Properties

Label 8034.2.a.w.1.5
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 1.5
Root \(1.38143\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.60911 q^{5} -1.00000 q^{6} +2.84255 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.60911 q^{5} -1.00000 q^{6} +2.84255 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.60911 q^{10} -4.57260 q^{11} +1.00000 q^{12} +1.00000 q^{13} -2.84255 q^{14} -1.60911 q^{15} +1.00000 q^{16} +6.96515 q^{17} -1.00000 q^{18} +5.10639 q^{19} -1.60911 q^{20} +2.84255 q^{21} +4.57260 q^{22} -7.39805 q^{23} -1.00000 q^{24} -2.41076 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.84255 q^{28} -7.17077 q^{29} +1.60911 q^{30} +4.44063 q^{31} -1.00000 q^{32} -4.57260 q^{33} -6.96515 q^{34} -4.57398 q^{35} +1.00000 q^{36} -0.817018 q^{37} -5.10639 q^{38} +1.00000 q^{39} +1.60911 q^{40} -11.4476 q^{41} -2.84255 q^{42} +4.69111 q^{43} -4.57260 q^{44} -1.60911 q^{45} +7.39805 q^{46} -6.40573 q^{47} +1.00000 q^{48} +1.08009 q^{49} +2.41076 q^{50} +6.96515 q^{51} +1.00000 q^{52} -8.74627 q^{53} -1.00000 q^{54} +7.35782 q^{55} -2.84255 q^{56} +5.10639 q^{57} +7.17077 q^{58} +0.862231 q^{59} -1.60911 q^{60} -5.94901 q^{61} -4.44063 q^{62} +2.84255 q^{63} +1.00000 q^{64} -1.60911 q^{65} +4.57260 q^{66} -5.12966 q^{67} +6.96515 q^{68} -7.39805 q^{69} +4.57398 q^{70} +5.91999 q^{71} -1.00000 q^{72} -1.48512 q^{73} +0.817018 q^{74} -2.41076 q^{75} +5.10639 q^{76} -12.9978 q^{77} -1.00000 q^{78} -16.1372 q^{79} -1.60911 q^{80} +1.00000 q^{81} +11.4476 q^{82} +7.18794 q^{83} +2.84255 q^{84} -11.2077 q^{85} -4.69111 q^{86} -7.17077 q^{87} +4.57260 q^{88} +13.9273 q^{89} +1.60911 q^{90} +2.84255 q^{91} -7.39805 q^{92} +4.44063 q^{93} +6.40573 q^{94} -8.21675 q^{95} -1.00000 q^{96} -2.58490 q^{97} -1.08009 q^{98} -4.57260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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
\(5\) −1.60911 −0.719616 −0.359808 0.933026i \(-0.617158\pi\)
−0.359808 + 0.933026i \(0.617158\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.84255 1.07438 0.537192 0.843460i \(-0.319485\pi\)
0.537192 + 0.843460i \(0.319485\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.60911 0.508846
\(11\) −4.57260 −1.37869 −0.689345 0.724433i \(-0.742102\pi\)
−0.689345 + 0.724433i \(0.742102\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −2.84255 −0.759704
\(15\) −1.60911 −0.415471
\(16\) 1.00000 0.250000
\(17\) 6.96515 1.68930 0.844648 0.535322i \(-0.179810\pi\)
0.844648 + 0.535322i \(0.179810\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.10639 1.17149 0.585743 0.810497i \(-0.300803\pi\)
0.585743 + 0.810497i \(0.300803\pi\)
\(20\) −1.60911 −0.359808
\(21\) 2.84255 0.620295
\(22\) 4.57260 0.974881
\(23\) −7.39805 −1.54260 −0.771300 0.636471i \(-0.780393\pi\)
−0.771300 + 0.636471i \(0.780393\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.41076 −0.482152
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 2.84255 0.537192
\(29\) −7.17077 −1.33158 −0.665789 0.746140i \(-0.731905\pi\)
−0.665789 + 0.746140i \(0.731905\pi\)
\(30\) 1.60911 0.293782
\(31\) 4.44063 0.797560 0.398780 0.917047i \(-0.369434\pi\)
0.398780 + 0.917047i \(0.369434\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.57260 −0.795987
\(34\) −6.96515 −1.19451
\(35\) −4.57398 −0.773144
\(36\) 1.00000 0.166667
\(37\) −0.817018 −0.134317 −0.0671584 0.997742i \(-0.521393\pi\)
−0.0671584 + 0.997742i \(0.521393\pi\)
\(38\) −5.10639 −0.828366
\(39\) 1.00000 0.160128
\(40\) 1.60911 0.254423
\(41\) −11.4476 −1.78782 −0.893911 0.448244i \(-0.852050\pi\)
−0.893911 + 0.448244i \(0.852050\pi\)
\(42\) −2.84255 −0.438615
\(43\) 4.69111 0.715387 0.357693 0.933839i \(-0.383563\pi\)
0.357693 + 0.933839i \(0.383563\pi\)
\(44\) −4.57260 −0.689345
\(45\) −1.60911 −0.239872
\(46\) 7.39805 1.09078
\(47\) −6.40573 −0.934372 −0.467186 0.884159i \(-0.654732\pi\)
−0.467186 + 0.884159i \(0.654732\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.08009 0.154299
\(50\) 2.41076 0.340933
\(51\) 6.96515 0.975315
\(52\) 1.00000 0.138675
\(53\) −8.74627 −1.20139 −0.600696 0.799477i \(-0.705110\pi\)
−0.600696 + 0.799477i \(0.705110\pi\)
\(54\) −1.00000 −0.136083
\(55\) 7.35782 0.992128
\(56\) −2.84255 −0.379852
\(57\) 5.10639 0.676358
\(58\) 7.17077 0.941568
\(59\) 0.862231 0.112253 0.0561265 0.998424i \(-0.482125\pi\)
0.0561265 + 0.998424i \(0.482125\pi\)
\(60\) −1.60911 −0.207735
\(61\) −5.94901 −0.761693 −0.380847 0.924638i \(-0.624367\pi\)
−0.380847 + 0.924638i \(0.624367\pi\)
\(62\) −4.44063 −0.563960
\(63\) 2.84255 0.358128
\(64\) 1.00000 0.125000
\(65\) −1.60911 −0.199586
\(66\) 4.57260 0.562848
\(67\) −5.12966 −0.626687 −0.313344 0.949640i \(-0.601449\pi\)
−0.313344 + 0.949640i \(0.601449\pi\)
\(68\) 6.96515 0.844648
\(69\) −7.39805 −0.890621
\(70\) 4.57398 0.546695
\(71\) 5.91999 0.702574 0.351287 0.936268i \(-0.385744\pi\)
0.351287 + 0.936268i \(0.385744\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.48512 −0.173821 −0.0869103 0.996216i \(-0.527699\pi\)
−0.0869103 + 0.996216i \(0.527699\pi\)
\(74\) 0.817018 0.0949764
