Properties

Label 384.7.b.c.319.5
Level $384$
Weight $7$
Character 384.319
Analytic conductor $88.341$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 106 x^{6} - 304 x^{5} + 4359 x^{4} - 8216 x^{3} + 73366 x^{2} - 69308 x + 604693\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.5
Root \(2.23205 - 6.41320i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.7.b.c.319.8

$q$-expansion

\(f(q)\) \(=\) \(q+15.5885 q^{3} -195.235i q^{5} -277.510i q^{7} +243.000 q^{9} +O(q^{10})\) \(q+15.5885 q^{3} -195.235i q^{5} -277.510i q^{7} +243.000 q^{9} -1753.29 q^{11} -1246.75i q^{13} -3043.41i q^{15} +6888.35 q^{17} +4401.67 q^{19} -4325.95i q^{21} -12728.7i q^{23} -22491.7 q^{25} +3788.00 q^{27} -8278.44i q^{29} -43940.1i q^{31} -27331.1 q^{33} -54179.6 q^{35} +12186.4i q^{37} -19434.9i q^{39} -54739.6 q^{41} +45453.4 q^{43} -47442.1i q^{45} -152336. i q^{47} +40637.3 q^{49} +107379. q^{51} +272550. i q^{53} +342303. i q^{55} +68615.3 q^{57} -213175. q^{59} -83964.8i q^{61} -67434.9i q^{63} -243409. q^{65} -373099. q^{67} -198420. i q^{69} +667812. i q^{71} -399457. q^{73} -350611. q^{75} +486555. i q^{77} +435376. i q^{79} +59049.0 q^{81} +246583. q^{83} -1.34485e6i q^{85} -129048. i q^{87} +80967.5 q^{89} -345985. q^{91} -684959. i q^{93} -859361. i q^{95} +877030. q^{97} -426049. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1944 q^{9} + O(q^{10}) \) \( 8 q + 1944 q^{9} - 6544 q^{17} - 56632 q^{25} - 33696 q^{33} - 499568 q^{41} - 414712 q^{49} - 375840 q^{57} - 36096 q^{65} - 1962640 q^{73} + 472392 q^{81} - 1694992 q^{89} + 7632752 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 15.5885 0.577350
\(4\) 0 0
\(5\) − 195.235i − 1.56188i −0.624606 0.780940i \(-0.714740\pi\)
0.624606 0.780940i \(-0.285260\pi\)
\(6\) 0 0
\(7\) − 277.510i − 0.809067i −0.914523 0.404533i \(-0.867434\pi\)
0.914523 0.404533i \(-0.132566\pi\)
\(8\) 0 0
\(9\) 243.000 0.333333
\(10\) 0 0
\(11\) −1753.29 −1.31727 −0.658636 0.752462i \(-0.728866\pi\)
−0.658636 + 0.752462i \(0.728866\pi\)
\(12\) 0 0
\(13\) − 1246.75i − 0.567478i −0.958902 0.283739i \(-0.908425\pi\)
0.958902 0.283739i \(-0.0915749\pi\)
\(14\) 0 0
\(15\) − 3043.41i − 0.901752i
\(16\) 0 0
\(17\) 6888.35 1.40207 0.701033 0.713128i \(-0.252722\pi\)
0.701033 + 0.713128i \(0.252722\pi\)
\(18\) 0 0
\(19\) 4401.67 0.641737 0.320869 0.947124i \(-0.396025\pi\)
0.320869 + 0.947124i \(0.396025\pi\)
\(20\) 0 0
\(21\) − 4325.95i − 0.467115i
\(22\) 0 0
\(23\) − 12728.7i − 1.04616i −0.852283 0.523082i \(-0.824782\pi\)
0.852283 0.523082i \(-0.175218\pi\)
\(24\) 0 0
\(25\) −22491.7 −1.43947
\(26\) 0 0
\(27\) 3788.00 0.192450
\(28\) 0 0
\(29\) − 8278.44i − 0.339433i −0.985493 0.169717i \(-0.945715\pi\)
0.985493 0.169717i \(-0.0542852\pi\)
\(30\) 0 0
\(31\) − 43940.1i − 1.47495i −0.675376 0.737474i \(-0.736019\pi\)
0.675376 0.737474i \(-0.263981\pi\)
\(32\) 0 0
\(33\) −27331.1 −0.760527
\(34\) 0 0
\(35\) −54179.6 −1.26367
\(36\) 0 0
\(37\) 12186.4i 0.240585i 0.992738 + 0.120293i \(0.0383833\pi\)
−0.992738 + 0.120293i \(0.961617\pi\)
\(38\) 0 0
\(39\) − 19434.9i − 0.327633i
\(40\) 0 0
\(41\) −54739.6 −0.794238 −0.397119 0.917767i \(-0.629990\pi\)
−0.397119 + 0.917767i \(0.629990\pi\)
\(42\) 0 0
\(43\) 45453.4 0.571691 0.285845 0.958276i \(-0.407726\pi\)
0.285845 + 0.958276i \(0.407726\pi\)
\(44\) 0 0
\(45\) − 47442.1i − 0.520627i
\(46\) 0 0
\(47\) − 152336.i − 1.46727i −0.679545 0.733634i \(-0.737823\pi\)
0.679545 0.733634i \(-0.262177\pi\)
\(48\) 0 0
\(49\) 40637.3 0.345411
\(50\) 0 0
\(51\) 107379. 0.809484
\(52\) 0 0
\(53\) 272550.i 1.83071i 0.402649 + 0.915354i \(0.368089\pi\)
−0.402649 + 0.915354i \(0.631911\pi\)
\(54\) 0 0
\(55\) 342303.i 2.05742i
\(56\) 0 0
\(57\) 68615.3 0.370507
\(58\) 0 0
\(59\) −213175. −1.03796 −0.518978 0.854787i \(-0.673687\pi\)
−0.518978 + 0.854787i \(0.673687\pi\)
\(60\) 0 0
\(61\) − 83964.8i − 0.369920i −0.982746 0.184960i \(-0.940784\pi\)
0.982746 0.184960i \(-0.0592156\pi\)
\(62\) 0 0
\(63\) − 67434.9i − 0.269689i
\(64\) 0 0
\(65\) −243409. −0.886332
\(66\) 0 0
\(67\) −373099. −1.24051 −0.620254 0.784401i \(-0.712970\pi\)
−0.620254 + 0.784401i \(0.712970\pi\)
\(68\) 0 0
\(69\) − 198420.i − 0.604003i
\(70\) 0 0
\(71\) 667812.i 1.86586i 0.360057 + 0.932930i \(0.382757\pi\)
−0.360057 + 0.932930i \(0.617243\pi\)
\(72\) 0 0
\(73\) −399457. −1.02684 −0.513419 0.858138i \(-0.671621\pi\)
−0.513419 + 0.858138i \(0.671621\pi\)
