Properties

Label 4002.2.a.x.1.2
Level 4002
Weight 2
Character 4002.1
Self dual yes
Analytic conductor 31.956
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +4.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} +4.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +4.89898 q^{11} +1.00000 q^{12} -6.89898 q^{13} +4.44949 q^{14} +4.00000 q^{15} +1.00000 q^{16} -4.89898 q^{17} +1.00000 q^{18} -4.44949 q^{19} +4.00000 q^{20} +4.44949 q^{21} +4.89898 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} -6.89898 q^{26} +1.00000 q^{27} +4.44949 q^{28} +1.00000 q^{29} +4.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +4.89898 q^{33} -4.89898 q^{34} +17.7980 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.44949 q^{38} -6.89898 q^{39} +4.00000 q^{40} -8.89898 q^{41} +4.44949 q^{42} +12.4495 q^{43} +4.89898 q^{44} +4.00000 q^{45} -1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +12.7980 q^{49} +11.0000 q^{50} -4.89898 q^{51} -6.89898 q^{52} -8.89898 q^{53} +1.00000 q^{54} +19.5959 q^{55} +4.44949 q^{56} -4.44949 q^{57} +1.00000 q^{58} -4.89898 q^{59} +4.00000 q^{60} +2.89898 q^{61} -6.00000 q^{62} +4.44949 q^{63} +1.00000 q^{64} -27.5959 q^{65} +4.89898 q^{66} +2.00000 q^{67} -4.89898 q^{68} -1.00000 q^{69} +17.7980 q^{70} -10.8990 q^{71} +1.00000 q^{72} -6.89898 q^{73} -2.00000 q^{74} +11.0000 q^{75} -4.44949 q^{76} +21.7980 q^{77} -6.89898 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -8.89898 q^{82} -1.55051 q^{83} +4.44949 q^{84} -19.5959 q^{85} +12.4495 q^{86} +1.00000 q^{87} +4.89898 q^{88} +12.8990 q^{89} +4.00000 q^{90} -30.6969 q^{91} -1.00000 q^{92} -6.00000 q^{93} +4.00000 q^{94} -17.7980 q^{95} +1.00000 q^{96} -19.1464 q^{97} +12.7980 q^{98} +4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 8q^{10} + 2q^{12} - 4q^{13} + 4q^{14} + 8q^{15} + 2q^{16} + 2q^{18} - 4q^{19} + 8q^{20} + 4q^{21} - 2q^{23} + 2q^{24} + 22q^{25} - 4q^{26} + 2q^{27} + 4q^{28} + 2q^{29} + 8q^{30} - 12q^{31} + 2q^{32} + 16q^{35} + 2q^{36} - 4q^{37} - 4q^{38} - 4q^{39} + 8q^{40} - 8q^{41} + 4q^{42} + 20q^{43} + 8q^{45} - 2q^{46} + 8q^{47} + 2q^{48} + 6q^{49} + 22q^{50} - 4q^{52} - 8q^{53} + 2q^{54} + 4q^{56} - 4q^{57} + 2q^{58} + 8q^{60} - 4q^{61} - 12q^{62} + 4q^{63} + 2q^{64} - 16q^{65} + 4q^{67} - 2q^{69} + 16q^{70} - 12q^{71} + 2q^{72} - 4q^{73} - 4q^{74} + 22q^{75} - 4q^{76} + 24q^{77} - 4q^{78} + 20q^{79} + 8q^{80} + 2q^{81} - 8q^{82} - 8q^{83} + 4q^{84} + 20q^{86} + 2q^{87} + 16q^{89} + 8q^{90} - 32q^{91} - 2q^{92} - 12q^{93} + 8q^{94} - 16q^{95} + 2q^{96} - 4q^{97} + 6q^{98} + O(q^{100}) \)

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
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.44949 1.68175 0.840875 0.541230i \(-0.182041\pi\)
0.840875 + 0.541230i \(0.182041\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.89898 −1.91343 −0.956716 0.291022i \(-0.906005\pi\)
−0.956716 + 0.291022i \(0.906005\pi\)
\(14\) 4.44949 1.18918
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.44949 −1.02078 −0.510391 0.859942i \(-0.670499\pi\)
−0.510391 + 0.859942i \(0.670499\pi\)
\(20\) 4.00000 0.894427
\(21\) 4.44949 0.970958
\(22\) 4.89898 1.04447
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) −6.89898 −1.35300
\(27\) 1.00000 0.192450
\(28\) 4.44949 0.840875
\(29\) 1.00000 0.185695
\(30\) 4.00000 0.730297
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.89898 0.852803
\(34\) −4.89898 −0.840168
\(35\) 17.7980 3.00840
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.44949 −0.721803
\(39\) −6.89898 −1.10472
\(40\) 4.00000 0.632456
\(41\) −8.89898 −1.38979 −0.694894 0.719113i \(-0.744549\pi\)
−0.694894 + 0.719113i \(0.744549\pi\)
\(42\) 4.44949 0.686571
\(43\) 12.4495 1.89853 0.949265 0.314478i \(-0.101830\pi\)
0.949265 + 0.314478i \(0.101830\pi\)
\(44\) 4.89898 0.738549
\(45\) 4.00000 0.596285
\(46\) −1.00000 −0.147442
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 12.7980 1.82828
\(50\) 11.0000 1.55563
\(51\) −4.89898 −0.685994
\(52\) −6.89898 −0.956716
\(53\) −8.89898 −1.22237 −0.611184 0.791488i \(-0.709307\pi\)
−0.611184 + 0.791488i \(0.709307\pi\)
\(54\) 1.00000 0.136083
\(55\) 19.5959 2.64231
\(56\) 4.44949 0.594588
\(57\) −4.44949 −0.589349
\(58\) 1.00000 0.131306
\(59\) −4.89898 −0.637793 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(60\) 4.00000 0.516398
\(61\) 2.89898 0.371176 0.185588 0.982628i \(-0.440581\pi\)
0.185588 + 0.982628i \(0.440581\pi\)
\(62\) −6.00000 −0.762001
\(63\) 4.44949 0.560583
\(64\) 1.00000 0.125000
\(65\) −27.5959 −3.42285
\(66\) 4.89898 0.603023
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −4.89898 −0.594089
\(69\) −1.00000 −0.120386
\(70\) 17.7980 2.12726
\(71\) −10.8990 −1.29347 −0.646735 0.762714i \(-0.723866\pi\)
−0.646735 + 0.762714i \(0.723866\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.89898 −0.807464 −0.403732 0.914877i \(-0.632287\pi\)
−0.403732 + 0.914877i \(0.632287\pi\)
