Properties

Label 722.2.a.m.1.3
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.90211\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.442463 q^{3} +1.00000 q^{4} -0.891491 q^{5} +0.442463 q^{6} +2.52015 q^{7} -1.00000 q^{8} -2.80423 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.442463 q^{3} +1.00000 q^{4} -0.891491 q^{5} +0.442463 q^{6} +2.52015 q^{7} -1.00000 q^{8} -2.80423 q^{9} +0.891491 q^{10} +1.95199 q^{11} -0.442463 q^{12} -6.45965 q^{13} -2.52015 q^{14} +0.394452 q^{15} +1.00000 q^{16} -3.42226 q^{17} +2.80423 q^{18} -0.891491 q^{20} -1.11507 q^{21} -1.95199 q^{22} +8.18504 q^{23} +0.442463 q^{24} -4.20524 q^{25} +6.45965 q^{26} +2.56816 q^{27} +2.52015 q^{28} +4.58064 q^{29} -0.394452 q^{30} -8.79360 q^{31} -1.00000 q^{32} -0.863684 q^{33} +3.42226 q^{34} -2.24669 q^{35} -2.80423 q^{36} -5.97980 q^{37} +2.85816 q^{39} +0.891491 q^{40} -3.48932 q^{41} +1.11507 q^{42} -6.24669 q^{43} +1.95199 q^{44} +2.49994 q^{45} -8.18504 q^{46} -10.5498 q^{47} -0.442463 q^{48} -0.648859 q^{49} +4.20524 q^{50} +1.51423 q^{51} -6.45965 q^{52} +3.76278 q^{53} -2.56816 q^{54} -1.74018 q^{55} -2.52015 q^{56} -4.58064 q^{58} +2.84162 q^{59} +0.394452 q^{60} -2.45495 q^{61} +8.79360 q^{62} -7.06706 q^{63} +1.00000 q^{64} +5.75872 q^{65} +0.863684 q^{66} -2.67261 q^{67} -3.42226 q^{68} -3.62158 q^{69} +2.24669 q^{70} +0.0564404 q^{71} +2.80423 q^{72} -6.96917 q^{73} +5.97980 q^{74} +1.86067 q^{75} +4.91930 q^{77} -2.85816 q^{78} -9.13162 q^{79} -0.891491 q^{80} +7.27636 q^{81} +3.48932 q^{82} -14.1239 q^{83} -1.11507 q^{84} +3.05092 q^{85} +6.24669 q^{86} -2.02677 q^{87} -1.95199 q^{88} +1.86774 q^{89} -2.49994 q^{90} -16.2793 q^{91} +8.18504 q^{92} +3.89085 q^{93} +10.5498 q^{94} +0.442463 q^{96} +7.82432 q^{97} +0.648859 q^{98} -5.47382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 18 q^{13} + 2 q^{14} - 4 q^{15} + 4 q^{16} + 6 q^{17} - 4 q^{18} - 2 q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} + 2 q^{24} + 6 q^{25} + 18 q^{26} + 4 q^{27} - 2 q^{28} + 2 q^{29} + 4 q^{30} - 26 q^{31} - 4 q^{32} - 16 q^{33} - 6 q^{34} + 6 q^{35} + 4 q^{36} - 4 q^{37} - 6 q^{39} + 2 q^{40} + 12 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{44} - 22 q^{45} + 10 q^{46} - 12 q^{47} - 2 q^{48} - 12 q^{49} - 6 q^{50} + 2 q^{51} - 18 q^{52} - 8 q^{53} - 4 q^{54} - 26 q^{55} + 2 q^{56} - 2 q^{58} + 8 q^{59} - 4 q^{60} + 26 q^{62} - 22 q^{63} + 4 q^{64} + 4 q^{65} + 16 q^{66} - 10 q^{67} + 6 q^{68} + 20 q^{69} - 6 q^{70} - 4 q^{72} - 14 q^{73} + 4 q^{74} - 8 q^{75} + 4 q^{77} + 6 q^{78} - 22 q^{79} - 2 q^{80} - 4 q^{81} - 12 q^{82} - 12 q^{83} - 4 q^{84} - 18 q^{85} + 10 q^{86} - 26 q^{87} - 2 q^{88} + 16 q^{89} + 22 q^{90} + 4 q^{91} - 10 q^{92} + 8 q^{93} + 12 q^{94} + 2 q^{96} - 28 q^{97} + 12 q^{98} + 22 q^{99} + O(q^{100}) \)

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
\(3\) −0.442463 −0.255456 −0.127728 0.991809i \(-0.540769\pi\)
−0.127728 + 0.991809i \(0.540769\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.891491 −0.398687 −0.199344 0.979930i \(-0.563881\pi\)
−0.199344 + 0.979930i \(0.563881\pi\)
\(6\) 0.442463 0.180635
\(7\) 2.52015 0.952526 0.476263 0.879303i \(-0.341991\pi\)
0.476263 + 0.879303i \(0.341991\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.80423 −0.934742
\(10\) 0.891491 0.281914
\(11\) 1.95199 0.588547 0.294273 0.955721i \(-0.404922\pi\)
0.294273 + 0.955721i \(0.404922\pi\)
\(12\) −0.442463 −0.127728
\(13\) −6.45965 −1.79158 −0.895792 0.444473i \(-0.853391\pi\)
−0.895792 + 0.444473i \(0.853391\pi\)
\(14\) −2.52015 −0.673538
\(15\) 0.394452 0.101847
\(16\) 1.00000 0.250000
\(17\) −3.42226 −0.830020 −0.415010 0.909817i \(-0.636222\pi\)
−0.415010 + 0.909817i \(0.636222\pi\)
\(18\) 2.80423 0.660962
\(19\) 0 0
\(20\) −0.891491 −0.199344
\(21\) −1.11507 −0.243329
\(22\) −1.95199 −0.416165
\(23\) 8.18504 1.70670 0.853349 0.521339i \(-0.174567\pi\)
0.853349 + 0.521339i \(0.174567\pi\)
\(24\) 0.442463 0.0903175
\(25\) −4.20524 −0.841049
\(26\) 6.45965 1.26684
\(27\) 2.56816 0.494242
\(28\) 2.52015 0.476263
\(29\) 4.58064 0.850604 0.425302 0.905051i \(-0.360168\pi\)
0.425302 + 0.905051i \(0.360168\pi\)
\(30\) −0.394452 −0.0720168
\(31\) −8.79360 −1.57938 −0.789689 0.613507i \(-0.789758\pi\)
−0.789689 + 0.613507i \(0.789758\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.863684 −0.150348
\(34\) 3.42226 0.586913
\(35\) −2.24669 −0.379760
\(36\) −2.80423 −0.467371
\(37\) −5.97980 −0.983073 −0.491536 0.870857i \(-0.663564\pi\)
−0.491536 + 0.870857i \(0.663564\pi\)
\(38\) 0 0
\(39\) 2.85816 0.457672
\(40\) 0.891491 0.140957
\(41\) −3.48932 −0.544941 −0.272470 0.962164i \(-0.587841\pi\)
−0.272470 + 0.962164i \(0.587841\pi\)
\(42\) 1.11507 0.172060
\(43\) −6.24669 −0.952611 −0.476306 0.879280i \(-0.658024\pi\)
−0.476306 + 0.879280i \(0.658024\pi\)
\(44\) 1.95199 0.294273
\(45\) 2.49994 0.372670
\(46\) −8.18504 −1.20682
\(47\) −10.5498 −1.53885 −0.769425 0.638738i \(-0.779457\pi\)
−0.769425 + 0.638738i \(0.779457\pi\)
\(48\) −0.442463 −0.0638641
\(49\) −0.648859 −0.0926941
\(50\) 4.20524 0.594711
\(51\) 1.51423 0.212034
\(52\) −6.45965 −0.895792
\(53\) 3.76278 0.516858 0.258429 0.966030i \(-0.416795\pi\)
0.258429 + 0.966030i \(0.416795\pi\)
\(54\) −2.56816 −0.349482
\(55\) −1.74018 −0.234646
