Properties

Label 7360.2.a.bd.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3680)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.30278 q^{3} +1.00000 q^{5} +0.302776 q^{7} +7.90833 q^{9} +O(q^{10})\) \(q-3.30278 q^{3} +1.00000 q^{5} +0.302776 q^{7} +7.90833 q^{9} +3.30278 q^{11} +2.30278 q^{13} -3.30278 q^{15} +7.30278 q^{17} -2.90833 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -16.2111 q^{27} -8.60555 q^{29} -4.30278 q^{31} -10.9083 q^{33} +0.302776 q^{35} -5.21110 q^{37} -7.60555 q^{39} -4.69722 q^{41} -4.00000 q^{43} +7.90833 q^{45} -3.39445 q^{47} -6.90833 q^{49} -24.1194 q^{51} +7.21110 q^{53} +3.30278 q^{55} +9.60555 q^{57} -1.39445 q^{59} -1.30278 q^{61} +2.39445 q^{63} +2.30278 q^{65} -9.21110 q^{67} -3.30278 q^{69} -5.30278 q^{71} +5.39445 q^{73} -3.30278 q^{75} +1.00000 q^{77} +4.00000 q^{79} +29.8167 q^{81} -11.2111 q^{83} +7.30278 q^{85} +28.4222 q^{87} +2.78890 q^{89} +0.697224 q^{91} +14.2111 q^{93} -2.90833 q^{95} -13.7250 q^{97} +26.1194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} + q^{13} - 3 q^{15} + 11 q^{17} + 5 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 18 q^{27} - 10 q^{29} - 5 q^{31} - 11 q^{33} - 3 q^{35} + 4 q^{37} - 8 q^{39} - 13 q^{41} - 8 q^{43} + 5 q^{45} - 14 q^{47} - 3 q^{49} - 23 q^{51} + 3 q^{55} + 12 q^{57} - 10 q^{59} + q^{61} + 12 q^{63} + q^{65} - 4 q^{67} - 3 q^{69} - 7 q^{71} + 18 q^{73} - 3 q^{75} + 2 q^{77} + 8 q^{79} + 38 q^{81} - 8 q^{83} + 11 q^{85} + 28 q^{87} + 20 q^{89} + 5 q^{91} + 14 q^{93} + 5 q^{95} + 5 q^{97} + 27 q^{99} + O(q^{100}) \)

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
\(3\) −3.30278 −1.90686 −0.953429 0.301617i \(-0.902474\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.302776 0.114438 0.0572192 0.998362i \(-0.481777\pi\)
0.0572192 + 0.998362i \(0.481777\pi\)
\(8\) 0 0
\(9\) 7.90833 2.63611
\(10\) 0 0
\(11\) 3.30278 0.995824 0.497912 0.867227i \(-0.334100\pi\)
0.497912 + 0.867227i \(0.334100\pi\)
\(12\) 0 0
\(13\) 2.30278 0.638675 0.319338 0.947641i \(-0.396540\pi\)
0.319338 + 0.947641i \(0.396540\pi\)
\(14\) 0 0
\(15\) −3.30278 −0.852773
\(16\) 0 0
\(17\) 7.30278 1.77118 0.885592 0.464465i \(-0.153753\pi\)
0.885592 + 0.464465i \(0.153753\pi\)
\(18\) 0 0
\(19\) −2.90833 −0.667216 −0.333608 0.942712i \(-0.608266\pi\)
−0.333608 + 0.942712i \(0.608266\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −16.2111 −3.11983
\(28\) 0 0
\(29\) −8.60555 −1.59801 −0.799005 0.601324i \(-0.794640\pi\)
−0.799005 + 0.601324i \(0.794640\pi\)
\(30\) 0 0
\(31\) −4.30278 −0.772801 −0.386401 0.922331i \(-0.626282\pi\)
−0.386401 + 0.922331i \(0.626282\pi\)
\(32\) 0 0
\(33\) −10.9083 −1.89890
\(34\) 0 0
\(35\) 0.302776 0.0511784
\(36\) 0 0
\(37\) −5.21110 −0.856700 −0.428350 0.903613i \(-0.640905\pi\)
−0.428350 + 0.903613i \(0.640905\pi\)
\(38\) 0 0
\(39\) −7.60555 −1.21786
\(40\) 0 0
\(41\) −4.69722 −0.733583 −0.366792 0.930303i \(-0.619544\pi\)
−0.366792 + 0.930303i \(0.619544\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 7.90833 1.17890
\(46\) 0 0
\(47\) −3.39445 −0.495131 −0.247566 0.968871i \(-0.579631\pi\)
−0.247566 + 0.968871i \(0.579631\pi\)
\(48\) 0 0
\(49\) −6.90833 −0.986904
\(50\) 0 0
\(51\) −24.1194 −3.37740
\(52\) 0 0
\(53\) 7.21110 0.990521 0.495261 0.868744i \(-0.335073\pi\)
0.495261 + 0.868744i \(0.335073\pi\)
\(54\) 0 0
\(55\) 3.30278 0.445346
\(56\) 0 0
\(57\) 9.60555 1.27229
\(58\) 0 0
\(59\) −1.39445 −0.181542 −0.0907709 0.995872i \(-0.528933\pi\)
−0.0907709 + 0.995872i \(0.528933\pi\)
\(60\) 0 0
\(61\) −1.30278 −0.166803 −0.0834017 0.996516i \(-0.526578\pi\)
−0.0834017 + 0.996516i \(0.526578\pi\)
\(62\) 0 0
\(63\) 2.39445 0.301672
\(64\) 0 0
\(65\) 2.30278 0.285624
\(66\) 0 0
\(67\) −9.21110 −1.12532 −0.562658 0.826690i \(-0.690221\pi\)
−0.562658 + 0.826690i \(0.690221\pi\)
\(68\) 0 0
\(69\) −3.30278 −0.397607
\(70\) 0 0
\(71\) −5.30278 −0.629324 −0.314662 0.949204i \(-0.601891\pi\)
−0.314662 + 0.949204i \(0.601891\pi\)
\(72\) 0 0
\(73\) 5.39445 0.631372 0.315686 0.948864i \(-0.397765\pi\)
