Properties

Label 1805.2.a.n.1.2
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.820249\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.820249 q^{2} +2.32719 q^{3} -1.32719 q^{4} +1.00000 q^{5} -1.90888 q^{6} -0.561717 q^{7} +2.72913 q^{8} +2.41582 q^{9} +O(q^{10})\) \(q-0.820249 q^{2} +2.32719 q^{3} -1.32719 q^{4} +1.00000 q^{5} -1.90888 q^{6} -0.561717 q^{7} +2.72913 q^{8} +2.41582 q^{9} -0.820249 q^{10} -0.111092 q^{11} -3.08863 q^{12} -6.89045 q^{13} +0.460748 q^{14} +2.32719 q^{15} +0.415819 q^{16} -3.81167 q^{17} -1.98157 q^{18} -1.32719 q^{20} -1.30722 q^{21} +0.0911232 q^{22} -4.04243 q^{23} +6.35120 q^{24} +1.00000 q^{25} +5.65189 q^{26} -1.35950 q^{27} +0.745506 q^{28} -9.29239 q^{29} -1.90888 q^{30} +4.18987 q^{31} -5.79933 q^{32} -0.258532 q^{33} +3.12652 q^{34} -0.561717 q^{35} -3.20625 q^{36} -2.68669 q^{37} -16.0354 q^{39} +2.72913 q^{40} +10.0988 q^{41} +1.07225 q^{42} -9.63192 q^{43} +0.147440 q^{44} +2.41582 q^{45} +3.31580 q^{46} +2.12652 q^{47} +0.967690 q^{48} -6.68447 q^{49} -0.820249 q^{50} -8.87048 q^{51} +9.14494 q^{52} +5.74455 q^{53} +1.11513 q^{54} -0.111092 q^{55} -1.53300 q^{56} +7.62207 q^{58} -7.89903 q^{59} -3.08863 q^{60} +5.56575 q^{61} -3.43674 q^{62} -1.35701 q^{63} +3.92526 q^{64} -6.89045 q^{65} +0.212061 q^{66} +10.6534 q^{67} +5.05881 q^{68} -9.40752 q^{69} +0.460748 q^{70} +6.64050 q^{71} +6.59307 q^{72} -8.45825 q^{73} +2.20376 q^{74} +2.32719 q^{75} +0.0624023 q^{77} +13.1530 q^{78} +6.27087 q^{79} +0.415819 q^{80} -10.4113 q^{81} -8.28349 q^{82} -8.16132 q^{83} +1.73493 q^{84} -3.81167 q^{85} +7.90057 q^{86} -21.6252 q^{87} -0.303184 q^{88} -1.16741 q^{89} -1.98157 q^{90} +3.87048 q^{91} +5.36508 q^{92} +9.75064 q^{93} -1.74428 q^{94} -13.4961 q^{96} +8.24746 q^{97} +5.48294 q^{98} -0.268378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} + 3 q^{4} + 4 q^{5} - 7 q^{6} - 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} + 3 q^{4} + 4 q^{5} - 7 q^{6} - 11 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} - 16 q^{12} - 2 q^{13} - 11 q^{14} + q^{15} - 3 q^{16} - 7 q^{17} + 17 q^{18} + 3 q^{20} + 2 q^{21} + q^{22} - 11 q^{23} - 13 q^{24} + 4 q^{25} + 9 q^{26} - 14 q^{27} - 13 q^{28} - 15 q^{29} - 7 q^{30} - q^{31} + 3 q^{32} + 12 q^{33} - 22 q^{34} - 11 q^{35} + 16 q^{36} - 11 q^{37} - 29 q^{39} + 6 q^{40} + 22 q^{41} + 19 q^{42} - 26 q^{43} - 12 q^{44} + 5 q^{45} + 10 q^{46} - 26 q^{47} - 13 q^{48} + 13 q^{49} + q^{50} - 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} - 8 q^{56} - 3 q^{58} - 10 q^{59} - 16 q^{60} + 2 q^{61} - 31 q^{62} - 17 q^{63} + 4 q^{64} - 2 q^{65} + 22 q^{66} + 3 q^{67} + 4 q^{68} + 14 q^{69} - 11 q^{70} + 18 q^{71} + 29 q^{72} - 24 q^{73} - 17 q^{74} + q^{75} - 6 q^{77} - 15 q^{78} + 30 q^{79} - 3 q^{80} - 4 q^{81} - 13 q^{82} - 12 q^{83} + 52 q^{84} - 7 q^{85} - 16 q^{86} - q^{87} - 23 q^{88} + 9 q^{89} + 17 q^{90} - 9 q^{91} - 25 q^{92} + 7 q^{93} - 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.820249 −0.580004 −0.290002 0.957026i \(-0.593656\pi\)
−0.290002 + 0.957026i \(0.593656\pi\)
\(3\) 2.32719 1.34360 0.671802 0.740731i \(-0.265520\pi\)
0.671802 + 0.740731i \(0.265520\pi\)
\(4\) −1.32719 −0.663596
\(5\) 1.00000 0.447214
\(6\) −1.90888 −0.779296
\(7\) −0.561717 −0.212309 −0.106154 0.994350i \(-0.533854\pi\)
−0.106154 + 0.994350i \(0.533854\pi\)
\(8\) 2.72913 0.964892
\(9\) 2.41582 0.805273
\(10\) −0.820249 −0.259386
\(11\) −0.111092 −0.0334955 −0.0167478 0.999860i \(-0.505331\pi\)
−0.0167478 + 0.999860i \(0.505331\pi\)
\(12\) −3.08863 −0.891610
\(13\) −6.89045 −1.91107 −0.955534 0.294882i \(-0.904720\pi\)
−0.955534 + 0.294882i \(0.904720\pi\)
\(14\) 0.460748 0.123140
\(15\) 2.32719 0.600878
\(16\) 0.415819 0.103955
\(17\) −3.81167 −0.924465 −0.462233 0.886759i \(-0.652952\pi\)
−0.462233 + 0.886759i \(0.652952\pi\)
\(18\) −1.98157 −0.467061
\(19\) 0 0
\(20\) −1.32719 −0.296769
\(21\) −1.30722 −0.285259
\(22\) 0.0911232 0.0194275
\(23\) −4.04243 −0.842906 −0.421453 0.906850i \(-0.638480\pi\)
−0.421453 + 0.906850i \(0.638480\pi\)
\(24\) 6.35120 1.29643
\(25\) 1.00000 0.200000
\(26\) 5.65189 1.10843
\(27\) −1.35950 −0.261636
\(28\) 0.745506 0.140887
\(29\) −9.29239 −1.72555 −0.862776 0.505586i \(-0.831276\pi\)
−0.862776 + 0.505586i \(0.831276\pi\)
\(30\) −1.90888 −0.348512
\(31\) 4.18987 0.752524 0.376262 0.926513i \(-0.377209\pi\)
0.376262 + 0.926513i \(0.377209\pi\)
\(32\) −5.79933 −1.02519
\(33\) −0.258532 −0.0450047
\(34\) 3.12652 0.536193
\(35\) −0.561717 −0.0949475
\(36\) −3.20625 −0.534376
\(37\) −2.68669 −0.441690 −0.220845 0.975309i \(-0.570881\pi\)
−0.220845 + 0.975309i \(0.570881\pi\)
\(38\) 0 0
\(39\) −16.0354 −2.56772
\(40\) 2.72913 0.431513
\(41\) 10.0988 1.57716 0.788580 0.614932i \(-0.210817\pi\)
0.788580 + 0.614932i \(0.210817\pi\)
\(42\) 1.07225 0.165451
\(43\) −9.63192 −1.46885 −0.734427 0.678688i \(-0.762549\pi\)
−0.734427 + 0.678688i \(0.762549\pi\)
\(44\) 0.147440 0.0222275
\(45\) 2.41582 0.360129
\(46\) 3.31580 0.488888
\(47\) 2.12652 0.310185 0.155092 0.987900i \(-0.450433\pi\)
0.155092 + 0.987900i \(0.450433\pi\)
\(48\) 0.967690 0.139674
\(49\) −6.68447 −0.954925
\(50\) −0.820249 −0.116001
