Properties

Label 4864.2.a.bn.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 \(2.67705\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.09554 q^{3} -3.36827 q^{5} +4.47116 q^{7} +1.39129 q^{9} +O(q^{10})\) \(q+2.09554 q^{3} -3.36827 q^{5} +4.47116 q^{7} +1.39129 q^{9} +0.608709 q^{11} -1.03922 q^{13} -7.05835 q^{15} -3.06367 q^{17} +1.00000 q^{19} +9.36950 q^{21} -8.50224 q^{23} +6.34525 q^{25} -3.37112 q^{27} -7.27343 q^{29} -4.02054 q^{31} +1.27558 q^{33} -15.0601 q^{35} -4.31265 q^{37} -2.17773 q^{39} +4.15770 q^{41} -6.27910 q^{43} -4.68624 q^{45} -4.73522 q^{47} +12.9913 q^{49} -6.42005 q^{51} -6.98132 q^{53} -2.05030 q^{55} +2.09554 q^{57} -2.64652 q^{59} +5.11145 q^{61} +6.22069 q^{63} +3.50037 q^{65} +2.62178 q^{67} -17.8168 q^{69} +12.0085 q^{71} +12.5175 q^{73} +13.2967 q^{75} +2.72164 q^{77} +0.913307 q^{79} -11.2382 q^{81} -0.887809 q^{83} +10.3193 q^{85} -15.2418 q^{87} +7.61792 q^{89} -4.64652 q^{91} -8.42521 q^{93} -3.36827 q^{95} -5.93426 q^{97} +0.846892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} - 4q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} - 4q^{7} + 12q^{9} + 4q^{11} - 8q^{13} - 4q^{17} + 8q^{19} - 16q^{21} + 12q^{25} - 28q^{29} - 8q^{31} - 12q^{35} - 4q^{37} + 4q^{39} - 8q^{41} - 4q^{43} - 24q^{45} - 12q^{47} + 12q^{49} + 12q^{51} - 32q^{53} + 8q^{55} + 12q^{59} - 8q^{61} + 16q^{63} + 8q^{65} - 4q^{67} - 28q^{69} + 24q^{71} - 24q^{77} + 24q^{79} - 8q^{81} + 40q^{83} - 24q^{85} - 24q^{87} + 8q^{89} - 4q^{91} - 32q^{93} - 8q^{95} + 16q^{97} - 76q^{99} + O(q^{100}) \)

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
\(3\) 2.09554 1.20986 0.604930 0.796278i \(-0.293201\pi\)
0.604930 + 0.796278i \(0.293201\pi\)
\(4\) 0 0
\(5\) −3.36827 −1.50634 −0.753168 0.657828i \(-0.771475\pi\)
−0.753168 + 0.657828i \(0.771475\pi\)
\(6\) 0 0
\(7\) 4.47116 1.68994 0.844970 0.534813i \(-0.179618\pi\)
0.844970 + 0.534813i \(0.179618\pi\)
\(8\) 0 0
\(9\) 1.39129 0.463764
\(10\) 0 0
\(11\) 0.608709 0.183533 0.0917664 0.995781i \(-0.470749\pi\)
0.0917664 + 0.995781i \(0.470749\pi\)
\(12\) 0 0
\(13\) −1.03922 −0.288228 −0.144114 0.989561i \(-0.546033\pi\)
−0.144114 + 0.989561i \(0.546033\pi\)
\(14\) 0 0
\(15\) −7.05835 −1.82246
\(16\) 0 0
\(17\) −3.06367 −0.743050 −0.371525 0.928423i \(-0.621165\pi\)
−0.371525 + 0.928423i \(0.621165\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 9.36950 2.04459
\(22\) 0 0
\(23\) −8.50224 −1.77284 −0.886419 0.462883i \(-0.846815\pi\)
−0.886419 + 0.462883i \(0.846815\pi\)
\(24\) 0 0
\(25\) 6.34525 1.26905
\(26\) 0 0
\(27\) −3.37112 −0.648772
\(28\) 0 0
\(29\) −7.27343 −1.35064 −0.675321 0.737524i \(-0.735995\pi\)
−0.675321 + 0.737524i \(0.735995\pi\)
\(30\) 0 0
\(31\) −4.02054 −0.722111 −0.361055 0.932544i \(-0.617583\pi\)
−0.361055 + 0.932544i \(0.617583\pi\)
\(32\) 0 0
\(33\) 1.27558 0.222049
\(34\) 0 0
\(35\) −15.0601 −2.54562
\(36\) 0 0
\(37\) −4.31265 −0.708995 −0.354498 0.935057i \(-0.615348\pi\)
−0.354498 + 0.935057i \(0.615348\pi\)
\(38\) 0 0
\(39\) −2.17773 −0.348715
\(40\) 0 0
\(41\) 4.15770 0.649323 0.324661 0.945830i \(-0.394750\pi\)
0.324661 + 0.945830i \(0.394750\pi\)
\(42\) 0 0
\(43\) −6.27910 −0.957554 −0.478777 0.877937i \(-0.658920\pi\)
−0.478777 + 0.877937i \(0.658920\pi\)
\(44\) 0 0
\(45\) −4.68624 −0.698584
\(46\) 0 0
\(47\) −4.73522 −0.690702 −0.345351 0.938474i \(-0.612240\pi\)
−0.345351 + 0.938474i \(0.612240\pi\)
\(48\) 0 0
\(49\) 12.9913 1.85590
\(50\) 0 0
\(51\) −6.42005 −0.898987
\(52\) 0 0
\(53\) −6.98132 −0.958958 −0.479479 0.877553i \(-0.659174\pi\)
−0.479479 + 0.877553i \(0.659174\pi\)
\(54\) 0 0
\(55\) −2.05030 −0.276462
\(56\) 0 0
\(57\) 2.09554 0.277561
\(58\) 0 0
\(59\) −2.64652 −0.344547 −0.172274 0.985049i \(-0.555111\pi\)
−0.172274 + 0.985049i \(0.555111\pi\)
\(60\) 0 0
\(61\) 5.11145 0.654454 0.327227 0.944946i \(-0.393886\pi\)
0.327227 + 0.944946i \(0.393886\pi\)
\(62\) 0 0
\(63\) 6.22069 0.783733
\(64\) 0 0
\(65\) 3.50037 0.434168
\(66\) 0 0
\(67\) 2.62178 0.320301 0.160150 0.987093i \(-0.448802\pi\)
0.160150 + 0.987093i \(0.448802\pi\)
\(68\) 0 0
\(69\) −17.8168 −2.14489
\(70\) 0 0
\(71\) 12.0085 1.42514 0.712572 0.701599i \(-0.247530\pi\)
0.712572 + 0.701599i \(0.247530\pi\)
\(72\) 0 0
