Properties

Label 4923.2.a.l.1.18
Level $4923$
Weight $2$
Character 4923.1
Self dual yes
Analytic conductor $39.310$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4923 = 3^{2} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4923.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.3103529151\)
Analytic rank: \(0\)
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 1.18
Root \(2.72204\) of defining polynomial
Character \(\chi\) \(=\) 4923.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.72204 q^{2} +5.40952 q^{4} -0.469688 q^{5} +1.03831 q^{7} +9.28087 q^{8} +O(q^{10})\) \(q+2.72204 q^{2} +5.40952 q^{4} -0.469688 q^{5} +1.03831 q^{7} +9.28087 q^{8} -1.27851 q^{10} +1.84194 q^{11} -0.700934 q^{13} +2.82633 q^{14} +14.4439 q^{16} -0.793975 q^{17} +3.65845 q^{19} -2.54079 q^{20} +5.01384 q^{22} +3.65159 q^{23} -4.77939 q^{25} -1.90797 q^{26} +5.61677 q^{28} -5.52848 q^{29} +6.13605 q^{31} +20.7552 q^{32} -2.16124 q^{34} -0.487683 q^{35} -2.73114 q^{37} +9.95846 q^{38} -4.35911 q^{40} +9.69220 q^{41} -4.25340 q^{43} +9.96402 q^{44} +9.93978 q^{46} +5.21084 q^{47} -5.92191 q^{49} -13.0097 q^{50} -3.79172 q^{52} +7.01107 q^{53} -0.865138 q^{55} +9.63643 q^{56} -15.0488 q^{58} -6.98691 q^{59} -3.87686 q^{61} +16.7026 q^{62} +27.6087 q^{64} +0.329220 q^{65} -5.10753 q^{67} -4.29503 q^{68} -1.32749 q^{70} -1.33192 q^{71} -7.83377 q^{73} -7.43429 q^{74} +19.7905 q^{76} +1.91251 q^{77} +4.36166 q^{79} -6.78412 q^{80} +26.3826 q^{82} +13.3348 q^{83} +0.372921 q^{85} -11.5779 q^{86} +17.0948 q^{88} +0.208861 q^{89} -0.727788 q^{91} +19.7533 q^{92} +14.1841 q^{94} -1.71833 q^{95} +3.08809 q^{97} -16.1197 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 4q^{2} + 16q^{4} + 27q^{5} - 11q^{7} + 12q^{8} + O(q^{10}) \) \( 18q + 4q^{2} + 16q^{4} + 27q^{5} - 11q^{7} + 12q^{8} - 5q^{10} - 2q^{11} - 25q^{13} + 7q^{14} + 8q^{16} + 30q^{17} + 4q^{19} + 41q^{20} - 24q^{22} + 26q^{23} + 31q^{25} + 18q^{26} - 16q^{28} + 18q^{29} - 5q^{31} + 28q^{32} + 5q^{34} + 9q^{35} - 18q^{37} + 45q^{38} + 7q^{40} + 17q^{41} + 8q^{43} - 12q^{44} + 30q^{46} + 52q^{47} + 29q^{49} - 13q^{50} - 14q^{52} + 60q^{53} + 11q^{55} - 7q^{56} + 14q^{58} + 8q^{59} - 26q^{61} - 4q^{62} + 44q^{64} + 6q^{65} + 12q^{67} + 61q^{68} + 35q^{70} + q^{71} - 2q^{73} - 16q^{74} + 66q^{76} + 73q^{77} + 18q^{79} + 32q^{80} + 44q^{82} + 43q^{83} + 51q^{85} - 4q^{86} - 17q^{88} + 28q^{89} - q^{91} + 68q^{92} + 78q^{94} + 18q^{95} - 34q^{97} - 34q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72204 1.92478 0.962388 0.271679i \(-0.0875790\pi\)
0.962388 + 0.271679i \(0.0875790\pi\)
\(3\) 0 0
\(4\) 5.40952 2.70476
\(5\) −0.469688 −0.210051 −0.105025 0.994470i \(-0.533492\pi\)
−0.105025 + 0.994470i \(0.533492\pi\)
\(6\) 0 0
\(7\) 1.03831 0.392445 0.196222 0.980559i \(-0.437133\pi\)
0.196222 + 0.980559i \(0.437133\pi\)
\(8\) 9.28087 3.28128
\(9\) 0 0
\(10\) −1.27851 −0.404301
\(11\) 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 \(0.530989\pi\)
\(14\) 2.82633 0.755368
\(15\) 0 0
\(16\) 14.4439 3.61097
\(17\) −0.793975 −0.192567 −0.0962837 0.995354i \(-0.530696\pi\)
−0.0962837 + 0.995354i \(0.530696\pi\)
\(18\) 0 0
\(19\) 3.65845 0.839306 0.419653 0.907685i \(-0.362152\pi\)
0.419653 + 0.907685i \(0.362152\pi\)
\(20\) −2.54079 −0.568138
\(21\) 0 0
\(22\) 5.01384 1.06895
\(23\) 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
\(26\) −1.90797 −0.374184
\(27\) 0 0
\(28\) 5.61677 1.06147
\(29\) −5.52848 −1.02661 −0.513307 0.858205i \(-0.671580\pi\)
−0.513307 + 0.858205i \(0.671580\pi\)
\(30\) 0 0
\(31\) 6.13605 1.10207 0.551034 0.834483i \(-0.314234\pi\)
0.551034 + 0.834483i \(0.314234\pi\)
\(32\) 20.7552 3.66903
\(33\) 0 0
\(34\) −2.16124 −0.370649
\(35\) −0.487683 −0.0824334
\(36\) 0 0
\(37\) −2.73114 −0.448997 −0.224498 0.974474i \(-0.572074\pi\)
−0.224498 + 0.974474i \(0.572074\pi\)
\(38\) 9.95846 1.61547
\(39\) 0 0
\(40\) −4.35911 −0.689237
\(41\) 9.69220 1.51367 0.756833 0.653608i \(-0.226745\pi\)
0.756833 + 0.653608i \(0.226745\pi\)
\(42\) 0 0
