Properties

Label 2800.2.g.s.449.3
Level $2800$
Weight $2$
Character 2800.449
Analytic conductor $22.358$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.s.449.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.23607i q^{3} -1.00000i q^{7} +1.47214 q^{9} +O(q^{10})\) \(q+1.23607i q^{3} -1.00000i q^{7} +1.47214 q^{9} -4.23607 q^{11} +3.23607i q^{13} -6.47214i q^{17} +4.47214 q^{19} +1.23607 q^{21} +1.76393i q^{23} +5.52786i q^{27} -5.00000 q^{29} +9.70820 q^{31} -5.23607i q^{33} +3.00000i q^{37} -4.00000 q^{39} +9.23607 q^{41} +6.23607i q^{43} +2.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} +0.472136i q^{53} +5.52786i q^{57} -1.70820 q^{59} +3.70820 q^{61} -1.47214i q^{63} -0.236068i q^{67} -2.18034 q^{69} +4.70820 q^{71} +13.2361i q^{73} +4.23607i q^{77} +11.1803 q^{79} -2.41641 q^{81} +5.70820i q^{83} -6.18034i q^{87} -12.7639 q^{89} +3.23607 q^{91} +12.0000i q^{93} +0.763932i q^{97} -6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + O(q^{10}) \) \( 4 q - 12 q^{9} - 8 q^{11} - 4 q^{21} - 20 q^{29} + 12 q^{31} - 16 q^{39} + 28 q^{41} - 4 q^{49} + 32 q^{51} + 20 q^{59} - 12 q^{61} + 36 q^{69} - 8 q^{71} + 44 q^{81} - 60 q^{89} + 4 q^{91} - 16 q^{99} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.23607i 0.713644i 0.934172 + 0.356822i \(0.116140\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 1.47214 0.490712
\(10\) 0 0
\(11\) −4.23607 −1.27722 −0.638611 0.769529i \(-0.720491\pi\)
−0.638611 + 0.769529i \(0.720491\pi\)
\(12\) 0 0
\(13\) 3.23607i 0.897524i 0.893651 + 0.448762i \(0.148135\pi\)
−0.893651 + 0.448762i \(0.851865\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.47214i − 1.56972i −0.619671 0.784862i \(-0.712734\pi\)
0.619671 0.784862i \(-0.287266\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) 1.76393i 0.367805i 0.982944 + 0.183903i \(0.0588731\pi\)
−0.982944 + 0.183903i \(0.941127\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.52786i 1.06384i
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 9.70820 1.74364 0.871822 0.489822i \(-0.162938\pi\)
0.871822 + 0.489822i \(0.162938\pi\)
\(32\) 0 0
\(33\) − 5.23607i − 0.911482i
\(34\) 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
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 9.23607 1.44243 0.721216 0.692711i \(-0.243584\pi\)
0.721216 + 0.692711i \(0.243584\pi\)
\(42\) 0 0
\(43\) 6.23607i 0.950991i 0.879718 + 0.475496i \(0.157731\pi\)
−0.879718 + 0.475496i \(0.842269\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 0.472136i 0.0648529i 0.999474 + 0.0324264i \(0.0103235\pi\)
−0.999474 + 0.0324264i \(0.989677\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.52786i 0.732183i
\(58\) 0 0
\(59\) −1.70820 −0.222389 −0.111195 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) 3.70820 0.474787 0.237393 0.971414i \(-0.423707\pi\)
0.237393 + 0.971414i \(0.423707\pi\)
\(62\) 0 0
\(63\) − 1.47214i − 0.185472i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 0.236068i − 0.0288403i −0.999896 0.0144201i \(-0.995410\pi\)
0.999896 0.0144201i \(-0.00459023\pi\)
\(68\) 0 0
\(69\) −2.18034 −0.262482
\(70\) 0 0
\(71\) 4.70820 0.558761 0.279381 0.960180i \(-0.409871\pi\)
0.279381 + 0.960180i \(0.409871\pi\)
\(72\) 0 0
\(73\) 13.2361i 1.54916i 0.632473 + 0.774582i \(0.282040\pi\)
−0.632473 + 0.774582i \(0.717960\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.23607i 0.482745i
\(78\) 0 0
\(79\) 11.1803 1.25789 0.628943 0.777451i \(-0.283488\pi\)
0.628943 + 0.777451i \(0.283488\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) 5.70820i 0.626557i 0.949661 + 0.313278i \(0.101427\pi\)
−0.949661 + 0.313278i \(0.898573\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.18034i − 0.662602i
\(88\) 0 0
\(89\) −12.7639 −1.35297 −0.676487 0.736455i \(-0.736499\pi\)
−0.676487 + 0.736455i \(0.736499\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 0 0
\(93\) 12.0000i 1.24434i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.763932i 0.0775655i 0.999248 + 0.0387828i \(0.0123480\pi\)
