Properties

Label 9408.2.a.ec.1.2
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.27492 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +4.27492 q^{5} +1.00000 q^{9} +4.27492 q^{11} +1.27492 q^{13} +4.27492 q^{15} -4.00000 q^{17} -1.27492 q^{19} +4.00000 q^{23} +13.2749 q^{25} +1.00000 q^{27} +2.27492 q^{29} -1.00000 q^{31} +4.27492 q^{33} -5.27492 q^{37} +1.27492 q^{39} +10.5498 q^{41} +7.27492 q^{43} +4.27492 q^{45} -6.00000 q^{47} -4.00000 q^{51} -1.72508 q^{53} +18.2749 q^{55} -1.27492 q^{57} -6.27492 q^{59} -10.0000 q^{61} +5.45017 q^{65} -7.27492 q^{67} +4.00000 q^{69} +2.00000 q^{71} +3.27492 q^{73} +13.2749 q^{75} -3.54983 q^{79} +1.00000 q^{81} +0.274917 q^{83} -17.0997 q^{85} +2.27492 q^{87} +4.54983 q^{89} -1.00000 q^{93} -5.45017 q^{95} +16.2749 q^{97} +4.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} + 2q^{9} + q^{11} - 5q^{13} + q^{15} - 8q^{17} + 5q^{19} + 8q^{23} + 19q^{25} + 2q^{27} - 3q^{29} - 2q^{31} + q^{33} - 3q^{37} - 5q^{39} + 6q^{41} + 7q^{43} + q^{45} - 12q^{47} - 8q^{51} - 11q^{53} + 29q^{55} + 5q^{57} - 5q^{59} - 20q^{61} + 26q^{65} - 7q^{67} + 8q^{69} + 4q^{71} - q^{73} + 19q^{75} + 8q^{79} + 2q^{81} - 7q^{83} - 4q^{85} - 3q^{87} - 6q^{89} - 2q^{93} - 26q^{95} + 25q^{97} + q^{99} + O(q^{100}) \)

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
\(5\) 4.27492 1.91180 0.955901 0.293691i \(-0.0948835\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.27492 1.28894 0.644468 0.764631i \(-0.277079\pi\)
0.644468 + 0.764631i \(0.277079\pi\)
\(12\) 0 0
\(13\) 1.27492 0.353598 0.176799 0.984247i \(-0.443426\pi\)
0.176799 + 0.984247i \(0.443426\pi\)
\(14\) 0 0
\(15\) 4.27492 1.10378
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −1.27492 −0.292486 −0.146243 0.989249i \(-0.546718\pi\)
−0.146243 + 0.989249i \(0.546718\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 13.2749 2.65498
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.27492 0.422442 0.211221 0.977438i \(-0.432256\pi\)
0.211221 + 0.977438i \(0.432256\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 4.27492 0.744168
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.27492 −0.867191 −0.433596 0.901108i \(-0.642755\pi\)
−0.433596 + 0.901108i \(0.642755\pi\)
\(38\) 0 0
\(39\) 1.27492 0.204150
\(40\) 0 0
\(41\) 10.5498 1.64761 0.823804 0.566875i \(-0.191848\pi\)
0.823804 + 0.566875i \(0.191848\pi\)
\(42\) 0 0
\(43\) 7.27492 1.10941 0.554707 0.832046i \(-0.312830\pi\)
0.554707 + 0.832046i \(0.312830\pi\)
\(44\) 0 0
\(45\) 4.27492 0.637267
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −1.72508 −0.236958 −0.118479 0.992957i \(-0.537802\pi\)
−0.118479 + 0.992957i \(0.537802\pi\)
\(54\) 0 0
\(55\) 18.2749 2.46419
\(56\) 0 0
\(57\) −1.27492 −0.168867
\(58\) 0 0
\(59\) −6.27492 −0.816925 −0.408462 0.912775i \(-0.633935\pi\)
−0.408462 + 0.912775i \(0.633935\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.45017 0.676010
\(66\) 0 0
\(67\) −7.27492 −0.888773 −0.444386 0.895835i \(-0.646578\pi\)
−0.444386 + 0.895835i \(0.646578\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 3.27492 0.383300 0.191650 0.981463i \(-0.438616\pi\)
0.191650 + 0.981463i \(0.438616\pi\)
\(74\) 0 0
\(75\) 13.2749 1.53286
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.54983 −0.399388 −0.199694 0.979858i \(-0.563995\pi\)
−0.199694 + 0.979858i \(0.563995\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.274917 0.0301761 0.0150880 0.999886i \(-0.495197\pi\)
0.0150880 + 0.999886i \(0.495197\pi\)
\(84\) 0 0
\(85\) −17.0997 −1.85472
\(86\) 0 0
\(87\) 2.27492 0.243897
\(88\) 0 0
\(89\) 4.54983 0.482281 0.241141 0.970490i \(-0.422478\pi\)
0.241141 + 0.970490i \(0.422478\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −5.45017 −0.559175
\(96\) 0 0
\(97\) 16.2749 1.65247 0.826234 0.563327i \(-0.190479\pi\)
0.826234 + 0.563327i \(0.190479\pi\)
\(98\) 0 0
\(99\) 4.27492 0.429645
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −11.8248 −1.16513 −0.582564 0.812785i \(-0.697950\pi\)
