Properties

Label 2001.2.a.i.1.4
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8\)
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 1.4
Root \(0.743347\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.743347 q^{2} +1.00000 q^{3} -1.44743 q^{4} -3.36180 q^{5} -0.743347 q^{6} -2.98340 q^{7} +2.56264 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.743347 q^{2} +1.00000 q^{3} -1.44743 q^{4} -3.36180 q^{5} -0.743347 q^{6} -2.98340 q^{7} +2.56264 q^{8} +1.00000 q^{9} +2.49898 q^{10} +1.73328 q^{11} -1.44743 q^{12} +0.418547 q^{13} +2.21770 q^{14} -3.36180 q^{15} +0.989937 q^{16} +3.04186 q^{17} -0.743347 q^{18} +6.79263 q^{19} +4.86598 q^{20} -2.98340 q^{21} -1.28843 q^{22} +1.00000 q^{23} +2.56264 q^{24} +6.30168 q^{25} -0.311125 q^{26} +1.00000 q^{27} +4.31828 q^{28} -1.00000 q^{29} +2.49898 q^{30} -1.26899 q^{31} -5.86115 q^{32} +1.73328 q^{33} -2.26116 q^{34} +10.0296 q^{35} -1.44743 q^{36} -8.40619 q^{37} -5.04928 q^{38} +0.418547 q^{39} -8.61508 q^{40} -5.67691 q^{41} +2.21770 q^{42} +2.38122 q^{43} -2.50882 q^{44} -3.36180 q^{45} -0.743347 q^{46} -1.01953 q^{47} +0.989937 q^{48} +1.90067 q^{49} -4.68433 q^{50} +3.04186 q^{51} -0.605819 q^{52} +1.08924 q^{53} -0.743347 q^{54} -5.82695 q^{55} -7.64538 q^{56} +6.79263 q^{57} +0.743347 q^{58} -0.434930 q^{59} +4.86598 q^{60} -11.8388 q^{61} +0.943302 q^{62} -2.98340 q^{63} +2.37699 q^{64} -1.40707 q^{65} -1.28843 q^{66} -4.29594 q^{67} -4.40289 q^{68} +1.00000 q^{69} -7.45546 q^{70} -7.01482 q^{71} +2.56264 q^{72} -9.28137 q^{73} +6.24872 q^{74} +6.30168 q^{75} -9.83189 q^{76} -5.17108 q^{77} -0.311125 q^{78} +2.43604 q^{79} -3.32797 q^{80} +1.00000 q^{81} +4.21991 q^{82} +15.3185 q^{83} +4.31828 q^{84} -10.2261 q^{85} -1.77007 q^{86} -1.00000 q^{87} +4.44179 q^{88} +11.8322 q^{89} +2.49898 q^{90} -1.24869 q^{91} -1.44743 q^{92} -1.26899 q^{93} +0.757862 q^{94} -22.8354 q^{95} -5.86115 q^{96} +4.37241 q^{97} -1.41286 q^{98} +1.73328 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + O(q^{10}) \) \( 7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + 3q^{10} - 4q^{11} + 5q^{12} - 18q^{13} - 2q^{14} - 3q^{15} - 7q^{16} - 3q^{17} - 3q^{18} - 4q^{19} - 2q^{20} - 5q^{21} - 26q^{22} + 7q^{23} - 6q^{24} - 8q^{25} - 7q^{26} + 7q^{27} - 6q^{28} - 7q^{29} + 3q^{30} - 22q^{31} + 5q^{32} - 4q^{33} + 9q^{34} + 3q^{35} + 5q^{36} - 25q^{37} + 14q^{38} - 18q^{39} - 10q^{40} - 13q^{41} - 2q^{42} - 2q^{43} + 4q^{44} - 3q^{45} - 3q^{46} - 25q^{47} - 7q^{48} - 8q^{49} + 19q^{50} - 3q^{51} - 12q^{52} - 5q^{53} - 3q^{54} - 15q^{55} + 18q^{56} - 4q^{57} + 3q^{58} + 11q^{59} - 2q^{60} - 33q^{61} + 28q^{62} - 5q^{63} - 14q^{64} - 2q^{65} - 26q^{66} + 8q^{67} + 12q^{68} + 7q^{69} - 22q^{70} - 6q^{71} - 6q^{72} + 15q^{73} + 34q^{74} - 8q^{75} - 28q^{76} - q^{77} - 7q^{78} - 15q^{79} - 12q^{80} + 7q^{81} - 14q^{82} + 21q^{83} - 6q^{84} - 28q^{85} - 12q^{86} - 7q^{87} - 13q^{88} + 8q^{89} + 3q^{90} + 6q^{91} + 5q^{92} - 22q^{93} - 35q^{94} - 25q^{95} + 5q^{96} + 13q^{97} + q^{98} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.743347 −0.525626 −0.262813 0.964847i \(-0.584650\pi\)
−0.262813 + 0.964847i \(0.584650\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.44743 −0.723717
\(5\) −3.36180 −1.50344 −0.751721 0.659482i \(-0.770776\pi\)
−0.751721 + 0.659482i \(0.770776\pi\)
\(6\) −0.743347 −0.303470
\(7\) −2.98340 −1.12762 −0.563809 0.825905i \(-0.690665\pi\)
−0.563809 + 0.825905i \(0.690665\pi\)
\(8\) 2.56264 0.906031
\(9\) 1.00000 0.333333
\(10\) 2.49898 0.790248
\(11\) 1.73328 0.522605 0.261302 0.965257i \(-0.415848\pi\)
0.261302 + 0.965257i \(0.415848\pi\)
\(12\) −1.44743 −0.417838
\(13\) 0.418547 0.116084 0.0580420 0.998314i \(-0.481514\pi\)
0.0580420 + 0.998314i \(0.481514\pi\)
\(14\) 2.21770 0.592706
\(15\) −3.36180 −0.868012
\(16\) 0.989937 0.247484
\(17\) 3.04186 0.737758 0.368879 0.929477i \(-0.379742\pi\)
0.368879 + 0.929477i \(0.379742\pi\)
\(18\) −0.743347 −0.175209
\(19\) 6.79263 1.55834 0.779168 0.626815i \(-0.215642\pi\)
0.779168 + 0.626815i \(0.215642\pi\)
\(20\) 4.86598 1.08807
\(21\) −2.98340 −0.651031
\(22\) −1.28843 −0.274695
\(23\) 1.00000 0.208514
\(24\) 2.56264 0.523097
\(25\) 6.30168 1.26034
\(26\) −0.311125 −0.0610167
\(27\) 1.00000 0.192450
\(28\) 4.31828 0.816077
\(29\) −1.00000 −0.185695
\(30\) 2.49898 0.456250
\(31\) −1.26899 −0.227918 −0.113959 0.993485i \(-0.536353\pi\)
−0.113959 + 0.993485i \(0.536353\pi\)
\(32\) −5.86115 −1.03611
\(33\) 1.73328 0.301726
\(34\) −2.26116 −0.387785
\(35\) 10.0296 1.69531
\(36\) −1.44743 −0.241239
\(37\) −8.40619 −1.38197 −0.690985 0.722869i \(-0.742823\pi\)
−0.690985 + 0.722869i \(0.742823\pi\)
\(38\) −5.04928 −0.819102
\(39\) 0.418547 0.0670211
\(40\) −8.61508 −1.36216
\(41\) −5.67691 −0.886584 −0.443292 0.896377i \(-0.646189\pi\)
−0.443292 + 0.896377i \(0.646189\pi\)
\(42\) 2.21770 0.342199
\(43\) 2.38122 0.363133 0.181566 0.983379i \(-0.441883\pi\)
0.181566 + 0.983379i \(0.441883\pi\)
\(44\) −2.50882 −0.378218
\(45\) −3.36180 −0.501147
\(46\) −0.743347 −0.109601
\(47\) −1.01953 −0.148713 −0.0743566 0.997232i \(-0.523690\pi\)
−0.0743566 + 0.997232i \(0.523690\pi\)
\(48\) 0.989937 0.142885
\(49\) 1.90067 0.271525
\(50\) −4.68433 −0.662465
\(51\) 3.04186 0.425945
