Properties

Label 547.2.a.b.1.6
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 1.6
Root \(1.35726\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.35726 q^{2} +0.387927 q^{3} -0.157846 q^{4} -1.46929 q^{5} -0.526517 q^{6} +0.194696 q^{7} +2.92876 q^{8} -2.84951 q^{9} +O(q^{10})\) \(q-1.35726 q^{2} +0.387927 q^{3} -0.157846 q^{4} -1.46929 q^{5} -0.526517 q^{6} +0.194696 q^{7} +2.92876 q^{8} -2.84951 q^{9} +1.99421 q^{10} +1.87259 q^{11} -0.0612327 q^{12} +6.15611 q^{13} -0.264253 q^{14} -0.569977 q^{15} -3.65939 q^{16} -5.37804 q^{17} +3.86753 q^{18} -3.78340 q^{19} +0.231922 q^{20} +0.0755279 q^{21} -2.54160 q^{22} -3.43075 q^{23} +1.13614 q^{24} -2.84119 q^{25} -8.35544 q^{26} -2.26918 q^{27} -0.0307320 q^{28} +0.0833365 q^{29} +0.773606 q^{30} -3.95043 q^{31} -0.890769 q^{32} +0.726429 q^{33} +7.29940 q^{34} -0.286065 q^{35} +0.449784 q^{36} +3.24239 q^{37} +5.13506 q^{38} +2.38812 q^{39} -4.30319 q^{40} -12.5362 q^{41} -0.102511 q^{42} +2.32367 q^{43} -0.295581 q^{44} +4.18676 q^{45} +4.65642 q^{46} -8.88803 q^{47} -1.41958 q^{48} -6.96209 q^{49} +3.85623 q^{50} -2.08629 q^{51} -0.971717 q^{52} +7.04808 q^{53} +3.07987 q^{54} -2.75138 q^{55} +0.570218 q^{56} -1.46768 q^{57} -0.113109 q^{58} +7.89744 q^{59} +0.0899686 q^{60} -2.95043 q^{61} +5.36176 q^{62} -0.554790 q^{63} +8.52779 q^{64} -9.04511 q^{65} -0.985952 q^{66} -1.41048 q^{67} +0.848903 q^{68} -1.33088 q^{69} +0.388265 q^{70} -11.1843 q^{71} -8.34553 q^{72} -0.418335 q^{73} -4.40076 q^{74} -1.10217 q^{75} +0.597195 q^{76} +0.364587 q^{77} -3.24130 q^{78} +0.962128 q^{79} +5.37671 q^{80} +7.66826 q^{81} +17.0149 q^{82} -15.7469 q^{83} -0.0119218 q^{84} +7.90190 q^{85} -3.15383 q^{86} +0.0323284 q^{87} +5.48437 q^{88} -9.44358 q^{89} -5.68252 q^{90} +1.19857 q^{91} +0.541530 q^{92} -1.53248 q^{93} +12.0634 q^{94} +5.55892 q^{95} -0.345553 q^{96} +11.6561 q^{97} +9.44937 q^{98} -5.33598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35726 −0.959728 −0.479864 0.877343i \(-0.659314\pi\)
−0.479864 + 0.877343i \(0.659314\pi\)
\(3\) 0.387927 0.223970 0.111985 0.993710i \(-0.464279\pi\)
0.111985 + 0.993710i \(0.464279\pi\)
\(4\) −0.157846 −0.0789230
\(5\) −1.46929 −0.657086 −0.328543 0.944489i \(-0.606558\pi\)
−0.328543 + 0.944489i \(0.606558\pi\)
\(6\) −0.526517 −0.214950
\(7\) 0.194696 0.0735883 0.0367941 0.999323i \(-0.488285\pi\)
0.0367941 + 0.999323i \(0.488285\pi\)
\(8\) 2.92876 1.03547
\(9\) −2.84951 −0.949838
\(10\) 1.99421 0.630624
\(11\) 1.87259 0.564608 0.282304 0.959325i \(-0.408901\pi\)
0.282304 + 0.959325i \(0.408901\pi\)
\(12\) −0.0612327 −0.0176764
\(13\) 6.15611 1.70740 0.853698 0.520768i \(-0.174354\pi\)
0.853698 + 0.520768i \(0.174354\pi\)
\(14\) −0.264253 −0.0706247
\(15\) −0.569977 −0.147167
\(16\) −3.65939 −0.914848
\(17\) −5.37804 −1.30437 −0.652183 0.758061i \(-0.726147\pi\)
−0.652183 + 0.758061i \(0.726147\pi\)
\(18\) 3.86753 0.911585
\(19\) −3.78340 −0.867972 −0.433986 0.900920i \(-0.642893\pi\)
−0.433986 + 0.900920i \(0.642893\pi\)
\(20\) 0.231922 0.0518593
\(21\) 0.0755279 0.0164815
\(22\) −2.54160 −0.541870
\(23\) −3.43075 −0.715361 −0.357680 0.933844i \(-0.616432\pi\)
−0.357680 + 0.933844i \(0.616432\pi\)
\(24\) 1.13614 0.231914
\(25\) −2.84119 −0.568237
\(26\) −8.35544 −1.63864
\(27\) −2.26918 −0.436704
\(28\) −0.0307320 −0.00580781
\(29\) 0.0833365 0.0154752 0.00773760 0.999970i \(-0.497537\pi\)
0.00773760 + 0.999970i \(0.497537\pi\)
\(30\) 0.773606 0.141241
\(31\) −3.95043 −0.709518 −0.354759 0.934958i \(-0.615437\pi\)
−0.354759 + 0.934958i \(0.615437\pi\)
\(32\) −0.890769 −0.157467
\(33\) 0.726429 0.126455
\(34\) 7.29940 1.25184
\(35\) −0.286065 −0.0483539
\(36\) 0.449784 0.0749641
\(37\) 3.24239 0.533045 0.266522 0.963829i \(-0.414125\pi\)
0.266522 + 0.963829i \(0.414125\pi\)
\(38\) 5.13506 0.833017
\(39\) 2.38812 0.382405
\(40\) −4.30319 −0.680395
\(41\) −12.5362 −1.95783 −0.978915 0.204269i \(-0.934518\pi\)
−0.978915 + 0.204269i \(0.934518\pi\)
\(42\) −0.102511 −0.0158178
\(43\) 2.32367 0.354357 0.177178 0.984179i \(-0.443303\pi\)
0.177178 + 0.984179i \(0.443303\pi\)
\(44\) −0.295581 −0.0445606
\(45\) 4.18676 0.624125
\(46\) 4.65642 0.686552
\(47\) −8.88803 −1.29645 −0.648226 0.761448i \(-0.724489\pi\)
−0.648226 + 0.761448i \(0.724489\pi\)
\(48\) −1.41958 −0.204898
\(49\) −6.96209 −0.994585
\(50\) 3.85623 0.545353
\(51\) −2.08629 −0.292138
\(52\) −0.971717 −0.134753
\(53\) 7.04808 0.968128 0.484064 0.875033i \(-0.339160\pi\)
0.484064 + 0.875033i \(0.339160\pi\)
\(54\) 3.07987 0.419117
\(55\) −2.75138 −0.370996
\(56\) 0.570218 0.0761986
\(57\) −1.46768 −0.194399
\(58\) −0.113109 −0.0148520
\(59\) 7.89744 1.02816 0.514080 0.857742i \(-0.328134\pi\)
0.514080 + 0.857742i \(0.328134\pi\)
\(60\) 0.0899686 0.0116149
\(61\) −2.95043 −0.377764 −0.188882 0.982000i \(-0.560486\pi\)
−0.188882 + 0.982000i \(0.560486\pi\)
\(62\) 5.36176 0.680944
\(63\) −0.554790 −0.0698969
\(64\) 8.52779 1.06597
\(65\) −9.04511 −1.12191
\(66\) −0.985952 −0.121362
\(67\) −1.41048 −0.172318 −0.0861589 0.996281i \(-0.527459\pi\)
−0.0861589 + 0.996281i \(0.527459\pi\)
\(68\) 0.848903 0.102945
\(69\) −1.33088 −0.160219
\(70\) 0.388265 0.0464065
