Properties

Label 1536.2.a.j.1.2
Level $1536$
Weight $2$
Character 1536.1
Self dual yes
Analytic conductor $12.265$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1536.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.41421 q^{5} +1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.41421 q^{5} +1.41421 q^{7} +1.00000 q^{9} +2.00000 q^{11} +1.41421 q^{15} +2.00000 q^{17} +4.00000 q^{19} +1.41421 q^{21} +2.82843 q^{23} -3.00000 q^{25} +1.00000 q^{27} -9.89949 q^{29} -7.07107 q^{31} +2.00000 q^{33} +2.00000 q^{35} +8.48528 q^{37} +6.00000 q^{41} +8.00000 q^{43} +1.41421 q^{45} +2.82843 q^{47} -5.00000 q^{49} +2.00000 q^{51} +1.41421 q^{53} +2.82843 q^{55} +4.00000 q^{57} +12.0000 q^{59} -14.1421 q^{61} +1.41421 q^{63} +8.00000 q^{67} +2.82843 q^{69} -14.1421 q^{71} -8.00000 q^{73} -3.00000 q^{75} +2.82843 q^{77} -4.24264 q^{79} +1.00000 q^{81} +6.00000 q^{83} +2.82843 q^{85} -9.89949 q^{87} +2.00000 q^{89} -7.07107 q^{93} +5.65685 q^{95} -14.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{9} + 4q^{11} + 4q^{17} + 8q^{19} - 6q^{25} + 2q^{27} + 4q^{33} + 4q^{35} + 12q^{41} + 16q^{43} - 10q^{49} + 4q^{51} + 8q^{57} + 24q^{59} + 16q^{67} - 16q^{73} - 6q^{75} + 2q^{81} + 12q^{83} + 4q^{89} - 28q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.89949 −1.83829 −0.919145 0.393919i \(-0.871119\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) −7.07107 −1.27000 −0.635001 0.772512i \(-0.719000\pi\)
−0.635001 + 0.772512i \(0.719000\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 1.41421 0.210819
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 1.41421 0.194257 0.0971286 0.995272i \(-0.469034\pi\)
0.0971286 + 0.995272i \(0.469034\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −14.1421 −1.81071 −0.905357 0.424650i \(-0.860397\pi\)
−0.905357 + 0.424650i \(0.860397\pi\)
\(62\) 0 0
\(63\) 1.41421 0.178174
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) −14.1421 −1.67836 −0.839181 0.543852i \(-0.816965\pi\)
−0.839181 + 0.543852i \(0.816965\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) −4.24264 −0.477334 −0.238667 0.971101i \(-0.576710\pi\)
−0.238667 + 0.971101i \(0.576710\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) −9.89949 −1.06134
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.07107 −0.733236
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 9.89949 0.985037 0.492518 0.870302i \(-0.336076\pi\)
0.492518 + 0.870302i \(0.336076\pi\)
\(102\) 0 0
\(103\) 12.7279 1.25412 0.627060 0.778971i \(-0.284258\pi\)
0.627060 + 0.778971i \(0.284258\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 11.3137 1.08366 0.541828 0.840489i \(-0.317732\pi\)
0.541828 + 0.840489i \(0.317732\pi\)
\(110\) 0 0
\(111\) 8.48528 0.805387
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.82843 0.259281
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −4.24264 −0.376473 −0.188237 0.982124i \(-0.560277\pi\)
−0.188237 + 0.982124i \(0.560277\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 5.65685 0.490511
\(134\) 0 0
\(135\) 1.41421 0.121716
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 2.82843 0.238197
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −14.0000 −1.16264
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 0 0
\(149\) −4.24264 −0.347571 −0.173785 0.984784i \(-0.555600\pi\)
−0.173785 + 0.984784i \(0.555600\pi\)
\(150\) 0 0
\(151\) 4.24264 0.345261 0.172631 0.984987i \(-0.444773\pi\)
