Properties

Label 5054.2.a.bg.1.7
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 1.7
Root \(-0.236286\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.04815 q^{3} +1.00000 q^{4} +3.90247 q^{5} -3.04815 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.29119 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.04815 q^{3} +1.00000 q^{4} +3.90247 q^{5} -3.04815 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.29119 q^{9} -3.90247 q^{10} -5.43763 q^{11} +3.04815 q^{12} +2.24823 q^{13} -1.00000 q^{14} +11.8953 q^{15} +1.00000 q^{16} +0.977224 q^{17} -6.29119 q^{18} +3.90247 q^{20} +3.04815 q^{21} +5.43763 q^{22} +0.640603 q^{23} -3.04815 q^{24} +10.2292 q^{25} -2.24823 q^{26} +10.0320 q^{27} +1.00000 q^{28} +6.85823 q^{29} -11.8953 q^{30} +3.00774 q^{31} -1.00000 q^{32} -16.5747 q^{33} -0.977224 q^{34} +3.90247 q^{35} +6.29119 q^{36} +0.824327 q^{37} +6.85294 q^{39} -3.90247 q^{40} +11.3245 q^{41} -3.04815 q^{42} -12.2093 q^{43} -5.43763 q^{44} +24.5512 q^{45} -0.640603 q^{46} -12.6901 q^{47} +3.04815 q^{48} +1.00000 q^{49} -10.2292 q^{50} +2.97872 q^{51} +2.24823 q^{52} -2.05679 q^{53} -10.0320 q^{54} -21.2202 q^{55} -1.00000 q^{56} -6.85823 q^{58} -3.71241 q^{59} +11.8953 q^{60} +3.56439 q^{61} -3.00774 q^{62} +6.29119 q^{63} +1.00000 q^{64} +8.77365 q^{65} +16.5747 q^{66} +1.14843 q^{67} +0.977224 q^{68} +1.95265 q^{69} -3.90247 q^{70} -4.58975 q^{71} -6.29119 q^{72} -6.31633 q^{73} -0.824327 q^{74} +31.1802 q^{75} -5.43763 q^{77} -6.85294 q^{78} +12.3096 q^{79} +3.90247 q^{80} +11.7055 q^{81} -11.3245 q^{82} -9.84224 q^{83} +3.04815 q^{84} +3.81358 q^{85} +12.2093 q^{86} +20.9049 q^{87} +5.43763 q^{88} -11.8845 q^{89} -24.5512 q^{90} +2.24823 q^{91} +0.640603 q^{92} +9.16802 q^{93} +12.6901 q^{94} -3.04815 q^{96} -7.94295 q^{97} -1.00000 q^{98} -34.2092 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) 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
\(3\) 3.04815 1.75985 0.879924 0.475115i \(-0.157593\pi\)
0.879924 + 0.475115i \(0.157593\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.90247 1.74524 0.872618 0.488404i \(-0.162421\pi\)
0.872618 + 0.488404i \(0.162421\pi\)
\(6\) −3.04815 −1.24440
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 6.29119 2.09706
\(10\) −3.90247 −1.23407
\(11\) −5.43763 −1.63951 −0.819754 0.572716i \(-0.805890\pi\)
−0.819754 + 0.572716i \(0.805890\pi\)
\(12\) 3.04815 0.879924
\(13\) 2.24823 0.623548 0.311774 0.950156i \(-0.399077\pi\)
0.311774 + 0.950156i \(0.399077\pi\)
\(14\) −1.00000 −0.267261
\(15\) 11.8953 3.07135
\(16\) 1.00000 0.250000
\(17\) 0.977224 0.237012 0.118506 0.992953i \(-0.462190\pi\)
0.118506 + 0.992953i \(0.462190\pi\)
\(18\) −6.29119 −1.48285
\(19\) 0 0
\(20\) 3.90247 0.872618
\(21\) 3.04815 0.665160
\(22\) 5.43763 1.15931
\(23\) 0.640603 0.133575 0.0667875 0.997767i \(-0.478725\pi\)
0.0667875 + 0.997767i \(0.478725\pi\)
\(24\) −3.04815 −0.622200
\(25\) 10.2292 2.04585
\(26\) −2.24823 −0.440915
\(27\) 10.0320 1.93067
\(28\) 1.00000 0.188982
\(29\) 6.85823 1.27354 0.636771 0.771053i \(-0.280270\pi\)
0.636771 + 0.771053i \(0.280270\pi\)
\(30\) −11.8953 −2.17177
\(31\) 3.00774 0.540205 0.270103 0.962832i \(-0.412942\pi\)
0.270103 + 0.962832i \(0.412942\pi\)
\(32\) −1.00000 −0.176777
\(33\) −16.5747 −2.88528
\(34\) −0.977224 −0.167592
\(35\) 3.90247 0.659637
\(36\) 6.29119 1.04853
\(37\) 0.824327 0.135518 0.0677592 0.997702i \(-0.478415\pi\)
0.0677592 + 0.997702i \(0.478415\pi\)
\(38\) 0 0
\(39\) 6.85294 1.09735
\(40\) −3.90247 −0.617034
\(41\) 11.3245 1.76859 0.884295 0.466928i \(-0.154639\pi\)
0.884295 + 0.466928i \(0.154639\pi\)
\(42\) −3.04815 −0.470339
\(43\) −12.2093 −1.86190 −0.930952 0.365142i \(-0.881020\pi\)
−0.930952 + 0.365142i \(0.881020\pi\)
\(44\) −5.43763 −0.819754
\(45\) 24.5512 3.65987
\(46\) −0.640603 −0.0944517
\(47\) −12.6901 −1.85104 −0.925518 0.378704i \(-0.876370\pi\)
−0.925518 + 0.378704i \(0.876370\pi\)
\(48\) 3.04815 0.439962
\(49\) 1.00000 0.142857
\(50\) −10.2292 −1.44663
\(51\) 2.97872 0.417104
\(52\) 2.24823 0.311774
\(53\) −2.05679 −0.282521 −0.141261 0.989972i \(-0.545116\pi\)
−0.141261 + 0.989972i \(0.545116\pi\)
\(54\) −10.0320 −1.36519
\(55\) −21.2202 −2.86133
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.85823 −0.900530
\(59\) −3.71241 −0.483315 −0.241658 0.970362i \(-0.577691\pi\)
−0.241658 + 0.970362i \(0.577691\pi\)
\(60\) 11.8953 1.53567
\(61\) 3.56439 0.456374 0.228187 0.973617i \(-0.426720\pi\)
0.228187 + 0.973617i \(0.426720\pi\)
\(62\) −3.00774 −0.381983
\(63\) 6.29119 0.792616
\(64\) 1.00000 0.125000
\(65\) 8.77365 1.08824
\(66\) 16.5747 2.04020
\(67\) 1.14843 0.140303 0.0701516 0.997536i \(-0.477652\pi\)
0.0701516 + 0.997536i \(0.477652\pi\)
\(68\) 0.977224 0.118506
\(69\) 1.95265 0.235072
\(70\) −3.90247 −0.466434
\(71\) −4.58975 −0.544703 −0.272352 0.962198i \(-0.587801\pi\)
−0.272352 + 0.962198i \(0.587801\pi\)
\(72\) −6.29119 −0.741424
\(73\) −6.31633 −0.739271 −0.369635 0.929177i \(-0.620517\pi\)
