} -2.16073 q^{77} +15.4164 q^{79} +2.70820 q^{81} +13.7295 q^{83} -3.90879 q^{87} +2.94427 q^{89} -15.4164 q^{91} -3.49613 q^{93} +13.9358 q^{97} +1.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 12 q^{11} - 4 q^{19} - 8 q^{21} - 16 q^{31} - 28 q^{39} - 24 q^{41} + 4 q^{49} - 8 q^{51} - 24 q^{59} - 20 q^{61} + 32 q^{69} - 24 q^{71} + 8 q^{79} - 16 q^{81} - 24 q^{89} - 8 q^{91} - 20 q^{99} + O(q^{100}) \)

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
\(3\) 0.874032 0.504623 0.252311 0.967646i \(-0.418809\pi\)
0.252311 + 0.967646i \(0.418809\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) −2.23607 −0.745356
\(10\) 0 0
\(11\) −0.763932 −0.230334 −0.115167 0.993346i \(-0.536740\pi\)
−0.115167 + 0.993346i \(0.536740\pi\)
\(12\) 0 0
\(13\) −5.45052 −1.51170 −0.755852 0.654743i \(-0.772777\pi\)
−0.755852 + 0.654743i \(0.772777\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.40492 −1.79596 −0.897978 0.440040i \(-0.854964\pi\)
−0.897978 + 0.440040i \(0.854964\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.47214 0.539464
\(22\) 0 0
\(23\) −1.08036 −0.225271 −0.112636 0.993636i \(-0.535929\pi\)
−0.112636 + 0.993636i \(0.535929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.57649 −0.880746
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −0.667701 −0.116232
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.62210 −0.431070 −0.215535 0.976496i \(-0.569150\pi\)
−0.215535 + 0.976496i \(0.569150\pi\)
\(38\) 0 0
\(39\) −4.76393 −0.762840
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.48528 1.23771 0.618853 0.785507i \(-0.287598\pi\)
0.618853 + 0.785507i \(0.287598\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.47214 −0.906280
\(52\) 0 0
\(53\) 2.62210 0.360173 0.180086 0.983651i \(-0.442362\pi\)
0.180086 + 0.983651i \(0.442362\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.874032 −0.115768
\(58\) 0 0
\(59\) −1.52786 −0.198911 −0.0994555 0.995042i \(-0.531710\pi\)
−0.0994555 + 0.995042i \(0.531710\pi\)
\(60\) 0 0
\(61\) −11.7082 −1.49908 −0.749541 0.661958i \(-0.769726\pi\)
−0.749541 + 0.661958i \(0.769726\pi\)
\(62\) 0 0
\(63\) −6.32456 −0.796819
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.1074 1.35698 0.678491 0.734609i \(-0.262634\pi\)
0.678491 + 0.734609i \(0.262634\pi\)
\(68\) 0 0
\(69\) −0.944272 −0.113677
\(70\) 0 0
\(71\) −10.4721 −1.24281 −0.621407 0.783488i \(-0.713439\pi\)
−0.621407 + 0.783488i \(0.713439\pi\)
\(72\) 0 0
\(73\) 5.24419 0.613786 0.306893 0.951744i \(-0.400711\pi\)
0.306893 + 0.951744i \(0.400711\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.16073 −0.246238
\(78\) 0 0
\(79\) 15.4164 1.73448 0.867241 0.497889i \(-0.165891\pi\)
0.867241 + 0.497889i \(0.165891\pi\)
\(80\) 0 0
\(81\) 2.70820 0.300912
\(82\) 0 0
\(83\) 13.7295 1.50701 0.753503 0.657445i \(-0.228363\pi\)
0.753503 + 0.657445i \(0.228363\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.90879 −0.419066
\(88\) 0 0
\(89\) 2.94427 0.312092 0.156046 0.987750i \(-0.450125\pi\)
0.156046 + 0.987750i \(0.450125\pi\)
\(90\) 0 0
\(91\) −15.4164 −1.61608
\(92\) 0 0
\(93\) −3.49613 −0.362532
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.9358 1.41497 0.707483 0.706730i \(-0.249830\pi\)
0.707483 + 0.706730i \(0.249830\pi\)
\(98\) 0 0
\(99\) 1.70820 0.171681
\(100\) 0 0
\(101\) 5.23607 0.521008 0.260504 0.965473i \(-0.416111\pi\)
0.260504 + 0.965473i \(0.416111\pi\)
\(102\) 0 0
\(103\) −5.86319 −0.577717 −0.288858 0.957372i \(-0.593276\pi\)
−0.288858 + 0.957372i \(0.593276\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.2681 −1.28268 −0.641338 0.767258i \(-0.721620\pi\)
−0.641338 + 0.767258i \(0.721620\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −2.29180 −0.217528
