Properties

Label 8820.2.a.bk.1.1
Level $8820$
Weight $2$
Character 8820.1
Self dual yes
Analytic conductor $70.428$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8820.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +O(q^{10})\) \(q+1.00000 q^{5} -4.24264 q^{11} -5.24264 q^{13} +4.24264 q^{17} -7.00000 q^{19} -4.24264 q^{23} +1.00000 q^{25} +10.2426 q^{29} +7.48528 q^{31} -5.24264 q^{37} -4.24264 q^{41} -5.24264 q^{43} +6.00000 q^{47} +8.48528 q^{53} -4.24264 q^{55} +1.75736 q^{59} -12.4853 q^{61} -5.24264 q^{65} +3.24264 q^{67} -12.7279 q^{71} +0.757359 q^{73} +11.0000 q^{79} +1.75736 q^{83} +4.24264 q^{85} +1.75736 q^{89} -7.00000 q^{95} +16.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{13} - 14 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{43} + 12 q^{47} + 12 q^{59} - 8 q^{61} - 2 q^{65} - 2 q^{67} + 10 q^{73} + 22 q^{79} + 12 q^{83} + 12 q^{89} - 14 q^{95} + 16 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.24264 −1.27920 −0.639602 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) −5.24264 −1.45405 −0.727023 0.686613i \(-0.759097\pi\)
−0.727023 + 0.686613i \(0.759097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.24264 1.02899 0.514496 0.857493i \(-0.327979\pi\)
0.514496 + 0.857493i \(0.327979\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.24264 −0.884652 −0.442326 0.896854i \(-0.645847\pi\)
−0.442326 + 0.896854i \(0.645847\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.2426 1.90201 0.951005 0.309175i \(-0.100053\pi\)
0.951005 + 0.309175i \(0.100053\pi\)
\(30\) 0 0
\(31\) 7.48528 1.34440 0.672198 0.740371i \(-0.265350\pi\)
0.672198 + 0.740371i \(0.265350\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.24264 −0.861885 −0.430942 0.902379i \(-0.641819\pi\)
−0.430942 + 0.902379i \(0.641819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 0 0
\(43\) −5.24264 −0.799495 −0.399748 0.916625i \(-0.630902\pi\)
−0.399748 + 0.916625i \(0.630902\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.48528 1.16554 0.582772 0.812636i \(-0.301968\pi\)
0.582772 + 0.812636i \(0.301968\pi\)
\(54\) 0 0
\(55\) −4.24264 −0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.75736 0.228789 0.114394 0.993435i \(-0.463507\pi\)
0.114394 + 0.993435i \(0.463507\pi\)
\(60\) 0 0
\(61\) −12.4853 −1.59858 −0.799288 0.600948i \(-0.794790\pi\)
−0.799288 + 0.600948i \(0.794790\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.24264 −0.650270
\(66\) 0 0
\(67\) 3.24264 0.396152 0.198076 0.980187i \(-0.436531\pi\)
0.198076 + 0.980187i \(0.436531\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.7279 −1.51053 −0.755263 0.655422i \(-0.772491\pi\)
−0.755263 + 0.655422i \(0.772491\pi\)
\(72\) 0 0
\(73\) 0.757359 0.0886422 0.0443211 0.999017i \(-0.485888\pi\)
0.0443211 + 0.999017i \(0.485888\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.75736 0.192895 0.0964476 0.995338i \(-0.469252\pi\)
0.0964476 + 0.995338i \(0.469252\pi\)
\(84\) 0 0
\(85\) 4.24264 0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.75736 0.186280 0.0931399 0.995653i \(-0.470310\pi\)
0.0931399 + 0.995653i \(0.470310\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) 16.4853 1.67383 0.836913 0.547335i \(-0.184358\pi\)
0.836913 + 0.547335i \(0.184358\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.2426 1.61620 0.808102 0.589043i \(-0.200495\pi\)
0.808102 + 0.589043i \(0.200495\pi\)
\(102\) 0 0
\(103\) −8.75736 −0.862888 −0.431444 0.902140i \(-0.641996\pi\)
−0.431444 + 0.902140i \(0.641996\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.7279 1.23045 0.615227 0.788350i \(-0.289064\pi\)
0.615227 + 0.788350i \(0.289064\pi\)
\(108\) 0 0
