Properties

Label 252.9.c.a.197.13
Level $252$
Weight $9$
Character 252.197
Analytic conductor $102.659$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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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 197.13
Root \(675.810i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.4

$q$-expansion

\(f(q)\) \(=\) \(q+675.810i q^{5} +907.493 q^{7} +O(q^{10})\) \(q+675.810i q^{5} +907.493 q^{7} +18942.7i q^{11} +6233.80 q^{13} -15144.2i q^{17} +181701. q^{19} +421094. i q^{23} -66094.2 q^{25} -190089. i q^{29} -198070. q^{31} +613293. i q^{35} +700460. q^{37} -628097. i q^{41} +5.91651e6 q^{43} +6.30001e6i q^{47} +823543. q^{49} -2.32657e6i q^{53} -1.28016e7 q^{55} -1.34161e7i q^{59} -9.18165e6 q^{61} +4.21286e6i q^{65} +6.29819e6 q^{67} -1.43966e7i q^{71} -4.04114e7 q^{73} +1.71903e7i q^{77} -3.00463e7 q^{79} +5.01957e7i q^{83} +1.02346e7 q^{85} +4.17353e6i q^{89} +5.65713e6 q^{91} +1.22795e8i q^{95} +1.41211e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

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
\(5\) 675.810i 1.08130i 0.841249 + 0.540648i \(0.181821\pi\)
−0.841249 + 0.540648i \(0.818179\pi\)
\(6\) 0 0
\(7\) 907.493 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18942.7i 1.29381i 0.762571 + 0.646905i \(0.223937\pi\)
−0.762571 + 0.646905i \(0.776063\pi\)
\(12\) 0 0
\(13\) 6233.80 0.218263 0.109131 0.994027i \(-0.465193\pi\)
0.109131 + 0.994027i \(0.465193\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 15144.2i − 0.181322i −0.995882 0.0906611i \(-0.971102\pi\)
0.995882 0.0906611i \(-0.0288980\pi\)
\(18\) 0 0
\(19\) 181701. 1.39425 0.697127 0.716947i \(-0.254461\pi\)
0.697127 + 0.716947i \(0.254461\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 421094.i 1.50476i 0.658728 + 0.752381i \(0.271095\pi\)
−0.658728 + 0.752381i \(0.728905\pi\)
\(24\) 0 0
\(25\) −66094.2 −0.169201
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 190089.i − 0.268760i −0.990930 0.134380i \(-0.957096\pi\)
0.990930 0.134380i \(-0.0429043\pi\)
\(30\) 0 0
\(31\) −198070. −0.214473 −0.107236 0.994234i \(-0.534200\pi\)
−0.107236 + 0.994234i \(0.534200\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 613293.i 0.408692i
\(36\) 0 0
\(37\) 700460. 0.373746 0.186873 0.982384i \(-0.440165\pi\)
0.186873 + 0.982384i \(0.440165\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 628097.i − 0.222275i −0.993805 0.111138i \(-0.964551\pi\)
0.993805 0.111138i \(-0.0354495\pi\)
\(42\) 0 0
\(43\) 5.91651e6 1.73058 0.865290 0.501271i \(-0.167134\pi\)
0.865290 + 0.501271i \(0.167134\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.30001e6i 1.29107i 0.763730 + 0.645535i \(0.223366\pi\)
−0.763730 + 0.645535i \(0.776634\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.32657e6i − 0.294858i −0.989073 0.147429i \(-0.952900\pi\)
0.989073 0.147429i \(-0.0470998\pi\)
\(54\) 0 0
\(55\) −1.28016e7 −1.39899
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1.34161e7i − 1.10718i −0.832788 0.553592i \(-0.813257\pi\)
0.832788 0.553592i \(-0.186743\pi\)
\(60\) 0 0
\(61\) −9.18165e6 −0.663134 −0.331567 0.943432i \(-0.607577\pi\)
−0.331567 + 0.943432i \(0.607577\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.21286e6i 0.236007i
\(66\) 0 0
\(67\) 6.29819e6 0.312548 0.156274 0.987714i \(-0.450052\pi\)
0.156274 + 0.987714i \(0.450052\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1.43966e7i − 0.566536i −0.959041 0.283268i \(-0.908581\pi\)
0.959041 0.283268i \(-0.0914186\pi\)
\(72\) 0 0
\(73\) −4.04114e7 −1.42302 −0.711512 0.702674i \(-0.751989\pi\)
−0.711512 + 0.702674i \(0.751989\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.71903e7i 0.489014i
\(78\) 0 0
\(79\) −3.00463e7 −0.771406 −0.385703 0.922623i \(-0.626041\pi\)
−0.385703 + 0.922623i \(0.626041\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.01957e7i 1.05768i 0.848722 + 0.528840i \(0.177373\pi\)
−0.848722 + 0.528840i \(0.822627\pi\)
\(84\) 0 0
\(85\) 1.02346e7 0.196063
