Properties

Label 5054.2.a.g.1.2
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +2.61803 q^{10} -4.85410 q^{11} +1.61803 q^{12} -4.47214 q^{13} -1.00000 q^{14} -4.23607 q^{15} +1.00000 q^{16} +0.763932 q^{17} +0.381966 q^{18} -2.61803 q^{20} +1.61803 q^{21} +4.85410 q^{22} +8.94427 q^{23} -1.61803 q^{24} +1.85410 q^{25} +4.47214 q^{26} -5.47214 q^{27} +1.00000 q^{28} +0.145898 q^{29} +4.23607 q^{30} +6.00000 q^{31} -1.00000 q^{32} -7.85410 q^{33} -0.763932 q^{34} -2.61803 q^{35} -0.381966 q^{36} +7.85410 q^{37} -7.23607 q^{39} +2.61803 q^{40} -9.56231 q^{41} -1.61803 q^{42} +3.85410 q^{43} -4.85410 q^{44} +1.00000 q^{45} -8.94427 q^{46} -10.8541 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.85410 q^{50} +1.23607 q^{51} -4.47214 q^{52} -5.14590 q^{53} +5.47214 q^{54} +12.7082 q^{55} -1.00000 q^{56} -0.145898 q^{58} -11.5623 q^{59} -4.23607 q^{60} -8.56231 q^{61} -6.00000 q^{62} -0.381966 q^{63} +1.00000 q^{64} +11.7082 q^{65} +7.85410 q^{66} +5.23607 q^{67} +0.763932 q^{68} +14.4721 q^{69} +2.61803 q^{70} +8.56231 q^{71} +0.381966 q^{72} -7.85410 q^{74} +3.00000 q^{75} -4.85410 q^{77} +7.23607 q^{78} +0.326238 q^{79} -2.61803 q^{80} -7.70820 q^{81} +9.56231 q^{82} +11.2361 q^{83} +1.61803 q^{84} -2.00000 q^{85} -3.85410 q^{86} +0.236068 q^{87} +4.85410 q^{88} -3.14590 q^{89} -1.00000 q^{90} -4.47214 q^{91} +8.94427 q^{92} +9.70820 q^{93} +10.8541 q^{94} -1.61803 q^{96} +7.14590 q^{97} -1.00000 q^{98} +1.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 3 q^{10} - 3 q^{11} + q^{12} - 2 q^{14} - 4 q^{15} + 2 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{20} + q^{21} + 3 q^{22} - q^{24} - 3 q^{25} - 2 q^{27} + 2 q^{28} + 7 q^{29} + 4 q^{30} + 12 q^{31} - 2 q^{32} - 9 q^{33} - 6 q^{34} - 3 q^{35} - 3 q^{36} + 9 q^{37} - 10 q^{39} + 3 q^{40} + q^{41} - q^{42} + q^{43} - 3 q^{44} + 2 q^{45} - 15 q^{47} + q^{48} + 2 q^{49} + 3 q^{50} - 2 q^{51} - 17 q^{53} + 2 q^{54} + 12 q^{55} - 2 q^{56} - 7 q^{58} - 3 q^{59} - 4 q^{60} + 3 q^{61} - 12 q^{62} - 3 q^{63} + 2 q^{64} + 10 q^{65} + 9 q^{66} + 6 q^{67} + 6 q^{68} + 20 q^{69} + 3 q^{70} - 3 q^{71} + 3 q^{72} - 9 q^{74} + 6 q^{75} - 3 q^{77} + 10 q^{78} - 15 q^{79} - 3 q^{80} - 2 q^{81} - q^{82} + 18 q^{83} + q^{84} - 4 q^{85} - q^{86} - 4 q^{87} + 3 q^{88} - 13 q^{89} - 2 q^{90} + 6 q^{93} + 15 q^{94} - q^{96} + 21 q^{97} - 2 q^{98} - 3 q^{99} + O(q^{100}) \)

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
\(3\) 1.61803 0.934172 0.467086 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.61803 −1.17082 −0.585410 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −0.381966 −0.127322
\(10\) 2.61803 0.827895
\(11\) −4.85410 −1.46357 −0.731783 0.681537i \(-0.761312\pi\)
−0.731783 + 0.681537i \(0.761312\pi\)
\(12\) 1.61803 0.467086
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) −1.00000 −0.267261
\(15\) −4.23607 −1.09375
\(16\) 1.00000 0.250000
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) 0.381966 0.0900303
\(19\) 0 0
\(20\) −2.61803 −0.585410
\(21\) 1.61803 0.353084
\(22\) 4.85410 1.03490
\(23\) 8.94427 1.86501 0.932505 0.361158i \(-0.117618\pi\)
0.932505 + 0.361158i \(0.117618\pi\)
\(24\) −1.61803 −0.330280
\(25\) 1.85410 0.370820
\(26\) 4.47214 0.877058
\(27\) −5.47214 −1.05311
\(28\) 1.00000 0.188982
\(29\) 0.145898 0.0270926 0.0135463 0.999908i \(-0.495688\pi\)
0.0135463 + 0.999908i \(0.495688\pi\)
\(30\) 4.23607 0.773397
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) −7.85410 −1.36722
\(34\) −0.763932 −0.131013
\(35\) −2.61803 −0.442529
\(36\) −0.381966 −0.0636610
\(37\) 7.85410 1.29121 0.645603 0.763673i \(-0.276606\pi\)
0.645603 + 0.763673i \(0.276606\pi\)
\(38\) 0 0
\(39\) −7.23607 −1.15870
\(40\) 2.61803 0.413948
\(41\) −9.56231 −1.49338 −0.746691 0.665171i \(-0.768358\pi\)
−0.746691 + 0.665171i \(0.768358\pi\)
\(42\) −1.61803 −0.249668
\(43\) 3.85410 0.587745 0.293873 0.955845i \(-0.405056\pi\)
0.293873 + 0.955845i \(0.405056\pi\)
\(44\) −4.85410 −0.731783
\(45\) 1.00000 0.149071
\(46\) −8.94427 −1.31876
\(47\) −10.8541 −1.58323 −0.791617 0.611018i \(-0.790760\pi\)
−0.791617 + 0.611018i \(0.790760\pi\)
\(48\) 1.61803 0.233543
\(49\) 1.00000 0.142857
\(50\) −1.85410 −0.262210
\(51\) 1.23607 0.173084
\(52\) −4.47214 −0.620174
\(53\) −5.14590 −0.706843 −0.353422 0.935464i \(-0.614982\pi\)
−0.353422 + 0.935464i \(0.614982\pi\)
\(54\) 5.47214 0.744663
\(55\) 12.7082 1.71357
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −0.145898 −0.0191574
\(59\) −11.5623 −1.50528 −0.752642 0.658430i \(-0.771221\pi\)
−0.752642 + 0.658430i \(0.771221\pi\)
\(60\) −4.23607 −0.546874
\(61\) −8.56231 −1.09629 −0.548145 0.836383i \(-0.684666\pi\)
−0.548145 + 0.836383i \(0.684666\pi\)
\(62\) −6.00000 −0.762001
\(63\) −0.381966 −0.0481232
\(64\) 1.00000 0.125000
\(65\) 11.7082 1.45222
\(66\) 7.85410 0.966773
\(67\) 5.23607 0.639688 0.319844 0.947470i \(-0.396370\pi\)
0.319844 + 0.947470i \(0.396370\pi\)
