Properties

Label 3920.2.a.ce.1.4
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.87996\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.87996 q^{3} -1.00000 q^{5} +5.29417 q^{9} +O(q^{10})\) \(q+2.87996 q^{3} -1.00000 q^{5} +5.29417 q^{9} -1.46575 q^{11} -2.22129 q^{13} -2.87996 q^{15} -7.26580 q^{17} -5.48709 q^{19} -2.51547 q^{23} +1.00000 q^{25} +6.60713 q^{27} -7.12260 q^{29} -6.51547 q^{31} -4.22129 q^{33} +6.90131 q^{37} -6.39724 q^{39} +11.3199 q^{41} -3.31733 q^{43} -5.29417 q^{45} -8.36705 q^{47} -20.9252 q^{51} -7.00437 q^{53} +1.46575 q^{55} -15.8026 q^{57} +9.07107 q^{59} -11.2143 q^{61} +2.22129 q^{65} +6.41240 q^{67} -7.24445 q^{69} +10.5316 q^{71} -10.5217 q^{73} +2.87996 q^{75} -10.1911 q^{79} +3.14576 q^{81} +16.4186 q^{83} +7.26580 q^{85} -20.5128 q^{87} -9.83024 q^{89} -18.7643 q^{93} +5.48709 q^{95} +2.09423 q^{97} -7.75992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{5} + 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{5} + 6q^{9} - 2q^{11} - 10q^{13} - 2q^{15} - 6q^{17} + 4q^{23} + 4q^{25} + 14q^{27} - 2q^{29} - 12q^{31} - 18q^{33} - 14q^{39} - 12q^{41} + 8q^{43} - 6q^{45} - 2q^{47} - 2q^{51} - 4q^{53} + 2q^{55} - 8q^{57} + 8q^{59} - 20q^{61} + 10q^{65} + 8q^{67} - 24q^{69} - 4q^{71} - 16q^{73} + 2q^{75} - 22q^{79} - 20q^{81} + 36q^{83} + 6q^{85} - 18q^{87} - 40q^{89} - 32q^{93} - 26q^{97} - 12q^{99} + O(q^{100}) \)

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
\(3\) 2.87996 1.66275 0.831373 0.555715i \(-0.187555\pi\)
0.831373 + 0.555715i \(0.187555\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.29417 1.76472
\(10\) 0 0
\(11\) −1.46575 −0.441939 −0.220970 0.975281i \(-0.570922\pi\)
−0.220970 + 0.975281i \(0.570922\pi\)
\(12\) 0 0
\(13\) −2.22129 −0.616076 −0.308038 0.951374i \(-0.599672\pi\)
−0.308038 + 0.951374i \(0.599672\pi\)
\(14\) 0 0
\(15\) −2.87996 −0.743603
\(16\) 0 0
\(17\) −7.26580 −1.76221 −0.881107 0.472916i \(-0.843201\pi\)
−0.881107 + 0.472916i \(0.843201\pi\)
\(18\) 0 0
\(19\) −5.48709 −1.25883 −0.629413 0.777071i \(-0.716705\pi\)
−0.629413 + 0.777071i \(0.716705\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.51547 −0.524512 −0.262256 0.964998i \(-0.584466\pi\)
−0.262256 + 0.964998i \(0.584466\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 6.60713 1.27154
\(28\) 0 0
\(29\) −7.12260 −1.32263 −0.661317 0.750107i \(-0.730002\pi\)
−0.661317 + 0.750107i \(0.730002\pi\)
\(30\) 0 0
\(31\) −6.51547 −1.17021 −0.585106 0.810957i \(-0.698947\pi\)
−0.585106 + 0.810957i \(0.698947\pi\)
\(32\) 0 0
\(33\) −4.22129 −0.734833
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.90131 1.13457 0.567284 0.823522i \(-0.307994\pi\)
0.567284 + 0.823522i \(0.307994\pi\)
\(38\) 0 0
\(39\) −6.39724 −1.02438
\(40\) 0 0
\(41\) 11.3199 1.76787 0.883935 0.467609i \(-0.154885\pi\)
0.883935 + 0.467609i \(0.154885\pi\)
\(42\) 0 0
\(43\) −3.31733 −0.505888 −0.252944 0.967481i \(-0.581399\pi\)
−0.252944 + 0.967481i \(0.581399\pi\)
\(44\) 0 0
\(45\) −5.29417 −0.789209
\(46\) 0 0
\(47\) −8.36705 −1.22046 −0.610230 0.792224i \(-0.708923\pi\)
−0.610230 + 0.792224i \(0.708923\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −20.9252 −2.93012
\(52\) 0 0
\(53\) −7.00437 −0.962125 −0.481062 0.876686i \(-0.659749\pi\)
−0.481062 + 0.876686i \(0.659749\pi\)
\(54\) 0 0
\(55\) 1.46575 0.197641
\(56\) 0 0
\(57\) −15.8026 −2.09311
\(58\) 0 0
\(59\) 9.07107 1.18095 0.590476 0.807055i \(-0.298940\pi\)
0.590476 + 0.807055i \(0.298940\pi\)
\(60\) 0 0
\(61\) −11.2143 −1.43584 −0.717920 0.696126i \(-0.754906\pi\)
−0.717920 + 0.696126i \(0.754906\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.22129 0.275518
\(66\) 0 0
\(67\) 6.41240 0.783400 0.391700 0.920093i \(-0.371887\pi\)
0.391700 + 0.920093i \(0.371887\pi\)
\(68\) 0 0
\(69\) −7.24445 −0.872130
\(70\) 0 0
\(71\) 10.5316 1.24987 0.624935 0.780677i \(-0.285125\pi\)
0.624935 + 0.780677i \(0.285125\pi\)
\(72\) 0 0
\(73\) −10.5217 −1.23147 −0.615733 0.787955i \(-0.711140\pi\)
−0.615733 + 0.787955i \(0.711140\pi\)
\(74\) 0 0
\(75\) 2.87996 0.332549
