Properties

Label 8470.2.a.br.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
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 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +2.73205 q^{12} +1.46410 q^{13} +1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} +3.46410 q^{17} -4.46410 q^{18} -6.73205 q^{19} -1.00000 q^{20} -2.73205 q^{21} -8.19615 q^{23} -2.73205 q^{24} +1.00000 q^{25} -1.46410 q^{26} +4.00000 q^{27} -1.00000 q^{28} +4.73205 q^{29} +2.73205 q^{30} +2.00000 q^{31} -1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} +4.46410 q^{36} +0.732051 q^{37} +6.73205 q^{38} +4.00000 q^{39} +1.00000 q^{40} +2.19615 q^{41} +2.73205 q^{42} -2.00000 q^{43} -4.46410 q^{45} +8.19615 q^{46} +6.92820 q^{47} +2.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} +9.46410 q^{51} +1.46410 q^{52} -7.26795 q^{53} -4.00000 q^{54} +1.00000 q^{56} -18.3923 q^{57} -4.73205 q^{58} +6.92820 q^{59} -2.73205 q^{60} +4.92820 q^{61} -2.00000 q^{62} -4.46410 q^{63} +1.00000 q^{64} -1.46410 q^{65} -4.00000 q^{67} +3.46410 q^{68} -22.3923 q^{69} -1.00000 q^{70} +9.46410 q^{71} -4.46410 q^{72} +14.3923 q^{73} -0.732051 q^{74} +2.73205 q^{75} -6.73205 q^{76} -4.00000 q^{78} +12.1962 q^{79} -1.00000 q^{80} -2.46410 q^{81} -2.19615 q^{82} +16.3923 q^{83} -2.73205 q^{84} -3.46410 q^{85} +2.00000 q^{86} +12.9282 q^{87} +3.46410 q^{89} +4.46410 q^{90} -1.46410 q^{91} -8.19615 q^{92} +5.46410 q^{93} -6.92820 q^{94} +6.73205 q^{95} -2.73205 q^{96} +14.5885 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} - 4q^{13} + 2q^{14} - 2q^{15} + 2q^{16} - 2q^{18} - 10q^{19} - 2q^{20} - 2q^{21} - 6q^{23} - 2q^{24} + 2q^{25} + 4q^{26} + 8q^{27} - 2q^{28} + 6q^{29} + 2q^{30} + 4q^{31} - 2q^{32} + 2q^{35} + 2q^{36} - 2q^{37} + 10q^{38} + 8q^{39} + 2q^{40} - 6q^{41} + 2q^{42} - 4q^{43} - 2q^{45} + 6q^{46} + 2q^{48} + 2q^{49} - 2q^{50} + 12q^{51} - 4q^{52} - 18q^{53} - 8q^{54} + 2q^{56} - 16q^{57} - 6q^{58} - 2q^{60} - 4q^{61} - 4q^{62} - 2q^{63} + 2q^{64} + 4q^{65} - 8q^{67} - 24q^{69} - 2q^{70} + 12q^{71} - 2q^{72} + 8q^{73} + 2q^{74} + 2q^{75} - 10q^{76} - 8q^{78} + 14q^{79} - 2q^{80} + 2q^{81} + 6q^{82} + 12q^{83} - 2q^{84} + 4q^{86} + 12q^{87} + 2q^{90} + 4q^{91} - 6q^{92} + 4q^{93} + 10q^{95} - 2q^{96} - 2q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.73205 −1.11536
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.46410 1.48803
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 2.73205 0.788675
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 1.00000 0.267261
\(15\) −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
\(21\) −2.73205 −0.596182
\(22\) 0 0
\(23\) −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
\(28\) −1.00000 −0.188982
\(29\) 4.73205 0.878720 0.439360 0.898311i \(-0.355205\pi\)
0.439360 + 0.898311i \(0.355205\pi\)
\(30\) 2.73205 0.498802
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) 1.00000 0.169031
\(36\) 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 \(0.445142\pi\)
\(42\) 2.73205 0.421565
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −4.46410 −0.665469
\(46\) 8.19615 1.20846
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 2.73205 0.394338
\(49\) 1.00000 0.142857
\(50\) −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
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −18.3923 −2.43612
\(58\) −4.73205 −0.621349
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) −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
