Properties

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

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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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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.m.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} +1.00000i q^{13} -3.00000i q^{17} -6.00000 q^{19} +1.00000 q^{21} +6.00000i q^{23} +5.00000i q^{27} +9.00000 q^{29} +5.00000i q^{33} -6.00000i q^{37} -1.00000 q^{39} +8.00000 q^{41} -6.00000i q^{43} +3.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} -12.0000i q^{53} -6.00000i q^{57} +8.00000 q^{59} -4.00000 q^{61} -2.00000i q^{63} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} +10.0000i q^{73} -5.00000i q^{77} -3.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +16.0000 q^{89} +1.00000 q^{91} -7.00000i q^{97} +10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} + 10q^{11} - 12q^{19} + 2q^{21} + 18q^{29} - 2q^{39} + 16q^{41} - 2q^{49} + 6q^{51} + 16q^{59} - 8q^{61} - 12q^{69} - 16q^{71} - 6q^{79} + 2q^{81} + 32q^{89} + 2q^{91} + 20q^{99} + O(q^{100}) \)

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
\(3\) 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
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 5.00000i 0.870388i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 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
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.00000i − 0.569803i
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.00000i 0.964901i
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 9.00000i 0.886796i 0.896325 + 0.443398i \(0.146227\pi\)
−0.896325 + 0.443398i \(0.853773\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.0000i 1.36697i 0.729964 + 0.683486i \(0.239537\pi\)
−0.729964 + 0.683486i \(0.760463\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) 5.00000i 0.418121i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) − 6.00000i − 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 9.00000i − 0.696441i −0.937413 0.348220i \(-0.886786\pi\)
0.937413 0.348220i \(-0.113214\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) 0 0
\(173\) 19.0000i 1.44454i 0.691609 + 0.722272i \(0.256902\pi\)
−0.691609 + 0.722272i \(0.743098\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.00000i 0.601317i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) − 4.00000i − 0.295689i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 15.0000i − 1.09691i
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 0 0
\(193\) 8.00000i 0.575853i 0.957653 + 0.287926i \(0.0929658\pi\)
−0.957653 + 0.287926i \(0.907034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) − 9.00000i − 0.631676i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0000i 0.834058i
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) − 8.00000i − 0.548151i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 25.0000i 1.67412i 0.547108 + 0.837062i \(0.315729\pi\)
−0.547108 + 0.837062i \(0.684271\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 13.0000i − 0.862840i −0.902151 0.431420i \(-0.858013\pi\)
0.902151 0.431420i \(-0.141987\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) 0 0
\(233\) 8.00000i 0.524097i 0.965055 + 0.262049i \(0.0843981\pi\)
−0.965055 + 0.262049i \(0.915602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 3.00000i − 0.194871i
\(238\) 0 0
\(239\) −7.00000 −0.452792 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.00000i − 0.381771i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 0 0
\(253\) 30.0000i 1.88608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) − 18.0000i − 1.10993i −0.831875 0.554964i \(-0.812732\pi\)
0.831875 0.554964i \(-0.187268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 16.0000i 0.979184i
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 1.00000i 0.0605228i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 14.0000i − 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) − 29.0000i − 1.72387i −0.507018 0.861936i \(-0.669252\pi\)
0.507018 0.861936i \(-0.330748\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.00000i − 0.472225i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) 0 0
\(293\) − 1.00000i − 0.0584206i −0.999573 0.0292103i \(-0.990701\pi\)
0.999573 0.0292103i \(-0.00929925\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 25.0000i 1.45065i
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) − 14.0000i − 0.804279i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 27.0000i − 1.54097i −0.637457 0.770486i \(-0.720014\pi\)