\(74\) 0 0
\(75\) −350611. −0.831078
\(76\) 0 0
\(77\) 486555.i 1.06576i
\(78\) 0 0
\(79\) 435376.i 0.883046i 0.897250 + 0.441523i \(0.145562\pi\)
−0.897250 + 0.441523i \(0.854438\pi\)
\(80\) 0 0
\(81\) 59049.0 0.111111
\(82\) 0 0
\(83\) 246583. 0.431250 0.215625 0.976476i \(-0.430821\pi\)
0.215625 + 0.976476i \(0.430821\pi\)
\(84\) 0 0
\(85\) − 1.34485e6i − 2.18986i
\(86\) 0 0
\(87\) − 129048.i − 0.195972i
\(88\) 0 0
\(89\) 80967.5 0.114853 0.0574263 0.998350i \(-0.481711\pi\)
0.0574263 + 0.998350i \(0.481711\pi\)
\(90\) 0 0
\(91\) −345985. −0.459127
\(92\) 0 0
\(93\) − 684959.i − 0.851561i
\(94\) 0 0
\(95\) − 859361.i − 1.00232i
\(96\) 0 0
\(97\) 877030. 0.960947 0.480474 0.877009i \(-0.340465\pi\)
0.480474 + 0.877009i \(0.340465\pi\)
\(98\) 0 0
\(99\) −426049. −0.439091
\(100\) 0 0
\(101\) 1.80491e6i 1.75183i 0.482470 + 0.875913i \(0.339740\pi\)
−0.482470 + 0.875913i \(0.660260\pi\)
\(102\) 0 0
\(103\) − 826267.i − 0.756151i −0.925775 0.378076i \(-0.876586\pi\)
0.925775 0.378076i \(-0.123414\pi\)
\(104\) 0 0
\(105\) −844577. −0.729577
\(106\) 0 0
\(107\) −1.34587e6 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(108\) 0 0
\(109\) 704622.i 0.544098i 0.962283 + 0.272049i \(0.0877012\pi\)
−0.962283 + 0.272049i \(0.912299\pi\)
\(110\) 0 0
\(111\) 189967.i 0.138902i
\(112\) 0 0
\(113\) 2.55273e6 1.76917 0.884583 0.466382i \(-0.154443\pi\)
0.884583 + 0.466382i \(0.154443\pi\)
\(114\) 0 0
\(115\) −2.48508e6 −1.63398
\(116\) 0 0
\(117\) − 302960.i − 0.189159i
\(118\) 0 0
\(119\) − 1.91159e6i − 1.13437i
\(120\) 0 0
\(121\) 1.30246e6 0.735205
\(122\) 0 0
\(123\) −853307. −0.458553
\(124\) 0 0
\(125\) 1.34062e6i 0.686399i
\(126\) 0 0
\(127\) − 670190.i − 0.327180i −0.986528 0.163590i \(-0.947693\pi\)
0.986528 0.163590i \(-0.0523074\pi\)
\(128\) 0 0
\(129\) 708549. 0.330066
\(130\) 0 0
\(131\) −1.73132e6 −0.770128 −0.385064 0.922890i \(-0.625821\pi\)
−0.385064 + 0.922890i \(0.625821\pi\)
\(132\) 0 0
\(133\) − 1.22151e6i − 0.519208i
\(134\) 0 0
\(135\) − 739549.i − 0.300584i
\(136\) 0 0
\(137\) −2.77355e6 −1.07863 −0.539317 0.842103i \(-0.681317\pi\)
−0.539317 + 0.842103i \(0.681317\pi\)
\(138\) 0 0
\(139\) −4.31042e6 −1.60500 −0.802500 0.596652i \(-0.796497\pi\)
−0.802500 + 0.596652i \(0.796497\pi\)
\(140\) 0 0
\(141\) − 2.37469e6i − 0.847128i
\(142\) 0 0
\(143\) 2.18591e6i 0.747522i
\(144\) 0 0
\(145\) −1.61624e6 −0.530154
\(146\) 0 0
\(147\) 633472. 0.199423
\(148\) 0 0
\(149\) 213236.i 0.0644616i 0.999480 + 0.0322308i \(0.0102612\pi\)
−0.999480 + 0.0322308i \(0.989739\pi\)
\(150\) 0 0
\(151\) − 5.33710e6i − 1.55015i −0.631868 0.775076i \(-0.717711\pi\)
0.631868 0.775076i \(-0.282289\pi\)
\(152\) 0 0
\(153\) 1.67387e6 0.467356
\(154\) 0 0
\(155\) −8.57866e6 −2.30369
\(156\) 0 0
\(157\) 4.08979e6i 1.05682i 0.848988 + 0.528412i \(0.177212\pi\)
−0.848988 + 0.528412i \(0.822788\pi\)
\(158\) 0 0
\(159\) 4.24864e6i 1.05696i
\(160\) 0 0
\(161\) −3.53233e6 −0.846416
\(162\) 0 0
\(163\) 1.54126e6 0.355888 0.177944 0.984041i \(-0.443055\pi\)
0.177944 + 0.984041i \(0.443055\pi\)
\(164\) 0 0
\(165\) 5.33598e6i 1.18785i
\(166\) 0 0
\(167\) 7.66555e6i 1.64586i 0.568140 + 0.822932i \(0.307663\pi\)
−0.568140 + 0.822932i \(0.692337\pi\)
\(168\) 0 0
\(169\) 3.27243e6 0.677969
\(170\) 0 0
\(171\) 1.06961e6 0.213912
\(172\) 0 0
\(173\) − 1.00380e7i − 1.93870i −0.245693 0.969348i \(-0.579016\pi\)
0.245693 0.969348i \(-0.420984\pi\)
\(174\) 0 0
\(175\) 6.24167e6i 1.16463i
\(176\) 0 0
\(177\) −3.32306e6 −0.599265
\(178\) 0 0
\(179\) −8.83296e6 −1.54009 −0.770047 0.637987i \(-0.779767\pi\)
−0.770047 + 0.637987i \(0.779767\pi\)
\(180\) 0 0
\(181\) − 6.00091e6i − 1.01200i −0.862533 0.506001i \(-0.831123\pi\)
0.862533 0.506001i \(-0.168877\pi\)
\(182\) 0 0
\(183\) − 1.30888e6i − 0.213573i
\(184\) 0 0
\(185\) 2.37920e6 0.375765
\(186\) 0 0
\(187\) −1.20773e7 −1.84690
\(188\) 0 0
\(189\) − 1.05121e6i − 0.155705i
\(190\) 0 0
\(191\) − 603085.i − 0.0865522i −0.999063 0.0432761i \(-0.986220\pi\)
0.999063 0.0432761i \(-0.0137795\pi\)
\(192\) 0 0
\(193\) 7.75593e6 1.07885 0.539426 0.842033i \(-0.318641\pi\)
0.539426 + 0.842033i \(0.318641\pi\)
\(194\) 0 0
\(195\) −3.79437e6 −0.511724
\(196\) 0 0
\(197\) − 1.91770e6i − 0.250832i −0.992104 0.125416i \(-0.959974\pi\)
0.992104 0.125416i \(-0.0400265\pi\)
\(198\) 0 0
\(199\) 1.01444e7i 1.28726i 0.765337 + 0.643629i \(0.222572\pi\)
−0.765337 + 0.643629i \(0.777428\pi\)