\(71\) −11.1843 −1.32733 −0.663667 0.748029i \(-0.731001\pi\)
−0.663667 + 0.748029i \(0.731001\pi\)
\(72\) −8.34553 −0.983530
\(73\) −0.418335 −0.0489624 −0.0244812 0.999700i \(-0.507793\pi\)
−0.0244812 + 0.999700i \(0.507793\pi\)
\(74\) −4.40076 −0.511578
\(75\) −1.10217 −0.127268
\(76\) 0.597195 0.0685030
\(77\) 0.364587 0.0415485
\(78\) −3.24130 −0.367004
\(79\) 0.962128 0.108248 0.0541239 0.998534i \(-0.482763\pi\)
0.0541239 + 0.998534i \(0.482763\pi\)
\(80\) 5.37671 0.601134
\(81\) 7.66826 0.852029
\(82\) 17.0149 1.87898
\(83\) −15.7469 −1.72844 −0.864222 0.503110i \(-0.832189\pi\)
−0.864222 + 0.503110i \(0.832189\pi\)
\(84\) −0.0119218 −0.00130077
\(85\) 7.90190 0.857082
\(86\) −3.15383 −0.340086
\(87\) 0.0323284 0.00346597
\(88\) 5.48437 0.584636
\(89\) −9.44358 −1.00102 −0.500509 0.865732i \(-0.666854\pi\)
−0.500509 + 0.865732i \(0.666854\pi\)
\(90\) −5.68252 −0.598990
\(91\) 1.19857 0.125644
\(92\) 0.541530 0.0564585
\(93\) −1.53248 −0.158910
\(94\) 12.0634 1.24424
\(95\) 5.55892 0.570333
\(96\) −0.345553 −0.0352679
\(97\) 11.6561 1.18350 0.591749 0.806122i \(-0.298438\pi\)
0.591749 + 0.806122i \(0.298438\pi\)
\(98\) 9.44937 0.954530
\(99\) −5.33598 −0.536286
\(100\) 0.448470 0.0448470
\(101\) −6.90248 −0.686822 −0.343411 0.939185i \(-0.611582\pi\)
−0.343411 + 0.939185i \(0.611582\pi\)
\(102\) 2.83163 0.280373
\(103\) 2.77350 0.273281 0.136640 0.990621i \(-0.456370\pi\)
0.136640 + 0.990621i \(0.456370\pi\)
\(104\) 18.0297 1.76796
\(105\) −0.110972 −0.0108298
\(106\) −9.56607 −0.929139
\(107\) −3.43676 −0.332244 −0.166122 0.986105i \(-0.553125\pi\)
−0.166122 + 0.986105i \(0.553125\pi\)
\(108\) 0.358181 0.0344660
\(109\) 14.3014 1.36982 0.684912 0.728626i \(-0.259841\pi\)
0.684912 + 0.728626i \(0.259841\pi\)
\(110\) 3.73434 0.356055
\(111\) 1.25781 0.119386
\(112\) −0.712470 −0.0673221
\(113\) 5.25192 0.494059 0.247029 0.969008i \(-0.420546\pi\)
0.247029 + 0.969008i \(0.420546\pi\)
\(114\) 1.99203 0.186570
\(115\) 5.04077 0.470054
\(116\) −0.0131543 −0.00122135
\(117\) −17.5419 −1.62175
\(118\) −10.7189 −0.986753
\(119\) −1.04708 −0.0959861
\(120\) −1.66932 −0.152388
\(121\) −7.49340 −0.681218
\(122\) 4.00450 0.362550
\(123\) −4.86314 −0.438494
\(124\) 0.623560 0.0559973
\(125\) 11.5210 1.03047
\(126\) 0.752994 0.0670820
\(127\) −6.29760 −0.558822 −0.279411 0.960172i \(-0.590139\pi\)
−0.279411 + 0.960172i \(0.590139\pi\)
\(128\) −9.79289 −0.865577
\(129\) 0.901415 0.0793652
\(130\) 12.2766 1.07673
\(131\) 13.4030 1.17102 0.585512 0.810663i \(-0.300893\pi\)
0.585512 + 0.810663i \(0.300893\pi\)
\(132\) −0.114664 −0.00998021
\(133\) −0.736615 −0.0638726
\(134\) 1.91439 0.165378
\(135\) 3.33409 0.286952
\(136\) −15.7510 −1.35064
\(137\) −0.171047 −0.0146136 −0.00730678 0.999973i \(-0.502326\pi\)
−0.00730678 + 0.999973i \(0.502326\pi\)
\(138\) 1.80635 0.153767
\(139\) 13.8883 1.17799 0.588995 0.808137i \(-0.299524\pi\)
0.588995 + 0.808137i \(0.299524\pi\)
\(140\) 0.0451543 0.00381623
\(141\) −3.44790 −0.290366
\(142\) 15.1800 1.27388
\(143\) 11.5279 0.964010
\(144\) 10.4275 0.868957
\(145\) −0.122445 −0.0101685
\(146\) 0.567790 0.0469906
\(147\) −2.70078 −0.222757
\(148\) −0.511798 −0.0420695
\(149\) −12.9947 −1.06457 −0.532286 0.846565i \(-0.678667\pi\)
−0.532286 + 0.846565i \(0.678667\pi\)
\(150\) 1.49593 0.122142
\(151\) 17.0458 1.38716 0.693582 0.720378i \(-0.256031\pi\)
0.693582 + 0.720378i \(0.256031\pi\)
\(152\) −11.0807 −0.898761
\(153\) 15.3248 1.23894
\(154\) −0.494839 −0.0398753
\(155\) 5.80433 0.466215
\(156\) −0.376955 −0.0301805
\(157\) 3.46375 0.276437 0.138219 0.990402i \(-0.455862\pi\)
0.138219 + 0.990402i \(0.455862\pi\)
\(158\) −1.30586 −0.103888
\(159\) 2.73414 0.216831
\(160\) 1.30880 0.103470
\(161\) −0.667954 −0.0526422
\(162\) −10.4078 −0.817716
\(163\) 8.68759 0.680464 0.340232 0.940341i \(-0.389494\pi\)
0.340232 + 0.940341i \(0.389494\pi\)
\(164\) 1.97879 0.154518
\(165\) −1.06733 −0.0830919
\(166\) 21.3726 1.65884
\(167\) 6.93900 0.536956 0.268478 0.963286i \(-0.413479\pi\)
0.268478 + 0.963286i \(0.413479\pi\)
\(168\) 0.221203 0.0170662
\(169\) 24.8977 1.91520
\(170\) −10.7249 −0.822565
\(171\) 10.7809 0.824433
\(172\) −0.366783 −0.0279669
\(173\) −8.06411 −0.613103 −0.306552 0.951854i \(-0.599175\pi\)
−0.306552 + 0.951854i \(0.599175\pi\)
\(174\) −0.0438781 −0.00332639
\(175\) −0.553169 −0.0418156
\(176\) −6.85255 −0.516531
\(177\) 3.06363 0.230276
\(178\) 12.8174 0.960704
\(179\) −4.43362 −0.331384 −0.165692 0.986178i \(-0.552986\pi\)
−0.165692 + 0.986178i \(0.552986\pi\)
\(180\) −0.660864 −0.0492579
\(181\) 15.5528 1.15603 0.578015 0.816026i \(-0.303828\pi\)
0.578015 + 0.816026i \(0.303828\pi\)
\(182\) −1.62677 −0.120584
\(183\) −1.14455 −0.0846076
\(184\) −10.0478 −0.740736
\(185\) −4.76400 −0.350257
\(186\) 2.07997 0.152511
\(187\) −10.0709 −0.736456
\(188\) 1.40294 0.102320
\(189\) −0.441801 −0.0321363
\(190\) −7.54489 −0.547364
\(191\) 22.5547 1.63200 0.816000 0.578052i \(-0.196187\pi\)
0.816000 + 0.578052i \(0.196187\pi\)
\(192\) 3.30816 0.238746
\(193\) −21.8042 −1.56950 −0.784751 0.619811i \(-0.787209\pi\)
−0.784751 + 0.619811i \(0.787209\pi\)
\(194\) −15.8204 −1.13584