\(63\) −4.46410 −0.562424
\(64\) 1.00000 0.125000
\(65\) −1.46410 −0.181599
\(66\) 0 0
\(67\) −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
\(70\) −1.00000 −0.119523
\(71\) 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 \(-0.259327\pi\)
0.686087 + 0.727519i \(0.259327\pi\)
\(80\) −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 \(0.143826\pi\)
\(84\) −2.73205 −0.298091
\(85\) −3.46410 −0.375735
\(86\) 2.00000 0.215666
\(87\) 12.9282 1.38605
\(88\) 0 0
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 4.46410 0.470558
\(91\) −1.46410 −0.153480
\(92\) −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 \(-0.234533\pi\)
0.740617 + 0.671928i \(0.234533\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 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 \(-0.290958\pi\)
0.610525 + 0.791997i \(0.290958\pi\)
\(104\) −1.46410 −0.143567
\(105\) 2.73205 0.266621
\(106\) 7.26795 0.705926
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) 4.00000 0.384900
\(109\) 15.6603 1.49998 0.749990 0.661449i \(-0.230058\pi\)
0.749990 + 0.661449i \(0.230058\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 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
\(119\) −3.46410 −0.317554
\(120\) 2.73205 0.249401
\(121\) 0 0
\(122\) −4.92820 −0.446179
\(123\) 6.00000 0.541002
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 4.46410 0.397694
\(127\) −1.07180 −0.0951066 −0.0475533 0.998869i \(-0.515142\pi\)
−0.0475533 + 0.998869i \(0.515142\pi\)
\(128\) −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 \(0.579533\pi\)
\(132\) 0 0
\(133\) 6.73205 0.583743
\(134\) 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 \(0.696683\pi\)
\(140\) 1.00000 0.0845154
\(141\) 18.9282 1.59404
\(142\) −9.46410 −0.794210
\(143\) 0 0
\(144\) 4.46410 0.372008
\(145\) −4.73205 −0.392975
\(146\) −14.3923 −1.19112
\(147\) 2.73205 0.225336
\(148\) 0.732051 0.0601742
\(149\) 7.26795 0.595414 0.297707 0.954657i \(-0.403778\pi\)
0.297707 + 0.954657i \(0.403778\pi\)
\(150\) −2.73205 −0.223071
\(151\) −11.1244 −0.905287 −0.452644 0.891692i \(-0.649519\pi\)
−0.452644 + 0.891692i \(0.649519\pi\)
\(152\) 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 \(-0.442066\pi\)
0.181002 + 0.983483i \(0.442066\pi\)
\(158\) −12.1962 −0.970274
\(159\) −19.8564 −1.57472
\(160\) 1.00000 0.0790569
\(161\) 8.19615 0.645947
\(162\) 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
\(165\) 0 0
\(166\) −16.3923 −1.27229
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 2.73205 0.210782
\(169\) −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
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 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 \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.46410 0.108526
\(183\) 13.4641 0.995295
\(184\) 8.19615 0.604228
\(185\) −0.732051 −0.0538214
\(186\) −5.46410 −0.400647
\(187\) 0 0
\(188\) 6.92820 0.505291
\(189\) −4.00000 −0.290957
\(190\) −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 \(-0.647155\pi\)
−0.446009 + 0.895029i \(0.647155\pi\)
\(194\) −14.5885 −1.04739
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) 24.2487 1.72765 0.863825 0.503793i \(-0.168062\pi\)
0.863825 + 0.503793i \(0.168062\pi\)
\(198\) 0 0
\(199\) 2.92820 0.207575 0.103787 0.994600i \(-0.466904\pi\)
0.103787 + 0.994600i \(0.466904\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −10.9282 −0.770816
