Properties

Label 731.2.a.c.1.5
Level 731
Weight 2
Character 731.1
Self dual yes
Analytic conductor 5.837
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.83706438776\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.2460365.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.668929\)
Character \(\chi\) = 731.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.55253 q^{2} +0.467968 q^{3} +0.410361 q^{4} -1.37639 q^{5} +0.726536 q^{6} -3.35157 q^{7} -2.46797 q^{8} -2.78101 q^{9} +O(q^{10})\) \(q+1.55253 q^{2} +0.467968 q^{3} +0.410361 q^{4} -1.37639 q^{5} +0.726536 q^{6} -3.35157 q^{7} -2.46797 q^{8} -2.78101 q^{9} -2.13690 q^{10} +2.05761 q^{11} +0.192036 q^{12} -1.60557 q^{13} -5.20343 q^{14} -0.644108 q^{15} -4.65233 q^{16} -1.00000 q^{17} -4.31761 q^{18} +4.43647 q^{19} -0.564818 q^{20} -1.56843 q^{21} +3.19450 q^{22} -2.95501 q^{23} -1.15493 q^{24} -3.10554 q^{25} -2.49271 q^{26} -2.70533 q^{27} -1.37535 q^{28} -3.13539 q^{29} -1.00000 q^{30} -4.72687 q^{31} -2.28696 q^{32} +0.962895 q^{33} -1.55253 q^{34} +4.61308 q^{35} -1.14122 q^{36} +6.73408 q^{37} +6.88777 q^{38} -0.751357 q^{39} +3.39690 q^{40} +9.28079 q^{41} -2.43504 q^{42} -1.00000 q^{43} +0.844362 q^{44} +3.82776 q^{45} -4.58775 q^{46} -8.52365 q^{47} -2.17714 q^{48} +4.23304 q^{49} -4.82146 q^{50} -0.467968 q^{51} -0.658865 q^{52} +9.33387 q^{53} -4.20011 q^{54} -2.83208 q^{55} +8.27157 q^{56} +2.07613 q^{57} -4.86779 q^{58} -9.83672 q^{59} -0.264317 q^{60} +2.75383 q^{61} -7.33862 q^{62} +9.32074 q^{63} +5.75407 q^{64} +2.20990 q^{65} +1.49493 q^{66} +8.84901 q^{67} -0.410361 q^{68} -1.38285 q^{69} +7.16197 q^{70} -1.37235 q^{71} +6.86343 q^{72} -0.237911 q^{73} +10.4549 q^{74} -1.45329 q^{75} +1.82055 q^{76} -6.89622 q^{77} -1.16651 q^{78} -7.25698 q^{79} +6.40343 q^{80} +7.07701 q^{81} +14.4087 q^{82} -2.15336 q^{83} -0.643622 q^{84} +1.37639 q^{85} -1.55253 q^{86} -1.46726 q^{87} -5.07811 q^{88} -13.4194 q^{89} +5.94272 q^{90} +5.38120 q^{91} -1.21262 q^{92} -2.21202 q^{93} -13.2332 q^{94} -6.10633 q^{95} -1.07022 q^{96} -12.3738 q^{97} +6.57194 q^{98} -5.72222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 3q^{3} + 5q^{4} + 3q^{5} - 7q^{6} - 7q^{7} - 9q^{8} - 3q^{9} + O(q^{10}) \) \( 6q - q^{2} - 3q^{3} + 5q^{4} + 3q^{5} - 7q^{6} - 7q^{7} - 9q^{8} - 3q^{9} - 4q^{10} + 4q^{11} + 11q^{12} - 10q^{13} - 7q^{14} + q^{15} - q^{16} - 6q^{17} + q^{18} - 20q^{19} + q^{20} - 8q^{21} + 2q^{22} - 3q^{23} - 9q^{24} - 7q^{25} - 3q^{26} - 6q^{27} - 11q^{28} - 15q^{29} - 6q^{30} + 12q^{31} + q^{32} - 2q^{33} + q^{34} - 9q^{35} - 16q^{36} - 14q^{37} + 27q^{38} + 5q^{39} - 7q^{40} - 2q^{41} + 19q^{42} - 6q^{43} - 12q^{44} - 18q^{45} + 14q^{46} - 11q^{47} - 6q^{48} + 3q^{49} + 7q^{50} + 3q^{51} - 5q^{52} + 3q^{53} + 25q^{54} + 6q^{55} + 22q^{56} + 11q^{57} - 21q^{58} + 2q^{59} - q^{60} - 20q^{61} + 3q^{62} + 23q^{63} - 39q^{64} - 34q^{65} + 7q^{66} - 2q^{67} - 5q^{68} - 17q^{69} - q^{70} + q^{71} + 21q^{72} + 13q^{73} + 28q^{74} - 5q^{75} - 29q^{76} - 11q^{77} - 26q^{79} + 12q^{80} + 2q^{81} - 9q^{82} + 10q^{83} - 3q^{84} - 3q^{85} + q^{86} + 12q^{87} - 6q^{88} - 15q^{89} + 15q^{90} + 8q^{91} - 9q^{92} - 11q^{93} - 33q^{94} - 21q^{95} + 25q^{96} - 22q^{97} - 3q^{98} - 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.55253 1.09781 0.548904 0.835886i \(-0.315045\pi\)
0.548904 + 0.835886i \(0.315045\pi\)
\(3\) 0.467968 0.270182 0.135091 0.990833i \(-0.456867\pi\)
0.135091 + 0.990833i \(0.456867\pi\)
\(4\) 0.410361 0.205180
\(5\) −1.37639 −0.615542 −0.307771 0.951460i \(-0.599583\pi\)
−0.307771 + 0.951460i \(0.599583\pi\)
\(6\) 0.726536 0.296607
\(7\) −3.35157 −1.26678 −0.633388 0.773835i \(-0.718336\pi\)
−0.633388 + 0.773835i \(0.718336\pi\)
\(8\) −2.46797 −0.872559
\(9\) −2.78101 −0.927002
\(10\) −2.13690 −0.675746
\(11\) 2.05761 0.620392 0.310196 0.950673i \(-0.399605\pi\)
0.310196 + 0.950673i \(0.399605\pi\)
\(12\) 0.192036 0.0554360
\(13\) −1.60557 −0.445306 −0.222653 0.974898i \(-0.571472\pi\)
−0.222653 + 0.974898i \(0.571472\pi\)
\(14\) −5.20343 −1.39068
\(15\) −0.644108 −0.166308
\(16\) −4.65233 −1.16308
\(17\) −1.00000 −0.242536
\(18\) −4.31761 −1.01767
\(19\) 4.43647 1.01780 0.508898 0.860827i \(-0.330053\pi\)
0.508898 + 0.860827i \(0.330053\pi\)
\(20\) −0.564818 −0.126297
\(21\) −1.56843 −0.342259
\(22\) 3.19450 0.681071
\(23\) −2.95501 −0.616162 −0.308081 0.951360i \(-0.599687\pi\)
−0.308081 + 0.951360i \(0.599687\pi\)
\(24\) −1.15493 −0.235749
\(25\) −3.10554 −0.621108
\(26\) −2.49271 −0.488860
\(27\) −2.70533 −0.520640
\(28\) −1.37535 −0.259918
\(29\) −3.13539 −0.582226 −0.291113 0.956689i \(-0.594026\pi\)
−0.291113 + 0.956689i \(0.594026\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.72687 −0.848970 −0.424485 0.905435i \(-0.639545\pi\)
−0.424485 + 0.905435i \(0.639545\pi\)
\(32\) −2.28696 −0.404281
\(33\) 0.962895 0.167618
\(34\) −1.55253 −0.266257
\(35\) 4.61308 0.779753
\(36\) −1.14122 −0.190203
\(37\) 6.73408 1.10708 0.553538 0.832824i \(-0.313277\pi\)
0.553538 + 0.832824i \(0.313277\pi\)
\(38\) 6.88777 1.11734
\(39\) −0.751357 −0.120313
\(40\) 3.39690 0.537096
\(41\) 9.28079 1.44942 0.724708 0.689056i \(-0.241975\pi\)
0.724708 + 0.689056i \(0.241975\pi\)
\(42\) −2.43504 −0.375735
\(43\) −1.00000 −0.152499
