Properties

Label 8280.2.a.bg.1.2
Level $8280$
Weight $2$
Character 8280.1
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 8280.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +3.44949 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +3.44949 q^{7} -2.44949 q^{11} -4.44949 q^{13} -0.550510 q^{17} +2.44949 q^{19} -1.00000 q^{23} +1.00000 q^{25} -7.89898 q^{29} -7.89898 q^{31} +3.44949 q^{35} +8.34847 q^{37} -1.89898 q^{41} -0.898979 q^{43} -1.55051 q^{47} +4.89898 q^{49} +7.44949 q^{53} -2.44949 q^{55} -1.00000 q^{59} +4.44949 q^{61} -4.44949 q^{65} -9.44949 q^{67} -5.89898 q^{71} -3.55051 q^{73} -8.44949 q^{77} -15.4495 q^{83} -0.550510 q^{85} -8.89898 q^{89} -15.3485 q^{91} +2.44949 q^{95} -1.10102 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} - 4 q^{13} - 6 q^{17} - 2 q^{23} + 2 q^{25} - 6 q^{29} - 6 q^{31} + 2 q^{35} + 2 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} + 10 q^{53} - 2 q^{59} + 4 q^{61} - 4 q^{65} - 14 q^{67} - 2 q^{71} - 12 q^{73} - 12 q^{77} - 26 q^{83} - 6 q^{85} - 8 q^{89} - 16 q^{91} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.44949 1.30378 0.651892 0.758312i \(-0.273975\pi\)
0.651892 + 0.758312i \(0.273975\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.44949 −0.738549 −0.369274 0.929320i \(-0.620394\pi\)
−0.369274 + 0.929320i \(0.620394\pi\)
\(12\) 0 0
\(13\) −4.44949 −1.23407 −0.617033 0.786937i \(-0.711666\pi\)
−0.617033 + 0.786937i \(0.711666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.550510 −0.133518 −0.0667592 0.997769i \(-0.521266\pi\)
−0.0667592 + 0.997769i \(0.521266\pi\)
\(18\) 0 0
\(19\) 2.44949 0.561951 0.280976 0.959715i \(-0.409342\pi\)
0.280976 + 0.959715i \(0.409342\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.89898 −1.46680 −0.733402 0.679795i \(-0.762069\pi\)
−0.733402 + 0.679795i \(0.762069\pi\)
\(30\) 0 0
\(31\) −7.89898 −1.41870 −0.709349 0.704857i \(-0.751011\pi\)
−0.709349 + 0.704857i \(0.751011\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44949 0.583070
\(36\) 0 0
\(37\) 8.34847 1.37248 0.686240 0.727375i \(-0.259260\pi\)
0.686240 + 0.727375i \(0.259260\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.89898 −0.296571 −0.148285 0.988945i \(-0.547375\pi\)
−0.148285 + 0.988945i \(0.547375\pi\)
\(42\) 0 0
\(43\) −0.898979 −0.137093 −0.0685465 0.997648i \(-0.521836\pi\)
−0.0685465 + 0.997648i \(0.521836\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.55051 −0.226165 −0.113083 0.993586i \(-0.536072\pi\)
−0.113083 + 0.993586i \(0.536072\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.44949 1.02327 0.511633 0.859204i \(-0.329041\pi\)
0.511633 + 0.859204i \(0.329041\pi\)
\(54\) 0 0
\(55\) −2.44949 −0.330289
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 0 0
\(61\) 4.44949 0.569699 0.284849 0.958572i \(-0.408056\pi\)
0.284849 + 0.958572i \(0.408056\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.44949 −0.551891
\(66\) 0 0
\(67\) −9.44949 −1.15444 −0.577219 0.816589i \(-0.695862\pi\)
−0.577219 + 0.816589i \(0.695862\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.89898 −0.700080 −0.350040 0.936735i \(-0.613832\pi\)
−0.350040 + 0.936735i \(0.613832\pi\)
\(72\) 0 0
\(73\) −3.55051 −0.415556 −0.207778 0.978176i \(-0.566623\pi\)
−0.207778 + 0.978176i \(0.566623\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.44949 −0.962909
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.4495 −1.69580 −0.847901 0.530155i \(-0.822134\pi\)
−0.847901 + 0.530155i \(0.822134\pi\)
\(84\) 0 0
\(85\) −0.550510 −0.0597112
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.89898 −0.943290 −0.471645 0.881789i \(-0.656340\pi\)
