Properties

Label 547.2.a.b.1.10
Level $547$
Weight $2$
Character 547.1
Self dual yes
Analytic conductor $4.368$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.36781699056\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(0.523506\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.523506 q^{2} -3.08736 q^{3} -1.72594 q^{4} -1.35183 q^{5} +1.61625 q^{6} +3.60927 q^{7} +1.95055 q^{8} +6.53179 q^{9} +O(q^{10})\) \(q-0.523506 q^{2} -3.08736 q^{3} -1.72594 q^{4} -1.35183 q^{5} +1.61625 q^{6} +3.60927 q^{7} +1.95055 q^{8} +6.53179 q^{9} +0.707691 q^{10} -0.337880 q^{11} +5.32860 q^{12} -0.968153 q^{13} -1.88948 q^{14} +4.17358 q^{15} +2.43075 q^{16} +1.24740 q^{17} -3.41944 q^{18} -5.50049 q^{19} +2.33318 q^{20} -11.1431 q^{21} +0.176882 q^{22} +3.13569 q^{23} -6.02206 q^{24} -3.17256 q^{25} +0.506834 q^{26} -10.9039 q^{27} -6.22939 q^{28} -7.18364 q^{29} -2.18490 q^{30} +10.2089 q^{31} -5.17362 q^{32} +1.04316 q^{33} -0.653022 q^{34} -4.87912 q^{35} -11.2735 q^{36} +5.45502 q^{37} +2.87954 q^{38} +2.98904 q^{39} -2.63682 q^{40} -7.24577 q^{41} +5.83349 q^{42} -6.48922 q^{43} +0.583161 q^{44} -8.82987 q^{45} -1.64155 q^{46} -7.68104 q^{47} -7.50462 q^{48} +6.02683 q^{49} +1.66085 q^{50} -3.85117 q^{51} +1.67097 q^{52} -4.50651 q^{53} +5.70827 q^{54} +0.456756 q^{55} +7.04008 q^{56} +16.9820 q^{57} +3.76068 q^{58} +2.41312 q^{59} -7.20336 q^{60} +6.37922 q^{61} -5.34441 q^{62} +23.5750 q^{63} -2.15308 q^{64} +1.30878 q^{65} -0.546100 q^{66} +0.799497 q^{67} -2.15294 q^{68} -9.68099 q^{69} +2.55425 q^{70} +2.93852 q^{71} +12.7406 q^{72} -15.3091 q^{73} -2.85574 q^{74} +9.79483 q^{75} +9.49353 q^{76} -1.21950 q^{77} -1.56478 q^{78} +10.3913 q^{79} -3.28597 q^{80} +14.0690 q^{81} +3.79321 q^{82} -13.3213 q^{83} +19.2324 q^{84} -1.68627 q^{85} +3.39715 q^{86} +22.1785 q^{87} -0.659054 q^{88} -4.42608 q^{89} +4.62249 q^{90} -3.49433 q^{91} -5.41201 q^{92} -31.5185 q^{93} +4.02107 q^{94} +7.43573 q^{95} +15.9728 q^{96} -6.98217 q^{97} -3.15508 q^{98} -2.20696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.523506 −0.370175 −0.185087 0.982722i \(-0.559257\pi\)
−0.185087 + 0.982722i \(0.559257\pi\)
\(3\) −3.08736 −1.78249 −0.891244 0.453524i \(-0.850167\pi\)
−0.891244 + 0.453524i \(0.850167\pi\)
\(4\) −1.72594 −0.862971
\(5\) −1.35183 −0.604556 −0.302278 0.953220i \(-0.597747\pi\)
−0.302278 + 0.953220i \(0.597747\pi\)
\(6\) 1.61625 0.659832
\(7\) 3.60927 1.36418 0.682088 0.731270i \(-0.261072\pi\)
0.682088 + 0.731270i \(0.261072\pi\)
\(8\) 1.95055 0.689625
\(9\) 6.53179 2.17726
\(10\) 0.707691 0.223792
\(11\) −0.337880 −0.101875 −0.0509374 0.998702i \(-0.516221\pi\)
−0.0509374 + 0.998702i \(0.516221\pi\)
\(12\) 5.32860 1.53823
\(13\) −0.968153 −0.268517 −0.134259 0.990946i \(-0.542865\pi\)
−0.134259 + 0.990946i \(0.542865\pi\)
\(14\) −1.88948 −0.504984
\(15\) 4.17358 1.07761
\(16\) 2.43075 0.607689
\(17\) 1.24740 0.302539 0.151270 0.988493i \(-0.451664\pi\)
0.151270 + 0.988493i \(0.451664\pi\)
\(18\) −3.41944 −0.805969
\(19\) −5.50049 −1.26190 −0.630950 0.775824i \(-0.717335\pi\)
−0.630950 + 0.775824i \(0.717335\pi\)
\(20\) 2.33318 0.521714
\(21\) −11.1431 −2.43163
\(22\) 0.176882 0.0377115
\(23\) 3.13569 0.653836 0.326918 0.945053i \(-0.393990\pi\)
0.326918 + 0.945053i \(0.393990\pi\)
\(24\) −6.02206 −1.22925
\(25\) −3.17256 −0.634511
\(26\) 0.506834 0.0993984
\(27\) −10.9039 −2.09846
\(28\) −6.22939 −1.17724
\(29\) −7.18364 −1.33397 −0.666984 0.745072i \(-0.732415\pi\)
−0.666984 + 0.745072i \(0.732415\pi\)
\(30\) −2.18490 −0.398906
\(31\) 10.2089 1.83357 0.916784 0.399384i \(-0.130776\pi\)
0.916784 + 0.399384i \(0.130776\pi\)
\(32\) −5.17362 −0.914576
\(33\) 1.04316 0.181591
\(34\) −0.653022 −0.111992
\(35\) −4.87912 −0.824721
\(36\) −11.2735 −1.87892
\(37\) 5.45502 0.896799 0.448400 0.893833i \(-0.351994\pi\)
0.448400 + 0.893833i \(0.351994\pi\)
\(38\) 2.87954 0.467124
\(39\) 2.98904 0.478629
\(40\) −2.63682 −0.416917
\(41\) −7.24577 −1.13160 −0.565800 0.824543i \(-0.691432\pi\)
−0.565800 + 0.824543i \(0.691432\pi\)
\(42\) 5.83349 0.900128
\(43\) −6.48922 −0.989597 −0.494798 0.869008i \(-0.664758\pi\)
−0.494798 + 0.869008i \(0.664758\pi\)
\(44\) 0.583161 0.0879149
\(45\) −8.82987 −1.31628
\(46\) −1.64155 −0.242034
\(47\) −7.68104 −1.12039 −0.560197 0.828359i \(-0.689275\pi\)
−0.560197 + 0.828359i \(0.689275\pi\)
\(48\) −7.50462 −1.08320
\(49\) 6.02683 0.860976
\(50\) 1.66085 0.234880
\(51\) −3.85117 −0.539272
\(52\) 1.67097 0.231723
\(53\) −4.50651 −0.619017 −0.309509 0.950897i \(-0.600165\pi\)
−0.309509 + 0.950897i \(0.600165\pi\)
\(54\) 5.70827 0.776797
\(55\) 0.456756 0.0615890
\(56\) 7.04008 0.940770
\(57\) 16.9820 2.24932
\(58\) 3.76068 0.493801
\(59\) 2.41312 0.314161 0.157081 0.987586i \(-0.449792\pi\)
0.157081 + 0.987586i \(0.449792\pi\)
\(60\) −7.20336 −0.929950
\(61\) 6.37922 0.816775 0.408387 0.912809i \(-0.366091\pi\)
0.408387 + 0.912809i \(0.366091\pi\)
\(62\) −5.34441 −0.678741
\(63\) 23.5750 2.97017
\(64\) −2.15308 −0.269136
\(65\) 1.30878 0.162334
\(66\) −0.546100 −0.0672203
\(67\) 0.799497 0.0976741 0.0488370 0.998807i \(-0.484449\pi\)
