Properties

Label 8034.2.a.bb.1.13
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-3.07028\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.07028 q^{5} -1.00000 q^{6} -1.20257 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.07028 q^{5} -1.00000 q^{6} -1.20257 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.07028 q^{10} +1.64327 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.20257 q^{14} +3.07028 q^{15} +1.00000 q^{16} +5.16756 q^{17} -1.00000 q^{18} -6.97237 q^{19} +3.07028 q^{20} -1.20257 q^{21} -1.64327 q^{22} -4.45098 q^{23} -1.00000 q^{24} +4.42660 q^{25} +1.00000 q^{26} +1.00000 q^{27} -1.20257 q^{28} -5.46012 q^{29} -3.07028 q^{30} -0.105312 q^{31} -1.00000 q^{32} +1.64327 q^{33} -5.16756 q^{34} -3.69223 q^{35} +1.00000 q^{36} -4.85429 q^{37} +6.97237 q^{38} -1.00000 q^{39} -3.07028 q^{40} -1.62153 q^{41} +1.20257 q^{42} -12.7227 q^{43} +1.64327 q^{44} +3.07028 q^{45} +4.45098 q^{46} -9.64077 q^{47} +1.00000 q^{48} -5.55382 q^{49} -4.42660 q^{50} +5.16756 q^{51} -1.00000 q^{52} -8.08025 q^{53} -1.00000 q^{54} +5.04531 q^{55} +1.20257 q^{56} -6.97237 q^{57} +5.46012 q^{58} -4.90361 q^{59} +3.07028 q^{60} -2.19611 q^{61} +0.105312 q^{62} -1.20257 q^{63} +1.00000 q^{64} -3.07028 q^{65} -1.64327 q^{66} -15.1928 q^{67} +5.16756 q^{68} -4.45098 q^{69} +3.69223 q^{70} +12.9797 q^{71} -1.00000 q^{72} +0.0855982 q^{73} +4.85429 q^{74} +4.42660 q^{75} -6.97237 q^{76} -1.97615 q^{77} +1.00000 q^{78} -11.5359 q^{79} +3.07028 q^{80} +1.00000 q^{81} +1.62153 q^{82} +5.79368 q^{83} -1.20257 q^{84} +15.8658 q^{85} +12.7227 q^{86} -5.46012 q^{87} -1.64327 q^{88} +4.12175 q^{89} -3.07028 q^{90} +1.20257 q^{91} -4.45098 q^{92} -0.105312 q^{93} +9.64077 q^{94} -21.4071 q^{95} -1.00000 q^{96} -3.96607 q^{97} +5.55382 q^{98} +1.64327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) 3.07028 1.37307 0.686535 0.727097i \(-0.259131\pi\)
0.686535 + 0.727097i \(0.259131\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.20257 −0.454529 −0.227265 0.973833i \(-0.572978\pi\)
−0.227265 + 0.973833i \(0.572978\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.07028 −0.970907
\(11\) 1.64327 0.495466 0.247733 0.968828i \(-0.420314\pi\)
0.247733 + 0.968828i \(0.420314\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.20257 0.321401
\(15\) 3.07028 0.792742
\(16\) 1.00000 0.250000
\(17\) 5.16756 1.25332 0.626659 0.779294i \(-0.284422\pi\)
0.626659 + 0.779294i \(0.284422\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.97237 −1.59957 −0.799786 0.600286i \(-0.795054\pi\)
−0.799786 + 0.600286i \(0.795054\pi\)
\(20\) 3.07028 0.686535
\(21\) −1.20257 −0.262423
\(22\) −1.64327 −0.350347
\(23\) −4.45098 −0.928094 −0.464047 0.885811i \(-0.653603\pi\)
−0.464047 + 0.885811i \(0.653603\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.42660 0.885320
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −1.20257 −0.227265
\(29\) −5.46012 −1.01392 −0.506959 0.861970i \(-0.669231\pi\)
−0.506959 + 0.861970i \(0.669231\pi\)
\(30\) −3.07028 −0.560553
\(31\) −0.105312 −0.0189146 −0.00945731 0.999955i \(-0.503010\pi\)
−0.00945731 + 0.999955i \(0.503010\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.64327 0.286057
\(34\) −5.16756 −0.886230
\(35\) −3.69223 −0.624101
\(36\) 1.00000 0.166667
\(37\) −4.85429 −0.798040 −0.399020 0.916942i \(-0.630650\pi\)
−0.399020 + 0.916942i \(0.630650\pi\)
\(38\) 6.97237 1.13107
\(39\) −1.00000 −0.160128
\(40\) −3.07028 −0.485453
\(41\) −1.62153 −0.253240 −0.126620 0.991951i \(-0.540413\pi\)
−0.126620 + 0.991951i \(0.540413\pi\)
\(42\) 1.20257 0.185561
\(43\) −12.7227 −1.94020 −0.970100 0.242707i \(-0.921964\pi\)
−0.970100 + 0.242707i \(0.921964\pi\)
\(44\) 1.64327 0.247733
\(45\) 3.07028 0.457690
\(46\) 4.45098 0.656261
\(47\) −9.64077 −1.40625 −0.703126 0.711066i \(-0.748213\pi\)
−0.703126 + 0.711066i \(0.748213\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.55382 −0.793403
\(50\) −4.42660 −0.626016
\(51\) 5.16756 0.723603
\(52\) −1.00000 −0.138675
\(53\) −8.08025 −1.10991 −0.554954 0.831881i \(-0.687264\pi\)
−0.554954 + 0.831881i \(0.687264\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.04531 0.680309
\(56\) 1.20257 0.160700
\(57\) −6.97237 −0.923513
\(58\) 5.46012 0.716949
\(59\) −4.90361 −0.638396 −0.319198 0.947688i \(-0.603413\pi\)
−0.319198 + 0.947688i \(0.603413\pi\)
\(60\) 3.07028 0.396371
\(61\) −2.19611 −0.281184 −0.140592 0.990068i \(-0.544901\pi\)
−0.140592 + 0.990068i \(0.544901\pi\)
\(62\) 0.105312 0.0133747
\(63\) −1.20257 −0.151510
\(64\) 1.00000 0.125000
\(65\) −3.07028 −0.380821
\(66\) −1.64327 −0.202273
\(67\) −15.1928 −1.85609 −0.928047 0.372463i \(-0.878513\pi\)
−0.928047 + 0.372463i \(0.878513\pi\)
\(68\) 5.16756 0.626659
\(69\) −4.45098 −0.535835
\(70\) 3.69223 0.441306
\(71\) 12.9797 1.54040 0.770201 0.637801i \(-0.220156\pi\)
0.770201 + 0.637801i \(0.220156\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0.0855982 0.0100185 0.00500925 0.999987i \(-0.498405\pi\)
0.00500925 + 0.999987i \(0.498405\pi\)
\(74\) 4.85429 0.564300
\(75\) 4.42660 0.511140
