Properties

Label 8034.2.a.w.1.11
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.65870\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.30797 q^{5} -1.00000 q^{6} +4.49849 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.30797 q^{5} -1.00000 q^{6} +4.49849 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.30797 q^{10} -3.07064 q^{11} +1.00000 q^{12} +1.00000 q^{13} -4.49849 q^{14} +2.30797 q^{15} +1.00000 q^{16} -6.42341 q^{17} -1.00000 q^{18} -7.58167 q^{19} +2.30797 q^{20} +4.49849 q^{21} +3.07064 q^{22} -9.27355 q^{23} -1.00000 q^{24} +0.326720 q^{25} -1.00000 q^{26} +1.00000 q^{27} +4.49849 q^{28} -3.29159 q^{29} -2.30797 q^{30} -5.32783 q^{31} -1.00000 q^{32} -3.07064 q^{33} +6.42341 q^{34} +10.3824 q^{35} +1.00000 q^{36} +2.93933 q^{37} +7.58167 q^{38} +1.00000 q^{39} -2.30797 q^{40} +7.89321 q^{41} -4.49849 q^{42} -0.660219 q^{43} -3.07064 q^{44} +2.30797 q^{45} +9.27355 q^{46} -10.3443 q^{47} +1.00000 q^{48} +13.2364 q^{49} -0.326720 q^{50} -6.42341 q^{51} +1.00000 q^{52} +3.71472 q^{53} -1.00000 q^{54} -7.08695 q^{55} -4.49849 q^{56} -7.58167 q^{57} +3.29159 q^{58} -8.12616 q^{59} +2.30797 q^{60} +4.61685 q^{61} +5.32783 q^{62} +4.49849 q^{63} +1.00000 q^{64} +2.30797 q^{65} +3.07064 q^{66} -6.12513 q^{67} -6.42341 q^{68} -9.27355 q^{69} -10.3824 q^{70} -8.35045 q^{71} -1.00000 q^{72} +16.0060 q^{73} -2.93933 q^{74} +0.326720 q^{75} -7.58167 q^{76} -13.8133 q^{77} -1.00000 q^{78} -14.8666 q^{79} +2.30797 q^{80} +1.00000 q^{81} -7.89321 q^{82} -12.9601 q^{83} +4.49849 q^{84} -14.8250 q^{85} +0.660219 q^{86} -3.29159 q^{87} +3.07064 q^{88} +1.15775 q^{89} -2.30797 q^{90} +4.49849 q^{91} -9.27355 q^{92} -5.32783 q^{93} +10.3443 q^{94} -17.4983 q^{95} -1.00000 q^{96} +5.29299 q^{97} -13.2364 q^{98} -3.07064 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.30797 1.03216 0.516078 0.856542i \(-0.327392\pi\)
0.516078 + 0.856542i \(0.327392\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.49849 1.70027 0.850134 0.526566i \(-0.176521\pi\)
0.850134 + 0.526566i \(0.176521\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.30797 −0.729844
\(11\) −3.07064 −0.925834 −0.462917 0.886402i \(-0.653197\pi\)
−0.462917 + 0.886402i \(0.653197\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −4.49849 −1.20227
\(15\) 2.30797 0.595915
\(16\) 1.00000 0.250000
\(17\) −6.42341 −1.55791 −0.778953 0.627083i \(-0.784249\pi\)
−0.778953 + 0.627083i \(0.784249\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.58167 −1.73935 −0.869677 0.493621i \(-0.835673\pi\)
−0.869677 + 0.493621i \(0.835673\pi\)
\(20\) 2.30797 0.516078
\(21\) 4.49849 0.981650
\(22\) 3.07064 0.654664
\(23\) −9.27355 −1.93367 −0.966835 0.255403i \(-0.917792\pi\)
−0.966835 + 0.255403i \(0.917792\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0.326720 0.0653440
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 4.49849 0.850134
\(29\) −3.29159 −0.611233 −0.305616 0.952155i \(-0.598863\pi\)
−0.305616 + 0.952155i \(0.598863\pi\)
\(30\) −2.30797 −0.421376
\(31\) −5.32783 −0.956906 −0.478453 0.878113i \(-0.658802\pi\)
−0.478453 + 0.878113i \(0.658802\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.07064 −0.534531
\(34\) 6.42341 1.10161
\(35\) 10.3824 1.75494
\(36\) 1.00000 0.166667
\(37\) 2.93933 0.483223 0.241611 0.970373i \(-0.422324\pi\)
0.241611 + 0.970373i \(0.422324\pi\)
\(38\) 7.58167 1.22991
\(39\) 1.00000 0.160128
\(40\) −2.30797 −0.364922
\(41\) 7.89321 1.23271 0.616356 0.787468i \(-0.288608\pi\)
0.616356 + 0.787468i \(0.288608\pi\)
\(42\) −4.49849 −0.694131
\(43\) −0.660219 −0.100683 −0.0503413 0.998732i \(-0.516031\pi\)
−0.0503413 + 0.998732i \(0.516031\pi\)
\(44\) −3.07064 −0.462917
\(45\) 2.30797 0.344052
\(46\) 9.27355 1.36731
\(47\) −10.3443 −1.50888 −0.754439 0.656371i \(-0.772091\pi\)
−0.754439 + 0.656371i \(0.772091\pi\)
\(48\) 1.00000 0.144338
\(49\) 13.2364 1.89091
\(50\) −0.326720 −0.0462052
\(51\) −6.42341 −0.899457
\(52\) 1.00000 0.138675
\(53\) 3.71472 0.510256 0.255128 0.966907i \(-0.417882\pi\)
0.255128 + 0.966907i \(0.417882\pi\)
\(54\) −1.00000 −0.136083
\(55\) −7.08695 −0.955604
\(56\) −4.49849 −0.601136
\(57\) −7.58167 −1.00422
\(58\) 3.29159 0.432207
\(59\) −8.12616 −1.05794 −0.528968 0.848642i \(-0.677421\pi\)
−0.528968 + 0.848642i \(0.677421\pi\)
\(60\) 2.30797 0.297957
\(61\) 4.61685 0.591127 0.295563 0.955323i \(-0.404493\pi\)
0.295563 + 0.955323i \(0.404493\pi\)
\(62\) 5.32783 0.676635
\(63\) 4.49849 0.566756
\(64\) 1.00000 0.125000
\(65\) 2.30797 0.286268
\(66\) 3.07064 0.377970
\(67\) −6.12513 −0.748304 −0.374152 0.927367i \(-0.622066\pi\)
−0.374152 + 0.927367i \(0.622066\pi\)
\(68\) −6.42341 −0.778953
\(69\) −9.27355 −1.11640
\(70\) −10.3824 −1.24093
\(71\) −8.35045 −0.991016 −0.495508 0.868603i \(-0.665018\pi\)
−0.495508 + 0.868603i \(0.665018\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.0060 1.87337 0.936683 0.350179i \(-0.113879\pi\)
0.936683 + 0.350179i \(0.113879\pi\)
\(74\) −2.93933 −0.341690
\(75\) 0.326720 0.0377264
\(76\) −7.58167 −0.869677
\(77\) −13.8133 −1.57417