−0.471645 + 0.881789i \(0.656340\pi\)
\(90\) 0 0
\(91\) −15.3485 −1.60896
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.44949 0.251312
\(96\) 0 0
\(97\) −1.10102 −0.111792 −0.0558958 0.998437i \(-0.517801\pi\)
−0.0558958 + 0.998437i \(0.517801\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) −13.7980 −1.35955 −0.679777 0.733419i \(-0.737923\pi\)
−0.679777 + 0.733419i \(0.737923\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.24745 0.700637 0.350319 0.936631i \(-0.386073\pi\)
0.350319 + 0.936631i \(0.386073\pi\)
\(108\) 0 0
\(109\) 2.44949 0.234619 0.117309 0.993095i \(-0.462573\pi\)
0.117309 + 0.993095i \(0.462573\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.34847 −0.597214 −0.298607 0.954376i \(-0.596522\pi\)
−0.298607 + 0.954376i \(0.596522\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.89898 −0.174079
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.2474 1.79667 0.898335 0.439311i \(-0.144777\pi\)
0.898335 + 0.439311i \(0.144777\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.8990 0.952248 0.476124 0.879378i \(-0.342041\pi\)
0.476124 + 0.879378i \(0.342041\pi\)
\(132\) 0 0
\(133\) 8.44949 0.732664
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.89898 −0.760291 −0.380146 0.924927i \(-0.624126\pi\)
−0.380146 + 0.924927i \(0.624126\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.8990 0.911418
\(144\) 0 0
\(145\) −7.89898 −0.655975
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.5505 −0.946255 −0.473127 0.880994i \(-0.656875\pi\)
−0.473127 + 0.880994i \(0.656875\pi\)
\(150\) 0 0
\(151\) −11.7980 −0.960104 −0.480052 0.877240i \(-0.659382\pi\)
−0.480052 + 0.877240i \(0.659382\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.89898 −0.634461
\(156\) 0 0
\(157\) 7.65153 0.610659 0.305329 0.952247i \(-0.401233\pi\)
0.305329 + 0.952247i \(0.401233\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.44949 −0.271858
\(162\) 0 0
\(163\) 2.89898 0.227066 0.113533 0.993534i \(-0.463783\pi\)
0.113533 + 0.993534i \(0.463783\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.44949 −0.499077 −0.249538 0.968365i \(-0.580279\pi\)
−0.249538 + 0.968365i \(0.580279\pi\)
\(168\) 0 0
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.79796 0.136696 0.0683481 0.997662i \(-0.478227\pi\)
0.0683481 + 0.997662i \(0.478227\pi\)
\(174\) 0 0
\(175\) 3.44949 0.260757
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.79796 0.732334 0.366167 0.930549i \(-0.380670\pi\)
0.366167 + 0.930549i \(0.380670\pi\)
\(180\) 0 0
\(181\) −7.10102 −0.527815 −0.263907 0.964548i \(-0.585011\pi\)
−0.263907 + 0.964548i \(0.585011\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.34847 0.613792
\(186\) 0 0
\(187\) 1.34847 0.0986098
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.2474 −1.03091 −0.515455 0.856917i \(-0.672377\pi\)
−0.515455 + 0.856917i \(0.672377\pi\)
\(192\) 0 0
\(193\) −13.7980 −0.993199 −0.496599 0.867980i \(-0.665418\pi\)
−0.496599 + 0.867980i \(0.665418\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.79796 0.270593 0.135297 0.990805i \(-0.456801\pi\)
0.135297 + 0.990805i \(0.456801\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −27.2474 −1.91240
\(204\) 0 0
\(205\) −1.89898 −0.132630
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 1.20204 0.0827519 0.0413760 0.999144i \(-0.486826\pi\)
0.0413760 + 0.999144i \(0.486826\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.898979 −0.0613099
\(216\) 0 0
\(217\) −27.2474 −1.84968
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.44949 0.164771
\(222\) 0 0
\(223\) 6.89898 0.461990 0.230995 0.972955i \(-0.425802\pi\)
