Properties

Label 7920.2.a.bz.1.1
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -2.00000 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{11} -1.46410 q^{13} +1.46410 q^{19} -6.92820 q^{23} +1.00000 q^{25} -3.46410 q^{29} -2.92820 q^{31} -2.00000 q^{35} +8.92820 q^{37} +3.46410 q^{41} -8.92820 q^{43} +6.92820 q^{47} -3.00000 q^{49} +12.9282 q^{53} -1.00000 q^{55} +6.92820 q^{59} +2.00000 q^{61} -1.46410 q^{65} -8.00000 q^{67} -13.8564 q^{71} +12.3923 q^{73} +2.00000 q^{77} +13.4641 q^{79} +15.4641 q^{83} +12.9282 q^{89} +2.92820 q^{91} +1.46410 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 2q + 2q^{5} - 4q^{7} - 2q^{11} + 4q^{13} - 4q^{19} + 2q^{25} + 8q^{31} - 4q^{35} + 4q^{37} - 4q^{43} - 6q^{49} + 12q^{53} - 2q^{55} + 4q^{61} + 4q^{65} - 16q^{67} + 4q^{73} + 4q^{77} + 20q^{79} + 24q^{83} + 12q^{89} - 8q^{91} - 4q^{95} - 20q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.46410 0.335888 0.167944 0.985797i \(-0.446287\pi\)
0.167944 + 0.985797i \(0.446287\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −2.92820 −0.525921 −0.262960 0.964807i \(-0.584699\pi\)
−0.262960 + 0.964807i \(0.584699\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) −8.92820 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.9282 1.77583 0.887913 0.460012i \(-0.152155\pi\)
0.887913 + 0.460012i \(0.152155\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.46410 −0.181599
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) 12.3923 1.45041 0.725205 0.688533i \(-0.241745\pi\)
0.725205 + 0.688533i \(0.241745\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 13.4641 1.51483 0.757415 0.652934i \(-0.226462\pi\)
0.757415 + 0.652934i \(0.226462\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.4641 1.69741 0.848703 0.528870i \(-0.177384\pi\)
0.848703 + 0.528870i \(0.177384\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) 2.92820 0.306959
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.46410 0.150214
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 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
\(106\) 0 0
\(107\) 15.4641 1.49497 0.747486 0.664278i \(-0.231261\pi\)
0.747486 + 0.664278i \(0.231261\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.928203 −0.0873180 −0.0436590 0.999046i \(-0.513902\pi\)
−0.0436590 + 0.999046i \(0.513902\pi\)
\(114\) 0 0
\(115\) −6.92820 −0.646058
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.92820 0.437307 0.218654 0.975803i \(-0.429834\pi\)
0.218654 + 0.975803i \(0.429834\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.07180 0.443125 0.221562 0.975146i \(-0.428884\pi\)
0.221562 + 0.975146i \(0.428884\pi\)
\(132\) 0 0
\(133\) −2.92820 −0.253907
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 8.39230 0.711826 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.46410 0.122434
\(144\) 0 0
\(145\) −3.46410 −0.287678
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.53590 0.699288 0.349644 0.936883i \(-0.386303\pi\)
0.349644 + 0.936883i \(0.386303\pi\)
\(150\) 0 0
\(151\) −0.392305 −0.0319253 −0.0159627 0.999873i \(-0.505081\pi\)
−0.0159627 + 0.999873i \(0.505081\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.92820 −0.235199
\(156\) 0 0
\(157\) −16.9282 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) 17.8564 1.39862 0.699311 0.714818i \(-0.253490\pi\)
0.699311 + 0.714818i \(0.253490\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.92820 0.656415
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) 0 0
\(193\) 3.60770 0.259688 0.129844 0.991534i \(-0.458552\pi\)
0.129844 + 0.991534i \(0.458552\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.46410 −0.101274