\(78\) −1.00000 −0.113228
\(79\) −14.8666 −1.67263 −0.836313 0.548253i \(-0.815293\pi\)
−0.836313 + 0.548253i \(0.815293\pi\)
\(80\) 2.30797 0.258039
\(81\) 1.00000 0.111111
\(82\) −7.89321 −0.871659
\(83\) −12.9601 −1.42255 −0.711276 0.702913i \(-0.751883\pi\)
−0.711276 + 0.702913i \(0.751883\pi\)
\(84\) 4.49849 0.490825
\(85\) −14.8250 −1.60800
\(86\) 0.660219 0.0711933
\(87\) −3.29159 −0.352895
\(88\) 3.07064 0.327332
\(89\) 1.15775 0.122721 0.0613607 0.998116i \(-0.480456\pi\)
0.0613607 + 0.998116i \(0.480456\pi\)
\(90\) −2.30797 −0.243281
\(91\) 4.49849 0.471569
\(92\) −9.27355 −0.966835
\(93\) −5.32783 −0.552470
\(94\) 10.3443 1.06694
\(95\) −17.4983 −1.79528
\(96\) −1.00000 −0.102062
\(97\) 5.29299 0.537421 0.268711 0.963221i \(-0.413402\pi\)
0.268711 + 0.963221i \(0.413402\pi\)
\(98\) −13.2364 −1.33708
\(99\) −3.07064 −0.308611
\(100\) 0.326720 0.0326720
\(101\) 2.72101 0.270751 0.135375 0.990794i \(-0.456776\pi\)
0.135375 + 0.990794i \(0.456776\pi\)
\(102\) 6.42341 0.636012
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 10.3824 1.01322
\(106\) −3.71472 −0.360806
\(107\) 3.60502 0.348511 0.174255 0.984700i \(-0.444248\pi\)
0.174255 + 0.984700i \(0.444248\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.4950 −1.19680 −0.598401 0.801196i \(-0.704197\pi\)
−0.598401 + 0.801196i \(0.704197\pi\)
\(110\) 7.08695 0.675714
\(111\) 2.93933 0.278989
\(112\) 4.49849 0.425067
\(113\) −2.67954 −0.252070 −0.126035 0.992026i \(-0.540225\pi\)
−0.126035 + 0.992026i \(0.540225\pi\)
\(114\) 7.58167 0.710089
\(115\) −21.4031 −1.99585
\(116\) −3.29159 −0.305616
\(117\) 1.00000 0.0924500
\(118\) 8.12616 0.748074
\(119\) −28.8956 −2.64886
\(120\) −2.30797 −0.210688
\(121\) −1.57114 −0.142831
\(122\) −4.61685 −0.417990
\(123\) 7.89321 0.711707
\(124\) −5.32783 −0.478453
\(125\) −10.7858 −0.964710
\(126\) −4.49849 −0.400757
\(127\) −12.4620 −1.10582 −0.552911 0.833240i \(-0.686483\pi\)
−0.552911 + 0.833240i \(0.686483\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.660219 −0.0581291
\(130\) −2.30797 −0.202422
\(131\) 16.5343 1.44461 0.722306 0.691574i \(-0.243082\pi\)
0.722306 + 0.691574i \(0.243082\pi\)
\(132\) −3.07064 −0.267265
\(133\) −34.1060 −2.95737
\(134\) 6.12513 0.529131
\(135\) 2.30797 0.198638
\(136\) 6.42341 0.550803
\(137\) 13.6535 1.16650 0.583248 0.812294i \(-0.301782\pi\)
0.583248 + 0.812294i \(0.301782\pi\)
\(138\) 9.27355 0.789417
\(139\) −7.77347 −0.659337 −0.329669 0.944097i \(-0.606937\pi\)
−0.329669 + 0.944097i \(0.606937\pi\)
\(140\) 10.3824 0.877470
\(141\) −10.3443 −0.871151
\(142\) 8.35045 0.700754
\(143\) −3.07064 −0.256780
\(144\) 1.00000 0.0833333
\(145\) −7.59689 −0.630887
\(146\) −16.0060 −1.32467
\(147\) 13.2364 1.09172
\(148\) 2.93933 0.241611
\(149\) −4.13520 −0.338768 −0.169384 0.985550i \(-0.554178\pi\)
−0.169384 + 0.985550i \(0.554178\pi\)
\(150\) −0.326720 −0.0266766
\(151\) 16.9804 1.38184 0.690922 0.722930i \(-0.257205\pi\)
0.690922 + 0.722930i \(0.257205\pi\)
\(152\) 7.58167 0.614955
\(153\) −6.42341 −0.519302
\(154\) 13.8133 1.11310
\(155\) −12.2965 −0.987676
\(156\) 1.00000 0.0800641
\(157\) 11.2088 0.894562 0.447281 0.894393i \(-0.352392\pi\)
0.447281 + 0.894393i \(0.352392\pi\)
\(158\) 14.8666 1.18272
\(159\) 3.71472 0.294597
\(160\) −2.30797 −0.182461
\(161\) −41.7170 −3.28776
\(162\) −1.00000 −0.0785674
\(163\) 16.1485 1.26485 0.632425 0.774622i \(-0.282060\pi\)
0.632425 + 0.774622i \(0.282060\pi\)
\(164\) 7.89321 0.616356
\(165\) −7.08695 −0.551718
\(166\) 12.9601 1.00590
\(167\) −13.4298 −1.03923 −0.519614 0.854401i \(-0.673924\pi\)
−0.519614 + 0.854401i \(0.673924\pi\)
\(168\) −4.49849 −0.347066
\(169\) 1.00000 0.0769231
\(170\) 14.8250 1.13703
\(171\) −7.58167 −0.579785
\(172\) −0.660219 −0.0503413
\(173\) 19.5550 1.48674 0.743369 0.668881i \(-0.233227\pi\)
0.743369 + 0.668881i \(0.233227\pi\)
\(174\) 3.29159 0.249535
\(175\) 1.46975 0.111102
\(176\) −3.07064 −0.231459
\(177\) −8.12616 −0.610800
\(178\) −1.15775 −0.0867771
\(179\) 20.1020 1.50249 0.751247 0.660022i \(-0.229453\pi\)
0.751247 + 0.660022i \(0.229453\pi\)
\(180\) 2.30797 0.172026
\(181\) 7.14284 0.530923 0.265462 0.964121i \(-0.414476\pi\)
0.265462 + 0.964121i \(0.414476\pi\)
\(182\) −4.49849 −0.333450
\(183\) 4.61685 0.341287
\(184\) 9.27355 0.683655
\(185\) 6.78388 0.498761
\(186\) 5.32783 0.390655
\(187\) 19.7240 1.44236
\(188\) −10.3443 −0.754439
\(189\) 4.49849 0.327217
\(190\) 17.4983 1.26946
\(191\) −5.32169 −0.385064 −0.192532 0.981291i \(-0.561670\pi\)
−0.192532 + 0.981291i \(0.561670\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0.0650024 0.00467897 0.00233949 0.999997i \(-0.499255\pi\)
0.00233949 + 0.999997i \(0.499255\pi\)
\(194\) −5.29299 −0.380014
\(195\) 2.30797 0.165277
\(196\) 13.2364 0.945456
\(197\) −14.4690 −1.03087 −0.515436 0.856928i \(-0.672370\pi\)
−0.515436 + 0.856928i \(0.672370\pi\)
\(198\) 3.07064 0.218221
\(199\) 5.34906 0.379185 0.189593 0.981863i \(-0.439283\pi\)
0.189593 + 0.981863i \(0.439283\pi\)
\(200\) −0.326720 −0.0231026
\(201\) −6.12513 −0.432033
