Properties

Label 1900.2.a.j.1.4
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: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
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 1.4
Root \(2.28825\) of defining polynomial
Character \(\chi\) \(=\) 1900.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} +O(q^{10})\) \(q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} -5.23607 q^{11} -4.03631 q^{13} +1.08036 q^{17} -1.00000 q^{19} -6.47214 q^{21} +7.40492 q^{23} -1.74806 q^{27} +4.47214 q^{29} -4.00000 q^{31} -11.9814 q^{33} -6.86474 q^{37} -9.23607 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} +2.47214 q^{51} +6.86474 q^{53} -2.28825 q^{57} -10.4721 q^{59} +1.70820 q^{61} -6.32456 q^{63} -1.62054 q^{67} +16.9443 q^{69} -1.52786 q^{71} +13.7295 q^{73} +14.8098 q^{77} -11.4164 q^{79} -10.7082 q^{81} +5.24419 q^{83} +10.2333 q^{87} -14.9443 q^{89} +11.4164 q^{91} -9.15298 q^{93} -4.44897 q^{97} -11.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{11} - 4 q^{19} - 8 q^{21} - 16 q^{31} - 28 q^{39} - 24 q^{41} + 4 q^{49} - 8 q^{51} - 24 q^{59} - 20 q^{61} + 32 q^{69} - 24 q^{71} + 8 q^{79} - 16 q^{81} - 24 q^{89} - 8 q^{91} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.28825 1.32112 0.660560 0.750774i \(-0.270319\pi\)
0.660560 + 0.750774i \(0.270319\pi\)
\(4\) 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
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 0 0
\(13\) −4.03631 −1.11947 −0.559735 0.828671i \(-0.689097\pi\)
−0.559735 + 0.828671i \(0.689097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.08036 0.262027 0.131013 0.991381i \(-0.458177\pi\)
0.131013 + 0.991381i \(0.458177\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −6.47214 −1.41234
\(22\) 0 0
\(23\) 7.40492 1.54403 0.772016 0.635603i \(-0.219248\pi\)
0.772016 + 0.635603i \(0.219248\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.74806 −0.336415
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −11.9814 −2.08570
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.86474 −1.12856 −0.564278 0.825585i \(-0.690845\pi\)
−0.564278 + 0.825585i \(0.690845\pi\)
\(38\) 0 0
\(39\) −9.23607 −1.47895
\(40\) 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
\(51\) 2.47214 0.346168
\(52\) 0 0
\(53\) 6.86474 0.942944 0.471472 0.881881i \(-0.343723\pi\)
0.471472 + 0.881881i \(0.343723\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.28825 −0.303086
\(58\) 0 0
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 0 0
\(61\) 1.70820 0.218713 0.109357 0.994003i \(-0.465121\pi\)
0.109357 + 0.994003i \(0.465121\pi\)
\(62\) 0 0
\(63\) −6.32456 −0.796819
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.62054 −0.197981 −0.0989905 0.995088i \(-0.531561\pi\)
−0.0989905 + 0.995088i \(0.531561\pi\)
\(68\) 0 0
\(69\) 16.9443 2.03985
\(70\) 0 0
\(71\) −1.52786 −0.181324 −0.0906621 0.995882i \(-0.528898\pi\)
−0.0906621 + 0.995882i \(0.528898\pi\)
\(72\) 0 0
\(73\) 13.7295 1.60691 0.803457 0.595363i \(-0.202992\pi\)
0.803457 + 0.595363i \(0.202992\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.8098 1.68774
\(78\) 0 0
\(79\) −11.4164 −1.28445 −0.642223 0.766518i \(-0.721988\pi\)
−0.642223 + 0.766518i \(0.721988\pi\)
\(80\) 0 0
\(81\) −10.7082 −1.18980
\(82\) 0 0
\(83\) 5.24419 0.575625 0.287812 0.957687i \(-0.407072\pi\)
0.287812 + 0.957687i \(0.407072\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.2333 1.09713
\(88\) 0 0
\(89\) −14.9443 −1.58409 −0.792045 0.610463i \(-0.790983\pi\)
−0.792045 + 0.610463i \(0.790983\pi\)
\(90\) 0 0
\(91\) 11.4164 1.19676
\(92\) 0 0
\(93\) −9.15298 −0.949120
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.44897 −0.451725 −0.225862 0.974159i \(-0.572520\pi\)
