Properties

Label 4864.2.a.bt.1.1
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.719482\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.46871 q^{3} -3.22672 q^{5} -4.04213 q^{7} +3.09455 q^{9} +O(q^{10})\) \(q-2.46871 q^{3} -3.22672 q^{5} -4.04213 q^{7} +3.09455 q^{9} +6.41171 q^{11} -0.815411 q^{13} +7.96585 q^{15} -2.62028 q^{17} +1.00000 q^{19} +9.97886 q^{21} -6.16717 q^{23} +5.41171 q^{25} -0.233424 q^{27} -3.85818 q^{29} -7.43670 q^{31} -15.8287 q^{33} +13.0428 q^{35} -5.13742 q^{37} +2.01302 q^{39} -11.5007 q^{41} -0.0837292 q^{43} -9.98525 q^{45} -3.99564 q^{47} +9.33880 q^{49} +6.46871 q^{51} -4.22608 q^{53} -20.6888 q^{55} -2.46871 q^{57} +8.53655 q^{59} -5.72239 q^{61} -12.5086 q^{63} +2.63110 q^{65} -11.4278 q^{67} +15.2250 q^{69} +1.10168 q^{71} -7.25458 q^{73} -13.3600 q^{75} -25.9169 q^{77} +13.9906 q^{79} -8.70740 q^{81} -6.51708 q^{83} +8.45489 q^{85} +9.52474 q^{87} -15.3867 q^{89} +3.29599 q^{91} +18.3591 q^{93} -3.22672 q^{95} -13.2654 q^{97} +19.8414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 4q^{3} + 14q^{9} + O(q^{10}) \) \( 10q + 4q^{3} + 14q^{9} + 20q^{11} + 4q^{17} + 10q^{19} + 10q^{25} + 28q^{27} - 8q^{33} + 36q^{35} - 12q^{41} - 4q^{43} + 26q^{49} + 36q^{51} + 4q^{57} + 52q^{59} - 24q^{65} + 12q^{67} + 12q^{73} - 12q^{75} + 34q^{81} + 16q^{83} - 20q^{89} + 60q^{91} - 28q^{97} + 60q^{99} + O(q^{100}) \)

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
\(3\) −2.46871 −1.42531 −0.712657 0.701513i \(-0.752508\pi\)
−0.712657 + 0.701513i \(0.752508\pi\)
\(4\) 0 0
\(5\) −3.22672 −1.44303 −0.721516 0.692398i \(-0.756554\pi\)
−0.721516 + 0.692398i \(0.756554\pi\)
\(6\) 0 0
\(7\) −4.04213 −1.52778 −0.763890 0.645346i \(-0.776713\pi\)
−0.763890 + 0.645346i \(0.776713\pi\)
\(8\) 0 0
\(9\) 3.09455 1.03152
\(10\) 0 0
\(11\) 6.41171 1.93320 0.966601 0.256286i \(-0.0824988\pi\)
0.966601 + 0.256286i \(0.0824988\pi\)
\(12\) 0 0
\(13\) −0.815411 −0.226154 −0.113077 0.993586i \(-0.536071\pi\)
−0.113077 + 0.993586i \(0.536071\pi\)
\(14\) 0 0
\(15\) 7.96585 2.05677
\(16\) 0 0
\(17\) −2.62028 −0.635510 −0.317755 0.948173i \(-0.602929\pi\)
−0.317755 + 0.948173i \(0.602929\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 9.97886 2.17757
\(22\) 0 0
\(23\) −6.16717 −1.28594 −0.642972 0.765890i \(-0.722299\pi\)
−0.642972 + 0.765890i \(0.722299\pi\)
\(24\) 0 0
\(25\) 5.41171 1.08234
\(26\) 0 0
\(27\) −0.233424 −0.0449225
\(28\) 0 0
\(29\) −3.85818 −0.716446 −0.358223 0.933636i \(-0.616617\pi\)
−0.358223 + 0.933636i \(0.616617\pi\)
\(30\) 0 0
\(31\) −7.43670 −1.33567 −0.667835 0.744309i \(-0.732779\pi\)
−0.667835 + 0.744309i \(0.732779\pi\)
\(32\) 0 0
\(33\) −15.8287 −2.75542
\(34\) 0 0
\(35\) 13.0428 2.20464
\(36\) 0 0
\(37\) −5.13742 −0.844586 −0.422293 0.906459i \(-0.638775\pi\)
−0.422293 + 0.906459i \(0.638775\pi\)
\(38\) 0 0
\(39\) 2.01302 0.322341
\(40\) 0 0
\(41\) −11.5007 −1.79611 −0.898054 0.439886i \(-0.855019\pi\)
−0.898054 + 0.439886i \(0.855019\pi\)
\(42\) 0 0
\(43\) −0.0837292 −0.0127686 −0.00638429 0.999980i \(-0.502032\pi\)
−0.00638429 + 0.999980i \(0.502032\pi\)
\(44\) 0 0
\(45\) −9.98525 −1.48851
\(46\) 0 0
\(47\) −3.99564 −0.582824 −0.291412 0.956598i \(-0.594125\pi\)
−0.291412 + 0.956598i \(0.594125\pi\)
\(48\) 0 0
\(49\) 9.33880 1.33411
\(50\) 0 0
\(51\) 6.46871 0.905801
\(52\) 0 0
\(53\) −4.22608 −0.580497 −0.290248 0.956951i \(-0.593738\pi\)
−0.290248 + 0.956951i \(0.593738\pi\)
\(54\) 0 0
\(55\) −20.6888 −2.78967
\(56\) 0 0
\(57\) −2.46871 −0.326989
\(58\) 0 0
\(59\) 8.53655 1.11136 0.555682 0.831395i \(-0.312457\pi\)
0.555682 + 0.831395i \(0.312457\pi\)
\(60\) 0 0
\(61\) −5.72239 −0.732677 −0.366339 0.930482i \(-0.619389\pi\)
−0.366339 + 0.930482i \(0.619389\pi\)
\(62\) 0 0
\(63\) −12.5086 −1.57593
\(64\) 0 0
\(65\) 2.63110 0.326348
\(66\) 0 0
\(67\) −11.4278 −1.39613 −0.698063 0.716036i \(-0.745955\pi\)
−0.698063 + 0.716036i \(0.745955\pi\)
\(68\) 0 0
\(69\) 15.2250 1.83287
\(70\) 0 0
\(71\) 1.10168 0.130745 0.0653725 0.997861i \(-0.479176\pi\)
0.0653725 + 0.997861i \(0.479176\pi\)
\(72\) 0 0
\(73\) −7.25458 −0.849085 −0.424542 0.905408i \(-0.639565\pi\)
−0.424542 + 0.905408i \(0.639565\pi\)