\(200\) 0 0
\(201\) −5.81603e6 −0.716207
\(202\) 0 0
\(203\) −2.29735e6 −0.274624
\(204\) 0 0
\(205\) 1.06871e7i 1.24050i
\(206\) 0 0
\(207\) − 3.09307e6i − 0.348721i
\(208\) 0 0
\(209\) −7.71741e6 −0.845342
\(210\) 0 0
\(211\) 9.54322e6 1.01589 0.507946 0.861389i \(-0.330405\pi\)
0.507946 + 0.861389i \(0.330405\pi\)
\(212\) 0 0
\(213\) 1.04102e7i 1.07726i
\(214\) 0 0
\(215\) − 8.87410e6i − 0.892912i
\(216\) 0 0
\(217\) −1.21938e7 −1.19333
\(218\) 0 0
\(219\) −6.22692e6 −0.592845
\(220\) 0 0
\(221\) − 8.58805e6i − 0.795642i
\(222\) 0 0
\(223\) − 1.58078e7i − 1.42547i −0.701434 0.712735i \(-0.747456\pi\)
0.701434 0.712735i \(-0.252544\pi\)
\(224\) 0 0
\(225\) −5.46549e6 −0.479823
\(226\) 0 0
\(227\) 1.48574e7 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(228\) 0 0
\(229\) 1.28465e6i 0.106974i 0.998569 + 0.0534872i \(0.0170336\pi\)
−0.998569 + 0.0534872i \(0.982966\pi\)
\(230\) 0 0
\(231\) 7.58464e6i 0.615317i
\(232\) 0 0
\(233\) 262969. 0.0207892 0.0103946 0.999946i \(-0.496691\pi\)
0.0103946 + 0.999946i \(0.496691\pi\)
\(234\) 0 0
\(235\) −2.97414e7 −2.29170
\(236\) 0 0
\(237\) 6.78684e6i 0.509827i
\(238\) 0 0
\(239\) − 1.89260e7i − 1.38633i −0.720781 0.693163i \(-0.756217\pi\)
0.720781 0.693163i \(-0.243783\pi\)
\(240\) 0 0
\(241\) −1.91538e6 −0.136837 −0.0684185 0.997657i \(-0.521795\pi\)
−0.0684185 + 0.997657i \(0.521795\pi\)
\(242\) 0 0
\(243\) 920483. 0.0641500
\(244\) 0 0
\(245\) − 7.93382e6i − 0.539491i
\(246\) 0 0
\(247\) − 5.48778e6i − 0.364172i
\(248\) 0 0
\(249\) 3.84385e6 0.248982
\(250\) 0 0
\(251\) −1.10676e7 −0.699895 −0.349948 0.936769i \(-0.613801\pi\)
−0.349948 + 0.936769i \(0.613801\pi\)
\(252\) 0 0
\(253\) 2.23170e7i 1.37808i
\(254\) 0 0
\(255\) − 2.09641e7i − 1.26432i
\(256\) 0 0
\(257\) 678718. 0.0399843 0.0199922 0.999800i \(-0.493636\pi\)
0.0199922 + 0.999800i \(0.493636\pi\)
\(258\) 0 0
\(259\) 3.38184e6 0.194650
\(260\) 0 0
\(261\) − 2.01166e6i − 0.113144i
\(262\) 0 0
\(263\) 1.11128e7i 0.610882i 0.952211 + 0.305441i \(0.0988039\pi\)
−0.952211 + 0.305441i \(0.901196\pi\)
\(264\) 0 0
\(265\) 5.32114e7 2.85935
\(266\) 0 0
\(267\) 1.26216e6 0.0663101
\(268\) 0 0
\(269\) − 1.00441e7i − 0.516007i −0.966144 0.258004i \(-0.916935\pi\)
0.966144 0.258004i \(-0.0830647\pi\)
\(270\) 0 0
\(271\) 6.62102e6i 0.332673i 0.986069 + 0.166336i \(0.0531938\pi\)
−0.986069 + 0.166336i \(0.946806\pi\)
\(272\) 0 0
\(273\) −5.39337e6 −0.265077
\(274\) 0 0
\(275\) 3.94345e7 1.89617
\(276\) 0 0
\(277\) − 2.44357e7i − 1.14970i −0.818257 0.574852i \(-0.805060\pi\)
0.818257 0.574852i \(-0.194940\pi\)
\(278\) 0 0
\(279\) − 1.06775e7i − 0.491649i
\(280\) 0 0
\(281\) −1.39627e7 −0.629288 −0.314644 0.949210i \(-0.601885\pi\)
−0.314644 + 0.949210i \(0.601885\pi\)
\(282\) 0 0
\(283\) 4.28834e7 1.89204 0.946018 0.324113i \(-0.105066\pi\)
0.946018 + 0.324113i \(0.105066\pi\)
\(284\) 0 0
\(285\) − 1.33961e7i − 0.578688i
\(286\) 0 0
\(287\) 1.51908e7i 0.642591i
\(288\) 0 0
\(289\) 2.33119e7 0.965792
\(290\) 0 0
\(291\) 1.36716e7 0.554803
\(292\) 0 0
\(293\) 5.00254e6i 0.198878i 0.995044 + 0.0994392i \(0.0317049\pi\)
−0.995044 + 0.0994392i \(0.968295\pi\)
\(294\) 0 0
\(295\) 4.16191e7i 1.62116i
\(296\) 0 0
\(297\) −6.64145e6 −0.253509
\(298\) 0 0
\(299\) −1.58695e7 −0.593675
\(300\) 0 0
\(301\) − 1.26138e7i − 0.462536i
\(302\) 0 0
\(303\) 2.81357e7i 1.01142i
\(304\) 0 0
\(305\) −1.63929e7 −0.577771
\(306\) 0 0
\(307\) 2.73510e7 0.945274 0.472637 0.881257i \(-0.343302\pi\)
0.472637 + 0.881257i \(0.343302\pi\)
\(308\) 0 0
\(309\) − 1.28802e7i − 0.436564i
\(310\) 0 0
\(311\) − 3.37365e7i − 1.12155i −0.827968 0.560775i \(-0.810503\pi\)
0.827968 0.560775i \(-0.189497\pi\)
\(312\) 0 0
\(313\) 4.91891e7 1.60412 0.802058 0.597246i \(-0.203738\pi\)
0.802058 + 0.597246i \(0.203738\pi\)
\(314\) 0 0
\(315\) −1.31657e7 −0.421222
\(316\) 0 0
\(317\) 1.97091e7i 0.618712i 0.950946 + 0.309356i \(0.100113\pi\)
−0.950946 + 0.309356i \(0.899887\pi\)
\(318\) 0 0
\(319\) 1.45145e7i 0.447126i
\(320\) 0 0
\(321\) −2.09800e7 −0.634293
\(322\) 0 0
\(323\) 3.03203e7 0.899758
\(324\) 0 0
\(325\) 2.80415e7i 0.816867i
\(326\) 0 0
\(327\) 1.09840e7i 0.314135i
\(328\) 0 0
\(329\) −4.22748e7 −1.18712
\(330\) 0 0
\(331\) −1.59711e7 −0.440403 −0.220202 0.975454i \(-0.570672\pi\)
−0.220202 + 0.975454i \(0.570672\pi\)
\(332\) 0 0
\(333\) 2.96129e6i 0.0801951i
\(334\) 0 0
\(335\) 7.28419e7i 1.93752i
\(336\) 0 0
