Properties

Label 8004.2.a.e.1.4
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.25739\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.650765 q^{5} -1.33403 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.650765 q^{5} -1.33403 q^{7} +1.00000 q^{9} +5.13354 q^{11} +1.17838 q^{13} +0.650765 q^{15} -0.132625 q^{17} -1.33048 q^{19} +1.33403 q^{21} -1.00000 q^{23} -4.57650 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.92836 q^{31} -5.13354 q^{33} +0.868143 q^{35} -2.05624 q^{37} -1.17838 q^{39} -10.6659 q^{41} +5.90191 q^{43} -0.650765 q^{45} -6.07807 q^{47} -5.22035 q^{49} +0.132625 q^{51} +13.1354 q^{53} -3.34073 q^{55} +1.33048 q^{57} -5.88852 q^{59} +9.63773 q^{61} -1.33403 q^{63} -0.766847 q^{65} -5.90527 q^{67} +1.00000 q^{69} -7.38396 q^{71} +4.38565 q^{73} +4.57650 q^{75} -6.84832 q^{77} -7.87595 q^{79} +1.00000 q^{81} +17.7631 q^{83} +0.0863074 q^{85} -1.00000 q^{87} +9.08883 q^{89} -1.57200 q^{91} +1.92836 q^{93} +0.865829 q^{95} -16.7482 q^{97} +5.13354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.650765 −0.291031 −0.145515 0.989356i \(-0.546484\pi\)
−0.145515 + 0.989356i \(0.546484\pi\)
\(6\) 0 0
\(7\) −1.33403 −0.504218 −0.252109 0.967699i \(-0.581124\pi\)
−0.252109 + 0.967699i \(0.581124\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.13354 1.54782 0.773911 0.633294i \(-0.218298\pi\)
0.773911 + 0.633294i \(0.218298\pi\)
\(12\) 0 0
\(13\) 1.17838 0.326823 0.163412 0.986558i \(-0.447750\pi\)
0.163412 + 0.986558i \(0.447750\pi\)
\(14\) 0 0
\(15\) 0.650765 0.168027
\(16\) 0 0
\(17\) −0.132625 −0.0321662 −0.0160831 0.999871i \(-0.505120\pi\)
−0.0160831 + 0.999871i \(0.505120\pi\)
\(18\) 0 0
\(19\) −1.33048 −0.305233 −0.152616 0.988286i \(-0.548770\pi\)
−0.152616 + 0.988286i \(0.548770\pi\)
\(20\) 0 0
\(21\) 1.33403 0.291110
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.57650 −0.915301
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.92836 −0.346343 −0.173172 0.984892i \(-0.555402\pi\)
−0.173172 + 0.984892i \(0.555402\pi\)
\(32\) 0 0
\(33\) −5.13354 −0.893635
\(34\) 0 0
\(35\) 0.868143 0.146743
\(36\) 0 0
\(37\) −2.05624 −0.338043 −0.169022 0.985612i \(-0.554061\pi\)
−0.169022 + 0.985612i \(0.554061\pi\)
\(38\) 0 0
\(39\) −1.17838 −0.188691
\(40\) 0 0
\(41\) −10.6659 −1.66574 −0.832868 0.553472i \(-0.813303\pi\)
−0.832868 + 0.553472i \(0.813303\pi\)
\(42\) 0 0
\(43\) 5.90191 0.900033 0.450016 0.893020i \(-0.351418\pi\)
0.450016 + 0.893020i \(0.351418\pi\)
\(44\) 0 0
\(45\) −0.650765 −0.0970103
\(46\) 0 0
\(47\) −6.07807 −0.886578 −0.443289 0.896379i \(-0.646188\pi\)
−0.443289 + 0.896379i \(0.646188\pi\)
\(48\) 0 0
\(49\) −5.22035 −0.745765
\(50\) 0 0
\(51\) 0.132625 0.0185712
\(52\) 0 0
\(53\) 13.1354 1.80428 0.902141 0.431442i \(-0.141995\pi\)
0.902141 + 0.431442i \(0.141995\pi\)
\(54\) 0 0
\(55\) −3.34073 −0.450464
\(56\) 0 0
\(57\) 1.33048 0.176226
\(58\) 0 0
\(59\) −5.88852 −0.766620 −0.383310 0.923620i \(-0.625216\pi\)
−0.383310 + 0.923620i \(0.625216\pi\)
\(60\) 0 0
\(61\) 9.63773 1.23399 0.616993 0.786969i \(-0.288351\pi\)
0.616993 + 0.786969i \(0.288351\pi\)
\(62\) 0 0
\(63\) −1.33403 −0.168073
\(64\) 0 0
\(65\) −0.766847 −0.0951156
\(66\) 0 0
\(67\) −5.90527 −0.721444 −0.360722 0.932673i \(-0.617470\pi\)
−0.360722 + 0.932673i \(0.617470\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −7.38396 −0.876314 −0.438157 0.898898i \(-0.644369\pi\)
−0.438157 + 0.898898i \(0.644369\pi\)
\(72\) 0 0
\(73\) 4.38565 0.513302 0.256651 0.966504i \(-0.417381\pi\)
0.256651 + 0.966504i \(0.417381\pi\)
\(74\) 0 0
\(75\) 4.57650 0.528449
\(76\) 0 0
\(77\) −6.84832 −0.780439
\(78\) 0 0
\(79\) −7.87595 −0.886114 −0.443057 0.896493i \(-0.646106\pi\)
−0.443057 + 0.896493i \(0.646106\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 17.7631 1.94975 0.974877 0.222744i \(-0.0715014\pi\)
0.974877 + 0.222744i \(0.0715014\pi\)
\(84\) 0 0
\(85\) 0.0863074 0.00936136
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 9.08883 0.963414 0.481707 0.876332i \(-0.340017\pi\)
0.481707 + 0.876332i \(0.340017\pi\)
