Properties

Label 920.2.a.g.1.1
Level $920$
Weight $2$
Character 920.1
Self dual yes
Analytic conductor $7.346$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.14510\) of defining polynomial
Character \(\chi\) \(=\) 920.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.14510 q^{3} -1.00000 q^{5} -1.14510 q^{7} +1.60147 q^{9} +O(q^{10})\) \(q-2.14510 q^{3} -1.00000 q^{5} -1.14510 q^{7} +1.60147 q^{9} +5.89167 q^{11} -4.89167 q^{13} +2.14510 q^{15} -5.89167 q^{17} -2.34803 q^{19} +2.45636 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} +3.74657 q^{29} +5.68874 q^{31} -12.6382 q^{33} +1.14510 q^{35} +4.00000 q^{37} +10.4931 q^{39} -1.05783 q^{41} +11.4931 q^{43} -1.60147 q^{45} +7.74657 q^{47} -5.68874 q^{49} +12.6382 q^{51} +12.9863 q^{53} -5.89167 q^{55} +5.03677 q^{57} +0.797069 q^{59} +13.8917 q^{61} -1.83384 q^{63} +4.89167 q^{65} -15.5667 q^{67} -2.14510 q^{69} +2.94217 q^{71} +6.32698 q^{73} -2.14510 q^{75} -6.74657 q^{77} -11.2397 q^{81} -0.912726 q^{83} +5.89167 q^{85} -8.03677 q^{87} -15.4931 q^{89} +5.60147 q^{91} -12.2029 q^{93} +2.34803 q^{95} +13.3848 q^{97} +9.43531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43} - 3 q^{45} + 15 q^{47} - 6 q^{49} + 15 q^{51} + 6 q^{53} - 3 q^{55} - 6 q^{57} + 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} + 6 q^{71} - 15 q^{73} - 12 q^{77} - 9 q^{81} - 12 q^{83} + 3 q^{85} - 3 q^{87} - 30 q^{89} + 15 q^{91} - 33 q^{93} - 3 q^{95} + 9 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.14510 −1.23848 −0.619238 0.785204i \(-0.712558\pi\)
−0.619238 + 0.785204i \(0.712558\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.14510 −0.432808 −0.216404 0.976304i \(-0.569433\pi\)
−0.216404 + 0.976304i \(0.569433\pi\)
\(8\) 0 0
\(9\) 1.60147 0.533822
\(10\) 0 0
\(11\) 5.89167 1.77641 0.888203 0.459452i \(-0.151954\pi\)
0.888203 + 0.459452i \(0.151954\pi\)
\(12\) 0 0
\(13\) −4.89167 −1.35671 −0.678353 0.734736i \(-0.737306\pi\)
−0.678353 + 0.734736i \(0.737306\pi\)
\(14\) 0 0
\(15\) 2.14510 0.553863
\(16\) 0 0
\(17\) −5.89167 −1.42894 −0.714470 0.699666i \(-0.753332\pi\)
−0.714470 + 0.699666i \(0.753332\pi\)
\(18\) 0 0
\(19\) −2.34803 −0.538676 −0.269338 0.963046i \(-0.586805\pi\)
−0.269338 + 0.963046i \(0.586805\pi\)
\(20\) 0 0
\(21\) 2.45636 0.536022
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.00000 0.577350
\(28\) 0 0
\(29\) 3.74657 0.695720 0.347860 0.937546i \(-0.386908\pi\)
0.347860 + 0.937546i \(0.386908\pi\)
\(30\) 0 0
\(31\) 5.68874 1.02173 0.510864 0.859662i \(-0.329326\pi\)
0.510864 + 0.859662i \(0.329326\pi\)
\(32\) 0 0
\(33\) −12.6382 −2.20004
\(34\) 0 0
\(35\) 1.14510 0.193558
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 10.4931 1.68025
\(40\) 0 0
\(41\) −1.05783 −0.165205 −0.0826025 0.996583i \(-0.526323\pi\)
−0.0826025 + 0.996583i \(0.526323\pi\)
\(42\) 0 0
\(43\) 11.4931 1.75269 0.876343 0.481687i \(-0.159976\pi\)
0.876343 + 0.481687i \(0.159976\pi\)
\(44\) 0 0
\(45\) −1.60147 −0.238732
\(46\) 0 0
\(47\) 7.74657 1.12995 0.564977 0.825107i \(-0.308885\pi\)
0.564977 + 0.825107i \(0.308885\pi\)
\(48\) 0 0
\(49\) −5.68874 −0.812677
\(50\) 0 0
\(51\) 12.6382 1.76971
\(52\) 0 0
\(53\) 12.9863 1.78380 0.891901 0.452231i \(-0.149372\pi\)
0.891901 + 0.452231i \(0.149372\pi\)
\(54\) 0 0
\(55\) −5.89167 −0.794433
\(56\) 0 0
\(57\) 5.03677 0.667137
\(58\) 0 0
\(59\) 0.797069 0.103770 0.0518848 0.998653i \(-0.483477\pi\)
0.0518848 + 0.998653i \(0.483477\pi\)
\(60\) 0 0
\(61\) 13.8917 1.77865 0.889323 0.457279i \(-0.151176\pi\)
0.889323 + 0.457279i \(0.151176\pi\)
\(62\) 0 0
\(63\) −1.83384 −0.231042
\(64\) 0 0
\(65\) 4.89167 0.606737
\(66\) 0 0
\(67\) −15.5667 −1.90177 −0.950887 0.309540i \(-0.899825\pi\)
−0.950887 + 0.309540i \(0.899825\pi\)
\(68\) 0 0
\(69\) −2.14510 −0.258240
\(70\) 0 0
\(71\) 2.94217 0.349172 0.174586 0.984642i \(-0.444141\pi\)
0.174586 + 0.984642i \(0.444141\pi\)
\(72\) 0 0
\(73\) 6.32698 0.740517 0.370258 0.928929i \(-0.379269\pi\)
0.370258 + 0.928929i \(0.379269\pi\)
\(74\) 0 0
\(75\) −2.14510 −0.247695
\(76\) 0 0
\(77\) −6.74657 −0.768843
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.2397 −1.24886