\(56\) −2.52015 −0.336769
\(57\) 0 0
\(58\) −4.58064 −0.601468
\(59\) 2.84162 0.369947 0.184973 0.982744i \(-0.440780\pi\)
0.184973 + 0.982744i \(0.440780\pi\)
\(60\) 0.394452 0.0509236
\(61\) −2.45495 −0.314324 −0.157162 0.987573i \(-0.550235\pi\)
−0.157162 + 0.987573i \(0.550235\pi\)
\(62\) 8.79360 1.11679
\(63\) −7.06706 −0.890366
\(64\) 1.00000 0.125000
\(65\) 5.75872 0.714282
\(66\) 0.863684 0.106312
\(67\) −2.67261 −0.326511 −0.163256 0.986584i \(-0.552200\pi\)
−0.163256 + 0.986584i \(0.552200\pi\)
\(68\) −3.42226 −0.415010
\(69\) −3.62158 −0.435987
\(70\) 2.24669 0.268531
\(71\) 0.0564404 0.00669824 0.00334912 0.999994i \(-0.498934\pi\)
0.00334912 + 0.999994i \(0.498934\pi\)
\(72\) 2.80423 0.330481
\(73\) −6.96917 −0.815680 −0.407840 0.913053i \(-0.633718\pi\)
−0.407840 + 0.913053i \(0.633718\pi\)
\(74\) 5.97980 0.695137
\(75\) 1.86067 0.214851
\(76\) 0 0
\(77\) 4.91930 0.560606
\(78\) −2.85816 −0.323623
\(79\) −9.13162 −1.02739 −0.513694 0.857974i \(-0.671723\pi\)
−0.513694 + 0.857974i \(0.671723\pi\)
\(80\) −0.891491 −0.0996718
\(81\) 7.27636 0.808485
\(82\) 3.48932 0.385331
\(83\) −14.1239 −1.55030 −0.775150 0.631778i \(-0.782326\pi\)
−0.775150 + 0.631778i \(0.782326\pi\)
\(84\) −1.11507 −0.121664
\(85\) 3.05092 0.330918
\(86\) 6.24669 0.673598
\(87\) −2.02677 −0.217292
\(88\) −1.95199 −0.208083
\(89\) 1.86774 0.197980 0.0989901 0.995088i \(-0.468439\pi\)
0.0989901 + 0.995088i \(0.468439\pi\)
\(90\) −2.49994 −0.263517
\(91\) −16.2793 −1.70653
\(92\) 8.18504 0.853349
\(93\) 3.89085 0.403462
\(94\) 10.5498 1.08813
\(95\) 0 0
\(96\) 0.442463 0.0451587
\(97\) 7.82432 0.794439 0.397219 0.917724i \(-0.369975\pi\)
0.397219 + 0.917724i \(0.369975\pi\)
\(98\) 0.648859 0.0655447
\(99\) −5.47382 −0.550139
\(100\) −4.20524 −0.420524
\(101\) −5.38487 −0.535815 −0.267907 0.963445i \(-0.586332\pi\)
−0.267907 + 0.963445i \(0.586332\pi\)
\(102\) −1.51423 −0.149931
\(103\) −4.69468 −0.462580 −0.231290 0.972885i \(-0.574295\pi\)
−0.231290 + 0.972885i \(0.574295\pi\)
\(104\) 6.45965 0.633421
\(105\) 0.994078 0.0970121
\(106\) −3.76278 −0.365473
\(107\) 16.0373 1.55038 0.775191 0.631727i \(-0.217654\pi\)
0.775191 + 0.631727i \(0.217654\pi\)
\(108\) 2.56816 0.247121
\(109\) 14.1008 1.35061 0.675305 0.737538i \(-0.264012\pi\)
0.675305 + 0.737538i \(0.264012\pi\)
\(110\) 1.74018 0.165920
\(111\) 2.64584 0.251132
\(112\) 2.52015 0.238132
\(113\) −12.7224 −1.19682 −0.598410 0.801190i \(-0.704201\pi\)
−0.598410 + 0.801190i \(0.704201\pi\)
\(114\) 0 0
\(115\) −7.29689 −0.680439
\(116\) 4.58064 0.425302
\(117\) 18.1143 1.67467
\(118\) −2.84162 −0.261592
\(119\) −8.62460 −0.790616
\(120\) −0.394452 −0.0360084
\(121\) −7.18974 −0.653613
\(122\) 2.45495 0.222261
\(123\) 1.54390 0.139209
\(124\) −8.79360 −0.789689
\(125\) 8.20640 0.734002
\(126\) 7.06706 0.629584
\(127\) 7.17442 0.636627 0.318313 0.947986i \(-0.396884\pi\)
0.318313 + 0.947986i \(0.396884\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.76393 0.243351
\(130\) −5.75872 −0.505073
\(131\) −3.52265 −0.307776 −0.153888 0.988088i \(-0.549179\pi\)
−0.153888 + 0.988088i \(0.549179\pi\)
\(132\) −0.863684 −0.0751740
\(133\) 0 0
\(134\) 2.67261 0.230878
\(135\) −2.28949 −0.197048
\(136\) 3.42226 0.293456
\(137\) −11.0035 −0.940096 −0.470048 0.882641i \(-0.655763\pi\)
−0.470048 + 0.882641i \(0.655763\pi\)
\(138\) 3.62158 0.308289
\(139\) 21.7586 1.84554 0.922771 0.385350i \(-0.125919\pi\)
0.922771 + 0.385350i \(0.125919\pi\)
\(140\) −2.24669 −0.189880
\(141\) 4.66791 0.393109
\(142\) −0.0564404 −0.00473637
\(143\) −12.6092 −1.05443
\(144\) −2.80423 −0.233686
\(145\) −4.08361 −0.339125
\(146\) 6.96917 0.576773
\(147\) 0.287096 0.0236793
\(148\) −5.97980 −0.491536
\(149\) 14.3896 1.17884 0.589420 0.807827i \(-0.299356\pi\)
0.589420 + 0.807827i \(0.299356\pi\)
\(150\) −1.86067 −0.151923
\(151\) 12.4068 1.00965 0.504824 0.863222i \(-0.331557\pi\)
0.504824 + 0.863222i \(0.331557\pi\)
\(152\) 0 0
\(153\) 9.59679 0.775855
\(154\) −4.91930 −0.396408
\(155\) 7.83942 0.629678
\(156\) 2.85816 0.228836
\(157\) −0.451220 −0.0360113 −0.0180056 0.999838i \(-0.505732\pi\)
−0.0180056 + 0.999838i \(0.505732\pi\)
\(158\) 9.13162 0.726472
\(159\) −1.66489 −0.132035
\(160\) 0.891491 0.0704786
\(161\) 20.6275 1.62567
\(162\) −7.27636 −0.571685
\(163\) −0.0692542 −0.00542441 −0.00271220 0.999996i \(-0.500863\pi\)
−0.00271220 + 0.999996i \(0.500863\pi\)
\(164\) −3.48932 −0.272470
\(165\) 0.769967 0.0599418
\(166\) 14.1239 1.09623
\(167\) 2.17371 0.168206 0.0841032 0.996457i \(-0.473197\pi\)
0.0841032 + 0.996457i \(0.473197\pi\)
\(168\) 1.11507 0.0860298
\(169\) 28.7271 2.20977
\(170\) −3.05092 −0.233995
\(171\) 0 0
\(172\) −6.24669 −0.476306
\(173\) −16.8493 −1.28103 −0.640514 0.767947i \(-0.721279\pi\)
−0.640514 + 0.767947i \(0.721279\pi\)
\(174\) 2.02677 0.153649
\(175\) −10.5978 −0.801121
\(176\) 1.95199 0.147137
\(177\) −1.25731 −0.0945053
\(178\) −1.86774 −0.139993
\(179\) 18.9655 1.41755 0.708775 0.705435i \(-0.249248\pi\)
0.708775 + 0.705435i \(0.249248\pi\)
\(180\) 2.49994 0.186335
\(181\) −1.98687 −0.147683 −0.0738415 0.997270i \(-0.523526\pi\)
−0.0738415 + 0.997270i \(0.523526\pi\)
\(182\) 16.2793 1.20670
\(183\) 1.08623 0.0802961
\(184\) −8.18504 −0.603409
\(185\) 5.33094 0.391938
\(186\) −3.89085 −0.285291
\(187\) −6.68021 −0.488506
