Properties

Label 882.2.a.d.1.1
Level $882$
Weight $2$
Character 882.1
Self dual yes
Analytic conductor $7.043$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.04280545828\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{16} -4.00000 q^{17} -8.00000 q^{19} +1.00000 q^{20} +5.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} +5.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{37} +8.00000 q^{38} -1.00000 q^{40} +2.00000 q^{43} -5.00000 q^{44} -4.00000 q^{46} -6.00000 q^{47} +4.00000 q^{50} +9.00000 q^{53} -5.00000 q^{55} -5.00000 q^{58} -11.0000 q^{59} +6.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} -4.00000 q^{68} -2.00000 q^{71} -10.0000 q^{73} +4.00000 q^{74} -8.00000 q^{76} +3.00000 q^{79} +1.00000 q^{80} -7.00000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +5.00000 q^{88} -6.00000 q^{89} +4.00000 q^{92} +6.00000 q^{94} -8.00000 q^{95} -7.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 5.00000 0.476731
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 11.0000 1.01263
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 9.00000 0.798621 0.399310 0.916816i \(-0.369250\pi\)
0.399310 + 0.916816i \(0.369250\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 1.00000 0.0873704 0.0436852 0.999045i \(-0.486090\pi\)
0.0436852 + 0.999045i \(0.486090\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 8.00000 0.648886
\(153\) 0 0
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −3.00000 −0.238667
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 20.0000 1.46254
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 7.00000 0.502571
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 0 0
\(209\) 40.0000 2.76686
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −5.00000 −0.337100
\(221\) 0 0
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 16.0000 1.06430
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −11.0000 −0.716039
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) −9.00000 −0.564710
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.00000 −0.0617802
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 31.0000 1.89010 0.945052 0.326921i \(-0.106011\pi\)
0.945052 + 0.326921i \(0.106011\pi\)
\(270\) 0 0
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −14.0000 −0.839664
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −5.00000 −0.293610
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) −11.0000 −0.640445
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −19.0000 −1.09333
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3.00000 0.170389
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) −3.00000 −0.168497 −0.0842484 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 0 0
\(319\) −25.0000 −1.39973
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 32.0000 1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −7.00000 −0.384175
\(333\) 0 0
\(334\) 14.0000 0.766046
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −17.0000 −0.887393 −0.443696 0.896177i \(-0.646333\pi\)
−0.443696 + 0.896177i \(0.646333\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) −20.0000 −1.03418
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 24.0000 1.22795
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) 0 0
\(388\) −7.00000 −0.355371
\(389\) 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 0
\(394\) 2.00000 0.100759
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) −36.0000 −1.80679 −0.903394 0.428811i \(-0.858933\pi\)
−0.903394 + 0.428811i \(0.858933\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −7.00000 −0.343616
\(416\) 0 0
\(417\) 0 0
\(418\) −40.0000 −1.95646
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 5.00000 0.238366
\(441\) 0 0
\(442\) 0 0
\(443\) −17.0000 −0.807694 −0.403847 0.914826i \(-0.632327\pi\)
−0.403847 + 0.914826i \(0.632327\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) −7.00000 −0.331460
\(447\) 0 0
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −16.0000 −0.752577
\(453\) 0 0
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) −20.0000 −0.934539
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) 11.0000 0.506316
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) 38.0000 1.73626 0.868132 0.496333i \(-0.165321\pi\)
0.868132 + 0.496333i \(0.165321\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −25.0000 −1.13872
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) 5.00000 0.226572 0.113286 0.993562i \(-0.463862\pi\)