\(75\) −2.41076 −0.278371
\(76\) 5.10639 0.585743
\(77\) −12.9978 −1.48124
\(78\) −1.00000 −0.113228
\(79\) −16.1372 −1.81557 −0.907786 0.419433i \(-0.862229\pi\)
−0.907786 + 0.419433i \(0.862229\pi\)
\(80\) −1.60911 −0.179904
\(81\) 1.00000 0.111111
\(82\) 11.4476 1.26418
\(83\) 7.18794 0.788979 0.394489 0.918900i \(-0.370921\pi\)
0.394489 + 0.918900i \(0.370921\pi\)
\(84\) 2.84255 0.310148
\(85\) −11.2077 −1.21564
\(86\) −4.69111 −0.505855
\(87\) −7.17077 −0.768787
\(88\) 4.57260 0.487441
\(89\) 13.9273 1.47629 0.738143 0.674645i \(-0.235703\pi\)
0.738143 + 0.674645i \(0.235703\pi\)
\(90\) 1.60911 0.169615
\(91\) 2.84255 0.297980
\(92\) −7.39805 −0.771300
\(93\) 4.44063 0.460472
\(94\) 6.40573 0.660701
\(95\) −8.21675 −0.843021
\(96\) −1.00000 −0.102062
\(97\) −2.58490 −0.262457 −0.131228 0.991352i \(-0.541892\pi\)
−0.131228 + 0.991352i \(0.541892\pi\)
\(98\) −1.08009 −0.109106
\(99\) −4.57260 −0.459564
\(100\) −2.41076 −0.241076
\(101\) 2.68909 0.267574 0.133787 0.991010i \(-0.457286\pi\)
0.133787 + 0.991010i \(0.457286\pi\)
\(102\) −6.96515 −0.689652
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −4.57398 −0.446375
\(106\) 8.74627 0.849513
\(107\) 15.4507 1.49367 0.746837 0.665008i \(-0.231572\pi\)
0.746837 + 0.665008i \(0.231572\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.02806 −0.290036 −0.145018 0.989429i \(-0.546324\pi\)
−0.145018 + 0.989429i \(0.546324\pi\)
\(110\) −7.35782 −0.701541
\(111\) −0.817018 −0.0775479
\(112\) 2.84255 0.268596
\(113\) −4.47739 −0.421198 −0.210599 0.977573i \(-0.567541\pi\)
−0.210599 + 0.977573i \(0.567541\pi\)
\(114\) −5.10639 −0.478257
\(115\) 11.9043 1.11008
\(116\) −7.17077 −0.665789
\(117\) 1.00000 0.0924500
\(118\) −0.862231 −0.0793748
\(119\) 19.7988 1.81495
\(120\) 1.60911 0.146891
\(121\) 9.90866 0.900788
\(122\) 5.94901 0.538598
\(123\) −11.4476 −1.03220
\(124\) 4.44063 0.398780
\(125\) 11.9247 1.06658
\(126\) −2.84255 −0.253235
\(127\) 1.94343 0.172452 0.0862260 0.996276i \(-0.472519\pi\)
0.0862260 + 0.996276i \(0.472519\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.69111 0.413029
\(130\) 1.60911 0.141128
\(131\) 5.44624 0.475841 0.237920 0.971285i \(-0.423534\pi\)
0.237920 + 0.971285i \(0.423534\pi\)
\(132\) −4.57260 −0.397994
\(133\) 14.5152 1.25862
\(134\) 5.12966 0.443135
\(135\) −1.60911 −0.138490
\(136\) −6.96515 −0.597256
\(137\) −5.97882 −0.510805 −0.255403 0.966835i \(-0.582208\pi\)
−0.255403 + 0.966835i \(0.582208\pi\)
\(138\) 7.39805 0.629764
\(139\) 0.970755 0.0823384 0.0411692 0.999152i \(-0.486892\pi\)
0.0411692 + 0.999152i \(0.486892\pi\)
\(140\) −4.57398 −0.386572
\(141\) −6.40573 −0.539460
\(142\) −5.91999 −0.496795
\(143\) −4.57260 −0.382380
\(144\) 1.00000 0.0833333
\(145\) 11.5386 0.958225
\(146\) 1.48512 0.122910
\(147\) 1.08009 0.0890846
\(148\) −0.817018 −0.0671584
\(149\) 3.63472 0.297768 0.148884 0.988855i \(-0.452432\pi\)
0.148884 + 0.988855i \(0.452432\pi\)
\(150\) 2.41076 0.196838
\(151\) −8.48578 −0.690563 −0.345281 0.938499i \(-0.612217\pi\)
−0.345281 + 0.938499i \(0.612217\pi\)
\(152\) −5.10639 −0.414183
\(153\) 6.96515 0.563099
\(154\) 12.9978 1.04740
\(155\) −7.14546 −0.573937
\(156\) 1.00000 0.0800641
\(157\) 1.63370 0.130383 0.0651917 0.997873i \(-0.479234\pi\)
0.0651917 + 0.997873i \(0.479234\pi\)
\(158\) 16.1372 1.28380
\(159\) −8.74627 −0.693624
\(160\) 1.60911 0.127211
\(161\) −21.0293 −1.65734
\(162\) −1.00000 −0.0785674
\(163\) −11.6549 −0.912881 −0.456441 0.889754i \(-0.650876\pi\)
−0.456441 + 0.889754i \(0.650876\pi\)
\(164\) −11.4476 −0.893911
\(165\) 7.35782 0.572806
\(166\) −7.18794 −0.557892
\(167\) −10.1875 −0.788333 −0.394167 0.919039i \(-0.628967\pi\)
−0.394167 + 0.919039i \(0.628967\pi\)
\(168\) −2.84255 −0.219308
\(169\) 1.00000 0.0769231
\(170\) 11.2077 0.859591
\(171\) 5.10639 0.390495
\(172\) 4.69111 0.357693
\(173\) −4.35726 −0.331276 −0.165638 0.986187i \(-0.552968\pi\)
−0.165638 + 0.986187i \(0.552968\pi\)
\(174\) 7.17077 0.543614
\(175\) −6.85271 −0.518016
\(176\) −4.57260 −0.344673
\(177\) 0.862231 0.0648093
\(178\) −13.9273 −1.04389
\(179\) 14.7457 1.10215 0.551074 0.834456i \(-0.314218\pi\)
0.551074 + 0.834456i \(0.314218\pi\)
\(180\) −1.60911 −0.119936
\(181\) −20.1298 −1.49624 −0.748118 0.663566i \(-0.769042\pi\)
−0.748118 + 0.663566i \(0.769042\pi\)
\(182\) −2.84255 −0.210704
\(183\) −5.94901 −0.439764
\(184\) 7.39805 0.545392
\(185\) 1.31467 0.0966566
\(186\) −4.44063 −0.325603
\(187\) −31.8488 −2.32902
\(188\) −6.40573 −0.467186
\(189\) 2.84255 0.206765
\(190\) 8.21675 0.596106
\(191\) −18.6504 −1.34949 −0.674747 0.738049i \(-0.735747\pi\)
−0.674747 + 0.738049i \(0.735747\pi\)
\(192\) 1.00000 0.0721688
\(193\) 3.50650 0.252403 0.126202 0.992005i \(-0.459721\pi\)
0.126202 + 0.992005i \(0.459721\pi\)
\(194\) 2.58490 0.185585
\(195\) −1.60911 −0.115231
\(196\) 1.08009 0.0771496
\(197\) −2.08818 −0.148776 −0.0743882 0.997229i \(-0.523700\pi\)
−0.0743882 + 0.997229i \(0.523700\pi\)
\(198\) 4.57260 0.324960
\(199\) −4.31102 −0.305600 −0.152800 0.988257i \(-0.548829\pi\)
−0.152800 + 0.988257i \(0.548829\pi\)
