Properties

Label 8004.2.a.i.1.15
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(4.02229\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.02229 q^{5} +4.04331 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.02229 q^{5} +4.04331 q^{7} +1.00000 q^{9} -3.42405 q^{11} +4.92009 q^{13} -4.02229 q^{15} +4.60807 q^{17} +7.63146 q^{19} -4.04331 q^{21} +1.00000 q^{23} +11.1788 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.21724 q^{31} +3.42405 q^{33} +16.2634 q^{35} +3.25376 q^{37} -4.92009 q^{39} -3.81231 q^{41} -4.87928 q^{43} +4.02229 q^{45} +4.28191 q^{47} +9.34836 q^{49} -4.60807 q^{51} +11.5936 q^{53} -13.7725 q^{55} -7.63146 q^{57} +4.70683 q^{59} -10.5681 q^{61} +4.04331 q^{63} +19.7900 q^{65} -15.2771 q^{67} -1.00000 q^{69} -7.01655 q^{71} -11.3905 q^{73} -11.1788 q^{75} -13.8445 q^{77} -5.09508 q^{79} +1.00000 q^{81} -0.476321 q^{83} +18.5350 q^{85} -1.00000 q^{87} +8.93310 q^{89} +19.8935 q^{91} +4.21724 q^{93} +30.6959 q^{95} +14.9323 q^{97} -3.42405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) 4.02229 1.79882 0.899411 0.437104i \(-0.143996\pi\)
0.899411 + 0.437104i \(0.143996\pi\)
\(6\) 0 0
\(7\) 4.04331 1.52823 0.764114 0.645081i \(-0.223177\pi\)
0.764114 + 0.645081i \(0.223177\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.42405 −1.03239 −0.516195 0.856471i \(-0.672652\pi\)
−0.516195 + 0.856471i \(0.672652\pi\)
\(12\) 0 0
\(13\) 4.92009 1.36459 0.682294 0.731078i \(-0.260982\pi\)
0.682294 + 0.731078i \(0.260982\pi\)
\(14\) 0 0
\(15\) −4.02229 −1.03855
\(16\) 0 0
\(17\) 4.60807 1.11762 0.558810 0.829295i \(-0.311258\pi\)
0.558810 + 0.829295i \(0.311258\pi\)
\(18\) 0 0
\(19\) 7.63146 1.75078 0.875388 0.483421i \(-0.160606\pi\)
0.875388 + 0.483421i \(0.160606\pi\)
\(20\) 0 0
\(21\) −4.04331 −0.882323
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 11.1788 2.23576
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.21724 −0.757438 −0.378719 0.925512i \(-0.623635\pi\)
−0.378719 + 0.925512i \(0.623635\pi\)
\(32\) 0 0
\(33\) 3.42405 0.596050
\(34\) 0 0
\(35\) 16.2634 2.74901
\(36\) 0 0
\(37\) 3.25376 0.534915 0.267458 0.963570i \(-0.413816\pi\)
0.267458 + 0.963570i \(0.413816\pi\)
\(38\) 0 0
\(39\) −4.92009 −0.787845
\(40\) 0 0
\(41\) −3.81231 −0.595383 −0.297692 0.954662i \(-0.596217\pi\)
−0.297692 + 0.954662i \(0.596217\pi\)
\(42\) 0 0
\(43\) −4.87928 −0.744083 −0.372041 0.928216i \(-0.621342\pi\)
−0.372041 + 0.928216i \(0.621342\pi\)
\(44\) 0 0
\(45\) 4.02229 0.599607
\(46\) 0 0
\(47\) 4.28191 0.624581 0.312290 0.949987i \(-0.398904\pi\)
0.312290 + 0.949987i \(0.398904\pi\)
\(48\) 0 0
\(49\) 9.34836 1.33548
\(50\) 0 0
\(51\) −4.60807 −0.645259
\(52\) 0 0
\(53\) 11.5936 1.59251 0.796253 0.604964i \(-0.206812\pi\)
0.796253 + 0.604964i \(0.206812\pi\)
\(54\) 0 0
\(55\) −13.7725 −1.85709
\(56\) 0 0
\(57\) −7.63146 −1.01081
\(58\) 0 0
\(59\) 4.70683 0.612777 0.306389 0.951907i \(-0.400879\pi\)
0.306389 + 0.951907i \(0.400879\pi\)
\(60\) 0 0
\(61\) −10.5681 −1.35311 −0.676554 0.736393i \(-0.736527\pi\)
−0.676554 + 0.736393i \(0.736527\pi\)
\(62\) 0 0
\(63\) 4.04331 0.509409
\(64\) 0 0
\(65\) 19.7900 2.45465
\(66\) 0 0
\(67\) −15.2771 −1.86639 −0.933197 0.359365i \(-0.882993\pi\)
−0.933197 + 0.359365i \(0.882993\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.01655 −0.832711 −0.416356 0.909202i \(-0.636693\pi\)
−0.416356 + 0.909202i \(0.636693\pi\)
\(72\) 0 0
\(73\) −11.3905 −1.33316 −0.666580 0.745434i \(-0.732242\pi\)
−0.666580 + 0.745434i \(0.732242\pi\)
\(74\) 0 0
\(75\) −11.1788 −1.29082
\(76\) 0 0
\(77\) −13.8445 −1.57773
\(78\) 0 0
\(79\) −5.09508 −0.573241 −0.286621 0.958044i \(-0.592532\pi\)
−0.286621 + 0.958044i \(0.592532\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.476321 −0.0522830 −0.0261415 0.999658i \(-0.508322\pi\)
−0.0261415 + 0.999658i \(0.508322\pi\)
\(84\) 0 0
\(85\) 18.5350 2.01040
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 8.93310 0.946907 0.473453 0.880819i \(-0.343007\pi\)
0.473453 + 0.880819i \(0.343007\pi\)
\(90\) 0 0