\(195\) −3.50884 −0.251273
\(196\) 1.09894 0.0784956
\(197\) −17.3331 −1.23493 −0.617465 0.786598i \(-0.711840\pi\)
−0.617465 + 0.786598i \(0.711840\pi\)
\(198\) 7.24231 0.514688
\(199\) −8.74283 −0.619763 −0.309881 0.950775i \(-0.600289\pi\)
−0.309881 + 0.950775i \(0.600289\pi\)
\(200\) −8.32115 −0.588394
\(201\) −0.547163 −0.0385939
\(202\) 9.36845 0.659162
\(203\) 0.0162253 0.00113879
\(204\) 0.329312 0.0230564
\(205\) 18.4194 1.28646
\(206\) −3.76435 −0.262275
\(207\) 9.77597 0.679477
\(208\) −22.5276 −1.56201
\(209\) −7.08477 −0.490064
\(210\) 0.150618 0.0103936
\(211\) −15.9290 −1.09659 −0.548297 0.836284i \(-0.684724\pi\)
−0.548297 + 0.836284i \(0.684724\pi\)
\(212\) −1.11251 −0.0764076
\(213\) −4.33869 −0.297282
\(214\) 4.66458 0.318864
\(215\) −3.41415 −0.232843
\(216\) −6.64588 −0.452195
\(217\) −0.769134 −0.0522122
\(218\) −19.4107 −1.31466
\(219\) −0.162283 −0.0109661
\(220\) 0.434295 0.0292802
\(221\) −33.1078 −2.22707
\(222\) −1.70717 −0.114578
\(223\) −14.0059 −0.937903 −0.468952 0.883224i \(-0.655368\pi\)
−0.468952 + 0.883224i \(0.655368\pi\)
\(224\) −0.173429 −0.0115877
\(225\) 8.09600 0.539733
\(226\) −7.12821 −0.474162
\(227\) −0.284670 −0.0188942 −0.00944712 0.999955i \(-0.503007\pi\)
−0.00944712 + 0.999955i \(0.503007\pi\)
\(228\) 0.231668 0.0153426
\(229\) −20.0021 −1.32177 −0.660887 0.750486i \(-0.729820\pi\)
−0.660887 + 0.750486i \(0.729820\pi\)
\(230\) −6.84163 −0.451124
\(231\) 0.141433 0.00930561
\(232\) 0.244072 0.0160241
\(233\) −11.1194 −0.728459 −0.364229 0.931309i \(-0.618668\pi\)
−0.364229 + 0.931309i \(0.618668\pi\)
\(234\) 23.8089 1.55644
\(235\) 13.0591 0.851881
\(236\) −1.24658 −0.0811454
\(237\) 0.373235 0.0242442
\(238\) 1.42117 0.0921205
\(239\) −2.37885 −0.153875 −0.0769375 0.997036i \(-0.524514\pi\)
−0.0769375 + 0.997036i \(0.524514\pi\)
\(240\) 2.08577 0.134636
\(241\) 9.34293 0.601831 0.300915 0.953651i \(-0.402708\pi\)
0.300915 + 0.953651i \(0.402708\pi\)
\(242\) 10.1705 0.653783
\(243\) 9.78227 0.627533
\(244\) 0.465714 0.0298143
\(245\) 10.2293 0.653528
\(246\) 6.60054 0.420835
\(247\) −23.2910 −1.48197
\(248\) −11.5698 −0.734686
\(249\) −6.10863 −0.387119
\(250\) −15.6370 −0.988968
\(251\) 18.0060 1.13653 0.568265 0.822846i \(-0.307615\pi\)
0.568265 + 0.822846i \(0.307615\pi\)
\(252\) 0.0875714 0.00551648
\(253\) −6.42440 −0.403899
\(254\) 8.54748 0.536316
\(255\) 3.06536 0.191960
\(256\) −3.76409 −0.235255
\(257\) −27.8759 −1.73885 −0.869425 0.494065i \(-0.835510\pi\)
−0.869425 + 0.494065i \(0.835510\pi\)
\(258\) −1.22345 −0.0761689
\(259\) 0.631280 0.0392259
\(260\) 1.42773 0.0885443
\(261\) −0.237468 −0.0146989
\(262\) −18.1913 −1.12386
\(263\) −7.92612 −0.488745 −0.244373 0.969681i \(-0.578582\pi\)
−0.244373 + 0.969681i \(0.578582\pi\)
\(264\) 2.12753 0.130941
\(265\) −10.3557 −0.636144
\(266\) 0.999777 0.0613003
\(267\) −3.66342 −0.224197
\(268\) 0.222639 0.0135998
\(269\) −11.9964 −0.731436 −0.365718 0.930726i \(-0.619177\pi\)
−0.365718 + 0.930726i \(0.619177\pi\)
\(270\) −4.52522 −0.275396
\(271\) 15.2592 0.926929 0.463465 0.886115i \(-0.346606\pi\)
0.463465 + 0.886115i \(0.346606\pi\)
\(272\) 19.6804 1.19330
\(273\) 0.464958 0.0281405
\(274\) 0.232156 0.0140250
\(275\) −5.32039 −0.320831
\(276\) 0.210074 0.0126450
\(277\) −12.3368 −0.741247 −0.370624 0.928783i \(-0.620856\pi\)
−0.370624 + 0.928783i \(0.620856\pi\)
\(278\) −18.8500 −1.13055
\(279\) 11.2568 0.673927
\(280\) −0.837816 −0.0500691
\(281\) 3.53084 0.210632 0.105316 0.994439i \(-0.466415\pi\)
0.105316 + 0.994439i \(0.466415\pi\)
\(282\) 4.67970 0.278672
\(283\) −19.8205 −1.17820 −0.589102 0.808059i \(-0.700518\pi\)
−0.589102 + 0.808059i \(0.700518\pi\)
\(284\) 1.76540 0.104757
\(285\) 2.15645 0.127737
\(286\) −15.6463 −0.925187
\(287\) −2.44076 −0.144073
\(288\) 2.53826 0.149568
\(289\) 11.9233 0.701372
\(290\) 0.166190 0.00975903
\(291\) 4.52171 0.265068
\(292\) 0.0660326 0.00386426
\(293\) 2.84565 0.166244 0.0831222 0.996539i \(-0.473511\pi\)
0.0831222 + 0.996539i \(0.473511\pi\)
\(294\) 3.66566 0.213786
\(295\) −11.6036 −0.675590
\(296\) 9.49616 0.551953
\(297\) −4.24925 −0.246567
\(298\) 17.6372 1.02170
\(299\) −21.1201 −1.22141
\(300\) 0.173974 0.0100444
\(301\) 0.452411 0.0260765
\(302\) −23.1355 −1.33130
\(303\) −2.67765 −0.153827
\(304\) 13.8450 0.794063
\(305\) 4.33504 0.248224
\(306\) −20.7997 −1.18904
\(307\) 14.8470 0.847362 0.423681 0.905811i \(-0.360738\pi\)
0.423681 + 0.905811i \(0.360738\pi\)
\(308\) −0.0575486 −0.00327914
\(309\) 1.07591 0.0612065
\(310\) −7.87798 −0.447439
\(311\) 27.9972 1.58758 0.793789 0.608194i \(-0.208106\pi\)
0.793789 + 0.608194i \(0.208106\pi\)
\(312\) 6.99422 0.395970
\(313\) 5.53601 0.312914 0.156457 0.987685i \(-0.449993\pi\)
0.156457 + 0.987685i \(0.449993\pi\)
\(314\) −4.70121 −0.265305
\(315\) 0.815147 0.0459283
\(316\) −0.151868 −0.00854325
\(317\) −4.87056 −0.273558 −0.136779 0.990602i \(-0.543675\pi\)
−0.136779 + 0.990602i \(0.543675\pi\)
\(318\) −3.71093 −0.208099
\(319\) 0.156055 0.00873742
\(320\) −12.5298 −0.700437
\(321\) −1.33321 −0.0744126
\(322\) 0.906588 0.0505222
\(323\) 20.3473 1.13215
\(324\) −1.21041 −0.0672447
\(325\) −17.4907 −0.970207