−0.369635 + 0.929177i \(0.620517\pi\)
\(74\) −0.824327 −0.0958260
\(75\) 31.1802 3.60038
\(76\) 0 0
\(77\) −5.43763 −0.619675
\(78\) −6.85294 −0.775943
\(79\) 12.3096 1.38494 0.692471 0.721446i \(-0.256522\pi\)
0.692471 + 0.721446i \(0.256522\pi\)
\(80\) 3.90247 0.436309
\(81\) 11.7055 1.30061
\(82\) −11.3245 −1.25058
\(83\) −9.84224 −1.08033 −0.540163 0.841561i \(-0.681637\pi\)
−0.540163 + 0.841561i \(0.681637\pi\)
\(84\) 3.04815 0.332580
\(85\) 3.81358 0.413641
\(86\) 12.2093 1.31656
\(87\) 20.9049 2.24124
\(88\) 5.43763 0.579653
\(89\) −11.8845 −1.25976 −0.629879 0.776693i \(-0.716896\pi\)
−0.629879 + 0.776693i \(0.716896\pi\)
\(90\) −24.5512 −2.58792
\(91\) 2.24823 0.235679
\(92\) 0.640603 0.0667875
\(93\) 9.16802 0.950679
\(94\) 12.6901 1.30888
\(95\) 0 0
\(96\) −3.04815 −0.311100
\(97\) −7.94295 −0.806485 −0.403242 0.915093i \(-0.632117\pi\)
−0.403242 + 0.915093i \(0.632117\pi\)
\(98\) −1.00000 −0.101015
\(99\) −34.2092 −3.43815
\(100\) 10.2292 1.02292
\(101\) 12.5382 1.24760 0.623799 0.781585i \(-0.285588\pi\)
0.623799 + 0.781585i \(0.285588\pi\)
\(102\) −2.97872 −0.294937
\(103\) −13.4463 −1.32490 −0.662450 0.749106i \(-0.730483\pi\)
−0.662450 + 0.749106i \(0.730483\pi\)
\(104\) −2.24823 −0.220457
\(105\) 11.8953 1.16086
\(106\) 2.05679 0.199773
\(107\) −11.5515 −1.11673 −0.558363 0.829597i \(-0.688570\pi\)
−0.558363 + 0.829597i \(0.688570\pi\)
\(108\) 10.0320 0.965333
\(109\) 18.8662 1.80706 0.903528 0.428528i \(-0.140968\pi\)
0.903528 + 0.428528i \(0.140968\pi\)
\(110\) 21.2202 2.02326
\(111\) 2.51267 0.238492
\(112\) 1.00000 0.0944911
\(113\) −2.61917 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(114\) 0 0
\(115\) 2.49993 0.233120
\(116\) 6.85823 0.636771
\(117\) 14.1441 1.30762
\(118\) 3.71241 0.341755
\(119\) 0.977224 0.0895819
\(120\) −11.8953 −1.08589
\(121\) 18.5678 1.68798
\(122\) −3.56439 −0.322705
\(123\) 34.5188 3.11245
\(124\) 3.00774 0.270103
\(125\) 20.4069 1.82525
\(126\) −6.29119 −0.560464
\(127\) 2.91792 0.258924 0.129462 0.991584i \(-0.458675\pi\)
0.129462 + 0.991584i \(0.458675\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −37.2158 −3.27667
\(130\) −8.77365 −0.769500
\(131\) 0.814699 0.0711805 0.0355903 0.999366i \(-0.488669\pi\)
0.0355903 + 0.999366i \(0.488669\pi\)
\(132\) −16.5747 −1.44264
\(133\) 0 0
\(134\) −1.14843 −0.0992094
\(135\) 39.1497 3.36947
\(136\) −0.977224 −0.0837962
\(137\) −7.30748 −0.624320 −0.312160 0.950029i \(-0.601053\pi\)
−0.312160 + 0.950029i \(0.601053\pi\)
\(138\) −1.95265 −0.166221
\(139\) 15.1627 1.28608 0.643040 0.765833i \(-0.277673\pi\)
0.643040 + 0.765833i \(0.277673\pi\)
\(140\) 3.90247 0.329819
\(141\) −38.6812 −3.25754
\(142\) 4.58975 0.385164
\(143\) −12.2251 −1.02231
\(144\) 6.29119 0.524266
\(145\) 26.7640 2.22263
\(146\) 6.31633 0.522743
\(147\) 3.04815 0.251407
\(148\) 0.824327 0.0677592
\(149\) −5.81139 −0.476088 −0.238044 0.971254i \(-0.576506\pi\)
−0.238044 + 0.971254i \(0.576506\pi\)
\(150\) −31.1802 −2.54585
\(151\) −0.480998 −0.0391430 −0.0195715 0.999808i \(-0.506230\pi\)
−0.0195715 + 0.999808i \(0.506230\pi\)
\(152\) 0 0
\(153\) 6.14790 0.497028
\(154\) 5.43763 0.438177
\(155\) 11.7376 0.942786
\(156\) 6.85294 0.548675
\(157\) −5.53300 −0.441581 −0.220791 0.975321i \(-0.570864\pi\)
−0.220791 + 0.975321i \(0.570864\pi\)
\(158\) −12.3096 −0.979302
\(159\) −6.26939 −0.497195
\(160\) −3.90247 −0.308517
\(161\) 0.640603 0.0504866
\(162\) −11.7055 −0.919673
\(163\) −2.43842 −0.190992 −0.0954958 0.995430i \(-0.530444\pi\)
−0.0954958 + 0.995430i \(0.530444\pi\)
\(164\) 11.3245 0.884295
\(165\) −64.6822 −5.03550
\(166\) 9.84224 0.763906
\(167\) 3.74745 0.289986 0.144993 0.989433i \(-0.453684\pi\)
0.144993 + 0.989433i \(0.453684\pi\)
\(168\) −3.04815 −0.235170
\(169\) −7.94545 −0.611188
\(170\) −3.81358 −0.292488
\(171\) 0 0
\(172\) −12.2093 −0.930952
\(173\) 6.84994 0.520792 0.260396 0.965502i \(-0.416147\pi\)
0.260396 + 0.965502i \(0.416147\pi\)
\(174\) −20.9049 −1.58480
\(175\) 10.2292 0.773258
\(176\) −5.43763 −0.409877
\(177\) −11.3160 −0.850561
\(178\) 11.8845 0.890784
\(179\) 13.2215 0.988222 0.494111 0.869399i \(-0.335494\pi\)
0.494111 + 0.869399i \(0.335494\pi\)
\(180\) 24.5512 1.82994
\(181\) 5.11234 0.379997 0.189998 0.981784i \(-0.439152\pi\)
0.189998 + 0.981784i \(0.439152\pi\)
\(182\) −2.24823 −0.166650
\(183\) 10.8648 0.803149
\(184\) −0.640603 −0.0472259
\(185\) 3.21691 0.236512
\(186\) −9.16802 −0.672232
\(187\) −5.31378 −0.388582
\(188\) −12.6901 −0.925518
\(189\) 10.0320 0.729723
\(190\) 0 0
\(191\) 26.4002 1.91025 0.955124 0.296206i \(-0.0957216\pi\)
0.955124 + 0.296206i \(0.0957216\pi\)
\(192\) 3.04815 0.219981
\(193\) 15.7941 1.13688 0.568442 0.822723i \(-0.307546\pi\)
0.568442 + 0.822723i \(0.307546\pi\)
\(194\) 7.94295 0.570271
\(195\) 26.7434 1.91513
\(196\) 1.00000 0.0714286
\(197\) −3.73311 −0.265973 −0.132987 0.991118i \(-0.542457\pi\)
−0.132987 + 0.991118i \(0.542457\pi\)
\(198\) 34.2092 2.43114
