Properties

Label 8004.2.a.g.1.12
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-3.54084\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.54084 q^{5} +2.31030 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.54084 q^{5} +2.31030 q^{7} +1.00000 q^{9} -0.0711814 q^{11} -0.620759 q^{13} -3.54084 q^{15} -4.97914 q^{17} -6.83275 q^{19} -2.31030 q^{21} +1.00000 q^{23} +7.53753 q^{25} -1.00000 q^{27} -1.00000 q^{29} -2.38311 q^{31} +0.0711814 q^{33} +8.18041 q^{35} -7.78652 q^{37} +0.620759 q^{39} -6.42713 q^{41} +1.04310 q^{43} +3.54084 q^{45} +5.51860 q^{47} -1.66249 q^{49} +4.97914 q^{51} -5.37715 q^{53} -0.252042 q^{55} +6.83275 q^{57} +0.604181 q^{59} -9.02854 q^{61} +2.31030 q^{63} -2.19801 q^{65} -12.1486 q^{67} -1.00000 q^{69} -3.52530 q^{71} -4.78770 q^{73} -7.53753 q^{75} -0.164451 q^{77} +1.72122 q^{79} +1.00000 q^{81} -12.0521 q^{83} -17.6303 q^{85} +1.00000 q^{87} -8.77465 q^{89} -1.43414 q^{91} +2.38311 q^{93} -24.1937 q^{95} +5.44485 q^{97} -0.0711814 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) 3.54084 1.58351 0.791755 0.610839i \(-0.209168\pi\)
0.791755 + 0.610839i \(0.209168\pi\)
\(6\) 0 0
\(7\) 2.31030 0.873213 0.436606 0.899653i \(-0.356180\pi\)
0.436606 + 0.899653i \(0.356180\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.0711814 −0.0214620 −0.0107310 0.999942i \(-0.503416\pi\)
−0.0107310 + 0.999942i \(0.503416\pi\)
\(12\) 0 0
\(13\) −0.620759 −0.172168 −0.0860838 0.996288i \(-0.527435\pi\)
−0.0860838 + 0.996288i \(0.527435\pi\)
\(14\) 0 0
\(15\) −3.54084 −0.914240
\(16\) 0 0
\(17\) −4.97914 −1.20762 −0.603810 0.797129i \(-0.706351\pi\)
−0.603810 + 0.797129i \(0.706351\pi\)
\(18\) 0 0
\(19\) −6.83275 −1.56754 −0.783771 0.621051i \(-0.786706\pi\)
−0.783771 + 0.621051i \(0.786706\pi\)
\(20\) 0 0
\(21\) −2.31030 −0.504150
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 7.53753 1.50751
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −2.38311 −0.428019 −0.214009 0.976832i \(-0.568652\pi\)
−0.214009 + 0.976832i \(0.568652\pi\)
\(32\) 0 0
\(33\) 0.0711814 0.0123911
\(34\) 0 0
\(35\) 8.18041 1.38274
\(36\) 0 0
\(37\) −7.78652 −1.28010 −0.640048 0.768335i \(-0.721086\pi\)
−0.640048 + 0.768335i \(0.721086\pi\)
\(38\) 0 0
\(39\) 0.620759 0.0994010
\(40\) 0 0
\(41\) −6.42713 −1.00375 −0.501874 0.864941i \(-0.667356\pi\)
−0.501874 + 0.864941i \(0.667356\pi\)
\(42\) 0 0
\(43\) 1.04310 0.159071 0.0795355 0.996832i \(-0.474656\pi\)
0.0795355 + 0.996832i \(0.474656\pi\)
\(44\) 0 0
\(45\) 3.54084 0.527837
\(46\) 0 0
\(47\) 5.51860 0.804970 0.402485 0.915427i \(-0.368147\pi\)
0.402485 + 0.915427i \(0.368147\pi\)
\(48\) 0 0
\(49\) −1.66249 −0.237499
\(50\) 0 0
\(51\) 4.97914 0.697219
\(52\) 0 0
\(53\) −5.37715 −0.738608 −0.369304 0.929309i \(-0.620404\pi\)
−0.369304 + 0.929309i \(0.620404\pi\)
\(54\) 0 0
\(55\) −0.252042 −0.0339853
\(56\) 0 0
\(57\) 6.83275 0.905020
\(58\) 0 0
\(59\) 0.604181 0.0786577 0.0393289 0.999226i \(-0.487478\pi\)
0.0393289 + 0.999226i \(0.487478\pi\)
\(60\) 0 0
\(61\) −9.02854 −1.15599 −0.577993 0.816042i \(-0.696164\pi\)
−0.577993 + 0.816042i \(0.696164\pi\)
\(62\) 0 0
\(63\) 2.31030 0.291071
\(64\) 0 0
\(65\) −2.19801 −0.272629
\(66\) 0 0
\(67\) −12.1486 −1.48418 −0.742091 0.670299i \(-0.766166\pi\)
−0.742091 + 0.670299i \(0.766166\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −3.52530 −0.418377 −0.209188 0.977875i \(-0.567082\pi\)
−0.209188 + 0.977875i \(0.567082\pi\)
\(72\) 0 0
\(73\) −4.78770 −0.560358 −0.280179 0.959948i \(-0.590394\pi\)
−0.280179 + 0.959948i \(0.590394\pi\)
\(74\) 0 0
\(75\) −7.53753 −0.870358
\(76\) 0 0
\(77\) −0.164451 −0.0187409
\(78\) 0 0
\(79\) 1.72122 0.193652 0.0968262 0.995301i \(-0.469131\pi\)
0.0968262 + 0.995301i \(0.469131\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0521 −1.32289 −0.661443 0.749996i \(-0.730055\pi\)
−0.661443 + 0.749996i \(0.730055\pi\)
\(84\) 0 0
\(85\) −17.6303 −1.91228
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −8.77465 −0.930111 −0.465056 0.885281i \(-0.653966\pi\)
−0.465056 + 0.885281i \(0.653966\pi\)
