Properties

Label 5054.2.a.bi.1.8
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.91631\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.35992 q^{3} +1.00000 q^{4} -1.35877 q^{5} +3.35992 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.28904 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.35992 q^{3} +1.00000 q^{4} -1.35877 q^{5} +3.35992 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.28904 q^{9} -1.35877 q^{10} +0.672249 q^{11} +3.35992 q^{12} -1.04304 q^{13} -1.00000 q^{14} -4.56537 q^{15} +1.00000 q^{16} -2.46184 q^{17} +8.28904 q^{18} -1.35877 q^{20} -3.35992 q^{21} +0.672249 q^{22} +9.22420 q^{23} +3.35992 q^{24} -3.15373 q^{25} -1.04304 q^{26} +17.7707 q^{27} -1.00000 q^{28} +9.26173 q^{29} -4.56537 q^{30} +7.38217 q^{31} +1.00000 q^{32} +2.25870 q^{33} -2.46184 q^{34} +1.35877 q^{35} +8.28904 q^{36} -6.98489 q^{37} -3.50451 q^{39} -1.35877 q^{40} +5.46835 q^{41} -3.35992 q^{42} +0.0812913 q^{43} +0.672249 q^{44} -11.2629 q^{45} +9.22420 q^{46} -0.226152 q^{47} +3.35992 q^{48} +1.00000 q^{49} -3.15373 q^{50} -8.27156 q^{51} -1.04304 q^{52} -6.14086 q^{53} +17.7707 q^{54} -0.913435 q^{55} -1.00000 q^{56} +9.26173 q^{58} -10.2103 q^{59} -4.56537 q^{60} -9.07391 q^{61} +7.38217 q^{62} -8.28904 q^{63} +1.00000 q^{64} +1.41725 q^{65} +2.25870 q^{66} -12.4558 q^{67} -2.46184 q^{68} +30.9925 q^{69} +1.35877 q^{70} +0.358308 q^{71} +8.28904 q^{72} +5.86115 q^{73} -6.98489 q^{74} -10.5963 q^{75} -0.672249 q^{77} -3.50451 q^{78} +10.0492 q^{79} -1.35877 q^{80} +34.8410 q^{81} +5.46835 q^{82} +2.73042 q^{83} -3.35992 q^{84} +3.34508 q^{85} +0.0812913 q^{86} +31.1186 q^{87} +0.672249 q^{88} +9.58916 q^{89} -11.2629 q^{90} +1.04304 q^{91} +9.22420 q^{92} +24.8035 q^{93} -0.226152 q^{94} +3.35992 q^{96} +13.2610 q^{97} +1.00000 q^{98} +5.57230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9} + 6 q^{10} + 4 q^{11} + 4 q^{12} + 6 q^{13} - 8 q^{14} + 8 q^{15} + 8 q^{16} + 2 q^{17} + 8 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} + 22 q^{27} - 8 q^{28} + 16 q^{29} + 8 q^{30} + 22 q^{31} + 8 q^{32} - 8 q^{33} + 2 q^{34} - 6 q^{35} + 8 q^{36} - 12 q^{37} + 8 q^{39} + 6 q^{40} + 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} + 20 q^{46} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} - 4 q^{51} + 6 q^{52} + 16 q^{53} + 22 q^{54} + 18 q^{55} - 8 q^{56} + 16 q^{58} + 14 q^{59} + 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} + 32 q^{65} - 8 q^{66} - 20 q^{67} + 2 q^{68} + 40 q^{69} - 6 q^{70} + 8 q^{72} - 18 q^{73} - 12 q^{74} + 16 q^{75} - 4 q^{77} + 8 q^{78} + 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} - 4 q^{84} - 16 q^{85} + 28 q^{87} + 4 q^{88} + 18 q^{89} - 4 q^{90} - 6 q^{91} + 20 q^{92} + 16 q^{93} + 6 q^{94} + 4 q^{96} + 26 q^{97} + 8 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 3.35992 1.93985 0.969924 0.243407i \(-0.0782651\pi\)
0.969924 + 0.243407i \(0.0782651\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.35877 −0.607663 −0.303831 0.952726i \(-0.598266\pi\)
−0.303831 + 0.952726i \(0.598266\pi\)
\(6\) 3.35992 1.37168
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 8.28904 2.76301
\(10\) −1.35877 −0.429682
\(11\) 0.672249 0.202691 0.101345 0.994851i \(-0.467685\pi\)
0.101345 + 0.994851i \(0.467685\pi\)
\(12\) 3.35992 0.969924
\(13\) −1.04304 −0.289286 −0.144643 0.989484i \(-0.546203\pi\)
−0.144643 + 0.989484i \(0.546203\pi\)
\(14\) −1.00000 −0.267261
\(15\) −4.56537 −1.17877
\(16\) 1.00000 0.250000
\(17\) −2.46184 −0.597083 −0.298542 0.954397i \(-0.596500\pi\)
−0.298542 + 0.954397i \(0.596500\pi\)
\(18\) 8.28904 1.95374
\(19\) 0 0
\(20\) −1.35877 −0.303831
\(21\) −3.35992 −0.733194
\(22\) 0.672249 0.143324
\(23\) 9.22420 1.92338 0.961689 0.274142i \(-0.0883938\pi\)
0.961689 + 0.274142i \(0.0883938\pi\)
\(24\) 3.35992 0.685840
\(25\) −3.15373 −0.630746
\(26\) −1.04304 −0.204556
\(27\) 17.7707 3.41998
\(28\) −1.00000 −0.188982
\(29\) 9.26173 1.71986 0.859930 0.510411i \(-0.170507\pi\)
0.859930 + 0.510411i \(0.170507\pi\)
\(30\) −4.56537 −0.833519
\(31\) 7.38217 1.32588 0.662938 0.748674i \(-0.269309\pi\)
0.662938 + 0.748674i \(0.269309\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.25870 0.393189
\(34\) −2.46184 −0.422201
\(35\) 1.35877 0.229675
\(36\) 8.28904 1.38151
\(37\) −6.98489 −1.14831 −0.574155 0.818747i \(-0.694669\pi\)
−0.574155 + 0.818747i \(0.694669\pi\)
\(38\) 0 0
\(39\) −3.50451 −0.561171
\(40\) −1.35877 −0.214841
\(41\) 5.46835 0.854012 0.427006 0.904249i \(-0.359568\pi\)
0.427006 + 0.904249i \(0.359568\pi\)
\(42\) −3.35992 −0.518446
\(43\) 0.0812913 0.0123968 0.00619841 0.999981i \(-0.498027\pi\)
0.00619841 + 0.999981i \(0.498027\pi\)
\(44\) 0.672249 0.101345
\(45\) −11.2629 −1.67898
\(46\) 9.22420 1.36003
\(47\) −0.226152 −0.0329876 −0.0164938 0.999864i \(-0.505250\pi\)
−0.0164938 + 0.999864i \(0.505250\pi\)
\(48\) 3.35992 0.484962
\(49\) 1.00000 0.142857
\(50\) −3.15373 −0.446005
\(51\) −8.27156 −1.15825
\(52\) −1.04304 −0.144643
\(53\) −6.14086 −0.843512 −0.421756 0.906709i \(-0.638586\pi\)
−0.421756 + 0.906709i \(0.638586\pi\)
\(54\) 17.7707 2.41829
