Properties

Label 5390.2.a.bs.1.1
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +1.00000 q^{11} -2.73205 q^{12} +1.46410 q^{13} -2.73205 q^{15} +1.00000 q^{16} +3.46410 q^{17} +4.46410 q^{18} -6.73205 q^{19} +1.00000 q^{20} +1.00000 q^{22} -8.19615 q^{23} -2.73205 q^{24} +1.00000 q^{25} +1.46410 q^{26} -4.00000 q^{27} -4.73205 q^{29} -2.73205 q^{30} -2.00000 q^{31} +1.00000 q^{32} -2.73205 q^{33} +3.46410 q^{34} +4.46410 q^{36} +0.732051 q^{37} -6.73205 q^{38} -4.00000 q^{39} +1.00000 q^{40} +2.19615 q^{41} +2.00000 q^{43} +1.00000 q^{44} +4.46410 q^{45} -8.19615 q^{46} -6.92820 q^{47} -2.73205 q^{48} +1.00000 q^{50} -9.46410 q^{51} +1.46410 q^{52} -7.26795 q^{53} -4.00000 q^{54} +1.00000 q^{55} +18.3923 q^{57} -4.73205 q^{58} -6.92820 q^{59} -2.73205 q^{60} +4.92820 q^{61} -2.00000 q^{62} +1.00000 q^{64} +1.46410 q^{65} -2.73205 q^{66} -4.00000 q^{67} +3.46410 q^{68} +22.3923 q^{69} +9.46410 q^{71} +4.46410 q^{72} +14.3923 q^{73} +0.732051 q^{74} -2.73205 q^{75} -6.73205 q^{76} -4.00000 q^{78} -12.1962 q^{79} +1.00000 q^{80} -2.46410 q^{81} +2.19615 q^{82} +16.3923 q^{83} +3.46410 q^{85} +2.00000 q^{86} +12.9282 q^{87} +1.00000 q^{88} -3.46410 q^{89} +4.46410 q^{90} -8.19615 q^{92} +5.46410 q^{93} -6.92820 q^{94} -6.73205 q^{95} -2.73205 q^{96} -14.5885 q^{97} +4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{15} + 2 q^{16} + 2 q^{18} - 10 q^{19} + 2 q^{20} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 4 q^{26} - 8 q^{27} - 6 q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{36} - 2 q^{37} - 10 q^{38} - 8 q^{39} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 2 q^{44} + 2 q^{45} - 6 q^{46} - 2 q^{48} + 2 q^{50} - 12 q^{51} - 4 q^{52} - 18 q^{53} - 8 q^{54} + 2 q^{55} + 16 q^{57} - 6 q^{58} - 2 q^{60} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 4 q^{65} - 2 q^{66} - 8 q^{67} + 24 q^{69} + 12 q^{71} + 2 q^{72} + 8 q^{73} - 2 q^{74} - 2 q^{75} - 10 q^{76} - 8 q^{78} - 14 q^{79} + 2 q^{80} + 2 q^{81} - 6 q^{82} + 12 q^{83} + 4 q^{86} + 12 q^{87} + 2 q^{88} + 2 q^{90} - 6 q^{92} + 4 q^{93} - 10 q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{99} + O(q^{100}) \)

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
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.73205 −1.11536
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 4.46410 1.48803
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −2.73205 −0.788675
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 0 0
\(15\) −2.73205 −0.705412
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 4.46410 1.05220
\(19\) −6.73205 −1.54444 −0.772219 0.635356i \(-0.780853\pi\)
−0.772219 + 0.635356i \(0.780853\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −8.19615 −1.70902 −0.854508 0.519438i \(-0.826141\pi\)
−0.854508 + 0.519438i \(0.826141\pi\)
\(24\) −2.73205 −0.557678
\(25\) 1.00000 0.200000
\(26\) 1.46410 0.287134
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −4.73205 −0.878720 −0.439360 0.898311i \(-0.644795\pi\)
−0.439360 + 0.898311i \(0.644795\pi\)
\(30\) −2.73205 −0.498802
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.73205 −0.475589
\(34\) 3.46410 0.594089
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) 0.732051 0.120348 0.0601742 0.998188i \(-0.480834\pi\)
0.0601742 + 0.998188i \(0.480834\pi\)
\(38\) −6.73205 −1.09208
\(39\) −4.00000 −0.640513
\(40\) 1.00000 0.158114
\(41\) 2.19615 0.342981 0.171491 0.985186i \(-0.445142\pi\)
0.171491 + 0.985186i \(0.445142\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 1.00000 0.150756
\(45\) 4.46410 0.665469
\(46\) −8.19615 −1.20846
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) −2.73205 −0.394338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −9.46410 −1.32524
\(52\) 1.46410 0.203034
\(53\) −7.26795 −0.998330 −0.499165 0.866507i \(-0.666360\pi\)
−0.499165 + 0.866507i \(0.666360\pi\)
\(54\) −4.00000 −0.544331
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 18.3923 2.43612
\(58\) −4.73205 −0.621349
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) −2.73205 −0.352706
\(61\) 4.92820 0.630992 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.46410 0.181599
\(66\) −2.73205 −0.336292
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.46410 0.420084
\(69\) 22.3923 2.69572
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 4.46410 0.526099
\(73\) 14.3923 1.68449 0.842246 0.539093i \(-0.181233\pi\)
0.842246 + 0.539093i \(0.181233\pi\)
\(74\) 0.732051 0.0850992
\(75\) −2.73205 −0.315470
\(76\) −6.73205 −0.772219
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −12.1962 −1.37217 −0.686087 0.727519i \(-0.740673\pi\)