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.1911 −1.14659 −0.573295 0.819349i \(-0.694335\pi\)
−0.573295 + 0.819349i \(0.694335\pi\)
\(80\) 0 0
\(81\) 3.14576 0.349529
\(82\) 0 0
\(83\) 16.4186 1.80217 0.901087 0.433638i \(-0.142770\pi\)
0.901087 + 0.433638i \(0.142770\pi\)
\(84\) 0 0
\(85\) 7.26580 0.788086
\(86\) 0 0
\(87\) −20.5128 −2.19920
\(88\) 0 0
\(89\) −9.83024 −1.04200 −0.521002 0.853556i \(-0.674441\pi\)
−0.521002 + 0.853556i \(0.674441\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −18.7643 −1.94577
\(94\) 0 0
\(95\) 5.48709 0.562964
\(96\) 0 0
\(97\) 2.09423 0.212636 0.106318 0.994332i \(-0.466094\pi\)
0.106318 + 0.994332i \(0.466094\pi\)
\(98\) 0 0
\(99\) −7.75992 −0.779901
\(100\) 0 0
\(101\) 13.6656 1.35978 0.679889 0.733315i \(-0.262028\pi\)
0.679889 + 0.733315i \(0.262028\pi\)
\(102\) 0 0
\(103\) 10.0239 0.987685 0.493843 0.869551i \(-0.335592\pi\)
0.493843 + 0.869551i \(0.335592\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.42040 0.910704 0.455352 0.890311i \(-0.349513\pi\)
0.455352 + 0.890311i \(0.349513\pi\)
\(108\) 0 0
\(109\) −8.98559 −0.860663 −0.430332 0.902671i \(-0.641603\pi\)
−0.430332 + 0.902671i \(0.641603\pi\)
\(110\) 0 0
\(111\) 19.8755 1.88650
\(112\) 0 0
\(113\) 10.9476 1.02987 0.514933 0.857231i \(-0.327817\pi\)
0.514933 + 0.857231i \(0.327817\pi\)
\(114\) 0 0
\(115\) 2.51547 0.234569
\(116\) 0 0
\(117\) −11.7599 −1.08721
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.85158 −0.804690
\(122\) 0 0
\(123\) 32.6009 2.93952
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 5.44696 0.483340 0.241670 0.970359i \(-0.422305\pi\)
0.241670 + 0.970359i \(0.422305\pi\)
\(128\) 0 0
\(129\) −9.55379 −0.841164
\(130\) 0 0
\(131\) 9.17414 0.801548 0.400774 0.916177i \(-0.368741\pi\)
0.400774 + 0.916177i \(0.368741\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −6.60713 −0.568651
\(136\) 0 0
\(137\) 11.3137 0.966595 0.483298 0.875456i \(-0.339439\pi\)
0.483298 + 0.875456i \(0.339439\pi\)
\(138\) 0 0
\(139\) −6.83099 −0.579397 −0.289698 0.957118i \(-0.593555\pi\)
−0.289698 + 0.957118i \(0.593555\pi\)
\(140\) 0 0
\(141\) −24.0968 −2.02932
\(142\) 0 0
\(143\) 3.25586 0.272268
\(144\) 0 0
\(145\) 7.12260 0.591500
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.96243 −0.160769 −0.0803844 0.996764i \(-0.525615\pi\)
−0.0803844 + 0.996764i \(0.525615\pi\)
\(150\) 0 0
\(151\) 7.32511 0.596109 0.298055 0.954549i \(-0.403662\pi\)
0.298055 + 0.954549i \(0.403662\pi\)
\(152\) 0 0
\(153\) −38.4664 −3.10982
\(154\) 0 0
\(155\) 6.51547 0.523335
\(156\) 0 0
\(157\) 7.46309 0.595620 0.297810 0.954625i \(-0.403744\pi\)
0.297810 + 0.954625i \(0.403744\pi\)
\(158\) 0 0
\(159\) −20.1723 −1.59977
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.2790 0.961767 0.480883 0.876785i \(-0.340316\pi\)
0.480883 + 0.876785i \(0.340316\pi\)
\(164\) 0 0
\(165\) 4.22129 0.328627
\(166\) 0 0
\(167\) −18.2300 −1.41068 −0.705342 0.708868i \(-0.749206\pi\)
−0.705342 + 0.708868i \(0.749206\pi\)
\(168\) 0 0
\(169\) −8.06585 −0.620450
\(170\) 0 0
\(171\) −29.0496 −2.22148
\(172\) 0 0
\(173\) −17.9245 −1.36277 −0.681386 0.731924i \(-0.738622\pi\)
−0.681386 + 0.731924i \(0.738622\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 26.1243 1.96362
\(178\) 0 0
\(179\) −0.0375672 −0.00280791 −0.00140395 0.999999i \(-0.500447\pi\)
−0.00140395 + 0.999999i \(0.500447\pi\)
\(180\) 0 0
\(181\) 6.82105 0.507004 0.253502 0.967335i \(-0.418417\pi\)
0.253502 + 0.967335i \(0.418417\pi\)
\(182\) 0 0
\(183\) −32.2966 −2.38744
\(184\) 0 0
\(185\) −6.90131 −0.507394
\(186\) 0 0
\(187\) 10.6498 0.778792
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.86202 0.424161 0.212080 0.977252i \(-0.431976\pi\)
0.212080 + 0.977252i \(0.431976\pi\)
\(192\) 0 0
\(193\) −4.77168 −0.343473 −0.171736 0.985143i \(-0.554938\pi\)
−0.171736 + 0.985143i \(0.554938\pi\)
\(194\) 0 0
\(195\) 6.39724 0.458116
\(196\) 0 0
\(197\) −9.28714 −0.661682 −0.330841 0.943687i \(-0.607332\pi\)
−0.330841 + 0.943687i \(0.607332\pi\)
\(198\) 0 0
\(199\) −3.20176 −0.226967 −0.113483 0.993540i \(-0.536201\pi\)
−0.113483 + 0.993540i \(0.536201\pi\)
\(200\) 0 0
