Properties

Label 547.2.a.b.1.9
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 1.9
Root \(0.763493\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.763493 q^{2} +1.83524 q^{3} -1.41708 q^{4} -1.51218 q^{5} -1.40119 q^{6} +1.20167 q^{7} +2.60892 q^{8} +0.368111 q^{9} +O(q^{10})\) \(q-0.763493 q^{2} +1.83524 q^{3} -1.41708 q^{4} -1.51218 q^{5} -1.40119 q^{6} +1.20167 q^{7} +2.60892 q^{8} +0.368111 q^{9} +1.15454 q^{10} -5.83960 q^{11} -2.60068 q^{12} -5.40780 q^{13} -0.917468 q^{14} -2.77522 q^{15} +0.842269 q^{16} +1.92787 q^{17} -0.281050 q^{18} +0.965947 q^{19} +2.14288 q^{20} +2.20536 q^{21} +4.45849 q^{22} -0.470051 q^{23} +4.78799 q^{24} -2.71331 q^{25} +4.12882 q^{26} -4.83015 q^{27} -1.70286 q^{28} -3.67331 q^{29} +2.11886 q^{30} +0.675583 q^{31} -5.86090 q^{32} -10.7171 q^{33} -1.47191 q^{34} -1.81715 q^{35} -0.521642 q^{36} +5.66915 q^{37} -0.737494 q^{38} -9.92463 q^{39} -3.94515 q^{40} -5.00667 q^{41} -1.68378 q^{42} +3.18500 q^{43} +8.27517 q^{44} -0.556651 q^{45} +0.358881 q^{46} -10.1509 q^{47} +1.54577 q^{48} -5.55598 q^{49} +2.07159 q^{50} +3.53810 q^{51} +7.66328 q^{52} +8.25219 q^{53} +3.68779 q^{54} +8.83053 q^{55} +3.13506 q^{56} +1.77275 q^{57} +2.80455 q^{58} -5.23947 q^{59} +3.93270 q^{60} -3.53188 q^{61} -0.515803 q^{62} +0.442349 q^{63} +2.79021 q^{64} +8.17758 q^{65} +8.18241 q^{66} +8.70291 q^{67} -2.73194 q^{68} -0.862657 q^{69} +1.38738 q^{70} -9.50861 q^{71} +0.960371 q^{72} +9.68659 q^{73} -4.32836 q^{74} -4.97957 q^{75} -1.36882 q^{76} -7.01728 q^{77} +7.57738 q^{78} -5.23633 q^{79} -1.27366 q^{80} -9.96883 q^{81} +3.82256 q^{82} +14.9853 q^{83} -3.12517 q^{84} -2.91529 q^{85} -2.43172 q^{86} -6.74142 q^{87} -15.2350 q^{88} +10.2143 q^{89} +0.424999 q^{90} -6.49841 q^{91} +0.666099 q^{92} +1.23986 q^{93} +7.75014 q^{94} -1.46069 q^{95} -10.7562 q^{96} -6.07529 q^{97} +4.24195 q^{98} -2.14962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.763493 −0.539871 −0.269935 0.962878i \(-0.587002\pi\)
−0.269935 + 0.962878i \(0.587002\pi\)
\(3\) 1.83524 1.05958 0.529789 0.848130i \(-0.322271\pi\)
0.529789 + 0.848130i \(0.322271\pi\)
\(4\) −1.41708 −0.708539
\(5\) −1.51218 −0.676268 −0.338134 0.941098i \(-0.609796\pi\)
−0.338134 + 0.941098i \(0.609796\pi\)
\(6\) −1.40119 −0.572035
\(7\) 1.20167 0.454189 0.227095 0.973873i \(-0.427077\pi\)
0.227095 + 0.973873i \(0.427077\pi\)
\(8\) 2.60892 0.922391
\(9\) 0.368111 0.122704
\(10\) 1.15454 0.365098
\(11\) −5.83960 −1.76070 −0.880352 0.474321i \(-0.842694\pi\)
−0.880352 + 0.474321i \(0.842694\pi\)
\(12\) −2.60068 −0.750752
\(13\) −5.40780 −1.49986 −0.749928 0.661520i \(-0.769912\pi\)
−0.749928 + 0.661520i \(0.769912\pi\)
\(14\) −0.917468 −0.245204
\(15\) −2.77522 −0.716558
\(16\) 0.842269 0.210567
\(17\) 1.92787 0.467577 0.233788 0.972288i \(-0.424888\pi\)
0.233788 + 0.972288i \(0.424888\pi\)
\(18\) −0.281050 −0.0662442
\(19\) 0.965947 0.221603 0.110802 0.993843i \(-0.464658\pi\)
0.110802 + 0.993843i \(0.464658\pi\)
\(20\) 2.14288 0.479163
\(21\) 2.20536 0.481249
\(22\) 4.45849 0.950553
\(23\) −0.470051 −0.0980124 −0.0490062 0.998798i \(-0.515605\pi\)
−0.0490062 + 0.998798i \(0.515605\pi\)
\(24\) 4.78799 0.977344
\(25\) −2.71331 −0.542661
\(26\) 4.12882 0.809728
\(27\) −4.83015 −0.929563
\(28\) −1.70286 −0.321811
\(29\) −3.67331 −0.682117 −0.341059 0.940042i \(-0.610785\pi\)
−0.341059 + 0.940042i \(0.610785\pi\)
\(30\) 2.11886 0.386849
\(31\) 0.675583 0.121338 0.0606691 0.998158i \(-0.480677\pi\)
0.0606691 + 0.998158i \(0.480677\pi\)
\(32\) −5.86090 −1.03607
\(33\) −10.7171 −1.86560
\(34\) −1.47191 −0.252431
\(35\) −1.81715 −0.307154
\(36\) −0.521642 −0.0869404
\(37\) 5.66915 0.932003 0.466002 0.884784i \(-0.345694\pi\)
0.466002 + 0.884784i \(0.345694\pi\)
\(38\) −0.737494 −0.119637
\(39\) −9.92463 −1.58921
\(40\) −3.94515 −0.623784
\(41\) −5.00667 −0.781911 −0.390955 0.920410i \(-0.627855\pi\)
−0.390955 + 0.920410i \(0.627855\pi\)
\(42\) −1.68378 −0.259812
\(43\) 3.18500 0.485707 0.242854 0.970063i \(-0.421917\pi\)
0.242854 + 0.970063i \(0.421917\pi\)
\(44\) 8.27517 1.24753
\(45\) −0.556651 −0.0829806
\(46\) 0.358881 0.0529141
\(47\) −10.1509 −1.48066 −0.740331 0.672243i \(-0.765331\pi\)
−0.740331 + 0.672243i \(0.765331\pi\)
\(48\) 1.54577 0.223112
\(49\) −5.55598 −0.793712
\(50\) 2.07159 0.292967
\(51\) 3.53810 0.495433
\(52\) 7.66328 1.06271
\(53\) 8.25219 1.13353 0.566763 0.823881i \(-0.308196\pi\)
0.566763 + 0.823881i \(0.308196\pi\)
\(54\) 3.68779 0.501844
\(55\) 8.83053 1.19071
\(56\) 3.13506 0.418940
\(57\) 1.77275 0.234806
\(58\) 2.80455 0.368255
\(59\) −5.23947 −0.682121 −0.341060 0.940041i \(-0.610786\pi\)
−0.341060 + 0.940041i \(0.610786\pi\)
\(60\) 3.93270 0.507710
\(61\) −3.53188 −0.452211 −0.226106 0.974103i \(-0.572599\pi\)
−0.226106 + 0.974103i \(0.572599\pi\)
\(62\) −0.515803 −0.0655070
\(63\) 0.442349 0.0557307
\(64\) 2.79021 0.348777
\(65\) 8.17758 1.01430
\(66\) 8.18241 1.00718
\(67\) 8.70291 1.06323 0.531615 0.846986i \(-0.321585\pi\)
0.531615 + 0.846986i \(0.321585\pi\)
\(68\) −2.73194 −0.331296
\(69\) −0.862657 −0.103852
\(70\) 1.38738 0.165823
