Properties

Label 3696.2.a.bo.1.2
Level $3696$
Weight $2$
Character 3696.1
Self dual yes
Analytic conductor $29.513$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 3696.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.39821 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.39821 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.04502 q^{13} -2.39821 q^{15} -6.36842 q^{17} +5.32340 q^{19} -1.00000 q^{21} -4.92520 q^{23} +0.751399 q^{25} -1.00000 q^{27} +5.04502 q^{29} +7.57201 q^{31} -1.00000 q^{33} +2.39821 q^{35} +4.24860 q^{37} -5.04502 q^{39} -0.646809 q^{41} +10.5180 q^{43} +2.39821 q^{45} -0.526989 q^{47} +1.00000 q^{49} +6.36842 q^{51} +3.72161 q^{53} +2.39821 q^{55} -5.32340 q^{57} -7.97021 q^{59} -2.00000 q^{61} +1.00000 q^{63} +12.0990 q^{65} -8.76663 q^{67} +4.92520 q^{69} +11.4432 q^{71} -13.0450 q^{73} -0.751399 q^{75} +1.00000 q^{77} -11.4432 q^{79} +1.00000 q^{81} -13.1648 q^{83} -15.2728 q^{85} -5.04502 q^{87} +11.8504 q^{89} +5.04502 q^{91} -7.57201 q^{93} +12.7666 q^{95} -1.87122 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99} + O(q^{100}) \)

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
\(5\) 2.39821 1.07251 0.536255 0.844056i \(-0.319838\pi\)
0.536255 + 0.844056i \(0.319838\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.04502 1.39924 0.699618 0.714517i \(-0.253354\pi\)
0.699618 + 0.714517i \(0.253354\pi\)
\(14\) 0 0
\(15\) −2.39821 −0.619214
\(16\) 0 0
\(17\) −6.36842 −1.54457 −0.772284 0.635277i \(-0.780886\pi\)
−0.772284 + 0.635277i \(0.780886\pi\)
\(18\) 0 0
\(19\) 5.32340 1.22127 0.610636 0.791911i \(-0.290914\pi\)
0.610636 + 0.791911i \(0.290914\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.92520 −1.02697 −0.513487 0.858097i \(-0.671647\pi\)
−0.513487 + 0.858097i \(0.671647\pi\)
\(24\) 0 0
\(25\) 0.751399 0.150280
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.04502 0.936836 0.468418 0.883507i \(-0.344824\pi\)
0.468418 + 0.883507i \(0.344824\pi\)
\(30\) 0 0
\(31\) 7.57201 1.35997 0.679986 0.733225i \(-0.261986\pi\)
0.679986 + 0.733225i \(0.261986\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 2.39821 0.405371
\(36\) 0 0
\(37\) 4.24860 0.698466 0.349233 0.937036i \(-0.386442\pi\)
0.349233 + 0.937036i \(0.386442\pi\)
\(38\) 0 0
\(39\) −5.04502 −0.807849
\(40\) 0 0
\(41\) −0.646809 −0.101015 −0.0505073 0.998724i \(-0.516084\pi\)
−0.0505073 + 0.998724i \(0.516084\pi\)
\(42\) 0 0
\(43\) 10.5180 1.60398 0.801992 0.597335i \(-0.203774\pi\)
0.801992 + 0.597335i \(0.203774\pi\)
\(44\) 0 0
\(45\) 2.39821 0.357504
\(46\) 0 0
\(47\) −0.526989 −0.0768693 −0.0384347 0.999261i \(-0.512237\pi\)
−0.0384347 + 0.999261i \(0.512237\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.36842 0.891757
\(52\) 0 0
\(53\) 3.72161 0.511203 0.255601 0.966782i \(-0.417727\pi\)
0.255601 + 0.966782i \(0.417727\pi\)
\(54\) 0 0
\(55\) 2.39821 0.323374
\(56\) 0 0
\(57\) −5.32340 −0.705102
\(58\) 0 0
\(59\) −7.97021 −1.03763 −0.518817 0.854886i \(-0.673627\pi\)
−0.518817 + 0.854886i \(0.673627\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 12.0990 1.50070
\(66\) 0 0
\(67\) −8.76663 −1.07101 −0.535507 0.844531i \(-0.679879\pi\)
−0.535507 + 0.844531i \(0.679879\pi\)
\(68\) 0 0
\(69\) 4.92520 0.592924
\(70\) 0 0
\(71\) 11.4432 1.35806 0.679030 0.734110i \(-0.262400\pi\)
0.679030 + 0.734110i \(0.262400\pi\)
\(72\) 0 0
\(73\) −13.0450 −1.52680 −0.763402 0.645924i \(-0.776472\pi\)
−0.763402 + 0.645924i \(0.776472\pi\)
\(74\) 0 0
\(75\) −0.751399 −0.0867641
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −11.4432 −1.28746 −0.643732 0.765251i \(-0.722615\pi\)
−0.643732 + 0.765251i \(0.722615\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.1648 −1.44503 −0.722514 0.691356i \(-0.757014\pi\)
−0.722514 + 0.691356i \(0.757014\pi\)
\(84\) 0 0
\(85\) −15.2728 −1.65657
\(86\) 0 0
\(87\) −5.04502 −0.540882
\(88\) 0 0
\(89\) 11.8504 1.25614 0.628070 0.778157i \(-0.283845\pi\)
0.628070 + 0.778157i \(0.283845\pi\)
\(90\) 0 0
\(91\) 5.04502 0.528861
\(92\) 0 0
\(93\) −7.57201 −0.785180