\(201\) 18.4675 1.30259
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −11.3199 −0.790616
\(206\) 0 0
\(207\) −13.3173 −0.925619
\(208\) 0 0
\(209\) 8.04269 0.556325
\(210\) 0 0
\(211\) 26.1055 1.79718 0.898590 0.438790i \(-0.144593\pi\)
0.898590 + 0.438790i \(0.144593\pi\)
\(212\) 0 0
\(213\) 30.3306 2.07822
\(214\) 0 0
\(215\) 3.31733 0.226240
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −30.3020 −2.04762
\(220\) 0 0
\(221\) 16.1395 1.08566
\(222\) 0 0
\(223\) −23.8214 −1.59520 −0.797599 0.603188i \(-0.793897\pi\)
−0.797599 + 0.603188i \(0.793897\pi\)
\(224\) 0 0
\(225\) 5.29417 0.352945
\(226\) 0 0
\(227\) −9.42199 −0.625360 −0.312680 0.949859i \(-0.601227\pi\)
−0.312680 + 0.949859i \(0.601227\pi\)
\(228\) 0 0
\(229\) −9.93587 −0.656581 −0.328290 0.944577i \(-0.606472\pi\)
−0.328290 + 0.944577i \(0.606472\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.23707 0.0810434 0.0405217 0.999179i \(-0.487098\pi\)
0.0405217 + 0.999179i \(0.487098\pi\)
\(234\) 0 0
\(235\) 8.36705 0.545806
\(236\) 0 0
\(237\) −29.3500 −1.90649
\(238\) 0 0
\(239\) −1.05710 −0.0683782 −0.0341891 0.999415i \(-0.510885\pi\)
−0.0341891 + 0.999415i \(0.510885\pi\)
\(240\) 0 0
\(241\) −4.52903 −0.291741 −0.145870 0.989304i \(-0.546598\pi\)
−0.145870 + 0.989304i \(0.546598\pi\)
\(242\) 0 0
\(243\) −10.7617 −0.690366
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.1885 0.775533
\(248\) 0 0
\(249\) 47.2849 2.99656
\(250\) 0 0
\(251\) 4.74198 0.299311 0.149656 0.988738i \(-0.452184\pi\)
0.149656 + 0.988738i \(0.452184\pi\)
\(252\) 0 0
\(253\) 3.68704 0.231802
\(254\) 0 0
\(255\) 20.9252 1.31039
\(256\) 0 0
\(257\) −15.9297 −0.993666 −0.496833 0.867846i \(-0.665504\pi\)
−0.496833 + 0.867846i \(0.665504\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −37.7083 −2.33408
\(262\) 0 0
\(263\) 13.2746 0.818549 0.409275 0.912411i \(-0.365782\pi\)
0.409275 + 0.912411i \(0.365782\pi\)
\(264\) 0 0
\(265\) 7.00437 0.430275
\(266\) 0 0
\(267\) −28.3107 −1.73259
\(268\) 0 0
\(269\) −24.8601 −1.51575 −0.757874 0.652401i \(-0.773762\pi\)
−0.757874 + 0.652401i \(0.773762\pi\)
\(270\) 0 0
\(271\) −30.4285 −1.84840 −0.924201 0.381907i \(-0.875267\pi\)
−0.924201 + 0.381907i \(0.875267\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.46575 −0.0883879
\(276\) 0 0
\(277\) −21.7297 −1.30561 −0.652807 0.757525i \(-0.726409\pi\)
−0.652807 + 0.757525i \(0.726409\pi\)
\(278\) 0 0
\(279\) −34.4940 −2.06510
\(280\) 0 0
\(281\) −0.285426 −0.0170271 −0.00851354 0.999964i \(-0.502710\pi\)
−0.00851354 + 0.999964i \(0.502710\pi\)
\(282\) 0 0
\(283\) 5.51463 0.327810 0.163905 0.986476i \(-0.447591\pi\)
0.163905 + 0.986476i \(0.447591\pi\)
\(284\) 0 0
\(285\) 15.8026 0.936066
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 35.7918 2.10540
\(290\) 0 0
\(291\) 6.03129 0.353560
\(292\) 0 0
\(293\) −17.5777 −1.02690 −0.513450 0.858120i \(-0.671633\pi\)
−0.513450 + 0.858120i \(0.671633\pi\)
\(294\) 0 0
\(295\) −9.07107 −0.528138
\(296\) 0 0
\(297\) −9.68439 −0.561945
\(298\) 0 0
\(299\) 5.58760 0.323139
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 39.3564 2.26097
\(304\) 0 0
\(305\) 11.2143 0.642127
\(306\) 0 0
\(307\) −25.4771 −1.45405 −0.727026 0.686610i \(-0.759098\pi\)
−0.727026 + 0.686610i \(0.759098\pi\)
\(308\) 0 0
\(309\) 28.8685 1.64227
\(310\) 0 0
\(311\) 21.5662 1.22290 0.611452 0.791281i \(-0.290586\pi\)
0.611452 + 0.791281i \(0.290586\pi\)
\(312\) 0 0
\(313\) −2.97940 −0.168406 −0.0842030 0.996449i \(-0.526834\pi\)
−0.0842030 + 0.996449i \(0.526834\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.99638 −0.449121 −0.224561 0.974460i \(-0.572095\pi\)
−0.224561 + 0.974460i \(0.572095\pi\)
\(318\) 0 0
\(319\) 10.4399 0.584524
\(320\) 0 0
\(321\) 27.1304 1.51427
\(322\) 0 0
\(323\) 39.8681 2.21832
\(324\) 0 0
\(325\) −2.22129 −0.123215
\(326\) 0 0
\(327\) −25.8781 −1.43106
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −34.2966 −1.88511 −0.942557 0.334045i \(-0.891586\pi\)
−0.942557 + 0.334045i \(0.891586\pi\)
\(332\) 0 0
\(333\) 36.5367 2.00220
\(334\) 0 0
\(335\) −6.41240 −0.350347
\(336\) 0 0
