Properties

Label 1400.2.g.k.449.1
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.k.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155i q^{3} +1.00000i q^{7} -3.56155 q^{9} +O(q^{10})\) \(q-2.56155i q^{3} +1.00000i q^{7} -3.56155 q^{9} -2.56155 q^{11} +5.68466i q^{13} +3.43845i q^{17} -1.12311 q^{19} +2.56155 q^{21} +5.12311i q^{23} +1.43845i q^{27} -4.56155 q^{29} -10.2462 q^{31} +6.56155i q^{33} +8.24621i q^{37} +14.5616 q^{39} +7.12311 q^{41} -1.12311i q^{43} +6.56155i q^{47} -1.00000 q^{49} +8.80776 q^{51} +4.87689i q^{53} +2.87689i q^{57} +4.00000 q^{59} -15.1231 q^{61} -3.56155i q^{63} -14.2462i q^{67} +13.1231 q^{69} -12.2462i q^{73} -2.56155i q^{77} +11.6847 q^{79} -7.00000 q^{81} -12.0000i q^{83} +11.6847i q^{87} +3.12311 q^{89} -5.68466 q^{91} +26.2462i q^{93} +13.6847i q^{97} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 2q^{11} + 12q^{19} + 2q^{21} - 10q^{29} - 8q^{31} + 50q^{39} + 12q^{41} - 4q^{49} - 6q^{51} + 16q^{59} - 44q^{61} + 36q^{69} + 22q^{79} - 28q^{81} - 4q^{89} + 2q^{91} + 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 2.56155i − 1.47891i −0.673204 0.739457i \(-0.735083\pi\)
0.673204 0.739457i \(-0.264917\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 5.68466i 1.57664i 0.615265 + 0.788320i \(0.289049\pi\)
−0.615265 + 0.788320i \(0.710951\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.43845i 0.833946i 0.908919 + 0.416973i \(0.136909\pi\)
−0.908919 + 0.416973i \(0.863091\pi\)
\(18\) 0 0
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 0 0
\(21\) 2.56155 0.558977
\(22\) 0 0
\(23\) 5.12311i 1.06824i 0.845408 + 0.534121i \(0.179357\pi\)
−0.845408 + 0.534121i \(0.820643\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.43845i 0.276829i
\(28\) 0 0
\(29\) −4.56155 −0.847059 −0.423530 0.905882i \(-0.639209\pi\)
−0.423530 + 0.905882i \(0.639209\pi\)
\(30\) 0 0
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) 0 0
\(33\) 6.56155i 1.14222i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24621i 1.35567i 0.735215 + 0.677834i \(0.237081\pi\)
−0.735215 + 0.677834i \(0.762919\pi\)
\(38\) 0 0
\(39\) 14.5616 2.33171
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) 0 0
\(43\) − 1.12311i − 0.171272i −0.996326 0.0856360i \(-0.972708\pi\)
0.996326 0.0856360i \(-0.0272922\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.56155i 0.957101i 0.878060 + 0.478550i \(0.158838\pi\)
−0.878060 + 0.478550i \(0.841162\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.80776 1.23333
\(52\) 0 0
\(53\) 4.87689i 0.669893i 0.942237 + 0.334946i \(0.108718\pi\)
−0.942237 + 0.334946i \(0.891282\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.87689i 0.381054i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −15.1231 −1.93632 −0.968158 0.250341i \(-0.919457\pi\)
−0.968158 + 0.250341i \(0.919457\pi\)
\(62\) 0 0
\(63\) − 3.56155i − 0.448713i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 14.2462i − 1.74045i −0.492653 0.870226i \(-0.663973\pi\)
0.492653 0.870226i \(-0.336027\pi\)
\(68\) 0 0
\(69\) 13.1231 1.57984
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 12.2462i − 1.43331i −0.697428 0.716655i \(-0.745672\pi\)
0.697428 0.716655i \(-0.254328\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.56155i − 0.291916i
\(78\) 0 0
\(79\) 11.6847 1.31463 0.657313 0.753617i \(-0.271693\pi\)
0.657313 + 0.753617i \(0.271693\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.6847i 1.25273i
\(88\) 0 0
\(89\) 3.12311 0.331049 0.165524 0.986206i \(-0.447068\pi\)
0.165524 + 0.986206i \(0.447068\pi\)
\(90\) 0 0
\(91\) −5.68466 −0.595914
\(92\) 0 0
\(93\) 26.2462i 2.72161i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.6847i 1.38947i 0.719267 + 0.694733i \(0.244478\pi\)
−0.719267 + 0.694733i \(0.755522\pi\)
\(98\) 0 0
\(99\) 9.12311 0.916907
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 8.80776i − 0.867855i −0.900948 0.433927i \(-0.857127\pi\)