\(199\) −16.1988 −1.14830 −0.574151 0.818749i \(-0.694668\pi\)
−0.574151 + 0.818749i \(0.694668\pi\)
\(200\) −10.2292 −0.723317
\(201\) 3.50059 0.246912
\(202\) −12.5382 −0.882185
\(203\) 6.85823 0.481353
\(204\) 2.97872 0.208552
\(205\) 44.1935 3.08661
\(206\) 13.4463 0.936845
\(207\) 4.03016 0.280115
\(208\) 2.24823 0.155887
\(209\) 0 0
\(210\) −11.8953 −0.820853
\(211\) 13.9798 0.962412 0.481206 0.876607i \(-0.340199\pi\)
0.481206 + 0.876607i \(0.340199\pi\)
\(212\) −2.05679 −0.141261
\(213\) −13.9902 −0.958595
\(214\) 11.5515 0.789644
\(215\) −47.6464 −3.24946
\(216\) −10.0320 −0.682594
\(217\) 3.00774 0.204178
\(218\) −18.8662 −1.27778
\(219\) −19.2531 −1.30100
\(220\) −21.2202 −1.43066
\(221\) 2.19703 0.147788
\(222\) −2.51267 −0.168639
\(223\) −11.9087 −0.797466 −0.398733 0.917067i \(-0.630550\pi\)
−0.398733 + 0.917067i \(0.630550\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 64.3541 4.29028
\(226\) 2.61917 0.174224
\(227\) −13.4459 −0.892438 −0.446219 0.894924i \(-0.647230\pi\)
−0.446219 + 0.894924i \(0.647230\pi\)
\(228\) 0 0
\(229\) −5.02164 −0.331839 −0.165920 0.986139i \(-0.553059\pi\)
−0.165920 + 0.986139i \(0.553059\pi\)
\(230\) −2.49993 −0.164841
\(231\) −16.5747 −1.09053
\(232\) −6.85823 −0.450265
\(233\) 3.80785 0.249460 0.124730 0.992191i \(-0.460193\pi\)
0.124730 + 0.992191i \(0.460193\pi\)
\(234\) −14.1441 −0.924627
\(235\) −49.5225 −3.23049
\(236\) −3.71241 −0.241658
\(237\) 37.5216 2.43729
\(238\) −0.977224 −0.0633440
\(239\) −2.48796 −0.160933 −0.0804663 0.996757i \(-0.525641\pi\)
−0.0804663 + 0.996757i \(0.525641\pi\)
\(240\) 11.8953 0.767837
\(241\) −5.53763 −0.356710 −0.178355 0.983966i \(-0.557078\pi\)
−0.178355 + 0.983966i \(0.557078\pi\)
\(242\) −18.5678 −1.19358
\(243\) 5.58406 0.358218
\(244\) 3.56439 0.228187
\(245\) 3.90247 0.249319
\(246\) −34.5188 −2.20084
\(247\) 0 0
\(248\) −3.00774 −0.190991
\(249\) −30.0006 −1.90121
\(250\) −20.4069 −1.29065
\(251\) −5.55332 −0.350523 −0.175261 0.984522i \(-0.556077\pi\)
−0.175261 + 0.984522i \(0.556077\pi\)
\(252\) 6.29119 0.396308
\(253\) −3.48336 −0.218997
\(254\) −2.91792 −0.183087
\(255\) 11.6244 0.727945
\(256\) 1.00000 0.0625000
\(257\) −21.9978 −1.37219 −0.686093 0.727514i \(-0.740676\pi\)
−0.686093 + 0.727514i \(0.740676\pi\)
\(258\) 37.2158 2.31695
\(259\) 0.824327 0.0512212
\(260\) 8.77365 0.544119
\(261\) 43.1464 2.67070
\(262\) −0.814699 −0.0503322
\(263\) −22.9548 −1.41546 −0.707728 0.706485i \(-0.750280\pi\)
−0.707728 + 0.706485i \(0.750280\pi\)
\(264\) 16.5747 1.02010
\(265\) −8.02654 −0.493066
\(266\) 0 0
\(267\) −36.2258 −2.21698
\(268\) 1.14843 0.0701516
\(269\) 18.7189 1.14131 0.570656 0.821189i \(-0.306689\pi\)
0.570656 + 0.821189i \(0.306689\pi\)
\(270\) −39.1497 −2.38257
\(271\) 13.1366 0.797994 0.398997 0.916952i \(-0.369358\pi\)
0.398997 + 0.916952i \(0.369358\pi\)
\(272\) 0.977224 0.0592529
\(273\) 6.85294 0.414759
\(274\) 7.30748 0.441461
\(275\) −55.6228 −3.35418
\(276\) 1.95265 0.117536
\(277\) 9.00748 0.541207 0.270604 0.962691i \(-0.412777\pi\)
0.270604 + 0.962691i \(0.412777\pi\)
\(278\) −15.1627 −0.909396
\(279\) 18.9223 1.13285
\(280\) −3.90247 −0.233217
\(281\) −13.8823 −0.828149 −0.414074 0.910243i \(-0.635895\pi\)
−0.414074 + 0.910243i \(0.635895\pi\)
\(282\) 38.6812 2.30343
\(283\) 3.52545 0.209566 0.104783 0.994495i \(-0.466585\pi\)
0.104783 + 0.994495i \(0.466585\pi\)
\(284\) −4.58975 −0.272352
\(285\) 0 0
\(286\) 12.2251 0.722883
\(287\) 11.3245 0.668465
\(288\) −6.29119 −0.370712
\(289\) −16.0450 −0.943826
\(290\) −26.7640 −1.57164
\(291\) −24.2113 −1.41929
\(292\) −6.31633 −0.369635
\(293\) 30.6355 1.78975 0.894873 0.446321i \(-0.147266\pi\)
0.894873 + 0.446321i \(0.147266\pi\)
\(294\) −3.04815 −0.177771
\(295\) −14.4876 −0.843499
\(296\) −0.824327 −0.0479130
\(297\) −54.5505 −3.16534
\(298\) 5.81139 0.336645
\(299\) 1.44022 0.0832903
\(300\) 31.1802 1.80019
\(301\) −12.2093 −0.703733
\(302\) 0.480998 0.0276783
\(303\) 38.2183 2.19558
\(304\) 0 0
\(305\) 13.9099 0.796480
\(306\) −6.14790 −0.351452
\(307\) −25.2223 −1.43951 −0.719756 0.694227i \(-0.755746\pi\)
−0.719756 + 0.694227i \(0.755746\pi\)
\(308\) −5.43763 −0.309838
\(309\) −40.9862 −2.33162
\(310\) −11.7376 −0.666650
\(311\) −5.26030 −0.298285 −0.149142 0.988816i \(-0.547651\pi\)
−0.149142 + 0.988816i \(0.547651\pi\)
\(312\) −6.85294 −0.387971
\(313\) −3.76787 −0.212972 −0.106486 0.994314i \(-0.533960\pi\)
−0.106486 + 0.994314i \(0.533960\pi\)
\(314\) 5.53300 0.312245
\(315\) 24.5512 1.38330
\(316\) 12.3096 0.692471
\(317\) 16.6180 0.933359 0.466679 0.884427i \(-0.345450\pi\)
0.466679 + 0.884427i \(0.345450\pi\)
\(318\) 6.26939 0.351570
\(319\) −37.2925 −2.08798
\(320\) 3.90247 0.218154
\(321\) −35.2106 −1.96527
\(322\) −0.640603 −0.0356994
\(323\) 0 0
\(324\) 11.7055 0.650307
\(325\) 22.9977 1.27568
\(326\) 2.43842 0.135052
\(327\) 57.5070 3.18014
\(328\) −11.3245 −0.625291
\(329\) −12.6901 −0.699626
\(330\) 64.6822 3.56064