\(202\) −2.72101 −0.191450
\(203\) −14.8072 −1.03926
\(204\) −6.42341 −0.449728
\(205\) 18.2173 1.27235
\(206\) 1.00000 0.0696733
\(207\) −9.27355 −0.644557
\(208\) 1.00000 0.0693375
\(209\) 23.2806 1.61035
\(210\) −10.3824 −0.716451
\(211\) 17.0007 1.17038 0.585189 0.810897i \(-0.301020\pi\)
0.585189 + 0.810897i \(0.301020\pi\)
\(212\) 3.71472 0.255128
\(213\) −8.35045 −0.572164
\(214\) −3.60502 −0.246434
\(215\) −1.52377 −0.103920
\(216\) −1.00000 −0.0680414
\(217\) −23.9672 −1.62700
\(218\) 12.4950 0.846267
\(219\) 16.0060 1.08159
\(220\) −7.08695 −0.477802
\(221\) −6.42341 −0.432085
\(222\) −2.93933 −0.197275
\(223\) 7.92415 0.530640 0.265320 0.964160i \(-0.414522\pi\)
0.265320 + 0.964160i \(0.414522\pi\)
\(224\) −4.49849 −0.300568
\(225\) 0.326720 0.0217813
\(226\) 2.67954 0.178240
\(227\) −22.2557 −1.47716 −0.738580 0.674166i \(-0.764503\pi\)
−0.738580 + 0.674166i \(0.764503\pi\)
\(228\) −7.58167 −0.502108
\(229\) −11.7515 −0.776559 −0.388280 0.921542i \(-0.626931\pi\)
−0.388280 + 0.921542i \(0.626931\pi\)
\(230\) 21.4031 1.41128
\(231\) −13.8133 −0.908845
\(232\) 3.29159 0.216103
\(233\) −26.8321 −1.75783 −0.878915 0.476978i \(-0.841732\pi\)
−0.878915 + 0.476978i \(0.841732\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −23.8744 −1.55740
\(236\) −8.12616 −0.528968
\(237\) −14.8666 −0.965691
\(238\) 28.8956 1.87302
\(239\) 21.8917 1.41606 0.708029 0.706183i \(-0.249585\pi\)
0.708029 + 0.706183i \(0.249585\pi\)
\(240\) 2.30797 0.148979
\(241\) −26.7387 −1.72239 −0.861197 0.508271i \(-0.830285\pi\)
−0.861197 + 0.508271i \(0.830285\pi\)
\(242\) 1.57114 0.100997
\(243\) 1.00000 0.0641500
\(244\) 4.61685 0.295563
\(245\) 30.5491 1.95171
\(246\) −7.89321 −0.503253
\(247\) −7.58167 −0.482410
\(248\) 5.32783 0.338318
\(249\) −12.9601 −0.821311
\(250\) 10.7858 0.682153
\(251\) −15.2561 −0.962957 −0.481479 0.876458i \(-0.659900\pi\)
−0.481479 + 0.876458i \(0.659900\pi\)
\(252\) 4.49849 0.283378
\(253\) 28.4758 1.79026
\(254\) 12.4620 0.781934
\(255\) −14.8250 −0.928379
\(256\) 1.00000 0.0625000
\(257\) 16.8820 1.05307 0.526534 0.850154i \(-0.323491\pi\)
0.526534 + 0.850154i \(0.323491\pi\)
\(258\) 0.660219 0.0411035
\(259\) 13.2225 0.821608
\(260\) 2.30797 0.143134
\(261\) −3.29159 −0.203744
\(262\) −16.5343 −1.02150
\(263\) 16.9537 1.04541 0.522706 0.852513i \(-0.324923\pi\)
0.522706 + 0.852513i \(0.324923\pi\)
\(264\) 3.07064 0.188985
\(265\) 8.57346 0.526664
\(266\) 34.1060 2.09118
\(267\) 1.15775 0.0708532
\(268\) −6.12513 −0.374152
\(269\) −15.1010 −0.920725 −0.460362 0.887731i \(-0.652280\pi\)
−0.460362 + 0.887731i \(0.652280\pi\)
\(270\) −2.30797 −0.140459
\(271\) 19.6361 1.19281 0.596405 0.802683i \(-0.296595\pi\)
0.596405 + 0.802683i \(0.296595\pi\)
\(272\) −6.42341 −0.389476
\(273\) 4.49849 0.272261
\(274\) −13.6535 −0.824837
\(275\) −1.00324 −0.0604977
\(276\) −9.27355 −0.558202
\(277\) −23.0876 −1.38720 −0.693599 0.720361i \(-0.743976\pi\)
−0.693599 + 0.720361i \(0.743976\pi\)
\(278\) 7.77347 0.466222
\(279\) −5.32783 −0.318969
\(280\) −10.3824 −0.620465
\(281\) 14.6426 0.873502 0.436751 0.899582i \(-0.356129\pi\)
0.436751 + 0.899582i \(0.356129\pi\)
\(282\) 10.3443 0.615997
\(283\) 7.91544 0.470524 0.235262 0.971932i \(-0.424405\pi\)
0.235262 + 0.971932i \(0.424405\pi\)
\(284\) −8.35045 −0.495508
\(285\) −17.4983 −1.03651
\(286\) 3.07064 0.181571
\(287\) 35.5075 2.09594
\(288\) −1.00000 −0.0589256
\(289\) 24.2602 1.42707
\(290\) 7.59689 0.446105
\(291\) 5.29299 0.310280
\(292\) 16.0060 0.936683
\(293\) 11.8360 0.691467 0.345734 0.938333i \(-0.387630\pi\)
0.345734 + 0.938333i \(0.387630\pi\)
\(294\) −13.2364 −0.771961
\(295\) −18.7549 −1.09195
\(296\) −2.93933 −0.170845
\(297\) −3.07064 −0.178177
\(298\) 4.13520 0.239545
\(299\) −9.27355 −0.536303
\(300\) 0.326720 0.0188632
\(301\) −2.96999 −0.171187
\(302\) −16.9804 −0.977111
\(303\) 2.72101 0.156318
\(304\) −7.58167 −0.434839
\(305\) 10.6555 0.610135
\(306\) 6.42341 0.367202
\(307\) −19.8623 −1.13360 −0.566802 0.823854i \(-0.691820\pi\)
−0.566802 + 0.823854i \(0.691820\pi\)
\(308\) −13.8133 −0.787083
\(309\) −1.00000 −0.0568880
\(310\) 12.2965 0.698392
\(311\) −5.31128 −0.301175 −0.150587 0.988597i \(-0.548117\pi\)
−0.150587 + 0.988597i \(0.548117\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −15.0309 −0.849598 −0.424799 0.905288i \(-0.639655\pi\)
−0.424799 + 0.905288i \(0.639655\pi\)
\(314\) −11.2088 −0.632551
\(315\) 10.3824 0.584980
\(316\) −14.8666 −0.836313
\(317\) 8.94002 0.502122 0.251061 0.967971i \(-0.419221\pi\)
0.251061 + 0.967971i \(0.419221\pi\)
\(318\) −3.71472 −0.208311
\(319\) 10.1073 0.565900
\(320\) 2.30797 0.129019
\(321\) 3.60502 0.201213
\(322\) 41.7170 2.32479
\(323\) 48.7002 2.70975
\(324\) 1.00000 0.0555556
\(325\) 0.326720 0.0181232
\(326\) −16.1485 −0.894383
\(327\) −12.4950 −0.690974
\(328\) −7.89321 −0.435830
\(329\) −46.5339 −2.56550
\(330\) 7.08695 0.390124
\(331\) −19.7105 −1.08339 −0.541693 0.840576i \(-0.682217\pi\)
−0.541693 + 0.840576i \(0.682217\pi\)
\(332\) −12.9601 −0.711276
\(333\) 2.93933 0.161074
