Properties

Label 8112.2.a.cf.1.3
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.80194 q^{5} -4.80194 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.80194 q^{5} -4.80194 q^{7} +1.00000 q^{9} +1.46681 q^{11} -2.80194 q^{15} -2.44504 q^{17} +2.54288 q^{19} +4.80194 q^{21} +3.51573 q^{23} +2.85086 q^{25} -1.00000 q^{27} +1.85086 q^{29} -7.63102 q^{31} -1.46681 q^{33} -13.4547 q^{35} +4.55496 q^{37} -1.24698 q^{41} -2.38404 q^{43} +2.80194 q^{45} +12.8170 q^{47} +16.0586 q^{49} +2.44504 q^{51} -8.85086 q^{53} +4.10992 q^{55} -2.54288 q^{57} +2.17629 q^{59} -7.82908 q^{61} -4.80194 q^{63} -3.58211 q^{67} -3.51573 q^{69} -8.83877 q^{71} +7.69202 q^{73} -2.85086 q^{75} -7.04354 q^{77} +4.02177 q^{79} +1.00000 q^{81} -0.652793 q^{83} -6.85086 q^{85} -1.85086 q^{87} -6.29590 q^{89} +7.63102 q^{93} +7.12498 q^{95} +10.0315 q^{97} +1.46681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 4q^{5} - 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 4q^{5} - 10q^{7} + 3q^{9} + q^{11} - 4q^{15} - 7q^{17} - 11q^{19} + 10q^{21} - 2q^{23} - 5q^{25} - 3q^{27} - 8q^{29} - 8q^{31} - q^{33} - 18q^{35} + 14q^{37} + q^{41} + 3q^{43} + 4q^{45} + 9q^{47} + 17q^{49} + 7q^{51} - 13q^{53} + 13q^{55} + 11q^{57} + 14q^{59} - 13q^{61} - 10q^{63} - 5q^{67} + 2q^{69} + 6q^{71} + 18q^{73} + 5q^{75} - 15q^{77} + 9q^{79} + 3q^{81} + 16q^{83} - 7q^{85} + 8q^{87} - 5q^{89} + 8q^{93} - 3q^{95} + 5q^{97} + q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.80194 1.25306 0.626532 0.779395i \(-0.284474\pi\)
0.626532 + 0.779395i \(0.284474\pi\)
\(6\) 0 0
\(7\) −4.80194 −1.81496 −0.907481 0.420093i \(-0.861997\pi\)
−0.907481 + 0.420093i \(0.861997\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.46681 0.442260 0.221130 0.975244i \(-0.429025\pi\)
0.221130 + 0.975244i \(0.429025\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.80194 −0.723457
\(16\) 0 0
\(17\) −2.44504 −0.593010 −0.296505 0.955031i \(-0.595821\pi\)
−0.296505 + 0.955031i \(0.595821\pi\)
\(18\) 0 0
\(19\) 2.54288 0.583376 0.291688 0.956514i \(-0.405783\pi\)
0.291688 + 0.956514i \(0.405783\pi\)
\(20\) 0 0
\(21\) 4.80194 1.04787
\(22\) 0 0
\(23\) 3.51573 0.733080 0.366540 0.930402i \(-0.380542\pi\)
0.366540 + 0.930402i \(0.380542\pi\)
\(24\) 0 0
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.85086 0.343695 0.171848 0.985124i \(-0.445026\pi\)
0.171848 + 0.985124i \(0.445026\pi\)
\(30\) 0 0
\(31\) −7.63102 −1.37057 −0.685286 0.728274i \(-0.740323\pi\)
−0.685286 + 0.728274i \(0.740323\pi\)
\(32\) 0 0
\(33\) −1.46681 −0.255339
\(34\) 0 0
\(35\) −13.4547 −2.27426
\(36\) 0 0
\(37\) 4.55496 0.748831 0.374415 0.927261i \(-0.377843\pi\)
0.374415 + 0.927261i \(0.377843\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.24698 −0.194745 −0.0973727 0.995248i \(-0.531044\pi\)
−0.0973727 + 0.995248i \(0.531044\pi\)
\(42\) 0 0
\(43\) −2.38404 −0.363563 −0.181782 0.983339i \(-0.558186\pi\)
−0.181782 + 0.983339i \(0.558186\pi\)
\(44\) 0 0
\(45\) 2.80194 0.417688
\(46\) 0 0
\(47\) 12.8170 1.86955 0.934776 0.355238i \(-0.115600\pi\)
0.934776 + 0.355238i \(0.115600\pi\)
\(48\) 0 0
\(49\) 16.0586 2.29409
\(50\) 0 0
\(51\) 2.44504 0.342374
\(52\) 0 0
\(53\) −8.85086 −1.21576 −0.607879 0.794030i \(-0.707980\pi\)
−0.607879 + 0.794030i \(0.707980\pi\)
\(54\) 0 0
\(55\) 4.10992 0.554181
\(56\) 0 0
\(57\) −2.54288 −0.336812
\(58\) 0 0
\(59\) 2.17629 0.283329 0.141665 0.989915i \(-0.454755\pi\)
0.141665 + 0.989915i \(0.454755\pi\)
\(60\) 0 0
\(61\) −7.82908 −1.00241 −0.501206 0.865328i \(-0.667110\pi\)
−0.501206 + 0.865328i \(0.667110\pi\)
\(62\) 0 0
\(63\) −4.80194 −0.604987
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.58211 −0.437624 −0.218812 0.975767i \(-0.570218\pi\)
−0.218812 + 0.975767i \(0.570218\pi\)
\(68\) 0 0
\(69\) −3.51573 −0.423244
\(70\) 0 0
\(71\) −8.83877 −1.04897 −0.524485 0.851420i \(-0.675742\pi\)
−0.524485 + 0.851420i \(0.675742\pi\)
\(72\) 0 0
\(73\) 7.69202 0.900283 0.450142 0.892957i \(-0.351374\pi\)
0.450142 + 0.892957i \(0.351374\pi\)
\(74\) 0 0
\(75\) −2.85086 −0.329188
\(76\) 0 0
\(77\) −7.04354 −0.802686
\(78\) 0 0
\(79\) 4.02177 0.452485 0.226242 0.974071i \(-0.427356\pi\)