\(337\) 23.8719 1.30038 0.650192 0.759770i \(-0.274689\pi\)
0.650192 + 0.759770i \(0.274689\pi\)
\(338\) 0 0
\(339\) 31.5287 1.71241
\(340\) 0 0
\(341\) 9.55003 0.517163
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 7.24445 0.390028
\(346\) 0 0
\(347\) −6.35565 −0.341189 −0.170595 0.985341i \(-0.554569\pi\)
−0.170595 + 0.985341i \(0.554569\pi\)
\(348\) 0 0
\(349\) 9.24821 0.495045 0.247523 0.968882i \(-0.420384\pi\)
0.247523 + 0.968882i \(0.420384\pi\)
\(350\) 0 0
\(351\) −14.6764 −0.783368
\(352\) 0 0
\(353\) −17.9608 −0.955959 −0.477979 0.878371i \(-0.658631\pi\)
−0.477979 + 0.878371i \(0.658631\pi\)
\(354\) 0 0
\(355\) −10.5316 −0.558959
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.2843 1.07056 0.535281 0.844674i \(-0.320206\pi\)
0.535281 + 0.844674i \(0.320206\pi\)
\(360\) 0 0
\(361\) 11.1082 0.584642
\(362\) 0 0
\(363\) −25.4922 −1.33799
\(364\) 0 0
\(365\) 10.5217 0.550729
\(366\) 0 0
\(367\) 25.1439 1.31250 0.656249 0.754544i \(-0.272142\pi\)
0.656249 + 0.754544i \(0.272142\pi\)
\(368\) 0 0
\(369\) 59.9295 3.11980
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.737732 0.0381983 0.0190992 0.999818i \(-0.493920\pi\)
0.0190992 + 0.999818i \(0.493920\pi\)
\(374\) 0 0
\(375\) −2.87996 −0.148721
\(376\) 0 0
\(377\) 15.8214 0.814843
\(378\) 0 0
\(379\) 7.11120 0.365278 0.182639 0.983180i \(-0.441536\pi\)
0.182639 + 0.983180i \(0.441536\pi\)
\(380\) 0 0
\(381\) 15.6870 0.803672
\(382\) 0 0
\(383\) 28.6842 1.46569 0.732846 0.680394i \(-0.238191\pi\)
0.732846 + 0.680394i \(0.238191\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −17.5625 −0.892754
\(388\) 0 0
\(389\) 13.5189 0.685434 0.342717 0.939439i \(-0.388653\pi\)
0.342717 + 0.939439i \(0.388653\pi\)
\(390\) 0 0
\(391\) 18.2769 0.924302
\(392\) 0 0
\(393\) 26.4212 1.33277
\(394\) 0 0
\(395\) 10.1911 0.512770
\(396\) 0 0
\(397\) −4.02391 −0.201954 −0.100977 0.994889i \(-0.532197\pi\)
−0.100977 + 0.994889i \(0.532197\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.3796 −1.21746 −0.608729 0.793379i \(-0.708320\pi\)
−0.608729 + 0.793379i \(0.708320\pi\)
\(402\) 0 0
\(403\) 14.4728 0.720940
\(404\) 0 0
\(405\) −3.14576 −0.156314
\(406\) 0 0
\(407\) −10.1156 −0.501410
\(408\) 0 0
\(409\) −8.49522 −0.420062 −0.210031 0.977695i \(-0.567356\pi\)
−0.210031 + 0.977695i \(0.567356\pi\)
\(410\) 0 0
\(411\) 32.5830 1.60720
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −16.4186 −0.805957
\(416\) 0 0
\(417\) −19.6730 −0.963390
\(418\) 0 0
\(419\) 3.07469 0.150209 0.0751043 0.997176i \(-0.476071\pi\)
0.0751043 + 0.997176i \(0.476071\pi\)
\(420\) 0 0
\(421\) 22.4790 1.09556 0.547780 0.836623i \(-0.315473\pi\)
0.547780 + 0.836623i \(0.315473\pi\)
\(422\) 0 0
\(423\) −44.2966 −2.15378
\(424\) 0 0
\(425\) −7.26580 −0.352443
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 9.37674 0.452713
\(430\) 0 0
\(431\) 5.17254 0.249153 0.124576 0.992210i \(-0.460243\pi\)
0.124576 + 0.992210i \(0.460243\pi\)
\(432\) 0 0
\(433\) −1.78030 −0.0855557 −0.0427779 0.999085i \(-0.513621\pi\)
−0.0427779 + 0.999085i \(0.513621\pi\)
\(434\) 0 0
\(435\) 20.5128 0.983514
\(436\) 0 0
\(437\) 13.8026 0.660269
\(438\) 0 0
\(439\) −25.7865 −1.23072 −0.615361 0.788245i \(-0.710990\pi\)
−0.615361 + 0.788245i \(0.710990\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.4426 0.591165 0.295583 0.955317i \(-0.404486\pi\)
0.295583 + 0.955317i \(0.404486\pi\)
\(444\) 0 0
\(445\) 9.83024 0.465998
\(446\) 0 0
\(447\) −5.65173 −0.267318
\(448\) 0 0
\(449\) −25.3574 −1.19669 −0.598344 0.801239i \(-0.704174\pi\)
−0.598344 + 0.801239i \(0.704174\pi\)
\(450\) 0 0
\(451\) −16.5921 −0.781292
\(452\) 0 0
\(453\) 21.0960 0.991178
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −34.7039 −1.62338 −0.811690 0.584088i \(-0.801452\pi\)
−0.811690 + 0.584088i \(0.801452\pi\)
\(458\) 0 0
\(459\) −48.0061 −2.24073
\(460\) 0 0
\(461\) 40.8762 1.90380 0.951898 0.306414i \(-0.0991293\pi\)
0.951898 + 0.306414i \(0.0991293\pi\)
\(462\) 0 0
\(463\) 25.1236 1.16759 0.583796 0.811901i \(-0.301567\pi\)
0.583796 + 0.811901i \(0.301567\pi\)
\(464\) 0 0
\(465\) 18.7643 0.870173
\(466\) 0 0