\(91\) 19.8935 2.08540
\(92\) 0 0
\(93\) 4.21724 0.437307
\(94\) 0 0
\(95\) 30.6959 3.14934
\(96\) 0 0
\(97\) 14.9323 1.51615 0.758075 0.652168i \(-0.226140\pi\)
0.758075 + 0.652168i \(0.226140\pi\)
\(98\) 0 0
\(99\) −3.42405 −0.344130
\(100\) 0 0
\(101\) −10.2354 −1.01847 −0.509233 0.860629i \(-0.670071\pi\)
−0.509233 + 0.860629i \(0.670071\pi\)
\(102\) 0 0
\(103\) −11.1878 −1.10237 −0.551183 0.834384i \(-0.685823\pi\)
−0.551183 + 0.834384i \(0.685823\pi\)
\(104\) 0 0
\(105\) −16.2634 −1.58714
\(106\) 0 0
\(107\) 15.6748 1.51534 0.757671 0.652637i \(-0.226337\pi\)
0.757671 + 0.652637i \(0.226337\pi\)
\(108\) 0 0
\(109\) −16.3071 −1.56194 −0.780968 0.624572i \(-0.785274\pi\)
−0.780968 + 0.624572i \(0.785274\pi\)
\(110\) 0 0
\(111\) −3.25376 −0.308833
\(112\) 0 0
\(113\) 15.1546 1.42562 0.712812 0.701355i \(-0.247421\pi\)
0.712812 + 0.701355i \(0.247421\pi\)
\(114\) 0 0
\(115\) 4.02229 0.375080
\(116\) 0 0
\(117\) 4.92009 0.454863
\(118\) 0 0
\(119\) 18.6319 1.70798
\(120\) 0 0
\(121\) 0.724108 0.0658280
\(122\) 0 0
\(123\) 3.81231 0.343745
\(124\) 0 0
\(125\) 24.8529 2.22291
\(126\) 0 0
\(127\) −19.7464 −1.75221 −0.876104 0.482123i \(-0.839866\pi\)
−0.876104 + 0.482123i \(0.839866\pi\)
\(128\) 0 0
\(129\) 4.87928 0.429597
\(130\) 0 0
\(131\) 1.99863 0.174621 0.0873107 0.996181i \(-0.472173\pi\)
0.0873107 + 0.996181i \(0.472173\pi\)
\(132\) 0 0
\(133\) 30.8564 2.67559
\(134\) 0 0
\(135\) −4.02229 −0.346183
\(136\) 0 0
\(137\) −3.85067 −0.328985 −0.164492 0.986378i \(-0.552599\pi\)
−0.164492 + 0.986378i \(0.552599\pi\)
\(138\) 0 0
\(139\) 5.64898 0.479140 0.239570 0.970879i \(-0.422993\pi\)
0.239570 + 0.970879i \(0.422993\pi\)
\(140\) 0 0
\(141\) −4.28191 −0.360602
\(142\) 0 0
\(143\) −16.8466 −1.40879
\(144\) 0 0
\(145\) 4.02229 0.334033
\(146\) 0 0
\(147\) −9.34836 −0.771040
\(148\) 0 0
\(149\) −14.9170 −1.22205 −0.611023 0.791613i \(-0.709242\pi\)
−0.611023 + 0.791613i \(0.709242\pi\)
\(150\) 0 0
\(151\) −5.91177 −0.481093 −0.240547 0.970638i \(-0.577327\pi\)
−0.240547 + 0.970638i \(0.577327\pi\)
\(152\) 0 0
\(153\) 4.60807 0.372540
\(154\) 0 0
\(155\) −16.9629 −1.36250
\(156\) 0 0
\(157\) −16.9191 −1.35029 −0.675146 0.737684i \(-0.735920\pi\)
−0.675146 + 0.737684i \(0.735920\pi\)
\(158\) 0 0
\(159\) −11.5936 −0.919434
\(160\) 0 0
\(161\) 4.04331 0.318658
\(162\) 0 0
\(163\) 12.0575 0.944412 0.472206 0.881488i \(-0.343458\pi\)
0.472206 + 0.881488i \(0.343458\pi\)
\(164\) 0 0
\(165\) 13.7725 1.07219
\(166\) 0 0
\(167\) −8.41234 −0.650966 −0.325483 0.945548i \(-0.605527\pi\)
−0.325483 + 0.945548i \(0.605527\pi\)
\(168\) 0 0
\(169\) 11.2073 0.862101
\(170\) 0 0
\(171\) 7.63146 0.583592
\(172\) 0 0
\(173\) −20.2440 −1.53913 −0.769563 0.638571i \(-0.779526\pi\)
−0.769563 + 0.638571i \(0.779526\pi\)
\(174\) 0 0
\(175\) 45.1994 3.41675
\(176\) 0 0
\(177\) −4.70683 −0.353787
\(178\) 0 0
\(179\) 7.01898 0.524623 0.262312 0.964983i \(-0.415515\pi\)
0.262312 + 0.964983i \(0.415515\pi\)
\(180\) 0 0
\(181\) −13.5088 −1.00410 −0.502050 0.864838i \(-0.667421\pi\)
−0.502050 + 0.864838i \(0.667421\pi\)
\(182\) 0 0
\(183\) 10.5681 0.781217
\(184\) 0 0
\(185\) 13.0876 0.962217
\(186\) 0 0
\(187\) −15.7783 −1.15382
\(188\) 0 0
\(189\) −4.04331 −0.294108
\(190\) 0 0
\(191\) −14.6319 −1.05872 −0.529362 0.848396i \(-0.677569\pi\)
−0.529362 + 0.848396i \(0.677569\pi\)
\(192\) 0 0
\(193\) −5.90206 −0.424840 −0.212420 0.977178i \(-0.568134\pi\)
−0.212420 + 0.977178i \(0.568134\pi\)
\(194\) 0 0
\(195\) −19.7900 −1.41719
\(196\) 0 0
\(197\) 3.58043 0.255095 0.127548 0.991832i \(-0.459289\pi\)
0.127548 + 0.991832i \(0.459289\pi\)
\(198\) 0 0
\(199\) −0.206157 −0.0146141 −0.00730703 0.999973i \(-0.502326\pi\)
−0.00730703 + 0.999973i \(0.502326\pi\)
\(200\) 0 0
\(201\) 15.2771 1.07756
\(202\) 0 0
\(203\) 4.04331 0.283785
\(204\) 0 0
\(205\) −15.3342 −1.07099
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −26.1305 −1.80748
\(210\) 0 0
\(211\) −26.7071 −1.83860 −0.919298 0.393563i \(-0.871242\pi\)
−0.919298 + 0.393563i \(0.871242\pi\)