\(43\) −4.25340 −0.648638 −0.324319 0.945948i \(-0.605135\pi\)
−0.324319 + 0.945948i \(0.605135\pi\)
\(44\) 9.96402 1.50213
\(45\) 0 0
\(46\) 9.93978 1.46554
\(47\) 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
\(50\) −13.0097 −1.83985
\(51\) 0 0
\(52\) −3.79172 −0.525817
\(53\) 7.01107 0.963044 0.481522 0.876434i \(-0.340084\pi\)
0.481522 + 0.876434i \(0.340084\pi\)
\(54\) 0 0
\(55\) −0.865138 −0.116655
\(56\) 9.63643 1.28772
\(57\) 0 0
\(58\) −15.0488 −1.97600
\(59\) −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 \(0.579836\pi\)
\(62\) 16.7026 2.12123
\(63\) 0 0
\(64\) 27.6087 3.45109
\(65\) 0.329220 0.0408348
\(66\) 0 0
\(67\) −5.10753 −0.623984 −0.311992 0.950085i \(-0.600996\pi\)
−0.311992 + 0.950085i \(0.600996\pi\)
\(68\) −4.29503 −0.520849
\(69\) 0 0
\(70\) −1.32749 −0.158666
\(71\) −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 \(0.651591\pi\)
\(74\) −7.43429 −0.864218
\(75\) 0 0
\(76\) 19.7905 2.27012
\(77\) 1.91251 0.217950
\(78\) 0 0
\(79\) 4.36166 0.490726 0.245363 0.969431i \(-0.421093\pi\)
0.245363 + 0.969431i \(0.421093\pi\)
\(80\) −6.78412 −0.758488
\(81\) 0 0
\(82\) 26.3826 2.91347
\(83\) 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
\(86\) −11.5779 −1.24848
\(87\) 0 0
\(88\) 17.0948 1.82231
\(89\) 0.208861 0.0221393 0.0110696 0.999939i \(-0.496476\pi\)
0.0110696 + 0.999939i \(0.496476\pi\)
\(90\) 0 0
\(91\) −0.727788 −0.0762929
\(92\) 19.7533 2.05943
\(93\) 0 0
\(94\) 14.1841 1.46298
\(95\) −1.71833 −0.176297
\(96\) 0 0
\(97\) 3.08809 0.313549 0.156774 0.987634i \(-0.449890\pi\)
0.156774 + 0.987634i \(0.449890\pi\)
\(98\) −16.1197 −1.62834
\(99\) 0 0
\(100\) −25.8542 −2.58542
\(101\) 8.00261 0.796290 0.398145 0.917323i \(-0.369654\pi\)
0.398145 + 0.917323i \(0.369654\pi\)
\(102\) 0 0
\(103\) −3.26470 −0.321681 −0.160840 0.986980i \(-0.551420\pi\)
−0.160840 + 0.986980i \(0.551420\pi\)
\(104\) −6.50528 −0.637895
\(105\) 0 0
\(106\) 19.0844 1.85364
\(107\) 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 \(0.852442\pi\)
\(110\) −2.35494 −0.224535
\(111\) 0 0
\(112\) 14.9973 1.41711
\(113\) −7.85874 −0.739288 −0.369644 0.929173i \(-0.620520\pi\)
−0.369644 + 0.929173i \(0.620520\pi\)
\(114\) 0 0
\(115\) −1.71511 −0.159935
\(116\) −29.9064 −2.77674
\(117\) 0 0
\(118\) −19.0187 −1.75081
\(119\) −0.824394 −0.0755720
\(120\) 0 0
\(121\) −7.60726 −0.691569
\(122\) −10.5530 −0.955422
\(123\) 0 0
\(124\) 33.1931 2.98083
\(125\) 4.59326 0.410834
\(126\) 0 0
\(127\) 9.84052 0.873205 0.436602 0.899655i \(-0.356182\pi\)
0.436602 + 0.899655i \(0.356182\pi\)
\(128\) 33.6417 2.97354
\(129\) 0 0
\(130\) 0.896152 0.0785978
\(131\) −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
\(134\) −13.9029 −1.20103
\(135\) 0 0
\(136\) −7.36878 −0.631868
\(137\) −16.8434 −1.43903 −0.719516 0.694476i \(-0.755636\pi\)
−0.719516 + 0.694476i \(0.755636\pi\)
\(138\) 0 0
\(139\) 11.2506 0.954260 0.477130 0.878833i \(-0.341677\pi\)
0.477130 + 0.878833i \(0.341677\pi\)
\(140\) −2.63813 −0.222963
\(141\) 0 0
\(142\) −3.62555 −0.304249
\(143\) −1.29108 −0.107965
\(144\) 0 0
\(145\) 2.59666 0.215641
\(146\) −21.3239 −1.76478
\(147\) 0 0
\(148\) −14.7742 −1.21443
\(149\) 18.8662 1.54558 0.772788 0.634664i \(-0.218862\pi\)
0.772788 + 0.634664i \(0.218862\pi\)
\(150\) 0 0
\(151\) −6.40292 −0.521062 −0.260531 0.965465i \(-0.583898\pi\)
−0.260531 + 0.965465i \(0.583898\pi\)
\(152\) 33.9536 2.75400
\(153\) 0 0
\(154\) 5.20593 0.419506
\(155\) −2.88203 −0.231490
\(156\) 0 0
\(157\) −14.9254 −1.19118 −0.595589 0.803290i \(-0.703081\pi\)
−0.595589 + 0.803290i \(0.703081\pi\)
\(158\) 11.8726 0.944537
\(159\) 0 0
\(160\) −9.74845 −0.770683
\(161\) 3.79148 0.298811
\(162\) 0 0
\(163\) −20.3834 −1.59655 −0.798274 0.602294i \(-0.794253\pi\)
−0.798274 + 0.602294i \(0.794253\pi\)
\(164\) 52.4302 4.09411
\(165\) 0 0
\(166\) 36.2979 2.81726
\(167\) −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
\(170\) 1.01511 0.0778551
\(171\) 0 0
\(172\) −23.0089 −1.75441
\(173\) −16.4110 −1.24771 −0.623854 0.781541i \(-0.714434\pi\)