\(52\) −0.605819 −0.0840120
\(53\) 1.08924 0.149619 0.0748095 0.997198i \(-0.476165\pi\)
0.0748095 + 0.997198i \(0.476165\pi\)
\(54\) −0.743347 −0.101157
\(55\) −5.82695 −0.785706
\(56\) −7.64538 −1.02166
\(57\) 6.79263 0.899706
\(58\) 0.743347 0.0976063
\(59\) −0.434930 −0.0566231 −0.0283115 0.999599i \(-0.509013\pi\)
−0.0283115 + 0.999599i \(0.509013\pi\)
\(60\) 4.86598 0.628195
\(61\) −11.8388 −1.51580 −0.757901 0.652369i \(-0.773775\pi\)
−0.757901 + 0.652369i \(0.773775\pi\)
\(62\) 0.943302 0.119800
\(63\) −2.98340 −0.375873
\(64\) 2.37699 0.297124
\(65\) −1.40707 −0.174525
\(66\) −1.28843 −0.158595
\(67\) −4.29594 −0.524833 −0.262416 0.964955i \(-0.584519\pi\)
−0.262416 + 0.964955i \(0.584519\pi\)
\(68\) −4.40289 −0.533929
\(69\) 1.00000 0.120386
\(70\) −7.45546 −0.891098
\(71\) −7.01482 −0.832506 −0.416253 0.909249i \(-0.636657\pi\)
−0.416253 + 0.909249i \(0.636657\pi\)
\(72\) 2.56264 0.302010
\(73\) −9.28137 −1.08630 −0.543151 0.839635i \(-0.682769\pi\)
−0.543151 + 0.839635i \(0.682769\pi\)
\(74\) 6.24872 0.726399
\(75\) 6.30168 0.727655
\(76\) −9.83189 −1.12780
\(77\) −5.17108 −0.589299
\(78\) −0.311125 −0.0352280
\(79\) 2.43604 0.274076 0.137038 0.990566i \(-0.456242\pi\)
0.137038 + 0.990566i \(0.456242\pi\)
\(80\) −3.32797 −0.372078
\(81\) 1.00000 0.111111
\(82\) 4.21991 0.466011
\(83\) 15.3185 1.68142 0.840712 0.541482i \(-0.182136\pi\)
0.840712 + 0.541482i \(0.182136\pi\)
\(84\) 4.31828 0.471163
\(85\) −10.2261 −1.10918
\(86\) −1.77007 −0.190872
\(87\) −1.00000 −0.107211
\(88\) 4.44179 0.473496
\(89\) 11.8322 1.25421 0.627105 0.778935i \(-0.284240\pi\)
0.627105 + 0.778935i \(0.284240\pi\)
\(90\) 2.49898 0.263416
\(91\) −1.24869 −0.130898
\(92\) −1.44743 −0.150906
\(93\) −1.26899 −0.131588
\(94\) 0.757862 0.0781675
\(95\) −22.8354 −2.34287
\(96\) −5.86115 −0.598201
\(97\) 4.37241 0.443951 0.221975 0.975052i \(-0.428750\pi\)
0.221975 + 0.975052i \(0.428750\pi\)
\(98\) −1.41286 −0.142720
\(99\) 1.73328 0.174202
\(100\) −9.12126 −0.912126
\(101\) −7.23477 −0.719886 −0.359943 0.932974i \(-0.617204\pi\)
−0.359943 + 0.932974i \(0.617204\pi\)
\(102\) −2.26116 −0.223888
\(103\) −17.9998 −1.77358 −0.886788 0.462176i \(-0.847069\pi\)
−0.886788 + 0.462176i \(0.847069\pi\)
\(104\) 1.07258 0.105176
\(105\) 10.0296 0.978787
\(106\) −0.809686 −0.0786437
\(107\) −5.18130 −0.500895 −0.250447 0.968130i \(-0.580578\pi\)
−0.250447 + 0.968130i \(0.580578\pi\)
\(108\) −1.44743 −0.139279
\(109\) −11.3146 −1.08375 −0.541873 0.840460i \(-0.682285\pi\)
−0.541873 + 0.840460i \(0.682285\pi\)
\(110\) 4.33145 0.412987
\(111\) −8.40619 −0.797880
\(112\) −2.95338 −0.279068
\(113\) 6.08524 0.572451 0.286226 0.958162i \(-0.407599\pi\)
0.286226 + 0.958162i \(0.407599\pi\)
\(114\) −5.04928 −0.472909
\(115\) −3.36180 −0.313489
\(116\) 1.44743 0.134391
\(117\) 0.418547 0.0386946
\(118\) 0.323304 0.0297625
\(119\) −9.07507 −0.831910
\(120\) −8.61508 −0.786445
\(121\) −7.99572 −0.726884
\(122\) 8.80034 0.796745
\(123\) −5.67691 −0.511869
\(124\) 1.83678 0.164948
\(125\) −4.37597 −0.391399
\(126\) 2.21770 0.197569
\(127\) −2.68174 −0.237966 −0.118983 0.992896i \(-0.537963\pi\)
−0.118983 + 0.992896i \(0.537963\pi\)
\(128\) 9.95537 0.879938
\(129\) 2.38122 0.209655
\(130\) 1.04594 0.0917350
\(131\) 5.26968 0.460414 0.230207 0.973142i \(-0.426060\pi\)
0.230207 + 0.973142i \(0.426060\pi\)
\(132\) −2.50882 −0.218364
\(133\) −20.2651 −1.75721
\(134\) 3.19338 0.275866
\(135\) −3.36180 −0.289337
\(136\) 7.79518 0.668432
\(137\) 11.0969 0.948071 0.474036 0.880506i \(-0.342797\pi\)
0.474036 + 0.880506i \(0.342797\pi\)
\(138\) −0.743347 −0.0632779
\(139\) −13.7752 −1.16840 −0.584200 0.811610i \(-0.698592\pi\)
−0.584200 + 0.811610i \(0.698592\pi\)
\(140\) −14.5172 −1.22692
\(141\) −1.01953 −0.0858596
\(142\) 5.21445 0.437587
\(143\) 0.725460 0.0606660
\(144\) 0.989937 0.0824948
\(145\) 3.36180 0.279182
\(146\) 6.89928 0.570989
\(147\) 1.90067 0.156765
\(148\) 12.1674 1.00016
\(149\) 2.15134 0.176244 0.0881222 0.996110i \(-0.471913\pi\)
0.0881222 + 0.996110i \(0.471913\pi\)
\(150\) −4.68433 −0.382474
\(151\) 18.3565 1.49383 0.746915 0.664920i \(-0.231534\pi\)
0.746915 + 0.664920i \(0.231534\pi\)
\(152\) 17.4071 1.41190
\(153\) 3.04186 0.245919
\(154\) 3.84391 0.309751
\(155\) 4.26610 0.342661
\(156\) −0.605819 −0.0485043
\(157\) −0.545808 −0.0435602 −0.0217801 0.999763i \(-0.506933\pi\)
−0.0217801 + 0.999763i \(0.506933\pi\)
\(158\) −1.81082 −0.144061
\(159\) 1.08924 0.0863826
\(160\) 19.7040 1.55774
\(161\) −2.98340 −0.235125
\(162\) −0.743347 −0.0584029
\(163\) −0.544969 −0.0426853 −0.0213426 0.999772i \(-0.506794\pi\)
−0.0213426 + 0.999772i \(0.506794\pi\)
\(164\) 8.21695 0.641636
\(165\) −5.82695 −0.453627
\(166\) −11.3870 −0.883800
\(167\) 13.5418 1.04790 0.523950 0.851749i \(-0.324458\pi\)
0.523950 + 0.851749i \(0.324458\pi\)
\(168\) −7.64538 −0.589854
\(169\) −12.8248 −0.986525
\(170\) 7.60154 0.583012
\(171\) 6.79263 0.519445
\(172\) −3.44666 −0.262806
\(173\) −2.48390 −0.188847 −0.0944236 0.995532i \(-0.530101\pi\)
−0.0944236 + 0.995532i \(0.530101\pi\)
\(174\) 0.743347 0.0563530
\(175\) −18.8004 −1.42118
\(176\) 1.71584 0.129337
\(177\) −0.434930 −0.0326913
\(178\) −8.79543 −0.659245
\(179\) −8.19708 −0.612679 −0.306339 0.951922i \(-0.599104\pi\)
