Properties

Label 8034.2.a.z.1.12
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: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-2.28948\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.28948 q^{5} +1.00000 q^{6} -1.19378 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.28948 q^{5} +1.00000 q^{6} -1.19378 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.28948 q^{10} +4.27905 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.19378 q^{14} -2.28948 q^{15} +1.00000 q^{16} -3.59570 q^{17} -1.00000 q^{18} -1.35770 q^{19} +2.28948 q^{20} +1.19378 q^{21} -4.27905 q^{22} -6.42459 q^{23} +1.00000 q^{24} +0.241736 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.19378 q^{28} -6.53525 q^{29} +2.28948 q^{30} -1.25937 q^{31} -1.00000 q^{32} -4.27905 q^{33} +3.59570 q^{34} -2.73314 q^{35} +1.00000 q^{36} -6.96050 q^{37} +1.35770 q^{38} -1.00000 q^{39} -2.28948 q^{40} +7.36952 q^{41} -1.19378 q^{42} +1.70506 q^{43} +4.27905 q^{44} +2.28948 q^{45} +6.42459 q^{46} +1.19672 q^{47} -1.00000 q^{48} -5.57489 q^{49} -0.241736 q^{50} +3.59570 q^{51} +1.00000 q^{52} +2.84069 q^{53} +1.00000 q^{54} +9.79682 q^{55} +1.19378 q^{56} +1.35770 q^{57} +6.53525 q^{58} +14.3018 q^{59} -2.28948 q^{60} +3.12763 q^{61} +1.25937 q^{62} -1.19378 q^{63} +1.00000 q^{64} +2.28948 q^{65} +4.27905 q^{66} +10.3747 q^{67} -3.59570 q^{68} +6.42459 q^{69} +2.73314 q^{70} +1.20531 q^{71} -1.00000 q^{72} +10.1318 q^{73} +6.96050 q^{74} -0.241736 q^{75} -1.35770 q^{76} -5.10824 q^{77} +1.00000 q^{78} -2.00621 q^{79} +2.28948 q^{80} +1.00000 q^{81} -7.36952 q^{82} +4.49374 q^{83} +1.19378 q^{84} -8.23231 q^{85} -1.70506 q^{86} +6.53525 q^{87} -4.27905 q^{88} +5.55082 q^{89} -2.28948 q^{90} -1.19378 q^{91} -6.42459 q^{92} +1.25937 q^{93} -1.19672 q^{94} -3.10843 q^{95} +1.00000 q^{96} +7.35332 q^{97} +5.57489 q^{98} +4.27905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.28948 1.02389 0.511944 0.859019i \(-0.328925\pi\)
0.511944 + 0.859019i \(0.328925\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.19378 −0.451206 −0.225603 0.974219i \(-0.572435\pi\)
−0.225603 + 0.974219i \(0.572435\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.28948 −0.723998
\(11\) 4.27905 1.29018 0.645091 0.764106i \(-0.276819\pi\)
0.645091 + 0.764106i \(0.276819\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 1.19378 0.319051
\(15\) −2.28948 −0.591142
\(16\) 1.00000 0.250000
\(17\) −3.59570 −0.872086 −0.436043 0.899926i \(-0.643621\pi\)
−0.436043 + 0.899926i \(0.643621\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.35770 −0.311478 −0.155739 0.987798i \(-0.549776\pi\)
−0.155739 + 0.987798i \(0.549776\pi\)
\(20\) 2.28948 0.511944
\(21\) 1.19378 0.260504
\(22\) −4.27905 −0.912297
\(23\) −6.42459 −1.33962 −0.669810 0.742532i \(-0.733625\pi\)
−0.669810 + 0.742532i \(0.733625\pi\)
\(24\) 1.00000 0.204124
\(25\) 0.241736 0.0483473
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −1.19378 −0.225603
\(29\) −6.53525 −1.21357 −0.606783 0.794867i \(-0.707540\pi\)
−0.606783 + 0.794867i \(0.707540\pi\)
\(30\) 2.28948 0.418001
\(31\) −1.25937 −0.226189 −0.113095 0.993584i \(-0.536076\pi\)
−0.113095 + 0.993584i \(0.536076\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.27905 −0.744887
\(34\) 3.59570 0.616658
\(35\) −2.73314 −0.461985
\(36\) 1.00000 0.166667
\(37\) −6.96050 −1.14430 −0.572149 0.820150i \(-0.693890\pi\)
−0.572149 + 0.820150i \(0.693890\pi\)
\(38\) 1.35770 0.220248
\(39\) −1.00000 −0.160128
\(40\) −2.28948 −0.361999
\(41\) 7.36952 1.15093 0.575463 0.817828i \(-0.304822\pi\)
0.575463 + 0.817828i \(0.304822\pi\)
\(42\) −1.19378 −0.184204
\(43\) 1.70506 0.260020 0.130010 0.991513i \(-0.458499\pi\)
0.130010 + 0.991513i \(0.458499\pi\)
\(44\) 4.27905 0.645091
\(45\) 2.28948 0.341296
\(46\) 6.42459 0.947255
\(47\) 1.19672 0.174560 0.0872800 0.996184i \(-0.472183\pi\)
0.0872800 + 0.996184i \(0.472183\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.57489 −0.796413
\(50\) −0.241736 −0.0341867
\(51\) 3.59570 0.503499
\(52\) 1.00000 0.138675
\(53\) 2.84069 0.390199 0.195100 0.980783i \(-0.437497\pi\)
0.195100 + 0.980783i \(0.437497\pi\)
\(54\) 1.00000 0.136083
\(55\) 9.79682 1.32100
\(56\) 1.19378 0.159525
\(57\) 1.35770 0.179832
\(58\) 6.53525 0.858121
\(59\) 14.3018 1.86194 0.930971 0.365093i \(-0.118963\pi\)
0.930971 + 0.365093i \(0.118963\pi\)
\(60\) −2.28948 −0.295571
\(61\) 3.12763 0.400453 0.200226 0.979750i \(-0.435832\pi\)
0.200226 + 0.979750i \(0.435832\pi\)
\(62\) 1.25937 0.159940
\(63\) −1.19378 −0.150402
\(64\) 1.00000 0.125000
\(65\) 2.28948 0.283976
\(66\) 4.27905 0.526715
\(67\) 10.3747 1.26748 0.633738 0.773548i \(-0.281520\pi\)
0.633738 + 0.773548i \(0.281520\pi\)
\(68\) −3.59570 −0.436043
\(69\) 6.42459 0.773430
\(70\) 2.73314 0.326672
\(71\) 1.20531 0.143044 0.0715222 0.997439i \(-0.477214\pi\)
0.0715222 + 0.997439i \(0.477214\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.1318 1.18584 0.592919 0.805262i \(-0.297975\pi\)
0.592919 + 0.805262i \(0.297975\pi\)
\(74\) 6.96050 0.809141
\(75\) −0.241736 −0.0279133