0.226242 + 0.974071i \(0.427356\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.652793 −0.0716533 −0.0358267 0.999358i \(-0.511406\pi\)
−0.0358267 + 0.999358i \(0.511406\pi\)
\(84\) 0 0
\(85\) −6.85086 −0.743080
\(86\) 0 0
\(87\) −1.85086 −0.198432
\(88\) 0 0
\(89\) −6.29590 −0.667364 −0.333682 0.942686i \(-0.608291\pi\)
−0.333682 + 0.942686i \(0.608291\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.63102 0.791300
\(94\) 0 0
\(95\) 7.12498 0.731008
\(96\) 0 0
\(97\) 10.0315 1.01854 0.509270 0.860607i \(-0.329915\pi\)
0.509270 + 0.860607i \(0.329915\pi\)
\(98\) 0 0
\(99\) 1.46681 0.147420
\(100\) 0 0
\(101\) 13.8877 1.38188 0.690938 0.722914i \(-0.257198\pi\)
0.690938 + 0.722914i \(0.257198\pi\)
\(102\) 0 0
\(103\) 17.4034 1.71481 0.857405 0.514642i \(-0.172075\pi\)
0.857405 + 0.514642i \(0.172075\pi\)
\(104\) 0 0
\(105\) 13.4547 1.31305
\(106\) 0 0
\(107\) −10.5526 −1.02015 −0.510077 0.860128i \(-0.670383\pi\)
−0.510077 + 0.860128i \(0.670383\pi\)
\(108\) 0 0
\(109\) −1.07069 −0.102553 −0.0512766 0.998684i \(-0.516329\pi\)
−0.0512766 + 0.998684i \(0.516329\pi\)
\(110\) 0 0
\(111\) −4.55496 −0.432337
\(112\) 0 0
\(113\) −16.5308 −1.55509 −0.777543 0.628830i \(-0.783534\pi\)
−0.777543 + 0.628830i \(0.783534\pi\)
\(114\) 0 0
\(115\) 9.85086 0.918597
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.7409 1.07629
\(120\) 0 0
\(121\) −8.84846 −0.804406
\(122\) 0 0
\(123\) 1.24698 0.112436
\(124\) 0 0
\(125\) −6.02177 −0.538604
\(126\) 0 0
\(127\) 9.53750 0.846316 0.423158 0.906056i \(-0.360921\pi\)
0.423158 + 0.906056i \(0.360921\pi\)
\(128\) 0 0
\(129\) 2.38404 0.209903
\(130\) 0 0
\(131\) 5.50902 0.481326 0.240663 0.970609i \(-0.422635\pi\)
0.240663 + 0.970609i \(0.422635\pi\)
\(132\) 0 0
\(133\) −12.2107 −1.05880
\(134\) 0 0
\(135\) −2.80194 −0.241152
\(136\) 0 0
\(137\) 16.1836 1.38266 0.691329 0.722540i \(-0.257026\pi\)
0.691329 + 0.722540i \(0.257026\pi\)
\(138\) 0 0
\(139\) 10.5090 0.891364 0.445682 0.895191i \(-0.352961\pi\)
0.445682 + 0.895191i \(0.352961\pi\)
\(140\) 0 0
\(141\) −12.8170 −1.07939
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.18598 0.430672
\(146\) 0 0
\(147\) −16.0586 −1.32449
\(148\) 0 0
\(149\) −14.3502 −1.17561 −0.587807 0.809001i \(-0.700008\pi\)
−0.587807 + 0.809001i \(0.700008\pi\)
\(150\) 0 0
\(151\) 1.96615 0.160003 0.0800014 0.996795i \(-0.474508\pi\)
0.0800014 + 0.996795i \(0.474508\pi\)
\(152\) 0 0
\(153\) −2.44504 −0.197670
\(154\) 0 0
\(155\) −21.3817 −1.71742
\(156\) 0 0
\(157\) 10.7017 0.854089 0.427045 0.904231i \(-0.359555\pi\)
0.427045 + 0.904231i \(0.359555\pi\)
\(158\) 0 0
\(159\) 8.85086 0.701918
\(160\) 0 0
\(161\) −16.8823 −1.33051
\(162\) 0 0
\(163\) 3.89977 0.305454 0.152727 0.988268i \(-0.451195\pi\)
0.152727 + 0.988268i \(0.451195\pi\)
\(164\) 0 0
\(165\) −4.10992 −0.319957
\(166\) 0 0
\(167\) 21.0194 1.62653 0.813264 0.581895i \(-0.197688\pi\)
0.813264 + 0.581895i \(0.197688\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.54288 0.194459
\(172\) 0 0
\(173\) 13.2349 1.00623 0.503115 0.864219i \(-0.332187\pi\)
0.503115 + 0.864219i \(0.332187\pi\)
\(174\) 0 0
\(175\) −13.6896 −1.03484
\(176\) 0 0
\(177\) −2.17629 −0.163580
\(178\) 0 0
\(179\) −8.52781 −0.637399 −0.318699 0.947856i \(-0.603246\pi\)
−0.318699 + 0.947856i \(0.603246\pi\)
\(180\) 0 0
\(181\) −3.63640 −0.270291 −0.135146 0.990826i \(-0.543150\pi\)
−0.135146 + 0.990826i \(0.543150\pi\)
\(182\) 0 0
\(183\) 7.82908 0.578743
\(184\) 0 0
\(185\) 12.7627 0.938333
\(186\) 0 0
\(187\) −3.58642 −0.262265
\(188\) 0 0
\(189\) 4.80194 0.349290
\(190\) 0 0
\(191\) −21.3817 −1.54712 −0.773561 0.633722i \(-0.781526\pi\)
−0.773561 + 0.633722i \(0.781526\pi\)
\(192\) 0 0
\(193\) 8.42758 0.606631 0.303315 0.952890i \(-0.401906\pi\)
0.303315 + 0.952890i \(0.401906\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.4765 1.88637 0.943186 0.332264i \(-0.107813\pi\)
0.943186 + 0.332264i \(0.107813\pi\)
\(198\) 0 0
\(199\) 14.2524 1.01032 0.505161 0.863025i \(-0.331433\pi\)
0.505161 + 0.863025i \(0.331433\pi\)
\(200\) 0 0
\(201\) 3.58211 0.252662
\(202\) 0 0
\(203\) −8.88769 −0.623794
\(204\) 0 0
\(205\) −3.49396 −0.244029
\(206\) 0 0
\(207\) 3.51573 0.244360
\(208\) 0 0
