Properties

Label 8281.2.a.cp.1.7
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 1.7
Root \(0.180824\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.180824 q^{2} -1.82601 q^{3} -1.96730 q^{4} +2.68664 q^{5} -0.330186 q^{6} -0.717383 q^{8} +0.334323 q^{9} +O(q^{10})\) \(q+0.180824 q^{2} -1.82601 q^{3} -1.96730 q^{4} +2.68664 q^{5} -0.330186 q^{6} -0.717383 q^{8} +0.334323 q^{9} +0.485809 q^{10} +2.69424 q^{11} +3.59232 q^{12} -4.90584 q^{15} +3.80489 q^{16} +4.76493 q^{17} +0.0604535 q^{18} +0.188424 q^{19} -5.28544 q^{20} +0.487183 q^{22} +4.39929 q^{23} +1.30995 q^{24} +2.21804 q^{25} +4.86756 q^{27} +7.08560 q^{29} -0.887093 q^{30} -3.69931 q^{31} +2.12278 q^{32} -4.91972 q^{33} +0.861613 q^{34} -0.657715 q^{36} +7.95413 q^{37} +0.0340716 q^{38} -1.92735 q^{40} +5.42958 q^{41} -8.01065 q^{43} -5.30039 q^{44} +0.898206 q^{45} +0.795496 q^{46} -1.84889 q^{47} -6.94777 q^{48} +0.401075 q^{50} -8.70083 q^{51} -7.07244 q^{53} +0.880171 q^{54} +7.23846 q^{55} -0.344066 q^{57} +1.28125 q^{58} +7.58888 q^{59} +9.65128 q^{60} -0.411564 q^{61} -0.668922 q^{62} -7.22592 q^{64} -0.889602 q^{66} +11.4010 q^{67} -9.37407 q^{68} -8.03315 q^{69} -3.34488 q^{71} -0.239838 q^{72} -14.2158 q^{73} +1.43830 q^{74} -4.05018 q^{75} -0.370688 q^{76} +9.11059 q^{79} +10.2224 q^{80} -9.89120 q^{81} +0.981797 q^{82} +16.5866 q^{83} +12.8017 q^{85} -1.44852 q^{86} -12.9384 q^{87} -1.93280 q^{88} -5.89165 q^{89} +0.162417 q^{90} -8.65473 q^{92} +6.75498 q^{93} -0.334323 q^{94} +0.506229 q^{95} -3.87622 q^{96} -0.451094 q^{97} +0.900747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + O(q^{10}) \) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + 24q^{10} - 2q^{12} + 16q^{16} + 34q^{17} + 30q^{22} + 6q^{23} - 10q^{25} + 12q^{27} + 2q^{29} + 22q^{30} - 26q^{36} + 38q^{38} + 2q^{40} + 22q^{43} - 38q^{48} + 8q^{51} + 16q^{53} + 30q^{55} - 10q^{61} + 82q^{62} - 2q^{64} + 68q^{66} + 22q^{68} + 14q^{69} + 66q^{74} + 2q^{75} + 70q^{79} - 28q^{81} + 10q^{82} - 20q^{87} - 28q^{88} - 66q^{92} - 2q^{94} + 4q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.180824 0.127862 0.0639308 0.997954i \(-0.479636\pi\)
0.0639308 + 0.997954i \(0.479636\pi\)
\(3\) −1.82601 −1.05425 −0.527125 0.849788i \(-0.676730\pi\)
−0.527125 + 0.849788i \(0.676730\pi\)
\(4\) −1.96730 −0.983651
\(5\) 2.68664 1.20150 0.600751 0.799436i \(-0.294868\pi\)
0.600751 + 0.799436i \(0.294868\pi\)
\(6\) −0.330186 −0.134798
\(7\) 0 0
\(8\) −0.717383 −0.253633
\(9\) 0.334323 0.111441
\(10\) 0.485809 0.153626
\(11\) 2.69424 0.812345 0.406172 0.913796i \(-0.366863\pi\)
0.406172 + 0.913796i \(0.366863\pi\)
\(12\) 3.59232 1.03701
\(13\) 0 0
\(14\) 0 0
\(15\) −4.90584 −1.26668
\(16\) 3.80489 0.951221
\(17\) 4.76493 1.15567 0.577833 0.816155i \(-0.303898\pi\)
0.577833 + 0.816155i \(0.303898\pi\)
\(18\) 0.0604535 0.0142490
\(19\) 0.188424 0.0432275 0.0216138 0.999766i \(-0.493120\pi\)
0.0216138 + 0.999766i \(0.493120\pi\)
\(20\) −5.28544 −1.18186
\(21\) 0 0
\(22\) 0.487183 0.103868
\(23\) 4.39929 0.917315 0.458657 0.888613i \(-0.348331\pi\)
0.458657 + 0.888613i \(0.348331\pi\)
\(24\) 1.30995 0.267392
\(25\) 2.21804 0.443609
\(26\) 0 0
\(27\) 4.86756 0.936762
\(28\) 0 0
\(29\) 7.08560 1.31576 0.657882 0.753121i \(-0.271453\pi\)
0.657882 + 0.753121i \(0.271453\pi\)
\(30\) −0.887093 −0.161960
\(31\) −3.69931 −0.664415 −0.332207 0.943206i \(-0.607793\pi\)
−0.332207 + 0.943206i \(0.607793\pi\)
\(32\) 2.12278 0.375258
\(33\) −4.91972 −0.856414
\(34\) 0.861613 0.147765
\(35\) 0 0
\(36\) −0.657715 −0.109619
\(37\) 7.95413 1.30765 0.653826 0.756645i \(-0.273163\pi\)
0.653826 + 0.756645i \(0.273163\pi\)
\(38\) 0.0340716 0.00552715
\(39\) 0 0
\(40\) −1.92735 −0.304741
\(41\) 5.42958 0.847958 0.423979 0.905672i \(-0.360633\pi\)
0.423979 + 0.905672i \(0.360633\pi\)
\(42\) 0 0
\(43\) −8.01065 −1.22161 −0.610807 0.791780i \(-0.709155\pi\)
−0.610807 + 0.791780i \(0.709155\pi\)
\(44\) −5.30039 −0.799064
\(45\) 0.898206 0.133897
\(46\) 0.795496 0.117289
\(47\) −1.84889 −0.269688 −0.134844 0.990867i \(-0.543053\pi\)
−0.134844 + 0.990867i \(0.543053\pi\)
\(48\) −6.94777 −1.00282
\(49\) 0 0
\(50\) 0.401075 0.0567206
\(51\) −8.70083 −1.21836
\(52\) 0 0
\(53\) −7.07244 −0.971474 −0.485737 0.874105i \(-0.661449\pi\)
−0.485737 + 0.874105i \(0.661449\pi\)
\(54\) 0.880171 0.119776
\(55\) 7.23846 0.976034
\(56\) 0 0
\(57\) −0.344066 −0.0455726
\(58\) 1.28125 0.168236
\(59\) 7.58888 0.987988 0.493994 0.869465i \(-0.335536\pi\)
0.493994 + 0.869465i \(0.335536\pi\)
\(60\) 9.65128 1.24597
\(61\) −0.411564 −0.0526954 −0.0263477 0.999653i \(-0.508388\pi\)
−0.0263477 + 0.999653i \(0.508388\pi\)
\(62\) −0.668922 −0.0849532
\(63\) 0 0
\(64\) −7.22592 −0.903240
\(65\) 0 0
\(66\) −0.889602 −0.109502
\(67\) 11.4010 1.39286 0.696429 0.717626i \(-0.254771\pi\)
0.696429 + 0.717626i \(0.254771\pi\)
\(68\) −9.37407 −1.13677
\(69\) −8.03315 −0.967078
\(70\) 0 0
\(71\) −3.34488 −0.396965 −0.198482 0.980104i \(-0.563601\pi\)