\(210\) 0 0
\(211\) −12.3923 −0.853121 −0.426561 0.904459i \(-0.640275\pi\)
−0.426561 + 0.904459i \(0.640275\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.92820 −0.608898
\(216\) 0 0
\(217\) 5.85641 0.397559
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.8564 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.53590 0.566547 0.283274 0.959039i \(-0.408580\pi\)
0.283274 + 0.959039i \(0.408580\pi\)
\(228\) 0 0
\(229\) 3.85641 0.254839 0.127419 0.991849i \(-0.459331\pi\)
0.127419 + 0.991849i \(0.459331\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −2.14359 −0.136394
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.8564 1.63204 0.816021 0.578022i \(-0.196175\pi\)
0.816021 + 0.578022i \(0.196175\pi\)
\(252\) 0 0
\(253\) 6.92820 0.435572
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.85641 −0.490069 −0.245035 0.969514i \(-0.578799\pi\)
−0.245035 + 0.969514i \(0.578799\pi\)
\(258\) 0 0
\(259\) −17.8564 −1.10954
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 27.4641 1.69351 0.846755 0.531984i \(-0.178553\pi\)
0.846755 + 0.531984i \(0.178553\pi\)
\(264\) 0 0
\(265\) 12.9282 0.794173
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.85641 0.479014 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(270\) 0 0
\(271\) 32.3923 1.96769 0.983846 0.179016i \(-0.0572913\pi\)
0.983846 + 0.179016i \(0.0572913\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 22.5359 1.35405 0.677025 0.735960i \(-0.263269\pi\)
0.677025 + 0.735960i \(0.263269\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 0 0
\(283\) −8.92820 −0.530727 −0.265363 0.964148i \(-0.585492\pi\)
−0.265363 + 0.964148i \(0.585492\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.1436 0.586619
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.9282 1.07332 0.536660 0.843799i \(-0.319686\pi\)
0.536660 + 0.843799i \(0.319686\pi\)
\(312\) 0 0
\(313\) 7.07180 0.399722 0.199861 0.979824i \(-0.435951\pi\)
0.199861 + 0.979824i \(0.435951\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.0718 0.621854 0.310927 0.950434i \(-0.399361\pi\)
0.310927 + 0.950434i \(0.399361\pi\)
\(318\) 0 0
\(319\) 3.46410 0.193952
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 17.8564 0.981477 0.490738 0.871307i \(-0.336727\pi\)
0.490738 + 0.871307i \(0.336727\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −29.1769 −1.58937 −0.794684 0.607023i \(-0.792363\pi\)
−0.794684 + 0.607023i \(0.792363\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.92820 0.158571
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.60770 0.0863056 0.0431528 0.999068i \(-0.486260\pi\)
0.0431528 + 0.999068i \(0.486260\pi\)
\(348\) 0 0
\(349\) −35.8564 −1.91935 −0.959675 0.281113i \(-0.909296\pi\)
−0.959675 + 0.281113i \(0.909296\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.928203 0.0494033 0.0247016 0.999695i \(-0.492136\pi\)
0.0247016 + 0.999695i \(0.492136\pi\)
\(354\) 0 0
\(355\) −13.8564 −0.735422
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7846 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.3923 0.648643
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −25.8564 −1.34240
\(372\) 0 0
\(373\) 0.392305 0.0203128 0.0101564 0.999948i \(-0.496767\pi\)
0.0101564 + 0.999948i \(0.496767\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.07180 0.261211
\(378\) 0 0
\(379\) −9.85641 −0.506290 −0.253145 0.967428i \(-0.581465\pi\)
−0.253145 + 0.967428i \(0.581465\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −24.9282 −1.26391 −0.631955 0.775005i \(-0.717747\pi\)
−0.631955 + 0.775005i \(0.717747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.4641 0.677452
