Properties

Label 3920.2.a.bs.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

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

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 \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 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 \(-0.718001\pi\)
−0.632574 + 0.774500i \(0.718001\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 0 0
\(17\) 4.56155 1.10634 0.553170 0.833069i \(-0.313418\pi\)
0.553170 + 0.833069i \(0.313418\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(35\) 0 0
\(36\) 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 \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 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 \(-0.413396\pi\)
0.268731 + 0.963215i \(0.413396\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 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 \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 9.36932 1.19962 0.599809 0.800143i \(-0.295243\pi\)
0.599809 + 0.800143i \(0.295243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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 \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) 0 0
\(78\) 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 \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 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 \(-0.623224\pi\)
−0.377524 + 0.926000i \(0.623224\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 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 \(-0.229210\pi\)
0.751750 + 0.659448i \(0.229210\pi\)
\(98\) 0 0
\(99\) −9.12311 −0.916907
\(100\) 0 0
\(101\) −0.246211 −0.0244989 −0.0122495 0.999925i \(-0.503899\pi\)
−0.0122495 + 0.999925i \(0.503899\pi\)
\(102\) 0 0
\(103\) 1.43845 0.141734 0.0708672 0.997486i \(-0.477423\pi\)
0.0708672 + 0.997486i \(0.477423\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 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
\(119\) 0 0
\(120\) 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 \(-0.630485\pi\)
−0.398545 + 0.917149i \(0.630485\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 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 \(-0.594203\pi\)
−0.291645 + 0.956527i \(0.594203\pi\)
\(140\) 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
\(147\) 0 0
\(148\) 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 \(-0.547861\pi\)
−0.149792 + 0.988717i \(0.547861\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 0 0
\(162\) 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 \(-0.177488\pi\)
0.848531 + 0.529146i \(0.177488\pi\)
\(168\) 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 \(-0.394480\pi\)
0.325461 + 0.945555i \(0.394480\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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 \(-0.840899\pi\)
−0.877664 + 0.479276i \(0.840899\pi\)
\(182\) 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
\(189\) 0 0
\(190\) 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 \(-0.723860\pi\)
−0.646720 + 0.762728i \(0.723860\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) 0 0
\(204\) 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 \(-0.570507\pi\)
−0.219697 + 0.975568i \(0.570507\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) 23.6847 1.57201 0.786003 0.618223i \(-0.212147\pi\)
0.786003 + 0.618223i \(0.212147\pi\)
\(228\) 0 0
\(229\) −19.1231 −1.26369 −0.631845 0.775095i \(-0.717702\pi\)
−0.631845 + 0.775095i \(0.717702\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 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 \(-0.629056\pi\)
−0.394424 + 0.918929i \(0.629056\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) 0 0
\(246\) 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 \(-0.681727\pi\)
−0.540400 + 0.841408i \(0.681727\pi\)
\(252\) 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 \(-0.747495\pi\)
−0.701519 + 0.712650i \(0.747495\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 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 \(-0.839872\pi\)
−0.876113 + 0.482106i \(0.839872\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 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 \(-0.748584\pi\)
−0.703953 + 0.710246i \(0.748584\pi\)
\(284\) 0 0
\(285\) 2.87689 0.170413
\(286\) 0 0
\(287\) 0 0
\(288\) 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 \(-0.591294\pi\)
−0.282892 + 0.959152i \(0.591294\pi\)
\(294\) 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
\(301\) 0 0
\(302\) 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 \(-0.859509\pi\)
−0.904169 + 0.427174i \(0.859509\pi\)
\(308\) 0 0
\(309\) −3.68466 −0.209613
\(310\) 0 0
\(311\) −9.61553 −0.545247 −0.272623 0.962121i \(-0.587891\pi\)
−0.272623 + 0.962121i \(0.587891\pi\)
\(312\) 0 0
\(313\) −31.3002 −1.76919 −0.884596 0.466359i \(-0.845566\pi\)
−0.884596 + 0.466359i \(0.845566\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 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
\(329\) 0 0