−0.623854 + 0.781541i \(0.714434\pi\)
\(174\) 0 0
\(175\) −4.96250 −0.375130
\(176\) 26.6048 2.00541
\(177\) 0 0
\(178\) 0.568530 0.0426131
\(179\) 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 \(0.625913\pi\)
\(182\) −1.98107 −0.146847
\(183\) 0 0
\(184\) 33.8899 2.49840
\(185\) 1.28279 0.0943122
\(186\) 0 0
\(187\) −1.46246 −0.106945
\(188\) 28.1881 2.05583
\(189\) 0 0
\(190\) −4.67737 −0.339332
\(191\) 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 \(0.169075\pi\)
\(194\) 8.40593 0.603511
\(195\) 0 0
\(196\) −32.0347 −2.28819
\(197\) −5.15127 −0.367013 −0.183506 0.983019i \(-0.558745\pi\)
−0.183506 + 0.983019i \(0.558745\pi\)
\(198\) 0 0
\(199\) 1.54311 0.109388 0.0546940 0.998503i \(-0.482582\pi\)
0.0546940 + 0.998503i \(0.482582\pi\)
\(200\) −44.3569 −3.13651
\(201\) 0 0
\(202\) 21.7835 1.53268
\(203\) −5.74028 −0.402889
\(204\) 0 0
\(205\) −4.55231 −0.317947
\(206\) −8.88666 −0.619163
\(207\) 0 0
\(208\) −10.1242 −0.701988
\(209\) 6.73864 0.466122
\(210\) 0 0
\(211\) 8.81178 0.606628 0.303314 0.952891i \(-0.401907\pi\)
0.303314 + 0.952891i \(0.401907\pi\)
\(212\) 37.9265 2.60480
\(213\) 0 0
\(214\) 50.3264 3.44024
\(215\) 1.99777 0.136247
\(216\) 0 0
\(217\) 6.37113 0.432501
\(218\) −50.8395 −3.44328
\(219\) 0 0
\(220\) −4.67998 −0.315524
\(221\) 0.556524 0.0374359
\(222\) 0 0
\(223\) 24.2050 1.62089 0.810445 0.585815i \(-0.199226\pi\)
0.810445 + 0.585815i \(0.199226\pi\)
\(224\) 21.5503 1.43989
\(225\) 0 0
\(226\) −21.3918 −1.42296
\(227\) 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 \(0.713562\pi\)
\(230\) −4.66860 −0.307838
\(231\) 0 0
\(232\) −51.3091 −3.36861
\(233\) −12.1897 −0.798575 −0.399288 0.916826i \(-0.630742\pi\)
−0.399288 + 0.916826i \(0.630742\pi\)
\(234\) 0 0
\(235\) −2.44747 −0.159655
\(236\) −37.7959 −2.46030
\(237\) 0 0
\(238\) −2.24404 −0.145459
\(239\) −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 \(0.575157\pi\)
\(242\) −20.7073 −1.33111
\(243\) 0 0
\(244\) −20.9720 −1.34259
\(245\) 2.78145 0.177700
\(246\) 0 0
\(247\) −2.56433 −0.163164
\(248\) 56.9479 3.61620
\(249\) 0 0
\(250\) 12.5031 0.790764
\(251\) 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
\(254\) 26.7863 1.68072
\(255\) 0 0
\(256\) 36.3569 2.27230
\(257\) 26.5017 1.65313 0.826564 0.562842i \(-0.190292\pi\)
0.826564 + 0.562842i \(0.190292\pi\)
\(258\) 0 0
\(259\) −2.83578 −0.176207
\(260\) 1.78093 0.110448
\(261\) 0 0
\(262\) −21.8333 −1.34886
\(263\) −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
\(266\) 10.3400 0.633985
\(267\) 0 0
\(268\) −27.6293 −1.68773
\(269\) −1.45914 −0.0889652 −0.0444826 0.999010i \(-0.514164\pi\)
−0.0444826 + 0.999010i \(0.514164\pi\)
\(270\) 0 0
\(271\) 5.16205 0.313572 0.156786 0.987633i \(-0.449887\pi\)
0.156786 + 0.987633i \(0.449887\pi\)
\(272\) −11.4681 −0.695355
\(273\) 0 0
\(274\) −45.8486 −2.76981
\(275\) −8.80336 −0.530862
\(276\) 0 0
\(277\) 14.0110 0.841837 0.420918 0.907098i \(-0.361708\pi\)
0.420918 + 0.907098i \(0.361708\pi\)
\(278\) 30.6245 1.83674
\(279\) 0 0
\(280\) −4.52612 −0.270487
\(281\) −16.3617 −0.976059 −0.488030 0.872827i \(-0.662284\pi\)
−0.488030 + 0.872827i \(0.662284\pi\)
\(282\) 0 0
\(283\) 18.5182 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(284\) −7.20506 −0.427542
\(285\) 0 0
\(286\) −3.51437 −0.207809
\(287\) 10.0635 0.594031
\(288\) 0 0
\(289\) −16.3696 −0.962918
\(290\) 7.06823 0.415061
\(291\) 0 0
\(292\) −42.3770 −2.47993
\(293\) −30.5239 −1.78323 −0.891613 0.452798i \(-0.850426\pi\)
−0.891613 + 0.452798i \(0.850426\pi\)
\(294\) 0 0
\(295\) 3.28167 0.191066
\(296\) −25.3474 −1.47329
\(297\) 0 0
\(298\) 51.3545 2.97489
\(299\) −2.55952 −0.148021
\(300\) 0 0
\(301\) −4.41636 −0.254555
\(302\) −17.4290 −1.00293
\(303\) 0 0
\(304\) 52.8422 3.03071
\(305\) 1.82092 0.104265
\(306\) 0 0
\(307\) 3.36035 0.191785 0.0958926 0.995392i \(-0.469429\pi\)
0.0958926 + 0.995392i \(0.469429\pi\)
\(308\) 10.3458 0.589504
\(309\) 0 0
\(310\) −7.84501 −0.445567
\(311\) −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 \(0.798020\pi\)
