Properties

Label 405.2.a.a.1.1
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{10} -5.00000 q^{11} +4.00000 q^{13} -4.00000 q^{16} +4.00000 q^{17} -5.00000 q^{19} -2.00000 q^{20} +10.0000 q^{22} -6.00000 q^{23} +1.00000 q^{25} -8.00000 q^{26} +5.00000 q^{29} -9.00000 q^{31} +8.00000 q^{32} -8.00000 q^{34} -10.0000 q^{37} +10.0000 q^{38} -7.00000 q^{41} -2.00000 q^{43} -10.0000 q^{44} +12.0000 q^{46} -2.00000 q^{47} -7.00000 q^{49} -2.00000 q^{50} +8.00000 q^{52} -8.00000 q^{53} +5.00000 q^{55} -10.0000 q^{58} +1.00000 q^{59} -2.00000 q^{61} +18.0000 q^{62} -8.00000 q^{64} -4.00000 q^{65} +6.00000 q^{67} +8.00000 q^{68} -1.00000 q^{71} -8.00000 q^{73} +20.0000 q^{74} -10.0000 q^{76} +12.0000 q^{79} +4.00000 q^{80} +14.0000 q^{82} -6.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} +9.00000 q^{89} -12.0000 q^{92} +4.00000 q^{94} +5.00000 q^{95} +14.0000 q^{97} +14.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 10.0000 2.13201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −8.00000 −1.56893
\(27\) 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
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 10.0000 1.62221
\(39\) 0 0
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) 8.00000 1.10940
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 1.00000 0.130189 0.0650945 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 18.0000 2.28600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 20.0000 2.32495
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 14.0000 1.54604
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 14.0000 1.41421
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 16.0000 1.55406
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) −10.0000 −0.953463
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −18.0000 −1.61645
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) −20.0000 −1.67248
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 16.0000 1.32417
\(147\) 0 0
\(148\) −20.0000 −1.64399
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −24.0000 −1.90934
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −14.0000 −1.09322
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −23.0000 −1.71910 −0.859550 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) −10.0000 −0.725476
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −28.0000 −2.01028
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −16.0000 −1.10940
\(209\) 25.0000 1.72929
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) −16.0000 −1.09888
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 10.0000 0.674200
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −32.0000 −2.12861
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −12.0000 −0.791257
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) −28.0000 −1.79991
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 7.00000 0.447214
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 0 0
\(249\) 0 0
\(250\) 2.00000 0.126491
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 30.0000 1.88608
\(254\) −32.0000 −2.00786
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −8.00000 −0.496139
\(261\) 0 0
\(262\) 30.0000 1.85341
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 31.0000 1.89010 0.945052 0.326921i \(-0.106011\pi\)
0.945052 + 0.326921i \(0.106011\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −16.0000 −0.970143
\(273\) 0 0
\(274\) 24.0000 1.44989
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 38.0000 2.27909
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 40.0000 2.36525
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 10.0000 0.587220
\(291\) 0 0
\(292\) −16.0000 −0.936329
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) −1.00000 −0.0582223
\(296\) 0 0
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 20.0000 1.14708
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −18.0000 −1.02233
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 24.0000 1.35011
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −25.0000 −1.39973
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) −20.0000 −1.11283
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −21.0000 −1.15426 −0.577132 0.816651i \(-0.695828\pi\)
−0.577132 + 0.816651i \(0.695828\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −6.00000 −0.326357
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) 45.0000 2.43689
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 13.0000 0.695874 0.347937 0.937518i \(-0.386882\pi\)
0.347937 + 0.937518i \(0.386882\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −40.0000 −2.13201
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 1.00000 0.0530745
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 46.0000 2.43118
\(359\) 27.0000 1.42501 0.712503 0.701669i \(-0.247562\pi\)
0.712503 + 0.701669i \(0.247562\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 50.0000 2.62794
\(363\) 0 0
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) −20.0000 −1.03975
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 40.0000 2.06835
\(375\) 0 0
\(376\) 0 0
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 10.0000 0.512989
\(381\) 0 0
\(382\) 2.00000 0.102329
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −52.0000 −2.64673
\(387\) 0 0
\(388\) 28.0000 1.42148
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 24.0000 1.20910