\(212\) 0 0
\(213\) 7.01655 0.480766
\(214\) 0 0
\(215\) −19.6259 −1.33847
\(216\) 0 0
\(217\) −17.0516 −1.15754
\(218\) 0 0
\(219\) 11.3905 0.769700
\(220\) 0 0
\(221\) 22.6721 1.52509
\(222\) 0 0
\(223\) 6.90704 0.462529 0.231265 0.972891i \(-0.425714\pi\)
0.231265 + 0.972891i \(0.425714\pi\)
\(224\) 0 0
\(225\) 11.1788 0.745254
\(226\) 0 0
\(227\) −0.0517430 −0.00343430 −0.00171715 0.999999i \(-0.500547\pi\)
−0.00171715 + 0.999999i \(0.500547\pi\)
\(228\) 0 0
\(229\) 3.67060 0.242560 0.121280 0.992618i \(-0.461300\pi\)
0.121280 + 0.992618i \(0.461300\pi\)
\(230\) 0 0
\(231\) 13.8445 0.910901
\(232\) 0 0
\(233\) 18.6446 1.22145 0.610723 0.791844i \(-0.290879\pi\)
0.610723 + 0.791844i \(0.290879\pi\)
\(234\) 0 0
\(235\) 17.2231 1.12351
\(236\) 0 0
\(237\) 5.09508 0.330961
\(238\) 0 0
\(239\) −17.1647 −1.11029 −0.555147 0.831752i \(-0.687338\pi\)
−0.555147 + 0.831752i \(0.687338\pi\)
\(240\) 0 0
\(241\) −15.5050 −0.998762 −0.499381 0.866383i \(-0.666439\pi\)
−0.499381 + 0.866383i \(0.666439\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 37.6018 2.40229
\(246\) 0 0
\(247\) 37.5475 2.38909
\(248\) 0 0
\(249\) 0.476321 0.0301856
\(250\) 0 0
\(251\) −2.04662 −0.129181 −0.0645907 0.997912i \(-0.520574\pi\)
−0.0645907 + 0.997912i \(0.520574\pi\)
\(252\) 0 0
\(253\) −3.42405 −0.215268
\(254\) 0 0
\(255\) −18.5350 −1.16071
\(256\) 0 0
\(257\) 25.5277 1.59238 0.796188 0.605049i \(-0.206847\pi\)
0.796188 + 0.605049i \(0.206847\pi\)
\(258\) 0 0
\(259\) 13.1560 0.817472
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −16.2098 −0.999539 −0.499770 0.866158i \(-0.666582\pi\)
−0.499770 + 0.866158i \(0.666582\pi\)
\(264\) 0 0
\(265\) 46.6329 2.86464
\(266\) 0 0
\(267\) −8.93310 −0.546697
\(268\) 0 0
\(269\) −16.0182 −0.976646 −0.488323 0.872663i \(-0.662391\pi\)
−0.488323 + 0.872663i \(0.662391\pi\)
\(270\) 0 0
\(271\) −3.64736 −0.221561 −0.110781 0.993845i \(-0.535335\pi\)
−0.110781 + 0.993845i \(0.535335\pi\)
\(272\) 0 0
\(273\) −19.8935 −1.20401
\(274\) 0 0
\(275\) −38.2768 −2.30818
\(276\) 0 0
\(277\) −8.19668 −0.492491 −0.246245 0.969208i \(-0.579197\pi\)
−0.246245 + 0.969208i \(0.579197\pi\)
\(278\) 0 0
\(279\) −4.21724 −0.252479
\(280\) 0 0
\(281\) −30.6032 −1.82563 −0.912817 0.408368i \(-0.866098\pi\)
−0.912817 + 0.408368i \(0.866098\pi\)
\(282\) 0 0
\(283\) −18.1893 −1.08124 −0.540622 0.841266i \(-0.681811\pi\)
−0.540622 + 0.841266i \(0.681811\pi\)
\(284\) 0 0
\(285\) −30.6959 −1.81827
\(286\) 0 0
\(287\) −15.4144 −0.909881
\(288\) 0 0
\(289\) 4.23430 0.249077
\(290\) 0 0
\(291\) −14.9323 −0.875349
\(292\) 0 0
\(293\) 25.5008 1.48977 0.744887 0.667191i \(-0.232503\pi\)
0.744887 + 0.667191i \(0.232503\pi\)
\(294\) 0 0
\(295\) 18.9322 1.10228
\(296\) 0 0
\(297\) 3.42405 0.198683
\(298\) 0 0
\(299\) 4.92009 0.284536
\(300\) 0 0
\(301\) −19.7284 −1.13713
\(302\) 0 0
\(303\) 10.2354 0.588011
\(304\) 0 0
\(305\) −42.5080 −2.43400
\(306\) 0 0
\(307\) 6.49238 0.370540 0.185270 0.982688i \(-0.440684\pi\)
0.185270 + 0.982688i \(0.440684\pi\)
\(308\) 0 0
\(309\) 11.1878 0.636451
\(310\) 0 0
\(311\) −31.0601 −1.76126 −0.880630 0.473805i \(-0.842880\pi\)
−0.880630 + 0.473805i \(0.842880\pi\)
\(312\) 0 0
\(313\) 21.1142 1.19344 0.596722 0.802448i \(-0.296470\pi\)
0.596722 + 0.802448i \(0.296470\pi\)
\(314\) 0 0
\(315\) 16.2634 0.916337
\(316\) 0 0
\(317\) 14.5266 0.815897 0.407948 0.913005i \(-0.366244\pi\)
0.407948 + 0.913005i \(0.366244\pi\)
\(318\) 0 0
\(319\) −3.42405 −0.191710
\(320\) 0 0
\(321\) −15.6748 −0.874883
\(322\) 0 0
\(323\) 35.1663 1.95670
\(324\) 0 0
\(325\) 55.0008 3.05089
\(326\) 0 0
\(327\) 16.3071 0.901784
\(328\) 0 0
\(329\) 17.3131 0.954502
\(330\) 0 0
\(331\) −17.6417 −0.969678 −0.484839 0.874603i \(-0.661122\pi\)
−0.484839 + 0.874603i \(0.661122\pi\)
\(332\) 0 0
\(333\) 3.25376 0.178305
\(334\) 0 0
\(335\) −61.4489 −3.35731
\(336\) 0 0
\(337\) −18.7050 −1.01893 −0.509464 0.860492i \(-0.670156\pi\)
−0.509464 + 0.860492i \(0.670156\pi\)
\(338\) 0 0
\(339\) −15.1546 −0.823085
\(340\) 0 0
