Properties

Label 5070.2.a.bu.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.35690 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.35690 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.29590 q^{11} -1.00000 q^{12} -1.35690 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.04892 q^{17} +1.00000 q^{18} +3.24698 q^{19} +1.00000 q^{20} +1.35690 q^{21} -5.29590 q^{22} +1.33513 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.35690 q^{28} -5.13706 q^{29} -1.00000 q^{30} +2.80194 q^{31} +1.00000 q^{32} +5.29590 q^{33} -2.04892 q^{34} -1.35690 q^{35} +1.00000 q^{36} -2.24698 q^{37} +3.24698 q^{38} +1.00000 q^{40} +10.6310 q^{41} +1.35690 q^{42} +4.41789 q^{43} -5.29590 q^{44} +1.00000 q^{45} +1.33513 q^{46} +4.82371 q^{47} -1.00000 q^{48} -5.15883 q^{49} +1.00000 q^{50} +2.04892 q^{51} +14.2349 q^{53} -1.00000 q^{54} -5.29590 q^{55} -1.35690 q^{56} -3.24698 q^{57} -5.13706 q^{58} +13.8998 q^{59} -1.00000 q^{60} -3.62565 q^{61} +2.80194 q^{62} -1.35690 q^{63} +1.00000 q^{64} +5.29590 q^{66} +9.03684 q^{67} -2.04892 q^{68} -1.33513 q^{69} -1.35690 q^{70} +11.7506 q^{71} +1.00000 q^{72} -12.1032 q^{73} -2.24698 q^{74} -1.00000 q^{75} +3.24698 q^{76} +7.18598 q^{77} +0.259061 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.6310 q^{82} +3.02177 q^{83} +1.35690 q^{84} -2.04892 q^{85} +4.41789 q^{86} +5.13706 q^{87} -5.29590 q^{88} -5.47650 q^{89} +1.00000 q^{90} +1.33513 q^{92} -2.80194 q^{93} +4.82371 q^{94} +3.24698 q^{95} -1.00000 q^{96} -5.85086 q^{97} -5.15883 q^{98} -5.29590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{10} - 2q^{11} - 3q^{12} - 3q^{15} + 3q^{16} + 3q^{17} + 3q^{18} + 5q^{19} + 3q^{20} - 2q^{22} + 3q^{23} - 3q^{24} + 3q^{25} - 3q^{27} - 10q^{29} - 3q^{30} + 4q^{31} + 3q^{32} + 2q^{33} + 3q^{34} + 3q^{36} - 2q^{37} + 5q^{38} + 3q^{40} + 17q^{41} + 19q^{43} - 2q^{44} + 3q^{45} + 3q^{46} + 7q^{47} - 3q^{48} - 7q^{49} + 3q^{50} - 3q^{51} + 19q^{53} - 3q^{54} - 2q^{55} - 5q^{57} - 10q^{58} + 19q^{59} - 3q^{60} + q^{61} + 4q^{62} + 3q^{64} + 2q^{66} - q^{67} + 3q^{68} - 3q^{69} - q^{71} + 3q^{72} - 15q^{73} - 2q^{74} - 3q^{75} + 5q^{76} + 7q^{77} + 15q^{79} + 3q^{80} + 3q^{81} + 17q^{82} + 6q^{83} + 3q^{85} + 19q^{86} + 10q^{87} - 2q^{88} + 9q^{89} + 3q^{90} + 3q^{92} - 4q^{93} + 7q^{94} + 5q^{95} - 3q^{96} - 4q^{97} - 7q^{98} - 2q^{99} + O(q^{100}) \)

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
\(7\) −1.35690 −0.512858 −0.256429 0.966563i \(-0.582546\pi\)
−0.256429 + 0.966563i \(0.582546\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −5.29590 −1.59677 −0.798387 0.602145i \(-0.794313\pi\)
−0.798387 + 0.602145i \(0.794313\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.35690 −0.362646
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.04892 −0.496935 −0.248468 0.968640i \(-0.579927\pi\)
−0.248468 + 0.968640i \(0.579927\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.24698 0.744908 0.372454 0.928051i \(-0.378516\pi\)
0.372454 + 0.928051i \(0.378516\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.35690 0.296099
\(22\) −5.29590 −1.12909
\(23\) 1.33513 0.278393 0.139196 0.990265i \(-0.455548\pi\)
0.139196 + 0.990265i \(0.455548\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.35690 −0.256429
\(29\) −5.13706 −0.953929 −0.476964 0.878923i \(-0.658263\pi\)
−0.476964 + 0.878923i \(0.658263\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.80194 0.503243 0.251621 0.967826i \(-0.419036\pi\)
0.251621 + 0.967826i \(0.419036\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.29590 0.921897
\(34\) −2.04892 −0.351386
\(35\) −1.35690 −0.229357
\(36\) 1.00000 0.166667
\(37\) −2.24698 −0.369401 −0.184701 0.982795i \(-0.559132\pi\)
−0.184701 + 0.982795i \(0.559132\pi\)
\(38\) 3.24698 0.526730
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 10.6310 1.66029 0.830143 0.557550i \(-0.188258\pi\)
0.830143 + 0.557550i \(0.188258\pi\)
\(42\) 1.35690 0.209374
\(43\) 4.41789 0.673723 0.336861 0.941554i \(-0.390635\pi\)
0.336861 + 0.941554i \(0.390635\pi\)
\(44\) −5.29590 −0.798387
\(45\) 1.00000 0.149071
\(46\) 1.33513 0.196854
\(47\) 4.82371 0.703610 0.351805 0.936073i \(-0.385568\pi\)
0.351805 + 0.936073i \(0.385568\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.15883 −0.736976
\(50\) 1.00000 0.141421
\(51\) 2.04892 0.286906
\(52\) 0 0
\(53\) 14.2349 1.95531 0.977657 0.210207i \(-0.0674139\pi\)
0.977657 + 0.210207i \(0.0674139\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.29590 −0.714099
\(56\) −1.35690 −0.181323
\(57\) −3.24698 −0.430073
\(58\) −5.13706 −0.674529
\(59\) 13.8998 1.80960 0.904798 0.425841i \(-0.140022\pi\)
0.904798 + 0.425841i \(0.140022\pi\)
\(60\) −1.00000 −0.129099
\(61\) −3.62565 −0.464216 −0.232108 0.972690i \(-0.574562\pi\)
−0.232108 + 0.972690i \(0.574562\pi\)
\(62\) 2.80194 0.355846
\(63\) −1.35690 −0.170953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.29590 0.651880
\(67\) 9.03684 1.10403 0.552013 0.833836i \(-0.313860\pi\)
0.552013 + 0.833836i \(0.313860\pi\)