−0.225862 + 0.974159i \(0.572520\pi\)
\(98\) 0 0
\(99\) −11.7082 −1.17672
\(100\) 0 0
\(101\) 0.763932 0.0760141 0.0380070 0.999277i \(-0.487899\pi\)
0.0380070 + 0.999277i \(0.487899\pi\)
\(102\) 0 0
\(103\) 15.3500 1.51248 0.756241 0.654293i \(-0.227034\pi\)
0.756241 + 0.654293i \(0.227034\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4304 1.58838 0.794192 0.607666i \(-0.207894\pi\)
0.794192 + 0.607666i \(0.207894\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −15.7082 −1.49096
\(112\) 0 0
\(113\) −12.1089 −1.13911 −0.569556 0.821952i \(-0.692885\pi\)
−0.569556 + 0.821952i \(0.692885\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.02546 −0.834404
\(118\) 0 0
\(119\) −3.05573 −0.280118
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) −13.7295 −1.23794
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.1058 0.896747 0.448374 0.893846i \(-0.352003\pi\)
0.448374 + 0.893846i \(0.352003\pi\)
\(128\) 0 0
\(129\) −19.4164 −1.70952
\(130\) 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
\(137\) 1.08036 0.0923016 0.0461508 0.998934i \(-0.485305\pi\)
0.0461508 + 0.998934i \(0.485305\pi\)
\(138\) 0 0
\(139\) 15.7082 1.33235 0.666176 0.745794i \(-0.267930\pi\)
0.666176 + 0.745794i \(0.267930\pi\)
\(140\) 0 0
\(141\) −19.4164 −1.63516
\(142\) 0 0
\(143\) 21.1344 1.76735
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.28825 0.188731
\(148\) 0 0
\(149\) 18.6525 1.52807 0.764035 0.645175i \(-0.223215\pi\)
0.764035 + 0.645175i \(0.223215\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 2.41577 0.195303
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.3863 1.54720 0.773599 0.633676i \(-0.218455\pi\)
0.773599 + 0.633676i \(0.218455\pi\)
\(158\) 0 0
\(159\) 15.7082 1.24574
\(160\) 0 0
\(161\) −20.9443 −1.65064
\(162\) 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
\(167\) 4.70401 0.364007 0.182004 0.983298i \(-0.441742\pi\)
0.182004 + 0.983298i \(0.441742\pi\)
\(168\) 0 0
\(169\) 3.29180 0.253215
\(170\) 0 0
\(171\) −2.23607 −0.170996
\(172\) 0 0
\(173\) 13.1893 1.00276 0.501382 0.865226i \(-0.332825\pi\)
0.501382 + 0.865226i \(0.332825\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −23.9628 −1.80116
\(178\) 0 0
\(179\) −13.5279 −1.01112 −0.505560 0.862791i \(-0.668714\pi\)
−0.505560 + 0.862791i \(0.668714\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 3.90879 0.288946
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.65685 −0.413670
\(188\) 0 0
\(189\) 4.94427 0.359643
\(190\) 0 0
\(191\) −20.9443 −1.51547 −0.757737 0.652560i \(-0.773695\pi\)
−0.757737 + 0.652560i \(0.773695\pi\)
\(192\) 0 0
\(193\) −4.44897 −0.320244 −0.160122 0.987097i \(-0.551189\pi\)
−0.160122 + 0.987097i \(0.551189\pi\)
\(194\) 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
\(201\) −3.70820 −0.261557
\(202\) 0 0
\(203\) −12.6491 −0.887794
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 16.5579 1.15085
\(208\) 0 0
\(209\) 5.23607 0.362186
\(210\) 0 0
\(211\) −11.4164 −0.785938 −0.392969 0.919552i \(-0.628552\pi\)
−0.392969 + 0.919552i \(0.628552\pi\)
\(212\) 0 0
\(213\) −3.49613 −0.239551
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 11.3137 0.768025
\(218\) 0 0
\(219\) 31.4164 2.12292
\(220\) 0 0
\(221\) −4.36068 −0.293331
\(222\) 0 0
\(223\) 28.6668 1.91967 0.959836 0.280560i \(-0.0905202\pi\)
0.959836 + 0.280560i \(0.0905202\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.3500 −1.01882 −0.509408 0.860525i \(-0.670136\pi\)
−0.509408 + 0.860525i \(0.670136\pi\)
\(228\) 0 0
\(229\) −5.70820 −0.377209 −0.188604 0.982053i \(-0.560396\pi\)
−0.188604 + 0.982053i \(0.560396\pi\)
\(230\) 0 0
