Properties

Label 2800.2.g.a.449.2
Level $2800$
Weight $2$
Character 2800.449
Analytic conductor $22.358$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.a.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} -1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -1.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} -6.00000i q^{13} +1.00000i q^{17} -3.00000 q^{19} +3.00000 q^{21} -9.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +15.0000i q^{33} -8.00000i q^{37} +18.0000 q^{39} +11.0000 q^{41} +8.00000i q^{43} +2.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} +4.00000i q^{53} -9.00000i q^{57} +4.00000 q^{59} -2.00000 q^{61} +6.00000i q^{63} +9.00000i q^{67} +10.0000 q^{71} -7.00000i q^{73} -5.00000i q^{77} -2.00000 q^{79} +9.00000 q^{81} -11.0000i q^{83} +18.0000i q^{87} +11.0000 q^{89} -6.00000 q^{91} +12.0000i q^{93} +10.0000i q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9} + O(q^{10}) \) \( 2 q - 12 q^{9} + 10 q^{11} - 6 q^{19} + 6 q^{21} + 12 q^{29} + 8 q^{31} + 36 q^{39} + 22 q^{41} - 2 q^{49} - 6 q^{51} + 8 q^{59} - 4 q^{61} + 20 q^{71} - 4 q^{79} + 18 q^{81} + 22 q^{89} - 12 q^{91} - 60 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000i 0.242536i 0.992620 + 0.121268i \(0.0386960\pi\)
−0.992620 + 0.121268i \(0.961304\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 9.00000i − 1.73205i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 15.0000i 2.61116i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 9.00000i − 1.19208i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 6.00000i 0.755929i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.00000i 1.09952i 0.835321 + 0.549762i \(0.185282\pi\)
−0.835321 + 0.549762i \(0.814718\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.00000i − 0.569803i
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 11.0000i − 1.20741i −0.797209 0.603703i \(-0.793691\pi\)
0.797209 0.603703i \(-0.206309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 18.0000i 1.92980i
\(88\) 0 0
\(89\) 11.0000 1.16600 0.582999 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 12.0000i 1.24434i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) − 1.00000i − 0.0940721i −0.998893 0.0470360i \(-0.985022\pi\)
0.998893 0.0470360i \(-0.0149776\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 36.0000i 3.32820i
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 33.0000i 2.97551i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 14.0000i − 1.24230i −0.783692 0.621150i \(-0.786666\pi\)
0.783692 0.621150i \(-0.213334\pi\)
\(128\) 0 0
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) 0 0
\(139\) −11.0000 −0.933008 −0.466504 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) − 30.0000i − 2.50873i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.00000i − 0.247436i
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.0000i 1.48819i 0.668071 + 0.744097i \(0.267120\pi\)
−0.668071 + 0.744097i \(0.732880\pi\)
\(164\) 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
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 18.0000 1.37649
\(172\) 0 0
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) − 6.00000i − 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.00000i 0.365636i
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) − 19.0000i − 1.36765i −0.729646 0.683825i \(-0.760315\pi\)
0.729646 0.683825i \(-0.239685\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) −27.0000 −1.90443
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 30.0000i 2.05557i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 0 0
\(219\) 21.0000 1.41905
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) − 22.0000i − 1.47323i −0.676313 0.736614i \(-0.736423\pi\)
0.676313 0.736614i \(-0.263577\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 28.0000i − 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 6.00000i − 0.389742i
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.0000i 1.14531i
\(248\) 0 0
\(249\) 33.0000 2.09129
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −36.0000 −2.22834
\(262\) 0 0
\(263\) 10.0000i 0.616626i 0.951285 + 0.308313i \(0.0997645\pi\)
−0.951285 + 0.308313i \(0.900236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 33.0000i 2.01957i
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 0 0
\(273\) − 18.0000i − 1.08941i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 30.0000i 1.80253i 0.433273 + 0.901263i \(0.357359\pi\)
−0.433273 + 0.901263i \(0.642641\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 0 0
\(283\) 13.0000i 0.772770i 0.922338 + 0.386385i \(0.126276\pi\)