\(76\) −6.97237 −0.799786
\(77\) −1.97615 −0.225204
\(78\) 1.00000 0.113228
\(79\) −11.5359 −1.29789 −0.648943 0.760837i \(-0.724789\pi\)
−0.648943 + 0.760837i \(0.724789\pi\)
\(80\) 3.07028 0.343267
\(81\) 1.00000 0.111111
\(82\) 1.62153 0.179068
\(83\) 5.79368 0.635939 0.317970 0.948101i \(-0.396999\pi\)
0.317970 + 0.948101i \(0.396999\pi\)
\(84\) −1.20257 −0.131211
\(85\) 15.8658 1.72089
\(86\) 12.7227 1.37193
\(87\) −5.46012 −0.585386
\(88\) −1.64327 −0.175174
\(89\) 4.12175 0.436905 0.218452 0.975848i \(-0.429899\pi\)
0.218452 + 0.975848i \(0.429899\pi\)
\(90\) −3.07028 −0.323636
\(91\) 1.20257 0.126064
\(92\) −4.45098 −0.464047
\(93\) −0.105312 −0.0109204
\(94\) 9.64077 0.994370
\(95\) −21.4071 −2.19632
\(96\) −1.00000 −0.102062
\(97\) −3.96607 −0.402694 −0.201347 0.979520i \(-0.564532\pi\)
−0.201347 + 0.979520i \(0.564532\pi\)
\(98\) 5.55382 0.561021
\(99\) 1.64327 0.165155
\(100\) 4.42660 0.442660
\(101\) 5.55432 0.552675 0.276338 0.961061i \(-0.410879\pi\)
0.276338 + 0.961061i \(0.410879\pi\)
\(102\) −5.16756 −0.511665
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −3.69223 −0.360325
\(106\) 8.08025 0.784824
\(107\) 3.11236 0.300883 0.150442 0.988619i \(-0.451930\pi\)
0.150442 + 0.988619i \(0.451930\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.7609 −1.03070 −0.515352 0.856978i \(-0.672339\pi\)
−0.515352 + 0.856978i \(0.672339\pi\)
\(110\) −5.04531 −0.481051
\(111\) −4.85429 −0.460749
\(112\) −1.20257 −0.113632
\(113\) 20.4678 1.92545 0.962726 0.270480i \(-0.0871823\pi\)
0.962726 + 0.270480i \(0.0871823\pi\)
\(114\) 6.97237 0.653022
\(115\) −13.6657 −1.27434
\(116\) −5.46012 −0.506959
\(117\) −1.00000 −0.0924500
\(118\) 4.90361 0.451414
\(119\) −6.21437 −0.569670
\(120\) −3.07028 −0.280277
\(121\) −8.29965 −0.754514
\(122\) 2.19611 0.198827
\(123\) −1.62153 −0.146208
\(124\) −0.105312 −0.00945731
\(125\) −1.76049 −0.157463
\(126\) 1.20257 0.107134
\(127\) 11.0748 0.982728 0.491364 0.870954i \(-0.336498\pi\)
0.491364 + 0.870954i \(0.336498\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.7227 −1.12017
\(130\) 3.07028 0.269281
\(131\) −4.30992 −0.376560 −0.188280 0.982115i \(-0.560291\pi\)
−0.188280 + 0.982115i \(0.560291\pi\)
\(132\) 1.64327 0.143029
\(133\) 8.38478 0.727052
\(134\) 15.1928 1.31246
\(135\) 3.07028 0.264247
\(136\) −5.16756 −0.443115
\(137\) 2.72360 0.232693 0.116346 0.993209i \(-0.462882\pi\)
0.116346 + 0.993209i \(0.462882\pi\)
\(138\) 4.45098 0.378893
\(139\) 12.1741 1.03259 0.516296 0.856410i \(-0.327311\pi\)
0.516296 + 0.856410i \(0.327311\pi\)
\(140\) −3.69223 −0.312050
\(141\) −9.64077 −0.811899
\(142\) −12.9797 −1.08923
\(143\) −1.64327 −0.137417
\(144\) 1.00000 0.0833333
\(145\) −16.7641 −1.39218
\(146\) −0.0855982 −0.00708416
\(147\) −5.55382 −0.458071
\(148\) −4.85429 −0.399020
\(149\) −11.2270 −0.919751 −0.459875 0.887984i \(-0.652106\pi\)
−0.459875 + 0.887984i \(0.652106\pi\)
\(150\) −4.42660 −0.361431
\(151\) −11.9752 −0.974529 −0.487265 0.873254i \(-0.662005\pi\)
−0.487265 + 0.873254i \(0.662005\pi\)
\(152\) 6.97237 0.565534
\(153\) 5.16756 0.417773
\(154\) 1.97615 0.159243
\(155\) −0.323338 −0.0259711
\(156\) −1.00000 −0.0800641
\(157\) 9.53725 0.761156 0.380578 0.924749i \(-0.375725\pi\)
0.380578 + 0.924749i \(0.375725\pi\)
\(158\) 11.5359 0.917744
\(159\) −8.08025 −0.640806
\(160\) −3.07028 −0.242727
\(161\) 5.35263 0.421846
\(162\) −1.00000 −0.0785674
\(163\) 17.6275 1.38070 0.690348 0.723477i \(-0.257457\pi\)
0.690348 + 0.723477i \(0.257457\pi\)
\(164\) −1.62153 −0.126620
\(165\) 5.04531 0.392776
\(166\) −5.79368 −0.449677
\(167\) −19.3747 −1.49926 −0.749630 0.661857i \(-0.769769\pi\)
−0.749630 + 0.661857i \(0.769769\pi\)
\(168\) 1.20257 0.0927804
\(169\) 1.00000 0.0769231
\(170\) −15.8658 −1.21686
\(171\) −6.97237 −0.533190
\(172\) −12.7227 −0.970100
\(173\) 18.6200 1.41565 0.707827 0.706386i \(-0.249675\pi\)
0.707827 + 0.706386i \(0.249675\pi\)
\(174\) 5.46012 0.413931
\(175\) −5.32331 −0.402404
\(176\) 1.64327 0.123866
\(177\) −4.90361 −0.368578
\(178\) −4.12175 −0.308938
\(179\) −1.68183 −0.125706 −0.0628528 0.998023i \(-0.520020\pi\)
−0.0628528 + 0.998023i \(0.520020\pi\)
\(180\) 3.07028 0.228845
\(181\) 0.182749 0.0135836 0.00679182 0.999977i \(-0.497838\pi\)
0.00679182 + 0.999977i \(0.497838\pi\)
\(182\) −1.20257 −0.0891406
\(183\) −2.19611 −0.162341
\(184\) 4.45098 0.328131
\(185\) −14.9040 −1.09577
\(186\) 0.105312 0.00772186
\(187\) 8.49172 0.620976
\(188\) −9.64077 −0.703126
\(189\) −1.20257 −0.0874742
\(190\) 21.4071 1.55303
\(191\) 10.0979 0.730660 0.365330 0.930878i \(-0.380956\pi\)
0.365330 + 0.930878i \(0.380956\pi\)
\(192\) 1.00000 0.0721688
\(193\) 17.7644 1.27871 0.639354 0.768913i \(-0.279202\pi\)
0.639354 + 0.768913i \(0.279202\pi\)
\(194\) 3.96607 0.284747
\(195\) −3.07028 −0.219867
\(196\) −5.55382 −0.396701
\(197\) 13.3735 0.952821 0.476410 0.879223i \(-0.341938\pi\)
0.476410 + 0.879223i \(0.341938\pi\)
\(198\) −1.64327 −0.116782
\(199\) 9.16019 0.649348 0.324674 0.945826i \(-0.394745\pi\)
0.324674 + 0.945826i \(0.394745\pi\)
\(200\) −4.42660 −0.313008
