Properties

Label 8281.2.a.co.1.4
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.34523\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.34523 q^{2} -2.05010 q^{3} -0.190366 q^{4} +3.56778 q^{5} +2.75785 q^{6} +2.94654 q^{8} +1.20292 q^{9} +O(q^{10})\) \(q-1.34523 q^{2} -2.05010 q^{3} -0.190366 q^{4} +3.56778 q^{5} +2.75785 q^{6} +2.94654 q^{8} +1.20292 q^{9} -4.79947 q^{10} -1.27867 q^{11} +0.390271 q^{12} -7.31431 q^{15} -3.58303 q^{16} -7.73920 q^{17} -1.61821 q^{18} -0.943878 q^{19} -0.679185 q^{20} +1.72010 q^{22} +1.64727 q^{23} -6.04071 q^{24} +7.72903 q^{25} +3.68419 q^{27} +4.04484 q^{29} +9.83940 q^{30} -5.15220 q^{31} -1.07309 q^{32} +2.62141 q^{33} +10.4110 q^{34} -0.228996 q^{36} +1.05608 q^{37} +1.26973 q^{38} +10.5126 q^{40} +4.19882 q^{41} +3.83065 q^{43} +0.243416 q^{44} +4.29176 q^{45} -2.21596 q^{46} -0.894217 q^{47} +7.34558 q^{48} -10.3973 q^{50} +15.8662 q^{51} -0.0799923 q^{53} -4.95607 q^{54} -4.56202 q^{55} +1.93505 q^{57} -5.44122 q^{58} +11.1847 q^{59} +1.39240 q^{60} +7.62392 q^{61} +6.93087 q^{62} +8.60961 q^{64} -3.52639 q^{66} -6.32103 q^{67} +1.47328 q^{68} -3.37708 q^{69} -11.4240 q^{71} +3.54446 q^{72} +0.760506 q^{73} -1.42067 q^{74} -15.8453 q^{75} +0.179683 q^{76} -2.85531 q^{79} -12.7834 q^{80} -11.1617 q^{81} -5.64837 q^{82} -2.32483 q^{83} -27.6117 q^{85} -5.15308 q^{86} -8.29233 q^{87} -3.76766 q^{88} -7.57626 q^{89} -5.77339 q^{90} -0.313586 q^{92} +10.5625 q^{93} +1.20292 q^{94} -3.36755 q^{95} +2.19995 q^{96} -0.478557 q^{97} -1.53815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{3} + 8q^{4} + 2q^{9} + O(q^{10}) \) \( 12q - 6q^{3} + 8q^{4} + 2q^{9} - 24q^{10} + 2q^{12} + 16q^{16} - 34q^{17} + 30q^{22} + 6q^{23} - 10q^{25} - 12q^{27} + 2q^{29} + 22q^{30} - 26q^{36} - 38q^{38} - 2q^{40} + 22q^{43} + 38q^{48} + 8q^{51} + 16q^{53} - 30q^{55} + 10q^{61} - 82q^{62} - 2q^{64} - 68q^{66} - 22q^{68} - 14q^{69} + 66q^{74} - 2q^{75} + 70q^{79} - 28q^{81} - 10q^{82} + 20q^{87} - 28q^{88} - 66q^{92} + 2q^{94} + 4q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34523 −0.951219 −0.475609 0.879657i \(-0.657772\pi\)
−0.475609 + 0.879657i \(0.657772\pi\)
\(3\) −2.05010 −1.18363 −0.591814 0.806075i \(-0.701588\pi\)
−0.591814 + 0.806075i \(0.701588\pi\)
\(4\) −0.190366 −0.0951832
\(5\) 3.56778 1.59556 0.797779 0.602950i \(-0.206008\pi\)
0.797779 + 0.602950i \(0.206008\pi\)
\(6\) 2.75785 1.12589
\(7\) 0 0
\(8\) 2.94654 1.04176
\(9\) 1.20292 0.400975
\(10\) −4.79947 −1.51772
\(11\) −1.27867 −0.385534 −0.192767 0.981245i \(-0.561746\pi\)
−0.192767 + 0.981245i \(0.561746\pi\)
\(12\) 0.390271 0.112661
\(13\) 0 0
\(14\) 0 0
\(15\) −7.31431 −1.88855
\(16\) −3.58303 −0.895757
\(17\) −7.73920 −1.87703 −0.938515 0.345238i \(-0.887798\pi\)
−0.938515 + 0.345238i \(0.887798\pi\)
\(18\) −1.61821 −0.381415
\(19\) −0.943878 −0.216540 −0.108270 0.994121i \(-0.534531\pi\)
−0.108270 + 0.994121i \(0.534531\pi\)
\(20\) −0.679185 −0.151870
\(21\) 0 0
\(22\) 1.72010 0.366727
\(23\) 1.64727 0.343481 0.171740 0.985142i \(-0.445061\pi\)
0.171740 + 0.985142i \(0.445061\pi\)
\(24\) −6.04071 −1.23305
\(25\) 7.72903 1.54581
\(26\) 0 0
\(27\) 3.68419 0.709023
\(28\) 0 0
\(29\) 4.04484 0.751107 0.375554 0.926801i \(-0.377453\pi\)
0.375554 + 0.926801i \(0.377453\pi\)
\(30\) 9.83940 1.79642
\(31\) −5.15220 −0.925362 −0.462681 0.886525i \(-0.653112\pi\)
−0.462681 + 0.886525i \(0.653112\pi\)
\(32\) −1.07309 −0.189698
\(33\) 2.62141 0.456329
\(34\) 10.4110 1.78547
\(35\) 0 0
\(36\) −0.228996 −0.0381661
\(37\) 1.05608 0.173619 0.0868094 0.996225i \(-0.472333\pi\)
0.0868094 + 0.996225i \(0.472333\pi\)
\(38\) 1.26973 0.205977
\(39\) 0 0
\(40\) 10.5126 1.66219
\(41\) 4.19882 0.655746 0.327873 0.944722i \(-0.393668\pi\)
0.327873 + 0.944722i \(0.393668\pi\)
\(42\) 0 0
\(43\) 3.83065 0.584168 0.292084 0.956393i \(-0.405651\pi\)
0.292084 + 0.956393i \(0.405651\pi\)
\(44\) 0.243416 0.0366964
\(45\) 4.29176 0.639778
\(46\) −2.21596 −0.326725
\(47\) −0.894217 −0.130435 −0.0652175 0.997871i \(-0.520774\pi\)
−0.0652175 + 0.997871i \(0.520774\pi\)
\(48\) 7.34558 1.06024
\(49\) 0 0
\(50\) −10.3973 −1.47040
\(51\) 15.8662 2.22171
\(52\) 0 0
\(53\) −0.0799923 −0.0109878 −0.00549389 0.999985i \(-0.501749\pi\)
−0.00549389 + 0.999985i \(0.501749\pi\)
\(54\) −4.95607 −0.674436
\(55\) −4.56202 −0.615142
\(56\) 0 0
\(57\) 1.93505 0.256303
\(58\) −5.44122 −0.714467
\(59\) 11.1847 1.45613 0.728064 0.685509i \(-0.240420\pi\)
0.728064 + 0.685509i \(0.240420\pi\)
\(60\) 1.39240 0.179758
\(61\) 7.62392 0.976143 0.488072 0.872804i \(-0.337701\pi\)
0.488072 + 0.872804i \(0.337701\pi\)
\(62\) 6.93087 0.880221
\(63\) 0 0
\(64\) 8.60961 1.07620
\(65\) 0 0
\(66\) −3.52639 −0.434069
\(67\) −6.32103 −0.772237 −0.386119 0.922449i \(-0.626184\pi\)
−0.386119 + 0.922449i \(0.626184\pi\)
\(68\) 1.47328 0.178662
\(69\) −3.37708 −0.406553
\(70\) 0 0
\(71\) −11.4240 −1.35578 −0.677889 0.735165i \(-0.737105\pi\)