\(71\) −9.50861 −1.12846 −0.564232 0.825616i \(-0.690828\pi\)
−0.564232 + 0.825616i \(0.690828\pi\)
\(72\) 0.960371 0.113181
\(73\) 9.68659 1.13373 0.566865 0.823811i \(-0.308156\pi\)
0.566865 + 0.823811i \(0.308156\pi\)
\(74\) −4.32836 −0.503161
\(75\) −4.97957 −0.574991
\(76\) −1.36882 −0.157015
\(77\) −7.01728 −0.799693
\(78\) 7.57738 0.857970
\(79\) −5.23633 −0.589133 −0.294567 0.955631i \(-0.595175\pi\)
−0.294567 + 0.955631i \(0.595175\pi\)
\(80\) −1.27366 −0.142400
\(81\) −9.96883 −1.10765
\(82\) 3.82256 0.422131
\(83\) 14.9853 1.64485 0.822427 0.568871i \(-0.192620\pi\)
0.822427 + 0.568871i \(0.192620\pi\)
\(84\) −3.12517 −0.340984
\(85\) −2.91529 −0.316207
\(86\) −2.43172 −0.262219
\(87\) −6.74142 −0.722756
\(88\) −15.2350 −1.62406
\(89\) 10.2143 1.08271 0.541357 0.840793i \(-0.317911\pi\)
0.541357 + 0.840793i \(0.317911\pi\)
\(90\) 0.424999 0.0447988
\(91\) −6.49841 −0.681218
\(92\) 0.666099 0.0694457
\(93\) 1.23986 0.128567
\(94\) 7.75014 0.799366
\(95\) −1.46069 −0.149863
\(96\) −10.7562 −1.09780
\(97\) −6.07529 −0.616853 −0.308426 0.951248i \(-0.599802\pi\)
−0.308426 + 0.951248i \(0.599802\pi\)
\(98\) 4.24195 0.428502
\(99\) −2.14962 −0.216045
\(100\) 3.84497 0.384497
\(101\) −2.36723 −0.235549 −0.117774 0.993040i \(-0.537576\pi\)
−0.117774 + 0.993040i \(0.537576\pi\)
\(102\) −2.70132 −0.267470
\(103\) 12.4607 1.22779 0.613895 0.789388i \(-0.289602\pi\)
0.613895 + 0.789388i \(0.289602\pi\)
\(104\) −14.1085 −1.38345
\(105\) −3.33490 −0.325453
\(106\) −6.30049 −0.611958
\(107\) −17.8981 −1.73027 −0.865137 0.501536i \(-0.832768\pi\)
−0.865137 + 0.501536i \(0.832768\pi\)
\(108\) 6.84470 0.658632
\(109\) −2.89181 −0.276985 −0.138492 0.990363i \(-0.544226\pi\)
−0.138492 + 0.990363i \(0.544226\pi\)
\(110\) −6.74205 −0.642829
\(111\) 10.4043 0.987529
\(112\) 1.01213 0.0956374
\(113\) 5.34990 0.503276 0.251638 0.967821i \(-0.419031\pi\)
0.251638 + 0.967821i \(0.419031\pi\)
\(114\) −1.35348 −0.126765
\(115\) 0.710803 0.0662827
\(116\) 5.20537 0.483307
\(117\) −1.99067 −0.184038
\(118\) 4.00030 0.368257
\(119\) 2.31666 0.212368
\(120\) −7.24031 −0.660947
\(121\) 23.1009 2.10008
\(122\) 2.69657 0.244136
\(123\) −9.18845 −0.828495
\(124\) −0.957354 −0.0859730
\(125\) 11.6639 1.04325
\(126\) −0.337730 −0.0300874
\(127\) 14.6006 1.29559 0.647795 0.761815i \(-0.275691\pi\)
0.647795 + 0.761815i \(0.275691\pi\)
\(128\) 9.59148 0.847775
\(129\) 5.84524 0.514644
\(130\) −6.24353 −0.547594
\(131\) −19.8795 −1.73688 −0.868441 0.495792i \(-0.834878\pi\)
−0.868441 + 0.495792i \(0.834878\pi\)
\(132\) 15.1869 1.32185
\(133\) 1.16075 0.100650
\(134\) −6.64461 −0.574007
\(135\) 7.30407 0.628634
\(136\) 5.02964 0.431288
\(137\) 19.2841 1.64755 0.823777 0.566914i \(-0.191863\pi\)
0.823777 + 0.566914i \(0.191863\pi\)
\(138\) 0.658633 0.0560665
\(139\) 11.2982 0.958297 0.479149 0.877734i \(-0.340946\pi\)
0.479149 + 0.877734i \(0.340946\pi\)
\(140\) 2.57504 0.217631
\(141\) −18.6294 −1.56887
\(142\) 7.25976 0.609225
\(143\) 31.5794 2.64080
\(144\) 0.310049 0.0258374
\(145\) 5.55472 0.461294
\(146\) −7.39564 −0.612067
\(147\) −10.1966 −0.840999
\(148\) −8.03364 −0.660361
\(149\) 0.559854 0.0458650 0.0229325 0.999737i \(-0.492700\pi\)
0.0229325 + 0.999737i \(0.492700\pi\)
\(150\) 3.80187 0.310421
\(151\) −7.77394 −0.632634 −0.316317 0.948653i \(-0.602446\pi\)
−0.316317 + 0.948653i \(0.602446\pi\)
\(152\) 2.52007 0.204405
\(153\) 0.709670 0.0573734
\(154\) 5.35764 0.431731
\(155\) −1.02160 −0.0820572
\(156\) 14.0640 1.12602
\(157\) −0.453089 −0.0361604 −0.0180802 0.999837i \(-0.505755\pi\)
−0.0180802 + 0.999837i \(0.505755\pi\)
\(158\) 3.99790 0.318056
\(159\) 15.1448 1.20106
\(160\) 8.86274 0.700661
\(161\) −0.564847 −0.0445162
\(162\) 7.61113 0.597987
\(163\) −7.12748 −0.558267 −0.279134 0.960252i \(-0.590047\pi\)
−0.279134 + 0.960252i \(0.590047\pi\)
\(164\) 7.09485 0.554014
\(165\) 16.2062 1.26165
\(166\) −11.4412 −0.888009
\(167\) −10.5555 −0.816808 −0.408404 0.912801i \(-0.633914\pi\)
−0.408404 + 0.912801i \(0.633914\pi\)
\(168\) 5.75359 0.443899
\(169\) 16.2443 1.24957
\(170\) 2.22580 0.170711
\(171\) 0.355576 0.0271916
\(172\) −4.51339 −0.344143
\(173\) −16.7583 −1.27411 −0.637055 0.770818i \(-0.719848\pi\)
−0.637055 + 0.770818i \(0.719848\pi\)
\(174\) 5.14702 0.390195
\(175\) −3.26050 −0.246471
\(176\) −4.91851 −0.370747
\(177\) −9.61569 −0.722760
\(178\) −7.79855 −0.584526
\(179\) 4.75191 0.355174 0.177587 0.984105i \(-0.443171\pi\)
0.177587 + 0.984105i \(0.443171\pi\)
\(180\) 0.788818 0.0587950
\(181\) −9.02008 −0.670457 −0.335229 0.942137i \(-0.608814\pi\)
−0.335229 + 0.942137i \(0.608814\pi\)
\(182\) 4.96149 0.367770
\(183\) −6.48186 −0.479153
\(184\) −1.22632 −0.0904058
\(185\) −8.57279 −0.630284
\(186\) −0.946623 −0.0694097
\(187\) −11.2580 −0.823264
\(188\) 14.3846 1.04911
\(189\) −5.80426 −0.422198
\(190\) 1.11522 0.0809069
\(191\) −24.6001 −1.78000 −0.890000 0.455960i \(-0.849296\pi\)
−0.890000 + 0.455960i \(0.849296\pi\)
\(192\) 5.12072 0.369556
\(193\) 15.5317 1.11800 0.558998 0.829169i \(-0.311186\pi\)
0.558998 + 0.829169i \(0.311186\pi\)
\(194\) 4.63844 0.333021