\(231\) 33.8885 2.22970
\(232\) 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
\(237\) −26.1235 −1.69691
\(238\) 0 0
\(239\) 5.88854 0.380898 0.190449 0.981697i \(-0.439006\pi\)
0.190449 + 0.981697i \(0.439006\pi\)
\(240\) 0 0
\(241\) 24.8328 1.59962 0.799811 0.600252i \(-0.204933\pi\)
0.799811 + 0.600252i \(0.204933\pi\)
\(242\) 0 0
\(243\) −19.2588 −1.23545
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.03631 0.256824
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 29.8885 1.88655 0.943274 0.332015i \(-0.107728\pi\)
0.943274 + 0.332015i \(0.107728\pi\)
\(252\) 0 0
\(253\) −38.7727 −2.43762
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.70401 −0.293428 −0.146714 0.989179i \(-0.546870\pi\)
−0.146714 + 0.989179i \(0.546870\pi\)
\(258\) 0 0
\(259\) 19.4164 1.20648
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) −9.56564 −0.589843 −0.294921 0.955522i \(-0.595293\pi\)
−0.294921 + 0.955522i \(0.595293\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −34.1962 −2.09277
\(268\) 0 0
\(269\) −2.94427 −0.179515 −0.0897577 0.995964i \(-0.528609\pi\)
−0.0897577 + 0.995964i \(0.528609\pi\)
\(270\) 0 0
\(271\) −19.1246 −1.16174 −0.580869 0.813997i \(-0.697287\pi\)
−0.580869 + 0.813997i \(0.697287\pi\)
\(272\) 0 0
\(273\) 26.1235 1.58107
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.24109 0.194738 0.0973691 0.995248i \(-0.468957\pi\)
0.0973691 + 0.995248i \(0.468957\pi\)
\(278\) 0 0
\(279\) −8.94427 −0.535480
\(280\) 0 0
\(281\) 1.41641 0.0844958 0.0422479 0.999107i \(-0.486548\pi\)
0.0422479 + 0.999107i \(0.486548\pi\)
\(282\) 0 0
\(283\) −16.5579 −0.984265 −0.492133 0.870520i \(-0.663782\pi\)
−0.492133 + 0.870520i \(0.663782\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.9706 1.00174
\(288\) 0 0
\(289\) −15.8328 −0.931342
\(290\) 0 0
\(291\) −10.1803 −0.596782
\(292\) 0 0
\(293\) 6.86474 0.401042 0.200521 0.979689i \(-0.435736\pi\)
0.200521 + 0.979689i \(0.435736\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.15298 0.531110
\(298\) 0 0
\(299\) −29.8885 −1.72850
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 1.74806 0.100424
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.03631 −0.230364 −0.115182 0.993344i \(-0.536745\pi\)
−0.115182 + 0.993344i \(0.536745\pi\)
\(308\) 0 0
\(309\) 35.1246 1.99817
\(310\) 0 0
\(311\) 3.70820 0.210273 0.105136 0.994458i \(-0.466472\pi\)
0.105136 + 0.994458i \(0.466472\pi\)
\(312\) 0 0
\(313\) −27.4589 −1.55207 −0.776036 0.630689i \(-0.782772\pi\)
−0.776036 + 0.630689i \(0.782772\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.1893 −0.740784 −0.370392 0.928875i \(-0.620777\pi\)
−0.370392 + 0.928875i \(0.620777\pi\)
\(318\) 0 0
\(319\) −23.4164 −1.31107
\(320\) 0 0
\(321\) 37.5967 2.09845
\(322\) 0 0
\(323\) −1.08036 −0.0601130
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.57649 0.253081
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −19.4164 −1.06722 −0.533611 0.845730i \(-0.679165\pi\)
−0.533611 + 0.845730i \(0.679165\pi\)
\(332\) 0 0
\(333\) −15.3500 −0.841176
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.27740 −0.396425 −0.198213 0.980159i \(-0.563514\pi\)
−0.198213 + 0.980159i \(0.563514\pi\)
\(338\) 0 0
\(339\) −27.7082 −1.50490
\(340\) 0 0
\(341\) 20.9443 1.13420
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −34.8639 −1.87159 −0.935795 0.352544i \(-0.885317\pi\)
−0.935795 + 0.352544i \(0.885317\pi\)
\(348\) 0 0
\(349\) −1.41641 −0.0758186 −0.0379093 0.999281i \(-0.512070\pi\)
−0.0379093 + 0.999281i \(0.512070\pi\)
\(350\) 0 0
\(351\) 7.05573 0.376607
\(352\) 0 0
\(353\) −15.8902 −0.845750 −0.422875 0.906188i \(-0.638979\pi\)
