Properties

Label 1900.2.c.g.1749.2
Level $1900$
Weight $2$
Character 1900.1749
Analytic conductor $15.172$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Defining polynomial: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.2
Root \(-2.19869i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.g.1749.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.19869i q^{3} +1.19869i q^{7} +1.56314 q^{9} +O(q^{10})\) \(q-1.19869i q^{3} +1.19869i q^{7} +1.56314 q^{9} -5.86718 q^{11} +0.364448i q^{13} +1.19869i q^{17} +1.00000 q^{19} +1.43686 q^{21} +8.23163i q^{23} -5.46980i q^{27} +7.86718 q^{29} +7.30404 q^{31} +7.03293i q^{33} +7.13828i q^{37} +0.436861 q^{39} +2.43686 q^{41} +7.39738i q^{43} -13.7014i q^{47} +5.56314 q^{49} +1.43686 q^{51} +7.39738i q^{53} -1.19869i q^{57} -12.8606 q^{59} -1.30404 q^{61} +1.87372i q^{63} +11.9330i q^{67} +9.86718 q^{69} +2.12628 q^{71} -2.50273i q^{73} -7.03293i q^{77} +7.74090 q^{79} -1.86718 q^{81} +3.02093i q^{83} -9.43032i q^{87} +5.68942 q^{89} -0.436861 q^{91} -8.75529i q^{93} +1.09334i q^{97} -9.17122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{9} + O(q^{10}) \) \( 6q - 2q^{9} - 2q^{11} + 6q^{19} + 20q^{21} + 14q^{29} + 22q^{31} + 14q^{39} + 26q^{41} + 22q^{49} + 20q^{51} + 12q^{59} + 14q^{61} + 26q^{69} - 10q^{71} + 36q^{79} + 22q^{81} - 14q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 1.19869i − 0.692065i −0.938223 0.346032i \(-0.887529\pi\)
0.938223 0.346032i \(-0.112471\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.19869i 0.453063i 0.974004 + 0.226531i \(0.0727386\pi\)
−0.974004 + 0.226531i \(0.927261\pi\)
\(8\) 0 0
\(9\) 1.56314 0.521046
\(10\) 0 0
\(11\) −5.86718 −1.76902 −0.884510 0.466521i \(-0.845507\pi\)
−0.884510 + 0.466521i \(0.845507\pi\)
\(12\) 0 0
\(13\) 0.364448i 0.101080i 0.998722 + 0.0505399i \(0.0160942\pi\)
−0.998722 + 0.0505399i \(0.983906\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.19869i 0.290725i 0.989378 + 0.145363i \(0.0464349\pi\)
−0.989378 + 0.145363i \(0.953565\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.43686 0.313549
\(22\) 0 0
\(23\) 8.23163i 1.71641i 0.513305 + 0.858206i \(0.328421\pi\)
−0.513305 + 0.858206i \(0.671579\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.46980i − 1.05266i
\(28\) 0 0
\(29\) 7.86718 1.46090 0.730449 0.682967i \(-0.239311\pi\)
0.730449 + 0.682967i \(0.239311\pi\)
\(30\) 0 0
\(31\) 7.30404 1.31184 0.655922 0.754829i \(-0.272280\pi\)
0.655922 + 0.754829i \(0.272280\pi\)
\(32\) 0 0
\(33\) 7.03293i 1.22428i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.13828i 1.17353i 0.809759 + 0.586763i \(0.199598\pi\)
−0.809759 + 0.586763i \(0.800402\pi\)
\(38\) 0 0
\(39\) 0.436861 0.0699537
\(40\) 0 0
\(41\) 2.43686 0.380574 0.190287 0.981729i \(-0.439058\pi\)
0.190287 + 0.981729i \(0.439058\pi\)
\(42\) 0 0
\(43\) 7.39738i 1.12809i 0.825744 + 0.564045i \(0.190756\pi\)
−0.825744 + 0.564045i \(0.809244\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13.7014i − 1.99856i −0.0379717 0.999279i \(-0.512090\pi\)
0.0379717 0.999279i \(-0.487910\pi\)
\(48\) 0 0
\(49\) 5.56314 0.794734
\(50\) 0 0
\(51\) 1.43686 0.201201
\(52\) 0 0
\(53\) 7.39738i 1.01611i 0.861325 + 0.508054i \(0.169635\pi\)
−0.861325 + 0.508054i \(0.830365\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.19869i − 0.158771i
\(58\) 0 0
\(59\) −12.8606 −1.67431 −0.837156 0.546964i \(-0.815783\pi\)
−0.837156 + 0.546964i \(0.815783\pi\)
\(60\) 0 0
\(61\) −1.30404 −0.166965 −0.0834825 0.996509i \(-0.526604\pi\)
−0.0834825 + 0.996509i \(0.526604\pi\)
\(62\) 0 0
\(63\) 1.87372i 0.236067i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.9330i 1.45785i 0.684592 + 0.728927i \(0.259981\pi\)
−0.684592 + 0.728927i \(0.740019\pi\)
\(68\) 0 0
\(69\) 9.86718 1.18787
\(70\) 0 0
\(71\) 2.12628 0.252343 0.126171 0.992008i \(-0.459731\pi\)
0.126171 + 0.992008i \(0.459731\pi\)
\(72\) 0 0
\(73\) − 2.50273i − 0.292922i −0.989216 0.146461i \(-0.953212\pi\)
0.989216 0.146461i \(-0.0467883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7.03293i − 0.801477i
\(78\) 0 0
\(79\) 7.74090 0.870919 0.435460 0.900208i \(-0.356586\pi\)
