Properties

Label 8752.2.a.s.1.13
Level $8752$
Weight $2$
Character 8752.1
Self dual yes
Analytic conductor $69.885$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8752 = 2^{4} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8752.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.8850718490\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.72204\) of defining polynomial
Character \(\chi\) \(=\) 8752.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.76734 q^{3} +0.469688 q^{5} -1.03831 q^{7} +0.123492 q^{9} +O(q^{10})\) \(q+1.76734 q^{3} +0.469688 q^{5} -1.03831 q^{7} +0.123492 q^{9} +1.84194 q^{11} -0.700934 q^{13} +0.830099 q^{15} +0.793975 q^{17} -3.65845 q^{19} -1.83505 q^{21} +3.65159 q^{23} -4.77939 q^{25} -5.08377 q^{27} +5.52848 q^{29} -6.13605 q^{31} +3.25534 q^{33} -0.487683 q^{35} -2.73114 q^{37} -1.23879 q^{39} -9.69220 q^{41} +4.25340 q^{43} +0.0580027 q^{45} +5.21084 q^{47} -5.92191 q^{49} +1.40322 q^{51} -7.01107 q^{53} +0.865138 q^{55} -6.46572 q^{57} -6.98691 q^{59} -3.87686 q^{61} -0.128223 q^{63} -0.329220 q^{65} +5.10753 q^{67} +6.45360 q^{69} -1.33192 q^{71} -7.83377 q^{73} -8.44681 q^{75} -1.91251 q^{77} -4.36166 q^{79} -9.35523 q^{81} +13.3348 q^{83} +0.372921 q^{85} +9.77071 q^{87} -0.208861 q^{89} +0.727788 q^{91} -10.8445 q^{93} -1.71833 q^{95} +3.08809 q^{97} +0.227465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 10q^{3} - 27q^{5} + 11q^{7} + 14q^{9} + O(q^{10}) \) \( 18q + 10q^{3} - 27q^{5} + 11q^{7} + 14q^{9} - 2q^{11} - 25q^{13} - 9q^{15} - 30q^{17} - 4q^{19} - 16q^{21} + 26q^{23} + 31q^{25} + 37q^{27} - 18q^{29} + 5q^{31} - 10q^{33} + 9q^{35} - 18q^{37} - 7q^{39} - 17q^{41} - 8q^{43} - 44q^{45} + 52q^{47} + 29q^{49} - 19q^{51} - 60q^{53} - 11q^{55} + 4q^{57} + 8q^{59} - 26q^{61} + q^{63} - 6q^{65} - 12q^{67} - 38q^{69} + q^{71} - 2q^{73} + 17q^{75} - 73q^{77} - 18q^{79} + 18q^{81} + 43q^{83} + 51q^{85} - 3q^{87} - 28q^{89} + q^{91} - 60q^{93} + 18q^{95} - 34q^{97} - 15q^{99} + O(q^{100}) \)

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
\(3\) 1.76734 1.02037 0.510187 0.860063i \(-0.329576\pi\)
0.510187 + 0.860063i \(0.329576\pi\)
\(4\) 0 0
\(5\) 0.469688 0.210051 0.105025 0.994470i \(-0.466508\pi\)
0.105025 + 0.994470i \(0.466508\pi\)
\(6\) 0 0
\(7\) −1.03831 −0.392445 −0.196222 0.980559i \(-0.562867\pi\)
−0.196222 + 0.980559i \(0.562867\pi\)
\(8\) 0 0
\(9\) 0.123492 0.0411640
\(10\) 0 0
\(11\) 1.84194 0.555366 0.277683 0.960673i \(-0.410434\pi\)
0.277683 + 0.960673i \(0.410434\pi\)
\(12\) 0 0
\(13\) −0.700934 −0.194404 −0.0972021 0.995265i \(-0.530989\pi\)
−0.0972021 + 0.995265i \(0.530989\pi\)
\(14\) 0 0
\(15\) 0.830099 0.214331
\(16\) 0 0
\(17\) 0.793975 0.192567 0.0962837 0.995354i \(-0.469304\pi\)
0.0962837 + 0.995354i \(0.469304\pi\)
\(18\) 0 0
\(19\) −3.65845 −0.839306 −0.419653 0.907685i \(-0.637848\pi\)
−0.419653 + 0.907685i \(0.637848\pi\)
\(20\) 0 0
\(21\) −1.83505 −0.400441
\(22\) 0 0
\(23\) 3.65159 0.761409 0.380704 0.924697i \(-0.375682\pi\)
0.380704 + 0.924697i \(0.375682\pi\)
\(24\) 0 0
\(25\) −4.77939 −0.955879
\(26\) 0 0
\(27\) −5.08377 −0.978372
\(28\) 0 0
\(29\) 5.52848 1.02661 0.513307 0.858205i \(-0.328420\pi\)
0.513307 + 0.858205i \(0.328420\pi\)
\(30\) 0 0
\(31\) −6.13605 −1.10207 −0.551034 0.834483i \(-0.685766\pi\)
−0.551034 + 0.834483i \(0.685766\pi\)
\(32\) 0 0
\(33\) 3.25534 0.566681
\(34\) 0 0
\(35\) −0.487683 −0.0824334
\(36\) 0 0
\(37\) −2.73114 −0.448997 −0.224498 0.974474i \(-0.572074\pi\)
−0.224498 + 0.974474i \(0.572074\pi\)
\(38\) 0 0
\(39\) −1.23879 −0.198365
\(40\) 0 0
\(41\) −9.69220 −1.51367 −0.756833 0.653608i \(-0.773255\pi\)
−0.756833 + 0.653608i \(0.773255\pi\)
\(42\) 0 0
\(43\) 4.25340 0.648638 0.324319 0.945948i \(-0.394865\pi\)
0.324319 + 0.945948i \(0.394865\pi\)
\(44\) 0 0
\(45\) 0.0580027 0.00864653
\(46\) 0 0
\(47\) 5.21084 0.760078 0.380039 0.924970i \(-0.375910\pi\)
0.380039 + 0.924970i \(0.375910\pi\)
\(48\) 0 0
\(49\) −5.92191 −0.845987
\(50\) 0 0
\(51\) 1.40322 0.196491
\(52\) 0 0
\(53\) −7.01107 −0.963044 −0.481522 0.876434i \(-0.659916\pi\)
−0.481522 + 0.876434i \(0.659916\pi\)
\(54\) 0 0
\(55\) 0.865138 0.116655
\(56\) 0 0
\(57\) −6.46572 −0.856406
\(58\) 0 0