−0.686087 + 0.727519i \(0.740673\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.46410 −0.273789
\(82\) 2.19615 0.242524
\(83\) 16.3923 1.79929 0.899645 0.436623i \(-0.143826\pi\)
0.899645 + 0.436623i \(0.143826\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) 2.00000 0.215666
\(87\) 12.9282 1.38605
\(88\) 1.00000 0.106600
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 4.46410 0.470558
\(91\) 0 0
\(92\) −8.19615 −0.854508
\(93\) 5.46410 0.566601
\(94\) −6.92820 −0.714590
\(95\) −6.73205 −0.690694
\(96\) −2.73205 −0.278839
\(97\) −14.5885 −1.48123 −0.740617 0.671928i \(-0.765467\pi\)
−0.740617 + 0.671928i \(0.765467\pi\)
\(98\) 0 0
\(99\) 4.46410 0.448659
\(100\) 1.00000 0.100000
\(101\) 7.85641 0.781742 0.390871 0.920446i \(-0.372174\pi\)
0.390871 + 0.920446i \(0.372174\pi\)
\(102\) −9.46410 −0.937086
\(103\) −12.3923 −1.22105 −0.610525 0.791997i \(-0.709042\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(104\) 1.46410 0.143567
\(105\) 0 0
\(106\) −7.26795 −0.705926
\(107\) 7.85641 0.759507 0.379754 0.925088i \(-0.376009\pi\)
0.379754 + 0.925088i \(0.376009\pi\)
\(108\) −4.00000 −0.384900
\(109\) −15.6603 −1.49998 −0.749990 0.661449i \(-0.769942\pi\)
−0.749990 + 0.661449i \(0.769942\pi\)
\(110\) 1.00000 0.0953463
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) 18.3923 1.72260
\(115\) −8.19615 −0.764295
\(116\) −4.73205 −0.439360
\(117\) 6.53590 0.604244
\(118\) −6.92820 −0.637793
\(119\) 0 0
\(120\) −2.73205 −0.249401
\(121\) 1.00000 0.0909091
\(122\) 4.92820 0.446179
\(123\) −6.00000 −0.541002
\(124\) −2.00000 −0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.07180 0.0951066 0.0475533 0.998869i \(-0.484858\pi\)
0.0475533 + 0.998869i \(0.484858\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.46410 −0.481087
\(130\) 1.46410 0.128410
\(131\) −5.66025 −0.494539 −0.247269 0.968947i \(-0.579533\pi\)
−0.247269 + 0.968947i \(0.579533\pi\)
\(132\) −2.73205 −0.237795
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −4.00000 −0.344265
\(136\) 3.46410 0.297044
\(137\) −0.928203 −0.0793018 −0.0396509 0.999214i \(-0.512625\pi\)
−0.0396509 + 0.999214i \(0.512625\pi\)
\(138\) 22.3923 1.90616
\(139\) −13.6603 −1.15865 −0.579324 0.815097i \(-0.696683\pi\)
−0.579324 + 0.815097i \(0.696683\pi\)
\(140\) 0 0
\(141\) 18.9282 1.59404
\(142\) 9.46410 0.794210
\(143\) 1.46410 0.122434
\(144\) 4.46410 0.372008
\(145\) −4.73205 −0.392975
\(146\) 14.3923 1.19112
\(147\) 0 0
\(148\) 0.732051 0.0601742
\(149\) −7.26795 −0.595414 −0.297707 0.954657i \(-0.596222\pi\)
−0.297707 + 0.954657i \(0.596222\pi\)
\(150\) −2.73205 −0.223071
\(151\) 11.1244 0.905287 0.452644 0.891692i \(-0.350481\pi\)
0.452644 + 0.891692i \(0.350481\pi\)
\(152\) −6.73205 −0.546041
\(153\) 15.4641 1.25020
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) −4.00000 −0.320256
\(157\) −4.53590 −0.362004 −0.181002 0.983483i \(-0.557934\pi\)
−0.181002 + 0.983483i \(0.557934\pi\)
\(158\) −12.1962 −0.970274
\(159\) 19.8564 1.57472
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −2.46410 −0.193598
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.19615 0.171491
\(165\) −2.73205 −0.212690
\(166\) 16.3923 1.27229
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 3.46410 0.265684
\(171\) −30.0526 −2.29818
\(172\) 2.00000 0.152499
\(173\) −0.928203 −0.0705700 −0.0352850 0.999377i \(-0.511234\pi\)
−0.0352850 + 0.999377i \(0.511234\pi\)
\(174\) 12.9282 0.980085
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 18.9282 1.42273
\(178\) −3.46410 −0.259645
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 4.46410 0.332734
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −13.4641 −0.995295
\(184\) −8.19615 −0.604228
\(185\) 0.732051 0.0538214
\(186\) 5.46410 0.400647
\(187\) 3.46410 0.253320
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) −6.73205 −0.488394
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −2.73205 −0.197169
\(193\) 12.3923 0.892018 0.446009 0.895029i \(-0.352845\pi\)
0.446009 + 0.895029i \(0.352845\pi\)
\(194\) −14.5885 −1.04739
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −24.2487 −1.72765 −0.863825 0.503793i \(-0.831938\pi\)
−0.863825 + 0.503793i \(0.831938\pi\)
\(198\) 4.46410 0.317250
\(199\) −2.92820 −0.207575 −0.103787 0.994600i \(-0.533096\pi\)
−0.103787 + 0.994600i \(0.533096\pi\)
\(200\) 1.00000 0.0707107
\(201\) 10.9282 0.770816
\(202\) 7.85641 0.552775
\(203\) 0 0
\(204\) −9.46410 −0.662620
\(205\) 2.19615 0.153386
\(206\) −12.3923 −0.863413
\(207\) −36.5885 −2.54307
\(208\) 1.46410 0.101517
\(209\) −6.73205 −0.465666