0.435460 + 0.900208i \(0.356586\pi\)
\(80\) 0 0
\(81\) −1.86718 −0.207464
\(82\) 0 0
\(83\) 3.02093i 0.331590i 0.986160 + 0.165795i \(0.0530191\pi\)
−0.986160 + 0.165795i \(0.946981\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 9.43032i − 1.01104i
\(88\) 0 0
\(89\) 5.68942 0.603077 0.301539 0.953454i \(-0.402500\pi\)
0.301539 + 0.953454i \(0.402500\pi\)
\(90\) 0 0
\(91\) −0.436861 −0.0457954
\(92\) 0 0
\(93\) − 8.75529i − 0.907881i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.09334i 0.111012i 0.998458 + 0.0555061i \(0.0176772\pi\)
−0.998458 + 0.0555061i \(0.982323\pi\)
\(98\) 0 0
\(99\) −9.17122 −0.921742
\(100\) 0 0
\(101\) 10.8672 1.08132 0.540662 0.841240i \(-0.318174\pi\)
0.540662 + 0.841240i \(0.318174\pi\)
\(102\) 0 0
\(103\) 9.74090i 0.959799i 0.877323 + 0.479900i \(0.159327\pi\)
−0.877323 + 0.479900i \(0.840673\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.3634i 1.58191i 0.611877 + 0.790953i \(0.290415\pi\)
−0.611877 + 0.790953i \(0.709585\pi\)
\(108\) 0 0
\(109\) −9.29749 −0.890538 −0.445269 0.895397i \(-0.646892\pi\)
−0.445269 + 0.895397i \(0.646892\pi\)
\(110\) 0 0
\(111\) 8.55660 0.812156
\(112\) 0 0
\(113\) 2.96707i 0.279118i 0.990214 + 0.139559i \(0.0445685\pi\)
−0.990214 + 0.139559i \(0.955432\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.569683i 0.0526672i
\(118\) 0 0
\(119\) −1.43686 −0.131717
\(120\) 0 0
\(121\) 23.4238 2.12943
\(122\) 0 0
\(123\) − 2.92104i − 0.263382i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 11.3579i − 1.00785i −0.863747 0.503926i \(-0.831889\pi\)
0.863747 0.503926i \(-0.168111\pi\)
\(128\) 0 0
\(129\) 8.86718 0.780711
\(130\) 0 0
\(131\) −1.56314 −0.136572 −0.0682861 0.997666i \(-0.521753\pi\)
−0.0682861 + 0.997666i \(0.521753\pi\)
\(132\) 0 0
\(133\) 1.19869i 0.103940i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 15.8397i − 1.35328i −0.736315 0.676639i \(-0.763436\pi\)
0.736315 0.676639i \(-0.236564\pi\)
\(138\) 0 0
\(139\) 19.8606 1.68456 0.842278 0.539043i \(-0.181214\pi\)
0.842278 + 0.539043i \(0.181214\pi\)
\(140\) 0 0
\(141\) −16.4238 −1.38313
\(142\) 0 0
\(143\) − 2.13828i − 0.178812i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 6.66849i − 0.550007i
\(148\) 0 0
\(149\) −2.68942 −0.220326 −0.110163 0.993914i \(-0.535137\pi\)
−0.110163 + 0.993914i \(0.535137\pi\)
\(150\) 0 0
\(151\) 17.1647 1.39684 0.698421 0.715688i \(-0.253887\pi\)
0.698421 + 0.715688i \(0.253887\pi\)
\(152\) 0 0
\(153\) 1.87372i 0.151481i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 16.3634i − 1.30594i −0.757384 0.652969i \(-0.773523\pi\)
0.757384 0.652969i \(-0.226477\pi\)
\(158\) 0 0
\(159\) 8.86718 0.703213
\(160\) 0 0
\(161\) −9.86718 −0.777643
\(162\) 0 0
\(163\) 8.28549i 0.648970i 0.945891 + 0.324485i \(0.105191\pi\)
−0.945891 + 0.324485i \(0.894809\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 7.49727i − 0.580156i −0.957003 0.290078i \(-0.906319\pi\)
0.957003 0.290078i \(-0.0936813\pi\)
\(168\) 0 0
\(169\) 12.8672 0.989783
\(170\) 0 0
\(171\) 1.56314 0.119536
\(172\) 0 0
\(173\) 14.7464i 1.12114i 0.828105 + 0.560572i \(0.189419\pi\)
−0.828105 + 0.560572i \(0.810581\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.4159i 1.15873i
\(178\) 0 0
\(179\) −13.2526 −0.990543 −0.495271 0.868738i \(-0.664931\pi\)
−0.495271 + 0.868738i \(0.664931\pi\)
\(180\) 0 0
\(181\) 16.7344 1.24385 0.621927 0.783075i \(-0.286350\pi\)
0.621927 + 0.783075i \(0.286350\pi\)
\(182\) 0 0
\(183\) 1.56314i 0.115551i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 7.03293i − 0.514299i
\(188\) 0 0
\(189\) 6.55660 0.476922
\(190\) 0 0
\(191\) −4.13282 −0.299041 −0.149520 0.988759i \(-0.547773\pi\)
−0.149520 + 0.988759i \(0.547773\pi\)
\(192\) 0 0
\(193\) − 7.08680i − 0.510119i −0.966925 0.255060i \(-0.917905\pi\)
0.966925 0.255060i \(-0.0820951\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.4753i 1.45880i 0.684087 + 0.729401i \(0.260201\pi\)
−0.684087 + 0.729401i \(0.739799\pi\)
\(198\) 0 0
\(199\) −8.74090 −0.619626 −0.309813 0.950798i \(-0.600266\pi\)
−0.309813 + 0.950798i \(0.600266\pi\)