−0.422875 + 0.906188i \(0.638979\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.99226 −0.370069
\(358\) 0 0
\(359\) −11.1246 −0.587135 −0.293567 0.955938i \(-0.594842\pi\)
−0.293567 + 0.955938i \(0.594842\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 37.5648 1.97164
\(364\) 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
\(371\) −19.4164 −1.00805
\(372\) 0 0
\(373\) 34.3237 1.77721 0.888606 0.458670i \(-0.151674\pi\)
0.888606 + 0.458670i \(0.151674\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −18.0509 −0.929670
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 23.1246 1.18471
\(382\) 0 0
\(383\) −23.6777 −1.20987 −0.604936 0.796274i \(-0.706801\pi\)
−0.604936 + 0.796274i \(0.706801\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −18.9737 −0.964486
\(388\) 0 0
\(389\) 2.94427 0.149281 0.0746403 0.997211i \(-0.476219\pi\)
0.0746403 + 0.997211i \(0.476219\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −30.7000 −1.54079 −0.770395 0.637566i \(-0.779941\pi\)
−0.770395 + 0.637566i \(0.779941\pi\)
\(398\) 0 0
\(399\) 6.47214 0.324012
\(400\) 0 0
\(401\) 4.47214 0.223328 0.111664 0.993746i \(-0.464382\pi\)
0.111664 + 0.993746i \(0.464382\pi\)
\(402\) 0 0
\(403\) 16.1452 0.804252
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 35.9442 1.78169
\(408\) 0 0
\(409\) −9.41641 −0.465611 −0.232806 0.972523i \(-0.574791\pi\)
−0.232806 + 0.972523i \(0.574791\pi\)
\(410\) 0 0
\(411\) 2.47214 0.121941
\(412\) 0 0
\(413\) 29.6197 1.45749
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 35.9442 1.76020
\(418\) 0 0
\(419\) −3.05573 −0.149282 −0.0746410 0.997210i \(-0.523781\pi\)
−0.0746410 + 0.997210i \(0.523781\pi\)
\(420\) 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
\(427\) −4.83153 −0.233814
\(428\) 0 0
\(429\) 48.3607 2.33488
\(430\) 0 0
\(431\) 10.4721 0.504425 0.252213 0.967672i \(-0.418842\pi\)
0.252213 + 0.967672i \(0.418842\pi\)
\(432\) 0 0
\(433\) 37.9774 1.82508 0.912540 0.408989i \(-0.134118\pi\)
0.912540 + 0.408989i \(0.134118\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.40492 −0.354225
\(438\) 0 0
\(439\) 34.8328 1.66248 0.831240 0.555914i \(-0.187632\pi\)
0.831240 + 0.555914i \(0.187632\pi\)
\(440\) 0 0
\(441\) 2.23607 0.106479
\(442\) 0 0
\(443\) −5.24419 −0.249159 −0.124580 0.992210i \(-0.539758\pi\)
−0.124580 + 0.992210i \(0.539758\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 42.6814 2.01876
\(448\) 0 0
\(449\) −7.52786 −0.355262 −0.177631 0.984097i \(-0.556843\pi\)
−0.177631 + 0.984097i \(0.556843\pi\)
\(450\) 0 0
\(451\) 31.4164 1.47934
\(452\) 0 0
\(453\) −18.3060 −0.860089
\(454\) 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
\(459\) −1.88854 −0.0881497
\(460\) 0 0
\(461\) −11.8885 −0.553705 −0.276852 0.960912i \(-0.589291\pi\)
−0.276852 + 0.960912i \(0.589291\pi\)
\(462\) 0 0
\(463\) −33.5285 −1.55820 −0.779100 0.626900i \(-0.784324\pi\)
−0.779100 + 0.626900i \(0.784324\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.7264 −0.542632 −0.271316 0.962490i \(-0.587459\pi\)
−0.271316 + 0.962490i \(0.587459\pi\)
\(468\) 0 0
\(469\) 4.58359 0.211651
\(470\) 0 0
\(471\) 44.3607 2.04403
\(472\) 0 0
\(473\) 44.4295 2.04287
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 15.3500 0.702829
\(478\) 0 0
\(479\) 6.76393 0.309052 0.154526 0.987989i \(-0.450615\pi\)
0.154526 + 0.987989i \(0.450615\pi\)
\(480\) 0 0
\(481\) 27.7082 1.26339
\(482\) 0 0
\(483\) −47.9256 −2.18069
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −17.7658 −0.805044 −0.402522 0.915410i \(-0.631866\pi\)
−0.402522 + 0.915410i \(0.631866\pi\)
\(488\) 0 0
\(489\) 19.4164 0.878040
\(490\) 0 0
\(491\) −29.8885 −1.34885 −0.674426 0.738343i \(-0.735609\pi\)