−0.582564 + 0.812785i \(0.697950\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.82475 −0.659774 −0.329887 0.944020i \(-0.607011\pi\)
−0.329887 + 0.944020i \(0.607011\pi\)
\(108\) 0 0
\(109\) 5.82475 0.557910 0.278955 0.960304i \(-0.410012\pi\)
0.278955 + 0.960304i \(0.410012\pi\)
\(110\) 0 0
\(111\) −5.27492 −0.500673
\(112\) 0 0
\(113\) 10.5498 0.992445 0.496222 0.868195i \(-0.334720\pi\)
0.496222 + 0.868195i \(0.334720\pi\)
\(114\) 0 0
\(115\) 17.0997 1.59455
\(116\) 0 0
\(117\) 1.27492 0.117866
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.27492 0.661356
\(122\) 0 0
\(123\) 10.5498 0.951247
\(124\) 0 0
\(125\) 35.3746 3.16400
\(126\) 0 0
\(127\) −21.5498 −1.91224 −0.956119 0.292978i \(-0.905354\pi\)
−0.956119 + 0.292978i \(0.905354\pi\)
\(128\) 0 0
\(129\) 7.27492 0.640521
\(130\) 0 0
\(131\) 0.274917 0.0240196 0.0120098 0.999928i \(-0.496177\pi\)
0.0120098 + 0.999928i \(0.496177\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.27492 0.367926
\(136\) 0 0
\(137\) 16.5498 1.41395 0.706974 0.707240i \(-0.250060\pi\)
0.706974 + 0.707240i \(0.250060\pi\)
\(138\) 0 0
\(139\) −0.725083 −0.0615007 −0.0307504 0.999527i \(-0.509790\pi\)
−0.0307504 + 0.999527i \(0.509790\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 5.45017 0.455766
\(144\) 0 0
\(145\) 9.72508 0.807624
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.549834 −0.0450442 −0.0225221 0.999746i \(-0.507170\pi\)
−0.0225221 + 0.999746i \(0.507170\pi\)
\(150\) 0 0
\(151\) −15.3746 −1.25117 −0.625583 0.780158i \(-0.715139\pi\)
−0.625583 + 0.780158i \(0.715139\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −4.27492 −0.343370
\(156\) 0 0
\(157\) −14.5498 −1.16120 −0.580602 0.814188i \(-0.697183\pi\)
−0.580602 + 0.814188i \(0.697183\pi\)
\(158\) 0 0
\(159\) −1.72508 −0.136808
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 18.2749 1.42270
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) −11.3746 −0.874968
\(170\) 0 0
\(171\) −1.27492 −0.0974954
\(172\) 0 0
\(173\) 7.45017 0.566426 0.283213 0.959057i \(-0.408600\pi\)
0.283213 + 0.959057i \(0.408600\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.27492 −0.471652
\(178\) 0 0
\(179\) 1.45017 0.108390 0.0541952 0.998530i \(-0.482741\pi\)
0.0541952 + 0.998530i \(0.482741\pi\)
\(180\) 0 0
\(181\) −3.82475 −0.284292 −0.142146 0.989846i \(-0.545400\pi\)
−0.142146 + 0.989846i \(0.545400\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −22.5498 −1.65790
\(186\) 0 0
\(187\) −17.0997 −1.25045
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.5498 −0.763359 −0.381680 0.924295i \(-0.624654\pi\)
−0.381680 + 0.924295i \(0.624654\pi\)
\(192\) 0 0
\(193\) 15.5498 1.11930 0.559651 0.828729i \(-0.310935\pi\)
0.559651 + 0.828729i \(0.310935\pi\)
\(194\) 0 0
\(195\) 5.45017 0.390294
\(196\) 0 0
\(197\) 16.5498 1.17913 0.589563 0.807722i \(-0.299300\pi\)
0.589563 + 0.807722i \(0.299300\pi\)
\(198\) 0 0
\(199\) 25.0997 1.77927 0.889634 0.456674i \(-0.150959\pi\)
0.889634 + 0.456674i \(0.150959\pi\)
\(200\) 0 0
\(201\) −7.27492 −0.513133
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 45.0997 3.14990
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −5.45017 −0.376996
\(210\) 0 0
\(211\) −17.6495 −1.21504 −0.607521 0.794304i \(-0.707836\pi\)
−0.607521 + 0.794304i \(0.707836\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 31.0997 2.12098
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.27492 0.221298
\(220\) 0 0
\(221\) −5.09967 −0.343041
\(222\) 0 0
\(223\) 6.27492 0.420200 0.210100 0.977680i \(-0.432621\pi\)
0.210100 + 0.977680i \(0.432621\pi\)
\(224\) 0 0
\(225\) 13.2749 0.884994
\(226\) 0 0
\(227\) 3.72508 0.247242 0.123621 0.992329i \(-0.460549\pi\)
0.123621 + 0.992329i \(0.460549\pi\)
\(228\) 0 0
\(229\) −10.7251 −0.708733 −0.354367 0.935107i \(-0.615304\pi\)
−0.354367 + 0.935107i \(0.615304\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.5498 0.953191 0.476596 0.879123i \(-0.341871\pi\)