\(74\) −2.00000 −0.232495
\(75\) 11.0000 1.27017
\(76\) −4.44949 −0.510391
\(77\) 21.7980 2.48411
\(78\) −6.89898 −0.781156
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −8.89898 −0.982728
\(83\) −1.55051 −0.170191 −0.0850953 0.996373i \(-0.527119\pi\)
−0.0850953 + 0.996373i \(0.527119\pi\)
\(84\) 4.44949 0.485479
\(85\) −19.5959 −2.12548
\(86\) 12.4495 1.34246
\(87\) 1.00000 0.107211
\(88\) 4.89898 0.522233
\(89\) 12.8990 1.36729 0.683645 0.729815i \(-0.260394\pi\)
0.683645 + 0.729815i \(0.260394\pi\)
\(90\) 4.00000 0.421637
\(91\) −30.6969 −3.21791
\(92\) −1.00000 −0.104257
\(93\) −6.00000 −0.622171
\(94\) 4.00000 0.412568
\(95\) −17.7980 −1.82603
\(96\) 1.00000 0.102062
\(97\) −19.1464 −1.94403 −0.972013 0.234929i \(-0.924514\pi\)
−0.972013 + 0.234929i \(0.924514\pi\)
\(98\) 12.7980 1.29279
\(99\) 4.89898 0.492366
\(100\) 11.0000 1.10000
\(101\) −5.10102 −0.507571 −0.253785 0.967261i \(-0.581676\pi\)
−0.253785 + 0.967261i \(0.581676\pi\)
\(102\) −4.89898 −0.485071
\(103\) 6.24745 0.615579 0.307790 0.951454i \(-0.400411\pi\)
0.307790 + 0.951454i \(0.400411\pi\)
\(104\) −6.89898 −0.676501
\(105\) 17.7980 1.73690
\(106\) −8.89898 −0.864345
\(107\) −9.55051 −0.923283 −0.461641 0.887067i \(-0.652739\pi\)
−0.461641 + 0.887067i \(0.652739\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.44949 −0.426184 −0.213092 0.977032i \(-0.568353\pi\)
−0.213092 + 0.977032i \(0.568353\pi\)
\(110\) 19.5959 1.86840
\(111\) −2.00000 −0.189832
\(112\) 4.44949 0.420437
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −4.44949 −0.416733
\(115\) −4.00000 −0.373002
\(116\) 1.00000 0.0928477
\(117\) −6.89898 −0.637811
\(118\) −4.89898 −0.450988
\(119\) −21.7980 −1.99822
\(120\) 4.00000 0.365148
\(121\) 13.0000 1.18182
\(122\) 2.89898 0.262461
\(123\) −8.89898 −0.802394
\(124\) −6.00000 −0.538816
\(125\) 24.0000 2.14663
\(126\) 4.44949 0.396392
\(127\) 5.79796 0.514486 0.257243 0.966347i \(-0.417186\pi\)
0.257243 + 0.966347i \(0.417186\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.4495 1.09612
\(130\) −27.5959 −2.42032
\(131\) 18.8990 1.65121 0.825606 0.564247i \(-0.190834\pi\)
0.825606 + 0.564247i \(0.190834\pi\)
\(132\) 4.89898 0.426401
\(133\) −19.7980 −1.71670
\(134\) 2.00000 0.172774
\(135\) 4.00000 0.344265
\(136\) −4.89898 −0.420084
\(137\) −5.79796 −0.495353 −0.247677 0.968843i \(-0.579667\pi\)
−0.247677 + 0.968843i \(0.579667\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 17.7980 1.50420
\(141\) 4.00000 0.336861
\(142\) −10.8990 −0.914622
\(143\) −33.7980 −2.82633
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −6.89898 −0.570964
\(147\) 12.7980 1.05556
\(148\) −2.00000 −0.164399
\(149\) −18.6969 −1.53171 −0.765856 0.643012i \(-0.777685\pi\)
−0.765856 + 0.643012i \(0.777685\pi\)
\(150\) 11.0000 0.898146
\(151\) −13.7980 −1.12286 −0.561431 0.827524i \(-0.689749\pi\)
−0.561431 + 0.827524i \(0.689749\pi\)
\(152\) −4.44949 −0.360901
\(153\) −4.89898 −0.396059
\(154\) 21.7980 1.75653
\(155\) −24.0000 −1.92773
\(156\) −6.89898 −0.552360
\(157\) −15.7980 −1.26081 −0.630407 0.776265i \(-0.717112\pi\)
−0.630407 + 0.776265i \(0.717112\pi\)
\(158\) 10.0000 0.795557
\(159\) −8.89898 −0.705735
\(160\) 4.00000 0.316228
\(161\) −4.44949 −0.350669
\(162\) 1.00000 0.0785674
\(163\) 0.898979 0.0704135 0.0352068 0.999380i \(-0.488791\pi\)
0.0352068 + 0.999380i \(0.488791\pi\)
\(164\) −8.89898 −0.694894
\(165\) 19.5959 1.52554
\(166\) −1.55051 −0.120343
\(167\) 9.79796 0.758189 0.379094 0.925358i \(-0.376236\pi\)
0.379094 + 0.925358i \(0.376236\pi\)
\(168\) 4.44949 0.343286
\(169\) 34.5959 2.66122
\(170\) −19.5959 −1.50294
\(171\) −4.44949 −0.340261
\(172\) 12.4495 0.949265
\(173\) −3.10102 −0.235766 −0.117883 0.993027i \(-0.537611\pi\)
−0.117883 + 0.993027i \(0.537611\pi\)
\(174\) 1.00000 0.0758098
\(175\) 48.9444 3.69985
\(176\) 4.89898 0.369274
\(177\) −4.89898 −0.368230
\(178\) 12.8990 0.966819
\(179\) 14.6969 1.09850 0.549250 0.835658i \(-0.314913\pi\)
0.549250 + 0.835658i \(0.314913\pi\)
\(180\) 4.00000 0.298142
\(181\) −25.3485 −1.88414 −0.942068 0.335421i \(-0.891122\pi\)
−0.942068 + 0.335421i \(0.891122\pi\)
\(182\) −30.6969 −2.27541
\(183\) 2.89898 0.214299
\(184\) −1.00000 −0.0737210
\(185\) −8.00000 −0.588172
\(186\) −6.00000 −0.439941
\(187\) −24.0000 −1.75505
\(188\) 4.00000 0.291730
\(189\) 4.44949 0.323653
\(190\) −17.7980 −1.29120
\(191\) 7.34847 0.531717 0.265858 0.964012i \(-0.414345\pi\)
0.265858 + 0.964012i \(0.414345\pi\)
\(192\) 1.00000 0.0721688
\(193\) 21.5959 1.55451 0.777254 0.629187i \(-0.216612\pi\)
0.777254 + 0.629187i \(0.216612\pi\)
\(194\) −19.1464 −1.37463
\(195\) −27.5959 −1.97618
\(196\) 12.7980 0.914140
\(197\) 3.79796 0.270593 0.135297 0.990805i \(-0.456801\pi\)
0.135297 + 0.990805i \(0.456801\pi\)
\(198\) 4.89898 0.348155
\(199\) −26.2474 −1.86063 −0.930316 0.366759i \(-0.880468\pi\)
−0.930316 + 0.366759i \(0.880468\pi\)
\(200\) 11.0000 0.777817