0.0488370 + 0.998807i \(0.484449\pi\)
\(68\) −2.15294 −0.261082
\(69\) −9.68099 −1.16545
\(70\) 2.55425 0.305291
\(71\) 2.93852 0.348738 0.174369 0.984680i \(-0.444211\pi\)
0.174369 + 0.984680i \(0.444211\pi\)
\(72\) 12.7406 1.50150
\(73\) −15.3091 −1.79180 −0.895899 0.444257i \(-0.853468\pi\)
−0.895899 + 0.444257i \(0.853468\pi\)
\(74\) −2.85574 −0.331973
\(75\) 9.79483 1.13101
\(76\) 9.49353 1.08898
\(77\) −1.21950 −0.138975
\(78\) −1.56478 −0.177176
\(79\) 10.3913 1.16911 0.584557 0.811353i \(-0.301268\pi\)
0.584557 + 0.811353i \(0.301268\pi\)
\(80\) −3.28597 −0.367382
\(81\) 14.0690 1.56322
\(82\) 3.79321 0.418890
\(83\) −13.3213 −1.46220 −0.731102 0.682269i \(-0.760993\pi\)
−0.731102 + 0.682269i \(0.760993\pi\)
\(84\) 19.2324 2.09842
\(85\) −1.68627 −0.182902
\(86\) 3.39715 0.366324
\(87\) 22.1785 2.37778
\(88\) −0.659054 −0.0702554
\(89\) −4.42608 −0.469163 −0.234582 0.972096i \(-0.575372\pi\)
−0.234582 + 0.972096i \(0.575372\pi\)
\(90\) 4.62249 0.487254
\(91\) −3.49433 −0.366305
\(92\) −5.41201 −0.564241
\(93\) −31.5185 −3.26831
\(94\) 4.02107 0.414742
\(95\) 7.43573 0.762890
\(96\) 15.9728 1.63022
\(97\) −6.98217 −0.708932 −0.354466 0.935069i \(-0.615337\pi\)
−0.354466 + 0.935069i \(0.615337\pi\)
\(98\) −3.15508 −0.318712
\(99\) −2.20696 −0.221808
\(100\) 5.47565 0.547565
\(101\) −3.92586 −0.390638 −0.195319 0.980740i \(-0.562574\pi\)
−0.195319 + 0.980740i \(0.562574\pi\)
\(102\) 2.01611 0.199625
\(103\) −14.0937 −1.38870 −0.694348 0.719639i \(-0.744307\pi\)
−0.694348 + 0.719639i \(0.744307\pi\)
\(104\) −1.88843 −0.185176
\(105\) 15.0636 1.47006
\(106\) 2.35919 0.229145
\(107\) 7.65150 0.739699 0.369849 0.929092i \(-0.379409\pi\)
0.369849 + 0.929092i \(0.379409\pi\)
\(108\) 18.8195 1.81091
\(109\) −4.57688 −0.438386 −0.219193 0.975682i \(-0.570342\pi\)
−0.219193 + 0.975682i \(0.570342\pi\)
\(110\) −0.239115 −0.0227987
\(111\) −16.8416 −1.59853
\(112\) 8.77325 0.828994
\(113\) −15.9981 −1.50497 −0.752487 0.658607i \(-0.771146\pi\)
−0.752487 + 0.658607i \(0.771146\pi\)
\(114\) −8.89019 −0.832642
\(115\) −4.23891 −0.395281
\(116\) 12.3985 1.15118
\(117\) −6.32378 −0.584633
\(118\) −1.26328 −0.116295
\(119\) 4.50220 0.412716
\(120\) 8.14080 0.743150
\(121\) −10.8858 −0.989622
\(122\) −3.33956 −0.302350
\(123\) 22.3703 2.01706
\(124\) −17.6199 −1.58231
\(125\) 11.0479 0.988154
\(126\) −12.3417 −1.09948
\(127\) 7.02400 0.623279 0.311640 0.950200i \(-0.399122\pi\)
0.311640 + 0.950200i \(0.399122\pi\)
\(128\) 11.4744 1.01420
\(129\) 20.0346 1.76394
\(130\) −0.685153 −0.0600919
\(131\) −3.36814 −0.294276 −0.147138 0.989116i \(-0.547006\pi\)
−0.147138 + 0.989116i \(0.547006\pi\)
\(132\) −1.80043 −0.156707
\(133\) −19.8528 −1.72145
\(134\) −0.418542 −0.0361565
\(135\) 14.7402 1.26864
\(136\) 2.43312 0.208638
\(137\) −21.8903 −1.87022 −0.935108 0.354362i \(-0.884698\pi\)
−0.935108 + 0.354362i \(0.884698\pi\)
\(138\) 5.06806 0.431422
\(139\) −7.81159 −0.662571 −0.331285 0.943531i \(-0.607482\pi\)
−0.331285 + 0.943531i \(0.607482\pi\)
\(140\) 8.42107 0.711710
\(141\) 23.7141 1.99709
\(142\) −1.53833 −0.129094
\(143\) 0.327120 0.0273551
\(144\) 15.8772 1.32310
\(145\) 9.71105 0.806459
\(146\) 8.01443 0.663279
\(147\) −18.6070 −1.53468
\(148\) −9.41504 −0.773911
\(149\) −1.73671 −0.142277 −0.0711385 0.997466i \(-0.522663\pi\)
−0.0711385 + 0.997466i \(0.522663\pi\)
\(150\) −5.12766 −0.418671
\(151\) 9.10500 0.740955 0.370477 0.928841i \(-0.379194\pi\)
0.370477 + 0.928841i \(0.379194\pi\)
\(152\) −10.7290 −0.870237
\(153\) 8.14776 0.658708
\(154\) 0.638417 0.0514451
\(155\) −13.8007 −1.10850
\(156\) −5.15890 −0.413043
\(157\) 10.4340 0.832727 0.416364 0.909198i \(-0.363304\pi\)
0.416364 + 0.909198i \(0.363304\pi\)
\(158\) −5.43992 −0.432777
\(159\) 13.9132 1.10339
\(160\) 6.99386 0.552913
\(161\) 11.3175 0.891947
\(162\) −7.36519 −0.578664
\(163\) 23.6947 1.85591 0.927954 0.372694i \(-0.121566\pi\)
0.927954 + 0.372694i \(0.121566\pi\)
\(164\) 12.5058 0.976537
\(165\) −1.41017 −0.109782
\(166\) 6.97379 0.541271
\(167\) −14.5777 −1.12806 −0.564029 0.825755i \(-0.690749\pi\)
−0.564029 + 0.825755i \(0.690749\pi\)
\(168\) −21.7353 −1.67691
\(169\) −12.0627 −0.927898
\(170\) 0.882774 0.0677057
\(171\) −35.9281 −2.74749
\(172\) 11.2000 0.853993
\(173\) −16.5647 −1.25939 −0.629695 0.776843i \(-0.716820\pi\)
−0.629695 + 0.776843i \(0.716820\pi\)
\(174\) −11.6106 −0.880195
\(175\) −11.4506 −0.865585
\(176\) −0.821304 −0.0619081
\(177\) −7.45017 −0.559989
\(178\) 2.31708 0.173672
\(179\) 14.0810 1.05246 0.526230 0.850342i \(-0.323605\pi\)
0.526230 + 0.850342i \(0.323605\pi\)
\(180\) 15.2398 1.13591
\(181\) −5.40061 −0.401424 −0.200712 0.979650i \(-0.564326\pi\)
−0.200712 + 0.979650i \(0.564326\pi\)
\(182\) 1.82930 0.135597
\(183\) −19.6949 −1.45589
\(184\) 6.11632 0.450901
\(185\) −7.37425 −0.542166
\(186\) 16.5001 1.20985
\(187\) −0.421472 −0.0308211
\(188\) 13.2570 0.966867
\(189\) −39.3552 −2.86267
\(190\) −3.89265 −0.282403
\(191\) 7.86102 0.568803 0.284402 0.958705i \(-0.408205\pi\)
0.284402 + 0.958705i \(0.408205\pi\)
\(192\) 6.64735 0.479731