0.476596 + 0.879123i \(0.341871\pi\)
\(234\) 0 0
\(235\) −25.6495 −1.67319
\(236\) 0 0
\(237\) −3.54983 −0.230587
\(238\) 0 0
\(239\) 30.5498 1.97610 0.988052 0.154119i \(-0.0492540\pi\)
0.988052 + 0.154119i \(0.0492540\pi\)
\(240\) 0 0
\(241\) −12.8248 −0.826115 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.62541 −0.103423
\(248\) 0 0
\(249\) 0.274917 0.0174222
\(250\) 0 0
\(251\) −19.3746 −1.22291 −0.611457 0.791278i \(-0.709416\pi\)
−0.611457 + 0.791278i \(0.709416\pi\)
\(252\) 0 0
\(253\) 17.0997 1.07505
\(254\) 0 0
\(255\) −17.0997 −1.07082
\(256\) 0 0
\(257\) −19.0997 −1.19140 −0.595702 0.803205i \(-0.703126\pi\)
−0.595702 + 0.803205i \(0.703126\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.27492 0.140814
\(262\) 0 0
\(263\) −24.5498 −1.51381 −0.756904 0.653526i \(-0.773289\pi\)
−0.756904 + 0.653526i \(0.773289\pi\)
\(264\) 0 0
\(265\) −7.37459 −0.453017
\(266\) 0 0
\(267\) 4.54983 0.278445
\(268\) 0 0
\(269\) −28.2749 −1.72395 −0.861976 0.506949i \(-0.830773\pi\)
−0.861976 + 0.506949i \(0.830773\pi\)
\(270\) 0 0
\(271\) −6.27492 −0.381174 −0.190587 0.981670i \(-0.561039\pi\)
−0.190587 + 0.981670i \(0.561039\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 56.7492 3.42210
\(276\) 0 0
\(277\) 4.17525 0.250866 0.125433 0.992102i \(-0.459968\pi\)
0.125433 + 0.992102i \(0.459968\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −11.4502 −0.683060 −0.341530 0.939871i \(-0.610945\pi\)
−0.341530 + 0.939871i \(0.610945\pi\)
\(282\) 0 0
\(283\) −26.9244 −1.60049 −0.800245 0.599673i \(-0.795297\pi\)
−0.800245 + 0.599673i \(0.795297\pi\)
\(284\) 0 0
\(285\) −5.45017 −0.322840
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 16.2749 0.954053
\(292\) 0 0
\(293\) 5.17525 0.302341 0.151171 0.988508i \(-0.451696\pi\)
0.151171 + 0.988508i \(0.451696\pi\)
\(294\) 0 0
\(295\) −26.8248 −1.56180
\(296\) 0 0
\(297\) 4.27492 0.248056
\(298\) 0 0
\(299\) 5.09967 0.294921
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −42.7492 −2.44781
\(306\) 0 0
\(307\) 26.3746 1.50528 0.752639 0.658434i \(-0.228781\pi\)
0.752639 + 0.658434i \(0.228781\pi\)
\(308\) 0 0
\(309\) −11.8248 −0.672687
\(310\) 0 0
\(311\) 10.5498 0.598226 0.299113 0.954218i \(-0.403309\pi\)
0.299113 + 0.954218i \(0.403309\pi\)
\(312\) 0 0
\(313\) −4.45017 −0.251538 −0.125769 0.992060i \(-0.540140\pi\)
−0.125769 + 0.992060i \(0.540140\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.17525 0.290671 0.145335 0.989382i \(-0.453574\pi\)
0.145335 + 0.989382i \(0.453574\pi\)
\(318\) 0 0
\(319\) 9.72508 0.544500
\(320\) 0 0
\(321\) −6.82475 −0.380920
\(322\) 0 0
\(323\) 5.09967 0.283753
\(324\) 0 0
\(325\) 16.9244 0.938798
\(326\) 0 0
\(327\) 5.82475 0.322110
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −23.8248 −1.30953 −0.654763 0.755834i \(-0.727232\pi\)
−0.654763 + 0.755834i \(0.727232\pi\)
\(332\) 0 0
\(333\) −5.27492 −0.289064
\(334\) 0 0
\(335\) −31.0997 −1.69916
\(336\) 0 0
\(337\) 6.09967 0.332270 0.166135 0.986103i \(-0.446871\pi\)
0.166135 + 0.986103i \(0.446871\pi\)
\(338\) 0 0
\(339\) 10.5498 0.572988
\(340\) 0 0
\(341\) −4.27492 −0.231500
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 17.0997 0.920615
\(346\) 0 0
\(347\) 30.1993 1.62119 0.810593 0.585610i \(-0.199145\pi\)
0.810593 + 0.585610i \(0.199145\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 1.27492 0.0680500
\(352\) 0 0
\(353\) −5.45017 −0.290083 −0.145042 0.989426i \(-0.546332\pi\)
−0.145042 + 0.989426i \(0.546332\pi\)
\(354\) 0 0
\(355\) 8.54983 0.453778
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.6495 1.35373 0.676865 0.736108i \(-0.263338\pi\)
0.676865 + 0.736108i \(0.263338\pi\)
\(360\) 0 0
\(361\) −17.3746 −0.914452
\(362\) 0 0
\(363\) 7.27492 0.381834
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −8.09967 −0.422799 −0.211400 0.977400i \(-0.567802\pi\)
−0.211400 + 0.977400i \(0.567802\pi\)
\(368\) 0 0