−0.198482 + 0.980104i \(0.563601\pi\)
\(72\) −0.239838 −0.0282651
\(73\) −14.2158 −1.66383 −0.831917 0.554900i \(-0.812756\pi\)
−0.831917 + 0.554900i \(0.812756\pi\)
\(74\) 1.43830 0.167199
\(75\) −4.05018 −0.467674
\(76\) −0.370688 −0.0425208
\(77\) 0 0
\(78\) 0 0
\(79\) 9.11059 1.02502 0.512511 0.858681i \(-0.328715\pi\)
0.512511 + 0.858681i \(0.328715\pi\)
\(80\) 10.2224 1.14290
\(81\) −9.89120 −1.09902
\(82\) 0.981797 0.108421
\(83\) 16.5866 1.82061 0.910307 0.413934i \(-0.135845\pi\)
0.910307 + 0.413934i \(0.135845\pi\)
\(84\) 0 0
\(85\) 12.8017 1.38854
\(86\) −1.44852 −0.156198
\(87\) −12.9384 −1.38714
\(88\) −1.93280 −0.206037
\(89\) −5.89165 −0.624513 −0.312257 0.949998i \(-0.601085\pi\)
−0.312257 + 0.949998i \(0.601085\pi\)
\(90\) 0.162417 0.0171203
\(91\) 0 0
\(92\) −8.65473 −0.902318
\(93\) 6.75498 0.700459
\(94\) −0.334323 −0.0344828
\(95\) 0.506229 0.0519380
\(96\) −3.87622 −0.395615
\(97\) −0.451094 −0.0458016 −0.0229008 0.999738i \(-0.507290\pi\)
−0.0229008 + 0.999738i \(0.507290\pi\)
\(98\) 0 0
\(99\) 0.900747 0.0905285
\(100\) −4.36356 −0.436356
\(101\) −7.65680 −0.761880 −0.380940 0.924600i \(-0.624400\pi\)
−0.380940 + 0.924600i \(0.624400\pi\)
\(102\) −1.57332 −0.155782
\(103\) 5.15740 0.508173 0.254087 0.967181i \(-0.418225\pi\)
0.254087 + 0.967181i \(0.418225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.27887 −0.124214
\(107\) 8.03289 0.776569 0.388284 0.921540i \(-0.373068\pi\)
0.388284 + 0.921540i \(0.373068\pi\)
\(108\) −9.57597 −0.921448
\(109\) 1.33356 0.127732 0.0638660 0.997958i \(-0.479657\pi\)
0.0638660 + 0.997958i \(0.479657\pi\)
\(110\) 1.30889 0.124797
\(111\) −14.5243 −1.37859
\(112\) 0 0
\(113\) −19.9383 −1.87564 −0.937821 0.347119i \(-0.887160\pi\)
−0.937821 + 0.347119i \(0.887160\pi\)
\(114\) −0.0622152 −0.00582699
\(115\) 11.8193 1.10216
\(116\) −13.9395 −1.29425
\(117\) 0 0
\(118\) 1.37225 0.126326
\(119\) 0 0
\(120\) 3.51937 0.321273
\(121\) −3.74106 −0.340096
\(122\) −0.0744205 −0.00673772
\(123\) −9.91448 −0.893959
\(124\) 7.27765 0.653553
\(125\) −7.47412 −0.668505
\(126\) 0 0
\(127\) −7.96722 −0.706976 −0.353488 0.935439i \(-0.615005\pi\)
−0.353488 + 0.935439i \(0.615005\pi\)
\(128\) −5.55218 −0.490748
\(129\) 14.6276 1.28788
\(130\) 0 0
\(131\) 10.0179 0.875271 0.437636 0.899152i \(-0.355816\pi\)
0.437636 + 0.899152i \(0.355816\pi\)
\(132\) 9.67858 0.842412
\(133\) 0 0
\(134\) 2.06158 0.178093
\(135\) 13.0774 1.12552
\(136\) −3.41828 −0.293115
\(137\) −5.06696 −0.432899 −0.216450 0.976294i \(-0.569448\pi\)
−0.216450 + 0.976294i \(0.569448\pi\)
\(138\) −1.45259 −0.123652
\(139\) 7.72578 0.655292 0.327646 0.944800i \(-0.393745\pi\)
0.327646 + 0.944800i \(0.393745\pi\)
\(140\) 0 0
\(141\) 3.37610 0.284319
\(142\) −0.604834 −0.0507566
\(143\) 0 0
\(144\) 1.27206 0.106005
\(145\) 19.0365 1.58089
\(146\) −2.57055 −0.212741
\(147\) 0 0
\(148\) −15.6482 −1.28627
\(149\) 14.3185 1.17301 0.586507 0.809944i \(-0.300502\pi\)
0.586507 + 0.809944i \(0.300502\pi\)
\(150\) −0.732368 −0.0597976
\(151\) −6.47249 −0.526724 −0.263362 0.964697i \(-0.584831\pi\)
−0.263362 + 0.964697i \(0.584831\pi\)
\(152\) −0.135172 −0.0109639
\(153\) 1.59303 0.128789
\(154\) 0 0
\(155\) −9.93871 −0.798296
\(156\) 0 0
\(157\) 15.9187 1.27045 0.635227 0.772326i \(-0.280907\pi\)
0.635227 + 0.772326i \(0.280907\pi\)
\(158\) 1.64741 0.131061
\(159\) 12.9144 1.02418
\(160\) 5.70315 0.450873
\(161\) 0 0
\(162\) −1.78856 −0.140523
\(163\) 4.78162 0.374525 0.187263 0.982310i \(-0.440038\pi\)
0.187263 + 0.982310i \(0.440038\pi\)
\(164\) −10.6816 −0.834095
\(165\) −13.2175 −1.02898
\(166\) 2.99925 0.232787
\(167\) 2.71042 0.209739 0.104869 0.994486i \(-0.466558\pi\)
0.104869 + 0.994486i \(0.466558\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 2.31485 0.177541
\(171\) 0.0629946 0.00481732
\(172\) 15.7594 1.20164
\(173\) −0.899816 −0.0684118 −0.0342059 0.999415i \(-0.510890\pi\)
−0.0342059 + 0.999415i \(0.510890\pi\)
\(174\) −2.33957 −0.177362
\(175\) 0 0
\(176\) 10.2513 0.772720
\(177\) −13.8574 −1.04159
\(178\) −1.06535 −0.0798513
\(179\) −11.0558 −0.826351 −0.413175 0.910651i \(-0.635580\pi\)
−0.413175 + 0.910651i \(0.635580\pi\)
\(180\) −1.76704 −0.131708
\(181\) 3.52898 0.262307 0.131153 0.991362i \(-0.458132\pi\)
0.131153 + 0.991362i \(0.458132\pi\)
\(182\) 0 0
\(183\) 0.751521 0.0555540
\(184\) −3.15597 −0.232661
\(185\) 21.3699 1.57115
\(186\) 1.22146 0.0895619
\(187\) 12.8379 0.938799
\(188\) 3.63732 0.265279
\(189\) 0 0
\(190\) 0.0915382 0.00664088
\(191\) −20.4004 −1.47612 −0.738059 0.674736i \(-0.764258\pi\)
−0.738059 + 0.674736i \(0.764258\pi\)
\(192\) 13.1946 0.952240
\(193\) −17.2646 −1.24273 −0.621365 0.783521i \(-0.713422\pi\)
−0.621365 + 0.783521i \(0.713422\pi\)
\(194\) −0.0815684 −0.00585627
\(195\) 0 0
\(196\) 0 0
\(197\) −4.95672 −0.353152 −0.176576 0.984287i \(-0.556502\pi\)
−0.176576 + 0.984287i \(0.556502\pi\)
\(198\) 0.162877 0.0115751