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.17353e6i 0.0665186i 0.999447 + 0.0332593i \(0.0105887\pi\)
−0.999447 + 0.0332593i \(0.989411\pi\)
\(90\) 0 0
\(91\) 5.65713e6 0.0824955
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.22795e8i 1.50760i
\(96\) 0 0
\(97\) 1.41211e8 1.59507 0.797536 0.603272i \(-0.206136\pi\)
0.797536 + 0.603272i \(0.206136\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 6.22032e6i − 0.0597761i −0.999553 0.0298881i \(-0.990485\pi\)
0.999553 0.0298881i \(-0.00951508\pi\)
\(102\) 0 0
\(103\) −5.46907e7 −0.485920 −0.242960 0.970036i \(-0.578118\pi\)
−0.242960 + 0.970036i \(0.578118\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.90127e7i − 0.526494i −0.964728 0.263247i \(-0.915207\pi\)
0.964728 0.263247i \(-0.0847935\pi\)
\(108\) 0 0
\(109\) 8.93853e7 0.633228 0.316614 0.948554i \(-0.397454\pi\)
0.316614 + 0.948554i \(0.397454\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.90457e8i 1.78143i 0.454565 + 0.890714i \(0.349795\pi\)
−0.454565 + 0.890714i \(0.650205\pi\)
\(114\) 0 0
\(115\) −2.84580e8 −1.62709
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 1.37433e7i − 0.0685334i
\(120\) 0 0
\(121\) −1.44466e8 −0.673943
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.19321e8i 0.898339i
\(126\) 0 0
\(127\) −3.37474e8 −1.29725 −0.648627 0.761107i \(-0.724656\pi\)
−0.648627 + 0.761107i \(0.724656\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.80523e8i 0.612982i 0.951874 + 0.306491i \(0.0991549\pi\)
−0.951874 + 0.306491i \(0.900845\pi\)
\(132\) 0 0
\(133\) 1.64892e8 0.526979
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.75533e8i 0.498283i 0.968467 + 0.249141i \(0.0801484\pi\)
−0.968467 + 0.249141i \(0.919852\pi\)
\(138\) 0 0
\(139\) −1.14176e8 −0.305856 −0.152928 0.988237i \(-0.548870\pi\)
−0.152928 + 0.988237i \(0.548870\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.18085e8i 0.282390i
\(144\) 0 0
\(145\) 1.28464e8 0.290609
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.90588e8i 1.19823i 0.800663 + 0.599115i \(0.204481\pi\)
−0.800663 + 0.599115i \(0.795519\pi\)
\(150\) 0 0
\(151\) −9.52935e8 −1.83297 −0.916485 0.400069i \(-0.868986\pi\)
−0.916485 + 0.400069i \(0.868986\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.33858e8i − 0.231908i
\(156\) 0 0
\(157\) −9.02887e8 −1.48605 −0.743027 0.669261i \(-0.766611\pi\)
−0.743027 + 0.669261i \(0.766611\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.82140e8i 0.568747i
\(162\) 0 0
\(163\) 1.63169e8 0.231146 0.115573 0.993299i \(-0.463130\pi\)
0.115573 + 0.993299i \(0.463130\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.25869e8i 0.418964i 0.977813 + 0.209482i \(0.0671777\pi\)
−0.977813 + 0.209482i \(0.932822\pi\)
\(168\) 0 0
\(169\) −7.76870e8 −0.952361
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.01042e9i 1.12802i 0.825768 + 0.564009i \(0.190742\pi\)
−0.825768 + 0.564009i \(0.809258\pi\)
\(174\) 0 0
\(175\) −5.99800e7 −0.0639520
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 8.92198e7i − 0.0869059i −0.999055 0.0434530i \(-0.986164\pi\)
0.999055 0.0434530i \(-0.0138359\pi\)
\(180\) 0 0
\(181\) −9.63231e8 −0.897463 −0.448731 0.893667i \(-0.648124\pi\)
−0.448731 + 0.893667i \(0.648124\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.73378e8i 0.404130i
\(186\) 0 0
\(187\) 2.86872e8 0.234597
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.36076e9i 1.77386i 0.461907 + 0.886928i \(0.347165\pi\)
−0.461907 + 0.886928i \(0.652835\pi\)
\(192\) 0 0
\(193\) 2.96485e8 0.213684 0.106842 0.994276i \(-0.465926\pi\)
0.106842 + 0.994276i \(0.465926\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.55058e9i − 1.02951i −0.857338 0.514754i \(-0.827883\pi\)
0.857338 0.514754i \(-0.172117\pi\)
\(198\) 0 0
\(199\) 2.31261e9 1.47465 0.737326 0.675537i \(-0.236088\pi\)
0.737326 + 0.675537i \(0.236088\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.72504e8i − 0.101582i
\(204\) 0 0
\(205\) 4.24474e8 0.240345
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.44190e9i 1.80390i
\(210\) 0 0
\(211\) −3.86336e9 −1.94910 −0.974552 0.224163i \(-0.928035\pi\)