0.315686 + 0.948864i \(0.397765\pi\)
\(74\) 0 0
\(75\) −3.30278 −0.381372
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 0 0
\(83\) −11.2111 −1.23058 −0.615289 0.788301i \(-0.710961\pi\)
−0.615289 + 0.788301i \(0.710961\pi\)
\(84\) 0 0
\(85\) 7.30278 0.792097
\(86\) 0 0
\(87\) 28.4222 3.04718
\(88\) 0 0
\(89\) 2.78890 0.295623 0.147811 0.989016i \(-0.452777\pi\)
0.147811 + 0.989016i \(0.452777\pi\)
\(90\) 0 0
\(91\) 0.697224 0.0730890
\(92\) 0 0
\(93\) 14.2111 1.47362
\(94\) 0 0
\(95\) −2.90833 −0.298388
\(96\) 0 0
\(97\) −13.7250 −1.39356 −0.696780 0.717285i \(-0.745385\pi\)
−0.696780 + 0.717285i \(0.745385\pi\)
\(98\) 0 0
\(99\) 26.1194 2.62510
\(100\) 0 0
\(101\) −12.6056 −1.25430 −0.627150 0.778899i \(-0.715779\pi\)
−0.627150 + 0.778899i \(0.715779\pi\)
\(102\) 0 0
\(103\) 9.11943 0.898564 0.449282 0.893390i \(-0.351680\pi\)
0.449282 + 0.893390i \(0.351680\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 1.81665 0.175623 0.0878113 0.996137i \(-0.472013\pi\)
0.0878113 + 0.996137i \(0.472013\pi\)
\(108\) 0 0
\(109\) −13.6972 −1.31196 −0.655978 0.754780i \(-0.727744\pi\)
−0.655978 + 0.754780i \(0.727744\pi\)
\(110\) 0 0
\(111\) 17.2111 1.63361
\(112\) 0 0
\(113\) −3.21110 −0.302075 −0.151038 0.988528i \(-0.548261\pi\)
−0.151038 + 0.988528i \(0.548261\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 18.2111 1.68362
\(118\) 0 0
\(119\) 2.21110 0.202691
\(120\) 0 0
\(121\) −0.0916731 −0.00833392
\(122\) 0 0
\(123\) 15.5139 1.39884
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.3944 −1.18857 −0.594283 0.804256i \(-0.702564\pi\)
−0.594283 + 0.804256i \(0.702564\pi\)
\(128\) 0 0
\(129\) 13.2111 1.16317
\(130\) 0 0
\(131\) −0.788897 −0.0689263 −0.0344631 0.999406i \(-0.510972\pi\)
−0.0344631 + 0.999406i \(0.510972\pi\)
\(132\) 0 0
\(133\) −0.880571 −0.0763551
\(134\) 0 0
\(135\) −16.2111 −1.39523
\(136\) 0 0
\(137\) 18.1194 1.54805 0.774024 0.633157i \(-0.218241\pi\)
0.774024 + 0.633157i \(0.218241\pi\)
\(138\) 0 0
\(139\) 11.8167 1.00228 0.501138 0.865368i \(-0.332915\pi\)
0.501138 + 0.865368i \(0.332915\pi\)
\(140\) 0 0
\(141\) 11.2111 0.944145
\(142\) 0 0
\(143\) 7.60555 0.636008
\(144\) 0 0
\(145\) −8.60555 −0.714652
\(146\) 0 0
\(147\) 22.8167 1.88189
\(148\) 0 0
\(149\) 2.90833 0.238259 0.119130 0.992879i \(-0.461990\pi\)
0.119130 + 0.992879i \(0.461990\pi\)
\(150\) 0 0
\(151\) −15.7250 −1.27968 −0.639840 0.768508i \(-0.721000\pi\)
−0.639840 + 0.768508i \(0.721000\pi\)
\(152\) 0 0
\(153\) 57.7527 4.66903
\(154\) 0 0
\(155\) −4.30278 −0.345607
\(156\) 0 0
\(157\) 3.39445 0.270907 0.135453 0.990784i \(-0.456751\pi\)
0.135453 + 0.990784i \(0.456751\pi\)
\(158\) 0 0
\(159\) −23.8167 −1.88878
\(160\) 0 0
\(161\) 0.302776 0.0238621
\(162\) 0 0
\(163\) −6.90833 −0.541102 −0.270551 0.962706i \(-0.587206\pi\)
−0.270551 + 0.962706i \(0.587206\pi\)
\(164\) 0 0
\(165\) −10.9083 −0.849212
\(166\) 0 0
\(167\) −13.2111 −1.02231 −0.511153 0.859490i \(-0.670781\pi\)
−0.511153 + 0.859490i \(0.670781\pi\)
\(168\) 0 0
\(169\) −7.69722 −0.592094
\(170\) 0 0
\(171\) −23.0000 −1.75885
\(172\) 0 0
\(173\) 14.5139 1.10347 0.551735 0.834020i \(-0.313966\pi\)
0.551735 + 0.834020i \(0.313966\pi\)
\(174\) 0 0
\(175\) 0.302776 0.0228877
\(176\) 0 0
\(177\) 4.60555 0.346174
\(178\) 0 0
\(179\) 23.0278 1.72118 0.860588 0.509302i \(-0.170097\pi\)
0.860588 + 0.509302i \(0.170097\pi\)
\(180\) 0 0
\(181\) −21.6972 −1.61274 −0.806371 0.591410i \(-0.798571\pi\)
−0.806371 + 0.591410i \(0.798571\pi\)
\(182\) 0 0
\(183\) 4.30278 0.318070
\(184\) 0 0
\(185\) −5.21110 −0.383128
\(186\) 0 0
\(187\) 24.1194 1.76379
\(188\) 0 0
\(189\) −4.90833 −0.357028
\(190\) 0 0
\(191\) 0.183346 0.0132665 0.00663323 0.999978i \(-0.497889\pi\)
0.00663323 + 0.999978i \(0.497889\pi\)
\(192\) 0 0
\(193\) 9.39445 0.676227 0.338114 0.941105i \(-0.390211\pi\)
0.338114 + 0.941105i \(0.390211\pi\)
\(194\) 0 0
\(195\) −7.60555 −0.544645
\(196\) 0 0
\(197\) −22.5139 −1.60405 −0.802024 0.597292i \(-0.796243\pi\)
−0.802024 + 0.597292i \(0.796243\pi\)
\(198\) 0 0