\(68\) −2.04892 −0.248468
\(69\) −1.33513 −0.160730
\(70\) −1.35690 −0.162180
\(71\) 11.7506 1.39454 0.697271 0.716807i \(-0.254397\pi\)
0.697271 + 0.716807i \(0.254397\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.1032 −1.41657 −0.708287 0.705925i \(-0.750532\pi\)
−0.708287 + 0.705925i \(0.750532\pi\)
\(74\) −2.24698 −0.261206
\(75\) −1.00000 −0.115470
\(76\) 3.24698 0.372454
\(77\) 7.18598 0.818919
\(78\) 0 0
\(79\) 0.259061 0.0291467 0.0145733 0.999894i \(-0.495361\pi\)
0.0145733 + 0.999894i \(0.495361\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.6310 1.17400
\(83\) 3.02177 0.331682 0.165841 0.986152i \(-0.446966\pi\)
0.165841 + 0.986152i \(0.446966\pi\)
\(84\) 1.35690 0.148049
\(85\) −2.04892 −0.222236
\(86\) 4.41789 0.476394
\(87\) 5.13706 0.550751
\(88\) −5.29590 −0.564545
\(89\) −5.47650 −0.580508 −0.290254 0.956950i \(-0.593740\pi\)
−0.290254 + 0.956950i \(0.593740\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.33513 0.139196
\(93\) −2.80194 −0.290547
\(94\) 4.82371 0.497527
\(95\) 3.24698 0.333133
\(96\) −1.00000 −0.102062
\(97\) −5.85086 −0.594064 −0.297032 0.954867i \(-0.595997\pi\)
−0.297032 + 0.954867i \(0.595997\pi\)
\(98\) −5.15883 −0.521121
\(99\) −5.29590 −0.532258
\(100\) 1.00000 0.100000
\(101\) −8.39612 −0.835446 −0.417723 0.908575i \(-0.637172\pi\)
−0.417723 + 0.908575i \(0.637172\pi\)
\(102\) 2.04892 0.202873
\(103\) 13.6353 1.34353 0.671765 0.740765i \(-0.265537\pi\)
0.671765 + 0.740765i \(0.265537\pi\)
\(104\) 0 0
\(105\) 1.35690 0.132419
\(106\) 14.2349 1.38262
\(107\) 3.92394 0.379341 0.189671 0.981848i \(-0.439258\pi\)
0.189671 + 0.981848i \(0.439258\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −0.554958 −0.0531553 −0.0265777 0.999647i \(-0.508461\pi\)
−0.0265777 + 0.999647i \(0.508461\pi\)
\(110\) −5.29590 −0.504944
\(111\) 2.24698 0.213274
\(112\) −1.35690 −0.128215
\(113\) 4.39373 0.413328 0.206664 0.978412i \(-0.433739\pi\)
0.206664 + 0.978412i \(0.433739\pi\)
\(114\) −3.24698 −0.304108
\(115\) 1.33513 0.124501
\(116\) −5.13706 −0.476964
\(117\) 0 0
\(118\) 13.8998 1.27958
\(119\) 2.78017 0.254858
\(120\) −1.00000 −0.0912871
\(121\) 17.0465 1.54968
\(122\) −3.62565 −0.328251
\(123\) −10.6310 −0.958567
\(124\) 2.80194 0.251621
\(125\) 1.00000 0.0894427
\(126\) −1.35690 −0.120882
\(127\) −2.23490 −0.198315 −0.0991576 0.995072i \(-0.531615\pi\)
−0.0991576 + 0.995072i \(0.531615\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.41789 −0.388974
\(130\) 0 0
\(131\) −8.51573 −0.744023 −0.372011 0.928228i \(-0.621332\pi\)
−0.372011 + 0.928228i \(0.621332\pi\)
\(132\) 5.29590 0.460949
\(133\) −4.40581 −0.382032
\(134\) 9.03684 0.780664
\(135\) −1.00000 −0.0860663
\(136\) −2.04892 −0.175693
\(137\) 13.1685 1.12506 0.562531 0.826776i \(-0.309828\pi\)
0.562531 + 0.826776i \(0.309828\pi\)
\(138\) −1.33513 −0.113653
\(139\) 7.17629 0.608685 0.304343 0.952563i \(-0.401563\pi\)
0.304343 + 0.952563i \(0.401563\pi\)
\(140\) −1.35690 −0.114679
\(141\) −4.82371 −0.406229
\(142\) 11.7506 0.986091
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.13706 −0.426610
\(146\) −12.1032 −1.00167
\(147\) 5.15883 0.425493
\(148\) −2.24698 −0.184701
\(149\) −9.66248 −0.791581 −0.395791 0.918341i \(-0.629529\pi\)
−0.395791 + 0.918341i \(0.629529\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 7.35152 0.598258 0.299129 0.954213i \(-0.403304\pi\)
0.299129 + 0.954213i \(0.403304\pi\)
\(152\) 3.24698 0.263365
\(153\) −2.04892 −0.165645
\(154\) 7.18598 0.579063
\(155\) 2.80194 0.225057
\(156\) 0 0
\(157\) 2.85325 0.227714 0.113857 0.993497i \(-0.463679\pi\)
0.113857 + 0.993497i \(0.463679\pi\)
\(158\) 0.259061 0.0206098
\(159\) −14.2349 −1.12890
\(160\) 1.00000 0.0790569
\(161\) −1.81163 −0.142776
\(162\) 1.00000 0.0785674
\(163\) 15.1075 1.18331 0.591656 0.806190i \(-0.298474\pi\)
0.591656 + 0.806190i \(0.298474\pi\)
\(164\) 10.6310 0.830143
\(165\) 5.29590 0.412285
\(166\) 3.02177 0.234535
\(167\) 16.0881 1.24494 0.622469 0.782644i \(-0.286130\pi\)
0.622469 + 0.782644i \(0.286130\pi\)
\(168\) 1.35690 0.104687
\(169\) 0 0
\(170\) −2.04892 −0.157145
\(171\) 3.24698 0.248303
\(172\) 4.41789 0.336861
\(173\) −15.4795 −1.17688 −0.588442 0.808540i \(-0.700258\pi\)
−0.588442 + 0.808540i \(0.700258\pi\)
\(174\) 5.13706 0.389440
\(175\) −1.35690 −0.102572
\(176\) −5.29590 −0.399193
\(177\) −13.8998 −1.04477
\(178\) −5.47650 −0.410481
\(179\) 19.0737 1.42563 0.712817 0.701351i \(-0.247419\pi\)
0.712817 + 0.701351i \(0.247419\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.16852 0.161185 0.0805925 0.996747i \(-0.474319\pi\)
0.0805925 + 0.996747i \(0.474319\pi\)
\(182\) 0 0
\(183\) 3.62565 0.268015
\(184\) 1.33513 0.0984268
\(185\) −2.24698 −0.165201
\(186\) −2.80194 −0.205448
\(187\) 10.8509 0.793493
\(188\) 4.82371 0.351805
\(189\) 1.35690 0.0986997
\(190\) 3.24698 0.235561
\(191\) −11.7409 −0.849545 −0.424772 0.905300i \(-0.639646\pi\)
−0.424772 + 0.905300i \(0.639646\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.6722 −1.77594 −0.887971 0.459900i \(-0.847885\pi\)