\(199\) 7.18195 0.509115 0.254557 0.967058i \(-0.418070\pi\)
0.254557 + 0.967058i \(0.418070\pi\)
\(200\) −1.59119 −0.112514
\(201\) −20.8184 −1.46842
\(202\) −1.38453 −0.0974153
\(203\) 0 0
\(204\) 17.1172 1.19844
\(205\) 14.5873 1.01882
\(206\) 0.932580 0.0649759
\(207\) 1.47078 0.102226
\(208\) 0 0
\(209\) 0.507661 0.0351157
\(210\) 0 0
\(211\) −17.5927 −1.21113 −0.605566 0.795795i \(-0.707053\pi\)
−0.605566 + 0.795795i \(0.707053\pi\)
\(212\) 13.9136 0.955592
\(213\) 6.10780 0.418499
\(214\) 1.45254 0.0992934
\(215\) −21.5218 −1.46777
\(216\) −3.49190 −0.237594
\(217\) 0 0
\(218\) 0.241140 0.0163320
\(219\) 25.9582 1.75410
\(220\) −14.2403 −0.960078
\(221\) 0 0
\(222\) −2.62635 −0.176269
\(223\) 14.1054 0.944569 0.472284 0.881446i \(-0.343430\pi\)
0.472284 + 0.881446i \(0.343430\pi\)
\(224\) 0 0
\(225\) 0.741543 0.0494362
\(226\) −3.60533 −0.239823
\(227\) 2.86877 0.190407 0.0952035 0.995458i \(-0.469650\pi\)
0.0952035 + 0.995458i \(0.469650\pi\)
\(228\) 0.676881 0.0448275
\(229\) 8.77411 0.579810 0.289905 0.957055i \(-0.406376\pi\)
0.289905 + 0.957055i \(0.406376\pi\)
\(230\) 2.13721 0.140924
\(231\) 0 0
\(232\) −5.08309 −0.333721
\(233\) 5.10743 0.334599 0.167299 0.985906i \(-0.446495\pi\)
0.167299 + 0.985906i \(0.446495\pi\)
\(234\) 0 0
\(235\) −4.96730 −0.324031
\(236\) −14.9296 −0.971836
\(237\) −16.6361 −1.08063
\(238\) 0 0
\(239\) 2.49797 0.161580 0.0807901 0.996731i \(-0.474256\pi\)
0.0807901 + 0.996731i \(0.474256\pi\)
\(240\) −18.6662 −1.20490
\(241\) −7.98512 −0.514367 −0.257183 0.966363i \(-0.582794\pi\)
−0.257183 + 0.966363i \(0.582794\pi\)
\(242\) −0.676472 −0.0434853
\(243\) 3.45877 0.221880
\(244\) 0.809671 0.0518339
\(245\) 0 0
\(246\) −1.79277 −0.114303
\(247\) 0 0
\(248\) 2.65382 0.168518
\(249\) −30.2873 −1.91938
\(250\) −1.35150 −0.0854762
\(251\) 25.2570 1.59421 0.797105 0.603841i \(-0.206364\pi\)
0.797105 + 0.603841i \(0.206364\pi\)
\(252\) 0 0
\(253\) 11.8527 0.745176
\(254\) −1.44066 −0.0903952
\(255\) −23.3760 −1.46386
\(256\) 13.4479 0.840493
\(257\) −3.37363 −0.210442 −0.105221 0.994449i \(-0.533555\pi\)
−0.105221 + 0.994449i \(0.533555\pi\)
\(258\) 2.64501 0.164671
\(259\) 0 0
\(260\) 0 0
\(261\) 2.36888 0.146630
\(262\) 1.81148 0.111914
\(263\) −0.158935 −0.00980037 −0.00490019 0.999988i \(-0.501560\pi\)
−0.00490019 + 0.999988i \(0.501560\pi\)
\(264\) 3.52932 0.217215
\(265\) −19.0011 −1.16723
\(266\) 0 0
\(267\) 10.7582 0.658392
\(268\) −22.4293 −1.37009
\(269\) −23.3266 −1.42225 −0.711124 0.703066i \(-0.751814\pi\)
−0.711124 + 0.703066i \(0.751814\pi\)
\(270\) 2.36470 0.143911
\(271\) 11.8210 0.718074 0.359037 0.933323i \(-0.383105\pi\)
0.359037 + 0.933323i \(0.383105\pi\)
\(272\) 18.1300 1.09929
\(273\) 0 0
\(274\) −0.916226 −0.0553513
\(275\) 5.97595 0.360363
\(276\) 15.8036 0.951268
\(277\) −27.3653 −1.64422 −0.822111 0.569327i \(-0.807204\pi\)
−0.822111 + 0.569327i \(0.807204\pi\)
\(278\) 1.39701 0.0837868
\(279\) −1.23676 −0.0740431
\(280\) 0 0
\(281\) 28.5383 1.70245 0.851225 0.524801i \(-0.175860\pi\)
0.851225 + 0.524801i \(0.175860\pi\)
\(282\) 0.610478 0.0363534
\(283\) 17.9721 1.06833 0.534165 0.845380i \(-0.320626\pi\)
0.534165 + 0.845380i \(0.320626\pi\)
\(284\) 6.58040 0.390475
\(285\) −0.924381 −0.0547556
\(286\) 0 0
\(287\) 0 0
\(288\) 0.709694 0.0418191
\(289\) 5.70459 0.335564
\(290\) 3.44225 0.202136
\(291\) 0.823703 0.0482863
\(292\) 27.9668 1.63663
\(293\) 14.8891 0.869828 0.434914 0.900472i \(-0.356779\pi\)
0.434914 + 0.900472i \(0.356779\pi\)
\(294\) 0 0
\(295\) 20.3886 1.18707
\(296\) −5.70616 −0.331664
\(297\) 13.1144 0.760974
\(298\) 2.58912 0.149984
\(299\) 0 0
\(300\) 7.96793 0.460028
\(301\) 0 0
\(302\) −1.17038 −0.0673478
\(303\) 13.9814 0.803211
\(304\) 0.716934 0.0411190
\(305\) −1.10572 −0.0633136
\(306\) 0.288057 0.0164671
\(307\) −23.5161 −1.34214 −0.671068 0.741396i \(-0.734164\pi\)
−0.671068 + 0.741396i \(0.734164\pi\)
\(308\) 0 0
\(309\) −9.41747 −0.535741
\(310\) −1.79715 −0.102072
\(311\) 1.63090 0.0924799 0.0462399 0.998930i \(-0.485276\pi\)
0.0462399 + 0.998930i \(0.485276\pi\)
\(312\) 0 0
\(313\) −0.696734 −0.0393817 −0.0196909 0.999806i \(-0.506268\pi\)
−0.0196909 + 0.999806i \(0.506268\pi\)
\(314\) 2.87849 0.162442
\(315\) 0 0
\(316\) −17.9233 −1.00826
\(317\) −21.4288 −1.20356 −0.601780 0.798662i \(-0.705542\pi\)
−0.601780 + 0.798662i \(0.705542\pi\)
\(318\) 2.33522 0.130953
\(319\) 19.0903 1.06885
\(320\) −19.4135 −1.08525
\(321\) −14.6682 −0.818697
\(322\) 0 0
\(323\) 0.897830 0.0499566
\(324\) 19.4590 1.08105
\(325\) 0 0
\(326\) 0.864630 0.0478874
\(327\) −2.43510 −0.134661
\(328\) −3.89509 −0.215070
\(329\) 0 0
\(330\) −2.39004 −0.131568
\(331\) −1.52046 −0.0835722 −0.0417861 0.999127i \(-0.513305\pi\)
−0.0417861 + 0.999127i \(0.513305\pi\)
\(332\) −32.6308 −1.79085
\(333\) 2.65925 0.145726
\(334\) 0.490108 0.0268175
\(335\) 30.6305 1.67352
\(336\) 0 0
\(337\) 32.2304 1.75570 0.877850 0.478936i \(-0.158977\pi\)