\(199\) 8.42221 0.597034 0.298517 0.954404i \(-0.403508\pi\)
0.298517 + 0.954404i \(0.403508\pi\)
\(200\) 0 0
\(201\) 30.4222 2.14582
\(202\) 0 0
\(203\) −2.60555 −0.182874
\(204\) 0 0
\(205\) −4.69722 −0.328068
\(206\) 0 0
\(207\) 7.90833 0.549667
\(208\) 0 0
\(209\) −9.60555 −0.664430
\(210\) 0 0
\(211\) −28.4222 −1.95667 −0.978333 0.207039i \(-0.933617\pi\)
−0.978333 + 0.207039i \(0.933617\pi\)
\(212\) 0 0
\(213\) 17.5139 1.20003
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −1.30278 −0.0884382
\(218\) 0 0
\(219\) −17.8167 −1.20394
\(220\) 0 0
\(221\) 16.8167 1.13121
\(222\) 0 0
\(223\) 14.4222 0.965782 0.482891 0.875680i \(-0.339587\pi\)
0.482891 + 0.875680i \(0.339587\pi\)
\(224\) 0 0
\(225\) 7.90833 0.527222
\(226\) 0 0
\(227\) 25.8167 1.71351 0.856756 0.515722i \(-0.172476\pi\)
0.856756 + 0.515722i \(0.172476\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −3.30278 −0.217307
\(232\) 0 0
\(233\) 22.2389 1.45692 0.728458 0.685090i \(-0.240237\pi\)
0.728458 + 0.685090i \(0.240237\pi\)
\(234\) 0 0
\(235\) −3.39445 −0.221429
\(236\) 0 0
\(237\) −13.2111 −0.858153
\(238\) 0 0
\(239\) −29.2111 −1.88951 −0.944755 0.327779i \(-0.893700\pi\)
−0.944755 + 0.327779i \(0.893700\pi\)
\(240\) 0 0
\(241\) −27.6333 −1.78002 −0.890009 0.455943i \(-0.849302\pi\)
−0.890009 + 0.455943i \(0.849302\pi\)
\(242\) 0 0
\(243\) −49.8444 −3.19752
\(244\) 0 0
\(245\) −6.90833 −0.441357
\(246\) 0 0
\(247\) −6.69722 −0.426134
\(248\) 0 0
\(249\) 37.0278 2.34654
\(250\) 0 0
\(251\) 8.48612 0.535639 0.267820 0.963469i \(-0.413697\pi\)
0.267820 + 0.963469i \(0.413697\pi\)
\(252\) 0 0
\(253\) 3.30278 0.207644
\(254\) 0 0
\(255\) −24.1194 −1.51042
\(256\) 0 0
\(257\) −26.6056 −1.65961 −0.829804 0.558054i \(-0.811548\pi\)
−0.829804 + 0.558054i \(0.811548\pi\)
\(258\) 0 0
\(259\) −1.57779 −0.0980394
\(260\) 0 0
\(261\) −68.0555 −4.21253
\(262\) 0 0
\(263\) −19.9083 −1.22760 −0.613800 0.789462i \(-0.710360\pi\)
−0.613800 + 0.789462i \(0.710360\pi\)
\(264\) 0 0
\(265\) 7.21110 0.442975
\(266\) 0 0
\(267\) −9.21110 −0.563710
\(268\) 0 0
\(269\) 23.0278 1.40403 0.702014 0.712164i \(-0.252285\pi\)
0.702014 + 0.712164i \(0.252285\pi\)
\(270\) 0 0
\(271\) −28.3028 −1.71927 −0.859636 0.510908i \(-0.829309\pi\)
−0.859636 + 0.510908i \(0.829309\pi\)
\(272\) 0 0
\(273\) −2.30278 −0.139370
\(274\) 0 0
\(275\) 3.30278 0.199165
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −34.0278 −2.03719
\(280\) 0 0
\(281\) −15.0278 −0.896481 −0.448240 0.893913i \(-0.647949\pi\)
−0.448240 + 0.893913i \(0.647949\pi\)
\(282\) 0 0
\(283\) 29.6333 1.76152 0.880759 0.473565i \(-0.157033\pi\)
0.880759 + 0.473565i \(0.157033\pi\)
\(284\) 0 0
\(285\) 9.60555 0.568984
\(286\) 0 0
\(287\) −1.42221 −0.0839501
\(288\) 0 0
\(289\) 36.3305 2.13709
\(290\) 0 0
\(291\) 45.3305 2.65732
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −1.39445 −0.0811879
\(296\) 0 0
\(297\) −53.5416 −3.10680
\(298\) 0 0
\(299\) 2.30278 0.133173
\(300\) 0 0
\(301\) −1.21110 −0.0698068
\(302\) 0 0
\(303\) 41.6333 2.39177
\(304\) 0 0
\(305\) −1.30278 −0.0745967
\(306\) 0 0
\(307\) −5.90833 −0.337206 −0.168603 0.985684i \(-0.553926\pi\)
−0.168603 + 0.985684i \(0.553926\pi\)
\(308\) 0 0
\(309\) −30.1194 −1.71343
\(310\) 0 0
\(311\) 10.4222 0.590989 0.295495 0.955344i \(-0.404516\pi\)
0.295495 + 0.955344i \(0.404516\pi\)
\(312\) 0 0
\(313\) 26.7250 1.51059 0.755293 0.655388i \(-0.227495\pi\)
0.755293 + 0.655388i \(0.227495\pi\)
\(314\) 0 0
\(315\) 2.39445 0.134912
\(316\) 0 0
\(317\) 30.3028 1.70197 0.850987 0.525187i \(-0.176005\pi\)
0.850987 + 0.525187i \(0.176005\pi\)
\(318\) 0 0
\(319\) −28.4222 −1.59134
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −21.2389 −1.18176
\(324\) 0 0
\(325\) 2.30278 0.127735
\(326\) 0 0
\(327\) 45.2389 2.50171
\(328\) 0 0
\(329\) −1.02776 −0.0566620
\(330\) 0 0
\(331\) 3.81665 0.209782 0.104891 0.994484i \(-0.466551\pi\)
0.104891 + 0.994484i \(0.466551\pi\)
\(332\) 0 0
\(333\) −41.2111 −2.25835
\(334\) 0 0
\(335\) −9.21110 −0.503256
\(336\) 0 0