\(73\) 12.5175 1.46507 0.732533 0.680731i \(-0.238338\pi\)
0.732533 + 0.680731i \(0.238338\pi\)
\(74\) 0 0
\(75\) 13.2967 1.53537
\(76\) 0 0
\(77\) 2.72164 0.310159
\(78\) 0 0
\(79\) 0.913307 0.102755 0.0513775 0.998679i \(-0.483639\pi\)
0.0513775 + 0.998679i \(0.483639\pi\)
\(80\) 0 0
\(81\) −11.2382 −1.24869
\(82\) 0 0
\(83\) −0.887809 −0.0974497 −0.0487249 0.998812i \(-0.515516\pi\)
−0.0487249 + 0.998812i \(0.515516\pi\)
\(84\) 0 0
\(85\) 10.3193 1.11928
\(86\) 0 0
\(87\) −15.2418 −1.63409
\(88\) 0 0
\(89\) 7.61792 0.807498 0.403749 0.914870i \(-0.367707\pi\)
0.403749 + 0.914870i \(0.367707\pi\)
\(90\) 0 0
\(91\) −4.64652 −0.487087
\(92\) 0 0
\(93\) −8.42521 −0.873653
\(94\) 0 0
\(95\) −3.36827 −0.345577
\(96\) 0 0
\(97\) −5.93426 −0.602533 −0.301267 0.953540i \(-0.597409\pi\)
−0.301267 + 0.953540i \(0.597409\pi\)
\(98\) 0 0
\(99\) 0.846892 0.0851158
\(100\) 0 0
\(101\) −16.5793 −1.64970 −0.824849 0.565353i \(-0.808740\pi\)
−0.824849 + 0.565353i \(0.808740\pi\)
\(102\) 0 0
\(103\) −11.6631 −1.14920 −0.574600 0.818434i \(-0.694842\pi\)
−0.574600 + 0.818434i \(0.694842\pi\)
\(104\) 0 0
\(105\) −31.5590 −3.07985
\(106\) 0 0
\(107\) −10.3213 −0.997801 −0.498900 0.866659i \(-0.666263\pi\)
−0.498900 + 0.866659i \(0.666263\pi\)
\(108\) 0 0
\(109\) −1.85592 −0.177765 −0.0888824 0.996042i \(-0.528330\pi\)
−0.0888824 + 0.996042i \(0.528330\pi\)
\(110\) 0 0
\(111\) −9.03734 −0.857786
\(112\) 0 0
\(113\) 18.4343 1.73415 0.867077 0.498174i \(-0.165996\pi\)
0.867077 + 0.498174i \(0.165996\pi\)
\(114\) 0 0
\(115\) 28.6378 2.67049
\(116\) 0 0
\(117\) −1.44586 −0.133669
\(118\) 0 0
\(119\) −13.6982 −1.25571
\(120\) 0 0
\(121\) −10.6295 −0.966316
\(122\) 0 0
\(123\) 8.71262 0.785591
\(124\) 0 0
\(125\) −4.53117 −0.405281
\(126\) 0 0
\(127\) 9.29686 0.824963 0.412481 0.910966i \(-0.364662\pi\)
0.412481 + 0.910966i \(0.364662\pi\)
\(128\) 0 0
\(129\) −13.1581 −1.15851
\(130\) 0 0
\(131\) −8.15016 −0.712083 −0.356042 0.934470i \(-0.615874\pi\)
−0.356042 + 0.934470i \(0.615874\pi\)
\(132\) 0 0
\(133\) 4.47116 0.387699
\(134\) 0 0
\(135\) 11.3548 0.977268
\(136\) 0 0
\(137\) −11.0756 −0.946254 −0.473127 0.880994i \(-0.656875\pi\)
−0.473127 + 0.880994i \(0.656875\pi\)
\(138\) 0 0
\(139\) 12.2091 1.03557 0.517783 0.855512i \(-0.326758\pi\)
0.517783 + 0.855512i \(0.326758\pi\)
\(140\) 0 0
\(141\) −9.92284 −0.835654
\(142\) 0 0
\(143\) −0.632582 −0.0528992
\(144\) 0 0
\(145\) 24.4989 2.03452
\(146\) 0 0
\(147\) 27.2238 2.24538
\(148\) 0 0
\(149\) 4.31060 0.353138 0.176569 0.984288i \(-0.443500\pi\)
0.176569 + 0.984288i \(0.443500\pi\)
\(150\) 0 0
\(151\) −9.89022 −0.804855 −0.402428 0.915452i \(-0.631833\pi\)
−0.402428 + 0.915452i \(0.631833\pi\)
\(152\) 0 0
\(153\) −4.26246 −0.344599
\(154\) 0 0
\(155\) 13.5423 1.08774
\(156\) 0 0
\(157\) 7.39359 0.590073 0.295036 0.955486i \(-0.404668\pi\)
0.295036 + 0.955486i \(0.404668\pi\)
\(158\) 0 0
\(159\) −14.6296 −1.16021
\(160\) 0 0
\(161\) −38.0149 −2.99599
\(162\) 0 0
\(163\) 12.5566 0.983509 0.491755 0.870734i \(-0.336356\pi\)
0.491755 + 0.870734i \(0.336356\pi\)
\(164\) 0 0
\(165\) −4.29648 −0.334481
\(166\) 0 0
\(167\) −0.00624861 −0.000483532 0 −0.000241766 1.00000i \(-0.500077\pi\)
−0.000241766 1.00000i \(0.500077\pi\)
\(168\) 0 0
\(169\) −11.9200 −0.916925
\(170\) 0 0
\(171\) 1.39129 0.106395
\(172\) 0 0
\(173\) −4.45345 −0.338589 −0.169295 0.985565i \(-0.554149\pi\)
−0.169295 + 0.985565i \(0.554149\pi\)
\(174\) 0 0
\(175\) 28.3707 2.14462
\(176\) 0 0
\(177\) −5.54589 −0.416854
\(178\) 0 0
\(179\) −9.80470 −0.732838 −0.366419 0.930450i \(-0.619416\pi\)
−0.366419 + 0.930450i \(0.619416\pi\)
\(180\) 0 0
\(181\) −17.9475 −1.33403 −0.667013 0.745046i \(-0.732427\pi\)
−0.667013 + 0.745046i \(0.732427\pi\)
\(182\) 0 0
\(183\) 10.7113 0.791799
\(184\) 0 0
\(185\) 14.5262 1.06799
\(186\) 0 0
\(187\) −1.86489 −0.136374
\(188\) 0 0
\(189\) −15.0728 −1.09639
\(190\) 0 0
\(191\) −3.85328 −0.278814 −0.139407 0.990235i \(-0.544520\pi\)
−0.139407 + 0.990235i \(0.544520\pi\)
\(192\) 0 0
\(193\) 19.5610 1.40803 0.704017 0.710183i \(-0.251388\pi\)
0.704017 + 0.710183i \(0.251388\pi\)
\(194\) 0 0
\(195\) 7.33517 0.525283
\(196\) 0 0
\(197\) 3.44831 0.245682 0.122841 0.992426i \(-0.460799\pi\)
