Properties

Label 4400.2.b.q.4049.2
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.q.4049.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{3} +2.00000i q^{7} -5.00000 q^{9} +O(q^{10})\) \(q-2.82843i q^{3} +2.00000i q^{7} -5.00000 q^{9} -1.00000 q^{11} +1.17157i q^{13} +6.82843i q^{17} +5.65685 q^{21} -2.82843i q^{23} +5.65685i q^{27} +3.65685 q^{29} +2.82843i q^{33} -7.65685i q^{37} +3.31371 q^{39} +6.00000 q^{41} -6.00000i q^{43} -2.82843i q^{47} +3.00000 q^{49} +19.3137 q^{51} -11.6569i q^{53} +1.65685 q^{59} -9.31371 q^{61} -10.0000i q^{63} -12.4853i q^{67} -8.00000 q^{69} -11.3137 q^{71} +1.17157i q^{73} -2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -10.3431i q^{87} +13.3137 q^{89} -2.34315 q^{91} +3.65685i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} + O(q^{10}) \) \( 4 q - 20 q^{9} - 4 q^{11} - 8 q^{29} - 32 q^{39} + 24 q^{41} + 12 q^{49} + 32 q^{51} - 16 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} + 4 q^{81} + 8 q^{89} - 32 q^{91} + 20 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 2.82843i − 1.63299i −0.577350 0.816497i \(-0.695913\pi\)
0.577350 0.816497i \(-0.304087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.17157i 0.324936i 0.986714 + 0.162468i \(0.0519454\pi\)
−0.986714 + 0.162468i \(0.948055\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.82843i 1.65614i 0.560627 + 0.828068i \(0.310560\pi\)
−0.560627 + 0.828068i \(0.689440\pi\)
\(18\) 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
\(23\) − 2.82843i − 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.82843i 0.492366i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.65685i − 1.25878i −0.777090 0.629390i \(-0.783305\pi\)
0.777090 0.629390i \(-0.216695\pi\)
\(38\) 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
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.82843i − 0.412568i −0.978492 0.206284i \(-0.933863\pi\)
0.978492 0.206284i \(-0.0661372\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 19.3137 2.70446
\(52\) 0 0
\(53\) − 11.6569i − 1.60119i −0.599204 0.800596i \(-0.704516\pi\)
0.599204 0.800596i \(-0.295484\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.65685 0.215704 0.107852 0.994167i \(-0.465603\pi\)
0.107852 + 0.994167i \(0.465603\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) − 10.0000i − 1.25988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.4853i − 1.52532i −0.646800 0.762660i \(-0.723893\pi\)
0.646800 0.762660i \(-0.276107\pi\)
\(68\) 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
\(73\) 1.17157i 0.137122i 0.997647 + 0.0685611i \(0.0218408\pi\)
−0.997647 + 0.0685611i \(0.978159\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 10.3431i − 1.10890i
\(88\) 0 0
\(89\) 13.3137 1.41125 0.705625 0.708585i \(-0.250666\pi\)
0.705625 + 0.708585i \(0.250666\pi\)
\(90\) 0 0
\(91\) −2.34315 −0.245628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.65685i 0.371297i 0.982616 + 0.185649i \(0.0594386\pi\)
−0.982616 + 0.185649i \(0.940561\pi\)
\(98\) 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
\(103\) 6.82843i 0.672825i 0.941715 + 0.336412i \(0.109214\pi\)
−0.941715 + 0.336412i \(0.890786\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.65685i − 0.740216i −0.928989 0.370108i \(-0.879321\pi\)
0.928989 0.370108i \(-0.120679\pi\)
\(108\) 0 0
\(109\) 7.65685 0.733394 0.366697 0.930341i \(-0.380489\pi\)
0.366697 + 0.930341i \(0.380489\pi\)
\(110\) 0 0
\(111\) −21.6569 −2.05558
\(112\) 0 0
\(113\) − 19.6569i − 1.84916i −0.380986 0.924581i \(-0.624416\pi\)
0.380986 0.924581i \(-0.375584\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 5.85786i − 0.541560i
\(118\) 0 0
\(119\) −13.6569 −1.25192
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 16.9706i − 1.53018i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.34315i − 0.385392i −0.981259 0.192696i \(-0.938277\pi\)