0.900948 0.433927i \(-0.142873\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.87689i 0.664814i 0.943136 + 0.332407i \(0.107861\pi\)
−0.943136 + 0.332407i \(0.892139\pi\)
\(108\) 0 0
\(109\) 8.56155 0.820048 0.410024 0.912075i \(-0.365520\pi\)
0.410024 + 0.912075i \(0.365520\pi\)
\(110\) 0 0
\(111\) 21.1231 2.00492
\(112\) 0 0
\(113\) 8.24621i 0.775738i 0.921714 + 0.387869i \(0.126789\pi\)
−0.921714 + 0.387869i \(0.873211\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 20.2462i − 1.87176i
\(118\) 0 0
\(119\) −3.43845 −0.315202
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 0 0
\(123\) − 18.2462i − 1.64521i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.2462i 0.909204i 0.890695 + 0.454602i \(0.150219\pi\)
−0.890695 + 0.454602i \(0.849781\pi\)
\(128\) 0 0
\(129\) −2.87689 −0.253296
\(130\) 0 0
\(131\) 6.87689 0.600837 0.300419 0.953807i \(-0.402874\pi\)
0.300419 + 0.953807i \(0.402874\pi\)
\(132\) 0 0
\(133\) − 1.12311i − 0.0973856i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.3693i 1.48396i 0.670422 + 0.741980i \(0.266113\pi\)
−0.670422 + 0.741980i \(0.733887\pi\)
\(138\) 0 0
\(139\) −3.36932 −0.285782 −0.142891 0.989738i \(-0.545640\pi\)
−0.142891 + 0.989738i \(0.545640\pi\)
\(140\) 0 0
\(141\) 16.8078 1.41547
\(142\) 0 0
\(143\) − 14.5616i − 1.21770i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.56155i 0.211273i
\(148\) 0 0
\(149\) −16.2462 −1.33094 −0.665471 0.746424i \(-0.731769\pi\)
−0.665471 + 0.746424i \(0.731769\pi\)
\(150\) 0 0
\(151\) 0.807764 0.0657349 0.0328675 0.999460i \(-0.489536\pi\)
0.0328675 + 0.999460i \(0.489536\pi\)
\(152\) 0 0
\(153\) − 12.2462i − 0.990048i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 12.2462i − 0.977354i −0.872465 0.488677i \(-0.837480\pi\)
0.872465 0.488677i \(-0.162520\pi\)
\(158\) 0 0
\(159\) 12.4924 0.990714
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) 21.6155i 1.69306i 0.532342 + 0.846529i \(0.321312\pi\)
−0.532342 + 0.846529i \(0.678688\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.8078i 1.91968i 0.280544 + 0.959841i \(0.409485\pi\)
−0.280544 + 0.959841i \(0.590515\pi\)
\(168\) 0 0
\(169\) −19.3153 −1.48580
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) − 22.1771i − 1.68609i −0.537841 0.843046i \(-0.680760\pi\)
0.537841 0.843046i \(-0.319240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 10.2462i − 0.770152i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 3.12311 0.232139 0.116069 0.993241i \(-0.462971\pi\)
0.116069 + 0.993241i \(0.462971\pi\)
\(182\) 0 0
\(183\) 38.7386i 2.86364i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.80776i − 0.644087i
\(188\) 0 0
\(189\) −1.43845 −0.104632
\(190\) 0 0
\(191\) 11.6847 0.845472 0.422736 0.906253i \(-0.361070\pi\)
0.422736 + 0.906253i \(0.361070\pi\)
\(192\) 0 0
\(193\) − 4.87689i − 0.351047i −0.984475 0.175523i \(-0.943838\pi\)
0.984475 0.175523i \(-0.0561617\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4.87689i − 0.347464i −0.984793 0.173732i \(-0.944417\pi\)
0.984793 0.173732i \(-0.0555827\pi\)
\(198\) 0 0
\(199\) −2.24621 −0.159230 −0.0796148 0.996826i \(-0.525369\pi\)
−0.0796148 + 0.996826i \(0.525369\pi\)
\(200\) 0 0
\(201\) −36.4924 −2.57398
\(202\) 0 0
\(203\) − 4.56155i − 0.320158i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 18.2462i − 1.26820i
\(208\) 0 0
\(209\) 2.87689 0.198999
\(210\) 0 0
\(211\) −13.4384 −0.925141 −0.462570 0.886583i \(-0.653073\pi\)
−0.462570 + 0.886583i \(0.653073\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 10.2462i − 0.695558i
\(218\) 0 0
\(219\) −31.3693 −2.11974
\(220\) 0 0
\(221\) −19.5464 −1.31483
\(222\) 0 0
\(223\) − 12.3153i − 0.824696i −0.911026 0.412348i \(-0.864709\pi\)
0.911026 0.412348i \(-0.135291\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 23.6847i − 1.57201i −0.618223 0.786003i \(-0.712147\pi\)
