Properties

Label 8034.2.a.ba.1.10
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.05309\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.05309 q^{5} +1.00000 q^{6} -0.189425 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.05309 q^{5} +1.00000 q^{6} -0.189425 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.05309 q^{10} -6.30119 q^{11} -1.00000 q^{12} -1.00000 q^{13} +0.189425 q^{14} -1.05309 q^{15} +1.00000 q^{16} -4.95903 q^{17} -1.00000 q^{18} -6.89185 q^{19} +1.05309 q^{20} +0.189425 q^{21} +6.30119 q^{22} +8.44585 q^{23} +1.00000 q^{24} -3.89100 q^{25} +1.00000 q^{26} -1.00000 q^{27} -0.189425 q^{28} -2.48934 q^{29} +1.05309 q^{30} -6.01703 q^{31} -1.00000 q^{32} +6.30119 q^{33} +4.95903 q^{34} -0.199481 q^{35} +1.00000 q^{36} +0.704737 q^{37} +6.89185 q^{38} +1.00000 q^{39} -1.05309 q^{40} -0.424853 q^{41} -0.189425 q^{42} -5.29921 q^{43} -6.30119 q^{44} +1.05309 q^{45} -8.44585 q^{46} +4.70034 q^{47} -1.00000 q^{48} -6.96412 q^{49} +3.89100 q^{50} +4.95903 q^{51} -1.00000 q^{52} -12.3270 q^{53} +1.00000 q^{54} -6.63573 q^{55} +0.189425 q^{56} +6.89185 q^{57} +2.48934 q^{58} -13.9362 q^{59} -1.05309 q^{60} +1.74461 q^{61} +6.01703 q^{62} -0.189425 q^{63} +1.00000 q^{64} -1.05309 q^{65} -6.30119 q^{66} +14.4705 q^{67} -4.95903 q^{68} -8.44585 q^{69} +0.199481 q^{70} +12.7428 q^{71} -1.00000 q^{72} +8.73221 q^{73} -0.704737 q^{74} +3.89100 q^{75} -6.89185 q^{76} +1.19360 q^{77} -1.00000 q^{78} +10.7464 q^{79} +1.05309 q^{80} +1.00000 q^{81} +0.424853 q^{82} -7.04418 q^{83} +0.189425 q^{84} -5.22231 q^{85} +5.29921 q^{86} +2.48934 q^{87} +6.30119 q^{88} -11.5379 q^{89} -1.05309 q^{90} +0.189425 q^{91} +8.44585 q^{92} +6.01703 q^{93} -4.70034 q^{94} -7.25774 q^{95} +1.00000 q^{96} -2.13606 q^{97} +6.96412 q^{98} -6.30119 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) 1.05309 0.470956 0.235478 0.971880i \(-0.424334\pi\)
0.235478 + 0.971880i \(0.424334\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.189425 −0.0715957 −0.0357979 0.999359i \(-0.511397\pi\)
−0.0357979 + 0.999359i \(0.511397\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.05309 −0.333016
\(11\) −6.30119 −1.89988 −0.949941 0.312430i \(-0.898857\pi\)
−0.949941 + 0.312430i \(0.898857\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0.189425 0.0506258
\(15\) −1.05309 −0.271907
\(16\) 1.00000 0.250000
\(17\) −4.95903 −1.20274 −0.601371 0.798970i \(-0.705379\pi\)
−0.601371 + 0.798970i \(0.705379\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.89185 −1.58110 −0.790550 0.612398i \(-0.790205\pi\)
−0.790550 + 0.612398i \(0.790205\pi\)
\(20\) 1.05309 0.235478
\(21\) 0.189425 0.0413358
\(22\) 6.30119 1.34342
\(23\) 8.44585 1.76108 0.880541 0.473970i \(-0.157179\pi\)
0.880541 + 0.473970i \(0.157179\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.89100 −0.778200
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −0.189425 −0.0357979
\(29\) −2.48934 −0.462259 −0.231130 0.972923i \(-0.574242\pi\)
−0.231130 + 0.972923i \(0.574242\pi\)
\(30\) 1.05309 0.192267
\(31\) −6.01703 −1.08069 −0.540345 0.841443i \(-0.681706\pi\)
−0.540345 + 0.841443i \(0.681706\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.30119 1.09690
\(34\) 4.95903 0.850467
\(35\) −0.199481 −0.0337185
\(36\) 1.00000 0.166667
\(37\) 0.704737 0.115858 0.0579290 0.998321i \(-0.481550\pi\)
0.0579290 + 0.998321i \(0.481550\pi\)
\(38\) 6.89185 1.11801
\(39\) 1.00000 0.160128
\(40\) −1.05309 −0.166508
\(41\) −0.424853 −0.0663509 −0.0331754 0.999450i \(-0.510562\pi\)
−0.0331754 + 0.999450i \(0.510562\pi\)
\(42\) −0.189425 −0.0292288
\(43\) −5.29921 −0.808122 −0.404061 0.914732i \(-0.632402\pi\)
−0.404061 + 0.914732i \(0.632402\pi\)
\(44\) −6.30119 −0.949941
\(45\) 1.05309 0.156985
\(46\) −8.44585 −1.24527
\(47\) 4.70034 0.685614 0.342807 0.939406i \(-0.388622\pi\)
0.342807 + 0.939406i \(0.388622\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.96412 −0.994874
\(50\) 3.89100 0.550271
\(51\) 4.95903 0.694403
\(52\) −1.00000 −0.138675
\(53\) −12.3270 −1.69324 −0.846621 0.532196i \(-0.821367\pi\)
−0.846621 + 0.532196i \(0.821367\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.63573 −0.894761
\(56\) 0.189425 0.0253129
\(57\) 6.89185 0.912848
\(58\) 2.48934 0.326867
\(59\) −13.9362 −1.81434 −0.907171 0.420761i \(-0.861763\pi\)
−0.907171 + 0.420761i \(0.861763\pi\)
\(60\) −1.05309 −0.135953
\(61\) 1.74461 0.223375 0.111687 0.993743i \(-0.464374\pi\)
0.111687 + 0.993743i \(0.464374\pi\)
\(62\) 6.01703 0.764164
\(63\) −0.189425 −0.0238652
\(64\) 1.00000 0.125000
\(65\) −1.05309 −0.130620
\(66\) −6.30119 −0.775623
\(67\) 14.4705 1.76785 0.883925 0.467628i \(-0.154891\pi\)
0.883925 + 0.467628i \(0.154891\pi\)
\(68\) −4.95903 −0.601371
\(69\) −8.44585 −1.01676
\(70\) 0.199481 0.0238425
\(71\) 12.7428 1.51229 0.756146 0.654404i \(-0.227080\pi\)
0.756146 + 0.654404i \(0.227080\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.73221 1.02203 0.511014 0.859572i \(-0.329270\pi\)
0.511014 + 0.859572i \(0.329270\pi\)
\(74\) −0.704737 −0.0819240
\(75\) 3.89100 0.449294
\(76\) −6.89185 −0.790550