0.981259 0.192696i \(-0.0617231\pi\)
\(128\) 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
\(137\) − 10.9706i − 0.937278i −0.883390 0.468639i \(-0.844744\pi\)
0.883390 0.468639i \(-0.155256\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) − 1.17157i − 0.0979718i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 8.48528i − 0.699854i
\(148\) 0 0
\(149\) −0.343146 −0.0281116 −0.0140558 0.999901i \(-0.504474\pi\)
−0.0140558 + 0.999901i \(0.504474\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) − 34.1421i − 2.76023i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) −32.9706 −2.61474
\(160\) 0 0
\(161\) 5.65685 0.445823
\(162\) 0 0
\(163\) 16.4853i 1.29123i 0.763664 + 0.645613i \(0.223398\pi\)
−0.763664 + 0.645613i \(0.776602\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.9706i 1.77752i 0.458377 + 0.888758i \(0.348431\pi\)
−0.458377 + 0.888758i \(0.651569\pi\)
\(168\) 0 0
\(169\) 11.6274 0.894417
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.1421i 1.68344i 0.539918 + 0.841718i \(0.318455\pi\)
−0.539918 + 0.841718i \(0.681545\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.68629i − 0.352243i
\(178\) 0 0
\(179\) 9.65685 0.721787 0.360894 0.932607i \(-0.382472\pi\)
0.360894 + 0.932607i \(0.382472\pi\)
\(180\) 0 0
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) 0 0
\(183\) 26.3431i 1.94734i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.82843i − 0.499344i
\(188\) 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
\(193\) 1.17157i 0.0843317i 0.999111 + 0.0421658i \(0.0134258\pi\)
−0.999111 + 0.0421658i \(0.986574\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.8284i − 0.771493i −0.922605 0.385747i \(-0.873944\pi\)
0.922605 0.385747i \(-0.126056\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) −35.3137 −2.49084
\(202\) 0 0
\(203\) 7.31371i 0.513322i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 14.1421i 0.982946i
\(208\) 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
\(213\) 32.0000i 2.19260i
\(214\) 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
\(223\) − 10.8284i − 0.725125i −0.931959 0.362563i \(-0.881902\pi\)
0.931959 0.362563i \(-0.118098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 25.3137i − 1.68013i −0.542486 0.840065i \(-0.682517\pi\)
0.542486 0.840065i \(-0.317483\pi\)
\(228\) 0 0
\(229\) −1.31371 −0.0868123 −0.0434062 0.999058i \(-0.513821\pi\)
−0.0434062 + 0.999058i \(0.513821\pi\)
\(230\) 0 0
\(231\) −5.65685 −0.372194
\(232\) 0 0
\(233\) 6.14214i 0.402385i 0.979552 + 0.201192i \(0.0644816\pi\)
−0.979552 + 0.201192i \(0.935518\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 11.3137i − 0.734904i
\(238\) 0 0
\(239\) −23.3137 −1.50804 −0.754019 0.656852i \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 14.1421i 0.907218i
\(244\) 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
\(253\) 2.82843i 0.177822i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.31371i − 0.580973i −0.956879 0.290487i \(-0.906183\pi\)
0.956879 0.290487i \(-0.0938172\pi\)
\(258\) 0 0
\(259\) 15.3137 0.951548
\(260\) 0 0
\(261\) −18.2843 −1.13177
\(262\) 0 0
\(263\) − 10.9706i − 0.676474i −0.941061 0.338237i \(-0.890169\pi\)
0.941061 0.338237i \(-0.109831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 37.6569i − 2.30456i
\(268\) 0 0
\(269\) −17.3137 −1.05564 −0.527818 0.849358i \(-0.676990\pi\)
−0.527818 + 0.849358i \(0.676990\pi\)
\(270\) 0 0
\(271\) −7.31371 −0.444276 −0.222138 0.975015i \(-0.571304\pi\)
−0.222138 + 0.975015i \(0.571304\pi\)
\(272\) 0 0
\(273\) 6.62742i 0.401110i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.82843i 0.410280i 0.978733 + 0.205140i \(0.0657650\pi\)
−0.978733 + 0.205140i \(0.934235\pi\)
\(278\) 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