\(201\) 2.00000 0.141069
\(202\) −5.10102 −0.358907
\(203\) 4.44949 0.312293
\(204\) −4.89898 −0.342997
\(205\) −35.5959 −2.48613
\(206\) 6.24745 0.435280
\(207\) −1.00000 −0.0695048
\(208\) −6.89898 −0.478358
\(209\) −21.7980 −1.50780
\(210\) 17.7980 1.22818
\(211\) 24.8990 1.71412 0.857058 0.515220i \(-0.172290\pi\)
0.857058 + 0.515220i \(0.172290\pi\)
\(212\) −8.89898 −0.611184
\(213\) −10.8990 −0.746786
\(214\) −9.55051 −0.652859
\(215\) 49.7980 3.39619
\(216\) 1.00000 0.0680414
\(217\) −26.6969 −1.81231
\(218\) −4.44949 −0.301357
\(219\) −6.89898 −0.466190
\(220\) 19.5959 1.32116
\(221\) 33.7980 2.27350
\(222\) −2.00000 −0.134231
\(223\) 13.7980 0.923980 0.461990 0.886885i \(-0.347136\pi\)
0.461990 + 0.886885i \(0.347136\pi\)
\(224\) 4.44949 0.297294
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 7.34847 0.487735 0.243868 0.969809i \(-0.421584\pi\)
0.243868 + 0.969809i \(0.421584\pi\)
\(228\) −4.44949 −0.294675
\(229\) −12.6969 −0.839037 −0.419519 0.907747i \(-0.637801\pi\)
−0.419519 + 0.907747i \(0.637801\pi\)
\(230\) −4.00000 −0.263752
\(231\) 21.7980 1.43420
\(232\) 1.00000 0.0656532
\(233\) 16.6969 1.09385 0.546926 0.837181i \(-0.315798\pi\)
0.546926 + 0.837181i \(0.315798\pi\)
\(234\) −6.89898 −0.451000
\(235\) 16.0000 1.04372
\(236\) −4.89898 −0.318896
\(237\) 10.0000 0.649570
\(238\) −21.7980 −1.41295
\(239\) −1.10102 −0.0712191 −0.0356095 0.999366i \(-0.511337\pi\)
−0.0356095 + 0.999366i \(0.511337\pi\)
\(240\) 4.00000 0.258199
\(241\) 20.6969 1.33321 0.666604 0.745412i \(-0.267747\pi\)
0.666604 + 0.745412i \(0.267747\pi\)
\(242\) 13.0000 0.835672
\(243\) 1.00000 0.0641500
\(244\) 2.89898 0.185588
\(245\) 51.1918 3.27053
\(246\) −8.89898 −0.567378
\(247\) 30.6969 1.95320
\(248\) −6.00000 −0.381000
\(249\) −1.55051 −0.0982596
\(250\) 24.0000 1.51789
\(251\) 3.10102 0.195735 0.0978673 0.995199i \(-0.468798\pi\)
0.0978673 + 0.995199i \(0.468798\pi\)
\(252\) 4.44949 0.280292
\(253\) −4.89898 −0.307996
\(254\) 5.79796 0.363796
\(255\) −19.5959 −1.22714
\(256\) 1.00000 0.0625000
\(257\) −1.10102 −0.0686798 −0.0343399 0.999410i \(-0.510933\pi\)
−0.0343399 + 0.999410i \(0.510933\pi\)
\(258\) 12.4495 0.775071
\(259\) −8.89898 −0.552956
\(260\) −27.5959 −1.71143
\(261\) 1.00000 0.0618984
\(262\) 18.8990 1.16758
\(263\) 28.2474 1.74181 0.870906 0.491449i \(-0.163533\pi\)
0.870906 + 0.491449i \(0.163533\pi\)
\(264\) 4.89898 0.301511
\(265\) −35.5959 −2.18664
\(266\) −19.7980 −1.21389
\(267\) 12.8990 0.789405
\(268\) 2.00000 0.122169
\(269\) 1.10102 0.0671304 0.0335652 0.999437i \(-0.489314\pi\)
0.0335652 + 0.999437i \(0.489314\pi\)
\(270\) 4.00000 0.243432
\(271\) 5.79796 0.352201 0.176100 0.984372i \(-0.443652\pi\)
0.176100 + 0.984372i \(0.443652\pi\)
\(272\) −4.89898 −0.297044
\(273\) −30.6969 −1.85786
\(274\) −5.79796 −0.350268
\(275\) 53.8888 3.24962
\(276\) −1.00000 −0.0601929
\(277\) −4.69694 −0.282212 −0.141106 0.989995i \(-0.545066\pi\)
−0.141106 + 0.989995i \(0.545066\pi\)
\(278\) 6.00000 0.359856
\(279\) −6.00000 −0.359211
\(280\) 17.7980 1.06363
\(281\) −21.5505 −1.28560 −0.642798 0.766036i \(-0.722226\pi\)
−0.642798 + 0.766036i \(0.722226\pi\)
\(282\) 4.00000 0.238197
\(283\) 17.5959 1.04597 0.522984 0.852342i \(-0.324819\pi\)
0.522984 + 0.852342i \(0.324819\pi\)
\(284\) −10.8990 −0.646735
\(285\) −17.7980 −1.05426
\(286\) −33.7980 −1.99852
\(287\) −39.5959 −2.33727
\(288\) 1.00000 0.0589256
\(289\) 7.00000 0.411765
\(290\) 4.00000 0.234888
\(291\) −19.1464 −1.12238
\(292\) −6.89898 −0.403732
\(293\) 25.1464 1.46907 0.734535 0.678571i \(-0.237400\pi\)
0.734535 + 0.678571i \(0.237400\pi\)
\(294\) 12.7980 0.746392
\(295\) −19.5959 −1.14092
\(296\) −2.00000 −0.116248
\(297\) 4.89898 0.284268
\(298\) −18.6969 −1.08308
\(299\) 6.89898 0.398978
\(300\) 11.0000 0.635085
\(301\) 55.3939 3.19285
\(302\) −13.7980 −0.793983
\(303\) −5.10102 −0.293046
\(304\) −4.44949 −0.255196
\(305\) 11.5959 0.663980
\(306\) −4.89898 −0.280056
\(307\) −22.6969 −1.29538 −0.647691 0.761903i \(-0.724265\pi\)
−0.647691 + 0.761903i \(0.724265\pi\)
\(308\) 21.7980 1.24205
\(309\) 6.24745 0.355405
\(310\) −24.0000 −1.36311
\(311\) −2.69694 −0.152929 −0.0764647 0.997072i \(-0.524363\pi\)
−0.0764647 + 0.997072i \(0.524363\pi\)
\(312\) −6.89898 −0.390578
\(313\) −21.5959 −1.22067 −0.610337 0.792142i \(-0.708966\pi\)
−0.610337 + 0.792142i \(0.708966\pi\)
\(314\) −15.7980 −0.891530
\(315\) 17.7980 1.00280
\(316\) 10.0000 0.562544
\(317\) 10.8990 0.612148 0.306074 0.952008i \(-0.400985\pi\)
0.306074 + 0.952008i \(0.400985\pi\)
\(318\) −8.89898 −0.499030
\(319\) 4.89898 0.274290
\(320\) 4.00000 0.223607
\(321\) −9.55051 −0.533058
\(322\) −4.44949 −0.247960
\(323\) 21.7980 1.21287
\(324\) 1.00000 0.0555556
\(325\) −75.8888 −4.20955
\(326\) 0.898979 0.0497899
\(327\) −4.44949 −0.246057
\(328\) −8.89898 −0.491364
\(329\) 17.7980 0.981233
\(330\) 19.5959 1.07872
\(331\) −11.1010 −0.610167 −0.305084 0.952326i \(-0.598684\pi\)