−0.674426 + 0.738343i \(0.735609\pi\)
\(492\) 0 0
\(493\) 4.83153 0.217601
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.32145 0.193844
\(498\) 0 0
\(499\) 23.1246 1.03520 0.517600 0.855623i \(-0.326826\pi\)
0.517600 + 0.855623i \(0.326826\pi\)
\(500\) 0 0
\(501\) 10.7639 0.480897
\(502\) 0 0
\(503\) −39.1853 −1.74719 −0.873593 0.486656i \(-0.838216\pi\)
−0.873593 + 0.486656i \(0.838216\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.53244 0.334527
\(508\) 0 0
\(509\) −37.4164 −1.65845 −0.829227 0.558913i \(-0.811219\pi\)
−0.829227 + 0.558913i \(0.811219\pi\)
\(510\) 0 0
\(511\) −38.8328 −1.71786
\(512\) 0 0
\(513\) 1.74806 0.0771789
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 44.4295 1.95401
\(518\) 0 0
\(519\) 30.1803 1.32477
\(520\) 0 0
\(521\) 35.8885 1.57231 0.786153 0.618032i \(-0.212070\pi\)
0.786153 + 0.618032i \(0.212070\pi\)
\(522\) 0 0
\(523\) −3.62365 −0.158451 −0.0792255 0.996857i \(-0.525245\pi\)
−0.0792255 + 0.996857i \(0.525245\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.32145 −0.188245
\(528\) 0 0
\(529\) 31.8328 1.38404
\(530\) 0 0
\(531\) −23.4164 −1.01619
\(532\) 0 0
\(533\) 24.2179 1.04899
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −30.9551 −1.33581
\(538\) 0 0
\(539\) −5.23607 −0.225533
\(540\) 0 0
\(541\) −1.70820 −0.0734414 −0.0367207 0.999326i \(-0.511691\pi\)
−0.0367207 + 0.999326i \(0.511691\pi\)
\(542\) 0 0
\(543\) 13.7295 0.589188
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.20788 −0.0516453 −0.0258227 0.999667i \(-0.508221\pi\)
−0.0258227 + 0.999667i \(0.508221\pi\)
\(548\) 0 0
\(549\) 3.81966 0.163019
\(550\) 0 0
\(551\) −4.47214 −0.190519
\(552\) 0 0
\(553\) 32.2905 1.37313
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.5687 0.490184 0.245092 0.969500i \(-0.421182\pi\)
0.245092 + 0.969500i \(0.421182\pi\)
\(558\) 0 0
\(559\) 34.2492 1.44859
\(560\) 0 0
\(561\) −12.9443 −0.546508
\(562\) 0 0
\(563\) 17.3531 0.731347 0.365673 0.930743i \(-0.380839\pi\)
0.365673 + 0.930743i \(0.380839\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 30.2874 1.27195
\(568\) 0 0
\(569\) −21.0557 −0.882702 −0.441351 0.897335i \(-0.645501\pi\)
−0.441351 + 0.897335i \(0.645501\pi\)
\(570\) 0 0
\(571\) −15.1246 −0.632945 −0.316473 0.948602i \(-0.602499\pi\)
−0.316473 + 0.948602i \(0.602499\pi\)
\(572\) 0 0
\(573\) −47.9256 −2.00212
\(574\) 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
\(579\) −10.1803 −0.423080
\(580\) 0 0
\(581\) −14.8328 −0.615369
\(582\) 0 0
\(583\) −35.9442 −1.48866
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −37.0246 −1.52817 −0.764084 0.645117i \(-0.776809\pi\)
−0.764084 + 0.645117i \(0.776809\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 28.9443 1.19061
\(592\) 0 0
\(593\) 14.8098 0.608167 0.304084 0.952645i \(-0.401650\pi\)
0.304084 + 0.952645i \(0.401650\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −36.6119 −1.49843
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −36.8328 −1.50244 −0.751221 0.660051i \(-0.770535\pi\)
−0.751221 + 0.660051i \(0.770535\pi\)
\(602\) 0 0
\(603\) −3.62365 −0.147566
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −12.9343 −0.524985 −0.262493 0.964934i \(-0.584545\pi\)
−0.262493 + 0.964934i \(0.584545\pi\)
\(608\) 0 0
\(609\) −28.9443 −1.17288
\(610\) 0 0
\(611\) 34.2492 1.38558
\(612\) 0 0
\(613\) 20.2117 0.816341 0.408170 0.912906i \(-0.366167\pi\)
0.408170 + 0.912906i \(0.366167\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.5393 1.14895 0.574475 0.818522i \(-0.305206\pi\)
0.574475 + 0.818522i \(0.305206\pi\)
\(618\) 0 0
\(619\) 0.875388 0.0351848 0.0175924 0.999845i \(-0.494400\pi\)
