Properties

Label 668.2.a.b.1.3
Level $668$
Weight $2$
Character 668.1
Self dual yes
Analytic conductor $5.334$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.826865.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1\)
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 1.3
Root \(-0.873948\) of defining polynomial
Character \(\chi\) \(=\) 668.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.36227 q^{3} +2.00000 q^{5} -1.42676 q^{7} -1.14423 q^{9} +O(q^{10})\) \(q+1.36227 q^{3} +2.00000 q^{5} -1.42676 q^{7} -1.14423 q^{9} +2.19055 q^{11} +1.74790 q^{13} +2.72453 q^{15} +3.26510 q^{17} +5.89213 q^{19} -1.94363 q^{21} +6.12900 q^{23} -1.00000 q^{25} -5.64555 q^{27} -0.321133 q^{29} +1.75308 q^{31} +2.98411 q^{33} -2.85353 q^{35} -2.47243 q^{37} +2.38110 q^{39} -5.98963 q^{41} -3.49579 q^{43} -2.28846 q^{45} +2.39634 q^{47} -4.96434 q^{49} +4.44793 q^{51} -1.51720 q^{53} +4.38110 q^{55} +8.02665 q^{57} +5.65526 q^{59} -4.51463 q^{61} +1.63255 q^{63} +3.49579 q^{65} -9.61442 q^{67} +8.34932 q^{69} -6.66956 q^{71} +2.86390 q^{73} -1.36227 q^{75} -3.12540 q^{77} +10.4997 q^{79} -4.25804 q^{81} -7.24369 q^{83} +6.53020 q^{85} -0.437468 q^{87} -0.256634 q^{89} -2.49384 q^{91} +2.38816 q^{93} +11.7843 q^{95} +5.83544 q^{97} -2.50650 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + 5q^{11} - 4q^{13} + 6q^{15} - 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} - 5q^{25} + 12q^{27} - 5q^{29} + 9q^{31} - 8q^{33} + 18q^{35} + 8q^{37} - 4q^{41} + 8q^{43} + 12q^{45} + 13q^{47} + 14q^{49} - 2q^{53} + 10q^{55} - 12q^{57} + 4q^{59} + 11q^{61} - q^{63} - 8q^{65} + 28q^{67} - 16q^{69} + 2q^{71} + 8q^{73} - 3q^{75} - 12q^{77} - 10q^{79} - 15q^{81} + 2q^{83} - 4q^{85} + 4q^{87} - 17q^{89} - 12q^{91} - 12q^{93} + 10q^{95} - 27q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) 1.36227 0.786504 0.393252 0.919431i \(-0.371350\pi\)
0.393252 + 0.919431i \(0.371350\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.42676 −0.539266 −0.269633 0.962963i \(-0.586902\pi\)
−0.269633 + 0.962963i \(0.586902\pi\)
\(8\) 0 0
\(9\) −1.14423 −0.381411
\(10\) 0 0
\(11\) 2.19055 0.660476 0.330238 0.943898i \(-0.392871\pi\)
0.330238 + 0.943898i \(0.392871\pi\)
\(12\) 0 0
\(13\) 1.74790 0.484779 0.242390 0.970179i \(-0.422069\pi\)
0.242390 + 0.970179i \(0.422069\pi\)
\(14\) 0 0
\(15\) 2.72453 0.703471
\(16\) 0 0
\(17\) 3.26510 0.791903 0.395951 0.918271i \(-0.370415\pi\)
0.395951 + 0.918271i \(0.370415\pi\)
\(18\) 0 0
\(19\) 5.89213 1.35175 0.675874 0.737018i \(-0.263767\pi\)
0.675874 + 0.737018i \(0.263767\pi\)
\(20\) 0 0
\(21\) −1.94363 −0.424135
\(22\) 0 0
\(23\) 6.12900 1.27798 0.638992 0.769213i \(-0.279352\pi\)
0.638992 + 0.769213i \(0.279352\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.64555 −1.08649
\(28\) 0 0
\(29\) −0.321133 −0.0596329 −0.0298164 0.999555i \(-0.509492\pi\)
−0.0298164 + 0.999555i \(0.509492\pi\)
\(30\) 0 0
\(31\) 1.75308 0.314863 0.157431 0.987530i \(-0.449679\pi\)
0.157431 + 0.987530i \(0.449679\pi\)
\(32\) 0 0
\(33\) 2.98411 0.519467
\(34\) 0 0
\(35\) −2.85353 −0.482334
\(36\) 0 0
\(37\) −2.47243 −0.406465 −0.203232 0.979131i \(-0.565145\pi\)
−0.203232 + 0.979131i \(0.565145\pi\)
\(38\) 0 0
\(39\) 2.38110 0.381281
\(40\) 0 0
\(41\) −5.98963 −0.935423 −0.467712 0.883881i \(-0.654921\pi\)
−0.467712 + 0.883881i \(0.654921\pi\)
\(42\) 0 0
\(43\) −3.49579 −0.533104 −0.266552 0.963821i \(-0.585884\pi\)
−0.266552 + 0.963821i \(0.585884\pi\)
\(44\) 0 0
\(45\) −2.28846 −0.341144
\(46\) 0 0
\(47\) 2.39634 0.349541 0.174771 0.984609i \(-0.444082\pi\)
0.174771 + 0.984609i \(0.444082\pi\)
\(48\) 0 0
\(49\) −4.96434 −0.709192
\(50\) 0 0
\(51\) 4.44793 0.622835
\(52\) 0 0
\(53\) −1.51720 −0.208404 −0.104202 0.994556i \(-0.533229\pi\)
−0.104202 + 0.994556i \(0.533229\pi\)
\(54\) 0 0
\(55\) 4.38110 0.590747
\(56\) 0 0
\(57\) 8.02665 1.06316
\(58\) 0 0
\(59\) 5.65526 0.736252 0.368126 0.929776i \(-0.379999\pi\)
0.368126 + 0.929776i \(0.379999\pi\)
\(60\) 0 0
\(61\) −4.51463 −0.578039 −0.289019 0.957323i \(-0.593329\pi\)
−0.289019 + 0.957323i \(0.593329\pi\)
\(62\) 0 0
\(63\) 1.63255 0.205682
\(64\) 0 0
\(65\) 3.49579 0.433600
\(66\) 0 0