\(210\) 0 0
\(211\) 26.9282 1.85381 0.926907 0.375291i \(-0.122457\pi\)
0.926907 + 0.375291i \(0.122457\pi\)
\(212\) −7.26795 −0.499165
\(213\) −25.8564 −1.77165
\(214\) 7.85641 0.537053
\(215\) 2.00000 0.136399
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) −15.6603 −1.06065
\(219\) −39.3205 −2.65703
\(220\) 1.00000 0.0674200
\(221\) 5.07180 0.341166
\(222\) −2.00000 −0.134231
\(223\) 25.4641 1.70520 0.852601 0.522562i \(-0.175024\pi\)
0.852601 + 0.522562i \(0.175024\pi\)
\(224\) 0 0
\(225\) 4.46410 0.297607
\(226\) 7.85641 0.522600
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) 18.3923 1.21806
\(229\) −24.3923 −1.61189 −0.805944 0.591991i \(-0.798342\pi\)
−0.805944 + 0.591991i \(0.798342\pi\)
\(230\) −8.19615 −0.540438
\(231\) 0 0
\(232\) −4.73205 −0.310674
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) 6.53590 0.427265
\(235\) −6.92820 −0.451946
\(236\) −6.92820 −0.450988
\(237\) 33.3205 2.16440
\(238\) 0 0
\(239\) 1.26795 0.0820168 0.0410084 0.999159i \(-0.486943\pi\)
0.0410084 + 0.999159i \(0.486943\pi\)
\(240\) −2.73205 −0.176353
\(241\) −3.26795 −0.210507 −0.105254 0.994445i \(-0.533565\pi\)
−0.105254 + 0.994445i \(0.533565\pi\)
\(242\) 1.00000 0.0642824
\(243\) 18.7321 1.20166
\(244\) 4.92820 0.315496
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −9.85641 −0.627148
\(248\) −2.00000 −0.127000
\(249\) −44.7846 −2.83811
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −8.19615 −0.515288
\(254\) 1.07180 0.0672505
\(255\) −9.46410 −0.592665
\(256\) 1.00000 0.0625000
\(257\) 23.6603 1.47589 0.737943 0.674863i \(-0.235797\pi\)
0.737943 + 0.674863i \(0.235797\pi\)
\(258\) −5.46410 −0.340180
\(259\) 0 0
\(260\) 1.46410 0.0907997
\(261\) −21.1244 −1.30756
\(262\) −5.66025 −0.349692
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −2.73205 −0.168146
\(265\) −7.26795 −0.446467
\(266\) 0 0
\(267\) 9.46410 0.579194
\(268\) −4.00000 −0.244339
\(269\) −28.3923 −1.73111 −0.865555 0.500814i \(-0.833034\pi\)
−0.865555 + 0.500814i \(0.833034\pi\)
\(270\) −4.00000 −0.243432
\(271\) −0.392305 −0.0238308 −0.0119154 0.999929i \(-0.503793\pi\)
−0.0119154 + 0.999929i \(0.503793\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) −0.928203 −0.0560748
\(275\) 1.00000 0.0603023
\(276\) 22.3923 1.34786
\(277\) 7.07180 0.424903 0.212452 0.977172i \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(278\) −13.6603 −0.819288
\(279\) −8.92820 −0.534518
\(280\) 0 0
\(281\) 22.3923 1.33581 0.667906 0.744245i \(-0.267191\pi\)
0.667906 + 0.744245i \(0.267191\pi\)
\(282\) 18.9282 1.12716
\(283\) 31.7128 1.88513 0.942566 0.334021i \(-0.108406\pi\)
0.942566 + 0.334021i \(0.108406\pi\)
\(284\) 9.46410 0.561591
\(285\) 18.3923 1.08947
\(286\) 1.46410 0.0865741
\(287\) 0 0
\(288\) 4.46410 0.263050
\(289\) −5.00000 −0.294118
\(290\) −4.73205 −0.277876
\(291\) 39.8564 2.33642
\(292\) 14.3923 0.842246
\(293\) −21.4641 −1.25395 −0.626973 0.779041i \(-0.715706\pi\)
−0.626973 + 0.779041i \(0.715706\pi\)
\(294\) 0 0
\(295\) −6.92820 −0.403376
\(296\) 0.732051 0.0425496
\(297\) −4.00000 −0.232104
\(298\) −7.26795 −0.421021
\(299\) −12.0000 −0.693978
\(300\) −2.73205 −0.157735
\(301\) 0 0
\(302\) 11.1244 0.640135
\(303\) −21.4641 −1.23308
\(304\) −6.73205 −0.386110
\(305\) 4.92820 0.282188
\(306\) 15.4641 0.884024
\(307\) −24.3923 −1.39214 −0.696071 0.717973i \(-0.745070\pi\)
−0.696071 + 0.717973i \(0.745070\pi\)
\(308\) 0 0
\(309\) 33.8564 1.92602
\(310\) −2.00000 −0.113592
\(311\) −24.9282 −1.41355 −0.706774 0.707439i \(-0.749850\pi\)
−0.706774 + 0.707439i \(0.749850\pi\)
\(312\) −4.00000 −0.226455
\(313\) −22.1962 −1.25460 −0.627300 0.778777i \(-0.715840\pi\)
−0.627300 + 0.778777i \(0.715840\pi\)
\(314\) −4.53590 −0.255976
\(315\) 0 0
\(316\) −12.1962 −0.686087
\(317\) 30.5885 1.71802 0.859009 0.511960i \(-0.171080\pi\)
0.859009 + 0.511960i \(0.171080\pi\)
\(318\) 19.8564 1.11349
\(319\) −4.73205 −0.264944
\(320\) 1.00000 0.0559017
\(321\) −21.4641 −1.19801
\(322\) 0 0
\(323\) −23.3205 −1.29759
\(324\) −2.46410 −0.136895
\(325\) 1.46410 0.0812137
\(326\) −4.00000 −0.221540
\(327\) 42.7846 2.36599
\(328\) 2.19615 0.121262
\(329\) 0 0
\(330\) −2.73205 −0.150394
\(331\) −18.7846 −1.03250 −0.516248 0.856439i \(-0.672672\pi\)
−0.516248 + 0.856439i \(0.672672\pi\)
\(332\) 16.3923 0.899645
\(333\) 3.26795 0.179083
\(334\) −13.8564 −0.758189
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 22.7846 1.24116 0.620578 0.784144i \(-0.286898\pi\)
0.620578 + 0.784144i \(0.286898\pi\)
\(338\) −10.8564 −0.590511
\(339\) −21.4641 −1.16577
\(340\) 3.46410 0.187867
\(341\) −2.00000 −0.108306