\(314\) −40.6276 −2.29275
\(315\) 0 0
\(316\) 23.5945 1.32730
\(317\) 8.26530 0.464225 0.232113 0.972689i \(-0.425436\pi\)
0.232113 + 0.972689i \(0.425436\pi\)
\(318\) 0 0
\(319\) −10.1831 −0.570146
\(320\) −12.9675 −0.724904
\(321\) 0 0
\(322\) 10.3206 0.575144
\(323\) −2.90472 −0.161623
\(324\) 0 0
\(325\) 3.35004 0.185827
\(326\) −55.4844 −3.07300
\(327\) 0 0
\(328\) 89.9520 4.96677
\(329\) 5.41047 0.298289
\(330\) 0 0
\(331\) 12.5162 0.687951 0.343976 0.938979i \(-0.388226\pi\)
0.343976 + 0.938979i \(0.388226\pi\)
\(332\) 72.1348 3.95891
\(333\) 0 0
\(334\) −43.2406 −2.36602
\(335\) 2.39895 0.131068
\(336\) 0 0
\(337\) −20.0461 −1.09198 −0.545990 0.837792i \(-0.683846\pi\)
−0.545990 + 0.837792i \(0.683846\pi\)
\(338\) −34.0492 −1.85203
\(339\) 0 0
\(340\) 2.01732 0.109405
\(341\) 11.3022 0.612051
\(342\) 0 0
\(343\) −13.4170 −0.724448
\(344\) −39.4753 −2.12836
\(345\) 0 0
\(346\) −44.6716 −2.40156
\(347\) −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 \(0.464405\pi\)
\(350\) −13.5081 −0.722040
\(351\) 0 0
\(352\) 38.2298 2.03765
\(353\) −28.9002 −1.53820 −0.769100 0.639128i \(-0.779295\pi\)
−0.769100 + 0.639128i \(0.779295\pi\)
\(354\) 0 0
\(355\) 0.625588 0.0332028
\(356\) 1.12984 0.0598814
\(357\) 0 0
\(358\) 55.6030 2.93871
\(359\) −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
\(362\) −28.2228 −1.48336
\(363\) 0 0
\(364\) −3.93698 −0.206354
\(365\) 3.67943 0.192590
\(366\) 0 0
\(367\) −3.07935 −0.160741 −0.0803705 0.996765i \(-0.525610\pi\)
−0.0803705 + 0.996765i \(0.525610\pi\)
\(368\) 52.7431 2.74943
\(369\) 0 0
\(370\) 3.49180 0.181530
\(371\) 7.27967 0.377942
\(372\) 0 0
\(373\) −23.8756 −1.23623 −0.618115 0.786088i \(-0.712103\pi\)
−0.618115 + 0.786088i \(0.712103\pi\)
\(374\) −3.98087 −0.205846
\(375\) 0 0
\(376\) 48.3611 2.49403
\(377\) 3.87510 0.199578
\(378\) 0 0
\(379\) 3.60479 0.185166 0.0925828 0.995705i \(-0.470488\pi\)
0.0925828 + 0.995705i \(0.470488\pi\)
\(380\) −9.29534 −0.476841
\(381\) 0 0
\(382\) 21.6534 1.10788
\(383\) 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
\(386\) 65.2109 3.31915
\(387\) 0 0
\(388\) 16.7051 0.848074
\(389\) −21.6735 −1.09889 −0.549445 0.835530i \(-0.685161\pi\)
−0.549445 + 0.835530i \(0.685161\pi\)
\(390\) 0 0
\(391\) −2.89927 −0.146622
\(392\) −54.9605 −2.77592
\(393\) 0 0
\(394\) −14.0220 −0.706417
\(395\) −2.04862 −0.103077
\(396\) 0 0
\(397\) 6.59119 0.330802 0.165401 0.986226i \(-0.447108\pi\)
0.165401 + 0.986226i \(0.447108\pi\)
\(398\) 4.20041 0.210547
\(399\) 0 0
\(400\) −69.0330 −3.45165
\(401\) 4.35340 0.217398 0.108699 0.994075i \(-0.465332\pi\)
0.108699 + 0.994075i \(0.465332\pi\)
\(402\) 0 0
\(403\) −4.30097 −0.214246
\(404\) 43.2903 2.15377
\(405\) 0 0
\(406\) −15.6253 −0.775471
\(407\) −5.03060 −0.249358
\(408\) 0 0
\(409\) 31.8412 1.57445 0.787223 0.616668i \(-0.211518\pi\)
0.787223 + 0.616668i \(0.211518\pi\)
\(410\) −12.3916 −0.611977
\(411\) 0 0
\(412\) −17.6605 −0.870069
\(413\) −7.25459 −0.356975
\(414\) 0 0
\(415\) −6.26319 −0.307448
\(416\) −14.5480 −0.713274
\(417\) 0 0
\(418\) 18.3429 0.897180
\(419\) 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 \(0.561908\pi\)
\(422\) 23.9860 1.16762
\(423\) 0 0
\(424\) 65.0688 3.16002
\(425\) 3.79472 0.184071
\(426\) 0 0
\(427\) −4.02539 −0.194802
\(428\) 100.014 4.83434
\(429\) 0 0
\(430\) 5.43803 0.262245
\(431\) 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 \(0.312231\pi\)
\(434\) 17.3425 0.832467
\(435\) 0 0
\(436\) −101.033 −4.83862
\(437\) 13.3591 0.639054
\(438\) 0 0
\(439\) 18.9654 0.905167 0.452584 0.891722i \(-0.350502\pi\)
0.452584 + 0.891722i \(0.350502\pi\)
\(440\) −8.02923 −0.382778
\(441\) 0 0
\(442\) 1.51488 0.0720557
\(443\) −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
\(446\) 65.8872 3.11985
\(447\) 0 0
\(448\) 28.6664 1.35436
\(449\) −15.1299 −0.714023 −0.357011 0.934100i \(-0.616204\pi\)
−0.357011 + 0.934100i \(0.616204\pi\)
\(450\) 0 0
\(451\) 17.8524 0.840639
\(452\) −42.5121 −1.99960
\(453\) 0 0
\(454\) 9.66767 0.453726
\(455\) 0.341833 0.0160254
