Properties

Label 547.2.a.b.1.7
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 1.7
Root \(1.04467\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.04467 q^{2} -2.71791 q^{3} -0.908666 q^{4} +0.714085 q^{5} +2.83932 q^{6} -2.03236 q^{7} +3.03859 q^{8} +4.38705 q^{9} +O(q^{10})\) \(q-1.04467 q^{2} -2.71791 q^{3} -0.908666 q^{4} +0.714085 q^{5} +2.83932 q^{6} -2.03236 q^{7} +3.03859 q^{8} +4.38705 q^{9} -0.745983 q^{10} +5.43392 q^{11} +2.46968 q^{12} -3.96770 q^{13} +2.12315 q^{14} -1.94082 q^{15} -1.35699 q^{16} -1.30793 q^{17} -4.58302 q^{18} +8.24470 q^{19} -0.648865 q^{20} +5.52379 q^{21} -5.67665 q^{22} -2.44824 q^{23} -8.25864 q^{24} -4.49008 q^{25} +4.14494 q^{26} -3.76989 q^{27} +1.84674 q^{28} +1.49405 q^{29} +2.02752 q^{30} -6.02301 q^{31} -4.65958 q^{32} -14.7689 q^{33} +1.36635 q^{34} -1.45128 q^{35} -3.98637 q^{36} -6.36158 q^{37} -8.61299 q^{38} +10.7839 q^{39} +2.16982 q^{40} -2.87189 q^{41} -5.77053 q^{42} -5.76713 q^{43} -4.93762 q^{44} +3.13273 q^{45} +2.55760 q^{46} +6.42848 q^{47} +3.68819 q^{48} -2.86949 q^{49} +4.69065 q^{50} +3.55483 q^{51} +3.60532 q^{52} -0.932545 q^{53} +3.93829 q^{54} +3.88028 q^{55} -6.17553 q^{56} -22.4084 q^{57} -1.56079 q^{58} +1.57369 q^{59} +1.76356 q^{60} -7.93215 q^{61} +6.29205 q^{62} -8.91609 q^{63} +7.58170 q^{64} -2.83328 q^{65} +15.4286 q^{66} +10.5294 q^{67} +1.18847 q^{68} +6.65410 q^{69} +1.51611 q^{70} -12.8929 q^{71} +13.3305 q^{72} +9.81833 q^{73} +6.64575 q^{74} +12.2037 q^{75} -7.49168 q^{76} -11.0437 q^{77} -11.2656 q^{78} -2.80129 q^{79} -0.969009 q^{80} -2.91492 q^{81} +3.00017 q^{82} -15.3935 q^{83} -5.01928 q^{84} -0.933971 q^{85} +6.02474 q^{86} -4.06069 q^{87} +16.5115 q^{88} -1.89811 q^{89} -3.27267 q^{90} +8.06382 q^{91} +2.22463 q^{92} +16.3700 q^{93} -6.71563 q^{94} +5.88742 q^{95} +12.6643 q^{96} +0.276333 q^{97} +2.99767 q^{98} +23.8389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.04467 −0.738693 −0.369346 0.929292i \(-0.620418\pi\)
−0.369346 + 0.929292i \(0.620418\pi\)
\(3\) −2.71791 −1.56919 −0.784594 0.620010i \(-0.787129\pi\)
−0.784594 + 0.620010i \(0.787129\pi\)
\(4\) −0.908666 −0.454333
\(5\) 0.714085 0.319349 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(6\) 2.83932 1.15915
\(7\) −2.03236 −0.768162 −0.384081 0.923299i \(-0.625482\pi\)
−0.384081 + 0.923299i \(0.625482\pi\)
\(8\) 3.03859 1.07431
\(9\) 4.38705 1.46235
\(10\) −0.745983 −0.235901
\(11\) 5.43392 1.63839 0.819194 0.573516i \(-0.194421\pi\)
0.819194 + 0.573516i \(0.194421\pi\)
\(12\) 2.46968 0.712934
\(13\) −3.96770 −1.10044 −0.550221 0.835019i \(-0.685457\pi\)
−0.550221 + 0.835019i \(0.685457\pi\)
\(14\) 2.12315 0.567435
\(15\) −1.94082 −0.501118
\(16\) −1.35699 −0.339248
\(17\) −1.30793 −0.317219 −0.158609 0.987341i \(-0.550701\pi\)
−0.158609 + 0.987341i \(0.550701\pi\)
\(18\) −4.58302 −1.08023
\(19\) 8.24470 1.89146 0.945732 0.324947i \(-0.105346\pi\)
0.945732 + 0.324947i \(0.105346\pi\)
\(20\) −0.648865 −0.145091
\(21\) 5.52379 1.20539
\(22\) −5.67665 −1.21027
\(23\) −2.44824 −0.510493 −0.255246 0.966876i \(-0.582157\pi\)
−0.255246 + 0.966876i \(0.582157\pi\)
\(24\) −8.25864 −1.68579
\(25\) −4.49008 −0.898016
\(26\) 4.14494 0.812889
\(27\) −3.76989 −0.725517
\(28\) 1.84674 0.349001
\(29\) 1.49405 0.277438 0.138719 0.990332i \(-0.455702\pi\)
0.138719 + 0.990332i \(0.455702\pi\)
\(30\) 2.02752 0.370172
\(31\) −6.02301 −1.08176 −0.540882 0.841098i \(-0.681909\pi\)
−0.540882 + 0.841098i \(0.681909\pi\)
\(32\) −4.65958 −0.823705
\(33\) −14.7689 −2.57094
\(34\) 1.36635 0.234327
\(35\) −1.45128 −0.245311
\(36\) −3.98637 −0.664395
\(37\) −6.36158 −1.04584 −0.522919 0.852383i \(-0.675157\pi\)
−0.522919 + 0.852383i \(0.675157\pi\)
\(38\) −8.61299 −1.39721
\(39\) 10.7839 1.72680
\(40\) 2.16982 0.343078
\(41\) −2.87189 −0.448514 −0.224257 0.974530i \(-0.571995\pi\)
−0.224257 + 0.974530i \(0.571995\pi\)
\(42\) −5.77053 −0.890413
\(43\) −5.76713 −0.879478 −0.439739 0.898125i \(-0.644929\pi\)
−0.439739 + 0.898125i \(0.644929\pi\)
\(44\) −4.93762 −0.744374
\(45\) 3.13273 0.467000
\(46\) 2.55760 0.377097
\(47\) 6.42848 0.937689 0.468845 0.883281i \(-0.344670\pi\)
0.468845 + 0.883281i \(0.344670\pi\)
\(48\) 3.68819 0.532344
\(49\) −2.86949 −0.409928
\(50\) 4.69065 0.663358
\(51\) 3.55483 0.497776
\(52\) 3.60532 0.499967
\(53\) −0.932545 −0.128095 −0.0640474 0.997947i \(-0.520401\pi\)
−0.0640474 + 0.997947i \(0.520401\pi\)
\(54\) 3.93829 0.535934
\(55\) 3.88028 0.523217
\(56\) −6.17553 −0.825240
\(57\) −22.4084 −2.96806
\(58\) −1.56079 −0.204941
\(59\) 1.57369 0.204877 0.102438 0.994739i \(-0.467336\pi\)
0.102438 + 0.994739i \(0.467336\pi\)
\(60\) 1.76356 0.227675
\(61\) −7.93215 −1.01561 −0.507804 0.861473i \(-0.669542\pi\)
−0.507804 + 0.861473i \(0.669542\pi\)
\(62\) 6.29205 0.799091
\(63\) −8.91609 −1.12332
\(64\) 7.58170 0.947713
\(65\) −2.83328 −0.351425
\(66\) 15.4286 1.89913
\(67\) 10.5294 1.28637 0.643186 0.765710i \(-0.277612\pi\)
0.643186 + 0.765710i \(0.277612\pi\)
\(68\) 1.18847 0.144123
\(69\) 6.65410 0.801059
\(70\) 1.51611 0.181210
\(71\) −12.8929 −1.53011 −0.765054 0.643966i \(-0.777288\pi\)