−0.306339 + 0.951922i \(0.599104\pi\)
\(180\) 4.86598 0.362689
\(181\) −8.21935 −0.610940 −0.305470 0.952202i \(-0.598814\pi\)
−0.305470 + 0.952202i \(0.598814\pi\)
\(182\) 0.928211 0.0688036
\(183\) −11.8388 −0.875149
\(184\) 2.56264 0.188920
\(185\) 28.2599 2.07771
\(186\) 0.943302 0.0691663
\(187\) 5.27240 0.385556
\(188\) 1.47570 0.107626
\(189\) −2.98340 −0.217010
\(190\) 16.9747 1.23147
\(191\) −14.0910 −1.01959 −0.509794 0.860296i \(-0.670278\pi\)
−0.509794 + 0.860296i \(0.670278\pi\)
\(192\) 2.37699 0.171545
\(193\) 0.272693 0.0196289 0.00981444 0.999952i \(-0.496876\pi\)
0.00981444 + 0.999952i \(0.496876\pi\)
\(194\) −3.25022 −0.233352
\(195\) −1.40707 −0.100762
\(196\) −2.75110 −0.196507
\(197\) 23.5938 1.68099 0.840495 0.541819i \(-0.182264\pi\)
0.840495 + 0.541819i \(0.182264\pi\)
\(198\) −1.28843 −0.0915649
\(199\) 3.51044 0.248848 0.124424 0.992229i \(-0.460292\pi\)
0.124424 + 0.992229i \(0.460292\pi\)
\(200\) 16.1489 1.14190
\(201\) −4.29594 −0.303012
\(202\) 5.37795 0.378391
\(203\) 2.98340 0.209394
\(204\) −4.40289 −0.308264
\(205\) 19.0846 1.33293
\(206\) 13.3801 0.932238
\(207\) 1.00000 0.0695048
\(208\) 0.414335 0.0287290
\(209\) 11.7736 0.814394
\(210\) −7.45546 −0.514476
\(211\) −0.138150 −0.00951064 −0.00475532 0.999989i \(-0.501514\pi\)
−0.00475532 + 0.999989i \(0.501514\pi\)
\(212\) −1.57661 −0.108282
\(213\) −7.01482 −0.480648
\(214\) 3.85150 0.263283
\(215\) −8.00518 −0.545949
\(216\) 2.56264 0.174366
\(217\) 3.78591 0.257004
\(218\) 8.41070 0.569645
\(219\) −9.28137 −0.627177
\(220\) 8.43413 0.568629
\(221\) 1.27316 0.0856419
\(222\) 6.24872 0.419387
\(223\) 19.7576 1.32306 0.661532 0.749917i \(-0.269907\pi\)
0.661532 + 0.749917i \(0.269907\pi\)
\(224\) 17.4862 1.16834
\(225\) 6.30168 0.420112
\(226\) −4.52345 −0.300895
\(227\) −21.2886 −1.41297 −0.706486 0.707727i \(-0.749721\pi\)
−0.706486 + 0.707727i \(0.749721\pi\)
\(228\) −9.83189 −0.651133
\(229\) −9.98323 −0.659711 −0.329855 0.944031i \(-0.607000\pi\)
−0.329855 + 0.944031i \(0.607000\pi\)
\(230\) 2.49898 0.164778
\(231\) −5.17108 −0.340232
\(232\) −2.56264 −0.168246
\(233\) −11.7610 −0.770490 −0.385245 0.922814i \(-0.625883\pi\)
−0.385245 + 0.922814i \(0.625883\pi\)
\(234\) −0.311125 −0.0203389
\(235\) 3.42744 0.223581
\(236\) 0.629533 0.0409791
\(237\) 2.43604 0.158238
\(238\) 6.74593 0.437274
\(239\) −14.7029 −0.951053 −0.475527 0.879701i \(-0.657742\pi\)
−0.475527 + 0.879701i \(0.657742\pi\)
\(240\) −3.32797 −0.214819
\(241\) −6.48179 −0.417529 −0.208764 0.977966i \(-0.566944\pi\)
−0.208764 + 0.977966i \(0.566944\pi\)
\(242\) 5.94360 0.382069
\(243\) 1.00000 0.0641500
\(244\) 17.1359 1.09701
\(245\) −6.38967 −0.408221
\(246\) 4.21991 0.269052
\(247\) 2.84303 0.180898
\(248\) −3.25197 −0.206501
\(249\) 15.3185 0.970771
\(250\) 3.25286 0.205729
\(251\) 5.51951 0.348389 0.174194 0.984711i \(-0.444268\pi\)
0.174194 + 0.984711i \(0.444268\pi\)
\(252\) 4.31828 0.272026
\(253\) 1.73328 0.108971
\(254\) 1.99347 0.125081
\(255\) −10.2261 −0.640383
\(256\) −12.1543 −0.759643
\(257\) −20.9651 −1.30776 −0.653882 0.756597i \(-0.726861\pi\)
−0.653882 + 0.756597i \(0.726861\pi\)
\(258\) −1.77007 −0.110200
\(259\) 25.0790 1.55833
\(260\) 2.03664 0.126307
\(261\) −1.00000 −0.0618984
\(262\) −3.91720 −0.242005
\(263\) 4.37099 0.269527 0.134763 0.990878i \(-0.456973\pi\)
0.134763 + 0.990878i \(0.456973\pi\)
\(264\) 4.44179 0.273373
\(265\) −3.66181 −0.224943
\(266\) 15.0640 0.923635
\(267\) 11.8322 0.724119
\(268\) 6.21809 0.379831
\(269\) −18.5392 −1.13035 −0.565177 0.824970i \(-0.691192\pi\)
−0.565177 + 0.824970i \(0.691192\pi\)
\(270\) 2.49898 0.152083
\(271\) −17.8222 −1.08262 −0.541311 0.840822i \(-0.682072\pi\)
−0.541311 + 0.840822i \(0.682072\pi\)
\(272\) 3.01125 0.182584
\(273\) −1.24869 −0.0755742
\(274\) −8.24884 −0.498331
\(275\) 10.9226 0.658657
\(276\) −1.44743 −0.0871253
\(277\) 0.652922 0.0392303 0.0196152 0.999808i \(-0.493756\pi\)
0.0196152 + 0.999808i \(0.493756\pi\)
\(278\) 10.2398 0.614142
\(279\) −1.26899 −0.0759726
\(280\) 25.7022 1.53600
\(281\) −24.5072 −1.46198 −0.730988 0.682390i \(-0.760940\pi\)
−0.730988 + 0.682390i \(0.760940\pi\)
\(282\) 0.757862 0.0451300
\(283\) −29.4332 −1.74962 −0.874810 0.484466i \(-0.839014\pi\)
−0.874810 + 0.484466i \(0.839014\pi\)
\(284\) 10.1535 0.602499
\(285\) −22.8354 −1.35265
\(286\) −0.539269 −0.0318876
\(287\) 16.9365 0.999729
\(288\) −5.86115 −0.345372
\(289\) −7.74711 −0.455713
\(290\) −2.49898 −0.146745
\(291\) 4.37241 0.256315
\(292\) 13.4342 0.786176
\(293\) 10.4542 0.610739 0.305369 0.952234i \(-0.401220\pi\)
0.305369 + 0.952234i \(0.401220\pi\)
\(294\) −1.41286 −0.0823996
\(295\) 1.46215 0.0851294
\(296\) −21.5421 −1.25211
\(297\) 1.73328 0.100575
\(298\) −1.59919 −0.0926387
\(299\) 0.418547 0.0242052
\(300\) −9.12126 −0.526616
\(301\) −7.10413 −0.409476
\(302\) −13.6452 −0.785195
\(303\) −7.23477 −0.415627
\(304\) 6.72428 0.385664
\(305\) 39.7996 2.27892
\(306\) −2.26116 −0.129262
\(307\) −29.3141 −1.67304 −0.836522 0.547933i \(-0.815415\pi\)
−0.836522 + 0.547933i \(0.815415\pi\)
\(308\) 7.48480 0.426486
\(309\) −17.9998 −1.02398
\(310\) −3.17119 −0.180112
\(311\) −14.2023 −0.805338 −0.402669 0.915346i \(-0.631917\pi\)
−0.402669 + 0.915346i \(0.631917\pi\)