\(201\) −15.1928 −1.07162
\(202\) −5.55432 −0.390800
\(203\) 6.56619 0.460856
\(204\) 5.16756 0.361802
\(205\) −4.97853 −0.347716
\(206\) −1.00000 −0.0696733
\(207\) −4.45098 −0.309365
\(208\) −1.00000 −0.0693375
\(209\) −11.4575 −0.792533
\(210\) 3.69223 0.254788
\(211\) −24.9722 −1.71915 −0.859577 0.511006i \(-0.829273\pi\)
−0.859577 + 0.511006i \(0.829273\pi\)
\(212\) −8.08025 −0.554954
\(213\) 12.9797 0.889351
\(214\) −3.11236 −0.212757
\(215\) −39.0623 −2.66403
\(216\) −1.00000 −0.0680414
\(217\) 0.126645 0.00859726
\(218\) 10.7609 0.728818
\(219\) 0.0855982 0.00578419
\(220\) 5.04531 0.340154
\(221\) −5.16756 −0.347608
\(222\) 4.85429 0.325799
\(223\) 19.9780 1.33782 0.668912 0.743341i \(-0.266760\pi\)
0.668912 + 0.743341i \(0.266760\pi\)
\(224\) 1.20257 0.0803502
\(225\) 4.42660 0.295107
\(226\) −20.4678 −1.36150
\(227\) −7.71833 −0.512283 −0.256142 0.966639i \(-0.582451\pi\)
−0.256142 + 0.966639i \(0.582451\pi\)
\(228\) −6.97237 −0.461756
\(229\) 16.6721 1.10172 0.550862 0.834596i \(-0.314299\pi\)
0.550862 + 0.834596i \(0.314299\pi\)
\(230\) 13.6657 0.901093
\(231\) −1.97615 −0.130021
\(232\) 5.46012 0.358474
\(233\) 10.8908 0.713481 0.356740 0.934204i \(-0.383888\pi\)
0.356740 + 0.934204i \(0.383888\pi\)
\(234\) 1.00000 0.0653720
\(235\) −29.5998 −1.93088
\(236\) −4.90361 −0.319198
\(237\) −11.5359 −0.749335
\(238\) 6.21437 0.402817
\(239\) −22.7544 −1.47186 −0.735929 0.677058i \(-0.763254\pi\)
−0.735929 + 0.677058i \(0.763254\pi\)
\(240\) 3.07028 0.198186
\(241\) 18.4300 1.18718 0.593592 0.804766i \(-0.297709\pi\)
0.593592 + 0.804766i \(0.297709\pi\)
\(242\) 8.29965 0.533522
\(243\) 1.00000 0.0641500
\(244\) −2.19611 −0.140592
\(245\) −17.0518 −1.08940
\(246\) 1.62153 0.103385
\(247\) 6.97237 0.443641
\(248\) 0.105312 0.00668733
\(249\) 5.79368 0.367160
\(250\) 1.76049 0.111343
\(251\) 3.06979 0.193763 0.0968816 0.995296i \(-0.469113\pi\)
0.0968816 + 0.995296i \(0.469113\pi\)
\(252\) −1.20257 −0.0757549
\(253\) −7.31418 −0.459839
\(254\) −11.0748 −0.694894
\(255\) 15.8658 0.993558
\(256\) 1.00000 0.0625000
\(257\) −6.34044 −0.395506 −0.197753 0.980252i \(-0.563364\pi\)
−0.197753 + 0.980252i \(0.563364\pi\)
\(258\) 12.7227 0.792083
\(259\) 5.83763 0.362733
\(260\) −3.07028 −0.190411
\(261\) −5.46012 −0.337973
\(262\) 4.30992 0.266268
\(263\) 3.14227 0.193761 0.0968803 0.995296i \(-0.469114\pi\)
0.0968803 + 0.995296i \(0.469114\pi\)
\(264\) −1.64327 −0.101136
\(265\) −24.8086 −1.52398
\(266\) −8.38478 −0.514104
\(267\) 4.12175 0.252247
\(268\) −15.1928 −0.928047
\(269\) 11.5731 0.705622 0.352811 0.935695i \(-0.385226\pi\)
0.352811 + 0.935695i \(0.385226\pi\)
\(270\) −3.07028 −0.186851
\(271\) 6.67362 0.405394 0.202697 0.979242i \(-0.435029\pi\)
0.202697 + 0.979242i \(0.435029\pi\)
\(272\) 5.16756 0.313329
\(273\) 1.20257 0.0727830
\(274\) −2.72360 −0.164539
\(275\) 7.27412 0.438646
\(276\) −4.45098 −0.267918
\(277\) 17.7381 1.06578 0.532891 0.846184i \(-0.321106\pi\)
0.532891 + 0.846184i \(0.321106\pi\)
\(278\) −12.1741 −0.730152
\(279\) −0.105312 −0.00630488
\(280\) 3.69223 0.220653
\(281\) −5.19281 −0.309777 −0.154889 0.987932i \(-0.549502\pi\)
−0.154889 + 0.987932i \(0.549502\pi\)
\(282\) 9.64077 0.574100
\(283\) −20.2995 −1.20668 −0.603341 0.797484i \(-0.706164\pi\)
−0.603341 + 0.797484i \(0.706164\pi\)
\(284\) 12.9797 0.770201
\(285\) −21.4071 −1.26805
\(286\) 1.64327 0.0971688
\(287\) 1.95000 0.115105
\(288\) −1.00000 −0.0589256
\(289\) 9.70370 0.570806
\(290\) 16.7641 0.984421
\(291\) −3.96607 −0.232495
\(292\) 0.0855982 0.00500925
\(293\) −19.9094 −1.16312 −0.581561 0.813503i \(-0.697558\pi\)
−0.581561 + 0.813503i \(0.697558\pi\)
\(294\) 5.55382 0.323905
\(295\) −15.0554 −0.876562
\(296\) 4.85429 0.282150
\(297\) 1.64327 0.0953524
\(298\) 11.2270 0.650362
\(299\) 4.45098 0.257407
\(300\) 4.42660 0.255570
\(301\) 15.3000 0.881878
\(302\) 11.9752 0.689096
\(303\) 5.55432 0.319087
\(304\) −6.97237 −0.399893
\(305\) −6.74268 −0.386085
\(306\) −5.16756 −0.295410
\(307\) 6.24787 0.356585 0.178292 0.983978i \(-0.442943\pi\)
0.178292 + 0.983978i \(0.442943\pi\)
\(308\) −1.97615 −0.112602
\(309\) 1.00000 0.0568880
\(310\) 0.323338 0.0183643
\(311\) −27.1527 −1.53969 −0.769845 0.638231i \(-0.779667\pi\)
−0.769845 + 0.638231i \(0.779667\pi\)
\(312\) 1.00000 0.0566139
\(313\) −15.7601 −0.890812 −0.445406 0.895329i \(-0.646941\pi\)
−0.445406 + 0.895329i \(0.646941\pi\)
\(314\) −9.53725 −0.538218
\(315\) −3.69223 −0.208034
\(316\) −11.5359 −0.648943
\(317\) −33.4872 −1.88083 −0.940415 0.340028i \(-0.889564\pi\)
−0.940415 + 0.340028i \(0.889564\pi\)
\(318\) 8.08025 0.453118
\(319\) −8.97247 −0.502362
\(320\) 3.07028 0.171634
\(321\) 3.11236 0.173715
\(322\) −5.35263 −0.298290
\(323\) −36.0302 −2.00477
\(324\) 1.00000 0.0555556
\(325\) −4.42660 −0.245544
\(326\) −17.6275 −0.976300
\(327\) −10.7609 −0.595078
\(328\) 1.62153 0.0895338
\(329\) 11.5937 0.639183
\(330\) −5.04531 −0.277735
\(331\) 21.6617 1.19063 0.595316 0.803492i \(-0.297027\pi\)
0.595316 + 0.803492i \(0.297027\pi\)
\(332\) 5.79368 0.317970
\(333\) −4.85429 −0.266013