\(337\) −7.09167 −0.386308 −0.193154 0.981168i \(-0.561872\pi\)
−0.193154 + 0.981168i \(0.561872\pi\)
\(338\) 0 0
\(339\) 10.6056 0.576014
\(340\) 0 0
\(341\) −14.2111 −0.769574
\(342\) 0 0
\(343\) −4.21110 −0.227378
\(344\) 0 0
\(345\) −3.30278 −0.177815
\(346\) 0 0
\(347\) −14.0917 −0.756481 −0.378240 0.925707i \(-0.623471\pi\)
−0.378240 + 0.925707i \(0.623471\pi\)
\(348\) 0 0
\(349\) 17.6333 0.943889 0.471945 0.881628i \(-0.343552\pi\)
0.471945 + 0.881628i \(0.343552\pi\)
\(350\) 0 0
\(351\) −37.3305 −1.99256
\(352\) 0 0
\(353\) −21.2111 −1.12895 −0.564477 0.825449i \(-0.690922\pi\)
−0.564477 + 0.825449i \(0.690922\pi\)
\(354\) 0 0
\(355\) −5.30278 −0.281442
\(356\) 0 0
\(357\) −7.30278 −0.386504
\(358\) 0 0
\(359\) 16.4222 0.866731 0.433365 0.901218i \(-0.357326\pi\)
0.433365 + 0.901218i \(0.357326\pi\)
\(360\) 0 0
\(361\) −10.5416 −0.554823
\(362\) 0 0
\(363\) 0.302776 0.0158916
\(364\) 0 0
\(365\) 5.39445 0.282358
\(366\) 0 0
\(367\) 21.2111 1.10721 0.553605 0.832779i \(-0.313252\pi\)
0.553605 + 0.832779i \(0.313252\pi\)
\(368\) 0 0
\(369\) −37.1472 −1.93381
\(370\) 0 0
\(371\) 2.18335 0.113354
\(372\) 0 0
\(373\) −26.6056 −1.37758 −0.688792 0.724959i \(-0.741859\pi\)
−0.688792 + 0.724959i \(0.741859\pi\)
\(374\) 0 0
\(375\) −3.30278 −0.170555
\(376\) 0 0
\(377\) −19.8167 −1.02061
\(378\) 0 0
\(379\) 37.5416 1.92838 0.964192 0.265205i \(-0.0854396\pi\)
0.964192 + 0.265205i \(0.0854396\pi\)
\(380\) 0 0
\(381\) 44.2389 2.26643
\(382\) 0 0
\(383\) −26.4222 −1.35011 −0.675056 0.737767i \(-0.735880\pi\)
−0.675056 + 0.737767i \(0.735880\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) −31.6333 −1.60801
\(388\) 0 0
\(389\) 18.5139 0.938691 0.469345 0.883015i \(-0.344490\pi\)
0.469345 + 0.883015i \(0.344490\pi\)
\(390\) 0 0
\(391\) 7.30278 0.369317
\(392\) 0 0
\(393\) 2.60555 0.131433
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) −10.5139 −0.527676 −0.263838 0.964567i \(-0.584989\pi\)
−0.263838 + 0.964567i \(0.584989\pi\)
\(398\) 0 0
\(399\) 2.90833 0.145598
\(400\) 0 0
\(401\) −19.8167 −0.989596 −0.494798 0.869008i \(-0.664758\pi\)
−0.494798 + 0.869008i \(0.664758\pi\)
\(402\) 0 0
\(403\) −9.90833 −0.493569
\(404\) 0 0
\(405\) 29.8167 1.48160
\(406\) 0 0
\(407\) −17.2111 −0.853123
\(408\) 0 0
\(409\) −15.0917 −0.746235 −0.373118 0.927784i \(-0.621711\pi\)
−0.373118 + 0.927784i \(0.621711\pi\)
\(410\) 0 0
\(411\) −59.8444 −2.95191
\(412\) 0 0
\(413\) −0.422205 −0.0207754
\(414\) 0 0
\(415\) −11.2111 −0.550331
\(416\) 0 0
\(417\) −39.0278 −1.91120
\(418\) 0 0
\(419\) 9.21110 0.449992 0.224996 0.974360i \(-0.427763\pi\)
0.224996 + 0.974360i \(0.427763\pi\)
\(420\) 0 0
\(421\) 1.09167 0.0532049 0.0266024 0.999646i \(-0.491531\pi\)
0.0266024 + 0.999646i \(0.491531\pi\)
\(422\) 0 0
\(423\) −26.8444 −1.30522
\(424\) 0 0
\(425\) 7.30278 0.354237
\(426\) 0 0
\(427\) −0.394449 −0.0190887
\(428\) 0 0
\(429\) −25.1194 −1.21278
\(430\) 0 0
\(431\) 8.60555 0.414515 0.207257 0.978286i \(-0.433546\pi\)
0.207257 + 0.978286i \(0.433546\pi\)
\(432\) 0 0
\(433\) 35.6972 1.71550 0.857750 0.514068i \(-0.171862\pi\)
0.857750 + 0.514068i \(0.171862\pi\)
\(434\) 0 0
\(435\) 28.4222 1.36274
\(436\) 0 0
\(437\) −2.90833 −0.139124
\(438\) 0 0
\(439\) −18.7250 −0.893695 −0.446847 0.894610i \(-0.647453\pi\)
−0.446847 + 0.894610i \(0.647453\pi\)
\(440\) 0 0
\(441\) −54.6333 −2.60159
\(442\) 0 0
\(443\) 1.33053 0.0632155 0.0316077 0.999500i \(-0.489937\pi\)
0.0316077 + 0.999500i \(0.489937\pi\)
\(444\) 0 0
\(445\) 2.78890 0.132206
\(446\) 0 0
\(447\) −9.60555 −0.454327
\(448\) 0 0
\(449\) −2.48612 −0.117327 −0.0586637 0.998278i \(-0.518684\pi\)
−0.0586637 + 0.998278i \(0.518684\pi\)
\(450\) 0 0
\(451\) −15.5139 −0.730520
\(452\) 0 0
\(453\) 51.9361 2.44017
\(454\) 0 0
\(455\) 0.697224 0.0326864
\(456\) 0 0
\(457\) 15.2111 0.711545 0.355773 0.934573i \(-0.384218\pi\)
0.355773 + 0.934573i \(0.384218\pi\)
\(458\) 0 0
\(459\) −118.386 −5.52579
\(460\) 0 0
\(461\) −35.4500 −1.65107 −0.825535 0.564351i \(-0.809126\pi\)
−0.825535 + 0.564351i \(0.809126\pi\)
\(462\) 0 0
