Properties

Label 722.2.a.j.1.2
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.64575 q^{3} +1.00000 q^{4} -1.64575 q^{5} +2.64575 q^{6} +3.64575 q^{7} +1.00000 q^{8} +4.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.64575 q^{3} +1.00000 q^{4} -1.64575 q^{5} +2.64575 q^{6} +3.64575 q^{7} +1.00000 q^{8} +4.00000 q^{9} -1.64575 q^{10} -4.64575 q^{11} +2.64575 q^{12} +2.00000 q^{13} +3.64575 q^{14} -4.35425 q^{15} +1.00000 q^{16} +4.00000 q^{18} -1.64575 q^{20} +9.64575 q^{21} -4.64575 q^{22} -1.64575 q^{23} +2.64575 q^{24} -2.29150 q^{25} +2.00000 q^{26} +2.64575 q^{27} +3.64575 q^{28} +1.64575 q^{29} -4.35425 q^{30} -5.64575 q^{31} +1.00000 q^{32} -12.2915 q^{33} -6.00000 q^{35} +4.00000 q^{36} +0.354249 q^{37} +5.29150 q^{39} -1.64575 q^{40} -0.291503 q^{41} +9.64575 q^{42} +11.2915 q^{43} -4.64575 q^{44} -6.58301 q^{45} -1.64575 q^{46} +4.35425 q^{47} +2.64575 q^{48} +6.29150 q^{49} -2.29150 q^{50} +2.00000 q^{52} -12.5830 q^{53} +2.64575 q^{54} +7.64575 q^{55} +3.64575 q^{56} +1.64575 q^{58} -7.93725 q^{59} -4.35425 q^{60} +0.937254 q^{61} -5.64575 q^{62} +14.5830 q^{63} +1.00000 q^{64} -3.29150 q^{65} -12.2915 q^{66} +0.645751 q^{67} -4.35425 q^{69} -6.00000 q^{70} -2.70850 q^{71} +4.00000 q^{72} +1.70850 q^{73} +0.354249 q^{74} -6.06275 q^{75} -16.9373 q^{77} +5.29150 q^{78} -4.00000 q^{79} -1.64575 q^{80} -5.00000 q^{81} -0.291503 q^{82} +7.93725 q^{83} +9.64575 q^{84} +11.2915 q^{86} +4.35425 q^{87} -4.64575 q^{88} -6.58301 q^{90} +7.29150 q^{91} -1.64575 q^{92} -14.9373 q^{93} +4.35425 q^{94} +2.64575 q^{96} -3.70850 q^{97} +6.29150 q^{98} -18.5830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + 2q^{8} + 8q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + 2q^{8} + 8q^{9} + 2q^{10} - 4q^{11} + 4q^{13} + 2q^{14} - 14q^{15} + 2q^{16} + 8q^{18} + 2q^{20} + 14q^{21} - 4q^{22} + 2q^{23} + 6q^{25} + 4q^{26} + 2q^{28} - 2q^{29} - 14q^{30} - 6q^{31} + 2q^{32} - 14q^{33} - 12q^{35} + 8q^{36} + 6q^{37} + 2q^{40} + 10q^{41} + 14q^{42} + 12q^{43} - 4q^{44} + 8q^{45} + 2q^{46} + 14q^{47} + 2q^{49} + 6q^{50} + 4q^{52} - 4q^{53} + 10q^{55} + 2q^{56} - 2q^{58} - 14q^{60} - 14q^{61} - 6q^{62} + 8q^{63} + 2q^{64} + 4q^{65} - 14q^{66} - 4q^{67} - 14q^{69} - 12q^{70} - 16q^{71} + 8q^{72} + 14q^{73} + 6q^{74} - 28q^{75} - 18q^{77} - 8q^{79} + 2q^{80} - 10q^{81} + 10q^{82} + 14q^{84} + 12q^{86} + 14q^{87} - 4q^{88} + 8q^{90} + 4q^{91} + 2q^{92} - 14q^{93} + 14q^{94} - 18q^{97} + 2q^{98} - 16q^{99} + O(q^{100}) \)

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
\(3\) 2.64575 1.52753 0.763763 0.645497i \(-0.223350\pi\)
0.763763 + 0.645497i \(0.223350\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.64575 −0.736002 −0.368001 0.929825i \(-0.619958\pi\)
−0.368001 + 0.929825i \(0.619958\pi\)
\(6\) 2.64575 1.08012
\(7\) 3.64575 1.37796 0.688982 0.724778i \(-0.258058\pi\)
0.688982 + 0.724778i \(0.258058\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.00000 1.33333
\(10\) −1.64575 −0.520432
\(11\) −4.64575 −1.40075 −0.700373 0.713777i \(-0.746983\pi\)
−0.700373 + 0.713777i \(0.746983\pi\)
\(12\) 2.64575 0.763763
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.64575 0.974368
\(15\) −4.35425 −1.12426
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 4.00000 0.942809
\(19\) 0 0
\(20\) −1.64575 −0.368001
\(21\) 9.64575 2.10488
\(22\) −4.64575 −0.990478
\(23\) −1.64575 −0.343163 −0.171581 0.985170i \(-0.554888\pi\)
−0.171581 + 0.985170i \(0.554888\pi\)
\(24\) 2.64575 0.540062
\(25\) −2.29150 −0.458301
\(26\) 2.00000 0.392232
\(27\) 2.64575 0.509175
\(28\) 3.64575 0.688982
\(29\) 1.64575 0.305608 0.152804 0.988256i \(-0.451170\pi\)
0.152804 + 0.988256i \(0.451170\pi\)
\(30\) −4.35425 −0.794973
\(31\) −5.64575 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.2915 −2.13968
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 4.00000 0.666667
\(37\) 0.354249 0.0582381 0.0291191 0.999576i \(-0.490730\pi\)
0.0291191 + 0.999576i \(0.490730\pi\)
\(38\) 0 0
\(39\) 5.29150 0.847319
\(40\) −1.64575 −0.260216
\(41\) −0.291503 −0.0455251 −0.0227625 0.999741i \(-0.507246\pi\)
−0.0227625 + 0.999741i \(0.507246\pi\)
\(42\) 9.64575 1.48837
\(43\) 11.2915 1.72194 0.860969 0.508657i \(-0.169858\pi\)
0.860969 + 0.508657i \(0.169858\pi\)
\(44\) −4.64575 −0.700373
\(45\) −6.58301 −0.981336
\(46\) −1.64575 −0.242653
\(47\) 4.35425 0.635132 0.317566 0.948236i \(-0.397134\pi\)
0.317566 + 0.948236i \(0.397134\pi\)
\(48\) 2.64575 0.381881
\(49\) 6.29150 0.898786
\(50\) −2.29150 −0.324067
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −12.5830 −1.72841 −0.864204 0.503141i \(-0.832178\pi\)
−0.864204 + 0.503141i \(0.832178\pi\)
\(54\) 2.64575 0.360041
\(55\) 7.64575 1.03095
\(56\) 3.64575 0.487184
\(57\) 0 0
\(58\) 1.64575 0.216098
\(59\) −7.93725 −1.03334 −0.516671 0.856184i \(-0.672829\pi\)
−0.516671 + 0.856184i \(0.672829\pi\)
\(60\) −4.35425 −0.562131
\(61\) 0.937254 0.120003 0.0600015 0.998198i \(-0.480889\pi\)
0.0600015 + 0.998198i \(0.480889\pi\)
\(62\) −5.64575 −0.717011
\(63\) 14.5830 1.83729
\(64\) 1.00000 0.125000
\(65\) −3.29150 −0.408261