\(337\) 9.18400e6 0.239962 0.119981 0.992776i \(-0.461717\pi\)
0.119981 + 0.992776i \(0.461717\pi\)
\(338\) 0 0
\(339\) 3.97930e7 1.02143
\(340\) 0 0
\(341\) 7.70398e7i 1.94291i
\(342\) 0 0
\(343\) − 4.39260e7i − 1.08853i
\(344\) 0 0
\(345\) −3.87386e7 −0.943380
\(346\) 0 0
\(347\) −3.95976e7 −0.947721 −0.473861 0.880600i \(-0.657140\pi\)
−0.473861 + 0.880600i \(0.657140\pi\)
\(348\) 0 0
\(349\) − 7.85281e7i − 1.84735i −0.383180 0.923674i \(-0.625171\pi\)
0.383180 0.923674i \(-0.374829\pi\)
\(350\) 0 0
\(351\) − 4.72268e6i − 0.109211i
\(352\) 0 0
\(353\) 6.58915e7 1.49798 0.748989 0.662582i \(-0.230540\pi\)
0.748989 + 0.662582i \(0.230540\pi\)
\(354\) 0 0
\(355\) 1.30380e8 2.91425
\(356\) 0 0
\(357\) − 2.97987e7i − 0.654926i
\(358\) 0 0
\(359\) − 9.24524e6i − 0.199818i −0.994997 0.0999090i \(-0.968145\pi\)
0.994997 0.0999090i \(-0.0318552\pi\)
\(360\) 0 0
\(361\) −2.76711e7 −0.588173
\(362\) 0 0
\(363\) 2.03033e7 0.424471
\(364\) 0 0
\(365\) 7.79880e7i 1.60380i
\(366\) 0 0
\(367\) − 1.05595e7i − 0.213622i −0.994279 0.106811i \(-0.965936\pi\)
0.994279 0.106811i \(-0.0340640\pi\)
\(368\) 0 0
\(369\) −1.33017e7 −0.264746
\(370\) 0 0
\(371\) 7.56354e7 1.48117
\(372\) 0 0
\(373\) 1.92271e7i 0.370499i 0.982691 + 0.185250i \(0.0593094\pi\)
−0.982691 + 0.185250i \(0.940691\pi\)
\(374\) 0 0
\(375\) 2.08982e7i 0.396292i
\(376\) 0 0
\(377\) −1.03211e7 −0.192621
\(378\) 0 0
\(379\) 1.53936e7 0.282764 0.141382 0.989955i \(-0.454845\pi\)
0.141382 + 0.989955i \(0.454845\pi\)
\(380\) 0 0
\(381\) − 1.04472e7i − 0.188897i
\(382\) 0 0
\(383\) − 1.18517e7i − 0.210953i −0.994422 0.105476i \(-0.966363\pi\)
0.994422 0.105476i \(-0.0336367\pi\)
\(384\) 0 0
\(385\) 9.49926e7 1.66459
\(386\) 0 0
\(387\) 1.10452e7 0.190564
\(388\) 0 0
\(389\) − 5.48941e7i − 0.932561i −0.884637 0.466281i \(-0.845594\pi\)
0.884637 0.466281i \(-0.154406\pi\)
\(390\) 0 0
\(391\) − 8.76796e7i − 1.46679i
\(392\) 0 0
\(393\) −2.69886e7 −0.444634
\(394\) 0 0
\(395\) 8.50007e7 1.37921
\(396\) 0 0
\(397\) − 5.07085e7i − 0.810419i −0.914224 0.405209i \(-0.867199\pi\)
0.914224 0.405209i \(-0.132801\pi\)
\(398\) 0 0
\(399\) − 1.90414e7i − 0.299765i
\(400\) 0 0
\(401\) −1.11684e8 −1.73204 −0.866021 0.500007i \(-0.833331\pi\)
−0.866021 + 0.500007i \(0.833331\pi\)
\(402\) 0 0
\(403\) −5.47823e7 −0.837000
\(404\) 0 0
\(405\) − 1.15284e7i − 0.173542i
\(406\) 0 0
\(407\) − 2.13662e7i − 0.316916i
\(408\) 0 0
\(409\) 8.03112e7 1.17383 0.586917 0.809647i \(-0.300342\pi\)
0.586917 + 0.809647i \(0.300342\pi\)
\(410\) 0 0
\(411\) −4.32353e7 −0.622749
\(412\) 0 0
\(413\) 5.91580e7i 0.839776i
\(414\) 0 0
\(415\) − 4.81417e7i − 0.673561i
\(416\) 0 0
\(417\) −6.71928e7 −0.926648
\(418\) 0 0
\(419\) 8.64533e7 1.17527 0.587637 0.809125i \(-0.300058\pi\)
0.587637 + 0.809125i \(0.300058\pi\)
\(420\) 0 0
\(421\) 1.94432e7i 0.260569i 0.991477 + 0.130284i \(0.0415890\pi\)
−0.991477 + 0.130284i \(0.958411\pi\)
\(422\) 0 0
\(423\) − 3.70177e7i − 0.489089i
\(424\) 0 0
\(425\) −1.54931e8 −2.01823
\(426\) 0 0
\(427\) −2.33011e7 −0.299290
\(428\) 0 0
\(429\) 3.40750e7i 0.431582i
\(430\) 0 0
\(431\) − 1.05362e8i − 1.31599i −0.753021 0.657996i \(-0.771404\pi\)
0.753021 0.657996i \(-0.228596\pi\)
\(432\) 0 0
\(433\) 1.32564e8 1.63291 0.816457 0.577407i \(-0.195935\pi\)
0.816457 + 0.577407i \(0.195935\pi\)
\(434\) 0 0
\(435\) −2.51947e7 −0.306085
\(436\) 0 0
\(437\) − 5.60275e7i − 0.671362i
\(438\) 0 0
\(439\) 5.71422e7i 0.675403i 0.941253 + 0.337702i \(0.109649\pi\)
−0.941253 + 0.337702i \(0.890351\pi\)
\(440\) 0 0
\(441\) 9.87485e6 0.115137
\(442\) 0 0
\(443\) 2.10479e7 0.242102 0.121051 0.992646i \(-0.461374\pi\)
0.121051 + 0.992646i \(0.461374\pi\)
\(444\) 0 0
\(445\) − 1.58077e7i − 0.179386i
\(446\) 0 0
\(447\) 3.32402e6i 0.0372169i
\(448\) 0 0
\(449\) −6.46748e7 −0.714490 −0.357245 0.934011i \(-0.616284\pi\)
−0.357245 + 0.934011i \(0.616284\pi\)
\(450\) 0 0
\(451\) 9.59744e7 1.04623
\(452\) 0 0
\(453\) − 8.31971e7i − 0.894981i
\(454\) 0 0
\(455\) 6.75484e7i 0.717102i
\(456\) 0 0
\(457\) 5.37459e7 0.563114 0.281557 0.959544i \(-0.409149\pi\)
0.281557 + 0.959544i \(0.409149\pi\)
\(458\) 0 0
\(459\) 2.60931e7 0.269828
\(460\) 0 0
\(461\) − 2.36640e7i − 0.241538i −0.992681 0.120769i \(-0.961464\pi\)
0.992681 0.120769i \(-0.0385360\pi\)
\(462\) 0 0
\(463\) 2.80452e7i 0.282563i 0.989969 + 0.141281i \(0.0451222\pi\)
−0.989969 + 0.141281i \(0.954878\pi\)
\(464\) 0 0