\(90\) 0 0
\(91\) −1.43414 −0.150339
\(92\) 0 0
\(93\) 2.38311 0.247117
\(94\) 0 0
\(95\) −24.1937 −2.48222
\(96\) 0 0
\(97\) 5.44485 0.552841 0.276420 0.961037i \(-0.410852\pi\)
0.276420 + 0.961037i \(0.410852\pi\)
\(98\) 0 0
\(99\) −0.0711814 −0.00715400
\(100\) 0 0
\(101\) −10.2508 −1.01999 −0.509994 0.860178i \(-0.670353\pi\)
−0.509994 + 0.860178i \(0.670353\pi\)
\(102\) 0 0
\(103\) 4.52130 0.445497 0.222748 0.974876i \(-0.428497\pi\)
0.222748 + 0.974876i \(0.428497\pi\)
\(104\) 0 0
\(105\) −8.18041 −0.798326
\(106\) 0 0
\(107\) −12.7966 −1.23709 −0.618545 0.785750i \(-0.712277\pi\)
−0.618545 + 0.785750i \(0.712277\pi\)
\(108\) 0 0
\(109\) 1.35997 0.130261 0.0651305 0.997877i \(-0.479254\pi\)
0.0651305 + 0.997877i \(0.479254\pi\)
\(110\) 0 0
\(111\) 7.78652 0.739064
\(112\) 0 0
\(113\) −5.37462 −0.505601 −0.252801 0.967518i \(-0.581352\pi\)
−0.252801 + 0.967518i \(0.581352\pi\)
\(114\) 0 0
\(115\) 3.54084 0.330185
\(116\) 0 0
\(117\) −0.620759 −0.0573892
\(118\) 0 0
\(119\) −11.5033 −1.05451
\(120\) 0 0
\(121\) −10.9949 −0.999539
\(122\) 0 0
\(123\) 6.42713 0.579515
\(124\) 0 0
\(125\) 8.98496 0.803639
\(126\) 0 0
\(127\) 20.1570 1.78864 0.894320 0.447427i \(-0.147660\pi\)
0.894320 + 0.447427i \(0.147660\pi\)
\(128\) 0 0
\(129\) −1.04310 −0.0918397
\(130\) 0 0
\(131\) 14.0047 1.22360 0.611799 0.791014i \(-0.290446\pi\)
0.611799 + 0.791014i \(0.290446\pi\)
\(132\) 0 0
\(133\) −15.7857 −1.36880
\(134\) 0 0
\(135\) −3.54084 −0.304747
\(136\) 0 0
\(137\) 6.14660 0.525140 0.262570 0.964913i \(-0.415430\pi\)
0.262570 + 0.964913i \(0.415430\pi\)
\(138\) 0 0
\(139\) 14.3036 1.21321 0.606607 0.795002i \(-0.292530\pi\)
0.606607 + 0.795002i \(0.292530\pi\)
\(140\) 0 0
\(141\) −5.51860 −0.464750
\(142\) 0 0
\(143\) 0.0441865 0.00369506
\(144\) 0 0
\(145\) −3.54084 −0.294050
\(146\) 0 0
\(147\) 1.66249 0.137120
\(148\) 0 0
\(149\) 20.3437 1.66662 0.833311 0.552805i \(-0.186442\pi\)
0.833311 + 0.552805i \(0.186442\pi\)
\(150\) 0 0
\(151\) −2.73246 −0.222365 −0.111182 0.993800i \(-0.535464\pi\)
−0.111182 + 0.993800i \(0.535464\pi\)
\(152\) 0 0
\(153\) −4.97914 −0.402540
\(154\) 0 0
\(155\) −8.43819 −0.677772
\(156\) 0 0
\(157\) −21.3377 −1.70293 −0.851465 0.524411i \(-0.824286\pi\)
−0.851465 + 0.524411i \(0.824286\pi\)
\(158\) 0 0
\(159\) 5.37715 0.426435
\(160\) 0 0
\(161\) 2.31030 0.182077
\(162\) 0 0
\(163\) 11.6539 0.912804 0.456402 0.889774i \(-0.349138\pi\)
0.456402 + 0.889774i \(0.349138\pi\)
\(164\) 0 0
\(165\) 0.252042 0.0196214
\(166\) 0 0
\(167\) −1.01179 −0.0782944 −0.0391472 0.999233i \(-0.512464\pi\)
−0.0391472 + 0.999233i \(0.512464\pi\)
\(168\) 0 0
\(169\) −12.6147 −0.970358
\(170\) 0 0
\(171\) −6.83275 −0.522514
\(172\) 0 0
\(173\) −1.34864 −0.102535 −0.0512677 0.998685i \(-0.516326\pi\)
−0.0512677 + 0.998685i \(0.516326\pi\)
\(174\) 0 0
\(175\) 17.4140 1.31637
\(176\) 0 0
\(177\) −0.604181 −0.0454131
\(178\) 0 0
\(179\) −11.9175 −0.890753 −0.445376 0.895343i \(-0.646930\pi\)
−0.445376 + 0.895343i \(0.646930\pi\)
\(180\) 0 0
\(181\) 14.2013 1.05557 0.527786 0.849377i \(-0.323022\pi\)
0.527786 + 0.849377i \(0.323022\pi\)
\(182\) 0 0
\(183\) 9.02854 0.667409
\(184\) 0 0
\(185\) −27.5708 −2.02705
\(186\) 0 0
\(187\) 0.354422 0.0259179
\(188\) 0 0
\(189\) −2.31030 −0.168050
\(190\) 0 0
\(191\) 13.8182 0.999851 0.499926 0.866068i \(-0.333361\pi\)
0.499926 + 0.866068i \(0.333361\pi\)
\(192\) 0 0
\(193\) −7.49114 −0.539224 −0.269612 0.962969i \(-0.586895\pi\)
−0.269612 + 0.962969i \(0.586895\pi\)
\(194\) 0 0
\(195\) 2.19801 0.157402
\(196\) 0 0
\(197\) 12.1766 0.867550 0.433775 0.901021i \(-0.357181\pi\)
0.433775 + 0.901021i \(0.357181\pi\)
\(198\) 0 0
\(199\) 23.0031 1.63065 0.815323 0.579007i \(-0.196559\pi\)
0.815323 + 0.579007i \(0.196559\pi\)
\(200\) 0 0
\(201\) 12.1486 0.856893
\(202\) 0 0
\(203\) −2.31030 −0.162152
\(204\) 0 0
\(205\) −22.7574 −1.58945
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0.486365 0.0336426
\(210\) 0 0
\(211\) −7.53452 −0.518698 −0.259349 0.965784i \(-0.583508\pi\)
−0.259349 + 0.965784i \(0.583508\pi\)