\(326\) −11.7913 −0.653060
\(327\) 5.54789 0.306799
\(328\) −36.7156 −2.02728
\(329\) −1.73047 −0.0954037
\(330\) 1.44865 0.0797455
\(331\) 33.0775 1.81810 0.909051 0.416686i \(-0.136808\pi\)
0.909051 + 0.416686i \(0.136808\pi\)
\(332\) 2.48558 0.136414
\(333\) −9.23922 −0.506306
\(334\) −9.41802 −0.515331
\(335\) 2.07241 0.113228
\(336\) −0.276386 −0.0150781
\(337\) −4.05736 −0.221018 −0.110509 0.993875i \(-0.535248\pi\)
−0.110509 + 0.993875i \(0.535248\pi\)
\(338\) −33.7926 −1.83807
\(339\) 2.03736 0.110654
\(340\) −1.24728 −0.0676435
\(341\) −7.39755 −0.400600
\(342\) −14.6324 −0.791231
\(343\) −2.71837 −0.146778
\(344\) 6.80548 0.366927
\(345\) 1.95545 0.105278
\(346\) 10.9451 0.588412
\(347\) 16.2823 0.874080 0.437040 0.899442i \(-0.356027\pi\)
0.437040 + 0.899442i \(0.356027\pi\)
\(348\) −0.00510292 −0.000273545 0
\(349\) 1.78004 0.0952835 0.0476418 0.998864i \(-0.484829\pi\)
0.0476418 + 0.998864i \(0.484829\pi\)
\(350\) 0.750793 0.0401316
\(351\) −13.9693 −0.745627
\(352\) −1.66805 −0.0889073
\(353\) 30.6765 1.63274 0.816372 0.577527i \(-0.195982\pi\)
0.816372 + 0.577527i \(0.195982\pi\)
\(354\) −4.15814 −0.221003
\(355\) 16.4330 0.872173
\(356\) 1.49063 0.0790033
\(357\) −0.406192 −0.0214980
\(358\) 6.01757 0.318039
\(359\) 33.7830 1.78300 0.891499 0.453024i \(-0.149655\pi\)
0.891499 + 0.453024i \(0.149655\pi\)
\(360\) 12.2620 0.646264
\(361\) −4.68586 −0.246624
\(362\) −21.1092 −1.10947
\(363\) −2.90689 −0.152572
\(364\) −0.189190 −0.00991624
\(365\) 0.614656 0.0321726
\(366\) 1.55345 0.0812002
\(367\) −35.3670 −1.84614 −0.923070 0.384632i \(-0.874328\pi\)
−0.923070 + 0.384632i \(0.874328\pi\)
\(368\) 12.5545 0.654447
\(369\) 35.7221 1.85962
\(370\) 6.46599 0.336151
\(371\) 1.37223 0.0712429
\(372\) 0.241895 0.0125417
\(373\) 31.9814 1.65594 0.827968 0.560775i \(-0.189497\pi\)
0.827968 + 0.560775i \(0.189497\pi\)
\(374\) 13.6688 0.706797
\(375\) 4.46929 0.230793
\(376\) −26.0309 −1.34244
\(377\) 0.513028 0.0264223
\(378\) 0.599639 0.0308421
\(379\) 33.1441 1.70250 0.851249 0.524761i \(-0.175845\pi\)
0.851249 + 0.524761i \(0.175845\pi\)
\(380\) −0.877453 −0.0450124
\(381\) −2.44301 −0.125159
\(382\) −30.6126 −1.56627
\(383\) −15.2158 −0.777490 −0.388745 0.921345i \(-0.627091\pi\)
−0.388745 + 0.921345i \(0.627091\pi\)
\(384\) −3.79892 −0.193863
\(385\) −0.535684 −0.0273010
\(386\) 29.5940 1.50629
\(387\) −6.62134 −0.336582
\(388\) −1.83987 −0.0934053
\(389\) 6.55899 0.332554 0.166277 0.986079i \(-0.446825\pi\)
0.166277 + 0.986079i \(0.446825\pi\)
\(390\) 4.76240 0.241154
\(391\) 18.4507 0.933093
\(392\) −20.3903 −1.02986
\(393\) 5.19938 0.262274
\(394\) 23.5255 1.18520
\(395\) −1.41365 −0.0711282
\(396\) 0.842263 0.0423253
\(397\) −19.5788 −0.982632 −0.491316 0.870982i \(-0.663484\pi\)
−0.491316 + 0.870982i \(0.663484\pi\)
\(398\) 11.8663 0.594803
\(399\) −0.285752 −0.0143055
\(400\) 10.3970 0.519851
\(401\) −36.0834 −1.80192 −0.900959 0.433905i \(-0.857135\pi\)
−0.900959 + 0.433905i \(0.857135\pi\)
\(402\) 0.742643 0.0370397
\(403\) −24.3193 −1.21143
\(404\) 1.08953 0.0542061
\(405\) −11.2669 −0.559857
\(406\) −0.0220219 −0.00109293
\(407\) 6.07167 0.300961
\(408\) −6.11022 −0.302501
\(409\) −19.0011 −0.939545 −0.469773 0.882787i \(-0.655664\pi\)
−0.469773 + 0.882787i \(0.655664\pi\)
\(410\) −24.9998 −1.23465
\(411\) −0.0663538 −0.00327299
\(412\) −0.437785 −0.0215681
\(413\) 1.53760 0.0756605
\(414\) −13.2685 −0.652113
\(415\) 23.1367 1.13574
\(416\) −5.48367 −0.268859
\(417\) 5.38764 0.263834
\(418\) 9.61588 0.470328
\(419\) 31.5429 1.54097 0.770485 0.637458i \(-0.220014\pi\)
0.770485 + 0.637458i \(0.220014\pi\)
\(420\) 0.0175165 0.000854720 0
\(421\) −22.9117 −1.11665 −0.558325 0.829623i \(-0.688556\pi\)
−0.558325 + 0.829623i \(0.688556\pi\)
\(422\) 21.6197 1.05243
\(423\) 25.3266 1.23142
\(424\) 20.6421 1.00247
\(425\) 15.2800 0.741190
\(426\) 5.88873 0.285310
\(427\) −0.574438 −0.0277990
\(428\) 0.542479 0.0262217
\(429\) 4.47197 0.215909
\(430\) 4.63389 0.223466
\(431\) −26.6379 −1.28310 −0.641552 0.767079i \(-0.721709\pi\)
−0.641552 + 0.767079i \(0.721709\pi\)
\(432\) 8.30383 0.399518
\(433\) −25.7160 −1.23583 −0.617916 0.786244i \(-0.712023\pi\)
−0.617916 + 0.786244i \(0.712023\pi\)
\(434\) 1.04391 0.0501095
\(435\) −0.0474998 −0.00227744
\(436\) −2.25742 −0.108111
\(437\) 12.9799 0.620913
\(438\) 0.220261 0.0105245
\(439\) −25.6733 −1.22532 −0.612661 0.790346i \(-0.709901\pi\)
−0.612661 + 0.790346i \(0.709901\pi\)
\(440\) −8.05813 −0.384156
\(441\) 19.8386 0.944694
\(442\) 44.9359 2.13738
\(443\) 34.7224 1.64971 0.824857 0.565342i \(-0.191256\pi\)
0.824857 + 0.565342i \(0.191256\pi\)
\(444\) −0.198540 −0.00942229
\(445\) 13.8754 0.657755
\(446\) 19.0096 0.900132
\(447\) −5.04101 −0.238431
\(448\) 1.66033 0.0784432
\(449\) 18.0522 0.851936 0.425968 0.904738i \(-0.359934\pi\)
0.425968 + 0.904738i \(0.359934\pi\)
\(450\) −10.9884 −0.517997
\(451\) −23.4753 −1.10541
\(452\) −0.828994 −0.0389926
\(453\) 6.61250 0.310682
\(454\) 0.386372 0.0181333
\(455\) −1.76105 −0.0825592
\(456\) −4.29849 −0.201295
\(457\) 13.7905 0.645092 0.322546 0.946554i \(-0.395461\pi\)
0.322546 + 0.946554i \(0.395461\pi\)