\(334\) 13.4298 0.734845
\(335\) −14.1366 −0.772365
\(336\) 4.49849 0.245413
\(337\) 10.6622 0.580807 0.290404 0.956904i \(-0.406210\pi\)
0.290404 + 0.956904i \(0.406210\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −2.67954 −0.145533
\(340\) −14.8250 −0.804000
\(341\) 16.3599 0.885937
\(342\) 7.58167 0.409970
\(343\) 28.0543 1.51479
\(344\) 0.660219 0.0355966
\(345\) −21.4031 −1.15230
\(346\) −19.5550 −1.05128
\(347\) 4.94453 0.265437 0.132718 0.991154i \(-0.457629\pi\)
0.132718 + 0.991154i \(0.457629\pi\)
\(348\) −3.29159 −0.176448
\(349\) 13.6533 0.730846 0.365423 0.930842i \(-0.380924\pi\)
0.365423 + 0.930842i \(0.380924\pi\)
\(350\) −1.46975 −0.0785612
\(351\) 1.00000 0.0533761
\(352\) 3.07064 0.163666
\(353\) 24.1245 1.28401 0.642007 0.766698i \(-0.278102\pi\)
0.642007 + 0.766698i \(0.278102\pi\)
\(354\) 8.12616 0.431901
\(355\) −19.2726 −1.02288
\(356\) 1.15775 0.0613607
\(357\) −28.8956 −1.52932
\(358\) −20.1020 −1.06242
\(359\) 32.9593 1.73952 0.869762 0.493471i \(-0.164272\pi\)
0.869762 + 0.493471i \(0.164272\pi\)
\(360\) −2.30797 −0.121641
\(361\) 38.4817 2.02535
\(362\) −7.14284 −0.375420
\(363\) −1.57114 −0.0824636
\(364\) 4.49849 0.235785
\(365\) 36.9415 1.93360
\(366\) −4.61685 −0.241327
\(367\) −19.8490 −1.03611 −0.518054 0.855348i \(-0.673344\pi\)
−0.518054 + 0.855348i \(0.673344\pi\)
\(368\) −9.27355 −0.483417
\(369\) 7.89321 0.410904
\(370\) −6.78388 −0.352677
\(371\) 16.7106 0.867572
\(372\) −5.32783 −0.276235
\(373\) 16.8027 0.870009 0.435004 0.900428i \(-0.356747\pi\)
0.435004 + 0.900428i \(0.356747\pi\)
\(374\) −19.7240 −1.01990
\(375\) −10.7858 −0.556976
\(376\) 10.3443 0.533469
\(377\) −3.29159 −0.169525
\(378\) −4.49849 −0.231377
\(379\) 15.2413 0.782895 0.391447 0.920200i \(-0.371974\pi\)
0.391447 + 0.920200i \(0.371974\pi\)
\(380\) −17.4983 −0.897642
\(381\) −12.4620 −0.638446
\(382\) 5.32169 0.272281
\(383\) −19.8328 −1.01341 −0.506704 0.862120i \(-0.669136\pi\)
−0.506704 + 0.862120i \(0.669136\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −31.8806 −1.62478
\(386\) −0.0650024 −0.00330853
\(387\) −0.660219 −0.0335608
\(388\) 5.29299 0.268711
\(389\) 4.41628 0.223914 0.111957 0.993713i \(-0.464288\pi\)
0.111957 + 0.993713i \(0.464288\pi\)
\(390\) −2.30797 −0.116869
\(391\) 59.5678 3.01247
\(392\) −13.2364 −0.668538
\(393\) 16.5343 0.834047
\(394\) 14.4690 0.728936
\(395\) −34.3117 −1.72641
\(396\) −3.07064 −0.154306
\(397\) 12.5999 0.632371 0.316185 0.948697i \(-0.397598\pi\)
0.316185 + 0.948697i \(0.397598\pi\)
\(398\) −5.34906 −0.268124
\(399\) −34.1060 −1.70744
\(400\) 0.326720 0.0163360
\(401\) 21.0919 1.05328 0.526641 0.850088i \(-0.323451\pi\)
0.526641 + 0.850088i \(0.323451\pi\)
\(402\) 6.12513 0.305494
\(403\) −5.32783 −0.265398
\(404\) 2.72101 0.135375
\(405\) 2.30797 0.114684
\(406\) 14.8072 0.734868
\(407\) −9.02564 −0.447384
\(408\) 6.42341 0.318006
\(409\) −28.0566 −1.38731 −0.693655 0.720308i \(-0.744001\pi\)
−0.693655 + 0.720308i \(0.744001\pi\)
\(410\) −18.2173 −0.899688
\(411\) 13.6535 0.673476
\(412\) −1.00000 −0.0492665
\(413\) −36.5554 −1.79878
\(414\) 9.27355 0.455770
\(415\) −29.9114 −1.46829
\(416\) −1.00000 −0.0490290
\(417\) −7.77347 −0.380669
\(418\) −23.2806 −1.13869
\(419\) −30.0038 −1.46578 −0.732890 0.680347i \(-0.761829\pi\)
−0.732890 + 0.680347i \(0.761829\pi\)
\(420\) 10.3824 0.506608
\(421\) −16.9752 −0.827319 −0.413660 0.910432i \(-0.635750\pi\)
−0.413660 + 0.910432i \(0.635750\pi\)
\(422\) −17.0007 −0.827582
\(423\) −10.3443 −0.502959
\(424\) −3.71472 −0.180403
\(425\) −2.09866 −0.101800
\(426\) 8.35045 0.404581
\(427\) 20.7688 1.00507
\(428\) 3.60502 0.174255
\(429\) −3.07064 −0.148252
\(430\) 1.52377 0.0734825
\(431\) 4.12181 0.198540 0.0992702 0.995061i \(-0.468349\pi\)
0.0992702 + 0.995061i \(0.468349\pi\)
\(432\) 1.00000 0.0481125
\(433\) 10.3207 0.495983 0.247991 0.968762i \(-0.420230\pi\)
0.247991 + 0.968762i \(0.420230\pi\)
\(434\) 23.9672 1.15046
\(435\) −7.59689 −0.364243
\(436\) −12.4950 −0.598401
\(437\) 70.3090 3.36334
\(438\) −16.0060 −0.764798
\(439\) 20.6072 0.983528 0.491764 0.870728i \(-0.336352\pi\)
0.491764 + 0.870728i \(0.336352\pi\)
\(440\) 7.08695 0.337857
\(441\) 13.2364 0.630304
\(442\) 6.42341 0.305530
\(443\) 11.9518 0.567845 0.283923 0.958847i \(-0.408364\pi\)
0.283923 + 0.958847i \(0.408364\pi\)
\(444\) 2.93933 0.139494
\(445\) 2.67205 0.126667
\(446\) −7.92415 −0.375219
\(447\) −4.13520 −0.195588
\(448\) 4.49849 0.212533
\(449\) −31.7564 −1.49868 −0.749339 0.662187i \(-0.769629\pi\)
−0.749339 + 0.662187i \(0.769629\pi\)
\(450\) −0.326720 −0.0154017
\(451\) −24.2372 −1.14129
\(452\) −2.67954 −0.126035
\(453\) 16.9804 0.797807
\(454\) 22.2557 1.04451
\(455\) 10.3824 0.486733
\(456\) 7.58167 0.355044
\(457\) 17.5169 0.819405 0.409702 0.912219i \(-0.365633\pi\)
0.409702 + 0.912219i \(0.365633\pi\)
\(458\) 11.7515 0.549110
\(459\) −6.42341 −0.299819
\(460\) −21.4031 −0.997923
\(461\) 29.9410 1.39449 0.697245 0.716832i \(-0.254409\pi\)
0.697245 + 0.716832i \(0.254409\pi\)
\(462\) 13.8133 0.642651