\(90\) 0 0
\(91\) −1.57200 −0.164790
\(92\) 0 0
\(93\) 1.92836 0.199961
\(94\) 0 0
\(95\) 0.865829 0.0888321
\(96\) 0 0
\(97\) −16.7482 −1.70053 −0.850263 0.526358i \(-0.823557\pi\)
−0.850263 + 0.526358i \(0.823557\pi\)
\(98\) 0 0
\(99\) 5.13354 0.515941
\(100\) 0 0
\(101\) 9.82078 0.977204 0.488602 0.872507i \(-0.337507\pi\)
0.488602 + 0.872507i \(0.337507\pi\)
\(102\) 0 0
\(103\) 7.80111 0.768667 0.384333 0.923194i \(-0.374431\pi\)
0.384333 + 0.923194i \(0.374431\pi\)
\(104\) 0 0
\(105\) −0.868143 −0.0847221
\(106\) 0 0
\(107\) −3.93159 −0.380081 −0.190041 0.981776i \(-0.560862\pi\)
−0.190041 + 0.981776i \(0.560862\pi\)
\(108\) 0 0
\(109\) −10.5756 −1.01296 −0.506480 0.862251i \(-0.669054\pi\)
−0.506480 + 0.862251i \(0.669054\pi\)
\(110\) 0 0
\(111\) 2.05624 0.195169
\(112\) 0 0
\(113\) −12.8588 −1.20966 −0.604829 0.796355i \(-0.706759\pi\)
−0.604829 + 0.796355i \(0.706759\pi\)
\(114\) 0 0
\(115\) 0.650765 0.0606841
\(116\) 0 0
\(117\) 1.17838 0.108941
\(118\) 0 0
\(119\) 0.176926 0.0162188
\(120\) 0 0
\(121\) 15.3533 1.39575
\(122\) 0 0
\(123\) 10.6659 0.961713
\(124\) 0 0
\(125\) 6.23205 0.557412
\(126\) 0 0
\(127\) 1.91677 0.170086 0.0850428 0.996377i \(-0.472897\pi\)
0.0850428 + 0.996377i \(0.472897\pi\)
\(128\) 0 0
\(129\) −5.90191 −0.519634
\(130\) 0 0
\(131\) 1.66409 0.145392 0.0726962 0.997354i \(-0.476840\pi\)
0.0726962 + 0.997354i \(0.476840\pi\)
\(132\) 0 0
\(133\) 1.77490 0.153904
\(134\) 0 0
\(135\) 0.650765 0.0560089
\(136\) 0 0
\(137\) 20.0439 1.71247 0.856233 0.516589i \(-0.172799\pi\)
0.856233 + 0.516589i \(0.172799\pi\)
\(138\) 0 0
\(139\) 10.0030 0.848445 0.424223 0.905558i \(-0.360547\pi\)
0.424223 + 0.905558i \(0.360547\pi\)
\(140\) 0 0
\(141\) 6.07807 0.511866
\(142\) 0 0
\(143\) 6.04925 0.505864
\(144\) 0 0
\(145\) −0.650765 −0.0540431
\(146\) 0 0
\(147\) 5.22035 0.430567
\(148\) 0 0
\(149\) 4.34210 0.355719 0.177859 0.984056i \(-0.443083\pi\)
0.177859 + 0.984056i \(0.443083\pi\)
\(150\) 0 0
\(151\) −5.85205 −0.476233 −0.238117 0.971237i \(-0.576530\pi\)
−0.238117 + 0.971237i \(0.576530\pi\)
\(152\) 0 0
\(153\) −0.132625 −0.0107221
\(154\) 0 0
\(155\) 1.25491 0.100797
\(156\) 0 0
\(157\) −0.295720 −0.0236010 −0.0118005 0.999930i \(-0.503756\pi\)
−0.0118005 + 0.999930i \(0.503756\pi\)
\(158\) 0 0
\(159\) −13.1354 −1.04170
\(160\) 0 0
\(161\) 1.33403 0.105137
\(162\) 0 0
\(163\) −2.66668 −0.208871 −0.104435 0.994532i \(-0.533304\pi\)
−0.104435 + 0.994532i \(0.533304\pi\)
\(164\) 0 0
\(165\) 3.34073 0.260076
\(166\) 0 0
\(167\) −19.6691 −1.52204 −0.761020 0.648728i \(-0.775301\pi\)
−0.761020 + 0.648728i \(0.775301\pi\)
\(168\) 0 0
\(169\) −11.6114 −0.893187
\(170\) 0 0
\(171\) −1.33048 −0.101744
\(172\) 0 0
\(173\) −12.4733 −0.948325 −0.474163 0.880437i \(-0.657249\pi\)
−0.474163 + 0.880437i \(0.657249\pi\)
\(174\) 0 0
\(175\) 6.10521 0.461511
\(176\) 0 0
\(177\) 5.88852 0.442608
\(178\) 0 0
\(179\) −4.84197 −0.361906 −0.180953 0.983492i \(-0.557918\pi\)
−0.180953 + 0.983492i \(0.557918\pi\)
\(180\) 0 0
\(181\) −4.99870 −0.371551 −0.185775 0.982592i \(-0.559480\pi\)
−0.185775 + 0.982592i \(0.559480\pi\)
\(182\) 0 0
\(183\) −9.63773 −0.712442
\(184\) 0 0
\(185\) 1.33813 0.0983811
\(186\) 0 0
\(187\) −0.680834 −0.0497875
\(188\) 0 0
\(189\) 1.33403 0.0970367
\(190\) 0 0
\(191\) −7.01032 −0.507249 −0.253624 0.967303i \(-0.581623\pi\)
−0.253624 + 0.967303i \(0.581623\pi\)
\(192\) 0 0
\(193\) −13.6972 −0.985947 −0.492973 0.870044i \(-0.664090\pi\)
−0.492973 + 0.870044i \(0.664090\pi\)
\(194\) 0 0
\(195\) 0.766847 0.0549150
\(196\) 0 0
\(197\) −8.40065 −0.598521 −0.299261 0.954171i \(-0.596740\pi\)
−0.299261 + 0.954171i \(0.596740\pi\)
\(198\) 0 0
\(199\) 16.5621 1.17406 0.587029 0.809566i \(-0.300298\pi\)
0.587029 + 0.809566i \(0.300298\pi\)
\(200\) 0 0
\(201\) 5.90527 0.416526
\(202\) 0 0
\(203\) −1.33403 −0.0936309
\(204\) 0 0
\(205\) 6.94100 0.484781
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −6.83007 −0.472446
\(210\) 0 0
\(211\) 8.47612 0.583520 0.291760 0.956492i \(-0.405759\pi\)
0.291760 + 0.956492i \(0.405759\pi\)