\(341\) 14.4400 0.781971
\(342\) 0 0
\(343\) 9.49516 0.512690
\(344\) 0 0
\(345\) −4.02229 −0.216553
\(346\) 0 0
\(347\) −23.6707 −1.27071 −0.635355 0.772220i \(-0.719146\pi\)
−0.635355 + 0.772220i \(0.719146\pi\)
\(348\) 0 0
\(349\) −20.5123 −1.09800 −0.548999 0.835823i \(-0.684991\pi\)
−0.548999 + 0.835823i \(0.684991\pi\)
\(350\) 0 0
\(351\) −4.92009 −0.262615
\(352\) 0 0
\(353\) 33.8808 1.80329 0.901646 0.432475i \(-0.142360\pi\)
0.901646 + 0.432475i \(0.142360\pi\)
\(354\) 0 0
\(355\) −28.2226 −1.49790
\(356\) 0 0
\(357\) −18.6319 −0.986102
\(358\) 0 0
\(359\) 5.37177 0.283511 0.141755 0.989902i \(-0.454725\pi\)
0.141755 + 0.989902i \(0.454725\pi\)
\(360\) 0 0
\(361\) 39.2391 2.06522
\(362\) 0 0
\(363\) −0.724108 −0.0380058
\(364\) 0 0
\(365\) −45.8160 −2.39812
\(366\) 0 0
\(367\) −23.2226 −1.21221 −0.606106 0.795384i \(-0.707269\pi\)
−0.606106 + 0.795384i \(0.707269\pi\)
\(368\) 0 0
\(369\) −3.81231 −0.198461
\(370\) 0 0
\(371\) 46.8766 2.43371
\(372\) 0 0
\(373\) 37.2210 1.92723 0.963615 0.267293i \(-0.0861290\pi\)
0.963615 + 0.267293i \(0.0861290\pi\)
\(374\) 0 0
\(375\) −24.8529 −1.28340
\(376\) 0 0
\(377\) 4.92009 0.253398
\(378\) 0 0
\(379\) 30.5601 1.56977 0.784884 0.619643i \(-0.212723\pi\)
0.784884 + 0.619643i \(0.212723\pi\)
\(380\) 0 0
\(381\) 19.7464 1.01164
\(382\) 0 0
\(383\) 25.5302 1.30453 0.652266 0.757990i \(-0.273818\pi\)
0.652266 + 0.757990i \(0.273818\pi\)
\(384\) 0 0
\(385\) −55.6865 −2.83805
\(386\) 0 0
\(387\) −4.87928 −0.248028
\(388\) 0 0
\(389\) −30.3878 −1.54072 −0.770362 0.637606i \(-0.779925\pi\)
−0.770362 + 0.637606i \(0.779925\pi\)
\(390\) 0 0
\(391\) 4.60807 0.233040
\(392\) 0 0
\(393\) −1.99863 −0.100818
\(394\) 0 0
\(395\) −20.4939 −1.03116
\(396\) 0 0
\(397\) 3.14774 0.157981 0.0789903 0.996875i \(-0.474830\pi\)
0.0789903 + 0.996875i \(0.474830\pi\)
\(398\) 0 0
\(399\) −30.8564 −1.54475
\(400\) 0 0
\(401\) −7.83957 −0.391489 −0.195745 0.980655i \(-0.562712\pi\)
−0.195745 + 0.980655i \(0.562712\pi\)
\(402\) 0 0
\(403\) −20.7492 −1.03359
\(404\) 0 0
\(405\) 4.02229 0.199869
\(406\) 0 0
\(407\) −11.1410 −0.552241
\(408\) 0 0
\(409\) −4.94580 −0.244554 −0.122277 0.992496i \(-0.539020\pi\)
−0.122277 + 0.992496i \(0.539020\pi\)
\(410\) 0 0
\(411\) 3.85067 0.189939
\(412\) 0 0
\(413\) 19.0312 0.936463
\(414\) 0 0
\(415\) −1.91590 −0.0940478
\(416\) 0 0
\(417\) −5.64898 −0.276632
\(418\) 0 0
\(419\) 21.0781 1.02973 0.514867 0.857270i \(-0.327841\pi\)
0.514867 + 0.857270i \(0.327841\pi\)
\(420\) 0 0
\(421\) −18.5556 −0.904343 −0.452172 0.891931i \(-0.649351\pi\)
−0.452172 + 0.891931i \(0.649351\pi\)
\(422\) 0 0
\(423\) 4.28191 0.208194
\(424\) 0 0
\(425\) 51.5127 2.49873
\(426\) 0 0
\(427\) −42.7301 −2.06786
\(428\) 0 0
\(429\) 16.8466 0.813363
\(430\) 0 0
\(431\) −0.248683 −0.0119786 −0.00598931 0.999982i \(-0.501906\pi\)
−0.00598931 + 0.999982i \(0.501906\pi\)
\(432\) 0 0
\(433\) −31.8676 −1.53146 −0.765731 0.643161i \(-0.777622\pi\)
−0.765731 + 0.643161i \(0.777622\pi\)
\(434\) 0 0
\(435\) −4.02229 −0.192854
\(436\) 0 0
\(437\) 7.63146 0.365062
\(438\) 0 0
\(439\) 17.1853 0.820208 0.410104 0.912039i \(-0.365492\pi\)
0.410104 + 0.912039i \(0.365492\pi\)
\(440\) 0 0
\(441\) 9.34836 0.445160
\(442\) 0 0
\(443\) 26.7251 1.26975 0.634875 0.772615i \(-0.281052\pi\)
0.634875 + 0.772615i \(0.281052\pi\)
\(444\) 0 0
\(445\) 35.9315 1.70332
\(446\) 0 0
\(447\) 14.9170 0.705548
\(448\) 0 0
\(449\) 25.0993 1.18451 0.592254 0.805751i \(-0.298238\pi\)
0.592254 + 0.805751i \(0.298238\pi\)
\(450\) 0 0
\(451\) 13.0535 0.614667
\(452\) 0 0
\(453\) 5.91177 0.277759
\(454\) 0 0
\(455\) 80.0173 3.75127
\(456\) 0 0
\(457\) 4.68062 0.218950 0.109475 0.993990i \(-0.465083\pi\)
0.109475 + 0.993990i \(0.465083\pi\)
\(458\) 0 0
\(459\) −4.60807 −0.215086
\(460\) 0 0
\(461\) −8.78731 −0.409266 −0.204633 0.978839i \(-0.565600\pi\)
−0.204633 + 0.978839i \(0.565600\pi\)
\(462\) 0 0
\(463\) 31.6797 1.47228 0.736139 0.676830i \(-0.236647\pi\)
0.736139 + 0.676830i \(0.236647\pi\)
\(464\) 0 0
\(465\) 16.9629 0.786637