\(76\) −1.35770 −0.155739
\(77\) −5.10824 −0.582138
\(78\) 1.00000 0.113228
\(79\) −2.00621 −0.225716 −0.112858 0.993611i \(-0.536000\pi\)
−0.112858 + 0.993611i \(0.536000\pi\)
\(80\) 2.28948 0.255972
\(81\) 1.00000 0.111111
\(82\) −7.36952 −0.813828
\(83\) 4.49374 0.493252 0.246626 0.969111i \(-0.420678\pi\)
0.246626 + 0.969111i \(0.420678\pi\)
\(84\) 1.19378 0.130252
\(85\) −8.23231 −0.892919
\(86\) −1.70506 −0.183862
\(87\) 6.53525 0.700653
\(88\) −4.27905 −0.456148
\(89\) 5.55082 0.588386 0.294193 0.955746i \(-0.404949\pi\)
0.294193 + 0.955746i \(0.404949\pi\)
\(90\) −2.28948 −0.241333
\(91\) −1.19378 −0.125142
\(92\) −6.42459 −0.669810
\(93\) 1.25937 0.130590
\(94\) −1.19672 −0.123433
\(95\) −3.10843 −0.318919
\(96\) 1.00000 0.102062
\(97\) 7.35332 0.746616 0.373308 0.927707i \(-0.378223\pi\)
0.373308 + 0.927707i \(0.378223\pi\)
\(98\) 5.57489 0.563149
\(99\) 4.27905 0.430061
\(100\) 0.241736 0.0241736
\(101\) 11.1470 1.10917 0.554584 0.832128i \(-0.312878\pi\)
0.554584 + 0.832128i \(0.312878\pi\)
\(102\) −3.59570 −0.356028
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 2.73314 0.266727
\(106\) −2.84069 −0.275912
\(107\) −4.78496 −0.462580 −0.231290 0.972885i \(-0.574295\pi\)
−0.231290 + 0.972885i \(0.574295\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.92640 0.280298 0.140149 0.990130i \(-0.455242\pi\)
0.140149 + 0.990130i \(0.455242\pi\)
\(110\) −9.79682 −0.934090
\(111\) 6.96050 0.660661
\(112\) −1.19378 −0.112801
\(113\) 2.55170 0.240044 0.120022 0.992771i \(-0.461704\pi\)
0.120022 + 0.992771i \(0.461704\pi\)
\(114\) −1.35770 −0.127160
\(115\) −14.7090 −1.37162
\(116\) −6.53525 −0.606783
\(117\) 1.00000 0.0924500
\(118\) −14.3018 −1.31659
\(119\) 4.29248 0.393491
\(120\) 2.28948 0.209000
\(121\) 7.31027 0.664570
\(122\) −3.12763 −0.283163
\(123\) −7.36952 −0.664487
\(124\) −1.25937 −0.113095
\(125\) −10.8940 −0.974386
\(126\) 1.19378 0.106350
\(127\) −3.49577 −0.310200 −0.155100 0.987899i \(-0.549570\pi\)
−0.155100 + 0.987899i \(0.549570\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.70506 −0.150123
\(130\) −2.28948 −0.200801
\(131\) −18.1097 −1.58226 −0.791128 0.611651i \(-0.790506\pi\)
−0.791128 + 0.611651i \(0.790506\pi\)
\(132\) −4.27905 −0.372444
\(133\) 1.62079 0.140541
\(134\) −10.3747 −0.896241
\(135\) −2.28948 −0.197047
\(136\) 3.59570 0.308329
\(137\) 0.772432 0.0659933 0.0329967 0.999455i \(-0.489495\pi\)
0.0329967 + 0.999455i \(0.489495\pi\)
\(138\) −6.42459 −0.546898
\(139\) 14.3030 1.21317 0.606583 0.795020i \(-0.292540\pi\)
0.606583 + 0.795020i \(0.292540\pi\)
\(140\) −2.73314 −0.230992
\(141\) −1.19672 −0.100782
\(142\) −1.20531 −0.101148
\(143\) 4.27905 0.357832
\(144\) 1.00000 0.0833333
\(145\) −14.9624 −1.24256
\(146\) −10.1318 −0.838515
\(147\) 5.57489 0.459809
\(148\) −6.96050 −0.572149
\(149\) 17.3427 1.42077 0.710385 0.703813i \(-0.248521\pi\)
0.710385 + 0.703813i \(0.248521\pi\)
\(150\) 0.241736 0.0197377
\(151\) 14.7147 1.19746 0.598731 0.800950i \(-0.295672\pi\)
0.598731 + 0.800950i \(0.295672\pi\)
\(152\) 1.35770 0.110124
\(153\) −3.59570 −0.290695
\(154\) 5.10824 0.411634
\(155\) −2.88331 −0.231593
\(156\) −1.00000 −0.0800641
\(157\) 22.0695 1.76134 0.880671 0.473729i \(-0.157092\pi\)
0.880671 + 0.473729i \(0.157092\pi\)
\(158\) 2.00621 0.159605
\(159\) −2.84069 −0.225282
\(160\) −2.28948 −0.181000
\(161\) 7.66954 0.604445
\(162\) −1.00000 −0.0785674
\(163\) −5.41189 −0.423892 −0.211946 0.977281i \(-0.567980\pi\)
−0.211946 + 0.977281i \(0.567980\pi\)
\(164\) 7.36952 0.575463
\(165\) −9.79682 −0.762681
\(166\) −4.49374 −0.348782
\(167\) 24.2697 1.87805 0.939025 0.343850i \(-0.111731\pi\)
0.939025 + 0.343850i \(0.111731\pi\)
\(168\) −1.19378 −0.0921020
\(169\) 1.00000 0.0769231
\(170\) 8.23231 0.631389
\(171\) −1.35770 −0.103826
\(172\) 1.70506 0.130010
\(173\) −9.74264 −0.740720 −0.370360 0.928888i \(-0.620766\pi\)
−0.370360 + 0.928888i \(0.620766\pi\)
\(174\) −6.53525 −0.495436
\(175\) −0.288580 −0.0218146
\(176\) 4.27905 0.322546
\(177\) −14.3018 −1.07499
\(178\) −5.55082 −0.416052
\(179\) 5.63260 0.421000 0.210500 0.977594i \(-0.432491\pi\)
0.210500 + 0.977594i \(0.432491\pi\)
\(180\) 2.28948 0.170648
\(181\) −6.35214 −0.472151 −0.236075 0.971735i \(-0.575861\pi\)
−0.236075 + 0.971735i \(0.575861\pi\)
\(182\) 1.19378 0.0884888
\(183\) −3.12763 −0.231201
\(184\) 6.42459 0.473627
\(185\) −15.9359 −1.17163
\(186\) −1.25937 −0.0923414
\(187\) −15.3862 −1.12515
\(188\) 1.19672 0.0872800
\(189\) 1.19378 0.0868346
\(190\) 3.10843 0.225509
\(191\) 0.0280475 0.00202944 0.00101472 0.999999i \(-0.499677\pi\)
0.00101472 + 0.999999i \(0.499677\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −16.8861 −1.21549 −0.607743 0.794134i \(-0.707925\pi\)
−0.607743 + 0.794134i \(0.707925\pi\)
\(194\) −7.35332 −0.527937
\(195\) −2.28948 −0.163953
\(196\) −5.57489 −0.398207
\(197\) 11.9854 0.853928 0.426964 0.904269i \(-0.359583\pi\)
0.426964 + 0.904269i \(0.359583\pi\)
\(198\) −4.27905 −0.304099
\(199\) 16.7843 1.18980 0.594902 0.803798i \(-0.297191\pi\)
0.594902 + 0.803798i \(0.297191\pi\)