\(202\) −7.85641 −0.552775
\(203\) −4.73205 −0.332125
\(204\) 9.46410 0.662620
\(205\) −2.19615 −0.153386
\(206\) −12.3923 −0.863413
\(207\) −36.5885 −2.54307
\(208\) 1.46410 0.101517
\(209\) 0 0
\(210\) −2.73205 −0.188529
\(211\) −26.9282 −1.85381 −0.926907 0.375291i \(-0.877543\pi\)
−0.926907 + 0.375291i \(0.877543\pi\)
\(212\) −7.26795 −0.499165
\(213\) 25.8564 1.77165
\(214\) 7.85641 0.537053
\(215\) 2.00000 0.136399
\(216\) −4.00000 −0.272166
\(217\) −2.00000 −0.135769
\(218\) −15.6603 −1.06065
\(219\) 39.3205 2.65703
\(220\) 0 0
\(221\) 5.07180 0.341166
\(222\) −2.00000 −0.134231
\(223\) −25.4641 −1.70520 −0.852601 0.522562i \(-0.824976\pi\)
−0.852601 + 0.522562i \(0.824976\pi\)
\(224\) 1.00000 0.0668153
\(225\) 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 \(-0.201658\pi\)
0.805944 + 0.591991i \(0.201658\pi\)
\(230\) −8.19615 −0.540438
\(231\) 0 0
\(232\) −4.73205 −0.310674
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) −6.53590 −0.427265
\(235\) −6.92820 −0.451946
\(236\) 6.92820 0.450988
\(237\) 33.3205 2.16440
\(238\) 3.46410 0.224544
\(239\) −1.26795 −0.0820168 −0.0410084 0.999159i \(-0.513057\pi\)
−0.0410084 + 0.999159i \(0.513057\pi\)
\(240\) −2.73205 −0.176353
\(241\) −3.26795 −0.210507 −0.105254 0.994445i \(-0.533565\pi\)
−0.105254 + 0.994445i \(0.533565\pi\)
\(242\) 0 0
\(243\) −18.7321 −1.20166
\(244\) 4.92820 0.315496
\(245\) −1.00000 −0.0638877
\(246\) −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 \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −4.46410 −0.281212
\(253\) 0 0
\(254\) 1.07180 0.0672505
\(255\) −9.46410 −0.592665
\(256\) 1.00000 0.0625000
\(257\) −23.6603 −1.47589 −0.737943 0.674863i \(-0.764203\pi\)
−0.737943 + 0.674863i \(0.764203\pi\)
\(258\) 5.46410 0.340180
\(259\) −0.732051 −0.0454874
\(260\) −1.46410 −0.0907997
\(261\) 21.1244 1.30756
\(262\) 5.66025 0.349692
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 7.26795 0.446467
\(266\) −6.73205 −0.412769
\(267\) 9.46410 0.579194
\(268\) −4.00000 −0.244339
\(269\) 28.3923 1.73111 0.865555 0.500814i \(-0.166966\pi\)
0.865555 + 0.500814i \(0.166966\pi\)
\(270\) 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
\(273\) −4.00000 −0.242091
\(274\) 0.928203 0.0560748
\(275\) 0 0
\(276\) −22.3923 −1.34786
\(277\) −7.07180 −0.424903 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(278\) 13.6603 0.819288
\(279\) 8.92820 0.534518
\(280\) −1.00000 −0.0597614
\(281\) −22.3923 −1.33581 −0.667906 0.744245i \(-0.732809\pi\)
−0.667906 + 0.744245i \(0.732809\pi\)
\(282\) −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
\(286\) 0 0
\(287\) −2.19615 −0.129635
\(288\) −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 \(0.715706\pi\)
\(294\) −2.73205 −0.159336
\(295\) −6.92820 −0.403376
\(296\) −0.732051 −0.0425496
\(297\) 0 0
\(298\) −7.26795 −0.421021
\(299\) −12.0000 −0.693978
\(300\) 2.73205 0.157735
\(301\) 2.00000 0.115278
\(302\) 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 \(-0.250150\pi\)
0.706774 + 0.707439i \(0.250150\pi\)
\(312\) −4.00000 −0.226455
\(313\) 22.1962 1.25460 0.627300 0.778777i \(-0.284160\pi\)
0.627300 + 0.778777i \(0.284160\pi\)
\(314\) −4.53590 −0.255976
\(315\) 4.46410 0.251524
\(316\) 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
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −21.4641 −1.19801
\(322\) −8.19615 −0.456754
\(323\) −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