\(109\) 7.48528 0.716960 0.358480 0.933537i \(-0.383295\pi\)
0.358480 + 0.933537i \(0.383295\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −4.24264 −0.395628
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.2426 −0.997623 −0.498812 0.866710i \(-0.666230\pi\)
−0.498812 + 0.866710i \(0.666230\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.48528 0.217140 0.108570 0.994089i \(-0.465373\pi\)
0.108570 + 0.994089i \(0.465373\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.24264 0.362473 0.181237 0.983440i \(-0.441990\pi\)
0.181237 + 0.983440i \(0.441990\pi\)
\(138\) 0 0
\(139\) 1.48528 0.125980 0.0629900 0.998014i \(-0.479936\pi\)
0.0629900 + 0.998014i \(0.479936\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 22.2426 1.86002
\(144\) 0 0
\(145\) 10.2426 0.850605
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 5.51472 0.448781 0.224391 0.974499i \(-0.427961\pi\)
0.224391 + 0.974499i \(0.427961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.48528 0.601232
\(156\) 0 0
\(157\) 10.4853 0.836817 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.72792 0.520622 0.260311 0.965525i \(-0.416175\pi\)
0.260311 + 0.965525i \(0.416175\pi\)
\(168\) 0 0
\(169\) 14.4853 1.11425
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.51472 0.267219 0.133610 0.991034i \(-0.457343\pi\)
0.133610 + 0.991034i \(0.457343\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.24264 −0.385447
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) 15.2426 1.09719 0.548595 0.836088i \(-0.315163\pi\)
0.548595 + 0.836088i \(0.315163\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.75736 0.552689 0.276344 0.961059i \(-0.410877\pi\)
0.276344 + 0.961059i \(0.410877\pi\)
\(198\) 0 0
\(199\) −6.48528 −0.459729 −0.229865 0.973223i \(-0.573828\pi\)
−0.229865 + 0.973223i \(0.573828\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.24264 −0.296319
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.6985 2.05429
\(210\) 0 0
\(211\) 5.51472 0.379649 0.189824 0.981818i \(-0.439208\pi\)
0.189824 + 0.981818i \(0.439208\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.24264 −0.357545
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −22.2426 −1.49620
\(222\) 0 0
\(223\) −24.4853 −1.63966 −0.819828 0.572610i \(-0.805931\pi\)
−0.819828 + 0.572610i \(0.805931\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −27.2132 −1.80620 −0.903102 0.429426i \(-0.858716\pi\)
−0.903102 + 0.429426i \(0.858716\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.48528 0.162816 0.0814081 0.996681i \(-0.474058\pi\)
0.0814081 + 0.996681i \(0.474058\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.9706 1.48584 0.742921 0.669379i \(-0.233440\pi\)
0.742921 + 0.669379i \(0.233440\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 36.6985 2.33507
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.72792 −0.424663 −0.212331 0.977198i \(-0.568106\pi\)
−0.212331 + 0.977198i \(0.568106\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.75736 −0.109621 −0.0548105 0.998497i \(-0.517455\pi\)
−0.0548105 + 0.998497i \(0.517455\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.48528 −0.523225 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.4853 −0.883183 −0.441592 0.897216i \(-0.645586\pi\)
−0.441592 + 0.897216i \(0.645586\pi\)
\(270\) 0 0
\(271\) 27.4558 1.66782 0.833912 0.551898i \(-0.186096\pi\)
0.833912 + 0.551898i \(0.186096\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.24264 −0.255841
\(276\) 0 0
\(277\) −7.72792 −0.464326 −0.232163 0.972677i \(-0.574580\pi\)
−0.232163 + 0.972677i \(0.574580\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.97056 0.296519 0.148259 0.988948i \(-0.452633\pi\)