0.230995 + 0.972955i \(0.425802\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.20204 0.411644 0.205822 0.978589i \(-0.434013\pi\)
0.205822 + 0.978589i \(0.434013\pi\)
\(228\) 0 0
\(229\) −5.79796 −0.383140 −0.191570 0.981479i \(-0.561358\pi\)
−0.191570 + 0.981479i \(0.561358\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.10102 −0.465203 −0.232602 0.972572i \(-0.574724\pi\)
−0.232602 + 0.972572i \(0.574724\pi\)
\(234\) 0 0
\(235\) −1.55051 −0.101144
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.79796 0.310354 0.155177 0.987887i \(-0.450405\pi\)
0.155177 + 0.987887i \(0.450405\pi\)
\(240\) 0 0
\(241\) −14.0454 −0.904744 −0.452372 0.891829i \(-0.649422\pi\)
−0.452372 + 0.891829i \(0.649422\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.89898 0.312984
\(246\) 0 0
\(247\) −10.8990 −0.693485
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.69694 0.296468 0.148234 0.988952i \(-0.452641\pi\)
0.148234 + 0.988952i \(0.452641\pi\)
\(252\) 0 0
\(253\) 2.44949 0.153998
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.1464 1.31908 0.659539 0.751670i \(-0.270752\pi\)
0.659539 + 0.751670i \(0.270752\pi\)
\(258\) 0 0
\(259\) 28.7980 1.78942
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −21.2474 −1.31017 −0.655087 0.755554i \(-0.727368\pi\)
−0.655087 + 0.755554i \(0.727368\pi\)
\(264\) 0 0
\(265\) 7.44949 0.457619
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.89898 −0.237725 −0.118862 0.992911i \(-0.537925\pi\)
−0.118862 + 0.992911i \(0.537925\pi\)
\(270\) 0 0
\(271\) 16.7980 1.02040 0.510202 0.860055i \(-0.329571\pi\)
0.510202 + 0.860055i \(0.329571\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.44949 −0.147710
\(276\) 0 0
\(277\) −18.8990 −1.13553 −0.567765 0.823191i \(-0.692192\pi\)
−0.567765 + 0.823191i \(0.692192\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.55051 −0.569736 −0.284868 0.958567i \(-0.591950\pi\)
−0.284868 + 0.958567i \(0.591950\pi\)
\(282\) 0 0
\(283\) 5.65153 0.335949 0.167974 0.985791i \(-0.446277\pi\)
0.167974 + 0.985791i \(0.446277\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.55051 −0.386664
\(288\) 0 0
\(289\) −16.6969 −0.982173
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.24745 0.189718 0.0948590 0.995491i \(-0.469760\pi\)
0.0948590 + 0.995491i \(0.469760\pi\)
\(294\) 0 0
\(295\) −1.00000 −0.0582223
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.44949 0.257321
\(300\) 0 0
\(301\) −3.10102 −0.178740
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.44949 0.254777
\(306\) 0 0
\(307\) 22.2474 1.26973 0.634864 0.772624i \(-0.281056\pi\)
0.634864 + 0.772624i \(0.281056\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.10102 −0.289252 −0.144626 0.989486i \(-0.546198\pi\)
−0.144626 + 0.989486i \(0.546198\pi\)
\(312\) 0 0
\(313\) 11.0454 0.624323 0.312162 0.950029i \(-0.398947\pi\)
0.312162 + 0.950029i \(0.398947\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.55051 −0.0870853 −0.0435427 0.999052i \(-0.513864\pi\)
−0.0435427 + 0.999052i \(0.513864\pi\)
\(318\) 0 0
\(319\) 19.3485 1.08331
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.34847 −0.0750308
\(324\) 0 0
\(325\) −4.44949 −0.246813
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.34847 −0.294871
\(330\) 0 0
\(331\) 25.6969 1.41243 0.706216 0.707997i \(-0.250401\pi\)
0.706216 + 0.707997i \(0.250401\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.44949 −0.516281
\(336\) 0 0
\(337\) −32.8990 −1.79212 −0.896061 0.443931i \(-0.853583\pi\)
−0.896061 + 0.443931i \(0.853583\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 19.3485 1.04778
\(342\) 0 0
\(343\) −7.24745 −0.391325
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 25.6969 1.37553 0.687763 0.725935i \(-0.258593\pi\)