\(55\) −0.913435 −0.123168
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 9.26173 1.21613
\(59\) −10.2103 −1.32926 −0.664632 0.747171i \(-0.731412\pi\)
−0.664632 + 0.747171i \(0.731412\pi\)
\(60\) −4.56537 −0.589387
\(61\) −9.07391 −1.16179 −0.580897 0.813977i \(-0.697298\pi\)
−0.580897 + 0.813977i \(0.697298\pi\)
\(62\) 7.38217 0.937536
\(63\) −8.28904 −1.04432
\(64\) 1.00000 0.125000
\(65\) 1.41725 0.175788
\(66\) 2.25870 0.278027
\(67\) −12.4558 −1.52172 −0.760862 0.648914i \(-0.775224\pi\)
−0.760862 + 0.648914i \(0.775224\pi\)
\(68\) −2.46184 −0.298542
\(69\) 30.9925 3.73106
\(70\) 1.35877 0.162405
\(71\) 0.358308 0.0425234 0.0212617 0.999774i \(-0.493232\pi\)
0.0212617 + 0.999774i \(0.493232\pi\)
\(72\) 8.28904 0.976872
\(73\) 5.86115 0.685996 0.342998 0.939336i \(-0.388558\pi\)
0.342998 + 0.939336i \(0.388558\pi\)
\(74\) −6.98489 −0.811978
\(75\) −10.5963 −1.22355
\(76\) 0 0
\(77\) −0.672249 −0.0766099
\(78\) −3.50451 −0.396808
\(79\) 10.0492 1.13062 0.565311 0.824878i \(-0.308756\pi\)
0.565311 + 0.824878i \(0.308756\pi\)
\(80\) −1.35877 −0.151916
\(81\) 34.8410 3.87122
\(82\) 5.46835 0.603878
\(83\) 2.73042 0.299702 0.149851 0.988709i \(-0.452121\pi\)
0.149851 + 0.988709i \(0.452121\pi\)
\(84\) −3.35992 −0.366597
\(85\) 3.34508 0.362825
\(86\) 0.0812913 0.00876587
\(87\) 31.1186 3.33627
\(88\) 0.672249 0.0716620
\(89\) 9.58916 1.01645 0.508225 0.861225i \(-0.330302\pi\)
0.508225 + 0.861225i \(0.330302\pi\)
\(90\) −11.2629 −1.18722
\(91\) 1.04304 0.109340
\(92\) 9.22420 0.961689
\(93\) 24.8035 2.57200
\(94\) −0.226152 −0.0233258
\(95\) 0 0
\(96\) 3.35992 0.342920
\(97\) 13.2610 1.34646 0.673228 0.739435i \(-0.264907\pi\)
0.673228 + 0.739435i \(0.264907\pi\)
\(98\) 1.00000 0.101015
\(99\) 5.57230 0.560037
\(100\) −3.15373 −0.315373
\(101\) 18.5225 1.84305 0.921527 0.388314i \(-0.126942\pi\)
0.921527 + 0.388314i \(0.126942\pi\)
\(102\) −8.27156 −0.819007
\(103\) −7.88935 −0.777361 −0.388681 0.921373i \(-0.627069\pi\)
−0.388681 + 0.921373i \(0.627069\pi\)
\(104\) −1.04304 −0.102278
\(105\) 4.56537 0.445534
\(106\) −6.14086 −0.596453
\(107\) −5.44998 −0.526869 −0.263435 0.964677i \(-0.584855\pi\)
−0.263435 + 0.964677i \(0.584855\pi\)
\(108\) 17.7707 1.70999
\(109\) 0.525792 0.0503617 0.0251809 0.999683i \(-0.491984\pi\)
0.0251809 + 0.999683i \(0.491984\pi\)
\(110\) −0.913435 −0.0870926
\(111\) −23.4687 −2.22755
\(112\) −1.00000 −0.0944911
\(113\) −17.3578 −1.63289 −0.816443 0.577426i \(-0.804057\pi\)
−0.816443 + 0.577426i \(0.804057\pi\)
\(114\) 0 0
\(115\) −12.5336 −1.16877
\(116\) 9.26173 0.859930
\(117\) −8.64576 −0.799301
\(118\) −10.2103 −0.939932
\(119\) 2.46184 0.225676
\(120\) −4.56537 −0.416759
\(121\) −10.5481 −0.958916
\(122\) −9.07391 −0.821513
\(123\) 18.3732 1.65665
\(124\) 7.38217 0.662938
\(125\) 11.0791 0.990943
\(126\) −8.28904 −0.738446
\(127\) −7.06798 −0.627182 −0.313591 0.949558i \(-0.601532\pi\)
−0.313591 + 0.949558i \(0.601532\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.273132 0.0240479
\(130\) 1.41725 0.124301
\(131\) 9.79698 0.855966 0.427983 0.903787i \(-0.359224\pi\)
0.427983 + 0.903787i \(0.359224\pi\)
\(132\) 2.25870 0.196595
\(133\) 0 0
\(134\) −12.4558 −1.07602
\(135\) −24.1464 −2.07819
\(136\) −2.46184 −0.211101
\(137\) −20.2972 −1.73411 −0.867054 0.498214i \(-0.833989\pi\)
−0.867054 + 0.498214i \(0.833989\pi\)
\(138\) 30.9925 2.63826
\(139\) 5.15852 0.437540 0.218770 0.975776i \(-0.429796\pi\)
0.218770 + 0.975776i \(0.429796\pi\)
\(140\) 1.35877 0.114837
\(141\) −0.759851 −0.0639910
\(142\) 0.358308 0.0300685
\(143\) −0.701180 −0.0586356
\(144\) 8.28904 0.690753
\(145\) −12.5846 −1.04509
\(146\) 5.86115 0.485073
\(147\) 3.35992 0.277121
\(148\) −6.98489 −0.574155
\(149\) 9.78505 0.801623 0.400811 0.916161i \(-0.368728\pi\)
0.400811 + 0.916161i \(0.368728\pi\)
\(150\) −10.5963 −0.865182
\(151\) 6.87697 0.559640 0.279820 0.960052i \(-0.409725\pi\)
0.279820 + 0.960052i \(0.409725\pi\)
\(152\) 0 0
\(153\) −20.4063 −1.64975
\(154\) −0.672249 −0.0541714
\(155\) −10.0307 −0.805685
\(156\) −3.50451 −0.280585
\(157\) 12.1056 0.966133 0.483066 0.875584i \(-0.339523\pi\)
0.483066 + 0.875584i \(0.339523\pi\)
\(158\) 10.0492 0.799470
\(159\) −20.6328 −1.63629
\(160\) −1.35877 −0.107421
\(161\) −9.22420 −0.726969
\(162\) 34.8410 2.73737
\(163\) −11.3384 −0.888089 −0.444045 0.896005i \(-0.646457\pi\)
−0.444045 + 0.896005i \(0.646457\pi\)
\(164\) 5.46835 0.427006
\(165\) −3.06907 −0.238926
\(166\) 2.73042 0.211922
\(167\) −11.8009 −0.913184 −0.456592 0.889676i \(-0.650930\pi\)
−0.456592 + 0.889676i \(0.650930\pi\)
\(168\) −3.35992 −0.259223
\(169\) −11.9121 −0.916314
\(170\) 3.34508 0.256556
\(171\) 0 0
\(172\) 0.0812913 0.00619841
\(173\) 18.0756 1.37426 0.687132 0.726533i \(-0.258869\pi\)
0.687132 + 0.726533i \(0.258869\pi\)
\(174\) 31.1186 2.35910
\(175\) 3.15373 0.238400
\(176\) 0.672249 0.0506727
\(177\) −34.3057 −2.57857
\(178\) 9.58916 0.718738
\(179\) 4.30866 0.322044 0.161022 0.986951i \(-0.448521\pi\)
0.161022 + 0.986951i \(0.448521\pi\)
\(180\) −11.2629 −0.839489
\(181\) 3.51816 0.261503 0.130751 0.991415i \(-0.458261\pi\)