\(200\) 0 0
\(201\) 14.3040 1.00893
\(202\) 0 0
\(203\) 9.43032i 0.661878i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.8672i 0.894331i
\(208\) 0 0
\(209\) −5.86718 −0.405841
\(210\) 0 0
\(211\) −18.7344 −1.28973 −0.644863 0.764298i \(-0.723086\pi\)
−0.644863 + 0.764298i \(0.723086\pi\)
\(212\) 0 0
\(213\) − 2.54875i − 0.174638i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.75529i 0.594348i
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) −0.436861 −0.0293864
\(222\) 0 0
\(223\) − 4.27110i − 0.286014i −0.989722 0.143007i \(-0.954323\pi\)
0.989722 0.143007i \(-0.0456772\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 21.5631i − 1.43120i −0.698512 0.715598i \(-0.746154\pi\)
0.698512 0.715598i \(-0.253846\pi\)
\(228\) 0 0
\(229\) −27.4172 −1.81178 −0.905891 0.423511i \(-0.860797\pi\)
−0.905891 + 0.423511i \(0.860797\pi\)
\(230\) 0 0
\(231\) −8.43032 −0.554674
\(232\) 0 0
\(233\) − 7.86718i − 0.515396i −0.966226 0.257698i \(-0.917036\pi\)
0.966226 0.257698i \(-0.0829640\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 9.27895i − 0.602732i
\(238\) 0 0
\(239\) −9.30404 −0.601828 −0.300914 0.953651i \(-0.597292\pi\)
−0.300914 + 0.953651i \(0.597292\pi\)
\(240\) 0 0
\(241\) −21.9056 −1.41106 −0.705531 0.708679i \(-0.749291\pi\)
−0.705531 + 0.708679i \(0.749291\pi\)
\(242\) 0 0
\(243\) − 14.1712i − 0.909084i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.364448i 0.0231893i
\(248\) 0 0
\(249\) 3.62116 0.229482
\(250\) 0 0
\(251\) −7.43032 −0.468997 −0.234499 0.972116i \(-0.575345\pi\)
−0.234499 + 0.972116i \(0.575345\pi\)
\(252\) 0 0
\(253\) − 48.2964i − 3.03637i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.05387i 0.315251i 0.987499 + 0.157626i \(0.0503840\pi\)
−0.987499 + 0.157626i \(0.949616\pi\)
\(258\) 0 0
\(259\) −8.55660 −0.531681
\(260\) 0 0
\(261\) 12.2975 0.761196
\(262\) 0 0
\(263\) 17.7673i 1.09558i 0.836617 + 0.547789i \(0.184530\pi\)
−0.836617 + 0.547789i \(0.815470\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.81986i − 0.417368i
\(268\) 0 0
\(269\) −9.17776 −0.559578 −0.279789 0.960062i \(-0.590264\pi\)
−0.279789 + 0.960062i \(0.590264\pi\)
\(270\) 0 0
\(271\) −20.7278 −1.25912 −0.629562 0.776950i \(-0.716766\pi\)
−0.629562 + 0.776950i \(0.716766\pi\)
\(272\) 0 0
\(273\) 0.523661i 0.0316934i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 24.5895i 1.47744i 0.674012 + 0.738721i \(0.264570\pi\)
−0.674012 + 0.738721i \(0.735430\pi\)
\(278\) 0 0
\(279\) 11.4172 0.683532
\(280\) 0 0
\(281\) 3.73436 0.222773 0.111386 0.993777i \(-0.464471\pi\)
0.111386 + 0.993777i \(0.464471\pi\)
\(282\) 0 0
\(283\) 13.0318i 0.774663i 0.921941 + 0.387332i \(0.126603\pi\)
−0.921941 + 0.387332i \(0.873397\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.92104i 0.172424i
\(288\) 0 0
\(289\) 15.5631 0.915479
\(290\) 0 0
\(291\) 1.31058 0.0768277
\(292\) 0 0
\(293\) − 19.9001i − 1.16258i −0.813698 0.581288i \(-0.802549\pi\)
0.813698 0.581288i \(-0.197451\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 32.0923i 1.86218i
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −8.86718 −0.511096
\(302\) 0 0
\(303\) − 13.0264i − 0.748347i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.02093i 0.457779i 0.973452 + 0.228889i \(0.0735094\pi\)
−0.973452 + 0.228889i \(0.926491\pi\)
\(308\) 0 0
\(309\) 11.6763 0.664243
\(310\) 0 0
\(311\) −16.1778 −0.917357 −0.458678 0.888602i \(-0.651677\pi\)
−0.458678 + 0.888602i \(0.651677\pi\)
\(312\) 0 0
\(313\) 7.13828i 0.403480i 0.979439 + 0.201740i \(0.0646595\pi\)
−0.979439 + 0.201740i \(0.935340\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.96052i 0.222445i 0.993796 + 0.111223i \(0.0354766\pi\)
−0.993796 + 0.111223i \(0.964523\pi\)
\(318\) 0 0
\(319\) −46.1581 −2.58436
\(320\) 0 0
\(321\) 19.6146 1.09478
\(322\) 0 0
\(323\) 1.19869i 0.0666970i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.1448i 0.616310i
\(328\) 0 0
\(329\) 16.4238 0.905472
\(330\) 0 0
\(331\) −5.94852 −0.326960 −0.163480 0.986547i \(-0.552272\pi\)
−0.163480 + 0.986547i \(0.552272\pi\)
\(332\) 0 0