−0.922338 + 0.386385i \(0.873724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 11.0000i − 0.649309i
\(288\) 0 0
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) −30.0000 −1.75863
\(292\) 0 0
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 45.0000i − 2.61116i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 13.0000i 0.741949i 0.928643 + 0.370975i \(0.120976\pi\)
−0.928643 + 0.370975i \(0.879024\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 4.00000i − 0.224662i −0.993671 0.112331i \(-0.964168\pi\)
0.993671 0.112331i \(-0.0358318\pi\)
\(318\) 0 0
\(319\) 30.0000 1.67968
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 54.0000i 2.98621i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 48.0000i 2.63038i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.0000i 1.57973i 0.613280 + 0.789865i \(0.289850\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(338\) 0 0
\(339\) 3.00000 0.162938
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.0000i 1.01997i 0.860182 + 0.509987i \(0.170350\pi\)
−0.860182 + 0.509987i \(0.829650\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) −54.0000 −2.88231
\(352\) 0 0
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.00000i 0.158777i
\(358\) 0 0
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 42.0000i 2.20443i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) −66.0000 −3.43582
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 36.0000i − 1.85409i
\(378\) 0 0
\(379\) 9.00000 0.462299 0.231149 0.972918i \(-0.425751\pi\)
0.231149 + 0.972918i \(0.425751\pi\)
\(380\) 0 0
\(381\) 42.0000 2.15173
\(382\) 0 0
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 48.0000i − 2.43998i
\(388\) 0 0
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 24.0000i − 1.21064i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) −9.00000 −0.450564
\(400\) 0 0
\(401\) 37.0000 1.84769 0.923846 0.382765i \(-0.125028\pi\)
0.923846 + 0.382765i \(0.125028\pi\)
\(402\) 0 0
\(403\) − 24.0000i − 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 40.0000i − 1.98273i
\(408\) 0 0
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 33.0000i − 1.61602i
\(418\) 0 0
\(419\) −39.0000 −1.90527 −0.952637 0.304109i \(-0.901641\pi\)
−0.952637 + 0.304109i \(0.901641\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) − 12.0000i − 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 0 0
\(429\) 90.0000 4.34524
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) − 1.00000i − 0.0480569i −0.999711 0.0240285i \(-0.992351\pi\)
0.999711 0.0240285i \(-0.00764923\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) − 37.0000i − 1.75792i −0.476893 0.878962i \(-0.658237\pi\)
0.476893 0.878962i \(-0.341763\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 36.0000i − 1.70274i
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) 55.0000 2.58985
\(452\) 0 0
\(453\) − 24.0000i − 1.12762i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 25.0000i − 1.16945i −0.811231 0.584725i \(-0.801202\pi\)
0.811231 0.584725i \(-0.198798\pi\)
\(458\) 0 0
\(459\) 9.00000 0.420084
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4.00000i − 0.185098i −0.995708 0.0925490i \(-0.970499\pi\)
0.995708 0.0925490i \(-0.0295015\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) 0 0
\(473\) 40.0000i 1.83920i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 24.0000i − 1.09888i
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 34.0000i − 1.54069i −0.637629 0.770344i \(-0.720085\pi\)
0.637629 0.770344i \(-0.279915\pi\)
\(488\) 0 0
\(489\) −57.0000 −2.57763
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 6.00000i 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.0000i − 0.448561i
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −36.0000 −1.60836
\(502\) 0 0
\(503\) 30.0000i 1.33763i 0.743427 + 0.668817i \(0.233199\pi\)
−0.743427 + 0.668817i \(0.766801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 69.0000i − 3.06440i
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) 0 0
\(513\) 27.0000i 1.19208i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 10.0000i 0.439799i
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 11.0000 0.481919 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(522\) 0 0
\(523\) − 13.0000i − 0.568450i −0.958758 0.284225i \(-0.908264\pi\)
0.958758 0.284225i \(-0.0917363\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) − 66.0000i − 2.85878i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.00000i 0.388379i
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 30.0000i 1.28742i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 27.0000i 1.15444i 0.816590 + 0.577218i \(0.195862\pi\)