\(94\) 0 0
\(95\) 12.7666 1.30983
\(96\) 0 0
\(97\) −1.87122 −0.189993 −0.0949967 0.995478i \(-0.530284\pi\)
−0.0949967 + 0.995478i \(0.530284\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 4.51803 0.449560 0.224780 0.974409i \(-0.427834\pi\)
0.224780 + 0.974409i \(0.427834\pi\)
\(102\) 0 0
\(103\) 10.6468 1.04906 0.524531 0.851392i \(-0.324241\pi\)
0.524531 + 0.851392i \(0.324241\pi\)
\(104\) 0 0
\(105\) −2.39821 −0.234041
\(106\) 0 0
\(107\) −15.9702 −1.54390 −0.771949 0.635684i \(-0.780718\pi\)
−0.771949 + 0.635684i \(0.780718\pi\)
\(108\) 0 0
\(109\) 12.7756 1.22368 0.611840 0.790982i \(-0.290430\pi\)
0.611840 + 0.790982i \(0.290430\pi\)
\(110\) 0 0
\(111\) −4.24860 −0.403259
\(112\) 0 0
\(113\) 18.7368 1.76261 0.881307 0.472544i \(-0.156664\pi\)
0.881307 + 0.472544i \(0.156664\pi\)
\(114\) 0 0
\(115\) −11.8116 −1.10144
\(116\) 0 0
\(117\) 5.04502 0.466412
\(118\) 0 0
\(119\) −6.36842 −0.583792
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0.646809 0.0583208
\(124\) 0 0
\(125\) −10.1890 −0.911334
\(126\) 0 0
\(127\) 2.27839 0.202174 0.101087 0.994878i \(-0.467768\pi\)
0.101087 + 0.994878i \(0.467768\pi\)
\(128\) 0 0
\(129\) −10.5180 −0.926061
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 5.32340 0.461598
\(134\) 0 0
\(135\) −2.39821 −0.206405
\(136\) 0 0
\(137\) −4.77559 −0.408006 −0.204003 0.978970i \(-0.565395\pi\)
−0.204003 + 0.978970i \(0.565395\pi\)
\(138\) 0 0
\(139\) 15.4432 1.30988 0.654939 0.755682i \(-0.272694\pi\)
0.654939 + 0.755682i \(0.272694\pi\)
\(140\) 0 0
\(141\) 0.526989 0.0443805
\(142\) 0 0
\(143\) 5.04502 0.421885
\(144\) 0 0
\(145\) 12.0990 1.00477
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −9.84143 −0.806241 −0.403121 0.915147i \(-0.632075\pi\)
−0.403121 + 0.915147i \(0.632075\pi\)
\(150\) 0 0
\(151\) 4.12878 0.335996 0.167998 0.985787i \(-0.446270\pi\)
0.167998 + 0.985787i \(0.446270\pi\)
\(152\) 0 0
\(153\) −6.36842 −0.514856
\(154\) 0 0
\(155\) 18.1592 1.45859
\(156\) 0 0
\(157\) −0.946021 −0.0755007 −0.0377504 0.999287i \(-0.512019\pi\)
−0.0377504 + 0.999287i \(0.512019\pi\)
\(158\) 0 0
\(159\) −3.72161 −0.295143
\(160\) 0 0
\(161\) −4.92520 −0.388160
\(162\) 0 0
\(163\) −8.76663 −0.686655 −0.343328 0.939216i \(-0.611554\pi\)
−0.343328 + 0.939216i \(0.611554\pi\)
\(164\) 0 0
\(165\) −2.39821 −0.186700
\(166\) 0 0
\(167\) 24.3684 1.88568 0.942842 0.333239i \(-0.108142\pi\)
0.942842 + 0.333239i \(0.108142\pi\)
\(168\) 0 0
\(169\) 12.4522 0.957860
\(170\) 0 0
\(171\) 5.32340 0.407091
\(172\) 0 0
\(173\) 12.3476 0.938770 0.469385 0.882994i \(-0.344476\pi\)
0.469385 + 0.882994i \(0.344476\pi\)
\(174\) 0 0
\(175\) 0.751399 0.0568004
\(176\) 0 0
\(177\) 7.97021 0.599078
\(178\) 0 0
\(179\) 5.59283 0.418028 0.209014 0.977913i \(-0.432975\pi\)
0.209014 + 0.977913i \(0.432975\pi\)
\(180\) 0 0
\(181\) 13.5720 1.00880 0.504400 0.863470i \(-0.331714\pi\)
0.504400 + 0.863470i \(0.331714\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 10.1890 0.749112
\(186\) 0 0
\(187\) −6.36842 −0.465705
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 9.42240 0.681781 0.340890 0.940103i \(-0.389271\pi\)
0.340890 + 0.940103i \(0.389271\pi\)
\(192\) 0 0
\(193\) −10.1288 −0.729086 −0.364543 0.931187i \(-0.618775\pi\)
−0.364543 + 0.931187i \(0.618775\pi\)
\(194\) 0 0
\(195\) −12.0990 −0.866427
\(196\) 0 0
\(197\) −2.25756 −0.160845 −0.0804224 0.996761i \(-0.525627\pi\)
−0.0804224 + 0.996761i \(0.525627\pi\)
\(198\) 0 0
\(199\) −3.07480 −0.217967 −0.108984 0.994044i \(-0.534760\pi\)
−0.108984 + 0.994044i \(0.534760\pi\)
\(200\) 0 0
\(201\) 8.76663 0.618350
\(202\) 0 0
\(203\) 5.04502 0.354091
\(204\) 0 0
\(205\) −1.55118 −0.108339
\(206\) 0 0
\(207\) −4.92520 −0.342325
\(208\) 0 0
\(209\) 5.32340 0.368228
\(210\) 0 0
\(211\) 14.6468 1.00833 0.504164 0.863608i \(-0.331801\pi\)
0.504164 + 0.863608i \(0.331801\pi\)
\(212\) 0 0
\(213\) −11.4432 −0.784077
\(214\) 0 0
\(215\) 25.2244 1.72029
\(216\) 0 0
\(217\) 7.57201 0.514021
\(218\) 0 0
\(219\) 13.0450 0.881500
\(220\) 0 0