\(212\) 0 0
\(213\) 3.52530 0.241550
\(214\) 0 0
\(215\) 3.69344 0.251891
\(216\) 0 0
\(217\) −5.50570 −0.373751
\(218\) 0 0
\(219\) 4.78770 0.323523
\(220\) 0 0
\(221\) 3.09085 0.207913
\(222\) 0 0
\(223\) −27.4689 −1.83945 −0.919726 0.392561i \(-0.871589\pi\)
−0.919726 + 0.392561i \(0.871589\pi\)
\(224\) 0 0
\(225\) 7.53753 0.502502
\(226\) 0 0
\(227\) 15.2865 1.01460 0.507299 0.861770i \(-0.330644\pi\)
0.507299 + 0.861770i \(0.330644\pi\)
\(228\) 0 0
\(229\) −10.2081 −0.674570 −0.337285 0.941403i \(-0.609509\pi\)
−0.337285 + 0.941403i \(0.609509\pi\)
\(230\) 0 0
\(231\) 0.164451 0.0108201
\(232\) 0 0
\(233\) 12.6697 0.830021 0.415010 0.909817i \(-0.363778\pi\)
0.415010 + 0.909817i \(0.363778\pi\)
\(234\) 0 0
\(235\) 19.5404 1.27468
\(236\) 0 0
\(237\) −1.72122 −0.111805
\(238\) 0 0
\(239\) 17.0568 1.10331 0.551657 0.834071i \(-0.313996\pi\)
0.551657 + 0.834071i \(0.313996\pi\)
\(240\) 0 0
\(241\) 25.6164 1.65010 0.825050 0.565060i \(-0.191147\pi\)
0.825050 + 0.565060i \(0.191147\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −5.88662 −0.376082
\(246\) 0 0
\(247\) 4.24149 0.269880
\(248\) 0 0
\(249\) 12.0521 0.763768
\(250\) 0 0
\(251\) 12.9643 0.818298 0.409149 0.912468i \(-0.365826\pi\)
0.409149 + 0.912468i \(0.365826\pi\)
\(252\) 0 0
\(253\) −0.0711814 −0.00447514
\(254\) 0 0
\(255\) 17.6303 1.10405
\(256\) 0 0
\(257\) −7.55820 −0.471467 −0.235734 0.971818i \(-0.575749\pi\)
−0.235734 + 0.971818i \(0.575749\pi\)
\(258\) 0 0
\(259\) −17.9892 −1.11780
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 4.86890 0.300229 0.150115 0.988669i \(-0.452036\pi\)
0.150115 + 0.988669i \(0.452036\pi\)
\(264\) 0 0
\(265\) −19.0396 −1.16959
\(266\) 0 0
\(267\) 8.77465 0.537000
\(268\) 0 0
\(269\) 1.51039 0.0920898 0.0460449 0.998939i \(-0.485338\pi\)
0.0460449 + 0.998939i \(0.485338\pi\)
\(270\) 0 0
\(271\) 10.4038 0.631986 0.315993 0.948762i \(-0.397662\pi\)
0.315993 + 0.948762i \(0.397662\pi\)
\(272\) 0 0
\(273\) 1.43414 0.0867982
\(274\) 0 0
\(275\) −0.536532 −0.0323541
\(276\) 0 0
\(277\) −6.22112 −0.373791 −0.186896 0.982380i \(-0.559843\pi\)
−0.186896 + 0.982380i \(0.559843\pi\)
\(278\) 0 0
\(279\) −2.38311 −0.142673
\(280\) 0 0
\(281\) 14.1239 0.842562 0.421281 0.906930i \(-0.361581\pi\)
0.421281 + 0.906930i \(0.361581\pi\)
\(282\) 0 0
\(283\) −14.4023 −0.856127 −0.428064 0.903749i \(-0.640804\pi\)
−0.428064 + 0.903749i \(0.640804\pi\)
\(284\) 0 0
\(285\) 24.1937 1.43311
\(286\) 0 0
\(287\) −14.8486 −0.876486
\(288\) 0 0
\(289\) 7.79184 0.458344
\(290\) 0 0
\(291\) −5.44485 −0.319183
\(292\) 0 0
\(293\) 10.4437 0.610127 0.305063 0.952332i \(-0.401322\pi\)
0.305063 + 0.952332i \(0.401322\pi\)
\(294\) 0 0
\(295\) 2.13931 0.124555
\(296\) 0 0
\(297\) 0.0711814 0.00413036
\(298\) 0 0
\(299\) −0.620759 −0.0358994
\(300\) 0 0
\(301\) 2.40988 0.138903
\(302\) 0 0
\(303\) 10.2508 0.588891
\(304\) 0 0
\(305\) −31.9686 −1.83052
\(306\) 0 0
\(307\) 7.71773 0.440474 0.220237 0.975446i \(-0.429317\pi\)
0.220237 + 0.975446i \(0.429317\pi\)
\(308\) 0 0
\(309\) −4.52130 −0.257208
\(310\) 0 0
\(311\) 21.7132 1.23124 0.615621 0.788042i \(-0.288905\pi\)
0.615621 + 0.788042i \(0.288905\pi\)
\(312\) 0 0
\(313\) 20.6727 1.16849 0.584244 0.811578i \(-0.301391\pi\)
0.584244 + 0.811578i \(0.301391\pi\)
\(314\) 0 0
\(315\) 8.18041 0.460914
\(316\) 0 0
\(317\) −29.2949 −1.64537 −0.822683 0.568501i \(-0.807524\pi\)
−0.822683 + 0.568501i \(0.807524\pi\)
\(318\) 0 0
\(319\) 0.0711814 0.00398539
\(320\) 0 0
\(321\) 12.7966 0.714234
\(322\) 0 0
\(323\) 34.0212 1.89299
\(324\) 0 0
\(325\) −4.67899 −0.259543
\(326\) 0 0
\(327\) −1.35997 −0.0752062
\(328\) 0 0
\(329\) 12.7496 0.702910
\(330\) 0 0
\(331\) −15.9585 −0.877156 −0.438578 0.898693i \(-0.644518\pi\)
−0.438578 + 0.898693i \(0.644518\pi\)
\(332\) 0 0
\(333\) −7.78652 −0.426699
\(334\) 0 0
\(335\) −43.0160 −2.35022
\(336\) 0 0
\(337\) 15.3538 0.836373 0.418186 0.908361i \(-0.362666\pi\)
0.418186 + 0.908361i \(0.362666\pi\)
\(338\) 0 0
\(339\) 5.37462 0.291909
\(340\) 0 0
\(341\) 0.169633 0.00918613
\(342\) 0 0
