Properties

Label 6840.2.a.w.1.2
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +1.64575 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +1.64575 q^{7} +0.354249 q^{11} +3.64575 q^{13} -1.00000 q^{19} -2.00000 q^{23} +1.00000 q^{25} -9.64575 q^{29} -2.00000 q^{31} -1.64575 q^{35} -6.93725 q^{37} -1.64575 q^{41} -4.93725 q^{43} -6.00000 q^{47} -4.29150 q^{49} -4.00000 q^{53} -0.354249 q^{55} -3.29150 q^{59} +11.2915 q^{61} -3.64575 q^{65} +4.00000 q^{67} +3.29150 q^{71} -2.00000 q^{73} +0.583005 q^{77} -8.00000 q^{79} +15.8745 q^{83} +12.2288 q^{89} +6.00000 q^{91} +1.00000 q^{95} +18.2288 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} + 6 q^{11} + 2 q^{13} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 14 q^{29} - 4 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{41} + 6 q^{43} - 12 q^{47} + 2 q^{49} - 8 q^{53} - 6 q^{55} + 4 q^{59} + 12 q^{61} - 2 q^{65} + 8 q^{67} - 4 q^{71} - 4 q^{73} - 20 q^{77} - 16 q^{79} - 2 q^{89} + 12 q^{91} + 2 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 1.64575 0.622036 0.311018 0.950404i \(-0.399330\pi\)
0.311018 + 0.950404i \(0.399330\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.354249 0.106810 0.0534050 0.998573i \(-0.482993\pi\)
0.0534050 + 0.998573i \(0.482993\pi\)
\(12\) 0 0
\(13\) 3.64575 1.01115 0.505575 0.862783i \(-0.331280\pi\)
0.505575 + 0.862783i \(0.331280\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.64575 −1.79117 −0.895586 0.444889i \(-0.853243\pi\)
−0.895586 + 0.444889i \(0.853243\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.64575 −0.278183
\(36\) 0 0
\(37\) −6.93725 −1.14048 −0.570239 0.821479i \(-0.693149\pi\)
−0.570239 + 0.821479i \(0.693149\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.64575 −0.257023 −0.128512 0.991708i \(-0.541020\pi\)
−0.128512 + 0.991708i \(0.541020\pi\)
\(42\) 0 0
\(43\) −4.93725 −0.752924 −0.376462 0.926432i \(-0.622859\pi\)
−0.376462 + 0.926432i \(0.622859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −4.29150 −0.613072
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −0.354249 −0.0477669
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.29150 −0.428517 −0.214259 0.976777i \(-0.568734\pi\)
−0.214259 + 0.976777i \(0.568734\pi\)
\(60\) 0 0
\(61\) 11.2915 1.44573 0.722864 0.690990i \(-0.242825\pi\)
0.722864 + 0.690990i \(0.242825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.64575 −0.452200
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.29150 0.390629 0.195315 0.980741i \(-0.437427\pi\)
0.195315 + 0.980741i \(0.437427\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.583005 0.0664396
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.8745 1.74245 0.871227 0.490881i \(-0.163325\pi\)
0.871227 + 0.490881i \(0.163325\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2288 1.29625 0.648123 0.761536i \(-0.275554\pi\)
0.648123 + 0.761536i \(0.275554\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 18.2288 1.85085 0.925425 0.378931i \(-0.123708\pi\)
0.925425 + 0.378931i \(0.123708\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.8745 −1.57957 −0.789786 0.613382i \(-0.789809\pi\)
−0.789786 + 0.613382i \(0.789809\pi\)
\(102\) 0 0
\(103\) 14.5830 1.43691 0.718453 0.695575i \(-0.244850\pi\)
0.718453 + 0.695575i \(0.244850\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.5830 −1.40979 −0.704896 0.709311i \(-0.749006\pi\)
−0.704896 + 0.709311i \(0.749006\pi\)
\(108\) 0 0
\(109\) 5.29150 0.506834 0.253417 0.967357i \(-0.418446\pi\)
0.253417 + 0.967357i \(0.418446\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −17.8745 −1.68149 −0.840746 0.541430i \(-0.817883\pi\)
−0.840746 + 0.541430i \(0.817883\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.8745 −0.988592