\(74\) 0 0
\(75\) −13.3600 −1.54268
\(76\) 0 0
\(77\) −25.9169 −2.95351
\(78\) 0 0
\(79\) 13.9906 1.57407 0.787033 0.616911i \(-0.211616\pi\)
0.787033 + 0.616911i \(0.211616\pi\)
\(80\) 0 0
\(81\) −8.70740 −0.967489
\(82\) 0 0
\(83\) −6.51708 −0.715343 −0.357671 0.933848i \(-0.616429\pi\)
−0.357671 + 0.933848i \(0.616429\pi\)
\(84\) 0 0
\(85\) 8.45489 0.917062
\(86\) 0 0
\(87\) 9.52474 1.02116
\(88\) 0 0
\(89\) −15.3867 −1.63098 −0.815492 0.578768i \(-0.803534\pi\)
−0.815492 + 0.578768i \(0.803534\pi\)
\(90\) 0 0
\(91\) 3.29599 0.345514
\(92\) 0 0
\(93\) 18.3591 1.90375
\(94\) 0 0
\(95\) −3.22672 −0.331054
\(96\) 0 0
\(97\) −13.2654 −1.34690 −0.673449 0.739234i \(-0.735188\pi\)
−0.673449 + 0.739234i \(0.735188\pi\)
\(98\) 0 0
\(99\) 19.8414 1.99413
\(100\) 0 0
\(101\) −2.76311 −0.274940 −0.137470 0.990506i \(-0.543897\pi\)
−0.137470 + 0.990506i \(0.543897\pi\)
\(102\) 0 0
\(103\) −0.529145 −0.0521382 −0.0260691 0.999660i \(-0.508299\pi\)
−0.0260691 + 0.999660i \(0.508299\pi\)
\(104\) 0 0
\(105\) −32.1990 −3.14230
\(106\) 0 0
\(107\) 0.211777 0.0204732 0.0102366 0.999948i \(-0.496742\pi\)
0.0102366 + 0.999948i \(0.496742\pi\)
\(108\) 0 0
\(109\) −9.73908 −0.932835 −0.466417 0.884565i \(-0.654456\pi\)
−0.466417 + 0.884565i \(0.654456\pi\)
\(110\) 0 0
\(111\) 12.6828 1.20380
\(112\) 0 0
\(113\) −6.09269 −0.573152 −0.286576 0.958058i \(-0.592517\pi\)
−0.286576 + 0.958058i \(0.592517\pi\)
\(114\) 0 0
\(115\) 19.8997 1.85566
\(116\) 0 0
\(117\) −2.52333 −0.233282
\(118\) 0 0
\(119\) 10.5915 0.970921
\(120\) 0 0
\(121\) 30.1100 2.73727
\(122\) 0 0
\(123\) 28.3919 2.56002
\(124\) 0 0
\(125\) −1.32846 −0.118821
\(126\) 0 0
\(127\) 6.64012 0.589215 0.294608 0.955618i \(-0.404811\pi\)
0.294608 + 0.955618i \(0.404811\pi\)
\(128\) 0 0
\(129\) 0.206704 0.0181992
\(130\) 0 0
\(131\) 3.61315 0.315682 0.157841 0.987465i \(-0.449547\pi\)
0.157841 + 0.987465i \(0.449547\pi\)
\(132\) 0 0
\(133\) −4.04213 −0.350497
\(134\) 0 0
\(135\) 0.753194 0.0648246
\(136\) 0 0
\(137\) −11.7468 −1.00360 −0.501799 0.864984i \(-0.667328\pi\)
−0.501799 + 0.864984i \(0.667328\pi\)
\(138\) 0 0
\(139\) −20.4757 −1.73672 −0.868362 0.495931i \(-0.834827\pi\)
−0.868362 + 0.495931i \(0.834827\pi\)
\(140\) 0 0
\(141\) 9.86410 0.830707
\(142\) 0 0
\(143\) −5.22817 −0.437202
\(144\) 0 0
\(145\) 12.4493 1.03385
\(146\) 0 0
\(147\) −23.0548 −1.90153
\(148\) 0 0
\(149\) 20.4213 1.67298 0.836490 0.547982i \(-0.184604\pi\)
0.836490 + 0.547982i \(0.184604\pi\)
\(150\) 0 0
\(151\) 13.1686 1.07164 0.535822 0.844331i \(-0.320002\pi\)
0.535822 + 0.844331i \(0.320002\pi\)
\(152\) 0 0
\(153\) −8.10858 −0.655540
\(154\) 0 0
\(155\) 23.9961 1.92742
\(156\) 0 0
\(157\) 15.0668 1.20246 0.601232 0.799074i \(-0.294677\pi\)
0.601232 + 0.799074i \(0.294677\pi\)
\(158\) 0 0
\(159\) 10.4330 0.827389
\(160\) 0 0
\(161\) 24.9285 1.96464
\(162\) 0 0
\(163\) 11.8860 0.930982 0.465491 0.885053i \(-0.345878\pi\)
0.465491 + 0.885053i \(0.345878\pi\)
\(164\) 0 0
\(165\) 51.0747 3.97616
\(166\) 0 0
\(167\) 19.2419 1.48898 0.744491 0.667632i \(-0.232692\pi\)
0.744491 + 0.667632i \(0.232692\pi\)
\(168\) 0 0
\(169\) −12.3351 −0.948854
\(170\) 0 0
\(171\) 3.09455 0.236646
\(172\) 0 0
\(173\) −1.53489 −0.116696 −0.0583478 0.998296i \(-0.518583\pi\)
−0.0583478 + 0.998296i \(0.518583\pi\)
\(174\) 0 0
\(175\) −21.8748 −1.65358
\(176\) 0 0
\(177\) −21.0743 −1.58404
\(178\) 0 0
\(179\) −3.82867 −0.286169 −0.143084 0.989711i \(-0.545702\pi\)
−0.143084 + 0.989711i \(0.545702\pi\)
\(180\) 0 0
\(181\) −5.89663 −0.438293 −0.219147 0.975692i \(-0.570327\pi\)
−0.219147 + 0.975692i \(0.570327\pi\)
\(182\) 0 0
\(183\) 14.1270 1.04429
\(184\) 0 0
\(185\) 16.5770 1.21877
\(186\) 0 0
\(187\) −16.8004 −1.22857
\(188\) 0 0
\(189\) 0.943531 0.0686318
\(190\) 0 0
\(191\) −19.8427 −1.43577 −0.717885 0.696162i \(-0.754890\pi\)
−0.717885 + 0.696162i \(0.754890\pi\)
\(192\) 0 0
\(193\) −3.62263 −0.260763 −0.130381 0.991464i \(-0.541620\pi\)
−0.130381 + 0.991464i \(0.541620\pi\)
\(194\) 0 0
\(195\) −6.49544 −0.465148
\(196\) 0 0
\(197\) 13.1190 0.934690 0.467345 0.884075i \(-0.345211\pi\)
0.467345 + 0.884075i \(0.345211\pi\)
\(198\) 0 0