\(188\) −10.5498 −0.769425
\(189\) 6.47214 0.470779
\(190\) 0 0
\(191\) −20.9259 −1.51415 −0.757074 0.653329i \(-0.773372\pi\)
−0.757074 + 0.653329i \(0.773372\pi\)
\(192\) −0.442463 −0.0319321
\(193\) 7.78879 0.560650 0.280325 0.959905i \(-0.409558\pi\)
0.280325 + 0.959905i \(0.409558\pi\)
\(194\) −7.82432 −0.561753
\(195\) −2.54802 −0.182468
\(196\) −0.648859 −0.0463471
\(197\) 1.82514 0.130036 0.0650180 0.997884i \(-0.479290\pi\)
0.0650180 + 0.997884i \(0.479290\pi\)
\(198\) 5.47382 0.389007
\(199\) 2.35926 0.167243 0.0836216 0.996498i \(-0.473351\pi\)
0.0836216 + 0.996498i \(0.473351\pi\)
\(200\) 4.20524 0.297356
\(201\) 1.18253 0.0834094
\(202\) 5.38487 0.378878
\(203\) 11.5439 0.810223
\(204\) 1.51423 0.106017
\(205\) 3.11070 0.217261
\(206\) 4.69468 0.327094
\(207\) −22.9527 −1.59532
\(208\) −6.45965 −0.447896
\(209\) 0 0
\(210\) −0.994078 −0.0685979
\(211\) −0.361050 −0.0248557 −0.0124279 0.999923i \(-0.503956\pi\)
−0.0124279 + 0.999923i \(0.503956\pi\)
\(212\) 3.76278 0.258429
\(213\) −0.0249728 −0.00171111
\(214\) −16.0373 −1.09629
\(215\) 5.56887 0.379794
\(216\) −2.56816 −0.174741
\(217\) −22.1612 −1.50440
\(218\) −14.1008 −0.955026
\(219\) 3.08361 0.208371
\(220\) −1.74018 −0.117323
\(221\) 22.1066 1.48705
\(222\) −2.64584 −0.177577
\(223\) 18.0486 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(224\) −2.52015 −0.168384
\(225\) 11.7925 0.786163
\(226\) 12.7224 0.846280
\(227\) 12.2845 0.815349 0.407675 0.913127i \(-0.366340\pi\)
0.407675 + 0.913127i \(0.366340\pi\)
\(228\) 0 0
\(229\) 6.79174 0.448811 0.224405 0.974496i \(-0.427956\pi\)
0.224405 + 0.974496i \(0.427956\pi\)
\(230\) 7.29689 0.481143
\(231\) −2.17661 −0.143210
\(232\) −4.58064 −0.300734
\(233\) 21.4294 1.40388 0.701942 0.712234i \(-0.252317\pi\)
0.701942 + 0.712234i \(0.252317\pi\)
\(234\) −18.1143 −1.18417
\(235\) 9.40507 0.613519
\(236\) 2.84162 0.184973
\(237\) 4.04041 0.262453
\(238\) 8.62460 0.559050
\(239\) −11.8376 −0.765713 −0.382856 0.923808i \(-0.625060\pi\)
−0.382856 + 0.923808i \(0.625060\pi\)
\(240\) 0.394452 0.0254618
\(241\) −21.7463 −1.40080 −0.700401 0.713749i \(-0.746996\pi\)
−0.700401 + 0.713749i \(0.746996\pi\)
\(242\) 7.18974 0.462174
\(243\) −10.9240 −0.700775
\(244\) −2.45495 −0.157162
\(245\) 0.578452 0.0369560
\(246\) −1.54390 −0.0984353
\(247\) 0 0
\(248\) 8.79360 0.558394
\(249\) 6.24931 0.396034
\(250\) −8.20640 −0.519018
\(251\) 23.6410 1.49221 0.746104 0.665829i \(-0.231922\pi\)
0.746104 + 0.665829i \(0.231922\pi\)
\(252\) −7.06706 −0.445183
\(253\) 15.9771 1.00447
\(254\) −7.17442 −0.450163
\(255\) −1.34992 −0.0845352
\(256\) 1.00000 0.0625000
\(257\) −0.923469 −0.0576044 −0.0288022 0.999585i \(-0.509169\pi\)
−0.0288022 + 0.999585i \(0.509169\pi\)
\(258\) −2.76393 −0.172075
\(259\) −15.0700 −0.936402
\(260\) 5.75872 0.357141
\(261\) −12.8452 −0.795096
\(262\) 3.52265 0.217630
\(263\) 4.31967 0.266362 0.133181 0.991092i \(-0.457481\pi\)
0.133181 + 0.991092i \(0.457481\pi\)
\(264\) 0.863684 0.0531561
\(265\) −3.35449 −0.206064
\(266\) 0 0
\(267\) −0.826407 −0.0505753
\(268\) −2.67261 −0.163256
\(269\) −10.7954 −0.658205 −0.329102 0.944294i \(-0.606746\pi\)
−0.329102 + 0.944294i \(0.606746\pi\)
\(270\) 2.28949 0.139334
\(271\) 0.807243 0.0490365 0.0245183 0.999699i \(-0.492195\pi\)
0.0245183 + 0.999699i \(0.492195\pi\)
\(272\) −3.42226 −0.207505
\(273\) 7.20298 0.435944
\(274\) 11.0035 0.664749
\(275\) −8.20859 −0.494996
\(276\) −3.62158 −0.217994
\(277\) −9.14869 −0.549691 −0.274846 0.961488i \(-0.588627\pi\)
−0.274846 + 0.961488i \(0.588627\pi\)
\(278\) −21.7586 −1.30499
\(279\) 24.6593 1.47631
\(280\) 2.24669 0.134265
\(281\) 21.2158 1.26563 0.632813 0.774305i \(-0.281900\pi\)
0.632813 + 0.774305i \(0.281900\pi\)
\(282\) −4.66791 −0.277970
\(283\) 5.18985 0.308505 0.154252 0.988031i \(-0.450703\pi\)
0.154252 + 0.988031i \(0.450703\pi\)
\(284\) 0.0564404 0.00334912
\(285\) 0 0
\(286\) 12.6092 0.745596
\(287\) −8.79360 −0.519070
\(288\) 2.80423 0.165241
\(289\) −5.28814 −0.311067
\(290\) 4.08361 0.239798
\(291\) −3.46197 −0.202945
\(292\) −6.96917 −0.407840
\(293\) −17.5660 −1.02621 −0.513107 0.858324i \(-0.671506\pi\)
−0.513107 + 0.858324i \(0.671506\pi\)
\(294\) −0.287096 −0.0167438
\(295\) −2.53328 −0.147493
\(296\) 5.97980 0.347569
\(297\) 5.01302 0.290885
\(298\) −14.3896 −0.833565
\(299\) −52.8725 −3.05769
\(300\) 1.86067 0.107426
\(301\) −15.7426 −0.907387
\(302\) −12.4068 −0.713929
\(303\) 2.38261 0.136877
\(304\) 0 0
\(305\) 2.18857 0.125317
\(306\) −9.59679 −0.548612
\(307\) −21.5684 −1.23097 −0.615486 0.788148i \(-0.711040\pi\)
−0.615486 + 0.788148i \(0.711040\pi\)
\(308\) 4.91930 0.280303
\(309\) 2.07722 0.118169
\(310\) −7.83942 −0.445249
\(311\) −2.57368 −0.145940 −0.0729701 0.997334i \(-0.523248\pi\)
−0.0729701 + 0.997334i \(0.523248\pi\)
\(312\) −2.85816 −0.161811
\(313\) −5.18389 −0.293011 −0.146505 0.989210i \(-0.546803\pi\)
−0.146505 + 0.989210i \(0.546803\pi\)
\(314\) 0.451220 0.0254638
\(315\) 6.30023 0.354977
\(316\) −9.13162 −0.513694
\(317\) −20.2562 −1.13770 −0.568850 0.822442i \(-0.692611\pi\)
−0.568850 + 0.822442i \(0.692611\pi\)
\(318\) 1.66489 0.0933625
\(319\) 8.94137 0.500620
\(320\) −0.891491 −0.0498359
\(321\) −7.09591 −0.396055
\(322\) −20.6275 −1.14953
\(323\) 0 0
\(324\) 7.27636 0.404242