\(312\) 1.07258 0.0607231
\(313\) −18.3008 −1.03443 −0.517213 0.855857i \(-0.673030\pi\)
−0.517213 + 0.855857i \(0.673030\pi\)
\(314\) 0.405725 0.0228964
\(315\) 10.0296 0.565103
\(316\) −3.52600 −0.198353
\(317\) 1.64584 0.0924397 0.0462198 0.998931i \(-0.485283\pi\)
0.0462198 + 0.998931i \(0.485283\pi\)
\(318\) −0.809686 −0.0454049
\(319\) −1.73328 −0.0970453
\(320\) −7.99097 −0.446709
\(321\) −5.18130 −0.289192
\(322\) 2.21770 0.123588
\(323\) 20.6622 1.14968
\(324\) −1.44743 −0.0804130
\(325\) 2.63754 0.146305
\(326\) 0.405101 0.0224365
\(327\) −11.3146 −0.625701
\(328\) −14.5479 −0.803272
\(329\) 3.04165 0.167692
\(330\) 4.33145 0.238438
\(331\) −14.4258 −0.792914 −0.396457 0.918053i \(-0.629760\pi\)
−0.396457 + 0.918053i \(0.629760\pi\)
\(332\) −22.1725 −1.21688
\(333\) −8.40619 −0.460656
\(334\) −10.0663 −0.550803
\(335\) 14.4421 0.789055
\(336\) −2.95338 −0.161120
\(337\) −7.79363 −0.424546 −0.212273 0.977210i \(-0.568087\pi\)
−0.212273 + 0.977210i \(0.568087\pi\)
\(338\) 9.53329 0.518543
\(339\) 6.08524 0.330505
\(340\) 14.8016 0.802730
\(341\) −2.19953 −0.119111
\(342\) −5.04928 −0.273034
\(343\) 15.2133 0.821443
\(344\) 6.10222 0.329009
\(345\) −3.36180 −0.180993
\(346\) 1.84640 0.0992630
\(347\) −27.5605 −1.47953 −0.739764 0.672867i \(-0.765063\pi\)
−0.739764 + 0.672867i \(0.765063\pi\)
\(348\) 1.44743 0.0775907
\(349\) −29.9855 −1.60509 −0.802543 0.596594i \(-0.796520\pi\)
−0.802543 + 0.596594i \(0.796520\pi\)
\(350\) 13.9752 0.747008
\(351\) 0.418547 0.0223404
\(352\) −10.1590 −0.541479
\(353\) −29.6920 −1.58034 −0.790172 0.612885i \(-0.790009\pi\)
−0.790172 + 0.612885i \(0.790009\pi\)
\(354\) 0.323304 0.0171834
\(355\) 23.5824 1.25162
\(356\) −17.1263 −0.907694
\(357\) −9.07507 −0.480304
\(358\) 6.09328 0.322040
\(359\) 2.09906 0.110784 0.0553921 0.998465i \(-0.482359\pi\)
0.0553921 + 0.998465i \(0.482359\pi\)
\(360\) −8.61508 −0.454054
\(361\) 27.1398 1.42841
\(362\) 6.10983 0.321126
\(363\) −7.99572 −0.419667
\(364\) 1.80740 0.0947335
\(365\) 31.2021 1.63319
\(366\) 8.80034 0.460001
\(367\) 19.4918 1.01746 0.508732 0.860925i \(-0.330114\pi\)
0.508732 + 0.860925i \(0.330114\pi\)
\(368\) 0.989937 0.0516041
\(369\) −5.67691 −0.295528
\(370\) −21.0069 −1.09210
\(371\) −3.24965 −0.168713
\(372\) 1.83678 0.0952328
\(373\) −33.4309 −1.73099 −0.865494 0.500919i \(-0.832995\pi\)
−0.865494 + 0.500919i \(0.832995\pi\)
\(374\) −3.91923 −0.202658
\(375\) −4.37597 −0.225974
\(376\) −2.61268 −0.134739
\(377\) −0.418547 −0.0215562
\(378\) 2.21770 0.114066
\(379\) 23.4822 1.20620 0.603100 0.797665i \(-0.293932\pi\)
0.603100 + 0.797665i \(0.293932\pi\)
\(380\) 33.0528 1.69557
\(381\) −2.68174 −0.137390
\(382\) 10.4745 0.535922
\(383\) 22.0092 1.12462 0.562310 0.826927i \(-0.309913\pi\)
0.562310 + 0.826927i \(0.309913\pi\)
\(384\) 9.95537 0.508033
\(385\) 17.3841 0.885977
\(386\) −0.202706 −0.0103174
\(387\) 2.38122 0.121044
\(388\) −6.32877 −0.321295
\(389\) 14.7724 0.748991 0.374496 0.927229i \(-0.377816\pi\)
0.374496 + 0.927229i \(0.377816\pi\)
\(390\) 1.04594 0.0529632
\(391\) 3.04186 0.153833
\(392\) 4.87074 0.246010
\(393\) 5.26968 0.265820
\(394\) −17.5384 −0.883572
\(395\) −8.18946 −0.412056
\(396\) −2.50882 −0.126073
\(397\) 21.7744 1.09283 0.546413 0.837516i \(-0.315993\pi\)
0.546413 + 0.837516i \(0.315993\pi\)
\(398\) −2.60947 −0.130801
\(399\) −20.2651 −1.01453
\(400\) 6.23826 0.311913
\(401\) −2.37651 −0.118677 −0.0593387 0.998238i \(-0.518899\pi\)
−0.0593387 + 0.998238i \(0.518899\pi\)
\(402\) 3.19338 0.159271
\(403\) −0.531133 −0.0264576
\(404\) 10.4719 0.520994
\(405\) −3.36180 −0.167049
\(406\) −2.21770 −0.110063
\(407\) −14.5703 −0.722224
\(408\) 7.79518 0.385919
\(409\) 0.143213 0.00708141 0.00354071 0.999994i \(-0.498873\pi\)
0.00354071 + 0.999994i \(0.498873\pi\)
\(410\) −14.1865 −0.700621
\(411\) 11.0969 0.547369
\(412\) 26.0536 1.28357
\(413\) 1.29757 0.0638492
\(414\) −0.743347 −0.0365335
\(415\) −51.4977 −2.52792
\(416\) −2.45316 −0.120276
\(417\) −13.7752 −0.674576
\(418\) −8.75185 −0.428067
\(419\) −1.85725 −0.0907324 −0.0453662 0.998970i \(-0.514445\pi\)
−0.0453662 + 0.998970i \(0.514445\pi\)
\(420\) −14.5172 −0.708365
\(421\) 16.7959 0.818584 0.409292 0.912403i \(-0.365776\pi\)
0.409292 + 0.912403i \(0.365776\pi\)
\(422\) 0.102693 0.00499904
\(423\) −1.01953 −0.0495710
\(424\) 2.79134 0.135559
\(425\) 19.1688 0.929823
\(426\) 5.21445 0.252641
\(427\) 35.3199 1.70925
\(428\) 7.49959 0.362506
\(429\) 0.725460 0.0350256
\(430\) 5.95063 0.286965
\(431\) 15.4833 0.745803 0.372901 0.927871i \(-0.378363\pi\)
0.372901 + 0.927871i \(0.378363\pi\)
\(432\) 0.989937 0.0476284
\(433\) −10.8387 −0.520873 −0.260436 0.965491i \(-0.583866\pi\)
−0.260436 + 0.965491i \(0.583866\pi\)
\(434\) −2.81425 −0.135088
\(435\) 3.36180 0.161186
\(436\) 16.3772 0.784326
\(437\) 6.79263 0.324936
\(438\) 6.89928 0.329661
\(439\) 13.0587 0.623260 0.311630 0.950204i \(-0.399125\pi\)
0.311630 + 0.950204i \(0.399125\pi\)
\(440\) −14.9324 −0.711873
\(441\) 1.90067 0.0905082
\(442\) −0.946399 −0.0450156
\(443\) −13.9516 −0.662861 −0.331431 0.943480i \(-0.607531\pi\)
−0.331431 + 0.943480i \(0.607531\pi\)
\(444\) 12.1674 0.577440
\(445\) −39.7774 −1.88563
\(446\) −14.6867 −0.695437
