Properties

Label 1440.2.a.o.1.1
Level $1440$
Weight $2$
Character 1440.1
Self dual yes
Analytic conductor $11.498$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.4984578911\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1440.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -2.82843 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -2.82843 q^{7} +5.65685 q^{11} -2.00000 q^{13} -2.00000 q^{17} -2.82843 q^{23} +1.00000 q^{25} -6.00000 q^{29} +5.65685 q^{31} +2.82843 q^{35} -10.0000 q^{37} -2.00000 q^{41} -8.48528 q^{43} +2.82843 q^{47} +1.00000 q^{49} -6.00000 q^{53} -5.65685 q^{55} -11.3137 q^{59} -2.00000 q^{61} +2.00000 q^{65} -2.82843 q^{67} -5.65685 q^{71} -6.00000 q^{73} -16.0000 q^{77} +11.3137 q^{79} -2.82843 q^{83} +2.00000 q^{85} -10.0000 q^{89} +5.65685 q^{91} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} - 4 q^{13} - 4 q^{17} + 2 q^{25} - 12 q^{29} - 20 q^{37} - 4 q^{41} + 2 q^{49} - 12 q^{53} - 4 q^{61} + 4 q^{65} - 12 q^{73} - 32 q^{77} + 4 q^{85} - 20 q^{89} + 4 q^{97} + O(q^{100}) \)

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
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −2.82843 −0.345547 −0.172774 0.984962i \(-0.555273\pi\)
−0.172774 + 0.984962i \(0.555273\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 5.65685 0.592999
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 14.1421 1.39347 0.696733 0.717331i \(-0.254636\pi\)
0.696733 + 0.717331i \(0.254636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1421 1.36717 0.683586 0.729870i \(-0.260419\pi\)
0.683586 + 0.729870i \(0.260419\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 11.3137 0.959616 0.479808 0.877373i \(-0.340706\pi\)
0.479808 + 0.877373i \(0.340706\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.3137 −0.946100
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.9706 −1.38104 −0.690522 0.723311i \(-0.742619\pi\)
−0.690522 + 0.723311i \(0.742619\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.65685 −0.454369
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 14.1421 1.10770 0.553849 0.832617i \(-0.313159\pi\)
0.553849 + 0.832617i \(0.313159\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.1421 1.09435 0.547176 0.837018i \(-0.315703\pi\)
0.547176 + 0.837018i \(0.315703\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −2.82843 −0.213809
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.3137 −0.845626 −0.422813 0.906217i \(-0.638957\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −11.3137 −0.827340
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −22.6274 −1.60402 −0.802008 0.597314i \(-0.796235\pi\)
−0.802008 + 0.597314i \(0.796235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.9706 1.19110
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.9706 −1.16830 −0.584151 0.811645i \(-0.698572\pi\)
−0.584151 + 0.811645i \(0.698572\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.48528 0.578691
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −8.48528 −0.568216 −0.284108 0.958792i \(-0.591698\pi\)
−0.284108 + 0.958792i \(0.591698\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.7990 −1.31411 −0.657053 0.753845i \(-0.728197\pi\)
−0.657053 + 0.753845i \(0.728197\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −2.82843 −0.184506
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.65685 0.357057 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 28.2843 1.75750
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.7990 1.22086 0.610429 0.792071i \(-0.290997\pi\)
0.610429 + 0.792071i \(0.290997\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.9706 1.03089 0.515444 0.856923i \(-0.327627\pi\)
0.515444 + 0.856923i \(0.327627\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.65685 0.341121
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −8.48528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.65685 0.333914
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 11.3137 0.658710
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −2.82843 −0.161427 −0.0807134 0.996737i \(-0.525720\pi\)
−0.0807134 + 0.996737i \(0.525720\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2843 1.60385 0.801927 0.597422i \(-0.203808\pi\)