−0.999248 + 0.0387828i \(0.987652\pi\)
\(98\) 0 0
\(99\) −6.23607 −0.626748
\(100\) 0 0
\(101\) 9.23607 0.919023 0.459512 0.888172i \(-0.348024\pi\)
0.459512 + 0.888172i \(0.348024\pi\)
\(102\) 0 0
\(103\) − 0.472136i − 0.0465209i −0.999729 0.0232605i \(-0.992595\pi\)
0.999729 0.0232605i \(-0.00740471\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 18.4164 1.76397 0.881986 0.471276i \(-0.156206\pi\)
0.881986 + 0.471276i \(0.156206\pi\)
\(110\) 0 0
\(111\) −3.70820 −0.351967
\(112\) 0 0
\(113\) − 12.4164i − 1.16804i −0.811740 0.584019i \(-0.801479\pi\)
0.811740 0.584019i \(-0.198521\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.76393i 0.440426i
\(118\) 0 0
\(119\) −6.47214 −0.593300
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) 0 0
\(123\) 11.4164i 1.02938i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 17.6525i 1.56640i 0.621767 + 0.783202i \(0.286415\pi\)
−0.621767 + 0.783202i \(0.713585\pi\)
\(128\) 0 0
\(129\) −7.70820 −0.678670
\(130\) 0 0
\(131\) −0.944272 −0.0825014 −0.0412507 0.999149i \(-0.513134\pi\)
−0.0412507 + 0.999149i \(0.513134\pi\)
\(132\) 0 0
\(133\) − 4.47214i − 0.387783i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.94427i 0.593289i 0.954988 + 0.296645i \(0.0958677\pi\)
−0.954988 + 0.296645i \(0.904132\pi\)
\(138\) 0 0
\(139\) −20.6525 −1.75172 −0.875860 0.482565i \(-0.839705\pi\)
−0.875860 + 0.482565i \(0.839705\pi\)
\(140\) 0 0
\(141\) −2.47214 −0.208191
\(142\) 0 0
\(143\) − 13.7082i − 1.14634i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.23607i − 0.101949i
\(148\) 0 0
\(149\) 13.9443 1.14236 0.571180 0.820825i \(-0.306486\pi\)
0.571180 + 0.820825i \(0.306486\pi\)
\(150\) 0 0
\(151\) 15.7639 1.28285 0.641425 0.767185i \(-0.278343\pi\)
0.641425 + 0.767185i \(0.278343\pi\)
\(152\) 0 0
\(153\) − 9.52786i − 0.770282i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.23607i 0.417884i 0.977928 + 0.208942i \(0.0670019\pi\)
−0.977928 + 0.208942i \(0.932998\pi\)
\(158\) 0 0
\(159\) −0.583592 −0.0462819
\(160\) 0 0
\(161\) 1.76393 0.139017
\(162\) 0 0
\(163\) − 10.4721i − 0.820241i −0.912031 0.410120i \(-0.865487\pi\)
0.912031 0.410120i \(-0.134513\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 0.763932i − 0.0591148i −0.999563 0.0295574i \(-0.990590\pi\)
0.999563 0.0295574i \(-0.00940979\pi\)
\(168\) 0 0
\(169\) 2.52786 0.194451
\(170\) 0 0
\(171\) 6.58359 0.503460
\(172\) 0 0
\(173\) 20.4721i 1.55647i 0.627975 + 0.778234i \(0.283884\pi\)
−0.627975 + 0.778234i \(0.716116\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2.11146i − 0.158707i
\(178\) 0 0
\(179\) 3.41641 0.255354 0.127677 0.991816i \(-0.459248\pi\)
0.127677 + 0.991816i \(0.459248\pi\)
\(180\) 0 0
\(181\) −14.1803 −1.05402 −0.527008 0.849860i \(-0.676686\pi\)
−0.527008 + 0.849860i \(0.676686\pi\)
\(182\) 0 0
\(183\) 4.58359i 0.338829i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 27.4164i 2.00489i
\(188\) 0 0
\(189\) 5.52786 0.402093
\(190\) 0 0
\(191\) 2.47214 0.178877 0.0894387 0.995992i \(-0.471493\pi\)
0.0894387 + 0.995992i \(0.471493\pi\)
\(192\) 0 0
\(193\) 14.4164i 1.03772i 0.854861 + 0.518858i \(0.173643\pi\)
−0.854861 + 0.518858i \(0.826357\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.47214i 0.532368i 0.963922 + 0.266184i \(0.0857628\pi\)
−0.963922 + 0.266184i \(0.914237\pi\)
\(198\) 0 0
\(199\) 2.76393 0.195930 0.0979650 0.995190i \(-0.468767\pi\)
0.0979650 + 0.995190i \(0.468767\pi\)
\(200\) 0 0
\(201\) 0.291796 0.0205817
\(202\) 0 0
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.59675i 0.180486i
\(208\) 0 0
\(209\) −18.9443 −1.31040
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 5.81966i 0.398757i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.70820i − 0.659036i
\(218\) 0 0
\(219\) −16.3607 −1.10555
\(220\) 0 0
\(221\) 20.9443 1.40886
\(222\) 0 0
\(223\) − 2.18034i − 0.146006i −0.997332 0.0730032i \(-0.976742\pi\)
