Properties

Label 4864.2.a.bo.1.7
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 68 x^{3} - 62 x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.77833\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.32921 q^{3} -3.13887 q^{5} +0.535658 q^{7} +2.42523 q^{9} +O(q^{10})\) \(q+2.32921 q^{3} -3.13887 q^{5} +0.535658 q^{7} +2.42523 q^{9} +0.425227 q^{11} -6.65786 q^{13} -7.31110 q^{15} +7.33239 q^{17} -1.00000 q^{19} +1.24766 q^{21} +5.90033 q^{23} +4.85252 q^{25} -1.33877 q^{27} -0.837037 q^{29} -3.16999 q^{31} +0.990444 q^{33} -1.68136 q^{35} -3.49490 q^{37} -15.5076 q^{39} +0.123059 q^{41} -5.39744 q^{43} -7.61248 q^{45} -2.02928 q^{47} -6.71307 q^{49} +17.0787 q^{51} +5.82785 q^{53} -1.33473 q^{55} -2.32921 q^{57} -5.56633 q^{59} +6.99606 q^{61} +1.29909 q^{63} +20.8982 q^{65} -12.3777 q^{67} +13.7431 q^{69} -12.1786 q^{71} +6.99382 q^{73} +11.3025 q^{75} +0.227776 q^{77} +2.07996 q^{79} -10.3940 q^{81} -11.8227 q^{83} -23.0154 q^{85} -1.94964 q^{87} -13.7684 q^{89} -3.56633 q^{91} -7.38358 q^{93} +3.13887 q^{95} +0.801240 q^{97} +1.03127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 4q^{7} + 12q^{9} - 4q^{11} - 8q^{13} - 4q^{17} - 8q^{19} - 16q^{21} + 12q^{25} - 28q^{29} + 8q^{31} + 12q^{35} - 4q^{37} - 4q^{39} - 8q^{41} + 4q^{43} - 24q^{45} + 12q^{47} + 12q^{49} - 12q^{51} - 32q^{53} - 8q^{55} - 12q^{59} - 8q^{61} - 16q^{63} + 8q^{65} + 4q^{67} - 28q^{69} - 24q^{71} - 24q^{77} - 24q^{79} - 8q^{81} - 40q^{83} - 24q^{85} + 24q^{87} + 8q^{89} + 4q^{91} - 32q^{93} + 8q^{95} + 16q^{97} + 76q^{99} + O(q^{100}) \)

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
\(3\) 2.32921 1.34477 0.672385 0.740201i \(-0.265270\pi\)
0.672385 + 0.740201i \(0.265270\pi\)
\(4\) 0 0
\(5\) −3.13887 −1.40375 −0.701873 0.712302i \(-0.747653\pi\)
−0.701873 + 0.712302i \(0.747653\pi\)
\(6\) 0 0
\(7\) 0.535658 0.202460 0.101230 0.994863i \(-0.467722\pi\)
0.101230 + 0.994863i \(0.467722\pi\)
\(8\) 0 0
\(9\) 2.42523 0.808409
\(10\) 0 0
\(11\) 0.425227 0.128211 0.0641054 0.997943i \(-0.479581\pi\)
0.0641054 + 0.997943i \(0.479581\pi\)
\(12\) 0 0
\(13\) −6.65786 −1.84656 −0.923279 0.384129i \(-0.874502\pi\)
−0.923279 + 0.384129i \(0.874502\pi\)
\(14\) 0 0
\(15\) −7.31110 −1.88772
\(16\) 0 0
\(17\) 7.33239 1.77837 0.889183 0.457552i \(-0.151273\pi\)
0.889183 + 0.457552i \(0.151273\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.24766 0.272262
\(22\) 0 0
\(23\) 5.90033 1.23030 0.615152 0.788408i \(-0.289095\pi\)
0.615152 + 0.788408i \(0.289095\pi\)
\(24\) 0 0
\(25\) 4.85252 0.970503
\(26\) 0 0
\(27\) −1.33877 −0.257646
\(28\) 0 0
\(29\) −0.837037 −0.155434 −0.0777170 0.996975i \(-0.524763\pi\)
−0.0777170 + 0.996975i \(0.524763\pi\)
\(30\) 0 0
\(31\) −3.16999 −0.569347 −0.284674 0.958625i \(-0.591885\pi\)
−0.284674 + 0.958625i \(0.591885\pi\)
\(32\) 0 0
\(33\) 0.990444 0.172414
\(34\) 0 0
\(35\) −1.68136 −0.284202
\(36\) 0 0
\(37\) −3.49490 −0.574558 −0.287279 0.957847i \(-0.592751\pi\)
−0.287279 + 0.957847i \(0.592751\pi\)
\(38\) 0 0
\(39\) −15.5076 −2.48320
\(40\) 0 0
\(41\) 0.123059 0.0192186 0.00960932 0.999954i \(-0.496941\pi\)
0.00960932 + 0.999954i \(0.496941\pi\)
\(42\) 0 0
\(43\) −5.39744 −0.823101 −0.411551 0.911387i \(-0.635013\pi\)
−0.411551 + 0.911387i \(0.635013\pi\)
\(44\) 0 0
\(45\) −7.61248 −1.13480
\(46\) 0 0
\(47\) −2.02928 −0.296001 −0.148000 0.988987i \(-0.547284\pi\)
−0.148000 + 0.988987i \(0.547284\pi\)
\(48\) 0 0
\(49\) −6.71307 −0.959010
\(50\) 0 0
\(51\) 17.0787 2.39150
\(52\) 0 0
\(53\) 5.82785 0.800517 0.400259 0.916402i \(-0.368920\pi\)
0.400259 + 0.916402i \(0.368920\pi\)
\(54\) 0 0
\(55\) −1.33473 −0.179975
\(56\) 0 0
\(57\) −2.32921 −0.308512
\(58\) 0 0
\(59\) −5.56633 −0.724675 −0.362337 0.932047i \(-0.618021\pi\)
−0.362337 + 0.932047i \(0.618021\pi\)
\(60\) 0 0
\(61\) 6.99606 0.895753 0.447877 0.894095i \(-0.352180\pi\)
0.447877 + 0.894095i \(0.352180\pi\)
\(62\) 0 0
\(63\) 1.29909 0.163670
\(64\) 0 0
\(65\) 20.8982 2.59210
\(66\) 0 0
\(67\) −12.3777 −1.51217 −0.756087 0.654471i \(-0.772891\pi\)
−0.756087 + 0.654471i \(0.772891\pi\)
\(68\) 0 0
\(69\) 13.7431 1.65448
\(70\) 0 0
\(71\) −12.1786 −1.44534 −0.722669 0.691194i \(-0.757085\pi\)
−0.722669 + 0.691194i \(0.757085\pi\)
\(72\) 0 0