−0.765054 + 0.643966i \(0.777288\pi\)
\(72\) 13.3305 1.57101
\(73\) 9.81833 1.14915 0.574574 0.818452i \(-0.305168\pi\)
0.574574 + 0.818452i \(0.305168\pi\)
\(74\) 6.64575 0.772552
\(75\) 12.2037 1.40916
\(76\) −7.49168 −0.859355
\(77\) −11.0437 −1.25855
\(78\) −11.2656 −1.27558
\(79\) −2.80129 −0.315170 −0.157585 0.987505i \(-0.550371\pi\)
−0.157585 + 0.987505i \(0.550371\pi\)
\(80\) −0.969009 −0.108338
\(81\) −2.91492 −0.323880
\(82\) 3.00017 0.331314
\(83\) −15.3935 −1.68966 −0.844829 0.535037i \(-0.820298\pi\)
−0.844829 + 0.535037i \(0.820298\pi\)
\(84\) −5.01928 −0.547649
\(85\) −0.933971 −0.101303
\(86\) 6.02474 0.649664
\(87\) −4.06069 −0.435352
\(88\) 16.5115 1.76013
\(89\) −1.89811 −0.201199 −0.100600 0.994927i \(-0.532076\pi\)
−0.100600 + 0.994927i \(0.532076\pi\)
\(90\) −3.27267 −0.344969
\(91\) 8.06382 0.845318
\(92\) 2.22463 0.231934
\(93\) 16.3700 1.69749
\(94\) −6.71563 −0.692664
\(95\) 5.88742 0.604037
\(96\) 12.6643 1.29255
\(97\) 0.276333 0.0280574 0.0140287 0.999902i \(-0.495534\pi\)
0.0140287 + 0.999902i \(0.495534\pi\)
\(98\) 2.99767 0.302811
\(99\) 23.8389 2.39590
\(100\) 4.07999 0.407999
\(101\) −17.0451 −1.69605 −0.848024 0.529958i \(-0.822208\pi\)
−0.848024 + 0.529958i \(0.822208\pi\)
\(102\) −3.71362 −0.367703
\(103\) 6.40415 0.631020 0.315510 0.948922i \(-0.397824\pi\)
0.315510 + 0.948922i \(0.397824\pi\)
\(104\) −12.0562 −1.18221
\(105\) 3.94446 0.384940
\(106\) 0.974201 0.0946227
\(107\) 6.34857 0.613739 0.306870 0.951752i \(-0.400718\pi\)
0.306870 + 0.951752i \(0.400718\pi\)
\(108\) 3.42558 0.329626
\(109\) 9.33636 0.894261 0.447130 0.894469i \(-0.352446\pi\)
0.447130 + 0.894469i \(0.352446\pi\)
\(110\) −4.05361 −0.386497
\(111\) 17.2902 1.64112
\(112\) 2.75790 0.260597
\(113\) −1.89369 −0.178144 −0.0890718 0.996025i \(-0.528390\pi\)
−0.0890718 + 0.996025i \(0.528390\pi\)
\(114\) 23.4094 2.19249
\(115\) −1.74825 −0.163025
\(116\) −1.35759 −0.126049
\(117\) −17.4065 −1.60923
\(118\) −1.64398 −0.151341
\(119\) 2.65818 0.243675
\(120\) −5.89737 −0.538354
\(121\) 18.5275 1.68432
\(122\) 8.28647 0.750222
\(123\) 7.80555 0.703802
\(124\) 5.47290 0.491481
\(125\) −6.77673 −0.606129
\(126\) 9.31437 0.829790
\(127\) 10.2086 0.905867 0.452933 0.891544i \(-0.350378\pi\)
0.452933 + 0.891544i \(0.350378\pi\)
\(128\) 1.39879 0.123636
\(129\) 15.6746 1.38007
\(130\) 2.95984 0.259595
\(131\) −13.9590 −1.21960 −0.609801 0.792554i \(-0.708751\pi\)
−0.609801 + 0.792554i \(0.708751\pi\)
\(132\) 13.4200 1.16806
\(133\) −16.7562 −1.45295
\(134\) −10.9997 −0.950233
\(135\) −2.69203 −0.231693
\(136\) −3.97426 −0.340790
\(137\) −2.19310 −0.187369 −0.0936845 0.995602i \(-0.529864\pi\)
−0.0936845 + 0.995602i \(0.529864\pi\)
\(138\) −6.95133 −0.591737
\(139\) −19.6789 −1.66914 −0.834572 0.550899i \(-0.814285\pi\)
−0.834572 + 0.550899i \(0.814285\pi\)
\(140\) 1.31873 0.111453
\(141\) −17.4720 −1.47141
\(142\) 13.4688 1.13028
\(143\) −21.5602 −1.80295
\(144\) −5.95320 −0.496100
\(145\) 1.06688 0.0885994
\(146\) −10.2569 −0.848868
\(147\) 7.79904 0.643254
\(148\) 5.78055 0.475159
\(149\) −11.0882 −0.908379 −0.454189 0.890905i \(-0.650071\pi\)
−0.454189 + 0.890905i \(0.650071\pi\)
\(150\) −12.7488 −1.04093
\(151\) 7.03425 0.572439 0.286219 0.958164i \(-0.407601\pi\)
0.286219 + 0.958164i \(0.407601\pi\)
\(152\) 25.0523 2.03201
\(153\) −5.73795 −0.463885
\(154\) 11.5370 0.929679
\(155\) −4.30094 −0.345460
\(156\) −9.79894 −0.784543
\(157\) −24.2151 −1.93257 −0.966287 0.257468i \(-0.917112\pi\)
−0.966287 + 0.257468i \(0.917112\pi\)
\(158\) 2.92642 0.232814
\(159\) 2.53458 0.201005
\(160\) −3.32734 −0.263049
\(161\) 4.97571 0.392141
\(162\) 3.04512 0.239247
\(163\) 0.672203 0.0526510 0.0263255 0.999653i \(-0.491619\pi\)
0.0263255 + 0.999653i \(0.491619\pi\)
\(164\) 2.60959 0.203775
\(165\) −10.5463 −0.821026
\(166\) 16.0811 1.24814
\(167\) −13.2184 −1.02287 −0.511436 0.859321i \(-0.670886\pi\)
−0.511436 + 0.859321i \(0.670886\pi\)
\(168\) 16.7846 1.29496
\(169\) 2.74265 0.210973
\(170\) 0.975691 0.0748321
\(171\) 36.1700 2.76599
\(172\) 5.24039 0.399576
\(173\) −18.2674 −1.38884 −0.694421 0.719569i \(-0.744339\pi\)
−0.694421 + 0.719569i \(0.744339\pi\)
\(174\) 4.24208 0.321591
\(175\) 9.12548 0.689822
\(176\) −7.37379 −0.555820
\(177\) −4.27715 −0.321490
\(178\) 1.98290 0.148624
\(179\) 21.6343 1.61702 0.808511 0.588481i \(-0.200274\pi\)
0.808511 + 0.588481i \(0.200274\pi\)
\(180\) −2.84661 −0.212174
\(181\) 0.765562 0.0569038 0.0284519 0.999595i \(-0.490942\pi\)
0.0284519 + 0.999595i \(0.490942\pi\)
\(182\) −8.42402 −0.624430
\(183\) 21.5589 1.59368
\(184\) −7.43920 −0.548425
\(185\) −4.54271 −0.333987
\(186\) −17.1012 −1.25392
\(187\) −7.10717 −0.519728
\(188\) −5.84134 −0.426023
\(189\) 7.66180 0.557314
\(190\) −6.15041 −0.446197
\(191\) −3.52515 −0.255071 −0.127535 0.991834i \(-0.540707\pi\)
−0.127535 + 0.991834i \(0.540707\pi\)
\(192\) −20.6064 −1.48714
\(193\) −8.94657 −0.643988 −0.321994 0.946742i \(-0.604353\pi\)
−0.321994 + 0.946742i \(0.604353\pi\)
\(194\) −0.288677 −0.0207258
\(195\) 7.70060 0.551452
