Properties

Label 560.2.a.i.1.2
Level $560$
Weight $2$
Character 560.1
Self dual yes
Analytic conductor $4.472$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 560.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q+2.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} -2.56155 q^{11} +4.56155 q^{13} +2.56155 q^{15} -4.56155 q^{17} -1.12311 q^{19} +2.56155 q^{21} +5.12311 q^{23} +1.00000 q^{25} +1.43845 q^{27} -5.68466 q^{29} -6.56155 q^{33} +1.00000 q^{35} +6.00000 q^{37} +11.6847 q^{39} -3.12311 q^{41} -9.12311 q^{43} +3.56155 q^{45} -3.68466 q^{47} +1.00000 q^{49} -11.6847 q^{51} +3.12311 q^{53} -2.56155 q^{55} -2.87689 q^{57} +4.00000 q^{59} -9.36932 q^{61} +3.56155 q^{63} +4.56155 q^{65} +6.24621 q^{67} +13.1231 q^{69} -8.00000 q^{71} +4.24621 q^{73} +2.56155 q^{75} -2.56155 q^{77} +6.56155 q^{79} -7.00000 q^{81} -4.00000 q^{83} -4.56155 q^{85} -14.5616 q^{87} +7.12311 q^{89} +4.56155 q^{91} -1.12311 q^{95} -14.8078 q^{97} -9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} + 2q^{7} + 3q^{9} - q^{11} + 5q^{13} + q^{15} - 5q^{17} + 6q^{19} + q^{21} + 2q^{23} + 2q^{25} + 7q^{27} + q^{29} - 9q^{33} + 2q^{35} + 12q^{37} + 11q^{39} + 2q^{41} - 10q^{43} + 3q^{45} + 5q^{47} + 2q^{49} - 11q^{51} - 2q^{53} - q^{55} - 14q^{57} + 8q^{59} + 6q^{61} + 3q^{63} + 5q^{65} - 4q^{67} + 18q^{69} - 16q^{71} - 8q^{73} + q^{75} - q^{77} + 9q^{79} - 14q^{81} - 8q^{83} - 5q^{85} - 25q^{87} + 6q^{89} + 5q^{91} + 6q^{95} - 9q^{97} - 10q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 4.56155 1.26515 0.632574 0.774500i \(-0.281999\pi\)
0.632574 + 0.774500i \(0.281999\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 0 0
\(17\) −4.56155 −1.10634 −0.553170 0.833069i \(-0.686582\pi\)
−0.553170 + 0.833069i \(0.686582\pi\)
\(18\) 0 0
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 0 0
\(21\) 2.56155 0.558977
\(22\) 0 0
\(23\) 5.12311 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −6.56155 −1.14222
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 11.6847 1.87104
\(40\) 0 0
\(41\) −3.12311 −0.487747 −0.243874 0.969807i \(-0.578418\pi\)
−0.243874 + 0.969807i \(0.578418\pi\)
\(42\) 0 0
\(43\) −9.12311 −1.39126 −0.695630 0.718400i \(-0.744875\pi\)
−0.695630 + 0.718400i \(0.744875\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) −3.68466 −0.537463 −0.268731 0.963215i \(-0.586604\pi\)
−0.268731 + 0.963215i \(0.586604\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −11.6847 −1.63618
\(52\) 0 0
\(53\) 3.12311 0.428992 0.214496 0.976725i \(-0.431189\pi\)
0.214496 + 0.976725i \(0.431189\pi\)
\(54\) 0 0
\(55\) −2.56155 −0.345400
\(56\) 0 0
\(57\) −2.87689 −0.381054
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −9.36932 −1.19962 −0.599809 0.800143i \(-0.704757\pi\)
−0.599809 + 0.800143i \(0.704757\pi\)
\(62\) 0 0
\(63\) 3.56155 0.448713
\(64\) 0 0
\(65\) 4.56155 0.565791
\(66\) 0 0
\(67\) 6.24621 0.763096 0.381548 0.924349i \(-0.375391\pi\)
0.381548 + 0.924349i \(0.375391\pi\)
\(68\) 0 0
\(69\) 13.1231 1.57984
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 4.24621 0.496981 0.248491 0.968634i \(-0.420065\pi\)
0.248491 + 0.968634i \(0.420065\pi\)
\(74\) 0 0
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) −2.56155 −0.291916
\(78\) 0 0
\(79\) 6.56155 0.738232 0.369116 0.929383i \(-0.379660\pi\)
0.369116 + 0.929383i \(0.379660\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.56155 −0.494770
\(86\) 0 0
\(87\) −14.5616 −1.56116
\(88\) 0 0
\(89\) 7.12311 0.755048 0.377524 0.926000i \(-0.376776\pi\)
0.377524 + 0.926000i \(0.376776\pi\)
\(90\) 0 0
\(91\) 4.56155 0.478181
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.12311 −0.115228
\(96\) 0 0
\(97\) −14.8078 −1.50350 −0.751750 0.659448i \(-0.770790\pi\)
−0.751750 + 0.659448i \(0.770790\pi\)
\(98\) 0 0
\(99\) −9.12311 −0.916907
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) −1.43845 −0.141734 −0.0708672 0.997486i \(-0.522577\pi\)