\(44\) 0.844362 0.127292
\(45\) 3.82776 0.570609
\(46\) −4.58775 −0.676427
\(47\) −8.52365 −1.24330 −0.621651 0.783295i \(-0.713538\pi\)
−0.621651 + 0.783295i \(0.713538\pi\)
\(48\) −2.17714 −0.314243
\(49\) 4.23304 0.604720
\(50\) −4.82146 −0.681857
\(51\) −0.467968 −0.0655287
\(52\) −0.658865 −0.0913681
\(53\) 9.33387 1.28211 0.641053 0.767497i \(-0.278498\pi\)
0.641053 + 0.767497i \(0.278498\pi\)
\(54\) −4.20011 −0.571563
\(55\) −2.83208 −0.381877
\(56\) 8.27157 1.10534
\(57\) 2.07613 0.274990
\(58\) −4.86779 −0.639172
\(59\) −9.83672 −1.28063 −0.640316 0.768111i \(-0.721197\pi\)
−0.640316 + 0.768111i \(0.721197\pi\)
\(60\) −0.264317 −0.0341232
\(61\) 2.75383 0.352591 0.176296 0.984337i \(-0.443589\pi\)
0.176296 + 0.984337i \(0.443589\pi\)
\(62\) −7.33862 −0.932006
\(63\) 9.32074 1.17430
\(64\) 5.75407 0.719259
\(65\) 2.20990 0.274104
\(66\) 1.49493 0.184013
\(67\) 8.84901 1.08108 0.540539 0.841319i \(-0.318220\pi\)
0.540539 + 0.841319i \(0.318220\pi\)
\(68\) −0.410361 −0.0497636
\(69\) −1.38285 −0.166476
\(70\) 7.16197 0.856019
\(71\) −1.37235 −0.162868 −0.0814342 0.996679i \(-0.525950\pi\)
−0.0814342 + 0.996679i \(0.525950\pi\)
\(72\) 6.86343 0.808863
\(73\) −0.237911 −0.0278454 −0.0139227 0.999903i \(-0.504432\pi\)
−0.0139227 + 0.999903i \(0.504432\pi\)
\(74\) 10.4549 1.21536
\(75\) −1.45329 −0.167812
\(76\) 1.82055 0.208832
\(77\) −6.89622 −0.785897
\(78\) −1.16651 −0.132081
\(79\) −7.25698 −0.816474 −0.408237 0.912876i \(-0.633856\pi\)
−0.408237 + 0.912876i \(0.633856\pi\)
\(80\) 6.40343 0.715925
\(81\) 7.07701 0.786334
\(82\) 14.4087 1.59118
\(83\) −2.15336 −0.236362 −0.118181 0.992992i \(-0.537706\pi\)
−0.118181 + 0.992992i \(0.537706\pi\)
\(84\) −0.643622 −0.0702249
\(85\) 1.37639 0.149291
\(86\) −1.55253 −0.167414
\(87\) −1.46726 −0.157307
\(88\) −5.07811 −0.541328
\(89\) −13.4194 −1.42246 −0.711229 0.702961i \(-0.751861\pi\)
−0.711229 + 0.702961i \(0.751861\pi\)
\(90\) 5.94272 0.626418
\(91\) 5.38120 0.564103
\(92\) −1.21262 −0.126424
\(93\) −2.21202 −0.229376
\(94\) −13.2332 −1.36491
\(95\) −6.10633 −0.626496
\(96\) −1.07022 −0.109229
\(97\) −12.3738 −1.25636 −0.628182 0.778066i \(-0.716201\pi\)
−0.628182 + 0.778066i \(0.716201\pi\)
\(98\) 6.57194 0.663866
\(99\) −5.72222 −0.575104
\(100\) −1.27439 −0.127439
\(101\) −16.9693 −1.68850 −0.844252 0.535946i \(-0.819955\pi\)
−0.844252 + 0.535946i \(0.819955\pi\)
\(102\) −0.726536 −0.0719378
\(103\) 1.00189 0.0987190 0.0493595 0.998781i \(-0.484282\pi\)
0.0493595 + 0.998781i \(0.484282\pi\)
\(104\) 3.96250 0.388555
\(105\) 2.15878 0.210675
\(106\) 14.4911 1.40750
\(107\) 17.2371 1.66637 0.833185 0.552994i \(-0.186515\pi\)
0.833185 + 0.552994i \(0.186515\pi\)
\(108\) −1.11016 −0.106825
\(109\) −9.48121 −0.908135 −0.454068 0.890967i \(-0.650028\pi\)
−0.454068 + 0.890967i \(0.650028\pi\)
\(110\) −4.39690 −0.419228
\(111\) 3.15134 0.299112
\(112\) 15.5926 1.47336
\(113\) −5.69643 −0.535875 −0.267937 0.963436i \(-0.586342\pi\)
−0.267937 + 0.963436i \(0.586342\pi\)
\(114\) 3.22326 0.301886
\(115\) 4.06725 0.379273
\(116\) −1.28664 −0.119462
\(117\) 4.46511 0.412799
\(118\) −15.2718 −1.40589
\(119\) 3.35157 0.307238
\(120\) 1.58964 0.145114
\(121\) −6.76625 −0.615114
\(122\) 4.27541 0.387077
\(123\) 4.34312 0.391606
\(124\) −1.93972 −0.174192
\(125\) 11.1564 0.997860
\(126\) 14.4708 1.28916
\(127\) −9.03729 −0.801930 −0.400965 0.916093i \(-0.631325\pi\)
−0.400965 + 0.916093i \(0.631325\pi\)
\(128\) 13.5073 1.19389
\(129\) −0.467968 −0.0412023
\(130\) 3.43095 0.300914
\(131\) 13.2079 1.15398 0.576990 0.816751i \(-0.304227\pi\)
0.576990 + 0.816751i \(0.304227\pi\)
\(132\) 0.395134 0.0343920
\(133\) −14.8691 −1.28932
\(134\) 13.7384 1.18682
\(135\) 3.72359 0.320476
\(136\) 2.46797 0.211627
\(137\) 0.498421 0.0425830 0.0212915 0.999773i \(-0.493222\pi\)
0.0212915 + 0.999773i \(0.493222\pi\)
\(138\) −2.14692 −0.182758
\(139\) 13.0119 1.10366 0.551829 0.833957i \(-0.313930\pi\)
0.551829 + 0.833957i \(0.313930\pi\)
\(140\) 1.89303 0.159990
\(141\) −3.98880 −0.335917
\(142\) −2.13062 −0.178798
\(143\) −3.30364 −0.276264
\(144\) 12.9381 1.07818
\(145\) 4.31552 0.358385
\(146\) −0.369365 −0.0305689
\(147\) 1.98093 0.163384
\(148\) 2.76341 0.227151
\(149\) −18.6715 −1.52963 −0.764813 0.644253i \(-0.777168\pi\)
−0.764813 + 0.644253i \(0.777168\pi\)
\(150\) −2.25629 −0.184225
\(151\) −14.9169 −1.21392 −0.606960 0.794733i \(-0.707611\pi\)
−0.606960 + 0.794733i \(0.707611\pi\)
\(152\) −10.9491 −0.888086
\(153\) 2.78101 0.224831
\(154\) −10.7066 −0.862764
\(155\) 6.50603 0.522577
\(156\) −0.308328 −0.0246860
\(157\) 19.2406 1.53556 0.767782 0.640711i \(-0.221360\pi\)
0.767782 + 0.640711i \(0.221360\pi\)
\(158\) −11.2667 −0.896331
\(159\) 4.36795 0.346401
\(160\) 3.14775 0.248852
\(161\) 9.90393 0.780539
\(162\) 10.9873 0.863244
\(163\) 6.92729 0.542587 0.271294 0.962497i \(-0.412549\pi\)
0.271294 + 0.962497i \(0.412549\pi\)
\(164\) 3.80848 0.297392
\(165\) −1.32532 −0.103176
\(166\) −3.34316 −0.259480
\(167\) 11.3060 0.874881 0.437441 0.899247i \(-0.355885\pi\)
0.437441 + 0.899247i \(0.355885\pi\)
\(168\) 3.87083 0.298641
\(169\) −10.4221 −0.801703
\(170\) 2.13690 0.163893
\(171\) −12.3378 −0.943499
\(172\) −0.410361 −0.0312897
\(173\) 18.8950 1.43656 0.718281 0.695754i \(-0.244929\pi\)