\(77\) 1.19360 0.136023
\(78\) −1.00000 −0.113228
\(79\) 10.7464 1.20906 0.604531 0.796581i \(-0.293360\pi\)
0.604531 + 0.796581i \(0.293360\pi\)
\(80\) 1.05309 0.117739
\(81\) 1.00000 0.111111
\(82\) 0.424853 0.0469171
\(83\) −7.04418 −0.773199 −0.386600 0.922248i \(-0.626350\pi\)
−0.386600 + 0.922248i \(0.626350\pi\)
\(84\) 0.189425 0.0206679
\(85\) −5.22231 −0.566439
\(86\) 5.29921 0.571429
\(87\) 2.48934 0.266885
\(88\) 6.30119 0.671710
\(89\) −11.5379 −1.22301 −0.611506 0.791240i \(-0.709436\pi\)
−0.611506 + 0.791240i \(0.709436\pi\)
\(90\) −1.05309 −0.111005
\(91\) 0.189425 0.0198571
\(92\) 8.44585 0.880541
\(93\) 6.01703 0.623937
\(94\) −4.70034 −0.484803
\(95\) −7.25774 −0.744629
\(96\) 1.00000 0.102062
\(97\) −2.13606 −0.216884 −0.108442 0.994103i \(-0.534586\pi\)
−0.108442 + 0.994103i \(0.534586\pi\)
\(98\) 6.96412 0.703482
\(99\) −6.30119 −0.633294
\(100\) −3.89100 −0.389100
\(101\) −6.03888 −0.600891 −0.300445 0.953799i \(-0.597135\pi\)
−0.300445 + 0.953799i \(0.597135\pi\)
\(102\) −4.95903 −0.491017
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 0.199481 0.0194674
\(106\) 12.3270 1.19730
\(107\) 3.17310 0.306755 0.153378 0.988168i \(-0.450985\pi\)
0.153378 + 0.988168i \(0.450985\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0.672224 0.0643873 0.0321937 0.999482i \(-0.489751\pi\)
0.0321937 + 0.999482i \(0.489751\pi\)
\(110\) 6.63573 0.632692
\(111\) −0.704737 −0.0668907
\(112\) −0.189425 −0.0178989
\(113\) −1.37079 −0.128953 −0.0644765 0.997919i \(-0.520538\pi\)
−0.0644765 + 0.997919i \(0.520538\pi\)
\(114\) −6.89185 −0.645481
\(115\) 8.89424 0.829393
\(116\) −2.48934 −0.231130
\(117\) −1.00000 −0.0924500
\(118\) 13.9362 1.28293
\(119\) 0.939362 0.0861112
\(120\) 1.05309 0.0961335
\(121\) 28.7050 2.60955
\(122\) −1.74461 −0.157950
\(123\) 0.424853 0.0383077
\(124\) −6.01703 −0.540345
\(125\) −9.36303 −0.837454
\(126\) 0.189425 0.0168753
\(127\) 17.8983 1.58822 0.794108 0.607777i \(-0.207938\pi\)
0.794108 + 0.607777i \(0.207938\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.29921 0.466570
\(130\) 1.05309 0.0923621
\(131\) −10.5649 −0.923062 −0.461531 0.887124i \(-0.652700\pi\)
−0.461531 + 0.887124i \(0.652700\pi\)
\(132\) 6.30119 0.548449
\(133\) 1.30549 0.113200
\(134\) −14.4705 −1.25006
\(135\) −1.05309 −0.0906356
\(136\) 4.95903 0.425233
\(137\) 20.2743 1.73215 0.866076 0.499912i \(-0.166634\pi\)
0.866076 + 0.499912i \(0.166634\pi\)
\(138\) 8.44585 0.718959
\(139\) 10.8071 0.916645 0.458323 0.888786i \(-0.348450\pi\)
0.458323 + 0.888786i \(0.348450\pi\)
\(140\) −0.199481 −0.0168592
\(141\) −4.70034 −0.395840
\(142\) −12.7428 −1.06935
\(143\) 6.30119 0.526932
\(144\) 1.00000 0.0833333
\(145\) −2.62150 −0.217704
\(146\) −8.73221 −0.722683
\(147\) 6.96412 0.574391
\(148\) 0.704737 0.0579290
\(149\) 16.2553 1.33168 0.665841 0.746094i \(-0.268073\pi\)
0.665841 + 0.746094i \(0.268073\pi\)
\(150\) −3.89100 −0.317699
\(151\) 18.2933 1.48869 0.744344 0.667797i \(-0.232762\pi\)
0.744344 + 0.667797i \(0.232762\pi\)
\(152\) 6.89185 0.559003
\(153\) −4.95903 −0.400914
\(154\) −1.19360 −0.0961831
\(155\) −6.33648 −0.508958
\(156\) 1.00000 0.0800641
\(157\) −15.3375 −1.22406 −0.612032 0.790833i \(-0.709648\pi\)
−0.612032 + 0.790833i \(0.709648\pi\)
\(158\) −10.7464 −0.854937
\(159\) 12.3270 0.977594
\(160\) −1.05309 −0.0832541
\(161\) −1.59985 −0.126086
\(162\) −1.00000 −0.0785674
\(163\) 19.0841 1.49478 0.747391 0.664384i \(-0.231306\pi\)
0.747391 + 0.664384i \(0.231306\pi\)
\(164\) −0.424853 −0.0331754
\(165\) 6.63573 0.516591
\(166\) 7.04418 0.546734
\(167\) 10.4370 0.807640 0.403820 0.914838i \(-0.367682\pi\)
0.403820 + 0.914838i \(0.367682\pi\)
\(168\) −0.189425 −0.0146144
\(169\) 1.00000 0.0769231
\(170\) 5.22231 0.400533
\(171\) −6.89185 −0.527033
\(172\) −5.29921 −0.404061
\(173\) −2.06672 −0.157130 −0.0785650 0.996909i \(-0.525034\pi\)
−0.0785650 + 0.996909i \(0.525034\pi\)
\(174\) −2.48934 −0.188716
\(175\) 0.737051 0.0557158
\(176\) −6.30119 −0.474970
\(177\) 13.9362 1.04751
\(178\) 11.5379 0.864801
\(179\) −8.42487 −0.629704 −0.314852 0.949141i \(-0.601955\pi\)
−0.314852 + 0.949141i \(0.601955\pi\)
\(180\) 1.05309 0.0784927
\(181\) −10.7366 −0.798045 −0.399022 0.916941i \(-0.630650\pi\)
−0.399022 + 0.916941i \(0.630650\pi\)
\(182\) −0.189425 −0.0140411
\(183\) −1.74461 −0.128966
\(184\) −8.44585 −0.622636
\(185\) 0.742151 0.0545640
\(186\) −6.01703 −0.441190
\(187\) 31.2478 2.28507
\(188\) 4.70034 0.342807
\(189\) 0.189425 0.0137786
\(190\) 7.25774 0.526532
\(191\) −1.14807 −0.0830715 −0.0415358 0.999137i \(-0.513225\pi\)
−0.0415358 + 0.999137i \(0.513225\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 3.72614 0.268214 0.134107 0.990967i \(-0.457183\pi\)
0.134107 + 0.990967i \(0.457183\pi\)
\(194\) 2.13606 0.153360
\(195\) 1.05309 0.0754134
\(196\) −6.96412 −0.497437
\(197\) −7.52027 −0.535797 −0.267898 0.963447i \(-0.586329\pi\)
−0.267898 + 0.963447i \(0.586329\pi\)
\(198\) 6.30119 0.447806
\(199\) −12.7348 −0.902746 −0.451373 0.892335i \(-0.649066\pi\)
−0.451373 + 0.892335i \(0.649066\pi\)
\(200\) 3.89100 0.275135