\(465\) −1.33728e8 −1.33004
\(466\) 0 0
\(467\) −3.61101e7 −0.354550 −0.177275 0.984161i \(-0.556728\pi\)
−0.177275 + 0.984161i \(0.556728\pi\)
\(468\) 0 0
\(469\) 1.03539e8i 1.00365i
\(470\) 0 0
\(471\) 6.37536e7i 0.610157i
\(472\) 0 0
\(473\) −7.96930e7 −0.753072
\(474\) 0 0
\(475\) −9.90012e7 −0.923761
\(476\) 0 0
\(477\) 6.62298e7i 0.610236i
\(478\) 0 0
\(479\) 4.70272e7i 0.427900i 0.976845 + 0.213950i \(0.0686330\pi\)
−0.976845 + 0.213950i \(0.931367\pi\)
\(480\) 0 0
\(481\) 1.51933e7 0.136527
\(482\) 0 0
\(483\) −5.50636e7 −0.488679
\(484\) 0 0
\(485\) − 1.71227e8i − 1.50088i
\(486\) 0 0
\(487\) 9.56947e7i 0.828516i 0.910160 + 0.414258i \(0.135959\pi\)
−0.910160 + 0.414258i \(0.864041\pi\)
\(488\) 0 0
\(489\) 2.40259e7 0.205472
\(490\) 0 0
\(491\) −9.69668e7 −0.819178 −0.409589 0.912270i \(-0.634328\pi\)
−0.409589 + 0.912270i \(0.634328\pi\)
\(492\) 0 0
\(493\) − 5.70248e7i − 0.475908i
\(494\) 0 0
\(495\) 8.31797e7i 0.685807i
\(496\) 0 0
\(497\) 1.85324e8 1.50961
\(498\) 0 0
\(499\) 5.16768e7 0.415905 0.207952 0.978139i \(-0.433320\pi\)
0.207952 + 0.978139i \(0.433320\pi\)
\(500\) 0 0
\(501\) 1.19494e8i 0.950240i
\(502\) 0 0
\(503\) 2.28417e8i 1.79483i 0.441185 + 0.897416i \(0.354558\pi\)
−0.441185 + 0.897416i \(0.645442\pi\)
\(504\) 0 0
\(505\) 3.52381e8 2.73614
\(506\) 0 0
\(507\) 5.10121e7 0.391426
\(508\) 0 0
\(509\) − 1.31906e8i − 1.00025i −0.865952 0.500127i \(-0.833287\pi\)
0.865952 0.500127i \(-0.166713\pi\)
\(510\) 0 0
\(511\) 1.10853e8i 0.830780i
\(512\) 0 0
\(513\) 1.66735e7 0.123502
\(514\) 0 0
\(515\) −1.61316e8 −1.18102
\(516\) 0 0
\(517\) 2.67089e8i 1.93279i
\(518\) 0 0
\(519\) − 1.56477e8i − 1.11931i
\(520\) 0 0
\(521\) 1.21687e7 0.0860462 0.0430231 0.999074i \(-0.486301\pi\)
0.0430231 + 0.999074i \(0.486301\pi\)
\(522\) 0 0
\(523\) −8.96786e7 −0.626879 −0.313440 0.949608i \(-0.601481\pi\)
−0.313440 + 0.949608i \(0.601481\pi\)
\(524\) 0 0
\(525\) 9.72980e7i 0.672398i
\(526\) 0 0
\(527\) − 3.02675e8i − 2.06797i
\(528\) 0 0
\(529\) −1.39832e7 −0.0944579
\(530\) 0 0
\(531\) −5.18014e7 −0.345986
\(532\) 0 0
\(533\) 6.82466e7i 0.450712i
\(534\) 0 0
\(535\) 2.62760e8i 1.71592i
\(536\) 0 0
\(537\) −1.37692e8 −0.889174
\(538\) 0 0
\(539\) −7.12488e7 −0.455000
\(540\) 0 0
\(541\) 2.72097e8i 1.71843i 0.511613 + 0.859216i \(0.329048\pi\)
−0.511613 + 0.859216i \(0.670952\pi\)
\(542\) 0 0
\(543\) − 9.35449e7i − 0.584279i
\(544\) 0 0
\(545\) 1.37567e8 0.849815
\(546\) 0 0
\(547\) −1.84749e8 −1.12881 −0.564404 0.825499i \(-0.690894\pi\)
−0.564404 + 0.825499i \(0.690894\pi\)
\(548\) 0 0
\(549\) − 2.04035e7i − 0.123307i
\(550\) 0 0
\(551\) − 3.64390e7i − 0.217827i
\(552\) 0 0
\(553\) 1.20821e8 0.714443
\(554\) 0 0
\(555\) 3.70881e7 0.216948
\(556\) 0 0
\(557\) − 9.60239e7i − 0.555666i −0.960629 0.277833i \(-0.910384\pi\)
0.960629 0.277833i \(-0.0896162\pi\)
\(558\) 0 0
\(559\) − 5.66690e7i − 0.324422i
\(560\) 0 0
\(561\) −1.88266e8 −1.06631
\(562\) 0 0
\(563\) −2.37375e8 −1.33018 −0.665089 0.746764i \(-0.731607\pi\)
−0.665089 + 0.746764i \(0.731607\pi\)
\(564\) 0 0
\(565\) − 4.98381e8i − 2.76323i
\(566\) 0 0
\(567\) − 1.63867e7i − 0.0898963i
\(568\) 0 0
\(569\) −3.26674e8 −1.77328 −0.886641 0.462458i \(-0.846968\pi\)
−0.886641 + 0.462458i \(0.846968\pi\)
\(570\) 0 0
\(571\) −971103. −0.00521623 −0.00260812 0.999997i \(-0.500830\pi\)
−0.00260812 + 0.999997i \(0.500830\pi\)
\(572\) 0 0
\(573\) − 9.40116e6i − 0.0499709i
\(574\) 0 0
\(575\) 2.86290e8i 1.50592i
\(576\) 0 0
\(577\) 1.87315e8 0.975092 0.487546 0.873097i \(-0.337892\pi\)
0.487546 + 0.873097i \(0.337892\pi\)
\(578\) 0 0
\(579\) 1.20903e8 0.622876
\(580\) 0 0
\(581\) − 6.84293e7i − 0.348910i
\(582\) 0 0
\(583\) − 4.77860e8i − 2.41154i
\(584\) 0 0
\(585\) −5.91484e7 −0.295444
\(586\) 0 0
\(587\) −8.47232e7 −0.418878 −0.209439 0.977822i \(-0.567164\pi\)
−0.209439 + 0.977822i \(0.567164\pi\)
\(588\) 0 0
\(589\) − 1.93410e8i − 0.946528i
\(590\) 0 0
\(591\) − 2.98940e7i − 0.144818i
\(592\) 0 0
\(593\) −2.30616e8 −1.10592 −0.552962 0.833206i \(-0.686503\pi\)
−0.552962 + 0.833206i \(0.686503\pi\)
\(594\) 0 0
\(595\) −3.73209e8 −1.77174
\(596\) 0 0
\(597\) 1.58135e8i 0.743199i
\(598\) 0 0
\(599\) − 3.65098e8i − 1.69875i −0.527791 0.849374i \(-0.676980\pi\)
0.527791 0.849374i \(-0.323020\pi\)
\(600\) 0 0
\(601\) 2.63307e8 1.21294 0.606469 0.795107i \(-0.292585\pi\)
0.606469 + 0.795107i \(0.292585\pi\)
\(602\) 0 0
\(603\) −9.06630e7 −0.413502
