Properties

Label 3800.2.a.bc.1.5
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 10 x^{4} + 16 x^{3} + 15 x^{2} - 14 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.93590\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.93590 q^{3} -1.24708 q^{7} +0.747704 q^{9} +O(q^{10})\) \(q+1.93590 q^{3} -1.24708 q^{7} +0.747704 q^{9} -0.513860 q^{11} -6.15670 q^{13} +4.51986 q^{17} -1.00000 q^{19} -2.41421 q^{21} +5.86084 q^{23} -4.36022 q^{27} +6.62700 q^{29} +6.41995 q^{31} -0.994780 q^{33} +1.40671 q^{37} -11.9187 q^{39} +10.6870 q^{41} +3.04878 q^{43} +1.99478 q^{47} -5.44480 q^{49} +8.74998 q^{51} +14.0848 q^{53} -1.93590 q^{57} -4.34261 q^{59} +10.7173 q^{61} -0.932444 q^{63} +9.89978 q^{67} +11.3460 q^{69} +7.42517 q^{71} -12.8079 q^{73} +0.640822 q^{77} +2.56138 q^{79} -10.6841 q^{81} +7.50864 q^{83} +12.8292 q^{87} -7.85353 q^{89} +7.67787 q^{91} +12.4284 q^{93} -6.74797 q^{97} -0.384215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + 3 q^{11} - q^{13} - 14 q^{17} - 6 q^{19} + 15 q^{21} + 12 q^{23} + 8 q^{27} + 9 q^{29} + 5 q^{31} + 2 q^{33} - 8 q^{37} + 12 q^{39} + 3 q^{41} + 15 q^{43} + 4 q^{47} + 22 q^{49} + 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} + 21 q^{63} - 3 q^{67} - 11 q^{69} + 19 q^{71} + 3 q^{73} - 36 q^{77} - 16 q^{79} + 26 q^{81} + 31 q^{83} - 25 q^{87} + 14 q^{89} + 42 q^{91} + 39 q^{93} - 11 q^{97} - 6 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.93590 1.11769 0.558846 0.829272i \(-0.311244\pi\)
0.558846 + 0.829272i \(0.311244\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.24708 −0.471350 −0.235675 0.971832i \(-0.575730\pi\)
−0.235675 + 0.971832i \(0.575730\pi\)
\(8\) 0 0
\(9\) 0.747704 0.249235
\(10\) 0 0
\(11\) −0.513860 −0.154935 −0.0774673 0.996995i \(-0.524683\pi\)
−0.0774673 + 0.996995i \(0.524683\pi\)
\(12\) 0 0
\(13\) −6.15670 −1.70756 −0.853780 0.520633i \(-0.825696\pi\)
−0.853780 + 0.520633i \(0.825696\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.51986 1.09623 0.548113 0.836404i \(-0.315346\pi\)
0.548113 + 0.836404i \(0.315346\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.41421 −0.526824
\(22\) 0 0
\(23\) 5.86084 1.22207 0.611035 0.791603i \(-0.290753\pi\)
0.611035 + 0.791603i \(0.290753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.36022 −0.839124
\(28\) 0 0
\(29\) 6.62700 1.23060 0.615302 0.788292i \(-0.289034\pi\)
0.615302 + 0.788292i \(0.289034\pi\)
\(30\) 0 0
\(31\) 6.41995 1.15306 0.576528 0.817077i \(-0.304407\pi\)
0.576528 + 0.817077i \(0.304407\pi\)
\(32\) 0 0
\(33\) −0.994780 −0.173169
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.40671 0.231263 0.115631 0.993292i \(-0.463111\pi\)
0.115631 + 0.993292i \(0.463111\pi\)
\(38\) 0 0
\(39\) −11.9187 −1.90853
\(40\) 0 0
\(41\) 10.6870 1.66903 0.834514 0.550987i \(-0.185749\pi\)
0.834514 + 0.550987i \(0.185749\pi\)
\(42\) 0 0
\(43\) 3.04878 0.464934 0.232467 0.972604i \(-0.425320\pi\)
0.232467 + 0.972604i \(0.425320\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.99478 0.290969 0.145484 0.989361i \(-0.453526\pi\)
0.145484 + 0.989361i \(0.453526\pi\)
\(48\) 0 0
\(49\) −5.44480 −0.777829
\(50\) 0 0
\(51\) 8.74998 1.22524
\(52\) 0 0
\(53\) 14.0848 1.93469 0.967346 0.253459i \(-0.0815685\pi\)
0.967346 + 0.253459i \(0.0815685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.93590 −0.256416
\(58\) 0 0
\(59\) −4.34261 −0.565360 −0.282680 0.959214i \(-0.591223\pi\)
−0.282680 + 0.959214i \(0.591223\pi\)
\(60\) 0 0
\(61\) 10.7173 1.37221 0.686106 0.727502i \(-0.259319\pi\)
0.686106 + 0.727502i \(0.259319\pi\)
\(62\) 0 0
\(63\) −0.932444 −0.117477
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.89978 1.20945 0.604725 0.796434i \(-0.293283\pi\)
0.604725 + 0.796434i \(0.293283\pi\)
\(68\) 0 0
\(69\) 11.3460 1.36590
\(70\) 0 0
\(71\) 7.42517 0.881205 0.440603 0.897702i \(-0.354765\pi\)
0.440603 + 0.897702i \(0.354765\pi\)
\(72\) 0 0
\(73\) −12.8079 −1.49905 −0.749525 0.661976i \(-0.769718\pi\)
−0.749525 + 0.661976i \(0.769718\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.640822 0.0730285