0.130751 + 0.991415i \(0.458261\pi\)
\(182\) 1.04304 0.0773149
\(183\) −30.4876 −2.25371
\(184\) 9.22420 0.680017
\(185\) 9.49090 0.697785
\(186\) 24.8035 1.81868
\(187\) −1.65497 −0.121023
\(188\) −0.226152 −0.0164938
\(189\) −17.7707 −1.29263
\(190\) 0 0
\(191\) 15.7576 1.14018 0.570090 0.821582i \(-0.306908\pi\)
0.570090 + 0.821582i \(0.306908\pi\)
\(192\) 3.35992 0.242481
\(193\) −11.2836 −0.812214 −0.406107 0.913826i \(-0.633114\pi\)
−0.406107 + 0.913826i \(0.633114\pi\)
\(194\) 13.2610 0.952088
\(195\) 4.76184 0.341003
\(196\) 1.00000 0.0714286
\(197\) −14.9533 −1.06538 −0.532691 0.846310i \(-0.678819\pi\)
−0.532691 + 0.846310i \(0.678819\pi\)
\(198\) 5.57230 0.396006
\(199\) −1.85467 −0.131474 −0.0657370 0.997837i \(-0.520940\pi\)
−0.0657370 + 0.997837i \(0.520940\pi\)
\(200\) −3.15373 −0.223002
\(201\) −41.8506 −2.95191
\(202\) 18.5225 1.30324
\(203\) −9.26173 −0.650046
\(204\) −8.27156 −0.579125
\(205\) −7.43025 −0.518951
\(206\) −7.88935 −0.549677
\(207\) 76.4597 5.31432
\(208\) −1.04304 −0.0723215
\(209\) 0 0
\(210\) 4.56537 0.315040
\(211\) −27.3986 −1.88620 −0.943100 0.332510i \(-0.892105\pi\)
−0.943100 + 0.332510i \(0.892105\pi\)
\(212\) −6.14086 −0.421756
\(213\) 1.20389 0.0824889
\(214\) −5.44998 −0.372553
\(215\) −0.110457 −0.00753308
\(216\) 17.7707 1.20914
\(217\) −7.38217 −0.501134
\(218\) 0.525792 0.0356111
\(219\) 19.6930 1.33073
\(220\) −0.913435 −0.0615838
\(221\) 2.56778 0.172728
\(222\) −23.4687 −1.57511
\(223\) −1.56448 −0.104766 −0.0523828 0.998627i \(-0.516682\pi\)
−0.0523828 + 0.998627i \(0.516682\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −26.1414 −1.74276
\(226\) −17.3578 −1.15463
\(227\) −24.2868 −1.61197 −0.805986 0.591935i \(-0.798364\pi\)
−0.805986 + 0.591935i \(0.798364\pi\)
\(228\) 0 0
\(229\) 12.8094 0.846471 0.423236 0.906020i \(-0.360894\pi\)
0.423236 + 0.906020i \(0.360894\pi\)
\(230\) −12.5336 −0.826442
\(231\) −2.25870 −0.148612
\(232\) 9.26173 0.608063
\(233\) −16.1943 −1.06092 −0.530460 0.847710i \(-0.677981\pi\)
−0.530460 + 0.847710i \(0.677981\pi\)
\(234\) −8.64576 −0.565191
\(235\) 0.307289 0.0200453
\(236\) −10.2103 −0.664632
\(237\) 33.7644 2.19323
\(238\) 2.46184 0.159577
\(239\) −11.1682 −0.722410 −0.361205 0.932487i \(-0.617634\pi\)
−0.361205 + 0.932487i \(0.617634\pi\)
\(240\) −4.56537 −0.294693
\(241\) −18.5754 −1.19655 −0.598274 0.801292i \(-0.704147\pi\)
−0.598274 + 0.801292i \(0.704147\pi\)
\(242\) −10.5481 −0.678056
\(243\) 63.7507 4.08961
\(244\) −9.07391 −0.580897
\(245\) −1.35877 −0.0868089
\(246\) 18.3732 1.17143
\(247\) 0 0
\(248\) 7.38217 0.468768
\(249\) 9.17397 0.581377
\(250\) 11.0791 0.700703
\(251\) −6.42172 −0.405335 −0.202668 0.979248i \(-0.564961\pi\)
−0.202668 + 0.979248i \(0.564961\pi\)
\(252\) −8.28904 −0.522160
\(253\) 6.20096 0.389851
\(254\) −7.06798 −0.443484
\(255\) 11.2392 0.703826
\(256\) 1.00000 0.0625000
\(257\) −5.64329 −0.352019 −0.176009 0.984389i \(-0.556319\pi\)
−0.176009 + 0.984389i \(0.556319\pi\)
\(258\) 0.273132 0.0170045
\(259\) 6.98489 0.434020
\(260\) 1.41725 0.0878941
\(261\) 76.7708 4.75200
\(262\) 9.79698 0.605259
\(263\) −6.80169 −0.419410 −0.209705 0.977765i \(-0.567250\pi\)
−0.209705 + 0.977765i \(0.567250\pi\)
\(264\) 2.25870 0.139013
\(265\) 8.34405 0.512571
\(266\) 0 0
\(267\) 32.2188 1.97176
\(268\) −12.4558 −0.760862
\(269\) 16.3966 0.999715 0.499858 0.866108i \(-0.333386\pi\)
0.499858 + 0.866108i \(0.333386\pi\)
\(270\) −24.1464 −1.46950
\(271\) −16.7984 −1.02043 −0.510216 0.860046i \(-0.670435\pi\)
−0.510216 + 0.860046i \(0.670435\pi\)
\(272\) −2.46184 −0.149271
\(273\) 3.50451 0.212103
\(274\) −20.2972 −1.22620
\(275\) −2.12009 −0.127846
\(276\) 30.9925 1.86553
\(277\) −18.1789 −1.09227 −0.546133 0.837699i \(-0.683901\pi\)
−0.546133 + 0.837699i \(0.683901\pi\)
\(278\) 5.15852 0.309387
\(279\) 61.1910 3.66341
\(280\) 1.35877 0.0812023
\(281\) −16.7577 −0.999682 −0.499841 0.866117i \(-0.666608\pi\)
−0.499841 + 0.866117i \(0.666608\pi\)
\(282\) −0.759851 −0.0452485
\(283\) −3.36785 −0.200198 −0.100099 0.994977i \(-0.531916\pi\)
−0.100099 + 0.994977i \(0.531916\pi\)
\(284\) 0.358308 0.0212617
\(285\) 0 0
\(286\) −0.701180 −0.0414616
\(287\) −5.46835 −0.322786
\(288\) 8.28904 0.488436
\(289\) −10.9394 −0.643492
\(290\) −12.5846 −0.738994
\(291\) 44.5560 2.61192
\(292\) 5.86115 0.342998
\(293\) −8.95240 −0.523005 −0.261502 0.965203i \(-0.584218\pi\)
−0.261502 + 0.965203i \(0.584218\pi\)
\(294\) 3.35992 0.195954
\(295\) 13.8735 0.807744
\(296\) −6.98489 −0.405989
\(297\) 11.9464 0.693198
\(298\) 9.78505 0.566833
\(299\) −9.62117 −0.556406
\(300\) −10.5963 −0.611776
\(301\) −0.0812913 −0.00468556
\(302\) 6.87697 0.395725
\(303\) 62.2339 3.57525
\(304\) 0 0
\(305\) 12.3294 0.705979
\(306\) −20.4063 −1.16655
\(307\) 10.2203 0.583303 0.291651 0.956525i \(-0.405795\pi\)
0.291651 + 0.956525i \(0.405795\pi\)
\(308\) −0.672249 −0.0383050
\(309\) −26.5076 −1.50796
\(310\) −10.0307 −0.569705
\(311\) −11.7748 −0.667688 −0.333844 0.942628i \(-0.608346\pi\)
−0.333844 + 0.942628i \(0.608346\pi\)
\(312\) −3.50451 −0.198404