\(82\) 0 0
\(83\) −0.912726 −0.100185 −0.0500923 0.998745i \(-0.515952\pi\)
−0.0500923 + 0.998745i \(0.515952\pi\)
\(84\) 0 0
\(85\) 5.89167 0.639041
\(86\) 0 0
\(87\) −8.03677 −0.861633
\(88\) 0 0
\(89\) −15.4931 −1.64227 −0.821135 0.570735i \(-0.806659\pi\)
−0.821135 + 0.570735i \(0.806659\pi\)
\(90\) 0 0
\(91\) 5.60147 0.587193
\(92\) 0 0
\(93\) −12.2029 −1.26539
\(94\) 0 0
\(95\) 2.34803 0.240903
\(96\) 0 0
\(97\) 13.3848 1.35902 0.679511 0.733666i \(-0.262192\pi\)
0.679511 + 0.733666i \(0.262192\pi\)
\(98\) 0 0
\(99\) 9.43531 0.948284
\(100\) 0 0
\(101\) 2.79707 0.278319 0.139159 0.990270i \(-0.455560\pi\)
0.139159 + 0.990270i \(0.455560\pi\)
\(102\) 0 0
\(103\) −9.89167 −0.974655 −0.487328 0.873219i \(-0.662028\pi\)
−0.487328 + 0.873219i \(0.662028\pi\)
\(104\) 0 0
\(105\) −2.45636 −0.239716
\(106\) 0 0
\(107\) 12.6961 1.22738 0.613688 0.789549i \(-0.289685\pi\)
0.613688 + 0.789549i \(0.289685\pi\)
\(108\) 0 0
\(109\) 3.65197 0.349795 0.174897 0.984587i \(-0.444041\pi\)
0.174897 + 0.984587i \(0.444041\pi\)
\(110\) 0 0
\(111\) −8.58041 −0.814417
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −7.83384 −0.724239
\(118\) 0 0
\(119\) 6.74657 0.618457
\(120\) 0 0
\(121\) 23.7118 2.15562
\(122\) 0 0
\(123\) 2.26915 0.204602
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.1524 1.07835 0.539177 0.842193i \(-0.318735\pi\)
0.539177 + 0.842193i \(0.318735\pi\)
\(128\) 0 0
\(129\) −24.6540 −2.17066
\(130\) 0 0
\(131\) −6.94950 −0.607181 −0.303590 0.952803i \(-0.598185\pi\)
−0.303590 + 0.952803i \(0.598185\pi\)
\(132\) 0 0
\(133\) 2.68874 0.233143
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −2.44904 −0.209235 −0.104618 0.994513i \(-0.533362\pi\)
−0.104618 + 0.994513i \(0.533362\pi\)
\(138\) 0 0
\(139\) 9.23970 0.783702 0.391851 0.920029i \(-0.371835\pi\)
0.391851 + 0.920029i \(0.371835\pi\)
\(140\) 0 0
\(141\) −16.6172 −1.39942
\(142\) 0 0
\(143\) −28.8201 −2.41006
\(144\) 0 0
\(145\) −3.74657 −0.311136
\(146\) 0 0
\(147\) 12.2029 1.00648
\(148\) 0 0
\(149\) −2.05783 −0.168584 −0.0842919 0.996441i \(-0.526863\pi\)
−0.0842919 + 0.996441i \(0.526863\pi\)
\(150\) 0 0
\(151\) 9.34803 0.760732 0.380366 0.924836i \(-0.375798\pi\)
0.380366 + 0.924836i \(0.375798\pi\)
\(152\) 0 0
\(153\) −9.43531 −0.762799
\(154\) 0 0
\(155\) −5.68874 −0.456931
\(156\) 0 0
\(157\) −3.60879 −0.288013 −0.144007 0.989577i \(-0.545999\pi\)
−0.144007 + 0.989577i \(0.545999\pi\)
\(158\) 0 0
\(159\) −27.8569 −2.20920
\(160\) 0 0
\(161\) −1.14510 −0.0902467
\(162\) 0 0
\(163\) 21.2554 1.66485 0.832427 0.554135i \(-0.186951\pi\)
0.832427 + 0.554135i \(0.186951\pi\)
\(164\) 0 0
\(165\) 12.6382 0.983886
\(166\) 0 0
\(167\) −10.9863 −0.850143 −0.425072 0.905160i \(-0.639751\pi\)
−0.425072 + 0.905160i \(0.639751\pi\)
\(168\) 0 0
\(169\) 10.9284 0.840650
\(170\) 0 0
\(171\) −3.76030 −0.287557
\(172\) 0 0
\(173\) −6.28288 −0.477678 −0.238839 0.971059i \(-0.576767\pi\)
−0.238839 + 0.971059i \(0.576767\pi\)
\(174\) 0 0
\(175\) −1.14510 −0.0865616
\(176\) 0 0
\(177\) −1.70979 −0.128516
\(178\) 0 0
\(179\) 10.1524 0.758828 0.379414 0.925227i \(-0.376126\pi\)
0.379414 + 0.925227i \(0.376126\pi\)
\(180\) 0 0
\(181\) 11.1451 0.828409 0.414204 0.910184i \(-0.364060\pi\)
0.414204 + 0.910184i \(0.364060\pi\)
\(182\) 0 0
\(183\) −29.7991 −2.20281
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −34.7118 −2.53838
\(188\) 0 0
\(189\) −3.43531 −0.249882
\(190\) 0 0
\(191\) −14.6961 −1.06337 −0.531685 0.846942i \(-0.678441\pi\)
−0.531685 + 0.846942i \(0.678441\pi\)
\(192\) 0 0
\(193\) −26.4005 −1.90035 −0.950176 0.311715i \(-0.899097\pi\)
−0.950176 + 0.311715i \(0.899097\pi\)
\(194\) 0 0
\(195\) −10.4931 −0.751429
\(196\) 0 0
\(197\) −4.26076 −0.303567 −0.151783 0.988414i \(-0.548502\pi\)
−0.151783 + 0.988414i \(0.548502\pi\)
\(198\) 0 0
\(199\) 18.0735 1.28120 0.640600 0.767875i \(-0.278686\pi\)
0.640600 + 0.767875i \(0.278686\pi\)
\(200\) 0 0
\(201\) 33.3921 2.35530
\(202\) 0 0
\(203\) −4.29021 −0.301113
\(204\) 0 0
\(205\) 1.05783 0.0738819
\(206\) 0 0
\(207\) 1.60147 0.111310
\(208\) 0 0
\(209\) −13.8338 −0.956907
\(210\) 0 0