\(331\) 12.6506 0.695341 0.347671 0.937617i \(-0.386973\pi\)
0.347671 + 0.937617i \(0.386973\pi\)
\(332\) −9.84224 −0.540163
\(333\) 5.18600 0.284191
\(334\) −3.74745 −0.205051
\(335\) 4.48172 0.244862
\(336\) 3.04815 0.166290
\(337\) −9.93403 −0.541141 −0.270570 0.962700i \(-0.587212\pi\)
−0.270570 + 0.962700i \(0.587212\pi\)
\(338\) 7.94545 0.432175
\(339\) −7.98360 −0.433610
\(340\) 3.81358 0.206821
\(341\) −16.3550 −0.885671
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.2093 0.658282
\(345\) 7.62015 0.410255
\(346\) −6.84994 −0.368255
\(347\) −13.2140 −0.709366 −0.354683 0.934987i \(-0.615411\pi\)
−0.354683 + 0.934987i \(0.615411\pi\)
\(348\) 20.9049 1.12062
\(349\) −2.44779 −0.131027 −0.0655137 0.997852i \(-0.520869\pi\)
−0.0655137 + 0.997852i \(0.520869\pi\)
\(350\) −10.2292 −0.546776
\(351\) 22.5544 1.20386
\(352\) 5.43763 0.289827
\(353\) 35.5373 1.89146 0.945729 0.324957i \(-0.105350\pi\)
0.945729 + 0.324957i \(0.105350\pi\)
\(354\) 11.3160 0.601438
\(355\) −17.9114 −0.950636
\(356\) −11.8845 −0.629879
\(357\) 2.97872 0.157651
\(358\) −13.2215 −0.698778
\(359\) −28.0462 −1.48022 −0.740112 0.672483i \(-0.765228\pi\)
−0.740112 + 0.672483i \(0.765228\pi\)
\(360\) −24.5512 −1.29396
\(361\) 0 0
\(362\) −5.11234 −0.268698
\(363\) 56.5974 2.97060
\(364\) 2.24823 0.117839
\(365\) −24.6493 −1.29020
\(366\) −10.8648 −0.567912
\(367\) 5.52498 0.288402 0.144201 0.989548i \(-0.453939\pi\)
0.144201 + 0.989548i \(0.453939\pi\)
\(368\) 0.640603 0.0333937
\(369\) 71.2447 3.70885
\(370\) −3.21691 −0.167239
\(371\) −2.05679 −0.106783
\(372\) 9.16802 0.475340
\(373\) −1.66659 −0.0862927 −0.0431464 0.999069i \(-0.513738\pi\)
−0.0431464 + 0.999069i \(0.513738\pi\)
\(374\) 5.31378 0.274769
\(375\) 62.2033 3.21216
\(376\) 12.6901 0.654440
\(377\) 15.4189 0.794114
\(378\) −10.0320 −0.515992
\(379\) 24.2234 1.24427 0.622137 0.782908i \(-0.286265\pi\)
0.622137 + 0.782908i \(0.286265\pi\)
\(380\) 0 0
\(381\) 8.89426 0.455667
\(382\) −26.4002 −1.35075
\(383\) −5.06432 −0.258774 −0.129387 0.991594i \(-0.541301\pi\)
−0.129387 + 0.991594i \(0.541301\pi\)
\(384\) −3.04815 −0.155550
\(385\) −21.2202 −1.08148
\(386\) −15.7941 −0.803898
\(387\) −76.8112 −3.90453
\(388\) −7.94295 −0.403242
\(389\) −23.5536 −1.19421 −0.597107 0.802162i \(-0.703683\pi\)
−0.597107 + 0.802162i \(0.703683\pi\)
\(390\) −26.7434 −1.35420
\(391\) 0.626012 0.0316588
\(392\) −1.00000 −0.0505076
\(393\) 2.48332 0.125267
\(394\) 3.73311 0.188072
\(395\) 48.0379 2.41705
\(396\) −34.2092 −1.71908
\(397\) −16.9499 −0.850691 −0.425345 0.905031i \(-0.639847\pi\)
−0.425345 + 0.905031i \(0.639847\pi\)
\(398\) 16.1988 0.811972
\(399\) 0 0
\(400\) 10.2292 0.511462
\(401\) −22.6365 −1.13041 −0.565207 0.824949i \(-0.691204\pi\)
−0.565207 + 0.824949i \(0.691204\pi\)
\(402\) −3.50059 −0.174593
\(403\) 6.76209 0.336844
\(404\) 12.5382 0.623799
\(405\) 45.6804 2.26988
\(406\) −6.85823 −0.340368
\(407\) −4.48238 −0.222183
\(408\) −2.97872 −0.147469
\(409\) −1.54005 −0.0761503 −0.0380752 0.999275i \(-0.512123\pi\)
−0.0380752 + 0.999275i \(0.512123\pi\)
\(410\) −44.1935 −2.18256
\(411\) −22.2743 −1.09871
\(412\) −13.4463 −0.662450
\(413\) −3.71241 −0.182676
\(414\) −4.03016 −0.198071
\(415\) −38.4090 −1.88542
\(416\) −2.24823 −0.110229
\(417\) 46.2180 2.26330
\(418\) 0 0
\(419\) −2.33850 −0.114243 −0.0571216 0.998367i \(-0.518192\pi\)
−0.0571216 + 0.998367i \(0.518192\pi\)
\(420\) 11.8953 0.580430
\(421\) −9.21012 −0.448874 −0.224437 0.974489i \(-0.572054\pi\)
−0.224437 + 0.974489i \(0.572054\pi\)
\(422\) −13.9798 −0.680528
\(423\) −79.8356 −3.88174
\(424\) 2.05679 0.0998864
\(425\) 9.99625 0.484890
\(426\) 13.9902 0.677829
\(427\) 3.56439 0.172493
\(428\) −11.5515 −0.558363
\(429\) −37.2638 −1.79911
\(430\) 47.6464 2.29772
\(431\) 7.77763 0.374635 0.187318 0.982299i \(-0.440021\pi\)
0.187318 + 0.982299i \(0.440021\pi\)
\(432\) 10.0320 0.482667
\(433\) −3.43962 −0.165298 −0.0826488 0.996579i \(-0.526338\pi\)
−0.0826488 + 0.996579i \(0.526338\pi\)
\(434\) −3.00774 −0.144376
\(435\) 81.5806 3.91149
\(436\) 18.8662 0.903528
\(437\) 0 0
\(438\) 19.2531 0.919949
\(439\) 27.4754 1.31133 0.655664 0.755053i \(-0.272389\pi\)
0.655664 + 0.755053i \(0.272389\pi\)
\(440\) 21.2202 1.01163
\(441\) 6.29119 0.299581
\(442\) −2.19703 −0.104502
\(443\) 29.7544 1.41367 0.706836 0.707377i \(-0.250122\pi\)
0.706836 + 0.707377i \(0.250122\pi\)
\(444\) 2.51267 0.119246
\(445\) −46.3790 −2.19858
\(446\) 11.9087 0.563894
\(447\) −17.7140 −0.837842
\(448\) 1.00000 0.0472456
\(449\) −27.8042 −1.31216 −0.656081 0.754690i \(-0.727787\pi\)
−0.656081 + 0.754690i \(0.727787\pi\)
\(450\) −64.3541 −3.03368
\(451\) −61.5785 −2.89962
\(452\) −2.61917 −0.123195
\(453\) −1.46615 −0.0688858
\(454\) 13.4459 0.631049
\(455\) 8.77365 0.411315
\(456\) 0 0
\(457\) −7.02276 −0.328511 −0.164255 0.986418i \(-0.552522\pi\)
−0.164255 + 0.986418i \(0.552522\pi\)
\(458\) 5.02164 0.234646
\(459\) 9.80354 0.457590
\(460\) 2.49993 0.116560