−0.887971 + 0.459900i \(0.847885\pi\)
\(194\) −5.85086 −0.420067
\(195\) 0 0
\(196\) −5.15883 −0.368488
\(197\) 26.2010 1.86675 0.933374 0.358906i \(-0.116850\pi\)
0.933374 + 0.358906i \(0.116850\pi\)
\(198\) −5.29590 −0.376363
\(199\) 19.6722 1.39452 0.697262 0.716817i \(-0.254402\pi\)
0.697262 + 0.716817i \(0.254402\pi\)
\(200\) 1.00000 0.0707107
\(201\) −9.03684 −0.637409
\(202\) −8.39612 −0.590749
\(203\) 6.97046 0.489230
\(204\) 2.04892 0.143453
\(205\) 10.6310 0.742503
\(206\) 13.6353 0.950019
\(207\) 1.33513 0.0927976
\(208\) 0 0
\(209\) −17.1957 −1.18945
\(210\) 1.35690 0.0936347
\(211\) −16.2935 −1.12169 −0.560846 0.827920i \(-0.689524\pi\)
−0.560846 + 0.827920i \(0.689524\pi\)
\(212\) 14.2349 0.977657
\(213\) −11.7506 −0.805140
\(214\) 3.92394 0.268235
\(215\) 4.41789 0.301298
\(216\) −1.00000 −0.0680414
\(217\) −3.80194 −0.258092
\(218\) −0.554958 −0.0375865
\(219\) 12.1032 0.817859
\(220\) −5.29590 −0.357049
\(221\) 0 0
\(222\) 2.24698 0.150807
\(223\) −2.28083 −0.152736 −0.0763679 0.997080i \(-0.524332\pi\)
−0.0763679 + 0.997080i \(0.524332\pi\)
\(224\) −1.35690 −0.0906614
\(225\) 1.00000 0.0666667
\(226\) 4.39373 0.292267
\(227\) −3.37196 −0.223805 −0.111902 0.993719i \(-0.535694\pi\)
−0.111902 + 0.993719i \(0.535694\pi\)
\(228\) −3.24698 −0.215036
\(229\) 18.3763 1.21434 0.607169 0.794573i \(-0.292305\pi\)
0.607169 + 0.794573i \(0.292305\pi\)
\(230\) 1.33513 0.0880356
\(231\) −7.18598 −0.472803
\(232\) −5.13706 −0.337265
\(233\) 10.0858 0.660740 0.330370 0.943852i \(-0.392827\pi\)
0.330370 + 0.943852i \(0.392827\pi\)
\(234\) 0 0
\(235\) 4.82371 0.314664
\(236\) 13.8998 0.904798
\(237\) −0.259061 −0.0168278
\(238\) 2.78017 0.180211
\(239\) −20.0834 −1.29908 −0.649542 0.760325i \(-0.725039\pi\)
−0.649542 + 0.760325i \(0.725039\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.99462 0.192901 0.0964503 0.995338i \(-0.469251\pi\)
0.0964503 + 0.995338i \(0.469251\pi\)
\(242\) 17.0465 1.09579
\(243\) −1.00000 −0.0641500
\(244\) −3.62565 −0.232108
\(245\) −5.15883 −0.329586
\(246\) −10.6310 −0.677809
\(247\) 0 0
\(248\) 2.80194 0.177923
\(249\) −3.02177 −0.191497
\(250\) 1.00000 0.0632456
\(251\) −7.20237 −0.454610 −0.227305 0.973824i \(-0.572991\pi\)
−0.227305 + 0.973824i \(0.572991\pi\)
\(252\) −1.35690 −0.0854764
\(253\) −7.07069 −0.444530
\(254\) −2.23490 −0.140230
\(255\) 2.04892 0.128308
\(256\) 1.00000 0.0625000
\(257\) −18.8291 −1.17453 −0.587263 0.809396i \(-0.699795\pi\)
−0.587263 + 0.809396i \(0.699795\pi\)
\(258\) −4.41789 −0.275046
\(259\) 3.04892 0.189451
\(260\) 0 0
\(261\) −5.13706 −0.317976
\(262\) −8.51573 −0.526104
\(263\) −17.9245 −1.10527 −0.552637 0.833422i \(-0.686378\pi\)
−0.552637 + 0.833422i \(0.686378\pi\)
\(264\) 5.29590 0.325940
\(265\) 14.2349 0.874443
\(266\) −4.40581 −0.270138
\(267\) 5.47650 0.335156
\(268\) 9.03684 0.552013
\(269\) −22.4045 −1.36603 −0.683013 0.730406i \(-0.739331\pi\)
−0.683013 + 0.730406i \(0.739331\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −23.2325 −1.41127 −0.705637 0.708573i \(-0.749339\pi\)
−0.705637 + 0.708573i \(0.749339\pi\)
\(272\) −2.04892 −0.124234
\(273\) 0 0
\(274\) 13.1685 0.795540
\(275\) −5.29590 −0.319355
\(276\) −1.33513 −0.0803651
\(277\) −9.78448 −0.587892 −0.293946 0.955822i \(-0.594969\pi\)
−0.293946 + 0.955822i \(0.594969\pi\)
\(278\) 7.17629 0.430405
\(279\) 2.80194 0.167748
\(280\) −1.35690 −0.0810900
\(281\) 21.6286 1.29026 0.645128 0.764075i \(-0.276804\pi\)
0.645128 + 0.764075i \(0.276804\pi\)
\(282\) −4.82371 −0.287248
\(283\) 3.38942 0.201480 0.100740 0.994913i \(-0.467879\pi\)
0.100740 + 0.994913i \(0.467879\pi\)
\(284\) 11.7506 0.697271
\(285\) −3.24698 −0.192334
\(286\) 0 0
\(287\) −14.4252 −0.851492
\(288\) 1.00000 0.0589256
\(289\) −12.8019 −0.753055
\(290\) −5.13706 −0.301659
\(291\) 5.85086 0.342983
\(292\) −12.1032 −0.708287
\(293\) 34.1280 1.99378 0.996889 0.0788243i \(-0.0251166\pi\)
0.996889 + 0.0788243i \(0.0251166\pi\)
\(294\) 5.15883 0.300869
\(295\) 13.8998 0.809276
\(296\) −2.24698 −0.130603
\(297\) 5.29590 0.307299
\(298\) −9.66248 −0.559733
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −5.99462 −0.345524
\(302\) 7.35152 0.423032
\(303\) 8.39612 0.482345
\(304\) 3.24698 0.186227
\(305\) −3.62565 −0.207604
\(306\) −2.04892 −0.117129
\(307\) −15.6601 −0.893768 −0.446884 0.894592i \(-0.647466\pi\)
−0.446884 + 0.894592i \(0.647466\pi\)
\(308\) 7.18598 0.409459
\(309\) −13.6353 −0.775687
\(310\) 2.80194 0.159139
\(311\) 25.7972 1.46282 0.731411 0.681937i \(-0.238862\pi\)
0.731411 + 0.681937i \(0.238862\pi\)
\(312\) 0 0
\(313\) −15.6243 −0.883139 −0.441569 0.897227i \(-0.645578\pi\)
−0.441569 + 0.897227i \(0.645578\pi\)
\(314\) 2.85325 0.161018
\(315\) −1.35690 −0.0764524
\(316\) 0.259061 0.0145733
\(317\) −9.25965 −0.520074 −0.260037 0.965599i \(-0.583735\pi\)
−0.260037 + 0.965599i \(0.583735\pi\)
\(318\) −14.2349 −0.798253
\(319\) 27.2054 1.52321
\(320\) 1.00000 0.0559017
\(321\) −3.92394 −0.219013
\(322\) −1.81163 −0.100958