\(463\) 12.7889 0.594350 0.297175 0.954823i \(-0.403955\pi\)
0.297175 + 0.954823i \(0.403955\pi\)
\(464\) 0 0
\(465\) 14.2111 0.659024
\(466\) 0 0
\(467\) 17.0278 0.787951 0.393975 0.919121i \(-0.371100\pi\)
0.393975 + 0.919121i \(0.371100\pi\)
\(468\) 0 0
\(469\) −2.78890 −0.128779
\(470\) 0 0
\(471\) −11.2111 −0.516580
\(472\) 0 0
\(473\) −13.2111 −0.607447
\(474\) 0 0
\(475\) −2.90833 −0.133443
\(476\) 0 0
\(477\) 57.0278 2.61112
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −13.7250 −0.623219
\(486\) 0 0
\(487\) −39.4500 −1.78765 −0.893824 0.448418i \(-0.851988\pi\)
−0.893824 + 0.448418i \(0.851988\pi\)
\(488\) 0 0
\(489\) 22.8167 1.03180
\(490\) 0 0
\(491\) −8.60555 −0.388363 −0.194182 0.980966i \(-0.562205\pi\)
−0.194182 + 0.980966i \(0.562205\pi\)
\(492\) 0 0
\(493\) −62.8444 −2.83037
\(494\) 0 0
\(495\) 26.1194 1.17398
\(496\) 0 0
\(497\) −1.60555 −0.0720188
\(498\) 0 0
\(499\) 13.2111 0.591410 0.295705 0.955279i \(-0.404445\pi\)
0.295705 + 0.955279i \(0.404445\pi\)
\(500\) 0 0
\(501\) 43.6333 1.94939
\(502\) 0 0
\(503\) 9.51388 0.424203 0.212101 0.977248i \(-0.431969\pi\)
0.212101 + 0.977248i \(0.431969\pi\)
\(504\) 0 0
\(505\) −12.6056 −0.560940
\(506\) 0 0
\(507\) 25.4222 1.12904
\(508\) 0 0
\(509\) −19.3944 −0.859644 −0.429822 0.902914i \(-0.641424\pi\)
−0.429822 + 0.902914i \(0.641424\pi\)
\(510\) 0 0
\(511\) 1.63331 0.0722533
\(512\) 0 0
\(513\) 47.1472 2.08160
\(514\) 0 0
\(515\) 9.11943 0.401850
\(516\) 0 0
\(517\) −11.2111 −0.493064
\(518\) 0 0
\(519\) −47.9361 −2.10416
\(520\) 0 0
\(521\) −36.4222 −1.59569 −0.797843 0.602865i \(-0.794026\pi\)
−0.797843 + 0.602865i \(0.794026\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) −31.4222 −1.36877
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.0278 −0.478564
\(532\) 0 0
\(533\) −10.8167 −0.468521
\(534\) 0 0
\(535\) 1.81665 0.0785408
\(536\) 0 0
\(537\) −76.0555 −3.28204
\(538\) 0 0
\(539\) −22.8167 −0.982783
\(540\) 0 0
\(541\) −26.4222 −1.13598 −0.567990 0.823036i \(-0.692279\pi\)
−0.567990 + 0.823036i \(0.692279\pi\)
\(542\) 0 0
\(543\) 71.6611 3.07527
\(544\) 0 0
\(545\) −13.6972 −0.586725
\(546\) 0 0
\(547\) 2.90833 0.124351 0.0621755 0.998065i \(-0.480196\pi\)
0.0621755 + 0.998065i \(0.480196\pi\)
\(548\) 0 0
\(549\) −10.3028 −0.439712
\(550\) 0 0
\(551\) 25.0278 1.06622
\(552\) 0 0
\(553\) 1.21110 0.0515013
\(554\) 0 0
\(555\) 17.2111 0.730571
\(556\) 0 0
\(557\) −5.21110 −0.220802 −0.110401 0.993887i \(-0.535213\pi\)
−0.110401 + 0.993887i \(0.535213\pi\)
\(558\) 0 0
\(559\) −9.21110 −0.389588
\(560\) 0 0
\(561\) −79.6611 −3.36329
\(562\) 0 0
\(563\) −2.78890 −0.117538 −0.0587690 0.998272i \(-0.518718\pi\)
−0.0587690 + 0.998272i \(0.518718\pi\)
\(564\) 0 0
\(565\) −3.21110 −0.135092
\(566\) 0 0
\(567\) 9.02776 0.379130
\(568\) 0 0
\(569\) −18.8444 −0.789999 −0.394999 0.918681i \(-0.629255\pi\)
−0.394999 + 0.918681i \(0.629255\pi\)
\(570\) 0 0
\(571\) 7.30278 0.305612 0.152806 0.988256i \(-0.451169\pi\)
0.152806 + 0.988256i \(0.451169\pi\)
\(572\) 0 0
\(573\) −0.605551 −0.0252973
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 7.21110 0.300202 0.150101 0.988671i \(-0.452040\pi\)
0.150101 + 0.988671i \(0.452040\pi\)
\(578\) 0 0
\(579\) −31.0278 −1.28947
\(580\) 0 0
\(581\) −3.39445 −0.140825
\(582\) 0 0
\(583\) 23.8167 0.986385
\(584\) 0 0
\(585\) 18.2111 0.752936
\(586\) 0 0
\(587\) −30.9361 −1.27687 −0.638434 0.769676i \(-0.720418\pi\)
−0.638434 + 0.769676i \(0.720418\pi\)
\(588\) 0 0
\(589\) 12.5139 0.515625
\(590\) 0 0
\(591\) 74.3583 3.05869
\(592\) 0 0
\(593\) 7.81665 0.320991 0.160496 0.987037i \(-0.448691\pi\)
0.160496 + 0.987037i \(0.448691\pi\)
\(594\) 0 0
\(595\) 2.21110 0.0906464
\(596\) 0 0
\(597\) −27.8167 −1.13846
\(598\) 0 0
\(599\) −15.9083 −0.649997 −0.324998 0.945715i \(-0.605364\pi\)
−0.324998 + 0.945715i \(0.605364\pi\)
\(600\) 0 0
\(601\) 44.5416 1.81689 0.908446 0.418003i \(-0.137270\pi\)
0.908446 + 0.418003i \(0.137270\pi\)
\(602\) 0 0
\(603\) −72.8444 −2.96645
\(604\) 0 0
\(605\) −0.0916731 −0.00372704