0.877850 + 0.478936i \(0.158977\pi\)
\(338\) 0 0
\(339\) 36.4077 1.97739
\(340\) −25.1848 −1.36584
\(341\) −9.96683 −0.539734
\(342\) 0.0113909 0.000615951 0
\(343\) 0 0
\(344\) 5.74670 0.309841
\(345\) −21.5822 −1.16195
\(346\) −0.162708 −0.00874724
\(347\) 8.18431 0.439357 0.219678 0.975572i \(-0.429499\pi\)
0.219678 + 0.975572i \(0.429499\pi\)
\(348\) 25.4538 1.36446
\(349\) 21.8493 1.16956 0.584782 0.811190i \(-0.301180\pi\)
0.584782 + 0.811190i \(0.301180\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.71928 0.304839
\(353\) 0.567179 0.0301879 0.0150940 0.999886i \(-0.495195\pi\)
0.0150940 + 0.999886i \(0.495195\pi\)
\(354\) −2.50575 −0.133179
\(355\) −8.98650 −0.476954
\(356\) 11.5907 0.614303
\(357\) 0 0
\(358\) −1.99915 −0.105659
\(359\) −32.4043 −1.71024 −0.855118 0.518434i \(-0.826515\pi\)
−0.855118 + 0.518434i \(0.826515\pi\)
\(360\) −0.644358 −0.0339606
\(361\) −18.9645 −0.998131
\(362\) 0.638123 0.0335390
\(363\) 6.83122 0.358546
\(364\) 0 0
\(365\) −38.1928 −1.99910
\(366\) 0.135893 0.00710323
\(367\) −7.86888 −0.410752 −0.205376 0.978683i \(-0.565842\pi\)
−0.205376 + 0.978683i \(0.565842\pi\)
\(368\) 16.7388 0.872569
\(369\) 1.81523 0.0944973
\(370\) 3.86419 0.200889
\(371\) 0 0
\(372\) −13.2891 −0.689007
\(373\) −2.09163 −0.108300 −0.0541502 0.998533i \(-0.517245\pi\)
−0.0541502 + 0.998533i \(0.517245\pi\)
\(374\) 2.32139 0.120036
\(375\) 13.6478 0.704771
\(376\) 1.32636 0.0684018
\(377\) 0 0
\(378\) 0 0
\(379\) 14.3163 0.735381 0.367691 0.929948i \(-0.380149\pi\)
0.367691 + 0.929948i \(0.380149\pi\)
\(380\) −0.995906 −0.0510889
\(381\) 14.5482 0.745329
\(382\) −3.68887 −0.188739
\(383\) 25.1873 1.28701 0.643507 0.765441i \(-0.277479\pi\)
0.643507 + 0.765441i \(0.277479\pi\)
\(384\) 10.1383 0.517370
\(385\) 0 0
\(386\) −3.12184 −0.158898
\(387\) −2.67815 −0.136138
\(388\) 0.887438 0.0450528
\(389\) 28.1023 1.42484 0.712422 0.701752i \(-0.247598\pi\)
0.712422 + 0.701752i \(0.247598\pi\)
\(390\) 0 0
\(391\) 20.9623 1.06011
\(392\) 0 0
\(393\) −18.2929 −0.922754
\(394\) −0.896292 −0.0451546
\(395\) 24.4769 1.23157
\(396\) −1.77204 −0.0890485
\(397\) 21.7765 1.09293 0.546465 0.837482i \(-0.315973\pi\)
0.546465 + 0.837482i \(0.315973\pi\)
\(398\) 1.29867 0.0650963
\(399\) 0 0
\(400\) 8.43941 0.421970
\(401\) −20.5290 −1.02517 −0.512584 0.858637i \(-0.671312\pi\)
−0.512584 + 0.858637i \(0.671312\pi\)
\(402\) −3.76447 −0.187754
\(403\) 0 0
\(404\) 15.0632 0.749424
\(405\) −26.5741 −1.32048
\(406\) 0 0
\(407\) 21.4304 1.06226
\(408\) 6.24182 0.309016
\(409\) 6.26862 0.309963 0.154982 0.987917i \(-0.450468\pi\)
0.154982 + 0.987917i \(0.450468\pi\)
\(410\) 2.63774 0.130269
\(411\) 9.25233 0.456384
\(412\) −10.1462 −0.499865
\(413\) 0 0
\(414\) 0.265953 0.0130709
\(415\) 44.5622 2.18747
\(416\) 0 0
\(417\) −14.1074 −0.690841
\(418\) 0.0917972 0.00448995
\(419\) 34.1635 1.66899 0.834497 0.551013i \(-0.185758\pi\)
0.834497 + 0.551013i \(0.185758\pi\)
\(420\) 0 0
\(421\) 11.5233 0.561613 0.280806 0.959764i \(-0.409398\pi\)
0.280806 + 0.959764i \(0.409398\pi\)
\(422\) −3.18118 −0.154858
\(423\) −0.618126 −0.0300543
\(424\) 5.07364 0.246398
\(425\) 10.5688 0.512664
\(426\) 1.10444 0.0535101
\(427\) 0 0
\(428\) −15.8031 −0.763873
\(429\) 0 0
\(430\) −3.89164 −0.187672
\(431\) 8.77001 0.422436 0.211218 0.977439i \(-0.432257\pi\)
0.211218 + 0.977439i \(0.432257\pi\)
\(432\) 18.5205 0.891069
\(433\) 22.1069 1.06239 0.531196 0.847249i \(-0.321743\pi\)
0.531196 + 0.847249i \(0.321743\pi\)
\(434\) 0 0
\(435\) −34.7609 −1.66666
\(436\) −2.62352 −0.125644
\(437\) 0.828933 0.0396533
\(438\) 4.69387 0.224282
\(439\) −10.3709 −0.494978 −0.247489 0.968891i \(-0.579605\pi\)
−0.247489 + 0.968891i \(0.579605\pi\)
\(440\) −5.19275 −0.247555
\(441\) 0 0
\(442\) 0 0
\(443\) 35.8137 1.70156 0.850780 0.525522i \(-0.176130\pi\)
0.850780 + 0.525522i \(0.176130\pi\)
\(444\) 28.5738 1.35605
\(445\) −15.8287 −0.750354
\(446\) 2.55059 0.120774
\(447\) −26.1457 −1.23665
\(448\) 0 0
\(449\) −22.7502 −1.07365 −0.536825 0.843693i \(-0.680377\pi\)
−0.536825 + 0.843693i \(0.680377\pi\)
\(450\) 0.134089 0.00632100
\(451\) 14.6286 0.688834
\(452\) 39.2248 1.84498
\(453\) 11.8188 0.555298
\(454\) 0.518742 0.0243458
\(455\) 0 0
\(456\) 0.246827 0.0115587
\(457\) 31.3172 1.46496 0.732478 0.680791i \(-0.238364\pi\)
0.732478 + 0.680791i \(0.238364\pi\)
\(458\) 1.58657 0.0741354
\(459\) 23.1936 1.08258
\(460\) −23.2522 −1.08414
\(461\) 8.40753 0.391578 0.195789 0.980646i \(-0.437273\pi\)
0.195789 + 0.980646i \(0.437273\pi\)
\(462\) 0 0
\(463\) 10.0392 0.466563 0.233281 0.972409i \(-0.425054\pi\)
0.233281 + 0.972409i \(0.425054\pi\)
\(464\) 26.9599 1.25158
\(465\) 18.1482 0.841603
\(466\) 0.923545 0.0427824
\(467\) −26.3513 −1.21939 −0.609696 0.792635i \(-0.708709\pi\)
−0.609696 + 0.792635i \(0.708709\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.898206 −0.0414312
\(471\) −29.0678 −1.33937
\(472\) −5.44413 −0.250586
\(473\) −21.5826 −0.992371