\(209\) 3.72992 0.258004
\(210\) 0 0
\(211\) −1.85086 −0.127418 −0.0637091 0.997969i \(-0.520293\pi\)
−0.0637091 + 0.997969i \(0.520293\pi\)
\(212\) 0 0
\(213\) 8.83877 0.605623
\(214\) 0 0
\(215\) −6.67994 −0.455568
\(216\) 0 0
\(217\) 36.6437 2.48754
\(218\) 0 0
\(219\) −7.69202 −0.519779
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.6504 1.24892 0.624462 0.781056i \(-0.285318\pi\)
0.624462 + 0.781056i \(0.285318\pi\)
\(224\) 0 0
\(225\) 2.85086 0.190057
\(226\) 0 0
\(227\) 9.75733 0.647617 0.323808 0.946123i \(-0.395037\pi\)
0.323808 + 0.946123i \(0.395037\pi\)
\(228\) 0 0
\(229\) 2.86294 0.189188 0.0945941 0.995516i \(-0.469845\pi\)
0.0945941 + 0.995516i \(0.469845\pi\)
\(230\) 0 0
\(231\) 7.04354 0.463431
\(232\) 0 0
\(233\) −5.78554 −0.379024 −0.189512 0.981878i \(-0.560691\pi\)
−0.189512 + 0.981878i \(0.560691\pi\)
\(234\) 0 0
\(235\) 35.9124 2.34267
\(236\) 0 0
\(237\) −4.02177 −0.261242
\(238\) 0 0
\(239\) 7.09246 0.458773 0.229386 0.973335i \(-0.426328\pi\)
0.229386 + 0.973335i \(0.426328\pi\)
\(240\) 0 0
\(241\) −3.89977 −0.251206 −0.125603 0.992081i \(-0.540087\pi\)
−0.125603 + 0.992081i \(0.540087\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 44.9952 2.87464
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.652793 0.0413691
\(250\) 0 0
\(251\) 2.44504 0.154330 0.0771648 0.997018i \(-0.475413\pi\)
0.0771648 + 0.997018i \(0.475413\pi\)
\(252\) 0 0
\(253\) 5.15691 0.324212
\(254\) 0 0
\(255\) 6.85086 0.429017
\(256\) 0 0
\(257\) −14.1304 −0.881428 −0.440714 0.897648i \(-0.645275\pi\)
−0.440714 + 0.897648i \(0.645275\pi\)
\(258\) 0 0
\(259\) −21.8726 −1.35910
\(260\) 0 0
\(261\) 1.85086 0.114565
\(262\) 0 0
\(263\) 23.7235 1.46285 0.731426 0.681921i \(-0.238855\pi\)
0.731426 + 0.681921i \(0.238855\pi\)
\(264\) 0 0
\(265\) −24.7995 −1.52342
\(266\) 0 0
\(267\) 6.29590 0.385303
\(268\) 0 0
\(269\) −5.91617 −0.360715 −0.180357 0.983601i \(-0.557725\pi\)
−0.180357 + 0.983601i \(0.557725\pi\)
\(270\) 0 0
\(271\) 3.19029 0.193796 0.0968982 0.995294i \(-0.469108\pi\)
0.0968982 + 0.995294i \(0.469108\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.18167 0.252164
\(276\) 0 0
\(277\) 21.9366 1.31804 0.659022 0.752124i \(-0.270971\pi\)
0.659022 + 0.752124i \(0.270971\pi\)
\(278\) 0 0
\(279\) −7.63102 −0.456857
\(280\) 0 0
\(281\) 11.9903 0.715282 0.357641 0.933859i \(-0.383581\pi\)
0.357641 + 0.933859i \(0.383581\pi\)
\(282\) 0 0
\(283\) 14.3666 0.854005 0.427002 0.904250i \(-0.359570\pi\)
0.427002 + 0.904250i \(0.359570\pi\)
\(284\) 0 0
\(285\) −7.12498 −0.422047
\(286\) 0 0
\(287\) 5.98792 0.353456
\(288\) 0 0
\(289\) −11.0218 −0.648339
\(290\) 0 0
\(291\) −10.0315 −0.588055
\(292\) 0 0
\(293\) 18.7584 1.09588 0.547939 0.836519i \(-0.315413\pi\)
0.547939 + 0.836519i \(0.315413\pi\)
\(294\) 0 0
\(295\) 6.09783 0.355030
\(296\) 0 0
\(297\) −1.46681 −0.0851131
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 11.4480 0.659853
\(302\) 0 0
\(303\) −13.8877 −0.797827
\(304\) 0 0
\(305\) −21.9366 −1.25609
\(306\) 0 0
\(307\) −25.6262 −1.46257 −0.731283 0.682074i \(-0.761078\pi\)
−0.731283 + 0.682074i \(0.761078\pi\)
\(308\) 0 0
\(309\) −17.4034 −0.990046
\(310\) 0 0
\(311\) −11.3817 −0.645394 −0.322697 0.946502i \(-0.604590\pi\)
−0.322697 + 0.946502i \(0.604590\pi\)
\(312\) 0 0
\(313\) 27.5743 1.55859 0.779297 0.626655i \(-0.215576\pi\)
0.779297 + 0.626655i \(0.215576\pi\)
\(314\) 0 0
\(315\) −13.4547 −0.758088
\(316\) 0 0
\(317\) −11.2597 −0.632405 −0.316203 0.948692i \(-0.602408\pi\)
−0.316203 + 0.948692i \(0.602408\pi\)
\(318\) 0 0
\(319\) 2.71486 0.152003
\(320\) 0 0
\(321\) 10.5526 0.588987
\(322\) 0 0
\(323\) −6.21744 −0.345948
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.07069 0.0592092
\(328\) 0 0
\(329\) −61.5465 −3.39317
\(330\) 0 0
\(331\) 11.9065 0.654439 0.327220 0.944948i \(-0.393888\pi\)
0.327220 + 0.944948i \(0.393888\pi\)
\(332\) 0 0
\(333\) 4.55496 0.249610
\(334\) 0 0
\(335\) −10.0368 −0.548371
\(336\) 0 0
\(337\) 17.1672 0.935157 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(338\) 0 0
\(339\) 16.5308 0.897830
\(340\) 0 0
\(341\) −11.1933 −0.606150
\(342\) 0 0
\(343\) −43.4989 −2.34872
\(344\) 0 0
\(345\) −9.85086 −0.530352