\(196\) 2.60741 0.186244
\(197\) −20.1670 −1.43684 −0.718418 0.695612i \(-0.755133\pi\)
−0.718418 + 0.695612i \(0.755133\pi\)
\(198\) −24.9038 −1.76983
\(199\) 3.72704 0.264203 0.132101 0.991236i \(-0.457828\pi\)
0.132101 + 0.991236i \(0.457828\pi\)
\(200\) −13.6435 −0.964744
\(201\) −28.6180 −2.01856
\(202\) 17.8065 1.25286
\(203\) −3.03645 −0.213117
\(204\) −3.23016 −0.226156
\(205\) −2.05077 −0.143232
\(206\) −6.69022 −0.466130
\(207\) −10.7406 −0.746520
\(208\) 5.38414 0.373323
\(209\) 44.8011 3.09895
\(210\) −4.12065 −0.284352
\(211\) 17.9812 1.23788 0.618940 0.785438i \(-0.287562\pi\)
0.618940 + 0.785438i \(0.287562\pi\)
\(212\) 0.847372 0.0581977
\(213\) 35.0419 2.40103
\(214\) −6.63215 −0.453365
\(215\) −4.11822 −0.280860
\(216\) −11.4552 −0.779426
\(217\) 12.2409 0.830970
\(218\) −9.75341 −0.660584
\(219\) −26.6854 −1.80323
\(220\) −3.52588 −0.237715
\(221\) 5.18946 0.349081
\(222\) −18.0626 −1.21228
\(223\) −19.2192 −1.28701 −0.643507 0.765440i \(-0.722521\pi\)
−0.643507 + 0.765440i \(0.722521\pi\)
\(224\) 9.46996 0.632739
\(225\) −19.6982 −1.31322
\(226\) 1.97828 0.131593
\(227\) 23.2733 1.54470 0.772350 0.635197i \(-0.219081\pi\)
0.772350 + 0.635197i \(0.219081\pi\)
\(228\) 20.3617 1.34849
\(229\) 1.04868 0.0692987 0.0346494 0.999400i \(-0.488969\pi\)
0.0346494 + 0.999400i \(0.488969\pi\)
\(230\) 1.82634 0.120426
\(231\) 30.0158 1.97490
\(232\) 4.53981 0.298053
\(233\) 10.4018 0.681441 0.340721 0.940165i \(-0.389329\pi\)
0.340721 + 0.940165i \(0.389329\pi\)
\(234\) 18.1841 1.18873
\(235\) 4.59048 0.299450
\(236\) −1.42996 −0.0930823
\(237\) 7.61367 0.494561
\(238\) −2.77692 −0.180001
\(239\) 5.21082 0.337060 0.168530 0.985697i \(-0.446098\pi\)
0.168530 + 0.985697i \(0.446098\pi\)
\(240\) 2.63368 0.170003
\(241\) −23.1270 −1.48974 −0.744871 0.667209i \(-0.767489\pi\)
−0.744871 + 0.667209i \(0.767489\pi\)
\(242\) −19.3551 −1.24419
\(243\) 19.2322 1.23374
\(244\) 7.20767 0.461424
\(245\) −2.04906 −0.130910
\(246\) −8.15421 −0.519894
\(247\) −32.7125 −2.08145
\(248\) −18.3015 −1.16214
\(249\) 41.8382 2.65139
\(250\) 7.07944 0.447743
\(251\) −18.2828 −1.15400 −0.577001 0.816744i \(-0.695777\pi\)
−0.577001 + 0.816744i \(0.695777\pi\)
\(252\) 8.10175 0.510363
\(253\) −13.3035 −0.836386
\(254\) −10.6646 −0.669157
\(255\) 2.53845 0.158964
\(256\) −16.6247 −1.03904
\(257\) 30.4924 1.90207 0.951033 0.309091i \(-0.100025\pi\)
0.951033 + 0.309091i \(0.100025\pi\)
\(258\) −16.3747 −1.01945
\(259\) 12.9291 0.803372
\(260\) 2.57450 0.159664
\(261\) 6.55447 0.405712
\(262\) 14.5825 0.900911
\(263\) −11.2828 −0.695730 −0.347865 0.937545i \(-0.613093\pi\)
−0.347865 + 0.937545i \(0.613093\pi\)
\(264\) −44.8768 −2.76197
\(265\) −0.665916 −0.0409069
\(266\) 17.5047 1.07328
\(267\) 5.15890 0.315720
\(268\) −9.56772 −0.584441
\(269\) −1.54283 −0.0940679 −0.0470339 0.998893i \(-0.514977\pi\)
−0.0470339 + 0.998893i \(0.514977\pi\)
\(270\) 2.81228 0.171150
\(271\) −23.6023 −1.43374 −0.716870 0.697207i \(-0.754426\pi\)
−0.716870 + 0.697207i \(0.754426\pi\)
\(272\) 1.77485 0.107616
\(273\) −21.9168 −1.32646
\(274\) 2.29106 0.138408
\(275\) −24.3987 −1.47130
\(276\) −6.04635 −0.363948
\(277\) 14.2694 0.857367 0.428684 0.903455i \(-0.358978\pi\)
0.428684 + 0.903455i \(0.358978\pi\)
\(278\) 20.5580 1.23298
\(279\) −26.4233 −1.58192
\(280\) −4.40986 −0.263539
\(281\) 22.6952 1.35388 0.676942 0.736037i \(-0.263305\pi\)
0.676942 + 0.736037i \(0.263305\pi\)
\(282\) 18.2525 1.08692
\(283\) −22.9783 −1.36592 −0.682958 0.730458i \(-0.739307\pi\)
−0.682958 + 0.730458i \(0.739307\pi\)
\(284\) 11.7154 0.695179
\(285\) −16.0015 −0.947847
\(286\) 22.5232 1.33183
\(287\) 5.83672 0.344531
\(288\) −20.4418 −1.20455
\(289\) −15.2893 −0.899372
\(290\) −1.11453 −0.0654477
\(291\) −0.751051 −0.0440274
\(292\) −8.92158 −0.522096
\(293\) −25.1176 −1.46739 −0.733694 0.679480i \(-0.762205\pi\)
−0.733694 + 0.679480i \(0.762205\pi\)
\(294\) −8.14742 −0.475167
\(295\) 1.12375 0.0654271
\(296\) −19.3303 −1.12355
\(297\) −20.4853 −1.18868
\(298\) 11.5835 0.671013
\(299\) 9.71388 0.561768
\(300\) −11.0891 −0.640227
\(301\) 11.7209 0.675582
\(302\) −7.34846 −0.422856
\(303\) 46.3270 2.66142
\(304\) −11.1880 −0.641676
\(305\) −5.66423 −0.324333
\(306\) 5.99425 0.342669
\(307\) −14.5944 −0.832947 −0.416473 0.909148i \(-0.636734\pi\)
−0.416473 + 0.909148i \(0.636734\pi\)
\(308\) 10.0350 0.571800
\(309\) −17.4059 −0.990188
\(310\) 4.49306 0.255189
\(311\) 8.99147 0.509860 0.254930 0.966960i \(-0.417948\pi\)
0.254930 + 0.966960i \(0.417948\pi\)
\(312\) 32.7678 1.85511
\(313\) −4.56361 −0.257951 −0.128975 0.991648i \(-0.541169\pi\)
−0.128975 + 0.991648i \(0.541169\pi\)
\(314\) 25.2967 1.42758
\(315\) −6.36685 −0.358731
\(316\) 2.54544 0.143192
\(317\) −7.39682 −0.415447 −0.207723 0.978188i \(-0.566605\pi\)
−0.207723 + 0.978188i \(0.566605\pi\)
\(318\) −2.64779 −0.148481
\(319\) 8.11854 0.454551
\(320\) 5.41398 0.302651
\(321\) −17.2549 −0.963072
\(322\) −5.19797 −0.289672
\(323\) −10.7835 −0.600008
\(324\) 2.64869 0.147149
\(325\) 17.8153 0.988215
\(326\) −0.702230 −0.0388929