0.172631 + 0.984987i \(0.444773\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −14.1421 −1.12867 −0.564333 0.825547i \(-0.690866\pi\)
−0.564333 + 0.825547i \(0.690866\pi\)
\(158\) 0 0
\(159\) 1.41421 0.112154
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 15.5563 1.18273 0.591364 0.806405i \(-0.298590\pi\)
0.591364 + 0.806405i \(0.298590\pi\)
\(174\) 0 0
\(175\) −4.24264 −0.320713
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −11.3137 −0.840941 −0.420471 0.907306i \(-0.638135\pi\)
−0.420471 + 0.907306i \(0.638135\pi\)
\(182\) 0 0
\(183\) −14.1421 −1.04542
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 1.41421 0.102869
\(190\) 0 0
\(191\) 22.6274 1.63726 0.818631 0.574320i \(-0.194733\pi\)
0.818631 + 0.574320i \(0.194733\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.07107 0.503793 0.251896 0.967754i \(-0.418946\pi\)
0.251896 + 0.967754i \(0.418946\pi\)
\(198\) 0 0
\(199\) 24.0416 1.70427 0.852133 0.523325i \(-0.175309\pi\)
0.852133 + 0.523325i \(0.175309\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −14.0000 −0.982607
\(204\) 0 0
\(205\) 8.48528 0.592638
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) −14.1421 −0.969003
\(214\) 0 0
\(215\) 11.3137 0.771589
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −12.7279 −0.852325 −0.426162 0.904647i \(-0.640135\pi\)
−0.426162 + 0.904647i \(0.640135\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 0 0
\(229\) −16.9706 −1.12145 −0.560723 0.828003i \(-0.689477\pi\)
−0.560723 + 0.828003i \(0.689477\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) −4.24264 −0.275589
\(238\) 0 0
\(239\) −22.6274 −1.46365 −0.731823 0.681495i \(-0.761330\pi\)
−0.731823 + 0.681495i \(0.761330\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −7.07107 −0.451754
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 5.65685 0.355643
\(254\) 0 0
\(255\) 2.82843 0.177123
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) −9.89949 −0.612763
\(262\) 0 0
\(263\) −28.2843 −1.74408 −0.872041 0.489432i \(-0.837204\pi\)
−0.872041 + 0.489432i \(0.837204\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) 0 0
\(269\) 21.2132 1.29339 0.646696 0.762748i \(-0.276150\pi\)
0.646696 + 0.762748i \(0.276150\pi\)
\(270\) 0 0
\(271\) −24.0416 −1.46043 −0.730213 0.683220i \(-0.760579\pi\)
−0.730213 + 0.683220i \(0.760579\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) −7.07107 −0.423334
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 5.65685 0.335083
\(286\) 0 0
\(287\) 8.48528 0.500870
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) −9.89949 −0.578335 −0.289167 0.957279i \(-0.593378\pi\)
−0.289167 + 0.957279i \(0.593378\pi\)
\(294\) 0 0
\(295\) 16.9706 0.988064
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 11.3137 0.652111
\(302\) 0 0
\(303\) 9.89949 0.568711
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 12.7279 0.724066
\(310\) 0 0
\(311\) 22.6274 1.28308 0.641542 0.767088i \(-0.278295\pi\)
0.641542 + 0.767088i \(0.278295\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 9.89949 0.556011 0.278006 0.960579i \(-0.410327\pi\)
0.278006 + 0.960579i \(0.410327\pi\)
\(318\) 0 0
\(319\) −19.7990 −1.10853
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.3137 0.625650
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 8.48528 0.464991
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −14.1421 −0.765840
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) 25.4558 1.36262 0.681310 0.731995i \(-0.261411\pi\)
0.681310 + 0.731995i \(0.261411\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −20.0000 −1.06149