\(346\) 0 0
\(347\) 24.2760 1.30321 0.651603 0.758560i \(-0.274097\pi\)
0.651603 + 0.758560i \(0.274097\pi\)
\(348\) 0 0
\(349\) −4.57242 −0.244756 −0.122378 0.992484i \(-0.539052\pi\)
−0.122378 + 0.992484i \(0.539052\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.07606 −0.323396 −0.161698 0.986840i \(-0.551697\pi\)
−0.161698 + 0.986840i \(0.551697\pi\)
\(354\) 0 0
\(355\) −24.7657 −1.31443
\(356\) 0 0
\(357\) −11.7409 −0.621396
\(358\) 0 0
\(359\) 14.9661 0.789883 0.394942 0.918706i \(-0.370765\pi\)
0.394942 + 0.918706i \(0.370765\pi\)
\(360\) 0 0
\(361\) −12.5338 −0.659673
\(362\) 0 0
\(363\) 8.84846 0.464424
\(364\) 0 0
\(365\) 21.5526 1.12811
\(366\) 0 0
\(367\) −37.0834 −1.93574 −0.967868 0.251459i \(-0.919089\pi\)
−0.967868 + 0.251459i \(0.919089\pi\)
\(368\) 0 0
\(369\) −1.24698 −0.0649152
\(370\) 0 0
\(371\) 42.5013 2.20656
\(372\) 0 0
\(373\) −36.5090 −1.89037 −0.945183 0.326542i \(-0.894117\pi\)
−0.945183 + 0.326542i \(0.894117\pi\)
\(374\) 0 0
\(375\) 6.02177 0.310963
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 26.5851 1.36558 0.682792 0.730613i \(-0.260765\pi\)
0.682792 + 0.730613i \(0.260765\pi\)
\(380\) 0 0
\(381\) −9.53750 −0.488621
\(382\) 0 0
\(383\) 14.3502 0.733261 0.366630 0.930367i \(-0.380511\pi\)
0.366630 + 0.930367i \(0.380511\pi\)
\(384\) 0 0
\(385\) −19.7356 −1.00582
\(386\) 0 0
\(387\) −2.38404 −0.121188
\(388\) 0 0
\(389\) −22.6582 −1.14881 −0.574407 0.818570i \(-0.694767\pi\)
−0.574407 + 0.818570i \(0.694767\pi\)
\(390\) 0 0
\(391\) −8.59611 −0.434724
\(392\) 0 0
\(393\) −5.50902 −0.277894
\(394\) 0 0
\(395\) 11.2687 0.566992
\(396\) 0 0
\(397\) 7.90754 0.396868 0.198434 0.980114i \(-0.436414\pi\)
0.198434 + 0.980114i \(0.436414\pi\)
\(398\) 0 0
\(399\) 12.2107 0.611301
\(400\) 0 0
\(401\) 2.93661 0.146647 0.0733236 0.997308i \(-0.476639\pi\)
0.0733236 + 0.997308i \(0.476639\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.80194 0.139229
\(406\) 0 0
\(407\) 6.68127 0.331178
\(408\) 0 0
\(409\) 11.7549 0.581244 0.290622 0.956838i \(-0.406138\pi\)
0.290622 + 0.956838i \(0.406138\pi\)
\(410\) 0 0
\(411\) −16.1836 −0.798278
\(412\) 0 0
\(413\) −10.4504 −0.514231
\(414\) 0 0
\(415\) −1.82908 −0.0897862
\(416\) 0 0
\(417\) −10.5090 −0.514629
\(418\) 0 0
\(419\) −7.34183 −0.358672 −0.179336 0.983788i \(-0.557395\pi\)
−0.179336 + 0.983788i \(0.557395\pi\)
\(420\) 0 0
\(421\) 25.6963 1.25236 0.626181 0.779677i \(-0.284617\pi\)
0.626181 + 0.779677i \(0.284617\pi\)
\(422\) 0 0
\(423\) 12.8170 0.623184
\(424\) 0 0
\(425\) −6.97046 −0.338117
\(426\) 0 0
\(427\) 37.5948 1.81934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.94198 −0.430720 −0.215360 0.976535i \(-0.569093\pi\)
−0.215360 + 0.976535i \(0.569093\pi\)
\(432\) 0 0
\(433\) 2.91484 0.140078 0.0700391 0.997544i \(-0.477688\pi\)
0.0700391 + 0.997544i \(0.477688\pi\)
\(434\) 0 0
\(435\) −5.18598 −0.248649
\(436\) 0 0
\(437\) 8.94007 0.427661
\(438\) 0 0
\(439\) −9.05861 −0.432344 −0.216172 0.976355i \(-0.569357\pi\)
−0.216172 + 0.976355i \(0.569357\pi\)
\(440\) 0 0
\(441\) 16.0586 0.764696
\(442\) 0 0
\(443\) −11.2325 −0.533672 −0.266836 0.963742i \(-0.585978\pi\)
−0.266836 + 0.963742i \(0.585978\pi\)
\(444\) 0 0
\(445\) −17.6407 −0.836250
\(446\) 0 0
\(447\) 14.3502 0.678741
\(448\) 0 0
\(449\) 28.7579 1.35717 0.678585 0.734522i \(-0.262593\pi\)
0.678585 + 0.734522i \(0.262593\pi\)
\(450\) 0 0
\(451\) −1.82908 −0.0861282
\(452\) 0 0
\(453\) −1.96615 −0.0923777
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0761 −0.892341 −0.446170 0.894948i \(-0.647212\pi\)
−0.446170 + 0.894948i \(0.647212\pi\)
\(458\) 0 0
\(459\) 2.44504 0.114125
\(460\) 0 0
\(461\) 31.7332 1.47796 0.738981 0.673727i \(-0.235308\pi\)
0.738981 + 0.673727i \(0.235308\pi\)
\(462\) 0 0
\(463\) 36.4784 1.69530 0.847648 0.530559i \(-0.178018\pi\)
0.847648 + 0.530559i \(0.178018\pi\)
\(464\) 0 0
\(465\) 21.3817 0.991550
\(466\) 0 0
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 17.2010 0.794271
\(470\) 0 0
\(471\) −10.7017 −0.493109
\(472\) 0 0
\(473\) −3.49694 −0.160790
\(474\) 0 0
\(475\) 7.24937 0.332624
\(476\) 0 0
\(477\) −8.85086 −0.405253
\(478\) 0 0
\(479\) −5.61655 −0.256627 −0.128313 0.991734i \(-0.540956\pi\)