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.58301 −0.229205 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.64575 −0.668012 −0.334006 0.942571i \(-0.608401\pi\)
−0.334006 + 0.942571i \(0.608401\pi\)
\(132\) 0 0
\(133\) −1.64575 −0.142705
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.583005 0.0498095 0.0249047 0.999690i \(-0.492072\pi\)
0.0249047 + 0.999690i \(0.492072\pi\)
\(138\) 0 0
\(139\) −17.8745 −1.51610 −0.758048 0.652199i \(-0.773847\pi\)
−0.758048 + 0.652199i \(0.773847\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.29150 0.108001
\(144\) 0 0
\(145\) 9.64575 0.801036
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 15.8745 1.26692 0.633462 0.773774i \(-0.281633\pi\)
0.633462 + 0.773774i \(0.281633\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.29150 −0.259407
\(162\) 0 0
\(163\) 9.64575 0.755514 0.377757 0.925905i \(-0.376696\pi\)
0.377757 + 0.925905i \(0.376696\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.29150 0.0999395 0.0499697 0.998751i \(-0.484088\pi\)
0.0499697 + 0.998751i \(0.484088\pi\)
\(168\) 0 0
\(169\) 0.291503 0.0224233
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.70850 −0.662095 −0.331047 0.943614i \(-0.607402\pi\)
−0.331047 + 0.943614i \(0.607402\pi\)
\(174\) 0 0
\(175\) 1.64575 0.124407
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.29150 0.544992 0.272496 0.962157i \(-0.412151\pi\)
0.272496 + 0.962157i \(0.412151\pi\)
\(180\) 0 0
\(181\) −5.29150 −0.393314 −0.196657 0.980472i \(-0.563009\pi\)
−0.196657 + 0.980472i \(0.563009\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.93725 0.510037
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.9373 −0.791392 −0.395696 0.918382i \(-0.629497\pi\)
−0.395696 + 0.918382i \(0.629497\pi\)
\(192\) 0 0
\(193\) 2.93725 0.211428 0.105714 0.994397i \(-0.466287\pi\)
0.105714 + 0.994397i \(0.466287\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −23.2915 −1.65109 −0.825545 0.564336i \(-0.809132\pi\)
−0.825545 + 0.564336i \(0.809132\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −15.8745 −1.11417
\(204\) 0 0
\(205\) 1.64575 0.114944
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.354249 −0.0245039
\(210\) 0 0
\(211\) −10.5830 −0.728564 −0.364282 0.931289i \(-0.618686\pi\)
−0.364282 + 0.931289i \(0.618686\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.93725 0.336718
\(216\) 0 0
\(217\) −3.29150 −0.223442
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.41699 −0.0948890 −0.0474445 0.998874i \(-0.515108\pi\)
−0.0474445 + 0.998874i \(0.515108\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −25.2915 −1.67866 −0.839328 0.543625i \(-0.817051\pi\)
−0.839328 + 0.543625i \(0.817051\pi\)
\(228\) 0 0
\(229\) 7.29150 0.481836 0.240918 0.970545i \(-0.422552\pi\)
0.240918 + 0.970545i \(0.422552\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.0627 0.844959 0.422479 0.906373i \(-0.361160\pi\)
0.422479 + 0.906373i \(0.361160\pi\)
\(240\) 0 0
\(241\) −4.58301 −0.295217 −0.147609 0.989046i \(-0.547158\pi\)
−0.147609 + 0.989046i \(0.547158\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.29150 0.274174
\(246\) 0 0
\(247\) −3.64575 −0.231974
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.22876 0.393156 0.196578 0.980488i \(-0.437017\pi\)
0.196578 + 0.980488i \(0.437017\pi\)
\(252\) 0 0
\(253\) −0.708497 −0.0445428
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.87451 0.615955 0.307977 0.951394i \(-0.400348\pi\)
0.307977 + 0.951394i \(0.400348\pi\)
\(258\) 0 0
\(259\) −11.4170 −0.709418
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.29150 0.0796375 0.0398187 0.999207i \(-0.487322\pi\)
0.0398187 + 0.999207i \(0.487322\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.93725 −0.301030 −0.150515 0.988608i \(-0.548093\pi\)