\(201\) −14.4705 −1.02067
\(202\) 6.03888 0.424894
\(203\) 0.471542 0.0330958
\(204\) 4.95903 0.347202
\(205\) −0.447408 −0.0312483
\(206\) −1.00000 −0.0696733
\(207\) 8.44585 0.587027
\(208\) −1.00000 −0.0693375
\(209\) 43.4269 3.00390
\(210\) −0.199481 −0.0137655
\(211\) −13.2352 −0.911147 −0.455574 0.890198i \(-0.650566\pi\)
−0.455574 + 0.890198i \(0.650566\pi\)
\(212\) −12.3270 −0.846621
\(213\) −12.7428 −0.873122
\(214\) −3.17310 −0.216909
\(215\) −5.58055 −0.380590
\(216\) 1.00000 0.0680414
\(217\) 1.13977 0.0773728
\(218\) −0.672224 −0.0455287
\(219\) −8.73221 −0.590068
\(220\) −6.63573 −0.447381
\(221\) 4.95903 0.333580
\(222\) 0.704737 0.0472988
\(223\) 0.794330 0.0531923 0.0265961 0.999646i \(-0.491533\pi\)
0.0265961 + 0.999646i \(0.491533\pi\)
\(224\) 0.189425 0.0126565
\(225\) −3.89100 −0.259400
\(226\) 1.37079 0.0911835
\(227\) −19.8316 −1.31627 −0.658134 0.752900i \(-0.728654\pi\)
−0.658134 + 0.752900i \(0.728654\pi\)
\(228\) 6.89185 0.456424
\(229\) 0.366382 0.0242112 0.0121056 0.999927i \(-0.496147\pi\)
0.0121056 + 0.999927i \(0.496147\pi\)
\(230\) −8.89424 −0.586469
\(231\) −1.19360 −0.0785331
\(232\) 2.48934 0.163433
\(233\) −19.6590 −1.28790 −0.643952 0.765066i \(-0.722706\pi\)
−0.643952 + 0.765066i \(0.722706\pi\)
\(234\) 1.00000 0.0653720
\(235\) 4.94988 0.322894
\(236\) −13.9362 −0.907171
\(237\) −10.7464 −0.698053
\(238\) −0.939362 −0.0608898
\(239\) 10.2458 0.662747 0.331374 0.943500i \(-0.392488\pi\)
0.331374 + 0.943500i \(0.392488\pi\)
\(240\) −1.05309 −0.0679767
\(241\) 4.66441 0.300461 0.150230 0.988651i \(-0.451998\pi\)
0.150230 + 0.988651i \(0.451998\pi\)
\(242\) −28.7050 −1.84523
\(243\) −1.00000 −0.0641500
\(244\) 1.74461 0.111687
\(245\) −7.33384 −0.468542
\(246\) −0.424853 −0.0270876
\(247\) 6.89185 0.438518
\(248\) 6.01703 0.382082
\(249\) 7.04418 0.446407
\(250\) 9.36303 0.592170
\(251\) 8.36406 0.527935 0.263968 0.964532i \(-0.414969\pi\)
0.263968 + 0.964532i \(0.414969\pi\)
\(252\) −0.189425 −0.0119326
\(253\) −53.2190 −3.34585
\(254\) −17.8983 −1.12304
\(255\) 5.22231 0.327033
\(256\) 1.00000 0.0625000
\(257\) −11.3987 −0.711030 −0.355515 0.934671i \(-0.615695\pi\)
−0.355515 + 0.934671i \(0.615695\pi\)
\(258\) −5.29921 −0.329914
\(259\) −0.133494 −0.00829494
\(260\) −1.05309 −0.0653099
\(261\) −2.48934 −0.154086
\(262\) 10.5649 0.652703
\(263\) −4.13508 −0.254980 −0.127490 0.991840i \(-0.540692\pi\)
−0.127490 + 0.991840i \(0.540692\pi\)
\(264\) −6.30119 −0.387812
\(265\) −12.9814 −0.797443
\(266\) −1.30549 −0.0800445
\(267\) 11.5379 0.706107
\(268\) 14.4705 0.883925
\(269\) −27.1252 −1.65385 −0.826927 0.562309i \(-0.809913\pi\)
−0.826927 + 0.562309i \(0.809913\pi\)
\(270\) 1.05309 0.0640890
\(271\) 15.3612 0.933127 0.466563 0.884488i \(-0.345492\pi\)
0.466563 + 0.884488i \(0.345492\pi\)
\(272\) −4.95903 −0.300685
\(273\) −0.189425 −0.0114645
\(274\) −20.2743 −1.22482
\(275\) 24.5180 1.47849
\(276\) −8.44585 −0.508381
\(277\) −11.8077 −0.709453 −0.354727 0.934970i \(-0.615426\pi\)
−0.354727 + 0.934970i \(0.615426\pi\)
\(278\) −10.8071 −0.648166
\(279\) −6.01703 −0.360230
\(280\) 0.199481 0.0119213
\(281\) 11.4689 0.684174 0.342087 0.939668i \(-0.388866\pi\)
0.342087 + 0.939668i \(0.388866\pi\)
\(282\) 4.70034 0.279901
\(283\) 25.0579 1.48954 0.744770 0.667321i \(-0.232559\pi\)
0.744770 + 0.667321i \(0.232559\pi\)
\(284\) 12.7428 0.756146
\(285\) 7.25774 0.429912
\(286\) −6.30119 −0.372597
\(287\) 0.0804775 0.00475044
\(288\) −1.00000 −0.0589256
\(289\) 7.59198 0.446587
\(290\) 2.62150 0.153940
\(291\) 2.13606 0.125218
\(292\) 8.73221 0.511014
\(293\) 0.545131 0.0318469 0.0159234 0.999873i \(-0.494931\pi\)
0.0159234 + 0.999873i \(0.494931\pi\)
\(294\) −6.96412 −0.406156
\(295\) −14.6761 −0.854476
\(296\) −0.704737 −0.0409620
\(297\) 6.30119 0.365632
\(298\) −16.2553 −0.941642
\(299\) −8.44585 −0.488436
\(300\) 3.89100 0.224647
\(301\) 1.00380 0.0578581
\(302\) −18.2933 −1.05266
\(303\) 6.03888 0.346924
\(304\) −6.89185 −0.395275
\(305\) 1.83724 0.105200
\(306\) 4.95903 0.283489
\(307\) 25.4222 1.45092 0.725460 0.688265i \(-0.241627\pi\)
0.725460 + 0.688265i \(0.241627\pi\)
\(308\) 1.19360 0.0680117
\(309\) −1.00000 −0.0568880
\(310\) 6.33648 0.359888
\(311\) −31.7263 −1.79903 −0.899516 0.436888i \(-0.856081\pi\)
−0.899516 + 0.436888i \(0.856081\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −13.1655 −0.744156 −0.372078 0.928201i \(-0.621355\pi\)
−0.372078 + 0.928201i \(0.621355\pi\)
\(314\) 15.3375 0.865544
\(315\) −0.199481 −0.0112395
\(316\) 10.7464 0.604531
\(317\) 24.2733 1.36332 0.681661 0.731668i \(-0.261258\pi\)
0.681661 + 0.731668i \(0.261258\pi\)
\(318\) −12.3270 −0.691263
\(319\) 15.6858 0.878238
\(320\) 1.05309 0.0588695
\(321\) −3.17310 −0.177105
\(322\) 1.59985 0.0891562
\(323\) 34.1769 1.90165
\(324\) 1.00000 0.0555556
\(325\) 3.89100 0.215834
\(326\) −19.0841 −1.05697
\(327\) −0.672224 −0.0371741
\(328\) 0.424853 0.0234586
\(329\) −0.890359 −0.0490871
\(330\) −6.63573 −0.365285
\(331\) −14.0156 −0.770367 −0.385183 0.922840i \(-0.625862\pi\)
−0.385183 + 0.922840i \(0.625862\pi\)
\(332\) −7.04418 −0.386600