0.122841 + 0.992426i \(0.460799\pi\)
\(198\) 0 0
\(199\) −7.64829 −0.542173 −0.271087 0.962555i \(-0.587383\pi\)
−0.271087 + 0.962555i \(0.587383\pi\)
\(200\) 0 0
\(201\) 5.49404 0.387520
\(202\) 0 0
\(203\) −32.5207 −2.28251
\(204\) 0 0
\(205\) −14.0042 −0.978099
\(206\) 0 0
\(207\) −11.8291 −0.822178
\(208\) 0 0
\(209\) 0.608709 0.0421053
\(210\) 0 0
\(211\) −14.4648 −0.995801 −0.497900 0.867234i \(-0.665895\pi\)
−0.497900 + 0.867234i \(0.665895\pi\)
\(212\) 0 0
\(213\) 25.1643 1.72423
\(214\) 0 0
\(215\) 21.1497 1.44240
\(216\) 0 0
\(217\) −17.9765 −1.22032
\(218\) 0 0
\(219\) 26.2310 1.77253
\(220\) 0 0
\(221\) 3.18383 0.214167
\(222\) 0 0
\(223\) −4.74734 −0.317906 −0.158953 0.987286i \(-0.550812\pi\)
−0.158953 + 0.987286i \(0.550812\pi\)
\(224\) 0 0
\(225\) 8.82809 0.588539
\(226\) 0 0
\(227\) −2.67149 −0.177313 −0.0886566 0.996062i \(-0.528257\pi\)
−0.0886566 + 0.996062i \(0.528257\pi\)
\(228\) 0 0
\(229\) −7.87392 −0.520323 −0.260162 0.965565i \(-0.583776\pi\)
−0.260162 + 0.965565i \(0.583776\pi\)
\(230\) 0 0
\(231\) 5.70330 0.375250
\(232\) 0 0
\(233\) 10.2320 0.670321 0.335160 0.942161i \(-0.391209\pi\)
0.335160 + 0.942161i \(0.391209\pi\)
\(234\) 0 0
\(235\) 15.9495 1.04043
\(236\) 0 0
\(237\) 1.91387 0.124319
\(238\) 0 0
\(239\) 15.1051 0.977067 0.488533 0.872545i \(-0.337532\pi\)
0.488533 + 0.872545i \(0.337532\pi\)
\(240\) 0 0
\(241\) 6.17274 0.397621 0.198811 0.980038i \(-0.436292\pi\)
0.198811 + 0.980038i \(0.436292\pi\)
\(242\) 0 0
\(243\) −13.4367 −0.861966
\(244\) 0 0
\(245\) −43.7582 −2.79561
\(246\) 0 0
\(247\) −1.03922 −0.0661239
\(248\) 0 0
\(249\) −1.86044 −0.117901
\(250\) 0 0
\(251\) −6.78315 −0.428149 −0.214074 0.976817i \(-0.568673\pi\)
−0.214074 + 0.976817i \(0.568673\pi\)
\(252\) 0 0
\(253\) −5.17539 −0.325374
\(254\) 0 0
\(255\) 21.6245 1.35418
\(256\) 0 0
\(257\) −26.6333 −1.66134 −0.830669 0.556766i \(-0.812042\pi\)
−0.830669 + 0.556766i \(0.812042\pi\)
\(258\) 0 0
\(259\) −19.2826 −1.19816
\(260\) 0 0
\(261\) −10.1195 −0.626379
\(262\) 0 0
\(263\) 14.6785 0.905117 0.452559 0.891735i \(-0.350511\pi\)
0.452559 + 0.891735i \(0.350511\pi\)
\(264\) 0 0
\(265\) 23.5150 1.44451
\(266\) 0 0
\(267\) 15.9637 0.976961
\(268\) 0 0
\(269\) −11.6590 −0.710862 −0.355431 0.934702i \(-0.615666\pi\)
−0.355431 + 0.934702i \(0.615666\pi\)
\(270\) 0 0
\(271\) −0.125029 −0.00759496 −0.00379748 0.999993i \(-0.501209\pi\)
−0.00379748 + 0.999993i \(0.501209\pi\)
\(272\) 0 0
\(273\) −9.73697 −0.589308
\(274\) 0 0
\(275\) 3.86241 0.232912
\(276\) 0 0
\(277\) −19.8203 −1.19088 −0.595442 0.803398i \(-0.703023\pi\)
−0.595442 + 0.803398i \(0.703023\pi\)
\(278\) 0 0
\(279\) −5.59374 −0.334889
\(280\) 0 0
\(281\) −9.24019 −0.551223 −0.275612 0.961269i \(-0.588880\pi\)
−0.275612 + 0.961269i \(0.588880\pi\)
\(282\) 0 0
\(283\) −30.3491 −1.80407 −0.902035 0.431664i \(-0.857927\pi\)
−0.902035 + 0.431664i \(0.857927\pi\)
\(284\) 0 0
\(285\) −7.05835 −0.418101
\(286\) 0 0
\(287\) 18.5897 1.09732
\(288\) 0 0
\(289\) −7.61391 −0.447877
\(290\) 0 0
\(291\) −12.4355 −0.728981
\(292\) 0 0
\(293\) 19.5780 1.14376 0.571880 0.820337i \(-0.306214\pi\)
0.571880 + 0.820337i \(0.306214\pi\)
\(294\) 0 0
\(295\) 8.91419 0.519004
\(296\) 0 0
\(297\) −2.05203 −0.119071
\(298\) 0 0
\(299\) 8.83569 0.510981
\(300\) 0 0
\(301\) −28.0749 −1.61821
\(302\) 0 0
\(303\) −34.7425 −1.99591
\(304\) 0 0
\(305\) −17.2168 −0.985829
\(306\) 0 0
\(307\) −1.54809 −0.0883541 −0.0441770 0.999024i \(-0.514067\pi\)
−0.0441770 + 0.999024i \(0.514067\pi\)
\(308\) 0 0
\(309\) −24.4405 −1.39037
\(310\) 0 0
\(311\) −3.34671 −0.189774 −0.0948872 0.995488i \(-0.530249\pi\)
−0.0948872 + 0.995488i \(0.530249\pi\)
\(312\) 0 0
\(313\) −6.40232 −0.361881 −0.180940 0.983494i \(-0.557914\pi\)
−0.180940 + 0.983494i \(0.557914\pi\)
\(314\) 0 0
\(315\) −20.9530 −1.18057
\(316\) 0 0
\(317\) 9.23830 0.518875 0.259437 0.965760i \(-0.416463\pi\)
0.259437 + 0.965760i \(0.416463\pi\)
\(318\) 0 0
\(319\) −4.42741 −0.247887
\(320\) 0 0
\(321\) −21.6288 −1.20720
\(322\) 0 0
\(323\) −3.06367 −0.170467
\(324\) 0 0
\(325\) −6.59411 −0.365775
\(326\) 0 0
\(327\) −3.88915 −0.215071
\(328\) 0 0
\(329\) −21.1719 −1.16725
\(330\) 0 0
\(331\) 4.65778 0.256015 0.128007 0.991773i \(-0.459142\pi\)