−0.305084 + 0.952326i \(0.598684\pi\)
\(332\) −1.55051 −0.0850953
\(333\) −2.00000 −0.109599
\(334\) 9.79796 0.536120
\(335\) 8.00000 0.437087
\(336\) 4.44949 0.242740
\(337\) 20.4495 1.11395 0.556977 0.830528i \(-0.311961\pi\)
0.556977 + 0.830528i \(0.311961\pi\)
\(338\) 34.5959 1.88177
\(339\) 0 0
\(340\) −19.5959 −1.06274
\(341\) −29.3939 −1.59177
\(342\) −4.44949 −0.240601
\(343\) 25.7980 1.39296
\(344\) 12.4495 0.671232
\(345\) −4.00000 −0.215353
\(346\) −3.10102 −0.166712
\(347\) −19.5959 −1.05196 −0.525982 0.850496i \(-0.676302\pi\)
−0.525982 + 0.850496i \(0.676302\pi\)
\(348\) 1.00000 0.0536056
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 48.9444 2.61619
\(351\) −6.89898 −0.368240
\(352\) 4.89898 0.261116
\(353\) −3.79796 −0.202145 −0.101072 0.994879i \(-0.532227\pi\)
−0.101072 + 0.994879i \(0.532227\pi\)
\(354\) −4.89898 −0.260378
\(355\) −43.5959 −2.31383
\(356\) 12.8990 0.683645
\(357\) −21.7980 −1.15367
\(358\) 14.6969 0.776757
\(359\) −10.4495 −0.551503 −0.275751 0.961229i \(-0.588927\pi\)
−0.275751 + 0.961229i \(0.588927\pi\)
\(360\) 4.00000 0.210819
\(361\) 0.797959 0.0419978
\(362\) −25.3485 −1.33229
\(363\) 13.0000 0.682323
\(364\) −30.6969 −1.60896
\(365\) −27.5959 −1.44444
\(366\) 2.89898 0.151532
\(367\) −30.0000 −1.56599 −0.782994 0.622030i \(-0.786308\pi\)
−0.782994 + 0.622030i \(0.786308\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −8.89898 −0.463262
\(370\) −8.00000 −0.415900
\(371\) −39.5959 −2.05572
\(372\) −6.00000 −0.311086
\(373\) 24.0454 1.24502 0.622512 0.782610i \(-0.286112\pi\)
0.622512 + 0.782610i \(0.286112\pi\)
\(374\) −24.0000 −1.24101
\(375\) 24.0000 1.23935
\(376\) 4.00000 0.206284
\(377\) −6.89898 −0.355316
\(378\) 4.44949 0.228857
\(379\) 2.65153 0.136200 0.0681000 0.997679i \(-0.478306\pi\)
0.0681000 + 0.997679i \(0.478306\pi\)
\(380\) −17.7980 −0.913016
\(381\) 5.79796 0.297038
\(382\) 7.34847 0.375980
\(383\) 9.79796 0.500652 0.250326 0.968162i \(-0.419462\pi\)
0.250326 + 0.968162i \(0.419462\pi\)
\(384\) 1.00000 0.0510310
\(385\) 87.1918 4.44371
\(386\) 21.5959 1.09920
\(387\) 12.4495 0.632843
\(388\) −19.1464 −0.972013
\(389\) 16.6515 0.844266 0.422133 0.906534i \(-0.361281\pi\)
0.422133 + 0.906534i \(0.361281\pi\)
\(390\) −27.5959 −1.39737
\(391\) 4.89898 0.247752
\(392\) 12.7980 0.646395
\(393\) 18.8990 0.953327
\(394\) 3.79796 0.191338
\(395\) 40.0000 2.01262
\(396\) 4.89898 0.246183
\(397\) 25.5959 1.28462 0.642311 0.766444i \(-0.277976\pi\)
0.642311 + 0.766444i \(0.277976\pi\)
\(398\) −26.2474 −1.31567
\(399\) −19.7980 −0.991138
\(400\) 11.0000 0.550000
\(401\) 7.75255 0.387144 0.193572 0.981086i \(-0.437993\pi\)
0.193572 + 0.981086i \(0.437993\pi\)
\(402\) 2.00000 0.0997509
\(403\) 41.3939 2.06198
\(404\) −5.10102 −0.253785
\(405\) 4.00000 0.198762
\(406\) 4.44949 0.220824
\(407\) −9.79796 −0.485667
\(408\) −4.89898 −0.242536
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −35.5959 −1.75796
\(411\) −5.79796 −0.285992
\(412\) 6.24745 0.307790
\(413\) −21.7980 −1.07261
\(414\) −1.00000 −0.0491473
\(415\) −6.20204 −0.304446
\(416\) −6.89898 −0.338250
\(417\) 6.00000 0.293821
\(418\) −21.7980 −1.06617
\(419\) −7.75255 −0.378737 −0.189368 0.981906i \(-0.560644\pi\)
−0.189368 + 0.981906i \(0.560644\pi\)
\(420\) 17.7980 0.868451
\(421\) 34.4949 1.68118 0.840589 0.541673i \(-0.182209\pi\)
0.840589 + 0.541673i \(0.182209\pi\)
\(422\) 24.8990 1.21206
\(423\) 4.00000 0.194487
\(424\) −8.89898 −0.432173
\(425\) −53.8888 −2.61399
\(426\) −10.8990 −0.528057
\(427\) 12.8990 0.624225
\(428\) −9.55051 −0.461641
\(429\) −33.7980 −1.63178
\(430\) 49.7980 2.40147
\(431\) −12.4949 −0.601858 −0.300929 0.953647i \(-0.597297\pi\)
−0.300929 + 0.953647i \(0.597297\pi\)
\(432\) 1.00000 0.0481125
\(433\) 1.75255 0.0842222 0.0421111 0.999113i \(-0.486592\pi\)
0.0421111 + 0.999113i \(0.486592\pi\)
\(434\) −26.6969 −1.28149
\(435\) 4.00000 0.191785
\(436\) −4.44949 −0.213092
\(437\) 4.44949 0.212848
\(438\) −6.89898 −0.329646
\(439\) 2.20204 0.105098 0.0525488 0.998618i \(-0.483265\pi\)
0.0525488 + 0.998618i \(0.483265\pi\)
\(440\) 19.5959 0.934199
\(441\) 12.7980 0.609427
\(442\) 33.7980 1.60760
\(443\) 9.10102 0.432403 0.216201 0.976349i \(-0.430633\pi\)
0.216201 + 0.976349i \(0.430633\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 51.5959 2.44588
\(446\) 13.7980 0.653352
\(447\) −18.6969 −0.884335
\(448\) 4.44949 0.210219
\(449\) 28.4949 1.34476 0.672379 0.740207i \(-0.265273\pi\)
0.672379 + 0.740207i \(0.265273\pi\)
\(450\) 11.0000 0.518545
\(451\) −43.5959 −2.05285
\(452\) 0 0
\(453\) −13.7980 −0.648285
\(454\) 7.34847 0.344881
\(455\) −122.788 −5.75638
\(456\) −4.44949 −0.208366
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −12.6969 −0.593289
\(459\) −4.89898 −0.228665
\(460\) −4.00000 −0.186501
\(461\) −20.6969 −0.963953 −0.481976 0.876184i \(-0.660081\pi\)
−0.481976 + 0.876184i \(0.660081\pi\)
\(462\) 21.7980 1.01413