\(59\) −6.98691 −0.909619 −0.454809 0.890589i \(-0.650293\pi\)
−0.454809 + 0.890589i \(0.650293\pi\)
\(60\) 0 0
\(61\) −3.87686 −0.496381 −0.248191 0.968711i \(-0.579836\pi\)
−0.248191 + 0.968711i \(0.579836\pi\)
\(62\) 0 0
\(63\) −0.128223 −0.0161546
\(64\) 0 0
\(65\) −0.329220 −0.0408348
\(66\) 0 0
\(67\) 5.10753 0.623984 0.311992 0.950085i \(-0.399004\pi\)
0.311992 + 0.950085i \(0.399004\pi\)
\(68\) 0 0
\(69\) 6.45360 0.776922
\(70\) 0 0
\(71\) −1.33192 −0.158070 −0.0790350 0.996872i \(-0.525184\pi\)
−0.0790350 + 0.996872i \(0.525184\pi\)
\(72\) 0 0
\(73\) −7.83377 −0.916874 −0.458437 0.888727i \(-0.651591\pi\)
−0.458437 + 0.888727i \(0.651591\pi\)
\(74\) 0 0
\(75\) −8.44681 −0.975354
\(76\) 0 0
\(77\) −1.91251 −0.217950
\(78\) 0 0
\(79\) −4.36166 −0.490726 −0.245363 0.969431i \(-0.578907\pi\)
−0.245363 + 0.969431i \(0.578907\pi\)
\(80\) 0 0
\(81\) −9.35523 −1.03947
\(82\) 0 0
\(83\) 13.3348 1.46368 0.731842 0.681475i \(-0.238661\pi\)
0.731842 + 0.681475i \(0.238661\pi\)
\(84\) 0 0
\(85\) 0.372921 0.0404489
\(86\) 0 0
\(87\) 9.77071 1.04753
\(88\) 0 0
\(89\) −0.208861 −0.0221393 −0.0110696 0.999939i \(-0.503524\pi\)
−0.0110696 + 0.999939i \(0.503524\pi\)
\(90\) 0 0
\(91\) 0.727788 0.0762929
\(92\) 0 0
\(93\) −10.8445 −1.12452
\(94\) 0 0
\(95\) −1.71833 −0.176297
\(96\) 0 0
\(97\) 3.08809 0.313549 0.156774 0.987634i \(-0.449890\pi\)
0.156774 + 0.987634i \(0.449890\pi\)
\(98\) 0 0
\(99\) 0.227465 0.0228611
\(100\) 0 0
\(101\) −8.00261 −0.796290 −0.398145 0.917323i \(-0.630346\pi\)
−0.398145 + 0.917323i \(0.630346\pi\)
\(102\) 0 0
\(103\) 3.26470 0.321681 0.160840 0.986980i \(-0.448580\pi\)
0.160840 + 0.986980i \(0.448580\pi\)
\(104\) 0 0
\(105\) −0.861901 −0.0841129
\(106\) 0 0
\(107\) 18.4884 1.78735 0.893673 0.448719i \(-0.148120\pi\)
0.893673 + 0.448719i \(0.148120\pi\)
\(108\) 0 0
\(109\) −18.6769 −1.78893 −0.894463 0.447141i \(-0.852442\pi\)
−0.894463 + 0.447141i \(0.852442\pi\)
\(110\) 0 0
\(111\) −4.82686 −0.458145
\(112\) 0 0
\(113\) 7.85874 0.739288 0.369644 0.929173i \(-0.379480\pi\)
0.369644 + 0.929173i \(0.379480\pi\)
\(114\) 0 0
\(115\) 1.71511 0.159935
\(116\) 0 0
\(117\) −0.0865597 −0.00800245
\(118\) 0 0
\(119\) −0.824394 −0.0755720
\(120\) 0 0
\(121\) −7.60726 −0.691569
\(122\) 0 0
\(123\) −17.1294 −1.54451
\(124\) 0 0
\(125\) −4.59326 −0.410834
\(126\) 0 0
\(127\) −9.84052 −0.873205 −0.436602 0.899655i \(-0.643818\pi\)
−0.436602 + 0.899655i \(0.643818\pi\)
\(128\) 0 0
\(129\) 7.51721 0.661854
\(130\) 0 0
\(131\) −8.02091 −0.700791 −0.350395 0.936602i \(-0.613953\pi\)
−0.350395 + 0.936602i \(0.613953\pi\)
\(132\) 0 0
\(133\) 3.79861 0.329381
\(134\) 0 0
\(135\) −2.38779 −0.205508
\(136\) 0 0
\(137\) 16.8434 1.43903 0.719516 0.694476i \(-0.244364\pi\)
0.719516 + 0.694476i \(0.244364\pi\)
\(138\) 0 0
\(139\) −11.2506 −0.954260 −0.477130 0.878833i \(-0.658323\pi\)
−0.477130 + 0.878833i \(0.658323\pi\)
\(140\) 0 0
\(141\) 9.20932 0.775565
\(142\) 0 0
\(143\) −1.29108 −0.107965
\(144\) 0 0
\(145\) 2.59666 0.215641
\(146\) 0 0
\(147\) −10.4660 −0.863224
\(148\) 0 0
\(149\) −18.8662 −1.54558 −0.772788 0.634664i \(-0.781138\pi\)
−0.772788 + 0.634664i \(0.781138\pi\)
\(150\) 0 0
\(151\) 6.40292 0.521062 0.260531 0.965465i \(-0.416102\pi\)
0.260531 + 0.965465i \(0.416102\pi\)
\(152\) 0 0
\(153\) 0.0980495 0.00792684
\(154\) 0 0
\(155\) −2.88203 −0.231490
\(156\) 0 0
\(157\) −14.9254 −1.19118 −0.595589 0.803290i \(-0.703081\pi\)
−0.595589 + 0.803290i \(0.703081\pi\)
\(158\) 0 0
\(159\) −12.3909 −0.982665
\(160\) 0 0
\(161\) −3.79148 −0.298811
\(162\) 0 0
\(163\) 20.3834 1.59655 0.798274 0.602294i \(-0.205747\pi\)
0.798274 + 0.602294i \(0.205747\pi\)
\(164\) 0 0
\(165\) 1.52899 0.119032
\(166\) 0 0
\(167\) −15.8853 −1.22924 −0.614622 0.788822i \(-0.710691\pi\)
−0.614622 + 0.788822i \(0.710691\pi\)
\(168\) 0 0
\(169\) −12.5087 −0.962207
\(170\) 0 0
\(171\) −0.451789 −0.0345492
\(172\) 0 0
\(173\) 16.4110 1.24771 0.623854 0.781541i \(-0.285566\pi\)
0.623854 + 0.781541i \(0.285566\pi\)
\(174\) 0 0
\(175\) 4.96250 0.375130
\(176\) 0 0
\(177\) −12.3483 −0.928151
\(178\) 0 0
\(179\) 20.4269 1.52678 0.763390 0.645938i \(-0.223534\pi\)
0.763390 + 0.645938i \(0.223534\pi\)
\(180\) 0 0
\(181\) −10.3683 −0.770666 −0.385333 0.922778i \(-0.625913\pi\)