0.997332 0.0730032i \(-0.0232583\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.41641i 0.359500i 0.983712 + 0.179750i \(0.0575288\pi\)
−0.983712 + 0.179750i \(0.942471\pi\)
\(228\) 0 0
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) −5.23607 −0.344508
\(232\) 0 0
\(233\) 9.94427i 0.651471i 0.945461 + 0.325735i \(0.105612\pi\)
−0.945461 + 0.325735i \(0.894388\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.8197i 0.897683i
\(238\) 0 0
\(239\) −14.4721 −0.936125 −0.468062 0.883695i \(-0.655048\pi\)
−0.468062 + 0.883695i \(0.655048\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) 0 0
\(243\) 13.5967i 0.872232i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.4721i 0.920840i
\(248\) 0 0
\(249\) −7.05573 −0.447139
\(250\) 0 0
\(251\) 2.47214 0.156040 0.0780199 0.996952i \(-0.475140\pi\)
0.0780199 + 0.996952i \(0.475140\pi\)
\(252\) 0 0
\(253\) − 7.47214i − 0.469769i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.6525i 1.16351i 0.813364 + 0.581755i \(0.197634\pi\)
−0.813364 + 0.581755i \(0.802366\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) −7.36068 −0.455615
\(262\) 0 0
\(263\) 11.7639i 0.725395i 0.931907 + 0.362698i \(0.118144\pi\)
−0.931907 + 0.362698i \(0.881856\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 15.7771i − 0.965542i
\(268\) 0 0
\(269\) −1.70820 −0.104151 −0.0520755 0.998643i \(-0.516584\pi\)
−0.0520755 + 0.998643i \(0.516584\pi\)
\(270\) 0 0
\(271\) −10.2918 −0.625182 −0.312591 0.949888i \(-0.601197\pi\)
−0.312591 + 0.949888i \(0.601197\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 15.8885i 0.954650i 0.878727 + 0.477325i \(0.158394\pi\)
−0.878727 + 0.477325i \(0.841606\pi\)
\(278\) 0 0
\(279\) 14.2918 0.855627
\(280\) 0 0
\(281\) 29.3607 1.75151 0.875756 0.482755i \(-0.160364\pi\)
0.875756 + 0.482755i \(0.160364\pi\)
\(282\) 0 0
\(283\) − 9.41641i − 0.559747i −0.960037 0.279874i \(-0.909707\pi\)
0.960037 0.279874i \(-0.0902926\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 9.23607i − 0.545188i
\(288\) 0 0
\(289\) −24.8885 −1.46403
\(290\) 0 0
\(291\) −0.944272 −0.0553542
\(292\) 0 0
\(293\) − 9.12461i − 0.533066i −0.963826 0.266533i \(-0.914122\pi\)
0.963826 0.266533i \(-0.0858781\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 23.4164i − 1.35876i
\(298\) 0 0
\(299\) −5.70820 −0.330114
\(300\) 0 0
\(301\) 6.23607 0.359441
\(302\) 0 0
\(303\) 11.4164i 0.655855i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 31.4164i − 1.79303i −0.443014 0.896515i \(-0.646091\pi\)
0.443014 0.896515i \(-0.353909\pi\)
\(308\) 0 0
\(309\) 0.583592 0.0331994
\(310\) 0 0
\(311\) 20.3607 1.15455 0.577274 0.816550i \(-0.304116\pi\)
0.577274 + 0.816550i \(0.304116\pi\)
\(312\) 0 0
\(313\) − 28.4721i − 1.60934i −0.593722 0.804670i \(-0.702342\pi\)
0.593722 0.804670i \(-0.297658\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 19.3607i − 1.08740i −0.839278 0.543702i \(-0.817022\pi\)
0.839278 0.543702i \(-0.182978\pi\)
\(318\) 0 0
\(319\) 21.1803 1.18587
\(320\) 0 0
\(321\) 9.88854 0.551925
\(322\) 0 0
\(323\) − 28.9443i − 1.61050i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 22.7639i 1.25885i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 11.2918 0.620653 0.310327 0.950630i \(-0.399562\pi\)
0.310327 + 0.950630i \(0.399562\pi\)
\(332\) 0 0
\(333\) 4.41641i 0.242018i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 7.52786i − 0.410069i −0.978755 0.205034i \(-0.934269\pi\)
0.978755 0.205034i \(-0.0657306\pi\)
\(338\) 0 0
\(339\) 15.3475 0.833563
\(340\) 0 0
\(341\) −41.1246 −2.22702
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 15.7639i − 0.846252i −0.906071 0.423126i \(-0.860933\pi\)
0.906071 0.423126i \(-0.139067\pi\)
\(348\) 0 0
\(349\) 4.47214 0.239388 0.119694 0.992811i \(-0.461809\pi\)
0.119694 + 0.992811i \(0.461809\pi\)
\(350\) 0 0
\(351\) −17.8885 −0.954820
\(352\) 0 0
\(353\) − 20.1803i − 1.07409i −0.843553 0.537046i \(-0.819540\pi\)
0.843553 0.537046i \(-0.180460\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 8.00000i − 0.423405i