\(463\) −10.2020 −0.474129 −0.237065 0.971494i \(-0.576185\pi\)
−0.237065 + 0.971494i \(0.576185\pi\)
\(464\) 1.00000 0.0464238
\(465\) −24.0000 −1.11297
\(466\) 16.6969 0.773471
\(467\) −1.79796 −0.0831996 −0.0415998 0.999134i \(-0.513245\pi\)
−0.0415998 + 0.999134i \(0.513245\pi\)
\(468\) −6.89898 −0.318905
\(469\) 8.89898 0.410917
\(470\) 16.0000 0.738025
\(471\) −15.7980 −0.727932
\(472\) −4.89898 −0.225494
\(473\) 60.9898 2.80431
\(474\) 10.0000 0.459315
\(475\) −48.9444 −2.24572
\(476\) −21.7980 −0.999108
\(477\) −8.89898 −0.407456
\(478\) −1.10102 −0.0503595
\(479\) −0.247449 −0.0113062 −0.00565311 0.999984i \(-0.501799\pi\)
−0.00565311 + 0.999984i \(0.501799\pi\)
\(480\) 4.00000 0.182574
\(481\) 13.7980 0.629133
\(482\) 20.6969 0.942720
\(483\) −4.44949 −0.202459
\(484\) 13.0000 0.590909
\(485\) −76.5857 −3.47758
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 2.89898 0.131231
\(489\) 0.898979 0.0406533
\(490\) 51.1918 2.31261
\(491\) −14.8990 −0.672382 −0.336191 0.941794i \(-0.609139\pi\)
−0.336191 + 0.941794i \(0.609139\pi\)
\(492\) −8.89898 −0.401197
\(493\) −4.89898 −0.220639
\(494\) 30.6969 1.38112
\(495\) 19.5959 0.880771
\(496\) −6.00000 −0.269408
\(497\) −48.4949 −2.17529
\(498\) −1.55051 −0.0694800
\(499\) −15.7980 −0.707214 −0.353607 0.935394i \(-0.615045\pi\)
−0.353607 + 0.935394i \(0.615045\pi\)
\(500\) 24.0000 1.07331
\(501\) 9.79796 0.437741
\(502\) 3.10102 0.138405
\(503\) 20.6515 0.920806 0.460403 0.887710i \(-0.347705\pi\)
0.460403 + 0.887710i \(0.347705\pi\)
\(504\) 4.44949 0.198196
\(505\) −20.4041 −0.907970
\(506\) −4.89898 −0.217786
\(507\) 34.5959 1.53646
\(508\) 5.79796 0.257243
\(509\) 3.10102 0.137450 0.0687252 0.997636i \(-0.478107\pi\)
0.0687252 + 0.997636i \(0.478107\pi\)
\(510\) −19.5959 −0.867722
\(511\) −30.6969 −1.35795
\(512\) 1.00000 0.0441942
\(513\) −4.44949 −0.196450
\(514\) −1.10102 −0.0485639
\(515\) 24.9898 1.10118
\(516\) 12.4495 0.548058
\(517\) 19.5959 0.861827
\(518\) −8.89898 −0.390999
\(519\) −3.10102 −0.136120
\(520\) −27.5959 −1.21016
\(521\) 13.5505 0.593659 0.296829 0.954931i \(-0.404071\pi\)
0.296829 + 0.954931i \(0.404071\pi\)
\(522\) 1.00000 0.0437688
\(523\) 0.202041 0.00883464 0.00441732 0.999990i \(-0.498594\pi\)
0.00441732 + 0.999990i \(0.498594\pi\)
\(524\) 18.8990 0.825606
\(525\) 48.9444 2.13611
\(526\) 28.2474 1.23165
\(527\) 29.3939 1.28042
\(528\) 4.89898 0.213201
\(529\) 1.00000 0.0434783
\(530\) −35.5959 −1.54619
\(531\) −4.89898 −0.212598
\(532\) −19.7980 −0.858350
\(533\) 61.3939 2.65926
\(534\) 12.8990 0.558193
\(535\) −38.2020 −1.65162
\(536\) 2.00000 0.0863868
\(537\) 14.6969 0.634220
\(538\) 1.10102 0.0474684
\(539\) 62.6969 2.70055
\(540\) 4.00000 0.172133
\(541\) 21.7980 0.937167 0.468584 0.883419i \(-0.344764\pi\)
0.468584 + 0.883419i \(0.344764\pi\)
\(542\) 5.79796 0.249044
\(543\) −25.3485 −1.08781
\(544\) −4.89898 −0.210042
\(545\) −17.7980 −0.762381
\(546\) −30.6969 −1.31371
\(547\) 8.20204 0.350694 0.175347 0.984507i \(-0.443895\pi\)
0.175347 + 0.984507i \(0.443895\pi\)
\(548\) −5.79796 −0.247677
\(549\) 2.89898 0.123725
\(550\) 53.8888 2.29783
\(551\) −4.44949 −0.189555
\(552\) −1.00000 −0.0425628
\(553\) 44.4949 1.89212
\(554\) −4.69694 −0.199554
\(555\) −8.00000 −0.339581
\(556\) 6.00000 0.254457
\(557\) 37.3939 1.58443 0.792215 0.610242i \(-0.208928\pi\)
0.792215 + 0.610242i \(0.208928\pi\)
\(558\) −6.00000 −0.254000
\(559\) −85.8888 −3.63271
\(560\) 17.7980 0.752101
\(561\) −24.0000 −1.01328
\(562\) −21.5505 −0.909053
\(563\) 16.8990 0.712207 0.356104 0.934447i \(-0.384105\pi\)
0.356104 + 0.934447i \(0.384105\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) 17.5959 0.739612
\(567\) 4.44949 0.186861
\(568\) −10.8990 −0.457311
\(569\) −17.7980 −0.746129 −0.373065 0.927805i \(-0.621693\pi\)
−0.373065 + 0.927805i \(0.621693\pi\)
\(570\) −17.7980 −0.745474
\(571\) −25.1010 −1.05045 −0.525223 0.850965i \(-0.676018\pi\)
−0.525223 + 0.850965i \(0.676018\pi\)
\(572\) −33.7980 −1.41316
\(573\) 7.34847 0.306987
\(574\) −39.5959 −1.65270
\(575\) −11.0000 −0.458732
\(576\) 1.00000 0.0416667
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 7.00000 0.291162
\(579\) 21.5959 0.897496
\(580\) 4.00000 0.166091
\(581\) −6.89898 −0.286218
\(582\) −19.1464 −0.793645
\(583\) −43.5959 −1.80556
\(584\) −6.89898 −0.285482
\(585\) −27.5959 −1.14095
\(586\) 25.1464 1.03879
\(587\) −31.1918 −1.28742 −0.643712 0.765267i \(-0.722607\pi\)
−0.643712 + 0.765267i \(0.722607\pi\)
\(588\) 12.7980 0.527779
\(589\) 26.6969 1.10003
\(590\) −19.5959 −0.806751
\(591\) 3.79796 0.156227
\(592\) −2.00000 −0.0821995
\(593\) 21.5959 0.886838 0.443419 0.896314i \(-0.353765\pi\)
0.443419 + 0.896314i \(0.353765\pi\)
\(594\) 4.89898 0.201008
\(595\) −87.1918 −3.57452
\(596\) −18.6969 −0.765856
\(597\) −26.2474 −1.07424
\(598\) 6.89898 0.282120
\(599\) 22.6969 0.927372 0.463686 0.886000i \(-0.346527\pi\)
0.463686 + 0.886000i \(0.346527\pi\)