\(343\) −20.0130 −1.08060
\(344\) 0 0
\(345\) −3.54084 −0.190632
\(346\) 0 0
\(347\) −7.09962 −0.381128 −0.190564 0.981675i \(-0.561032\pi\)
−0.190564 + 0.981675i \(0.561032\pi\)
\(348\) 0 0
\(349\) −6.79974 −0.363982 −0.181991 0.983300i \(-0.558254\pi\)
−0.181991 + 0.983300i \(0.558254\pi\)
\(350\) 0 0
\(351\) 0.620759 0.0331337
\(352\) 0 0
\(353\) 14.8482 0.790288 0.395144 0.918619i \(-0.370695\pi\)
0.395144 + 0.918619i \(0.370695\pi\)
\(354\) 0 0
\(355\) −12.4825 −0.662504
\(356\) 0 0
\(357\) 11.5033 0.608821
\(358\) 0 0
\(359\) −31.6435 −1.67008 −0.835040 0.550190i \(-0.814555\pi\)
−0.835040 + 0.550190i \(0.814555\pi\)
\(360\) 0 0
\(361\) 27.6865 1.45718
\(362\) 0 0
\(363\) 10.9949 0.577084
\(364\) 0 0
\(365\) −16.9525 −0.887332
\(366\) 0 0
\(367\) 10.9055 0.569261 0.284631 0.958637i \(-0.408129\pi\)
0.284631 + 0.958637i \(0.408129\pi\)
\(368\) 0 0
\(369\) −6.42713 −0.334583
\(370\) 0 0
\(371\) −12.4228 −0.644962
\(372\) 0 0
\(373\) 22.4072 1.16020 0.580101 0.814545i \(-0.303013\pi\)
0.580101 + 0.814545i \(0.303013\pi\)
\(374\) 0 0
\(375\) −8.98496 −0.463981
\(376\) 0 0
\(377\) 0.620759 0.0319707
\(378\) 0 0
\(379\) 22.0565 1.13297 0.566483 0.824074i \(-0.308304\pi\)
0.566483 + 0.824074i \(0.308304\pi\)
\(380\) 0 0
\(381\) −20.1570 −1.03267
\(382\) 0 0
\(383\) −14.8911 −0.760898 −0.380449 0.924802i \(-0.624231\pi\)
−0.380449 + 0.924802i \(0.624231\pi\)
\(384\) 0 0
\(385\) −0.582293 −0.0296764
\(386\) 0 0
\(387\) 1.04310 0.0530237
\(388\) 0 0
\(389\) 8.34025 0.422867 0.211434 0.977392i \(-0.432187\pi\)
0.211434 + 0.977392i \(0.432187\pi\)
\(390\) 0 0
\(391\) −4.97914 −0.251806
\(392\) 0 0
\(393\) −14.0047 −0.706444
\(394\) 0 0
\(395\) 6.09456 0.306651
\(396\) 0 0
\(397\) 18.1769 0.912272 0.456136 0.889910i \(-0.349233\pi\)
0.456136 + 0.889910i \(0.349233\pi\)
\(398\) 0 0
\(399\) 15.7857 0.790275
\(400\) 0 0
\(401\) −31.2426 −1.56018 −0.780090 0.625668i \(-0.784827\pi\)
−0.780090 + 0.625668i \(0.784827\pi\)
\(402\) 0 0
\(403\) 1.47933 0.0736909
\(404\) 0 0
\(405\) 3.54084 0.175946
\(406\) 0 0
\(407\) 0.554256 0.0274734
\(408\) 0 0
\(409\) −7.88288 −0.389783 −0.194892 0.980825i \(-0.562436\pi\)
−0.194892 + 0.980825i \(0.562436\pi\)
\(410\) 0 0
\(411\) −6.14660 −0.303190
\(412\) 0 0
\(413\) 1.39584 0.0686849
\(414\) 0 0
\(415\) −42.6744 −2.09480
\(416\) 0 0
\(417\) −14.3036 −0.700449
\(418\) 0 0
\(419\) 26.0481 1.27253 0.636266 0.771470i \(-0.280478\pi\)
0.636266 + 0.771470i \(0.280478\pi\)
\(420\) 0 0
\(421\) −16.0576 −0.782597 −0.391299 0.920264i \(-0.627974\pi\)
−0.391299 + 0.920264i \(0.627974\pi\)
\(422\) 0 0
\(423\) 5.51860 0.268323
\(424\) 0 0
\(425\) −37.5304 −1.82049
\(426\) 0 0
\(427\) −20.8587 −1.00942
\(428\) 0 0
\(429\) −0.0441865 −0.00213334
\(430\) 0 0
\(431\) −1.85883 −0.0895368 −0.0447684 0.998997i \(-0.514255\pi\)
−0.0447684 + 0.998997i \(0.514255\pi\)
\(432\) 0 0
\(433\) 40.4890 1.94578 0.972890 0.231270i \(-0.0742880\pi\)
0.972890 + 0.231270i \(0.0742880\pi\)
\(434\) 0 0
\(435\) 3.54084 0.169770
\(436\) 0 0
\(437\) −6.83275 −0.326855
\(438\) 0 0
\(439\) −12.6232 −0.602471 −0.301235 0.953550i \(-0.597399\pi\)
−0.301235 + 0.953550i \(0.597399\pi\)
\(440\) 0 0
\(441\) −1.66249 −0.0791664
\(442\) 0 0
\(443\) −1.67057 −0.0793712 −0.0396856 0.999212i \(-0.512636\pi\)
−0.0396856 + 0.999212i \(0.512636\pi\)
\(444\) 0 0
\(445\) −31.0696 −1.47284
\(446\) 0 0
\(447\) −20.3437 −0.962225
\(448\) 0 0
\(449\) −34.0388 −1.60639 −0.803196 0.595715i \(-0.796869\pi\)
−0.803196 + 0.595715i \(0.796869\pi\)
\(450\) 0 0
\(451\) 0.457492 0.0215425
\(452\) 0 0
\(453\) 2.73246 0.128382
\(454\) 0 0
\(455\) −5.07806 −0.238063
\(456\) 0 0
\(457\) −22.0105 −1.02961 −0.514805 0.857308i \(-0.672136\pi\)
−0.514805 + 0.857308i \(0.672136\pi\)
\(458\) 0 0
\(459\) 4.97914 0.232406
\(460\) 0 0
\(461\) −34.2152 −1.59356 −0.796779 0.604270i \(-0.793465\pi\)
−0.796779 + 0.604270i \(0.793465\pi\)
\(462\) 0 0
\(463\) −19.5768 −0.909810 −0.454905 0.890540i \(-0.650327\pi\)
−0.454905 + 0.890540i \(0.650327\pi\)
\(464\) 0 0
\(465\) 8.43819 0.391312
\(466\) 0 0