\(66\) −12.2915 −1.51298
\(67\) 0.645751 0.0788911 0.0394455 0.999222i \(-0.487441\pi\)
0.0394455 + 0.999222i \(0.487441\pi\)
\(68\) 0 0
\(69\) −4.35425 −0.524190
\(70\) −6.00000 −0.717137
\(71\) −2.70850 −0.321440 −0.160720 0.987000i \(-0.551382\pi\)
−0.160720 + 0.987000i \(0.551382\pi\)
\(72\) 4.00000 0.471405
\(73\) 1.70850 0.199964 0.0999822 0.994989i \(-0.468121\pi\)
0.0999822 + 0.994989i \(0.468121\pi\)
\(74\) 0.354249 0.0411806
\(75\) −6.06275 −0.700066
\(76\) 0 0
\(77\) −16.9373 −1.93018
\(78\) 5.29150 0.599145
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.64575 −0.184001
\(81\) −5.00000 −0.555556
\(82\) −0.291503 −0.0321911
\(83\) 7.93725 0.871227 0.435613 0.900134i \(-0.356531\pi\)
0.435613 + 0.900134i \(0.356531\pi\)
\(84\) 9.64575 1.05244
\(85\) 0 0
\(86\) 11.2915 1.21759
\(87\) 4.35425 0.466824
\(88\) −4.64575 −0.495239
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −6.58301 −0.693910
\(91\) 7.29150 0.764357
\(92\) −1.64575 −0.171581
\(93\) −14.9373 −1.54892
\(94\) 4.35425 0.449106
\(95\) 0 0
\(96\) 2.64575 0.270031
\(97\) −3.70850 −0.376541 −0.188270 0.982117i \(-0.560288\pi\)
−0.188270 + 0.982117i \(0.560288\pi\)
\(98\) 6.29150 0.635538
\(99\) −18.5830 −1.86766
\(100\) −2.29150 −0.229150
\(101\) −13.6458 −1.35780 −0.678902 0.734229i \(-0.737544\pi\)
−0.678902 + 0.734229i \(0.737544\pi\)
\(102\) 0 0
\(103\) −13.2915 −1.30965 −0.654825 0.755780i \(-0.727258\pi\)
−0.654825 + 0.755780i \(0.727258\pi\)
\(104\) 2.00000 0.196116
\(105\) −15.8745 −1.54919
\(106\) −12.5830 −1.22217
\(107\) 15.2915 1.47829 0.739143 0.673549i \(-0.235231\pi\)
0.739143 + 0.673549i \(0.235231\pi\)
\(108\) 2.64575 0.254588
\(109\) 14.5830 1.39680 0.698399 0.715708i \(-0.253896\pi\)
0.698399 + 0.715708i \(0.253896\pi\)
\(110\) 7.64575 0.728994
\(111\) 0.937254 0.0889602
\(112\) 3.64575 0.344491
\(113\) −15.5830 −1.46593 −0.732963 0.680269i \(-0.761863\pi\)
−0.732963 + 0.680269i \(0.761863\pi\)
\(114\) 0 0
\(115\) 2.70850 0.252569
\(116\) 1.64575 0.152804
\(117\) 8.00000 0.739600
\(118\) −7.93725 −0.730683
\(119\) 0 0
\(120\) −4.35425 −0.397487
\(121\) 10.5830 0.962091
\(122\) 0.937254 0.0848550
\(123\) −0.771243 −0.0695407
\(124\) −5.64575 −0.507003
\(125\) 12.0000 1.07331
\(126\) 14.5830 1.29916
\(127\) −13.2915 −1.17943 −0.589715 0.807611i \(-0.700760\pi\)
−0.589715 + 0.807611i \(0.700760\pi\)
\(128\) 1.00000 0.0883883
\(129\) 29.8745 2.63030
\(130\) −3.29150 −0.288684
\(131\) 1.93725 0.169259 0.0846293 0.996413i \(-0.473029\pi\)
0.0846293 + 0.996413i \(0.473029\pi\)
\(132\) −12.2915 −1.06984
\(133\) 0 0
\(134\) 0.645751 0.0557844
\(135\) −4.35425 −0.374754
\(136\) 0 0
\(137\) 15.5830 1.33135 0.665673 0.746244i \(-0.268145\pi\)
0.665673 + 0.746244i \(0.268145\pi\)
\(138\) −4.35425 −0.370658
\(139\) 18.6458 1.58151 0.790756 0.612131i \(-0.209688\pi\)
0.790756 + 0.612131i \(0.209688\pi\)
\(140\) −6.00000 −0.507093
\(141\) 11.5203 0.970181
\(142\) −2.70850 −0.227292
\(143\) −9.29150 −0.776994
\(144\) 4.00000 0.333333
\(145\) −2.70850 −0.224928
\(146\) 1.70850 0.141396
\(147\) 16.6458 1.37292
\(148\) 0.354249 0.0291191
\(149\) 10.9373 0.896015 0.448007 0.894030i \(-0.352134\pi\)
0.448007 + 0.894030i \(0.352134\pi\)
\(150\) −6.06275 −0.495021
\(151\) 12.9373 1.05282 0.526409 0.850231i \(-0.323538\pi\)
0.526409 + 0.850231i \(0.323538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −16.9373 −1.36484
\(155\) 9.29150 0.746311
\(156\) 5.29150 0.423659
\(157\) −10.5830 −0.844616 −0.422308 0.906452i \(-0.638780\pi\)
−0.422308 + 0.906452i \(0.638780\pi\)
\(158\) −4.00000 −0.318223
\(159\) −33.2915 −2.64019
\(160\) −1.64575 −0.130108
\(161\) −6.00000 −0.472866
\(162\) −5.00000 −0.392837
\(163\) 3.93725 0.308390 0.154195 0.988040i \(-0.450722\pi\)
0.154195 + 0.988040i \(0.450722\pi\)
\(164\) −0.291503 −0.0227625
\(165\) 20.2288 1.57481
\(166\) 7.93725 0.616050
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 9.64575 0.744186
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 11.2915 0.860969
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.35425 0.330095
\(175\) −8.35425 −0.631522
\(176\) −4.64575 −0.350187
\(177\) −21.0000 −1.57846
\(178\) 0 0
\(179\) 4.06275 0.303664 0.151832 0.988406i \(-0.451483\pi\)
0.151832 + 0.988406i \(0.451483\pi\)
\(180\) −6.58301 −0.490668
\(181\) 22.2288 1.65225 0.826125 0.563487i \(-0.190540\pi\)
0.826125 + 0.563487i \(0.190540\pi\)
\(182\) 7.29150 0.540482
\(183\) 2.47974 0.183308
\(184\) −1.64575 −0.121326
\(185\) −0.583005 −0.0428634
\(186\) −14.9373 −1.09525
\(187\) 0 0
\(188\) 4.35425 0.317566
\(189\) 9.64575 0.701625
\(190\) 0 0
\(191\) 6.58301 0.476330 0.238165 0.971225i \(-0.423454\pi\)
0.238165 + 0.971225i \(0.423454\pi\)
\(192\) 2.64575 0.190941
\(193\) 14.5830 1.04971 0.524854 0.851192i \(-0.324120\pi\)
0.524854 + 0.851192i \(0.324120\pi\)
\(194\) −3.70850 −0.266255
\(195\) −8.70850 −0.623628
\(196\) 6.29150 0.449393
\(197\) −7.64575 −0.544737 −0.272369 0.962193i \(-0.587807\pi\)
−0.272369 + 0.962193i \(0.587807\pi\)
\(198\) −18.5830 −1.32064
\(199\) −19.8745 −1.40887 −0.704433 0.709770i \(-0.748799\pi\)