\(333\) 0.704737 0.0386193
\(334\) −10.4370 −0.571088
\(335\) 15.2387 0.832580
\(336\) 0.189425 0.0103340
\(337\) −33.6272 −1.83179 −0.915895 0.401418i \(-0.868517\pi\)
−0.915895 + 0.401418i \(0.868517\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 1.37079 0.0744510
\(340\) −5.22231 −0.283219
\(341\) 37.9145 2.05318
\(342\) 6.89185 0.372669
\(343\) 2.64515 0.142824
\(344\) 5.29921 0.285714
\(345\) −8.89424 −0.478850
\(346\) 2.06672 0.111108
\(347\) −8.14151 −0.437059 −0.218530 0.975830i \(-0.570126\pi\)
−0.218530 + 0.975830i \(0.570126\pi\)
\(348\) 2.48934 0.133443
\(349\) 27.9084 1.49390 0.746952 0.664878i \(-0.231517\pi\)
0.746952 + 0.664878i \(0.231517\pi\)
\(350\) −0.737051 −0.0393970
\(351\) 1.00000 0.0533761
\(352\) 6.30119 0.335855
\(353\) −9.11644 −0.485219 −0.242610 0.970124i \(-0.578003\pi\)
−0.242610 + 0.970124i \(0.578003\pi\)
\(354\) −13.9362 −0.740702
\(355\) 13.4193 0.712223
\(356\) −11.5379 −0.611506
\(357\) −0.939362 −0.0497163
\(358\) 8.42487 0.445268
\(359\) −16.4159 −0.866398 −0.433199 0.901298i \(-0.642615\pi\)
−0.433199 + 0.901298i \(0.642615\pi\)
\(360\) −1.05309 −0.0555027
\(361\) 28.4976 1.49988
\(362\) 10.7366 0.564303
\(363\) −28.7050 −1.50662
\(364\) 0.189425 0.00992854
\(365\) 9.19581 0.481331
\(366\) 1.74461 0.0911925
\(367\) −33.3974 −1.74333 −0.871666 0.490101i \(-0.836960\pi\)
−0.871666 + 0.490101i \(0.836960\pi\)
\(368\) 8.44585 0.440270
\(369\) −0.424853 −0.0221170
\(370\) −0.742151 −0.0385826
\(371\) 2.33503 0.121229
\(372\) 6.01703 0.311968
\(373\) 9.39484 0.486446 0.243223 0.969970i \(-0.421795\pi\)
0.243223 + 0.969970i \(0.421795\pi\)
\(374\) −31.2478 −1.61579
\(375\) 9.36303 0.483505
\(376\) −4.70034 −0.242401
\(377\) 2.48934 0.128208
\(378\) −0.189425 −0.00974295
\(379\) 18.9930 0.975607 0.487803 0.872953i \(-0.337798\pi\)
0.487803 + 0.872953i \(0.337798\pi\)
\(380\) −7.25774 −0.372314
\(381\) −17.8983 −0.916957
\(382\) 1.14807 0.0587404
\(383\) 11.2538 0.575044 0.287522 0.957774i \(-0.407169\pi\)
0.287522 + 0.957774i \(0.407169\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.25697 0.0640611
\(386\) −3.72614 −0.189656
\(387\) −5.29921 −0.269374
\(388\) −2.13606 −0.108442
\(389\) 25.8266 1.30946 0.654730 0.755863i \(-0.272782\pi\)
0.654730 + 0.755863i \(0.272782\pi\)
\(390\) −1.05309 −0.0533253
\(391\) −41.8832 −2.11813
\(392\) 6.96412 0.351741
\(393\) 10.5649 0.532930
\(394\) 7.52027 0.378866
\(395\) 11.3169 0.569416
\(396\) −6.30119 −0.316647
\(397\) 4.09534 0.205539 0.102770 0.994705i \(-0.467230\pi\)
0.102770 + 0.994705i \(0.467230\pi\)
\(398\) 12.7348 0.638338
\(399\) −1.30549 −0.0653560
\(400\) −3.89100 −0.194550
\(401\) 4.12362 0.205924 0.102962 0.994685i \(-0.467168\pi\)
0.102962 + 0.994685i \(0.467168\pi\)
\(402\) 14.4705 0.721722
\(403\) 6.01703 0.299730
\(404\) −6.03888 −0.300445
\(405\) 1.05309 0.0523285
\(406\) −0.471542 −0.0234022
\(407\) −4.44068 −0.220116
\(408\) −4.95903 −0.245509
\(409\) −28.7895 −1.42355 −0.711773 0.702409i \(-0.752108\pi\)
−0.711773 + 0.702409i \(0.752108\pi\)
\(410\) 0.447408 0.0220959
\(411\) −20.2743 −1.00006
\(412\) 1.00000 0.0492665
\(413\) 2.63986 0.129899
\(414\) −8.44585 −0.415091
\(415\) −7.41816 −0.364143
\(416\) 1.00000 0.0490290
\(417\) −10.8071 −0.529225
\(418\) −43.4269 −2.12408
\(419\) 2.93488 0.143378 0.0716891 0.997427i \(-0.477161\pi\)
0.0716891 + 0.997427i \(0.477161\pi\)
\(420\) 0.199481 0.00973368
\(421\) 17.2512 0.840771 0.420386 0.907346i \(-0.361895\pi\)
0.420386 + 0.907346i \(0.361895\pi\)
\(422\) 13.2352 0.644278
\(423\) 4.70034 0.228538
\(424\) 12.3270 0.598652
\(425\) 19.2956 0.935974
\(426\) 12.7428 0.617390
\(427\) −0.330473 −0.0159927
\(428\) 3.17310 0.153378
\(429\) −6.30119 −0.304225
\(430\) 5.58055 0.269118
\(431\) 28.0145 1.34941 0.674706 0.738087i \(-0.264270\pi\)
0.674706 + 0.738087i \(0.264270\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −0.102646 −0.00493285 −0.00246642 0.999997i \(-0.500785\pi\)
−0.00246642 + 0.999997i \(0.500785\pi\)
\(434\) −1.13977 −0.0547109
\(435\) 2.62150 0.125691
\(436\) 0.672224 0.0321937
\(437\) −58.2076 −2.78445
\(438\) 8.73221 0.417241
\(439\) 23.1977 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(440\) 6.63573 0.316346
\(441\) −6.96412 −0.331625
\(442\) −4.95903 −0.235877
\(443\) 5.64862 0.268374 0.134187 0.990956i \(-0.457158\pi\)
0.134187 + 0.990956i \(0.457158\pi\)
\(444\) −0.704737 −0.0334453
\(445\) −12.1504 −0.575985
\(446\) −0.794330 −0.0376126
\(447\) −16.2553 −0.768847
\(448\) −0.189425 −0.00894947
\(449\) −12.3808 −0.584288 −0.292144 0.956374i \(-0.594369\pi\)
−0.292144 + 0.956374i \(0.594369\pi\)
\(450\) 3.89100 0.183424
\(451\) 2.67708 0.126059
\(452\) −1.37079 −0.0644765
\(453\) −18.2933 −0.859494
\(454\) 19.8316 0.930743
\(455\) 0.199481 0.00935182
\(456\) −6.89185 −0.322741
\(457\) 19.3974 0.907372 0.453686 0.891162i \(-0.350109\pi\)
0.453686 + 0.891162i \(0.350109\pi\)
\(458\) −0.366382 −0.0171199
\(459\) 4.95903 0.231468
\(460\) 8.89424 0.414696
\(461\) −36.2967 −1.69051 −0.845253 0.534366i \(-0.820551\pi\)
−0.845253 + 0.534366i \(0.820551\pi\)
\(462\) 1.19360 0.0555313