\(221\) −32.1288 −2.16122
\(222\) 0 0
\(223\) 1.90997 0.127901 0.0639505 0.997953i \(-0.479630\pi\)
0.0639505 + 0.997953i \(0.479630\pi\)
\(224\) 0 0
\(225\) 0.751399 0.0500933
\(226\) 0 0
\(227\) 3.20359 0.212629 0.106315 0.994333i \(-0.466095\pi\)
0.106315 + 0.994333i \(0.466095\pi\)
\(228\) 0 0
\(229\) −18.3088 −1.20988 −0.604941 0.796270i \(-0.706803\pi\)
−0.604941 + 0.796270i \(0.706803\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −16.5872 −1.08667 −0.543333 0.839517i \(-0.682838\pi\)
−0.543333 + 0.839517i \(0.682838\pi\)
\(234\) 0 0
\(235\) −1.26383 −0.0824432
\(236\) 0 0
\(237\) 11.4432 0.743317
\(238\) 0 0
\(239\) −2.91623 −0.188635 −0.0943177 0.995542i \(-0.530067\pi\)
−0.0943177 + 0.995542i \(0.530067\pi\)
\(240\) 0 0
\(241\) −6.09899 −0.392871 −0.196435 0.980517i \(-0.562937\pi\)
−0.196435 + 0.980517i \(0.562937\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.39821 0.153216
\(246\) 0 0
\(247\) 26.8567 1.70885
\(248\) 0 0
\(249\) 13.1648 0.834288
\(250\) 0 0
\(251\) −1.62262 −0.102419 −0.0512093 0.998688i \(-0.516308\pi\)
−0.0512093 + 0.998688i \(0.516308\pi\)
\(252\) 0 0
\(253\) −4.92520 −0.309644
\(254\) 0 0
\(255\) 15.2728 0.956419
\(256\) 0 0
\(257\) −6.89541 −0.430124 −0.215062 0.976600i \(-0.568995\pi\)
−0.215062 + 0.976600i \(0.568995\pi\)
\(258\) 0 0
\(259\) 4.24860 0.263995
\(260\) 0 0
\(261\) 5.04502 0.312279
\(262\) 0 0
\(263\) −5.08377 −0.313478 −0.156739 0.987640i \(-0.550098\pi\)
−0.156739 + 0.987640i \(0.550098\pi\)
\(264\) 0 0
\(265\) 8.92520 0.548270
\(266\) 0 0
\(267\) −11.8504 −0.725232
\(268\) 0 0
\(269\) −0.886447 −0.0540476 −0.0270238 0.999635i \(-0.508603\pi\)
−0.0270238 + 0.999635i \(0.508603\pi\)
\(270\) 0 0
\(271\) 25.3234 1.53829 0.769144 0.639076i \(-0.220683\pi\)
0.769144 + 0.639076i \(0.220683\pi\)
\(272\) 0 0
\(273\) −5.04502 −0.305338
\(274\) 0 0
\(275\) 0.751399 0.0453111
\(276\) 0 0
\(277\) −24.8269 −1.49170 −0.745851 0.666113i \(-0.767957\pi\)
−0.745851 + 0.666113i \(0.767957\pi\)
\(278\) 0 0
\(279\) 7.57201 0.453324
\(280\) 0 0
\(281\) 1.90101 0.113404 0.0567022 0.998391i \(-0.481941\pi\)
0.0567022 + 0.998391i \(0.481941\pi\)
\(282\) 0 0
\(283\) 22.3178 1.32666 0.663328 0.748329i \(-0.269143\pi\)
0.663328 + 0.748329i \(0.269143\pi\)
\(284\) 0 0
\(285\) −12.7666 −0.756230
\(286\) 0 0
\(287\) −0.646809 −0.0381799
\(288\) 0 0
\(289\) 23.5568 1.38569
\(290\) 0 0
\(291\) 1.87122 0.109693
\(292\) 0 0
\(293\) 12.0900 0.706307 0.353154 0.935565i \(-0.385109\pi\)
0.353154 + 0.935565i \(0.385109\pi\)
\(294\) 0 0
\(295\) −19.1142 −1.11287
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −24.8477 −1.43698
\(300\) 0 0
\(301\) 10.5180 0.606249
\(302\) 0 0
\(303\) −4.51803 −0.259554
\(304\) 0 0
\(305\) −4.79641 −0.274642
\(306\) 0 0
\(307\) 13.5928 0.775784 0.387892 0.921705i \(-0.373203\pi\)
0.387892 + 0.921705i \(0.373203\pi\)
\(308\) 0 0
\(309\) −10.6468 −0.605676
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 14.9252 0.843622 0.421811 0.906684i \(-0.361395\pi\)
0.421811 + 0.906684i \(0.361395\pi\)
\(314\) 0 0
\(315\) 2.39821 0.135124
\(316\) 0 0
\(317\) 3.97918 0.223493 0.111746 0.993737i \(-0.464356\pi\)
0.111746 + 0.993737i \(0.464356\pi\)
\(318\) 0 0
\(319\) 5.04502 0.282467
\(320\) 0 0
\(321\) 15.9702 0.891370
\(322\) 0 0
\(323\) −33.9017 −1.88634
\(324\) 0 0
\(325\) 3.79082 0.210277
\(326\) 0 0
\(327\) −12.7756 −0.706492
\(328\) 0 0
\(329\) −0.526989 −0.0290539
\(330\) 0 0
\(331\) −23.4432 −1.28856 −0.644278 0.764791i \(-0.722842\pi\)
−0.644278 + 0.764791i \(0.722842\pi\)
\(332\) 0 0
\(333\) 4.24860 0.232822
\(334\) 0 0
\(335\) −21.0242 −1.14867
\(336\) 0 0
\(337\) 11.1648 0.608187 0.304094 0.952642i \(-0.401646\pi\)
0.304094 + 0.952642i \(0.401646\pi\)
\(338\) 0 0
\(339\) −18.7368 −1.01765
\(340\) 0 0
\(341\) 7.57201 0.410047
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 11.8116 0.635918
\(346\) 0 0
\(347\) 22.5872 1.21255 0.606273 0.795256i \(-0.292664\pi\)
0.606273 + 0.795256i \(0.292664\pi\)
\(348\) 0 0
\(349\) 27.9315 1.49514 0.747568 0.664185i \(-0.231221\pi\)