\(283\) 32.6274i 1.93950i 0.244103 + 0.969749i \(0.421507\pi\)
−0.244103 + 0.969749i \(0.578493\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) −29.6274 −1.74279
\(290\) 0 0
\(291\) 10.3431 0.606326
\(292\) 0 0
\(293\) 9.17157i 0.535809i 0.963445 + 0.267905i \(0.0863312\pi\)
−0.963445 + 0.267905i \(0.913669\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.65685i − 0.328244i
\(298\) 0 0
\(299\) 3.31371 0.191637
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) − 26.3431i − 1.51337i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.3431i 0.932753i 0.884586 + 0.466376i \(0.154441\pi\)
−0.884586 + 0.466376i \(0.845559\pi\)
\(308\) 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
\(313\) 1.31371i 0.0742552i 0.999311 + 0.0371276i \(0.0118208\pi\)
−0.999311 + 0.0371276i \(0.988179\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.31371i − 0.0737852i −0.999319 0.0368926i \(-0.988254\pi\)
0.999319 0.0368926i \(-0.0117459\pi\)
\(318\) 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
\(327\) − 21.6569i − 1.19763i
\(328\) 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
\(333\) 38.2843i 2.09797i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 20.4853i − 1.11590i −0.829873 0.557952i \(-0.811587\pi\)
0.829873 0.557952i \(-0.188413\pi\)
\(338\) 0 0
\(339\) −55.5980 −3.01967
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.9706i − 0.588931i −0.955662 0.294465i \(-0.904858\pi\)
0.955662 0.294465i \(-0.0951416\pi\)
\(348\) 0 0
\(349\) −26.9706 −1.44370 −0.721851 0.692049i \(-0.756708\pi\)
−0.721851 + 0.692049i \(0.756708\pi\)
\(350\) 0 0
\(351\) −6.62742 −0.353745
\(352\) 0 0
\(353\) − 21.3137i − 1.13441i −0.823575 0.567207i \(-0.808024\pi\)
0.823575 0.567207i \(-0.191976\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 38.6274i 2.04438i
\(358\) 0 0
\(359\) 0.686292 0.0362211 0.0181105 0.999836i \(-0.494235\pi\)
0.0181105 + 0.999836i \(0.494235\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 2.82843i − 0.148454i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) 23.3137 1.21039
\(372\) 0 0
\(373\) − 35.7990i − 1.85360i −0.375554 0.926801i \(-0.622547\pi\)
0.375554 0.926801i \(-0.377453\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.28427i 0.220651i
\(378\) 0 0
\(379\) 33.6569 1.72884 0.864418 0.502773i \(-0.167687\pi\)
0.864418 + 0.502773i \(0.167687\pi\)
\(380\) 0 0
\(381\) −12.2843 −0.629342
\(382\) 0 0
\(383\) − 5.85786i − 0.299323i −0.988737 0.149661i \(-0.952182\pi\)
0.988737 0.149661i \(-0.0478184\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 30.0000i 1.52499i
\(388\) 0 0
\(389\) −20.6274 −1.04585 −0.522926 0.852378i \(-0.675160\pi\)
−0.522926 + 0.852378i \(0.675160\pi\)
\(390\) 0 0
\(391\) 19.3137 0.976736
\(392\) 0 0
\(393\) − 32.0000i − 1.61419i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 9.31371i − 0.467442i −0.972304 0.233721i \(-0.924910\pi\)
0.972304 0.233721i \(-0.0750902\pi\)
\(398\) 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
\(407\) 7.65685i 0.379536i
\(408\) 0 0
\(409\) −1.02944 −0.0509024 −0.0254512 0.999676i \(-0.508102\pi\)
−0.0254512 + 0.999676i \(0.508102\pi\)
\(410\) 0 0
\(411\) −31.0294 −1.53057
\(412\) 0 0
\(413\) 3.31371i 0.163057i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.3137i 0.554035i
\(418\) 0 0
\(419\) −25.6569 −1.25342 −0.626710 0.779253i \(-0.715599\pi\)
−0.626710 + 0.779253i \(0.715599\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 14.1421i 0.687614i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 18.6274i − 0.901444i
\(428\) 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
\(433\) 7.65685i 0.367965i 0.982930 + 0.183982i \(0.0588990\pi\)
−0.982930 + 0.183982i \(0.941101\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) − 26.8284i − 1.27466i −0.770592 0.637329i \(-0.780039\pi\)