\(73\) 6.99382 0.818565 0.409282 0.912408i \(-0.365779\pi\)
0.409282 + 0.912408i \(0.365779\pi\)
\(74\) 0 0
\(75\) 11.3025 1.30510
\(76\) 0 0
\(77\) 0.227776 0.0259575
\(78\) 0 0
\(79\) 2.07996 0.234014 0.117007 0.993131i \(-0.462670\pi\)
0.117007 + 0.993131i \(0.462670\pi\)
\(80\) 0 0
\(81\) −10.3940 −1.15488
\(82\) 0 0
\(83\) −11.8227 −1.29771 −0.648853 0.760914i \(-0.724751\pi\)
−0.648853 + 0.760914i \(0.724751\pi\)
\(84\) 0 0
\(85\) −23.0154 −2.49637
\(86\) 0 0
\(87\) −1.94964 −0.209023
\(88\) 0 0
\(89\) −13.7684 −1.45945 −0.729723 0.683743i \(-0.760351\pi\)
−0.729723 + 0.683743i \(0.760351\pi\)
\(90\) 0 0
\(91\) −3.56633 −0.373853
\(92\) 0 0
\(93\) −7.38358 −0.765641
\(94\) 0 0
\(95\) 3.13887 0.322041
\(96\) 0 0
\(97\) 0.801240 0.0813536 0.0406768 0.999172i \(-0.487049\pi\)
0.0406768 + 0.999172i \(0.487049\pi\)
\(98\) 0 0
\(99\) 1.03127 0.103647
\(100\) 0 0
\(101\) −12.4171 −1.23555 −0.617773 0.786356i \(-0.711965\pi\)
−0.617773 + 0.786356i \(0.711965\pi\)
\(102\) 0 0
\(103\) 2.84869 0.280690 0.140345 0.990103i \(-0.455179\pi\)
0.140345 + 0.990103i \(0.455179\pi\)
\(104\) 0 0
\(105\) −3.91624 −0.382186
\(106\) 0 0
\(107\) 0.288142 0.0278557 0.0139278 0.999903i \(-0.495566\pi\)
0.0139278 + 0.999903i \(0.495566\pi\)
\(108\) 0 0
\(109\) 1.91873 0.183781 0.0918905 0.995769i \(-0.470709\pi\)
0.0918905 + 0.995769i \(0.470709\pi\)
\(110\) 0 0
\(111\) −8.14036 −0.772649
\(112\) 0 0
\(113\) −13.4370 −1.26405 −0.632024 0.774949i \(-0.717776\pi\)
−0.632024 + 0.774949i \(0.717776\pi\)
\(114\) 0 0
\(115\) −18.5204 −1.72704
\(116\) 0 0
\(117\) −16.1468 −1.49277
\(118\) 0 0
\(119\) 3.92765 0.360047
\(120\) 0 0
\(121\) −10.8192 −0.983562
\(122\) 0 0
\(123\) 0.286631 0.0258447
\(124\) 0 0
\(125\) 0.462931 0.0414058
\(126\) 0 0
\(127\) −18.9992 −1.68591 −0.842953 0.537987i \(-0.819185\pi\)
−0.842953 + 0.537987i \(0.819185\pi\)
\(128\) 0 0
\(129\) −12.5718 −1.10688
\(130\) 0 0
\(131\) 10.1330 0.885322 0.442661 0.896689i \(-0.354035\pi\)
0.442661 + 0.896689i \(0.354035\pi\)
\(132\) 0 0
\(133\) −0.535658 −0.0464474
\(134\) 0 0
\(135\) 4.20222 0.361670
\(136\) 0 0
\(137\) 3.28221 0.280418 0.140209 0.990122i \(-0.455222\pi\)
0.140209 + 0.990122i \(0.455222\pi\)
\(138\) 0 0
\(139\) 10.5345 0.893527 0.446763 0.894652i \(-0.352577\pi\)
0.446763 + 0.894652i \(0.352577\pi\)
\(140\) 0 0
\(141\) −4.72662 −0.398053
\(142\) 0 0
\(143\) −2.83110 −0.236749
\(144\) 0 0
\(145\) 2.62735 0.218190
\(146\) 0 0
\(147\) −15.6362 −1.28965
\(148\) 0 0
\(149\) −5.93244 −0.486005 −0.243002 0.970026i \(-0.578132\pi\)
−0.243002 + 0.970026i \(0.578132\pi\)
\(150\) 0 0
\(151\) −4.57907 −0.372639 −0.186320 0.982489i \(-0.559656\pi\)
−0.186320 + 0.982489i \(0.559656\pi\)
\(152\) 0 0
\(153\) 17.7827 1.43765
\(154\) 0 0
\(155\) 9.95019 0.799219
\(156\) 0 0
\(157\) −3.80861 −0.303960 −0.151980 0.988384i \(-0.548565\pi\)
−0.151980 + 0.988384i \(0.548565\pi\)
\(158\) 0 0
\(159\) 13.5743 1.07651
\(160\) 0 0
\(161\) 3.16056 0.249087
\(162\) 0 0
\(163\) 17.9293 1.40433 0.702164 0.712016i \(-0.252217\pi\)
0.702164 + 0.712016i \(0.252217\pi\)
\(164\) 0 0
\(165\) −3.10888 −0.242026
\(166\) 0 0
\(167\) −18.8189 −1.45625 −0.728123 0.685446i \(-0.759607\pi\)
−0.728123 + 0.685446i \(0.759607\pi\)
\(168\) 0 0
\(169\) 31.3271 2.40978
\(170\) 0 0
\(171\) −2.42523 −0.185462
\(172\) 0 0
\(173\) −5.87750 −0.446858 −0.223429 0.974720i \(-0.571725\pi\)
−0.223429 + 0.974720i \(0.571725\pi\)
\(174\) 0 0
\(175\) 2.59929 0.196488
\(176\) 0 0
\(177\) −12.9652 −0.974522
\(178\) 0 0
\(179\) 18.9245 1.41448 0.707241 0.706973i \(-0.249940\pi\)
0.707241 + 0.706973i \(0.249940\pi\)
\(180\) 0 0
\(181\) 10.5330 0.782914 0.391457 0.920196i \(-0.371971\pi\)
0.391457 + 0.920196i \(0.371971\pi\)
\(182\) 0 0
\(183\) 16.2953 1.20458
\(184\) 0 0
\(185\) 10.9700 0.806534
\(186\) 0 0
\(187\) 3.11793 0.228006
\(188\) 0 0
\(189\) −0.717121 −0.0521629
\(190\) 0 0
\(191\) 1.60481 0.116120 0.0580600 0.998313i \(-0.481509\pi\)
0.0580600 + 0.998313i \(0.481509\pi\)
\(192\) 0 0
\(193\) −13.2889 −0.956554 −0.478277 0.878209i \(-0.658739\pi\)
−0.478277 + 0.878209i \(0.658739\pi\)
\(194\) 0 0
\(195\) 48.6763 3.48578
\(196\) 0 0
\(197\) −4.06689 −0.289754 −0.144877 0.989450i \(-0.546279\pi\)
−0.144877 + 0.989450i \(0.546279\pi\)