\(195\) 15.0078 1.07473
\(196\) 7.87327 0.562376
\(197\) −15.1172 −1.07705 −0.538527 0.842608i \(-0.681019\pi\)
−0.538527 + 0.842608i \(0.681019\pi\)
\(198\) 1.64122 0.116636
\(199\) 13.6303 0.966227 0.483114 0.875558i \(-0.339506\pi\)
0.483114 + 0.875558i \(0.339506\pi\)
\(200\) −7.07879 −0.500546
\(201\) 15.9719 1.12657
\(202\) 1.80737 0.127166
\(203\) −4.41412 −0.309810
\(204\) −5.01377 −0.351034
\(205\) 7.57100 0.528781
\(206\) −9.51366 −0.662848
\(207\) −0.173031 −0.0120265
\(208\) −4.55483 −0.315820
\(209\) −5.64074 −0.390178
\(210\) 2.54617 0.175703
\(211\) 11.3226 0.779478 0.389739 0.920925i \(-0.372565\pi\)
0.389739 + 0.920925i \(0.372565\pi\)
\(212\) −11.6940 −0.803147
\(213\) −17.4506 −1.19570
\(214\) 13.6651 0.934124
\(215\) −4.81629 −0.328468
\(216\) −12.6015 −0.857420
\(217\) 0.811829 0.0551106
\(218\) 2.20787 0.149536
\(219\) 17.7772 1.20127
\(220\) −12.5136 −0.843664
\(221\) −10.4255 −0.701297
\(222\) −7.94358 −0.533138
\(223\) 5.49894 0.368236 0.184118 0.982904i \(-0.441057\pi\)
0.184118 + 0.982904i \(0.441057\pi\)
\(224\) −7.04288 −0.470572
\(225\) −0.998798 −0.0665866
\(226\) −4.08461 −0.271704
\(227\) −2.09244 −0.138880 −0.0694402 0.997586i \(-0.522121\pi\)
−0.0694402 + 0.997586i \(0.522121\pi\)
\(228\) −2.51212 −0.166369
\(229\) −16.4218 −1.08518 −0.542592 0.839996i \(-0.682557\pi\)
−0.542592 + 0.839996i \(0.682557\pi\)
\(230\) −0.542693 −0.0357841
\(231\) −12.8784 −0.847337
\(232\) −9.58336 −0.629179
\(233\) −10.0033 −0.655338 −0.327669 0.944793i \(-0.606263\pi\)
−0.327669 + 0.944793i \(0.606263\pi\)
\(234\) 1.51986 0.0993567
\(235\) 15.3500 1.00132
\(236\) 7.42474 0.483309
\(237\) −9.60993 −0.624232
\(238\) −1.76876 −0.114651
\(239\) 12.8814 0.833227 0.416614 0.909084i \(-0.363217\pi\)
0.416614 + 0.909084i \(0.363217\pi\)
\(240\) −2.33748 −0.150884
\(241\) 9.58394 0.617356 0.308678 0.951167i \(-0.400114\pi\)
0.308678 + 0.951167i \(0.400114\pi\)
\(242\) −17.6374 −1.13377
\(243\) −3.80475 −0.244075
\(244\) 5.00496 0.320409
\(245\) 8.40166 0.536762
\(246\) 7.01532 0.447280
\(247\) −5.22365 −0.332373
\(248\) 1.76254 0.111921
\(249\) 27.5017 1.74285
\(250\) −8.90532 −0.563222
\(251\) −7.09126 −0.447596 −0.223798 0.974636i \(-0.571846\pi\)
−0.223798 + 0.974636i \(0.571846\pi\)
\(252\) −0.626843 −0.0394874
\(253\) 2.74491 0.172571
\(254\) −11.1474 −0.699452
\(255\) −5.35025 −0.335046
\(256\) −12.9035 −0.806466
\(257\) −7.64221 −0.476708 −0.238354 0.971178i \(-0.576608\pi\)
−0.238354 + 0.971178i \(0.576608\pi\)
\(258\) −4.46280 −0.277842
\(259\) 6.81246 0.423306
\(260\) −11.5883 −0.718675
\(261\) −1.35219 −0.0836983
\(262\) 15.1779 0.937693
\(263\) −18.7853 −1.15835 −0.579176 0.815202i \(-0.696626\pi\)
−0.579176 + 0.815202i \(0.696626\pi\)
\(264\) −27.9599 −1.72081
\(265\) −12.4788 −0.766567
\(266\) −0.886226 −0.0543380
\(267\) 18.7457 1.14722
\(268\) −12.3327 −0.753340
\(269\) −17.1967 −1.04850 −0.524251 0.851564i \(-0.675654\pi\)
−0.524251 + 0.851564i \(0.675654\pi\)
\(270\) −5.57660 −0.339381
\(271\) −13.7691 −0.836416 −0.418208 0.908351i \(-0.637342\pi\)
−0.418208 + 0.908351i \(0.637342\pi\)
\(272\) 1.62378 0.0984563
\(273\) −11.9261 −0.721803
\(274\) −14.7233 −0.889467
\(275\) 15.8446 0.955466
\(276\) 1.22245 0.0735830
\(277\) −18.7759 −1.12813 −0.564066 0.825730i \(-0.690764\pi\)
−0.564066 + 0.825730i \(0.690764\pi\)
\(278\) −8.62606 −0.517357
\(279\) 0.248690 0.0148887
\(280\) −4.74078 −0.283316
\(281\) 9.00392 0.537129 0.268564 0.963262i \(-0.413451\pi\)
0.268564 + 0.963262i \(0.413451\pi\)
\(282\) 14.2234 0.846990
\(283\) 14.0938 0.837789 0.418894 0.908035i \(-0.362418\pi\)
0.418894 + 0.908035i \(0.362418\pi\)
\(284\) 13.4744 0.799561
\(285\) −2.68071 −0.158792
\(286\) −24.1106 −1.42569
\(287\) −6.01638 −0.355135
\(288\) −2.15746 −0.127130
\(289\) −13.2833 −0.781372
\(290\) −4.24099 −0.249039
\(291\) −11.1496 −0.653603
\(292\) −13.7267 −0.803292
\(293\) −25.2452 −1.47484 −0.737421 0.675434i \(-0.763956\pi\)
−0.737421 + 0.675434i \(0.763956\pi\)
\(294\) 7.78501 0.454031
\(295\) 7.92303 0.461297
\(296\) 14.7903 0.859671
\(297\) 28.2061 1.63669
\(298\) −0.427444 −0.0247612
\(299\) 2.54194 0.147004
\(300\) 7.05645 0.407404
\(301\) 3.82732 0.220603
\(302\) 5.93535 0.341541
\(303\) −4.34445 −0.249582
\(304\) 0.813587 0.0466624
\(305\) 5.34085 0.305816
\(306\) −0.541828 −0.0309742
\(307\) 16.6513 0.950342 0.475171 0.879893i \(-0.342386\pi\)
0.475171 + 0.879893i \(0.342386\pi\)
\(308\) 9.94404 0.566614
\(309\) 22.8684 1.30094
\(310\) 0.779988 0.0443003
\(311\) −3.26046 −0.184884 −0.0924418 0.995718i \(-0.529467\pi\)
−0.0924418 + 0.995718i \(0.529467\pi\)
\(312\) −25.8925 −1.46587
\(313\) −5.48703 −0.310145 −0.155073 0.987903i \(-0.549561\pi\)
−0.155073 + 0.987903i \(0.549561\pi\)
\(314\) 0.345930 0.0195220
\(315\) −0.668912 −0.0376889
\(316\) 7.42029 0.417424
\(317\) −22.3722 −1.25655 −0.628275 0.777992i \(-0.716239\pi\)
−0.628275 + 0.777992i \(0.716239\pi\)
\(318\) −11.5629 −0.648416
\(319\) 21.4507 1.20101
\(320\) −4.21931 −0.235867
\(321\) −32.8473 −1.83336
\(322\) 0.431257 0.0240330
\(323\) 1.86222 0.103617
\(324\) 14.1266 0.784812
\(325\) 14.6730 0.813913