\(67\) −9.61442 −1.17459 −0.587294 0.809374i \(-0.699807\pi\)
−0.587294 + 0.809374i \(0.699807\pi\)
\(68\) 0 0
\(69\) 8.34932 1.00514
\(70\) 0 0
\(71\) −6.66956 −0.791532 −0.395766 0.918351i \(-0.629521\pi\)
−0.395766 + 0.918351i \(0.629521\pi\)
\(72\) 0 0
\(73\) 2.86390 0.335194 0.167597 0.985856i \(-0.446399\pi\)
0.167597 + 0.985856i \(0.446399\pi\)
\(74\) 0 0
\(75\) −1.36227 −0.157301
\(76\) 0 0
\(77\) −3.12540 −0.356172
\(78\) 0 0
\(79\) 10.4997 1.18131 0.590656 0.806924i \(-0.298869\pi\)
0.590656 + 0.806924i \(0.298869\pi\)
\(80\) 0 0
\(81\) −4.25804 −0.473115
\(82\) 0 0
\(83\) −7.24369 −0.795098 −0.397549 0.917581i \(-0.630139\pi\)
−0.397549 + 0.917581i \(0.630139\pi\)
\(84\) 0 0
\(85\) 6.53020 0.708299
\(86\) 0 0
\(87\) −0.437468 −0.0469015
\(88\) 0 0
\(89\) −0.256634 −0.0272032 −0.0136016 0.999907i \(-0.504330\pi\)
−0.0136016 + 0.999907i \(0.504330\pi\)
\(90\) 0 0
\(91\) −2.49384 −0.261425
\(92\) 0 0
\(93\) 2.38816 0.247641
\(94\) 0 0
\(95\) 11.7843 1.20904
\(96\) 0 0
\(97\) 5.83544 0.592499 0.296250 0.955111i \(-0.404264\pi\)
0.296250 + 0.955111i \(0.404264\pi\)
\(98\) 0 0
\(99\) −2.50650 −0.251913
\(100\) 0 0
\(101\) −5.58843 −0.556069 −0.278035 0.960571i \(-0.589683\pi\)
−0.278035 + 0.960571i \(0.589683\pi\)
\(102\) 0 0
\(103\) 6.17359 0.608302 0.304151 0.952624i \(-0.401627\pi\)
0.304151 + 0.952624i \(0.401627\pi\)
\(104\) 0 0
\(105\) −3.88726 −0.379358
\(106\) 0 0
\(107\) −11.9570 −1.15592 −0.577962 0.816064i \(-0.696152\pi\)
−0.577962 + 0.816064i \(0.696152\pi\)
\(108\) 0 0
\(109\) −15.1784 −1.45382 −0.726911 0.686731i \(-0.759045\pi\)
−0.726911 + 0.686731i \(0.759045\pi\)
\(110\) 0 0
\(111\) −3.36810 −0.319686
\(112\) 0 0
\(113\) −0.710405 −0.0668293 −0.0334146 0.999442i \(-0.510638\pi\)
−0.0334146 + 0.999442i \(0.510638\pi\)
\(114\) 0 0
\(115\) 12.2580 1.14306
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −4.65853 −0.427046
\(120\) 0 0
\(121\) −6.20149 −0.563772
\(122\) 0 0
\(123\) −8.15947 −0.735714
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 11.5165 1.02192 0.510962 0.859603i \(-0.329289\pi\)
0.510962 + 0.859603i \(0.329289\pi\)
\(128\) 0 0
\(129\) −4.76220 −0.419288
\(130\) 0 0
\(131\) 8.02599 0.701234 0.350617 0.936519i \(-0.385972\pi\)
0.350617 + 0.936519i \(0.385972\pi\)
\(132\) 0 0
\(133\) −8.40668 −0.728951
\(134\) 0 0
\(135\) −11.2911 −0.971782
\(136\) 0 0
\(137\) −15.6656 −1.33840 −0.669201 0.743081i \(-0.733364\pi\)
−0.669201 + 0.743081i \(0.733364\pi\)
\(138\) 0 0
\(139\) −2.22163 −0.188436 −0.0942182 0.995552i \(-0.530035\pi\)
−0.0942182 + 0.995552i \(0.530035\pi\)
\(140\) 0 0
\(141\) 3.26445 0.274916
\(142\) 0 0
\(143\) 3.82886 0.320185
\(144\) 0 0
\(145\) −0.642266 −0.0533373
\(146\) 0 0
\(147\) −6.76276 −0.557783
\(148\) 0 0
\(149\) 6.02599 0.493668 0.246834 0.969058i \(-0.420610\pi\)
0.246834 + 0.969058i \(0.420610\pi\)
\(150\) 0 0
\(151\) −10.9684 −0.892596 −0.446298 0.894884i \(-0.647258\pi\)
−0.446298 + 0.894884i \(0.647258\pi\)
\(152\) 0 0
\(153\) −3.73603 −0.302040
\(154\) 0 0
\(155\) 3.50616 0.281622
\(156\) 0 0
\(157\) 20.0474 1.59996 0.799980 0.600027i \(-0.204843\pi\)
0.799980 + 0.600027i \(0.204843\pi\)
\(158\) 0 0
\(159\) −2.06683 −0.163910
\(160\) 0 0
\(161\) −8.74463 −0.689174
\(162\) 0 0
\(163\) 5.21770 0.408682 0.204341 0.978900i \(-0.434495\pi\)
0.204341 + 0.978900i \(0.434495\pi\)
\(164\) 0 0
\(165\) 5.96822 0.464625
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −9.94486 −0.764989
\(170\) 0 0
\(171\) −6.74196 −0.515571
\(172\) 0 0
\(173\) −21.5165 −1.63587 −0.817935 0.575311i \(-0.804881\pi\)
−0.817935 + 0.575311i \(0.804881\pi\)
\(174\) 0 0
\(175\) 1.42676 0.107853
\(176\) 0 0
\(177\) 7.70397 0.579066
\(178\) 0 0
\(179\) 10.1501 0.758656 0.379328 0.925262i \(-0.376155\pi\)
0.379328 + 0.925262i \(0.376155\pi\)
\(180\) 0 0
\(181\) −3.64003 −0.270561 −0.135281 0.990807i \(-0.543194\pi\)
−0.135281 + 0.990807i \(0.543194\pi\)
\(182\) 0 0
\(183\) −6.15012 −0.454630
\(184\) 0 0
\(185\) −4.94486 −0.363553
\(186\) 0 0
\(187\) 7.15236 0.523033
\(188\) 0 0
\(189\) 8.05486 0.585905
\(190\) 0 0
\(191\) −17.4861 −1.26525 −0.632626 0.774457i \(-0.718023\pi\)
−0.632626 + 0.774457i \(0.718023\pi\)
\(192\) 0 0