\(606\) 0 0
\(607\) −35.6333 −1.44631 −0.723156 0.690685i \(-0.757309\pi\)
−0.723156 + 0.690685i \(0.757309\pi\)
\(608\) 0 0
\(609\) 8.60555 0.348715
\(610\) 0 0
\(611\) −7.81665 −0.316228
\(612\) 0 0
\(613\) −36.0555 −1.45627 −0.728134 0.685435i \(-0.759612\pi\)
−0.728134 + 0.685435i \(0.759612\pi\)
\(614\) 0 0
\(615\) 15.5139 0.625580
\(616\) 0 0
\(617\) 8.51388 0.342756 0.171378 0.985205i \(-0.445178\pi\)
0.171378 + 0.985205i \(0.445178\pi\)
\(618\) 0 0
\(619\) −28.7250 −1.15455 −0.577277 0.816548i \(-0.695885\pi\)
−0.577277 + 0.816548i \(0.695885\pi\)
\(620\) 0 0
\(621\) −16.2111 −0.650529
\(622\) 0 0
\(623\) 0.844410 0.0338306
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 31.7250 1.26697
\(628\) 0 0
\(629\) −38.0555 −1.51737
\(630\) 0 0
\(631\) 1.39445 0.0555121 0.0277561 0.999615i \(-0.491164\pi\)
0.0277561 + 0.999615i \(0.491164\pi\)
\(632\) 0 0
\(633\) 93.8722 3.73108
\(634\) 0 0
\(635\) −13.3944 −0.531542
\(636\) 0 0
\(637\) −15.9083 −0.630311
\(638\) 0 0
\(639\) −41.9361 −1.65897
\(640\) 0 0
\(641\) 15.6333 0.617479 0.308739 0.951147i \(-0.400093\pi\)
0.308739 + 0.951147i \(0.400093\pi\)
\(642\) 0 0
\(643\) 8.23886 0.324909 0.162454 0.986716i \(-0.448059\pi\)
0.162454 + 0.986716i \(0.448059\pi\)
\(644\) 0 0
\(645\) 13.2111 0.520187
\(646\) 0 0
\(647\) 45.6333 1.79403 0.897015 0.442000i \(-0.145731\pi\)
0.897015 + 0.442000i \(0.145731\pi\)
\(648\) 0 0
\(649\) −4.60555 −0.180784
\(650\) 0 0
\(651\) 4.30278 0.168639
\(652\) 0 0
\(653\) −0.486122 −0.0190234 −0.00951171 0.999955i \(-0.503028\pi\)
−0.00951171 + 0.999955i \(0.503028\pi\)
\(654\) 0 0
\(655\) −0.788897 −0.0308248
\(656\) 0 0
\(657\) 42.6611 1.66437
\(658\) 0 0
\(659\) 26.0555 1.01498 0.507489 0.861658i \(-0.330574\pi\)
0.507489 + 0.861658i \(0.330574\pi\)
\(660\) 0 0
\(661\) −0.0916731 −0.00356567 −0.00178283 0.999998i \(-0.500567\pi\)
−0.00178283 + 0.999998i \(0.500567\pi\)
\(662\) 0 0
\(663\) −55.5416 −2.15706
\(664\) 0 0
\(665\) −0.880571 −0.0341471
\(666\) 0 0
\(667\) −8.60555 −0.333208
\(668\) 0 0
\(669\) −47.6333 −1.84161
\(670\) 0 0
\(671\) −4.30278 −0.166107
\(672\) 0 0
\(673\) −26.8444 −1.03478 −0.517388 0.855751i \(-0.673096\pi\)
−0.517388 + 0.855751i \(0.673096\pi\)
\(674\) 0 0
\(675\) −16.2111 −0.623966
\(676\) 0 0
\(677\) 12.7889 0.491517 0.245759 0.969331i \(-0.420963\pi\)
0.245759 + 0.969331i \(0.420963\pi\)
\(678\) 0 0
\(679\) −4.15559 −0.159477
\(680\) 0 0
\(681\) −85.2666 −3.26742
\(682\) 0 0
\(683\) 11.8806 0.454597 0.227299 0.973825i \(-0.427011\pi\)
0.227299 + 0.973825i \(0.427011\pi\)
\(684\) 0 0
\(685\) 18.1194 0.692308
\(686\) 0 0
\(687\) 46.2389 1.76412
\(688\) 0 0
\(689\) 16.6056 0.632621
\(690\) 0 0
\(691\) −7.39445 −0.281298 −0.140649 0.990060i \(-0.544919\pi\)
−0.140649 + 0.990060i \(0.544919\pi\)
\(692\) 0 0
\(693\) 7.90833 0.300412
\(694\) 0 0
\(695\) 11.8167 0.448231
\(696\) 0 0
\(697\) −34.3028 −1.29931
\(698\) 0 0
\(699\) −73.4500 −2.77813
\(700\) 0 0
\(701\) −17.9361 −0.677437 −0.338718 0.940888i \(-0.609993\pi\)
−0.338718 + 0.940888i \(0.609993\pi\)
\(702\) 0 0
\(703\) 15.1556 0.571604
\(704\) 0 0
\(705\) 11.2111 0.422235
\(706\) 0 0
\(707\) −3.81665 −0.143540
\(708\) 0 0
\(709\) −27.5416 −1.03435 −0.517174 0.855880i \(-0.673016\pi\)
−0.517174 + 0.855880i \(0.673016\pi\)
\(710\) 0 0
\(711\) 31.6333 1.18634
\(712\) 0 0
\(713\) −4.30278 −0.161140
\(714\) 0 0
\(715\) 7.60555 0.284431
\(716\) 0 0
\(717\) 96.4777 3.60303
\(718\) 0 0
\(719\) 31.9361 1.19101 0.595507 0.803350i \(-0.296951\pi\)
0.595507 + 0.803350i \(0.296951\pi\)
\(720\) 0 0
\(721\) 2.76114 0.102830
\(722\) 0 0
\(723\) 91.2666 3.39424
\(724\) 0 0
\(725\) −8.60555 −0.319602
\(726\) 0 0
\(727\) 10.6972 0.396738 0.198369 0.980127i \(-0.436436\pi\)
0.198369 + 0.980127i \(0.436436\pi\)
\(728\) 0 0
\(729\) 75.1749 2.78426
\(730\) 0 0
\(731\) −29.2111 −1.08041
\(732\) 0 0
\(733\) 40.0555 1.47948 0.739742 0.672891i \(-0.234948\pi\)
0.739742 + 0.672891i \(0.234948\pi\)
\(734\) 0 0
\(735\) 22.8167 0.841605
\(736\) 0 0
\(737\) −30.4222 −1.12062