\(112\) 0 0
\(113\) −16.3516 −1.53823 −0.769113 0.639113i \(-0.779302\pi\)
−0.769113 + 0.639113i \(0.779302\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.1877 1.12676
\(118\) 0 0
\(119\) −20.9443 −1.91996
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0 0
\(123\) −5.24419 −0.472853
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −19.5927 −1.73857 −0.869284 0.494314i \(-0.835419\pi\)
−0.869284 + 0.494314i \(0.835419\pi\)
\(128\) 0 0
\(129\) 7.41641 0.652978
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −2.82843 −0.245256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.40492 −0.632645 −0.316322 0.948652i \(-0.602448\pi\)
−0.316322 + 0.948652i \(0.602448\pi\)
\(138\) 0 0
\(139\) 2.29180 0.194388 0.0971938 0.995265i \(-0.469013\pi\)
0.0971938 + 0.995265i \(0.469013\pi\)
\(140\) 0 0
\(141\) 7.41641 0.624574
\(142\) 0 0
\(143\) 4.16383 0.348197
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.874032 0.0720889
\(148\) 0 0
\(149\) −12.6525 −1.03653 −0.518266 0.855220i \(-0.673422\pi\)
−0.518266 + 0.855220i \(0.673422\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 16.5579 1.33863
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.412662 −0.0329340 −0.0164670 0.999864i \(-0.505242\pi\)
−0.0164670 + 0.999864i \(0.505242\pi\)
\(158\) 0 0
\(159\) 2.29180 0.181751
\(160\) 0 0
\(161\) −3.05573 −0.240825
\(162\) 0 0
\(163\) −8.48528 −0.664619 −0.332309 0.943170i \(-0.607828\pi\)
−0.332309 + 0.943170i \(0.607828\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.4319 1.34892 0.674462 0.738310i \(-0.264376\pi\)
0.674462 + 0.738310i \(0.264376\pi\)
\(168\) 0 0
\(169\) 16.7082 1.28525
\(170\) 0 0
\(171\) 2.23607 0.170996
\(172\) 0 0
\(173\) 8.94665 0.680201 0.340101 0.940389i \(-0.389539\pi\)
0.340101 + 0.940389i \(0.389539\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.33540 −0.100375
\(178\) 0 0
\(179\) −22.4721 −1.67965 −0.839823 0.542860i \(-0.817341\pi\)
−0.839823 + 0.542860i \(0.817341\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −10.2333 −0.756471
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) −12.9443 −0.941557
\(190\) 0 0
\(191\) −3.05573 −0.221105 −0.110552 0.993870i \(-0.535262\pi\)
−0.110552 + 0.993870i \(0.535262\pi\)
\(192\) 0 0
\(193\) 13.9358 1.00312 0.501561 0.865123i \(-0.332759\pi\)
0.501561 + 0.865123i \(0.332759\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.6491 0.901212 0.450606 0.892723i \(-0.351208\pi\)
0.450606 + 0.892723i \(0.351208\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 9.70820 0.684764
\(202\) 0 0
\(203\) −12.6491 −0.887794
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.41577 0.167907
\(208\) 0 0
\(209\) 0.763932 0.0528423
\(210\) 0 0
\(211\) 15.4164 1.06131 0.530655 0.847588i \(-0.321946\pi\)
0.530655 + 0.847588i \(0.321946\pi\)
\(212\) 0 0
\(213\) −9.15298 −0.627152
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) 0 0
\(219\) 4.58359 0.309730
\(220\) 0 0
\(221\) 40.3607 2.71495
\(222\) 0 0
\(223\) 18.7673 1.25675 0.628377 0.777909i \(-0.283720\pi\)
0.628377 + 0.777909i \(0.283720\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.86319 0.389153 0.194577 0.980887i \(-0.437667\pi\)
0.194577 + 0.980887i \(0.437667\pi\)
\(228\) 0 0
\(229\) 7.70820 0.509372 0.254686 0.967024i \(-0.418028\pi\)
0.254686 + 0.967024i \(0.418028\pi\)
\(230\) 0 0
\(231\) −1.88854 −0.124257
\(232\) 0 0
\(233\) −12.6491 −0.828671 −0.414335 0.910124i \(-0.635986\pi\)
−0.414335 + 0.910124i \(0.635986\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.4744 0.875259
\(238\) 0 0
\(239\) −29.8885 −1.93333 −0.966665 0.256046i \(-0.917580\pi\)
−0.966665 + 0.256046i \(0.917580\pi\)
\(240\) 0 0
\(241\) −28.8328 −1.85728 −0.928642 0.370976i \(-0.879023\pi\)