\(193\) 23.5348 1.69407 0.847036 0.531536i \(-0.178385\pi\)
0.847036 + 0.531536i \(0.178385\pi\)
\(194\) 3.65521 0.262429
\(195\) −4.04067 −0.289358
\(196\) −10.4020 −0.742997
\(197\) −1.91517 −0.136450 −0.0682251 0.997670i \(-0.521734\pi\)
−0.0682251 + 0.997670i \(0.521734\pi\)
\(198\) 1.15536 0.0821079
\(199\) 2.24700 0.159286 0.0796430 0.996823i \(-0.474622\pi\)
0.0796430 + 0.996823i \(0.474622\pi\)
\(200\) −6.18824 −0.437575
\(201\) −2.46833 −0.174103
\(202\) 2.05521 0.144604
\(203\) −25.9277 −1.81977
\(204\) 6.64690 0.465376
\(205\) 9.79505 0.684116
\(206\) 7.37816 0.514061
\(207\) 20.4817 1.42357
\(208\) −2.35334 −0.163175
\(209\) 1.85851 0.128556
\(210\) −7.88589 −0.544178
\(211\) 14.5078 0.998758 0.499379 0.866384i \(-0.333561\pi\)
0.499379 + 0.866384i \(0.333561\pi\)
\(212\) 7.77797 0.534193
\(213\) −9.07226 −0.621621
\(214\) −4.00561 −0.273818
\(215\) 8.77232 0.598267
\(216\) −21.2687 −1.44715
\(217\) 36.8466 2.50131
\(218\) 2.39603 0.162279
\(219\) 47.2648 3.19386
\(220\) −0.788335 −0.0531495
\(221\) −1.20767 −0.0812370
\(222\) 8.81669 0.591737
\(223\) −25.0935 −1.68038 −0.840191 0.542290i \(-0.817557\pi\)
−0.840191 + 0.542290i \(0.817557\pi\)
\(224\) −18.6730 −1.24764
\(225\) −20.7225 −1.38150
\(226\) 8.37511 0.557104
\(227\) −20.1566 −1.33784 −0.668922 0.743333i \(-0.733244\pi\)
−0.668922 + 0.743333i \(0.733244\pi\)
\(228\) −29.3099 −1.94110
\(229\) 24.6507 1.62897 0.814483 0.580188i \(-0.197021\pi\)
0.814483 + 0.580188i \(0.197021\pi\)
\(230\) 2.21910 0.146323
\(231\) 3.76504 0.247721
\(232\) −14.0121 −0.919938
\(233\) −22.2305 −1.45637 −0.728185 0.685380i \(-0.759636\pi\)
−0.728185 + 0.685380i \(0.759636\pi\)
\(234\) 3.31054 0.216417
\(235\) 10.3834 0.677342
\(236\) −4.16490 −0.271112
\(237\) −32.0817 −2.08393
\(238\) −2.35693 −0.152777
\(239\) −5.88759 −0.380837 −0.190418 0.981703i \(-0.560984\pi\)
−0.190418 + 0.981703i \(0.560984\pi\)
\(240\) 10.1450 0.654854
\(241\) 7.97674 0.513827 0.256913 0.966434i \(-0.417294\pi\)
0.256913 + 0.966434i \(0.417294\pi\)
\(242\) 5.69881 0.366333
\(243\) −10.7242 −0.687955
\(244\) −11.0102 −0.704853
\(245\) −8.14725 −0.520508
\(246\) −11.7110 −0.746666
\(247\) 5.32532 0.338842
\(248\) 19.9130 1.26447
\(249\) 41.1277 2.60636
\(250\) −5.78365 −0.365790
\(251\) −20.2547 −1.27847 −0.639233 0.769013i \(-0.720748\pi\)
−0.639233 + 0.769013i \(0.720748\pi\)
\(252\) −40.6891 −2.56317
\(253\) −1.05949 −0.0666093
\(254\) −3.67711 −0.230722
\(255\) 5.20613 0.326021
\(256\) −1.70075 −0.106297
\(257\) −15.9624 −0.995705 −0.497853 0.867262i \(-0.665878\pi\)
−0.497853 + 0.867262i \(0.665878\pi\)
\(258\) −10.4882 −0.652968
\(259\) 19.6886 1.22339
\(260\) −2.25887 −0.140089
\(261\) −46.9220 −2.90440
\(262\) 1.76324 0.108933
\(263\) 26.2647 1.61955 0.809774 0.586741i \(-0.199589\pi\)
0.809774 + 0.586741i \(0.199589\pi\)
\(264\) 2.03474 0.125229
\(265\) 6.09204 0.374231
\(266\) 10.3930 0.637239
\(267\) 13.6649 0.836278
\(268\) −1.37988 −0.0842898
\(269\) 7.56302 0.461125 0.230563 0.973057i \(-0.425943\pi\)
0.230563 + 0.973057i \(0.425943\pi\)
\(270\) −7.71661 −0.469618
\(271\) 0.855122 0.0519450 0.0259725 0.999663i \(-0.491732\pi\)
0.0259725 + 0.999663i \(0.491732\pi\)
\(272\) 3.03212 0.183850
\(273\) 10.7882 0.652934
\(274\) 11.4597 0.692307
\(275\) 1.07194 0.0646407
\(276\) 16.7088 1.00575
\(277\) −7.71774 −0.463714 −0.231857 0.972750i \(-0.574480\pi\)
−0.231857 + 0.972750i \(0.574480\pi\)
\(278\) 4.08942 0.245267
\(279\) 66.6823 3.99216
\(280\) −9.51698 −0.568748
\(281\) 5.04416 0.300910 0.150455 0.988617i \(-0.451926\pi\)
0.150455 + 0.988617i \(0.451926\pi\)
\(282\) −12.4145 −0.739272
\(283\) 1.44488 0.0858894 0.0429447 0.999077i \(-0.486326\pi\)
0.0429447 + 0.999077i \(0.486326\pi\)
\(284\) −5.07171 −0.300950
\(285\) −22.9568 −1.35984
\(286\) −0.171249 −0.0101262
\(287\) −26.1519 −1.54370
\(288\) −33.7930 −1.99127
\(289\) −15.4440 −0.908470
\(290\) −5.08380 −0.298531
\(291\) 21.5565 1.26366
\(292\) 26.4227 1.54627
\(293\) 11.9975 0.700902 0.350451 0.936581i \(-0.386028\pi\)
0.350451 + 0.936581i \(0.386028\pi\)
\(294\) 9.74088 0.568100
\(295\) −3.26213 −0.189928
\(296\) 10.6403 0.618455
\(297\) 3.68422 0.213780
\(298\) 0.909180 0.0526674
\(299\) −3.03582 −0.175566
\(300\) −16.9053 −0.976028
\(301\) −23.4213 −1.34998
\(302\) −4.76653 −0.274283
\(303\) 12.1205 0.696307
\(304\) −13.3703 −0.766842
\(305\) −8.62361 −0.493787
\(306\) −4.26541 −0.243837
\(307\) 14.7380 0.841140 0.420570 0.907260i \(-0.361830\pi\)
0.420570 + 0.907260i \(0.361830\pi\)
\(308\) 2.10479 0.119931
\(309\) 43.5124 2.47534
\(310\) 7.22473 0.410337
\(311\) −25.6282 −1.45324 −0.726621 0.687038i \(-0.758910\pi\)
−0.726621 + 0.687038i \(0.758910\pi\)
\(312\) 5.83028 0.330074
\(313\) −3.29616 −0.186310 −0.0931551 0.995652i \(-0.529695\pi\)
−0.0931551 + 0.995652i \(0.529695\pi\)
\(314\) −5.46229 −0.308255
\(315\) −31.8694 −1.79564
\(316\) −17.9348 −1.00891
\(317\) 21.5851 1.21234 0.606170 0.795335i \(-0.292705\pi\)
0.606170 + 0.795335i \(0.292705\pi\)
\(318\) −7.28366 −0.408448
\(319\) 2.42721 0.135898
\(320\) 2.91060 0.162708
\(321\) −23.6230 −1.31850
\(322\) −5.92480 −0.330176