0.148259 + 0.988948i \(0.452633\pi\)
\(282\) 0 0
\(283\) −1.72792 −0.102714 −0.0513572 0.998680i \(-0.516355\pi\)
−0.0513572 + 0.998680i \(0.516355\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −28.9706 −1.69248 −0.846239 0.532803i \(-0.821139\pi\)
−0.846239 + 0.532803i \(0.821139\pi\)
\(294\) 0 0
\(295\) 1.75736 0.102317
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.2426 1.28633
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −12.4853 −0.714905
\(306\) 0 0
\(307\) −5.24264 −0.299213 −0.149607 0.988746i \(-0.547801\pi\)
−0.149607 + 0.988746i \(0.547801\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.2132 1.20289 0.601445 0.798914i \(-0.294592\pi\)
0.601445 + 0.798914i \(0.294592\pi\)
\(312\) 0 0
\(313\) −1.72792 −0.0976679 −0.0488340 0.998807i \(-0.515551\pi\)
−0.0488340 + 0.998807i \(0.515551\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.727922 −0.0408842 −0.0204421 0.999791i \(-0.506507\pi\)
−0.0204421 + 0.999791i \(0.506507\pi\)
\(318\) 0 0
\(319\) −43.4558 −2.43306
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −29.6985 −1.65247
\(324\) 0 0
\(325\) −5.24264 −0.290809
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.24264 0.177164
\(336\) 0 0
\(337\) 11.7279 0.638861 0.319430 0.947610i \(-0.396508\pi\)
0.319430 + 0.947610i \(0.396508\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −31.7574 −1.71976
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.75736 0.0935348 0.0467674 0.998906i \(-0.485108\pi\)
0.0467674 + 0.998906i \(0.485108\pi\)
\(354\) 0 0
\(355\) −12.7279 −0.675528
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.2426 −0.540586 −0.270293 0.962778i \(-0.587121\pi\)
−0.270293 + 0.962778i \(0.587121\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.757359 0.0396420
\(366\) 0 0
\(367\) 10.2721 0.536198 0.268099 0.963391i \(-0.413605\pi\)
0.268099 + 0.963391i \(0.413605\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.72792 −0.400137 −0.200068 0.979782i \(-0.564116\pi\)
−0.200068 + 0.979782i \(0.564116\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −53.6985 −2.76561
\(378\) 0 0
\(379\) 30.4558 1.56441 0.782206 0.623020i \(-0.214095\pi\)
0.782206 + 0.623020i \(0.214095\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 25.4558 1.30073 0.650366 0.759621i \(-0.274615\pi\)
0.650366 + 0.759621i \(0.274615\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −21.2132 −1.07555 −0.537776 0.843088i \(-0.680735\pi\)
−0.537776 + 0.843088i \(0.680735\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −17.2426 −0.865383 −0.432692 0.901542i \(-0.642436\pi\)
−0.432692 + 0.901542i \(0.642436\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.4853 1.02299 0.511493 0.859287i \(-0.329093\pi\)
0.511493 + 0.859287i \(0.329093\pi\)
\(402\) 0 0
\(403\) −39.2426 −1.95482
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.2426 1.10253
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.75736 0.0862654
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.48528 −0.121414 −0.0607070 0.998156i \(-0.519336\pi\)
−0.0607070 + 0.998156i \(0.519336\pi\)
\(420\) 0 0
\(421\) 14.5147 0.707404 0.353702 0.935358i \(-0.384923\pi\)
0.353702 + 0.935358i \(0.384923\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24264 0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.4558 1.51517 0.757587 0.652734i \(-0.226378\pi\)
0.757587 + 0.652734i \(0.226378\pi\)
\(432\) 0 0
\(433\) 24.7574 1.18976 0.594881 0.803814i \(-0.297199\pi\)
0.594881 + 0.803814i \(0.297199\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 29.6985 1.42067