\(198\) 0 0
\(199\) −12.8273 −0.909307 −0.454653 0.890668i \(-0.650237\pi\)
−0.454653 + 0.890668i \(0.650237\pi\)
\(200\) 0 0
\(201\) −28.8302 −2.03353
\(202\) 0 0
\(203\) −0.448365 −0.0314691
\(204\) 0 0
\(205\) −0.386268 −0.0269781
\(206\) 0 0
\(207\) 14.3096 0.994589
\(208\) 0 0
\(209\) −0.425227 −0.0294136
\(210\) 0 0
\(211\) −15.9906 −1.10084 −0.550420 0.834888i \(-0.685532\pi\)
−0.550420 + 0.834888i \(0.685532\pi\)
\(212\) 0 0
\(213\) −28.3666 −1.94365
\(214\) 0 0
\(215\) 16.9419 1.15543
\(216\) 0 0
\(217\) −1.69803 −0.115270
\(218\) 0 0
\(219\) 16.2901 1.10078
\(220\) 0 0
\(221\) −48.8181 −3.28386
\(222\) 0 0
\(223\) 21.1424 1.41580 0.707899 0.706314i \(-0.249643\pi\)
0.707899 + 0.706314i \(0.249643\pi\)
\(224\) 0 0
\(225\) 11.7685 0.784564
\(226\) 0 0
\(227\) −22.2056 −1.47384 −0.736918 0.675982i \(-0.763720\pi\)
−0.736918 + 0.675982i \(0.763720\pi\)
\(228\) 0 0
\(229\) −12.6331 −0.834816 −0.417408 0.908719i \(-0.637061\pi\)
−0.417408 + 0.908719i \(0.637061\pi\)
\(230\) 0 0
\(231\) 0.530539 0.0349069
\(232\) 0 0
\(233\) −3.01852 −0.197750 −0.0988749 0.995100i \(-0.531524\pi\)
−0.0988749 + 0.995100i \(0.531524\pi\)
\(234\) 0 0
\(235\) 6.36965 0.415510
\(236\) 0 0
\(237\) 4.84468 0.314695
\(238\) 0 0
\(239\) 12.7156 0.822504 0.411252 0.911522i \(-0.365092\pi\)
0.411252 + 0.911522i \(0.365092\pi\)
\(240\) 0 0
\(241\) −1.05734 −0.0681091 −0.0340545 0.999420i \(-0.510842\pi\)
−0.0340545 + 0.999420i \(0.510842\pi\)
\(242\) 0 0
\(243\) −20.1934 −1.29541
\(244\) 0 0
\(245\) 21.0715 1.34621
\(246\) 0 0
\(247\) 6.65786 0.423630
\(248\) 0 0
\(249\) −27.5375 −1.74512
\(250\) 0 0
\(251\) −13.4689 −0.850151 −0.425076 0.905158i \(-0.639752\pi\)
−0.425076 + 0.905158i \(0.639752\pi\)
\(252\) 0 0
\(253\) 2.50898 0.157738
\(254\) 0 0
\(255\) −53.6078 −3.35705
\(256\) 0 0
\(257\) 19.1631 1.19536 0.597680 0.801734i \(-0.296089\pi\)
0.597680 + 0.801734i \(0.296089\pi\)
\(258\) 0 0
\(259\) −1.87207 −0.116325
\(260\) 0 0
\(261\) −2.03001 −0.125654
\(262\) 0 0
\(263\) 2.55940 0.157819 0.0789097 0.996882i \(-0.474856\pi\)
0.0789097 + 0.996882i \(0.474856\pi\)
\(264\) 0 0
\(265\) −18.2929 −1.12372
\(266\) 0 0
\(267\) −32.0695 −1.96262
\(268\) 0 0
\(269\) 4.88596 0.297902 0.148951 0.988845i \(-0.452410\pi\)
0.148951 + 0.988845i \(0.452410\pi\)
\(270\) 0 0
\(271\) 32.1299 1.95175 0.975875 0.218329i \(-0.0700604\pi\)
0.975875 + 0.218329i \(0.0700604\pi\)
\(272\) 0 0
\(273\) −8.30675 −0.502747
\(274\) 0 0
\(275\) 2.06342 0.124429
\(276\) 0 0
\(277\) 7.46639 0.448612 0.224306 0.974519i \(-0.427988\pi\)
0.224306 + 0.974519i \(0.427988\pi\)
\(278\) 0 0
\(279\) −7.68795 −0.460265
\(280\) 0 0
\(281\) −18.9770 −1.13208 −0.566038 0.824379i \(-0.691524\pi\)
−0.566038 + 0.824379i \(0.691524\pi\)
\(282\) 0 0
\(283\) −1.24321 −0.0739012 −0.0369506 0.999317i \(-0.511764\pi\)
−0.0369506 + 0.999317i \(0.511764\pi\)
\(284\) 0 0
\(285\) 7.31110 0.433072
\(286\) 0 0
\(287\) 0.0659177 0.00389100
\(288\) 0 0
\(289\) 36.7640 2.16259
\(290\) 0 0
\(291\) 1.86626 0.109402
\(292\) 0 0
\(293\) −5.35018 −0.312561 −0.156280 0.987713i \(-0.549950\pi\)
−0.156280 + 0.987713i \(0.549950\pi\)
\(294\) 0 0
\(295\) 17.4720 1.01726
\(296\) 0 0
\(297\) −0.569280 −0.0330330
\(298\) 0 0
\(299\) −39.2836 −2.27183
\(300\) 0 0
\(301\) −2.89118 −0.166645
\(302\) 0 0
\(303\) −28.9220 −1.66153
\(304\) 0 0
\(305\) −21.9597 −1.25741
\(306\) 0 0
\(307\) −10.4800 −0.598123 −0.299062 0.954234i \(-0.596674\pi\)
−0.299062 + 0.954234i \(0.596674\pi\)
\(308\) 0 0
\(309\) 6.63521 0.377464
\(310\) 0 0
\(311\) 27.3720 1.55212 0.776061 0.630658i \(-0.217215\pi\)
0.776061 + 0.630658i \(0.217215\pi\)
\(312\) 0 0
\(313\) 16.9214 0.956451 0.478226 0.878237i \(-0.341280\pi\)
0.478226 + 0.878237i \(0.341280\pi\)
\(314\) 0 0
\(315\) −4.07768 −0.229751
\(316\) 0 0
\(317\) −3.82118 −0.214619 −0.107309 0.994226i \(-0.534224\pi\)
−0.107309 + 0.994226i \(0.534224\pi\)
\(318\) 0 0
\(319\) −0.355931 −0.0199283
\(320\) 0 0
\(321\) 0.671143 0.0374595
\(322\) 0 0
\(323\) −7.33239 −0.407985
\(324\) 0 0
\(325\) −32.3074 −1.79209
\(326\) 0 0
\(327\) 4.46913 0.247143
\(328\) 0 0
\(329\) −1.08700 −0.0599282
\(330\) 0 0
\(331\) 26.6082 1.46252 0.731258 0.682101i \(-0.238933\pi\)
0.731258 + 0.682101i \(0.238933\pi\)