\(369\) 10.5498 0.549202
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.27492 −0.0660127 −0.0330064 0.999455i \(-0.510508\pi\)
−0.0330064 + 0.999455i \(0.510508\pi\)
\(374\) 0 0
\(375\) 35.3746 1.82674
\(376\) 0 0
\(377\) 2.90033 0.149375
\(378\) 0 0
\(379\) −35.8248 −1.84019 −0.920097 0.391691i \(-0.871890\pi\)
−0.920097 + 0.391691i \(0.871890\pi\)
\(380\) 0 0
\(381\) −21.5498 −1.10403
\(382\) 0 0
\(383\) −4.54983 −0.232486 −0.116243 0.993221i \(-0.537085\pi\)
−0.116243 + 0.993221i \(0.537085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.27492 0.369805
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0.274917 0.0138677
\(394\) 0 0
\(395\) −15.1752 −0.763550
\(396\) 0 0
\(397\) −34.3746 −1.72521 −0.862606 0.505877i \(-0.831169\pi\)
−0.862606 + 0.505877i \(0.831169\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −1.27492 −0.0635082
\(404\) 0 0
\(405\) 4.27492 0.212422
\(406\) 0 0
\(407\) −22.5498 −1.11775
\(408\) 0 0
\(409\) 25.5498 1.26336 0.631679 0.775230i \(-0.282366\pi\)
0.631679 + 0.775230i \(0.282366\pi\)
\(410\) 0 0
\(411\) 16.5498 0.816343
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.17525 0.0576907
\(416\) 0 0
\(417\) −0.725083 −0.0355075
\(418\) 0 0
\(419\) −13.4502 −0.657084 −0.328542 0.944489i \(-0.606557\pi\)
−0.328542 + 0.944489i \(0.606557\pi\)
\(420\) 0 0
\(421\) 13.8248 0.673777 0.336889 0.941545i \(-0.390625\pi\)
0.336889 + 0.941545i \(0.390625\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) −53.0997 −2.57571
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5.45017 0.263136
\(430\) 0 0
\(431\) 27.6495 1.33183 0.665915 0.746028i \(-0.268041\pi\)
0.665915 + 0.746028i \(0.268041\pi\)
\(432\) 0 0
\(433\) 25.8248 1.24106 0.620529 0.784183i \(-0.286918\pi\)
0.620529 + 0.784183i \(0.286918\pi\)
\(434\) 0 0
\(435\) 9.72508 0.466282
\(436\) 0 0
\(437\) −5.09967 −0.243950
\(438\) 0 0
\(439\) 9.72508 0.464153 0.232076 0.972698i \(-0.425448\pi\)
0.232076 + 0.972698i \(0.425448\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.3746 −1.49065 −0.745326 0.666700i \(-0.767706\pi\)
−0.745326 + 0.666700i \(0.767706\pi\)
\(444\) 0 0
\(445\) 19.4502 0.922026
\(446\) 0 0
\(447\) −0.549834 −0.0260063
\(448\) 0 0
\(449\) 5.45017 0.257209 0.128605 0.991696i \(-0.458950\pi\)
0.128605 + 0.991696i \(0.458950\pi\)
\(450\) 0 0
\(451\) 45.0997 2.12366
\(452\) 0 0
\(453\) −15.3746 −0.722361
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.64950 −0.404607 −0.202303 0.979323i \(-0.564843\pi\)
−0.202303 + 0.979323i \(0.564843\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −41.6495 −1.93981 −0.969905 0.243482i \(-0.921710\pi\)
−0.969905 + 0.243482i \(0.921710\pi\)
\(462\) 0 0
\(463\) 35.8248 1.66492 0.832459 0.554087i \(-0.186933\pi\)
0.832459 + 0.554087i \(0.186933\pi\)
\(464\) 0 0
\(465\) −4.27492 −0.198245
\(466\) 0 0
\(467\) −25.4502 −1.17769 −0.588847 0.808245i \(-0.700418\pi\)
−0.588847 + 0.808245i \(0.700418\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.5498 −0.670421
\(472\) 0 0
\(473\) 31.0997 1.42996
\(474\) 0 0
\(475\) −16.9244 −0.776546
\(476\) 0 0
\(477\) −1.72508 −0.0789861
\(478\) 0 0
\(479\) −20.5498 −0.938946 −0.469473 0.882947i \(-0.655556\pi\)
−0.469473 + 0.882947i \(0.655556\pi\)
\(480\) 0 0
\(481\) −6.72508 −0.306637
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 69.5739 3.15919
\(486\) 0 0
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 15.9244 0.718659 0.359330 0.933211i \(-0.383005\pi\)
0.359330 + 0.933211i \(0.383005\pi\)
\(492\) 0 0
\(493\) −9.09967 −0.409828
\(494\) 0 0
\(495\) 18.2749 0.821396
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −24.7251 −1.10685 −0.553423 0.832900i \(-0.686679\pi\)
−0.553423 + 0.832900i \(0.686679\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 7.64950 0.341074 0.170537 0.985351i \(-0.445450\pi\)
0.170537 + 0.985351i \(0.445450\pi\)
\(504\) 0 0
\(505\) 25.6495 1.14139
\(506\) 0 0
\(507\) −11.3746 −0.505163
\(508\) 0 0