\(358\) 0 0
\(359\) 10.1246 0.534357 0.267178 0.963647i \(-0.413909\pi\)
0.267178 + 0.963647i \(0.413909\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 8.58359i 0.450522i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.12461i − 0.163103i −0.996669 0.0815517i \(-0.974012\pi\)
0.996669 0.0815517i \(-0.0259876\pi\)
\(368\) 0 0
\(369\) 13.5967 0.707818
\(370\) 0 0
\(371\) 0.472136 0.0245121
\(372\) 0 0
\(373\) − 15.8328i − 0.819792i −0.912132 0.409896i \(-0.865565\pi\)
0.912132 0.409896i \(-0.134435\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 16.1803i − 0.833330i
\(378\) 0 0
\(379\) 11.1803 0.574295 0.287148 0.957886i \(-0.407293\pi\)
0.287148 + 0.957886i \(0.407293\pi\)
\(380\) 0 0
\(381\) −21.8197 −1.11786
\(382\) 0 0
\(383\) − 28.7639i − 1.46977i −0.678193 0.734884i \(-0.737237\pi\)
0.678193 0.734884i \(-0.262763\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.18034i 0.466663i
\(388\) 0 0
\(389\) 32.8885 1.66752 0.833758 0.552131i \(-0.186185\pi\)
0.833758 + 0.552131i \(0.186185\pi\)
\(390\) 0 0
\(391\) 11.4164 0.577353
\(392\) 0 0
\(393\) − 1.16718i − 0.0588767i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 26.9443i 1.35229i 0.736767 + 0.676147i \(0.236352\pi\)
−0.736767 + 0.676147i \(0.763648\pi\)
\(398\) 0 0
\(399\) 5.52786 0.276739
\(400\) 0 0
\(401\) 11.4721 0.572891 0.286446 0.958097i \(-0.407526\pi\)
0.286446 + 0.958097i \(0.407526\pi\)
\(402\) 0 0
\(403\) 31.4164i 1.56496i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 12.7082i − 0.629922i
\(408\) 0 0
\(409\) −15.5279 −0.767803 −0.383902 0.923374i \(-0.625420\pi\)
−0.383902 + 0.923374i \(0.625420\pi\)
\(410\) 0 0
\(411\) −8.58359 −0.423397
\(412\) 0 0
\(413\) 1.70820i 0.0840552i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 25.5279i − 1.25010i
\(418\) 0 0
\(419\) −3.81966 −0.186603 −0.0933013 0.995638i \(-0.529742\pi\)
−0.0933013 + 0.995638i \(0.529742\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 0 0
\(423\) 2.94427i 0.143155i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3.70820i − 0.179453i
\(428\) 0 0
\(429\) 16.9443 0.818077
\(430\) 0 0
\(431\) −26.4721 −1.27512 −0.637559 0.770402i \(-0.720056\pi\)
−0.637559 + 0.770402i \(0.720056\pi\)
\(432\) 0 0
\(433\) − 16.3607i − 0.786244i −0.919486 0.393122i \(-0.871395\pi\)
0.919486 0.393122i \(-0.128605\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.88854i 0.377360i
\(438\) 0 0
\(439\) −21.7082 −1.03608 −0.518038 0.855358i \(-0.673337\pi\)
−0.518038 + 0.855358i \(0.673337\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) 0 0
\(443\) 7.41641i 0.352364i 0.984358 + 0.176182i \(0.0563748\pi\)
−0.984358 + 0.176182i \(0.943625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 17.2361i 0.815238i
\(448\) 0 0
\(449\) −29.4721 −1.39088 −0.695438 0.718586i \(-0.744790\pi\)
−0.695438 + 0.718586i \(0.744790\pi\)
\(450\) 0 0
\(451\) −39.1246 −1.84231
\(452\) 0 0
\(453\) 19.4853i 0.915499i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 21.4721i − 1.00442i −0.864744 0.502212i \(-0.832520\pi\)
0.864744 0.502212i \(-0.167480\pi\)
\(458\) 0 0
\(459\) 35.7771 1.66993
\(460\) 0 0
\(461\) 8.18034 0.380996 0.190498 0.981688i \(-0.438990\pi\)
0.190498 + 0.981688i \(0.438990\pi\)
\(462\) 0 0
\(463\) 21.8885i 1.01725i 0.860989 + 0.508623i \(0.169845\pi\)
−0.860989 + 0.508623i \(0.830155\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.9443i 0.506441i 0.967409 + 0.253220i \(0.0814897\pi\)
−0.967409 + 0.253220i \(0.918510\pi\)
\(468\) 0 0
\(469\) −0.236068 −0.0109006
\(470\) 0 0
\(471\) −6.47214 −0.298220
\(472\) 0 0
\(473\) − 26.4164i − 1.21463i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.695048i 0.0318241i
\(478\) 0 0
\(479\) −3.81966 −0.174525 −0.0872624 0.996185i \(-0.527812\pi\)
−0.0872624 + 0.996185i \(0.527812\pi\)
\(480\) 0 0
\(481\) −9.70820 −0.442656
\(482\) 0 0
\(483\) 2.18034i 0.0992089i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 10.2361i − 0.463841i −0.972735 0.231920i \(-0.925499\pi\)