\(456\) 0 0
\(457\) −7.87882 −0.368556 −0.184278 0.982874i \(-0.558995\pi\)
−0.184278 + 0.982874i \(0.558995\pi\)
\(458\) −51.2190 −2.39331
\(459\) 0 0
\(460\) −9.27791 −0.432585
\(461\) 33.2729 1.54967 0.774836 0.632162i \(-0.217832\pi\)
0.774836 + 0.632162i \(0.217832\pi\)
\(462\) 0 0
\(463\) 31.9493 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(464\) −79.8528 −3.70707
\(465\) 0 0
\(466\) −33.1810 −1.53708
\(467\) −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
\(470\) −6.66211 −0.307300
\(471\) 0 0
\(472\) −64.8446 −2.98472
\(473\) −7.83451 −0.360231
\(474\) 0 0
\(475\) −17.4852 −0.802274
\(476\) −4.45958 −0.204404
\(477\) 0 0
\(478\) −83.4096 −3.81507
\(479\) −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
\(482\) −19.7700 −0.900501
\(483\) 0 0
\(484\) −41.1516 −1.87053
\(485\) −1.45044 −0.0658612
\(486\) 0 0
\(487\) −23.3591 −1.05850 −0.529251 0.848465i \(-0.677527\pi\)
−0.529251 + 0.848465i \(0.677527\pi\)
\(488\) −35.9806 −1.62877
\(489\) 0 0
\(490\) 7.57123 0.342033
\(491\) 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
\(494\) −6.98022 −0.314055
\(495\) 0 0
\(496\) 88.6285 3.97954
\(497\) −1.38295 −0.0620338
\(498\) 0 0
\(499\) −12.0079 −0.537549 −0.268775 0.963203i \(-0.586619\pi\)
−0.268775 + 0.963203i \(0.586619\pi\)
\(500\) 24.8474 1.11121
\(501\) 0 0
\(502\) 68.8169 3.07145
\(503\) 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
\(506\) 18.3085 0.813911
\(507\) 0 0
\(508\) 53.2325 2.36181
\(509\) 11.9642 0.530302 0.265151 0.964207i \(-0.414578\pi\)
0.265151 + 0.964207i \(0.414578\pi\)
\(510\) 0 0
\(511\) −8.13390 −0.359822
\(512\) 31.6816 1.40014
\(513\) 0 0
\(514\) 72.1387 3.18190
\(515\) 1.53339 0.0675693
\(516\) 0 0
\(517\) 9.59805 0.422122
\(518\) −7.71911 −0.339158
\(519\) 0 0
\(520\) 3.05545 0.133990
\(521\) 5.47217 0.239740 0.119870 0.992790i \(-0.461752\pi\)
0.119870 + 0.992790i \(0.461752\pi\)
\(522\) 0 0
\(523\) −31.1363 −1.36149 −0.680747 0.732519i \(-0.738345\pi\)
−0.680747 + 0.732519i \(0.738345\pi\)
\(524\) −43.3893 −1.89547
\(525\) 0 0
\(526\) −34.2931 −1.49525
\(527\) −4.87187 −0.212222
\(528\) 0 0
\(529\) −9.66591 −0.420257
\(530\) −8.96373 −0.389360
\(531\) 0 0
\(532\) 20.5487 0.890897
\(533\) −6.79359 −0.294263
\(534\) 0 0
\(535\) −8.68380 −0.375434
\(536\) −47.4023 −2.04747
\(537\) 0 0
\(538\) −3.97183 −0.171238
\(539\) −10.9078 −0.469832
\(540\) 0 0
\(541\) −15.0578 −0.647387 −0.323694 0.946162i \(-0.604925\pi\)
−0.323694 + 0.946162i \(0.604925\pi\)
\(542\) 14.0513 0.603556
\(543\) 0 0
\(544\) −16.4791 −0.706535
\(545\) 8.77234 0.375766
\(546\) 0 0
\(547\) −1.00000 −0.0427569
\(548\) −91.1150 −3.89224
\(549\) 0 0
\(550\) −23.9631 −1.02179
\(551\) −20.2257 −0.861642
\(552\) 0 0
\(553\) 4.52877 0.192583
\(554\) 38.1384 1.62035
\(555\) 0 0
\(556\) 60.8602 2.58105
\(557\) −23.5488 −0.997796 −0.498898 0.866661i \(-0.666262\pi\)
−0.498898 + 0.866661i \(0.666262\pi\)
\(558\) 0 0
\(559\) 2.98136 0.126098
\(560\) −7.04403 −0.297665
\(561\) 0 0
\(562\) −44.5374 −1.87869
\(563\) 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
\(566\) 50.4074 2.11878
\(567\) 0 0
\(568\) −12.3614 −0.518673
\(569\) 0.227367 0.00953174 0.00476587 0.999989i \(-0.498483\pi\)
0.00476587 + 0.999989i \(0.498483\pi\)
\(570\) 0 0
\(571\) 0.874566 0.0365994 0.0182997 0.999833i \(-0.494175\pi\)
0.0182997 + 0.999833i \(0.494175\pi\)
\(572\) −6.98412 −0.292021
\(573\) 0 0
\(574\) 27.3933 1.14338
\(575\) −17.4524 −0.727814
\(576\) 0 0
\(577\) −29.2997 −1.21976 −0.609882 0.792492i \(-0.708783\pi\)
−0.609882 + 0.792492i \(0.708783\pi\)
\(578\) −44.5588 −1.85340
\(579\) 0 0
\(580\) 14.0467 0.583257
\(581\) 13.8457 0.574415
\(582\) 0 0
\(583\) 12.9140 0.534842
\(584\) −72.7042 −3.00852
\(585\) 0 0
\(586\) −83.0875 −3.43231
\(587\) 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
\(590\) 8.93285 0.367760
\(591\) 0 0
\(592\) −39.4483 −1.62132
\(593\) 42.7926 1.75728 0.878641 0.477483i \(-0.158451\pi\)
0.878641 + 0.477483i \(0.158451\pi\)
\(594\) 0 0
\(595\) 0.387208 0.0158740
\(596\) 102.057 4.18042