\(604\) 0 0
\(605\) − 2.54286e8i − 1.14830i
\(606\) 0 0
\(607\) − 1.38978e8i − 0.621413i −0.950506 0.310707i \(-0.899434\pi\)
0.950506 0.310707i \(-0.100566\pi\)
\(608\) 0 0
\(609\) −3.58121e7 −0.158554
\(610\) 0 0
\(611\) −1.89925e8 −0.832642
\(612\) 0 0
\(613\) 4.20201e8i 1.82421i 0.409953 + 0.912106i \(0.365545\pi\)
−0.409953 + 0.912106i \(0.634455\pi\)
\(614\) 0 0
\(615\) 1.66595e8i 0.716205i
\(616\) 0 0
\(617\) 1.25856e7 0.0535818 0.0267909 0.999641i \(-0.491471\pi\)
0.0267909 + 0.999641i \(0.491471\pi\)
\(618\) 0 0
\(619\) 2.43243e8 1.02558 0.512789 0.858515i \(-0.328612\pi\)
0.512789 + 0.858515i \(0.328612\pi\)
\(620\) 0 0
\(621\) − 4.82161e7i − 0.201334i
\(622\) 0 0
\(623\) − 2.24693e7i − 0.0929234i
\(624\) 0 0
\(625\) −8.96966e7 −0.367397
\(626\) 0 0
\(627\) −1.20302e8 −0.488059
\(628\) 0 0
\(629\) 8.39440e7i 0.337317i
\(630\) 0 0
\(631\) − 5.45064e7i − 0.216950i −0.994099 0.108475i \(-0.965403\pi\)
0.994099 0.108475i \(-0.0345967\pi\)
\(632\) 0 0
\(633\) 1.48764e8 0.586526
\(634\) 0 0
\(635\) −1.30844e8 −0.511016
\(636\) 0 0
\(637\) − 5.06644e7i − 0.196013i
\(638\) 0 0
\(639\) 1.62278e8i 0.621954i
\(640\) 0 0
\(641\) −1.69420e8 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(642\) 0 0
\(643\) −4.20644e8 −1.58227 −0.791137 0.611639i \(-0.790510\pi\)
−0.791137 + 0.611639i \(0.790510\pi\)
\(644\) 0 0
\(645\) − 1.38333e8i − 0.515523i
\(646\) 0 0
\(647\) − 3.19909e8i − 1.18117i −0.806974 0.590587i \(-0.798896\pi\)
0.806974 0.590587i \(-0.201104\pi\)
\(648\) 0 0
\(649\) 3.73756e8 1.36727
\(650\) 0 0
\(651\) −1.90083e8 −0.688970
\(652\) 0 0
\(653\) 2.98645e8i 1.07254i 0.844045 + 0.536272i \(0.180168\pi\)
−0.844045 + 0.536272i \(0.819832\pi\)
\(654\) 0 0
\(655\) 3.38014e8i 1.20285i
\(656\) 0 0
\(657\) −9.70681e7 −0.342279
\(658\) 0 0
\(659\) −3.27357e8 −1.14384 −0.571920 0.820309i \(-0.693801\pi\)
−0.571920 + 0.820309i \(0.693801\pi\)
\(660\) 0 0
\(661\) − 4.03255e8i − 1.39629i −0.715956 0.698145i \(-0.754009\pi\)
0.715956 0.698145i \(-0.245991\pi\)
\(662\) 0 0
\(663\) − 1.33874e8i − 0.459364i
\(664\) 0 0
\(665\) −2.38481e8 −0.810941
\(666\) 0 0
\(667\) −1.05373e8 −0.355103
\(668\) 0 0
\(669\) − 2.46420e8i − 0.822995i
\(670\) 0 0
\(671\) 1.47215e8i 0.487285i
\(672\) 0 0
\(673\) −1.48356e8 −0.486697 −0.243349 0.969939i \(-0.578246\pi\)
−0.243349 + 0.969939i \(0.578246\pi\)
\(674\) 0 0
\(675\) −8.51985e7 −0.277026
\(676\) 0 0
\(677\) 4.50729e8i 1.45261i 0.687371 + 0.726306i \(0.258765\pi\)
−0.687371 + 0.726306i \(0.741235\pi\)
\(678\) 0 0
\(679\) − 2.43385e8i − 0.777470i
\(680\) 0 0
\(681\) 2.31605e8 0.733341
\(682\) 0 0
\(683\) 5.93405e8 1.86247 0.931235 0.364418i \(-0.118732\pi\)
0.931235 + 0.364418i \(0.118732\pi\)
\(684\) 0 0
\(685\) 5.41493e8i 1.68470i
\(686\) 0 0
\(687\) 2.00258e7i 0.0617617i
\(688\) 0 0
\(689\) 3.39802e8 1.03889
\(690\) 0 0
\(691\) −2.84794e8 −0.863170 −0.431585 0.902072i \(-0.642045\pi\)
−0.431585 + 0.902072i \(0.642045\pi\)
\(692\) 0 0
\(693\) 1.18233e8i 0.355254i
\(694\) 0 0
\(695\) 8.41545e8i 2.50682i
\(696\) 0 0
\(697\) −3.77066e8 −1.11357
\(698\) 0 0
\(699\) 4.09928e6 0.0120026
\(700\) 0 0
\(701\) 1.51280e8i 0.439165i 0.975594 + 0.219583i \(0.0704695\pi\)
−0.975594 + 0.219583i \(0.929530\pi\)
\(702\) 0 0
\(703\) 5.36404e7i 0.154392i
\(704\) 0 0
\(705\) −4.63622e8 −1.32311
\(706\) 0 0
\(707\) 5.00880e8 1.41734
\(708\) 0 0
\(709\) − 6.04474e8i − 1.69605i −0.529956 0.848025i \(-0.677792\pi\)
0.529956 0.848025i \(-0.322208\pi\)
\(710\) 0 0
\(711\) 1.05796e8i 0.294349i
\(712\) 0 0
\(713\) −5.59300e8 −1.54304
\(714\) 0 0
\(715\) 4.26766e8 1.16754
\(716\) 0 0
\(717\) − 2.95027e8i − 0.800396i
\(718\) 0 0
\(719\) 1.92148e8i 0.516951i 0.966018 + 0.258476i \(0.0832201\pi\)
−0.966018 + 0.258476i \(0.916780\pi\)
\(720\) 0 0
\(721\) −2.29297e8 −0.611777
\(722\) 0 0
\(723\) −2.98578e7 −0.0790029
\(724\) 0 0
\(725\) 1.86196e8i 0.488604i
\(726\) 0 0
\(727\) 1.33711e8i 0.347989i 0.984747 + 0.173994i \(0.0556675\pi\)
−0.984747 + 0.173994i \(0.944333\pi\)
\(728\) 0 0
\(729\) 1.43489e7 0.0370370
\(730\) 0 0
\(731\) 3.13099e8 0.801549
\(732\) 0 0
\(733\) 5.99936e8i 1.52333i 0.647973 + 0.761663i \(0.275617\pi\)
−0.647973 + 0.761663i \(0.724383\pi\)
\(734\) 0 0
\(735\) − 1.23676e8i − 0.311475i
\(736\) 0 0
\(737\) 6.54150e8 1.63409
\(738\) 0 0
\(739\) 4.83646e7 0.119838 0.0599190 0.998203i \(-0.480916\pi\)
0.0599190 + 0.998203i \(0.480916\pi\)
\(740\) 0 0