\(474\) −3.00819 −0.138171
\(475\) 0.417934 0.0191761
\(476\) 0 0
\(477\) −2.36448 −0.108262
\(478\) 0.451692 0.0206599
\(479\) −8.58414 −0.392220 −0.196110 0.980582i \(-0.562831\pi\)
−0.196110 + 0.980582i \(0.562831\pi\)
\(480\) −10.4140 −0.475333
\(481\) 0 0
\(482\) −1.44390 −0.0657678
\(483\) 0 0
\(484\) 7.35979 0.334536
\(485\) −1.21193 −0.0550308
\(486\) 0.625428 0.0283700
\(487\) 21.2562 0.963212 0.481606 0.876388i \(-0.340054\pi\)
0.481606 + 0.876388i \(0.340054\pi\)
\(488\) 0.295249 0.0133653
\(489\) −8.73130 −0.394843
\(490\) 0 0
\(491\) −22.4535 −1.01331 −0.506657 0.862148i \(-0.669119\pi\)
−0.506657 + 0.862148i \(0.669119\pi\)
\(492\) 19.5048 0.879344
\(493\) 33.7624 1.52058
\(494\) 0 0
\(495\) 2.41999 0.108770
\(496\) −14.0754 −0.632006
\(497\) 0 0
\(498\) −5.47667 −0.245415
\(499\) 38.8780 1.74042 0.870210 0.492681i \(-0.163983\pi\)
0.870210 + 0.492681i \(0.163983\pi\)
\(500\) 14.7039 0.657576
\(501\) −4.94926 −0.221117
\(502\) 4.56707 0.203838
\(503\) 5.45701 0.243316 0.121658 0.992572i \(-0.461179\pi\)
0.121658 + 0.992572i \(0.461179\pi\)
\(504\) 0 0
\(505\) −20.5711 −0.915401
\(506\) 2.14326 0.0952794
\(507\) 0 0
\(508\) 15.6739 0.695418
\(509\) −10.8925 −0.482800 −0.241400 0.970426i \(-0.577607\pi\)
−0.241400 + 0.970426i \(0.577607\pi\)
\(510\) −4.22694 −0.187172
\(511\) 0 0
\(512\) 13.5360 0.598214
\(513\) 0.917168 0.0404939
\(514\) −0.610033 −0.0269074
\(515\) 13.8561 0.610572
\(516\) −28.7768 −1.26683
\(517\) −4.98136 −0.219080
\(518\) 0 0
\(519\) 1.64308 0.0721230
\(520\) 0 0
\(521\) 27.8961 1.22215 0.611074 0.791573i \(-0.290738\pi\)
0.611074 + 0.791573i \(0.290738\pi\)
\(522\) 0.428350 0.0187484
\(523\) 16.7236 0.731272 0.365636 0.930758i \(-0.380852\pi\)
0.365636 + 0.930758i \(0.380852\pi\)
\(524\) −19.7083 −0.860962
\(525\) 0 0
\(526\) −0.0287393 −0.00125309
\(527\) −17.6269 −0.767842
\(528\) −18.7190 −0.814639
\(529\) −3.64627 −0.158534
\(530\) −3.43585 −0.149244
\(531\) 2.53714 0.110102
\(532\) 0 0
\(533\) 0 0
\(534\) 1.94534 0.0841832
\(535\) 21.5815 0.933049
\(536\) −8.17890 −0.353275
\(537\) 20.1881 0.871179
\(538\) −4.21800 −0.181851
\(539\) 0 0
\(540\) −25.7272 −1.10712
\(541\) 11.1605 0.479828 0.239914 0.970794i \(-0.422881\pi\)
0.239914 + 0.970794i \(0.422881\pi\)
\(542\) 2.13751 0.0918141
\(543\) −6.44396 −0.276537
\(544\) 10.1149 0.433673
\(545\) 3.58280 0.153470
\(546\) 0 0
\(547\) 36.6556 1.56728 0.783640 0.621215i \(-0.213361\pi\)
0.783640 + 0.621215i \(0.213361\pi\)
\(548\) 9.96824 0.425822
\(549\) −0.137595 −0.00587242
\(550\) 1.08059 0.0460767
\(551\) 1.33510 0.0568772
\(552\) 5.76285 0.245283
\(553\) 0 0
\(554\) −4.94830 −0.210233
\(555\) −39.0217 −1.65638
\(556\) −15.1990 −0.644579
\(557\) −33.0776 −1.40154 −0.700772 0.713386i \(-0.747161\pi\)
−0.700772 + 0.713386i \(0.747161\pi\)
\(558\) −0.223636 −0.00946727
\(559\) 0 0
\(560\) 0 0
\(561\) −23.4421 −0.989728
\(562\) 5.16039 0.217678
\(563\) 17.7967 0.750043 0.375021 0.927016i \(-0.377635\pi\)
0.375021 + 0.927016i \(0.377635\pi\)
\(564\) −6.64180 −0.279670
\(565\) −53.5672 −2.25359
\(566\) 3.24978 0.136598
\(567\) 0 0
\(568\) 2.39956 0.100683
\(569\) −8.22094 −0.344640 −0.172320 0.985041i \(-0.555126\pi\)
−0.172320 + 0.985041i \(0.555126\pi\)
\(570\) −0.167150 −0.00700114
\(571\) −25.7553 −1.07782 −0.538912 0.842362i \(-0.681165\pi\)
−0.538912 + 0.842362i \(0.681165\pi\)
\(572\) 0 0
\(573\) 37.2513 1.55620
\(574\) 0 0
\(575\) 9.75781 0.406929
\(576\) −2.41579 −0.100658
\(577\) −0.769393 −0.0320302 −0.0160151 0.999872i \(-0.505098\pi\)
−0.0160151 + 0.999872i \(0.505098\pi\)
\(578\) 1.03152 0.0429058
\(579\) 31.5253 1.31015
\(580\) −37.4505 −1.55505
\(581\) 0 0
\(582\) 0.148945 0.00617397
\(583\) −19.0549 −0.789172
\(584\) 10.1982 0.422003
\(585\) 0 0
\(586\) 2.69229 0.111218
\(587\) −12.0929 −0.499127 −0.249563 0.968358i \(-0.580287\pi\)
−0.249563 + 0.968358i \(0.580287\pi\)
\(588\) 0 0
\(589\) −0.697040 −0.0287210
\(590\) 3.68674 0.151781
\(591\) 9.05103 0.372310
\(592\) 30.2646 1.24387
\(593\) −15.9481 −0.654911 −0.327456 0.944867i \(-0.606191\pi\)
−0.327456 + 0.944867i \(0.606191\pi\)
\(594\) 2.37139 0.0972994
\(595\) 0 0
\(596\) −28.1688 −1.15384
\(597\) −13.1143 −0.536734
\(598\) 0 0
\(599\) −7.11022 −0.290516 −0.145258 0.989394i \(-0.546401\pi\)
−0.145258 + 0.989394i \(0.546401\pi\)
\(600\) 2.90553 0.118618
\(601\) 20.7905 0.848064 0.424032 0.905647i \(-0.360614\pi\)
0.424032 + 0.905647i \(0.360614\pi\)
\(602\) 0 0
\(603\) 3.81163 0.155221
\(604\) 12.7333 0.518112
\(605\) −10.0509 −0.408626
\(606\) 2.52817 0.102700
\(607\) −7.71405 −0.313104 −0.156552 0.987670i \(-0.550038\pi\)
−0.156552 + 0.987670i \(0.550038\pi\)
\(608\) 0.399983 0.0162215
\(609\) 0 0
\(610\) −0.199941 −0.00809539
\(611\) 0 0
\(612\) −3.13397 −0.126683
\(613\) 20.4378 0.825476 0.412738 0.910850i \(-0.364572\pi\)
0.412738 + 0.910850i \(0.364572\pi\)
\(614\) −4.25227 −0.171608
\(615\) −26.6367 −1.07409
\(616\) 0 0