\(600\) 11.0000 0.449073
\(601\) 39.3939 1.60691 0.803455 0.595366i \(-0.202993\pi\)
0.803455 + 0.595366i \(0.202993\pi\)
\(602\) 55.3939 2.25769
\(603\) 2.00000 0.0814463
\(604\) −13.7980 −0.561431
\(605\) 52.0000 2.11410
\(606\) −5.10102 −0.207215
\(607\) 15.7980 0.641219 0.320610 0.947211i \(-0.396112\pi\)
0.320610 + 0.947211i \(0.396112\pi\)
\(608\) −4.44949 −0.180451
\(609\) 4.44949 0.180302
\(610\) 11.5959 0.469505
\(611\) −27.5959 −1.11641
\(612\) −4.89898 −0.198030
\(613\) −8.04541 −0.324951 −0.162475 0.986713i \(-0.551948\pi\)
−0.162475 + 0.986713i \(0.551948\pi\)
\(614\) −22.6969 −0.915974
\(615\) −35.5959 −1.43537
\(616\) 21.7980 0.878265
\(617\) 19.5959 0.788902 0.394451 0.918917i \(-0.370935\pi\)
0.394451 + 0.918917i \(0.370935\pi\)
\(618\) 6.24745 0.251309
\(619\) 20.0454 0.805693 0.402846 0.915268i \(-0.368021\pi\)
0.402846 + 0.915268i \(0.368021\pi\)
\(620\) −24.0000 −0.963863
\(621\) −1.00000 −0.0401286
\(622\) −2.69694 −0.108137
\(623\) 57.3939 2.29944
\(624\) −6.89898 −0.276180
\(625\) 41.0000 1.64000
\(626\) −21.5959 −0.863146
\(627\) −21.7980 −0.870527
\(628\) −15.7980 −0.630407
\(629\) 9.79796 0.390670
\(630\) 17.7980 0.709088
\(631\) −3.55051 −0.141344 −0.0706718 0.997500i \(-0.522514\pi\)
−0.0706718 + 0.997500i \(0.522514\pi\)
\(632\) 10.0000 0.397779
\(633\) 24.8990 0.989646
\(634\) 10.8990 0.432854
\(635\) 23.1918 0.920340
\(636\) −8.89898 −0.352867
\(637\) −88.2929 −3.49829
\(638\) 4.89898 0.193952
\(639\) −10.8990 −0.431157
\(640\) 4.00000 0.158114
\(641\) −39.5959 −1.56394 −0.781972 0.623313i \(-0.785786\pi\)
−0.781972 + 0.623313i \(0.785786\pi\)
\(642\) −9.55051 −0.376929
\(643\) −28.2020 −1.11218 −0.556090 0.831122i \(-0.687699\pi\)
−0.556090 + 0.831122i \(0.687699\pi\)
\(644\) −4.44949 −0.175334
\(645\) 49.7980 1.96079
\(646\) 21.7980 0.857629
\(647\) −5.10102 −0.200542 −0.100271 0.994960i \(-0.531971\pi\)
−0.100271 + 0.994960i \(0.531971\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) −75.8888 −2.97660
\(651\) −26.6969 −1.04634
\(652\) 0.898979 0.0352068
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −4.44949 −0.173989
\(655\) 75.5959 2.95378
\(656\) −8.89898 −0.347447
\(657\) −6.89898 −0.269155
\(658\) 17.7980 0.693837
\(659\) −4.40408 −0.171559 −0.0857793 0.996314i \(-0.527338\pi\)
−0.0857793 + 0.996314i \(0.527338\pi\)
\(660\) 19.5959 0.762770
\(661\) −18.6515 −0.725460 −0.362730 0.931894i \(-0.618155\pi\)
−0.362730 + 0.931894i \(0.618155\pi\)
\(662\) −11.1010 −0.431453
\(663\) 33.7980 1.31260
\(664\) −1.55051 −0.0601715
\(665\) −79.1918 −3.07093
\(666\) −2.00000 −0.0774984
\(667\) −1.00000 −0.0387202
\(668\) 9.79796 0.379094
\(669\) 13.7980 0.533460
\(670\) 8.00000 0.309067
\(671\) 14.2020 0.548264
\(672\) 4.44949 0.171643
\(673\) −2.40408 −0.0926706 −0.0463353 0.998926i \(-0.514754\pi\)
−0.0463353 + 0.998926i \(0.514754\pi\)
\(674\) 20.4495 0.787685
\(675\) 11.0000 0.423390
\(676\) 34.5959 1.33061
\(677\) −19.3485 −0.743622 −0.371811 0.928308i \(-0.621263\pi\)
−0.371811 + 0.928308i \(0.621263\pi\)
\(678\) 0 0
\(679\) −85.1918 −3.26936
\(680\) −19.5959 −0.751469
\(681\) 7.34847 0.281594
\(682\) −29.3939 −1.12555
\(683\) 19.5959 0.749817 0.374908 0.927062i \(-0.377674\pi\)
0.374908 + 0.927062i \(0.377674\pi\)
\(684\) −4.44949 −0.170130
\(685\) −23.1918 −0.886115
\(686\) 25.7980 0.984971
\(687\) −12.6969 −0.484418
\(688\) 12.4495 0.474632
\(689\) 61.3939 2.33892
\(690\) −4.00000 −0.152277
\(691\) −29.3939 −1.11820 −0.559098 0.829102i \(-0.688852\pi\)
−0.559098 + 0.829102i \(0.688852\pi\)
\(692\) −3.10102 −0.117883
\(693\) 21.7980 0.828036
\(694\) −19.5959 −0.743851
\(695\) 24.0000 0.910372
\(696\) 1.00000 0.0379049
\(697\) 43.5959 1.65131
\(698\) 14.0000 0.529908
\(699\) 16.6969 0.631536
\(700\) 48.9444 1.84992
\(701\) 25.7980 0.974375 0.487188 0.873297i \(-0.338023\pi\)
0.487188 + 0.873297i \(0.338023\pi\)
\(702\) −6.89898 −0.260385
\(703\) 8.89898 0.335631
\(704\) 4.89898 0.184637
\(705\) 16.0000 0.602595
\(706\) −3.79796 −0.142938
\(707\) −22.6969 −0.853606
\(708\) −4.89898 −0.184115
\(709\) 6.24745 0.234628 0.117314 0.993095i \(-0.462572\pi\)
0.117314 + 0.993095i \(0.462572\pi\)
\(710\) −43.5959 −1.63613
\(711\) 10.0000 0.375029
\(712\) 12.8990 0.483410
\(713\) 6.00000 0.224702
\(714\) −21.7980 −0.815768
\(715\) −135.192 −5.05589
\(716\) 14.6969 0.549250
\(717\) −1.10102 −0.0411184
\(718\) −10.4495 −0.389971
\(719\) −21.3939 −0.797857 −0.398928 0.916982i \(-0.630618\pi\)
−0.398928 + 0.916982i \(0.630618\pi\)
\(720\) 4.00000 0.149071
\(721\) 27.7980 1.03525
\(722\) 0.797959 0.0296970
\(723\) 20.6969 0.769727
\(724\) −25.3485 −0.942068
\(725\) 11.0000 0.408530
\(726\) 13.0000 0.482475
\(727\) 19.7980 0.734266 0.367133 0.930169i \(-0.380339\pi\)
0.367133 + 0.930169i \(0.380339\pi\)
\(728\) −30.6969 −1.13770
\(729\) 1.00000 0.0370370
\(730\) −27.5959 −1.02137
\(731\) −60.9898 −2.25579
\(732\) 2.89898 0.107149
