Properties

Label 1110.2.a.r.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.12311 q^{11} -1.00000 q^{12} +5.68466 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +0.561553 q^{19} -1.00000 q^{20} +1.00000 q^{21} -6.12311 q^{22} -6.56155 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.68466 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.68466 q^{29} +1.00000 q^{30} -3.56155 q^{31} +1.00000 q^{32} +6.12311 q^{33} -5.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +0.561553 q^{38} -5.68466 q^{39} -1.00000 q^{40} +8.68466 q^{41} +1.00000 q^{42} -12.6847 q^{43} -6.12311 q^{44} -1.00000 q^{45} -6.56155 q^{46} -8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +5.00000 q^{51} +5.68466 q^{52} +5.24621 q^{53} -1.00000 q^{54} +6.12311 q^{55} -1.00000 q^{56} -0.561553 q^{57} -6.68466 q^{58} +4.24621 q^{59} +1.00000 q^{60} -13.8078 q^{61} -3.56155 q^{62} -1.00000 q^{63} +1.00000 q^{64} -5.68466 q^{65} +6.12311 q^{66} +13.1231 q^{67} -5.00000 q^{68} +6.56155 q^{69} +1.00000 q^{70} +1.12311 q^{71} +1.00000 q^{72} -6.56155 q^{73} +1.00000 q^{74} -1.00000 q^{75} +0.561553 q^{76} +6.12311 q^{77} -5.68466 q^{78} +14.2462 q^{79} -1.00000 q^{80} +1.00000 q^{81} +8.68466 q^{82} -7.43845 q^{83} +1.00000 q^{84} +5.00000 q^{85} -12.6847 q^{86} +6.68466 q^{87} -6.12311 q^{88} +5.68466 q^{89} -1.00000 q^{90} -5.68466 q^{91} -6.56155 q^{92} +3.56155 q^{93} -8.00000 q^{94} -0.561553 q^{95} -1.00000 q^{96} +3.31534 q^{97} -6.00000 q^{98} -6.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} - q^{13} - 2q^{14} + 2q^{15} + 2q^{16} - 10q^{17} + 2q^{18} - 3q^{19} - 2q^{20} + 2q^{21} - 4q^{22} - 9q^{23} - 2q^{24} + 2q^{25} - q^{26} - 2q^{27} - 2q^{28} - q^{29} + 2q^{30} - 3q^{31} + 2q^{32} + 4q^{33} - 10q^{34} + 2q^{35} + 2q^{36} + 2q^{37} - 3q^{38} + q^{39} - 2q^{40} + 5q^{41} + 2q^{42} - 13q^{43} - 4q^{44} - 2q^{45} - 9q^{46} - 16q^{47} - 2q^{48} - 12q^{49} + 2q^{50} + 10q^{51} - q^{52} - 6q^{53} - 2q^{54} + 4q^{55} - 2q^{56} + 3q^{57} - q^{58} - 8q^{59} + 2q^{60} - 7q^{61} - 3q^{62} - 2q^{63} + 2q^{64} + q^{65} + 4q^{66} + 18q^{67} - 10q^{68} + 9q^{69} + 2q^{70} - 6q^{71} + 2q^{72} - 9q^{73} + 2q^{74} - 2q^{75} - 3q^{76} + 4q^{77} + q^{78} + 12q^{79} - 2q^{80} + 2q^{81} + 5q^{82} - 19q^{83} + 2q^{84} + 10q^{85} - 13q^{86} + q^{87} - 4q^{88} - q^{89} - 2q^{90} + q^{91} - 9q^{92} + 3q^{93} - 16q^{94} + 3q^{95} - 2q^{96} + 19q^{97} - 12q^{98} - 4q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −6.12311 −1.84619 −0.923093 0.384577i \(-0.874347\pi\)
−0.923093 + 0.384577i \(0.874347\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.68466 1.57664 0.788320 0.615265i \(-0.210951\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.561553 0.128829 0.0644145 0.997923i \(-0.479482\pi\)
0.0644145 + 0.997923i \(0.479482\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −6.12311 −1.30545
\(23\) −6.56155 −1.36818 −0.684089 0.729398i \(-0.739800\pi\)
−0.684089 + 0.729398i \(0.739800\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 5.68466 1.11485
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 1.00000 0.182574
\(31\) −3.56155 −0.639674 −0.319837 0.947473i \(-0.603628\pi\)
−0.319837 + 0.947473i \(0.603628\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.12311 1.06590
\(34\) −5.00000 −0.857493
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 0.561553 0.0910959
\(39\) −5.68466 −0.910274
\(40\) −1.00000 −0.158114
\(41\) 8.68466 1.35632 0.678158 0.734916i \(-0.262779\pi\)
0.678158 + 0.734916i \(0.262779\pi\)
\(42\) 1.00000 0.154303
\(43\) −12.6847 −1.93439 −0.967196 0.254031i \(-0.918244\pi\)
−0.967196 + 0.254031i \(0.918244\pi\)
\(44\) −6.12311 −0.923093
\(45\) −1.00000 −0.149071
\(46\) −6.56155 −0.967448
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 5.00000 0.700140
\(52\) 5.68466 0.788320
\(53\) 5.24621 0.720623 0.360311 0.932832i \(-0.382670\pi\)
0.360311 + 0.932832i \(0.382670\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.12311 0.825639
\(56\) −1.00000 −0.133631
\(57\) −0.561553 −0.0743795
\(58\) −6.68466 −0.877739
\(59\) 4.24621 0.552810 0.276405 0.961041i \(-0.410857\pi\)
0.276405 + 0.961041i \(0.410857\pi\)
\(60\) 1.00000 0.129099
\(61\) −13.8078 −1.76790 −0.883952 0.467579i \(-0.845126\pi\)
−0.883952 + 0.467579i \(0.845126\pi\)
\(62\) −3.56155 −0.452318
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −5.68466 −0.705095
\(66\) 6.12311 0.753702
\(67\) 13.1231 1.60324 0.801621 0.597832i \(-0.203971\pi\)
0.801621 + 0.597832i \(0.203971\pi\)
\(68\) −5.00000 −0.606339
\(69\) 6.56155 0.789918
\(70\) 1.00000 0.119523
\(71\) 1.12311 0.133288 0.0666441 0.997777i \(-0.478771\pi\)
0.0666441 + 0.997777i \(0.478771\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.56155 −0.767972 −0.383986 0.923339i \(-0.625449\pi\)
−0.383986 + 0.923339i \(0.625449\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) 0.561553 0.0644145
\(77\) 6.12311 0.697793
\(78\) −5.68466 −0.643661
\(79\) 14.2462 1.60282 0.801412 0.598113i \(-0.204082\pi\)