\(325\) 27.1644 1.50681
\(326\) 0.0692542 0.00383564
\(327\) −6.23909 −0.345022
\(328\) 3.48932 0.192666
\(329\) −26.5871 −1.46579
\(330\) −0.769967 −0.0423853
\(331\) −21.3280 −1.17229 −0.586146 0.810205i \(-0.699356\pi\)
−0.586146 + 0.810205i \(0.699356\pi\)
\(332\) −14.1239 −0.775150
\(333\) 16.7687 0.918919
\(334\) −2.17371 −0.118940
\(335\) 2.38261 0.130176
\(336\) −1.11507 −0.0608322
\(337\) −22.7275 −1.23804 −0.619022 0.785374i \(-0.712471\pi\)
−0.619022 + 0.785374i \(0.712471\pi\)
\(338\) −28.7271 −1.56255
\(339\) 5.62919 0.305735
\(340\) 3.05092 0.165459
\(341\) −17.1650 −0.929538
\(342\) 0 0
\(343\) −19.2762 −1.04082
\(344\) 6.24669 0.336799
\(345\) 3.22861 0.173822
\(346\) 16.8493 0.905823
\(347\) −4.92855 −0.264579 −0.132289 0.991211i \(-0.542233\pi\)
−0.132289 + 0.991211i \(0.542233\pi\)
\(348\) −2.02677 −0.108646
\(349\) 26.1615 1.40039 0.700197 0.713950i \(-0.253096\pi\)
0.700197 + 0.713950i \(0.253096\pi\)
\(350\) 10.5978 0.566478
\(351\) −16.5894 −0.885477
\(352\) −1.95199 −0.104041
\(353\) 2.70813 0.144139 0.0720697 0.997400i \(-0.477040\pi\)
0.0720697 + 0.997400i \(0.477040\pi\)
\(354\) 1.25731 0.0668253
\(355\) −0.0503161 −0.00267050
\(356\) 1.86774 0.0989901
\(357\) 3.81607 0.201968
\(358\) −18.9655 −1.00236
\(359\) −9.31002 −0.491364 −0.245682 0.969350i \(-0.579012\pi\)
−0.245682 + 0.969350i \(0.579012\pi\)
\(360\) −2.49994 −0.131759
\(361\) 0 0
\(362\) 1.98687 0.104428
\(363\) 3.18120 0.166970
\(364\) −16.2793 −0.853265
\(365\) 6.21296 0.325201
\(366\) −1.08623 −0.0567779
\(367\) 7.75442 0.404777 0.202389 0.979305i \(-0.435130\pi\)
0.202389 + 0.979305i \(0.435130\pi\)
\(368\) 8.18504 0.426675
\(369\) 9.78485 0.509379
\(370\) −5.33094 −0.277142
\(371\) 9.48276 0.492320
\(372\) 3.89085 0.201731
\(373\) −25.0124 −1.29509 −0.647546 0.762027i \(-0.724205\pi\)
−0.647546 + 0.762027i \(0.724205\pi\)
\(374\) 6.68021 0.345426
\(375\) −3.63103 −0.187506
\(376\) 10.5498 0.544065
\(377\) −29.5894 −1.52393
\(378\) −6.47214 −0.332891
\(379\) 19.1802 0.985222 0.492611 0.870250i \(-0.336043\pi\)
0.492611 + 0.870250i \(0.336043\pi\)
\(380\) 0 0
\(381\) −3.17442 −0.162630
\(382\) 20.9259 1.07066
\(383\) −4.83319 −0.246964 −0.123482 0.992347i \(-0.539406\pi\)
−0.123482 + 0.992347i \(0.539406\pi\)
\(384\) 0.442463 0.0225794
\(385\) −4.38551 −0.223506
\(386\) −7.78879 −0.396439
\(387\) 17.5171 0.890446
\(388\) 7.82432 0.397219
\(389\) 21.7291 1.10171 0.550855 0.834601i \(-0.314302\pi\)
0.550855 + 0.834601i \(0.314302\pi\)
\(390\) 2.54802 0.129024
\(391\) −28.0113 −1.41659
\(392\) 0.648859 0.0327723
\(393\) 1.55865 0.0786233
\(394\) −1.82514 −0.0919493
\(395\) 8.14076 0.409606
\(396\) −5.47382 −0.275070
\(397\) 11.7140 0.587909 0.293955 0.955819i \(-0.405029\pi\)
0.293955 + 0.955819i \(0.405029\pi\)
\(398\) −2.35926 −0.118259
\(399\) 0 0
\(400\) −4.20524 −0.210262
\(401\) −11.8123 −0.589880 −0.294940 0.955516i \(-0.595300\pi\)
−0.294940 + 0.955516i \(0.595300\pi\)
\(402\) −1.18253 −0.0589793
\(403\) 56.8036 2.82959
\(404\) −5.38487 −0.267907
\(405\) −6.48681 −0.322332
\(406\) −11.5439 −0.572914
\(407\) −11.6725 −0.578584
\(408\) −1.51423 −0.0749653
\(409\) 25.2663 1.24934 0.624668 0.780890i \(-0.285234\pi\)
0.624668 + 0.780890i \(0.285234\pi\)
\(410\) −3.11070 −0.153627
\(411\) 4.86867 0.240154
\(412\) −4.69468 −0.231290
\(413\) 7.16129 0.352384
\(414\) 22.9527 1.12806
\(415\) 12.5913 0.618084
\(416\) 6.45965 0.316710
\(417\) −9.62739 −0.471455
\(418\) 0 0
\(419\) 31.6143 1.54446 0.772229 0.635344i \(-0.219142\pi\)
0.772229 + 0.635344i \(0.219142\pi\)
\(420\) 0.994078 0.0485060
\(421\) −2.18972 −0.106720 −0.0533602 0.998575i \(-0.516993\pi\)
−0.0533602 + 0.998575i \(0.516993\pi\)
\(422\) 0.361050 0.0175757
\(423\) 29.5841 1.43843
\(424\) −3.76278 −0.182737
\(425\) 14.3914 0.698087
\(426\) 0.0249728 0.00120994
\(427\) −6.18683 −0.299402
\(428\) 16.0373 0.775191
\(429\) 5.57909 0.269361
\(430\) −5.56887 −0.268555
\(431\) −1.80372 −0.0868819 −0.0434410 0.999056i \(-0.513832\pi\)
−0.0434410 + 0.999056i \(0.513832\pi\)
\(432\) 2.56816 0.123561
\(433\) −6.57233 −0.315846 −0.157923 0.987451i \(-0.550480\pi\)
−0.157923 + 0.987451i \(0.550480\pi\)
\(434\) 22.1612 1.06377
\(435\) 1.80685 0.0866316
\(436\) 14.1008 0.675305
\(437\) 0 0
\(438\) −3.08361 −0.147340
\(439\) −9.13060 −0.435780 −0.217890 0.975973i \(-0.569917\pi\)
−0.217890 + 0.975973i \(0.569917\pi\)
\(440\) 1.74018 0.0829599
\(441\) 1.81955 0.0866451
\(442\) −22.1066 −1.05150
\(443\) −27.4686 −1.30507 −0.652536 0.757757i \(-0.726295\pi\)
−0.652536 + 0.757757i \(0.726295\pi\)
\(444\) 2.64584 0.125566
\(445\) −1.66508 −0.0789321
\(446\) −18.0486 −0.854626
\(447\) −6.36686 −0.301142
\(448\) 2.52015 0.119066
\(449\) −7.57293 −0.357389 −0.178694 0.983905i \(-0.557187\pi\)
−0.178694 + 0.983905i \(0.557187\pi\)
\(450\) −11.7925 −0.555902
\(451\) −6.81112 −0.320723
\(452\) −12.7224 −0.598410
\(453\) −5.48954 −0.257921
\(454\) −12.2845 −0.576539
\(455\) 14.5128 0.680372
\(456\) 0 0
\(457\) −21.7675 −1.01824 −0.509120 0.860696i \(-0.670029\pi\)
−0.509120 + 0.860696i \(0.670029\pi\)
\(458\) −6.79174 −0.317357
\(459\) −8.78890 −0.410231
\(460\) −7.29689 −0.340219
\(461\) 11.9444 0.556305 0.278153 0.960537i \(-0.410278\pi\)
0.278153 + 0.960537i \(0.410278\pi\)
\(462\) 2.17661 0.101265