\(467\) −27.1766 −1.25758 −0.628792 0.777574i \(-0.716450\pi\)
−0.628792 + 0.777574i \(0.716450\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 21.4934 0.990364
\(472\) 0 0
\(473\) 4.86237 0.223572
\(474\) 0 0
\(475\) −5.48709 −0.251765
\(476\) 0 0
\(477\) −37.0824 −1.69789
\(478\) 0 0
\(479\) −2.07288 −0.0947123 −0.0473561 0.998878i \(-0.515080\pi\)
−0.0473561 + 0.998878i \(0.515080\pi\)
\(480\) 0 0
\(481\) −15.3298 −0.698980
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.09423 −0.0950939
\(486\) 0 0
\(487\) −33.3490 −1.51119 −0.755594 0.655040i \(-0.772652\pi\)
−0.755594 + 0.655040i \(0.772652\pi\)
\(488\) 0 0
\(489\) 35.3631 1.59917
\(490\) 0 0
\(491\) −36.4487 −1.64491 −0.822453 0.568833i \(-0.807395\pi\)
−0.822453 + 0.568833i \(0.807395\pi\)
\(492\) 0 0
\(493\) 51.7514 2.33077
\(494\) 0 0
\(495\) 7.75992 0.348783
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7.49368 0.335463 0.167732 0.985833i \(-0.446356\pi\)
0.167732 + 0.985833i \(0.446356\pi\)
\(500\) 0 0
\(501\) −52.5018 −2.34561
\(502\) 0 0
\(503\) −15.6381 −0.697267 −0.348634 0.937259i \(-0.613354\pi\)
−0.348634 + 0.937259i \(0.613354\pi\)
\(504\) 0 0
\(505\) −13.6656 −0.608111
\(506\) 0 0
\(507\) −23.2293 −1.03165
\(508\) 0 0
\(509\) −7.21051 −0.319600 −0.159800 0.987149i \(-0.551085\pi\)
−0.159800 + 0.987149i \(0.551085\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −36.2540 −1.60065
\(514\) 0 0
\(515\) −10.0239 −0.441706
\(516\) 0 0
\(517\) 12.2640 0.539370
\(518\) 0 0
\(519\) −51.6218 −2.26594
\(520\) 0 0
\(521\) −14.8236 −0.649434 −0.324717 0.945811i \(-0.605269\pi\)
−0.324717 + 0.945811i \(0.605269\pi\)
\(522\) 0 0
\(523\) −4.27283 −0.186838 −0.0934189 0.995627i \(-0.529780\pi\)
−0.0934189 + 0.995627i \(0.529780\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 47.3401 2.06217
\(528\) 0 0
\(529\) −16.6724 −0.724888
\(530\) 0 0
\(531\) 48.0238 2.08406
\(532\) 0 0
\(533\) −25.1448 −1.08914
\(534\) 0 0
\(535\) −9.42040 −0.407279
\(536\) 0 0
\(537\) −0.108192 −0.00466883
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.942899 −0.0405384 −0.0202692 0.999795i \(-0.506452\pi\)
−0.0202692 + 0.999795i \(0.506452\pi\)
\(542\) 0 0
\(543\) 19.6444 0.843020
\(544\) 0 0
\(545\) 8.98559 0.384900
\(546\) 0 0
\(547\) 11.1871 0.478327 0.239164 0.970979i \(-0.423127\pi\)
0.239164 + 0.970979i \(0.423127\pi\)
\(548\) 0 0
\(549\) −59.3703 −2.53386
\(550\) 0 0
\(551\) 39.0824 1.66497
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −19.8755 −0.843668
\(556\) 0 0
\(557\) 28.7305 1.21735 0.608675 0.793420i \(-0.291701\pi\)
0.608675 + 0.793420i \(0.291701\pi\)
\(558\) 0 0
\(559\) 7.36877 0.311666
\(560\) 0 0
\(561\) 30.6711 1.29493
\(562\) 0 0
\(563\) 2.44741 0.103146 0.0515729 0.998669i \(-0.483577\pi\)
0.0515729 + 0.998669i \(0.483577\pi\)
\(564\) 0 0
\(565\) −10.9476 −0.460570
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.5058 1.48848 0.744240 0.667912i \(-0.232812\pi\)
0.744240 + 0.667912i \(0.232812\pi\)
\(570\) 0 0
\(571\) −5.78874 −0.242251 −0.121126 0.992637i \(-0.538650\pi\)
−0.121126 + 0.992637i \(0.538650\pi\)
\(572\) 0 0
\(573\) 16.8824 0.705272
\(574\) 0 0
\(575\) −2.51547 −0.104902
\(576\) 0 0
\(577\) −2.20773 −0.0919090 −0.0459545 0.998944i \(-0.514633\pi\)
−0.0459545 + 0.998944i \(0.514633\pi\)
\(578\) 0 0
\(579\) −13.7422 −0.571108
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 10.2666 0.425201
\(584\) 0 0
\(585\) 11.7599 0.486213
\(586\) 0 0
\(587\) 39.6719 1.63744 0.818718 0.574196i \(-0.194685\pi\)
0.818718 + 0.574196i \(0.194685\pi\)
\(588\) 0 0
\(589\) 35.7510 1.47309
\(590\) 0 0
\(591\) −26.7466 −1.10021
\(592\) 0 0
\(593\) 8.93309 0.366838 0.183419 0.983035i \(-0.441283\pi\)
0.183419 + 0.983035i \(0.441283\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.22095 −0.377388
\(598\) 0 0
\(599\) −18.3900 −0.751395 −0.375697 0.926742i \(-0.622597\pi\)
−0.375697 + 0.926742i \(0.622597\pi\)
\(600\) 0 0
\(601\) −17.5063 −0.714096 −0.357048 0.934086i \(-0.616217\pi\)
−0.357048 + 0.934086i \(0.616217\pi\)
\(602\) 0 0
\(603\) 33.9484 1.38248
\(604\) 0 0
\(605\) 8.85158 0.359868