\(323\) −6.65279 −0.370171
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 15.1075 0.836728
\(327\) 0.554958 0.0306893
\(328\) 10.6310 0.587000
\(329\) −6.54527 −0.360852
\(330\) 5.29590 0.291530
\(331\) 21.8582 1.20143 0.600716 0.799462i \(-0.294882\pi\)
0.600716 + 0.799462i \(0.294882\pi\)
\(332\) 3.02177 0.165841
\(333\) −2.24698 −0.123134
\(334\) 16.0881 0.880304
\(335\) 9.03684 0.493735
\(336\) 1.35690 0.0740247
\(337\) 14.5013 0.789934 0.394967 0.918695i \(-0.370756\pi\)
0.394967 + 0.918695i \(0.370756\pi\)
\(338\) 0 0
\(339\) −4.39373 −0.238635
\(340\) −2.04892 −0.111118
\(341\) −14.8388 −0.803565
\(342\) 3.24698 0.175577
\(343\) 16.4983 0.890823
\(344\) 4.41789 0.238197
\(345\) −1.33513 −0.0718807
\(346\) −15.4795 −0.832182
\(347\) 7.84117 0.420936 0.210468 0.977601i \(-0.432501\pi\)
0.210468 + 0.977601i \(0.432501\pi\)
\(348\) 5.13706 0.275375
\(349\) 20.4252 1.09334 0.546668 0.837350i \(-0.315896\pi\)
0.546668 + 0.837350i \(0.315896\pi\)
\(350\) −1.35690 −0.0725291
\(351\) 0 0
\(352\) −5.29590 −0.282272
\(353\) 13.2905 0.707383 0.353692 0.935362i \(-0.384926\pi\)
0.353692 + 0.935362i \(0.384926\pi\)
\(354\) −13.8998 −0.738765
\(355\) 11.7506 0.623659
\(356\) −5.47650 −0.290254
\(357\) −2.78017 −0.147142
\(358\) 19.0737 1.00807
\(359\) −12.8267 −0.676967 −0.338483 0.940972i \(-0.609914\pi\)
−0.338483 + 0.940972i \(0.609914\pi\)
\(360\) 1.00000 0.0527046
\(361\) −8.45712 −0.445112
\(362\) 2.16852 0.113975
\(363\) −17.0465 −0.894710
\(364\) 0 0
\(365\) −12.1032 −0.633511
\(366\) 3.62565 0.189516
\(367\) −17.9554 −0.937264 −0.468632 0.883393i \(-0.655253\pi\)
−0.468632 + 0.883393i \(0.655253\pi\)
\(368\) 1.33513 0.0695982
\(369\) 10.6310 0.553429
\(370\) −2.24698 −0.116815
\(371\) −19.3153 −1.00280
\(372\) −2.80194 −0.145274
\(373\) 27.8732 1.44322 0.721610 0.692300i \(-0.243402\pi\)
0.721610 + 0.692300i \(0.243402\pi\)
\(374\) 10.8509 0.561084
\(375\) −1.00000 −0.0516398
\(376\) 4.82371 0.248764
\(377\) 0 0
\(378\) 1.35690 0.0697912
\(379\) −13.8702 −0.712466 −0.356233 0.934397i \(-0.615939\pi\)
−0.356233 + 0.934397i \(0.615939\pi\)
\(380\) 3.24698 0.166567
\(381\) 2.23490 0.114497
\(382\) −11.7409 −0.600719
\(383\) 26.8984 1.37445 0.687223 0.726446i \(-0.258829\pi\)
0.687223 + 0.726446i \(0.258829\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 7.18598 0.366231
\(386\) −24.6722 −1.25578
\(387\) 4.41789 0.224574
\(388\) −5.85086 −0.297032
\(389\) −0.187309 −0.00949693 −0.00474846 0.999989i \(-0.501511\pi\)
−0.00474846 + 0.999989i \(0.501511\pi\)
\(390\) 0 0
\(391\) −2.73556 −0.138343
\(392\) −5.15883 −0.260560
\(393\) 8.51573 0.429562
\(394\) 26.2010 1.31999
\(395\) 0.259061 0.0130348
\(396\) −5.29590 −0.266129
\(397\) 8.58211 0.430724 0.215362 0.976534i \(-0.430907\pi\)
0.215362 + 0.976534i \(0.430907\pi\)
\(398\) 19.6722 0.986077
\(399\) 4.40581 0.220567
\(400\) 1.00000 0.0500000
\(401\) 23.9748 1.19724 0.598621 0.801032i \(-0.295715\pi\)
0.598621 + 0.801032i \(0.295715\pi\)
\(402\) −9.03684 −0.450716
\(403\) 0 0
\(404\) −8.39612 −0.417723
\(405\) 1.00000 0.0496904
\(406\) 6.97046 0.345938
\(407\) 11.8998 0.589850
\(408\) 2.04892 0.101437
\(409\) −8.37926 −0.414328 −0.207164 0.978306i \(-0.566423\pi\)
−0.207164 + 0.978306i \(0.566423\pi\)
\(410\) 10.6310 0.525029
\(411\) −13.1685 −0.649555
\(412\) 13.6353 0.671765
\(413\) −18.8605 −0.928067
\(414\) 1.33513 0.0656178
\(415\) 3.02177 0.148333
\(416\) 0 0
\(417\) −7.17629 −0.351425
\(418\) −17.1957 −0.841068
\(419\) 37.0887 1.81190 0.905952 0.423381i \(-0.139157\pi\)
0.905952 + 0.423381i \(0.139157\pi\)
\(420\) 1.35690 0.0662097
\(421\) 4.90515 0.239062 0.119531 0.992830i \(-0.461861\pi\)
0.119531 + 0.992830i \(0.461861\pi\)
\(422\) −16.2935 −0.793155
\(423\) 4.82371 0.234537
\(424\) 14.2349 0.691308
\(425\) −2.04892 −0.0993871
\(426\) −11.7506 −0.569320
\(427\) 4.91962 0.238077
\(428\) 3.92394 0.189671
\(429\) 0 0
\(430\) 4.41789 0.213050
\(431\) −6.06531 −0.292156 −0.146078 0.989273i \(-0.546665\pi\)
−0.146078 + 0.989273i \(0.546665\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.8877 −0.715457 −0.357728 0.933826i \(-0.616449\pi\)
−0.357728 + 0.933826i \(0.616449\pi\)
\(434\) −3.80194 −0.182499
\(435\) 5.13706 0.246303
\(436\) −0.554958 −0.0265777
\(437\) 4.33513 0.207377
\(438\) 12.1032 0.578314
\(439\) −21.1336 −1.00865 −0.504326 0.863513i \(-0.668259\pi\)
−0.504326 + 0.863513i \(0.668259\pi\)
\(440\) −5.29590 −0.252472
\(441\) −5.15883 −0.245659
\(442\) 0 0
\(443\) 28.0683 1.33356 0.666782 0.745252i \(-0.267671\pi\)
0.666782 + 0.745252i \(0.267671\pi\)
\(444\) 2.24698 0.106637
\(445\) −5.47650 −0.259611
\(446\) −2.28083 −0.108000
\(447\) 9.66248 0.457020
\(448\) −1.35690 −0.0641073
\(449\) 19.2446 0.908208 0.454104 0.890949i \(-0.349959\pi\)
0.454104 + 0.890949i \(0.349959\pi\)
\(450\) 1.00000 0.0471405
\(451\) −56.3008 −2.65110
\(452\) 4.39373 0.206664
\(453\) −7.35152 −0.345404
\(454\) −3.37196 −0.158254
\(455\) 0 0
\(456\) −3.24698 −0.152054
\(457\) −37.8485 −1.77048 −0.885238 0.465138i \(-0.846005\pi\)