0.718281 + 0.695754i \(0.244929\pi\)
\(174\) −2.27797 −0.172693
\(175\) 10.4084 0.786805
\(176\) −9.57266 −0.721566
\(177\) −4.60327 −0.346003
\(178\) −20.8341 −1.56158
\(179\) −16.1432 −1.20660 −0.603300 0.797515i \(-0.706148\pi\)
−0.603300 + 0.797515i \(0.706148\pi\)
\(180\) 1.57076 0.117078
\(181\) −17.6672 −1.31319 −0.656596 0.754242i \(-0.728004\pi\)
−0.656596 + 0.754242i \(0.728004\pi\)
\(182\) 8.35449 0.619276
\(183\) 1.28870 0.0952637
\(184\) 7.29287 0.537637
\(185\) −9.26875 −0.681452
\(186\) −3.43424 −0.251811
\(187\) −2.05761 −0.150467
\(188\) −3.49777 −0.255101
\(189\) 9.06710 0.659534
\(190\) −9.48028 −0.687772
\(191\) −5.21087 −0.377045 −0.188523 0.982069i \(-0.560370\pi\)
−0.188523 + 0.982069i \(0.560370\pi\)
\(192\) 2.69272 0.194331
\(193\) −10.0275 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(194\) −19.2107 −1.37925
\(195\) 1.03416 0.0740580
\(196\) 1.73707 0.124077
\(197\) −8.23341 −0.586606 −0.293303 0.956020i \(-0.594754\pi\)
−0.293303 + 0.956020i \(0.594754\pi\)
\(198\) −8.88394 −0.631354
\(199\) −0.596092 −0.0422559 −0.0211279 0.999777i \(-0.506726\pi\)
−0.0211279 + 0.999777i \(0.506726\pi\)
\(200\) 7.66438 0.541953
\(201\) 4.14105 0.292087
\(202\) −26.3454 −1.85365
\(203\) 10.5085 0.737550
\(204\) −0.192036 −0.0134452
\(205\) −12.7740 −0.892176
\(206\) 1.55547 0.108374
\(207\) 8.21790 0.571183
\(208\) 7.46965 0.517927
\(209\) 9.12851 0.631432
\(210\) 3.35157 0.231280
\(211\) −2.11726 −0.145758 −0.0728790 0.997341i \(-0.523219\pi\)
−0.0728790 + 0.997341i \(0.523219\pi\)
\(212\) 3.83026 0.263063
\(213\) −0.642217 −0.0440040
\(214\) 26.7611 1.82935
\(215\) 1.37639 0.0938693
\(216\) 6.67666 0.454289
\(217\) 15.8424 1.07545
\(218\) −14.7199 −0.996958
\(219\) −0.111335 −0.00752332
\(220\) −1.16217 −0.0783538
\(221\) 1.60557 0.108003
\(222\) 4.89256 0.328367
\(223\) −21.3509 −1.42976 −0.714881 0.699247i \(-0.753519\pi\)
−0.714881 + 0.699247i \(0.753519\pi\)
\(224\) 7.66490 0.512133
\(225\) 8.63653 0.575768
\(226\) −8.84389 −0.588287
\(227\) 17.2277 1.14344 0.571720 0.820449i \(-0.306276\pi\)
0.571720 + 0.820449i \(0.306276\pi\)
\(228\) 0.851961 0.0564225
\(229\) −0.830329 −0.0548697 −0.0274348 0.999624i \(-0.508734\pi\)
−0.0274348 + 0.999624i \(0.508734\pi\)
\(230\) 6.31455 0.416369
\(231\) −3.22721 −0.212335
\(232\) 7.73803 0.508027
\(233\) −17.7159 −1.16061 −0.580303 0.814401i \(-0.697066\pi\)
−0.580303 + 0.814401i \(0.697066\pi\)
\(234\) 6.93223 0.453174
\(235\) 11.7319 0.765304
\(236\) −4.03661 −0.262761
\(237\) −3.39604 −0.220596
\(238\) 5.20343 0.337288
\(239\) −20.7223 −1.34041 −0.670207 0.742174i \(-0.733795\pi\)
−0.670207 + 0.742174i \(0.733795\pi\)
\(240\) 2.99660 0.193430
\(241\) 8.01971 0.516595 0.258297 0.966065i \(-0.416839\pi\)
0.258297 + 0.966065i \(0.416839\pi\)
\(242\) −10.5048 −0.675276
\(243\) 11.4278 0.733093
\(244\) 1.13006 0.0723449
\(245\) −5.82633 −0.372230
\(246\) 6.74283 0.429907
\(247\) −7.12308 −0.453231
\(248\) 11.6658 0.740776
\(249\) −1.00770 −0.0638606
\(250\) 17.3207 1.09546
\(251\) 12.0601 0.761228 0.380614 0.924734i \(-0.375713\pi\)
0.380614 + 0.924734i \(0.375713\pi\)
\(252\) 3.82487 0.240944
\(253\) −6.08025 −0.382262
\(254\) −14.0307 −0.880365
\(255\) 0.644108 0.0403356
\(256\) 9.46240 0.591400
\(257\) −0.113536 −0.00708217 −0.00354108 0.999994i \(-0.501127\pi\)
−0.00354108 + 0.999994i \(0.501127\pi\)
\(258\) −0.726536 −0.0452322
\(259\) −22.5698 −1.40242
\(260\) 0.906857 0.0562409
\(261\) 8.71952 0.539725
\(262\) 20.5057 1.26685
\(263\) −1.04330 −0.0643325 −0.0321663 0.999483i \(-0.510241\pi\)
−0.0321663 + 0.999483i \(0.510241\pi\)
\(264\) −2.37639 −0.146257
\(265\) −12.8471 −0.789190
\(266\) −23.0849 −1.41542
\(267\) −6.27987 −0.384322
\(268\) 3.63129 0.221816
\(269\) −3.25815 −0.198653 −0.0993265 0.995055i \(-0.531669\pi\)
−0.0993265 + 0.995055i \(0.531669\pi\)
\(270\) 5.78101 0.351821
\(271\) 2.16125 0.131286 0.0656432 0.997843i \(-0.479090\pi\)
0.0656432 + 0.997843i \(0.479090\pi\)
\(272\) 4.65233 0.282089
\(273\) 2.51823 0.152410
\(274\) 0.773816 0.0467479
\(275\) −6.38998 −0.385330
\(276\) −0.567468 −0.0341575
\(277\) −2.06077 −0.123820 −0.0619098 0.998082i \(-0.519719\pi\)
−0.0619098 + 0.998082i \(0.519719\pi\)
\(278\) 20.2015 1.21160
\(279\) 13.1454 0.786997
\(280\) −11.3849 −0.680380
\(281\) 19.6254 1.17075 0.585375 0.810763i \(-0.300947\pi\)
0.585375 + 0.810763i \(0.300947\pi\)
\(282\) −6.19274 −0.368772
\(283\) 29.8412 1.77388 0.886939 0.461888i \(-0.152828\pi\)
0.886939 + 0.461888i \(0.152828\pi\)
\(284\) −0.563160 −0.0334174
\(285\) −2.85757 −0.169268
\(286\) −5.12901 −0.303285
\(287\) −31.1053 −1.83608
\(288\) 6.36004 0.374769
\(289\) 1.00000 0.0588235
\(290\) 6.70000 0.393437
\(291\) −5.79052 −0.339447
\(292\) −0.0976295 −0.00571334
\(293\) 9.76035 0.570206 0.285103 0.958497i \(-0.407972\pi\)
0.285103 + 0.958497i \(0.407972\pi\)
\(294\) 3.07546 0.179364
\(295\) 13.5392 0.788283
\(296\) −16.6195 −0.965989
\(297\) −5.56650 −0.323001
\(298\) −28.9881 −1.67923
\(299\) 4.74448 0.274381
\(300\) −0.596375 −0.0344317
\(301\) 3.35157 0.193181
\(302\) −23.1590 −1.33265
\(303\) −7.94108 −0.456203
\(304\) −20.6399 −1.18378
\(305\) −3.79035 −0.217035
\(306\) 4.31761 0.246821
\(307\) 5.45230 0.311179 0.155590 0.987822i \(-0.450272\pi\)