\(597\) 0 0
\(598\) −6.96713 −0.284907
\(599\) −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 \(0.631952\pi\)
\(602\) −12.0215 −0.489960
\(603\) 0 0
\(604\) −34.6367 −1.40935
\(605\) 3.57304 0.145265
\(606\) 0 0
\(607\) −0.928802 −0.0376989 −0.0188495 0.999822i \(-0.506000\pi\)
−0.0188495 + 0.999822i \(0.506000\pi\)
\(608\) 75.9317 3.07944
\(609\) 0 0
\(610\) 4.95661 0.200687
\(611\) −3.65245 −0.147762
\(612\) 0 0
\(613\) 20.1162 0.812485 0.406242 0.913765i \(-0.366839\pi\)
0.406242 + 0.913765i \(0.366839\pi\)
\(614\) 9.14702 0.369144
\(615\) 0 0
\(616\) 17.7497 0.715157
\(617\) 13.2513 0.533479 0.266739 0.963769i \(-0.414054\pi\)
0.266739 + 0.963769i \(0.414054\pi\)
\(618\) 0 0
\(619\) −29.2870 −1.17714 −0.588572 0.808445i \(-0.700310\pi\)
−0.588572 + 0.808445i \(0.700310\pi\)
\(620\) −15.5904 −0.626126
\(621\) 0 0
\(622\) −70.9450 −2.84464
\(623\) 0.216863 0.00868844
\(624\) 0 0
\(625\) 21.7396 0.869583
\(626\) −77.5674 −3.10022
\(627\) 0 0
\(628\) −80.7393 −3.22185
\(629\) 2.16846 0.0864621
\(630\) 0 0
\(631\) −46.5766 −1.85418 −0.927092 0.374834i \(-0.877700\pi\)
−0.927092 + 0.374834i \(0.877700\pi\)
\(632\) 40.4800 1.61021
\(633\) 0 0
\(634\) 22.4985 0.893530
\(635\) −4.62197 −0.183417
\(636\) 0 0
\(637\) 4.15087 0.164463
\(638\) −27.7189 −1.09740
\(639\) 0 0
\(640\) −15.8011 −0.624594
\(641\) 5.11165 0.201898 0.100949 0.994892i \(-0.467812\pi\)
0.100949 + 0.994892i \(0.467812\pi\)
\(642\) 0 0
\(643\) −4.32994 −0.170756 −0.0853781 0.996349i \(-0.527210\pi\)
−0.0853781 + 0.996349i \(0.527210\pi\)
\(644\) 20.5101 0.808212
\(645\) 0 0
\(646\) −7.90677 −0.311088
\(647\) 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
\(650\) 9.11895 0.357675
\(651\) 0 0
\(652\) −110.264 −4.31828
\(653\) −39.4366 −1.54327 −0.771636 0.636064i \(-0.780561\pi\)
−0.771636 + 0.636064i \(0.780561\pi\)
\(654\) 0 0
\(655\) 3.76733 0.147202
\(656\) 139.993 5.46581
\(657\) 0 0
\(658\) 14.7275 0.574139
\(659\) −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 \(0.532179\pi\)
\(662\) 34.0696 1.32415
\(663\) 0 0
\(664\) 123.758 4.80276
\(665\) −1.78416 −0.0691868
\(666\) 0 0
\(667\) −20.1877 −0.781672
\(668\) −85.9321 −3.32481
\(669\) 0 0
\(670\) 6.53003 0.252277
\(671\) −7.14094 −0.275673
\(672\) 0 0
\(673\) 37.7606 1.45557 0.727783 0.685808i \(-0.240551\pi\)
0.727783 + 0.685808i \(0.240551\pi\)
\(674\) −54.5663 −2.10182
\(675\) 0 0
\(676\) −67.6661 −2.60254
\(677\) 36.6368 1.40806 0.704032 0.710168i \(-0.251381\pi\)
0.704032 + 0.710168i \(0.251381\pi\)
\(678\) 0 0
\(679\) 3.20640 0.123050
\(680\) 3.46103 0.132724
\(681\) 0 0
\(682\) 30.7652 1.17806
\(683\) −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
\(686\) −36.5216 −1.39440
\(687\) 0 0
\(688\) −61.4357 −2.34221
\(689\) −4.91430 −0.187220
\(690\) 0 0
\(691\) 18.9407 0.720540 0.360270 0.932848i \(-0.382685\pi\)
0.360270 + 0.932848i \(0.382685\pi\)
\(692\) −88.7759 −3.37475
\(693\) 0 0
\(694\) −2.03678 −0.0773153
\(695\) −5.28426 −0.200443
\(696\) 0 0
\(697\) −7.69537 −0.291483
\(698\) 11.3493 0.429578
\(699\) 0 0
\(700\) −26.8447 −1.01464
\(701\) −42.8730 −1.61929 −0.809646 0.586919i \(-0.800341\pi\)
−0.809646 + 0.586919i \(0.800341\pi\)
\(702\) 0 0
\(703\) −9.99174 −0.376846
\(704\) 50.8535 1.91662
\(705\) 0 0
\(706\) −78.6675 −2.96069
\(707\) 8.30921 0.312500
\(708\) 0 0
\(709\) 35.4811 1.33252 0.666260 0.745719i \(-0.267894\pi\)
0.666260 + 0.745719i \(0.267894\pi\)
\(710\) 1.70288 0.0639079
\(711\) 0 0
\(712\) 1.93842 0.0726452
\(713\) 22.4063 0.839124
\(714\) 0 0
\(715\) 0.606404 0.0226782
\(716\) 110.500 4.12957
\(717\) 0 0
\(718\) −61.3107 −2.28810
\(719\) 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
\(722\) −15.2863 −0.568899
\(723\) 0 0
\(724\) −56.0873 −2.08447
\(725\) 26.4228 0.981317
\(726\) 0 0
\(727\) −28.7796 −1.06737 −0.533687 0.845682i \(-0.679194\pi\)
−0.533687 + 0.845682i \(0.679194\pi\)
\(728\) −6.75450 −0.250339
\(729\) 0 0
\(730\) 10.0156 0.370693
\(731\) 3.37710 0.124906
\(732\) 0 0
\(733\) 23.0494 0.851350 0.425675 0.904876i \(-0.360037\pi\)