\(334\) 19.3747 1.06014
\(335\) −46.6461 −2.54855
\(336\) −1.20257 −0.0656057
\(337\) −3.74274 −0.203880 −0.101940 0.994791i \(-0.532505\pi\)
−0.101940 + 0.994791i \(0.532505\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 20.4678 1.11166
\(340\) 15.8658 0.860446
\(341\) −0.173057 −0.00937155
\(342\) 6.97237 0.377023
\(343\) 15.0969 0.815155
\(344\) 12.7227 0.685964
\(345\) −13.6657 −0.735739
\(346\) −18.6200 −1.00102
\(347\) 24.1538 1.29664 0.648321 0.761367i \(-0.275472\pi\)
0.648321 + 0.761367i \(0.275472\pi\)
\(348\) −5.46012 −0.292693
\(349\) 33.4838 1.79234 0.896172 0.443707i \(-0.146337\pi\)
0.896172 + 0.443707i \(0.146337\pi\)
\(350\) 5.32331 0.284543
\(351\) −1.00000 −0.0533761
\(352\) −1.64327 −0.0875868
\(353\) 15.3361 0.816257 0.408129 0.912924i \(-0.366181\pi\)
0.408129 + 0.912924i \(0.366181\pi\)
\(354\) 4.90361 0.260624
\(355\) 39.8511 2.11508
\(356\) 4.12175 0.218452
\(357\) −6.21437 −0.328899
\(358\) 1.68183 0.0888873
\(359\) −19.2016 −1.01342 −0.506711 0.862116i \(-0.669139\pi\)
−0.506711 + 0.862116i \(0.669139\pi\)
\(360\) −3.07028 −0.161818
\(361\) 29.6139 1.55863
\(362\) −0.182749 −0.00960508
\(363\) −8.29965 −0.435619
\(364\) 1.20257 0.0630319
\(365\) 0.262810 0.0137561
\(366\) 2.19611 0.114793
\(367\) 11.5935 0.605177 0.302588 0.953121i \(-0.402149\pi\)
0.302588 + 0.953121i \(0.402149\pi\)
\(368\) −4.45098 −0.232023
\(369\) −1.62153 −0.0844133
\(370\) 14.9040 0.774823
\(371\) 9.71709 0.504486
\(372\) −0.105312 −0.00546018
\(373\) 21.8239 1.13000 0.565000 0.825091i \(-0.308876\pi\)
0.565000 + 0.825091i \(0.308876\pi\)
\(374\) −8.49172 −0.439096
\(375\) −1.76049 −0.0909113
\(376\) 9.64077 0.497185
\(377\) 5.46012 0.281210
\(378\) 1.20257 0.0618536
\(379\) 9.10148 0.467512 0.233756 0.972295i \(-0.424898\pi\)
0.233756 + 0.972295i \(0.424898\pi\)
\(380\) −21.4071 −1.09816
\(381\) 11.0748 0.567378
\(382\) −10.0979 −0.516655
\(383\) 32.9427 1.68329 0.841647 0.540029i \(-0.181587\pi\)
0.841647 + 0.540029i \(0.181587\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.06734 −0.309220
\(386\) −17.7644 −0.904183
\(387\) −12.7227 −0.646733
\(388\) −3.96607 −0.201347
\(389\) 23.9395 1.21378 0.606890 0.794786i \(-0.292417\pi\)
0.606890 + 0.794786i \(0.292417\pi\)
\(390\) 3.07028 0.155470
\(391\) −23.0007 −1.16320
\(392\) 5.55382 0.280510
\(393\) −4.30992 −0.217407
\(394\) −13.3735 −0.673746
\(395\) −35.4183 −1.78209
\(396\) 1.64327 0.0825776
\(397\) −37.8751 −1.90090 −0.950449 0.310882i \(-0.899376\pi\)
−0.950449 + 0.310882i \(0.899376\pi\)
\(398\) −9.16019 −0.459159
\(399\) 8.38478 0.419764
\(400\) 4.42660 0.221330
\(401\) −16.3655 −0.817254 −0.408627 0.912701i \(-0.633992\pi\)
−0.408627 + 0.912701i \(0.633992\pi\)
\(402\) 15.1928 0.757747
\(403\) 0.105312 0.00524597
\(404\) 5.55432 0.276338
\(405\) 3.07028 0.152563
\(406\) −6.56619 −0.325874
\(407\) −7.97693 −0.395402
\(408\) −5.16756 −0.255832
\(409\) −31.5789 −1.56148 −0.780738 0.624858i \(-0.785157\pi\)
−0.780738 + 0.624858i \(0.785157\pi\)
\(410\) 4.97853 0.245872
\(411\) 2.72360 0.134345
\(412\) 1.00000 0.0492665
\(413\) 5.89694 0.290170
\(414\) 4.45098 0.218754
\(415\) 17.7882 0.873189
\(416\) 1.00000 0.0490290
\(417\) 12.1741 0.596167
\(418\) 11.4575 0.560405
\(419\) 22.0491 1.07717 0.538584 0.842572i \(-0.318960\pi\)
0.538584 + 0.842572i \(0.318960\pi\)
\(420\) −3.69223 −0.180162
\(421\) 24.6640 1.20205 0.601024 0.799231i \(-0.294759\pi\)
0.601024 + 0.799231i \(0.294759\pi\)
\(422\) 24.9722 1.21563
\(423\) −9.64077 −0.468750
\(424\) 8.08025 0.392412
\(425\) 22.8747 1.10959
\(426\) −12.9797 −0.628866
\(427\) 2.64098 0.127806
\(428\) 3.11236 0.150442
\(429\) −1.64327 −0.0793380
\(430\) 39.0623 1.88375
\(431\) −39.9767 −1.92561 −0.962804 0.270201i \(-0.912910\pi\)
−0.962804 + 0.270201i \(0.912910\pi\)
\(432\) 1.00000 0.0481125
\(433\) −7.46041 −0.358524 −0.179262 0.983801i \(-0.557371\pi\)
−0.179262 + 0.983801i \(0.557371\pi\)
\(434\) −0.126645 −0.00607918
\(435\) −16.7641 −0.803776
\(436\) −10.7609 −0.515352
\(437\) 31.0339 1.48455
\(438\) −0.0855982 −0.00409004
\(439\) −24.6346 −1.17574 −0.587872 0.808954i \(-0.700034\pi\)
−0.587872 + 0.808954i \(0.700034\pi\)
\(440\) −5.04531 −0.240525
\(441\) −5.55382 −0.264468
\(442\) 5.16756 0.245796
\(443\) 39.3345 1.86884 0.934420 0.356173i \(-0.115918\pi\)
0.934420 + 0.356173i \(0.115918\pi\)
\(444\) −4.85429 −0.230374
\(445\) 12.6549 0.599900
\(446\) −19.9780 −0.945985
\(447\) −11.2270 −0.531018
\(448\) −1.20257 −0.0568162
\(449\) −11.6656 −0.550535 −0.275267 0.961368i \(-0.588766\pi\)
−0.275267 + 0.961368i \(0.588766\pi\)
\(450\) −4.42660 −0.208672
\(451\) −2.66461 −0.125472
\(452\) 20.4678 0.962726
\(453\) −11.9752 −0.562645
\(454\) 7.71833 0.362239
\(455\) 3.69223 0.173094
\(456\) 6.97237 0.326511
\(457\) 25.6576 1.20021 0.600106 0.799920i \(-0.295125\pi\)
0.600106 + 0.799920i \(0.295125\pi\)
\(458\) −16.6721 −0.779037
\(459\) 5.16756 0.241201
\(460\) −13.6657 −0.637169
\(461\) 17.6951 0.824145 0.412072 0.911151i \(-0.364805\pi\)
0.412072 + 0.911151i \(0.364805\pi\)
\(462\) 1.97615 0.0919390