\(51\) −8.87048 −1.24212
\(52\) 9.14494 1.26818
\(53\) 5.74455 0.789075 0.394537 0.918880i \(-0.370905\pi\)
0.394537 + 0.918880i \(0.370905\pi\)
\(54\) 1.11513 0.151750
\(55\) −0.111092 −0.0149797
\(56\) −1.53300 −0.204855
\(57\) 0 0
\(58\) 7.62207 1.00083
\(59\) −7.89903 −1.02837 −0.514183 0.857680i \(-0.671905\pi\)
−0.514183 + 0.857680i \(0.671905\pi\)
\(60\) −3.08863 −0.398740
\(61\) 5.56575 0.712622 0.356311 0.934367i \(-0.384034\pi\)
0.356311 + 0.934367i \(0.384034\pi\)
\(62\) −3.43674 −0.436467
\(63\) −1.35701 −0.170967
\(64\) 3.92526 0.490657
\(65\) −6.89045 −0.854655
\(66\) 0.212061 0.0261029
\(67\) 10.6534 1.30152 0.650762 0.759282i \(-0.274450\pi\)
0.650762 + 0.759282i \(0.274450\pi\)
\(68\) 5.05881 0.613471
\(69\) −9.40752 −1.13253
\(70\) 0.460748 0.0550699
\(71\) 6.64050 0.788082 0.394041 0.919093i \(-0.371077\pi\)
0.394041 + 0.919093i \(0.371077\pi\)
\(72\) 6.59307 0.777001
\(73\) −8.45825 −0.989964 −0.494982 0.868903i \(-0.664825\pi\)
−0.494982 + 0.868903i \(0.664825\pi\)
\(74\) 2.20376 0.256182
\(75\) 2.32719 0.268721
\(76\) 0 0
\(77\) 0.0624023 0.00711140
\(78\) 13.1530 1.48929
\(79\) 6.27087 0.705528 0.352764 0.935712i \(-0.385242\pi\)
0.352764 + 0.935712i \(0.385242\pi\)
\(80\) 0.415819 0.0464899
\(81\) −10.4113 −1.15681
\(82\) −8.28349 −0.914759
\(83\) −8.16132 −0.895822 −0.447911 0.894078i \(-0.647832\pi\)
−0.447911 + 0.894078i \(0.647832\pi\)
\(84\) 1.73493 0.189297
\(85\) −3.81167 −0.413434
\(86\) 7.90057 0.851941
\(87\) −21.6252 −2.31846
\(88\) −0.303184 −0.0323196
\(89\) −1.16741 −0.123745 −0.0618726 0.998084i \(-0.519707\pi\)
−0.0618726 + 0.998084i \(0.519707\pi\)
\(90\) −1.98157 −0.208876
\(91\) 3.87048 0.405737
\(92\) 5.36508 0.559348
\(93\) 9.75064 1.01109
\(94\) −1.74428 −0.179908
\(95\) 0 0
\(96\) −13.4961 −1.37744
\(97\) 8.24746 0.837402 0.418701 0.908124i \(-0.362485\pi\)
0.418701 + 0.908124i \(0.362485\pi\)
\(98\) 5.48294 0.553860
\(99\) −0.268378 −0.0269730
\(100\) −1.32719 −0.132719
\(101\) −11.7400 −1.16817 −0.584087 0.811691i \(-0.698547\pi\)
−0.584087 + 0.811691i \(0.698547\pi\)
\(102\) 7.27601 0.720432
\(103\) 19.4676 1.91820 0.959099 0.283069i \(-0.0913526\pi\)
0.959099 + 0.283069i \(0.0913526\pi\)
\(104\) −18.8049 −1.84397
\(105\) −1.30722 −0.127572
\(106\) −4.71196 −0.457666
\(107\) −16.5648 −1.60138 −0.800690 0.599079i \(-0.795534\pi\)
−0.800690 + 0.599079i \(0.795534\pi\)
\(108\) 1.80432 0.173621
\(109\) −14.0588 −1.34659 −0.673295 0.739374i \(-0.735122\pi\)
−0.673295 + 0.739374i \(0.735122\pi\)
\(110\) 0.0911232 0.00868826
\(111\) −6.25245 −0.593456
\(112\) −0.233572 −0.0220705
\(113\) −8.88950 −0.836254 −0.418127 0.908389i \(-0.637313\pi\)
−0.418127 + 0.908389i \(0.637313\pi\)
\(114\) 0 0
\(115\) −4.04243 −0.376959
\(116\) 12.3328 1.14507
\(117\) −16.6461 −1.53893
\(118\) 6.47917 0.596456
\(119\) 2.14108 0.196272
\(120\) 6.35120 0.579782
\(121\) −10.9877 −0.998878
\(122\) −4.56531 −0.413323
\(123\) 23.5017 2.11908
\(124\) −5.56076 −0.499371
\(125\) 1.00000 0.0894427
\(126\) 1.11308 0.0991613
\(127\) 7.76469 0.689005 0.344503 0.938785i \(-0.388048\pi\)
0.344503 + 0.938785i \(0.388048\pi\)
\(128\) 8.37897 0.740603
\(129\) −22.4153 −1.97356
\(130\) 5.65189 0.495703
\(131\) −4.58042 −0.400193 −0.200097 0.979776i \(-0.564126\pi\)
−0.200097 + 0.979776i \(0.564126\pi\)
\(132\) 0.343122 0.0298649
\(133\) 0 0
\(134\) −8.73847 −0.754889
\(135\) −1.35950 −0.117007
\(136\) −10.4025 −0.892009
\(137\) 1.19518 0.102111 0.0510554 0.998696i \(-0.483741\pi\)
0.0510554 + 0.998696i \(0.483741\pi\)
\(138\) 7.71651 0.656873
\(139\) 8.06036 0.683670 0.341835 0.939760i \(-0.388952\pi\)
0.341835 + 0.939760i \(0.388952\pi\)
\(140\) 0.745506 0.0630067
\(141\) 4.94881 0.416765
\(142\) −5.44686 −0.457091
\(143\) 0.765474 0.0640122
\(144\) 1.00454 0.0837119
\(145\) −9.29239 −0.771691
\(146\) 6.93787 0.574183
\(147\) −15.5560 −1.28304
\(148\) 3.56575 0.293103
\(149\) 9.89499 0.810629 0.405315 0.914177i \(-0.367162\pi\)
0.405315 + 0.914177i \(0.367162\pi\)
\(150\) −1.90888 −0.155859
\(151\) 20.7413 1.68790 0.843951 0.536421i \(-0.180224\pi\)
0.843951 + 0.536421i \(0.180224\pi\)
\(152\) 0 0
\(153\) −9.20830 −0.744447
\(154\) −0.0511854 −0.00412464
\(155\) 4.18987 0.336539
\(156\) 21.2820 1.70393
\(157\) −12.9092 −1.03026 −0.515131 0.857111i \(-0.672257\pi\)
−0.515131 + 0.857111i \(0.672257\pi\)
\(158\) −5.14368 −0.409209
\(159\) 13.3687 1.06020
\(160\) −5.79933 −0.458477
\(161\) 2.27070 0.178956
\(162\) 8.53984 0.670953
\(163\) −7.93134 −0.621231 −0.310615 0.950536i \(-0.600535\pi\)
−0.310615 + 0.950536i \(0.600535\pi\)
\(164\) −13.4030 −1.04660
\(165\) −0.258532 −0.0201267
\(166\) 6.69432 0.519580
\(167\) −5.69132 −0.440408 −0.220204 0.975454i \(-0.570672\pi\)
−0.220204 + 0.975454i \(0.570672\pi\)
\(168\) −3.56757 −0.275244
\(169\) 34.4783 2.65218
\(170\) 3.12652 0.239793
\(171\) 0 0
\(172\) 12.7834 0.974725
\(173\) 7.12043 0.541357 0.270678 0.962670i \(-0.412752\pi\)
0.270678 + 0.962670i \(0.412752\pi\)
\(174\) 17.7380 1.34472
\(175\) −0.561717 −0.0424618
\(176\) −0.0461942 −0.00348202
\(177\) −18.3826 −1.38172
\(178\) 0.957567 0.0717727
\(179\) 2.37647 0.177626 0.0888129 0.996048i \(-0.471693\pi\)
