Properties

Label 1900.2.a.j.1.2
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 1.2
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 \(-0.581191\pi\)
−0.252311 + 0.967646i \(0.581191\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 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 \(-0.227223\pi\)
0.755852 + 0.654743i \(0.227223\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.40492 1.79596 0.897978 0.440040i \(-0.145036\pi\)
0.897978 + 0.440040i \(0.145036\pi\)
\(18\) 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 \(-0.464071\pi\)
0.112636 + 0.993636i \(0.464071\pi\)
\(24\) 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 \(-0.430850\pi\)
0.215535 + 0.976496i \(0.430850\pi\)
\(38\) 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 \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.48528 −1.23771 −0.618853 0.785507i \(-0.712402\pi\)
−0.618853 + 0.785507i \(0.712402\pi\)
\(48\) 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 \(-0.557638\pi\)
−0.180086 + 0.983651i \(0.557638\pi\)
\(54\) 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 \(-0.737366\pi\)
−0.678491 + 0.734609i \(0.737366\pi\)
\(68\) 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 \(-0.599289\pi\)
−0.306893 + 0.951744i \(0.599289\pi\)
\(74\) 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 \(-0.771637\pi\)
−0.753503 + 0.657445i \(0.771637\pi\)
\(84\) 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 \(-0.750170\pi\)
−0.707483 + 0.706730i \(0.750170\pi\)
\(98\) 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 \(-0.406724\pi\)
0.288858 + 0.957372i \(0.406724\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.2681 1.28268 0.641338 0.767258i \(-0.278380\pi\)
0.641338 + 0.767258i \(0.278380\pi\)
\(108\) 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 \(-0.220698\pi\)
0.769113 + 0.639113i \(0.220698\pi\)
\(114\) 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 \(-0.164581\pi\)
0.869284 + 0.494314i \(0.164581\pi\)
\(128\) 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 \(-0.397552\pi\)
0.316322 + 0.948652i \(0.397552\pi\)
\(138\) 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 \(-0.494758\pi\)
0.0164670 + 0.999864i \(0.494758\pi\)
\(158\) 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 \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.4319 −1.34892 −0.674462 0.738310i \(-0.735624\pi\)
−0.674462 + 0.738310i \(0.735624\pi\)
\(168\) 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 \(-0.610461\pi\)
−0.340101 + 0.940389i \(0.610461\pi\)
\(174\) 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 \(-0.667241\pi\)
−0.501561 + 0.865123i \(0.667241\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.6491 −0.901212 −0.450606 0.892723i \(-0.648792\pi\)
−0.450606 + 0.892723i \(0.648792\pi\)
\(198\) 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 \(-0.716280\pi\)
−0.628377 + 0.777909i \(0.716280\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.86319 −0.389153 −0.194577 0.980887i \(-0.562333\pi\)
−0.194577 + 0.980887i \(0.562333\pi\)
\(228\) 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 \(-0.364014\pi\)
0.414335 + 0.910124i \(0.364014\pi\)
\(234\) 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 \(-0.317028\pi\)
0.543687 + 0.839288i \(0.317028\pi\)
\(258\) 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 \(-0.662973\pi\)
−0.489916 + 0.871770i \(0.662973\pi\)
\(264\) 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 \(-0.267417\pi\)
0.667378 + 0.744719i \(0.267417\pi\)
\(278\) 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 \(-0.477125\pi\)
0.0718012 + 0.997419i \(0.477125\pi\)
\(284\) 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 \(-0.524404\pi\)
−0.0765922 + 0.997062i \(0.524404\pi\)
\(294\) 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 \(-0.450289\pi\)
0.155539 + 0.987830i \(0.450289\pi\)
\(308\) 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 \(-0.404207\pi\)
0.296419 + 0.955058i \(0.404207\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.94665 0.502494 0.251247 0.967923i \(-0.419159\pi\)
0.251247 + 0.967923i \(0.419159\pi\)
\(318\) 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 \(-0.650934\pi\)
−0.456603 + 0.889671i \(0.650934\pi\)
\(338\) 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 \(-0.418739\pi\)
0.252525 + 0.967590i \(0.418739\pi\)
\(348\) 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 \(-0.581932\pi\)
−0.254564 + 0.967056i \(0.581932\pi\)
\(354\) 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 \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 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 \(-0.610230\pi\)
−0.339417 + 0.940636i \(0.610230\pi\)
\(374\) 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 \(-0.119703\pi\)
0.930120 + 0.367257i \(0.119703\pi\)
\(384\) 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 \(-0.595075\pi\)
−0.294265 + 0.955724i \(0.595075\pi\)
\(398\) 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 \(-0.259983\pi\)
0.684585 + 0.728933i \(0.259983\pi\)
\(434\) 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 \(-0.394247\pi\)
0.326153 + 0.945317i \(0.394247\pi\)
\(444\) 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 \(-0.847595\pi\)
−0.887551 + 0.460710i \(0.847595\pi\)
\(458\) 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 \(-0.609821\pi\)
−0.338209 + 0.941071i \(0.609821\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.7000 −1.42063 −0.710314 0.703885i \(-0.751447\pi\)
−0.710314 + 0.703885i \(0.751447\pi\)