\(323\) −6.86132 −0.381774
\(324\) −24.2822 −1.34901
\(325\) 3.07152 0.170377
\(326\) −12.4043 −0.687011
\(327\) 14.1305 0.781417
\(328\) −14.1333 −0.780379
\(329\) −27.7229 −1.52841
\(330\) 0.738234 0.0406384
\(331\) −19.2871 −1.06011 −0.530057 0.847962i \(-0.677829\pi\)
−0.530057 + 0.847962i \(0.677829\pi\)
\(332\) 22.9918 1.26184
\(333\) 35.6310 1.95257
\(334\) 7.63153 0.417579
\(335\) −1.08078 −0.0590495
\(336\) −27.0862 −1.47767
\(337\) 0.183297 0.00998481 0.00499241 0.999988i \(-0.498411\pi\)
0.00499241 + 0.999988i \(0.498411\pi\)
\(338\) 6.31489 0.343485
\(339\) 49.3919 2.68260
\(340\) 2.91041 0.157839
\(341\) −3.44938 −0.186794
\(342\) 18.8086 1.01705
\(343\) −3.51243 −0.189654
\(344\) −12.6576 −0.682451
\(345\) 13.0871 0.704583
\(346\) 8.67172 0.466194
\(347\) −24.5220 −1.31641 −0.658206 0.752838i \(-0.728684\pi\)
−0.658206 + 0.752838i \(0.728684\pi\)
\(348\) −38.2787 −2.05196
\(349\) −8.19072 −0.438439 −0.219220 0.975676i \(-0.570351\pi\)
−0.219220 + 0.975676i \(0.570351\pi\)
\(350\) 5.99447 0.320418
\(351\) 10.5567 0.563473
\(352\) 1.74807 0.0931722
\(353\) 17.5573 0.934483 0.467242 0.884130i \(-0.345248\pi\)
0.467242 + 0.884130i \(0.345248\pi\)
\(354\) 3.90021 0.207294
\(355\) −3.97237 −0.210832
\(356\) 7.63915 0.404874
\(357\) −13.8999 −0.735662
\(358\) −7.37147 −0.389595
\(359\) 20.7724 1.09632 0.548162 0.836372i \(-0.315328\pi\)
0.548162 + 0.836372i \(0.315328\pi\)
\(360\) −17.2231 −0.907739
\(361\) 11.2554 0.592391
\(362\) 2.82726 0.148597
\(363\) 33.6085 1.76399
\(364\) 6.03100 0.316110
\(365\) 20.6953 1.08324
\(366\) 10.3104 0.538935
\(367\) −34.8797 −1.82070 −0.910352 0.413835i \(-0.864189\pi\)
−0.910352 + 0.413835i \(0.864189\pi\)
\(368\) 7.62208 0.397329
\(369\) −47.3279 −2.46379
\(370\) 3.86047 0.200696
\(371\) −16.2652 −0.844448
\(372\) 54.3990 2.82046
\(373\) −9.73592 −0.504107 −0.252054 0.967713i \(-0.581106\pi\)
−0.252054 + 0.967713i \(0.581106\pi\)
\(374\) 0.220643 0.0114092
\(375\) −34.1089 −1.76137
\(376\) −14.9823 −0.772652
\(377\) 6.95486 0.358193
\(378\) 20.6027 1.05969
\(379\) 17.7671 0.912634 0.456317 0.889817i \(-0.349168\pi\)
0.456317 + 0.889817i \(0.349168\pi\)
\(380\) −12.8336 −0.658351
\(381\) −21.6856 −1.11099
\(382\) −4.11529 −0.210557
\(383\) 12.7845 0.653257 0.326628 0.945153i \(-0.394087\pi\)
0.326628 + 0.945153i \(0.394087\pi\)
\(384\) −35.4256 −1.80781
\(385\) 1.64856 0.0840183
\(386\) −12.3206 −0.627103
\(387\) −42.3862 −2.15461
\(388\) 12.0508 0.611787
\(389\) −10.8081 −0.547992 −0.273996 0.961731i \(-0.588345\pi\)
−0.273996 + 0.961731i \(0.588345\pi\)
\(390\) 2.11532 0.107113
\(391\) 3.91146 0.197811
\(392\) 11.7557 0.593750
\(393\) 10.3987 0.524543
\(394\) 1.00260 0.0505105
\(395\) −14.0473 −0.706796
\(396\) 3.80909 0.191414
\(397\) −2.02404 −0.101584 −0.0507919 0.998709i \(-0.516175\pi\)
−0.0507919 + 0.998709i \(0.516175\pi\)
\(398\) −1.17632 −0.0589636
\(399\) 61.2926 3.06847
\(400\) −7.71171 −0.385585
\(401\) −7.98951 −0.398977 −0.199488 0.979900i \(-0.563928\pi\)
−0.199488 + 0.979900i \(0.563928\pi\)
\(402\) 1.29219 0.0644485
\(403\) −9.88375 −0.492345
\(404\) 6.77581 0.337109
\(405\) −19.0188 −0.945053
\(406\) 13.5733 0.673632
\(407\) −1.84314 −0.0913612
\(408\) −7.51192 −0.371896
\(409\) −17.7234 −0.876366 −0.438183 0.898886i \(-0.644378\pi\)
−0.438183 + 0.898886i \(0.644378\pi\)
\(410\) −5.12777 −0.253243
\(411\) 67.5833 3.33364
\(412\) 24.3250 1.19840
\(413\) 8.70960 0.428571
\(414\) −10.7223 −0.526971
\(415\) 18.0081 0.883984
\(416\) 5.00886 0.245579
\(417\) 24.1172 1.18102
\(418\) −0.972941 −0.0475881
\(419\) 23.2376 1.13523 0.567615 0.823294i \(-0.307866\pi\)
0.567615 + 0.823294i \(0.307866\pi\)
\(420\) −25.9989 −1.26862
\(421\) 16.0445 0.781961 0.390980 0.920399i \(-0.372136\pi\)
0.390980 + 0.920399i \(0.372136\pi\)
\(422\) −7.59493 −0.369715
\(423\) −50.1709 −2.43939
\(424\) −8.79020 −0.426890
\(425\) −3.95745 −0.191964
\(426\) 4.74939 0.230108
\(427\) 23.0243 1.11422
\(428\) −13.2060 −0.638338
\(429\) −1.00994 −0.0487602
\(430\) −4.59236 −0.221463
\(431\) 16.8476 0.811519 0.405759 0.913980i \(-0.367007\pi\)
0.405759 + 0.913980i \(0.367007\pi\)
\(432\) −26.5048 −1.27521
\(433\) 13.9110 0.668518 0.334259 0.942481i \(-0.391514\pi\)
0.334259 + 0.942481i \(0.391514\pi\)
\(434\) −19.2894 −0.925922
\(435\) −29.9815 −1.43750
\(436\) 7.89942 0.378314
\(437\) −17.2478 −0.825075
\(438\) −24.7434 −1.18229
\(439\) 36.1726 1.72643 0.863213 0.504840i \(-0.168448\pi\)
0.863213 + 0.504840i \(0.168448\pi\)
\(440\) 0.890928 0.0424733
\(441\) 39.3660 1.87457
\(442\) 0.632225 0.0300719
\(443\) −33.8300 −1.60731 −0.803655 0.595096i \(-0.797114\pi\)
−0.803655 + 0.595096i \(0.797114\pi\)
\(444\) 29.0676 1.37949
\(445\) 5.98330 0.283636
\(446\) 13.1366 0.622036
\(447\) 5.36186 0.253607
\(448\) −7.77106 −0.367148
\(449\) −35.4973 −1.67522 −0.837611 0.546267i \(-0.816049\pi\)
−0.837611 + 0.546267i \(0.816049\pi\)
\(450\) 10.8484 0.511396
\(451\) 2.44820 0.115281
\(452\) 27.6118 1.29875
\(453\) −28.1104 −1.32074
\(454\) 10.5521 0.495236
\(455\) 4.72373 0.221452
\(456\) 33.1243 1.55119
\(457\) −27.6267 −1.29232 −0.646162 0.763201i \(-0.723627\pi\)