\(78\) 0 0
\(79\) 2.56138 0.288178 0.144089 0.989565i \(-0.453975\pi\)
0.144089 + 0.989565i \(0.453975\pi\)
\(80\) 0 0
\(81\) −10.6841 −1.18712
\(82\) 0 0
\(83\) 7.50864 0.824180 0.412090 0.911143i \(-0.364799\pi\)
0.412090 + 0.911143i \(0.364799\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.8292 1.37543
\(88\) 0 0
\(89\) −7.85353 −0.832473 −0.416236 0.909256i \(-0.636651\pi\)
−0.416236 + 0.909256i \(0.636651\pi\)
\(90\) 0 0
\(91\) 7.67787 0.804859
\(92\) 0 0
\(93\) 12.4284 1.28876
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.74797 −0.685152 −0.342576 0.939490i \(-0.611299\pi\)
−0.342576 + 0.939490i \(0.611299\pi\)
\(98\) 0 0
\(99\) −0.384215 −0.0386151
\(100\) 0 0
\(101\) 17.6503 1.75627 0.878134 0.478415i \(-0.158788\pi\)
0.878134 + 0.478415i \(0.158788\pi\)
\(102\) 0 0
\(103\) 5.84147 0.575577 0.287789 0.957694i \(-0.407080\pi\)
0.287789 + 0.957694i \(0.407080\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.14743 0.400947 0.200474 0.979699i \(-0.435752\pi\)
0.200474 + 0.979699i \(0.435752\pi\)
\(108\) 0 0
\(109\) 5.69767 0.545738 0.272869 0.962051i \(-0.412027\pi\)
0.272869 + 0.962051i \(0.412027\pi\)
\(110\) 0 0
\(111\) 2.72326 0.258480
\(112\) 0 0
\(113\) 5.36073 0.504295 0.252148 0.967689i \(-0.418863\pi\)
0.252148 + 0.967689i \(0.418863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −4.60339 −0.425584
\(118\) 0 0
\(119\) −5.63660 −0.516707
\(120\) 0 0
\(121\) −10.7359 −0.975995
\(122\) 0 0
\(123\) 20.6889 1.86546
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −18.3991 −1.63266 −0.816328 0.577589i \(-0.803994\pi\)
−0.816328 + 0.577589i \(0.803994\pi\)
\(128\) 0 0
\(129\) 5.90212 0.519653
\(130\) 0 0
\(131\) −14.4958 −1.26650 −0.633252 0.773946i \(-0.718280\pi\)
−0.633252 + 0.773946i \(0.718280\pi\)
\(132\) 0 0
\(133\) 1.24708 0.108135
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.40281 −0.290722 −0.145361 0.989379i \(-0.546434\pi\)
−0.145361 + 0.989379i \(0.546434\pi\)
\(138\) 0 0
\(139\) −4.01062 −0.340176 −0.170088 0.985429i \(-0.554405\pi\)
−0.170088 + 0.985429i \(0.554405\pi\)
\(140\) 0 0
\(141\) 3.86169 0.325213
\(142\) 0 0
\(143\) 3.16368 0.264560
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.5406 −0.869373
\(148\) 0 0
\(149\) 9.42255 0.771926 0.385963 0.922514i \(-0.373869\pi\)
0.385963 + 0.922514i \(0.373869\pi\)
\(150\) 0 0
\(151\) 1.60111 0.130297 0.0651483 0.997876i \(-0.479248\pi\)
0.0651483 + 0.997876i \(0.479248\pi\)
\(152\) 0 0
\(153\) 3.37952 0.273218
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.7241 −1.49434 −0.747172 0.664631i \(-0.768589\pi\)
−0.747172 + 0.664631i \(0.768589\pi\)
\(158\) 0 0
\(159\) 27.2667 2.16239
\(160\) 0 0
\(161\) −7.30892 −0.576024
\(162\) 0 0
\(163\) −15.6370 −1.22479 −0.612394 0.790553i \(-0.709793\pi\)
−0.612394 + 0.790553i \(0.709793\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.5865 −1.36088 −0.680442 0.732801i \(-0.738212\pi\)
−0.680442 + 0.732801i \(0.738212\pi\)
\(168\) 0 0
\(169\) 24.9049 1.91576
\(170\) 0 0
\(171\) −0.747704 −0.0571784
\(172\) 0 0
\(173\) −18.7799 −1.42781 −0.713903 0.700245i \(-0.753074\pi\)
−0.713903 + 0.700245i \(0.753074\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.40686 −0.631898
\(178\) 0 0
\(179\) −4.22047 −0.315453 −0.157726 0.987483i \(-0.550416\pi\)
−0.157726 + 0.987483i \(0.550416\pi\)
\(180\) 0 0
\(181\) 11.8180 0.878424 0.439212 0.898383i \(-0.355258\pi\)
0.439212 + 0.898383i \(0.355258\pi\)
\(182\) 0 0
\(183\) 20.7476 1.53371
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.32257 −0.169843
\(188\) 0 0
\(189\) 5.43752 0.395521
\(190\) 0 0
\(191\) 26.7184 1.93327 0.966636 0.256153i \(-0.0824551\pi\)
0.966636 + 0.256153i \(0.0824551\pi\)
\(192\) 0 0
\(193\) −9.25224 −0.665990 −0.332995 0.942929i \(-0.608059\pi\)
−0.332995 + 0.942929i \(0.608059\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.32896 −0.237179 −0.118589 0.992943i \(-0.537837\pi\)
−0.118589 + 0.992943i \(0.537837\pi\)
\(198\) 0 0
\(199\) −0.0700969 −0.00496903 −0.00248452 0.999997i \(-0.500791\pi\)