−0.150515 + 0.988608i \(0.548093\pi\)
\(270\) 0 0
\(271\) 12.7085 0.771986 0.385993 0.922502i \(-0.373859\pi\)
0.385993 + 0.922502i \(0.373859\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.354249 0.0213620
\(276\) 0 0
\(277\) −24.5830 −1.47705 −0.738525 0.674226i \(-0.764477\pi\)
−0.738525 + 0.674226i \(0.764477\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.93725 0.294532 0.147266 0.989097i \(-0.452953\pi\)
0.147266 + 0.989097i \(0.452953\pi\)
\(282\) 0 0
\(283\) −3.77124 −0.224177 −0.112089 0.993698i \(-0.535754\pi\)
−0.112089 + 0.993698i \(0.535754\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.70850 −0.159878
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.8745 1.04424 0.522120 0.852872i \(-0.325141\pi\)
0.522120 + 0.852872i \(0.325141\pi\)
\(294\) 0 0
\(295\) 3.29150 0.191639
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.29150 −0.421678
\(300\) 0 0
\(301\) −8.12549 −0.468346
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.2915 −0.646550
\(306\) 0 0
\(307\) 12.7085 0.725312 0.362656 0.931923i \(-0.381870\pi\)
0.362656 + 0.931923i \(0.381870\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.35425 0.246907 0.123453 0.992350i \(-0.460603\pi\)
0.123453 + 0.992350i \(0.460603\pi\)
\(312\) 0 0
\(313\) −18.7085 −1.05747 −0.528733 0.848788i \(-0.677333\pi\)
−0.528733 + 0.848788i \(0.677333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.5830 −1.26839 −0.634194 0.773174i \(-0.718668\pi\)
−0.634194 + 0.773174i \(0.718668\pi\)
\(318\) 0 0
\(319\) −3.41699 −0.191315
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.64575 0.202230
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.87451 −0.544399
\(330\) 0 0
\(331\) 16.5830 0.911484 0.455742 0.890112i \(-0.349374\pi\)
0.455742 + 0.890112i \(0.349374\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −22.2288 −1.21088 −0.605439 0.795892i \(-0.707002\pi\)
−0.605439 + 0.795892i \(0.707002\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −0.708497 −0.0383673
\(342\) 0 0
\(343\) −18.5830 −1.00339
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.29150 −0.498794 −0.249397 0.968401i \(-0.580232\pi\)
−0.249397 + 0.968401i \(0.580232\pi\)
\(348\) 0 0
\(349\) −27.1660 −1.45416 −0.727082 0.686551i \(-0.759124\pi\)
−0.727082 + 0.686551i \(0.759124\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 35.1660 1.87170 0.935849 0.352401i \(-0.114635\pi\)
0.935849 + 0.352401i \(0.114635\pi\)
\(354\) 0 0
\(355\) −3.29150 −0.174695
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.2288 0.539853 0.269927 0.962881i \(-0.413001\pi\)
0.269927 + 0.962881i \(0.413001\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 0.228757 0.0119410 0.00597050 0.999982i \(-0.498100\pi\)
0.00597050 + 0.999982i \(0.498100\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.58301 −0.341773
\(372\) 0 0
\(373\) −8.35425 −0.432567 −0.216283 0.976331i \(-0.569393\pi\)
−0.216283 + 0.976331i \(0.569393\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −35.1660 −1.81114
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.41699 0.276796 0.138398 0.990377i \(-0.455805\pi\)
0.138398 + 0.990377i \(0.455805\pi\)
\(384\) 0 0
\(385\) −0.583005 −0.0297127
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.35425 −0.317316 −0.158658 0.987334i \(-0.550717\pi\)
−0.158658 + 0.987334i \(0.550717\pi\)
\(402\) 0 0
\(403\) −7.29150 −0.363216
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.45751 −0.121814
\(408\) 0 0
\(409\) −6.70850 −0.331714 −0.165857 0.986150i \(-0.553039\pi\)
−0.165857 + 0.986150i \(0.553039\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.41699 −0.266553
\(414\) 0 0
\(415\) −15.8745 −0.779249