−0.385333 + 0.922778i \(0.625913\pi\)
\(182\) 0 0
\(183\) −6.85173 −0.506495
\(184\) 0 0
\(185\) −1.28279 −0.0943122
\(186\) 0 0
\(187\) 1.46246 0.106945
\(188\) 0 0
\(189\) 5.27853 0.383957
\(190\) 0 0
\(191\) 7.95483 0.575591 0.287796 0.957692i \(-0.407078\pi\)
0.287796 + 0.957692i \(0.407078\pi\)
\(192\) 0 0
\(193\) 23.9566 1.72443 0.862217 0.506539i \(-0.169075\pi\)
0.862217 + 0.506539i \(0.169075\pi\)
\(194\) 0 0
\(195\) −0.581845 −0.0416668
\(196\) 0 0
\(197\) 5.15127 0.367013 0.183506 0.983019i \(-0.441255\pi\)
0.183506 + 0.983019i \(0.441255\pi\)
\(198\) 0 0
\(199\) −1.54311 −0.109388 −0.0546940 0.998503i \(-0.517418\pi\)
−0.0546940 + 0.998503i \(0.517418\pi\)
\(200\) 0 0
\(201\) 9.02674 0.636697
\(202\) 0 0
\(203\) −5.74028 −0.402889
\(204\) 0 0
\(205\) −4.55231 −0.317947
\(206\) 0 0
\(207\) 0.450942 0.0313426
\(208\) 0 0
\(209\) −6.73864 −0.466122
\(210\) 0 0
\(211\) −8.81178 −0.606628 −0.303314 0.952891i \(-0.598093\pi\)
−0.303314 + 0.952891i \(0.598093\pi\)
\(212\) 0 0
\(213\) −2.35396 −0.161291
\(214\) 0 0
\(215\) 1.99777 0.136247
\(216\) 0 0
\(217\) 6.37113 0.432501
\(218\) 0 0
\(219\) −13.8449 −0.935555
\(220\) 0 0
\(221\) −0.556524 −0.0374359
\(222\) 0 0
\(223\) −24.2050 −1.62089 −0.810445 0.585815i \(-0.800774\pi\)
−0.810445 + 0.585815i \(0.800774\pi\)
\(224\) 0 0
\(225\) −0.590216 −0.0393478
\(226\) 0 0
\(227\) 3.55162 0.235729 0.117865 0.993030i \(-0.462395\pi\)
0.117865 + 0.993030i \(0.462395\pi\)
\(228\) 0 0
\(229\) −18.8164 −1.24342 −0.621711 0.783247i \(-0.713562\pi\)
−0.621711 + 0.783247i \(0.713562\pi\)
\(230\) 0 0
\(231\) −3.38005 −0.222391
\(232\) 0 0
\(233\) 12.1897 0.798575 0.399288 0.916826i \(-0.369258\pi\)
0.399288 + 0.916826i \(0.369258\pi\)
\(234\) 0 0
\(235\) 2.44747 0.159655
\(236\) 0 0
\(237\) −7.70855 −0.500724
\(238\) 0 0
\(239\) −30.6423 −1.98208 −0.991042 0.133553i \(-0.957361\pi\)
−0.991042 + 0.133553i \(0.957361\pi\)
\(240\) 0 0
\(241\) −7.26294 −0.467847 −0.233924 0.972255i \(-0.575157\pi\)
−0.233924 + 0.972255i \(0.575157\pi\)
\(242\) 0 0
\(243\) −1.28256 −0.0822763
\(244\) 0 0
\(245\) −2.78145 −0.177700
\(246\) 0 0
\(247\) 2.56433 0.163164
\(248\) 0 0
\(249\) 23.5671 1.49350
\(250\) 0 0
\(251\) 25.2813 1.59574 0.797871 0.602828i \(-0.205959\pi\)
0.797871 + 0.602828i \(0.205959\pi\)
\(252\) 0 0
\(253\) 6.72601 0.422860
\(254\) 0 0
\(255\) 0.659078 0.0412731
\(256\) 0 0
\(257\) −26.5017 −1.65313 −0.826564 0.562842i \(-0.809708\pi\)
−0.826564 + 0.562842i \(0.809708\pi\)
\(258\) 0 0
\(259\) 2.83578 0.176207
\(260\) 0 0
\(261\) 0.682723 0.0422595
\(262\) 0 0
\(263\) −12.5983 −0.776845 −0.388422 0.921481i \(-0.626980\pi\)
−0.388422 + 0.921481i \(0.626980\pi\)
\(264\) 0 0
\(265\) −3.29301 −0.202288
\(266\) 0 0
\(267\) −0.369129 −0.0225903
\(268\) 0 0
\(269\) 1.45914 0.0889652 0.0444826 0.999010i \(-0.485836\pi\)
0.0444826 + 0.999010i \(0.485836\pi\)
\(270\) 0 0
\(271\) −5.16205 −0.313572 −0.156786 0.987633i \(-0.550113\pi\)
−0.156786 + 0.987633i \(0.550113\pi\)
\(272\) 0 0
\(273\) 1.28625 0.0778473
\(274\) 0 0
\(275\) −8.80336 −0.530862
\(276\) 0 0
\(277\) 14.0110 0.841837 0.420918 0.907098i \(-0.361708\pi\)
0.420918 + 0.907098i \(0.361708\pi\)
\(278\) 0 0
\(279\) −0.757753 −0.0453655
\(280\) 0 0
\(281\) 16.3617 0.976059 0.488030 0.872827i \(-0.337716\pi\)
0.488030 + 0.872827i \(0.337716\pi\)
\(282\) 0 0
\(283\) −18.5182 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(284\) 0 0
\(285\) −3.03687 −0.179889
\(286\) 0 0
\(287\) 10.0635 0.594031
\(288\) 0 0
\(289\) −16.3696 −0.962918
\(290\) 0 0
\(291\) 5.45771 0.319937
\(292\) 0 0
\(293\) 30.5239 1.78323 0.891613 0.452798i \(-0.149574\pi\)
0.891613 + 0.452798i \(0.149574\pi\)
\(294\) 0 0
\(295\) −3.28167 −0.191066
\(296\) 0 0
\(297\) −9.36400 −0.543354
\(298\) 0 0
\(299\) −2.55952 −0.148021
\(300\) 0 0
\(301\) −4.41636 −0.254555
\(302\) 0 0
\(303\) −14.1433 −0.812514
\(304\) 0 0
\(305\) −1.82092 −0.104265
\(306\) 0 0
\(307\) −3.36035 −0.191785 −0.0958926 0.995392i \(-0.530571\pi\)
−0.0958926 + 0.995392i \(0.530571\pi\)
\(308\) 0 0
\(309\) 5.76984 0.328235
\(310\) 0 0
\(311\) −26.0631 −1.47791 −0.738953 0.673757i \(-0.764679\pi\)
−0.738953 + 0.673757i \(0.764679\pi\)
\(312\) 0 0