−0.646162 + 0.763201i \(0.723627\pi\)
\(458\) −12.9048 −0.603002
\(459\) −13.6016 −0.634866
\(460\) 7.31611 0.341116
\(461\) −1.39808 −0.0651150 −0.0325575 0.999470i \(-0.510365\pi\)
−0.0325575 + 0.999470i \(0.510365\pi\)
\(462\) −1.97102 −0.0917003
\(463\) −16.4657 −0.765225 −0.382613 0.923909i \(-0.624976\pi\)
−0.382613 + 0.923909i \(0.624976\pi\)
\(464\) −17.4617 −0.810637
\(465\) 42.6076 1.97588
\(466\) 11.6378 0.539112
\(467\) 19.2949 0.892863 0.446432 0.894818i \(-0.352695\pi\)
0.446432 + 0.894818i \(0.352695\pi\)
\(468\) 10.9145 0.504521
\(469\) 2.88560 0.133245
\(470\) −5.43580 −0.250735
\(471\) −32.2137 −1.48433
\(472\) 4.70692 0.216654
\(473\) 2.19258 0.100815
\(474\) 16.7950 0.771420
\(475\) 17.4506 0.800690
\(476\) −7.77054 −0.356162
\(477\) −29.4356 −1.34776
\(478\) 3.08219 0.140976
\(479\) −26.2032 −1.19725 −0.598626 0.801028i \(-0.704287\pi\)
−0.598626 + 0.801028i \(0.704287\pi\)
\(480\) −21.5926 −0.985561
\(481\) −5.28129 −0.240806
\(482\) −4.17587 −0.190206
\(483\) −34.9413 −1.58989
\(484\) 18.7883 0.854014
\(485\) 9.43870 0.428589
\(486\) 5.61417 0.254664
\(487\) 30.4664 1.38057 0.690283 0.723539i \(-0.257486\pi\)
0.690283 + 0.723539i \(0.257486\pi\)
\(488\) 12.4430 0.563268
\(489\) −73.1539 −3.30814
\(490\) 4.26514 0.192679
\(491\) 36.3956 1.64251 0.821255 0.570562i \(-0.193274\pi\)
0.821255 + 0.570562i \(0.193274\pi\)
\(492\) −38.6098 −1.74067
\(493\) −8.96087 −0.403577
\(494\) −2.78784 −0.125431
\(495\) 2.98344 0.134096
\(496\) 24.8153 1.11424
\(497\) 10.6059 0.475740
\(498\) −21.5306 −0.964809
\(499\) −2.03599 −0.0911435 −0.0455717 0.998961i \(-0.514511\pi\)
−0.0455717 + 0.998961i \(0.514511\pi\)
\(500\) −19.0680 −0.852748
\(501\) 45.0067 2.01075
\(502\) 10.6035 0.473256
\(503\) 23.3684 1.04195 0.520973 0.853573i \(-0.325569\pi\)
0.520973 + 0.853573i \(0.325569\pi\)
\(504\) 45.9843 2.04830
\(505\) 5.30709 0.236163
\(506\) 0.554648 0.0246571
\(507\) 37.2418 1.65397
\(508\) −12.1230 −0.537872
\(509\) 32.6744 1.44827 0.724133 0.689661i \(-0.242240\pi\)
0.724133 + 0.689661i \(0.242240\pi\)
\(510\) −2.72544 −0.120685
\(511\) −55.2548 −2.44433
\(512\) −22.0584 −0.974855
\(513\) 59.9769 2.64805
\(514\) 8.35640 0.368585
\(515\) 19.0523 0.839546
\(516\) −34.5785 −1.52223
\(517\) 2.59527 0.114140
\(518\) −10.3071 −0.452869
\(519\) 51.1412 2.24485
\(520\) 2.55284 0.111949
\(521\) 8.73632 0.382745 0.191373 0.981517i \(-0.438706\pi\)
0.191373 + 0.981517i \(0.438706\pi\)
\(522\) 24.5640 1.07514
\(523\) 15.6850 0.685855 0.342928 0.939362i \(-0.388581\pi\)
0.342928 + 0.939362i \(0.388581\pi\)
\(524\) 5.81321 0.253951
\(525\) 35.3522 1.54290
\(526\) −13.7497 −0.599516
\(527\) 12.7346 0.554726
\(528\) 2.53566 0.110351
\(529\) −13.1675 −0.572499
\(530\) −3.18922 −0.138531
\(531\) 15.7620 0.684013
\(532\) 34.2647 1.48556
\(533\) 7.01502 0.303854
\(534\) −7.15366 −0.309569
\(535\) −10.3435 −0.447190
\(536\) 1.55946 0.0673585
\(537\) −43.4730 −1.87600
\(538\) −3.95929 −0.170697
\(539\) −2.03635 −0.0877117
\(540\) −25.4408 −1.09480
\(541\) 28.4519 1.22324 0.611621 0.791151i \(-0.290518\pi\)
0.611621 + 0.791151i \(0.290518\pi\)
\(542\) −0.447662 −0.0192287
\(543\) 16.6736 0.715534
\(544\) −6.45358 −0.276695
\(545\) 6.18716 0.265029
\(546\) −5.64771 −0.241700
\(547\) −1.00000 −0.0427569
\(548\) 37.7814 1.61394
\(549\) 41.6677 1.77834
\(550\) −0.561170 −0.0239284
\(551\) 39.5135 1.68333
\(552\) −18.8833 −0.803727
\(553\) 37.5051 1.59488
\(554\) 4.04029 0.171655
\(555\) 22.7670 0.966404
\(556\) 13.4824 0.571779
\(557\) 32.4488 1.37490 0.687450 0.726231i \(-0.258730\pi\)
0.687450 + 0.726231i \(0.258730\pi\)
\(558\) −34.9086 −1.47780
\(559\) 6.28256 0.265724
\(560\) −11.8599 −0.501174
\(561\) 1.30124 0.0549382
\(562\) −2.64065 −0.111389
\(563\) 23.7803 1.00222 0.501111 0.865383i \(-0.332925\pi\)
0.501111 + 0.865383i \(0.332925\pi\)
\(564\) −40.9292 −1.72343
\(565\) 21.6267 0.909842
\(566\) −0.756406 −0.0317941
\(567\) 50.7786 2.13250
\(568\) 5.73173 0.240498
\(569\) 42.0011 1.76078 0.880389 0.474252i \(-0.157281\pi\)
0.880389 + 0.474252i \(0.157281\pi\)
\(570\) 12.0180 0.503379
\(571\) −39.1170 −1.63699 −0.818497 0.574510i \(-0.805193\pi\)
−0.818497 + 0.574510i \(0.805193\pi\)
\(572\) −0.564589 −0.0236067
\(573\) −24.2698 −1.01389
\(574\) 13.6907 0.571439
\(575\) −9.94814 −0.414866
\(576\) −14.0635 −0.585979
\(577\) −20.9523 −0.872256 −0.436128 0.899885i \(-0.643651\pi\)
−0.436128 + 0.899885i \(0.643651\pi\)
\(578\) 8.08503 0.336293
\(579\) −72.6604 −3.01966
\(580\) −16.7607 −0.695950
\(581\) −48.0802 −1.99470
\(582\) −11.2850 −0.467776
\(583\) 1.52266 0.0630622
\(584\) −29.8613 −1.23567
\(585\) 8.54867 0.353444
\(586\) −6.28077 −0.259456
\(587\) −37.1311 −1.53257 −0.766283 0.642503i \(-0.777896\pi\)
−0.766283 + 0.642503i \(0.777896\pi\)
\(588\) 32.1146 1.32438
\(589\) −56.1538 −2.31378
\(590\) 1.70774 0.0703067
\(591\) 5.91282 0.243221
\(592\) 13.2598 0.544975
\(593\) 19.1878 0.787949 0.393974 0.919121i \(-0.371100\pi\)
0.393974 + 0.919121i \(0.371100\pi\)
\(594\) −1.92871 −0.0791360
\(595\) −6.08621 −0.249510
\(596\) 2.99746 0.122781