0.801412 + 0.598113i \(0.204082\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 8.68466 0.959060
\(83\) −7.43845 −0.816476 −0.408238 0.912876i \(-0.633857\pi\)
−0.408238 + 0.912876i \(0.633857\pi\)
\(84\) 1.00000 0.109109
\(85\) 5.00000 0.542326
\(86\) −12.6847 −1.36782
\(87\) 6.68466 0.716671
\(88\) −6.12311 −0.652725
\(89\) 5.68466 0.602573 0.301286 0.953534i \(-0.402584\pi\)
0.301286 + 0.953534i \(0.402584\pi\)
\(90\) −1.00000 −0.105409
\(91\) −5.68466 −0.595914
\(92\) −6.56155 −0.684089
\(93\) 3.56155 0.369316
\(94\) −8.00000 −0.825137
\(95\) −0.561553 −0.0576141
\(96\) −1.00000 −0.102062
\(97\) 3.31534 0.336622 0.168311 0.985734i \(-0.446169\pi\)
0.168311 + 0.985734i \(0.446169\pi\)
\(98\) −6.00000 −0.606092
\(99\) −6.12311 −0.615395
\(100\) 1.00000 0.100000
\(101\) 10.8769 1.08229 0.541146 0.840929i \(-0.317991\pi\)
0.541146 + 0.840929i \(0.317991\pi\)
\(102\) 5.00000 0.495074
\(103\) −16.4924 −1.62505 −0.812523 0.582929i \(-0.801907\pi\)
−0.812523 + 0.582929i \(0.801907\pi\)
\(104\) 5.68466 0.557427
\(105\) −1.00000 −0.0975900
\(106\) 5.24621 0.509557
\(107\) −10.8078 −1.04483 −0.522413 0.852693i \(-0.674968\pi\)
−0.522413 + 0.852693i \(0.674968\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.12311 −0.203357 −0.101678 0.994817i \(-0.532421\pi\)
−0.101678 + 0.994817i \(0.532421\pi\)
\(110\) 6.12311 0.583815
\(111\) −1.00000 −0.0949158
\(112\) −1.00000 −0.0944911
\(113\) −10.6847 −1.00513 −0.502564 0.864540i \(-0.667610\pi\)
−0.502564 + 0.864540i \(0.667610\pi\)
\(114\) −0.561553 −0.0525942
\(115\) 6.56155 0.611868
\(116\) −6.68466 −0.620655
\(117\) 5.68466 0.525547
\(118\) 4.24621 0.390895
\(119\) 5.00000 0.458349
\(120\) 1.00000 0.0912871
\(121\) 26.4924 2.40840
\(122\) −13.8078 −1.25010
\(123\) −8.68466 −0.783069
\(124\) −3.56155 −0.319837
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 14.5616 1.29213 0.646064 0.763283i \(-0.276414\pi\)
0.646064 + 0.763283i \(0.276414\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.6847 1.11682
\(130\) −5.68466 −0.498578
\(131\) 7.12311 0.622349 0.311174 0.950353i \(-0.399278\pi\)
0.311174 + 0.950353i \(0.399278\pi\)
\(132\) 6.12311 0.532948
\(133\) −0.561553 −0.0486928
\(134\) 13.1231 1.13366
\(135\) 1.00000 0.0860663
\(136\) −5.00000 −0.428746
\(137\) −4.24621 −0.362778 −0.181389 0.983411i \(-0.558059\pi\)
−0.181389 + 0.983411i \(0.558059\pi\)
\(138\) 6.56155 0.558556
\(139\) 13.8078 1.17116 0.585580 0.810615i \(-0.300867\pi\)
0.585580 + 0.810615i \(0.300867\pi\)
\(140\) 1.00000 0.0845154
\(141\) 8.00000 0.673722
\(142\) 1.12311 0.0942489
\(143\) −34.8078 −2.91077
\(144\) 1.00000 0.0833333
\(145\) 6.68466 0.555131
\(146\) −6.56155 −0.543038
\(147\) 6.00000 0.494872
\(148\) 1.00000 0.0821995
\(149\) 7.75379 0.635215 0.317608 0.948222i \(-0.397121\pi\)
0.317608 + 0.948222i \(0.397121\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −15.4384 −1.25636 −0.628182 0.778067i \(-0.716200\pi\)
−0.628182 + 0.778067i \(0.716200\pi\)
\(152\) 0.561553 0.0455479
\(153\) −5.00000 −0.404226
\(154\) 6.12311 0.493414
\(155\) 3.56155 0.286071
\(156\) −5.68466 −0.455137
\(157\) −18.0540 −1.44086 −0.720432 0.693526i \(-0.756056\pi\)
−0.720432 + 0.693526i \(0.756056\pi\)
\(158\) 14.2462 1.13337
\(159\) −5.24621 −0.416052
\(160\) −1.00000 −0.0790569
\(161\) 6.56155 0.517123
\(162\) 1.00000 0.0785674
\(163\) 19.4924 1.52676 0.763382 0.645947i \(-0.223537\pi\)
0.763382 + 0.645947i \(0.223537\pi\)
\(164\) 8.68466 0.678158
\(165\) −6.12311 −0.476683
\(166\) −7.43845 −0.577335
\(167\) 19.0540 1.47444 0.737220 0.675652i \(-0.236138\pi\)
0.737220 + 0.675652i \(0.236138\pi\)
\(168\) 1.00000 0.0771517
\(169\) 19.3153 1.48580
\(170\) 5.00000 0.383482
\(171\) 0.561553 0.0429430
\(172\) −12.6847 −0.967196
\(173\) −2.12311 −0.161417 −0.0807084 0.996738i \(-0.525718\pi\)
−0.0807084 + 0.996738i \(0.525718\pi\)
\(174\) 6.68466 0.506763
\(175\) −1.00000 −0.0755929
\(176\) −6.12311 −0.461546
\(177\) −4.24621 −0.319165
\(178\) 5.68466 0.426083
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 20.2462 1.50489 0.752445 0.658656i \(-0.228875\pi\)
0.752445 + 0.658656i \(0.228875\pi\)
\(182\) −5.68466 −0.421375
\(183\) 13.8078 1.02070
\(184\) −6.56155 −0.483724
\(185\) −1.00000 −0.0735215
\(186\) 3.56155 0.261146
\(187\) 30.6155 2.23883
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) −0.561553 −0.0407393
\(191\) 14.3693 1.03973 0.519864 0.854249i \(-0.325983\pi\)
0.519864 + 0.854249i \(0.325983\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 3.31534 0.238028
\(195\) 5.68466 0.407087
\(196\) −6.00000 −0.428571
\(197\) −2.31534 −0.164961 −0.0824806 0.996593i \(-0.526284\pi\)
−0.0824806 + 0.996593i \(0.526284\pi\)
\(198\) −6.12311 −0.435150
\(199\) −16.4924 −1.16912 −0.584558 0.811352i \(-0.698732\pi\)
−0.584558 + 0.811352i \(0.698732\pi\)
\(200\) 1.00000 0.0707107
\(201\) −13.1231 −0.925633
\(202\) 10.8769 0.765296
\(203\) 6.68466 0.469171
\(204\) 5.00000 0.350070
\(205\) −8.68466 −0.606563
\(206\) −16.4924 −1.14908
\(207\) −6.56155 −0.456059
\(208\) 5.68466 0.394160