−0.928642 + 0.370976i \(0.879023\pi\)
\(242\) 0 0
\(243\) 16.0965 1.03259
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.45052 0.346808
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −5.88854 −0.371682 −0.185841 0.982580i \(-0.559501\pi\)
−0.185841 + 0.982580i \(0.559501\pi\)
\(252\) 0 0
\(253\) 0.825324 0.0518877
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −17.4319 −1.08737 −0.543687 0.839288i \(-0.682972\pi\)
−0.543687 + 0.839288i \(0.682972\pi\)
\(258\) 0 0
\(259\) −7.41641 −0.460833
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 15.8902 0.979832 0.489916 0.871770i \(-0.337027\pi\)
0.489916 + 0.871770i \(0.337027\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.57339 0.157489
\(268\) 0 0
\(269\) 14.9443 0.911168 0.455584 0.890193i \(-0.349430\pi\)
0.455584 + 0.890193i \(0.349430\pi\)
\(270\) 0 0
\(271\) 21.1246 1.28323 0.641614 0.767027i \(-0.278265\pi\)
0.641614 + 0.767027i \(0.278265\pi\)
\(272\) 0 0
\(273\) −13.4744 −0.815510
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −22.2148 −1.33476 −0.667378 0.744719i \(-0.732583\pi\)
−0.667378 + 0.744719i \(0.732583\pi\)
\(278\) 0 0
\(279\) 8.94427 0.535480
\(280\) 0 0
\(281\) −25.4164 −1.51622 −0.758108 0.652129i \(-0.773876\pi\)
−0.758108 + 0.652129i \(0.773876\pi\)
\(282\) 0 0
\(283\) −2.41577 −0.143602 −0.0718012 0.997419i \(-0.522875\pi\)
−0.0718012 + 0.997419i \(0.522875\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −16.9706 −1.00174
\(288\) 0 0
\(289\) 37.8328 2.22546
\(290\) 0 0
\(291\) 12.1803 0.714024
\(292\) 0 0
\(293\) 2.62210 0.153184 0.0765922 0.997062i \(-0.475596\pi\)
0.0765922 + 0.997062i \(0.475596\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.49613 0.202866
\(298\) 0 0
\(299\) 5.88854 0.340543
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 4.57649 0.262913
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.45052 −0.311078 −0.155539 0.987830i \(-0.549711\pi\)
−0.155539 + 0.987830i \(0.549711\pi\)
\(308\) 0 0
\(309\) −5.12461 −0.291529
\(310\) 0 0
\(311\) −9.70820 −0.550502 −0.275251 0.961372i \(-0.588761\pi\)
−0.275251 + 0.961372i \(0.588761\pi\)
\(312\) 0 0
\(313\) −10.4884 −0.592839 −0.296419 0.955058i \(-0.595793\pi\)
−0.296419 + 0.955058i \(0.595793\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.94665 −0.502494 −0.251247 0.967923i \(-0.580841\pi\)
−0.251247 + 0.967923i \(0.580841\pi\)
\(318\) 0 0
\(319\) 3.41641 0.191282
\(320\) 0 0
\(321\) −11.5967 −0.647267
\(322\) 0 0
\(323\) 7.40492 0.412021
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.74806 0.0966682
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 7.41641 0.407643 0.203821 0.979008i \(-0.434664\pi\)
0.203821 + 0.979008i \(0.434664\pi\)
\(332\) 0 0
\(333\) 5.86319 0.321301
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.7642 0.913206 0.456603 0.889671i \(-0.349066\pi\)
0.456603 + 0.889671i \(0.349066\pi\)
\(338\) 0 0
\(339\) −14.2918 −0.776224
\(340\) 0 0
\(341\) 3.05573 0.165477
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.40802 −0.505049 −0.252525 0.967590i \(-0.581261\pi\)
−0.252525 + 0.967590i \(0.581261\pi\)
\(348\) 0 0
\(349\) 25.4164 1.36051 0.680255 0.732976i \(-0.261869\pi\)
0.680255 + 0.732976i \(0.261869\pi\)
\(350\) 0 0
\(351\) 24.9443 1.33143
\(352\) 0 0
\(353\) 9.56564 0.509128 0.254564 0.967056i \(-0.418068\pi\)
0.254564 + 0.967056i \(0.418068\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −18.3060 −0.968854
\(358\) 0 0
\(359\) 29.1246 1.53714 0.768569 0.639767i \(-0.220969\pi\)
0.768569 + 0.639767i \(0.220969\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.10427 −0.477850
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528 0.442928 0.221464 0.975169i \(-0.428916\pi\)