\(212\) 0 0
\(213\) 7.38396 0.505940
\(214\) 0 0
\(215\) −3.84076 −0.261937
\(216\) 0 0
\(217\) 2.57249 0.174632
\(218\) 0 0
\(219\) −4.38565 −0.296355
\(220\) 0 0
\(221\) −0.156282 −0.0105127
\(222\) 0 0
\(223\) 14.2177 0.952085 0.476042 0.879422i \(-0.342071\pi\)
0.476042 + 0.879422i \(0.342071\pi\)
\(224\) 0 0
\(225\) −4.57650 −0.305100
\(226\) 0 0
\(227\) 5.21362 0.346040 0.173020 0.984918i \(-0.444647\pi\)
0.173020 + 0.984918i \(0.444647\pi\)
\(228\) 0 0
\(229\) −19.0310 −1.25761 −0.628803 0.777565i \(-0.716455\pi\)
−0.628803 + 0.777565i \(0.716455\pi\)
\(230\) 0 0
\(231\) 6.84832 0.450587
\(232\) 0 0
\(233\) −26.7954 −1.75543 −0.877713 0.479186i \(-0.840932\pi\)
−0.877713 + 0.479186i \(0.840932\pi\)
\(234\) 0 0
\(235\) 3.95540 0.258022
\(236\) 0 0
\(237\) 7.87595 0.511598
\(238\) 0 0
\(239\) 2.26026 0.146204 0.0731020 0.997324i \(-0.476710\pi\)
0.0731020 + 0.997324i \(0.476710\pi\)
\(240\) 0 0
\(241\) 6.92524 0.446094 0.223047 0.974808i \(-0.428400\pi\)
0.223047 + 0.974808i \(0.428400\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.39722 0.217041
\(246\) 0 0
\(247\) −1.56781 −0.0997571
\(248\) 0 0
\(249\) −17.7631 −1.12569
\(250\) 0 0
\(251\) 15.1859 0.958527 0.479264 0.877671i \(-0.340904\pi\)
0.479264 + 0.877671i \(0.340904\pi\)
\(252\) 0 0
\(253\) −5.13354 −0.322743
\(254\) 0 0
\(255\) −0.0863074 −0.00540478
\(256\) 0 0
\(257\) −15.2246 −0.949688 −0.474844 0.880070i \(-0.657495\pi\)
−0.474844 + 0.880070i \(0.657495\pi\)
\(258\) 0 0
\(259\) 2.74309 0.170447
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −5.21935 −0.321839 −0.160920 0.986968i \(-0.551446\pi\)
−0.160920 + 0.986968i \(0.551446\pi\)
\(264\) 0 0
\(265\) −8.54804 −0.525102
\(266\) 0 0
\(267\) −9.08883 −0.556227
\(268\) 0 0
\(269\) −29.4706 −1.79685 −0.898426 0.439125i \(-0.855289\pi\)
−0.898426 + 0.439125i \(0.855289\pi\)
\(270\) 0 0
\(271\) 1.04180 0.0632850 0.0316425 0.999499i \(-0.489926\pi\)
0.0316425 + 0.999499i \(0.489926\pi\)
\(272\) 0 0
\(273\) 1.57200 0.0951415
\(274\) 0 0
\(275\) −23.4937 −1.41672
\(276\) 0 0
\(277\) −10.2229 −0.614233 −0.307117 0.951672i \(-0.599364\pi\)
−0.307117 + 0.951672i \(0.599364\pi\)
\(278\) 0 0
\(279\) −1.92836 −0.115448
\(280\) 0 0
\(281\) −13.3230 −0.794783 −0.397392 0.917649i \(-0.630085\pi\)
−0.397392 + 0.917649i \(0.630085\pi\)
\(282\) 0 0
\(283\) −18.7477 −1.11443 −0.557217 0.830367i \(-0.688131\pi\)
−0.557217 + 0.830367i \(0.688131\pi\)
\(284\) 0 0
\(285\) −0.865829 −0.0512873
\(286\) 0 0
\(287\) 14.2287 0.839893
\(288\) 0 0
\(289\) −16.9824 −0.998965
\(290\) 0 0
\(291\) 16.7482 0.981799
\(292\) 0 0
\(293\) −6.11769 −0.357399 −0.178700 0.983904i \(-0.557189\pi\)
−0.178700 + 0.983904i \(0.557189\pi\)
\(294\) 0 0
\(295\) 3.83204 0.223110
\(296\) 0 0
\(297\) −5.13354 −0.297878
\(298\) 0 0
\(299\) −1.17838 −0.0681473
\(300\) 0 0
\(301\) −7.87335 −0.453812
\(302\) 0 0
\(303\) −9.82078 −0.564189
\(304\) 0 0
\(305\) −6.27190 −0.359128
\(306\) 0 0
\(307\) 9.66557 0.551643 0.275821 0.961209i \(-0.411050\pi\)
0.275821 + 0.961209i \(0.411050\pi\)
\(308\) 0 0
\(309\) −7.80111 −0.443790
\(310\) 0 0
\(311\) −22.9305 −1.30027 −0.650134 0.759820i \(-0.725287\pi\)
−0.650134 + 0.759820i \(0.725287\pi\)
\(312\) 0 0
\(313\) 28.5054 1.61122 0.805610 0.592446i \(-0.201837\pi\)
0.805610 + 0.592446i \(0.201837\pi\)
\(314\) 0 0
\(315\) 0.868143 0.0489143
\(316\) 0 0
\(317\) −23.3183 −1.30968 −0.654842 0.755766i \(-0.727265\pi\)
−0.654842 + 0.755766i \(0.727265\pi\)
\(318\) 0 0
\(319\) 5.13354 0.287423
\(320\) 0 0
\(321\) 3.93159 0.219440
\(322\) 0 0
\(323\) 0.176454 0.00981817
\(324\) 0 0
\(325\) −5.39285 −0.299141
\(326\) 0 0
\(327\) 10.5756 0.584833
\(328\) 0 0
\(329\) 8.10836 0.447028
\(330\) 0 0
\(331\) 13.7752 0.757153 0.378577 0.925570i \(-0.376414\pi\)
0.378577 + 0.925570i \(0.376414\pi\)
\(332\) 0 0
\(333\) −2.05624 −0.112681
\(334\) 0 0
\(335\) 3.84294 0.209962
\(336\) 0 0
\(337\) −12.1266 −0.660580 −0.330290 0.943879i \(-0.607147\pi\)
−0.330290 + 0.943879i \(0.607147\pi\)
\(338\) 0 0
\(339\) 12.8588 0.698397
\(340\) 0 0