\(200\) 2.41076 0.170467
\(201\) −5.12966 −0.361818
\(202\) −2.68909 −0.189204
\(203\) −20.3833 −1.43062
\(204\) 6.96515 0.487658
\(205\) 18.4205 1.28655
\(206\) 1.00000 0.0696733
\(207\) −7.39805 −0.514200
\(208\) 1.00000 0.0693375
\(209\) −23.3495 −1.61512
\(210\) 4.57398 0.315635
\(211\) −14.1685 −0.975401 −0.487701 0.873011i \(-0.662164\pi\)
−0.487701 + 0.873011i \(0.662164\pi\)
\(212\) −8.74627 −0.600696
\(213\) 5.91999 0.405631
\(214\) −15.4507 −1.05619
\(215\) −7.54851 −0.514804
\(216\) −1.00000 −0.0680414
\(217\) 12.6227 0.856885
\(218\) 3.02806 0.205086
\(219\) −1.48512 −0.100355
\(220\) 7.35782 0.496064
\(221\) 6.96515 0.468526
\(222\) 0.817018 0.0548346
\(223\) −6.11406 −0.409428 −0.204714 0.978822i \(-0.565626\pi\)
−0.204714 + 0.978822i \(0.565626\pi\)
\(224\) −2.84255 −0.189926
\(225\) −2.41076 −0.160717
\(226\) 4.47739 0.297832
\(227\) 11.5741 0.768202 0.384101 0.923291i \(-0.374511\pi\)
0.384101 + 0.923291i \(0.374511\pi\)
\(228\) 5.10639 0.338179
\(229\) −3.87756 −0.256237 −0.128118 0.991759i \(-0.540894\pi\)
−0.128118 + 0.991759i \(0.540894\pi\)
\(230\) −11.9043 −0.784946
\(231\) −12.9978 −0.855195
\(232\) 7.17077 0.470784
\(233\) −0.0308552 −0.00202139 −0.00101069 0.999999i \(-0.500322\pi\)
−0.00101069 + 0.999999i \(0.500322\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 10.3075 0.672389
\(236\) 0.862231 0.0561265
\(237\) −16.1372 −1.04822
\(238\) −19.7988 −1.28336
\(239\) 0.243671 0.0157618 0.00788089 0.999969i \(-0.497491\pi\)
0.00788089 + 0.999969i \(0.497491\pi\)
\(240\) −1.60911 −0.103868
\(241\) 4.60369 0.296550 0.148275 0.988946i \(-0.452628\pi\)
0.148275 + 0.988946i \(0.452628\pi\)
\(242\) −9.90866 −0.636953
\(243\) 1.00000 0.0641500
\(244\) −5.94901 −0.380847
\(245\) −1.73799 −0.111036
\(246\) 11.4476 0.729875
\(247\) 5.10639 0.324912
\(248\) −4.44063 −0.281980
\(249\) 7.18794 0.455517
\(250\) −11.9247 −0.754187
\(251\) −18.3295 −1.15695 −0.578475 0.815700i \(-0.696352\pi\)
−0.578475 + 0.815700i \(0.696352\pi\)
\(252\) 2.84255 0.179064
\(253\) 33.8283 2.12677
\(254\) −1.94343 −0.121942
\(255\) −11.2077 −0.701853
\(256\) 1.00000 0.0625000
\(257\) −7.93770 −0.495140 −0.247570 0.968870i \(-0.579632\pi\)
−0.247570 + 0.968870i \(0.579632\pi\)
\(258\) −4.69111 −0.292055
\(259\) −2.32241 −0.144308
\(260\) −1.60911 −0.0997928
\(261\) −7.17077 −0.443859
\(262\) −5.44624 −0.336470
\(263\) 0.633427 0.0390588 0.0195294 0.999809i \(-0.493783\pi\)
0.0195294 + 0.999809i \(0.493783\pi\)
\(264\) 4.57260 0.281424
\(265\) 14.0737 0.864542
\(266\) −14.5152 −0.889982
\(267\) 13.9273 0.852334
\(268\) −5.12966 −0.313344
\(269\) 1.06226 0.0647672 0.0323836 0.999476i \(-0.489690\pi\)
0.0323836 + 0.999476i \(0.489690\pi\)
\(270\) 1.60911 0.0979274
\(271\) 25.8546 1.57056 0.785279 0.619142i \(-0.212520\pi\)
0.785279 + 0.619142i \(0.212520\pi\)
\(272\) 6.96515 0.422324
\(273\) 2.84255 0.172039
\(274\) 5.97882 0.361194
\(275\) 11.0234 0.664739
\(276\) −7.39805 −0.445310
\(277\) −24.9779 −1.50077 −0.750387 0.660998i \(-0.770133\pi\)
−0.750387 + 0.660998i \(0.770133\pi\)
\(278\) −0.970755 −0.0582220
\(279\) 4.44063 0.265853
\(280\) 4.57398 0.273348
\(281\) 6.09885 0.363827 0.181913 0.983315i \(-0.441771\pi\)
0.181913 + 0.983315i \(0.441771\pi\)
\(282\) 6.40573 0.381456
\(283\) 12.1903 0.724636 0.362318 0.932055i \(-0.381985\pi\)
0.362318 + 0.932055i \(0.381985\pi\)
\(284\) 5.91999 0.351287
\(285\) −8.21675 −0.486718
\(286\) 4.57260 0.270383
\(287\) −32.5405 −1.92081
\(288\) −1.00000 −0.0589256
\(289\) 31.5133 1.85372
\(290\) −11.5386 −0.677568
\(291\) −2.58490 −0.151530
\(292\) −1.48512 −0.0869103
\(293\) −15.6402 −0.913711 −0.456855 0.889541i \(-0.651024\pi\)
−0.456855 + 0.889541i \(0.651024\pi\)
\(294\) −1.08009 −0.0629923
\(295\) −1.38743 −0.0807790
\(296\) 0.817018 0.0474882
\(297\) −4.57260 −0.265329
\(298\) −3.63472 −0.210554
\(299\) −7.39805 −0.427840
\(300\) −2.41076 −0.139185
\(301\) 13.3347 0.768600
\(302\) 8.48578 0.488302
\(303\) 2.68909 0.154484
\(304\) 5.10639 0.292872
\(305\) 9.57262 0.548127
\(306\) −6.96515 −0.398171
\(307\) −3.23081 −0.184392 −0.0921960 0.995741i \(-0.529389\pi\)
−0.0921960 + 0.995741i \(0.529389\pi\)
\(308\) −12.9978 −0.740621
\(309\) −1.00000 −0.0568880
\(310\) 7.14546 0.405835
\(311\) −21.8255 −1.23761 −0.618807 0.785543i \(-0.712384\pi\)
−0.618807 + 0.785543i \(0.712384\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −15.6711 −0.885781 −0.442890 0.896576i \(-0.646047\pi\)
−0.442890 + 0.896576i \(0.646047\pi\)
\(314\) −1.63370 −0.0921950
\(315\) −4.57398 −0.257715
\(316\) −16.1372 −0.907786
\(317\) 4.78367 0.268677 0.134339 0.990935i \(-0.457109\pi\)
0.134339 + 0.990935i \(0.457109\pi\)
\(318\) 8.74627 0.490466
\(319\) 32.7890 1.83583
\(320\) −1.60911 −0.0899520
\(321\) 15.4507 0.862373
\(322\) 21.0293 1.17192
\(323\) 35.5667 1.97899
\(324\) 1.00000 0.0555556
\(325\) −2.41076 −0.133725
\(326\) 11.6549 0.645505
\(327\) −3.02806 −0.167452
\(328\) 11.4476 0.632091
\(329\) −18.2086 −1.00387
\(330\) −7.35782 −0.405035
\(331\) −5.01013 −0.275382 −0.137691 0.990475i \(-0.543968\pi\)