\(313\) −28.4960 −1.61069 −0.805345 0.592807i \(-0.798020\pi\)
−0.805345 + 0.592807i \(0.798020\pi\)
\(314\) 0 0
\(315\) −0.0602249 −0.00339329
\(316\) 0 0
\(317\) −8.26530 −0.464225 −0.232113 0.972689i \(-0.574564\pi\)
−0.232113 + 0.972689i \(0.574564\pi\)
\(318\) 0 0
\(319\) 10.1831 0.570146
\(320\) 0 0
\(321\) 32.6754 1.82376
\(322\) 0 0
\(323\) −2.90472 −0.161623
\(324\) 0 0
\(325\) 3.35004 0.185827
\(326\) 0 0
\(327\) −33.0085 −1.82537
\(328\) 0 0
\(329\) −5.41047 −0.298289
\(330\) 0 0
\(331\) −12.5162 −0.687951 −0.343976 0.938979i \(-0.611774\pi\)
−0.343976 + 0.938979i \(0.611774\pi\)
\(332\) 0 0
\(333\) −0.337274 −0.0184825
\(334\) 0 0
\(335\) 2.39895 0.131068
\(336\) 0 0
\(337\) −20.0461 −1.09198 −0.545990 0.837792i \(-0.683846\pi\)
−0.545990 + 0.837792i \(0.683846\pi\)
\(338\) 0 0
\(339\) 13.8891 0.754351
\(340\) 0 0
\(341\) −11.3022 −0.612051
\(342\) 0 0
\(343\) 13.4170 0.724448
\(344\) 0 0
\(345\) 3.03118 0.163193
\(346\) 0 0
\(347\) −0.748255 −0.0401684 −0.0200842 0.999798i \(-0.506393\pi\)
−0.0200842 + 0.999798i \(0.506393\pi\)
\(348\) 0 0
\(349\) 4.16941 0.223183 0.111592 0.993754i \(-0.464405\pi\)
0.111592 + 0.993754i \(0.464405\pi\)
\(350\) 0 0
\(351\) 3.56339 0.190200
\(352\) 0 0
\(353\) 28.9002 1.53820 0.769100 0.639128i \(-0.220705\pi\)
0.769100 + 0.639128i \(0.220705\pi\)
\(354\) 0 0
\(355\) −0.625588 −0.0332028
\(356\) 0 0
\(357\) −1.45698 −0.0771118
\(358\) 0 0
\(359\) −22.5238 −1.18876 −0.594380 0.804185i \(-0.702602\pi\)
−0.594380 + 0.804185i \(0.702602\pi\)
\(360\) 0 0
\(361\) −5.61576 −0.295566
\(362\) 0 0
\(363\) −13.4446 −0.705659
\(364\) 0 0
\(365\) −3.67943 −0.192590
\(366\) 0 0
\(367\) 3.07935 0.160741 0.0803705 0.996765i \(-0.474390\pi\)
0.0803705 + 0.996765i \(0.474390\pi\)
\(368\) 0 0
\(369\) −1.19691 −0.0623085
\(370\) 0 0
\(371\) 7.27967 0.377942
\(372\) 0 0
\(373\) −23.8756 −1.23623 −0.618115 0.786088i \(-0.712103\pi\)
−0.618115 + 0.786088i \(0.712103\pi\)
\(374\) 0 0
\(375\) −8.11786 −0.419205
\(376\) 0 0
\(377\) −3.87510 −0.199578
\(378\) 0 0
\(379\) −3.60479 −0.185166 −0.0925828 0.995705i \(-0.529512\pi\)
−0.0925828 + 0.995705i \(0.529512\pi\)
\(380\) 0 0
\(381\) −17.3915 −0.890996
\(382\) 0 0
\(383\) 3.63594 0.185788 0.0928939 0.995676i \(-0.470388\pi\)
0.0928939 + 0.995676i \(0.470388\pi\)
\(384\) 0 0
\(385\) −0.898282 −0.0457807
\(386\) 0 0
\(387\) 0.525261 0.0267005
\(388\) 0 0
\(389\) 21.6735 1.09889 0.549445 0.835530i \(-0.314839\pi\)
0.549445 + 0.835530i \(0.314839\pi\)
\(390\) 0 0
\(391\) 2.89927 0.146622
\(392\) 0 0
\(393\) −14.1757 −0.715069
\(394\) 0 0
\(395\) −2.04862 −0.103077
\(396\) 0 0
\(397\) 6.59119 0.330802 0.165401 0.986226i \(-0.447108\pi\)
0.165401 + 0.986226i \(0.447108\pi\)
\(398\) 0 0
\(399\) 6.71343 0.336092
\(400\) 0 0
\(401\) −4.35340 −0.217398 −0.108699 0.994075i \(-0.534668\pi\)
−0.108699 + 0.994075i \(0.534668\pi\)
\(402\) 0 0
\(403\) 4.30097 0.214246
\(404\) 0 0
\(405\) −4.39404 −0.218342
\(406\) 0 0
\(407\) −5.03060 −0.249358
\(408\) 0 0
\(409\) 31.8412 1.57445 0.787223 0.616668i \(-0.211518\pi\)
0.787223 + 0.616668i \(0.211518\pi\)
\(410\) 0 0
\(411\) 29.7681 1.46835
\(412\) 0 0
\(413\) 7.25459 0.356975
\(414\) 0 0
\(415\) 6.26319 0.307448
\(416\) 0 0
\(417\) −19.8836 −0.973703
\(418\) 0 0
\(419\) 29.8225 1.45692 0.728462 0.685087i \(-0.240236\pi\)
0.728462 + 0.685087i \(0.240236\pi\)
\(420\) 0 0
\(421\) −7.93092 −0.386530 −0.193265 0.981147i \(-0.561908\pi\)
−0.193265 + 0.981147i \(0.561908\pi\)
\(422\) 0 0
\(423\) 0.643496 0.0312879
\(424\) 0 0
\(425\) −3.79472 −0.184071
\(426\) 0 0
\(427\) 4.02539 0.194802
\(428\) 0 0
\(429\) −2.28178 −0.110165
\(430\) 0 0
\(431\) 24.9537 1.20198 0.600988 0.799258i \(-0.294774\pi\)
0.600988 + 0.799258i \(0.294774\pi\)
\(432\) 0 0
\(433\) 23.1505 1.11254 0.556272 0.831000i \(-0.312231\pi\)
0.556272 + 0.831000i \(0.312231\pi\)
\(434\) 0 0
\(435\) 4.58918 0.220035
\(436\) 0 0
\(437\) −13.3591 −0.639054
\(438\) 0 0
\(439\) −18.9654 −0.905167 −0.452584 0.891722i \(-0.649498\pi\)
−0.452584 + 0.891722i \(0.649498\pi\)
\(440\) 0 0
\(441\) −0.731308 −0.0348242
\(442\) 0 0
\(443\) −21.6203 −1.02721 −0.513606 0.858026i \(-0.671691\pi\)
−0.513606 + 0.858026i \(0.671691\pi\)
\(444\) 0 0
\(445\) −0.0980997 −0.00465037
\(446\) 0 0