−0.677889 + 0.735165i \(0.737105\pi\)
\(72\) 3.54446 0.417719
\(73\) 0.760506 0.0890105 0.0445052 0.999009i \(-0.485829\pi\)
0.0445052 + 0.999009i \(0.485829\pi\)
\(74\) −1.42067 −0.165149
\(75\) −15.8453 −1.82966
\(76\) 0.179683 0.0206110
\(77\) 0 0
\(78\) 0 0
\(79\) −2.85531 −0.321247 −0.160624 0.987016i \(-0.551351\pi\)
−0.160624 + 0.987016i \(0.551351\pi\)
\(80\) −12.7834 −1.42923
\(81\) −11.1617 −1.24019
\(82\) −5.64837 −0.623758
\(83\) −2.32483 −0.255183 −0.127591 0.991827i \(-0.540725\pi\)
−0.127591 + 0.991827i \(0.540725\pi\)
\(84\) 0 0
\(85\) −27.6117 −2.99491
\(86\) −5.15308 −0.555671
\(87\) −8.29233 −0.889032
\(88\) −3.76766 −0.401634
\(89\) −7.57626 −0.803082 −0.401541 0.915841i \(-0.631525\pi\)
−0.401541 + 0.915841i \(0.631525\pi\)
\(90\) −5.77339 −0.608569
\(91\) 0 0
\(92\) −0.313586 −0.0326936
\(93\) 10.5625 1.09528
\(94\) 1.20292 0.124072
\(95\) −3.36755 −0.345503
\(96\) 2.19995 0.224532
\(97\) −0.478557 −0.0485901 −0.0242951 0.999705i \(-0.507734\pi\)
−0.0242951 + 0.999705i \(0.507734\pi\)
\(98\) 0 0
\(99\) −1.53815 −0.154589
\(100\) −1.47135 −0.147135
\(101\) −2.87836 −0.286407 −0.143204 0.989693i \(-0.545740\pi\)
−0.143204 + 0.989693i \(0.545740\pi\)
\(102\) −21.3436 −2.11333
\(103\) 11.3351 1.11688 0.558441 0.829544i \(-0.311400\pi\)
0.558441 + 0.829544i \(0.311400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.107608 0.0104518
\(107\) −6.57206 −0.635345 −0.317673 0.948200i \(-0.602901\pi\)
−0.317673 + 0.948200i \(0.602901\pi\)
\(108\) −0.701346 −0.0674871
\(109\) −5.83914 −0.559288 −0.279644 0.960104i \(-0.590216\pi\)
−0.279644 + 0.960104i \(0.590216\pi\)
\(110\) 6.13694 0.585135
\(111\) −2.16508 −0.205500
\(112\) 0 0
\(113\) 6.53233 0.614510 0.307255 0.951627i \(-0.400590\pi\)
0.307255 + 0.951627i \(0.400590\pi\)
\(114\) −2.60308 −0.243800
\(115\) 5.87711 0.548043
\(116\) −0.770001 −0.0714928
\(117\) 0 0
\(118\) −15.0460 −1.38510
\(119\) 0 0
\(120\) −21.5519 −1.96741
\(121\) −9.36500 −0.851363
\(122\) −10.2559 −0.928525
\(123\) −8.60802 −0.776159
\(124\) 0.980805 0.0880789
\(125\) 9.73656 0.870865
\(126\) 0 0
\(127\) 14.7164 1.30586 0.652932 0.757416i \(-0.273539\pi\)
0.652932 + 0.757416i \(0.273539\pi\)
\(128\) −9.43568 −0.834005
\(129\) −7.85322 −0.691437
\(130\) 0 0
\(131\) −11.1867 −0.977386 −0.488693 0.872456i \(-0.662526\pi\)
−0.488693 + 0.872456i \(0.662526\pi\)
\(132\) −0.499028 −0.0434349
\(133\) 0 0
\(134\) 8.50322 0.734566
\(135\) 13.1444 1.13129
\(136\) −22.8038 −1.95541
\(137\) 17.6308 1.50630 0.753151 0.657848i \(-0.228533\pi\)
0.753151 + 0.657848i \(0.228533\pi\)
\(138\) 4.54294 0.386721
\(139\) 5.85710 0.496793 0.248396 0.968658i \(-0.420096\pi\)
0.248396 + 0.968658i \(0.420096\pi\)
\(140\) 0 0
\(141\) 1.83324 0.154386
\(142\) 15.3678 1.28964
\(143\) 0 0
\(144\) −4.31011 −0.359176
\(145\) 14.4311 1.19844
\(146\) −1.02305 −0.0846684
\(147\) 0 0
\(148\) −0.201043 −0.0165256
\(149\) 10.4790 0.858470 0.429235 0.903193i \(-0.358783\pi\)
0.429235 + 0.903193i \(0.358783\pi\)
\(150\) 21.3155 1.74041
\(151\) 4.71406 0.383625 0.191812 0.981432i \(-0.438564\pi\)
0.191812 + 0.981432i \(0.438564\pi\)
\(152\) −2.78117 −0.225583
\(153\) −9.30967 −0.752642
\(154\) 0 0
\(155\) −18.3819 −1.47647
\(156\) 0 0
\(157\) −9.00210 −0.718445 −0.359223 0.933252i \(-0.616958\pi\)
−0.359223 + 0.933252i \(0.616958\pi\)
\(158\) 3.84103 0.305576
\(159\) 0.163992 0.0130054
\(160\) −3.82856 −0.302674
\(161\) 0 0
\(162\) 15.0151 1.17970
\(163\) −12.0324 −0.942449 −0.471224 0.882013i \(-0.656188\pi\)
−0.471224 + 0.882013i \(0.656188\pi\)
\(164\) −0.799315 −0.0624160
\(165\) 9.35261 0.728099
\(166\) 3.12742 0.242735
\(167\) −19.4220 −1.50292 −0.751459 0.659780i \(-0.770650\pi\)
−0.751459 + 0.659780i \(0.770650\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 37.1440 2.84882
\(171\) −1.13541 −0.0868273
\(172\) −0.729226 −0.0556030
\(173\) −14.3795 −1.09325 −0.546627 0.837376i \(-0.684088\pi\)
−0.546627 + 0.837376i \(0.684088\pi\)
\(174\) 11.1551 0.845663
\(175\) 0 0
\(176\) 4.58152 0.345345
\(177\) −22.9299 −1.72351
\(178\) 10.1918 0.763907
\(179\) 5.42606 0.405563 0.202781 0.979224i \(-0.435002\pi\)
0.202781 + 0.979224i \(0.435002\pi\)
\(180\) −0.817008 −0.0608962
\(181\) −15.4902 −1.15138 −0.575688 0.817669i \(-0.695266\pi\)
−0.575688 + 0.817669i \(0.695266\pi\)
\(182\) 0 0
\(183\) −15.6298 −1.15539
\(184\) 4.85376 0.357824
\(185\) 3.76786 0.277019
\(186\) −14.2090 −1.04185
\(187\) 9.89589 0.723659
\(188\) 0.170229 0.0124152
\(189\) 0 0
\(190\) 4.53011 0.328649
\(191\) 4.74622 0.343425 0.171712 0.985147i \(-0.445070\pi\)
0.171712 + 0.985147i \(0.445070\pi\)
\(192\) −17.6506 −1.27382
\(193\) 21.0391 1.51443 0.757215 0.653166i \(-0.226559\pi\)
0.757215 + 0.653166i \(0.226559\pi\)
\(194\) 0.643768 0.0462198
\(195\) 0 0
\(196\) 0 0
\(197\) 5.81209 0.414094 0.207047 0.978331i \(-0.433615\pi\)
0.207047 + 0.978331i \(0.433615\pi\)
\(198\) 2.06915 0.147048
\(199\) −10.6182 −0.752703 −0.376352 0.926477i \(-0.622821\pi\)