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 22.1194 0.812578
\(742\) 0 0
\(743\) −12.9083 −0.473561 −0.236780 0.971563i \(-0.576092\pi\)
−0.236780 + 0.971563i \(0.576092\pi\)
\(744\) 0 0
\(745\) 2.90833 0.106553
\(746\) 0 0
\(747\) −88.6611 −3.24394
\(748\) 0 0
\(749\) 0.550039 0.0200980
\(750\) 0 0
\(751\) 47.8167 1.74485 0.872427 0.488744i \(-0.162545\pi\)
0.872427 + 0.488744i \(0.162545\pi\)
\(752\) 0 0
\(753\) −28.0278 −1.02139
\(754\) 0 0
\(755\) −15.7250 −0.572291
\(756\) 0 0
\(757\) −47.2666 −1.71793 −0.858967 0.512031i \(-0.828893\pi\)
−0.858967 + 0.512031i \(0.828893\pi\)
\(758\) 0 0
\(759\) −10.9083 −0.395947
\(760\) 0 0
\(761\) 22.4861 0.815121 0.407561 0.913178i \(-0.366380\pi\)
0.407561 + 0.913178i \(0.366380\pi\)
\(762\) 0 0
\(763\) −4.14719 −0.150138
\(764\) 0 0
\(765\) 57.7527 2.08805
\(766\) 0 0
\(767\) −3.21110 −0.115946
\(768\) 0 0
\(769\) −40.0555 −1.44444 −0.722219 0.691664i \(-0.756878\pi\)
−0.722219 + 0.691664i \(0.756878\pi\)
\(770\) 0 0
\(771\) 87.8722 3.16464
\(772\) 0 0
\(773\) 26.4222 0.950341 0.475170 0.879894i \(-0.342386\pi\)
0.475170 + 0.879894i \(0.342386\pi\)
\(774\) 0 0
\(775\) −4.30278 −0.154560
\(776\) 0 0
\(777\) 5.21110 0.186947
\(778\) 0 0
\(779\) 13.6611 0.489458
\(780\) 0 0
\(781\) −17.5139 −0.626696
\(782\) 0 0
\(783\) 139.505 4.98552
\(784\) 0 0
\(785\) 3.39445 0.121153
\(786\) 0 0
\(787\) 12.1833 0.434289 0.217145 0.976139i \(-0.430326\pi\)
0.217145 + 0.976139i \(0.430326\pi\)
\(788\) 0 0
\(789\) 65.7527 2.34086
\(790\) 0 0
\(791\) −0.972244 −0.0345690
\(792\) 0 0
\(793\) −3.00000 −0.106533
\(794\) 0 0
\(795\) −23.8167 −0.844690
\(796\) 0 0
\(797\) 25.3944 0.899518 0.449759 0.893150i \(-0.351510\pi\)
0.449759 + 0.893150i \(0.351510\pi\)
\(798\) 0 0
\(799\) −24.7889 −0.876968
\(800\) 0 0
\(801\) 22.0555 0.779293
\(802\) 0 0
\(803\) 17.8167 0.628736
\(804\) 0 0
\(805\) 0.302776 0.0106714
\(806\) 0 0
\(807\) −76.0555 −2.67728
\(808\) 0 0
\(809\) 9.90833 0.348358 0.174179 0.984714i \(-0.444273\pi\)
0.174179 + 0.984714i \(0.444273\pi\)
\(810\) 0 0
\(811\) 45.8167 1.60884 0.804420 0.594061i \(-0.202476\pi\)
0.804420 + 0.594061i \(0.202476\pi\)
\(812\) 0 0
\(813\) 93.4777 3.27841
\(814\) 0 0
\(815\) −6.90833 −0.241988
\(816\) 0 0
\(817\) 11.6333 0.406998
\(818\) 0 0
\(819\) 5.51388 0.192670
\(820\) 0 0
\(821\) −7.21110 −0.251669 −0.125835 0.992051i \(-0.540161\pi\)
−0.125835 + 0.992051i \(0.540161\pi\)
\(822\) 0 0
\(823\) −27.2111 −0.948519 −0.474260 0.880385i \(-0.657284\pi\)
−0.474260 + 0.880385i \(0.657284\pi\)
\(824\) 0 0
\(825\) −10.9083 −0.379779
\(826\) 0 0
\(827\) 39.4500 1.37181 0.685905 0.727691i \(-0.259407\pi\)
0.685905 + 0.727691i \(0.259407\pi\)
\(828\) 0 0
\(829\) 10.8444 0.376642 0.188321 0.982108i \(-0.439695\pi\)
0.188321 + 0.982108i \(0.439695\pi\)
\(830\) 0 0
\(831\) 6.60555 0.229144
\(832\) 0 0
\(833\) −50.4500 −1.74799
\(834\) 0 0
\(835\) −13.2111 −0.457189
\(836\) 0 0
\(837\) 69.7527 2.41101
\(838\) 0 0
\(839\) 19.3944 0.669571 0.334785 0.942294i \(-0.391336\pi\)
0.334785 + 0.942294i \(0.391336\pi\)
\(840\) 0 0
\(841\) 45.0555 1.55364
\(842\) 0 0
\(843\) 49.6333 1.70946
\(844\) 0 0
\(845\) −7.69722 −0.264793
\(846\) 0 0
\(847\) −0.0277564 −0.000953720 0
\(848\) 0 0
\(849\) −97.8722 −3.35896
\(850\) 0 0
\(851\) −5.21110 −0.178634
\(852\) 0 0
\(853\) 2.06392 0.0706672 0.0353336 0.999376i \(-0.488751\pi\)
0.0353336 + 0.999376i \(0.488751\pi\)
\(854\) 0 0
\(855\) −23.0000 −0.786583
\(856\) 0 0
\(857\) −45.6333 −1.55880 −0.779402 0.626524i \(-0.784477\pi\)
−0.779402 + 0.626524i \(0.784477\pi\)
\(858\) 0 0
\(859\) −9.81665 −0.334940 −0.167470 0.985877i \(-0.553560\pi\)
−0.167470 + 0.985877i \(0.553560\pi\)
\(860\) 0 0
\(861\) 4.69722 0.160081
\(862\) 0 0
\(863\) −54.2389 −1.84631 −0.923156 0.384425i \(-0.874400\pi\)
−0.923156 + 0.384425i \(0.874400\pi\)
\(864\) 0 0
\(865\) 14.5139 0.493487
\(866\) 0 0
\(867\) −119.992 −4.07513
\(868\) 0 0
\(869\) 13.2111 0.448156
\(870\) 0 0
\(871\) −21.2111 −0.718711
\(872\) 0 0
\(873\) −108.542 −3.67358
\(874\) 0 0
\(875\) 0.302776 0.0102357
