Properties

Label 6400.2.a.bn.1.2
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{3} +4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} +4.24264 q^{7} -1.00000 q^{9} -5.65685 q^{11} +2.00000 q^{13} -6.00000 q^{17} +2.82843 q^{19} +6.00000 q^{21} -7.07107 q^{23} -5.65685 q^{27} -4.00000 q^{29} +2.82843 q^{31} -8.00000 q^{33} -2.00000 q^{37} +2.82843 q^{39} +8.00000 q^{41} -1.41421 q^{43} -1.41421 q^{47} +11.0000 q^{49} -8.48528 q^{51} +2.00000 q^{53} +4.00000 q^{57} -2.82843 q^{59} -14.0000 q^{61} -4.24264 q^{63} -4.24264 q^{67} -10.0000 q^{69} +2.82843 q^{71} -6.00000 q^{73} -24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} -12.7279 q^{83} -5.65685 q^{87} +6.00000 q^{89} +8.48528 q^{91} +4.00000 q^{93} -10.0000 q^{97} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{13} - 12q^{17} + 12q^{21} - 8q^{29} - 16q^{33} - 4q^{37} + 16q^{41} + 22q^{49} + 4q^{53} + 8q^{57} - 28q^{61} - 20q^{69} - 12q^{73} - 48q^{77} - 10q^{81} + 12q^{89} + 8q^{93} - 20q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.24264 1.60357 0.801784 0.597614i \(-0.203885\pi\)
0.801784 + 0.597614i \(0.203885\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) −7.07107 −1.47442 −0.737210 0.675664i \(-0.763857\pi\)
−0.737210 + 0.675664i \(0.763857\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.41421 −0.215666 −0.107833 0.994169i \(-0.534391\pi\)
−0.107833 + 0.994169i \(0.534391\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.41421 −0.206284 −0.103142 0.994667i \(-0.532890\pi\)
−0.103142 + 0.994667i \(0.532890\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) −8.48528 −1.18818
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −2.82843 −0.368230 −0.184115 0.982905i \(-0.558942\pi\)
−0.184115 + 0.982905i \(0.558942\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −4.24264 −0.534522
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.24264 −0.518321 −0.259161 0.965834i \(-0.583446\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) 0 0
\(69\) −10.0000 −1.20386
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 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
\(77\) −24.0000 −2.73505
\(78\) 0 0
\(79\) −16.9706 −1.90934 −0.954669 0.297670i \(-0.903790\pi\)
−0.954669 + 0.297670i \(0.903790\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −12.7279 −1.39707 −0.698535 0.715575i \(-0.746165\pi\)
−0.698535 + 0.715575i \(0.746165\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.65685 −0.606478
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.48528 0.889499
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 5.65685 0.568535
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 9.89949 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.07107 −0.683586 −0.341793 0.939775i \(-0.611034\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −2.82843 −0.268462
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −25.4558 −2.33353
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 11.3137 1.02012
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −9.89949 −0.878438 −0.439219 0.898380i \(-0.644745\pi\)
−0.439219 + 0.898380i \(0.644745\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 8.48528 0.719712 0.359856 0.933008i \(-0.382826\pi\)
0.359856 + 0.933008i \(0.382826\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) −11.3137 −0.946100
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.5563 1.28307
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 14.1421 1.15087 0.575435 0.817847i \(-0.304833\pi\)
0.575435 + 0.817847i \(0.304833\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −30.0000 −2.36433
\(162\) 0 0
\(163\) 18.3848 1.44001 0.720003 0.693971i \(-0.244140\pi\)
0.720003 + 0.693971i \(0.244140\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.89949 −0.766046 −0.383023 0.923739i \(-0.625117\pi\)
−0.383023 + 0.923739i \(0.625117\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −19.7990 −1.47985 −0.739923 0.672692i \(-0.765138\pi\)
−0.739923 + 0.672692i \(0.765138\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) −19.7990 −1.46358
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 33.9411 2.48202
\(188\) 0 0
\(189\) −24.0000 −1.74574
\(190\) 0 0
\(191\) −2.82843 −0.204658 −0.102329 0.994751i \(-0.532629\pi\)
−0.102329 + 0.994751i \(0.532629\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 11.3137 0.802008 0.401004 0.916076i \(-0.368661\pi\)