0.0175924 + 0.999845i \(0.494400\pi\)
\(620\) 0 0
\(621\) −12.9443 −0.519436
\(622\) 0 0
\(623\) 42.2688 1.69346
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 11.9814 0.478491
\(628\) 0 0
\(629\) −7.41641 −0.295712
\(630\) 0 0
\(631\) 11.7082 0.466096 0.233048 0.972465i \(-0.425130\pi\)
0.233048 + 0.972465i \(0.425130\pi\)
\(632\) 0 0
\(633\) −26.1235 −1.03832
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.03631 −0.159924
\(638\) 0 0
\(639\) −3.41641 −0.135151
\(640\) 0 0
\(641\) −47.8885 −1.89148 −0.945742 0.324919i \(-0.894663\pi\)
−0.945742 + 0.324919i \(0.894663\pi\)
\(642\) 0 0
\(643\) 11.7264 0.462443 0.231221 0.972901i \(-0.425728\pi\)
0.231221 + 0.972901i \(0.425728\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −34.8639 −1.37064 −0.685320 0.728242i \(-0.740338\pi\)
−0.685320 + 0.728242i \(0.740338\pi\)
\(648\) 0 0
\(649\) 54.8328 2.15238
\(650\) 0 0
\(651\) 25.8885 1.01465
\(652\) 0 0
\(653\) 3.24109 0.126834 0.0634168 0.997987i \(-0.479800\pi\)
0.0634168 + 0.997987i \(0.479800\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 30.7000 1.19772
\(658\) 0 0
\(659\) −27.0557 −1.05394 −0.526971 0.849883i \(-0.676672\pi\)
−0.526971 + 0.849883i \(0.676672\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) −9.97831 −0.387525
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 33.1158 1.28225
\(668\) 0 0
\(669\) 65.5967 2.53612
\(670\) 0 0
\(671\) −8.94427 −0.345290
\(672\) 0 0
\(673\) −42.8090 −1.65016 −0.825082 0.565013i \(-0.808871\pi\)
−0.825082 + 0.565013i \(0.808871\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.0022 1.15308 0.576540 0.817069i \(-0.304403\pi\)
0.576540 + 0.817069i \(0.304403\pi\)
\(678\) 0 0
\(679\) 12.5836 0.482914
\(680\) 0 0
\(681\) −35.1246 −1.34598
\(682\) 0 0
\(683\) −10.1058 −0.386689 −0.193344 0.981131i \(-0.561933\pi\)
−0.193344 + 0.981131i \(0.561933\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13.0618 −0.498338
\(688\) 0 0
\(689\) −27.7082 −1.05560
\(690\) 0 0
\(691\) −15.1246 −0.575367 −0.287684 0.957725i \(-0.592885\pi\)
−0.287684 + 0.957725i \(0.592885\pi\)
\(692\) 0 0
\(693\) 33.1158 1.25797
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6.48218 −0.245530
\(698\) 0 0
\(699\) −28.9443 −1.09477
\(700\) 0 0
\(701\) 41.1246 1.55326 0.776628 0.629960i \(-0.216929\pi\)
0.776628 + 0.629960i \(0.216929\pi\)
\(702\) 0 0
\(703\) 6.86474 0.258908
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.16073 −0.0812625
\(708\) 0 0
\(709\) 6.58359 0.247252 0.123626 0.992329i \(-0.460548\pi\)
0.123626 + 0.992329i \(0.460548\pi\)
\(710\) 0 0
\(711\) −25.5279 −0.957370
\(712\) 0 0
\(713\) −29.6197 −1.10927
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13.4744 0.503212
\(718\) 0 0
\(719\) −21.5967 −0.805423 −0.402711 0.915327i \(-0.631932\pi\)
−0.402711 + 0.915327i \(0.631932\pi\)
\(720\) 0 0
\(721\) −43.4164 −1.61691
\(722\) 0 0
\(723\) 56.8236 2.11329
\(724\) 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
\(729\) −11.9443 −0.442380
\(730\) 0 0
\(731\) −9.16718 −0.339061
\(732\) 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
\(739\) 34.8328 1.28135 0.640673 0.767814i \(-0.278655\pi\)
0.640673 + 0.767814i \(0.278655\pi\)
\(740\) 0 0
\(741\) 9.23607 0.339295
\(742\) 0 0
\(743\) −6.86474 −0.251843 −0.125921 0.992040i \(-0.540189\pi\)
−0.125921 + 0.992040i \(0.540189\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.7264 0.429045
\(748\) 0 0
\(749\) −46.4721 −1.69805
\(750\) 0 0
\(751\) 47.4164 1.73025 0.865125 0.501557i \(-0.167239\pi\)
0.865125 + 0.501557i \(0.167239\pi\)
\(752\) 0 0
\(753\) 68.3923 2.49236
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.825324 0.0299969 0.0149985 0.999888i \(-0.495226\pi\)