0.113286 + 0.993562i \(0.463862\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) −21.0000 −0.937276
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 20.0000 0.889108
\(507\) 0 0
\(508\) 9.00000 0.399310
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 1.00000 0.0436852
\(525\) 0 0
\(526\) −30.0000 −1.30806
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) −31.0000 −1.33650
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 15.0000 0.644305
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 2.00000 0.0854358
\(549\) 0 0
\(550\) −20.0000 −0.852803
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) 0 0
\(554\) 16.0000 0.679775
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 23.0000 0.974541 0.487271 0.873251i \(-0.337993\pi\)
0.487271 + 0.873251i \(0.337993\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 17.0000 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −30.0000 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −31.0000 −1.29055 −0.645273 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 5.00000 0.207614
\(581\) 0 0
\(582\) 0 0
\(583\) −45.0000 −1.86371
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) 35.0000 1.44460 0.722302 0.691577i \(-0.243084\pi\)
0.722302 + 0.691577i \(0.243084\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 11.0000 0.452863
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 19.0000 0.773099
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 27.0000 1.09590 0.547948 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) 0 0
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −3.00000 −0.120483
\(621\) 0 0
\(622\) 32.0000 1.28308
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) −3.00000 −0.119334
\(633\) 0 0
\(634\) 3.00000 0.119145
\(635\) 9.00000 0.357154
\(636\) 0 0
\(637\) 0 0
\(638\) 25.0000 0.989759
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −32.0000 −1.25902
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 55.0000 2.15894
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) 1.00000 0.0390732
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) −14.0000 −0.541676
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) −9.00000 −0.346667
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) −15.0000 −0.574380
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 2.00000 0.0750587
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 10.0000 0.373197
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 17.0000 0.627481
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) 20.0000 0.731272
\(749\) 0 0
\(750\) 0 0
\(751\) 45.0000 1.64207 0.821037 0.570875i \(-0.193396\pi\)
0.821037 + 0.570875i \(0.193396\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 34.0000 1.22847
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 7.00000 0.251285
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) 0 0
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) −3.00000 −0.106735
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 36.0000 1.27759
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 24.0000 0.847469
\(803\) 50.0000 1.76446
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) −40.0000 −1.40633 −0.703163 0.711029i \(-0.748229\pi\)
−0.703163 + 0.711029i \(0.748229\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −20.0000 −0.701000
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −25.0000 −0.874105
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000 0.872506 0.436253 0.899824i \(-0.356305\pi\)
0.436253 + 0.899824i \(0.356305\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 0 0
\(829\) 32.0000 1.11141 0.555703 0.831381i \(-0.312449\pi\)
0.555703 + 0.831381i \(0.312449\pi\)
\(830\) 7.00000 0.242974
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −14.0000 −0.484490
\(836\) 40.0000 1.38343
\(837\) 0 0
\(838\) 0 0
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −30.0000 −1.03387
\(843\) 0 0
\(844\) 2.00000 0.0688428
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) −16.0000 −0.548795
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 15.0000 0.506225
\(879\) 0 0
\(880\) −5.00000 −0.168550
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 17.0000 0.571126
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 7.00000 0.234377
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 16.0000 0.533927
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 16.0000 0.532152