\(606\) 0 0
\(607\) −45.1439 −1.83233 −0.916166 0.400798i \(-0.868733\pi\)
−0.916166 + 0.400798i \(0.868733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.5857 0.751897
\(612\) 0 0
\(613\) −11.8557 −0.478849 −0.239424 0.970915i \(-0.576959\pi\)
−0.239424 + 0.970915i \(0.576959\pi\)
\(614\) 0 0
\(615\) −32.6009 −1.31459
\(616\) 0 0
\(617\) −20.8174 −0.838078 −0.419039 0.907968i \(-0.637633\pi\)
−0.419039 + 0.907968i \(0.637633\pi\)
\(618\) 0 0
\(619\) −8.78149 −0.352958 −0.176479 0.984304i \(-0.556471\pi\)
−0.176479 + 0.984304i \(0.556471\pi\)
\(620\) 0 0
\(621\) −16.6200 −0.666939
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 23.1626 0.925027
\(628\) 0 0
\(629\) −50.1435 −1.99935
\(630\) 0 0
\(631\) −22.1004 −0.879803 −0.439902 0.898046i \(-0.644987\pi\)
−0.439902 + 0.898046i \(0.644987\pi\)
\(632\) 0 0
\(633\) 75.1829 2.98825
\(634\) 0 0
\(635\) −5.44696 −0.216156
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 55.7561 2.20568
\(640\) 0 0
\(641\) −43.8577 −1.73227 −0.866137 0.499807i \(-0.833404\pi\)
−0.866137 + 0.499807i \(0.833404\pi\)
\(642\) 0 0
\(643\) 37.9895 1.49816 0.749079 0.662480i \(-0.230496\pi\)
0.749079 + 0.662480i \(0.230496\pi\)
\(644\) 0 0
\(645\) 9.55379 0.376180
\(646\) 0 0
\(647\) 33.3705 1.31193 0.655964 0.754792i \(-0.272262\pi\)
0.655964 + 0.754792i \(0.272262\pi\)
\(648\) 0 0
\(649\) −13.2959 −0.521909
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.0632 0.589468 0.294734 0.955579i \(-0.404769\pi\)
0.294734 + 0.955579i \(0.404769\pi\)
\(654\) 0 0
\(655\) −9.17414 −0.358463
\(656\) 0 0
\(657\) −55.7035 −2.17320
\(658\) 0 0
\(659\) 0.153540 0.00598106 0.00299053 0.999996i \(-0.499048\pi\)
0.00299053 + 0.999996i \(0.499048\pi\)
\(660\) 0 0
\(661\) −12.3056 −0.478632 −0.239316 0.970942i \(-0.576923\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(662\) 0 0
\(663\) 46.4811 1.80518
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 17.9167 0.693737
\(668\) 0 0
\(669\) −68.6047 −2.65241
\(670\) 0 0
\(671\) 16.4373 0.634554
\(672\) 0 0
\(673\) 4.82480 0.185983 0.0929913 0.995667i \(-0.470357\pi\)
0.0929913 + 0.995667i \(0.470357\pi\)
\(674\) 0 0
\(675\) 6.60713 0.254309
\(676\) 0 0
\(677\) −16.5041 −0.634303 −0.317151 0.948375i \(-0.602726\pi\)
−0.317151 + 0.948375i \(0.602726\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −27.1350 −1.03981
\(682\) 0 0
\(683\) −32.3621 −1.23830 −0.619152 0.785272i \(-0.712523\pi\)
−0.619152 + 0.785272i \(0.712523\pi\)
\(684\) 0 0
\(685\) −11.3137 −0.432275
\(686\) 0 0
\(687\) −28.6149 −1.09173
\(688\) 0 0
\(689\) 15.5588 0.592742
\(690\) 0 0
\(691\) 2.32446 0.0884265 0.0442132 0.999022i \(-0.485922\pi\)
0.0442132 + 0.999022i \(0.485922\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.83099 0.259114
\(696\) 0 0
\(697\) −82.2481 −3.11537
\(698\) 0 0
\(699\) 3.56272 0.134755
\(700\) 0 0
\(701\) −6.77071 −0.255726 −0.127863 0.991792i \(-0.540812\pi\)
−0.127863 + 0.991792i \(0.540812\pi\)
\(702\) 0 0
\(703\) −37.8681 −1.42822
\(704\) 0 0
\(705\) 24.0968 0.907538
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.23742 −0.234251 −0.117126 0.993117i \(-0.537368\pi\)
−0.117126 + 0.993117i \(0.537368\pi\)
\(710\) 0 0
\(711\) −53.9535 −2.02341
\(712\) 0 0
\(713\) 16.3895 0.613790
\(714\) 0 0
\(715\) −3.25586 −0.121762
\(716\) 0 0
\(717\) −3.04441 −0.113696
\(718\) 0 0
\(719\) 4.94931 0.184578 0.0922891 0.995732i \(-0.470582\pi\)
0.0922891 + 0.995732i \(0.470582\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −13.0434 −0.485091
\(724\) 0 0
\(725\) −7.12260 −0.264527
\(726\) 0 0
\(727\) −18.5847 −0.689269 −0.344635 0.938737i \(-0.611997\pi\)
−0.344635 + 0.938737i \(0.611997\pi\)
\(728\) 0 0
\(729\) −40.4306 −1.49743
\(730\) 0 0
\(731\) 24.1031 0.891484
\(732\) 0 0
\(733\) 24.5696 0.907498 0.453749 0.891130i \(-0.350086\pi\)
0.453749 + 0.891130i \(0.350086\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.39896 −0.346215
\(738\) 0 0
\(739\) 8.42950 0.310084 0.155042 0.987908i \(-0.450449\pi\)
0.155042 + 0.987908i \(0.450449\pi\)
\(740\) 0 0
\(741\) 35.1023 1.28951
\(742\) 0 0