\(193\) 4.75052 0.341950 0.170975 0.985275i \(-0.445308\pi\)
0.170975 + 0.985275i \(0.445308\pi\)
\(194\) 0 0
\(195\) 4.76220 0.341028
\(196\) 0 0
\(197\) −2.67780 −0.190785 −0.0953927 0.995440i \(-0.530411\pi\)
−0.0953927 + 0.995440i \(0.530411\pi\)
\(198\) 0 0
\(199\) 25.6656 1.81939 0.909693 0.415281i \(-0.136317\pi\)
0.909693 + 0.415281i \(0.136317\pi\)
\(200\) 0 0
\(201\) −13.0974 −0.923819
\(202\) 0 0
\(203\) 0.458181 0.0321580
\(204\) 0 0
\(205\) −11.9793 −0.836668
\(206\) 0 0
\(207\) −7.01300 −0.487437
\(208\) 0 0
\(209\) 12.9070 0.892796
\(210\) 0 0
\(211\) 9.73234 0.670002 0.335001 0.942218i \(-0.391263\pi\)
0.335001 + 0.942218i \(0.391263\pi\)
\(212\) 0 0
\(213\) −9.08572 −0.622543
\(214\) 0 0
\(215\) −6.99159 −0.476822
\(216\) 0 0
\(217\) −2.50123 −0.169795
\(218\) 0 0
\(219\) 3.90139 0.263631
\(220\) 0 0
\(221\) 5.70706 0.383898
\(222\) 0 0
\(223\) 21.5995 1.44641 0.723205 0.690633i \(-0.242668\pi\)
0.723205 + 0.690633i \(0.242668\pi\)
\(224\) 0 0
\(225\) 1.14423 0.0762822
\(226\) 0 0
\(227\) −14.0401 −0.931875 −0.465938 0.884818i \(-0.654283\pi\)
−0.465938 + 0.884818i \(0.654283\pi\)
\(228\) 0 0
\(229\) 5.77020 0.381305 0.190653 0.981658i \(-0.438940\pi\)
0.190653 + 0.981658i \(0.438940\pi\)
\(230\) 0 0
\(231\) −4.25762 −0.280131
\(232\) 0 0
\(233\) −19.4705 −1.27556 −0.637778 0.770221i \(-0.720146\pi\)
−0.637778 + 0.770221i \(0.720146\pi\)
\(234\) 0 0
\(235\) 4.79267 0.312639
\(236\) 0 0
\(237\) 14.3034 0.929107
\(238\) 0 0
\(239\) −14.7014 −0.950954 −0.475477 0.879728i \(-0.657725\pi\)
−0.475477 + 0.879728i \(0.657725\pi\)
\(240\) 0 0
\(241\) −21.6813 −1.39661 −0.698306 0.715799i \(-0.746063\pi\)
−0.698306 + 0.715799i \(0.746063\pi\)
\(242\) 0 0
\(243\) 11.1361 0.714379
\(244\) 0 0
\(245\) −9.92869 −0.634321
\(246\) 0 0
\(247\) 10.2988 0.655299
\(248\) 0 0
\(249\) −9.86783 −0.625348
\(250\) 0 0
\(251\) −6.71130 −0.423613 −0.211807 0.977312i \(-0.567935\pi\)
−0.211807 + 0.977312i \(0.567935\pi\)
\(252\) 0 0
\(253\) 13.4259 0.844077
\(254\) 0 0
\(255\) 8.89586 0.557081
\(256\) 0 0
\(257\) −26.1000 −1.62807 −0.814037 0.580813i \(-0.802735\pi\)
−0.814037 + 0.580813i \(0.802735\pi\)
\(258\) 0 0
\(259\) 3.52757 0.219193
\(260\) 0 0
\(261\) 0.367451 0.0227446
\(262\) 0 0
\(263\) 16.1595 0.996439 0.498219 0.867051i \(-0.333987\pi\)
0.498219 + 0.867051i \(0.333987\pi\)
\(264\) 0 0
\(265\) −3.03440 −0.186402
\(266\) 0 0
\(267\) −0.349604 −0.0213954
\(268\) 0 0
\(269\) 10.9418 0.667131 0.333566 0.942727i \(-0.391748\pi\)
0.333566 + 0.942727i \(0.391748\pi\)
\(270\) 0 0
\(271\) 9.49579 0.576828 0.288414 0.957506i \(-0.406872\pi\)
0.288414 + 0.957506i \(0.406872\pi\)
\(272\) 0 0
\(273\) −3.39727 −0.205612
\(274\) 0 0
\(275\) −2.19055 −0.132095
\(276\) 0 0
\(277\) 1.86063 0.111795 0.0558973 0.998437i \(-0.482198\pi\)
0.0558973 + 0.998437i \(0.482198\pi\)
\(278\) 0 0
\(279\) −2.00593 −0.120092
\(280\) 0 0
\(281\) 4.01042 0.239242 0.119621 0.992820i \(-0.461832\pi\)
0.119621 + 0.992820i \(0.461832\pi\)
\(282\) 0 0
\(283\) −7.71234 −0.458451 −0.229225 0.973373i \(-0.573619\pi\)
−0.229225 + 0.973373i \(0.573619\pi\)
\(284\) 0 0
\(285\) 16.0533 0.950915
\(286\) 0 0
\(287\) 8.54579 0.504442
\(288\) 0 0
\(289\) −6.33913 −0.372890
\(290\) 0 0
\(291\) 7.94942 0.466003
\(292\) 0 0
\(293\) 9.05585 0.529048 0.264524 0.964379i \(-0.414785\pi\)
0.264524 + 0.964379i \(0.414785\pi\)
\(294\) 0 0
\(295\) 11.3105 0.658524
\(296\) 0 0
\(297\) −12.3668 −0.717597
\(298\) 0 0
\(299\) 10.7129 0.619540
\(300\) 0 0
\(301\) 4.98767 0.287485
\(302\) 0 0
\(303\) −7.61293 −0.437351
\(304\) 0 0
\(305\) −9.02926 −0.517014
\(306\) 0 0
\(307\) 12.1632 0.694192 0.347096 0.937830i \(-0.387168\pi\)
0.347096 + 0.937830i \(0.387168\pi\)
\(308\) 0 0
\(309\) 8.41008 0.478432
\(310\) 0 0
\(311\) 24.4952 1.38900 0.694499 0.719494i \(-0.255626\pi\)
0.694499 + 0.719494i \(0.255626\pi\)
\(312\) 0 0
\(313\) 22.5593 1.27513 0.637563 0.770398i \(-0.279943\pi\)
0.637563 + 0.770398i \(0.279943\pi\)
\(314\) 0 0
\(315\) 3.26510 0.183967
\(316\) 0 0
\(317\) 13.9546 0.783770 0.391885 0.920014i \(-0.371823\pi\)
0.391885 + 0.920014i \(0.371823\pi\)
\(318\) 0 0
\(319\) −0.703457 −0.0393861
\(320\) 0 0