\(509\) 3.72508 0.165111 0.0825557 0.996586i \(-0.473692\pi\)
0.0825557 + 0.996586i \(0.473692\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.27492 −0.0562890
\(514\) 0 0
\(515\) −50.5498 −2.22749
\(516\) 0 0
\(517\) −25.6495 −1.12806
\(518\) 0 0
\(519\) 7.45017 0.327026
\(520\) 0 0
\(521\) 0.549834 0.0240887 0.0120443 0.999927i \(-0.496166\pi\)
0.0120443 + 0.999927i \(0.496166\pi\)
\(522\) 0 0
\(523\) 25.2749 1.10519 0.552597 0.833448i \(-0.313637\pi\)
0.552597 + 0.833448i \(0.313637\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −6.27492 −0.272308
\(532\) 0 0
\(533\) 13.4502 0.582591
\(534\) 0 0
\(535\) −29.1752 −1.26136
\(536\) 0 0
\(537\) 1.45017 0.0625793
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.725083 0.0311737 0.0155869 0.999879i \(-0.495038\pi\)
0.0155869 + 0.999879i \(0.495038\pi\)
\(542\) 0 0
\(543\) −3.82475 −0.164136
\(544\) 0 0
\(545\) 24.9003 1.06661
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −2.90033 −0.123558
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −22.5498 −0.957187
\(556\) 0 0
\(557\) 7.17525 0.304025 0.152013 0.988379i \(-0.451425\pi\)
0.152013 + 0.988379i \(0.451425\pi\)
\(558\) 0 0
\(559\) 9.27492 0.392287
\(560\) 0 0
\(561\) −17.0997 −0.721949
\(562\) 0 0
\(563\) −7.72508 −0.325573 −0.162787 0.986661i \(-0.552048\pi\)
−0.162787 + 0.986661i \(0.552048\pi\)
\(564\) 0 0
\(565\) 45.0997 1.89736
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.5498 1.11303 0.556513 0.830839i \(-0.312139\pi\)
0.556513 + 0.830839i \(0.312139\pi\)
\(570\) 0 0
\(571\) −0.725083 −0.0303438 −0.0151719 0.999885i \(-0.504830\pi\)
−0.0151719 + 0.999885i \(0.504830\pi\)
\(572\) 0 0
\(573\) −10.5498 −0.440726
\(574\) 0 0
\(575\) 53.0997 2.21441
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) 15.5498 0.646229
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −7.37459 −0.305424
\(584\) 0 0
\(585\) 5.45017 0.225337
\(586\) 0 0
\(587\) −1.72508 −0.0712018 −0.0356009 0.999366i \(-0.511335\pi\)
−0.0356009 + 0.999366i \(0.511335\pi\)
\(588\) 0 0
\(589\) 1.27492 0.0525320
\(590\) 0 0
\(591\) 16.5498 0.680769
\(592\) 0 0
\(593\) 14.5498 0.597490 0.298745 0.954333i \(-0.403432\pi\)
0.298745 + 0.954333i \(0.403432\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 25.0997 1.02726
\(598\) 0 0
\(599\) 7.45017 0.304406 0.152203 0.988349i \(-0.451363\pi\)
0.152203 + 0.988349i \(0.451363\pi\)
\(600\) 0 0
\(601\) −26.0997 −1.06463 −0.532314 0.846547i \(-0.678677\pi\)
−0.532314 + 0.846547i \(0.678677\pi\)
\(602\) 0 0
\(603\) −7.27492 −0.296258
\(604\) 0 0
\(605\) 31.0997 1.26438
\(606\) 0 0
\(607\) −7.00000 −0.284121 −0.142061 0.989858i \(-0.545373\pi\)
−0.142061 + 0.989858i \(0.545373\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.64950 −0.309466
\(612\) 0 0
\(613\) −6.54983 −0.264545 −0.132273 0.991213i \(-0.542227\pi\)
−0.132273 + 0.991213i \(0.542227\pi\)
\(614\) 0 0
\(615\) 45.0997 1.81859
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −6.17525 −0.248204 −0.124102 0.992269i \(-0.539605\pi\)
−0.124102 + 0.992269i \(0.539605\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 84.8488 3.39395
\(626\) 0 0
\(627\) −5.45017 −0.217659
\(628\) 0 0
\(629\) 21.0997 0.841299
\(630\) 0 0
\(631\) −2.82475 −0.112452 −0.0562258 0.998418i \(-0.517907\pi\)
−0.0562258 + 0.998418i \(0.517907\pi\)
\(632\) 0 0
\(633\) −17.6495 −0.701505
\(634\) 0 0
\(635\) −92.1238 −3.65582
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 41.6495 1.64506 0.822528 0.568724i \(-0.192563\pi\)
0.822528 + 0.568724i \(0.192563\pi\)
\(642\) 0 0
\(643\) 32.3746 1.27673 0.638365 0.769734i \(-0.279611\pi\)
0.638365 + 0.769734i \(0.279611\pi\)
\(644\) 0 0
\(645\) 31.0997 1.22455
\(646\) 0 0
\(647\) −34.0000 −1.33668 −0.668339 0.743857i \(-0.732994\pi\)
−0.668339 + 0.743857i \(0.732994\pi\)
\(648\) 0 0
\(649\) −26.8248 −1.05296
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −45.9244 −1.79716 −0.898581 0.438808i \(-0.855401\pi\)