−0.0708672 + 0.997486i \(0.522577\pi\)
\(104\) 0 0
\(105\) 2.56155 0.249982
\(106\) 0 0
\(107\) 11.3693 1.09911 0.549557 0.835456i \(-0.314797\pi\)
0.549557 + 0.835456i \(0.314797\pi\)
\(108\) 0 0
\(109\) 17.6847 1.69388 0.846942 0.531686i \(-0.178441\pi\)
0.846942 + 0.531686i \(0.178441\pi\)
\(110\) 0 0
\(111\) 15.3693 1.45879
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 5.12311 0.477732
\(116\) 0 0
\(117\) 16.2462 1.50196
\(118\) 0 0
\(119\) −4.56155 −0.418157
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.2462 −0.909204 −0.454602 0.890695i \(-0.650219\pi\)
−0.454602 + 0.890695i \(0.650219\pi\)
\(128\) 0 0
\(129\) −23.3693 −2.05755
\(130\) 0 0
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) 0 0
\(133\) −1.12311 −0.0973856
\(134\) 0 0
\(135\) 1.43845 0.123802
\(136\) 0 0
\(137\) −8.87689 −0.758404 −0.379202 0.925314i \(-0.623801\pi\)
−0.379202 + 0.925314i \(0.623801\pi\)
\(138\) 0 0
\(139\) 6.87689 0.583291 0.291645 0.956527i \(-0.405797\pi\)
0.291645 + 0.956527i \(0.405797\pi\)
\(140\) 0 0
\(141\) −9.43845 −0.794861
\(142\) 0 0
\(143\) −11.6847 −0.977120
\(144\) 0 0
\(145\) −5.68466 −0.472085
\(146\) 0 0
\(147\) 2.56155 0.211273
\(148\) 0 0
\(149\) −4.24621 −0.347863 −0.173932 0.984758i \(-0.555647\pi\)
−0.173932 + 0.984758i \(0.555647\pi\)
\(150\) 0 0
\(151\) −21.9309 −1.78471 −0.892354 0.451335i \(-0.850948\pi\)
−0.892354 + 0.451335i \(0.850948\pi\)
\(152\) 0 0
\(153\) −16.2462 −1.31343
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.75379 0.299585 0.149792 0.988717i \(-0.452139\pi\)
0.149792 + 0.988717i \(0.452139\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 5.12311 0.403757
\(162\) 0 0
\(163\) −1.12311 −0.0879684 −0.0439842 0.999032i \(-0.514005\pi\)
−0.0439842 + 0.999032i \(0.514005\pi\)
\(164\) 0 0
\(165\) −6.56155 −0.510816
\(166\) 0 0
\(167\) −21.9309 −1.69706 −0.848531 0.529146i \(-0.822512\pi\)
−0.848531 + 0.529146i \(0.822512\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −8.56155 −0.650923 −0.325461 0.945555i \(-0.605520\pi\)
−0.325461 + 0.945555i \(0.605520\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 10.2462 0.770152
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 23.6155 1.75533 0.877664 0.479276i \(-0.159101\pi\)
0.877664 + 0.479276i \(0.159101\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 11.6847 0.854467
\(188\) 0 0
\(189\) 1.43845 0.104632
\(190\) 0 0
\(191\) 9.43845 0.682942 0.341471 0.939892i \(-0.389075\pi\)
0.341471 + 0.939892i \(0.389075\pi\)
\(192\) 0 0
\(193\) −5.36932 −0.386492 −0.193246 0.981150i \(-0.561902\pi\)
−0.193246 + 0.981150i \(0.561902\pi\)
\(194\) 0 0
\(195\) 11.6847 0.836756
\(196\) 0 0
\(197\) −7.12311 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(198\) 0 0
\(199\) 18.2462 1.29344 0.646720 0.762728i \(-0.276140\pi\)
0.646720 + 0.762728i \(0.276140\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) −5.68466 −0.398985
\(204\) 0 0
\(205\) −3.12311 −0.218127
\(206\) 0 0
\(207\) 18.2462 1.26820
\(208\) 0 0
\(209\) 2.87689 0.198999
\(210\) 0 0
\(211\) 23.0540 1.58710 0.793551 0.608504i \(-0.208230\pi\)
0.793551 + 0.608504i \(0.208230\pi\)
\(212\) 0 0
\(213\) −20.4924 −1.40412
\(214\) 0 0
\(215\) −9.12311 −0.622191
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.8769 0.734992
\(220\) 0 0
\(221\) −20.8078 −1.39968
\(222\) 0 0
\(223\) 6.56155 0.439394 0.219697 0.975568i \(-0.429493\pi\)
0.219697 + 0.975568i \(0.429493\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) −23.6847 −1.57201 −0.786003 0.618223i \(-0.787853\pi\)
−0.786003 + 0.618223i \(0.787853\pi\)
\(228\) 0 0
\(229\) 19.1231 1.26369 0.631845 0.775095i \(-0.282298\pi\)
0.631845 + 0.775095i \(0.282298\pi\)
\(230\) 0 0
\(231\) −6.56155 −0.431718
\(232\) 0 0
\(233\) −3.12311 −0.204601 −0.102301 0.994754i \(-0.532620\pi\)
−0.102301 + 0.994754i \(0.532620\pi\)
\(234\) 0 0
\(235\) −3.68466 −0.240361
\(236\) 0 0