−0.704433 + 0.709770i \(0.748799\pi\)
\(200\) −2.29150 −0.162034
\(201\) 1.70850 0.120508
\(202\) −13.6458 −0.960112
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 0.479741 0.0335066
\(206\) −13.2915 −0.926063
\(207\) −6.58301 −0.457550
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −15.8745 −1.09545
\(211\) −13.2915 −0.915025 −0.457512 0.889203i \(-0.651259\pi\)
−0.457512 + 0.889203i \(0.651259\pi\)
\(212\) −12.5830 −0.864204
\(213\) −7.16601 −0.491007
\(214\) 15.2915 1.04531
\(215\) −18.5830 −1.26735
\(216\) 2.64575 0.180021
\(217\) −20.5830 −1.39727
\(218\) 14.5830 0.987686
\(219\) 4.52026 0.305451
\(220\) 7.64575 0.515476
\(221\) 0 0
\(222\) 0.937254 0.0629044
\(223\) −18.8118 −1.25973 −0.629864 0.776705i \(-0.716890\pi\)
−0.629864 + 0.776705i \(0.716890\pi\)
\(224\) 3.64575 0.243592
\(225\) −9.16601 −0.611067
\(226\) −15.5830 −1.03657
\(227\) −7.35425 −0.488119 −0.244059 0.969760i \(-0.578479\pi\)
−0.244059 + 0.969760i \(0.578479\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 2.70850 0.178593
\(231\) −44.8118 −2.94840
\(232\) 1.64575 0.108049
\(233\) 18.8745 1.23651 0.618255 0.785978i \(-0.287840\pi\)
0.618255 + 0.785978i \(0.287840\pi\)
\(234\) 8.00000 0.522976
\(235\) −7.16601 −0.467459
\(236\) −7.93725 −0.516671
\(237\) −10.5830 −0.687440
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −4.35425 −0.281066
\(241\) −7.58301 −0.488464 −0.244232 0.969717i \(-0.578536\pi\)
−0.244232 + 0.969717i \(0.578536\pi\)
\(242\) 10.5830 0.680301
\(243\) −21.1660 −1.35780
\(244\) 0.937254 0.0600015
\(245\) −10.3542 −0.661509
\(246\) −0.771243 −0.0491727
\(247\) 0 0
\(248\) −5.64575 −0.358506
\(249\) 21.0000 1.33082
\(250\) 12.0000 0.758947
\(251\) 29.2288 1.84490 0.922451 0.386113i \(-0.126183\pi\)
0.922451 + 0.386113i \(0.126183\pi\)
\(252\) 14.5830 0.918643
\(253\) 7.64575 0.480684
\(254\) −13.2915 −0.833983
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.2915 0.766723 0.383361 0.923598i \(-0.374766\pi\)
0.383361 + 0.923598i \(0.374766\pi\)
\(258\) 29.8745 1.85991
\(259\) 1.29150 0.0802501
\(260\) −3.29150 −0.204130
\(261\) 6.58301 0.407478
\(262\) 1.93725 0.119684
\(263\) −10.9373 −0.674420 −0.337210 0.941429i \(-0.609483\pi\)
−0.337210 + 0.941429i \(0.609483\pi\)
\(264\) −12.2915 −0.756490
\(265\) 20.7085 1.27211
\(266\) 0 0
\(267\) 0 0
\(268\) 0.645751 0.0394455
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −4.35425 −0.264991
\(271\) 12.3542 0.750467 0.375234 0.926930i \(-0.377562\pi\)
0.375234 + 0.926930i \(0.377562\pi\)
\(272\) 0 0
\(273\) 19.2915 1.16757
\(274\) 15.5830 0.941404
\(275\) 10.6458 0.641963
\(276\) −4.35425 −0.262095
\(277\) −27.5203 −1.65353 −0.826766 0.562546i \(-0.809822\pi\)
−0.826766 + 0.562546i \(0.809822\pi\)
\(278\) 18.6458 1.11830
\(279\) −22.5830 −1.35201
\(280\) −6.00000 −0.358569
\(281\) 25.4575 1.51867 0.759334 0.650701i \(-0.225525\pi\)
0.759334 + 0.650701i \(0.225525\pi\)
\(282\) 11.5203 0.686021
\(283\) 30.6458 1.82170 0.910850 0.412737i \(-0.135427\pi\)
0.910850 + 0.412737i \(0.135427\pi\)
\(284\) −2.70850 −0.160720
\(285\) 0 0
\(286\) −9.29150 −0.549418
\(287\) −1.06275 −0.0627319
\(288\) 4.00000 0.235702
\(289\) −17.0000 −1.00000
\(290\) −2.70850 −0.159048
\(291\) −9.81176 −0.575176
\(292\) 1.70850 0.0999822
\(293\) 28.9373 1.69053 0.845266 0.534345i \(-0.179442\pi\)
0.845266 + 0.534345i \(0.179442\pi\)
\(294\) 16.6458 0.970800
\(295\) 13.0627 0.760542
\(296\) 0.354249 0.0205903
\(297\) −12.2915 −0.713225
\(298\) 10.9373 0.633578
\(299\) −3.29150 −0.190353
\(300\) −6.06275 −0.350033
\(301\) 41.1660 2.37277
\(302\) 12.9373 0.744455
\(303\) −36.1033 −2.07408
\(304\) 0 0
\(305\) −1.54249 −0.0883225
\(306\) 0 0
\(307\) 0.645751 0.0368550 0.0184275 0.999830i \(-0.494134\pi\)
0.0184275 + 0.999830i \(0.494134\pi\)
\(308\) −16.9373 −0.965090
\(309\) −35.1660 −2.00052
\(310\) 9.29150 0.527722
\(311\) 13.6458 0.773780 0.386890 0.922126i \(-0.373549\pi\)
0.386890 + 0.922126i \(0.373549\pi\)
\(312\) 5.29150 0.299572
\(313\) 8.87451 0.501617 0.250808 0.968037i \(-0.419304\pi\)
0.250808 + 0.968037i \(0.419304\pi\)
\(314\) −10.5830 −0.597234
\(315\) −24.0000 −1.35225
\(316\) −4.00000 −0.225018
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −33.2915 −1.86689
\(319\) −7.64575 −0.428080
\(320\) −1.64575 −0.0920003
\(321\) 40.4575 2.25812
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) −5.00000 −0.277778
\(325\) −4.58301 −0.254219
\(326\) 3.93725 0.218064
\(327\) 38.5830 2.13365
\(328\) −0.291503 −0.0160955
\(329\) 15.8745 0.875190
\(330\) 20.2288 1.11356
\(331\) 19.8118 1.08895 0.544476 0.838776i \(-0.316728\pi\)
0.544476 + 0.838776i \(0.316728\pi\)
\(332\) 7.93725 0.435613
\(333\) 1.41699 0.0776508
\(334\) −12.0000 −0.656611
\(335\) −1.06275 −0.0580640
\(336\) 9.64575 0.526219
\(337\) −9.70850 −0.528856 −0.264428 0.964405i \(-0.585183\pi\)
−0.264428 + 0.964405i \(0.585183\pi\)
\(338\) −9.00000 −0.489535
\(339\) −41.2288 −2.23924
\(340\) 0 0
\(341\) 26.2288 1.42037
\(342\) 0 0
\(343\) −2.58301 −0.139469
\(344\) 11.2915 0.608797
\(345\) 7.16601 0.385805
\(346\) −6.00000 −0.322562