\(447\) −33.3429 −1.57707
\(448\) 0 0
\(449\) 15.1299 0.714023 0.357011 0.934100i \(-0.383796\pi\)
0.357011 + 0.934100i \(0.383796\pi\)
\(450\) 0 0
\(451\) −17.8524 −0.840639
\(452\) 0 0
\(453\) 11.3161 0.531678
\(454\) 0 0
\(455\) 0.341833 0.0160254
\(456\) 0 0
\(457\) −7.87882 −0.368556 −0.184278 0.982874i \(-0.558995\pi\)
−0.184278 + 0.982874i \(0.558995\pi\)
\(458\) 0 0
\(459\) −4.03639 −0.188402
\(460\) 0 0
\(461\) −33.2729 −1.54967 −0.774836 0.632162i \(-0.782168\pi\)
−0.774836 + 0.632162i \(0.782168\pi\)
\(462\) 0 0
\(463\) −31.9493 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(464\) 0 0
\(465\) −5.09353 −0.236207
\(466\) 0 0
\(467\) −6.03218 −0.279136 −0.139568 0.990213i \(-0.544571\pi\)
−0.139568 + 0.990213i \(0.544571\pi\)
\(468\) 0 0
\(469\) −5.30320 −0.244879
\(470\) 0 0
\(471\) −26.3783 −1.21545
\(472\) 0 0
\(473\) 7.83451 0.360231
\(474\) 0 0
\(475\) 17.4852 0.802274
\(476\) 0 0
\(477\) −0.865810 −0.0396427
\(478\) 0 0
\(479\) −18.1368 −0.828693 −0.414347 0.910119i \(-0.635990\pi\)
−0.414347 + 0.910119i \(0.635990\pi\)
\(480\) 0 0
\(481\) 1.91435 0.0872869
\(482\) 0 0
\(483\) −6.70084 −0.304899
\(484\) 0 0
\(485\) 1.45044 0.0658612
\(486\) 0 0
\(487\) 23.3591 1.05850 0.529251 0.848465i \(-0.322473\pi\)
0.529251 + 0.848465i \(0.322473\pi\)
\(488\) 0 0
\(489\) 36.0243 1.62908
\(490\) 0 0
\(491\) 4.28889 0.193555 0.0967775 0.995306i \(-0.469146\pi\)
0.0967775 + 0.995306i \(0.469146\pi\)
\(492\) 0 0
\(493\) 4.38948 0.197692
\(494\) 0 0
\(495\) 0.106838 0.00480199
\(496\) 0 0
\(497\) 1.38295 0.0620338
\(498\) 0 0
\(499\) 12.0079 0.537549 0.268775 0.963203i \(-0.413381\pi\)
0.268775 + 0.963203i \(0.413381\pi\)
\(500\) 0 0
\(501\) −28.0748 −1.25429
\(502\) 0 0
\(503\) 22.0435 0.982872 0.491436 0.870914i \(-0.336472\pi\)
0.491436 + 0.870914i \(0.336472\pi\)
\(504\) 0 0
\(505\) −3.75873 −0.167261
\(506\) 0 0
\(507\) −22.1071 −0.981811
\(508\) 0 0
\(509\) −11.9642 −0.530302 −0.265151 0.964207i \(-0.585422\pi\)
−0.265151 + 0.964207i \(0.585422\pi\)
\(510\) 0 0
\(511\) 8.13390 0.359822
\(512\) 0 0
\(513\) 18.5987 0.821153
\(514\) 0 0
\(515\) 1.53339 0.0675693
\(516\) 0 0
\(517\) 9.59805 0.422122
\(518\) 0 0
\(519\) 29.0039 1.27313
\(520\) 0 0
\(521\) −5.47217 −0.239740 −0.119870 0.992790i \(-0.538248\pi\)
−0.119870 + 0.992790i \(0.538248\pi\)
\(522\) 0 0
\(523\) 31.1363 1.36149 0.680747 0.732519i \(-0.261655\pi\)
0.680747 + 0.732519i \(0.261655\pi\)
\(524\) 0 0
\(525\) 8.77042 0.382773
\(526\) 0 0
\(527\) −4.87187 −0.212222
\(528\) 0 0
\(529\) −9.66591 −0.420257
\(530\) 0 0
\(531\) −0.862827 −0.0374435
\(532\) 0 0
\(533\) 6.79359 0.294263
\(534\) 0 0
\(535\) 8.68380 0.375434
\(536\) 0 0
\(537\) 36.1013 1.55789
\(538\) 0 0
\(539\) −10.9078 −0.469832
\(540\) 0 0
\(541\) −15.0578 −0.647387 −0.323694 0.946162i \(-0.604925\pi\)
−0.323694 + 0.946162i \(0.604925\pi\)
\(542\) 0 0
\(543\) −18.3242 −0.786368
\(544\) 0 0
\(545\) −8.77234 −0.375766
\(546\) 0 0
\(547\) 1.00000 0.0427569
\(548\) 0 0
\(549\) −0.478761 −0.0204330
\(550\) 0 0
\(551\) −20.2257 −0.861642
\(552\) 0 0
\(553\) 4.52877 0.192583
\(554\) 0 0
\(555\) −2.26712 −0.0962338
\(556\) 0 0
\(557\) 23.5488 0.997796 0.498898 0.866661i \(-0.333738\pi\)
0.498898 + 0.866661i \(0.333738\pi\)
\(558\) 0 0
\(559\) −2.98136 −0.126098
\(560\) 0 0
\(561\) 2.58466 0.109124
\(562\) 0 0
\(563\) 29.0653 1.22496 0.612478 0.790488i \(-0.290173\pi\)
0.612478 + 0.790488i \(0.290173\pi\)
\(564\) 0 0
\(565\) 3.69116 0.155288
\(566\) 0 0
\(567\) 9.71364 0.407934
\(568\) 0 0
\(569\) −0.227367 −0.00953174 −0.00476587 0.999989i \(-0.501517\pi\)
−0.00476587 + 0.999989i \(0.501517\pi\)
\(570\) 0 0
\(571\) −0.874566 −0.0365994 −0.0182997 0.999833i \(-0.505825\pi\)
−0.0182997 + 0.999833i \(0.505825\pi\)
\(572\) 0 0
\(573\) 14.0589 0.587319
\(574\) 0 0
\(575\) −17.4524 −0.727814
\(576\) 0 0
\(577\) −29.2997 −1.21976 −0.609882 0.792492i \(-0.708783\pi\)
−0.609882 + 0.792492i \(0.708783\pi\)
\(578\) 0 0
\(579\) 42.3395 1.75957
\(580\) 0 0
\(581\) −13.8457 −0.574415
\(582\) 0 0
\(583\) −12.9140 −0.534842
\(584\) 0 0
\(585\) −0.0406561 −0.00168092
\(586\) 0 0
\(587\) 10.9789 0.453149 0.226574 0.973994i \(-0.427247\pi\)
0.226574 + 0.973994i \(0.427247\pi\)
\(588\) 0 0
\(589\) 22.4484 0.924971