\(447\) 2.15134 0.101755
\(448\) −7.09153 −0.335043
\(449\) 0.108496 0.00512024 0.00256012 0.999997i \(-0.499185\pi\)
0.00256012 + 0.999997i \(0.499185\pi\)
\(450\) −4.68433 −0.220822
\(451\) −9.83969 −0.463333
\(452\) −8.80799 −0.414293
\(453\) 18.3565 0.862463
\(454\) 15.8248 0.742695
\(455\) 4.19785 0.196798
\(456\) 17.4071 0.815161
\(457\) 28.6297 1.33924 0.669621 0.742703i \(-0.266457\pi\)
0.669621 + 0.742703i \(0.266457\pi\)
\(458\) 7.42101 0.346761
\(459\) 3.04186 0.141982
\(460\) 4.86598 0.226878
\(461\) 27.3672 1.27462 0.637310 0.770608i \(-0.280047\pi\)
0.637310 + 0.770608i \(0.280047\pi\)
\(462\) 3.84391 0.178835
\(463\) −9.24333 −0.429574 −0.214787 0.976661i \(-0.568906\pi\)
−0.214787 + 0.976661i \(0.568906\pi\)
\(464\) −0.989937 −0.0459567
\(465\) 4.26610 0.197835
\(466\) 8.74253 0.404990
\(467\) 27.1433 1.25604 0.628022 0.778196i \(-0.283865\pi\)
0.628022 + 0.778196i \(0.283865\pi\)
\(468\) −0.605819 −0.0280040
\(469\) 12.8165 0.591811
\(470\) −2.54778 −0.117520
\(471\) −0.545808 −0.0251495
\(472\) −1.11457 −0.0513022
\(473\) 4.12733 0.189775
\(474\) −1.81082 −0.0831738
\(475\) 42.8050 1.96403
\(476\) 13.1356 0.602068
\(477\) 1.08924 0.0498730
\(478\) 10.9294 0.499898
\(479\) −15.3490 −0.701315 −0.350657 0.936504i \(-0.614042\pi\)
−0.350657 + 0.936504i \(0.614042\pi\)
\(480\) 19.7040 0.899360
\(481\) −3.51838 −0.160424
\(482\) 4.81822 0.219464
\(483\) −2.98340 −0.135749
\(484\) 11.5733 0.526059
\(485\) −14.6991 −0.667454
\(486\) −0.743347 −0.0337189
\(487\) −23.2996 −1.05581 −0.527904 0.849304i \(-0.677022\pi\)
−0.527904 + 0.849304i \(0.677022\pi\)
\(488\) −30.3386 −1.37336
\(489\) −0.544969 −0.0246443
\(490\) 4.74974 0.214572
\(491\) −31.9633 −1.44248 −0.721242 0.692683i \(-0.756429\pi\)
−0.721242 + 0.692683i \(0.756429\pi\)
\(492\) 8.21695 0.370449
\(493\) −3.04186 −0.136998
\(494\) −2.11336 −0.0950846
\(495\) −5.82695 −0.261902
\(496\) −1.25622 −0.0564061
\(497\) 20.9280 0.938750
\(498\) −11.3870 −0.510262
\(499\) 20.4662 0.916194 0.458097 0.888902i \(-0.348531\pi\)
0.458097 + 0.888902i \(0.348531\pi\)
\(500\) 6.33393 0.283262
\(501\) 13.5418 0.605005
\(502\) −4.10292 −0.183122
\(503\) −0.0476041 −0.00212256 −0.00106128 0.999999i \(-0.500338\pi\)
−0.00106128 + 0.999999i \(0.500338\pi\)
\(504\) −7.64538 −0.340552
\(505\) 24.3218 1.08231
\(506\) −1.28843 −0.0572778
\(507\) −12.8248 −0.569570
\(508\) 3.88165 0.172220
\(509\) 11.6626 0.516935 0.258467 0.966020i \(-0.416783\pi\)
0.258467 + 0.966020i \(0.416783\pi\)
\(510\) 7.60154 0.336602
\(511\) 27.6900 1.22494
\(512\) −10.8759 −0.480651
\(513\) 6.79263 0.299902
\(514\) 15.5843 0.687395
\(515\) 60.5118 2.66647
\(516\) −3.44666 −0.151731
\(517\) −1.76713 −0.0777182
\(518\) −18.6424 −0.819101
\(519\) −2.48390 −0.109031
\(520\) −3.60581 −0.158125
\(521\) −20.4685 −0.896743 −0.448371 0.893847i \(-0.647996\pi\)
−0.448371 + 0.893847i \(0.647996\pi\)
\(522\) 0.743347 0.0325354
\(523\) −16.5007 −0.721523 −0.360762 0.932658i \(-0.617483\pi\)
−0.360762 + 0.932658i \(0.617483\pi\)
\(524\) −7.62752 −0.333210
\(525\) −18.8004 −0.820517
\(526\) −3.24916 −0.141670
\(527\) −3.86009 −0.168148
\(528\) 1.71584 0.0746725
\(529\) 1.00000 0.0434783
\(530\) 2.72200 0.118236
\(531\) −0.434930 −0.0188744
\(532\) 29.3325 1.27172
\(533\) −2.37605 −0.102918
\(534\) −8.79543 −0.380615
\(535\) 17.4185 0.753066
\(536\) −11.0090 −0.475514
\(537\) −8.19708 −0.353730
\(538\) 13.7811 0.594144
\(539\) 3.29441 0.141900
\(540\) 4.86598 0.209398
\(541\) −7.89169 −0.339290 −0.169645 0.985505i \(-0.554262\pi\)
−0.169645 + 0.985505i \(0.554262\pi\)
\(542\) 13.2481 0.569055
\(543\) −8.21935 −0.352726
\(544\) −17.8288 −0.764402
\(545\) 38.0375 1.62935
\(546\) 0.928211 0.0397238
\(547\) −34.3963 −1.47068 −0.735340 0.677698i \(-0.762978\pi\)
−0.735340 + 0.677698i \(0.762978\pi\)
\(548\) −16.0620 −0.686136
\(549\) −11.8388 −0.505268
\(550\) −8.11928 −0.346207
\(551\) −6.79263 −0.289376
\(552\) 2.56264 0.109073
\(553\) −7.26767 −0.309053
\(554\) −0.485348 −0.0206205
\(555\) 28.2599 1.19957
\(556\) 19.9388 0.845592
\(557\) 44.5893 1.88931 0.944655 0.328066i \(-0.106397\pi\)
0.944655 + 0.328066i \(0.106397\pi\)
\(558\) 0.943302 0.0399332
\(559\) 0.996652 0.0421539
\(560\) 9.92866 0.419562
\(561\) 5.27240 0.222601
\(562\) 18.2173 0.768452
\(563\) −1.52081 −0.0640943 −0.0320472 0.999486i \(-0.510203\pi\)
−0.0320472 + 0.999486i \(0.510203\pi\)
\(564\) 1.47570 0.0621381
\(565\) −20.4573 −0.860647
\(566\) 21.8791 0.919646
\(567\) −2.98340 −0.125291
\(568\) −17.9765 −0.754276
\(569\) 2.96412 0.124262 0.0621311 0.998068i \(-0.480210\pi\)
0.0621311 + 0.998068i \(0.480210\pi\)
\(570\) 16.9747 0.710990
\(571\) 22.2318 0.930371 0.465185 0.885213i \(-0.345988\pi\)
0.465185 + 0.885213i \(0.345988\pi\)
\(572\) −1.05006 −0.0439051
\(573\) −14.0910 −0.588660
\(574\) −12.5897 −0.525483
\(575\) 6.30168 0.262798
\(576\) 2.37699 0.0990415
\(577\) 1.31813 0.0548745 0.0274373 0.999624i \(-0.491265\pi\)
0.0274373 + 0.999624i \(0.491265\pi\)
\(578\) 5.75880 0.239534
\(579\) 0.272693 0.0113327
\(580\) −4.86598 −0.202049
\(581\) −45.7012 −1.89601
\(582\) −3.25022 −0.134726
\(583\) 1.88797 0.0781917
\(584\) −23.7848 −0.984223
\(585\) −1.40707 −0.0581751
\(586\) −7.77107 −0.321020