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.8564 −0.991582 −0.495791 0.868442i \(-0.665122\pi\)
−0.495791 + 0.868442i \(0.665122\pi\)
\(402\) 0 0
\(403\) 4.28719 0.213560
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.92820 −0.442555
\(408\) 0 0
\(409\) 34.7846 1.71999 0.859994 0.510304i \(-0.170467\pi\)
0.859994 + 0.510304i \(0.170467\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) 15.4641 0.759103
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.0718 0.834012 0.417006 0.908904i \(-0.363079\pi\)
0.417006 + 0.908904i \(0.363079\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.7846 −1.57918 −0.789590 0.613635i \(-0.789706\pi\)
−0.789590 + 0.613635i \(0.789706\pi\)
\(432\) 0 0
\(433\) 27.8564 1.33869 0.669347 0.742950i \(-0.266574\pi\)
0.669347 + 0.742950i \(0.266574\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.1436 −0.485234
\(438\) 0 0
\(439\) 29.1769 1.39254 0.696269 0.717781i \(-0.254842\pi\)
0.696269 + 0.717781i \(0.254842\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 12.9282 0.612856
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −14.7846 −0.697729 −0.348864 0.937173i \(-0.613433\pi\)
−0.348864 + 0.937173i \(0.613433\pi\)
\(450\) 0 0
\(451\) −3.46410 −0.163118
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.92820 0.137276
\(456\) 0 0
\(457\) −8.39230 −0.392575 −0.196288 0.980546i \(-0.562889\pi\)
−0.196288 + 0.980546i \(0.562889\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.2487 0.570479 0.285240 0.958456i \(-0.407927\pi\)
0.285240 + 0.958456i \(0.407927\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.9282 0.875893 0.437946 0.899001i \(-0.355706\pi\)
0.437946 + 0.899001i \(0.355706\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.92820 0.410519
\(474\) 0 0
\(475\) 1.46410 0.0671776
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −13.0718 −0.596023
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −23.7128 −1.07453 −0.537265 0.843413i \(-0.680543\pi\)
−0.537265 + 0.843413i \(0.680543\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.0718 −0.770439 −0.385220 0.922825i \(-0.625874\pi\)
−0.385220 + 0.922825i \(0.625874\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.7128 1.24309
\(498\) 0 0
\(499\) 12.7846 0.572318 0.286159 0.958182i \(-0.407621\pi\)
0.286159 + 0.958182i \(0.407621\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) −10.3923 −0.462451
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.85641 0.348229 0.174115 0.984725i \(-0.444294\pi\)
0.174115 + 0.984725i \(0.444294\pi\)
\(510\) 0 0
\(511\) −24.7846 −1.09641
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −6.92820 −0.304702
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.07180 −0.219684
\(534\) 0 0
\(535\) 15.4641 0.668571
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.07180 −0.216066
\(552\) 0 0
\(553\) −26.9282 −1.14510
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −44.7846 −1.89758 −0.948792 0.315900i \(-0.897694\pi\)
−0.948792 + 0.315900i \(0.897694\pi\)
\(558\) 0 0
\(559\) 13.0718 0.552878
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.3923 −0.437983 −0.218992 0.975727i \(-0.570277\pi\)
−0.218992 + 0.975727i \(0.570277\pi\)
\(564\) 0 0
\(565\) −0.928203 −0.0390498
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 29.3205 1.22918 0.614590 0.788847i \(-0.289322\pi\)
0.614590 + 0.788847i \(0.289322\pi\)
\(570\) 0 0
\(571\) −24.3923 −1.02079 −0.510393 0.859941i \(-0.670500\pi\)
−0.510393 + 0.859941i \(0.670500\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) 22.7846 0.948536 0.474268 0.880381i \(-0.342713\pi\)