\(356\) 0 0
\(357\) 2.82843 0.149696
\(358\) 0 0
\(359\) −19.7990 −1.04495 −0.522475 0.852654i \(-0.674991\pi\)
−0.522475 + 0.852654i \(0.674991\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −11.3137 −0.592187
\(366\) 0 0
\(367\) 4.24264 0.221464 0.110732 0.993850i \(-0.464680\pi\)
0.110732 + 0.993850i \(0.464680\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −19.7990 −1.02515 −0.512576 0.858642i \(-0.671309\pi\)
−0.512576 + 0.858642i \(0.671309\pi\)
\(374\) 0 0
\(375\) −11.3137 −0.584237
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −4.24264 −0.217357
\(382\) 0 0
\(383\) −5.65685 −0.289052 −0.144526 0.989501i \(-0.546166\pi\)
−0.144526 + 0.989501i \(0.546166\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) −24.0416 −1.21896 −0.609480 0.792802i \(-0.708622\pi\)
−0.609480 + 0.792802i \(0.708622\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) −2.82843 −0.141955 −0.0709773 0.997478i \(-0.522612\pi\)
−0.0709773 + 0.997478i \(0.522612\pi\)
\(398\) 0 0
\(399\) 5.65685 0.283197
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.41421 0.0702728
\(406\) 0 0
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 16.9706 0.835067
\(414\) 0 0
\(415\) 8.48528 0.416526
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −5.65685 −0.275698 −0.137849 0.990453i \(-0.544019\pi\)
−0.137849 + 0.990453i \(0.544019\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.82843 0.136241 0.0681203 0.997677i \(-0.478300\pi\)
0.0681203 + 0.997677i \(0.478300\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −14.0000 −0.671249
\(436\) 0 0
\(437\) 11.3137 0.541208
\(438\) 0 0
\(439\) −4.24264 −0.202490 −0.101245 0.994862i \(-0.532283\pi\)
−0.101245 + 0.994862i \(0.532283\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) 30.0000 1.42534 0.712672 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) 0 0
\(445\) 2.82843 0.134080
\(446\) 0 0
\(447\) −4.24264 −0.200670
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 4.24264 0.199337
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −12.7279 −0.592798 −0.296399 0.955064i \(-0.595786\pi\)
−0.296399 + 0.955064i \(0.595786\pi\)
\(462\) 0 0
\(463\) 35.3553 1.64310 0.821551 0.570135i \(-0.193109\pi\)
0.821551 + 0.570135i \(0.193109\pi\)
\(464\) 0 0
\(465\) −10.0000 −0.463739
\(466\) 0 0
\(467\) −22.0000 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(468\) 0 0
\(469\) 11.3137 0.522419
\(470\) 0 0
\(471\) −14.1421 −0.651635
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) 1.41421 0.0647524
\(478\) 0 0
\(479\) 25.4558 1.16311 0.581554 0.813508i \(-0.302445\pi\)
0.581554 + 0.813508i \(0.302445\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) −19.7990 −0.899026
\(486\) 0 0
\(487\) −4.24264 −0.192252 −0.0961262 0.995369i \(-0.530645\pi\)
−0.0961262 + 0.995369i \(0.530645\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −19.7990 −0.891702
\(494\) 0 0
\(495\) 2.82843 0.127128
\(496\) 0 0
\(497\) −20.0000 −0.897123
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −11.3137 −0.505459
\(502\) 0 0
\(503\) 2.82843 0.126113 0.0630567 0.998010i \(-0.479915\pi\)
0.0630567 + 0.998010i \(0.479915\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 24.0416 1.06563 0.532813 0.846233i \(-0.321135\pi\)
0.532813 + 0.846233i \(0.321135\pi\)
\(510\) 0 0
\(511\) −11.3137 −0.500489
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) 15.5563 0.682848
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) −4.24264 −0.185164