\(211\) −11.8990 −0.819161 −0.409580 0.912274i \(-0.634325\pi\)
−0.409580 + 0.912274i \(0.634325\pi\)
\(212\) 0 0
\(213\) −6.31126 −0.432440
\(214\) 0 0
\(215\) −11.4931 −0.783825
\(216\) 0 0
\(217\) −6.51419 −0.442212
\(218\) 0 0
\(219\) −13.5720 −0.917112
\(220\) 0 0
\(221\) 28.8201 1.93865
\(222\) 0 0
\(223\) 4.58041 0.306727 0.153363 0.988170i \(-0.450989\pi\)
0.153363 + 0.988170i \(0.450989\pi\)
\(224\) 0 0
\(225\) 1.60147 0.106764
\(226\) 0 0
\(227\) 2.21666 0.147125 0.0735624 0.997291i \(-0.476563\pi\)
0.0735624 + 0.997291i \(0.476563\pi\)
\(228\) 0 0
\(229\) −19.1608 −1.26618 −0.633091 0.774077i \(-0.718214\pi\)
−0.633091 + 0.774077i \(0.718214\pi\)
\(230\) 0 0
\(231\) 14.4721 0.952193
\(232\) 0 0
\(233\) −21.7466 −1.42467 −0.712333 0.701842i \(-0.752361\pi\)
−0.712333 + 0.701842i \(0.752361\pi\)
\(234\) 0 0
\(235\) −7.74657 −0.505330
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.9495 1.35511 0.677555 0.735472i \(-0.263039\pi\)
0.677555 + 0.735472i \(0.263039\pi\)
\(240\) 0 0
\(241\) 24.0735 1.55071 0.775357 0.631523i \(-0.217570\pi\)
0.775357 + 0.631523i \(0.217570\pi\)
\(242\) 0 0
\(243\) 15.1103 0.969328
\(244\) 0 0
\(245\) 5.68874 0.363440
\(246\) 0 0
\(247\) 11.4858 0.730825
\(248\) 0 0
\(249\) 1.95789 0.124076
\(250\) 0 0
\(251\) −7.82651 −0.494005 −0.247003 0.969015i \(-0.579446\pi\)
−0.247003 + 0.969015i \(0.579446\pi\)
\(252\) 0 0
\(253\) 5.89167 0.370406
\(254\) 0 0
\(255\) −12.6382 −0.791437
\(256\) 0 0
\(257\) 9.82012 0.612562 0.306281 0.951941i \(-0.400915\pi\)
0.306281 + 0.951941i \(0.400915\pi\)
\(258\) 0 0
\(259\) −4.58041 −0.284613
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −14.6887 −0.905746 −0.452873 0.891575i \(-0.649601\pi\)
−0.452873 + 0.891575i \(0.649601\pi\)
\(264\) 0 0
\(265\) −12.9863 −0.797740
\(266\) 0 0
\(267\) 33.2344 2.03391
\(268\) 0 0
\(269\) −0.253432 −0.0154520 −0.00772600 0.999970i \(-0.502459\pi\)
−0.00772600 + 0.999970i \(0.502459\pi\)
\(270\) 0 0
\(271\) −19.6099 −1.19121 −0.595607 0.803276i \(-0.703088\pi\)
−0.595607 + 0.803276i \(0.703088\pi\)
\(272\) 0 0
\(273\) −12.0157 −0.727224
\(274\) 0 0
\(275\) 5.89167 0.355281
\(276\) 0 0
\(277\) 29.1976 1.75431 0.877157 0.480204i \(-0.159437\pi\)
0.877157 + 0.480204i \(0.159437\pi\)
\(278\) 0 0
\(279\) 9.11032 0.545421
\(280\) 0 0
\(281\) −27.1755 −1.62115 −0.810577 0.585633i \(-0.800846\pi\)
−0.810577 + 0.585633i \(0.800846\pi\)
\(282\) 0 0
\(283\) 18.1471 1.07873 0.539366 0.842071i \(-0.318664\pi\)
0.539366 + 0.842071i \(0.318664\pi\)
\(284\) 0 0
\(285\) −5.03677 −0.298353
\(286\) 0 0
\(287\) 1.21132 0.0715021
\(288\) 0 0
\(289\) 17.7118 1.04187
\(290\) 0 0
\(291\) −28.7118 −1.68311
\(292\) 0 0
\(293\) 0.681412 0.0398085 0.0199043 0.999802i \(-0.493664\pi\)
0.0199043 + 0.999802i \(0.493664\pi\)
\(294\) 0 0
\(295\) −0.797069 −0.0464071
\(296\) 0 0
\(297\) 17.6750 1.02561
\(298\) 0 0
\(299\) −4.89167 −0.282893
\(300\) 0 0
\(301\) −13.1608 −0.758577
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −13.8917 −0.795435
\(306\) 0 0
\(307\) 26.5877 1.51744 0.758721 0.651415i \(-0.225824\pi\)
0.758721 + 0.651415i \(0.225824\pi\)
\(308\) 0 0
\(309\) 21.2186 1.20709
\(310\) 0 0
\(311\) 13.5299 0.767211 0.383605 0.923497i \(-0.374682\pi\)
0.383605 + 0.923497i \(0.374682\pi\)
\(312\) 0 0
\(313\) 19.8412 1.12149 0.560745 0.827989i \(-0.310515\pi\)
0.560745 + 0.827989i \(0.310515\pi\)
\(314\) 0 0
\(315\) 1.83384 0.103325
\(316\) 0 0
\(317\) 3.55096 0.199442 0.0997210 0.995015i \(-0.468205\pi\)
0.0997210 + 0.995015i \(0.468205\pi\)
\(318\) 0 0
\(319\) 22.0735 1.23588
\(320\) 0 0
\(321\) −27.2344 −1.52007
\(322\) 0 0
\(323\) 13.8338 0.769736
\(324\) 0 0
\(325\) −4.89167 −0.271341
\(326\) 0 0
\(327\) −7.83384 −0.433212
\(328\) 0 0
\(329\) −8.87062 −0.489053
\(330\) 0 0
\(331\) −13.3133 −0.731763 −0.365881 0.930662i \(-0.619232\pi\)
−0.365881 + 0.930662i \(0.619232\pi\)
\(332\) 0 0
\(333\) 6.40586 0.351039
\(334\) 0 0
\(335\) 15.5667 0.850499
\(336\) 0 0
\(337\) −20.4951 −1.11644 −0.558220 0.829693i \(-0.688516\pi\)
−0.558220 + 0.829693i \(0.688516\pi\)
\(338\) 0 0
\(339\) 21.4510 1.16506
\(340\) 0 0
\(341\) 33.5162 1.81500