−0.898581 + 0.438808i \(0.855401\pi\)
\(654\) 0 0
\(655\) 1.17525 0.0459208
\(656\) 0 0
\(657\) 3.27492 0.127767
\(658\) 0 0
\(659\) −18.1993 −0.708946 −0.354473 0.935066i \(-0.615340\pi\)
−0.354473 + 0.935066i \(0.615340\pi\)
\(660\) 0 0
\(661\) −29.8248 −1.16005 −0.580024 0.814599i \(-0.696957\pi\)
−0.580024 + 0.814599i \(0.696957\pi\)
\(662\) 0 0
\(663\) −5.09967 −0.198055
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.09967 0.352341
\(668\) 0 0
\(669\) 6.27492 0.242602
\(670\) 0 0
\(671\) −42.7492 −1.65031
\(672\) 0 0
\(673\) 26.4502 1.01958 0.509789 0.860299i \(-0.329723\pi\)
0.509789 + 0.860299i \(0.329723\pi\)
\(674\) 0 0
\(675\) 13.2749 0.510952
\(676\) 0 0
\(677\) 5.72508 0.220033 0.110016 0.993930i \(-0.464910\pi\)
0.110016 + 0.993930i \(0.464910\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 3.72508 0.142745
\(682\) 0 0
\(683\) −34.8248 −1.33253 −0.666266 0.745714i \(-0.732108\pi\)
−0.666266 + 0.745714i \(0.732108\pi\)
\(684\) 0 0
\(685\) 70.7492 2.70319
\(686\) 0 0
\(687\) −10.7251 −0.409187
\(688\) 0 0
\(689\) −2.19934 −0.0837881
\(690\) 0 0
\(691\) −11.8248 −0.449835 −0.224917 0.974378i \(-0.572211\pi\)
−0.224917 + 0.974378i \(0.572211\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.09967 −0.117577
\(696\) 0 0
\(697\) −42.1993 −1.59841
\(698\) 0 0
\(699\) 14.5498 0.550325
\(700\) 0 0
\(701\) −39.9244 −1.50792 −0.753962 0.656918i \(-0.771860\pi\)
−0.753962 + 0.656918i \(0.771860\pi\)
\(702\) 0 0
\(703\) 6.72508 0.253641
\(704\) 0 0
\(705\) −25.6495 −0.966016
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −32.1993 −1.20927 −0.604636 0.796502i \(-0.706681\pi\)
−0.604636 + 0.796502i \(0.706681\pi\)
\(710\) 0 0
\(711\) −3.54983 −0.133129
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 23.2990 0.871333
\(716\) 0 0
\(717\) 30.5498 1.14090
\(718\) 0 0
\(719\) −32.1993 −1.20083 −0.600416 0.799688i \(-0.704998\pi\)
−0.600416 + 0.799688i \(0.704998\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −12.8248 −0.476958
\(724\) 0 0
\(725\) 30.1993 1.12158
\(726\) 0 0
\(727\) 16.4502 0.610103 0.305051 0.952336i \(-0.401326\pi\)
0.305051 + 0.952336i \(0.401326\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −29.0997 −1.07629
\(732\) 0 0
\(733\) 46.9244 1.73319 0.866597 0.499010i \(-0.166303\pi\)
0.866597 + 0.499010i \(0.166303\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −31.0997 −1.14557
\(738\) 0 0
\(739\) 36.3746 1.33806 0.669030 0.743235i \(-0.266710\pi\)
0.669030 + 0.743235i \(0.266710\pi\)
\(740\) 0 0
\(741\) −1.62541 −0.0597111
\(742\) 0 0
\(743\) 16.1993 0.594296 0.297148 0.954831i \(-0.403965\pi\)
0.297148 + 0.954831i \(0.403965\pi\)
\(744\) 0 0
\(745\) −2.35050 −0.0861155
\(746\) 0 0
\(747\) 0.274917 0.0100587
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −25.5498 −0.932327 −0.466163 0.884699i \(-0.654364\pi\)
−0.466163 + 0.884699i \(0.654364\pi\)
\(752\) 0 0
\(753\) −19.3746 −0.706049
\(754\) 0 0
\(755\) −65.7251 −2.39198
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 17.0997 0.620679
\(760\) 0 0
\(761\) 5.09967 0.184863 0.0924314 0.995719i \(-0.470536\pi\)
0.0924314 + 0.995719i \(0.470536\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −17.0997 −0.618240
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 12.6495 0.456153 0.228076 0.973643i \(-0.426756\pi\)
0.228076 + 0.973643i \(0.426756\pi\)
\(770\) 0 0
\(771\) −19.0997 −0.687858
\(772\) 0 0
\(773\) 27.0997 0.974707 0.487354 0.873205i \(-0.337962\pi\)
0.487354 + 0.873205i \(0.337962\pi\)
\(774\) 0 0
\(775\) −13.2749 −0.476849
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.4502 −0.481902
\(780\) 0 0
\(781\) 8.54983 0.305937
\(782\) 0 0
\(783\) 2.27492 0.0812989
\(784\) 0 0
\(785\) −62.1993 −2.21999
\(786\) 0 0
\(787\) 12.5498 0.447353 0.223677 0.974663i \(-0.428194\pi\)
0.223677 + 0.974663i \(0.428194\pi\)
\(788\) 0 0
\(789\) −24.5498 −0.873997
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.7492 −0.452736
\(794\) 0 0