\(461\) −11.0511 −0.514699 −0.257350 0.966318i \(-0.582849\pi\)
−0.257350 + 0.966318i \(0.582849\pi\)
\(462\) 16.5747 0.771124
\(463\) −11.9667 −0.556138 −0.278069 0.960561i \(-0.589694\pi\)
−0.278069 + 0.960561i \(0.589694\pi\)
\(464\) 6.85823 0.318385
\(465\) 35.7779 1.65916
\(466\) −3.80785 −0.176395
\(467\) −33.3755 −1.54444 −0.772218 0.635358i \(-0.780853\pi\)
−0.772218 + 0.635358i \(0.780853\pi\)
\(468\) 14.1441 0.653810
\(469\) 1.14843 0.0530297
\(470\) 49.5225 2.28430
\(471\) −16.8654 −0.777116
\(472\) 3.71241 0.170878
\(473\) 66.3898 3.05260
\(474\) −37.5216 −1.72342
\(475\) 0 0
\(476\) 0.977224 0.0447910
\(477\) −12.9396 −0.592465
\(478\) 2.48796 0.113797
\(479\) 28.5407 1.30406 0.652029 0.758194i \(-0.273918\pi\)
0.652029 + 0.758194i \(0.273918\pi\)
\(480\) −11.8953 −0.542943
\(481\) 1.85328 0.0845022
\(482\) 5.53763 0.252232
\(483\) 1.95265 0.0888487
\(484\) 18.5678 0.843992
\(485\) −30.9971 −1.40751
\(486\) −5.58406 −0.253298
\(487\) 27.5006 1.24617 0.623087 0.782153i \(-0.285878\pi\)
0.623087 + 0.782153i \(0.285878\pi\)
\(488\) −3.56439 −0.161353
\(489\) −7.43266 −0.336116
\(490\) −3.90247 −0.176295
\(491\) 16.7667 0.756671 0.378336 0.925669i \(-0.376497\pi\)
0.378336 + 0.925669i \(0.376497\pi\)
\(492\) 34.5188 1.55623
\(493\) 6.70202 0.301844
\(494\) 0 0
\(495\) −133.500 −6.00039
\(496\) 3.00774 0.135051
\(497\) −4.58975 −0.205879
\(498\) 30.0006 1.34436
\(499\) −39.8615 −1.78444 −0.892222 0.451597i \(-0.850854\pi\)
−0.892222 + 0.451597i \(0.850854\pi\)
\(500\) 20.4069 0.912626
\(501\) 11.4228 0.510331
\(502\) 5.55332 0.247857
\(503\) 16.5023 0.735800 0.367900 0.929865i \(-0.380077\pi\)
0.367900 + 0.929865i \(0.380077\pi\)
\(504\) −6.29119 −0.280232
\(505\) 48.9299 2.17735
\(506\) 3.48336 0.154854
\(507\) −24.2189 −1.07560
\(508\) 2.91792 0.129462
\(509\) −27.8470 −1.23429 −0.617147 0.786848i \(-0.711712\pi\)
−0.617147 + 0.786848i \(0.711712\pi\)
\(510\) −11.6244 −0.514735
\(511\) −6.31633 −0.279418
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 21.9978 0.970282
\(515\) −52.4736 −2.31226
\(516\) −37.2158 −1.63833
\(517\) 69.0039 3.03479
\(518\) −0.824327 −0.0362188
\(519\) 20.8796 0.916514
\(520\) −8.77365 −0.384750
\(521\) −28.0944 −1.23084 −0.615418 0.788201i \(-0.711013\pi\)
−0.615418 + 0.788201i \(0.711013\pi\)
\(522\) −43.1464 −1.88847
\(523\) −14.6392 −0.640129 −0.320064 0.947396i \(-0.603705\pi\)
−0.320064 + 0.947396i \(0.603705\pi\)
\(524\) 0.814699 0.0355903
\(525\) 31.1802 1.36082
\(526\) 22.9548 1.00088
\(527\) 2.93923 0.128035
\(528\) −16.5747 −0.721321
\(529\) −22.5896 −0.982158
\(530\) 8.02654 0.348651
\(531\) −23.3555 −1.01354
\(532\) 0 0
\(533\) 25.4601 1.10280
\(534\) 36.2258 1.56764
\(535\) −45.0793 −1.94895
\(536\) −1.14843 −0.0496047
\(537\) 40.3011 1.73912
\(538\) −18.7189 −0.807030
\(539\) −5.43763 −0.234215
\(540\) 39.1497 1.68473
\(541\) −10.3054 −0.443064 −0.221532 0.975153i \(-0.571106\pi\)
−0.221532 + 0.975153i \(0.571106\pi\)
\(542\) −13.1366 −0.564267
\(543\) 15.5831 0.668737
\(544\) −0.977224 −0.0418981
\(545\) 73.6248 3.15374
\(546\) −6.85294 −0.293279
\(547\) −24.8564 −1.06278 −0.531392 0.847126i \(-0.678331\pi\)
−0.531392 + 0.847126i \(0.678331\pi\)
\(548\) −7.30748 −0.312160
\(549\) 22.4243 0.957045
\(550\) 55.6228 2.37177
\(551\) 0 0
\(552\) −1.95265 −0.0831103
\(553\) 12.3096 0.523459
\(554\) −9.00748 −0.382691
\(555\) 9.80560 0.416225
\(556\) 15.1627 0.643040
\(557\) −36.8917 −1.56315 −0.781575 0.623812i \(-0.785583\pi\)
−0.781575 + 0.623812i \(0.785583\pi\)
\(558\) −18.9223 −0.801043
\(559\) −27.4494 −1.16099
\(560\) 3.90247 0.164909
\(561\) −16.1972 −0.683845
\(562\) 13.8823 0.585589
\(563\) 7.28154 0.306880 0.153440 0.988158i \(-0.450965\pi\)
0.153440 + 0.988158i \(0.450965\pi\)
\(564\) −38.6812 −1.62877
\(565\) −10.2212 −0.430009
\(566\) −3.52545 −0.148186
\(567\) 11.7055 0.491586
\(568\) 4.58975 0.192582
\(569\) 11.4629 0.480551 0.240276 0.970705i \(-0.422762\pi\)
0.240276 + 0.970705i \(0.422762\pi\)
\(570\) 0 0
\(571\) 20.3517 0.851691 0.425846 0.904796i \(-0.359977\pi\)
0.425846 + 0.904796i \(0.359977\pi\)
\(572\) −12.2251 −0.511156
\(573\) 80.4715 3.36175
\(574\) −11.3245 −0.472676
\(575\) 6.55288 0.273274
\(576\) 6.29119 0.262133
\(577\) 37.2381 1.55024 0.775122 0.631811i \(-0.217688\pi\)
0.775122 + 0.631811i \(0.217688\pi\)
\(578\) 16.0450 0.667385
\(579\) 48.1427 2.00074
\(580\) 26.7640 1.11131
\(581\) −9.84224 −0.408325
\(582\) 24.2113 1.00359
\(583\) 11.1840 0.463196
\(584\) 6.31633 0.261372
\(585\) 55.1968 2.28210
\(586\) −30.6355 −1.26554
\(587\) 11.2034 0.462416 0.231208 0.972904i \(-0.425732\pi\)
0.231208 + 0.972904i \(0.425732\pi\)
\(588\) 3.04815 0.125703
\(589\) 0 0
\(590\) 14.4876 0.596444
\(591\) −11.3791 −0.468073
\(592\) 0.824327 0.0338796
\(593\) −37.0804 −1.52271 −0.761354 0.648337i \(-0.775465\pi\)
−0.761354 + 0.648337i \(0.775465\pi\)
\(594\) 54.5505 2.23823
\(595\) 3.81358 0.156342
\(596\) −5.81139 −0.238044
\(597\) −49.3763 −2.02084