\(342\) 0 0
\(343\) 14.5299 0.784541
\(344\) 0 0
\(345\) 2.14510 0.115488
\(346\) 0 0
\(347\) −7.60147 −0.408068 −0.204034 0.978964i \(-0.565405\pi\)
−0.204034 + 0.978964i \(0.565405\pi\)
\(348\) 0 0
\(349\) 5.55736 0.297479 0.148739 0.988876i \(-0.452478\pi\)
0.148739 + 0.988876i \(0.452478\pi\)
\(350\) 0 0
\(351\) −14.6750 −0.783294
\(352\) 0 0
\(353\) 1.96323 0.104492 0.0522460 0.998634i \(-0.483362\pi\)
0.0522460 + 0.998634i \(0.483362\pi\)
\(354\) 0 0
\(355\) −2.94217 −0.156154
\(356\) 0 0
\(357\) −14.4721 −0.765944
\(358\) 0 0
\(359\) 10.5069 0.554531 0.277266 0.960793i \(-0.410572\pi\)
0.277266 + 0.960793i \(0.410572\pi\)
\(360\) 0 0
\(361\) −13.4867 −0.709828
\(362\) 0 0
\(363\) −50.8642 −2.66968
\(364\) 0 0
\(365\) −6.32698 −0.331169
\(366\) 0 0
\(367\) 33.7412 1.76128 0.880639 0.473788i \(-0.157114\pi\)
0.880639 + 0.473788i \(0.157114\pi\)
\(368\) 0 0
\(369\) −1.69408 −0.0881901
\(370\) 0 0
\(371\) −14.8706 −0.772044
\(372\) 0 0
\(373\) −17.4510 −0.903580 −0.451790 0.892124i \(-0.649214\pi\)
−0.451790 + 0.892124i \(0.649214\pi\)
\(374\) 0 0
\(375\) 2.14510 0.110773
\(376\) 0 0
\(377\) −18.3270 −0.943887
\(378\) 0 0
\(379\) −33.4584 −1.71864 −0.859320 0.511438i \(-0.829113\pi\)
−0.859320 + 0.511438i \(0.829113\pi\)
\(380\) 0 0
\(381\) −26.0682 −1.33551
\(382\) 0 0
\(383\) 13.0873 0.668728 0.334364 0.942444i \(-0.391478\pi\)
0.334364 + 0.942444i \(0.391478\pi\)
\(384\) 0 0
\(385\) 6.74657 0.343837
\(386\) 0 0
\(387\) 18.4059 0.935623
\(388\) 0 0
\(389\) 14.1083 0.715321 0.357660 0.933852i \(-0.383575\pi\)
0.357660 + 0.933852i \(0.383575\pi\)
\(390\) 0 0
\(391\) −5.89167 −0.297955
\(392\) 0 0
\(393\) 14.9074 0.751978
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.6245 0.834360 0.417180 0.908824i \(-0.363018\pi\)
0.417180 + 0.908824i \(0.363018\pi\)
\(398\) 0 0
\(399\) −5.76762 −0.288742
\(400\) 0 0
\(401\) 22.2628 1.11175 0.555874 0.831266i \(-0.312384\pi\)
0.555874 + 0.831266i \(0.312384\pi\)
\(402\) 0 0
\(403\) −27.8274 −1.38618
\(404\) 0 0
\(405\) 11.2397 0.558505
\(406\) 0 0
\(407\) 23.5667 1.16816
\(408\) 0 0
\(409\) 15.6887 0.775758 0.387879 0.921710i \(-0.373208\pi\)
0.387879 + 0.921710i \(0.373208\pi\)
\(410\) 0 0
\(411\) 5.25343 0.259133
\(412\) 0 0
\(413\) −0.912726 −0.0449123
\(414\) 0 0
\(415\) 0.912726 0.0448039
\(416\) 0 0
\(417\) −19.8201 −0.970595
\(418\) 0 0
\(419\) −5.66769 −0.276885 −0.138442 0.990371i \(-0.544210\pi\)
−0.138442 + 0.990371i \(0.544210\pi\)
\(420\) 0 0
\(421\) 15.0716 0.734543 0.367271 0.930114i \(-0.380292\pi\)
0.367271 + 0.930114i \(0.380292\pi\)
\(422\) 0 0
\(423\) 12.4059 0.603194
\(424\) 0 0
\(425\) −5.89167 −0.285788
\(426\) 0 0
\(427\) −15.9074 −0.769813
\(428\) 0 0
\(429\) 61.8221 2.98480
\(430\) 0 0
\(431\) 29.3500 1.41374 0.706870 0.707343i \(-0.250106\pi\)
0.706870 + 0.707343i \(0.250106\pi\)
\(432\) 0 0
\(433\) 29.6245 1.42366 0.711832 0.702350i \(-0.247866\pi\)
0.711832 + 0.702350i \(0.247866\pi\)
\(434\) 0 0
\(435\) 8.03677 0.385334
\(436\) 0 0
\(437\) −2.34803 −0.112322
\(438\) 0 0
\(439\) −20.7907 −0.992285 −0.496142 0.868241i \(-0.665251\pi\)
−0.496142 + 0.868241i \(0.665251\pi\)
\(440\) 0 0
\(441\) −9.11032 −0.433825
\(442\) 0 0
\(443\) 21.2975 1.01188 0.505938 0.862570i \(-0.331146\pi\)
0.505938 + 0.862570i \(0.331146\pi\)
\(444\) 0 0
\(445\) 15.4931 0.734445
\(446\) 0 0
\(447\) 4.41425 0.208787
\(448\) 0 0
\(449\) −19.3197 −0.911751 −0.455875 0.890044i \(-0.650674\pi\)
−0.455875 + 0.890044i \(0.650674\pi\)
\(450\) 0 0
\(451\) −6.23238 −0.293471
\(452\) 0 0
\(453\) −20.0525 −0.942148
\(454\) 0 0
\(455\) −5.60147 −0.262601
\(456\) 0 0
\(457\) 0.912726 0.0426955 0.0213478 0.999772i \(-0.493204\pi\)
0.0213478 + 0.999772i \(0.493204\pi\)
\(458\) 0 0
\(459\) −17.6750 −0.824999
\(460\) 0 0
\(461\) 2.83384 0.131985 0.0659926 0.997820i \(-0.478979\pi\)
0.0659926 + 0.997820i \(0.478979\pi\)
\(462\) 0 0
\(463\) −17.2618 −0.802225 −0.401112 0.916029i \(-0.631376\pi\)
−0.401112 + 0.916029i \(0.631376\pi\)
\(464\) 0 0
\(465\) 12.2029 0.565897
\(466\) 0 0
\(467\) −21.4510 −0.992635 −0.496318 0.868141i \(-0.665315\pi\)
−0.496318 + 0.868141i \(0.665315\pi\)
