Properties

Label 252.9.c.a.197.16
Level $252$
Weight $9$
Character 252.197
Analytic conductor $102.659$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + 5613923146579405376 x^{10} + 2733728904154246859079616 x^{8} + 757873148017661341349205888000 x^{6} + 113644318422397913452531577312640000 x^{4} + 8098650340007618970326973663348480000000 x^{2} + 199066230990417435753898292645889849600000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{37}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.16
Root \(1013.87i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.1

$q$-expansion

\(f(q)\) \(=\) \(q+1013.87i q^{5} -907.493 q^{7} +O(q^{10})\) \(q+1013.87i q^{5} -907.493 q^{7} -9545.60i q^{11} +1998.60 q^{13} +46137.9i q^{17} -34120.9 q^{19} -97009.8i q^{23} -637298. q^{25} -833372. i q^{29} -1.03301e6 q^{31} -920075. i q^{35} -1.96878e6 q^{37} -1.24983e6i q^{41} +1.48518e6 q^{43} +1.69893e6i q^{47} +823543. q^{49} -6.91772e6i q^{53} +9.67796e6 q^{55} +1.43342e7i q^{59} +2.12764e6 q^{61} +2.02631e6i q^{65} +696928. q^{67} -1.02684e6i q^{71} +3.22586e7 q^{73} +8.66257e6i q^{77} +4.61512e7 q^{79} +5.08432e6i q^{83} -4.67776e7 q^{85} -5.60351e7i q^{89} -1.81371e6 q^{91} -3.45940e7i q^{95} -4.04787e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 1013.87i 1.62218i 0.584918 + 0.811092i \(0.301127\pi\)
−0.584918 + 0.811092i \(0.698873\pi\)
\(6\) 0 0
\(7\) −907.493 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 9545.60i − 0.651978i −0.945374 0.325989i \(-0.894303\pi\)
0.945374 0.325989i \(-0.105697\pi\)
\(12\) 0 0
\(13\) 1998.60 0.0699765 0.0349883 0.999388i \(-0.488861\pi\)
0.0349883 + 0.999388i \(0.488861\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 46137.9i 0.552411i 0.961099 + 0.276205i \(0.0890770\pi\)
−0.961099 + 0.276205i \(0.910923\pi\)
\(18\) 0 0
\(19\) −34120.9 −0.261822 −0.130911 0.991394i \(-0.541790\pi\)
−0.130911 + 0.991394i \(0.541790\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 97009.8i − 0.346660i −0.984864 0.173330i \(-0.944547\pi\)
0.984864 0.173330i \(-0.0554528\pi\)
\(24\) 0 0
\(25\) −637298. −1.63148
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 833372.i − 1.17828i −0.808033 0.589138i \(-0.799467\pi\)
0.808033 0.589138i \(-0.200533\pi\)
\(30\) 0 0
\(31\) −1.03301e6 −1.11856 −0.559280 0.828979i \(-0.688922\pi\)
−0.559280 + 0.828979i \(0.688922\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 920075.i − 0.613128i
\(36\) 0 0
\(37\) −1.96878e6 −1.05048 −0.525242 0.850953i \(-0.676025\pi\)
−0.525242 + 0.850953i \(0.676025\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.24983e6i − 0.442298i −0.975240 0.221149i \(-0.929019\pi\)
0.975240 0.221149i \(-0.0709808\pi\)
\(42\) 0 0
\(43\) 1.48518e6 0.434414 0.217207 0.976126i \(-0.430305\pi\)
0.217207 + 0.976126i \(0.430305\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.69893e6i 0.348165i 0.984731 + 0.174082i \(0.0556959\pi\)
−0.984731 + 0.174082i \(0.944304\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.91772e6i − 0.876718i −0.898800 0.438359i \(-0.855560\pi\)
0.898800 0.438359i \(-0.144440\pi\)
\(54\) 0 0
\(55\) 9.67796e6 1.05763
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.43342e7i 1.18295i 0.806324 + 0.591474i \(0.201454\pi\)
−0.806324 + 0.591474i \(0.798546\pi\)
\(60\) 0 0
\(61\) 2.12764e6 0.153666 0.0768332 0.997044i \(-0.475519\pi\)
0.0768332 + 0.997044i \(0.475519\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.02631e6i 0.113515i
\(66\) 0 0
\(67\) 696928. 0.0345851 0.0172925 0.999850i \(-0.494495\pi\)
0.0172925 + 0.999850i \(0.494495\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1.02684e6i − 0.0404080i −0.999796 0.0202040i \(-0.993568\pi\)
0.999796 0.0202040i \(-0.00643158\pi\)
\(72\) 0 0
\(73\) 3.22586e7 1.13594 0.567969 0.823050i \(-0.307729\pi\)
0.567969 + 0.823050i \(0.307729\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.66257e6i 0.246424i
\(78\) 0 0
\(79\) 4.61512e7 1.18488 0.592440 0.805614i \(-0.298165\pi\)
0.592440 + 0.805614i \(0.298165\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.08432e6i 0.107132i 0.998564 + 0.0535662i \(0.0170588\pi\)