\(327\) −25.3754 −1.40326
\(328\) −8.72650 −0.481841
\(329\) −13.0650 −0.720297
\(330\) 11.0174 0.606486
\(331\) −23.5847 −1.29633 −0.648166 0.761499i \(-0.724464\pi\)
−0.648166 + 0.761499i \(0.724464\pi\)
\(332\) 13.9876 0.767667
\(333\) −27.9086 −1.52938
\(334\) 13.8089 0.755588
\(335\) 7.51890 0.410801
\(336\) −7.49575 −0.408926
\(337\) 25.0637 1.36531 0.682653 0.730743i \(-0.260826\pi\)
0.682653 + 0.730743i \(0.260826\pi\)
\(338\) −2.86517 −0.155845
\(339\) 5.14689 0.279541
\(340\) 0.848668 0.0460255
\(341\) −32.7285 −1.77235
\(342\) −37.7856 −2.04321
\(343\) 20.0584 1.08305
\(344\) −17.5240 −0.944828
\(345\) 4.75159 0.255817
\(346\) 19.0834 1.02593
\(347\) 11.1374 0.597888 0.298944 0.954271i \(-0.403366\pi\)
0.298944 + 0.954271i \(0.403366\pi\)
\(348\) 3.68982 0.197795
\(349\) 34.3068 1.83640 0.918202 0.396114i \(-0.129641\pi\)
0.918202 + 0.396114i \(0.129641\pi\)
\(350\) −9.53311 −0.509566
\(351\) 14.9578 0.798389
\(352\) −25.3198 −1.34955
\(353\) 18.3887 0.978731 0.489366 0.872079i \(-0.337228\pi\)
0.489366 + 0.872079i \(0.337228\pi\)
\(354\) 4.46821 0.237482
\(355\) −9.20665 −0.488638
\(356\) 1.72475 0.0914115
\(357\) −7.22471 −0.382372
\(358\) −22.6007 −1.19448
\(359\) 16.8572 0.889688 0.444844 0.895608i \(-0.353259\pi\)
0.444844 + 0.895608i \(0.353259\pi\)
\(360\) 9.51910 0.501701
\(361\) 48.9751 2.57764
\(362\) −0.799759 −0.0420344
\(363\) −50.3561 −2.64301
\(364\) −7.32732 −0.384056
\(365\) 7.01112 0.366979
\(366\) −22.5219 −1.17724
\(367\) 20.9807 1.09518 0.547592 0.836745i \(-0.315544\pi\)
0.547592 + 0.836745i \(0.315544\pi\)
\(368\) 3.32224 0.173184
\(369\) −12.5991 −0.655885
\(370\) 4.74563 0.246714
\(371\) 1.89527 0.0983975
\(372\) −14.8749 −0.771227
\(373\) 8.57191 0.443837 0.221918 0.975065i \(-0.428768\pi\)
0.221918 + 0.975065i \(0.428768\pi\)
\(374\) 7.42464 0.383919
\(375\) 18.4186 0.951130
\(376\) 19.5335 1.00736
\(377\) −5.92794 −0.305304
\(378\) −8.00405 −0.411684
\(379\) −24.0731 −1.23655 −0.618275 0.785962i \(-0.712168\pi\)
−0.618275 + 0.785962i \(0.712168\pi\)
\(380\) −5.34970 −0.274434
\(381\) −27.7461 −1.42148
\(382\) 3.68261 0.188419
\(383\) −28.2472 −1.44336 −0.721682 0.692225i \(-0.756631\pi\)
−0.721682 + 0.692225i \(0.756631\pi\)
\(384\) −3.80178 −0.194009
\(385\) −7.88615 −0.401915
\(386\) 9.34621 0.475709
\(387\) −25.3007 −1.28611
\(388\) −0.251095 −0.0127474
\(389\) 20.2718 1.02782 0.513911 0.857843i \(-0.328196\pi\)
0.513911 + 0.857843i \(0.328196\pi\)
\(390\) −8.04458 −0.407353
\(391\) 3.20211 0.161938
\(392\) −8.71923 −0.440388
\(393\) 37.9393 1.91379
\(394\) 21.0678 1.06138
\(395\) −2.00036 −0.100649
\(396\) −21.6616 −1.08854
\(397\) 9.39156 0.471349 0.235674 0.971832i \(-0.424270\pi\)
0.235674 + 0.971832i \(0.424270\pi\)
\(398\) −3.89352 −0.195165
\(399\) 45.5420 2.27995
\(400\) 6.09301 0.304650
\(401\) 8.35327 0.417143 0.208571 0.978007i \(-0.433119\pi\)
0.208571 + 0.978007i \(0.433119\pi\)
\(402\) 29.8964 1.49110
\(403\) 23.8975 1.19042
\(404\) 15.4883 0.770571
\(405\) −2.08150 −0.103431
\(406\) 3.17209 0.157428
\(407\) −34.5683 −1.71349
\(408\) 10.8017 0.534763
\(409\) −9.04325 −0.447160 −0.223580 0.974686i \(-0.571774\pi\)
−0.223580 + 0.974686i \(0.571774\pi\)
\(410\) 2.14238 0.105805
\(411\) 5.96065 0.294017
\(412\) −5.81923 −0.286693
\(413\) −3.19831 −0.157378
\(414\) 11.2203 0.551449
\(415\) −10.9923 −0.539590
\(416\) 18.4878 0.906440
\(417\) 53.4856 2.61920
\(418\) −46.8023 −2.28917
\(419\) −21.2265 −1.03698 −0.518491 0.855083i \(-0.673506\pi\)
−0.518491 + 0.855083i \(0.673506\pi\)
\(420\) −3.58420 −0.174891
\(421\) 16.3175 0.795269 0.397634 0.917544i \(-0.369831\pi\)
0.397634 + 0.917544i \(0.369831\pi\)
\(422\) −18.7844 −0.914412
\(423\) 28.2021 1.37123
\(424\) −2.83362 −0.137613
\(425\) 5.87270 0.284868
\(426\) −36.6071 −1.77362
\(427\) 16.1210 0.780151
\(428\) −5.76873 −0.278842
\(429\) 58.5987 2.82917
\(430\) 4.30218 0.207469
\(431\) 17.2192 0.829420 0.414710 0.909954i \(-0.363883\pi\)
0.414710 + 0.909954i \(0.363883\pi\)
\(432\) 5.11572 0.246130
\(433\) 0.155714 0.00748312 0.00374156 0.999993i \(-0.498809\pi\)
0.00374156 + 0.999993i \(0.498809\pi\)
\(434\) −12.7877 −0.613831
\(435\) −2.89968 −0.139029
\(436\) −8.48363 −0.406292
\(437\) −20.1850 −0.965579
\(438\) 27.8774 1.33203
\(439\) 23.1300 1.10394 0.551968 0.833865i \(-0.313877\pi\)
0.551968 + 0.833865i \(0.313877\pi\)
\(440\) 11.7906 0.562095
\(441\) −12.5886 −0.599459
\(442\) −5.42127 −0.257864
\(443\) 12.5268 0.595168 0.297584 0.954696i \(-0.403819\pi\)
0.297584 + 0.954696i \(0.403819\pi\)
\(444\) −15.7110 −0.745613
\(445\) −1.35541 −0.0642527
\(446\) 20.0777 0.950708
\(447\) 30.1367 1.42542
\(448\) −15.4088 −0.727997
\(449\) −38.0487 −1.79563 −0.897814 0.440375i \(-0.854846\pi\)
−0.897814 + 0.440375i \(0.854846\pi\)
\(450\) 20.5781 0.970063
\(451\) −15.6056 −0.734840
\(452\) 1.72074 0.0809366
\(453\) −19.1185 −0.898264
\(454\) −24.3129 −1.14106
\(455\) 5.75825 0.269951
\(456\) −68.0900 −3.18861
\(457\) 25.4956 1.19263 0.596317 0.802749i \(-0.296630\pi\)
0.596317 + 0.802749i \(0.296630\pi\)
\(458\) −1.09552 −0.0511905