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) −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
\(336\) −2.73205 −0.149046
\(337\) −22.7846 −1.24116 −0.620578 0.784144i \(-0.713102\pi\)
−0.620578 + 0.784144i \(0.713102\pi\)
\(338\) 10.8564 0.590511
\(339\) 21.4641 1.16577
\(340\) −3.46410 −0.187867
\(341\) 0 0
\(342\) 30.0526 1.62506
\(343\) −1.00000 −0.0539949
\(344\) 2.00000 0.107833
\(345\) 22.3923 1.20556
\(346\) 0.928203 0.0499005
\(347\) 23.0718 1.23856 0.619279 0.785171i \(-0.287425\pi\)
0.619279 + 0.785171i \(0.287425\pi\)
\(348\) 12.9282 0.693024
\(349\) 5.60770 0.300173 0.150087 0.988673i \(-0.452045\pi\)
0.150087 + 0.988673i \(0.452045\pi\)
\(350\) 1.00000 0.0534522
\(351\) 5.85641 0.312592
\(352\) 0 0
\(353\) 14.1962 0.755585 0.377792 0.925890i \(-0.376683\pi\)
0.377792 + 0.925890i \(0.376683\pi\)
\(354\) −18.9282 −1.00602
\(355\) −9.46410 −0.502302
\(356\) 3.46410 0.183597
\(357\) −9.46410 −0.500893
\(358\) 6.00000 0.317110
\(359\) 34.0526 1.79723 0.898613 0.438743i \(-0.144576\pi\)
0.898613 + 0.438743i \(0.144576\pi\)
\(360\) 4.46410 0.235279
\(361\) 26.3205 1.38529
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) −1.46410 −0.0767398
\(365\) −14.3923 −0.753328
\(366\) −13.4641 −0.703780
\(367\) 12.3923 0.646873 0.323437 0.946250i \(-0.395162\pi\)
0.323437 + 0.946250i \(0.395162\pi\)
\(368\) −8.19615 −0.427254
\(369\) 9.80385 0.510368
\(370\) 0.732051 0.0380575
\(371\) 7.26795 0.377333
\(372\) 5.46410 0.283300
\(373\) −18.3923 −0.952317 −0.476159 0.879359i \(-0.657971\pi\)
−0.476159 + 0.879359i \(0.657971\pi\)
\(374\) 0 0
\(375\) −2.73205 −0.141082
\(376\) −6.92820 −0.357295
\(377\) 6.92820 0.356821
\(378\) 4.00000 0.205738
\(379\) 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 \(-0.826423\pi\)
−0.854968 + 0.518681i \(0.826423\pi\)
\(384\) −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
\(392\) −1.00000 −0.0505076
\(393\) −15.4641 −0.780061
\(394\) −24.2487 −1.22163
\(395\) −12.1962 −0.613655
\(396\) 0 0
\(397\) 30.3923 1.52535 0.762673 0.646784i \(-0.223887\pi\)
0.762673 + 0.646784i \(0.223887\pi\)
\(398\) −2.92820 −0.146778
\(399\) 18.3923 0.920767
\(400\) 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
\(406\) 4.73205 0.234848
\(407\) 0 0
\(408\) −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
\(413\) −6.92820 −0.340915
\(414\) 36.5885 1.79822
\(415\) −16.3923 −0.804667
\(416\) −1.46410 −0.0717835
\(417\) −37.3205 −1.82759
\(418\) 0 0
\(419\) 30.9282 1.51094 0.755471 0.655182i \(-0.227408\pi\)
0.755471 + 0.655182i \(0.227408\pi\)
\(420\) 2.73205 0.133310
\(421\) −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
\(427\) −4.92820 −0.238492
\(428\) −7.85641 −0.379754
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −3.12436 −0.150495 −0.0752475 0.997165i \(-0.523975\pi\)
−0.0752475 + 0.997165i \(0.523975\pi\)
\(432\) 4.00000 0.192450
\(433\) −18.1962 −0.874451 −0.437226 0.899352i \(-0.644039\pi\)
−0.437226 + 0.899352i \(0.644039\pi\)
\(434\) 2.00000 0.0960031
\(435\) −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 \(0.715469\pi\)
\(440\) 0 0
\(441\) 4.46410 0.212576
\(442\) −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
\(448\) −1.00000 −0.0472456
\(449\) 40.3923 1.90623 0.953115 0.302607i \(-0.0978570\pi\)
0.953115 + 0.302607i \(0.0978570\pi\)
\(450\) −4.46410 −0.210440
\(451\) 0 0
\(452\) 7.85641 0.369534
\(453\) −30.3923 −1.42796
\(454\) 6.92820 0.325157