\(463\) 27.3038 1.26892 0.634458 0.772958i \(-0.281223\pi\)
0.634458 + 0.772958i \(0.281223\pi\)
\(464\) −5.46012 −0.253480
\(465\) −0.323338 −0.0149944
\(466\) −10.8908 −0.504507
\(467\) 1.21442 0.0561969 0.0280984 0.999605i \(-0.491055\pi\)
0.0280984 + 0.999605i \(0.491055\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 18.2704 0.843649
\(470\) 29.5998 1.36534
\(471\) 9.53725 0.439453
\(472\) 4.90361 0.225707
\(473\) −20.9069 −0.961302
\(474\) 11.5359 0.529860
\(475\) −30.8639 −1.41613
\(476\) −6.21437 −0.284835
\(477\) −8.08025 −0.369969
\(478\) 22.7544 1.04076
\(479\) −39.2516 −1.79345 −0.896725 0.442589i \(-0.854060\pi\)
−0.896725 + 0.442589i \(0.854060\pi\)
\(480\) −3.07028 −0.140138
\(481\) 4.85429 0.221337
\(482\) −18.4300 −0.839466
\(483\) 5.35263 0.243553
\(484\) −8.29965 −0.377257
\(485\) −12.1769 −0.552926
\(486\) −1.00000 −0.0453609
\(487\) −21.0855 −0.955477 −0.477739 0.878502i \(-0.658543\pi\)
−0.477739 + 0.878502i \(0.658543\pi\)
\(488\) 2.19611 0.0994134
\(489\) 17.6275 0.797145
\(490\) 17.0518 0.770320
\(491\) −3.30888 −0.149328 −0.0746638 0.997209i \(-0.523788\pi\)
−0.0746638 + 0.997209i \(0.523788\pi\)
\(492\) −1.62153 −0.0731040
\(493\) −28.2155 −1.27076
\(494\) −6.97237 −0.313702
\(495\) 5.04531 0.226770
\(496\) −0.105312 −0.00472866
\(497\) −15.6090 −0.700158
\(498\) −5.79368 −0.259621
\(499\) 35.6352 1.59525 0.797625 0.603154i \(-0.206090\pi\)
0.797625 + 0.603154i \(0.206090\pi\)
\(500\) −1.76049 −0.0787315
\(501\) −19.3747 −0.865599
\(502\) −3.06979 −0.137011
\(503\) 24.5682 1.09544 0.547722 0.836660i \(-0.315495\pi\)
0.547722 + 0.836660i \(0.315495\pi\)
\(504\) 1.20257 0.0535668
\(505\) 17.0533 0.758862
\(506\) 7.31418 0.325155
\(507\) 1.00000 0.0444116
\(508\) 11.0748 0.491364
\(509\) 30.2639 1.34142 0.670711 0.741719i \(-0.265989\pi\)
0.670711 + 0.741719i \(0.265989\pi\)
\(510\) −15.8658 −0.702552
\(511\) −0.102938 −0.00455371
\(512\) −1.00000 −0.0441942
\(513\) −6.97237 −0.307838
\(514\) 6.34044 0.279665
\(515\) 3.07028 0.135293
\(516\) −12.7227 −0.560087
\(517\) −15.8424 −0.696749
\(518\) −5.83763 −0.256491
\(519\) 18.6200 0.817329
\(520\) 3.07028 0.134641
\(521\) −2.64228 −0.115760 −0.0578802 0.998324i \(-0.518434\pi\)
−0.0578802 + 0.998324i \(0.518434\pi\)
\(522\) 5.46012 0.238983
\(523\) 6.62048 0.289493 0.144747 0.989469i \(-0.453763\pi\)
0.144747 + 0.989469i \(0.453763\pi\)
\(524\) −4.30992 −0.188280
\(525\) −5.32331 −0.232328
\(526\) −3.14227 −0.137009
\(527\) −0.544207 −0.0237060
\(528\) 1.64327 0.0715143
\(529\) −3.18876 −0.138642
\(530\) 24.8086 1.07762
\(531\) −4.90361 −0.212799
\(532\) 8.38478 0.363526
\(533\) 1.62153 0.0702361
\(534\) −4.12175 −0.178366
\(535\) 9.55582 0.413134
\(536\) 15.1928 0.656228
\(537\) −1.68183 −0.0725762
\(538\) −11.5731 −0.498950
\(539\) −9.12645 −0.393104
\(540\) 3.07028 0.132124
\(541\) 13.1057 0.563459 0.281730 0.959494i \(-0.409092\pi\)
0.281730 + 0.959494i \(0.409092\pi\)
\(542\) −6.67362 −0.286657
\(543\) 0.182749 0.00784252
\(544\) −5.16756 −0.221557
\(545\) −33.0389 −1.41523
\(546\) −1.20257 −0.0514653
\(547\) 18.0492 0.771729 0.385865 0.922555i \(-0.373903\pi\)
0.385865 + 0.922555i \(0.373903\pi\)
\(548\) 2.72360 0.116346
\(549\) −2.19611 −0.0937278
\(550\) −7.27412 −0.310169
\(551\) 38.0700 1.62184
\(552\) 4.45098 0.189446
\(553\) 13.8727 0.589928
\(554\) −17.7381 −0.753621
\(555\) −14.9040 −0.632640
\(556\) 12.1741 0.516296
\(557\) 6.55802 0.277872 0.138936 0.990301i \(-0.455632\pi\)
0.138936 + 0.990301i \(0.455632\pi\)
\(558\) 0.105312 0.00445822
\(559\) 12.7227 0.538114
\(560\) −3.69223 −0.156025
\(561\) 8.49172 0.358521
\(562\) 5.19281 0.219046
\(563\) −21.4589 −0.904385 −0.452192 0.891920i \(-0.649358\pi\)
−0.452192 + 0.891920i \(0.649358\pi\)
\(564\) −9.64077 −0.405950
\(565\) 62.8419 2.64378
\(566\) 20.2995 0.853252
\(567\) −1.20257 −0.0505033
\(568\) −12.9797 −0.544614
\(569\) 11.2933 0.473440 0.236720 0.971578i \(-0.423928\pi\)
0.236720 + 0.971578i \(0.423928\pi\)
\(570\) 21.4071 0.896645
\(571\) 32.6803 1.36763 0.683813 0.729657i \(-0.260320\pi\)
0.683813 + 0.729657i \(0.260320\pi\)
\(572\) −1.64327 −0.0687087
\(573\) 10.0979 0.421847
\(574\) −1.95000 −0.0813915
\(575\) −19.7027 −0.821661
\(576\) 1.00000 0.0416667
\(577\) 10.4058 0.433201 0.216600 0.976260i \(-0.430503\pi\)
0.216600 + 0.976260i \(0.430503\pi\)
\(578\) −9.70370 −0.403621
\(579\) 17.7644 0.738262
\(580\) −16.7641 −0.696091
\(581\) −6.96732 −0.289053
\(582\) 3.96607 0.164399
\(583\) −13.2781 −0.549921
\(584\) −0.0855982 −0.00354208
\(585\) −3.07028 −0.126940
\(586\) 19.9094 0.822451
\(587\) 14.2361 0.587586 0.293793 0.955869i \(-0.405082\pi\)
0.293793 + 0.955869i \(0.405082\pi\)
\(588\) −5.55382 −0.229036
\(589\) 0.734276 0.0302553
\(590\) 15.0554 0.619823
\(591\) 13.3735 0.550111
\(592\) −4.85429 −0.199510
\(593\) −47.7002 −1.95881 −0.979405 0.201903i \(-0.935287\pi\)
−0.979405 + 0.201903i \(0.935287\pi\)
\(594\) −1.64327 −0.0674243
\(595\) −19.0798 −0.782197
\(596\) −11.2270 −0.459875
\(597\) 9.16019 0.374902
\(598\) −4.45098 −0.182014
\(599\) −11.1070 −0.453821 −0.226910 0.973916i \(-0.572862\pi\)