\(199\) −15.9156 −1.12823 −0.564113 0.825697i \(-0.690782\pi\)
−0.564113 + 0.825697i \(0.690782\pi\)
\(200\) 0 0
\(201\) 28.2120 1.98992
\(202\) 0 0
\(203\) 15.5953 1.09457
\(204\) 0 0
\(205\) 37.1095 2.59184
\(206\) 0 0
\(207\) −19.0846 −1.32647
\(208\) 0 0
\(209\) 6.41171 0.443507
\(210\) 0 0
\(211\) 5.00713 0.344705 0.172352 0.985035i \(-0.444863\pi\)
0.172352 + 0.985035i \(0.444863\pi\)
\(212\) 0 0
\(213\) −2.71973 −0.186352
\(214\) 0 0
\(215\) 0.270170 0.0184255
\(216\) 0 0
\(217\) 30.0601 2.04061
\(218\) 0 0
\(219\) 17.9095 1.21021
\(220\) 0 0
\(221\) 2.13660 0.143723
\(222\) 0 0
\(223\) −8.30156 −0.555913 −0.277957 0.960594i \(-0.589657\pi\)
−0.277957 + 0.960594i \(0.589657\pi\)
\(224\) 0 0
\(225\) 16.7468 1.11645
\(226\) 0 0
\(227\) −26.8177 −1.77995 −0.889976 0.456007i \(-0.849279\pi\)
−0.889976 + 0.456007i \(0.849279\pi\)
\(228\) 0 0
\(229\) −6.15107 −0.406474 −0.203237 0.979130i \(-0.565146\pi\)
−0.203237 + 0.979130i \(0.565146\pi\)
\(230\) 0 0
\(231\) 63.9815 4.20968
\(232\) 0 0
\(233\) 7.79905 0.510933 0.255466 0.966818i \(-0.417771\pi\)
0.255466 + 0.966818i \(0.417771\pi\)
\(234\) 0 0
\(235\) 12.8928 0.841034
\(236\) 0 0
\(237\) −34.5388 −2.24354
\(238\) 0 0
\(239\) 27.7964 1.79800 0.898999 0.437951i \(-0.144296\pi\)
0.898999 + 0.437951i \(0.144296\pi\)
\(240\) 0 0
\(241\) −5.53347 −0.356442 −0.178221 0.983990i \(-0.557034\pi\)
−0.178221 + 0.983990i \(0.557034\pi\)
\(242\) 0 0
\(243\) 22.1964 1.42390
\(244\) 0 0
\(245\) −30.1337 −1.92517
\(246\) 0 0
\(247\) −0.815411 −0.0518833
\(248\) 0 0
\(249\) 16.0888 1.01959
\(250\) 0 0
\(251\) −14.9928 −0.946334 −0.473167 0.880973i \(-0.656889\pi\)
−0.473167 + 0.880973i \(0.656889\pi\)
\(252\) 0 0
\(253\) −39.5421 −2.48599
\(254\) 0 0
\(255\) −20.8727 −1.30710
\(256\) 0 0
\(257\) 13.6003 0.848365 0.424182 0.905577i \(-0.360562\pi\)
0.424182 + 0.905577i \(0.360562\pi\)
\(258\) 0 0
\(259\) 20.7661 1.29034
\(260\) 0 0
\(261\) −11.9393 −0.739026
\(262\) 0 0
\(263\) −26.4415 −1.63046 −0.815228 0.579140i \(-0.803388\pi\)
−0.815228 + 0.579140i \(0.803388\pi\)
\(264\) 0 0
\(265\) 13.6364 0.837675
\(266\) 0 0
\(267\) 37.9853 2.32466
\(268\) 0 0
\(269\) −19.6429 −1.19765 −0.598825 0.800880i \(-0.704365\pi\)
−0.598825 + 0.800880i \(0.704365\pi\)
\(270\) 0 0
\(271\) 14.0146 0.851327 0.425663 0.904882i \(-0.360041\pi\)
0.425663 + 0.904882i \(0.360041\pi\)
\(272\) 0 0
\(273\) −8.13687 −0.492466
\(274\) 0 0
\(275\) 34.6983 2.09238
\(276\) 0 0
\(277\) −4.92581 −0.295963 −0.147982 0.988990i \(-0.547278\pi\)
−0.147982 + 0.988990i \(0.547278\pi\)
\(278\) 0 0
\(279\) −23.0133 −1.37777
\(280\) 0 0
\(281\) 4.90466 0.292587 0.146294 0.989241i \(-0.453266\pi\)
0.146294 + 0.989241i \(0.453266\pi\)
\(282\) 0 0
\(283\) −2.07261 −0.123204 −0.0616018 0.998101i \(-0.519621\pi\)
−0.0616018 + 0.998101i \(0.519621\pi\)
\(284\) 0 0
\(285\) 7.96585 0.471856
\(286\) 0 0
\(287\) 46.4873 2.74406
\(288\) 0 0
\(289\) −10.1342 −0.596127
\(290\) 0 0
\(291\) 32.7485 1.91975
\(292\) 0 0
\(293\) 10.4606 0.611117 0.305559 0.952173i \(-0.401157\pi\)
0.305559 + 0.952173i \(0.401157\pi\)
\(294\) 0 0
\(295\) −27.5450 −1.60373
\(296\) 0 0
\(297\) −1.49665 −0.0868443
\(298\) 0 0
\(299\) 5.02878 0.290822
\(300\) 0 0
\(301\) 0.338444 0.0195076
\(302\) 0 0
\(303\) 6.82134 0.391875
\(304\) 0 0
\(305\) 18.4645 1.05728
\(306\) 0 0
\(307\) 5.87873 0.335517 0.167758 0.985828i \(-0.446347\pi\)
0.167758 + 0.985828i \(0.446347\pi\)
\(308\) 0 0
\(309\) 1.30631 0.0743132
\(310\) 0 0
\(311\) 12.6861 0.719365 0.359683 0.933075i \(-0.382885\pi\)
0.359683 + 0.933075i \(0.382885\pi\)
\(312\) 0 0
\(313\) 6.25089 0.353321 0.176661 0.984272i \(-0.443471\pi\)
0.176661 + 0.984272i \(0.443471\pi\)
\(314\) 0 0
\(315\) 40.3617 2.27412
\(316\) 0 0
\(317\) −4.72978 −0.265651 −0.132826 0.991139i \(-0.542405\pi\)
−0.132826 + 0.991139i \(0.542405\pi\)
\(318\) 0 0
\(319\) −24.7375 −1.38503
\(320\) 0 0
\(321\) −0.522816 −0.0291808
\(322\) 0 0
\(323\) −2.62028 −0.145796
\(324\) 0 0
\(325\) −4.41276 −0.244776
\(326\) 0 0
\(327\) 24.0430 1.32958
\(328\) 0 0
\(329\) 16.1509 0.890428
\(330\) 0 0
\(331\) 19.8060 1.08864 0.544318 0.838879i \(-0.316788\pi\)
0.544318 + 0.838879i \(0.316788\pi\)
\(332\) 0 0
