Properties

Label 2800.2.g.o.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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
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.o.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{7} +3.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} +4.00000i q^{17} -2.00000 q^{19} +5.00000i q^{23} -1.00000 q^{29} +2.00000 q^{31} +3.00000i q^{37} +12.0000 q^{41} +11.0000i q^{43} -2.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} -10.0000 q^{59} +4.00000 q^{61} +3.00000i q^{63} -1.00000i q^{67} +3.00000 q^{71} -1.00000i q^{77} -9.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} +6.00000 q^{89} +2.00000 q^{91} +14.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{9} + O(q^{10}) \) \( 2q + 6q^{9} - 2q^{11} - 4q^{19} - 2q^{29} + 4q^{31} + 24q^{41} - 2q^{49} - 20q^{59} + 8q^{61} + 6q^{71} - 18q^{79} + 18q^{81} + 12q^{89} + 4q^{91} - 6q^{99} + O(q^{100}) \)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00000i 1.04257i 0.853382 + 0.521286i \(0.174548\pi\)
−0.853382 + 0.521286i \(0.825452\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 11.0000i 1.67748i 0.544529 + 0.838742i \(0.316708\pi\)
−0.544529 + 0.838742i \(0.683292\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.00000i − 0.122169i −0.998133 0.0610847i \(-0.980544\pi\)
0.998133 0.0610847i \(-0.0194560\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.00000i − 0.113961i
\(78\) 0 0
\(79\) −9.00000 −1.01258 −0.506290 0.862364i \(-0.668983\pi\)
−0.506290 + 0.862364i \(0.668983\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 2.00000i − 0.219529i −0.993958 0.109764i \(-0.964990\pi\)
0.993958 0.109764i \(-0.0350096\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.00000i 0.282216i 0.989994 + 0.141108i \(0.0450665\pi\)
−0.989994 + 0.141108i \(0.954933\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 6.00000i − 0.554700i
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.00000i 0.266207i 0.991102 + 0.133103i \(0.0424943\pi\)
−0.991102 + 0.133103i \(0.957506\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.0000i 1.19610i 0.801459 + 0.598050i \(0.204058\pi\)
−0.801459 + 0.598050i \(0.795942\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 12.0000i 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.00000i 0.638470i 0.947676 + 0.319235i \(0.103426\pi\)
−0.947676 + 0.319235i \(0.896574\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.00000 −0.394055
\(162\) 0 0
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.00000i − 0.464294i −0.972681 0.232147i \(-0.925425\pi\)
0.972681 0.232147i \(-0.0745750\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 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
\(177\) 0 0
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.00000i − 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 19.0000i 1.36765i 0.729646 + 0.683825i \(0.239685\pi\)
−0.729646 + 0.683825i \(0.760315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 19.0000i − 1.35369i −0.736124 0.676847i \(-0.763346\pi\)
0.736124 0.676847i \(-0.236654\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.00000i − 0.0701862i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 15.0000i 1.04257i
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) − 20.0000i − 1.33930i −0.742677 0.669650i \(-0.766444\pi\)
0.742677 0.669650i \(-0.233556\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 3.00000i − 0.196537i −0.995160 0.0982683i \(-0.968670\pi\)
0.995160 0.0982683i \(-0.0313303\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) − 5.00000i − 0.314347i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) − 21.0000i − 1.29492i −0.762101 0.647458i \(-0.775832\pi\)
0.762101 0.647458i \(-0.224168\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 30.0000 1.82237 0.911185 0.411997i \(-0.135169\pi\)
0.911185 + 0.411997i \(0.135169\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −11.0000 −0.656205 −0.328102 0.944642i \(-0.606409\pi\)
−0.328102 + 0.944642i \(0.606409\pi\)
\(282\) 0 0
\(283\) − 14.0000i − 0.832214i −0.909316 0.416107i \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 32.0000i − 1.82634i −0.407583 0.913168i \(-0.633628\pi\)
0.407583 0.913168i \(-0.366372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.0000i − 0.842484i −0.906948 0.421242i \(-0.861594\pi\)
0.906948 0.421242i \(-0.138406\pi\)
\(318\) 0 0
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 9.00000 0.494685 0.247342 0.968928i \(-0.420443\pi\)
0.247342 + 0.968928i \(0.420443\pi\)
\(332\) 0 0
\(333\) 9.00000i 0.493197i
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.00000i − 0.375780i −0.982190 0.187890i \(-0.939835\pi\)
0.982190 0.187890i \(-0.0601648\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 0 0
\(369\) 36.0000 1.87409
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) − 29.0000i − 1.50156i −0.660551 0.750782i \(-0.729677\pi\)
0.660551 0.750782i \(-0.270323\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) 31.0000 1.59236 0.796182 0.605058i \(-0.206850\pi\)