−0.885238 + 0.465138i \(0.846005\pi\)
\(458\) 18.3763 0.858667
\(459\) 2.04892 0.0956353
\(460\) 1.33513 0.0622506
\(461\) −11.2228 −0.522699 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(462\) −7.18598 −0.334322
\(463\) 25.0726 1.16522 0.582611 0.812751i \(-0.302031\pi\)
0.582611 + 0.812751i \(0.302031\pi\)
\(464\) −5.13706 −0.238482
\(465\) −2.80194 −0.129937
\(466\) 10.0858 0.467213
\(467\) 36.9657 1.71057 0.855284 0.518160i \(-0.173383\pi\)
0.855284 + 0.518160i \(0.173383\pi\)
\(468\) 0 0
\(469\) −12.2620 −0.566209
\(470\) 4.82371 0.222501
\(471\) −2.85325 −0.131471
\(472\) 13.8998 0.639789
\(473\) −23.3967 −1.07578
\(474\) −0.259061 −0.0118991
\(475\) 3.24698 0.148982
\(476\) 2.78017 0.127429
\(477\) 14.2349 0.651771
\(478\) −20.0834 −0.918592
\(479\) 31.4590 1.43740 0.718700 0.695320i \(-0.244737\pi\)
0.718700 + 0.695320i \(0.244737\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 2.99462 0.136401
\(483\) 1.81163 0.0824319
\(484\) 17.0465 0.774842
\(485\) −5.85086 −0.265674
\(486\) −1.00000 −0.0453609
\(487\) 18.8713 0.855140 0.427570 0.903982i \(-0.359370\pi\)
0.427570 + 0.903982i \(0.359370\pi\)
\(488\) −3.62565 −0.164125
\(489\) −15.1075 −0.683186
\(490\) −5.15883 −0.233052
\(491\) −16.3961 −0.739947 −0.369973 0.929042i \(-0.620633\pi\)
−0.369973 + 0.929042i \(0.620633\pi\)
\(492\) −10.6310 −0.479284
\(493\) 10.5254 0.474041
\(494\) 0 0
\(495\) −5.29590 −0.238033
\(496\) 2.80194 0.125811
\(497\) −15.9444 −0.715203
\(498\) −3.02177 −0.135409
\(499\) 40.2452 1.80162 0.900811 0.434212i \(-0.142973\pi\)
0.900811 + 0.434212i \(0.142973\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.0881 −0.718765
\(502\) −7.20237 −0.321458
\(503\) 18.4004 0.820435 0.410217 0.911988i \(-0.365453\pi\)
0.410217 + 0.911988i \(0.365453\pi\)
\(504\) −1.35690 −0.0604409
\(505\) −8.39612 −0.373623
\(506\) −7.07069 −0.314330
\(507\) 0 0
\(508\) −2.23490 −0.0991576
\(509\) −35.1377 −1.55745 −0.778725 0.627366i \(-0.784133\pi\)
−0.778725 + 0.627366i \(0.784133\pi\)
\(510\) 2.04892 0.0907276
\(511\) 16.4228 0.726502
\(512\) 1.00000 0.0441942
\(513\) −3.24698 −0.143358
\(514\) −18.8291 −0.830515
\(515\) 13.6353 0.600845
\(516\) −4.41789 −0.194487
\(517\) −25.5459 −1.12351
\(518\) 3.04892 0.133962
\(519\) 15.4795 0.679474
\(520\) 0 0
\(521\) −35.2610 −1.54481 −0.772406 0.635129i \(-0.780947\pi\)
−0.772406 + 0.635129i \(0.780947\pi\)
\(522\) −5.13706 −0.224843
\(523\) 14.3043 0.625482 0.312741 0.949839i \(-0.398753\pi\)
0.312741 + 0.949839i \(0.398753\pi\)
\(524\) −8.51573 −0.372011
\(525\) 1.35690 0.0592198
\(526\) −17.9245 −0.781546
\(527\) −5.74094 −0.250079
\(528\) 5.29590 0.230474
\(529\) −21.2174 −0.922497
\(530\) 14.2349 0.618324
\(531\) 13.8998 0.603199
\(532\) −4.40581 −0.191016
\(533\) 0 0
\(534\) 5.47650 0.236991
\(535\) 3.92394 0.169647
\(536\) 9.03684 0.390332
\(537\) −19.0737 −0.823090
\(538\) −22.4045 −0.965926
\(539\) 27.3207 1.17678
\(540\) −1.00000 −0.0430331
\(541\) −44.2422 −1.90212 −0.951060 0.309006i \(-0.900004\pi\)
−0.951060 + 0.309006i \(0.900004\pi\)
\(542\) −23.2325 −0.997922
\(543\) −2.16852 −0.0930602
\(544\) −2.04892 −0.0878466
\(545\) −0.554958 −0.0237718
\(546\) 0 0
\(547\) 6.34481 0.271285 0.135642 0.990758i \(-0.456690\pi\)
0.135642 + 0.990758i \(0.456690\pi\)
\(548\) 13.1685 0.562531
\(549\) −3.62565 −0.154739
\(550\) −5.29590 −0.225818
\(551\) −16.6799 −0.710589
\(552\) −1.33513 −0.0568267
\(553\) −0.351519 −0.0149481
\(554\) −9.78448 −0.415703
\(555\) 2.24698 0.0953790
\(556\) 7.17629 0.304343
\(557\) 31.3884 1.32997 0.664984 0.746858i \(-0.268438\pi\)
0.664984 + 0.746858i \(0.268438\pi\)
\(558\) 2.80194 0.118615
\(559\) 0 0
\(560\) −1.35690 −0.0573393
\(561\) −10.8509 −0.458123
\(562\) 21.6286 0.912349
\(563\) −44.4989 −1.87540 −0.937702 0.347441i \(-0.887051\pi\)
−0.937702 + 0.347441i \(0.887051\pi\)
\(564\) −4.82371 −0.203115
\(565\) 4.39373 0.184846
\(566\) 3.38942 0.142468
\(567\) −1.35690 −0.0569843
\(568\) 11.7506 0.493045
\(569\) 30.5394 1.28028 0.640140 0.768259i \(-0.278877\pi\)
0.640140 + 0.768259i \(0.278877\pi\)
\(570\) −3.24698 −0.136001
\(571\) −32.7614 −1.37102 −0.685511 0.728063i \(-0.740421\pi\)
−0.685511 + 0.728063i \(0.740421\pi\)
\(572\) 0 0
\(573\) 11.7409 0.490485
\(574\) −14.4252 −0.602096
\(575\) 1.33513 0.0556786
\(576\) 1.00000 0.0416667
\(577\) 36.5870 1.52314 0.761569 0.648084i \(-0.224430\pi\)
0.761569 + 0.648084i \(0.224430\pi\)
\(578\) −12.8019 −0.532490
\(579\) 24.6722 1.02534
\(580\) −5.13706 −0.213305
\(581\) −4.10023 −0.170106
\(582\) 5.85086 0.242526
\(583\) −75.3866 −3.12219
\(584\) −12.1032 −0.500834
\(585\) 0 0
\(586\) 34.1280 1.40981
\(587\) 22.4873 0.928148 0.464074 0.885796i \(-0.346387\pi\)
0.464074 + 0.885796i \(0.346387\pi\)
\(588\) 5.15883 0.212747
\(589\) 9.09783 0.374870
\(590\) 13.8998 0.572245
\(591\) −26.2010 −1.07777
\(592\) −2.24698 −0.0923503
\(593\) 39.5749 1.62515 0.812574 0.582858i \(-0.198066\pi\)
0.812574 + 0.582858i \(0.198066\pi\)
\(594\) 5.29590 0.217293
\(595\) 2.78017 0.113976
\(596\) −9.66248 −0.395791