\(209\) −3.43845 −0.237842
\(210\) −1.00000 −0.0690066
\(211\) 4.43845 0.305555 0.152778 0.988261i \(-0.451178\pi\)
0.152778 + 0.988261i \(0.451178\pi\)
\(212\) 5.24621 0.360311
\(213\) −1.12311 −0.0769539
\(214\) −10.8078 −0.738804
\(215\) 12.6847 0.865087
\(216\) −1.00000 −0.0680414
\(217\) 3.56155 0.241774
\(218\) −2.12311 −0.143795
\(219\) 6.56155 0.443389
\(220\) 6.12311 0.412820
\(221\) −28.4233 −1.91196
\(222\) −1.00000 −0.0671156
\(223\) −19.8078 −1.32643 −0.663213 0.748431i \(-0.730808\pi\)
−0.663213 + 0.748431i \(0.730808\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −10.6847 −0.710733
\(227\) −10.9309 −0.725507 −0.362754 0.931885i \(-0.618163\pi\)
−0.362754 + 0.931885i \(0.618163\pi\)
\(228\) −0.561553 −0.0371897
\(229\) −5.12311 −0.338544 −0.169272 0.985569i \(-0.554142\pi\)
−0.169272 + 0.985569i \(0.554142\pi\)
\(230\) 6.56155 0.432656
\(231\) −6.12311 −0.402871
\(232\) −6.68466 −0.438869
\(233\) 11.3693 0.744829 0.372414 0.928067i \(-0.378530\pi\)
0.372414 + 0.928067i \(0.378530\pi\)
\(234\) 5.68466 0.371618
\(235\) 8.00000 0.521862
\(236\) 4.24621 0.276405
\(237\) −14.2462 −0.925391
\(238\) 5.00000 0.324102
\(239\) 7.80776 0.505042 0.252521 0.967591i \(-0.418740\pi\)
0.252521 + 0.967591i \(0.418740\pi\)
\(240\) 1.00000 0.0645497
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 26.4924 1.70300
\(243\) −1.00000 −0.0641500
\(244\) −13.8078 −0.883952
\(245\) 6.00000 0.383326
\(246\) −8.68466 −0.553714
\(247\) 3.19224 0.203117
\(248\) −3.56155 −0.226159
\(249\) 7.43845 0.471392
\(250\) −1.00000 −0.0632456
\(251\) 25.6155 1.61684 0.808419 0.588608i \(-0.200324\pi\)
0.808419 + 0.588608i \(0.200324\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 40.1771 2.52591
\(254\) 14.5616 0.913673
\(255\) −5.00000 −0.313112
\(256\) 1.00000 0.0625000
\(257\) 10.8078 0.674170 0.337085 0.941474i \(-0.390559\pi\)
0.337085 + 0.941474i \(0.390559\pi\)
\(258\) 12.6847 0.789712
\(259\) −1.00000 −0.0621370
\(260\) −5.68466 −0.352548
\(261\) −6.68466 −0.413770
\(262\) 7.12311 0.440067
\(263\) −6.43845 −0.397012 −0.198506 0.980100i \(-0.563609\pi\)
−0.198506 + 0.980100i \(0.563609\pi\)
\(264\) 6.12311 0.376851
\(265\) −5.24621 −0.322272
\(266\) −0.561553 −0.0344310
\(267\) −5.68466 −0.347895
\(268\) 13.1231 0.801621
\(269\) −16.8078 −1.02479 −0.512394 0.858751i \(-0.671241\pi\)
−0.512394 + 0.858751i \(0.671241\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.4924 −1.00184 −0.500922 0.865493i \(-0.667006\pi\)
−0.500922 + 0.865493i \(0.667006\pi\)
\(272\) −5.00000 −0.303170
\(273\) 5.68466 0.344051
\(274\) −4.24621 −0.256523
\(275\) −6.12311 −0.369237
\(276\) 6.56155 0.394959
\(277\) 27.6847 1.66341 0.831705 0.555218i \(-0.187365\pi\)
0.831705 + 0.555218i \(0.187365\pi\)
\(278\) 13.8078 0.828135
\(279\) −3.56155 −0.213225
\(280\) 1.00000 0.0597614
\(281\) 22.1771 1.32297 0.661487 0.749957i \(-0.269926\pi\)
0.661487 + 0.749957i \(0.269926\pi\)
\(282\) 8.00000 0.476393
\(283\) 9.93087 0.590329 0.295164 0.955446i \(-0.404626\pi\)
0.295164 + 0.955446i \(0.404626\pi\)
\(284\) 1.12311 0.0666441
\(285\) 0.561553 0.0332635
\(286\) −34.8078 −2.05823
\(287\) −8.68466 −0.512639
\(288\) 1.00000 0.0589256
\(289\) 8.00000 0.470588
\(290\) 6.68466 0.392537
\(291\) −3.31534 −0.194349
\(292\) −6.56155 −0.383986
\(293\) −16.6155 −0.970690 −0.485345 0.874323i \(-0.661306\pi\)
−0.485345 + 0.874323i \(0.661306\pi\)
\(294\) 6.00000 0.349927
\(295\) −4.24621 −0.247224
\(296\) 1.00000 0.0581238
\(297\) 6.12311 0.355299
\(298\) 7.75379 0.449165
\(299\) −37.3002 −2.15713
\(300\) −1.00000 −0.0577350
\(301\) 12.6847 0.731132
\(302\) −15.4384 −0.888383
\(303\) −10.8769 −0.624861
\(304\) 0.561553 0.0322073
\(305\) 13.8078 0.790630
\(306\) −5.00000 −0.285831
\(307\) −0.246211 −0.0140520 −0.00702601 0.999975i \(-0.502236\pi\)
−0.00702601 + 0.999975i \(0.502236\pi\)
\(308\) 6.12311 0.348896
\(309\) 16.4924 0.938221
\(310\) 3.56155 0.202283
\(311\) −21.1771 −1.20084 −0.600421 0.799684i \(-0.705000\pi\)
−0.600421 + 0.799684i \(0.705000\pi\)
\(312\) −5.68466 −0.321830
\(313\) −24.8769 −1.40613 −0.703063 0.711128i \(-0.748185\pi\)
−0.703063 + 0.711128i \(0.748185\pi\)
\(314\) −18.0540 −1.01884
\(315\) 1.00000 0.0563436
\(316\) 14.2462 0.801412
\(317\) 16.9309 0.950932 0.475466 0.879734i \(-0.342279\pi\)
0.475466 + 0.879734i \(0.342279\pi\)
\(318\) −5.24621 −0.294193
\(319\) 40.9309 2.29169
\(320\) −1.00000 −0.0559017
\(321\) 10.8078 0.603231
\(322\) 6.56155 0.365661
\(323\) −2.80776 −0.156228
\(324\) 1.00000 0.0555556
\(325\) 5.68466 0.315328
\(326\) 19.4924 1.07959
\(327\) 2.12311 0.117408
\(328\) 8.68466 0.479530
\(329\) 8.00000 0.441054
\(330\) −6.12311 −0.337066
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −7.43845 −0.408238
\(333\) 1.00000 0.0547997
\(334\) 19.0540 1.04259
\(335\) −13.1231 −0.716992
\(336\) 1.00000 0.0545545
\(337\) 4.56155 0.248484 0.124242 0.992252i \(-0.460350\pi\)
0.124242 + 0.992252i \(0.460350\pi\)
\(338\) 19.3153 1.05062
\(339\) 10.6847 0.580311
\(340\) 5.00000 0.271163
\(341\) 21.8078 1.18096