0.687763 + 0.725935i \(0.258593\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −14.0454 −0.747562 −0.373781 0.927517i \(-0.621939\pi\)
−0.373781 + 0.927517i \(0.621939\pi\)
\(354\) 0 0
\(355\) −5.89898 −0.313085
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.24745 −0.329728 −0.164864 0.986316i \(-0.552718\pi\)
−0.164864 + 0.986316i \(0.552718\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.55051 −0.185842
\(366\) 0 0
\(367\) −23.0454 −1.20296 −0.601480 0.798888i \(-0.705422\pi\)
−0.601480 + 0.798888i \(0.705422\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 25.6969 1.33412
\(372\) 0 0
\(373\) 19.7980 1.02510 0.512550 0.858658i \(-0.328701\pi\)
0.512550 + 0.858658i \(0.328701\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 35.1464 1.81013
\(378\) 0 0
\(379\) 5.79796 0.297821 0.148911 0.988851i \(-0.452423\pi\)
0.148911 + 0.988851i \(0.452423\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 29.9444 1.53009 0.765043 0.643979i \(-0.222718\pi\)
0.765043 + 0.643979i \(0.222718\pi\)
\(384\) 0 0
\(385\) −8.44949 −0.430626
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.8990 −1.56664 −0.783320 0.621618i \(-0.786475\pi\)
−0.783320 + 0.621618i \(0.786475\pi\)
\(390\) 0 0
\(391\) 0.550510 0.0278405
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.5959 1.18425 0.592123 0.805848i \(-0.298290\pi\)
0.592123 + 0.805848i \(0.298290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.4949 1.52284 0.761421 0.648257i \(-0.224502\pi\)
0.761421 + 0.648257i \(0.224502\pi\)
\(402\) 0 0
\(403\) 35.1464 1.75077
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.4495 −1.01364
\(408\) 0 0
\(409\) 16.7980 0.830606 0.415303 0.909683i \(-0.363676\pi\)
0.415303 + 0.909683i \(0.363676\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.44949 −0.169738
\(414\) 0 0
\(415\) −15.4495 −0.758386
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0454 0.881576 0.440788 0.897611i \(-0.354699\pi\)
0.440788 + 0.897611i \(0.354699\pi\)
\(420\) 0 0
\(421\) −6.44949 −0.314329 −0.157164 0.987572i \(-0.550235\pi\)
−0.157164 + 0.987572i \(0.550235\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.550510 −0.0267037
\(426\) 0 0
\(427\) 15.3485 0.742764
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −15.4495 −0.742455 −0.371228 0.928542i \(-0.621063\pi\)
−0.371228 + 0.928542i \(0.621063\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.44949 −0.117175
\(438\) 0 0
\(439\) −2.89898 −0.138361 −0.0691804 0.997604i \(-0.522038\pi\)
−0.0691804 + 0.997604i \(0.522038\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.2474 0.581894 0.290947 0.956739i \(-0.406030\pi\)
0.290947 + 0.956739i \(0.406030\pi\)
\(444\) 0 0
\(445\) −8.89898 −0.421852
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −40.5959 −1.91584 −0.957920 0.287036i \(-0.907330\pi\)
−0.957920 + 0.287036i \(0.907330\pi\)
\(450\) 0 0
\(451\) 4.65153 0.219032
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.3485 −0.719547
\(456\) 0 0
\(457\) 23.2474 1.08747 0.543735 0.839257i \(-0.317010\pi\)
0.543735 + 0.839257i \(0.317010\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.6969 −1.42970 −0.714849 0.699278i \(-0.753505\pi\)
−0.714849 + 0.699278i \(0.753505\pi\)
\(462\) 0 0
\(463\) 29.3485 1.36394 0.681970 0.731381i \(-0.261124\pi\)
0.681970 + 0.731381i \(0.261124\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.14643 −0.376972 −0.188486 0.982076i \(-0.560358\pi\)
−0.188486 + 0.982076i \(0.560358\pi\)
\(468\) 0 0
\(469\) −32.5959 −1.50514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.20204 0.101250
\(474\) 0 0
\(475\) 2.44949 0.112390