0.972735 0.231920i \(-0.0745008\pi\)
\(488\) 0 0
\(489\) 12.9443 0.585360
\(490\) 0 0
\(491\) 10.2361 0.461947 0.230974 0.972960i \(-0.425809\pi\)
0.230974 + 0.972960i \(0.425809\pi\)
\(492\) 0 0
\(493\) 32.3607i 1.45745i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 4.70820i − 0.211192i
\(498\) 0 0
\(499\) 28.9443 1.29572 0.647862 0.761758i \(-0.275663\pi\)
0.647862 + 0.761758i \(0.275663\pi\)
\(500\) 0 0
\(501\) 0.944272 0.0421870
\(502\) 0 0
\(503\) − 43.8885i − 1.95689i −0.206499 0.978447i \(-0.566207\pi\)
0.206499 0.978447i \(-0.433793\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.12461i 0.138769i
\(508\) 0 0
\(509\) −9.34752 −0.414322 −0.207161 0.978307i \(-0.566422\pi\)
−0.207161 + 0.978307i \(0.566422\pi\)
\(510\) 0 0
\(511\) 13.2361 0.585529
\(512\) 0 0
\(513\) 24.7214i 1.09147i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.47214i − 0.372604i
\(518\) 0 0
\(519\) −25.3050 −1.11076
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) − 28.3607i − 1.24013i −0.784552 0.620063i \(-0.787107\pi\)
0.784552 0.620063i \(-0.212893\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 62.8328i − 2.73704i
\(528\) 0 0
\(529\) 19.8885 0.864719
\(530\) 0 0
\(531\) −2.51471 −0.109129
\(532\) 0 0
\(533\) 29.8885i 1.29462i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.22291i 0.182232i
\(538\) 0 0
\(539\) 4.23607 0.182460
\(540\) 0 0
\(541\) −1.94427 −0.0835908 −0.0417954 0.999126i \(-0.513308\pi\)
−0.0417954 + 0.999126i \(0.513308\pi\)
\(542\) 0 0
\(543\) − 17.5279i − 0.752193i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.2361i 0.608690i 0.952562 + 0.304345i \(0.0984376\pi\)
−0.952562 + 0.304345i \(0.901562\pi\)
\(548\) 0 0
\(549\) 5.45898 0.232984
\(550\) 0 0
\(551\) −22.3607 −0.952597
\(552\) 0 0
\(553\) − 11.1803i − 0.475436i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 44.8885i − 1.90199i −0.309208 0.950994i \(-0.600064\pi\)
0.309208 0.950994i \(-0.399936\pi\)
\(558\) 0 0
\(559\) −20.1803 −0.853537
\(560\) 0 0
\(561\) −33.8885 −1.43078
\(562\) 0 0
\(563\) − 9.41641i − 0.396854i −0.980116 0.198427i \(-0.936417\pi\)
0.980116 0.198427i \(-0.0635833\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.41641i 0.101480i
\(568\) 0 0
\(569\) −13.9443 −0.584574 −0.292287 0.956331i \(-0.594416\pi\)
−0.292287 + 0.956331i \(0.594416\pi\)
\(570\) 0 0
\(571\) 12.5967 0.527157 0.263579 0.964638i \(-0.415097\pi\)
0.263579 + 0.964638i \(0.415097\pi\)
\(572\) 0 0
\(573\) 3.05573i 0.127655i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) −17.8197 −0.740560
\(580\) 0 0
\(581\) 5.70820 0.236816
\(582\) 0 0
\(583\) − 2.00000i − 0.0828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.2361i 1.20670i 0.797476 + 0.603351i \(0.206168\pi\)
−0.797476 + 0.603351i \(0.793832\pi\)
\(588\) 0 0
\(589\) 43.4164 1.78894
\(590\) 0 0
\(591\) −9.23607 −0.379921
\(592\) 0 0
\(593\) − 25.3050i − 1.03915i −0.854425 0.519575i \(-0.826090\pi\)
0.854425 0.519575i \(-0.173910\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.41641i 0.139824i
\(598\) 0 0
\(599\) 11.1803 0.456816 0.228408 0.973565i \(-0.426648\pi\)
0.228408 + 0.973565i \(0.426648\pi\)
\(600\) 0 0
\(601\) −19.0557 −0.777299 −0.388650 0.921386i \(-0.627058\pi\)
−0.388650 + 0.921386i \(0.627058\pi\)
\(602\) 0 0
\(603\) − 0.347524i − 0.0141523i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 33.1246i − 1.34449i −0.740330 0.672243i \(-0.765331\pi\)
0.740330 0.672243i \(-0.234669\pi\)
\(608\) 0 0
\(609\) −6.18034 −0.250440
\(610\) 0 0
\(611\) −6.47214 −0.261835
\(612\) 0 0
\(613\) 17.5836i 0.710195i 0.934829 + 0.355097i \(0.115552\pi\)
−0.934829 + 0.355097i \(0.884448\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.9443i 0.480858i 0.970667 + 0.240429i \(0.0772882\pi\)
−0.970667 + 0.240429i \(0.922712\pi\)
\(618\) 0 0
\(619\) 1.70820 0.0686585 0.0343293 0.999411i \(-0.489071\pi\)
0.0343293 + 0.999411i \(0.489071\pi\)
\(620\) 0 0
\(621\) −9.75078 −0.391285
\(622\) 0 0