\(342\) 0.561553 0.0303653
\(343\) 13.0000 0.701934
\(344\) −12.6847 −0.683911
\(345\) −6.56155 −0.353262
\(346\) −2.12311 −0.114139
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 6.68466 0.358335
\(349\) 14.2462 0.762582 0.381291 0.924455i \(-0.375480\pi\)
0.381291 + 0.924455i \(0.375480\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −5.68466 −0.303425
\(352\) −6.12311 −0.326363
\(353\) −29.8078 −1.58651 −0.793254 0.608891i \(-0.791615\pi\)
−0.793254 + 0.608891i \(0.791615\pi\)
\(354\) −4.24621 −0.225684
\(355\) −1.12311 −0.0596083
\(356\) 5.68466 0.301286
\(357\) −5.00000 −0.264628
\(358\) −4.00000 −0.211407
\(359\) −20.8769 −1.10184 −0.550920 0.834558i \(-0.685723\pi\)
−0.550920 + 0.834558i \(0.685723\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.6847 −0.983403
\(362\) 20.2462 1.06412
\(363\) −26.4924 −1.39049
\(364\) −5.68466 −0.297957
\(365\) 6.56155 0.343447
\(366\) 13.8078 0.721743
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) −6.56155 −0.342045
\(369\) 8.68466 0.452105
\(370\) −1.00000 −0.0519875
\(371\) −5.24621 −0.272370
\(372\) 3.56155 0.184658
\(373\) −30.4924 −1.57884 −0.789419 0.613855i \(-0.789618\pi\)
−0.789419 + 0.613855i \(0.789618\pi\)
\(374\) 30.6155 1.58309
\(375\) 1.00000 0.0516398
\(376\) −8.00000 −0.412568
\(377\) −38.0000 −1.95710
\(378\) 1.00000 0.0514344
\(379\) −2.87689 −0.147776 −0.0738881 0.997267i \(-0.523541\pi\)
−0.0738881 + 0.997267i \(0.523541\pi\)
\(380\) −0.561553 −0.0288071
\(381\) −14.5616 −0.746011
\(382\) 14.3693 0.735198
\(383\) −27.6847 −1.41462 −0.707310 0.706904i \(-0.750091\pi\)
−0.707310 + 0.706904i \(0.750091\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.12311 −0.312062
\(386\) −14.0000 −0.712581
\(387\) −12.6847 −0.644797
\(388\) 3.31534 0.168311
\(389\) 20.9309 1.06124 0.530619 0.847611i \(-0.321960\pi\)
0.530619 + 0.847611i \(0.321960\pi\)
\(390\) 5.68466 0.287854
\(391\) 32.8078 1.65916
\(392\) −6.00000 −0.303046
\(393\) −7.12311 −0.359313
\(394\) −2.31534 −0.116645
\(395\) −14.2462 −0.716805
\(396\) −6.12311 −0.307698
\(397\) 27.1231 1.36127 0.680635 0.732623i \(-0.261704\pi\)
0.680635 + 0.732623i \(0.261704\pi\)
\(398\) −16.4924 −0.826690
\(399\) 0.561553 0.0281128
\(400\) 1.00000 0.0500000
\(401\) −14.8078 −0.739464 −0.369732 0.929138i \(-0.620551\pi\)
−0.369732 + 0.929138i \(0.620551\pi\)
\(402\) −13.1231 −0.654521
\(403\) −20.2462 −1.00854
\(404\) 10.8769 0.541146
\(405\) −1.00000 −0.0496904
\(406\) 6.68466 0.331754
\(407\) −6.12311 −0.303511
\(408\) 5.00000 0.247537
\(409\) −0.876894 −0.0433596 −0.0216798 0.999765i \(-0.506901\pi\)
−0.0216798 + 0.999765i \(0.506901\pi\)
\(410\) −8.68466 −0.428905
\(411\) 4.24621 0.209450
\(412\) −16.4924 −0.812523
\(413\) −4.24621 −0.208942
\(414\) −6.56155 −0.322483
\(415\) 7.43845 0.365139
\(416\) 5.68466 0.278713
\(417\) −13.8078 −0.676169
\(418\) −3.43845 −0.168180
\(419\) −10.5616 −0.515966 −0.257983 0.966150i \(-0.583058\pi\)
−0.257983 + 0.966150i \(0.583058\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 30.9848 1.51011 0.755054 0.655662i \(-0.227610\pi\)
0.755054 + 0.655662i \(0.227610\pi\)
\(422\) 4.43845 0.216060
\(423\) −8.00000 −0.388973
\(424\) 5.24621 0.254779
\(425\) −5.00000 −0.242536
\(426\) −1.12311 −0.0544146
\(427\) 13.8078 0.668205
\(428\) −10.8078 −0.522413
\(429\) 34.8078 1.68053
\(430\) 12.6847 0.611709
\(431\) −29.0000 −1.39688 −0.698440 0.715668i \(-0.746122\pi\)
−0.698440 + 0.715668i \(0.746122\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.4233 −1.46205 −0.731025 0.682351i \(-0.760958\pi\)
−0.731025 + 0.682351i \(0.760958\pi\)
\(434\) 3.56155 0.170960
\(435\) −6.68466 −0.320505
\(436\) −2.12311 −0.101678
\(437\) −3.68466 −0.176261
\(438\) 6.56155 0.313523
\(439\) −14.0540 −0.670760 −0.335380 0.942083i \(-0.608865\pi\)
−0.335380 + 0.942083i \(0.608865\pi\)
\(440\) 6.12311 0.291908
\(441\) −6.00000 −0.285714
\(442\) −28.4233 −1.35196
\(443\) 33.8617 1.60882 0.804410 0.594075i \(-0.202482\pi\)
0.804410 + 0.594075i \(0.202482\pi\)
\(444\) −1.00000 −0.0474579
\(445\) −5.68466 −0.269479
\(446\) −19.8078 −0.937925
\(447\) −7.75379 −0.366742
\(448\) −1.00000 −0.0472456
\(449\) −1.61553 −0.0762415 −0.0381207 0.999273i \(-0.512137\pi\)
−0.0381207 + 0.999273i \(0.512137\pi\)
\(450\) 1.00000 0.0471405
\(451\) −53.1771 −2.50401
\(452\) −10.6847 −0.502564
\(453\) 15.4384 0.725362
\(454\) −10.9309 −0.513011
\(455\) 5.68466 0.266501
\(456\) −0.561553 −0.0262971
\(457\) −30.9309 −1.44689 −0.723443 0.690385i \(-0.757441\pi\)
−0.723443 + 0.690385i \(0.757441\pi\)
\(458\) −5.12311 −0.239387
\(459\) 5.00000 0.233380
\(460\) 6.56155 0.305934
\(461\) 0.438447 0.0204205 0.0102103 0.999948i \(-0.496750\pi\)
0.0102103 + 0.999948i \(0.496750\pi\)
\(462\) −6.12311 −0.284873
\(463\) 4.87689 0.226649 0.113324 0.993558i \(-0.463850\pi\)
0.113324 + 0.993558i \(0.463850\pi\)
\(464\) −6.68466 −0.310327
\(465\) −3.56155 −0.165163
\(466\) 11.3693 0.526673
\(467\) 2.43845 0.112838 0.0564189 0.998407i \(-0.482032\pi\)
0.0564189 + 0.998407i \(0.482032\pi\)