\(341\) −9.89931 −0.536077
\(342\) 0 0
\(343\) 16.3024 0.880245
\(344\) 0 0
\(345\) −0.650765 −0.0350360
\(346\) 0 0
\(347\) −27.6195 −1.48269 −0.741346 0.671123i \(-0.765812\pi\)
−0.741346 + 0.671123i \(0.765812\pi\)
\(348\) 0 0
\(349\) −20.5169 −1.09825 −0.549123 0.835742i \(-0.685038\pi\)
−0.549123 + 0.835742i \(0.685038\pi\)
\(350\) 0 0
\(351\) −1.17838 −0.0628971
\(352\) 0 0
\(353\) 32.1730 1.71240 0.856198 0.516648i \(-0.172821\pi\)
0.856198 + 0.516648i \(0.172821\pi\)
\(354\) 0 0
\(355\) 4.80522 0.255035
\(356\) 0 0
\(357\) −0.176926 −0.00936390
\(358\) 0 0
\(359\) 5.54317 0.292557 0.146279 0.989243i \(-0.453270\pi\)
0.146279 + 0.989243i \(0.453270\pi\)
\(360\) 0 0
\(361\) −17.2298 −0.906833
\(362\) 0 0
\(363\) −15.3533 −0.805838
\(364\) 0 0
\(365\) −2.85403 −0.149387
\(366\) 0 0
\(367\) −20.1473 −1.05168 −0.525840 0.850584i \(-0.676249\pi\)
−0.525840 + 0.850584i \(0.676249\pi\)
\(368\) 0 0
\(369\) −10.6659 −0.555245
\(370\) 0 0
\(371\) −17.5230 −0.909751
\(372\) 0 0
\(373\) 1.00565 0.0520704 0.0260352 0.999661i \(-0.491712\pi\)
0.0260352 + 0.999661i \(0.491712\pi\)
\(374\) 0 0
\(375\) −6.23205 −0.321822
\(376\) 0 0
\(377\) 1.17838 0.0606895
\(378\) 0 0
\(379\) 5.99274 0.307827 0.153913 0.988084i \(-0.450812\pi\)
0.153913 + 0.988084i \(0.450812\pi\)
\(380\) 0 0
\(381\) −1.91677 −0.0981989
\(382\) 0 0
\(383\) −11.6728 −0.596453 −0.298226 0.954495i \(-0.596395\pi\)
−0.298226 + 0.954495i \(0.596395\pi\)
\(384\) 0 0
\(385\) 4.45665 0.227132
\(386\) 0 0
\(387\) 5.90191 0.300011
\(388\) 0 0
\(389\) 14.8000 0.750387 0.375194 0.926946i \(-0.377576\pi\)
0.375194 + 0.926946i \(0.377576\pi\)
\(390\) 0 0
\(391\) 0.132625 0.00670711
\(392\) 0 0
\(393\) −1.66409 −0.0839424
\(394\) 0 0
\(395\) 5.12539 0.257887
\(396\) 0 0
\(397\) 21.7493 1.09157 0.545783 0.837927i \(-0.316232\pi\)
0.545783 + 0.837927i \(0.316232\pi\)
\(398\) 0 0
\(399\) −1.77490 −0.0888563
\(400\) 0 0
\(401\) −30.6226 −1.52922 −0.764610 0.644494i \(-0.777068\pi\)
−0.764610 + 0.644494i \(0.777068\pi\)
\(402\) 0 0
\(403\) −2.27233 −0.113193
\(404\) 0 0
\(405\) −0.650765 −0.0323368
\(406\) 0 0
\(407\) −10.5558 −0.523231
\(408\) 0 0
\(409\) −3.73481 −0.184674 −0.0923372 0.995728i \(-0.529434\pi\)
−0.0923372 + 0.995728i \(0.529434\pi\)
\(410\) 0 0
\(411\) −20.0439 −0.988693
\(412\) 0 0
\(413\) 7.85549 0.386543
\(414\) 0 0
\(415\) −11.5596 −0.567439
\(416\) 0 0
\(417\) −10.0030 −0.489850
\(418\) 0 0
\(419\) 19.6995 0.962383 0.481192 0.876615i \(-0.340204\pi\)
0.481192 + 0.876615i \(0.340204\pi\)
\(420\) 0 0
\(421\) −11.9228 −0.581080 −0.290540 0.956863i \(-0.593835\pi\)
−0.290540 + 0.956863i \(0.593835\pi\)
\(422\) 0 0
\(423\) −6.07807 −0.295526
\(424\) 0 0
\(425\) 0.606957 0.0294417
\(426\) 0 0
\(427\) −12.8571 −0.622197
\(428\) 0 0
\(429\) −6.04925 −0.292061
\(430\) 0 0
\(431\) −25.0991 −1.20898 −0.604489 0.796613i \(-0.706623\pi\)
−0.604489 + 0.796613i \(0.706623\pi\)
\(432\) 0 0
\(433\) 1.59603 0.0767003 0.0383502 0.999264i \(-0.487790\pi\)
0.0383502 + 0.999264i \(0.487790\pi\)
\(434\) 0 0
\(435\) 0.650765 0.0312018
\(436\) 0 0
\(437\) 1.33048 0.0636454
\(438\) 0 0
\(439\) 9.17612 0.437953 0.218976 0.975730i \(-0.429728\pi\)
0.218976 + 0.975730i \(0.429728\pi\)
\(440\) 0 0
\(441\) −5.22035 −0.248588
\(442\) 0 0
\(443\) −40.2215 −1.91098 −0.955490 0.295025i \(-0.904672\pi\)
−0.955490 + 0.295025i \(0.904672\pi\)
\(444\) 0 0
\(445\) −5.91469 −0.280383
\(446\) 0 0
\(447\) −4.34210 −0.205374
\(448\) 0 0
\(449\) −37.2563 −1.75823 −0.879117 0.476605i \(-0.841867\pi\)
−0.879117 + 0.476605i \(0.841867\pi\)
\(450\) 0 0
\(451\) −54.7539 −2.57826
\(452\) 0 0
\(453\) 5.85205 0.274953
\(454\) 0 0
\(455\) 1.02300 0.0479590
\(456\) 0 0
\(457\) −6.82461 −0.319242 −0.159621 0.987178i \(-0.551027\pi\)
−0.159621 + 0.987178i \(0.551027\pi\)
\(458\) 0 0
\(459\) 0.132625 0.00619039
\(460\) 0 0
\(461\) −12.4719 −0.580873 −0.290436 0.956894i \(-0.593800\pi\)
−0.290436 + 0.956894i \(0.593800\pi\)
\(462\) 0 0
\(463\) −7.66529 −0.356236 −0.178118 0.984009i \(-0.557001\pi\)
−0.178118 + 0.984009i \(0.557001\pi\)
\(464\) 0 0