−0.137691 + 0.990475i \(0.543968\pi\)
\(332\) 7.18794 0.394489
\(333\) −0.817018 −0.0447723
\(334\) 10.1875 0.557436
\(335\) 8.25419 0.450974
\(336\) 2.84255 0.155074
\(337\) −11.2909 −0.615055 −0.307528 0.951539i \(-0.599502\pi\)
−0.307528 + 0.951539i \(0.599502\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −4.47739 −0.243179
\(340\) −11.2077 −0.607822
\(341\) −20.3052 −1.09959
\(342\) −5.10639 −0.276122
\(343\) −16.8276 −0.908607
\(344\) −4.69111 −0.252927
\(345\) 11.9043 0.640905
\(346\) 4.35726 0.234247
\(347\) 0.556221 0.0298595 0.0149297 0.999889i \(-0.495248\pi\)
0.0149297 + 0.999889i \(0.495248\pi\)
\(348\) −7.17077 −0.384393
\(349\) 20.7543 1.11095 0.555475 0.831533i \(-0.312536\pi\)
0.555475 + 0.831533i \(0.312536\pi\)
\(350\) 6.85271 0.366293
\(351\) 1.00000 0.0533761
\(352\) 4.57260 0.243720
\(353\) −16.4735 −0.876795 −0.438398 0.898781i \(-0.644454\pi\)
−0.438398 + 0.898781i \(0.644454\pi\)
\(354\) −0.862231 −0.0458271
\(355\) −9.52592 −0.505583
\(356\) 13.9273 0.738143
\(357\) 19.7988 1.04786
\(358\) −14.7457 −0.779337
\(359\) 27.2326 1.43728 0.718640 0.695382i \(-0.244765\pi\)
0.718640 + 0.695382i \(0.244765\pi\)
\(360\) 1.60911 0.0848076
\(361\) 7.07521 0.372380
\(362\) 20.1298 1.05800
\(363\) 9.90866 0.520070
\(364\) 2.84255 0.148990
\(365\) 2.38973 0.125084
\(366\) 5.94901 0.310960
\(367\) −23.2016 −1.21111 −0.605556 0.795803i \(-0.707049\pi\)
−0.605556 + 0.795803i \(0.707049\pi\)
\(368\) −7.39805 −0.385650
\(369\) −11.4476 −0.595941
\(370\) −1.31467 −0.0683465
\(371\) −24.8617 −1.29076
\(372\) 4.44063 0.230236
\(373\) 8.85675 0.458585 0.229293 0.973358i \(-0.426359\pi\)
0.229293 + 0.973358i \(0.426359\pi\)
\(374\) 31.8488 1.64686
\(375\) 11.9247 0.615791
\(376\) 6.40573 0.330350
\(377\) −7.17077 −0.369313
\(378\) −2.84255 −0.146205
\(379\) −16.6386 −0.854668 −0.427334 0.904094i \(-0.640547\pi\)
−0.427334 + 0.904094i \(0.640547\pi\)
\(380\) −8.21675 −0.421510
\(381\) 1.94343 0.0995652
\(382\) 18.6504 0.954236
\(383\) −11.0662 −0.565459 −0.282729 0.959200i \(-0.591240\pi\)
−0.282729 + 0.959200i \(0.591240\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 20.9150 1.06593
\(386\) −3.50650 −0.178476
\(387\) 4.69111 0.238462
\(388\) −2.58490 −0.131228
\(389\) −7.44643 −0.377549 −0.188775 0.982020i \(-0.560452\pi\)
−0.188775 + 0.982020i \(0.560452\pi\)
\(390\) 1.60911 0.0814805
\(391\) −51.5285 −2.60591
\(392\) −1.08009 −0.0545530
\(393\) 5.44624 0.274727
\(394\) 2.08818 0.105201
\(395\) 25.9665 1.30652
\(396\) −4.57260 −0.229782
\(397\) −8.13122 −0.408094 −0.204047 0.978961i \(-0.565410\pi\)
−0.204047 + 0.978961i \(0.565410\pi\)
\(398\) 4.31102 0.216092
\(399\) 14.5152 0.726667
\(400\) −2.41076 −0.120538
\(401\) −30.5947 −1.52782 −0.763912 0.645320i \(-0.776724\pi\)
−0.763912 + 0.645320i \(0.776724\pi\)
\(402\) 5.12966 0.255844
\(403\) 4.44063 0.221203
\(404\) 2.68909 0.133787
\(405\) −1.60911 −0.0799574
\(406\) 20.3833 1.01160
\(407\) 3.73589 0.185181
\(408\) −6.96515 −0.344826
\(409\) 15.2968 0.756376 0.378188 0.925729i \(-0.376547\pi\)
0.378188 + 0.925729i \(0.376547\pi\)
\(410\) −18.4205 −0.909726
\(411\) −5.97882 −0.294913
\(412\) −1.00000 −0.0492665
\(413\) 2.45094 0.120603
\(414\) 7.39805 0.363594
\(415\) −11.5662 −0.567762
\(416\) −1.00000 −0.0490290
\(417\) 0.970755 0.0475381
\(418\) 23.3495 1.14206
\(419\) −18.8495 −0.920857 −0.460428 0.887697i \(-0.652304\pi\)
−0.460428 + 0.887697i \(0.652304\pi\)
\(420\) −4.57398 −0.223187
\(421\) 39.5238 1.92627 0.963137 0.269011i \(-0.0866968\pi\)
0.963137 + 0.269011i \(0.0866968\pi\)
\(422\) 14.1685 0.689713
\(423\) −6.40573 −0.311457
\(424\) 8.74627 0.424756
\(425\) −16.7913 −0.814498
\(426\) −5.91999 −0.286824
\(427\) −16.9104 −0.818350
\(428\) 15.4507 0.746837
\(429\) −4.57260 −0.220767
\(430\) 7.54851 0.364021
\(431\) −9.74194 −0.469253 −0.234626 0.972086i \(-0.575387\pi\)
−0.234626 + 0.972086i \(0.575387\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0803 −1.25334 −0.626670 0.779285i \(-0.715582\pi\)
−0.626670 + 0.779285i \(0.715582\pi\)
\(434\) −12.6227 −0.605909
\(435\) 11.5386 0.553232
\(436\) −3.02806 −0.145018
\(437\) −37.7773 −1.80714
\(438\) 1.48512 0.0709619
\(439\) −28.2846 −1.34995 −0.674975 0.737840i \(-0.735846\pi\)
−0.674975 + 0.737840i \(0.735846\pi\)
\(440\) −7.35782 −0.350770
\(441\) 1.08009 0.0514330
\(442\) −6.96515 −0.331298
\(443\) −1.32483 −0.0629448 −0.0314724 0.999505i \(-0.510020\pi\)
−0.0314724 + 0.999505i \(0.510020\pi\)
\(444\) −0.817018 −0.0387739
\(445\) −22.4105 −1.06236
\(446\) 6.11406 0.289509
\(447\) 3.63472 0.171916
\(448\) 2.84255 0.134298
\(449\) 36.8887 1.74088 0.870442 0.492271i \(-0.163833\pi\)
0.870442 + 0.492271i \(0.163833\pi\)
\(450\) 2.41076 0.113644
\(451\) 52.3455 2.46485
\(452\) −4.47739 −0.210599
\(453\) −8.48578 −0.398697
\(454\) −11.5741 −0.543201
\(455\) −4.57398 −0.214431
\(456\) −5.10639 −0.239129
\(457\) −24.7797 −1.15914 −0.579572 0.814921i \(-0.696780\pi\)
−0.579572 + 0.814921i \(0.696780\pi\)
\(458\) 3.87756 0.181187
\(459\) 6.96515 0.325105
\(460\) 11.9043 0.555040
\(461\) 17.9360 0.835365 0.417682 0.908593i \(-0.362842\pi\)