\(333\) −15.8980 −0.871206
\(334\) 0 0
\(335\) 36.8743 2.01466
\(336\) 0 0
\(337\) 9.40107 0.512109 0.256054 0.966662i \(-0.417577\pi\)
0.256054 + 0.966662i \(0.417577\pi\)
\(338\) 0 0
\(339\) 15.0411 0.816921
\(340\) 0 0
\(341\) −47.6819 −2.58212
\(342\) 0 0
\(343\) −9.45373 −0.510454
\(344\) 0 0
\(345\) −49.1267 −2.64489
\(346\) 0 0
\(347\) −25.2991 −1.35813 −0.679063 0.734080i \(-0.737614\pi\)
−0.679063 + 0.734080i \(0.737614\pi\)
\(348\) 0 0
\(349\) 28.9343 1.54882 0.774408 0.632687i \(-0.218048\pi\)
0.774408 + 0.632687i \(0.218048\pi\)
\(350\) 0 0
\(351\) 0.190337 0.0101594
\(352\) 0 0
\(353\) −22.4897 −1.19701 −0.598503 0.801121i \(-0.704238\pi\)
−0.598503 + 0.801121i \(0.704238\pi\)
\(354\) 0 0
\(355\) −3.55480 −0.188669
\(356\) 0 0
\(357\) −26.1474 −1.38387
\(358\) 0 0
\(359\) −25.7745 −1.36033 −0.680164 0.733060i \(-0.738091\pi\)
−0.680164 + 0.733060i \(0.738091\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −74.3330 −3.90147
\(364\) 0 0
\(365\) 23.4085 1.22526
\(366\) 0 0
\(367\) 3.91733 0.204483 0.102242 0.994760i \(-0.467399\pi\)
0.102242 + 0.994760i \(0.467399\pi\)
\(368\) 0 0
\(369\) −35.5895 −1.85272
\(370\) 0 0
\(371\) 17.0824 0.886871
\(372\) 0 0
\(373\) −7.17543 −0.371530 −0.185765 0.982594i \(-0.559476\pi\)
−0.185765 + 0.982594i \(0.559476\pi\)
\(374\) 0 0
\(375\) 3.27959 0.169357
\(376\) 0 0
\(377\) 3.14600 0.162027
\(378\) 0 0
\(379\) −9.11028 −0.467964 −0.233982 0.972241i \(-0.575176\pi\)
−0.233982 + 0.972241i \(0.575176\pi\)
\(380\) 0 0
\(381\) −16.3926 −0.839816
\(382\) 0 0
\(383\) 28.4774 1.45513 0.727564 0.686040i \(-0.240652\pi\)
0.727564 + 0.686040i \(0.240652\pi\)
\(384\) 0 0
\(385\) 83.6266 4.26201
\(386\) 0 0
\(387\) −0.259104 −0.0131710
\(388\) 0 0
\(389\) −27.5101 −1.39482 −0.697409 0.716674i \(-0.745664\pi\)
−0.697409 + 0.716674i \(0.745664\pi\)
\(390\) 0 0
\(391\) 16.1597 0.817231
\(392\) 0 0
\(393\) −8.91983 −0.449946
\(394\) 0 0
\(395\) −45.1437 −2.27143
\(396\) 0 0
\(397\) 32.7251 1.64242 0.821212 0.570624i \(-0.193298\pi\)
0.821212 + 0.570624i \(0.193298\pi\)
\(398\) 0 0
\(399\) 9.97886 0.499568
\(400\) 0 0
\(401\) 13.5472 0.676515 0.338257 0.941054i \(-0.390163\pi\)
0.338257 + 0.941054i \(0.390163\pi\)
\(402\) 0 0
\(403\) 6.06397 0.302068
\(404\) 0 0
\(405\) 28.0963 1.39612
\(406\) 0 0
\(407\) −32.9396 −1.63276
\(408\) 0 0
\(409\) 27.0909 1.33956 0.669779 0.742561i \(-0.266389\pi\)
0.669779 + 0.742561i \(0.266389\pi\)
\(410\) 0 0
\(411\) 28.9995 1.43044
\(412\) 0 0
\(413\) −34.5058 −1.69792
\(414\) 0 0
\(415\) 21.0288 1.03226
\(416\) 0 0
\(417\) 50.5486 2.47538
\(418\) 0 0
\(419\) 12.0175 0.587092 0.293546 0.955945i \(-0.405165\pi\)
0.293546 + 0.955945i \(0.405165\pi\)
\(420\) 0 0
\(421\) −21.6253 −1.05395 −0.526977 0.849880i \(-0.676675\pi\)
−0.526977 + 0.849880i \(0.676675\pi\)
\(422\) 0 0
\(423\) −12.3647 −0.601194
\(424\) 0 0
\(425\) −14.1802 −0.687839
\(426\) 0 0
\(427\) 23.1306 1.11937
\(428\) 0 0
\(429\) 12.9069 0.623150
\(430\) 0 0
\(431\) 0.466340 0.0224628 0.0112314 0.999937i \(-0.496425\pi\)
0.0112314 + 0.999937i \(0.496425\pi\)
\(432\) 0 0
\(433\) −12.1789 −0.585281 −0.292640 0.956223i \(-0.594534\pi\)
−0.292640 + 0.956223i \(0.594534\pi\)
\(434\) 0 0
\(435\) −30.7337 −1.47357
\(436\) 0 0
\(437\) −6.16717 −0.295016
\(438\) 0 0
\(439\) −22.9016 −1.09303 −0.546517 0.837448i \(-0.684047\pi\)
−0.546517 + 0.837448i \(0.684047\pi\)
\(440\) 0 0
\(441\) 28.8994 1.37616
\(442\) 0 0
\(443\) 4.15864 0.197583 0.0987914 0.995108i \(-0.468502\pi\)
0.0987914 + 0.995108i \(0.468502\pi\)
\(444\) 0 0
\(445\) 49.6485 2.35356
\(446\) 0 0
\(447\) −50.4144 −2.38452
\(448\) 0 0
\(449\) −13.4154 −0.633112 −0.316556 0.948574i \(-0.602527\pi\)
−0.316556 + 0.948574i \(0.602527\pi\)
\(450\) 0 0
\(451\) −73.7391 −3.47224
\(452\) 0 0
\(453\) −32.5095 −1.52743
\(454\) 0 0
\(455\) −10.6352 −0.498588
\(456\) 0 0
\(457\) −23.9359 −1.11968 −0.559838 0.828602i \(-0.689136\pi\)
−0.559838 + 0.828602i \(0.689136\pi\)
\(458\) 0 0
\(459\) 0.611636 0.0285487
\(460\) 0 0
\(461\) 31.0335 1.44537 0.722686 0.691176i \(-0.242907\pi\)
0.722686 + 0.691176i \(0.242907\pi\)
\(462\) 0 0
\(463\) −31.3472 −1.45683 −0.728415 0.685136i \(-0.759743\pi\)