0.155590 + 0.987822i \(0.450272\pi\)
\(308\) −2.82994 −0.161251
\(309\) 0.468852 0.0266721
\(310\) 10.1008 0.573689
\(311\) 6.91146 0.391913 0.195956 0.980613i \(-0.437219\pi\)
0.195956 + 0.980613i \(0.437219\pi\)
\(312\) 1.85433 0.104981
\(313\) −30.2823 −1.71166 −0.855830 0.517258i \(-0.826953\pi\)
−0.855830 + 0.517258i \(0.826953\pi\)
\(314\) 29.8716 1.68575
\(315\) −12.8290 −0.722833
\(316\) −2.97798 −0.167525
\(317\) −1.05835 −0.0594431 −0.0297215 0.999558i \(-0.509462\pi\)
−0.0297215 + 0.999558i \(0.509462\pi\)
\(318\) 6.78140 0.380282
\(319\) −6.45139 −0.361209
\(320\) −7.91987 −0.442734
\(321\) 8.06640 0.450223
\(322\) 15.3762 0.856881
\(323\) −4.43647 −0.246852
\(324\) 2.90413 0.161340
\(325\) 4.98617 0.276583
\(326\) 10.7549 0.595656
\(327\) −4.43691 −0.245361
\(328\) −22.9047 −1.26470
\(329\) 28.5676 1.57498
\(330\) −2.05761 −0.113268
\(331\) 6.60561 0.363077 0.181539 0.983384i \(-0.441892\pi\)
0.181539 + 0.983384i \(0.441892\pi\)
\(332\) −0.883654 −0.0484968
\(333\) −18.7275 −1.02626
\(334\) 17.5529 0.960451
\(335\) −12.1797 −0.665449
\(336\) 7.29684 0.398076
\(337\) −7.79916 −0.424847 −0.212424 0.977178i \(-0.568136\pi\)
−0.212424 + 0.977178i \(0.568136\pi\)
\(338\) −16.1807 −0.880115
\(339\) −2.66575 −0.144783
\(340\) 0.564818 0.0306316
\(341\) −9.72604 −0.526694
\(342\) −19.1549 −1.03578
\(343\) 9.27367 0.500731
\(344\) 2.46797 0.133064
\(345\) 1.90335 0.102473
\(346\) 29.3351 1.57707
\(347\) −9.84344 −0.528424 −0.264212 0.964465i \(-0.585112\pi\)
−0.264212 + 0.964465i \(0.585112\pi\)
\(348\) −0.602106 −0.0322763
\(349\) −30.2937 −1.62158 −0.810791 0.585335i \(-0.800963\pi\)
−0.810791 + 0.585335i \(0.800963\pi\)
\(350\) 16.1595 0.863760
\(351\) 4.34360 0.231844
\(352\) −4.70566 −0.250812
\(353\) −3.33858 −0.177695 −0.0888473 0.996045i \(-0.528318\pi\)
−0.0888473 + 0.996045i \(0.528318\pi\)
\(354\) −7.14674 −0.379845
\(355\) 1.88890 0.100252
\(356\) −5.50681 −0.291861
\(357\) 1.56843 0.0830101
\(358\) −25.0629 −1.32461
\(359\) 32.9629 1.73972 0.869858 0.493301i \(-0.164210\pi\)
0.869858 + 0.493301i \(0.164210\pi\)
\(360\) −9.44679 −0.497889
\(361\) 0.682261 0.0359085
\(362\) −27.4289 −1.44163
\(363\) −3.16639 −0.166192
\(364\) 2.20823 0.115743
\(365\) 0.327460 0.0171400
\(366\) 2.00075 0.104581
\(367\) 19.8773 1.03759 0.518793 0.854900i \(-0.326382\pi\)
0.518793 + 0.854900i \(0.326382\pi\)
\(368\) 13.7477 0.716646
\(369\) −25.8099 −1.34361
\(370\) −14.3900 −0.748103
\(371\) −31.2831 −1.62414
\(372\) −0.907728 −0.0470635
\(373\) 31.8286 1.64802 0.824011 0.566574i \(-0.191732\pi\)
0.824011 + 0.566574i \(0.191732\pi\)
\(374\) −3.19450 −0.165184
\(375\) 5.22085 0.269603
\(376\) 21.0361 1.08485
\(377\) 5.03409 0.259269
\(378\) 14.0770 0.724042
\(379\) −18.3892 −0.944590 −0.472295 0.881440i \(-0.656574\pi\)
−0.472295 + 0.881440i \(0.656574\pi\)
\(380\) −2.50580 −0.128545
\(381\) −4.22917 −0.216667
\(382\) −8.09005 −0.413923
\(383\) 31.4493 1.60699 0.803493 0.595314i \(-0.202972\pi\)
0.803493 + 0.595314i \(0.202972\pi\)
\(384\) 6.32099 0.322567
\(385\) 9.49191 0.483753
\(386\) −15.5681 −0.792394
\(387\) 2.78101 0.141366
\(388\) −5.07771 −0.257782
\(389\) −36.1960 −1.83521 −0.917604 0.397496i \(-0.869879\pi\)
−0.917604 + 0.397496i \(0.869879\pi\)
\(390\) 1.60557 0.0813014
\(391\) 2.95501 0.149441
\(392\) −10.4470 −0.527654
\(393\) 6.18088 0.311784
\(394\) −12.7826 −0.643980
\(395\) 9.98846 0.502574
\(396\) −2.34817 −0.118000
\(397\) −16.0708 −0.806569 −0.403284 0.915075i \(-0.632131\pi\)
−0.403284 + 0.915075i \(0.632131\pi\)
\(398\) −0.925454 −0.0463888
\(399\) −6.95829 −0.348350
\(400\) 14.4480 0.722399
\(401\) 18.3382 0.915766 0.457883 0.889013i \(-0.348608\pi\)
0.457883 + 0.889013i \(0.348608\pi\)
\(402\) 6.42913 0.320656
\(403\) 7.58933 0.378052
\(404\) −6.96352 −0.346448
\(405\) −9.74075 −0.484022
\(406\) 16.3148 0.809688
\(407\) 13.8561 0.686821
\(408\) 1.15493 0.0571776
\(409\) −21.6593 −1.07098 −0.535491 0.844541i \(-0.679874\pi\)
−0.535491 + 0.844541i \(0.679874\pi\)
\(410\) −19.8321 −0.979438
\(411\) 0.233245 0.0115051
\(412\) 0.411136 0.0202552
\(413\) 32.9685 1.62227
\(414\) 12.7586 0.627049
\(415\) 2.96387 0.145491
\(416\) 3.67188 0.180029
\(417\) 6.08918 0.298188
\(418\) 14.1723 0.693191
\(419\) −0.799500 −0.0390581 −0.0195291 0.999809i \(-0.506217\pi\)
−0.0195291 + 0.999809i \(0.506217\pi\)
\(420\) 0.885877 0.0432264
\(421\) −21.4459 −1.04521 −0.522604 0.852576i \(-0.675039\pi\)
−0.522604 + 0.852576i \(0.675039\pi\)
\(422\) −3.28711 −0.160014
\(423\) 23.7043 1.15254
\(424\) −23.0357 −1.11871
\(425\) 3.10554 0.150641
\(426\) −0.997064 −0.0483079
\(427\) −9.22965 −0.446654
\(428\) 7.07342 0.341907
\(429\) −1.54600 −0.0746415
\(430\) 2.13690 0.103050
\(431\) −12.5374 −0.603905 −0.301953 0.953323i \(-0.597638\pi\)
−0.301953 + 0.953323i \(0.597638\pi\)
\(432\) 12.5861 0.605547
\(433\) 0.316199 0.0151956 0.00759778 0.999971i \(-0.497582\pi\)
0.00759778 + 0.999971i \(0.497582\pi\)
\(434\) 24.5959 1.18064
\(435\) 2.01953 0.0968290
\(436\) −3.89072 −0.186332
\(437\) −13.1098 −0.627127
\(438\) −0.172851 −0.00825915
\(439\) −39.0690 −1.86466 −0.932331 0.361607i \(-0.882228\pi\)
−0.932331 + 0.361607i \(0.882228\pi\)
\(440\) 6.98948 0.333210
\(441\) −11.7721 −0.560577
\(442\) 2.49271 0.118566
