Properties

Label 4400.2.a.bn.1.1
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +1.17157 q^{13} -6.82843 q^{17} +5.65685 q^{21} -2.82843 q^{23} -5.65685 q^{27} -3.65685 q^{29} +2.82843 q^{33} +7.65685 q^{37} -3.31371 q^{39} +6.00000 q^{41} -6.00000 q^{43} +2.82843 q^{47} -3.00000 q^{49} +19.3137 q^{51} -11.6569 q^{53} -1.65685 q^{59} -9.31371 q^{61} -10.0000 q^{63} +12.4853 q^{67} +8.00000 q^{69} -11.3137 q^{71} +1.17157 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +10.3431 q^{87} -13.3137 q^{89} -2.34315 q^{91} -3.65685 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + 10 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69} + 8 q^{73} + 4 q^{77} - 8 q^{79} + 2 q^{81} - 12 q^{83} + 32 q^{87} - 4 q^{89} - 16 q^{91} + 4 q^{97} - 10 q^{99} + O(q^{100}) \)

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
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.17157 0.324936 0.162468 0.986714i \(-0.448055\pi\)
0.162468 + 0.986714i \(0.448055\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 5.65685 1.23443
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) 0 0
\(39\) −3.31371 −0.530618
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 19.3137 2.70446
\(52\) 0 0
\(53\) −11.6569 −1.60119 −0.800596 0.599204i \(-0.795484\pi\)
−0.800596 + 0.599204i \(0.795484\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) −10.0000 −1.25988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.4853 1.52532 0.762660 0.646800i \(-0.223893\pi\)
0.762660 + 0.646800i \(0.223893\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) 1.17157 0.137122 0.0685611 0.997647i \(-0.478159\pi\)
0.0685611 + 0.997647i \(0.478159\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.3431 1.10890
\(88\) 0 0
\(89\) −13.3137 −1.41125 −0.705625 0.708585i \(-0.749334\pi\)
−0.705625 + 0.708585i \(0.749334\pi\)
\(90\) 0 0
\(91\) −2.34315 −0.245628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) 0 0
\(103\) 6.82843 0.672825 0.336412 0.941715i \(-0.390786\pi\)
0.336412 + 0.941715i \(0.390786\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.65685 0.740216 0.370108 0.928989i \(-0.379321\pi\)
0.370108 + 0.928989i \(0.379321\pi\)
\(108\) 0 0
\(109\) −7.65685 −0.733394 −0.366697 0.930341i \(-0.619511\pi\)
−0.366697 + 0.930341i \(0.619511\pi\)
\(110\) 0 0
\(111\) −21.6569 −2.05558
\(112\) 0 0
\(113\) −19.6569 −1.84916 −0.924581 0.380986i \(-0.875584\pi\)
−0.924581 + 0.380986i \(0.875584\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.85786 0.541560
\(118\) 0 0
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −16.9706 −1.53018
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.34315 0.385392 0.192696 0.981259i \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) 0 0
\(129\) 16.9706 1.49417
\(130\) 0 0
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.9706 0.937278 0.468639 0.883390i \(-0.344744\pi\)
0.468639 + 0.883390i \(0.344744\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −1.17157 −0.0979718
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.48528 0.699854
\(148\) 0 0
\(149\) 0.343146 0.0281116 0.0140558 0.999901i \(-0.495526\pi\)
0.0140558 + 0.999901i \(0.495526\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −34.1421 −2.76023
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 32.9706 2.61474