−0.128313 + 0.991734i \(0.540956\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 16.8823 0.768172
\(484\) 0 0
\(485\) 28.1075 1.27630
\(486\) 0 0
\(487\) 9.75733 0.442147 0.221073 0.975257i \(-0.429044\pi\)
0.221073 + 0.975257i \(0.429044\pi\)
\(488\) 0 0
\(489\) −3.89977 −0.176354
\(490\) 0 0
\(491\) 7.38835 0.333432 0.166716 0.986005i \(-0.446684\pi\)
0.166716 + 0.986005i \(0.446684\pi\)
\(492\) 0 0
\(493\) −4.52542 −0.203815
\(494\) 0 0
\(495\) 4.10992 0.184727
\(496\) 0 0
\(497\) 42.4432 1.90384
\(498\) 0 0
\(499\) 43.2814 1.93754 0.968771 0.247956i \(-0.0797589\pi\)
0.968771 + 0.247956i \(0.0797589\pi\)
\(500\) 0 0
\(501\) −21.0194 −0.939077
\(502\) 0 0
\(503\) −10.7670 −0.480078 −0.240039 0.970763i \(-0.577160\pi\)
−0.240039 + 0.970763i \(0.577160\pi\)
\(504\) 0 0
\(505\) 38.9124 1.73158
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −41.5448 −1.84144 −0.920720 0.390223i \(-0.872398\pi\)
−0.920720 + 0.390223i \(0.872398\pi\)
\(510\) 0 0
\(511\) −36.9366 −1.63398
\(512\) 0 0
\(513\) −2.54288 −0.112271
\(514\) 0 0
\(515\) 48.7633 2.14877
\(516\) 0 0
\(517\) 18.8001 0.826829
\(518\) 0 0
\(519\) −13.2349 −0.580948
\(520\) 0 0
\(521\) 25.7198 1.12680 0.563402 0.826183i \(-0.309492\pi\)
0.563402 + 0.826183i \(0.309492\pi\)
\(522\) 0 0
\(523\) −8.59286 −0.375739 −0.187870 0.982194i \(-0.560158\pi\)
−0.187870 + 0.982194i \(0.560158\pi\)
\(524\) 0 0
\(525\) 13.6896 0.597464
\(526\) 0 0
\(527\) 18.6582 0.812763
\(528\) 0 0
\(529\) −10.6396 −0.462593
\(530\) 0 0
\(531\) 2.17629 0.0944430
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −29.5676 −1.27832
\(536\) 0 0
\(537\) 8.52781 0.368002
\(538\) 0 0
\(539\) 23.5550 1.01458
\(540\) 0 0
\(541\) 31.3534 1.34799 0.673995 0.738736i \(-0.264577\pi\)
0.673995 + 0.738736i \(0.264577\pi\)
\(542\) 0 0
\(543\) 3.63640 0.156053
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) −19.9342 −0.852325 −0.426163 0.904647i \(-0.640135\pi\)
−0.426163 + 0.904647i \(0.640135\pi\)
\(548\) 0 0
\(549\) −7.82908 −0.334137
\(550\) 0 0
\(551\) 4.70650 0.200503
\(552\) 0 0
\(553\) −19.3123 −0.821242
\(554\) 0 0
\(555\) −12.7627 −0.541747
\(556\) 0 0
\(557\) 33.2349 1.40821 0.704104 0.710097i \(-0.251349\pi\)
0.704104 + 0.710097i \(0.251349\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 3.58642 0.151419
\(562\) 0 0
\(563\) −3.87130 −0.163156 −0.0815779 0.996667i \(-0.525996\pi\)
−0.0815779 + 0.996667i \(0.525996\pi\)
\(564\) 0 0
\(565\) −46.3183 −1.94862
\(566\) 0 0
\(567\) −4.80194 −0.201662
\(568\) 0 0
\(569\) 20.1457 0.844551 0.422276 0.906468i \(-0.361231\pi\)
0.422276 + 0.906468i \(0.361231\pi\)
\(570\) 0 0
\(571\) 32.1269 1.34447 0.672234 0.740338i \(-0.265335\pi\)
0.672234 + 0.740338i \(0.265335\pi\)
\(572\) 0 0
\(573\) 21.3817 0.893231
\(574\) 0 0
\(575\) 10.0228 0.417981
\(576\) 0 0
\(577\) −16.7506 −0.697338 −0.348669 0.937246i \(-0.613366\pi\)
−0.348669 + 0.937246i \(0.613366\pi\)
\(578\) 0 0
\(579\) −8.42758 −0.350238
\(580\) 0 0
\(581\) 3.13467 0.130048
\(582\) 0 0
\(583\) −12.9825 −0.537682
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.73795 0.278105 0.139053 0.990285i \(-0.455594\pi\)
0.139053 + 0.990285i \(0.455594\pi\)
\(588\) 0 0
\(589\) −19.4047 −0.799559
\(590\) 0 0
\(591\) −26.4765 −1.08910
\(592\) 0 0
\(593\) 18.1172 0.743985 0.371992 0.928236i \(-0.378675\pi\)
0.371992 + 0.928236i \(0.378675\pi\)
\(594\) 0 0
\(595\) 32.8974 1.34866
\(596\) 0 0
\(597\) −14.2524 −0.583310
\(598\) 0 0
\(599\) 26.7851 1.09441 0.547204 0.836999i \(-0.315692\pi\)
0.547204 + 0.836999i \(0.315692\pi\)
\(600\) 0 0
\(601\) −4.70171 −0.191787 −0.0958934 0.995392i \(-0.530571\pi\)
−0.0958934 + 0.995392i \(0.530571\pi\)
\(602\) 0 0
\(603\) −3.58211 −0.145875
\(604\) 0 0
\(605\) −24.7928 −1.00797
\(606\) 0 0
\(607\) −27.6396 −1.12186 −0.560929 0.827864i \(-0.689556\pi\)
−0.560929 + 0.827864i \(0.689556\pi\)
\(608\) 0 0
\(609\) 8.88769 0.360147
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 48.1782 1.94590 0.972950 0.231017i \(-0.0742052\pi\)
0.972950 + 0.231017i \(0.0742052\pi\)
\(614\) 0 0
\(615\) 3.49396 0.140890
\(616\) 0 0
\(617\) 30.3043 1.22000 0.610002 0.792400i \(-0.291169\pi\)