\(326\) 5.44178 0.301392
\(327\) −5.30716 −0.293487
\(328\) −13.0620 −0.721227
\(329\) −12.1981 −0.672501
\(330\) −12.3733 −0.681127
\(331\) −33.2750 −1.82896 −0.914479 0.404634i \(-0.867399\pi\)
−0.914479 + 0.404634i \(0.867399\pi\)
\(332\) −21.2354 −1.16544
\(333\) 2.08688 0.114360
\(334\) 8.05903 0.440971
\(335\) −13.1604 −0.719028
\(336\) 1.85751 0.101335
\(337\) 4.21753 0.229743 0.114872 0.993380i \(-0.463354\pi\)
0.114872 + 0.993380i \(0.463354\pi\)
\(338\) −12.4024 −0.674604
\(339\) 9.81835 0.533260
\(340\) 4.13119 0.224045
\(341\) −3.94513 −0.213641
\(342\) −0.271480 −0.0146799
\(343\) −15.0882 −0.814685
\(344\) 8.30939 0.448012
\(345\) 1.30449 0.0702316
\(346\) 12.7948 0.687855
\(347\) −17.8458 −0.958013 −0.479007 0.877811i \(-0.659003\pi\)
−0.479007 + 0.877811i \(0.659003\pi\)
\(348\) 9.55312 0.512101
\(349\) −4.70118 −0.251648 −0.125824 0.992053i \(-0.540157\pi\)
−0.125824 + 0.992053i \(0.540157\pi\)
\(350\) 2.48937 0.133063
\(351\) 26.1205 1.39421
\(352\) 34.2253 1.82421
\(353\) −25.5243 −1.35852 −0.679261 0.733896i \(-0.737700\pi\)
−0.679261 + 0.733896i \(0.737700\pi\)
\(354\) 7.34151 0.390197
\(355\) 14.3787 0.763145
\(356\) −14.4745 −0.767145
\(357\) 4.25164 0.225021
\(358\) −3.62805 −0.191748
\(359\) 32.0017 1.68898 0.844492 0.535568i \(-0.179903\pi\)
0.844492 + 0.535568i \(0.179903\pi\)
\(360\) −1.45226 −0.0765406
\(361\) −18.0669 −0.950892
\(362\) 6.88676 0.361960
\(363\) 42.3957 2.22520
\(364\) 9.20875 0.482670
\(365\) −14.6479 −0.766705
\(366\) 4.94885 0.258681
\(367\) 14.7807 0.771546 0.385773 0.922594i \(-0.373935\pi\)
0.385773 + 0.922594i \(0.373935\pi\)
\(368\) −0.395909 −0.0206382
\(369\) −1.84301 −0.0959433
\(370\) 6.54526 0.340272
\(371\) 9.91643 0.514835
\(372\) −1.75698 −0.0910950
\(373\) −30.5239 −1.58047 −0.790235 0.612804i \(-0.790042\pi\)
−0.790235 + 0.612804i \(0.790042\pi\)
\(374\) 8.59538 0.444456
\(375\) 21.4061 1.10541
\(376\) −26.4828 −1.36575
\(377\) 19.8646 1.02308
\(378\) 4.43151 0.227932
\(379\) −13.3030 −0.683329 −0.341665 0.939822i \(-0.610991\pi\)
−0.341665 + 0.939822i \(0.610991\pi\)
\(380\) 2.06991 0.106184
\(381\) 26.7956 1.37278
\(382\) 18.7820 0.960971
\(383\) 20.6416 1.05474 0.527369 0.849636i \(-0.323179\pi\)
0.527369 + 0.849636i \(0.323179\pi\)
\(384\) 17.6027 0.898284
\(385\) 10.6114 0.540807
\(386\) −11.8583 −0.603573
\(387\) 1.17243 0.0595981
\(388\) 8.60917 0.437064
\(389\) 18.0385 0.914590 0.457295 0.889315i \(-0.348818\pi\)
0.457295 + 0.889315i \(0.348818\pi\)
\(390\) −11.4584 −0.580218
\(391\) −0.906196 −0.0458283
\(392\) −14.4951 −0.732113
\(393\) −36.4837 −1.84036
\(394\) 11.5419 0.581470
\(395\) 7.91828 0.398412
\(396\) 3.04618 0.153076
\(397\) −24.2893 −1.21904 −0.609521 0.792770i \(-0.708638\pi\)
−0.609521 + 0.792770i \(0.708638\pi\)
\(398\) −10.4066 −0.521638
\(399\) 2.13026 0.106646
\(400\) −2.28533 −0.114267
\(401\) 11.4436 0.571464 0.285732 0.958310i \(-0.407763\pi\)
0.285732 + 0.958310i \(0.407763\pi\)
\(402\) −12.1945 −0.608204
\(403\) −3.65342 −0.181990
\(404\) 3.35456 0.166895
\(405\) 15.0747 0.749067
\(406\) 3.37015 0.167258
\(407\) −33.1056 −1.64098
\(408\) 9.23061 0.456983
\(409\) 4.34200 0.214698 0.107349 0.994221i \(-0.465764\pi\)
0.107349 + 0.994221i \(0.465764\pi\)
\(410\) −5.78040 −0.285474
\(411\) 35.3910 1.74571
\(412\) −17.6578 −0.869938
\(413\) −6.29613 −0.309812
\(414\) 0.132108 0.00649275
\(415\) −22.6605 −1.11236
\(416\) 31.6946 1.55395
\(417\) 20.7348 1.01539
\(418\) 4.30667 0.210646
\(419\) −35.0918 −1.71434 −0.857172 0.515030i \(-0.827781\pi\)
−0.857172 + 0.515030i \(0.827781\pi\)
\(420\) 4.72582 0.230596
\(421\) −36.2929 −1.76881 −0.884405 0.466720i \(-0.845436\pi\)
−0.884405 + 0.466720i \(0.845436\pi\)
\(422\) −8.64471 −0.420818
\(423\) −3.73666 −0.181683
\(424\) 21.5293 1.04555
\(425\) −5.23089 −0.253736
\(426\) 13.3234 0.645521
\(427\) −4.24417 −0.205390
\(428\) 25.3630 1.22597
\(429\) 57.9558 2.79813
\(430\) 3.67721 0.177331
\(431\) 8.92616 0.429958 0.214979 0.976619i \(-0.431032\pi\)
0.214979 + 0.976619i \(0.431032\pi\)
\(432\) −4.06829 −0.195736
\(433\) 40.1476 1.92937 0.964684 0.263408i \(-0.0848466\pi\)
0.964684 + 0.263408i \(0.0848466\pi\)
\(434\) −0.619826 −0.0297526
\(435\) 10.1942 0.488777
\(436\) 4.09792 0.196255
\(437\) −0.454045 −0.0217199
\(438\) −13.5728 −0.648533
\(439\) −16.4516 −0.785194 −0.392597 0.919711i \(-0.628423\pi\)
−0.392597 + 0.919711i \(0.628423\pi\)
\(440\) 23.0381 1.09830
\(441\) −2.04522 −0.0973914
\(442\) 7.95982 0.378610
\(443\) 14.1442 0.672012 0.336006 0.941860i \(-0.390924\pi\)
0.336006 + 0.941860i \(0.390924\pi\)
\(444\) −14.7437 −0.699703
\(445\) −15.4459 −0.732205
\(446\) −4.19840 −0.198800
\(447\) 1.02747 0.0485975
\(448\) 3.35292 0.158411
\(449\) 6.96741 0.328813 0.164406 0.986393i \(-0.447429\pi\)
0.164406 + 0.986393i \(0.447429\pi\)
\(450\) 0.762575 0.0359482
\(451\) 29.2369 1.37671
\(452\) −7.58122 −0.356591
\(453\) −14.2671 −0.670325
\(454\) 1.59757 0.0749775
\(455\) 9.82677 0.460686
\(456\) 4.62494 0.216583
\(457\) −16.6490 −0.778809 −0.389404 0.921067i \(-0.627319\pi\)
−0.389404 + 0.921067i \(0.627319\pi\)
