Properties

Label 4400.2.b.k.4049.2
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.k.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +1.00000 q^{11} +4.00000i q^{13} +6.00000i q^{17} +8.00000 q^{19} +2.00000 q^{21} -3.00000i q^{23} +5.00000i q^{27} -5.00000 q^{31} +1.00000i q^{33} -1.00000i q^{37} -4.00000 q^{39} -10.0000i q^{43} +3.00000 q^{49} -6.00000 q^{51} +6.00000i q^{53} +8.00000i q^{57} +3.00000 q^{59} -4.00000 q^{61} -4.00000i q^{63} +1.00000i q^{67} +3.00000 q^{69} -15.0000 q^{71} +4.00000i q^{73} -2.00000i q^{77} +2.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +9.00000 q^{89} +8.00000 q^{91} -5.00000i q^{93} -7.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 2 q^{11} + 16 q^{19} + 4 q^{21} - 10 q^{31} - 8 q^{39} + 6 q^{49} - 12 q^{51} + 6 q^{59} - 8 q^{61} + 6 q^{69} - 30 q^{71} + 4 q^{79} + 2 q^{81} + 18 q^{89} + 16 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000i 0.122169i 0.998133 + 0.0610847i \(0.0194560\pi\)
−0.998133 + 0.0610847i \(0.980544\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) − 5.00000i − 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.00000i − 0.710742i −0.934725 0.355371i \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) 15.0000i 1.41108i 0.708669 + 0.705541i \(0.249296\pi\)
−0.708669 + 0.705541i \(0.750704\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.00000i 0.739600i
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 12.0000i 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.00000i 0.399043i 0.979893 + 0.199522i \(0.0639388\pi\)
−0.979893 + 0.199522i \(0.936061\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 16.0000 1.22355
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.00000i 0.225494i
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 0 0
\(183\) − 4.00000i − 0.295689i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) 0 0
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) 21.0000 1.51951 0.759753 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 0 0
\(193\) − 20.0000i − 1.43963i −0.694165 0.719816i \(-0.744226\pi\)
0.694165 0.719816i \(-0.255774\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.00000i − 0.417029i
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) − 15.0000i − 1.02778i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.0000i 0.678844i
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 17.0000i 1.13840i 0.822198 + 0.569202i \(0.192748\pi\)
−0.822198 + 0.569202i \(0.807252\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 6.00000i − 0.398234i −0.979976 0.199117i \(-0.936193\pi\)
0.979976 0.199117i \(-0.0638074\pi\)
\(228\) 0 0
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.00000i 0.129914i
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 32.0000i 2.03611i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) − 3.00000i − 0.188608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.00000i 0.550791i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 8.00000i 0.484182i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 0 0
\(303\) 18.0000i 1.03407i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 1.00000i 0.0565233i 0.999601 + 0.0282617i \(0.00899717\pi\)
−0.999601 + 0.0282617i \(0.991003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 33.0000i 1.85346i 0.375722 + 0.926732i \(0.377395\pi\)
−0.375722 + 0.926732i \(0.622605\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 48.0000i 2.67079i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.00000i − 0.110600i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −20.0000 −1.06752
\(352\) 0 0
\(353\) 21.0000i 1.11772i 0.829263 + 0.558859i \(0.188761\pi\)
−0.829263 + 0.558859i \(0.811239\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.0000i 0.635107i
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 1.00000i 0.0524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.0000i 0.991792i 0.868382 + 0.495896i \(0.165160\pi\)
−0.868382 + 0.495896i \(0.834840\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) − 27.0000i − 1.37964i −0.723983 0.689818i \(-0.757691\pi\)