−0.998564 + 0.0535662i \(0.982941\pi\)
\(84\) 0 0
\(85\) −4.67776e7 −0.896112
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 5.60351e7i − 0.893100i −0.894759 0.446550i \(-0.852653\pi\)
0.894759 0.446550i \(-0.147347\pi\)
\(90\) 0 0
\(91\) −1.81371e6 −0.0264486
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 3.45940e7i − 0.424723i
\(96\) 0 0
\(97\) −4.04787e7 −0.457235 −0.228617 0.973516i \(-0.573420\pi\)
−0.228617 + 0.973516i \(0.573420\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1.32286e8i − 1.27125i −0.772000 0.635623i \(-0.780743\pi\)
0.772000 0.635623i \(-0.219257\pi\)
\(102\) 0 0
\(103\) 7.57094e7 0.672668 0.336334 0.941743i \(-0.390813\pi\)
0.336334 + 0.941743i \(0.390813\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.42571e7i − 0.719083i −0.933129 0.359541i \(-0.882933\pi\)
0.933129 0.359541i \(-0.117067\pi\)
\(108\) 0 0
\(109\) −1.33918e8 −0.948711 −0.474355 0.880333i \(-0.657319\pi\)
−0.474355 + 0.880333i \(0.657319\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 2.65742e8i − 1.62984i −0.579570 0.814922i \(-0.696779\pi\)
0.579570 0.814922i \(-0.303221\pi\)
\(114\) 0 0
\(115\) 9.83549e7 0.562347
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 4.18698e7i − 0.208792i
\(120\) 0 0
\(121\) 1.23240e8 0.574925
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.50093e8i − 1.02438i
\(126\) 0 0
\(127\) −9.05365e7 −0.348024 −0.174012 0.984744i \(-0.555673\pi\)
−0.174012 + 0.984744i \(0.555673\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.17212e8i 1.07712i 0.842587 + 0.538561i \(0.181032\pi\)
−0.842587 + 0.538561i \(0.818968\pi\)
\(132\) 0 0
\(133\) 3.09645e7 0.0989594
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 3.82637e8i − 1.08619i −0.839672 0.543094i \(-0.817253\pi\)
0.839672 0.543094i \(-0.182747\pi\)
\(138\) 0 0
\(139\) 6.25504e8 1.67560 0.837801 0.545976i \(-0.183841\pi\)
0.837801 + 0.545976i \(0.183841\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 1.90778e7i − 0.0456231i
\(144\) 0 0
\(145\) 8.44927e8 1.91138
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.20466e8i 0.650185i 0.945682 + 0.325092i \(0.105395\pi\)
−0.945682 + 0.325092i \(0.894605\pi\)
\(150\) 0 0
\(151\) 8.82464e8 1.69742 0.848710 0.528859i \(-0.177380\pi\)
0.848710 + 0.528859i \(0.177380\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.04734e9i − 1.81451i
\(156\) 0 0
\(157\) −4.34353e8 −0.714898 −0.357449 0.933933i \(-0.616353\pi\)
−0.357449 + 0.933933i \(0.616353\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.80357e7i 0.131025i
\(162\) 0 0
\(163\) 8.00912e8 1.13458 0.567289 0.823519i \(-0.307992\pi\)
0.567289 + 0.823519i \(0.307992\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.38508e8i 0.435214i 0.976036 + 0.217607i \(0.0698251\pi\)
−0.976036 + 0.217607i \(0.930175\pi\)
\(168\) 0 0
\(169\) −8.11736e8 −0.995103
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.12498e9i − 1.25592i −0.778245 0.627960i \(-0.783890\pi\)
0.778245 0.627960i \(-0.216110\pi\)
\(174\) 0 0
\(175\) 5.78343e8 0.616643
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 8.77566e8i − 0.854806i −0.904061 0.427403i \(-0.859429\pi\)
0.904061 0.427403i \(-0.140571\pi\)
\(180\) 0 0
\(181\) 9.66077e8 0.900114 0.450057 0.893000i \(-0.351404\pi\)
0.450057 + 0.893000i \(0.351404\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 1.99607e9i − 1.70408i
\(186\) 0 0
\(187\) 4.40414e8 0.360159
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 1.74119e9i − 1.30832i −0.756357 0.654159i \(-0.773023\pi\)
0.756357 0.654159i \(-0.226977\pi\)
\(192\) 0 0
\(193\) −1.34770e9 −0.971323 −0.485661 0.874147i \(-0.661421\pi\)
−0.485661 + 0.874147i \(0.661421\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.96552e9i − 1.30500i −0.757787 0.652502i \(-0.773719\pi\)
0.757787 0.652502i \(-0.226281\pi\)
\(198\) 0 0
\(199\) −5.20469e8 −0.331881 −0.165940 0.986136i \(-0.553066\pi\)
−0.165940 + 0.986136i \(0.553066\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.56279e8i 0.445346i
\(204\) 0 0
\(205\) 1.26716e9 0.717490