\(463\) −8.75371 −0.406819 −0.203410 0.979094i \(-0.565202\pi\)
−0.203410 + 0.979094i \(0.565202\pi\)
\(464\) 4.58064 0.212651
\(465\) −3.46866 −0.160855
\(466\) −21.4294 −0.992696
\(467\) 29.3568 1.35847 0.679236 0.733920i \(-0.262311\pi\)
0.679236 + 0.733920i \(0.262311\pi\)
\(468\) 18.1143 0.837335
\(469\) −6.73537 −0.311010
\(470\) −9.40507 −0.433824
\(471\) 0.199648 0.00919932
\(472\) −2.84162 −0.130796
\(473\) −12.1935 −0.560656
\(474\) −4.04041 −0.185582
\(475\) 0 0
\(476\) −8.62460 −0.395308
\(477\) −10.5517 −0.483128
\(478\) 11.8376 0.541441
\(479\) −24.8705 −1.13636 −0.568180 0.822904i \(-0.692352\pi\)
−0.568180 + 0.822904i \(0.692352\pi\)
\(480\) −0.394452 −0.0180042
\(481\) 38.6274 1.76126
\(482\) 21.7463 0.990517
\(483\) −9.12692 −0.415289
\(484\) −7.18974 −0.326806
\(485\) −6.97531 −0.316733
\(486\) 10.9240 0.495523
\(487\) −8.53766 −0.386878 −0.193439 0.981112i \(-0.561964\pi\)
−0.193439 + 0.981112i \(0.561964\pi\)
\(488\) 2.45495 0.111130
\(489\) 0.0306425 0.00138570
\(490\) −0.578452 −0.0261318
\(491\) −11.5397 −0.520780 −0.260390 0.965504i \(-0.583851\pi\)
−0.260390 + 0.965504i \(0.583851\pi\)
\(492\) 1.54390 0.0696043
\(493\) −15.6762 −0.706019
\(494\) 0 0
\(495\) 4.87986 0.219333
\(496\) −8.79360 −0.394844
\(497\) 0.142238 0.00638025
\(498\) −6.24931 −0.280038
\(499\) −29.6091 −1.32548 −0.662742 0.748848i \(-0.730607\pi\)
−0.662742 + 0.748848i \(0.730607\pi\)
\(500\) 8.20640 0.367001
\(501\) −0.961785 −0.0429694
\(502\) −23.6410 −1.05515
\(503\) 5.86827 0.261653 0.130827 0.991405i \(-0.458237\pi\)
0.130827 + 0.991405i \(0.458237\pi\)
\(504\) 7.06706 0.314792
\(505\) 4.80057 0.213622
\(506\) −15.9771 −0.710269
\(507\) −12.7107 −0.564501
\(508\) 7.17442 0.318313
\(509\) 25.5687 1.13331 0.566656 0.823954i \(-0.308237\pi\)
0.566656 + 0.823954i \(0.308237\pi\)
\(510\) 1.34992 0.0597754
\(511\) −17.5633 −0.776957
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 0.923469 0.0407325
\(515\) 4.18527 0.184425
\(516\) 2.76393 0.121675
\(517\) −20.5931 −0.905685
\(518\) 15.0700 0.662136
\(519\) 7.45520 0.327247
\(520\) −5.75872 −0.252537
\(521\) 14.0332 0.614807 0.307403 0.951579i \(-0.400540\pi\)
0.307403 + 0.951579i \(0.400540\pi\)
\(522\) 12.8452 0.562217
\(523\) −31.0784 −1.35896 −0.679482 0.733692i \(-0.737795\pi\)
−0.679482 + 0.733692i \(0.737795\pi\)
\(524\) −3.52265 −0.153888
\(525\) 4.68915 0.204651
\(526\) −4.31967 −0.188347
\(527\) 30.0940 1.31092
\(528\) −0.863684 −0.0375870
\(529\) 43.9949 1.91282
\(530\) 3.35449 0.145710
\(531\) −7.96853 −0.345805
\(532\) 0 0
\(533\) 22.5398 0.976307
\(534\) 0.826407 0.0357621
\(535\) −14.2971 −0.618117
\(536\) 2.67261 0.115439
\(537\) −8.39155 −0.362122
\(538\) 10.7954 0.465421
\(539\) −1.26657 −0.0545548
\(540\) −2.28949 −0.0985240
\(541\) 2.89608 0.124512 0.0622561 0.998060i \(-0.480170\pi\)
0.0622561 + 0.998060i \(0.480170\pi\)
\(542\) −0.807243 −0.0346741
\(543\) 0.879118 0.0377266
\(544\) 3.42226 0.146728
\(545\) −12.5707 −0.538471
\(546\) −7.20298 −0.308259
\(547\) −30.3861 −1.29922 −0.649608 0.760269i \(-0.725067\pi\)
−0.649608 + 0.760269i \(0.725067\pi\)
\(548\) −11.0035 −0.470048
\(549\) 6.88423 0.293812
\(550\) 8.20859 0.350015
\(551\) 0 0
\(552\) 3.62158 0.154145
\(553\) −23.0130 −0.978613
\(554\) 9.14869 0.388691
\(555\) −2.35875 −0.100123
\(556\) 21.7586 0.922771
\(557\) −6.13412 −0.259911 −0.129956 0.991520i \(-0.541483\pi\)
−0.129956 + 0.991520i \(0.541483\pi\)
\(558\) −24.6593 −1.04391
\(559\) 40.3514 1.70668
\(560\) −2.24669 −0.0949400
\(561\) 2.95575 0.124792
\(562\) −21.2158 −0.894932
\(563\) 28.1917 1.18814 0.594069 0.804414i \(-0.297521\pi\)
0.594069 + 0.804414i \(0.297521\pi\)
\(564\) 4.66791 0.196554
\(565\) 11.3419 0.477157
\(566\) −5.18985 −0.218146
\(567\) 18.3375 0.770103
\(568\) −0.0564404 −0.00236819
\(569\) 42.1145 1.76553 0.882766 0.469812i \(-0.155678\pi\)
0.882766 + 0.469812i \(0.155678\pi\)
\(570\) 0 0
\(571\) 0.166927 0.00698566 0.00349283 0.999994i \(-0.498888\pi\)
0.00349283 + 0.999994i \(0.498888\pi\)
\(572\) −12.6092 −0.527216
\(573\) 9.25896 0.386799
\(574\) 8.79360 0.367038
\(575\) −34.4201 −1.43542
\(576\) −2.80423 −0.116843
\(577\) −19.9631 −0.831076 −0.415538 0.909576i \(-0.636407\pi\)
−0.415538 + 0.909576i \(0.636407\pi\)
\(578\) 5.28814 0.219957
\(579\) −3.44626 −0.143222
\(580\) −4.08361 −0.169562
\(581\) −35.5943 −1.47670
\(582\) 3.46197 0.143503
\(583\) 7.34490 0.304195
\(584\) 6.96917 0.288387
\(585\) −16.1488 −0.667669
\(586\) 17.5660 0.725643
\(587\) 10.1440 0.418689 0.209345 0.977842i \(-0.432867\pi\)
0.209345 + 0.977842i \(0.432867\pi\)
\(588\) 0.287096 0.0118397
\(589\) 0 0
\(590\) 2.53328 0.104293
\(591\) −0.807559 −0.0332185
\(592\) −5.97980 −0.245768
\(593\) 7.45315 0.306064 0.153032 0.988221i \(-0.451096\pi\)
0.153032 + 0.988221i \(0.451096\pi\)
\(594\) −5.01302 −0.205687
\(595\) 7.68876 0.315208
\(596\) 14.3896 0.589420
\(597\) −1.04388 −0.0427233
\(598\) 52.8725 2.16212
\(599\) 6.37141 0.260329 0.130164 0.991492i \(-0.458449\pi\)
0.130164 + 0.991492i \(0.458449\pi\)
\(600\) −1.86067 −0.0759614
\(601\) −1.74631 −0.0712333 −0.0356166 0.999366i \(-0.511340\pi\)
−0.0356166 + 0.999366i \(0.511340\pi\)
\(602\) 15.7426 0.641620
\(603\) 7.49460 0.305204
\(604\) 12.4068 0.504824
\(605\) 6.40959 0.260587
\(606\) −2.38261 −0.0967869