\(342\) −30.0526 −1.62506
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 22.3923 1.20556
\(346\) −0.928203 −0.0499005
\(347\) −23.0718 −1.23856 −0.619279 0.785171i \(-0.712575\pi\)
−0.619279 + 0.785171i \(0.712575\pi\)
\(348\) 12.9282 0.693024
\(349\) 5.60770 0.300173 0.150087 0.988673i \(-0.452045\pi\)
0.150087 + 0.988673i \(0.452045\pi\)
\(350\) 0 0
\(351\) −5.85641 −0.312592
\(352\) 1.00000 0.0533002
\(353\) −14.1962 −0.755585 −0.377792 0.925890i \(-0.623317\pi\)
−0.377792 + 0.925890i \(0.623317\pi\)
\(354\) 18.9282 1.00602
\(355\) 9.46410 0.502302
\(356\) −3.46410 −0.183597
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) −34.0526 −1.79723 −0.898613 0.438743i \(-0.855424\pi\)
−0.898613 + 0.438743i \(0.855424\pi\)
\(360\) 4.46410 0.235279
\(361\) 26.3205 1.38529
\(362\) −14.0000 −0.735824
\(363\) −2.73205 −0.143395
\(364\) 0 0
\(365\) 14.3923 0.753328
\(366\) −13.4641 −0.703780
\(367\) −12.3923 −0.646873 −0.323437 0.946250i \(-0.604838\pi\)
−0.323437 + 0.946250i \(0.604838\pi\)
\(368\) −8.19615 −0.427254
\(369\) 9.80385 0.510368
\(370\) 0.732051 0.0380575
\(371\) 0 0
\(372\) 5.46410 0.283300
\(373\) 18.3923 0.952317 0.476159 0.879359i \(-0.342029\pi\)
0.476159 + 0.879359i \(0.342029\pi\)
\(374\) 3.46410 0.179124
\(375\) −2.73205 −0.141082
\(376\) −6.92820 −0.357295
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) 6.14359 0.315575 0.157788 0.987473i \(-0.449564\pi\)
0.157788 + 0.987473i \(0.449564\pi\)
\(380\) −6.73205 −0.345347
\(381\) −2.92820 −0.150016
\(382\) −12.0000 −0.613973
\(383\) 33.4641 1.70994 0.854968 0.518681i \(-0.173577\pi\)
0.854968 + 0.518681i \(0.173577\pi\)
\(384\) −2.73205 −0.139419
\(385\) 0 0
\(386\) 12.3923 0.630752
\(387\) 8.92820 0.453846
\(388\) −14.5885 −0.740617
\(389\) −1.60770 −0.0815134 −0.0407567 0.999169i \(-0.512977\pi\)
−0.0407567 + 0.999169i \(0.512977\pi\)
\(390\) −4.00000 −0.202548
\(391\) −28.3923 −1.43586
\(392\) 0 0
\(393\) 15.4641 0.780061
\(394\) −24.2487 −1.22163
\(395\) −12.1962 −0.613655
\(396\) 4.46410 0.224330
\(397\) −30.3923 −1.52535 −0.762673 0.646784i \(-0.776113\pi\)
−0.762673 + 0.646784i \(0.776113\pi\)
\(398\) −2.92820 −0.146778
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.53590 0.126637 0.0633184 0.997993i \(-0.479832\pi\)
0.0633184 + 0.997993i \(0.479832\pi\)
\(402\) 10.9282 0.545049
\(403\) −2.92820 −0.145864
\(404\) 7.85641 0.390871
\(405\) −2.46410 −0.122442
\(406\) 0 0
\(407\) 0.732051 0.0362864
\(408\) −9.46410 −0.468543
\(409\) −10.8756 −0.537766 −0.268883 0.963173i \(-0.586654\pi\)
−0.268883 + 0.963173i \(0.586654\pi\)
\(410\) 2.19615 0.108460
\(411\) 2.53590 0.125087
\(412\) −12.3923 −0.610525
\(413\) 0 0
\(414\) −36.5885 −1.79822
\(415\) 16.3923 0.804667
\(416\) 1.46410 0.0717835
\(417\) 37.3205 1.82759
\(418\) −6.73205 −0.329275
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) −35.8564 −1.74753 −0.873767 0.486344i \(-0.838330\pi\)
−0.873767 + 0.486344i \(0.838330\pi\)
\(422\) 26.9282 1.31084
\(423\) −30.9282 −1.50378
\(424\) −7.26795 −0.352963
\(425\) 3.46410 0.168034
\(426\) −25.8564 −1.25275
\(427\) 0 0
\(428\) 7.85641 0.379754
\(429\) −4.00000 −0.193122
\(430\) 2.00000 0.0964486
\(431\) 3.12436 0.150495 0.0752475 0.997165i \(-0.476025\pi\)
0.0752475 + 0.997165i \(0.476025\pi\)
\(432\) −4.00000 −0.192450
\(433\) 18.1962 0.874451 0.437226 0.899352i \(-0.355961\pi\)
0.437226 + 0.899352i \(0.355961\pi\)
\(434\) 0 0
\(435\) 12.9282 0.619860
\(436\) −15.6603 −0.749990
\(437\) 55.1769 2.63947
\(438\) −39.3205 −1.87881
\(439\) −26.2487 −1.25278 −0.626391 0.779509i \(-0.715469\pi\)
−0.626391 + 0.779509i \(0.715469\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 5.07180 0.241241
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −3.46410 −0.164214
\(446\) 25.4641 1.20576
\(447\) 19.8564 0.939176
\(448\) 0 0
\(449\) 40.3923 1.90623 0.953115 0.302607i \(-0.0978570\pi\)
0.953115 + 0.302607i \(0.0978570\pi\)
\(450\) 4.46410 0.210440
\(451\) 2.19615 0.103413
\(452\) 7.85641 0.369534
\(453\) −30.3923 −1.42796
\(454\) −6.92820 −0.325157
\(455\) 0 0
\(456\) 18.3923 0.861299
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −24.3923 −1.13978
\(459\) −13.8564 −0.646762
\(460\) −8.19615 −0.382148
\(461\) −22.3923 −1.04291 −0.521457 0.853278i \(-0.674611\pi\)
−0.521457 + 0.853278i \(0.674611\pi\)
\(462\) 0 0
\(463\) 27.5167 1.27881 0.639404 0.768871i \(-0.279181\pi\)
0.639404 + 0.768871i \(0.279181\pi\)
\(464\) −4.73205 −0.219680
\(465\) 5.46410 0.253392
\(466\) −7.85641 −0.363941
\(467\) 34.0526 1.57576 0.787882 0.615826i \(-0.211178\pi\)
0.787882 + 0.615826i \(0.211178\pi\)
\(468\) 6.53590 0.302122