\(587\) 9.07620 0.374615 0.187307 0.982301i \(-0.440024\pi\)
0.187307 + 0.982301i \(0.440024\pi\)
\(588\) −2.75110 −0.113453
\(589\) −8.61980 −0.355173
\(590\) −1.08688 −0.0447462
\(591\) 23.5938 0.970520
\(592\) −8.32160 −0.342016
\(593\) −40.3794 −1.65818 −0.829092 0.559112i \(-0.811142\pi\)
−0.829092 + 0.559112i \(0.811142\pi\)
\(594\) −1.28843 −0.0528650
\(595\) 30.5085 1.25073
\(596\) −3.11392 −0.127551
\(597\) 3.51044 0.143673
\(598\) −0.311125 −0.0127229
\(599\) 13.4887 0.551132 0.275566 0.961282i \(-0.411135\pi\)
0.275566 + 0.961282i \(0.411135\pi\)
\(600\) 16.1489 0.659277
\(601\) −11.4881 −0.468608 −0.234304 0.972163i \(-0.575281\pi\)
−0.234304 + 0.972163i \(0.575281\pi\)
\(602\) 5.28084 0.215231
\(603\) −4.29594 −0.174944
\(604\) −26.5698 −1.08111
\(605\) 26.8800 1.09283
\(606\) 5.37795 0.218464
\(607\) −3.94753 −0.160225 −0.0801126 0.996786i \(-0.525528\pi\)
−0.0801126 + 0.996786i \(0.525528\pi\)
\(608\) −39.8126 −1.61462
\(609\) 2.98340 0.120893
\(610\) −29.5849 −1.19786
\(611\) −0.426719 −0.0172632
\(612\) −4.40289 −0.177976
\(613\) 4.36193 0.176177 0.0880884 0.996113i \(-0.471924\pi\)
0.0880884 + 0.996113i \(0.471924\pi\)
\(614\) 21.7906 0.879396
\(615\) 19.0846 0.769565
\(616\) −13.2516 −0.533923
\(617\) 20.0408 0.806813 0.403407 0.915021i \(-0.367826\pi\)
0.403407 + 0.915021i \(0.367826\pi\)
\(618\) 13.3801 0.538228
\(619\) 31.6150 1.27071 0.635357 0.772219i \(-0.280853\pi\)
0.635357 + 0.772219i \(0.280853\pi\)
\(620\) −6.17490 −0.247990
\(621\) 1.00000 0.0401286
\(622\) 10.5572 0.423306
\(623\) −35.3002 −1.41427
\(624\) 0.414335 0.0165867
\(625\) −16.7973 −0.671891
\(626\) 13.6039 0.543721
\(627\) 11.7736 0.470191
\(628\) 0.790021 0.0315253
\(629\) −25.5704 −1.01956
\(630\) −7.45546 −0.297033
\(631\) 17.0088 0.677110 0.338555 0.940947i \(-0.390062\pi\)
0.338555 + 0.940947i \(0.390062\pi\)
\(632\) 6.24269 0.248321
\(633\) −0.138150 −0.00549097
\(634\) −1.22343 −0.0485887
\(635\) 9.01548 0.357768
\(636\) −1.57661 −0.0625166
\(637\) 0.795520 0.0315196
\(638\) 1.28843 0.0510095
\(639\) −7.01482 −0.277502
\(640\) −33.4679 −1.32294
\(641\) 17.9247 0.707982 0.353991 0.935249i \(-0.384824\pi\)
0.353991 + 0.935249i \(0.384824\pi\)
\(642\) 3.85150 0.152007
\(643\) 19.9092 0.785143 0.392571 0.919722i \(-0.371586\pi\)
0.392571 + 0.919722i \(0.371586\pi\)
\(644\) 4.31828 0.170164
\(645\) −8.00518 −0.315204
\(646\) −15.3592 −0.604299
\(647\) 43.0673 1.69315 0.846574 0.532270i \(-0.178661\pi\)
0.846574 + 0.532270i \(0.178661\pi\)
\(648\) 2.56264 0.100670
\(649\) −0.753857 −0.0295915
\(650\) −1.96061 −0.0769015
\(651\) 3.78591 0.148382
\(652\) 0.788807 0.0308921
\(653\) 1.82134 0.0712744 0.0356372 0.999365i \(-0.488654\pi\)
0.0356372 + 0.999365i \(0.488654\pi\)
\(654\) 8.41070 0.328885
\(655\) −17.7156 −0.692205
\(656\) −5.61978 −0.219416
\(657\) −9.28137 −0.362101
\(658\) −2.26100 −0.0881431
\(659\) 26.8403 1.04555 0.522774 0.852471i \(-0.324897\pi\)
0.522774 + 0.852471i \(0.324897\pi\)
\(660\) 8.43413 0.328298
\(661\) 4.26124 0.165743 0.0828715 0.996560i \(-0.473591\pi\)
0.0828715 + 0.996560i \(0.473591\pi\)
\(662\) 10.7234 0.416776
\(663\) 1.27316 0.0494454
\(664\) 39.2558 1.52342
\(665\) 68.1272 2.64186
\(666\) 6.24872 0.242133
\(667\) −1.00000 −0.0387202
\(668\) −19.6009 −0.758383
\(669\) 19.7576 0.763871
\(670\) −10.7355 −0.414748
\(671\) −20.5200 −0.792166
\(672\) 17.4862 0.674543
\(673\) 7.95990 0.306832 0.153416 0.988162i \(-0.450973\pi\)
0.153416 + 0.988162i \(0.450973\pi\)
\(674\) 5.79337 0.223152
\(675\) 6.30168 0.242552
\(676\) 18.5631 0.713965
\(677\) 45.3855 1.74431 0.872154 0.489232i \(-0.162723\pi\)
0.872154 + 0.489232i \(0.162723\pi\)
\(678\) −4.52345 −0.173722
\(679\) −13.0446 −0.500607
\(680\) −26.2058 −1.00495
\(681\) −21.2886 −0.815780
\(682\) 1.63501 0.0626078
\(683\) 10.6867 0.408914 0.204457 0.978876i \(-0.434457\pi\)
0.204457 + 0.978876i \(0.434457\pi\)
\(684\) −9.83189 −0.375932
\(685\) −37.3055 −1.42537
\(686\) −11.3088 −0.431772
\(687\) −9.98323 −0.380884
\(688\) 2.35726 0.0898697
\(689\) 0.455899 0.0173684
\(690\) 2.49898 0.0951346
\(691\) −28.8789 −1.09861 −0.549303 0.835623i \(-0.685107\pi\)
−0.549303 + 0.835623i \(0.685107\pi\)
\(692\) 3.59528 0.136672
\(693\) −5.17108 −0.196433
\(694\) 20.4871 0.777678
\(695\) 46.3096 1.75662
\(696\) −2.56264 −0.0971367
\(697\) −17.2683 −0.654085
\(698\) 22.2896 0.843675
\(699\) −11.7610 −0.444843
\(700\) 27.2124 1.02853
\(701\) −39.1416 −1.47836 −0.739179 0.673510i \(-0.764786\pi\)
−0.739179 + 0.673510i \(0.764786\pi\)
\(702\) −0.311125 −0.0117427
\(703\) −57.1002 −2.15357
\(704\) 4.12001 0.155279
\(705\) 3.42744 0.129085
\(706\) 22.0714 0.830670
\(707\) 21.5842 0.811758
\(708\) 0.629533 0.0236593
\(709\) −38.0885 −1.43045 −0.715223 0.698897i \(-0.753675\pi\)
−0.715223 + 0.698897i \(0.753675\pi\)
\(710\) −17.5299 −0.657886
\(711\) 2.43604 0.0913585
\(712\) 30.3217 1.13635
\(713\) −1.26899 −0.0475242
\(714\) 6.74593 0.252460
\(715\) −2.43885 −0.0912078
\(716\) 11.8647 0.443406
\(717\) −14.7029 −0.549091
\(718\) −1.56033 −0.0582311
\(719\) 9.52549 0.355241 0.177620 0.984099i \(-0.443160\pi\)
0.177620 + 0.984099i \(0.443160\pi\)
\(720\) −3.32797 −0.124026
\(721\) 53.7007 1.99992
\(722\) −20.1743 −0.750810
\(723\) −6.48179 −0.241060
\(724\) 11.8970 0.442148