\(459\) 4.93075 0.230147
\(460\) 1.58858 0.0740677
\(461\) 22.5869 1.05198 0.525988 0.850492i \(-0.323696\pi\)
0.525988 + 0.850492i \(0.323696\pi\)
\(462\) −31.3566 −1.45884
\(463\) 42.2153 1.96191 0.980956 0.194232i \(-0.0622214\pi\)
0.980956 + 0.194232i \(0.0622214\pi\)
\(464\) −2.02741 −0.0941203
\(465\) 11.6896 0.542092
\(466\) −10.8664 −0.503376
\(467\) 4.29006 0.198520 0.0992601 0.995062i \(-0.468352\pi\)
0.0992601 + 0.995062i \(0.468352\pi\)
\(468\) 15.8167 0.731128
\(469\) −21.3996 −0.988141
\(470\) −4.79553 −0.221201
\(471\) 65.8145 3.03257
\(472\) 4.78180 0.220100
\(473\) −31.3381 −1.44093
\(474\) −7.95376 −0.365328
\(475\) −37.0194 −1.69857
\(476\) −2.41540 −0.110710
\(477\) −4.09112 −0.187320
\(478\) −5.44358 −0.248984
\(479\) −7.74802 −0.354016 −0.177008 0.984209i \(-0.556642\pi\)
−0.177008 + 0.984209i \(0.556642\pi\)
\(480\) 9.04342 0.412774
\(481\) 25.2409 1.15088
\(482\) 24.1601 1.10046
\(483\) −13.5236 −0.615343
\(484\) −16.8353 −0.765241
\(485\) 0.197326 0.00896010
\(486\) −20.0913 −0.911358
\(487\) 3.12687 0.141692 0.0708459 0.997487i \(-0.477430\pi\)
0.0708459 + 0.997487i \(0.477430\pi\)
\(488\) −24.1026 −1.09107
\(489\) −1.82699 −0.0826193
\(490\) 2.14059 0.0967022
\(491\) −17.8547 −0.805771 −0.402886 0.915250i \(-0.631993\pi\)
−0.402886 + 0.915250i \(0.631993\pi\)
\(492\) −7.09264 −0.319761
\(493\) −1.95411 −0.0880085
\(494\) 34.1738 1.53755
\(495\) 17.0230 0.765127
\(496\) 8.17318 0.366987
\(497\) 26.2031 1.17537
\(498\) −43.7071 −1.95856
\(499\) 2.04004 0.0913246 0.0456623 0.998957i \(-0.485460\pi\)
0.0456623 + 0.998957i \(0.485460\pi\)
\(500\) 6.15778 0.275385
\(501\) 35.9265 1.60508
\(502\) 19.0995 0.852452
\(503\) 7.07972 0.315669 0.157835 0.987466i \(-0.449549\pi\)
0.157835 + 0.987466i \(0.449549\pi\)
\(504\) −27.0924 −1.20679
\(505\) −12.1716 −0.541631
\(506\) 13.8978 0.617832
\(507\) −7.45430 −0.331057
\(508\) −9.27621 −0.411565
\(509\) 17.9368 0.795037 0.397518 0.917594i \(-0.369871\pi\)
0.397518 + 0.917594i \(0.369871\pi\)
\(510\) −2.65184 −0.117426
\(511\) −19.9544 −0.882732
\(512\) 14.5697 0.643897
\(513\) −31.0817 −1.37229
\(514\) −31.8545 −1.40504
\(515\) 4.57311 0.201515
\(516\) −14.2429 −0.627010
\(517\) 34.9318 1.53630
\(518\) −13.5066 −0.593445
\(519\) 49.6491 2.17935
\(520\) −8.60918 −0.377538
\(521\) −3.19375 −0.139921 −0.0699603 0.997550i \(-0.522287\pi\)
−0.0699603 + 0.997550i \(0.522287\pi\)
\(522\) −6.84726 −0.299696
\(523\) −15.7713 −0.689630 −0.344815 0.938671i \(-0.612058\pi\)
−0.344815 + 0.938671i \(0.612058\pi\)
\(524\) 12.6841 0.554106
\(525\) −24.8023 −1.08246
\(526\) 11.7868 0.513931
\(527\) 7.87765 0.343156
\(528\) 20.0413 0.872187
\(529\) −17.0061 −0.739397
\(530\) 0.695662 0.0302176
\(531\) 6.90386 0.299602
\(532\) 15.2258 0.660123
\(533\) 11.3948 0.493563
\(534\) −5.38935 −0.233220
\(535\) 4.53342 0.195997
\(536\) 31.9946 1.38196
\(537\) −58.8001 −2.53741
\(538\) 1.61175 0.0694873
\(539\) −15.5926 −0.671621
\(540\) 2.44615 0.105266
\(541\) 28.1775 1.21144 0.605722 0.795676i \(-0.292884\pi\)
0.605722 + 0.795676i \(0.292884\pi\)
\(542\) 24.6566 1.05909
\(543\) −2.08073 −0.0892927
\(544\) 6.09439 0.261295
\(545\) 6.66696 0.285581
\(546\) 22.8958 0.979848
\(547\) −1.00000 −0.0427569
\(548\) 1.99279 0.0851280
\(549\) −34.7988 −1.48517
\(550\) 25.4886 1.08684
\(551\) 12.3180 0.524764
\(552\) 20.2191 0.860582
\(553\) 5.69324 0.242101
\(554\) −14.9068 −0.633331
\(555\) 12.3467 0.524088
\(556\) 17.8816 0.758348
\(557\) −11.6421 −0.493290 −0.246645 0.969106i \(-0.579328\pi\)
−0.246645 + 0.969106i \(0.579328\pi\)
\(558\) 27.6036 1.16855
\(559\) 22.8822 0.967815
\(560\) 1.96938 0.0832214
\(561\) 19.3167 0.815550
\(562\) −23.7090 −1.00010
\(563\) 35.1631 1.48195 0.740974 0.671534i \(-0.234364\pi\)
0.740974 + 0.671534i \(0.234364\pi\)
\(564\) 15.8763 0.668511
\(565\) −1.35226 −0.0568899
\(566\) 24.0047 1.00899
\(567\) 5.92417 0.248792
\(568\) −39.1764 −1.64380
\(569\) −5.54994 −0.232665 −0.116333 0.993210i \(-0.537114\pi\)
−0.116333 + 0.993210i \(0.537114\pi\)
\(570\) 16.7163 0.700168
\(571\) −12.2975 −0.514633 −0.257317 0.966327i \(-0.582838\pi\)
−0.257317 + 0.966327i \(0.582838\pi\)
\(572\) 19.5910 0.819141
\(573\) 9.58105 0.400254
\(574\) −6.09745 −0.254502
\(575\) 10.9928 0.458431
\(576\) 33.2614 1.38589
\(577\) −21.7654 −0.906105 −0.453052 0.891484i \(-0.649665\pi\)
−0.453052 + 0.891484i \(0.649665\pi\)
\(578\) 15.9723 0.664360
\(579\) 24.3160 1.01054
\(580\) −0.969436 −0.0402536
\(581\) 31.2852 1.29793
\(582\) 0.784599 0.0325227
\(583\) −5.06737 −0.209869
\(584\) 29.8339 1.23454
\(585\) −12.4297 −0.513907
\(586\) 26.2396 1.08395
\(587\) −1.32577 −0.0547206 −0.0273603 0.999626i \(-0.508710\pi\)
−0.0273603 + 0.999626i \(0.508710\pi\)
\(588\) −7.08672 −0.292252
\(589\) −49.6579 −2.04612
\(590\) −1.17394 −0.0483305
\(591\) 54.8120 2.25467
\(592\) 8.63262 0.354799
\(593\) −0.00527691 −0.000216697 0 −0.000108348 1.00000i \(-0.500034\pi\)
−0.000108348 1.00000i \(0.500034\pi\)
\(594\) 21.4004 0.878068
\(595\) 1.89817 0.0778174
\(596\) 10.0755 0.412707
\(597\) −10.1298 −0.414584
\(598\) −10.1478 −0.414974