\(597\) −6.93731 −0.283925
\(598\) 1.58927 0.0649902
\(599\) 31.5960 1.29098 0.645488 0.763770i \(-0.276654\pi\)
0.645488 + 0.763770i \(0.276654\pi\)
\(600\) 19.1053 0.779972
\(601\) 16.9578 0.691725 0.345862 0.938285i \(-0.387586\pi\)
0.345862 + 0.938285i \(0.387586\pi\)
\(602\) 12.2612 0.499730
\(603\) 5.22215 0.212662
\(604\) −15.7147 −0.639422
\(605\) 14.7158 0.598282
\(606\) −6.34519 −0.257756
\(607\) −10.5096 −0.426572 −0.213286 0.976990i \(-0.568417\pi\)
−0.213286 + 0.976990i \(0.568417\pi\)
\(608\) 28.4575 1.15410
\(609\) 80.0481 3.24371
\(610\) 4.51452 0.182787
\(611\) 7.43642 0.300845
\(612\) −14.0626 −0.568445
\(613\) −14.8071 −0.598055 −0.299027 0.954245i \(-0.596662\pi\)
−0.299027 + 0.954245i \(0.596662\pi\)
\(614\) −7.71542 −0.311369
\(615\) −30.2408 −1.21943
\(616\) −2.37870 −0.0958407
\(617\) −27.3637 −1.10162 −0.550811 0.834630i \(-0.685681\pi\)
−0.550811 + 0.834630i \(0.685681\pi\)
\(618\) −22.7790 −0.916307
\(619\) 4.81738 0.193627 0.0968134 0.995303i \(-0.469135\pi\)
0.0968134 + 0.995303i \(0.469135\pi\)
\(620\) 23.8191 0.956599
\(621\) −34.1913 −1.37205
\(622\) 13.4165 0.537954
\(623\) −15.9749 −0.640021
\(624\) 7.26562 0.290857
\(625\) 0.927908 0.0371163
\(626\) 1.72556 0.0689674
\(627\) −5.73788 −0.229149
\(628\) −18.0085 −0.718619
\(629\) 6.80459 0.271317
\(630\) 16.6838 0.664700
\(631\) 29.9291 1.19146 0.595730 0.803185i \(-0.296863\pi\)
0.595730 + 0.803185i \(0.296863\pi\)
\(632\) 20.2688 0.806250
\(633\) −44.7908 −1.78028
\(634\) −11.2999 −0.448778
\(635\) −9.49525 −0.376808
\(636\) −24.0134 −0.952194
\(637\) −5.83489 −0.231187
\(638\) −1.27066 −0.0503059
\(639\) 19.1938 0.759294
\(640\) −15.5114 −0.613143
\(641\) 2.88534 0.113964 0.0569821 0.998375i \(-0.481852\pi\)
0.0569821 + 0.998375i \(0.481852\pi\)
\(642\) 12.3668 0.488077
\(643\) −32.7470 −1.29142 −0.645708 0.763584i \(-0.723438\pi\)
−0.645708 + 0.763584i \(0.723438\pi\)
\(644\) −19.5334 −0.769724
\(645\) −27.0833 −1.06640
\(646\) 3.59194 0.141323
\(647\) 20.0383 0.787787 0.393894 0.919156i \(-0.371128\pi\)
0.393894 + 0.919156i \(0.371128\pi\)
\(648\) 27.4422 1.07803
\(649\) −0.815346 −0.0320051
\(650\) −1.60796 −0.0630694
\(651\) −113.759 −4.45855
\(652\) −40.8956 −1.60159
\(653\) −8.58315 −0.335885 −0.167942 0.985797i \(-0.553712\pi\)
−0.167942 + 0.985797i \(0.553712\pi\)
\(654\) −7.39740 −0.289261
\(655\) 4.55315 0.177906
\(656\) −17.6127 −0.687660
\(657\) −99.9961 −3.90122
\(658\) 14.5131 0.565781
\(659\) −34.5534 −1.34601 −0.673004 0.739638i \(-0.734997\pi\)
−0.673004 + 0.739638i \(0.734997\pi\)
\(660\) 2.43387 0.0947384
\(661\) −18.6496 −0.725386 −0.362693 0.931909i \(-0.618143\pi\)
−0.362693 + 0.931909i \(0.618143\pi\)
\(662\) 10.0969 0.392428
\(663\) 3.72853 0.144804
\(664\) −25.9839 −1.00837
\(665\) 26.8375 1.04072
\(666\) −18.6531 −0.722792
\(667\) −22.5256 −0.872196
\(668\) 25.1603 0.973480
\(669\) 77.4726 2.99526
\(670\) 0.565797 0.0218586
\(671\) −2.15541 −0.0832087
\(672\) 57.6503 2.22391
\(673\) 23.0907 0.890083 0.445041 0.895510i \(-0.353189\pi\)
0.445041 + 0.895510i \(0.353189\pi\)
\(674\) −0.0959571 −0.00369613
\(675\) 34.5933 1.33150
\(676\) 20.8195 0.800749
\(677\) −41.7475 −1.60449 −0.802243 0.596997i \(-0.796360\pi\)
−0.802243 + 0.596997i \(0.796360\pi\)
\(678\) −25.8570 −0.993031
\(679\) −25.2005 −0.967108
\(680\) −3.28917 −0.126134
\(681\) 62.2308 2.38469
\(682\) 1.80577 0.0691465
\(683\) −46.2392 −1.76930 −0.884648 0.466260i \(-0.845601\pi\)
−0.884648 + 0.466260i \(0.845601\pi\)
\(684\) 62.0098 2.37100
\(685\) 29.5920 1.13065
\(686\) 1.83878 0.0702050
\(687\) −76.1056 −2.90361
\(688\) −15.7737 −0.601367
\(689\) 4.36299 0.166217
\(690\) −6.85115 −0.260819
\(691\) −51.8556 −1.97268 −0.986340 0.164724i \(-0.947327\pi\)
−0.986340 + 0.164724i \(0.947327\pi\)
\(692\) 28.5897 1.08682
\(693\) −7.96553 −0.302585
\(694\) 12.8374 0.487303
\(695\) 10.5599 0.400561
\(696\) 43.2603 1.63978
\(697\) −9.03838 −0.342353
\(698\) 4.28790 0.162299
\(699\) 68.6337 2.59596
\(700\) 19.7631 0.746975
\(701\) 46.8098 1.76798 0.883992 0.467503i \(-0.154846\pi\)
0.883992 + 0.467503i \(0.154846\pi\)
\(702\) −5.52648 −0.208584
\(703\) −30.0053 −1.13167
\(704\) 0.727485 0.0274181
\(705\) −32.0574 −1.20735
\(706\) −9.19138 −0.345922
\(707\) −14.1695 −0.532899
\(708\) 12.8586 0.483254
\(709\) 2.30760 0.0866637 0.0433318 0.999061i \(-0.486203\pi\)
0.0433318 + 0.999061i \(0.486203\pi\)
\(710\) 2.07956 0.0780446
\(711\) 67.8739 2.54547
\(712\) −8.63330 −0.323547
\(713\) 32.0118 1.19885
\(714\) 7.27670 0.272324
\(715\) −0.442210 −0.0165377
\(716\) −24.3029 −0.908242
\(717\) 18.1771 0.678837
\(718\) −10.8745 −0.405831
\(719\) 1.47404 0.0549723 0.0274862 0.999622i \(-0.491250\pi\)
0.0274862 + 0.999622i \(0.491250\pi\)
\(720\) −21.4632 −0.799888
\(721\) −50.8681 −1.89443
\(722\) −5.89229 −0.219288
\(723\) −24.6271 −0.915890
\(724\) 9.32114 0.346417
\(725\) 22.7905 0.846418
\(726\) −17.5943 −0.652984
\(727\) −11.7670 −0.436412 −0.218206 0.975903i \(-0.570021\pi\)
−0.218206 + 0.975903i \(0.570021\pi\)
\(728\) −6.81587 −0.252613
\(729\) −9.09751 −0.336945
\(730\) −10.8341 −0.400990
\(731\) −8.09466 −0.299392