0.747568 + 0.664185i \(0.231221\pi\)
\(350\) 0 0
\(351\) −5.04502 −0.269283
\(352\) 0 0
\(353\) −16.5478 −0.880751 −0.440376 0.897814i \(-0.645155\pi\)
−0.440376 + 0.897814i \(0.645155\pi\)
\(354\) 0 0
\(355\) 27.4432 1.45654
\(356\) 0 0
\(357\) 6.36842 0.337053
\(358\) 0 0
\(359\) −22.0305 −1.16272 −0.581362 0.813645i \(-0.697480\pi\)
−0.581362 + 0.813645i \(0.697480\pi\)
\(360\) 0 0
\(361\) 9.33863 0.491507
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −31.2847 −1.63751
\(366\) 0 0
\(367\) 19.3836 1.01182 0.505909 0.862587i \(-0.331157\pi\)
0.505909 + 0.862587i \(0.331157\pi\)
\(368\) 0 0
\(369\) −0.646809 −0.0336715
\(370\) 0 0
\(371\) 3.72161 0.193216
\(372\) 0 0
\(373\) 29.2549 1.51476 0.757380 0.652975i \(-0.226479\pi\)
0.757380 + 0.652975i \(0.226479\pi\)
\(374\) 0 0
\(375\) 10.1890 0.526159
\(376\) 0 0
\(377\) 25.4522 1.31085
\(378\) 0 0
\(379\) −12.5270 −0.643468 −0.321734 0.946830i \(-0.604266\pi\)
−0.321734 + 0.946830i \(0.604266\pi\)
\(380\) 0 0
\(381\) −2.27839 −0.116725
\(382\) 0 0
\(383\) 17.5928 0.898952 0.449476 0.893293i \(-0.351611\pi\)
0.449476 + 0.893293i \(0.351611\pi\)
\(384\) 0 0
\(385\) 2.39821 0.122224
\(386\) 0 0
\(387\) 10.5180 0.534661
\(388\) 0 0
\(389\) 20.0900 1.01861 0.509303 0.860588i \(-0.329903\pi\)
0.509303 + 0.860588i \(0.329903\pi\)
\(390\) 0 0
\(391\) 31.3657 1.58623
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −27.4432 −1.38082
\(396\) 0 0
\(397\) −35.1053 −1.76188 −0.880941 0.473226i \(-0.843090\pi\)
−0.880941 + 0.473226i \(0.843090\pi\)
\(398\) 0 0
\(399\) −5.32340 −0.266504
\(400\) 0 0
\(401\) 9.57201 0.478003 0.239002 0.971019i \(-0.423180\pi\)
0.239002 + 0.971019i \(0.423180\pi\)
\(402\) 0 0
\(403\) 38.2009 1.90292
\(404\) 0 0
\(405\) 2.39821 0.119168
\(406\) 0 0
\(407\) 4.24860 0.210595
\(408\) 0 0
\(409\) 38.1801 1.88788 0.943941 0.330113i \(-0.107087\pi\)
0.943941 + 0.330113i \(0.107087\pi\)
\(410\) 0 0
\(411\) 4.77559 0.235563
\(412\) 0 0
\(413\) −7.97021 −0.392189
\(414\) 0 0
\(415\) −31.5720 −1.54981
\(416\) 0 0
\(417\) −15.4432 −0.756258
\(418\) 0 0
\(419\) −7.17380 −0.350463 −0.175231 0.984527i \(-0.556067\pi\)
−0.175231 + 0.984527i \(0.556067\pi\)
\(420\) 0 0
\(421\) 15.1530 0.738511 0.369255 0.929328i \(-0.379613\pi\)
0.369255 + 0.929328i \(0.379613\pi\)
\(422\) 0 0
\(423\) −0.526989 −0.0256231
\(424\) 0 0
\(425\) −4.78522 −0.232117
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −5.04502 −0.243576
\(430\) 0 0
\(431\) −5.56304 −0.267962 −0.133981 0.990984i \(-0.542776\pi\)
−0.133981 + 0.990984i \(0.542776\pi\)
\(432\) 0 0
\(433\) −25.6412 −1.23224 −0.616119 0.787653i \(-0.711296\pi\)
−0.616119 + 0.787653i \(0.711296\pi\)
\(434\) 0 0
\(435\) −12.0990 −0.580102
\(436\) 0 0
\(437\) −26.2188 −1.25422
\(438\) 0 0
\(439\) −23.6710 −1.12976 −0.564878 0.825175i \(-0.691077\pi\)
−0.564878 + 0.825175i \(0.691077\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.0305 0.856653 0.428326 0.903624i \(-0.359103\pi\)
0.428326 + 0.903624i \(0.359103\pi\)
\(444\) 0 0
\(445\) 28.4197 1.34722
\(446\) 0 0
\(447\) 9.84143 0.465484
\(448\) 0 0
\(449\) −34.9765 −1.65064 −0.825321 0.564664i \(-0.809006\pi\)
−0.825321 + 0.564664i \(0.809006\pi\)
\(450\) 0 0
\(451\) −0.646809 −0.0304570
\(452\) 0 0
\(453\) −4.12878 −0.193987
\(454\) 0 0
\(455\) 12.0990 0.567210
\(456\) 0 0
\(457\) 4.53595 0.212183 0.106091 0.994356i \(-0.466166\pi\)
0.106091 + 0.994356i \(0.466166\pi\)
\(458\) 0 0
\(459\) 6.36842 0.297252
\(460\) 0 0
\(461\) −2.79641 −0.130242 −0.0651210 0.997877i \(-0.520743\pi\)
−0.0651210 + 0.997877i \(0.520743\pi\)
\(462\) 0 0
\(463\) −38.3595 −1.78272 −0.891358 0.453301i \(-0.850246\pi\)
−0.891358 + 0.453301i \(0.850246\pi\)
\(464\) 0 0
\(465\) −18.1592 −0.842115
\(466\) 0 0
\(467\) 20.4674 0.947119 0.473560 0.880762i \(-0.342969\pi\)
0.473560 + 0.880762i \(0.342969\pi\)
\(468\) 0 0
\(469\) −8.76663 −0.404805
\(470\) 0 0
\(471\) 0.946021 0.0435904
\(472\) 0 0
\(473\) 10.5180 0.483619
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 3.72161 0.170401
\(478\) 0 0