\(598\) −1.44022 −0.0588952
\(599\) 31.7534 1.29741 0.648705 0.761040i \(-0.275311\pi\)
0.648705 + 0.761040i \(0.275311\pi\)
\(600\) −31.1802 −1.27293
\(601\) −19.7428 −0.805324 −0.402662 0.915349i \(-0.631915\pi\)
−0.402662 + 0.915349i \(0.631915\pi\)
\(602\) 12.2093 0.497615
\(603\) 7.22501 0.294225
\(604\) −0.480998 −0.0195715
\(605\) 72.4603 2.94593
\(606\) −38.2183 −1.55251
\(607\) 21.2159 0.861128 0.430564 0.902560i \(-0.358315\pi\)
0.430564 + 0.902560i \(0.358315\pi\)
\(608\) 0 0
\(609\) 20.9049 0.847109
\(610\) −13.9099 −0.563196
\(611\) −28.5302 −1.15421
\(612\) 6.14790 0.248514
\(613\) 19.9161 0.804405 0.402203 0.915551i \(-0.368245\pi\)
0.402203 + 0.915551i \(0.368245\pi\)
\(614\) 25.2223 1.01789
\(615\) 134.708 5.43196
\(616\) 5.43763 0.219088
\(617\) 18.0683 0.727402 0.363701 0.931516i \(-0.381513\pi\)
0.363701 + 0.931516i \(0.381513\pi\)
\(618\) 40.9862 1.64871
\(619\) 31.5626 1.26861 0.634303 0.773084i \(-0.281287\pi\)
0.634303 + 0.773084i \(0.281287\pi\)
\(620\) 11.7376 0.471393
\(621\) 6.42655 0.257889
\(622\) 5.26030 0.210919
\(623\) −11.8845 −0.476144
\(624\) 6.85294 0.274337
\(625\) 28.4912 1.13965
\(626\) 3.76787 0.150594
\(627\) 0 0
\(628\) −5.53300 −0.220791
\(629\) 0.805551 0.0321194
\(630\) −24.5512 −0.978142
\(631\) 25.1232 1.00014 0.500070 0.865985i \(-0.333308\pi\)
0.500070 + 0.865985i \(0.333308\pi\)
\(632\) −12.3096 −0.489651
\(633\) 42.6126 1.69370
\(634\) −16.6180 −0.659984
\(635\) 11.3871 0.451883
\(636\) −6.26939 −0.248597
\(637\) 2.24823 0.0890782
\(638\) 37.2925 1.47642
\(639\) −28.8750 −1.14228
\(640\) −3.90247 −0.154259
\(641\) 3.23583 0.127808 0.0639039 0.997956i \(-0.479645\pi\)
0.0639039 + 0.997956i \(0.479645\pi\)
\(642\) 35.2106 1.38965
\(643\) −19.3127 −0.761619 −0.380809 0.924654i \(-0.624355\pi\)
−0.380809 + 0.924654i \(0.624355\pi\)
\(644\) 0.640603 0.0252433
\(645\) −145.233 −5.71856
\(646\) 0 0
\(647\) −11.5212 −0.452943 −0.226472 0.974018i \(-0.572719\pi\)
−0.226472 + 0.974018i \(0.572719\pi\)
\(648\) −11.7055 −0.459837
\(649\) 20.1867 0.792399
\(650\) −22.9977 −0.902045
\(651\) 9.16802 0.359323
\(652\) −2.43842 −0.0954958
\(653\) −43.5279 −1.70338 −0.851689 0.524047i \(-0.824421\pi\)
−0.851689 + 0.524047i \(0.824421\pi\)
\(654\) −57.5070 −2.24870
\(655\) 3.17933 0.124227
\(656\) 11.3245 0.442148
\(657\) −39.7373 −1.55030
\(658\) 12.6901 0.494710
\(659\) −5.02462 −0.195731 −0.0978657 0.995200i \(-0.531202\pi\)
−0.0978657 + 0.995200i \(0.531202\pi\)
\(660\) −64.6822 −2.51775
\(661\) 16.0652 0.624864 0.312432 0.949940i \(-0.398856\pi\)
0.312432 + 0.949940i \(0.398856\pi\)
\(662\) −12.6506 −0.491680
\(663\) 6.69686 0.260084
\(664\) 9.84224 0.381953
\(665\) 0 0
\(666\) −5.18600 −0.200953
\(667\) 4.39340 0.170113
\(668\) 3.74745 0.144993
\(669\) −36.2995 −1.40342
\(670\) −4.48172 −0.173144
\(671\) −19.3819 −0.748228
\(672\) −3.04815 −0.117585
\(673\) 1.32631 0.0511255 0.0255627 0.999673i \(-0.491862\pi\)
0.0255627 + 0.999673i \(0.491862\pi\)
\(674\) 9.93403 0.382644
\(675\) 102.620 3.94985
\(676\) −7.94545 −0.305594
\(677\) −17.6403 −0.677971 −0.338986 0.940792i \(-0.610084\pi\)
−0.338986 + 0.940792i \(0.610084\pi\)
\(678\) 7.98360 0.306608
\(679\) −7.94295 −0.304823
\(680\) −3.81358 −0.146244
\(681\) −40.9852 −1.57056
\(682\) 16.3550 0.626264
\(683\) 17.9010 0.684965 0.342482 0.939524i \(-0.388732\pi\)
0.342482 + 0.939524i \(0.388732\pi\)
\(684\) 0 0
\(685\) −28.5172 −1.08959
\(686\) −1.00000 −0.0381802
\(687\) −15.3067 −0.583986
\(688\) −12.2093 −0.465476
\(689\) −4.62414 −0.176166
\(690\) −7.62015 −0.290094
\(691\) 10.4222 0.396480 0.198240 0.980153i \(-0.436477\pi\)
0.198240 + 0.980153i \(0.436477\pi\)
\(692\) 6.84994 0.260396
\(693\) −34.2092 −1.29950
\(694\) 13.2140 0.501598
\(695\) 59.1718 2.24451
\(696\) −20.9049 −0.792398
\(697\) 11.0666 0.419176
\(698\) 2.44779 0.0926503
\(699\) 11.6069 0.439012
\(700\) 10.2292 0.386629
\(701\) −45.9994 −1.73737 −0.868686 0.495362i \(-0.835035\pi\)
−0.868686 + 0.495362i \(0.835035\pi\)
\(702\) −22.5544 −0.851259
\(703\) 0 0
\(704\) −5.43763 −0.204938
\(705\) −150.952 −5.68518
\(706\) −35.5373 −1.33746
\(707\) 12.5382 0.471548
\(708\) −11.3160 −0.425281
\(709\) 4.31058 0.161887 0.0809436 0.996719i \(-0.474207\pi\)
0.0809436 + 0.996719i \(0.474207\pi\)
\(710\) 17.9114 0.672201
\(711\) 77.4423 2.90431
\(712\) 11.8845 0.445392
\(713\) 1.92676 0.0721579
\(714\) −2.97872 −0.111476
\(715\) −47.7079 −1.78417
\(716\) 13.2215 0.494111
\(717\) −7.58366 −0.283217
\(718\) 28.0462 1.04668
\(719\) 32.1996 1.20084 0.600421 0.799684i \(-0.295000\pi\)
0.600421 + 0.799684i \(0.295000\pi\)
\(720\) 24.5512 0.914968
\(721\) −13.4463 −0.500765
\(722\) 0 0
\(723\) −16.8795 −0.627755
\(724\) 5.11234 0.189998
\(725\) 70.1545 2.60547
\(726\) −56.5974 −2.10053
\(727\) 15.3908 0.570814 0.285407 0.958406i \(-0.407871\pi\)
0.285407 + 0.958406i \(0.407871\pi\)
\(728\) −2.24823 −0.0833251
\(729\) −18.0956 −0.670206
\(730\) 24.6493 0.912310
\(731\) −11.9312 −0.441293
\(732\) 10.8648 0.401574