\(455\) 1.46410 0.0686381
\(456\) 18.3923 0.861299
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −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 \(-0.788822\pi\)
−0.787882 + 0.615826i \(0.788822\pi\)
\(468\) 6.53590 0.302122
\(469\) 4.00000 0.184703
\(470\) 6.92820 0.319574
\(471\) 12.3923 0.571007
\(472\) −6.92820 −0.318896
\(473\) 0 0
\(474\) −33.3205 −1.53046
\(475\) −6.73205 −0.308888
\(476\) −3.46410 −0.158777
\(477\) −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
\(483\) 22.3923 1.01889
\(484\) 0 0
\(485\) −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
\(490\) 1.00000 0.0451754
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 6.00000 0.270501
\(493\) 16.3923 0.738272
\(494\) 9.85641 0.443461
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −9.46410 −0.424523
\(498\) −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 \(0.562744\pi\)
\(504\) 4.46410 0.198847
\(505\) −7.85641 −0.349605
\(506\) 0 0
\(507\) −29.6603 −1.31726
\(508\) −1.07180 −0.0475533
\(509\) 11.0718 0.490749 0.245374 0.969428i \(-0.421089\pi\)
0.245374 + 0.969428i \(0.421089\pi\)
\(510\) 9.46410 0.419077
\(511\) −14.3923 −0.636678
\(512\) −1.00000 −0.0441942
\(513\) −26.9282 −1.18891
\(514\) 23.6603 1.04361
\(515\) −12.3923 −0.546070
\(516\) −5.46410 −0.240544
\(517\) 0 0
\(518\) 0.732051 0.0321645
\(519\) −2.53590 −0.111314
\(520\) 1.46410 0.0642051
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) −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
\(525\) −2.73205 −0.119236
\(526\) −24.0000 −1.04645
\(527\) 6.92820 0.301797
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) −7.26795 −0.315700
\(531\) 30.9282 1.34217
\(532\) 6.73205 0.291871
\(533\) 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 \(-0.352965\pi\)
0.445670 + 0.895197i \(0.352965\pi\)
\(542\) 0.392305 0.0168509
\(543\) 38.2487 1.64141
\(544\) −3.46410 −0.148522
\(545\) −15.6603 −0.670812
\(546\) 4.00000 0.171184
\(547\) −28.7846 −1.23074 −0.615371 0.788238i \(-0.710994\pi\)
−0.615371 + 0.788238i \(0.710994\pi\)
\(548\) −0.928203 −0.0396509
\(549\) 22.0000 0.938937
\(550\) 0 0
\(551\) −31.8564 −1.35713
\(552\) 22.3923 0.953080
\(553\) −12.1962 −0.518633
\(554\) 7.07180 0.300452
\(555\) −2.00000 −0.0848953
\(556\) −13.6603 −0.579324
\(557\) 25.6077 1.08503 0.542516 0.840045i \(-0.317472\pi\)
0.542516 + 0.840045i \(0.317472\pi\)
\(558\) −8.92820 −0.377961
\(559\) −2.92820 −0.123850
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 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
\(567\) 2.46410 0.103483
\(568\) −9.46410 −0.397105
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −18.3923 −0.770369
\(571\) −3.60770 −0.150977 −0.0754887 0.997147i \(-0.524052\pi\)
−0.0754887 + 0.997147i \(0.524052\pi\)
\(572\) 0 0
\(573\) −32.7846 −1.36960
\(574\) 2.19615 0.0916656
\(575\) −8.19615 −0.341803
\(576\) 4.46410 0.186004
\(577\) −34.5885 −1.43994 −0.719968 0.694007i \(-0.755844\pi\)
−0.719968 + 0.694007i \(0.755844\pi\)
\(578\) 5.00000 0.207973
\(579\) −33.8564 −1.40702
\(580\) −4.73205 −0.196488
\(581\) −16.3923 −0.680067
\(582\) −39.8564 −1.65210
\(583\) 0 0
\(584\) −14.3923 −0.595558
\(585\) −6.53590 −0.270226
\(586\) 21.4641 0.886674
\(587\) −11.4115 −0.471005 −0.235502 0.971874i \(-0.575674\pi\)
−0.235502 + 0.971874i \(0.575674\pi\)
\(588\) 2.73205 0.112668