\(321\) −16.2886 −0.909139
\(322\) 0 0
\(323\) 19.2384 1.07045
\(324\) 0 0
\(325\) −1.74790 −0.0969559
\(326\) 0 0
\(327\) −20.6770 −1.14344
\(328\) 0 0
\(329\) −3.41901 −0.188496
\(330\) 0 0
\(331\) −21.5826 −1.18629 −0.593145 0.805096i \(-0.702114\pi\)
−0.593145 + 0.805096i \(0.702114\pi\)
\(332\) 0 0
\(333\) 2.82903 0.155030
\(334\) 0 0
\(335\) −19.2288 −1.05058
\(336\) 0 0
\(337\) 16.7494 0.912399 0.456199 0.889878i \(-0.349210\pi\)
0.456199 + 0.889878i \(0.349210\pi\)
\(338\) 0 0
\(339\) −0.967760 −0.0525615
\(340\) 0 0
\(341\) 3.84021 0.207959
\(342\) 0 0
\(343\) 17.0703 0.921709
\(344\) 0 0
\(345\) 16.6986 0.899025
\(346\) 0 0
\(347\) −12.7273 −0.683239 −0.341620 0.939838i \(-0.610975\pi\)
−0.341620 + 0.939838i \(0.610975\pi\)
\(348\) 0 0
\(349\) 21.8665 1.17049 0.585244 0.810857i \(-0.300999\pi\)
0.585244 + 0.810857i \(0.300999\pi\)
\(350\) 0 0
\(351\) −9.86783 −0.526706
\(352\) 0 0
\(353\) 27.7352 1.47620 0.738099 0.674692i \(-0.235724\pi\)
0.738099 + 0.674692i \(0.235724\pi\)
\(354\) 0 0
\(355\) −13.3391 −0.707967
\(356\) 0 0
\(357\) −6.34615 −0.335874
\(358\) 0 0
\(359\) −8.30402 −0.438269 −0.219135 0.975695i \(-0.570323\pi\)
−0.219135 + 0.975695i \(0.570323\pi\)
\(360\) 0 0
\(361\) 15.7172 0.827220
\(362\) 0 0
\(363\) −8.44808 −0.443409
\(364\) 0 0
\(365\) 5.72780 0.299807
\(366\) 0 0
\(367\) 12.4528 0.650031 0.325015 0.945709i \(-0.394631\pi\)
0.325015 + 0.945709i \(0.394631\pi\)
\(368\) 0 0
\(369\) 6.85353 0.356780
\(370\) 0 0
\(371\) 2.16469 0.112385
\(372\) 0 0
\(373\) 17.6189 0.912272 0.456136 0.889910i \(-0.349233\pi\)
0.456136 + 0.889910i \(0.349233\pi\)
\(374\) 0 0
\(375\) −16.3472 −0.844165
\(376\) 0 0
\(377\) −0.561307 −0.0289088
\(378\) 0 0
\(379\) 12.1749 0.625383 0.312691 0.949855i \(-0.398769\pi\)
0.312691 + 0.949855i \(0.398769\pi\)
\(380\) 0 0
\(381\) 15.6885 0.803748
\(382\) 0 0
\(383\) 16.9672 0.866981 0.433491 0.901158i \(-0.357282\pi\)
0.433491 + 0.901158i \(0.357282\pi\)
\(384\) 0 0
\(385\) −6.25080 −0.318570
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 14.9358 0.757275 0.378637 0.925545i \(-0.376393\pi\)
0.378637 + 0.925545i \(0.376393\pi\)
\(390\) 0 0
\(391\) 20.0118 1.01204
\(392\) 0 0
\(393\) 10.9335 0.551524
\(394\) 0 0
\(395\) 20.9995 1.05660
\(396\) 0 0
\(397\) 27.3638 1.37335 0.686674 0.726966i \(-0.259070\pi\)
0.686674 + 0.726966i \(0.259070\pi\)
\(398\) 0 0
\(399\) −11.4521 −0.573324
\(400\) 0 0
\(401\) 16.3411 0.816035 0.408017 0.912974i \(-0.366220\pi\)
0.408017 + 0.912974i \(0.366220\pi\)
\(402\) 0 0
\(403\) 3.06421 0.152639
\(404\) 0 0
\(405\) −8.51607 −0.423167
\(406\) 0 0
\(407\) −5.41598 −0.268460
\(408\) 0 0
\(409\) −19.2236 −0.950544 −0.475272 0.879839i \(-0.657650\pi\)
−0.475272 + 0.879839i \(0.657650\pi\)
\(410\) 0 0
\(411\) −21.3407 −1.05266
\(412\) 0 0
\(413\) −8.06872 −0.397036
\(414\) 0 0
\(415\) −14.4874 −0.711158
\(416\) 0 0
\(417\) −3.02645 −0.148206
\(418\) 0 0
\(419\) 1.15541 0.0564456 0.0282228 0.999602i \(-0.491015\pi\)
0.0282228 + 0.999602i \(0.491015\pi\)
\(420\) 0 0
\(421\) 4.80899 0.234376 0.117188 0.993110i \(-0.462612\pi\)
0.117188 + 0.993110i \(0.462612\pi\)
\(422\) 0 0
\(423\) −2.74196 −0.133319
\(424\) 0 0
\(425\) −3.26510 −0.158381
\(426\) 0 0
\(427\) 6.44131 0.311717
\(428\) 0 0
\(429\) 5.21592 0.251827
\(430\) 0 0
\(431\) −34.3708 −1.65558 −0.827791 0.561036i \(-0.810403\pi\)
−0.827791 + 0.561036i \(0.810403\pi\)
\(432\) 0 0
\(433\) −20.0626 −0.964145 −0.482072 0.876131i \(-0.660116\pi\)
−0.482072 + 0.876131i \(0.660116\pi\)
\(434\) 0 0
\(435\) −0.874936 −0.0419500
\(436\) 0 0
\(437\) 36.1128 1.72751
\(438\) 0 0
\(439\) 1.90270 0.0908108 0.0454054 0.998969i \(-0.485542\pi\)
0.0454054 + 0.998969i \(0.485542\pi\)
\(440\) 0 0
\(441\) 5.68036 0.270493
\(442\) 0 0
\(443\) −13.8782 −0.659373 −0.329687 0.944090i \(-0.606943\pi\)
−0.329687 + 0.944090i \(0.606943\pi\)
\(444\) 0 0
\(445\) −0.513269 −0.0243313
\(446\) 0 0
\(447\) 8.20900 0.388272
\(448\) 0 0
\(449\) 14.6316 0.690509 0.345254 0.938509i \(-0.387793\pi\)
0.345254 + 0.938509i \(0.387793\pi\)
\(450\) 0 0
\(451\) −13.1206 −0.617824
\(452\) 0 0
\(453\) −14.9419 −0.702030
\(454\) 0 0
\(455\) −4.98767 −0.233826
\(456\) 0 0