0.0888129 + 0.996048i \(0.471693\pi\)
\(180\) −3.20625 −0.238980
\(181\) −20.1766 −1.49971 −0.749857 0.661600i \(-0.769878\pi\)
−0.749857 + 0.661600i \(0.769878\pi\)
\(182\) −3.17476 −0.235329
\(183\) 12.9526 0.957482
\(184\) −11.0323 −0.813313
\(185\) −2.68669 −0.197530
\(186\) −7.99795 −0.586438
\(187\) 0.423446 0.0309655
\(188\) −2.82230 −0.205837
\(189\) 0.763655 0.0555477
\(190\) 0 0
\(191\) 9.88950 0.715579 0.357789 0.933802i \(-0.383531\pi\)
0.357789 + 0.933802i \(0.383531\pi\)
\(192\) 9.13482 0.659249
\(193\) 4.55092 0.327582 0.163791 0.986495i \(-0.447628\pi\)
0.163791 + 0.986495i \(0.447628\pi\)
\(194\) −6.76497 −0.485697
\(195\) −16.0354 −1.14832
\(196\) 8.87158 0.633684
\(197\) −12.4701 −0.888457 −0.444229 0.895914i \(-0.646522\pi\)
−0.444229 + 0.895914i \(0.646522\pi\)
\(198\) 0.220137 0.0156445
\(199\) −19.9932 −1.41728 −0.708642 0.705569i \(-0.750692\pi\)
−0.708642 + 0.705569i \(0.750692\pi\)
\(200\) 2.72913 0.192978
\(201\) 24.7926 1.74873
\(202\) 9.62973 0.677546
\(203\) 5.21969 0.366350
\(204\) 11.7728 0.824263
\(205\) 10.0988 0.705327
\(206\) −15.9683 −1.11256
\(207\) −9.76579 −0.678769
\(208\) −2.86518 −0.198664
\(209\) 0 0
\(210\) 1.07225 0.0739921
\(211\) −16.8102 −1.15726 −0.578631 0.815589i \(-0.696413\pi\)
−0.578631 + 0.815589i \(0.696413\pi\)
\(212\) −7.62412 −0.523627
\(213\) 15.4537 1.05887
\(214\) 13.5873 0.928806
\(215\) −9.63192 −0.656891
\(216\) −3.71025 −0.252451
\(217\) −2.35352 −0.159768
\(218\) 11.5317 0.781027
\(219\) −19.6840 −1.33012
\(220\) 0.147440 0.00994043
\(221\) 26.2641 1.76672
\(222\) 5.12857 0.344207
\(223\) 1.88641 0.126324 0.0631618 0.998003i \(-0.479882\pi\)
0.0631618 + 0.998003i \(0.479882\pi\)
\(224\) 3.25758 0.217656
\(225\) 2.41582 0.161055
\(226\) 7.29160 0.485030
\(227\) 10.7044 0.710479 0.355239 0.934775i \(-0.384399\pi\)
0.355239 + 0.934775i \(0.384399\pi\)
\(228\) 0 0
\(229\) 8.34388 0.551379 0.275690 0.961247i \(-0.411094\pi\)
0.275690 + 0.961247i \(0.411094\pi\)
\(230\) 3.31580 0.218638
\(231\) 0.145222 0.00955491
\(232\) −25.3601 −1.66497
\(233\) −2.14436 −0.140481 −0.0702407 0.997530i \(-0.522377\pi\)
−0.0702407 + 0.997530i \(0.522377\pi\)
\(234\) 13.6539 0.892586
\(235\) 2.12652 0.138719
\(236\) 10.4835 0.682419
\(237\) 14.5935 0.947951
\(238\) −1.75622 −0.113839
\(239\) 25.0796 1.62227 0.811134 0.584861i \(-0.198851\pi\)
0.811134 + 0.584861i \(0.198851\pi\)
\(240\) 0.967690 0.0624641
\(241\) −11.3527 −0.731294 −0.365647 0.930754i \(-0.619152\pi\)
−0.365647 + 0.930754i \(0.619152\pi\)
\(242\) 9.01262 0.579353
\(243\) −20.1505 −1.29266
\(244\) −7.38682 −0.472893
\(245\) −6.68447 −0.427055
\(246\) −19.2773 −1.22907
\(247\) 0 0
\(248\) 11.4347 0.726104
\(249\) −18.9930 −1.20363
\(250\) −0.820249 −0.0518771
\(251\) −6.92245 −0.436941 −0.218471 0.975844i \(-0.570107\pi\)
−0.218471 + 0.975844i \(0.570107\pi\)
\(252\) 1.80101 0.113453
\(253\) 0.449082 0.0282336
\(254\) −6.36898 −0.399626
\(255\) −8.87048 −0.555491
\(256\) −14.7234 −0.920210
\(257\) −1.89721 −0.118345 −0.0591724 0.998248i \(-0.518846\pi\)
−0.0591724 + 0.998248i \(0.518846\pi\)
\(258\) 18.3861 1.14467
\(259\) 1.50916 0.0937747
\(260\) 9.14494 0.567145
\(261\) −22.4487 −1.38954
\(262\) 3.75709 0.232114
\(263\) −11.5446 −0.711868 −0.355934 0.934511i \(-0.615837\pi\)
−0.355934 + 0.934511i \(0.615837\pi\)
\(264\) −0.705568 −0.0434247
\(265\) 5.74455 0.352885
\(266\) 0 0
\(267\) −2.71678 −0.166265
\(268\) −14.1391 −0.863685
\(269\) 3.02056 0.184167 0.0920833 0.995751i \(-0.470647\pi\)
0.0920833 + 0.995751i \(0.470647\pi\)
\(270\) 1.11513 0.0678647
\(271\) 8.81150 0.535260 0.267630 0.963522i \(-0.413759\pi\)
0.267630 + 0.963522i \(0.413759\pi\)
\(272\) −1.58496 −0.0961025
\(273\) 9.00735 0.545150
\(274\) −0.980343 −0.0592247
\(275\) −0.111092 −0.00669910
\(276\) 12.4856 0.751543
\(277\) −9.68919 −0.582167 −0.291083 0.956698i \(-0.594016\pi\)
−0.291083 + 0.956698i \(0.594016\pi\)
\(278\) −6.61150 −0.396531
\(279\) 10.1220 0.605987
\(280\) −1.53300 −0.0916140
\(281\) 18.6762 1.11413 0.557063 0.830470i \(-0.311928\pi\)
0.557063 + 0.830470i \(0.311928\pi\)
\(282\) −4.05926 −0.241726
\(283\) −7.46451 −0.443719 −0.221859 0.975079i \(-0.571213\pi\)
−0.221859 + 0.975079i \(0.571213\pi\)
\(284\) −8.81321 −0.522968
\(285\) 0 0
\(286\) −0.627880 −0.0371273
\(287\) −5.67264 −0.334845
\(288\) −14.0101 −0.825554
\(289\) −2.47118 −0.145364
\(290\) 7.62207 0.447583
\(291\) 19.1934 1.12514
\(292\) 11.2257 0.656935
\(293\) 3.73771 0.218359 0.109180 0.994022i \(-0.465178\pi\)
0.109180 + 0.994022i \(0.465178\pi\)
\(294\) 12.7598 0.744169
\(295\) −7.89903 −0.459899
\(296\) −7.33232 −0.426183
\(297\) 0.151030 0.00876364
\(298\) −8.11636 −0.470168
\(299\) 27.8542 1.61085
\(300\) −3.08863 −0.178322
\(301\) 5.41041 0.311851
\(302\) −17.0130 −0.978989
\(303\) −27.3212 −1.56956
\(304\) 0 0
\(305\) 5.56575 0.318694
\(306\) 7.55310 0.431782
\(307\) −23.2825 −1.32880 −0.664402 0.747375i \(-0.731314\pi\)
−0.664402 + 0.747375i \(0.731314\pi\)
\(308\) −0.0828198 −0.00471909
\(309\) 45.3048 2.57730
\(310\) −3.43674 −0.195194
\(311\) 23.4553 1.33003 0.665013 0.746832i \(-0.268426\pi\)
0.665013 + 0.746832i \(0.268426\pi\)