0.637457 0.770486i \(-0.279986\pi\)
\(308\) 0 0
\(309\) −9.00000 −0.511992
\(310\) 0 0
\(311\) 14.0000 0.793867 0.396934 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 0 0
\(313\) 29.0000i 1.63918i 0.572953 + 0.819588i \(0.305798\pi\)
−0.572953 + 0.819588i \(0.694202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 0 0
\(319\) 45.0000 2.51952
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) 18.0000i 1.00155i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.0000i 0.608301i
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) − 12.0000i − 0.657596i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.0000i − 0.536828i −0.963304 0.268414i \(-0.913500\pi\)
0.963304 0.268414i \(-0.0864995\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) − 33.0000i − 1.75641i −0.478282 0.878206i \(-0.658740\pi\)
0.478282 0.878206i \(-0.341260\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3.00000i − 0.158777i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 29.0000i − 1.51379i −0.653538 0.756894i \(-0.726716\pi\)
0.653538 0.756894i \(-0.273284\pi\)
\(368\) 0 0
\(369\) 16.0000 0.832927
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.00000i 0.463524i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 12.0000i − 0.609994i
\(388\) 0 0
\(389\) −25.0000 −1.26755 −0.633775 0.773517i \(-0.718496\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 29.0000i 1.45547i 0.685859 + 0.727734i \(0.259427\pi\)
−0.685859 + 0.727734i \(0.740573\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 9.00000 0.449439 0.224719 0.974424i \(-0.427853\pi\)
0.224719 + 0.974424i \(0.427853\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 30.0000i − 1.48704i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) 0 0
\(413\) − 8.00000i − 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 18.0000i 0.881464i
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) −23.0000 −1.10787 −0.553936 0.832560i \(-0.686875\pi\)
−0.553936 + 0.832560i \(0.686875\pi\)
\(432\) 0 0
\(433\) − 30.0000i − 1.44171i −0.693087 0.720854i \(-0.743750\pi\)
0.693087 0.720854i \(-0.256250\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 36.0000i − 1.72211i
\(438\) 0 0
\(439\) −34.0000 −1.62273 −0.811366 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) − 30.0000i − 1.42534i −0.701498 0.712672i \(-0.747485\pi\)
0.701498 0.712672i \(-0.252515\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.0000i 0.662177i
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 0 0
\(453\) 19.0000i 0.892698i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.0000i 1.52706i 0.645774 + 0.763529i \(0.276535\pi\)
−0.645774 + 0.763529i \(0.723465\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) − 30.0000i − 1.37940i
\(474\) 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
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 0 0
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) 0 0
\(493\) − 27.0000i − 1.21602i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) 31.0000i 1.38222i 0.722749 + 0.691111i \(0.242878\pi\)
−0.722749 + 0.691111i \(0.757122\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000i 0.532939i
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) − 30.0000i − 1.32453i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 15.0000i 0.659699i
\(518\) 0 0
\(519\) −19.0000 −0.834007
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 16.0000 0.694341
\(532\) 0 0
\(533\) 8.00000i 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −9.00000 −0.386940 −0.193470 0.981106i \(-0.561974\pi\)
−0.193470 + 0.981106i \(0.561974\pi\)
\(542\) 0 0
\(543\) − 20.0000i − 0.858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −54.0000 −2.30048
\(552\) 0 0
\(553\) 3.00000i 0.127573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.0000i − 1.01691i −0.861088 0.508456i \(-0.830216\pi\)
0.861088 0.508456i \(-0.169784\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 0 0
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) − 11.0000i − 0.459532i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 17.0000i − 0.707719i −0.935299 0.353860i \(-0.884869\pi\)
0.935299 0.353860i \(-0.115131\pi\)
\(578\) 0 0
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) − 60.0000i − 2.48495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) − 7.00000i − 0.287456i −0.989617 0.143728i \(-0.954091\pi\)
0.989617 0.143728i \(-0.0459090\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −33.0000 −1.34834 −0.674172 0.738575i \(-0.735499\pi\)