\(457\) −15.1270 −0.707613 −0.353807 0.935319i \(-0.615113\pi\)
−0.353807 + 0.935319i \(0.615113\pi\)
\(458\) 0 0
\(459\) −18.4333 −0.860391
\(460\) 0 0
\(461\) −0.617968 −0.0287816 −0.0143908 0.999896i \(-0.504581\pi\)
−0.0143908 + 0.999896i \(0.504581\pi\)
\(462\) 0 0
\(463\) −6.12321 −0.284570 −0.142285 0.989826i \(-0.545445\pi\)
−0.142285 + 0.989826i \(0.545445\pi\)
\(464\) 0 0
\(465\) 4.77633 0.221497
\(466\) 0 0
\(467\) −22.0319 −1.01952 −0.509758 0.860318i \(-0.670265\pi\)
−0.509758 + 0.860318i \(0.670265\pi\)
\(468\) 0 0
\(469\) 13.7175 0.633416
\(470\) 0 0
\(471\) 27.3099 1.25838
\(472\) 0 0
\(473\) −7.65771 −0.352102
\(474\) 0 0
\(475\) −5.89213 −0.270349
\(476\) 0 0
\(477\) 1.73603 0.0794874
\(478\) 0 0
\(479\) −6.88989 −0.314807 −0.157404 0.987534i \(-0.550312\pi\)
−0.157404 + 0.987534i \(0.550312\pi\)
\(480\) 0 0
\(481\) −4.32155 −0.197046
\(482\) 0 0
\(483\) −11.9125 −0.542038
\(484\) 0 0
\(485\) 11.6709 0.529948
\(486\) 0 0
\(487\) −20.9486 −0.949272 −0.474636 0.880182i \(-0.657420\pi\)
−0.474636 + 0.880182i \(0.657420\pi\)
\(488\) 0 0
\(489\) 7.10789 0.321430
\(490\) 0 0
\(491\) 13.3184 0.601051 0.300525 0.953774i \(-0.402838\pi\)
0.300525 + 0.953774i \(0.402838\pi\)
\(492\) 0 0
\(493\) −1.04853 −0.0472234
\(494\) 0 0
\(495\) −5.01300 −0.225317
\(496\) 0 0
\(497\) 9.51589 0.426846
\(498\) 0 0
\(499\) 15.2496 0.682665 0.341333 0.939943i \(-0.389122\pi\)
0.341333 + 0.939943i \(0.389122\pi\)
\(500\) 0 0
\(501\) −1.36227 −0.0608615
\(502\) 0 0
\(503\) −26.7387 −1.19222 −0.596110 0.802903i \(-0.703288\pi\)
−0.596110 + 0.802903i \(0.703288\pi\)
\(504\) 0 0
\(505\) −11.1769 −0.497364
\(506\) 0 0
\(507\) −13.5475 −0.601667
\(508\) 0 0
\(509\) 0.156638 0.00694286 0.00347143 0.999994i \(-0.498895\pi\)
0.00347143 + 0.999994i \(0.498895\pi\)
\(510\) 0 0
\(511\) −4.08611 −0.180759
\(512\) 0 0
\(513\) −33.2643 −1.46865
\(514\) 0 0
\(515\) 12.3472 0.544082
\(516\) 0 0
\(517\) 5.24929 0.230864
\(518\) 0 0
\(519\) −29.3112 −1.28662
\(520\) 0 0
\(521\) −32.1200 −1.40720 −0.703602 0.710594i \(-0.748426\pi\)
−0.703602 + 0.710594i \(0.748426\pi\)
\(522\) 0 0
\(523\) 4.73691 0.207131 0.103565 0.994623i \(-0.466975\pi\)
0.103565 + 0.994623i \(0.466975\pi\)
\(524\) 0 0
\(525\) 1.94363 0.0848270
\(526\) 0 0
\(527\) 5.72399 0.249341
\(528\) 0 0
\(529\) 14.5646 0.633244
\(530\) 0 0
\(531\) −6.47093 −0.280815
\(532\) 0 0
\(533\) −10.4693 −0.453474
\(534\) 0 0
\(535\) −23.9139 −1.03389
\(536\) 0 0
\(537\) 13.8272 0.596686
\(538\) 0 0
\(539\) −10.8746 −0.468404
\(540\) 0 0
\(541\) 15.3824 0.661342 0.330671 0.943746i \(-0.392725\pi\)
0.330671 + 0.943746i \(0.392725\pi\)
\(542\) 0 0
\(543\) −4.95868 −0.212797
\(544\) 0 0
\(545\) −30.3567 −1.30034
\(546\) 0 0
\(547\) 19.2682 0.823848 0.411924 0.911218i \(-0.364857\pi\)
0.411924 + 0.911218i \(0.364857\pi\)
\(548\) 0 0
\(549\) 5.16578 0.220470
\(550\) 0 0
\(551\) −1.89216 −0.0806086
\(552\) 0 0
\(553\) −14.9806 −0.637041
\(554\) 0 0
\(555\) −6.73621 −0.285936
\(556\) 0 0
\(557\) −10.0313 −0.425039 −0.212519 0.977157i \(-0.568167\pi\)
−0.212519 + 0.977157i \(0.568167\pi\)
\(558\) 0 0
\(559\) −6.11029 −0.258438
\(560\) 0 0
\(561\) 9.74342 0.411367
\(562\) 0 0
\(563\) −8.61733 −0.363177 −0.181589 0.983375i \(-0.558124\pi\)
−0.181589 + 0.983375i \(0.558124\pi\)
\(564\) 0 0
\(565\) −1.42081 −0.0597739
\(566\) 0 0
\(567\) 6.07521 0.255135
\(568\) 0 0
\(569\) −22.8782 −0.959105 −0.479552 0.877513i \(-0.659201\pi\)
−0.479552 + 0.877513i \(0.659201\pi\)
\(570\) 0 0
\(571\) −36.9981 −1.54832 −0.774162 0.632987i \(-0.781829\pi\)
−0.774162 + 0.632987i \(0.781829\pi\)
\(572\) 0 0
\(573\) −23.8208 −0.995126
\(574\) 0 0
\(575\) −6.12900 −0.255597
\(576\) 0 0
\(577\) 9.42378 0.392317 0.196158 0.980572i \(-0.437153\pi\)
0.196158 + 0.980572i \(0.437153\pi\)
\(578\) 0 0
\(579\) 6.47147 0.268945
\(580\) 0 0
\(581\) 10.3350 0.428770
\(582\) 0 0
\(583\) −3.32351 −0.137646
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −6.06497 −0.250328 −0.125164 0.992136i \(-0.539946\pi\)
−0.125164 + 0.992136i \(0.539946\pi\)
\(588\) 0 0
\(589\) 10.3294 0.425615
\(590\) 0 0
\(591\) −3.64788 −0.150054
\(592\) 0 0
\(593\) 30.0669 1.23470 0.617351 0.786688i \(-0.288206\pi\)
