Properties

Label 1216.2.a.a.1.1
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} -2.00000 q^{11} +1.00000 q^{13} +3.00000 q^{17} +1.00000 q^{19} +3.00000 q^{21} +3.00000 q^{23} -5.00000 q^{25} -9.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} +6.00000 q^{33} +10.0000 q^{37} -3.00000 q^{39} -12.0000 q^{41} -8.00000 q^{43} -8.00000 q^{47} -6.00000 q^{49} -9.00000 q^{51} +9.00000 q^{53} -3.00000 q^{57} +5.00000 q^{59} -10.0000 q^{61} -6.00000 q^{63} -7.00000 q^{67} -9.00000 q^{69} -10.0000 q^{71} +1.00000 q^{73} +15.0000 q^{75} +2.00000 q^{77} -14.0000 q^{79} +9.00000 q^{81} -6.00000 q^{83} +9.00000 q^{87} -4.00000 q^{89} -1.00000 q^{91} -24.0000 q^{93} -6.00000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 15.0000 1.73205
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −24.0000 −2.48868
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.00000 −0.483368 −0.241684 0.970355i \(-0.577700\pi\)
−0.241684 + 0.970355i \(0.577700\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) −30.0000 −2.84747
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 36.0000 3.24601
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 18.0000 1.48461
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 18.0000 1.45521
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −27.0000 −2.14124
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) −15.0000 −1.12747
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 30.0000 2.21766
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) −21.0000 −1.51951 −0.759753 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 21.0000 1.48123
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 0 0
\(213\) 30.0000 2.05557
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) −30.0000 −2.00000
\(226\) 0 0
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 42.0000 2.72819
\(238\) 0 0
\(239\) 27.0000 1.74648 0.873242 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) 18.0000 1.14070
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 10.0000 0.603023
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 0 0
\(279\) 48.0000 2.87368
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 0 0
\(293\) −1.00000 −0.0584206 −0.0292103 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.0000 1.04447
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 30.0000 1.72345
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −42.0000 −2.38930
\(310\) 0 0
\(311\) −5.00000 −0.283524 −0.141762 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 0 0
\(313\) −3.00000 −0.169570 −0.0847850 0.996399i \(-0.527020\pi\)
−0.0847850 + 0.996399i \(0.527020\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −5.00000 −0.277350
\(326\) 0 0
\(327\) 27.0000 1.49310
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 0 0
\(333\) 60.0000 3.28798
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) 9.00000 0.479022 0.239511 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.0000 1.10221
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) −72.0000 −3.74817
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 0 0
\(381\) −54.0000 −2.76650
\(382\) 0 0
\(383\) −38.0000 −1.94171 −0.970855 0.239669i \(-0.922961\pi\)
−0.970855 + 0.239669i \(0.922961\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −48.0000 −2.43998
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) −24.0000 −1.21064
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) 51.0000 2.51564
\(412\) 0 0
\(413\) −5.00000 −0.246034
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −48.0000 −2.35057
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 37.0000 1.80327 0.901635 0.432498i \(-0.142368\pi\)
0.901635 + 0.432498i \(0.142368\pi\)
\(422\) 0 0
\(423\) −48.0000 −2.33384
\(424\) 0 0
\(425\) −15.0000 −0.727607
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −26.0000 −1.25238 −0.626188 0.779672i \(-0.715386\pi\)
−0.626188 + 0.779672i \(0.715386\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) −30.0000 −1.40952
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 0 0
\(459\) −27.0000 −1.26025
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 54.0000 2.47249