\(465\) −1.25491 −0.0581949
\(466\) 0 0
\(467\) −0.864313 −0.0399956 −0.0199978 0.999800i \(-0.506366\pi\)
−0.0199978 + 0.999800i \(0.506366\pi\)
\(468\) 0 0
\(469\) 7.87783 0.363765
\(470\) 0 0
\(471\) 0.295720 0.0136260
\(472\) 0 0
\(473\) 30.2977 1.39309
\(474\) 0 0
\(475\) 6.08894 0.279380
\(476\) 0 0
\(477\) 13.1354 0.601427
\(478\) 0 0
\(479\) −1.22116 −0.0557960 −0.0278980 0.999611i \(-0.508881\pi\)
−0.0278980 + 0.999611i \(0.508881\pi\)
\(480\) 0 0
\(481\) −2.42302 −0.110480
\(482\) 0 0
\(483\) −1.33403 −0.0607007
\(484\) 0 0
\(485\) 10.8992 0.494906
\(486\) 0 0
\(487\) 30.8011 1.39573 0.697865 0.716229i \(-0.254134\pi\)
0.697865 + 0.716229i \(0.254134\pi\)
\(488\) 0 0
\(489\) 2.66668 0.120592
\(490\) 0 0
\(491\) −17.1501 −0.773973 −0.386986 0.922085i \(-0.626484\pi\)
−0.386986 + 0.922085i \(0.626484\pi\)
\(492\) 0 0
\(493\) −0.132625 −0.00597311
\(494\) 0 0
\(495\) −3.34073 −0.150155
\(496\) 0 0
\(497\) 9.85045 0.441853
\(498\) 0 0
\(499\) 9.43136 0.422206 0.211103 0.977464i \(-0.432294\pi\)
0.211103 + 0.977464i \(0.432294\pi\)
\(500\) 0 0
\(501\) 19.6691 0.878750
\(502\) 0 0
\(503\) 28.4809 1.26990 0.634950 0.772553i \(-0.281021\pi\)
0.634950 + 0.772553i \(0.281021\pi\)
\(504\) 0 0
\(505\) −6.39102 −0.284397
\(506\) 0 0
\(507\) 11.6114 0.515682
\(508\) 0 0
\(509\) 37.5818 1.66578 0.832892 0.553436i \(-0.186684\pi\)
0.832892 + 0.553436i \(0.186684\pi\)
\(510\) 0 0
\(511\) −5.85061 −0.258816
\(512\) 0 0
\(513\) 1.33048 0.0587420
\(514\) 0 0
\(515\) −5.07669 −0.223706
\(516\) 0 0
\(517\) −31.2021 −1.37226
\(518\) 0 0
\(519\) 12.4733 0.547516
\(520\) 0 0
\(521\) −34.7827 −1.52386 −0.761929 0.647660i \(-0.775748\pi\)
−0.761929 + 0.647660i \(0.775748\pi\)
\(522\) 0 0
\(523\) 14.9186 0.652345 0.326172 0.945310i \(-0.394241\pi\)
0.326172 + 0.945310i \(0.394241\pi\)
\(524\) 0 0
\(525\) −6.10521 −0.266453
\(526\) 0 0
\(527\) 0.255748 0.0111405
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.88852 −0.255540
\(532\) 0 0
\(533\) −12.5685 −0.544401
\(534\) 0 0
\(535\) 2.55854 0.110615
\(536\) 0 0
\(537\) 4.84197 0.208946
\(538\) 0 0
\(539\) −26.7989 −1.15431
\(540\) 0 0
\(541\) −6.89030 −0.296237 −0.148119 0.988970i \(-0.547322\pi\)
−0.148119 + 0.988970i \(0.547322\pi\)
\(542\) 0 0
\(543\) 4.99870 0.214515
\(544\) 0 0
\(545\) 6.88224 0.294803
\(546\) 0 0
\(547\) 19.7438 0.844185 0.422092 0.906553i \(-0.361296\pi\)
0.422092 + 0.906553i \(0.361296\pi\)
\(548\) 0 0
\(549\) 9.63773 0.411328
\(550\) 0 0
\(551\) −1.33048 −0.0566803
\(552\) 0 0
\(553\) 10.5068 0.446794
\(554\) 0 0
\(555\) −1.33813 −0.0568004
\(556\) 0 0
\(557\) 2.04602 0.0866928 0.0433464 0.999060i \(-0.486198\pi\)
0.0433464 + 0.999060i \(0.486198\pi\)
\(558\) 0 0
\(559\) 6.95467 0.294151
\(560\) 0 0
\(561\) 0.680834 0.0287448
\(562\) 0 0
\(563\) −43.4542 −1.83138 −0.915688 0.401890i \(-0.868353\pi\)
−0.915688 + 0.401890i \(0.868353\pi\)
\(564\) 0 0
\(565\) 8.36809 0.352048
\(566\) 0 0
\(567\) −1.33403 −0.0560242
\(568\) 0 0
\(569\) −2.78371 −0.116699 −0.0583496 0.998296i \(-0.518584\pi\)
−0.0583496 + 0.998296i \(0.518584\pi\)
\(570\) 0 0
\(571\) 17.4200 0.729004 0.364502 0.931203i \(-0.381239\pi\)
0.364502 + 0.931203i \(0.381239\pi\)
\(572\) 0 0
\(573\) 7.01032 0.292860
\(574\) 0 0
\(575\) 4.57650 0.190853
\(576\) 0 0
\(577\) 25.1597 1.04741 0.523706 0.851899i \(-0.324549\pi\)
0.523706 + 0.851899i \(0.324549\pi\)
\(578\) 0 0
\(579\) 13.6972 0.569237
\(580\) 0 0
\(581\) −23.6966 −0.983100
\(582\) 0 0
\(583\) 67.4310 2.79271
\(584\) 0 0
\(585\) −0.766847 −0.0317052
\(586\) 0 0
\(587\) −11.2993 −0.466373 −0.233187 0.972432i \(-0.574915\pi\)
−0.233187 + 0.972432i \(0.574915\pi\)
\(588\) 0 0
\(589\) 2.56564 0.105715
\(590\) 0 0
\(591\) 8.40065 0.345556
\(592\) 0 0
\(593\) 21.8453 0.897077 0.448539 0.893763i \(-0.351945\pi\)
0.448539 + 0.893763i \(0.351945\pi\)
\(594\) 0 0
\(595\) −0.115137 −0.00472016
\(596\) 0 0
\(597\) −16.5621 −0.677842
\(598\) 0 0
\(599\) −13.5125 −0.552104 −0.276052 0.961143i \(-0.589026\pi\)
−0.276052 + 0.961143i \(0.589026\pi\)
\(600\) 0 0
\(601\) −34.8960 −1.42344 −0.711719 0.702464i \(-0.752083\pi\)