0.474268 + 0.880381i \(0.342713\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −30.9282 −1.28312
\(582\) 0 0
\(583\) −12.9282 −0.535431
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.07180 −0.209335 −0.104668 0.994507i \(-0.533378\pi\)
−0.104668 + 0.994507i \(0.533378\pi\)
\(588\) 0 0
\(589\) −4.28719 −0.176650
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.7846 1.34630 0.673151 0.739505i \(-0.264940\pi\)
0.673151 + 0.739505i \(0.264940\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10.1436 −0.414456 −0.207228 0.978293i \(-0.566444\pi\)
−0.207228 + 0.978293i \(0.566444\pi\)
\(600\) 0 0
\(601\) 36.6410 1.49462 0.747309 0.664477i \(-0.231345\pi\)
0.747309 + 0.664477i \(0.231345\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −22.7846 −0.924799 −0.462399 0.886672i \(-0.653011\pi\)
−0.462399 + 0.886672i \(0.653011\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 0 0
\(613\) 0.392305 0.0158450 0.00792252 0.999969i \(-0.497478\pi\)
0.00792252 + 0.999969i \(0.497478\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.0718 0.928836 0.464418 0.885616i \(-0.346264\pi\)
0.464418 + 0.885616i \(0.346264\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −25.8564 −1.03592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 34.9282 1.39047 0.695235 0.718783i \(-0.255300\pi\)
0.695235 + 0.718783i \(0.255300\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.92820 0.195570
\(636\) 0 0
\(637\) 4.39230 0.174029
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.928203 −0.0366618 −0.0183309 0.999832i \(-0.505835\pi\)
−0.0183309 + 0.999832i \(0.505835\pi\)
\(642\) 0 0
\(643\) 45.5692 1.79707 0.898537 0.438897i \(-0.144631\pi\)
0.898537 + 0.438897i \(0.144631\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.85641 0.307445 0.153722 0.988114i \(-0.450874\pi\)
0.153722 + 0.988114i \(0.450874\pi\)
\(654\) 0 0
\(655\) 5.07180 0.198171
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 39.7128 1.54699 0.773496 0.633801i \(-0.218506\pi\)
0.773496 + 0.633801i \(0.218506\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.92820 −0.113551
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 31.3205 1.20732 0.603658 0.797243i \(-0.293709\pi\)
0.603658 + 0.797243i \(0.293709\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32.7846 1.26001 0.630007 0.776589i \(-0.283052\pi\)
0.630007 + 0.776589i \(0.283052\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.78461 0.336134 0.168067 0.985776i \(-0.446248\pi\)
0.168067 + 0.985776i \(0.446248\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −18.9282 −0.721107
\(690\) 0 0
\(691\) 7.71281 0.293409 0.146705 0.989180i \(-0.453133\pi\)
0.146705 + 0.989180i \(0.453133\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.39230 0.318338
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −32.5359 −1.22886 −0.614432 0.788970i \(-0.710615\pi\)
−0.614432 + 0.788970i \(0.710615\pi\)
\(702\) 0 0
\(703\) 13.0718 0.493012
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) 15.8564 0.595500 0.297750 0.954644i \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.2872 0.759761
\(714\) 0 0
\(715\) 1.46410 0.0547543
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.9282 −0.705903 −0.352951 0.935642i \(-0.614822\pi\)
−0.352951 + 0.935642i \(0.614822\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −49.9615 −1.84537 −0.922686 0.385553i \(-0.874011\pi\)
−0.922686 + 0.385553i \(0.874011\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −10.5359 −0.387569 −0.193785 0.981044i \(-0.562076\pi\)
−0.193785 + 0.981044i \(0.562076\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 46.3923 1.70197 0.850984 0.525191i \(-0.176006\pi\)