0.417682 + 0.908593i \(0.362842\pi\)
\(462\) 12.9978 0.604714
\(463\) −6.09532 −0.283273 −0.141637 0.989919i \(-0.545236\pi\)
−0.141637 + 0.989919i \(0.545236\pi\)
\(464\) −7.17077 −0.332894
\(465\) −7.14546 −0.331363
\(466\) 0.0308552 0.00142934
\(467\) 32.7750 1.51664 0.758322 0.651880i \(-0.226019\pi\)
0.758322 + 0.651880i \(0.226019\pi\)
\(468\) 1.00000 0.0462250
\(469\) −14.5813 −0.673302
\(470\) −10.3075 −0.475451
\(471\) 1.63370 0.0752769
\(472\) −0.862231 −0.0396874
\(473\) −21.4505 −0.986297
\(474\) 16.1372 0.741205
\(475\) −12.3103 −0.564835
\(476\) 19.7988 0.907476
\(477\) −8.74627 −0.400464
\(478\) −0.243671 −0.0111453
\(479\) 27.7395 1.26745 0.633725 0.773558i \(-0.281525\pi\)
0.633725 + 0.773558i \(0.281525\pi\)
\(480\) 1.60911 0.0734455
\(481\) −0.817018 −0.0372528
\(482\) −4.60369 −0.209692
\(483\) −21.0293 −0.956868
\(484\) 9.90866 0.450394
\(485\) 4.15939 0.188868
\(486\) −1.00000 −0.0453609
\(487\) −35.9493 −1.62902 −0.814510 0.580150i \(-0.802994\pi\)
−0.814510 + 0.580150i \(0.802994\pi\)
\(488\) 5.94901 0.269299
\(489\) −11.6549 −0.527052
\(490\) 1.73799 0.0785144
\(491\) −29.2751 −1.32117 −0.660583 0.750753i \(-0.729691\pi\)
−0.660583 + 0.750753i \(0.729691\pi\)
\(492\) −11.4476 −0.516100
\(493\) −49.9454 −2.24943
\(494\) −5.10639 −0.229747
\(495\) 7.35782 0.330709
\(496\) 4.44063 0.199390
\(497\) 16.8279 0.754833
\(498\) −7.18794 −0.322099
\(499\) −2.29770 −0.102859 −0.0514297 0.998677i \(-0.516378\pi\)
−0.0514297 + 0.998677i \(0.516378\pi\)
\(500\) 11.9247 0.533291
\(501\) −10.1875 −0.455145
\(502\) 18.3295 0.818087
\(503\) −12.8249 −0.571835 −0.285917 0.958254i \(-0.592298\pi\)
−0.285917 + 0.958254i \(0.592298\pi\)
\(504\) −2.84255 −0.126617
\(505\) −4.32704 −0.192551
\(506\) −33.8283 −1.50385
\(507\) 1.00000 0.0444116
\(508\) 1.94343 0.0862260
\(509\) −18.0905 −0.801849 −0.400924 0.916111i \(-0.631311\pi\)
−0.400924 + 0.916111i \(0.631311\pi\)
\(510\) 11.2077 0.496285
\(511\) −4.22154 −0.186750
\(512\) −1.00000 −0.0441942
\(513\) 5.10639 0.225453
\(514\) 7.93770 0.350117
\(515\) 1.60911 0.0709059
\(516\) 4.69111 0.206514
\(517\) 29.2908 1.28821
\(518\) 2.32241 0.102041
\(519\) −4.35726 −0.191262
\(520\) 1.60911 0.0705642
\(521\) 20.9542 0.918019 0.459010 0.888431i \(-0.348204\pi\)
0.459010 + 0.888431i \(0.348204\pi\)
\(522\) 7.17077 0.313856
\(523\) −5.27308 −0.230576 −0.115288 0.993332i \(-0.536779\pi\)
−0.115288 + 0.993332i \(0.536779\pi\)
\(524\) 5.44624 0.237920
\(525\) −6.85271 −0.299077
\(526\) −0.633427 −0.0276187
\(527\) 30.9296 1.34732
\(528\) −4.57260 −0.198997
\(529\) 31.7312 1.37962
\(530\) −14.0737 −0.611323
\(531\) 0.862231 0.0374176
\(532\) 14.5152 0.629312
\(533\) −11.4476 −0.495853
\(534\) −13.9273 −0.602691
\(535\) −24.8619 −1.07487
\(536\) 5.12966 0.221567
\(537\) 14.7457 0.636326
\(538\) −1.06226 −0.0457973
\(539\) −4.93884 −0.212731
\(540\) −1.60911 −0.0692451
\(541\) 11.2091 0.481919 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(542\) −25.8546 −1.11055
\(543\) −20.1298 −0.863852
\(544\) −6.96515 −0.298628
\(545\) 4.87249 0.208715
\(546\) −2.84255 −0.121650
\(547\) 18.3150 0.783095 0.391547 0.920158i \(-0.371940\pi\)
0.391547 + 0.920158i \(0.371940\pi\)
\(548\) −5.97882 −0.255403
\(549\) −5.94901 −0.253898
\(550\) −11.0234 −0.470041
\(551\) −36.6167 −1.55993
\(552\) 7.39805 0.314882
\(553\) −45.8707 −1.95062
\(554\) 24.9779 1.06121
\(555\) 1.31467 0.0558047
\(556\) 0.970755 0.0411692
\(557\) 19.5113 0.826719 0.413360 0.910568i \(-0.364355\pi\)
0.413360 + 0.910568i \(0.364355\pi\)
\(558\) −4.44063 −0.187987
\(559\) 4.69111 0.198413
\(560\) −4.57398 −0.193286
\(561\) −31.8488 −1.34466
\(562\) −6.09885 −0.257264
\(563\) 30.4444 1.28308 0.641540 0.767090i \(-0.278296\pi\)
0.641540 + 0.767090i \(0.278296\pi\)
\(564\) −6.40573 −0.269730
\(565\) 7.20463 0.303101
\(566\) −12.1903 −0.512395
\(567\) 2.84255 0.119376
\(568\) −5.91999 −0.248397
\(569\) 44.5917 1.86938 0.934690 0.355463i \(-0.115677\pi\)
0.934690 + 0.355463i \(0.115677\pi\)
\(570\) 8.21675 0.344162
\(571\) −37.2634 −1.55942 −0.779712 0.626139i \(-0.784634\pi\)
−0.779712 + 0.626139i \(0.784634\pi\)
\(572\) −4.57260 −0.191190
\(573\) −18.6504 −0.779131
\(574\) 32.5405 1.35822
\(575\) 17.8349 0.743768
\(576\) 1.00000 0.0416667
\(577\) −4.87519 −0.202957 −0.101479 0.994838i \(-0.532357\pi\)
−0.101479 + 0.994838i \(0.532357\pi\)
\(578\) −31.5133 −1.31078
\(579\) 3.50650 0.145725
\(580\) 11.5386 0.479113
\(581\) 20.4321 0.847666
\(582\) 2.58490 0.107148
\(583\) 39.9932 1.65635
\(584\) 1.48512 0.0614548
\(585\) −1.60911 −0.0665286
\(586\) 15.6402 0.646091
\(587\) −3.09476 −0.127735 −0.0638673 0.997958i \(-0.520343\pi\)
−0.0638673 + 0.997958i \(0.520343\pi\)
\(588\) 1.08009 0.0445423
\(589\) 22.6756 0.934331
\(590\) 1.38743 0.0571194
\(591\) −2.08818 −0.0858961
\(592\) −0.817018 −0.0335792
\(593\) −17.1003 −0.702227 −0.351114 0.936333i \(-0.614197\pi\)
−0.351114 + 0.936333i \(0.614197\pi\)
\(594\) 4.57260 0.187616
\(595\) −31.8584 −1.30607
\(596\) 3.63472 0.148884
\(597\) −4.31102 −0.176438
\(598\) 7.39805 0.302529