\(741\) − 8.55461e7i − 0.210255i
\(742\) 0 0
\(743\) − 3.69904e7i − 0.0901826i −0.998983 0.0450913i \(-0.985642\pi\)
0.998983 0.0450913i \(-0.0143579\pi\)
\(744\) 0 0
\(745\) 4.16311e7 0.100681
\(746\) 0 0
\(747\) 5.99197e7 0.143750
\(748\) 0 0
\(749\) 3.73491e8i 0.888863i
\(750\) 0 0
\(751\) − 1.99551e8i − 0.471123i −0.971859 0.235562i \(-0.924307\pi\)
0.971859 0.235562i \(-0.0756929\pi\)
\(752\) 0 0
\(753\) −1.72527e8 −0.404085
\(754\) 0 0
\(755\) −1.04199e9 −2.42115
\(756\) 0 0
\(757\) − 4.79819e8i − 1.10609i −0.833152 0.553045i \(-0.813466\pi\)
0.833152 0.553045i \(-0.186534\pi\)
\(758\) 0 0
\(759\) 3.47888e8i 0.795636i
\(760\) 0 0
\(761\) −2.43069e8 −0.551537 −0.275769 0.961224i \(-0.588932\pi\)
−0.275769 + 0.961224i \(0.588932\pi\)
\(762\) 0 0
\(763\) 1.95540e8 0.440211
\(764\) 0 0
\(765\) − 3.26798e8i − 0.729953i
\(766\) 0 0
\(767\) 2.65775e8i 0.589017i
\(768\) 0 0
\(769\) 4.11830e8 0.905605 0.452802 0.891611i \(-0.350424\pi\)
0.452802 + 0.891611i \(0.350424\pi\)
\(770\) 0 0
\(771\) 1.05802e7 0.0230850
\(772\) 0 0
\(773\) − 9.17077e7i − 0.198549i −0.995060 0.0992744i \(-0.968348\pi\)
0.995060 0.0992744i \(-0.0316522\pi\)
\(774\) 0 0
\(775\) 9.88289e8i 2.12314i
\(776\) 0 0
\(777\) 5.27176e7 0.112381
\(778\) 0 0
\(779\) −2.40946e8 −0.509692
\(780\) 0 0
\(781\) − 1.17087e9i − 2.45785i
\(782\) 0 0
\(783\) − 3.13587e7i − 0.0653240i
\(784\) 0 0
\(785\) 7.98471e8 1.65063
\(786\) 0 0
\(787\) −5.23454e8 −1.07388 −0.536938 0.843622i \(-0.680419\pi\)
−0.536938 + 0.843622i \(0.680419\pi\)
\(788\) 0 0
\(789\) 1.73232e8i 0.352693i
\(790\) 0 0
\(791\) − 7.08406e8i − 1.43137i
\(792\) 0 0
\(793\) −1.04683e8 −0.209921
\(794\) 0 0
\(795\) 8.29483e8 1.65085
\(796\) 0 0
\(797\) 1.58874e7i 0.0313818i 0.999877 + 0.0156909i \(0.00499477\pi\)
−0.999877 + 0.0156909i \(0.995005\pi\)
\(798\) 0 0
\(799\) − 1.04935e9i − 2.05721i
\(800\) 0 0
\(801\) 1.96751e7 0.0382842
\(802\) 0 0
\(803\) 7.00364e8 1.35262
\(804\) 0 0
\(805\) 6.89635e8i 1.32200i
\(806\) 0 0
\(807\) − 1.56573e8i − 0.297917i
\(808\) 0 0
\(809\) −5.17580e8 −0.977534 −0.488767 0.872414i \(-0.662553\pi\)
−0.488767 + 0.872414i \(0.662553\pi\)
\(810\) 0 0
\(811\) 9.43642e8 1.76907 0.884534 0.466476i \(-0.154476\pi\)
0.884534 + 0.466476i \(0.154476\pi\)
\(812\) 0 0
\(813\) 1.03212e8i 0.192069i
\(814\) 0 0
\(815\) − 3.00908e8i − 0.555855i
\(816\) 0 0
\(817\) 2.00071e8 0.366875
\(818\) 0 0
\(819\) −8.40744e7 −0.153042
\(820\) 0 0
\(821\) 6.51239e8i 1.17682i 0.808562 + 0.588411i \(0.200246\pi\)
−0.808562 + 0.588411i \(0.799754\pi\)
\(822\) 0 0
\(823\) − 1.12175e8i − 0.201231i −0.994925 0.100615i \(-0.967919\pi\)
0.994925 0.100615i \(-0.0320812\pi\)
\(824\) 0 0
\(825\) 6.14722e8 1.09476
\(826\) 0 0
\(827\) 6.24440e8 1.10401 0.552007 0.833840i \(-0.313862\pi\)
0.552007 + 0.833840i \(0.313862\pi\)
\(828\) 0 0
\(829\) 5.96016e8i 1.04615i 0.852286 + 0.523076i \(0.175216\pi\)
−0.852286 + 0.523076i \(0.824784\pi\)
\(830\) 0 0
\(831\) − 3.80915e8i − 0.663782i
\(832\) 0 0
\(833\) 2.79924e8 0.484289
\(834\) 0 0
\(835\) 1.49658e9 2.57064
\(836\) 0 0
\(837\) − 1.66445e8i − 0.283854i
\(838\) 0 0
\(839\) − 4.46780e7i − 0.0756497i −0.999284 0.0378249i \(-0.987957\pi\)
0.999284 0.0378249i \(-0.0120429\pi\)
\(840\) 0 0
\(841\) 5.26291e8 0.884785
\(842\) 0 0
\(843\) −2.17656e8 −0.363320
\(844\) 0 0
\(845\) − 6.38892e8i − 1.05891i
\(846\) 0 0
\(847\) − 3.61446e8i − 0.594830i
\(848\) 0 0
\(849\) 6.68485e8 1.09237
\(850\) 0 0
\(851\) 1.55116e8 0.251691
\(852\) 0 0
\(853\) − 1.12761e9i − 1.81683i −0.418072 0.908414i \(-0.637294\pi\)
0.418072 0.908414i \(-0.362706\pi\)
\(854\) 0 0
\(855\) − 2.08825e8i − 0.334105i
\(856\) 0 0
\(857\) 5.47949e8 0.870557 0.435279 0.900296i \(-0.356650\pi\)
0.435279 + 0.900296i \(0.356650\pi\)
\(858\) 0 0
\(859\) −6.34544e8 −1.00111 −0.500556 0.865704i \(-0.666871\pi\)
−0.500556 + 0.865704i \(0.666871\pi\)
\(860\) 0 0
\(861\) 2.36801e8i 0.371000i
\(862\) 0 0
\(863\) − 1.20080e9i − 1.86826i −0.356928 0.934132i \(-0.616176\pi\)
0.356928 0.934132i \(-0.383824\pi\)
\(864\) 0 0
\(865\) −1.95977e9 −3.02801
\(866\) 0 0
\(867\) 3.63396e8 0.557600
\(868\) 0 0
\(869\) − 7.63340e8i − 1.16321i
\(870\) 0 0
\(871\) 4.65160e8i 0.703960i
\(872\) 0 0
\(873\) 2.13118e8 0.320316
\(874\) 0 0
\(875\) 3.72036e8 0.555342
\(876\) 0 0
\(877\) − 6.11977e6i − 0.00907270i −0.999990 0.00453635i \(-0.998556\pi\)
0.999990 0.00453635i \(-0.00144397\pi\)
\(878\) 0 0