−0.974552 + 0.224163i \(0.928035\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.99844e9i 1.87127i
\(216\) 0 0
\(217\) −1.79747e8 −0.0810630
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 9.44060e7i − 0.0395759i
\(222\) 0 0
\(223\) 3.06779e9 1.24053 0.620264 0.784393i \(-0.287025\pi\)
0.620264 + 0.784393i \(0.287025\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.63319e8i 0.212154i 0.994358 + 0.106077i \(0.0338290\pi\)
−0.994358 + 0.106077i \(0.966171\pi\)
\(228\) 0 0
\(229\) 1.34829e9 0.490278 0.245139 0.969488i \(-0.421166\pi\)
0.245139 + 0.969488i \(0.421166\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.61906e8i − 0.224581i −0.993675 0.112290i \(-0.964181\pi\)
0.993675 0.112290i \(-0.0358187\pi\)
\(234\) 0 0
\(235\) −4.25761e9 −1.39603
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 4.96165e9i − 1.52067i −0.649531 0.760335i \(-0.725035\pi\)
0.649531 0.760335i \(-0.274965\pi\)
\(240\) 0 0
\(241\) −3.38994e9 −1.00490 −0.502452 0.864605i \(-0.667569\pi\)
−0.502452 + 0.864605i \(0.667569\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.56559e8i 0.154471i
\(246\) 0 0
\(247\) 1.13269e9 0.304314
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 2.26441e9i − 0.570507i −0.958452 0.285253i \(-0.907922\pi\)
0.958452 0.285253i \(-0.0920777\pi\)
\(252\) 0 0
\(253\) −7.97665e9 −1.94688
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 6.00346e9i − 1.37616i −0.725635 0.688080i \(-0.758454\pi\)
0.725635 0.688080i \(-0.241546\pi\)
\(258\) 0 0
\(259\) 6.35662e8 0.141263
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 5.31075e8i − 0.111003i −0.998459 0.0555013i \(-0.982324\pi\)
0.998459 0.0555013i \(-0.0176757\pi\)
\(264\) 0 0
\(265\) 1.57232e9 0.318829
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.46616e8i 0.0661972i 0.999452 + 0.0330986i \(0.0105375\pi\)
−0.999452 + 0.0330986i \(0.989462\pi\)
\(270\) 0 0
\(271\) 7.88625e9 1.46216 0.731078 0.682294i \(-0.239018\pi\)
0.731078 + 0.682294i \(0.239018\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.25200e9i − 0.218914i
\(276\) 0 0
\(277\) 2.59726e9 0.441161 0.220580 0.975369i \(-0.429205\pi\)
0.220580 + 0.975369i \(0.429205\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.57542e9i 0.573459i 0.958012 + 0.286729i \(0.0925681\pi\)
−0.958012 + 0.286729i \(0.907432\pi\)
\(282\) 0 0
\(283\) −3.40830e9 −0.531365 −0.265682 0.964061i \(-0.585597\pi\)
−0.265682 + 0.964061i \(0.585597\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 5.69994e8i − 0.0840122i
\(288\) 0 0
\(289\) 6.74641e9 0.967122
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 6.14339e9i − 0.833562i −0.909007 0.416781i \(-0.863158\pi\)
0.909007 0.416781i \(-0.136842\pi\)
\(294\) 0 0
\(295\) 9.06676e9 1.19719
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.62502e9i 0.328433i
\(300\) 0 0
\(301\) 5.36919e9 0.654098
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 6.20505e9i − 0.717045i
\(306\) 0 0
\(307\) 5.89041e9 0.663120 0.331560 0.943434i \(-0.392425\pi\)
0.331560 + 0.943434i \(0.392425\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 1.47415e10i − 1.57580i −0.615803 0.787900i \(-0.711168\pi\)
0.615803 0.787900i \(-0.288832\pi\)
\(312\) 0 0
\(313\) −2.41159e9 −0.251262 −0.125631 0.992077i \(-0.540096\pi\)
−0.125631 + 0.992077i \(0.540096\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.73533e9i − 0.270877i −0.990786 0.135438i \(-0.956756\pi\)
0.990786 0.135438i \(-0.0432443\pi\)
\(318\) 0 0
\(319\) 3.60079e9 0.347725
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 2.75171e9i − 0.252809i
\(324\) 0 0
\(325\) −4.12018e8 −0.0369303
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.71722e9i 0.487979i
\(330\) 0 0
\(331\) 1.97672e9 0.164677 0.0823387 0.996604i \(-0.473761\pi\)
0.0823387 + 0.996604i \(0.473761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.25638e9i 0.337957i
\(336\) 0 0
\(337\) 5.00212e9 0.387824 0.193912 0.981019i \(-0.437882\pi\)
0.193912 + 0.981019i \(0.437882\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 3.75197e9i − 0.277487i