0.128007 + 0.991773i \(0.459142\pi\)
\(332\) 0 0
\(333\) −6.00015 −0.328806
\(334\) 0 0
\(335\) −8.83085 −0.482481
\(336\) 0 0
\(337\) −9.34371 −0.508984 −0.254492 0.967075i \(-0.581908\pi\)
−0.254492 + 0.967075i \(0.581908\pi\)
\(338\) 0 0
\(339\) 38.6298 2.09808
\(340\) 0 0
\(341\) −2.44734 −0.132531
\(342\) 0 0
\(343\) 26.7881 1.44642
\(344\) 0 0
\(345\) 60.0117 3.23092
\(346\) 0 0
\(347\) −30.9185 −1.65979 −0.829897 0.557917i \(-0.811601\pi\)
−0.829897 + 0.557917i \(0.811601\pi\)
\(348\) 0 0
\(349\) 25.2445 1.35131 0.675653 0.737220i \(-0.263862\pi\)
0.675653 + 0.737220i \(0.263862\pi\)
\(350\) 0 0
\(351\) 3.50333 0.186994
\(352\) 0 0
\(353\) −2.15727 −0.114820 −0.0574099 0.998351i \(-0.518284\pi\)
−0.0574099 + 0.998351i \(0.518284\pi\)
\(354\) 0 0
\(355\) −40.4478 −2.14675
\(356\) 0 0
\(357\) −28.7051 −1.51923
\(358\) 0 0
\(359\) 32.0925 1.69378 0.846889 0.531770i \(-0.178473\pi\)
0.846889 + 0.531770i \(0.178473\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −22.2745 −1.16911
\(364\) 0 0
\(365\) −42.1624 −2.20688
\(366\) 0 0
\(367\) −10.3206 −0.538729 −0.269365 0.963038i \(-0.586814\pi\)
−0.269365 + 0.963038i \(0.586814\pi\)
\(368\) 0 0
\(369\) 5.78456 0.301132
\(370\) 0 0
\(371\) −31.2146 −1.62058
\(372\) 0 0
\(373\) 20.3031 1.05125 0.525627 0.850715i \(-0.323831\pi\)
0.525627 + 0.850715i \(0.323831\pi\)
\(374\) 0 0
\(375\) −9.49526 −0.490333
\(376\) 0 0
\(377\) 7.55869 0.389292
\(378\) 0 0
\(379\) 15.4355 0.792867 0.396433 0.918063i \(-0.370248\pi\)
0.396433 + 0.918063i \(0.370248\pi\)
\(380\) 0 0
\(381\) 19.4819 0.998090
\(382\) 0 0
\(383\) 37.7837 1.93066 0.965328 0.261041i \(-0.0840658\pi\)
0.965328 + 0.261041i \(0.0840658\pi\)
\(384\) 0 0
\(385\) −9.16722 −0.467205
\(386\) 0 0
\(387\) −8.73605 −0.444079
\(388\) 0 0
\(389\) 13.7943 0.699399 0.349700 0.936862i \(-0.386284\pi\)
0.349700 + 0.936862i \(0.386284\pi\)
\(390\) 0 0
\(391\) 26.0481 1.31731
\(392\) 0 0
\(393\) −17.0790 −0.861522
\(394\) 0 0
\(395\) −3.07627 −0.154784
\(396\) 0 0
\(397\) 33.1665 1.66458 0.832288 0.554343i \(-0.187030\pi\)
0.832288 + 0.554343i \(0.187030\pi\)
\(398\) 0 0
\(399\) 9.36950 0.469062
\(400\) 0 0
\(401\) −8.24620 −0.411796 −0.205898 0.978573i \(-0.566011\pi\)
−0.205898 + 0.978573i \(0.566011\pi\)
\(402\) 0 0
\(403\) 4.17822 0.208132
\(404\) 0 0
\(405\) 37.8532 1.88094
\(406\) 0 0
\(407\) −2.62515 −0.130124
\(408\) 0 0
\(409\) −30.4291 −1.50462 −0.752312 0.658807i \(-0.771061\pi\)
−0.752312 + 0.658807i \(0.771061\pi\)
\(410\) 0 0
\(411\) −23.2094 −1.14484
\(412\) 0 0
\(413\) −11.8330 −0.582264
\(414\) 0 0
\(415\) 2.99038 0.146792
\(416\) 0 0
\(417\) 25.5847 1.25289
\(418\) 0 0
\(419\) 5.11734 0.249999 0.124999 0.992157i \(-0.460107\pi\)
0.124999 + 0.992157i \(0.460107\pi\)
\(420\) 0 0
\(421\) 28.8854 1.40779 0.703894 0.710305i \(-0.251443\pi\)
0.703894 + 0.710305i \(0.251443\pi\)
\(422\) 0 0
\(423\) −6.58806 −0.320323
\(424\) 0 0
\(425\) −19.4398 −0.942967
\(426\) 0 0
\(427\) 22.8541 1.10599
\(428\) 0 0
\(429\) −1.32560 −0.0640007
\(430\) 0 0
\(431\) 19.3963 0.934288 0.467144 0.884181i \(-0.345283\pi\)
0.467144 + 0.884181i \(0.345283\pi\)
\(432\) 0 0
\(433\) 30.5791 1.46954 0.734770 0.678317i \(-0.237290\pi\)
0.734770 + 0.678317i \(0.237290\pi\)
\(434\) 0 0
\(435\) 51.3384 2.46149
\(436\) 0 0
\(437\) −8.50224 −0.406717
\(438\) 0 0
\(439\) −21.4056 −1.02163 −0.510817 0.859690i \(-0.670657\pi\)
−0.510817 + 0.859690i \(0.670657\pi\)
\(440\) 0 0
\(441\) 18.0747 0.860699
\(442\) 0 0
\(443\) 8.83665 0.419842 0.209921 0.977718i \(-0.432679\pi\)
0.209921 + 0.977718i \(0.432679\pi\)
\(444\) 0 0
\(445\) −25.6592 −1.21636
\(446\) 0 0
\(447\) 9.03303 0.427248
\(448\) 0 0
\(449\) 17.9257 0.845968 0.422984 0.906137i \(-0.360983\pi\)
0.422984 + 0.906137i \(0.360983\pi\)
\(450\) 0 0
\(451\) 2.53083 0.119172
\(452\) 0 0
\(453\) −20.7254 −0.973763
\(454\) 0 0
\(455\) 15.6507 0.733718
\(456\) 0 0
\(457\) 11.8630 0.554930 0.277465 0.960736i \(-0.410506\pi\)
0.277465 + 0.960736i \(0.410506\pi\)
\(458\) 0 0
\(459\) 10.3280 0.482069
\(460\) 0 0
\(461\) −7.03036 −0.327436 −0.163718 0.986507i \(-0.552349\pi\)
−0.163718 + 0.986507i \(0.552349\pi\)
\(462\) 0 0
\(463\) 0.523381 0.0243236 0.0121618 0.999926i \(-0.496129\pi\)