\(599\) −26.4628 −1.08124 −0.540620 0.841267i \(-0.681810\pi\)
−0.540620 + 0.841267i \(0.681810\pi\)
\(600\) 37.0820 1.51386
\(601\) −17.3745 −0.708719 −0.354360 0.935109i \(-0.615301\pi\)
−0.354360 + 0.935109i \(0.615301\pi\)
\(602\) −12.2445 −0.499047
\(603\) 46.1931 1.88113
\(604\) −6.39178 −0.260078
\(605\) 13.2302 0.537884
\(606\) −48.3964 −1.96597
\(607\) −28.3811 −1.15195 −0.575977 0.817466i \(-0.695378\pi\)
−0.575977 + 0.817466i \(0.695378\pi\)
\(608\) −38.4168 −1.55801
\(609\) 8.25281 0.334421
\(610\) 5.91725 0.239582
\(611\) −25.5063 −1.03187
\(612\) 5.21388 0.210759
\(613\) −31.8417 −1.28607 −0.643036 0.765836i \(-0.722326\pi\)
−0.643036 + 0.765836i \(0.722326\pi\)
\(614\) 15.2463 0.615292
\(615\) 5.57383 0.224758
\(616\) −33.5573 −1.35206
\(617\) −41.2687 −1.66141 −0.830707 0.556710i \(-0.812064\pi\)
−0.830707 + 0.556710i \(0.812064\pi\)
\(618\) 18.1834 0.731445
\(619\) 36.6519 1.47316 0.736582 0.676348i \(-0.236439\pi\)
0.736582 + 0.676348i \(0.236439\pi\)
\(620\) 3.90812 0.156954
\(621\) 9.22960 0.370371
\(622\) −9.39311 −0.376630
\(623\) 3.85765 0.154554
\(624\) −14.6336 −0.585814
\(625\) 17.6112 0.704450
\(626\) 4.76747 0.190546
\(627\) −121.765 −4.86284
\(628\) 22.0034 0.878032
\(629\) 8.32048 0.331759
\(630\) 6.65125 0.264992
\(631\) −33.4006 −1.32966 −0.664829 0.746996i \(-0.731495\pi\)
−0.664829 + 0.746996i \(0.731495\pi\)
\(632\) −8.51199 −0.338589
\(633\) −48.8715 −1.94247
\(634\) 7.72723 0.306887
\(635\) 7.28981 0.289287
\(636\) −2.30308 −0.0913232
\(637\) 11.3853 0.451102
\(638\) −8.48119 −0.335773
\(639\) −56.5620 −2.23756
\(640\) 0.998852 0.0394831
\(641\) 6.94918 0.274476 0.137238 0.990538i \(-0.456177\pi\)
0.137238 + 0.990538i \(0.456177\pi\)
\(642\) 18.0256 0.711414
\(643\) 4.52617 0.178495 0.0892474 0.996009i \(-0.471554\pi\)
0.0892474 + 0.996009i \(0.471554\pi\)
\(644\) −4.52126 −0.178163
\(645\) 11.1930 0.440723
\(646\) 11.2652 0.443222
\(647\) 25.3439 0.996371 0.498185 0.867071i \(-0.334000\pi\)
0.498185 + 0.867071i \(0.334000\pi\)
\(648\) −8.85725 −0.347946
\(649\) 8.55130 0.335668
\(650\) −18.6111 −0.729987
\(651\) −33.2698 −1.30395
\(652\) −0.610808 −0.0239211
\(653\) −8.49304 −0.332358 −0.166179 0.986096i \(-0.553143\pi\)
−0.166179 + 0.986096i \(0.553143\pi\)
\(654\) 26.5089 1.03658
\(655\) −9.96791 −0.389478
\(656\) 3.89713 0.152157
\(657\) 43.0735 1.68046
\(658\) 13.6486 0.532078
\(659\) 18.2273 0.710036 0.355018 0.934859i \(-0.384475\pi\)
0.355018 + 0.934859i \(0.384475\pi\)
\(660\) 9.58304 0.373019
\(661\) −16.2494 −0.632027 −0.316013 0.948755i \(-0.602344\pi\)
−0.316013 + 0.948755i \(0.602344\pi\)
\(662\) 24.6382 0.957591
\(663\) −14.1045 −0.547774
\(664\) −46.7746 −1.81521
\(665\) −11.9654 −0.463998
\(666\) 29.1553 1.12974
\(667\) −3.65779 −0.141630
\(668\) 12.0111 0.464725
\(669\) 52.2362 2.01957
\(670\) −7.85476 −0.303456
\(671\) −43.1027 −1.66396
\(672\) −25.7385 −0.992886
\(673\) 39.3776 1.51789 0.758947 0.651153i \(-0.225714\pi\)
0.758947 + 0.651153i \(0.225714\pi\)
\(674\) −26.1832 −1.00854
\(675\) 16.9271 0.651526
\(676\) −2.49216 −0.0958522
\(677\) −39.4458 −1.51602 −0.758012 0.652241i \(-0.773829\pi\)
−0.758012 + 0.652241i \(0.773829\pi\)
\(678\) −5.37680 −0.206495
\(679\) −0.561610 −0.0215526
\(680\) −2.83796 −0.108831
\(681\) −63.2547 −2.42393
\(682\) 34.1905 1.30922
\(683\) 25.8506 0.989144 0.494572 0.869137i \(-0.335325\pi\)
0.494572 + 0.869137i \(0.335325\pi\)
\(684\) −32.8664 −1.25668
\(685\) −1.56606 −0.0598360
\(686\) −20.9544 −0.800043
\(687\) −2.85022 −0.108743
\(688\) 7.82595 0.298362
\(689\) 3.70006 0.140961
\(690\) −4.96384 −0.188970
\(691\) −34.6845 −1.31946 −0.659729 0.751503i \(-0.729329\pi\)
−0.659729 + 0.751503i \(0.729329\pi\)
\(692\) 16.5989 0.630997
\(693\) −48.4493 −1.84044
\(694\) −11.6349 −0.441656
\(695\) −14.0524 −0.533039
\(696\) −12.3388 −0.467701
\(697\) 3.75622 0.142277
\(698\) −35.8393 −1.35654
\(699\) −28.2711 −1.06931
\(700\) −8.29202 −0.313409
\(701\) −38.7984 −1.46540 −0.732698 0.680553i \(-0.761739\pi\)
−0.732698 + 0.680553i \(0.761739\pi\)
\(702\) −15.6260 −0.589764
\(703\) −52.4493 −1.97816
\(704\) 41.1984 1.55272
\(705\) −12.4765 −0.469893
\(706\) −19.2101 −0.722982
\(707\) 34.6418 1.30284
\(708\) 3.88650 0.146064
\(709\) −40.9056 −1.53624 −0.768121 0.640305i \(-0.778808\pi\)
−0.768121 + 0.640305i \(0.778808\pi\)
\(710\) 9.61790 0.360953
\(711\) −12.2894 −0.460889
\(712\) −5.76759 −0.216150
\(713\) 14.7458 0.552233
\(714\) 7.54744 0.282456
\(715\) −15.3958 −0.575770
\(716\) −19.6583 −0.734667
\(717\) −14.1626 −0.528910
\(718\) −17.6102 −0.657206
\(719\) 4.65393 0.173562 0.0867812 0.996227i \(-0.472342\pi\)
0.0867812 + 0.996227i \(0.472342\pi\)
\(720\) −4.25109 −0.158429
\(721\) −13.0156 −0.484725
\(722\) −51.1628 −1.90408
\(723\) 62.8572 2.33768
\(724\) −0.695640 −0.0258533
\(725\) −6.70840 −0.249144
\(726\) 52.6055 1.95237
\(727\) 8.00848 0.297018 0.148509 0.988911i \(-0.452553\pi\)
0.148509 + 0.988911i \(0.452553\pi\)
\(728\) 24.5027 0.908129
\(729\) −43.5266 −1.61210
\(730\) −7.32431 −0.271085
\(731\) 7.54298 0.278987