\(237\) 16.8078 1.09178
\(238\) 0 0
\(239\) 0.807764 0.0522499 0.0261250 0.999659i \(-0.491683\pi\)
0.0261250 + 0.999659i \(0.491683\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 0 0
\(243\) −22.2462 −1.42710
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −5.12311 −0.325975
\(248\) 0 0
\(249\) −10.2462 −0.649327
\(250\) 0 0
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) 0 0
\(253\) −13.1231 −0.825043
\(254\) 0 0
\(255\) −11.6847 −0.731722
\(256\) 0 0
\(257\) 22.4924 1.40304 0.701519 0.712650i \(-0.252505\pi\)
0.701519 + 0.712650i \(0.252505\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −20.2462 −1.25321
\(262\) 0 0
\(263\) 21.1231 1.30251 0.651253 0.758860i \(-0.274244\pi\)
0.651253 + 0.758860i \(0.274244\pi\)
\(264\) 0 0
\(265\) 3.12311 0.191851
\(266\) 0 0
\(267\) 18.2462 1.11665
\(268\) 0 0
\(269\) 28.7386 1.75223 0.876113 0.482106i \(-0.160128\pi\)
0.876113 + 0.482106i \(0.160128\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 11.6847 0.707188
\(274\) 0 0
\(275\) −2.56155 −0.154467
\(276\) 0 0
\(277\) 16.2462 0.976140 0.488070 0.872804i \(-0.337701\pi\)
0.488070 + 0.872804i \(0.337701\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.5616 0.987979 0.493990 0.869468i \(-0.335538\pi\)
0.493990 + 0.869468i \(0.335538\pi\)
\(282\) 0 0
\(283\) 23.6847 1.40791 0.703953 0.710246i \(-0.251416\pi\)
0.703953 + 0.710246i \(0.251416\pi\)
\(284\) 0 0
\(285\) −2.87689 −0.170413
\(286\) 0 0
\(287\) −3.12311 −0.184351
\(288\) 0 0
\(289\) 3.80776 0.223986
\(290\) 0 0
\(291\) −37.9309 −2.22355
\(292\) 0 0
\(293\) 9.68466 0.565784 0.282892 0.959152i \(-0.408706\pi\)
0.282892 + 0.959152i \(0.408706\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) −3.68466 −0.213806
\(298\) 0 0
\(299\) 23.3693 1.35148
\(300\) 0 0
\(301\) −9.12311 −0.525847
\(302\) 0 0
\(303\) 0.630683 0.0362318
\(304\) 0 0
\(305\) −9.36932 −0.536486
\(306\) 0 0
\(307\) 31.6847 1.80834 0.904169 0.427174i \(-0.140491\pi\)
0.904169 + 0.427174i \(0.140491\pi\)
\(308\) 0 0
\(309\) −3.68466 −0.209613
\(310\) 0 0
\(311\) 9.61553 0.545247 0.272623 0.962121i \(-0.412109\pi\)
0.272623 + 0.962121i \(0.412109\pi\)
\(312\) 0 0
\(313\) 31.3002 1.76919 0.884596 0.466359i \(-0.154434\pi\)
0.884596 + 0.466359i \(0.154434\pi\)
\(314\) 0 0
\(315\) 3.56155 0.200671
\(316\) 0 0
\(317\) −22.4924 −1.26330 −0.631650 0.775254i \(-0.717622\pi\)
−0.631650 + 0.775254i \(0.717622\pi\)
\(318\) 0 0
\(319\) 14.5616 0.815290
\(320\) 0 0
\(321\) 29.1231 1.62549
\(322\) 0 0
\(323\) 5.12311 0.285057
\(324\) 0 0
\(325\) 4.56155 0.253029
\(326\) 0 0
\(327\) 45.3002 2.50511
\(328\) 0 0
\(329\) −3.68466 −0.203142
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 21.3693 1.17103
\(334\) 0 0
\(335\) 6.24621 0.341267
\(336\) 0 0
\(337\) −34.4924 −1.87892 −0.939461 0.342656i \(-0.888674\pi\)
−0.939461 + 0.342656i \(0.888674\pi\)
\(338\) 0 0
\(339\) −35.8617 −1.94774
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 13.1231 0.706524
\(346\) 0 0
\(347\) 1.12311 0.0602915 0.0301457 0.999546i \(-0.490403\pi\)
0.0301457 + 0.999546i \(0.490403\pi\)
\(348\) 0 0
\(349\) −22.4924 −1.20399 −0.601996 0.798499i \(-0.705628\pi\)
−0.601996 + 0.798499i \(0.705628\pi\)
\(350\) 0 0
\(351\) 6.56155 0.350230
\(352\) 0 0
\(353\) −14.8078 −0.788138 −0.394069 0.919081i \(-0.628933\pi\)
−0.394069 + 0.919081i \(0.628933\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) −11.6847 −0.618418
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) −11.3693 −0.596734
\(364\) 0 0
\(365\) 4.24621 0.222257
\(366\) 0 0
\(367\) −3.68466 −0.192338 −0.0961688 0.995365i \(-0.530659\pi\)
−0.0961688 + 0.995365i \(0.530659\pi\)
\(368\) 0 0
\(369\) −11.1231 −0.579046
\(370\) 0 0
\(371\) 3.12311 0.162144
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) 0 0
\(375\) 2.56155 0.132278
\(376\) 0 0
\(377\) −25.9309 −1.33551
\(378\) 0 0
\(379\) −16.4924 −0.847159 −0.423579 0.905859i \(-0.639227\pi\)