0.401004 + 0.916076i \(0.368661\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) −16.9706 −1.19110
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.07107 0.491473
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 16.9706 1.16830 0.584151 0.811645i \(-0.301428\pi\)
0.584151 + 0.811645i \(0.301428\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) −8.48528 −0.573382
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 15.5563 1.04173 0.520865 0.853639i \(-0.325609\pi\)
0.520865 + 0.853639i \(0.325609\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.07107 −0.469323 −0.234662 0.972077i \(-0.575398\pi\)
−0.234662 + 0.972077i \(0.575398\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) −33.9411 −2.23316
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −24.0000 −1.55897
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.65685 0.359937
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 22.6274 1.42823 0.714115 0.700028i \(-0.246829\pi\)
0.714115 + 0.700028i \(0.246829\pi\)
\(252\) 0 0
\(253\) 40.0000 2.51478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −8.48528 −0.527250
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 0 0
\(263\) 7.07107 0.436021 0.218010 0.975946i \(-0.430043\pi\)
0.218010 + 0.975946i \(0.430043\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.48528 0.519291
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 25.4558 1.54633 0.773166 0.634203i \(-0.218672\pi\)
0.773166 + 0.634203i \(0.218672\pi\)
\(272\) 0 0
\(273\) 12.0000 0.726273
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −9.89949 −0.588464 −0.294232 0.955734i \(-0.595064\pi\)
−0.294232 + 0.955734i \(0.595064\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 33.9411 2.00348
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −14.1421 −0.829027
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 32.0000 1.85683
\(298\) 0 0
\(299\) −14.1421 −0.817861
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −15.5563 −0.887848 −0.443924 0.896065i \(-0.646414\pi\)
−0.443924 + 0.896065i \(0.646414\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 2.82843 0.160385 0.0801927 0.996779i \(-0.474446\pi\)
0.0801927 + 0.996779i \(0.474446\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 22.6274 1.26689
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) −16.9706 −0.944267
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.48528 0.469237
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −22.6274 −1.24372 −0.621858 0.783130i \(-0.713622\pi\)
−0.621858 + 0.783130i \(0.713622\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 14.1421 0.768095
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.24264 −0.227757 −0.113878 0.993495i \(-0.536327\pi\)
−0.113878 + 0.993495i \(0.536327\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) −11.3137 −0.603881
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −36.0000 −1.90532
\(358\) 0 0
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 29.6985 1.55877
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.24264 −0.221464 −0.110732 0.993850i \(-0.535320\pi\)
−0.110732 + 0.993850i \(0.535320\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 8.48528 0.440534
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) 31.1127 1.59815 0.799076 0.601230i \(-0.205322\pi\)
0.799076 + 0.601230i \(0.205322\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) 0 0
\(383\) −7.07107 −0.361315 −0.180657 0.983546i \(-0.557823\pi\)
−0.180657 + 0.983546i \(0.557823\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.41421 0.0718885
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 42.4264 2.14560
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 16.9706 0.849591
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 5.65685 0.281788
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.3137 0.560800
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 2.82843 0.139516
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 8.48528 0.414533 0.207267 0.978285i \(-0.433543\pi\)
0.207267 + 0.978285i \(0.433543\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 1.41421 0.0687614
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −59.3970 −2.87442
\(428\) 0 0
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) −31.1127 −1.49865 −0.749323 0.662205i \(-0.769621\pi\)
−0.749323 + 0.662205i \(0.769621\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20.0000 −0.956730
\(438\) 0 0