\(469\) 0 0
\(470\) −6.92820 −0.319574
\(471\) 12.3923 0.571007
\(472\) −6.92820 −0.318896
\(473\) 2.00000 0.0919601
\(474\) 33.3205 1.53046
\(475\) −6.73205 −0.308888
\(476\) 0 0
\(477\) −32.4449 −1.48555
\(478\) 1.26795 0.0579946
\(479\) 32.7846 1.49797 0.748984 0.662589i \(-0.230542\pi\)
0.748984 + 0.662589i \(0.230542\pi\)
\(480\) −2.73205 −0.124700
\(481\) 1.07180 0.0488697
\(482\) −3.26795 −0.148851
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −14.5885 −0.662428
\(486\) 18.7321 0.849703
\(487\) 4.19615 0.190146 0.0950729 0.995470i \(-0.469692\pi\)
0.0950729 + 0.995470i \(0.469692\pi\)
\(488\) 4.92820 0.223089
\(489\) 10.9282 0.494190
\(490\) 0 0
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) −6.00000 −0.270501
\(493\) −16.3923 −0.738272
\(494\) −9.85641 −0.443461
\(495\) 4.46410 0.200646
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −44.7846 −2.00685
\(499\) 12.1436 0.543622 0.271811 0.962351i \(-0.412377\pi\)
0.271811 + 0.962351i \(0.412377\pi\)
\(500\) 1.00000 0.0447214
\(501\) 37.8564 1.69130
\(502\) −12.0000 −0.535586
\(503\) −8.78461 −0.391686 −0.195843 0.980635i \(-0.562744\pi\)
−0.195843 + 0.980635i \(0.562744\pi\)
\(504\) 0 0
\(505\) 7.85641 0.349605
\(506\) −8.19615 −0.364363
\(507\) 29.6603 1.31726
\(508\) 1.07180 0.0475533
\(509\) −11.0718 −0.490749 −0.245374 0.969428i \(-0.578911\pi\)
−0.245374 + 0.969428i \(0.578911\pi\)
\(510\) −9.46410 −0.419077
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 26.9282 1.18891
\(514\) 23.6603 1.04361
\(515\) −12.3923 −0.546070
\(516\) −5.46410 −0.240544
\(517\) −6.92820 −0.304702
\(518\) 0 0
\(519\) 2.53590 0.111314
\(520\) 1.46410 0.0642051
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) −21.1244 −0.924588
\(523\) 29.1769 1.27582 0.637909 0.770112i \(-0.279800\pi\)
0.637909 + 0.770112i \(0.279800\pi\)
\(524\) −5.66025 −0.247269
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −6.92820 −0.301797
\(528\) −2.73205 −0.118897
\(529\) 44.1769 1.92074
\(530\) −7.26795 −0.315700
\(531\) −30.9282 −1.34217
\(532\) 0 0
\(533\) 3.21539 0.139274
\(534\) 9.46410 0.409552
\(535\) 7.85641 0.339662
\(536\) −4.00000 −0.172774
\(537\) 16.3923 0.707380
\(538\) −28.3923 −1.22408
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) −20.7321 −0.891340 −0.445670 0.895197i \(-0.647035\pi\)
−0.445670 + 0.895197i \(0.647035\pi\)
\(542\) −0.392305 −0.0168509
\(543\) 38.2487 1.64141
\(544\) 3.46410 0.148522
\(545\) −15.6603 −0.670812
\(546\) 0 0
\(547\) 28.7846 1.23074 0.615371 0.788238i \(-0.289006\pi\)
0.615371 + 0.788238i \(0.289006\pi\)
\(548\) −0.928203 −0.0396509
\(549\) 22.0000 0.938937
\(550\) 1.00000 0.0426401
\(551\) 31.8564 1.35713
\(552\) 22.3923 0.953080
\(553\) 0 0
\(554\) 7.07180 0.300452
\(555\) −2.00000 −0.0848953
\(556\) −13.6603 −0.579324
\(557\) −25.6077 −1.08503 −0.542516 0.840045i \(-0.682528\pi\)
−0.542516 + 0.840045i \(0.682528\pi\)
\(558\) −8.92820 −0.377961
\(559\) 2.92820 0.123850
\(560\) 0 0
\(561\) −9.46410 −0.399575
\(562\) 22.3923 0.944562
\(563\) −5.07180 −0.213751 −0.106875 0.994272i \(-0.534085\pi\)
−0.106875 + 0.994272i \(0.534085\pi\)
\(564\) 18.9282 0.797021
\(565\) 7.85641 0.330522
\(566\) 31.7128 1.33299
\(567\) 0 0
\(568\) 9.46410 0.397105
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 18.3923 0.770369
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) 1.46410 0.0612172
\(573\) 32.7846 1.36960
\(574\) 0 0
\(575\) −8.19615 −0.341803
\(576\) 4.46410 0.186004
\(577\) 34.5885 1.43994 0.719968 0.694007i \(-0.244156\pi\)
0.719968 + 0.694007i \(0.244156\pi\)
\(578\) −5.00000 −0.207973
\(579\) −33.8564 −1.40702
\(580\) −4.73205 −0.196488
\(581\) 0 0
\(582\) 39.8564 1.65210
\(583\) −7.26795 −0.301008
\(584\) 14.3923 0.595558
\(585\) 6.53590 0.270226
\(586\) −21.4641 −0.886674
\(587\) 11.4115 0.471005 0.235502 0.971874i \(-0.424326\pi\)
0.235502 + 0.971874i \(0.424326\pi\)
\(588\) 0 0
\(589\) 13.4641 0.554779
\(590\) −6.92820 −0.285230
\(591\) 66.2487 2.72511
\(592\) 0.732051 0.0300871
\(593\) −0.248711 −0.0102133 −0.00510667 0.999987i \(-0.501626\pi\)
−0.00510667 + 0.999987i \(0.501626\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −7.26795 −0.297707
\(597\) 8.00000 0.327418
\(598\) −12.0000 −0.490716
\(599\) 37.1769 1.51901 0.759504 0.650503i \(-0.225442\pi\)
0.759504 + 0.650503i \(0.225442\pi\)
\(600\) −2.73205 −0.111536
\(601\) 5.51666 0.225029 0.112515 0.993650i \(-0.464109\pi\)
0.112515 + 0.993650i \(0.464109\pi\)
\(602\) 0 0
\(603\) −17.8564 −0.727169
\(604\) 11.1244 0.452644
\(605\) 1.00000 0.0406558
\(606\) −21.4641 −0.871920
\(607\) −20.9282 −0.849450 −0.424725 0.905323i \(-0.639629\pi\)