−0.423579 + 0.905859i \(0.639227\pi\)
\(380\) 0 0
\(381\) −26.2462 −1.34463
\(382\) 0 0
\(383\) 10.2462 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(384\) 0 0
\(385\) −2.56155 −0.130549
\(386\) 0 0
\(387\) −32.4924 −1.65168
\(388\) 0 0
\(389\) 3.93087 0.199303 0.0996515 0.995022i \(-0.468227\pi\)
0.0996515 + 0.995022i \(0.468227\pi\)
\(390\) 0 0
\(391\) −23.3693 −1.18184
\(392\) 0 0
\(393\) 23.3693 1.17883
\(394\) 0 0
\(395\) 6.56155 0.330148
\(396\) 0 0
\(397\) 23.4384 1.17634 0.588171 0.808737i \(-0.299848\pi\)
0.588171 + 0.808737i \(0.299848\pi\)
\(398\) 0 0
\(399\) −2.87689 −0.144025
\(400\) 0 0
\(401\) 27.4384 1.37021 0.685105 0.728444i \(-0.259756\pi\)
0.685105 + 0.728444i \(0.259756\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) −15.3693 −0.761829
\(408\) 0 0
\(409\) −26.4924 −1.30997 −0.654983 0.755644i \(-0.727324\pi\)
−0.654983 + 0.755644i \(0.727324\pi\)
\(410\) 0 0
\(411\) −22.7386 −1.12161
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 17.6155 0.862636
\(418\) 0 0
\(419\) −9.75379 −0.476504 −0.238252 0.971203i \(-0.576574\pi\)
−0.238252 + 0.971203i \(0.576574\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) 0 0
\(423\) −13.1231 −0.638067
\(424\) 0 0
\(425\) −4.56155 −0.221268
\(426\) 0 0
\(427\) −9.36932 −0.453413
\(428\) 0 0
\(429\) −29.9309 −1.44508
\(430\) 0 0
\(431\) −0.807764 −0.0389086 −0.0194543 0.999811i \(-0.506193\pi\)
−0.0194543 + 0.999811i \(0.506193\pi\)
\(432\) 0 0
\(433\) −8.24621 −0.396288 −0.198144 0.980173i \(-0.563491\pi\)
−0.198144 + 0.980173i \(0.563491\pi\)
\(434\) 0 0
\(435\) −14.5616 −0.698173
\(436\) 0 0
\(437\) −5.75379 −0.275241
\(438\) 0 0
\(439\) 15.3693 0.733537 0.366769 0.930312i \(-0.380464\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(440\) 0 0
\(441\) 3.56155 0.169598
\(442\) 0 0
\(443\) 27.3693 1.30036 0.650178 0.759782i \(-0.274694\pi\)
0.650178 + 0.759782i \(0.274694\pi\)
\(444\) 0 0
\(445\) 7.12311 0.337668
\(446\) 0 0
\(447\) −10.8769 −0.514459
\(448\) 0 0
\(449\) 18.8078 0.887593 0.443797 0.896128i \(-0.353631\pi\)
0.443797 + 0.896128i \(0.353631\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) −56.1771 −2.63943
\(454\) 0 0
\(455\) 4.56155 0.213849
\(456\) 0 0
\(457\) −8.87689 −0.415244 −0.207622 0.978209i \(-0.566572\pi\)
−0.207622 + 0.978209i \(0.566572\pi\)
\(458\) 0 0
\(459\) −6.56155 −0.306267
\(460\) 0 0
\(461\) −4.87689 −0.227140 −0.113570 0.993530i \(-0.536229\pi\)
−0.113570 + 0.993530i \(0.536229\pi\)
\(462\) 0 0
\(463\) 20.4924 0.952364 0.476182 0.879347i \(-0.342020\pi\)
0.476182 + 0.879347i \(0.342020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −26.5616 −1.22912 −0.614561 0.788869i \(-0.710667\pi\)
−0.614561 + 0.788869i \(0.710667\pi\)
\(468\) 0 0
\(469\) 6.24621 0.288423
\(470\) 0 0
\(471\) 9.61553 0.443060
\(472\) 0 0
\(473\) 23.3693 1.07452
\(474\) 0 0
\(475\) −1.12311 −0.0515316
\(476\) 0 0
\(477\) 11.1231 0.509292
\(478\) 0 0
\(479\) −13.1231 −0.599610 −0.299805 0.954001i \(-0.596922\pi\)
−0.299805 + 0.954001i \(0.596922\pi\)
\(480\) 0 0
\(481\) 27.3693 1.24793
\(482\) 0 0
\(483\) 13.1231 0.597122
\(484\) 0 0
\(485\) −14.8078 −0.672386
\(486\) 0 0
\(487\) −5.12311 −0.232150 −0.116075 0.993240i \(-0.537031\pi\)
−0.116075 + 0.993240i \(0.537031\pi\)
\(488\) 0 0
\(489\) −2.87689 −0.130098
\(490\) 0 0
\(491\) −4.17708 −0.188509 −0.0942545 0.995548i \(-0.530047\pi\)
−0.0942545 + 0.995548i \(0.530047\pi\)
\(492\) 0 0
\(493\) 25.9309 1.16787
\(494\) 0 0
\(495\) −9.12311 −0.410053
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 4.17708 0.186992 0.0934959 0.995620i \(-0.470196\pi\)
0.0934959 + 0.995620i \(0.470196\pi\)
\(500\) 0 0
\(501\) −56.1771 −2.50981
\(502\) 0 0
\(503\) −10.0691 −0.448960 −0.224480 0.974479i \(-0.572068\pi\)
−0.224480 + 0.974479i \(0.572068\pi\)
\(504\) 0 0
\(505\) 0.246211 0.0109563
\(506\) 0 0
\(507\) 20.0000 0.888231
\(508\) 0 0
\(509\) −28.2462 −1.25199 −0.625996 0.779827i \(-0.715307\pi\)