\(200\) −0.241736 −0.0170933
\(201\) −10.3747 −0.731778
\(202\) −11.1470 −0.784300
\(203\) 7.80165 0.547568
\(204\) 3.59570 0.251750
\(205\) 16.8724 1.17842
\(206\) 1.00000 0.0696733
\(207\) −6.42459 −0.446540
\(208\) 1.00000 0.0693375
\(209\) −5.80967 −0.401863
\(210\) −2.73314 −0.188604
\(211\) −5.85650 −0.403178 −0.201589 0.979470i \(-0.564611\pi\)
−0.201589 + 0.979470i \(0.564611\pi\)
\(212\) 2.84069 0.195100
\(213\) −1.20531 −0.0825867
\(214\) 4.78496 0.327093
\(215\) 3.90372 0.266231
\(216\) 1.00000 0.0680414
\(217\) 1.50341 0.102058
\(218\) −2.92640 −0.198201
\(219\) −10.1318 −0.684644
\(220\) 9.79682 0.660501
\(221\) −3.59570 −0.241873
\(222\) −6.96050 −0.467158
\(223\) 2.27388 0.152270 0.0761351 0.997098i \(-0.475742\pi\)
0.0761351 + 0.997098i \(0.475742\pi\)
\(224\) 1.19378 0.0797627
\(225\) 0.241736 0.0161158
\(226\) −2.55170 −0.169736
\(227\) 24.4840 1.62506 0.812530 0.582919i \(-0.198090\pi\)
0.812530 + 0.582919i \(0.198090\pi\)
\(228\) 1.35770 0.0899159
\(229\) 18.1528 1.19957 0.599786 0.800161i \(-0.295252\pi\)
0.599786 + 0.800161i \(0.295252\pi\)
\(230\) 14.7090 0.969883
\(231\) 5.10824 0.336097
\(232\) 6.53525 0.429060
\(233\) −12.6906 −0.831392 −0.415696 0.909504i \(-0.636462\pi\)
−0.415696 + 0.909504i \(0.636462\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 2.73988 0.178730
\(236\) 14.3018 0.930971
\(237\) 2.00621 0.130317
\(238\) −4.29248 −0.278240
\(239\) 17.9914 1.16377 0.581883 0.813273i \(-0.302316\pi\)
0.581883 + 0.813273i \(0.302316\pi\)
\(240\) −2.28948 −0.147786
\(241\) 3.62121 0.233262 0.116631 0.993175i \(-0.462790\pi\)
0.116631 + 0.993175i \(0.462790\pi\)
\(242\) −7.31027 −0.469922
\(243\) −1.00000 −0.0641500
\(244\) 3.12763 0.200226
\(245\) −12.7636 −0.815438
\(246\) 7.36952 0.469864
\(247\) −1.35770 −0.0863884
\(248\) 1.25937 0.0799700
\(249\) −4.49374 −0.284779
\(250\) 10.8940 0.688995
\(251\) 9.61212 0.606712 0.303356 0.952877i \(-0.401893\pi\)
0.303356 + 0.952877i \(0.401893\pi\)
\(252\) −1.19378 −0.0752010
\(253\) −27.4912 −1.72835
\(254\) 3.49577 0.219344
\(255\) 8.23231 0.515527
\(256\) 1.00000 0.0625000
\(257\) −4.38700 −0.273654 −0.136827 0.990595i \(-0.543690\pi\)
−0.136827 + 0.990595i \(0.543690\pi\)
\(258\) 1.70506 0.106153
\(259\) 8.30929 0.516314
\(260\) 2.28948 0.141988
\(261\) −6.53525 −0.404522
\(262\) 18.1097 1.11882
\(263\) −28.0116 −1.72727 −0.863635 0.504118i \(-0.831818\pi\)
−0.863635 + 0.504118i \(0.831818\pi\)
\(264\) 4.27905 0.263357
\(265\) 6.50372 0.399520
\(266\) −1.62079 −0.0993773
\(267\) −5.55082 −0.339705
\(268\) 10.3747 0.633738
\(269\) −7.77286 −0.473920 −0.236960 0.971519i \(-0.576151\pi\)
−0.236960 + 0.971519i \(0.576151\pi\)
\(270\) 2.28948 0.139334
\(271\) 11.6289 0.706406 0.353203 0.935547i \(-0.385093\pi\)
0.353203 + 0.935547i \(0.385093\pi\)
\(272\) −3.59570 −0.218022
\(273\) 1.19378 0.0722508
\(274\) −0.772432 −0.0466643
\(275\) 1.03440 0.0623768
\(276\) 6.42459 0.386715
\(277\) 15.6095 0.937883 0.468941 0.883229i \(-0.344636\pi\)
0.468941 + 0.883229i \(0.344636\pi\)
\(278\) −14.3030 −0.857837
\(279\) −1.25937 −0.0753965
\(280\) 2.73314 0.163336
\(281\) −14.6040 −0.871199 −0.435599 0.900141i \(-0.643464\pi\)
−0.435599 + 0.900141i \(0.643464\pi\)
\(282\) 1.19672 0.0712638
\(283\) 14.0678 0.836245 0.418122 0.908391i \(-0.362688\pi\)
0.418122 + 0.908391i \(0.362688\pi\)
\(284\) 1.20531 0.0715222
\(285\) 3.10843 0.184128
\(286\) −4.27905 −0.253026
\(287\) −8.79758 −0.519305
\(288\) −1.00000 −0.0589256
\(289\) −4.07091 −0.239465
\(290\) 14.9624 0.878620
\(291\) −7.35332 −0.431059
\(292\) 10.1318 0.592919
\(293\) −24.4565 −1.42876 −0.714381 0.699757i \(-0.753292\pi\)
−0.714381 + 0.699757i \(0.753292\pi\)
\(294\) −5.57489 −0.325134
\(295\) 32.7438 1.90642
\(296\) 6.96050 0.404571
\(297\) −4.27905 −0.248296
\(298\) −17.3427 −1.00464
\(299\) −6.42459 −0.371544
\(300\) −0.241736 −0.0139566
\(301\) −2.03547 −0.117323
\(302\) −14.7147 −0.846733
\(303\) −11.1470 −0.640378
\(304\) −1.35770 −0.0778695
\(305\) 7.16067 0.410019
\(306\) 3.59570 0.205553
\(307\) −4.95540 −0.282820 −0.141410 0.989951i \(-0.545164\pi\)
−0.141410 + 0.989951i \(0.545164\pi\)
\(308\) −5.10824 −0.291069
\(309\) 1.00000 0.0568880
\(310\) 2.88331 0.163761
\(311\) 0.541516 0.0307066 0.0153533 0.999882i \(-0.495113\pi\)
0.0153533 + 0.999882i \(0.495113\pi\)
\(312\) 1.00000 0.0566139
\(313\) −21.4984 −1.21516 −0.607580 0.794258i \(-0.707860\pi\)
−0.607580 + 0.794258i \(0.707860\pi\)
\(314\) −22.0695 −1.24546
\(315\) −2.73314 −0.153995
\(316\) −2.00621 −0.112858
\(317\) 1.44767 0.0813094 0.0406547 0.999173i \(-0.487056\pi\)
0.0406547 + 0.999173i \(0.487056\pi\)
\(318\) 2.84069 0.159298
\(319\) −27.9647 −1.56572
\(320\) 2.28948 0.127986
\(321\) 4.78496 0.267070
\(322\) −7.66954 −0.427407
\(323\) 4.88189 0.271636
\(324\) 1.00000 0.0555556
\(325\) 0.241736 0.0134091
\(326\) 5.41189 0.299737
\(327\) −2.92640 −0.161830
\(328\) −7.36952 −0.406914
\(329\) −1.42862 −0.0787625
\(330\) 9.79682 0.539297
\(331\) 28.5941 1.57167 0.785836 0.618435i \(-0.212233\pi\)
0.785836 + 0.618435i \(0.212233\pi\)
\(332\) 4.49374 0.246626