0.221464 + 0.975169i \(0.428916\pi\)
\(368\) 0 0
\(369\) 13.4164 0.698430
\(370\) 0 0
\(371\) 7.41641 0.385041
\(372\) 0 0
\(373\) 13.1105 0.678835 0.339417 0.940636i \(-0.389770\pi\)
0.339417 + 0.940636i \(0.389770\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.3755 1.25540
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −17.1246 −0.877320
\(382\) 0 0
\(383\) −36.4056 −1.86024 −0.930120 0.367257i \(-0.880297\pi\)
−0.930120 + 0.367257i \(0.880297\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −18.9737 −0.964486
\(388\) 0 0
\(389\) −14.9443 −0.757705 −0.378852 0.925457i \(-0.623681\pi\)
−0.378852 + 0.925457i \(0.623681\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.7264 0.588530 0.294265 0.955724i \(-0.404925\pi\)
0.294265 + 0.955724i \(0.404925\pi\)
\(398\) 0 0
\(399\) −2.47214 −0.123762
\(400\) 0 0
\(401\) −4.47214 −0.223328 −0.111664 0.993746i \(-0.535618\pi\)
−0.111664 + 0.993746i \(0.535618\pi\)
\(402\) 0 0
\(403\) 21.8021 1.08604
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00310 0.0992901
\(408\) 0 0
\(409\) 17.4164 0.861186 0.430593 0.902546i \(-0.358304\pi\)
0.430593 + 0.902546i \(0.358304\pi\)
\(410\) 0 0
\(411\) −6.47214 −0.319247
\(412\) 0 0
\(413\) −4.32145 −0.212645
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.00310 0.0980924
\(418\) 0 0
\(419\) −20.9443 −1.02319 −0.511597 0.859225i \(-0.670946\pi\)
−0.511597 + 0.859225i \(0.670946\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) −18.9737 −0.922531
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −33.1158 −1.60259
\(428\) 0 0
\(429\) 3.63932 0.175708
\(430\) 0 0
\(431\) 1.52786 0.0735946 0.0367973 0.999323i \(-0.488284\pi\)
0.0367973 + 0.999323i \(0.488284\pi\)
\(432\) 0 0
\(433\) −28.4906 −1.36917 −0.684585 0.728933i \(-0.740017\pi\)
−0.684585 + 0.728933i \(0.740017\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.08036 0.0516808
\(438\) 0 0
\(439\) −18.8328 −0.898841 −0.449421 0.893320i \(-0.648370\pi\)
−0.449421 + 0.893320i \(0.648370\pi\)
\(440\) 0 0
\(441\) −2.23607 −0.106479
\(442\) 0 0
\(443\) −13.7295 −0.652307 −0.326153 0.945317i \(-0.605753\pi\)
−0.326153 + 0.945317i \(0.605753\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −11.0587 −0.523057
\(448\) 0 0
\(449\) −16.4721 −0.777368 −0.388684 0.921371i \(-0.627070\pi\)
−0.388684 + 0.921371i \(0.627070\pi\)
\(450\) 0 0
\(451\) 4.58359 0.215833
\(452\) 0 0
\(453\) −6.99226 −0.328525
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.9473 1.77510 0.887551 0.460710i \(-0.152405\pi\)
0.887551 + 0.460710i \(0.152405\pi\)
\(458\) 0 0
\(459\) 33.8885 1.58178
\(460\) 0 0
\(461\) 23.8885 1.11260 0.556300 0.830981i \(-0.312220\pi\)
0.556300 + 0.830981i \(0.312220\pi\)
\(462\) 0 0
\(463\) 14.5548 0.676419 0.338209 0.941071i \(-0.390179\pi\)
0.338209 + 0.941071i \(0.390179\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.7000 1.42063 0.710314 0.703885i \(-0.248553\pi\)
0.710314 + 0.703885i \(0.248553\pi\)
\(468\) 0 0
\(469\) 31.4164 1.45067
\(470\) 0 0
\(471\) −0.360680 −0.0166192
\(472\) 0 0
\(473\) −6.48218 −0.298051
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.86319 −0.268457
\(478\) 0 0
\(479\) 11.2361 0.513389 0.256695 0.966493i \(-0.417367\pi\)
0.256695 + 0.966493i \(0.417367\pi\)
\(480\) 0 0
\(481\) 14.2918 0.651650
\(482\) 0 0
\(483\) −2.67080 −0.121526
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10.6947 −0.484624 −0.242312 0.970198i \(-0.577906\pi\)
−0.242312 + 0.970198i \(0.577906\pi\)
\(488\) 0 0
\(489\) −7.41641 −0.335382
\(490\) 0 0
\(491\) 5.88854 0.265746 0.132873 0.991133i \(-0.457580\pi\)
0.132873 + 0.991133i \(0.457580\pi\)
\(492\) 0 0
\(493\) 33.1158 1.49146
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −29.6197 −1.32862