\(607\) −32.1612 −1.30538 −0.652691 0.757624i \(-0.726360\pi\)
−0.652691 + 0.757624i \(0.726360\pi\)
\(608\) 0 0
\(609\) −5.10775 −0.206977
\(610\) −2.18857 −0.0886125
\(611\) 68.1481 2.75698
\(612\) 9.59679 0.387927
\(613\) 12.7645 0.515552 0.257776 0.966205i \(-0.417010\pi\)
0.257776 + 0.966205i \(0.417010\pi\)
\(614\) 21.5684 0.870428
\(615\) −1.37637 −0.0555007
\(616\) −4.91930 −0.198204
\(617\) −1.68323 −0.0677643 −0.0338822 0.999426i \(-0.510787\pi\)
−0.0338822 + 0.999426i \(0.510787\pi\)
\(618\) −2.07722 −0.0835582
\(619\) 31.5285 1.26724 0.633618 0.773646i \(-0.281569\pi\)
0.633618 + 0.773646i \(0.281569\pi\)
\(620\) 7.83942 0.314839
\(621\) 21.0205 0.843523
\(622\) 2.57368 0.103195
\(623\) 4.70698 0.188581
\(624\) 2.85816 0.114418
\(625\) 13.7103 0.548411
\(626\) 5.18389 0.207190
\(627\) 0 0
\(628\) −0.451220 −0.0180056
\(629\) 20.4644 0.815970
\(630\) −6.30023 −0.251007
\(631\) −31.7662 −1.26459 −0.632297 0.774726i \(-0.717888\pi\)
−0.632297 + 0.774726i \(0.717888\pi\)
\(632\) 9.13162 0.363236
\(633\) 0.159752 0.00634955
\(634\) 20.2562 0.804475
\(635\) −6.39593 −0.253815
\(636\) −1.66489 −0.0660173
\(637\) 4.19140 0.166069
\(638\) −8.94137 −0.353992
\(639\) −0.158272 −0.00626113
\(640\) 0.891491 0.0352393
\(641\) 40.4437 1.59743 0.798715 0.601709i \(-0.205513\pi\)
0.798715 + 0.601709i \(0.205513\pi\)
\(642\) 7.09591 0.280053
\(643\) −26.1033 −1.02941 −0.514706 0.857367i \(-0.672099\pi\)
−0.514706 + 0.857367i \(0.672099\pi\)
\(644\) 20.6275 0.812837
\(645\) −2.46402 −0.0970208
\(646\) 0 0
\(647\) −48.7713 −1.91740 −0.958699 0.284423i \(-0.908198\pi\)
−0.958699 + 0.284423i \(0.908198\pi\)
\(648\) −7.27636 −0.285842
\(649\) 5.54680 0.217731
\(650\) −27.1644 −1.06548
\(651\) 9.80551 0.384308
\(652\) −0.0692542 −0.00271220
\(653\) −43.1831 −1.68988 −0.844942 0.534858i \(-0.820365\pi\)
−0.844942 + 0.534858i \(0.820365\pi\)
\(654\) 6.23909 0.243968
\(655\) 3.14042 0.122706
\(656\) −3.48932 −0.136235
\(657\) 19.5431 0.762451
\(658\) 26.5871 1.03647
\(659\) 14.2769 0.556148 0.278074 0.960560i \(-0.410304\pi\)
0.278074 + 0.960560i \(0.410304\pi\)
\(660\) 0.769967 0.0299709
\(661\) −35.7497 −1.39050 −0.695251 0.718768i \(-0.744707\pi\)
−0.695251 + 0.718768i \(0.744707\pi\)
\(662\) 21.3280 0.828936
\(663\) −9.78136 −0.379877
\(664\) 14.1239 0.548114
\(665\) 0 0
\(666\) −16.7687 −0.649774
\(667\) 37.4928 1.45173
\(668\) 2.17371 0.0841032
\(669\) −7.98585 −0.308751
\(670\) −2.38261 −0.0920482
\(671\) −4.79204 −0.184994
\(672\) 1.11507 0.0430149
\(673\) −10.3574 −0.399248 −0.199624 0.979873i \(-0.563972\pi\)
−0.199624 + 0.979873i \(0.563972\pi\)
\(674\) 22.7275 0.875429
\(675\) −10.7997 −0.415682
\(676\) 28.7271 1.10489
\(677\) 10.7171 0.411891 0.205945 0.978564i \(-0.433973\pi\)
0.205945 + 0.978564i \(0.433973\pi\)
\(678\) −5.62919 −0.216188
\(679\) 19.7184 0.756724
\(680\) −3.05092 −0.116997
\(681\) −5.43543 −0.208286
\(682\) 17.1650 0.657283
\(683\) −0.122930 −0.00470377 −0.00235188 0.999997i \(-0.500749\pi\)
−0.00235188 + 0.999997i \(0.500749\pi\)
\(684\) 0 0
\(685\) 9.80957 0.374804
\(686\) 19.2762 0.735971
\(687\) −3.00510 −0.114652
\(688\) −6.24669 −0.238153
\(689\) −24.3062 −0.925994
\(690\) −3.22861 −0.122911
\(691\) −25.2815 −0.961752 −0.480876 0.876789i \(-0.659681\pi\)
−0.480876 + 0.876789i \(0.659681\pi\)
\(692\) −16.8493 −0.640514
\(693\) −13.7948 −0.524022
\(694\) 4.92855 0.187085
\(695\) −19.3976 −0.735793
\(696\) 2.02677 0.0768244
\(697\) 11.9414 0.452312
\(698\) −26.1615 −0.990228
\(699\) −9.48171 −0.358631
\(700\) −10.5978 −0.400560
\(701\) −5.17287 −0.195377 −0.0976883 0.995217i \(-0.531145\pi\)
−0.0976883 + 0.995217i \(0.531145\pi\)
\(702\) 16.5894 0.626127
\(703\) 0 0
\(704\) 1.95199 0.0735684
\(705\) −4.16140 −0.156727
\(706\) −2.70813 −0.101922
\(707\) −13.5707 −0.510377
\(708\) −1.25731 −0.0472526
\(709\) −29.9907 −1.12632 −0.563162 0.826346i \(-0.690415\pi\)
−0.563162 + 0.826346i \(0.690415\pi\)
\(710\) 0.0503161 0.00188833
\(711\) 25.6071 0.960342
\(712\) −1.86774 −0.0699965
\(713\) −71.9760 −2.69552
\(714\) −3.81607 −0.142813
\(715\) 11.2410 0.420388
\(716\) 18.9655 0.708775
\(717\) 5.23772 0.195606
\(718\) 9.31002 0.347447
\(719\) −22.7033 −0.846688 −0.423344 0.905969i \(-0.639144\pi\)
−0.423344 + 0.905969i \(0.639144\pi\)
\(720\) 2.49994 0.0931674
\(721\) −11.8313 −0.440620
\(722\) 0 0
\(723\) 9.62195 0.357844
\(724\) −1.98687 −0.0738415
\(725\) −19.2627 −0.715400
\(726\) −3.18120 −0.118065
\(727\) 36.5937 1.35719 0.678593 0.734514i \(-0.262590\pi\)
0.678593 + 0.734514i \(0.262590\pi\)
\(728\) 16.2793 0.603350
\(729\) −16.9956 −0.629467
\(730\) −6.21296 −0.229952
\(731\) 21.3778 0.790686
\(732\) 1.08623 0.0401481
\(733\) 21.1522 0.781275 0.390637 0.920545i \(-0.372255\pi\)
0.390637 + 0.920545i \(0.372255\pi\)
\(734\) −7.75442 −0.286221
\(735\) −0.255944 −0.00944064
\(736\) −8.18504 −0.301705
\(737\) −5.21690 −0.192167
\(738\) −9.78485 −0.360185
\(739\) 48.3066 1.77699 0.888494 0.458888i \(-0.151752\pi\)
0.888494 + 0.458888i \(0.151752\pi\)
\(740\) 5.33094 0.195969
\(741\) 0 0
\(742\) −9.48276 −0.348123
\(743\) −31.8598 −1.16882 −0.584412 0.811457i \(-0.698675\pi\)
−0.584412 + 0.811457i \(0.698675\pi\)
\(744\) −3.89085 −0.142645
\(745\) −12.8282 −0.469988
\(746\) 25.0124 0.915768
\(747\) 39.6066 1.44913
\(748\) −6.68021 −0.244253