\(599\) 20.3018 0.829510 0.414755 0.909933i \(-0.363867\pi\)
0.414755 + 0.909933i \(0.363867\pi\)
\(600\) 2.41076 0.0984189
\(601\) −42.8024 −1.74595 −0.872973 0.487769i \(-0.837811\pi\)
−0.872973 + 0.487769i \(0.837811\pi\)
\(602\) −13.3347 −0.543482
\(603\) −5.12966 −0.208896
\(604\) −8.48578 −0.345281
\(605\) −15.9441 −0.648221
\(606\) −2.68909 −0.109237
\(607\) 28.4394 1.15432 0.577161 0.816631i \(-0.304161\pi\)
0.577161 + 0.816631i \(0.304161\pi\)
\(608\) −5.10639 −0.207091
\(609\) −20.3833 −0.825972
\(610\) −9.57262 −0.387584
\(611\) −6.40573 −0.259148
\(612\) 6.96515 0.281549
\(613\) 42.3147 1.70907 0.854537 0.519391i \(-0.173841\pi\)
0.854537 + 0.519391i \(0.173841\pi\)
\(614\) 3.23081 0.130385
\(615\) 18.4205 0.742788
\(616\) 12.9978 0.523698
\(617\) 5.20542 0.209562 0.104781 0.994495i \(-0.466586\pi\)
0.104781 + 0.994495i \(0.466586\pi\)
\(618\) 1.00000 0.0402259
\(619\) −40.1514 −1.61382 −0.806910 0.590674i \(-0.798862\pi\)
−0.806910 + 0.590674i \(0.798862\pi\)
\(620\) −7.14546 −0.286969
\(621\) −7.39805 −0.296874
\(622\) 21.8255 0.875125
\(623\) 39.5889 1.58610
\(624\) 1.00000 0.0400320
\(625\) −7.13442 −0.285377
\(626\) 15.6711 0.626341
\(627\) −23.3495 −0.932488
\(628\) 1.63370 0.0651917
\(629\) −5.69065 −0.226901
\(630\) 4.57398 0.182232
\(631\) 33.9516 1.35159 0.675795 0.737090i \(-0.263800\pi\)
0.675795 + 0.737090i \(0.263800\pi\)
\(632\) 16.1372 0.641902
\(633\) −14.1685 −0.563148
\(634\) −4.78367 −0.189984
\(635\) −3.12720 −0.124099
\(636\) −8.74627 −0.346812
\(637\) 1.08009 0.0427949
\(638\) −32.7890 −1.29813
\(639\) 5.91999 0.234191
\(640\) 1.60911 0.0636057
\(641\) −25.8159 −1.01967 −0.509834 0.860273i \(-0.670293\pi\)
−0.509834 + 0.860273i \(0.670293\pi\)
\(642\) −15.4507 −0.609790
\(643\) −18.0616 −0.712279 −0.356140 0.934433i \(-0.615907\pi\)
−0.356140 + 0.934433i \(0.615907\pi\)
\(644\) −21.0293 −0.828672
\(645\) −7.54851 −0.297222
\(646\) −35.5667 −1.39935
\(647\) −34.2875 −1.34798 −0.673991 0.738739i \(-0.735421\pi\)
−0.673991 + 0.738739i \(0.735421\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −3.94264 −0.154762
\(650\) 2.41076 0.0945578
\(651\) 12.6227 0.494723
\(652\) −11.6549 −0.456441
\(653\) −19.3922 −0.758876 −0.379438 0.925217i \(-0.623883\pi\)
−0.379438 + 0.925217i \(0.623883\pi\)
\(654\) 3.02806 0.118407
\(655\) −8.76361 −0.342423
\(656\) −11.4476 −0.446956
\(657\) −1.48512 −0.0579402
\(658\) 18.2086 0.709846
\(659\) −8.93576 −0.348088 −0.174044 0.984738i \(-0.555683\pi\)
−0.174044 + 0.984738i \(0.555683\pi\)
\(660\) 7.35782 0.286403
\(661\) 26.2423 1.02071 0.510354 0.859965i \(-0.329515\pi\)
0.510354 + 0.859965i \(0.329515\pi\)
\(662\) 5.01013 0.194724
\(663\) 6.96515 0.270504
\(664\) −7.18794 −0.278946
\(665\) −23.3565 −0.905727
\(666\) 0.817018 0.0316588
\(667\) 53.0497 2.05409
\(668\) −10.1875 −0.394167
\(669\) −6.11406 −0.236383
\(670\) −8.25419 −0.318887
\(671\) 27.2025 1.05014
\(672\) −2.84255 −0.109654
\(673\) −25.8503 −0.996456 −0.498228 0.867046i \(-0.666016\pi\)
−0.498228 + 0.867046i \(0.666016\pi\)
\(674\) 11.2909 0.434910
\(675\) −2.41076 −0.0927903
\(676\) 1.00000 0.0384615
\(677\) −8.54439 −0.328387 −0.164194 0.986428i \(-0.552502\pi\)
−0.164194 + 0.986428i \(0.552502\pi\)
\(678\) 4.47739 0.171953
\(679\) −7.34771 −0.281979
\(680\) 11.2077 0.429795
\(681\) 11.5741 0.443522
\(682\) 20.3052 0.777527
\(683\) −4.47036 −0.171053 −0.0855267 0.996336i \(-0.527257\pi\)
−0.0855267 + 0.996336i \(0.527257\pi\)
\(684\) 5.10639 0.195248
\(685\) 9.62059 0.367584
\(686\) 16.8276 0.642482
\(687\) −3.87756 −0.147938
\(688\) 4.69111 0.178847
\(689\) −8.74627 −0.333206
\(690\) −11.9043 −0.453189
\(691\) 4.28713 0.163090 0.0815451 0.996670i \(-0.474015\pi\)
0.0815451 + 0.996670i \(0.474015\pi\)
\(692\) −4.35726 −0.165638
\(693\) −12.9978 −0.493747
\(694\) −0.556221 −0.0211138
\(695\) −1.56205 −0.0592521
\(696\) 7.17077 0.271807
\(697\) −79.7345 −3.02016
\(698\) −20.7543 −0.785560
\(699\) −0.0308552 −0.00116705
\(700\) −6.85271 −0.259008
\(701\) −42.7745 −1.61557 −0.807786 0.589476i \(-0.799334\pi\)
−0.807786 + 0.589476i \(0.799334\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −4.17201 −0.157350
\(704\) −4.57260 −0.172336
\(705\) 10.3075 0.388204
\(706\) 16.4735 0.619988
\(707\) 7.64387 0.287477
\(708\) 0.862231 0.0324046
\(709\) −35.9901 −1.35164 −0.675819 0.737067i \(-0.736210\pi\)
−0.675819 + 0.737067i \(0.736210\pi\)
\(710\) 9.52592 0.357501
\(711\) −16.1372 −0.605191
\(712\) −13.9273 −0.521946
\(713\) −32.8520 −1.23032
\(714\) −19.7988 −0.740951
\(715\) 7.35782 0.275167
\(716\) 14.7457 0.551074
\(717\) 0.243671 0.00910007
\(718\) −27.2326 −1.01631
\(719\) 35.1165 1.30962 0.654812 0.755791i \(-0.272748\pi\)
0.654812 + 0.755791i \(0.272748\pi\)
\(720\) −1.60911 −0.0599680
\(721\) −2.84255 −0.105862
\(722\) −7.07521 −0.263312
\(723\) 4.60369 0.171213
\(724\) −20.1298 −0.748118
\(725\) 17.2870 0.642023
\(726\) −9.90866 −0.367745
\(727\) 25.7215 0.953959 0.476980 0.878914i \(-0.341732\pi\)
0.476980 + 0.878914i \(0.341732\pi\)
\(728\) −2.84255 −0.105352
\(729\) 1.00000 0.0370370
\(730\) −2.38973 −0.0884478