\(333\) −6.96050 −0.381433
\(334\) −24.2697 −1.32798
\(335\) 23.7528 1.29775
\(336\) 1.19378 0.0651260
\(337\) −32.2295 −1.75565 −0.877827 0.478978i \(-0.841007\pi\)
−0.877827 + 0.478978i \(0.841007\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −2.55170 −0.138589
\(340\) −8.23231 −0.446460
\(341\) −5.38890 −0.291825
\(342\) 1.35770 0.0734160
\(343\) 15.0116 0.810552
\(344\) −1.70506 −0.0919309
\(345\) 14.7090 0.791906
\(346\) 9.74264 0.523768
\(347\) 22.3484 1.19973 0.599863 0.800103i \(-0.295222\pi\)
0.599863 + 0.800103i \(0.295222\pi\)
\(348\) 6.53525 0.350326
\(349\) 9.93972 0.532061 0.266030 0.963965i \(-0.414288\pi\)
0.266030 + 0.963965i \(0.414288\pi\)
\(350\) 0.288580 0.0154252
\(351\) −1.00000 −0.0533761
\(352\) −4.27905 −0.228074
\(353\) −17.0214 −0.905958 −0.452979 0.891521i \(-0.649639\pi\)
−0.452979 + 0.891521i \(0.649639\pi\)
\(354\) 14.3018 0.760135
\(355\) 2.75955 0.146462
\(356\) 5.55082 0.294193
\(357\) −4.29248 −0.227182
\(358\) −5.63260 −0.297692
\(359\) 0.00561075 0.000296124 0 0.000148062 1.00000i \(-0.499953\pi\)
0.000148062 1.00000i \(0.499953\pi\)
\(360\) −2.28948 −0.120666
\(361\) −17.1566 −0.902982
\(362\) 6.35214 0.333861
\(363\) −7.31027 −0.383690
\(364\) −1.19378 −0.0625710
\(365\) 23.1966 1.21417
\(366\) 3.12763 0.163484
\(367\) −4.96085 −0.258954 −0.129477 0.991582i \(-0.541330\pi\)
−0.129477 + 0.991582i \(0.541330\pi\)
\(368\) −6.42459 −0.334905
\(369\) 7.36952 0.383642
\(370\) 15.9359 0.828470
\(371\) −3.39116 −0.176060
\(372\) 1.25937 0.0652952
\(373\) 11.5261 0.596801 0.298400 0.954441i \(-0.403547\pi\)
0.298400 + 0.954441i \(0.403547\pi\)
\(374\) 15.3862 0.795601
\(375\) 10.8940 0.562562
\(376\) −1.19672 −0.0617163
\(377\) −6.53525 −0.336583
\(378\) −1.19378 −0.0614014
\(379\) 19.0110 0.976532 0.488266 0.872695i \(-0.337630\pi\)
0.488266 + 0.872695i \(0.337630\pi\)
\(380\) −3.10843 −0.159459
\(381\) 3.49577 0.179094
\(382\) −0.0280475 −0.00143503
\(383\) 8.99836 0.459795 0.229897 0.973215i \(-0.426161\pi\)
0.229897 + 0.973215i \(0.426161\pi\)
\(384\) 1.00000 0.0510310
\(385\) −11.6952 −0.596044
\(386\) 16.8861 0.859478
\(387\) 1.70506 0.0866733
\(388\) 7.35332 0.373308
\(389\) 34.9429 1.77168 0.885838 0.463995i \(-0.153584\pi\)
0.885838 + 0.463995i \(0.153584\pi\)
\(390\) 2.28948 0.115933
\(391\) 23.1009 1.16826
\(392\) 5.57489 0.281575
\(393\) 18.1097 0.913516
\(394\) −11.9854 −0.603818
\(395\) −4.59318 −0.231108
\(396\) 4.27905 0.215030
\(397\) 7.48202 0.375512 0.187756 0.982216i \(-0.439879\pi\)
0.187756 + 0.982216i \(0.439879\pi\)
\(398\) −16.7843 −0.841319
\(399\) −1.62079 −0.0811412
\(400\) 0.241736 0.0120868
\(401\) −13.7786 −0.688071 −0.344036 0.938957i \(-0.611794\pi\)
−0.344036 + 0.938957i \(0.611794\pi\)
\(402\) 10.3747 0.517445
\(403\) −1.25937 −0.0627336
\(404\) 11.1470 0.554584
\(405\) 2.28948 0.113765
\(406\) −7.80165 −0.387189
\(407\) −29.7843 −1.47635
\(408\) −3.59570 −0.178014
\(409\) 18.5833 0.918885 0.459443 0.888207i \(-0.348049\pi\)
0.459443 + 0.888207i \(0.348049\pi\)
\(410\) −16.8724 −0.833269
\(411\) −0.772432 −0.0381013
\(412\) −1.00000 −0.0492665
\(413\) −17.0732 −0.840119
\(414\) 6.42459 0.315752
\(415\) 10.2883 0.505035
\(416\) −1.00000 −0.0490290
\(417\) −14.3030 −0.700421
\(418\) 5.80967 0.284160
\(419\) −16.9895 −0.829991 −0.414996 0.909823i \(-0.636217\pi\)
−0.414996 + 0.909823i \(0.636217\pi\)
\(420\) 2.73314 0.133363
\(421\) 26.6550 1.29908 0.649542 0.760326i \(-0.274961\pi\)
0.649542 + 0.760326i \(0.274961\pi\)
\(422\) 5.85650 0.285090
\(423\) 1.19672 0.0581867
\(424\) −2.84069 −0.137956
\(425\) −0.869212 −0.0421630
\(426\) 1.20531 0.0583977
\(427\) −3.73370 −0.180687
\(428\) −4.78496 −0.231290
\(429\) −4.27905 −0.206594
\(430\) −3.90372 −0.188254
\(431\) −31.1412 −1.50002 −0.750011 0.661426i \(-0.769952\pi\)
−0.750011 + 0.661426i \(0.769952\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.81223 −0.231261 −0.115631 0.993292i \(-0.536889\pi\)
−0.115631 + 0.993292i \(0.536889\pi\)
\(434\) −1.50341 −0.0721659
\(435\) 14.9624 0.717390
\(436\) 2.92640 0.140149
\(437\) 8.72268 0.417262
\(438\) 10.1318 0.484117
\(439\) 40.6140 1.93840 0.969200 0.246275i \(-0.0792067\pi\)
0.969200 + 0.246275i \(0.0792067\pi\)
\(440\) −9.79682 −0.467045
\(441\) −5.57489 −0.265471
\(442\) 3.59570 0.171030
\(443\) −12.7887 −0.607611 −0.303806 0.952734i \(-0.598257\pi\)
−0.303806 + 0.952734i \(0.598257\pi\)
\(444\) 6.96050 0.330331
\(445\) 12.7085 0.602442
\(446\) −2.27388 −0.107671
\(447\) −17.3427 −0.820282
\(448\) −1.19378 −0.0564007
\(449\) −41.2330 −1.94591 −0.972953 0.231002i \(-0.925800\pi\)
−0.972953 + 0.231002i \(0.925800\pi\)
\(450\) −0.241736 −0.0113956
\(451\) 31.5346 1.48490
\(452\) 2.55170 0.120022
\(453\) −14.7147 −0.691355
\(454\) −24.4840 −1.14909
\(455\) −2.73314 −0.128131
\(456\) −1.35770 −0.0635802
\(457\) −13.7388 −0.642675 −0.321338 0.946965i \(-0.604132\pi\)
−0.321338 + 0.946965i \(0.604132\pi\)
\(458\) −18.1528 −0.848225
\(459\) 3.59570 0.167833
\(460\) −14.7090 −0.685811
\(461\) −10.4692 −0.487601 −0.243801 0.969825i \(-0.578394\pi\)
−0.243801 + 0.969825i \(0.578394\pi\)