0.0149985 + 0.999888i \(0.495226\pi\)
\(758\) 0 0
\(759\) −88.7214 −3.22038
\(760\) 0 0
\(761\) 24.5410 0.889611 0.444806 0.895627i \(-0.353273\pi\)
0.444806 + 0.895627i \(0.353273\pi\)
\(762\) 0 0
\(763\) −5.65685 −0.204792
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 42.2688 1.52624
\(768\) 0 0
\(769\) 28.5410 1.02922 0.514608 0.857426i \(-0.327938\pi\)
0.514608 + 0.857426i \(0.327938\pi\)
\(770\) 0 0
\(771\) −10.7639 −0.387654
\(772\) 0 0
\(773\) −15.3500 −0.552102 −0.276051 0.961143i \(-0.589026\pi\)
−0.276051 + 0.961143i \(0.589026\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 44.4295 1.59390
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) −7.81758 −0.279378
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.62365 0.129169 0.0645845 0.997912i \(-0.479428\pi\)
0.0645845 + 0.997912i \(0.479428\pi\)
\(788\) 0 0
\(789\) −21.8885 −0.779253
\(790\) 0 0
\(791\) 34.2492 1.21776
\(792\) 0 0
\(793\) −6.89484 −0.244843
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −14.1120 −0.499874 −0.249937 0.968262i \(-0.580410\pi\)
−0.249937 + 0.968262i \(0.580410\pi\)
\(798\) 0 0
\(799\) −9.16718 −0.324312
\(800\) 0 0
\(801\) −33.4164 −1.18071
\(802\) 0 0
\(803\) −71.8885 −2.53689
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.73722 −0.237161
\(808\) 0 0
\(809\) −16.4721 −0.579129 −0.289565 0.957158i \(-0.593511\pi\)
−0.289565 + 0.957158i \(0.593511\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) −43.7618 −1.53479
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.48528 0.296862
\(818\) 0 0
\(819\) 25.5279 0.892016
\(820\) 0 0
\(821\) 21.0557 0.734850 0.367425 0.930053i \(-0.380239\pi\)
0.367425 + 0.930053i \(0.380239\pi\)
\(822\) 0 0
\(823\) −2.00310 −0.0698238 −0.0349119 0.999390i \(-0.511115\pi\)
−0.0349119 + 0.999390i \(0.511115\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.5942 −0.716131 −0.358065 0.933696i \(-0.616564\pi\)
−0.358065 + 0.933696i \(0.616564\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) 7.41641 0.257272
\(832\) 0 0
\(833\) 1.08036 0.0374324
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.99226 0.241688
\(838\) 0 0
\(839\) −13.3050 −0.459338 −0.229669 0.973269i \(-0.573764\pi\)
−0.229669 + 0.973269i \(0.573764\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 3.24109 0.111629
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −46.4326 −1.59544
\(848\) 0 0
\(849\) −37.8885 −1.30033
\(850\) 0 0
\(851\) −50.8328 −1.74253
\(852\) 0 0
\(853\) −10.4884 −0.359115 −0.179558 0.983747i \(-0.557467\pi\)
−0.179558 + 0.983747i \(0.557467\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.8059 −1.39390 −0.696951 0.717119i \(-0.745460\pi\)
−0.696951 + 0.717119i \(0.745460\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 38.8328 1.32342
\(862\) 0 0
\(863\) 20.5942 0.701035 0.350518 0.936556i \(-0.386006\pi\)
0.350518 + 0.936556i \(0.386006\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −36.2294 −1.23041
\(868\) 0 0
\(869\) 59.7771 2.02780
\(870\) 0 0
\(871\) 6.54102 0.221634
\(872\) 0 0
\(873\) −9.94820 −0.336696
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.44897 −0.150231 −0.0751155 0.997175i \(-0.523933\pi\)
−0.0751155 + 0.997175i \(0.523933\pi\)
\(878\) 0 0
\(879\) 15.7082 0.529825
\(880\) 0 0
\(881\) −30.6525 −1.03271 −0.516354 0.856375i \(-0.672711\pi\)
−0.516354 + 0.856375i \(0.672711\pi\)
\(882\) 0 0
\(883\) 14.9675 0.503695 0.251848 0.967767i \(-0.418962\pi\)
0.251848 + 0.967767i \(0.418962\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.3268 −1.21973 −0.609867 0.792504i \(-0.708777\pi\)
−0.609867 + 0.792504i \(0.708777\pi\)
\(888\) 0 0
\(889\) −28.5836 −0.958663