\(458\) 12.5379 0.585859
\(459\) −9.31189 −0.434642
\(460\) −1.00726 −0.0469639
\(461\) 18.1015 0.843072 0.421536 0.906812i \(-0.361491\pi\)
0.421536 + 0.906812i \(0.361491\pi\)
\(462\) 9.83257 0.457452
\(463\) 33.1519 1.54070 0.770350 0.637621i \(-0.220081\pi\)
0.770350 + 0.637621i \(0.220081\pi\)
\(464\) −3.09392 −0.143632
\(465\) −1.87489 −0.0869460
\(466\) 7.63745 0.353798
\(467\) 3.70462 0.171430 0.0857148 0.996320i \(-0.472683\pi\)
0.0857148 + 0.996320i \(0.472683\pi\)
\(468\) 2.82094 0.130398
\(469\) 10.4580 0.482907
\(470\) −11.7196 −0.540586
\(471\) −0.831527 −0.0383148
\(472\) −13.6693 −0.629182
\(473\) −18.5991 −0.855187
\(474\) 7.33711 0.337005
\(475\) −2.62091 −0.120256
\(476\) −3.28290 −0.150471
\(477\) 3.03772 0.139088
\(478\) −9.83484 −0.449835
\(479\) 13.6667 0.624449 0.312225 0.950008i \(-0.398926\pi\)
0.312225 + 0.950008i \(0.398926\pi\)
\(480\) 16.2653 0.742405
\(481\) −30.6577 −1.39787
\(482\) −7.31727 −0.333292
\(483\) −1.03663 −0.0471683
\(484\) −32.7358 −1.48799
\(485\) 9.18695 0.417158
\(486\) 2.90490 0.131769
\(487\) 26.6571 1.20795 0.603974 0.797004i \(-0.293583\pi\)
0.603974 + 0.797004i \(0.293583\pi\)
\(488\) −9.21438 −0.417116
\(489\) −13.0806 −0.591527
\(490\) −6.41461 −0.289782
\(491\) 8.27214 0.373317 0.186658 0.982425i \(-0.440234\pi\)
0.186658 + 0.982425i \(0.440234\pi\)
\(492\) 13.0208 0.587021
\(493\) −7.08166 −0.318942
\(494\) 3.98822 0.179439
\(495\) 3.25062 0.146104
\(496\) 0.569023 0.0255499
\(497\) −11.4262 −0.512537
\(498\) −20.9974 −0.940914
\(499\) −32.1496 −1.43921 −0.719607 0.694382i \(-0.755678\pi\)
−0.719607 + 0.694382i \(0.755678\pi\)
\(500\) −16.5287 −0.739186
\(501\) −19.3719 −0.865471
\(502\) 5.41412 0.241644
\(503\) −4.95942 −0.221130 −0.110565 0.993869i \(-0.535266\pi\)
−0.110565 + 0.993869i \(0.535266\pi\)
\(504\) 1.15405 0.0514055
\(505\) 3.57969 0.159294
\(506\) −2.09572 −0.0931660
\(507\) 29.8123 1.32401
\(508\) −20.6901 −0.917977
\(509\) 13.2112 0.585577 0.292788 0.956177i \(-0.405417\pi\)
0.292788 + 0.956177i \(0.405417\pi\)
\(510\) 4.08488 0.180882
\(511\) 11.6401 0.514928
\(512\) −9.33127 −0.412388
\(513\) −4.66567 −0.205994
\(514\) 5.83477 0.257361
\(515\) −18.8429 −0.830315
\(516\) −8.28316 −0.364646
\(517\) 59.2772 2.60701
\(518\) −5.20127 −0.228531
\(519\) −30.7555 −1.35002
\(520\) 21.3346 0.935585
\(521\) −7.65030 −0.335166 −0.167583 0.985858i \(-0.553596\pi\)
−0.167583 + 0.985858i \(0.553596\pi\)
\(522\) 1.03239 0.0451863
\(523\) 38.3821 1.67833 0.839165 0.543876i \(-0.183044\pi\)
0.839165 + 0.543876i \(0.183044\pi\)
\(524\) 28.1709 1.23065
\(525\) −5.98381 −0.261155
\(526\) 14.3425 0.625361
\(527\) 1.30243 0.0567349
\(528\) −9.02665 −0.392835
\(529\) −22.7791 −0.990394
\(530\) 9.52748 0.413847
\(531\) −1.92871 −0.0836988
\(532\) −1.64488 −0.0713144
\(533\) 27.0751 1.17275
\(534\) −14.3122 −0.619350
\(535\) 27.0652 1.17013
\(536\) 22.7051 0.980713
\(537\) 8.72090 0.376335
\(538\) 13.1296 0.566055
\(539\) 32.4447 1.39749
\(540\) −10.3504 −0.445412
\(541\) −15.6994 −0.674969 −0.337484 0.941331i \(-0.609576\pi\)
−0.337484 + 0.941331i \(0.609576\pi\)
\(542\) 10.5126 0.451557
\(543\) −16.5540 −0.710401
\(544\) −11.2990 −0.484442
\(545\) 4.37294 0.187316
\(546\) 9.10553 0.389681
\(547\) −1.00000 −0.0427569
\(548\) −27.3271 −1.16736
\(549\) −1.30013 −0.0554880
\(550\) −12.0972 −0.515828
\(551\) −3.54823 −0.151160
\(552\) −2.25060 −0.0957919
\(553\) −6.29235 −0.267578
\(554\) 14.3352 0.609046
\(555\) −15.7331 −0.667835
\(556\) −16.0104 −0.678991
\(557\) −25.9854 −1.10104 −0.550518 0.834823i \(-0.685570\pi\)
−0.550518 + 0.834823i \(0.685570\pi\)
\(558\) −0.189873 −0.00803796
\(559\) −17.2238 −0.728491
\(560\) −1.53053 −0.0646765
\(561\) −20.6611 −0.872312
\(562\) −6.87443 −0.289980
\(563\) 11.0040 0.463762 0.231881 0.972744i \(-0.425512\pi\)
0.231881 + 0.972744i \(0.425512\pi\)
\(564\) 26.3993 1.11161
\(565\) −8.09002 −0.340350
\(566\) −10.7605 −0.452298
\(567\) −11.9793 −0.503082
\(568\) −24.8072 −1.04089
\(569\) −2.32372 −0.0974152 −0.0487076 0.998813i \(-0.515510\pi\)
−0.0487076 + 0.998813i \(0.515510\pi\)
\(570\) 2.04671 0.0857271
\(571\) −6.08147 −0.254502 −0.127251 0.991871i \(-0.540615\pi\)
−0.127251 + 0.991871i \(0.540615\pi\)
\(572\) −44.7505 −1.87111
\(573\) −45.1471 −1.88605
\(574\) 4.59346 0.191727
\(575\) 1.27539 0.0531875
\(576\) 1.02711 0.0427962
\(577\) −43.2965 −1.80246 −0.901228 0.433346i \(-0.857333\pi\)
−0.901228 + 0.433346i \(0.857333\pi\)
\(578\) 10.1417 0.421840
\(579\) 28.5044 1.18460
\(580\) −7.87147 −0.326845
\(581\) 18.0075 0.747075
\(582\) 8.51266 0.352861
\(583\) −48.1895 −1.99580
\(584\) 25.2715 1.04574
\(585\) 3.01026 0.124459
\(586\) 19.2745 0.796224
\(587\) 34.3433 1.41750 0.708749 0.705460i \(-0.249260\pi\)
0.708749 + 0.705460i \(0.249260\pi\)
\(588\) 14.4493 0.595881
\(589\) 0.652578 0.0268890
\(590\) −6.04918 −0.249041
\(591\) −27.7437 −1.14122
\(592\) 4.77495 0.196249
\(593\) 12.4576 0.511572 0.255786 0.966733i \(-0.417666\pi\)
0.255786 + 0.966733i \(0.417666\pi\)
\(594\) −21.5352 −0.883599
\(595\) −3.50322 −0.143618
\(596\) −0.793357 −0.0324972
\(597\) 25.0149 1.02379