0.801927 + 0.597422i \(0.203808\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) −33.9411 −1.90034
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 5.65685 0.310929 0.155464 0.987841i \(-0.450313\pi\)
0.155464 + 0.987841i \(0.450313\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.82843 0.154533
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.48528 −0.455514 −0.227757 0.973718i \(-0.573139\pi\)
−0.227757 + 0.973718i \(0.573139\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.6274 −1.19423 −0.597115 0.802156i \(-0.703686\pi\)
−0.597115 + 0.802156i \(0.703686\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 8.48528 0.442928 0.221464 0.975169i \(-0.428916\pi\)
0.221464 + 0.975169i \(0.428916\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 16.9706 0.881068
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −22.6274 −1.16229 −0.581146 0.813799i \(-0.697396\pi\)
−0.581146 + 0.813799i \(0.697396\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −36.7696 −1.87884 −0.939418 0.342773i \(-0.888634\pi\)
−0.939418 + 0.342773i \(0.888634\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.3137 −0.569254
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −11.3137 −0.563576
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −56.5685 −2.80400
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 32.0000 1.57462
\(414\) 0 0
\(415\) 2.82843 0.138842
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.3137 0.552711 0.276355 0.961056i \(-0.410873\pi\)
0.276355 + 0.961056i \(0.410873\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 5.65685 0.273754
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685 0.272481 0.136241 0.990676i \(-0.456498\pi\)
0.136241 + 0.990676i \(0.456498\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 22.6274 1.07995 0.539974 0.841682i \(-0.318434\pi\)
0.539974 + 0.841682i \(0.318434\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.82843 −0.134383 −0.0671913 0.997740i \(-0.521404\pi\)
−0.0671913 + 0.997740i \(0.521404\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −11.3137 −0.532742
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.65685 −0.265197
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −8.48528 −0.394344 −0.197172 0.980369i \(-0.563176\pi\)
−0.197172 + 0.980369i \(0.563176\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.1421 0.654420 0.327210 0.944952i \(-0.393892\pi\)
0.327210 + 0.944952i \(0.393892\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.9411 1.55081 0.775405 0.631464i \(-0.217546\pi\)
0.775405 + 0.631464i \(0.217546\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 31.1127 1.40985 0.704925 0.709281i \(-0.250980\pi\)
0.704925 + 0.709281i \(0.250980\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −39.5980 −1.78703 −0.893516 0.449032i \(-0.851769\pi\)
−0.893516 + 0.449032i \(0.851769\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.48528 0.378340 0.189170 0.981944i \(-0.439420\pi\)
0.189170 + 0.981944i \(0.439420\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) 16.9706 0.750733
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −14.1421 −0.623177
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 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
\(523\) −8.48528 −0.371035 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.3137 −0.492833
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −14.1421 −0.611418
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.65685 0.243658
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) 42.4264 1.81402 0.907011 0.421107i \(-0.138358\pi\)
0.907011 + 0.421107i \(0.138358\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 16.9706 0.717778
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.7990 0.834428 0.417214 0.908808i \(-0.363007\pi\)
0.417214 + 0.908808i \(0.363007\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 28.2843 1.18366 0.591830 0.806063i \(-0.298406\pi\)
0.591830 + 0.806063i \(0.298406\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.82843 −0.117954
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −33.9411 −1.40570