\(890\) 0 0
\(891\) 56.0689 1.87838
\(892\) 0 0
\(893\) 8.48528 0.283949
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −68.3923 −2.28355
\(898\) 0 0
\(899\) −17.8885 −0.596616
\(900\) 0 0
\(901\) 7.41641 0.247076
\(902\) 0 0
\(903\) 54.9179 1.82755
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8.86784 −0.294452 −0.147226 0.989103i \(-0.547034\pi\)
−0.147226 + 0.989103i \(0.547034\pi\)
\(908\) 0 0
\(909\) 1.70820 0.0566575
\(910\) 0 0
\(911\) −37.5279 −1.24335 −0.621677 0.783274i \(-0.713548\pi\)
−0.621677 + 0.783274i \(0.713548\pi\)
\(912\) 0 0
\(913\) −27.4589 −0.908759
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −26.8328 −0.885133 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(920\) 0 0
\(921\) −9.23607 −0.304339
\(922\) 0 0
\(923\) 6.16693 0.202987
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 34.3237 1.12734
\(928\) 0 0
\(929\) 16.4721 0.540433 0.270217 0.962800i \(-0.412905\pi\)
0.270217 + 0.962800i \(0.412905\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) 8.48528 0.277796
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −51.6768 −1.68821 −0.844104 0.536180i \(-0.819867\pi\)
−0.844104 + 0.536180i \(0.819867\pi\)
\(938\) 0 0
\(939\) −62.8328 −2.05047
\(940\) 0 0
\(941\) −21.0557 −0.686397 −0.343199 0.939263i \(-0.611510\pi\)
−0.343199 + 0.939263i \(0.611510\pi\)
\(942\) 0 0
\(943\) −44.4295 −1.44682
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.6969 0.932525 0.466263 0.884646i \(-0.345600\pi\)
0.466263 + 0.884646i \(0.345600\pi\)
\(948\) 0 0
\(949\) −55.4164 −1.79889
\(950\) 0 0
\(951\) −30.1803 −0.978665
\(952\) 0 0
\(953\) 27.0764 0.877090 0.438545 0.898709i \(-0.355494\pi\)
0.438545 + 0.898709i \(0.355494\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −53.5825 −1.73208
\(958\) 0 0
\(959\) −3.05573 −0.0986746
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 36.7394 1.18391
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −16.5579 −0.532466 −0.266233 0.963909i \(-0.585779\pi\)
−0.266233 + 0.963909i \(0.585779\pi\)
\(968\) 0 0
\(969\) −2.47214 −0.0794164
\(970\) 0 0
\(971\) 4.36068 0.139941 0.0699704 0.997549i \(-0.477710\pi\)
0.0699704 + 0.997549i \(0.477710\pi\)
\(972\) 0 0
\(973\) −44.4295 −1.42434
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −47.1304 −1.50784 −0.753918 0.656969i \(-0.771838\pi\)
−0.753918 + 0.656969i \(0.771838\pi\)
\(978\) 0 0
\(979\) 78.2492 2.50086
\(980\) 0 0
\(981\) 4.47214 0.142784
\(982\) 0 0
\(983\) 33.4009 1.06532 0.532662 0.846328i \(-0.321192\pi\)
0.532662 + 0.846328i \(0.321192\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 54.9179 1.74806
\(988\) 0 0
\(989\) −62.8328 −1.99797
\(990\) 0 0
\(991\) −14.8328 −0.471180 −0.235590 0.971853i \(-0.575702\pi\)
−0.235590 + 0.971853i \(0.575702\pi\)
\(992\) 0 0
\(993\) −44.4295 −1.40993
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 30.7000 0.972280 0.486140 0.873881i \(-0.338405\pi\)
0.486140 + 0.873881i \(0.338405\pi\)
\(998\) 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.1.4 4
4.3 odd 2 7600.2.a.ce.1.1 4
5.2 odd 4 380.2.c.a.229.1 4
5.3 odd 4 380.2.c.a.229.4 yes 4
5.4 even 2 inner 1900.2.a.j.1.1 4
15.2 even 4 3420.2.f.a.1369.3 4
15.8 even 4 3420.2.f.a.1369.4 4
20.3 even 4 1520.2.d.f.609.1 4
20.7 even 4 1520.2.d.f.609.4 4
20.19 odd 2 7600.2.a.ce.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.1 4 5.2 odd 4
380.2.c.a.229.4 yes 4 5.3 odd 4
1520.2.d.f.609.1 4 20.3 even 4
1520.2.d.f.609.4 4 20.7 even 4
1900.2.a.j.1.1 4 5.4 even 2 inner
1900.2.a.j.1.4 4 1.1 even 1 trivial
3420.2.f.a.1369.3 4 15.2 even 4
3420.2.f.a.1369.4 4 15.8 even 4
7600.2.a.ce.1.1 4 4.3 odd 2
7600.2.a.ce.1.4 4 20.19 odd 2