\(468\) 5.68466 0.262773
\(469\) −13.1231 −0.605969
\(470\) 8.00000 0.369012
\(471\) 18.0540 0.831883
\(472\) 4.24621 0.195448
\(473\) 77.6695 3.57125
\(474\) −14.2462 −0.654350
\(475\) 0.561553 0.0257658
\(476\) 5.00000 0.229175
\(477\) 5.24621 0.240208
\(478\) 7.80776 0.357119
\(479\) −6.56155 −0.299805 −0.149903 0.988701i \(-0.547896\pi\)
−0.149903 + 0.988701i \(0.547896\pi\)
\(480\) 1.00000 0.0456435
\(481\) 5.68466 0.259198
\(482\) −2.00000 −0.0910975
\(483\) −6.56155 −0.298561
\(484\) 26.4924 1.20420
\(485\) −3.31534 −0.150542
\(486\) −1.00000 −0.0453609
\(487\) 16.7386 0.758500 0.379250 0.925294i \(-0.376182\pi\)
0.379250 + 0.925294i \(0.376182\pi\)
\(488\) −13.8078 −0.625048
\(489\) −19.4924 −0.881478
\(490\) 6.00000 0.271052
\(491\) −12.1771 −0.549544 −0.274772 0.961509i \(-0.588602\pi\)
−0.274772 + 0.961509i \(0.588602\pi\)
\(492\) −8.68466 −0.391535
\(493\) 33.4233 1.50531
\(494\) 3.19224 0.143625
\(495\) 6.12311 0.275213
\(496\) −3.56155 −0.159918
\(497\) −1.12311 −0.0503782
\(498\) 7.43845 0.333325
\(499\) 32.8078 1.46868 0.734339 0.678783i \(-0.237492\pi\)
0.734339 + 0.678783i \(0.237492\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −19.0540 −0.851269
\(502\) 25.6155 1.14328
\(503\) −18.2462 −0.813558 −0.406779 0.913527i \(-0.633348\pi\)
−0.406779 + 0.913527i \(0.633348\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −10.8769 −0.484015
\(506\) 40.1771 1.78609
\(507\) −19.3153 −0.857824
\(508\) 14.5616 0.646064
\(509\) 23.0540 1.02185 0.510925 0.859625i \(-0.329303\pi\)
0.510925 + 0.859625i \(0.329303\pi\)
\(510\) −5.00000 −0.221404
\(511\) 6.56155 0.290266
\(512\) 1.00000 0.0441942
\(513\) −0.561553 −0.0247932
\(514\) 10.8078 0.476710
\(515\) 16.4924 0.726743
\(516\) 12.6847 0.558411
\(517\) 48.9848 2.15435
\(518\) −1.00000 −0.0439375
\(519\) 2.12311 0.0931940
\(520\) −5.68466 −0.249289
\(521\) −18.4384 −0.807803 −0.403902 0.914802i \(-0.632346\pi\)
−0.403902 + 0.914802i \(0.632346\pi\)
\(522\) −6.68466 −0.292580
\(523\) 30.7386 1.34411 0.672053 0.740503i \(-0.265413\pi\)
0.672053 + 0.740503i \(0.265413\pi\)
\(524\) 7.12311 0.311174
\(525\) 1.00000 0.0436436
\(526\) −6.43845 −0.280730
\(527\) 17.8078 0.775718
\(528\) 6.12311 0.266474
\(529\) 20.0540 0.871912
\(530\) −5.24621 −0.227881
\(531\) 4.24621 0.184270
\(532\) −0.561553 −0.0243464
\(533\) 49.3693 2.13842
\(534\) −5.68466 −0.245999
\(535\) 10.8078 0.467260
\(536\) 13.1231 0.566832
\(537\) 4.00000 0.172613
\(538\) −16.8078 −0.724634
\(539\) 36.7386 1.58244
\(540\) 1.00000 0.0430331
\(541\) −20.4233 −0.878066 −0.439033 0.898471i \(-0.644679\pi\)
−0.439033 + 0.898471i \(0.644679\pi\)
\(542\) −16.4924 −0.708410
\(543\) −20.2462 −0.868848
\(544\) −5.00000 −0.214373
\(545\) 2.12311 0.0909439
\(546\) 5.68466 0.243281
\(547\) −39.7386 −1.69910 −0.849551 0.527507i \(-0.823127\pi\)
−0.849551 + 0.527507i \(0.823127\pi\)
\(548\) −4.24621 −0.181389
\(549\) −13.8078 −0.589301
\(550\) −6.12311 −0.261090
\(551\) −3.75379 −0.159917
\(552\) 6.56155 0.279278
\(553\) −14.2462 −0.605811
\(554\) 27.6847 1.17621
\(555\) 1.00000 0.0424476
\(556\) 13.8078 0.585580
\(557\) 7.61553 0.322680 0.161340 0.986899i \(-0.448418\pi\)
0.161340 + 0.986899i \(0.448418\pi\)
\(558\) −3.56155 −0.150773
\(559\) −72.1080 −3.04984
\(560\) 1.00000 0.0422577
\(561\) −30.6155 −1.29259
\(562\) 22.1771 0.935484
\(563\) 4.93087 0.207811 0.103906 0.994587i \(-0.466866\pi\)
0.103906 + 0.994587i \(0.466866\pi\)
\(564\) 8.00000 0.336861
\(565\) 10.6847 0.449507
\(566\) 9.93087 0.417426
\(567\) −1.00000 −0.0419961
\(568\) 1.12311 0.0471245
\(569\) 2.31534 0.0970642 0.0485321 0.998822i \(-0.484546\pi\)
0.0485321 + 0.998822i \(0.484546\pi\)
\(570\) 0.561553 0.0235209
\(571\) −5.80776 −0.243047 −0.121524 0.992589i \(-0.538778\pi\)
−0.121524 + 0.992589i \(0.538778\pi\)
\(572\) −34.8078 −1.45539
\(573\) −14.3693 −0.600287
\(574\) −8.68466 −0.362491
\(575\) −6.56155 −0.273636
\(576\) 1.00000 0.0416667
\(577\) −14.4924 −0.603327 −0.301664 0.953414i \(-0.597542\pi\)
−0.301664 + 0.953414i \(0.597542\pi\)
\(578\) 8.00000 0.332756
\(579\) 14.0000 0.581820
\(580\) 6.68466 0.277565
\(581\) 7.43845 0.308599
\(582\) −3.31534 −0.137425
\(583\) −32.1231 −1.33040
\(584\) −6.56155 −0.271519
\(585\) −5.68466 −0.235032
\(586\) −16.6155 −0.686381
\(587\) −12.3002 −0.507683 −0.253842 0.967246i \(-0.581694\pi\)
−0.253842 + 0.967246i \(0.581694\pi\)
\(588\) 6.00000 0.247436
\(589\) −2.00000 −0.0824086
\(590\) −4.24621 −0.174814
\(591\) 2.31534 0.0952404
\(592\) 1.00000 0.0410997
\(593\) 4.87689 0.200270 0.100135 0.994974i \(-0.468073\pi\)
0.100135 + 0.994974i \(0.468073\pi\)
\(594\) 6.12311 0.251234
\(595\) −5.00000 −0.204980
\(596\) 7.75379 0.317608
\(597\) 16.4924 0.674990
\(598\) −37.3002 −1.52532
\(599\) 26.7386 1.09251 0.546255 0.837619i \(-0.316053\pi\)
0.546255 + 0.837619i \(0.316053\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −42.8617 −1.74837 −0.874183 0.485596i \(-0.838603\pi\)
−0.874183 + 0.485596i \(0.838603\pi\)
\(602\) 12.6847 0.516988
\(603\) 13.1231 0.534414