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 38.9373 1.90221 0.951105 0.308869i \(-0.0999504\pi\)
0.951105 + 0.308869i \(0.0999504\pi\)
\(420\) 0 0
\(421\) −0.583005 −0.0284139 −0.0142070 0.999899i \(-0.504522\pi\)
−0.0142070 + 0.999899i \(0.504522\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.5830 0.899295
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 32.4575 1.56342 0.781712 0.623640i \(-0.214347\pi\)
0.781712 + 0.623640i \(0.214347\pi\)
\(432\) 0 0
\(433\) −26.2288 −1.26047 −0.630237 0.776403i \(-0.717042\pi\)
−0.630237 + 0.776403i \(0.717042\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) 5.16601 0.246560 0.123280 0.992372i \(-0.460659\pi\)
0.123280 + 0.992372i \(0.460659\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.58301 −0.407791 −0.203895 0.978993i \(-0.565360\pi\)
−0.203895 + 0.978993i \(0.565360\pi\)
\(444\) 0 0
\(445\) −12.2288 −0.579699
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.35425 0.111104 0.0555519 0.998456i \(-0.482308\pi\)
0.0555519 + 0.998456i \(0.482308\pi\)
\(450\) 0 0
\(451\) −0.583005 −0.0274526
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 9.29150 0.434638 0.217319 0.976101i \(-0.430269\pi\)
0.217319 + 0.976101i \(0.430269\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −7.77124 −0.361160 −0.180580 0.983560i \(-0.557798\pi\)
−0.180580 + 0.983560i \(0.557798\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.583005 0.0269783 0.0134891 0.999909i \(-0.495706\pi\)
0.0134891 + 0.999909i \(0.495706\pi\)
\(468\) 0 0
\(469\) 6.58301 0.303975
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.74902 −0.0804198
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.5203 0.617756 0.308878 0.951102i \(-0.400046\pi\)
0.308878 + 0.951102i \(0.400046\pi\)
\(480\) 0 0
\(481\) −25.2915 −1.15319
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −18.2288 −0.827725
\(486\) 0 0
\(487\) 3.29150 0.149152 0.0745761 0.997215i \(-0.476240\pi\)
0.0745761 + 0.997215i \(0.476240\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.8118 1.48077 0.740387 0.672181i \(-0.234642\pi\)
0.740387 + 0.672181i \(0.234642\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.41699 0.242985
\(498\) 0 0
\(499\) 19.2915 0.863606 0.431803 0.901968i \(-0.357878\pi\)
0.431803 + 0.901968i \(0.357878\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.29150 0.0575853 0.0287926 0.999585i \(-0.490834\pi\)
0.0287926 + 0.999585i \(0.490834\pi\)
\(504\) 0 0
\(505\) 15.8745 0.706406
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −34.3542 −1.52273 −0.761363 0.648326i \(-0.775469\pi\)
−0.761363 + 0.648326i \(0.775469\pi\)
\(510\) 0 0
\(511\) −3.29150 −0.145608
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −14.5830 −0.642604
\(516\) 0 0
\(517\) −2.12549 −0.0934790
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.8118 −1.17464 −0.587322 0.809353i \(-0.699818\pi\)
−0.587322 + 0.809353i \(0.699818\pi\)
\(522\) 0 0
\(523\) 6.12549 0.267849 0.133925 0.990992i \(-0.457242\pi\)
0.133925 + 0.990992i \(0.457242\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 14.5830 0.630478
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.52026 −0.0654822
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.29150 −0.226663
\(546\) 0 0
\(547\) −40.4575 −1.72984 −0.864919 0.501911i \(-0.832630\pi\)
−0.864919 + 0.501911i \(0.832630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.64575 0.410923
\(552\) 0 0
\(553\) −13.1660 −0.559876
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.87451 0.163291 0.0816455 0.996661i \(-0.473982\pi\)
0.0816455 + 0.996661i \(0.473982\pi\)
\(564\) 0 0
\(565\) 17.8745 0.751986
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.479741 −0.0201118 −0.0100559 0.999949i \(-0.503201\pi\)