−0.00248452 + 0.999997i \(0.500791\pi\)
\(200\) 0 0
\(201\) 19.1650 1.35179
\(202\) 0 0
\(203\) −8.26437 −0.580045
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.38218 0.304583
\(208\) 0 0
\(209\) 0.513860 0.0355444
\(210\) 0 0
\(211\) 15.5952 1.07362 0.536810 0.843703i \(-0.319629\pi\)
0.536810 + 0.843703i \(0.319629\pi\)
\(212\) 0 0
\(213\) 14.3744 0.984916
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00616 −0.543494
\(218\) 0 0
\(219\) −24.7948 −1.67547
\(220\) 0 0
\(221\) −27.8274 −1.87187
\(222\) 0 0
\(223\) −22.4019 −1.50014 −0.750071 0.661358i \(-0.769980\pi\)
−0.750071 + 0.661358i \(0.769980\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.3464 1.61593 0.807963 0.589234i \(-0.200570\pi\)
0.807963 + 0.589234i \(0.200570\pi\)
\(228\) 0 0
\(229\) −5.27650 −0.348681 −0.174340 0.984685i \(-0.555779\pi\)
−0.174340 + 0.984685i \(0.555779\pi\)
\(230\) 0 0
\(231\) 1.24057 0.0816233
\(232\) 0 0
\(233\) −4.30335 −0.281922 −0.140961 0.990015i \(-0.545019\pi\)
−0.140961 + 0.990015i \(0.545019\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.95857 0.322094
\(238\) 0 0
\(239\) −1.44454 −0.0934395 −0.0467197 0.998908i \(-0.514877\pi\)
−0.0467197 + 0.998908i \(0.514877\pi\)
\(240\) 0 0
\(241\) 5.20083 0.335015 0.167507 0.985871i \(-0.446428\pi\)
0.167507 + 0.985871i \(0.446428\pi\)
\(242\) 0 0
\(243\) −7.60259 −0.487707
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.15670 0.391741
\(248\) 0 0
\(249\) 14.5360 0.921180
\(250\) 0 0
\(251\) 1.57046 0.0991267 0.0495634 0.998771i \(-0.484217\pi\)
0.0495634 + 0.998771i \(0.484217\pi\)
\(252\) 0 0
\(253\) −3.01165 −0.189341
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.2831 −1.51474 −0.757368 0.652988i \(-0.773515\pi\)
−0.757368 + 0.652988i \(0.773515\pi\)
\(258\) 0 0
\(259\) −1.75428 −0.109006
\(260\) 0 0
\(261\) 4.95504 0.306709
\(262\) 0 0
\(263\) 7.50729 0.462919 0.231460 0.972844i \(-0.425650\pi\)
0.231460 + 0.972844i \(0.425650\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −15.2036 −0.930448
\(268\) 0 0
\(269\) 9.37783 0.571776 0.285888 0.958263i \(-0.407711\pi\)
0.285888 + 0.958263i \(0.407711\pi\)
\(270\) 0 0
\(271\) 16.8103 1.02115 0.510577 0.859832i \(-0.329432\pi\)
0.510577 + 0.859832i \(0.329432\pi\)
\(272\) 0 0
\(273\) 14.8636 0.899585
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.55337 −0.0933328 −0.0466664 0.998911i \(-0.514860\pi\)
−0.0466664 + 0.998911i \(0.514860\pi\)
\(278\) 0 0
\(279\) 4.80022 0.287382
\(280\) 0 0
\(281\) −5.68294 −0.339016 −0.169508 0.985529i \(-0.554218\pi\)
−0.169508 + 0.985529i \(0.554218\pi\)
\(282\) 0 0
\(283\) 28.2237 1.67773 0.838864 0.544342i \(-0.183220\pi\)
0.838864 + 0.544342i \(0.183220\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.3275 −0.786697
\(288\) 0 0
\(289\) 3.42909 0.201711
\(290\) 0 0
\(291\) −13.0634 −0.765789
\(292\) 0 0
\(293\) −5.92000 −0.345850 −0.172925 0.984935i \(-0.555322\pi\)
−0.172925 + 0.984935i \(0.555322\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.24054 0.130009
\(298\) 0 0
\(299\) −36.0835 −2.08676
\(300\) 0 0
\(301\) −3.80206 −0.219147
\(302\) 0 0
\(303\) 34.1691 1.96297
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.37023 0.306495 0.153248 0.988188i \(-0.451027\pi\)
0.153248 + 0.988188i \(0.451027\pi\)
\(308\) 0 0
\(309\) 11.3085 0.643318
\(310\) 0 0
\(311\) −3.90761 −0.221580 −0.110790 0.993844i \(-0.535338\pi\)
−0.110790 + 0.993844i \(0.535338\pi\)
\(312\) 0 0
\(313\) −30.1038 −1.70156 −0.850782 0.525518i \(-0.823871\pi\)
−0.850782 + 0.525518i \(0.823871\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.7722 −0.942019 −0.471010 0.882128i \(-0.656110\pi\)
−0.471010 + 0.882128i \(0.656110\pi\)
\(318\) 0 0
\(319\) −3.40535 −0.190663
\(320\) 0 0
\(321\) 8.02900 0.448135
\(322\) 0 0
\(323\) −4.51986 −0.251491
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.0301 0.609967
\(328\) 0 0
\(329\) −2.48764 −0.137148
\(330\) 0 0
\(331\) −24.4039 −1.34136 −0.670680 0.741747i \(-0.733998\pi\)
−0.670680 + 0.741747i \(0.733998\pi\)