−0.376352 + 0.926477i \(0.622821\pi\)
\(200\) 22.7739 1.61036
\(201\) 12.9588 0.914041
\(202\) 3.87204 0.272436
\(203\) 0 0
\(204\) −3.02038 −0.211469
\(205\) 14.9805 1.04628
\(206\) −15.2483 −1.06240
\(207\) 1.98155 0.137727
\(208\) 0 0
\(209\) 1.20691 0.0834837
\(210\) 0 0
\(211\) −4.66549 −0.321186 −0.160593 0.987021i \(-0.551341\pi\)
−0.160593 + 0.987021i \(0.551341\pi\)
\(212\) 0.0152278 0.00104585
\(213\) 23.4203 1.60474
\(214\) 8.84091 0.604352
\(215\) 13.6669 0.932074
\(216\) 10.8556 0.738631
\(217\) 0 0
\(218\) 7.85497 0.532005
\(219\) −1.55912 −0.105355
\(220\) 0.868455 0.0585512
\(221\) 0 0
\(222\) 2.91252 0.195475
\(223\) −24.2254 −1.62225 −0.811126 0.584871i \(-0.801145\pi\)
−0.811126 + 0.584871i \(0.801145\pi\)
\(224\) 0 0
\(225\) 9.29744 0.619829
\(226\) −8.78747 −0.584534
\(227\) 15.3753 1.02049 0.510247 0.860028i \(-0.329554\pi\)
0.510247 + 0.860028i \(0.329554\pi\)
\(228\) −0.368368 −0.0243958
\(229\) 16.3515 1.08054 0.540268 0.841493i \(-0.318323\pi\)
0.540268 + 0.841493i \(0.318323\pi\)
\(230\) −7.90604 −0.521309
\(231\) 0 0
\(232\) 11.9183 0.782473
\(233\) −29.1107 −1.90711 −0.953554 0.301223i \(-0.902605\pi\)
−0.953554 + 0.301223i \(0.902605\pi\)
\(234\) 0 0
\(235\) −3.19037 −0.208117
\(236\) −2.12920 −0.138599
\(237\) 5.85368 0.380237
\(238\) 0 0
\(239\) −8.65409 −0.559787 −0.279893 0.960031i \(-0.590299\pi\)
−0.279893 + 0.960031i \(0.590299\pi\)
\(240\) 26.2074 1.69168
\(241\) 18.1982 1.17225 0.586124 0.810222i \(-0.300653\pi\)
0.586124 + 0.810222i \(0.300653\pi\)
\(242\) 12.5980 0.809833
\(243\) 11.8302 0.758905
\(244\) −1.45134 −0.0929124
\(245\) 0 0
\(246\) 11.5797 0.738297
\(247\) 0 0
\(248\) −15.1811 −0.964003
\(249\) 4.76614 0.302042
\(250\) −13.0979 −0.828383
\(251\) −15.8720 −1.00183 −0.500915 0.865497i \(-0.667003\pi\)
−0.500915 + 0.865497i \(0.667003\pi\)
\(252\) 0 0
\(253\) −2.10632 −0.132423
\(254\) −19.7968 −1.24216
\(255\) 56.6069 3.54486
\(256\) −4.52609 −0.282880
\(257\) 24.3267 1.51746 0.758730 0.651406i \(-0.225820\pi\)
0.758730 + 0.651406i \(0.225820\pi\)
\(258\) 10.5644 0.657708
\(259\) 0 0
\(260\) 0 0
\(261\) 4.86563 0.301175
\(262\) 15.0486 0.929708
\(263\) 15.4345 0.951734 0.475867 0.879517i \(-0.342134\pi\)
0.475867 + 0.879517i \(0.342134\pi\)
\(264\) 7.72409 0.475385
\(265\) −0.285395 −0.0175317
\(266\) 0 0
\(267\) 15.5321 0.950550
\(268\) 1.20331 0.0735040
\(269\) 13.0407 0.795106 0.397553 0.917579i \(-0.369859\pi\)
0.397553 + 0.917579i \(0.369859\pi\)
\(270\) −17.6822 −1.07610
\(271\) −26.9706 −1.63835 −0.819174 0.573544i \(-0.805568\pi\)
−0.819174 + 0.573544i \(0.805568\pi\)
\(272\) 27.7298 1.68136
\(273\) 0 0
\(274\) −23.7174 −1.43282
\(275\) −9.88289 −0.595961
\(276\) 0.642883 0.0386970
\(277\) 12.7015 0.763156 0.381578 0.924337i \(-0.375381\pi\)
0.381578 + 0.924337i \(0.375381\pi\)
\(278\) −7.87912 −0.472558
\(279\) −6.19770 −0.371047
\(280\) 0 0
\(281\) 26.7216 1.59408 0.797038 0.603930i \(-0.206399\pi\)
0.797038 + 0.603930i \(0.206399\pi\)
\(282\) −2.46612 −0.146855
\(283\) −14.7423 −0.876336 −0.438168 0.898893i \(-0.644373\pi\)
−0.438168 + 0.898893i \(0.644373\pi\)
\(284\) 2.17474 0.129047
\(285\) 6.90382 0.408947
\(286\) 0 0
\(287\) 0 0
\(288\) −1.29085 −0.0760641
\(289\) 42.8952 2.52324
\(290\) −19.4131 −1.13997
\(291\) 0.981092 0.0575126
\(292\) −0.144775 −0.00847230
\(293\) 11.5831 0.676689 0.338345 0.941022i \(-0.390133\pi\)
0.338345 + 0.941022i \(0.390133\pi\)
\(294\) 0 0
\(295\) 39.9046 2.32334
\(296\) 3.11179 0.180869
\(297\) −4.71087 −0.273353
\(298\) −14.0966 −0.816593
\(299\) 0 0
\(300\) 3.01641 0.174153
\(301\) 0 0
\(302\) −6.34147 −0.364911
\(303\) 5.90093 0.339000
\(304\) 3.38194 0.193968
\(305\) 27.2004 1.55749
\(306\) 12.5236 0.715927
\(307\) −29.3335 −1.67415 −0.837076 0.547086i \(-0.815737\pi\)
−0.837076 + 0.547086i \(0.815737\pi\)
\(308\) 0 0
\(309\) −23.2381 −1.32197
\(310\) 24.7278 1.40444
\(311\) 0.150654 0.00854282 0.00427141 0.999991i \(-0.498640\pi\)
0.00427141 + 0.999991i \(0.498640\pi\)
\(312\) 0 0
\(313\) 10.5211 0.594690 0.297345 0.954770i \(-0.403899\pi\)
0.297345 + 0.954770i \(0.403899\pi\)
\(314\) 12.1099 0.683399
\(315\) 0 0
\(316\) 0.543555 0.0305773
\(317\) −1.50676 −0.0846281 −0.0423140 0.999104i \(-0.513473\pi\)
−0.0423140 + 0.999104i \(0.513473\pi\)
\(318\) −0.220607 −0.0123710
\(319\) −5.17202 −0.289578
\(320\) 30.7172 1.71714
\(321\) 13.4734 0.752012
\(322\) 0 0
\(323\) 7.30486 0.406453
\(324\) 2.12482 0.118046
\(325\) 0 0
\(326\) 16.1863 0.896475
\(327\) 11.9708 0.661989
\(328\) 12.3720 0.683129
\(329\) 0 0
\(330\) −12.5814 −0.692582
\(331\) −25.2509 −1.38791 −0.693957 0.720017i \(-0.744134\pi\)
−0.693957 + 0.720017i \(0.744134\pi\)
\(332\) 0.442569 0.0242891
\(333\) 1.27039 0.0696167
\(334\) 26.1270 1.42960
\(335\) −22.5520 −1.23215
\(336\) 0 0
\(337\) −32.1811 −1.75302 −0.876509 0.481386i \(-0.840134\pi\)
−0.876509 + 0.481386i \(0.840134\pi\)
\(338\) 0 0