\(330\) 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
\(343\) 0 0
\(344\) 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 \(-0.294372\pi\)
0.601996 + 0.798499i \(0.294372\pi\)
\(350\) 0 0
\(351\) 6.56155 0.350230
\(352\) 0 0
\(353\) 14.8078 0.788138 0.394069 0.919081i \(-0.371067\pi\)
0.394069 + 0.919081i \(0.371067\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 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 \(-0.469341\pi\)
0.0961688 + 0.995365i \(0.469341\pi\)
\(368\) 0 0
\(369\) 11.1231 0.579046
\(370\) 0 0
\(371\) 0 0
\(372\) 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 \(-0.584309\pi\)
−0.261778 + 0.965128i \(0.584309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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 \(-0.700152\pi\)
−0.588171 + 0.808737i \(0.700152\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 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 \(-0.272676\pi\)
0.654983 + 0.755644i \(0.272676\pi\)
\(410\) 0 0
\(411\) 22.7386 1.12161
\(412\) 0 0
\(413\) 0 0
\(414\) 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 \(-0.423426\pi\)
0.238252 + 0.971203i \(0.423426\pi\)
\(420\) 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
\(427\) 0 0
\(428\) 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 \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 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 \(-0.619536\pi\)
−0.366769 + 0.930312i \(0.619536\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 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
\(455\) 0 0
\(456\) 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 \(-0.463771\pi\)
0.113570 + 0.993530i \(0.463771\pi\)
\(462\) 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 \(-0.289333\pi\)
0.614561 + 0.788869i \(0.289333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 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 \(-0.403078\pi\)
0.299805 + 0.954001i \(0.403078\pi\)
\(480\) 0 0
\(481\) −27.3693 −1.24793
\(482\) 0 0
\(483\) 0 0
\(484\) 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
\(497\) 0 0
\(498\) 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 \(-0.427932\pi\)
0.224480 + 0.974479i \(0.427932\pi\)
\(504\) 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 \(-0.284693\pi\)
0.625996 + 0.779827i \(0.284693\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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 \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 7.50758 0.328283 0.164142 0.986437i \(-0.447515\pi\)
0.164142 + 0.986437i \(0.447515\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 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
\(539\) 0 0
\(540\) 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
\(553\) 0 0
\(554\) 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 \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 0 0
\(567\) 0 0
\(568\) 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 \(-0.587593\pi\)
−0.271722 + 0.962376i \(0.587593\pi\)
\(578\) 0 0
\(579\) 13.7538 0.571588
\(580\) 0 0
\(581\) 0 0
\(582\) 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 \(-0.564514\pi\)
−0.201291 + 0.979531i \(0.564514\pi\)
\(588\) 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 \(-0.340183\pi\)
0.481251 + 0.876583i \(0.340183\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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 \(-0.318302\pi\)
0.540324 + 0.841457i \(0.318302\pi\)
\(602\) 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 \(-0.532005\pi\)
−0.100376 + 0.994950i \(0.532005\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 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 \(-0.821134\pi\)
−0.846231 + 0.532817i \(0.821134\pi\)
\(620\) 0 0
\(621\) −7.36932 −0.295720
\(622\) 0 0
\(623\) 0 0
\(624\) 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
\(637\) 0 0
\(638\) 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 \(-0.516084\pi\)
−0.0505089 + 0.998724i \(0.516084\pi\)
\(644\) 0 0
\(645\) −23.3693 −0.920166
\(646\) 0 0
\(647\) 3.50758 0.137897 0.0689486 0.997620i \(-0.478036\pi\)
0.0689486 + 0.997620i \(0.478036\pi\)
\(648\) 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 \(-0.519345\pi\)
−0.0607374 + 0.998154i \(0.519345\pi\)
\(662\) 0 0
\(663\) 53.3002 2.07001
\(664\) 0 0
\(665\) 0 0
\(666\) 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 \(-0.347896\pi\)
0.459869 + 0.887987i \(0.347896\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 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 \(-0.448356\pi\)
0.161533 + 0.986867i \(0.448356\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 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 0
\(708\) 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 \(-0.123919\pi\)
0.925173 + 0.379544i \(0.123919\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 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 \(-0.382180\pi\)
0.361748 + 0.932276i \(0.382180\pi\)