\(498\) 0 0
\(499\) −17.1246 −0.766603 −0.383301 0.923623i \(-0.625213\pi\)
−0.383301 + 0.923623i \(0.625213\pi\)
\(500\) 0 0
\(501\) 15.2361 0.680697
\(502\) 0 0
\(503\) 20.2117 0.901193 0.450597 0.892728i \(-0.351211\pi\)
0.450597 + 0.892728i \(0.351211\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.6035 0.648564
\(508\) 0 0
\(509\) −10.5836 −0.469109 −0.234555 0.972103i \(-0.575363\pi\)
−0.234555 + 0.972103i \(0.575363\pi\)
\(510\) 0 0
\(511\) 14.8328 0.656165
\(512\) 0 0
\(513\) 4.57649 0.202057
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.48218 −0.285086
\(518\) 0 0
\(519\) 7.81966 0.343245
\(520\) 0 0
\(521\) 0.111456 0.00488298 0.00244149 0.999997i \(-0.499223\pi\)
0.00244149 + 0.999997i \(0.499223\pi\)
\(522\) 0 0
\(523\) −24.8369 −1.08604 −0.543020 0.839720i \(-0.682719\pi\)
−0.543020 + 0.839720i \(0.682719\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.6197 1.29025
\(528\) 0 0
\(529\) −21.8328 −0.949253
\(530\) 0 0
\(531\) 3.41641 0.148259
\(532\) 0 0
\(533\) 32.7031 1.41653
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −19.6414 −0.847588
\(538\) 0 0
\(539\) −0.763932 −0.0329049
\(540\) 0 0
\(541\) 11.7082 0.503375 0.251688 0.967809i \(-0.419014\pi\)
0.251688 + 0.967809i \(0.419014\pi\)
\(542\) 0 0
\(543\) 5.24419 0.225050
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.27895 −0.353982 −0.176991 0.984212i \(-0.556636\pi\)
−0.176991 + 0.984212i \(0.556636\pi\)
\(548\) 0 0
\(549\) 26.1803 1.11735
\(550\) 0 0
\(551\) 4.47214 0.190519
\(552\) 0 0
\(553\) 43.6042 1.85424
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.0540 0.849716 0.424858 0.905260i \(-0.360324\pi\)
0.424858 + 0.905260i \(0.360324\pi\)
\(558\) 0 0
\(559\) −46.2492 −1.95613
\(560\) 0 0
\(561\) 4.94427 0.208747
\(562\) 0 0
\(563\) 30.0810 1.26776 0.633882 0.773429i \(-0.281460\pi\)
0.633882 + 0.773429i \(0.281460\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.65996 0.321688
\(568\) 0 0
\(569\) −38.9443 −1.63263 −0.816314 0.577608i \(-0.803986\pi\)
−0.816314 + 0.577608i \(0.803986\pi\)
\(570\) 0 0
\(571\) 25.1246 1.05143 0.525716 0.850660i \(-0.323797\pi\)
0.525716 + 0.850660i \(0.323797\pi\)
\(572\) 0 0
\(573\) −2.67080 −0.111574
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.65685 −0.235498 −0.117749 0.993043i \(-0.537568\pi\)
−0.117749 + 0.993043i \(0.537568\pi\)
\(578\) 0 0
\(579\) 12.1803 0.506198
\(580\) 0 0
\(581\) 38.8328 1.61106
\(582\) 0 0
\(583\) −2.00310 −0.0829601
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.40182 0.222957 0.111478 0.993767i \(-0.464441\pi\)
0.111478 + 0.993767i \(0.464441\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 11.0557 0.454772
\(592\) 0 0
\(593\) −2.16073 −0.0887304 −0.0443652 0.999015i \(-0.514127\pi\)
−0.0443652 + 0.999015i \(0.514127\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −13.9845 −0.572348
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 16.8328 0.686625 0.343312 0.939221i \(-0.388451\pi\)
0.343312 + 0.939221i \(0.388451\pi\)
\(602\) 0 0
\(603\) −24.8369 −1.01143
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22.4211 0.910044 0.455022 0.890480i \(-0.349631\pi\)
0.455022 + 0.890480i \(0.349631\pi\)
\(608\) 0 0
\(609\) −11.0557 −0.448001
\(610\) 0 0
\(611\) −46.2492 −1.87104
\(612\) 0 0
\(613\) −39.1853 −1.58268 −0.791340 0.611376i \(-0.790616\pi\)
−0.791340 + 0.611376i \(0.790616\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.08347 0.124136 0.0620678 0.998072i \(-0.480230\pi\)
0.0620678 + 0.998072i \(0.480230\pi\)
\(618\) 0 0
\(619\) 41.1246 1.65294 0.826469 0.562982i \(-0.190346\pi\)
0.826469 + 0.562982i \(0.190346\pi\)
\(620\) 0 0
\(621\) 4.94427 0.198407
\(622\) 0 0
\(623\) 8.32766 0.333641