\(462\) −5.10824 −0.237657
\(463\) 19.0331 0.884545 0.442272 0.896881i \(-0.354172\pi\)
0.442272 + 0.896881i \(0.354172\pi\)
\(464\) −6.53525 −0.303392
\(465\) 2.88331 0.133710
\(466\) 12.6906 0.587883
\(467\) −4.60131 −0.212923 −0.106462 0.994317i \(-0.533952\pi\)
−0.106462 + 0.994317i \(0.533952\pi\)
\(468\) 1.00000 0.0462250
\(469\) −12.3851 −0.571893
\(470\) −2.73988 −0.126381
\(471\) −22.0695 −1.01691
\(472\) −14.3018 −0.658296
\(473\) 7.29606 0.335473
\(474\) −2.00621 −0.0921481
\(475\) −0.328206 −0.0150591
\(476\) 4.29248 0.196745
\(477\) 2.84069 0.130066
\(478\) −17.9914 −0.822907
\(479\) 8.25442 0.377154 0.188577 0.982058i \(-0.439613\pi\)
0.188577 + 0.982058i \(0.439613\pi\)
\(480\) 2.28948 0.104500
\(481\) −6.96050 −0.317371
\(482\) −3.62121 −0.164941
\(483\) −7.66954 −0.348976
\(484\) 7.31027 0.332285
\(485\) 16.8353 0.764451
\(486\) 1.00000 0.0453609
\(487\) 30.9469 1.40234 0.701169 0.712996i \(-0.252662\pi\)
0.701169 + 0.712996i \(0.252662\pi\)
\(488\) −3.12763 −0.141581
\(489\) 5.41189 0.244734
\(490\) 12.7636 0.576602
\(491\) 2.62104 0.118286 0.0591429 0.998250i \(-0.481163\pi\)
0.0591429 + 0.998250i \(0.481163\pi\)
\(492\) −7.36952 −0.332244
\(493\) 23.4988 1.05833
\(494\) 1.35770 0.0610858
\(495\) 9.79682 0.440334
\(496\) −1.25937 −0.0565473
\(497\) −1.43888 −0.0645425
\(498\) 4.49374 0.201369
\(499\) −10.9828 −0.491657 −0.245829 0.969313i \(-0.579060\pi\)
−0.245829 + 0.969313i \(0.579060\pi\)
\(500\) −10.8940 −0.487193
\(501\) −24.2697 −1.08429
\(502\) −9.61212 −0.429010
\(503\) −2.99356 −0.133476 −0.0667382 0.997771i \(-0.521259\pi\)
−0.0667382 + 0.997771i \(0.521259\pi\)
\(504\) 1.19378 0.0531751
\(505\) 25.5209 1.13566
\(506\) 27.4912 1.22213
\(507\) −1.00000 −0.0444116
\(508\) −3.49577 −0.155100
\(509\) 19.6098 0.869188 0.434594 0.900626i \(-0.356892\pi\)
0.434594 + 0.900626i \(0.356892\pi\)
\(510\) −8.23231 −0.364533
\(511\) −12.0951 −0.535058
\(512\) −1.00000 −0.0441942
\(513\) 1.35770 0.0599440
\(514\) 4.38700 0.193502
\(515\) −2.28948 −0.100887
\(516\) −1.70506 −0.0750613
\(517\) 5.12084 0.225214
\(518\) −8.30929 −0.365089
\(519\) 9.74264 0.427655
\(520\) −2.28948 −0.100401
\(521\) 22.8030 0.999017 0.499509 0.866309i \(-0.333514\pi\)
0.499509 + 0.866309i \(0.333514\pi\)
\(522\) 6.53525 0.286040
\(523\) −25.2714 −1.10504 −0.552522 0.833499i \(-0.686334\pi\)
−0.552522 + 0.833499i \(0.686334\pi\)
\(524\) −18.1097 −0.791128
\(525\) 0.288580 0.0125946
\(526\) 28.0116 1.22136
\(527\) 4.52832 0.197257
\(528\) −4.27905 −0.186222
\(529\) 18.2754 0.794583
\(530\) −6.50372 −0.282504
\(531\) 14.3018 0.620647
\(532\) 1.62079 0.0702703
\(533\) 7.36952 0.319209
\(534\) 5.55082 0.240208
\(535\) −10.9551 −0.473630
\(536\) −10.3747 −0.448121
\(537\) −5.63260 −0.243065
\(538\) 7.77286 0.335112
\(539\) −23.8552 −1.02752
\(540\) −2.28948 −0.0985237
\(541\) 4.86481 0.209154 0.104577 0.994517i \(-0.466651\pi\)
0.104577 + 0.994517i \(0.466651\pi\)
\(542\) −11.6289 −0.499504
\(543\) 6.35214 0.272596
\(544\) 3.59570 0.154165
\(545\) 6.69995 0.286994
\(546\) −1.19378 −0.0510890
\(547\) 25.2611 1.08009 0.540043 0.841638i \(-0.318408\pi\)
0.540043 + 0.841638i \(0.318408\pi\)
\(548\) 0.772432 0.0329967
\(549\) 3.12763 0.133484
\(550\) −1.03440 −0.0441070
\(551\) 8.87292 0.377999
\(552\) −6.42459 −0.273449
\(553\) 2.39497 0.101844
\(554\) −15.6095 −0.663183
\(555\) 15.9359 0.676443
\(556\) 14.3030 0.606583
\(557\) 10.1532 0.430205 0.215102 0.976591i \(-0.430991\pi\)
0.215102 + 0.976591i \(0.430991\pi\)
\(558\) 1.25937 0.0533133
\(559\) 1.70506 0.0721166
\(560\) −2.73314 −0.115496
\(561\) 15.3862 0.649606
\(562\) 14.6040 0.616031
\(563\) −5.09473 −0.214717 −0.107359 0.994220i \(-0.534239\pi\)
−0.107359 + 0.994220i \(0.534239\pi\)
\(564\) −1.19672 −0.0503911
\(565\) 5.84207 0.245778
\(566\) −14.0678 −0.591314
\(567\) −1.19378 −0.0501340
\(568\) −1.20531 −0.0505738
\(569\) −42.8195 −1.79509 −0.897543 0.440926i \(-0.854650\pi\)
−0.897543 + 0.440926i \(0.854650\pi\)
\(570\) −3.10843 −0.130198
\(571\) 32.0885 1.34286 0.671430 0.741068i \(-0.265680\pi\)
0.671430 + 0.741068i \(0.265680\pi\)
\(572\) 4.27905 0.178916
\(573\) −0.0280475 −0.00117170
\(574\) 8.79758 0.367204
\(575\) −1.55306 −0.0647670
\(576\) 1.00000 0.0416667
\(577\) 13.9671 0.581458 0.290729 0.956805i \(-0.406102\pi\)
0.290729 + 0.956805i \(0.406102\pi\)
\(578\) 4.07091 0.169327
\(579\) 16.8861 0.701761
\(580\) −14.9624 −0.621278
\(581\) −5.36453 −0.222558
\(582\) 7.35332 0.304805
\(583\) 12.1555 0.503428
\(584\) −10.1318 −0.419257
\(585\) 2.28948 0.0946585
\(586\) 24.4565 1.01029
\(587\) −13.4259 −0.554147 −0.277073 0.960849i \(-0.589365\pi\)
−0.277073 + 0.960849i \(0.589365\pi\)
\(588\) 5.57489 0.229905
\(589\) 1.70985 0.0704530
\(590\) −32.7438 −1.34804
\(591\) −11.9854 −0.493016
\(592\) −6.96050 −0.286075
\(593\) −31.2213 −1.28210 −0.641052 0.767497i \(-0.721502\pi\)
−0.641052 + 0.767497i \(0.721502\pi\)
\(594\) 4.27905 0.175572
\(595\) 9.82755 0.402890
\(596\) 17.3427 0.710385
\(597\) −16.7843 −0.686934
\(598\) 6.42459 0.262721
\(599\) −30.1210 −1.23071 −0.615356 0.788250i \(-0.710988\pi\)