\(342\) 0 0
\(343\) 7.47359e8 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1.03079e9i − 0.0710971i −0.999368 0.0355485i \(-0.988682\pi\)
0.999368 0.0355485i \(-0.0113178\pi\)
\(348\) 0 0
\(349\) 5.68272e9 0.383049 0.191525 0.981488i \(-0.438657\pi\)
0.191525 + 0.981488i \(0.438657\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 7.38729e9i − 0.475758i −0.971295 0.237879i \(-0.923548\pi\)
0.971295 0.237879i \(-0.0764522\pi\)
\(354\) 0 0
\(355\) 9.72939e9 0.612593
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 1.75128e10i − 1.05434i −0.849761 0.527168i \(-0.823254\pi\)
0.849761 0.527168i \(-0.176746\pi\)
\(360\) 0 0
\(361\) 1.60316e10 0.943946
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 2.73104e10i − 1.53871i
\(366\) 0 0
\(367\) −9.61052e9 −0.529764 −0.264882 0.964281i \(-0.585333\pi\)
−0.264882 + 0.964281i \(0.585333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 2.11135e9i − 0.111446i
\(372\) 0 0
\(373\) −1.74568e10 −0.901837 −0.450919 0.892565i \(-0.648904\pi\)
−0.450919 + 0.892565i \(0.648904\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.18498e9i − 0.0586603i
\(378\) 0 0
\(379\) 3.39385e10 1.64489 0.822444 0.568846i \(-0.192610\pi\)
0.822444 + 0.568846i \(0.192610\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 3.21374e10i − 1.49354i −0.665084 0.746769i \(-0.731604\pi\)
0.665084 0.746769i \(-0.268396\pi\)
\(384\) 0 0
\(385\) −1.16174e10 −0.528769
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.89590e10i 0.827977i 0.910282 + 0.413988i \(0.135865\pi\)
−0.910282 + 0.413988i \(0.864135\pi\)
\(390\) 0 0
\(391\) 6.37714e9 0.272847
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 2.03056e10i − 0.834119i
\(396\) 0 0
\(397\) −2.73123e10 −1.09950 −0.549751 0.835328i \(-0.685278\pi\)
−0.549751 + 0.835328i \(0.685278\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.05653e10i 0.795349i 0.917527 + 0.397675i \(0.130183\pi\)
−0.917527 + 0.397675i \(0.869817\pi\)
\(402\) 0 0
\(403\) −1.23473e9 −0.0468113
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.32686e10i 0.483556i
\(408\) 0 0
\(409\) 4.15906e10 1.48629 0.743143 0.669133i \(-0.233334\pi\)
0.743143 + 0.669133i \(0.233334\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1.21751e10i − 0.418476i
\(414\) 0 0
\(415\) −3.39228e10 −1.14366
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 8.09475e9i − 0.262632i −0.991341 0.131316i \(-0.958080\pi\)
0.991341 0.131316i \(-0.0419202\pi\)
\(420\) 0 0
\(421\) −1.32776e10 −0.422660 −0.211330 0.977415i \(-0.567780\pi\)
−0.211330 + 0.977415i \(0.567780\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00095e9i 0.0306800i
\(426\) 0 0
\(427\) −8.33228e9 −0.250641
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 4.74178e10i − 1.37415i −0.726589 0.687073i \(-0.758895\pi\)
0.726589 0.687073i \(-0.241105\pi\)
\(432\) 0 0
\(433\) −4.31302e10 −1.22696 −0.613479 0.789711i \(-0.710230\pi\)
−0.613479 + 0.789711i \(0.710230\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.65131e10i 2.09802i
\(438\) 0 0
\(439\) 9.81569e9 0.264279 0.132140 0.991231i \(-0.457815\pi\)
0.132140 + 0.991231i \(0.457815\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 2.98569e10i − 0.775229i −0.921822 0.387614i \(-0.873299\pi\)
0.921822 0.387614i \(-0.126701\pi\)
\(444\) 0 0
\(445\) −2.82051e9 −0.0719263
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.45404e10i 1.58798i 0.607928 + 0.793992i \(0.292001\pi\)
−0.607928 + 0.793992i \(0.707999\pi\)
\(450\) 0 0
\(451\) 1.18978e10 0.287582
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.82314e9i 0.0892021i
\(456\) 0 0
\(457\) 7.79905e10 1.78804 0.894019 0.448029i \(-0.147874\pi\)
0.894019 + 0.448029i \(0.147874\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 3.74346e10i − 0.828838i −0.910086 0.414419i \(-0.863985\pi\)
0.910086 0.414419i \(-0.136015\pi\)
\(462\) 0 0
\(463\) 1.63866e10 0.356586 0.178293 0.983977i \(-0.442943\pi\)
0.178293 + 0.983977i \(0.442943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.06367e10i 0.223634i 0.993729 + 0.111817i \(0.0356671\pi\)