\(590\) 0 0
\(591\) 9.10404 0.374490
\(592\) 0 0
\(593\) −42.7926 −1.75728 −0.878641 0.477483i \(-0.841549\pi\)
−0.878641 + 0.477483i \(0.841549\pi\)
\(594\) 0 0
\(595\) −0.387208 −0.0158740
\(596\) 0 0
\(597\) −2.72720 −0.111617
\(598\) 0 0
\(599\) −0.841737 −0.0343925 −0.0171962 0.999852i \(-0.505474\pi\)
−0.0171962 + 0.999852i \(0.505474\pi\)
\(600\) 0 0
\(601\) −19.7479 −0.805536 −0.402768 0.915302i \(-0.631952\pi\)
−0.402768 + 0.915302i \(0.631952\pi\)
\(602\) 0 0
\(603\) 0.630738 0.0256857
\(604\) 0 0
\(605\) −3.57304 −0.145265
\(606\) 0 0
\(607\) 0.928802 0.0376989 0.0188495 0.999822i \(-0.494000\pi\)
0.0188495 + 0.999822i \(0.494000\pi\)
\(608\) 0 0
\(609\) −10.1450 −0.411098
\(610\) 0 0
\(611\) −3.65245 −0.147762
\(612\) 0 0
\(613\) 20.1162 0.812485 0.406242 0.913765i \(-0.366839\pi\)
0.406242 + 0.913765i \(0.366839\pi\)
\(614\) 0 0
\(615\) −8.04548 −0.324425
\(616\) 0 0
\(617\) −13.2513 −0.533479 −0.266739 0.963769i \(-0.585946\pi\)
−0.266739 + 0.963769i \(0.585946\pi\)
\(618\) 0 0
\(619\) 29.2870 1.17714 0.588572 0.808445i \(-0.299690\pi\)
0.588572 + 0.808445i \(0.299690\pi\)
\(620\) 0 0
\(621\) −18.5638 −0.744941
\(622\) 0 0
\(623\) 0.216863 0.00868844
\(624\) 0 0
\(625\) 21.7396 0.869583
\(626\) 0 0
\(627\) −11.9095 −0.475619
\(628\) 0 0
\(629\) −2.16846 −0.0864621
\(630\) 0 0
\(631\) 46.5766 1.85418 0.927092 0.374834i \(-0.122300\pi\)
0.927092 + 0.374834i \(0.122300\pi\)
\(632\) 0 0
\(633\) −15.5734 −0.618987
\(634\) 0 0
\(635\) −4.62197 −0.183417
\(636\) 0 0
\(637\) 4.15087 0.164463
\(638\) 0 0
\(639\) −0.164482 −0.00650679
\(640\) 0 0
\(641\) −5.11165 −0.201898 −0.100949 0.994892i \(-0.532188\pi\)
−0.100949 + 0.994892i \(0.532188\pi\)
\(642\) 0 0
\(643\) 4.32994 0.170756 0.0853781 0.996349i \(-0.472790\pi\)
0.0853781 + 0.996349i \(0.472790\pi\)
\(644\) 0 0
\(645\) 3.53074 0.139023
\(646\) 0 0
\(647\) 12.0401 0.473343 0.236672 0.971590i \(-0.423943\pi\)
0.236672 + 0.971590i \(0.423943\pi\)
\(648\) 0 0
\(649\) −12.8695 −0.505171
\(650\) 0 0
\(651\) 11.2600 0.441313
\(652\) 0 0
\(653\) 39.4366 1.54327 0.771636 0.636064i \(-0.219439\pi\)
0.771636 + 0.636064i \(0.219439\pi\)
\(654\) 0 0
\(655\) −3.76733 −0.147202
\(656\) 0 0
\(657\) −0.967408 −0.0377422
\(658\) 0 0
\(659\) −43.8625 −1.70864 −0.854320 0.519748i \(-0.826026\pi\)
−0.854320 + 0.519748i \(0.826026\pi\)
\(660\) 0 0
\(661\) −5.18941 −0.201845 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(662\) 0 0
\(663\) −0.983568 −0.0381986
\(664\) 0 0
\(665\) 1.78416 0.0691868
\(666\) 0 0
\(667\) 20.1877 0.781672
\(668\) 0 0
\(669\) −42.7785 −1.65391
\(670\) 0 0
\(671\) −7.14094 −0.275673
\(672\) 0 0
\(673\) 37.7606 1.45557 0.727783 0.685808i \(-0.240551\pi\)
0.727783 + 0.685808i \(0.240551\pi\)
\(674\) 0 0
\(675\) 24.2973 0.935205
\(676\) 0 0
\(677\) −36.6368 −1.40806 −0.704032 0.710168i \(-0.748619\pi\)
−0.704032 + 0.710168i \(0.748619\pi\)
\(678\) 0 0
\(679\) −3.20640 −0.123050
\(680\) 0 0
\(681\) 6.27692 0.240532
\(682\) 0 0
\(683\) −31.5096 −1.20568 −0.602841 0.797861i \(-0.705965\pi\)
−0.602841 + 0.797861i \(0.705965\pi\)
\(684\) 0 0
\(685\) 7.91116 0.302270
\(686\) 0 0
\(687\) −33.2549 −1.26876
\(688\) 0 0
\(689\) 4.91430 0.187220
\(690\) 0 0
\(691\) −18.9407 −0.720540 −0.360270 0.932848i \(-0.617315\pi\)
−0.360270 + 0.932848i \(0.617315\pi\)
\(692\) 0 0
\(693\) −0.236179 −0.00897171
\(694\) 0 0
\(695\) −5.28426 −0.200443
\(696\) 0 0
\(697\) −7.69537 −0.291483
\(698\) 0 0
\(699\) 21.5434 0.814846
\(700\) 0 0
\(701\) 42.8730 1.61929 0.809646 0.586919i \(-0.199659\pi\)
0.809646 + 0.586919i \(0.199659\pi\)
\(702\) 0 0
\(703\) 9.99174 0.376846
\(704\) 0 0
\(705\) 4.32551 0.162908
\(706\) 0 0
\(707\) 8.30921 0.312500
\(708\) 0 0
\(709\) 35.4811 1.33252 0.666260 0.745719i \(-0.267894\pi\)
0.666260 + 0.745719i \(0.267894\pi\)
\(710\) 0 0
\(711\) −0.538630 −0.0202002
\(712\) 0 0
\(713\) −22.4063 −0.839124
\(714\) 0 0
\(715\) −0.606404 −0.0226782
\(716\) 0 0
\(717\) −54.1553 −2.02247
\(718\) 0 0
\(719\) 24.4790 0.912913 0.456456 0.889746i \(-0.349118\pi\)
0.456456 + 0.889746i \(0.349118\pi\)
\(720\) 0 0
\(721\) −3.38978 −0.126242
\(722\) 0 0
\(723\) −12.8361 −0.477379
\(724\) 0 0
\(725\) −26.4228 −0.981317
\(726\) 0 0