\(526\) 0 0
\(527\) −14.1421 −0.616041
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.65685 0.244567
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) 28.2843 1.21604 0.608018 0.793923i \(-0.291965\pi\)
0.608018 + 0.793923i \(0.291965\pi\)
\(542\) 0 0
\(543\) −11.3137 −0.485518
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) −14.1421 −0.603572
\(550\) 0 0
\(551\) −39.5980 −1.68693
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 12.0000 0.509372
\(556\) 0 0
\(557\) −38.1838 −1.61790 −0.808949 0.587879i \(-0.799963\pi\)
−0.808949 + 0.587879i \(0.799963\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) 8.48528 0.356978
\(566\) 0 0
\(567\) 1.41421 0.0593914
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 22.6274 0.945274
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 8.48528 0.352029
\(582\) 0 0
\(583\) 2.82843 0.117141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) −28.2843 −1.16543
\(590\) 0 0
\(591\) 7.07107 0.290865
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 24.0416 0.983958
\(598\) 0 0
\(599\) 36.7696 1.50236 0.751182 0.660096i \(-0.229484\pi\)
0.751182 + 0.660096i \(0.229484\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) −9.89949 −0.402472
\(606\) 0 0
\(607\) −4.24264 −0.172203 −0.0861017 0.996286i \(-0.527441\pi\)
−0.0861017 + 0.996286i \(0.527441\pi\)
\(608\) 0 0
\(609\) −14.0000 −0.567309
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.48528 0.342717 0.171359 0.985209i \(-0.445184\pi\)
0.171359 + 0.985209i \(0.445184\pi\)
\(614\) 0 0
\(615\) 8.48528 0.342160
\(616\) 0 0
\(617\) 46.0000 1.85189 0.925945 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 0 0
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) 2.82843 0.113501
\(622\) 0 0
\(623\) 2.82843 0.113319
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 8.00000 0.319489
\(628\) 0 0
\(629\) 16.9706 0.676661
\(630\) 0 0
\(631\) −15.5563 −0.619288 −0.309644 0.950852i \(-0.600210\pi\)
−0.309644 + 0.950852i \(0.600210\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −14.1421 −0.559454
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 11.3137 0.445477
\(646\) 0 0
\(647\) −14.1421 −0.555985 −0.277992 0.960583i \(-0.589669\pi\)
−0.277992 + 0.960583i \(0.589669\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 0 0
\(653\) −29.6985 −1.16219 −0.581096 0.813835i \(-0.697376\pi\)
−0.581096 + 0.813835i \(0.697376\pi\)
\(654\) 0 0
\(655\) 16.9706 0.663095
\(656\) 0 0
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) 19.7990 0.770091 0.385046 0.922897i \(-0.374186\pi\)
0.385046 + 0.922897i \(0.374186\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −28.0000 −1.08416
\(668\) 0 0
\(669\) −12.7279 −0.492090
\(670\) 0 0
\(671\) −28.2843 −1.09190
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) −3.00000 −0.115470
\(676\) 0 0
\(677\) 24.0416 0.923995 0.461997 0.886881i \(-0.347133\pi\)
0.461997 + 0.886881i \(0.347133\pi\)
\(678\) 0 0
\(679\) −19.7990 −0.759815
\(680\) 0 0
\(681\) −22.0000 −0.843042
\(682\) 0 0
\(683\) 26.0000 0.994862 0.497431 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) −14.1421 −0.540343
\(686\) 0 0
\(687\) −16.9706 −0.647467
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 2.82843 0.107443
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 4.24264 0.160242 0.0801212 0.996785i \(-0.474469\pi\)
0.0801212 + 0.996785i \(0.474469\pi\)
\(702\) 0 0
\(703\) 33.9411 1.28011
\(704\) 0 0
\(705\) 4.00000 0.150649
\(706\) 0 0