\(589\) −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 \(0.501626\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) 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 \(0.464109\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −17.8564 −0.727169
\(604\) −11.1244 −0.452644
\(605\) 0 0
\(606\) −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
\(609\) −12.9282 −0.523877
\(610\) 4.92820 0.199537
\(611\) 10.1436 0.410366
\(612\) 15.4641 0.625099
\(613\) 35.1769 1.42078 0.710391 0.703807i \(-0.248518\pi\)
0.710391 + 0.703807i \(0.248518\pi\)
\(614\) 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 \(-0.303648\pi\)
0.578476 + 0.815700i \(0.303648\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −32.7846 −1.31560
\(622\) −24.9282 −0.999530
\(623\) −3.46410 −0.138786
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) −22.1962 −0.887137
\(627\) 0 0
\(628\) 4.53590 0.181002
\(629\) 2.53590 0.101113
\(630\) −4.46410 −0.177854
\(631\) −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
\(637\) 1.46410 0.0580098
\(638\) 0 0
\(639\) 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 \(-0.272256\pi\)
0.655980 + 0.754778i \(0.272256\pi\)
\(644\) 8.19615 0.322974
\(645\) 5.46410 0.215149
\(646\) 23.3205 0.917533
\(647\) −26.5359 −1.04323 −0.521617 0.853180i \(-0.674671\pi\)
−0.521617 + 0.853180i \(0.674671\pi\)
\(648\) 2.46410 0.0967991
\(649\) 0 0
\(650\) −1.46410 −0.0574268
\(651\) −5.46410 −0.214155
\(652\) −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
\(658\) 6.92820 0.270089
\(659\) −30.2487 −1.17832 −0.589161 0.808015i \(-0.700542\pi\)
−0.589161 + 0.808015i \(0.700542\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 18.7846 0.730085
\(663\) 13.8564 0.538138
\(664\) −16.3923 −0.636145
\(665\) −6.73205 −0.261058
\(666\) −3.26795 −0.126630
\(667\) −38.7846 −1.50175
\(668\) −13.8564 −0.536120
\(669\) −69.5692 −2.68970
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) 2.73205 0.105391
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 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
\(679\) −14.5885 −0.559854
\(680\) 3.46410 0.132842
\(681\) −18.9282 −0.725330
\(682\) 0 0
\(683\) 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
\(686\) 1.00000 0.0381802
\(687\) 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 \(-0.582437\pi\)
−0.256099 + 0.966650i \(0.582437\pi\)
\(692\) −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
\(700\) −1.00000 −0.0377964
\(701\) −41.9090 −1.58288 −0.791440 0.611247i \(-0.790668\pi\)
−0.791440 + 0.611247i \(0.790668\pi\)
\(702\) −5.85641 −0.221036
\(703\) −4.92820 −0.185871
\(704\) 0 0
\(705\) −18.9282 −0.712877
\(706\) −14.1962 −0.534279
\(707\) −7.85641 −0.295471
\(708\) 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
\(714\) 9.46410 0.354185
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) −3.46410 −0.129369
\(718\) −34.0526 −1.27083
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) −4.46410 −0.166367
\(721\) −12.3923 −0.461514
\(722\) −26.3205 −0.979548
\(723\) −8.92820 −0.332043
\(724\) 14.0000 0.520306
\(725\) 4.73205 0.175744
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) 1.46410 0.0542632
\(729\) −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
\(735\) −2.73205 −0.100773
\(736\) 8.19615 0.302114
\(737\) 0 0
\(738\) −9.80385 −0.360885
\(739\) −11.7128 −0.430863 −0.215431 0.976519i \(-0.569116\pi\)
−0.215431 + 0.976519i \(0.569116\pi\)
\(740\) −0.732051 −0.0269107
\(741\) −26.9282 −0.989232
\(742\) −7.26795 −0.266815