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.667701 0.0266654
\(628\) 0 0
\(629\) 19.4164 0.774183
\(630\) 0 0
\(631\) −1.70820 −0.0680025 −0.0340013 0.999422i \(-0.510825\pi\)
−0.0340013 + 0.999422i \(0.510825\pi\)
\(632\) 0 0
\(633\) 13.4744 0.535561
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.45052 −0.215958
\(638\) 0 0
\(639\) 23.4164 0.926339
\(640\) 0 0
\(641\) −12.1115 −0.478374 −0.239187 0.970974i \(-0.576881\pi\)
−0.239187 + 0.970974i \(0.576881\pi\)
\(642\) 0 0
\(643\) −30.7000 −1.21069 −0.605346 0.795963i \(-0.706965\pi\)
−0.605346 + 0.795963i \(0.706965\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.40802 −0.369867 −0.184934 0.982751i \(-0.559207\pi\)
−0.184934 + 0.982751i \(0.559207\pi\)
\(648\) 0 0
\(649\) 1.16718 0.0458160
\(650\) 0 0
\(651\) −9.88854 −0.387563
\(652\) 0 0
\(653\) −22.2148 −0.869331 −0.434665 0.900592i \(-0.643133\pi\)
−0.434665 + 0.900592i \(0.643133\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −11.7264 −0.457489
\(658\) 0 0
\(659\) −44.9443 −1.75078 −0.875390 0.483417i \(-0.839395\pi\)
−0.875390 + 0.483417i \(0.839395\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) 35.2765 1.37003
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.83153 0.187078
\(668\) 0 0
\(669\) 16.4033 0.634186
\(670\) 0 0
\(671\) 8.94427 0.345290
\(672\) 0 0
\(673\) −4.62520 −0.178288 −0.0891442 0.996019i \(-0.528413\pi\)
−0.0891442 + 0.996019i \(0.528413\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.7302 1.64225 0.821127 0.570746i \(-0.193346\pi\)
0.821127 + 0.570746i \(0.193346\pi\)
\(678\) 0 0
\(679\) 39.4164 1.51266
\(680\) 0 0
\(681\) 5.12461 0.196376
\(682\) 0 0
\(683\) 19.5927 0.749692 0.374846 0.927087i \(-0.377696\pi\)
0.374846 + 0.927087i \(0.377696\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6.73722 0.257041
\(688\) 0 0
\(689\) −14.2918 −0.544474
\(690\) 0 0
\(691\) 25.1246 0.955785 0.477893 0.878418i \(-0.341401\pi\)
0.477893 + 0.878418i \(0.341401\pi\)
\(692\) 0 0
\(693\) 4.83153 0.183535
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 44.4295 1.68289
\(698\) 0 0
\(699\) −11.0557 −0.418166
\(700\) 0 0
\(701\) 0.875388 0.0330630 0.0165315 0.999863i \(-0.494738\pi\)
0.0165315 + 0.999863i \(0.494738\pi\)
\(702\) 0 0
\(703\) 2.62210 0.0988942
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.8098 0.556981
\(708\) 0 0
\(709\) 33.4164 1.25498 0.627490 0.778625i \(-0.284082\pi\)
0.627490 + 0.778625i \(0.284082\pi\)
\(710\) 0 0
\(711\) −34.4721 −1.29281
\(712\) 0 0
\(713\) 4.32145 0.161840
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −26.1235 −0.975602
\(718\) 0 0
\(719\) 27.5967 1.02919 0.514593 0.857435i \(-0.327943\pi\)
0.514593 + 0.857435i \(0.327943\pi\)
\(720\) 0 0
\(721\) −16.5836 −0.617605
\(722\) 0 0
\(723\) −25.2008 −0.937228
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.48528 0.314702 0.157351 0.987543i \(-0.449705\pi\)
0.157351 + 0.987543i \(0.449705\pi\)
\(728\) 0 0
\(729\) 5.94427 0.220158
\(730\) 0 0
\(731\) −62.8328 −2.32396
\(732\) 0 0
\(733\) −33.9411 −1.25364 −0.626822 0.779162i \(-0.715645\pi\)
−0.626822 + 0.779162i \(0.715645\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.48528 −0.312559
\(738\) 0 0
\(739\) −18.8328 −0.692776 −0.346388 0.938091i \(-0.612592\pi\)
−0.346388 + 0.938091i \(0.612592\pi\)
\(740\) 0 0
\(741\) 4.76393 0.175007
\(742\) 0 0
\(743\) −2.62210 −0.0961954 −0.0480977 0.998843i \(-0.515316\pi\)
−0.0480977 + 0.998843i \(0.515316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −30.7000 −1.12326
\(748\) 0 0
\(749\) −37.5279 −1.37124
\(750\) 0 0
\(751\) 20.5836 0.751106 0.375553 0.926801i \(-0.377453\pi\)
0.375553 + 0.926801i \(0.377453\pi\)
\(752\) 0 0
\(753\) −5.14678 −0.187559