\(479\) 11.6137 0.530641 0.265321 0.964160i \(-0.414522\pi\)
0.265321 + 0.964160i \(0.414522\pi\)
\(480\) 0 0
\(481\) 21.4343 0.977318
\(482\) 0 0
\(483\) 4.92520 0.224104
\(484\) 0 0
\(485\) −4.48757 −0.203770
\(486\) 0 0
\(487\) −32.4793 −1.47178 −0.735888 0.677103i \(-0.763235\pi\)
−0.735888 + 0.677103i \(0.763235\pi\)
\(488\) 0 0
\(489\) 8.76663 0.396441
\(490\) 0 0
\(491\) −26.6766 −1.20390 −0.601949 0.798535i \(-0.705609\pi\)
−0.601949 + 0.798535i \(0.705609\pi\)
\(492\) 0 0
\(493\) −32.1288 −1.44701
\(494\) 0 0
\(495\) 2.39821 0.107791
\(496\) 0 0
\(497\) 11.4432 0.513299
\(498\) 0 0
\(499\) 41.2459 1.84642 0.923210 0.384296i \(-0.125556\pi\)
0.923210 + 0.384296i \(0.125556\pi\)
\(500\) 0 0
\(501\) −24.3684 −1.08870
\(502\) 0 0
\(503\) −30.5180 −1.36073 −0.680366 0.732873i \(-0.738179\pi\)
−0.680366 + 0.732873i \(0.738179\pi\)
\(504\) 0 0
\(505\) 10.8352 0.482159
\(506\) 0 0
\(507\) −12.4522 −0.553021
\(508\) 0 0
\(509\) −18.9944 −0.841912 −0.420956 0.907081i \(-0.638305\pi\)
−0.420956 + 0.907081i \(0.638305\pi\)
\(510\) 0 0
\(511\) −13.0450 −0.577078
\(512\) 0 0
\(513\) −5.32340 −0.235034
\(514\) 0 0
\(515\) 25.5333 1.12513
\(516\) 0 0
\(517\) −0.526989 −0.0231770
\(518\) 0 0
\(519\) −12.3476 −0.541999
\(520\) 0 0
\(521\) 25.2430 1.10592 0.552958 0.833209i \(-0.313499\pi\)
0.552958 + 0.833209i \(0.313499\pi\)
\(522\) 0 0
\(523\) 2.93416 0.128302 0.0641509 0.997940i \(-0.479566\pi\)
0.0641509 + 0.997940i \(0.479566\pi\)
\(524\) 0 0
\(525\) −0.751399 −0.0327937
\(526\) 0 0
\(527\) −48.2217 −2.10057
\(528\) 0 0
\(529\) 1.25756 0.0546767
\(530\) 0 0
\(531\) −7.97021 −0.345878
\(532\) 0 0
\(533\) −3.26316 −0.141343
\(534\) 0 0
\(535\) −38.2999 −1.65585
\(536\) 0 0
\(537\) −5.59283 −0.241348
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 8.90437 0.382829 0.191414 0.981509i \(-0.438693\pi\)
0.191414 + 0.981509i \(0.438693\pi\)
\(542\) 0 0
\(543\) −13.5720 −0.582430
\(544\) 0 0
\(545\) 30.6385 1.31241
\(546\) 0 0
\(547\) 29.4737 1.26020 0.630102 0.776513i \(-0.283013\pi\)
0.630102 + 0.776513i \(0.283013\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 26.8567 1.14413
\(552\) 0 0
\(553\) −11.4432 −0.486615
\(554\) 0 0
\(555\) −10.1890 −0.432500
\(556\) 0 0
\(557\) 14.8954 0.631139 0.315569 0.948903i \(-0.397804\pi\)
0.315569 + 0.948903i \(0.397804\pi\)
\(558\) 0 0
\(559\) 53.0636 2.24435
\(560\) 0 0
\(561\) 6.36842 0.268875
\(562\) 0 0
\(563\) 7.81164 0.329222 0.164611 0.986359i \(-0.447363\pi\)
0.164611 + 0.986359i \(0.447363\pi\)
\(564\) 0 0
\(565\) 44.9348 1.89042
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −25.5512 −1.06928 −0.534642 0.845079i \(-0.679553\pi\)
−0.534642 + 0.845079i \(0.679553\pi\)
\(572\) 0 0
\(573\) −9.42240 −0.393626
\(574\) 0 0
\(575\) −3.70079 −0.154334
\(576\) 0 0
\(577\) −29.5124 −1.22862 −0.614309 0.789065i \(-0.710565\pi\)
−0.614309 + 0.789065i \(0.710565\pi\)
\(578\) 0 0
\(579\) 10.1288 0.420938
\(580\) 0 0
\(581\) −13.1648 −0.546169
\(582\) 0 0
\(583\) 3.72161 0.154133
\(584\) 0 0
\(585\) 12.0990 0.500232
\(586\) 0 0
\(587\) −31.3955 −1.29583 −0.647916 0.761712i \(-0.724359\pi\)
−0.647916 + 0.761712i \(0.724359\pi\)
\(588\) 0 0
\(589\) 40.3088 1.66090
\(590\) 0 0
\(591\) 2.25756 0.0928638
\(592\) 0 0
\(593\) −7.90997 −0.324823 −0.162412 0.986723i \(-0.551927\pi\)
−0.162412 + 0.986723i \(0.551927\pi\)
\(594\) 0 0
\(595\) −15.2728 −0.626123
\(596\) 0 0
\(597\) 3.07480 0.125843
\(598\) 0 0
\(599\) −27.4432 −1.12130 −0.560650 0.828053i \(-0.689449\pi\)
−0.560650 + 0.828053i \(0.689449\pi\)
\(600\) 0 0
\(601\) 31.9910 1.30494 0.652471 0.757814i \(-0.273732\pi\)
0.652471 + 0.757814i \(0.273732\pi\)
\(602\) 0 0
\(603\) −8.76663 −0.357005
\(604\) 0 0
\(605\) 2.39821 0.0975010
\(606\) 0 0
\(607\) −7.41344 −0.300902 −0.150451 0.988617i \(-0.548073\pi\)
−0.150451 + 0.988617i \(0.548073\pi\)
\(608\) 0 0
\(609\) −5.04502 −0.204434
\(610\) 0 0
\(611\) −2.65867 −0.107558
\(612\) 0 0
\(613\) −33.9917 −1.37291 −0.686456 0.727171i \(-0.740835\pi\)
−0.686456 + 0.727171i \(0.740835\pi\)
\(614\) 0 0
