Properties

Label 384.10.d.a.193.3
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.3
Root \(53.8781i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.a.193.6

$q$-expansion

\(f(q)\) \(=\) \(q-81.0000i q^{3} +473.533i q^{5} -574.939 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q-81.0000i q^{3} +473.533i q^{5} -574.939 q^{7} -6561.00 q^{9} +72774.7i q^{11} -14829.2i q^{13} +38356.2 q^{15} +109932. q^{17} -497573. i q^{19} +46570.1i q^{21} -104764. q^{23} +1.72889e6 q^{25} +531441. i q^{27} -7.06390e6i q^{29} -2.30578e6 q^{31} +5.89475e6 q^{33} -272253. i q^{35} +5.20960e6i q^{37} -1.20116e6 q^{39} -1.08234e7 q^{41} -2.04081e7i q^{43} -3.10685e6i q^{45} +3.47476e7 q^{47} -4.00231e7 q^{49} -8.90445e6i q^{51} +9.34226e7i q^{53} -3.44612e7 q^{55} -4.03034e7 q^{57} +9.04786e7i q^{59} -2.00269e7i q^{61} +3.77218e6 q^{63} +7.02210e6 q^{65} -1.09151e8i q^{67} +8.48588e6i q^{69} +2.65931e8 q^{71} +5.05525e7 q^{73} -1.40040e8i q^{75} -4.18410e7i q^{77} -1.24238e7 q^{79} +4.30467e7 q^{81} +2.72920e8i q^{83} +5.20562e7i q^{85} -5.72176e8 q^{87} -2.70637e8 q^{89} +8.52588e6i q^{91} +1.86768e8i q^{93} +2.35617e8 q^{95} -4.06136e8 q^{97} -4.77475e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) − 81.0000i − 0.577350i
\(4\) 0 0
\(5\) 473.533i 0.338833i 0.985545 + 0.169416i \(0.0541882\pi\)
−0.985545 + 0.169416i \(0.945812\pi\)
\(6\) 0 0
\(7\) −574.939 −0.0905067 −0.0452534 0.998976i \(-0.514410\pi\)
−0.0452534 + 0.998976i \(0.514410\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 72774.7i 1.49869i 0.662177 + 0.749347i \(0.269633\pi\)
−0.662177 + 0.749347i \(0.730367\pi\)
\(12\) 0 0
\(13\) − 14829.2i − 0.144003i −0.997405 0.0720016i \(-0.977061\pi\)
0.997405 0.0720016i \(-0.0229387\pi\)
\(14\) 0 0
\(15\) 38356.2 0.195625
\(16\) 0 0
\(17\) 109932. 0.319229 0.159614 0.987179i \(-0.448975\pi\)
0.159614 + 0.987179i \(0.448975\pi\)
\(18\) 0 0
\(19\) − 497573.i − 0.875922i −0.898994 0.437961i \(-0.855701\pi\)
0.898994 0.437961i \(-0.144299\pi\)
\(20\) 0 0
\(21\) 46570.1i 0.0522541i
\(22\) 0 0
\(23\) −104764. −0.0780614 −0.0390307 0.999238i \(-0.512427\pi\)
−0.0390307 + 0.999238i \(0.512427\pi\)
\(24\) 0 0
\(25\) 1.72889e6 0.885192
\(26\) 0 0
\(27\) 531441.i 0.192450i
\(28\) 0 0
\(29\) − 7.06390e6i − 1.85461i −0.374301 0.927307i \(-0.622117\pi\)
0.374301 0.927307i \(-0.377883\pi\)
\(30\) 0 0
\(31\) −2.30578e6 −0.448426 −0.224213 0.974540i \(-0.571981\pi\)
−0.224213 + 0.974540i \(0.571981\pi\)
\(32\) 0 0
\(33\) 5.89475e6 0.865272
\(34\) 0 0