\(467\) −4.68553 −0.216820 −0.108410 0.994106i \(-0.534576\pi\)
−0.108410 + 0.994106i \(0.534576\pi\)
\(468\) 0 0
\(469\) −28.0669 −1.29601
\(470\) 0 0
\(471\) 21.3377 0.983187
\(472\) 0 0
\(473\) −0.0742492 −0.00341398
\(474\) 0 0
\(475\) −51.5020 −2.36308
\(476\) 0 0
\(477\) −5.37715 −0.246203
\(478\) 0 0
\(479\) −9.30781 −0.425285 −0.212642 0.977130i \(-0.568207\pi\)
−0.212642 + 0.977130i \(0.568207\pi\)
\(480\) 0 0
\(481\) 4.83355 0.220391
\(482\) 0 0
\(483\) −2.31030 −0.105122
\(484\) 0 0
\(485\) 19.2793 0.875429
\(486\) 0 0
\(487\) 25.9057 1.17390 0.586950 0.809623i \(-0.300329\pi\)
0.586950 + 0.809623i \(0.300329\pi\)
\(488\) 0 0
\(489\) −11.6539 −0.527007
\(490\) 0 0
\(491\) −25.0263 −1.12942 −0.564710 0.825290i \(-0.691012\pi\)
−0.564710 + 0.825290i \(0.691012\pi\)
\(492\) 0 0
\(493\) 4.97914 0.224249
\(494\) 0 0
\(495\) −0.252042 −0.0113284
\(496\) 0 0
\(497\) −8.14453 −0.365332
\(498\) 0 0
\(499\) 12.6605 0.566761 0.283381 0.959008i \(-0.408544\pi\)
0.283381 + 0.959008i \(0.408544\pi\)
\(500\) 0 0
\(501\) 1.01179 0.0452033
\(502\) 0 0
\(503\) 19.6887 0.877876 0.438938 0.898517i \(-0.355355\pi\)
0.438938 + 0.898517i \(0.355355\pi\)
\(504\) 0 0
\(505\) −36.2963 −1.61516
\(506\) 0 0
\(507\) 12.6147 0.560237
\(508\) 0 0
\(509\) 13.9225 0.617106 0.308553 0.951207i \(-0.400155\pi\)
0.308553 + 0.951207i \(0.400155\pi\)
\(510\) 0 0
\(511\) −11.0610 −0.489311
\(512\) 0 0
\(513\) 6.83275 0.301673
\(514\) 0 0
\(515\) 16.0092 0.705449
\(516\) 0 0
\(517\) −0.392821 −0.0172763
\(518\) 0 0
\(519\) 1.34864 0.0591988
\(520\) 0 0
\(521\) −14.1358 −0.619301 −0.309651 0.950850i \(-0.600212\pi\)
−0.309651 + 0.950850i \(0.600212\pi\)
\(522\) 0 0
\(523\) 1.66533 0.0728197 0.0364099 0.999337i \(-0.488408\pi\)
0.0364099 + 0.999337i \(0.488408\pi\)
\(524\) 0 0
\(525\) −17.4140 −0.760008
\(526\) 0 0
\(527\) 11.8658 0.516883
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0.604181 0.0262192
\(532\) 0 0
\(533\) 3.98970 0.172813
\(534\) 0 0
\(535\) −45.3105 −1.95894
\(536\) 0 0
\(537\) 11.9175 0.514276
\(538\) 0 0
\(539\) 0.118339 0.00509721
\(540\) 0 0
\(541\) −3.43339 −0.147613 −0.0738065 0.997273i \(-0.523515\pi\)
−0.0738065 + 0.997273i \(0.523515\pi\)
\(542\) 0 0
\(543\) −14.2013 −0.609435
\(544\) 0 0
\(545\) 4.81541 0.206270
\(546\) 0 0
\(547\) 0.855283 0.0365693 0.0182846 0.999833i \(-0.494179\pi\)
0.0182846 + 0.999833i \(0.494179\pi\)
\(548\) 0 0
\(549\) −9.02854 −0.385329
\(550\) 0 0
\(551\) 6.83275 0.291085
\(552\) 0 0
\(553\) 3.97654 0.169100
\(554\) 0 0
\(555\) 27.5708 1.17032
\(556\) 0 0
\(557\) −6.10647 −0.258739 −0.129370 0.991596i \(-0.541295\pi\)
−0.129370 + 0.991596i \(0.541295\pi\)
\(558\) 0 0
\(559\) −0.647513 −0.0273869
\(560\) 0 0
\(561\) −0.354422 −0.0149637
\(562\) 0 0
\(563\) −12.8657 −0.542224 −0.271112 0.962548i \(-0.587391\pi\)
−0.271112 + 0.962548i \(0.587391\pi\)
\(564\) 0 0
\(565\) −19.0306 −0.800625
\(566\) 0 0
\(567\) 2.31030 0.0970237
\(568\) 0 0
\(569\) −2.30747 −0.0967342 −0.0483671 0.998830i \(-0.515402\pi\)
−0.0483671 + 0.998830i \(0.515402\pi\)
\(570\) 0 0
\(571\) 36.7084 1.53620 0.768100 0.640330i \(-0.221203\pi\)
0.768100 + 0.640330i \(0.221203\pi\)
\(572\) 0 0
\(573\) −13.8182 −0.577264
\(574\) 0 0
\(575\) 7.53753 0.314337
\(576\) 0 0
\(577\) −8.21186 −0.341864 −0.170932 0.985283i \(-0.554678\pi\)
−0.170932 + 0.985283i \(0.554678\pi\)
\(578\) 0 0
\(579\) 7.49114 0.311321
\(580\) 0 0
\(581\) −27.8439 −1.15516
\(582\) 0 0
\(583\) 0.382753 0.0158520
\(584\) 0 0
\(585\) −2.19801 −0.0908764
\(586\) 0 0
\(587\) 27.7611 1.14582 0.572911 0.819617i \(-0.305814\pi\)
0.572911 + 0.819617i \(0.305814\pi\)
\(588\) 0 0
\(589\) 16.2832 0.670937
\(590\) 0 0
\(591\) −12.1766 −0.500880
\(592\) 0 0
\(593\) −4.40184 −0.180762 −0.0903809 0.995907i \(-0.528808\pi\)
−0.0903809 + 0.995907i \(0.528808\pi\)
\(594\) 0 0
\(595\) −40.7314 −1.66983
\(596\) 0 0
\(597\) −23.0031 −0.941454
\(598\) 0 0
\(599\) −38.1021 −1.55681 −0.778404 0.627763i \(-0.783971\pi\)
−0.778404 + 0.627763i \(0.783971\pi\)
\(600\) 0 0
\(601\) 8.67479 0.353852 0.176926 0.984224i \(-0.443385\pi\)