\(623\) 12.7639i 0.511376i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 23.4164i − 0.935161i
\(628\) 0 0
\(629\) 19.4164 0.774183
\(630\) 0 0
\(631\) 3.65248 0.145403 0.0727014 0.997354i \(-0.476838\pi\)
0.0727014 + 0.997354i \(0.476838\pi\)
\(632\) 0 0
\(633\) − 14.8328i − 0.589551i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.23607i − 0.128218i
\(638\) 0 0
\(639\) 6.93112 0.274191
\(640\) 0 0
\(641\) −9.83282 −0.388373 −0.194186 0.980965i \(-0.562207\pi\)
−0.194186 + 0.980965i \(0.562207\pi\)
\(642\) 0 0
\(643\) 9.52786i 0.375742i 0.982194 + 0.187871i \(0.0601587\pi\)
−0.982194 + 0.187871i \(0.939841\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 15.8885i − 0.624643i −0.949976 0.312322i \(-0.898893\pi\)
0.949976 0.312322i \(-0.101107\pi\)
\(648\) 0 0
\(649\) 7.23607 0.284041
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) − 42.9443i − 1.68054i −0.542169 0.840270i \(-0.682397\pi\)
0.542169 0.840270i \(-0.317603\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 19.4853i 0.760194i
\(658\) 0 0
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) 46.7214 1.81725 0.908625 0.417613i \(-0.137133\pi\)
0.908625 + 0.417613i \(0.137133\pi\)
\(662\) 0 0
\(663\) 25.8885i 1.00543i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 8.81966i − 0.341499i
\(668\) 0 0
\(669\) 2.69505 0.104197
\(670\) 0 0
\(671\) −15.7082 −0.606408
\(672\) 0 0
\(673\) − 28.4721i − 1.09752i −0.835980 0.548760i \(-0.815100\pi\)
0.835980 0.548760i \(-0.184900\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.3607i 1.16686i 0.812165 + 0.583428i \(0.198289\pi\)
−0.812165 + 0.583428i \(0.801711\pi\)
\(678\) 0 0
\(679\) 0.763932 0.0293170
\(680\) 0 0
\(681\) −6.69505 −0.256555
\(682\) 0 0
\(683\) − 26.1246i − 0.999630i −0.866132 0.499815i \(-0.833401\pi\)
0.866132 0.499815i \(-0.166599\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 5.52786i − 0.210901i
\(688\) 0 0
\(689\) −1.52786 −0.0582070
\(690\) 0 0
\(691\) −18.1803 −0.691613 −0.345806 0.938306i \(-0.612395\pi\)
−0.345806 + 0.938306i \(0.612395\pi\)
\(692\) 0 0
\(693\) 6.23607i 0.236889i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 59.7771i − 2.26422i
\(698\) 0 0
\(699\) −12.2918 −0.464918
\(700\) 0 0
\(701\) −46.9443 −1.77306 −0.886530 0.462670i \(-0.846891\pi\)
−0.886530 + 0.462670i \(0.846891\pi\)
\(702\) 0 0
\(703\) 13.4164i 0.506009i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.23607i − 0.347358i
\(708\) 0 0
\(709\) 47.8885 1.79849 0.899246 0.437443i \(-0.144116\pi\)
0.899246 + 0.437443i \(0.144116\pi\)
\(710\) 0 0
\(711\) 16.4590 0.617260
\(712\) 0 0
\(713\) 17.1246i 0.641322i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 17.8885i − 0.668060i
\(718\) 0 0
\(719\) −6.18034 −0.230488 −0.115244 0.993337i \(-0.536765\pi\)
−0.115244 + 0.993337i \(0.536765\pi\)
\(720\) 0 0
\(721\) −0.472136 −0.0175833
\(722\) 0 0
\(723\) − 15.4164i − 0.573342i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20.9443i 0.776780i 0.921495 + 0.388390i \(0.126969\pi\)
−0.921495 + 0.388390i \(0.873031\pi\)
\(728\) 0 0
\(729\) −24.0557 −0.890953
\(730\) 0 0
\(731\) 40.3607 1.49279
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000i 0.0368355i
\(738\) 0 0
\(739\) −5.65248 −0.207930 −0.103965 0.994581i \(-0.533153\pi\)
−0.103965 + 0.994581i \(0.533153\pi\)
\(740\) 0 0
\(741\) −17.8885 −0.657152
\(742\) 0 0
\(743\) − 1.52786i − 0.0560519i −0.999607 0.0280259i \(-0.991078\pi\)
0.999607 0.0280259i \(-0.00892210\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.40325i 0.307459i
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −20.9443 −0.764267 −0.382134 0.924107i \(-0.624811\pi\)
−0.382134 + 0.924107i \(0.624811\pi\)
\(752\) 0 0
\(753\) 3.05573i 0.111357i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 46.4164i 1.68703i 0.537103 + 0.843517i \(0.319519\pi\)
−0.537103 + 0.843517i \(0.680481\pi\)
\(758\) 0 0
\(759\) 9.23607 0.335248
\(760\) 0 0
\(761\) −43.7771 −1.58692 −0.793459 0.608624i \(-0.791722\pi\)