−0.711719 + 0.702464i \(0.752083\pi\)
\(602\) 0 0
\(603\) −5.90527 −0.240481
\(604\) 0 0
\(605\) −9.99137 −0.406207
\(606\) 0 0
\(607\) 24.2382 0.983798 0.491899 0.870652i \(-0.336303\pi\)
0.491899 + 0.870652i \(0.336303\pi\)
\(608\) 0 0
\(609\) 1.33403 0.0540578
\(610\) 0 0
\(611\) −7.16226 −0.289754
\(612\) 0 0
\(613\) −31.7548 −1.28256 −0.641282 0.767305i \(-0.721597\pi\)
−0.641282 + 0.767305i \(0.721597\pi\)
\(614\) 0 0
\(615\) −6.94100 −0.279888
\(616\) 0 0
\(617\) 10.6331 0.428071 0.214036 0.976826i \(-0.431339\pi\)
0.214036 + 0.976826i \(0.431339\pi\)
\(618\) 0 0
\(619\) 30.3614 1.22033 0.610163 0.792276i \(-0.291104\pi\)
0.610163 + 0.792276i \(0.291104\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −12.1248 −0.485770
\(624\) 0 0
\(625\) 18.8269 0.753077
\(626\) 0 0
\(627\) 6.83007 0.272767
\(628\) 0 0
\(629\) 0.272708 0.0108736
\(630\) 0 0
\(631\) 22.5240 0.896667 0.448334 0.893866i \(-0.352018\pi\)
0.448334 + 0.893866i \(0.352018\pi\)
\(632\) 0 0
\(633\) −8.47612 −0.336895
\(634\) 0 0
\(635\) −1.24736 −0.0495001
\(636\) 0 0
\(637\) −6.15154 −0.243733
\(638\) 0 0
\(639\) −7.38396 −0.292105
\(640\) 0 0
\(641\) −15.2156 −0.600981 −0.300491 0.953785i \(-0.597150\pi\)
−0.300491 + 0.953785i \(0.597150\pi\)
\(642\) 0 0
\(643\) −24.7646 −0.976621 −0.488311 0.872670i \(-0.662387\pi\)
−0.488311 + 0.872670i \(0.662387\pi\)
\(644\) 0 0
\(645\) 3.84076 0.151230
\(646\) 0 0
\(647\) 29.8978 1.17541 0.587703 0.809077i \(-0.300032\pi\)
0.587703 + 0.809077i \(0.300032\pi\)
\(648\) 0 0
\(649\) −30.2290 −1.18659
\(650\) 0 0
\(651\) −2.57249 −0.100824
\(652\) 0 0
\(653\) −15.5766 −0.609558 −0.304779 0.952423i \(-0.598583\pi\)
−0.304779 + 0.952423i \(0.598583\pi\)
\(654\) 0 0
\(655\) −1.08293 −0.0423137
\(656\) 0 0
\(657\) 4.38565 0.171101
\(658\) 0 0
\(659\) −28.9820 −1.12898 −0.564490 0.825440i \(-0.690927\pi\)
−0.564490 + 0.825440i \(0.690927\pi\)
\(660\) 0 0
\(661\) 22.7430 0.884601 0.442301 0.896867i \(-0.354162\pi\)
0.442301 + 0.896867i \(0.354162\pi\)
\(662\) 0 0
\(663\) 0.156282 0.00606948
\(664\) 0 0
\(665\) −1.15505 −0.0447907
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −14.2177 −0.549686
\(670\) 0 0
\(671\) 49.4757 1.90999
\(672\) 0 0
\(673\) −20.2767 −0.781611 −0.390805 0.920473i \(-0.627803\pi\)
−0.390805 + 0.920473i \(0.627803\pi\)
\(674\) 0 0
\(675\) 4.57650 0.176150
\(676\) 0 0
\(677\) −2.62787 −0.100997 −0.0504987 0.998724i \(-0.516081\pi\)
−0.0504987 + 0.998724i \(0.516081\pi\)
\(678\) 0 0
\(679\) 22.3427 0.857435
\(680\) 0 0
\(681\) −5.21362 −0.199786
\(682\) 0 0
\(683\) 21.8387 0.835633 0.417817 0.908531i \(-0.362795\pi\)
0.417817 + 0.908531i \(0.362795\pi\)
\(684\) 0 0
\(685\) −13.0439 −0.498381
\(686\) 0 0
\(687\) 19.0310 0.726079
\(688\) 0 0
\(689\) 15.4784 0.589681
\(690\) 0 0
\(691\) −34.7850 −1.32328 −0.661642 0.749820i \(-0.730140\pi\)
−0.661642 + 0.749820i \(0.730140\pi\)
\(692\) 0 0
\(693\) −6.84832 −0.260146
\(694\) 0 0
\(695\) −6.50961 −0.246924
\(696\) 0 0
\(697\) 1.41456 0.0535804
\(698\) 0 0
\(699\) 26.7954 1.01350
\(700\) 0 0
\(701\) −8.72029 −0.329361 −0.164680 0.986347i \(-0.552659\pi\)
−0.164680 + 0.986347i \(0.552659\pi\)
\(702\) 0 0
\(703\) 2.73578 0.103182
\(704\) 0 0
\(705\) −3.95540 −0.148969
\(706\) 0 0
\(707\) −13.1013 −0.492723
\(708\) 0 0
\(709\) −2.64918 −0.0994922 −0.0497461 0.998762i \(-0.515841\pi\)
−0.0497461 + 0.998762i \(0.515841\pi\)
\(710\) 0 0
\(711\) −7.87595 −0.295371
\(712\) 0 0
\(713\) 1.92836 0.0722175
\(714\) 0 0
\(715\) −3.93664 −0.147222
\(716\) 0 0
\(717\) −2.26026 −0.0844109
\(718\) 0 0
\(719\) 14.3115 0.533728 0.266864 0.963734i \(-0.414013\pi\)
0.266864 + 0.963734i \(0.414013\pi\)
\(720\) 0 0
\(721\) −10.4070 −0.387575
\(722\) 0 0
\(723\) −6.92524 −0.257553
\(724\) 0 0
\(725\) −4.57650 −0.169967
\(726\) 0 0
\(727\) 19.5436 0.724831 0.362415 0.932017i \(-0.381952\pi\)
0.362415 + 0.932017i \(0.381952\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.782738 −0.0289506
\(732\) 0 0
\(733\) 48.5435 1.79299 0.896497 0.443050i \(-0.146104\pi\)