\(725\) −6.30168 −0.234038
\(726\) 5.94360 0.220588
\(727\) −6.13864 −0.227669 −0.113835 0.993500i \(-0.536313\pi\)
−0.113835 + 0.993500i \(0.536313\pi\)
\(728\) −3.19995 −0.118598
\(729\) 1.00000 0.0370370
\(730\) −23.1940 −0.858448
\(731\) 7.24333 0.267904
\(732\) 17.1359 0.633361
\(733\) 37.0637 1.36898 0.684489 0.729023i \(-0.260025\pi\)
0.684489 + 0.729023i \(0.260025\pi\)
\(734\) −14.4892 −0.534806
\(735\) −6.38967 −0.235687
\(736\) −5.86115 −0.216045
\(737\) −7.44609 −0.274280
\(738\) 4.21991 0.155337
\(739\) 27.6507 1.01715 0.508573 0.861019i \(-0.330173\pi\)
0.508573 + 0.861019i \(0.330173\pi\)
\(740\) −40.9044 −1.50367
\(741\) 2.84303 0.104441
\(742\) 2.41562 0.0886801
\(743\) −23.5165 −0.862738 −0.431369 0.902176i \(-0.641969\pi\)
−0.431369 + 0.902176i \(0.641969\pi\)
\(744\) −3.25197 −0.119223
\(745\) −7.23236 −0.264973
\(746\) 24.8508 0.909852
\(747\) 15.3185 0.560475
\(748\) −7.63146 −0.279034
\(749\) 15.4579 0.564819
\(750\) 3.25286 0.118778
\(751\) −39.7925 −1.45205 −0.726025 0.687669i \(-0.758634\pi\)
−0.726025 + 0.687669i \(0.758634\pi\)
\(752\) −1.00927 −0.0368042
\(753\) 5.51951 0.201142
\(754\) 0.311125 0.0113305
\(755\) −61.7108 −2.24588
\(756\) 4.31828 0.157054
\(757\) 19.1170 0.694818 0.347409 0.937714i \(-0.387062\pi\)
0.347409 + 0.937714i \(0.387062\pi\)
\(758\) −17.4554 −0.634010
\(759\) 1.73328 0.0629142
\(760\) −58.5190 −2.12271
\(761\) −23.8344 −0.863996 −0.431998 0.901874i \(-0.642191\pi\)
−0.431998 + 0.901874i \(0.642191\pi\)
\(762\) 1.99347 0.0722157
\(763\) 33.7561 1.22205
\(764\) 20.3958 0.737894
\(765\) −10.2261 −0.369725
\(766\) −16.3605 −0.591129
\(767\) −0.182038 −0.00657303
\(768\) −12.1543 −0.438580
\(769\) 36.2669 1.30782 0.653909 0.756573i \(-0.273128\pi\)
0.653909 + 0.756573i \(0.273128\pi\)
\(770\) −12.9224 −0.465692
\(771\) −20.9651 −0.755038
\(772\) −0.394706 −0.0142058
\(773\) 31.7295 1.14123 0.570616 0.821217i \(-0.306704\pi\)
0.570616 + 0.821217i \(0.306704\pi\)
\(774\) −1.77007 −0.0636240
\(775\) −7.99678 −0.287253
\(776\) 11.2049 0.402233
\(777\) 25.0790 0.899705
\(778\) −10.9810 −0.393689
\(779\) −38.5611 −1.38160
\(780\) 2.03664 0.0729234
\(781\) −12.1587 −0.435072
\(782\) −2.26116 −0.0808587
\(783\) −1.00000 −0.0357371
\(784\) 1.88155 0.0671981
\(785\) 1.83489 0.0654902
\(786\) −3.91720 −0.139722
\(787\) −4.23836 −0.151081 −0.0755405 0.997143i \(-0.524068\pi\)
−0.0755405 + 0.997143i \(0.524068\pi\)
\(788\) −34.1505 −1.21656
\(789\) 4.37099 0.155611
\(790\) 6.08761 0.216588
\(791\) −18.1547 −0.645507
\(792\) 4.44179 0.157832
\(793\) −4.95509 −0.175960
\(794\) −16.1859 −0.574418
\(795\) −3.66181 −0.129871
\(796\) −5.08113 −0.180096
\(797\) −24.4161 −0.864863 −0.432432 0.901667i \(-0.642344\pi\)
−0.432432 + 0.901667i \(0.642344\pi\)
\(798\) 15.0640 0.533261
\(799\) −3.10125 −0.109714
\(800\) −36.9351 −1.30585
\(801\) 11.8322 0.418070
\(802\) 1.76657 0.0623799
\(803\) −16.0873 −0.567707
\(804\) 6.21809 0.219295
\(805\) 10.0296 0.353496
\(806\) 0.394816 0.0139068
\(807\) −18.5392 −0.652610
\(808\) −18.5401 −0.652239
\(809\) 27.0005 0.949287 0.474644 0.880178i \(-0.342577\pi\)
0.474644 + 0.880178i \(0.342577\pi\)
\(810\) 2.49898 0.0878053
\(811\) 26.8385 0.942428 0.471214 0.882019i \(-0.343816\pi\)
0.471214 + 0.882019i \(0.343816\pi\)
\(812\) −4.31828 −0.151542
\(813\) −17.8222 −0.625053
\(814\) 10.8308 0.379620
\(815\) 1.83207 0.0641748
\(816\) 3.01125 0.105415
\(817\) 16.1748 0.565883
\(818\) −0.106457 −0.00372217
\(819\) −1.24869 −0.0436328
\(820\) −27.6237 −0.964662
\(821\) −53.0026 −1.84980 −0.924901 0.380208i \(-0.875852\pi\)
−0.924901 + 0.380208i \(0.875852\pi\)
\(822\) −8.24884 −0.287711
\(823\) −5.65489 −0.197117 −0.0985585 0.995131i \(-0.531423\pi\)
−0.0985585 + 0.995131i \(0.531423\pi\)
\(824\) −46.1271 −1.60691
\(825\) 10.9226 0.380276
\(826\) −0.964545 −0.0335608
\(827\) −39.2613 −1.36525 −0.682624 0.730770i \(-0.739161\pi\)
−0.682624 + 0.730770i \(0.739161\pi\)
\(828\) −1.44743 −0.0503018
\(829\) 30.1229 1.04621 0.523106 0.852268i \(-0.324773\pi\)
0.523106 + 0.852268i \(0.324773\pi\)
\(830\) 38.2807 1.32874
\(831\) 0.652922 0.0226496
\(832\) 0.994883 0.0344914
\(833\) 5.78157 0.200320
\(834\) 10.2398 0.354575
\(835\) −45.5249 −1.57546
\(836\) −17.0415 −0.589391
\(837\) −1.26899 −0.0438628
\(838\) 1.38058 0.0476913
\(839\) 2.39338 0.0826286 0.0413143 0.999146i \(-0.486845\pi\)
0.0413143 + 0.999146i \(0.486845\pi\)
\(840\) 25.7022 0.886811
\(841\) 1.00000 0.0344828
\(842\) −12.4852 −0.430269
\(843\) −24.5072 −0.844072
\(844\) 0.199963 0.00688301
\(845\) 43.1144 1.48318
\(846\) 0.757862 0.0260558
\(847\) 23.8544 0.819648
\(848\) 1.07828 0.0370284
\(849\) −29.4332 −1.01014
\(850\) −14.2491 −0.488739
\(851\) −8.40619 −0.288161
\(852\) 10.1535 0.347853
\(853\) 20.8704 0.714588 0.357294 0.933992i \(-0.383699\pi\)
0.357294 + 0.933992i \(0.383699\pi\)
\(854\) −26.2549 −0.898425
\(855\) −22.8354 −0.780956
\(856\) −13.2778 −0.453826
\(857\) 45.3139 1.54789 0.773946 0.633251i \(-0.218280\pi\)
0.773946 + 0.633251i \(0.218280\pi\)
\(858\) −0.539269 −0.0184103
\(859\) −35.5885 −1.21427 −0.607133 0.794600i \(-0.707680\pi\)
−0.607133 + 0.794600i \(0.707680\pi\)
\(860\) 11.5870 0.395113
\(861\) 16.9365 0.577194
\(862\) −11.5094 −0.392013