0.723983 0.689818i \(-0.242309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 20.0000i − 1.01666i
\(388\) 0 0
\(389\) 27.0000 1.36895 0.684477 0.729034i \(-0.260031\pi\)
0.684477 + 0.729034i \(0.260031\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 34.0000i − 1.70641i −0.521575 0.853206i \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) − 20.0000i − 0.996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.00000i − 0.0495682i
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) − 6.00000i − 0.295241i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000i 0.685583i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) − 29.0000i − 1.39365i −0.717241 0.696826i \(-0.754595\pi\)
0.717241 0.696826i \(-0.245405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) − 21.0000i − 0.997740i −0.866677 0.498870i \(-0.833748\pi\)
0.866677 0.498870i \(-0.166252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 6.00000i − 0.283790i
\(448\) 0 0
\(449\) −3.00000 −0.141579 −0.0707894 0.997491i \(-0.522552\pi\)
−0.0707894 + 0.997491i \(0.522552\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 10.0000i 0.469841i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 28.0000i − 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) 0 0
\(459\) −30.0000 −1.40028
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 23.0000i 1.06890i 0.845200 + 0.534450i \(0.179481\pi\)
−0.845200 + 0.534450i \(0.820519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 0 0
\(473\) − 10.0000i − 0.459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) − 6.00000i − 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 29.0000i − 1.31412i −0.753840 0.657058i \(-0.771801\pi\)
0.753840 0.657058i \(-0.228199\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 30.0000i 1.34568i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) − 30.0000i − 1.33763i −0.743427 0.668817i \(-0.766801\pi\)
0.743427 0.668817i \(-0.233199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 3.00000i − 0.133235i
\(508\) 0 0
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 40.0000i 1.76604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 30.0000i − 1.30682i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.00000i − 0.388379i
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 0 0
\(543\) − 13.0000i − 0.557883i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 4.00000i − 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 0 0
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.00000i − 0.0839921i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) 21.0000i 0.877288i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.0000i 0.707719i 0.935299 + 0.353860i \(0.115131\pi\)
−0.935299 + 0.353860i \(0.884869\pi\)
\(578\) 0 0
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 6.00000i 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 36.0000i 1.47834i 0.673517 + 0.739171i \(0.264783\pi\)
−0.673517 + 0.739171i \(0.735217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 14.0000i − 0.568242i −0.958788 0.284121i \(-0.908298\pi\)
0.958788 0.284121i \(-0.0917018\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 32.0000i − 1.29247i −0.763139 0.646234i \(-0.776343\pi\)
0.763139 0.646234i \(-0.223657\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 0 0
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) − 18.0000i − 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 43.0000 1.71180 0.855901 0.517139i \(-0.173003\pi\)
0.855901 + 0.517139i \(0.173003\pi\)
\(632\) 0 0
\(633\) − 20.0000i − 0.794929i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) −30.0000 −1.18678
\(640\) 0 0
\(641\) 39.0000 1.54041 0.770204 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) − 13.0000i − 0.512670i −0.966588 0.256335i \(-0.917485\pi\)
0.966588 0.256335i \(-0.0825150\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 3.00000i − 0.117942i −0.998260 0.0589711i \(-0.981218\pi\)