\(313\) −8.34493 −0.471683 −0.235842 0.971792i \(-0.575785\pi\)
−0.235842 + 0.971792i \(0.575785\pi\)
\(314\) 12.1056 0.683159
\(315\) 11.2629 0.634594
\(316\) 10.0492 0.565311
\(317\) −21.3660 −1.20003 −0.600016 0.799988i \(-0.704839\pi\)
−0.600016 + 0.799988i \(0.704839\pi\)
\(318\) −20.6328 −1.15703
\(319\) 6.22619 0.348600
\(320\) −1.35877 −0.0759578
\(321\) −18.3115 −1.02205
\(322\) −9.22420 −0.514045
\(323\) 0 0
\(324\) 34.8410 1.93561
\(325\) 3.28945 0.182466
\(326\) −11.3384 −0.627974
\(327\) 1.76662 0.0976941
\(328\) 5.46835 0.301939
\(329\) 0.226152 0.0124681
\(330\) −3.06907 −0.168947
\(331\) 12.2945 0.675769 0.337885 0.941188i \(-0.390289\pi\)
0.337885 + 0.941188i \(0.390289\pi\)
\(332\) 2.73042 0.149851
\(333\) −57.8980 −3.17279
\(334\) −11.8009 −0.645718
\(335\) 16.9247 0.924695
\(336\) −3.35992 −0.183298
\(337\) 29.3162 1.59695 0.798477 0.602025i \(-0.205639\pi\)
0.798477 + 0.602025i \(0.205639\pi\)
\(338\) −11.9121 −0.647932
\(339\) −58.3208 −3.16755
\(340\) 3.34508 0.181413
\(341\) 4.96265 0.268743
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0.0812913 0.00438294
\(345\) −42.1119 −2.26723
\(346\) 18.0756 0.971752
\(347\) 4.66549 0.250457 0.125228 0.992128i \(-0.460034\pi\)
0.125228 + 0.992128i \(0.460034\pi\)
\(348\) 31.1186 1.66813
\(349\) 10.9979 0.588705 0.294353 0.955697i \(-0.404896\pi\)
0.294353 + 0.955697i \(0.404896\pi\)
\(350\) 3.15373 0.168574
\(351\) −18.5355 −0.989351
\(352\) 0.672249 0.0358310
\(353\) −8.33356 −0.443551 −0.221775 0.975098i \(-0.571185\pi\)
−0.221775 + 0.975098i \(0.571185\pi\)
\(354\) −34.3057 −1.82333
\(355\) −0.486860 −0.0258398
\(356\) 9.58916 0.508225
\(357\) 8.27156 0.437778
\(358\) 4.30866 0.227720
\(359\) −4.47179 −0.236012 −0.118006 0.993013i \(-0.537650\pi\)
−0.118006 + 0.993013i \(0.537650\pi\)
\(360\) −11.2629 −0.593609
\(361\) 0 0
\(362\) 3.51816 0.184910
\(363\) −35.4407 −1.86015
\(364\) 1.04304 0.0546699
\(365\) −7.96399 −0.416854
\(366\) −30.4876 −1.59361
\(367\) −6.11166 −0.319026 −0.159513 0.987196i \(-0.550992\pi\)
−0.159513 + 0.987196i \(0.550992\pi\)
\(368\) 9.22420 0.480845
\(369\) 45.3273 2.35965
\(370\) 9.49090 0.493408
\(371\) 6.14086 0.318818
\(372\) 24.8035 1.28600
\(373\) 12.6405 0.654502 0.327251 0.944937i \(-0.393878\pi\)
0.327251 + 0.944937i \(0.393878\pi\)
\(374\) −1.65497 −0.0855763
\(375\) 37.2248 1.92228
\(376\) −0.226152 −0.0116629
\(377\) −9.66032 −0.497532
\(378\) −17.7707 −0.914027
\(379\) 1.93824 0.0995607 0.0497803 0.998760i \(-0.484148\pi\)
0.0497803 + 0.998760i \(0.484148\pi\)
\(380\) 0 0
\(381\) −23.7478 −1.21664
\(382\) 15.7576 0.806229
\(383\) −24.3608 −1.24478 −0.622388 0.782709i \(-0.713837\pi\)
−0.622388 + 0.782709i \(0.713837\pi\)
\(384\) 3.35992 0.171460
\(385\) 0.913435 0.0465530
\(386\) −11.2836 −0.574322
\(387\) 0.673827 0.0342525
\(388\) 13.2610 0.673228
\(389\) −10.0046 −0.507252 −0.253626 0.967302i \(-0.581623\pi\)
−0.253626 + 0.967302i \(0.581623\pi\)
\(390\) 4.76184 0.241125
\(391\) −22.7085 −1.14842
\(392\) 1.00000 0.0505076
\(393\) 32.9170 1.66044
\(394\) −14.9533 −0.753339
\(395\) −13.6546 −0.687036
\(396\) 5.57230 0.280019
\(397\) −31.1380 −1.56277 −0.781385 0.624050i \(-0.785486\pi\)
−0.781385 + 0.624050i \(0.785486\pi\)
\(398\) −1.85467 −0.0929661
\(399\) 0 0
\(400\) −3.15373 −0.157687
\(401\) 4.25458 0.212464 0.106232 0.994341i \(-0.466121\pi\)
0.106232 + 0.994341i \(0.466121\pi\)
\(402\) −41.8506 −2.08732
\(403\) −7.69986 −0.383557
\(404\) 18.5225 0.921527
\(405\) −47.3411 −2.35240
\(406\) −9.26173 −0.459652
\(407\) −4.69559 −0.232752
\(408\) −8.27156 −0.409503
\(409\) 35.8582 1.77307 0.886537 0.462658i \(-0.153104\pi\)
0.886537 + 0.462658i \(0.153104\pi\)
\(410\) −7.43025 −0.366954
\(411\) −68.1969 −3.36391
\(412\) −7.88935 −0.388681
\(413\) 10.2103 0.502415
\(414\) 76.4597 3.75779
\(415\) −3.71002 −0.182118
\(416\) −1.04304 −0.0511390
\(417\) 17.3322 0.848761
\(418\) 0 0
\(419\) −21.8254 −1.06624 −0.533121 0.846039i \(-0.678981\pi\)
−0.533121 + 0.846039i \(0.678981\pi\)
\(420\) 4.56537 0.222767
\(421\) 27.7218 1.35108 0.675540 0.737324i \(-0.263911\pi\)
0.675540 + 0.737324i \(0.263911\pi\)
\(422\) −27.3986 −1.33374
\(423\) −1.87458 −0.0911452
\(424\) −6.14086 −0.298227
\(425\) 7.76397 0.376608
\(426\) 1.20389 0.0583284
\(427\) 9.07391 0.439117
\(428\) −5.44998 −0.263435
\(429\) −2.35591 −0.113744
\(430\) −0.110457 −0.00532669
\(431\) −28.6460 −1.37983 −0.689915 0.723890i \(-0.742352\pi\)
−0.689915 + 0.723890i \(0.742352\pi\)
\(432\) 17.7707 0.854994
\(433\) 3.35163 0.161069 0.0805345 0.996752i \(-0.474337\pi\)
0.0805345 + 0.996752i \(0.474337\pi\)
\(434\) −7.38217 −0.354355
\(435\) −42.2832 −2.02733
\(436\) 0.525792 0.0251809
\(437\) 0 0
\(438\) 19.6930 0.940967
\(439\) 34.8282 1.66226 0.831130 0.556078i \(-0.187694\pi\)
0.831130 + 0.556078i \(0.187694\pi\)
\(440\) −0.913435 −0.0435463
\(441\) 8.28904 0.394716
\(442\) 2.56778 0.122137
\(443\) 27.7162 1.31683 0.658417 0.752653i \(-0.271226\pi\)
0.658417 + 0.752653i \(0.271226\pi\)
\(444\) −23.4687 −1.11377
\(445\) −13.0295 −0.617658
\(446\) −1.56448 −0.0740804
\(447\) 32.8770 1.55503
\(448\) −1.00000 −0.0472456