\(68\) 0.763932 0.0926404
\(69\) 14.4721 1.74224
\(70\) 2.61803 0.312915
\(71\) 8.56231 1.01616 0.508079 0.861310i \(-0.330356\pi\)
0.508079 + 0.861310i \(0.330356\pi\)
\(72\) 0.381966 0.0450151
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −7.85410 −0.913021
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) −4.85410 −0.553176
\(78\) 7.23607 0.819323
\(79\) 0.326238 0.0367046 0.0183523 0.999832i \(-0.494158\pi\)
0.0183523 + 0.999832i \(0.494158\pi\)
\(80\) −2.61803 −0.292705
\(81\) −7.70820 −0.856467
\(82\) 9.56231 1.05598
\(83\) 11.2361 1.23332 0.616659 0.787230i \(-0.288486\pi\)
0.616659 + 0.787230i \(0.288486\pi\)
\(84\) 1.61803 0.176542
\(85\) −2.00000 −0.216930
\(86\) −3.85410 −0.415599
\(87\) 0.236068 0.0253091
\(88\) 4.85410 0.517449
\(89\) −3.14590 −0.333465 −0.166732 0.986002i \(-0.553322\pi\)
−0.166732 + 0.986002i \(0.553322\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.47214 −0.468807
\(92\) 8.94427 0.932505
\(93\) 9.70820 1.00669
\(94\) 10.8541 1.11952
\(95\) 0 0
\(96\) −1.61803 −0.165140
\(97\) 7.14590 0.725556 0.362778 0.931876i \(-0.381828\pi\)
0.362778 + 0.931876i \(0.381828\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.85410 0.186344
\(100\) 1.85410 0.185410
\(101\) 2.94427 0.292966 0.146483 0.989213i \(-0.453205\pi\)
0.146483 + 0.989213i \(0.453205\pi\)
\(102\) −1.23607 −0.122389
\(103\) 8.94427 0.881305 0.440653 0.897678i \(-0.354747\pi\)
0.440653 + 0.897678i \(0.354747\pi\)
\(104\) 4.47214 0.438529
\(105\) −4.23607 −0.413398
\(106\) 5.14590 0.499814
\(107\) 19.7082 1.90526 0.952632 0.304125i \(-0.0983642\pi\)
0.952632 + 0.304125i \(0.0983642\pi\)
\(108\) −5.47214 −0.526557
\(109\) 10.8541 1.03963 0.519817 0.854278i \(-0.326000\pi\)
0.519817 + 0.854278i \(0.326000\pi\)
\(110\) −12.7082 −1.21168
\(111\) 12.7082 1.20621
\(112\) 1.00000 0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −23.4164 −2.18359
\(116\) 0.145898 0.0135463
\(117\) 1.70820 0.157924
\(118\) 11.5623 1.06440
\(119\) 0.763932 0.0700295
\(120\) 4.23607 0.386698
\(121\) 12.5623 1.14203
\(122\) 8.56231 0.775195
\(123\) −15.4721 −1.39508
\(124\) 6.00000 0.538816
\(125\) 8.23607 0.736656
\(126\) 0.381966 0.0340282
\(127\) 22.0902 1.96019 0.980093 0.198540i \(-0.0636199\pi\)
0.980093 + 0.198540i \(0.0636199\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.23607 0.549055
\(130\) −11.7082 −1.02688
\(131\) −4.47214 −0.390732 −0.195366 0.980730i \(-0.562590\pi\)
−0.195366 + 0.980730i \(0.562590\pi\)
\(132\) −7.85410 −0.683612
\(133\) 0 0
\(134\) −5.23607 −0.452327
\(135\) 14.3262 1.23301
\(136\) −0.763932 −0.0655066
\(137\) 3.38197 0.288941 0.144470 0.989509i \(-0.453852\pi\)
0.144470 + 0.989509i \(0.453852\pi\)
\(138\) −14.4721 −1.23195
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −2.61803 −0.221264
\(141\) −17.5623 −1.47901
\(142\) −8.56231 −0.718533
\(143\) 21.7082 1.81533
\(144\) −0.381966 −0.0318305
\(145\) −0.381966 −0.0317206
\(146\) 0 0
\(147\) 1.61803 0.133453
\(148\) 7.85410 0.645603
\(149\) −17.2361 −1.41203 −0.706017 0.708195i \(-0.749510\pi\)
−0.706017 + 0.708195i \(0.749510\pi\)
\(150\) −3.00000 −0.244949
\(151\) 3.05573 0.248672 0.124336 0.992240i \(-0.460320\pi\)
0.124336 + 0.992240i \(0.460320\pi\)
\(152\) 0 0
\(153\) −0.291796 −0.0235903
\(154\) 4.85410 0.391155
\(155\) −15.7082 −1.26171
\(156\) −7.23607 −0.579349
\(157\) 2.85410 0.227782 0.113891 0.993493i \(-0.463669\pi\)
0.113891 + 0.993493i \(0.463669\pi\)
\(158\) −0.326238 −0.0259541
\(159\) −8.32624 −0.660314
\(160\) 2.61803 0.206974
\(161\) 8.94427 0.704907
\(162\) 7.70820 0.605614
\(163\) −20.2705 −1.58771 −0.793854 0.608108i \(-0.791929\pi\)
−0.793854 + 0.608108i \(0.791929\pi\)
\(164\) −9.56231 −0.746691
\(165\) 20.5623 1.60077
\(166\) −11.2361 −0.872088
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) −1.61803 −0.124834
\(169\) 7.00000 0.538462
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 3.85410 0.293873
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −0.236068 −0.0178963
\(175\) 1.85410 0.140157
\(176\) −4.85410 −0.365892
\(177\) −18.7082 −1.40619
\(178\) 3.14590 0.235795
\(179\) 17.7082 1.32357 0.661787 0.749692i \(-0.269798\pi\)
0.661787 + 0.749692i \(0.269798\pi\)
\(180\) 1.00000 0.0745356
\(181\) −7.41641 −0.551257 −0.275629 0.961264i \(-0.588886\pi\)
−0.275629 + 0.961264i \(0.588886\pi\)
\(182\) 4.47214 0.331497
\(183\) −13.8541 −1.02412
\(184\) −8.94427 −0.659380
\(185\) −20.5623 −1.51177
\(186\) −9.70820 −0.711840
\(187\) −3.70820 −0.271171
\(188\) −10.8541 −0.791617
\(189\) −5.47214 −0.398039
\(190\) 0 0
\(191\) 14.9443 1.08133 0.540665 0.841238i \(-0.318173\pi\)
0.540665 + 0.841238i \(0.318173\pi\)
\(192\) 1.61803 0.116772
\(193\) −20.9443 −1.50760 −0.753801 0.657103i \(-0.771782\pi\)
−0.753801 + 0.657103i \(0.771782\pi\)
\(194\) −7.14590 −0.513046
\(195\) 18.9443 1.35663
\(196\) 1.00000 0.0714286
\(197\) −5.23607 −0.373054 −0.186527 0.982450i \(-0.559723\pi\)