\(35\) − 272253.i − 0.0306666i
\(36\) 0 0
\(37\) 5.20960e6i 0.456979i 0.973546 + 0.228490i \(0.0733787\pi\)
−0.973546 + 0.228490i \(0.926621\pi\)
\(38\) 0 0
\(39\) −1.20116e6 −0.0831402
\(40\) 0 0
\(41\) −1.08234e7 −0.598185 −0.299092 0.954224i \(-0.596684\pi\)
−0.299092 + 0.954224i \(0.596684\pi\)
\(42\) 0 0
\(43\) − 2.04081e7i − 0.910319i −0.890410 0.455160i \(-0.849582\pi\)
0.890410 0.455160i \(-0.150418\pi\)
\(44\) 0 0
\(45\) − 3.10685e6i − 0.112944i
\(46\) 0 0
\(47\) 3.47476e7 1.03869 0.519343 0.854566i \(-0.326177\pi\)
0.519343 + 0.854566i \(0.326177\pi\)
\(48\) 0 0
\(49\) −4.00231e7 −0.991809
\(50\) 0 0
\(51\) − 8.90445e6i − 0.184307i
\(52\) 0 0
\(53\) 9.34226e7i 1.62634i 0.582028 + 0.813169i \(0.302259\pi\)
−0.582028 + 0.813169i \(0.697741\pi\)
\(54\) 0 0
\(55\) −3.44612e7 −0.507807
\(56\) 0 0
\(57\) −4.03034e7 −0.505714
\(58\) 0 0
\(59\) 9.04786e7i 0.972102i 0.873930 + 0.486051i \(0.161563\pi\)
−0.873930 + 0.486051i \(0.838437\pi\)
\(60\) 0 0
\(61\) − 2.00269e7i − 0.185195i −0.995704 0.0925976i \(-0.970483\pi\)
0.995704 0.0925976i \(-0.0295170\pi\)
\(62\) 0 0
\(63\) 3.77218e6 0.0301689
\(64\) 0 0
\(65\) 7.02210e6 0.0487930
\(66\) 0 0
\(67\) − 1.09151e8i − 0.661745i −0.943676 0.330872i \(-0.892657\pi\)
0.943676 0.330872i \(-0.107343\pi\)
\(68\) 0 0
\(69\) 8.48588e6i 0.0450688i
\(70\) 0 0
\(71\) 2.65931e8 1.24196 0.620978 0.783828i \(-0.286736\pi\)
0.620978 + 0.783828i \(0.286736\pi\)
\(72\) 0 0
\(73\) 5.05525e7 0.208348 0.104174 0.994559i \(-0.466780\pi\)
0.104174 + 0.994559i \(0.466780\pi\)
\(74\) 0 0
\(75\) − 1.40040e8i − 0.511066i
\(76\) 0 0
\(77\) − 4.18410e7i − 0.135642i
\(78\) 0 0
\(79\) −1.24238e7 −0.0358865 −0.0179433 0.999839i \(-0.505712\pi\)
−0.0179433 + 0.999839i \(0.505712\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) 2.72920e8i 0.631225i 0.948888 + 0.315613i \(0.102210\pi\)
−0.948888 + 0.315613i \(0.897790\pi\)
\(84\) 0 0
\(85\) 5.20562e7i 0.108165i
\(86\) 0 0
\(87\) −5.72176e8 −1.07076
\(88\) 0 0
\(89\) −2.70637e8 −0.457227 −0.228613 0.973517i \(-0.573419\pi\)
−0.228613 + 0.973517i \(0.573419\pi\)
\(90\) 0 0
\(91\) 8.52588e6i 0.0130333i
\(92\) 0 0
\(93\) 1.86768e8i 0.258899i
\(94\) 0 0
\(95\) 2.35617e8 0.296791
\(96\) 0 0
\(97\) −4.06136e8 −0.465799 −0.232899 0.972501i \(-0.574821\pi\)
−0.232899 + 0.972501i \(0.574821\pi\)
\(98\) 0 0
\(99\) − 4.77475e8i − 0.499565i
\(100\) 0 0
\(101\) 3.58458e8i 0.342762i 0.985205 + 0.171381i \(0.0548229\pi\)
−0.985205 + 0.171381i \(0.945177\pi\)
\(102\) 0 0