\(347\) −23.2288 −1.24698 −0.623492 0.781829i \(-0.714287\pi\)
−0.623492 + 0.781829i \(0.714287\pi\)
\(348\) 4.35425 0.233412
\(349\) 21.1660 1.13299 0.566495 0.824065i \(-0.308299\pi\)
0.566495 + 0.824065i \(0.308299\pi\)
\(350\) −8.35425 −0.446553
\(351\) 5.29150 0.282440
\(352\) −4.64575 −0.247619
\(353\) 12.8745 0.685241 0.342620 0.939474i \(-0.388686\pi\)
0.342620 + 0.939474i \(0.388686\pi\)
\(354\) −21.0000 −1.11614
\(355\) 4.45751 0.236580
\(356\) 0 0
\(357\) 0 0
\(358\) 4.06275 0.214723
\(359\) −4.93725 −0.260578 −0.130289 0.991476i \(-0.541591\pi\)
−0.130289 + 0.991476i \(0.541591\pi\)
\(360\) −6.58301 −0.346955
\(361\) 0 0
\(362\) 22.2288 1.16832
\(363\) 28.0000 1.46962
\(364\) 7.29150 0.382179
\(365\) −2.81176 −0.147174
\(366\) 2.47974 0.129618
\(367\) 16.2288 0.847134 0.423567 0.905865i \(-0.360778\pi\)
0.423567 + 0.905865i \(0.360778\pi\)
\(368\) −1.64575 −0.0857907
\(369\) −1.16601 −0.0607001
\(370\) −0.583005 −0.0303090
\(371\) −45.8745 −2.38169
\(372\) −14.9373 −0.774461
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 31.7490 1.63951
\(376\) 4.35425 0.224553
\(377\) 3.29150 0.169521
\(378\) 9.64575 0.496124
\(379\) 10.7085 0.550059 0.275029 0.961436i \(-0.411312\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(380\) 0 0
\(381\) −35.1660 −1.80161
\(382\) 6.58301 0.336816
\(383\) 5.52026 0.282072 0.141036 0.990004i \(-0.454957\pi\)
0.141036 + 0.990004i \(0.454957\pi\)
\(384\) 2.64575 0.135015
\(385\) 27.8745 1.42062
\(386\) 14.5830 0.742255
\(387\) 45.1660 2.29592
\(388\) −3.70850 −0.188270
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) −8.70850 −0.440972
\(391\) 0 0
\(392\) 6.29150 0.317769
\(393\) 5.12549 0.258547
\(394\) −7.64575 −0.385187
\(395\) 6.58301 0.331227
\(396\) −18.5830 −0.933831
\(397\) 21.0627 1.05711 0.528554 0.848899i \(-0.322734\pi\)
0.528554 + 0.848899i \(0.322734\pi\)
\(398\) −19.8745 −0.996219
\(399\) 0 0
\(400\) −2.29150 −0.114575
\(401\) 27.5830 1.37743 0.688715 0.725032i \(-0.258175\pi\)
0.688715 + 0.725032i \(0.258175\pi\)
\(402\) 1.70850 0.0852121
\(403\) −11.2915 −0.562470
\(404\) −13.6458 −0.678902
\(405\) 8.22876 0.408890
\(406\) 6.00000 0.297775
\(407\) −1.64575 −0.0815769
\(408\) 0 0
\(409\) −7.58301 −0.374955 −0.187478 0.982269i \(-0.560031\pi\)
−0.187478 + 0.982269i \(0.560031\pi\)
\(410\) 0.479741 0.0236927
\(411\) 41.2288 2.03366
\(412\) −13.2915 −0.654825
\(413\) −28.9373 −1.42391
\(414\) −6.58301 −0.323537
\(415\) −13.0627 −0.641225
\(416\) 2.00000 0.0980581
\(417\) 49.3320 2.41580
\(418\) 0 0
\(419\) −31.7490 −1.55104 −0.775520 0.631322i \(-0.782512\pi\)
−0.775520 + 0.631322i \(0.782512\pi\)
\(420\) −15.8745 −0.774597
\(421\) −24.8118 −1.20925 −0.604626 0.796510i \(-0.706677\pi\)
−0.604626 + 0.796510i \(0.706677\pi\)
\(422\) −13.2915 −0.647020
\(423\) 17.4170 0.846843
\(424\) −12.5830 −0.611085
\(425\) 0 0
\(426\) −7.16601 −0.347194
\(427\) 3.41699 0.165360
\(428\) 15.2915 0.739143
\(429\) −24.5830 −1.18688
\(430\) −18.5830 −0.896152
\(431\) 27.8745 1.34267 0.671334 0.741155i \(-0.265722\pi\)
0.671334 + 0.741155i \(0.265722\pi\)
\(432\) 2.64575 0.127294
\(433\) 17.8745 0.858994 0.429497 0.903068i \(-0.358691\pi\)
0.429497 + 0.903068i \(0.358691\pi\)
\(434\) −20.5830 −0.988016
\(435\) −7.16601 −0.343584
\(436\) 14.5830 0.698399
\(437\) 0 0
\(438\) 4.52026 0.215986
\(439\) 10.8118 0.516017 0.258009 0.966143i \(-0.416934\pi\)
0.258009 + 0.966143i \(0.416934\pi\)
\(440\) 7.64575 0.364497
\(441\) 25.1660 1.19838
\(442\) 0 0
\(443\) 10.6458 0.505795 0.252897 0.967493i \(-0.418616\pi\)
0.252897 + 0.967493i \(0.418616\pi\)
\(444\) 0.937254 0.0444801
\(445\) 0 0
\(446\) −18.8118 −0.890763
\(447\) 28.9373 1.36869
\(448\) 3.64575 0.172246
\(449\) −24.2915 −1.14639 −0.573193 0.819420i \(-0.694296\pi\)
−0.573193 + 0.819420i \(0.694296\pi\)
\(450\) −9.16601 −0.432090
\(451\) 1.35425 0.0637691
\(452\) −15.5830 −0.732963
\(453\) 34.2288 1.60821
\(454\) −7.35425 −0.345152
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 32.8745 1.53780 0.768902 0.639366i \(-0.220803\pi\)
0.768902 + 0.639366i \(0.220803\pi\)
\(458\) 20.0000 0.934539
\(459\) 0 0
\(460\) 2.70850 0.126284
\(461\) 19.1660 0.892650 0.446325 0.894871i \(-0.352733\pi\)
0.446325 + 0.894871i \(0.352733\pi\)
\(462\) −44.8118 −2.08483
\(463\) −38.4575 −1.78727 −0.893636 0.448792i \(-0.851854\pi\)
−0.893636 + 0.448792i \(0.851854\pi\)
\(464\) 1.64575 0.0764021
\(465\) 24.5830 1.14001
\(466\) 18.8745 0.874345
\(467\) −19.3542 −0.895608 −0.447804 0.894132i \(-0.647794\pi\)
−0.447804 + 0.894132i \(0.647794\pi\)
\(468\) 8.00000 0.369800
\(469\) 2.35425 0.108709
\(470\) −7.16601 −0.330543
\(471\) −28.0000 −1.29017
\(472\) −7.93725 −0.365342
\(473\) −52.4575 −2.41200
\(474\) −10.5830 −0.486094
\(475\) 0 0
\(476\) 0 0
\(477\) −50.3320 −2.30454
\(478\) −12.0000 −0.548867
\(479\) 6.58301 0.300785 0.150393 0.988626i \(-0.451946\pi\)
0.150393 + 0.988626i \(0.451946\pi\)
\(480\) −4.35425 −0.198743
\(481\) 0.708497 0.0323047
\(482\) −7.58301 −0.345396
\(483\) −15.8745 −0.722315
\(484\) 10.5830 0.481046
\(485\) 6.10326 0.277135
\(486\) −21.1660 −0.960110