\(449\) −2.10506 −0.0993438 −0.0496719 0.998766i \(-0.515818\pi\)
−0.0496719 + 0.998766i \(0.515818\pi\)
\(450\) −26.1414 −1.23232
\(451\) 3.67609 0.173100
\(452\) −17.3578 −0.816443
\(453\) 23.1061 1.08562
\(454\) −24.2868 −1.13984
\(455\) −1.41725 −0.0664417
\(456\) 0 0
\(457\) −13.1492 −0.615093 −0.307547 0.951533i \(-0.599508\pi\)
−0.307547 + 0.951533i \(0.599508\pi\)
\(458\) 12.8094 0.598546
\(459\) −43.7486 −2.04201
\(460\) −12.5336 −0.584383
\(461\) −13.9604 −0.650200 −0.325100 0.945680i \(-0.605398\pi\)
−0.325100 + 0.945680i \(0.605398\pi\)
\(462\) −2.25870 −0.105084
\(463\) −5.45782 −0.253647 −0.126823 0.991925i \(-0.540478\pi\)
−0.126823 + 0.991925i \(0.540478\pi\)
\(464\) 9.26173 0.429965
\(465\) −33.7023 −1.56291
\(466\) −16.1943 −0.750184
\(467\) 40.0931 1.85529 0.927643 0.373469i \(-0.121832\pi\)
0.927643 + 0.373469i \(0.121832\pi\)
\(468\) −8.64576 −0.399650
\(469\) 12.4558 0.575158
\(470\) 0.307289 0.0141742
\(471\) 40.6738 1.87415
\(472\) −10.2103 −0.469966
\(473\) 0.0546480 0.00251272
\(474\) 33.7644 1.55085
\(475\) 0 0
\(476\) 2.46184 0.112838
\(477\) −50.9018 −2.33063
\(478\) −11.1682 −0.510821
\(479\) −14.5050 −0.662751 −0.331375 0.943499i \(-0.607513\pi\)
−0.331375 + 0.943499i \(0.607513\pi\)
\(480\) −4.56537 −0.208380
\(481\) 7.28549 0.332190
\(482\) −18.5754 −0.846087
\(483\) −30.9925 −1.41021
\(484\) −10.5481 −0.479458
\(485\) −18.0188 −0.818190
\(486\) 63.7507 2.89179
\(487\) −6.25818 −0.283585 −0.141793 0.989896i \(-0.545287\pi\)
−0.141793 + 0.989896i \(0.545287\pi\)
\(488\) −9.07391 −0.410756
\(489\) −38.0959 −1.72276
\(490\) −1.35877 −0.0613832
\(491\) −24.5017 −1.10575 −0.552873 0.833266i \(-0.686468\pi\)
−0.552873 + 0.833266i \(0.686468\pi\)
\(492\) 18.3732 0.828327
\(493\) −22.8009 −1.02690
\(494\) 0 0
\(495\) −7.57150 −0.340314
\(496\) 7.38217 0.331469
\(497\) −0.358308 −0.0160723
\(498\) 9.17397 0.411096
\(499\) 3.36082 0.150451 0.0752255 0.997167i \(-0.476032\pi\)
0.0752255 + 0.997167i \(0.476032\pi\)
\(500\) 11.0791 0.495472
\(501\) −39.6501 −1.77144
\(502\) −6.42172 −0.286615
\(503\) −17.1646 −0.765332 −0.382666 0.923887i \(-0.624994\pi\)
−0.382666 + 0.923887i \(0.624994\pi\)
\(504\) −8.28904 −0.369223
\(505\) −25.1679 −1.11996
\(506\) 6.20096 0.275666
\(507\) −40.0236 −1.77751
\(508\) −7.06798 −0.313591
\(509\) 34.1490 1.51363 0.756813 0.653631i \(-0.226755\pi\)
0.756813 + 0.653631i \(0.226755\pi\)
\(510\) 11.2392 0.497680
\(511\) −5.86115 −0.259282
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −5.64329 −0.248915
\(515\) 10.7199 0.472373
\(516\) 0.273132 0.0120240
\(517\) −0.152030 −0.00668629
\(518\) 6.98489 0.306899
\(519\) 60.7326 2.66586
\(520\) 1.41725 0.0621505
\(521\) −5.10628 −0.223710 −0.111855 0.993725i \(-0.535679\pi\)
−0.111855 + 0.993725i \(0.535679\pi\)
\(522\) 76.7708 3.36017
\(523\) −9.09139 −0.397539 −0.198769 0.980046i \(-0.563694\pi\)
−0.198769 + 0.980046i \(0.563694\pi\)
\(524\) 9.79698 0.427983
\(525\) 10.5963 0.462459
\(526\) −6.80169 −0.296568
\(527\) −18.1737 −0.791658
\(528\) 2.25870 0.0982973
\(529\) 62.0858 2.69938
\(530\) 8.34405 0.362442
\(531\) −84.6333 −3.67277
\(532\) 0 0
\(533\) −5.70368 −0.247054
\(534\) 32.2188 1.39424
\(535\) 7.40529 0.320159
\(536\) −12.4558 −0.538011
\(537\) 14.4767 0.624717
\(538\) 16.3966 0.706906
\(539\) 0.672249 0.0289558
\(540\) −24.1464 −1.03910
\(541\) 26.4006 1.13505 0.567525 0.823356i \(-0.307901\pi\)
0.567525 + 0.823356i \(0.307901\pi\)
\(542\) −16.7984 −0.721555
\(543\) 11.8207 0.507276
\(544\) −2.46184 −0.105550
\(545\) −0.714433 −0.0306029
\(546\) 3.50451 0.149979
\(547\) −7.01748 −0.300046 −0.150023 0.988683i \(-0.547935\pi\)
−0.150023 + 0.988683i \(0.547935\pi\)
\(548\) −20.2972 −0.867054
\(549\) −75.2139 −3.21005
\(550\) −2.12009 −0.0904011
\(551\) 0 0
\(552\) 30.9925 1.31913
\(553\) −10.0492 −0.427335
\(554\) −18.1789 −0.772348
\(555\) 31.8886 1.35360
\(556\) 5.15852 0.218770
\(557\) 15.3486 0.650342 0.325171 0.945655i \(-0.394578\pi\)
0.325171 + 0.945655i \(0.394578\pi\)
\(558\) 61.1910 2.59042
\(559\) −0.0847898 −0.00358622
\(560\) 1.35877 0.0574187
\(561\) −5.56055 −0.234767
\(562\) −16.7577 −0.706882
\(563\) 5.92162 0.249567 0.124783 0.992184i \(-0.460176\pi\)
0.124783 + 0.992184i \(0.460176\pi\)
\(564\) −0.759851 −0.0319955
\(565\) 23.5854 0.992244
\(566\) −3.36785 −0.141561
\(567\) −34.8410 −1.46318
\(568\) 0.358308 0.0150343
\(569\) −23.2519 −0.974770 −0.487385 0.873187i \(-0.662049\pi\)
−0.487385 + 0.873187i \(0.662049\pi\)
\(570\) 0 0
\(571\) −19.4012 −0.811916 −0.405958 0.913892i \(-0.633062\pi\)
−0.405958 + 0.913892i \(0.633062\pi\)
\(572\) −0.701180 −0.0293178
\(573\) 52.9442 2.21178
\(574\) −5.46835 −0.228244
\(575\) −29.0906 −1.21316
\(576\) 8.28904 0.345376
\(577\) −2.39813 −0.0998353 −0.0499177 0.998753i \(-0.515896\pi\)
−0.0499177 + 0.998753i \(0.515896\pi\)
\(578\) −10.9394 −0.455017
\(579\) −37.9121 −1.57557
\(580\) −12.5846 −0.522547
\(581\) −2.73042 −0.113277
\(582\) 44.5560 1.84691
\(583\) −4.12819 −0.170972
\(584\) 5.86115 0.242536
\(585\) 11.7476 0.485705
\(586\) −8.95240 −0.369820
\(587\) −2.32808 −0.0960903 −0.0480452 0.998845i \(-0.515299\pi\)