0.618223 0.786003i \(-0.287853\pi\)
\(228\) 0 0
\(229\) −13.3693 −0.883469 −0.441735 0.897146i \(-0.645637\pi\)
−0.441735 + 0.897146i \(0.645637\pi\)
\(230\) 0 0
\(231\) −6.56155 −0.431718
\(232\) 0 0
\(233\) 19.1231i 1.25280i 0.779503 + 0.626398i \(0.215472\pi\)
−0.779503 + 0.626398i \(0.784528\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 29.9309i − 1.94422i
\(238\) 0 0
\(239\) −19.0540 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 22.2462i 1.42710i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.38447i − 0.406234i
\(248\) 0 0
\(249\) −30.7386 −1.94798
\(250\) 0 0
\(251\) −11.3693 −0.717625 −0.358812 0.933410i \(-0.616818\pi\)
−0.358812 + 0.933410i \(0.616818\pi\)
\(252\) 0 0
\(253\) − 13.1231i − 0.825043i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.4924i 0.904012i 0.892015 + 0.452006i \(0.149292\pi\)
−0.892015 + 0.452006i \(0.850708\pi\)
\(258\) 0 0
\(259\) −8.24621 −0.512395
\(260\) 0 0
\(261\) 16.2462 1.00562
\(262\) 0 0
\(263\) 15.3693i 0.947713i 0.880602 + 0.473856i \(0.157138\pi\)
−0.880602 + 0.473856i \(0.842862\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 8.00000i − 0.489592i
\(268\) 0 0
\(269\) 12.2462 0.746665 0.373332 0.927698i \(-0.378215\pi\)
0.373332 + 0.927698i \(0.378215\pi\)
\(270\) 0 0
\(271\) 10.2462 0.622413 0.311207 0.950342i \(-0.399267\pi\)
0.311207 + 0.950342i \(0.399267\pi\)
\(272\) 0 0
\(273\) 14.5616i 0.881305i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.24621i 0.495467i 0.968828 + 0.247733i \(0.0796857\pi\)
−0.968828 + 0.247733i \(0.920314\pi\)
\(278\) 0 0
\(279\) 36.4924 2.18474
\(280\) 0 0
\(281\) −15.4384 −0.920981 −0.460490 0.887665i \(-0.652326\pi\)
−0.460490 + 0.887665i \(0.652326\pi\)
\(282\) 0 0
\(283\) − 7.68466i − 0.456806i −0.973567 0.228403i \(-0.926650\pi\)
0.973567 0.228403i \(-0.0733503\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.12311i 0.420464i
\(288\) 0 0
\(289\) 5.17708 0.304534
\(290\) 0 0
\(291\) 35.0540 2.05490
\(292\) 0 0
\(293\) 5.05398i 0.295256i 0.989043 + 0.147628i \(0.0471639\pi\)
−0.989043 + 0.147628i \(0.952836\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 3.68466i − 0.213806i
\(298\) 0 0
\(299\) −29.1231 −1.68423
\(300\) 0 0
\(301\) 1.12311 0.0647347
\(302\) 0 0
\(303\) − 15.3693i − 0.882944i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.3153i − 0.931166i −0.885004 0.465583i \(-0.845845\pi\)
0.885004 0.465583i \(-0.154155\pi\)
\(308\) 0 0
\(309\) −22.5616 −1.28348
\(310\) 0 0
\(311\) −21.1231 −1.19778 −0.598891 0.800831i \(-0.704392\pi\)
−0.598891 + 0.800831i \(0.704392\pi\)
\(312\) 0 0
\(313\) − 14.3153i − 0.809151i −0.914505 0.404575i \(-0.867419\pi\)
0.914505 0.404575i \(-0.132581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 10.0000i − 0.561656i −0.959758 0.280828i \(-0.909391\pi\)
0.959758 0.280828i \(-0.0906090\pi\)
\(318\) 0 0
\(319\) 11.6847 0.654215
\(320\) 0 0
\(321\) 17.6155 0.983203
\(322\) 0 0
\(323\) − 3.86174i − 0.214873i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 21.9309i − 1.21278i
\(328\) 0 0
\(329\) −6.56155 −0.361750
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) − 29.3693i − 1.60943i
\(334\) 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
\(339\) 21.1231 1.14725
\(340\) 0 0
\(341\) 26.2462 1.42131
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.87689i 0.369171i 0.982816 + 0.184586i \(0.0590943\pi\)
−0.982816 + 0.184586i \(0.940906\pi\)
\(348\) 0 0
\(349\) 28.2462 1.51199 0.755993 0.654580i \(-0.227155\pi\)
0.755993 + 0.654580i \(0.227155\pi\)
\(350\) 0 0
\(351\) −8.17708 −0.436460
\(352\) 0 0
\(353\) 22.8078i 1.21393i 0.794727 + 0.606967i \(0.207614\pi\)
−0.794727 + 0.606967i \(0.792386\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.80776i 0.466156i
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) 11.3693i 0.596734i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 13.9309i − 0.727185i −0.931558 0.363593i \(-0.881550\pi\)