0.770592 0.637329i \(-0.219961\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.970563i 0.0459060i
\(448\) 0 0
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) − 33.9411i − 1.59469i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.485281i 0.0227005i 0.999936 + 0.0113503i \(0.00361298\pi\)
−0.999936 + 0.0113503i \(0.996387\pi\)
\(458\) 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
\(463\) − 6.14214i − 0.285449i −0.989762 0.142725i \(-0.954414\pi\)
0.989762 0.142725i \(-0.0455863\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.8284i 0.686178i 0.939303 + 0.343089i \(0.111473\pi\)
−0.939303 + 0.343089i \(0.888527\pi\)
\(468\) 0 0
\(469\) 24.9706 1.15303
\(470\) 0 0
\(471\) −39.5980 −1.82458
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 58.2843i 2.66865i
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 8.97056 0.409022
\(482\) 0 0
\(483\) − 16.0000i − 0.728025i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 24.4853i 1.10953i 0.832006 + 0.554767i \(0.187193\pi\)
−0.832006 + 0.554767i \(0.812807\pi\)
\(488\) 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
\(493\) 24.9706i 1.12462i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 22.6274i − 1.01498i
\(498\) 0 0
\(499\) 9.65685 0.432300 0.216150 0.976360i \(-0.430650\pi\)
0.216150 + 0.976360i \(0.430650\pi\)
\(500\) 0 0
\(501\) 64.9706 2.90267
\(502\) 0 0
\(503\) 16.6274i 0.741380i 0.928757 + 0.370690i \(0.120879\pi\)
−0.928757 + 0.370690i \(0.879121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 32.8873i − 1.46058i
\(508\) 0 0
\(509\) 13.3137 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(510\) 0 0
\(511\) −2.34315 −0.103655
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.82843i 0.124394i
\(518\) 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
\(523\) − 41.5980i − 1.81895i −0.415756 0.909476i \(-0.636483\pi\)
0.415756 0.909476i \(-0.363517\pi\)
\(524\) 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
\(533\) 7.02944i 0.304479i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 27.3137i − 1.17867i
\(538\) 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
\(543\) − 60.2843i − 2.58705i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 34.0000i 1.45374i 0.686778 + 0.726868i \(0.259025\pi\)
−0.686778 + 0.726868i \(0.740975\pi\)
\(548\) 0 0
\(549\) 46.5685 1.98750
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.85786i 0.417691i 0.977949 + 0.208846i \(0.0669706\pi\)
−0.977949 + 0.208846i \(0.933029\pi\)
\(558\) 0 0
\(559\) 7.02944 0.297314
\(560\) 0 0
\(561\) −19.3137 −0.815425
\(562\) 0 0
\(563\) 0.343146i 0.0144619i 0.999974 + 0.00723093i \(0.00230170\pi\)
−0.999974 + 0.00723093i \(0.997698\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) 21.9411 0.918208 0.459104 0.888383i \(-0.348171\pi\)
0.459104 + 0.888383i \(0.348171\pi\)
\(572\) 0 0
\(573\) 9.37258i 0.391545i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 26.9706i − 1.12280i −0.827545 0.561400i \(-0.810263\pi\)
0.827545 0.561400i \(-0.189737\pi\)
\(578\) 0 0
\(579\) 3.31371 0.137713
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 11.6569i 0.482778i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.14214i 0.0884154i 0.999022 + 0.0442077i \(0.0140763\pi\)
−0.999022 + 0.0442077i \(0.985924\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −30.6274 −1.25984
\(592\) 0 0
\(593\) − 3.51472i − 0.144332i −0.997393 0.0721661i \(-0.977009\pi\)
0.997393 0.0721661i \(-0.0229912\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 29.2548i − 1.19732i
\(598\) 0 0
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 0 0
\(601\) 23.9411 0.976579 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) 0 0
\(603\) 62.4264i 2.54220i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 38.2843i − 1.55391i −0.629556 0.776955i \(-0.716763\pi\)