−0.615356 + 0.788250i \(0.710988\pi\)
\(600\) 0.241736 0.00986884
\(601\) 32.1632 1.31197 0.655983 0.754776i \(-0.272254\pi\)
0.655983 + 0.754776i \(0.272254\pi\)
\(602\) 2.03547 0.0829596
\(603\) 10.3747 0.422492
\(604\) 14.7147 0.598731
\(605\) 16.7367 0.680445
\(606\) 11.1470 0.452816
\(607\) 26.2385 1.06499 0.532494 0.846434i \(-0.321255\pi\)
0.532494 + 0.846434i \(0.321255\pi\)
\(608\) 1.35770 0.0550620
\(609\) −7.80165 −0.316139
\(610\) −7.16067 −0.289927
\(611\) 1.19672 0.0484142
\(612\) −3.59570 −0.145348
\(613\) −15.6293 −0.631262 −0.315631 0.948882i \(-0.602216\pi\)
−0.315631 + 0.948882i \(0.602216\pi\)
\(614\) 4.95540 0.199984
\(615\) −16.8724 −0.680361
\(616\) 5.10824 0.205817
\(617\) −1.25068 −0.0503505 −0.0251753 0.999683i \(-0.508014\pi\)
−0.0251753 + 0.999683i \(0.508014\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 31.7077 1.27444 0.637221 0.770681i \(-0.280084\pi\)
0.637221 + 0.770681i \(0.280084\pi\)
\(620\) −2.88331 −0.115796
\(621\) 6.42459 0.257810
\(622\) −0.541516 −0.0217128
\(623\) −6.62646 −0.265483
\(624\) −1.00000 −0.0400320
\(625\) −26.1502 −1.04601
\(626\) 21.4984 0.859248
\(627\) 5.80967 0.232016
\(628\) 22.0695 0.880671
\(629\) 25.0279 0.997927
\(630\) 2.73314 0.108891
\(631\) −15.6313 −0.622273 −0.311137 0.950365i \(-0.600710\pi\)
−0.311137 + 0.950365i \(0.600710\pi\)
\(632\) 2.00621 0.0798026
\(633\) 5.85650 0.232775
\(634\) −1.44767 −0.0574944
\(635\) −8.00352 −0.317610
\(636\) −2.84069 −0.112641
\(637\) −5.57489 −0.220885
\(638\) 27.9647 1.10713
\(639\) 1.20531 0.0476815
\(640\) −2.28948 −0.0904998
\(641\) −33.4222 −1.32010 −0.660048 0.751223i \(-0.729464\pi\)
−0.660048 + 0.751223i \(0.729464\pi\)
\(642\) −4.78496 −0.188847
\(643\) −8.92919 −0.352133 −0.176067 0.984378i \(-0.556337\pi\)
−0.176067 + 0.984378i \(0.556337\pi\)
\(644\) 7.66954 0.302222
\(645\) −3.90372 −0.153709
\(646\) −4.88189 −0.192075
\(647\) −42.6542 −1.67691 −0.838455 0.544970i \(-0.816541\pi\)
−0.838455 + 0.544970i \(0.816541\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 61.1983 2.40224
\(650\) −0.241736 −0.00948168
\(651\) −1.50341 −0.0589232
\(652\) −5.41189 −0.211946
\(653\) 15.3018 0.598806 0.299403 0.954127i \(-0.403213\pi\)
0.299403 + 0.954127i \(0.403213\pi\)
\(654\) 2.92640 0.114431
\(655\) −41.4620 −1.62005
\(656\) 7.36952 0.287732
\(657\) 10.1318 0.395280
\(658\) 1.42862 0.0556935
\(659\) 4.82352 0.187898 0.0939489 0.995577i \(-0.470051\pi\)
0.0939489 + 0.995577i \(0.470051\pi\)
\(660\) −9.79682 −0.381341
\(661\) 24.6784 0.959880 0.479940 0.877301i \(-0.340658\pi\)
0.479940 + 0.877301i \(0.340658\pi\)
\(662\) −28.5941 −1.11134
\(663\) 3.59570 0.139646
\(664\) −4.49374 −0.174391
\(665\) 3.71078 0.143898
\(666\) 6.96050 0.269714
\(667\) 41.9864 1.62572
\(668\) 24.2697 0.939025
\(669\) −2.27388 −0.0879133
\(670\) −23.7528 −0.917651
\(671\) 13.3833 0.516657
\(672\) −1.19378 −0.0460510
\(673\) −7.82552 −0.301652 −0.150826 0.988560i \(-0.548193\pi\)
−0.150826 + 0.988560i \(0.548193\pi\)
\(674\) 32.2295 1.24143
\(675\) −0.241736 −0.00930443
\(676\) 1.00000 0.0384615
\(677\) 20.1050 0.772699 0.386350 0.922352i \(-0.373736\pi\)
0.386350 + 0.922352i \(0.373736\pi\)
\(678\) 2.55170 0.0979974
\(679\) −8.77823 −0.336878
\(680\) 8.23231 0.315695
\(681\) −24.4840 −0.938229
\(682\) 5.38890 0.206352
\(683\) 4.60842 0.176336 0.0881681 0.996106i \(-0.471899\pi\)
0.0881681 + 0.996106i \(0.471899\pi\)
\(684\) −1.35770 −0.0519130
\(685\) 1.76847 0.0675698
\(686\) −15.0116 −0.573147
\(687\) −18.1528 −0.692573
\(688\) 1.70506 0.0650050
\(689\) 2.84069 0.108222
\(690\) −14.7090 −0.559962
\(691\) −2.68257 −0.102050 −0.0510249 0.998697i \(-0.516249\pi\)
−0.0510249 + 0.998697i \(0.516249\pi\)
\(692\) −9.74264 −0.370360
\(693\) −5.10824 −0.194046
\(694\) −22.3484 −0.848335
\(695\) 32.7465 1.24215
\(696\) −6.53525 −0.247718
\(697\) −26.4986 −1.00371
\(698\) −9.93972 −0.376224
\(699\) 12.6906 0.480004
\(700\) −0.288580 −0.0109073
\(701\) −35.8678 −1.35471 −0.677354 0.735657i \(-0.736873\pi\)
−0.677354 + 0.735657i \(0.736873\pi\)
\(702\) 1.00000 0.0377426
\(703\) 9.45027 0.356424
\(704\) 4.27905 0.161273
\(705\) −2.73988 −0.103190
\(706\) 17.0214 0.640609
\(707\) −13.3070 −0.500463
\(708\) −14.3018 −0.537496
\(709\) 4.94712 0.185793 0.0928965 0.995676i \(-0.470387\pi\)
0.0928965 + 0.995676i \(0.470387\pi\)
\(710\) −2.75955 −0.103564
\(711\) −2.00621 −0.0752386
\(712\) −5.55082 −0.208026
\(713\) 8.09093 0.303008
\(714\) 4.29248 0.160642
\(715\) 9.79682 0.366380
\(716\) 5.63260 0.210500
\(717\) −17.9914 −0.671900
\(718\) −0.00561075 −0.000209391 0
\(719\) 11.8733 0.442800 0.221400 0.975183i \(-0.428937\pi\)
0.221400 + 0.975183i \(0.428937\pi\)
\(720\) 2.28948 0.0853240
\(721\) 1.19378 0.0444586
\(722\) 17.1566 0.638504
\(723\) −3.62121 −0.134674
\(724\) −6.35214 −0.236075
\(725\) −1.57981 −0.0586726
\(726\) 7.31027 0.271310
\(727\) 30.5078 1.13147 0.565736 0.824587i \(-0.308592\pi\)
0.565736 + 0.824587i \(0.308592\pi\)
\(728\) 1.19378 0.0442444
\(729\) 1.00000 0.0370370
\(730\) −23.1966 −0.858545
\(731\) −6.13091 −0.226760