\(312\) −43.7626 −2.47757
\(313\) −15.0079 −0.848297 −0.424148 0.905593i \(-0.639427\pi\)
−0.424148 + 0.905593i \(0.639427\pi\)
\(314\) 10.5887 0.597556
\(315\) −1.35701 −0.0764586
\(316\) −8.32265 −0.468186
\(317\) 19.3000 1.08400 0.541998 0.840380i \(-0.317668\pi\)
0.541998 + 0.840380i \(0.317668\pi\)
\(318\) −10.9656 −0.614923
\(319\) 1.03231 0.0577983
\(320\) 3.92526 0.219428
\(321\) −38.5495 −2.15162
\(322\) −1.86254 −0.103795
\(323\) 0 0
\(324\) 13.8178 0.767653
\(325\) −6.89045 −0.382213
\(326\) 6.50568 0.360316
\(327\) −32.7175 −1.80928
\(328\) 27.5608 1.52179
\(329\) −1.19450 −0.0658550
\(330\) 0.212061 0.0116736
\(331\) −19.1303 −1.05150 −0.525748 0.850641i \(-0.676214\pi\)
−0.525748 + 0.850641i \(0.676214\pi\)
\(332\) 10.8316 0.594463
\(333\) −6.49056 −0.355681
\(334\) 4.66830 0.255438
\(335\) 10.6534 0.582059
\(336\) −0.543568 −0.0296540
\(337\) −25.5100 −1.38962 −0.694810 0.719193i \(-0.744511\pi\)
−0.694810 + 0.719193i \(0.744511\pi\)
\(338\) −28.2808 −1.53827
\(339\) −20.6876 −1.12359
\(340\) 5.05881 0.274353
\(341\) −0.465462 −0.0252062
\(342\) 0 0
\(343\) 7.68680 0.415048
\(344\) −26.2867 −1.41728
\(345\) −9.40752 −0.506484
\(346\) −5.84053 −0.313989
\(347\) −34.3847 −1.84587 −0.922933 0.384960i \(-0.874215\pi\)
−0.922933 + 0.384960i \(0.874215\pi\)
\(348\) 28.7007 1.53852
\(349\) −28.6428 −1.53321 −0.766607 0.642117i \(-0.778057\pi\)
−0.766607 + 0.642117i \(0.778057\pi\)
\(350\) 0.460748 0.0246280
\(351\) 9.36758 0.500004
\(352\) 0.644259 0.0343391
\(353\) 23.5203 1.25186 0.625930 0.779879i \(-0.284719\pi\)
0.625930 + 0.779879i \(0.284719\pi\)
\(354\) 15.0783 0.801401
\(355\) 6.64050 0.352441
\(356\) 1.54938 0.0821167
\(357\) 4.98270 0.263712
\(358\) −1.94930 −0.103024
\(359\) 9.77437 0.515871 0.257936 0.966162i \(-0.416958\pi\)
0.257936 + 0.966162i \(0.416958\pi\)
\(360\) 6.59307 0.347486
\(361\) 0 0
\(362\) 16.5498 0.869839
\(363\) −25.5704 −1.34210
\(364\) −5.13687 −0.269245
\(365\) −8.45825 −0.442725
\(366\) −10.6243 −0.555343
\(367\) −2.03857 −0.106412 −0.0532062 0.998584i \(-0.516944\pi\)
−0.0532062 + 0.998584i \(0.516944\pi\)
\(368\) −1.68092 −0.0876240
\(369\) 24.3968 1.27004
\(370\) 2.20376 0.114568
\(371\) −3.22681 −0.167528
\(372\) −12.9410 −0.670958
\(373\) 12.9613 0.671110 0.335555 0.942021i \(-0.391076\pi\)
0.335555 + 0.942021i \(0.391076\pi\)
\(374\) −0.347331 −0.0179601
\(375\) 2.32719 0.120176
\(376\) 5.80354 0.299295
\(377\) 64.0287 3.29765
\(378\) −0.626387 −0.0322179
\(379\) 6.82911 0.350788 0.175394 0.984498i \(-0.443880\pi\)
0.175394 + 0.984498i \(0.443880\pi\)
\(380\) 0 0
\(381\) 18.0699 0.925750
\(382\) −8.11185 −0.415038
\(383\) 17.1311 0.875356 0.437678 0.899132i \(-0.355801\pi\)
0.437678 + 0.899132i \(0.355801\pi\)
\(384\) 19.4995 0.995077
\(385\) 0.0624023 0.00318031
\(386\) −3.73289 −0.189999
\(387\) −23.2690 −1.18283
\(388\) −10.9460 −0.555696
\(389\) 18.4370 0.934794 0.467397 0.884048i \(-0.345192\pi\)
0.467397 + 0.884048i \(0.345192\pi\)
\(390\) 13.1530 0.666029
\(391\) 15.4084 0.779237
\(392\) −18.2428 −0.921399
\(393\) −10.6595 −0.537701
\(394\) 10.2286 0.515308
\(395\) 6.27087 0.315522
\(396\) 0.356189 0.0178992
\(397\) −3.44622 −0.172961 −0.0864805 0.996254i \(-0.527562\pi\)
−0.0864805 + 0.996254i \(0.527562\pi\)
\(398\) 16.3994 0.822030
\(399\) 0 0
\(400\) 0.415819 0.0207909
\(401\) 12.4272 0.620585 0.310293 0.950641i \(-0.399573\pi\)
0.310293 + 0.950641i \(0.399573\pi\)
\(402\) −20.3361 −1.01427
\(403\) −28.8701 −1.43812
\(404\) 15.5812 0.775196
\(405\) −10.4113 −0.517340
\(406\) −4.28145 −0.212485
\(407\) 0.298470 0.0147946
\(408\) −24.2087 −1.19851
\(409\) 2.53190 0.125194 0.0625972 0.998039i \(-0.480062\pi\)
0.0625972 + 0.998039i \(0.480062\pi\)
\(410\) −8.28349 −0.409093
\(411\) 2.78141 0.137197
\(412\) −25.8372 −1.27291
\(413\) 4.43702 0.218331
\(414\) 8.01038 0.393689
\(415\) −8.16132 −0.400624
\(416\) 39.9600 1.95920
\(417\) 18.7580 0.918583
\(418\) 0 0
\(419\) 0.647537 0.0316342 0.0158171 0.999875i \(-0.494965\pi\)
0.0158171 + 0.999875i \(0.494965\pi\)
\(420\) 1.73493 0.0846561
\(421\) 4.37148 0.213053 0.106526 0.994310i \(-0.466027\pi\)
0.106526 + 0.994310i \(0.466027\pi\)
\(422\) 13.7886 0.671217
\(423\) 5.13728 0.249783
\(424\) 15.6776 0.761372
\(425\) −3.81167 −0.184893
\(426\) −12.6759 −0.614149
\(427\) −3.12638 −0.151296
\(428\) 21.9847 1.06267
\(429\) 1.78141 0.0860071
\(430\) 7.90057 0.380999
\(431\) 30.0094 1.44550 0.722752 0.691107i \(-0.242877\pi\)
0.722752 + 0.691107i \(0.242877\pi\)
\(432\) −0.565306 −0.0271983
\(433\) 17.6077 0.846173 0.423086 0.906089i \(-0.360947\pi\)
0.423086 + 0.906089i \(0.360947\pi\)
\(434\) 1.93047 0.0926658
\(435\) −21.6252 −1.03685
\(436\) 18.6587 0.893591
\(437\) 0 0
\(438\) 16.1458 0.771474
\(439\) 25.8827 1.23532 0.617658 0.786447i \(-0.288082\pi\)
0.617658 + 0.786447i \(0.288082\pi\)
\(440\) −0.303184 −0.0144537
\(441\) −16.1485 −0.768975
\(442\) −21.5431 −1.02470
\(443\) 11.8766 0.564273 0.282136 0.959374i \(-0.408957\pi\)
0.282136 + 0.959374i \(0.408957\pi\)
\(444\) 8.29819 0.393815
\(445\) −1.16741 −0.0553405
\(446\) −1.54733 −0.0732681
\(447\) 23.0275 1.08917
\(448\) −2.20488 −0.104171
\(449\) 29.4524 1.38995 0.694973 0.719035i \(-0.255416\pi\)