0.0121618 + 0.999926i \(0.496129\pi\)
\(464\) 0 0
\(465\) 28.3784 1.31602
\(466\) 0 0
\(467\) −15.1010 −0.698789 −0.349395 0.936976i \(-0.613613\pi\)
−0.349395 + 0.936976i \(0.613613\pi\)
\(468\) 0 0
\(469\) 11.7224 0.541290
\(470\) 0 0
\(471\) 15.4936 0.713906
\(472\) 0 0
\(473\) −3.82215 −0.175742
\(474\) 0 0
\(475\) 6.34525 0.291140
\(476\) 0 0
\(477\) −9.71305 −0.444730
\(478\) 0 0
\(479\) −42.8653 −1.95857 −0.979284 0.202493i \(-0.935096\pi\)
−0.979284 + 0.202493i \(0.935096\pi\)
\(480\) 0 0
\(481\) 4.48179 0.204352
\(482\) 0 0
\(483\) −79.6617 −3.62473
\(484\) 0 0
\(485\) 19.9882 0.907618
\(486\) 0 0
\(487\) −26.1098 −1.18315 −0.591573 0.806251i \(-0.701493\pi\)
−0.591573 + 0.806251i \(0.701493\pi\)
\(488\) 0 0
\(489\) 26.3129 1.18991
\(490\) 0 0
\(491\) 31.5875 1.42552 0.712761 0.701407i \(-0.247444\pi\)
0.712761 + 0.701407i \(0.247444\pi\)
\(492\) 0 0
\(493\) 22.2834 1.00359
\(494\) 0 0
\(495\) −2.85256 −0.128213
\(496\) 0 0
\(497\) 53.6919 2.40841
\(498\) 0 0
\(499\) 16.1119 0.721268 0.360634 0.932707i \(-0.382560\pi\)
0.360634 + 0.932707i \(0.382560\pi\)
\(500\) 0 0
\(501\) −0.0130942 −0.000585007 0
\(502\) 0 0
\(503\) 6.81093 0.303684 0.151842 0.988405i \(-0.451479\pi\)
0.151842 + 0.988405i \(0.451479\pi\)
\(504\) 0 0
\(505\) 55.8435 2.48500
\(506\) 0 0
\(507\) −24.9789 −1.10935
\(508\) 0 0
\(509\) −15.5299 −0.688350 −0.344175 0.938905i \(-0.611841\pi\)
−0.344175 + 0.938905i \(0.611841\pi\)
\(510\) 0 0
\(511\) 55.9679 2.47588
\(512\) 0 0
\(513\) −3.37112 −0.148838
\(514\) 0 0
\(515\) 39.2845 1.73108
\(516\) 0 0
\(517\) −2.88237 −0.126766
\(518\) 0 0
\(519\) −9.33238 −0.409646
\(520\) 0 0
\(521\) 4.33054 0.189724 0.0948622 0.995490i \(-0.469759\pi\)
0.0948622 + 0.995490i \(0.469759\pi\)
\(522\) 0 0
\(523\) −5.17275 −0.226189 −0.113094 0.993584i \(-0.536076\pi\)
−0.113094 + 0.993584i \(0.536076\pi\)
\(524\) 0 0
\(525\) 59.4519 2.59469
\(526\) 0 0
\(527\) 12.3176 0.536564
\(528\) 0 0
\(529\) 49.2880 2.14296
\(530\) 0 0
\(531\) −3.68208 −0.159788
\(532\) 0 0
\(533\) −4.32076 −0.187153
\(534\) 0 0
\(535\) 34.7650 1.50302
\(536\) 0 0
\(537\) −20.5462 −0.886632
\(538\) 0 0
\(539\) 7.90792 0.340618
\(540\) 0 0
\(541\) −18.2195 −0.783316 −0.391658 0.920111i \(-0.628098\pi\)
−0.391658 + 0.920111i \(0.628098\pi\)
\(542\) 0 0
\(543\) −37.6097 −1.61399
\(544\) 0 0
\(545\) 6.25124 0.267774
\(546\) 0 0
\(547\) 39.0195 1.66835 0.834176 0.551499i \(-0.185944\pi\)
0.834176 + 0.551499i \(0.185944\pi\)
\(548\) 0 0
\(549\) 7.11152 0.303512
\(550\) 0 0
\(551\) −7.27343 −0.309859
\(552\) 0 0
\(553\) 4.08354 0.173650
\(554\) 0 0
\(555\) 30.4402 1.29211
\(556\) 0 0
\(557\) −17.3116 −0.733516 −0.366758 0.930316i \(-0.619532\pi\)
−0.366758 + 0.930316i \(0.619532\pi\)
\(558\) 0 0
\(559\) 6.52536 0.275993
\(560\) 0 0
\(561\) −3.90794 −0.164994
\(562\) 0 0
\(563\) 1.53556 0.0647162 0.0323581 0.999476i \(-0.489698\pi\)
0.0323581 + 0.999476i \(0.489698\pi\)
\(564\) 0 0
\(565\) −62.0917 −2.61222
\(566\) 0 0
\(567\) −50.2477 −2.11021
\(568\) 0 0
\(569\) −25.4671 −1.06764 −0.533818 0.845600i \(-0.679243\pi\)
−0.533818 + 0.845600i \(0.679243\pi\)
\(570\) 0 0
\(571\) −31.1021 −1.30158 −0.650792 0.759256i \(-0.725563\pi\)
−0.650792 + 0.759256i \(0.725563\pi\)
\(572\) 0 0
\(573\) −8.07471 −0.337326
\(574\) 0 0
\(575\) −53.9488 −2.24982
\(576\) 0 0
\(577\) 6.93064 0.288526 0.144263 0.989539i \(-0.453919\pi\)
0.144263 + 0.989539i \(0.453919\pi\)
\(578\) 0 0
\(579\) 40.9909 1.70353
\(580\) 0 0
\(581\) −3.96954 −0.164684
\(582\) 0 0
\(583\) −4.24960 −0.176000
\(584\) 0 0
\(585\) 4.87004 0.201351
\(586\) 0 0
\(587\) 14.2948 0.590009 0.295005 0.955496i \(-0.404679\pi\)
0.295005 + 0.955496i \(0.404679\pi\)
\(588\) 0 0
\(589\) −4.02054 −0.165664
\(590\) 0 0
\(591\) 7.22608 0.297241
\(592\) 0 0
\(593\) −21.2046 −0.870770 −0.435385 0.900244i \(-0.643388\pi\)
−0.435385 + 0.900244i \(0.643388\pi\)
\(594\) 0 0
\(595\) 46.1392 1.89152
\(596\) 0 0
\(597\) −16.0273 −0.655954
\(598\) 0 0
\(599\) 6.52684 0.266679 0.133340 0.991070i \(-0.457430\pi\)
0.133340 + 0.991070i \(0.457430\pi\)
\(600\) 0 0
\(601\) 37.6306 1.53498 0.767492 0.641059i \(-0.221505\pi\)
0.767492 + 0.641059i \(0.221505\pi\)
\(602\) 0 0
\(603\) 3.64765 0.148544
\(604\) 0 0
\(605\) 35.8029 1.45560