\(463\) 2.46778 0.114687 0.0573437 0.998354i \(-0.481737\pi\)
0.0573437 + 0.998354i \(0.481737\pi\)
\(464\) −3.29159 −0.152808
\(465\) −12.2965 −0.570235
\(466\) 26.8321 1.24297
\(467\) −7.56075 −0.349870 −0.174935 0.984580i \(-0.555972\pi\)
−0.174935 + 0.984580i \(0.555972\pi\)
\(468\) 1.00000 0.0462250
\(469\) −27.5538 −1.27232
\(470\) 23.8744 1.10124
\(471\) 11.2088 0.516476
\(472\) 8.12616 0.374037
\(473\) 2.02730 0.0932153
\(474\) 14.8666 0.682846
\(475\) −2.47708 −0.113656
\(476\) −28.8956 −1.32443
\(477\) 3.71472 0.170085
\(478\) −21.8917 −1.00130
\(479\) 13.0787 0.597579 0.298789 0.954319i \(-0.403417\pi\)
0.298789 + 0.954319i \(0.403417\pi\)
\(480\) −2.30797 −0.105344
\(481\) 2.93933 0.134022
\(482\) 26.7387 1.21792
\(483\) −41.7170 −1.89819
\(484\) −1.57114 −0.0714156
\(485\) 12.2160 0.554702
\(486\) −1.00000 −0.0453609
\(487\) 14.0815 0.638092 0.319046 0.947739i \(-0.396638\pi\)
0.319046 + 0.947739i \(0.396638\pi\)
\(488\) −4.61685 −0.208995
\(489\) 16.1485 0.730261
\(490\) −30.5491 −1.38007
\(491\) 25.6762 1.15875 0.579375 0.815061i \(-0.303297\pi\)
0.579375 + 0.815061i \(0.303297\pi\)
\(492\) 7.89321 0.355853
\(493\) 21.1432 0.952243
\(494\) 7.58167 0.341115
\(495\) −7.08695 −0.318535
\(496\) −5.32783 −0.239227
\(497\) −37.5644 −1.68499
\(498\) 12.9601 0.580755
\(499\) −11.4674 −0.513349 −0.256675 0.966498i \(-0.582627\pi\)
−0.256675 + 0.966498i \(0.582627\pi\)
\(500\) −10.7858 −0.482355
\(501\) −13.4298 −0.599999
\(502\) 15.2561 0.680913
\(503\) −13.5082 −0.602299 −0.301150 0.953577i \(-0.597370\pi\)
−0.301150 + 0.953577i \(0.597370\pi\)
\(504\) −4.49849 −0.200379
\(505\) 6.28000 0.279457
\(506\) −28.4758 −1.26590
\(507\) 1.00000 0.0444116
\(508\) −12.4620 −0.552911
\(509\) 8.23158 0.364859 0.182429 0.983219i \(-0.441604\pi\)
0.182429 + 0.983219i \(0.441604\pi\)
\(510\) 14.8250 0.656463
\(511\) 72.0030 3.18522
\(512\) −1.00000 −0.0441942
\(513\) −7.58167 −0.334739
\(514\) −16.8820 −0.744632
\(515\) −2.30797 −0.101701
\(516\) −0.660219 −0.0290645
\(517\) 31.7638 1.39697
\(518\) −13.2225 −0.580965
\(519\) 19.5550 0.858369
\(520\) −2.30797 −0.101211
\(521\) −6.99521 −0.306466 −0.153233 0.988190i \(-0.548968\pi\)
−0.153233 + 0.988190i \(0.548968\pi\)
\(522\) 3.29159 0.144069
\(523\) −14.8102 −0.647603 −0.323802 0.946125i \(-0.604961\pi\)
−0.323802 + 0.946125i \(0.604961\pi\)
\(524\) 16.5343 0.722306
\(525\) 1.46975 0.0641450
\(526\) −16.9537 −0.739218
\(527\) 34.2228 1.49077
\(528\) −3.07064 −0.133633
\(529\) 62.9988 2.73908
\(530\) −8.57346 −0.372407
\(531\) −8.12616 −0.352645
\(532\) −34.1060 −1.47868
\(533\) 7.89321 0.341893
\(534\) −1.15775 −0.0501008
\(535\) 8.32028 0.359717
\(536\) 6.12513 0.264565
\(537\) 20.1020 0.867465
\(538\) 15.1010 0.651051
\(539\) −40.6442 −1.75067
\(540\) 2.30797 0.0993192
\(541\) −42.2998 −1.81861 −0.909306 0.416128i \(-0.863387\pi\)
−0.909306 + 0.416128i \(0.863387\pi\)
\(542\) −19.6361 −0.843444
\(543\) 7.14284 0.306529
\(544\) 6.42341 0.275401
\(545\) −28.8380 −1.23529
\(546\) −4.49849 −0.192517
\(547\) −36.3830 −1.55563 −0.777813 0.628496i \(-0.783671\pi\)
−0.777813 + 0.628496i \(0.783671\pi\)
\(548\) 13.6535 0.583248
\(549\) 4.61685 0.197042
\(550\) 1.00324 0.0427784
\(551\) 24.9557 1.06315
\(552\) 9.27355 0.394709
\(553\) −66.8773 −2.84391
\(554\) 23.0876 0.980897
\(555\) 6.78388 0.287960
\(556\) −7.77347 −0.329669
\(557\) −41.6451 −1.76456 −0.882280 0.470725i \(-0.843992\pi\)
−0.882280 + 0.470725i \(0.843992\pi\)
\(558\) 5.32783 0.225545
\(559\) −0.660219 −0.0279243
\(560\) 10.3824 0.438735
\(561\) 19.7240 0.832748
\(562\) −14.6426 −0.617659
\(563\) 1.32087 0.0556680 0.0278340 0.999613i \(-0.491139\pi\)
0.0278340 + 0.999613i \(0.491139\pi\)
\(564\) −10.3443 −0.435575
\(565\) −6.18430 −0.260175
\(566\) −7.91544 −0.332711
\(567\) 4.49849 0.188919
\(568\) 8.35045 0.350377
\(569\) 7.64593 0.320534 0.160267 0.987074i \(-0.448764\pi\)
0.160267 + 0.987074i \(0.448764\pi\)
\(570\) 17.4983 0.732921
\(571\) −43.4082 −1.81658 −0.908288 0.418345i \(-0.862610\pi\)
−0.908288 + 0.418345i \(0.862610\pi\)
\(572\) −3.07064 −0.128390
\(573\) −5.32169 −0.222317
\(574\) −35.5075 −1.48205
\(575\) −3.02986 −0.126354
\(576\) 1.00000 0.0416667
\(577\) −16.3677 −0.681395 −0.340698 0.940173i \(-0.610663\pi\)
−0.340698 + 0.940173i \(0.610663\pi\)
\(578\) −24.2602 −1.00909
\(579\) 0.0650024 0.00270141
\(580\) −7.59689 −0.315444
\(581\) −58.3007 −2.41872
\(582\) −5.29299 −0.219401
\(583\) −11.4066 −0.472413
\(584\) −16.0060 −0.662335
\(585\) 2.30797 0.0954228
\(586\) −11.8360 −0.488941
\(587\) 11.4748 0.473614 0.236807 0.971557i \(-0.423899\pi\)
0.236807 + 0.971557i \(0.423899\pi\)
\(588\) 13.2364 0.545859
\(589\) 40.3939 1.66440
\(590\) 18.7549 0.772128
\(591\) −14.4690 −0.595174
\(592\) 2.93933 0.120806
\(593\) −38.6695 −1.58796 −0.793982 0.607941i \(-0.791996\pi\)
−0.793982 + 0.607941i \(0.791996\pi\)
\(594\) 3.07064 0.125990
\(595\) −66.6902 −2.73403
\(596\) −4.13520 −0.169384
\(597\) 5.34906 0.218923
\(598\) 9.27355 0.379224
\(599\) −38.0549 −1.55488 −0.777440 0.628958i \(-0.783482\pi\)