\(604\) −15.4384 −0.628182
\(605\) −26.4924 −1.07707
\(606\) −10.8769 −0.441844
\(607\) −26.4924 −1.07529 −0.537647 0.843170i \(-0.680687\pi\)
−0.537647 + 0.843170i \(0.680687\pi\)
\(608\) 0.561553 0.0227740
\(609\) −6.68466 −0.270876
\(610\) 13.8078 0.559060
\(611\) −45.4773 −1.83981
\(612\) −5.00000 −0.202113
\(613\) 8.43845 0.340826 0.170413 0.985373i \(-0.445490\pi\)
0.170413 + 0.985373i \(0.445490\pi\)
\(614\) −0.246211 −0.00993628
\(615\) 8.68466 0.350199
\(616\) 6.12311 0.246707
\(617\) 14.4924 0.583443 0.291721 0.956503i \(-0.405772\pi\)
0.291721 + 0.956503i \(0.405772\pi\)
\(618\) 16.4924 0.663423
\(619\) −35.8078 −1.43924 −0.719618 0.694370i \(-0.755683\pi\)
−0.719618 + 0.694370i \(0.755683\pi\)
\(620\) 3.56155 0.143035
\(621\) 6.56155 0.263306
\(622\) −21.1771 −0.849124
\(623\) −5.68466 −0.227751
\(624\) −5.68466 −0.227568
\(625\) 1.00000 0.0400000
\(626\) −24.8769 −0.994281
\(627\) 3.43845 0.137318
\(628\) −18.0540 −0.720432
\(629\) −5.00000 −0.199363
\(630\) 1.00000 0.0398410
\(631\) 15.3153 0.609694 0.304847 0.952401i \(-0.401395\pi\)
0.304847 + 0.952401i \(0.401395\pi\)
\(632\) 14.2462 0.566684
\(633\) −4.43845 −0.176412
\(634\) 16.9309 0.672411
\(635\) −14.5616 −0.577858
\(636\) −5.24621 −0.208026
\(637\) −34.1080 −1.35141
\(638\) 40.9309 1.62047
\(639\) 1.12311 0.0444294
\(640\) −1.00000 −0.0395285
\(641\) 1.94602 0.0768634 0.0384317 0.999261i \(-0.487764\pi\)
0.0384317 + 0.999261i \(0.487764\pi\)
\(642\) 10.8078 0.426548
\(643\) −34.1231 −1.34568 −0.672842 0.739786i \(-0.734927\pi\)
−0.672842 + 0.739786i \(0.734927\pi\)
\(644\) 6.56155 0.258561
\(645\) −12.6847 −0.499458
\(646\) −2.80776 −0.110470
\(647\) −36.8078 −1.44706 −0.723531 0.690292i \(-0.757482\pi\)
−0.723531 + 0.690292i \(0.757482\pi\)
\(648\) 1.00000 0.0392837
\(649\) −26.0000 −1.02059
\(650\) 5.68466 0.222971
\(651\) −3.56155 −0.139588
\(652\) 19.4924 0.763382
\(653\) −28.4924 −1.11499 −0.557497 0.830179i \(-0.688238\pi\)
−0.557497 + 0.830179i \(0.688238\pi\)
\(654\) 2.12311 0.0830200
\(655\) −7.12311 −0.278323
\(656\) 8.68466 0.339079
\(657\) −6.56155 −0.255991
\(658\) 8.00000 0.311872
\(659\) −18.2462 −0.710771 −0.355386 0.934720i \(-0.615650\pi\)
−0.355386 + 0.934720i \(0.615650\pi\)
\(660\) −6.12311 −0.238342
\(661\) 32.8617 1.27817 0.639087 0.769135i \(-0.279312\pi\)
0.639087 + 0.769135i \(0.279312\pi\)
\(662\) −12.0000 −0.466393
\(663\) 28.4233 1.10387
\(664\) −7.43845 −0.288668
\(665\) 0.561553 0.0217761
\(666\) 1.00000 0.0387492
\(667\) 43.8617 1.69833
\(668\) 19.0540 0.737220
\(669\) 19.8078 0.765812
\(670\) −13.1231 −0.506990
\(671\) 84.5464 3.26388
\(672\) 1.00000 0.0385758
\(673\) −18.5616 −0.715495 −0.357748 0.933818i \(-0.616455\pi\)
−0.357748 + 0.933818i \(0.616455\pi\)
\(674\) 4.56155 0.175704
\(675\) −1.00000 −0.0384900
\(676\) 19.3153 0.742898
\(677\) 35.9309 1.38094 0.690468 0.723363i \(-0.257405\pi\)
0.690468 + 0.723363i \(0.257405\pi\)
\(678\) 10.6847 0.410342
\(679\) −3.31534 −0.127231
\(680\) 5.00000 0.191741
\(681\) 10.9309 0.418872
\(682\) 21.8078 0.835062
\(683\) −30.4384 −1.16469 −0.582347 0.812940i \(-0.697866\pi\)
−0.582347 + 0.812940i \(0.697866\pi\)
\(684\) 0.561553 0.0214715
\(685\) 4.24621 0.162239
\(686\) 13.0000 0.496342
\(687\) 5.12311 0.195459
\(688\) −12.6847 −0.483598
\(689\) 29.8229 1.13616
\(690\) −6.56155 −0.249794
\(691\) −5.31534 −0.202205 −0.101103 0.994876i \(-0.532237\pi\)
−0.101103 + 0.994876i \(0.532237\pi\)
\(692\) −2.12311 −0.0807084
\(693\) 6.12311 0.232598
\(694\) −28.0000 −1.06287
\(695\) −13.8078 −0.523758
\(696\) 6.68466 0.253381
\(697\) −43.4233 −1.64477
\(698\) 14.2462 0.539227
\(699\) −11.3693 −0.430027
\(700\) −1.00000 −0.0377964
\(701\) 9.36932 0.353874 0.176937 0.984222i \(-0.443381\pi\)
0.176937 + 0.984222i \(0.443381\pi\)
\(702\) −5.68466 −0.214554
\(703\) 0.561553 0.0211794
\(704\) −6.12311 −0.230773
\(705\) −8.00000 −0.301297
\(706\) −29.8078 −1.12183
\(707\) −10.8769 −0.409068
\(708\) −4.24621 −0.159582
\(709\) −4.50758 −0.169286 −0.0846428 0.996411i \(-0.526975\pi\)
−0.0846428 + 0.996411i \(0.526975\pi\)
\(710\) −1.12311 −0.0421494
\(711\) 14.2462 0.534275
\(712\) 5.68466 0.213042
\(713\) 23.3693 0.875188
\(714\) −5.00000 −0.187120
\(715\) 34.8078 1.30174
\(716\) −4.00000 −0.149487
\(717\) −7.80776 −0.291586
\(718\) −20.8769 −0.779119
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.4924 0.614210
\(722\) −18.6847 −0.695371
\(723\) 2.00000 0.0743808
\(724\) 20.2462 0.752445
\(725\) −6.68466 −0.248262
\(726\) −26.4924 −0.983226
\(727\) 50.9848 1.89092 0.945462 0.325734i \(-0.105611\pi\)
0.945462 + 0.325734i \(0.105611\pi\)
\(728\) −5.68466 −0.210687
\(729\) 1.00000 0.0370370
\(730\) 6.56155 0.242854
\(731\) 63.4233 2.34580
\(732\) 13.8078 0.510350
\(733\) −22.3002 −0.823676 −0.411838 0.911257i \(-0.635113\pi\)
−0.411838 + 0.911257i \(0.635113\pi\)
\(734\) −17.0000 −0.627481
\(735\) −6.00000 −0.221313
\(736\) −6.56155 −0.241862
\(737\) −80.3542 −2.95988
\(738\) 8.68466 0.319687
\(739\) −39.1771 −1.44115 −0.720576 0.693376i \(-0.756123\pi\)