−0.728415 + 0.685136i \(0.759743\pi\)
\(464\) 0 0
\(465\) −59.2396 −2.74717
\(466\) 0 0
\(467\) 7.14257 0.330519 0.165259 0.986250i \(-0.447154\pi\)
0.165259 + 0.986250i \(0.447154\pi\)
\(468\) 0 0
\(469\) 46.1926 2.13298
\(470\) 0 0
\(471\) −37.1957 −1.71389
\(472\) 0 0
\(473\) −0.536847 −0.0246843
\(474\) 0 0
\(475\) 5.41171 0.248306
\(476\) 0 0
\(477\) −13.0778 −0.598792
\(478\) 0 0
\(479\) −26.5447 −1.21286 −0.606430 0.795137i \(-0.707399\pi\)
−0.606430 + 0.795137i \(0.707399\pi\)
\(480\) 0 0
\(481\) 4.18911 0.191007
\(482\) 0 0
\(483\) −61.5413 −2.80023
\(484\) 0 0
\(485\) 42.8037 1.94362
\(486\) 0 0
\(487\) 9.74569 0.441619 0.220810 0.975317i \(-0.429130\pi\)
0.220810 + 0.975317i \(0.429130\pi\)
\(488\) 0 0
\(489\) −29.3431 −1.32694
\(490\) 0 0
\(491\) 25.6150 1.15599 0.577995 0.816040i \(-0.303835\pi\)
0.577995 + 0.816040i \(0.303835\pi\)
\(492\) 0 0
\(493\) 10.1095 0.455309
\(494\) 0 0
\(495\) −64.0225 −2.87760
\(496\) 0 0
\(497\) −4.45312 −0.199750
\(498\) 0 0
\(499\) −5.48801 −0.245677 −0.122838 0.992427i \(-0.539200\pi\)
−0.122838 + 0.992427i \(0.539200\pi\)
\(500\) 0 0
\(501\) −47.5028 −2.12227
\(502\) 0 0
\(503\) −9.00013 −0.401296 −0.200648 0.979663i \(-0.564305\pi\)
−0.200648 + 0.979663i \(0.564305\pi\)
\(504\) 0 0
\(505\) 8.91578 0.396747
\(506\) 0 0
\(507\) 30.4519 1.35241
\(508\) 0 0
\(509\) −28.2818 −1.25357 −0.626784 0.779193i \(-0.715629\pi\)
−0.626784 + 0.779193i \(0.715629\pi\)
\(510\) 0 0
\(511\) 29.3240 1.29722
\(512\) 0 0
\(513\) −0.233424 −0.0103059
\(514\) 0 0
\(515\) 1.70740 0.0752371
\(516\) 0 0
\(517\) −25.6189 −1.12672
\(518\) 0 0
\(519\) 3.78921 0.166328
\(520\) 0 0
\(521\) −4.19618 −0.183838 −0.0919190 0.995766i \(-0.529300\pi\)
−0.0919190 + 0.995766i \(0.529300\pi\)
\(522\) 0 0
\(523\) −33.6631 −1.47198 −0.735992 0.676990i \(-0.763284\pi\)
−0.735992 + 0.676990i \(0.763284\pi\)
\(524\) 0 0
\(525\) 54.0027 2.35687
\(526\) 0 0
\(527\) 19.4862 0.848833
\(528\) 0 0
\(529\) 15.0340 0.653651
\(530\) 0 0
\(531\) 26.4168 1.14639
\(532\) 0 0
\(533\) 9.37779 0.406197
\(534\) 0 0
\(535\) −0.683344 −0.0295435
\(536\) 0 0
\(537\) 9.45191 0.407880
\(538\) 0 0
\(539\) 59.8776 2.57911
\(540\) 0 0
\(541\) 18.3995 0.791056 0.395528 0.918454i \(-0.370562\pi\)
0.395528 + 0.918454i \(0.370562\pi\)
\(542\) 0 0
\(543\) 14.5571 0.624705
\(544\) 0 0
\(545\) 31.4253 1.34611
\(546\) 0 0
\(547\) −5.83156 −0.249340 −0.124670 0.992198i \(-0.539787\pi\)
−0.124670 + 0.992198i \(0.539787\pi\)
\(548\) 0 0
\(549\) −17.7082 −0.755770
\(550\) 0 0
\(551\) −3.85818 −0.164364
\(552\) 0 0
\(553\) −56.5518 −2.40483
\(554\) 0 0
\(555\) −40.9239 −1.73712
\(556\) 0 0
\(557\) 30.9517 1.31146 0.655732 0.754993i \(-0.272360\pi\)
0.655732 + 0.754993i \(0.272360\pi\)
\(558\) 0 0
\(559\) 0.0682737 0.00288767
\(560\) 0 0
\(561\) 41.4755 1.75110
\(562\) 0 0
\(563\) 43.3007 1.82490 0.912452 0.409183i \(-0.134186\pi\)
0.912452 + 0.409183i \(0.134186\pi\)
\(564\) 0 0
\(565\) 19.6594 0.827076
\(566\) 0 0
\(567\) 35.1964 1.47811
\(568\) 0 0
\(569\) −8.32798 −0.349127 −0.174563 0.984646i \(-0.555851\pi\)
−0.174563 + 0.984646i \(0.555851\pi\)
\(570\) 0 0
\(571\) 21.5676 0.902574 0.451287 0.892379i \(-0.350965\pi\)
0.451287 + 0.892379i \(0.350965\pi\)
\(572\) 0 0
\(573\) 48.9861 2.04642
\(574\) 0 0
\(575\) −33.3749 −1.39183
\(576\) 0 0
\(577\) −31.5716 −1.31434 −0.657172 0.753741i \(-0.728247\pi\)
−0.657172 + 0.753741i \(0.728247\pi\)
\(578\) 0 0
\(579\) 8.94324 0.371669
\(580\) 0 0
\(581\) 26.3429 1.09289
\(582\) 0 0
\(583\) −27.0964 −1.12222
\(584\) 0 0
\(585\) 8.14208 0.336634
\(586\) 0 0
\(587\) 28.9679 1.19563 0.597817 0.801633i \(-0.296035\pi\)
0.597817 + 0.801633i \(0.296035\pi\)
\(588\) 0 0
\(589\) −7.43670 −0.306424
\(590\) 0 0
\(591\) −32.3871 −1.33223
\(592\) 0 0
\(593\) 23.8431 0.979117 0.489559 0.871970i \(-0.337158\pi\)
0.489559 + 0.871970i \(0.337158\pi\)
\(594\) 0 0
\(595\) −34.1758 −1.40107
\(596\) 0 0
\(597\) 39.2911 1.60808
\(598\) 0 0
\(599\) 24.0716 0.983537 0.491769 0.870726i \(-0.336351\pi\)
0.491769 + 0.870726i \(0.336351\pi\)
\(600\) 0 0
\(601\) −1.72226 −0.0702523 −0.0351262 0.999383i \(-0.511183\pi\)
−0.0351262 + 0.999383i \(0.511183\pi\)
\(602\) 0 0