\(466\) 0 0
\(467\) 17.0114 0.787196 0.393598 0.919283i \(-0.371230\pi\)
0.393598 + 0.919283i \(0.371230\pi\)
\(468\) 0 0
\(469\) −61.7701 −2.85228
\(470\) 0 0
\(471\) 16.9191 0.779592
\(472\) 0 0
\(473\) 16.7069 0.768183
\(474\) 0 0
\(475\) 85.3106 3.91432
\(476\) 0 0
\(477\) 11.5936 0.530835
\(478\) 0 0
\(479\) 28.3153 1.29376 0.646880 0.762592i \(-0.276073\pi\)
0.646880 + 0.762592i \(0.276073\pi\)
\(480\) 0 0
\(481\) 16.0088 0.729939
\(482\) 0 0
\(483\) −4.04331 −0.183977
\(484\) 0 0
\(485\) 60.0622 2.72728
\(486\) 0 0
\(487\) 19.5510 0.885942 0.442971 0.896536i \(-0.353925\pi\)
0.442971 + 0.896536i \(0.353925\pi\)
\(488\) 0 0
\(489\) −12.0575 −0.545257
\(490\) 0 0
\(491\) −30.8233 −1.39103 −0.695517 0.718510i \(-0.744825\pi\)
−0.695517 + 0.718510i \(0.744825\pi\)
\(492\) 0 0
\(493\) 4.60807 0.207537
\(494\) 0 0
\(495\) −13.7725 −0.619028
\(496\) 0 0
\(497\) −28.3701 −1.27257
\(498\) 0 0
\(499\) −32.3081 −1.44631 −0.723154 0.690687i \(-0.757308\pi\)
−0.723154 + 0.690687i \(0.757308\pi\)
\(500\) 0 0
\(501\) 8.41234 0.375836
\(502\) 0 0
\(503\) −21.6976 −0.967449 −0.483725 0.875220i \(-0.660716\pi\)
−0.483725 + 0.875220i \(0.660716\pi\)
\(504\) 0 0
\(505\) −41.1699 −1.83204
\(506\) 0 0
\(507\) −11.2073 −0.497734
\(508\) 0 0
\(509\) −6.16165 −0.273110 −0.136555 0.990632i \(-0.543603\pi\)
−0.136555 + 0.990632i \(0.543603\pi\)
\(510\) 0 0
\(511\) −46.0554 −2.03737
\(512\) 0 0
\(513\) −7.63146 −0.336937
\(514\) 0 0
\(515\) −45.0005 −1.98296
\(516\) 0 0
\(517\) −14.6615 −0.644810
\(518\) 0 0
\(519\) 20.2440 0.888615
\(520\) 0 0
\(521\) −8.32842 −0.364875 −0.182437 0.983217i \(-0.558399\pi\)
−0.182437 + 0.983217i \(0.558399\pi\)
\(522\) 0 0
\(523\) 13.4234 0.586965 0.293482 0.955964i \(-0.405186\pi\)
0.293482 + 0.955964i \(0.405186\pi\)
\(524\) 0 0
\(525\) −45.1994 −1.97266
\(526\) 0 0
\(527\) −19.4333 −0.846528
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.70683 0.204259
\(532\) 0 0
\(533\) −18.7569 −0.812453
\(534\) 0 0
\(535\) 63.0486 2.72583
\(536\) 0 0
\(537\) −7.01898 −0.302892
\(538\) 0 0
\(539\) −32.0092 −1.37874
\(540\) 0 0
\(541\) −20.0066 −0.860153 −0.430076 0.902792i \(-0.641513\pi\)
−0.430076 + 0.902792i \(0.641513\pi\)
\(542\) 0 0
\(543\) 13.5088 0.579718
\(544\) 0 0
\(545\) −65.5918 −2.80964
\(546\) 0 0
\(547\) 29.1367 1.24580 0.622898 0.782303i \(-0.285955\pi\)
0.622898 + 0.782303i \(0.285955\pi\)
\(548\) 0 0
\(549\) −10.5681 −0.451036
\(550\) 0 0
\(551\) 7.63146 0.325111
\(552\) 0 0
\(553\) −20.6010 −0.876043
\(554\) 0 0
\(555\) −13.0876 −0.555536
\(556\) 0 0
\(557\) 36.2619 1.53646 0.768232 0.640171i \(-0.221137\pi\)
0.768232 + 0.640171i \(0.221137\pi\)
\(558\) 0 0
\(559\) −24.0065 −1.01537
\(560\) 0 0
\(561\) 15.7783 0.666158
\(562\) 0 0
\(563\) 23.3050 0.982190 0.491095 0.871106i \(-0.336597\pi\)
0.491095 + 0.871106i \(0.336597\pi\)
\(564\) 0 0
\(565\) 60.9562 2.56444
\(566\) 0 0
\(567\) 4.04331 0.169803
\(568\) 0 0
\(569\) −16.6105 −0.696350 −0.348175 0.937430i \(-0.613198\pi\)
−0.348175 + 0.937430i \(0.613198\pi\)
\(570\) 0 0
\(571\) 18.2191 0.762444 0.381222 0.924483i \(-0.375503\pi\)
0.381222 + 0.924483i \(0.375503\pi\)
\(572\) 0 0
\(573\) 14.6319 0.611254
\(574\) 0 0
\(575\) 11.1788 0.466188
\(576\) 0 0
\(577\) 26.5487 1.10524 0.552619 0.833434i \(-0.313628\pi\)
0.552619 + 0.833434i \(0.313628\pi\)
\(578\) 0 0
\(579\) 5.90206 0.245281
\(580\) 0 0
\(581\) −1.92591 −0.0799003
\(582\) 0 0
\(583\) −39.6971 −1.64409
\(584\) 0 0
\(585\) 19.7900 0.818217
\(586\) 0 0
\(587\) −6.48638 −0.267722 −0.133861 0.991000i \(-0.542738\pi\)
−0.133861 + 0.991000i \(0.542738\pi\)
\(588\) 0 0
\(589\) −32.1837 −1.32610
\(590\) 0 0
\(591\) −3.58043 −0.147279
\(592\) 0 0
\(593\) 37.2027 1.52773 0.763866 0.645375i \(-0.223299\pi\)
0.763866 + 0.645375i \(0.223299\pi\)
\(594\) 0 0
\(595\) 74.9427 3.07235
\(596\) 0 0
\(597\) 0.206157 0.00843744
\(598\) 0 0
\(599\) −47.8509 −1.95514 −0.977568 0.210618i \(-0.932452\pi\)
−0.977568 + 0.210618i \(0.932452\pi\)
\(600\) 0 0
\(601\) 18.4176 0.751270 0.375635 0.926768i \(-0.377425\pi\)