\(443\) 2.92193 0.138825 0.0694125 0.997588i \(-0.477888\pi\)
0.0694125 + 0.997588i \(0.477888\pi\)
\(444\) 1.29319 0.0613719
\(445\) 18.4704 0.875582
\(446\) −33.1480 −1.56960
\(447\) −8.73765 −0.413277
\(448\) −19.2852 −0.911140
\(449\) 3.14960 0.148639 0.0743195 0.997234i \(-0.476322\pi\)
0.0743195 + 0.997234i \(0.476322\pi\)
\(450\) 13.4085 0.632083
\(451\) 19.0962 0.899206
\(452\) −2.33759 −0.109951
\(453\) −6.98063 −0.327979
\(454\) 26.7465 1.25528
\(455\) −7.40664 −0.347229
\(456\) −5.12381 −0.239945
\(457\) −31.7688 −1.48608 −0.743040 0.669247i \(-0.766617\pi\)
−0.743040 + 0.669247i \(0.766617\pi\)
\(458\) −1.28911 −0.0602363
\(459\) 2.70533 0.126274
\(460\) 1.66904 0.0778195
\(461\) 27.5045 1.28101 0.640505 0.767954i \(-0.278725\pi\)
0.640505 + 0.767954i \(0.278725\pi\)
\(462\) −5.01035 −0.233103
\(463\) −6.28353 −0.292020 −0.146010 0.989283i \(-0.546643\pi\)
−0.146010 + 0.989283i \(0.546643\pi\)
\(464\) 14.5868 0.677177
\(465\) 3.04462 0.141191
\(466\) −27.5045 −1.27412
\(467\) −15.1305 −0.700158 −0.350079 0.936720i \(-0.613845\pi\)
−0.350079 + 0.936720i \(0.613845\pi\)
\(468\) 1.83231 0.0846984
\(469\) −29.6581 −1.36948
\(470\) 18.2142 0.840156
\(471\) 9.00397 0.414881
\(472\) 24.2767 1.11743
\(473\) −2.05761 −0.0946089
\(474\) −5.27246 −0.242172
\(475\) −13.7776 −0.632161
\(476\) 1.37535 0.0630393
\(477\) −25.9575 −1.18851
\(478\) −32.1721 −1.47152
\(479\) 18.3840 0.839984 0.419992 0.907528i \(-0.362033\pi\)
0.419992 + 0.907528i \(0.362033\pi\)
\(480\) 1.47305 0.0672351
\(481\) −10.8121 −0.492988
\(482\) 12.4509 0.567121
\(483\) 4.63472 0.210887
\(484\) −2.77661 −0.126209
\(485\) 17.0312 0.773345
\(486\) 17.7420 0.804795
\(487\) −25.0292 −1.13418 −0.567091 0.823655i \(-0.691931\pi\)
−0.567091 + 0.823655i \(0.691931\pi\)
\(488\) −6.79635 −0.307657
\(489\) 3.24175 0.146597
\(490\) −9.04557 −0.408637
\(491\) 8.38193 0.378271 0.189136 0.981951i \(-0.439431\pi\)
0.189136 + 0.981951i \(0.439431\pi\)
\(492\) 1.78225 0.0803498
\(493\) 3.13539 0.141211
\(494\) −11.0588 −0.497560
\(495\) 7.87602 0.354001
\(496\) 21.9909 0.987422
\(497\) 4.59954 0.206318
\(498\) −1.56449 −0.0701066
\(499\) 12.4436 0.557050 0.278525 0.960429i \(-0.410154\pi\)
0.278525 + 0.960429i \(0.410154\pi\)
\(500\) 4.57816 0.204741
\(501\) 5.29083 0.236377
\(502\) 18.7238 0.835682
\(503\) −16.3260 −0.727942 −0.363971 0.931410i \(-0.618579\pi\)
−0.363971 + 0.931410i \(0.618579\pi\)
\(504\) −23.0033 −1.02465
\(505\) 23.3564 1.03935
\(506\) −9.43979 −0.419650
\(507\) −4.87723 −0.216605
\(508\) −3.70855 −0.164540
\(509\) 18.4749 0.818887 0.409443 0.912336i \(-0.365723\pi\)
0.409443 + 0.912336i \(0.365723\pi\)
\(510\) 1.00000 0.0442807
\(511\) 0.797377 0.0352739
\(512\) −12.3239 −0.544645
\(513\) −12.0021 −0.529906
\(514\) −0.176268 −0.00777485
\(515\) −1.37899 −0.0607657
\(516\) −0.192036 −0.00845391
\(517\) −17.5383 −0.771334
\(518\) −35.0403 −1.53958
\(519\) 8.84226 0.388132
\(520\) −5.45396 −0.239172
\(521\) 16.3339 0.715600 0.357800 0.933798i \(-0.383527\pi\)
0.357800 + 0.933798i \(0.383527\pi\)
\(522\) 13.5374 0.592514
\(523\) −10.8949 −0.476399 −0.238200 0.971216i \(-0.576557\pi\)
−0.238200 + 0.971216i \(0.576557\pi\)
\(524\) 5.42001 0.236774
\(525\) 4.87082 0.212580
\(526\) −1.61976 −0.0706247
\(527\) 4.72687 0.205906
\(528\) −4.47970 −0.194954
\(529\) −14.2679 −0.620345
\(530\) −19.9455 −0.866378
\(531\) 27.3560 1.18715
\(532\) −6.10172 −0.264543
\(533\) −14.9010 −0.645434
\(534\) −9.74971 −0.421911
\(535\) −23.7250 −1.02572
\(536\) −21.8391 −0.943304
\(537\) −7.55450 −0.326001
\(538\) −5.05839 −0.218083
\(539\) 8.70993 0.375163
\(540\) 1.52802 0.0657554
\(541\) 14.6315 0.629055 0.314528 0.949248i \(-0.398154\pi\)
0.314528 + 0.949248i \(0.398154\pi\)
\(542\) 3.35541 0.144127
\(543\) −8.26769 −0.354800
\(544\) 2.28696 0.0980524
\(545\) 13.0499 0.558995
\(546\) 3.90963 0.167317
\(547\) 38.8321 1.66034 0.830170 0.557510i \(-0.188243\pi\)
0.830170 + 0.557510i \(0.188243\pi\)
\(548\) 0.204533 0.00873720
\(549\) −7.65841 −0.326853
\(550\) −9.92066 −0.423019
\(551\) −13.9100 −0.592588
\(552\) 3.41283 0.145260
\(553\) 24.3223 1.03429
\(554\) −3.19941 −0.135930
\(555\) −4.33748 −0.184116
\(556\) 5.33960 0.226449
\(557\) −15.0377 −0.637167 −0.318583 0.947895i \(-0.603207\pi\)
−0.318583 + 0.947895i \(0.603207\pi\)
\(558\) 20.4087 0.863971
\(559\) 1.60557 0.0679085
\(560\) −21.4616 −0.906917
\(561\) −0.962895 −0.0406534
\(562\) 30.4690 1.28526
\(563\) 25.9281 1.09274 0.546370 0.837544i \(-0.316009\pi\)
0.546370 + 0.837544i \(0.316009\pi\)
\(564\) −1.63685 −0.0689236
\(565\) 7.84052 0.329853
\(566\) 46.3295 1.94737
\(567\) −23.7191 −0.996109
\(568\) 3.38692 0.142112
\(569\) 33.5425 1.40617 0.703086 0.711104i \(-0.251805\pi\)
0.703086 + 0.711104i \(0.251805\pi\)
\(570\) −4.43647 −0.185823
\(571\) −1.78040 −0.0745073 −0.0372536 0.999306i \(-0.511861\pi\)
−0.0372536 + 0.999306i \(0.511861\pi\)
\(572\) −1.35568 −0.0566840
\(573\) −2.43852 −0.101871
\(574\) −48.2920 −2.01567
\(575\) 9.17690 0.382703
\(576\) −16.0021 −0.666755
\(577\) −5.31057 −0.221082 −0.110541 0.993872i \(-0.535258\pi\)
−0.110541 + 0.993872i \(0.535258\pi\)
\(578\) 1.55253 0.0645769
\(579\) −4.69256 −0.195016
\(580\) 1.77092 0.0735336
\(581\) 7.21714 0.299417
\(582\) −8.98998 −0.372647
\(583\) 19.2054 0.795408