\(458\) 27.1480 1.26854
\(459\) 12.2038 0.569622
\(460\) −0.795665 −0.0370981
\(461\) −4.41465 −0.205611 −0.102805 0.994701i \(-0.532782\pi\)
−0.102805 + 0.994701i \(0.532782\pi\)
\(462\) −0.191961 −0.00893085
\(463\) −31.5089 −1.46434 −0.732171 0.681121i \(-0.761493\pi\)
−0.732171 + 0.681121i \(0.761493\pi\)
\(464\) −0.304961 −0.0141575
\(465\) 2.25165 0.104418
\(466\) 15.0920 0.699122
\(467\) 13.6967 0.633810 0.316905 0.948457i \(-0.397356\pi\)
0.316905 + 0.948457i \(0.397356\pi\)
\(468\) 2.76892 0.127993
\(469\) −0.274616 −0.0126806
\(470\) −17.7246 −0.817574
\(471\) 1.34368 0.0619135
\(472\) 23.1297 1.06463
\(473\) 4.35130 0.200073
\(474\) −0.506577 −0.0232679
\(475\) 10.7494 0.493214
\(476\) 0.165278 0.00757551
\(477\) −20.0836 −0.919564
\(478\) 3.22871 0.147678
\(479\) 24.1143 1.10181 0.550905 0.834568i \(-0.314283\pi\)
0.550905 + 0.834568i \(0.314283\pi\)
\(480\) 0.507718 0.0231740
\(481\) 19.9605 0.910119
\(482\) −12.6808 −0.577594
\(483\) −0.259117 −0.0117902
\(484\) 1.18280 0.0537638
\(485\) −17.1262 −0.777661
\(486\) −13.2771 −0.602260
\(487\) 9.17271 0.415655 0.207828 0.978165i \(-0.433361\pi\)
0.207828 + 0.978165i \(0.433361\pi\)
\(488\) −8.64110 −0.391164
\(489\) 3.37015 0.152403
\(490\) −13.8839 −0.627209
\(491\) −32.9154 −1.48545 −0.742727 0.669595i \(-0.766468\pi\)
−0.742727 + 0.669595i \(0.766468\pi\)
\(492\) 0.767627 0.0346073
\(493\) −0.448187 −0.0201853
\(494\) 31.6120 1.42229
\(495\) 7.84010 0.352386
\(496\) 14.4562 0.649101
\(497\) −2.17754 −0.0976762
\(498\) 8.29100 0.371529
\(499\) −1.24204 −0.0556016 −0.0278008 0.999613i \(-0.508850\pi\)
−0.0278008 + 0.999613i \(0.508850\pi\)
\(500\) −1.81854 −0.0813276
\(501\) 2.69182 0.120262
\(502\) −24.4388 −1.09076
\(503\) 27.6666 1.23359 0.616796 0.787123i \(-0.288430\pi\)
0.616796 + 0.787123i \(0.288430\pi\)
\(504\) −1.62484 −0.0723763
\(505\) 10.1417 0.451301
\(506\) 8.71958 0.387633
\(507\) 9.65846 0.428947
\(508\) 0.994051 0.0441039
\(509\) −17.4776 −0.774682 −0.387341 0.921937i \(-0.626606\pi\)
−0.387341 + 0.921937i \(0.626606\pi\)
\(510\) −4.16049 −0.184229
\(511\) −0.0814483 −0.00360306
\(512\) 24.6946 1.09136
\(513\) 8.58523 0.379047
\(514\) 37.8348 1.66882
\(515\) −4.07507 −0.179569
\(516\) −0.142285 −0.00626374
\(517\) −16.6437 −0.731988
\(518\) −0.856811 −0.0376461
\(519\) −3.12828 −0.137316
\(520\) −26.4909 −1.16170
\(521\) −25.9169 −1.13544 −0.567719 0.823222i \(-0.692174\pi\)
−0.567719 + 0.823222i \(0.692174\pi\)
\(522\) 0.322306 0.0141070
\(523\) 2.02186 0.0884096 0.0442048 0.999022i \(-0.485925\pi\)
0.0442048 + 0.999022i \(0.485925\pi\)
\(524\) −2.11561 −0.0924208
\(525\) −0.214589 −0.00936542
\(526\) 10.7578 0.469062
\(527\) 21.2456 0.925472
\(528\) −2.65829 −0.115687
\(529\) −11.2299 −0.488259
\(530\) 14.0553 0.610525
\(531\) −22.5039 −0.976584
\(532\) 0.116272 0.00504102
\(533\) −77.1744 −3.34279
\(534\) 4.97221 0.215168
\(535\) 5.04960 0.218313
\(536\) −4.13096 −0.178430
\(537\) −1.71992 −0.0742200
\(538\) 16.2823 0.701979
\(539\) −13.0372 −0.561551
\(540\) −0.526272 −0.0226472
\(541\) −5.39281 −0.231855 −0.115927 0.993258i \(-0.536984\pi\)
−0.115927 + 0.993258i \(0.536984\pi\)
\(542\) −20.7107 −0.889599
\(543\) 6.03334 0.258915
\(544\) 4.79059 0.205395
\(545\) −21.0129 −0.900093
\(546\) −0.631068 −0.0270072
\(547\) −1.00000 −0.0427569
\(548\) 0.0269991 0.00115335
\(549\) 8.40729 0.358814
\(550\) 7.22115 0.307911
\(551\) −0.315295 −0.0134320
\(552\) −3.89782 −0.165902
\(553\) 0.187323 0.00796578
\(554\) 16.7443 0.711396
\(555\) −1.84808 −0.0784468
\(556\) −2.19221 −0.0929705
\(557\) −24.8723 −1.05387 −0.526937 0.849904i \(-0.676660\pi\)
−0.526937 + 0.849904i \(0.676660\pi\)
\(558\) −15.2784 −0.646786
\(559\) 14.3048 0.605028
\(560\) 1.04683 0.0442364
\(561\) −3.90676 −0.164944
\(562\) −4.79227 −0.202150
\(563\) −32.9750 −1.38973 −0.694865 0.719140i \(-0.744536\pi\)
−0.694865 + 0.719140i \(0.744536\pi\)
\(564\) 0.544238 0.0229166
\(565\) −7.71659 −0.324639
\(566\) 26.9015 1.13075
\(567\) 1.49298 0.0626994
\(568\) −32.7561 −1.37442
\(569\) 31.7599 1.33145 0.665723 0.746199i \(-0.268123\pi\)
0.665723 + 0.746199i \(0.268123\pi\)
\(570\) −2.92686 −0.122593
\(571\) 23.0226 0.963466 0.481733 0.876318i \(-0.340008\pi\)
0.481733 + 0.876318i \(0.340008\pi\)
\(572\) −1.81963 −0.0760826
\(573\) 8.74956 0.365518
\(574\) 3.31274 0.138271
\(575\) 9.74741 0.406495
\(576\) −24.3000 −1.01250
\(577\) 9.46259 0.393933 0.196966 0.980410i \(-0.436891\pi\)
0.196966 + 0.980410i \(0.436891\pi\)
\(578\) −16.1831 −0.673126
\(579\) −8.45844 −0.351521
\(580\) 0.0193275 0.000802532 0
\(581\) −3.06586 −0.127193
\(582\) −6.13714 −0.254393
\(583\) 13.1982 0.546613
\(584\) −1.22520 −0.0506992
\(585\) 25.7741 1.06563
\(586\) −3.86228 −0.159549
\(587\) 32.6862 1.34911 0.674553 0.738227i \(-0.264336\pi\)
0.674553 + 0.738227i \(0.264336\pi\)
\(588\) 0.426308 0.0175806
\(589\) 14.9461 0.615842
\(590\) 15.7491 0.648382
\(591\) −6.72396 −0.276587
\(592\) −11.8652 −0.487655
\(593\) 26.7627 1.09901 0.549506 0.835490i \(-0.314816\pi\)
0.549506 + 0.835490i \(0.314816\pi\)
\(594\) 5.76734 0.236637
\(595\) 1.53847 0.0630712
\(596\) 2.05117 0.0840192
\(597\) −3.39158 −0.138808