\(332\) 0 0
\(333\) −8.47592 −0.464478
\(334\) 0 0
\(335\) 38.8520 2.12271
\(336\) 0 0
\(337\) −11.6884 −0.636707 −0.318354 0.947972i \(-0.603130\pi\)
−0.318354 + 0.947972i \(0.603130\pi\)
\(338\) 0 0
\(339\) −31.2976 −1.69985
\(340\) 0 0
\(341\) −1.34797 −0.0729964
\(342\) 0 0
\(343\) −7.34551 −0.396620
\(344\) 0 0
\(345\) −43.1379 −2.32247
\(346\) 0 0
\(347\) 16.2573 0.872739 0.436370 0.899767i \(-0.356264\pi\)
0.436370 + 0.899767i \(0.356264\pi\)
\(348\) 0 0
\(349\) −22.5885 −1.20913 −0.604567 0.796554i \(-0.706654\pi\)
−0.604567 + 0.796554i \(0.706654\pi\)
\(350\) 0 0
\(351\) 8.91333 0.475758
\(352\) 0 0
\(353\) −10.2804 −0.547169 −0.273584 0.961848i \(-0.588209\pi\)
−0.273584 + 0.961848i \(0.588209\pi\)
\(354\) 0 0
\(355\) 38.2272 2.02889
\(356\) 0 0
\(357\) 9.14833 0.484181
\(358\) 0 0
\(359\) 16.1694 0.853386 0.426693 0.904397i \(-0.359679\pi\)
0.426693 + 0.904397i \(0.359679\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −25.2002 −1.32267
\(364\) 0 0
\(365\) −21.9527 −1.14906
\(366\) 0 0
\(367\) 19.5371 1.01983 0.509914 0.860225i \(-0.329677\pi\)
0.509914 + 0.860225i \(0.329677\pi\)
\(368\) 0 0
\(369\) 0.298447 0.0155365
\(370\) 0 0
\(371\) 3.12173 0.162072
\(372\) 0 0
\(373\) −12.9672 −0.671418 −0.335709 0.941966i \(-0.608976\pi\)
−0.335709 + 0.941966i \(0.608976\pi\)
\(374\) 0 0
\(375\) 1.07826 0.0556813
\(376\) 0 0
\(377\) 5.57288 0.287018
\(378\) 0 0
\(379\) 0.229262 0.0117764 0.00588821 0.999983i \(-0.498126\pi\)
0.00588821 + 0.999983i \(0.498126\pi\)
\(380\) 0 0
\(381\) −44.2532 −2.26716
\(382\) 0 0
\(383\) −18.9168 −0.966601 −0.483301 0.875455i \(-0.660562\pi\)
−0.483301 + 0.875455i \(0.660562\pi\)
\(384\) 0 0
\(385\) −0.714960 −0.0364377
\(386\) 0 0
\(387\) −13.0900 −0.665403
\(388\) 0 0
\(389\) −9.38404 −0.475790 −0.237895 0.971291i \(-0.576457\pi\)
−0.237895 + 0.971291i \(0.576457\pi\)
\(390\) 0 0
\(391\) 43.2636 2.18793
\(392\) 0 0
\(393\) 23.6018 1.19055
\(394\) 0 0
\(395\) −6.52874 −0.328496
\(396\) 0 0
\(397\) 18.3307 0.919990 0.459995 0.887921i \(-0.347851\pi\)
0.459995 + 0.887921i \(0.347851\pi\)
\(398\) 0 0
\(399\) −1.24766 −0.0624611
\(400\) 0 0
\(401\) −22.7518 −1.13617 −0.568085 0.822970i \(-0.692316\pi\)
−0.568085 + 0.822970i \(0.692316\pi\)
\(402\) 0 0
\(403\) 21.1054 1.05133
\(404\) 0 0
\(405\) 32.6253 1.62116
\(406\) 0 0
\(407\) −1.48613 −0.0736645
\(408\) 0 0
\(409\) 19.3401 0.956305 0.478152 0.878277i \(-0.341307\pi\)
0.478152 + 0.878277i \(0.341307\pi\)
\(410\) 0 0
\(411\) 7.64497 0.377099
\(412\) 0 0
\(413\) −2.98165 −0.146717
\(414\) 0 0
\(415\) 37.1098 1.82165
\(416\) 0 0
\(417\) 24.5371 1.20159
\(418\) 0 0
\(419\) 28.0225 1.36899 0.684495 0.729017i \(-0.260023\pi\)
0.684495 + 0.729017i \(0.260023\pi\)
\(420\) 0 0
\(421\) 15.5487 0.757798 0.378899 0.925438i \(-0.376303\pi\)
0.378899 + 0.925438i \(0.376303\pi\)
\(422\) 0 0
\(423\) −4.92146 −0.239290
\(424\) 0 0
\(425\) 35.5806 1.72591
\(426\) 0 0
\(427\) 3.74749 0.181354
\(428\) 0 0
\(429\) −6.59424 −0.318373
\(430\) 0 0
\(431\) 15.5573 0.749368 0.374684 0.927153i \(-0.377751\pi\)
0.374684 + 0.927153i \(0.377751\pi\)
\(432\) 0 0
\(433\) 4.16219 0.200022 0.100011 0.994986i \(-0.468112\pi\)
0.100011 + 0.994986i \(0.468112\pi\)
\(434\) 0 0
\(435\) 6.11966 0.293415
\(436\) 0 0
\(437\) −5.90033 −0.282251
\(438\) 0 0
\(439\) −24.6600 −1.17696 −0.588478 0.808513i \(-0.700273\pi\)
−0.588478 + 0.808513i \(0.700273\pi\)
\(440\) 0 0
\(441\) −16.2807 −0.775272
\(442\) 0 0
\(443\) 19.7595 0.938801 0.469401 0.882985i \(-0.344470\pi\)
0.469401 + 0.882985i \(0.344470\pi\)
\(444\) 0 0
\(445\) 43.2172 2.04869
\(446\) 0 0
\(447\) −13.8179 −0.653565
\(448\) 0 0
\(449\) −16.4826 −0.777862 −0.388931 0.921267i \(-0.627156\pi\)
−0.388931 + 0.921267i \(0.627156\pi\)
\(450\) 0 0
\(451\) 0.0523282 0.00246404
\(452\) 0 0
\(453\) −10.6656 −0.501115
\(454\) 0 0
\(455\) 11.1943 0.524795
\(456\) 0 0
\(457\) −7.44504 −0.348264 −0.174132 0.984722i \(-0.555712\pi\)
−0.174132 + 0.984722i \(0.555712\pi\)
\(458\) 0 0
\(459\) −9.81637 −0.458189
\(460\) 0 0
\(461\) 14.9337 0.695532 0.347766 0.937581i \(-0.386940\pi\)
0.347766 + 0.937581i \(0.386940\pi\)
\(462\) 0 0
\(463\) −23.6042 −1.09698 −0.548491 0.836156i \(-0.684798\pi\)
−0.548491 + 0.836156i \(0.684798\pi\)
\(464\) 0 0