0.694973 + 0.719035i \(0.255416\pi\)
\(450\) −1.98157 −0.0934123
\(451\) −1.12189 −0.0528278
\(452\) 11.7981 0.554934
\(453\) 48.2689 2.26787
\(454\) −8.78031 −0.412080
\(455\) 3.87048 0.181451
\(456\) 0 0
\(457\) 24.3329 1.13825 0.569123 0.822253i \(-0.307283\pi\)
0.569123 + 0.822253i \(0.307283\pi\)
\(458\) −6.84406 −0.319802
\(459\) 5.18197 0.241874
\(460\) 5.36508 0.250148
\(461\) 34.5042 1.60702 0.803511 0.595290i \(-0.202963\pi\)
0.803511 + 0.595290i \(0.202963\pi\)
\(462\) −0.119118 −0.00554188
\(463\) −4.89328 −0.227410 −0.113705 0.993515i \(-0.536272\pi\)
−0.113705 + 0.993515i \(0.536272\pi\)
\(464\) −3.86395 −0.179379
\(465\) 9.75064 0.452175
\(466\) 1.75891 0.0814798
\(467\) 15.6840 0.725769 0.362885 0.931834i \(-0.381792\pi\)
0.362885 + 0.931834i \(0.381792\pi\)
\(468\) 22.0925 1.02123
\(469\) −5.98421 −0.276325
\(470\) −1.74428 −0.0804574
\(471\) −30.0421 −1.38427
\(472\) −21.5575 −0.992262
\(473\) 1.07003 0.0492000
\(474\) −11.9703 −0.549815
\(475\) 0 0
\(476\) −2.84162 −0.130245
\(477\) 13.8778 0.635421
\(478\) −20.5716 −0.940921
\(479\) 15.1666 0.692981 0.346490 0.938054i \(-0.387373\pi\)
0.346490 + 0.938054i \(0.387373\pi\)
\(480\) −13.4961 −0.616012
\(481\) 18.5125 0.844098
\(482\) 9.31208 0.424153
\(483\) 5.28436 0.240447
\(484\) 14.5827 0.662851
\(485\) 8.24746 0.374498
\(486\) 16.5285 0.749746
\(487\) 14.8366 0.672312 0.336156 0.941806i \(-0.390873\pi\)
0.336156 + 0.941806i \(0.390873\pi\)
\(488\) 15.1896 0.687603
\(489\) −18.4577 −0.834688
\(490\) 5.48294 0.247694
\(491\) 21.7022 0.979408 0.489704 0.871889i \(-0.337105\pi\)
0.489704 + 0.871889i \(0.337105\pi\)
\(492\) −31.1913 −1.40621
\(493\) 35.4195 1.59521
\(494\) 0 0
\(495\) −0.268378 −0.0120627
\(496\) 1.74223 0.0782283
\(497\) −3.73008 −0.167317
\(498\) 15.5790 0.698110
\(499\) −20.5764 −0.921124 −0.460562 0.887627i \(-0.652352\pi\)
−0.460562 + 0.887627i \(0.652352\pi\)
\(500\) −1.32719 −0.0593538
\(501\) −13.2448 −0.591734
\(502\) 5.67813 0.253427
\(503\) −34.8559 −1.55415 −0.777074 0.629409i \(-0.783297\pi\)
−0.777074 + 0.629409i \(0.783297\pi\)
\(504\) −3.70344 −0.164964
\(505\) −11.7400 −0.522424
\(506\) −0.368359 −0.0163756
\(507\) 80.2376 3.56348
\(508\) −10.3052 −0.457221
\(509\) −30.7078 −1.36110 −0.680551 0.732701i \(-0.738259\pi\)
−0.680551 + 0.732701i \(0.738259\pi\)
\(510\) 7.27601 0.322187
\(511\) 4.75114 0.210178
\(512\) −4.68111 −0.206878
\(513\) 0 0
\(514\) 1.55619 0.0686404
\(515\) 19.4676 0.857845
\(516\) 29.7494 1.30964
\(517\) −0.236239 −0.0103898
\(518\) −1.23789 −0.0543897
\(519\) 16.5706 0.727369
\(520\) −18.8049 −0.824650
\(521\) −10.9533 −0.479874 −0.239937 0.970788i \(-0.577127\pi\)
−0.239937 + 0.970788i \(0.577127\pi\)
\(522\) 18.4135 0.805939
\(523\) −4.44485 −0.194360 −0.0971799 0.995267i \(-0.530982\pi\)
−0.0971799 + 0.995267i \(0.530982\pi\)
\(524\) 6.07909 0.265566
\(525\) −1.30722 −0.0570519
\(526\) 9.46941 0.412886
\(527\) −15.9704 −0.695682
\(528\) −0.107503 −0.00467845
\(529\) −6.65873 −0.289510
\(530\) −4.71196 −0.204675
\(531\) −19.0826 −0.828115
\(532\) 0 0
\(533\) −69.5849 −3.01406
\(534\) 2.22844 0.0964341
\(535\) −16.5648 −0.716159
\(536\) 29.0746 1.25583
\(537\) 5.53050 0.238659
\(538\) −2.47761 −0.106817
\(539\) 0.742592 0.0319857
\(540\) 1.80432 0.0776455
\(541\) 6.38522 0.274522 0.137261 0.990535i \(-0.456170\pi\)
0.137261 + 0.990535i \(0.456170\pi\)
\(542\) −7.22762 −0.310453
\(543\) −46.9548 −2.01502
\(544\) 22.1051 0.947749
\(545\) −14.0588 −0.602213
\(546\) −7.38827 −0.316189
\(547\) 10.6627 0.455904 0.227952 0.973672i \(-0.426797\pi\)
0.227952 + 0.973672i \(0.426797\pi\)
\(548\) −1.58623 −0.0677603
\(549\) 13.4459 0.573855
\(550\) 0.0911232 0.00388551
\(551\) 0 0
\(552\) −25.6743 −1.09277
\(553\) −3.52245 −0.149790
\(554\) 7.94755 0.337659
\(555\) −6.25245 −0.265402
\(556\) −10.6976 −0.453681
\(557\) −3.06585 −0.129904 −0.0649521 0.997888i \(-0.520689\pi\)
−0.0649521 + 0.997888i \(0.520689\pi\)
\(558\) −8.30254 −0.351475
\(559\) 66.3683 2.80708
\(560\) −0.233572 −0.00987023
\(561\) 0.985440 0.0416053
\(562\) −15.3191 −0.646198
\(563\) −35.1561 −1.48165 −0.740827 0.671696i \(-0.765566\pi\)
−0.740827 + 0.671696i \(0.765566\pi\)
\(564\) −6.56802 −0.276564
\(565\) −8.88950 −0.373984
\(566\) 6.12276 0.257359
\(567\) 5.84819 0.245601
\(568\) 18.1228 0.760414
\(569\) −42.2302 −1.77038 −0.885190 0.465229i \(-0.845972\pi\)
−0.885190 + 0.465229i \(0.845972\pi\)
\(570\) 0 0
\(571\) −24.3462 −1.01886 −0.509429 0.860513i \(-0.670143\pi\)
−0.509429 + 0.860513i \(0.670143\pi\)
\(572\) −1.01593 −0.0424782
\(573\) 23.0148 0.961455
\(574\) 4.65298 0.194211
\(575\) −4.04243 −0.168581
\(576\) 9.48271 0.395113
\(577\) −9.74424 −0.405658 −0.202829 0.979214i \(-0.565014\pi\)
−0.202829 + 0.979214i \(0.565014\pi\)
\(578\) 2.02699 0.0843115
\(579\) 10.5909 0.440141
\(580\) 12.3328 0.512090
\(581\) 4.58435 0.190191
\(582\) −15.7434 −0.652584
\(583\) −0.638174 −0.0264305
\(584\) −23.0836 −0.955208
\(585\) −16.6461 −0.688231
\(586\) −3.06585 −0.126649
\(587\) 15.9708 0.659186 0.329593 0.944123i \(-0.393089\pi\)
0.329593 + 0.944123i \(0.393089\pi\)
\(588\) 20.6459 0.851421
\(589\) 0 0