−0.720576 + 0.693376i \(0.756123\pi\)
\(740\) −1.00000 −0.0367607
\(741\) −3.19224 −0.117270
\(742\) −5.24621 −0.192594
\(743\) 21.4233 0.785944 0.392972 0.919550i \(-0.371447\pi\)
0.392972 + 0.919550i \(0.371447\pi\)
\(744\) 3.56155 0.130573
\(745\) −7.75379 −0.284077
\(746\) −30.4924 −1.11641
\(747\) −7.43845 −0.272159
\(748\) 30.6155 1.11941
\(749\) 10.8078 0.394907
\(750\) 1.00000 0.0365148
\(751\) 22.7386 0.829745 0.414872 0.909880i \(-0.363826\pi\)
0.414872 + 0.909880i \(0.363826\pi\)
\(752\) −8.00000 −0.291730
\(753\) −25.6155 −0.933482
\(754\) −38.0000 −1.38388
\(755\) 15.4384 0.561863
\(756\) 1.00000 0.0363696
\(757\) −14.9460 −0.543223 −0.271611 0.962407i \(-0.587556\pi\)
−0.271611 + 0.962407i \(0.587556\pi\)
\(758\) −2.87689 −0.104494
\(759\) −40.1771 −1.45834
\(760\) −0.561553 −0.0203697
\(761\) 17.3153 0.627681 0.313840 0.949476i \(-0.398384\pi\)
0.313840 + 0.949476i \(0.398384\pi\)
\(762\) −14.5616 −0.527509
\(763\) 2.12311 0.0768616
\(764\) 14.3693 0.519864
\(765\) 5.00000 0.180775
\(766\) −27.6847 −1.00029
\(767\) 24.1383 0.871582
\(768\) −1.00000 −0.0360844
\(769\) −9.50758 −0.342852 −0.171426 0.985197i \(-0.554837\pi\)
−0.171426 + 0.985197i \(0.554837\pi\)
\(770\) −6.12311 −0.220661
\(771\) −10.8078 −0.389232
\(772\) −14.0000 −0.503871
\(773\) −3.49242 −0.125614 −0.0628069 0.998026i \(-0.520005\pi\)
−0.0628069 + 0.998026i \(0.520005\pi\)
\(774\) −12.6847 −0.455941
\(775\) −3.56155 −0.127935
\(776\) 3.31534 0.119014
\(777\) 1.00000 0.0358748
\(778\) 20.9309 0.750408
\(779\) 4.87689 0.174733
\(780\) 5.68466 0.203543
\(781\) −6.87689 −0.246075
\(782\) 32.8078 1.17320
\(783\) 6.68466 0.238890
\(784\) −6.00000 −0.214286
\(785\) 18.0540 0.644374
\(786\) −7.12311 −0.254073
\(787\) 26.8769 0.958058 0.479029 0.877799i \(-0.340989\pi\)
0.479029 + 0.877799i \(0.340989\pi\)
\(788\) −2.31534 −0.0824806
\(789\) 6.43845 0.229215
\(790\) −14.2462 −0.506857
\(791\) 10.6847 0.379903
\(792\) −6.12311 −0.217575
\(793\) −78.4924 −2.78735
\(794\) 27.1231 0.962563
\(795\) 5.24621 0.186064
\(796\) −16.4924 −0.584558
\(797\) −25.6155 −0.907349 −0.453674 0.891168i \(-0.649887\pi\)
−0.453674 + 0.891168i \(0.649887\pi\)
\(798\) 0.561553 0.0198788
\(799\) 40.0000 1.41510
\(800\) 1.00000 0.0353553
\(801\) 5.68466 0.200858
\(802\) −14.8078 −0.522880
\(803\) 40.1771 1.41782
\(804\) −13.1231 −0.462816
\(805\) −6.56155 −0.231264
\(806\) −20.2462 −0.713142
\(807\) 16.8078 0.591661
\(808\) 10.8769 0.382648
\(809\) −25.4384 −0.894368 −0.447184 0.894442i \(-0.647573\pi\)
−0.447184 + 0.894442i \(0.647573\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 2.73863 0.0961664 0.0480832 0.998843i \(-0.484689\pi\)
0.0480832 + 0.998843i \(0.484689\pi\)
\(812\) 6.68466 0.234586
\(813\) 16.4924 0.578415
\(814\) −6.12311 −0.214615
\(815\) −19.4924 −0.682790
\(816\) 5.00000 0.175035
\(817\) −7.12311 −0.249206
\(818\) −0.876894 −0.0306599
\(819\) −5.68466 −0.198638
\(820\) −8.68466 −0.303281
\(821\) 23.4384 0.818007 0.409004 0.912533i \(-0.365876\pi\)
0.409004 + 0.912533i \(0.365876\pi\)
\(822\) 4.24621 0.148104
\(823\) 31.5464 1.09964 0.549819 0.835284i \(-0.314697\pi\)
0.549819 + 0.835284i \(0.314697\pi\)
\(824\) −16.4924 −0.574541
\(825\) 6.12311 0.213179
\(826\) −4.24621 −0.147745
\(827\) −10.4384 −0.362980 −0.181490 0.983393i \(-0.558092\pi\)
−0.181490 + 0.983393i \(0.558092\pi\)
\(828\) −6.56155 −0.228030
\(829\) −22.1231 −0.768367 −0.384184 0.923257i \(-0.625517\pi\)
−0.384184 + 0.923257i \(0.625517\pi\)
\(830\) 7.43845 0.258192
\(831\) −27.6847 −0.960370
\(832\) 5.68466 0.197080
\(833\) 30.0000 1.03944
\(834\) −13.8078 −0.478124
\(835\) −19.0540 −0.659390
\(836\) −3.43845 −0.118921
\(837\) 3.56155 0.123105
\(838\) −10.5616 −0.364843
\(839\) 28.2462 0.975168 0.487584 0.873076i \(-0.337878\pi\)
0.487584 + 0.873076i \(0.337878\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 15.6847 0.540850
\(842\) 30.9848 1.06781
\(843\) −22.1771 −0.763819
\(844\) 4.43845 0.152778
\(845\) −19.3153 −0.664468
\(846\) −8.00000 −0.275046
\(847\) −26.4924 −0.910290
\(848\) 5.24621 0.180156
\(849\) −9.93087 −0.340827
\(850\) −5.00000 −0.171499
\(851\) −6.56155 −0.224927
\(852\) −1.12311 −0.0384770
\(853\) 7.93087 0.271548 0.135774 0.990740i \(-0.456648\pi\)
0.135774 + 0.990740i \(0.456648\pi\)
\(854\) 13.8078 0.472492
\(855\) −0.561553 −0.0192047
\(856\) −10.8078 −0.369402
\(857\) −8.75379 −0.299024 −0.149512 0.988760i \(-0.547770\pi\)
−0.149512 + 0.988760i \(0.547770\pi\)
\(858\) 34.8078 1.18832
\(859\) −2.80776 −0.0957997 −0.0478998 0.998852i \(-0.515253\pi\)
−0.0478998 + 0.998852i \(0.515253\pi\)
\(860\) 12.6847 0.432543
\(861\) 8.68466 0.295972
\(862\) −29.0000 −0.987744
\(863\) 20.1922 0.687352 0.343676 0.939088i \(-0.388328\pi\)
0.343676 + 0.939088i \(0.388328\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 2.12311 0.0721878
\(866\) −30.4233 −1.03383
\(867\) −8.00000 −0.271694
\(868\) 3.56155 0.120887
\(869\) −87.2311 −2.95911
\(870\) −6.68466 −0.226631
\(871\) 74.6004 2.52774
\(872\) −2.12311 −0.0718974