\(749\) 40.4163 1.47678
\(750\) 3.63103 0.132586
\(751\) 40.9962 1.49597 0.747987 0.663713i \(-0.231020\pi\)
0.747987 + 0.663713i \(0.231020\pi\)
\(752\) −10.5498 −0.384712
\(753\) −10.4603 −0.381194
\(754\) 29.5894 1.07758
\(755\) −11.0605 −0.402533
\(756\) 6.47214 0.235389
\(757\) −12.8215 −0.466006 −0.233003 0.972476i \(-0.574855\pi\)
−0.233003 + 0.972476i \(0.574855\pi\)
\(758\) −19.1802 −0.696657
\(759\) −7.06929 −0.256599
\(760\) 0 0
\(761\) 35.6382 1.29188 0.645942 0.763386i \(-0.276465\pi\)
0.645942 + 0.763386i \(0.276465\pi\)
\(762\) 3.17442 0.114997
\(763\) 35.5361 1.28649
\(764\) −20.9259 −0.757074
\(765\) −8.55546 −0.309323
\(766\) 4.83319 0.174630
\(767\) −18.3558 −0.662791
\(768\) −0.442463 −0.0159660
\(769\) −5.57401 −0.201004 −0.100502 0.994937i \(-0.532045\pi\)
−0.100502 + 0.994937i \(0.532045\pi\)
\(770\) 4.38551 0.158043
\(771\) 0.408601 0.0147154
\(772\) 7.78879 0.280325
\(773\) 36.4119 1.30965 0.654823 0.755782i \(-0.272743\pi\)
0.654823 + 0.755782i \(0.272743\pi\)
\(774\) −17.5171 −0.629640
\(775\) 36.9792 1.32833
\(776\) −7.82432 −0.280877
\(777\) 6.66791 0.239210
\(778\) −21.7291 −0.779027
\(779\) 0 0
\(780\) −2.54802 −0.0912339
\(781\) 0.110171 0.00394223
\(782\) 28.0113 1.00168
\(783\) 11.7638 0.420405
\(784\) −0.648859 −0.0231735
\(785\) 0.402259 0.0143572
\(786\) −1.55865 −0.0555951
\(787\) 31.9046 1.13728 0.568638 0.822588i \(-0.307471\pi\)
0.568638 + 0.822588i \(0.307471\pi\)
\(788\) 1.82514 0.0650180
\(789\) −1.91130 −0.0680440
\(790\) −8.14076 −0.289635
\(791\) −32.0622 −1.14000
\(792\) 5.47382 0.194504
\(793\) 15.8581 0.563138
\(794\) −11.7140 −0.415715
\(795\) 1.48424 0.0526405
\(796\) 2.35926 0.0836216
\(797\) −2.09165 −0.0740900 −0.0370450 0.999314i \(-0.511794\pi\)
−0.0370450 + 0.999314i \(0.511794\pi\)
\(798\) 0 0
\(799\) 36.1042 1.27728
\(800\) 4.20524 0.148678
\(801\) −5.23757 −0.185060
\(802\) 11.8123 0.417108
\(803\) −13.6038 −0.480066
\(804\) 1.18253 0.0417047
\(805\) −18.3892 −0.648136
\(806\) −56.8036 −2.00082
\(807\) 4.77655 0.168143
\(808\) 5.38487 0.189439
\(809\) −30.0719 −1.05727 −0.528636 0.848849i \(-0.677296\pi\)
−0.528636 + 0.848849i \(0.677296\pi\)
\(810\) 6.48681 0.227923
\(811\) 0.823592 0.0289202 0.0144601 0.999895i \(-0.495397\pi\)
0.0144601 + 0.999895i \(0.495397\pi\)
\(812\) 11.5439 0.405111
\(813\) −0.357176 −0.0125267
\(814\) 11.6725 0.409121
\(815\) 0.0617395 0.00216264
\(816\) 1.51423 0.0530085
\(817\) 0 0
\(818\) −25.2663 −0.883414
\(819\) 45.6507 1.59517
\(820\) 3.11070 0.108630
\(821\) 7.71124 0.269124 0.134562 0.990905i \(-0.457037\pi\)
0.134562 + 0.990905i \(0.457037\pi\)
\(822\) −4.86867 −0.169814
\(823\) −24.8120 −0.864893 −0.432447 0.901660i \(-0.642350\pi\)
−0.432447 + 0.901660i \(0.642350\pi\)
\(824\) 4.69468 0.163547
\(825\) 3.63200 0.126450
\(826\) −7.16129 −0.249173
\(827\) 10.6285 0.369588 0.184794 0.982777i \(-0.440838\pi\)
0.184794 + 0.982777i \(0.440838\pi\)
\(828\) −22.9527 −0.797662
\(829\) 21.4710 0.745718 0.372859 0.927888i \(-0.378377\pi\)
0.372859 + 0.927888i \(0.378377\pi\)
\(830\) −12.5913 −0.437052
\(831\) 4.04796 0.140422
\(832\) −6.45965 −0.223948
\(833\) 2.22056 0.0769380
\(834\) 9.62739 0.333369
\(835\) −1.93784 −0.0670617
\(836\) 0 0
\(837\) −22.5834 −0.780595
\(838\) −31.6143 −1.09210
\(839\) −48.8239 −1.68559 −0.842794 0.538237i \(-0.819091\pi\)
−0.842794 + 0.538237i \(0.819091\pi\)
\(840\) −0.994078 −0.0342990
\(841\) −8.01770 −0.276472
\(842\) 2.18972 0.0754627
\(843\) −9.38720 −0.323312
\(844\) −0.361050 −0.0124279
\(845\) −25.6099 −0.881009
\(846\) −29.5841 −1.01712
\(847\) −18.1192 −0.622583
\(848\) 3.76278 0.129214
\(849\) −2.29632 −0.0788095
\(850\) −14.3914 −0.493622
\(851\) −48.9449 −1.67781
\(852\) −0.0249728 −0.000855554 0
\(853\) −21.0110 −0.719404 −0.359702 0.933067i \(-0.617122\pi\)
−0.359702 + 0.933067i \(0.617122\pi\)
\(854\) 6.18683 0.211709
\(855\) 0 0
\(856\) −16.0373 −0.548143
\(857\) 32.2392 1.10127 0.550635 0.834746i \(-0.314386\pi\)
0.550635 + 0.834746i \(0.314386\pi\)
\(858\) −5.57909 −0.190467
\(859\) −41.2274 −1.40666 −0.703331 0.710863i \(-0.748305\pi\)
−0.703331 + 0.710863i \(0.748305\pi\)
\(860\) 5.56887 0.189897
\(861\) 3.89085 0.132600
\(862\) 1.80372 0.0614348
\(863\) −42.0649 −1.43191 −0.715953 0.698148i \(-0.754008\pi\)
−0.715953 + 0.698148i \(0.754008\pi\)
\(864\) −2.56816 −0.0873705
\(865\) 15.0210 0.510729
\(866\) 6.57233 0.223337
\(867\) 2.33981 0.0794640
\(868\) −22.1612 −0.752199
\(869\) −17.8248 −0.604665
\(870\) −1.80685 −0.0612578
\(871\) 17.2641 0.584972
\(872\) −14.1008 −0.477513
\(873\) −21.9412 −0.742596
\(874\) 0 0
\(875\) 20.6813 0.699156
\(876\) 3.08361 0.104185
\(877\) −2.35551 −0.0795400 −0.0397700 0.999209i \(-0.512663\pi\)
−0.0397700 + 0.999209i \(0.512663\pi\)
\(878\) 9.13060 0.308143
\(879\) 7.77230 0.262153
\(880\) −1.74018 −0.0586615
\(881\) 23.1354 0.779450 0.389725 0.920931i \(-0.372570\pi\)
0.389725 + 0.920931i \(0.372570\pi\)
\(882\) −1.81955 −0.0612673
\(883\) 7.01191 0.235969 0.117985 0.993015i \(-0.462357\pi\)
0.117985 + 0.993015i \(0.462357\pi\)
\(884\) 22.1066 0.743525
\(885\) 1.12088 0.0376780
\(886\) 27.4686 0.922826
\(887\) −48.9293 −1.64289 −0.821443 0.570291i \(-0.806830\pi\)
−0.821443 + 0.570291i \(0.806830\pi\)
\(888\) −2.64584 −0.0887886
\(889\) 18.0806 0.606403
\(890\) 1.66508 0.0558134