−0.625996 + 0.779827i \(0.715307\pi\)
\(510\) 0 0
\(511\) 4.24621 0.187841
\(512\) 0 0
\(513\) −1.61553 −0.0713273
\(514\) 0 0
\(515\) −1.43845 −0.0633856
\(516\) 0 0
\(517\) 9.43845 0.415102
\(518\) 0 0
\(519\) −21.9309 −0.962658
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) −7.50758 −0.328283 −0.164142 0.986437i \(-0.552485\pi\)
−0.164142 + 0.986437i \(0.552485\pi\)
\(524\) 0 0
\(525\) 2.56155 0.111795
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 0 0
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) −14.2462 −0.617072
\(534\) 0 0
\(535\) 11.3693 0.491538
\(536\) 0 0
\(537\) −51.2311 −2.21078
\(538\) 0 0
\(539\) −2.56155 −0.110334
\(540\) 0 0
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) 0 0
\(543\) 60.4924 2.59598
\(544\) 0 0
\(545\) 17.6847 0.757528
\(546\) 0 0
\(547\) −14.2462 −0.609124 −0.304562 0.952493i \(-0.598510\pi\)
−0.304562 + 0.952493i \(0.598510\pi\)
\(548\) 0 0
\(549\) −33.3693 −1.42417
\(550\) 0 0
\(551\) 6.38447 0.271988
\(552\) 0 0
\(553\) 6.56155 0.279026
\(554\) 0 0
\(555\) 15.3693 0.652391
\(556\) 0 0
\(557\) −4.87689 −0.206641 −0.103320 0.994648i \(-0.532947\pi\)
−0.103320 + 0.994648i \(0.532947\pi\)
\(558\) 0 0
\(559\) −41.6155 −1.76015
\(560\) 0 0
\(561\) 29.9309 1.26368
\(562\) 0 0
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 0 0
\(567\) −7.00000 −0.293972
\(568\) 0 0
\(569\) 34.9848 1.46664 0.733320 0.679883i \(-0.237969\pi\)
0.733320 + 0.679883i \(0.237969\pi\)
\(570\) 0 0
\(571\) −7.50758 −0.314182 −0.157091 0.987584i \(-0.550212\pi\)
−0.157091 + 0.987584i \(0.550212\pi\)
\(572\) 0 0
\(573\) 24.1771 1.01001
\(574\) 0 0
\(575\) 5.12311 0.213648
\(576\) 0 0
\(577\) 13.0540 0.543444 0.271722 0.962376i \(-0.412407\pi\)
0.271722 + 0.962376i \(0.412407\pi\)
\(578\) 0 0
\(579\) −13.7538 −0.571588
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) 16.2462 0.671698
\(586\) 0 0
\(587\) 9.75379 0.402582 0.201291 0.979531i \(-0.435486\pi\)
0.201291 + 0.979531i \(0.435486\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −18.2462 −0.750549
\(592\) 0 0
\(593\) −23.4384 −0.962502 −0.481251 0.876583i \(-0.659817\pi\)
−0.481251 + 0.876583i \(0.659817\pi\)
\(594\) 0 0
\(595\) −4.56155 −0.187005
\(596\) 0 0
\(597\) 46.7386 1.91288
\(598\) 0 0
\(599\) −8.80776 −0.359875 −0.179938 0.983678i \(-0.557590\pi\)
−0.179938 + 0.983678i \(0.557590\pi\)
\(600\) 0 0
\(601\) −26.4924 −1.08065 −0.540324 0.841457i \(-0.681698\pi\)
−0.540324 + 0.841457i \(0.681698\pi\)
\(602\) 0 0
\(603\) 22.2462 0.905936
\(604\) 0 0
\(605\) −4.43845 −0.180449
\(606\) 0 0
\(607\) 4.94602 0.200753 0.100376 0.994950i \(-0.467995\pi\)
0.100376 + 0.994950i \(0.467995\pi\)
\(608\) 0 0
\(609\) −14.5616 −0.590064
\(610\) 0 0
\(611\) −16.8078 −0.679969
\(612\) 0 0
\(613\) −8.73863 −0.352950 −0.176475 0.984305i \(-0.556469\pi\)
−0.176475 + 0.984305i \(0.556469\pi\)
\(614\) 0 0
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) 15.7538 0.634224 0.317112 0.948388i \(-0.397287\pi\)
0.317112 + 0.948388i \(0.397287\pi\)
\(618\) 0 0
\(619\) 42.1080 1.69246 0.846231 0.532817i \(-0.178866\pi\)
0.846231 + 0.532817i \(0.178866\pi\)
\(620\) 0 0
\(621\) 7.36932 0.295720
\(622\) 0 0
\(623\) 7.12311 0.285381
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.36932 0.294302
\(628\) 0 0
\(629\) −27.3693 −1.09129
\(630\) 0 0
\(631\) −8.80776 −0.350632 −0.175316 0.984512i \(-0.556095\pi\)
−0.175316 + 0.984512i \(0.556095\pi\)
\(632\) 0 0
\(633\) 59.0540 2.34718
\(634\) 0 0
\(635\) −10.2462 −0.406608
\(636\) 0 0
\(637\) 4.56155 0.180735
\(638\) 0 0
\(639\) −28.4924 −1.12714
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 2.56155 0.101018 0.0505089 0.998724i \(-0.483916\pi\)
0.0505089 + 0.998724i \(0.483916\pi\)
\(644\) 0 0
\(645\) −23.3693 −0.920166
\(646\) 0 0
\(647\) −3.50758 −0.137897 −0.0689486 0.997620i \(-0.521964\pi\)
−0.0689486 + 0.997620i \(0.521964\pi\)
\(648\) 0 0