\(339\) −13.3920 −0.727351
\(340\) 5.25634 0.285065
\(341\) 6.58797 0.356759
\(342\) 1.52739 0.0825917
\(343\) 0 0
\(344\) 11.2871 0.608562
\(345\) −12.0487 −0.648679
\(346\) 19.3437 1.03992
\(347\) 24.7638 1.32939 0.664695 0.747115i \(-0.268562\pi\)
0.664695 + 0.747115i \(0.268562\pi\)
\(348\) 1.57858 0.0846209
\(349\) −11.5556 −0.618559 −0.309280 0.950971i \(-0.600088\pi\)
−0.309280 + 0.950971i \(0.600088\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.37213 0.0731350
\(353\) −20.0884 −1.06920 −0.534599 0.845106i \(-0.679537\pi\)
−0.534599 + 0.845106i \(0.679537\pi\)
\(354\) 30.8459 1.63944
\(355\) −40.7582 −2.16322
\(356\) 1.44227 0.0764399
\(357\) 0 0
\(358\) −7.29928 −0.385779
\(359\) 15.0510 0.794363 0.397181 0.917740i \(-0.369988\pi\)
0.397181 + 0.917740i \(0.369988\pi\)
\(360\) 12.6458 0.666495
\(361\) −18.1091 −0.953110
\(362\) 20.8378 1.09521
\(363\) 19.1992 1.00770
\(364\) 0 0
\(365\) 2.71331 0.142021
\(366\) 21.0257 1.09903
\(367\) −9.00355 −0.469982 −0.234991 0.971998i \(-0.575506\pi\)
−0.234991 + 0.971998i \(0.575506\pi\)
\(368\) −5.90223 −0.307675
\(369\) 5.05087 0.262938
\(370\) −5.06863 −0.263506
\(371\) 0 0
\(372\) −2.01075 −0.104253
\(373\) −16.1391 −0.835649 −0.417824 0.908528i \(-0.637207\pi\)
−0.417824 + 0.908528i \(0.637207\pi\)
\(374\) −13.3122 −0.688358
\(375\) −19.9610 −1.03078
\(376\) −2.63484 −0.135882
\(377\) 0 0
\(378\) 0 0
\(379\) −15.6655 −0.804685 −0.402342 0.915489i \(-0.631804\pi\)
−0.402342 + 0.915489i \(0.631804\pi\)
\(380\) 0.641068 0.0328861
\(381\) −30.1700 −1.54566
\(382\) −6.38474 −0.326672
\(383\) −24.6328 −1.25868 −0.629339 0.777131i \(-0.716674\pi\)
−0.629339 + 0.777131i \(0.716674\pi\)
\(384\) 19.3441 0.987151
\(385\) 0 0
\(386\) −28.3024 −1.44055
\(387\) 4.60798 0.234237
\(388\) 0.0911013 0.00462497
\(389\) −18.8567 −0.956071 −0.478036 0.878340i \(-0.658651\pi\)
−0.478036 + 0.878340i \(0.658651\pi\)
\(390\) 0 0
\(391\) −12.7486 −0.644724
\(392\) 0 0
\(393\) 22.9339 1.15686
\(394\) −7.81857 −0.393894
\(395\) −10.1871 −0.512569
\(396\) 0.292811 0.0147143
\(397\) −14.5030 −0.727884 −0.363942 0.931422i \(-0.618569\pi\)
−0.363942 + 0.931422i \(0.618569\pi\)
\(398\) 14.2839 0.715985
\(399\) 0 0
\(400\) −27.6933 −1.38467
\(401\) 20.9889 1.04814 0.524069 0.851676i \(-0.324413\pi\)
0.524069 + 0.851676i \(0.324413\pi\)
\(402\) −17.4325 −0.869453
\(403\) 0 0
\(404\) 0.547943 0.0272612
\(405\) −39.8226 −1.97880
\(406\) 0 0
\(407\) −1.35038 −0.0669360
\(408\) 46.7502 2.31448
\(409\) 21.4276 1.05953 0.529763 0.848146i \(-0.322281\pi\)
0.529763 + 0.848146i \(0.322281\pi\)
\(410\) −20.1521 −0.995242
\(411\) −36.1450 −1.78290
\(412\) −2.15782 −0.106308
\(413\) 0 0
\(414\) −2.66563 −0.131009
\(415\) −8.29446 −0.407159
\(416\) 0 0
\(417\) −12.0077 −0.588018
\(418\) −1.62357 −0.0794113
\(419\) −7.96406 −0.389070 −0.194535 0.980896i \(-0.562320\pi\)
−0.194535 + 0.980896i \(0.562320\pi\)
\(420\) 0 0
\(421\) 2.81786 0.137334 0.0686670 0.997640i \(-0.478125\pi\)
0.0686670 + 0.997640i \(0.478125\pi\)
\(422\) 6.27614 0.305518
\(423\) −1.07568 −0.0523011
\(424\) −0.235700 −0.0114466
\(425\) −59.8165 −2.90152
\(426\) −31.5057 −1.52645
\(427\) 0 0
\(428\) 1.25110 0.0604742
\(429\) 0 0
\(430\) −18.3851 −0.886606
\(431\) −5.73626 −0.276306 −0.138153 0.990411i \(-0.544117\pi\)
−0.138153 + 0.990411i \(0.544117\pi\)
\(432\) −13.2006 −0.635112
\(433\) 24.5257 1.17863 0.589314 0.807904i \(-0.299398\pi\)
0.589314 + 0.807904i \(0.299398\pi\)
\(434\) 0 0
\(435\) −29.5852 −1.41850
\(436\) 1.11158 0.0532349
\(437\) −1.55483 −0.0743774
\(438\) 2.09736 0.100216
\(439\) −36.6423 −1.74884 −0.874420 0.485169i \(-0.838758\pi\)
−0.874420 + 0.485169i \(0.838758\pi\)
\(440\) −13.4422 −0.640830
\(441\) 0 0
\(442\) 0 0
\(443\) 27.0933 1.28724 0.643622 0.765344i \(-0.277431\pi\)
0.643622 + 0.765344i \(0.277431\pi\)
\(444\) 0.412158 0.0195602
\(445\) −27.0304 −1.28136
\(446\) 32.5886 1.54312
\(447\) −21.4830 −1.01611
\(448\) 0 0
\(449\) 27.4324 1.29461 0.647307 0.762229i \(-0.275895\pi\)
0.647307 + 0.762229i \(0.275895\pi\)
\(450\) −12.5072 −0.589593
\(451\) −5.36892 −0.252812
\(452\) −1.24354 −0.0584911
\(453\) −9.66431 −0.454069
\(454\) −20.6832 −0.970712
\(455\) 0 0
\(456\) 5.70169 0.267006
\(457\) 39.6639 1.85540 0.927700 0.373327i \(-0.121783\pi\)
0.927700 + 0.373327i \(0.121783\pi\)
\(458\) −21.9964 −1.02783
\(459\) −28.5127 −1.33086
\(460\) −1.11880 −0.0521645
\(461\) −4.89580 −0.228020 −0.114010 0.993480i \(-0.536370\pi\)
−0.114010 + 0.993480i \(0.536370\pi\)
\(462\) 0 0
\(463\) 4.71193 0.218982 0.109491 0.993988i \(-0.465078\pi\)
0.109491 + 0.993988i \(0.465078\pi\)
\(464\) −14.4928 −0.672810
\(465\) 37.6848 1.74759
\(466\) 39.1605 1.81408
\(467\) −32.0161 −1.48153 −0.740765 0.671764i \(-0.765537\pi\)
−0.740765 + 0.671764i \(0.765537\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 4.29176 0.197964
\(471\) 18.4552 0.850372
\(472\) 32.9563 1.51693
\(473\) −4.89814 −0.225217