\(584\) 0.587158 0.0242968
\(585\) −6.14575 −0.254095
\(586\) 15.1533 0.625976
\(587\) −13.6765 −0.564488 −0.282244 0.959343i \(-0.591079\pi\)
−0.282244 + 0.959343i \(0.591079\pi\)
\(588\) 0.812896 0.0335232
\(589\) −20.9706 −0.864079
\(590\) 21.0201 0.865383
\(591\) −3.85297 −0.158490
\(592\) −31.3292 −1.28762
\(593\) −23.1903 −0.952312 −0.476156 0.879361i \(-0.657970\pi\)
−0.476156 + 0.879361i \(0.657970\pi\)
\(594\) −8.64218 −0.354593
\(595\) −4.61308 −0.189118
\(596\) −7.66204 −0.313849
\(597\) −0.278952 −0.0114168
\(598\) 7.36597 0.301217
\(599\) 3.62388 0.148068 0.0740339 0.997256i \(-0.476413\pi\)
0.0740339 + 0.997256i \(0.476413\pi\)
\(600\) 3.58668 0.146426
\(601\) 10.7756 0.439548 0.219774 0.975551i \(-0.429468\pi\)
0.219774 + 0.975551i \(0.429468\pi\)
\(602\) 5.20343 0.212076
\(603\) −24.6091 −1.00216
\(604\) −6.12131 −0.249073
\(605\) 9.31303 0.378628
\(606\) −12.3288 −0.500823
\(607\) 11.1257 0.451578 0.225789 0.974176i \(-0.427504\pi\)
0.225789 + 0.974176i \(0.427504\pi\)
\(608\) −10.1460 −0.411475
\(609\) 4.91763 0.199272
\(610\) −5.88464 −0.238262
\(611\) 13.6853 0.553650
\(612\) 1.14122 0.0461309
\(613\) −33.5257 −1.35409 −0.677046 0.735941i \(-0.736740\pi\)
−0.677046 + 0.735941i \(0.736740\pi\)
\(614\) 8.46488 0.341615
\(615\) −5.97784 −0.241050
\(616\) 17.0197 0.685741
\(617\) 40.9696 1.64937 0.824686 0.565591i \(-0.191352\pi\)
0.824686 + 0.565591i \(0.191352\pi\)
\(618\) 0.727909 0.0292808
\(619\) −18.3291 −0.736709 −0.368355 0.929685i \(-0.620079\pi\)
−0.368355 + 0.929685i \(0.620079\pi\)
\(620\) 2.66982 0.107223
\(621\) 7.99426 0.320799
\(622\) 10.7303 0.430245
\(623\) 44.9762 1.80193
\(624\) 3.49556 0.139934
\(625\) 0.172088 0.00688353
\(626\) −47.0144 −1.87907
\(627\) 4.27185 0.170601
\(628\) 7.89558 0.315068
\(629\) −6.73408 −0.268506
\(630\) −19.9175 −0.793531
\(631\) 4.31547 0.171796 0.0858981 0.996304i \(-0.472624\pi\)
0.0858981 + 0.996304i \(0.472624\pi\)
\(632\) 17.9100 0.712422
\(633\) −0.990809 −0.0393811
\(634\) −1.64313 −0.0652570
\(635\) 12.4389 0.493622
\(636\) 1.79244 0.0710748
\(637\) −6.79646 −0.269285
\(638\) −10.0160 −0.396537
\(639\) 3.81652 0.150979
\(640\) −18.5914 −0.734888
\(641\) 39.4483 1.55811 0.779057 0.626953i \(-0.215698\pi\)
0.779057 + 0.626953i \(0.215698\pi\)
\(642\) 12.5234 0.494258
\(643\) 44.4275 1.75205 0.876024 0.482267i \(-0.160186\pi\)
0.876024 + 0.482267i \(0.160186\pi\)
\(644\) 4.06418 0.160151
\(645\) 0.644108 0.0253617
\(646\) −6.88777 −0.270996
\(647\) −14.5542 −0.572183 −0.286091 0.958202i \(-0.592356\pi\)
−0.286091 + 0.958202i \(0.592356\pi\)
\(648\) −17.4658 −0.686123
\(649\) −20.2401 −0.794494
\(650\) 7.74120 0.303635
\(651\) 7.41376 0.290568
\(652\) 2.84269 0.111328
\(653\) −5.75768 −0.225315 −0.112658 0.993634i \(-0.535936\pi\)
−0.112658 + 0.993634i \(0.535936\pi\)
\(654\) −6.88845 −0.269360
\(655\) −18.1793 −0.710323
\(656\) −43.1773 −1.68579
\(657\) 0.661633 0.0258128
\(658\) 44.3522 1.72903
\(659\) 34.6474 1.34967 0.674835 0.737969i \(-0.264215\pi\)
0.674835 + 0.737969i \(0.264215\pi\)
\(660\) −0.543860 −0.0211697
\(661\) −2.04505 −0.0795432 −0.0397716 0.999209i \(-0.512663\pi\)
−0.0397716 + 0.999209i \(0.512663\pi\)
\(662\) 10.2554 0.398589
\(663\) 0.751357 0.0291803
\(664\) 5.31442 0.206240
\(665\) 20.4658 0.793630
\(666\) −29.0751 −1.12664
\(667\) 9.26509 0.358746
\(668\) 4.63952 0.179509
\(669\) −9.99154 −0.386295
\(670\) −18.9094 −0.730535
\(671\) 5.66629 0.218745
\(672\) 3.58693 0.138369
\(673\) −37.3113 −1.43825 −0.719123 0.694883i \(-0.755456\pi\)
−0.719123 + 0.694883i \(0.755456\pi\)
\(674\) −12.1085 −0.466400
\(675\) 8.40150 0.323374
\(676\) −4.27684 −0.164494
\(677\) 0.572224 0.0219924 0.0109962 0.999940i \(-0.496500\pi\)
0.0109962 + 0.999940i \(0.496500\pi\)
\(678\) −4.13866 −0.158944
\(679\) 41.4715 1.59153
\(680\) −3.39690 −0.130265
\(681\) 8.06199 0.308936
\(682\) −15.1000 −0.578209
\(683\) −40.0033 −1.53069 −0.765343 0.643623i \(-0.777430\pi\)
−0.765343 + 0.643623i \(0.777430\pi\)
\(684\) −5.06297 −0.193588
\(685\) −0.686024 −0.0262116
\(686\) 14.3977 0.549706
\(687\) −0.388568 −0.0148248
\(688\) 4.65233 0.177368
\(689\) −14.9862 −0.570929
\(690\) 2.95501 0.112495
\(691\) 19.8850 0.756461 0.378231 0.925711i \(-0.376533\pi\)
0.378231 + 0.925711i \(0.376533\pi\)
\(692\) 7.75378 0.294754
\(693\) 19.1784 0.728528
\(694\) −15.2823 −0.580107
\(695\) −17.9096 −0.679348
\(696\) 3.62115 0.137259
\(697\) −9.28079 −0.351535
\(698\) −47.0319 −1.78019
\(699\) −8.29046 −0.313574
\(700\) 4.27122 0.161437
\(701\) 19.6778 0.743222 0.371611 0.928389i \(-0.378806\pi\)
0.371611 + 0.928389i \(0.378806\pi\)
\(702\) 6.74359 0.254520
\(703\) 29.8756 1.12678
\(704\) 11.8396 0.446223
\(705\) 5.49015 0.206771
\(706\) −5.18325 −0.195074
\(707\) 56.8737 2.13896
\(708\) −1.88900 −0.0709931
\(709\) 19.6415 0.737653 0.368827 0.929498i \(-0.379760\pi\)
0.368827 + 0.929498i \(0.379760\pi\)
\(710\) 2.93258 0.110058
\(711\) 20.1817 0.756873
\(712\) 33.1187 1.24118
\(713\) 13.9679 0.523103
\(714\) 2.43504 0.0911291
\(715\) 4.54711 0.170052
\(716\) −6.62454 −0.247571
\(717\) −9.69738 −0.362155
\(718\) 51.1761 1.90987
\(719\) 37.4428 1.39638 0.698191 0.715912i \(-0.253989\pi\)
0.698191 + 0.715912i \(0.253989\pi\)
\(720\) −17.8080 −0.663664
\(721\) −3.35790 −0.125055