\(617\) 4.59812 0.185113 0.0925567 0.995707i \(-0.470496\pi\)
0.0925567 + 0.995707i \(0.470496\pi\)
\(618\) −1.70290 −0.0685008
\(619\) 10.0528 0.404057 0.202028 0.979380i \(-0.435247\pi\)
0.202028 + 0.979380i \(0.435247\pi\)
\(620\) 19.5525 0.785245
\(621\) 21.4138 0.859306
\(622\) 0.294906 0.0118246
\(623\) 0 0
\(624\) 0 0
\(625\) −31.1705 −1.24682
\(626\) −0.125986 −0.00503541
\(627\) −0.926996 −0.0370207
\(628\) −31.3170 −1.24968
\(629\) 37.9009 1.51121
\(630\) 0 0
\(631\) −7.27372 −0.289562 −0.144781 0.989464i \(-0.546248\pi\)
−0.144781 + 0.989464i \(0.546248\pi\)
\(632\) −6.53578 −0.259979
\(633\) 32.1245 1.27684
\(634\) −3.87483 −0.153889
\(635\) −21.4051 −0.849434
\(636\) −25.4065 −1.00743
\(637\) 0 0
\(638\) 3.45199 0.136665
\(639\) −1.11827 −0.0442381
\(640\) −14.9167 −0.589635
\(641\) 3.85033 0.152079 0.0760394 0.997105i \(-0.475773\pi\)
0.0760394 + 0.997105i \(0.475773\pi\)
\(642\) −2.65235 −0.104680
\(643\) −2.87709 −0.113461 −0.0567307 0.998390i \(-0.518068\pi\)
−0.0567307 + 0.998390i \(0.518068\pi\)
\(644\) 0 0
\(645\) 39.2990 1.54740
\(646\) 0.162349 0.00638754
\(647\) 37.1001 1.45856 0.729278 0.684218i \(-0.239856\pi\)
0.729278 + 0.684218i \(0.239856\pi\)
\(648\) 7.09577 0.278748
\(649\) 20.4463 0.802587
\(650\) 0 0
\(651\) 0 0
\(652\) −9.40689 −0.368402
\(653\) 20.0950 0.786377 0.393189 0.919458i \(-0.371372\pi\)
0.393189 + 0.919458i \(0.371372\pi\)
\(654\) −0.440324 −0.0172180
\(655\) 26.9146 1.05164
\(656\) 20.6589 0.806596
\(657\) −4.75267 −0.185419
\(658\) 0 0
\(659\) 9.91058 0.386061 0.193031 0.981193i \(-0.438168\pi\)
0.193031 + 0.981193i \(0.438168\pi\)
\(660\) 26.0029 1.01216
\(661\) 47.2266 1.83690 0.918450 0.395537i \(-0.129442\pi\)
0.918450 + 0.395537i \(0.129442\pi\)
\(662\) −0.274936 −0.0106857
\(663\) 0 0
\(664\) −11.8989 −0.461768
\(665\) 0 0
\(666\) 0.480855 0.0186328
\(667\) 31.1716 1.20697
\(668\) −5.33222 −0.206310
\(669\) −25.7567 −0.995811
\(670\) 5.53872 0.213979
\(671\) −1.10885 −0.0428068
\(672\) 0 0
\(673\) 6.91689 0.266627 0.133313 0.991074i \(-0.457438\pi\)
0.133313 + 0.991074i \(0.457438\pi\)
\(674\) 5.82801 0.224487
\(675\) 10.7965 0.415556
\(676\) 0 0
\(677\) −12.3291 −0.473844 −0.236922 0.971529i \(-0.576139\pi\)
−0.236922 + 0.971529i \(0.576139\pi\)
\(678\) 6.58337 0.252833
\(679\) 0 0
\(680\) −9.18369 −0.352179
\(681\) −5.23841 −0.200736
\(682\) −1.80224 −0.0690113
\(683\) −24.5364 −0.938859 −0.469430 0.882970i \(-0.655540\pi\)
−0.469430 + 0.882970i \(0.655540\pi\)
\(684\) −0.123930 −0.00473856
\(685\) −13.6131 −0.520130
\(686\) 0 0
\(687\) −16.0216 −0.611264
\(688\) −30.4796 −1.16202
\(689\) 0 0
\(690\) −3.90258 −0.148569
\(691\) 9.10716 0.346453 0.173226 0.984882i \(-0.444581\pi\)
0.173226 + 0.984882i \(0.444581\pi\)
\(692\) 1.77021 0.0672933
\(693\) 0 0
\(694\) 1.47992 0.0561769
\(695\) 20.7564 0.787336
\(696\) 9.28179 0.351825
\(697\) 25.8716 0.979956
\(698\) 3.95087 0.149542
\(699\) −9.32623 −0.352750
\(700\) 0 0
\(701\) −0.286950 −0.0108380 −0.00541898 0.999985i \(-0.501725\pi\)
−0.00541898 + 0.999985i \(0.501725\pi\)
\(702\) 0 0
\(703\) 1.49875 0.0565265
\(704\) −19.4684 −0.733742
\(705\) 9.07036 0.341609
\(706\) 0.102559 0.00385988
\(707\) 0 0
\(708\) 27.2617 1.02456
\(709\) −18.5848 −0.697967 −0.348984 0.937129i \(-0.613473\pi\)
−0.348984 + 0.937129i \(0.613473\pi\)
\(710\) −1.62497 −0.0609841
\(711\) 3.04588 0.114229
\(712\) 4.22656 0.158397
\(713\) −16.2743 −0.609478
\(714\) 0 0
\(715\) 0 0
\(716\) 21.7501 0.812841
\(717\) −4.56132 −0.170346
\(718\) −5.85947 −0.218674
\(719\) 41.6949 1.55496 0.777479 0.628909i \(-0.216498\pi\)
0.777479 + 0.628909i \(0.216498\pi\)
\(720\) 3.41757 0.127365
\(721\) 0 0
\(722\) −3.42923 −0.127623
\(723\) 14.5809 0.542271
\(724\) −6.94257 −0.258019
\(725\) 15.7162 0.583684
\(726\) 1.23525 0.0458443
\(727\) −32.7039 −1.21292 −0.606461 0.795113i \(-0.707411\pi\)
−0.606461 + 0.795113i \(0.707411\pi\)
\(728\) 0 0
\(729\) 23.3578 0.865105
\(730\) −6.90616 −0.255608
\(731\) −38.1702 −1.41178
\(732\) −1.47847 −0.0546458
\(733\) −9.93531 −0.366969 −0.183484 0.983023i \(-0.558738\pi\)
−0.183484 + 0.983023i \(0.558738\pi\)
\(734\) −1.42288 −0.0525195
\(735\) 0 0
\(736\) 9.33871 0.344230
\(737\) 30.7171 1.13148
\(738\) 0.328237 0.0120826
\(739\) −10.4022 −0.382649 −0.191325 0.981527i \(-0.561278\pi\)
−0.191325 + 0.981527i \(0.561278\pi\)
\(740\) −42.0411 −1.54546
\(741\) 0 0
\(742\) 0 0
\(743\) −1.70863 −0.0626837 −0.0313419 0.999509i \(-0.509978\pi\)
−0.0313419 + 0.999509i \(0.509978\pi\)
\(744\) −4.84590 −0.177659
\(745\) 38.4686 1.40938
\(746\) −0.378216 −0.0138475
\(747\) 5.54528 0.202891
\(748\) −25.2560 −0.923451
\(749\) 0 0
\(750\) 2.46785 0.0901132
\(751\) 29.9812 1.09403 0.547015 0.837123i \(-0.315764\pi\)
0.547015 + 0.837123i \(0.315764\pi\)
\(752\) −7.03481 −0.256533
\(753\) −46.1197 −1.68069
\(754\) 0 0
\(755\) −17.3893 −0.632860
\(756\) 0 0
\(757\) 8.40458 0.305470 0.152735 0.988267i \(-0.451192\pi\)
0.152735 + 0.988267i \(0.451192\pi\)