\(863\) −20.3394 −0.692362 −0.346181 0.938168i \(-0.612522\pi\)
−0.346181 + 0.938168i \(0.612522\pi\)
\(864\) −5.86115 −0.199400
\(865\) 8.35036 0.283921
\(866\) 8.05689 0.273784
\(867\) −7.74711 −0.263106
\(868\) −5.47986 −0.185999
\(869\) 4.22235 0.143233
\(870\) −2.49898 −0.0847234
\(871\) −1.79805 −0.0609246
\(872\) −28.9954 −0.981907
\(873\) 4.37241 0.147984
\(874\) −5.04928 −0.170795
\(875\) 13.0553 0.441348
\(876\) 13.4342 0.453899
\(877\) −47.6368 −1.60858 −0.804290 0.594238i \(-0.797454\pi\)
−0.804290 + 0.594238i \(0.797454\pi\)
\(878\) −9.70718 −0.327601
\(879\) 10.4542 0.352610
\(880\) −5.76832 −0.194450
\(881\) 22.4664 0.756911 0.378455 0.925620i \(-0.376455\pi\)
0.378455 + 0.925620i \(0.376455\pi\)
\(882\) −1.41286 −0.0475734
\(883\) 47.1244 1.58586 0.792931 0.609311i \(-0.208554\pi\)
0.792931 + 0.609311i \(0.208554\pi\)
\(884\) −1.84281 −0.0619805
\(885\) 1.46215 0.0491495
\(886\) 10.3709 0.348417
\(887\) 24.4228 0.820036 0.410018 0.912077i \(-0.365522\pi\)
0.410018 + 0.912077i \(0.365522\pi\)
\(888\) −21.5421 −0.722904
\(889\) 8.00071 0.268335
\(890\) 29.5684 0.991136
\(891\) 1.73328 0.0580672
\(892\) −28.5978 −0.957524
\(893\) −6.92526 −0.231745
\(894\) −1.59919 −0.0534850
\(895\) 27.5569 0.921126
\(896\) −29.7008 −0.992235
\(897\) 0.418547 0.0139749
\(898\) −0.0806501 −0.00269133
\(899\) 1.26899 0.0423233
\(900\) −9.12126 −0.304042
\(901\) 3.31332 0.110383
\(902\) 7.31431 0.243540
\(903\) −7.10413 −0.236411
\(904\) 15.5943 0.518658
\(905\) 27.6318 0.918512
\(906\) −13.6452 −0.453333
\(907\) −38.1574 −1.26700 −0.633498 0.773744i \(-0.718381\pi\)
−0.633498 + 0.773744i \(0.718381\pi\)
\(908\) 30.8138 1.02259
\(909\) −7.23477 −0.239962
\(910\) −3.12046 −0.103442
\(911\) −21.7734 −0.721386 −0.360693 0.932685i \(-0.617460\pi\)
−0.360693 + 0.932685i \(0.617460\pi\)
\(912\) 6.72428 0.222663
\(913\) 26.5513 0.878721
\(914\) −21.2818 −0.703940
\(915\) 39.7996 1.31574
\(916\) 14.4501 0.477444
\(917\) −15.7216 −0.519171
\(918\) −2.26116 −0.0746292
\(919\) 18.5974 0.613472 0.306736 0.951795i \(-0.400763\pi\)
0.306736 + 0.951795i \(0.400763\pi\)
\(920\) −8.61508 −0.284031
\(921\) −29.3141 −0.965933
\(922\) −20.3434 −0.669973
\(923\) −2.93603 −0.0966406
\(924\) 7.48480 0.246232
\(925\) −52.9731 −1.74174
\(926\) 6.87100 0.225795
\(927\) −17.9998 −0.591192
\(928\) 5.86115 0.192402
\(929\) −8.38393 −0.275068 −0.137534 0.990497i \(-0.543918\pi\)
−0.137534 + 0.990497i \(0.543918\pi\)
\(930\) −3.17119 −0.103987
\(931\) 12.9106 0.423127
\(932\) 17.0233 0.557617
\(933\) −14.2023 −0.464962
\(934\) −20.1769 −0.660209
\(935\) −17.7247 −0.579661
\(936\) 1.07258 0.0350585
\(937\) 46.3141 1.51301 0.756507 0.653985i \(-0.226904\pi\)
0.756507 + 0.653985i \(0.226904\pi\)
\(938\) −9.52711 −0.311071
\(939\) −18.3008 −0.597226
\(940\) −4.96099 −0.161810
\(941\) 38.1202 1.24268 0.621342 0.783540i \(-0.286588\pi\)
0.621342 + 0.783540i \(0.286588\pi\)
\(942\) 0.405725 0.0132192
\(943\) −5.67691 −0.184865
\(944\) −0.430553 −0.0140133
\(945\) 10.0296 0.326262
\(946\) −3.06804 −0.0997507
\(947\) −25.5915 −0.831612 −0.415806 0.909453i \(-0.636500\pi\)
−0.415806 + 0.909453i \(0.636500\pi\)
\(948\) −3.52600 −0.114519
\(949\) −3.88469 −0.126102
\(950\) −31.8189 −1.03234
\(951\) 1.64584 0.0533701
\(952\) −23.2561 −0.753736
\(953\) −9.53149 −0.308755 −0.154378 0.988012i \(-0.549337\pi\)
−0.154378 + 0.988012i \(0.549337\pi\)
\(954\) −0.809686 −0.0262146
\(955\) 47.3710 1.53289
\(956\) 21.2815 0.688294
\(957\) −1.73328 −0.0560291
\(958\) 11.4097 0.368629
\(959\) −33.1064 −1.06906
\(960\) −7.99097 −0.257908
\(961\) −29.3897 −0.948053
\(962\) 2.61538 0.0843232
\(963\) −5.18130 −0.166965
\(964\) 9.38197 0.302173
\(965\) −0.916739 −0.0295109
\(966\) 2.21770 0.0713534
\(967\) 57.4719 1.84817 0.924086 0.382184i \(-0.124828\pi\)
0.924086 + 0.382184i \(0.124828\pi\)
\(968\) −20.4902 −0.658579
\(969\) 20.6622 0.663766
\(970\) 10.9266 0.350831
\(971\) 2.22999 0.0715636 0.0357818 0.999360i \(-0.488608\pi\)
0.0357818 + 0.999360i \(0.488608\pi\)
\(972\) −1.44743 −0.0464265
\(973\) 41.0970 1.31751
\(974\) 17.3197 0.554960
\(975\) 2.63754 0.0844690
\(976\) −11.7197 −0.375137
\(977\) 16.0703 0.514133 0.257067 0.966394i \(-0.417244\pi\)
0.257067 + 0.966394i \(0.417244\pi\)
\(978\) 0.405101 0.0129537
\(979\) 20.5086 0.655456
\(980\) 9.24863 0.295437
\(981\) −11.3146 −0.361249
\(982\) 23.7598 0.758207
\(983\) −43.1593 −1.37657 −0.688284 0.725441i \(-0.741636\pi\)
−0.688284 + 0.725441i \(0.741636\pi\)
\(984\) −14.5479 −0.463769
\(985\) −79.3176 −2.52727
\(986\) 2.26116 0.0720098
\(987\) 3.04165 0.0968169
\(988\) −4.11510 −0.130919
\(989\) 2.38122 0.0757184
\(990\) 4.33145 0.137662
\(991\) 28.6889 0.911334 0.455667 0.890150i \(-0.349401\pi\)
0.455667 + 0.890150i \(0.349401\pi\)
\(992\) 7.43776 0.236149
\(993\) −14.4258 −0.457789
\(994\) −15.5568 −0.493431
\(995\) −11.8014 −0.374129
\(996\) −22.1725 −0.702564
\(997\) 51.9493 1.64525 0.822625 0.568584i \(-0.192509\pi\)
0.822625 + 0.568584i \(0.192509\pi\)
\(998\) −15.2135 −0.481575
\(999\) −8.40619 −0.265960
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 2001.2.a.i.1.4 7
3.2 odd 2 6003.2.a.j.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.i.1.4 7 1.1 even 1 trivial
6003.2.a.j.1.4 7 3.2 odd 2