−0.0480452 + 0.998845i \(0.515299\pi\)
\(588\) 3.35992 0.138561
\(589\) 0 0
\(590\) 13.8735 0.571161
\(591\) −50.2420 −2.06668
\(592\) −6.98489 −0.287077
\(593\) 9.50805 0.390449 0.195224 0.980759i \(-0.437457\pi\)
0.195224 + 0.980759i \(0.437457\pi\)
\(594\) 11.9464 0.490165
\(595\) −3.34508 −0.137135
\(596\) 9.78505 0.400811
\(597\) −6.23153 −0.255040
\(598\) −9.62117 −0.393439
\(599\) −11.9202 −0.487045 −0.243523 0.969895i \(-0.578303\pi\)
−0.243523 + 0.969895i \(0.578303\pi\)
\(600\) −10.5963 −0.432591
\(601\) −35.8525 −1.46245 −0.731227 0.682134i \(-0.761052\pi\)
−0.731227 + 0.682134i \(0.761052\pi\)
\(602\) −0.0812913 −0.00331319
\(603\) −103.247 −4.20454
\(604\) 6.87697 0.279820
\(605\) 14.3325 0.582698
\(606\) 62.2339 2.52808
\(607\) 16.0032 0.649549 0.324774 0.945792i \(-0.394712\pi\)
0.324774 + 0.945792i \(0.394712\pi\)
\(608\) 0 0
\(609\) −31.1186 −1.26099
\(610\) 12.3294 0.499203
\(611\) 0.235884 0.00954286
\(612\) −20.4063 −0.824874
\(613\) 14.1692 0.572290 0.286145 0.958186i \(-0.407626\pi\)
0.286145 + 0.958186i \(0.407626\pi\)
\(614\) 10.2203 0.412457
\(615\) −24.9650 −1.00669
\(616\) −0.672249 −0.0270857
\(617\) −27.0031 −1.08710 −0.543552 0.839375i \(-0.682921\pi\)
−0.543552 + 0.839375i \(0.682921\pi\)
\(618\) −26.5076 −1.06629
\(619\) −31.9044 −1.28235 −0.641173 0.767397i \(-0.721552\pi\)
−0.641173 + 0.767397i \(0.721552\pi\)
\(620\) −10.0307 −0.402843
\(621\) 163.921 6.57791
\(622\) −11.7748 −0.472126
\(623\) −9.58916 −0.384182
\(624\) −3.50451 −0.140293
\(625\) 0.714673 0.0285869
\(626\) −8.34493 −0.333530
\(627\) 0 0
\(628\) 12.1056 0.483066
\(629\) 17.1957 0.685636
\(630\) 11.2629 0.448726
\(631\) −19.9626 −0.794699 −0.397350 0.917667i \(-0.630070\pi\)
−0.397350 + 0.917667i \(0.630070\pi\)
\(632\) 10.0492 0.399735
\(633\) −92.0571 −3.65894
\(634\) −21.3660 −0.848551
\(635\) 9.60379 0.381115
\(636\) −20.6328 −0.818143
\(637\) −1.04304 −0.0413266
\(638\) 6.22619 0.246497
\(639\) 2.97003 0.117493
\(640\) −1.35877 −0.0537103
\(641\) 14.7339 0.581953 0.290976 0.956730i \(-0.406020\pi\)
0.290976 + 0.956730i \(0.406020\pi\)
\(642\) −18.3115 −0.722696
\(643\) 31.6423 1.24785 0.623926 0.781484i \(-0.285537\pi\)
0.623926 + 0.781484i \(0.285537\pi\)
\(644\) −9.22420 −0.363484
\(645\) −0.371125 −0.0146130
\(646\) 0 0
\(647\) 4.39065 0.172614 0.0863072 0.996269i \(-0.472493\pi\)
0.0863072 + 0.996269i \(0.472493\pi\)
\(648\) 34.8410 1.36868
\(649\) −6.86385 −0.269430
\(650\) 3.28945 0.129023
\(651\) −24.8035 −0.972124
\(652\) −11.3384 −0.444045
\(653\) 9.91627 0.388054 0.194027 0.980996i \(-0.437845\pi\)
0.194027 + 0.980996i \(0.437845\pi\)
\(654\) 1.76662 0.0690802
\(655\) −13.3119 −0.520138
\(656\) 5.46835 0.213503
\(657\) 48.5833 1.89542
\(658\) 0.226152 0.00881631
\(659\) 19.9626 0.777633 0.388816 0.921315i \(-0.372884\pi\)
0.388816 + 0.921315i \(0.372884\pi\)
\(660\) −3.06907 −0.119463
\(661\) −9.49464 −0.369299 −0.184649 0.982804i \(-0.559115\pi\)
−0.184649 + 0.982804i \(0.559115\pi\)
\(662\) 12.2945 0.477841
\(663\) 8.62753 0.335066
\(664\) 2.73042 0.105961
\(665\) 0 0
\(666\) −57.8980 −2.24350
\(667\) 85.4321 3.30794
\(668\) −11.8009 −0.456592
\(669\) −5.25653 −0.203229
\(670\) 16.9247 0.653858
\(671\) −6.09993 −0.235485
\(672\) −3.35992 −0.129612
\(673\) 48.6884 1.87680 0.938400 0.345552i \(-0.112308\pi\)
0.938400 + 0.345552i \(0.112308\pi\)
\(674\) 29.3162 1.12922
\(675\) −56.0441 −2.15714
\(676\) −11.9121 −0.458157
\(677\) 35.6077 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(678\) −58.3208 −2.23980
\(679\) −13.2610 −0.508912
\(680\) 3.34508 0.128278
\(681\) −81.6016 −3.12698
\(682\) 4.96265 0.190030
\(683\) −1.59061 −0.0608629 −0.0304314 0.999537i \(-0.509688\pi\)
−0.0304314 + 0.999537i \(0.509688\pi\)
\(684\) 0 0
\(685\) 27.5793 1.05375
\(686\) −1.00000 −0.0381802
\(687\) 43.0386 1.64203
\(688\) 0.0812913 0.00309920
\(689\) 6.40514 0.244016
\(690\) −42.1119 −1.60317
\(691\) 7.59300 0.288851 0.144426 0.989516i \(-0.453867\pi\)
0.144426 + 0.989516i \(0.453867\pi\)
\(692\) 18.0756 0.687132
\(693\) −5.57230 −0.211674
\(694\) 4.66549 0.177099
\(695\) −7.00926 −0.265877
\(696\) 31.1186 1.17955
\(697\) −13.4622 −0.509916
\(698\) 10.9979 0.416278
\(699\) −54.4113 −2.05803
\(700\) 3.15373 0.119200
\(701\) 20.5770 0.777183 0.388592 0.921410i \(-0.372962\pi\)
0.388592 + 0.921410i \(0.372962\pi\)
\(702\) −18.5355 −0.699577
\(703\) 0 0
\(704\) 0.672249 0.0253363
\(705\) 1.03247 0.0388849
\(706\) −8.33356 −0.313638
\(707\) −18.5225 −0.696609
\(708\) −34.3057 −1.28929
\(709\) 16.2673 0.610931 0.305465 0.952203i \(-0.401188\pi\)
0.305465 + 0.952203i \(0.401188\pi\)
\(710\) −0.486860 −0.0182715
\(711\) 83.2980 3.12392
\(712\) 9.58916 0.359369
\(713\) 68.0946 2.55016
\(714\) 8.27156 0.309555
\(715\) 0.952745 0.0356307
\(716\) 4.30866 0.161022
\(717\) −37.5242 −1.40137
\(718\) −4.47179 −0.166886
\(719\) 49.1755 1.83394 0.916969 0.398959i \(-0.130628\pi\)
0.916969 + 0.398959i \(0.130628\pi\)
\(720\) −11.2629 −0.419745
\(721\) 7.88935 0.293815
\(722\) 0 0
\(723\) −62.4119 −2.32112
\(724\) 3.51816 0.130751
\(725\) −29.2090 −1.08480