\(649\) −10.2462 −0.402199
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 49.2311 1.92656 0.963280 0.268499i \(-0.0865275\pi\)
0.963280 + 0.268499i \(0.0865275\pi\)
\(654\) 0 0
\(655\) 9.12311 0.356469
\(656\) 0 0
\(657\) 15.1231 0.590009
\(658\) 0 0
\(659\) 36.1771 1.40926 0.704629 0.709575i \(-0.251113\pi\)
0.704629 + 0.709575i \(0.251113\pi\)
\(660\) 0 0
\(661\) 3.12311 0.121475 0.0607374 0.998154i \(-0.480655\pi\)
0.0607374 + 0.998154i \(0.480655\pi\)
\(662\) 0 0
\(663\) −53.3002 −2.07001
\(664\) 0 0
\(665\) −1.12311 −0.0435522
\(666\) 0 0
\(667\) −29.1231 −1.12765
\(668\) 0 0
\(669\) 16.8078 0.649826
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −25.8617 −0.996897 −0.498448 0.866919i \(-0.666097\pi\)
−0.498448 + 0.866919i \(0.666097\pi\)
\(674\) 0 0
\(675\) 1.43845 0.0553659
\(676\) 0 0
\(677\) −23.9309 −0.919738 −0.459869 0.887987i \(-0.652104\pi\)
−0.459869 + 0.887987i \(0.652104\pi\)
\(678\) 0 0
\(679\) −14.8078 −0.568270
\(680\) 0 0
\(681\) −60.6695 −2.32486
\(682\) 0 0
\(683\) −42.7386 −1.63535 −0.817674 0.575681i \(-0.804737\pi\)
−0.817674 + 0.575681i \(0.804737\pi\)
\(684\) 0 0
\(685\) −8.87689 −0.339169
\(686\) 0 0
\(687\) 48.9848 1.86889
\(688\) 0 0
\(689\) 14.2462 0.542737
\(690\) 0 0
\(691\) −8.49242 −0.323067 −0.161533 0.986867i \(-0.551644\pi\)
−0.161533 + 0.986867i \(0.551644\pi\)
\(692\) 0 0
\(693\) −9.12311 −0.346558
\(694\) 0 0
\(695\) 6.87689 0.260855
\(696\) 0 0
\(697\) 14.2462 0.539614
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 0.0691303 0.00261102 0.00130551 0.999999i \(-0.499584\pi\)
0.00130551 + 0.999999i \(0.499584\pi\)
\(702\) 0 0
\(703\) −6.73863 −0.254152
\(704\) 0 0
\(705\) −9.43845 −0.355472
\(706\) 0 0
\(707\) 0.246211 0.00925973
\(708\) 0 0
\(709\) −18.1771 −0.682655 −0.341327 0.939945i \(-0.610876\pi\)
−0.341327 + 0.939945i \(0.610876\pi\)
\(710\) 0 0
\(711\) 23.3693 0.876418
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −11.6847 −0.436981
\(716\) 0 0
\(717\) 2.06913 0.0772731
\(718\) 0 0
\(719\) −49.6155 −1.85035 −0.925173 0.379544i \(-0.876081\pi\)
−0.925173 + 0.379544i \(0.876081\pi\)
\(720\) 0 0
\(721\) −1.43845 −0.0535706
\(722\) 0 0
\(723\) 31.3693 1.16664
\(724\) 0 0
\(725\) −5.68466 −0.211123
\(726\) 0 0
\(727\) −19.5076 −0.723496 −0.361748 0.932276i \(-0.617820\pi\)
−0.361748 + 0.932276i \(0.617820\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 41.6155 1.53921
\(732\) 0 0
\(733\) −5.68466 −0.209968 −0.104984 0.994474i \(-0.533479\pi\)
−0.104984 + 0.994474i \(0.533479\pi\)
\(734\) 0 0
\(735\) 2.56155 0.0944843
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −6.06913 −0.223257 −0.111628 0.993750i \(-0.535607\pi\)
−0.111628 + 0.993750i \(0.535607\pi\)
\(740\) 0 0
\(741\) −13.1231 −0.482089
\(742\) 0 0
\(743\) −32.9848 −1.21010 −0.605048 0.796189i \(-0.706846\pi\)
−0.605048 + 0.796189i \(0.706846\pi\)
\(744\) 0 0
\(745\) −4.24621 −0.155569
\(746\) 0 0
\(747\) −14.2462 −0.521242
\(748\) 0 0
\(749\) 11.3693 0.415426
\(750\) 0 0
\(751\) −45.9309 −1.67604 −0.838021 0.545639i \(-0.816287\pi\)
−0.838021 + 0.545639i \(0.816287\pi\)
\(752\) 0 0
\(753\) 43.8617 1.59841
\(754\) 0 0
\(755\) −21.9309 −0.798146
\(756\) 0 0
\(757\) 14.6307 0.531761 0.265881 0.964006i \(-0.414337\pi\)
0.265881 + 0.964006i \(0.414337\pi\)
\(758\) 0 0
\(759\) −33.6155 −1.22017
\(760\) 0 0
\(761\) 31.7538 1.15107 0.575537 0.817776i \(-0.304793\pi\)
0.575537 + 0.817776i \(0.304793\pi\)
\(762\) 0 0
\(763\) 17.6847 0.640228
\(764\) 0 0
\(765\) −16.2462 −0.587383
\(766\) 0 0
\(767\) 18.2462 0.658833
\(768\) 0 0
\(769\) −9.50758 −0.342852 −0.171426 0.985197i \(-0.554837\pi\)
−0.171426 + 0.985197i \(0.554837\pi\)
\(770\) 0 0
\(771\) 57.6155 2.07497
\(772\) 0 0
\(773\) 8.06913 0.290226 0.145113 0.989415i \(-0.453645\pi\)
0.145113 + 0.989415i \(0.453645\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 15.3693 0.551371
\(778\) 0 0
\(779\) 3.50758 0.125672