0.896497 + 0.443050i \(0.146104\pi\)
\(734\) 0 0
\(735\) −3.39722 −0.125308
\(736\) 0 0
\(737\) −30.3150 −1.11667
\(738\) 0 0
\(739\) −8.43523 −0.310295 −0.155148 0.987891i \(-0.549585\pi\)
−0.155148 + 0.987891i \(0.549585\pi\)
\(740\) 0 0
\(741\) 1.56781 0.0575948
\(742\) 0 0
\(743\) −52.5713 −1.92865 −0.964326 0.264717i \(-0.914721\pi\)
−0.964326 + 0.264717i \(0.914721\pi\)
\(744\) 0 0
\(745\) −2.82569 −0.103525
\(746\) 0 0
\(747\) 17.7631 0.649918
\(748\) 0 0
\(749\) 5.24488 0.191644
\(750\) 0 0
\(751\) 44.3346 1.61779 0.808897 0.587951i \(-0.200065\pi\)
0.808897 + 0.587951i \(0.200065\pi\)
\(752\) 0 0
\(753\) −15.1859 −0.553406
\(754\) 0 0
\(755\) 3.80831 0.138599
\(756\) 0 0
\(757\) −31.5742 −1.14759 −0.573793 0.819001i \(-0.694528\pi\)
−0.573793 + 0.819001i \(0.694528\pi\)
\(758\) 0 0
\(759\) 5.13354 0.186336
\(760\) 0 0
\(761\) −17.7051 −0.641811 −0.320905 0.947111i \(-0.603987\pi\)
−0.320905 + 0.947111i \(0.603987\pi\)
\(762\) 0 0
\(763\) 14.1082 0.510753
\(764\) 0 0
\(765\) 0.0863074 0.00312045
\(766\) 0 0
\(767\) −6.93890 −0.250549
\(768\) 0 0
\(769\) −16.5687 −0.597481 −0.298741 0.954334i \(-0.596567\pi\)
−0.298741 + 0.954334i \(0.596567\pi\)
\(770\) 0 0
\(771\) 15.2246 0.548302
\(772\) 0 0
\(773\) −10.9271 −0.393021 −0.196511 0.980502i \(-0.562961\pi\)
−0.196511 + 0.980502i \(0.562961\pi\)
\(774\) 0 0
\(775\) 8.82514 0.317008
\(776\) 0 0
\(777\) −2.74309 −0.0984079
\(778\) 0 0
\(779\) 14.1908 0.508437
\(780\) 0 0
\(781\) −37.9059 −1.35638
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 0.192444 0.00686862
\(786\) 0 0
\(787\) 22.0867 0.787307 0.393654 0.919259i \(-0.371211\pi\)
0.393654 + 0.919259i \(0.371211\pi\)
\(788\) 0 0
\(789\) 5.21935 0.185814
\(790\) 0 0
\(791\) 17.1541 0.609931
\(792\) 0 0
\(793\) 11.3569 0.403295
\(794\) 0 0
\(795\) 8.54804 0.303168
\(796\) 0 0
\(797\) 41.7258 1.47800 0.739001 0.673704i \(-0.235298\pi\)
0.739001 + 0.673704i \(0.235298\pi\)
\(798\) 0 0
\(799\) 0.806102 0.0285178
\(800\) 0 0
\(801\) 9.08883 0.321138
\(802\) 0 0
\(803\) 22.5139 0.794500
\(804\) 0 0
\(805\) −0.868143 −0.0305980
\(806\) 0 0
\(807\) 29.4706 1.03741
\(808\) 0 0
\(809\) 7.73463 0.271935 0.135967 0.990713i \(-0.456586\pi\)
0.135967 + 0.990713i \(0.456586\pi\)
\(810\) 0 0
\(811\) −38.0623 −1.33655 −0.668274 0.743915i \(-0.732967\pi\)
−0.668274 + 0.743915i \(0.732967\pi\)
\(812\) 0 0
\(813\) −1.04180 −0.0365376
\(814\) 0 0
\(815\) 1.73538 0.0607878
\(816\) 0 0
\(817\) −7.85236 −0.274719
\(818\) 0 0
\(819\) −1.57200 −0.0549300
\(820\) 0 0
\(821\) −27.6350 −0.964469 −0.482234 0.876042i \(-0.660175\pi\)
−0.482234 + 0.876042i \(0.660175\pi\)
\(822\) 0 0
\(823\) 14.1245 0.492349 0.246174 0.969226i \(-0.420826\pi\)
0.246174 + 0.969226i \(0.420826\pi\)
\(824\) 0 0
\(825\) 23.4937 0.817945
\(826\) 0 0
\(827\) 18.3713 0.638834 0.319417 0.947614i \(-0.396513\pi\)
0.319417 + 0.947614i \(0.396513\pi\)
\(828\) 0 0
\(829\) −30.7412 −1.06769 −0.533844 0.845583i \(-0.679253\pi\)
−0.533844 + 0.845583i \(0.679253\pi\)
\(830\) 0 0
\(831\) 10.2229 0.354628
\(832\) 0 0
\(833\) 0.692347 0.0239884
\(834\) 0 0
\(835\) 12.8000 0.442961
\(836\) 0 0
\(837\) 1.92836 0.0666538
\(838\) 0 0
\(839\) −29.4426 −1.01647 −0.508235 0.861218i \(-0.669702\pi\)
−0.508235 + 0.861218i \(0.669702\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 13.3230 0.458868
\(844\) 0 0
\(845\) 7.55631 0.259945
\(846\) 0 0
\(847\) −20.4818 −0.703763
\(848\) 0 0
\(849\) 18.7477 0.643418
\(850\) 0 0
\(851\) 2.05624 0.0704869
\(852\) 0 0
\(853\) −6.03614 −0.206673 −0.103337 0.994646i \(-0.532952\pi\)
−0.103337 + 0.994646i \(0.532952\pi\)
\(854\) 0 0
\(855\) 0.865829 0.0296107
\(856\) 0 0
\(857\) −40.2469 −1.37481 −0.687404 0.726276i \(-0.741250\pi\)
−0.687404 + 0.726276i \(0.741250\pi\)
\(858\) 0 0
\(859\) −12.2606 −0.418326 −0.209163 0.977881i \(-0.567074\pi\)
−0.209163 + 0.977881i \(0.567074\pi\)
\(860\) 0 0
\(861\) −14.2287 −0.484913
\(862\) 0 0
\(863\) 48.5646 1.65316 0.826579 0.562820i \(-0.190284\pi\)
0.826579 + 0.562820i \(0.190284\pi\)
\(864\) 0 0
\(865\) 8.11717 0.275992
\(866\) 0 0