\(487\) 4.22876 0.191623 0.0958116 0.995399i \(-0.469455\pi\)
0.0958116 + 0.995399i \(0.469455\pi\)
\(488\) 0.937254 0.0424275
\(489\) 10.4170 0.471073
\(490\) −10.3542 −0.467757
\(491\) 39.2915 1.77320 0.886600 0.462536i \(-0.153061\pi\)
0.886600 + 0.462536i \(0.153061\pi\)
\(492\) −0.771243 −0.0347703
\(493\) 0 0
\(494\) 0 0
\(495\) 30.5830 1.37460
\(496\) −5.64575 −0.253502
\(497\) −9.87451 −0.442932
\(498\) 21.0000 0.941033
\(499\) −4.77124 −0.213590 −0.106795 0.994281i \(-0.534059\pi\)
−0.106795 + 0.994281i \(0.534059\pi\)
\(500\) 12.0000 0.536656
\(501\) −31.7490 −1.41844
\(502\) 29.2288 1.30454
\(503\) −40.9373 −1.82530 −0.912651 0.408740i \(-0.865968\pi\)
−0.912651 + 0.408740i \(0.865968\pi\)
\(504\) 14.5830 0.649579
\(505\) 22.4575 0.999346
\(506\) 7.64575 0.339895
\(507\) −23.8118 −1.05752
\(508\) −13.2915 −0.589715
\(509\) −31.7490 −1.40725 −0.703625 0.710571i \(-0.748437\pi\)
−0.703625 + 0.710571i \(0.748437\pi\)
\(510\) 0 0
\(511\) 6.22876 0.275544
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 12.2915 0.542155
\(515\) 21.8745 0.963906
\(516\) 29.8745 1.31515
\(517\) −20.2288 −0.889660
\(518\) 1.29150 0.0567454
\(519\) −15.8745 −0.696814
\(520\) −3.29150 −0.144342
\(521\) −11.7085 −0.512959 −0.256479 0.966550i \(-0.582563\pi\)
−0.256479 + 0.966550i \(0.582563\pi\)
\(522\) 6.58301 0.288130
\(523\) −1.87451 −0.0819665 −0.0409833 0.999160i \(-0.513049\pi\)
−0.0409833 + 0.999160i \(0.513049\pi\)
\(524\) 1.93725 0.0846293
\(525\) −22.1033 −0.964666
\(526\) −10.9373 −0.476887
\(527\) 0 0
\(528\) −12.2915 −0.534919
\(529\) −20.2915 −0.882239
\(530\) 20.7085 0.899520
\(531\) −31.7490 −1.37779
\(532\) 0 0
\(533\) −0.583005 −0.0252528
\(534\) 0 0
\(535\) −25.1660 −1.08802
\(536\) 0.645751 0.0278922
\(537\) 10.7490 0.463854
\(538\) 0 0
\(539\) −29.2288 −1.25897
\(540\) −4.35425 −0.187377
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 12.3542 0.530660
\(543\) 58.8118 2.52385
\(544\) 0 0
\(545\) −24.0000 −1.02805
\(546\) 19.2915 0.825600
\(547\) 11.2915 0.482790 0.241395 0.970427i \(-0.422395\pi\)
0.241395 + 0.970427i \(0.422395\pi\)
\(548\) 15.5830 0.665673
\(549\) 3.74902 0.160004
\(550\) 10.6458 0.453936
\(551\) 0 0
\(552\) −4.35425 −0.185329
\(553\) −14.5830 −0.620132
\(554\) −27.5203 −1.16922
\(555\) −1.54249 −0.0654749
\(556\) 18.6458 0.790756
\(557\) −5.41699 −0.229525 −0.114763 0.993393i \(-0.536611\pi\)
−0.114763 + 0.993393i \(0.536611\pi\)
\(558\) −22.5830 −0.956015
\(559\) 22.5830 0.955159
\(560\) −6.00000 −0.253546
\(561\) 0 0
\(562\) 25.4575 1.07386
\(563\) 10.0627 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(564\) 11.5203 0.485090
\(565\) 25.6458 1.07892
\(566\) 30.6458 1.28814
\(567\) −18.2288 −0.765536
\(568\) −2.70850 −0.113646
\(569\) 6.58301 0.275974 0.137987 0.990434i \(-0.455937\pi\)
0.137987 + 0.990434i \(0.455937\pi\)
\(570\) 0 0
\(571\) 7.81176 0.326912 0.163456 0.986551i \(-0.447736\pi\)
0.163456 + 0.986551i \(0.447736\pi\)
\(572\) −9.29150 −0.388497
\(573\) 17.4170 0.727605
\(574\) −1.06275 −0.0443582
\(575\) 3.77124 0.157272
\(576\) 4.00000 0.166667
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) −17.0000 −0.707107
\(579\) 38.5830 1.60345
\(580\) −2.70850 −0.112464
\(581\) 28.9373 1.20052
\(582\) −9.81176 −0.406711
\(583\) 58.4575 2.42106
\(584\) 1.70850 0.0706981
\(585\) −13.1660 −0.544348
\(586\) 28.9373 1.19539
\(587\) −14.1255 −0.583021 −0.291511 0.956568i \(-0.594158\pi\)
−0.291511 + 0.956568i \(0.594158\pi\)
\(588\) 16.6458 0.686459
\(589\) 0 0
\(590\) 13.0627 0.537785
\(591\) −20.2288 −0.832100
\(592\) 0.354249 0.0145595
\(593\) 29.7085 1.21998 0.609991 0.792408i \(-0.291173\pi\)
0.609991 + 0.792408i \(0.291173\pi\)
\(594\) −12.2915 −0.504326
\(595\) 0 0
\(596\) 10.9373 0.448007
\(597\) −52.5830 −2.15208
\(598\) −3.29150 −0.134600
\(599\) −16.9373 −0.692037 −0.346019 0.938228i \(-0.612467\pi\)
−0.346019 + 0.938228i \(0.612467\pi\)
\(600\) −6.06275 −0.247511
\(601\) 10.4170 0.424918 0.212459 0.977170i \(-0.431853\pi\)
0.212459 + 0.977170i \(0.431853\pi\)
\(602\) 41.1660 1.67780
\(603\) 2.58301 0.105188
\(604\) 12.9373 0.526409
\(605\) −17.4170 −0.708102
\(606\) −36.1033 −1.46659
\(607\) 6.93725 0.281574 0.140787 0.990040i \(-0.455037\pi\)
0.140787 + 0.990040i \(0.455037\pi\)
\(608\) 0 0
\(609\) 15.8745 0.643268
\(610\) −1.54249 −0.0624535
\(611\) 8.70850 0.352308
\(612\) 0 0
\(613\) 7.41699 0.299570 0.149785 0.988719i \(-0.452142\pi\)
0.149785 + 0.988719i \(0.452142\pi\)
\(614\) 0.645751 0.0260604
\(615\) 1.26927 0.0511821
\(616\) −16.9373 −0.682421
\(617\) −30.8745 −1.24296 −0.621480 0.783430i \(-0.713468\pi\)
−0.621480 + 0.783430i \(0.713468\pi\)
\(618\) −35.1660 −1.41458
\(619\) −8.45751 −0.339936 −0.169968 0.985450i \(-0.554366\pi\)
−0.169968 + 0.985450i \(0.554366\pi\)
\(620\) 9.29150 0.373156
\(621\) −4.35425 −0.174730
\(622\) 13.6458 0.547145
\(623\) 0 0
\(624\) 5.29150 0.211830
\(625\) −8.29150 −0.331660
\(626\) 8.87451 0.354697
\(627\) 0 0
\(628\) −10.5830 −0.422308
\(629\) 0 0
\(630\) −24.0000 −0.956183
\(631\) −24.8118 −0.987741 −0.493870 0.869536i \(-0.664418\pi\)
−0.493870 + 0.869536i \(0.664418\pi\)