\(160\) 0 0
\(161\) 5.65685 0.445823
\(162\) 0 0
\(163\) 16.4853 1.29123 0.645613 0.763664i \(-0.276602\pi\)
0.645613 + 0.763664i \(0.276602\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.9706 −1.77752 −0.888758 0.458377i \(-0.848431\pi\)
−0.888758 + 0.458377i \(0.848431\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.1421 1.68344 0.841718 0.539918i \(-0.181545\pi\)
0.841718 + 0.539918i \(0.181545\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.68629 0.352243
\(178\) 0 0
\(179\) −9.65685 −0.721787 −0.360894 0.932607i \(-0.617528\pi\)
−0.360894 + 0.932607i \(0.617528\pi\)
\(180\) 0 0
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) 0 0
\(183\) 26.3431 1.94734
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.82843 0.499344
\(188\) 0 0
\(189\) 11.3137 0.822951
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) 1.17157 0.0843317 0.0421658 0.999111i \(-0.486574\pi\)
0.0421658 + 0.999111i \(0.486574\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.8284 0.771493 0.385747 0.922605i \(-0.373944\pi\)
0.385747 + 0.922605i \(0.373944\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) −35.3137 −2.49084
\(202\) 0 0
\(203\) 7.31371 0.513322
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −14.1421 −0.982946
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 32.0000 2.19260
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.31371 −0.223920
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −10.8284 −0.725125 −0.362563 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 25.3137 1.68013 0.840065 0.542486i \(-0.182517\pi\)
0.840065 + 0.542486i \(0.182517\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 0 0
\(231\) −5.65685 −0.372194
\(232\) 0 0
\(233\) 6.14214 0.402385 0.201192 0.979552i \(-0.435518\pi\)
0.201192 + 0.979552i \(0.435518\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 16.9706 1.07547
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.82843 0.177822
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.31371 0.580973 0.290487 0.956879i \(-0.406183\pi\)
0.290487 + 0.956879i \(0.406183\pi\)
\(258\) 0 0
\(259\) −15.3137 −0.951548
\(260\) 0 0
\(261\) −18.2843 −1.13177
\(262\) 0 0
\(263\) −10.9706 −0.676474 −0.338237 0.941061i \(-0.609831\pi\)
−0.338237 + 0.941061i \(0.609831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 37.6569 2.30456
\(268\) 0 0
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) 0 0
\(271\) −7.31371 −0.444276 −0.222138 0.975015i \(-0.571304\pi\)
−0.222138 + 0.975015i \(0.571304\pi\)
\(272\) 0 0
\(273\) 6.62742 0.401110
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6.82843 −0.410280 −0.205140 0.978733i \(-0.565765\pi\)
−0.205140 + 0.978733i \(0.565765\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.3137 1.03285 0.516425 0.856333i \(-0.327263\pi\)
0.516425 + 0.856333i \(0.327263\pi\)
\(282\) 0 0
\(283\) 32.6274 1.93950 0.969749 0.244103i \(-0.0784935\pi\)
0.969749 + 0.244103i \(0.0784935\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 10.3431 0.606326
\(292\) 0 0
\(293\) 9.17157 0.535809 0.267905 0.963445i \(-0.413669\pi\)
0.267905 + 0.963445i \(0.413669\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.65685 0.328244
\(298\) 0 0
\(299\) −3.31371 −0.191637