\(590\) 6.47917 0.266743
\(591\) −29.0203 −1.19373
\(592\) −1.11718 −0.0459157
\(593\) −0.404153 −0.0165966 −0.00829829 0.999966i \(-0.502641\pi\)
−0.00829829 + 0.999966i \(0.502641\pi\)
\(594\) −0.123882 −0.00508294
\(595\) 2.14108 0.0877756
\(596\) −13.1325 −0.537930
\(597\) −46.5281 −1.90427
\(598\) −22.8474 −0.934299
\(599\) 17.0426 0.696342 0.348171 0.937431i \(-0.386803\pi\)
0.348171 + 0.937431i \(0.386803\pi\)
\(600\) 6.35120 0.259287
\(601\) −0.819125 −0.0334128 −0.0167064 0.999860i \(-0.505318\pi\)
−0.0167064 + 0.999860i \(0.505318\pi\)
\(602\) −4.43788 −0.180875
\(603\) 25.7368 1.04808
\(604\) −27.5276 −1.12008
\(605\) −10.9877 −0.446712
\(606\) 22.4102 0.910353
\(607\) 18.2675 0.741455 0.370728 0.928742i \(-0.379108\pi\)
0.370728 + 0.928742i \(0.379108\pi\)
\(608\) 0 0
\(609\) 12.1472 0.492230
\(610\) −4.56531 −0.184844
\(611\) −14.6527 −0.592784
\(612\) 12.2212 0.494012
\(613\) 27.6822 1.11807 0.559037 0.829142i \(-0.311171\pi\)
0.559037 + 0.829142i \(0.311171\pi\)
\(614\) 19.0975 0.770712
\(615\) 23.5017 0.947681
\(616\) 0.170304 0.00686173
\(617\) −40.1940 −1.61815 −0.809075 0.587706i \(-0.800031\pi\)
−0.809075 + 0.587706i \(0.800031\pi\)
\(618\) −37.1612 −1.49484
\(619\) 46.3862 1.86442 0.932209 0.361920i \(-0.117879\pi\)
0.932209 + 0.361920i \(0.117879\pi\)
\(620\) −5.56076 −0.223326
\(621\) 5.49569 0.220535
\(622\) −19.2392 −0.771420
\(623\) 0.655753 0.0262722
\(624\) −6.66782 −0.266926
\(625\) 1.00000 0.0400000
\(626\) 12.3102 0.492015
\(627\) 0 0
\(628\) 17.1329 0.683678
\(629\) 10.2408 0.408327
\(630\) 1.11308 0.0443463
\(631\) −47.5023 −1.89104 −0.945519 0.325568i \(-0.894444\pi\)
−0.945519 + 0.325568i \(0.894444\pi\)
\(632\) 17.1140 0.680759
\(633\) −39.1206 −1.55490
\(634\) −15.8308 −0.628722
\(635\) 7.76469 0.308132
\(636\) −17.7428 −0.703547
\(637\) 46.0590 1.82493
\(638\) −0.846752 −0.0335232
\(639\) 16.0422 0.634621
\(640\) 8.37897 0.331208
\(641\) 13.1626 0.519891 0.259946 0.965623i \(-0.416295\pi\)
0.259946 + 0.965623i \(0.416295\pi\)
\(642\) 31.6202 1.24795
\(643\) −35.4538 −1.39816 −0.699081 0.715042i \(-0.746407\pi\)
−0.699081 + 0.715042i \(0.746407\pi\)
\(644\) −3.01366 −0.118755
\(645\) −22.4153 −0.882602
\(646\) 0 0
\(647\) 3.38283 0.132993 0.0664964 0.997787i \(-0.478818\pi\)
0.0664964 + 0.997787i \(0.478818\pi\)
\(648\) −28.4137 −1.11619
\(649\) 0.877520 0.0344457
\(650\) 5.65189 0.221685
\(651\) −5.47710 −0.214664
\(652\) 10.5264 0.412246
\(653\) −0.621795 −0.0243327 −0.0121664 0.999926i \(-0.503873\pi\)
−0.0121664 + 0.999926i \(0.503873\pi\)
\(654\) 26.8365 1.04939
\(655\) −4.58042 −0.178972
\(656\) 4.19925 0.163953
\(657\) −20.4336 −0.797191
\(658\) 0.979789 0.0381961
\(659\) 11.6203 0.452664 0.226332 0.974050i \(-0.427327\pi\)
0.226332 + 0.974050i \(0.427327\pi\)
\(660\) 0.343122 0.0133560
\(661\) 13.2897 0.516911 0.258456 0.966023i \(-0.416786\pi\)
0.258456 + 0.966023i \(0.416786\pi\)
\(662\) 15.6916 0.609871
\(663\) 61.1216 2.37377
\(664\) −22.2733 −0.864371
\(665\) 0 0
\(666\) 5.32388 0.206296
\(667\) 37.5638 1.45448
\(668\) 7.55347 0.292253
\(669\) 4.39004 0.169729
\(670\) −8.73847 −0.337596
\(671\) −0.618311 −0.0238696
\(672\) 7.58101 0.292444
\(673\) 23.9351 0.922630 0.461315 0.887236i \(-0.347378\pi\)
0.461315 + 0.887236i \(0.347378\pi\)
\(674\) 20.9246 0.805985
\(675\) −1.35950 −0.0523272
\(676\) −45.7593 −1.75997
\(677\) 3.71727 0.142866 0.0714331 0.997445i \(-0.477243\pi\)
0.0714331 + 0.997445i \(0.477243\pi\)
\(678\) 16.9690 0.651689
\(679\) −4.63273 −0.177788
\(680\) −10.4025 −0.398919
\(681\) 24.9113 0.954603
\(682\) 0.381795 0.0146197
\(683\) 1.40634 0.0538120 0.0269060 0.999638i \(-0.491435\pi\)
0.0269060 + 0.999638i \(0.491435\pi\)
\(684\) 0 0
\(685\) 1.19518 0.0456654
\(686\) −6.30509 −0.240729
\(687\) 19.4178 0.740836
\(688\) −4.00513 −0.152694
\(689\) −39.5826 −1.50798
\(690\) 7.71651 0.293762
\(691\) 2.20762 0.0839820 0.0419910 0.999118i \(-0.486630\pi\)
0.0419910 + 0.999118i \(0.486630\pi\)
\(692\) −9.45018 −0.359242
\(693\) 0.150753 0.00572662
\(694\) 28.2040 1.07061
\(695\) 8.06036 0.305747
\(696\) −59.0178 −2.23706
\(697\) −38.4931 −1.45803
\(698\) 23.4942 0.889270
\(699\) −4.99033 −0.188751
\(700\) 0.745506 0.0281775
\(701\) −3.95566 −0.149403 −0.0747016 0.997206i \(-0.523800\pi\)
−0.0747016 + 0.997206i \(0.523800\pi\)
\(702\) −7.68375 −0.290004
\(703\) 0 0
\(704\) −0.436065 −0.0164348
\(705\) 4.94881 0.186383
\(706\) −19.2925 −0.726084
\(707\) 6.59456 0.248014
\(708\) 24.3972 0.916902
\(709\) −31.8658 −1.19674 −0.598372 0.801218i \(-0.704186\pi\)
−0.598372 + 0.801218i \(0.704186\pi\)
\(710\) −5.44686 −0.204417
\(711\) 15.1493 0.568143
\(712\) −3.18601 −0.119401
\(713\) −16.9373 −0.634306
\(714\) −4.08705 −0.152954
\(715\) 0.765474 0.0286271
\(716\) −3.15403 −0.117872
\(717\) 58.3651 2.17969
\(718\) −8.01742 −0.299207
\(719\) −34.2360 −1.27679 −0.638393 0.769710i \(-0.720401\pi\)
−0.638393 + 0.769710i \(0.720401\pi\)
\(720\) 1.00454 0.0374371
\(721\) −10.9353 −0.407251
\(722\) 0 0
\(723\) −26.4200 −0.982570
\(724\) 26.7782 0.995203
\(725\) −9.29239 −0.345111
\(726\) 20.9741 0.778421
\(727\) −29.5181 −1.09477 −0.547383 0.836882i \(-0.684376\pi\)