\(478\) 0 0
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 9.00000 0.409514
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 0 0
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.0000 0.448561
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 48.0000 2.14448
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.0000 1.59882
\(508\) 0 0
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −1.00000 −0.0442374
\(512\) 0 0
\(513\) −9.00000 −0.397360
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −16.0000 −0.700973 −0.350486 0.936568i \(-0.613984\pi\)
−0.350486 + 0.936568i \(0.613984\pi\)
\(522\) 0 0
\(523\) −1.00000 −0.0437269 −0.0218635 0.999761i \(-0.506960\pi\)
−0.0218635 + 0.999761i \(0.506960\pi\)
\(524\) 0 0
\(525\) −15.0000 −0.654654
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 30.0000 1.30189
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 72.0000 3.10703
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) −54.0000 −2.31736
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −60.0000 −2.56074
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.0000 1.35588 0.677942 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 63.0000 2.63186
\(574\) 0 0
\(575\) −15.0000 −0.625543
\(576\) 0 0
\(577\) 43.0000 1.79011 0.895057 0.445952i \(-0.147135\pi\)
0.895057 + 0.445952i \(0.147135\pi\)
\(578\) 0 0
\(579\) −30.0000 −1.24676
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −33.0000 −1.35060
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 40.0000 1.63163 0.815817 0.578310i \(-0.196288\pi\)
0.815817 + 0.578310i \(0.196288\pi\)
\(602\) 0 0
\(603\) −42.0000 −1.71037
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) −27.0000 −1.08347
\(622\) 0 0
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) −60.0000 −2.37356
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) −50.0000 −1.97181 −0.985904 0.167313i \(-0.946491\pi\)
−0.985904 + 0.167313i \(0.946491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 24.0000 0.940634
\(652\) 0 0
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 31.0000 1.20759 0.603794 0.797140i \(-0.293655\pi\)
0.603794 + 0.797140i \(0.293655\pi\)
\(660\) 0 0
\(661\) 39.0000 1.51692 0.758462 0.651717i \(-0.225951\pi\)
0.758462 + 0.651717i \(0.225951\pi\)
\(662\) 0 0
\(663\) −9.00000 −0.349531
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) 0 0
\(675\) 45.0000 1.73205
\(676\) 0 0
\(677\) 11.0000 0.422764 0.211382 0.977403i \(-0.432204\pi\)
0.211382 + 0.977403i \(0.432204\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) −63.0000 −2.41417
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 66.0000 2.51806
\(688\) 0 0
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) −78.0000 −2.95023
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −84.0000 −3.15025
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −81.0000 −3.02500
\(718\) 0 0
\(719\) 15.0000 0.559406 0.279703 0.960087i \(-0.409764\pi\)
0.279703 + 0.960087i \(0.409764\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) −60.0000 −2.23142
\(724\) 0 0
\(725\) 15.0000 0.557086
\(726\) 0 0
\(727\) 11.0000 0.407967 0.203984 0.978974i \(-0.434611\pi\)
0.203984 + 0.978974i \(0.434611\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 24.0000 0.886460 0.443230 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.0000 0.515697
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 0 0
\(741\) −3.00000 −0.110208
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 5.00000 0.182696
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −29.0000 −1.05125 −0.525625 0.850717i \(-0.676168\pi\)
−0.525625 + 0.850717i \(0.676168\pi\)
\(762\) 0 0
\(763\) 9.00000 0.325822
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.00000 0.180540
\(768\) 0 0
\(769\) 13.0000 0.468792 0.234396 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 0 0
\(773\) 31.0000 1.11499 0.557496 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) 0 0
\(777\) 30.0000 1.07624
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 0 0
\(783\) 27.0000 0.964901
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 53.0000 1.88925 0.944623 0.328158i \(-0.106428\pi\)