\(795\) −7.37459 −0.261550
\(796\) 0 0
\(797\) −36.4743 −1.29198 −0.645992 0.763344i \(-0.723556\pi\)
−0.645992 + 0.763344i \(0.723556\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 4.54983 0.160760
\(802\) 0 0
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −28.2749 −0.995324
\(808\) 0 0
\(809\) 15.6495 0.550207 0.275104 0.961415i \(-0.411288\pi\)
0.275104 + 0.961415i \(0.411288\pi\)
\(810\) 0 0
\(811\) 27.4502 0.963906 0.481953 0.876197i \(-0.339928\pi\)
0.481953 + 0.876197i \(0.339928\pi\)
\(812\) 0 0
\(813\) −6.27492 −0.220071
\(814\) 0 0
\(815\) 51.2990 1.79693
\(816\) 0 0
\(817\) −9.27492 −0.324488
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.2749 −0.707599 −0.353800 0.935321i \(-0.615111\pi\)
−0.353800 + 0.935321i \(0.615111\pi\)
\(822\) 0 0
\(823\) 26.1993 0.913252 0.456626 0.889659i \(-0.349058\pi\)
0.456626 + 0.889659i \(0.349058\pi\)
\(824\) 0 0
\(825\) 56.7492 1.97575
\(826\) 0 0
\(827\) −39.0241 −1.35700 −0.678500 0.734600i \(-0.737370\pi\)
−0.678500 + 0.734600i \(0.737370\pi\)
\(828\) 0 0
\(829\) −22.7251 −0.789275 −0.394637 0.918837i \(-0.629130\pi\)
−0.394637 + 0.918837i \(0.629130\pi\)
\(830\) 0 0
\(831\) 4.17525 0.144838
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −25.6495 −0.887638
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 30.1993 1.04260 0.521298 0.853374i \(-0.325448\pi\)
0.521298 + 0.853374i \(0.325448\pi\)
\(840\) 0 0
\(841\) −23.8248 −0.821543
\(842\) 0 0
\(843\) −11.4502 −0.394365
\(844\) 0 0
\(845\) −48.6254 −1.67277
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −26.9244 −0.924044
\(850\) 0 0
\(851\) −21.0997 −0.723287
\(852\) 0 0
\(853\) 24.3746 0.834570 0.417285 0.908776i \(-0.362982\pi\)
0.417285 + 0.908776i \(0.362982\pi\)
\(854\) 0 0
\(855\) −5.45017 −0.186392
\(856\) 0 0
\(857\) −28.5498 −0.975244 −0.487622 0.873055i \(-0.662136\pi\)
−0.487622 + 0.873055i \(0.662136\pi\)
\(858\) 0 0
\(859\) 10.3505 0.353154 0.176577 0.984287i \(-0.443497\pi\)
0.176577 + 0.984287i \(0.443497\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.0997 −0.377837 −0.188919 0.981993i \(-0.560498\pi\)
−0.188919 + 0.981993i \(0.560498\pi\)
\(864\) 0 0
\(865\) 31.8488 1.08289
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −15.1752 −0.514785
\(870\) 0 0
\(871\) −9.27492 −0.314269
\(872\) 0 0
\(873\) 16.2749 0.550822
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.7492 0.970791 0.485395 0.874295i \(-0.338676\pi\)
0.485395 + 0.874295i \(0.338676\pi\)
\(878\) 0 0
\(879\) 5.17525 0.174557
\(880\) 0 0
\(881\) 24.5498 0.827105 0.413552 0.910480i \(-0.364288\pi\)
0.413552 + 0.910480i \(0.364288\pi\)
\(882\) 0 0
\(883\) 30.3746 1.02219 0.511093 0.859525i \(-0.329241\pi\)
0.511093 + 0.859525i \(0.329241\pi\)
\(884\) 0 0
\(885\) −26.8248 −0.901704
\(886\) 0 0
\(887\) −11.6495 −0.391152 −0.195576 0.980689i \(-0.562658\pi\)
−0.195576 + 0.980689i \(0.562658\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.27492 0.143215
\(892\) 0 0
\(893\) 7.64950 0.255981
\(894\) 0 0
\(895\) 6.19934 0.207221
\(896\) 0 0
\(897\) 5.09967 0.170273
\(898\) 0 0
\(899\) −2.27492 −0.0758727
\(900\) 0 0
\(901\) 6.90033 0.229883
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16.3505 −0.543509
\(906\) 0 0
\(907\) 37.2749 1.23769 0.618847 0.785512i \(-0.287600\pi\)
0.618847 + 0.785512i \(0.287600\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −51.8488 −1.71783 −0.858914 0.512119i \(-0.828861\pi\)
−0.858914 + 0.512119i \(0.828861\pi\)
\(912\) 0 0
\(913\) 1.17525 0.0388950
\(914\) 0 0
\(915\) −42.7492 −1.41324
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 19.8248 0.653958 0.326979 0.945032i \(-0.393969\pi\)
0.326979 + 0.945032i \(0.393969\pi\)
\(920\) 0 0
\(921\) 26.3746 0.869072
\(922\) 0 0
\(923\) 2.54983 0.0839288
\(924\) 0 0
\(925\) −70.0241 −2.30238
\(926\) 0 0
\(927\) −11.8248 −0.388376
\(928\) 0 0
\(929\) 20.1993 0.662719 0.331359 0.943505i \(-0.392493\pi\)