\(103\) 3.12727e8 0.273777 0.136889 0.990586i \(-0.456290\pi\)
0.136889 + 0.990586i \(0.456290\pi\)
\(104\) 0 0
\(105\) −2.20525e7 −0.0177054
\(106\) 0 0
\(107\) 9.28960e8i 0.685125i 0.939495 + 0.342562i \(0.111295\pi\)
−0.939495 + 0.342562i \(0.888705\pi\)
\(108\) 0 0
\(109\) 1.54301e9i 1.04701i 0.852024 + 0.523503i \(0.175375\pi\)
−0.852024 + 0.523503i \(0.824625\pi\)
\(110\) 0 0
\(111\) 4.21977e8 0.263837
\(112\) 0 0
\(113\) −5.98385e8 −0.345245 −0.172623 0.984988i \(-0.555224\pi\)
−0.172623 + 0.984988i \(0.555224\pi\)
\(114\) 0 0
\(115\) − 4.96092e7i − 0.0264498i
\(116\) 0 0
\(117\) 9.72942e7i 0.0480010i
\(118\) 0 0
\(119\) −6.32039e7 −0.0288924
\(120\) 0 0
\(121\) −2.93820e9 −1.24609
\(122\) 0 0
\(123\) 8.76693e8i 0.345362i
\(124\) 0 0
\(125\) 1.74356e9i 0.638765i
\(126\) 0 0
\(127\) −2.52986e9 −0.862940 −0.431470 0.902127i \(-0.642005\pi\)
−0.431470 + 0.902127i \(0.642005\pi\)
\(128\) 0 0
\(129\) −1.65305e9 −0.525573
\(130\) 0 0
\(131\) − 1.70203e8i − 0.0504948i −0.999681 0.0252474i \(-0.991963\pi\)
0.999681 0.0252474i \(-0.00803734\pi\)
\(132\) 0 0
\(133\) 2.86074e8i 0.0792768i
\(134\) 0 0
\(135\) −2.51655e8 −0.0652084
\(136\) 0 0
\(137\) 2.59035e9 0.628226 0.314113 0.949386i \(-0.398293\pi\)
0.314113 + 0.949386i \(0.398293\pi\)
\(138\) 0 0
\(139\) 2.75040e9i 0.624928i 0.949930 + 0.312464i \(0.101154\pi\)
−0.949930 + 0.312464i \(0.898846\pi\)
\(140\) 0 0
\(141\) − 2.81456e9i − 0.599686i
\(142\) 0 0
\(143\) 1.07919e9 0.215817
\(144\) 0 0
\(145\) 3.34499e9 0.628404
\(146\) 0 0
\(147\) 3.24187e9i 0.572621i
\(148\) 0 0
\(149\) 2.97566e9i 0.494590i 0.968940 + 0.247295i \(0.0795416\pi\)
−0.968940 + 0.247295i \(0.920458\pi\)
\(150\) 0 0
\(151\) 1.10799e10 1.73435 0.867177 0.498000i \(-0.165932\pi\)
0.867177 + 0.498000i \(0.165932\pi\)
\(152\) 0 0
\(153\) −7.21261e8 −0.106410
\(154\) 0 0
\(155\) − 1.09186e9i − 0.151941i
\(156\) 0 0
\(157\) − 1.20320e10i − 1.58048i −0.612797 0.790240i \(-0.709956\pi\)
0.612797 0.790240i \(-0.290044\pi\)
\(158\) 0 0
\(159\) 7.56723e9 0.938966
\(160\) 0 0
\(161\) 6.02329e7 0.00706508
\(162\) 0 0
\(163\) 1.21499e9i 0.134812i 0.997726 + 0.0674062i \(0.0214724\pi\)
−0.997726 + 0.0674062i \(0.978528\pi\)
\(164\) 0 0
\(165\) 2.79136e9i 0.293182i
\(166\) 0 0
\(167\) −1.01298e10 −1.00780 −0.503902 0.863761i \(-0.668103\pi\)
−0.503902 + 0.863761i \(0.668103\pi\)
\(168\) 0 0
\(169\) 1.03846e10 0.979263
\(170\) 0 0
\(171\) 3.26458e9i 0.291974i
\(172\) 0 0
\(173\) − 1.12366e10i − 0.953738i −0.878974 0.476869i \(-0.841772\pi\)