\(632\) −4.00000 −0.159111
\(633\) −35.1660 −1.39772
\(634\) −6.00000 −0.238290
\(635\) 21.8745 0.868063
\(636\) −33.2915 −1.32009
\(637\) 12.5830 0.498557
\(638\) −7.64575 −0.302698
\(639\) −10.8340 −0.428586
\(640\) −1.64575 −0.0650540
\(641\) 12.8745 0.508512 0.254256 0.967137i \(-0.418169\pi\)
0.254256 + 0.967137i \(0.418169\pi\)
\(642\) 40.4575 1.59673
\(643\) −6.52026 −0.257134 −0.128567 0.991701i \(-0.541038\pi\)
−0.128567 + 0.991701i \(0.541038\pi\)
\(644\) −6.00000 −0.236433
\(645\) −49.1660 −1.93591
\(646\) 0 0
\(647\) −22.4575 −0.882896 −0.441448 0.897287i \(-0.645535\pi\)
−0.441448 + 0.897287i \(0.645535\pi\)
\(648\) −5.00000 −0.196419
\(649\) 36.8745 1.44745
\(650\) −4.58301 −0.179760
\(651\) −54.4575 −2.13436
\(652\) 3.93725 0.154195
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 38.5830 1.50871
\(655\) −3.18824 −0.124575
\(656\) −0.291503 −0.0113813
\(657\) 6.83399 0.266619
\(658\) 15.8745 0.618853
\(659\) −18.5830 −0.723891 −0.361946 0.932199i \(-0.617887\pi\)
−0.361946 + 0.932199i \(0.617887\pi\)
\(660\) 20.2288 0.787403
\(661\) 16.2288 0.631225 0.315613 0.948888i \(-0.397790\pi\)
0.315613 + 0.948888i \(0.397790\pi\)
\(662\) 19.8118 0.770006
\(663\) 0 0
\(664\) 7.93725 0.308025
\(665\) 0 0
\(666\) 1.41699 0.0549074
\(667\) −2.70850 −0.104873
\(668\) −12.0000 −0.464294
\(669\) −49.7712 −1.92427
\(670\) −1.06275 −0.0410575
\(671\) −4.35425 −0.168094
\(672\) 9.64575 0.372093
\(673\) −13.8745 −0.534823 −0.267411 0.963582i \(-0.586168\pi\)
−0.267411 + 0.963582i \(0.586168\pi\)
\(674\) −9.70850 −0.373957
\(675\) −6.06275 −0.233355
\(676\) −9.00000 −0.346154
\(677\) −11.4170 −0.438791 −0.219395 0.975636i \(-0.570408\pi\)
−0.219395 + 0.975636i \(0.570408\pi\)
\(678\) −41.2288 −1.58338
\(679\) −13.5203 −0.518860
\(680\) 0 0
\(681\) −19.4575 −0.745614
\(682\) 26.2288 1.00435
\(683\) −5.41699 −0.207276 −0.103638 0.994615i \(-0.533048\pi\)
−0.103638 + 0.994615i \(0.533048\pi\)
\(684\) 0 0
\(685\) −25.6458 −0.979874
\(686\) −2.58301 −0.0986196
\(687\) 52.9150 2.01883
\(688\) 11.2915 0.430485
\(689\) −25.1660 −0.958749
\(690\) 7.16601 0.272805
\(691\) 2.58301 0.0982622 0.0491311 0.998792i \(-0.484355\pi\)
0.0491311 + 0.998792i \(0.484355\pi\)
\(692\) −6.00000 −0.228086
\(693\) −67.7490 −2.57357
\(694\) −23.2288 −0.881752
\(695\) −30.6863 −1.16400
\(696\) 4.35425 0.165047
\(697\) 0 0
\(698\) 21.1660 0.801145
\(699\) 49.9373 1.88880
\(700\) −8.35425 −0.315761
\(701\) 10.3542 0.391075 0.195537 0.980696i \(-0.437355\pi\)
0.195537 + 0.980696i \(0.437355\pi\)
\(702\) 5.29150 0.199715
\(703\) 0 0
\(704\) −4.64575 −0.175093
\(705\) −18.9595 −0.714055
\(706\) 12.8745 0.484538
\(707\) −49.7490 −1.87100
\(708\) −21.0000 −0.789228
\(709\) 3.64575 0.136919 0.0684595 0.997654i \(-0.478192\pi\)
0.0684595 + 0.997654i \(0.478192\pi\)
\(710\) 4.45751 0.167287
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 9.29150 0.347970
\(714\) 0 0
\(715\) 15.2915 0.571870
\(716\) 4.06275 0.151832
\(717\) −31.7490 −1.18569
\(718\) −4.93725 −0.184257
\(719\) 2.70850 0.101010 0.0505050 0.998724i \(-0.483917\pi\)
0.0505050 + 0.998724i \(0.483917\pi\)
\(720\) −6.58301 −0.245334
\(721\) −48.4575 −1.80465
\(722\) 0 0
\(723\) −20.0627 −0.746142
\(724\) 22.2288 0.826125
\(725\) −3.77124 −0.140060
\(726\) 28.0000 1.03918
\(727\) 1.41699 0.0525534 0.0262767 0.999655i \(-0.491635\pi\)
0.0262767 + 0.999655i \(0.491635\pi\)
\(728\) 7.29150 0.270241
\(729\) −41.0000 −1.51852
\(730\) −2.81176 −0.104068
\(731\) 0 0
\(732\) 2.47974 0.0916539
\(733\) −16.1033 −0.594788 −0.297394 0.954755i \(-0.596117\pi\)
−0.297394 + 0.954755i \(0.596117\pi\)
\(734\) 16.2288 0.599014
\(735\) −27.3948 −1.01047
\(736\) −1.64575 −0.0606632
\(737\) −3.00000 −0.110506
\(738\) −1.16601 −0.0429214
\(739\) −33.8118 −1.24379 −0.621893 0.783102i \(-0.713636\pi\)
−0.621893 + 0.783102i \(0.713636\pi\)
\(740\) −0.583005 −0.0214317
\(741\) 0 0
\(742\) −45.8745 −1.68411
\(743\) −47.5203 −1.74335 −0.871675 0.490085i \(-0.836966\pi\)
−0.871675 + 0.490085i \(0.836966\pi\)
\(744\) −14.9373 −0.547626
\(745\) −18.0000 −0.659469
\(746\) −4.00000 −0.146450
\(747\) 31.7490 1.16164
\(748\) 0 0
\(749\) 55.7490 2.03702
\(750\) 31.7490 1.15931
\(751\) −7.87451 −0.287345 −0.143672 0.989625i \(-0.545891\pi\)
−0.143672 + 0.989625i \(0.545891\pi\)
\(752\) 4.35425 0.158783
\(753\) 77.3320 2.81814
\(754\) 3.29150 0.119869
\(755\) −21.2915 −0.774877
\(756\) 9.64575 0.350813
\(757\) −4.58301 −0.166572 −0.0832861 0.996526i \(-0.526542\pi\)
−0.0832861 + 0.996526i \(0.526542\pi\)
\(758\) 10.7085 0.388950
\(759\) 20.2288 0.734257
\(760\) 0 0
\(761\) 42.8745 1.55420 0.777100 0.629377i \(-0.216690\pi\)
0.777100 + 0.629377i \(0.216690\pi\)
\(762\) −35.1660 −1.27393
\(763\) 53.1660 1.92474
\(764\) 6.58301 0.238165
\(765\) 0 0
\(766\) 5.52026 0.199455
\(767\) −15.8745 −0.573195
\(768\) 2.64575 0.0954703
\(769\) 35.2915 1.27264 0.636322 0.771424i \(-0.280455\pi\)
0.636322 + 0.771424i \(0.280455\pi\)
\(770\) 27.8745 1.00453
\(771\) 32.5203 1.17119
\(772\) 14.5830 0.524854
\(773\) −4.93725 −0.177581 −0.0887903 0.996050i \(-0.528300\pi\)