−0.993729 + 0.111817i \(0.964333\pi\)
\(468\) 0 0
\(469\) 5.71556e9 0.118132
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.12074e11i 2.23904i
\(474\) 0 0
\(475\) −1.20094e10 −0.235910
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.94703e10i 1.31964i 0.751422 + 0.659822i \(0.229368\pi\)
−0.751422 + 0.659822i \(0.770632\pi\)
\(480\) 0 0
\(481\) 4.36653e9 0.0815747
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.54315e10i 1.72474i
\(486\) 0 0
\(487\) −5.49782e10 −0.977405 −0.488703 0.872450i \(-0.662530\pi\)
−0.488703 + 0.872450i \(0.662530\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.04762e10i 0.696424i 0.937416 + 0.348212i \(0.113211\pi\)
−0.937416 + 0.348212i \(0.886789\pi\)
\(492\) 0 0
\(493\) −2.87875e9 −0.0487322
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.30648e10i − 0.214130i
\(498\) 0 0
\(499\) −4.67410e10 −0.753869 −0.376935 0.926240i \(-0.623022\pi\)
−0.376935 + 0.926240i \(0.623022\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 4.28912e10i − 0.670033i −0.942212 0.335017i \(-0.891258\pi\)
0.942212 0.335017i \(-0.108742\pi\)
\(504\) 0 0
\(505\) 4.20376e9 0.0646357
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 6.68892e10i − 0.996517i −0.867028 0.498259i \(-0.833973\pi\)
0.867028 0.498259i \(-0.166027\pi\)
\(510\) 0 0
\(511\) −3.66730e10 −0.537853
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 3.69606e10i − 0.525424i
\(516\) 0 0
\(517\) −1.19339e11 −1.67040
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.09751e10i 0.420399i 0.977659 + 0.210199i \(0.0674113\pi\)
−0.977659 + 0.210199i \(0.932589\pi\)
\(522\) 0 0
\(523\) 1.58369e10 0.211672 0.105836 0.994384i \(-0.466248\pi\)
0.105836 + 0.994384i \(0.466248\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.99961e9i 0.0388886i
\(528\) 0 0
\(529\) −9.90093e10 −1.26431
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 3.91543e9i − 0.0485144i
\(534\) 0 0
\(535\) 4.66395e10 0.569296
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.56001e10i 0.184830i
\(540\) 0 0
\(541\) −1.46603e11 −1.71141 −0.855705 0.517464i \(-0.826876\pi\)
−0.855705 + 0.517464i \(0.826876\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.04075e10i 0.684707i
\(546\) 0 0
\(547\) 8.52569e10 0.952314 0.476157 0.879360i \(-0.342029\pi\)
0.476157 + 0.879360i \(0.342029\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 3.45393e10i − 0.374720i
\(552\) 0 0
\(553\) −2.72668e10 −0.291564
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.05196e11i 1.09289i 0.837495 + 0.546445i \(0.184019\pi\)
−0.837495 + 0.546445i \(0.815981\pi\)
\(558\) 0 0
\(559\) 3.68823e10 0.377721
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.36005e10i 0.433969i 0.976175 + 0.216984i \(0.0696220\pi\)
−0.976175 + 0.216984i \(0.930378\pi\)
\(564\) 0 0
\(565\) −1.96294e11 −1.92625
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 1.63036e10i − 0.155538i −0.996971 0.0777688i \(-0.975220\pi\)
0.996971 0.0777688i \(-0.0247796\pi\)
\(570\) 0 0
\(571\) 6.58110e10 0.619090 0.309545 0.950885i \(-0.399823\pi\)
0.309545 + 0.950885i \(0.399823\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 2.78319e10i − 0.254608i
\(576\) 0 0
\(577\) −1.39333e11 −1.25705 −0.628523 0.777791i \(-0.716340\pi\)
−0.628523 + 0.777791i \(0.716340\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.55522e10i 0.399765i
\(582\) 0 0
\(583\) 4.40715e10 0.381490
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.50365e11i 1.26647i 0.773961 + 0.633233i \(0.218272\pi\)
−0.773961 + 0.633233i \(0.781728\pi\)
\(588\) 0 0
\(589\) −3.59894e10 −0.299029
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.73734e11i 1.40497i 0.711700 + 0.702484i \(0.247926\pi\)
−0.711700 + 0.702484i \(0.752074\pi\)
\(594\) 0 0
\(595\) 9.28784e9 0.0741049
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.75985e10i 0.214377i 0.994239 + 0.107189i \(0.0341849\pi\)
−0.994239 + 0.107189i \(0.965815\pi\)
\(600\) 0 0
\(601\) 1.52880e11 1.17180 0.585898 0.810385i \(-0.300742\pi\)