\(743\) 11.4705 0.420811 0.210405 0.977614i \(-0.432522\pi\)
0.210405 + 0.977614i \(0.432522\pi\)
\(744\) 0 0
\(745\) 1.96243 0.0718980
\(746\) 0 0
\(747\) 86.9229 3.18034
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.3029 0.594902 0.297451 0.954737i \(-0.403863\pi\)
0.297451 + 0.954737i \(0.403863\pi\)
\(752\) 0 0
\(753\) 13.6567 0.497679
\(754\) 0 0
\(755\) −7.32511 −0.266588
\(756\) 0 0
\(757\) 25.0875 0.911821 0.455910 0.890026i \(-0.349314\pi\)
0.455910 + 0.890026i \(0.349314\pi\)
\(758\) 0 0
\(759\) 10.6185 0.385428
\(760\) 0 0
\(761\) −49.0849 −1.77933 −0.889664 0.456616i \(-0.849061\pi\)
−0.889664 + 0.456616i \(0.849061\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 38.4664 1.39076
\(766\) 0 0
\(767\) −20.1495 −0.727557
\(768\) 0 0
\(769\) 27.0548 0.975619 0.487810 0.872950i \(-0.337796\pi\)
0.487810 + 0.872950i \(0.337796\pi\)
\(770\) 0 0
\(771\) −45.8769 −1.65221
\(772\) 0 0
\(773\) 41.0828 1.47764 0.738822 0.673900i \(-0.235382\pi\)
0.738822 + 0.673900i \(0.235382\pi\)
\(774\) 0 0
\(775\) −6.51547 −0.234043
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −62.1133 −2.22544
\(780\) 0 0
\(781\) −15.4367 −0.552367
\(782\) 0 0
\(783\) −47.0600 −1.68179
\(784\) 0 0
\(785\) −7.46309 −0.266369
\(786\) 0 0
\(787\) −19.7475 −0.703921 −0.351960 0.936015i \(-0.614485\pi\)
−0.351960 + 0.936015i \(0.614485\pi\)
\(788\) 0 0
\(789\) 38.2304 1.36104
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24.9102 0.884587
\(794\) 0 0
\(795\) 20.1723 0.715439
\(796\) 0 0
\(797\) 35.9488 1.27337 0.636685 0.771124i \(-0.280305\pi\)
0.636685 + 0.771124i \(0.280305\pi\)
\(798\) 0 0
\(799\) 60.7933 2.15071
\(800\) 0 0
\(801\) −52.0430 −1.83885
\(802\) 0 0
\(803\) 15.4221 0.544234
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −71.5962 −2.52030
\(808\) 0 0
\(809\) 39.1137 1.37516 0.687582 0.726107i \(-0.258672\pi\)
0.687582 + 0.726107i \(0.258672\pi\)
\(810\) 0 0
\(811\) 43.4017 1.52404 0.762019 0.647555i \(-0.224208\pi\)
0.762019 + 0.647555i \(0.224208\pi\)
\(812\) 0 0
\(813\) −87.6330 −3.07342
\(814\) 0 0
\(815\) −12.2790 −0.430115
\(816\) 0 0
\(817\) 18.2025 0.636825
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.0135 0.454175 0.227088 0.973874i \(-0.427080\pi\)
0.227088 + 0.973874i \(0.427080\pi\)
\(822\) 0 0
\(823\) 0.0590066 0.00205684 0.00102842 0.999999i \(-0.499673\pi\)
0.00102842 + 0.999999i \(0.499673\pi\)
\(824\) 0 0
\(825\) −4.22129 −0.146967
\(826\) 0 0
\(827\) −0.519093 −0.0180506 −0.00902531 0.999959i \(-0.502873\pi\)
−0.00902531 + 0.999959i \(0.502873\pi\)
\(828\) 0 0
\(829\) −50.1574 −1.74204 −0.871019 0.491250i \(-0.836540\pi\)
−0.871019 + 0.491250i \(0.836540\pi\)
\(830\) 0 0
\(831\) −62.5808 −2.17090
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 18.2300 0.630877
\(836\) 0 0
\(837\) −43.0486 −1.48798
\(838\) 0 0
\(839\) 9.44696 0.326145 0.163073 0.986614i \(-0.447859\pi\)
0.163073 + 0.986614i \(0.447859\pi\)
\(840\) 0 0
\(841\) 21.7315 0.749360
\(842\) 0 0
\(843\) −0.822016 −0.0283117
\(844\) 0 0
\(845\) 8.06585 0.277474
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 15.8819 0.545066
\(850\) 0 0
\(851\) −17.3600 −0.595094
\(852\) 0 0
\(853\) −21.2482 −0.727525 −0.363762 0.931492i \(-0.618508\pi\)
−0.363762 + 0.931492i \(0.618508\pi\)
\(854\) 0 0
\(855\) 29.0496 0.993476
\(856\) 0 0
\(857\) −6.56435 −0.224234 −0.112117 0.993695i \(-0.535763\pi\)
−0.112117 + 0.993695i \(0.535763\pi\)
\(858\) 0 0
\(859\) 21.2506 0.725062 0.362531 0.931972i \(-0.381913\pi\)
0.362531 + 0.931972i \(0.381913\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.43459 0.0488341 0.0244170 0.999702i \(-0.492227\pi\)
0.0244170 + 0.999702i \(0.492227\pi\)
\(864\) 0 0
\(865\) 17.9245 0.609450
\(866\) 0 0
\(867\) 103.079 3.50075
\(868\) 0 0
\(869\) 14.9376 0.506723
\(870\) 0 0
\(871\) −14.2438 −0.482634
\(872\) 0 0
\(873\) 11.0872 0.375245
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.6039 0.763280 0.381640 0.924311i \(-0.375359\pi\)
0.381640 + 0.924311i \(0.375359\pi\)
\(878\) 0 0
\(879\) −50.6231 −1.70747
\(880\) 0 0
\(881\) −25.2167 −0.849572 −0.424786 0.905294i \(-0.639651\pi\)