\(463\) −27.7766 −1.29089 −0.645445 0.763807i \(-0.723328\pi\)
−0.645445 + 0.763807i \(0.723328\pi\)
\(464\) −2.48934 −0.115565
\(465\) 6.33648 0.293847
\(466\) 19.6590 0.910686
\(467\) −24.9476 −1.15444 −0.577218 0.816590i \(-0.695862\pi\)
−0.577218 + 0.816590i \(0.695862\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −2.74106 −0.126571
\(470\) −4.94988 −0.228321
\(471\) 15.3375 0.706714
\(472\) 13.9362 0.641467
\(473\) 33.3914 1.53534
\(474\) 10.7464 0.493598
\(475\) 26.8162 1.23041
\(476\) 0.939362 0.0430556
\(477\) −12.3270 −0.564414
\(478\) −10.2458 −0.468633
\(479\) 20.9084 0.955328 0.477664 0.878543i \(-0.341484\pi\)
0.477664 + 0.878543i \(0.341484\pi\)
\(480\) 1.05309 0.0480668
\(481\) −0.704737 −0.0321332
\(482\) −4.66441 −0.212458
\(483\) 1.59985 0.0727958
\(484\) 28.7050 1.30477
\(485\) −2.24947 −0.102143
\(486\) 1.00000 0.0453609
\(487\) −8.35360 −0.378538 −0.189269 0.981925i \(-0.560612\pi\)
−0.189269 + 0.981925i \(0.560612\pi\)
\(488\) −1.74461 −0.0789750
\(489\) −19.0841 −0.863013
\(490\) 7.33384 0.331309
\(491\) 15.5291 0.700817 0.350408 0.936597i \(-0.386043\pi\)
0.350408 + 0.936597i \(0.386043\pi\)
\(492\) 0.424853 0.0191538
\(493\) 12.3447 0.555978
\(494\) −6.89185 −0.310079
\(495\) −6.63573 −0.298254
\(496\) −6.01703 −0.270173
\(497\) −2.41380 −0.108274
\(498\) −7.04418 −0.315657
\(499\) −2.73506 −0.122438 −0.0612190 0.998124i \(-0.519499\pi\)
−0.0612190 + 0.998124i \(0.519499\pi\)
\(500\) −9.36303 −0.418727
\(501\) −10.4370 −0.466291
\(502\) −8.36406 −0.373306
\(503\) 1.42393 0.0634897 0.0317448 0.999496i \(-0.489894\pi\)
0.0317448 + 0.999496i \(0.489894\pi\)
\(504\) 0.189425 0.00843764
\(505\) −6.35948 −0.282993
\(506\) 53.2190 2.36587
\(507\) −1.00000 −0.0444116
\(508\) 17.8983 0.794108
\(509\) 14.0771 0.623957 0.311978 0.950089i \(-0.399008\pi\)
0.311978 + 0.950089i \(0.399008\pi\)
\(510\) −5.22231 −0.231248
\(511\) −1.65410 −0.0731729
\(512\) −1.00000 −0.0441942
\(513\) 6.89185 0.304283
\(514\) 11.3987 0.502774
\(515\) 1.05309 0.0464047
\(516\) 5.29921 0.233285
\(517\) −29.6177 −1.30259
\(518\) 0.133494 0.00586541
\(519\) 2.06672 0.0907191
\(520\) 1.05309 0.0461811
\(521\) −31.7922 −1.39284 −0.696421 0.717633i \(-0.745225\pi\)
−0.696421 + 0.717633i \(0.745225\pi\)
\(522\) 2.48934 0.108956
\(523\) 41.2495 1.80371 0.901857 0.432035i \(-0.142204\pi\)
0.901857 + 0.432035i \(0.142204\pi\)
\(524\) −10.5649 −0.461531
\(525\) −0.737051 −0.0321675
\(526\) 4.13508 0.180298
\(527\) 29.8386 1.29979
\(528\) 6.30119 0.274224
\(529\) 48.3324 2.10141
\(530\) 12.9814 0.563877
\(531\) −13.9362 −0.604781
\(532\) 1.30549 0.0566000
\(533\) 0.424853 0.0184024
\(534\) −11.5379 −0.499293
\(535\) 3.34156 0.144468
\(536\) −14.4705 −0.625030
\(537\) 8.42487 0.363560
\(538\) 27.1252 1.16945
\(539\) 43.8823 1.89014
\(540\) −1.05309 −0.0453178
\(541\) −19.3016 −0.829843 −0.414921 0.909857i \(-0.636191\pi\)
−0.414921 + 0.909857i \(0.636191\pi\)
\(542\) −15.3612 −0.659820
\(543\) 10.7366 0.460751
\(544\) 4.95903 0.212617
\(545\) 0.707912 0.0303236
\(546\) 0.189425 0.00810662
\(547\) 37.9160 1.62117 0.810584 0.585622i \(-0.199150\pi\)
0.810584 + 0.585622i \(0.199150\pi\)
\(548\) 20.2743 0.866076
\(549\) 1.74461 0.0744583
\(550\) −24.5180 −1.04545
\(551\) 17.1562 0.730878
\(552\) 8.44585 0.359479
\(553\) −2.03563 −0.0865637
\(554\) 11.8077 0.501659
\(555\) −0.742151 −0.0315026
\(556\) 10.8071 0.458323
\(557\) −38.2536 −1.62086 −0.810428 0.585838i \(-0.800766\pi\)
−0.810428 + 0.585838i \(0.800766\pi\)
\(558\) 6.01703 0.254721
\(559\) 5.29921 0.224133
\(560\) −0.199481 −0.00842961
\(561\) −31.2478 −1.31928
\(562\) −11.4689 −0.483784
\(563\) −33.2804 −1.40260 −0.701301 0.712866i \(-0.747397\pi\)
−0.701301 + 0.712866i \(0.747397\pi\)
\(564\) −4.70034 −0.197920
\(565\) −1.44356 −0.0607312
\(566\) −25.0579 −1.05326
\(567\) −0.189425 −0.00795508
\(568\) −12.7428 −0.534676
\(569\) 34.3452 1.43983 0.719913 0.694064i \(-0.244181\pi\)
0.719913 + 0.694064i \(0.244181\pi\)
\(570\) −7.25774 −0.303993
\(571\) 3.88244 0.162475 0.0812376 0.996695i \(-0.474113\pi\)
0.0812376 + 0.996695i \(0.474113\pi\)
\(572\) 6.30119 0.263466
\(573\) 1.14807 0.0479614
\(574\) −0.0804775 −0.00335907
\(575\) −32.8628 −1.37047
\(576\) 1.00000 0.0416667
\(577\) 36.7291 1.52905 0.764527 0.644592i \(-0.222973\pi\)
0.764527 + 0.644592i \(0.222973\pi\)
\(578\) −7.59198 −0.315785
\(579\) −3.72614 −0.154853
\(580\) −2.62150 −0.108852
\(581\) 1.33434 0.0553578
\(582\) −2.13606 −0.0885426
\(583\) 77.6748 3.21696
\(584\) −8.73221 −0.361342
\(585\) −1.05309 −0.0435399
\(586\) −0.545131 −0.0225191
\(587\) 18.4319 0.760767 0.380384 0.924829i \(-0.375792\pi\)
0.380384 + 0.924829i \(0.375792\pi\)
\(588\) 6.96412 0.287195
\(589\) 41.4685 1.70868
\(590\) 14.6761 0.604206
\(591\) 7.52027 0.309342
\(592\) 0.704737 0.0289645
\(593\) 18.8293 0.773228 0.386614 0.922242i \(-0.373645\pi\)
0.386614 + 0.922242i \(0.373645\pi\)
\(594\) −6.30119 −0.258541
\(595\) 0.989233 0.0405546
\(596\) 16.2553 0.665841
\(597\) 12.7348 0.521201
\(598\) 8.44585 0.345377
\(599\) −19.8400 −0.810641 −0.405320 0.914175i \(-0.632840\pi\)