0.331359 + 0.943505i \(0.392493\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 10.5498 0.345386
\(934\) 0 0
\(935\) −73.0997 −2.39061
\(936\) 0 0
\(937\) 24.0997 0.787302 0.393651 0.919260i \(-0.371212\pi\)
0.393651 + 0.919260i \(0.371212\pi\)
\(938\) 0 0
\(939\) −4.45017 −0.145226
\(940\) 0 0
\(941\) −29.1752 −0.951086 −0.475543 0.879693i \(-0.657748\pi\)
−0.475543 + 0.879693i \(0.657748\pi\)
\(942\) 0 0
\(943\) 42.1993 1.37420
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 34.5498 1.12272 0.561359 0.827572i \(-0.310279\pi\)
0.561359 + 0.827572i \(0.310279\pi\)
\(948\) 0 0
\(949\) 4.17525 0.135534
\(950\) 0 0
\(951\) 5.17525 0.167819
\(952\) 0 0
\(953\) 10.3505 0.335285 0.167643 0.985848i \(-0.446384\pi\)
0.167643 + 0.985848i \(0.446384\pi\)
\(954\) 0 0
\(955\) −45.0997 −1.45939
\(956\) 0 0
\(957\) 9.72508 0.314367
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −6.82475 −0.219925
\(964\) 0 0
\(965\) 66.4743 2.13988
\(966\) 0 0
\(967\) −38.4502 −1.23647 −0.618237 0.785992i \(-0.712153\pi\)
−0.618237 + 0.785992i \(0.712153\pi\)
\(968\) 0 0
\(969\) 5.09967 0.163825
\(970\) 0 0
\(971\) 41.5739 1.33417 0.667085 0.744981i \(-0.267542\pi\)
0.667085 + 0.744981i \(0.267542\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 16.9244 0.542015
\(976\) 0 0
\(977\) 39.0997 1.25091 0.625455 0.780261i \(-0.284914\pi\)
0.625455 + 0.780261i \(0.284914\pi\)
\(978\) 0 0
\(979\) 19.4502 0.621630
\(980\) 0 0
\(981\) 5.82475 0.185970
\(982\) 0 0
\(983\) −51.2990 −1.63618 −0.818092 0.575087i \(-0.804968\pi\)
−0.818092 + 0.575087i \(0.804968\pi\)
\(984\) 0 0
\(985\) 70.7492 2.25426
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29.0997 0.925316
\(990\) 0 0
\(991\) 16.0997 0.511423 0.255711 0.966753i \(-0.417690\pi\)
0.255711 + 0.966753i \(0.417690\pi\)
\(992\) 0 0
\(993\) −23.8248 −0.756056
\(994\) 0 0
\(995\) 107.299 3.40161
\(996\) 0 0
\(997\) 22.1752 0.702297 0.351149 0.936320i \(-0.385791\pi\)
0.351149 + 0.936320i \(0.385791\pi\)
\(998\) 0 0
\(999\) −5.27492 −0.166891
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 9408.2.a.ec.1.2 2
4.3 odd 2 9408.2.a.dp.1.2 2
7.2 even 3 1344.2.q.w.193.1 4
7.4 even 3 1344.2.q.w.961.1 4
7.6 odd 2 9408.2.a.dj.1.1 2
8.3 odd 2 2352.2.a.bf.1.1 2
8.5 even 2 1176.2.a.k.1.1 2
24.5 odd 2 3528.2.a.bk.1.2 2
24.11 even 2 7056.2.a.cu.1.2 2
28.11 odd 6 1344.2.q.x.961.1 4
28.23 odd 6 1344.2.q.x.193.1 4
28.27 even 2 9408.2.a.dw.1.1 2
56.3 even 6 2352.2.q.bf.961.1 4
56.5 odd 6 1176.2.q.l.361.1 4
56.11 odd 6 336.2.q.g.289.2 4
56.13 odd 2 1176.2.a.n.1.2 2
56.19 even 6 2352.2.q.bf.1537.1 4
56.27 even 2 2352.2.a.ba.1.2 2
56.37 even 6 168.2.q.c.25.2 4
56.45 odd 6 1176.2.q.l.961.1 4
56.51 odd 6 336.2.q.g.193.2 4
56.53 even 6 168.2.q.c.121.2 yes 4
168.5 even 6 3528.2.s.bk.361.2 4
168.11 even 6 1008.2.s.r.289.1 4
168.53 odd 6 504.2.s.i.289.1 4
168.83 odd 2 7056.2.a.ch.1.1 2
168.101 even 6 3528.2.s.bk.3313.2 4
168.107 even 6 1008.2.s.r.865.1 4
168.125 even 2 3528.2.a.bd.1.1 2
168.149 odd 6 504.2.s.i.361.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.c.25.2 4 56.37 even 6
168.2.q.c.121.2 yes 4 56.53 even 6
336.2.q.g.193.2 4 56.51 odd 6
336.2.q.g.289.2 4 56.11 odd 6
504.2.s.i.289.1 4 168.53 odd 6
504.2.s.i.361.1 4 168.149 odd 6
1008.2.s.r.289.1 4 168.11 even 6
1008.2.s.r.865.1 4 168.107 even 6
1176.2.a.k.1.1 2 8.5 even 2
1176.2.a.n.1.2 2 56.13 odd 2
1176.2.q.l.361.1 4 56.5 odd 6
1176.2.q.l.961.1 4 56.45 odd 6
1344.2.q.w.193.1 4 7.2 even 3
1344.2.q.w.961.1 4 7.4 even 3
1344.2.q.x.193.1 4 28.23 odd 6
1344.2.q.x.961.1 4 28.11 odd 6
2352.2.a.ba.1.2 2 56.27 even 2
2352.2.a.bf.1.1 2 8.3 odd 2
2352.2.q.bf.961.1 4 56.3 even 6
2352.2.q.bf.1537.1 4 56.19 even 6
3528.2.a.bd.1.1 2 168.125 even 2
3528.2.a.bk.1.2 2 24.5 odd 2
3528.2.s.bk.361.2 4 168.5 even 6
3528.2.s.bk.3313.2 4 168.101 even 6
7056.2.a.ch.1.1 2 168.83 odd 2
7056.2.a.cu.1.2 2 24.11 even 2
9408.2.a.dj.1.1 2 7.6 odd 2
9408.2.a.dp.1.2 2 4.3 odd 2
9408.2.a.dw.1.1 2 28.27 even 2
9408.2.a.ec.1.2 2 1.1 even 1 trivial