\(891\) 14.2034 0.475831
\(892\) 18.0486 0.604312
\(893\) 0 0
\(894\) 6.36686 0.212940
\(895\) −16.9076 −0.565159
\(896\) −2.52015 −0.0841922
\(897\) 23.3941 0.781108
\(898\) 7.57293 0.252712
\(899\) −40.2804 −1.34343
\(900\) 11.7925 0.393082
\(901\) −12.8772 −0.429002
\(902\) 6.81112 0.226785
\(903\) 6.96552 0.231798
\(904\) 12.7224 0.423140
\(905\) 1.77128 0.0588793
\(906\) 5.48954 0.182378
\(907\) 1.19304 0.0396143 0.0198071 0.999804i \(-0.493695\pi\)
0.0198071 + 0.999804i \(0.493695\pi\)
\(908\) 12.2845 0.407675
\(909\) 15.1004 0.500848
\(910\) −14.5128 −0.481096
\(911\) −37.6344 −1.24688 −0.623442 0.781870i \(-0.714266\pi\)
−0.623442 + 0.781870i \(0.714266\pi\)
\(912\) 0 0
\(913\) −27.5697 −0.912424
\(914\) 21.7675 0.720004
\(915\) −0.968361 −0.0320130
\(916\) 6.79174 0.224405
\(917\) −8.87761 −0.293164
\(918\) 8.78890 0.290077
\(919\) 26.9476 0.888921 0.444460 0.895799i \(-0.353395\pi\)
0.444460 + 0.895799i \(0.353395\pi\)
\(920\) 7.29689 0.240571
\(921\) 9.54321 0.314460
\(922\) −11.9444 −0.393367
\(923\) −0.364585 −0.0120005
\(924\) −2.17661 −0.0716052
\(925\) 25.1465 0.826812
\(926\) 8.75371 0.287665
\(927\) 13.1649 0.432393
\(928\) −4.58064 −0.150367
\(929\) −17.8550 −0.585804 −0.292902 0.956142i \(-0.594621\pi\)
−0.292902 + 0.956142i \(0.594621\pi\)
\(930\) 3.46866 0.113742
\(931\) 0 0
\(932\) 21.4294 0.701942
\(933\) 1.13876 0.0372814
\(934\) −29.3568 −0.960585
\(935\) 5.95535 0.194761
\(936\) −18.1143 −0.592085
\(937\) −23.8326 −0.778576 −0.389288 0.921116i \(-0.627279\pi\)
−0.389288 + 0.921116i \(0.627279\pi\)
\(938\) 6.73537 0.219918
\(939\) 2.29368 0.0748514
\(940\) 9.40507 0.306760
\(941\) −20.3651 −0.663882 −0.331941 0.943300i \(-0.607704\pi\)
−0.331941 + 0.943300i \(0.607704\pi\)
\(942\) −0.199648 −0.00650490
\(943\) −28.5602 −0.930049
\(944\) 2.84162 0.0924867
\(945\) −5.76985 −0.187693
\(946\) 12.1935 0.396444
\(947\) −3.41590 −0.111002 −0.0555009 0.998459i \(-0.517676\pi\)
−0.0555009 + 0.998459i \(0.517676\pi\)
\(948\) 4.04041 0.131226
\(949\) 45.0184 1.46136
\(950\) 0 0
\(951\) 8.96261 0.290633
\(952\) 8.62460 0.279525
\(953\) 17.7393 0.574634 0.287317 0.957836i \(-0.407237\pi\)
0.287317 + 0.957836i \(0.407237\pi\)
\(954\) 10.5517 0.341623
\(955\) 18.6553 0.603671
\(956\) −11.8376 −0.382856
\(957\) −3.95623 −0.127887
\(958\) 24.8705 0.803528
\(959\) −27.7306 −0.895466
\(960\) 0.394452 0.0127309
\(961\) 46.3275 1.49443
\(962\) −38.6274 −1.24540
\(963\) −44.9721 −1.44921
\(964\) −21.7463 −0.700401
\(965\) −6.94364 −0.223524
\(966\) 9.12692 0.293654
\(967\) −42.7551 −1.37491 −0.687456 0.726226i \(-0.741273\pi\)
−0.687456 + 0.726226i \(0.741273\pi\)
\(968\) 7.18974 0.231087
\(969\) 0 0
\(970\) 6.97531 0.223964
\(971\) −53.0487 −1.70241 −0.851207 0.524830i \(-0.824129\pi\)
−0.851207 + 0.524830i \(0.824129\pi\)
\(972\) −10.9240 −0.350387
\(973\) 54.8349 1.75793
\(974\) 8.53766 0.273564
\(975\) −12.0193 −0.384924
\(976\) −2.45495 −0.0785810
\(977\) 12.3264 0.394356 0.197178 0.980368i \(-0.436822\pi\)
0.197178 + 0.980368i \(0.436822\pi\)
\(978\) −0.0306425 −0.000979838 0
\(979\) 3.64581 0.116521
\(980\) 0.578452 0.0184780
\(981\) −39.5418 −1.26247
\(982\) 11.5397 0.368247
\(983\) 32.2554 1.02879 0.514394 0.857554i \(-0.328017\pi\)
0.514394 + 0.857554i \(0.328017\pi\)
\(984\) −1.54390 −0.0492177
\(985\) −1.62710 −0.0518437
\(986\) 15.6762 0.499231
\(987\) 11.7638 0.374446
\(988\) 0 0
\(989\) −51.1294 −1.62582
\(990\) −4.87986 −0.155092
\(991\) −27.8366 −0.884258 −0.442129 0.896951i \(-0.645777\pi\)
−0.442129 + 0.896951i \(0.645777\pi\)
\(992\) 8.79360 0.279197
\(993\) 9.43686 0.299470
\(994\) −0.142238 −0.00451152
\(995\) −2.10326 −0.0666777
\(996\) 6.24931 0.198017
\(997\) −43.3313 −1.37232 −0.686159 0.727452i \(-0.740704\pi\)
−0.686159 + 0.727452i \(0.740704\pi\)
\(998\) 29.6091 0.937258
\(999\) −15.3571 −0.485876
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 722.2.a.m.1.3 4
3.2 odd 2 6498.2.a.ca.1.2 4
4.3 odd 2 5776.2.a.bv.1.2 4
19.2 odd 18 722.2.e.r.99.2 24
19.3 odd 18 722.2.e.r.389.2 24
19.4 even 9 722.2.e.s.415.2 24
19.5 even 9 722.2.e.s.595.2 24
19.6 even 9 722.2.e.s.245.3 24
19.7 even 3 722.2.c.n.429.2 8
19.8 odd 6 722.2.c.m.653.3 8
19.9 even 9 722.2.e.s.423.3 24
19.10 odd 18 722.2.e.r.423.2 24
19.11 even 3 722.2.c.n.653.2 8
19.12 odd 6 722.2.c.m.429.3 8
19.13 odd 18 722.2.e.r.245.2 24
19.14 odd 18 722.2.e.r.595.3 24
19.15 odd 18 722.2.e.r.415.3 24
19.16 even 9 722.2.e.s.389.3 24
19.17 even 9 722.2.e.s.99.3 24
19.18 odd 2 722.2.a.n.1.2 yes 4
57.56 even 2 6498.2.a.bx.1.2 4
76.75 even 2 5776.2.a.bt.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.m.1.3 4 1.1 even 1 trivial
722.2.a.n.1.2 yes 4 19.18 odd 2
722.2.c.m.429.3 8 19.12 odd 6
722.2.c.m.653.3 8 19.8 odd 6
722.2.c.n.429.2 8 19.7 even 3
722.2.c.n.653.2 8 19.11 even 3
722.2.e.r.99.2 24 19.2 odd 18
722.2.e.r.245.2 24 19.13 odd 18
722.2.e.r.389.2 24 19.3 odd 18
722.2.e.r.415.3 24 19.15 odd 18
722.2.e.r.423.2 24 19.10 odd 18
722.2.e.r.595.3 24 19.14 odd 18
722.2.e.s.99.3 24 19.17 even 9
722.2.e.s.245.3 24 19.6 even 9
722.2.e.s.389.3 24 19.16 even 9
722.2.e.s.415.2 24 19.4 even 9
722.2.e.s.423.3 24 19.9 even 9
722.2.e.s.595.2 24 19.5 even 9
5776.2.a.bt.1.3 4 76.75 even 2
5776.2.a.bv.1.2 4 4.3 odd 2
6498.2.a.bx.1.2 4 57.56 even 2
6498.2.a.ca.1.2 4 3.2 odd 2