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) −11.0000 −0.523810
\(442\) 0 0
\(443\) −15.5563 −0.739104 −0.369552 0.929210i \(-0.620489\pi\)
−0.369552 + 0.929210i \(0.620489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −31.1127 −1.47158
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −45.2548 −2.13097
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 0 0
\(459\) 33.9411 1.58424
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) 7.07107 0.328620 0.164310 0.986409i \(-0.447460\pi\)
0.164310 + 0.986409i \(0.447460\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.89949 0.458094 0.229047 0.973415i \(-0.426439\pi\)
0.229047 + 0.973415i \(0.426439\pi\)
\(468\) 0 0
\(469\) −18.0000 −0.831163
\(470\) 0 0
\(471\) −25.4558 −1.17294
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −33.9411 −1.55081 −0.775405 0.631464i \(-0.782454\pi\)
−0.775405 + 0.631464i \(0.782454\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) −42.4264 −1.93047
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.7279 −0.576757 −0.288379 0.957516i \(-0.593116\pi\)
−0.288379 + 0.957516i \(0.593116\pi\)
\(488\) 0 0
\(489\) 26.0000 1.17576
\(490\) 0 0
\(491\) −16.9706 −0.765871 −0.382935 0.923775i \(-0.625087\pi\)
−0.382935 + 0.923775i \(0.625087\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 25.4558 1.13956 0.569780 0.821797i \(-0.307028\pi\)
0.569780 + 0.821797i \(0.307028\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) 0 0
\(503\) 32.5269 1.45030 0.725152 0.688589i \(-0.241770\pi\)
0.725152 + 0.688589i \(0.241770\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.7279 −0.565267
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −25.4558 −1.12610
\(512\) 0 0
\(513\) −16.0000 −0.706417
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −25.4558 −1.11739
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 12.7279 0.556553 0.278277 0.960501i \(-0.410237\pi\)
0.278277 + 0.960501i \(0.410237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.9706 −0.739249
\(528\) 0 0
\(529\) 27.0000 1.17391
\(530\) 0 0
\(531\) 2.82843 0.122743
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −28.0000 −1.20829
\(538\) 0 0
\(539\) −62.2254 −2.68024
\(540\) 0 0
\(541\) 40.0000 1.71973 0.859867 0.510518i \(-0.170546\pi\)
0.859867 + 0.510518i \(0.170546\pi\)
\(542\) 0 0
\(543\) 22.6274 0.971035
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.24264 0.181402 0.0907011 0.995878i \(-0.471089\pi\)
0.0907011 + 0.995878i \(0.471089\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) −72.0000 −3.06175
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −2.82843 −0.119630
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) 35.3553 1.49005 0.745025 0.667037i \(-0.232438\pi\)
0.745025 + 0.667037i \(0.232438\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −21.2132 −0.890871
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) −22.6274 −0.946928 −0.473464 0.880813i \(-0.656997\pi\)
−0.473464 + 0.880813i \(0.656997\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) −8.48528 −0.352636
\(580\) 0 0
\(581\) −54.0000 −2.24030
\(582\) 0 0
\(583\) −11.3137 −0.468566
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.7279 0.525338 0.262669 0.964886i \(-0.415397\pi\)
0.262669 + 0.964886i \(0.415397\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 39.5980 1.61793 0.808965 0.587857i \(-0.200028\pi\)
0.808965 + 0.587857i \(0.200028\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) 4.24264 0.172774
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 9.89949 0.401808 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) −2.82843 −0.114426
\(612\) 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
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) −19.7990 −0.795789 −0.397894 0.917431i \(-0.630259\pi\)
−0.397894 + 0.917431i \(0.630259\pi\)
\(620\) 0 0
\(621\) 40.0000 1.60514
\(622\) 0 0
\(623\) 25.4558 1.01987
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −22.6274 −0.903652
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −36.7696 −1.46377 −0.731886 0.681427i \(-0.761360\pi\)
−0.731886 + 0.681427i \(0.761360\pi\)
\(632\) 0 0
\(633\) 24.0000 0.953914
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22.0000 0.871672
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −7.07107 −0.278856 −0.139428 0.990232i \(-0.544526\pi\)
−0.139428 + 0.990232i \(0.544526\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.5269 1.27876 0.639382 0.768889i \(-0.279190\pi\)