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) −33.9411 −1.27469 −0.637343 0.770580i \(-0.719966\pi\)
−0.637343 + 0.770580i \(0.719966\pi\)
\(710\) 0 0
\(711\) −4.24264 −0.159111
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −22.6274 −0.845036
\(718\) 0 0
\(719\) −25.4558 −0.949343 −0.474671 0.880163i \(-0.657433\pi\)
−0.474671 + 0.880163i \(0.657433\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.6985 1.10297
\(726\) 0 0
\(727\) −24.0416 −0.891655 −0.445827 0.895119i \(-0.647090\pi\)
−0.445827 + 0.895119i \(0.647090\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) 11.3137 0.417881 0.208941 0.977928i \(-0.432998\pi\)
0.208941 + 0.977928i \(0.432998\pi\)
\(734\) 0 0
\(735\) −7.07107 −0.260820
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.2843 1.03765 0.518825 0.854881i \(-0.326370\pi\)
0.518825 + 0.854881i \(0.326370\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 5.65685 0.206697
\(750\) 0 0
\(751\) 12.7279 0.464448 0.232224 0.972662i \(-0.425400\pi\)
0.232224 + 0.972662i \(0.425400\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 0 0
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 50.9117 1.85042 0.925208 0.379459i \(-0.123890\pi\)
0.925208 + 0.379459i \(0.123890\pi\)
\(758\) 0 0
\(759\) 5.65685 0.205331
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 2.82843 0.102262
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 15.5563 0.559523 0.279761 0.960070i \(-0.409745\pi\)
0.279761 + 0.960070i \(0.409745\pi\)
\(774\) 0 0
\(775\) 21.2132 0.762001
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −28.2843 −1.01209
\(782\) 0 0
\(783\) −9.89949 −0.353779
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) −28.2843 −1.00695
\(790\) 0 0
\(791\) 8.48528 0.301702
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.00000 0.0709327
\(796\) 0 0
\(797\) 1.41421 0.0500940 0.0250470 0.999686i \(-0.492026\pi\)
0.0250470 + 0.999686i \(0.492026\pi\)
\(798\) 0 0
\(799\) 5.65685 0.200125
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) −16.0000 −0.564628
\(804\) 0 0
\(805\) 5.65685 0.199378
\(806\) 0 0
\(807\) 21.2132 0.746740
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −24.0416 −0.843177
\(814\) 0 0
\(815\) −22.6274 −0.792604
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.89949 0.345495 0.172747 0.984966i \(-0.444736\pi\)
0.172747 + 0.984966i \(0.444736\pi\)
\(822\) 0 0
\(823\) −21.2132 −0.739446 −0.369723 0.929142i \(-0.620547\pi\)
−0.369723 + 0.929142i \(0.620547\pi\)
\(824\) 0 0
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) −11.3137 −0.392941 −0.196471 0.980510i \(-0.562948\pi\)
−0.196471 + 0.980510i \(0.562948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −10.0000 −0.346479
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −7.07107 −0.244412
\(838\) 0 0
\(839\) 14.1421 0.488241 0.244120 0.969745i \(-0.421501\pi\)
0.244120 + 0.969745i \(0.421501\pi\)
\(840\) 0 0
\(841\) 69.0000 2.37931
\(842\) 0 0
\(843\) −2.00000 −0.0688837
\(844\) 0 0
\(845\) −18.3848 −0.632456
\(846\) 0 0
\(847\) −9.89949 −0.340151
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −14.1421 −0.484218 −0.242109 0.970249i \(-0.577839\pi\)
−0.242109 + 0.970249i \(0.577839\pi\)
\(854\) 0 0
\(855\) 5.65685 0.193460
\(856\) 0 0
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 8.48528 0.289178
\(862\) 0 0
\(863\) 22.6274 0.770246 0.385123 0.922865i \(-0.374159\pi\)
0.385123 + 0.922865i \(0.374159\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) −16.0000 −0.540899