\(333\) 11.1581i 0.611462i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.90666i 0.485176i 0.970129 + 0.242588i \(0.0779964\pi\)
−0.970129 + 0.242588i \(0.922004\pi\)
\(338\) 0 0
\(339\) 3.55660 0.193168
\(340\) 0 0
\(341\) −42.8541 −2.32068
\(342\) 0 0
\(343\) 15.0593i 0.813127i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.5961i − 0.998290i −0.866519 0.499145i \(-0.833648\pi\)
0.866519 0.499145i \(-0.166352\pi\)
\(348\) 0 0
\(349\) 14.3040 0.765678 0.382839 0.923815i \(-0.374946\pi\)
0.382839 + 0.923815i \(0.374946\pi\)
\(350\) 0 0
\(351\) 1.99346 0.106403
\(352\) 0 0
\(353\) 12.7673i 0.679534i 0.940510 + 0.339767i \(0.110348\pi\)
−0.940510 + 0.339767i \(0.889652\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.72235i 0.0911566i
\(358\) 0 0
\(359\) −2.69596 −0.142287 −0.0711437 0.997466i \(-0.522665\pi\)
−0.0711437 + 0.997466i \(0.522665\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 28.0779i − 1.47371i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.43140i 0.179118i 0.995982 + 0.0895589i \(0.0285457\pi\)
−0.995982 + 0.0895589i \(0.971454\pi\)
\(368\) 0 0
\(369\) 3.80915 0.198297
\(370\) 0 0
\(371\) −8.86718 −0.460361
\(372\) 0 0
\(373\) − 15.5697i − 0.806168i −0.915163 0.403084i \(-0.867938\pi\)
0.915163 0.403084i \(-0.132062\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.86718i 0.147667i
\(378\) 0 0
\(379\) 0.164672 0.00845864 0.00422932 0.999991i \(-0.498654\pi\)
0.00422932 + 0.999991i \(0.498654\pi\)
\(380\) 0 0
\(381\) −13.6146 −0.697498
\(382\) 0 0
\(383\) − 22.3514i − 1.14210i −0.820915 0.571051i \(-0.806536\pi\)
0.820915 0.571051i \(-0.193464\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.5631i 0.587787i
\(388\) 0 0
\(389\) 9.34243 0.473680 0.236840 0.971549i \(-0.423888\pi\)
0.236840 + 0.971549i \(0.423888\pi\)
\(390\) 0 0
\(391\) −9.86718 −0.499005
\(392\) 0 0
\(393\) 1.87372i 0.0945167i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.83533i 0.242678i 0.992611 + 0.121339i \(0.0387188\pi\)
−0.992611 + 0.121339i \(0.961281\pi\)
\(398\) 0 0
\(399\) 1.43686 0.0719330
\(400\) 0 0
\(401\) 37.4622 1.87077 0.935386 0.353629i \(-0.115053\pi\)
0.935386 + 0.353629i \(0.115053\pi\)
\(402\) 0 0
\(403\) 2.66194i 0.132601i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 41.8816i − 2.07599i
\(408\) 0 0
\(409\) −9.42377 −0.465976 −0.232988 0.972480i \(-0.574850\pi\)
−0.232988 + 0.972480i \(0.574850\pi\)
\(410\) 0 0
\(411\) −18.9869 −0.936555
\(412\) 0 0
\(413\) − 15.4159i − 0.758568i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 23.8068i − 1.16582i
\(418\) 0 0
\(419\) −27.4303 −1.34006 −0.670029 0.742335i \(-0.733718\pi\)
−0.670029 + 0.742335i \(0.733718\pi\)
\(420\) 0 0
\(421\) 18.4303 0.898239 0.449119 0.893472i \(-0.351738\pi\)
0.449119 + 0.893472i \(0.351738\pi\)
\(422\) 0 0
\(423\) − 21.4172i − 1.04134i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.56314i − 0.0756456i
\(428\) 0 0
\(429\) −2.56314 −0.123750
\(430\) 0 0
\(431\) −15.5117 −0.747170 −0.373585 0.927596i \(-0.621872\pi\)
−0.373585 + 0.927596i \(0.621872\pi\)
\(432\) 0 0
\(433\) 31.8870i 1.53239i 0.642607 + 0.766196i \(0.277853\pi\)
−0.642607 + 0.766196i \(0.722147\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.23163i 0.393772i
\(438\) 0 0
\(439\) −14.7278 −0.702920 −0.351460 0.936203i \(-0.614315\pi\)
−0.351460 + 0.936203i \(0.614315\pi\)
\(440\) 0 0
\(441\) 8.69596 0.414093
\(442\) 0 0
\(443\) 6.46087i 0.306965i 0.988151 + 0.153483i \(0.0490489\pi\)
−0.988151 + 0.153483i \(0.950951\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.22378i 0.152480i
\(448\) 0 0
\(449\) −21.2910 −1.00478 −0.502391 0.864641i \(-0.667546\pi\)
−0.502391 + 0.864641i \(0.667546\pi\)
\(450\) 0 0
\(451\) −14.2975 −0.673243
\(452\) 0 0
\(453\) − 20.5751i − 0.966705i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.6465i 0.919023i 0.888172 + 0.459512i \(0.151976\pi\)
−0.888172 + 0.459512i \(0.848024\pi\)
\(458\) 0 0
\(459\) 6.55660 0.306036
\(460\) 0 0
\(461\) 1.16467 0.0542442 0.0271221 0.999632i \(-0.491366\pi\)
0.0271221 + 0.999632i \(0.491366\pi\)
\(462\) 0 0
\(463\) − 5.15375i − 0.239515i −0.992803 0.119758i \(-0.961788\pi\)