−0.0100559 + 0.999949i \(0.503201\pi\)
\(570\) 0 0
\(571\) 13.8745 0.580630 0.290315 0.956931i \(-0.406240\pi\)
0.290315 + 0.956931i \(0.406240\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) 27.8745 1.16043 0.580215 0.814463i \(-0.302968\pi\)
0.580215 + 0.814463i \(0.302968\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 26.1255 1.08387
\(582\) 0 0
\(583\) −1.41699 −0.0586859
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.5830 −1.50994 −0.754971 0.655758i \(-0.772349\pi\)
−0.754971 + 0.655758i \(0.772349\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.4170 0.633100 0.316550 0.948576i \(-0.397475\pi\)
0.316550 + 0.948576i \(0.397475\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.41699 0.221332 0.110666 0.993858i \(-0.464702\pi\)
0.110666 + 0.993858i \(0.464702\pi\)
\(600\) 0 0
\(601\) −15.8745 −0.647535 −0.323767 0.946137i \(-0.604950\pi\)
−0.323767 + 0.946137i \(0.604950\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.8745 0.442112
\(606\) 0 0
\(607\) −12.4575 −0.505635 −0.252817 0.967514i \(-0.581357\pi\)
−0.252817 + 0.967514i \(0.581357\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21.8745 −0.884948
\(612\) 0 0
\(613\) −14.4575 −0.583933 −0.291967 0.956428i \(-0.594310\pi\)
−0.291967 + 0.956428i \(0.594310\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −37.8745 −1.52231 −0.761153 0.648573i \(-0.775366\pi\)
−0.761153 + 0.648573i \(0.775366\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 20.1255 0.806311
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 37.1660 1.47956 0.739778 0.672851i \(-0.234931\pi\)
0.739778 + 0.672851i \(0.234931\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.58301 0.102503
\(636\) 0 0
\(637\) −15.6458 −0.619907
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.2288 0.798988 0.399494 0.916736i \(-0.369186\pi\)
0.399494 + 0.916736i \(0.369186\pi\)
\(642\) 0 0
\(643\) 44.9373 1.77215 0.886076 0.463540i \(-0.153421\pi\)
0.886076 + 0.463540i \(0.153421\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.2915 0.994312 0.497156 0.867661i \(-0.334378\pi\)
0.497156 + 0.867661i \(0.334378\pi\)
\(648\) 0 0
\(649\) −1.16601 −0.0457699
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 0 0
\(655\) 7.64575 0.298744
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13.1660 −0.512875 −0.256437 0.966561i \(-0.582549\pi\)
−0.256437 + 0.966561i \(0.582549\pi\)
\(660\) 0 0
\(661\) −5.29150 −0.205816 −0.102908 0.994691i \(-0.532815\pi\)
−0.102908 + 0.994691i \(0.532815\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.64575 0.0638195
\(666\) 0 0
\(667\) 19.2915 0.746970
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 9.52026 0.366979 0.183490 0.983022i \(-0.441261\pi\)
0.183490 + 0.983022i \(0.441261\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.00000 0.153732 0.0768662 0.997041i \(-0.475509\pi\)
0.0768662 + 0.997041i \(0.475509\pi\)
\(678\) 0 0
\(679\) 30.0000 1.15129
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) −0.583005 −0.0222755
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.5830 −0.555568
\(690\) 0 0
\(691\) 31.2915 1.19038 0.595192 0.803583i \(-0.297076\pi\)
0.595192 + 0.803583i \(0.297076\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.8745 0.678019
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) 6.93725 0.261643
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −26.1255 −0.982550
\(708\) 0 0
\(709\) −32.4575 −1.21897 −0.609484 0.792799i \(-0.708623\pi\)
−0.609484 + 0.792799i \(0.708623\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) −1.29150 −0.0482995
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.64575 0.285138 0.142569 0.989785i \(-0.454464\pi\)