\(876\) 0 0
\(877\) −42.4264 −1.43264 −0.716319 0.697773i \(-0.754174\pi\)
−0.716319 + 0.697773i \(0.754174\pi\)
\(878\) 0 0
\(879\) −9.89949 −0.333902
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 0 0
\(885\) 16.9706 0.570459
\(886\) 0 0
\(887\) 50.9117 1.70945 0.854724 0.519083i \(-0.173727\pi\)
0.854724 + 0.519083i \(0.173727\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 11.3137 0.378599
\(894\) 0 0
\(895\) −5.65685 −0.189088
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 70.0000 2.33463
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) 0 0
\(903\) 11.3137 0.376497
\(904\) 0 0
\(905\) −16.0000 −0.531858
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 0 0
\(909\) 9.89949 0.328346
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) −20.0000 −0.661180
\(916\) 0 0
\(917\) 16.9706 0.560417
\(918\) 0 0
\(919\) −21.2132 −0.699759 −0.349880 0.936795i \(-0.613777\pi\)
−0.349880 + 0.936795i \(0.613777\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −25.4558 −0.836983
\(926\) 0 0
\(927\) 12.7279 0.418040
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −20.0000 −0.655474
\(932\) 0 0
\(933\) 22.6274 0.740788
\(934\) 0 0
\(935\) 5.65685 0.184999
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) −1.41421 −0.0461020 −0.0230510 0.999734i \(-0.507338\pi\)
−0.0230510 + 0.999734i \(0.507338\pi\)
\(942\) 0 0
\(943\) 16.9706 0.552638
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 9.89949 0.321013
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) −19.7990 −0.640010
\(958\) 0 0
\(959\) −14.1421 −0.456673
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −5.65685 −0.182101
\(966\) 0 0
\(967\) −52.3259 −1.68269 −0.841344 0.540500i \(-0.818235\pi\)
−0.841344 + 0.540500i \(0.818235\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.0000 1.59964 0.799821 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) 11.3137 0.361219
\(982\) 0 0
\(983\) −50.9117 −1.62383 −0.811915 0.583775i \(-0.801575\pi\)
−0.811915 + 0.583775i \(0.801575\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 4.00000 0.127321
\(988\) 0 0
\(989\) 22.6274 0.719510
\(990\) 0 0
\(991\) 46.6690 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(992\) 0 0
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) 34.0000 1.07787
\(996\) 0 0
\(997\) 31.1127 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(998\) 0 0
\(999\) 8.48528 0.268462
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 1536.2.a.j.1.2 yes 2
3.2 odd 2 4608.2.a.h.1.1 2
4.3 odd 2 1536.2.a.c.1.2 yes 2
8.3 odd 2 inner 1536.2.a.j.1.1 yes 2
8.5 even 2 1536.2.a.c.1.1 2
12.11 even 2 4608.2.a.j.1.1 2
16.3 odd 4 1536.2.d.d.769.1 4
16.5 even 4 1536.2.d.d.769.2 4
16.11 odd 4 1536.2.d.d.769.4 4
16.13 even 4 1536.2.d.d.769.3 4
24.5 odd 2 4608.2.a.j.1.2 2
24.11 even 2 4608.2.a.h.1.2 2
48.5 odd 4 4608.2.d.g.2305.1 4
48.11 even 4 4608.2.d.g.2305.2 4
48.29 odd 4 4608.2.d.g.2305.3 4
48.35 even 4 4608.2.d.g.2305.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.c.1.1 2 8.5 even 2
1536.2.a.c.1.2 yes 2 4.3 odd 2
1536.2.a.j.1.1 yes 2 8.3 odd 2 inner
1536.2.a.j.1.2 yes 2 1.1 even 1 trivial
1536.2.d.d.769.1 4 16.3 odd 4
1536.2.d.d.769.2 4 16.5 even 4
1536.2.d.d.769.3 4 16.13 even 4
1536.2.d.d.769.4 4 16.11 odd 4
4608.2.a.h.1.1 2 3.2 odd 2
4608.2.a.h.1.2 2 24.11 even 2
4608.2.a.j.1.1 2 12.11 even 2
4608.2.a.j.1.2 2 24.5 odd 2
4608.2.d.g.2305.1 4 48.5 odd 4
4608.2.d.g.2305.2 4 48.11 even 4
4608.2.d.g.2305.3 4 48.29 odd 4
4608.2.d.g.2305.4 4 48.35 even 4