−0.777440 + 0.628958i \(0.783482\pi\)
\(600\) −0.326720 −0.0133383
\(601\) 7.28609 0.297206 0.148603 0.988897i \(-0.452522\pi\)
0.148603 + 0.988897i \(0.452522\pi\)
\(602\) 2.96999 0.121048
\(603\) −6.12513 −0.249435
\(604\) 16.9804 0.690922
\(605\) −3.62615 −0.147424
\(606\) −2.72101 −0.110533
\(607\) 24.2090 0.982611 0.491306 0.870987i \(-0.336520\pi\)
0.491306 + 0.870987i \(0.336520\pi\)
\(608\) 7.58167 0.307477
\(609\) −14.8072 −0.600017
\(610\) −10.6555 −0.431430
\(611\) −10.3443 −0.418487
\(612\) −6.42341 −0.259651
\(613\) −6.72585 −0.271654 −0.135827 0.990733i \(-0.543369\pi\)
−0.135827 + 0.990733i \(0.543369\pi\)
\(614\) 19.8623 0.801579
\(615\) 18.2173 0.734592
\(616\) 13.8133 0.556552
\(617\) −14.2561 −0.573929 −0.286965 0.957941i \(-0.592646\pi\)
−0.286965 + 0.957941i \(0.592646\pi\)
\(618\) 1.00000 0.0402259
\(619\) 14.7719 0.593732 0.296866 0.954919i \(-0.404059\pi\)
0.296866 + 0.954919i \(0.404059\pi\)
\(620\) −12.2965 −0.493838
\(621\) −9.27355 −0.372135
\(622\) 5.31128 0.212963
\(623\) 5.20813 0.208659
\(624\) 1.00000 0.0400320
\(625\) −26.5269 −1.06107
\(626\) 15.0309 0.600756
\(627\) 23.2806 0.929738
\(628\) 11.2088 0.447281
\(629\) −18.8805 −0.752815
\(630\) −10.3824 −0.413643
\(631\) 5.01834 0.199777 0.0998885 0.994999i \(-0.468151\pi\)
0.0998885 + 0.994999i \(0.468151\pi\)
\(632\) 14.8666 0.591362
\(633\) 17.0007 0.675718
\(634\) −8.94002 −0.355054
\(635\) −28.7619 −1.14138
\(636\) 3.71472 0.147298
\(637\) 13.2364 0.524444
\(638\) −10.1073 −0.400152
\(639\) −8.35045 −0.330339
\(640\) −2.30797 −0.0912305
\(641\) 3.33394 0.131683 0.0658414 0.997830i \(-0.479027\pi\)
0.0658414 + 0.997830i \(0.479027\pi\)
\(642\) −3.60502 −0.142279
\(643\) −4.71017 −0.185751 −0.0928756 0.995678i \(-0.529606\pi\)
−0.0928756 + 0.995678i \(0.529606\pi\)
\(644\) −41.7170 −1.64388
\(645\) −1.52377 −0.0599982
\(646\) −48.7002 −1.91608
\(647\) −38.2893 −1.50531 −0.752653 0.658417i \(-0.771226\pi\)
−0.752653 + 0.658417i \(0.771226\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 24.9526 0.979473
\(650\) −0.326720 −0.0128150
\(651\) −23.9672 −0.939347
\(652\) 16.1485 0.632425
\(653\) −28.1788 −1.10272 −0.551362 0.834266i \(-0.685892\pi\)
−0.551362 + 0.834266i \(0.685892\pi\)
\(654\) 12.4950 0.488593
\(655\) 38.1607 1.49106
\(656\) 7.89321 0.308178
\(657\) 16.0060 0.624455
\(658\) 46.5339 1.81408
\(659\) 50.0023 1.94781 0.973906 0.226950i \(-0.0728754\pi\)
0.973906 + 0.226950i \(0.0728754\pi\)
\(660\) −7.08695 −0.275859
\(661\) 27.3898 1.06534 0.532670 0.846323i \(-0.321189\pi\)
0.532670 + 0.846323i \(0.321189\pi\)
\(662\) 19.7105 0.766070
\(663\) −6.42341 −0.249464
\(664\) 12.9601 0.502948
\(665\) −78.7157 −3.05246
\(666\) −2.93933 −0.113897
\(667\) 30.5247 1.18192
\(668\) −13.4298 −0.519614
\(669\) 7.92415 0.306365
\(670\) 14.1366 0.546145
\(671\) −14.1767 −0.547286
\(672\) −4.49849 −0.173533
\(673\) 19.7615 0.761750 0.380875 0.924626i \(-0.375623\pi\)
0.380875 + 0.924626i \(0.375623\pi\)
\(674\) −10.6622 −0.410693
\(675\) 0.326720 0.0125755
\(676\) 1.00000 0.0384615
\(677\) 4.08334 0.156935 0.0784677 0.996917i \(-0.474997\pi\)
0.0784677 + 0.996917i \(0.474997\pi\)
\(678\) 2.67954 0.102907
\(679\) 23.8104 0.913760
\(680\) 14.8250 0.568514
\(681\) −22.2557 −0.852838
\(682\) −16.3599 −0.626452
\(683\) 19.9022 0.761535 0.380768 0.924671i \(-0.375660\pi\)
0.380768 + 0.924671i \(0.375660\pi\)
\(684\) −7.58167 −0.289892
\(685\) 31.5118 1.20400
\(686\) −28.0543 −1.07112
\(687\) −11.7515 −0.448347
\(688\) −0.660219 −0.0251706
\(689\) 3.71472 0.141520
\(690\) 21.4031 0.814801
\(691\) −10.7309 −0.408222 −0.204111 0.978948i \(-0.565430\pi\)
−0.204111 + 0.978948i \(0.565430\pi\)
\(692\) 19.5550 0.743369
\(693\) −13.8133 −0.524722
\(694\) −4.94453 −0.187692
\(695\) −17.9409 −0.680538
\(696\) 3.29159 0.124767
\(697\) −50.7013 −1.92045
\(698\) −13.6533 −0.516786
\(699\) −26.8321 −1.01488
\(700\) 1.46975 0.0555512
\(701\) −42.0313 −1.58750 −0.793750 0.608244i \(-0.791874\pi\)
−0.793750 + 0.608244i \(0.791874\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −22.2850 −0.840496
\(704\) −3.07064 −0.115729
\(705\) −23.8744 −0.899163
\(706\) −24.1245 −0.907936
\(707\) 12.2404 0.460348
\(708\) −8.12616 −0.305400
\(709\) 34.0159 1.27749 0.638747 0.769417i \(-0.279453\pi\)
0.638747 + 0.769417i \(0.279453\pi\)
\(710\) 19.2726 0.723287
\(711\) −14.8666 −0.557542
\(712\) −1.15775 −0.0433886
\(713\) 49.4079 1.85034
\(714\) 28.8956 1.08139
\(715\) −7.08695 −0.265037
\(716\) 20.1020 0.751247
\(717\) 21.8917 0.817561
\(718\) −32.9593 −1.23003
\(719\) −26.7071 −0.996007 −0.498003 0.867175i \(-0.665933\pi\)
−0.498003 + 0.867175i \(0.665933\pi\)
\(720\) 2.30797 0.0860129
\(721\) −4.49849 −0.167532
\(722\) −38.4817 −1.43214
\(723\) −26.7387 −0.994425
\(724\) 7.14284 0.265462
\(725\) −1.07543 −0.0399404
\(726\) 1.57114 0.0583106
\(727\) 41.4609 1.53770 0.768850 0.639429i \(-0.220829\pi\)
0.768850 + 0.639429i \(0.220829\pi\)
\(728\) −4.49849 −0.166725
\(729\) 1.00000 0.0370370
\(730\) −36.9415 −1.36726
\(731\) 4.24086 0.156854