−0.405320 + 0.914175i \(0.632840\pi\)
\(600\) −3.89100 −0.158849
\(601\) −40.2792 −1.64302 −0.821512 0.570192i \(-0.806869\pi\)
−0.821512 + 0.570192i \(0.806869\pi\)
\(602\) −1.00380 −0.0409119
\(603\) 14.4705 0.589284
\(604\) 18.2933 0.744344
\(605\) 30.2290 1.22898
\(606\) −6.03888 −0.245313
\(607\) −22.1552 −0.899253 −0.449626 0.893217i \(-0.648443\pi\)
−0.449626 + 0.893217i \(0.648443\pi\)
\(608\) 6.89185 0.279502
\(609\) −0.471542 −0.0191079
\(610\) −1.83724 −0.0743875
\(611\) −4.70034 −0.190155
\(612\) −4.95903 −0.200457
\(613\) −15.1016 −0.609946 −0.304973 0.952361i \(-0.598647\pi\)
−0.304973 + 0.952361i \(0.598647\pi\)
\(614\) −25.4222 −1.02595
\(615\) 0.447408 0.0180412
\(616\) −1.19360 −0.0480915
\(617\) −7.10984 −0.286231 −0.143116 0.989706i \(-0.545712\pi\)
−0.143116 + 0.989706i \(0.545712\pi\)
\(618\) 1.00000 0.0402259
\(619\) 38.9573 1.56582 0.782912 0.622132i \(-0.213733\pi\)
0.782912 + 0.622132i \(0.213733\pi\)
\(620\) −6.33648 −0.254479
\(621\) −8.44585 −0.338920
\(622\) 31.7263 1.27211
\(623\) 2.18556 0.0875625
\(624\) 1.00000 0.0400320
\(625\) 9.59490 0.383796
\(626\) 13.1655 0.526198
\(627\) −43.4269 −1.73430
\(628\) −15.3375 −0.612032
\(629\) −3.49481 −0.139347
\(630\) 0.199481 0.00794752
\(631\) −18.8358 −0.749839 −0.374920 0.927057i \(-0.622330\pi\)
−0.374920 + 0.927057i \(0.622330\pi\)
\(632\) −10.7464 −0.427468
\(633\) 13.2352 0.526051
\(634\) −24.2733 −0.964014
\(635\) 18.8485 0.747980
\(636\) 12.3270 0.488797
\(637\) 6.96412 0.275928
\(638\) −15.6858 −0.621008
\(639\) 12.7428 0.504097
\(640\) −1.05309 −0.0416270
\(641\) 48.3133 1.90826 0.954130 0.299392i \(-0.0967838\pi\)
0.954130 + 0.299392i \(0.0967838\pi\)
\(642\) 3.17310 0.125232
\(643\) −11.6209 −0.458284 −0.229142 0.973393i \(-0.573592\pi\)
−0.229142 + 0.973393i \(0.573592\pi\)
\(644\) −1.59985 −0.0630430
\(645\) 5.58055 0.219734
\(646\) −34.1769 −1.34467
\(647\) 40.7120 1.60055 0.800277 0.599630i \(-0.204686\pi\)
0.800277 + 0.599630i \(0.204686\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 87.8149 3.44704
\(650\) −3.89100 −0.152618
\(651\) −1.13977 −0.0446712
\(652\) 19.0841 0.747391
\(653\) −15.4355 −0.604037 −0.302018 0.953302i \(-0.597660\pi\)
−0.302018 + 0.953302i \(0.597660\pi\)
\(654\) 0.672224 0.0262860
\(655\) −11.1258 −0.434722
\(656\) −0.424853 −0.0165877
\(657\) 8.73221 0.340676
\(658\) 0.890359 0.0347098
\(659\) −35.0061 −1.36364 −0.681821 0.731519i \(-0.738812\pi\)
−0.681821 + 0.731519i \(0.738812\pi\)
\(660\) 6.63573 0.258295
\(661\) −5.46991 −0.212755 −0.106377 0.994326i \(-0.533925\pi\)
−0.106377 + 0.994326i \(0.533925\pi\)
\(662\) 14.0156 0.544732
\(663\) −4.95903 −0.192593
\(664\) 7.04418 0.273367
\(665\) 1.37479 0.0533122
\(666\) −0.704737 −0.0273080
\(667\) −21.0246 −0.814076
\(668\) 10.4370 0.403820
\(669\) −0.794330 −0.0307106
\(670\) −15.2387 −0.588723
\(671\) −10.9932 −0.424386
\(672\) −0.189425 −0.00730721
\(673\) 9.16405 0.353248 0.176624 0.984278i \(-0.443482\pi\)
0.176624 + 0.984278i \(0.443482\pi\)
\(674\) 33.6272 1.29527
\(675\) 3.89100 0.149765
\(676\) 1.00000 0.0384615
\(677\) 41.6779 1.60181 0.800906 0.598789i \(-0.204351\pi\)
0.800906 + 0.598789i \(0.204351\pi\)
\(678\) −1.37079 −0.0526448
\(679\) 0.404622 0.0155280
\(680\) 5.22231 0.200266
\(681\) 19.8316 0.759948
\(682\) −37.9145 −1.45182
\(683\) −22.7126 −0.869072 −0.434536 0.900654i \(-0.643088\pi\)
−0.434536 + 0.900654i \(0.643088\pi\)
\(684\) −6.89185 −0.263517
\(685\) 21.3507 0.815768
\(686\) −2.64515 −0.100992
\(687\) −0.366382 −0.0139784
\(688\) −5.29921 −0.202031
\(689\) 12.3270 0.469621
\(690\) 8.89424 0.338598
\(691\) 18.2678 0.694938 0.347469 0.937691i \(-0.387041\pi\)
0.347469 + 0.937691i \(0.387041\pi\)
\(692\) −2.06672 −0.0785650
\(693\) 1.19360 0.0453411
\(694\) 8.14151 0.309048
\(695\) 11.3808 0.431700
\(696\) −2.48934 −0.0943582
\(697\) 2.10686 0.0798029
\(698\) −27.9084 −1.05635
\(699\) 19.6590 0.743572
\(700\) 0.737051 0.0278579
\(701\) −6.76574 −0.255538 −0.127769 0.991804i \(-0.540782\pi\)
−0.127769 + 0.991804i \(0.540782\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −4.85694 −0.183183
\(704\) −6.30119 −0.237485
\(705\) −4.94988 −0.186423
\(706\) 9.11644 0.343102
\(707\) 1.14391 0.0430212
\(708\) 13.9362 0.523756
\(709\) 15.1334 0.568348 0.284174 0.958773i \(-0.408281\pi\)
0.284174 + 0.958773i \(0.408281\pi\)
\(710\) −13.4193 −0.503618
\(711\) 10.7464 0.403021
\(712\) 11.5379 0.432400
\(713\) −50.8189 −1.90318
\(714\) 0.939362 0.0351547
\(715\) 6.63573 0.248162
\(716\) −8.42487 −0.314852
\(717\) −10.2458 −0.382637
\(718\) 16.4159 0.612636
\(719\) −4.18830 −0.156197 −0.0780987 0.996946i \(-0.524885\pi\)
−0.0780987 + 0.996946i \(0.524885\pi\)
\(720\) 1.05309 0.0392464
\(721\) −0.189425 −0.00705454
\(722\) −28.4976 −1.06057
\(723\) −4.66441 −0.173471
\(724\) −10.7366 −0.399022
\(725\) 9.68603 0.359730
\(726\) 28.7050 1.06534
\(727\) 24.6242 0.913262 0.456631 0.889656i \(-0.349056\pi\)
0.456631 + 0.889656i \(0.349056\pi\)
\(728\) −0.189425 −0.00702054
\(729\) 1.00000 0.0370370
\(730\) −9.19581 −0.340352
\(731\) 26.2789 0.971962
\(732\) −1.74461 −0.0644828