−0.816590 + 0.577218i \(0.804138\pi\)
\(548\) 0 0
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.00000i − 0.169485i −0.996403 0.0847427i \(-0.972993\pi\)
0.996403 0.0847427i \(-0.0270068\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 9.00000i − 0.377964i
\(568\) 0 0
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 13.0000i − 0.541197i −0.962692 0.270599i \(-0.912778\pi\)
0.962692 0.270599i \(-0.0872216\pi\)
\(578\) 0 0
\(579\) 57.0000 2.36884
\(580\) 0 0
\(581\) −11.0000 −0.456357
\(582\) 0 0
\(583\) 20.0000i 0.828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 13.0000i − 0.536567i −0.963340 0.268284i \(-0.913544\pi\)
0.963340 0.268284i \(-0.0864565\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −66.0000 −2.71488
\(592\) 0 0
\(593\) − 39.0000i − 1.60154i −0.598973 0.800769i \(-0.704424\pi\)
0.598973 0.800769i \(-0.295576\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 30.0000i − 1.22782i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) 0 0
\(603\) − 54.0000i − 2.19905i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) 0 0
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 14.0000i − 0.563619i −0.959470 0.281809i \(-0.909065\pi\)
0.959470 0.281809i \(-0.0909346\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 11.0000i − 0.440706i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 45.0000i − 1.79713i
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) − 3.00000i − 0.119239i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) −60.0000 −2.37356
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 8.00000i − 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) − 28.0000i − 1.09572i −0.836569 0.547862i \(-0.815442\pi\)
0.836569 0.547862i \(-0.184558\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 42.0000i 1.63858i
\(658\) 0 0
\(659\) 1.00000 0.0389545 0.0194772 0.999810i \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 0 0
\(663\) 18.0000i 0.699062i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 66.0000 2.55171
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 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
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 84.0000 3.21889
\(682\) 0 0
\(683\) 13.0000i 0.497431i 0.968577 + 0.248716i \(0.0800084\pi\)
−0.968577 + 0.248716i \(0.919992\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 42.0000i 1.60240i
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −49.0000 −1.86405 −0.932024 0.362397i \(-0.881959\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 30.0000i 1.13961i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 11.0000i 0.416655i
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 0 0
\(703\) 24.0000i 0.905177i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) − 15.0000i − 0.557856i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.00000i 0.222528i 0.993791 + 0.111264i \(0.0354899\pi\)
−0.993791 + 0.111264i \(0.964510\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 40.0000i 1.47743i 0.674016 + 0.738717i \(0.264568\pi\)
−0.674016 + 0.738717i \(0.735432\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45.0000i 1.65760i
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −54.0000 −1.98374
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 66.0000i 2.41481i
\(748\) 0 0
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) − 81.0000i − 2.95180i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) − 18.0000i − 0.651644i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 36.0000i 1.29483i 0.762138 + 0.647415i \(0.224150\pi\)
−0.762138 + 0.647415i \(0.775850\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 24.0000i − 0.860995i
\(778\) 0 0
\(779\) −33.0000 −1.18235
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) 0 0
\(783\) − 54.0000i − 1.92980i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 52.0000i 1.85360i 0.375555 + 0.926800i \(0.377452\pi\)
−0.375555 + 0.926800i \(0.622548\pi\)
\(788\) 0 0
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) 0 0
\(801\) −66.0000 −2.33200
\(802\) 0 0
\(803\) − 35.0000i − 1.23512i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 54.0000i − 1.90089i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 18.0000i 0.631288i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 24.0000i − 0.839654i
\(818\) 0 0
\(819\) 36.0000 1.25794
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) − 10.0000i − 0.348578i −0.984695 0.174289i \(-0.944237\pi\)