0.998260 0.0589711i \(-0.0187820\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 0 0
\(653\) − 3.00000i − 0.117399i −0.998276 0.0586995i \(-0.981305\pi\)
0.998276 0.0586995i \(-0.0186954\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 0 0
\(663\) − 24.0000i − 0.932083i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −17.0000 −0.657258
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) 48.0000i 1.83667i 0.395805 + 0.918334i \(0.370466\pi\)
−0.395805 + 0.918334i \(0.629534\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.0000i 0.495981i
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 1.00000 0.0380418 0.0190209 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(692\) 0 0
\(693\) − 4.00000i − 0.151947i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) − 8.00000i − 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 36.0000i − 1.35392i
\(708\) 0 0
\(709\) 37.0000 1.38956 0.694782 0.719220i \(-0.255501\pi\)
0.694782 + 0.719220i \(0.255501\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 15.0000i 0.561754i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000i 0.224074i
\(718\) 0 0
\(719\) 45.0000 1.67822 0.839108 0.543964i \(-0.183077\pi\)
0.839108 + 0.543964i \(0.183077\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 17.0000i − 0.630495i −0.949009 0.315248i \(-0.897912\pi\)
0.949009 0.315248i \(-0.102088\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 60.0000 2.21918
\(732\) 0 0
\(733\) 4.00000i 0.147743i 0.997268 + 0.0738717i \(0.0235355\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000i 0.0368355i
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −35.0000 −1.27717 −0.638584 0.769552i \(-0.720480\pi\)
−0.638584 + 0.769552i \(0.720480\pi\)
\(752\) 0 0
\(753\) 9.00000i 0.327978i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −44.0000 −1.58668 −0.793340 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.00000i − 0.0717496i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −15.0000 −0.536742
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 30.0000 1.06668
\(792\) 0 0
\(793\) − 16.0000i − 0.568177i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.00000i 0.318796i 0.987214 + 0.159398i \(0.0509554\pi\)
−0.987214 + 0.159398i \(0.949045\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 4.00000i 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000i 0.211210i
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) − 20.0000i − 0.701431i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 80.0000i − 2.79885i
\(818\) 0 0
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) − 43.0000i − 1.49889i −0.662069 0.749443i \(-0.730321\pi\)
0.662069 0.749443i \(-0.269679\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 25.0000i − 0.864126i
\(838\) 0 0
\(839\) −39.0000 −1.34643 −0.673215 0.739447i \(-0.735087\pi\)
−0.673215 + 0.739447i \(0.735087\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) − 18.0000i − 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) − 38.0000i − 1.30110i −0.759465 0.650548i \(-0.774539\pi\)
0.759465 0.650548i \(-0.225461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0000i 0.819824i 0.912125 + 0.409912i \(0.134441\pi\)
−0.912125 + 0.409912i \(0.865559\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) − 14.0000i − 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 52.0000i − 1.75592i −0.478738 0.877958i \(-0.658906\pi\)
0.478738 0.877958i \(-0.341094\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) − 4.00000i − 0.134611i −0.997732 0.0673054i \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 30.0000i − 1.00730i −0.863907 0.503651i \(-0.831990\pi\)
0.863907 0.503651i \(-0.168010\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 12.0000i 0.400668i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 20.0000i − 0.665558i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 6.00000i 0.198571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 12.0000i − 0.396275i
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) − 60.0000i − 1.97492i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.0000i 0.525509i