\(474\) −7.87452 −0.361689
\(475\) −7.29526 −0.334730
\(476\) 0 0
\(477\) −0.0962247 −0.00440582
\(478\) 11.6417 0.532480
\(479\) −18.0245 −0.823560 −0.411780 0.911283i \(-0.635093\pi\)
−0.411780 + 0.911283i \(0.635093\pi\)
\(480\) 7.84894 0.358253
\(481\) 0 0
\(482\) −24.4807 −1.11506
\(483\) 0 0
\(484\) 1.78278 0.0810355
\(485\) −1.70739 −0.0775284
\(486\) −15.9142 −0.721885
\(487\) −17.6004 −0.797550 −0.398775 0.917049i \(-0.630565\pi\)
−0.398775 + 0.917049i \(0.630565\pi\)
\(488\) 22.4642 1.01691
\(489\) 24.6676 1.11551
\(490\) 0 0
\(491\) 3.86360 0.174362 0.0871810 0.996192i \(-0.472214\pi\)
0.0871810 + 0.996192i \(0.472214\pi\)
\(492\) 1.63868 0.0738773
\(493\) −31.3038 −1.40985
\(494\) 0 0
\(495\) −5.48776 −0.246656
\(496\) 18.4605 0.828899
\(497\) 0 0
\(498\) −6.41153 −0.287308
\(499\) 12.6473 0.566169 0.283084 0.959095i \(-0.408642\pi\)
0.283084 + 0.959095i \(0.408642\pi\)
\(500\) −1.85351 −0.0828917
\(501\) 39.8171 1.77890
\(502\) 21.3514 0.952959
\(503\) −22.0360 −0.982537 −0.491268 0.871008i \(-0.663467\pi\)
−0.491268 + 0.871008i \(0.663467\pi\)
\(504\) 0 0
\(505\) −10.2693 −0.456980
\(506\) 2.83348 0.125964
\(507\) 0 0
\(508\) −2.80150 −0.124296
\(509\) −15.6702 −0.694568 −0.347284 0.937760i \(-0.612896\pi\)
−0.347284 + 0.937760i \(0.612896\pi\)
\(510\) −76.1491 −3.37194
\(511\) 0 0
\(512\) 24.9600 1.10309
\(513\) −3.47743 −0.153532
\(514\) −32.7249 −1.44344
\(515\) 40.4411 1.78205
\(516\) 1.49499 0.0658132
\(517\) 1.14341 0.0502871
\(518\) 0 0
\(519\) 29.4795 1.29401
\(520\) 0 0
\(521\) 25.2415 1.10585 0.552925 0.833231i \(-0.313512\pi\)
0.552925 + 0.833231i \(0.313512\pi\)
\(522\) −6.54538 −0.286483
\(523\) 13.2477 0.579279 0.289640 0.957136i \(-0.406465\pi\)
0.289640 + 0.957136i \(0.406465\pi\)
\(524\) 2.12957 0.0930307
\(525\) 0 0
\(526\) −20.7629 −0.905307
\(527\) 39.8738 1.73693
\(528\) −9.39259 −0.408760
\(529\) −20.2865 −0.882021
\(530\) 0.383920 0.0166764
\(531\) 13.4544 0.583871
\(532\) 0 0
\(533\) 0 0
\(534\) −20.8942 −0.904181
\(535\) −23.4477 −1.01373
\(536\) −18.6252 −0.804485
\(537\) −11.1240 −0.480036
\(538\) −17.5427 −0.756320
\(539\) 0 0
\(540\) −2.50225 −0.107680
\(541\) −14.4034 −0.619250 −0.309625 0.950859i \(-0.600204\pi\)
−0.309625 + 0.950859i \(0.600204\pi\)
\(542\) 36.2816 1.55843
\(543\) 31.7565 1.36280
\(544\) 8.30488 0.356069
\(545\) −20.8328 −0.892377
\(546\) 0 0
\(547\) 2.00679 0.0858042 0.0429021 0.999079i \(-0.486340\pi\)
0.0429021 + 0.999079i \(0.486340\pi\)
\(548\) −3.35631 −0.143375
\(549\) 9.17100 0.391409
\(550\) 13.2947 0.566889
\(551\) −3.81783 −0.162645
\(552\) −9.95071 −0.423530
\(553\) 0 0
\(554\) −17.0863 −0.725928
\(555\) −7.72451 −0.327887
\(556\) −1.11499 −0.0472863
\(557\) −8.57916 −0.363511 −0.181755 0.983344i \(-0.558178\pi\)
−0.181755 + 0.983344i \(0.558178\pi\)
\(558\) 8.33731 0.352946
\(559\) 0 0
\(560\) 0 0
\(561\) −20.2876 −0.856543
\(562\) −35.9466 −1.51631
\(563\) 12.7744 0.538375 0.269188 0.963088i \(-0.413245\pi\)
0.269188 + 0.963088i \(0.413245\pi\)
\(564\) −0.348987 −0.0146950
\(565\) 23.3059 0.980487
\(566\) 19.8317 0.833587
\(567\) 0 0
\(568\) −33.6612 −1.41239
\(569\) −5.79116 −0.242778 −0.121389 0.992605i \(-0.538735\pi\)
−0.121389 + 0.992605i \(0.538735\pi\)
\(570\) −9.28720 −0.388998
\(571\) −44.1332 −1.84692 −0.923458 0.383700i \(-0.874650\pi\)
−0.923458 + 0.383700i \(0.874650\pi\)
\(572\) 0 0
\(573\) −9.73025 −0.406487
\(574\) 0 0
\(575\) 12.7318 0.530954
\(576\) 10.3567 0.431529
\(577\) −11.9330 −0.496776 −0.248388 0.968661i \(-0.579901\pi\)
−0.248388 + 0.968661i \(0.579901\pi\)
\(578\) −57.7037 −2.40016
\(579\) −43.1324 −1.79252
\(580\) −2.74719 −0.114071
\(581\) 0 0
\(582\) −1.31979 −0.0547071
\(583\) 0.102284 0.00423617
\(584\) 2.24086 0.0927274
\(585\) 0 0
\(586\) −15.5818 −0.643679
\(587\) −20.3516 −0.840000 −0.420000 0.907524i \(-0.637970\pi\)
−0.420000 + 0.907524i \(0.637970\pi\)
\(588\) 0 0
\(589\) 4.86304 0.200378
\(590\) −53.6808 −2.21000
\(591\) −11.9154 −0.490133
\(592\) −3.78397 −0.155520
\(593\) −18.1800 −0.746563 −0.373282 0.927718i \(-0.621768\pi\)
−0.373282 + 0.927718i \(0.621768\pi\)
\(594\) 6.33719 0.260018
\(595\) 0 0
\(596\) −1.99484 −0.0817120
\(597\) 21.7684 0.890920
\(598\) 0 0
\(599\) −38.2682 −1.56359 −0.781797 0.623532i \(-0.785697\pi\)
−0.781797 + 0.623532i \(0.785697\pi\)
\(600\) −46.6888 −1.90606
\(601\) 26.8719 1.09613 0.548064 0.836436i \(-0.315365\pi\)
0.548064 + 0.836436i \(0.315365\pi\)
\(602\) 0 0
\(603\) −7.60372 −0.309648
\(604\) −0.897398 −0.0365146
\(605\) −33.4122 −1.35840
\(606\) −7.93809 −0.322463
\(607\) 9.40209 0.381619 0.190810 0.981627i \(-0.438889\pi\)
0.190810 + 0.981627i \(0.438889\pi\)
\(608\) 1.01287 0.0410773
\(609\) 0 0
\(610\) −36.5908 −1.48152
\(611\) 0 0
\(612\) 1.77225 0.0716389
\(613\) −13.2894 −0.536753 −0.268376 0.963314i \(-0.586487\pi\)
−0.268376 + 0.963314i \(0.586487\pi\)
\(614\) 39.4602 1.59248
\(615\) −30.7115 −1.23841
\(616\) 0 0