\(598\) 28.6654 1.17222
\(599\) −28.9681 −1.18361 −0.591803 0.806083i \(-0.701584\pi\)
−0.591803 + 0.806083i \(0.701584\pi\)
\(600\) −3.22799 −0.131782
\(601\) 37.2161 1.51808 0.759038 0.651046i \(-0.225670\pi\)
0.759038 + 0.651046i \(0.225670\pi\)
\(602\) −0.614039 −0.0250264
\(603\) 4.01919 0.163674
\(604\) −2.69061 −0.109479
\(605\) 11.0100 0.447619
\(606\) 3.63427 0.147632
\(607\) −21.4512 −0.870678 −0.435339 0.900267i \(-0.643371\pi\)
−0.435339 + 0.900267i \(0.643371\pi\)
\(608\) 3.37014 0.136677
\(609\) 0.00629423 0.000255055 0
\(610\) −5.88377 −0.238227
\(611\) −54.7157 −2.21356
\(612\) −2.41896 −0.0977806
\(613\) 38.1315 1.54012 0.770058 0.637973i \(-0.220227\pi\)
0.770058 + 0.637973i \(0.220227\pi\)
\(614\) −20.1512 −0.813237
\(615\) 7.14536 0.288129
\(616\) 1.06779 0.0430224
\(617\) 3.35306 0.134989 0.0674946 0.997720i \(-0.478499\pi\)
0.0674946 + 0.997720i \(0.478499\pi\)
\(618\) −1.46029 −0.0587416
\(619\) −20.8491 −0.837995 −0.418998 0.907987i \(-0.637618\pi\)
−0.418998 + 0.907987i \(0.637618\pi\)
\(620\) −0.916190 −0.0367951
\(621\) 7.78500 0.312401
\(622\) −37.9995 −1.52364
\(623\) −1.83863 −0.0736631
\(624\) −8.73906 −0.349842
\(625\) −2.72172 −0.108869
\(626\) −7.51380 −0.300312
\(627\) −2.74837 −0.109759
\(628\) −0.546739 −0.0218173
\(629\) −17.4377 −0.695286
\(630\) −1.10637 −0.0440787
\(631\) 44.4501 1.76953 0.884765 0.466037i \(-0.154319\pi\)
0.884765 + 0.466037i \(0.154319\pi\)
\(632\) 2.81784 0.112088
\(633\) −6.17926 −0.245604
\(634\) 6.61061 0.262541
\(635\) 9.25300 0.367194
\(636\) −0.431573 −0.0171130
\(637\) −42.8594 −1.69815
\(638\) −0.211808 −0.00838554
\(639\) 31.8698 1.26075
\(640\) 14.3886 0.568759
\(641\) 15.0918 0.596092 0.298046 0.954551i \(-0.403665\pi\)
0.298046 + 0.954551i \(0.403665\pi\)
\(642\) 1.80951 0.0714158
\(643\) −23.5111 −0.927186 −0.463593 0.886048i \(-0.653440\pi\)
−0.463593 + 0.886048i \(0.653440\pi\)
\(644\) 0.105434 0.00415468
\(645\) −1.32444 −0.0521498
\(646\) −27.6166 −1.08656
\(647\) 26.2736 1.03292 0.516461 0.856311i \(-0.327249\pi\)
0.516461 + 0.856311i \(0.327249\pi\)
\(648\) 22.4585 0.882253
\(649\) 14.7887 0.580507
\(650\) 23.7394 0.931134
\(651\) −0.298367 −0.0116939
\(652\) −1.37130 −0.0537043
\(653\) −5.29541 −0.207225 −0.103613 0.994618i \(-0.533040\pi\)
−0.103613 + 0.994618i \(0.533040\pi\)
\(654\) −7.52993 −0.294443
\(655\) −19.6929 −0.769465
\(656\) 45.8750 1.79112
\(657\) 1.19205 0.0465064
\(658\) 2.34869 0.0915616
\(659\) −24.9443 −0.971693 −0.485846 0.874044i \(-0.661489\pi\)
−0.485846 + 0.874044i \(0.661489\pi\)
\(660\) 0.168475 0.00655786
\(661\) −15.4362 −0.600400 −0.300200 0.953876i \(-0.597053\pi\)
−0.300200 + 0.953876i \(0.597053\pi\)
\(662\) −44.8947 −1.74488
\(663\) −12.8434 −0.498796
\(664\) −46.1188 −1.78976
\(665\) 1.08230 0.0419698
\(666\) 12.5400 0.485916
\(667\) −0.285907 −0.0110703
\(668\) −1.09529 −0.0423782
\(669\) −5.43325 −0.210062
\(670\) −2.81279 −0.108668
\(671\) −5.52496 −0.213289
\(672\) −0.0672779 −0.00259530
\(673\) −8.36186 −0.322326 −0.161163 0.986928i \(-0.551524\pi\)
−0.161163 + 0.986928i \(0.551524\pi\)
\(674\) 5.50689 0.212117
\(675\) 6.44717 0.248152
\(676\) −3.93000 −0.151154
\(677\) 23.5347 0.904513 0.452257 0.891888i \(-0.350619\pi\)
0.452257 + 0.891888i \(0.350619\pi\)
\(678\) −2.76522 −0.106198
\(679\) 2.26940 0.0870916
\(680\) 23.1428 0.887484
\(681\) −0.110431 −0.00423173
\(682\) 10.0404 0.384466
\(683\) 42.2912 1.61823 0.809114 0.587651i \(-0.199947\pi\)
0.809114 + 0.587651i \(0.199947\pi\)
\(684\) −1.70172 −0.0650667
\(685\) 0.251318 0.00960237
\(686\) 3.68953 0.140867
\(687\) −7.75933 −0.296037
\(688\) −8.50323 −0.324183
\(689\) 43.3887 1.65298
\(690\) −2.65405 −0.101038
\(691\) 7.89039 0.300165 0.150082 0.988674i \(-0.452046\pi\)
0.150082 + 0.988674i \(0.452046\pi\)
\(692\) 1.27289 0.0483880
\(693\) −1.03890 −0.0394644
\(694\) −22.0993 −0.838879
\(695\) −20.4059 −0.774041
\(696\) 0.0946821 0.00358892
\(697\) 67.4204 2.55373
\(698\) −2.41598 −0.0914462
\(699\) −4.31353 −0.163153
\(700\) 0.0873155 0.00330022
\(701\) −7.76122 −0.293137 −0.146569 0.989200i \(-0.546823\pi\)
−0.146569 + 0.989200i \(0.546823\pi\)
\(702\) 18.9600 0.715599
\(703\) −12.2673 −0.462668
\(704\) 15.9691 0.601857
\(705\) 5.06597 0.190795
\(706\) −41.6359 −1.56699
\(707\) −1.34389 −0.0505421
\(708\) −0.483582 −0.0181741
\(709\) −29.9358 −1.12426 −0.562131 0.827048i \(-0.690019\pi\)
−0.562131 + 0.827048i \(0.690019\pi\)
\(710\) −22.3038 −0.837048
\(711\) −2.74160 −0.102818
\(712\) −27.6580 −1.03653
\(713\) 13.5529 0.507561
\(714\) 0.551308 0.0206322
\(715\) −16.9378 −0.633438
\(716\) 0.699830 0.0261539
\(717\) −0.922818 −0.0344633
\(718\) −45.8523 −1.71119
\(719\) −38.2117 −1.42506 −0.712528 0.701644i \(-0.752450\pi\)
−0.712528 + 0.701644i \(0.752450\pi\)
\(720\) −15.3210 −0.570980
\(721\) 0.539989 0.0201103
\(722\) 6.35993 0.236692
\(723\) 3.62437 0.134792
\(724\) −2.45495 −0.0912373
\(725\) −0.236774 −0.00879358
\(726\) 3.94540 0.146428
\(727\) −34.4040 −1.27597 −0.637986 0.770048i \(-0.720232\pi\)
−0.637986 + 0.770048i \(0.720232\pi\)
\(728\) 3.51032 0.130101
\(729\) −19.2100 −0.711481
\(730\) −0.834248 −0.0308769