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.25705e8i 0.170702i
\(210\) 0 0
\(211\) −3.68752e9 −1.86039 −0.930195 0.367065i \(-0.880363\pi\)
−0.930195 + 0.367065i \(0.880363\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.50577e9i 0.704700i
\(216\) 0 0
\(217\) 9.37453e8 0.422776
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.22112e7i 0.0386558i
\(222\) 0 0
\(223\) 3.02281e9 1.22234 0.611170 0.791499i \(-0.290699\pi\)
0.611170 + 0.791499i \(0.290699\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.46138e9i − 1.68022i −0.542418 0.840108i \(-0.682491\pi\)
0.542418 0.840108i \(-0.317509\pi\)
\(228\) 0 0
\(229\) 4.98782e9 1.81372 0.906858 0.421437i \(-0.138474\pi\)
0.906858 + 0.421437i \(0.138474\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.24117e9i 0.760416i 0.924901 + 0.380208i \(0.124148\pi\)
−0.924901 + 0.380208i \(0.875852\pi\)
\(234\) 0 0
\(235\) −1.72249e9 −0.564787
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 1.50773e9i − 0.462095i −0.972943 0.231047i \(-0.925785\pi\)
0.972943 0.231047i \(-0.0742152\pi\)
\(240\) 0 0
\(241\) −3.10570e9 −0.920644 −0.460322 0.887752i \(-0.652266\pi\)
−0.460322 + 0.887752i \(0.652266\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.34962e8i 0.231741i
\(246\) 0 0
\(247\) −6.81940e7 −0.0183214
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.08641e9i 0.777605i 0.921321 + 0.388802i \(0.127111\pi\)
−0.921321 + 0.388802i \(0.872889\pi\)
\(252\) 0 0
\(253\) −9.26017e8 −0.226015
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.74391e9i 1.31666i 0.752727 + 0.658332i \(0.228738\pi\)
−0.752727 + 0.658332i \(0.771262\pi\)
\(258\) 0 0
\(259\) 1.78665e9 0.397045
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 2.29821e9i − 0.480361i −0.970728 0.240180i \(-0.922793\pi\)
0.970728 0.240180i \(-0.0772066\pi\)
\(264\) 0 0
\(265\) 7.01364e9 1.42220
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.76923e9i 1.10182i 0.834566 + 0.550908i \(0.185718\pi\)
−0.834566 + 0.550908i \(0.814282\pi\)
\(270\) 0 0
\(271\) 2.15973e9 0.400426 0.200213 0.979752i \(-0.435837\pi\)
0.200213 + 0.979752i \(0.435837\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.08339e9i 1.06369i
\(276\) 0 0
\(277\) 2.42498e9 0.411898 0.205949 0.978563i \(-0.433972\pi\)
0.205949 + 0.978563i \(0.433972\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.33664e9i − 1.33711i −0.743665 0.668553i \(-0.766914\pi\)
0.743665 0.668553i \(-0.233086\pi\)
\(282\) 0 0
\(283\) −7.43925e9 −1.15980 −0.579900 0.814688i \(-0.696908\pi\)
−0.579900 + 0.814688i \(0.696908\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.13421e9i 0.167173i
\(288\) 0 0
\(289\) 4.84705e9 0.694842
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.71851e9i 0.775913i 0.921678 + 0.387956i \(0.126819\pi\)
−0.921678 + 0.387956i \(0.873181\pi\)
\(294\) 0 0
\(295\) −1.45330e10 −1.91896
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.93884e8i − 0.0242581i
\(300\) 0 0
\(301\) −1.34779e9 −0.164193
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.15714e9i 0.249275i
\(306\) 0 0
\(307\) −1.59969e10 −1.80087 −0.900437 0.434986i \(-0.856753\pi\)
−0.900437 + 0.434986i \(0.856753\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.54565e9i 0.913489i 0.889598 + 0.456745i \(0.150985\pi\)
−0.889598 + 0.456745i \(0.849015\pi\)
\(312\) 0 0
\(313\) −5.44401e9 −0.567207 −0.283604 0.958942i \(-0.591530\pi\)
−0.283604 + 0.958942i \(0.591530\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.70932e8i − 0.0268302i −0.999910 0.0134151i \(-0.995730\pi\)
0.999910 0.0134151i \(-0.00427028\pi\)
\(318\) 0 0
\(319\) −7.95504e9 −0.768209
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 1.57427e9i − 0.144633i
\(324\) 0 0
\(325\) −1.27370e9 −0.114166
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 1.54177e9i − 0.131594i
\(330\) 0 0
\(331\) −8.60297e9 −0.716698 −0.358349 0.933588i \(-0.616660\pi\)
−0.358349 + 0.933588i \(0.616660\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.06591e8i 0.0561034i
\(336\) 0 0
\(337\) −1.62536e10 −1.26017 −0.630086 0.776526i \(-0.716980\pi\)