0.425675 + 0.904876i \(0.360037\pi\)
\(734\) −8.38213 −0.309390
\(735\) 0 0
\(736\) 75.7893 2.79363
\(737\) −9.40776 −0.346539
\(738\) 0 0
\(739\) −9.02906 −0.332139 −0.166070 0.986114i \(-0.553108\pi\)
−0.166070 + 0.986114i \(0.553108\pi\)
\(740\) 6.93925 0.255092
\(741\) 0 0
\(742\) 19.8156 0.727453
\(743\) 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
\(746\) −64.9903 −2.37947
\(747\) 0 0
\(748\) −7.91118 −0.289262
\(749\) 19.1968 0.701434
\(750\) 0 0
\(751\) −20.5388 −0.749473 −0.374737 0.927131i \(-0.622267\pi\)
−0.374737 + 0.927131i \(0.622267\pi\)
\(752\) 75.2647 2.74462
\(753\) 0 0
\(754\) 10.5482 0.384143
\(755\) 3.00737 0.109450
\(756\) 0 0
\(757\) 36.3077 1.31962 0.659812 0.751430i \(-0.270636\pi\)
0.659812 + 0.751430i \(0.270636\pi\)
\(758\) 9.81240 0.356402
\(759\) 0 0
\(760\) −15.9476 −0.578480
\(761\) −23.0597 −0.835915 −0.417957 0.908467i \(-0.637254\pi\)
−0.417957 + 0.908467i \(0.637254\pi\)
\(762\) 0 0
\(763\) −19.3925 −0.702055
\(764\) 43.0318 1.55684
\(765\) 0 0
\(766\) 9.89719 0.357600
\(767\) 4.89737 0.176834
\(768\) 0 0
\(769\) 4.40403 0.158813 0.0794066 0.996842i \(-0.474697\pi\)
0.0794066 + 0.996842i \(0.474697\pi\)
\(770\) −2.44516 −0.0881176
\(771\) 0 0
\(772\) 129.594 4.66418
\(773\) 25.6259 0.921699 0.460849 0.887478i \(-0.347545\pi\)
0.460849 + 0.887478i \(0.347545\pi\)
\(774\) 0 0
\(775\) −29.3266 −1.05344
\(776\) 28.6602 1.02884
\(777\) 0 0
\(778\) −58.9962 −2.11512
\(779\) 35.4584 1.27043
\(780\) 0 0
\(781\) −2.45332 −0.0877867
\(782\) −7.89194 −0.282215
\(783\) 0 0
\(784\) −85.5354 −3.05484
\(785\) 7.01029 0.250208
\(786\) 0 0
\(787\) 46.4038 1.65412 0.827059 0.562115i \(-0.190012\pi\)
0.827059 + 0.562115i \(0.190012\pi\)
\(788\) −27.8659 −0.992681
\(789\) 0 0
\(790\) −5.57644 −0.198401
\(791\) −8.15982 −0.290130
\(792\) 0 0
\(793\) 2.71742 0.0964985
\(794\) 17.9415 0.636720
\(795\) 0 0
\(796\) 8.34748 0.295868
\(797\) 33.6692 1.19262 0.596312 0.802753i \(-0.296632\pi\)
0.596312 + 0.802753i \(0.296632\pi\)
\(798\) 0 0
\(799\) −4.13727 −0.146366
\(800\) −99.1971 −3.50715
\(801\) 0 0
\(802\) 11.8501 0.418443
\(803\) −14.4293 −0.509200
\(804\) 0 0
\(805\) −1.78082 −0.0627655
\(806\) −11.7074 −0.412376
\(807\) 0 0
\(808\) 74.2712 2.61285
\(809\) −32.2939 −1.13539 −0.567696 0.823238i \(-0.692165\pi\)
−0.567696 + 0.823238i \(0.692165\pi\)
\(810\) 0 0
\(811\) 27.4278 0.963119 0.481559 0.876413i \(-0.340071\pi\)
0.481559 + 0.876413i \(0.340071\pi\)
\(812\) −31.0522 −1.08972
\(813\) 0 0
\(814\) −13.6935 −0.479957
\(815\) 9.57382 0.335356
\(816\) 0 0
\(817\) −15.5609 −0.544405
\(818\) 86.6732 3.03046
\(819\) 0 0
\(820\) −24.6258 −0.859971
\(821\) 17.5646 0.613008 0.306504 0.951869i \(-0.400841\pi\)
0.306504 + 0.951869i \(0.400841\pi\)
\(822\) 0 0
\(823\) −7.94705 −0.277017 −0.138508 0.990361i \(-0.544231\pi\)
−0.138508 + 0.990361i \(0.544231\pi\)
\(824\) −30.2993 −1.05552
\(825\) 0 0
\(826\) −19.7473 −0.687097
\(827\) 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 \(0.613264\pi\)
\(830\) −17.0487 −0.591768
\(831\) 0 0
\(832\) −19.3519 −0.670905
\(833\) 4.70185 0.162909
\(834\) 0 0
\(835\) 7.46115 0.258204
\(836\) 36.4528 1.26075
\(837\) 0 0
\(838\) 81.1781 2.80425
\(839\) −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
\(842\) −21.5883 −0.743983
\(843\) 0 0
\(844\) 47.6675 1.64078
\(845\) 5.87518 0.202112
\(846\) 0 0
\(847\) −7.89870 −0.271403
\(848\) 101.267 3.47753
\(849\) 0 0
\(850\) 10.3294 0.354295
\(851\) −9.97300 −0.341870
\(852\) 0 0
\(853\) 41.2623 1.41280 0.706398 0.707815i \(-0.250319\pi\)
0.706398 + 0.707815i \(0.250319\pi\)
\(854\) −10.9573 −0.374950
\(855\) 0 0
\(856\) 171.589 5.86479
\(857\) −4.93880 −0.168706 −0.0843532 0.996436i \(-0.526882\pi\)
−0.0843532 + 0.996436i \(0.526882\pi\)
\(858\) 0 0
\(859\) −38.1044 −1.30011 −0.650053 0.759889i \(-0.725253\pi\)
−0.650053 + 0.759889i \(0.725253\pi\)
\(860\) 10.8070 0.368516
\(861\) 0 0
\(862\) 67.9250 2.31353
\(863\) 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
\(866\) 63.0168 2.14140
\(867\) 0 0