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.55051 0.344992 0.172496 0.985010i \(-0.444817\pi\)
0.172496 + 0.985010i \(0.444817\pi\)
\(480\) 0 0
\(481\) −37.1464 −1.69373
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.10102 −0.0499948
\(486\) 0 0
\(487\) −25.5505 −1.15780 −0.578902 0.815397i \(-0.696519\pi\)
−0.578902 + 0.815397i \(0.696519\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.8990 −0.717511 −0.358755 0.933432i \(-0.616799\pi\)
−0.358755 + 0.933432i \(0.616799\pi\)
\(492\) 0 0
\(493\) 4.34847 0.195845
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −20.3485 −0.912754
\(498\) 0 0
\(499\) 41.4949 1.85757 0.928783 0.370623i \(-0.120856\pi\)
0.928783 + 0.370623i \(0.120856\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.2474 −1.39325 −0.696627 0.717433i \(-0.745317\pi\)
−0.696627 + 0.717433i \(0.745317\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.79796 0.168342 0.0841708 0.996451i \(-0.473176\pi\)
0.0841708 + 0.996451i \(0.473176\pi\)
\(510\) 0 0
\(511\) −12.2474 −0.541795
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −13.7980 −0.608011
\(516\) 0 0
\(517\) 3.79796 0.167034
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.4495 −0.457800 −0.228900 0.973450i \(-0.573513\pi\)
−0.228900 + 0.973450i \(0.573513\pi\)
\(522\) 0 0
\(523\) 20.2020 0.883374 0.441687 0.897169i \(-0.354380\pi\)
0.441687 + 0.897169i \(0.354380\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.34847 0.189422
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.44949 0.365988
\(534\) 0 0
\(535\) 7.24745 0.313335
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 34.6969 1.49174 0.745869 0.666093i \(-0.232035\pi\)
0.745869 + 0.666093i \(0.232035\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.44949 0.104925
\(546\) 0 0
\(547\) 25.3939 1.08576 0.542882 0.839809i \(-0.317333\pi\)
0.542882 + 0.839809i \(0.317333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.3485 −0.824273
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −41.7423 −1.76868 −0.884340 0.466843i \(-0.845391\pi\)
−0.884340 + 0.466843i \(0.845391\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.65153 −0.153894 −0.0769468 0.997035i \(-0.524517\pi\)
−0.0769468 + 0.997035i \(0.524517\pi\)
\(564\) 0 0
\(565\) −6.34847 −0.267082
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.79796 0.159219 0.0796094 0.996826i \(-0.474633\pi\)
0.0796094 + 0.996826i \(0.474633\pi\)
\(570\) 0 0
\(571\) −10.2474 −0.428842 −0.214421 0.976741i \(-0.568787\pi\)
−0.214421 + 0.976741i \(0.568787\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −4.89898 −0.203947 −0.101974 0.994787i \(-0.532516\pi\)
−0.101974 + 0.994787i \(0.532516\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −53.2929 −2.21096
\(582\) 0 0
\(583\) −18.2474 −0.755732
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.7980 −1.14734 −0.573672 0.819085i \(-0.694482\pi\)
−0.573672 + 0.819085i \(0.694482\pi\)
\(588\) 0 0
\(589\) −19.3485 −0.797240
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.44949 −0.264849 −0.132424 0.991193i \(-0.542276\pi\)
−0.132424 + 0.991193i \(0.542276\pi\)
\(594\) 0 0
\(595\) −1.89898 −0.0778506
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −6.59592 −0.269053 −0.134527 0.990910i \(-0.542951\pi\)
−0.134527 + 0.990910i \(0.542951\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −10.4495 −0.424132 −0.212066 0.977255i \(-0.568019\pi\)
−0.212066 + 0.977255i \(0.568019\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.89898 0.279103
\(612\) 0 0
\(613\) 3.10102 0.125249 0.0626245 0.998037i \(-0.480053\pi\)