0.585898 + 0.810385i \(0.300742\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 9.76314e10i − 0.728732i
\(606\) 0 0
\(607\) 1.35603e11 0.998884 0.499442 0.866347i \(-0.333538\pi\)
0.499442 + 0.866347i \(0.333538\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.92730e10i 0.281793i
\(612\) 0 0
\(613\) 1.37425e11 0.973249 0.486625 0.873611i \(-0.338228\pi\)
0.486625 + 0.873611i \(0.338228\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 7.23941e10i − 0.499531i −0.968306 0.249765i \(-0.919647\pi\)
0.968306 0.249765i \(-0.0803535\pi\)
\(618\) 0 0
\(619\) −1.64909e10 −0.112327 −0.0561633 0.998422i \(-0.517887\pi\)
−0.0561633 + 0.998422i \(0.517887\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.78744e9i 0.0251417i
\(624\) 0 0
\(625\) −1.74038e11 −1.14057
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1.06079e10i − 0.0677684i
\(630\) 0 0
\(631\) 3.52602e10 0.222417 0.111208 0.993797i \(-0.464528\pi\)
0.111208 + 0.993797i \(0.464528\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 2.28068e11i − 1.40272i
\(636\) 0 0
\(637\) 5.13380e9 0.0311804
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.76394e11i 1.04484i 0.852687 + 0.522422i \(0.174971\pi\)
−0.852687 + 0.522422i \(0.825029\pi\)
\(642\) 0 0
\(643\) 1.16256e11 0.680095 0.340048 0.940408i \(-0.389557\pi\)
0.340048 + 0.940408i \(0.389557\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 7.89390e10i − 0.450479i −0.974303 0.225239i \(-0.927684\pi\)
0.974303 0.225239i \(-0.0723164\pi\)
\(648\) 0 0
\(649\) 2.54138e11 1.43248
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.76500e11i 0.970713i 0.874316 + 0.485357i \(0.161310\pi\)
−0.874316 + 0.485357i \(0.838690\pi\)
\(654\) 0 0
\(655\) −1.21999e11 −0.662815
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 2.00334e11i − 1.06222i −0.847304 0.531108i \(-0.821776\pi\)
0.847304 0.531108i \(-0.178224\pi\)
\(660\) 0 0
\(661\) −3.06069e11 −1.60330 −0.801648 0.597796i \(-0.796043\pi\)
−0.801648 + 0.597796i \(0.796043\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.11436e11i 0.569820i
\(666\) 0 0
\(667\) 8.00454e10 0.404420
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 1.73925e11i − 0.857970i
\(672\) 0 0
\(673\) 1.06648e11 0.519867 0.259934 0.965626i \(-0.416299\pi\)
0.259934 + 0.965626i \(0.416299\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 3.42785e11i − 1.63180i −0.578192 0.815901i \(-0.696241\pi\)
0.578192 0.815901i \(-0.303759\pi\)
\(678\) 0 0
\(679\) 1.28148e11 0.602880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 1.20025e11i − 0.551555i −0.961222 0.275777i \(-0.911065\pi\)
0.961222 0.275777i \(-0.0889352\pi\)
\(684\) 0 0
\(685\) −1.18627e11 −0.538791
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1.45034e10i − 0.0643565i
\(690\) 0 0
\(691\) 2.21637e11 0.972144 0.486072 0.873919i \(-0.338429\pi\)
0.486072 + 0.873919i \(0.338429\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 7.71615e10i − 0.330721i
\(696\) 0 0
\(697\) −9.51204e9 −0.0403035
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.28038e11i 0.530233i 0.964216 + 0.265116i \(0.0854104\pi\)
−0.964216 + 0.265116i \(0.914590\pi\)
\(702\) 0 0
\(703\) 1.27274e11 0.521097
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 5.64490e9i − 0.0225932i
\(708\) 0 0
\(709\) 1.99426e11 0.789217 0.394608 0.918849i \(-0.370880\pi\)
0.394608 + 0.918849i \(0.370880\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 8.34061e10i − 0.322730i
\(714\) 0 0
\(715\) −7.98029e10 −0.305348
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 3.72421e11i − 1.39354i −0.717296 0.696769i \(-0.754620\pi\)
0.717296 0.696769i \(-0.245380\pi\)
\(720\) 0 0
\(721\) −4.96314e10 −0.183661
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.25638e10i 0.0454746i
\(726\) 0 0
\(727\) 3.45884e11 1.23821 0.619103 0.785310i \(-0.287496\pi\)
0.619103 + 0.785310i \(0.287496\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 8.96009e10i − 0.313793i
\(732\) 0 0
\(733\) 5.42390e11 1.87887 0.939433 0.342733i \(-0.111353\pi\)
0.939433 + 0.342733i \(0.111353\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.19305e11i 0.404378i