\(758\) 2.58874 0.0940271
\(759\) −21.6433 −0.785601
\(760\) −0.363160 −0.0131732
\(761\) −51.0590 −1.85089 −0.925444 0.378885i \(-0.876308\pi\)
−0.925444 + 0.378885i \(0.876308\pi\)
\(762\) 2.63067 0.0952990
\(763\) 0 0
\(764\) 40.1337 1.45199
\(765\) 4.27989 0.154740
\(766\) 4.55447 0.164560
\(767\) 0 0
\(768\) −24.5560 −0.886088
\(769\) −0.704439 −0.0254027 −0.0127014 0.999919i \(-0.504043\pi\)
−0.0127014 + 0.999919i \(0.504043\pi\)
\(770\) 0 0
\(771\) 6.16030 0.221858
\(772\) 33.9646 1.22241
\(773\) −1.26521 −0.0455066 −0.0227533 0.999741i \(-0.507243\pi\)
−0.0227533 + 0.999741i \(0.507243\pi\)
\(774\) −0.484272 −0.0174068
\(775\) −8.20522 −0.294740
\(776\) 0.323607 0.0116168
\(777\) 0 0
\(778\) 5.08156 0.182183
\(779\) 1.02307 0.0366551
\(780\) 0 0
\(781\) −9.01193 −0.322472
\(782\) 3.79048 0.135547
\(783\) 34.4896 1.23256
\(784\) 0 0
\(785\) 42.7679 1.52645
\(786\) −3.30779 −0.117985
\(787\) −43.7969 −1.56119 −0.780595 0.625037i \(-0.785084\pi\)
−0.780595 + 0.625037i \(0.785084\pi\)
\(788\) 9.75137 0.347378
\(789\) 0.290218 0.0103320
\(790\) 4.42600 0.157470
\(791\) 0 0
\(792\) −0.646180 −0.0229610
\(793\) 0 0
\(794\) 3.93770 0.139744
\(795\) 34.6963 1.23055
\(796\) −14.1291 −0.500792
\(797\) 13.6837 0.484700 0.242350 0.970189i \(-0.422082\pi\)
0.242350 + 0.970189i \(0.422082\pi\)
\(798\) 0 0
\(799\) −8.80983 −0.311669
\(800\) 4.70842 0.166468
\(801\) −1.96971 −0.0695964
\(802\) −3.71212 −0.131080
\(803\) −38.3008 −1.35161
\(804\) 40.9561 1.44441
\(805\) 0 0
\(806\) 0 0
\(807\) 42.5947 1.49940
\(808\) 5.49285 0.193238
\(809\) 9.11375 0.320422 0.160211 0.987083i \(-0.448782\pi\)
0.160211 + 0.987083i \(0.448782\pi\)
\(810\) −4.80523 −0.168839
\(811\) −2.31899 −0.0814309 −0.0407154 0.999171i \(-0.512964\pi\)
−0.0407154 + 0.999171i \(0.512964\pi\)
\(812\) 0 0
\(813\) −21.5853 −0.757028
\(814\) 3.87512 0.135823
\(815\) 12.8465 0.449993
\(816\) −33.1057 −1.15893
\(817\) −1.50940 −0.0528073
\(818\) 1.13352 0.0396325
\(819\) 0 0
\(820\) −28.6977 −1.00217
\(821\) −10.1447 −0.354053 −0.177026 0.984206i \(-0.556648\pi\)
−0.177026 + 0.984206i \(0.556648\pi\)
\(822\) 1.67304 0.0583540
\(823\) −26.8178 −0.934811 −0.467405 0.884043i \(-0.654811\pi\)
−0.467405 + 0.884043i \(0.654811\pi\)
\(824\) −3.69983 −0.128890
\(825\) −10.9122 −0.379913
\(826\) 0 0
\(827\) −33.6015 −1.16844 −0.584219 0.811596i \(-0.698599\pi\)
−0.584219 + 0.811596i \(0.698599\pi\)
\(828\) −2.89348 −0.100555
\(829\) 11.8666 0.412142 0.206071 0.978537i \(-0.433932\pi\)
0.206071 + 0.978537i \(0.433932\pi\)
\(830\) 8.05791 0.279694
\(831\) 49.9694 1.73342
\(832\) 0 0
\(833\) 0 0
\(834\) −2.55095 −0.0883322
\(835\) 7.28193 0.252001
\(836\) −0.998723 −0.0345416
\(837\) −18.0066 −0.622399
\(838\) 6.17756 0.213400
\(839\) −34.6177 −1.19514 −0.597568 0.801818i \(-0.703866\pi\)
−0.597568 + 0.801818i \(0.703866\pi\)
\(840\) 0 0
\(841\) 21.2058 0.731234
\(842\) 2.08369 0.0718088
\(843\) −52.1112 −1.79481
\(844\) 34.6102 1.19133
\(845\) 0 0
\(846\) −0.111772 −0.00384280
\(847\) 0 0
\(848\) −26.9098 −0.924087
\(849\) −32.8173 −1.12629
\(850\) 1.91110 0.0655500
\(851\) 34.9925 1.19953
\(852\) −12.0159 −0.411658
\(853\) 29.1897 0.999436 0.499718 0.866188i \(-0.333437\pi\)
0.499718 + 0.866188i \(0.333437\pi\)
\(854\) 0 0
\(855\) 0.169244 0.00578802
\(856\) −5.76265 −0.196963
\(857\) −25.3198 −0.864909 −0.432455 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(858\) 0 0
\(859\) 26.0849 0.890004 0.445002 0.895530i \(-0.353203\pi\)
0.445002 + 0.895530i \(0.353203\pi\)
\(860\) 42.3398 1.44378
\(861\) 0 0
\(862\) 1.58583 0.0540134
\(863\) −36.0337 −1.22660 −0.613302 0.789849i \(-0.710159\pi\)
−0.613302 + 0.789849i \(0.710159\pi\)
\(864\) 10.3328 0.351527
\(865\) −2.41748 −0.0821969
\(866\) 3.99746 0.135839
\(867\) −10.4166 −0.353768
\(868\) 0 0
\(869\) 24.5461 0.832671
\(870\) −6.28559 −0.213101
\(871\) 0 0
\(872\) −0.956674 −0.0323971
\(873\) −0.150811 −0.00510418
\(874\) 0.149891 0.00507013
\(875\) 0 0
\(876\) −51.0677 −1.72542
\(877\) 9.12168 0.308017 0.154009 0.988070i \(-0.450782\pi\)
0.154009 + 0.988070i \(0.450782\pi\)
\(878\) −1.87531 −0.0632887
\(879\) −27.1876 −0.917015
\(880\) 27.5415 0.928425
\(881\) 13.0331 0.439095 0.219548 0.975602i \(-0.429542\pi\)
0.219548 + 0.975602i \(0.429542\pi\)
\(882\) 0 0
\(883\) 2.13222 0.0717548 0.0358774 0.999356i \(-0.488577\pi\)
0.0358774 + 0.999356i \(0.488577\pi\)
\(884\) 0 0
\(885\) −37.2299 −1.25147
\(886\) 6.47597 0.217564
\(887\) −47.1715 −1.58386 −0.791932 0.610610i \(-0.790924\pi\)
−0.791932 + 0.610610i \(0.790924\pi\)
\(888\) 10.4195 0.349656
\(889\) 0 0
\(890\) −2.86221 −0.0959416
\(891\) −26.6493 −0.892785
\(892\) −27.7496 −0.929126
\(893\) −0.348376 −0.0116580
\(894\) −4.72777 −0.158120
\(895\) −29.7030 −0.992863
\(896\) 0 0
\(897\) 0 0
\(898\) −4.11378 −0.137279
\(899\) −26.2118 −0.874213
\(900\) −1.45884 −0.0486280
\(901\) −33.6997 −1.12270
\(902\) 2.64520 0.0880755
\(903\) 0 0
\(904\) 14.3034 0.475725