−0.186527 + 0.982450i \(0.559723\pi\)
\(198\) −1.85410 −0.131765
\(199\) 1.56231 0.110749 0.0553745 0.998466i \(-0.482365\pi\)
0.0553745 + 0.998466i \(0.482365\pi\)
\(200\) −1.85410 −0.131105
\(201\) 8.47214 0.597578
\(202\) −2.94427 −0.207158
\(203\) 0.145898 0.0102400
\(204\) 1.23607 0.0865421
\(205\) 25.0344 1.74848
\(206\) −8.94427 −0.623177
\(207\) −3.41641 −0.237457
\(208\) −4.47214 −0.310087
\(209\) 0 0
\(210\) 4.23607 0.292316
\(211\) 17.2361 1.18658 0.593290 0.804989i \(-0.297829\pi\)
0.593290 + 0.804989i \(0.297829\pi\)
\(212\) −5.14590 −0.353422
\(213\) 13.8541 0.949267
\(214\) −19.7082 −1.34723
\(215\) −10.0902 −0.688144
\(216\) 5.47214 0.372332
\(217\) 6.00000 0.407307
\(218\) −10.8541 −0.735133
\(219\) 0 0
\(220\) 12.7082 0.856787
\(221\) −3.41641 −0.229812
\(222\) −12.7082 −0.852919
\(223\) −5.23607 −0.350633 −0.175317 0.984512i \(-0.556095\pi\)
−0.175317 + 0.984512i \(0.556095\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −0.708204 −0.0472136
\(226\) −10.0000 −0.665190
\(227\) 19.4164 1.28871 0.644356 0.764726i \(-0.277125\pi\)
0.644356 + 0.764726i \(0.277125\pi\)
\(228\) 0 0
\(229\) 23.8541 1.57632 0.788162 0.615468i \(-0.211033\pi\)
0.788162 + 0.615468i \(0.211033\pi\)
\(230\) 23.4164 1.54403
\(231\) −7.85410 −0.516762
\(232\) −0.145898 −0.00957868
\(233\) 4.09017 0.267956 0.133978 0.990984i \(-0.457225\pi\)
0.133978 + 0.990984i \(0.457225\pi\)
\(234\) −1.70820 −0.111669
\(235\) 28.4164 1.85368
\(236\) −11.5623 −0.752642
\(237\) 0.527864 0.0342885
\(238\) −0.763932 −0.0495184
\(239\) 14.1803 0.917250 0.458625 0.888630i \(-0.348342\pi\)
0.458625 + 0.888630i \(0.348342\pi\)
\(240\) −4.23607 −0.273437
\(241\) 17.5623 1.13129 0.565644 0.824650i \(-0.308628\pi\)
0.565644 + 0.824650i \(0.308628\pi\)
\(242\) −12.5623 −0.807536
\(243\) 3.94427 0.253025
\(244\) −8.56231 −0.548145
\(245\) −2.61803 −0.167260
\(246\) 15.4721 0.986467
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) 18.1803 1.15213
\(250\) −8.23607 −0.520895
\(251\) 9.05573 0.571592 0.285796 0.958290i \(-0.407742\pi\)
0.285796 + 0.958290i \(0.407742\pi\)
\(252\) −0.381966 −0.0240616
\(253\) −43.4164 −2.72957
\(254\) −22.0902 −1.38606
\(255\) −3.23607 −0.202650
\(256\) 1.00000 0.0625000
\(257\) 9.85410 0.614682 0.307341 0.951599i \(-0.400561\pi\)
0.307341 + 0.951599i \(0.400561\pi\)
\(258\) −6.23607 −0.388241
\(259\) 7.85410 0.488030
\(260\) 11.7082 0.726112
\(261\) −0.0557281 −0.00344948
\(262\) 4.47214 0.276289
\(263\) 12.6525 0.780185 0.390093 0.920776i \(-0.372443\pi\)
0.390093 + 0.920776i \(0.372443\pi\)
\(264\) 7.85410 0.483387
\(265\) 13.4721 0.827587
\(266\) 0 0
\(267\) −5.09017 −0.311513
\(268\) 5.23607 0.319844
\(269\) −4.58359 −0.279467 −0.139733 0.990189i \(-0.544625\pi\)
−0.139733 + 0.990189i \(0.544625\pi\)
\(270\) −14.3262 −0.871867
\(271\) 2.43769 0.148079 0.0740397 0.997255i \(-0.476411\pi\)
0.0740397 + 0.997255i \(0.476411\pi\)
\(272\) 0.763932 0.0463202
\(273\) −7.23607 −0.437947
\(274\) −3.38197 −0.204312
\(275\) −9.00000 −0.542720
\(276\) 14.4721 0.871120
\(277\) −25.4164 −1.52712 −0.763562 0.645735i \(-0.776551\pi\)
−0.763562 + 0.645735i \(0.776551\pi\)
\(278\) −6.00000 −0.359856
\(279\) −2.29180 −0.137206
\(280\) 2.61803 0.156457
\(281\) −23.4164 −1.39691 −0.698453 0.715656i \(-0.746128\pi\)
−0.698453 + 0.715656i \(0.746128\pi\)
\(282\) 17.5623 1.04582
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 8.56231 0.508079
\(285\) 0 0
\(286\) −21.7082 −1.28363
\(287\) −9.56231 −0.564445
\(288\) 0.381966 0.0225076
\(289\) −16.4164 −0.965671
\(290\) 0.381966 0.0224298
\(291\) 11.5623 0.677794
\(292\) 0 0
\(293\) 31.4164 1.83537 0.917683 0.397313i \(-0.130057\pi\)
0.917683 + 0.397313i \(0.130057\pi\)
\(294\) −1.61803 −0.0943657
\(295\) 30.2705 1.76242
\(296\) −7.85410 −0.456510
\(297\) 26.5623 1.54130
\(298\) 17.2361 0.998459
\(299\) −40.0000 −2.31326
\(300\) 3.00000 0.173205
\(301\) 3.85410 0.222147
\(302\) −3.05573 −0.175837
\(303\) 4.76393 0.273681
\(304\) 0 0
\(305\) 22.4164 1.28356
\(306\) 0.291796 0.0166809
\(307\) 18.9787 1.08317 0.541586 0.840645i \(-0.317824\pi\)
0.541586 + 0.840645i \(0.317824\pi\)
\(308\) −4.85410 −0.276588
\(309\) 14.4721 0.823291
\(310\) 15.7082 0.892166
\(311\) −21.3262 −1.20930 −0.604650 0.796491i \(-0.706687\pi\)
−0.604650 + 0.796491i \(0.706687\pi\)
\(312\) 7.23607 0.409662
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −2.85410 −0.161066
\(315\) 1.00000 0.0563436
\(316\) 0.326238 0.0183523
\(317\) −25.6869 −1.44272 −0.721361 0.692560i \(-0.756483\pi\)
−0.721361 + 0.692560i \(0.756483\pi\)
\(318\) 8.32624 0.466912
\(319\) −0.708204 −0.0396518
\(320\) −2.61803 −0.146353
\(321\) 31.8885 1.77984
\(322\) −8.94427 −0.498445
\(323\) 0 0
\(324\) −7.70820 −0.428234
\(325\) −8.29180 −0.459946
\(326\) 20.2705 1.12268
\(327\) 17.5623 0.971198
\(328\) 9.56231 0.527990
\(329\) −10.8541 −0.598406
\(330\) −20.5623 −1.13192
\(331\) −3.70820 −0.203821 −0.101911 0.994794i \(-0.532496\pi\)