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −26.3431 −1.51337
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −16.3431 −0.932753 −0.466376 0.884586i \(-0.654441\pi\)
−0.466376 + 0.884586i \(0.654441\pi\)
\(308\) 0 0
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) −4.68629 −0.265735 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(312\) 0 0
\(313\) 1.31371 0.0742552 0.0371276 0.999311i \(-0.488179\pi\)
0.0371276 + 0.999311i \(0.488179\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.31371 0.0737852 0.0368926 0.999319i \(-0.488254\pi\)
0.0368926 + 0.999319i \(0.488254\pi\)
\(318\) 0 0
\(319\) 3.65685 0.204745
\(320\) 0 0
\(321\) −21.6569 −1.20877
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 21.6569 1.19763
\(328\) 0 0
\(329\) −5.65685 −0.311872
\(330\) 0 0
\(331\) 7.31371 0.401998 0.200999 0.979591i \(-0.435581\pi\)
0.200999 + 0.979591i \(0.435581\pi\)
\(332\) 0 0
\(333\) 38.2843 2.09797
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.4853 1.11590 0.557952 0.829873i \(-0.311587\pi\)
0.557952 + 0.829873i \(0.311587\pi\)
\(338\) 0 0
\(339\) 55.5980 3.01967
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.9706 0.588931 0.294465 0.955662i \(-0.404858\pi\)
0.294465 + 0.955662i \(0.404858\pi\)
\(348\) 0 0
\(349\) 26.9706 1.44370 0.721851 0.692049i \(-0.243292\pi\)
0.721851 + 0.692049i \(0.243292\pi\)
\(350\) 0 0
\(351\) −6.62742 −0.353745
\(352\) 0 0
\(353\) −21.3137 −1.13441 −0.567207 0.823575i \(-0.691976\pi\)
−0.567207 + 0.823575i \(0.691976\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −38.6274 −2.04438
\(358\) 0 0
\(359\) −0.686292 −0.0362211 −0.0181105 0.999836i \(-0.505765\pi\)
−0.0181105 + 0.999836i \(0.505765\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −2.82843 −0.148454
\(364\) 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
\(369\) 30.0000 1.56174
\(370\) 0 0
\(371\) 23.3137 1.21039
\(372\) 0 0
\(373\) −35.7990 −1.85360 −0.926801 0.375554i \(-0.877453\pi\)
−0.926801 + 0.375554i \(0.877453\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.28427 −0.220651
\(378\) 0 0
\(379\) −33.6569 −1.72884 −0.864418 0.502773i \(-0.832313\pi\)
−0.864418 + 0.502773i \(0.832313\pi\)
\(380\) 0 0
\(381\) −12.2843 −0.629342
\(382\) 0 0
\(383\) −5.85786 −0.299323 −0.149661 0.988737i \(-0.547818\pi\)
−0.149661 + 0.988737i \(0.547818\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −30.0000 −1.52499
\(388\) 0 0
\(389\) 20.6274 1.04585 0.522926 0.852378i \(-0.324840\pi\)
0.522926 + 0.852378i \(0.324840\pi\)
\(390\) 0 0
\(391\) 19.3137 0.976736
\(392\) 0 0
\(393\) −32.0000 −1.61419
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.31371 0.467442 0.233721 0.972304i \(-0.424910\pi\)
0.233721 + 0.972304i \(0.424910\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.31371 −0.265354 −0.132677 0.991159i \(-0.542357\pi\)
−0.132677 + 0.991159i \(0.542357\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.65685 −0.379536
\(408\) 0 0
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) 0 0
\(411\) −31.0294 −1.53057
\(412\) 0 0
\(413\) 3.31371 0.163057
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.3137 −0.554035
\(418\) 0 0
\(419\) 25.6569 1.25342 0.626710 0.779253i \(-0.284401\pi\)
0.626710 + 0.779253i \(0.284401\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 14.1421 0.687614