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 0 0
\(933\) − 12.0000i − 0.392862i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 32.0000i 1.04539i 0.852518 + 0.522697i \(0.175074\pi\)
−0.852518 + 0.522697i \(0.824926\pi\)
\(938\) 0 0
\(939\) −1.00000 −0.0326338
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 27.0000i − 0.877382i −0.898638 0.438691i \(-0.855442\pi\)
0.898638 0.438691i \(-0.144558\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) −33.0000 −1.07010
\(952\) 0 0
\(953\) − 42.0000i − 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) 0 0
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) − 28.0000i − 0.897639i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.0000i 1.43968i 0.694141 + 0.719839i \(0.255784\pi\)
−0.694141 + 0.719839i \(0.744216\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 45.0000i 1.43528i 0.696416 + 0.717639i \(0.254777\pi\)
−0.696416 + 0.717639i \(0.745223\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 7.00000i 0.222138i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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.k.4049.2 2
4.3 odd 2 1100.2.b.c.749.1 2
5.2 odd 4 4400.2.a.v.1.1 1
5.3 odd 4 176.2.a.a.1.1 1
5.4 even 2 inner 4400.2.b.k.4049.1 2
12.11 even 2 9900.2.c.g.5149.2 2
15.8 even 4 1584.2.a.p.1.1 1
20.3 even 4 44.2.a.a.1.1 1
20.7 even 4 1100.2.a.b.1.1 1
20.19 odd 2 1100.2.b.c.749.2 2
35.13 even 4 8624.2.a.w.1.1 1
40.3 even 4 704.2.a.f.1.1 1
40.13 odd 4 704.2.a.i.1.1 1
55.43 even 4 1936.2.a.c.1.1 1
60.23 odd 4 396.2.a.c.1.1 1
60.47 odd 4 9900.2.a.h.1.1 1
60.59 even 2 9900.2.c.g.5149.1 2
80.3 even 4 2816.2.c.e.1409.2 2
80.13 odd 4 2816.2.c.k.1409.1 2
80.43 even 4 2816.2.c.e.1409.1 2
80.53 odd 4 2816.2.c.k.1409.2 2
120.53 even 4 6336.2.a.i.1.1 1
120.83 odd 4 6336.2.a.j.1.1 1
140.3 odd 12 2156.2.i.c.177.1 2
140.23 even 12 2156.2.i.b.1145.1 2
140.83 odd 4 2156.2.a.a.1.1 1
140.103 odd 12 2156.2.i.c.1145.1 2
140.123 even 12 2156.2.i.b.177.1 2
180.23 odd 12 3564.2.i.a.1189.1 2
180.43 even 12 3564.2.i.j.2377.1 2
180.83 odd 12 3564.2.i.a.2377.1 2
180.103 even 12 3564.2.i.j.1189.1 2
220.3 even 20 484.2.e.a.9.1 4
220.43 odd 4 484.2.a.a.1.1 1
220.63 odd 20 484.2.e.b.9.1 4
220.83 odd 20 484.2.e.b.245.1 4
220.103 even 20 484.2.e.a.269.1 4
220.123 odd 20 484.2.e.b.81.1 4
220.163 even 20 484.2.e.a.81.1 4
220.183 odd 20 484.2.e.b.269.1 4
220.203 even 20 484.2.e.a.245.1 4
260.103 even 4 7436.2.a.d.1.1 1
440.43 odd 4 7744.2.a.m.1.1 1
440.373 even 4 7744.2.a.bc.1.1 1
660.263 even 4 4356.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.2.a.a.1.1 1 20.3 even 4
176.2.a.a.1.1 1 5.3 odd 4
396.2.a.c.1.1 1 60.23 odd 4
484.2.a.a.1.1 1 220.43 odd 4
484.2.e.a.9.1 4 220.3 even 20
484.2.e.a.81.1 4 220.163 even 20
484.2.e.a.245.1 4 220.203 even 20
484.2.e.a.269.1 4 220.103 even 20
484.2.e.b.9.1 4 220.63 odd 20
484.2.e.b.81.1 4 220.123 odd 20
484.2.e.b.245.1 4 220.83 odd 20
484.2.e.b.269.1 4 220.183 odd 20
704.2.a.f.1.1 1 40.3 even 4
704.2.a.i.1.1 1 40.13 odd 4
1100.2.a.b.1.1 1 20.7 even 4
1100.2.b.c.749.1 2 4.3 odd 2
1100.2.b.c.749.2 2 20.19 odd 2
1584.2.a.p.1.1 1 15.8 even 4
1936.2.a.c.1.1 1 55.43 even 4
2156.2.a.a.1.1 1 140.83 odd 4
2156.2.i.b.177.1 2 140.123 even 12
2156.2.i.b.1145.1 2 140.23 even 12
2156.2.i.c.177.1 2 140.3 odd 12
2156.2.i.c.1145.1 2 140.103 odd 12
2816.2.c.e.1409.1 2 80.43 even 4
2816.2.c.e.1409.2 2 80.3 even 4
2816.2.c.k.1409.1 2 80.13 odd 4
2816.2.c.k.1409.2 2 80.53 odd 4
3564.2.i.a.1189.1 2 180.23 odd 12
3564.2.i.a.2377.1 2 180.83 odd 12
3564.2.i.j.1189.1 2 180.103 even 12
3564.2.i.j.2377.1 2 180.43 even 12
4356.2.a.j.1.1 1 660.263 even 4
4400.2.a.v.1.1 1 5.2 odd 4
4400.2.b.k.4049.1 2 5.4 even 2 inner
4400.2.b.k.4049.2 2 1.1 even 1 trivial
6336.2.a.i.1.1 1 120.53 even 4
6336.2.a.j.1.1 1 120.83 odd 4
7436.2.a.d.1.1 1 260.103 even 4
7744.2.a.m.1.1 1 440.43 odd 4
7744.2.a.bc.1.1 1 440.373 even 4
8624.2.a.w.1.1 1 35.13 even 4
9900.2.a.h.1.1 1 60.47 odd 4
9900.2.c.g.5149.1 2 60.59 even 2
9900.2.c.g.5149.2 2 12.11 even 2