−0.101911 + 0.994794i \(0.532496\pi\)
\(332\) 11.2361 0.616659
\(333\) −3.00000 −0.164399
\(334\) 10.0000 0.547176
\(335\) −13.7082 −0.748959
\(336\) 1.61803 0.0882710
\(337\) 9.70820 0.528840 0.264420 0.964408i \(-0.414820\pi\)
0.264420 + 0.964408i \(0.414820\pi\)
\(338\) −7.00000 −0.380750
\(339\) 16.1803 0.878795
\(340\) −2.00000 −0.108465
\(341\) −29.1246 −1.57719
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −3.85410 −0.207799
\(345\) −37.8885 −2.03985
\(346\) −18.0000 −0.967686
\(347\) 8.94427 0.480154 0.240077 0.970754i \(-0.422827\pi\)
0.240077 + 0.970754i \(0.422827\pi\)
\(348\) 0.236068 0.0126546
\(349\) −16.8328 −0.901040 −0.450520 0.892766i \(-0.648761\pi\)
−0.450520 + 0.892766i \(0.648761\pi\)
\(350\) −1.85410 −0.0991059
\(351\) 24.4721 1.30623
\(352\) 4.85410 0.258725
\(353\) 13.4164 0.714083 0.357042 0.934088i \(-0.383785\pi\)
0.357042 + 0.934088i \(0.383785\pi\)
\(354\) 18.7082 0.994330
\(355\) −22.4164 −1.18974
\(356\) −3.14590 −0.166732
\(357\) 1.23607 0.0654197
\(358\) −17.7082 −0.935908
\(359\) 2.29180 0.120956 0.0604782 0.998170i \(-0.480737\pi\)
0.0604782 + 0.998170i \(0.480737\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 0 0
\(362\) 7.41641 0.389798
\(363\) 20.3262 1.06685
\(364\) −4.47214 −0.234404
\(365\) 0 0
\(366\) 13.8541 0.724166
\(367\) −6.27051 −0.327318 −0.163659 0.986517i \(-0.552330\pi\)
−0.163659 + 0.986517i \(0.552330\pi\)
\(368\) 8.94427 0.466252
\(369\) 3.65248 0.190140
\(370\) 20.5623 1.06898
\(371\) −5.14590 −0.267162
\(372\) 9.70820 0.503347
\(373\) 14.6180 0.756893 0.378447 0.925623i \(-0.376458\pi\)
0.378447 + 0.925623i \(0.376458\pi\)
\(374\) 3.70820 0.191747
\(375\) 13.3262 0.688164
\(376\) 10.8541 0.559758
\(377\) −0.652476 −0.0336042
\(378\) 5.47214 0.281456
\(379\) 32.1803 1.65299 0.826497 0.562942i \(-0.190331\pi\)
0.826497 + 0.562942i \(0.190331\pi\)
\(380\) 0 0
\(381\) 35.7426 1.83115
\(382\) −14.9443 −0.764615
\(383\) −13.1246 −0.670636 −0.335318 0.942105i \(-0.608844\pi\)
−0.335318 + 0.942105i \(0.608844\pi\)
\(384\) −1.61803 −0.0825700
\(385\) 12.7082 0.647670
\(386\) 20.9443 1.06604
\(387\) −1.47214 −0.0748329
\(388\) 7.14590 0.362778
\(389\) −37.3050 −1.89144 −0.945718 0.324988i \(-0.894640\pi\)
−0.945718 + 0.324988i \(0.894640\pi\)
\(390\) −18.9443 −0.959280
\(391\) 6.83282 0.345550
\(392\) −1.00000 −0.0505076
\(393\) −7.23607 −0.365011
\(394\) 5.23607 0.263789
\(395\) −0.854102 −0.0429745
\(396\) 1.85410 0.0931721
\(397\) −28.5623 −1.43350 −0.716751 0.697330i \(-0.754371\pi\)
−0.716751 + 0.697330i \(0.754371\pi\)
\(398\) −1.56231 −0.0783113
\(399\) 0 0
\(400\) 1.85410 0.0927051
\(401\) −25.4164 −1.26923 −0.634617 0.772826i \(-0.718842\pi\)
−0.634617 + 0.772826i \(0.718842\pi\)
\(402\) −8.47214 −0.422552
\(403\) −26.8328 −1.33664
\(404\) 2.94427 0.146483
\(405\) 20.1803 1.00277
\(406\) −0.145898 −0.00724080
\(407\) −38.1246 −1.88977
\(408\) −1.23607 −0.0611945
\(409\) −1.79837 −0.0889239 −0.0444619 0.999011i \(-0.514157\pi\)
−0.0444619 + 0.999011i \(0.514157\pi\)
\(410\) −25.0344 −1.23636
\(411\) 5.47214 0.269921
\(412\) 8.94427 0.440653
\(413\) −11.5623 −0.568944
\(414\) 3.41641 0.167907
\(415\) −29.4164 −1.44399
\(416\) 4.47214 0.219265
\(417\) 9.70820 0.475413
\(418\) 0 0
\(419\) −12.6525 −0.618114 −0.309057 0.951044i \(-0.600013\pi\)
−0.309057 + 0.951044i \(0.600013\pi\)
\(420\) −4.23607 −0.206699
\(421\) 19.5279 0.951730 0.475865 0.879518i \(-0.342135\pi\)
0.475865 + 0.879518i \(0.342135\pi\)
\(422\) −17.2361 −0.839039
\(423\) 4.14590 0.201580
\(424\) 5.14590 0.249907
\(425\) 1.41641 0.0687059
\(426\) −13.8541 −0.671233
\(427\) −8.56231 −0.414359
\(428\) 19.7082 0.952632
\(429\) 35.1246 1.69583
\(430\) 10.0902 0.486591
\(431\) 22.8541 1.10084 0.550422 0.834887i \(-0.314467\pi\)
0.550422 + 0.834887i \(0.314467\pi\)
\(432\) −5.47214 −0.263278
\(433\) −3.38197 −0.162527 −0.0812635 0.996693i \(-0.525896\pi\)
−0.0812635 + 0.996693i \(0.525896\pi\)
\(434\) −6.00000 −0.288009
\(435\) −0.618034 −0.0296325
\(436\) 10.8541 0.519817
\(437\) 0 0
\(438\) 0 0
\(439\) −2.18034 −0.104062 −0.0520310 0.998645i \(-0.516569\pi\)
−0.0520310 + 0.998645i \(0.516569\pi\)
\(440\) −12.7082 −0.605840
\(441\) −0.381966 −0.0181889
\(442\) 3.41641 0.162502
\(443\) 1.09017 0.0517955 0.0258978 0.999665i \(-0.491756\pi\)
0.0258978 + 0.999665i \(0.491756\pi\)
\(444\) 12.7082 0.603105
\(445\) 8.23607 0.390427
\(446\) 5.23607 0.247935
\(447\) −27.8885 −1.31908
\(448\) 1.00000 0.0472456
\(449\) −0.291796 −0.0137707 −0.00688535 0.999976i \(-0.502192\pi\)
−0.00688535 + 0.999976i \(0.502192\pi\)
\(450\) 0.708204 0.0333851
\(451\) 46.4164 2.18566
\(452\) 10.0000 0.470360
\(453\) 4.94427 0.232302
\(454\) −19.4164 −0.911257
\(455\) 11.7082 0.548889
\(456\) 0 0
\(457\) 31.1459 1.45694 0.728472 0.685076i \(-0.240231\pi\)
0.728472 + 0.685076i \(0.240231\pi\)
\(458\) −23.8541 −1.11463
\(459\) −4.18034 −0.195122
\(460\) −23.4164 −1.09180