\(332\) 0 0
\(333\) 1.05181 0.0576387
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11.0027 −0.599355 −0.299678 0.954041i \(-0.596879\pi\)
−0.299678 + 0.954041i \(0.596879\pi\)
\(338\) 0 0
\(339\) 10.3778 0.563646
\(340\) 0 0
\(341\) −3.29895 −0.178648
\(342\) 0 0
\(343\) 15.5196 0.837980
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.2447 0.711014 0.355507 0.934674i \(-0.384308\pi\)
0.355507 + 0.934674i \(0.384308\pi\)
\(348\) 0 0
\(349\) 1.03534 0.0554206 0.0277103 0.999616i \(-0.491178\pi\)
0.0277103 + 0.999616i \(0.491178\pi\)
\(350\) 0 0
\(351\) 26.8445 1.43286
\(352\) 0 0
\(353\) −5.37333 −0.285994 −0.142997 0.989723i \(-0.545674\pi\)
−0.142997 + 0.989723i \(0.545674\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.9119 −0.577519
\(358\) 0 0
\(359\) 23.6189 1.24656 0.623280 0.781999i \(-0.285800\pi\)
0.623280 + 0.781999i \(0.285800\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −20.7837 −1.09086
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.75683 0.404903 0.202452 0.979292i \(-0.435109\pi\)
0.202452 + 0.979292i \(0.435109\pi\)
\(368\) 0 0
\(369\) 7.99071 0.415980
\(370\) 0 0
\(371\) −17.5648 −0.911918
\(372\) 0 0
\(373\) 32.2543 1.67006 0.835032 0.550201i \(-0.185449\pi\)
0.835032 + 0.550201i \(0.185449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −40.8004 −2.10133
\(378\) 0 0
\(379\) 28.7810 1.47838 0.739191 0.673496i \(-0.235208\pi\)
0.739191 + 0.673496i \(0.235208\pi\)
\(380\) 0 0
\(381\) −35.6188 −1.82481
\(382\) 0 0
\(383\) −11.0718 −0.565743 −0.282871 0.959158i \(-0.591287\pi\)
−0.282871 + 0.959158i \(0.591287\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.27958 0.115878
\(388\) 0 0
\(389\) −4.65488 −0.236012 −0.118006 0.993013i \(-0.537650\pi\)
−0.118006 + 0.993013i \(0.537650\pi\)
\(390\) 0 0
\(391\) 26.4902 1.33967
\(392\) 0 0
\(393\) −28.0624 −1.41556
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.573392 0.0287777 0.0143889 0.999896i \(-0.495420\pi\)
0.0143889 + 0.999896i \(0.495420\pi\)
\(398\) 0 0
\(399\) 2.41421 0.120862
\(400\) 0 0
\(401\) 35.9965 1.79758 0.898791 0.438378i \(-0.144447\pi\)
0.898791 + 0.438378i \(0.144447\pi\)
\(402\) 0 0
\(403\) −39.5257 −1.96891
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.722854 −0.0358305
\(408\) 0 0
\(409\) 18.3961 0.909626 0.454813 0.890587i \(-0.349706\pi\)
0.454813 + 0.890587i \(0.349706\pi\)
\(410\) 0 0
\(411\) −6.58750 −0.324937
\(412\) 0 0
\(413\) 5.41557 0.266483
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.76415 −0.380212
\(418\) 0 0
\(419\) 14.0610 0.686923 0.343461 0.939167i \(-0.388401\pi\)
0.343461 + 0.939167i \(0.388401\pi\)
\(420\) 0 0
\(421\) −36.7041 −1.78885 −0.894425 0.447218i \(-0.852415\pi\)
−0.894425 + 0.447218i \(0.852415\pi\)
\(422\) 0 0
\(423\) 1.49151 0.0725195
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −13.3653 −0.646793
\(428\) 0 0
\(429\) 6.12456 0.295697
\(430\) 0 0
\(431\) 21.2682 1.02445 0.512226 0.858851i \(-0.328821\pi\)
0.512226 + 0.858851i \(0.328821\pi\)
\(432\) 0 0
\(433\) 11.8229 0.568172 0.284086 0.958799i \(-0.408310\pi\)
0.284086 + 0.958799i \(0.408310\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.86084 −0.280362
\(438\) 0 0
\(439\) −8.03728 −0.383599 −0.191799 0.981434i \(-0.561432\pi\)
−0.191799 + 0.981434i \(0.561432\pi\)
\(440\) 0 0
\(441\) −4.07110 −0.193862
\(442\) 0 0
\(443\) 10.3632 0.492368 0.246184 0.969223i \(-0.420823\pi\)
0.246184 + 0.969223i \(0.420823\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 18.2411 0.862775
\(448\) 0 0
\(449\) −33.4637 −1.57925 −0.789625 0.613590i \(-0.789725\pi\)
−0.789625 + 0.613590i \(0.789725\pi\)
\(450\) 0 0
\(451\) −5.49161 −0.258590
\(452\) 0 0
\(453\) 3.09959 0.145631
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 27.6968 1.29560 0.647801 0.761809i \(-0.275689\pi\)
0.647801 + 0.761809i \(0.275689\pi\)
\(458\) 0 0
\(459\) −19.7075 −0.919870
\(460\) 0 0
\(461\) 4.70500 0.219134 0.109567 0.993979i \(-0.465054\pi\)
0.109567 + 0.993979i \(0.465054\pi\)
\(462\) 0 0