\(732\) 4.61685 0.170644
\(733\) 45.1082 1.66611 0.833054 0.553192i \(-0.186590\pi\)
0.833054 + 0.553192i \(0.186590\pi\)
\(734\) 19.8490 0.732640
\(735\) 30.5491 1.12682
\(736\) 9.27355 0.341828
\(737\) 18.8081 0.692805
\(738\) −7.89321 −0.290553
\(739\) 7.71378 0.283756 0.141878 0.989884i \(-0.454686\pi\)
0.141878 + 0.989884i \(0.454686\pi\)
\(740\) 6.78388 0.249380
\(741\) −7.58167 −0.278520
\(742\) −16.7106 −0.613466
\(743\) 19.7540 0.724705 0.362353 0.932041i \(-0.381974\pi\)
0.362353 + 0.932041i \(0.381974\pi\)
\(744\) 5.32783 0.195328
\(745\) −9.54390 −0.349662
\(746\) −16.8027 −0.615189
\(747\) −12.9601 −0.474184
\(748\) 19.7240 0.721181
\(749\) 16.2171 0.592562
\(750\) 10.7858 0.393841
\(751\) −17.9708 −0.655763 −0.327881 0.944719i \(-0.606335\pi\)
−0.327881 + 0.944719i \(0.606335\pi\)
\(752\) −10.3443 −0.377219
\(753\) −15.2561 −0.555963
\(754\) 3.29159 0.119873
\(755\) 39.1902 1.42628
\(756\) 4.49849 0.163608
\(757\) −21.0114 −0.763672 −0.381836 0.924230i \(-0.624708\pi\)
−0.381836 + 0.924230i \(0.624708\pi\)
\(758\) −15.2413 −0.553590
\(759\) 28.4758 1.03361
\(760\) 17.4983 0.634729
\(761\) −9.48934 −0.343988 −0.171994 0.985098i \(-0.555021\pi\)
−0.171994 + 0.985098i \(0.555021\pi\)
\(762\) 12.4620 0.451450
\(763\) −56.2085 −2.03489
\(764\) −5.32169 −0.192532
\(765\) −14.8250 −0.536000
\(766\) 19.8328 0.716588
\(767\) −8.12616 −0.293419
\(768\) 1.00000 0.0360844
\(769\) 47.3283 1.70670 0.853350 0.521338i \(-0.174567\pi\)
0.853350 + 0.521338i \(0.174567\pi\)
\(770\) 31.8806 1.14890
\(771\) 16.8820 0.607989
\(772\) 0.0650024 0.00233949
\(773\) −46.6715 −1.67866 −0.839328 0.543625i \(-0.817051\pi\)
−0.839328 + 0.543625i \(0.817051\pi\)
\(774\) 0.660219 0.0237311
\(775\) −1.74071 −0.0625281
\(776\) −5.29299 −0.190007
\(777\) 13.2225 0.474356
\(778\) −4.41628 −0.158331
\(779\) −59.8437 −2.14412
\(780\) 2.30797 0.0826385
\(781\) 25.6413 0.917517
\(782\) −59.5678 −2.13014
\(783\) −3.29159 −0.117632
\(784\) 13.2364 0.472728
\(785\) 25.8696 0.923327
\(786\) −16.5343 −0.589760
\(787\) −44.3381 −1.58048 −0.790242 0.612795i \(-0.790045\pi\)
−0.790242 + 0.612795i \(0.790045\pi\)
\(788\) −14.4690 −0.515436
\(789\) 16.9537 0.603569
\(790\) 34.3117 1.22076
\(791\) −12.0539 −0.428587
\(792\) 3.07064 0.109111
\(793\) 4.61685 0.163949
\(794\) −12.5999 −0.447153
\(795\) 8.57346 0.304069
\(796\) 5.34906 0.189593
\(797\) 27.9432 0.989799 0.494900 0.868950i \(-0.335205\pi\)
0.494900 + 0.868950i \(0.335205\pi\)
\(798\) 34.1060 1.20734
\(799\) 66.4459 2.35069
\(800\) −0.326720 −0.0115513
\(801\) 1.15775 0.0409071
\(802\) −21.0919 −0.744782
\(803\) −49.1489 −1.73443
\(804\) −6.12513 −0.216017
\(805\) −96.2814 −3.39347
\(806\) 5.32783 0.187665
\(807\) −15.1010 −0.531581
\(808\) −2.72101 −0.0957248
\(809\) −49.8493 −1.75261 −0.876305 0.481757i \(-0.839999\pi\)
−0.876305 + 0.481757i \(0.839999\pi\)
\(810\) −2.30797 −0.0810938
\(811\) −4.89957 −0.172047 −0.0860235 0.996293i \(-0.527416\pi\)
−0.0860235 + 0.996293i \(0.527416\pi\)
\(812\) −14.8072 −0.519630
\(813\) 19.6361 0.688670
\(814\) 9.02564 0.316348
\(815\) 37.2703 1.30552
\(816\) −6.42341 −0.224864
\(817\) 5.00557 0.175123
\(818\) 28.0566 0.980976
\(819\) 4.49849 0.157190
\(820\) 18.2173 0.636175
\(821\) −22.7975 −0.795637 −0.397818 0.917464i \(-0.630233\pi\)
−0.397818 + 0.917464i \(0.630233\pi\)
\(822\) −13.6535 −0.476220
\(823\) −42.6615 −1.48709 −0.743543 0.668688i \(-0.766856\pi\)
−0.743543 + 0.668688i \(0.766856\pi\)
\(824\) 1.00000 0.0348367
\(825\) −1.00324 −0.0349284
\(826\) 36.5554 1.27193
\(827\) 9.16292 0.318626 0.159313 0.987228i \(-0.449072\pi\)
0.159313 + 0.987228i \(0.449072\pi\)
\(828\) −9.27355 −0.322278
\(829\) 25.5976 0.889041 0.444520 0.895769i \(-0.353374\pi\)
0.444520 + 0.895769i \(0.353374\pi\)
\(830\) 29.9114 1.03824
\(831\) −23.0876 −0.800899
\(832\) 1.00000 0.0346688
\(833\) −85.0226 −2.94586
\(834\) 7.77347 0.269173
\(835\) −30.9955 −1.07264
\(836\) 23.2806 0.805177
\(837\) −5.32783 −0.184157
\(838\) 30.0038 1.03646
\(839\) 9.62037 0.332132 0.166066 0.986115i \(-0.446894\pi\)
0.166066 + 0.986115i \(0.446894\pi\)
\(840\) −10.3824 −0.358226
\(841\) −18.1654 −0.626394
\(842\) 16.9752 0.585003
\(843\) 14.6426 0.504317
\(844\) 17.0007 0.585189
\(845\) 2.30797 0.0793965
\(846\) 10.3443 0.355646
\(847\) −7.06777 −0.242851
\(848\) 3.71472 0.127564
\(849\) 7.91544 0.271657
\(850\) 2.09866 0.0719833
\(851\) −27.2580 −0.934393
\(852\) −8.35045 −0.286082
\(853\) 8.42538 0.288479 0.144240 0.989543i \(-0.453926\pi\)
0.144240 + 0.989543i \(0.453926\pi\)
\(854\) −20.7688 −0.710695
\(855\) −17.4983 −0.598428
\(856\) −3.60502 −0.123217
\(857\) −45.2094 −1.54432 −0.772161 0.635427i \(-0.780824\pi\)
−0.772161 + 0.635427i \(0.780824\pi\)
\(858\) 3.07064 0.104830
\(859\) 57.5928 1.96504 0.982521 0.186151i \(-0.0596013\pi\)
0.982521 + 0.186151i \(0.0596013\pi\)
\(860\) −1.52377 −0.0519600
\(861\) 35.5075 1.21009
\(862\) −4.12181 −0.140389
\(863\) 50.1018 1.70548 0.852742 0.522332i \(-0.174938\pi\)
0.852742 + 0.522332i \(0.174938\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 45.1323 1.53454