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.51472 −0.166989 −0.0834947 0.996508i \(-0.526608\pi\)
−0.0834947 + 0.996508i \(0.526608\pi\)
\(444\) 0 0
\(445\) 1.75736 0.0833068
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.2132 −0.477754 −0.238877 0.971050i \(-0.576779\pi\)
−0.238877 + 0.971050i \(0.576779\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.72792 0.313351 0.156675 0.987650i \(-0.449922\pi\)
0.156675 + 0.987650i \(0.449922\pi\)
\(462\) 0 0
\(463\) 35.7279 1.66042 0.830209 0.557453i \(-0.188221\pi\)
0.830209 + 0.557453i \(0.188221\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.9706 −1.06295 −0.531475 0.847074i \(-0.678362\pi\)
−0.531475 + 0.847074i \(0.678362\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 22.2426 1.02272
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 27.4853 1.25322
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.4853 0.748558
\(486\) 0 0
\(487\) −12.2721 −0.556101 −0.278050 0.960566i \(-0.589688\pi\)
−0.278050 + 0.960566i \(0.589688\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −32.4853 −1.46604 −0.733020 0.680207i \(-0.761890\pi\)
−0.733020 + 0.680207i \(0.761890\pi\)
\(492\) 0 0
\(493\) 43.4558 1.95715
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.51472 0.112574 0.0562871 0.998415i \(-0.482074\pi\)
0.0562871 + 0.998415i \(0.482074\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.51472 0.424240 0.212120 0.977244i \(-0.431963\pi\)
0.212120 + 0.977244i \(0.431963\pi\)
\(504\) 0 0
\(505\) 16.2426 0.722788
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −31.4558 −1.39426 −0.697128 0.716947i \(-0.745539\pi\)
−0.697128 + 0.716947i \(0.745539\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.75736 −0.385895
\(516\) 0 0
\(517\) −25.4558 −1.11955
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −37.4558 −1.64097 −0.820485 0.571668i \(-0.806297\pi\)
−0.820485 + 0.571668i \(0.806297\pi\)
\(522\) 0 0
\(523\) 3.24264 0.141791 0.0708954 0.997484i \(-0.477414\pi\)
0.0708954 + 0.997484i \(0.477414\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.7574 1.38337
\(528\) 0 0
\(529\) −5.00000 −0.217391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 22.2426 0.963436
\(534\) 0 0
\(535\) 12.7279 0.550276
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 26.9411 1.15829 0.579145 0.815225i \(-0.303387\pi\)
0.579145 + 0.815225i \(0.303387\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.48528 0.320634
\(546\) 0 0
\(547\) −17.4558 −0.746358 −0.373179 0.927759i \(-0.621732\pi\)
−0.373179 + 0.927759i \(0.621732\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −71.6985 −3.05446
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 27.4853 1.16250
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17.6985 −0.741959 −0.370980 0.928641i \(-0.620978\pi\)
−0.370980 + 0.928641i \(0.620978\pi\)
\(570\) 0 0
\(571\) 38.9411 1.62964 0.814818 0.579717i \(-0.196837\pi\)
0.814818 + 0.579717i \(0.196837\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.24264 −0.176930
\(576\) 0 0
\(577\) 21.2426 0.884343 0.442171 0.896931i \(-0.354208\pi\)
0.442171 + 0.896931i \(0.354208\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.78680 −0.115023 −0.0575117 0.998345i \(-0.518317\pi\)
−0.0575117 + 0.998345i \(0.518317\pi\)
\(588\) 0 0
\(589\) −52.3970 −2.15898
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.2132 1.61029 0.805147 0.593076i \(-0.202087\pi\)
0.805147 + 0.593076i \(0.202087\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.5147 0.633914 0.316957 0.948440i \(-0.397339\pi\)
0.316957 + 0.948440i \(0.397339\pi\)
\(600\) 0 0
\(601\) 13.4853 0.550076 0.275038 0.961433i \(-0.411310\pi\)