\(615\) 1.55118 0.0625497
\(616\) 0 0
\(617\) −44.0305 −1.77260 −0.886300 0.463112i \(-0.846733\pi\)
−0.886300 + 0.463112i \(0.846733\pi\)
\(618\) 0 0
\(619\) −40.0096 −1.60812 −0.804061 0.594546i \(-0.797332\pi\)
−0.804061 + 0.594546i \(0.797332\pi\)
\(620\) 0 0
\(621\) 4.92520 0.197641
\(622\) 0 0
\(623\) 11.8504 0.474776
\(624\) 0 0
\(625\) −28.1924 −1.12770
\(626\) 0 0
\(627\) −5.32340 −0.212596
\(628\) 0 0
\(629\) −27.0569 −1.07883
\(630\) 0 0
\(631\) −28.5568 −1.13683 −0.568414 0.822743i \(-0.692443\pi\)
−0.568414 + 0.822743i \(0.692443\pi\)
\(632\) 0 0
\(633\) −14.6468 −0.582158
\(634\) 0 0
\(635\) 5.46405 0.216834
\(636\) 0 0
\(637\) 5.04502 0.199891
\(638\) 0 0
\(639\) 11.4432 0.452687
\(640\) 0 0
\(641\) −31.1053 −1.22858 −0.614292 0.789079i \(-0.710558\pi\)
−0.614292 + 0.789079i \(0.710558\pi\)
\(642\) 0 0
\(643\) 5.48197 0.216188 0.108094 0.994141i \(-0.465525\pi\)
0.108094 + 0.994141i \(0.465525\pi\)
\(644\) 0 0
\(645\) −25.2244 −0.993210
\(646\) 0 0
\(647\) 9.26383 0.364199 0.182099 0.983280i \(-0.441711\pi\)
0.182099 + 0.983280i \(0.441711\pi\)
\(648\) 0 0
\(649\) −7.97021 −0.312858
\(650\) 0 0
\(651\) −7.57201 −0.296770
\(652\) 0 0
\(653\) −29.9821 −1.17329 −0.586645 0.809844i \(-0.699551\pi\)
−0.586645 + 0.809844i \(0.699551\pi\)
\(654\) 0 0
\(655\) −9.59283 −0.374823
\(656\) 0 0
\(657\) −13.0450 −0.508935
\(658\) 0 0
\(659\) −23.9702 −0.933747 −0.466873 0.884324i \(-0.654620\pi\)
−0.466873 + 0.884324i \(0.654620\pi\)
\(660\) 0 0
\(661\) −40.4585 −1.57365 −0.786826 0.617175i \(-0.788277\pi\)
−0.786826 + 0.617175i \(0.788277\pi\)
\(662\) 0 0
\(663\) 32.1288 1.24778
\(664\) 0 0
\(665\) 12.7666 0.495069
\(666\) 0 0
\(667\) −24.8477 −0.962107
\(668\) 0 0
\(669\) −1.90997 −0.0738436
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 21.8712 0.843073 0.421537 0.906811i \(-0.361491\pi\)
0.421537 + 0.906811i \(0.361491\pi\)
\(674\) 0 0
\(675\) −0.751399 −0.0289214
\(676\) 0 0
\(677\) −1.26316 −0.0485472 −0.0242736 0.999705i \(-0.507727\pi\)
−0.0242736 + 0.999705i \(0.507727\pi\)
\(678\) 0 0
\(679\) −1.87122 −0.0718108
\(680\) 0 0
\(681\) −3.20359 −0.122762
\(682\) 0 0
\(683\) −37.6441 −1.44041 −0.720206 0.693760i \(-0.755953\pi\)
−0.720206 + 0.693760i \(0.755953\pi\)
\(684\) 0 0
\(685\) −11.4529 −0.437591
\(686\) 0 0
\(687\) 18.3088 0.698526
\(688\) 0 0
\(689\) 18.7756 0.715293
\(690\) 0 0
\(691\) −14.3892 −0.547393 −0.273696 0.961816i \(-0.588246\pi\)
−0.273696 + 0.961816i \(0.588246\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 37.0361 1.40486
\(696\) 0 0
\(697\) 4.11915 0.156024
\(698\) 0 0
\(699\) 16.5872 0.627387
\(700\) 0 0
\(701\) −39.2936 −1.48410 −0.742050 0.670345i \(-0.766146\pi\)
−0.742050 + 0.670345i \(0.766146\pi\)
\(702\) 0 0
\(703\) 22.6170 0.853017
\(704\) 0 0
\(705\) 1.26383 0.0475986
\(706\) 0 0
\(707\) 4.51803 0.169918
\(708\) 0 0
\(709\) −49.2430 −1.84936 −0.924680 0.380745i \(-0.875667\pi\)
−0.924680 + 0.380745i \(0.875667\pi\)
\(710\) 0 0
\(711\) −11.4432 −0.429154
\(712\) 0 0
\(713\) −37.2936 −1.39666
\(714\) 0 0
\(715\) 12.0990 0.452477
\(716\) 0 0
\(717\) 2.91623 0.108909
\(718\) 0 0
\(719\) 7.41344 0.276475 0.138237 0.990399i \(-0.455856\pi\)
0.138237 + 0.990399i \(0.455856\pi\)
\(720\) 0 0
\(721\) 10.6468 0.396508
\(722\) 0 0
\(723\) 6.09899 0.226824
\(724\) 0 0
\(725\) 3.79082 0.140787
\(726\) 0 0
\(727\) −18.9557 −0.703026 −0.351513 0.936183i \(-0.614333\pi\)
−0.351513 + 0.936183i \(0.614333\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −66.9832 −2.47746
\(732\) 0 0
\(733\) −3.59283 −0.132704 −0.0663521 0.997796i \(-0.521136\pi\)
−0.0663521 + 0.997796i \(0.521136\pi\)
\(734\) 0 0
\(735\) −2.39821 −0.0884592
\(736\) 0 0
\(737\) −8.76663 −0.322923
\(738\) 0 0
\(739\) 26.7756 0.984956 0.492478 0.870325i \(-0.336091\pi\)
0.492478 + 0.870325i \(0.336091\pi\)
\(740\) 0 0
\(741\) −26.8567 −0.986604
\(742\) 0 0
\(743\) 33.8027 1.24010 0.620050 0.784562i \(-0.287112\pi\)
0.620050 + 0.784562i \(0.287112\pi\)
\(744\) 0 0
\(745\) −23.6018 −0.864703
\(746\) 0 0
\(747\) −13.1648 −0.481676