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −32.0000 −1.60402
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −36.0000 −1.79329
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 50.0000 2.47841
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −14.0000 −0.691411
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 32.0000 1.56893
\(417\) 0 0
\(418\) −50.0000 −2.44558
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) −22.0000 −1.07094
\(423\) 0 0
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 30.0000 1.43509
\(438\) 0 0
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −32.0000 −1.52208
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) −17.0000 −0.802280 −0.401140 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) 35.0000 1.64809
\(452\) 32.0000 1.50515
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) 12.0000 0.559503
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) −32.0000 −1.46365
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) 28.0000 1.27273
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −14.0000 −0.632456
\(491\) −43.0000 −1.94056 −0.970281 0.241979i \(-0.922203\pi\)
−0.970281 + 0.241979i \(0.922203\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 40.0000 1.79969
\(495\) 0 0
\(496\) 36.0000 1.61645
\(497\) 0 0
\(498\) 0 0
\(499\) 7.00000 0.313363 0.156682 0.987649i \(-0.449920\pi\)
0.156682 + 0.987649i \(0.449920\pi\)
\(500\) −2.00000 −0.0894427
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) −60.0000 −2.66733
\(507\) 0 0
\(508\) 32.0000 1.41977
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −30.0000 −1.31056
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) −36.0000 −1.56818
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −16.0000 −0.694996
\(531\) 0 0
\(532\) 0 0
\(533\) −28.0000 −1.21281
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) −62.0000 −2.67301
\(539\) 35.0000 1.50756
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 32.0000 1.37199
\(545\) 1.00000 0.0428353
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −24.0000 −1.02523
\(549\) 0 0
\(550\) 10.0000 0.426401
\(551\) −25.0000 −1.06504
\(552\) 0 0
\(553\) 0 0
\(554\) 36.0000 1.52949
\(555\) 0 0
\(556\) −38.0000 −1.61156
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 0.125767 0.0628833 0.998021i \(-0.479970\pi\)
0.0628833 + 0.998021i \(0.479970\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) −40.0000 −1.67248
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 2.00000 0.0831890
\(579\) 0 0
\(580\) −10.0000 −0.415227
\(581\) 0 0
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) 0 0
\(585\) 0 0
\(586\) 36.0000 1.48715
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) 45.0000 1.85419
\(590\) 2.00000 0.0823387
\(591\) 0 0
\(592\) 40.0000 1.64399
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 48.0000 1.96287
\(599\) 17.0000 0.694601 0.347301 0.937754i \(-0.387098\pi\)
0.347301 + 0.937754i \(0.387098\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 18.0000 0.722897
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −17.0000 −0.676759 −0.338380 0.941010i \(-0.609879\pi\)
−0.338380 + 0.941010i \(0.609879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −4.00000 −0.158860
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −28.0000 −1.10940
\(638\) 50.0000 1.97952
\(639\) 0 0
\(640\) 0 0
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) 0 0
\(643\) 6.00000 0.236617 0.118308 0.992977i \(-0.462253\pi\)
0.118308 + 0.992977i \(0.462253\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) 10.0000 0.393141 0.196570 0.980490i \(-0.437020\pi\)
0.196570 + 0.980490i \(0.437020\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 8.00000 0.313064 0.156532 0.987673i \(-0.449969\pi\)
0.156532 + 0.987673i \(0.449969\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 28.0000 1.09322
\(657\) 0 0
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 42.0000 1.63238
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −30.0000 −1.16160
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 42.0000 1.61898 0.809491 0.587133i \(-0.199743\pi\)
0.809491 + 0.587133i \(0.199743\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −90.0000 −3.44628
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −32.0000 −1.21910
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 40.0000 1.51838
\(695\) 19.0000 0.720711
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) −26.0000 −0.984115
\(699\) 0 0
\(700\) 0 0
\(701\) −19.0000 −0.717620 −0.358810 0.933411i \(-0.616817\pi\)
−0.358810 + 0.933411i \(0.616817\pi\)
\(702\) 0 0
\(703\) 50.0000 1.88579
\(704\) 40.0000 1.50756
\(705\) 0 0
\(706\) −36.0000 −1.35488
\(707\) 0 0
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 0 0
\(712\) 0 0
\(713\) 54.0000 2.02232
\(714\) 0 0
\(715\) 20.0000 0.747958
\(716\) −46.0000 −1.71910
\(717\) 0 0
\(718\) −54.0000 −2.01526
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −12.0000 −0.446594
\(723\) 0 0
\(724\) −50.0000 −1.85824
\(725\) 5.00000 0.185695
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) −35.0000 −1.28750 −0.643748 0.765238i \(-0.722621\pi\)
−0.643748 + 0.765238i \(0.722621\pi\)
\(740\) 20.0000 0.735215
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) −40.0000 −1.46254
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −40.0000 −1.45671
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.00000 −0.0723575