\(597\) −19.6722 −0.805128
\(598\) 0 0
\(599\) −31.3424 −1.28062 −0.640308 0.768118i \(-0.721193\pi\)
−0.640308 + 0.768118i \(0.721193\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −32.9487 −1.34401 −0.672003 0.740549i \(-0.734566\pi\)
−0.672003 + 0.740549i \(0.734566\pi\)
\(602\) −5.99462 −0.244323
\(603\) 9.03684 0.368008
\(604\) 7.35152 0.299129
\(605\) 17.0465 0.693040
\(606\) 8.39612 0.341069
\(607\) −40.1739 −1.63061 −0.815304 0.579033i \(-0.803430\pi\)
−0.815304 + 0.579033i \(0.803430\pi\)
\(608\) 3.24698 0.131682
\(609\) −6.97046 −0.282457
\(610\) −3.62565 −0.146798
\(611\) 0 0
\(612\) −2.04892 −0.0828226
\(613\) 11.3588 0.458778 0.229389 0.973335i \(-0.426327\pi\)
0.229389 + 0.973335i \(0.426327\pi\)
\(614\) −15.6601 −0.631990
\(615\) −10.6310 −0.428684
\(616\) 7.18598 0.289531
\(617\) 20.9172 0.842096 0.421048 0.907038i \(-0.361662\pi\)
0.421048 + 0.907038i \(0.361662\pi\)
\(618\) −13.6353 −0.548494
\(619\) −14.0562 −0.564967 −0.282483 0.959272i \(-0.591158\pi\)
−0.282483 + 0.959272i \(0.591158\pi\)
\(620\) 2.80194 0.112529
\(621\) −1.33513 −0.0535767
\(622\) 25.7972 1.03437
\(623\) 7.43104 0.297718
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −15.6243 −0.624473
\(627\) 17.1957 0.686729
\(628\) 2.85325 0.113857
\(629\) 4.60388 0.183569
\(630\) −1.35690 −0.0540600
\(631\) −40.6601 −1.61865 −0.809326 0.587359i \(-0.800168\pi\)
−0.809326 + 0.587359i \(0.800168\pi\)
\(632\) 0.259061 0.0103049
\(633\) 16.2935 0.647609
\(634\) −9.25965 −0.367748
\(635\) −2.23490 −0.0886892
\(636\) −14.2349 −0.564450
\(637\) 0 0
\(638\) 27.2054 1.07707
\(639\) 11.7506 0.464848
\(640\) 1.00000 0.0395285
\(641\) −1.92633 −0.0760854 −0.0380427 0.999276i \(-0.512112\pi\)
−0.0380427 + 0.999276i \(0.512112\pi\)
\(642\) −3.92394 −0.154865
\(643\) 3.37973 0.133284 0.0666418 0.997777i \(-0.478772\pi\)
0.0666418 + 0.997777i \(0.478772\pi\)
\(644\) −1.81163 −0.0713881
\(645\) −4.41789 −0.173954
\(646\) −6.65279 −0.261751
\(647\) −10.8181 −0.425302 −0.212651 0.977128i \(-0.568210\pi\)
−0.212651 + 0.977128i \(0.568210\pi\)
\(648\) 1.00000 0.0392837
\(649\) −73.6118 −2.88951
\(650\) 0 0
\(651\) 3.80194 0.149010
\(652\) 15.1075 0.591656
\(653\) 33.7646 1.32131 0.660656 0.750689i \(-0.270278\pi\)
0.660656 + 0.750689i \(0.270278\pi\)
\(654\) 0.554958 0.0217006
\(655\) −8.51573 −0.332737
\(656\) 10.6310 0.415072
\(657\) −12.1032 −0.472191
\(658\) −6.54527 −0.255161
\(659\) −16.8592 −0.656742 −0.328371 0.944549i \(-0.606500\pi\)
−0.328371 + 0.944549i \(0.606500\pi\)
\(660\) 5.29590 0.206143
\(661\) −4.71140 −0.183252 −0.0916261 0.995793i \(-0.529206\pi\)
−0.0916261 + 0.995793i \(0.529206\pi\)
\(662\) 21.8582 0.849541
\(663\) 0 0
\(664\) 3.02177 0.117267
\(665\) −4.40581 −0.170850
\(666\) −2.24698 −0.0870687
\(667\) −6.85862 −0.265567
\(668\) 16.0881 0.622469
\(669\) 2.28083 0.0881820
\(670\) 9.03684 0.349123
\(671\) 19.2010 0.741248
\(672\) 1.35690 0.0523434
\(673\) 3.12797 0.120574 0.0602871 0.998181i \(-0.480798\pi\)
0.0602871 + 0.998181i \(0.480798\pi\)
\(674\) 14.5013 0.558567
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −1.50737 −0.0579329 −0.0289664 0.999580i \(-0.509222\pi\)
−0.0289664 + 0.999580i \(0.509222\pi\)
\(678\) −4.39373 −0.168740
\(679\) 7.93900 0.304671
\(680\) −2.04892 −0.0785724
\(681\) 3.37196 0.129214
\(682\) −14.8388 −0.568206
\(683\) −31.0726 −1.18896 −0.594480 0.804110i \(-0.702642\pi\)
−0.594480 + 0.804110i \(0.702642\pi\)
\(684\) 3.24698 0.124151
\(685\) 13.1685 0.503143
\(686\) 16.4983 0.629907
\(687\) −18.3763 −0.701099
\(688\) 4.41789 0.168431
\(689\) 0 0
\(690\) −1.33513 −0.0508274
\(691\) 26.7469 1.01750 0.508750 0.860914i \(-0.330108\pi\)
0.508750 + 0.860914i \(0.330108\pi\)
\(692\) −15.4795 −0.588442
\(693\) 7.18598 0.272973
\(694\) 7.84117 0.297647
\(695\) 7.17629 0.272212
\(696\) 5.13706 0.194720
\(697\) −21.7821 −0.825055
\(698\) 20.4252 0.773105
\(699\) −10.0858 −0.381478
\(700\) −1.35690 −0.0512858
\(701\) 11.2054 0.423221 0.211610 0.977354i \(-0.432129\pi\)
0.211610 + 0.977354i \(0.432129\pi\)
\(702\) 0 0
\(703\) −7.29590 −0.275170
\(704\) −5.29590 −0.199597
\(705\) −4.82371 −0.181671
\(706\) 13.2905 0.500195
\(707\) 11.3927 0.428465
\(708\) −13.8998 −0.522385
\(709\) −26.9933 −1.01375 −0.506877 0.862018i \(-0.669200\pi\)
−0.506877 + 0.862018i \(0.669200\pi\)
\(710\) 11.7506 0.440993
\(711\) 0.259061 0.00971555
\(712\) −5.47650 −0.205241
\(713\) 3.74094 0.140099
\(714\) −2.78017 −0.104045
\(715\) 0 0
\(716\) 19.0737 0.712817
\(717\) 20.0834 0.750027
\(718\) −12.8267 −0.478688
\(719\) −15.4601 −0.576565 −0.288282 0.957545i \(-0.593084\pi\)
−0.288282 + 0.957545i \(0.593084\pi\)
\(720\) 1.00000 0.0372678
\(721\) −18.5017 −0.689040
\(722\) −8.45712 −0.314742
\(723\) −2.99462 −0.111371
\(724\) 2.16852 0.0805925
\(725\) −5.13706 −0.190786
\(726\) −17.0465 −0.632656
\(727\) −10.9148 −0.404809 −0.202404 0.979302i \(-0.564876\pi\)
−0.202404 + 0.979302i \(0.564876\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.1032 −0.447960
\(731\) −9.05190 −0.334797