0.796182 + 0.605058i \(0.206850\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 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
\(387\) 33.0000i 1.67748i
\(388\) 0 0
\(389\) −33.0000 −1.67317 −0.836583 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 34.0000i − 1.70641i −0.521575 0.853206i \(-0.674655\pi\)
0.521575 0.853206i \(-0.325345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.00000i − 0.148704i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 10.0000i − 0.492068i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −21.0000 −1.02348 −0.511739 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 20.0000i 0.961139i 0.876957 + 0.480569i \(0.159570\pi\)
−0.876957 + 0.480569i \(0.840430\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 10.0000i − 0.478365i
\(438\) 0 0
\(439\) 34.0000 1.62273 0.811366 0.584539i \(-0.198725\pi\)
0.811366 + 0.584539i \(0.198725\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000i 0.795226i 0.917553 + 0.397613i \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.0000i − 1.01804i −0.860755 0.509019i \(-0.830008\pi\)
0.860755 0.509019i \(-0.169992\pi\)
\(468\) 0 0
\(469\) 1.00000 0.0461757
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 11.0000i − 0.505781i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 18.0000i − 0.824163i
\(478\) 0 0
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.00000i 0.0453143i 0.999743 + 0.0226572i \(0.00721262\pi\)
−0.999743 + 0.0226572i \(0.992787\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 27.0000 1.21849 0.609246 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.00000i 0.134568i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 30.0000i − 1.33763i −0.743427 0.668817i \(-0.766801\pi\)
0.743427 0.668817i \(-0.233199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.00000i 0.0879599i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) −30.0000 −1.30189
\(532\) 0 0
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −41.0000 −1.76273 −0.881364 0.472438i \(-0.843374\pi\)
−0.881364 + 0.472438i \(0.843374\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.00000i − 0.0427569i −0.999771 0.0213785i \(-0.993195\pi\)
0.999771 0.0213785i \(-0.00680549\pi\)
\(548\) 0 0
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) − 9.00000i − 0.382719i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 11.0000i − 0.466085i −0.972467 0.233042i \(-0.925132\pi\)
0.972467 0.233042i \(-0.0748681\pi\)
\(558\) 0 0
\(559\) 22.0000 0.930501
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.00000i 0.0842900i 0.999112 + 0.0421450i \(0.0134191\pi\)
−0.999112 + 0.0421450i \(0.986581\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) 19.0000 0.796521 0.398261 0.917272i \(-0.369614\pi\)
0.398261 + 0.917272i \(0.369614\pi\)
\(570\) 0 0
\(571\) −43.0000 −1.79949 −0.899747 0.436412i \(-0.856249\pi\)
−0.899747 + 0.436412i \(0.856249\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 46.0000i 1.91501i 0.288425 + 0.957503i \(0.406868\pi\)
−0.288425 + 0.957503i \(0.593132\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) 0 0
\(583\) 6.00000i 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 36.0000i − 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 24.0000i − 0.985562i −0.870153 0.492781i \(-0.835980\pi\)
0.870153 0.492781i \(-0.164020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) 0 0
\(601\) 16.0000 0.652654 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(602\) 0 0
\(603\) − 3.00000i − 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 42.0000i − 1.70473i −0.522949 0.852364i \(-0.675168\pi\)
0.522949 0.852364i \(-0.324832\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) 0 0
\(613\) − 27.0000i − 1.09052i −0.838267 0.545260i \(-0.816431\pi\)
0.838267 0.545260i \(-0.183569\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 33.0000i − 1.32853i −0.747497 0.664265i \(-0.768745\pi\)
0.747497 0.664265i \(-0.231255\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 9.00000 0.356034
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 18.0000i − 0.707653i −0.935311 0.353827i \(-0.884880\pi\)
0.935311 0.353827i \(-0.115120\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.00000i − 0.193601i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) − 6.00000i − 0.231283i −0.993291 0.115642i \(-0.963108\pi\)
0.993291 0.115642i \(-0.0368924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 46.0000i − 1.76792i −0.467559 0.883962i \(-0.654866\pi\)
0.467559 0.883962i \(-0.345134\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 9.00000i − 0.344375i −0.985064 0.172188i \(-0.944916\pi\)
0.985064 0.172188i \(-0.0550836\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) − 3.00000i − 0.113961i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0000i 1.81813i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) − 6.00000i − 0.226294i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −27.0000 −1.01258
\(712\) 0 0
\(713\) 10.0000i 0.374503i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) 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