\(463\) 5.44945 0.253257 0.126629 0.991950i \(-0.459584\pi\)
0.126629 + 0.991950i \(0.459584\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.9687 −0.507571 −0.253786 0.967260i \(-0.581676\pi\)
−0.253786 + 0.967260i \(0.581676\pi\)
\(468\) 0 0
\(469\) −12.3458 −0.570075
\(470\) 0 0
\(471\) −36.2479 −1.67022
\(472\) 0 0
\(473\) −1.56664 −0.0720343
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.5312 0.482193
\(478\) 0 0
\(479\) −14.4663 −0.660980 −0.330490 0.943809i \(-0.607214\pi\)
−0.330490 + 0.943809i \(0.607214\pi\)
\(480\) 0 0
\(481\) −8.66072 −0.394895
\(482\) 0 0
\(483\) −14.1493 −0.643817
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −26.1973 −1.18711 −0.593556 0.804793i \(-0.702276\pi\)
−0.593556 + 0.804793i \(0.702276\pi\)
\(488\) 0 0
\(489\) −30.2717 −1.36894
\(490\) 0 0
\(491\) 12.3251 0.556224 0.278112 0.960549i \(-0.410291\pi\)
0.278112 + 0.960549i \(0.410291\pi\)
\(492\) 0 0
\(493\) 29.9531 1.34902
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.25975 −0.415356
\(498\) 0 0
\(499\) 27.4827 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(500\) 0 0
\(501\) −34.0457 −1.52105
\(502\) 0 0
\(503\) −21.0989 −0.940756 −0.470378 0.882465i \(-0.655882\pi\)
−0.470378 + 0.882465i \(0.655882\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 48.2134 2.14123
\(508\) 0 0
\(509\) −14.3975 −0.638156 −0.319078 0.947728i \(-0.603373\pi\)
−0.319078 + 0.947728i \(0.603373\pi\)
\(510\) 0 0
\(511\) 15.9724 0.706577
\(512\) 0 0
\(513\) 4.36022 0.192508
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.02504 −0.0450811
\(518\) 0 0
\(519\) −36.3559 −1.59585
\(520\) 0 0
\(521\) 9.60225 0.420682 0.210341 0.977628i \(-0.432543\pi\)
0.210341 + 0.977628i \(0.432543\pi\)
\(522\) 0 0
\(523\) −1.07927 −0.0471933 −0.0235966 0.999722i \(-0.507512\pi\)
−0.0235966 + 0.999722i \(0.507512\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.0172 1.26401
\(528\) 0 0
\(529\) 11.3495 0.493457
\(530\) 0 0
\(531\) −3.24699 −0.140907
\(532\) 0 0
\(533\) −65.7966 −2.84997
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.17040 −0.352579
\(538\) 0 0
\(539\) 2.79786 0.120513
\(540\) 0 0
\(541\) −3.55082 −0.152662 −0.0763309 0.997083i \(-0.524321\pi\)
−0.0763309 + 0.997083i \(0.524321\pi\)
\(542\) 0 0
\(543\) 22.8784 0.981807
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.5233 1.90368 0.951840 0.306596i \(-0.0991902\pi\)
0.951840 + 0.306596i \(0.0991902\pi\)
\(548\) 0 0
\(549\) 8.01339 0.342003
\(550\) 0 0
\(551\) −6.62700 −0.282320
\(552\) 0 0
\(553\) −3.19424 −0.135833
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.2157 −0.475227 −0.237613 0.971360i \(-0.576365\pi\)
−0.237613 + 0.971360i \(0.576365\pi\)
\(558\) 0 0
\(559\) −18.7704 −0.793903
\(560\) 0 0
\(561\) −4.49626 −0.189832
\(562\) 0 0
\(563\) 36.1900 1.52523 0.762614 0.646854i \(-0.223916\pi\)
0.762614 + 0.646854i \(0.223916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 13.3238 0.559548
\(568\) 0 0
\(569\) −9.60022 −0.402462 −0.201231 0.979544i \(-0.564494\pi\)
−0.201231 + 0.979544i \(0.564494\pi\)
\(570\) 0 0
\(571\) 29.4150 1.23098 0.615490 0.788145i \(-0.288958\pi\)
0.615490 + 0.788145i \(0.288958\pi\)
\(572\) 0 0
\(573\) 51.7240 2.16080
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 20.9070 0.870371 0.435186 0.900341i \(-0.356683\pi\)
0.435186 + 0.900341i \(0.356683\pi\)
\(578\) 0 0
\(579\) −17.9114 −0.744372
\(580\) 0 0
\(581\) −9.36384 −0.388478
\(582\) 0 0
\(583\) −7.23760 −0.299751
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.2156 1.70115 0.850576 0.525853i \(-0.176254\pi\)
0.850576 + 0.525853i \(0.176254\pi\)
\(588\) 0 0
\(589\) −6.41995 −0.264529
\(590\) 0 0
\(591\) −6.44453 −0.265093
\(592\) 0 0
\(593\) −32.0201 −1.31491 −0.657454 0.753495i \(-0.728367\pi\)
−0.657454 + 0.753495i \(0.728367\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.135700 −0.00555385
\(598\) 0 0
\(599\) 46.7364 1.90960 0.954798 0.297256i \(-0.0960713\pi\)
0.954798 + 0.297256i \(0.0960713\pi\)
\(600\) 0 0
\(601\) −22.9230 −0.935050 −0.467525 0.883980i \(-0.654854\pi\)