−0.0887903 + 0.996050i \(0.528300\pi\)
\(774\) 45.1660 1.62346
\(775\) 12.9373 0.464720
\(776\) −3.70850 −0.133127
\(777\) 3.41699 0.122584
\(778\) 12.0000 0.430221
\(779\) 0 0
\(780\) −8.70850 −0.311814
\(781\) 12.5830 0.450255
\(782\) 0 0
\(783\) 4.35425 0.155608
\(784\) 6.29150 0.224697
\(785\) 17.4170 0.621639
\(786\) 5.12549 0.182820
\(787\) −42.5203 −1.51568 −0.757842 0.652438i \(-0.773746\pi\)
−0.757842 + 0.652438i \(0.773746\pi\)
\(788\) −7.64575 −0.272369
\(789\) −28.9373 −1.03019
\(790\) 6.58301 0.234213
\(791\) −56.8118 −2.01999
\(792\) −18.5830 −0.660318
\(793\) 1.87451 0.0665657
\(794\) 21.0627 0.747489
\(795\) 54.7895 1.94318
\(796\) −19.8745 −0.704433
\(797\) −44.8118 −1.58731 −0.793657 0.608365i \(-0.791826\pi\)
−0.793657 + 0.608365i \(0.791826\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.29150 −0.0810169
\(801\) 0 0
\(802\) 27.5830 0.973990
\(803\) −7.93725 −0.280100
\(804\) 1.70850 0.0602541
\(805\) 9.87451 0.348031
\(806\) −11.2915 −0.397726
\(807\) 0 0
\(808\) −13.6458 −0.480056
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 8.22876 0.289129
\(811\) −31.2915 −1.09879 −0.549397 0.835562i \(-0.685142\pi\)
−0.549397 + 0.835562i \(0.685142\pi\)
\(812\) 6.00000 0.210559
\(813\) 32.6863 1.14636
\(814\) −1.64575 −0.0576836
\(815\) −6.47974 −0.226975
\(816\) 0 0
\(817\) 0 0
\(818\) −7.58301 −0.265134
\(819\) 29.1660 1.01914
\(820\) 0.479741 0.0167533
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 41.2288 1.43802
\(823\) −31.8745 −1.11108 −0.555538 0.831491i \(-0.687488\pi\)
−0.555538 + 0.831491i \(0.687488\pi\)
\(824\) −13.2915 −0.463031
\(825\) 28.1660 0.980615
\(826\) −28.9373 −1.00686
\(827\) −52.6458 −1.83067 −0.915336 0.402691i \(-0.868075\pi\)
−0.915336 + 0.402691i \(0.868075\pi\)
\(828\) −6.58301 −0.228775
\(829\) −17.1660 −0.596200 −0.298100 0.954535i \(-0.596353\pi\)
−0.298100 + 0.954535i \(0.596353\pi\)
\(830\) −13.0627 −0.453415
\(831\) −72.8118 −2.52581
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 49.3320 1.70823
\(835\) 19.7490 0.683443
\(836\) 0 0
\(837\) −14.9373 −0.516307
\(838\) −31.7490 −1.09675
\(839\) −41.5203 −1.43344 −0.716719 0.697362i \(-0.754357\pi\)
−0.716719 + 0.697362i \(0.754357\pi\)
\(840\) −15.8745 −0.547723
\(841\) −26.2915 −0.906604
\(842\) −24.8118 −0.855070
\(843\) 67.3542 2.31980
\(844\) −13.2915 −0.457512
\(845\) 14.8118 0.509540
\(846\) 17.4170 0.598809
\(847\) 38.5830 1.32573
\(848\) −12.5830 −0.432102
\(849\) 81.0810 2.78269
\(850\) 0 0
\(851\) −0.583005 −0.0199852
\(852\) −7.16601 −0.245503
\(853\) 8.58301 0.293877 0.146938 0.989146i \(-0.453058\pi\)
0.146938 + 0.989146i \(0.453058\pi\)
\(854\) 3.41699 0.116927
\(855\) 0 0
\(856\) 15.2915 0.522653
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) −24.5830 −0.839250
\(859\) 13.2288 0.451359 0.225680 0.974202i \(-0.427540\pi\)
0.225680 + 0.974202i \(0.427540\pi\)
\(860\) −18.5830 −0.633675
\(861\) −2.81176 −0.0958246
\(862\) 27.8745 0.949410
\(863\) −31.0627 −1.05739 −0.528694 0.848812i \(-0.677318\pi\)
−0.528694 + 0.848812i \(0.677318\pi\)
\(864\) 2.64575 0.0900103
\(865\) 9.87451 0.335743
\(866\) 17.8745 0.607401
\(867\) −44.9778 −1.52753
\(868\) −20.5830 −0.698633
\(869\) 18.5830 0.630385
\(870\) −7.16601 −0.242951
\(871\) 1.29150 0.0437609
\(872\) 14.5830 0.493843
\(873\) −14.8340 −0.502054
\(874\) 0 0
\(875\) 43.7490 1.47899
\(876\) 4.52026 0.152725
\(877\) −41.6458 −1.40628 −0.703139 0.711053i \(-0.748219\pi\)
−0.703139 + 0.711053i \(0.748219\pi\)
\(878\) 10.8118 0.364879
\(879\) 76.5608 2.58233
\(880\) 7.64575 0.257738
\(881\) −36.8745 −1.24233 −0.621167 0.783678i \(-0.713341\pi\)
−0.621167 + 0.783678i \(0.713341\pi\)
\(882\) 25.1660 0.847384
\(883\) −28.3948 −0.955560 −0.477780 0.878480i \(-0.658558\pi\)
−0.477780 + 0.878480i \(0.658558\pi\)
\(884\) 0 0
\(885\) 34.5608 1.16175
\(886\) 10.6458 0.357651
\(887\) −17.4170 −0.584805 −0.292403 0.956295i \(-0.594455\pi\)
−0.292403 + 0.956295i \(0.594455\pi\)
\(888\) 0.937254 0.0314522
\(889\) −48.4575 −1.62521
\(890\) 0 0
\(891\) 23.2288 0.778193
\(892\) −18.8118 −0.629864
\(893\) 0 0
\(894\) 28.9373 0.967807
\(895\) −6.68627 −0.223497
\(896\) 3.64575 0.121796
\(897\) −8.70850 −0.290768
\(898\) −24.2915 −0.810618
\(899\) −9.29150 −0.309889
\(900\) −9.16601 −0.305534
\(901\) 0 0
\(902\) 1.35425 0.0450915
\(903\) 108.915 3.62447
\(904\) −15.5830 −0.518283
\(905\) −36.5830 −1.21606
\(906\) 34.2288 1.13717
\(907\) 39.9373 1.32609 0.663047 0.748577i \(-0.269263\pi\)
0.663047 + 0.748577i \(0.269263\pi\)
\(908\) −7.35425 −0.244059
\(909\) −54.5830 −1.81040
\(910\) −12.0000 −0.397796
\(911\) 16.9373 0.561156 0.280578 0.959831i \(-0.409474\pi\)
0.280578 + 0.959831i \(0.409474\pi\)
\(912\) 0 0
\(913\) −36.8745 −1.22037
\(914\) 32.8745 1.08739
\(915\) −4.08104 −0.134915
\(916\) 20.0000 0.660819
\(917\) 7.06275 0.233232
\(918\) 0 0
\(919\) −19.8745 −0.655600 −0.327800 0.944747i \(-0.606307\pi\)
−0.327800 + 0.944747i \(0.606307\pi\)
\(920\) 2.70850 0.0892965
\(921\) 1.70850 0.0562969
\(922\) 19.1660 0.631199
\(923\) −5.41699 −0.178303
\(924\) −44.8118 −1.47420
\(925\) −0.811762 −0.0266906