\(733\) −1.87547 −0.0692719 −0.0346360 0.999400i \(-0.511027\pi\)
−0.0346360 + 0.999400i \(0.511027\pi\)
\(734\) −5.52498 −0.203931
\(735\) 11.8953 0.438764
\(736\) −0.640603 −0.0236129
\(737\) −6.24475 −0.230028
\(738\) −71.2447 −2.62255
\(739\) −23.4196 −0.861502 −0.430751 0.902471i \(-0.641751\pi\)
−0.430751 + 0.902471i \(0.641751\pi\)
\(740\) 3.21691 0.118256
\(741\) 0 0
\(742\) 2.05679 0.0755070
\(743\) −18.5235 −0.679561 −0.339780 0.940505i \(-0.610353\pi\)
−0.339780 + 0.940505i \(0.610353\pi\)
\(744\) −9.16802 −0.336116
\(745\) −22.6788 −0.830886
\(746\) 1.66659 0.0610182
\(747\) −61.9194 −2.26551
\(748\) −5.31378 −0.194291
\(749\) −11.5515 −0.422082
\(750\) −62.2033 −2.27134
\(751\) −1.49895 −0.0546974 −0.0273487 0.999626i \(-0.508706\pi\)
−0.0273487 + 0.999626i \(0.508706\pi\)
\(752\) −12.6901 −0.462759
\(753\) −16.9273 −0.616867
\(754\) −15.4189 −0.561523
\(755\) −1.87708 −0.0683138
\(756\) 10.0320 0.364862
\(757\) 28.5048 1.03603 0.518013 0.855373i \(-0.326672\pi\)
0.518013 + 0.855373i \(0.326672\pi\)
\(758\) −24.2234 −0.879835
\(759\) −10.6178 −0.385401
\(760\) 0 0
\(761\) 37.9783 1.37671 0.688356 0.725373i \(-0.258333\pi\)
0.688356 + 0.725373i \(0.258333\pi\)
\(762\) −8.89426 −0.322205
\(763\) 18.8662 0.683003
\(764\) 26.4002 0.955124
\(765\) 23.9920 0.867432
\(766\) 5.06432 0.182981
\(767\) −8.34637 −0.301370
\(768\) 3.04815 0.109990
\(769\) −32.6697 −1.17810 −0.589049 0.808097i \(-0.700498\pi\)
−0.589049 + 0.808097i \(0.700498\pi\)
\(770\) 21.2202 0.764722
\(771\) −67.0526 −2.41484
\(772\) 15.7941 0.568442
\(773\) −19.9190 −0.716436 −0.358218 0.933638i \(-0.616615\pi\)
−0.358218 + 0.933638i \(0.616615\pi\)
\(774\) 76.8112 2.76092
\(775\) 30.7669 1.10518
\(776\) 7.94295 0.285135
\(777\) 2.51267 0.0901414
\(778\) 23.5536 0.844437
\(779\) 0 0
\(780\) 26.7434 0.957566
\(781\) 24.9574 0.893045
\(782\) −0.626012 −0.0223861
\(783\) 68.8020 2.45878
\(784\) 1.00000 0.0357143
\(785\) −21.5923 −0.770664
\(786\) −2.48332 −0.0885771
\(787\) 6.83078 0.243491 0.121745 0.992561i \(-0.461151\pi\)
0.121745 + 0.992561i \(0.461151\pi\)
\(788\) −3.73311 −0.132987
\(789\) −69.9697 −2.49099
\(790\) −48.0379 −1.70911
\(791\) −2.61917 −0.0931268
\(792\) 34.2092 1.21557
\(793\) 8.01359 0.284571
\(794\) 16.9499 0.601529
\(795\) −24.4661 −0.867722
\(796\) −16.1988 −0.574151
\(797\) 14.9421 0.529275 0.264638 0.964348i \(-0.414748\pi\)
0.264638 + 0.964348i \(0.414748\pi\)
\(798\) 0 0
\(799\) −12.4010 −0.438717
\(800\) −10.2292 −0.361658
\(801\) −74.7679 −2.64179
\(802\) 22.6365 0.799324
\(803\) 34.3459 1.21204
\(804\) 3.50059 0.123456
\(805\) 2.49993 0.0881110
\(806\) −6.76209 −0.238185
\(807\) 57.0580 2.00854
\(808\) −12.5382 −0.441092
\(809\) 14.6592 0.515390 0.257695 0.966226i \(-0.417037\pi\)
0.257695 + 0.966226i \(0.417037\pi\)
\(810\) −45.6804 −1.60505
\(811\) 49.7049 1.74537 0.872687 0.488280i \(-0.162375\pi\)
0.872687 + 0.488280i \(0.162375\pi\)
\(812\) 6.85823 0.240677
\(813\) 40.0424 1.40435
\(814\) 4.48238 0.157107
\(815\) −9.51585 −0.333326
\(816\) 2.97872 0.104276
\(817\) 0 0
\(818\) 1.54005 0.0538464
\(819\) 14.1441 0.494234
\(820\) 44.1935 1.54330
\(821\) −26.7176 −0.932449 −0.466225 0.884666i \(-0.654386\pi\)
−0.466225 + 0.884666i \(0.654386\pi\)
\(822\) 22.2743 0.776905
\(823\) 3.16141 0.110200 0.0550999 0.998481i \(-0.482452\pi\)
0.0550999 + 0.998481i \(0.482452\pi\)
\(824\) 13.4463 0.468423
\(825\) −169.546 −5.90285
\(826\) 3.71241 0.129171
\(827\) −24.2855 −0.844488 −0.422244 0.906482i \(-0.638757\pi\)
−0.422244 + 0.906482i \(0.638757\pi\)
\(828\) 4.03016 0.140058
\(829\) 22.9819 0.798195 0.399098 0.916908i \(-0.369323\pi\)
0.399098 + 0.916908i \(0.369323\pi\)
\(830\) 38.4090 1.33320
\(831\) 27.4561 0.952442
\(832\) 2.24823 0.0779435
\(833\) 0.977224 0.0338588
\(834\) −46.2180 −1.60040
\(835\) 14.6243 0.506094
\(836\) 0 0
\(837\) 30.1737 1.04296
\(838\) 2.33850 0.0807821
\(839\) 7.80775 0.269553 0.134777 0.990876i \(-0.456968\pi\)
0.134777 + 0.990876i \(0.456968\pi\)
\(840\) −11.8953 −0.410426
\(841\) 18.0353 0.621907
\(842\) 9.21012 0.317402
\(843\) −42.3153 −1.45742
\(844\) 13.9798 0.481206
\(845\) −31.0068 −1.06667
\(846\) 79.8356 2.74481
\(847\) 18.5678 0.637998
\(848\) −2.05679 −0.0706303
\(849\) 10.7461 0.368805
\(850\) −9.99625 −0.342869
\(851\) 0.528066 0.0181019
\(852\) −13.9902 −0.479298
\(853\) −0.148294 −0.00507747 −0.00253874 0.999997i \(-0.500808\pi\)
−0.00253874 + 0.999997i \(0.500808\pi\)
\(854\) −3.56439 −0.121971
\(855\) 0 0
\(856\) 11.5515 0.394822
\(857\) −17.4431 −0.595845 −0.297923 0.954590i \(-0.596294\pi\)
−0.297923 + 0.954590i \(0.596294\pi\)
\(858\) 37.2638 1.27216
\(859\) 44.9110 1.53234 0.766171 0.642636i \(-0.222159\pi\)
0.766171 + 0.642636i \(0.222159\pi\)
\(860\) −47.6464 −1.62473
\(861\) 34.5188 1.17640
\(862\) −7.77763 −0.264907
\(863\) −4.73021 −0.161018 −0.0805091 0.996754i \(-0.525655\pi\)
−0.0805091 + 0.996754i \(0.525655\pi\)
\(864\) −10.0320 −0.341297
\(865\) 26.7317 0.908904