\(598\) −1.94076 −0.0793634
\(599\) 41.2358 1.68485 0.842424 0.538815i \(-0.181128\pi\)
0.842424 + 0.538815i \(0.181128\pi\)
\(600\) −12.9913 −0.530367
\(601\) 22.9872 0.937668 0.468834 0.883286i \(-0.344674\pi\)
0.468834 + 0.883286i \(0.344674\pi\)
\(602\) −2.92213 −0.119097
\(603\) 3.20364 0.130462
\(604\) 11.0163 0.448246
\(605\) −34.9327 −1.42022
\(606\) 3.31695 0.134742
\(607\) −44.6075 −1.81056 −0.905282 0.424812i \(-0.860340\pi\)
−0.905282 + 0.424812i \(0.860340\pi\)
\(608\) −5.66132 −0.229597
\(609\) −8.10097 −0.328268
\(610\) −4.07770 −0.165101
\(611\) 54.8941 2.22078
\(612\) −1.00566 −0.0406513
\(613\) −21.3775 −0.863428 −0.431714 0.902010i \(-0.642091\pi\)
−0.431714 + 0.902010i \(0.642091\pi\)
\(614\) −12.7132 −0.513062
\(615\) 13.8946 0.560285
\(616\) −18.3075 −0.737630
\(617\) 12.0119 0.483583 0.241791 0.970328i \(-0.422265\pi\)
0.241791 + 0.970328i \(0.422265\pi\)
\(618\) −17.4599 −0.702339
\(619\) 19.5487 0.785727 0.392863 0.919597i \(-0.371484\pi\)
0.392863 + 0.919597i \(0.371484\pi\)
\(620\) 1.44769 0.0581408
\(621\) 2.27042 0.0911087
\(622\) 2.48934 0.0998133
\(623\) 12.2742 0.491757
\(624\) −8.35921 −0.334636
\(625\) −4.07144 −0.162857
\(626\) 4.18931 0.167438
\(627\) −10.3521 −0.413424
\(628\) 0.642063 0.0256211
\(629\) 10.9294 0.435783
\(630\) 0.510710 0.0203472
\(631\) 40.4106 1.60872 0.804360 0.594142i \(-0.202508\pi\)
0.804360 + 0.594142i \(0.202508\pi\)
\(632\) −13.6611 −0.543411
\(633\) 20.7797 0.825917
\(634\) 17.0810 0.678375
\(635\) −22.0787 −0.876166
\(636\) −21.4613 −0.850997
\(637\) 30.0457 1.19045
\(638\) −16.3774 −0.648389
\(639\) −3.50023 −0.138467
\(640\) −14.5041 −0.573324
\(641\) 22.6243 0.893607 0.446803 0.894632i \(-0.352562\pi\)
0.446803 + 0.894632i \(0.352562\pi\)
\(642\) 25.0787 0.989777
\(643\) 40.2337 1.58666 0.793330 0.608792i \(-0.208345\pi\)
0.793330 + 0.608792i \(0.208345\pi\)
\(644\) 0.800433 0.0315415
\(645\) −8.83906 −0.348038
\(646\) −1.42179 −0.0559396
\(647\) −26.0029 −1.02228 −0.511139 0.859498i \(-0.670776\pi\)
−0.511139 + 0.859498i \(0.670776\pi\)
\(648\) −26.0078 −1.02168
\(649\) 30.5964 1.20101
\(650\) −11.2028 −0.439408
\(651\) 1.48990 0.0583939
\(652\) 10.1002 0.395554
\(653\) −20.6624 −0.808584 −0.404292 0.914630i \(-0.632482\pi\)
−0.404292 + 0.914630i \(0.632482\pi\)
\(654\) 4.05198 0.158445
\(655\) 30.0615 1.17460
\(656\) −4.21696 −0.164645
\(657\) 3.56574 0.139113
\(658\) 9.31313 0.363064
\(659\) 5.80002 0.225937 0.112968 0.993599i \(-0.463964\pi\)
0.112968 + 0.993599i \(0.463964\pi\)
\(660\) −22.9654 −0.893927
\(661\) 2.81113 0.109340 0.0546702 0.998504i \(-0.482589\pi\)
0.0546702 + 0.998504i \(0.482589\pi\)
\(662\) 25.4052 0.987401
\(663\) −19.1334 −0.743078
\(664\) 39.0955 1.51720
\(665\) −1.75527 −0.0680664
\(666\) −1.59332 −0.0617398
\(667\) 1.72665 0.0668560
\(668\) 14.9579 0.578740
\(669\) 10.0919 0.390175
\(670\) 10.0479 0.388183
\(671\) 20.6248 0.796210
\(672\) −12.9254 −0.498607
\(673\) −4.40770 −0.169904 −0.0849521 0.996385i \(-0.527074\pi\)
−0.0849521 + 0.996385i \(0.527074\pi\)
\(674\) −3.22005 −0.124032
\(675\) 13.1057 0.504438
\(676\) −23.0195 −0.885366
\(677\) −10.0686 −0.386969 −0.193484 0.981103i \(-0.561979\pi\)
−0.193484 + 0.981103i \(0.561979\pi\)
\(678\) −7.49624 −0.287891
\(679\) −7.30051 −0.280168
\(680\) −7.60573 −0.291667
\(681\) −3.84014 −0.147154
\(682\) 3.01208 0.115339
\(683\) 43.3757 1.65973 0.829863 0.557968i \(-0.188419\pi\)
0.829863 + 0.557968i \(0.188419\pi\)
\(684\) −0.503879 −0.0192663
\(685\) −29.1611 −1.11419
\(686\) 11.5197 0.439825
\(687\) −30.1380 −1.14984
\(688\) 2.68262 0.102274
\(689\) −44.6262 −1.70012
\(690\) −0.995972 −0.0379160
\(691\) 11.2784 0.429052 0.214526 0.976718i \(-0.431179\pi\)
0.214526 + 0.976718i \(0.431179\pi\)
\(692\) 23.7478 0.902757
\(693\) −2.58314 −0.0981253
\(694\) 13.6252 0.517204
\(695\) −17.0849 −0.648066
\(696\) −17.5878 −0.666663
\(697\) −9.65220 −0.365603
\(698\) 3.58931 0.135858
\(699\) −18.3585 −0.694382
\(700\) 4.62039 0.174634
\(701\) 24.4031 0.921691 0.460845 0.887480i \(-0.347546\pi\)
0.460845 + 0.887480i \(0.347546\pi\)
\(702\) −19.9428 −0.752693
\(703\) 5.47610 0.206535
\(704\) −16.2937 −0.614093
\(705\) 28.1710 1.06098
\(706\) 19.4876 0.733427
\(707\) −2.84464 −0.106984
\(708\) 13.6262 0.512104
\(709\) 20.1044 0.755037 0.377518 0.926002i \(-0.376777\pi\)
0.377518 + 0.926002i \(0.376777\pi\)
\(710\) −10.9781 −0.412000
\(711\) −1.92755 −0.0722888
\(712\) 26.6483 0.998685
\(713\) −0.317559 −0.0118927
\(714\) −3.24610 −0.121482
\(715\) −47.7538 −1.78589
\(716\) −6.73383 −0.251655
\(717\) 23.6404 0.882869
\(718\) −24.4331 −0.911833
\(719\) −26.8520 −1.00141 −0.500705 0.865618i \(-0.666926\pi\)
−0.500705 + 0.865618i \(0.666926\pi\)
\(720\) −0.468850 −0.0174730
\(721\) 14.9737 0.557649
\(722\) 13.7940 0.513359
\(723\) 17.5888 0.654136
\(724\) 12.7822 0.475045
\(725\) 9.96683 0.370159
\(726\) −32.3688 −1.20132
\(727\) 43.5422 1.61489 0.807446 0.589941i \(-0.200849\pi\)
0.807446 + 0.589941i \(0.200849\pi\)
\(728\) −16.9538 −0.628349
\(729\) 22.9238 0.849031
\(730\) 11.1836 0.413922
\(731\) 6.14025 0.227105
\(732\) 9.18530 0.339499