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.7727 −1.40922 −0.704608 0.709597i \(-0.748877\pi\)
−0.704608 + 0.709597i \(0.748877\pi\)
\(758\) 0 0
\(759\) 0.721360 0.0261837
\(760\) 0 0
\(761\) −42.5410 −1.54211 −0.771055 0.636768i \(-0.780271\pi\)
−0.771055 + 0.636768i \(0.780271\pi\)
\(762\) 0 0
\(763\) 5.65685 0.204792
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.32766 0.300694
\(768\) 0 0
\(769\) −38.5410 −1.38982 −0.694912 0.719094i \(-0.744557\pi\)
−0.694912 + 0.719094i \(0.744557\pi\)
\(770\) 0 0
\(771\) −15.2361 −0.548714
\(772\) 0 0
\(773\) 5.86319 0.210884 0.105442 0.994425i \(-0.466374\pi\)
0.105442 + 0.994425i \(0.466374\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.48218 −0.232547
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 20.4667 0.731420
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 24.8369 0.885338 0.442669 0.896685i \(-0.354032\pi\)
0.442669 + 0.896685i \(0.354032\pi\)
\(788\) 0 0
\(789\) 13.8885 0.494445
\(790\) 0 0
\(791\) −46.2492 −1.64443
\(792\) 0 0
\(793\) 63.8158 2.26617
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −52.2958 −1.85241 −0.926206 0.377018i \(-0.876950\pi\)
−0.926206 + 0.377018i \(0.876950\pi\)
\(798\) 0 0
\(799\) −62.8328 −2.22287
\(800\) 0 0
\(801\) −6.58359 −0.232620
\(802\) 0 0
\(803\) −4.00621 −0.141376
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.0618 0.459796
\(808\) 0 0
\(809\) −7.52786 −0.264666 −0.132333 0.991205i \(-0.542247\pi\)
−0.132333 + 0.991205i \(0.542247\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 18.4636 0.647546
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.48528 −0.296862
\(818\) 0 0
\(819\) 34.4721 1.20455
\(820\) 0 0
\(821\) 38.9443 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(822\) 0 0
\(823\) −35.9442 −1.25294 −0.626469 0.779447i \(-0.715500\pi\)
−0.626469 + 0.779447i \(0.715500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.86629 −0.273538 −0.136769 0.990603i \(-0.543672\pi\)
−0.136769 + 0.990603i \(0.543672\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) −19.4164 −0.673548
\(832\) 0 0
\(833\) −7.40492 −0.256565
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 18.3060 0.632747
\(838\) 0 0
\(839\) 49.3050 1.70220 0.851098 0.525007i \(-0.175937\pi\)
0.851098 + 0.525007i \(0.175937\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) −22.2148 −0.765117
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −29.4621 −1.01233
\(848\) 0 0
\(849\) −2.11146 −0.0724650
\(850\) 0 0
\(851\) 2.83282 0.0971077
\(852\) 0 0
\(853\) −27.4589 −0.940176 −0.470088 0.882619i \(-0.655778\pi\)
−0.470088 + 0.882619i \(0.655778\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31.3190 1.06984 0.534919 0.844903i \(-0.320342\pi\)
0.534919 + 0.844903i \(0.320342\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) −14.8328 −0.505501
\(862\) 0 0
\(863\) 7.86629 0.267772 0.133886 0.990997i \(-0.457254\pi\)
0.133886 + 0.990997i \(0.457254\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 33.0671 1.12302
\(868\) 0 0
\(869\) −11.7771 −0.399510
\(870\) 0 0
\(871\) −60.5410 −2.05135
\(872\) 0 0
\(873\) −31.1614 −1.05465
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.9358 0.470579 0.235289 0.971925i \(-0.424396\pi\)
0.235289 + 0.971925i \(0.424396\pi\)
\(878\) 0 0
\(879\) 2.29180 0.0773004
\(880\) 0 0
\(881\) 0.652476 0.0219825 0.0109912 0.999940i \(-0.496501\pi\)
0.0109912 + 0.999940i \(0.496501\pi\)
\(882\) 0 0
\(883\) −52.9148 −1.78072 −0.890362 0.455253i \(-0.849549\pi\)
−0.890362 + 0.455253i \(0.849549\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −49.0547 −1.64710 −0.823548 0.567247i \(-0.808009\pi\)
−0.823548 + 0.567247i \(0.808009\pi\)
\(888\) 0 0
\(889\) −55.4164 −1.85861
\(890\) 0 0