−0.547383 + 0.836882i \(0.684376\pi\)
\(728\) 10.5630 0.391492
\(729\) −15.6603 −0.580011
\(730\) 6.93787 0.256782
\(731\) 36.7137 1.35790
\(732\) −17.1905 −0.635381
\(733\) 28.2212 1.04237 0.521187 0.853442i \(-0.325489\pi\)
0.521187 + 0.853442i \(0.325489\pi\)
\(734\) 1.67213 0.0617195
\(735\) −15.5560 −0.573794
\(736\) 23.4434 0.864135
\(737\) −1.18351 −0.0435952
\(738\) −20.0114 −0.736630
\(739\) −24.0957 −0.886373 −0.443187 0.896429i \(-0.646152\pi\)
−0.443187 + 0.896429i \(0.646152\pi\)
\(740\) 3.56575 0.131080
\(741\) 0 0
\(742\) 2.64679 0.0971667
\(743\) −12.9103 −0.473632 −0.236816 0.971554i \(-0.576104\pi\)
−0.236816 + 0.971554i \(0.576104\pi\)
\(744\) 26.6107 0.975596
\(745\) 9.89499 0.362524
\(746\) −10.6315 −0.389247
\(747\) −19.7163 −0.721381
\(748\) −0.561994 −0.0205485
\(749\) 9.30473 0.339987
\(750\) −1.90888 −0.0697023
\(751\) −37.7129 −1.37616 −0.688082 0.725633i \(-0.741547\pi\)
−0.688082 + 0.725633i \(0.741547\pi\)
\(752\) 0.884246 0.0322451
\(753\) −16.1099 −0.587076
\(754\) −52.5195 −1.91265
\(755\) 20.7413 0.754852
\(756\) −1.01352 −0.0368612
\(757\) −34.7933 −1.26458 −0.632292 0.774730i \(-0.717886\pi\)
−0.632292 + 0.774730i \(0.717886\pi\)
\(758\) −5.60157 −0.203458
\(759\) 1.04510 0.0379347
\(760\) 0 0
\(761\) 4.00019 0.145007 0.0725034 0.997368i \(-0.476901\pi\)
0.0725034 + 0.997368i \(0.476901\pi\)
\(762\) −14.8218 −0.536939
\(763\) 7.89707 0.285893
\(764\) −13.1253 −0.474855
\(765\) −9.20830 −0.332927
\(766\) −14.0517 −0.507710
\(767\) 54.4279 1.96528
\(768\) −34.2641 −1.23640
\(769\) −9.59479 −0.345997 −0.172998 0.984922i \(-0.555346\pi\)
−0.172998 + 0.984922i \(0.555346\pi\)
\(770\) −0.0511854 −0.00184459
\(771\) −4.41517 −0.159009
\(772\) −6.03994 −0.217382
\(773\) −2.23562 −0.0804096 −0.0402048 0.999191i \(-0.512801\pi\)
−0.0402048 + 0.999191i \(0.512801\pi\)
\(774\) 19.0864 0.686045
\(775\) 4.18987 0.150505
\(776\) 22.5083 0.808003
\(777\) 3.51210 0.125996
\(778\) −15.1230 −0.542184
\(779\) 0 0
\(780\) 21.2820 0.762019
\(781\) −0.737707 −0.0263972
\(782\) −12.6387 −0.451960
\(783\) 12.6330 0.451467
\(784\) −2.77953 −0.0992689
\(785\) −12.9092 −0.460747
\(786\) 8.74346 0.311869
\(787\) −19.6480 −0.700376 −0.350188 0.936679i \(-0.613882\pi\)
−0.350188 + 0.936679i \(0.613882\pi\)
\(788\) 16.5502 0.589576
\(789\) −26.8664 −0.956469
\(790\) −5.14368 −0.183004
\(791\) 4.99338 0.177544
\(792\) −0.732438 −0.0260261
\(793\) −38.3506 −1.36187
\(794\) 2.82676 0.100318
\(795\) 13.3687 0.474138
\(796\) 26.5349 0.940503
\(797\) 15.8851 0.562680 0.281340 0.959608i \(-0.409221\pi\)
0.281340 + 0.959608i \(0.409221\pi\)
\(798\) 0 0
\(799\) −8.10558 −0.286755
\(800\) −5.79933 −0.205037
\(801\) −2.82025 −0.0996486
\(802\) −10.1934 −0.359942
\(803\) 0.939645 0.0331593
\(804\) −32.9045 −1.16045
\(805\) 2.27070 0.0800318
\(806\) 23.6807 0.834117
\(807\) 7.02942 0.247447
\(808\) −32.0400 −1.12716
\(809\) 9.82989 0.345600 0.172800 0.984957i \(-0.444718\pi\)
0.172800 + 0.984957i \(0.444718\pi\)
\(810\) 8.53984 0.300059
\(811\) −29.0667 −1.02067 −0.510335 0.859976i \(-0.670479\pi\)
−0.510335 + 0.859976i \(0.670479\pi\)
\(812\) −6.92752 −0.243108
\(813\) 20.5060 0.719178
\(814\) −0.244820 −0.00858094
\(815\) −7.93134 −0.277823
\(816\) −3.68851 −0.129124
\(817\) 0 0
\(818\) −2.07679 −0.0726133
\(819\) 9.35038 0.326729
\(820\) −13.4030 −0.468052
\(821\) 9.14446 0.319144 0.159572 0.987186i \(-0.448989\pi\)
0.159572 + 0.987186i \(0.448989\pi\)
\(822\) −2.28145 −0.0795746
\(823\) 25.8030 0.899436 0.449718 0.893171i \(-0.351524\pi\)
0.449718 + 0.893171i \(0.351524\pi\)
\(824\) 53.1295 1.85085
\(825\) −0.258532 −0.00900095
\(826\) −3.63946 −0.126633
\(827\) −4.28552 −0.149022 −0.0745110 0.997220i \(-0.523740\pi\)
−0.0745110 + 0.997220i \(0.523740\pi\)
\(828\) 12.9611 0.450428
\(829\) −5.70557 −0.198163 −0.0990813 0.995079i \(-0.531590\pi\)
−0.0990813 + 0.995079i \(0.531590\pi\)
\(830\) 6.69432 0.232363
\(831\) −22.5486 −0.782202
\(832\) −27.0468 −0.937679
\(833\) 25.4790 0.882795
\(834\) −15.3862 −0.532781
\(835\) −5.69132 −0.196956
\(836\) 0 0
\(837\) −5.69614 −0.196887
\(838\) −0.531142 −0.0183480
\(839\) 15.5188 0.535769 0.267884 0.963451i \(-0.413675\pi\)
0.267884 + 0.963451i \(0.413675\pi\)
\(840\) −3.56757 −0.123093
\(841\) 57.3484 1.97753
\(842\) −3.58570 −0.123571
\(843\) 43.4630 1.49695
\(844\) 22.3104 0.767954
\(845\) 34.4783 1.18609
\(846\) −4.21385 −0.144875
\(847\) 6.17195 0.212071
\(848\) 2.38869 0.0820280
\(849\) −17.3713 −0.596183
\(850\) 3.12652 0.107239
\(851\) 10.8608 0.372303
\(852\) −20.5100 −0.702662
\(853\) −49.8235 −1.70592 −0.852962 0.521972i \(-0.825196\pi\)
−0.852962 + 0.521972i \(0.825196\pi\)
\(854\) 2.56441 0.0877523
\(855\) 0 0
\(856\) −45.2074 −1.54516
\(857\) 43.1178 1.47288 0.736438 0.676505i \(-0.236506\pi\)
0.736438 + 0.676505i \(0.236506\pi\)
\(858\) −1.46120 −0.0498844
\(859\) 17.9279 0.611691 0.305845 0.952081i \(-0.401061\pi\)
0.305845 + 0.952081i \(0.401061\pi\)
\(860\) 12.7834 0.435910
\(861\) −13.2013 −0.449900
\(862\) −24.6152 −0.838398
\(863\) −55.6045 −1.89280 −0.946399 0.322999i \(-0.895309\pi\)
−0.946399 + 0.322999i \(0.895309\pi\)
\(864\) 7.88419 0.268226
\(865\) 7.12043 0.242102