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 0 0
\(783\) −8.17708 −0.292225
\(784\) 0 0
\(785\) 3.75379 0.133978
\(786\) 0 0
\(787\) 3.82292 0.136272 0.0681362 0.997676i \(-0.478295\pi\)
0.0681362 + 0.997676i \(0.478295\pi\)
\(788\) 0 0
\(789\) 54.1080 1.92629
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −42.7386 −1.51769
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) −13.0540 −0.462396 −0.231198 0.972907i \(-0.574264\pi\)
−0.231198 + 0.972907i \(0.574264\pi\)
\(798\) 0 0
\(799\) 16.8078 0.594616
\(800\) 0 0
\(801\) 25.3693 0.896381
\(802\) 0 0
\(803\) −10.8769 −0.383837
\(804\) 0 0
\(805\) 5.12311 0.180566
\(806\) 0 0
\(807\) 73.6155 2.59139
\(808\) 0 0
\(809\) −53.5464 −1.88259 −0.941296 0.337584i \(-0.890390\pi\)
−0.941296 + 0.337584i \(0.890390\pi\)
\(810\) 0 0
\(811\) 21.6155 0.759024 0.379512 0.925187i \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(812\) 0 0
\(813\) 40.9848 1.43740
\(814\) 0 0
\(815\) −1.12311 −0.0393407
\(816\) 0 0
\(817\) 10.2462 0.358470
\(818\) 0 0
\(819\) 16.2462 0.567689
\(820\) 0 0
\(821\) 40.4233 1.41078 0.705391 0.708818i \(-0.250771\pi\)
0.705391 + 0.708818i \(0.250771\pi\)
\(822\) 0 0
\(823\) 3.50758 0.122266 0.0611332 0.998130i \(-0.480529\pi\)
0.0611332 + 0.998130i \(0.480529\pi\)
\(824\) 0 0
\(825\) −6.56155 −0.228444
\(826\) 0 0
\(827\) −19.3693 −0.673537 −0.336769 0.941587i \(-0.609334\pi\)
−0.336769 + 0.941587i \(0.609334\pi\)
\(828\) 0 0
\(829\) 43.1231 1.49773 0.748864 0.662724i \(-0.230600\pi\)
0.748864 + 0.662724i \(0.230600\pi\)
\(830\) 0 0
\(831\) 41.6155 1.44363
\(832\) 0 0
\(833\) −4.56155 −0.158048
\(834\) 0 0
\(835\) −21.9309 −0.758949
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 37.1231 1.28163 0.640816 0.767695i \(-0.278596\pi\)
0.640816 + 0.767695i \(0.278596\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 0 0
\(843\) 42.4233 1.46114
\(844\) 0 0
\(845\) 7.80776 0.268595
\(846\) 0 0
\(847\) −4.43845 −0.152507
\(848\) 0 0
\(849\) 60.6695 2.08217
\(850\) 0 0
\(851\) 30.7386 1.05371
\(852\) 0 0
\(853\) −56.7386 −1.94269 −0.971347 0.237666i \(-0.923618\pi\)
−0.971347 + 0.237666i \(0.923618\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −32.2462 −1.10151 −0.550755 0.834667i \(-0.685660\pi\)
−0.550755 + 0.834667i \(0.685660\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 42.2462 1.43808 0.719039 0.694970i \(-0.244582\pi\)
0.719039 + 0.694970i \(0.244582\pi\)
\(864\) 0 0
\(865\) −8.56155 −0.291102
\(866\) 0 0
\(867\) 9.75379 0.331256
\(868\) 0 0
\(869\) −16.8078 −0.570164
\(870\) 0 0
\(871\) 28.4924 0.965429
\(872\) 0 0
\(873\) −52.7386 −1.78493
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −23.7538 −0.802108 −0.401054 0.916054i \(-0.631356\pi\)
−0.401054 + 0.916054i \(0.631356\pi\)
\(878\) 0 0
\(879\) 24.8078 0.836745
\(880\) 0 0
\(881\) 45.8617 1.54512 0.772561 0.634941i \(-0.218976\pi\)
0.772561 + 0.634941i \(0.218976\pi\)
\(882\) 0 0
\(883\) −24.4924 −0.824236 −0.412118 0.911131i \(-0.635211\pi\)
−0.412118 + 0.911131i \(0.635211\pi\)
\(884\) 0 0
\(885\) 10.2462 0.344423
\(886\) 0 0
\(887\) 12.4924 0.419454 0.209727 0.977760i \(-0.432742\pi\)
0.209727 + 0.977760i \(0.432742\pi\)
\(888\) 0 0
\(889\) −10.2462 −0.343647
\(890\) 0 0
\(891\) 17.9309 0.600707
\(892\) 0 0
\(893\) 4.13826 0.138482
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 59.8617 1.99873
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −14.2462 −0.474610
\(902\) 0 0
\(903\) −23.3693 −0.777682
\(904\) 0 0
\(905\) 23.6155 0.785007
\(906\) 0 0
\(907\) −50.1080 −1.66381 −0.831904 0.554920i \(-0.812749\pi\)
−0.831904 + 0.554920i \(0.812749\pi\)
\(908\) 0 0
\(909\) 0.876894 0.0290848
\(910\) 0 0
\(911\) −4.49242 −0.148841 −0.0744203 0.997227i \(-0.523711\pi\)
−0.0744203 + 0.997227i \(0.523711\pi\)
\(912\) 0 0
\(913\) 10.2462 0.339100
\(914\) 0 0
\(915\) −24.0000 −0.793416
\(916\) 0 0
\(917\) 9.12311 0.301271
\(918\) 0 0