0.142569 + 0.989785i \(0.454464\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −9.64575 −0.358234
\(726\) 0 0
\(727\) −39.9778 −1.48269 −0.741347 0.671122i \(-0.765813\pi\)
−0.741347 + 0.671122i \(0.765813\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −32.5830 −1.20348 −0.601740 0.798692i \(-0.705526\pi\)
−0.601740 + 0.798692i \(0.705526\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41699 0.0521957
\(738\) 0 0
\(739\) −10.5830 −0.389302 −0.194651 0.980873i \(-0.562357\pi\)
−0.194651 + 0.980873i \(0.562357\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.7490 1.16476 0.582379 0.812917i \(-0.302122\pi\)
0.582379 + 0.812917i \(0.302122\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 36.3320 1.32577 0.662887 0.748719i \(-0.269331\pi\)
0.662887 + 0.748719i \(0.269331\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −19.8745 −0.722351 −0.361176 0.932498i \(-0.617625\pi\)
−0.361176 + 0.932498i \(0.617625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11.1660 0.404768 0.202384 0.979306i \(-0.435131\pi\)
0.202384 + 0.979306i \(0.435131\pi\)
\(762\) 0 0
\(763\) 8.70850 0.315269
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 21.7490 0.784290 0.392145 0.919904i \(-0.371733\pi\)
0.392145 + 0.919904i \(0.371733\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.64575 0.0589652
\(780\) 0 0
\(781\) 1.16601 0.0417231
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −15.8745 −0.566585
\(786\) 0 0
\(787\) 11.2915 0.402499 0.201249 0.979540i \(-0.435500\pi\)
0.201249 + 0.979540i \(0.435500\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −29.4170 −1.04595
\(792\) 0 0
\(793\) 41.1660 1.46185
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −39.2915 −1.39178 −0.695888 0.718150i \(-0.744989\pi\)
−0.695888 + 0.718150i \(0.744989\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.708497 −0.0250023
\(804\) 0 0
\(805\) 3.29150 0.116010
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.58301 0.301762 0.150881 0.988552i \(-0.451789\pi\)
0.150881 + 0.988552i \(0.451789\pi\)
\(810\) 0 0
\(811\) 5.41699 0.190216 0.0951082 0.995467i \(-0.469680\pi\)
0.0951082 + 0.995467i \(0.469680\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9.64575 −0.337876
\(816\) 0 0
\(817\) 4.93725 0.172733
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14.7085 −0.513330 −0.256665 0.966500i \(-0.582624\pi\)
−0.256665 + 0.966500i \(0.582624\pi\)
\(822\) 0 0
\(823\) −43.2693 −1.50827 −0.754136 0.656718i \(-0.771944\pi\)
−0.754136 + 0.656718i \(0.771944\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.7085 −0.789652 −0.394826 0.918756i \(-0.629195\pi\)
−0.394826 + 0.918756i \(0.629195\pi\)
\(828\) 0 0
\(829\) −27.8745 −0.968122 −0.484061 0.875034i \(-0.660839\pi\)
−0.484061 + 0.875034i \(0.660839\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.29150 −0.0446943
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 28.7085 0.991127 0.495564 0.868572i \(-0.334962\pi\)
0.495564 + 0.868572i \(0.334962\pi\)
\(840\) 0 0
\(841\) 64.0405 2.20829
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.291503 −0.0100280
\(846\) 0 0
\(847\) −17.8967 −0.614939
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.8745 0.475612
\(852\) 0 0
\(853\) −45.2915 −1.55075 −0.775376 0.631500i \(-0.782440\pi\)
−0.775376 + 0.631500i \(0.782440\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.7490 0.811251 0.405625 0.914039i \(-0.367054\pi\)
0.405625 + 0.914039i \(0.367054\pi\)
\(858\) 0 0
\(859\) 2.83399 0.0966945 0.0483472 0.998831i \(-0.484605\pi\)
0.0483472 + 0.998831i \(0.484605\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.16601 −0.0396915 −0.0198457 0.999803i \(-0.506318\pi\)
−0.0198457 + 0.999803i \(0.506318\pi\)
\(864\) 0 0
\(865\) 8.70850 0.296098