\(738\) 0 0
\(739\) −5.62333e11 −1.88545 −0.942727 0.333565i \(-0.891748\pi\)
−0.942727 + 0.333565i \(0.891748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 4.74150e11i − 1.55582i −0.628374 0.777911i \(-0.716279\pi\)
0.628374 0.777911i \(-0.283721\pi\)
\(744\) 0 0
\(745\) −3.99126e11 −1.29564
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 6.26285e10i − 0.198996i
\(750\) 0 0
\(751\) 1.37525e11 0.432336 0.216168 0.976356i \(-0.430644\pi\)
0.216168 + 0.976356i \(0.430644\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 6.44003e11i − 1.98198i
\(756\) 0 0
\(757\) −6.09476e11 −1.85598 −0.927990 0.372606i \(-0.878464\pi\)
−0.927990 + 0.372606i \(0.878464\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.88452e11i 1.15824i 0.815242 + 0.579121i \(0.196604\pi\)
−0.815242 + 0.579121i \(0.803396\pi\)
\(762\) 0 0
\(763\) 8.11166e10 0.239338
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 8.36335e10i − 0.241657i
\(768\) 0 0
\(769\) 6.73576e10 0.192611 0.0963056 0.995352i \(-0.469297\pi\)
0.0963056 + 0.995352i \(0.469297\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.57274e10i 0.100065i 0.998748 + 0.0500327i \(0.0159325\pi\)
−0.998748 + 0.0500327i \(0.984067\pi\)
\(774\) 0 0
\(775\) 1.30913e10 0.0362890
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1.14126e11i − 0.309908i
\(780\) 0 0
\(781\) 2.72711e11 0.732990
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 6.10180e11i − 1.60686i
\(786\) 0 0
\(787\) 5.51060e11 1.43648 0.718240 0.695795i \(-0.244948\pi\)
0.718240 + 0.695795i \(0.244948\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.63588e11i 0.673316i
\(792\) 0 0
\(793\) −5.72366e10 −0.144737
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.25775e11i 1.55090i 0.631406 + 0.775452i \(0.282478\pi\)
−0.631406 + 0.775452i \(0.717522\pi\)
\(798\) 0 0
\(799\) 9.54088e10 0.234100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 7.65499e11i − 1.84112i
\(804\) 0 0
\(805\) −2.58254e11 −0.614984
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 3.34395e10i − 0.0780667i −0.999238 0.0390334i \(-0.987572\pi\)
0.999238 0.0390334i \(-0.0124279\pi\)
\(810\) 0 0
\(811\) 5.98789e11 1.38417 0.692086 0.721815i \(-0.256692\pi\)
0.692086 + 0.721815i \(0.256692\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.10271e11i 0.249938i
\(816\) 0 0
\(817\) 1.07503e12 2.41287
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.98864e11i 1.09802i 0.835816 + 0.549009i \(0.184995\pi\)
−0.835816 + 0.549009i \(0.815005\pi\)
\(822\) 0 0
\(823\) −1.03490e11 −0.225580 −0.112790 0.993619i \(-0.535979\pi\)
−0.112790 + 0.993619i \(0.535979\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 6.44947e11i − 1.37880i −0.724380 0.689401i \(-0.757874\pi\)
0.724380 0.689401i \(-0.242126\pi\)
\(828\) 0 0
\(829\) 3.99994e11 0.846906 0.423453 0.905918i \(-0.360818\pi\)
0.423453 + 0.905918i \(0.360818\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.24719e10i − 0.0259032i
\(834\) 0 0
\(835\) −2.20225e11 −0.453024
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.99830e11i 0.806914i 0.914999 + 0.403457i \(0.132191\pi\)
−0.914999 + 0.403457i \(0.867809\pi\)
\(840\) 0 0
\(841\) 4.64113e11 0.927768
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 5.25017e11i − 1.02978i
\(846\) 0 0
\(847\) −1.31102e11 −0.254727
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.94959e11i 0.562398i
\(852\) 0 0
\(853\) −1.79501e11 −0.339055 −0.169527 0.985525i \(-0.554224\pi\)
−0.169527 + 0.985525i \(0.554224\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.54874e11i 1.77020i 0.465398 + 0.885101i \(0.345911\pi\)
−0.465398 + 0.885101i \(0.654089\pi\)
\(858\) 0 0
\(859\) −3.02178e11 −0.554997 −0.277499 0.960726i \(-0.589505\pi\)
−0.277499 + 0.960726i \(0.589505\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.97426e11i 1.79820i 0.437745 + 0.899099i \(0.355777\pi\)
−0.437745 + 0.899099i \(0.644223\pi\)
\(864\) 0 0
\(865\) −6.82850e11 −1.21972
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 5.69158e11i − 0.998053i