\(733\) −19.7980 −0.731254 −0.365627 0.930761i \(-0.619145\pi\)
−0.365627 + 0.930761i \(0.619145\pi\)
\(734\) −30.0000 −1.10732
\(735\) 51.1918 1.88824
\(736\) −1.00000 −0.0368605
\(737\) 9.79796 0.360912
\(738\) −8.89898 −0.327576
\(739\) −33.7980 −1.24328 −0.621639 0.783304i \(-0.713533\pi\)
−0.621639 + 0.783304i \(0.713533\pi\)
\(740\) −8.00000 −0.294086
\(741\) 30.6969 1.12768
\(742\) −39.5959 −1.45361
\(743\) 33.1464 1.21602 0.608012 0.793928i \(-0.291967\pi\)
0.608012 + 0.793928i \(0.291967\pi\)
\(744\) −6.00000 −0.219971
\(745\) −74.7878 −2.74001
\(746\) 24.0454 0.880365
\(747\) −1.55051 −0.0567302
\(748\) −24.0000 −0.877527
\(749\) −42.4949 −1.55273
\(750\) 24.0000 0.876356
\(751\) 7.79796 0.284552 0.142276 0.989827i \(-0.454558\pi\)
0.142276 + 0.989827i \(0.454558\pi\)
\(752\) 4.00000 0.145865
\(753\) 3.10102 0.113007
\(754\) −6.89898 −0.251246
\(755\) −55.1918 −2.00864
\(756\) 4.44949 0.161826
\(757\) 6.49490 0.236061 0.118031 0.993010i \(-0.462342\pi\)
0.118031 + 0.993010i \(0.462342\pi\)
\(758\) 2.65153 0.0963079
\(759\) −4.89898 −0.177822
\(760\) −17.7980 −0.645600
\(761\) 11.3939 0.413028 0.206514 0.978444i \(-0.433788\pi\)
0.206514 + 0.978444i \(0.433788\pi\)
\(762\) 5.79796 0.210038
\(763\) −19.7980 −0.716734
\(764\) 7.34847 0.265858
\(765\) −19.5959 −0.708492
\(766\) 9.79796 0.354015
\(767\) 33.7980 1.22037
\(768\) 1.00000 0.0360844
\(769\) 9.75255 0.351686 0.175843 0.984418i \(-0.443735\pi\)
0.175843 + 0.984418i \(0.443735\pi\)
\(770\) 87.1918 3.14218
\(771\) −1.10102 −0.0396523
\(772\) 21.5959 0.777254
\(773\) 49.5505 1.78221 0.891104 0.453799i \(-0.149932\pi\)
0.891104 + 0.453799i \(0.149932\pi\)
\(774\) 12.4495 0.447488
\(775\) −66.0000 −2.37079
\(776\) −19.1464 −0.687317
\(777\) −8.89898 −0.319249
\(778\) 16.6515 0.596986
\(779\) 39.5959 1.41867
\(780\) −27.5959 −0.988092
\(781\) −53.3939 −1.91058
\(782\) 4.89898 0.175187
\(783\) 1.00000 0.0357371
\(784\) 12.7980 0.457070
\(785\) −63.1918 −2.25541
\(786\) 18.8990 0.674104
\(787\) 34.0000 1.21197 0.605985 0.795476i \(-0.292779\pi\)
0.605985 + 0.795476i \(0.292779\pi\)
\(788\) 3.79796 0.135297
\(789\) 28.2474 1.00564
\(790\) 40.0000 1.42314
\(791\) 0 0
\(792\) 4.89898 0.174078
\(793\) −20.0000 −0.710221
\(794\) 25.5959 0.908365
\(795\) −35.5959 −1.26246
\(796\) −26.2474 −0.930316
\(797\) 23.8434 0.844575 0.422288 0.906462i \(-0.361227\pi\)
0.422288 + 0.906462i \(0.361227\pi\)
\(798\) −19.7980 −0.700840
\(799\) −19.5959 −0.693254
\(800\) 11.0000 0.388909
\(801\) 12.8990 0.455763
\(802\) 7.75255 0.273752
\(803\) −33.7980 −1.19270
\(804\) 2.00000 0.0705346
\(805\) −17.7980 −0.627296
\(806\) 41.3939 1.45804
\(807\) 1.10102 0.0387578
\(808\) −5.10102 −0.179453
\(809\) −28.4949 −1.00183 −0.500914 0.865497i \(-0.667003\pi\)
−0.500914 + 0.865497i \(0.667003\pi\)
\(810\) 4.00000 0.140546
\(811\) −6.40408 −0.224878 −0.112439 0.993659i \(-0.535866\pi\)
−0.112439 + 0.993659i \(0.535866\pi\)
\(812\) 4.44949 0.156146
\(813\) 5.79796 0.203343
\(814\) −9.79796 −0.343418
\(815\) 3.59592 0.125960
\(816\) −4.89898 −0.171499
\(817\) −55.3939 −1.93799
\(818\) −6.00000 −0.209785
\(819\) −30.6969 −1.07264
\(820\) −35.5959 −1.24306
\(821\) 23.1010 0.806231 0.403116 0.915149i \(-0.367927\pi\)
0.403116 + 0.915149i \(0.367927\pi\)
\(822\) −5.79796 −0.202227
\(823\) −45.7980 −1.59642 −0.798208 0.602382i \(-0.794219\pi\)
−0.798208 + 0.602382i \(0.794219\pi\)
\(824\) 6.24745 0.217640
\(825\) 53.8888 1.87617
\(826\) −21.7980 −0.758448
\(827\) −31.5959 −1.09870 −0.549349 0.835593i \(-0.685124\pi\)
−0.549349 + 0.835593i \(0.685124\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −6.20204 −0.215276
\(831\) −4.69694 −0.162935
\(832\) −6.89898 −0.239179
\(833\) −62.6969 −2.17232
\(834\) 6.00000 0.207763
\(835\) 39.1918 1.35629
\(836\) −21.7980 −0.753898
\(837\) −6.00000 −0.207390
\(838\) −7.75255 −0.267807
\(839\) −20.7423 −0.716105 −0.358053 0.933701i \(-0.616559\pi\)
−0.358053 + 0.933701i \(0.616559\pi\)
\(840\) 17.7980 0.614088
\(841\) 1.00000 0.0344828
\(842\) 34.4949 1.18877
\(843\) −21.5505 −0.742239
\(844\) 24.8990 0.857058
\(845\) 138.384 4.76054
\(846\) 4.00000 0.137523
\(847\) 57.8434 1.98752
\(848\) −8.89898 −0.305592
\(849\) 17.5959 0.603890
\(850\) −53.8888 −1.84837
\(851\) 2.00000 0.0685591
\(852\) −10.8990 −0.373393
\(853\) 53.1918 1.82125 0.910627 0.413230i \(-0.135599\pi\)
0.910627 + 0.413230i \(0.135599\pi\)
\(854\) 12.8990 0.441394
\(855\) −17.7980 −0.608677
\(856\) −9.55051 −0.326430
\(857\) 25.1010 0.857435 0.428717 0.903439i \(-0.358966\pi\)
0.428717 + 0.903439i \(0.358966\pi\)
\(858\) −33.7980 −1.15384
\(859\) −35.5959 −1.21452 −0.607259 0.794504i \(-0.707731\pi\)
−0.607259 + 0.794504i \(0.707731\pi\)
\(860\) 49.7980 1.69810
\(861\) −39.5959 −1.34943
\(862\) −12.4949 −0.425578
\(863\) −1.79796 −0.0612032 −0.0306016 0.999532i \(-0.509742\pi\)
−0.0306016 + 0.999532i \(0.509742\pi\)
\(864\) 1.00000 0.0340207
\(865\) −12.4041 −0.421751
\(866\) 1.75255 0.0595541