\(732\) 33.9923 1.25639
\(733\) 20.5086 0.757502 0.378751 0.925499i \(-0.376354\pi\)
0.378751 + 0.925499i \(0.376354\pi\)
\(734\) 18.2597 0.673979
\(735\) 25.1535 0.927800
\(736\) −16.2229 −0.597983
\(737\) −0.270134 −0.00995052
\(738\) 24.7765 0.912034
\(739\) −31.6389 −1.16385 −0.581927 0.813241i \(-0.697701\pi\)
−0.581927 + 0.813241i \(0.697701\pi\)
\(740\) 12.7275 0.467873
\(741\) −16.4412 −0.603982
\(742\) 8.51495 0.312594
\(743\) −40.5825 −1.48883 −0.744414 0.667718i \(-0.767271\pi\)
−0.744414 + 0.667718i \(0.767271\pi\)
\(744\) −61.4785 −2.25391
\(745\) 2.34774 0.0860145
\(746\) 5.09682 0.186608
\(747\) −87.0120 −3.18360
\(748\) 0.727436 0.0265977
\(749\) 27.6163 1.00908
\(750\) 17.8562 0.652016
\(751\) 27.4293 1.00091 0.500454 0.865763i \(-0.333166\pi\)
0.500454 + 0.865763i \(0.333166\pi\)
\(752\) −18.6707 −0.680851
\(753\) 62.5336 2.27885
\(754\) −3.64091 −0.132594
\(755\) −12.3084 −0.447949
\(756\) 67.9247 2.47040
\(757\) −19.8595 −0.721805 −0.360903 0.932603i \(-0.617531\pi\)
−0.360903 + 0.932603i \(0.617531\pi\)
\(758\) −9.30119 −0.337834
\(759\) 3.27102 0.118730
\(760\) 14.5038 0.526108
\(761\) −30.0646 −1.08984 −0.544920 0.838488i \(-0.683440\pi\)
−0.544920 + 0.838488i \(0.683440\pi\)
\(762\) 11.3526 0.411260
\(763\) −16.5192 −0.598035
\(764\) −13.5677 −0.490861
\(765\) −11.0144 −0.398226
\(766\) −6.69276 −0.241819
\(767\) −2.33627 −0.0843578
\(768\) 5.25084 0.189473
\(769\) −27.2458 −0.982510 −0.491255 0.871016i \(-0.663462\pi\)
−0.491255 + 0.871016i \(0.663462\pi\)
\(770\) −0.863030 −0.0311015
\(771\) 49.2816 1.77483
\(772\) −40.6197 −1.46193
\(773\) −42.4735 −1.52767 −0.763833 0.645414i \(-0.776685\pi\)
−0.763833 + 0.645414i \(0.776685\pi\)
\(774\) 22.1895 0.797584
\(775\) −32.3882 −1.16342
\(776\) −13.6191 −0.488897
\(777\) −60.7859 −2.18068
\(778\) 5.65810 0.202853
\(779\) 39.8553 1.42796
\(780\) 6.97395 0.249708
\(781\) −0.992867 −0.0355276
\(782\) −2.04767 −0.0732246
\(783\) 78.3298 2.79928
\(784\) 14.6497 0.523205
\(785\) −14.1050 −0.503431
\(786\) −5.44376 −0.194173
\(787\) 45.4020 1.61841 0.809203 0.587529i \(-0.199899\pi\)
0.809203 + 0.587529i \(0.199899\pi\)
\(788\) 3.30547 0.117753
\(789\) −81.0885 −2.88683
\(790\) 7.35384 0.261638
\(791\) −57.7414 −2.05305
\(792\) −4.30480 −0.152965
\(793\) −6.17606 −0.219318
\(794\) 1.05960 0.0376038
\(795\) −18.8083 −0.667062
\(796\) −3.87820 −0.137459
\(797\) 27.6932 0.980944 0.490472 0.871457i \(-0.336824\pi\)
0.490472 + 0.871457i \(0.336824\pi\)
\(798\) −32.0871 −1.13587
\(799\) −9.58133 −0.338963
\(800\) 16.4136 0.580309
\(801\) −28.9102 −1.02149
\(802\) 4.18256 0.147691
\(803\) 5.17265 0.182539
\(804\) 4.26020 0.150246
\(805\) −15.2994 −0.539232
\(806\) 5.17421 0.182254
\(807\) −23.3498 −0.821950
\(808\) −7.65760 −0.269394
\(809\) 12.6422 0.444475 0.222238 0.974993i \(-0.428664\pi\)
0.222238 + 0.974993i \(0.428664\pi\)
\(810\) 9.95647 0.349835
\(811\) −7.30807 −0.256621 −0.128310 0.991734i \(-0.540955\pi\)
−0.128310 + 0.991734i \(0.540955\pi\)
\(812\) 44.7497 1.57041
\(813\) −2.64007 −0.0925913
\(814\) 0.964897 0.0338196
\(815\) −32.0311 −1.12200
\(816\) −9.36126 −0.327710
\(817\) 35.6939 1.24877
\(818\) 9.27831 0.324409
\(819\) −22.8242 −0.797543
\(820\) −16.9057 −0.590372
\(821\) 36.6884 1.28043 0.640217 0.768194i \(-0.278844\pi\)
0.640217 + 0.768194i \(0.278844\pi\)
\(822\) −35.3803 −1.23403
\(823\) −4.83046 −0.168379 −0.0841897 0.996450i \(-0.526830\pi\)
−0.0841897 + 0.996450i \(0.526830\pi\)
\(824\) −27.4906 −0.957680
\(825\) −3.30948 −0.115221
\(826\) −4.55953 −0.158646
\(827\) −23.1753 −0.805883 −0.402942 0.915226i \(-0.632012\pi\)
−0.402942 + 0.915226i \(0.632012\pi\)
\(828\) −35.3501 −1.22850
\(829\) 27.0086 0.938046 0.469023 0.883186i \(-0.344606\pi\)
0.469023 + 0.883186i \(0.344606\pi\)
\(830\) −9.42737 −0.327229
\(831\) 23.8274 0.826565
\(832\) 2.08451 0.0722676
\(833\) 7.51787 0.260479
\(834\) −12.6255 −0.437186
\(835\) 19.7066 0.681974
\(836\) −3.20768 −0.110940
\(837\) −111.317 −3.84767
\(838\) −12.1650 −0.420234
\(839\) −19.5336 −0.674373 −0.337187 0.941438i \(-0.609475\pi\)
−0.337187 + 0.941438i \(0.609475\pi\)
\(840\) 29.3824 1.01379
\(841\) 22.6046 0.779470
\(842\) −8.39939 −0.289462
\(843\) −15.5732 −0.536368
\(844\) −25.0396 −0.861899
\(845\) 16.3067 0.560967
\(846\) 26.2648 0.903003
\(847\) −39.2899 −1.35002
\(848\) −10.9542 −0.376170
\(849\) −4.46088 −0.153097
\(850\) 2.07175 0.0710604
\(851\) 17.1052 0.586359
\(852\) 15.6582 0.536441
\(853\) 38.7260 1.32595 0.662977 0.748640i \(-0.269293\pi\)
0.662977 + 0.748640i \(0.269293\pi\)
\(854\) −12.0534 −0.412458
\(855\) 48.5686 1.66101
\(856\) 14.9247 0.510115
\(857\) 35.3753 1.20840 0.604199 0.796834i \(-0.293493\pi\)
0.604199 + 0.796834i \(0.293493\pi\)
\(858\) 0.528708 0.0180498
\(859\) 7.81176 0.266534 0.133267 0.991080i \(-0.457453\pi\)
0.133267 + 0.991080i \(0.457453\pi\)
\(860\) −15.1405 −0.516287
\(861\) 80.7405 2.75163
\(862\) −8.81981 −0.300404
\(863\) 7.24097 0.246486 0.123243 0.992377i \(-0.460671\pi\)
0.123243 + 0.992377i \(0.460671\pi\)
\(864\) 56.4128 1.91920
\(865\) 22.3926 0.761372