0.992803 0.119758i \(-0.0382117\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.17122i 0.193021i 0.995332 + 0.0965104i \(0.0307681\pi\)
−0.995332 + 0.0965104i \(0.969232\pi\)
\(468\) 0 0
\(469\) −14.3040 −0.660499
\(470\) 0 0
\(471\) −19.6146 −0.903794
\(472\) 0 0
\(473\) − 43.4018i − 1.99561i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.5631i 0.529440i
\(478\) 0 0
\(479\) 2.35552 0.107626 0.0538132 0.998551i \(-0.482862\pi\)
0.0538132 + 0.998551i \(0.482862\pi\)
\(480\) 0 0
\(481\) −2.60153 −0.118620
\(482\) 0 0
\(483\) 11.8277i 0.538179i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 19.9056i − 0.902008i −0.892522 0.451004i \(-0.851066\pi\)
0.892522 0.451004i \(-0.148934\pi\)
\(488\) 0 0
\(489\) 9.93175 0.449129
\(490\) 0 0
\(491\) 7.90557 0.356773 0.178387 0.983960i \(-0.442912\pi\)
0.178387 + 0.983960i \(0.442912\pi\)
\(492\) 0 0
\(493\) 9.43032i 0.424720i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.54875i 0.114327i
\(498\) 0 0
\(499\) 35.7147 1.59881 0.799405 0.600792i \(-0.205148\pi\)
0.799405 + 0.600792i \(0.205148\pi\)
\(500\) 0 0
\(501\) −8.98691 −0.401506
\(502\) 0 0
\(503\) − 30.7991i − 1.37327i −0.727004 0.686633i \(-0.759088\pi\)
0.727004 0.686633i \(-0.240912\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 15.4238i − 0.684994i
\(508\) 0 0
\(509\) −17.5631 −0.778472 −0.389236 0.921138i \(-0.627261\pi\)
−0.389236 + 0.921138i \(0.627261\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) − 5.46980i − 0.241497i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 80.3887i 3.53549i
\(518\) 0 0
\(519\) 17.6763 0.775905
\(520\) 0 0
\(521\) −24.6081 −1.07810 −0.539050 0.842274i \(-0.681217\pi\)
−0.539050 + 0.842274i \(0.681217\pi\)
\(522\) 0 0
\(523\) − 43.7147i − 1.91151i −0.294162 0.955756i \(-0.595040\pi\)
0.294162 0.955756i \(-0.404960\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.75529i 0.381386i
\(528\) 0 0
\(529\) −44.7597 −1.94607
\(530\) 0 0
\(531\) −20.1030 −0.872394
\(532\) 0 0
\(533\) 0.888109i 0.0384683i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.8857i 0.685520i
\(538\) 0 0
\(539\) −32.6399 −1.40590
\(540\) 0 0
\(541\) −8.38538 −0.360516 −0.180258 0.983619i \(-0.557693\pi\)
−0.180258 + 0.983619i \(0.557693\pi\)
\(542\) 0 0
\(543\) − 20.0593i − 0.860828i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 26.3040i − 1.12468i −0.826906 0.562340i \(-0.809901\pi\)
0.826906 0.562340i \(-0.190099\pi\)
\(548\) 0 0
\(549\) −2.03839 −0.0869965
\(550\) 0 0
\(551\) 7.86718 0.335153
\(552\) 0 0
\(553\) 9.27895i 0.394581i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 2.66740i − 0.113021i −0.998402 0.0565107i \(-0.982002\pi\)
0.998402 0.0565107i \(-0.0179975\pi\)
\(558\) 0 0
\(559\) −2.69596 −0.114027
\(560\) 0 0
\(561\) −8.43032 −0.355928
\(562\) 0 0
\(563\) 37.7751i 1.59203i 0.605276 + 0.796016i \(0.293063\pi\)
−0.605276 + 0.796016i \(0.706937\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.23817i − 0.0939943i
\(568\) 0 0
\(569\) −38.8606 −1.62912 −0.814561 0.580078i \(-0.803022\pi\)
−0.814561 + 0.580078i \(0.803022\pi\)
\(570\) 0 0
\(571\) 4.17122 0.174560 0.0872800 0.996184i \(-0.472183\pi\)
0.0872800 + 0.996184i \(0.472183\pi\)
\(572\) 0 0
\(573\) 4.95398i 0.206955i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 39.3413i − 1.63780i −0.573935 0.818901i \(-0.694584\pi\)
0.573935 0.818901i \(-0.305416\pi\)
\(578\) 0 0
\(579\) −8.49489 −0.353035
\(580\) 0 0
\(581\) −3.62116 −0.150231
\(582\) 0 0
\(583\) − 43.4018i − 1.79752i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.8672i 0.737457i 0.929537 + 0.368729i \(0.120207\pi\)
−0.929537 + 0.368729i \(0.879793\pi\)
\(588\) 0 0
\(589\) 7.30404 0.300958
\(590\) 0 0
\(591\) 24.5435 1.00959
\(592\) 0 0
\(593\) − 13.0803i − 0.537142i −0.963260 0.268571i \(-0.913449\pi\)
0.963260 0.268571i \(-0.0865514\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.4776i 0.428821i
\(598\) 0 0
\(599\) −10.7278 −0.438326 −0.219163 0.975688i \(-0.570333\pi\)
−0.219163 + 0.975688i \(0.570333\pi\)
\(600\) 0 0
\(601\) 39.8026 1.62358 0.811791 0.583948i \(-0.198493\pi\)