\(733\) −29.4062 −1.08614 −0.543072 0.839686i \(-0.682739\pi\)
−0.543072 + 0.839686i \(0.682739\pi\)
\(734\) −11.2850 −0.416535
\(735\) 15.4191 0.568741
\(736\) 2.75492 0.101548
\(737\) −50.8215 −1.87203
\(738\) 1.40713 0.0517970
\(739\) −10.0339 −0.369103 −0.184552 0.982823i \(-0.559083\pi\)
−0.184552 + 0.982823i \(0.559083\pi\)
\(740\) 12.1483 0.446581
\(741\) −9.58667 −0.352175
\(742\) −7.57112 −0.277945
\(743\) 32.7364 1.20098 0.600492 0.799631i \(-0.294972\pi\)
0.600492 + 0.799631i \(0.294972\pi\)
\(744\) 3.23468 0.118589
\(745\) −0.846601 −0.0310171
\(746\) 23.3048 0.853250
\(747\) 5.51627 0.201830
\(748\) 15.9534 0.583315
\(749\) −21.5076 −0.785872
\(750\) −16.3434 −0.596777
\(751\) −29.0656 −1.06062 −0.530309 0.847804i \(-0.677924\pi\)
−0.530309 + 0.847804i \(0.677924\pi\)
\(752\) −8.54979 −0.311779
\(753\) −13.0142 −0.474263
\(754\) −15.1665 −0.552330
\(755\) 11.7556 0.427831
\(756\) 8.22509 0.299144
\(757\) −49.0178 −1.78158 −0.890791 0.454414i \(-0.849849\pi\)
−0.890791 + 0.454414i \(0.849849\pi\)
\(758\) 10.1567 0.368910
\(759\) 5.03757 0.182852
\(760\) −3.81081 −0.138233
\(761\) 42.6524 1.54615 0.773073 0.634317i \(-0.218718\pi\)
0.773073 + 0.634317i \(0.218718\pi\)
\(762\) −20.4582 −0.741123
\(763\) −3.47500 −0.125804
\(764\) 34.8603 1.26120
\(765\) −1.07315 −0.0387998
\(766\) −15.7597 −0.569423
\(767\) 28.3340 1.02308
\(768\) −23.6810 −0.854513
\(769\) 23.5755 0.850156 0.425078 0.905157i \(-0.360247\pi\)
0.425078 + 0.905157i \(0.360247\pi\)
\(770\) −8.10173 −0.291966
\(771\) −14.0253 −0.505109
\(772\) −22.0096 −0.792144
\(773\) −9.61811 −0.345939 −0.172970 0.984927i \(-0.555336\pi\)
−0.172970 + 0.984927i \(0.555336\pi\)
\(774\) −0.895144 −0.0321753
\(775\) −1.83306 −0.0658456
\(776\) −15.8499 −0.568979
\(777\) 12.5025 0.448525
\(778\) −13.7723 −0.493761
\(779\) −4.83618 −0.173274
\(780\) −21.2673 −0.761491
\(781\) 55.5264 1.98689
\(782\) 0.691874 0.0247414
\(783\) 17.7427 0.634071
\(784\) −4.67963 −0.167130
\(785\) 0.685153 0.0244541
\(786\) 27.8551 0.993558
\(787\) −27.3656 −0.975479 −0.487740 0.872989i \(-0.662178\pi\)
−0.487740 + 0.872989i \(0.662178\pi\)
\(788\) 21.4222 0.763135
\(789\) −34.4756 −1.22736
\(790\) −6.04555 −0.215091
\(791\) 6.42882 0.228583
\(792\) −5.60818 −0.199278
\(793\) 19.0997 0.678251
\(794\) 18.5447 0.658126
\(795\) −22.9016 −0.812237
\(796\) −19.3152 −0.684610
\(797\) −10.5153 −0.372471 −0.186236 0.982505i \(-0.559629\pi\)
−0.186236 + 0.982505i \(0.559629\pi\)
\(798\) −1.62644 −0.0575753
\(799\) −19.5696 −0.692322
\(800\) 15.9024 0.562235
\(801\) 3.76000 0.132853
\(802\) −8.73707 −0.308517
\(803\) −56.5657 −1.99616
\(804\) −22.6335 −0.798222
\(805\) 0.854152 0.0301049
\(806\) 2.78936 0.0982510
\(807\) −31.5601 −1.11097
\(808\) −6.17591 −0.217268
\(809\) 9.73359 0.342215 0.171107 0.985252i \(-0.445266\pi\)
0.171107 + 0.985252i \(0.445266\pi\)
\(810\) −11.5094 −0.404399
\(811\) 11.8190 0.415020 0.207510 0.978233i \(-0.433464\pi\)
0.207510 + 0.978233i \(0.433464\pi\)
\(812\) 6.25515 0.219513
\(813\) −25.2697 −0.886247
\(814\) 25.2759 0.885919
\(815\) 10.7780 0.377538
\(816\) 2.98003 0.104322
\(817\) 3.07654 0.107634
\(818\) −3.31509 −0.115909
\(819\) −2.39214 −0.0835880
\(820\) −10.7287 −0.374662
\(821\) 20.8740 0.728507 0.364253 0.931300i \(-0.381324\pi\)
0.364253 + 0.931300i \(0.381324\pi\)
\(822\) −27.0208 −0.942459
\(823\) 19.2428 0.670762 0.335381 0.942083i \(-0.391135\pi\)
0.335381 + 0.942083i \(0.391135\pi\)
\(824\) 32.5089 1.13250
\(825\) 29.0787 1.01239
\(826\) 4.80705 0.167259
\(827\) −31.8820 −1.10864 −0.554322 0.832302i \(-0.687022\pi\)
−0.554322 + 0.832302i \(0.687022\pi\)
\(828\) 0.245199 0.00852124
\(829\) −40.5150 −1.40714 −0.703572 0.710624i \(-0.748413\pi\)
−0.703572 + 0.710624i \(0.748413\pi\)
\(830\) 17.3012 0.600532
\(831\) −34.4582 −1.19534
\(832\) −15.0889 −0.523115
\(833\) −10.7112 −0.371121
\(834\) −15.8309 −0.548179
\(835\) 15.9618 0.552381
\(836\) 7.99337 0.276457
\(837\) −3.26317 −0.112792
\(838\) 26.7923 0.925525
\(839\) −30.1260 −1.04006 −0.520032 0.854147i \(-0.674080\pi\)
−0.520032 + 0.854147i \(0.674080\pi\)
\(840\) −8.70048 −0.300195
\(841\) −15.5068 −0.534716
\(842\) 27.7094 0.954929
\(843\) 16.5244 0.569129
\(844\) −16.0450 −0.552291
\(845\) −24.5644 −0.845041
\(846\) 2.85291 0.0980852
\(847\) 27.7597 0.953834
\(848\) 6.95057 0.238683
\(849\) 25.8655 0.887702
\(850\) 3.99375 0.136985
\(851\) −2.66479 −0.0913479
\(852\) 24.7289 0.847197
\(853\) 9.30329 0.318539 0.159269 0.987235i \(-0.449086\pi\)
0.159269 + 0.987235i \(0.449086\pi\)
\(854\) 3.24039 0.110884
\(855\) −0.537695 −0.0183888
\(856\) −46.6946 −1.59599
\(857\) 46.7828 1.59807 0.799035 0.601284i \(-0.205344\pi\)
0.799035 + 0.601284i \(0.205344\pi\)
\(858\) −44.2488 −1.51063
\(859\) −11.8562 −0.404528 −0.202264 0.979331i \(-0.564830\pi\)
−0.202264 + 0.979331i \(0.564830\pi\)
\(860\) 6.82507 0.232733
\(861\) −11.0415 −0.376293
\(862\) −6.81506 −0.232122
\(863\) −34.0665 −1.15964 −0.579819 0.814746i \(-0.696877\pi\)
−0.579819 + 0.814746i \(0.696877\pi\)
\(864\) 28.3090 0.963092
\(865\) 25.3416 0.861640
\(866\) −30.6524 −1.04161