0.275038 + 0.961433i \(0.411310\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −20.7574 −0.842515 −0.421258 0.906941i \(-0.638411\pi\)
−0.421258 + 0.906941i \(0.638411\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.4558 −1.27257
\(612\) 0 0
\(613\) 45.4558 1.83594 0.917972 0.396645i \(-0.129826\pi\)
0.917972 + 0.396645i \(0.129826\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.51472 0.383048 0.191524 0.981488i \(-0.438657\pi\)
0.191524 + 0.981488i \(0.438657\pi\)
\(618\) 0 0
\(619\) 21.9706 0.883071 0.441536 0.897244i \(-0.354434\pi\)
0.441536 + 0.897244i \(0.354434\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −22.2426 −0.886872
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.2426 −0.446151
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −25.7574 −1.01735 −0.508677 0.860957i \(-0.669865\pi\)
−0.508677 + 0.860957i \(0.669865\pi\)
\(642\) 0 0
\(643\) 5.72792 0.225887 0.112944 0.993601i \(-0.463972\pi\)
0.112944 + 0.993601i \(0.463972\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.75736 −0.0690889 −0.0345445 0.999403i \(-0.510998\pi\)
−0.0345445 + 0.999403i \(0.510998\pi\)
\(648\) 0 0
\(649\) −7.45584 −0.292667
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.75736 −0.0687708 −0.0343854 0.999409i \(-0.510947\pi\)
−0.0343854 + 0.999409i \(0.510947\pi\)
\(654\) 0 0
\(655\) 2.48528 0.0971080
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.02944 −0.273828 −0.136914 0.990583i \(-0.543718\pi\)
−0.136914 + 0.990583i \(0.543718\pi\)
\(660\) 0 0
\(661\) −35.9706 −1.39909 −0.699546 0.714587i \(-0.746615\pi\)
−0.699546 + 0.714587i \(0.746615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −43.4558 −1.68262
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 52.9706 2.04491
\(672\) 0 0
\(673\) 4.27208 0.164677 0.0823383 0.996604i \(-0.473761\pi\)
0.0823383 + 0.996604i \(0.473761\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.7279 0.489174 0.244587 0.969627i \(-0.421348\pi\)
0.244587 + 0.969627i \(0.421348\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.21320 0.352533 0.176267 0.984342i \(-0.443598\pi\)
0.176267 + 0.984342i \(0.443598\pi\)
\(684\) 0 0
\(685\) 4.24264 0.162103
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −44.4853 −1.69475
\(690\) 0 0
\(691\) −40.9411 −1.55747 −0.778737 0.627351i \(-0.784139\pi\)
−0.778737 + 0.627351i \(0.784139\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.48528 0.0563399
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −51.2132 −1.93430 −0.967148 0.254214i \(-0.918183\pi\)
−0.967148 + 0.254214i \(0.918183\pi\)
\(702\) 0 0
\(703\) 36.6985 1.38411
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.5147 −0.732891 −0.366445 0.930440i \(-0.619425\pi\)
−0.366445 + 0.930440i \(0.619425\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −31.7574 −1.18932
\(714\) 0 0
\(715\) 22.2426 0.831828
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9.51472 −0.354839 −0.177420 0.984135i \(-0.556775\pi\)
−0.177420 + 0.984135i \(0.556775\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.2426 0.380402
\(726\) 0 0
\(727\) 9.24264 0.342791 0.171395 0.985202i \(-0.445172\pi\)
0.171395 + 0.985202i \(0.445172\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −22.2426 −0.822674
\(732\) 0 0
\(733\) 44.2132 1.63305 0.816526 0.577309i \(-0.195897\pi\)
0.816526 + 0.577309i \(0.195897\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.7574 −0.506759
\(738\) 0 0
\(739\) 1.48528 0.0546370 0.0273185 0.999627i \(-0.491303\pi\)
0.0273185 + 0.999627i \(0.491303\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.2132 0.998356 0.499178 0.866500i \(-0.333635\pi\)