0.610002 + 0.792400i \(0.291169\pi\)
\(618\) 0 0
\(619\) 10.9041 0.438272 0.219136 0.975694i \(-0.429676\pi\)
0.219136 + 0.975694i \(0.429676\pi\)
\(620\) 0 0
\(621\) −3.51573 −0.141081
\(622\) 0 0
\(623\) 30.2325 1.21124
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 0 0
\(627\) −3.72992 −0.148959
\(628\) 0 0
\(629\) −11.1371 −0.444064
\(630\) 0 0
\(631\) 10.4523 0.416101 0.208050 0.978118i \(-0.433288\pi\)
0.208050 + 0.978118i \(0.433288\pi\)
\(632\) 0 0
\(633\) 1.85086 0.0735649
\(634\) 0 0
\(635\) 26.7235 1.06049
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.83877 −0.349656
\(640\) 0 0
\(641\) −17.5942 −0.694929 −0.347464 0.937693i \(-0.612957\pi\)
−0.347464 + 0.937693i \(0.612957\pi\)
\(642\) 0 0
\(643\) −22.6058 −0.891486 −0.445743 0.895161i \(-0.647060\pi\)
−0.445743 + 0.895161i \(0.647060\pi\)
\(644\) 0 0
\(645\) 6.67994 0.263022
\(646\) 0 0
\(647\) 24.7918 0.974665 0.487333 0.873216i \(-0.337970\pi\)
0.487333 + 0.873216i \(0.337970\pi\)
\(648\) 0 0
\(649\) 3.19221 0.125305
\(650\) 0 0
\(651\) −36.6437 −1.43618
\(652\) 0 0
\(653\) −21.8106 −0.853513 −0.426757 0.904367i \(-0.640344\pi\)
−0.426757 + 0.904367i \(0.640344\pi\)
\(654\) 0 0
\(655\) 15.4359 0.603132
\(656\) 0 0
\(657\) 7.69202 0.300094
\(658\) 0 0
\(659\) −16.5526 −0.644796 −0.322398 0.946604i \(-0.604489\pi\)
−0.322398 + 0.946604i \(0.604489\pi\)
\(660\) 0 0
\(661\) −15.9541 −0.620541 −0.310271 0.950648i \(-0.600420\pi\)
−0.310271 + 0.950648i \(0.600420\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −34.2137 −1.32675
\(666\) 0 0
\(667\) 6.50711 0.251956
\(668\) 0 0
\(669\) −18.6504 −0.721066
\(670\) 0 0
\(671\) −11.4838 −0.443327
\(672\) 0 0
\(673\) −7.38835 −0.284800 −0.142400 0.989809i \(-0.545482\pi\)
−0.142400 + 0.989809i \(0.545482\pi\)
\(674\) 0 0
\(675\) −2.85086 −0.109729
\(676\) 0 0
\(677\) −22.1454 −0.851118 −0.425559 0.904931i \(-0.639922\pi\)
−0.425559 + 0.904931i \(0.639922\pi\)
\(678\) 0 0
\(679\) −48.1704 −1.84861
\(680\) 0 0
\(681\) −9.75733 −0.373902
\(682\) 0 0
\(683\) −9.10023 −0.348211 −0.174105 0.984727i \(-0.555703\pi\)
−0.174105 + 0.984727i \(0.555703\pi\)
\(684\) 0 0
\(685\) 45.3454 1.73256
\(686\) 0 0
\(687\) −2.86294 −0.109228
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −13.7711 −0.523876 −0.261938 0.965085i \(-0.584362\pi\)
−0.261938 + 0.965085i \(0.584362\pi\)
\(692\) 0 0
\(693\) −7.04354 −0.267562
\(694\) 0 0
\(695\) 29.4456 1.11694
\(696\) 0 0
\(697\) 3.04892 0.115486
\(698\) 0 0
\(699\) 5.78554 0.218829
\(700\) 0 0
\(701\) 46.5090 1.75662 0.878311 0.478090i \(-0.158671\pi\)
0.878311 + 0.478090i \(0.158671\pi\)
\(702\) 0 0
\(703\) 11.5827 0.436850
\(704\) 0 0
\(705\) −35.9124 −1.35254
\(706\) 0 0
\(707\) −66.6878 −2.50805
\(708\) 0 0
\(709\) 7.24565 0.272116 0.136058 0.990701i \(-0.456557\pi\)
0.136058 + 0.990701i \(0.456557\pi\)
\(710\) 0 0
\(711\) 4.02177 0.150828
\(712\) 0 0
\(713\) −26.8286 −1.00474
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.09246 −0.264873
\(718\) 0 0
\(719\) −25.5147 −0.951536 −0.475768 0.879571i \(-0.657830\pi\)
−0.475768 + 0.879571i \(0.657830\pi\)
\(720\) 0 0
\(721\) −83.5701 −3.11231
\(722\) 0 0
\(723\) 3.89977 0.145034
\(724\) 0 0
\(725\) 5.27652 0.195965
\(726\) 0 0
\(727\) 14.4873 0.537303 0.268651 0.963238i \(-0.413422\pi\)
0.268651 + 0.963238i \(0.413422\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.82908 0.215596
\(732\) 0 0
\(733\) 37.5036 1.38523 0.692614 0.721308i \(-0.256459\pi\)
0.692614 + 0.721308i \(0.256459\pi\)
\(734\) 0 0
\(735\) −44.9952 −1.65967
\(736\) 0 0
\(737\) −5.25428 −0.193544
\(738\) 0 0
\(739\) −43.1876 −1.58868 −0.794341 0.607472i \(-0.792184\pi\)
−0.794341 + 0.607472i \(0.792184\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.4765 −0.494405 −0.247202 0.968964i \(-0.579511\pi\)
−0.247202 + 0.968964i \(0.579511\pi\)
\(744\) 0 0
\(745\) −40.2083 −1.47312
\(746\) 0 0
\(747\) −0.652793 −0.0238844
\(748\) 0 0
\(749\) 50.6728 1.85154
\(750\) 0 0
\(751\) −35.5894 −1.29868 −0.649338 0.760500i \(-0.724954\pi\)
−0.649338 + 0.760500i \(0.724954\pi\)
\(752\) 0 0
\(753\) −2.44504 −0.0891023
\(754\) 0 0
\(755\) 5.50902 0.200494
\(756\) 0 0
\(757\) −12.2107 −0.443807 −0.221903 0.975069i \(-0.571227\pi\)