\(603\) −35.3639 −1.44013
\(604\) 0 0
\(605\) −97.1564 −3.94997
\(606\) 0 0
\(607\) 17.2198 0.698928 0.349464 0.936950i \(-0.386364\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(608\) 0 0
\(609\) −38.5002 −1.56011
\(610\) 0 0
\(611\) 3.25809 0.131808
\(612\) 0 0
\(613\) −32.5924 −1.31639 −0.658197 0.752845i \(-0.728681\pi\)
−0.658197 + 0.752845i \(0.728681\pi\)
\(614\) 0 0
\(615\) −91.6128 −3.69418
\(616\) 0 0
\(617\) 6.44448 0.259445 0.129722 0.991550i \(-0.458591\pi\)
0.129722 + 0.991550i \(0.458591\pi\)
\(618\) 0 0
\(619\) −31.8637 −1.28071 −0.640356 0.768078i \(-0.721213\pi\)
−0.640356 + 0.768078i \(0.721213\pi\)
\(620\) 0 0
\(621\) 1.43957 0.0577678
\(622\) 0 0
\(623\) 62.1949 2.49179
\(624\) 0 0
\(625\) −22.7720 −0.910879
\(626\) 0 0
\(627\) −15.8287 −0.632136
\(628\) 0 0
\(629\) 13.4615 0.536743
\(630\) 0 0
\(631\) −7.36348 −0.293135 −0.146568 0.989201i \(-0.546823\pi\)
−0.146568 + 0.989201i \(0.546823\pi\)
\(632\) 0 0
\(633\) −12.3612 −0.491312
\(634\) 0 0
\(635\) −21.4258 −0.850257
\(636\) 0 0
\(637\) −7.61496 −0.301716
\(638\) 0 0
\(639\) 3.40920 0.134866
\(640\) 0 0
\(641\) 39.4715 1.55903 0.779514 0.626384i \(-0.215466\pi\)
0.779514 + 0.626384i \(0.215466\pi\)
\(642\) 0 0
\(643\) 15.7411 0.620768 0.310384 0.950611i \(-0.399542\pi\)
0.310384 + 0.950611i \(0.399542\pi\)
\(644\) 0 0
\(645\) −0.666974 −0.0262621
\(646\) 0 0
\(647\) 17.8689 0.702499 0.351250 0.936282i \(-0.385757\pi\)
0.351250 + 0.936282i \(0.385757\pi\)
\(648\) 0 0
\(649\) 54.7338 2.14849
\(650\) 0 0
\(651\) −74.2098 −2.90851
\(652\) 0 0
\(653\) −1.83114 −0.0716581 −0.0358291 0.999358i \(-0.511407\pi\)
−0.0358291 + 0.999358i \(0.511407\pi\)
\(654\) 0 0
\(655\) −11.6586 −0.455540
\(656\) 0 0
\(657\) −22.4497 −0.875846
\(658\) 0 0
\(659\) 17.9446 0.699021 0.349510 0.936932i \(-0.386348\pi\)
0.349510 + 0.936932i \(0.386348\pi\)
\(660\) 0 0
\(661\) 0.256100 0.00996114 0.00498057 0.999988i \(-0.498415\pi\)
0.00498057 + 0.999988i \(0.498415\pi\)
\(662\) 0 0
\(663\) −5.27466 −0.204851
\(664\) 0 0
\(665\) 13.0428 0.505778
\(666\) 0 0
\(667\) 23.7940 0.921309
\(668\) 0 0
\(669\) 20.4942 0.792350
\(670\) 0 0
\(671\) −36.6903 −1.41641
\(672\) 0 0
\(673\) −3.46127 −0.133422 −0.0667111 0.997772i \(-0.521251\pi\)
−0.0667111 + 0.997772i \(0.521251\pi\)
\(674\) 0 0
\(675\) −1.26322 −0.0486215
\(676\) 0 0
\(677\) 13.7402 0.528080 0.264040 0.964512i \(-0.414945\pi\)
0.264040 + 0.964512i \(0.414945\pi\)
\(678\) 0 0
\(679\) 53.6205 2.05777
\(680\) 0 0
\(681\) 66.2052 2.53699
\(682\) 0 0
\(683\) −5.31127 −0.203230 −0.101615 0.994824i \(-0.532401\pi\)
−0.101615 + 0.994824i \(0.532401\pi\)
\(684\) 0 0
\(685\) 37.9036 1.44822
\(686\) 0 0
\(687\) 15.1852 0.579353
\(688\) 0 0
\(689\) 3.44599 0.131282
\(690\) 0 0
\(691\) 14.0583 0.534802 0.267401 0.963585i \(-0.413835\pi\)
0.267401 + 0.963585i \(0.413835\pi\)
\(692\) 0 0
\(693\) −80.2013 −3.04660
\(694\) 0 0
\(695\) 66.0692 2.50615
\(696\) 0 0
\(697\) 30.1350 1.14144
\(698\) 0 0
\(699\) −19.2536 −0.728239
\(700\) 0 0
\(701\) 47.7193 1.80233 0.901167 0.433473i \(-0.142712\pi\)
0.901167 + 0.433473i \(0.142712\pi\)
\(702\) 0 0
\(703\) −5.13742 −0.193761
\(704\) 0 0
\(705\) −31.8287 −1.19874
\(706\) 0 0
\(707\) 11.1689 0.420048
\(708\) 0 0
\(709\) −12.8747 −0.483519 −0.241759 0.970336i \(-0.577724\pi\)
−0.241759 + 0.970336i \(0.577724\pi\)
\(710\) 0 0
\(711\) 43.2947 1.62368
\(712\) 0 0
\(713\) 45.8634 1.71760
\(714\) 0 0
\(715\) 16.8698 0.630896
\(716\) 0 0
\(717\) −68.6213 −2.56271
\(718\) 0 0
\(719\) −6.30671 −0.235201 −0.117600 0.993061i \(-0.537520\pi\)
−0.117600 + 0.993061i \(0.537520\pi\)
\(720\) 0 0
\(721\) 2.13887 0.0796557
\(722\) 0 0
\(723\) 13.6606 0.508042
\(724\) 0 0
\(725\) −20.8793 −0.775439
\(726\) 0 0
\(727\) −24.8092 −0.920123 −0.460061 0.887887i \(-0.652173\pi\)
−0.460061 + 0.887887i \(0.652173\pi\)
\(728\) 0 0
\(729\) −28.6743 −1.06201
\(730\) 0 0
\(731\) 0.219394 0.00811457
\(732\) 0 0
\(733\) 20.8961 0.771816 0.385908 0.922537i \(-0.373888\pi\)
0.385908 + 0.922537i \(0.373888\pi\)
\(734\) 0 0
\(735\) 74.3914 2.74397
\(736\) 0 0
\(737\) −73.2716 −2.69900
\(738\) 0 0
\(739\) 9.57101 0.352075 0.176038 0.984383i \(-0.443672\pi\)