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.6274 0.901444
\(428\) 0 0
\(429\) 3.31371 0.159987
\(430\) 0 0
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 0 0
\(433\) 7.65685 0.367965 0.183982 0.982930i \(-0.441101\pi\)
0.183982 + 0.982930i \(0.441101\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) −26.8284 −1.27466 −0.637329 0.770592i \(-0.719961\pi\)
−0.637329 + 0.770592i \(0.719961\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.970563 −0.0459060
\(448\) 0 0
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) −33.9411 −1.59469
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.485281 −0.0227005 −0.0113503 0.999936i \(-0.503613\pi\)
−0.0113503 + 0.999936i \(0.503613\pi\)
\(458\) 0 0
\(459\) 38.6274 1.80297
\(460\) 0 0
\(461\) 12.6274 0.588117 0.294059 0.955787i \(-0.404994\pi\)
0.294059 + 0.955787i \(0.404994\pi\)
\(462\) 0 0
\(463\) −6.14214 −0.285449 −0.142725 0.989762i \(-0.545586\pi\)
−0.142725 + 0.989762i \(0.545586\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.8284 −0.686178 −0.343089 0.939303i \(-0.611473\pi\)
−0.343089 + 0.939303i \(0.611473\pi\)
\(468\) 0 0
\(469\) −24.9706 −1.15303
\(470\) 0 0
\(471\) −39.5980 −1.82458
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −58.2843 −2.66865
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 8.97056 0.409022
\(482\) 0 0
\(483\) −16.0000 −0.728025
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −24.4853 −1.10953 −0.554767 0.832006i \(-0.687193\pi\)
−0.554767 + 0.832006i \(0.687193\pi\)
\(488\) 0 0
\(489\) −46.6274 −2.10856
\(490\) 0 0
\(491\) 0.686292 0.0309719 0.0154860 0.999880i \(-0.495070\pi\)
0.0154860 + 0.999880i \(0.495070\pi\)
\(492\) 0 0
\(493\) 24.9706 1.12462
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.6274 1.01498
\(498\) 0 0
\(499\) −9.65685 −0.432300 −0.216150 0.976360i \(-0.569350\pi\)
−0.216150 + 0.976360i \(0.569350\pi\)
\(500\) 0 0
\(501\) 64.9706 2.90267
\(502\) 0 0
\(503\) 16.6274 0.741380 0.370690 0.928757i \(-0.379121\pi\)
0.370690 + 0.928757i \(0.379121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 32.8873 1.46058
\(508\) 0 0
\(509\) −13.3137 −0.590120 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(510\) 0 0
\(511\) −2.34315 −0.103655
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −2.82843 −0.124394
\(518\) 0 0
\(519\) −62.6274 −2.74904
\(520\) 0 0
\(521\) 25.3137 1.10901 0.554507 0.832179i \(-0.312907\pi\)
0.554507 + 0.832179i \(0.312907\pi\)
\(522\) 0 0
\(523\) −41.5980 −1.81895 −0.909476 0.415756i \(-0.863517\pi\)
−0.909476 + 0.415756i \(0.863517\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) −8.28427 −0.359507
\(532\) 0 0
\(533\) 7.02944 0.304479
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 27.3137 1.17867
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 0 0
\(543\) −60.2843 −2.58705
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 0 0
\(549\) −46.5685 −1.98750
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.85786 −0.417691 −0.208846 0.977949i \(-0.566971\pi\)
−0.208846 + 0.977949i \(0.566971\pi\)
\(558\) 0 0
\(559\) −7.02944 −0.297314
\(560\) 0 0
\(561\) −19.3137 −0.815425
\(562\) 0 0
\(563\) 0.343146 0.0144619 0.00723093 0.999974i \(-0.497698\pi\)