−0.221903 + 0.975069i \(0.571227\pi\)
\(758\) 0 0
\(759\) −5.15691 −0.187184
\(760\) 0 0
\(761\) 30.9071 1.12038 0.560190 0.828364i \(-0.310728\pi\)
0.560190 + 0.828364i \(0.310728\pi\)
\(762\) 0 0
\(763\) 5.14138 0.186130
\(764\) 0 0
\(765\) −6.85086 −0.247693
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 11.9892 0.432343 0.216172 0.976355i \(-0.430643\pi\)
0.216172 + 0.976355i \(0.430643\pi\)
\(770\) 0 0
\(771\) 14.1304 0.508892
\(772\) 0 0
\(773\) 43.9661 1.58135 0.790676 0.612235i \(-0.209729\pi\)
0.790676 + 0.612235i \(0.209729\pi\)
\(774\) 0 0
\(775\) −21.7549 −0.781460
\(776\) 0 0
\(777\) 21.8726 0.784676
\(778\) 0 0
\(779\) −3.17092 −0.113610
\(780\) 0 0
\(781\) −12.9648 −0.463918
\(782\) 0 0
\(783\) −1.85086 −0.0661442
\(784\) 0 0
\(785\) 29.9855 1.07023
\(786\) 0 0
\(787\) −30.3763 −1.08280 −0.541399 0.840766i \(-0.682105\pi\)
−0.541399 + 0.840766i \(0.682105\pi\)
\(788\) 0 0
\(789\) −23.7235 −0.844578
\(790\) 0 0
\(791\) 79.3798 2.82242
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 24.7995 0.879549
\(796\) 0 0
\(797\) 52.5763 1.86235 0.931173 0.364577i \(-0.118786\pi\)
0.931173 + 0.364577i \(0.118786\pi\)
\(798\) 0 0
\(799\) −31.3381 −1.10866
\(800\) 0 0
\(801\) −6.29590 −0.222455
\(802\) 0 0
\(803\) 11.2828 0.398160
\(804\) 0 0
\(805\) −47.3032 −1.66722
\(806\) 0 0
\(807\) 5.91617 0.208259
\(808\) 0 0
\(809\) 49.4215 1.73757 0.868783 0.495193i \(-0.164903\pi\)
0.868783 + 0.495193i \(0.164903\pi\)
\(810\) 0 0
\(811\) 1.36526 0.0479406 0.0239703 0.999713i \(-0.492369\pi\)
0.0239703 + 0.999713i \(0.492369\pi\)
\(812\) 0 0
\(813\) −3.19029 −0.111888
\(814\) 0 0
\(815\) 10.9269 0.382753
\(816\) 0 0
\(817\) −6.06233 −0.212094
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.665939 0.0232414 0.0116207 0.999932i \(-0.496301\pi\)
0.0116207 + 0.999932i \(0.496301\pi\)
\(822\) 0 0
\(823\) 10.0592 0.350642 0.175321 0.984511i \(-0.443904\pi\)
0.175321 + 0.984511i \(0.443904\pi\)
\(824\) 0 0
\(825\) −4.18167 −0.145587
\(826\) 0 0
\(827\) 37.3038 1.29718 0.648590 0.761138i \(-0.275359\pi\)
0.648590 + 0.761138i \(0.275359\pi\)
\(828\) 0 0
\(829\) −42.6209 −1.48028 −0.740142 0.672451i \(-0.765242\pi\)
−0.740142 + 0.672451i \(0.765242\pi\)
\(830\) 0 0
\(831\) −21.9366 −0.760973
\(832\) 0 0
\(833\) −39.2640 −1.36042
\(834\) 0 0
\(835\) 58.8950 2.03815
\(836\) 0 0
\(837\) 7.63102 0.263767
\(838\) 0 0
\(839\) −4.14005 −0.142930 −0.0714652 0.997443i \(-0.522767\pi\)
−0.0714652 + 0.997443i \(0.522767\pi\)
\(840\) 0 0
\(841\) −25.5743 −0.881874
\(842\) 0 0
\(843\) −11.9903 −0.412968
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 42.4898 1.45997
\(848\) 0 0
\(849\) −14.3666 −0.493060
\(850\) 0 0
\(851\) 16.0140 0.548953
\(852\) 0 0
\(853\) 17.3502 0.594059 0.297030 0.954868i \(-0.404004\pi\)
0.297030 + 0.954868i \(0.404004\pi\)
\(854\) 0 0
\(855\) 7.12498 0.243669
\(856\) 0 0
\(857\) 41.0180 1.40115 0.700575 0.713579i \(-0.252927\pi\)
0.700575 + 0.713579i \(0.252927\pi\)
\(858\) 0 0
\(859\) −6.59286 −0.224945 −0.112473 0.993655i \(-0.535877\pi\)
−0.112473 + 0.993655i \(0.535877\pi\)
\(860\) 0 0
\(861\) −5.98792 −0.204068
\(862\) 0 0
\(863\) 16.6455 0.566619 0.283310 0.959028i \(-0.408568\pi\)
0.283310 + 0.959028i \(0.408568\pi\)
\(864\) 0 0
\(865\) 37.0834 1.26087
\(866\) 0 0
\(867\) 11.0218 0.374319
\(868\) 0 0
\(869\) 5.89918 0.200116
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0315 0.339513
\(874\) 0 0
\(875\) 28.9162 0.977545
\(876\) 0 0
\(877\) 54.4965 1.84022 0.920108 0.391666i \(-0.128101\pi\)
0.920108 + 0.391666i \(0.128101\pi\)
\(878\) 0 0
\(879\) −18.7584 −0.632705
\(880\) 0 0
\(881\) −9.00670 −0.303444 −0.151722 0.988423i \(-0.548482\pi\)
−0.151722 + 0.988423i \(0.548482\pi\)
\(882\) 0 0
\(883\) −18.8907 −0.635722 −0.317861 0.948137i \(-0.602965\pi\)
−0.317861 + 0.948137i \(0.602965\pi\)
\(884\) 0 0
\(885\) −6.09783 −0.204976
\(886\) 0 0
\(887\) 46.9124 1.57517 0.787583 0.616209i \(-0.211332\pi\)
0.787583 + 0.616209i \(0.211332\pi\)
\(888\) 0 0
\(889\) −45.7985 −1.53603
\(890\) 0 0
\(891\) 1.46681 0.0491401
\(892\) 0 0
\(893\) 32.5921 1.09065
\(894\) 0 0
\(895\) −23.8944 −0.798702