\(728\) 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 \(-0.466521\pi\)
0.104984 + 0.994474i \(0.466521\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 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
\(749\) 0 0
\(750\) 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 \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 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 \(-0.445163\pi\)
0.171426 + 0.985197i \(0.445163\pi\)
\(770\) 0 0
\(771\) 57.6155 2.07497
\(772\) 0 0
\(773\) −8.06913 −0.290226 −0.145113 0.989415i \(-0.546355\pi\)
−0.145113 + 0.989415i \(0.546355\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 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 \(-0.521705\pi\)
−0.0681362 + 0.997676i \(0.521705\pi\)
\(788\) 0 0
\(789\) −54.1080 −1.92629
\(790\) 0 0
\(791\) 0 0
\(792\) 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 \(-0.425736\pi\)
0.231198 + 0.972907i \(0.425736\pi\)
\(798\) 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
\(805\) 0 0
\(806\) 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 \(-0.623908\pi\)
−0.379512 + 0.925187i \(0.623908\pi\)
\(812\) 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
\(819\) 0 0
\(820\) 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 \(-0.769400\pi\)
−0.748864 + 0.662724i \(0.769400\pi\)
\(830\) 0 0
\(831\) −41.6155 −1.44363
\(832\) 0 0
\(833\) 0 0
\(834\) 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 \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(840\) 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
\(847\) 0 0
\(848\) 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 \(-0.0763824\pi\)
0.971347 + 0.237666i \(0.0763824\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 32.2462 1.10151 0.550755 0.834667i \(-0.314340\pi\)
0.550755 + 0.834667i \(0.314340\pi\)
\(858\) 0 0
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 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
\(875\) 0 0
\(876\) 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 \(-0.781024\pi\)
−0.772561 + 0.634941i \(0.781024\pi\)
\(882\) 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 \(-0.567258\pi\)
−0.209727 + 0.977760i \(0.567258\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 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
\(903\) 0 0
\(904\) 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
\(917\) 0 0
\(918\) 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 \(-0.173678\pi\)
0.854803 + 0.518952i \(0.173678\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 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 \(-0.620742\pi\)
−0.370290 + 0.928916i \(0.620742\pi\)
\(938\) 0 0
\(939\) 80.1771 2.61648
\(940\) 0 0
\(941\) 13.8617 0.451880 0.225940 0.974141i \(-0.427455\pi\)
0.225940 + 0.974141i \(0.427455\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 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
\(959\) 0 0
\(960\) 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 \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 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 \(-0.553158\pi\)
−0.166226 + 0.986088i \(0.553158\pi\)
\(984\) 0 0
\(985\) 7.12311 0.226961
\(986\) 0 0
\(987\) 0 0
\(988\) 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 \(-0.549009\pi\)
−0.153358 + 0.988171i \(0.549009\pi\)
\(998\) 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 3920.2.a.bs.1.1 2
4.3 odd 2 245.2.a.d.1.2 2
7.6 odd 2 560.2.a.i.1.2 2
12.11 even 2 2205.2.a.x.1.1 2
20.3 even 4 1225.2.b.f.99.2 4
20.7 even 4 1225.2.b.f.99.3 4
20.19 odd 2 1225.2.a.s.1.1 2
21.20 even 2 5040.2.a.bt.1.2 2
28.3 even 6 245.2.e.i.226.1 4
28.11 odd 6 245.2.e.h.226.1 4
28.19 even 6 245.2.e.i.116.1 4
28.23 odd 6 245.2.e.h.116.1 4
28.27 even 2 35.2.a.b.1.2 2
35.13 even 4 2800.2.g.t.449.4 4
35.27 even 4 2800.2.g.t.449.1 4
35.34 odd 2 2800.2.a.bi.1.1 2
56.13 odd 2 2240.2.a.bd.1.1 2
56.27 even 2 2240.2.a.bh.1.2 2
84.83 odd 2 315.2.a.e.1.1 2
140.27 odd 4 175.2.b.b.99.3 4
140.83 odd 4 175.2.b.b.99.2 4
140.139 even 2 175.2.a.f.1.1 2
308.307 odd 2 4235.2.a.m.1.1 2
364.363 even 2 5915.2.a.l.1.1 2
420.83 even 4 1575.2.d.e.1324.3 4
420.167 even 4 1575.2.d.e.1324.2 4
420.419 odd 2 1575.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 28.27 even 2
175.2.a.f.1.1 2 140.139 even 2
175.2.b.b.99.2 4 140.83 odd 4
175.2.b.b.99.3 4 140.27 odd 4
245.2.a.d.1.2 2 4.3 odd 2
245.2.e.h.116.1 4 28.23 odd 6
245.2.e.h.226.1 4 28.11 odd 6
245.2.e.i.116.1 4 28.19 even 6
245.2.e.i.226.1 4 28.3 even 6
315.2.a.e.1.1 2 84.83 odd 2
560.2.a.i.1.2 2 7.6 odd 2
1225.2.a.s.1.1 2 20.19 odd 2
1225.2.b.f.99.2 4 20.3 even 4
1225.2.b.f.99.3 4 20.7 even 4
1575.2.a.p.1.2 2 420.419 odd 2
1575.2.d.e.1324.2 4 420.167 even 4
1575.2.d.e.1324.3 4 420.83 even 4
2205.2.a.x.1.1 2 12.11 even 2
2240.2.a.bd.1.1 2 56.13 odd 2
2240.2.a.bh.1.2 2 56.27 even 2
2800.2.a.bi.1.1 2 35.34 odd 2
2800.2.g.t.449.1 4 35.27 even 4
2800.2.g.t.449.4 4 35.13 even 4
3920.2.a.bs.1.1 2 1.1 even 1 trivial
4235.2.a.m.1.1 2 308.307 odd 2
5040.2.a.bt.1.2 2 21.20 even 2
5915.2.a.l.1.1 2 364.363 even 2