0.375635 + 0.926768i \(0.377425\pi\)
\(602\) 0 0
\(603\) −15.2771 −0.622131
\(604\) 0 0
\(605\) 2.91257 0.118413
\(606\) 0 0
\(607\) 16.4814 0.668959 0.334480 0.942403i \(-0.391439\pi\)
0.334480 + 0.942403i \(0.391439\pi\)
\(608\) 0 0
\(609\) −4.04331 −0.163843
\(610\) 0 0
\(611\) 21.0674 0.852295
\(612\) 0 0
\(613\) −19.2142 −0.776056 −0.388028 0.921648i \(-0.626844\pi\)
−0.388028 + 0.921648i \(0.626844\pi\)
\(614\) 0 0
\(615\) 15.3342 0.618336
\(616\) 0 0
\(617\) −5.92586 −0.238566 −0.119283 0.992860i \(-0.538060\pi\)
−0.119283 + 0.992860i \(0.538060\pi\)
\(618\) 0 0
\(619\) 43.0096 1.72870 0.864351 0.502889i \(-0.167729\pi\)
0.864351 + 0.502889i \(0.167729\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 36.1193 1.44709
\(624\) 0 0
\(625\) 44.0717 1.76287
\(626\) 0 0
\(627\) 26.1305 1.04355
\(628\) 0 0
\(629\) 14.9936 0.597832
\(630\) 0 0
\(631\) −29.8343 −1.18769 −0.593843 0.804581i \(-0.702390\pi\)
−0.593843 + 0.804581i \(0.702390\pi\)
\(632\) 0 0
\(633\) 26.7071 1.06151
\(634\) 0 0
\(635\) −79.4256 −3.15191
\(636\) 0 0
\(637\) 45.9948 1.82238
\(638\) 0 0
\(639\) −7.01655 −0.277570
\(640\) 0 0
\(641\) 34.1137 1.34741 0.673705 0.739000i \(-0.264702\pi\)
0.673705 + 0.739000i \(0.264702\pi\)
\(642\) 0 0
\(643\) 26.9368 1.06229 0.531143 0.847282i \(-0.321763\pi\)
0.531143 + 0.847282i \(0.321763\pi\)
\(644\) 0 0
\(645\) 19.6259 0.772768
\(646\) 0 0
\(647\) 3.38305 0.133001 0.0665006 0.997786i \(-0.478817\pi\)
0.0665006 + 0.997786i \(0.478817\pi\)
\(648\) 0 0
\(649\) −16.1164 −0.632625
\(650\) 0 0
\(651\) 17.0516 0.668305
\(652\) 0 0
\(653\) 20.1857 0.789929 0.394964 0.918696i \(-0.370757\pi\)
0.394964 + 0.918696i \(0.370757\pi\)
\(654\) 0 0
\(655\) 8.03908 0.314113
\(656\) 0 0
\(657\) −11.3905 −0.444386
\(658\) 0 0
\(659\) 18.3517 0.714881 0.357441 0.933936i \(-0.383649\pi\)
0.357441 + 0.933936i \(0.383649\pi\)
\(660\) 0 0
\(661\) 29.7589 1.15749 0.578744 0.815509i \(-0.303543\pi\)
0.578744 + 0.815509i \(0.303543\pi\)
\(662\) 0 0
\(663\) −22.6721 −0.880513
\(664\) 0 0
\(665\) 124.113 4.81290
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −6.90704 −0.267041
\(670\) 0 0
\(671\) 36.1857 1.39693
\(672\) 0 0
\(673\) −5.25389 −0.202523 −0.101261 0.994860i \(-0.532288\pi\)
−0.101261 + 0.994860i \(0.532288\pi\)
\(674\) 0 0
\(675\) −11.1788 −0.430272
\(676\) 0 0
\(677\) 34.9045 1.34149 0.670745 0.741688i \(-0.265975\pi\)
0.670745 + 0.741688i \(0.265975\pi\)
\(678\) 0 0
\(679\) 60.3761 2.31702
\(680\) 0 0
\(681\) 0.0517430 0.00198280
\(682\) 0 0
\(683\) −9.35781 −0.358067 −0.179033 0.983843i \(-0.557297\pi\)
−0.179033 + 0.983843i \(0.557297\pi\)
\(684\) 0 0
\(685\) −15.4885 −0.591785
\(686\) 0 0
\(687\) −3.67060 −0.140042
\(688\) 0 0
\(689\) 57.0417 2.17312
\(690\) 0 0
\(691\) 4.96336 0.188815 0.0944076 0.995534i \(-0.469904\pi\)
0.0944076 + 0.995534i \(0.469904\pi\)
\(692\) 0 0
\(693\) −13.8445 −0.525909
\(694\) 0 0
\(695\) 22.7218 0.861888
\(696\) 0 0
\(697\) −17.5674 −0.665413
\(698\) 0 0
\(699\) −18.6446 −0.705202
\(700\) 0 0
\(701\) 18.6860 0.705762 0.352881 0.935668i \(-0.385202\pi\)
0.352881 + 0.935668i \(0.385202\pi\)
\(702\) 0 0
\(703\) 24.8309 0.936517
\(704\) 0 0
\(705\) −17.2231 −0.648659
\(706\) 0 0
\(707\) −41.3851 −1.55645
\(708\) 0 0
\(709\) 33.8457 1.27110 0.635551 0.772059i \(-0.280773\pi\)
0.635551 + 0.772059i \(0.280773\pi\)
\(710\) 0 0
\(711\) −5.09508 −0.191080
\(712\) 0 0
\(713\) −4.21724 −0.157937
\(714\) 0 0
\(715\) −67.7620 −2.53416
\(716\) 0 0
\(717\) 17.1647 0.641029
\(718\) 0 0
\(719\) 9.94275 0.370802 0.185401 0.982663i \(-0.440642\pi\)
0.185401 + 0.982663i \(0.440642\pi\)
\(720\) 0 0
\(721\) −45.2357 −1.68467
\(722\) 0 0
\(723\) 15.5050 0.576635
\(724\) 0 0
\(725\) 11.1788 0.415170
\(726\) 0 0
\(727\) −5.49962 −0.203970 −0.101985 0.994786i \(-0.532519\pi\)
−0.101985 + 0.994786i \(0.532519\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −22.4841 −0.831603
\(732\) 0 0
\(733\) 35.7108 1.31901 0.659504 0.751701i \(-0.270766\pi\)
0.659504 + 0.751701i \(0.270766\pi\)