0.00723093 + 0.999974i \(0.497698\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 31.6569 1.32712 0.663562 0.748121i \(-0.269044\pi\)
0.663562 + 0.748121i \(0.269044\pi\)
\(570\) 0 0
\(571\) 21.9411 0.918208 0.459104 0.888383i \(-0.348171\pi\)
0.459104 + 0.888383i \(0.348171\pi\)
\(572\) 0 0
\(573\) 9.37258 0.391545
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 26.9706 1.12280 0.561400 0.827545i \(-0.310263\pi\)
0.561400 + 0.827545i \(0.310263\pi\)
\(578\) 0 0
\(579\) −3.31371 −0.137713
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 11.6569 0.482778
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.14214 −0.0884154 −0.0442077 0.999022i \(-0.514076\pi\)
−0.0442077 + 0.999022i \(0.514076\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −30.6274 −1.25984
\(592\) 0 0
\(593\) −3.51472 −0.144332 −0.0721661 0.997393i \(-0.522991\pi\)
−0.0721661 + 0.997393i \(0.522991\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.2548 1.19732
\(598\) 0 0
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) 23.9411 0.976579 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) 0 0
\(603\) 62.4264 2.54220
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 38.2843 1.55391 0.776955 0.629556i \(-0.216763\pi\)
0.776955 + 0.629556i \(0.216763\pi\)
\(608\) 0 0
\(609\) −20.6863 −0.838251
\(610\) 0 0
\(611\) 3.31371 0.134058
\(612\) 0 0
\(613\) 25.4558 1.02815 0.514076 0.857745i \(-0.328135\pi\)
0.514076 + 0.857745i \(0.328135\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.343146 −0.0138145 −0.00690726 0.999976i \(-0.502199\pi\)
−0.00690726 + 0.999976i \(0.502199\pi\)
\(618\) 0 0
\(619\) 14.3431 0.576500 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) 26.6274 1.06680
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −52.2843 −2.08471
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −45.2548 −1.79872
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.51472 −0.139258
\(638\) 0 0
\(639\) −56.5685 −2.23782
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −1.45584 −0.0574129 −0.0287064 0.999588i \(-0.509139\pi\)
−0.0287064 + 0.999588i \(0.509139\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.1127 1.06591 0.532955 0.846144i \(-0.321081\pi\)
0.532955 + 0.846144i \(0.321081\pi\)
\(648\) 0 0
\(649\) 1.65685 0.0650372
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.6569 −0.456168 −0.228084 0.973641i \(-0.573246\pi\)
−0.228084 + 0.973641i \(0.573246\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.85786 0.228537
\(658\) 0 0
\(659\) −45.9411 −1.78961 −0.894806 0.446455i \(-0.852686\pi\)
−0.894806 + 0.446455i \(0.852686\pi\)
\(660\) 0 0
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) 0 0
\(663\) 22.6274 0.878776
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.3431 0.400488
\(668\) 0 0
\(669\) 30.6274 1.18412
\(670\) 0 0
\(671\) 9.31371 0.359552
\(672\) 0 0
\(673\) 12.4853 0.481272 0.240636 0.970615i \(-0.422644\pi\)
0.240636 + 0.970615i \(0.422644\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.8284 −0.877368 −0.438684 0.898641i \(-0.644555\pi\)
−0.438684 + 0.898641i \(0.644555\pi\)
\(678\) 0 0
\(679\) 7.31371 0.280674
\(680\) 0 0
\(681\) −71.5980 −2.74364
\(682\) 0 0
\(683\) −7.79899 −0.298420 −0.149210 0.988806i \(-0.547673\pi\)
−0.149210 + 0.988806i \(0.547673\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −3.71573 −0.141764