\(732\) −19.5898 −0.724061
\(733\) 20.7022 0.764652 0.382326 0.924028i \(-0.375123\pi\)
0.382326 + 0.924028i \(0.375123\pi\)
\(734\) −21.9179 −0.809005
\(735\) 5.56918 0.205422
\(736\) 11.4078 0.420495
\(737\) 57.2160 2.10758
\(738\) 13.1619 0.484497
\(739\) 10.3160 0.379479 0.189740 0.981834i \(-0.439236\pi\)
0.189740 + 0.981834i \(0.439236\pi\)
\(740\) 4.12781 0.151741
\(741\) 88.9098 3.26618
\(742\) −1.97993 −0.0726855
\(743\) −27.5334 −1.01010 −0.505051 0.863090i \(-0.668526\pi\)
−0.505051 + 0.863090i \(0.668526\pi\)
\(744\) 49.7418 1.82362
\(745\) −7.91790 −0.290090
\(746\) −8.95481 −0.327859
\(747\) −67.5322 −2.47087
\(748\) 6.45804 0.236129
\(749\) −12.9026 −0.471451
\(750\) −19.2413 −0.702593
\(751\) 30.6714 1.11922 0.559608 0.828758i \(-0.310952\pi\)
0.559608 + 0.828758i \(0.310952\pi\)
\(752\) −8.72340 −0.318110
\(753\) 49.6911 1.81084
\(754\) 6.19273 0.225526
\(755\) 5.02305 0.182808
\(756\) −6.96202 −0.253206
\(757\) 29.0720 1.05664 0.528320 0.849046i \(-0.322822\pi\)
0.528320 + 0.849046i \(0.322822\pi\)
\(758\) 25.1484 0.913431
\(759\) 36.1578 1.31245
\(760\) 17.8895 0.648920
\(761\) −46.1332 −1.67233 −0.836163 0.548481i \(-0.815206\pi\)
−0.836163 + 0.548481i \(0.815206\pi\)
\(762\) 28.9855 1.05003
\(763\) −18.9749 −0.686937
\(764\) 3.20318 0.115887
\(765\) −4.09738 −0.148141
\(766\) 29.5090 1.06620
\(767\) −6.24393 −0.225455
\(768\) 45.1844 1.63045
\(769\) 39.6177 1.42865 0.714326 0.699813i \(-0.246733\pi\)
0.714326 + 0.699813i \(0.246733\pi\)
\(770\) 8.23842 0.296892
\(771\) −82.8758 −2.98470
\(772\) 8.12945 0.292585
\(773\) −2.07610 −0.0746722 −0.0373361 0.999303i \(-0.511887\pi\)
−0.0373361 + 0.999303i \(0.511887\pi\)
\(774\) 26.4309 0.950038
\(775\) 27.0438 0.971442
\(776\) 0.839665 0.0301422
\(777\) −35.1400 −1.26064
\(778\) −21.1774 −0.759245
\(779\) −23.6779 −0.848348
\(780\) −6.99728 −0.250543
\(781\) −70.0591 −2.50691
\(782\) −3.34515 −0.119622
\(783\) −5.63241 −0.201286
\(784\) 3.89388 0.139067
\(785\) −17.2916 −0.617165
\(786\) −39.6340 −1.41370
\(787\) −7.33613 −0.261505 −0.130752 0.991415i \(-0.541739\pi\)
−0.130752 + 0.991415i \(0.541739\pi\)
\(788\) 18.3250 0.652802
\(789\) 30.6658 1.09173
\(790\) 2.08972 0.0743487
\(791\) 3.84867 0.136843
\(792\) 72.4367 2.57393
\(793\) 31.4724 1.11762
\(794\) −9.81107 −0.348182
\(795\) 1.80990 0.0641907
\(796\) −3.38663 −0.120036
\(797\) −19.2052 −0.680283 −0.340141 0.940374i \(-0.610475\pi\)
−0.340141 + 0.940374i \(0.610475\pi\)
\(798\) −47.5763 −1.68418
\(799\) −8.40797 −0.297453
\(800\) 20.9219 0.739701
\(801\) −8.32712 −0.294224
\(802\) −8.72641 −0.308140
\(803\) 53.3520 1.88275
\(804\) 26.0042 0.917098
\(805\) 3.55308 0.125230
\(806\) −24.9650 −0.879354
\(807\) 4.19327 0.147610
\(808\) −51.7931 −1.82207
\(809\) 15.0621 0.529554 0.264777 0.964310i \(-0.414702\pi\)
0.264777 + 0.964310i \(0.414702\pi\)
\(810\) 2.17448 0.0764034
\(811\) −3.10727 −0.109111 −0.0545554 0.998511i \(-0.517374\pi\)
−0.0545554 + 0.998511i \(0.517374\pi\)
\(812\) 2.75912 0.0968262
\(813\) 64.1491 2.24981
\(814\) 36.1125 1.26574
\(815\) 0.480010 0.0168140
\(816\) −4.82388 −0.168870
\(817\) −47.5482 −1.66350
\(818\) 9.44720 0.330314
\(819\) 35.3764 1.23615
\(820\) 1.86347 0.0650752
\(821\) −39.3045 −1.37174 −0.685869 0.727725i \(-0.740577\pi\)
−0.685869 + 0.727725i \(0.740577\pi\)
\(822\) −6.22691 −0.217188
\(823\) −24.7300 −0.862033 −0.431017 0.902344i \(-0.641845\pi\)
−0.431017 + 0.902344i \(0.641845\pi\)
\(824\) 19.4596 0.677908
\(825\) 66.3137 2.30875
\(826\) 3.34117 0.116254
\(827\) −37.2644 −1.29581 −0.647905 0.761721i \(-0.724355\pi\)
−0.647905 + 0.761721i \(0.724355\pi\)
\(828\) 9.75958 0.339169
\(829\) 52.3014 1.81650 0.908251 0.418425i \(-0.137418\pi\)
0.908251 + 0.418425i \(0.137418\pi\)
\(830\) 11.4833 0.398591
\(831\) −38.7831 −1.34537
\(832\) −30.0819 −1.04290
\(833\) 3.75309 0.130037
\(834\) −55.8748 −1.93478
\(835\) −9.43908 −0.326653
\(836\) −40.7092 −1.40796
\(837\) 22.7061 0.784838
\(838\) 22.1747 0.766011
\(839\) 18.0759 0.624050 0.312025 0.950074i \(-0.398993\pi\)
0.312025 + 0.950074i \(0.398993\pi\)
\(840\) 11.9856 0.413543
\(841\) −26.7678 −0.923028
\(842\) −17.0464 −0.587459
\(843\) −61.6836 −2.12450
\(844\) −16.3389 −0.562410
\(845\) 1.95849 0.0673741
\(846\) −29.4618 −1.01292
\(847\) −37.6546 −1.29383
\(848\) 1.26546 0.0434560
\(849\) 62.4529 2.14338
\(850\) −6.13503 −0.210430
\(851\) 15.5747 0.533892
\(852\) −31.8414 −1.09087
\(853\) 2.26471 0.0775422 0.0387711 0.999248i \(-0.487656\pi\)
0.0387711 + 0.999248i \(0.487656\pi\)
\(854\) −16.8411 −0.576291
\(855\) 25.8284 0.883314
\(856\) 19.2907 0.659343
\(857\) −34.0377 −1.16271 −0.581353 0.813652i \(-0.697476\pi\)
−0.581353 + 0.813652i \(0.697476\pi\)
\(858\) −61.2162 −2.08989
\(859\) 41.4218 1.41329 0.706646 0.707567i \(-0.250207\pi\)
0.706646 + 0.707567i \(0.250207\pi\)
\(860\) 3.74209 0.127604
\(861\) −15.8637 −0.540634
\(862\) −17.9884 −0.612686
\(863\) 27.5129 0.936551 0.468275 0.883583i \(-0.344876\pi\)
0.468275 + 0.883583i \(0.344876\pi\)
\(864\) 17.5661 0.597612
\(865\) −13.0445 −0.443525
\(866\) −0.162669 −0.00552773