\(748\) 0 0
\(749\) −15.9702 −0.583539
\(750\) 0 0
\(751\) −35.3955 −1.29160 −0.645800 0.763506i \(-0.723476\pi\)
−0.645800 + 0.763506i \(0.723476\pi\)
\(752\) 0 0
\(753\) 1.62262 0.0591314
\(754\) 0 0
\(755\) 9.90168 0.360359
\(756\) 0 0
\(757\) −29.3442 −1.06653 −0.533267 0.845947i \(-0.679036\pi\)
−0.533267 + 0.845947i \(0.679036\pi\)
\(758\) 0 0
\(759\) 4.92520 0.178773
\(760\) 0 0
\(761\) 12.9044 0.467783 0.233892 0.972263i \(-0.424854\pi\)
0.233892 + 0.972263i \(0.424854\pi\)
\(762\) 0 0
\(763\) 12.7756 0.462507
\(764\) 0 0
\(765\) −15.2728 −0.552189
\(766\) 0 0
\(767\) −40.2099 −1.45189
\(768\) 0 0
\(769\) 9.78186 0.352743 0.176371 0.984324i \(-0.443564\pi\)
0.176371 + 0.984324i \(0.443564\pi\)
\(770\) 0 0
\(771\) 6.89541 0.248332
\(772\) 0 0
\(773\) 29.7223 1.06904 0.534518 0.845157i \(-0.320493\pi\)
0.534518 + 0.845157i \(0.320493\pi\)
\(774\) 0 0
\(775\) 5.68960 0.204376
\(776\) 0 0
\(777\) −4.24860 −0.152418
\(778\) 0 0
\(779\) −3.44322 −0.123366
\(780\) 0 0
\(781\) 11.4432 0.409471
\(782\) 0 0
\(783\) −5.04502 −0.180294
\(784\) 0 0
\(785\) −2.26875 −0.0809753
\(786\) 0 0
\(787\) −17.0242 −0.606847 −0.303423 0.952856i \(-0.598130\pi\)
−0.303423 + 0.952856i \(0.598130\pi\)
\(788\) 0 0
\(789\) 5.08377 0.180987
\(790\) 0 0
\(791\) 18.7368 0.666205
\(792\) 0 0
\(793\) −10.0900 −0.358308
\(794\) 0 0
\(795\) −8.92520 −0.316544
\(796\) 0 0
\(797\) −36.4287 −1.29037 −0.645185 0.764027i \(-0.723220\pi\)
−0.645185 + 0.764027i \(0.723220\pi\)
\(798\) 0 0
\(799\) 3.35609 0.118730
\(800\) 0 0
\(801\) 11.8504 0.418713
\(802\) 0 0
\(803\) −13.0450 −0.460349
\(804\) 0 0
\(805\) −11.8116 −0.416306
\(806\) 0 0
\(807\) 0.886447 0.0312044
\(808\) 0 0
\(809\) 44.4882 1.56412 0.782062 0.623201i \(-0.214168\pi\)
0.782062 + 0.623201i \(0.214168\pi\)
\(810\) 0 0
\(811\) −7.65307 −0.268736 −0.134368 0.990932i \(-0.542900\pi\)
−0.134368 + 0.990932i \(0.542900\pi\)
\(812\) 0 0
\(813\) −25.3234 −0.888131
\(814\) 0 0
\(815\) −21.0242 −0.736445
\(816\) 0 0
\(817\) 55.9917 1.95890
\(818\) 0 0
\(819\) 5.04502 0.176287
\(820\) 0 0
\(821\) −44.3691 −1.54849 −0.774246 0.632885i \(-0.781871\pi\)
−0.774246 + 0.632885i \(0.781871\pi\)
\(822\) 0 0
\(823\) −6.61702 −0.230655 −0.115327 0.993328i \(-0.536792\pi\)
−0.115327 + 0.993328i \(0.536792\pi\)
\(824\) 0 0
\(825\) −0.751399 −0.0261604
\(826\) 0 0
\(827\) −39.7126 −1.38094 −0.690472 0.723359i \(-0.742597\pi\)
−0.690472 + 0.723359i \(0.742597\pi\)
\(828\) 0 0
\(829\) −3.90997 −0.135799 −0.0678994 0.997692i \(-0.521630\pi\)
−0.0678994 + 0.997692i \(0.521630\pi\)
\(830\) 0 0
\(831\) 24.8269 0.861235
\(832\) 0 0
\(833\) −6.36842 −0.220653
\(834\) 0 0
\(835\) 58.4405 2.02242
\(836\) 0 0
\(837\) −7.57201 −0.261727
\(838\) 0 0
\(839\) 9.58097 0.330772 0.165386 0.986229i \(-0.447113\pi\)
0.165386 + 0.986229i \(0.447113\pi\)
\(840\) 0 0
\(841\) −3.54781 −0.122338
\(842\) 0 0
\(843\) −1.90101 −0.0654741
\(844\) 0 0
\(845\) 29.8629 1.02732
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −22.3178 −0.765945
\(850\) 0 0
\(851\) −20.9252 −0.717307
\(852\) 0 0
\(853\) −14.5568 −0.498415 −0.249207 0.968450i \(-0.580170\pi\)
−0.249207 + 0.968450i \(0.580170\pi\)
\(854\) 0 0
\(855\) 12.7666 0.436609
\(856\) 0 0
\(857\) 10.4793 0.357965 0.178983 0.983852i \(-0.442719\pi\)
0.178983 + 0.983852i \(0.442719\pi\)
\(858\) 0 0
\(859\) −2.88645 −0.0984843 −0.0492421 0.998787i \(-0.515681\pi\)
−0.0492421 + 0.998787i \(0.515681\pi\)
\(860\) 0 0
\(861\) 0.646809 0.0220432
\(862\) 0 0
\(863\) 20.5485 0.699479 0.349739 0.936847i \(-0.386270\pi\)
0.349739 + 0.936847i \(0.386270\pi\)
\(864\) 0 0
\(865\) 29.6121 1.00684
\(866\) 0 0
\(867\) −23.5568 −0.800030
\(868\) 0 0
\(869\) −11.4432 −0.388185
\(870\) 0 0
\(871\) −44.2278 −1.49860
\(872\) 0 0
\(873\) −1.87122 −0.0633311
\(874\) 0 0
\(875\) −10.1890 −0.344452
\(876\) 0 0
\(877\) 59.1149 1.99617 0.998084 0.0618724i \(-0.0197072\pi\)
0.998084 + 0.0618724i \(0.0197072\pi\)
\(878\) 0 0
\(879\) −12.0900 −0.407787
\(880\) 0 0
\(881\) 30.3982 1.02414 0.512071 0.858943i \(-0.328879\pi\)