\(617\) 11.2261 0.451947 0.225973 0.974133i \(-0.427444\pi\)
0.225973 + 0.974133i \(0.427444\pi\)
\(618\) 31.2606 1.25748
\(619\) −9.28505 −0.373198 −0.186599 0.982436i \(-0.559746\pi\)
−0.186599 + 0.982436i \(0.559746\pi\)
\(620\) 3.49929 0.140535
\(621\) 6.06888 0.243536
\(622\) −0.202664 −0.00812608
\(623\) 0 0
\(624\) 0 0
\(625\) −3.90726 −0.156290
\(626\) −14.1533 −0.565680
\(627\) −2.47429 −0.0988137
\(628\) 1.71370 0.0683839
\(629\) −8.17322 −0.325888
\(630\) 0 0
\(631\) 10.4026 0.414122 0.207061 0.978328i \(-0.433610\pi\)
0.207061 + 0.978328i \(0.433610\pi\)
\(632\) −8.41327 −0.334662
\(633\) 9.56474 0.380164
\(634\) 2.02693 0.0804998
\(635\) 52.5046 2.08358
\(636\) −0.0312187 −0.00123790
\(637\) 0 0
\(638\) 6.95754 0.275452
\(639\) −13.7422 −0.543632
\(640\) −33.6644 −1.33070
\(641\) 14.8591 0.586899 0.293449 0.955975i \(-0.405197\pi\)
0.293449 + 0.955975i \(0.405197\pi\)
\(642\) −18.1248 −0.715328
\(643\) 2.29722 0.0905935 0.0452968 0.998974i \(-0.485577\pi\)
0.0452968 + 0.998974i \(0.485577\pi\)
\(644\) 0 0
\(645\) −28.0185 −1.10323
\(646\) −9.82669 −0.386626
\(647\) −7.99865 −0.314459 −0.157230 0.987562i \(-0.550256\pi\)
−0.157230 + 0.987562i \(0.550256\pi\)
\(648\) −32.8885 −1.29198
\(649\) −14.3016 −0.561387
\(650\) 0 0
\(651\) 0 0
\(652\) 2.29056 0.0897053
\(653\) 3.98444 0.155923 0.0779615 0.996956i \(-0.475159\pi\)
0.0779615 + 0.996956i \(0.475159\pi\)
\(654\) −16.1035 −0.629696
\(655\) −39.9116 −1.55948
\(656\) −15.0445 −0.587389
\(657\) 0.914831 0.0356910
\(658\) 0 0
\(659\) −27.5003 −1.07126 −0.535629 0.844453i \(-0.679925\pi\)
−0.535629 + 0.844453i \(0.679925\pi\)
\(660\) −1.78042 −0.0693028
\(661\) −6.98621 −0.271732 −0.135866 0.990727i \(-0.543382\pi\)
−0.135866 + 0.990727i \(0.543382\pi\)
\(662\) 33.9681 1.32021
\(663\) 0 0
\(664\) −6.85019 −0.265839
\(665\) 0 0
\(666\) −1.70896 −0.0662207
\(667\) 6.66296 0.257991
\(668\) 3.69729 0.143053
\(669\) 49.6646 1.92014
\(670\) 30.3376 1.17204
\(671\) −9.74849 −0.376336
\(672\) 0 0
\(673\) 5.45566 0.210300 0.105150 0.994456i \(-0.466468\pi\)
0.105150 + 0.994456i \(0.466468\pi\)
\(674\) 43.2909 1.66750
\(675\) 28.4752 1.09601
\(676\) 0 0
\(677\) −33.7922 −1.29874 −0.649371 0.760472i \(-0.724968\pi\)
−0.649371 + 0.760472i \(0.724968\pi\)
\(678\) 18.0152 0.691870
\(679\) 0 0
\(680\) −81.3590 −3.11997
\(681\) −31.5209 −1.20788
\(682\) −8.86231 −0.339355
\(683\) 12.2988 0.470602 0.235301 0.971923i \(-0.424392\pi\)
0.235301 + 0.971923i \(0.424392\pi\)
\(684\) 0.216145 0.00826450
\(685\) 62.9028 2.40339
\(686\) 0 0
\(687\) −33.5222 −1.27895
\(688\) −13.7253 −0.523272
\(689\) 0 0
\(690\) 16.2082 0.617036
\(691\) 11.0897 0.421871 0.210935 0.977500i \(-0.432349\pi\)
0.210935 + 0.977500i \(0.432349\pi\)
\(692\) 2.73738 0.104059
\(693\) 0 0
\(694\) −33.3129 −1.26454
\(695\) 20.8968 0.792662
\(696\) −24.4337 −0.926156
\(697\) −32.4955 −1.23086
\(698\) 15.5449 0.588385
\(699\) 59.6800 2.25731
\(700\) 0 0
\(701\) −10.6470 −0.402133 −0.201066 0.979578i \(-0.564441\pi\)
−0.201066 + 0.979578i \(0.564441\pi\)
\(702\) 0 0
\(703\) −0.996813 −0.0375955
\(704\) −11.0089 −0.414912
\(705\) 6.54058 0.246333
\(706\) 27.0235 1.01704
\(707\) 0 0
\(708\) 4.36508 0.164050
\(709\) −40.7069 −1.52878 −0.764391 0.644754i \(-0.776960\pi\)
−0.764391 + 0.644754i \(0.776960\pi\)
\(710\) 54.8290 2.05770
\(711\) −3.43472 −0.128812
\(712\) −22.3237 −0.836618
\(713\) −8.48708 −0.317844
\(714\) 0 0
\(715\) 0 0
\(716\) −1.03294 −0.0386028
\(717\) 17.7418 0.662579
\(718\) −20.2470 −0.755612
\(719\) 9.77537 0.364560 0.182280 0.983247i \(-0.441652\pi\)
0.182280 + 0.983247i \(0.441652\pi\)
\(720\) −15.3775 −0.573086
\(721\) 0 0
\(722\) 24.3608 0.906616
\(723\) −37.3081 −1.38750
\(724\) 2.94881 0.109592
\(725\) 31.2627 1.16107
\(726\) −25.8273 −0.958540
\(727\) −12.2091 −0.452811 −0.226406 0.974033i \(-0.572697\pi\)
−0.226406 + 0.974033i \(0.572697\pi\)
\(728\) 0 0
\(729\) 9.23219 0.341933
\(730\) −3.65002 −0.135093
\(731\) −29.6461 −1.09650
\(732\) 2.97539 0.109974
\(733\) −22.3153 −0.824236 −0.412118 0.911131i \(-0.635211\pi\)
−0.412118 + 0.911131i \(0.635211\pi\)
\(734\) 12.1118 0.447055
\(735\) 0 0
\(736\) −1.76768 −0.0651576
\(737\) 8.08253 0.297724
\(738\) −6.79456 −0.250111
\(739\) −42.3729 −1.55871 −0.779357 0.626580i \(-0.784454\pi\)
−0.779357 + 0.626580i \(0.784454\pi\)
\(740\) −0.717275 −0.0263675
\(741\) 0 0
\(742\) 0 0
\(743\) −30.9801 −1.13655 −0.568276 0.822838i \(-0.692389\pi\)
−0.568276 + 0.822838i \(0.692389\pi\)
\(744\) 31.1229 1.14102
\(745\) 37.3866 1.36974
\(746\) 21.7107 0.794885
\(747\) −2.79659 −0.102322
\(748\) −1.88385 −0.0688802
\(749\) 0 0
\(750\) 26.8520 0.980497
\(751\) 22.5660 0.823444 0.411722 0.911309i \(-0.364927\pi\)
0.411722 + 0.911309i \(0.364927\pi\)
\(752\) 3.20400 0.116838
\(753\) 32.5392 1.18579
\(754\) 0 0
\(755\) 16.8187 0.612095
\(756\) 0 0
\(757\) 32.2808 1.17327 0.586633 0.809853i \(-0.300453\pi\)
0.586633 + 0.809853i \(0.300453\pi\)