\(866\) −14.4427 −0.490784
\(867\) −5.75091 −0.195311
\(868\) 3.12357 0.106021
\(869\) −0.696644 −0.0236320
\(870\) 17.7380 0.601375
\(871\) −73.4069 −2.48730
\(872\) −38.3683 −1.29931
\(873\) 19.9244 0.674337
\(874\) 0 0
\(875\) −0.561717 −0.0189895
\(876\) 26.1244 0.882661
\(877\) −12.1639 −0.410744 −0.205372 0.978684i \(-0.565840\pi\)
−0.205372 + 0.978684i \(0.565840\pi\)
\(878\) −21.2303 −0.716488
\(879\) 8.69836 0.293388
\(880\) −0.0461942 −0.00155720
\(881\) 28.8330 0.971408 0.485704 0.874123i \(-0.338563\pi\)
0.485704 + 0.874123i \(0.338563\pi\)
\(882\) 13.2458 0.446009
\(883\) −0.648525 −0.0218246 −0.0109123 0.999940i \(-0.503474\pi\)
−0.0109123 + 0.999940i \(0.503474\pi\)
\(884\) −34.8575 −1.17238
\(885\) −18.3826 −0.617923
\(886\) −9.74174 −0.327280
\(887\) −45.2362 −1.51888 −0.759441 0.650577i \(-0.774527\pi\)
−0.759441 + 0.650577i \(0.774527\pi\)
\(888\) −17.0637 −0.572621
\(889\) −4.36156 −0.146282
\(890\) 0.957567 0.0320977
\(891\) 1.15661 0.0387479
\(892\) −2.50363 −0.0838277
\(893\) 0 0
\(894\) −18.8883 −0.631720
\(895\) 2.37647 0.0794366
\(896\) −4.70661 −0.157237
\(897\) 64.8220 2.16434
\(898\) −24.1583 −0.806174
\(899\) −38.9339 −1.29852
\(900\) −3.20625 −0.106875
\(901\) −21.8963 −0.729472
\(902\) 0.920230 0.0306403
\(903\) 12.5911 0.419004
\(904\) −24.2606 −0.806894
\(905\) −20.1766 −0.670692
\(906\) −39.5925 −1.31537
\(907\) 15.1528 0.503142 0.251571 0.967839i \(-0.419053\pi\)
0.251571 + 0.967839i \(0.419053\pi\)
\(908\) −14.2068 −0.471471
\(909\) −28.3617 −0.940699
\(910\) −3.17476 −0.105242
\(911\) −31.7373 −1.05150 −0.525752 0.850638i \(-0.676216\pi\)
−0.525752 + 0.850638i \(0.676216\pi\)
\(912\) 0 0
\(913\) 0.906658 0.0300060
\(914\) −19.9591 −0.660187
\(915\) 12.9526 0.428199
\(916\) −11.0739 −0.365893
\(917\) 2.57290 0.0849646
\(918\) −4.25051 −0.140288
\(919\) 5.31327 0.175269 0.0876343 0.996153i \(-0.472069\pi\)
0.0876343 + 0.996153i \(0.472069\pi\)
\(920\) −11.0323 −0.363725
\(921\) −54.1829 −1.78539
\(922\) −28.3021 −0.932079
\(923\) −45.7560 −1.50608
\(924\) −0.192737 −0.00634059
\(925\) −2.68669 −0.0883379
\(926\) 4.01371 0.131899
\(927\) 47.0302 1.54467
\(928\) 53.8896 1.76901
\(929\) −3.54458 −0.116294 −0.0581469 0.998308i \(-0.518519\pi\)
−0.0581469 + 0.998308i \(0.518519\pi\)
\(930\) −7.99795 −0.262263
\(931\) 0 0
\(932\) 2.84597 0.0932228
\(933\) 54.5849 1.78703
\(934\) −12.8648 −0.420949
\(935\) 0.423446 0.0138482
\(936\) −45.4292 −1.48490
\(937\) 31.2800 1.02187 0.510936 0.859619i \(-0.329299\pi\)
0.510936 + 0.859619i \(0.329299\pi\)
\(938\) 4.90854 0.160270
\(939\) −34.9263 −1.13978
\(940\) −2.82230 −0.0920532
\(941\) −45.5106 −1.48360 −0.741802 0.670619i \(-0.766029\pi\)
−0.741802 + 0.670619i \(0.766029\pi\)
\(942\) 24.6420 0.802879
\(943\) −40.8235 −1.32940
\(944\) −3.28457 −0.106903
\(945\) 0.763655 0.0248417
\(946\) −0.877691 −0.0285362
\(947\) −21.0071 −0.682640 −0.341320 0.939947i \(-0.610874\pi\)
−0.341320 + 0.939947i \(0.610874\pi\)
\(948\) −19.3684 −0.629056
\(949\) 58.2812 1.89189
\(950\) 0 0
\(951\) 44.9148 1.45646
\(952\) 5.84327 0.189382
\(953\) 32.6629 1.05806 0.529028 0.848604i \(-0.322557\pi\)
0.529028 + 0.848604i \(0.322557\pi\)
\(954\) −11.3833 −0.368546
\(955\) 9.88950 0.320017
\(956\) −33.2855 −1.07653
\(957\) 2.40238 0.0776580
\(958\) −12.4404 −0.401932
\(959\) −0.671351 −0.0216791
\(960\) 9.13482 0.294825
\(961\) −13.4450 −0.433708
\(962\) −15.1849 −0.489580
\(963\) −40.0176 −1.28955
\(964\) 15.0673 0.485284
\(965\) 4.55092 0.146499
\(966\) −4.33449 −0.139460
\(967\) −21.8080 −0.701299 −0.350650 0.936507i \(-0.614039\pi\)
−0.350650 + 0.936507i \(0.614039\pi\)
\(968\) −29.9867 −0.963809
\(969\) 0 0
\(970\) −6.76497 −0.217210
\(971\) −55.8370 −1.79189 −0.895947 0.444162i \(-0.853502\pi\)
−0.895947 + 0.444162i \(0.853502\pi\)
\(972\) 26.7436 0.857801
\(973\) −4.52764 −0.145149
\(974\) −12.1697 −0.389943
\(975\) −16.0354 −0.513544
\(976\) 2.31435 0.0740804
\(977\) −0.543879 −0.0174002 −0.00870012 0.999962i \(-0.502769\pi\)
−0.00870012 + 0.999962i \(0.502769\pi\)
\(978\) 15.1400 0.484122
\(979\) 0.129690 0.00414491
\(980\) 8.87158 0.283392
\(981\) −33.9635 −1.08437
\(982\) −17.8012 −0.568060
\(983\) −11.0176 −0.351407 −0.175704 0.984443i \(-0.556220\pi\)
−0.175704 + 0.984443i \(0.556220\pi\)
\(984\) 64.1392 2.04468
\(985\) −12.4701 −0.397330
\(986\) −29.0528 −0.925230
\(987\) −2.77983 −0.0884830
\(988\) 0 0
\(989\) 38.9364 1.23811
\(990\) 0.220137 0.00699642
\(991\) −28.1970 −0.895707 −0.447854 0.894107i \(-0.647811\pi\)
−0.447854 + 0.894107i \(0.647811\pi\)
\(992\) −24.2984 −0.771477
\(993\) −44.5198 −1.41279
\(994\) 3.05959 0.0970445
\(995\) −19.9932 −0.633828
\(996\) 25.2073 0.798724
\(997\) −30.5688 −0.968123 −0.484062 0.875034i \(-0.660839\pi\)
−0.484062 + 0.875034i \(0.660839\pi\)
\(998\) 16.8777 0.534256
\(999\) 3.65256 0.115562
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 1805.2.a.n.1.2 yes 4
5.4 even 2 9025.2.a.bh.1.3 4
19.18 odd 2 1805.2.a.j.1.3 4
95.94 odd 2 9025.2.a.bo.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.j.1.3 4 19.18 odd 2
1805.2.a.n.1.2 yes 4 1.1 even 1 trivial
9025.2.a.bh.1.3 4 5.4 even 2
9025.2.a.bo.1.2 4 95.94 odd 2