−0.424786 + 0.905294i \(0.639651\pi\)
\(882\) 0 0
\(883\) 42.0869 1.41634 0.708168 0.706044i \(-0.249522\pi\)
0.708168 + 0.706044i \(0.249522\pi\)
\(884\) 0 0
\(885\) −26.1243 −0.878159
\(886\) 0 0
\(887\) 23.0721 0.774686 0.387343 0.921936i \(-0.373393\pi\)
0.387343 + 0.921936i \(0.373393\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.61089 −0.154471
\(892\) 0 0
\(893\) 45.9108 1.53635
\(894\) 0 0
\(895\) 0.0375672 0.00125573
\(896\) 0 0
\(897\) 16.0921 0.537298
\(898\) 0 0
\(899\) 46.4071 1.54776
\(900\) 0 0
\(901\) 50.8924 1.69547
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.82105 −0.226739
\(906\) 0 0
\(907\) −22.3923 −0.743525 −0.371763 0.928328i \(-0.621246\pi\)
−0.371763 + 0.928328i \(0.621246\pi\)
\(908\) 0 0
\(909\) 72.3481 2.39963
\(910\) 0 0
\(911\) 54.7584 1.81423 0.907114 0.420886i \(-0.138281\pi\)
0.907114 + 0.420886i \(0.138281\pi\)
\(912\) 0 0
\(913\) −24.0655 −0.796452
\(914\) 0 0
\(915\) 32.2966 1.06769
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −56.0590 −1.84922 −0.924608 0.380919i \(-0.875608\pi\)
−0.924608 + 0.380919i \(0.875608\pi\)
\(920\) 0 0
\(921\) −73.3729 −2.41772
\(922\) 0 0
\(923\) −23.3938 −0.770016
\(924\) 0 0
\(925\) 6.90131 0.226914
\(926\) 0 0
\(927\) 53.0683 1.74299
\(928\) 0 0
\(929\) −37.5879 −1.23322 −0.616610 0.787269i \(-0.711494\pi\)
−0.616610 + 0.787269i \(0.711494\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 62.1097 2.03338
\(934\) 0 0
\(935\) −10.6498 −0.348286
\(936\) 0 0
\(937\) 55.4322 1.81089 0.905446 0.424461i \(-0.139536\pi\)
0.905446 + 0.424461i \(0.139536\pi\)
\(938\) 0 0
\(939\) −8.58057 −0.280016
\(940\) 0 0
\(941\) −15.1539 −0.494003 −0.247001 0.969015i \(-0.579445\pi\)
−0.247001 + 0.969015i \(0.579445\pi\)
\(942\) 0 0
\(943\) −28.4748 −0.927269
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19.6907 0.639861 0.319930 0.947441i \(-0.396340\pi\)
0.319930 + 0.947441i \(0.396340\pi\)
\(948\) 0 0
\(949\) 23.3717 0.758677
\(950\) 0 0
\(951\) −23.0293 −0.746775
\(952\) 0 0
\(953\) 44.8194 1.45184 0.725921 0.687778i \(-0.241414\pi\)
0.725921 + 0.687778i \(0.241414\pi\)
\(954\) 0 0
\(955\) −5.86202 −0.189691
\(956\) 0 0
\(957\) 30.0666 0.971915
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 11.4513 0.369398
\(962\) 0 0
\(963\) 49.8732 1.60714
\(964\) 0 0
\(965\) 4.77168 0.153606
\(966\) 0 0
\(967\) −32.3195 −1.03932 −0.519662 0.854372i \(-0.673942\pi\)
−0.519662 + 0.854372i \(0.673942\pi\)
\(968\) 0 0
\(969\) 114.819 3.68851
\(970\) 0 0
\(971\) 12.5807 0.403733 0.201866 0.979413i \(-0.435299\pi\)
0.201866 + 0.979413i \(0.435299\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −6.39724 −0.204876
\(976\) 0 0
\(977\) 42.1169 1.34744 0.673720 0.738987i \(-0.264695\pi\)
0.673720 + 0.738987i \(0.264695\pi\)
\(978\) 0 0
\(979\) 14.4086 0.460502
\(980\) 0 0
\(981\) −47.5713 −1.51883
\(982\) 0 0
\(983\) 26.7102 0.851923 0.425962 0.904741i \(-0.359936\pi\)
0.425962 + 0.904741i \(0.359936\pi\)
\(984\) 0 0
\(985\) 9.28714 0.295913
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.34465 0.265344
\(990\) 0 0
\(991\) 7.44903 0.236626 0.118313 0.992976i \(-0.462251\pi\)
0.118313 + 0.992976i \(0.462251\pi\)
\(992\) 0 0
\(993\) −98.7730 −3.13447
\(994\) 0 0
\(995\) 3.20176 0.101503
\(996\) 0 0
\(997\) −53.9068 −1.70724 −0.853622 0.520892i \(-0.825599\pi\)
−0.853622 + 0.520892i \(0.825599\pi\)
\(998\) 0 0
\(999\) 45.5978 1.44265
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 3920.2.a.ce.1.4 4
4.3 odd 2 1960.2.a.x.1.1 4
7.6 odd 2 3920.2.a.cd.1.1 4
20.19 odd 2 9800.2.a.cs.1.4 4
28.3 even 6 1960.2.q.x.961.1 8
28.11 odd 6 1960.2.q.y.961.4 8
28.19 even 6 1960.2.q.x.361.1 8
28.23 odd 6 1960.2.q.y.361.4 8
28.27 even 2 1960.2.a.y.1.4 yes 4
140.139 even 2 9800.2.a.cl.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.a.x.1.1 4 4.3 odd 2
1960.2.a.y.1.4 yes 4 28.27 even 2
1960.2.q.x.361.1 8 28.19 even 6
1960.2.q.x.961.1 8 28.3 even 6
1960.2.q.y.361.4 8 28.23 odd 6
1960.2.q.y.961.4 8 28.11 odd 6
3920.2.a.cd.1.1 4 7.6 odd 2
3920.2.a.ce.1.4 4 1.1 even 1 trivial
9800.2.a.cl.1.1 4 140.139 even 2
9800.2.a.cs.1.4 4 20.19 odd 2