Properties

Label 8281.2.a.cp.1.4
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
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: \(0\)
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 \(-0.298412\pi\)
0.591814 + 0.806075i \(0.298412\pi\)
\(4\) −0.190366 −0.0951832
\(5\) −3.56778 −1.59556 −0.797779 0.602950i \(-0.793992\pi\)
−0.797779 + 0.602950i \(0.793992\pi\)
\(6\) −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 \(-0.112202\pi\)
0.938515 + 0.345238i \(0.112202\pi\)
\(18\) −1.61821 −0.381415
\(19\) 0.943878 0.216540 0.108270 0.994121i \(-0.465469\pi\)
0.108270 + 0.994121i \(0.465469\pi\)
\(20\) 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 \(-0.346888\pi\)
0.462681 + 0.886525i \(0.346888\pi\)
\(32\) −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 \(-0.606332\pi\)
−0.327873 + 0.944722i \(0.606332\pi\)
\(42\) 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 \(-0.479226\pi\)
0.0652175 + 0.997871i \(0.479226\pi\)
\(48\) −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 \(-0.759580\pi\)
−0.728064 + 0.685509i \(0.759580\pi\)
\(60\) 1.39240 0.179758
\(61\) −7.62392 −0.976143 −0.488072 0.872804i \(-0.662299\pi\)
−0.488072 + 0.872804i \(0.662299\pi\)
\(62\) −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 \(-0.514171\pi\)
−0.0445052 + 0.999009i \(0.514171\pi\)
\(74\) −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 \(-0.459275\pi\)
0.127591 + 0.991827i \(0.459275\pi\)
\(84\) 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 \(-0.368475\pi\)
0.401541 + 0.915841i \(0.368475\pi\)
\(90\) 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 \(-0.492266\pi\)
0.0242951 + 0.999705i \(0.492266\pi\)
\(98\) 0 0
\(99\) −1.53815 −0.154589
\(100\) −1.47135 −0.147135
\(101\) 2.87836 0.286407 0.143204 0.989693i \(-0.454260\pi\)
0.143204 + 0.989693i \(0.454260\pi\)
\(102\) −21.3436 −2.11333
\(103\) −11.3351 −1.11688 −0.558441 0.829544i \(-0.688600\pi\)
−0.558441 + 0.829544i \(0.688600\pi\)
\(104\) 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 \(-0.337474\pi\)
0.488693 + 0.872456i \(0.337474\pi\)
\(132\) 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 \(-0.579904\pi\)
−0.248396 + 0.968658i \(0.579904\pi\)
\(140\) 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 \(-0.383042\pi\)
0.359223 + 0.933252i \(0.383042\pi\)
\(158\) 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 \(-0.229350\pi\)
0.751459 + 0.659780i \(0.229350\pi\)
\(168\) 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 \(-0.315912\pi\)
0.546627 + 0.837376i \(0.315912\pi\)
\(174\) −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 \(-0.304734\pi\)
0.575688 + 0.817669i \(0.304734\pi\)
\(182\) 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 \(-0.377179\pi\)
0.376352 + 0.926477i \(0.377179\pi\)
\(200\) 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 \(-0.198855\pi\)
0.811126 + 0.584871i \(0.198855\pi\)
\(224\) 0 0
\(225\) 9.29744 0.619829
\(226\) −8.78747 −0.584534
\(227\) −15.3753 −1.02049 −0.510247 0.860028i \(-0.670446\pi\)
−0.510247 + 0.860028i \(0.670446\pi\)
\(228\) −0.368368 −0.0243958
\(229\) −16.3515 −1.08054 −0.540268 0.841493i \(-0.681677\pi\)
−0.540268 + 0.841493i \(0.681677\pi\)
\(230\) 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 \(-0.699347\pi\)
−0.586124 + 0.810222i \(0.699347\pi\)
\(242\) 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 \(-0.332997\pi\)
0.500915 + 0.865497i \(0.332997\pi\)
\(252\) 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 \(-0.774180\pi\)
−0.758730 + 0.651406i \(0.774180\pi\)
\(258\) −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 \(-0.630141\pi\)
−0.397553 + 0.917579i \(0.630141\pi\)
\(270\) −17.6822 −1.07610
\(271\) 26.9706 1.63835 0.819174 0.573544i \(-0.194432\pi\)
0.819174 + 0.573544i \(0.194432\pi\)
\(272\) −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 \(-0.355627\pi\)
0.438168 + 0.898893i \(0.355627\pi\)
\(284\) 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 \(-0.609867\pi\)
−0.338345 + 0.941022i \(0.609867\pi\)