\(879\) 7.79819e7i 0.114823i
\(880\) 0 0
\(881\) 5.39118e8 0.788417 0.394209 0.919021i \(-0.371019\pi\)
0.394209 + 0.919021i \(0.371019\pi\)
\(882\) 0 0
\(883\) 4.91469e8 0.713861 0.356931 0.934131i \(-0.383823\pi\)
0.356931 + 0.934131i \(0.383823\pi\)
\(884\) 0 0
\(885\) 6.48778e8i 0.935980i
\(886\) 0 0
\(887\) − 6.95698e8i − 0.996897i −0.866919 0.498448i \(-0.833903\pi\)
0.866919 0.498448i \(-0.166097\pi\)
\(888\) 0 0
\(889\) −1.85984e8 −0.264710
\(890\) 0 0
\(891\) −1.03530e8 −0.146364
\(892\) 0 0
\(893\) − 6.70534e8i − 0.941600i
\(894\) 0 0
\(895\) 1.72450e9i 2.40544i
\(896\) 0 0
\(897\) −2.47380e8 −0.342758
\(898\) 0 0
\(899\) −3.63756e8 −0.500646
\(900\) 0 0
\(901\) 1.87742e9i 2.56678i
\(902\) 0 0
\(903\) − 1.96629e8i − 0.267045i
\(904\) 0 0
\(905\) −1.17159e9 −1.58063
\(906\) 0 0
\(907\) 1.10192e9 1.47682 0.738411 0.674351i \(-0.235577\pi\)
0.738411 + 0.674351i \(0.235577\pi\)
\(908\) 0 0
\(909\) 4.38592e8i 0.583942i
\(910\) 0 0
\(911\) 3.89543e8i 0.515229i 0.966248 + 0.257615i \(0.0829365\pi\)
−0.966248 + 0.257615i \(0.917063\pi\)
\(912\) 0 0
\(913\) −4.32331e8 −0.568073
\(914\) 0 0
\(915\) −2.55540e8 −0.333576
\(916\) 0 0
\(917\) 4.80458e8i 0.623085i
\(918\) 0 0
\(919\) − 1.13853e9i − 1.46690i −0.679746 0.733448i \(-0.737910\pi\)
0.679746 0.733448i \(-0.262090\pi\)
\(920\) 0 0
\(921\) 4.26360e8 0.545754
\(922\) 0 0
\(923\) 8.32594e8 1.05883
\(924\) 0 0
\(925\) − 2.74092e8i − 0.346315i
\(926\) 0 0
\(927\) − 2.00783e8i − 0.252050i
\(928\) 0 0
\(929\) 5.92474e8 0.738962 0.369481 0.929238i \(-0.379535\pi\)
0.369481 + 0.929238i \(0.379535\pi\)
\(930\) 0 0
\(931\) 1.78872e8 0.221663
\(932\) 0 0
\(933\) − 5.25900e8i − 0.647528i
\(934\) 0 0
\(935\) 2.35791e9i 2.88464i
\(936\) 0 0
\(937\) −6.16456e7 −0.0749348 −0.0374674 0.999298i \(-0.511929\pi\)
−0.0374674 + 0.999298i \(0.511929\pi\)
\(938\) 0 0
\(939\) 7.66782e8 0.926137
\(940\) 0 0
\(941\) 3.10725e8i 0.372913i 0.982463 + 0.186457i \(0.0597004\pi\)
−0.982463 + 0.186457i \(0.940300\pi\)
\(942\) 0 0
\(943\) 6.96763e8i 0.830902i
\(944\) 0 0
\(945\) −2.05232e8 −0.243192
\(946\) 0 0
\(947\) 1.13722e8 0.133904 0.0669522 0.997756i \(-0.478672\pi\)
0.0669522 + 0.997756i \(0.478672\pi\)
\(948\) 0 0
\(949\) 4.98023e8i 0.582707i
\(950\) 0 0
\(951\) 3.07234e8i 0.357214i
\(952\) 0 0
\(953\) −5.25518e8 −0.607168 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(954\) 0 0
\(955\) −1.17743e8 −0.135184
\(956\) 0 0
\(957\) 2.26258e8i 0.258148i
\(958\) 0 0
\(959\) 7.69687e8i 0.872686i
\(960\) 0 0
\(961\) −1.04323e9 −1.17547
\(962\) 0 0
\(963\) −3.27045e8 −0.366209
\(964\) 0 0
\(965\) − 1.51423e9i − 1.68504i
\(966\) 0 0
\(967\) − 5.50448e8i − 0.608747i −0.952553 0.304373i \(-0.901553\pi\)
0.952553 0.304373i \(-0.0984470\pi\)
\(968\) 0 0
\(969\) 4.72647e8 0.519476
\(970\) 0 0
\(971\) 1.54851e9 1.69144 0.845722 0.533624i \(-0.179170\pi\)
0.845722 + 0.533624i \(0.179170\pi\)
\(972\) 0 0
\(973\) 1.19618e9i 1.29855i
\(974\) 0 0
\(975\) 4.37124e8i 0.471618i
\(976\) 0 0
\(977\) 1.73673e9 1.86229 0.931146 0.364645i \(-0.118810\pi\)
0.931146 + 0.364645i \(0.118810\pi\)
\(978\) 0 0
\(979\) −1.41959e8 −0.151292
\(980\) 0 0
\(981\) 1.71223e8i 0.181366i
\(982\) 0 0
\(983\) − 7.96734e8i − 0.838789i −0.907804 0.419395i \(-0.862242\pi\)
0.907804 0.419395i \(-0.137758\pi\)
\(984\) 0 0
\(985\) −3.74402e8 −0.391769
\(986\) 0 0
\(987\) −6.58999e8 −0.685383
\(988\) 0 0
\(989\) − 5.78562e8i − 0.598082i
\(990\) 0 0
\(991\) − 3.33810e8i − 0.342988i −0.985185 0.171494i \(-0.945141\pi\)
0.985185 0.171494i \(-0.0548594\pi\)
\(992\) 0 0
\(993\) −2.48965e8 −0.254267
\(994\) 0 0
\(995\) 1.98054e9 2.01054
\(996\) 0 0
\(997\) − 4.81174e8i − 0.485531i −0.970085 0.242765i \(-0.921945\pi\)
0.970085 0.242765i \(-0.0780545\pi\)
\(998\) 0 0
\(999\) 4.61619e7i 0.0463007i
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 384.7.b.c.319.5 yes 8
4.3 odd 2 inner 384.7.b.c.319.1 8
8.3 odd 2 inner 384.7.b.c.319.8 yes 8
8.5 even 2 inner 384.7.b.c.319.4 yes 8
16.3 odd 4 768.7.g.g.511.1 8
16.5 even 4 768.7.g.g.511.4 8
16.11 odd 4 768.7.g.g.511.8 8
16.13 even 4 768.7.g.g.511.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.c.319.1 8 4.3 odd 2 inner
384.7.b.c.319.4 yes 8 8.5 even 2 inner
384.7.b.c.319.5 yes 8 1.1 even 1 trivial
384.7.b.c.319.8 yes 8 8.3 odd 2 inner
768.7.g.g.511.1 8 16.3 odd 4
768.7.g.g.511.4 8 16.5 even 4
768.7.g.g.511.5 8 16.13 even 4
768.7.g.g.511.8 8 16.11 odd 4