−0.630086 + 0.776526i \(0.716980\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.86074e9i 0.729276i
\(342\) 0 0
\(343\) −7.47359e8 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.12989e10i 1.46906i 0.678577 + 0.734529i \(0.262597\pi\)
−0.678577 + 0.734529i \(0.737403\pi\)
\(348\) 0 0
\(349\) 3.42208e9 0.230669 0.115334 0.993327i \(-0.463206\pi\)
0.115334 + 0.993327i \(0.463206\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1.70364e10i − 1.09719i −0.836090 0.548593i \(-0.815164\pi\)
0.836090 0.548593i \(-0.184836\pi\)
\(354\) 0 0
\(355\) 1.04107e9 0.0655493
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.00228e10i 1.20545i 0.797950 + 0.602723i \(0.205918\pi\)
−0.797950 + 0.602723i \(0.794082\pi\)
\(360\) 0 0
\(361\) −1.58193e10 −0.931449
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.27059e10i 1.84270i
\(366\) 0 0
\(367\) 2.18172e10 1.20264 0.601320 0.799008i \(-0.294642\pi\)
0.601320 + 0.799008i \(0.294642\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.27778e9i 0.331368i
\(372\) 0 0
\(373\) −3.07371e9 −0.158792 −0.0793958 0.996843i \(-0.525299\pi\)
−0.0793958 + 0.996843i \(0.525299\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.66558e9i − 0.0824517i
\(378\) 0 0
\(379\) 3.44231e10 1.66837 0.834187 0.551481i \(-0.185937\pi\)
0.834187 + 0.551481i \(0.185937\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 4.13176e9i − 0.192017i −0.995381 0.0960086i \(-0.969392\pi\)
0.995381 0.0960086i \(-0.0306076\pi\)
\(384\) 0 0
\(385\) −8.78268e9 −0.399746
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 5.47550e9i − 0.239125i −0.992827 0.119563i \(-0.961851\pi\)
0.992827 0.119563i \(-0.0381493\pi\)
\(390\) 0 0
\(391\) 4.47583e9 0.191499
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.67911e10i 1.92210i
\(396\) 0 0
\(397\) 8.24142e9 0.331772 0.165886 0.986145i \(-0.446952\pi\)
0.165886 + 0.986145i \(0.446952\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 2.16226e10i − 0.836240i −0.908392 0.418120i \(-0.862689\pi\)
0.908392 0.418120i \(-0.137311\pi\)
\(402\) 0 0
\(403\) −2.06458e9 −0.0782730
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.87931e10i 0.684892i
\(408\) 0 0
\(409\) −5.39407e9 −0.192763 −0.0963815 0.995344i \(-0.530727\pi\)
−0.0963815 + 0.995344i \(0.530727\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1.30082e10i − 0.447112i
\(414\) 0 0
\(415\) −5.15482e9 −0.173789
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.42840e9i 0.0787887i 0.999224 + 0.0393944i \(0.0125429\pi\)
−0.999224 + 0.0393944i \(0.987457\pi\)
\(420\) 0 0
\(421\) 4.69140e10 1.49339 0.746696 0.665165i \(-0.231639\pi\)
0.746696 + 0.665165i \(0.231639\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 2.94036e10i − 0.901249i
\(426\) 0 0
\(427\) −1.93082e9 −0.0580804
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 1.98912e9i − 0.0576436i −0.999585 0.0288218i \(-0.990824\pi\)
0.999585 0.0288218i \(-0.00917553\pi\)
\(432\) 0 0
\(433\) −5.78975e10 −1.64706 −0.823528 0.567276i \(-0.807997\pi\)
−0.823528 + 0.567276i \(0.807997\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.31006e9i 0.0907633i
\(438\) 0 0
\(439\) −6.87960e10 −1.85227 −0.926137 0.377188i \(-0.876891\pi\)
−0.926137 + 0.377188i \(0.876891\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.88688e9i 0.0489924i 0.999700 + 0.0244962i \(0.00779816\pi\)
−0.999700 + 0.0244962i \(0.992202\pi\)
\(444\) 0 0
\(445\) 5.68120e10 1.44877
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 2.03669e10i − 0.501117i −0.968101 0.250559i \(-0.919386\pi\)
0.968101 0.250559i \(-0.0806143\pi\)
\(450\) 0 0
\(451\) −1.19304e10 −0.288369
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.83886e9i − 0.0429046i
\(456\) 0 0
\(457\) −6.31506e10 −1.44781 −0.723907 0.689898i \(-0.757655\pi\)
−0.723907 + 0.689898i \(0.757655\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 5.65698e10i − 1.25251i −0.779619 0.626255i \(-0.784587\pi\)
0.779619 0.626255i \(-0.215413\pi\)
\(462\) 0 0
\(463\) 4.67294e10 1.01687 0.508436 0.861100i \(-0.330224\pi\)