\(294\) 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 \(-0.184263\pi\)
0.837076 + 0.547086i \(0.184263\pi\)
\(308\) 0 0
\(309\) −23.2381 −1.32197
\(310\) 24.7278 1.40444
\(311\) −0.150654 −0.00854282 −0.00427141 0.999991i \(-0.501360\pi\)
−0.00427141 + 0.999991i \(0.501360\pi\)
\(312\) 0 0
\(313\) −10.5211 −0.594690 −0.297345 0.954770i \(-0.596101\pi\)
−0.297345 + 0.954770i \(0.596101\pi\)
\(314\) −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 \(-0.399912\pi\)
0.309280 + 0.950971i \(0.399912\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.37213 0.0731350
\(353\) 20.0884 1.06920 0.534599 0.845106i \(-0.320463\pi\)
0.534599 + 0.845106i \(0.320463\pi\)
\(354\) 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 \(-0.424494\pi\)
0.234991 + 0.971998i \(0.424494\pi\)
\(368\) −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 \(-0.283326\pi\)
0.629339 + 0.777131i \(0.283326\pi\)
\(384\) −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 \(-0.381431\pi\)
0.363942 + 0.931422i \(0.381431\pi\)
\(398\) −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 \(-0.677719\pi\)
−0.529763 + 0.848146i \(0.677719\pi\)
\(410\) −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 \(-0.437680\pi\)
0.194535 + 0.980896i \(0.437680\pi\)
\(420\) 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 \(-0.700602\pi\)
−0.589314 + 0.807904i \(0.700602\pi\)
\(434\) 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 \(-0.161242\pi\)
0.874420 + 0.485169i \(0.161242\pi\)
\(440\) 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 \(-0.463630\pi\)
0.114010 + 0.993480i \(0.463630\pi\)
\(462\) 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 \(-0.234463\pi\)
0.740765 + 0.671764i \(0.234463\pi\)
\(468\) 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 \(-0.364907\pi\)
0.411780 + 0.911283i \(0.364907\pi\)
\(480\) 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 \(-0.336533\pi\)
0.491268 + 0.871008i \(0.336533\pi\)
\(504\) 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 \(-0.387104\pi\)
0.347284 + 0.937760i \(0.387104\pi\)
\(510\) 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 \(-0.686488\pi\)
−0.552925 + 0.833231i \(0.686488\pi\)
\(522\) −6.54538 −0.286483
\(523\) −13.2477 −0.579279 −0.289640 0.957136i \(-0.593535\pi\)
−0.289640 + 0.957136i \(0.593535\pi\)
\(524\) −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 \(-0.586755\pi\)
−0.269188 + 0.963088i \(0.586755\pi\)
\(564\) −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 \(-0.420099\pi\)
0.248388 + 0.968661i \(0.420099\pi\)
\(578\) −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 \(-0.362030\pi\)
0.420000 + 0.907524i \(0.362030\pi\)
\(588\) 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 \(-0.378232\pi\)
0.373282 + 0.927718i \(0.378232\pi\)
\(594\) −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 \(-0.684635\pi\)
−0.548064 + 0.836436i \(0.684635\pi\)
\(602\) 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 \(-0.561111\pi\)
−0.190810 + 0.981627i \(0.561111\pi\)
\(608\) −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 \(-0.440254\pi\)
0.186599 + 0.982436i \(0.440254\pi\)
\(620\) 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 \(-0.514423\pi\)
−0.0452968 + 0.998974i \(0.514423\pi\)
\(644\) 0 0
\(645\) −28.0185 −1.10323
\(646\) −9.82669 −0.386626
\(647\) 7.99865 0.314459 0.157230 0.987562i \(-0.449744\pi\)
0.157230 + 0.987562i \(0.449744\pi\)
\(648\) −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 \(-0.456618\pi\)
0.135866 + 0.990727i \(0.456618\pi\)
\(662\) 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 \(-0.275032\pi\)
0.649371 + 0.760472i \(0.275032\pi\)
\(678\) −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 \(-0.567651\pi\)
−0.210935 + 0.977500i \(0.567651\pi\)
\(692\) −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 \(-0.558348\pi\)
−0.182280 + 0.983247i \(0.558348\pi\)
\(720\) 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 \(-0.427303\pi\)
0.226406 + 0.974033i \(0.427303\pi\)
\(728\) 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 \(-0.364789\pi\)