\(465\) 23.1761 1.07477
\(466\) 0 0
\(467\) 0.808996 0.0374359 0.0187179 0.999825i \(-0.494042\pi\)
0.0187179 + 0.999825i \(0.494042\pi\)
\(468\) 0 0
\(469\) −6.63020 −0.306154
\(470\) 0 0
\(471\) −8.87106 −0.408757
\(472\) 0 0
\(473\) −2.29514 −0.105530
\(474\) 0 0
\(475\) −4.85252 −0.222649
\(476\) 0 0
\(477\) 14.1339 0.647145
\(478\) 0 0
\(479\) −12.2547 −0.559933 −0.279966 0.960010i \(-0.590323\pi\)
−0.279966 + 0.960010i \(0.590323\pi\)
\(480\) 0 0
\(481\) 23.2686 1.06095
\(482\) 0 0
\(483\) 7.36161 0.334965
\(484\) 0 0
\(485\) −2.51499 −0.114200
\(486\) 0 0
\(487\) −23.1692 −1.04990 −0.524948 0.851134i \(-0.675915\pi\)
−0.524948 + 0.851134i \(0.675915\pi\)
\(488\) 0 0
\(489\) 41.7610 1.88850
\(490\) 0 0
\(491\) −20.6669 −0.932684 −0.466342 0.884605i \(-0.654428\pi\)
−0.466342 + 0.884605i \(0.654428\pi\)
\(492\) 0 0
\(493\) −6.13749 −0.276418
\(494\) 0 0
\(495\) −3.23703 −0.145494
\(496\) 0 0
\(497\) −6.52358 −0.292623
\(498\) 0 0
\(499\) −5.36134 −0.240007 −0.120003 0.992773i \(-0.538291\pi\)
−0.120003 + 0.992773i \(0.538291\pi\)
\(500\) 0 0
\(501\) −43.8331 −1.95832
\(502\) 0 0
\(503\) 25.6878 1.14536 0.572681 0.819778i \(-0.305903\pi\)
0.572681 + 0.819778i \(0.305903\pi\)
\(504\) 0 0
\(505\) 38.9757 1.73439
\(506\) 0 0
\(507\) 72.9675 3.24060
\(508\) 0 0
\(509\) 16.2273 0.719261 0.359631 0.933095i \(-0.382903\pi\)
0.359631 + 0.933095i \(0.382903\pi\)
\(510\) 0 0
\(511\) 3.74629 0.165726
\(512\) 0 0
\(513\) 1.33877 0.0591080
\(514\) 0 0
\(515\) −8.94168 −0.394017
\(516\) 0 0
\(517\) −0.862904 −0.0379505
\(518\) 0 0
\(519\) −13.6899 −0.600922
\(520\) 0 0
\(521\) −32.4863 −1.42325 −0.711626 0.702559i \(-0.752041\pi\)
−0.711626 + 0.702559i \(0.752041\pi\)
\(522\) 0 0
\(523\) 11.1406 0.487143 0.243571 0.969883i \(-0.421681\pi\)
0.243571 + 0.969883i \(0.421681\pi\)
\(524\) 0 0
\(525\) 6.05429 0.264231
\(526\) 0 0
\(527\) −23.2436 −1.01251
\(528\) 0 0
\(529\) 11.8139 0.513649
\(530\) 0 0
\(531\) −13.4996 −0.585834
\(532\) 0 0
\(533\) −0.819312 −0.0354884
\(534\) 0 0
\(535\) −0.904439 −0.0391023
\(536\) 0 0
\(537\) 44.0791 1.90215
\(538\) 0 0
\(539\) −2.85458 −0.122955
\(540\) 0 0
\(541\) 1.33136 0.0572394 0.0286197 0.999590i \(-0.490889\pi\)
0.0286197 + 0.999590i \(0.490889\pi\)
\(542\) 0 0
\(543\) 24.5336 1.05284
\(544\) 0 0
\(545\) −6.02264 −0.257982
\(546\) 0 0
\(547\) 21.1345 0.903644 0.451822 0.892108i \(-0.350774\pi\)
0.451822 + 0.892108i \(0.350774\pi\)
\(548\) 0 0
\(549\) 16.9670 0.724135
\(550\) 0 0
\(551\) 0.837037 0.0356590
\(552\) 0 0
\(553\) 1.11415 0.0473784
\(554\) 0 0
\(555\) 25.5515 1.08460
\(556\) 0 0
\(557\) −6.60875 −0.280022 −0.140011 0.990150i \(-0.544714\pi\)
−0.140011 + 0.990150i \(0.544714\pi\)
\(558\) 0 0
\(559\) 35.9354 1.51991
\(560\) 0 0
\(561\) 7.26232 0.306615
\(562\) 0 0
\(563\) −4.42073 −0.186312 −0.0931559 0.995652i \(-0.529695\pi\)
−0.0931559 + 0.995652i \(0.529695\pi\)
\(564\) 0 0
\(565\) 42.1771 1.77440
\(566\) 0 0
\(567\) −5.56760 −0.233817
\(568\) 0 0
\(569\) −8.59442 −0.360297 −0.180149 0.983639i \(-0.557658\pi\)
−0.180149 + 0.983639i \(0.557658\pi\)
\(570\) 0 0
\(571\) −34.6802 −1.45132 −0.725661 0.688052i \(-0.758466\pi\)
−0.725661 + 0.688052i \(0.758466\pi\)
\(572\) 0 0
\(573\) 3.73794 0.156155
\(574\) 0 0
\(575\) 28.6315 1.19401
\(576\) 0 0
\(577\) 26.2877 1.09437 0.547185 0.837012i \(-0.315699\pi\)
0.547185 + 0.837012i \(0.315699\pi\)
\(578\) 0 0
\(579\) −30.9526 −1.28635
\(580\) 0 0
\(581\) −6.33290 −0.262733
\(582\) 0 0
\(583\) 2.47816 0.102635
\(584\) 0 0
\(585\) 50.6828 2.09548
\(586\) 0 0
\(587\) 7.01067 0.289361 0.144681 0.989478i \(-0.453785\pi\)
0.144681 + 0.989478i \(0.453785\pi\)
\(588\) 0 0
\(589\) 3.16999 0.130617
\(590\) 0 0
\(591\) −9.47265 −0.389653
\(592\) 0 0
\(593\) 34.7115 1.42543 0.712716 0.701453i \(-0.247465\pi\)
0.712716 + 0.701453i \(0.247465\pi\)
\(594\) 0 0
\(595\) −12.3284 −0.505415
\(596\) 0 0
\(597\) −29.8776 −1.22281
\(598\) 0 0
\(599\) 32.4456 1.32569 0.662845 0.748757i \(-0.269349\pi\)
0.662845 + 0.748757i \(0.269349\pi\)
\(600\) 0 0
\(601\) 36.2901 1.48030 0.740152 0.672440i \(-0.234754\pi\)
0.740152 + 0.672440i \(0.234754\pi\)
\(602\) 0 0
\(603\) −30.0187 −1.22246
\(604\) 0 0
\(605\) 33.9600 1.38067
\(606\) 0 0