0.508436 + 0.861100i \(0.330224\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8.15355e10i − 1.71427i −0.515091 0.857135i \(-0.672242\pi\)
0.515091 0.857135i \(-0.327758\pi\)
\(468\) 0 0
\(469\) −6.32457e8 −0.0130719
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 1.41769e10i − 0.283228i
\(474\) 0 0
\(475\) 2.17452e10 0.427158
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.57462e9i 0.105894i 0.998597 + 0.0529472i \(0.0168615\pi\)
−0.998597 + 0.0529472i \(0.983138\pi\)
\(480\) 0 0
\(481\) −3.93479e9 −0.0735092
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 4.10399e10i − 0.741719i
\(486\) 0 0
\(487\) 5.40461e9 0.0960834 0.0480417 0.998845i \(-0.484702\pi\)
0.0480417 + 0.998845i \(0.484702\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.58242e10i 0.616383i 0.951324 + 0.308191i \(0.0997237\pi\)
−0.951324 + 0.308191i \(0.900276\pi\)
\(492\) 0 0
\(493\) 3.84500e10 0.650892
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.31846e8i 0.0152728i
\(498\) 0 0
\(499\) −9.29005e9 −0.149836 −0.0749179 0.997190i \(-0.523869\pi\)
−0.0749179 + 0.997190i \(0.523869\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 3.48297e10i − 0.544099i −0.962283 0.272050i \(-0.912299\pi\)
0.962283 0.272050i \(-0.0877015\pi\)
\(504\) 0 0
\(505\) 1.34120e11 2.06219
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 7.35332e10i − 1.09550i −0.836642 0.547750i \(-0.815484\pi\)
0.836642 0.547750i \(-0.184516\pi\)
\(510\) 0 0
\(511\) −2.92745e10 −0.429344
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.67591e10i 1.09119i
\(516\) 0 0
\(517\) 1.62173e10 0.226996
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 8.62629e10i − 1.17077i −0.810754 0.585387i \(-0.800943\pi\)
0.810754 0.585387i \(-0.199057\pi\)
\(522\) 0 0
\(523\) −1.40847e10 −0.188252 −0.0941260 0.995560i \(-0.530006\pi\)
−0.0941260 + 0.995560i \(0.530006\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 4.76611e10i − 0.617905i
\(528\) 0 0
\(529\) 6.89001e10 0.879827
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 2.49791e9i − 0.0309505i
\(534\) 0 0
\(535\) 9.55640e10 1.16648
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 7.86122e9i − 0.0931397i
\(540\) 0 0
\(541\) 9.32399e10 1.08846 0.544230 0.838936i \(-0.316822\pi\)
0.544230 + 0.838936i \(0.316822\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1.35775e11i − 1.53898i
\(546\) 0 0
\(547\) 2.51789e10 0.281247 0.140623 0.990063i \(-0.455089\pi\)
0.140623 + 0.990063i \(0.455089\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.84354e10i 0.308498i
\(552\) 0 0
\(553\) −4.18819e10 −0.447843
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.43202e10i 0.668231i 0.942532 + 0.334115i \(0.108438\pi\)
−0.942532 + 0.334115i \(0.891562\pi\)
\(558\) 0 0
\(559\) 2.96827e9 0.0303988
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1.65235e10i − 0.164463i −0.996613 0.0822314i \(-0.973795\pi\)
0.996613 0.0822314i \(-0.0262047\pi\)
\(564\) 0 0
\(565\) 2.69427e11 2.64391
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.28229e10i 0.313132i 0.987667 + 0.156566i \(0.0500424\pi\)
−0.987667 + 0.156566i \(0.949958\pi\)
\(570\) 0 0
\(571\) −1.16352e11 −1.09454 −0.547269 0.836957i \(-0.684332\pi\)
−0.547269 + 0.836957i \(0.684332\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.18242e10i 0.565571i
\(576\) 0 0
\(577\) 1.44383e10 0.130260 0.0651302 0.997877i \(-0.479254\pi\)
0.0651302 + 0.997877i \(0.479254\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 4.61399e9i − 0.0404922i
\(582\) 0 0
\(583\) −6.60339e10 −0.571600
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.46230e11i 1.23164i 0.787886 + 0.615821i \(0.211175\pi\)
−0.787886 + 0.615821i \(0.788825\pi\)
\(588\) 0 0
\(589\) 3.52474e10 0.292864
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.76245e11i − 1.42527i −0.701534 0.712636i \(-0.747501\pi\)
0.701534 0.712636i \(-0.252499\pi\)
\(594\) 0 0
\(595\) 4.24503e10 0.338699
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1.77720e11i − 1.38047i −0.723583 0.690237i \(-0.757506\pi\)
0.723583 0.690237i \(-0.242494\pi\)
\(600\) 0 0