\(732\) 3.62565 0.134008
\(733\) 13.5539 0.500624 0.250312 0.968165i \(-0.419467\pi\)
0.250312 + 0.968165i \(0.419467\pi\)
\(734\) −17.9554 −0.662746
\(735\) 5.15883 0.190286
\(736\) 1.33513 0.0492134
\(737\) −47.8582 −1.76288
\(738\) 10.6310 0.391333
\(739\) −15.4679 −0.568995 −0.284498 0.958677i \(-0.591827\pi\)
−0.284498 + 0.958677i \(0.591827\pi\)
\(740\) −2.24698 −0.0826006
\(741\) 0 0
\(742\) −19.3153 −0.709086
\(743\) −40.2040 −1.47494 −0.737471 0.675378i \(-0.763980\pi\)
−0.737471 + 0.675378i \(0.763980\pi\)
\(744\) −2.80194 −0.102724
\(745\) −9.66248 −0.354006
\(746\) 27.8732 1.02051
\(747\) 3.02177 0.110561
\(748\) 10.8509 0.396747
\(749\) −5.32437 −0.194548
\(750\) −1.00000 −0.0365148
\(751\) −19.8465 −0.724211 −0.362105 0.932137i \(-0.617942\pi\)
−0.362105 + 0.932137i \(0.617942\pi\)
\(752\) 4.82371 0.175903
\(753\) 7.20237 0.262469
\(754\) 0 0
\(755\) 7.35152 0.267549
\(756\) 1.35690 0.0493498
\(757\) 39.8883 1.44976 0.724882 0.688873i \(-0.241894\pi\)
0.724882 + 0.688873i \(0.241894\pi\)
\(758\) −13.8702 −0.503790
\(759\) 7.07069 0.256650
\(760\) 3.24698 0.117780
\(761\) −31.8756 −1.15549 −0.577745 0.816217i \(-0.696067\pi\)
−0.577745 + 0.816217i \(0.696067\pi\)
\(762\) 2.23490 0.0809618
\(763\) 0.753020 0.0272612
\(764\) −11.7409 −0.424772
\(765\) −2.04892 −0.0740788
\(766\) 26.8984 0.971880
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 43.0672 1.55304 0.776522 0.630090i \(-0.216982\pi\)
0.776522 + 0.630090i \(0.216982\pi\)
\(770\) 7.18598 0.258965
\(771\) 18.8291 0.678113
\(772\) −24.6722 −0.887971
\(773\) −33.1282 −1.19154 −0.595770 0.803155i \(-0.703153\pi\)
−0.595770 + 0.803155i \(0.703153\pi\)
\(774\) 4.41789 0.158798
\(775\) 2.80194 0.100649
\(776\) −5.85086 −0.210033
\(777\) −3.04892 −0.109379
\(778\) −0.187309 −0.00671534
\(779\) 34.5187 1.23676
\(780\) 0 0
\(781\) −62.2301 −2.22677
\(782\) −2.73556 −0.0978235
\(783\) 5.13706 0.183584
\(784\) −5.15883 −0.184244
\(785\) 2.85325 0.101837
\(786\) 8.51573 0.303746
\(787\) −35.9138 −1.28019 −0.640094 0.768297i \(-0.721105\pi\)
−0.640094 + 0.768297i \(0.721105\pi\)
\(788\) 26.2010 0.933374
\(789\) 17.9245 0.638130
\(790\) 0.259061 0.00921698
\(791\) −5.96184 −0.211978
\(792\) −5.29590 −0.188182
\(793\) 0 0
\(794\) 8.58211 0.304568
\(795\) −14.2349 −0.504860
\(796\) 19.6722 0.697262
\(797\) −29.7918 −1.05528 −0.527639 0.849468i \(-0.676923\pi\)
−0.527639 + 0.849468i \(0.676923\pi\)
\(798\) 4.40581 0.155964
\(799\) −9.88338 −0.349649
\(800\) 1.00000 0.0353553
\(801\) −5.47650 −0.193503
\(802\) 23.9748 0.846579
\(803\) 64.0974 2.26195
\(804\) −9.03684 −0.318705
\(805\) −1.81163 −0.0638514
\(806\) 0 0
\(807\) 22.4045 0.788675
\(808\) −8.39612 −0.295375
\(809\) 1.52648 0.0536683 0.0268341 0.999640i \(-0.491457\pi\)
0.0268341 + 0.999640i \(0.491457\pi\)
\(810\) 1.00000 0.0351364
\(811\) −51.6039 −1.81206 −0.906029 0.423217i \(-0.860901\pi\)
−0.906029 + 0.423217i \(0.860901\pi\)
\(812\) 6.97046 0.244615
\(813\) 23.2325 0.814800
\(814\) 11.8998 0.417087
\(815\) 15.1075 0.529193
\(816\) 2.04892 0.0717265
\(817\) 14.3448 0.501862
\(818\) −8.37926 −0.292974
\(819\) 0 0
\(820\) 10.6310 0.371251
\(821\) 17.4795 0.610038 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(822\) −13.1685 −0.459305
\(823\) −25.5566 −0.890848 −0.445424 0.895320i \(-0.646947\pi\)
−0.445424 + 0.895320i \(0.646947\pi\)
\(824\) 13.6353 0.475009
\(825\) 5.29590 0.184379
\(826\) −18.8605 −0.656242
\(827\) −9.16182 −0.318588 −0.159294 0.987231i \(-0.550922\pi\)
−0.159294 + 0.987231i \(0.550922\pi\)
\(828\) 1.33513 0.0463988
\(829\) 48.6493 1.68966 0.844831 0.535034i \(-0.179701\pi\)
0.844831 + 0.535034i \(0.179701\pi\)
\(830\) 3.02177 0.104887
\(831\) 9.78448 0.339420
\(832\) 0 0
\(833\) 10.5700 0.366230
\(834\) −7.17629 −0.248495
\(835\) 16.0881 0.556753
\(836\) −17.1957 −0.594725
\(837\) −2.80194 −0.0968491
\(838\) 37.0887 1.28121
\(839\) −8.71246 −0.300788 −0.150394 0.988626i \(-0.548054\pi\)
−0.150394 + 0.988626i \(0.548054\pi\)
\(840\) 1.35690 0.0468174
\(841\) −2.61058 −0.0900200
\(842\) 4.90515 0.169043
\(843\) −21.6286 −0.744930
\(844\) −16.2935 −0.560846
\(845\) 0 0
\(846\) 4.82371 0.165842
\(847\) −23.1304 −0.794769
\(848\) 14.2349 0.488828
\(849\) −3.38942 −0.116325
\(850\) −2.04892 −0.0702773
\(851\) −3.00000 −0.102839
\(852\) −11.7506 −0.402570
\(853\) −5.41060 −0.185255 −0.0926277 0.995701i \(-0.529527\pi\)
−0.0926277 + 0.995701i \(0.529527\pi\)
\(854\) 4.91962 0.168346
\(855\) 3.24698 0.111044
\(856\) 3.92394 0.134117
\(857\) 0.165275 0.00564570 0.00282285 0.999996i \(-0.499101\pi\)
0.00282285 + 0.999996i \(0.499101\pi\)
\(858\) 0 0
\(859\) 47.0853 1.60653 0.803264 0.595622i \(-0.203095\pi\)
0.803264 + 0.595622i \(0.203095\pi\)
\(860\) 4.41789 0.150649
\(861\) 14.4252 0.491609
\(862\) −6.06531 −0.206585
\(863\) −31.2553 −1.06394 −0.531972 0.846762i \(-0.678549\pi\)
−0.531972 + 0.846762i \(0.678549\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −15.4795 −0.526318
\(866\) −14.8877 −0.505904
\(867\) 12.8019 0.434777