\(607\) 5.54708 0.225149 0.112575 0.993643i \(-0.464090\pi\)
0.112575 + 0.993643i \(0.464090\pi\)
\(608\) 0 0
\(609\) −1.04434 −0.0423187
\(610\) 0 0
\(611\) 13.5107 0.546583
\(612\) 0 0
\(613\) 41.7239 1.68521 0.842606 0.538530i \(-0.181020\pi\)
0.842606 + 0.538530i \(0.181020\pi\)
\(614\) 0 0
\(615\) −0.899699 −0.0362794
\(616\) 0 0
\(617\) −16.0859 −0.647593 −0.323797 0.946127i \(-0.604959\pi\)
−0.323797 + 0.946127i \(0.604959\pi\)
\(618\) 0 0
\(619\) −2.93257 −0.117870 −0.0589349 0.998262i \(-0.518770\pi\)
−0.0589349 + 0.998262i \(0.518770\pi\)
\(620\) 0 0
\(621\) −7.89918 −0.316983
\(622\) 0 0
\(623\) −7.37514 −0.295479
\(624\) 0 0
\(625\) −25.7157 −1.02863
\(626\) 0 0
\(627\) −0.990444 −0.0395545
\(628\) 0 0
\(629\) −25.6260 −1.02177
\(630\) 0 0
\(631\) 23.0095 0.915994 0.457997 0.888954i \(-0.348567\pi\)
0.457997 + 0.888954i \(0.348567\pi\)
\(632\) 0 0
\(633\) −37.2455 −1.48038
\(634\) 0 0
\(635\) 59.6361 2.36658
\(636\) 0 0
\(637\) 44.6947 1.77087
\(638\) 0 0
\(639\) −29.5360 −1.16842
\(640\) 0 0
\(641\) −0.00476234 −0.000188101 0 −9.40506e−5 1.00000i \(-0.500030\pi\)
−9.40506e−5 1.00000i \(0.500030\pi\)
\(642\) 0 0
\(643\) −37.6587 −1.48511 −0.742557 0.669783i \(-0.766387\pi\)
−0.742557 + 0.669783i \(0.766387\pi\)
\(644\) 0 0
\(645\) 39.4612 1.55378
\(646\) 0 0
\(647\) −25.3744 −0.997570 −0.498785 0.866726i \(-0.666220\pi\)
−0.498785 + 0.866726i \(0.666220\pi\)
\(648\) 0 0
\(649\) −2.36696 −0.0929111
\(650\) 0 0
\(651\) −3.95507 −0.155011
\(652\) 0 0
\(653\) −22.5548 −0.882637 −0.441319 0.897350i \(-0.645489\pi\)
−0.441319 + 0.897350i \(0.645489\pi\)
\(654\) 0 0
\(655\) −31.8061 −1.24277
\(656\) 0 0
\(657\) 16.9616 0.661735
\(658\) 0 0
\(659\) −18.3990 −0.716724 −0.358362 0.933583i \(-0.616665\pi\)
−0.358362 + 0.933583i \(0.616665\pi\)
\(660\) 0 0
\(661\) 39.9383 1.55342 0.776710 0.629859i \(-0.216887\pi\)
0.776710 + 0.629859i \(0.216887\pi\)
\(662\) 0 0
\(663\) −113.708 −4.41604
\(664\) 0 0
\(665\) 1.68136 0.0652004
\(666\) 0 0
\(667\) −4.93880 −0.191231
\(668\) 0 0
\(669\) 49.2451 1.90392
\(670\) 0 0
\(671\) 2.97491 0.114845
\(672\) 0 0
\(673\) −4.04663 −0.155986 −0.0779931 0.996954i \(-0.524851\pi\)
−0.0779931 + 0.996954i \(0.524851\pi\)
\(674\) 0 0
\(675\) −6.49639 −0.250046
\(676\) 0 0
\(677\) −42.8041 −1.64509 −0.822547 0.568697i \(-0.807448\pi\)
−0.822547 + 0.568697i \(0.807448\pi\)
\(678\) 0 0
\(679\) 0.429190 0.0164708
\(680\) 0 0
\(681\) −51.7215 −1.98197
\(682\) 0 0
\(683\) −6.33556 −0.242423 −0.121212 0.992627i \(-0.538678\pi\)
−0.121212 + 0.992627i \(0.538678\pi\)
\(684\) 0 0
\(685\) −10.3024 −0.393636
\(686\) 0 0
\(687\) −29.4251 −1.12264
\(688\) 0 0
\(689\) −38.8010 −1.47820
\(690\) 0 0
\(691\) −15.3700 −0.584702 −0.292351 0.956311i \(-0.594438\pi\)
−0.292351 + 0.956311i \(0.594438\pi\)
\(692\) 0 0
\(693\) 0.552409 0.0209843
\(694\) 0 0
\(695\) −33.0665 −1.25428
\(696\) 0 0
\(697\) 0.902320 0.0341778
\(698\) 0 0
\(699\) −7.03077 −0.265928
\(700\) 0 0
\(701\) −36.0602 −1.36197 −0.680987 0.732296i \(-0.738449\pi\)
−0.680987 + 0.732296i \(0.738449\pi\)
\(702\) 0 0
\(703\) 3.49490 0.131813
\(704\) 0 0
\(705\) 14.8363 0.558766
\(706\) 0 0
\(707\) −6.65131 −0.250148
\(708\) 0 0
\(709\) 18.1426 0.681360 0.340680 0.940179i \(-0.389343\pi\)
0.340680 + 0.940179i \(0.389343\pi\)
\(710\) 0 0
\(711\) 5.04438 0.189179
\(712\) 0 0
\(713\) −18.7040 −0.700470
\(714\) 0 0
\(715\) 8.88647 0.332335
\(716\) 0 0
\(717\) 29.6173 1.10608
\(718\) 0 0
\(719\) −38.2801 −1.42761 −0.713804 0.700345i \(-0.753029\pi\)
−0.713804 + 0.700345i \(0.753029\pi\)
\(720\) 0 0
\(721\) 1.52592 0.0568284
\(722\) 0 0
\(723\) −2.46276 −0.0915911
\(724\) 0 0
\(725\) −4.06174 −0.150849
\(726\) 0 0
\(727\) −23.4550 −0.869897 −0.434949 0.900455i \(-0.643233\pi\)
−0.434949 + 0.900455i \(0.643233\pi\)
\(728\) 0 0
\(729\) −15.8529 −0.587144
\(730\) 0 0
\(731\) −39.5761 −1.46378
\(732\) 0 0
\(733\) 19.3814 0.715870 0.357935 0.933747i \(-0.383481\pi\)
0.357935 + 0.933747i \(0.383481\pi\)
\(734\) 0 0
\(735\) 49.0799 1.81034
\(736\) 0 0
\(737\) −5.26332 −0.193877
\(738\) 0 0
\(739\) 12.1161 0.445697 0.222848 0.974853i \(-0.428464\pi\)
0.222848 + 0.974853i \(0.428464\pi\)
\(740\) 0 0
\(741\) 15.5076 0.569685
\(742\) 0 0
\(743\) 29.1148 1.06812 0.534060 0.845447i \(-0.320666\pi\)