\(731\) −12.4968 −0.462211
\(732\) 0.180663 0.00667749
\(733\) −32.2174 −1.18998 −0.594988 0.803735i \(-0.702843\pi\)
−0.594988 + 0.803735i \(0.702843\pi\)
\(734\) 48.0021 1.77179
\(735\) 3.96823 0.146370
\(736\) 3.05601 0.112646
\(737\) −2.64126 −0.0972920
\(738\) −48.4842 −1.78473
\(739\) 39.5093 1.45337 0.726686 0.686969i \(-0.241059\pi\)
0.726686 + 0.686969i \(0.241059\pi\)
\(740\) 0.751979 0.0276433
\(741\) −9.03521 −0.331917
\(742\) −1.86248 −0.0683737
\(743\) −5.63707 −0.206804 −0.103402 0.994640i \(-0.532973\pi\)
−0.103402 + 0.994640i \(0.532973\pi\)
\(744\) −4.48825 −0.164547
\(745\) 19.0930 0.699515
\(746\) −43.4071 −1.58925
\(747\) 44.8709 1.64174
\(748\) 1.58965 0.0581233
\(749\) −0.669125 −0.0244493
\(750\) −6.06599 −0.221499
\(751\) −30.4166 −1.10992 −0.554958 0.831878i \(-0.687266\pi\)
−0.554958 + 0.831878i \(0.687266\pi\)
\(752\) 32.5248 1.18606
\(753\) 6.98501 0.254548
\(754\) −0.696312 −0.0253582
\(755\) −25.0452 −0.911487
\(756\) 0.0697366 0.00253630
\(757\) −5.69640 −0.207039 −0.103520 0.994627i \(-0.533010\pi\)
−0.103520 + 0.994627i \(0.533010\pi\)
\(758\) −44.9852 −1.63394
\(759\) −2.49220 −0.0904610
\(760\) 16.2807 0.590564
\(761\) −46.0415 −1.66900 −0.834501 0.551007i \(-0.814244\pi\)
−0.834501 + 0.551007i \(0.814244\pi\)
\(762\) 3.31579 0.120119
\(763\) 2.78443 0.100803
\(764\) −3.56017 −0.128802
\(765\) −22.5166 −0.814088
\(766\) 20.6518 0.746178
\(767\) 48.6175 1.75548
\(768\) −1.46019 −0.0526901
\(769\) 24.4948 0.883304 0.441652 0.897186i \(-0.354393\pi\)
0.441652 + 0.897186i \(0.354393\pi\)
\(770\) 0.727062 0.0262015
\(771\) −10.8138 −0.389449
\(772\) 3.44171 0.123870
\(773\) 9.93793 0.357442 0.178721 0.983900i \(-0.442804\pi\)
0.178721 + 0.983900i \(0.442804\pi\)
\(774\) 8.98688 0.323027
\(775\) 11.2239 0.403175
\(776\) 34.1379 1.22548
\(777\) 0.244890 0.00878540
\(778\) −8.90225 −0.319161
\(779\) 47.4296 1.69934
\(780\) 0.553856 0.0198312
\(781\) −20.9437 −0.749423
\(782\) −25.0424 −0.895515
\(783\) −0.189106 −0.00675808
\(784\) 25.4770 0.909894
\(785\) −5.08925 −0.181643
\(786\) −7.05691 −0.251711
\(787\) −29.5730 −1.05416 −0.527082 0.849814i \(-0.676714\pi\)
−0.527082 + 0.849814i \(0.676714\pi\)
\(788\) 2.73596 0.0974644
\(789\) −3.07475 −0.109464
\(790\) 1.91868 0.0682637
\(791\) 1.02253 0.0363569
\(792\) −15.6278 −0.555309
\(793\) −18.1632 −0.644993
\(794\) 26.5735 0.943059
\(795\) −4.01724 −0.142477
\(796\) 1.38002 0.0489136
\(797\) −18.9234 −0.670302 −0.335151 0.942165i \(-0.608787\pi\)
−0.335151 + 0.942165i \(0.608787\pi\)
\(798\) 0.387840 0.0137294
\(799\) 47.8002 1.69105
\(800\) 2.53084 0.0894788
\(801\) 26.9096 0.950804
\(802\) 48.9745 1.72935
\(803\) −0.783372 −0.0276446
\(804\) 0.0863676 0.00304595
\(805\) 0.981419 0.0345905
\(806\) 33.0076 1.16264
\(807\) −4.65374 −0.163819
\(808\) −20.2157 −0.711185
\(809\) 16.3223 0.573860 0.286930 0.957952i \(-0.407365\pi\)
0.286930 + 0.957952i \(0.407365\pi\)
\(810\) 15.2921 0.537310
\(811\) 37.4100 1.31364 0.656821 0.754047i \(-0.271901\pi\)
0.656821 + 0.754047i \(0.271901\pi\)
\(812\) −0.00256110 −8.98770e−5 0
\(813\) 5.91944 0.207604
\(814\) −8.24083 −0.288841
\(815\) −12.7646 −0.447124
\(816\) 7.63454 0.267262
\(817\) −8.79140 −0.307572
\(818\) 25.7895 0.901708
\(819\) −3.41534 −0.119342
\(820\) −2.90742 −0.101532
\(821\) 24.6390 0.859906 0.429953 0.902851i \(-0.358530\pi\)
0.429953 + 0.902851i \(0.358530\pi\)
\(822\) 0.0900594 0.00314118
\(823\) −13.5452 −0.472155 −0.236078 0.971734i \(-0.575862\pi\)
−0.236078 + 0.971734i \(0.575862\pi\)
\(824\) 8.12290 0.282975
\(825\) −2.06392 −0.0718565
\(826\) −2.08693 −0.0726134
\(827\) 17.7461 0.617093 0.308547 0.951209i \(-0.400157\pi\)
0.308547 + 0.951209i \(0.400157\pi\)
\(828\) −1.54310 −0.0536264
\(829\) −26.4419 −0.918365 −0.459182 0.888342i \(-0.651858\pi\)
−0.459182 + 0.888342i \(0.651858\pi\)
\(830\) −31.4026 −1.09000
\(831\) −4.78578 −0.166017
\(832\) 52.4980 1.82004
\(833\) 37.4424 1.29730
\(834\) −7.31242 −0.253209
\(835\) −10.1954 −0.352826
\(836\) 1.11830 0.0386773
\(837\) 8.96424 0.309849
\(838\) −42.8119 −1.47891
\(839\) 5.72384 0.197609 0.0988044 0.995107i \(-0.468498\pi\)
0.0988044 + 0.995107i \(0.468498\pi\)
\(840\) −0.325011 −0.0112139
\(841\) −28.9931 −0.999761
\(842\) 31.0972 1.07168
\(843\) 1.36971 0.0471752
\(844\) 2.51432 0.0865465
\(845\) −36.5819 −1.25845
\(846\) −34.3747 −1.18183
\(847\) −1.45894 −0.0501296
\(848\) −25.7917 −0.885690
\(849\) −7.68888 −0.263882
\(850\) −20.7390 −0.711340
\(851\) −11.1238 −0.381319
\(852\) 0.684845 0.0234624
\(853\) 5.06635 0.173469 0.0867343 0.996231i \(-0.472357\pi\)
0.0867343 + 0.996231i \(0.472357\pi\)
\(854\) 0.779661 0.0266795
\(855\) −15.8402 −0.541724
\(856\) −10.0654 −0.344030
\(857\) −52.1200 −1.78038 −0.890192 0.455586i \(-0.849430\pi\)
−0.890192 + 0.455586i \(0.849430\pi\)
\(858\) −6.06963 −0.207214
\(859\) 9.02910 0.308069 0.154034 0.988065i \(-0.450773\pi\)
0.154034 + 0.988065i \(0.450773\pi\)
\(860\) 0.538910 0.0183767
\(861\) −0.946835 −0.0322680
\(862\) 36.1546 1.23143
\(863\) 11.6103 0.395220 0.197610 0.980281i \(-0.436682\pi\)
0.197610 + 0.980281i \(0.436682\pi\)
\(864\) 2.02132 0.0687666
\(865\) 11.8485 0.402862