\(688\) 0 0
\(689\) −13.6569 −0.520285
\(690\) 0 0
\(691\) 39.3137 1.49556 0.747782 0.663944i \(-0.231119\pi\)
0.747782 + 0.663944i \(0.231119\pi\)
\(692\) 0 0
\(693\) 10.0000 0.379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −40.9706 −1.55187
\(698\) 0 0
\(699\) −17.3726 −0.657091
\(700\) 0 0
\(701\) −12.6274 −0.476931 −0.238465 0.971151i \(-0.576644\pi\)
−0.238465 + 0.971151i \(0.576644\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.6274 −0.700556
\(708\) 0 0
\(709\) 24.6274 0.924902 0.462451 0.886645i \(-0.346970\pi\)
0.462451 + 0.886645i \(0.346970\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −65.9411 −2.46262
\(718\) 0 0
\(719\) −18.3431 −0.684084 −0.342042 0.939685i \(-0.611118\pi\)
−0.342042 + 0.939685i \(0.611118\pi\)
\(720\) 0 0
\(721\) −13.6569 −0.508608
\(722\) 0 0
\(723\) −16.9706 −0.631142
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −19.5147 −0.723761 −0.361880 0.932225i \(-0.617865\pi\)
−0.361880 + 0.932225i \(0.617865\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) 40.9706 1.51535
\(732\) 0 0
\(733\) 17.4558 0.644746 0.322373 0.946613i \(-0.395519\pi\)
0.322373 + 0.946613i \(0.395519\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.4853 −0.459901
\(738\) 0 0
\(739\) −29.9411 −1.10140 −0.550701 0.834703i \(-0.685640\pi\)
−0.550701 + 0.834703i \(0.685640\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −49.5980 −1.81957 −0.909787 0.415076i \(-0.863755\pi\)
−0.909787 + 0.415076i \(0.863755\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −30.0000 −1.09764
\(748\) 0 0
\(749\) −15.3137 −0.559551
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 33.9411 1.23688
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −13.3137 −0.483895 −0.241947 0.970289i \(-0.577786\pi\)
−0.241947 + 0.970289i \(0.577786\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 15.3137 0.554393
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.94113 −0.0700900
\(768\) 0 0
\(769\) −18.9706 −0.684096 −0.342048 0.939682i \(-0.611121\pi\)
−0.342048 + 0.939682i \(0.611121\pi\)
\(770\) 0 0
\(771\) −26.3431 −0.948725
\(772\) 0 0
\(773\) −26.2843 −0.945380 −0.472690 0.881229i \(-0.656717\pi\)
−0.472690 + 0.881229i \(0.656717\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 43.3137 1.55387
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 11.3137 0.404836
\(782\) 0 0
\(783\) 20.6863 0.739268
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14.9706 −0.533643 −0.266821 0.963746i \(-0.585973\pi\)
−0.266821 + 0.963746i \(0.585973\pi\)
\(788\) 0 0
\(789\) 31.0294 1.10468
\(790\) 0 0
\(791\) 39.3137 1.39783
\(792\) 0 0
\(793\) −10.9117 −0.387485
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −32.6274 −1.15572 −0.577861 0.816135i \(-0.696113\pi\)
−0.577861 + 0.816135i \(0.696113\pi\)
\(798\) 0 0
\(799\) −19.3137 −0.683270
\(800\) 0 0
\(801\) −66.5685 −2.35208
\(802\) 0 0
\(803\) −1.17157 −0.0413439
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −48.9706 −1.72385
\(808\) 0 0
\(809\) −10.9706 −0.385704 −0.192852 0.981228i \(-0.561774\pi\)
−0.192852 + 0.981228i \(0.561774\pi\)
\(810\) 0 0
\(811\) −53.9411 −1.89413 −0.947065 0.321043i \(-0.895967\pi\)
−0.947065 + 0.321043i \(0.895967\pi\)
\(812\) 0 0
\(813\) 20.6863 0.725500