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −44.0000 −1.62740
\(732\) 0 0
\(733\) − 36.0000i − 1.32969i −0.746981 0.664845i \(-0.768498\pi\)
0.746981 0.664845i \(-0.231502\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000i 0.0368355i
\(738\) 0 0
\(739\) −3.00000 −0.110357 −0.0551784 0.998477i \(-0.517573\pi\)
−0.0551784 + 0.998477i \(0.517573\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 17.0000i − 0.617876i −0.951082 0.308938i \(-0.900027\pi\)
0.951082 0.308938i \(-0.0999735\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) − 5.00000i − 0.181012i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.0000i 0.722158i
\(768\) 0 0
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) − 8.00000i − 0.284088i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) −26.0000 −0.912983 −0.456492 0.889728i \(-0.650894\pi\)
−0.456492 + 0.889728i \(0.650894\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 22.0000i − 0.769683i
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) − 31.0000i − 1.08059i −0.841475 0.540296i \(-0.818312\pi\)
0.841475 0.540296i \(-0.181688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 27.0000i − 0.938882i −0.882964 0.469441i \(-0.844455\pi\)
0.882964 0.469441i \(-0.155545\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 4.00000i − 0.138592i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.0000i − 0.343604i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −15.0000 −0.514193
\(852\) 0 0
\(853\) − 16.0000i − 0.547830i −0.961754 0.273915i \(-0.911681\pi\)
0.961754 0.273915i \(-0.0883186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 12.0000i − 0.409912i −0.978771 0.204956i \(-0.934295\pi\)
0.978771 0.204956i \(-0.0657052\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 51.0000i 1.73606i 0.496512 + 0.868030i \(0.334614\pi\)
−0.496512 + 0.868030i \(0.665386\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) 0 0
\(871\) −2.00000 −0.0677674
\(872\) 0 0
\(873\) 42.0000i 1.42148i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.0000i 0.877958i 0.898497 + 0.438979i \(0.144660\pi\)
−0.898497 + 0.438979i \(0.855340\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 25.0000i 0.841317i 0.907219 + 0.420658i \(0.138201\pi\)
−0.907219 + 0.420658i \(0.861799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 44.0000i − 1.47738i −0.674048 0.738688i \(-0.735446\pi\)
0.674048 0.738688i \(-0.264554\pi\)
\(888\) 0 0
\(889\) −3.00000 −0.100617
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 0 0
\(893\) 4.00000i 0.133855i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 0 0
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) 41.0000 1.35839 0.679195 0.733958i \(-0.262329\pi\)
0.679195 + 0.733958i \(0.262329\pi\)
\(912\) 0 0
\(913\) 2.00000i 0.0661903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −25.0000 −0.824674 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 6.00000i − 0.197492i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 42.0000i 1.37946i
\(928\) 0 0
\(929\) 20.0000 0.656179 0.328089 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.0000i 0.653372i 0.945133 + 0.326686i \(0.105932\pi\)
−0.945133 + 0.326686i \(0.894068\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) 60.0000i 1.95387i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 9.00000i − 0.291539i −0.989319 0.145769i \(-0.953434\pi\)
0.989319 0.145769i \(-0.0465657\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 36.0000i 1.16008i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 24.0000i − 0.771788i −0.922543 0.385894i \(-0.873893\pi\)
0.922543 0.385894i \(-0.126107\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.0000 0.706014 0.353007 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 0 0
\(973\) 12.0000i 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.00000i 0.223950i 0.993711 + 0.111975i \(0.0357176\pi\)
−0.993711 + 0.111975i \(0.964282\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −15.0000 −0.478913
\(982\) 0 0
\(983\) − 30.0000i − 0.956851i −0.878128 0.478426i \(-0.841208\pi\)
0.878128 0.478426i \(-0.158792\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −55.0000 −1.74890
\(990\) 0 0
\(991\) 61.0000 1.93773 0.968864 0.247592i \(-0.0796392\pi\)
0.968864 + 0.247592i \(0.0796392\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.00000i 0.190022i 0.995476 + 0.0950110i \(0.0302886\pi\)
−0.995476 + 0.0950110i \(0.969711\pi\)
\(998\) 0 0
\(999\) 0 0
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.o.449.2 2
4.3 odd 2 1400.2.g.h.449.1 2
5.2 odd 4 2800.2.a.n.1.1 1
5.3 odd 4 2800.2.a.r.1.1 1
5.4 even 2 inner 2800.2.g.o.449.1 2
20.3 even 4 1400.2.a.f.1.1 1
20.7 even 4 1400.2.a.h.1.1 yes 1
20.19 odd 2 1400.2.g.h.449.2 2
140.27 odd 4 9800.2.a.w.1.1 1
140.83 odd 4 9800.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.f.1.1 1 20.3 even 4
1400.2.a.h.1.1 yes 1 20.7 even 4
1400.2.g.h.449.1 2 4.3 odd 2
1400.2.g.h.449.2 2 20.19 odd 2
2800.2.a.n.1.1 1 5.2 odd 4
2800.2.a.r.1.1 1 5.3 odd 4
2800.2.g.o.449.1 2 5.4 even 2 inner
2800.2.g.o.449.2 2 1.1 even 1 trivial
9800.2.a.v.1.1 1 140.83 odd 4
9800.2.a.w.1.1 1 140.27 odd 4