0.629556 0.776955i \(-0.283237\pi\)
\(608\) 0 0
\(609\) 20.6863 0.838251
\(610\) 0 0
\(611\) 3.31371 0.134058
\(612\) 0 0
\(613\) 25.4558i 1.02815i 0.857745 + 0.514076i \(0.171865\pi\)
−0.857745 + 0.514076i \(0.828135\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.343146i 0.0138145i 0.999976 + 0.00690726i \(0.00219867\pi\)
−0.999976 + 0.00690726i \(0.997801\pi\)
\(618\) 0 0
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) 26.6274i 1.06680i
\(624\) 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
\(633\) − 45.2548i − 1.79872i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.51472i 0.139258i
\(638\) 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
\(643\) − 1.45584i − 0.0574129i −0.999588 0.0287064i \(-0.990861\pi\)
0.999588 0.0287064i \(-0.00913880\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 27.1127i − 1.06591i −0.846144 0.532955i \(-0.821081\pi\)
0.846144 0.532955i \(-0.178919\pi\)
\(648\) 0 0
\(649\) −1.65685 −0.0650372
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 11.6569i − 0.456168i −0.973641 0.228084i \(-0.926754\pi\)
0.973641 0.228084i \(-0.0732461\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 5.85786i − 0.228537i
\(658\) 0 0
\(659\) 45.9411 1.78961 0.894806 0.446455i \(-0.147314\pi\)
0.894806 + 0.446455i \(0.147314\pi\)
\(660\) 0 0
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) 0 0
\(663\) 22.6274i 0.878776i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 10.3431i − 0.400488i
\(668\) 0 0
\(669\) −30.6274 −1.18412
\(670\) 0 0
\(671\) 9.31371 0.359552
\(672\) 0 0
\(673\) 12.4853i 0.481272i 0.970615 + 0.240636i \(0.0773560\pi\)
−0.970615 + 0.240636i \(0.922644\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.8284i 0.877368i 0.898641 + 0.438684i \(0.144555\pi\)
−0.898641 + 0.438684i \(0.855445\pi\)
\(678\) 0 0
\(679\) −7.31371 −0.280674
\(680\) 0 0
\(681\) −71.5980 −2.74364
\(682\) 0 0
\(683\) − 7.79899i − 0.298420i −0.988806 0.149210i \(-0.952327\pi\)
0.988806 0.149210i \(-0.0476731\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.71573i 0.141764i
\(688\) 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
\(693\) 10.0000i 0.379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 40.9706i 1.55187i
\(698\) 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
\(707\) 18.6274i 0.700556i
\(708\) 0 0
\(709\) −24.6274 −0.924902 −0.462451 0.886645i \(-0.653030\pi\)
−0.462451 + 0.886645i \(0.653030\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 65.9411i 2.46262i
\(718\) 0 0
\(719\) 18.3431 0.684084 0.342042 0.939685i \(-0.388882\pi\)
0.342042 + 0.939685i \(0.388882\pi\)
\(720\) 0 0
\(721\) −13.6569 −0.508608
\(722\) 0 0
\(723\) − 16.9706i − 0.631142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 19.5147i 0.723761i 0.932225 + 0.361880i \(0.117865\pi\)
−0.932225 + 0.361880i \(0.882135\pi\)
\(728\) 0 0
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) 40.9706 1.51535
\(732\) 0 0
\(733\) 17.4558i 0.644746i 0.946613 + 0.322373i \(0.104481\pi\)
−0.946613 + 0.322373i \(0.895519\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.4853i 0.459901i
\(738\) 0 0
\(739\) 29.9411 1.10140 0.550701 0.834703i \(-0.314360\pi\)
0.550701 + 0.834703i \(0.314360\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 49.5980i − 1.81957i −0.415076 0.909787i \(-0.636245\pi\)
0.415076 0.909787i \(-0.363755\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 30.0000i 1.09764i
\(748\) 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
\(753\) 33.9411i 1.23688i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.3137i 0.483895i 0.970289 + 0.241947i \(0.0777862\pi\)
−0.970289 + 0.241947i \(0.922214\pi\)
\(758\) 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
\(763\) 15.3137i 0.554393i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.94113i 0.0700900i