0.0626245 + 0.998037i \(0.480053\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.0454 1.24984 0.624921 0.780688i \(-0.285131\pi\)
0.624921 + 0.780688i \(0.285131\pi\)
\(618\) 0 0
\(619\) −25.7980 −1.03691 −0.518454 0.855106i \(-0.673492\pi\)
−0.518454 + 0.855106i \(0.673492\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −30.6969 −1.22985
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.59592 −0.183251
\(630\) 0 0
\(631\) −8.04541 −0.320283 −0.160141 0.987094i \(-0.551195\pi\)
−0.160141 + 0.987094i \(0.551195\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.2474 0.803495
\(636\) 0 0
\(637\) −21.7980 −0.863667
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.9444 −0.906249 −0.453124 0.891447i \(-0.649691\pi\)
−0.453124 + 0.891447i \(0.649691\pi\)
\(642\) 0 0
\(643\) −14.5505 −0.573816 −0.286908 0.957958i \(-0.592627\pi\)
−0.286908 + 0.957958i \(0.592627\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.04541 −0.159041 −0.0795207 0.996833i \(-0.525339\pi\)
−0.0795207 + 0.996833i \(0.525339\pi\)
\(648\) 0 0
\(649\) 2.44949 0.0961509
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.3485 −0.991962 −0.495981 0.868333i \(-0.665191\pi\)
−0.495981 + 0.868333i \(0.665191\pi\)
\(654\) 0 0
\(655\) 10.8990 0.425858
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.44949 0.251236 0.125618 0.992079i \(-0.459909\pi\)
0.125618 + 0.992079i \(0.459909\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.44949 0.327657
\(666\) 0 0
\(667\) 7.89898 0.305850
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.8990 −0.420750
\(672\) 0 0
\(673\) −10.4495 −0.402798 −0.201399 0.979509i \(-0.564549\pi\)
−0.201399 + 0.979509i \(0.564549\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.44949 −0.286307 −0.143154 0.989700i \(-0.545724\pi\)
−0.143154 + 0.989700i \(0.545724\pi\)
\(678\) 0 0
\(679\) −3.79796 −0.145752
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −42.0454 −1.60882 −0.804411 0.594073i \(-0.797519\pi\)
−0.804411 + 0.594073i \(0.797519\pi\)
\(684\) 0 0
\(685\) −8.89898 −0.340013
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −33.1464 −1.26278
\(690\) 0 0
\(691\) −16.2020 −0.616355 −0.308177 0.951329i \(-0.599719\pi\)
−0.308177 + 0.951329i \(0.599719\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.0000 0.493118
\(696\) 0 0
\(697\) 1.04541 0.0395976
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.2474 −1.14243 −0.571215 0.820800i \(-0.693528\pi\)
−0.571215 + 0.820800i \(0.693528\pi\)
\(702\) 0 0
\(703\) 20.4495 0.771267
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −51.7423 −1.94597
\(708\) 0 0
\(709\) −8.04541 −0.302152 −0.151076 0.988522i \(-0.548274\pi\)
−0.151076 + 0.988522i \(0.548274\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.89898 0.295819
\(714\) 0 0
\(715\) 10.8990 0.407599
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.79796 −0.178934 −0.0894668 0.995990i \(-0.528516\pi\)
−0.0894668 + 0.995990i \(0.528516\pi\)
\(720\) 0 0
\(721\) −47.5959 −1.77256
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.89898 −0.293361
\(726\) 0 0
\(727\) −17.9444 −0.665520 −0.332760 0.943011i \(-0.607980\pi\)
−0.332760 + 0.943011i \(0.607980\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.494897 0.0183044
\(732\) 0 0
\(733\) 31.2474 1.15415 0.577075 0.816691i \(-0.304194\pi\)
0.577075 + 0.816691i \(0.304194\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.1464 0.852610
\(738\) 0 0
\(739\) 29.0000 1.06678 0.533391 0.845869i \(-0.320917\pi\)
0.533391 + 0.845869i \(0.320917\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.1918 1.43781 0.718905 0.695109i \(-0.244644\pi\)
0.718905 + 0.695109i \(0.244644\pi\)
\(744\) 0 0
\(745\) −11.5505 −0.423178