0.931558 0.363593i \(-0.118450\pi\)
\(368\) 0 0
\(369\) −25.3693 −1.32067
\(370\) 0 0
\(371\) −4.87689 −0.253196
\(372\) 0 0
\(373\) 9.36932i 0.485125i 0.970136 + 0.242562i \(0.0779879\pi\)
−0.970136 + 0.242562i \(0.922012\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 25.9309i − 1.33551i
\(378\) 0 0
\(379\) 0.492423 0.0252940 0.0126470 0.999920i \(-0.495974\pi\)
0.0126470 + 0.999920i \(0.495974\pi\)
\(380\) 0 0
\(381\) 26.2462 1.34463
\(382\) 0 0
\(383\) 26.2462i 1.34112i 0.741856 + 0.670559i \(0.233946\pi\)
−0.741856 + 0.670559i \(0.766054\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 31.3002 1.58698 0.793491 0.608582i \(-0.208261\pi\)
0.793491 + 0.608582i \(0.208261\pi\)
\(390\) 0 0
\(391\) −17.6155 −0.890856
\(392\) 0 0
\(393\) − 17.6155i − 0.888586i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.1771i 1.11304i 0.830836 + 0.556518i \(0.187863\pi\)
−0.830836 + 0.556518i \(0.812137\pi\)
\(398\) 0 0
\(399\) −2.87689 −0.144025
\(400\) 0 0
\(401\) 15.9309 0.795550 0.397775 0.917483i \(-0.369783\pi\)
0.397775 + 0.917483i \(0.369783\pi\)
\(402\) 0 0
\(403\) − 58.2462i − 2.90145i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 21.1231i − 1.04703i
\(408\) 0 0
\(409\) −14.4924 −0.716604 −0.358302 0.933606i \(-0.616644\pi\)
−0.358302 + 0.933606i \(0.616644\pi\)
\(410\) 0 0
\(411\) 44.4924 2.19465
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.63068i 0.422646i
\(418\) 0 0
\(419\) 1.75379 0.0856782 0.0428391 0.999082i \(-0.486360\pi\)
0.0428391 + 0.999082i \(0.486360\pi\)
\(420\) 0 0
\(421\) −0.561553 −0.0273684 −0.0136842 0.999906i \(-0.504356\pi\)
−0.0136842 + 0.999906i \(0.504356\pi\)
\(422\) 0 0
\(423\) − 23.3693i − 1.13626i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 15.1231i − 0.731858i
\(428\) 0 0
\(429\) −37.3002 −1.80087
\(430\) 0 0
\(431\) 5.93087 0.285680 0.142840 0.989746i \(-0.454377\pi\)
0.142840 + 0.989746i \(0.454377\pi\)
\(432\) 0 0
\(433\) − 36.2462i − 1.74188i −0.491388 0.870941i \(-0.663510\pi\)
0.491388 0.870941i \(-0.336490\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.75379i − 0.275241i
\(438\) 0 0
\(439\) −10.8769 −0.519126 −0.259563 0.965726i \(-0.583578\pi\)
−0.259563 + 0.965726i \(0.583578\pi\)
\(440\) 0 0
\(441\) 3.56155 0.169598
\(442\) 0 0
\(443\) − 21.6155i − 1.02698i −0.858094 0.513492i \(-0.828351\pi\)
0.858094 0.513492i \(-0.171649\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 41.6155i 1.96835i
\(448\) 0 0
\(449\) 33.6847 1.58968 0.794839 0.606821i \(-0.207555\pi\)
0.794839 + 0.606821i \(0.207555\pi\)
\(450\) 0 0
\(451\) −18.2462 −0.859181
\(452\) 0 0
\(453\) − 2.06913i − 0.0972162i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 33.3693i 1.56095i 0.625186 + 0.780475i \(0.285023\pi\)
−0.625186 + 0.780475i \(0.714977\pi\)
\(458\) 0 0
\(459\) −4.94602 −0.230861
\(460\) 0 0
\(461\) 27.1231 1.26325 0.631624 0.775274i \(-0.282388\pi\)
0.631624 + 0.775274i \(0.282388\pi\)
\(462\) 0 0
\(463\) − 10.2462i − 0.476182i −0.971243 0.238091i \(-0.923478\pi\)
0.971243 0.238091i \(-0.0765216\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.4384i 0.992053i 0.868307 + 0.496027i \(0.165208\pi\)
−0.868307 + 0.496027i \(0.834792\pi\)
\(468\) 0 0
\(469\) 14.2462 0.657829
\(470\) 0 0
\(471\) −31.3693 −1.44542
\(472\) 0 0
\(473\) 2.87689i 0.132280i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 17.3693i − 0.795286i
\(478\) 0 0
\(479\) 7.36932 0.336713 0.168356 0.985726i \(-0.446154\pi\)
0.168356 + 0.985726i \(0.446154\pi\)
\(480\) 0 0
\(481\) −46.8769 −2.13740
\(482\) 0 0
\(483\) 13.1231i 0.597122i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.12311i − 0.232150i −0.993240 0.116075i \(-0.962969\pi\)
0.993240 0.116075i \(-0.0370313\pi\)
\(488\) 0 0
\(489\) 55.3693 2.50389
\(490\) 0 0
\(491\) −15.6847 −0.707839 −0.353919 0.935276i \(-0.615151\pi\)
−0.353919 + 0.935276i \(0.615151\pi\)
\(492\) 0 0