0.617351 + 0.786688i \(0.288206\pi\)
\(594\) 0 0
\(595\) −9.31705 −0.381962
\(596\) 0 0
\(597\) 34.9634 1.43096
\(598\) 0 0
\(599\) −8.74168 −0.357175 −0.178588 0.983924i \(-0.557153\pi\)
−0.178588 + 0.983924i \(0.557153\pi\)
\(600\) 0 0
\(601\) −5.47200 −0.223208 −0.111604 0.993753i \(-0.535599\pi\)
−0.111604 + 0.993753i \(0.535599\pi\)
\(602\) 0 0
\(603\) 11.0011 0.448001
\(604\) 0 0
\(605\) −12.4030 −0.504253
\(606\) 0 0
\(607\) 8.28118 0.336123 0.168061 0.985777i \(-0.446249\pi\)
0.168061 + 0.985777i \(0.446249\pi\)
\(608\) 0 0
\(609\) 0.624164 0.0252924
\(610\) 0 0
\(611\) 4.18855 0.169450
\(612\) 0 0
\(613\) 46.5063 1.87837 0.939186 0.343408i \(-0.111581\pi\)
0.939186 + 0.343408i \(0.111581\pi\)
\(614\) 0 0
\(615\) −16.3189 −0.658043
\(616\) 0 0
\(617\) 34.1883 1.37637 0.688185 0.725535i \(-0.258408\pi\)
0.688185 + 0.725535i \(0.258408\pi\)
\(618\) 0 0
\(619\) −23.3216 −0.937373 −0.468686 0.883365i \(-0.655273\pi\)
−0.468686 + 0.883365i \(0.655273\pi\)
\(620\) 0 0
\(621\) −34.6015 −1.38851
\(622\) 0 0
\(623\) 0.366157 0.0146698
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 17.5828 0.702188
\(628\) 0 0
\(629\) −8.07272 −0.321881
\(630\) 0 0
\(631\) −29.0863 −1.15791 −0.578953 0.815361i \(-0.696538\pi\)
−0.578953 + 0.815361i \(0.696538\pi\)
\(632\) 0 0
\(633\) 13.2580 0.526960
\(634\) 0 0
\(635\) 23.0330 0.914037
\(636\) 0 0
\(637\) −8.67716 −0.343802
\(638\) 0 0
\(639\) 7.63153 0.301899
\(640\) 0 0
\(641\) 31.0603 1.22681 0.613404 0.789769i \(-0.289800\pi\)
0.613404 + 0.789769i \(0.289800\pi\)
\(642\) 0 0
\(643\) 14.9145 0.588169 0.294085 0.955779i \(-0.404985\pi\)
0.294085 + 0.955779i \(0.404985\pi\)
\(644\) 0 0
\(645\) −9.52440 −0.375023
\(646\) 0 0
\(647\) 28.9951 1.13991 0.569957 0.821675i \(-0.306960\pi\)
0.569957 + 0.821675i \(0.306960\pi\)
\(648\) 0 0
\(649\) 12.3881 0.486277
\(650\) 0 0
\(651\) −3.40735 −0.133544
\(652\) 0 0
\(653\) 7.42152 0.290426 0.145213 0.989400i \(-0.453613\pi\)
0.145213 + 0.989400i \(0.453613\pi\)
\(654\) 0 0
\(655\) 16.0520 0.627203
\(656\) 0 0
\(657\) −3.27696 −0.127847
\(658\) 0 0
\(659\) −1.40447 −0.0547102 −0.0273551 0.999626i \(-0.508708\pi\)
−0.0273551 + 0.999626i \(0.508708\pi\)
\(660\) 0 0
\(661\) 13.1593 0.511837 0.255919 0.966698i \(-0.417622\pi\)
0.255919 + 0.966698i \(0.417622\pi\)
\(662\) 0 0
\(663\) 7.77453 0.301938
\(664\) 0 0
\(665\) −16.8134 −0.651994
\(666\) 0 0
\(667\) −1.96822 −0.0762099
\(668\) 0 0
\(669\) 29.4243 1.13761
\(670\) 0 0
\(671\) −9.88952 −0.381781
\(672\) 0 0
\(673\) −24.1775 −0.931974 −0.465987 0.884791i \(-0.654301\pi\)
−0.465987 + 0.884791i \(0.654301\pi\)
\(674\) 0 0
\(675\) 5.64555 0.217297
\(676\) 0 0
\(677\) 13.6811 0.525806 0.262903 0.964822i \(-0.415320\pi\)
0.262903 + 0.964822i \(0.415320\pi\)
\(678\) 0 0
\(679\) −8.32580 −0.319515
\(680\) 0 0
\(681\) −19.1264 −0.732924
\(682\) 0 0
\(683\) −14.3047 −0.547355 −0.273678 0.961822i \(-0.588240\pi\)
−0.273678 + 0.961822i \(0.588240\pi\)
\(684\) 0 0
\(685\) −31.3312 −1.19710
\(686\) 0 0
\(687\) 7.86054 0.299898
\(688\) 0 0
\(689\) −2.65191 −0.101030
\(690\) 0 0
\(691\) 27.8109 1.05798 0.528989 0.848629i \(-0.322571\pi\)
0.528989 + 0.848629i \(0.322571\pi\)
\(692\) 0 0
\(693\) 3.57618 0.135848
\(694\) 0 0
\(695\) −4.44326 −0.168543
\(696\) 0 0
\(697\) −19.5567 −0.740764
\(698\) 0 0
\(699\) −26.5240 −1.00323
\(700\) 0 0
\(701\) 10.7440 0.405796 0.202898 0.979200i \(-0.434964\pi\)
0.202898 + 0.979200i \(0.434964\pi\)
\(702\) 0 0
\(703\) −14.5679 −0.549437
\(704\) 0 0
\(705\) 6.52889 0.245892
\(706\) 0 0
\(707\) 7.97337 0.299869
\(708\) 0 0
\(709\) −39.8043 −1.49488 −0.747440 0.664329i \(-0.768717\pi\)
−0.747440 + 0.664329i \(0.768717\pi\)
\(710\) 0 0
\(711\) −12.0141 −0.450565
\(712\) 0 0
\(713\) 10.7446 0.402390
\(714\) 0 0
\(715\) 7.65771 0.286382
\(716\) 0 0
\(717\) −20.0272 −0.747930
\(718\) 0 0
\(719\) 23.3327 0.870162 0.435081 0.900391i \(-0.356720\pi\)
0.435081 + 0.900391i \(0.356720\pi\)
\(720\) 0 0
\(721\) −8.80826 −0.328037
\(722\) 0 0
\(723\) −29.5356 −1.09844
\(724\) 0 0
\(725\) 0.321133 0.0119266
\(726\) 0 0
\(727\) −21.2593 −0.788464 −0.394232 0.919011i \(-0.628989\pi\)
−0.394232 + 0.919011i \(0.628989\pi\)