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 25.0000 0.913480
\(750\) 0 0
\(751\) 35.3485 1.28988 0.644942 0.764232i \(-0.276882\pi\)
0.644942 + 0.764232i \(0.276882\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.7980 −0.429372
\(756\) 0 0
\(757\) −17.4495 −0.634212 −0.317106 0.948390i \(-0.602711\pi\)
−0.317106 + 0.948390i \(0.602711\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.6969 1.00401 0.502007 0.864864i \(-0.332595\pi\)
0.502007 + 0.864864i \(0.332595\pi\)
\(762\) 0 0
\(763\) 8.44949 0.305892
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.44949 0.160662
\(768\) 0 0
\(769\) −53.8434 −1.94164 −0.970821 0.239806i \(-0.922916\pi\)
−0.970821 + 0.239806i \(0.922916\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.6969 1.75151 0.875754 0.482758i \(-0.160365\pi\)
0.875754 + 0.482758i \(0.160365\pi\)
\(774\) 0 0
\(775\) −7.89898 −0.283740
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.65153 −0.166658
\(780\) 0 0
\(781\) 14.4495 0.517043
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.65153 0.273095
\(786\) 0 0
\(787\) −0.348469 −0.0124216 −0.00621079 0.999981i \(-0.501977\pi\)
−0.00621079 + 0.999981i \(0.501977\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −21.8990 −0.778638
\(792\) 0 0
\(793\) −19.7980 −0.703046
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −15.2474 −0.540092 −0.270046 0.962847i \(-0.587039\pi\)
−0.270046 + 0.962847i \(0.587039\pi\)
\(798\) 0 0
\(799\) 0.853572 0.0301972
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.69694 0.306908
\(804\) 0 0
\(805\) −3.44949 −0.121579
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 26.5959 0.935063 0.467531 0.883976i \(-0.345144\pi\)
0.467531 + 0.883976i \(0.345144\pi\)
\(810\) 0 0
\(811\) 47.2929 1.66068 0.830338 0.557259i \(-0.188147\pi\)
0.830338 + 0.557259i \(0.188147\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.89898 0.101547
\(816\) 0 0
\(817\) −2.20204 −0.0770397
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 45.1918 1.57721 0.788603 0.614903i \(-0.210805\pi\)
0.788603 + 0.614903i \(0.210805\pi\)
\(822\) 0 0
\(823\) 14.6969 0.512303 0.256152 0.966637i \(-0.417545\pi\)
0.256152 + 0.966637i \(0.417545\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.9444 −0.415347 −0.207674 0.978198i \(-0.566589\pi\)
−0.207674 + 0.978198i \(0.566589\pi\)
\(828\) 0 0
\(829\) −32.3939 −1.12509 −0.562543 0.826768i \(-0.690177\pi\)
−0.562543 + 0.826768i \(0.690177\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.69694 −0.0934434
\(834\) 0 0
\(835\) −6.44949 −0.223194
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 39.7980 1.37398 0.686989 0.726668i \(-0.258932\pi\)
0.686989 + 0.726668i \(0.258932\pi\)
\(840\) 0 0
\(841\) 33.3939 1.15151
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.79796 0.233857
\(846\) 0 0
\(847\) −17.2474 −0.592629
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.34847 −0.286182
\(852\) 0 0
\(853\) −29.1918 −0.999509 −0.499755 0.866167i \(-0.666577\pi\)
−0.499755 + 0.866167i \(0.666577\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.6969 −1.39018 −0.695090 0.718922i \(-0.744636\pi\)
−0.695090 + 0.718922i \(0.744636\pi\)
\(858\) 0 0
\(859\) −55.0908 −1.87967 −0.939837 0.341623i \(-0.889024\pi\)
−0.939837 + 0.341623i \(0.889024\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21.5959 0.735134 0.367567 0.929997i \(-0.380191\pi\)
0.367567 + 0.929997i \(0.380191\pi\)
\(864\) 0 0
\(865\) 1.79796 0.0611324
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 42.0454 1.42465
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.44949 0.116614
\(876\) 0 0