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 3.00000 0.0995585
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 35.0000 1.15833
\(914\) −31.0000 −1.02539
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 0 0
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) 16.0000 0.526077
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 4.00000 0.131024
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 20.0000 0.654070
\(936\) 0 0
\(937\) −35.0000 −1.14340 −0.571700 0.820463i \(-0.693716\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) −11.0000 −0.358590 −0.179295 0.983795i \(-0.557382\pi\)
−0.179295 + 0.983795i \(0.557382\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −11.0000 −0.358020
\(945\) 0 0
\(946\) 10.0000 0.325128
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −32.0000 −1.03822
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −38.0000 −1.22772
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 0 0
\(964\) 25.0000 0.805196
\(965\) 5.00000 0.160956
\(966\) 0 0
\(967\) −61.0000 −1.96163 −0.980814 0.194946i \(-0.937547\pi\)
−0.980814 + 0.194946i \(0.937547\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) 7.00000 0.224756
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −5.00000 −0.160210
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) 9.00000 0.287202
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 20.0000 0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 3.00000 0.0952501
\(993\) 0 0
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −10.0000 −0.316544
\(999\) 0 0
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 882.2.a.d.1.1 1
3.2 odd 2 294.2.a.f.1.1 1
4.3 odd 2 7056.2.a.bl.1.1 1
7.2 even 3 882.2.g.i.361.1 2
7.3 odd 6 126.2.g.c.37.1 2
7.4 even 3 882.2.g.i.667.1 2
7.5 odd 6 126.2.g.c.109.1 2
7.6 odd 2 882.2.a.c.1.1 1
12.11 even 2 2352.2.a.f.1.1 1
15.14 odd 2 7350.2.a.q.1.1 1
21.2 odd 6 294.2.e.b.67.1 2
21.5 even 6 42.2.e.a.25.1 2
21.11 odd 6 294.2.e.b.79.1 2
21.17 even 6 42.2.e.a.37.1 yes 2
21.20 even 2 294.2.a.e.1.1 1
24.5 odd 2 9408.2.a.z.1.1 1
24.11 even 2 9408.2.a.cr.1.1 1
28.3 even 6 1008.2.s.k.289.1 2
28.19 even 6 1008.2.s.k.865.1 2
28.27 even 2 7056.2.a.w.1.1 1
63.5 even 6 1134.2.e.l.865.1 2
63.31 odd 6 1134.2.h.l.541.1 2
63.38 even 6 1134.2.e.l.919.1 2
63.40 odd 6 1134.2.e.e.865.1 2
63.47 even 6 1134.2.h.e.109.1 2
63.52 odd 6 1134.2.e.e.919.1 2
63.59 even 6 1134.2.h.e.541.1 2
63.61 odd 6 1134.2.h.l.109.1 2
84.11 even 6 2352.2.q.u.961.1 2
84.23 even 6 2352.2.q.u.1537.1 2
84.47 odd 6 336.2.q.b.193.1 2
84.59 odd 6 336.2.q.b.289.1 2
84.83 odd 2 2352.2.a.t.1.1 1
105.17 odd 12 1050.2.o.a.499.1 4
105.38 odd 12 1050.2.o.a.499.2 4
105.47 odd 12 1050.2.o.a.949.2 4
105.59 even 6 1050.2.i.l.751.1 2
105.68 odd 12 1050.2.o.a.949.1 4
105.89 even 6 1050.2.i.l.151.1 2
105.104 even 2 7350.2.a.bl.1.1 1
168.5 even 6 1344.2.q.g.193.1 2
168.59 odd 6 1344.2.q.s.961.1 2
168.83 odd 2 9408.2.a.q.1.1 1
168.101 even 6 1344.2.q.g.961.1 2
168.125 even 2 9408.2.a.ce.1.1 1
168.131 odd 6 1344.2.q.s.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.a.25.1 2 21.5 even 6
42.2.e.a.37.1 yes 2 21.17 even 6
126.2.g.c.37.1 2 7.3 odd 6
126.2.g.c.109.1 2 7.5 odd 6
294.2.a.e.1.1 1 21.20 even 2
294.2.a.f.1.1 1 3.2 odd 2
294.2.e.b.67.1 2 21.2 odd 6
294.2.e.b.79.1 2 21.11 odd 6
336.2.q.b.193.1 2 84.47 odd 6
336.2.q.b.289.1 2 84.59 odd 6
882.2.a.c.1.1 1 7.6 odd 2
882.2.a.d.1.1 1 1.1 even 1 trivial
882.2.g.i.361.1 2 7.2 even 3
882.2.g.i.667.1 2 7.4 even 3
1008.2.s.k.289.1 2 28.3 even 6
1008.2.s.k.865.1 2 28.19 even 6
1050.2.i.l.151.1 2 105.89 even 6
1050.2.i.l.751.1 2 105.59 even 6
1050.2.o.a.499.1 4 105.17 odd 12
1050.2.o.a.499.2 4 105.38 odd 12
1050.2.o.a.949.1 4 105.68 odd 12
1050.2.o.a.949.2 4 105.47 odd 12
1134.2.e.e.865.1 2 63.40 odd 6
1134.2.e.e.919.1 2 63.52 odd 6
1134.2.e.l.865.1 2 63.5 even 6
1134.2.e.l.919.1 2 63.38 even 6
1134.2.h.e.109.1 2 63.47 even 6
1134.2.h.e.541.1 2 63.59 even 6
1134.2.h.l.109.1 2 63.61 odd 6
1134.2.h.l.541.1 2 63.31 odd 6
1344.2.q.g.193.1 2 168.5 even 6
1344.2.q.g.961.1 2 168.101 even 6
1344.2.q.s.193.1 2 168.131 odd 6
1344.2.q.s.961.1 2 168.59 odd 6
2352.2.a.f.1.1 1 12.11 even 2
2352.2.a.t.1.1 1 84.83 odd 2
2352.2.q.u.961.1 2 84.11 even 6
2352.2.q.u.1537.1 2 84.23 even 6
7056.2.a.w.1.1 1 28.27 even 2
7056.2.a.bl.1.1 1 4.3 odd 2
7350.2.a.q.1.1 1 15.14 odd 2
7350.2.a.bl.1.1 1 105.104 even 2
9408.2.a.q.1.1 1 168.83 odd 2
9408.2.a.z.1.1 1 24.5 odd 2
9408.2.a.ce.1.1 1 168.125 even 2
9408.2.a.cr.1.1 1 24.11 even 2