\(758\) 21.0737 0.765431
\(759\) 4.31818 0.156740
\(760\) −9.92260 −0.359931
\(761\) −29.7517 −1.07850 −0.539249 0.842147i \(-0.681292\pi\)
−0.539249 + 0.842147i \(0.681292\pi\)
\(762\) 40.5855 1.47026
\(763\) 0 0
\(764\) −0.903521 −0.0326883
\(765\) −33.2148 −1.20088
\(766\) 33.1367 1.19728
\(767\) 0 0
\(768\) 9.27895 0.334825
\(769\) 41.8105 1.50773 0.753863 0.657032i \(-0.228188\pi\)
0.753863 + 0.657032i \(0.228188\pi\)
\(770\) 0 0
\(771\) −49.8723 −1.79611
\(772\) −4.00514 −0.144148
\(773\) −41.4336 −1.49026 −0.745132 0.666917i \(-0.767613\pi\)
−0.745132 + 0.666917i \(0.767613\pi\)
\(774\) −6.19877 −0.222810
\(775\) −39.8215 −1.43043
\(776\) −1.41009 −0.0506192
\(777\) 0 0
\(778\) 25.3665 0.909433
\(779\) −3.96318 −0.141996
\(780\) 0 0
\(781\) 14.6075 0.522698
\(782\) 17.1497 0.613273
\(783\) 14.9020 0.532552
\(784\) 0 0
\(785\) −32.1175 −1.14632
\(786\) −30.8513 −1.10043
\(787\) 23.8627 0.850612 0.425306 0.905050i \(-0.360167\pi\)
0.425306 + 0.905050i \(0.360167\pi\)
\(788\) −1.10643 −0.0394148
\(789\) −31.6424 −1.12650
\(790\) 13.7040 0.487565
\(791\) 0 0
\(792\) −4.53221 −0.161045
\(793\) 0 0
\(794\) 19.5098 0.692377
\(795\) 0.585088 0.0207509
\(796\) 2.02135 0.0716447
\(797\) −50.8231 −1.80025 −0.900123 0.435636i \(-0.856524\pi\)
−0.900123 + 0.435636i \(0.856524\pi\)
\(798\) 0 0
\(799\) 6.92052 0.244830
\(800\) −8.29397 −0.293236
\(801\) −9.11367 −0.322016
\(802\) −28.2349 −0.997008
\(803\) −0.972438 −0.0343166
\(804\) −2.46691 −0.0870014
\(805\) 0 0
\(806\) 0 0
\(807\) −26.7348 −0.941110
\(808\) −8.48119 −0.298367
\(809\) 4.41176 0.155109 0.0775547 0.996988i \(-0.475289\pi\)
0.0775547 + 0.996988i \(0.475289\pi\)
\(810\) 53.5704 1.88227
\(811\) 17.6493 0.619750 0.309875 0.950777i \(-0.399713\pi\)
0.309875 + 0.950777i \(0.399713\pi\)
\(812\) 0 0
\(813\) 55.2926 1.93920
\(814\) 1.81657 0.0636707
\(815\) −42.9288 −1.50373
\(816\) −56.8489 −1.99011
\(817\) −3.61566 −0.126496
\(818\) −28.8249 −1.00784
\(819\) 0 0
\(820\) −2.85178 −0.0995884
\(821\) −3.56043 −0.124260 −0.0621299 0.998068i \(-0.519789\pi\)
−0.0621299 + 0.998068i \(0.519789\pi\)
\(822\) 48.6232 1.69593
\(823\) 21.8665 0.762217 0.381109 0.924530i \(-0.375542\pi\)
0.381109 + 0.924530i \(0.375542\pi\)
\(824\) 33.3993 1.16352
\(825\) 20.2610 0.705396
\(826\) 0 0
\(827\) 18.1361 0.630653 0.315327 0.948983i \(-0.397886\pi\)
0.315327 + 0.948983i \(0.397886\pi\)
\(828\) −0.377220 −0.0131093
\(829\) −30.8994 −1.07318 −0.536590 0.843843i \(-0.680288\pi\)
−0.536590 + 0.843843i \(0.680288\pi\)
\(830\) 11.1579 0.387297
\(831\) −26.0393 −0.903293
\(832\) 0 0
\(833\) 0 0
\(834\) 16.1530 0.559333
\(835\) −69.2933 −2.39799
\(836\) −0.229755 −0.00794625
\(837\) −18.9817 −0.656103
\(838\) 10.7135 0.370090
\(839\) 15.3959 0.531526 0.265763 0.964038i \(-0.414376\pi\)
0.265763 + 0.964038i \(0.414376\pi\)
\(840\) 0 0
\(841\) −12.6393 −0.435838
\(842\) −3.79065 −0.130635
\(843\) −54.7820 −1.88679
\(844\) 0.888153 0.0305715
\(845\) 0 0
\(846\) 1.44703 0.0497498
\(847\) 0 0
\(848\) 0.286615 0.00984239
\(849\) 30.2231 1.03726
\(850\) 80.4667 2.75998
\(851\) 1.73966 0.0596347
\(852\) −4.45845 −0.152744
\(853\) 23.7772 0.814116 0.407058 0.913402i \(-0.366555\pi\)
0.407058 + 0.913402i \(0.366555\pi\)
\(854\) 0 0
\(855\) −4.05090 −0.138538
\(856\) −19.3648 −0.661877
\(857\) 30.1050 1.02837 0.514184 0.857680i \(-0.328095\pi\)
0.514184 + 0.857680i \(0.328095\pi\)
\(858\) 0 0
\(859\) 15.1343 0.516377 0.258188 0.966095i \(-0.416874\pi\)
0.258188 + 0.966095i \(0.416874\pi\)
\(860\) −2.60172 −0.0887178
\(861\) 0 0
\(862\) 7.71657 0.262827
\(863\) −18.2657 −0.621773 −0.310886 0.950447i \(-0.600626\pi\)
−0.310886 + 0.950447i \(0.600626\pi\)
\(864\) −3.95348 −0.134500
\(865\) −51.3029 −1.74435
\(866\) −32.9925 −1.12113
\(867\) −87.9395 −2.98658
\(868\) 0 0
\(869\) 3.65100 0.123852
\(870\) 39.7988 1.34931
\(871\) 0 0
\(872\) −17.2053 −0.582643
\(873\) −0.575668 −0.0194834
\(874\) 2.09159 0.0707492
\(875\) 0 0
\(876\) 0.296803 0.0100281
\(877\) −6.99639 −0.236251 −0.118126 0.992999i \(-0.537689\pi\)
−0.118126 + 0.992999i \(0.537689\pi\)
\(878\) 49.2922 1.66353
\(879\) −23.7465 −0.800948
\(880\) 16.3458 0.551018
\(881\) −25.7746 −0.868368 −0.434184 0.900824i \(-0.642963\pi\)
−0.434184 + 0.900824i \(0.642963\pi\)
\(882\) 0 0
\(883\) 16.4526 0.553674 0.276837 0.960917i \(-0.410714\pi\)
0.276837 + 0.960917i \(0.410714\pi\)
\(884\) 0 0
\(885\) −81.8086 −2.74997
\(886\) −36.4467 −1.22445
\(887\) −55.2455 −1.85496 −0.927481 0.373871i \(-0.878030\pi\)
−0.927481 + 0.373871i \(0.878030\pi\)
\(888\) −6.37948 −0.214081
\(889\) 0 0
\(890\) 36.3620 1.21886
\(891\) 14.2722 0.478137
\(892\) 4.61170 0.154411
\(893\) 0.844032 0.0282444
\(894\) 28.8995 0.966542
\(895\) 19.3590 0.647099
\(896\) 0 0
\(897\) 0 0
\(898\) −36.9028 −1.23146
\(899\) −20.8398 −0.695046
\(900\) −1.76992 −0.0589973
\(901\) 0.619076 0.0206244
\(902\) 7.22241 0.240480
\(903\) 0 0
\(904\) 19.2478 0.640171