\(468\) 0 0
\(469\) 17.8255 0.823103
\(470\) 0 0
\(471\) 7.74123 0.356697
\(472\) 0 0
\(473\) 67.7138 3.11348
\(474\) 0 0
\(475\) −2.34803 −0.107735
\(476\) 0 0
\(477\) 20.7971 0.952232
\(478\) 0 0
\(479\) 9.49314 0.433752 0.216876 0.976199i \(-0.430413\pi\)
0.216876 + 0.976199i \(0.430413\pi\)
\(480\) 0 0
\(481\) −19.5667 −0.892164
\(482\) 0 0
\(483\) 2.45636 0.111768
\(484\) 0 0
\(485\) −13.3848 −0.607773
\(486\) 0 0
\(487\) 38.3270 1.73676 0.868381 0.495898i \(-0.165161\pi\)
0.868381 + 0.495898i \(0.165161\pi\)
\(488\) 0 0
\(489\) −45.5951 −2.06188
\(490\) 0 0
\(491\) −13.9947 −0.631570 −0.315785 0.948831i \(-0.602268\pi\)
−0.315785 + 0.948831i \(0.602268\pi\)
\(492\) 0 0
\(493\) −22.0735 −0.994143
\(494\) 0 0
\(495\) −9.43531 −0.424086
\(496\) 0 0
\(497\) −3.36909 −0.151124
\(498\) 0 0
\(499\) −0.0642277 −0.00287522 −0.00143761 0.999999i \(-0.500458\pi\)
−0.00143761 + 0.999999i \(0.500458\pi\)
\(500\) 0 0
\(501\) 23.5667 1.05288
\(502\) 0 0
\(503\) −24.2785 −1.08252 −0.541262 0.840854i \(-0.682053\pi\)
−0.541262 + 0.840854i \(0.682053\pi\)
\(504\) 0 0
\(505\) −2.79707 −0.124468
\(506\) 0 0
\(507\) −23.4426 −1.04112
\(508\) 0 0
\(509\) −2.65929 −0.117871 −0.0589356 0.998262i \(-0.518771\pi\)
−0.0589356 + 0.998262i \(0.518771\pi\)
\(510\) 0 0
\(511\) −7.24504 −0.320502
\(512\) 0 0
\(513\) −7.04410 −0.311005
\(514\) 0 0
\(515\) 9.89167 0.435879
\(516\) 0 0
\(517\) 45.6402 2.00726
\(518\) 0 0
\(519\) 13.4774 0.591593
\(520\) 0 0
\(521\) −17.0598 −0.747404 −0.373702 0.927549i \(-0.621912\pi\)
−0.373702 + 0.927549i \(0.621912\pi\)
\(522\) 0 0
\(523\) −17.5667 −0.768137 −0.384069 0.923305i \(-0.625477\pi\)
−0.384069 + 0.923305i \(0.625477\pi\)
\(524\) 0 0
\(525\) 2.45636 0.107204
\(526\) 0 0
\(527\) −33.5162 −1.45999
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.27648 0.0553944
\(532\) 0 0
\(533\) 5.17455 0.224135
\(534\) 0 0
\(535\) −12.6961 −0.548899
\(536\) 0 0
\(537\) −21.7780 −0.939790
\(538\) 0 0
\(539\) −33.5162 −1.44364
\(540\) 0 0
\(541\) −8.36909 −0.359815 −0.179908 0.983684i \(-0.557580\pi\)
−0.179908 + 0.983684i \(0.557580\pi\)
\(542\) 0 0
\(543\) −23.9074 −1.02596
\(544\) 0 0
\(545\) −3.65197 −0.156433
\(546\) 0 0
\(547\) −16.7486 −0.716117 −0.358058 0.933699i \(-0.616561\pi\)
−0.358058 + 0.933699i \(0.616561\pi\)
\(548\) 0 0
\(549\) 22.2470 0.949480
\(550\) 0 0
\(551\) −8.79707 −0.374768
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.58041 0.364218
\(556\) 0 0
\(557\) −7.26182 −0.307693 −0.153847 0.988095i \(-0.549166\pi\)
−0.153847 + 0.988095i \(0.549166\pi\)
\(558\) 0 0
\(559\) −56.2206 −2.37788
\(560\) 0 0
\(561\) 74.4603 3.14372
\(562\) 0 0
\(563\) 34.5530 1.45623 0.728117 0.685453i \(-0.240396\pi\)
0.728117 + 0.685453i \(0.240396\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 12.8706 0.540515
\(568\) 0 0
\(569\) 34.7275 1.45585 0.727926 0.685655i \(-0.240484\pi\)
0.727926 + 0.685655i \(0.240484\pi\)
\(570\) 0 0
\(571\) 14.7034 0.615318 0.307659 0.951497i \(-0.400454\pi\)
0.307659 + 0.951497i \(0.400454\pi\)
\(572\) 0 0
\(573\) 31.5246 1.31696
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −6.13777 −0.255519 −0.127759 0.991805i \(-0.540779\pi\)
−0.127759 + 0.991805i \(0.540779\pi\)
\(578\) 0 0
\(579\) 56.6318 2.35354
\(580\) 0 0
\(581\) 1.04516 0.0433607
\(582\) 0 0
\(583\) 76.5108 3.16876
\(584\) 0 0
\(585\) 7.83384 0.323890
\(586\) 0 0
\(587\) −1.33338 −0.0550344 −0.0275172 0.999621i \(-0.508760\pi\)
−0.0275172 + 0.999621i \(0.508760\pi\)
\(588\) 0 0
\(589\) −13.3574 −0.550380
\(590\) 0 0
\(591\) 9.13977 0.375960
\(592\) 0 0
\(593\) 1.88434 0.0773807 0.0386903 0.999251i \(-0.487681\pi\)
0.0386903 + 0.999251i \(0.487681\pi\)
\(594\) 0 0
\(595\) −6.74657 −0.276582
\(596\) 0 0
\(597\) −38.7696 −1.58673
\(598\) 0 0
\(599\) −43.7632 −1.78812 −0.894058 0.447951i \(-0.852154\pi\)
−0.894058 + 0.447951i \(0.852154\pi\)
\(600\) 0 0
\(601\) 4.50046 0.183578 0.0917889 0.995778i \(-0.470742\pi\)
0.0917889 + 0.995778i \(0.470742\pi\)
\(602\) 0 0
\(603\) −24.9295 −1.01521
\(604\) 0 0
\(605\) −23.7118 −0.964021
\(606\) 0 0
\(607\) 6.17455 0.250617 0.125309 0.992118i \(-0.460008\pi\)