\(461\) −41.5066 −1.93315 −0.966577 0.256376i \(-0.917471\pi\)
−0.966577 + 0.256376i \(0.917471\pi\)
\(462\) 7.85410 0.365406
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 0.145898 0.00677315
\(465\) −25.4164 −1.17866
\(466\) −4.09017 −0.189473
\(467\) −40.3607 −1.86767 −0.933835 0.357705i \(-0.883559\pi\)
−0.933835 + 0.357705i \(0.883559\pi\)
\(468\) 1.70820 0.0789618
\(469\) 5.23607 0.241779
\(470\) −28.4164 −1.31075
\(471\) 4.61803 0.212788
\(472\) 11.5623 0.532198
\(473\) −18.7082 −0.860204
\(474\) −0.527864 −0.0242456
\(475\) 0 0
\(476\) 0.763932 0.0350148
\(477\) 1.96556 0.0899967
\(478\) −14.1803 −0.648594
\(479\) 36.3262 1.65979 0.829894 0.557921i \(-0.188401\pi\)
0.829894 + 0.557921i \(0.188401\pi\)
\(480\) 4.23607 0.193349
\(481\) −35.1246 −1.60154
\(482\) −17.5623 −0.799941
\(483\) 14.4721 0.658505
\(484\) 12.5623 0.571014
\(485\) −18.7082 −0.849496
\(486\) −3.94427 −0.178916
\(487\) 16.9098 0.766258 0.383129 0.923695i \(-0.374847\pi\)
0.383129 + 0.923695i \(0.374847\pi\)
\(488\) 8.56231 0.387597
\(489\) −32.7984 −1.48319
\(490\) 2.61803 0.118271
\(491\) −40.3607 −1.82145 −0.910726 0.413011i \(-0.864477\pi\)
−0.910726 + 0.413011i \(0.864477\pi\)
\(492\) −15.4721 −0.697538
\(493\) 0.111456 0.00501973
\(494\) 0 0
\(495\) −4.85410 −0.218176
\(496\) 6.00000 0.269408
\(497\) 8.56231 0.384072
\(498\) −18.1803 −0.814681
\(499\) −18.6869 −0.836541 −0.418271 0.908322i \(-0.637364\pi\)
−0.418271 + 0.908322i \(0.637364\pi\)
\(500\) 8.23607 0.368328
\(501\) −16.1803 −0.722884
\(502\) −9.05573 −0.404177
\(503\) 30.2148 1.34721 0.673605 0.739091i \(-0.264745\pi\)
0.673605 + 0.739091i \(0.264745\pi\)
\(504\) 0.381966 0.0170141
\(505\) −7.70820 −0.343011
\(506\) 43.4164 1.93009
\(507\) 11.3262 0.503016
\(508\) 22.0902 0.980093
\(509\) −35.1246 −1.55687 −0.778436 0.627725i \(-0.783986\pi\)
−0.778436 + 0.627725i \(0.783986\pi\)
\(510\) 3.23607 0.143295
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −9.85410 −0.434646
\(515\) −23.4164 −1.03185
\(516\) 6.23607 0.274528
\(517\) 52.6869 2.31717
\(518\) −7.85410 −0.345089
\(519\) 29.1246 1.27843
\(520\) −11.7082 −0.513439
\(521\) −21.4164 −0.938270 −0.469135 0.883127i \(-0.655434\pi\)
−0.469135 + 0.883127i \(0.655434\pi\)
\(522\) 0.0557281 0.00243915
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) −4.47214 −0.195366
\(525\) 3.00000 0.130931
\(526\) −12.6525 −0.551674
\(527\) 4.58359 0.199664
\(528\) −7.85410 −0.341806
\(529\) 57.0000 2.47826
\(530\) −13.4721 −0.585192
\(531\) 4.41641 0.191656
\(532\) 0 0
\(533\) 42.7639 1.85231
\(534\) 5.09017 0.220273
\(535\) −51.5967 −2.23072
\(536\) −5.23607 −0.226164
\(537\) 28.6525 1.23645
\(538\) 4.58359 0.197613
\(539\) −4.85410 −0.209081
\(540\) 14.3262 0.616503
\(541\) 19.1246 0.822231 0.411116 0.911583i \(-0.365139\pi\)
0.411116 + 0.911583i \(0.365139\pi\)
\(542\) −2.43769 −0.104708
\(543\) −12.0000 −0.514969
\(544\) −0.763932 −0.0327533
\(545\) −28.4164 −1.21723
\(546\) 7.23607 0.309675
\(547\) 12.6525 0.540981 0.270490 0.962723i \(-0.412814\pi\)
0.270490 + 0.962723i \(0.412814\pi\)
\(548\) 3.38197 0.144470
\(549\) 3.27051 0.139582
\(550\) 9.00000 0.383761
\(551\) 0 0
\(552\) −14.4721 −0.615975
\(553\) 0.326238 0.0138730
\(554\) 25.4164 1.07984
\(555\) −33.2705 −1.41225
\(556\) 6.00000 0.254457
\(557\) −32.0689 −1.35880 −0.679401 0.733767i \(-0.737760\pi\)
−0.679401 + 0.733767i \(0.737760\pi\)
\(558\) 2.29180 0.0970195
\(559\) −17.2361 −0.729008
\(560\) −2.61803 −0.110632
\(561\) −6.00000 −0.253320
\(562\) 23.4164 0.987762
\(563\) −6.56231 −0.276568 −0.138284 0.990393i \(-0.544159\pi\)
−0.138284 + 0.990393i \(0.544159\pi\)
\(564\) −17.5623 −0.739506
\(565\) −26.1803 −1.10142
\(566\) −8.00000 −0.336265
\(567\) −7.70820 −0.323714
\(568\) −8.56231 −0.359266
\(569\) −25.7082 −1.07774 −0.538872 0.842388i \(-0.681149\pi\)
−0.538872 + 0.842388i \(0.681149\pi\)
\(570\) 0 0
\(571\) 15.6869 0.656477 0.328239 0.944595i \(-0.393545\pi\)
0.328239 + 0.944595i \(0.393545\pi\)
\(572\) 21.7082 0.907666
\(573\) 24.1803 1.01015
\(574\) 9.56231 0.399123
\(575\) 16.5836 0.691584
\(576\) −0.381966 −0.0159153
\(577\) −13.1246 −0.546385 −0.273192 0.961959i \(-0.588080\pi\)
−0.273192 + 0.961959i \(0.588080\pi\)
\(578\) 16.4164 0.682833
\(579\) −33.8885 −1.40836
\(580\) −0.381966 −0.0158603
\(581\) 11.2361 0.466151
\(582\) −11.5623 −0.479273
\(583\) 24.9787 1.03451
\(584\) 0 0
\(585\) −4.47214 −0.184900
\(586\) −31.4164 −1.29780
\(587\) 19.5279 0.806001 0.403001 0.915200i \(-0.367967\pi\)
0.403001 + 0.915200i \(0.367967\pi\)
\(588\) 1.61803 0.0667266
\(589\) 0 0
\(590\) −30.2705 −1.24622
\(591\) −8.47214 −0.348497
\(592\) 7.85410 0.322802
\(593\) 16.4721 0.676430 0.338215 0.941069i \(-0.390177\pi\)
0.338215 + 0.941069i \(0.390177\pi\)
\(594\) −26.5623 −1.08986
\(595\) −2.00000 −0.0819920
\(596\) −17.2361 −0.706017
\(597\) 2.52786 0.103459
\(598\) 40.0000 1.63572
\(599\) 11.9787 0.489437 0.244718 0.969594i \(-0.421304\pi\)