\(468\) 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 \(-0.422094\pi\)
0.242312 + 0.970198i \(0.422094\pi\)
\(488\) 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 \(-0.648789\pi\)
−0.450597 + 0.892728i \(0.648789\pi\)
\(504\) 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 \(-0.317281\pi\)
0.543020 + 0.839720i \(0.317281\pi\)
\(524\) 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 \(-0.443364\pi\)
0.176991 + 0.984212i \(0.443364\pi\)
\(548\) 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 \(-0.639676\pi\)
−0.424858 + 0.905260i \(0.639676\pi\)
\(558\) 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 \(-0.718540\pi\)
−0.633882 + 0.773429i \(0.718540\pi\)
\(564\) 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 \(-0.462432\pi\)
0.117749 + 0.993043i \(0.462432\pi\)
\(578\) 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 \(-0.535559\pi\)
−0.111478 + 0.993767i \(0.535559\pi\)
\(588\) 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 \(-0.485873\pi\)
0.0443652 + 0.999015i \(0.485873\pi\)
\(594\) 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 \(-0.650369\pi\)
−0.455022 + 0.890480i \(0.650369\pi\)
\(608\) 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 \(-0.209384\pi\)
0.791340 + 0.611376i \(0.209384\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.08347 −0.124136 −0.0620678 0.998072i \(-0.519770\pi\)
−0.0620678 + 0.998072i \(0.519770\pi\)
\(618\) 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 \(-0.293035\pi\)
0.605346 + 0.795963i \(0.293035\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.40802 0.369867 0.184934 0.982751i \(-0.440793\pi\)
0.184934 + 0.982751i \(0.440793\pi\)
\(648\) 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 \(-0.356867\pi\)
0.434665 + 0.900592i \(0.356867\pi\)
\(654\) 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 \(-0.471587\pi\)
0.0891442 + 0.996019i \(0.471587\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −42.7302 −1.64225 −0.821127 0.570746i \(-0.806654\pi\)
−0.821127 + 0.570746i \(0.806654\pi\)
\(678\) 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 \(-0.622304\pi\)
−0.374846 + 0.927087i \(0.622304\pi\)
\(684\) 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 \(-0.550295\pi\)
−0.157351 + 0.987543i \(0.550295\pi\)
\(728\) 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 \(-0.284355\pi\)
0.626822 + 0.779162i \(0.284355\pi\)
\(734\) 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 \(-0.484684\pi\)
0.0480977 + 0.998843i \(0.484684\pi\)
\(744\) 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 \(-0.251123\pi\)
0.704608 + 0.709597i \(0.251123\pi\)
\(758\) 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 \(-0.533626\pi\)
−0.105442 + 0.994425i \(0.533626\pi\)
\(774\) 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 \(-0.645968\pi\)
−0.442669 + 0.896685i \(0.645968\pi\)
\(788\) 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 \(-0.123050\pi\)
0.926206 + 0.377018i \(0.123050\pi\)
\(798\) 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 \(-0.284500\pi\)
0.626469 + 0.779447i \(0.284500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.86629 0.273538 0.136769 0.990603i \(-0.456328\pi\)
0.136769 + 0.990603i \(0.456328\pi\)
\(828\) 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 \(-0.344222\pi\)
0.470088 + 0.882619i \(0.344222\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −31.3190 −1.06984 −0.534919 0.844903i \(-0.679658\pi\)
−0.534919 + 0.844903i \(0.679658\pi\)
\(858\) 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 \(-0.542746\pi\)
−0.133886 + 0.990997i \(0.542746\pi\)
\(864\) 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 \(-0.575604\pi\)
−0.235289 + 0.971925i \(0.575604\pi\)
\(878\) 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 \(-0.150451\pi\)
0.890362 + 0.455253i \(0.150451\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.0547 1.64710 0.823548 0.567247i \(-0.191991\pi\)
0.823548 + 0.567247i \(0.191991\pi\)
\(888\) 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 \(-0.278815\pi\)
0.640287 + 0.768136i \(0.278815\pi\)
\(908\) 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 \(-0.250721\pi\)
0.705503 + 0.708707i \(0.250721\pi\)
\(938\) 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 \(-0.217980\pi\)
0.774543 + 0.632521i \(0.217980\pi\)
\(948\) 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 \(-0.298260\pi\)
0.592199 + 0.805791i \(0.298260\pi\)
\(954\) 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 \(-0.487633\pi\)
0.0388429 + 0.999245i \(0.487633\pi\)
\(968\) 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 \(-0.630928\pi\)
−0.399822 + 0.916593i \(0.630928\pi\)
\(978\) 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 \(-0.339826\pi\)
0.482232 + 0.876044i \(0.339826\pi\)
\(984\) 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 \(-0.440548\pi\)
0.185689 + 0.982609i \(0.440548\pi\)
\(998\) 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.1.2 4
4.3 odd 2 7600.2.a.ce.1.3 4
5.2 odd 4 380.2.c.a.229.3 yes 4
5.3 odd 4 380.2.c.a.229.2 4
5.4 even 2 inner 1900.2.a.j.1.3 4
15.2 even 4 3420.2.f.a.1369.1 4
15.8 even 4 3420.2.f.a.1369.2 4
20.3 even 4 1520.2.d.f.609.3 4
20.7 even 4 1520.2.d.f.609.2 4
20.19 odd 2 7600.2.a.ce.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.2 4 5.3 odd 4
380.2.c.a.229.3 yes 4 5.2 odd 4
1520.2.d.f.609.2 4 20.7 even 4
1520.2.d.f.609.3 4 20.3 even 4
1900.2.a.j.1.2 4 1.1 even 1 trivial
1900.2.a.j.1.3 4 5.4 even 2 inner
3420.2.f.a.1369.1 4 15.2 even 4
3420.2.f.a.1369.2 4 15.8 even 4
7600.2.a.ce.1.2 4 20.19 odd 2
7600.2.a.ce.1.3 4 4.3 odd 2