−0.424725 + 0.905323i \(0.639629\pi\)
\(608\) −6.73205 −0.273021
\(609\) 0 0
\(610\) 4.92820 0.199537
\(611\) −10.1436 −0.410366
\(612\) 15.4641 0.625099
\(613\) −35.1769 −1.42078 −0.710391 0.703807i \(-0.751482\pi\)
−0.710391 + 0.703807i \(0.751482\pi\)
\(614\) −24.3923 −0.984393
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 17.3205 0.697297 0.348649 0.937253i \(-0.386641\pi\)
0.348649 + 0.937253i \(0.386641\pi\)
\(618\) 33.8564 1.36190
\(619\) −28.7846 −1.15695 −0.578476 0.815700i \(-0.696352\pi\)
−0.578476 + 0.815700i \(0.696352\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 32.7846 1.31560
\(622\) −24.9282 −0.999530
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −22.1962 −0.887137
\(627\) 18.3923 0.734518
\(628\) −4.53590 −0.181002
\(629\) 2.53590 0.101113
\(630\) 0 0
\(631\) −46.9282 −1.86818 −0.934091 0.357035i \(-0.883788\pi\)
−0.934091 + 0.357035i \(0.883788\pi\)
\(632\) −12.1962 −0.485137
\(633\) −73.5692 −2.92411
\(634\) 30.5885 1.21482
\(635\) 1.07180 0.0425330
\(636\) 19.8564 0.787358
\(637\) 0 0
\(638\) −4.73205 −0.187344
\(639\) 42.2487 1.67133
\(640\) 1.00000 0.0395285
\(641\) −35.5692 −1.40490 −0.702450 0.711733i \(-0.747910\pi\)
−0.702450 + 0.711733i \(0.747910\pi\)
\(642\) −21.4641 −0.847121
\(643\) −33.2679 −1.31196 −0.655980 0.754778i \(-0.727744\pi\)
−0.655980 + 0.754778i \(0.727744\pi\)
\(644\) 0 0
\(645\) −5.46410 −0.215149
\(646\) −23.3205 −0.917533
\(647\) 26.5359 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(648\) −2.46410 −0.0967991
\(649\) −6.92820 −0.271956
\(650\) 1.46410 0.0574268
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 11.6603 0.456301 0.228151 0.973626i \(-0.426732\pi\)
0.228151 + 0.973626i \(0.426732\pi\)
\(654\) 42.7846 1.67301
\(655\) −5.66025 −0.221164
\(656\) 2.19615 0.0857453
\(657\) 64.2487 2.50658
\(658\) 0 0
\(659\) 30.2487 1.17832 0.589161 0.808015i \(-0.299458\pi\)
0.589161 + 0.808015i \(0.299458\pi\)
\(660\) −2.73205 −0.106345
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −18.7846 −0.730085
\(663\) −13.8564 −0.538138
\(664\) 16.3923 0.636145
\(665\) 0 0
\(666\) 3.26795 0.126630
\(667\) 38.7846 1.50175
\(668\) −13.8564 −0.536120
\(669\) −69.5692 −2.68970
\(670\) −4.00000 −0.154533
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 22.7846 0.877630
\(675\) −4.00000 −0.153960
\(676\) −10.8564 −0.417554
\(677\) 4.14359 0.159251 0.0796256 0.996825i \(-0.474628\pi\)
0.0796256 + 0.996825i \(0.474628\pi\)
\(678\) −21.4641 −0.824324
\(679\) 0 0
\(680\) 3.46410 0.132842
\(681\) 18.9282 0.725330
\(682\) −2.00000 −0.0765840
\(683\) 6.24871 0.239100 0.119550 0.992828i \(-0.461855\pi\)
0.119550 + 0.992828i \(0.461855\pi\)
\(684\) −30.0526 −1.14909
\(685\) −0.928203 −0.0354648
\(686\) 0 0
\(687\) 66.6410 2.54251
\(688\) 2.00000 0.0762493
\(689\) −10.6410 −0.405390
\(690\) 22.3923 0.852460
\(691\) 13.4641 0.512199 0.256099 0.966650i \(-0.417563\pi\)
0.256099 + 0.966650i \(0.417563\pi\)
\(692\) −0.928203 −0.0352850
\(693\) 0 0
\(694\) −23.0718 −0.875793
\(695\) −13.6603 −0.518163
\(696\) 12.9282 0.490042
\(697\) 7.60770 0.288162
\(698\) 5.60770 0.212254
\(699\) 21.4641 0.811847
\(700\) 0 0
\(701\) 41.9090 1.58288 0.791440 0.611247i \(-0.209332\pi\)
0.791440 + 0.611247i \(0.209332\pi\)
\(702\) −5.85641 −0.221036
\(703\) −4.92820 −0.185871
\(704\) 1.00000 0.0376889
\(705\) 18.9282 0.712877
\(706\) −14.1962 −0.534279
\(707\) 0 0
\(708\) 18.9282 0.711365
\(709\) 25.3205 0.950932 0.475466 0.879734i \(-0.342280\pi\)
0.475466 + 0.879734i \(0.342280\pi\)
\(710\) 9.46410 0.355181
\(711\) −54.4449 −2.04184
\(712\) −3.46410 −0.129823
\(713\) 16.3923 0.613897
\(714\) 0 0
\(715\) 1.46410 0.0547543
\(716\) −6.00000 −0.224231
\(717\) −3.46410 −0.129369
\(718\) −34.0526 −1.27083
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 4.46410 0.166367
\(721\) 0 0
\(722\) 26.3205 0.979548
\(723\) 8.92820 0.332043
\(724\) −14.0000 −0.520306
\(725\) −4.73205 −0.175744
\(726\) −2.73205 −0.101396
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 14.3923 0.532683
\(731\) 6.92820 0.256249
\(732\) −13.4641 −0.497648
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −12.3923 −0.457408
\(735\) 0 0
\(736\) −8.19615 −0.302114
\(737\) −4.00000 −0.147342
\(738\) 9.80385 0.360885
\(739\) 11.7128 0.430863 0.215431 0.976519i \(-0.430884\pi\)
0.215431 + 0.976519i \(0.430884\pi\)
\(740\) 0.732051 0.0269107
\(741\) 26.9282 0.989232
\(742\) 0 0
\(743\) −32.7846 −1.20275 −0.601375 0.798967i \(-0.705380\pi\)
−0.601375 + 0.798967i \(0.705380\pi\)
\(744\) 5.46410 0.200324
\(745\) −7.26795 −0.266277
\(746\) 18.3923 0.673390