\(728\) 0 0
\(729\) 27.9444 1.03498
\(730\) 0 0
\(731\) −11.4141 −0.422166
\(732\) 0 0
\(733\) −40.8733 −1.50969 −0.754846 0.655902i \(-0.772288\pi\)
−0.754846 + 0.655902i \(0.772288\pi\)
\(734\) 0 0
\(735\) −13.5255 −0.498896
\(736\) 0 0
\(737\) −21.0609 −0.775787
\(738\) 0 0
\(739\) 38.5912 1.41960 0.709799 0.704404i \(-0.248786\pi\)
0.709799 + 0.704404i \(0.248786\pi\)
\(740\) 0 0
\(741\) 14.0297 0.515396
\(742\) 0 0
\(743\) 18.3330 0.672572 0.336286 0.941760i \(-0.390829\pi\)
0.336286 + 0.941760i \(0.390829\pi\)
\(744\) 0 0
\(745\) 12.0520 0.441551
\(746\) 0 0
\(747\) 8.28846 0.303259
\(748\) 0 0
\(749\) 17.0598 0.623350
\(750\) 0 0
\(751\) −41.3572 −1.50915 −0.754573 0.656216i \(-0.772156\pi\)
−0.754573 + 0.656216i \(0.772156\pi\)
\(752\) 0 0
\(753\) −9.14257 −0.333174
\(754\) 0 0
\(755\) −21.9368 −0.798362
\(756\) 0 0
\(757\) 14.2655 0.518490 0.259245 0.965812i \(-0.416526\pi\)
0.259245 + 0.965812i \(0.416526\pi\)
\(758\) 0 0
\(759\) 18.2896 0.663871
\(760\) 0 0
\(761\) −41.8123 −1.51569 −0.757847 0.652432i \(-0.773749\pi\)
−0.757847 + 0.652432i \(0.773749\pi\)
\(762\) 0 0
\(763\) 21.6559 0.783997
\(764\) 0 0
\(765\) −7.47206 −0.270153
\(766\) 0 0
\(767\) 9.88481 0.356920
\(768\) 0 0
\(769\) 3.27230 0.118002 0.0590010 0.998258i \(-0.481208\pi\)
0.0590010 + 0.998258i \(0.481208\pi\)
\(770\) 0 0
\(771\) −35.5552 −1.28049
\(772\) 0 0
\(773\) 23.3460 0.839696 0.419848 0.907594i \(-0.362083\pi\)
0.419848 + 0.907594i \(0.362083\pi\)
\(774\) 0 0
\(775\) −1.75308 −0.0629726
\(776\) 0 0
\(777\) 4.80549 0.172396
\(778\) 0 0
\(779\) −35.2917 −1.26446
\(780\) 0 0
\(781\) −14.6100 −0.522787
\(782\) 0 0
\(783\) 1.81297 0.0647903
\(784\) 0 0
\(785\) 40.0949 1.43105
\(786\) 0 0
\(787\) 44.5309 1.58736 0.793678 0.608338i \(-0.208163\pi\)
0.793678 + 0.608338i \(0.208163\pi\)
\(788\) 0 0
\(789\) 22.0136 0.783703
\(790\) 0 0
\(791\) 1.01358 0.0360388
\(792\) 0 0
\(793\) −7.89110 −0.280221
\(794\) 0 0
\(795\) −4.13366 −0.146606
\(796\) 0 0
\(797\) 7.60786 0.269484 0.134742 0.990881i \(-0.456979\pi\)
0.134742 + 0.990881i \(0.456979\pi\)
\(798\) 0 0
\(799\) 7.82427 0.276803
\(800\) 0 0
\(801\) 0.293649 0.0103756
\(802\) 0 0
\(803\) 6.27351 0.221387
\(804\) 0 0
\(805\) −17.4893 −0.616416
\(806\) 0 0
\(807\) 14.9056 0.524702
\(808\) 0 0
\(809\) 7.40132 0.260217 0.130108 0.991500i \(-0.458468\pi\)
0.130108 + 0.991500i \(0.458468\pi\)
\(810\) 0 0
\(811\) −5.69610 −0.200017 −0.100009 0.994987i \(-0.531887\pi\)
−0.100009 + 0.994987i \(0.531887\pi\)
\(812\) 0 0
\(813\) 12.9358 0.453678
\(814\) 0 0
\(815\) 10.4354 0.365536
\(816\) 0 0
\(817\) −20.5977 −0.720621
\(818\) 0 0
\(819\) 2.85353 0.0997103
\(820\) 0 0
\(821\) −45.1962 −1.57736 −0.788680 0.614804i \(-0.789235\pi\)
−0.788680 + 0.614804i \(0.789235\pi\)
\(822\) 0 0
\(823\) 13.9450 0.486093 0.243047 0.970015i \(-0.421853\pi\)
0.243047 + 0.970015i \(0.421853\pi\)
\(824\) 0 0
\(825\) −2.98411 −0.103893
\(826\) 0 0
\(827\) 2.90541 0.101031 0.0505155 0.998723i \(-0.483914\pi\)
0.0505155 + 0.998723i \(0.483914\pi\)
\(828\) 0 0
\(829\) 31.2634 1.08582 0.542912 0.839790i \(-0.317322\pi\)
0.542912 + 0.839790i \(0.317322\pi\)
\(830\) 0 0
\(831\) 2.53468 0.0879270
\(832\) 0 0
\(833\) −16.2091 −0.561611
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) −9.89710 −0.342094
\(838\) 0 0
\(839\) −44.4886 −1.53592 −0.767958 0.640500i \(-0.778727\pi\)
−0.767958 + 0.640500i \(0.778727\pi\)
\(840\) 0 0
\(841\) −28.8969 −0.996444
\(842\) 0 0
\(843\) 5.46326 0.188165
\(844\) 0 0
\(845\) −19.8897 −0.684227
\(846\) 0 0
\(847\) 8.84806 0.304023
\(848\) 0 0
\(849\) −10.5062 −0.360574
\(850\) 0 0
\(851\) −15.1535 −0.519455
\(852\) 0 0
\(853\) −17.2717 −0.591371 −0.295686 0.955285i \(-0.595548\pi\)
−0.295686 + 0.955285i \(0.595548\pi\)
\(854\) 0 0
\(855\) −13.4839 −0.461141
\(856\) 0 0
\(857\) −35.0334 −1.19672 −0.598359 0.801228i \(-0.704180\pi\)
−0.598359 + 0.801228i \(0.704180\pi\)
\(858\) 0 0
\(859\) 38.9649 1.32947 0.664733 0.747081i \(-0.268546\pi\)
0.664733 + 0.747081i \(0.268546\pi\)
\(860\) 0 0
\(861\) 11.6416 0.396746
\(862\) 0 0
\(863\) 32.0638 1.09146 0.545732 0.837960i \(-0.316252\pi\)
0.545732 + 0.837960i \(0.316252\pi\)
\(864\) 0 0