0.499178 + 0.866500i \(0.333635\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −22.9411 −0.837134 −0.418567 0.908186i \(-0.637467\pi\)
−0.418567 + 0.908186i \(0.637467\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.51472 0.200701
\(756\) 0 0
\(757\) 42.9706 1.56179 0.780896 0.624661i \(-0.214763\pi\)
0.780896 + 0.624661i \(0.214763\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −37.7574 −1.36870 −0.684352 0.729152i \(-0.739915\pi\)
−0.684352 + 0.729152i \(0.739915\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.21320 −0.332669
\(768\) 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
\(772\) 0 0
\(773\) 47.6985 1.71560 0.857798 0.513988i \(-0.171832\pi\)
0.857798 + 0.513988i \(0.171832\pi\)
\(774\) 0 0
\(775\) 7.48528 0.268879
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 29.6985 1.06406
\(780\) 0 0
\(781\) 54.0000 1.93227
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.4853 0.374236
\(786\) 0 0
\(787\) 11.5147 0.410455 0.205228 0.978714i \(-0.434207\pi\)
0.205228 + 0.978714i \(0.434207\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 65.4558 2.32441
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.2426 1.42547 0.712734 0.701435i \(-0.247457\pi\)
0.712734 + 0.701435i \(0.247457\pi\)
\(798\) 0 0
\(799\) 25.4558 0.900563
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.21320 −0.113391
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 39.9411 1.40426 0.702128 0.712051i \(-0.252234\pi\)
0.702128 + 0.712051i \(0.252234\pi\)
\(810\) 0 0
\(811\) −55.9411 −1.96436 −0.982179 0.187946i \(-0.939817\pi\)
−0.982179 + 0.187946i \(0.939817\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 36.6985 1.28392
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 53.6985 1.87409 0.937045 0.349209i \(-0.113550\pi\)
0.937045 + 0.349209i \(0.113550\pi\)
\(822\) 0 0
\(823\) −5.02944 −0.175315 −0.0876576 0.996151i \(-0.527938\pi\)
−0.0876576 + 0.996151i \(0.527938\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.45584 0.259265 0.129633 0.991562i \(-0.458620\pi\)
0.129633 + 0.991562i \(0.458620\pi\)
\(828\) 0 0
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 6.72792 0.232829
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.2426 −1.18219 −0.591094 0.806603i \(-0.701304\pi\)
−0.591094 + 0.806603i \(0.701304\pi\)
\(840\) 0 0
\(841\) 75.9117 2.61764
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.4853 0.498309
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 22.2426 0.762468
\(852\) 0 0
\(853\) −0.272078 −0.00931577 −0.00465789 0.999989i \(-0.501483\pi\)
−0.00465789 + 0.999989i \(0.501483\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.24264 0.144926 0.0724629 0.997371i \(-0.476914\pi\)
0.0724629 + 0.997371i \(0.476914\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.4853 1.31005 0.655027 0.755605i \(-0.272657\pi\)
0.655027 + 0.755605i \(0.272657\pi\)
\(864\) 0 0
\(865\) 3.51472 0.119504
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −46.6690 −1.58314
\(870\) 0 0
\(871\) −17.0000 −0.576023
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7.02944 −0.236828 −0.118414 0.992964i \(-0.537781\pi\)
−0.118414 + 0.992964i \(0.537781\pi\)
\(882\) 0 0
\(883\) −19.7279 −0.663897 −0.331949 0.943297i \(-0.607706\pi\)
−0.331949 + 0.943297i \(0.607706\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.2132 0.913730 0.456865 0.889536i \(-0.348972\pi\)
0.456865 + 0.889536i \(0.348972\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −42.0000 −1.40548
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 76.6690 2.55706
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13.0000 −0.432135
\(906\) 0 0