\(601\) −3.20993e10 −0.246035 −0.123018 0.992404i \(-0.539257\pi\)
−0.123018 + 0.992404i \(0.539257\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.24949e11i 0.932635i
\(606\) 0 0
\(607\) −1.41329e11 −1.04107 −0.520533 0.853842i \(-0.674267\pi\)
−0.520533 + 0.853842i \(0.674267\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.39549e9i 0.0243633i
\(612\) 0 0
\(613\) 5.44012e10 0.385271 0.192636 0.981270i \(-0.438296\pi\)
0.192636 + 0.981270i \(0.438296\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.79721e11i − 1.93012i −0.262029 0.965060i \(-0.584392\pi\)
0.262029 0.965060i \(-0.415608\pi\)
\(618\) 0 0
\(619\) −1.53977e11 −1.04880 −0.524401 0.851471i \(-0.675711\pi\)
−0.524401 + 0.851471i \(0.675711\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.08514e10i 0.337560i
\(624\) 0 0
\(625\) 4.61634e9 0.0302536
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 9.08352e10i − 0.580298i
\(630\) 0 0
\(631\) −1.78193e11 −1.12402 −0.562008 0.827132i \(-0.689971\pi\)
−0.562008 + 0.827132i \(0.689971\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 9.17918e10i − 0.564558i
\(636\) 0 0
\(637\) 1.64593e9 0.00999665
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 4.15099e9i − 0.0245878i −0.999924 0.0122939i \(-0.996087\pi\)
0.999924 0.0122939i \(-0.00391337\pi\)
\(642\) 0 0
\(643\) −2.08912e10 −0.122213 −0.0611067 0.998131i \(-0.519463\pi\)
−0.0611067 + 0.998131i \(0.519463\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.96707e11i 1.12254i 0.827632 + 0.561272i \(0.189688\pi\)
−0.827632 + 0.561272i \(0.810312\pi\)
\(648\) 0 0
\(649\) 1.36829e11 0.771255
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.19791e11i 0.658826i 0.944186 + 0.329413i \(0.106851\pi\)
−0.944186 + 0.329413i \(0.893149\pi\)
\(654\) 0 0
\(655\) −3.21611e11 −1.74729
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.57428e11i − 0.834718i −0.908742 0.417359i \(-0.862956\pi\)
0.908742 0.417359i \(-0.137044\pi\)
\(660\) 0 0
\(661\) −1.18188e11 −0.619108 −0.309554 0.950882i \(-0.600180\pi\)
−0.309554 + 0.950882i \(0.600180\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.13938e10i 0.160530i
\(666\) 0 0
\(667\) −8.08453e10 −0.408462
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 2.03096e10i − 0.100187i
\(672\) 0 0
\(673\) −2.81868e11 −1.37400 −0.686999 0.726658i \(-0.741072\pi\)
−0.686999 + 0.726658i \(0.741072\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.16668e11i 0.555388i 0.960670 + 0.277694i \(0.0895701\pi\)
−0.960670 + 0.277694i \(0.910430\pi\)
\(678\) 0 0
\(679\) 3.67341e10 0.172818
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.48480e11i 0.682316i 0.940006 + 0.341158i \(0.110819\pi\)
−0.940006 + 0.341158i \(0.889181\pi\)
\(684\) 0 0
\(685\) 3.87943e11 1.76200
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1.38258e10i − 0.0613497i
\(690\) 0 0
\(691\) 1.22512e11 0.537363 0.268681 0.963229i \(-0.413412\pi\)
0.268681 + 0.963229i \(0.413412\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.34177e11i 2.71813i
\(696\) 0 0
\(697\) 5.76645e10 0.244330
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 2.62323e11i − 1.08634i −0.839623 0.543169i \(-0.817224\pi\)
0.839623 0.543169i \(-0.182776\pi\)
\(702\) 0 0
\(703\) 6.71764e10 0.275040
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.20049e11i 0.480486i
\(708\) 0 0
\(709\) 1.26919e11 0.502276 0.251138 0.967951i \(-0.419195\pi\)
0.251138 + 0.967951i \(0.419195\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.00212e11i 0.387761i
\(714\) 0 0
\(715\) 1.93424e10 0.0740091
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.80440e10i 0.329446i 0.986340 + 0.164723i \(0.0526730\pi\)
−0.986340 + 0.164723i \(0.947327\pi\)
\(720\) 0 0
\(721\) −6.87057e10 −0.254245
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.31106e11i 1.92234i
\(726\) 0 0
\(727\) 1.70730e11 0.611185 0.305593 0.952162i \(-0.401145\pi\)
0.305593 + 0.952162i \(0.401145\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.85229e10i 0.239975i
\(732\) 0 0
\(733\) 3.96253e11 1.37264 0.686321 0.727299i \(-0.259225\pi\)