−0.226910 + 0.973916i \(0.572862\pi\)
\(600\) −4.42660 −0.180715
\(601\) 20.3015 0.828116 0.414058 0.910250i \(-0.364111\pi\)
0.414058 + 0.910250i \(0.364111\pi\)
\(602\) −15.3000 −0.623582
\(603\) −15.1928 −0.618698
\(604\) −11.9752 −0.487265
\(605\) −25.4822 −1.03600
\(606\) −5.55432 −0.225629
\(607\) −23.1722 −0.940530 −0.470265 0.882525i \(-0.655842\pi\)
−0.470265 + 0.882525i \(0.655842\pi\)
\(608\) 6.97237 0.282767
\(609\) 6.56619 0.266075
\(610\) 6.74268 0.273003
\(611\) 9.64077 0.390024
\(612\) 5.16756 0.208886
\(613\) −19.8141 −0.800284 −0.400142 0.916453i \(-0.631039\pi\)
−0.400142 + 0.916453i \(0.631039\pi\)
\(614\) −6.24787 −0.252143
\(615\) −4.97853 −0.200754
\(616\) 1.97615 0.0796215
\(617\) −31.1091 −1.25241 −0.626203 0.779660i \(-0.715392\pi\)
−0.626203 + 0.779660i \(0.715392\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −32.2240 −1.29519 −0.647595 0.761985i \(-0.724225\pi\)
−0.647595 + 0.761985i \(0.724225\pi\)
\(620\) −0.323338 −0.0129856
\(621\) −4.45098 −0.178612
\(622\) 27.1527 1.08873
\(623\) −4.95670 −0.198586
\(624\) −1.00000 −0.0400320
\(625\) −27.5382 −1.10153
\(626\) 15.7601 0.629899
\(627\) −11.4575 −0.457569
\(628\) 9.53725 0.380578
\(629\) −25.0848 −1.00020
\(630\) 3.69223 0.147102
\(631\) −24.4836 −0.974678 −0.487339 0.873213i \(-0.662032\pi\)
−0.487339 + 0.873213i \(0.662032\pi\)
\(632\) 11.5359 0.458872
\(633\) −24.9722 −0.992554
\(634\) 33.4872 1.32995
\(635\) 34.0027 1.34935
\(636\) −8.08025 −0.320403
\(637\) 5.55382 0.220050
\(638\) 8.97247 0.355223
\(639\) 12.9797 0.513467
\(640\) −3.07028 −0.121363
\(641\) 37.3835 1.47656 0.738279 0.674496i \(-0.235639\pi\)
0.738279 + 0.674496i \(0.235639\pi\)
\(642\) −3.11236 −0.122835
\(643\) 2.66560 0.105121 0.0525605 0.998618i \(-0.483262\pi\)
0.0525605 + 0.998618i \(0.483262\pi\)
\(644\) 5.35263 0.210923
\(645\) −39.0623 −1.53808
\(646\) 36.0302 1.41759
\(647\) −35.3663 −1.39039 −0.695195 0.718821i \(-0.744682\pi\)
−0.695195 + 0.718821i \(0.744682\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.05797 −0.316303
\(650\) 4.42660 0.173626
\(651\) 0.126645 0.00496363
\(652\) 17.6275 0.690348
\(653\) −16.9690 −0.664049 −0.332025 0.943271i \(-0.607732\pi\)
−0.332025 + 0.943271i \(0.607732\pi\)
\(654\) 10.7609 0.420784
\(655\) −13.2327 −0.517042
\(656\) −1.62153 −0.0633100
\(657\) 0.0855982 0.00333950
\(658\) −11.5937 −0.451970
\(659\) 32.3090 1.25858 0.629289 0.777171i \(-0.283346\pi\)
0.629289 + 0.777171i \(0.283346\pi\)
\(660\) 5.04531 0.196388
\(661\) −7.72472 −0.300457 −0.150228 0.988651i \(-0.548001\pi\)
−0.150228 + 0.988651i \(0.548001\pi\)
\(662\) −21.6617 −0.841904
\(663\) −5.16756 −0.200691
\(664\) −5.79368 −0.224838
\(665\) 25.7436 0.998294
\(666\) 4.85429 0.188100
\(667\) 24.3029 0.941012
\(668\) −19.3747 −0.749630
\(669\) 19.9780 0.772393
\(670\) 46.6461 1.80209
\(671\) −3.60882 −0.139317
\(672\) 1.20257 0.0463902
\(673\) −14.3304 −0.552397 −0.276198 0.961101i \(-0.589075\pi\)
−0.276198 + 0.961101i \(0.589075\pi\)
\(674\) 3.74274 0.144165
\(675\) 4.42660 0.170380
\(676\) 1.00000 0.0384615
\(677\) −15.9596 −0.613377 −0.306688 0.951810i \(-0.599221\pi\)
−0.306688 + 0.951810i \(0.599221\pi\)
\(678\) −20.4678 −0.786062
\(679\) 4.76949 0.183036
\(680\) −15.8658 −0.608428
\(681\) −7.71833 −0.295767
\(682\) 0.173057 0.00662669
\(683\) 14.1424 0.541142 0.270571 0.962700i \(-0.412787\pi\)
0.270571 + 0.962700i \(0.412787\pi\)
\(684\) −6.97237 −0.266595
\(685\) 8.36221 0.319504
\(686\) −15.0969 −0.576401
\(687\) 16.6721 0.636081
\(688\) −12.7227 −0.485050
\(689\) 8.08025 0.307833
\(690\) 13.6657 0.520246
\(691\) 17.1292 0.651627 0.325813 0.945434i \(-0.394362\pi\)
0.325813 + 0.945434i \(0.394362\pi\)
\(692\) 18.6200 0.707827
\(693\) −1.97615 −0.0750679
\(694\) −24.1538 −0.916864
\(695\) 37.3778 1.41782
\(696\) 5.46012 0.206965
\(697\) −8.37934 −0.317390
\(698\) −33.4838 −1.26738
\(699\) 10.8908 0.411928
\(700\) −5.32331 −0.201202
\(701\) −23.6808 −0.894412 −0.447206 0.894431i \(-0.647581\pi\)
−0.447206 + 0.894431i \(0.647581\pi\)
\(702\) 1.00000 0.0377426
\(703\) 33.8459 1.27652
\(704\) 1.64327 0.0619332
\(705\) −29.5998 −1.11479
\(706\) −15.3361 −0.577181
\(707\) −6.67947 −0.251207
\(708\) −4.90361 −0.184289
\(709\) 40.3891 1.51684 0.758422 0.651764i \(-0.225971\pi\)
0.758422 + 0.651764i \(0.225971\pi\)
\(710\) −39.8511 −1.49559
\(711\) −11.5359 −0.432629
\(712\) −4.12175 −0.154469
\(713\) 0.468743 0.0175546
\(714\) 6.21437 0.232567
\(715\) −5.04531 −0.188684
\(716\) −1.68183 −0.0628528
\(717\) −22.7544 −0.849778
\(718\) 19.2016 0.716598
\(719\) 18.8348 0.702420 0.351210 0.936297i \(-0.385770\pi\)
0.351210 + 0.936297i \(0.385770\pi\)
\(720\) 3.07028 0.114422
\(721\) −1.20257 −0.0447861
\(722\) −29.6139 −1.10212
\(723\) 18.4300 0.685421
\(724\) 0.182749 0.00679182
\(725\) −24.1698 −0.897643
\(726\) 8.29965 0.308029
\(727\) −23.0428 −0.854611 −0.427306 0.904107i \(-0.640537\pi\)
−0.427306 + 0.904107i \(0.640537\pi\)
\(728\) −1.20257 −0.0445703
\(729\) 1.00000 0.0370370
\(730\) −0.262810 −0.00972704
\(731\) −65.7455 −2.43169
\(732\) −2.19611 −0.0811707