\(493\) − 15.6847i − 0.706401i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 20.8078 0.931483 0.465742 0.884921i \(-0.345788\pi\)
0.465742 + 0.884921i \(0.345788\pi\)
\(500\) 0 0
\(501\) 63.5464 2.83904
\(502\) 0 0
\(503\) − 16.1771i − 0.721300i −0.932701 0.360650i \(-0.882555\pi\)
0.932701 0.360650i \(-0.117445\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 49.4773i 2.19736i
\(508\) 0 0
\(509\) 16.7386 0.741927 0.370963 0.928647i \(-0.379028\pi\)
0.370963 + 0.928647i \(0.379028\pi\)
\(510\) 0 0
\(511\) 12.2462 0.541740
\(512\) 0 0
\(513\) − 1.61553i − 0.0713273i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.8078i − 0.739205i
\(518\) 0 0
\(519\) −56.8078 −2.49358
\(520\) 0 0
\(521\) 4.24621 0.186030 0.0930149 0.995665i \(-0.470350\pi\)
0.0930149 + 0.995665i \(0.470350\pi\)
\(522\) 0 0
\(523\) 12.0000i 0.524723i 0.964970 + 0.262362i \(0.0845013\pi\)
−0.964970 + 0.262362i \(0.915499\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 35.2311i − 1.53469i
\(528\) 0 0
\(529\) −3.24621 −0.141140
\(530\) 0 0
\(531\) −14.2462 −0.618233
\(532\) 0 0
\(533\) 40.4924i 1.75392i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 10.2462i − 0.442157i
\(538\) 0 0
\(539\) 2.56155 0.110334
\(540\) 0 0
\(541\) −20.4233 −0.878066 −0.439033 0.898471i \(-0.644679\pi\)
−0.439033 + 0.898471i \(0.644679\pi\)
\(542\) 0 0
\(543\) − 8.00000i − 0.343313i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 0 0
\(549\) 53.8617 2.29876
\(550\) 0 0
\(551\) 5.12311 0.218252
\(552\) 0 0
\(553\) 11.6847i 0.496882i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.6155i 1.00062i 0.865846 + 0.500311i \(0.166781\pi\)
−0.865846 + 0.500311i \(0.833219\pi\)
\(558\) 0 0
\(559\) 6.38447 0.270034
\(560\) 0 0
\(561\) −22.5616 −0.952550
\(562\) 0 0
\(563\) 40.4924i 1.70655i 0.521459 + 0.853276i \(0.325388\pi\)
−0.521459 + 0.853276i \(0.674612\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.00000i − 0.293972i
\(568\) 0 0
\(569\) 46.9848 1.96971 0.984854 0.173388i \(-0.0554715\pi\)
0.984854 + 0.173388i \(0.0554715\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) − 29.9309i − 1.25038i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.561553i 0.0233777i 0.999932 + 0.0116889i \(0.00372077\pi\)
−0.999932 + 0.0116889i \(0.996279\pi\)
\(578\) 0 0
\(579\) −12.4924 −0.519167
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) − 12.4924i − 0.517383i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 6.24621i − 0.257809i −0.991657 0.128904i \(-0.958854\pi\)
0.991657 0.128904i \(-0.0411460\pi\)
\(588\) 0 0
\(589\) 11.5076 0.474161
\(590\) 0 0
\(591\) −12.4924 −0.513870
\(592\) 0 0
\(593\) 24.4233i 1.00294i 0.865174 + 0.501472i \(0.167208\pi\)
−0.865174 + 0.501472i \(0.832792\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.75379i 0.235487i
\(598\) 0 0
\(599\) 18.0691 0.738285 0.369142 0.929373i \(-0.379651\pi\)
0.369142 + 0.929373i \(0.379651\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 50.7386i 2.06624i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.68466i 0.149556i 0.997200 + 0.0747778i \(0.0238248\pi\)
−0.997200 + 0.0747778i \(0.976175\pi\)
\(608\) 0 0
\(609\) −11.6847 −0.473486
\(610\) 0 0
\(611\) −37.3002 −1.50900
\(612\) 0 0
\(613\) 32.7386i 1.32230i 0.750253 + 0.661150i \(0.229932\pi\)
−0.750253 + 0.661150i \(0.770068\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 32.2462i − 1.29818i −0.760710 0.649092i \(-0.775149\pi\)
0.760710 0.649092i \(-0.224851\pi\)
\(618\) 0 0
\(619\) −35.3693 −1.42161 −0.710806 0.703388i \(-0.751670\pi\)
−0.710806 + 0.703388i \(0.751670\pi\)
\(620\) 0 0
\(621\) −7.36932 −0.295720
\(622\) 0 0
\(623\) 3.12311i 0.125125i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 7.36932i − 0.294302i
\(628\) 0 0
\(629\) −28.3542 −1.13055
\(630\) 0 0
\(631\) 34.4233 1.37037 0.685185 0.728369i \(-0.259721\pi\)
0.685185 + 0.728369i \(0.259721\pi\)
\(632\) 0 0