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.4558 1.05068 0.525338 0.850894i \(-0.323939\pi\)
0.525338 + 0.850894i \(0.323939\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −5.65685 −0.231908
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.3137 −0.462266 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.0000 −0.853771
\(606\) 0 0
\(607\) −2.82843 −0.114802 −0.0574012 0.998351i \(-0.518281\pi\)
−0.0574012 + 0.998351i \(0.518281\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.65685 −0.228852
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −45.2548 −1.81895 −0.909473 0.415764i \(-0.863514\pi\)
−0.909473 + 0.415764i \(0.863514\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 28.2843 1.13319
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −16.9706 −0.675587 −0.337794 0.941220i \(-0.609681\pi\)
−0.337794 + 0.941220i \(0.609681\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.82843 0.112243
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) −31.1127 −1.22697 −0.613483 0.789708i \(-0.710232\pi\)
−0.613483 + 0.789708i \(0.710232\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 0 0
\(649\) −64.0000 −2.51222
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 33.9411 1.32216 0.661079 0.750316i \(-0.270099\pi\)
0.661079 + 0.750316i \(0.270099\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9706 0.657103
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.3137 −0.436761
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) −5.65685 −0.217090
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14.1421 −0.541134 −0.270567 0.962701i \(-0.587211\pi\)
−0.270567 + 0.962701i \(0.587211\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −28.2843 −1.07598 −0.537992 0.842950i \(-0.680817\pi\)
−0.537992 + 0.842950i \(0.680817\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.3137 −0.429153
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.65685 −0.212748
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 11.3137 0.423109
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −25.4558 −0.944105 −0.472052 0.881570i \(-0.656487\pi\)
−0.472052 + 0.881570i \(0.656487\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −11.3137 −0.416181 −0.208091 0.978110i \(-0.566725\pi\)
−0.208091 + 0.978110i \(0.566725\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.1421 −0.518825 −0.259412 0.965767i \(-0.583529\pi\)
−0.259412 + 0.965767i \(0.583529\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −40.0000 −1.46157
\(750\) 0 0
\(751\) −16.9706 −0.619265 −0.309632 0.950856i \(-0.600206\pi\)
−0.309632 + 0.950856i \(0.600206\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.9706 0.617622
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 50.9117 1.84313
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.6274 0.817029
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 5.65685 0.203200
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 8.48528 0.302468 0.151234 0.988498i \(-0.451675\pi\)
0.151234 + 0.988498i \(0.451675\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.65685 0.201135
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −5.65685 −0.200125
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −33.9411 −1.19776
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −5.65685 −0.198639 −0.0993195 0.995056i \(-0.531667\pi\)
−0.0993195 + 0.995056i \(0.531667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −14.1421 −0.495377
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 25.4558 0.887335 0.443667 0.896191i \(-0.353677\pi\)
0.443667 + 0.896191i \(0.353677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31.1127 −1.08189 −0.540947 0.841057i \(-0.681934\pi\)
−0.540947 + 0.841057i \(0.681934\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −14.1421 −0.489409
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.3137 −0.390593 −0.195296 0.980744i \(-0.562567\pi\)
−0.195296 + 0.980744i \(0.562567\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −59.3970 −2.04090
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 28.2843 0.969572