\(734\) 0 0
\(735\) −37.6018 −1.38696
\(736\) 0 0
\(737\) 52.3095 1.92685
\(738\) 0 0
\(739\) 40.9990 1.50817 0.754087 0.656775i \(-0.228080\pi\)
0.754087 + 0.656775i \(0.228080\pi\)
\(740\) 0 0
\(741\) −37.5475 −1.37934
\(742\) 0 0
\(743\) −3.54115 −0.129912 −0.0649562 0.997888i \(-0.520691\pi\)
−0.0649562 + 0.997888i \(0.520691\pi\)
\(744\) 0 0
\(745\) −60.0004 −2.19824
\(746\) 0 0
\(747\) −0.476321 −0.0174277
\(748\) 0 0
\(749\) 63.3782 2.31579
\(750\) 0 0
\(751\) 0.915056 0.0333909 0.0166954 0.999861i \(-0.494685\pi\)
0.0166954 + 0.999861i \(0.494685\pi\)
\(752\) 0 0
\(753\) 2.04662 0.0745829
\(754\) 0 0
\(755\) −23.7789 −0.865401
\(756\) 0 0
\(757\) 2.75926 0.100287 0.0501435 0.998742i \(-0.484032\pi\)
0.0501435 + 0.998742i \(0.484032\pi\)
\(758\) 0 0
\(759\) 3.42405 0.124285
\(760\) 0 0
\(761\) −23.4124 −0.848698 −0.424349 0.905499i \(-0.639497\pi\)
−0.424349 + 0.905499i \(0.639497\pi\)
\(762\) 0 0
\(763\) −65.9346 −2.38699
\(764\) 0 0
\(765\) 18.5350 0.670134
\(766\) 0 0
\(767\) 23.1580 0.836189
\(768\) 0 0
\(769\) −11.9546 −0.431095 −0.215547 0.976493i \(-0.569154\pi\)
−0.215547 + 0.976493i \(0.569154\pi\)
\(770\) 0 0
\(771\) −25.5277 −0.919359
\(772\) 0 0
\(773\) −24.0495 −0.864999 −0.432499 0.901634i \(-0.642368\pi\)
−0.432499 + 0.901634i \(0.642368\pi\)
\(774\) 0 0
\(775\) −47.1437 −1.69345
\(776\) 0 0
\(777\) −13.1560 −0.471968
\(778\) 0 0
\(779\) −29.0935 −1.04238
\(780\) 0 0
\(781\) 24.0250 0.859682
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) −68.0536 −2.42894
\(786\) 0 0
\(787\) −37.7208 −1.34460 −0.672300 0.740279i \(-0.734693\pi\)
−0.672300 + 0.740279i \(0.734693\pi\)
\(788\) 0 0
\(789\) 16.2098 0.577084
\(790\) 0 0
\(791\) 61.2747 2.17868
\(792\) 0 0
\(793\) −51.9961 −1.84643
\(794\) 0 0
\(795\) −46.6329 −1.65390
\(796\) 0 0
\(797\) −20.6036 −0.729817 −0.364908 0.931043i \(-0.618900\pi\)
−0.364908 + 0.931043i \(0.618900\pi\)
\(798\) 0 0
\(799\) 19.7313 0.698044
\(800\) 0 0
\(801\) 8.93310 0.315636
\(802\) 0 0
\(803\) 39.0017 1.37634
\(804\) 0 0
\(805\) 16.2634 0.573208
\(806\) 0 0
\(807\) 16.0182 0.563867
\(808\) 0 0
\(809\) −45.8191 −1.61091 −0.805457 0.592654i \(-0.798080\pi\)
−0.805457 + 0.592654i \(0.798080\pi\)
\(810\) 0 0
\(811\) 41.2866 1.44977 0.724884 0.688871i \(-0.241893\pi\)
0.724884 + 0.688871i \(0.241893\pi\)
\(812\) 0 0
\(813\) 3.64736 0.127918
\(814\) 0 0
\(815\) 48.4985 1.69883
\(816\) 0 0
\(817\) −37.2360 −1.30272
\(818\) 0 0
\(819\) 19.8935 0.695134
\(820\) 0 0
\(821\) 50.1840 1.75144 0.875718 0.482824i \(-0.160389\pi\)
0.875718 + 0.482824i \(0.160389\pi\)
\(822\) 0 0
\(823\) −47.5287 −1.65675 −0.828374 0.560176i \(-0.810734\pi\)
−0.828374 + 0.560176i \(0.810734\pi\)
\(824\) 0 0
\(825\) 38.2768 1.33263
\(826\) 0 0
\(827\) −14.5045 −0.504370 −0.252185 0.967679i \(-0.581149\pi\)
−0.252185 + 0.967679i \(0.581149\pi\)
\(828\) 0 0
\(829\) 36.4074 1.26448 0.632240 0.774772i \(-0.282136\pi\)
0.632240 + 0.774772i \(0.282136\pi\)
\(830\) 0 0
\(831\) 8.19668 0.284340
\(832\) 0 0
\(833\) 43.0779 1.49256
\(834\) 0 0
\(835\) −33.8369 −1.17097
\(836\) 0 0
\(837\) 4.21724 0.145769
\(838\) 0 0
\(839\) 5.28119 0.182327 0.0911635 0.995836i \(-0.470941\pi\)
0.0911635 + 0.995836i \(0.470941\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 30.6032 1.05403
\(844\) 0 0
\(845\) 45.0790 1.55077
\(846\) 0 0
\(847\) 2.92779 0.100600
\(848\) 0 0
\(849\) 18.1893 0.624256
\(850\) 0 0
\(851\) 3.25376 0.111538
\(852\) 0 0
\(853\) −34.7810 −1.19088 −0.595439 0.803401i \(-0.703022\pi\)
−0.595439 + 0.803401i \(0.703022\pi\)
\(854\) 0 0
\(855\) 30.6959 1.04978
\(856\) 0 0
\(857\) −8.18147 −0.279474 −0.139737 0.990189i \(-0.544626\pi\)
−0.139737 + 0.990189i \(0.544626\pi\)
\(858\) 0 0
\(859\) 3.39184 0.115728 0.0578641 0.998324i \(-0.481571\pi\)
0.0578641 + 0.998324i \(0.481571\pi\)
\(860\) 0 0
\(861\) 15.4144 0.525320
\(862\) 0 0
\(863\) 4.40961 0.150105 0.0750525 0.997180i \(-0.476088\pi\)
0.0750525 + 0.997180i \(0.476088\pi\)
\(864\) 0 0
\(865\) −81.4274 −2.76861
\(866\) 0 0