0.412118 + 0.911131i \(0.364789\pi\)
\(734\) −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 \(-0.318708\pi\)
0.539249 + 0.842147i \(0.318708\pi\)
\(762\) −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 \(-0.771812\pi\)
−0.753863 + 0.657032i \(0.771812\pi\)
\(770\) 0 0
\(771\) −49.8723 −1.79611
\(772\) −4.00514 −0.144148
\(773\) 41.4336 1.49026 0.745132 0.666917i \(-0.232387\pi\)
0.745132 + 0.666917i \(0.232387\pi\)
\(774\) −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 \(-0.639833\pi\)
−0.425306 + 0.905050i \(0.639833\pi\)
\(788\) −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 \(-0.143476\pi\)
0.900123 + 0.435636i \(0.143476\pi\)
\(798\) 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 \(-0.600287\pi\)
−0.309875 + 0.950777i \(0.600287\pi\)
\(812\) 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 \(-0.319712\pi\)
0.536590 + 0.843843i \(0.319712\pi\)
\(830\) 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 \(-0.585624\pi\)
−0.265763 + 0.964038i \(0.585624\pi\)
\(840\) 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 \(-0.633445\pi\)
−0.407058 + 0.913402i \(0.633445\pi\)
\(854\) 0 0
\(855\) −4.05090 −0.138538
\(856\) −19.3648 −0.661877
\(857\) −30.1050 −1.02837 −0.514184 0.857680i \(-0.671905\pi\)
−0.514184 + 0.857680i \(0.671905\pi\)
\(858\) 0 0
\(859\) −15.1343 −0.516377 −0.258188 0.966095i \(-0.583126\pi\)
−0.258188 + 0.966095i \(0.583126\pi\)
\(860\) 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 \(-0.357037\pi\)
0.434184 + 0.900824i \(0.357037\pi\)
\(882\) 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 \(-0.121970\pi\)
0.927481 + 0.373871i \(0.121970\pi\)
\(888\) 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 \(-0.432758\pi\)
0.209680 + 0.977770i \(0.432758\pi\)
\(930\) 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 \(-0.414468\pi\)
0.265486 + 0.964115i \(0.414468\pi\)
\(938\) 0 0
\(939\) −21.5694 −0.703891
\(940\) 0.607339 0.0198092
\(941\) 45.1488 1.47181 0.735905 0.677085i \(-0.236757\pi\)
0.735905 + 0.677085i \(0.236757\pi\)
\(942\) −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 \(-0.617347\pi\)
−0.360362 + 0.932813i \(0.617347\pi\)
\(972\) 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 \(-0.359248\pi\)
0.427916 + 0.903818i \(0.359248\pi\)
\(984\) −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 \(-0.825628\pi\)
−0.853669 + 0.520816i \(0.825628\pi\)
\(998\) −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.cp.1.4 12
7.2 even 3 1183.2.e.j.508.9 24
7.4 even 3 1183.2.e.j.170.9 24
7.6 odd 2 8281.2.a.co.1.4 12
13.2 odd 12 637.2.q.g.589.2 12
13.7 odd 12 637.2.q.g.491.2 12
13.12 even 2 inner 8281.2.a.cp.1.9 12
91.2 odd 12 91.2.k.b.4.5 12
91.20 even 12 637.2.q.i.491.2 12
91.25 even 6 1183.2.e.j.170.4 24
91.33 even 12 637.2.u.g.361.5 12
91.41 even 12 637.2.q.i.589.2 12
91.46 odd 12 91.2.k.b.23.2 yes 12
91.51 even 6 1183.2.e.j.508.4 24
91.54 even 12 637.2.k.i.459.5 12
91.59 even 12 637.2.k.i.569.2 12
91.67 odd 12 91.2.u.b.30.5 yes 12
91.72 odd 12 91.2.u.b.88.5 yes 12
91.80 even 12 637.2.u.g.30.5 12
91.90 odd 2 8281.2.a.co.1.9 12
273.2 even 12 819.2.bm.f.550.2 12
273.137 even 12 819.2.bm.f.478.5 12
273.158 even 12 819.2.do.e.667.2 12
273.254 even 12 819.2.do.e.361.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.5 12 91.2 odd 12
91.2.k.b.23.2 yes 12 91.46 odd 12
91.2.u.b.30.5 yes 12 91.67 odd 12
91.2.u.b.88.5 yes 12 91.72 odd 12
637.2.k.i.459.5 12 91.54 even 12
637.2.k.i.569.2 12 91.59 even 12
637.2.q.g.491.2 12 13.7 odd 12
637.2.q.g.589.2 12 13.2 odd 12
637.2.q.i.491.2 12 91.20 even 12
637.2.q.i.589.2 12 91.41 even 12
637.2.u.g.30.5 12 91.80 even 12
637.2.u.g.361.5 12 91.33 even 12
819.2.bm.f.478.5 12 273.137 even 12
819.2.bm.f.550.2 12 273.2 even 12
819.2.do.e.361.2 12 273.254 even 12
819.2.do.e.667.2 12 273.158 even 12
1183.2.e.j.170.4 24 91.25 even 6
1183.2.e.j.170.9 24 7.4 even 3
1183.2.e.j.508.4 24 91.51 even 6
1183.2.e.j.508.9 24 7.2 even 3
8281.2.a.co.1.4 12 7.6 odd 2
8281.2.a.co.1.9 12 91.90 odd 2
8281.2.a.cp.1.4 12 1.1 even 1 trivial
8281.2.a.cp.1.9 12 13.12 even 2 inner