0.244718 + 0.969594i \(0.421304\pi\)
\(600\) −3.00000 −0.122474
\(601\) 10.3607 0.422621 0.211310 0.977419i \(-0.432227\pi\)
0.211310 + 0.977419i \(0.432227\pi\)
\(602\) −3.85410 −0.157081
\(603\) −2.00000 −0.0814463
\(604\) 3.05573 0.124336
\(605\) −32.8885 −1.33711
\(606\) −4.76393 −0.193522
\(607\) 16.3607 0.664060 0.332030 0.943269i \(-0.392267\pi\)
0.332030 + 0.943269i \(0.392267\pi\)
\(608\) 0 0
\(609\) 0.236068 0.00956596
\(610\) −22.4164 −0.907614
\(611\) 48.5410 1.96376
\(612\) −0.291796 −0.0117952
\(613\) 12.2918 0.496461 0.248230 0.968701i \(-0.420151\pi\)
0.248230 + 0.968701i \(0.420151\pi\)
\(614\) −18.9787 −0.765919
\(615\) 40.5066 1.63338
\(616\) 4.85410 0.195577
\(617\) 33.9230 1.36569 0.682844 0.730564i \(-0.260743\pi\)
0.682844 + 0.730564i \(0.260743\pi\)
\(618\) −14.4721 −0.582155
\(619\) −35.4164 −1.42351 −0.711753 0.702430i \(-0.752098\pi\)
−0.711753 + 0.702430i \(0.752098\pi\)
\(620\) −15.7082 −0.630857
\(621\) −48.9443 −1.96407
\(622\) 21.3262 0.855104
\(623\) −3.14590 −0.126038
\(624\) −7.23607 −0.289675
\(625\) −30.8328 −1.23331
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 2.85410 0.113891
\(629\) 6.00000 0.239236
\(630\) −1.00000 −0.0398410
\(631\) 22.8328 0.908960 0.454480 0.890757i \(-0.349825\pi\)
0.454480 + 0.890757i \(0.349825\pi\)
\(632\) −0.326238 −0.0129770
\(633\) 27.8885 1.10847
\(634\) 25.6869 1.02016
\(635\) −57.8328 −2.29503
\(636\) −8.32624 −0.330157
\(637\) −4.47214 −0.177192
\(638\) 0.708204 0.0280381
\(639\) −3.27051 −0.129379
\(640\) 2.61803 0.103487
\(641\) −7.41641 −0.292930 −0.146465 0.989216i \(-0.546790\pi\)
−0.146465 + 0.989216i \(0.546790\pi\)
\(642\) −31.8885 −1.25854
\(643\) 21.4164 0.844581 0.422290 0.906461i \(-0.361226\pi\)
0.422290 + 0.906461i \(0.361226\pi\)
\(644\) 8.94427 0.352454
\(645\) −16.3262 −0.642845
\(646\) 0 0
\(647\) −0.326238 −0.0128257 −0.00641287 0.999979i \(-0.502041\pi\)
−0.00641287 + 0.999979i \(0.502041\pi\)
\(648\) 7.70820 0.302807
\(649\) 56.1246 2.20308
\(650\) 8.29180 0.325231
\(651\) 9.70820 0.380495
\(652\) −20.2705 −0.793854
\(653\) −13.5279 −0.529386 −0.264693 0.964333i \(-0.585271\pi\)
−0.264693 + 0.964333i \(0.585271\pi\)
\(654\) −17.5623 −0.686741
\(655\) 11.7082 0.457477
\(656\) −9.56231 −0.373345
\(657\) 0 0
\(658\) 10.8541 0.423137
\(659\) −0.875388 −0.0341003 −0.0170501 0.999855i \(-0.505427\pi\)
−0.0170501 + 0.999855i \(0.505427\pi\)
\(660\) 20.5623 0.800387
\(661\) −26.8328 −1.04368 −0.521838 0.853045i \(-0.674753\pi\)
−0.521838 + 0.853045i \(0.674753\pi\)
\(662\) 3.70820 0.144123
\(663\) −5.52786 −0.214684
\(664\) −11.2361 −0.436044
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 1.30495 0.0505279
\(668\) −10.0000 −0.386912
\(669\) −8.47214 −0.327552
\(670\) 13.7082 0.529594
\(671\) 41.5623 1.60450
\(672\) −1.61803 −0.0624170
\(673\) −3.05573 −0.117790 −0.0588948 0.998264i \(-0.518758\pi\)
−0.0588948 + 0.998264i \(0.518758\pi\)
\(674\) −9.70820 −0.373946
\(675\) −10.1459 −0.390516
\(676\) 7.00000 0.269231
\(677\) −41.7082 −1.60298 −0.801488 0.598011i \(-0.795958\pi\)
−0.801488 + 0.598011i \(0.795958\pi\)
\(678\) −16.1803 −0.621402
\(679\) 7.14590 0.274234
\(680\) 2.00000 0.0766965
\(681\) 31.4164 1.20388
\(682\) 29.1246 1.11524
\(683\) −2.29180 −0.0876931 −0.0438466 0.999038i \(-0.513961\pi\)
−0.0438466 + 0.999038i \(0.513961\pi\)
\(684\) 0 0
\(685\) −8.85410 −0.338298
\(686\) −1.00000 −0.0381802
\(687\) 38.5967 1.47256
\(688\) 3.85410 0.146936
\(689\) 23.0132 0.876731
\(690\) 37.8885 1.44239
\(691\) 20.5410 0.781417 0.390709 0.920514i \(-0.372230\pi\)
0.390709 + 0.920514i \(0.372230\pi\)
\(692\) 18.0000 0.684257
\(693\) 1.85410 0.0704315
\(694\) −8.94427 −0.339520
\(695\) −15.7082 −0.595846
\(696\) −0.236068 −0.00894813
\(697\) −7.30495 −0.276695
\(698\) 16.8328 0.637131
\(699\) 6.61803 0.250317
\(700\) 1.85410 0.0700785
\(701\) −5.23607 −0.197764 −0.0988818 0.995099i \(-0.531527\pi\)
−0.0988818 + 0.995099i \(0.531527\pi\)
\(702\) −24.4721 −0.923641
\(703\) 0 0
\(704\) −4.85410 −0.182946
\(705\) 45.9787 1.73166
\(706\) −13.4164 −0.504933
\(707\) 2.94427 0.110731
\(708\) −18.7082 −0.703097
\(709\) 9.12461 0.342682 0.171341 0.985212i \(-0.445190\pi\)
0.171341 + 0.985212i \(0.445190\pi\)
\(710\) 22.4164 0.841273
\(711\) −0.124612 −0.00467331
\(712\) 3.14590 0.117898
\(713\) 53.6656 2.00979
\(714\) −1.23607 −0.0462587
\(715\) −56.8328 −2.12543
\(716\) 17.7082 0.661787
\(717\) 22.9443 0.856870
\(718\) −2.29180 −0.0855291
\(719\) 16.3607 0.610150 0.305075 0.952328i \(-0.401318\pi\)
0.305075 + 0.952328i \(0.401318\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.94427 0.333102
\(722\) 0 0
\(723\) 28.4164 1.05682
\(724\) −7.41641 −0.275629
\(725\) 0.270510 0.0100465
\(726\) −20.3262 −0.754377
\(727\) 28.5623 1.05932 0.529659 0.848211i \(-0.322320\pi\)
0.529659 + 0.848211i \(0.322320\pi\)
\(728\) 4.47214 0.165748
\(729\) 29.5066 1.09284
\(730\) 0 0
\(731\) 2.94427 0.108898