\(732\) −3.12763 −0.115601
\(733\) −4.82880 −0.178356 −0.0891779 0.996016i \(-0.528424\pi\)
−0.0891779 + 0.996016i \(0.528424\pi\)
\(734\) 4.96085 0.183108
\(735\) 12.7636 0.470793
\(736\) 6.42459 0.236814
\(737\) 44.3940 1.63528
\(738\) −7.36952 −0.271276
\(739\) −3.09730 −0.113936 −0.0569681 0.998376i \(-0.518143\pi\)
−0.0569681 + 0.998376i \(0.518143\pi\)
\(740\) −15.9359 −0.585817
\(741\) 1.35770 0.0498764
\(742\) 3.39116 0.124493
\(743\) 16.3166 0.598597 0.299299 0.954159i \(-0.403247\pi\)
0.299299 + 0.954159i \(0.403247\pi\)
\(744\) −1.25937 −0.0461707
\(745\) 39.7059 1.45471
\(746\) −11.5261 −0.422002
\(747\) 4.49374 0.164417
\(748\) −15.3862 −0.562575
\(749\) 5.71218 0.208719
\(750\) −10.8940 −0.397791
\(751\) 39.3121 1.43452 0.717259 0.696807i \(-0.245396\pi\)
0.717259 + 0.696807i \(0.245396\pi\)
\(752\) 1.19672 0.0436400
\(753\) −9.61212 −0.350285
\(754\) 6.53525 0.238000
\(755\) 33.6890 1.22607
\(756\) 1.19378 0.0434173
\(757\) −41.8503 −1.52107 −0.760537 0.649294i \(-0.775064\pi\)
−0.760537 + 0.649294i \(0.775064\pi\)
\(758\) −19.0110 −0.690512
\(759\) 27.4912 0.997866
\(760\) 3.10843 0.112755
\(761\) −34.0486 −1.23426 −0.617131 0.786861i \(-0.711705\pi\)
−0.617131 + 0.786861i \(0.711705\pi\)
\(762\) −3.49577 −0.126639
\(763\) −3.49348 −0.126472
\(764\) 0.0280475 0.00101472
\(765\) −8.23231 −0.297640
\(766\) −8.99836 −0.325124
\(767\) 14.3018 0.516410
\(768\) −1.00000 −0.0360844
\(769\) 10.7045 0.386014 0.193007 0.981197i \(-0.438176\pi\)
0.193007 + 0.981197i \(0.438176\pi\)
\(770\) 11.6952 0.421467
\(771\) 4.38700 0.157994
\(772\) −16.8861 −0.607743
\(773\) 23.6267 0.849793 0.424896 0.905242i \(-0.360311\pi\)
0.424896 + 0.905242i \(0.360311\pi\)
\(774\) −1.70506 −0.0612873
\(775\) −0.304435 −0.0109356
\(776\) −7.35332 −0.263969
\(777\) −8.30929 −0.298094
\(778\) −34.9429 −1.25276
\(779\) −10.0056 −0.358488
\(780\) −2.28948 −0.0819767
\(781\) 5.15760 0.184553
\(782\) −23.1009 −0.826088
\(783\) 6.53525 0.233551
\(784\) −5.57489 −0.199103
\(785\) 50.5279 1.80342
\(786\) −18.1097 −0.645953
\(787\) 19.1971 0.684304 0.342152 0.939645i \(-0.388844\pi\)
0.342152 + 0.939645i \(0.388844\pi\)
\(788\) 11.9854 0.426964
\(789\) 28.0116 0.997240
\(790\) 4.59318 0.163418
\(791\) −3.04616 −0.108309
\(792\) −4.27905 −0.152049
\(793\) 3.12763 0.111066
\(794\) −7.48202 −0.265527
\(795\) −6.50372 −0.230663
\(796\) 16.7843 0.594902
\(797\) −42.4158 −1.50245 −0.751223 0.660049i \(-0.770536\pi\)
−0.751223 + 0.660049i \(0.770536\pi\)
\(798\) 1.62079 0.0573755
\(799\) −4.30306 −0.152231
\(800\) −0.241736 −0.00854667
\(801\) 5.55082 0.196129
\(802\) 13.7786 0.486540
\(803\) 43.3545 1.52995
\(804\) −10.3747 −0.365889
\(805\) 17.5593 0.618884
\(806\) 1.25937 0.0443594
\(807\) 7.77286 0.273618
\(808\) −11.1470 −0.392150
\(809\) −5.02509 −0.176673 −0.0883363 0.996091i \(-0.528155\pi\)
−0.0883363 + 0.996091i \(0.528155\pi\)
\(810\) −2.28948 −0.0804443
\(811\) 22.9949 0.807462 0.403731 0.914878i \(-0.367713\pi\)
0.403731 + 0.914878i \(0.367713\pi\)
\(812\) 7.80165 0.273784
\(813\) −11.6289 −0.407844
\(814\) 29.7843 1.04394
\(815\) −12.3904 −0.434018
\(816\) 3.59570 0.125875
\(817\) −2.31497 −0.0809905
\(818\) −18.5833 −0.649750
\(819\) −1.19378 −0.0417140
\(820\) 16.8724 0.589210
\(821\) 21.0749 0.735519 0.367760 0.929921i \(-0.380125\pi\)
0.367760 + 0.929921i \(0.380125\pi\)
\(822\) 0.772432 0.0269417
\(823\) −42.8530 −1.49376 −0.746881 0.664958i \(-0.768449\pi\)
−0.746881 + 0.664958i \(0.768449\pi\)
\(824\) 1.00000 0.0348367
\(825\) −1.03440 −0.0360132
\(826\) 17.0732 0.594054
\(827\) 35.1982 1.22396 0.611980 0.790873i \(-0.290373\pi\)
0.611980 + 0.790873i \(0.290373\pi\)
\(828\) −6.42459 −0.223270
\(829\) −16.5382 −0.574395 −0.287197 0.957871i \(-0.592724\pi\)
−0.287197 + 0.957871i \(0.592724\pi\)
\(830\) −10.2883 −0.357113
\(831\) −15.6095 −0.541487
\(832\) 1.00000 0.0346688
\(833\) 20.0457 0.694541
\(834\) 14.3030 0.495273
\(835\) 55.5652 1.92291
\(836\) −5.80967 −0.200932
\(837\) 1.25937 0.0435302
\(838\) 16.9895 0.586892
\(839\) 1.93922 0.0669493 0.0334747 0.999440i \(-0.489343\pi\)
0.0334747 + 0.999440i \(0.489343\pi\)
\(840\) −2.73314 −0.0943022
\(841\) 13.7095 0.472743
\(842\) −26.6550 −0.918591
\(843\) 14.6040 0.502987
\(844\) −5.85650 −0.201589
\(845\) 2.28948 0.0787606
\(846\) −1.19672 −0.0411442
\(847\) −8.72685 −0.299858
\(848\) 2.84069 0.0975498
\(849\) −14.0678 −0.482806
\(850\) 0.869212 0.0298137
\(851\) 44.7184 1.53293
\(852\) −1.20531 −0.0412934
\(853\) −18.4017 −0.630064 −0.315032 0.949081i \(-0.602015\pi\)
−0.315032 + 0.949081i \(0.602015\pi\)
\(854\) 3.73370 0.127765
\(855\) −3.10843 −0.106306
\(856\) 4.78496 0.163547
\(857\) 22.7555 0.777314 0.388657 0.921383i \(-0.372939\pi\)
0.388657 + 0.921383i \(0.372939\pi\)
\(858\) 4.27905 0.146084
\(859\) −35.4905 −1.21092 −0.605461 0.795875i \(-0.707011\pi\)
−0.605461 + 0.795875i \(0.707011\pi\)
\(860\) 3.90372 0.133116
\(861\) 8.79758 0.299821
\(862\) 31.1412 1.06068
\(863\) 16.4055 0.558449 0.279224 0.960226i \(-0.409923\pi\)
0.279224 + 0.960226i \(0.409923\pi\)
\(864\) 1.00000 0.0340207