0.125309 + 0.992118i \(0.460008\pi\)
\(608\) 0 0
\(609\) 9.20293 0.372922
\(610\) 0 0
\(611\) −37.8937 −1.53301
\(612\) 0 0
\(613\) −9.72445 −0.392767 −0.196383 0.980527i \(-0.562920\pi\)
−0.196383 + 0.980527i \(0.562920\pi\)
\(614\) 0 0
\(615\) −2.26915 −0.0915010
\(616\) 0 0
\(617\) −0.730849 −0.0294229 −0.0147114 0.999892i \(-0.504683\pi\)
−0.0147114 + 0.999892i \(0.504683\pi\)
\(618\) 0 0
\(619\) 11.1598 0.448549 0.224274 0.974526i \(-0.427999\pi\)
0.224274 + 0.974526i \(0.427999\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 17.7412 0.710787
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 29.6750 1.18511
\(628\) 0 0
\(629\) −23.5667 −0.939665
\(630\) 0 0
\(631\) −4.87062 −0.193896 −0.0969481 0.995289i \(-0.530908\pi\)
−0.0969481 + 0.995289i \(0.530908\pi\)
\(632\) 0 0
\(633\) 25.5246 1.01451
\(634\) 0 0
\(635\) −12.1524 −0.482254
\(636\) 0 0
\(637\) 27.8274 1.10256
\(638\) 0 0
\(639\) 4.71179 0.186395
\(640\) 0 0
\(641\) 32.6540 1.28975 0.644877 0.764286i \(-0.276909\pi\)
0.644877 + 0.764286i \(0.276909\pi\)
\(642\) 0 0
\(643\) 47.9579 1.89127 0.945637 0.325223i \(-0.105439\pi\)
0.945637 + 0.325223i \(0.105439\pi\)
\(644\) 0 0
\(645\) 24.6540 0.970749
\(646\) 0 0
\(647\) 12.1103 0.476106 0.238053 0.971252i \(-0.423491\pi\)
0.238053 + 0.971252i \(0.423491\pi\)
\(648\) 0 0
\(649\) 4.69607 0.184337
\(650\) 0 0
\(651\) 13.9736 0.547669
\(652\) 0 0
\(653\) −43.2133 −1.69107 −0.845534 0.533922i \(-0.820718\pi\)
−0.845534 + 0.533922i \(0.820718\pi\)
\(654\) 0 0
\(655\) 6.94950 0.271539
\(656\) 0 0
\(657\) 10.1324 0.395304
\(658\) 0 0
\(659\) 20.4333 0.795969 0.397984 0.917392i \(-0.369710\pi\)
0.397984 + 0.917392i \(0.369710\pi\)
\(660\) 0 0
\(661\) 48.1398 1.87242 0.936210 0.351441i \(-0.114308\pi\)
0.936210 + 0.351441i \(0.114308\pi\)
\(662\) 0 0
\(663\) −61.8221 −2.40097
\(664\) 0 0
\(665\) −2.68874 −0.104265
\(666\) 0 0
\(667\) 3.74657 0.145068
\(668\) 0 0
\(669\) −9.82545 −0.379874
\(670\) 0 0
\(671\) 81.8452 3.15960
\(672\) 0 0
\(673\) 16.5436 0.637710 0.318855 0.947803i \(-0.396702\pi\)
0.318855 + 0.947803i \(0.396702\pi\)
\(674\) 0 0
\(675\) 3.00000 0.115470
\(676\) 0 0
\(677\) −48.9863 −1.88270 −0.941348 0.337438i \(-0.890440\pi\)
−0.941348 + 0.337438i \(0.890440\pi\)
\(678\) 0 0
\(679\) −15.3270 −0.588195
\(680\) 0 0
\(681\) −4.75496 −0.182210
\(682\) 0 0
\(683\) −21.8779 −0.837136 −0.418568 0.908185i \(-0.637468\pi\)
−0.418568 + 0.908185i \(0.637468\pi\)
\(684\) 0 0
\(685\) 2.44904 0.0935728
\(686\) 0 0
\(687\) 41.1019 1.56814
\(688\) 0 0
\(689\) −63.5246 −2.42009
\(690\) 0 0
\(691\) −3.30393 −0.125688 −0.0628438 0.998023i \(-0.520017\pi\)
−0.0628438 + 0.998023i \(0.520017\pi\)
\(692\) 0 0
\(693\) −10.8044 −0.410425
\(694\) 0 0
\(695\) −9.23970 −0.350482
\(696\) 0 0
\(697\) 6.23238 0.236068
\(698\) 0 0
\(699\) 46.6486 1.76441
\(700\) 0 0
\(701\) −4.46369 −0.168591 −0.0842956 0.996441i \(-0.526864\pi\)
−0.0842956 + 0.996441i \(0.526864\pi\)
\(702\) 0 0
\(703\) −9.39214 −0.354231
\(704\) 0 0
\(705\) 16.6172 0.625839
\(706\) 0 0
\(707\) −3.20293 −0.120459
\(708\) 0 0
\(709\) −2.39853 −0.0900789 −0.0450394 0.998985i \(-0.514341\pi\)
−0.0450394 + 0.998985i \(0.514341\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.68874 0.213045
\(714\) 0 0
\(715\) 28.8201 1.07781
\(716\) 0 0
\(717\) −44.9388 −1.67827
\(718\) 0 0
\(719\) 12.0725 0.450228 0.225114 0.974332i \(-0.427725\pi\)
0.225114 + 0.974332i \(0.427725\pi\)
\(720\) 0 0
\(721\) 11.3270 0.421839
\(722\) 0 0
\(723\) −51.6402 −1.92052
\(724\) 0 0
\(725\) 3.74657 0.139144
\(726\) 0 0
\(727\) 11.4269 0.423801 0.211900 0.977291i \(-0.432035\pi\)
0.211900 + 0.977291i \(0.432035\pi\)
\(728\) 0 0
\(729\) 1.30592 0.0483676
\(730\) 0 0
\(731\) −67.7138 −2.50448
\(732\) 0 0
\(733\) −3.08727 −0.114031 −0.0570155 0.998373i \(-0.518158\pi\)
−0.0570155 + 0.998373i \(0.518158\pi\)
\(734\) 0 0
\(735\) −12.2029 −0.450112
\(736\) 0 0
\(737\) −91.7138 −3.37832
\(738\) 0 0
\(739\) −8.36909 −0.307862 −0.153931 0.988082i \(-0.549193\pi\)
−0.153931 + 0.988082i \(0.549193\pi\)
\(740\) 0 0
\(741\) −24.6382 −0.905108
\(742\) 0 0
\(743\) 15.1598 0.556158 0.278079 0.960558i \(-0.410302\pi\)