−0.467525 + 0.883980i \(0.654854\pi\)
\(602\) 0 0
\(603\) 7.40211 0.301437
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 38.7571 1.57310 0.786552 0.617525i \(-0.211864\pi\)
0.786552 + 0.617525i \(0.211864\pi\)
\(608\) 0 0
\(609\) −15.9990 −0.648312
\(610\) 0 0
\(611\) −12.2813 −0.496847
\(612\) 0 0
\(613\) −13.4487 −0.543187 −0.271593 0.962412i \(-0.587551\pi\)
−0.271593 + 0.962412i \(0.587551\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.9727 −0.441744 −0.220872 0.975303i \(-0.570890\pi\)
−0.220872 + 0.975303i \(0.570890\pi\)
\(618\) 0 0
\(619\) 46.1568 1.85520 0.927600 0.373575i \(-0.121868\pi\)
0.927600 + 0.373575i \(0.121868\pi\)
\(620\) 0 0
\(621\) −25.5546 −1.02547
\(622\) 0 0
\(623\) 9.79395 0.392386
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.994780 0.0397277
\(628\) 0 0
\(629\) 6.35815 0.253516
\(630\) 0 0
\(631\) 43.3700 1.72653 0.863266 0.504749i \(-0.168415\pi\)
0.863266 + 0.504749i \(0.168415\pi\)
\(632\) 0 0
\(633\) 30.1908 1.19998
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 33.5220 1.32819
\(638\) 0 0
\(639\) 5.55183 0.219627
\(640\) 0 0
\(641\) 16.6952 0.659420 0.329710 0.944082i \(-0.393049\pi\)
0.329710 + 0.944082i \(0.393049\pi\)
\(642\) 0 0
\(643\) −23.4243 −0.923765 −0.461882 0.886941i \(-0.652826\pi\)
−0.461882 + 0.886941i \(0.652826\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.3397 −0.681694 −0.340847 0.940119i \(-0.610714\pi\)
−0.340847 + 0.940119i \(0.610714\pi\)
\(648\) 0 0
\(649\) 2.23149 0.0875938
\(650\) 0 0
\(651\) −15.4991 −0.607458
\(652\) 0 0
\(653\) 18.3541 0.718251 0.359126 0.933289i \(-0.383075\pi\)
0.359126 + 0.933289i \(0.383075\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −9.57651 −0.373615
\(658\) 0 0
\(659\) −23.4398 −0.913085 −0.456543 0.889702i \(-0.650912\pi\)
−0.456543 + 0.889702i \(0.650912\pi\)
\(660\) 0 0
\(661\) −20.7743 −0.808027 −0.404014 0.914753i \(-0.632385\pi\)
−0.404014 + 0.914753i \(0.632385\pi\)
\(662\) 0 0
\(663\) −53.8710 −2.09218
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 38.8398 1.50388
\(668\) 0 0
\(669\) −43.3678 −1.67670
\(670\) 0 0
\(671\) −5.50720 −0.212603
\(672\) 0 0
\(673\) −6.74273 −0.259913 −0.129957 0.991520i \(-0.541484\pi\)
−0.129957 + 0.991520i \(0.541484\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.1589 0.966934 0.483467 0.875363i \(-0.339377\pi\)
0.483467 + 0.875363i \(0.339377\pi\)
\(678\) 0 0
\(679\) 8.41523 0.322947
\(680\) 0 0
\(681\) 47.1321 1.80611
\(682\) 0 0
\(683\) −13.7289 −0.525321 −0.262661 0.964888i \(-0.584600\pi\)
−0.262661 + 0.964888i \(0.584600\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −10.2148 −0.389718
\(688\) 0 0
\(689\) −86.7157 −3.30360
\(690\) 0 0
\(691\) 26.8531 1.02154 0.510770 0.859718i \(-0.329361\pi\)
0.510770 + 0.859718i \(0.329361\pi\)
\(692\) 0 0
\(693\) 0.479145 0.0182012
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 48.3037 1.82963
\(698\) 0 0
\(699\) −8.33085 −0.315102
\(700\) 0 0
\(701\) −20.1359 −0.760524 −0.380262 0.924879i \(-0.624166\pi\)
−0.380262 + 0.924879i \(0.624166\pi\)
\(702\) 0 0
\(703\) −1.40671 −0.0530553
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −22.0112 −0.827818
\(708\) 0 0
\(709\) 23.4008 0.878837 0.439419 0.898282i \(-0.355184\pi\)
0.439419 + 0.898282i \(0.355184\pi\)
\(710\) 0 0
\(711\) 1.91515 0.0718239
\(712\) 0 0
\(713\) 37.6263 1.40912
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2.79648 −0.104437
\(718\) 0 0
\(719\) 10.5133 0.392082 0.196041 0.980596i \(-0.437191\pi\)
0.196041 + 0.980596i \(0.437191\pi\)
\(720\) 0 0
\(721\) −7.28476 −0.271299
\(722\) 0 0
\(723\) 10.0683 0.374443
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −35.6031 −1.32044 −0.660222 0.751070i \(-0.729538\pi\)
−0.660222 + 0.751070i \(0.729538\pi\)
\(728\) 0 0
\(729\) 17.3343 0.642011
\(730\) 0 0
\(731\) 13.7800 0.509673
\(732\) 0 0
\(733\) −37.2476 −1.37577 −0.687886 0.725818i \(-0.741461\pi\)
−0.687886 + 0.725818i \(0.741461\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.08710 −0.187386