\(733\) 17.7081 0.654065 0.327032 0.945013i \(-0.393951\pi\)
0.327032 + 0.945013i \(0.393951\pi\)
\(734\) 33.3974 1.23272
\(735\) 7.33384 0.270513
\(736\) −8.44585 −0.311318
\(737\) −91.1813 −3.35871
\(738\) 0.424853 0.0156390
\(739\) −32.5190 −1.19623 −0.598114 0.801411i \(-0.704083\pi\)
−0.598114 + 0.801411i \(0.704083\pi\)
\(740\) 0.742151 0.0272820
\(741\) −6.89185 −0.253179
\(742\) −2.33503 −0.0857218
\(743\) 44.1268 1.61886 0.809428 0.587219i \(-0.199777\pi\)
0.809428 + 0.587219i \(0.199777\pi\)
\(744\) −6.01703 −0.220595
\(745\) 17.1182 0.627164
\(746\) −9.39484 −0.343970
\(747\) −7.04418 −0.257733
\(748\) 31.2478 1.14253
\(749\) −0.601063 −0.0219624
\(750\) −9.36303 −0.341889
\(751\) 35.2019 1.28453 0.642267 0.766481i \(-0.277994\pi\)
0.642267 + 0.766481i \(0.277994\pi\)
\(752\) 4.70034 0.171404
\(753\) −8.36406 −0.304803
\(754\) −2.48934 −0.0906565
\(755\) 19.2645 0.701107
\(756\) 0.189425 0.00688930
\(757\) −8.50362 −0.309069 −0.154535 0.987987i \(-0.549388\pi\)
−0.154535 + 0.987987i \(0.549388\pi\)
\(758\) −18.9930 −0.689858
\(759\) 53.2190 1.93173
\(760\) 7.25774 0.263266
\(761\) 20.9888 0.760845 0.380422 0.924813i \(-0.375779\pi\)
0.380422 + 0.924813i \(0.375779\pi\)
\(762\) 17.8983 0.648387
\(763\) −0.127336 −0.00460986
\(764\) −1.14807 −0.0415358
\(765\) −5.22231 −0.188813
\(766\) −11.2538 −0.406618
\(767\) 13.9362 0.503208
\(768\) −1.00000 −0.0360844
\(769\) −19.0681 −0.687614 −0.343807 0.939040i \(-0.611717\pi\)
−0.343807 + 0.939040i \(0.611717\pi\)
\(770\) −1.25697 −0.0452980
\(771\) 11.3987 0.410514
\(772\) 3.72614 0.134107
\(773\) −29.6344 −1.06587 −0.532937 0.846155i \(-0.678912\pi\)
−0.532937 + 0.846155i \(0.678912\pi\)
\(774\) 5.29921 0.190476
\(775\) 23.4123 0.840994
\(776\) 2.13606 0.0766801
\(777\) 0.133494 0.00478909
\(778\) −25.8266 −0.925928
\(779\) 2.92802 0.104907
\(780\) 1.05309 0.0377067
\(781\) −80.2948 −2.87317
\(782\) 41.8832 1.49774
\(783\) 2.48934 0.0889618
\(784\) −6.96412 −0.248719
\(785\) −16.1518 −0.576481
\(786\) −10.5649 −0.376838
\(787\) −28.8046 −1.02677 −0.513387 0.858157i \(-0.671609\pi\)
−0.513387 + 0.858157i \(0.671609\pi\)
\(788\) −7.52027 −0.267898
\(789\) 4.13508 0.147213
\(790\) −11.3169 −0.402638
\(791\) 0.259661 0.00923248
\(792\) 6.30119 0.223903
\(793\) −1.74461 −0.0619531
\(794\) −4.09534 −0.145338
\(795\) 12.9814 0.460404
\(796\) −12.7348 −0.451373
\(797\) −44.8138 −1.58739 −0.793693 0.608319i \(-0.791844\pi\)
−0.793693 + 0.608319i \(0.791844\pi\)
\(798\) 1.30549 0.0462137
\(799\) −23.3091 −0.824617
\(800\) 3.89100 0.137568
\(801\) −11.5379 −0.407671
\(802\) −4.12362 −0.145610
\(803\) −55.0234 −1.94173
\(804\) −14.4705 −0.510335
\(805\) −1.68479 −0.0593810
\(806\) −6.01703 −0.211941
\(807\) 27.1252 0.954853
\(808\) 6.03888 0.212447
\(809\) 50.2362 1.76621 0.883105 0.469175i \(-0.155449\pi\)
0.883105 + 0.469175i \(0.155449\pi\)
\(810\) −1.05309 −0.0370018
\(811\) 19.0399 0.668581 0.334290 0.942470i \(-0.391503\pi\)
0.334290 + 0.942470i \(0.391503\pi\)
\(812\) 0.471542 0.0165479
\(813\) −15.3612 −0.538741
\(814\) 4.44068 0.155646
\(815\) 20.0973 0.703977
\(816\) 4.95903 0.173601
\(817\) 36.5214 1.27772
\(818\) 28.7895 1.00660
\(819\) 0.189425 0.00661903
\(820\) −0.447408 −0.0156242
\(821\) 37.2395 1.29967 0.649834 0.760076i \(-0.274838\pi\)
0.649834 + 0.760076i \(0.274838\pi\)
\(822\) 20.2743 0.707148
\(823\) −5.46725 −0.190576 −0.0952882 0.995450i \(-0.530377\pi\)
−0.0952882 + 0.995450i \(0.530377\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −24.5180 −0.853606
\(826\) −2.63986 −0.0918526
\(827\) 38.4335 1.33647 0.668233 0.743952i \(-0.267051\pi\)
0.668233 + 0.743952i \(0.267051\pi\)
\(828\) 8.44585 0.293514
\(829\) −7.35497 −0.255449 −0.127724 0.991810i \(-0.540767\pi\)
−0.127724 + 0.991810i \(0.540767\pi\)
\(830\) 7.41816 0.257488
\(831\) 11.8077 0.409603
\(832\) −1.00000 −0.0346688
\(833\) 34.5353 1.19658
\(834\) 10.8071 0.374219
\(835\) 10.9911 0.380363
\(836\) 43.4269 1.50195
\(837\) 6.01703 0.207979
\(838\) −2.93488 −0.101384
\(839\) −50.6337 −1.74807 −0.874035 0.485862i \(-0.838506\pi\)
−0.874035 + 0.485862i \(0.838506\pi\)
\(840\) −0.199481 −0.00688275
\(841\) −22.8032 −0.786317
\(842\) −17.2512 −0.594515
\(843\) −11.4689 −0.395008
\(844\) −13.2352 −0.455574
\(845\) 1.05309 0.0362274
\(846\) −4.70034 −0.161601
\(847\) −5.43744 −0.186833
\(848\) −12.3270 −0.423311
\(849\) −25.0579 −0.859986
\(850\) −19.2956 −0.661833
\(851\) 5.95210 0.204035
\(852\) −12.7428 −0.436561
\(853\) −5.22926 −0.179046 −0.0895232 0.995985i \(-0.528534\pi\)
−0.0895232 + 0.995985i \(0.528534\pi\)
\(854\) 0.330473 0.0113085
\(855\) −7.25774 −0.248210
\(856\) −3.17310 −0.108454
\(857\) 4.92641 0.168283 0.0841415 0.996454i \(-0.473185\pi\)
0.0841415 + 0.996454i \(0.473185\pi\)
\(858\) 6.30119 0.215119
\(859\) −17.2776 −0.589506 −0.294753 0.955574i \(-0.595237\pi\)
−0.294753 + 0.955574i \(0.595237\pi\)
\(860\) −5.58055 −0.190295
\(861\) −0.0804775 −0.00274267
\(862\) −28.0145 −0.954178
\(863\) −48.6896 −1.65741 −0.828707 0.559682i \(-0.810923\pi\)
−0.828707 + 0.559682i \(0.810923\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.17645 −0.0740014