0.278079 + 0.960558i \(0.410302\pi\)
\(744\) 0 0
\(745\) 2.05783 0.0753930
\(746\) 0 0
\(747\) −1.46170 −0.0534808
\(748\) 0 0
\(749\) −14.5383 −0.531218
\(750\) 0 0
\(751\) −33.6823 −1.22909 −0.614543 0.788883i \(-0.710660\pi\)
−0.614543 + 0.788883i \(0.710660\pi\)
\(752\) 0 0
\(753\) 16.7887 0.611813
\(754\) 0 0
\(755\) −9.34803 −0.340210
\(756\) 0 0
\(757\) −29.1608 −1.05987 −0.529934 0.848039i \(-0.677783\pi\)
−0.529934 + 0.848039i \(0.677783\pi\)
\(758\) 0 0
\(759\) −12.6382 −0.458739
\(760\) 0 0
\(761\) −6.91912 −0.250818 −0.125409 0.992105i \(-0.540024\pi\)
−0.125409 + 0.992105i \(0.540024\pi\)
\(762\) 0 0
\(763\) −4.18188 −0.151394
\(764\) 0 0
\(765\) 9.43531 0.341134
\(766\) 0 0
\(767\) −3.89900 −0.140785
\(768\) 0 0
\(769\) −14.3491 −0.517442 −0.258721 0.965952i \(-0.583301\pi\)
−0.258721 + 0.965952i \(0.583301\pi\)
\(770\) 0 0
\(771\) −21.0652 −0.758643
\(772\) 0 0
\(773\) 50.6265 1.82091 0.910454 0.413609i \(-0.135732\pi\)
0.910454 + 0.413609i \(0.135732\pi\)
\(774\) 0 0
\(775\) 5.68874 0.204346
\(776\) 0 0
\(777\) 9.82545 0.352486
\(778\) 0 0
\(779\) 2.48382 0.0889920
\(780\) 0 0
\(781\) 17.3343 0.620270
\(782\) 0 0
\(783\) 11.2397 0.401674
\(784\) 0 0
\(785\) 3.60879 0.128803
\(786\) 0 0
\(787\) 20.9442 0.746579 0.373289 0.927715i \(-0.378230\pi\)
0.373289 + 0.927715i \(0.378230\pi\)
\(788\) 0 0
\(789\) 31.5089 1.12174
\(790\) 0 0
\(791\) 11.4510 0.407152
\(792\) 0 0
\(793\) −67.9535 −2.41310
\(794\) 0 0
\(795\) 27.8569 0.987982
\(796\) 0 0
\(797\) 19.0009 0.673047 0.336524 0.941675i \(-0.390749\pi\)
0.336524 + 0.941675i \(0.390749\pi\)
\(798\) 0 0
\(799\) −45.6402 −1.61464
\(800\) 0 0
\(801\) −24.8117 −0.876679
\(802\) 0 0
\(803\) 37.2765 1.31546
\(804\) 0 0
\(805\) 1.14510 0.0403596
\(806\) 0 0
\(807\) 0.543637 0.0191369
\(808\) 0 0
\(809\) 18.4300 0.647963 0.323982 0.946063i \(-0.394978\pi\)
0.323982 + 0.946063i \(0.394978\pi\)
\(810\) 0 0
\(811\) −21.1387 −0.742280 −0.371140 0.928577i \(-0.621033\pi\)
−0.371140 + 0.928577i \(0.621033\pi\)
\(812\) 0 0
\(813\) 42.0652 1.47529
\(814\) 0 0
\(815\) −21.2554 −0.744545
\(816\) 0 0
\(817\) −26.9863 −0.944130
\(818\) 0 0
\(819\) 8.97055 0.313457
\(820\) 0 0
\(821\) −16.7550 −0.584752 −0.292376 0.956303i \(-0.594446\pi\)
−0.292376 + 0.956303i \(0.594446\pi\)
\(822\) 0 0
\(823\) 24.9220 0.868728 0.434364 0.900737i \(-0.356973\pi\)
0.434364 + 0.900737i \(0.356973\pi\)
\(824\) 0 0
\(825\) −12.6382 −0.440007
\(826\) 0 0
\(827\) −6.62252 −0.230288 −0.115144 0.993349i \(-0.536733\pi\)
−0.115144 + 0.993349i \(0.536733\pi\)
\(828\) 0 0
\(829\) 52.7001 1.83035 0.915174 0.403058i \(-0.132053\pi\)
0.915174 + 0.403058i \(0.132053\pi\)
\(830\) 0 0
\(831\) −62.6318 −2.17267
\(832\) 0 0
\(833\) 33.5162 1.16127
\(834\) 0 0
\(835\) 10.9863 0.380196
\(836\) 0 0
\(837\) 17.0662 0.589895
\(838\) 0 0
\(839\) 42.5951 1.47054 0.735272 0.677772i \(-0.237054\pi\)
0.735272 + 0.677772i \(0.237054\pi\)
\(840\) 0 0
\(841\) −14.9632 −0.515973
\(842\) 0 0
\(843\) 58.2942 2.00776
\(844\) 0 0
\(845\) −10.9284 −0.375950
\(846\) 0 0
\(847\) −27.1524 −0.932969
\(848\) 0 0
\(849\) −38.9274 −1.33598
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) −57.7255 −1.97648 −0.988242 0.152898i \(-0.951139\pi\)
−0.988242 + 0.152898i \(0.951139\pi\)
\(854\) 0 0
\(855\) 3.76030 0.128599
\(856\) 0 0
\(857\) −55.3721 −1.89148 −0.945738 0.324930i \(-0.894659\pi\)
−0.945738 + 0.324930i \(0.894659\pi\)
\(858\) 0 0
\(859\) −14.3544 −0.489767 −0.244883 0.969553i \(-0.578750\pi\)
−0.244883 + 0.969553i \(0.578750\pi\)
\(860\) 0 0
\(861\) −2.59841 −0.0885536
\(862\) 0 0
\(863\) 15.9211 0.541961 0.270981 0.962585i \(-0.412652\pi\)
0.270981 + 0.962585i \(0.412652\pi\)
\(864\) 0 0
\(865\) 6.28288 0.213624
\(866\) 0 0
\(867\) −37.9936 −1.29033
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 76.1471 2.58015
\(872\) 0 0
\(873\) 21.4353 0.725475
\(874\) 0 0
\(875\) 1.14510 0.0387115
\(876\) 0 0
\(877\) −30.2975 −1.02308 −0.511538 0.859261i \(-0.670924\pi\)
−0.511538 + 0.859261i \(0.670924\pi\)
\(878\) 0 0
\(879\) −1.46170 −0.0493019
\(880\) 0 0
\(881\) −11.1883 −0.376943 −0.188471 0.982079i \(-0.560353\pi\)