\(765\) 0 0
\(766\) −72.0000 −2.60147
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 52.0000 1.87152
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) −9.00000 −0.323290
\(776\) 0 0
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) 35.0000 1.25401
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) 48.0000 1.71648
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −24.0000 −0.854965
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) 0 0
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 76.0000 2.69714
\(795\) 0 0
\(796\) 32.0000 1.13421
\(797\) −4.00000 −0.141687 −0.0708436 0.997487i \(-0.522569\pi\)
−0.0708436 + 0.997487i \(0.522569\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 8.00000 0.282843
\(801\) 0 0
\(802\) 60.0000 2.11867
\(803\) 40.0000 1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) 72.0000 2.53609
\(807\) 0 0
\(808\) 0 0
\(809\) −7.00000 −0.246107 −0.123053 0.992400i \(-0.539269\pi\)
−0.123053 + 0.992400i \(0.539269\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −100.000 −3.50500
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 10.0000 0.349856
\(818\) −28.0000 −0.978997
\(819\) 0 0
\(820\) 14.0000 0.488901
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) −46.0000 −1.60346 −0.801730 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) 27.0000 0.937749 0.468874 0.883265i \(-0.344660\pi\)
0.468874 + 0.883265i \(0.344660\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −32.0000 −1.10940
\(833\) −28.0000 −0.970143
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 50.0000 1.72929
\(837\) 0 0
\(838\) 32.0000 1.10542
\(839\) −13.0000 −0.448810 −0.224405 0.974496i \(-0.572044\pi\)
−0.224405 + 0.974496i \(0.572044\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) 22.0000 0.757271
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) 32.0000 1.09888
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 55.0000 1.87658 0.938288 0.345855i \(-0.112411\pi\)
0.938288 + 0.345855i \(0.112411\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) 44.0000 1.49778 0.748889 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 52.0000 1.76703
\(867\) 0 0
\(868\) 0 0
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 0 0
\(874\) −60.0000 −2.02953
\(875\) 0 0
\(876\) 0 0
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) 58.0000 1.95741
\(879\) 0 0
\(880\) −20.0000 −0.674200
\(881\) 25.0000 0.842271 0.421136 0.906998i \(-0.361632\pi\)
0.421136 + 0.906998i \(0.361632\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 32.0000 1.07628
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 10.0000 0.334637
\(894\) 0 0
\(895\) 23.0000 0.768805
\(896\) 0 0
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) −45.0000 −1.50083
\(900\) 0 0
\(901\) −32.0000 −1.06607
\(902\) −70.0000 −2.33075
\(903\) 0 0
\(904\) 0 0
\(905\) 25.0000 0.831028
\(906\) 0 0
\(907\) −18.0000 −0.597680 −0.298840 0.954303i \(-0.596600\pi\)
−0.298840 + 0.954303i \(0.596600\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 49.0000 1.62344 0.811721 0.584045i \(-0.198531\pi\)
0.811721 + 0.584045i \(0.198531\pi\)
\(912\) 0 0
\(913\) 30.0000 0.992855
\(914\) 76.0000 2.51386
\(915\) 0 0
\(916\) 12.0000 0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) 40.0000 1.31306
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) 35.0000 1.14708
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) 20.0000 0.654070
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 42.0000 1.36771
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 10.0000 0.324443
\(951\) 0 0
\(952\) 0 0
\(953\) 16.0000 0.518291 0.259145 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) 0 0
\(955\) 1.00000 0.0323592
\(956\) 32.0000 1.03495
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 80.0000 2.57930
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) −26.0000 −0.836970
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 28.0000 0.899026
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) 14.0000 0.447214
\(981\) 0 0
\(982\) 86.0000 2.74437
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) −40.0000 −1.27386
\(987\) 0 0
\(988\) −40.0000 −1.27257
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −1.00000 −0.0317660 −0.0158830 0.999874i \(-0.505056\pi\)
−0.0158830 + 0.999874i \(0.505056\pi\)
\(992\) −72.0000 −2.28600
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 48.0000 1.52018 0.760088 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) −14.0000 −0.443162
\(999\) 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 405.2.a.a.1.1 1
3.2 odd 2 405.2.a.f.1.1 yes 1
4.3 odd 2 6480.2.a.f.1.1 1
5.2 odd 4 2025.2.b.a.649.1 2
5.3 odd 4 2025.2.b.a.649.2 2
5.4 even 2 2025.2.a.f.1.1 1
9.2 odd 6 405.2.e.a.271.1 2
9.4 even 3 405.2.e.g.136.1 2
9.5 odd 6 405.2.e.a.136.1 2
9.7 even 3 405.2.e.g.271.1 2
12.11 even 2 6480.2.a.r.1.1 1
15.2 even 4 2025.2.b.b.649.2 2
15.8 even 4 2025.2.b.b.649.1 2
15.14 odd 2 2025.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.a.1.1 1 1.1 even 1 trivial
405.2.a.f.1.1 yes 1 3.2 odd 2
405.2.e.a.136.1 2 9.5 odd 6
405.2.e.a.271.1 2 9.2 odd 6
405.2.e.g.136.1 2 9.4 even 3
405.2.e.g.271.1 2 9.7 even 3
2025.2.a.a.1.1 1 15.14 odd 2
2025.2.a.f.1.1 1 5.4 even 2
2025.2.b.a.649.1 2 5.2 odd 4
2025.2.b.a.649.2 2 5.3 odd 4
2025.2.b.b.649.1 2 15.8 even 4
2025.2.b.b.649.2 2 15.2 even 4
6480.2.a.f.1.1 1 4.3 odd 2
6480.2.a.r.1.1 1 12.11 even 2