\(733\) 23.7889 0.878665 0.439332 0.898325i \(-0.355215\pi\)
0.439332 + 0.898325i \(0.355215\pi\)
\(734\) −11.5935 −0.427925
\(735\) −17.0518 −0.628964
\(736\) 4.45098 0.164065
\(737\) −24.9659 −0.919631
\(738\) 1.62153 0.0596892
\(739\) −15.4492 −0.568308 −0.284154 0.958779i \(-0.591713\pi\)
−0.284154 + 0.958779i \(0.591713\pi\)
\(740\) −14.9040 −0.547883
\(741\) 6.97237 0.256136
\(742\) −9.71709 −0.356725
\(743\) −32.4360 −1.18996 −0.594981 0.803740i \(-0.702840\pi\)
−0.594981 + 0.803740i \(0.702840\pi\)
\(744\) 0.105312 0.00386093
\(745\) −34.4700 −1.26288
\(746\) −21.8239 −0.799031
\(747\) 5.79368 0.211980
\(748\) 8.49172 0.310488
\(749\) −3.74284 −0.136760
\(750\) 1.76049 0.0642840
\(751\) 46.0229 1.67940 0.839700 0.543051i \(-0.182731\pi\)
0.839700 + 0.543051i \(0.182731\pi\)
\(752\) −9.64077 −0.351563
\(753\) 3.06979 0.111869
\(754\) −5.46012 −0.198846
\(755\) −36.7672 −1.33810
\(756\) −1.20257 −0.0437371
\(757\) −23.7709 −0.863970 −0.431985 0.901881i \(-0.642187\pi\)
−0.431985 + 0.901881i \(0.642187\pi\)
\(758\) −9.10148 −0.330581
\(759\) −7.31418 −0.265488
\(760\) 21.4071 0.776517
\(761\) 9.66927 0.350511 0.175255 0.984523i \(-0.443925\pi\)
0.175255 + 0.984523i \(0.443925\pi\)
\(762\) −11.0748 −0.401197
\(763\) 12.9407 0.468486
\(764\) 10.0979 0.365330
\(765\) 15.8658 0.573631
\(766\) −32.9427 −1.19027
\(767\) 4.90361 0.177059
\(768\) 1.00000 0.0360844
\(769\) −42.4028 −1.52908 −0.764541 0.644575i \(-0.777034\pi\)
−0.764541 + 0.644575i \(0.777034\pi\)
\(770\) 6.06734 0.218652
\(771\) −6.34044 −0.228345
\(772\) 17.7644 0.639354
\(773\) −26.9094 −0.967865 −0.483933 0.875105i \(-0.660792\pi\)
−0.483933 + 0.875105i \(0.660792\pi\)
\(774\) 12.7227 0.457309
\(775\) −0.466175 −0.0167455
\(776\) 3.96607 0.142374
\(777\) 5.83763 0.209424
\(778\) −23.9395 −0.858272
\(779\) 11.3059 0.405075
\(780\) −3.07028 −0.109934
\(781\) 21.3291 0.763216
\(782\) 23.0007 0.822504
\(783\) −5.46012 −0.195129
\(784\) −5.55382 −0.198351
\(785\) 29.2820 1.04512
\(786\) 4.30992 0.153730
\(787\) −54.1803 −1.93132 −0.965660 0.259810i \(-0.916340\pi\)
−0.965660 + 0.259810i \(0.916340\pi\)
\(788\) 13.3735 0.476410
\(789\) 3.14227 0.111868
\(790\) 35.4183 1.26013
\(791\) −24.6140 −0.875174
\(792\) −1.64327 −0.0583912
\(793\) 2.19611 0.0779863
\(794\) 37.8751 1.34414
\(795\) −24.8086 −0.879871
\(796\) 9.16019 0.324674
\(797\) −38.6249 −1.36816 −0.684082 0.729405i \(-0.739797\pi\)
−0.684082 + 0.729405i \(0.739797\pi\)
\(798\) −8.38478 −0.296818
\(799\) −49.8193 −1.76248
\(800\) −4.42660 −0.156504
\(801\) 4.12175 0.145635
\(802\) 16.3655 0.577886
\(803\) 0.140661 0.00496383
\(804\) −15.1928 −0.535808
\(805\) 16.4340 0.579224
\(806\) −0.105312 −0.00370946
\(807\) 11.5731 0.407391
\(808\) −5.55432 −0.195400
\(809\) 0.561972 0.0197579 0.00987895 0.999951i \(-0.496855\pi\)
0.00987895 + 0.999951i \(0.496855\pi\)
\(810\) −3.07028 −0.107879
\(811\) −8.60816 −0.302273 −0.151137 0.988513i \(-0.548293\pi\)
−0.151137 + 0.988513i \(0.548293\pi\)
\(812\) 6.56619 0.230428
\(813\) 6.67362 0.234054
\(814\) 7.97693 0.279591
\(815\) 54.1215 1.89579
\(816\) 5.16756 0.180901
\(817\) 88.7076 3.10349
\(818\) 31.5789 1.10413
\(819\) 1.20257 0.0420213
\(820\) −4.97853 −0.173858
\(821\) 43.7122 1.52557 0.762783 0.646654i \(-0.223832\pi\)
0.762783 + 0.646654i \(0.223832\pi\)
\(822\) −2.72360 −0.0949965
\(823\) −47.8699 −1.66864 −0.834320 0.551280i \(-0.814139\pi\)
−0.834320 + 0.551280i \(0.814139\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 7.27412 0.253252
\(826\) −5.89694 −0.205181
\(827\) 44.4387 1.54529 0.772643 0.634841i \(-0.218934\pi\)
0.772643 + 0.634841i \(0.218934\pi\)
\(828\) −4.45098 −0.154682
\(829\) −2.52204 −0.0875939 −0.0437970 0.999040i \(-0.513945\pi\)
−0.0437970 + 0.999040i \(0.513945\pi\)
\(830\) −17.7882 −0.617438
\(831\) 17.7381 0.615329
\(832\) −1.00000 −0.0346688
\(833\) −28.6997 −0.994386
\(834\) −12.1741 −0.421554
\(835\) −59.4858 −2.05859
\(836\) −11.4575 −0.396266
\(837\) −0.105312 −0.00364012
\(838\) −22.0491 −0.761672
\(839\) −32.9801 −1.13860 −0.569300 0.822130i \(-0.692786\pi\)
−0.569300 + 0.822130i \(0.692786\pi\)
\(840\) 3.69223 0.127394
\(841\) 0.812907 0.0280313
\(842\) −24.6640 −0.849977
\(843\) −5.19281 −0.178850
\(844\) −24.9722 −0.859577
\(845\) 3.07028 0.105621
\(846\) 9.64077 0.331457
\(847\) 9.98093 0.342949
\(848\) −8.08025 −0.277477
\(849\) −20.2995 −0.696678
\(850\) −22.8747 −0.784597
\(851\) 21.6064 0.740656
\(852\) 12.9797 0.444676
\(853\) −45.2186 −1.54826 −0.774128 0.633029i \(-0.781812\pi\)
−0.774128 + 0.633029i \(0.781812\pi\)
\(854\) −2.64098 −0.0903726
\(855\) −21.4071 −0.732108
\(856\) −3.11236 −0.106378
\(857\) −34.9215 −1.19289 −0.596447 0.802652i \(-0.703421\pi\)
−0.596447 + 0.802652i \(0.703421\pi\)
\(858\) 1.64327 0.0561004
\(859\) −50.5313 −1.72411 −0.862053 0.506819i \(-0.830821\pi\)
−0.862053 + 0.506819i \(0.830821\pi\)
\(860\) −39.0623 −1.33201
\(861\) 1.95000 0.0664559
\(862\) 39.9767 1.36161
\(863\) 4.09305 0.139329 0.0696645 0.997570i \(-0.477807\pi\)