\(867\) −4.23430 −0.143804
\(868\) 0 0
\(869\) 17.4458 0.591808
\(870\) 0 0
\(871\) −75.1647 −2.54686
\(872\) 0 0
\(873\) 14.9323 0.505383
\(874\) 0 0
\(875\) 100.488 3.39712
\(876\) 0 0
\(877\) 27.5955 0.931835 0.465917 0.884828i \(-0.345724\pi\)
0.465917 + 0.884828i \(0.345724\pi\)
\(878\) 0 0
\(879\) −25.5008 −0.860121
\(880\) 0 0
\(881\) −1.53612 −0.0517533 −0.0258767 0.999665i \(-0.508238\pi\)
−0.0258767 + 0.999665i \(0.508238\pi\)
\(882\) 0 0
\(883\) −5.50815 −0.185364 −0.0926820 0.995696i \(-0.529544\pi\)
−0.0926820 + 0.995696i \(0.529544\pi\)
\(884\) 0 0
\(885\) −18.9322 −0.636400
\(886\) 0 0
\(887\) 17.7856 0.597182 0.298591 0.954381i \(-0.403483\pi\)
0.298591 + 0.954381i \(0.403483\pi\)
\(888\) 0 0
\(889\) −79.8407 −2.67777
\(890\) 0 0
\(891\) −3.42405 −0.114710
\(892\) 0 0
\(893\) 32.6772 1.09350
\(894\) 0 0
\(895\) 28.2324 0.943704
\(896\) 0 0
\(897\) −4.92009 −0.164277
\(898\) 0 0
\(899\) −4.21724 −0.140653
\(900\) 0 0
\(901\) 53.4242 1.77982
\(902\) 0 0
\(903\) 19.7284 0.656521
\(904\) 0 0
\(905\) −54.3363 −1.80620
\(906\) 0 0
\(907\) 28.5259 0.947186 0.473593 0.880744i \(-0.342957\pi\)
0.473593 + 0.880744i \(0.342957\pi\)
\(908\) 0 0
\(909\) −10.2354 −0.339488
\(910\) 0 0
\(911\) −49.1530 −1.62851 −0.814255 0.580507i \(-0.802854\pi\)
−0.814255 + 0.580507i \(0.802854\pi\)
\(912\) 0 0
\(913\) 1.63095 0.0539764
\(914\) 0 0
\(915\) 42.5080 1.40527
\(916\) 0 0
\(917\) 8.08110 0.266861
\(918\) 0 0
\(919\) 43.9059 1.44832 0.724162 0.689630i \(-0.242227\pi\)
0.724162 + 0.689630i \(0.242227\pi\)
\(920\) 0 0
\(921\) −6.49238 −0.213931
\(922\) 0 0
\(923\) −34.5221 −1.13631
\(924\) 0 0
\(925\) 36.3732 1.19594
\(926\) 0 0
\(927\) −11.1878 −0.367455
\(928\) 0 0
\(929\) 13.3119 0.436749 0.218375 0.975865i \(-0.429925\pi\)
0.218375 + 0.975865i \(0.429925\pi\)
\(930\) 0 0
\(931\) 71.3416 2.33813
\(932\) 0 0
\(933\) 31.0601 1.01686
\(934\) 0 0
\(935\) −63.4647 −2.07552
\(936\) 0 0
\(937\) −16.2816 −0.531896 −0.265948 0.963987i \(-0.585685\pi\)
−0.265948 + 0.963987i \(0.585685\pi\)
\(938\) 0 0
\(939\) −21.1142 −0.689035
\(940\) 0 0
\(941\) −17.7132 −0.577434 −0.288717 0.957415i \(-0.593229\pi\)
−0.288717 + 0.957415i \(0.593229\pi\)
\(942\) 0 0
\(943\) −3.81231 −0.124146
\(944\) 0 0
\(945\) −16.2634 −0.529047
\(946\) 0 0
\(947\) 15.9640 0.518761 0.259381 0.965775i \(-0.416482\pi\)
0.259381 + 0.965775i \(0.416482\pi\)
\(948\) 0 0
\(949\) −56.0424 −1.81921
\(950\) 0 0
\(951\) −14.5266 −0.471058
\(952\) 0 0
\(953\) 54.4117 1.76257 0.881284 0.472587i \(-0.156680\pi\)
0.881284 + 0.472587i \(0.156680\pi\)
\(954\) 0 0
\(955\) −58.8535 −1.90446
\(956\) 0 0
\(957\) 3.42405 0.110684
\(958\) 0 0
\(959\) −15.5694 −0.502763
\(960\) 0 0
\(961\) −13.2149 −0.426288
\(962\) 0 0
\(963\) 15.6748 0.505114
\(964\) 0 0
\(965\) −23.7398 −0.764211
\(966\) 0 0
\(967\) 6.88545 0.221421 0.110711 0.993853i \(-0.464687\pi\)
0.110711 + 0.993853i \(0.464687\pi\)
\(968\) 0 0
\(969\) −35.1663 −1.12970
\(970\) 0 0
\(971\) −3.07284 −0.0986123 −0.0493061 0.998784i \(-0.515701\pi\)
−0.0493061 + 0.998784i \(0.515701\pi\)
\(972\) 0 0
\(973\) 22.8406 0.732236
\(974\) 0 0
\(975\) −55.0008 −1.76143
\(976\) 0 0
\(977\) 39.3904 1.26021 0.630106 0.776509i \(-0.283012\pi\)
0.630106 + 0.776509i \(0.283012\pi\)
\(978\) 0 0
\(979\) −30.5874 −0.977577
\(980\) 0 0
\(981\) −16.3071 −0.520645
\(982\) 0 0
\(983\) 45.5701 1.45346 0.726731 0.686922i \(-0.241039\pi\)
0.726731 + 0.686922i \(0.241039\pi\)
\(984\) 0 0
\(985\) 14.4015 0.458871
\(986\) 0 0
\(987\) −17.3131 −0.551082
\(988\) 0 0
\(989\) −4.87928 −0.155152
\(990\) 0 0
\(991\) −11.3994 −0.362114 −0.181057 0.983473i \(-0.557952\pi\)
−0.181057 + 0.983473i \(0.557952\pi\)
\(992\) 0 0
\(993\) 17.6417 0.559844
\(994\) 0 0
\(995\) −0.829222 −0.0262881
\(996\) 0 0
\(997\) 36.4117 1.15317 0.576585 0.817037i \(-0.304385\pi\)
0.576585 + 0.817037i \(0.304385\pi\)
\(998\) 0 0
\(999\) −3.25376 −0.102944
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.i.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.15 16 1.1 even 1 trivial