\(633\) 34.4233i 1.36820i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 5.68466i − 0.225234i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.9848 1.69780 0.848900 0.528554i \(-0.177266\pi\)
0.848900 + 0.528554i \(0.177266\pi\)
\(642\) 0 0
\(643\) − 7.05398i − 0.278182i −0.990280 0.139091i \(-0.955582\pi\)
0.990280 0.139091i \(-0.0444180\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000i 0.314512i 0.987558 + 0.157256i \(0.0502649\pi\)
−0.987558 + 0.157256i \(0.949735\pi\)
\(648\) 0 0
\(649\) −10.2462 −0.402199
\(650\) 0 0
\(651\) −26.2462 −1.02867
\(652\) 0 0
\(653\) 5.50758i 0.215528i 0.994176 + 0.107764i \(0.0343691\pi\)
−0.994176 + 0.107764i \(0.965631\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 43.6155i 1.70160i
\(658\) 0 0
\(659\) 20.8078 0.810555 0.405278 0.914194i \(-0.367175\pi\)
0.405278 + 0.914194i \(0.367175\pi\)
\(660\) 0 0
\(661\) 23.6155 0.918538 0.459269 0.888297i \(-0.348111\pi\)
0.459269 + 0.888297i \(0.348111\pi\)
\(662\) 0 0
\(663\) 50.0691i 1.94452i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 23.3693i − 0.904864i
\(668\) 0 0
\(669\) −31.5464 −1.21965
\(670\) 0 0
\(671\) 38.7386 1.49549
\(672\) 0 0
\(673\) − 0.384472i − 0.0148203i −0.999973 0.00741015i \(-0.997641\pi\)
0.999973 0.00741015i \(-0.00235875\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 13.6847i − 0.525944i −0.964803 0.262972i \(-0.915297\pi\)
0.964803 0.262972i \(-0.0847027\pi\)
\(678\) 0 0
\(679\) −13.6847 −0.525169
\(680\) 0 0
\(681\) −60.6695 −2.32486
\(682\) 0 0
\(683\) 20.9848i 0.802963i 0.915867 + 0.401481i \(0.131505\pi\)
−0.915867 + 0.401481i \(0.868495\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 34.2462i 1.30657i
\(688\) 0 0
\(689\) −27.7235 −1.05618
\(690\) 0 0
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) 0 0
\(693\) 9.12311i 0.346558i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 24.4924i 0.927717i
\(698\) 0 0
\(699\) 48.9848 1.85278
\(700\) 0 0
\(701\) −21.6847 −0.819018 −0.409509 0.912306i \(-0.634300\pi\)
−0.409509 + 0.912306i \(0.634300\pi\)
\(702\) 0 0
\(703\) − 9.26137i − 0.349299i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) −33.0540 −1.24137 −0.620684 0.784061i \(-0.713145\pi\)
−0.620684 + 0.784061i \(0.713145\pi\)
\(710\) 0 0
\(711\) −41.6155 −1.56070
\(712\) 0 0
\(713\) − 52.4924i − 1.96586i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 48.8078i 1.82276i
\(718\) 0 0
\(719\) −17.6155 −0.656948 −0.328474 0.944513i \(-0.606534\pi\)
−0.328474 + 0.944513i \(0.606534\pi\)
\(720\) 0 0
\(721\) 8.80776 0.328018
\(722\) 0 0
\(723\) − 5.12311i − 0.190530i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 16.9848i − 0.629933i −0.949103 0.314967i \(-0.898007\pi\)
0.949103 0.314967i \(-0.101993\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) 3.86174 0.142832
\(732\) 0 0
\(733\) − 20.5616i − 0.759458i −0.925098 0.379729i \(-0.876017\pi\)
0.925098 0.379729i \(-0.123983\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.4924i 1.34422i
\(738\) 0 0
\(739\) 10.5616 0.388513 0.194257 0.980951i \(-0.437771\pi\)
0.194257 + 0.980951i \(0.437771\pi\)
\(740\) 0 0
\(741\) −16.3542 −0.600785
\(742\) 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
\(747\) 42.7386i 1.56372i
\(748\) 0 0
\(749\) −6.87689 −0.251276
\(750\) 0 0
\(751\) −43.6847 −1.59408 −0.797038 0.603929i \(-0.793601\pi\)
−0.797038 + 0.603929i \(0.793601\pi\)
\(752\) 0 0
\(753\) 29.1231i 1.06130i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 9.36932i − 0.340534i −0.985398 0.170267i \(-0.945537\pi\)
0.985398 0.170267i \(-0.0544630\pi\)
\(758\) 0 0
\(759\) −33.6155 −1.22017
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 8.56155i 0.309949i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.7386i 0.821044i
\(768\) 0 0
\(769\) 22.9848 0.828855 0.414427 0.910082i \(-0.363982\pi\)
0.414427 + 0.910082i \(0.363982\pi\)
\(770\) 0 0