0.176038 + 0.984383i \(0.443672\pi\)
\(740\) 0 0
\(741\) 2.01302 0.0739500
\(742\) 0 0
\(743\) −10.3553 −0.379900 −0.189950 0.981794i \(-0.560833\pi\)
−0.189950 + 0.981794i \(0.560833\pi\)
\(744\) 0 0
\(745\) −65.8939 −2.41416
\(746\) 0 0
\(747\) −20.1675 −0.737889
\(748\) 0 0
\(749\) −0.856029 −0.0312786
\(750\) 0 0
\(751\) −30.4868 −1.11248 −0.556239 0.831022i \(-0.687756\pi\)
−0.556239 + 0.831022i \(0.687756\pi\)
\(752\) 0 0
\(753\) 37.0128 1.34882
\(754\) 0 0
\(755\) −42.4913 −1.54642
\(756\) 0 0
\(757\) −13.0841 −0.475549 −0.237774 0.971320i \(-0.576418\pi\)
−0.237774 + 0.971320i \(0.576418\pi\)
\(758\) 0 0
\(759\) 97.6181 3.54331
\(760\) 0 0
\(761\) 32.3638 1.17319 0.586594 0.809881i \(-0.300468\pi\)
0.586594 + 0.809881i \(0.300468\pi\)
\(762\) 0 0
\(763\) 39.3666 1.42517
\(764\) 0 0
\(765\) 26.1641 0.945965
\(766\) 0 0
\(767\) −6.96079 −0.251340
\(768\) 0 0
\(769\) 10.9234 0.393908 0.196954 0.980413i \(-0.436895\pi\)
0.196954 + 0.980413i \(0.436895\pi\)
\(770\) 0 0
\(771\) −33.5753 −1.20919
\(772\) 0 0
\(773\) −34.9008 −1.25530 −0.627648 0.778497i \(-0.715982\pi\)
−0.627648 + 0.778497i \(0.715982\pi\)
\(774\) 0 0
\(775\) −40.2452 −1.44565
\(776\) 0 0
\(777\) −51.2656 −1.83914
\(778\) 0 0
\(779\) −11.5007 −0.412055
\(780\) 0 0
\(781\) 7.06363 0.252756
\(782\) 0 0
\(783\) 0.900592 0.0321845
\(784\) 0 0
\(785\) −48.6164 −1.73519
\(786\) 0 0
\(787\) −5.72739 −0.204159 −0.102080 0.994776i \(-0.532550\pi\)
−0.102080 + 0.994776i \(0.532550\pi\)
\(788\) 0 0
\(789\) 65.2766 2.32391
\(790\) 0 0
\(791\) 24.6274 0.875650
\(792\) 0 0
\(793\) 4.66610 0.165698
\(794\) 0 0
\(795\) −33.6643 −1.19395
\(796\) 0 0
\(797\) 45.1622 1.59973 0.799863 0.600183i \(-0.204906\pi\)
0.799863 + 0.600183i \(0.204906\pi\)
\(798\) 0 0
\(799\) 10.4697 0.370391
\(800\) 0 0
\(801\) −47.6149 −1.68239
\(802\) 0 0
\(803\) −46.5143 −1.64145
\(804\) 0 0
\(805\) −80.4372 −2.83504
\(806\) 0 0
\(807\) 48.4928 1.70703
\(808\) 0 0
\(809\) 35.4870 1.24766 0.623828 0.781562i \(-0.285577\pi\)
0.623828 + 0.781562i \(0.285577\pi\)
\(810\) 0 0
\(811\) −20.2334 −0.710492 −0.355246 0.934773i \(-0.615603\pi\)
−0.355246 + 0.934773i \(0.615603\pi\)
\(812\) 0 0
\(813\) −34.5981 −1.21341
\(814\) 0 0
\(815\) −38.3527 −1.34344
\(816\) 0 0
\(817\) −0.0837292 −0.00292931
\(818\) 0 0
\(819\) 10.1996 0.356404
\(820\) 0 0
\(821\) 52.0305 1.81588 0.907938 0.419104i \(-0.137656\pi\)
0.907938 + 0.419104i \(0.137656\pi\)
\(822\) 0 0
\(823\) −5.60310 −0.195312 −0.0976559 0.995220i \(-0.531134\pi\)
−0.0976559 + 0.995220i \(0.531134\pi\)
\(824\) 0 0
\(825\) −85.6601 −2.98230
\(826\) 0 0
\(827\) 28.3875 0.987129 0.493565 0.869709i \(-0.335694\pi\)
0.493565 + 0.869709i \(0.335694\pi\)
\(828\) 0 0
\(829\) 32.5931 1.13201 0.566003 0.824403i \(-0.308489\pi\)
0.566003 + 0.824403i \(0.308489\pi\)
\(830\) 0 0
\(831\) 12.1604 0.421841
\(832\) 0 0
\(833\) −24.4702 −0.847844
\(834\) 0 0
\(835\) −62.0882 −2.14865
\(836\) 0 0
\(837\) 1.73591 0.0600017
\(838\) 0 0
\(839\) −45.0055 −1.55376 −0.776881 0.629647i \(-0.783199\pi\)
−0.776881 + 0.629647i \(0.783199\pi\)
\(840\) 0 0
\(841\) −14.1145 −0.486706
\(842\) 0 0
\(843\) −12.1082 −0.417029
\(844\) 0 0
\(845\) 39.8019 1.36923
\(846\) 0 0
\(847\) −121.708 −4.18195
\(848\) 0 0
\(849\) 5.11667 0.175604
\(850\) 0 0
\(851\) 31.6833 1.08609
\(852\) 0 0
\(853\) −21.6381 −0.740875 −0.370438 0.928857i \(-0.620792\pi\)
−0.370438 + 0.928857i \(0.620792\pi\)
\(854\) 0 0
\(855\) −9.98525 −0.341488
\(856\) 0 0
\(857\) −11.4432 −0.390892 −0.195446 0.980714i \(-0.562616\pi\)
−0.195446 + 0.980714i \(0.562616\pi\)
\(858\) 0 0
\(859\) 7.57422 0.258429 0.129215 0.991617i \(-0.458754\pi\)
0.129215 + 0.991617i \(0.458754\pi\)
\(860\) 0 0
\(861\) −114.764 −3.91114
\(862\) 0 0
\(863\) 34.6063 1.17801 0.589007 0.808128i \(-0.299519\pi\)
0.589007 + 0.808128i \(0.299519\pi\)
\(864\) 0 0
\(865\) 4.95266 0.168396
\(866\) 0 0
\(867\) 25.0183 0.849667
\(868\) 0 0
\(869\) 89.7036 3.04299
\(870\) 0 0
\(871\) 9.31834 0.315740
\(872\) 0 0
\(873\) −41.0505 −1.38935
\(874\) 0 0
\(875\) 5.36981 0.181533
\(876\) 0 0
\(877\) −29.4493 −0.994434 −0.497217 0.867626i \(-0.665645\pi\)
−0.497217 + 0.867626i \(0.665645\pi\)
\(878\) 0 0