0.984695 0.174289i \(-0.0557627\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 41.0000i 1.42571i 0.701312 + 0.712855i \(0.252598\pi\)
−0.701312 + 0.712855i \(0.747402\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −90.0000 −3.12207
\(832\) 0 0
\(833\) − 1.00000i − 0.0346479i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 36.0000i − 1.24434i
\(838\) 0 0
\(839\) 2.00000 0.0690477 0.0345238 0.999404i \(-0.489009\pi\)
0.0345238 + 0.999404i \(0.489009\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 42.0000i 1.44656i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 14.0000i − 0.481046i
\(848\) 0 0
\(849\) −39.0000 −1.33848
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 34.0000i − 1.16414i −0.813139 0.582069i \(-0.802243\pi\)
0.813139 0.582069i \(-0.197757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 3.00000i − 0.102478i −0.998686 0.0512390i \(-0.983683\pi\)
0.998686 0.0512390i \(-0.0163170\pi\)
\(858\) 0 0
\(859\) −51.0000 −1.74010 −0.870049 0.492966i \(-0.835913\pi\)
−0.870049 + 0.492966i \(0.835913\pi\)
\(860\) 0 0
\(861\) 33.0000 1.12464
\(862\) 0 0
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 48.0000i 1.63017i
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) 54.0000 1.82972
\(872\) 0 0
\(873\) − 60.0000i − 2.03069i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 0 0
\(879\) 42.0000 1.41662
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) − 15.0000i − 0.504790i −0.967624 0.252395i \(-0.918782\pi\)
0.967624 0.252395i \(-0.0812183\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 34.0000i − 1.14161i −0.821086 0.570804i \(-0.806632\pi\)
0.821086 0.570804i \(-0.193368\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 45.0000 1.50756
\(892\) 0 0
\(893\) − 6.00000i − 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 24.0000i 0.798670i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 4.00000i − 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) − 55.0000i − 1.82023i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.00000i 0.264183i
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −39.0000 −1.28509
\(922\) 0 0
\(923\) − 60.0000i − 1.97492i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 24.0000i 0.788263i
\(928\) 0 0
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 0 0
\(933\) − 18.0000i − 0.589294i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) 56.0000 1.82555 0.912774 0.408465i \(-0.133936\pi\)
0.912774 + 0.408465i \(0.133936\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) −42.0000 −1.36338
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 9.00000i 0.291539i 0.989319 + 0.145769i \(0.0465657\pi\)
−0.989319 + 0.145769i \(0.953434\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 90.0000i 2.90929i
\(958\) 0 0
\(959\) 3.00000 0.0968751
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 18.0000i − 0.580042i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 2.00000i − 0.0643157i −0.999483 0.0321578i \(-0.989762\pi\)
0.999483 0.0321578i \(-0.0102379\pi\)
\(968\) 0 0
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) 11.0000i 0.352644i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 21.0000i − 0.671850i −0.941889 0.335925i \(-0.890951\pi\)
0.941889 0.335925i \(-0.109049\pi\)
\(978\) 0 0
\(979\) 55.0000 1.75781
\(980\) 0 0
\(981\) −108.000 −3.44817
\(982\) 0 0
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 51.0000i 1.61844i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 58.0000i 1.83688i 0.395562 + 0.918439i \(0.370550\pi\)
−0.395562 + 0.918439i \(0.629450\pi\)
\(998\) 0 0
\(999\) −72.0000 −2.27798
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 2800.2.g.a.449.2 2
4.3 odd 2 350.2.c.a.99.1 2
5.2 odd 4 2800.2.a.bg.1.1 1
5.3 odd 4 2800.2.a.b.1.1 1
5.4 even 2 inner 2800.2.g.a.449.1 2
12.11 even 2 3150.2.g.v.2899.2 2
20.3 even 4 350.2.a.c.1.1 1
20.7 even 4 350.2.a.d.1.1 yes 1
20.19 odd 2 350.2.c.a.99.2 2
28.27 even 2 2450.2.c.r.99.1 2
60.23 odd 4 3150.2.a.bq.1.1 1
60.47 odd 4 3150.2.a.j.1.1 1
60.59 even 2 3150.2.g.v.2899.1 2
140.27 odd 4 2450.2.a.bg.1.1 1
140.83 odd 4 2450.2.a.a.1.1 1
140.139 even 2 2450.2.c.r.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.c.1.1 1 20.3 even 4
350.2.a.d.1.1 yes 1 20.7 even 4
350.2.c.a.99.1 2 4.3 odd 2
350.2.c.a.99.2 2 20.19 odd 2
2450.2.a.a.1.1 1 140.83 odd 4
2450.2.a.bg.1.1 1 140.27 odd 4
2450.2.c.r.99.1 2 28.27 even 2
2450.2.c.r.99.2 2 140.139 even 2
2800.2.a.b.1.1 1 5.3 odd 4
2800.2.a.bg.1.1 1 5.2 odd 4
2800.2.g.a.449.1 2 5.4 even 2 inner
2800.2.g.a.449.2 2 1.1 even 1 trivial
3150.2.a.j.1.1 1 60.47 odd 4
3150.2.a.bq.1.1 1 60.23 odd 4
3150.2.g.v.2899.1 2 60.59 even 2
3150.2.g.v.2899.2 2 12.11 even 2