\(771\) 37.1231 1.33696
\(772\) 0 0
\(773\) 9.19224i 0.330622i 0.986242 + 0.165311i \(0.0528627\pi\)
−0.986242 + 0.165311i \(0.947137\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 21.1231i 0.757787i
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 6.56155i − 0.234491i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 12.1771i − 0.434066i −0.976164 0.217033i \(-0.930362\pi\)
0.976164 0.217033i \(-0.0696379\pi\)
\(788\) 0 0
\(789\) 39.3693 1.40158
\(790\) 0 0
\(791\) −8.24621 −0.293202
\(792\) 0 0
\(793\) − 85.9697i − 3.05287i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 2.80776i − 0.0994561i −0.998763 0.0497281i \(-0.984165\pi\)
0.998763 0.0497281i \(-0.0158355\pi\)
\(798\) 0 0
\(799\) −22.5616 −0.798170
\(800\) 0 0
\(801\) −11.1231 −0.393016
\(802\) 0 0
\(803\) 31.3693i 1.10700i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 31.3693i − 1.10425i
\(808\) 0 0
\(809\) 33.0540 1.16212 0.581058 0.813862i \(-0.302639\pi\)
0.581058 + 0.813862i \(0.302639\pi\)
\(810\) 0 0
\(811\) 13.6155 0.478106 0.239053 0.971007i \(-0.423163\pi\)
0.239053 + 0.971007i \(0.423163\pi\)
\(812\) 0 0
\(813\) − 26.2462i − 0.920495i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.26137i 0.0441296i
\(818\) 0 0
\(819\) 20.2462 0.707460
\(820\) 0 0
\(821\) 0.699813 0.0244237 0.0122118 0.999925i \(-0.496113\pi\)
0.0122118 + 0.999925i \(0.496113\pi\)
\(822\) 0 0
\(823\) 34.2462i 1.19375i 0.802335 + 0.596874i \(0.203591\pi\)
−0.802335 + 0.596874i \(0.796409\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.1231i 1.15180i 0.817519 + 0.575902i \(0.195349\pi\)
−0.817519 + 0.575902i \(0.804651\pi\)
\(828\) 0 0
\(829\) 35.6155 1.23698 0.618489 0.785793i \(-0.287745\pi\)
0.618489 + 0.785793i \(0.287745\pi\)
\(830\) 0 0
\(831\) 21.1231 0.732752
\(832\) 0 0
\(833\) − 3.43845i − 0.119135i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 14.7386i − 0.509442i
\(838\) 0 0
\(839\) −10.8769 −0.375512 −0.187756 0.982216i \(-0.560121\pi\)
−0.187756 + 0.982216i \(0.560121\pi\)
\(840\) 0 0
\(841\) −8.19224 −0.282491
\(842\) 0 0
\(843\) 39.5464i 1.36205i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.43845i − 0.152507i
\(848\) 0 0
\(849\) −19.6847 −0.675576
\(850\) 0 0
\(851\) −42.2462 −1.44818
\(852\) 0 0
\(853\) − 32.2462i − 1.10409i −0.833815 0.552045i \(-0.813848\pi\)
0.833815 0.552045i \(-0.186152\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.26137i 0.111406i 0.998447 + 0.0557031i \(0.0177400\pi\)
−0.998447 + 0.0557031i \(0.982260\pi\)
\(858\) 0 0
\(859\) 28.9848 0.988950 0.494475 0.869192i \(-0.335360\pi\)
0.494475 + 0.869192i \(0.335360\pi\)
\(860\) 0 0
\(861\) 18.2462 0.621829
\(862\) 0 0
\(863\) 4.49242i 0.152924i 0.997073 + 0.0764619i \(0.0243624\pi\)
−0.997073 + 0.0764619i \(0.975638\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.2614i − 0.450380i
\(868\) 0 0
\(869\) −29.9309 −1.01534
\(870\) 0 0
\(871\) 80.9848 2.74407
\(872\) 0 0
\(873\) − 48.7386i − 1.64955i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 46.4924i − 1.56994i −0.619535 0.784969i \(-0.712679\pi\)
0.619535 0.784969i \(-0.287321\pi\)
\(878\) 0 0
\(879\) 12.9460 0.436659
\(880\) 0 0
\(881\) −20.1080 −0.677454 −0.338727 0.940885i \(-0.609996\pi\)
−0.338727 + 0.940885i \(0.609996\pi\)
\(882\) 0 0
\(883\) − 38.2462i − 1.28709i −0.765409 0.643544i \(-0.777463\pi\)
0.765409 0.643544i \(-0.222537\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 3.50758i − 0.117773i −0.998265 0.0588865i \(-0.981245\pi\)
0.998265 0.0588865i \(-0.0187550\pi\)
\(888\) 0 0
\(889\) −10.2462 −0.343647
\(890\) 0 0
\(891\) 17.9309 0.600707
\(892\) 0 0
\(893\) − 7.36932i − 0.246605i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 74.6004i 2.49083i
\(898\) 0 0
\(899\) 46.7386 1.55882
\(900\) 0 0
\(901\) −16.7689 −0.558655
\(902\) 0 0
\(903\) − 2.87689i − 0.0957371i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 26.1080i 0.866900i 0.901178 + 0.433450i \(0.142704\pi\)