\(865\) −43.0330 −1.46317
\(866\) 0 0
\(867\) −8.63558 −0.293280
\(868\) 0 0
\(869\) 23.0002 0.780228
\(870\) 0 0
\(871\) −16.8050 −0.569416
\(872\) 0 0
\(873\) −6.67710 −0.225986
\(874\) 0 0
\(875\) 17.1212 0.578801
\(876\) 0 0
\(877\) 53.9033 1.82018 0.910092 0.414406i \(-0.136011\pi\)
0.910092 + 0.414406i \(0.136011\pi\)
\(878\) 0 0
\(879\) 12.3365 0.416099
\(880\) 0 0
\(881\) 49.8614 1.67987 0.839937 0.542684i \(-0.182592\pi\)
0.839937 + 0.542684i \(0.182592\pi\)
\(882\) 0 0
\(883\) 12.9835 0.436930 0.218465 0.975845i \(-0.429895\pi\)
0.218465 + 0.975845i \(0.429895\pi\)
\(884\) 0 0
\(885\) 15.4079 0.517932
\(886\) 0 0
\(887\) 11.8245 0.397027 0.198513 0.980098i \(-0.436389\pi\)
0.198513 + 0.980098i \(0.436389\pi\)
\(888\) 0 0
\(889\) −16.4313 −0.551089
\(890\) 0 0
\(891\) −9.32744 −0.312481
\(892\) 0 0
\(893\) 14.1195 0.472492
\(894\) 0 0
\(895\) 20.3002 0.678562
\(896\) 0 0
\(897\) 14.5938 0.487271
\(898\) 0 0
\(899\) −0.562972 −0.0187762
\(900\) 0 0
\(901\) −4.95381 −0.165036
\(902\) 0 0
\(903\) 6.79454 0.226108
\(904\) 0 0
\(905\) −7.28005 −0.241997
\(906\) 0 0
\(907\) 1.08780 0.0361198 0.0180599 0.999837i \(-0.494251\pi\)
0.0180599 + 0.999837i \(0.494251\pi\)
\(908\) 0 0
\(909\) 6.39446 0.212091
\(910\) 0 0
\(911\) −9.16245 −0.303566 −0.151783 0.988414i \(-0.548501\pi\)
−0.151783 + 0.988414i \(0.548501\pi\)
\(912\) 0 0
\(913\) −15.8677 −0.525143
\(914\) 0 0
\(915\) −12.3002 −0.406634
\(916\) 0 0
\(917\) −11.4512 −0.378152
\(918\) 0 0
\(919\) −23.2997 −0.768587 −0.384293 0.923211i \(-0.625555\pi\)
−0.384293 + 0.923211i \(0.625555\pi\)
\(920\) 0 0
\(921\) 16.5695 0.545985
\(922\) 0 0
\(923\) −11.6577 −0.383718
\(924\) 0 0
\(925\) 2.47243 0.0812929
\(926\) 0 0
\(927\) −7.06403 −0.232013
\(928\) 0 0
\(929\) 48.9106 1.60471 0.802353 0.596850i \(-0.203581\pi\)
0.802353 + 0.596850i \(0.203581\pi\)
\(930\) 0 0
\(931\) −29.2506 −0.958648
\(932\) 0 0
\(933\) 33.3690 1.09245
\(934\) 0 0
\(935\) 14.3047 0.467815
\(936\) 0 0
\(937\) 29.3682 0.959418 0.479709 0.877428i \(-0.340742\pi\)
0.479709 + 0.877428i \(0.340742\pi\)
\(938\) 0 0
\(939\) 30.7317 1.00289
\(940\) 0 0
\(941\) −33.9716 −1.10744 −0.553721 0.832702i \(-0.686793\pi\)
−0.553721 + 0.832702i \(0.686793\pi\)
\(942\) 0 0
\(943\) −36.7104 −1.19546
\(944\) 0 0
\(945\) 16.1097 0.524049
\(946\) 0 0
\(947\) 36.7945 1.19566 0.597830 0.801623i \(-0.296030\pi\)
0.597830 + 0.801623i \(0.296030\pi\)
\(948\) 0 0
\(949\) 5.00580 0.162495
\(950\) 0 0
\(951\) 19.0099 0.616439
\(952\) 0 0
\(953\) −5.19613 −0.168319 −0.0841596 0.996452i \(-0.526821\pi\)
−0.0841596 + 0.996452i \(0.526821\pi\)
\(954\) 0 0
\(955\) −34.9723 −1.13168
\(956\) 0 0
\(957\) −0.958296 −0.0309773
\(958\) 0 0
\(959\) 22.3511 0.721755
\(960\) 0 0
\(961\) −27.9267 −0.900861
\(962\) 0 0
\(963\) 13.6815 0.440882
\(964\) 0 0
\(965\) 9.50105 0.305849
\(966\) 0 0
\(967\) 49.3421 1.58673 0.793367 0.608743i \(-0.208326\pi\)
0.793367 + 0.608743i \(0.208326\pi\)
\(968\) 0 0
\(969\) 26.2078 0.841916
\(970\) 0 0
\(971\) −43.3773 −1.39204 −0.696021 0.718021i \(-0.745048\pi\)
−0.696021 + 0.718021i \(0.745048\pi\)
\(972\) 0 0
\(973\) 3.16975 0.101617
\(974\) 0 0
\(975\) −2.38110 −0.0762562
\(976\) 0 0
\(977\) 55.2452 1.76745 0.883725 0.468006i \(-0.155028\pi\)
0.883725 + 0.468006i \(0.155028\pi\)
\(978\) 0 0
\(979\) −0.562170 −0.0179670
\(980\) 0 0
\(981\) 17.3676 0.554504
\(982\) 0 0
\(983\) −12.0299 −0.383694 −0.191847 0.981425i \(-0.561448\pi\)
−0.191847 + 0.981425i \(0.561448\pi\)
\(984\) 0 0
\(985\) −5.35560 −0.170644
\(986\) 0 0
\(987\) −4.65759 −0.148253
\(988\) 0 0
\(989\) −21.4257 −0.681298
\(990\) 0 0
\(991\) 44.9737 1.42864 0.714318 0.699822i \(-0.246737\pi\)
0.714318 + 0.699822i \(0.246737\pi\)
\(992\) 0 0
\(993\) −29.4013 −0.933022
\(994\) 0 0
\(995\) 51.3312 1.62731
\(996\) 0 0
\(997\) 44.0823 1.39610 0.698050 0.716049i \(-0.254051\pi\)
0.698050 + 0.716049i \(0.254051\pi\)
\(998\) 0 0
\(999\) 13.9582 0.441618
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 668.2.a.b.1.3 5
3.2 odd 2 6012.2.a.d.1.1 5
4.3 odd 2 2672.2.a.j.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.3 5 1.1 even 1 trivial
2672.2.a.j.1.3 5 4.3 odd 2
6012.2.a.d.1.1 5 3.2 odd 2