\(907\) −55.7279 −1.85042 −0.925208 0.379461i \(-0.876109\pi\)
−0.925208 + 0.379461i \(0.876109\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.78680 −0.291120 −0.145560 0.989349i \(-0.546498\pi\)
−0.145560 + 0.989349i \(0.546498\pi\)
\(912\) 0 0
\(913\) −7.45584 −0.246752
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.0294373 0.000971045 0 0.000485523 1.00000i \(-0.499845\pi\)
0.000485523 1.00000i \(0.499845\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 66.7279 2.19638
\(924\) 0 0
\(925\) −5.24264 −0.172377
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −11.2721 −0.369825 −0.184912 0.982755i \(-0.559200\pi\)
−0.184912 + 0.982755i \(0.559200\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) 40.6985 1.32956 0.664781 0.747039i \(-0.268525\pi\)
0.664781 + 0.747039i \(0.268525\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.6985 −0.381360 −0.190680 0.981652i \(-0.561069\pi\)
−0.190680 + 0.981652i \(0.561069\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −39.2132 −1.27426 −0.637129 0.770757i \(-0.719878\pi\)
−0.637129 + 0.770757i \(0.719878\pi\)
\(948\) 0 0
\(949\) −3.97056 −0.128890
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44.4853 1.44102 0.720510 0.693445i \(-0.243908\pi\)
0.720510 + 0.693445i \(0.243908\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 25.0294 0.807401
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.2426 0.490678
\(966\) 0 0
\(967\) −26.7574 −0.860459 −0.430229 0.902720i \(-0.641567\pi\)
−0.430229 + 0.902720i \(0.641567\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.0294 −0.610684 −0.305342 0.952243i \(-0.598771\pi\)
−0.305342 + 0.952243i \(0.598771\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51.2132 1.63846 0.819228 0.573468i \(-0.194402\pi\)
0.819228 + 0.573468i \(0.194402\pi\)
\(978\) 0 0
\(979\) −7.45584 −0.238290
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.6985 −0.564494 −0.282247 0.959342i \(-0.591080\pi\)
−0.282247 + 0.959342i \(0.591080\pi\)
\(984\) 0 0
\(985\) 7.75736 0.247170
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 22.2426 0.707275
\(990\) 0 0
\(991\) −52.9411 −1.68173 −0.840865 0.541245i \(-0.817953\pi\)
−0.840865 + 0.541245i \(0.817953\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.48528 −0.205597
\(996\) 0 0
\(997\) −0.272078 −0.00861679 −0.00430840 0.999991i \(-0.501371\pi\)
−0.00430840 + 0.999991i \(0.501371\pi\)
\(998\) 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 8820.2.a.bk.1.1 2
3.2 odd 2 2940.2.a.r.1.2 2
7.2 even 3 1260.2.s.e.361.1 4
7.4 even 3 1260.2.s.e.541.1 4
7.6 odd 2 8820.2.a.bf.1.1 2
21.2 odd 6 420.2.q.d.361.1 yes 4
21.5 even 6 2940.2.q.q.361.1 4
21.11 odd 6 420.2.q.d.121.1 4
21.17 even 6 2940.2.q.q.961.1 4
21.20 even 2 2940.2.a.p.1.2 2
84.11 even 6 1680.2.bg.t.961.2 4
84.23 even 6 1680.2.bg.t.1201.2 4
105.2 even 12 2100.2.bc.f.949.3 8
105.23 even 12 2100.2.bc.f.949.2 8
105.32 even 12 2100.2.bc.f.1549.2 8
105.44 odd 6 2100.2.q.k.1201.2 4
105.53 even 12 2100.2.bc.f.1549.3 8
105.74 odd 6 2100.2.q.k.1801.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.d.121.1 4 21.11 odd 6
420.2.q.d.361.1 yes 4 21.2 odd 6
1260.2.s.e.361.1 4 7.2 even 3
1260.2.s.e.541.1 4 7.4 even 3
1680.2.bg.t.961.2 4 84.11 even 6
1680.2.bg.t.1201.2 4 84.23 even 6
2100.2.q.k.1201.2 4 105.44 odd 6
2100.2.q.k.1801.2 4 105.74 odd 6
2100.2.bc.f.949.2 8 105.23 even 12
2100.2.bc.f.949.3 8 105.2 even 12
2100.2.bc.f.1549.2 8 105.32 even 12
2100.2.bc.f.1549.3 8 105.53 even 12
2940.2.a.p.1.2 2 21.20 even 2
2940.2.a.r.1.2 2 3.2 odd 2
2940.2.q.q.361.1 4 21.5 even 6
2940.2.q.q.961.1 4 21.17 even 6
8820.2.a.bf.1.1 2 7.6 odd 2
8820.2.a.bk.1.1 2 1.1 even 1 trivial