\(747\) 73.1769 2.67740
\(748\) 3.46410 0.126660
\(749\) 0 0
\(750\) −2.73205 −0.0997604
\(751\) −11.6077 −0.423571 −0.211785 0.977316i \(-0.567928\pi\)
−0.211785 + 0.977316i \(0.567928\pi\)
\(752\) −6.92820 −0.252646
\(753\) 32.7846 1.19474
\(754\) −6.92820 −0.252310
\(755\) 11.1244 0.404857
\(756\) 0 0
\(757\) −43.3731 −1.57642 −0.788210 0.615406i \(-0.788992\pi\)
−0.788210 + 0.615406i \(0.788992\pi\)
\(758\) 6.14359 0.223145
\(759\) 22.3923 0.812789
\(760\) −6.73205 −0.244197
\(761\) 12.3397 0.447315 0.223658 0.974668i \(-0.428200\pi\)
0.223658 + 0.974668i \(0.428200\pi\)
\(762\) −2.92820 −0.106078
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 15.4641 0.559106
\(766\) 33.4641 1.20911
\(767\) −10.1436 −0.366264
\(768\) −2.73205 −0.0985844
\(769\) 18.1962 0.656170 0.328085 0.944648i \(-0.393597\pi\)
0.328085 + 0.944648i \(0.393597\pi\)
\(770\) 0 0
\(771\) −64.6410 −2.32799
\(772\) 12.3923 0.446009
\(773\) 0.928203 0.0333851 0.0166926 0.999861i \(-0.494686\pi\)
0.0166926 + 0.999861i \(0.494686\pi\)
\(774\) 8.92820 0.320918
\(775\) −2.00000 −0.0718421
\(776\) −14.5885 −0.523695
\(777\) 0 0
\(778\) −1.60770 −0.0576387
\(779\) −14.7846 −0.529714
\(780\) −4.00000 −0.143223
\(781\) 9.46410 0.338652
\(782\) −28.3923 −1.01531
\(783\) 18.9282 0.676439
\(784\) 0 0
\(785\) −4.53590 −0.161893
\(786\) 15.4641 0.551586
\(787\) −18.1436 −0.646749 −0.323375 0.946271i \(-0.604817\pi\)
−0.323375 + 0.946271i \(0.604817\pi\)
\(788\) −24.2487 −0.863825
\(789\) 65.5692 2.33433
\(790\) −12.1962 −0.433920
\(791\) 0 0
\(792\) 4.46410 0.158625
\(793\) 7.21539 0.256226
\(794\) −30.3923 −1.07858
\(795\) 19.8564 0.704234
\(796\) −2.92820 −0.103787
\(797\) 25.6077 0.907071 0.453536 0.891238i \(-0.350163\pi\)
0.453536 + 0.891238i \(0.350163\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) −15.4641 −0.546397
\(802\) 2.53590 0.0895457
\(803\) 14.3923 0.507893
\(804\) 10.9282 0.385408
\(805\) 0 0
\(806\) −2.92820 −0.103142
\(807\) 77.5692 2.73057
\(808\) 7.85641 0.276387
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) −2.46410 −0.0865797
\(811\) 43.1244 1.51430 0.757150 0.653241i \(-0.226591\pi\)
0.757150 + 0.653241i \(0.226591\pi\)
\(812\) 0 0
\(813\) 1.07180 0.0375896
\(814\) 0.732051 0.0256584
\(815\) −4.00000 −0.140114
\(816\) −9.46410 −0.331310
\(817\) −13.4641 −0.471049
\(818\) −10.8756 −0.380258
\(819\) 0 0
\(820\) 2.19615 0.0766930
\(821\) 17.9090 0.625027 0.312514 0.949913i \(-0.398829\pi\)
0.312514 + 0.949913i \(0.398829\pi\)
\(822\) 2.53590 0.0884496
\(823\) −0.875644 −0.0305230 −0.0152615 0.999884i \(-0.504858\pi\)
−0.0152615 + 0.999884i \(0.504858\pi\)
\(824\) −12.3923 −0.431706
\(825\) −2.73205 −0.0951178
\(826\) 0 0
\(827\) −25.8564 −0.899115 −0.449558 0.893251i \(-0.648418\pi\)
−0.449558 + 0.893251i \(0.648418\pi\)
\(828\) −36.5885 −1.27154
\(829\) −2.24871 −0.0781010 −0.0390505 0.999237i \(-0.512433\pi\)
−0.0390505 + 0.999237i \(0.512433\pi\)
\(830\) 16.3923 0.568985
\(831\) −19.3205 −0.670221
\(832\) 1.46410 0.0507586
\(833\) 0 0
\(834\) 37.3205 1.29230
\(835\) −13.8564 −0.479521
\(836\) −6.73205 −0.232833
\(837\) 8.00000 0.276520
\(838\) −30.9282 −1.06840
\(839\) 31.8564 1.09981 0.549903 0.835229i \(-0.314665\pi\)
0.549903 + 0.835229i \(0.314665\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) −35.8564 −1.23569
\(843\) −61.1769 −2.10704
\(844\) 26.9282 0.926907
\(845\) −10.8564 −0.373472
\(846\) −30.9282 −1.06333
\(847\) 0 0
\(848\) −7.26795 −0.249582
\(849\) −86.6410 −2.97351
\(850\) 3.46410 0.118818
\(851\) −6.00000 −0.205677
\(852\) −25.8564 −0.885826
\(853\) 54.7846 1.87579 0.937895 0.346920i \(-0.112773\pi\)
0.937895 + 0.346920i \(0.112773\pi\)
\(854\) 0 0
\(855\) −30.0526 −1.02778
\(856\) 7.85641 0.268526
\(857\) 27.4641 0.938156 0.469078 0.883157i \(-0.344586\pi\)
0.469078 + 0.883157i \(0.344586\pi\)
\(858\) −4.00000 −0.136558
\(859\) 12.7846 0.436205 0.218103 0.975926i \(-0.430013\pi\)
0.218103 + 0.975926i \(0.430013\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 3.12436 0.106416
\(863\) −7.51666 −0.255870 −0.127935 0.991783i \(-0.540835\pi\)
−0.127935 + 0.991783i \(0.540835\pi\)
\(864\) −4.00000 −0.136083
\(865\) −0.928203 −0.0315599
\(866\) 18.1962 0.618330
\(867\) 13.6603 0.463927
\(868\) 0 0
\(869\) −12.1962 −0.413726
\(870\) 12.9282 0.438307
\(871\) −5.85641 −0.198437
\(872\) −15.6603 −0.530323
\(873\) −65.1244 −2.20413
\(874\) 55.1769 1.86639
\(875\) 0 0
\(876\) −39.3205 −1.32852
\(877\) −21.3205 −0.719942 −0.359971 0.932963i \(-0.617213\pi\)
−0.359971 + 0.932963i \(0.617213\pi\)
\(878\) −26.2487 −0.885851
\(879\) 58.6410 1.97791
\(880\) 1.00000 0.0337100