\(867\) 7.00000 0.237732
\(868\) −26.6969 −0.906153
\(869\) 48.9898 1.66186
\(870\) 4.00000 0.135613
\(871\) −13.7980 −0.467526
\(872\) −4.44949 −0.150679
\(873\) −19.1464 −0.648008
\(874\) 4.44949 0.150506
\(875\) 106.788 3.61008
\(876\) −6.89898 −0.233095
\(877\) −16.6969 −0.563816 −0.281908 0.959442i \(-0.590967\pi\)
−0.281908 + 0.959442i \(0.590967\pi\)
\(878\) 2.20204 0.0743153
\(879\) 25.1464 0.848168
\(880\) 19.5959 0.660578
\(881\) −20.8990 −0.704105 −0.352052 0.935980i \(-0.614516\pi\)
−0.352052 + 0.935980i \(0.614516\pi\)
\(882\) 12.7980 0.430930
\(883\) 1.79796 0.0605061 0.0302531 0.999542i \(-0.490369\pi\)
0.0302531 + 0.999542i \(0.490369\pi\)
\(884\) 33.7980 1.13675
\(885\) −19.5959 −0.658710
\(886\) 9.10102 0.305755
\(887\) 4.89898 0.164492 0.0822458 0.996612i \(-0.473791\pi\)
0.0822458 + 0.996612i \(0.473791\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 25.7980 0.865236
\(890\) 51.5959 1.72950
\(891\) 4.89898 0.164122
\(892\) 13.7980 0.461990
\(893\) −17.7980 −0.595586
\(894\) −18.6969 −0.625319
\(895\) 58.7878 1.96506
\(896\) 4.44949 0.148647
\(897\) 6.89898 0.230350
\(898\) 28.4949 0.950887
\(899\) −6.00000 −0.200111
\(900\) 11.0000 0.366667
\(901\) 43.5959 1.45239
\(902\) −43.5959 −1.45159
\(903\) 55.3939 1.84339
\(904\) 0 0
\(905\) −101.394 −3.37045
\(906\) −13.7980 −0.458406
\(907\) 10.6515 0.353678 0.176839 0.984240i \(-0.443413\pi\)
0.176839 + 0.984240i \(0.443413\pi\)
\(908\) 7.34847 0.243868
\(909\) −5.10102 −0.169190
\(910\) −122.788 −4.07037
\(911\) 35.8434 1.18754 0.593772 0.804633i \(-0.297638\pi\)
0.593772 + 0.804633i \(0.297638\pi\)
\(912\) −4.44949 −0.147337
\(913\) −7.59592 −0.251388
\(914\) −22.0000 −0.727695
\(915\) 11.5959 0.383349
\(916\) −12.6969 −0.419519
\(917\) 84.0908 2.77692
\(918\) −4.89898 −0.161690
\(919\) −1.75255 −0.0578113 −0.0289057 0.999582i \(-0.509202\pi\)
−0.0289057 + 0.999582i \(0.509202\pi\)
\(920\) −4.00000 −0.131876
\(921\) −22.6969 −0.747890
\(922\) −20.6969 −0.681617
\(923\) 75.1918 2.47497
\(924\) 21.7980 0.717100
\(925\) −22.0000 −0.723356
\(926\) −10.2020 −0.335260
\(927\) 6.24745 0.205193
\(928\) 1.00000 0.0328266
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) −24.0000 −0.786991
\(931\) −56.9444 −1.86628
\(932\) 16.6969 0.546926
\(933\) −2.69694 −0.0882938
\(934\) −1.79796 −0.0588310
\(935\) −96.0000 −3.13954
\(936\) −6.89898 −0.225500
\(937\) −11.7980 −0.385423 −0.192711 0.981256i \(-0.561728\pi\)
−0.192711 + 0.981256i \(0.561728\pi\)
\(938\) 8.89898 0.290562
\(939\) −21.5959 −0.704756
\(940\) 16.0000 0.521862
\(941\) −43.1918 −1.40801 −0.704007 0.710193i \(-0.748608\pi\)
−0.704007 + 0.710193i \(0.748608\pi\)
\(942\) −15.7980 −0.514725
\(943\) 8.89898 0.289791
\(944\) −4.89898 −0.159448
\(945\) 17.7980 0.578968
\(946\) 60.9898 1.98295
\(947\) 18.8990 0.614134 0.307067 0.951688i \(-0.400652\pi\)
0.307067 + 0.951688i \(0.400652\pi\)
\(948\) 10.0000 0.324785
\(949\) 47.5959 1.54503
\(950\) −48.9444 −1.58797
\(951\) 10.8990 0.353424
\(952\) −21.7980 −0.706476
\(953\) −1.05561 −0.0341947 −0.0170973 0.999854i \(-0.505443\pi\)
−0.0170973 + 0.999854i \(0.505443\pi\)
\(954\) −8.89898 −0.288115
\(955\) 29.3939 0.951164
\(956\) −1.10102 −0.0356095
\(957\) 4.89898 0.158362
\(958\) −0.247449 −0.00799471
\(959\) −25.7980 −0.833060
\(960\) 4.00000 0.129099
\(961\) 5.00000 0.161290
\(962\) 13.7980 0.444864
\(963\) −9.55051 −0.307761
\(964\) 20.6969 0.666604
\(965\) 86.3837 2.78079
\(966\) −4.44949 −0.143160
\(967\) −1.39388 −0.0448241 −0.0224120 0.999749i \(-0.507135\pi\)
−0.0224120 + 0.999749i \(0.507135\pi\)
\(968\) 13.0000 0.417836
\(969\) 21.7980 0.700251
\(970\) −76.5857 −2.45902
\(971\) −30.2929 −0.972144 −0.486072 0.873919i \(-0.661571\pi\)
−0.486072 + 0.873919i \(0.661571\pi\)
\(972\) 1.00000 0.0320750
\(973\) 26.6969 0.855865
\(974\) −32.0000 −1.02535
\(975\) −75.8888 −2.43039
\(976\) 2.89898 0.0927941
\(977\) 28.7423 0.919549 0.459775 0.888036i \(-0.347930\pi\)
0.459775 + 0.888036i \(0.347930\pi\)
\(978\) 0.898979 0.0287462
\(979\) 63.1918 2.01962
\(980\) 51.1918 1.63526
\(981\) −4.44949 −0.142061
\(982\) −14.8990 −0.475446
\(983\) 38.9444 1.24213 0.621067 0.783758i \(-0.286700\pi\)
0.621067 + 0.783758i \(0.286700\pi\)
\(984\) −8.89898 −0.283689
\(985\) 15.1918 0.484052
\(986\) −4.89898 −0.156015
\(987\) 17.7980 0.566515
\(988\) 30.6969 0.976600
\(989\) −12.4495 −0.395871
\(990\) 19.5959 0.622799
\(991\) −52.8990 −1.68039 −0.840196 0.542283i \(-0.817560\pi\)
−0.840196 + 0.542283i \(0.817560\pi\)
\(992\) −6.00000 −0.190500
\(993\) −11.1010 −0.352280
\(994\) −48.4949 −1.53816
\(995\) −104.990 −3.32840
\(996\) −1.55051 −0.0491298
\(997\) 37.5959 1.19067 0.595337 0.803476i \(-0.297018\pi\)
0.595337 + 0.803476i \(0.297018\pi\)
\(998\) −15.7980 −0.500076
\(999\) −2.00000 −0.0632772
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 4002.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.x.1.2 2 1.1 even 1 trivial