\(732\) −13.8541 −0.512062
\(733\) −19.5623 −0.722550 −0.361275 0.932459i \(-0.617658\pi\)
−0.361275 + 0.932459i \(0.617658\pi\)
\(734\) 6.27051 0.231449
\(735\) −4.23607 −0.156250
\(736\) −8.94427 −0.329690
\(737\) −25.4164 −0.936225
\(738\) −3.65248 −0.134449
\(739\) 35.2705 1.29745 0.648723 0.761024i \(-0.275303\pi\)
0.648723 + 0.761024i \(0.275303\pi\)
\(740\) −20.5623 −0.755885
\(741\) 0 0
\(742\) 5.14590 0.188912
\(743\) 24.9787 0.916380 0.458190 0.888854i \(-0.348498\pi\)
0.458190 + 0.888854i \(0.348498\pi\)
\(744\) −9.70820 −0.355920
\(745\) 45.1246 1.65324
\(746\) −14.6180 −0.535204
\(747\) −4.29180 −0.157029
\(748\) −3.70820 −0.135585
\(749\) 19.7082 0.720122
\(750\) −13.3262 −0.486605
\(751\) −0.326238 −0.0119046 −0.00595230 0.999982i \(-0.501895\pi\)
−0.00595230 + 0.999982i \(0.501895\pi\)
\(752\) −10.8541 −0.395808
\(753\) 14.6525 0.533966
\(754\) 0.652476 0.0237618
\(755\) −8.00000 −0.291150
\(756\) −5.47214 −0.199020
\(757\) 46.2492 1.68096 0.840478 0.541845i \(-0.182274\pi\)
0.840478 + 0.541845i \(0.182274\pi\)
\(758\) −32.1803 −1.16884
\(759\) −70.2492 −2.54989
\(760\) 0 0
\(761\) −17.7771 −0.644419 −0.322209 0.946668i \(-0.604426\pi\)
−0.322209 + 0.946668i \(0.604426\pi\)
\(762\) −35.7426 −1.29482
\(763\) 10.8541 0.392945
\(764\) 14.9443 0.540665
\(765\) 0.763932 0.0276200
\(766\) 13.1246 0.474212
\(767\) 51.7082 1.86708
\(768\) 1.61803 0.0583858
\(769\) 12.2918 0.443254 0.221627 0.975132i \(-0.428863\pi\)
0.221627 + 0.975132i \(0.428863\pi\)
\(770\) −12.7082 −0.457972
\(771\) 15.9443 0.574219
\(772\) −20.9443 −0.753801
\(773\) 1.70820 0.0614398 0.0307199 0.999528i \(-0.490220\pi\)
0.0307199 + 0.999528i \(0.490220\pi\)
\(774\) 1.47214 0.0529148
\(775\) 11.1246 0.399608
\(776\) −7.14590 −0.256523
\(777\) 12.7082 0.455904
\(778\) 37.3050 1.33745
\(779\) 0 0
\(780\) 18.9443 0.678314
\(781\) −41.5623 −1.48722
\(782\) −6.83282 −0.244341
\(783\) −0.798374 −0.0285316
\(784\) 1.00000 0.0357143
\(785\) −7.47214 −0.266692
\(786\) 7.23607 0.258102
\(787\) −27.2705 −0.972089 −0.486044 0.873934i \(-0.661561\pi\)
−0.486044 + 0.873934i \(0.661561\pi\)
\(788\) −5.23607 −0.186527
\(789\) 20.4721 0.728827
\(790\) 0.854102 0.0303876
\(791\) 10.0000 0.355559
\(792\) −1.85410 −0.0658826
\(793\) 38.2918 1.35978
\(794\) 28.5623 1.01364
\(795\) 21.7984 0.773109
\(796\) 1.56231 0.0553745
\(797\) −44.8328 −1.58806 −0.794030 0.607879i \(-0.792021\pi\)
−0.794030 + 0.607879i \(0.792021\pi\)
\(798\) 0 0
\(799\) −8.29180 −0.293343
\(800\) −1.85410 −0.0655524
\(801\) 1.20163 0.0424574
\(802\) 25.4164 0.897485
\(803\) 0 0
\(804\) 8.47214 0.298789
\(805\) −23.4164 −0.825320
\(806\) 26.8328 0.945146
\(807\) −7.41641 −0.261070
\(808\) −2.94427 −0.103579
\(809\) −16.0344 −0.563741 −0.281870 0.959452i \(-0.590955\pi\)
−0.281870 + 0.959452i \(0.590955\pi\)
\(810\) −20.1803 −0.709065
\(811\) 2.02129 0.0709770 0.0354885 0.999370i \(-0.488701\pi\)
0.0354885 + 0.999370i \(0.488701\pi\)
\(812\) 0.145898 0.00512002
\(813\) 3.94427 0.138332
\(814\) 38.1246 1.33627
\(815\) 53.0689 1.85892
\(816\) 1.23607 0.0432710
\(817\) 0 0
\(818\) 1.79837 0.0628787
\(819\) 1.70820 0.0596895
\(820\) 25.0344 0.874241
\(821\) 6.87539 0.239953 0.119976 0.992777i \(-0.461718\pi\)
0.119976 + 0.992777i \(0.461718\pi\)
\(822\) −5.47214 −0.190863
\(823\) −2.29180 −0.0798870 −0.0399435 0.999202i \(-0.512718\pi\)
−0.0399435 + 0.999202i \(0.512718\pi\)
\(824\) −8.94427 −0.311588
\(825\) −14.5623 −0.506994
\(826\) 11.5623 0.402304
\(827\) −43.9574 −1.52855 −0.764275 0.644891i \(-0.776903\pi\)
−0.764275 + 0.644891i \(0.776903\pi\)
\(828\) −3.41641 −0.118728
\(829\) −35.1246 −1.21993 −0.609964 0.792429i \(-0.708816\pi\)
−0.609964 + 0.792429i \(0.708816\pi\)
\(830\) 29.4164 1.02106
\(831\) −41.1246 −1.42660
\(832\) −4.47214 −0.155043
\(833\) 0.763932 0.0264687
\(834\) −9.70820 −0.336168
\(835\) 26.1803 0.906008
\(836\) 0 0
\(837\) −32.8328 −1.13487
\(838\) 12.6525 0.437073
\(839\) −4.83282 −0.166847 −0.0834237 0.996514i \(-0.526585\pi\)
−0.0834237 + 0.996514i \(0.526585\pi\)
\(840\) 4.23607 0.146158
\(841\) −28.9787 −0.999266
\(842\) −19.5279 −0.672975
\(843\) −37.8885 −1.30495
\(844\) 17.2361 0.593290
\(845\) −18.3262 −0.630442
\(846\) −4.14590 −0.142539
\(847\) 12.5623 0.431646
\(848\) −5.14590 −0.176711
\(849\) 12.9443 0.444246
\(850\) −1.41641 −0.0485824
\(851\) 70.2492 2.40811
\(852\) 13.8541 0.474634
\(853\) −23.2705 −0.796767 −0.398384 0.917219i \(-0.630429\pi\)
−0.398384 + 0.917219i \(0.630429\pi\)
\(854\) 8.56231 0.292996
\(855\) 0 0
\(856\) −19.7082 −0.673613
\(857\) 32.2492 1.10161 0.550806 0.834633i \(-0.314320\pi\)
0.550806 + 0.834633i \(0.314320\pi\)
\(858\) −35.1246 −1.19913
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) −10.0902 −0.344072
\(861\) −15.4721 −0.527289
\(862\) −22.8541 −0.778414
\(863\) 48.9787 1.66725 0.833627 0.552327i \(-0.186260\pi\)
0.833627 + 0.552327i \(0.186260\pi\)
\(864\) 5.47214 0.186166
\(865\) −47.1246 −1.60228