\(866\) −10.3207 −0.350713
\(867\) 24.2602 0.823918
\(868\) −23.9672 −0.813499
\(869\) 45.6501 1.54857
\(870\) 7.59689 0.257559
\(871\) −6.12513 −0.207542
\(872\) 12.4950 0.423134
\(873\) 5.29299 0.179140
\(874\) −70.3090 −2.37824
\(875\) −48.5197 −1.64027
\(876\) 16.0060 0.540794
\(877\) −12.1544 −0.410423 −0.205212 0.978718i \(-0.565788\pi\)
−0.205212 + 0.978718i \(0.565788\pi\)
\(878\) −20.6072 −0.695459
\(879\) 11.8360 0.399219
\(880\) −7.08695 −0.238901
\(881\) 46.8341 1.57788 0.788940 0.614470i \(-0.210630\pi\)
0.788940 + 0.614470i \(0.210630\pi\)
\(882\) −13.2364 −0.445692
\(883\) 8.24979 0.277628 0.138814 0.990318i \(-0.455671\pi\)
0.138814 + 0.990318i \(0.455671\pi\)
\(884\) −6.42341 −0.216043
\(885\) −18.7549 −0.630440
\(886\) −11.9518 −0.401527
\(887\) −19.1402 −0.642666 −0.321333 0.946966i \(-0.604131\pi\)
−0.321333 + 0.946966i \(0.604131\pi\)
\(888\) −2.93933 −0.0986374
\(889\) −56.0600 −1.88019
\(890\) −2.67205 −0.0895674
\(891\) −3.07064 −0.102870
\(892\) 7.92415 0.265320
\(893\) 78.4274 2.62447
\(894\) 4.13520 0.138302
\(895\) 46.3948 1.55081
\(896\) −4.49849 −0.150284
\(897\) −9.27355 −0.309635
\(898\) 31.7564 1.05973
\(899\) 17.5370 0.584893
\(900\) 0.326720 0.0108907
\(901\) −23.8612 −0.794931
\(902\) 24.2372 0.807012
\(903\) −2.96999 −0.0988350
\(904\) 2.67954 0.0891202
\(905\) 16.4855 0.547995
\(906\) −16.9804 −0.564135
\(907\) −29.1930 −0.969338 −0.484669 0.874698i \(-0.661060\pi\)
−0.484669 + 0.874698i \(0.661060\pi\)
\(908\) −22.2557 −0.738580
\(909\) 2.72101 0.0902502
\(910\) −10.3824 −0.344172
\(911\) −25.1881 −0.834519 −0.417259 0.908787i \(-0.637009\pi\)
−0.417259 + 0.908787i \(0.637009\pi\)
\(912\) −7.58167 −0.251054
\(913\) 39.7958 1.31705
\(914\) −17.5169 −0.579407
\(915\) 10.6555 0.352261
\(916\) −11.7515 −0.388280
\(917\) 74.3795 2.45623
\(918\) 6.42341 0.212004
\(919\) 48.1378 1.58792 0.793960 0.607970i \(-0.208016\pi\)
0.793960 + 0.607970i \(0.208016\pi\)
\(920\) 21.4031 0.705638
\(921\) −19.8623 −0.654486
\(922\) −29.9410 −0.986054
\(923\) −8.35045 −0.274858
\(924\) −13.8133 −0.454423
\(925\) 0.960338 0.0315757
\(926\) −2.46778 −0.0810962
\(927\) −1.00000 −0.0328443
\(928\) 3.29159 0.108052
\(929\) −51.1981 −1.67975 −0.839877 0.542776i \(-0.817373\pi\)
−0.839877 + 0.542776i \(0.817373\pi\)
\(930\) 12.2965 0.403217
\(931\) −100.354 −3.28896
\(932\) −26.8321 −0.878915
\(933\) −5.31128 −0.173883
\(934\) 7.56075 0.247395
\(935\) 45.5224 1.48874
\(936\) −1.00000 −0.0326860
\(937\) 1.68937 0.0551894 0.0275947 0.999619i \(-0.491215\pi\)
0.0275947 + 0.999619i \(0.491215\pi\)
\(938\) 27.5538 0.899664
\(939\) −15.0309 −0.490515
\(940\) −23.8744 −0.778698
\(941\) 6.17306 0.201236 0.100618 0.994925i \(-0.467918\pi\)
0.100618 + 0.994925i \(0.467918\pi\)
\(942\) −11.2088 −0.365204
\(943\) −73.1981 −2.38366
\(944\) −8.12616 −0.264484
\(945\) 10.3824 0.337738
\(946\) −2.02730 −0.0659132
\(947\) −27.0779 −0.879912 −0.439956 0.898019i \(-0.645006\pi\)
−0.439956 + 0.898019i \(0.645006\pi\)
\(948\) −14.8666 −0.482845
\(949\) 16.0060 0.519578
\(950\) 2.47708 0.0803672
\(951\) 8.94002 0.289900
\(952\) 28.8956 0.936512
\(953\) −2.10774 −0.0682765 −0.0341383 0.999417i \(-0.510869\pi\)
−0.0341383 + 0.999417i \(0.510869\pi\)
\(954\) −3.71472 −0.120269
\(955\) −12.2823 −0.397446
\(956\) 21.8917 0.708029
\(957\) 10.1073 0.326723
\(958\) −13.0787 −0.422552
\(959\) 61.4200 1.98335
\(960\) 2.30797 0.0744894
\(961\) −2.61423 −0.0843300
\(962\) −2.93933 −0.0947678
\(963\) 3.60502 0.116170
\(964\) −26.7387 −0.861197
\(965\) 0.150023 0.00482943
\(966\) 41.7170 1.34222
\(967\) 26.1368 0.840502 0.420251 0.907408i \(-0.361942\pi\)
0.420251 + 0.907408i \(0.361942\pi\)
\(968\) 1.57114 0.0504984
\(969\) 48.7002 1.56447
\(970\) −12.2160 −0.392234
\(971\) 38.0132 1.21990 0.609951 0.792439i \(-0.291189\pi\)
0.609951 + 0.792439i \(0.291189\pi\)
\(972\) 1.00000 0.0320750
\(973\) −34.9689 −1.12105
\(974\) −14.0815 −0.451199
\(975\) 0.326720 0.0104634
\(976\) 4.61685 0.147782
\(977\) 23.1277 0.739920 0.369960 0.929048i \(-0.379371\pi\)
0.369960 + 0.929048i \(0.379371\pi\)
\(978\) −16.1485 −0.516372
\(979\) −3.55504 −0.113620
\(980\) 30.5491 0.975857
\(981\) −12.4950 −0.398934
\(982\) −25.6762 −0.819360
\(983\) −6.20487 −0.197905 −0.0989523 0.995092i \(-0.531549\pi\)
−0.0989523 + 0.995092i \(0.531549\pi\)
\(984\) −7.89321 −0.251626
\(985\) −33.3939 −1.06402
\(986\) −21.1432 −0.673337
\(987\) −46.5339 −1.48119
\(988\) −7.58167 −0.241205
\(989\) 6.12258 0.194687
\(990\) 7.08695 0.225238
\(991\) 28.9708 0.920287 0.460143 0.887845i \(-0.347798\pi\)
0.460143 + 0.887845i \(0.347798\pi\)
\(992\) 5.32783 0.169159
\(993\) −19.7105 −0.625493
\(994\) 37.5644 1.19147
\(995\) 12.3455 0.391378
\(996\) −12.9601 −0.410656
\(997\) −27.3521 −0.866250 −0.433125 0.901334i \(-0.642589\pi\)
−0.433125 + 0.901334i \(0.642589\pi\)
\(998\) 11.4674 0.362993
\(999\) 2.93933 0.0929963
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 8034.2.a.w.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.11 12 1.1 even 1 trivial