0.0696645 + 0.997570i \(0.477807\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 57.1686 1.94379
\(866\) 7.46041 0.253515
\(867\) 9.70370 0.329555
\(868\) 0.126645 0.00429863
\(869\) −18.9566 −0.643058
\(870\) 16.7641 0.568356
\(871\) 15.1928 0.514788
\(872\) 10.7609 0.364409
\(873\) −3.96607 −0.134231
\(874\) −31.0339 −1.04974
\(875\) 2.11712 0.0715716
\(876\) 0.0855982 0.00289209
\(877\) −12.6943 −0.428657 −0.214329 0.976762i \(-0.568756\pi\)
−0.214329 + 0.976762i \(0.568756\pi\)
\(878\) 24.6346 0.831376
\(879\) −19.9094 −0.671529
\(880\) 5.04531 0.170077
\(881\) −14.7407 −0.496628 −0.248314 0.968680i \(-0.579876\pi\)
−0.248314 + 0.968680i \(0.579876\pi\)
\(882\) 5.55382 0.187007
\(883\) −6.77196 −0.227895 −0.113947 0.993487i \(-0.536350\pi\)
−0.113947 + 0.993487i \(0.536350\pi\)
\(884\) −5.16756 −0.173804
\(885\) −15.0554 −0.506083
\(886\) −39.3345 −1.32147
\(887\) −9.33024 −0.313279 −0.156639 0.987656i \(-0.550066\pi\)
−0.156639 + 0.987656i \(0.550066\pi\)
\(888\) 4.85429 0.162899
\(889\) −13.3182 −0.446679
\(890\) −12.6549 −0.424194
\(891\) 1.64327 0.0550517
\(892\) 19.9780 0.668912
\(893\) 67.2190 2.24940
\(894\) 11.2270 0.375487
\(895\) −5.16368 −0.172603
\(896\) 1.20257 0.0401751
\(897\) 4.45098 0.148614
\(898\) 11.6656 0.389287
\(899\) 0.575017 0.0191779
\(900\) 4.42660 0.147553
\(901\) −41.7552 −1.39107
\(902\) 2.66461 0.0887218
\(903\) 15.3000 0.509152
\(904\) −20.4678 −0.680750
\(905\) 0.561091 0.0186513
\(906\) 11.9752 0.397850
\(907\) −36.4415 −1.21002 −0.605010 0.796218i \(-0.706831\pi\)
−0.605010 + 0.796218i \(0.706831\pi\)
\(908\) −7.71833 −0.256142
\(909\) 5.55432 0.184225
\(910\) −3.69223 −0.122396
\(911\) −17.0168 −0.563790 −0.281895 0.959445i \(-0.590963\pi\)
−0.281895 + 0.959445i \(0.590963\pi\)
\(912\) −6.97237 −0.230878
\(913\) 9.52060 0.315086
\(914\) −25.6576 −0.848678
\(915\) −6.74268 −0.222906
\(916\) 16.6721 0.550862
\(917\) 5.18299 0.171157
\(918\) −5.16756 −0.170555
\(919\) −14.4786 −0.477606 −0.238803 0.971068i \(-0.576755\pi\)
−0.238803 + 0.971068i \(0.576755\pi\)
\(920\) 13.6657 0.450546
\(921\) 6.24787 0.205874
\(922\) −17.6951 −0.582759
\(923\) −12.9797 −0.427231
\(924\) −1.97615 −0.0650107
\(925\) −21.4880 −0.706521
\(926\) −27.3038 −0.897259
\(927\) 1.00000 0.0328443
\(928\) 5.46012 0.179237
\(929\) 33.7938 1.10874 0.554369 0.832271i \(-0.312960\pi\)
0.554369 + 0.832271i \(0.312960\pi\)
\(930\) 0.323338 0.0106027
\(931\) 38.7233 1.26910
\(932\) 10.8908 0.356740
\(933\) −27.1527 −0.888941
\(934\) −1.21442 −0.0397372
\(935\) 26.0719 0.852643
\(936\) 1.00000 0.0326860
\(937\) 12.9072 0.421660 0.210830 0.977523i \(-0.432383\pi\)
0.210830 + 0.977523i \(0.432383\pi\)
\(938\) −18.2704 −0.596550
\(939\) −15.7601 −0.514310
\(940\) −29.5998 −0.965440
\(941\) −45.2959 −1.47660 −0.738302 0.674470i \(-0.764372\pi\)
−0.738302 + 0.674470i \(0.764372\pi\)
\(942\) −9.53725 −0.310741
\(943\) 7.21738 0.235030
\(944\) −4.90361 −0.159599
\(945\) −3.69223 −0.120108
\(946\) 20.9069 0.679743
\(947\) −14.8623 −0.482960 −0.241480 0.970406i \(-0.577633\pi\)
−0.241480 + 0.970406i \(0.577633\pi\)
\(948\) −11.5359 −0.374668
\(949\) −0.0855982 −0.00277863
\(950\) 30.8639 1.00136
\(951\) −33.4872 −1.08590
\(952\) 6.21437 0.201409
\(953\) −24.5583 −0.795522 −0.397761 0.917489i \(-0.630213\pi\)
−0.397761 + 0.917489i \(0.630213\pi\)
\(954\) 8.08025 0.261608
\(955\) 31.0034 1.00325
\(956\) −22.7544 −0.735929
\(957\) −8.97247 −0.290039
\(958\) 39.2516 1.26816
\(959\) −3.27533 −0.105766
\(960\) 3.07028 0.0990928
\(961\) −30.9889 −0.999642
\(962\) −4.85429 −0.156509
\(963\) 3.11236 0.100294
\(964\) 18.4300 0.593592
\(965\) 54.5416 1.75576
\(966\) −5.35263 −0.172218
\(967\) −36.6901 −1.17988 −0.589938 0.807449i \(-0.700848\pi\)
−0.589938 + 0.807449i \(0.700848\pi\)
\(968\) 8.29965 0.266761
\(969\) −36.0302 −1.15746
\(970\) 12.1769 0.390978
\(971\) −57.1794 −1.83498 −0.917488 0.397764i \(-0.869786\pi\)
−0.917488 + 0.397764i \(0.869786\pi\)
\(972\) 1.00000 0.0320750
\(973\) −14.6402 −0.469343
\(974\) 21.0855 0.675624
\(975\) −4.42660 −0.141765
\(976\) −2.19611 −0.0702959
\(977\) 57.7285 1.84690 0.923450 0.383719i \(-0.125357\pi\)
0.923450 + 0.383719i \(0.125357\pi\)
\(978\) −17.6275 −0.563667
\(979\) 6.77316 0.216471
\(980\) −17.0518 −0.544699
\(981\) −10.7609 −0.343568
\(982\) 3.30888 0.105591
\(983\) −21.7251 −0.692924 −0.346462 0.938064i \(-0.612617\pi\)
−0.346462 + 0.938064i \(0.612617\pi\)
\(984\) 1.62153 0.0516924
\(985\) 41.0603 1.30829
\(986\) 28.2155 0.898565
\(987\) 11.5937 0.369032
\(988\) 6.97237 0.221821
\(989\) 56.6287 1.80069
\(990\) −5.04531 −0.160350
\(991\) −36.8378 −1.17019 −0.585095 0.810965i \(-0.698943\pi\)
−0.585095 + 0.810965i \(0.698943\pi\)
\(992\) 0.105312 0.00334367
\(993\) 21.6617 0.687412
\(994\) 15.6090 0.495087
\(995\) 28.1243 0.891601
\(996\) 5.79368 0.183580
\(997\) 19.7875 0.626677 0.313339 0.949641i \(-0.398553\pi\)
0.313339 + 0.949641i \(0.398553\pi\)
\(998\) −35.6352 −1.12801
\(999\) −4.85429 −0.153583
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.bb.1.13 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.13 14 1.1 even 1 trivial