\(891\) −2.06888 −0.0693102
\(892\) 0 0
\(893\) −8.48528 −0.283949
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.14678 0.171846
\(898\) 0 0
\(899\) 17.8885 0.596616
\(900\) 0 0
\(901\) −19.4164 −0.646854
\(902\) 0 0
\(903\) 20.9768 0.698063
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −38.5663 −1.28057 −0.640287 0.768136i \(-0.721185\pi\)
−0.640287 + 0.768136i \(0.721185\pi\)
\(908\) 0 0
\(909\) −11.7082 −0.388337
\(910\) 0 0
\(911\) −46.4721 −1.53969 −0.769845 0.638231i \(-0.779667\pi\)
−0.769845 + 0.638231i \(0.779667\pi\)
\(912\) 0 0
\(913\) −10.4884 −0.347115
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 26.8328 0.885133 0.442566 0.896736i \(-0.354068\pi\)
0.442566 + 0.896736i \(0.354068\pi\)
\(920\) 0 0
\(921\) −4.76393 −0.156977
\(922\) 0 0
\(923\) 57.0786 1.87877
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 13.1105 0.430605
\(928\) 0 0
\(929\) 7.52786 0.246981 0.123491 0.992346i \(-0.460591\pi\)
0.123491 + 0.992346i \(0.460591\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −8.48528 −0.277796
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −43.1915 −1.41101 −0.705503 0.708707i \(-0.749279\pi\)
−0.705503 + 0.708707i \(0.749279\pi\)
\(938\) 0 0
\(939\) −9.16718 −0.299160
\(940\) 0 0
\(941\) −38.9443 −1.26955 −0.634773 0.772698i \(-0.718907\pi\)
−0.634773 + 0.772698i \(0.718907\pi\)
\(942\) 0 0
\(943\) 6.48218 0.211089
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.6706 −1.54909 −0.774543 0.632521i \(-0.782020\pi\)
−0.774543 + 0.632521i \(0.782020\pi\)
\(948\) 0 0
\(949\) −28.5836 −0.927863
\(950\) 0 0
\(951\) −7.81966 −0.253570
\(952\) 0 0
\(953\) −36.5632 −1.18440 −0.592199 0.805791i \(-0.701740\pi\)
−0.592199 + 0.805791i \(0.701740\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.98605 0.0965253
\(958\) 0 0
\(959\) −20.9443 −0.676326
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 29.6684 0.956050
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.41577 −0.0776858 −0.0388429 0.999245i \(-0.512367\pi\)
−0.0388429 + 0.999245i \(0.512367\pi\)
\(968\) 0 0
\(969\) 6.47214 0.207915
\(970\) 0 0
\(971\) −40.3607 −1.29524 −0.647618 0.761965i \(-0.724235\pi\)
−0.647618 + 0.761965i \(0.724235\pi\)
\(972\) 0 0
\(973\) 6.48218 0.207809
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.9945 0.799644 0.399822 0.916593i \(-0.369072\pi\)
0.399822 + 0.916593i \(0.369072\pi\)
\(978\) 0 0
\(979\) −2.24922 −0.0718855
\(980\) 0 0
\(981\) −4.47214 −0.142784
\(982\) 0 0
\(983\) −30.2387 −0.964464 −0.482232 0.876044i \(-0.660174\pi\)
−0.482232 + 0.876044i \(0.660174\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 20.9768 0.667698
\(988\) 0 0
\(989\) −9.16718 −0.291500
\(990\) 0 0
\(991\) 38.8328 1.23357 0.616783 0.787134i \(-0.288436\pi\)
0.616783 + 0.787134i \(0.288436\pi\)
\(992\) 0 0
\(993\) 6.48218 0.205706
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −11.7264 −0.371378 −0.185689 0.982609i \(-0.559452\pi\)
−0.185689 + 0.982609i \(0.559452\pi\)
\(998\) 0 0
\(999\) 12.0000 0.379663
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1900.2.a.j.1.3 4
4.3 odd 2 7600.2.a.ce.1.2 4
5.2 odd 4 380.2.c.a.229.2 4
5.3 odd 4 380.2.c.a.229.3 yes 4
5.4 even 2 inner 1900.2.a.j.1.2 4
15.2 even 4 3420.2.f.a.1369.2 4
15.8 even 4 3420.2.f.a.1369.1 4
20.3 even 4 1520.2.d.f.609.2 4
20.7 even 4 1520.2.d.f.609.3 4
20.19 odd 2 7600.2.a.ce.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.2 4 5.2 odd 4
380.2.c.a.229.3 yes 4 5.3 odd 4
1520.2.d.f.609.2 4 20.3 even 4
1520.2.d.f.609.3 4 20.7 even 4
1900.2.a.j.1.2 4 5.4 even 2 inner
1900.2.a.j.1.3 4 1.1 even 1 trivial
3420.2.f.a.1369.1 4 15.8 even 4
3420.2.f.a.1369.2 4 15.2 even 4
7600.2.a.ce.1.2 4 4.3 odd 2
7600.2.a.ce.1.3 4 20.19 odd 2