\(879\) −25.8243 −0.871033
\(880\) 0 0
\(881\) 51.5583 1.73704 0.868522 0.495651i \(-0.165070\pi\)
0.868522 + 0.495651i \(0.165070\pi\)
\(882\) 0 0
\(883\) 25.2991 0.851382 0.425691 0.904869i \(-0.360031\pi\)
0.425691 + 0.904869i \(0.360031\pi\)
\(884\) 0 0
\(885\) 68.0008 2.28582
\(886\) 0 0
\(887\) 1.92340 0.0645815 0.0322908 0.999479i \(-0.489720\pi\)
0.0322908 + 0.999479i \(0.489720\pi\)
\(888\) 0 0
\(889\) −26.8402 −0.900192
\(890\) 0 0
\(891\) −55.8293 −1.87035
\(892\) 0 0
\(893\) −3.99564 −0.133709
\(894\) 0 0
\(895\) 12.3541 0.412950
\(896\) 0 0
\(897\) −12.4146 −0.414512
\(898\) 0 0
\(899\) 28.6921 0.956936
\(900\) 0 0
\(901\) 11.0735 0.368912
\(902\) 0 0
\(903\) −0.835522 −0.0278044
\(904\) 0 0
\(905\) 19.0268 0.632471
\(906\) 0 0
\(907\) −27.7093 −0.920071 −0.460036 0.887900i \(-0.652163\pi\)
−0.460036 + 0.887900i \(0.652163\pi\)
\(908\) 0 0
\(909\) −8.55060 −0.283605
\(910\) 0 0
\(911\) −51.4735 −1.70539 −0.852696 0.522407i \(-0.825034\pi\)
−0.852696 + 0.522407i \(0.825034\pi\)
\(912\) 0 0
\(913\) −41.7856 −1.38290
\(914\) 0 0
\(915\) −45.5837 −1.50695
\(916\) 0 0
\(917\) −14.6048 −0.482293
\(918\) 0 0
\(919\) 25.4158 0.838390 0.419195 0.907896i \(-0.362312\pi\)
0.419195 + 0.907896i \(0.362312\pi\)
\(920\) 0 0
\(921\) −14.5129 −0.478216
\(922\) 0 0
\(923\) −0.898319 −0.0295685
\(924\) 0 0
\(925\) −27.8022 −0.914131
\(926\) 0 0
\(927\) −1.63747 −0.0537815
\(928\) 0 0
\(929\) −41.2485 −1.35332 −0.676661 0.736295i \(-0.736574\pi\)
−0.676661 + 0.736295i \(0.736574\pi\)
\(930\) 0 0
\(931\) 9.33880 0.306067
\(932\) 0 0
\(933\) −31.3185 −1.02532
\(934\) 0 0
\(935\) 54.2103 1.77287
\(936\) 0 0
\(937\) −36.1849 −1.18211 −0.591054 0.806632i \(-0.701288\pi\)
−0.591054 + 0.806632i \(0.701288\pi\)
\(938\) 0 0
\(939\) −15.4317 −0.503593
\(940\) 0 0
\(941\) −17.2262 −0.561559 −0.280779 0.959772i \(-0.590593\pi\)
−0.280779 + 0.959772i \(0.590593\pi\)
\(942\) 0 0
\(943\) 70.9267 2.30969
\(944\) 0 0
\(945\) −3.04451 −0.0990378
\(946\) 0 0
\(947\) 17.7823 0.577847 0.288924 0.957352i \(-0.406703\pi\)
0.288924 + 0.957352i \(0.406703\pi\)
\(948\) 0 0
\(949\) 5.91546 0.192024
\(950\) 0 0
\(951\) 11.6765 0.378636
\(952\) 0 0
\(953\) −47.3478 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(954\) 0 0
\(955\) 64.0269 2.07186
\(956\) 0 0
\(957\) 61.0698 1.97411
\(958\) 0 0
\(959\) 47.4821 1.53328
\(960\) 0 0
\(961\) 24.3045 0.784017
\(962\) 0 0
\(963\) 0.655354 0.0211185
\(964\) 0 0
\(965\) 11.6892 0.376289
\(966\) 0 0
\(967\) 25.3504 0.815213 0.407606 0.913158i \(-0.366364\pi\)
0.407606 + 0.913158i \(0.366364\pi\)
\(968\) 0 0
\(969\) 6.46871 0.207805
\(970\) 0 0
\(971\) 45.3901 1.45664 0.728319 0.685238i \(-0.240302\pi\)
0.728319 + 0.685238i \(0.240302\pi\)
\(972\) 0 0
\(973\) 82.7653 2.65333
\(974\) 0 0
\(975\) 10.8939 0.348883
\(976\) 0 0
\(977\) 27.3398 0.874679 0.437339 0.899297i \(-0.355921\pi\)
0.437339 + 0.899297i \(0.355921\pi\)
\(978\) 0 0
\(979\) −98.6549 −3.15302
\(980\) 0 0
\(981\) −30.1381 −0.962236
\(982\) 0 0
\(983\) 4.84750 0.154611 0.0773056 0.997007i \(-0.475368\pi\)
0.0773056 + 0.997007i \(0.475368\pi\)
\(984\) 0 0
\(985\) −42.3313 −1.34879
\(986\) 0 0
\(987\) −39.8720 −1.26914
\(988\) 0 0
\(989\) 0.516372 0.0164197
\(990\) 0 0
\(991\) −28.1022 −0.892696 −0.446348 0.894859i \(-0.647276\pi\)
−0.446348 + 0.894859i \(0.647276\pi\)
\(992\) 0 0
\(993\) −48.8954 −1.55165
\(994\) 0 0
\(995\) 51.3551 1.62807
\(996\) 0 0
\(997\) −6.99262 −0.221458 −0.110729 0.993851i \(-0.535319\pi\)
−0.110729 + 0.993851i \(0.535319\pi\)
\(998\) 0 0
\(999\) 1.19920 0.0379409
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.bt.1.1 10
4.3 odd 2 4864.2.a.bs.1.9 10
8.3 odd 2 inner 4864.2.a.bt.1.2 10
8.5 even 2 4864.2.a.bs.1.10 10
16.3 odd 4 2432.2.c.j.1217.18 yes 20
16.5 even 4 2432.2.c.j.1217.17 yes 20
16.11 odd 4 2432.2.c.j.1217.3 20
16.13 even 4 2432.2.c.j.1217.4 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.3 20 16.11 odd 4
2432.2.c.j.1217.4 yes 20 16.13 even 4
2432.2.c.j.1217.17 yes 20 16.5 even 4
2432.2.c.j.1217.18 yes 20 16.3 odd 4
4864.2.a.bs.1.9 10 4.3 odd 2
4864.2.a.bs.1.10 10 8.5 even 2
4864.2.a.bt.1.1 10 1.1 even 1 trivial
4864.2.a.bt.1.2 10 8.3 odd 2 inner