\(722\) 1.05923 0.0394206
\(723\) 3.75297 0.139574
\(724\) −7.24993 −0.269442
\(725\) 9.73707 0.361626
\(726\) −4.91593 −0.182447
\(727\) 37.6367 1.39587 0.697933 0.716163i \(-0.254103\pi\)
0.697933 + 0.716163i \(0.254103\pi\)
\(728\) −13.2806 −0.492213
\(729\) −15.8832 −0.588266
\(730\) 0.508392 0.0188164
\(731\) 1.00000 0.0369863
\(732\) 0.528833 0.0195462
\(733\) 2.42520 0.0895766 0.0447883 0.998996i \(-0.485739\pi\)
0.0447883 + 0.998996i \(0.485739\pi\)
\(734\) 30.8602 1.13907
\(735\) −2.72654 −0.100570
\(736\) 6.75798 0.249102
\(737\) 18.2078 0.670692
\(738\) −40.0708 −1.47503
\(739\) −9.87099 −0.363110 −0.181555 0.983381i \(-0.558113\pi\)
−0.181555 + 0.983381i \(0.558113\pi\)
\(740\) −3.80353 −0.139821
\(741\) −3.33337 −0.122455
\(742\) −48.5681 −1.78299
\(743\) 25.0556 0.919202 0.459601 0.888125i \(-0.347992\pi\)
0.459601 + 0.888125i \(0.347992\pi\)
\(744\) 5.45920 0.200144
\(745\) 25.6993 0.941548
\(746\) 49.4149 1.80921
\(747\) 5.98850 0.219108
\(748\) −0.844362 −0.0308729
\(749\) −57.7713 −2.11092
\(750\) 8.10554 0.295973
\(751\) −17.8673 −0.651987 −0.325993 0.945372i \(-0.605699\pi\)
−0.325993 + 0.945372i \(0.605699\pi\)
\(752\) 39.6548 1.44606
\(753\) 5.64375 0.205670
\(754\) 7.81560 0.284627
\(755\) 20.5315 0.747218
\(756\) 3.72078 0.135324
\(757\) −29.2824 −1.06429 −0.532144 0.846654i \(-0.678614\pi\)
−0.532144 + 0.846654i \(0.678614\pi\)
\(758\) −28.5499 −1.03698
\(759\) −2.84536 −0.103280
\(760\) 15.0702 0.546654
\(761\) 29.8059 1.08046 0.540231 0.841517i \(-0.318337\pi\)
0.540231 + 0.841517i \(0.318337\pi\)
\(762\) −6.56592 −0.237858
\(763\) 31.7770 1.15040
\(764\) −2.13834 −0.0773623
\(765\) −3.82776 −0.138393
\(766\) 48.8262 1.76416
\(767\) 15.7936 0.570273
\(768\) 4.42810 0.159785
\(769\) −6.62324 −0.238840 −0.119420 0.992844i \(-0.538104\pi\)
−0.119420 + 0.992844i \(0.538104\pi\)
\(770\) 14.7365 0.531067
\(771\) −0.0531311 −0.00191347
\(772\) −4.11490 −0.148099
\(773\) 42.7240 1.53667 0.768337 0.640045i \(-0.221084\pi\)
0.768337 + 0.640045i \(0.221084\pi\)
\(774\) 4.31761 0.155193
\(775\) 14.6795 0.527302
\(776\) 30.5380 1.09625
\(777\) −10.5619 −0.378907
\(778\) −56.1954 −2.01470
\(779\) 41.1740 1.47521
\(780\) 0.424380 0.0151952
\(781\) −2.82376 −0.101042
\(782\) 4.58775 0.164058
\(783\) 8.48224 0.303131
\(784\) −19.6935 −0.703339
\(785\) −26.4826 −0.945204
\(786\) 9.59602 0.342279
\(787\) −54.3543 −1.93752 −0.968760 0.247999i \(-0.920227\pi\)
−0.968760 + 0.247999i \(0.920227\pi\)
\(788\) −3.37867 −0.120360
\(789\) −0.488231 −0.0173815
\(790\) 15.5074 0.551730
\(791\) 19.0920 0.678833
\(792\) 14.1223 0.501812
\(793\) −4.42147 −0.157011
\(794\) −24.9504 −0.885457
\(795\) −6.01202 −0.213225
\(796\) −0.244613 −0.00867008
\(797\) 39.0251 1.38234 0.691171 0.722692i \(-0.257095\pi\)
0.691171 + 0.722692i \(0.257095\pi\)
\(798\) −10.8030 −0.382421
\(799\) 8.52365 0.301545
\(800\) 7.10224 0.251102
\(801\) 37.3195 1.31862
\(802\) 28.4707 1.00533
\(803\) −0.489528 −0.0172751
\(804\) 1.69933 0.0599307
\(805\) −13.6317 −0.480454
\(806\) 11.7827 0.415028
\(807\) −1.52471 −0.0536724
\(808\) 41.8796 1.47332
\(809\) 1.18490 0.0416590 0.0208295 0.999783i \(-0.493369\pi\)
0.0208295 + 0.999783i \(0.493369\pi\)
\(810\) −15.1228 −0.531363
\(811\) 10.4542 0.367098 0.183549 0.983011i \(-0.441241\pi\)
0.183549 + 0.983011i \(0.441241\pi\)
\(812\) 4.31227 0.151331
\(813\) 1.01139 0.0354712
\(814\) 21.5121 0.753997
\(815\) −9.53468 −0.333985
\(816\) 2.17714 0.0762152
\(817\) −4.43647 −0.155212
\(818\) −33.6268 −1.17573
\(819\) −14.9651 −0.522924
\(820\) −5.24196 −0.183057
\(821\) 13.1726 0.459727 0.229863 0.973223i \(-0.426172\pi\)
0.229863 + 0.973223i \(0.426172\pi\)
\(822\) 0.362121 0.0126304
\(823\) −31.6599 −1.10360 −0.551798 0.833978i \(-0.686058\pi\)
−0.551798 + 0.833978i \(0.686058\pi\)
\(824\) −2.47263 −0.0861381
\(825\) −2.99031 −0.104109
\(826\) 51.1847 1.78094
\(827\) 3.05427 0.106208 0.0531038 0.998589i \(-0.483089\pi\)
0.0531038 + 0.998589i \(0.483089\pi\)
\(828\) 3.37230 0.117196
\(829\) −1.45481 −0.0505276 −0.0252638 0.999681i \(-0.508043\pi\)
−0.0252638 + 0.999681i \(0.508043\pi\)
\(830\) 4.60151 0.159721
\(831\) −0.964374 −0.0334538
\(832\) −9.23859 −0.320290
\(833\) −4.23304 −0.146666
\(834\) 9.45365 0.327353
\(835\) −15.5614 −0.538526
\(836\) 3.74598 0.129558
\(837\) 12.7877 0.442008
\(838\) −1.24125 −0.0428783
\(839\) −6.85018 −0.236494 −0.118247 0.992984i \(-0.537728\pi\)
−0.118247 + 0.992984i \(0.537728\pi\)
\(840\) −5.32779 −0.183826
\(841\) −19.1694 −0.661012
\(842\) −33.2954 −1.14744
\(843\) 9.18404 0.316315
\(844\) −0.868840 −0.0299067
\(845\) 14.3450 0.493482
\(846\) 36.8017 1.26527
\(847\) 22.6776 0.779211
\(848\) −43.4242 −1.49119
\(849\) 13.9647 0.479269
\(850\) 4.82146 0.165375
\(851\) −19.8993 −0.682138
\(852\) −0.263541 −0.00902877
\(853\) −17.5620 −0.601310 −0.300655 0.953733i \(-0.597205\pi\)
−0.300655 + 0.953733i \(0.597205\pi\)
\(854\) −14.3293 −0.490340
\(855\) 16.9817 0.580763
\(856\) −42.5406 −1.45401
\(857\) −6.67420 −0.227986 −0.113993 0.993482i \(-0.536364\pi\)
−0.113993 + 0.993482i \(0.536364\pi\)
\(858\) −2.40021 −0.0819420
\(859\) −19.3448 −0.660036 −0.330018 0.943975i \(-0.607055\pi\)
−0.330018 + 0.943975i \(0.607055\pi\)
\(860\) 0.564818 0.0192601
\(861\) −14.5563 −0.496076