0.850984 + 0.525191i \(0.176006\pi\)
\(744\) 0 0
\(745\) 8.53590 0.312731
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −30.9282 −1.13009
\(750\) 0 0
\(751\) −13.0718 −0.476997 −0.238498 0.971143i \(-0.576655\pi\)
−0.238498 + 0.971143i \(0.576655\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.392305 −0.0142774
\(756\) 0 0
\(757\) −6.78461 −0.246591 −0.123295 0.992370i \(-0.539346\pi\)
−0.123295 + 0.992370i \(0.539346\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39.4641 1.43057 0.715286 0.698832i \(-0.246296\pi\)
0.715286 + 0.698832i \(0.246296\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.1436 −0.366264
\(768\) 0 0
\(769\) −46.4974 −1.67674 −0.838370 0.545102i \(-0.816491\pi\)
−0.838370 + 0.545102i \(0.816491\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 31.8564 1.14580 0.572898 0.819627i \(-0.305819\pi\)
0.572898 + 0.819627i \(0.305819\pi\)
\(774\) 0 0
\(775\) −2.92820 −0.105184
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.07180 0.181716
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −16.9282 −0.604193
\(786\) 0 0
\(787\) 18.7846 0.669599 0.334800 0.942289i \(-0.391331\pi\)
0.334800 + 0.942289i \(0.391331\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.85641 0.0660062
\(792\) 0 0
\(793\) −2.92820 −0.103984
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.6410 −0.589455 −0.294728 0.955581i \(-0.595229\pi\)
−0.294728 + 0.955581i \(0.595229\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −12.3923 −0.437315
\(804\) 0 0
\(805\) 13.8564 0.488374
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.53590 0.300106 0.150053 0.988678i \(-0.452056\pi\)
0.150053 + 0.988678i \(0.452056\pi\)
\(810\) 0 0
\(811\) 8.39230 0.294694 0.147347 0.989085i \(-0.452927\pi\)
0.147347 + 0.989085i \(0.452927\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17.8564 0.625483
\(816\) 0 0
\(817\) −13.0718 −0.457324
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.4641 −0.958504 −0.479252 0.877677i \(-0.659092\pi\)
−0.479252 + 0.877677i \(0.659092\pi\)
\(822\) 0 0
\(823\) −49.5692 −1.72787 −0.863937 0.503600i \(-0.832009\pi\)
−0.863937 + 0.503600i \(0.832009\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.60770 0.0559050 0.0279525 0.999609i \(-0.491101\pi\)
0.0279525 + 0.999609i \(0.491101\pi\)
\(828\) 0 0
\(829\) −25.7128 −0.893043 −0.446521 0.894773i \(-0.647337\pi\)
−0.446521 + 0.894773i \(0.647337\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 10.3923 0.359641
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15.2154 0.525294 0.262647 0.964892i \(-0.415405\pi\)
0.262647 + 0.964892i \(0.415405\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.8564 −0.373472
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −61.8564 −2.12041
\(852\) 0 0
\(853\) 24.3923 0.835177 0.417588 0.908636i \(-0.362875\pi\)
0.417588 + 0.908636i \(0.362875\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.1436 0.346499 0.173249 0.984878i \(-0.444573\pi\)
0.173249 + 0.984878i \(0.444573\pi\)
\(858\) 0 0
\(859\) −47.7128 −1.62794 −0.813970 0.580907i \(-0.802698\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.1436 −0.345292 −0.172646 0.984984i \(-0.555232\pi\)
−0.172646 + 0.984984i \(0.555232\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.4641 −0.456738
\(870\) 0 0
\(871\) 11.7128 0.396874
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 14.2487 0.481145 0.240572 0.970631i \(-0.422665\pi\)
0.240572 + 0.970631i \(0.422665\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.9282 −0.435562 −0.217781 0.975998i \(-0.569882\pi\)