0.944623 + 0.328158i \(0.106428\pi\)
\(788\) 0 0
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.0000 1.31061 0.655304 0.755366i \(-0.272541\pi\)
0.655304 + 0.755366i \(0.272541\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −24.0000 −0.847998
\(802\) 0 0
\(803\) −2.00000 −0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 66.0000 2.32331
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 39.0000 1.36948 0.684738 0.728790i \(-0.259917\pi\)
0.684738 + 0.728790i \(0.259917\pi\)
\(812\) 0 0
\(813\) −9.00000 −0.315644
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) 41.0000 1.42917 0.714585 0.699549i \(-0.246616\pi\)
0.714585 + 0.699549i \(0.246616\pi\)
\(824\) 0 0
\(825\) −30.0000 −1.04447
\(826\) 0 0
\(827\) −51.0000 −1.77344 −0.886722 0.462303i \(-0.847023\pi\)
−0.886722 + 0.462303i \(0.847023\pi\)
\(828\) 0 0
\(829\) 23.0000 0.798823 0.399412 0.916772i \(-0.369214\pi\)
0.399412 + 0.916772i \(0.369214\pi\)
\(830\) 0 0
\(831\) 36.0000 1.24883
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −72.0000 −2.48868
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.0000 0.683187 0.341593 0.939848i \(-0.389033\pi\)
0.341593 + 0.939848i \(0.389033\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 0 0
\(861\) −36.0000 −1.22688
\(862\) 0 0
\(863\) 38.0000 1.29354 0.646768 0.762687i \(-0.276120\pi\)
0.646768 + 0.762687i \(0.276120\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 24.0000 0.815083
\(868\) 0 0
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 0 0
\(873\) −36.0000 −1.21842
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −37.0000 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(878\) 0 0
\(879\) 3.00000 0.101187
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.0000 −0.335767 −0.167884 0.985807i \(-0.553693\pi\)
−0.167884 + 0.985807i \(0.553693\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −9.00000 −0.300501
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 27.0000 0.899500
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −33.0000 −1.09575 −0.547874 0.836561i \(-0.684562\pi\)
−0.547874 + 0.836561i \(0.684562\pi\)
\(908\) 0 0
\(909\) −60.0000 −1.99007
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) 49.0000 1.61636 0.808180 0.588935i \(-0.200453\pi\)
0.808180 + 0.588935i \(0.200453\pi\)
\(920\) 0 0
\(921\) 36.0000 1.18624
\(922\) 0 0
\(923\) −10.0000 −0.329154
\(924\) 0 0
\(925\) −50.0000 −1.64399
\(926\) 0 0
\(927\) 84.0000 2.75892
\(928\) 0 0
\(929\) −39.0000 −1.27955 −0.639774 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 15.0000 0.491078
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −23.0000 −0.751377 −0.375689 0.926746i \(-0.622594\pi\)
−0.375689 + 0.926746i \(0.622594\pi\)
\(938\) 0 0
\(939\) 9.00000 0.293704
\(940\) 0 0
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.0000 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) 0 0
\(949\) 1.00000 0.0324614
\(950\) 0 0
\(951\) 63.0000 2.04291
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −18.0000 −0.581857
\(958\) 0 0
\(959\) 17.0000 0.548959
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −30.0000 −0.966736
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) −9.00000 −0.289122
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) 15.0000 0.480384
\(976\) 0 0
\(977\) 16.0000 0.511885 0.255943 0.966692i \(-0.417614\pi\)
0.255943 + 0.966692i \(0.417614\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −54.0000 −1.72409
\(982\) 0 0
\(983\) −14.0000 −0.446531 −0.223265 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 0 0
\(993\) 39.0000 1.23763
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) 0 0
\(999\) −90.0000 −2.84747
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 1216.2.a.a.1.1 1
4.3 odd 2 1216.2.a.r.1.1 1
8.3 odd 2 608.2.a.a.1.1 1
8.5 even 2 608.2.a.f.1.1 yes 1
24.5 odd 2 5472.2.a.i.1.1 1
24.11 even 2 5472.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.a.1.1 1 8.3 odd 2
608.2.a.f.1.1 yes 1 8.5 even 2
1216.2.a.a.1.1 1 1.1 even 1 trivial
1216.2.a.r.1.1 1 4.3 odd 2
5472.2.a.i.1.1 1 24.5 odd 2
5472.2.a.l.1.1 1 24.11 even 2