0.639382 + 0.768889i \(0.279190\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 16.9706 0.665129
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −25.4558 −0.991619 −0.495809 0.868431i \(-0.665129\pi\)
−0.495809 + 0.868431i \(0.665129\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) −16.9706 −0.659082
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.2843 1.09517
\(668\) 0 0
\(669\) 22.0000 0.850569
\(670\) 0 0
\(671\) 79.1960 3.05733
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 0 0
\(679\) −42.4264 −1.62818
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 0 0
\(683\) −35.3553 −1.35283 −0.676417 0.736519i \(-0.736468\pi\)
−0.676417 + 0.736519i \(0.736468\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −28.2843 −1.07911
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 22.6274 0.860788 0.430394 0.902641i \(-0.358375\pi\)
0.430394 + 0.902641i \(0.358375\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) 0 0
\(699\) 8.48528 0.320943
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −5.65685 −0.213352
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 16.9706 0.636446
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 0 0
\(723\) 11.3137 0.420761
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.24264 0.157351 0.0786754 0.996900i \(-0.474931\pi\)
0.0786754 + 0.996900i \(0.474931\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 8.48528 0.313839
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) 25.4558 0.936408 0.468204 0.883620i \(-0.344901\pi\)
0.468204 + 0.883620i \(0.344901\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) −32.5269 −1.19330 −0.596648 0.802503i \(-0.703501\pi\)
−0.596648 + 0.802503i \(0.703501\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.7279 0.465690
\(748\) 0 0
\(749\) −30.0000 −1.09618
\(750\) 0 0
\(751\) −8.48528 −0.309632 −0.154816 0.987943i \(-0.549479\pi\)
−0.154816 + 0.987943i \(0.549479\pi\)
\(752\) 0 0
\(753\) 32.0000 1.16614
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) 56.5685 2.05331
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 25.4558 0.921563
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.65685 −0.204257
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 25.4558 0.916770
\(772\) 0 0
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) 0 0
\(779\) 22.6274 0.810711
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 22.6274 0.808638
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.6690 1.66357 0.831786 0.555097i \(-0.187319\pi\)
0.831786 + 0.555097i \(0.187319\pi\)
\(788\) 0 0
\(789\) 10.0000 0.356009
\(790\) 0 0
\(791\) 42.4264 1.50851
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 8.48528 0.300188
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 33.9411 1.19776
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −8.48528 −0.298696
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 36.0000 1.26258
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) −8.48528 −0.296500
\(820\) 0 0
\(821\) −26.0000 −0.907406 −0.453703 0.891153i \(-0.649897\pi\)
−0.453703 + 0.891153i \(0.649897\pi\)
\(822\) 0 0
\(823\) 26.8701 0.936631 0.468316 0.883561i \(-0.344861\pi\)
0.468316 + 0.883561i \(0.344861\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.07107 0.245885 0.122943 0.992414i \(-0.460767\pi\)
0.122943 + 0.992414i \(0.460767\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −42.4264 −1.47176
\(832\) 0 0
\(833\) −66.0000 −2.28676
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −28.2843 −0.976481 −0.488241 0.872709i \(-0.662361\pi\)
−0.488241 + 0.872709i \(0.662361\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −11.3137 −0.389665
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 89.0955 3.06136
\(848\) 0 0
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 14.1421 0.484786
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −2.82843 −0.0965047 −0.0482523 0.998835i \(-0.515365\pi\)
−0.0482523 + 0.998835i \(0.515365\pi\)
\(860\) 0 0
\(861\) 48.0000 1.63584
\(862\) 0 0
\(863\) −55.1543 −1.87748 −0.938738 0.344633i \(-0.888003\pi\)
−0.938738 + 0.344633i \(0.888003\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 26.8701 0.912555
\(868\) 0 0
\(869\) 96.0000 3.25658
\(870\) 0 0
\(871\) −8.48528 −0.287513
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 8.48528 0.286201
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) −21.2132 −0.713881 −0.356941 0.934127i \(-0.616180\pi\)