\(868\) −3.80194 −0.129046
\(869\) −1.37196 −0.0465406
\(870\) 5.13706 0.174163
\(871\) 0 0
\(872\) −0.554958 −0.0187933
\(873\) −5.85086 −0.198021
\(874\) 4.33513 0.146638
\(875\) −1.35690 −0.0458715
\(876\) 12.1032 0.408930
\(877\) 16.9898 0.573706 0.286853 0.957975i \(-0.407391\pi\)
0.286853 + 0.957975i \(0.407391\pi\)
\(878\) −21.1336 −0.713225
\(879\) −34.1280 −1.15111
\(880\) −5.29590 −0.178525
\(881\) −3.29159 −0.110896 −0.0554482 0.998462i \(-0.517659\pi\)
−0.0554482 + 0.998462i \(0.517659\pi\)
\(882\) −5.15883 −0.173707
\(883\) 17.3157 0.582721 0.291361 0.956613i \(-0.405892\pi\)
0.291361 + 0.956613i \(0.405892\pi\)
\(884\) 0 0
\(885\) −13.8998 −0.467236
\(886\) 28.0683 0.942973
\(887\) 15.6775 0.526401 0.263200 0.964741i \(-0.415222\pi\)
0.263200 + 0.964741i \(0.415222\pi\)
\(888\) 2.24698 0.0754037
\(889\) 3.03252 0.101708
\(890\) −5.47650 −0.183573
\(891\) −5.29590 −0.177419
\(892\) −2.28083 −0.0763679
\(893\) 15.6625 0.524125
\(894\) 9.66248 0.323162
\(895\) 19.0737 0.637563
\(896\) −1.35690 −0.0453307
\(897\) 0 0
\(898\) 19.2446 0.642200
\(899\) −14.3937 −0.480058
\(900\) 1.00000 0.0333333
\(901\) −29.1661 −0.971665
\(902\) −56.3008 −1.87461
\(903\) 5.99462 0.199489
\(904\) 4.39373 0.146133
\(905\) 2.16852 0.0720841
\(906\) −7.35152 −0.244238
\(907\) −25.8877 −0.859587 −0.429793 0.902927i \(-0.641414\pi\)
−0.429793 + 0.902927i \(0.641414\pi\)
\(908\) −3.37196 −0.111902
\(909\) −8.39612 −0.278482
\(910\) 0 0
\(911\) −37.5220 −1.24316 −0.621579 0.783351i \(-0.713509\pi\)
−0.621579 + 0.783351i \(0.713509\pi\)
\(912\) −3.24698 −0.107518
\(913\) −16.0030 −0.529621
\(914\) −37.8485 −1.25192
\(915\) 3.62565 0.119860
\(916\) 18.3763 0.607169
\(917\) 11.5550 0.381578
\(918\) 2.04892 0.0676243
\(919\) −5.56358 −0.183526 −0.0917628 0.995781i \(-0.529250\pi\)
−0.0917628 + 0.995781i \(0.529250\pi\)
\(920\) 1.33513 0.0440178
\(921\) 15.6601 0.516017
\(922\) −11.2228 −0.369604
\(923\) 0 0
\(924\) −7.18598 −0.236401
\(925\) −2.24698 −0.0738802
\(926\) 25.0726 0.823937
\(927\) 13.6353 0.447843
\(928\) −5.13706 −0.168632
\(929\) 40.2602 1.32090 0.660448 0.750872i \(-0.270366\pi\)
0.660448 + 0.750872i \(0.270366\pi\)
\(930\) −2.80194 −0.0918792
\(931\) −16.7506 −0.548980
\(932\) 10.0858 0.330370
\(933\) −25.7972 −0.844561
\(934\) 36.9657 1.20955
\(935\) 10.8509 0.354861
\(936\) 0 0
\(937\) −30.5356 −0.997554 −0.498777 0.866730i \(-0.666217\pi\)
−0.498777 + 0.866730i \(0.666217\pi\)
\(938\) −12.2620 −0.400370
\(939\) 15.6243 0.509880
\(940\) 4.82371 0.157332
\(941\) −41.6461 −1.35762 −0.678812 0.734312i \(-0.737505\pi\)
−0.678812 + 0.734312i \(0.737505\pi\)
\(942\) −2.85325 −0.0929638
\(943\) 14.1938 0.462212
\(944\) 13.8998 0.452399
\(945\) 1.35690 0.0441398
\(946\) −23.3967 −0.760693
\(947\) −21.7724 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(948\) −0.259061 −0.00841392
\(949\) 0 0
\(950\) 3.24698 0.105346
\(951\) 9.25965 0.300265
\(952\) 2.78017 0.0901057
\(953\) −41.9560 −1.35909 −0.679544 0.733635i \(-0.737822\pi\)
−0.679544 + 0.733635i \(0.737822\pi\)
\(954\) 14.2349 0.460872
\(955\) −11.7409 −0.379928
\(956\) −20.0834 −0.649542
\(957\) −27.2054 −0.879424
\(958\) 31.4590 1.01640
\(959\) −17.8683 −0.576998
\(960\) −1.00000 −0.0322749
\(961\) −23.1491 −0.746747
\(962\) 0 0
\(963\) 3.92394 0.126447
\(964\) 2.99462 0.0964503
\(965\) −24.6722 −0.794225
\(966\) 1.81163 0.0582881
\(967\) −16.7041 −0.537168 −0.268584 0.963256i \(-0.586556\pi\)
−0.268584 + 0.963256i \(0.586556\pi\)
\(968\) 17.0465 0.547896
\(969\) 6.65279 0.213718
\(970\) −5.85086 −0.187860
\(971\) −20.7275 −0.665178 −0.332589 0.943072i \(-0.607922\pi\)
−0.332589 + 0.943072i \(0.607922\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −9.73748 −0.312169
\(974\) 18.8713 0.604675
\(975\) 0 0
\(976\) −3.62565 −0.116054
\(977\) 33.4547 1.07031 0.535156 0.844753i \(-0.320253\pi\)
0.535156 + 0.844753i \(0.320253\pi\)
\(978\) −15.1075 −0.483085
\(979\) 29.0030 0.926939
\(980\) −5.15883 −0.164793
\(981\) −0.554958 −0.0177184
\(982\) −16.3961 −0.523221
\(983\) −31.8605 −1.01619 −0.508097 0.861300i \(-0.669651\pi\)
−0.508097 + 0.861300i \(0.669651\pi\)
\(984\) −10.6310 −0.338905
\(985\) 26.2010 0.834835
\(986\) 10.5254 0.335198
\(987\) 6.54527 0.208338
\(988\) 0 0
\(989\) 5.89844 0.187560
\(990\) −5.29590 −0.168315
\(991\) −21.0925 −0.670024 −0.335012 0.942214i \(-0.608740\pi\)
−0.335012 + 0.942214i \(0.608740\pi\)
\(992\) 2.80194 0.0889616
\(993\) −21.8582 −0.693647
\(994\) −15.9444 −0.505725
\(995\) 19.6722 0.623650
\(996\) −3.02177 −0.0957485
\(997\) 17.7832 0.563198 0.281599 0.959532i \(-0.409135\pi\)
0.281599 + 0.959532i \(0.409135\pi\)
\(998\) 40.2452 1.27394
\(999\) 2.24698 0.0710913
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 5070.2.a.bu.1.2 yes 3
13.5 odd 4 5070.2.b.t.1351.2 6
13.8 odd 4 5070.2.b.t.1351.5 6
13.12 even 2 5070.2.a.bj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bj.1.2 3 13.12 even 2
5070.2.a.bu.1.2 yes 3 1.1 even 1 trivial
5070.2.b.t.1351.2 6 13.5 odd 4
5070.2.b.t.1351.5 6 13.8 odd 4