\(866\) −7.28247 −0.247468
\(867\) 47.6812 1.61934
\(868\) −63.5950 −2.15856
\(869\) −3.51102 −0.119103
\(870\) 15.6955 0.532128
\(871\) −0.774035 −0.0262272
\(872\) −8.92745 −0.302322
\(873\) −45.6061 −1.54353
\(874\) 9.02934 0.305422
\(875\) 39.8749 1.34802
\(876\) −81.5763 −2.75621
\(877\) −27.0100 −0.912062 −0.456031 0.889964i \(-0.650729\pi\)
−0.456031 + 0.889964i \(0.650729\pi\)
\(878\) −18.9366 −0.639080
\(879\) −37.0406 −1.24935
\(880\) 1.11026 0.0374270
\(881\) 20.0865 0.676731 0.338366 0.941015i \(-0.390126\pi\)
0.338366 + 0.941015i \(0.390126\pi\)
\(882\) −20.6084 −0.693920
\(883\) 11.1980 0.376842 0.188421 0.982088i \(-0.439663\pi\)
0.188421 + 0.982088i \(0.439663\pi\)
\(884\) 2.08437 0.0701051
\(885\) 10.0714 0.338545
\(886\) 17.7102 0.594986
\(887\) 56.2189 1.88765 0.943824 0.330450i \(-0.107200\pi\)
0.943824 + 0.330450i \(0.107200\pi\)
\(888\) −32.8505 −1.10239
\(889\) 25.3515 0.850263
\(890\) −3.13230 −0.104995
\(891\) −4.75362 −0.159252
\(892\) 43.3098 1.45012
\(893\) 42.2495 1.41382
\(894\) −2.80697 −0.0938790
\(895\) −19.0351 −0.636272
\(896\) 41.4142 1.38355
\(897\) 9.37268 0.312945
\(898\) 18.5831 0.620125
\(899\) −73.3368 −2.44592
\(900\) 35.7658 1.19219
\(901\) −5.62143 −0.187277
\(902\) −1.28165 −0.0426743
\(903\) 72.3101 2.40633
\(904\) −31.2051 −1.03787
\(905\) 7.30071 0.242684
\(906\) 14.7160 0.488906
\(907\) −41.1969 −1.36792 −0.683960 0.729520i \(-0.739744\pi\)
−0.683960 + 0.729520i \(0.739744\pi\)
\(908\) 34.7892 1.15452
\(909\) −25.6429 −0.850522
\(910\) −2.47290 −0.0819760
\(911\) 26.4943 0.877795 0.438897 0.898537i \(-0.355369\pi\)
0.438897 + 0.898537i \(0.355369\pi\)
\(912\) 41.2791 1.36689
\(913\) 4.50101 0.148962
\(914\) 14.4628 0.478386
\(915\) 26.6242 0.880169
\(916\) −42.5457 −1.40575
\(917\) −12.1565 −0.401444
\(918\) 7.12050 0.235012
\(919\) −29.7090 −0.980008 −0.490004 0.871720i \(-0.663005\pi\)
−0.490004 + 0.871720i \(0.663005\pi\)
\(920\) −8.26823 −0.272595
\(921\) −45.5014 −1.49932
\(922\) 0.731902 0.0241039
\(923\) −2.84493 −0.0936421
\(924\) −6.49824 −0.213776
\(925\) −17.3064 −0.569029
\(926\) 8.61989 0.283267
\(927\) −92.0574 −3.02356
\(928\) 37.1654 1.22002
\(929\) 31.6127 1.03718 0.518589 0.855024i \(-0.326457\pi\)
0.518589 + 0.855024i \(0.326457\pi\)
\(930\) −22.3053 −0.731421
\(931\) −33.1505 −1.08646
\(932\) 38.3686 1.25680
\(933\) 79.1235 2.59039
\(934\) −10.1010 −0.330516
\(935\) 0.569758 0.0186331
\(936\) −12.3349 −0.403178
\(937\) 25.5752 0.835506 0.417753 0.908561i \(-0.362818\pi\)
0.417753 + 0.908561i \(0.362818\pi\)
\(938\) −1.51063 −0.0493238
\(939\) 10.1764 0.332096
\(940\) −17.9212 −0.584526
\(941\) −38.4356 −1.25296 −0.626482 0.779436i \(-0.715506\pi\)
−0.626482 + 0.779436i \(0.715506\pi\)
\(942\) 16.8641 0.549461
\(943\) −22.7205 −0.739880
\(944\) 5.86570 0.190912
\(945\) 53.2015 1.73065
\(946\) −1.14783 −0.0373191
\(947\) 39.7014 1.29012 0.645060 0.764132i \(-0.276832\pi\)
0.645060 + 0.764132i \(0.276832\pi\)
\(948\) 55.3712 1.79837
\(949\) 14.8216 0.481129
\(950\) −9.13552 −0.296395
\(951\) −66.6409 −2.16098
\(952\) 8.78179 0.284620
\(953\) −22.9371 −0.743007 −0.371504 0.928432i \(-0.621158\pi\)
−0.371504 + 0.928432i \(0.621158\pi\)
\(954\) 15.4097 0.498908
\(955\) −10.6268 −0.343874
\(956\) 10.1616 0.328651
\(957\) −7.49367 −0.242236
\(958\) 13.7175 0.443193
\(959\) −79.0081 −2.55130
\(960\) −8.98608 −0.290024
\(961\) 73.2211 2.36197
\(962\) 2.76479 0.0891404
\(963\) 49.9780 1.61052
\(964\) −13.7674 −0.443417
\(965\) −31.8150 −1.02416
\(966\) 18.2920 0.588536
\(967\) −26.6775 −0.857891 −0.428946 0.903330i \(-0.641115\pi\)
−0.428946 + 0.903330i \(0.641115\pi\)
\(968\) −21.2334 −0.682468
\(969\) 21.1834 0.680508
\(970\) −4.94122 −0.158653
\(971\) −9.36106 −0.300411 −0.150205 0.988655i \(-0.547993\pi\)
−0.150205 + 0.988655i \(0.547993\pi\)
\(972\) 18.5093 0.593685
\(973\) −28.1942 −0.903863
\(974\) −15.9494 −0.511051
\(975\) −9.48289 −0.303696
\(976\) 15.5063 0.496345
\(977\) 17.6428 0.564445 0.282222 0.959349i \(-0.408928\pi\)
0.282222 + 0.959349i \(0.408928\pi\)
\(978\) 38.2966 1.22459
\(979\) 1.49548 0.0477959
\(980\) 14.0617 0.449183
\(981\) −29.8952 −0.954481
\(982\) −19.0533 −0.608016
\(983\) −1.76510 −0.0562979 −0.0281490 0.999604i \(-0.508961\pi\)
−0.0281490 + 0.999604i \(0.508961\pi\)
\(984\) 43.6345 1.39102
\(985\) 2.58898 0.0824919
\(986\) 4.69107 0.149394
\(987\) 85.5907 2.72438
\(988\) −9.19119 −0.292411
\(989\) −20.3482 −0.647034
\(990\) −1.56185 −0.0496388
\(991\) −33.2753 −1.05702 −0.528512 0.848926i \(-0.677250\pi\)
−0.528512 + 0.848926i \(0.677250\pi\)
\(992\) −52.8169 −1.67694
\(993\) 59.5462 1.88964
\(994\) −5.55226 −0.176107
\(995\) −3.03757 −0.0962973
\(996\) −70.9839 −2.24921
\(997\) 10.1732 0.322187 0.161094 0.986939i \(-0.448498\pi\)
0.161094 + 0.986939i \(0.448498\pi\)
\(998\) 1.06585 0.0337390
\(999\) −59.4811 −1.88190
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 547.2.a.b.1.10 18
3.2 odd 2 4923.2.a.l.1.9 18
4.3 odd 2 8752.2.a.s.1.17 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.10 18 1.1 even 1 trivial
4923.2.a.l.1.9 18 3.2 odd 2
8752.2.a.s.1.17 18 4.3 odd 2