\(926\) −38.4575 −1.26379
\(927\) −53.1660 −1.74620
\(928\) 1.64575 0.0540244
\(929\) −9.58301 −0.314408 −0.157204 0.987566i \(-0.550248\pi\)
−0.157204 + 0.987566i \(0.550248\pi\)
\(930\) 24.5830 0.806108
\(931\) 0 0
\(932\) 18.8745 0.618255
\(933\) 36.1033 1.18197
\(934\) −19.3542 −0.633290
\(935\) 0 0
\(936\) 8.00000 0.261488
\(937\) 7.12549 0.232780 0.116390 0.993204i \(-0.462868\pi\)
0.116390 + 0.993204i \(0.462868\pi\)
\(938\) 2.35425 0.0768689
\(939\) 23.4797 0.766232
\(940\) −7.16601 −0.233729
\(941\) 16.8340 0.548772 0.274386 0.961620i \(-0.411525\pi\)
0.274386 + 0.961620i \(0.411525\pi\)
\(942\) −28.0000 −0.912289
\(943\) 0.479741 0.0156225
\(944\) −7.93725 −0.258336
\(945\) −15.8745 −0.516398
\(946\) −52.4575 −1.70554
\(947\) −7.74902 −0.251809 −0.125905 0.992042i \(-0.540183\pi\)
−0.125905 + 0.992042i \(0.540183\pi\)
\(948\) −10.5830 −0.343720
\(949\) 3.41699 0.110920
\(950\) 0 0
\(951\) −15.8745 −0.514766
\(952\) 0 0
\(953\) −13.4575 −0.435932 −0.217966 0.975956i \(-0.569942\pi\)
−0.217966 + 0.975956i \(0.569942\pi\)
\(954\) −50.3320 −1.62956
\(955\) −10.8340 −0.350580
\(956\) −12.0000 −0.388108
\(957\) −20.2288 −0.653903
\(958\) 6.58301 0.212687
\(959\) 56.8118 1.83455
\(960\) −4.35425 −0.140533
\(961\) 0.874508 0.0282099
\(962\) 0.708497 0.0228429
\(963\) 61.1660 1.97105
\(964\) −7.58301 −0.244232
\(965\) −24.0000 −0.772587
\(966\) −15.8745 −0.510754
\(967\) −13.2915 −0.427426 −0.213713 0.976897i \(-0.568556\pi\)
−0.213713 + 0.976897i \(0.568556\pi\)
\(968\) 10.5830 0.340151
\(969\) 0 0
\(970\) 6.10326 0.195964
\(971\) −54.3948 −1.74561 −0.872806 0.488068i \(-0.837702\pi\)
−0.872806 + 0.488068i \(0.837702\pi\)
\(972\) −21.1660 −0.678900
\(973\) 67.9778 2.17927
\(974\) 4.22876 0.135498
\(975\) −12.1255 −0.388327
\(976\) 0.937254 0.0300008
\(977\) −7.45751 −0.238587 −0.119293 0.992859i \(-0.538063\pi\)
−0.119293 + 0.992859i \(0.538063\pi\)
\(978\) 10.4170 0.333099
\(979\) 0 0
\(980\) −10.3542 −0.330754
\(981\) 58.3320 1.86240
\(982\) 39.2915 1.25384
\(983\) 31.7490 1.01264 0.506318 0.862347i \(-0.331006\pi\)
0.506318 + 0.862347i \(0.331006\pi\)
\(984\) −0.771243 −0.0245863
\(985\) 12.5830 0.400928
\(986\) 0 0
\(987\) 42.0000 1.33687
\(988\) 0 0
\(989\) −18.5830 −0.590905
\(990\) 30.5830 0.971992
\(991\) −2.83399 −0.0900246 −0.0450123 0.998986i \(-0.514333\pi\)
−0.0450123 + 0.998986i \(0.514333\pi\)
\(992\) −5.64575 −0.179253
\(993\) 52.4170 1.66340
\(994\) −9.87451 −0.313200
\(995\) 32.7085 1.03693
\(996\) 21.0000 0.665410
\(997\) 16.2288 0.513970 0.256985 0.966415i \(-0.417271\pi\)
0.256985 + 0.966415i \(0.417271\pi\)
\(998\) −4.77124 −0.151031
\(999\) 0.937254 0.0296534
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 722.2.a.j.1.2 2
3.2 odd 2 6498.2.a.ba.1.2 2
4.3 odd 2 5776.2.a.ba.1.1 2
19.2 odd 18 722.2.e.o.99.1 12
19.3 odd 18 722.2.e.o.389.1 12
19.4 even 9 722.2.e.n.415.1 12
19.5 even 9 722.2.e.n.595.1 12
19.6 even 9 722.2.e.n.245.2 12
19.7 even 3 38.2.c.b.11.1 yes 4
19.8 odd 6 722.2.c.j.653.2 4
19.9 even 9 722.2.e.n.423.2 12
19.10 odd 18 722.2.e.o.423.1 12
19.11 even 3 38.2.c.b.7.1 4
19.12 odd 6 722.2.c.j.429.2 4
19.13 odd 18 722.2.e.o.245.1 12
19.14 odd 18 722.2.e.o.595.2 12
19.15 odd 18 722.2.e.o.415.2 12
19.16 even 9 722.2.e.n.389.2 12
19.17 even 9 722.2.e.n.99.2 12
19.18 odd 2 722.2.a.g.1.1 2
57.11 odd 6 342.2.g.f.235.1 4
57.26 odd 6 342.2.g.f.163.1 4
57.56 even 2 6498.2.a.bg.1.2 2
76.7 odd 6 304.2.i.e.49.2 4
76.11 odd 6 304.2.i.e.273.2 4
76.75 even 2 5776.2.a.z.1.2 2
95.7 odd 12 950.2.j.g.49.2 8
95.49 even 6 950.2.e.k.501.2 4
95.64 even 6 950.2.e.k.201.2 4
95.68 odd 12 950.2.j.g.349.2 8
95.83 odd 12 950.2.j.g.49.3 8
95.87 odd 12 950.2.j.g.349.3 8
152.11 odd 6 1216.2.i.k.577.1 4
152.45 even 6 1216.2.i.l.961.2 4
152.83 odd 6 1216.2.i.k.961.1 4
152.125 even 6 1216.2.i.l.577.2 4
228.11 even 6 2736.2.s.v.577.1 4
228.83 even 6 2736.2.s.v.1873.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.c.b.7.1 4 19.11 even 3
38.2.c.b.11.1 yes 4 19.7 even 3
304.2.i.e.49.2 4 76.7 odd 6
304.2.i.e.273.2 4 76.11 odd 6
342.2.g.f.163.1 4 57.26 odd 6
342.2.g.f.235.1 4 57.11 odd 6
722.2.a.g.1.1 2 19.18 odd 2
722.2.a.j.1.2 2 1.1 even 1 trivial
722.2.c.j.429.2 4 19.12 odd 6
722.2.c.j.653.2 4 19.8 odd 6
722.2.e.n.99.2 12 19.17 even 9
722.2.e.n.245.2 12 19.6 even 9
722.2.e.n.389.2 12 19.16 even 9
722.2.e.n.415.1 12 19.4 even 9
722.2.e.n.423.2 12 19.9 even 9
722.2.e.n.595.1 12 19.5 even 9
722.2.e.o.99.1 12 19.2 odd 18
722.2.e.o.245.1 12 19.13 odd 18
722.2.e.o.389.1 12 19.3 odd 18
722.2.e.o.415.2 12 19.15 odd 18
722.2.e.o.423.1 12 19.10 odd 18
722.2.e.o.595.2 12 19.14 odd 18
950.2.e.k.201.2 4 95.64 even 6
950.2.e.k.501.2 4 95.49 even 6
950.2.j.g.49.2 8 95.7 odd 12
950.2.j.g.49.3 8 95.83 odd 12
950.2.j.g.349.2 8 95.68 odd 12
950.2.j.g.349.3 8 95.87 odd 12
1216.2.i.k.577.1 4 152.11 odd 6
1216.2.i.k.961.1 4 152.83 odd 6
1216.2.i.l.577.2 4 152.125 even 6
1216.2.i.l.961.2 4 152.45 even 6
2736.2.s.v.577.1 4 228.11 even 6
2736.2.s.v.1873.1 4 228.83 even 6
5776.2.a.z.1.2 2 76.75 even 2
5776.2.a.ba.1.1 2 4.3 odd 2
6498.2.a.ba.1.2 2 3.2 odd 2
6498.2.a.bg.1.2 2 57.56 even 2