\(867\) 16.9824 0.576753
\(868\) 0 0
\(869\) −40.4316 −1.37155
\(870\) 0 0
\(871\) −6.95864 −0.235784
\(872\) 0 0
\(873\) −16.7482 −0.566842
\(874\) 0 0
\(875\) −8.31377 −0.281057
\(876\) 0 0
\(877\) 49.9111 1.68538 0.842688 0.538402i \(-0.180972\pi\)
0.842688 + 0.538402i \(0.180972\pi\)
\(878\) 0 0
\(879\) 6.11769 0.206345
\(880\) 0 0
\(881\) −34.6328 −1.16681 −0.583405 0.812181i \(-0.698280\pi\)
−0.583405 + 0.812181i \(0.698280\pi\)
\(882\) 0 0
\(883\) 56.3365 1.89587 0.947936 0.318460i \(-0.103166\pi\)
0.947936 + 0.318460i \(0.103166\pi\)
\(884\) 0 0
\(885\) −3.83204 −0.128813
\(886\) 0 0
\(887\) −15.9989 −0.537192 −0.268596 0.963253i \(-0.586560\pi\)
−0.268596 + 0.963253i \(0.586560\pi\)
\(888\) 0 0
\(889\) −2.55703 −0.0857601
\(890\) 0 0
\(891\) 5.13354 0.171980
\(892\) 0 0
\(893\) 8.08674 0.270613
\(894\) 0 0
\(895\) 3.15099 0.105326
\(896\) 0 0
\(897\) 1.17838 0.0393449
\(898\) 0 0
\(899\) −1.92836 −0.0643143
\(900\) 0 0
\(901\) −1.74207 −0.0580369
\(902\) 0 0
\(903\) 7.87335 0.262009
\(904\) 0 0
\(905\) 3.25298 0.108133
\(906\) 0 0
\(907\) −38.8639 −1.29046 −0.645228 0.763990i \(-0.723238\pi\)
−0.645228 + 0.763990i \(0.723238\pi\)
\(908\) 0 0
\(909\) 9.82078 0.325735
\(910\) 0 0
\(911\) −36.2327 −1.20044 −0.600222 0.799834i \(-0.704921\pi\)
−0.600222 + 0.799834i \(0.704921\pi\)
\(912\) 0 0
\(913\) 91.1877 3.01787
\(914\) 0 0
\(915\) 6.27190 0.207343
\(916\) 0 0
\(917\) −2.21996 −0.0733094
\(918\) 0 0
\(919\) −46.8721 −1.54617 −0.773083 0.634304i \(-0.781287\pi\)
−0.773083 + 0.634304i \(0.781287\pi\)
\(920\) 0 0
\(921\) −9.66557 −0.318491
\(922\) 0 0
\(923\) −8.70109 −0.286400
\(924\) 0 0
\(925\) 9.41038 0.309412
\(926\) 0 0
\(927\) 7.80111 0.256222
\(928\) 0 0
\(929\) 27.8959 0.915233 0.457617 0.889150i \(-0.348703\pi\)
0.457617 + 0.889150i \(0.348703\pi\)
\(930\) 0 0
\(931\) 6.94556 0.227632
\(932\) 0 0
\(933\) 22.9305 0.750710
\(934\) 0 0
\(935\) 0.443063 0.0144897
\(936\) 0 0
\(937\) −43.6319 −1.42539 −0.712697 0.701472i \(-0.752526\pi\)
−0.712697 + 0.701472i \(0.752526\pi\)
\(938\) 0 0
\(939\) −28.5054 −0.930239
\(940\) 0 0
\(941\) −41.5445 −1.35431 −0.677155 0.735840i \(-0.736787\pi\)
−0.677155 + 0.735840i \(0.736787\pi\)
\(942\) 0 0
\(943\) 10.6659 0.347330
\(944\) 0 0
\(945\) −0.868143 −0.0282407
\(946\) 0 0
\(947\) 52.4514 1.70444 0.852220 0.523184i \(-0.175256\pi\)
0.852220 + 0.523184i \(0.175256\pi\)
\(948\) 0 0
\(949\) 5.16795 0.167759
\(950\) 0 0
\(951\) 23.3183 0.756146
\(952\) 0 0
\(953\) −6.34579 −0.205560 −0.102780 0.994704i \(-0.532774\pi\)
−0.102780 + 0.994704i \(0.532774\pi\)
\(954\) 0 0
\(955\) 4.56207 0.147625
\(956\) 0 0
\(957\) −5.13354 −0.165944
\(958\) 0 0
\(959\) −26.7393 −0.863456
\(960\) 0 0
\(961\) −27.2814 −0.880046
\(962\) 0 0
\(963\) −3.93159 −0.126694
\(964\) 0 0
\(965\) 8.91366 0.286941
\(966\) 0 0
\(967\) −0.221567 −0.00712511 −0.00356255 0.999994i \(-0.501134\pi\)
−0.00356255 + 0.999994i \(0.501134\pi\)
\(968\) 0 0
\(969\) −0.176454 −0.00566852
\(970\) 0 0
\(971\) −19.4630 −0.624598 −0.312299 0.949984i \(-0.601099\pi\)
−0.312299 + 0.949984i \(0.601099\pi\)
\(972\) 0 0
\(973\) −13.3444 −0.427801
\(974\) 0 0
\(975\) 5.39285 0.172709
\(976\) 0 0
\(977\) −17.3891 −0.556328 −0.278164 0.960534i \(-0.589726\pi\)
−0.278164 + 0.960534i \(0.589726\pi\)
\(978\) 0 0
\(979\) 46.6579 1.49119
\(980\) 0 0
\(981\) −10.5756 −0.337654
\(982\) 0 0
\(983\) 34.9182 1.11372 0.556859 0.830607i \(-0.312007\pi\)
0.556859 + 0.830607i \(0.312007\pi\)
\(984\) 0 0
\(985\) 5.46685 0.174188
\(986\) 0 0
\(987\) −8.10836 −0.258092
\(988\) 0 0
\(989\) −5.90191 −0.187670
\(990\) 0 0
\(991\) 39.5383 1.25598 0.627988 0.778223i \(-0.283879\pi\)
0.627988 + 0.778223i \(0.283879\pi\)
\(992\) 0 0
\(993\) −13.7752 −0.437143
\(994\) 0 0
\(995\) −10.7780 −0.341687
\(996\) 0 0
\(997\) 29.9035 0.947053 0.473526 0.880780i \(-0.342981\pi\)
0.473526 + 0.880780i \(0.342981\pi\)
\(998\) 0 0
\(999\) 2.05624 0.0650565
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 8004.2.a.e.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.4 9 1.1 even 1 trivial