\(606\) 0 0
\(607\) 24.8911 1.01030 0.505150 0.863032i \(-0.331437\pi\)
0.505150 + 0.863032i \(0.331437\pi\)
\(608\) 0 0
\(609\) −68.1484 −2.76151
\(610\) 0 0
\(611\) 4.92093 0.199079
\(612\) 0 0
\(613\) −0.491774 −0.0198626 −0.00993128 0.999951i \(-0.503161\pi\)
−0.00993128 + 0.999951i \(0.503161\pi\)
\(614\) 0 0
\(615\) −29.3465 −1.18336
\(616\) 0 0
\(617\) −14.2755 −0.574711 −0.287356 0.957824i \(-0.592776\pi\)
−0.287356 + 0.957824i \(0.592776\pi\)
\(618\) 0 0
\(619\) −17.6251 −0.708411 −0.354206 0.935168i \(-0.615249\pi\)
−0.354206 + 0.935168i \(0.615249\pi\)
\(620\) 0 0
\(621\) 28.6620 1.15017
\(622\) 0 0
\(623\) 34.0610 1.36462
\(624\) 0 0
\(625\) −16.4640 −0.658561
\(626\) 0 0
\(627\) 1.27558 0.0509416
\(628\) 0 0
\(629\) 13.2125 0.526819
\(630\) 0 0
\(631\) −0.532668 −0.0212052 −0.0106026 0.999944i \(-0.503375\pi\)
−0.0106026 + 0.999944i \(0.503375\pi\)
\(632\) 0 0
\(633\) −30.3117 −1.20478
\(634\) 0 0
\(635\) −31.3143 −1.24267
\(636\) 0 0
\(637\) −13.5008 −0.534921
\(638\) 0 0
\(639\) 16.7073 0.660930
\(640\) 0 0
\(641\) 16.5745 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(642\) 0 0
\(643\) −9.17617 −0.361873 −0.180936 0.983495i \(-0.557913\pi\)
−0.180936 + 0.983495i \(0.557913\pi\)
\(644\) 0 0
\(645\) 44.3201 1.74510
\(646\) 0 0
\(647\) −20.1025 −0.790311 −0.395155 0.918614i \(-0.629309\pi\)
−0.395155 + 0.918614i \(0.629309\pi\)
\(648\) 0 0
\(649\) −1.61096 −0.0632357
\(650\) 0 0
\(651\) −37.6705 −1.47642
\(652\) 0 0
\(653\) 42.6830 1.67031 0.835157 0.550012i \(-0.185377\pi\)
0.835157 + 0.550012i \(0.185377\pi\)
\(654\) 0 0
\(655\) 27.4520 1.07264
\(656\) 0 0
\(657\) 17.4155 0.679444
\(658\) 0 0
\(659\) 34.7270 1.35277 0.676385 0.736548i \(-0.263546\pi\)
0.676385 + 0.736548i \(0.263546\pi\)
\(660\) 0 0
\(661\) 0.0670508 0.00260797 0.00130399 0.999999i \(-0.499585\pi\)
0.00130399 + 0.999999i \(0.499585\pi\)
\(662\) 0 0
\(663\) 6.67184 0.259113
\(664\) 0 0
\(665\) −15.0601 −0.584005
\(666\) 0 0
\(667\) 61.8404 2.39447
\(668\) 0 0
\(669\) −9.94825 −0.384622
\(670\) 0 0
\(671\) 3.11139 0.120114
\(672\) 0 0
\(673\) −26.4049 −1.01783 −0.508917 0.860816i \(-0.669954\pi\)
−0.508917 + 0.860816i \(0.669954\pi\)
\(674\) 0 0
\(675\) −21.3906 −0.823324
\(676\) 0 0
\(677\) 17.9627 0.690361 0.345181 0.938536i \(-0.387818\pi\)
0.345181 + 0.938536i \(0.387818\pi\)
\(678\) 0 0
\(679\) −26.5331 −1.01825
\(680\) 0 0
\(681\) −5.59822 −0.214524
\(682\) 0 0
\(683\) 32.3268 1.23695 0.618475 0.785805i \(-0.287751\pi\)
0.618475 + 0.785805i \(0.287751\pi\)
\(684\) 0 0
\(685\) 37.3057 1.42538
\(686\) 0 0
\(687\) −16.5001 −0.629519
\(688\) 0 0
\(689\) 7.25512 0.276398
\(690\) 0 0
\(691\) −1.75007 −0.0665757 −0.0332878 0.999446i \(-0.510598\pi\)
−0.0332878 + 0.999446i \(0.510598\pi\)
\(692\) 0 0
\(693\) 3.78659 0.143841
\(694\) 0 0
\(695\) −41.1237 −1.55991
\(696\) 0 0
\(697\) −12.7378 −0.482479
\(698\) 0 0
\(699\) 21.4416 0.810995
\(700\) 0 0
\(701\) −27.7248 −1.04715 −0.523576 0.851979i \(-0.675403\pi\)
−0.523576 + 0.851979i \(0.675403\pi\)
\(702\) 0 0
\(703\) −4.31265 −0.162655
\(704\) 0 0
\(705\) 33.4228 1.25878
\(706\) 0 0
\(707\) −74.1286 −2.78789
\(708\) 0 0
\(709\) −50.2818 −1.88837 −0.944186 0.329412i \(-0.893150\pi\)
−0.944186 + 0.329412i \(0.893150\pi\)
\(710\) 0 0
\(711\) 1.27068 0.0476541
\(712\) 0 0
\(713\) 34.1836 1.28019
\(714\) 0 0
\(715\) 2.13071 0.0796840
\(716\) 0 0
\(717\) 31.6533 1.18211
\(718\) 0 0
\(719\) 13.1202 0.489300 0.244650 0.969611i \(-0.421327\pi\)
0.244650 + 0.969611i \(0.421327\pi\)
\(720\) 0 0
\(721\) −52.1477 −1.94208
\(722\) 0 0
\(723\) 12.9352 0.481066
\(724\) 0 0
\(725\) −46.1518 −1.71403
\(726\) 0 0
\(727\) 32.5922 1.20878 0.604389 0.796690i \(-0.293417\pi\)
0.604389 + 0.796690i \(0.293417\pi\)
\(728\) 0 0
\(729\) 5.55735 0.205828
\(730\) 0 0
\(731\) 19.2371 0.711510
\(732\) 0 0
\(733\) −8.28509 −0.306017 −0.153008 0.988225i \(-0.548896\pi\)
−0.153008 + 0.988225i \(0.548896\pi\)
\(734\) 0 0
\(735\) −91.6971 −3.38230
\(736\) 0 0
\(737\) 1.59590 0.0587857
\(738\) 0 0
\(739\) −48.5291 −1.78517 −0.892587 0.450876i \(-0.851112\pi\)
−0.892587 + 0.450876i \(0.851112\pi\)
\(740\) 0 0
\(741\) −2.17773 −0.0800008
\(742\) 0 0
\(743\) 3.81374 0.139913 0.0699563 0.997550i \(-0.477714\pi\)