0.811791 + 0.583948i \(0.198493\pi\)
\(602\) 0 0
\(603\) 18.6530i 0.759609i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 43.0318i − 1.74661i −0.487175 0.873304i \(-0.661973\pi\)
0.487175 0.873304i \(-0.338027\pi\)
\(608\) 0 0
\(609\) 11.3040 0.458063
\(610\) 0 0
\(611\) 4.99346 0.202014
\(612\) 0 0
\(613\) − 15.4722i − 0.624915i −0.949932 0.312458i \(-0.898848\pi\)
0.949932 0.312458i \(-0.101152\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 35.5237i − 1.43013i −0.699059 0.715064i \(-0.746397\pi\)
0.699059 0.715064i \(-0.253603\pi\)
\(618\) 0 0
\(619\) 15.0449 0.604707 0.302354 0.953196i \(-0.402228\pi\)
0.302354 + 0.953196i \(0.402228\pi\)
\(620\) 0 0
\(621\) 45.0253 1.80680
\(622\) 0 0
\(623\) 6.81986i 0.273232i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.03293i 0.280868i
\(628\) 0 0
\(629\) −8.55660 −0.341174
\(630\) 0 0
\(631\) 35.6399 1.41880 0.709402 0.704805i \(-0.248965\pi\)
0.709402 + 0.704805i \(0.248965\pi\)
\(632\) 0 0
\(633\) 22.4567i 0.892574i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.02748i 0.0803315i
\(638\) 0 0
\(639\) 3.32367 0.131482
\(640\) 0 0
\(641\) 6.98691 0.275966 0.137983 0.990435i \(-0.455938\pi\)
0.137983 + 0.990435i \(0.455938\pi\)
\(642\) 0 0
\(643\) 3.74744i 0.147785i 0.997266 + 0.0738924i \(0.0235421\pi\)
−0.997266 + 0.0738924i \(0.976458\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 34.1658i − 1.34319i −0.740916 0.671597i \(-0.765609\pi\)
0.740916 0.671597i \(-0.234391\pi\)
\(648\) 0 0
\(649\) 75.4556 2.96189
\(650\) 0 0
\(651\) 10.4949 0.411327
\(652\) 0 0
\(653\) − 13.5446i − 0.530041i −0.964243 0.265020i \(-0.914621\pi\)
0.964243 0.265020i \(-0.0853787\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 3.91211i − 0.152626i
\(658\) 0 0
\(659\) 22.6334 0.881671 0.440836 0.897588i \(-0.354682\pi\)
0.440836 + 0.897588i \(0.354682\pi\)
\(660\) 0 0
\(661\) −3.96161 −0.154089 −0.0770443 0.997028i \(-0.524548\pi\)
−0.0770443 + 0.997028i \(0.524548\pi\)
\(662\) 0 0
\(663\) 0.523661i 0.0203373i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 64.7597i 2.50750i
\(668\) 0 0
\(669\) −5.11973 −0.197940
\(670\) 0 0
\(671\) 7.65102 0.295365
\(672\) 0 0
\(673\) 28.9605i 1.11635i 0.829725 + 0.558173i \(0.188497\pi\)
−0.829725 + 0.558173i \(0.811503\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 24.7518i − 0.951290i −0.879637 0.475645i \(-0.842215\pi\)
0.879637 0.475645i \(-0.157785\pi\)
\(678\) 0 0
\(679\) −1.31058 −0.0502955
\(680\) 0 0
\(681\) −25.8475 −0.990480
\(682\) 0 0
\(683\) 40.0593i 1.53283i 0.642347 + 0.766414i \(0.277961\pi\)
−0.642347 + 0.766414i \(0.722039\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 32.8648i 1.25387i
\(688\) 0 0
\(689\) −2.69596 −0.102708
\(690\) 0 0
\(691\) 35.2162 1.33969 0.669843 0.742503i \(-0.266361\pi\)
0.669843 + 0.742503i \(0.266361\pi\)
\(692\) 0 0
\(693\) − 10.9935i − 0.417607i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.92104i 0.110642i
\(698\) 0 0
\(699\) −9.43032 −0.356687
\(700\) 0 0
\(701\) −38.1283 −1.44008 −0.720042 0.693930i \(-0.755878\pi\)
−0.720042 + 0.693930i \(0.755878\pi\)
\(702\) 0 0
\(703\) 7.13828i 0.269225i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0264i 0.489908i
\(708\) 0 0
\(709\) 6.08789 0.228635 0.114318 0.993444i \(-0.463532\pi\)
0.114318 + 0.993444i \(0.463532\pi\)
\(710\) 0 0
\(711\) 12.1001 0.453789
\(712\) 0 0
\(713\) 60.1241i 2.25167i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.1527i 0.416504i
\(718\) 0 0
\(719\) 12.1647 0.453666 0.226833 0.973934i \(-0.427163\pi\)
0.226833 + 0.973934i \(0.427163\pi\)
\(720\) 0 0
\(721\) −11.6763 −0.434849
\(722\) 0 0
\(723\) 26.2580i 0.976546i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 25.6859i 0.952639i 0.879272 + 0.476320i \(0.158029\pi\)
−0.879272 + 0.476320i \(0.841971\pi\)
\(728\) 0 0
\(729\) −22.5884 −0.836609
\(730\) 0 0
\(731\) −8.86718 −0.327964
\(732\) 0 0
\(733\) − 43.4502i − 1.60487i −0.596741 0.802434i \(-0.703538\pi\)
0.596741 0.802434i \(-0.296462\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 70.0133i − 2.57897i
\(738\) 0 0