0.176926 + 0.984224i \(0.443385\pi\)
\(602\) 0 0
\(603\) −12.1486 −0.494727
\(604\) 0 0
\(605\) −38.9313 −1.58278
\(606\) 0 0
\(607\) 25.5908 1.03870 0.519349 0.854562i \(-0.326174\pi\)
0.519349 + 0.854562i \(0.326174\pi\)
\(608\) 0 0
\(609\) 2.31030 0.0936183
\(610\) 0 0
\(611\) −3.42572 −0.138590
\(612\) 0 0
\(613\) 15.1226 0.610795 0.305398 0.952225i \(-0.401211\pi\)
0.305398 + 0.952225i \(0.401211\pi\)
\(614\) 0 0
\(615\) 22.7574 0.917667
\(616\) 0 0
\(617\) −16.7383 −0.673859 −0.336930 0.941530i \(-0.609388\pi\)
−0.336930 + 0.941530i \(0.609388\pi\)
\(618\) 0 0
\(619\) 31.7185 1.27487 0.637436 0.770503i \(-0.279995\pi\)
0.637436 + 0.770503i \(0.279995\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −20.2721 −0.812185
\(624\) 0 0
\(625\) −5.87334 −0.234934
\(626\) 0 0
\(627\) −0.486365 −0.0194235
\(628\) 0 0
\(629\) 38.7702 1.54587
\(630\) 0 0
\(631\) 14.1918 0.564965 0.282482 0.959272i \(-0.408842\pi\)
0.282482 + 0.959272i \(0.408842\pi\)
\(632\) 0 0
\(633\) 7.53452 0.299470
\(634\) 0 0
\(635\) 71.3725 2.83233
\(636\) 0 0
\(637\) 1.03201 0.0408896
\(638\) 0 0
\(639\) −3.52530 −0.139459
\(640\) 0 0
\(641\) −13.3742 −0.528247 −0.264124 0.964489i \(-0.585083\pi\)
−0.264124 + 0.964489i \(0.585083\pi\)
\(642\) 0 0
\(643\) 4.83486 0.190668 0.0953341 0.995445i \(-0.469608\pi\)
0.0953341 + 0.995445i \(0.469608\pi\)
\(644\) 0 0
\(645\) −3.69344 −0.145429
\(646\) 0 0
\(647\) −10.6970 −0.420542 −0.210271 0.977643i \(-0.567435\pi\)
−0.210271 + 0.977643i \(0.567435\pi\)
\(648\) 0 0
\(649\) −0.0430065 −0.00168815
\(650\) 0 0
\(651\) 5.50570 0.215785
\(652\) 0 0
\(653\) −23.0895 −0.903563 −0.451781 0.892129i \(-0.649211\pi\)
−0.451781 + 0.892129i \(0.649211\pi\)
\(654\) 0 0
\(655\) 49.5884 1.93758
\(656\) 0 0
\(657\) −4.78770 −0.186786
\(658\) 0 0
\(659\) −18.9876 −0.739653 −0.369826 0.929101i \(-0.620583\pi\)
−0.369826 + 0.929101i \(0.620583\pi\)
\(660\) 0 0
\(661\) 25.3945 0.987730 0.493865 0.869539i \(-0.335584\pi\)
0.493865 + 0.869539i \(0.335584\pi\)
\(662\) 0 0
\(663\) −3.09085 −0.120039
\(664\) 0 0
\(665\) −55.8947 −2.16750
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 27.4689 1.06201
\(670\) 0 0
\(671\) 0.642664 0.0248098
\(672\) 0 0
\(673\) −12.3188 −0.474857 −0.237428 0.971405i \(-0.576304\pi\)
−0.237428 + 0.971405i \(0.576304\pi\)
\(674\) 0 0
\(675\) −7.53753 −0.290119
\(676\) 0 0
\(677\) −26.0942 −1.00288 −0.501440 0.865193i \(-0.667196\pi\)
−0.501440 + 0.865193i \(0.667196\pi\)
\(678\) 0 0
\(679\) 12.5793 0.482748
\(680\) 0 0
\(681\) −15.2865 −0.585779
\(682\) 0 0
\(683\) −10.5226 −0.402635 −0.201318 0.979526i \(-0.564522\pi\)
−0.201318 + 0.979526i \(0.564522\pi\)
\(684\) 0 0
\(685\) 21.7641 0.831564
\(686\) 0 0
\(687\) 10.2081 0.389463
\(688\) 0 0
\(689\) 3.33791 0.127164
\(690\) 0 0
\(691\) 20.7754 0.790332 0.395166 0.918610i \(-0.370687\pi\)
0.395166 + 0.918610i \(0.370687\pi\)
\(692\) 0 0
\(693\) −0.164451 −0.00624697
\(694\) 0 0
\(695\) 50.6466 1.92114
\(696\) 0 0
\(697\) 32.0016 1.21215
\(698\) 0 0
\(699\) −12.6697 −0.479213
\(700\) 0 0
\(701\) −4.61954 −0.174478 −0.0872389 0.996187i \(-0.527804\pi\)
−0.0872389 + 0.996187i \(0.527804\pi\)
\(702\) 0 0
\(703\) 53.2034 2.00660
\(704\) 0 0
\(705\) −19.5404 −0.735936
\(706\) 0 0
\(707\) −23.6824 −0.890667
\(708\) 0 0
\(709\) −38.3197 −1.43913 −0.719564 0.694426i \(-0.755658\pi\)
−0.719564 + 0.694426i \(0.755658\pi\)
\(710\) 0 0
\(711\) 1.72122 0.0645508
\(712\) 0 0
\(713\) −2.38311 −0.0892480
\(714\) 0 0
\(715\) 0.156457 0.00585117
\(716\) 0 0
\(717\) −17.0568 −0.636999
\(718\) 0 0
\(719\) −39.9640 −1.49040 −0.745202 0.666838i \(-0.767647\pi\)
−0.745202 + 0.666838i \(0.767647\pi\)
\(720\) 0 0
\(721\) 10.4456 0.389013
\(722\) 0 0
\(723\) −25.6164 −0.952686
\(724\) 0 0
\(725\) −7.53753 −0.279937
\(726\) 0 0
\(727\) 11.1402 0.413165 0.206583 0.978429i \(-0.433766\pi\)
0.206583 + 0.978429i \(0.433766\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.19373 −0.192097
\(732\) 0 0
\(733\) −30.9033 −1.14144 −0.570719 0.821145i \(-0.693335\pi\)
−0.570719 + 0.821145i \(0.693335\pi\)