\(866\) 3.43962 0.116883
\(867\) −48.9076 −1.66099
\(868\) 3.00774 0.102089
\(869\) −66.9352 −2.27062
\(870\) −81.5806 −2.76584
\(871\) 2.58194 0.0874858
\(872\) −18.8662 −0.638891
\(873\) −49.9707 −1.69125
\(874\) 0 0
\(875\) 20.4069 0.689880
\(876\) −19.2531 −0.650502
\(877\) −22.3767 −0.755608 −0.377804 0.925886i \(-0.623321\pi\)
−0.377804 + 0.925886i \(0.623321\pi\)
\(878\) −27.4754 −0.927249
\(879\) 93.3815 3.14968
\(880\) −21.2202 −0.715332
\(881\) 17.3063 0.583064 0.291532 0.956561i \(-0.405835\pi\)
0.291532 + 0.956561i \(0.405835\pi\)
\(882\) −6.29119 −0.211835
\(883\) 22.2407 0.748459 0.374230 0.927336i \(-0.377907\pi\)
0.374230 + 0.927336i \(0.377907\pi\)
\(884\) 2.19703 0.0738940
\(885\) −44.1602 −1.48443
\(886\) −29.7544 −0.999618
\(887\) 31.6781 1.06365 0.531823 0.846856i \(-0.321507\pi\)
0.531823 + 0.846856i \(0.321507\pi\)
\(888\) −2.51267 −0.0843196
\(889\) 2.91792 0.0978640
\(890\) 46.3790 1.55463
\(891\) −63.6504 −2.13237
\(892\) −11.9087 −0.398733
\(893\) 0 0
\(894\) 17.7140 0.592444
\(895\) 51.5965 1.72468
\(896\) −1.00000 −0.0334077
\(897\) 4.39001 0.146578
\(898\) 27.8042 0.927839
\(899\) 20.6277 0.687974
\(900\) 64.3541 2.14514
\(901\) −2.00994 −0.0669608
\(902\) 61.5785 2.05034
\(903\) −37.2158 −1.23846
\(904\) 2.61917 0.0871122
\(905\) 19.9507 0.663184
\(906\) 1.46615 0.0487096
\(907\) −35.7705 −1.18774 −0.593870 0.804561i \(-0.702401\pi\)
−0.593870 + 0.804561i \(0.702401\pi\)
\(908\) −13.4459 −0.446219
\(909\) 78.8803 2.61629
\(910\) −8.77365 −0.290844
\(911\) 1.00147 0.0331803 0.0165902 0.999862i \(-0.494719\pi\)
0.0165902 + 0.999862i \(0.494719\pi\)
\(912\) 0 0
\(913\) 53.5184 1.77120
\(914\) 7.02276 0.232292
\(915\) 42.3995 1.40168
\(916\) −5.02164 −0.165920
\(917\) 0.814699 0.0269037
\(918\) −9.80354 −0.323565
\(919\) 27.1273 0.894847 0.447423 0.894322i \(-0.352342\pi\)
0.447423 + 0.894322i \(0.352342\pi\)
\(920\) −2.49993 −0.0824203
\(921\) −76.8812 −2.53332
\(922\) 11.0511 0.363947
\(923\) −10.3188 −0.339649
\(924\) −16.5747 −0.545267
\(925\) 8.43223 0.277250
\(926\) 11.9667 0.393249
\(927\) −84.5930 −2.77840
\(928\) −6.85823 −0.225132
\(929\) 2.19665 0.0720697 0.0360349 0.999351i \(-0.488527\pi\)
0.0360349 + 0.999351i \(0.488527\pi\)
\(930\) −35.7779 −1.17320
\(931\) 0 0
\(932\) 3.80785 0.124730
\(933\) −16.0342 −0.524935
\(934\) 33.3755 1.09208
\(935\) −20.7368 −0.678167
\(936\) −14.1441 −0.462313
\(937\) −16.3687 −0.534742 −0.267371 0.963594i \(-0.586155\pi\)
−0.267371 + 0.963594i \(0.586155\pi\)
\(938\) −1.14843 −0.0374976
\(939\) −11.4850 −0.374799
\(940\) −49.5225 −1.61525
\(941\) 49.5617 1.61566 0.807832 0.589413i \(-0.200641\pi\)
0.807832 + 0.589413i \(0.200641\pi\)
\(942\) 16.8654 0.549504
\(943\) 7.25451 0.236239
\(944\) −3.71241 −0.120829
\(945\) 39.1497 1.27354
\(946\) −66.3898 −2.15852
\(947\) 32.3241 1.05039 0.525196 0.850981i \(-0.323992\pi\)
0.525196 + 0.850981i \(0.323992\pi\)
\(948\) 37.5216 1.21864
\(949\) −14.2006 −0.460971
\(950\) 0 0
\(951\) 50.6540 1.64257
\(952\) −0.977224 −0.0316720
\(953\) −29.6712 −0.961143 −0.480572 0.876955i \(-0.659571\pi\)
−0.480572 + 0.876955i \(0.659571\pi\)
\(954\) 12.9396 0.418936
\(955\) 103.026 3.33383
\(956\) −2.48796 −0.0804663
\(957\) −113.673 −3.67453
\(958\) −28.5407 −0.922109
\(959\) −7.30748 −0.235971
\(960\) 11.8953 0.383919
\(961\) −21.9535 −0.708178
\(962\) −1.85328 −0.0597521
\(963\) −72.6727 −2.34184
\(964\) −5.53763 −0.178355
\(965\) 61.6359 1.98413
\(966\) −1.95265 −0.0628255
\(967\) 23.6909 0.761847 0.380923 0.924607i \(-0.375606\pi\)
0.380923 + 0.924607i \(0.375606\pi\)
\(968\) −18.5678 −0.596792
\(969\) 0 0
\(970\) 30.9971 0.995257
\(971\) −29.8444 −0.957753 −0.478876 0.877882i \(-0.658956\pi\)
−0.478876 + 0.877882i \(0.658956\pi\)
\(972\) 5.58406 0.179109
\(973\) 15.1627 0.486093
\(974\) −27.5006 −0.881177
\(975\) 70.1004 2.24501
\(976\) 3.56439 0.114093
\(977\) −55.2708 −1.76827 −0.884135 0.467232i \(-0.845251\pi\)
−0.884135 + 0.467232i \(0.845251\pi\)
\(978\) 7.43266 0.237670
\(979\) 64.6237 2.06538
\(980\) 3.90247 0.124660
\(981\) 118.691 3.78951
\(982\) −16.7667 −0.535047
\(983\) −24.8474 −0.792508 −0.396254 0.918141i \(-0.629690\pi\)
−0.396254 + 0.918141i \(0.629690\pi\)
\(984\) −34.5188 −1.10042
\(985\) −14.5683 −0.464186
\(986\) −6.70202 −0.213436
\(987\) −38.6812 −1.23123
\(988\) 0 0
\(989\) −7.82132 −0.248704
\(990\) 133.500 4.24291
\(991\) −37.4253 −1.18885 −0.594427 0.804150i \(-0.702621\pi\)
−0.594427 + 0.804150i \(0.702621\pi\)
\(992\) −3.00774 −0.0954957
\(993\) 38.5609 1.22369
\(994\) 4.58975 0.145578
\(995\) −63.2152 −2.00406
\(996\) −30.0006 −0.950604
\(997\) 33.0295 1.04605 0.523027 0.852316i \(-0.324803\pi\)
0.523027 + 0.852316i \(0.324803\pi\)
\(998\) 39.8615 1.26179
\(999\) 8.26967 0.261641
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.bg.1.7 8
19.18 odd 2 5054.2.a.bh.1.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bg.1.7 8 1.1 even 1 trivial
5054.2.a.bh.1.2 yes 8 19.18 odd 2