−0.217781 + 0.975998i \(0.569882\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.2487 −1.21711 −0.608556 0.793511i \(-0.708251\pi\)
−0.608556 + 0.793511i \(0.708251\pi\)
\(888\) 0 0
\(889\) −9.85641 −0.330573
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10.1436 0.339442
\(894\) 0 0
\(895\) −6.92820 −0.231584
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.1436 0.338308
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.8564 −0.394120
\(906\) 0 0
\(907\) −45.8564 −1.52264 −0.761318 0.648378i \(-0.775448\pi\)
−0.761318 + 0.648378i \(0.775448\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.07180 0.168036 0.0840181 0.996464i \(-0.473225\pi\)
0.0840181 + 0.996464i \(0.473225\pi\)
\(912\) 0 0
\(913\) −15.4641 −0.511787
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.1436 −0.334971
\(918\) 0 0
\(919\) 11.6077 0.382903 0.191451 0.981502i \(-0.438681\pi\)
0.191451 + 0.981502i \(0.438681\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 20.2872 0.667761
\(924\) 0 0
\(925\) 8.92820 0.293558
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 38.7846 1.27248 0.636241 0.771490i \(-0.280488\pi\)
0.636241 + 0.771490i \(0.280488\pi\)
\(930\) 0 0
\(931\) −4.39230 −0.143952
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.392305 0.0128160 0.00640802 0.999979i \(-0.497960\pi\)
0.00640802 + 0.999979i \(0.497960\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.5359 −0.669451 −0.334726 0.942316i \(-0.608644\pi\)
−0.334726 + 0.942316i \(0.608644\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.07180 −0.164811 −0.0824056 0.996599i \(-0.526260\pi\)
−0.0824056 + 0.996599i \(0.526260\pi\)
\(948\) 0 0
\(949\) −18.1436 −0.588966
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44.7846 1.45072 0.725358 0.688372i \(-0.241674\pi\)
0.725358 + 0.688372i \(0.241674\pi\)
\(954\) 0 0
\(955\) −5.07180 −0.164119
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.60770 0.116136
\(966\) 0 0
\(967\) 18.7846 0.604072 0.302036 0.953296i \(-0.402334\pi\)
0.302036 + 0.953296i \(0.402334\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.85641 −0.0595749 −0.0297875 0.999556i \(-0.509483\pi\)
−0.0297875 + 0.999556i \(0.509483\pi\)
\(972\) 0 0
\(973\) −16.7846 −0.538090
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.5692 1.13796 0.568980 0.822351i \(-0.307338\pi\)
0.568980 + 0.822351i \(0.307338\pi\)
\(978\) 0 0
\(979\) −12.9282 −0.413187
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 61.8564 1.96692
\(990\) 0 0
\(991\) 48.7846 1.54969 0.774847 0.632149i \(-0.217827\pi\)
0.774847 + 0.632149i \(0.217827\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16.7846 −0.532108
\(996\) 0 0
\(997\) 48.3923 1.53260 0.766300 0.642483i \(-0.222096\pi\)
0.766300 + 0.642483i \(0.222096\pi\)
\(998\) 0 0
\(999\) 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 7920.2.a.bz.1.1 2
3.2 odd 2 2640.2.a.x.1.1 2
4.3 odd 2 495.2.a.c.1.1 2
12.11 even 2 165.2.a.b.1.2 2
20.3 even 4 2475.2.c.n.199.3 4
20.7 even 4 2475.2.c.n.199.2 4
20.19 odd 2 2475.2.a.r.1.2 2
44.43 even 2 5445.2.a.s.1.2 2
60.23 odd 4 825.2.c.c.199.2 4
60.47 odd 4 825.2.c.c.199.3 4
60.59 even 2 825.2.a.e.1.1 2
84.83 odd 2 8085.2.a.bd.1.2 2
132.131 odd 2 1815.2.a.i.1.1 2
660.659 odd 2 9075.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.2 2 12.11 even 2
495.2.a.c.1.1 2 4.3 odd 2
825.2.a.e.1.1 2 60.59 even 2
825.2.c.c.199.2 4 60.23 odd 4
825.2.c.c.199.3 4 60.47 odd 4
1815.2.a.i.1.1 2 132.131 odd 2
2475.2.a.r.1.2 2 20.19 odd 2
2475.2.c.n.199.2 4 20.7 even 4
2475.2.c.n.199.3 4 20.3 even 4
2640.2.a.x.1.1 2 3.2 odd 2
5445.2.a.s.1.2 2 44.43 even 2
7920.2.a.bz.1.1 2 1.1 even 1 trivial
8085.2.a.bd.1.2 2 84.83 odd 2
9075.2.a.bh.1.2 2 660.659 odd 2