\(726\) −35.4407 −1.31533
\(727\) −37.8702 −1.40453 −0.702264 0.711917i \(-0.747827\pi\)
−0.702264 + 0.711917i \(0.747827\pi\)
\(728\) 1.04304 0.0386575
\(729\) 109.674 4.06200
\(730\) −7.96399 −0.294760
\(731\) −0.200126 −0.00740193
\(732\) −30.4876 −1.12685
\(733\) −46.4071 −1.71409 −0.857044 0.515244i \(-0.827701\pi\)
−0.857044 + 0.515244i \(0.827701\pi\)
\(734\) −6.11166 −0.225585
\(735\) −4.56537 −0.168396
\(736\) 9.22420 0.340008
\(737\) −8.37343 −0.308439
\(738\) 45.3273 1.66852
\(739\) 36.1093 1.32830 0.664151 0.747598i \(-0.268793\pi\)
0.664151 + 0.747598i \(0.268793\pi\)
\(740\) 9.49090 0.348892
\(741\) 0 0
\(742\) 6.14086 0.225438
\(743\) 21.2678 0.780241 0.390120 0.920764i \(-0.372433\pi\)
0.390120 + 0.920764i \(0.372433\pi\)
\(744\) 24.8035 0.909339
\(745\) −13.2957 −0.487116
\(746\) 12.6405 0.462803
\(747\) 22.6325 0.828081
\(748\) −1.65497 −0.0605116
\(749\) 5.44998 0.199138
\(750\) 37.2248 1.35926
\(751\) 7.42960 0.271110 0.135555 0.990770i \(-0.456718\pi\)
0.135555 + 0.990770i \(0.456718\pi\)
\(752\) −0.226152 −0.00824690
\(753\) −21.5764 −0.786289
\(754\) −9.66032 −0.351808
\(755\) −9.34426 −0.340072
\(756\) −17.7707 −0.646315
\(757\) −30.1366 −1.09533 −0.547666 0.836697i \(-0.684484\pi\)
−0.547666 + 0.836697i \(0.684484\pi\)
\(758\) 1.93824 0.0704000
\(759\) 20.8347 0.756252
\(760\) 0 0
\(761\) −35.5985 −1.29044 −0.645222 0.763995i \(-0.723235\pi\)
−0.645222 + 0.763995i \(0.723235\pi\)
\(762\) −23.7478 −0.860292
\(763\) −0.525792 −0.0190349
\(764\) 15.7576 0.570090
\(765\) 27.7275 1.00249
\(766\) −24.3608 −0.880190
\(767\) 10.6497 0.384537
\(768\) 3.35992 0.121241
\(769\) 29.7011 1.07105 0.535525 0.844519i \(-0.320114\pi\)
0.535525 + 0.844519i \(0.320114\pi\)
\(770\) 0.913435 0.0329179
\(771\) −18.9610 −0.682863
\(772\) −11.2836 −0.406107
\(773\) 22.7091 0.816790 0.408395 0.912805i \(-0.366089\pi\)
0.408395 + 0.912805i \(0.366089\pi\)
\(774\) 0.673827 0.0242202
\(775\) −23.2814 −0.836291
\(776\) 13.2610 0.476044
\(777\) 23.4687 0.841933
\(778\) −10.0046 −0.358682
\(779\) 0 0
\(780\) 4.76184 0.170501
\(781\) 0.240872 0.00861909
\(782\) −22.7085 −0.812053
\(783\) 164.588 5.88188
\(784\) 1.00000 0.0357143
\(785\) −16.4488 −0.587083
\(786\) 32.9170 1.17411
\(787\) −9.88614 −0.352403 −0.176201 0.984354i \(-0.556381\pi\)
−0.176201 + 0.984354i \(0.556381\pi\)
\(788\) −14.9533 −0.532691
\(789\) −22.8531 −0.813592
\(790\) −13.6546 −0.485808
\(791\) 17.3578 0.617173
\(792\) 5.57230 0.198003
\(793\) 9.46440 0.336091
\(794\) −31.1380 −1.10504
\(795\) 28.0353 0.994310
\(796\) −1.85467 −0.0657370
\(797\) −13.4346 −0.475879 −0.237939 0.971280i \(-0.576472\pi\)
−0.237939 + 0.971280i \(0.576472\pi\)
\(798\) 0 0
\(799\) 0.556749 0.0196963
\(800\) −3.15373 −0.111501
\(801\) 79.4849 2.80846
\(802\) 4.25458 0.150234
\(803\) 3.94016 0.139045
\(804\) −41.8506 −1.47596
\(805\) 12.5336 0.441752
\(806\) −7.69986 −0.271216
\(807\) 55.0910 1.93930
\(808\) 18.5225 0.651618
\(809\) −29.6717 −1.04320 −0.521601 0.853190i \(-0.674665\pi\)
−0.521601 + 0.853190i \(0.674665\pi\)
\(810\) −47.3411 −1.66340
\(811\) 46.9132 1.64734 0.823672 0.567067i \(-0.191922\pi\)
0.823672 + 0.567067i \(0.191922\pi\)
\(812\) −9.26173 −0.325023
\(813\) −56.4413 −1.97948
\(814\) −4.69559 −0.164580
\(815\) 15.4063 0.539658
\(816\) −8.27156 −0.289563
\(817\) 0 0
\(818\) 35.8582 1.25375
\(819\) 8.64576 0.302107
\(820\) −7.43025 −0.259476
\(821\) 8.20066 0.286205 0.143103 0.989708i \(-0.454292\pi\)
0.143103 + 0.989708i \(0.454292\pi\)
\(822\) −68.1969 −2.37864
\(823\) −13.2650 −0.462388 −0.231194 0.972908i \(-0.574263\pi\)
−0.231194 + 0.972908i \(0.574263\pi\)
\(824\) −7.88935 −0.274839
\(825\) −7.12334 −0.248003
\(826\) 10.2103 0.355261
\(827\) 7.03178 0.244519 0.122260 0.992498i \(-0.460986\pi\)
0.122260 + 0.992498i \(0.460986\pi\)
\(828\) 76.4597 2.65716
\(829\) −9.85280 −0.342202 −0.171101 0.985254i \(-0.554732\pi\)
−0.171101 + 0.985254i \(0.554732\pi\)
\(830\) −3.71002 −0.128777
\(831\) −61.0796 −2.11883
\(832\) −1.04304 −0.0361607
\(833\) −2.46184 −0.0852976
\(834\) 17.3322 0.600164
\(835\) 16.0348 0.554907
\(836\) 0 0
\(837\) 131.186 4.53446
\(838\) −21.8254 −0.753946
\(839\) −23.8269 −0.822595 −0.411298 0.911501i \(-0.634924\pi\)
−0.411298 + 0.911501i \(0.634924\pi\)
\(840\) 4.56537 0.157520
\(841\) 56.7797 1.95792
\(842\) 27.7218 0.955358
\(843\) −56.3046 −1.93923
\(844\) −27.3986 −0.943100
\(845\) 16.1858 0.556809
\(846\) −1.87458 −0.0644494
\(847\) 10.5481 0.362436
\(848\) −6.14086 −0.210878
\(849\) −11.3157 −0.388353
\(850\) 7.76397 0.266302
\(851\) −64.4301 −2.20863
\(852\) 1.20389 0.0412444
\(853\) −16.3690 −0.560465 −0.280233 0.959932i \(-0.590412\pi\)
−0.280233 + 0.959932i \(0.590412\pi\)
\(854\) 9.07391 0.310503
\(855\) 0 0
\(856\) −5.44998 −0.186276
\(857\) −14.3213 −0.489207 −0.244604 0.969623i \(-0.578658\pi\)
−0.244604 + 0.969623i \(0.578658\pi\)
\(858\) −2.35591 −0.0804293
\(859\) −6.06335 −0.206879 −0.103439 0.994636i \(-0.532985\pi\)
−0.103439 + 0.994636i \(0.532985\pi\)
\(860\) −0.110457 −0.00376654
\(861\) −18.3732 −0.626156
\(862\) −28.6460 −0.975688
\(863\) −34.3964 −1.17087 −0.585434 0.810720i \(-0.699076\pi\)