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.83399 −0.0961365
\(870\) 0 0
\(871\) 14.5830 0.494126
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.64575 −0.0556365
\(876\) 0 0
\(877\) −16.8118 −0.567693 −0.283846 0.958870i \(-0.591611\pi\)
−0.283846 + 0.958870i \(0.591611\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 38.7085 1.30412 0.652061 0.758166i \(-0.273905\pi\)
0.652061 + 0.758166i \(0.273905\pi\)
\(882\) 0 0
\(883\) 25.6458 0.863048 0.431524 0.902101i \(-0.357976\pi\)
0.431524 + 0.902101i \(0.357976\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43.7490 −1.46895 −0.734474 0.678637i \(-0.762571\pi\)
−0.734474 + 0.678637i \(0.762571\pi\)
\(888\) 0 0
\(889\) −4.25098 −0.142573
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −7.29150 −0.243728
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19.2915 0.643408
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.29150 0.175895
\(906\) 0 0
\(907\) 53.8745 1.78887 0.894437 0.447194i \(-0.147577\pi\)
0.894437 + 0.447194i \(0.147577\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.7490 −0.786840 −0.393420 0.919359i \(-0.628708\pi\)
−0.393420 + 0.919359i \(0.628708\pi\)
\(912\) 0 0
\(913\) 5.62352 0.186111
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.5830 −0.415527
\(918\) 0 0
\(919\) 18.5830 0.612997 0.306498 0.951871i \(-0.400843\pi\)
0.306498 + 0.951871i \(0.400843\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −6.93725 −0.228096
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.8745 0.520826 0.260413 0.965497i \(-0.416141\pi\)
0.260413 + 0.965497i \(0.416141\pi\)
\(930\) 0 0
\(931\) 4.29150 0.140648
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17.2915 0.564889 0.282444 0.959284i \(-0.408855\pi\)
0.282444 + 0.959284i \(0.408855\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 16.2288 0.529042 0.264521 0.964380i \(-0.414786\pi\)
0.264521 + 0.964380i \(0.414786\pi\)
\(942\) 0 0
\(943\) 3.29150 0.107186
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −26.7085 −0.867910 −0.433955 0.900935i \(-0.642882\pi\)
−0.433955 + 0.900935i \(0.642882\pi\)
\(948\) 0 0
\(949\) −7.29150 −0.236692
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 28.7085 0.929959 0.464980 0.885321i \(-0.346062\pi\)
0.464980 + 0.885321i \(0.346062\pi\)
\(954\) 0 0
\(955\) 10.9373 0.353921
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.959482 0.0309833
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.93725 −0.0945535
\(966\) 0 0
\(967\) 51.5203 1.65678 0.828390 0.560152i \(-0.189257\pi\)
0.828390 + 0.560152i \(0.189257\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 11.7490 0.377044 0.188522 0.982069i \(-0.439630\pi\)
0.188522 + 0.982069i \(0.439630\pi\)
\(972\) 0 0
\(973\) −29.4170 −0.943066
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.3320 1.61026 0.805132 0.593096i \(-0.202094\pi\)
0.805132 + 0.593096i \(0.202094\pi\)
\(978\) 0 0
\(979\) 4.33202 0.138452
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −33.2915 −1.06183 −0.530917 0.847424i \(-0.678152\pi\)
−0.530917 + 0.847424i \(0.678152\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.87451 0.313991
\(990\) 0 0
\(991\) −3.74902 −0.119091 −0.0595457 0.998226i \(-0.518965\pi\)
−0.0595457 + 0.998226i \(0.518965\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 23.2915 0.738390
\(996\) 0 0
\(997\) −53.2915 −1.68776 −0.843879 0.536533i \(-0.819734\pi\)
−0.843879 + 0.536533i \(0.819734\pi\)
\(998\) 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 6840.2.a.w.1.2 2
3.2 odd 2 2280.2.a.n.1.2 2
12.11 even 2 4560.2.a.bp.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.n.1.2 2 3.2 odd 2
4560.2.a.bp.1.1 2 12.11 even 2
6840.2.a.w.1.2 2 1.1 even 1 trivial