\(865\) −22.3056 −0.758414
\(866\) 4.81223 0.163526
\(867\) 4.07091 0.138255
\(868\) 1.50341 0.0510290
\(869\) −8.58466 −0.291215
\(870\) −14.9624 −0.507271
\(871\) 10.3747 0.351535
\(872\) −2.92640 −0.0991005
\(873\) 7.35332 0.248872
\(874\) −8.72268 −0.295049
\(875\) 13.0050 0.439649
\(876\) −10.1318 −0.342322
\(877\) 4.73131 0.159765 0.0798825 0.996804i \(-0.474545\pi\)
0.0798825 + 0.996804i \(0.474545\pi\)
\(878\) −40.6140 −1.37066
\(879\) 24.4565 0.824896
\(880\) 9.79682 0.330251
\(881\) −23.2448 −0.783138 −0.391569 0.920149i \(-0.628067\pi\)
−0.391569 + 0.920149i \(0.628067\pi\)
\(882\) 5.57489 0.187716
\(883\) −11.6759 −0.392925 −0.196462 0.980511i \(-0.562945\pi\)
−0.196462 + 0.980511i \(0.562945\pi\)
\(884\) −3.59570 −0.120937
\(885\) −32.7438 −1.10067
\(886\) 12.7887 0.429646
\(887\) −18.4731 −0.620268 −0.310134 0.950693i \(-0.600374\pi\)
−0.310134 + 0.950693i \(0.600374\pi\)
\(888\) −6.96050 −0.233579
\(889\) 4.17318 0.139964
\(890\) −12.7085 −0.425991
\(891\) 4.27905 0.143354
\(892\) 2.27388 0.0761351
\(893\) −1.62479 −0.0543716
\(894\) 17.3427 0.580027
\(895\) 12.8957 0.431057
\(896\) 1.19378 0.0398813
\(897\) 6.42459 0.214511
\(898\) 41.2330 1.37596
\(899\) 8.23030 0.274496
\(900\) 0.241736 0.00805788
\(901\) −10.2143 −0.340287
\(902\) −31.5346 −1.04999
\(903\) 2.03547 0.0677362
\(904\) −2.55170 −0.0848682
\(905\) −14.5431 −0.483430
\(906\) 14.7147 0.488862
\(907\) 11.3470 0.376770 0.188385 0.982095i \(-0.439675\pi\)
0.188385 + 0.982095i \(0.439675\pi\)
\(908\) 24.4840 0.812530
\(909\) 11.1470 0.369723
\(910\) 2.73314 0.0906026
\(911\) 0.347605 0.0115167 0.00575834 0.999983i \(-0.498167\pi\)
0.00575834 + 0.999983i \(0.498167\pi\)
\(912\) 1.35770 0.0449580
\(913\) 19.2289 0.636385
\(914\) 13.7388 0.454440
\(915\) −7.16067 −0.236724
\(916\) 18.1528 0.599786
\(917\) 21.6190 0.713923
\(918\) −3.59570 −0.118676
\(919\) 56.7382 1.87162 0.935810 0.352505i \(-0.114670\pi\)
0.935810 + 0.352505i \(0.114670\pi\)
\(920\) 14.7090 0.484941
\(921\) 4.95540 0.163286
\(922\) 10.4692 0.344786
\(923\) 1.20531 0.0396734
\(924\) 5.10824 0.168049
\(925\) −1.68260 −0.0553237
\(926\) −19.0331 −0.625468
\(927\) −1.00000 −0.0328443
\(928\) 6.53525 0.214530
\(929\) 29.7076 0.974675 0.487337 0.873214i \(-0.337968\pi\)
0.487337 + 0.873214i \(0.337968\pi\)
\(930\) −2.88331 −0.0945473
\(931\) 7.56904 0.248065
\(932\) −12.6906 −0.415696
\(933\) −0.541516 −0.0177284
\(934\) 4.60131 0.150560
\(935\) −35.2265 −1.15203
\(936\) −1.00000 −0.0326860
\(937\) 0.370005 0.0120875 0.00604377 0.999982i \(-0.498076\pi\)
0.00604377 + 0.999982i \(0.498076\pi\)
\(938\) 12.3851 0.404389
\(939\) 21.4984 0.701573
\(940\) 2.73988 0.0893650
\(941\) −10.8330 −0.353146 −0.176573 0.984288i \(-0.556501\pi\)
−0.176573 + 0.984288i \(0.556501\pi\)
\(942\) 22.0695 0.719065
\(943\) −47.3462 −1.54180
\(944\) 14.3018 0.465486
\(945\) 2.73314 0.0889090
\(946\) −7.29606 −0.237215
\(947\) 6.65466 0.216247 0.108124 0.994137i \(-0.465516\pi\)
0.108124 + 0.994137i \(0.465516\pi\)
\(948\) 2.00621 0.0651585
\(949\) 10.1318 0.328893
\(950\) 0.328206 0.0106484
\(951\) −1.44767 −0.0469440
\(952\) −4.29248 −0.139120
\(953\) 24.0843 0.780168 0.390084 0.920779i \(-0.372446\pi\)
0.390084 + 0.920779i \(0.372446\pi\)
\(954\) −2.84069 −0.0919708
\(955\) 0.0642142 0.00207792
\(956\) 17.9914 0.581883
\(957\) 27.9647 0.903970
\(958\) −8.25442 −0.266688
\(959\) −0.922113 −0.0297766
\(960\) −2.28948 −0.0738928
\(961\) −29.4140 −0.948838
\(962\) 6.96050 0.224415
\(963\) −4.78496 −0.154193
\(964\) 3.62121 0.116631
\(965\) −38.6604 −1.24452
\(966\) 7.66954 0.246764
\(967\) 41.6001 1.33777 0.668885 0.743366i \(-0.266772\pi\)
0.668885 + 0.743366i \(0.266772\pi\)
\(968\) −7.31027 −0.234961
\(969\) −4.88189 −0.156829
\(970\) −16.8353 −0.540549
\(971\) 28.7867 0.923808 0.461904 0.886930i \(-0.347166\pi\)
0.461904 + 0.886930i \(0.347166\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −17.0746 −0.547387
\(974\) −30.9469 −0.991602
\(975\) −0.241736 −0.00774176
\(976\) 3.12763 0.100113
\(977\) −17.8877 −0.572277 −0.286138 0.958188i \(-0.592372\pi\)
−0.286138 + 0.958188i \(0.592372\pi\)
\(978\) −5.41189 −0.173053
\(979\) 23.7523 0.759125
\(980\) −12.7636 −0.407719
\(981\) 2.92640 0.0934328
\(982\) −2.62104 −0.0836407
\(983\) 7.03181 0.224280 0.112140 0.993692i \(-0.464229\pi\)
0.112140 + 0.993692i \(0.464229\pi\)
\(984\) 7.36952 0.234932
\(985\) 27.4405 0.874327
\(986\) −23.4988 −0.748356
\(987\) 1.42862 0.0454736
\(988\) −1.35770 −0.0431942
\(989\) −10.9543 −0.348328
\(990\) −9.79682 −0.311363
\(991\) 1.50489 0.0478044 0.0239022 0.999714i \(-0.492391\pi\)
0.0239022 + 0.999714i \(0.492391\pi\)
\(992\) 1.25937 0.0399850
\(993\) −28.5941 −0.907405
\(994\) 1.43888 0.0456384
\(995\) 38.4273 1.21823
\(996\) −4.49374 −0.142390
\(997\) 26.0217 0.824116 0.412058 0.911158i \(-0.364810\pi\)
0.412058 + 0.911158i \(0.364810\pi\)
\(998\) 10.9828 0.347654
\(999\) 6.96050 0.220220
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.z.1.12 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.12 14 1.1 even 1 trivial