−0.674172 + 0.738575i \(0.735499\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17.0000i 0.690009i 0.938601 + 0.345004i \(0.112123\pi\)
−0.938601 + 0.345004i \(0.887877\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) − 22.0000i − 0.888572i −0.895885 0.444286i \(-0.853457\pi\)
0.895885 0.444286i \(-0.146543\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 10.0000i − 0.402585i −0.979531 0.201292i \(-0.935486\pi\)
0.979531 0.201292i \(-0.0645141\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) −30.0000 −1.20386
\(622\) 0 0
\(623\) − 16.0000i − 0.641026i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 30.0000i − 1.19808i
\(628\) 0 0
\(629\) −18.0000 −0.717707
\(630\) 0 0
\(631\) 9.00000 0.358284 0.179142 0.983823i \(-0.442668\pi\)
0.179142 + 0.983823i \(0.442668\pi\)
\(632\) 0 0
\(633\) − 13.0000i − 0.516704i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.00000i − 0.0396214i
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) − 47.0000i − 1.85350i −0.375680 0.926750i \(-0.622591\pi\)
0.375680 0.926750i \(-0.377409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 20.0000i 0.780274i
\(658\) 0 0
\(659\) −25.0000 −0.973862 −0.486931 0.873441i \(-0.661884\pi\)
−0.486931 + 0.873441i \(0.661884\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 0 0
\(663\) 3.00000i 0.116510i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 54.0000i 2.09089i
\(668\) 0 0
\(669\) −25.0000 −0.966556
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 32.0000i 1.23351i 0.787155 + 0.616755i \(0.211553\pi\)
−0.787155 + 0.616755i \(0.788447\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 33.0000i − 1.26829i −0.773213 0.634147i \(-0.781352\pi\)
0.773213 0.634147i \(-0.218648\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 13.0000 0.498161
\(682\) 0 0
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 16.0000i − 0.610438i
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) − 10.0000i − 0.379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 24.0000i − 0.909065i
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) 36.0000i 1.35777i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.0000i 0.526524i
\(708\) 0 0
\(709\) 41.0000 1.53979 0.769894 0.638172i \(-0.220309\pi\)
0.769894 + 0.638172i \(0.220309\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 7.00000i − 0.261420i
\(718\) 0 0
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) 9.00000 0.335178
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) 5.00000i 0.184679i 0.995728 + 0.0923396i \(0.0294345\pi\)
−0.995728 + 0.0923396i \(0.970565\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 20.0000i − 0.736709i
\(738\) 0 0
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.0000i 0.878114i
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 35.0000 1.27717 0.638584 0.769552i \(-0.279520\pi\)
0.638584 + 0.769552i \(0.279520\pi\)
\(752\) 0 0
\(753\) − 14.0000i − 0.510188i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 8.00000i − 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) 0 0
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) −46.0000 −1.66750 −0.833749 0.552143i \(-0.813810\pi\)
−0.833749 + 0.552143i \(0.813810\pi\)
\(762\) 0 0
\(763\) − 11.0000i − 0.398227i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000i 0.288863i
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 1.00000i 0.0359675i 0.999838 + 0.0179838i \(0.00572471\pi\)
−0.999838 + 0.0179838i \(0.994275\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 6.00000i − 0.215249i
\(778\) 0 0
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) 0 0
\(783\) 45.0000i 1.60817i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 11.0000i 0.392108i 0.980593 + 0.196054i \(0.0628127\pi\)
−0.980593 + 0.196054i \(0.937187\pi\)
\(788\) 0 0
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) − 4.00000i − 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 35.0000i − 1.23976i −0.784695 0.619882i \(-0.787181\pi\)
0.784695 0.619882i \(-0.212819\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 32.0000 1.13066
\(802\) 0 0
\(803\) 50.0000i 1.76446i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 10.0000i − 0.352017i
\(808\) 0 0
\(809\) 23.0000 0.808637 0.404318 0.914618i \(-0.367509\pi\)
0.404318 + 0.914618i \(0.367509\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) − 4.00000i − 0.140286i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 36.0000i 1.25948i
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −7.00000 −0.244302 −0.122151 0.992512i \(-0.538979\pi\)
−0.122151 + 0.992512i \(0.538979\pi\)
\(822\) 0 0
\(823\) − 32.0000i − 1.11545i −0.830026 0.557725i \(-0.811674\pi\)