\(873\) 3.31534 0.112207
\(874\) −3.68466 −0.124635
\(875\) 1.00000 0.0338062
\(876\) 6.56155 0.221694
\(877\) 46.9309 1.58474 0.792371 0.610039i \(-0.208846\pi\)
0.792371 + 0.610039i \(0.208846\pi\)
\(878\) −14.0540 −0.474299
\(879\) 16.6155 0.560428
\(880\) 6.12311 0.206410
\(881\) 34.9309 1.17685 0.588425 0.808551i \(-0.299748\pi\)
0.588425 + 0.808551i \(0.299748\pi\)
\(882\) −6.00000 −0.202031
\(883\) −14.3693 −0.483566 −0.241783 0.970330i \(-0.577732\pi\)
−0.241783 + 0.970330i \(0.577732\pi\)
\(884\) −28.4233 −0.955979
\(885\) 4.24621 0.142735
\(886\) 33.8617 1.13761
\(887\) −17.8078 −0.597926 −0.298963 0.954265i \(-0.596641\pi\)
−0.298963 + 0.954265i \(0.596641\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −14.5616 −0.488379
\(890\) −5.68466 −0.190550
\(891\) −6.12311 −0.205132
\(892\) −19.8078 −0.663213
\(893\) −4.49242 −0.150333
\(894\) −7.75379 −0.259325
\(895\) 4.00000 0.133705
\(896\) −1.00000 −0.0334077
\(897\) 37.3002 1.24542
\(898\) −1.61553 −0.0539109
\(899\) 23.8078 0.794033
\(900\) 1.00000 0.0333333
\(901\) −26.2311 −0.873883
\(902\) −53.1771 −1.77060
\(903\) −12.6847 −0.422119
\(904\) −10.6847 −0.355366
\(905\) −20.2462 −0.673007
\(906\) 15.4384 0.512908
\(907\) 45.4384 1.50876 0.754379 0.656439i \(-0.227938\pi\)
0.754379 + 0.656439i \(0.227938\pi\)
\(908\) −10.9309 −0.362754
\(909\) 10.8769 0.360764
\(910\) 5.68466 0.188445
\(911\) 20.4924 0.678944 0.339472 0.940616i \(-0.389752\pi\)
0.339472 + 0.940616i \(0.389752\pi\)
\(912\) −0.561553 −0.0185949
\(913\) 45.5464 1.50737
\(914\) −30.9309 −1.02310
\(915\) −13.8078 −0.456471
\(916\) −5.12311 −0.169272
\(917\) −7.12311 −0.235226
\(918\) 5.00000 0.165025
\(919\) 6.73863 0.222287 0.111144 0.993804i \(-0.464549\pi\)
0.111144 + 0.993804i \(0.464549\pi\)
\(920\) 6.56155 0.216328
\(921\) 0.246211 0.00811294
\(922\) 0.438447 0.0144395
\(923\) 6.38447 0.210147
\(924\) −6.12311 −0.201435
\(925\) 1.00000 0.0328798
\(926\) 4.87689 0.160265
\(927\) −16.4924 −0.541682
\(928\) −6.68466 −0.219435
\(929\) −5.80776 −0.190547 −0.0952733 0.995451i \(-0.530372\pi\)
−0.0952733 + 0.995451i \(0.530372\pi\)
\(930\) −3.56155 −0.116788
\(931\) −3.36932 −0.110425
\(932\) 11.3693 0.372414
\(933\) 21.1771 0.693307
\(934\) 2.43845 0.0797884
\(935\) −30.6155 −1.00123
\(936\) 5.68466 0.185809
\(937\) −0.630683 −0.0206035 −0.0103018 0.999947i \(-0.503279\pi\)
−0.0103018 + 0.999947i \(0.503279\pi\)
\(938\) −13.1231 −0.428485
\(939\) 24.8769 0.811827
\(940\) 8.00000 0.260931
\(941\) −8.24621 −0.268819 −0.134409 0.990926i \(-0.542914\pi\)
−0.134409 + 0.990926i \(0.542914\pi\)
\(942\) 18.0540 0.588230
\(943\) −56.9848 −1.85568
\(944\) 4.24621 0.138202
\(945\) −1.00000 −0.0325300
\(946\) 77.6695 2.52525
\(947\) 16.3002 0.529685 0.264842 0.964292i \(-0.414680\pi\)
0.264842 + 0.964292i \(0.414680\pi\)
\(948\) −14.2462 −0.462695
\(949\) −37.3002 −1.21082
\(950\) 0.561553 0.0182192
\(951\) −16.9309 −0.549021
\(952\) 5.00000 0.162051
\(953\) −1.61553 −0.0523321 −0.0261660 0.999658i \(-0.508330\pi\)
−0.0261660 + 0.999658i \(0.508330\pi\)
\(954\) 5.24621 0.169852
\(955\) −14.3693 −0.464980
\(956\) 7.80776 0.252521
\(957\) −40.9309 −1.32311
\(958\) −6.56155 −0.211994
\(959\) 4.24621 0.137117
\(960\) 1.00000 0.0322749
\(961\) −18.3153 −0.590817
\(962\) 5.68466 0.183281
\(963\) −10.8078 −0.348275
\(964\) −2.00000 −0.0644157
\(965\) 14.0000 0.450676
\(966\) −6.56155 −0.211115
\(967\) 32.6307 1.04933 0.524666 0.851308i \(-0.324190\pi\)
0.524666 + 0.851308i \(0.324190\pi\)
\(968\) 26.4924 0.851499
\(969\) 2.80776 0.0901984
\(970\) −3.31534 −0.106449
\(971\) −15.8078 −0.507295 −0.253648 0.967297i \(-0.581630\pi\)
−0.253648 + 0.967297i \(0.581630\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −13.8078 −0.442657
\(974\) 16.7386 0.536340
\(975\) −5.68466 −0.182055
\(976\) −13.8078 −0.441976
\(977\) −12.5076 −0.400153 −0.200076 0.979780i \(-0.564119\pi\)
−0.200076 + 0.979780i \(0.564119\pi\)
\(978\) −19.4924 −0.623299
\(979\) −34.8078 −1.11246
\(980\) 6.00000 0.191663
\(981\) −2.12311 −0.0677855
\(982\) −12.1771 −0.388586
\(983\) −24.9309 −0.795171 −0.397586 0.917565i \(-0.630152\pi\)
−0.397586 + 0.917565i \(0.630152\pi\)
\(984\) −8.68466 −0.276857
\(985\) 2.31534 0.0737729
\(986\) 33.4233 1.06441
\(987\) −8.00000 −0.254643
\(988\) 3.19224 0.101559
\(989\) 83.2311 2.64659
\(990\) 6.12311 0.194605
\(991\) 3.42329 0.108744 0.0543722 0.998521i \(-0.482684\pi\)
0.0543722 + 0.998521i \(0.482684\pi\)
\(992\) −3.56155 −0.113079
\(993\) 12.0000 0.380808
\(994\) −1.12311 −0.0356227
\(995\) 16.4924 0.522845
\(996\) 7.43845 0.235696
\(997\) 19.3002 0.611243 0.305622 0.952153i \(-0.401136\pi\)
0.305622 + 0.952153i \(0.401136\pi\)
\(998\) 32.8078 1.03851
\(999\) −1.00000 −0.0316386
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 1110.2.a.r.1.1 2
3.2 odd 2 3330.2.a.bc.1.2 2
4.3 odd 2 8880.2.a.bp.1.2 2
5.4 even 2 5550.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.r.1.1 2 1.1 even 1 trivial
3330.2.a.bc.1.2 2 3.2 odd 2
5550.2.a.bw.1.1 2 5.4 even 2
8880.2.a.bp.1.2 2 4.3 odd 2