\(768\) 0 0
\(769\) 18.9706 0.684096 0.342048 0.939682i \(-0.388879\pi\)
0.342048 + 0.939682i \(0.388879\pi\)
\(770\) 0 0
\(771\) −26.3431 −0.948725
\(772\) 0 0
\(773\) − 26.2843i − 0.945380i −0.881229 0.472690i \(-0.843283\pi\)
0.881229 0.472690i \(-0.156717\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 43.3137i − 1.55387i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 11.3137 0.404836
\(782\) 0 0
\(783\) 20.6863i 0.739268i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 14.9706i 0.533643i 0.963746 + 0.266821i \(0.0859734\pi\)
−0.963746 + 0.266821i \(0.914027\pi\)
\(788\) 0 0
\(789\) −31.0294 −1.10468
\(790\) 0 0
\(791\) 39.3137 1.39783
\(792\) 0 0
\(793\) − 10.9117i − 0.387485i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.6274i 1.15572i 0.816135 + 0.577861i \(0.196113\pi\)
−0.816135 + 0.577861i \(0.803887\pi\)
\(798\) 0 0
\(799\) 19.3137 0.683270
\(800\) 0 0
\(801\) −66.5685 −2.35208
\(802\) 0 0
\(803\) − 1.17157i − 0.0413439i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 48.9706i 1.72385i
\(808\) 0 0
\(809\) 10.9706 0.385704 0.192852 0.981228i \(-0.438226\pi\)
0.192852 + 0.981228i \(0.438226\pi\)
\(810\) 0 0
\(811\) −53.9411 −1.89413 −0.947065 0.321043i \(-0.895967\pi\)
−0.947065 + 0.321043i \(0.895967\pi\)
\(812\) 0 0
\(813\) 20.6863i 0.725500i
\(814\) 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
\(823\) 19.5147i 0.680240i 0.940382 + 0.340120i \(0.110468\pi\)
−0.940382 + 0.340120i \(0.889532\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 22.2843i − 0.774900i −0.921891 0.387450i \(-0.873356\pi\)
0.921891 0.387450i \(-0.126644\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 19.3137 0.669985
\(832\) 0 0
\(833\) 20.4853i 0.709773i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 26.3431 0.909466 0.454733 0.890628i \(-0.349735\pi\)
0.454733 + 0.890628i \(0.349735\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) − 48.9706i − 1.68664i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 92.2843 3.16719
\(850\) 0 0
\(851\) −21.6569 −0.742387
\(852\) 0 0
\(853\) 15.5147i 0.531214i 0.964081 + 0.265607i \(0.0855723\pi\)
−0.964081 + 0.265607i \(0.914428\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.7696i 0.846112i 0.906104 + 0.423056i \(0.139043\pi\)
−0.906104 + 0.423056i \(0.860957\pi\)
\(858\) 0 0
\(859\) 24.2843 0.828569 0.414284 0.910148i \(-0.364032\pi\)
0.414284 + 0.910148i \(0.364032\pi\)
\(860\) 0 0
\(861\) 33.9411 1.15671
\(862\) 0 0
\(863\) − 9.17157i − 0.312204i −0.987741 0.156102i \(-0.950107\pi\)
0.987741 0.156102i \(-0.0498929\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 83.7990i 2.84596i
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) 14.6274 0.495631
\(872\) 0 0
\(873\) − 18.2843i − 0.618829i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 49.4558i − 1.67001i −0.550246 0.835003i \(-0.685466\pi\)
0.550246 0.835003i \(-0.314534\pi\)
\(878\) 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
\(883\) 37.1716i 1.25092i 0.780255 + 0.625462i \(0.215089\pi\)
−0.780255 + 0.625462i \(0.784911\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 38.2843i − 1.28546i −0.766093 0.642730i \(-0.777802\pi\)
0.766093 0.642730i \(-0.222198\pi\)
\(888\) 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
\(897\) − 9.37258i − 0.312941i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 79.5980 2.65179
\(902\) 0 0
\(903\) − 33.9411i − 1.12949i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 27.5147i 0.913611i 0.889567 + 0.456806i \(0.151007\pi\)
−0.889567 + 0.456806i \(0.848993\pi\)
\(908\) 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
\(913\) 6.00000i 0.198571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 22.6274i 0.747223i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 46.2254 1.52318