\(738\) 0 0
\(739\) −26.1515 −0.961998 −0.480999 0.876721i \(-0.659726\pi\)
−0.480999 + 0.876721i \(0.659726\pi\)
\(740\) 0 0
\(741\) 11.9187 0.437846
\(742\) 0 0
\(743\) 16.4700 0.604226 0.302113 0.953272i \(-0.402308\pi\)
0.302113 + 0.953272i \(0.402308\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.61424 0.205414
\(748\) 0 0
\(749\) −5.17216 −0.188987
\(750\) 0 0
\(751\) −24.0052 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(752\) 0 0
\(753\) 3.04026 0.110793
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.5457 1.25558 0.627792 0.778381i \(-0.283959\pi\)
0.627792 + 0.778381i \(0.283959\pi\)
\(758\) 0 0
\(759\) −5.83025 −0.211625
\(760\) 0 0
\(761\) −34.6920 −1.25758 −0.628792 0.777573i \(-0.716450\pi\)
−0.628792 + 0.777573i \(0.716450\pi\)
\(762\) 0 0
\(763\) −7.10543 −0.257234
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 26.7362 0.965387
\(768\) 0 0
\(769\) −30.1152 −1.08598 −0.542991 0.839738i \(-0.682708\pi\)
−0.542991 + 0.839738i \(0.682708\pi\)
\(770\) 0 0
\(771\) −47.0096 −1.69301
\(772\) 0 0
\(773\) 29.9879 1.07859 0.539295 0.842117i \(-0.318691\pi\)
0.539295 + 0.842117i \(0.318691\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3.39611 −0.121835
\(778\) 0 0
\(779\) −10.6870 −0.382901
\(780\) 0 0
\(781\) −3.81549 −0.136529
\(782\) 0 0
\(783\) −28.8952 −1.03263
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.3894 0.370342 0.185171 0.982706i \(-0.440716\pi\)
0.185171 + 0.982706i \(0.440716\pi\)
\(788\) 0 0
\(789\) 14.5334 0.517401
\(790\) 0 0
\(791\) −6.68524 −0.237700
\(792\) 0 0
\(793\) −65.9833 −2.34314
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.8854 1.16486 0.582430 0.812881i \(-0.302102\pi\)
0.582430 + 0.812881i \(0.302102\pi\)
\(798\) 0 0
\(799\) 9.01612 0.318967
\(800\) 0 0
\(801\) −5.87212 −0.207481
\(802\) 0 0
\(803\) 6.58145 0.232254
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.1545 0.639070
\(808\) 0 0
\(809\) −16.5347 −0.581329 −0.290664 0.956825i \(-0.593876\pi\)
−0.290664 + 0.956825i \(0.593876\pi\)
\(810\) 0 0
\(811\) 2.88477 0.101298 0.0506489 0.998717i \(-0.483871\pi\)
0.0506489 + 0.998717i \(0.483871\pi\)
\(812\) 0 0
\(813\) 32.5431 1.14134
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.04878 −0.106663
\(818\) 0 0
\(819\) 5.74078 0.200599
\(820\) 0 0
\(821\) −10.0122 −0.349428 −0.174714 0.984619i \(-0.555900\pi\)
−0.174714 + 0.984619i \(0.555900\pi\)
\(822\) 0 0
\(823\) 26.1520 0.911602 0.455801 0.890082i \(-0.349353\pi\)
0.455801 + 0.890082i \(0.349353\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25.1880 −0.875873 −0.437936 0.899006i \(-0.644290\pi\)
−0.437936 + 0.899006i \(0.644290\pi\)
\(828\) 0 0
\(829\) 3.32822 0.115594 0.0577969 0.998328i \(-0.481592\pi\)
0.0577969 + 0.998328i \(0.481592\pi\)
\(830\) 0 0
\(831\) −3.00716 −0.104317
\(832\) 0 0
\(833\) −24.6097 −0.852676
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −27.9924 −0.967557
\(838\) 0 0
\(839\) −50.5339 −1.74462 −0.872312 0.488949i \(-0.837380\pi\)
−0.872312 + 0.488949i \(0.837380\pi\)
\(840\) 0 0
\(841\) 14.9171 0.514384
\(842\) 0 0
\(843\) −11.0016 −0.378915
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 13.3885 0.460036
\(848\) 0 0
\(849\) 54.6383 1.87518
\(850\) 0 0
\(851\) 8.24454 0.282619
\(852\) 0 0
\(853\) −54.6636 −1.87165 −0.935823 0.352470i \(-0.885342\pi\)
−0.935823 + 0.352470i \(0.885342\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.8578 0.951603 0.475802 0.879553i \(-0.342158\pi\)
0.475802 + 0.879553i \(0.342158\pi\)
\(858\) 0 0
\(859\) −55.2474 −1.88502 −0.942509 0.334181i \(-0.891540\pi\)
−0.942509 + 0.334181i \(0.891540\pi\)
\(860\) 0 0
\(861\) −25.8007 −0.879285
\(862\) 0 0
\(863\) −27.3258 −0.930181 −0.465090 0.885263i \(-0.653978\pi\)
−0.465090 + 0.885263i \(0.653978\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.63838 0.225451
\(868\) 0 0
\(869\) −1.31619 −0.0446487
\(870\) 0 0
\(871\) −60.9500 −2.06521
\(872\) 0 0
\(873\) −5.04548 −0.170764
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13.3409 −0.450489 −0.225245 0.974302i \(-0.572318\pi\)