−0.793459 + 0.608624i \(0.791722\pi\)
\(762\) 0 0
\(763\) − 18.4164i − 0.666719i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5.52786i − 0.199600i
\(768\) 0 0
\(769\) −33.0132 −1.19048 −0.595242 0.803546i \(-0.702944\pi\)
−0.595242 + 0.803546i \(0.702944\pi\)
\(770\) 0 0
\(771\) −23.0557 −0.830332
\(772\) 0 0
\(773\) − 27.8197i − 1.00060i −0.865851 0.500302i \(-0.833222\pi\)
0.865851 0.500302i \(-0.166778\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.70820i 0.133031i
\(778\) 0 0
\(779\) 41.3050 1.47990
\(780\) 0 0
\(781\) −19.9443 −0.713662
\(782\) 0 0
\(783\) − 27.6393i − 0.987749i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 45.2361i − 1.61249i −0.591581 0.806246i \(-0.701496\pi\)
0.591581 0.806246i \(-0.298504\pi\)
\(788\) 0 0
\(789\) −14.5410 −0.517674
\(790\) 0 0
\(791\) −12.4164 −0.441477
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 8.58359i − 0.304046i −0.988377 0.152023i \(-0.951421\pi\)
0.988377 0.152023i \(-0.0485789\pi\)
\(798\) 0 0
\(799\) 12.9443 0.457935
\(800\) 0 0
\(801\) −18.7902 −0.663921
\(802\) 0 0
\(803\) − 56.0689i − 1.97863i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2.11146i − 0.0743268i
\(808\) 0 0
\(809\) −20.5279 −0.721721 −0.360861 0.932620i \(-0.617517\pi\)
−0.360861 + 0.932620i \(0.617517\pi\)
\(810\) 0 0
\(811\) −46.7214 −1.64061 −0.820304 0.571927i \(-0.806196\pi\)
−0.820304 + 0.571927i \(0.806196\pi\)
\(812\) 0 0
\(813\) − 12.7214i − 0.446158i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 27.8885i 0.975697i
\(818\) 0 0
\(819\) 4.76393 0.166465
\(820\) 0 0
\(821\) −24.8328 −0.866671 −0.433336 0.901233i \(-0.642664\pi\)
−0.433336 + 0.901233i \(0.642664\pi\)
\(822\) 0 0
\(823\) − 0.347524i − 0.0121139i −0.999982 0.00605697i \(-0.998072\pi\)
0.999982 0.00605697i \(-0.00192800\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.5410i 0.888148i 0.895990 + 0.444074i \(0.146467\pi\)
−0.895990 + 0.444074i \(0.853533\pi\)
\(828\) 0 0
\(829\) 52.3607 1.81856 0.909281 0.416183i \(-0.136632\pi\)
0.909281 + 0.416183i \(0.136632\pi\)
\(830\) 0 0
\(831\) −19.6393 −0.681280
\(832\) 0 0
\(833\) 6.47214i 0.224246i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 53.6656i 1.85496i
\(838\) 0 0
\(839\) 0.652476 0.0225260 0.0112630 0.999937i \(-0.496415\pi\)
0.0112630 + 0.999937i \(0.496415\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 36.2918i 1.24996i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 6.94427i − 0.238608i
\(848\) 0 0
\(849\) 11.6393 0.399460
\(850\) 0 0
\(851\) −5.29180 −0.181400
\(852\) 0 0
\(853\) − 0.583592i − 0.0199818i −0.999950 0.00999091i \(-0.996820\pi\)
0.999950 0.00999091i \(-0.00318026\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 38.1803i − 1.30422i −0.758126 0.652108i \(-0.773885\pi\)
0.758126 0.652108i \(-0.226115\pi\)
\(858\) 0 0
\(859\) −22.3607 −0.762937 −0.381468 0.924382i \(-0.624581\pi\)
−0.381468 + 0.924382i \(0.624581\pi\)
\(860\) 0 0
\(861\) 11.4164 0.389070
\(862\) 0 0
\(863\) 49.6525i 1.69019i 0.534616 + 0.845095i \(0.320456\pi\)
−0.534616 + 0.845095i \(0.679544\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 30.7639i − 1.04480i
\(868\) 0 0
\(869\) −47.3607 −1.60660
\(870\) 0 0
\(871\) 0.763932 0.0258848
\(872\) 0 0
\(873\) 1.12461i 0.0380623i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 14.3607i − 0.484926i −0.970161 0.242463i \(-0.922045\pi\)
0.970161 0.242463i \(-0.0779553\pi\)
\(878\) 0 0
\(879\) 11.2786 0.380419
\(880\) 0 0
\(881\) 28.1803 0.949420 0.474710 0.880142i \(-0.342553\pi\)
0.474710 + 0.880142i \(0.342553\pi\)
\(882\) 0 0
\(883\) − 50.5967i − 1.70272i −0.524585 0.851358i \(-0.675780\pi\)
0.524585 0.851358i \(-0.324220\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 52.6525i 1.76790i 0.467584 + 0.883949i \(0.345124\pi\)
−0.467584 + 0.883949i \(0.654876\pi\)
\(888\) 0 0
\(889\) 17.6525 0.592045
\(890\) 0 0
\(891\) 10.2361 0.342921
\(892\) 0 0
\(893\) 8.94427i 0.299309i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 7.05573i − 0.235584i