−0.585434 + 0.810720i \(0.699076\pi\)
\(864\) 17.7707 0.604572
\(865\) −24.5607 −0.835089
\(866\) 3.35163 0.113893
\(867\) −36.7553 −1.24828
\(868\) −7.38217 −0.250567
\(869\) 6.75555 0.229166
\(870\) −42.2832 −1.43354
\(871\) 12.9919 0.440213
\(872\) 0.525792 0.0178056
\(873\) 109.921 3.72027
\(874\) 0 0
\(875\) −11.0791 −0.374541
\(876\) 19.6930 0.665364
\(877\) 9.81095 0.331292 0.165646 0.986185i \(-0.447029\pi\)
0.165646 + 0.986185i \(0.447029\pi\)
\(878\) 34.8282 1.17540
\(879\) −30.0793 −1.01455
\(880\) −0.913435 −0.0307919
\(881\) −29.8319 −1.00506 −0.502531 0.864559i \(-0.667598\pi\)
−0.502531 + 0.864559i \(0.667598\pi\)
\(882\) 8.28904 0.279106
\(883\) −25.5339 −0.859283 −0.429642 0.902999i \(-0.641360\pi\)
−0.429642 + 0.902999i \(0.641360\pi\)
\(884\) 2.56778 0.0863639
\(885\) 46.6137 1.56690
\(886\) 27.7162 0.931143
\(887\) 1.66452 0.0558893 0.0279446 0.999609i \(-0.491104\pi\)
0.0279446 + 0.999609i \(0.491104\pi\)
\(888\) −23.4687 −0.787557
\(889\) 7.06798 0.237052
\(890\) −13.0295 −0.436750
\(891\) 23.4218 0.784661
\(892\) −1.56448 −0.0523828
\(893\) 0 0
\(894\) 32.8770 1.09957
\(895\) −5.85450 −0.195694
\(896\) −1.00000 −0.0334077
\(897\) −32.3263 −1.07934
\(898\) −2.10506 −0.0702467
\(899\) 68.3716 2.28032
\(900\) −26.1414 −0.871380
\(901\) 15.1178 0.503647
\(902\) 3.67609 0.122400
\(903\) −0.273132 −0.00908927
\(904\) −17.3578 −0.577313
\(905\) −4.78039 −0.158906
\(906\) 23.1061 0.767647
\(907\) 42.3441 1.40601 0.703007 0.711183i \(-0.251840\pi\)
0.703007 + 0.711183i \(0.251840\pi\)
\(908\) −24.2868 −0.805986
\(909\) 153.533 5.09238
\(910\) −1.41725 −0.0469814
\(911\) 2.81275 0.0931905 0.0465953 0.998914i \(-0.485163\pi\)
0.0465953 + 0.998914i \(0.485163\pi\)
\(912\) 0 0
\(913\) 1.83552 0.0607469
\(914\) −13.1492 −0.434937
\(915\) 41.4257 1.36949
\(916\) 12.8094 0.423236
\(917\) −9.79698 −0.323525
\(918\) −43.7486 −1.44392
\(919\) 42.3008 1.39537 0.697687 0.716403i \(-0.254213\pi\)
0.697687 + 0.716403i \(0.254213\pi\)
\(920\) −12.5336 −0.413221
\(921\) 34.3393 1.13152
\(922\) −13.9604 −0.459761
\(923\) −0.373728 −0.0123014
\(924\) −2.25870 −0.0743058
\(925\) 22.0285 0.724292
\(926\) −5.45782 −0.179355
\(927\) −65.3951 −2.14786
\(928\) 9.26173 0.304031
\(929\) 23.2939 0.764247 0.382123 0.924111i \(-0.375193\pi\)
0.382123 + 0.924111i \(0.375193\pi\)
\(930\) −33.7023 −1.10514
\(931\) 0 0
\(932\) −16.1943 −0.530460
\(933\) −39.5623 −1.29521
\(934\) 40.0931 1.31188
\(935\) 2.24873 0.0735413
\(936\) −8.64576 −0.282595
\(937\) −59.3078 −1.93750 −0.968750 0.248038i \(-0.920214\pi\)
−0.968750 + 0.248038i \(0.920214\pi\)
\(938\) 12.4558 0.406698
\(939\) −28.0382 −0.914994
\(940\) 0.307289 0.0100227
\(941\) −0.727116 −0.0237033 −0.0118517 0.999930i \(-0.503773\pi\)
−0.0118517 + 0.999930i \(0.503773\pi\)
\(942\) 40.6738 1.32522
\(943\) 50.4411 1.64259
\(944\) −10.2103 −0.332316
\(945\) 24.1464 0.785483
\(946\) 0.0546480 0.00177676
\(947\) 22.0588 0.716814 0.358407 0.933565i \(-0.383320\pi\)
0.358407 + 0.933565i \(0.383320\pi\)
\(948\) 33.7644 1.09662
\(949\) −6.11339 −0.198449
\(950\) 0 0
\(951\) −71.7879 −2.32788
\(952\) 2.46184 0.0797886
\(953\) 32.6219 1.05673 0.528364 0.849018i \(-0.322806\pi\)
0.528364 + 0.849018i \(0.322806\pi\)
\(954\) −50.9018 −1.64801
\(955\) −21.4110 −0.692844
\(956\) −11.1682 −0.361205
\(957\) 20.9195 0.676231
\(958\) −14.5050 −0.468636
\(959\) 20.2972 0.655431
\(960\) −4.56537 −0.147347
\(961\) 23.4964 0.757947
\(962\) 7.28549 0.234894
\(963\) −45.1751 −1.45575
\(964\) −18.5754 −0.598274
\(965\) 15.3319 0.493552
\(966\) −30.9925 −0.997168
\(967\) −34.4438 −1.10764 −0.553820 0.832637i \(-0.686830\pi\)
−0.553820 + 0.832637i \(0.686830\pi\)
\(968\) −10.5481 −0.339028
\(969\) 0 0
\(970\) −18.0188 −0.578548
\(971\) 10.0122 0.321307 0.160653 0.987011i \(-0.448640\pi\)
0.160653 + 0.987011i \(0.448640\pi\)
\(972\) 63.7507 2.04481
\(973\) −5.15852 −0.165374
\(974\) −6.25818 −0.200525
\(975\) 11.0523 0.353956
\(976\) −9.07391 −0.290449
\(977\) −0.947408 −0.0303103 −0.0151551 0.999885i \(-0.504824\pi\)
−0.0151551 + 0.999885i \(0.504824\pi\)
\(978\) −38.0959 −1.21817
\(979\) 6.44631 0.206025
\(980\) −1.35877 −0.0434045
\(981\) 4.35831 0.139150
\(982\) −24.5017 −0.781880
\(983\) −19.4148 −0.619237 −0.309619 0.950861i \(-0.600201\pi\)
−0.309619 + 0.950861i \(0.600201\pi\)
\(984\) 18.3732 0.585716
\(985\) 20.3182 0.647393
\(986\) −22.8009 −0.726128
\(987\) 0.759851 0.0241863
\(988\) 0 0
\(989\) 0.749848 0.0238438
\(990\) −7.57150 −0.240638
\(991\) 31.2746 0.993471 0.496736 0.867902i \(-0.334532\pi\)
0.496736 + 0.867902i \(0.334532\pi\)
\(992\) 7.38217 0.234384
\(993\) 41.3086 1.31089
\(994\) −0.358308 −0.0113648
\(995\) 2.52008 0.0798918
\(996\) 9.17397 0.290689
\(997\) 20.5619 0.651202 0.325601 0.945507i \(-0.394433\pi\)
0.325601 + 0.945507i \(0.394433\pi\)
\(998\) 3.36082 0.106385
\(999\) −124.127 −3.92719
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 5054.2.a.bi.1.8 yes 8
19.18 odd 2 5054.2.a.bf.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bf.1.1 8 19.18 odd 2
5054.2.a.bi.1.8 yes 8 1.1 even 1 trivial