\(743\) 32.7846 1.20275 0.601375 0.798967i \(-0.294620\pi\)
0.601375 + 0.798967i \(0.294620\pi\)
\(744\) −5.46410 −0.200324
\(745\) −7.26795 −0.266277
\(746\) 18.3923 0.673390
\(747\) 73.1769 2.67740
\(748\) 0 0
\(749\) 7.85641 0.287067
\(750\) 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
\(756\) −4.00000 −0.145479
\(757\) −43.3731 −1.57642 −0.788210 0.615406i \(-0.788992\pi\)
−0.788210 + 0.615406i \(0.788992\pi\)
\(758\) −6.14359 −0.223145
\(759\) 0 0
\(760\) −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
\(763\) −15.6603 −0.566939
\(764\) −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 \(-0.505314\pi\)
−0.0166926 + 0.999861i \(0.505314\pi\)
\(774\) 8.92820 0.320918
\(775\) 2.00000 0.0718421
\(776\) −14.5885 −0.523695
\(777\) −2.00000 −0.0717496
\(778\) 1.60770 0.0576387
\(779\) −14.7846 −0.529714
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 28.3923 1.01531
\(783\) 18.9282 0.676439
\(784\) 1.00000 0.0357143
\(785\) −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
\(791\) −7.85641 −0.279342
\(792\) 0 0
\(793\) 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 \(-0.649837\pi\)
−0.453536 + 0.891238i \(0.649837\pi\)
\(798\) −18.3923 −0.651081
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 15.4641 0.546397
\(802\) −2.53590 −0.0895457
\(803\) 0 0
\(804\) −10.9282 −0.385408
\(805\) −8.19615 −0.288876
\(806\) −2.92820 −0.103142
\(807\) 77.5692 2.73057
\(808\) −7.85641 −0.276387
\(809\) 31.1769 1.09612 0.548061 0.836438i \(-0.315366\pi\)
0.548061 + 0.836438i \(0.315366\pi\)
\(810\) −2.46410 −0.0865797
\(811\) 43.1244 1.51430 0.757150 0.653241i \(-0.226591\pi\)
0.757150 + 0.653241i \(0.226591\pi\)
\(812\) −4.73205 −0.166062
\(813\) −1.07180 −0.0375896
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 9.46410 0.331310
\(817\) 13.4641 0.471049
\(818\) 10.8756 0.380258
\(819\) −6.53590 −0.228383
\(820\) −2.19615 −0.0766930
\(821\) −17.9090 −0.625027 −0.312514 0.949913i \(-0.601171\pi\)
−0.312514 + 0.949913i \(0.601171\pi\)
\(822\) 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
\(825\) 0 0
\(826\) 6.92820 0.241063
\(827\) 25.8564 0.899115 0.449558 0.893251i \(-0.351582\pi\)
0.449558 + 0.893251i \(0.351582\pi\)
\(828\) −36.5885 −1.27154
\(829\) 2.24871 0.0781010 0.0390505 0.999237i \(-0.487567\pi\)
0.0390505 + 0.999237i \(0.487567\pi\)
\(830\) 16.3923 0.568985
\(831\) −19.3205 −0.670221
\(832\) 1.46410 0.0507586
\(833\) 3.46410 0.120024
\(834\) 37.3205 1.29230
\(835\) 13.8564 0.479521
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −30.9282 −1.06840
\(839\) −31.8564 −1.09981 −0.549903 0.835229i \(-0.685335\pi\)
−0.549903 + 0.835229i \(0.685335\pi\)
\(840\) −2.73205 −0.0942647
\(841\) −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 \(0.112773\pi\)
\(854\) 4.92820 0.168640
\(855\) 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 \(0.344586\pi\)
\(858\) 0 0
\(859\) −12.7846 −0.436205 −0.218103 0.975926i \(-0.569987\pi\)
−0.218103 + 0.975926i \(0.569987\pi\)
\(860\) 2.00000 0.0681994
\(861\) −6.00000 −0.204479
\(862\) 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
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 12.9282 0.438307
\(871\) −5.85641 −0.198437
\(872\) −15.6603 −0.530323
\(873\) 65.1244 2.20413
\(874\) −55.1769 −1.86639
\(875\) 1.00000 0.0338062
\(876\) 39.3205 1.32852
\(877\) 21.3205 0.719942 0.359971 0.932963i \(-0.382787\pi\)
0.359971 + 0.932963i \(0.382787\pi\)