−0.901178 + 0.433450i \(0.857296\pi\)
\(908\) 0 0
\(909\) −21.3693 −0.708776
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 30.7386i 1.01730i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.87689i 0.227095i
\(918\) 0 0
\(919\) −6.56155 −0.216446 −0.108223 0.994127i \(-0.534516\pi\)
−0.108223 + 0.994127i \(0.534516\pi\)
\(920\) 0 0
\(921\) −41.7926 −1.37711
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 31.3693i 1.03030i
\(928\) 0 0
\(929\) 11.1231 0.364937 0.182469 0.983212i \(-0.441591\pi\)
0.182469 + 0.983212i \(0.441591\pi\)
\(930\) 0 0
\(931\) 1.12311 0.0368083
\(932\) 0 0
\(933\) 54.1080i 1.77141i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17.1922i 0.561646i 0.959760 + 0.280823i \(0.0906074\pi\)
−0.959760 + 0.280823i \(0.909393\pi\)
\(938\) 0 0
\(939\) −36.6695 −1.19666
\(940\) 0 0
\(941\) 53.3693 1.73979 0.869895 0.493237i \(-0.164186\pi\)
0.869895 + 0.493237i \(0.164186\pi\)
\(942\) 0 0
\(943\) 36.4924i 1.18836i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.2311i 1.53480i 0.641167 + 0.767402i \(0.278451\pi\)
−0.641167 + 0.767402i \(0.721549\pi\)
\(948\) 0 0
\(949\) 69.6155 2.25982
\(950\) 0 0
\(951\) −25.6155 −0.830640
\(952\) 0 0
\(953\) − 5.86174i − 0.189880i −0.995483 0.0949402i \(-0.969734\pi\)
0.995483 0.0949402i \(-0.0302660\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 29.9309i − 0.967528i
\(958\) 0 0
\(959\) −17.3693 −0.560884
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) − 24.4924i − 0.789257i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 30.1080i − 0.968206i −0.875011 0.484103i \(-0.839146\pi\)
0.875011 0.484103i \(-0.160854\pi\)
\(968\) 0 0
\(969\) −9.89205 −0.317778
\(970\) 0 0
\(971\) 26.7386 0.858084 0.429042 0.903285i \(-0.358851\pi\)
0.429042 + 0.903285i \(0.358851\pi\)
\(972\) 0 0
\(973\) − 3.36932i − 0.108015i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 54.9848i − 1.75912i −0.475787 0.879561i \(-0.657837\pi\)
0.475787 0.879561i \(-0.342163\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −30.4924 −0.973548
\(982\) 0 0
\(983\) − 16.1771i − 0.515969i −0.966149 0.257984i \(-0.916942\pi\)
0.966149 0.257984i \(-0.0830583\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.8078i 0.534997i
\(988\) 0 0
\(989\) 5.75379 0.182960
\(990\) 0 0
\(991\) 12.4924 0.396835 0.198417 0.980118i \(-0.436420\pi\)
0.198417 + 0.980118i \(0.436420\pi\)
\(992\) 0 0
\(993\) 30.7386i 0.975461i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 28.0691i − 0.888958i −0.895789 0.444479i \(-0.853389\pi\)
0.895789 0.444479i \(-0.146611\pi\)
\(998\) 0 0
\(999\) −11.8617 −0.375289
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 1400.2.g.k.449.1 4
4.3 odd 2 2800.2.g.u.449.4 4
5.2 odd 4 1400.2.a.p.1.1 2
5.3 odd 4 280.2.a.d.1.2 2
5.4 even 2 inner 1400.2.g.k.449.4 4
15.8 even 4 2520.2.a.w.1.2 2
20.3 even 4 560.2.a.g.1.1 2
20.7 even 4 2800.2.a.bn.1.2 2
20.19 odd 2 2800.2.g.u.449.1 4
35.3 even 12 1960.2.q.u.961.2 4
35.13 even 4 1960.2.a.r.1.1 2
35.18 odd 12 1960.2.q.s.961.1 4
35.23 odd 12 1960.2.q.s.361.1 4
35.27 even 4 9800.2.a.by.1.2 2
35.33 even 12 1960.2.q.u.361.2 4
40.3 even 4 2240.2.a.bi.1.2 2
40.13 odd 4 2240.2.a.be.1.1 2
60.23 odd 4 5040.2.a.bq.1.1 2
140.83 odd 4 3920.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.2 2 5.3 odd 4
560.2.a.g.1.1 2 20.3 even 4
1400.2.a.p.1.1 2 5.2 odd 4
1400.2.g.k.449.1 4 1.1 even 1 trivial
1400.2.g.k.449.4 4 5.4 even 2 inner
1960.2.a.r.1.1 2 35.13 even 4
1960.2.q.s.361.1 4 35.23 odd 12
1960.2.q.s.961.1 4 35.18 odd 12
1960.2.q.u.361.2 4 35.33 even 12
1960.2.q.u.961.2 4 35.3 even 12
2240.2.a.be.1.1 2 40.13 odd 4
2240.2.a.bi.1.2 2 40.3 even 4
2520.2.a.w.1.2 2 15.8 even 4
2800.2.a.bn.1.2 2 20.7 even 4
2800.2.g.u.449.1 4 20.19 odd 2
2800.2.g.u.449.4 4 4.3 odd 2
3920.2.a.bu.1.2 2 140.83 odd 4
5040.2.a.bq.1.1 2 60.23 odd 4
9800.2.a.by.1.2 2 35.27 even 4