\(731\) 32.6742 1.20850
\(732\) −5.94901 −0.219882
\(733\) 9.22246 0.340639 0.170320 0.985389i \(-0.445520\pi\)
0.170320 + 0.985389i \(0.445520\pi\)
\(734\) 23.2016 0.856386
\(735\) −1.73799 −0.0641068
\(736\) 7.39805 0.272696
\(737\) 23.4559 0.864008
\(738\) 11.4476 0.421394
\(739\) 25.3614 0.932934 0.466467 0.884539i \(-0.345527\pi\)
0.466467 + 0.884539i \(0.345527\pi\)
\(740\) 1.31467 0.0483283
\(741\) 5.10639 0.187588
\(742\) 24.8617 0.912702
\(743\) −0.554445 −0.0203406 −0.0101703 0.999948i \(-0.503237\pi\)
−0.0101703 + 0.999948i \(0.503237\pi\)
\(744\) −4.44063 −0.162801
\(745\) −5.84867 −0.214279
\(746\) −8.85675 −0.324269
\(747\) 7.18794 0.262993
\(748\) −31.8488 −1.16451
\(749\) 43.9193 1.60478
\(750\) −11.9247 −0.435430
\(751\) −5.17316 −0.188771 −0.0943856 0.995536i \(-0.530089\pi\)
−0.0943856 + 0.995536i \(0.530089\pi\)
\(752\) −6.40573 −0.233593
\(753\) −18.3295 −0.667965
\(754\) 7.17077 0.261144
\(755\) 13.6546 0.496940
\(756\) 2.84255 0.103383
\(757\) 24.3558 0.885228 0.442614 0.896712i \(-0.354051\pi\)
0.442614 + 0.896712i \(0.354051\pi\)
\(758\) 16.6386 0.604342
\(759\) 33.8283 1.22789
\(760\) 8.21675 0.298053
\(761\) −14.6861 −0.532372 −0.266186 0.963922i \(-0.585764\pi\)
−0.266186 + 0.963922i \(0.585764\pi\)
\(762\) −1.94343 −0.0704032
\(763\) −8.60743 −0.311610
\(764\) −18.6504 −0.674747
\(765\) −11.2077 −0.405215
\(766\) 11.0662 0.399840
\(767\) 0.862231 0.0311334
\(768\) 1.00000 0.0360844
\(769\) 39.7035 1.43174 0.715872 0.698231i \(-0.246029\pi\)
0.715872 + 0.698231i \(0.246029\pi\)
\(770\) −20.9150 −0.753723
\(771\) −7.93770 −0.285869
\(772\) 3.50650 0.126202
\(773\) −40.0586 −1.44081 −0.720403 0.693555i \(-0.756043\pi\)
−0.720403 + 0.693555i \(0.756043\pi\)
\(774\) −4.69111 −0.168618
\(775\) −10.7053 −0.384546
\(776\) 2.58490 0.0927925
\(777\) −2.32241 −0.0833161
\(778\) 7.44643 0.266968
\(779\) −58.4562 −2.09441
\(780\) −1.60911 −0.0576154
\(781\) −27.0697 −0.968632
\(782\) 51.5285 1.84266
\(783\) −7.17077 −0.256262
\(784\) 1.08009 0.0385748
\(785\) −2.62880 −0.0938260
\(786\) −5.44624 −0.194261
\(787\) 55.0910 1.96378 0.981891 0.189446i \(-0.0606692\pi\)
0.981891 + 0.189446i \(0.0606692\pi\)
\(788\) −2.08818 −0.0743882
\(789\) 0.633427 0.0225506
\(790\) −25.9665 −0.923846
\(791\) −12.7272 −0.452528
\(792\) 4.57260 0.162480
\(793\) −5.94901 −0.211256
\(794\) 8.13122 0.288566
\(795\) 14.0737 0.499143
\(796\) −4.31102 −0.152800
\(797\) −54.7705 −1.94007 −0.970035 0.242964i \(-0.921880\pi\)
−0.970035 + 0.242964i \(0.921880\pi\)
\(798\) −14.5152 −0.513831
\(799\) −44.6168 −1.57843
\(800\) 2.41076 0.0852333
\(801\) 13.9273 0.492095
\(802\) 30.5947 1.08034
\(803\) 6.79087 0.239645
\(804\) −5.12966 −0.180909
\(805\) 33.8385 1.19265
\(806\) −4.44063 −0.156414
\(807\) 1.06226 0.0373933
\(808\) −2.68909 −0.0946018
\(809\) −23.2987 −0.819139 −0.409569 0.912279i \(-0.634321\pi\)
−0.409569 + 0.912279i \(0.634321\pi\)
\(810\) 1.60911 0.0565384
\(811\) 28.6759 1.00695 0.503474 0.864010i \(-0.332055\pi\)
0.503474 + 0.864010i \(0.332055\pi\)
\(812\) −20.3833 −0.715312
\(813\) 25.8546 0.906762
\(814\) −3.73589 −0.130943
\(815\) 18.7540 0.656924
\(816\) 6.96515 0.243829
\(817\) 23.9546 0.838066
\(818\) −15.2968 −0.534839
\(819\) 2.84255 0.0993268
\(820\) 18.4205 0.643273
\(821\) 2.70362 0.0943568 0.0471784 0.998886i \(-0.484977\pi\)
0.0471784 + 0.998886i \(0.484977\pi\)
\(822\) 5.97882 0.208535
\(823\) −17.3732 −0.605592 −0.302796 0.953055i \(-0.597920\pi\)
−0.302796 + 0.953055i \(0.597920\pi\)
\(824\) 1.00000 0.0348367
\(825\) 11.0234 0.383787
\(826\) −2.45094 −0.0852789
\(827\) −44.1877 −1.53656 −0.768279 0.640115i \(-0.778887\pi\)
−0.768279 + 0.640115i \(0.778887\pi\)
\(828\) −7.39805 −0.257100
\(829\) −37.5709 −1.30489 −0.652446 0.757835i \(-0.726257\pi\)
−0.652446 + 0.757835i \(0.726257\pi\)
\(830\) 11.5662 0.401468
\(831\) −24.9779 −0.866473
\(832\) 1.00000 0.0346688
\(833\) 7.52301 0.260657
\(834\) −0.970755 −0.0336145
\(835\) 16.3928 0.567298
\(836\) −23.3495 −0.807558
\(837\) 4.44063 0.153491
\(838\) 18.8495 0.651144
\(839\) −28.3089 −0.977333 −0.488667 0.872471i \(-0.662517\pi\)
−0.488667 + 0.872471i \(0.662517\pi\)
\(840\) 4.57398 0.157817
\(841\) 22.4199 0.773100
\(842\) −39.5238 −1.36208
\(843\) 6.09885 0.210055
\(844\) −14.1685 −0.487701
\(845\) −1.60911 −0.0553551
\(846\) 6.40573 0.220234
\(847\) 28.1659 0.967791
\(848\) −8.74627 −0.300348
\(849\) 12.1903 0.418369
\(850\) 16.7913 0.575937
\(851\) 6.04434 0.207197
\(852\) 5.91999 0.202816
\(853\) 2.80830 0.0961544 0.0480772 0.998844i \(-0.484691\pi\)
0.0480772 + 0.998844i \(0.484691\pi\)
\(854\) 16.9104 0.578661
\(855\) −8.21675 −0.281007
\(856\) −15.4507 −0.528093
\(857\) 52.8361 1.80485 0.902424 0.430849i \(-0.141786\pi\)
0.902424 + 0.430849i \(0.141786\pi\)
\(858\) 4.57260 0.156106
\(859\) 22.0375 0.751908 0.375954 0.926638i \(-0.377315\pi\)
0.375954 + 0.926638i \(0.377315\pi\)
\(860\) −7.54851 −0.257402
\(861\) −32.5405 −1.10898
\(862\) 9.74194 0.331812
\(863\) −43.1974 −1.47046 −0.735229 0.677819i \(-0.762925\pi\)