\(905\) 9.48110 0.315162
\(906\) 2.13713 0.0710013
\(907\) 16.0431 0.532702 0.266351 0.963876i \(-0.414182\pi\)
0.266351 + 0.963876i \(0.414182\pi\)
\(908\) −5.64374 −0.187294
\(909\) −2.55984 −0.0849047
\(910\) 0 0
\(911\) −24.4319 −0.809466 −0.404733 0.914435i \(-0.632636\pi\)
−0.404733 + 0.914435i \(0.632636\pi\)
\(912\) −1.30913 −0.0433496
\(913\) 44.6883 1.47897
\(914\) 5.66289 0.187312
\(915\) 2.01907 0.0667483
\(916\) −17.2613 −0.570331
\(917\) 0 0
\(918\) 4.19395 0.138421
\(919\) 15.9160 0.525020 0.262510 0.964929i \(-0.415450\pi\)
0.262510 + 0.964929i \(0.415450\pi\)
\(920\) −8.47897 −0.279543
\(921\) 42.9407 1.41494
\(922\) 1.52028 0.0500678
\(923\) 0 0
\(924\) 0 0
\(925\) 17.6426 0.580086
\(926\) 1.81533 0.0596555
\(927\) 1.72424 0.0566314
\(928\) 15.0412 0.493751
\(929\) −47.3235 −1.55263 −0.776317 0.630342i \(-0.782914\pi\)
−0.776317 + 0.630342i \(0.782914\pi\)
\(930\) 3.28163 0.107609
\(931\) 0 0
\(932\) −10.0479 −0.329129
\(933\) −2.97805 −0.0974968
\(934\) −4.76494 −0.155914
\(935\) 34.4908 1.12797
\(936\) 0 0
\(937\) −29.7044 −0.970401 −0.485200 0.874403i \(-0.661253\pi\)
−0.485200 + 0.874403i \(0.661253\pi\)
\(938\) 0 0
\(939\) 1.27224 0.0415181
\(940\) 9.77219 0.318734
\(941\) 40.4224 1.31773 0.658866 0.752260i \(-0.271036\pi\)
0.658866 + 0.752260i \(0.271036\pi\)
\(942\) −5.25615 −0.171255
\(943\) 23.8863 0.777844
\(944\) 28.8748 0.939795
\(945\) 0 0
\(946\) −3.90265 −0.126886
\(947\) 45.9228 1.49229 0.746146 0.665783i \(-0.231902\pi\)
0.746146 + 0.665783i \(0.231902\pi\)
\(948\) 32.7282 1.06296
\(949\) 0 0
\(950\) 0.0755724 0.00245189
\(951\) 39.1292 1.26885
\(952\) 0 0
\(953\) 12.3893 0.401329 0.200664 0.979660i \(-0.435690\pi\)
0.200664 + 0.979660i \(0.435690\pi\)
\(954\) −0.427554 −0.0138426
\(955\) −54.8085 −1.77356
\(956\) −4.91426 −0.158939
\(957\) −34.8592 −1.12684
\(958\) −1.55222 −0.0501499
\(959\) 0 0
\(960\) 35.4492 1.14412
\(961\) −17.3151 −0.558553
\(962\) 0 0
\(963\) 2.68558 0.0865416
\(964\) 15.7091 0.505958
\(965\) −46.3837 −1.49314
\(966\) 0 0
\(967\) 9.44932 0.303870 0.151935 0.988391i \(-0.451450\pi\)
0.151935 + 0.988391i \(0.451450\pi\)
\(968\) 2.68377 0.0862596
\(969\) −1.63945 −0.0526667
\(970\) −0.219145 −0.00703633
\(971\) −20.9763 −0.673163 −0.336581 0.941654i \(-0.609271\pi\)
−0.336581 + 0.941654i \(0.609271\pi\)
\(972\) −6.80445 −0.218253
\(973\) 0 0
\(974\) 3.84363 0.123158
\(975\) 0 0
\(976\) −1.56595 −0.0501249
\(977\) 13.1843 0.421802 0.210901 0.977507i \(-0.432360\pi\)
0.210901 + 0.977507i \(0.432360\pi\)
\(978\) −1.57883 −0.0504853
\(979\) −15.8735 −0.507320
\(980\) 0 0
\(981\) 0.445840 0.0142346
\(982\) −4.06013 −0.129564
\(983\) 23.3588 0.745032 0.372516 0.928026i \(-0.378495\pi\)
0.372516 + 0.928026i \(0.378495\pi\)
\(984\) 7.11248 0.226738
\(985\) −13.3169 −0.424313
\(986\) 6.10505 0.194424
\(987\) 0 0
\(988\) 0 0
\(989\) −35.2412 −1.12060
\(990\) 0.437591 0.0139076
\(991\) 39.4929 1.25453 0.627267 0.778804i \(-0.284173\pi\)
0.627267 + 0.778804i \(0.284173\pi\)
\(992\) −7.85281 −0.249327
\(993\) 2.77639 0.0881060
\(994\) 0 0
\(995\) 19.2953 0.611703
\(996\) 59.5843 1.88800
\(997\) −51.4791 −1.63036 −0.815180 0.579208i \(-0.803362\pi\)
−0.815180 + 0.579208i \(0.803362\pi\)
\(998\) 7.03007 0.222533
\(999\) 38.7172 1.22496
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.1.7 12
7.2 even 3 1183.2.e.j.508.6 24
7.4 even 3 1183.2.e.j.170.6 24
7.6 odd 2 8281.2.a.co.1.7 12
13.2 odd 12 637.2.q.g.589.4 12
13.7 odd 12 637.2.q.g.491.4 12
13.12 even 2 inner 8281.2.a.cp.1.6 12
91.2 odd 12 91.2.k.b.4.3 12
91.20 even 12 637.2.q.i.491.4 12
91.25 even 6 1183.2.e.j.170.7 24
91.33 even 12 637.2.u.g.361.3 12
91.41 even 12 637.2.q.i.589.4 12
91.46 odd 12 91.2.k.b.23.4 yes 12
91.51 even 6 1183.2.e.j.508.7 24
91.54 even 12 637.2.k.i.459.3 12
91.59 even 12 637.2.k.i.569.4 12
91.67 odd 12 91.2.u.b.30.3 yes 12
91.72 odd 12 91.2.u.b.88.3 yes 12
91.80 even 12 637.2.u.g.30.3 12
91.90 odd 2 8281.2.a.co.1.6 12
273.2 even 12 819.2.bm.f.550.4 12
273.137 even 12 819.2.bm.f.478.3 12
273.158 even 12 819.2.do.e.667.4 12
273.254 even 12 819.2.do.e.361.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.3 12 91.2 odd 12
91.2.k.b.23.4 yes 12 91.46 odd 12
91.2.u.b.30.3 yes 12 91.67 odd 12
91.2.u.b.88.3 yes 12 91.72 odd 12
637.2.k.i.459.3 12 91.54 even 12
637.2.k.i.569.4 12 91.59 even 12
637.2.q.g.491.4 12 13.7 odd 12
637.2.q.g.589.4 12 13.2 odd 12
637.2.q.i.491.4 12 91.20 even 12
637.2.q.i.589.4 12 91.41 even 12
637.2.u.g.30.3 12 91.80 even 12
637.2.u.g.361.3 12 91.33 even 12
819.2.bm.f.478.3 12 273.137 even 12
819.2.bm.f.550.4 12 273.2 even 12
819.2.do.e.361.4 12 273.254 even 12
819.2.do.e.667.4 12 273.158 even 12
1183.2.e.j.170.6 24 7.4 even 3
1183.2.e.j.170.7 24 91.25 even 6
1183.2.e.j.508.6 24 7.2 even 3
1183.2.e.j.508.7 24 91.51 even 6
8281.2.a.co.1.6 12 91.90 odd 2
8281.2.a.co.1.7 12 7.6 odd 2
8281.2.a.cp.1.6 12 13.12 even 2 inner
8281.2.a.cp.1.7 12 1.1 even 1 trivial