\(727\) 28.7796 1.06737 0.533687 0.845682i \(-0.320806\pi\)
0.533687 + 0.845682i \(0.320806\pi\)
\(728\) 0 0
\(729\) 25.7990 0.955517
\(730\) 0 0
\(731\) 3.37710 0.124906
\(732\) 0 0
\(733\) 23.0494 0.851350 0.425675 0.904876i \(-0.360037\pi\)
0.425675 + 0.904876i \(0.360037\pi\)
\(734\) 0 0
\(735\) −4.91577 −0.181321
\(736\) 0 0
\(737\) 9.40776 0.346539
\(738\) 0 0
\(739\) 9.02906 0.332139 0.166070 0.986114i \(-0.446892\pi\)
0.166070 + 0.986114i \(0.446892\pi\)
\(740\) 0 0
\(741\) 4.53205 0.166489
\(742\) 0 0
\(743\) 47.5028 1.74271 0.871355 0.490653i \(-0.163242\pi\)
0.871355 + 0.490653i \(0.163242\pi\)
\(744\) 0 0
\(745\) −8.86121 −0.324650
\(746\) 0 0
\(747\) 1.64674 0.0602510
\(748\) 0 0
\(749\) −19.1968 −0.701434
\(750\) 0 0
\(751\) 20.5388 0.749473 0.374737 0.927131i \(-0.377733\pi\)
0.374737 + 0.927131i \(0.377733\pi\)
\(752\) 0 0
\(753\) 44.6807 1.62825
\(754\) 0 0
\(755\) 3.00737 0.109450
\(756\) 0 0
\(757\) 36.3077 1.31962 0.659812 0.751430i \(-0.270636\pi\)
0.659812 + 0.751430i \(0.270636\pi\)
\(758\) 0 0
\(759\) 11.8871 0.431476
\(760\) 0 0
\(761\) 23.0597 0.835915 0.417957 0.908467i \(-0.362746\pi\)
0.417957 + 0.908467i \(0.362746\pi\)
\(762\) 0 0
\(763\) 19.3925 0.702055
\(764\) 0 0
\(765\) 0.0460527 0.00166504
\(766\) 0 0
\(767\) 4.89737 0.176834
\(768\) 0 0
\(769\) 4.40403 0.158813 0.0794066 0.996842i \(-0.474697\pi\)
0.0794066 + 0.996842i \(0.474697\pi\)
\(770\) 0 0
\(771\) −46.8375 −1.68681
\(772\) 0 0
\(773\) −25.6259 −0.921699 −0.460849 0.887478i \(-0.652455\pi\)
−0.460849 + 0.887478i \(0.652455\pi\)
\(774\) 0 0
\(775\) 29.3266 1.05344
\(776\) 0 0
\(777\) 5.01178 0.179797
\(778\) 0 0
\(779\) 35.4584 1.27043
\(780\) 0 0
\(781\) −2.45332 −0.0877867
\(782\) 0 0
\(783\) −28.1055 −1.00441
\(784\) 0 0
\(785\) −7.01029 −0.250208
\(786\) 0 0
\(787\) −46.4038 −1.65412 −0.827059 0.562115i \(-0.809988\pi\)
−0.827059 + 0.562115i \(0.809988\pi\)
\(788\) 0 0
\(789\) −22.2655 −0.792673
\(790\) 0 0
\(791\) −8.15982 −0.290130
\(792\) 0 0
\(793\) 2.71742 0.0964985
\(794\) 0 0
\(795\) −5.81988 −0.206410
\(796\) 0 0
\(797\) −33.6692 −1.19262 −0.596312 0.802753i \(-0.703368\pi\)
−0.596312 + 0.802753i \(0.703368\pi\)
\(798\) 0 0
\(799\) 4.13727 0.146366
\(800\) 0 0
\(801\) −0.0257927 −0.000911340 0
\(802\) 0 0
\(803\) −14.4293 −0.509200
\(804\) 0 0
\(805\) −1.78082 −0.0627655
\(806\) 0 0
\(807\) 2.57879 0.0907778
\(808\) 0 0
\(809\) 32.2939 1.13539 0.567696 0.823238i \(-0.307835\pi\)
0.567696 + 0.823238i \(0.307835\pi\)
\(810\) 0 0
\(811\) −27.4278 −0.963119 −0.481559 0.876413i \(-0.659929\pi\)
−0.481559 + 0.876413i \(0.659929\pi\)
\(812\) 0 0
\(813\) −9.12310 −0.319961
\(814\) 0 0
\(815\) 9.57382 0.335356
\(816\) 0 0
\(817\) −15.5609 −0.544405
\(818\) 0 0
\(819\) 0.0898759 0.00314052
\(820\) 0 0
\(821\) −17.5646 −0.613008 −0.306504 0.951869i \(-0.599159\pi\)
−0.306504 + 0.951869i \(0.599159\pi\)
\(822\) 0 0
\(823\) 7.94705 0.277017 0.138508 0.990361i \(-0.455769\pi\)
0.138508 + 0.990361i \(0.455769\pi\)
\(824\) 0 0
\(825\) −15.5585 −0.541678
\(826\) 0 0
\(827\) 5.16367 0.179558 0.0897792 0.995962i \(-0.471384\pi\)
0.0897792 + 0.995962i \(0.471384\pi\)
\(828\) 0 0
\(829\) −20.0606 −0.696734 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(830\) 0 0
\(831\) 24.7621 0.858989
\(832\) 0 0
\(833\) −4.70185 −0.162909
\(834\) 0 0
\(835\) −7.46115 −0.258204
\(836\) 0 0
\(837\) 31.1943 1.07823
\(838\) 0 0
\(839\) −39.3019 −1.35685 −0.678426 0.734669i \(-0.737338\pi\)
−0.678426 + 0.734669i \(0.737338\pi\)
\(840\) 0 0
\(841\) 1.56409 0.0539343
\(842\) 0 0
\(843\) 28.9168 0.995946
\(844\) 0 0
\(845\) −5.87518 −0.202112
\(846\) 0 0
\(847\) 7.89870 0.271403
\(848\) 0 0
\(849\) −32.7280 −1.12322
\(850\) 0 0
\(851\) −9.97300 −0.341870
\(852\) 0 0
\(853\) 41.2623 1.41280 0.706398 0.707815i \(-0.250319\pi\)
0.706398 + 0.707815i \(0.250319\pi\)
\(854\) 0 0
\(855\) −0.212200 −0.00725708
\(856\) 0 0
\(857\) 4.93880 0.168706 0.0843532 0.996436i \(-0.473118\pi\)
0.0843532 + 0.996436i \(0.473118\pi\)
\(858\) 0 0
\(859\) 38.1044 1.30011 0.650053 0.759889i \(-0.274747\pi\)
0.650053 + 0.759889i \(0.274747\pi\)
\(860\) 0 0
\(861\) 17.7857 0.606134
\(862\) 0 0
\(863\) 1.71273 0.0583019 0.0291510 0.999575i \(-0.490720\pi\)
0.0291510 + 0.999575i \(0.490720\pi\)