0.830026 0.557725i \(-0.188326\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.00000i 0.0695468i 0.999395 + 0.0347734i \(0.0110710\pi\)
−0.999395 + 0.0347734i \(0.988929\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 0 0
\(833\) 3.00000i 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) − 13.0000i − 0.447744i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 14.0000i − 0.481046i
\(848\) 0 0
\(849\) 29.0000 0.995277
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.0000i 0.478231i 0.970991 + 0.239115i \(0.0768574\pi\)
−0.970991 + 0.239115i \(0.923143\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) − 20.0000i − 0.680808i −0.940279 0.340404i \(-0.889436\pi\)
0.940279 0.340404i \(-0.110564\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) − 14.0000i − 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.0000i 0.337676i 0.985644 + 0.168838i \(0.0540015\pi\)
−0.985644 + 0.168838i \(0.945999\pi\)
\(878\) 0 0
\(879\) 1.00000 0.0337292
\(880\) 0 0
\(881\) −16.0000 −0.539054 −0.269527 0.962993i \(-0.586867\pi\)
−0.269527 + 0.962993i \(0.586867\pi\)
\(882\) 0 0
\(883\) 16.0000i 0.538443i 0.963078 + 0.269221i \(0.0867663\pi\)
−0.963078 + 0.269221i \(0.913234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) − 18.0000i − 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 6.00000i − 0.200334i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 6.00000i − 0.199667i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 30.0000i − 0.996134i −0.867139 0.498067i \(-0.834043\pi\)
0.867139 0.498067i \(-0.165957\pi\)
\(908\) 0 0
\(909\) −28.0000 −0.928701
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 60.0000i 1.98571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6.00000i − 0.198137i
\(918\) 0 0
\(919\) 25.0000 0.824674 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(920\) 0 0
\(921\) 27.0000 0.889680
\(922\) 0 0
\(923\) − 8.00000i − 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 18.0000i 0.591198i
\(928\) 0 0
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 14.0000i 0.458339i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 7.00000i − 0.228680i −0.993442 0.114340i \(-0.963525\pi\)
0.993442 0.114340i \(-0.0364753\pi\)
\(938\) 0 0
\(939\) −29.0000 −0.946379
\(940\) 0 0
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 0 0
\(943\) 48.0000i 1.56310i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 20.0000i 0.647864i 0.946080 + 0.323932i \(0.105005\pi\)
−0.946080 + 0.323932i \(0.894995\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 45.0000i 1.45464i
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 22.0000i 0.707472i 0.935345 + 0.353736i \(0.115089\pi\)
−0.935345 + 0.353736i \(0.884911\pi\)
\(968\) 0 0
\(969\) −18.0000 −0.578243
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) − 18.0000i − 0.577054i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 0 0
\(979\) 80.0000 2.55681
\(980\) 0 0
\(981\) 22.0000 0.702406
\(982\) 0 0
\(983\) − 9.00000i − 0.287055i −0.989646 0.143528i \(-0.954155\pi\)
0.989646 0.143528i \(-0.0458446\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.00000i 0.0954911i
\(988\) 0 0
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.00000i 0.0316703i 0.999875 + 0.0158352i \(0.00504070\pi\)
−0.999875 + 0.0158352i \(0.994959\pi\)
\(998\) 0 0
\(999\) 30.0000 0.949158
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.m.449.2 2
4.3 odd 2 1400.2.g.e.449.1 2
5.2 odd 4 560.2.a.e.1.1 1
5.3 odd 4 2800.2.a.i.1.1 1
5.4 even 2 inner 2800.2.g.m.449.1 2
15.2 even 4 5040.2.a.be.1.1 1
20.3 even 4 1400.2.a.k.1.1 1
20.7 even 4 280.2.a.b.1.1 1
20.19 odd 2 1400.2.g.e.449.2 2
35.27 even 4 3920.2.a.r.1.1 1
40.27 even 4 2240.2.a.v.1.1 1
40.37 odd 4 2240.2.a.j.1.1 1
60.47 odd 4 2520.2.a.p.1.1 1
140.27 odd 4 1960.2.a.k.1.1 1
140.47 odd 12 1960.2.q.e.361.1 2
140.67 even 12 1960.2.q.m.961.1 2
140.83 odd 4 9800.2.a.n.1.1 1
140.87 odd 12 1960.2.q.e.961.1 2
140.107 even 12 1960.2.q.m.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.b.1.1 1 20.7 even 4
560.2.a.e.1.1 1 5.2 odd 4
1400.2.a.k.1.1 1 20.3 even 4
1400.2.g.e.449.1 2 4.3 odd 2
1400.2.g.e.449.2 2 20.19 odd 2
1960.2.a.k.1.1 1 140.27 odd 4
1960.2.q.e.361.1 2 140.47 odd 12
1960.2.q.e.961.1 2 140.87 odd 12
1960.2.q.m.361.1 2 140.107 even 12
1960.2.q.m.961.1 2 140.67 even 12
2240.2.a.j.1.1 1 40.37 odd 4
2240.2.a.v.1.1 1 40.27 even 4
2520.2.a.p.1.1 1 60.47 odd 4
2800.2.a.i.1.1 1 5.3 odd 4
2800.2.g.m.449.1 2 5.4 even 2 inner
2800.2.g.m.449.2 2 1.1 even 1 trivial
3920.2.a.r.1.1 1 35.27 even 4
5040.2.a.be.1.1 1 15.2 even 4
9800.2.a.n.1.1 1 140.83 odd 4