\(862\) −19.4647 −0.662971
\(863\) −30.8248 −1.04929 −0.524645 0.851321i \(-0.675802\pi\)
−0.524645 + 0.851321i \(0.675802\pi\)
\(864\) 6.18696 0.210485
\(865\) −26.0070 −0.884264
\(866\) 0.490910 0.0166818
\(867\) 0.467968 0.0158930
\(868\) 6.50112 0.220662
\(869\) −14.9320 −0.506534
\(870\) 3.13539 0.106300
\(871\) −14.2077 −0.481411
\(872\) 23.3993 0.792401
\(873\) 34.4115 1.16465
\(874\) −20.3534 −0.688464
\(875\) −37.3915 −1.26406
\(876\) −0.0456875 −0.00154364
\(877\) −9.29617 −0.313909 −0.156955 0.987606i \(-0.550168\pi\)
−0.156955 + 0.987606i \(0.550168\pi\)
\(878\) −60.6559 −2.04704
\(879\) 4.56753 0.154059
\(880\) 13.1757 0.444154
\(881\) 10.8604 0.365896 0.182948 0.983123i \(-0.441436\pi\)
0.182948 + 0.983123i \(0.441436\pi\)
\(882\) −18.2766 −0.615405
\(883\) 5.78586 0.194710 0.0973549 0.995250i \(-0.468962\pi\)
0.0973549 + 0.995250i \(0.468962\pi\)
\(884\) 0.658865 0.0221600
\(885\) 6.33592 0.212980
\(886\) 4.53639 0.152403
\(887\) 23.4572 0.787615 0.393807 0.919193i \(-0.371158\pi\)
0.393807 + 0.919193i \(0.371158\pi\)
\(888\) −7.77740 −0.260992
\(889\) 30.2891 1.01587
\(890\) 28.6760 0.961220
\(891\) 14.5617 0.487836
\(892\) −8.76157 −0.293359
\(893\) −37.8149 −1.26543
\(894\) −13.5655 −0.453698
\(895\) 22.2194 0.742713
\(896\) −45.2707 −1.51239
\(897\) 2.22027 0.0741326
\(898\) 4.88986 0.163177
\(899\) 14.8206 0.494293
\(900\) 3.54409 0.118136
\(901\) −9.33387 −0.310956
\(902\) 29.6475 0.987155
\(903\) 1.56843 0.0521941
\(904\) 14.0586 0.467582
\(905\) 24.3170 0.808325
\(906\) −10.8377 −0.360057
\(907\) −24.4247 −0.811009 −0.405504 0.914093i \(-0.632904\pi\)
−0.405504 + 0.914093i \(0.632904\pi\)
\(908\) 7.06956 0.234611
\(909\) 47.1916 1.56525
\(910\) −11.4991 −0.381190
\(911\) −29.5612 −0.979406 −0.489703 0.871889i \(-0.662895\pi\)
−0.489703 + 0.871889i \(0.662895\pi\)
\(912\) −9.65882 −0.319835
\(913\) −4.43077 −0.146637
\(914\) −49.3221 −1.63143
\(915\) −1.77376 −0.0586388
\(916\) −0.340735 −0.0112582
\(917\) −44.2672 −1.46183
\(918\) 4.20011 0.138624
\(919\) −8.03175 −0.264943 −0.132471 0.991187i \(-0.542291\pi\)
−0.132471 + 0.991187i \(0.542291\pi\)
\(920\) −10.0379 −0.330938
\(921\) 2.55150 0.0840749
\(922\) 42.7016 1.40630
\(923\) 2.20341 0.0725262
\(924\) −1.32432 −0.0435670
\(925\) −20.9130 −0.687614
\(926\) −9.75539 −0.320582
\(927\) −2.78626 −0.0915127
\(928\) 7.17049 0.235383
\(929\) 34.6810 1.13785 0.568924 0.822390i \(-0.307360\pi\)
0.568924 + 0.822390i \(0.307360\pi\)
\(930\) 4.72687 0.155000
\(931\) 18.7798 0.615481
\(932\) −7.26990 −0.238134
\(933\) 3.23434 0.105888
\(934\) −23.4907 −0.768638
\(935\) 2.83208 0.0926188
\(936\) −11.0197 −0.360192
\(937\) 35.9363 1.17399 0.586994 0.809591i \(-0.300311\pi\)
0.586994 + 0.809591i \(0.300311\pi\)
\(938\) −46.0452 −1.50343
\(939\) −14.1712 −0.462459
\(940\) 4.81431 0.157025
\(941\) −48.1664 −1.57018 −0.785090 0.619382i \(-0.787383\pi\)
−0.785090 + 0.619382i \(0.787383\pi\)
\(942\) 13.9790 0.455460
\(943\) −27.4248 −0.893075
\(944\) 45.7636 1.48948
\(945\) −12.4799 −0.405971
\(946\) −3.19450 −0.103862
\(947\) 50.0891 1.62768 0.813839 0.581091i \(-0.197374\pi\)
0.813839 + 0.581091i \(0.197374\pi\)
\(948\) −1.39360 −0.0452621
\(949\) 0.381984 0.0123997
\(950\) −21.3902 −0.693991
\(951\) −0.495276 −0.0160604
\(952\) −8.27157 −0.268083
\(953\) −47.1986 −1.52891 −0.764457 0.644675i \(-0.776993\pi\)
−0.764457 + 0.644675i \(0.776993\pi\)
\(954\) −40.3000 −1.30476
\(955\) 7.17221 0.232087
\(956\) −8.50363 −0.275027
\(957\) −3.01905 −0.0975919
\(958\) 28.5417 0.922141
\(959\) −1.67050 −0.0539431
\(960\) −3.70625 −0.119619
\(961\) −8.65672 −0.279249
\(962\) −16.7861 −0.541206
\(963\) −47.9364 −1.54473
\(964\) 3.29097 0.105995
\(965\) 13.8018 0.444296
\(966\) 7.19556 0.231513
\(967\) 50.7117 1.63078 0.815390 0.578912i \(-0.196523\pi\)
0.815390 + 0.578912i \(0.196523\pi\)
\(968\) 16.6989 0.536723
\(969\) −2.07613 −0.0666948
\(970\) 26.4414 0.848984
\(971\) 14.3657 0.461017 0.230508 0.973070i \(-0.425961\pi\)
0.230508 + 0.973070i \(0.425961\pi\)
\(972\) 4.68952 0.150416
\(973\) −43.6105 −1.39809
\(974\) −38.8587 −1.24511
\(975\) 2.33337 0.0747277
\(976\) −12.8117 −0.410092
\(977\) 3.90889 0.125056 0.0625282 0.998043i \(-0.480084\pi\)
0.0625282 + 0.998043i \(0.480084\pi\)
\(978\) 5.03293 0.160935
\(979\) −27.6119 −0.882481
\(980\) −2.39090 −0.0763744
\(981\) 26.3673 0.841843
\(982\) 13.0132 0.415269
\(983\) 46.8969 1.49578 0.747889 0.663824i \(-0.231067\pi\)
0.747889 + 0.663824i \(0.231067\pi\)
\(984\) −10.7187 −0.341699
\(985\) 11.3324 0.361081
\(986\) 4.86779 0.155022
\(987\) 13.3687 0.425532
\(988\) −2.92303 −0.0929941
\(989\) 2.95501 0.0939638
\(990\) 12.2278 0.388625
\(991\) −31.8168 −1.01069 −0.505347 0.862916i \(-0.668635\pi\)
−0.505347 + 0.862916i \(0.668635\pi\)
\(992\) 10.8101 0.343222
\(993\) 3.09121 0.0980967
\(994\) 7.14094 0.226497
\(995\) 0.820458 0.0260103
\(996\) −0.413522 −0.0131029
\(997\) −19.3075 −0.611473 −0.305737 0.952116i \(-0.598903\pi\)
−0.305737 + 0.952116i \(0.598903\pi\)
\(998\) 19.3191 0.611534
\(999\) −18.2179 −0.576389
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 731.2.a.c.1.5 6
3.2 odd 2 6579.2.a.j.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.c.1.5 6 1.1 even 1 trivial
6579.2.a.j.1.2 6 3.2 odd 2