−0.735229 + 0.677819i \(0.762925\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 7.01131 0.238392
\(866\) 26.0803 0.886245
\(867\) 31.5133 1.07025
\(868\) 12.6227 0.428443
\(869\) 73.7888 2.50311
\(870\) −11.5386 −0.391194
\(871\) −5.12966 −0.173812
\(872\) 3.02806 0.102543
\(873\) −2.58490 −0.0874856
\(874\) 37.7773 1.27784
\(875\) 33.8967 1.14592
\(876\) −1.48512 −0.0501777
\(877\) −2.44950 −0.0827137 −0.0413569 0.999144i \(-0.513168\pi\)
−0.0413569 + 0.999144i \(0.513168\pi\)
\(878\) 28.2846 0.954559
\(879\) −15.6402 −0.527531
\(880\) 7.35782 0.248032
\(881\) −3.04447 −0.102571 −0.0512855 0.998684i \(-0.516332\pi\)
−0.0512855 + 0.998684i \(0.516332\pi\)
\(882\) −1.08009 −0.0363686
\(883\) −18.2491 −0.614131 −0.307066 0.951688i \(-0.599347\pi\)
−0.307066 + 0.951688i \(0.599347\pi\)
\(884\) 6.96515 0.234263
\(885\) −1.38743 −0.0466378
\(886\) 1.32483 0.0445087
\(887\) 20.3594 0.683602 0.341801 0.939772i \(-0.388963\pi\)
0.341801 + 0.939772i \(0.388963\pi\)
\(888\) 0.817018 0.0274173
\(889\) 5.52431 0.185279
\(890\) 22.4105 0.751201
\(891\) −4.57260 −0.153188
\(892\) −6.11406 −0.204714
\(893\) −32.7102 −1.09460
\(894\) −3.63472 −0.121563
\(895\) −23.7275 −0.793124
\(896\) −2.84255 −0.0949629
\(897\) −7.39805 −0.247014
\(898\) −36.8887 −1.23099
\(899\) −31.8427 −1.06201
\(900\) −2.41076 −0.0803587
\(901\) −60.9190 −2.02951
\(902\) −52.3455 −1.74291
\(903\) 13.3347 0.443751
\(904\) 4.47739 0.148916
\(905\) 32.3911 1.07672
\(906\) 8.48578 0.281921
\(907\) −42.0999 −1.39790 −0.698952 0.715168i \(-0.746350\pi\)
−0.698952 + 0.715168i \(0.746350\pi\)
\(908\) 11.5741 0.384101
\(909\) 2.68909 0.0891914
\(910\) 4.57398 0.151626
\(911\) 31.7160 1.05080 0.525400 0.850856i \(-0.323916\pi\)
0.525400 + 0.850856i \(0.323916\pi\)
\(912\) 5.10639 0.169089
\(913\) −32.8676 −1.08776
\(914\) 24.7797 0.819639
\(915\) 9.57262 0.316461
\(916\) −3.87756 −0.128118
\(917\) 15.4812 0.511235
\(918\) −6.96515 −0.229884
\(919\) 47.6436 1.57162 0.785808 0.618470i \(-0.212247\pi\)
0.785808 + 0.618470i \(0.212247\pi\)
\(920\) −11.9043 −0.392473
\(921\) −3.23081 −0.106459
\(922\) −17.9360 −0.590692
\(923\) 5.91999 0.194859
\(924\) −12.9978 −0.427598
\(925\) 1.96963 0.0647612
\(926\) 6.09532 0.200305
\(927\) −1.00000 −0.0328443
\(928\) 7.17077 0.235392
\(929\) −26.1267 −0.857190 −0.428595 0.903497i \(-0.640991\pi\)
−0.428595 + 0.903497i \(0.640991\pi\)
\(930\) 7.14546 0.234309
\(931\) 5.51538 0.180759
\(932\) −0.0308552 −0.00101069
\(933\) −21.8255 −0.714536
\(934\) −32.7750 −1.07243
\(935\) 51.2483 1.67600
\(936\) −1.00000 −0.0326860
\(937\) −15.1510 −0.494962 −0.247481 0.968893i \(-0.579603\pi\)
−0.247481 + 0.968893i \(0.579603\pi\)
\(938\) 14.5813 0.476097
\(939\) −15.6711 −0.511406
\(940\) 10.3075 0.336195
\(941\) −37.7111 −1.22935 −0.614674 0.788781i \(-0.710712\pi\)
−0.614674 + 0.788781i \(0.710712\pi\)
\(942\) −1.63370 −0.0532288
\(943\) 84.6903 2.75790
\(944\) 0.862231 0.0280632
\(945\) −4.57398 −0.148792
\(946\) 21.4505 0.697417
\(947\) 32.6297 1.06032 0.530162 0.847896i \(-0.322131\pi\)
0.530162 + 0.847896i \(0.322131\pi\)
\(948\) −16.1372 −0.524111
\(949\) −1.48512 −0.0482091
\(950\) 12.3103 0.399398
\(951\) 4.78367 0.155121
\(952\) −19.7988 −0.641682
\(953\) 14.2906 0.462918 0.231459 0.972845i \(-0.425650\pi\)
0.231459 + 0.972845i \(0.425650\pi\)
\(954\) 8.74627 0.283171
\(955\) 30.0105 0.971118
\(956\) 0.243671 0.00788089
\(957\) 32.7890 1.05992
\(958\) −27.7395 −0.896223
\(959\) −16.9951 −0.548800
\(960\) −1.60911 −0.0519338
\(961\) −11.2808 −0.363898
\(962\) 0.817018 0.0263417
\(963\) 15.4507 0.497891
\(964\) 4.60369 0.148275
\(965\) −5.64234 −0.181633
\(966\) 21.0293 0.676608
\(967\) 5.69742 0.183217 0.0916084 0.995795i \(-0.470799\pi\)
0.0916084 + 0.995795i \(0.470799\pi\)
\(968\) −9.90866 −0.318477
\(969\) 35.5667 1.14257
\(970\) −4.15939 −0.133550
\(971\) −5.23681 −0.168057 −0.0840287 0.996463i \(-0.526779\pi\)
−0.0840287 + 0.996463i \(0.526779\pi\)
\(972\) 1.00000 0.0320750
\(973\) 2.75942 0.0884630
\(974\) 35.9493 1.15189
\(975\) −2.41076 −0.0772062
\(976\) −5.94901 −0.190423
\(977\) −32.9126 −1.05297 −0.526483 0.850185i \(-0.676490\pi\)
−0.526483 + 0.850185i \(0.676490\pi\)
\(978\) 11.6549 0.372682
\(979\) −63.6837 −2.03534
\(980\) −1.73799 −0.0555181
\(981\) −3.02806 −0.0966787
\(982\) 29.2751 0.934206
\(983\) 25.3432 0.808322 0.404161 0.914688i \(-0.367564\pi\)
0.404161 + 0.914688i \(0.367564\pi\)
\(984\) 11.4476 0.364938
\(985\) 3.36011 0.107062
\(986\) 49.9454 1.59059
\(987\) −18.2086 −0.579587
\(988\) 5.10639 0.162456
\(989\) −34.7050 −1.10356
\(990\) −7.35782 −0.233847
\(991\) −32.4714 −1.03149 −0.515744 0.856743i \(-0.672485\pi\)
−0.515744 + 0.856743i \(0.672485\pi\)
\(992\) −4.44063 −0.140990
\(993\) −5.01013 −0.158992
\(994\) −16.8279 −0.533748
\(995\) 6.93691 0.219915
\(996\) 7.18794 0.227759
\(997\) 51.4453 1.62929 0.814645 0.579961i \(-0.196932\pi\)
0.814645 + 0.579961i \(0.196932\pi\)
\(998\) 2.29770 0.0727326
\(999\) −0.817018 −0.0258493
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.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.5 12 1.1 even 1 trivial