\(905\) −55.2655 −1.83709
\(906\) 13.0007 0.431919
\(907\) 47.8424 1.58858 0.794290 0.607538i \(-0.207843\pi\)
0.794290 + 0.607538i \(0.207843\pi\)
\(908\) −2.92694 −0.0971338
\(909\) −3.46245 −0.114842
\(910\) 0 0
\(911\) −23.0711 −0.764380 −0.382190 0.924084i \(-0.624830\pi\)
−0.382190 + 0.924084i \(0.624830\pi\)
\(912\) −6.93333 −0.229585
\(913\) 2.97269 0.0983817
\(914\) −53.3569 −1.76489
\(915\) −55.7637 −1.84349
\(916\) −3.11277 −0.102849
\(917\) 0 0
\(918\) 38.3560 1.26594
\(919\) −43.4368 −1.43285 −0.716424 0.697665i \(-0.754222\pi\)
−0.716424 + 0.697665i \(0.754222\pi\)
\(920\) 17.3171 0.570929
\(921\) 60.1367 1.98157
\(922\) 6.58595 0.216897
\(923\) 0 0
\(924\) 0 0
\(925\) 8.16249 0.268381
\(926\) −6.33861 −0.208300
\(927\) 13.6353 0.447841
\(928\) −4.34049 −0.142484
\(929\) −12.7819 −0.419361 −0.209680 0.977770i \(-0.567242\pi\)
−0.209680 + 0.977770i \(0.567242\pi\)
\(930\) −50.6945 −1.66234
\(931\) 0 0
\(932\) 5.54171 0.181525
\(933\) −0.308857 −0.0101115
\(934\) 43.0689 1.40926
\(935\) 35.3063 1.15464
\(936\) 0 0
\(937\) −16.2533 −0.530971 −0.265486 0.964115i \(-0.585532\pi\)
−0.265486 + 0.964115i \(0.585532\pi\)
\(938\) 0 0
\(939\) −21.5694 −0.703891
\(940\) 0.607339 0.0198092
\(941\) −45.1488 −1.47181 −0.735905 0.677085i \(-0.763243\pi\)
−0.735905 + 0.677085i \(0.763243\pi\)
\(942\) −24.8265 −0.808890
\(943\) 6.91662 0.225236
\(944\) −40.0752 −1.30434
\(945\) 0 0
\(946\) 6.58911 0.214230
\(947\) 19.8557 0.645225 0.322612 0.946531i \(-0.395439\pi\)
0.322612 + 0.946531i \(0.395439\pi\)
\(948\) −1.11434 −0.0361922
\(949\) 0 0
\(950\) 9.81378 0.318401
\(951\) 3.08901 0.100168
\(952\) 0 0
\(953\) 15.7287 0.509501 0.254751 0.967007i \(-0.418007\pi\)
0.254751 + 0.967007i \(0.418007\pi\)
\(954\) 0.129444 0.00419090
\(955\) 16.9335 0.547954
\(956\) 1.64745 0.0532823
\(957\) 10.6032 0.342752
\(958\) 24.2470 0.783385
\(959\) 0 0
\(960\) −62.9734 −2.03246
\(961\) −4.45488 −0.143706
\(962\) 0 0
\(963\) −7.90569 −0.254757
\(964\) −3.46432 −0.111578
\(965\) 75.0629 2.41636
\(966\) 0 0
\(967\) 52.1912 1.67835 0.839177 0.543858i \(-0.183037\pi\)
0.839177 + 0.543858i \(0.183037\pi\)
\(968\) −27.5943 −0.886915
\(969\) −14.9757 −0.481089
\(970\) 2.29682 0.0737465
\(971\) 22.4584 0.720724 0.360362 0.932813i \(-0.382653\pi\)
0.360362 + 0.932813i \(0.382653\pi\)
\(972\) −2.25207 −0.0722350
\(973\) 0 0
\(974\) 23.6765 0.758645
\(975\) 0 0
\(976\) −27.3167 −0.874387
\(977\) 41.0345 1.31281 0.656405 0.754409i \(-0.272076\pi\)
0.656405 + 0.754409i \(0.272076\pi\)
\(978\) −33.1835 −1.06109
\(979\) 9.68755 0.309616
\(980\) 0 0
\(981\) −7.02404 −0.224260
\(982\) −5.19742 −0.165856
\(983\) −26.8328 −0.855832 −0.427916 0.903818i \(-0.640752\pi\)
−0.427916 + 0.903818i \(0.640752\pi\)
\(984\) −25.3639 −0.808571
\(985\) 20.7362 0.660711
\(986\) 42.1107 1.34108
\(987\) 0 0
\(988\) 0 0
\(989\) 6.31013 0.200650
\(990\) 7.38228 0.234624
\(991\) 10.3751 0.329576 0.164788 0.986329i \(-0.447306\pi\)
0.164788 + 0.986329i \(0.447306\pi\)
\(992\) 5.52879 0.175539
\(993\) 51.7669 1.64277
\(994\) 0 0
\(995\) −37.8833 −1.20098
\(996\) −0.907312 −0.0287493
\(997\) 53.9097 1.70734 0.853669 0.520816i \(-0.174372\pi\)
0.853669 + 0.520816i \(0.174372\pi\)
\(998\) −17.0134 −0.538550
\(999\) 3.89081 0.123100
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 8281.2.a.co.1.4 12
7.3 odd 6 1183.2.e.j.170.9 24
7.5 odd 6 1183.2.e.j.508.9 24
7.6 odd 2 8281.2.a.cp.1.4 12
13.2 odd 12 637.2.q.i.589.2 12
13.7 odd 12 637.2.q.i.491.2 12
13.12 even 2 inner 8281.2.a.co.1.9 12
91.2 odd 12 637.2.k.i.459.5 12
91.12 odd 6 1183.2.e.j.508.4 24
91.20 even 12 637.2.q.g.491.2 12
91.33 even 12 91.2.u.b.88.5 yes 12
91.38 odd 6 1183.2.e.j.170.4 24
91.41 even 12 637.2.q.g.589.2 12
91.46 odd 12 637.2.k.i.569.2 12
91.54 even 12 91.2.k.b.4.5 12
91.59 even 12 91.2.k.b.23.2 yes 12
91.67 odd 12 637.2.u.g.30.5 12
91.72 odd 12 637.2.u.g.361.5 12
91.80 even 12 91.2.u.b.30.5 yes 12
91.90 odd 2 8281.2.a.cp.1.9 12
273.59 odd 12 819.2.bm.f.478.5 12
273.80 odd 12 819.2.do.e.667.2 12
273.215 odd 12 819.2.do.e.361.2 12
273.236 odd 12 819.2.bm.f.550.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.5 12 91.54 even 12
91.2.k.b.23.2 yes 12 91.59 even 12
91.2.u.b.30.5 yes 12 91.80 even 12
91.2.u.b.88.5 yes 12 91.33 even 12
637.2.k.i.459.5 12 91.2 odd 12
637.2.k.i.569.2 12 91.46 odd 12
637.2.q.g.491.2 12 91.20 even 12
637.2.q.g.589.2 12 91.41 even 12
637.2.q.i.491.2 12 13.7 odd 12
637.2.q.i.589.2 12 13.2 odd 12
637.2.u.g.30.5 12 91.67 odd 12
637.2.u.g.361.5 12 91.72 odd 12
819.2.bm.f.478.5 12 273.59 odd 12
819.2.bm.f.550.2 12 273.236 odd 12
819.2.do.e.361.2 12 273.215 odd 12
819.2.do.e.667.2 12 273.80 odd 12
1183.2.e.j.170.4 24 91.38 odd 6
1183.2.e.j.170.9 24 7.3 odd 6
1183.2.e.j.508.4 24 91.12 odd 6
1183.2.e.j.508.9 24 7.5 odd 6
8281.2.a.co.1.4 12 1.1 even 1 trivial
8281.2.a.co.1.9 12 13.12 even 2 inner
8281.2.a.cp.1.4 12 7.6 odd 2
8281.2.a.cp.1.9 12 91.90 odd 2