\(919\) 13.3002 0.438733 0.219366 0.975643i \(-0.429601\pi\)
0.219366 + 0.975643i \(0.429601\pi\)
\(920\) 0 0
\(921\) 81.1619 2.67438
\(922\) 0 0
\(923\) −36.4924 −1.20116
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −5.12311 −0.168265
\(928\) 0 0
\(929\) −52.1080 −1.70961 −0.854803 0.518952i \(-0.826322\pi\)
−0.854803 + 0.518952i \(0.826322\pi\)
\(930\) 0 0
\(931\) −1.12311 −0.0368083
\(932\) 0 0
\(933\) 24.6307 0.806372
\(934\) 0 0
\(935\) 11.6847 0.382129
\(936\) 0 0
\(937\) 22.6695 0.740580 0.370290 0.928916i \(-0.379258\pi\)
0.370290 + 0.928916i \(0.379258\pi\)
\(938\) 0 0
\(939\) 80.1771 2.61648
\(940\) 0 0
\(941\) −13.8617 −0.451880 −0.225940 0.974141i \(-0.572545\pi\)
−0.225940 + 0.974141i \(0.572545\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 1.43845 0.0467927
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 19.3693 0.628755
\(950\) 0 0
\(951\) −57.6155 −1.86831
\(952\) 0 0
\(953\) −24.8769 −0.805842 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(954\) 0 0
\(955\) 9.43845 0.305421
\(956\) 0 0
\(957\) 37.3002 1.20574
\(958\) 0 0
\(959\) −8.87689 −0.286650
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 40.4924 1.30485
\(964\) 0 0
\(965\) −5.36932 −0.172844
\(966\) 0 0
\(967\) 26.8769 0.864303 0.432151 0.901801i \(-0.357755\pi\)
0.432151 + 0.901801i \(0.357755\pi\)
\(968\) 0 0
\(969\) 13.1231 0.421575
\(970\) 0 0
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 0 0
\(973\) 6.87689 0.220463
\(974\) 0 0
\(975\) 11.6847 0.374209
\(976\) 0 0
\(977\) −49.2311 −1.57504 −0.787521 0.616288i \(-0.788636\pi\)
−0.787521 + 0.616288i \(0.788636\pi\)
\(978\) 0 0
\(979\) −18.2462 −0.583151
\(980\) 0 0
\(981\) 62.9848 2.01095
\(982\) 0 0
\(983\) 10.4233 0.332451 0.166226 0.986088i \(-0.446842\pi\)
0.166226 + 0.986088i \(0.446842\pi\)
\(984\) 0 0
\(985\) −7.12311 −0.226961
\(986\) 0 0
\(987\) −9.43845 −0.300429
\(988\) 0 0
\(989\) −46.7386 −1.48620
\(990\) 0 0
\(991\) 20.4924 0.650963 0.325482 0.945548i \(-0.394474\pi\)
0.325482 + 0.945548i \(0.394474\pi\)
\(992\) 0 0
\(993\) −30.7386 −0.975461
\(994\) 0 0
\(995\) 18.2462 0.578444
\(996\) 0 0
\(997\) 9.68466 0.306716 0.153358 0.988171i \(-0.450991\pi\)
0.153358 + 0.988171i \(0.450991\pi\)
\(998\) 0 0
\(999\) 8.63068 0.273063
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 560.2.a.i.1.2 2
3.2 odd 2 5040.2.a.bt.1.2 2
4.3 odd 2 35.2.a.b.1.2 2
5.2 odd 4 2800.2.g.t.449.1 4
5.3 odd 4 2800.2.g.t.449.4 4
5.4 even 2 2800.2.a.bi.1.1 2
7.6 odd 2 3920.2.a.bs.1.1 2
8.3 odd 2 2240.2.a.bh.1.2 2
8.5 even 2 2240.2.a.bd.1.1 2
12.11 even 2 315.2.a.e.1.1 2
20.3 even 4 175.2.b.b.99.2 4
20.7 even 4 175.2.b.b.99.3 4
20.19 odd 2 175.2.a.f.1.1 2
28.3 even 6 245.2.e.h.226.1 4
28.11 odd 6 245.2.e.i.226.1 4
28.19 even 6 245.2.e.h.116.1 4
28.23 odd 6 245.2.e.i.116.1 4
28.27 even 2 245.2.a.d.1.2 2
44.43 even 2 4235.2.a.m.1.1 2
52.51 odd 2 5915.2.a.l.1.1 2
60.23 odd 4 1575.2.d.e.1324.3 4
60.47 odd 4 1575.2.d.e.1324.2 4
60.59 even 2 1575.2.a.p.1.2 2
84.83 odd 2 2205.2.a.x.1.1 2
140.27 odd 4 1225.2.b.f.99.3 4
140.83 odd 4 1225.2.b.f.99.2 4
140.139 even 2 1225.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 4.3 odd 2
175.2.a.f.1.1 2 20.19 odd 2
175.2.b.b.99.2 4 20.3 even 4
175.2.b.b.99.3 4 20.7 even 4
245.2.a.d.1.2 2 28.27 even 2
245.2.e.h.116.1 4 28.19 even 6
245.2.e.h.226.1 4 28.3 even 6
245.2.e.i.116.1 4 28.23 odd 6
245.2.e.i.226.1 4 28.11 odd 6
315.2.a.e.1.1 2 12.11 even 2
560.2.a.i.1.2 2 1.1 even 1 trivial
1225.2.a.s.1.1 2 140.139 even 2
1225.2.b.f.99.2 4 140.83 odd 4
1225.2.b.f.99.3 4 140.27 odd 4
1575.2.a.p.1.2 2 60.59 even 2
1575.2.d.e.1324.2 4 60.47 odd 4
1575.2.d.e.1324.3 4 60.23 odd 4
2205.2.a.x.1.1 2 84.83 odd 2
2240.2.a.bd.1.1 2 8.5 even 2
2240.2.a.bh.1.2 2 8.3 odd 2
2800.2.a.bi.1.1 2 5.4 even 2
2800.2.g.t.449.1 4 5.2 odd 4
2800.2.g.t.449.4 4 5.3 odd 4
3920.2.a.bs.1.1 2 7.6 odd 2
4235.2.a.m.1.1 2 44.43 even 2
5040.2.a.bt.1.2 2 3.2 odd 2
5915.2.a.l.1.1 2 52.51 odd 2