\(866\) 34.9033 1.18606
\(867\) 4.62538 0.157086
\(868\) 0.121405 0.00412075
\(869\) 1.80167 0.0611176
\(870\) 0.0644696 0.00218572
\(871\) −8.68308 −0.294215
\(872\) 41.8853 1.41842
\(873\) −33.2142 −1.12413
\(874\) −17.6171 −0.595908
\(875\) 2.24309 0.0758303
\(876\) 0.0256158 0.000865477 0
\(877\) −7.52717 −0.254174 −0.127087 0.991892i \(-0.540563\pi\)
−0.127087 + 0.991892i \(0.540563\pi\)
\(878\) 34.8454 1.17597
\(879\) 1.10390 0.0372337
\(880\) 10.0684 0.339405
\(881\) 33.5613 1.13071 0.565354 0.824848i \(-0.308739\pi\)
0.565354 + 0.824848i \(0.308739\pi\)
\(882\) −26.9261 −0.906649
\(883\) 12.5358 0.421863 0.210932 0.977501i \(-0.432350\pi\)
0.210932 + 0.977501i \(0.432350\pi\)
\(884\) 5.22594 0.175767
\(885\) −4.50136 −0.151311
\(886\) −47.1274 −1.58328
\(887\) 3.55658 0.119418 0.0597091 0.998216i \(-0.480983\pi\)
0.0597091 + 0.998216i \(0.480983\pi\)
\(888\) 3.68381 0.123621
\(889\) −1.22612 −0.0411227
\(890\) −18.8325 −0.631266
\(891\) 14.3595 0.481063
\(892\) 2.21077 0.0740222
\(893\) 33.6270 1.12528
\(894\) 6.84196 0.228829
\(895\) 6.51427 0.217748
\(896\) −1.90664 −0.0636963
\(897\) −8.19304 −0.273558
\(898\) −24.5015 −0.817626
\(899\) −0.329215 −0.0109799
\(900\) −1.27792 −0.0425974
\(901\) −37.9049 −1.26279
\(902\) 31.8620 1.06089
\(903\) 0.175502 0.00584035
\(904\) 15.3816 0.511584
\(905\) −22.8515 −0.759611
\(906\) −8.97488 −0.298171
\(907\) −5.91904 −0.196539 −0.0982693 0.995160i \(-0.531331\pi\)
−0.0982693 + 0.995160i \(0.531331\pi\)
\(908\) 0.0449341 0.00149119
\(909\) 19.6687 0.652369
\(910\) 2.39020 0.0792344
\(911\) 14.3654 0.475948 0.237974 0.971271i \(-0.423517\pi\)
0.237974 + 0.971271i \(0.423517\pi\)
\(912\) 5.37083 0.177846
\(913\) −29.4875 −0.975894
\(914\) −18.7173 −0.619112
\(915\) 1.68168 0.0555945
\(916\) 3.15725 0.104318
\(917\) 2.60951 0.0861737
\(918\) −16.5637 −0.546682
\(919\) 7.41560 0.244618 0.122309 0.992492i \(-0.460970\pi\)
0.122309 + 0.992492i \(0.460970\pi\)
\(920\) 14.7632 0.486728
\(921\) 5.75954 0.189783
\(922\) 5.99183 0.197330
\(923\) −68.8518 −2.26628
\(924\) −0.0223246 −0.000734427 0
\(925\) −9.21222 −0.302896
\(926\) 42.7657 1.40537
\(927\) −7.90311 −0.259572
\(928\) −0.0742335 −0.00243684
\(929\) 9.71228 0.318650 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(930\) −3.05608 −0.100213
\(931\) 26.3404 0.863272
\(932\) 1.75516 0.0574922
\(933\) 10.8609 0.355569
\(934\) −18.5900 −0.608285
\(935\) 14.7970 0.483915
\(936\) −51.3760 −1.67928
\(937\) −28.3847 −0.927287 −0.463643 0.886022i \(-0.653458\pi\)
−0.463643 + 0.886022i \(0.653458\pi\)
\(938\) 0.372725 0.0121699
\(939\) 2.14757 0.0700832
\(940\) −2.06133 −0.0672331
\(941\) −7.54245 −0.245877 −0.122938 0.992414i \(-0.539232\pi\)
−0.122938 + 0.992414i \(0.539232\pi\)
\(942\) −1.82372 −0.0594201
\(943\) 43.0087 1.40056
\(944\) −28.8998 −0.940610
\(945\) 0.649134 0.0211163
\(946\) −5.90584 −0.192015
\(947\) −32.4877 −1.05571 −0.527854 0.849335i \(-0.677003\pi\)
−0.527854 + 0.849335i \(0.677003\pi\)
\(948\) −0.0589137 −0.00191343
\(949\) −2.57532 −0.0835983
\(950\) −14.5897 −0.473351
\(951\) −1.88942 −0.0612686
\(952\) −3.06666 −0.0993909
\(953\) −6.79285 −0.220042 −0.110021 0.993929i \(-0.535092\pi\)
−0.110021 + 0.993929i \(0.535092\pi\)
\(954\) 27.2586 0.882531
\(955\) −33.1394 −1.07236
\(956\) 0.375492 0.0121443
\(957\) 0.0605380 0.00195692
\(958\) −32.7293 −1.05744
\(959\) −0.0333023 −0.00107539
\(960\) −4.86064 −0.156877
\(961\) −15.3941 −0.496584
\(962\) −27.0915 −0.873466
\(963\) 9.79310 0.315578
\(964\) −1.47474 −0.0474983
\(965\) 32.0367 1.03130
\(966\) 0.351689 0.0113154
\(967\) −3.50671 −0.112768 −0.0563840 0.998409i \(-0.517957\pi\)
−0.0563840 + 0.998409i \(0.517957\pi\)
\(968\) −21.9463 −0.705382
\(969\) 7.89326 0.253568
\(970\) 23.2447 0.746342
\(971\) −31.1697 −1.00028 −0.500142 0.865943i \(-0.666719\pi\)
−0.500142 + 0.865943i \(0.666719\pi\)
\(972\) −1.54409 −0.0495268
\(973\) 2.70400 0.0866862
\(974\) −12.4498 −0.398916
\(975\) −6.78509 −0.217297
\(976\) 10.7968 0.345597
\(977\) −27.2061 −0.870400 −0.435200 0.900334i \(-0.643322\pi\)
−0.435200 + 0.900334i \(0.643322\pi\)
\(978\) −4.57416 −0.146266
\(979\) −17.6840 −0.565182
\(980\) −1.61466 −0.0515784
\(981\) −40.7520 −1.30111
\(982\) 44.6748 1.42563
\(983\) 9.22401 0.294200 0.147100 0.989122i \(-0.453006\pi\)
0.147100 + 0.989122i \(0.453006\pi\)
\(984\) −14.2429 −0.454049
\(985\) 25.4673 0.811456
\(986\) 0.608306 0.0193724
\(987\) −0.671294 −0.0213675
\(988\) 3.67640 0.116962
\(989\) −7.97195 −0.253493
\(990\) −10.6411 −0.338195
\(991\) −37.7461 −1.19904 −0.599522 0.800358i \(-0.704643\pi\)
−0.599522 + 0.800358i \(0.704643\pi\)
\(992\) 3.51892 0.111726
\(993\) 12.8316 0.407199
\(994\) 2.95549 0.0937425
\(995\) 12.8458 0.407238
\(996\) 0.964224 0.0305526
\(997\) −45.6962 −1.44721 −0.723606 0.690213i \(-0.757517\pi\)
−0.723606 + 0.690213i \(0.757517\pi\)
\(998\) 1.68578 0.0533623
\(999\) −7.35756 −0.232783
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.1.6 18
3.2 odd 2 4923.2.a.l.1.13 18
4.3 odd 2 8752.2.a.s.1.7 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.6 18 1.1 even 1 trivial
4923.2.a.l.1.13 18 3.2 odd 2
8752.2.a.s.1.7 18 4.3 odd 2