\(877\) −51.7980 −1.74909 −0.874546 0.484942i \(-0.838841\pi\)
−0.874546 + 0.484942i \(0.838841\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.34847 −0.0454311 −0.0227155 0.999742i \(-0.507231\pi\)
−0.0227155 + 0.999742i \(0.507231\pi\)
\(882\) 0 0
\(883\) 24.7423 0.832646 0.416323 0.909217i \(-0.363319\pi\)
0.416323 + 0.909217i \(0.363319\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.79796 −0.127523 −0.0637615 0.997965i \(-0.520310\pi\)
−0.0637615 + 0.997965i \(0.520310\pi\)
\(888\) 0 0
\(889\) 69.8434 2.34247
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.79796 −0.127094
\(894\) 0 0
\(895\) 9.79796 0.327510
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 62.3939 2.08095
\(900\) 0 0
\(901\) −4.10102 −0.136625
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.10102 −0.236046
\(906\) 0 0
\(907\) −15.4495 −0.512992 −0.256496 0.966545i \(-0.582568\pi\)
−0.256496 + 0.966545i \(0.582568\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.4949 0.679026 0.339513 0.940601i \(-0.389738\pi\)
0.339513 + 0.940601i \(0.389738\pi\)
\(912\) 0 0
\(913\) 37.8434 1.25243
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 37.5959 1.24153
\(918\) 0 0
\(919\) 13.3031 0.438828 0.219414 0.975632i \(-0.429586\pi\)
0.219414 + 0.975632i \(0.429586\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 26.2474 0.863945
\(924\) 0 0
\(925\) 8.34847 0.274496
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 20.1918 0.662473 0.331236 0.943548i \(-0.392534\pi\)
0.331236 + 0.943548i \(0.392534\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.34847 0.0440997
\(936\) 0 0
\(937\) −42.8990 −1.40145 −0.700724 0.713432i \(-0.747140\pi\)
−0.700724 + 0.713432i \(0.747140\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.34847 −0.0439588 −0.0219794 0.999758i \(-0.506997\pi\)
−0.0219794 + 0.999758i \(0.506997\pi\)
\(942\) 0 0
\(943\) 1.89898 0.0618393
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.1010 −0.360735 −0.180367 0.983599i \(-0.557729\pi\)
−0.180367 + 0.983599i \(0.557729\pi\)
\(948\) 0 0
\(949\) 15.7980 0.512823
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 23.3939 0.757802 0.378901 0.925437i \(-0.376302\pi\)
0.378901 + 0.925437i \(0.376302\pi\)
\(954\) 0 0
\(955\) −14.2474 −0.461037
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −30.6969 −0.991256
\(960\) 0 0
\(961\) 31.3939 1.01271
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13.7980 −0.444172
\(966\) 0 0
\(967\) −27.3485 −0.879467 −0.439734 0.898128i \(-0.644927\pi\)
−0.439734 + 0.898128i \(0.644927\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 46.6969 1.49858 0.749288 0.662244i \(-0.230396\pi\)
0.749288 + 0.662244i \(0.230396\pi\)
\(972\) 0 0
\(973\) 44.8434 1.43761
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 22.3485 0.714991 0.357495 0.933915i \(-0.383631\pi\)
0.357495 + 0.933915i \(0.383631\pi\)
\(978\) 0 0
\(979\) 21.7980 0.696666
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34.1464 1.08910 0.544551 0.838728i \(-0.316700\pi\)
0.544551 + 0.838728i \(0.316700\pi\)
\(984\) 0 0
\(985\) 3.79796 0.121013
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.898979 0.0285859
\(990\) 0 0
\(991\) 30.1918 0.959075 0.479538 0.877521i \(-0.340804\pi\)
0.479538 + 0.877521i \(0.340804\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.0000 0.443830
\(996\) 0 0
\(997\) −32.0000 −1.01345 −0.506725 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) 0 0
\(999\) 0 0
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 8280.2.a.bg.1.2 yes 2
3.2 odd 2 8280.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8280.2.a.ba.1.2 2 3.2 odd 2
8280.2.a.bg.1.2 yes 2 1.1 even 1 trivial