\(881\) −11.0718 −0.373018 −0.186509 0.982453i \(-0.559717\pi\)
−0.186509 + 0.982453i \(0.559717\pi\)
\(882\) 0 0
\(883\) −35.6077 −1.19829 −0.599147 0.800639i \(-0.704494\pi\)
−0.599147 + 0.800639i \(0.704494\pi\)
\(884\) 5.07180 0.170583
\(885\) 18.9282 0.636265
\(886\) 0 0
\(887\) −13.8564 −0.465253 −0.232626 0.972566i \(-0.574732\pi\)
−0.232626 + 0.972566i \(0.574732\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) −3.46410 −0.116117
\(891\) −2.46410 −0.0825505
\(892\) 25.4641 0.852601
\(893\) 46.6410 1.56078
\(894\) 19.8564 0.664098
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) 32.7846 1.09465
\(898\) 40.3923 1.34791
\(899\) 9.46410 0.315645
\(900\) 4.46410 0.148803
\(901\) −25.1769 −0.838765
\(902\) 2.19615 0.0731239
\(903\) 0 0
\(904\) 7.85641 0.261300
\(905\) −14.0000 −0.465376
\(906\) −30.3923 −1.00972
\(907\) −31.0333 −1.03044 −0.515222 0.857057i \(-0.672291\pi\)
−0.515222 + 0.857057i \(0.672291\pi\)
\(908\) −6.92820 −0.229920
\(909\) 35.0718 1.16326
\(910\) 0 0
\(911\) 9.46410 0.313560 0.156780 0.987634i \(-0.449889\pi\)
0.156780 + 0.987634i \(0.449889\pi\)
\(912\) 18.3923 0.609030
\(913\) 16.3923 0.542506
\(914\) 2.00000 0.0661541
\(915\) −13.4641 −0.445109
\(916\) −24.3923 −0.805944
\(917\) 0 0
\(918\) −13.8564 −0.457330
\(919\) −25.3731 −0.836980 −0.418490 0.908221i \(-0.637441\pi\)
−0.418490 + 0.908221i \(0.637441\pi\)
\(920\) −8.19615 −0.270219
\(921\) 66.6410 2.19590
\(922\) −22.3923 −0.737451
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 0.732051 0.0240697
\(926\) 27.5167 0.904254
\(927\) −55.3205 −1.81696
\(928\) −4.73205 −0.155337
\(929\) 25.6077 0.840161 0.420081 0.907487i \(-0.362002\pi\)
0.420081 + 0.907487i \(0.362002\pi\)
\(930\) 5.46410 0.179175
\(931\) 0 0
\(932\) −7.85641 −0.257345
\(933\) 68.1051 2.22966
\(934\) 34.0526 1.11423
\(935\) 3.46410 0.113288
\(936\) 6.53590 0.213633
\(937\) 1.21539 0.0397051 0.0198525 0.999803i \(-0.493680\pi\)
0.0198525 + 0.999803i \(0.493680\pi\)
\(938\) 0 0
\(939\) 60.6410 1.97894
\(940\) −6.92820 −0.225973
\(941\) 14.7846 0.481965 0.240982 0.970530i \(-0.422530\pi\)
0.240982 + 0.970530i \(0.422530\pi\)
\(942\) 12.3923 0.403763
\(943\) −18.0000 −0.586161
\(944\) −6.92820 −0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) 47.3205 1.53771 0.768855 0.639423i \(-0.220827\pi\)
0.768855 + 0.639423i \(0.220827\pi\)
\(948\) 33.3205 1.08220
\(949\) 21.0718 0.684019
\(950\) −6.73205 −0.218417
\(951\) −83.5692 −2.70992
\(952\) 0 0
\(953\) 26.5359 0.859582 0.429791 0.902928i \(-0.358587\pi\)
0.429791 + 0.902928i \(0.358587\pi\)
\(954\) −32.4449 −1.05044
\(955\) −12.0000 −0.388311
\(956\) 1.26795 0.0410084
\(957\) 12.9282 0.417909
\(958\) 32.7846 1.05922
\(959\) 0 0
\(960\) −2.73205 −0.0881766
\(961\) −27.0000 −0.870968
\(962\) 1.07180 0.0345561
\(963\) 35.0718 1.13017
\(964\) −3.26795 −0.105254
\(965\) 12.3923 0.398922
\(966\) 0 0
\(967\) 26.9282 0.865953 0.432976 0.901405i \(-0.357463\pi\)
0.432976 + 0.901405i \(0.357463\pi\)
\(968\) 1.00000 0.0321412
\(969\) 63.7128 2.04675
\(970\) −14.5885 −0.468407
\(971\) −42.9282 −1.37763 −0.688816 0.724936i \(-0.741869\pi\)
−0.688816 + 0.724936i \(0.741869\pi\)
\(972\) 18.7321 0.600831
\(973\) 0 0
\(974\) 4.19615 0.134453
\(975\) −4.00000 −0.128103
\(976\) 4.92820 0.157748
\(977\) −33.7128 −1.07857 −0.539284 0.842124i \(-0.681305\pi\)
−0.539284 + 0.842124i \(0.681305\pi\)
\(978\) 10.9282 0.349445
\(979\) −3.46410 −0.110713
\(980\) 0 0
\(981\) −69.9090 −2.23202
\(982\) −27.7128 −0.884351
\(983\) 37.1769 1.18576 0.592880 0.805291i \(-0.297991\pi\)
0.592880 + 0.805291i \(0.297991\pi\)
\(984\) −6.00000 −0.191273
\(985\) −24.2487 −0.772628
\(986\) −16.3923 −0.522037
\(987\) 0 0
\(988\) −9.85641 −0.313574
\(989\) −16.3923 −0.521245
\(990\) 4.46410 0.141878
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 51.3205 1.62861
\(994\) 0 0
\(995\) −2.92820 −0.0928303
\(996\) −44.7846 −1.41905
\(997\) −17.7128 −0.560970 −0.280485 0.959858i \(-0.590495\pi\)
−0.280485 + 0.959858i \(0.590495\pi\)
\(998\) 12.1436 0.384399
\(999\) −2.92820 −0.0926443
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 5390.2.a.bs.1.1 2
7.6 odd 2 770.2.a.j.1.2 2
21.20 even 2 6930.2.a.bv.1.1 2
28.27 even 2 6160.2.a.t.1.1 2
35.13 even 4 3850.2.c.x.1849.2 4
35.27 even 4 3850.2.c.x.1849.3 4
35.34 odd 2 3850.2.a.bd.1.1 2
77.76 even 2 8470.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 7.6 odd 2
3850.2.a.bd.1.1 2 35.34 odd 2
3850.2.c.x.1849.2 4 35.13 even 4
3850.2.c.x.1849.3 4 35.27 even 4
5390.2.a.bs.1.1 2 1.1 even 1 trivial
6160.2.a.t.1.1 2 28.27 even 2
6930.2.a.bv.1.1 2 21.20 even 2
8470.2.a.br.1.2 2 77.76 even 2