\(734\) 0 0
\(735\) 5.88662 0.217131
\(736\) 0 0
\(737\) 0.864751 0.0318535
\(738\) 0 0
\(739\) −5.38902 −0.198238 −0.0991191 0.995076i \(-0.531602\pi\)
−0.0991191 + 0.995076i \(0.531602\pi\)
\(740\) 0 0
\(741\) −4.24149 −0.155815
\(742\) 0 0
\(743\) 4.30318 0.157868 0.0789342 0.996880i \(-0.474848\pi\)
0.0789342 + 0.996880i \(0.474848\pi\)
\(744\) 0 0
\(745\) 72.0338 2.63911
\(746\) 0 0
\(747\) −12.0521 −0.440962
\(748\) 0 0
\(749\) −29.5639 −1.08024
\(750\) 0 0
\(751\) −43.2393 −1.57783 −0.788913 0.614505i \(-0.789356\pi\)
−0.788913 + 0.614505i \(0.789356\pi\)
\(752\) 0 0
\(753\) −12.9643 −0.472444
\(754\) 0 0
\(755\) −9.67520 −0.352117
\(756\) 0 0
\(757\) −38.0765 −1.38391 −0.691957 0.721939i \(-0.743251\pi\)
−0.691957 + 0.721939i \(0.743251\pi\)
\(758\) 0 0
\(759\) 0.0711814 0.00258372
\(760\) 0 0
\(761\) 28.1001 1.01863 0.509314 0.860581i \(-0.329899\pi\)
0.509314 + 0.860581i \(0.329899\pi\)
\(762\) 0 0
\(763\) 3.14193 0.113746
\(764\) 0 0
\(765\) −17.6303 −0.637426
\(766\) 0 0
\(767\) −0.375051 −0.0135423
\(768\) 0 0
\(769\) 27.2065 0.981091 0.490546 0.871416i \(-0.336798\pi\)
0.490546 + 0.871416i \(0.336798\pi\)
\(770\) 0 0
\(771\) 7.55820 0.272202
\(772\) 0 0
\(773\) 36.2723 1.30462 0.652312 0.757951i \(-0.273799\pi\)
0.652312 + 0.757951i \(0.273799\pi\)
\(774\) 0 0
\(775\) −17.9627 −0.645240
\(776\) 0 0
\(777\) 17.9892 0.645360
\(778\) 0 0
\(779\) 43.9150 1.57342
\(780\) 0 0
\(781\) 0.250936 0.00897920
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −75.5532 −2.69661
\(786\) 0 0
\(787\) 7.29110 0.259900 0.129950 0.991521i \(-0.458518\pi\)
0.129950 + 0.991521i \(0.458518\pi\)
\(788\) 0 0
\(789\) −4.86890 −0.173338
\(790\) 0 0
\(791\) −12.4170 −0.441498
\(792\) 0 0
\(793\) 5.60455 0.199023
\(794\) 0 0
\(795\) 19.0396 0.675265
\(796\) 0 0
\(797\) 4.32204 0.153095 0.0765473 0.997066i \(-0.475610\pi\)
0.0765473 + 0.997066i \(0.475610\pi\)
\(798\) 0 0
\(799\) −27.4779 −0.972097
\(800\) 0 0
\(801\) −8.77465 −0.310037
\(802\) 0 0
\(803\) 0.340795 0.0120264
\(804\) 0 0
\(805\) 8.18041 0.288322
\(806\) 0 0
\(807\) −1.51039 −0.0531681
\(808\) 0 0
\(809\) 6.45744 0.227032 0.113516 0.993536i \(-0.463789\pi\)
0.113516 + 0.993536i \(0.463789\pi\)
\(810\) 0 0
\(811\) −22.0673 −0.774888 −0.387444 0.921893i \(-0.626642\pi\)
−0.387444 + 0.921893i \(0.626642\pi\)
\(812\) 0 0
\(813\) −10.4038 −0.364877
\(814\) 0 0
\(815\) 41.2645 1.44543
\(816\) 0 0
\(817\) −7.12723 −0.249350
\(818\) 0 0
\(819\) −1.43414 −0.0501130
\(820\) 0 0
\(821\) −18.3813 −0.641513 −0.320756 0.947162i \(-0.603937\pi\)
−0.320756 + 0.947162i \(0.603937\pi\)
\(822\) 0 0
\(823\) 23.1189 0.805874 0.402937 0.915228i \(-0.367989\pi\)
0.402937 + 0.915228i \(0.367989\pi\)
\(824\) 0 0
\(825\) 0.536532 0.0186796
\(826\) 0 0
\(827\) 27.4701 0.955229 0.477614 0.878570i \(-0.341502\pi\)
0.477614 + 0.878570i \(0.341502\pi\)
\(828\) 0 0
\(829\) −16.9519 −0.588763 −0.294382 0.955688i \(-0.595114\pi\)
−0.294382 + 0.955688i \(0.595114\pi\)
\(830\) 0 0
\(831\) 6.22112 0.215808
\(832\) 0 0
\(833\) 8.27779 0.286808
\(834\) 0 0
\(835\) −3.58257 −0.123980
\(836\) 0 0
\(837\) 2.38311 0.0823722
\(838\) 0 0
\(839\) 11.7048 0.404095 0.202048 0.979376i \(-0.435240\pi\)
0.202048 + 0.979376i \(0.435240\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −14.1239 −0.486454
\(844\) 0 0
\(845\) −44.6664 −1.53657
\(846\) 0 0
\(847\) −25.4016 −0.872811
\(848\) 0 0
\(849\) 14.4023 0.494285
\(850\) 0 0
\(851\) −7.78652 −0.266919
\(852\) 0 0
\(853\) −5.09938 −0.174599 −0.0872997 0.996182i \(-0.527824\pi\)
−0.0872997 + 0.996182i \(0.527824\pi\)
\(854\) 0 0
\(855\) −24.1937 −0.827406
\(856\) 0 0
\(857\) −6.81602 −0.232831 −0.116415 0.993201i \(-0.537140\pi\)
−0.116415 + 0.993201i \(0.537140\pi\)
\(858\) 0 0
\(859\) −22.6219 −0.771850 −0.385925 0.922530i \(-0.626118\pi\)
−0.385925 + 0.922530i \(0.626118\pi\)
\(860\) 0 0
\(861\) 14.8486 0.506040
\(862\) 0 0
\(863\) −47.7217 −1.62446 −0.812232 0.583334i \(-0.801748\pi\)
−0.812232 + 0.583334i \(0.801748\pi\)
\(864\) 0 0
\(865\) −4.77532 −0.162366
\(866\) 0 0