−0.225245 + 0.974302i \(0.572318\pi\)
\(878\) 0 0
\(879\) −11.4605 −0.386554
\(880\) 0 0
\(881\) −44.5458 −1.50079 −0.750393 0.660992i \(-0.770136\pi\)
−0.750393 + 0.660992i \(0.770136\pi\)
\(882\) 0 0
\(883\) −23.4498 −0.789148 −0.394574 0.918864i \(-0.629108\pi\)
−0.394574 + 0.918864i \(0.629108\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.1362 −1.34764 −0.673821 0.738895i \(-0.735348\pi\)
−0.673821 + 0.738895i \(0.735348\pi\)
\(888\) 0 0
\(889\) 22.9451 0.769553
\(890\) 0 0
\(891\) 5.49010 0.183925
\(892\) 0 0
\(893\) −1.99478 −0.0667528
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −69.8539 −2.33235
\(898\) 0 0
\(899\) 42.5450 1.41895
\(900\) 0 0
\(901\) 63.6611 2.12086
\(902\) 0 0
\(903\) −7.36040 −0.244939
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −35.8062 −1.18893 −0.594463 0.804123i \(-0.702636\pi\)
−0.594463 + 0.804123i \(0.702636\pi\)
\(908\) 0 0
\(909\) 13.1972 0.437723
\(910\) 0 0
\(911\) −29.5302 −0.978381 −0.489190 0.872177i \(-0.662708\pi\)
−0.489190 + 0.872177i \(0.662708\pi\)
\(912\) 0 0
\(913\) −3.85839 −0.127694
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.0774 0.596967
\(918\) 0 0
\(919\) 11.7570 0.387829 0.193914 0.981018i \(-0.437882\pi\)
0.193914 + 0.981018i \(0.437882\pi\)
\(920\) 0 0
\(921\) 10.3962 0.342567
\(922\) 0 0
\(923\) −45.7145 −1.50471
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.36769 0.143454
\(928\) 0 0
\(929\) −21.5189 −0.706012 −0.353006 0.935621i \(-0.614841\pi\)
−0.353006 + 0.935621i \(0.614841\pi\)
\(930\) 0 0
\(931\) 5.44480 0.178446
\(932\) 0 0
\(933\) −7.56473 −0.247658
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 31.7690 1.03785 0.518924 0.854820i \(-0.326333\pi\)
0.518924 + 0.854820i \(0.326333\pi\)
\(938\) 0 0
\(939\) −58.2778 −1.90182
\(940\) 0 0
\(941\) −0.730603 −0.0238170 −0.0119085 0.999929i \(-0.503791\pi\)
−0.0119085 + 0.999929i \(0.503791\pi\)
\(942\) 0 0
\(943\) 62.6348 2.03967
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.2171 1.33938 0.669688 0.742643i \(-0.266428\pi\)
0.669688 + 0.742643i \(0.266428\pi\)
\(948\) 0 0
\(949\) 78.8543 2.55972
\(950\) 0 0
\(951\) −32.4692 −1.05289
\(952\) 0 0
\(953\) −39.9343 −1.29360 −0.646800 0.762660i \(-0.723893\pi\)
−0.646800 + 0.762660i \(0.723893\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.59241 −0.213102
\(958\) 0 0
\(959\) 4.24357 0.137032
\(960\) 0 0
\(961\) 10.2157 0.329539
\(962\) 0 0
\(963\) 3.10105 0.0999300
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.73989 0.281056 0.140528 0.990077i \(-0.455120\pi\)
0.140528 + 0.990077i \(0.455120\pi\)
\(968\) 0 0
\(969\) −8.74998 −0.281090
\(970\) 0 0
\(971\) 7.49072 0.240389 0.120194 0.992750i \(-0.461648\pi\)
0.120194 + 0.992750i \(0.461648\pi\)
\(972\) 0 0
\(973\) 5.00155 0.160342
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −51.6069 −1.65105 −0.825526 0.564364i \(-0.809121\pi\)
−0.825526 + 0.564364i \(0.809121\pi\)
\(978\) 0 0
\(979\) 4.03561 0.128979
\(980\) 0 0
\(981\) 4.26017 0.136017
\(982\) 0 0
\(983\) 57.9017 1.84678 0.923389 0.383865i \(-0.125407\pi\)
0.923389 + 0.383865i \(0.125407\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.81582 −0.153289
\(988\) 0 0
\(989\) 17.8684 0.568182
\(990\) 0 0
\(991\) −18.2369 −0.579314 −0.289657 0.957131i \(-0.593541\pi\)
−0.289657 + 0.957131i \(0.593541\pi\)
\(992\) 0 0
\(993\) −47.2435 −1.49923
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.7461 0.340332 0.170166 0.985415i \(-0.445570\pi\)
0.170166 + 0.985415i \(0.445570\pi\)
\(998\) 0 0
\(999\) −6.13358 −0.194058
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 3800.2.a.bc.1.5 yes 6
4.3 odd 2 7600.2.a.ch.1.2 6
5.2 odd 4 3800.2.d.q.3649.3 12
5.3 odd 4 3800.2.d.q.3649.10 12
5.4 even 2 3800.2.a.ba.1.2 6
20.19 odd 2 7600.2.a.cl.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.2 6 5.4 even 2
3800.2.a.bc.1.5 yes 6 1.1 even 1 trivial
3800.2.d.q.3649.3 12 5.2 odd 4
3800.2.d.q.3649.10 12 5.3 odd 4
7600.2.a.ch.1.2 6 4.3 odd 2
7600.2.a.cl.1.5 6 20.19 odd 2