0.512071 + 0.858943i \(0.328879\pi\)
\(882\) 0 0
\(883\) 35.6114 1.19842 0.599210 0.800592i \(-0.295481\pi\)
0.599210 + 0.800592i \(0.295481\pi\)
\(884\) 0 0
\(885\) 19.1142 0.642518
\(886\) 0 0
\(887\) −22.9736 −0.771377 −0.385689 0.922629i \(-0.626036\pi\)
−0.385689 + 0.922629i \(0.626036\pi\)
\(888\) 0 0
\(889\) 2.27839 0.0764147
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −2.80538 −0.0938784
\(894\) 0 0
\(895\) 13.4128 0.448339
\(896\) 0 0
\(897\) 24.8477 0.829640
\(898\) 0 0
\(899\) 38.2009 1.27407
\(900\) 0 0
\(901\) −23.7008 −0.789588
\(902\) 0 0
\(903\) −10.5180 −0.350018
\(904\) 0 0
\(905\) 32.5485 1.08195
\(906\) 0 0
\(907\) 57.1745 1.89845 0.949224 0.314602i \(-0.101871\pi\)
0.949224 + 0.314602i \(0.101871\pi\)
\(908\) 0 0
\(909\) 4.51803 0.149853
\(910\) 0 0
\(911\) −6.82687 −0.226184 −0.113092 0.993584i \(-0.536076\pi\)
−0.113092 + 0.993584i \(0.536076\pi\)
\(912\) 0 0
\(913\) −13.1648 −0.435692
\(914\) 0 0
\(915\) 4.79641 0.158565
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −12.0692 −0.398126 −0.199063 0.979987i \(-0.563790\pi\)
−0.199063 + 0.979987i \(0.563790\pi\)
\(920\) 0 0
\(921\) −13.5928 −0.447899
\(922\) 0 0
\(923\) 57.7312 1.90025
\(924\) 0 0
\(925\) 3.19239 0.104965
\(926\) 0 0
\(927\) 10.6468 0.349687
\(928\) 0 0
\(929\) 26.8954 0.882410 0.441205 0.897406i \(-0.354551\pi\)
0.441205 + 0.897406i \(0.354551\pi\)
\(930\) 0 0
\(931\) 5.32340 0.174468
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) −15.2728 −0.499474
\(936\) 0 0
\(937\) −14.9944 −0.489846 −0.244923 0.969543i \(-0.578763\pi\)
−0.244923 + 0.969543i \(0.578763\pi\)
\(938\) 0 0
\(939\) −14.9252 −0.487065
\(940\) 0 0
\(941\) −30.1205 −0.981900 −0.490950 0.871188i \(-0.663350\pi\)
−0.490950 + 0.871188i \(0.663350\pi\)
\(942\) 0 0
\(943\) 3.18566 0.103739
\(944\) 0 0
\(945\) −2.39821 −0.0780137
\(946\) 0 0
\(947\) 17.3532 0.563903 0.281951 0.959429i \(-0.409018\pi\)
0.281951 + 0.959429i \(0.409018\pi\)
\(948\) 0 0
\(949\) −65.8123 −2.13636
\(950\) 0 0
\(951\) −3.97918 −0.129034
\(952\) 0 0
\(953\) −2.14064 −0.0693422 −0.0346711 0.999399i \(-0.511038\pi\)
−0.0346711 + 0.999399i \(0.511038\pi\)
\(954\) 0 0
\(955\) 22.5969 0.731217
\(956\) 0 0
\(957\) −5.04502 −0.163082
\(958\) 0 0
\(959\) −4.77559 −0.154212
\(960\) 0 0
\(961\) 26.3353 0.849525
\(962\) 0 0
\(963\) −15.9702 −0.514633
\(964\) 0 0
\(965\) −24.2909 −0.781952
\(966\) 0 0
\(967\) 1.53326 0.0493062 0.0246531 0.999696i \(-0.492152\pi\)
0.0246531 + 0.999696i \(0.492152\pi\)
\(968\) 0 0
\(969\) 33.9017 1.08908
\(970\) 0 0
\(971\) 26.5574 0.852269 0.426135 0.904660i \(-0.359875\pi\)
0.426135 + 0.904660i \(0.359875\pi\)
\(972\) 0 0
\(973\) 15.4432 0.495087
\(974\) 0 0
\(975\) −3.79082 −0.121403
\(976\) 0 0
\(977\) −55.9017 −1.78845 −0.894227 0.447615i \(-0.852274\pi\)
−0.894227 + 0.447615i \(0.852274\pi\)
\(978\) 0 0
\(979\) 11.8504 0.378740
\(980\) 0 0
\(981\) 12.7756 0.407893
\(982\) 0 0
\(983\) 53.0361 1.69159 0.845794 0.533510i \(-0.179127\pi\)
0.845794 + 0.533510i \(0.179127\pi\)
\(984\) 0 0
\(985\) −5.41411 −0.172508
\(986\) 0 0
\(987\) 0.526989 0.0167743
\(988\) 0 0
\(989\) −51.8034 −1.64725
\(990\) 0 0
\(991\) −14.7362 −0.468110 −0.234055 0.972223i \(-0.575200\pi\)
−0.234055 + 0.972223i \(0.575200\pi\)
\(992\) 0 0
\(993\) 23.4432 0.743948
\(994\) 0 0
\(995\) −7.37402 −0.233772
\(996\) 0 0
\(997\) −45.0665 −1.42727 −0.713635 0.700517i \(-0.752953\pi\)
−0.713635 + 0.700517i \(0.752953\pi\)
\(998\) 0 0
\(999\) −4.24860 −0.134420
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 3696.2.a.bo.1.2 3
4.3 odd 2 231.2.a.e.1.2 3
12.11 even 2 693.2.a.l.1.2 3
20.19 odd 2 5775.2.a.bp.1.2 3
28.27 even 2 1617.2.a.t.1.2 3
44.43 even 2 2541.2.a.bg.1.2 3
84.83 odd 2 4851.2.a.bi.1.2 3
132.131 odd 2 7623.2.a.cd.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 4.3 odd 2
693.2.a.l.1.2 3 12.11 even 2
1617.2.a.t.1.2 3 28.27 even 2
2541.2.a.bg.1.2 3 44.43 even 2
3696.2.a.bo.1.2 3 1.1 even 1 trivial
4851.2.a.bi.1.2 3 84.83 odd 2
5775.2.a.bp.1.2 3 20.19 odd 2
7623.2.a.cd.1.2 3 132.131 odd 2