0.534060 + 0.845447i \(0.320666\pi\)
\(744\) 0 0
\(745\) 18.6212 0.682227
\(746\) 0 0
\(747\) −28.6726 −1.04908
\(748\) 0 0
\(749\) 0.154345 0.00563965
\(750\) 0 0
\(751\) −2.09618 −0.0764906 −0.0382453 0.999268i \(-0.512177\pi\)
−0.0382453 + 0.999268i \(0.512177\pi\)
\(752\) 0 0
\(753\) −31.3720 −1.14326
\(754\) 0 0
\(755\) 14.3731 0.523091
\(756\) 0 0
\(757\) 36.7044 1.33404 0.667022 0.745038i \(-0.267569\pi\)
0.667022 + 0.745038i \(0.267569\pi\)
\(758\) 0 0
\(759\) 5.84395 0.212122
\(760\) 0 0
\(761\) −26.8362 −0.972812 −0.486406 0.873733i \(-0.661692\pi\)
−0.486406 + 0.873733i \(0.661692\pi\)
\(762\) 0 0
\(763\) 1.02778 0.0372082
\(764\) 0 0
\(765\) −55.8177 −2.01809
\(766\) 0 0
\(767\) 37.0599 1.33815
\(768\) 0 0
\(769\) 15.3434 0.553299 0.276649 0.960971i \(-0.410776\pi\)
0.276649 + 0.960971i \(0.410776\pi\)
\(770\) 0 0
\(771\) 44.6349 1.60749
\(772\) 0 0
\(773\) −54.9160 −1.97519 −0.987596 0.157017i \(-0.949812\pi\)
−0.987596 + 0.157017i \(0.949812\pi\)
\(774\) 0 0
\(775\) −15.3824 −0.552553
\(776\) 0 0
\(777\) −4.36045 −0.156430
\(778\) 0 0
\(779\) −0.123059 −0.00440906
\(780\) 0 0
\(781\) −5.17869 −0.185308
\(782\) 0 0
\(783\) 1.12060 0.0400469
\(784\) 0 0
\(785\) 11.9547 0.426683
\(786\) 0 0
\(787\) −16.2826 −0.580413 −0.290206 0.956964i \(-0.593724\pi\)
−0.290206 + 0.956964i \(0.593724\pi\)
\(788\) 0 0
\(789\) 5.96139 0.212231
\(790\) 0 0
\(791\) −7.19764 −0.255918
\(792\) 0 0
\(793\) −46.5788 −1.65406
\(794\) 0 0
\(795\) −42.6080 −1.51115
\(796\) 0 0
\(797\) 20.2945 0.718867 0.359434 0.933171i \(-0.382970\pi\)
0.359434 + 0.933171i \(0.382970\pi\)
\(798\) 0 0
\(799\) −14.8795 −0.526398
\(800\) 0 0
\(801\) −33.3915 −1.17983
\(802\) 0 0
\(803\) 2.97396 0.104949
\(804\) 0 0
\(805\) −9.92059 −0.349655
\(806\) 0 0
\(807\) 11.3804 0.400610
\(808\) 0 0
\(809\) 5.57944 0.196163 0.0980813 0.995178i \(-0.468729\pi\)
0.0980813 + 0.995178i \(0.468729\pi\)
\(810\) 0 0
\(811\) 28.4695 0.999699 0.499849 0.866112i \(-0.333389\pi\)
0.499849 + 0.866112i \(0.333389\pi\)
\(812\) 0 0
\(813\) 74.8373 2.62466
\(814\) 0 0
\(815\) −56.2776 −1.97132
\(816\) 0 0
\(817\) 5.39744 0.188832
\(818\) 0 0
\(819\) −8.64917 −0.302226
\(820\) 0 0
\(821\) 3.41291 0.119111 0.0595556 0.998225i \(-0.481032\pi\)
0.0595556 + 0.998225i \(0.481032\pi\)
\(822\) 0 0
\(823\) 12.8444 0.447729 0.223865 0.974620i \(-0.428133\pi\)
0.223865 + 0.974620i \(0.428133\pi\)
\(824\) 0 0
\(825\) 4.80615 0.167328
\(826\) 0 0
\(827\) −11.4599 −0.398501 −0.199251 0.979949i \(-0.563851\pi\)
−0.199251 + 0.979949i \(0.563851\pi\)
\(828\) 0 0
\(829\) 15.6953 0.545119 0.272559 0.962139i \(-0.412130\pi\)
0.272559 + 0.962139i \(0.412130\pi\)
\(830\) 0 0
\(831\) 17.3908 0.603280
\(832\) 0 0
\(833\) −49.2229 −1.70547
\(834\) 0 0
\(835\) 59.0700 2.04420
\(836\) 0 0
\(837\) 4.24388 0.146690
\(838\) 0 0
\(839\) −40.4629 −1.39693 −0.698467 0.715642i \(-0.746134\pi\)
−0.698467 + 0.715642i \(0.746134\pi\)
\(840\) 0 0
\(841\) −28.2994 −0.975840
\(842\) 0 0
\(843\) −44.2016 −1.52238
\(844\) 0 0
\(845\) −98.3318 −3.38272
\(846\) 0 0
\(847\) −5.79538 −0.199131
\(848\) 0 0
\(849\) −2.89570 −0.0993802
\(850\) 0 0
\(851\) −20.6211 −0.706881
\(852\) 0 0
\(853\) −1.22071 −0.0417962 −0.0208981 0.999782i \(-0.506653\pi\)
−0.0208981 + 0.999782i \(0.506653\pi\)
\(854\) 0 0
\(855\) 7.61248 0.260341
\(856\) 0 0
\(857\) 49.3646 1.68626 0.843131 0.537709i \(-0.180710\pi\)
0.843131 + 0.537709i \(0.180710\pi\)
\(858\) 0 0
\(859\) 41.6828 1.42220 0.711099 0.703092i \(-0.248198\pi\)
0.711099 + 0.703092i \(0.248198\pi\)
\(860\) 0 0
\(861\) 0.153536 0.00523250
\(862\) 0 0
\(863\) −22.0429 −0.750348 −0.375174 0.926954i \(-0.622417\pi\)
−0.375174 + 0.926954i \(0.622417\pi\)
\(864\) 0 0
\(865\) 18.4487 0.627275
\(866\) 0 0
\(867\) 85.6311 2.90818
\(868\) 0 0
\(869\) 0.884457 0.0300031
\(870\) 0 0
\(871\) 82.4089 2.79232
\(872\) 0 0
\(873\) 1.94319 0.0657670
\(874\) 0 0
\(875\) 0.247973 0.00838300
\(876\) 0 0
\(877\) −33.7237 −1.13877 −0.569385 0.822071i \(-0.692819\pi\)
−0.569385 + 0.822071i \(0.692819\pi\)
\(878\) 0 0
\(879\) −12.4617 −0.420323
\(880\) 0 0
\(881\) 7.24781 0.244185 0.122092 0.992519i \(-0.461040\pi\)
0.122092 + 0.992519i \(0.461040\pi\)
\(882\) 0 0
\(883\) −24.1525 −0.812797 −0.406398 0.913696i \(-0.633215\pi\)