\(922\) 0 0
\(923\) − 13.2548i − 0.436288i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 34.1421i − 1.12137i
\(928\) 0 0
\(929\) −5.31371 −0.174337 −0.0871686 0.996194i \(-0.527782\pi\)
−0.0871686 + 0.996194i \(0.527782\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13.2548i 0.433944i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.45584i − 0.0475604i −0.999717 0.0237802i \(-0.992430\pi\)
0.999717 0.0237802i \(-0.00757018\pi\)
\(938\) 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
\(943\) − 16.9706i − 0.552638i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 41.1716i − 1.33790i −0.743309 0.668948i \(-0.766745\pi\)
0.743309 0.668948i \(-0.233255\pi\)
\(948\) 0 0
\(949\) −1.37258 −0.0445559
\(950\) 0 0
\(951\) −3.71573 −0.120491
\(952\) 0 0
\(953\) − 53.1716i − 1.72240i −0.508269 0.861198i \(-0.669715\pi\)
0.508269 0.861198i \(-0.330285\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.3431i 0.334346i
\(958\) 0 0
\(959\) 21.9411 0.708516
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 38.2843i 1.23369i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.9706i 0.481421i 0.970597 + 0.240710i \(0.0773804\pi\)
−0.970597 + 0.240710i \(0.922620\pi\)
\(968\) 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
\(973\) − 8.00000i − 0.256468i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.3431i 1.03475i 0.855759 + 0.517374i \(0.173091\pi\)
−0.855759 + 0.517374i \(0.826909\pi\)
\(978\) 0 0
\(979\) −13.3137 −0.425508
\(980\) 0 0
\(981\) −38.2843 −1.22232
\(982\) 0 0
\(983\) − 21.8579i − 0.697158i −0.937279 0.348579i \(-0.886664\pi\)
0.937279 0.348579i \(-0.113336\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 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
\(993\) − 20.6863i − 0.656460i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 41.4558i − 1.31292i −0.754361 0.656460i \(-0.772053\pi\)
0.754361 0.656460i \(-0.227947\pi\)
\(998\) 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.b.q.4049.2 4
4.3 odd 2 275.2.b.d.199.4 4
5.2 odd 4 4400.2.a.bn.1.1 2
5.3 odd 4 880.2.a.m.1.2 2
5.4 even 2 inner 4400.2.b.q.4049.3 4
12.11 even 2 2475.2.c.l.199.1 4
15.8 even 4 7920.2.a.ch.1.2 2
20.3 even 4 55.2.a.b.1.2 2
20.7 even 4 275.2.a.c.1.1 2
20.19 odd 2 275.2.b.d.199.1 4
40.3 even 4 3520.2.a.bn.1.2 2
40.13 odd 4 3520.2.a.bo.1.1 2
55.43 even 4 9680.2.a.bn.1.2 2
60.23 odd 4 495.2.a.b.1.1 2
60.47 odd 4 2475.2.a.x.1.2 2
60.59 even 2 2475.2.c.l.199.4 4
140.83 odd 4 2695.2.a.f.1.2 2
220.3 even 20 605.2.g.f.251.2 8
220.43 odd 4 605.2.a.d.1.1 2
220.63 odd 20 605.2.g.l.251.1 8
220.83 odd 20 605.2.g.l.366.2 8
220.87 odd 4 3025.2.a.o.1.2 2
220.103 even 20 605.2.g.f.511.2 8
220.123 odd 20 605.2.g.l.81.2 8
220.163 even 20 605.2.g.f.81.1 8
220.183 odd 20 605.2.g.l.511.1 8
220.203 even 20 605.2.g.f.366.1 8
260.103 even 4 9295.2.a.g.1.1 2
660.263 even 4 5445.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 20.3 even 4
275.2.a.c.1.1 2 20.7 even 4
275.2.b.d.199.1 4 20.19 odd 2
275.2.b.d.199.4 4 4.3 odd 2
495.2.a.b.1.1 2 60.23 odd 4
605.2.a.d.1.1 2 220.43 odd 4
605.2.g.f.81.1 8 220.163 even 20
605.2.g.f.251.2 8 220.3 even 20
605.2.g.f.366.1 8 220.203 even 20
605.2.g.f.511.2 8 220.103 even 20
605.2.g.l.81.2 8 220.123 odd 20
605.2.g.l.251.1 8 220.63 odd 20
605.2.g.l.366.2 8 220.83 odd 20
605.2.g.l.511.1 8 220.183 odd 20
880.2.a.m.1.2 2 5.3 odd 4
2475.2.a.x.1.2 2 60.47 odd 4
2475.2.c.l.199.1 4 12.11 even 2
2475.2.c.l.199.4 4 60.59 even 2
2695.2.a.f.1.2 2 140.83 odd 4
3025.2.a.o.1.2 2 220.87 odd 4
3520.2.a.bn.1.2 2 40.3 even 4
3520.2.a.bo.1.1 2 40.13 odd 4
4400.2.a.bn.1.1 2 5.2 odd 4
4400.2.b.q.4049.2 4 1.1 even 1 trivial
4400.2.b.q.4049.3 4 5.4 even 2 inner
5445.2.a.y.1.2 2 660.263 even 4
7920.2.a.ch.1.2 2 15.8 even 4
9295.2.a.g.1.1 2 260.103 even 4
9680.2.a.bn.1.2 2 55.43 even 4