0.0699563 + 0.997550i \(0.477714\pi\)
\(744\) 0 0
\(745\) −14.5193 −0.531945
\(746\) 0 0
\(747\) −1.23520 −0.0451936
\(748\) 0 0
\(749\) −46.1483 −1.68622
\(750\) 0 0
\(751\) −46.7083 −1.70441 −0.852205 0.523208i \(-0.824735\pi\)
−0.852205 + 0.523208i \(0.824735\pi\)
\(752\) 0 0
\(753\) −14.2144 −0.518000
\(754\) 0 0
\(755\) 33.3130 1.21238
\(756\) 0 0
\(757\) 38.8886 1.41343 0.706716 0.707498i \(-0.250176\pi\)
0.706716 + 0.707498i \(0.250176\pi\)
\(758\) 0 0
\(759\) −10.8452 −0.393657
\(760\) 0 0
\(761\) −36.7676 −1.33283 −0.666413 0.745583i \(-0.732171\pi\)
−0.666413 + 0.745583i \(0.732171\pi\)
\(762\) 0 0
\(763\) −8.29812 −0.300412
\(764\) 0 0
\(765\) 14.3571 0.519083
\(766\) 0 0
\(767\) 2.75031 0.0993080
\(768\) 0 0
\(769\) 52.3408 1.88746 0.943730 0.330718i \(-0.107291\pi\)
0.943730 + 0.330718i \(0.107291\pi\)
\(770\) 0 0
\(771\) −55.8111 −2.00999
\(772\) 0 0
\(773\) 13.8632 0.498626 0.249313 0.968423i \(-0.419795\pi\)
0.249313 + 0.968423i \(0.419795\pi\)
\(774\) 0 0
\(775\) −25.5113 −0.916395
\(776\) 0 0
\(777\) −40.4074 −1.44961
\(778\) 0 0
\(779\) 4.15770 0.148965
\(780\) 0 0
\(781\) 7.30967 0.261561
\(782\) 0 0
\(783\) 24.5196 0.876258
\(784\) 0 0
\(785\) −24.9036 −0.888848
\(786\) 0 0
\(787\) −7.53623 −0.268638 −0.134319 0.990938i \(-0.542885\pi\)
−0.134319 + 0.990938i \(0.542885\pi\)
\(788\) 0 0
\(789\) 30.7595 1.09507
\(790\) 0 0
\(791\) 82.4228 2.93062
\(792\) 0 0
\(793\) −5.31192 −0.188632
\(794\) 0 0
\(795\) 49.2766 1.74766
\(796\) 0 0
\(797\) −44.5030 −1.57638 −0.788189 0.615433i \(-0.788981\pi\)
−0.788189 + 0.615433i \(0.788981\pi\)
\(798\) 0 0
\(799\) 14.5071 0.513226
\(800\) 0 0
\(801\) 10.5987 0.374488
\(802\) 0 0
\(803\) 7.61954 0.268888
\(804\) 0 0
\(805\) 128.044 4.51297
\(806\) 0 0
\(807\) −24.4319 −0.860044
\(808\) 0 0
\(809\) 3.45607 0.121509 0.0607544 0.998153i \(-0.480649\pi\)
0.0607544 + 0.998153i \(0.480649\pi\)
\(810\) 0 0
\(811\) 14.9470 0.524860 0.262430 0.964951i \(-0.415476\pi\)
0.262430 + 0.964951i \(0.415476\pi\)
\(812\) 0 0
\(813\) −0.262003 −0.00918885
\(814\) 0 0
\(815\) −42.2941 −1.48150
\(816\) 0 0
\(817\) −6.27910 −0.219678
\(818\) 0 0
\(819\) −6.46466 −0.225893
\(820\) 0 0
\(821\) 3.72440 0.129982 0.0649912 0.997886i \(-0.479298\pi\)
0.0649912 + 0.997886i \(0.479298\pi\)
\(822\) 0 0
\(823\) 15.0687 0.525262 0.262631 0.964896i \(-0.415410\pi\)
0.262631 + 0.964896i \(0.415410\pi\)
\(824\) 0 0
\(825\) 8.09385 0.281792
\(826\) 0 0
\(827\) −37.8202 −1.31514 −0.657568 0.753395i \(-0.728415\pi\)
−0.657568 + 0.753395i \(0.728415\pi\)
\(828\) 0 0
\(829\) 38.8523 1.34940 0.674698 0.738094i \(-0.264274\pi\)
0.674698 + 0.738094i \(0.264274\pi\)
\(830\) 0 0
\(831\) −41.5342 −1.44080
\(832\) 0 0
\(833\) −39.8011 −1.37903
\(834\) 0 0
\(835\) 0.0210470 0.000728362 0
\(836\) 0 0
\(837\) 13.5537 0.468485
\(838\) 0 0
\(839\) −15.8272 −0.546417 −0.273209 0.961955i \(-0.588085\pi\)
−0.273209 + 0.961955i \(0.588085\pi\)
\(840\) 0 0
\(841\) 23.9028 0.824235
\(842\) 0 0
\(843\) −19.3632 −0.666904
\(844\) 0 0
\(845\) 40.1499 1.38120
\(846\) 0 0
\(847\) −47.5261 −1.63302
\(848\) 0 0
\(849\) −63.5979 −2.18267
\(850\) 0 0
\(851\) 36.6672 1.25693
\(852\) 0 0
\(853\) −6.18601 −0.211805 −0.105902 0.994377i \(-0.533773\pi\)
−0.105902 + 0.994377i \(0.533773\pi\)
\(854\) 0 0
\(855\) −4.68624 −0.160266
\(856\) 0 0
\(857\) −12.2246 −0.417583 −0.208792 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(858\) 0 0
\(859\) −44.9215 −1.53270 −0.766350 0.642423i \(-0.777929\pi\)
−0.766350 + 0.642423i \(0.777929\pi\)
\(860\) 0 0
\(861\) 38.9555 1.32760
\(862\) 0 0
\(863\) −21.2268 −0.722568 −0.361284 0.932456i \(-0.617661\pi\)
−0.361284 + 0.932456i \(0.617661\pi\)
\(864\) 0 0
\(865\) 15.0004 0.510029
\(866\) 0 0
\(867\) −15.9553 −0.541869
\(868\) 0 0
\(869\) 0.555938 0.0188589
\(870\) 0 0
\(871\) −2.72460 −0.0923196
\(872\) 0 0
\(873\) −8.25628 −0.279433
\(874\) 0 0
\(875\) −20.2596 −0.684900
\(876\) 0 0
\(877\) 15.4462 0.521582 0.260791 0.965395i \(-0.416017\pi\)
0.260791 + 0.965395i \(0.416017\pi\)
\(878\) 0 0
\(879\) 41.0266 1.38379
\(880\) 0 0
\(881\) −25.4049 −0.855912 −0.427956 0.903800i \(-0.640766\pi\)
−0.427956 + 0.903800i \(0.640766\pi\)
\(882\) 0 0
\(883\) 29.9928 1.00934 0.504669 0.863313i \(-0.331615\pi\)