\(867\) −24.3781 −0.827924
\(868\) −1.15043 −0.0390480
\(869\) 30.5781 1.03729
\(870\) −7.78324 −0.263876
\(871\) −47.0636 −1.59469
\(872\) −7.54448 −0.255488
\(873\) −2.23638 −0.0756901
\(874\) 0.346660 0.0117259
\(875\) 14.0162 0.473834
\(876\) −25.1917 −0.851149
\(877\) 6.26125 0.211427 0.105714 0.994397i \(-0.466287\pi\)
0.105714 + 0.994397i \(0.466287\pi\)
\(878\) 12.5607 0.423903
\(879\) −46.3311 −1.56271
\(880\) 7.43768 0.250724
\(881\) 3.16036 0.106475 0.0532377 0.998582i \(-0.483046\pi\)
0.0532377 + 0.998582i \(0.483046\pi\)
\(882\) 1.56151 0.0525788
\(883\) −31.3346 −1.05449 −0.527247 0.849712i \(-0.676776\pi\)
−0.527247 + 0.849712i \(0.676776\pi\)
\(884\) 14.7738 0.496897
\(885\) 14.5407 0.488779
\(886\) −10.7990 −0.362800
\(887\) −12.6168 −0.423632 −0.211816 0.977310i \(-0.567938\pi\)
−0.211816 + 0.977310i \(0.567938\pi\)
\(888\) 27.1438 0.910888
\(889\) 17.5451 0.588443
\(890\) 11.7928 0.395296
\(891\) 58.2139 1.95024
\(892\) −7.79243 −0.260910
\(893\) −9.80523 −0.328120
\(894\) −0.784464 −0.0262364
\(895\) −7.18575 −0.240193
\(896\) 11.5258 0.385051
\(897\) 4.66508 0.155763
\(898\) −5.31957 −0.177516
\(899\) −2.48163 −0.0827669
\(900\) 1.41538 0.0471792
\(901\) 15.9091 0.530010
\(902\) −22.3222 −0.743248
\(903\) 7.02406 0.233746
\(904\) 13.9574 0.464217
\(905\) 13.6400 0.453409
\(906\) 10.8928 0.361889
\(907\) −34.0345 −1.13010 −0.565049 0.825057i \(-0.691143\pi\)
−0.565049 + 0.825057i \(0.691143\pi\)
\(908\) 2.96516 0.0984022
\(909\) −0.871405 −0.0289027
\(910\) −7.50267 −0.248711
\(911\) −41.1212 −1.36241 −0.681203 0.732095i \(-0.738543\pi\)
−0.681203 + 0.732095i \(0.738543\pi\)
\(912\) 1.49313 0.0494425
\(913\) −87.5083 −2.89610
\(914\) 12.7114 0.420456
\(915\) 9.80175 0.324036
\(916\) 23.2710 0.768896
\(917\) −23.8887 −0.788874
\(918\) 7.10956 0.234651
\(919\) −0.391473 −0.0129135 −0.00645675 0.999979i \(-0.502055\pi\)
−0.00645675 + 0.999979i \(0.502055\pi\)
\(920\) 1.85442 0.0611385
\(921\) 30.5592 1.00696
\(922\) −13.8204 −0.455150
\(923\) 51.4207 1.69253
\(924\) 18.2497 0.600371
\(925\) −15.3822 −0.505762
\(926\) −25.3113 −0.831779
\(927\) 4.58693 0.150654
\(928\) 21.5289 0.706721
\(929\) 20.6457 0.677365 0.338682 0.940901i \(-0.390019\pi\)
0.338682 + 0.940901i \(0.390019\pi\)
\(930\) 1.43147 0.0469396
\(931\) −5.36679 −0.175889
\(932\) 14.1755 0.464333
\(933\) −5.98373 −0.195898
\(934\) −2.82845 −0.0925499
\(935\) 17.0241 0.556747
\(936\) −5.19350 −0.169755
\(937\) −44.5315 −1.45478 −0.727390 0.686225i \(-0.759267\pi\)
−0.727390 + 0.686225i \(0.759267\pi\)
\(938\) −7.98464 −0.260708
\(939\) −10.0700 −0.328623
\(940\) −21.7522 −0.709477
\(941\) −40.1815 −1.30988 −0.654940 0.755681i \(-0.727306\pi\)
−0.654940 + 0.755681i \(0.727306\pi\)
\(942\) 0.634865 0.0206850
\(943\) 2.35339 0.0766369
\(944\) −4.41304 −0.143632
\(945\) 8.77709 0.285519
\(946\) 14.2003 0.461691
\(947\) −54.0978 −1.75794 −0.878972 0.476874i \(-0.841770\pi\)
−0.878972 + 0.476874i \(0.841770\pi\)
\(948\) 13.6180 0.442293
\(949\) −52.3832 −1.70043
\(950\) 2.00105 0.0649225
\(951\) −41.0584 −1.33141
\(952\) 6.04398 0.195887
\(953\) −18.8291 −0.609933 −0.304967 0.952363i \(-0.598645\pi\)
−0.304967 + 0.952363i \(0.598645\pi\)
\(954\) −2.31928 −0.0750895
\(955\) 37.1998 1.20376
\(956\) −18.2539 −0.590374
\(957\) 39.3672 1.27256
\(958\) −10.4345 −0.337122
\(959\) 23.1732 0.748302
\(960\) −7.74346 −0.249919
\(961\) −30.5436 −0.985277
\(962\) 23.4069 0.754669
\(963\) −6.58849 −0.212311
\(964\) −13.5812 −0.437421
\(965\) −23.4867 −0.756065
\(966\) 0.791460 0.0254648
\(967\) 53.0693 1.70659 0.853296 0.521426i \(-0.174600\pi\)
0.853296 + 0.521426i \(0.174600\pi\)
\(968\) 60.2682 1.93709
\(969\) 3.41762 0.109790
\(970\) −7.01417 −0.225211
\(971\) 40.4523 1.29818 0.649088 0.760713i \(-0.275151\pi\)
0.649088 + 0.760713i \(0.275151\pi\)
\(972\) 5.39163 0.172937
\(973\) 13.5767 0.435248
\(974\) −20.3525 −0.652136
\(975\) 26.9286 0.862404
\(976\) −2.97480 −0.0952209
\(977\) −10.6610 −0.341075 −0.170537 0.985351i \(-0.554550\pi\)
−0.170537 + 0.985351i \(0.554550\pi\)
\(978\) 9.98698 0.319348
\(979\) −59.6474 −1.90634
\(980\) −11.9058 −0.380317
\(981\) −1.06451 −0.0339871
\(982\) −6.31572 −0.201543
\(983\) 50.7688 1.61927 0.809637 0.586931i \(-0.199664\pi\)
0.809637 + 0.586931i \(0.199664\pi\)
\(984\) −23.9719 −0.764196
\(985\) 22.8599 0.728377
\(986\) 5.40680 0.172188
\(987\) −22.3864 −0.712566
\(988\) 7.40233 0.235499
\(989\) −1.49711 −0.0476054
\(990\) −2.48182 −0.0788775
\(991\) −31.6640 −1.00584 −0.502919 0.864333i \(-0.667741\pi\)
−0.502919 + 0.864333i \(0.667741\pi\)
\(992\) −3.95952 −0.125715
\(993\) −61.0676 −1.93792
\(994\) 8.72385 0.276704
\(995\) −20.6115 −0.653429
\(996\) −38.9721 −1.23488
\(997\) 46.8252 1.48297 0.741485 0.670970i \(-0.234122\pi\)
0.741485 + 0.670970i \(0.234122\pi\)
\(998\) 24.5460 0.776990
\(999\) −27.3829 −0.866356
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.1.9 18
3.2 odd 2 4923.2.a.l.1.10 18
4.3 odd 2 8752.2.a.s.1.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.9 18 1.1 even 1 trivial
4923.2.a.l.1.10 18 3.2 odd 2
8752.2.a.s.1.4 18 4.3 odd 2