\(870\) 0 0
\(871\) 3.92617e10 0.0682176
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.99032e11i 0.339540i
\(876\) 0 0
\(877\) −1.40216e11 −0.237028 −0.118514 0.992952i \(-0.537813\pi\)
−0.118514 + 0.992952i \(0.537813\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 1.02525e12i − 1.70187i −0.525268 0.850937i \(-0.676035\pi\)
0.525268 0.850937i \(-0.323965\pi\)
\(882\) 0 0
\(883\) 3.77728e11 0.621350 0.310675 0.950516i \(-0.399445\pi\)
0.310675 + 0.950516i \(0.399445\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 4.25328e11i − 0.687115i −0.939132 0.343557i \(-0.888368\pi\)
0.939132 0.343557i \(-0.111632\pi\)
\(888\) 0 0
\(889\) −3.06255e11 −0.490316
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.14472e12i 1.80008i
\(894\) 0 0
\(895\) 6.02957e10 0.0939710
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.76509e10i 0.0576417i
\(900\) 0 0
\(901\) −3.52341e10 −0.0534644
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 6.50961e11i − 0.970423i
\(906\) 0 0
\(907\) −2.30935e11 −0.341241 −0.170621 0.985337i \(-0.554577\pi\)
−0.170621 + 0.985337i \(0.554577\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.01079e11i 0.146753i 0.997304 + 0.0733764i \(0.0233774\pi\)
−0.997304 + 0.0733764i \(0.976623\pi\)
\(912\) 0 0
\(913\) −9.50841e11 −1.36844
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.63823e11i 0.231685i
\(918\) 0 0
\(919\) −5.67287e11 −0.795317 −0.397659 0.917533i \(-0.630177\pi\)
−0.397659 + 0.917533i \(0.630177\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 8.97457e10i − 0.123654i
\(924\) 0 0
\(925\) −4.62963e10 −0.0632382
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.04274e10i 0.107979i 0.998541 + 0.0539897i \(0.0171938\pi\)
−0.998541 + 0.0539897i \(0.982806\pi\)
\(930\) 0 0
\(931\) 1.49638e11 0.199179
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.93871e11i 0.253668i
\(936\) 0 0
\(937\) −1.25692e12 −1.63061 −0.815306 0.579030i \(-0.803431\pi\)
−0.815306 + 0.579030i \(0.803431\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 2.50994e11i − 0.320115i −0.987108 0.160057i \(-0.948832\pi\)
0.987108 0.160057i \(-0.0511679\pi\)
\(942\) 0 0
\(943\) 2.64488e11 0.334472
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.27838e11i 0.407623i 0.979010 + 0.203812i \(0.0653330\pi\)
−0.979010 + 0.203812i \(0.934667\pi\)
\(948\) 0 0
\(949\) −2.51916e11 −0.310593
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.04603e12i 1.26816i 0.773268 + 0.634080i \(0.218621\pi\)
−0.773268 + 0.634080i \(0.781379\pi\)
\(954\) 0 0
\(955\) −1.59543e12 −1.91806
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.59295e11i 0.188333i
\(960\) 0 0
\(961\) −8.13659e11 −0.954002
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.00367e11i 0.231056i
\(966\) 0 0
\(967\) 6.58893e11 0.753545 0.376773 0.926306i \(-0.377034\pi\)
0.376773 + 0.926306i \(0.377034\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1.55166e12i − 1.74550i −0.488170 0.872749i \(-0.662335\pi\)
0.488170 0.872749i \(-0.337665\pi\)
\(972\) 0 0
\(973\) −1.03614e11 −0.115603
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.77778e11i 0.743891i 0.928255 + 0.371945i \(0.121309\pi\)
−0.928255 + 0.371945i \(0.878691\pi\)
\(978\) 0 0
\(979\) −7.90577e10 −0.0860624
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.15349e12i 1.23538i 0.786423 + 0.617688i \(0.211930\pi\)
−0.786423 + 0.617688i \(0.788070\pi\)
\(984\) 0 0
\(985\) 1.04790e12 1.11320
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.49141e12i 2.60411i
\(990\) 0 0
\(991\) −6.26653e11 −0.649730 −0.324865 0.945760i \(-0.605319\pi\)
−0.324865 + 0.945760i \(0.605319\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.56288e12i 1.59454i
\(996\) 0 0
\(997\) 5.43417e11 0.549987 0.274994 0.961446i \(-0.411324\pi\)
0.274994 + 0.961446i \(0.411324\pi\)
\(998\) 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.197.13 yes 16
3.2 odd 2 inner 252.9.c.a.197.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.4 16 3.2 odd 2 inner
252.9.c.a.197.13 yes 16 1.1 even 1 trivial