\(864\) 0 0
\(865\) 7.70808 0.262082
\(866\) 0 0
\(867\) −28.9307 −0.982537
\(868\) 0 0
\(869\) −8.03393 −0.272532
\(870\) 0 0
\(871\) −3.58004 −0.121305
\(872\) 0 0
\(873\) 0.381355 0.0129069
\(874\) 0 0
\(875\) 4.76924 0.161230
\(876\) 0 0
\(877\) 19.2114 0.648724 0.324362 0.945933i \(-0.394850\pi\)
0.324362 + 0.945933i \(0.394850\pi\)
\(878\) 0 0
\(879\) 53.9462 1.81956
\(880\) 0 0
\(881\) 0.0458679 0.00154533 0.000772664 1.00000i \(-0.499754\pi\)
0.000772664 1.00000i \(0.499754\pi\)
\(882\) 0 0
\(883\) −42.1763 −1.41935 −0.709674 0.704531i \(-0.751158\pi\)
−0.709674 + 0.704531i \(0.751158\pi\)
\(884\) 0 0
\(885\) −5.79983 −0.194959
\(886\) 0 0
\(887\) −36.1371 −1.21337 −0.606683 0.794944i \(-0.707500\pi\)
−0.606683 + 0.794944i \(0.707500\pi\)
\(888\) 0 0
\(889\) 10.2175 0.342685
\(890\) 0 0
\(891\) −17.2318 −0.577286
\(892\) 0 0
\(893\) −19.0636 −0.637938
\(894\) 0 0
\(895\) 9.59428 0.320701
\(896\) 0 0
\(897\) −4.52355 −0.151037
\(898\) 0 0
\(899\) −33.9230 −1.13140
\(900\) 0 0
\(901\) −5.56661 −0.185451
\(902\) 0 0
\(903\) −7.80520 −0.259741
\(904\) 0 0
\(905\) −4.86985 −0.161879
\(906\) 0 0
\(907\) 21.2700 0.706259 0.353129 0.935574i \(-0.385118\pi\)
0.353129 + 0.935574i \(0.385118\pi\)
\(908\) 0 0
\(909\) −0.988258 −0.0327785
\(910\) 0 0
\(911\) 10.2529 0.339694 0.169847 0.985470i \(-0.445673\pi\)
0.169847 + 0.985470i \(0.445673\pi\)
\(912\) 0 0
\(913\) 24.5619 0.812880
\(914\) 0 0
\(915\) −3.21818 −0.106390
\(916\) 0 0
\(917\) 8.32821 0.275022
\(918\) 0 0
\(919\) 39.3681 1.29864 0.649318 0.760517i \(-0.275055\pi\)
0.649318 + 0.760517i \(0.275055\pi\)
\(920\) 0 0
\(921\) −5.93888 −0.195693
\(922\) 0 0
\(923\) 0.933589 0.0307295
\(924\) 0 0
\(925\) 13.0532 0.429187
\(926\) 0 0
\(927\) 0.403164 0.0132416
\(928\) 0 0
\(929\) 48.6956 1.59765 0.798826 0.601562i \(-0.205455\pi\)
0.798826 + 0.601562i \(0.205455\pi\)
\(930\) 0 0
\(931\) 21.6650 0.710042
\(932\) 0 0
\(933\) −46.0625 −1.50802
\(934\) 0 0
\(935\) 0.686898 0.0224640
\(936\) 0 0
\(937\) −12.4806 −0.407723 −0.203862 0.979000i \(-0.565349\pi\)
−0.203862 + 0.979000i \(0.565349\pi\)
\(938\) 0 0
\(939\) −50.3621 −1.64351
\(940\) 0 0
\(941\) −44.4531 −1.44913 −0.724565 0.689207i \(-0.757959\pi\)
−0.724565 + 0.689207i \(0.757959\pi\)
\(942\) 0 0
\(943\) −35.3919 −1.15252
\(944\) 0 0
\(945\) 2.47927 0.0806505
\(946\) 0 0
\(947\) 7.49160 0.243444 0.121722 0.992564i \(-0.461158\pi\)
0.121722 + 0.992564i \(0.461158\pi\)
\(948\) 0 0
\(949\) 5.49096 0.178244
\(950\) 0 0
\(951\) −14.6076 −0.473684
\(952\) 0 0
\(953\) −14.6341 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(954\) 0 0
\(955\) 3.73629 0.120903
\(956\) 0 0
\(957\) 17.9971 0.581762
\(958\) 0 0
\(959\) −17.4887 −0.564741
\(960\) 0 0
\(961\) 6.65114 0.214553
\(962\) 0 0
\(963\) 2.28317 0.0735743
\(964\) 0 0
\(965\) 11.2521 0.362219
\(966\) 0 0
\(967\) −1.24602 −0.0400692 −0.0200346 0.999799i \(-0.506378\pi\)
−0.0200346 + 0.999799i \(0.506378\pi\)
\(968\) 0 0
\(969\) −5.13362 −0.164916
\(970\) 0 0
\(971\) −15.4693 −0.496434 −0.248217 0.968704i \(-0.579845\pi\)
−0.248217 + 0.968704i \(0.579845\pi\)
\(972\) 0 0
\(973\) 11.6816 0.374494
\(974\) 0 0
\(975\) 5.92066 0.189613
\(976\) 0 0
\(977\) 29.2177 0.934757 0.467379 0.884057i \(-0.345199\pi\)
0.467379 + 0.884057i \(0.345199\pi\)
\(978\) 0 0
\(979\) −0.384710 −0.0122954
\(980\) 0 0
\(981\) −2.30645 −0.0736393
\(982\) 0 0
\(983\) 2.39930 0.0765259 0.0382629 0.999268i \(-0.487818\pi\)
0.0382629 + 0.999268i \(0.487818\pi\)
\(984\) 0 0
\(985\) 2.41949 0.0770913
\(986\) 0 0
\(987\) −9.56214 −0.304366
\(988\) 0 0
\(989\) 15.5317 0.493878
\(990\) 0 0
\(991\) −51.4773 −1.63523 −0.817614 0.575766i \(-0.804704\pi\)
−0.817614 + 0.575766i \(0.804704\pi\)
\(992\) 0 0
\(993\) −22.1203 −0.701968
\(994\) 0 0
\(995\) −0.724779 −0.0229771
\(996\) 0 0
\(997\) −41.2171 −1.30536 −0.652680 0.757634i \(-0.726355\pi\)
−0.652680 + 0.757634i \(0.726355\pi\)
\(998\) 0 0
\(999\) 13.8845 0.439286
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 8752.2.a.s.1.13 18
4.3 odd 2 547.2.a.b.1.1 18
12.11 even 2 4923.2.a.l.1.18 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.1 18 4.3 odd 2
4923.2.a.l.1.18 18 12.11 even 2
8752.2.a.s.1.13 18 1.1 even 1 trivial