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) 33.9411 1.15806 0.579028 0.815308i \(-0.303432\pi\)
0.579028 + 0.815308i \(0.303432\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 42.4264 1.44421 0.722106 0.691783i \(-0.243174\pi\)
0.722106 + 0.691783i \(0.243174\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 5.65685 0.191675
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 2.82843 0.0951842 0.0475921 0.998867i \(-0.484845\pi\)
0.0475921 + 0.998867i \(0.484845\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.82843 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 11.3137 0.378176
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −33.9411 −1.13200
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −25.4558 −0.845247 −0.422624 0.906305i \(-0.638891\pi\)
−0.422624 + 0.906305i \(0.638891\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 33.9411 1.11961 0.559807 0.828623i \(-0.310875\pi\)
0.559807 + 0.828623i \(0.310875\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 11.3137 0.369998
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 0 0
\(943\) 5.65685 0.184213
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −42.4264 −1.37867 −0.689336 0.724441i \(-0.742098\pi\)
−0.689336 + 0.724441i \(0.742098\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.9706 0.549155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −16.9706 −0.548008
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) 42.4264 1.36434 0.682171 0.731193i \(-0.261036\pi\)
0.682171 + 0.731193i \(0.261036\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.2843 −0.907685 −0.453843 0.891082i \(-0.649947\pi\)
−0.453843 + 0.891082i \(0.649947\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −56.5685 −1.80794
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.82843 −0.0902128 −0.0451064 0.998982i \(-0.514363\pi\)
−0.0451064 + 0.998982i \(0.514363\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −16.9706 −0.539088 −0.269544 0.962988i \(-0.586873\pi\)
−0.269544 + 0.962988i \(0.586873\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.6274 0.717337
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 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 1440.2.a.o.1.1 2
3.2 odd 2 160.2.a.c.1.2 yes 2
4.3 odd 2 inner 1440.2.a.o.1.2 2
5.2 odd 4 7200.2.f.bh.6049.2 4
5.3 odd 4 7200.2.f.bh.6049.4 4
5.4 even 2 7200.2.a.cm.1.2 2
8.3 odd 2 2880.2.a.bk.1.2 2
8.5 even 2 2880.2.a.bk.1.1 2
12.11 even 2 160.2.a.c.1.1 2
15.2 even 4 800.2.c.f.449.1 4
15.8 even 4 800.2.c.f.449.3 4
15.14 odd 2 800.2.a.m.1.1 2
20.3 even 4 7200.2.f.bh.6049.1 4
20.7 even 4 7200.2.f.bh.6049.3 4
20.19 odd 2 7200.2.a.cm.1.1 2
21.20 even 2 7840.2.a.bf.1.1 2
24.5 odd 2 320.2.a.g.1.1 2
24.11 even 2 320.2.a.g.1.2 2
48.5 odd 4 1280.2.d.l.641.2 4
48.11 even 4 1280.2.d.l.641.4 4
48.29 odd 4 1280.2.d.l.641.3 4
48.35 even 4 1280.2.d.l.641.1 4
60.23 odd 4 800.2.c.f.449.2 4
60.47 odd 4 800.2.c.f.449.4 4
60.59 even 2 800.2.a.m.1.2 2
84.83 odd 2 7840.2.a.bf.1.2 2
120.29 odd 2 1600.2.a.bc.1.2 2
120.53 even 4 1600.2.c.n.449.2 4
120.59 even 2 1600.2.a.bc.1.1 2
120.77 even 4 1600.2.c.n.449.4 4
120.83 odd 4 1600.2.c.n.449.3 4
120.107 odd 4 1600.2.c.n.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.c.1.1 2 12.11 even 2
160.2.a.c.1.2 yes 2 3.2 odd 2
320.2.a.g.1.1 2 24.5 odd 2
320.2.a.g.1.2 2 24.11 even 2
800.2.a.m.1.1 2 15.14 odd 2
800.2.a.m.1.2 2 60.59 even 2
800.2.c.f.449.1 4 15.2 even 4
800.2.c.f.449.2 4 60.23 odd 4
800.2.c.f.449.3 4 15.8 even 4
800.2.c.f.449.4 4 60.47 odd 4
1280.2.d.l.641.1 4 48.35 even 4
1280.2.d.l.641.2 4 48.5 odd 4
1280.2.d.l.641.3 4 48.29 odd 4
1280.2.d.l.641.4 4 48.11 even 4
1440.2.a.o.1.1 2 1.1 even 1 trivial
1440.2.a.o.1.2 2 4.3 odd 2 inner
1600.2.a.bc.1.1 2 120.59 even 2
1600.2.a.bc.1.2 2 120.29 odd 2
1600.2.c.n.449.1 4 120.107 odd 4
1600.2.c.n.449.2 4 120.53 even 4
1600.2.c.n.449.3 4 120.83 odd 4
1600.2.c.n.449.4 4 120.77 even 4
2880.2.a.bk.1.1 2 8.5 even 2
2880.2.a.bk.1.2 2 8.3 odd 2
7200.2.a.cm.1.1 2 20.19 odd 2
7200.2.a.cm.1.2 2 5.4 even 2
7200.2.f.bh.6049.1 4 20.3 even 4
7200.2.f.bh.6049.2 4 5.2 odd 4
7200.2.f.bh.6049.3 4 20.7 even 4
7200.2.f.bh.6049.4 4 5.3 odd 4
7840.2.a.bf.1.1 2 21.20 even 2
7840.2.a.bf.1.2 2 84.83 odd 2