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −11.7157 −0.409381
\(820\) 0 0
\(821\) −41.3137 −1.44186 −0.720929 0.693009i \(-0.756285\pi\)
−0.720929 + 0.693009i \(0.756285\pi\)
\(822\) 0 0
\(823\) 19.5147 0.680240 0.340120 0.940382i \(-0.389532\pi\)
0.340120 + 0.940382i \(0.389532\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.2843 0.774900 0.387450 0.921891i \(-0.373356\pi\)
0.387450 + 0.921891i \(0.373356\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) 19.3137 0.669985
\(832\) 0 0
\(833\) 20.4853 0.709773
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.3431 −0.909466 −0.454733 0.890628i \(-0.650265\pi\)
−0.454733 + 0.890628i \(0.650265\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) −48.9706 −1.68664
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) −92.2843 −3.16719
\(850\) 0 0
\(851\) −21.6569 −0.742387
\(852\) 0 0
\(853\) 15.5147 0.531214 0.265607 0.964081i \(-0.414428\pi\)
0.265607 + 0.964081i \(0.414428\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.7696 −0.846112 −0.423056 0.906104i \(-0.639043\pi\)
−0.423056 + 0.906104i \(0.639043\pi\)
\(858\) 0 0
\(859\) −24.2843 −0.828569 −0.414284 0.910148i \(-0.635968\pi\)
−0.414284 + 0.910148i \(0.635968\pi\)
\(860\) 0 0
\(861\) 33.9411 1.15671
\(862\) 0 0
\(863\) −9.17157 −0.312204 −0.156102 0.987741i \(-0.549893\pi\)
−0.156102 + 0.987741i \(0.549893\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −83.7990 −2.84596
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 14.6274 0.495631
\(872\) 0 0
\(873\) −18.2843 −0.618829
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 49.4558 1.67001 0.835003 0.550246i \(-0.185466\pi\)
0.835003 + 0.550246i \(0.185466\pi\)
\(878\) 0 0
\(879\) −25.9411 −0.874972
\(880\) 0 0
\(881\) −7.37258 −0.248389 −0.124194 0.992258i \(-0.539635\pi\)
−0.124194 + 0.992258i \(0.539635\pi\)
\(882\) 0 0
\(883\) 37.1716 1.25092 0.625462 0.780255i \(-0.284911\pi\)
0.625462 + 0.780255i \(0.284911\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.2843 1.28546 0.642730 0.766093i \(-0.277802\pi\)
0.642730 + 0.766093i \(0.277802\pi\)
\(888\) 0 0
\(889\) −8.68629 −0.291329
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.37258 0.312941
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 79.5980 2.65179
\(902\) 0 0
\(903\) −33.9411 −1.12949
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −27.5147 −0.913611 −0.456806 0.889567i \(-0.651007\pi\)
−0.456806 + 0.889567i \(0.651007\pi\)
\(908\) 0 0
\(909\) 46.5685 1.54458
\(910\) 0 0
\(911\) 9.94113 0.329364 0.164682 0.986347i \(-0.447340\pi\)
0.164682 + 0.986347i \(0.447340\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 46.2254 1.52318
\(922\) 0 0
\(923\) −13.2548 −0.436288
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 34.1421 1.12137
\(928\) 0 0
\(929\) 5.31371 0.174337 0.0871686 0.996194i \(-0.472218\pi\)
0.0871686 + 0.996194i \(0.472218\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13.2548 0.433944
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.45584 0.0475604 0.0237802 0.999717i \(-0.492430\pi\)
0.0237802 + 0.999717i \(0.492430\pi\)
\(938\) 0 0
\(939\) −3.71573 −0.121258
\(940\) 0 0
\(941\) −6.68629 −0.217967 −0.108983 0.994044i \(-0.534760\pi\)
−0.108983 + 0.994044i \(0.534760\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.1716 1.33790 0.668948 0.743309i \(-0.266745\pi\)