−0.356941 + 0.934127i \(0.616180\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.8701 0.902208 0.451104 0.892471i \(-0.351030\pi\)
0.451104 + 0.892471i \(0.351030\pi\)
\(888\) 0 0
\(889\) −42.0000 −1.40863
\(890\) 0 0
\(891\) 28.2843 0.947559
\(892\) 0 0
\(893\) −4.00000 −0.133855
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −20.0000 −0.667781
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) −8.48528 −0.282372
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.41421 −0.0469582 −0.0234791 0.999724i \(-0.507474\pi\)
−0.0234791 + 0.999724i \(0.507474\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −31.1127 −1.03081 −0.515405 0.856947i \(-0.672358\pi\)
−0.515405 + 0.856947i \(0.672358\pi\)
\(912\) 0 0
\(913\) 72.0000 2.38285
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) −39.5980 −1.30622 −0.653108 0.757264i \(-0.726535\pi\)
−0.653108 + 0.757264i \(0.726535\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 0 0
\(923\) 5.65685 0.186198
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.89949 −0.325142
\(928\) 0 0
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 0 0
\(931\) 31.1127 1.01968
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 8.48528 0.276907
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 0 0
\(943\) −56.5685 −1.84213
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.5563 −0.505513 −0.252757 0.967530i \(-0.581337\pi\)
−0.252757 + 0.967530i \(0.581337\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −2.82843 −0.0917180
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 32.0000 1.03441
\(958\) 0 0
\(959\) 8.48528 0.274004
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 7.07107 0.227862
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −4.24264 −0.136434 −0.0682171 0.997671i \(-0.521731\pi\)
−0.0682171 + 0.997671i \(0.521731\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 28.2843 0.907685 0.453843 0.891082i \(-0.350053\pi\)
0.453843 + 0.891082i \(0.350053\pi\)
\(972\) 0 0
\(973\) 36.0000 1.15411
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −33.9411 −1.08476
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 18.3848 0.586383 0.293192 0.956054i \(-0.405283\pi\)
0.293192 + 0.956054i \(0.405283\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8.48528 −0.270089
\(988\) 0 0
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) 8.48528 0.269544 0.134772 0.990877i \(-0.456970\pi\)
0.134772 + 0.990877i \(0.456970\pi\)
\(992\) 0 0
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) 11.3137 0.357950
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 6400.2.a.bn.1.2 2
4.3 odd 2 inner 6400.2.a.bn.1.1 2
5.4 even 2 1280.2.a.j.1.1 2
8.3 odd 2 6400.2.a.bl.1.2 2
8.5 even 2 6400.2.a.bl.1.1 2
16.3 odd 4 3200.2.d.s.1601.2 4
16.5 even 4 3200.2.d.s.1601.1 4
16.11 odd 4 3200.2.d.s.1601.4 4
16.13 even 4 3200.2.d.s.1601.3 4
20.19 odd 2 1280.2.a.j.1.2 2
40.19 odd 2 1280.2.a.f.1.1 2
40.29 even 2 1280.2.a.f.1.2 2
80.3 even 4 3200.2.f.i.449.3 4
80.13 odd 4 3200.2.f.i.449.2 4
80.19 odd 4 640.2.d.d.321.3 yes 4
80.27 even 4 3200.2.f.i.449.4 4
80.29 even 4 640.2.d.d.321.1 4
80.37 odd 4 3200.2.f.i.449.1 4
80.43 even 4 3200.2.f.j.449.1 4
80.53 odd 4 3200.2.f.j.449.4 4
80.59 odd 4 640.2.d.d.321.2 yes 4
80.67 even 4 3200.2.f.j.449.2 4
80.69 even 4 640.2.d.d.321.4 yes 4
80.77 odd 4 3200.2.f.j.449.3 4
240.29 odd 4 5760.2.k.o.2881.4 4
240.59 even 4 5760.2.k.o.2881.1 4
240.149 odd 4 5760.2.k.o.2881.2 4
240.179 even 4 5760.2.k.o.2881.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.d.321.1 4 80.29 even 4
640.2.d.d.321.2 yes 4 80.59 odd 4
640.2.d.d.321.3 yes 4 80.19 odd 4
640.2.d.d.321.4 yes 4 80.69 even 4
1280.2.a.f.1.1 2 40.19 odd 2
1280.2.a.f.1.2 2 40.29 even 2
1280.2.a.j.1.1 2 5.4 even 2
1280.2.a.j.1.2 2 20.19 odd 2
3200.2.d.s.1601.1 4 16.5 even 4
3200.2.d.s.1601.2 4 16.3 odd 4
3200.2.d.s.1601.3 4 16.13 even 4
3200.2.d.s.1601.4 4 16.11 odd 4
3200.2.f.i.449.1 4 80.37 odd 4
3200.2.f.i.449.2 4 80.13 odd 4
3200.2.f.i.449.3 4 80.3 even 4
3200.2.f.i.449.4 4 80.27 even 4
3200.2.f.j.449.1 4 80.43 even 4
3200.2.f.j.449.2 4 80.67 even 4
3200.2.f.j.449.3 4 80.77 odd 4
3200.2.f.j.449.4 4 80.53 odd 4
5760.2.k.o.2881.1 4 240.59 even 4
5760.2.k.o.2881.2 4 240.149 odd 4
5760.2.k.o.2881.3 4 240.179 even 4
5760.2.k.o.2881.4 4 240.29 odd 4
6400.2.a.bl.1.1 2 8.5 even 2
6400.2.a.bl.1.2 2 8.3 odd 2
6400.2.a.bn.1.1 2 4.3 odd 2 inner
6400.2.a.bn.1.2 2 1.1 even 1 trivial