\(867\) −7.79184 −0.264625
\(868\) 0 0
\(869\) −0.122519 −0.00415617
\(870\) 0 0
\(871\) 7.54132 0.255528
\(872\) 0 0
\(873\) 5.44485 0.184280
\(874\) 0 0
\(875\) 20.7580 0.701748
\(876\) 0 0
\(877\) −17.9010 −0.604475 −0.302237 0.953233i \(-0.597734\pi\)
−0.302237 + 0.953233i \(0.597734\pi\)
\(878\) 0 0
\(879\) −10.4437 −0.352257
\(880\) 0 0
\(881\) −21.9553 −0.739693 −0.369846 0.929093i \(-0.620590\pi\)
−0.369846 + 0.929093i \(0.620590\pi\)
\(882\) 0 0
\(883\) 37.8079 1.27234 0.636169 0.771550i \(-0.280518\pi\)
0.636169 + 0.771550i \(0.280518\pi\)
\(884\) 0 0
\(885\) −2.13931 −0.0719120
\(886\) 0 0
\(887\) 17.2550 0.579366 0.289683 0.957123i \(-0.406450\pi\)
0.289683 + 0.957123i \(0.406450\pi\)
\(888\) 0 0
\(889\) 46.5687 1.56186
\(890\) 0 0
\(891\) −0.0711814 −0.00238467
\(892\) 0 0
\(893\) −37.7072 −1.26182
\(894\) 0 0
\(895\) −42.1978 −1.41052
\(896\) 0 0
\(897\) 0.620759 0.0207265
\(898\) 0 0
\(899\) 2.38311 0.0794811
\(900\) 0 0
\(901\) 26.7736 0.891957
\(902\) 0 0
\(903\) −2.40988 −0.0801956
\(904\) 0 0
\(905\) 50.2844 1.67151
\(906\) 0 0
\(907\) −20.1323 −0.668481 −0.334240 0.942488i \(-0.608480\pi\)
−0.334240 + 0.942488i \(0.608480\pi\)
\(908\) 0 0
\(909\) −10.2508 −0.339996
\(910\) 0 0
\(911\) 16.6963 0.553174 0.276587 0.960989i \(-0.410797\pi\)
0.276587 + 0.960989i \(0.410797\pi\)
\(912\) 0 0
\(913\) 0.857882 0.0283918
\(914\) 0 0
\(915\) 31.9686 1.05685
\(916\) 0 0
\(917\) 32.3551 1.06846
\(918\) 0 0
\(919\) 18.5764 0.612778 0.306389 0.951906i \(-0.400879\pi\)
0.306389 + 0.951906i \(0.400879\pi\)
\(920\) 0 0
\(921\) −7.71773 −0.254308
\(922\) 0 0
\(923\) 2.18836 0.0720309
\(924\) 0 0
\(925\) −58.6911 −1.92975
\(926\) 0 0
\(927\) 4.52130 0.148499
\(928\) 0 0
\(929\) −1.42951 −0.0469008 −0.0234504 0.999725i \(-0.507465\pi\)
−0.0234504 + 0.999725i \(0.507465\pi\)
\(930\) 0 0
\(931\) 11.3594 0.372290
\(932\) 0 0
\(933\) −21.7132 −0.710858
\(934\) 0 0
\(935\) 1.25495 0.0410413
\(936\) 0 0
\(937\) 1.77221 0.0578955 0.0289478 0.999581i \(-0.490784\pi\)
0.0289478 + 0.999581i \(0.490784\pi\)
\(938\) 0 0
\(939\) −20.6727 −0.674627
\(940\) 0 0
\(941\) −52.1196 −1.69905 −0.849526 0.527548i \(-0.823112\pi\)
−0.849526 + 0.527548i \(0.823112\pi\)
\(942\) 0 0
\(943\) −6.42713 −0.209296
\(944\) 0 0
\(945\) −8.18041 −0.266109
\(946\) 0 0
\(947\) −41.9737 −1.36396 −0.681981 0.731369i \(-0.738882\pi\)
−0.681981 + 0.731369i \(0.738882\pi\)
\(948\) 0 0
\(949\) 2.97201 0.0964754
\(950\) 0 0
\(951\) 29.2949 0.949952
\(952\) 0 0
\(953\) 16.9920 0.550423 0.275212 0.961384i \(-0.411252\pi\)
0.275212 + 0.961384i \(0.411252\pi\)
\(954\) 0 0
\(955\) 48.9281 1.58327
\(956\) 0 0
\(957\) −0.0711814 −0.00230097
\(958\) 0 0
\(959\) 14.2005 0.458559
\(960\) 0 0
\(961\) −25.3208 −0.816800
\(962\) 0 0
\(963\) −12.7966 −0.412363
\(964\) 0 0
\(965\) −26.5249 −0.853867
\(966\) 0 0
\(967\) 14.0030 0.450306 0.225153 0.974323i \(-0.427712\pi\)
0.225153 + 0.974323i \(0.427712\pi\)
\(968\) 0 0
\(969\) −34.0212 −1.09292
\(970\) 0 0
\(971\) −10.6948 −0.343212 −0.171606 0.985166i \(-0.554896\pi\)
−0.171606 + 0.985166i \(0.554896\pi\)
\(972\) 0 0
\(973\) 33.0456 1.05939
\(974\) 0 0
\(975\) 4.67899 0.149847
\(976\) 0 0
\(977\) −35.3413 −1.13067 −0.565334 0.824862i \(-0.691253\pi\)
−0.565334 + 0.824862i \(0.691253\pi\)
\(978\) 0 0
\(979\) 0.624592 0.0199621
\(980\) 0 0
\(981\) 1.35997 0.0434203
\(982\) 0 0
\(983\) −46.6747 −1.48869 −0.744346 0.667794i \(-0.767239\pi\)
−0.744346 + 0.667794i \(0.767239\pi\)
\(984\) 0 0
\(985\) 43.1155 1.37377
\(986\) 0 0
\(987\) −12.7496 −0.405825
\(988\) 0 0
\(989\) 1.04310 0.0331686
\(990\) 0 0
\(991\) 11.8635 0.376857 0.188428 0.982087i \(-0.439661\pi\)
0.188428 + 0.982087i \(0.439661\pi\)
\(992\) 0 0
\(993\) 15.9585 0.506426
\(994\) 0 0
\(995\) 81.4502 2.58214
\(996\) 0 0
\(997\) −14.9795 −0.474404 −0.237202 0.971460i \(-0.576230\pi\)
−0.237202 + 0.971460i \(0.576230\pi\)
\(998\) 0 0
\(999\) 7.78652 0.246355
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 8004.2.a.g.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.12 12 1.1 even 1 trivial