\(739\) 43.4172 1.59713 0.798564 0.601910i \(-0.205593\pi\)
0.798564 + 0.601910i \(0.205593\pi\)
\(740\) 0 0
\(741\) 0.436861 0.0160485
\(742\) 0 0
\(743\) 26.6709i 0.978459i 0.872155 + 0.489230i \(0.162722\pi\)
−0.872155 + 0.489230i \(0.837278\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.72214i 0.172774i
\(748\) 0 0
\(749\) −19.6146 −0.716703
\(750\) 0 0
\(751\) −1.52674 −0.0557114 −0.0278557 0.999612i \(-0.508868\pi\)
−0.0278557 + 0.999612i \(0.508868\pi\)
\(752\) 0 0
\(753\) 8.90666i 0.324577i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.38191i 0.340991i 0.985358 + 0.170496i \(0.0545369\pi\)
−0.985358 + 0.170496i \(0.945463\pi\)
\(758\) 0 0
\(759\) −57.8925 −2.10136
\(760\) 0 0
\(761\) −6.70251 −0.242966 −0.121483 0.992594i \(-0.538765\pi\)
−0.121483 + 0.992594i \(0.538765\pi\)
\(762\) 0 0
\(763\) − 11.1448i − 0.403470i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 4.68703i − 0.169239i
\(768\) 0 0
\(769\) 36.0637 1.30049 0.650245 0.759724i \(-0.274666\pi\)
0.650245 + 0.759724i \(0.274666\pi\)
\(770\) 0 0
\(771\) 6.05802 0.218174
\(772\) 0 0
\(773\) − 11.6763i − 0.419968i −0.977705 0.209984i \(-0.932659\pi\)
0.977705 0.209984i \(-0.0673413\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.2567i 0.367958i
\(778\) 0 0
\(779\) 2.43686 0.0873096
\(780\) 0 0
\(781\) −12.4753 −0.446400
\(782\) 0 0
\(783\) − 43.0318i − 1.53783i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 43.1880i − 1.53949i −0.638354 0.769743i \(-0.720384\pi\)
0.638354 0.769743i \(-0.279616\pi\)
\(788\) 0 0
\(789\) 21.2975 0.758211
\(790\) 0 0
\(791\) −3.55660 −0.126458
\(792\) 0 0
\(793\) − 0.475254i − 0.0168768i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 29.1658i − 1.03310i −0.856256 0.516552i \(-0.827215\pi\)
0.856256 0.516552i \(-0.172785\pi\)
\(798\) 0 0
\(799\) 16.4238 0.581031
\(800\) 0 0
\(801\) 8.89335 0.314231
\(802\) 0 0
\(803\) 14.6840i 0.518186i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11.0013i 0.387264i
\(808\) 0 0
\(809\) −2.43032 −0.0854454 −0.0427227 0.999087i \(-0.513603\pi\)
−0.0427227 + 0.999087i \(0.513603\pi\)
\(810\) 0 0
\(811\) −42.2779 −1.48458 −0.742288 0.670081i \(-0.766259\pi\)
−0.742288 + 0.670081i \(0.766259\pi\)
\(812\) 0 0
\(813\) 24.8462i 0.871396i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.39738i 0.258802i
\(818\) 0 0
\(819\) −0.682874 −0.0238616
\(820\) 0 0
\(821\) 52.5136 1.83274 0.916369 0.400334i \(-0.131106\pi\)
0.916369 + 0.400334i \(0.131106\pi\)
\(822\) 0 0
\(823\) − 36.3150i − 1.26586i −0.774209 0.632930i \(-0.781852\pi\)
0.774209 0.632930i \(-0.218148\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 46.3150i − 1.61053i −0.592916 0.805264i \(-0.702023\pi\)
0.592916 0.805264i \(-0.297977\pi\)
\(828\) 0 0
\(829\) −20.0899 −0.697750 −0.348875 0.937169i \(-0.613436\pi\)
−0.348875 + 0.937169i \(0.613436\pi\)
\(830\) 0 0
\(831\) 29.4753 1.02249
\(832\) 0 0
\(833\) 6.66849i 0.231049i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 39.9516i − 1.38093i
\(838\) 0 0
\(839\) 10.5930 0.365711 0.182855 0.983140i \(-0.441466\pi\)
0.182855 + 0.983140i \(0.441466\pi\)
\(840\) 0 0
\(841\) 32.8925 1.13422
\(842\) 0 0
\(843\) − 4.47634i − 0.154173i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 28.0779i 0.964767i
\(848\) 0 0
\(849\) 15.6212 0.536117
\(850\) 0 0
\(851\) −58.7597 −2.01426
\(852\) 0 0
\(853\) 40.0648i 1.37179i 0.727700 + 0.685896i \(0.240590\pi\)
−0.727700 + 0.685896i \(0.759410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.3019i 1.30837i 0.756336 + 0.654183i \(0.226988\pi\)
−0.756336 + 0.654183i \(0.773012\pi\)
\(858\) 0 0
\(859\) 39.9056 1.36156 0.680780 0.732488i \(-0.261641\pi\)
0.680780 + 0.732488i \(0.261641\pi\)
\(860\) 0 0
\(861\) 3.50143 0.119328
\(862\) 0 0
\(863\) 23.3315i 0.794214i 0.917772 + 0.397107i \(0.129986\pi\)
−0.917772 + 0.397107i \(0.870014\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 18.6554i − 0.633571i
\(868\) 0 0
\(869\) −45.4172 −1.54067
\(870\) 0 0
\(871\) −4.34898 −0.147359
\(872\) 0 0
\(873\) 1.70905i 0.0578426i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.4105i 0.351537i 0.984432 + 0.175768i \(0.0562410\pi\)