\(876\) 0 0
\(877\) −24.0917 −0.813518 −0.406759 0.913536i \(-0.633341\pi\)
−0.406759 + 0.913536i \(0.633341\pi\)
\(878\) 0 0
\(879\) −59.4500 −2.00520
\(880\) 0 0
\(881\) −40.8444 −1.37608 −0.688042 0.725671i \(-0.741529\pi\)
−0.688042 + 0.725671i \(0.741529\pi\)
\(882\) 0 0
\(883\) 26.9083 0.905537 0.452769 0.891628i \(-0.350436\pi\)
0.452769 + 0.891628i \(0.350436\pi\)
\(884\) 0 0
\(885\) 4.60555 0.154814
\(886\) 0 0
\(887\) −47.6333 −1.59937 −0.799685 0.600420i \(-0.795000\pi\)
−0.799685 + 0.600420i \(0.795000\pi\)
\(888\) 0 0
\(889\) −4.05551 −0.136018
\(890\) 0 0
\(891\) 98.4777 3.29913
\(892\) 0 0
\(893\) 9.87217 0.330359
\(894\) 0 0
\(895\) 23.0278 0.769733
\(896\) 0 0
\(897\) −7.60555 −0.253942
\(898\) 0 0
\(899\) 37.0278 1.23494
\(900\) 0 0
\(901\) 52.6611 1.75439
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −21.6972 −0.721240
\(906\) 0 0
\(907\) −15.3944 −0.511164 −0.255582 0.966787i \(-0.582267\pi\)
−0.255582 + 0.966787i \(0.582267\pi\)
\(908\) 0 0
\(909\) −99.6888 −3.30647
\(910\) 0 0
\(911\) 45.8167 1.51797 0.758987 0.651106i \(-0.225695\pi\)
0.758987 + 0.651106i \(0.225695\pi\)
\(912\) 0 0
\(913\) −37.0278 −1.22544
\(914\) 0 0
\(915\) 4.30278 0.142245
\(916\) 0 0
\(917\) −0.238859 −0.00788782
\(918\) 0 0
\(919\) 51.2666 1.69113 0.845565 0.533873i \(-0.179264\pi\)
0.845565 + 0.533873i \(0.179264\pi\)
\(920\) 0 0
\(921\) 19.5139 0.643004
\(922\) 0 0
\(923\) −12.2111 −0.401933
\(924\) 0 0
\(925\) −5.21110 −0.171340
\(926\) 0 0
\(927\) 72.1194 2.36871
\(928\) 0 0
\(929\) 3.21110 0.105353 0.0526764 0.998612i \(-0.483225\pi\)
0.0526764 + 0.998612i \(0.483225\pi\)
\(930\) 0 0
\(931\) 20.0917 0.658478
\(932\) 0 0
\(933\) −34.4222 −1.12693
\(934\) 0 0
\(935\) 24.1194 0.788790
\(936\) 0 0
\(937\) 44.5416 1.45511 0.727556 0.686048i \(-0.240656\pi\)
0.727556 + 0.686048i \(0.240656\pi\)
\(938\) 0 0
\(939\) −88.2666 −2.88047
\(940\) 0 0
\(941\) −28.9361 −0.943289 −0.471645 0.881789i \(-0.656339\pi\)
−0.471645 + 0.881789i \(0.656339\pi\)
\(942\) 0 0
\(943\) −4.69722 −0.152963
\(944\) 0 0
\(945\) −4.90833 −0.159668
\(946\) 0 0
\(947\) −19.1472 −0.622200 −0.311100 0.950377i \(-0.600697\pi\)
−0.311100 + 0.950377i \(0.600697\pi\)
\(948\) 0 0
\(949\) 12.4222 0.403242
\(950\) 0 0
\(951\) −100.083 −3.24542
\(952\) 0 0
\(953\) 19.4861 0.631217 0.315609 0.948889i \(-0.397791\pi\)
0.315609 + 0.948889i \(0.397791\pi\)
\(954\) 0 0
\(955\) 0.183346 0.00593294
\(956\) 0 0
\(957\) 93.8722 3.03446
\(958\) 0 0
\(959\) 5.48612 0.177156
\(960\) 0 0
\(961\) −12.4861 −0.402778
\(962\) 0 0
\(963\) 14.3667 0.462960
\(964\) 0 0
\(965\) 9.39445 0.302418
\(966\) 0 0
\(967\) −3.81665 −0.122735 −0.0613677 0.998115i \(-0.519546\pi\)
−0.0613677 + 0.998115i \(0.519546\pi\)
\(968\) 0 0
\(969\) 70.1472 2.25345
\(970\) 0 0
\(971\) 19.1194 0.613572 0.306786 0.951779i \(-0.400746\pi\)
0.306786 + 0.951779i \(0.400746\pi\)
\(972\) 0 0
\(973\) 3.57779 0.114699
\(974\) 0 0
\(975\) −7.60555 −0.243573
\(976\) 0 0
\(977\) 2.72498 0.0871799 0.0435899 0.999050i \(-0.486120\pi\)
0.0435899 + 0.999050i \(0.486120\pi\)
\(978\) 0 0
\(979\) 9.21110 0.294388
\(980\) 0 0
\(981\) −108.322 −3.45846
\(982\) 0 0
\(983\) −36.0917 −1.15115 −0.575573 0.817751i \(-0.695221\pi\)
−0.575573 + 0.817751i \(0.695221\pi\)
\(984\) 0 0
\(985\) −22.5139 −0.717352
\(986\) 0 0
\(987\) 3.39445 0.108046
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 57.3028 1.82028 0.910141 0.414298i \(-0.135973\pi\)
0.910141 + 0.414298i \(0.135973\pi\)
\(992\) 0 0
\(993\) −12.6056 −0.400025
\(994\) 0 0
\(995\) 8.42221 0.267002
\(996\) 0 0
\(997\) 24.4222 0.773459 0.386729 0.922193i \(-0.373605\pi\)
0.386729 + 0.922193i \(0.373605\pi\)
\(998\) 0 0
\(999\) 84.4777 2.67276
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 7360.2.a.bd.1.1 2
4.3 odd 2 7360.2.a.bv.1.2 2
8.3 odd 2 3680.2.a.k.1.1 2
8.5 even 2 3680.2.a.p.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3680.2.a.k.1.1 2 8.3 odd 2
3680.2.a.p.1.2 yes 2 8.5 even 2
7360.2.a.bd.1.1 2 1.1 even 1 trivial
7360.2.a.bv.1.2 2 4.3 odd 2