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −14.1239 −0.471059
\(900\) 0 0
\(901\) 21.6407 0.720957
\(902\) 0 0
\(903\) −11.4480 −0.380966
\(904\) 0 0
\(905\) −10.1890 −0.338693
\(906\) 0 0
\(907\) 35.3013 1.17216 0.586080 0.810253i \(-0.300671\pi\)
0.586080 + 0.810253i \(0.300671\pi\)
\(908\) 0 0
\(909\) 13.8877 0.460626
\(910\) 0 0
\(911\) 9.80731 0.324931 0.162465 0.986714i \(-0.448055\pi\)
0.162465 + 0.986714i \(0.448055\pi\)
\(912\) 0 0
\(913\) −0.957524 −0.0316894
\(914\) 0 0
\(915\) 21.9366 0.725202
\(916\) 0 0
\(917\) −26.4540 −0.873588
\(918\) 0 0
\(919\) −18.4655 −0.609120 −0.304560 0.952493i \(-0.598509\pi\)
−0.304560 + 0.952493i \(0.598509\pi\)
\(920\) 0 0
\(921\) 25.6262 0.844413
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 12.9855 0.426961
\(926\) 0 0
\(927\) 17.4034 0.571603
\(928\) 0 0
\(929\) −25.6267 −0.840785 −0.420393 0.907342i \(-0.638108\pi\)
−0.420393 + 0.907342i \(0.638108\pi\)
\(930\) 0 0
\(931\) 40.8351 1.33831
\(932\) 0 0
\(933\) 11.3817 0.372618
\(934\) 0 0
\(935\) −10.0489 −0.328635
\(936\) 0 0
\(937\) 7.54932 0.246625 0.123313 0.992368i \(-0.460648\pi\)
0.123313 + 0.992368i \(0.460648\pi\)
\(938\) 0 0
\(939\) −27.5743 −0.899854
\(940\) 0 0
\(941\) −12.6418 −0.412110 −0.206055 0.978540i \(-0.566063\pi\)
−0.206055 + 0.978540i \(0.566063\pi\)
\(942\) 0 0
\(943\) −4.38404 −0.142764
\(944\) 0 0
\(945\) 13.4547 0.437682
\(946\) 0 0
\(947\) −27.9801 −0.909233 −0.454616 0.890687i \(-0.650224\pi\)
−0.454616 + 0.890687i \(0.650224\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 11.2597 0.365119
\(952\) 0 0
\(953\) 4.00239 0.129650 0.0648251 0.997897i \(-0.479351\pi\)
0.0648251 + 0.997897i \(0.479351\pi\)
\(954\) 0 0
\(955\) −59.9101 −1.93864
\(956\) 0 0
\(957\) −2.71486 −0.0877589
\(958\) 0 0
\(959\) −77.7126 −2.50947
\(960\) 0 0
\(961\) 27.2325 0.878468
\(962\) 0 0
\(963\) −10.5526 −0.340052
\(964\) 0 0
\(965\) 23.6136 0.760148
\(966\) 0 0
\(967\) −12.2239 −0.393094 −0.196547 0.980494i \(-0.562973\pi\)
−0.196547 + 0.980494i \(0.562973\pi\)
\(968\) 0 0
\(969\) 6.21744 0.199733
\(970\) 0 0
\(971\) 23.6401 0.758648 0.379324 0.925264i \(-0.376157\pi\)
0.379324 + 0.925264i \(0.376157\pi\)
\(972\) 0 0
\(973\) −50.4637 −1.61779
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.7313 0.599266 0.299633 0.954055i \(-0.403136\pi\)
0.299633 + 0.954055i \(0.403136\pi\)
\(978\) 0 0
\(979\) −9.23490 −0.295149
\(980\) 0 0
\(981\) −1.07069 −0.0341844
\(982\) 0 0
\(983\) −12.0954 −0.385785 −0.192892 0.981220i \(-0.561787\pi\)
−0.192892 + 0.981220i \(0.561787\pi\)
\(984\) 0 0
\(985\) 74.1855 2.36375
\(986\) 0 0
\(987\) 61.5465 1.95905
\(988\) 0 0
\(989\) −8.38165 −0.266521
\(990\) 0 0
\(991\) 28.5526 0.907002 0.453501 0.891256i \(-0.350175\pi\)
0.453501 + 0.891256i \(0.350175\pi\)
\(992\) 0 0
\(993\) −11.9065 −0.377841
\(994\) 0 0
\(995\) 39.9342 1.26600
\(996\) 0 0
\(997\) −23.9347 −0.758019 −0.379010 0.925393i \(-0.623735\pi\)
−0.379010 + 0.925393i \(0.623735\pi\)
\(998\) 0 0
\(999\) −4.55496 −0.144112
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 8112.2.a.cf.1.3 3
4.3 odd 2 507.2.a.k.1.1 yes 3
12.11 even 2 1521.2.a.p.1.3 3
13.12 even 2 8112.2.a.by.1.1 3
52.3 odd 6 507.2.e.j.22.3 6
52.7 even 12 507.2.j.h.361.2 12
52.11 even 12 507.2.j.h.316.5 12
52.15 even 12 507.2.j.h.316.2 12
52.19 even 12 507.2.j.h.361.5 12
52.23 odd 6 507.2.e.k.22.1 6
52.31 even 4 507.2.b.g.337.5 6
52.35 odd 6 507.2.e.j.484.3 6
52.43 odd 6 507.2.e.k.484.1 6
52.47 even 4 507.2.b.g.337.2 6
52.51 odd 2 507.2.a.j.1.3 3
156.47 odd 4 1521.2.b.m.1351.5 6
156.83 odd 4 1521.2.b.m.1351.2 6
156.155 even 2 1521.2.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.3 3 52.51 odd 2
507.2.a.k.1.1 yes 3 4.3 odd 2
507.2.b.g.337.2 6 52.47 even 4
507.2.b.g.337.5 6 52.31 even 4
507.2.e.j.22.3 6 52.3 odd 6
507.2.e.j.484.3 6 52.35 odd 6
507.2.e.k.22.1 6 52.23 odd 6
507.2.e.k.484.1 6 52.43 odd 6
507.2.j.h.316.2 12 52.15 even 12
507.2.j.h.316.5 12 52.11 even 12
507.2.j.h.361.2 12 52.7 even 12
507.2.j.h.361.5 12 52.19 even 12
1521.2.a.p.1.3 3 12.11 even 2
1521.2.a.q.1.1 3 156.155 even 2
1521.2.b.m.1351.2 6 156.83 odd 4
1521.2.b.m.1351.5 6 156.47 odd 4
8112.2.a.by.1.1 3 13.12 even 2
8112.2.a.cf.1.3 3 1.1 even 1 trivial