\(866\) 3.38197 0.114924
\(867\) −26.5623 −0.902103
\(868\) 6.00000 0.203653
\(869\) −1.58359 −0.0537197
\(870\) 0.618034 0.0209533
\(871\) −23.4164 −0.793435
\(872\) −10.8541 −0.367566
\(873\) −2.72949 −0.0923792
\(874\) 0 0
\(875\) 8.23607 0.278430
\(876\) 0 0
\(877\) −47.3951 −1.60042 −0.800210 0.599720i \(-0.795279\pi\)
−0.800210 + 0.599720i \(0.795279\pi\)
\(878\) 2.18034 0.0735829
\(879\) 50.8328 1.71455
\(880\) 12.7082 0.428393
\(881\) 9.81966 0.330833 0.165416 0.986224i \(-0.447103\pi\)
0.165416 + 0.986224i \(0.447103\pi\)
\(882\) 0.381966 0.0128615
\(883\) −27.6869 −0.931739 −0.465869 0.884853i \(-0.654258\pi\)
−0.465869 + 0.884853i \(0.654258\pi\)
\(884\) −3.41641 −0.114906
\(885\) 48.9787 1.64640
\(886\) −1.09017 −0.0366250
\(887\) 2.83282 0.0951166 0.0475583 0.998868i \(-0.484856\pi\)
0.0475583 + 0.998868i \(0.484856\pi\)
\(888\) −12.7082 −0.426459
\(889\) 22.0902 0.740881
\(890\) −8.23607 −0.276074
\(891\) 37.4164 1.25350
\(892\) −5.23607 −0.175317
\(893\) 0 0
\(894\) 27.8885 0.932732
\(895\) −46.3607 −1.54967
\(896\) −1.00000 −0.0334077
\(897\) −64.7214 −2.16098
\(898\) 0.291796 0.00973736
\(899\) 0.875388 0.0291958
\(900\) −0.708204 −0.0236068
\(901\) −3.93112 −0.130964
\(902\) −46.4164 −1.54550
\(903\) 6.23607 0.207523
\(904\) −10.0000 −0.332595
\(905\) 19.4164 0.645423
\(906\) −4.94427 −0.164262
\(907\) 48.0000 1.59381 0.796907 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) 19.4164 0.644356
\(909\) −1.12461 −0.0373010
\(910\) −11.7082 −0.388123
\(911\) −49.8541 −1.65174 −0.825870 0.563861i \(-0.809316\pi\)
−0.825870 + 0.563861i \(0.809316\pi\)
\(912\) 0 0
\(913\) −54.5410 −1.80504
\(914\) −31.1459 −1.03021
\(915\) 36.2705 1.19907
\(916\) 23.8541 0.788162
\(917\) −4.47214 −0.147683
\(918\) 4.18034 0.137972
\(919\) −12.2918 −0.405469 −0.202734 0.979234i \(-0.564983\pi\)
−0.202734 + 0.979234i \(0.564983\pi\)
\(920\) 23.4164 0.772016
\(921\) 30.7082 1.01187
\(922\) 41.5066 1.36695
\(923\) −38.2918 −1.26039
\(924\) −7.85410 −0.258381
\(925\) 14.5623 0.478806
\(926\) 12.0000 0.394344
\(927\) −3.41641 −0.112210
\(928\) −0.145898 −0.00478934
\(929\) 42.7639 1.40304 0.701520 0.712650i \(-0.252505\pi\)
0.701520 + 0.712650i \(0.252505\pi\)
\(930\) 25.4164 0.833437
\(931\) 0 0
\(932\) 4.09017 0.133978
\(933\) −34.5066 −1.12969
\(934\) 40.3607 1.32064
\(935\) 9.70820 0.317492
\(936\) −1.70820 −0.0558344
\(937\) 13.7082 0.447828 0.223914 0.974609i \(-0.428117\pi\)
0.223914 + 0.974609i \(0.428117\pi\)
\(938\) −5.23607 −0.170964
\(939\) 16.1803 0.528025
\(940\) 28.4164 0.926841
\(941\) −33.7082 −1.09886 −0.549428 0.835541i \(-0.685154\pi\)
−0.549428 + 0.835541i \(0.685154\pi\)
\(942\) −4.61803 −0.150464
\(943\) −85.5279 −2.78517
\(944\) −11.5623 −0.376321
\(945\) 14.3262 0.466033
\(946\) 18.7082 0.608256
\(947\) −43.1459 −1.40205 −0.701027 0.713135i \(-0.747275\pi\)
−0.701027 + 0.713135i \(0.747275\pi\)
\(948\) 0.527864 0.0171442
\(949\) 0 0
\(950\) 0 0
\(951\) −41.5623 −1.34775
\(952\) −0.763932 −0.0247592
\(953\) 2.29180 0.0742386 0.0371193 0.999311i \(-0.488182\pi\)
0.0371193 + 0.999311i \(0.488182\pi\)
\(954\) −1.96556 −0.0636373
\(955\) −39.1246 −1.26604
\(956\) 14.1803 0.458625
\(957\) −1.14590 −0.0370416
\(958\) −36.3262 −1.17365
\(959\) 3.38197 0.109209
\(960\) −4.23607 −0.136719
\(961\) 5.00000 0.161290
\(962\) 35.1246 1.13246
\(963\) −7.52786 −0.242582
\(964\) 17.5623 0.565644
\(965\) 54.8328 1.76513
\(966\) −14.4721 −0.465633
\(967\) 54.2492 1.74454 0.872269 0.489027i \(-0.162648\pi\)
0.872269 + 0.489027i \(0.162648\pi\)
\(968\) −12.5623 −0.403768
\(969\) 0 0
\(970\) 18.7082 0.600684
\(971\) 11.7295 0.376417 0.188209 0.982129i \(-0.439732\pi\)
0.188209 + 0.982129i \(0.439732\pi\)
\(972\) 3.94427 0.126513
\(973\) 6.00000 0.192351
\(974\) −16.9098 −0.541826
\(975\) −13.4164 −0.429669
\(976\) −8.56231 −0.274073
\(977\) 35.7082 1.14241 0.571203 0.820809i \(-0.306477\pi\)
0.571203 + 0.820809i \(0.306477\pi\)
\(978\) 32.7984 1.04878
\(979\) 15.2705 0.488048
\(980\) −2.61803 −0.0836300
\(981\) −4.14590 −0.132368
\(982\) 40.3607 1.28796
\(983\) 19.1246 0.609980 0.304990 0.952355i \(-0.401347\pi\)
0.304990 + 0.952355i \(0.401347\pi\)
\(984\) 15.4721 0.493234
\(985\) 13.7082 0.436780
\(986\) −0.111456 −0.00354949
\(987\) −17.5623 −0.559014
\(988\) 0 0
\(989\) 34.4721 1.09615
\(990\) 4.85410 0.154273
\(991\) 47.3951 1.50556 0.752778 0.658275i \(-0.228713\pi\)
0.752778 + 0.658275i \(0.228713\pi\)
\(992\) −6.00000 −0.190500
\(993\) −6.00000 −0.190404
\(994\) −8.56231 −0.271580
\(995\) −4.09017 −0.129667
\(996\) 18.1803 0.576066
\(997\) −51.8541 −1.64224 −0.821118 0.570759i \(-0.806649\pi\)
−0.821118 + 0.570759i \(0.806649\pi\)
\(998\) 18.6869 0.591524
\(999\) −42.9787 −1.35979
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 5054.2.a.g.1.2 2
19.18 odd 2 5054.2.a.l.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.g.1.2 2 1.1 even 1 trivial
5054.2.a.l.1.1 yes 2 19.18 odd 2