\(898\) 0 0
\(899\) −48.5410 −1.61893
\(900\) 0 0
\(901\) 3.05573 0.101801
\(902\) 0 0
\(903\) 7.70820i 0.256513i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.8328i 0.625333i 0.949863 + 0.312667i \(0.101222\pi\)
−0.949863 + 0.312667i \(0.898778\pi\)
\(908\) 0 0
\(909\) 13.5967 0.450976
\(910\) 0 0
\(911\) −23.1803 −0.767999 −0.383999 0.923333i \(-0.625454\pi\)
−0.383999 + 0.923333i \(0.625454\pi\)
\(912\) 0 0
\(913\) − 24.1803i − 0.800252i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.944272i 0.0311826i
\(918\) 0 0
\(919\) −32.2361 −1.06337 −0.531685 0.846942i \(-0.678441\pi\)
−0.531685 + 0.846942i \(0.678441\pi\)
\(920\) 0 0
\(921\) 38.8328 1.27958
\(922\) 0 0
\(923\) 15.2361i 0.501501i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 0.695048i − 0.0228284i
\(928\) 0 0
\(929\) −51.7082 −1.69649 −0.848246 0.529603i \(-0.822341\pi\)
−0.848246 + 0.529603i \(0.822341\pi\)
\(930\) 0 0
\(931\) −4.47214 −0.146568
\(932\) 0 0
\(933\) 25.1672i 0.823937i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.7639i 1.00501i 0.864573 + 0.502507i \(0.167589\pi\)
−0.864573 + 0.502507i \(0.832411\pi\)
\(938\) 0 0
\(939\) 35.1935 1.14850
\(940\) 0 0
\(941\) −0.763932 −0.0249035 −0.0124517 0.999922i \(-0.503964\pi\)
−0.0124517 + 0.999922i \(0.503964\pi\)
\(942\) 0 0
\(943\) 16.2918i 0.530534i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.8328i 0.611984i 0.952034 + 0.305992i \(0.0989881\pi\)
−0.952034 + 0.305992i \(0.901012\pi\)
\(948\) 0 0
\(949\) −42.8328 −1.39041
\(950\) 0 0
\(951\) 23.9311 0.776020
\(952\) 0 0
\(953\) 5.47214i 0.177260i 0.996065 + 0.0886299i \(0.0282489\pi\)
−0.996065 + 0.0886299i \(0.971751\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 26.1803i 0.846290i
\(958\) 0 0
\(959\) 6.94427 0.224242
\(960\) 0 0
\(961\) 63.2492 2.04030
\(962\) 0 0
\(963\) − 11.7771i − 0.379511i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 49.8885i 1.60431i 0.597118 + 0.802154i \(0.296313\pi\)
−0.597118 + 0.802154i \(0.703687\pi\)
\(968\) 0 0
\(969\) 35.7771 1.14933
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 20.6525i 0.662088i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 2.52786i − 0.0808735i −0.999182 0.0404368i \(-0.987125\pi\)
0.999182 0.0404368i \(-0.0128749\pi\)
\(978\) 0 0
\(979\) 54.0689 1.72805
\(980\) 0 0
\(981\) 27.1115 0.865602
\(982\) 0 0
\(983\) 32.5410i 1.03790i 0.854805 + 0.518949i \(0.173676\pi\)
−0.854805 + 0.518949i \(0.826324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.47214i 0.0786890i
\(988\) 0 0
\(989\) −11.0000 −0.349780
\(990\) 0 0
\(991\) 9.18034 0.291623 0.145812 0.989312i \(-0.453421\pi\)
0.145812 + 0.989312i \(0.453421\pi\)
\(992\) 0 0
\(993\) 13.9574i 0.442926i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 18.5836i − 0.588548i −0.955721 0.294274i \(-0.904922\pi\)
0.955721 0.294274i \(-0.0950779\pi\)
\(998\) 0 0
\(999\) −16.5836 −0.524682
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.s.449.3 4
4.3 odd 2 175.2.b.c.99.4 4
5.2 odd 4 2800.2.a.bh.1.2 2
5.3 odd 4 2800.2.a.bp.1.1 2
5.4 even 2 inner 2800.2.g.s.449.2 4
12.11 even 2 1575.2.d.k.1324.1 4
20.3 even 4 175.2.a.e.1.2 yes 2
20.7 even 4 175.2.a.d.1.1 2
20.19 odd 2 175.2.b.c.99.1 4
28.27 even 2 1225.2.b.k.99.4 4
60.23 odd 4 1575.2.a.n.1.1 2
60.47 odd 4 1575.2.a.s.1.2 2
60.59 even 2 1575.2.d.k.1324.4 4
140.27 odd 4 1225.2.a.n.1.1 2
140.83 odd 4 1225.2.a.u.1.2 2
140.139 even 2 1225.2.b.k.99.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 2 20.7 even 4
175.2.a.e.1.2 yes 2 20.3 even 4
175.2.b.c.99.1 4 20.19 odd 2
175.2.b.c.99.4 4 4.3 odd 2
1225.2.a.n.1.1 2 140.27 odd 4
1225.2.a.u.1.2 2 140.83 odd 4
1225.2.b.k.99.1 4 140.139 even 2
1225.2.b.k.99.4 4 28.27 even 2
1575.2.a.n.1.1 2 60.23 odd 4
1575.2.a.s.1.2 2 60.47 odd 4
1575.2.d.k.1324.1 4 12.11 even 2
1575.2.d.k.1324.4 4 60.59 even 2
2800.2.a.bh.1.2 2 5.2 odd 4
2800.2.a.bp.1.1 2 5.3 odd 4
2800.2.g.s.449.2 4 5.4 even 2 inner
2800.2.g.s.449.3 4 1.1 even 1 trivial