−0.984432 + 0.175768i \(0.943759\pi\)
\(878\) 0 0
\(879\) −23.8541 −0.804578
\(880\) 0 0
\(881\) −23.7662 −0.800704 −0.400352 0.916361i \(-0.631112\pi\)
−0.400352 + 0.916361i \(0.631112\pi\)
\(882\) 0 0
\(883\) 28.3908i 0.955428i 0.878515 + 0.477714i \(0.158534\pi\)
−0.878515 + 0.477714i \(0.841466\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 32.5644i − 1.09341i −0.837326 0.546703i \(-0.815883\pi\)
0.837326 0.546703i \(-0.184117\pi\)
\(888\) 0 0
\(889\) 13.6146 0.456620
\(890\) 0 0
\(891\) 10.9551 0.367008
\(892\) 0 0
\(893\) − 13.7014i − 0.458501i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.59607i 0.120069i
\(898\) 0 0
\(899\) 57.4622 1.91647
\(900\) 0 0
\(901\) −8.86718 −0.295409
\(902\) 0 0
\(903\) 10.6290i 0.353711i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 57.2109i 1.89966i 0.312771 + 0.949829i \(0.398743\pi\)
−0.312771 + 0.949829i \(0.601257\pi\)
\(908\) 0 0
\(909\) 16.9869 0.563420
\(910\) 0 0
\(911\) −42.0617 −1.39357 −0.696783 0.717282i \(-0.745386\pi\)
−0.696783 + 0.717282i \(0.745386\pi\)
\(912\) 0 0
\(913\) − 17.7243i − 0.586590i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.87372i − 0.0618757i
\(918\) 0 0
\(919\) −5.68287 −0.187461 −0.0937304 0.995598i \(-0.529879\pi\)
−0.0937304 + 0.995598i \(0.529879\pi\)
\(920\) 0 0
\(921\) 9.61462 0.316813
\(922\) 0 0
\(923\) 0.774918i 0.0255067i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 15.2264i 0.500100i
\(928\) 0 0
\(929\) 30.9289 1.01474 0.507372 0.861727i \(-0.330617\pi\)
0.507372 + 0.861727i \(0.330617\pi\)
\(930\) 0 0
\(931\) 5.56314 0.182325
\(932\) 0 0
\(933\) 19.3921i 0.634870i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44.5357i 1.45492i 0.686152 + 0.727458i \(0.259299\pi\)
−0.686152 + 0.727458i \(0.740701\pi\)
\(938\) 0 0
\(939\) 8.55660 0.279234
\(940\) 0 0
\(941\) −2.08134 −0.0678498 −0.0339249 0.999424i \(-0.510801\pi\)
−0.0339249 + 0.999424i \(0.510801\pi\)
\(942\) 0 0
\(943\) 20.0593i 0.653221i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 52.7189i − 1.71313i −0.516036 0.856567i \(-0.672593\pi\)
0.516036 0.856567i \(-0.327407\pi\)
\(948\) 0 0
\(949\) 0.912115 0.0296085
\(950\) 0 0
\(951\) 4.74744 0.153946
\(952\) 0 0
\(953\) 3.22071i 0.104329i 0.998639 + 0.0521645i \(0.0166120\pi\)
−0.998639 + 0.0521645i \(0.983388\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 55.3293i 1.78854i
\(958\) 0 0
\(959\) 18.9869 0.613119
\(960\) 0 0
\(961\) 22.3490 0.720935
\(962\) 0 0
\(963\) 25.5782i 0.824247i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 2.65956i − 0.0855256i −0.999085 0.0427628i \(-0.986384\pi\)
0.999085 0.0427628i \(-0.0136160\pi\)
\(968\) 0 0
\(969\) 1.43686 0.0461586
\(970\) 0 0
\(971\) 35.1263 1.12726 0.563628 0.826029i \(-0.309405\pi\)
0.563628 + 0.826029i \(0.309405\pi\)
\(972\) 0 0
\(973\) 23.8068i 0.763210i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 18.1668i − 0.581209i −0.956843 0.290604i \(-0.906144\pi\)
0.956843 0.290604i \(-0.0938563\pi\)
\(978\) 0 0
\(979\) −33.3808 −1.06686
\(980\) 0 0
\(981\) −14.5333 −0.464012
\(982\) 0 0
\(983\) 48.4556i 1.54549i 0.634714 + 0.772747i \(0.281118\pi\)
−0.634714 + 0.772747i \(0.718882\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 19.6870i − 0.626645i
\(988\) 0 0
\(989\) −60.8925 −1.93627
\(990\) 0 0
\(991\) 29.8157 0.947127 0.473563 0.880760i \(-0.342967\pi\)
0.473563 + 0.880760i \(0.342967\pi\)
\(992\) 0 0
\(993\) 7.13044i 0.226278i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 29.6894i − 0.940273i −0.882594 0.470137i \(-0.844205\pi\)
0.882594 0.470137i \(-0.155795\pi\)
\(998\) 0 0
\(999\) 39.0449 1.23533
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 1900.2.c.g.1749.2 6
5.2 odd 4 1900.2.a.h.1.1 yes 3
5.3 odd 4 1900.2.a.f.1.3 3
5.4 even 2 inner 1900.2.c.g.1749.5 6
20.3 even 4 7600.2.a.by.1.1 3
20.7 even 4 7600.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.3 3 5.3 odd 4
1900.2.a.h.1.1 yes 3 5.2 odd 4
1900.2.c.g.1749.2 6 1.1 even 1 trivial
1900.2.c.g.1749.5 6 5.4 even 2 inner
7600.2.a.bj.1.3 3 20.7 even 4
7600.2.a.by.1.1 3 20.3 even 4