0.668948 + 0.743309i \(0.266745\pi\)
\(948\) 0 0
\(949\) 1.37258 0.0445559
\(950\) 0 0
\(951\) −3.71573 −0.120491
\(952\) 0 0
\(953\) −53.1716 −1.72240 −0.861198 0.508269i \(-0.830285\pi\)
−0.861198 + 0.508269i \(0.830285\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −10.3431 −0.334346
\(958\) 0 0
\(959\) −21.9411 −0.708516
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 38.2843 1.23369
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −14.9706 −0.481421 −0.240710 0.970597i \(-0.577380\pi\)
−0.240710 + 0.970597i \(0.577380\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.68629 0.278756 0.139378 0.990239i \(-0.455490\pi\)
0.139378 + 0.990239i \(0.455490\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −32.3431 −1.03475 −0.517374 0.855759i \(-0.673091\pi\)
−0.517374 + 0.855759i \(0.673091\pi\)
\(978\) 0 0
\(979\) 13.3137 0.425508
\(980\) 0 0
\(981\) −38.2843 −1.22232
\(982\) 0 0
\(983\) −21.8579 −0.697158 −0.348579 0.937279i \(-0.613336\pi\)
−0.348579 + 0.937279i \(0.613336\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) 57.9411 1.84056 0.920280 0.391260i \(-0.127961\pi\)
0.920280 + 0.391260i \(0.127961\pi\)
\(992\) 0 0
\(993\) −20.6863 −0.656460
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 41.4558 1.31292 0.656460 0.754361i \(-0.272053\pi\)
0.656460 + 0.754361i \(0.272053\pi\)
\(998\) 0 0
\(999\) −43.3137 −1.37039
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 4400.2.a.bn.1.1 2
4.3 odd 2 275.2.a.c.1.1 2
5.2 odd 4 4400.2.b.q.4049.3 4
5.3 odd 4 4400.2.b.q.4049.2 4
5.4 even 2 880.2.a.m.1.2 2
12.11 even 2 2475.2.a.x.1.2 2
15.14 odd 2 7920.2.a.ch.1.2 2
20.3 even 4 275.2.b.d.199.4 4
20.7 even 4 275.2.b.d.199.1 4
20.19 odd 2 55.2.a.b.1.2 2
40.19 odd 2 3520.2.a.bn.1.2 2
40.29 even 2 3520.2.a.bo.1.1 2
44.43 even 2 3025.2.a.o.1.2 2
55.54 odd 2 9680.2.a.bn.1.2 2
60.23 odd 4 2475.2.c.l.199.1 4
60.47 odd 4 2475.2.c.l.199.4 4
60.59 even 2 495.2.a.b.1.1 2
140.139 even 2 2695.2.a.f.1.2 2
220.19 even 10 605.2.g.l.251.1 8
220.39 even 10 605.2.g.l.366.2 8
220.59 odd 10 605.2.g.f.511.2 8
220.79 even 10 605.2.g.l.81.2 8
220.119 odd 10 605.2.g.f.81.1 8
220.139 even 10 605.2.g.l.511.1 8
220.159 odd 10 605.2.g.f.366.1 8
220.179 odd 10 605.2.g.f.251.2 8
220.219 even 2 605.2.a.d.1.1 2
260.259 odd 2 9295.2.a.g.1.1 2
660.659 odd 2 5445.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 20.19 odd 2
275.2.a.c.1.1 2 4.3 odd 2
275.2.b.d.199.1 4 20.7 even 4
275.2.b.d.199.4 4 20.3 even 4
495.2.a.b.1.1 2 60.59 even 2
605.2.a.d.1.1 2 220.219 even 2
605.2.g.f.81.1 8 220.119 odd 10
605.2.g.f.251.2 8 220.179 odd 10
605.2.g.f.366.1 8 220.159 odd 10
605.2.g.f.511.2 8 220.59 odd 10
605.2.g.l.81.2 8 220.79 even 10
605.2.g.l.251.1 8 220.19 even 10
605.2.g.l.366.2 8 220.39 even 10
605.2.g.l.511.1 8 220.139 even 10
880.2.a.m.1.2 2 5.4 even 2
2475.2.a.x.1.2 2 12.11 even 2
2475.2.c.l.199.1 4 60.23 odd 4
2475.2.c.l.199.4 4 60.47 odd 4
2695.2.a.f.1.2 2 140.139 even 2
3025.2.a.o.1.2 2 44.43 even 2
3520.2.a.bn.1.2 2 40.19 odd 2
3520.2.a.bo.1.1 2 40.29 even 2
4400.2.a.bn.1.1 2 1.1 even 1 trivial
4400.2.b.q.4049.2 4 5.3 odd 4
4400.2.b.q.4049.3 4 5.2 odd 4
5445.2.a.y.1.2 2 660.659 odd 2
7920.2.a.ch.1.2 2 15.14 odd 2
9295.2.a.g.1.1 2 260.259 odd 2
9680.2.a.bn.1.2 2 55.54 odd 2