\(867\) 41.5551 1.41128
\(868\) −11.1229 −0.377537
\(869\) −15.2220 −0.516371
\(870\) 3.02921 0.102700
\(871\) −41.7775 −1.41558
\(872\) 28.3694 0.960709
\(873\) 1.21229 0.0410298
\(874\) 21.0866 0.713266
\(875\) 13.7728 0.465605
\(876\) 24.2481 0.819267
\(877\) 24.8045 0.837589 0.418794 0.908081i \(-0.362453\pi\)
0.418794 + 0.908081i \(0.362453\pi\)
\(878\) −24.1632 −0.815469
\(879\) 68.2675 2.30261
\(880\) −5.26552 −0.177501
\(881\) 25.2629 0.851128 0.425564 0.904928i \(-0.360076\pi\)
0.425564 + 0.904928i \(0.360076\pi\)
\(882\) 13.1510 0.442816
\(883\) −19.5417 −0.657631 −0.328815 0.944394i \(-0.606649\pi\)
−0.328815 + 0.944394i \(0.606649\pi\)
\(884\) −4.71549 −0.158599
\(885\) −3.05425 −0.102667
\(886\) −13.0864 −0.439646
\(887\) −34.5222 −1.15914 −0.579571 0.814922i \(-0.696780\pi\)
−0.579571 + 0.814922i \(0.696780\pi\)
\(888\) 52.5380 1.76306
\(889\) −20.7476 −0.695852
\(890\) 1.41596 0.0474630
\(891\) −15.8394 −0.530641
\(892\) 17.4639 0.584733
\(893\) 53.0009 1.77361
\(894\) −31.4829 −1.05295
\(895\) 15.4487 0.516394
\(896\) −2.84284 −0.0949727
\(897\) −26.4015 −0.881520
\(898\) 39.7483 1.32642
\(899\) −8.99867 −0.300122
\(900\) 17.8991 0.596637
\(901\) 1.21970 0.0406341
\(902\) 16.3027 0.542821
\(903\) −31.8564 −1.06011
\(904\) −5.75416 −0.191381
\(905\) 0.546677 0.0181721
\(906\) 19.9725 0.663541
\(907\) 48.4647 1.60924 0.804622 0.593787i \(-0.202368\pi\)
0.804622 + 0.593787i \(0.202368\pi\)
\(908\) −21.1476 −0.701809
\(909\) −74.7777 −2.48022
\(910\) −6.01547 −0.199411
\(911\) 14.4271 0.477990 0.238995 0.971021i \(-0.423182\pi\)
0.238995 + 0.971021i \(0.423182\pi\)
\(912\) 30.4080 1.00691
\(913\) −83.6471 −2.76832
\(914\) −26.6345 −0.880990
\(915\) 15.3949 0.508939
\(916\) −0.952900 −0.0314847
\(917\) 28.3698 0.936852
\(918\) −5.15100 −0.170008
\(919\) 14.9071 0.491741 0.245871 0.969303i \(-0.420926\pi\)
0.245871 + 0.969303i \(0.420926\pi\)
\(920\) −5.31222 −0.175139
\(921\) 39.6663 1.30705
\(922\) −23.5958 −0.777088
\(923\) 51.1553 1.68380
\(924\) −27.2744 −0.897261
\(925\) 28.5640 0.939179
\(926\) −44.1010 −1.44925
\(927\) 28.0954 0.922772
\(928\) −6.96164 −0.228527
\(929\) −19.0507 −0.625033 −0.312517 0.949912i \(-0.601172\pi\)
−0.312517 + 0.949912i \(0.601172\pi\)
\(930\) −12.2118 −0.400439
\(931\) −23.6581 −0.775364
\(932\) −9.45172 −0.309601
\(933\) −24.4380 −0.800066
\(934\) −4.48169 −0.146645
\(935\) −5.07512 −0.165974
\(936\) −52.8914 −1.72881
\(937\) 1.62214 0.0529931 0.0264965 0.999649i \(-0.491565\pi\)
0.0264965 + 0.999649i \(0.491565\pi\)
\(938\) 22.3555 0.729933
\(939\) 12.4035 0.404773
\(940\) −4.17121 −0.136050
\(941\) −1.42371 −0.0464115 −0.0232058 0.999731i \(-0.507387\pi\)
−0.0232058 + 0.999731i \(0.507387\pi\)
\(942\) −68.7544 −2.24014
\(943\) 7.03107 0.228963
\(944\) −2.13548 −0.0695041
\(945\) 5.47118 0.177977
\(946\) 32.7379 1.06440
\(947\) −42.2934 −1.37435 −0.687175 0.726492i \(-0.741150\pi\)
−0.687175 + 0.726492i \(0.741150\pi\)
\(948\) −6.91828 −0.224695
\(949\) −38.9562 −1.26457
\(950\) 38.6730 1.25472
\(951\) 20.1039 0.651914
\(952\) 8.07714 0.261782
\(953\) −30.0275 −0.972686 −0.486343 0.873768i \(-0.661670\pi\)
−0.486343 + 0.873768i \(0.661670\pi\)
\(954\) 4.27387 0.138372
\(955\) −2.51726 −0.0814565
\(956\) −4.73490 −0.153137
\(957\) −22.0655 −0.713276
\(958\) 8.09412 0.261509
\(959\) 4.45717 0.143930
\(960\) −14.7147 −0.474916
\(961\) 5.27662 0.170213
\(962\) −26.3683 −0.850149
\(963\) 27.8515 0.897502
\(964\) 21.0147 0.676839
\(965\) −6.38862 −0.205657
\(966\) 14.1276 0.454549
\(967\) −26.6788 −0.857933 −0.428966 0.903320i \(-0.641122\pi\)
−0.428966 + 0.903320i \(0.641122\pi\)
\(968\) 56.2975 1.80947
\(969\) 29.3085 0.941526
\(970\) −0.206140 −0.00661876
\(971\) 11.1042 0.356351 0.178175 0.983999i \(-0.442981\pi\)
0.178175 + 0.983999i \(0.442981\pi\)
\(972\) −17.4756 −0.560531
\(973\) 39.9947 1.28217
\(974\) −3.26654 −0.104667
\(975\) −48.4205 −1.55070
\(976\) 10.7639 0.344543
\(977\) 16.4245 0.525467 0.262733 0.964868i \(-0.415376\pi\)
0.262733 + 0.964868i \(0.415376\pi\)
\(978\) 1.90860 0.0610303
\(979\) −10.3142 −0.329643
\(980\) 1.86192 0.0594767
\(981\) 40.9591 1.30772
\(982\) 18.6523 0.595217
\(983\) 8.61161 0.274668 0.137334 0.990525i \(-0.456147\pi\)
0.137334 + 0.990525i \(0.456147\pi\)
\(984\) 23.7179 0.756099
\(985\) −14.4009 −0.458852
\(986\) 2.04139 0.0650112
\(987\) 35.5096 1.13028
\(988\) 29.7248 0.945671
\(989\) 14.1193 0.448967
\(990\) −17.7834 −0.565194
\(991\) 30.1651 0.958227 0.479114 0.877753i \(-0.340958\pi\)
0.479114 + 0.877753i \(0.340958\pi\)
\(992\) 28.0647 0.891055
\(993\) 64.1012 2.03419
\(994\) −27.3736 −0.868238
\(995\) 2.66142 0.0843728
\(996\) −38.0170 −1.20461
\(997\) 6.84091 0.216654 0.108327 0.994115i \(-0.465451\pi\)
0.108327 + 0.994115i \(0.465451\pi\)
\(998\) −2.13116 −0.0674608
\(999\) 23.9825 0.758772
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.1.7 18
3.2 odd 2 4923.2.a.l.1.12 18
4.3 odd 2 8752.2.a.s.1.15 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.7 18 1.1 even 1 trivial
4923.2.a.l.1.12 18 3.2 odd 2
8752.2.a.s.1.15 18 4.3 odd 2