0.686321 + 0.727299i \(0.259225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 6.65260e9i − 0.0225487i
\(738\) 0 0
\(739\) −5.07867e11 −1.70283 −0.851417 0.524489i \(-0.824256\pi\)
−0.851417 + 0.524489i \(0.824256\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.71694e11i 0.891508i 0.895155 + 0.445754i \(0.147064\pi\)
−0.895155 + 0.445754i \(0.852936\pi\)
\(744\) 0 0
\(745\) −3.24909e11 −1.05472
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.55376e10i 0.271788i
\(750\) 0 0
\(751\) −2.61830e11 −0.823115 −0.411557 0.911384i \(-0.635015\pi\)
−0.411557 + 0.911384i \(0.635015\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.94699e11i 2.75353i
\(756\) 0 0
\(757\) −4.81228e11 −1.46544 −0.732720 0.680531i \(-0.761749\pi\)
−0.732720 + 0.680531i \(0.761749\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.99450e10i 0.0892863i 0.999003 + 0.0446432i \(0.0142151\pi\)
−0.999003 + 0.0446432i \(0.985785\pi\)
\(762\) 0 0
\(763\) 1.21530e11 0.358579
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.86483e10i 0.0827786i
\(768\) 0 0
\(769\) −3.60756e11 −1.03159 −0.515796 0.856712i \(-0.672504\pi\)
−0.515796 + 0.856712i \(0.672504\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 6.41322e11i − 1.79621i −0.439776 0.898107i \(-0.644942\pi\)
0.439776 0.898107i \(-0.355058\pi\)
\(774\) 0 0
\(775\) 6.58338e11 1.82491
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.26453e10i 0.115803i
\(780\) 0 0
\(781\) −9.80177e9 −0.0263451
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 4.40376e11i − 1.15970i
\(786\) 0 0
\(787\) −6.13315e11 −1.59876 −0.799382 0.600823i \(-0.794840\pi\)
−0.799382 + 0.600823i \(0.794840\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.41159e11i 0.616023i
\(792\) 0 0
\(793\) 4.25230e9 0.0107530
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 3.02614e11i − 0.749990i −0.927027 0.374995i \(-0.877644\pi\)
0.927027 0.374995i \(-0.122356\pi\)
\(798\) 0 0
\(799\) −7.83852e10 −0.192330
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 3.07928e11i − 0.740606i
\(804\) 0 0
\(805\) −8.92563e10 −0.212547
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.95071e11i 0.922320i 0.887317 + 0.461160i \(0.152567\pi\)
−0.887317 + 0.461160i \(0.847433\pi\)
\(810\) 0 0
\(811\) 5.52971e11 1.27826 0.639129 0.769099i \(-0.279295\pi\)
0.639129 + 0.769099i \(0.279295\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.12017e11i 1.84049i
\(816\) 0 0
\(817\) −5.06755e10 −0.113739
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.37182e11i 0.522047i 0.965332 + 0.261024i \(0.0840601\pi\)
−0.965332 + 0.261024i \(0.915940\pi\)
\(822\) 0 0
\(823\) −6.39357e11 −1.39362 −0.696809 0.717256i \(-0.745398\pi\)
−0.696809 + 0.717256i \(0.745398\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 2.61243e11i − 0.558500i −0.960218 0.279250i \(-0.909914\pi\)
0.960218 0.279250i \(-0.0900859\pi\)
\(828\) 0 0
\(829\) −5.75999e11 −1.21956 −0.609781 0.792570i \(-0.708742\pi\)
−0.609781 + 0.792570i \(0.708742\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.79965e10i 0.0789158i
\(834\) 0 0
\(835\) −3.43201e11 −0.705997
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 6.46156e11i − 1.30404i −0.758203 0.652018i \(-0.773923\pi\)
0.758203 0.652018i \(-0.226077\pi\)
\(840\) 0 0
\(841\) −1.94263e11 −0.388334
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 8.22991e11i − 1.61424i
\(846\) 0 0
\(847\) −1.11840e11 −0.217301
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.90991e11i 0.364161i
\(852\) 0 0
\(853\) −6.56405e11 −1.23987 −0.619934 0.784654i \(-0.712841\pi\)
−0.619934 + 0.784654i \(0.712841\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.41157e11i − 0.261686i −0.991403 0.130843i \(-0.958232\pi\)
0.991403 0.130843i \(-0.0417684\pi\)
\(858\) 0 0
\(859\) 2.85464e11 0.524299 0.262149 0.965027i \(-0.415569\pi\)
0.262149 + 0.965027i \(0.415569\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.38850e11i 0.791176i 0.918428 + 0.395588i \(0.129459\pi\)
−0.918428 + 0.395588i \(0.870541\pi\)
\(864\) 0 0
\(865\) 1.14058e12 2.03733
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 4.40541e11i − 0.772516i