−0.406398 + 0.913696i \(0.633215\pi\)
\(884\) 0 0
\(885\) 40.6960 1.36798
\(886\) 0 0
\(887\) −20.3957 −0.684821 −0.342411 0.939550i \(-0.611243\pi\)
−0.342411 + 0.939550i \(0.611243\pi\)
\(888\) 0 0
\(889\) −10.1771 −0.341328
\(890\) 0 0
\(891\) −4.41979 −0.148069
\(892\) 0 0
\(893\) 2.02928 0.0679072
\(894\) 0 0
\(895\) −59.4015 −1.98557
\(896\) 0 0
\(897\) −91.4998 −3.05509
\(898\) 0 0
\(899\) 2.65340 0.0884959
\(900\) 0 0
\(901\) 42.7321 1.42361
\(902\) 0 0
\(903\) −6.73416 −0.224099
\(904\) 0 0
\(905\) −33.0618 −1.09901
\(906\) 0 0
\(907\) −6.56562 −0.218008 −0.109004 0.994041i \(-0.534766\pi\)
−0.109004 + 0.994041i \(0.534766\pi\)
\(908\) 0 0
\(909\) −30.1143 −0.998827
\(910\) 0 0
\(911\) 6.98339 0.231370 0.115685 0.993286i \(-0.463094\pi\)
0.115685 + 0.993286i \(0.463094\pi\)
\(912\) 0 0
\(913\) −5.02732 −0.166380
\(914\) 0 0
\(915\) −51.1488 −1.69093
\(916\) 0 0
\(917\) 5.42780 0.179242
\(918\) 0 0
\(919\) −12.4237 −0.409819 −0.204910 0.978781i \(-0.565690\pi\)
−0.204910 + 0.978781i \(0.565690\pi\)
\(920\) 0 0
\(921\) −24.4101 −0.804339
\(922\) 0 0
\(923\) 81.0837 2.66890
\(924\) 0 0
\(925\) −16.9591 −0.557610
\(926\) 0 0
\(927\) 6.90873 0.226912
\(928\) 0 0
\(929\) −23.0649 −0.756735 −0.378367 0.925655i \(-0.623514\pi\)
−0.378367 + 0.925655i \(0.623514\pi\)
\(930\) 0 0
\(931\) 6.71307 0.220012
\(932\) 0 0
\(933\) 63.7551 2.08725
\(934\) 0 0
\(935\) −9.78679 −0.320062
\(936\) 0 0
\(937\) 20.2747 0.662347 0.331173 0.943570i \(-0.392555\pi\)
0.331173 + 0.943570i \(0.392555\pi\)
\(938\) 0 0
\(939\) 39.4134 1.28621
\(940\) 0 0
\(941\) 33.8390 1.10312 0.551559 0.834136i \(-0.314033\pi\)
0.551559 + 0.834136i \(0.314033\pi\)
\(942\) 0 0
\(943\) 0.726091 0.0236448
\(944\) 0 0
\(945\) 2.25095 0.0732234
\(946\) 0 0
\(947\) 55.4208 1.80093 0.900466 0.434926i \(-0.143225\pi\)
0.900466 + 0.434926i \(0.143225\pi\)
\(948\) 0 0
\(949\) −46.5639 −1.51153
\(950\) 0 0
\(951\) −8.90033 −0.288613
\(952\) 0 0
\(953\) 19.8530 0.643102 0.321551 0.946892i \(-0.395796\pi\)
0.321551 + 0.946892i \(0.395796\pi\)
\(954\) 0 0
\(955\) −5.03729 −0.163003
\(956\) 0 0
\(957\) −0.829039 −0.0267990
\(958\) 0 0
\(959\) 1.75814 0.0567734
\(960\) 0 0
\(961\) −20.9512 −0.675844
\(962\) 0 0
\(963\) 0.698809 0.0225188
\(964\) 0 0
\(965\) 41.7121 1.34276
\(966\) 0 0
\(967\) 5.95336 0.191447 0.0957236 0.995408i \(-0.469484\pi\)
0.0957236 + 0.995408i \(0.469484\pi\)
\(968\) 0 0
\(969\) −17.0787 −0.548647
\(970\) 0 0
\(971\) 5.33267 0.171134 0.0855668 0.996332i \(-0.472730\pi\)
0.0855668 + 0.996332i \(0.472730\pi\)
\(972\) 0 0
\(973\) 5.64290 0.180903
\(974\) 0 0
\(975\) −75.2507 −2.40995
\(976\) 0 0
\(977\) −12.6063 −0.403312 −0.201656 0.979456i \(-0.564632\pi\)
−0.201656 + 0.979456i \(0.564632\pi\)
\(978\) 0 0
\(979\) −5.85469 −0.187117
\(980\) 0 0
\(981\) 4.65335 0.148570
\(982\) 0 0
\(983\) 19.8397 0.632788 0.316394 0.948628i \(-0.397528\pi\)
0.316394 + 0.948628i \(0.397528\pi\)
\(984\) 0 0
\(985\) 12.7655 0.406741
\(986\) 0 0
\(987\) −2.53185 −0.0805897
\(988\) 0 0
\(989\) −31.8467 −1.01267
\(990\) 0 0
\(991\) −53.9425 −1.71354 −0.856770 0.515699i \(-0.827532\pi\)
−0.856770 + 0.515699i \(0.827532\pi\)
\(992\) 0 0
\(993\) 61.9760 1.96675
\(994\) 0 0
\(995\) 40.2634 1.27644
\(996\) 0 0
\(997\) −20.6213 −0.653084 −0.326542 0.945183i \(-0.605884\pi\)
−0.326542 + 0.945183i \(0.605884\pi\)
\(998\) 0 0
\(999\) 4.67886 0.148033
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 4864.2.a.bo.1.7 8
4.3 odd 2 4864.2.a.bn.1.2 8
8.3 odd 2 4864.2.a.bp.1.7 8
8.5 even 2 4864.2.a.bq.1.2 8
16.3 odd 4 608.2.c.b.305.3 16
16.5 even 4 152.2.c.b.77.14 yes 16
16.11 odd 4 608.2.c.b.305.14 16
16.13 even 4 152.2.c.b.77.13 16
48.5 odd 4 1368.2.g.b.685.3 16
48.11 even 4 5472.2.g.b.2737.14 16
48.29 odd 4 1368.2.g.b.685.4 16
48.35 even 4 5472.2.g.b.2737.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.13 16 16.13 even 4
152.2.c.b.77.14 yes 16 16.5 even 4
608.2.c.b.305.3 16 16.3 odd 4
608.2.c.b.305.14 16 16.11 odd 4
1368.2.g.b.685.3 16 48.5 odd 4
1368.2.g.b.685.4 16 48.29 odd 4
4864.2.a.bn.1.2 8 4.3 odd 2
4864.2.a.bo.1.7 8 1.1 even 1 trivial
4864.2.a.bp.1.7 8 8.3 odd 2
4864.2.a.bq.1.2 8 8.5 even 2
5472.2.g.b.2737.3 16 48.35 even 4
5472.2.g.b.2737.14 16 48.11 even 4