\(866\) 0.102646 0.00348805
\(867\) −7.59198 −0.257837
\(868\) 1.13977 0.0386864
\(869\) −67.7151 −2.29708
\(870\) −2.62150 −0.0888772
\(871\) −14.4705 −0.490314
\(872\) −0.672224 −0.0227644
\(873\) −2.13606 −0.0722947
\(874\) 58.2076 1.96890
\(875\) 1.77359 0.0599582
\(876\) −8.73221 −0.295034
\(877\) −34.5015 −1.16503 −0.582516 0.812819i \(-0.697932\pi\)
−0.582516 + 0.812819i \(0.697932\pi\)
\(878\) −23.1977 −0.782883
\(879\) −0.545131 −0.0183868
\(880\) −6.63573 −0.223690
\(881\) 23.8168 0.802408 0.401204 0.915989i \(-0.368592\pi\)
0.401204 + 0.915989i \(0.368592\pi\)
\(882\) 6.96412 0.234494
\(883\) 41.0927 1.38288 0.691440 0.722434i \(-0.256977\pi\)
0.691440 + 0.722434i \(0.256977\pi\)
\(884\) 4.95903 0.166790
\(885\) 14.6761 0.493332
\(886\) −5.64862 −0.189769
\(887\) 45.8192 1.53846 0.769229 0.638973i \(-0.220640\pi\)
0.769229 + 0.638973i \(0.220640\pi\)
\(888\) 0.704737 0.0236494
\(889\) −3.39037 −0.113709
\(890\) 12.1504 0.407283
\(891\) −6.30119 −0.211098
\(892\) 0.794330 0.0265961
\(893\) −32.3940 −1.08402
\(894\) 16.2553 0.543657
\(895\) −8.87214 −0.296563
\(896\) 0.189425 0.00632823
\(897\) 8.44585 0.281999
\(898\) 12.3808 0.413154
\(899\) 14.9784 0.499559
\(900\) −3.89100 −0.129700
\(901\) 61.1299 2.03653
\(902\) −2.67708 −0.0891370
\(903\) −1.00380 −0.0334044
\(904\) 1.37079 0.0455918
\(905\) −11.3066 −0.375844
\(906\) 18.2933 0.607754
\(907\) 21.1713 0.702980 0.351490 0.936192i \(-0.385675\pi\)
0.351490 + 0.936192i \(0.385675\pi\)
\(908\) −19.8316 −0.658134
\(909\) −6.03888 −0.200297
\(910\) −0.199481 −0.00661273
\(911\) −43.2041 −1.43142 −0.715708 0.698400i \(-0.753896\pi\)
−0.715708 + 0.698400i \(0.753896\pi\)
\(912\) 6.89185 0.228212
\(913\) 44.3867 1.46899
\(914\) −19.3974 −0.641609
\(915\) −1.83724 −0.0607372
\(916\) 0.366382 0.0121056
\(917\) 2.00126 0.0660873
\(918\) −4.95903 −0.163672
\(919\) 54.5527 1.79953 0.899764 0.436378i \(-0.143739\pi\)
0.899764 + 0.436378i \(0.143739\pi\)
\(920\) −8.89424 −0.293235
\(921\) −25.4222 −0.837689
\(922\) 36.2967 1.19537
\(923\) −12.7428 −0.419434
\(924\) −1.19360 −0.0392666
\(925\) −2.74213 −0.0901607
\(926\) 27.7766 0.912797
\(927\) 1.00000 0.0328443
\(928\) 2.48934 0.0817166
\(929\) −15.0501 −0.493778 −0.246889 0.969044i \(-0.579408\pi\)
−0.246889 + 0.969044i \(0.579408\pi\)
\(930\) −6.33648 −0.207781
\(931\) 47.9957 1.57299
\(932\) −19.6590 −0.643952
\(933\) 31.7263 1.03867
\(934\) 24.9476 0.816310
\(935\) 32.9068 1.07617
\(936\) 1.00000 0.0326860
\(937\) 8.25479 0.269672 0.134836 0.990868i \(-0.456949\pi\)
0.134836 + 0.990868i \(0.456949\pi\)
\(938\) 2.74106 0.0894989
\(939\) 13.1655 0.429639
\(940\) 4.94988 0.161447
\(941\) −34.1684 −1.11386 −0.556928 0.830561i \(-0.688020\pi\)
−0.556928 + 0.830561i \(0.688020\pi\)
\(942\) −15.3375 −0.499722
\(943\) −3.58824 −0.116849
\(944\) −13.9362 −0.453586
\(945\) 0.199481 0.00648912
\(946\) −33.3914 −1.08565
\(947\) 16.4384 0.534175 0.267087 0.963672i \(-0.413939\pi\)
0.267087 + 0.963672i \(0.413939\pi\)
\(948\) −10.7464 −0.349026
\(949\) −8.73221 −0.283460
\(950\) −26.8162 −0.870033
\(951\) −24.2733 −0.787114
\(952\) −0.939362 −0.0304449
\(953\) 49.3439 1.59840 0.799202 0.601062i \(-0.205256\pi\)
0.799202 + 0.601062i \(0.205256\pi\)
\(954\) 12.3270 0.399101
\(955\) −1.20902 −0.0391231
\(956\) 10.2458 0.331374
\(957\) −15.6858 −0.507051
\(958\) −20.9084 −0.675519
\(959\) −3.84045 −0.124015
\(960\) −1.05309 −0.0339883
\(961\) 5.20465 0.167892
\(962\) 0.704737 0.0227216
\(963\) 3.17310 0.102252
\(964\) 4.66441 0.150230
\(965\) 3.92396 0.126317
\(966\) −1.59985 −0.0514744
\(967\) −33.3715 −1.07316 −0.536578 0.843851i \(-0.680283\pi\)
−0.536578 + 0.843851i \(0.680283\pi\)
\(968\) −28.7050 −0.922615
\(969\) −34.1769 −1.09792
\(970\) 2.24947 0.0722260
\(971\) −32.1683 −1.03233 −0.516165 0.856489i \(-0.672641\pi\)
−0.516165 + 0.856489i \(0.672641\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −2.04713 −0.0656279
\(974\) 8.35360 0.267667
\(975\) −3.89100 −0.124612
\(976\) 1.74461 0.0558437
\(977\) −42.1490 −1.34847 −0.674233 0.738519i \(-0.735526\pi\)
−0.674233 + 0.738519i \(0.735526\pi\)
\(978\) 19.0841 0.610242
\(979\) 72.7024 2.32358
\(980\) −7.33384 −0.234271
\(981\) 0.672224 0.0214624
\(982\) −15.5291 −0.495552
\(983\) −6.49059 −0.207018 −0.103509 0.994629i \(-0.533007\pi\)
−0.103509 + 0.994629i \(0.533007\pi\)
\(984\) −0.424853 −0.0135438
\(985\) −7.91952 −0.252337
\(986\) −12.3447 −0.393136
\(987\) 0.890359 0.0283404
\(988\) 6.89185 0.219259
\(989\) −44.7564 −1.42317
\(990\) 6.63573 0.210897
\(991\) 32.1833 1.02233 0.511167 0.859481i \(-0.329213\pi\)
0.511167 + 0.859481i \(0.329213\pi\)
\(992\) 6.01703 0.191041
\(993\) 14.0156 0.444772
\(994\) 2.41380 0.0765610
\(995\) −13.4109 −0.425154
\(996\) 7.04418 0.223203
\(997\) 30.7553 0.974032 0.487016 0.873393i \(-0.338085\pi\)
0.487016 + 0.873393i \(0.338085\pi\)
\(998\) 2.73506 0.0865767
\(999\) −0.704737 −0.0222969
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.ba.1.10 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.10 14 1.1 even 1 trivial