\(870\) 0 0
\(871\) 1.39288e9 0.00242014
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.26958e11i 0.387180i
\(876\) 0 0
\(877\) 1.30388e11 0.220414 0.110207 0.993909i \(-0.464849\pi\)
0.110207 + 0.993909i \(0.464849\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 2.22271e11i − 0.368960i −0.982836 0.184480i \(-0.940940\pi\)
0.982836 0.184480i \(-0.0590600\pi\)
\(882\) 0 0
\(883\) 7.44193e11 1.22417 0.612086 0.790791i \(-0.290330\pi\)
0.612086 + 0.790791i \(0.290330\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 6.60994e11i − 1.06783i −0.845538 0.533916i \(-0.820720\pi\)
0.845538 0.533916i \(-0.179280\pi\)
\(888\) 0 0
\(889\) 8.21612e10 0.131541
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 5.79691e10i − 0.0911571i
\(894\) 0 0
\(895\) 8.89734e11 1.38665
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.60885e11i 1.31797i
\(900\) 0 0
\(901\) 3.19169e11 0.484308
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.79473e11i 1.46015i
\(906\) 0 0
\(907\) −1.10739e12 −1.63634 −0.818168 0.574980i \(-0.805010\pi\)
−0.818168 + 0.574980i \(0.805010\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 9.17243e10i − 0.133171i −0.997781 0.0665857i \(-0.978789\pi\)
0.997781 0.0665857i \(-0.0212106\pi\)
\(912\) 0 0
\(913\) 4.85330e10 0.0698479
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.87868e11i − 0.407114i
\(918\) 0 0
\(919\) −8.14021e11 −1.14123 −0.570615 0.821218i \(-0.693295\pi\)
−0.570615 + 0.821218i \(0.693295\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 2.05223e9i − 0.00282761i
\(924\) 0 0
\(925\) 1.25470e12 1.71385
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 8.69774e11i − 1.16773i −0.811849 0.583867i \(-0.801539\pi\)
0.811849 0.583867i \(-0.198461\pi\)
\(930\) 0 0
\(931\) −2.81000e10 −0.0374031
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.46521e11i 0.584245i
\(936\) 0 0
\(937\) 9.17180e11 1.18986 0.594930 0.803777i \(-0.297180\pi\)
0.594930 + 0.803777i \(0.297180\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.00191e11i 0.382859i 0.981506 + 0.191430i \(0.0613123\pi\)
−0.981506 + 0.191430i \(0.938688\pi\)
\(942\) 0 0
\(943\) −1.21246e11 −0.153327
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7.39412e11i − 0.919362i −0.888084 0.459681i \(-0.847964\pi\)
0.888084 0.459681i \(-0.152036\pi\)
\(948\) 0 0
\(949\) 6.44721e10 0.0794890
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1.49547e11i − 0.181303i −0.995883 0.0906516i \(-0.971105\pi\)
0.995883 0.0906516i \(-0.0288950\pi\)
\(954\) 0 0
\(955\) 1.76534e12 2.12233
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.47240e11i 0.410540i
\(960\) 0 0
\(961\) 2.14227e11 0.251177
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1.36639e12i − 1.57566i
\(966\) 0 0
\(967\) 1.07075e12 1.22456 0.612282 0.790639i \(-0.290252\pi\)
0.612282 + 0.790639i \(0.290252\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1.54708e12i − 1.74035i −0.492743 0.870175i \(-0.664006\pi\)
0.492743 0.870175i \(-0.335994\pi\)
\(972\) 0 0
\(973\) −5.67640e11 −0.633318
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.50352e12i 1.65017i 0.565006 + 0.825087i \(0.308874\pi\)
−0.565006 + 0.825087i \(0.691126\pi\)
\(978\) 0 0
\(979\) −5.34889e11 −0.582281
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 5.92520e11i − 0.634584i −0.948328 0.317292i \(-0.897227\pi\)
0.948328 0.317292i \(-0.102773\pi\)
\(984\) 0 0
\(985\) 1.99277e12 2.11696
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1.44077e11i − 0.150594i
\(990\) 0 0
\(991\) −7.60363e11 −0.788363 −0.394182 0.919033i \(-0.628972\pi\)
−0.394182 + 0.919033i \(0.628972\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 5.27685e11i − 0.538372i
\(996\) 0 0
\(997\) −5.93177e10 −0.0600349 −0.0300175 0.999549i \(-0.509556\pi\)
−0.0300175 + 0.999549i \(0.509556\pi\)
\(998\) 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 252.9.c.a.197.16 yes 16
3.2 odd 2 inner 252.9.c.a.197.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.1 16 3.2 odd 2 inner
252.9.c.a.197.16 yes 16 1.1 even 1 trivial