\(868\) 34.4648 1.16981
\(869\) 8.03393 0.272532
\(870\) 0 0
\(871\) 3.58004 0.121305
\(872\) −173.338 −5.86997
\(873\) 0 0
\(874\) 36.3642 1.23004
\(875\) 4.76924 0.161230
\(876\) 0 0
\(877\) 19.2114 0.648724 0.324362 0.945933i \(-0.394850\pi\)
0.324362 + 0.945933i \(0.394850\pi\)
\(878\) 51.6245 1.74224
\(879\) 0 0
\(880\) −12.4960 −0.421238
\(881\) −0.0458679 −0.00154533 −0.000772664 1.00000i \(-0.500246\pi\)
−0.000772664 1.00000i \(0.500246\pi\)
\(882\) 0 0
\(883\) 42.1763 1.41935 0.709674 0.704531i \(-0.248842\pi\)
0.709674 + 0.704531i \(0.248842\pi\)
\(884\) 3.01053 0.101255
\(885\) 0 0
\(886\) −58.8515 −1.97715
\(887\) −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
\(890\) −0.267032 −0.00895093
\(891\) 0 0
\(892\) 130.938 4.38412
\(893\) 19.0636 0.637938
\(894\) 0 0
\(895\) −9.59428 −0.320701
\(896\) 34.9306 1.16695
\(897\) 0 0
\(898\) −41.1842 −1.37433
\(899\) −33.9230 −1.13140
\(900\) 0 0
\(901\) −5.56661 −0.185451
\(902\) 48.5951 1.61804
\(903\) 0 0
\(904\) −72.9360 −2.42581
\(905\) 4.86985 0.161879
\(906\) 0 0
\(907\) −21.2700 −0.706259 −0.353129 0.935574i \(-0.614882\pi\)
−0.353129 + 0.935574i \(0.614882\pi\)
\(908\) 19.2126 0.637592
\(909\) 0 0
\(910\) 0.930485 0.0308453
\(911\) 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
\(914\) −21.4465 −0.709387
\(915\) 0 0
\(916\) −101.788 −3.36316
\(917\) −8.32821 −0.275022
\(918\) 0 0
\(919\) −39.3681 −1.29864 −0.649318 0.760517i \(-0.724945\pi\)
−0.649318 + 0.760517i \(0.724945\pi\)
\(920\) −15.9177 −0.524791
\(921\) 0 0
\(922\) 90.5703 2.98277
\(923\) 0.933589 0.0307295
\(924\) 0 0
\(925\) 13.0532 0.429187
\(926\) 86.9674 2.85793
\(927\) 0 0
\(928\) −114.745 −3.76667
\(929\) −48.6956 −1.59765 −0.798826 0.601562i \(-0.794545\pi\)
−0.798826 + 0.601562i \(0.794545\pi\)
\(930\) 0 0
\(931\) −21.6650 −0.710042
\(932\) −65.9406 −2.15996
\(933\) 0 0
\(934\) −16.4198 −0.537274
\(935\) 0.686898 0.0224640
\(936\) 0 0
\(937\) −12.4806 −0.407723 −0.203862 0.979000i \(-0.565349\pi\)
−0.203862 + 0.979000i \(0.565349\pi\)
\(938\) −14.4356 −0.471338
\(939\) 0 0
\(940\) −13.2396 −0.431829
\(941\) 44.4531 1.44913 0.724565 0.689207i \(-0.242041\pi\)
0.724565 + 0.689207i \(0.242041\pi\)
\(942\) 0 0
\(943\) 35.3919 1.15252
\(944\) −100.918 −3.28461
\(945\) 0 0
\(946\) −21.3259 −0.693365
\(947\) 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
\(950\) −47.5954 −1.54420
\(951\) 0 0
\(952\) −7.65109 −0.247973
\(953\) 14.6341 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(954\) 0 0
\(955\) −3.73629 −0.120903
\(956\) −165.760 −5.36106
\(957\) 0 0
\(958\) −49.3693 −1.59505
\(959\) −17.4887 −0.564741
\(960\) 0 0
\(961\) 6.65114 0.214553
\(962\) 5.21095 0.168008
\(963\) 0 0
\(964\) −39.2891 −1.26542
\(965\) −11.2521 −0.362219
\(966\) 0 0
\(967\) 1.24602 0.0400692 0.0200346 0.999799i \(-0.493622\pi\)
0.0200346 + 0.999799i \(0.493622\pi\)
\(968\) −70.6020 −2.26923
\(969\) 0 0
\(970\) −3.94817 −0.126768
\(971\) −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
\(974\) −63.5846 −2.03738
\(975\) 0 0
\(976\) −55.9969 −1.79242
\(977\) −29.2177 −0.934757 −0.467379 0.884057i \(-0.654801\pi\)
−0.467379 + 0.884057i \(0.654801\pi\)
\(978\) 0 0
\(979\) 0.384710 0.0122954
\(980\) 15.0463 0.480637
\(981\) 0 0
\(982\) 11.6745 0.372550
\(983\) 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
\(986\) 11.9483 0.380513
\(987\) 0 0
\(988\) −13.8718 −0.441321
\(989\) −15.5317 −0.493878
\(990\) 0 0
\(991\) 51.4773 1.63523 0.817614 0.575766i \(-0.195296\pi\)
0.817614 + 0.575766i \(0.195296\pi\)
\(992\) 127.355 4.04352
\(993\) 0 0
\(994\) −3.76445 −0.119401
\(995\) −0.724779 −0.0229771
\(996\) 0 0
\(997\) −41.2171 −1.30536 −0.652680 0.757634i \(-0.726355\pi\)
−0.652680 + 0.757634i \(0.726355\pi\)
\(998\) −32.6862 −1.03466
\(999\) 0 0
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 4923.2.a.l.1.18 18
3.2 odd 2 547.2.a.b.1.1 18
12.11 even 2 8752.2.a.s.1.13 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.1 18 3.2 odd 2
4923.2.a.l.1.18 18 1.1 even 1 trivial
8752.2.a.s.1.13 18 12.11 even 2