−0.188471 + 0.982079i \(0.560353\pi\)
\(882\) 0 0
\(883\) 25.2882 0.851016 0.425508 0.904955i \(-0.360095\pi\)
0.425508 + 0.904955i \(0.360095\pi\)
\(884\) 0 0
\(885\) 1.70979 0.0574741
\(886\) 0 0
\(887\) −32.2849 −1.08402 −0.542010 0.840372i \(-0.682336\pi\)
−0.542010 + 0.840372i \(0.682336\pi\)
\(888\) 0 0
\(889\) −13.9158 −0.466720
\(890\) 0 0
\(891\) −66.2206 −2.21847
\(892\) 0 0
\(893\) −18.1892 −0.608679
\(894\) 0 0
\(895\) −10.1524 −0.339358
\(896\) 0 0
\(897\) 10.4931 0.350356
\(898\) 0 0
\(899\) 21.3133 0.710837
\(900\) 0 0
\(901\) −76.5108 −2.54895
\(902\) 0 0
\(903\) 28.2313 0.939479
\(904\) 0 0
\(905\) −11.1451 −0.370476
\(906\) 0 0
\(907\) −1.47848 −0.0490922 −0.0245461 0.999699i \(-0.507814\pi\)
−0.0245461 + 0.999699i \(0.507814\pi\)
\(908\) 0 0
\(909\) 4.47941 0.148573
\(910\) 0 0
\(911\) 5.70979 0.189174 0.0945870 0.995517i \(-0.469847\pi\)
0.0945870 + 0.995517i \(0.469847\pi\)
\(912\) 0 0
\(913\) −5.37748 −0.177969
\(914\) 0 0
\(915\) 29.7991 0.985127
\(916\) 0 0
\(917\) 7.95789 0.262793
\(918\) 0 0
\(919\) 22.6265 0.746379 0.373190 0.927755i \(-0.378264\pi\)
0.373190 + 0.927755i \(0.378264\pi\)
\(920\) 0 0
\(921\) −57.0334 −1.87932
\(922\) 0 0
\(923\) −14.3921 −0.473723
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 0 0
\(927\) −15.8412 −0.520292
\(928\) 0 0
\(929\) −50.0829 −1.64317 −0.821583 0.570089i \(-0.806909\pi\)
−0.821583 + 0.570089i \(0.806909\pi\)
\(930\) 0 0
\(931\) 13.3574 0.437770
\(932\) 0 0
\(933\) −29.0230 −0.950172
\(934\) 0 0
\(935\) 34.7118 1.13520
\(936\) 0 0
\(937\) 9.98428 0.326172 0.163086 0.986612i \(-0.447855\pi\)
0.163086 + 0.986612i \(0.447855\pi\)
\(938\) 0 0
\(939\) −42.5613 −1.38894
\(940\) 0 0
\(941\) −36.0809 −1.17620 −0.588101 0.808787i \(-0.700124\pi\)
−0.588101 + 0.808787i \(0.700124\pi\)
\(942\) 0 0
\(943\) −1.05783 −0.0344476
\(944\) 0 0
\(945\) 3.43531 0.111751
\(946\) 0 0
\(947\) −32.4730 −1.05523 −0.527616 0.849483i \(-0.676914\pi\)
−0.527616 + 0.849483i \(0.676914\pi\)
\(948\) 0 0
\(949\) −30.9495 −1.00466
\(950\) 0 0
\(951\) −7.61718 −0.247004
\(952\) 0 0
\(953\) −39.3427 −1.27443 −0.637217 0.770684i \(-0.719915\pi\)
−0.637217 + 0.770684i \(0.719915\pi\)
\(954\) 0 0
\(955\) 14.6961 0.475554
\(956\) 0 0
\(957\) −47.3500 −1.53061
\(958\) 0 0
\(959\) 2.80440 0.0905587
\(960\) 0 0
\(961\) 1.36176 0.0439278
\(962\) 0 0
\(963\) 20.3323 0.655200
\(964\) 0 0
\(965\) 26.4005 0.849863
\(966\) 0 0
\(967\) 23.5720 0.758025 0.379013 0.925392i \(-0.376264\pi\)
0.379013 + 0.925392i \(0.376264\pi\)
\(968\) 0 0
\(969\) −29.6750 −0.953299
\(970\) 0 0
\(971\) 12.9368 0.415163 0.207581 0.978218i \(-0.433441\pi\)
0.207581 + 0.978218i \(0.433441\pi\)
\(972\) 0 0
\(973\) −10.5804 −0.339192
\(974\) 0 0
\(975\) 10.4931 0.336049
\(976\) 0 0
\(977\) −22.7118 −0.726614 −0.363307 0.931669i \(-0.618352\pi\)
−0.363307 + 0.931669i \(0.618352\pi\)
\(978\) 0 0
\(979\) −91.2805 −2.91734
\(980\) 0 0
\(981\) 5.84850 0.186728
\(982\) 0 0
\(983\) 26.1819 0.835072 0.417536 0.908660i \(-0.362894\pi\)
0.417536 + 0.908660i \(0.362894\pi\)
\(984\) 0 0
\(985\) 4.26076 0.135759
\(986\) 0 0
\(987\) 19.0284 0.605680
\(988\) 0 0
\(989\) 11.4931 0.365460
\(990\) 0 0
\(991\) −56.3143 −1.78888 −0.894442 0.447185i \(-0.852427\pi\)
−0.894442 + 0.447185i \(0.852427\pi\)
\(992\) 0 0
\(993\) 28.5583 0.906270
\(994\) 0 0
\(995\) −18.0735 −0.572970
\(996\) 0 0
\(997\) −3.74123 −0.118486 −0.0592430 0.998244i \(-0.518869\pi\)
−0.0592430 + 0.998244i \(0.518869\pi\)
\(998\) 0 0
\(999\) 12.0000 0.379663
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 920.2.a.g.1.1 3
3.2 odd 2 8280.2.a.bo.1.1 3
4.3 odd 2 1840.2.a.t.1.3 3
5.2 odd 4 4600.2.e.r.4049.5 6
5.3 odd 4 4600.2.e.r.4049.2 6
5.4 even 2 4600.2.a.y.1.3 3
8.3 odd 2 7360.2.a.ca.1.1 3
8.5 even 2 7360.2.a.cb.1.3 3
20.19 odd 2 9200.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.1 3 1.1 even 1 trivial
1840.2.a.t.1.3 3 4.3 odd 2
4600.2.a.y.1.3 3 5.4 even 2
4600.2.e.r.4049.2 6 5.3 odd 4
4600.2.e.r.4049.5 6 5.2 odd 4
7360.2.a.ca.1.1 3 8.3 odd 2
7360.2.a.cb.1.3 3 8.5 even 2
8280.2.a.bo.1.1 3 3.2 odd 2
9200.2.a.cd.1.1 3 20.19 odd 2