\(878\) 26.2487 0.885851
\(879\) −58.6410 −1.97791
\(880\) 0 0
\(881\) 11.0718 0.373018 0.186509 0.982453i \(-0.440283\pi\)
0.186509 + 0.982453i \(0.440283\pi\)
\(882\) −4.46410 −0.150314
\(883\) −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
\(889\) 1.07180 0.0359469
\(890\) 3.46410 0.116117
\(891\) 0 0
\(892\) −25.4641 −0.852601
\(893\) −46.6410 −1.56078
\(894\) −19.8564 −0.664098
\(895\) 6.00000 0.200558
\(896\) 1.00000 0.0334077
\(897\) −32.7846 −1.09465
\(898\) −40.3923 −1.34791
\(899\) 9.46410 0.315645
\(900\) 4.46410 0.148803
\(901\) −25.1769 −0.838765
\(902\) 0 0
\(903\) 5.46410 0.181834
\(904\) −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
\(910\) −1.46410 −0.0485345
\(911\) 9.46410 0.313560 0.156780 0.987634i \(-0.449889\pi\)
0.156780 + 0.987634i \(0.449889\pi\)
\(912\) −18.3923 −0.609030
\(913\) 0 0
\(914\) 2.00000 0.0661541
\(915\) −13.4641 −0.445109
\(916\) 24.3923 0.805944
\(917\) 5.66025 0.186918
\(918\) −13.8564 −0.457330
\(919\) 25.3731 0.836980 0.418490 0.908221i \(-0.362559\pi\)
0.418490 + 0.908221i \(0.362559\pi\)
\(920\) −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 \(-0.637998\pi\)
−0.420081 + 0.907487i \(0.637998\pi\)
\(930\) 5.46410 0.179175
\(931\) −6.73205 −0.220634
\(932\) 7.85641 0.257345
\(933\) 68.1051 2.22966
\(934\) 34.0526 1.11423
\(935\) 0 0
\(936\) −6.53590 −0.213633
\(937\) 1.21539 0.0397051 0.0198525 0.999803i \(-0.493680\pi\)
0.0198525 + 0.999803i \(0.493680\pi\)
\(938\) −4.00000 −0.130605
\(939\) 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
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 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
\(952\) 3.46410 0.112272
\(953\) −26.5359 −0.859582 −0.429791 0.902928i \(-0.641413\pi\)
−0.429791 + 0.902928i \(0.641413\pi\)
\(954\) 32.4449 1.05044
\(955\) 12.0000 0.388311
\(956\) −1.26795 −0.0410084
\(957\) 0 0
\(958\) −32.7846 −1.05922
\(959\) 0.928203 0.0299732
\(960\) −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
\(966\) −22.3923 −0.720461
\(967\) −26.9282 −0.865953 −0.432976 0.901405i \(-0.642537\pi\)
−0.432976 + 0.901405i \(0.642537\pi\)
\(968\) 0 0
\(969\) −63.7128 −2.04675
\(970\) 14.5885 0.468407
\(971\) 42.9282 1.37763 0.688816 0.724936i \(-0.258131\pi\)
0.688816 + 0.724936i \(0.258131\pi\)
\(972\) −18.7321 −0.600831
\(973\) 13.6603 0.437928
\(974\) −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
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 69.9090 2.23202
\(982\) −27.7128 −0.884351
\(983\) −37.1769 −1.18576 −0.592880 0.805291i \(-0.702009\pi\)
−0.592880 + 0.805291i \(0.702009\pi\)
\(984\) −6.00000 −0.191273
\(985\) −24.2487 −0.772628
\(986\) −16.3923 −0.522037
\(987\) −18.9282 −0.602491
\(988\) −9.85641 −0.313574
\(989\) 16.3923 0.521245
\(990\) 0 0
\(991\) −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
\(994\) 9.46410 0.300183
\(995\) −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 8470.2.a.br.1.2 2
11.10 odd 2 770.2.a.j.1.2 2
33.32 even 2 6930.2.a.bv.1.1 2
44.43 even 2 6160.2.a.t.1.1 2
55.32 even 4 3850.2.c.x.1849.3 4
55.43 even 4 3850.2.c.x.1849.2 4
55.54 odd 2 3850.2.a.bd.1.1 2
77.76 even 2 5390.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 11.10 odd 2
3850.2.a.bd.1.1 2 55.54 odd 2
3850.2.c.x.1849.2 4 55.43 even 4
3850.2.c.x.1849.3 4 55.32 even 4
5390.2.a.bs.1.1 2 77.76 even 2
6160.2.a.t.1.1 2 44.43 even 2
6930.2.a.bv.1.1 2 33.32 even 2
8470.2.a.br.1.2 2 1.1 even 1 trivial