0.504669 + 0.863313i \(0.331615\pi\)
\(884\) 0 0
\(885\) 18.6800 0.627923
\(886\) 0 0
\(887\) −27.5162 −0.923903 −0.461952 0.886905i \(-0.652851\pi\)
−0.461952 + 0.886905i \(0.652851\pi\)
\(888\) 0 0
\(889\) 41.5678 1.39414
\(890\) 0 0
\(891\) −6.84079 −0.229175
\(892\) 0 0
\(893\) −4.73522 −0.158458
\(894\) 0 0
\(895\) 33.0249 1.10390
\(896\) 0 0
\(897\) 18.5155 0.618216
\(898\) 0 0
\(899\) 29.2431 0.975313
\(900\) 0 0
\(901\) 21.3885 0.712554
\(902\) 0 0
\(903\) −58.8321 −1.95781
\(904\) 0 0
\(905\) 60.4520 2.00949
\(906\) 0 0
\(907\) −26.8589 −0.891837 −0.445918 0.895074i \(-0.647123\pi\)
−0.445918 + 0.895074i \(0.647123\pi\)
\(908\) 0 0
\(909\) −23.0666 −0.765070
\(910\) 0 0
\(911\) 34.5330 1.14413 0.572065 0.820208i \(-0.306142\pi\)
0.572065 + 0.820208i \(0.306142\pi\)
\(912\) 0 0
\(913\) −0.540418 −0.0178852
\(914\) 0 0
\(915\) −36.0784 −1.19272
\(916\) 0 0
\(917\) −36.4407 −1.20338
\(918\) 0 0
\(919\) −20.0300 −0.660729 −0.330365 0.943853i \(-0.607172\pi\)
−0.330365 + 0.943853i \(0.607172\pi\)
\(920\) 0 0
\(921\) −3.24408 −0.106896
\(922\) 0 0
\(923\) −12.4794 −0.410766
\(924\) 0 0
\(925\) −27.3649 −0.899751
\(926\) 0 0
\(927\) −16.2268 −0.532957
\(928\) 0 0
\(929\) 2.91198 0.0955389 0.0477695 0.998858i \(-0.484789\pi\)
0.0477695 + 0.998858i \(0.484789\pi\)
\(930\) 0 0
\(931\) 12.9913 0.425773
\(932\) 0 0
\(933\) −7.01316 −0.229601
\(934\) 0 0
\(935\) 6.28144 0.205425
\(936\) 0 0
\(937\) −4.03574 −0.131842 −0.0659210 0.997825i \(-0.520999\pi\)
−0.0659210 + 0.997825i \(0.520999\pi\)
\(938\) 0 0
\(939\) −13.4163 −0.437825
\(940\) 0 0
\(941\) −51.2751 −1.67152 −0.835760 0.549096i \(-0.814972\pi\)
−0.835760 + 0.549096i \(0.814972\pi\)
\(942\) 0 0
\(943\) −35.3497 −1.15114
\(944\) 0 0
\(945\) 50.7693 1.65153
\(946\) 0 0
\(947\) 37.7009 1.22511 0.612557 0.790426i \(-0.290141\pi\)
0.612557 + 0.790426i \(0.290141\pi\)
\(948\) 0 0
\(949\) −13.0085 −0.422272
\(950\) 0 0
\(951\) 19.3592 0.627766
\(952\) 0 0
\(953\) −41.4066 −1.34129 −0.670645 0.741779i \(-0.733983\pi\)
−0.670645 + 0.741779i \(0.733983\pi\)
\(954\) 0 0
\(955\) 12.9789 0.419987
\(956\) 0 0
\(957\) −9.27781 −0.299909
\(958\) 0 0
\(959\) −49.5209 −1.59911
\(960\) 0 0
\(961\) −14.8352 −0.478556
\(962\) 0 0
\(963\) −14.3600 −0.462744
\(964\) 0 0
\(965\) −65.8869 −2.12097
\(966\) 0 0
\(967\) 29.9529 0.963220 0.481610 0.876386i \(-0.340052\pi\)
0.481610 + 0.876386i \(0.340052\pi\)
\(968\) 0 0
\(969\) −6.42005 −0.206242
\(970\) 0 0
\(971\) 53.5633 1.71893 0.859465 0.511195i \(-0.170797\pi\)
0.859465 + 0.511195i \(0.170797\pi\)
\(972\) 0 0
\(973\) 54.5890 1.75004
\(974\) 0 0
\(975\) −13.8182 −0.442537
\(976\) 0 0
\(977\) −26.2092 −0.838506 −0.419253 0.907869i \(-0.637708\pi\)
−0.419253 + 0.907869i \(0.637708\pi\)
\(978\) 0 0
\(979\) 4.63710 0.148202
\(980\) 0 0
\(981\) −2.58212 −0.0824408
\(982\) 0 0
\(983\) 6.77344 0.216039 0.108020 0.994149i \(-0.465549\pi\)
0.108020 + 0.994149i \(0.465549\pi\)
\(984\) 0 0
\(985\) −11.6149 −0.370080
\(986\) 0 0
\(987\) −44.3666 −1.41221
\(988\) 0 0
\(989\) 53.3864 1.69759
\(990\) 0 0
\(991\) 8.71348 0.276793 0.138396 0.990377i \(-0.455805\pi\)
0.138396 + 0.990377i \(0.455805\pi\)
\(992\) 0 0
\(993\) 9.76058 0.309743
\(994\) 0 0
\(995\) 25.7615 0.816695
\(996\) 0 0
\(997\) 58.7540 1.86076 0.930379 0.366599i \(-0.119478\pi\)
0.930379 + 0.366599i \(0.119478\pi\)
\(998\) 0 0
\(999\) 14.5384 0.459976
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.bn.1.7 8
4.3 odd 2 4864.2.a.bo.1.2 8
8.3 odd 2 4864.2.a.bq.1.7 8
8.5 even 2 4864.2.a.bp.1.2 8
16.3 odd 4 152.2.c.b.77.10 yes 16
16.5 even 4 608.2.c.b.305.4 16
16.11 odd 4 152.2.c.b.77.9 16
16.13 even 4 608.2.c.b.305.13 16
48.5 odd 4 5472.2.g.b.2737.15 16
48.11 even 4 1368.2.g.b.685.8 16
48.29 odd 4 5472.2.g.b.2737.2 16
48.35 even 4 1368.2.g.b.685.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.9 16 16.11 odd 4
152.2.c.b.77.10 yes 16 16.3 odd 4
608.2.c.b.305.4 16 16.5 even 4
608.2.c.b.305.13 16 16.13 even 4
1368.2.g.b.685.7 16 48.35 even 4
1368.2.g.b.685.8 16 48.11 even 4
4864.2.a.bn.1.7 8 1.1 even 1 trivial
4864.2.a.bo.1.2 8 4.3 odd 2
4864.2.a.bp.1.2 8 8.5 even 2
4864.2.a.bq.1.7 8 8.3 odd 2
5472.2.g.b.2737.2 16 48.29 odd 4
5472.2.g.b.2737.15 16 48.5 odd 4