0.878974 0.476869i \(-0.158228\pi\)
\(174\) 0 0
\(175\) −9.94008e8 −0.0801159
\(176\) 0 0
\(177\) 7.32877e9 0.561244
\(178\) 0 0
\(179\) − 3.72939e9i − 0.271518i −0.990742 0.135759i \(-0.956653\pi\)
0.990742 0.135759i \(-0.0433473\pi\)
\(180\) 0 0
\(181\) 1.47052e10i 1.01840i 0.860649 + 0.509198i \(0.170058\pi\)
−0.860649 + 0.509198i \(0.829942\pi\)
\(182\) 0 0
\(183\) −1.62218e9 −0.106922
\(184\) 0 0
\(185\) −2.46692e9 −0.154839
\(186\) 0 0
\(187\) 8.00023e9i 0.478426i
\(188\) 0 0
\(189\) − 3.05546e8i − 0.0174180i
\(190\) 0 0
\(191\) 2.16788e10 1.17865 0.589325 0.807896i \(-0.299394\pi\)
0.589325 + 0.807896i \(0.299394\pi\)
\(192\) 0 0
\(193\) −2.45321e10 −1.27270 −0.636350 0.771400i \(-0.719557\pi\)
−0.636350 + 0.771400i \(0.719557\pi\)
\(194\) 0 0
\(195\) − 5.68790e8i − 0.0281706i
\(196\) 0 0
\(197\) 1.44518e10i 0.683632i 0.939767 + 0.341816i \(0.111042\pi\)
−0.939767 + 0.341816i \(0.888958\pi\)
\(198\) 0 0
\(199\) 2.71655e10 1.22794 0.613972 0.789328i \(-0.289571\pi\)
0.613972 + 0.789328i \(0.289571\pi\)
\(200\) 0 0
\(201\) −8.84122e9 −0.382058
\(202\) 0 0
\(203\) 4.06132e9i 0.167855i
\(204\) 0 0
\(205\) − 5.12522e9i − 0.202684i
\(206\) 0 0
\(207\) 6.87356e8 0.0260205
\(208\) 0 0
\(209\) 3.62107e10 1.31274
\(210\) 0 0
\(211\) 3.52449e10i 1.22412i 0.790810 + 0.612062i \(0.209660\pi\)
−0.790810 + 0.612062i \(0.790340\pi\)
\(212\) 0 0
\(213\) − 2.15404e10i − 0.717044i
\(214\) 0 0
\(215\) 9.66389e9 0.308446
\(216\) 0 0
\(217\) 1.32568e9 0.0405855
\(218\) 0 0
\(219\) − 4.09475e9i − 0.120290i
\(220\) 0 0
\(221\) − 1.63019e9i − 0.0459699i
\(222\) 0 0
\(223\) 1.45878e10 0.395019 0.197510 0.980301i \(-0.436715\pi\)
0.197510 + 0.980301i \(0.436715\pi\)
\(224\) 0 0
\(225\) −1.13433e10 −0.295064
\(226\) 0 0
\(227\) 6.30500e10i 1.57605i 0.615646 + 0.788023i \(0.288895\pi\)
−0.615646 + 0.788023i \(0.711105\pi\)
\(228\) 0 0
\(229\) − 4.91705e10i − 1.18153i −0.806843 0.590765i \(-0.798826\pi\)
0.806843 0.590765i \(-0.201174\pi\)
\(230\) 0 0
\(231\) −3.38912e9 −0.0783129
\(232\) 0 0
\(233\) 3.30380e10 0.734366 0.367183 0.930149i \(-0.380322\pi\)
0.367183 + 0.930149i \(0.380322\pi\)
\(234\) 0 0
\(235\) 1.64541e10i 0.351941i
\(236\) 0 0
\(237\) 1.00633e9i 0.0207191i
\(238\) 0 0
\(239\) 7.33801e10 1.45475 0.727374 0.686241i \(-0.240740\pi\)
0.727374 + 0.686241i \(0.240740\pi\)
\(240\) 0 0
\(241\) 5.53816e10 1.05752 0.528760 0.848771i \(-0.322657\pi\)
0.528760 + 0.848771i \(0.322657\pi\)
\(242\) 0 0
\(243\) − 3.48678e9i − 0.0641500i
\(244\) 0 0
\(245\) − 1.89522e10i − 0.336057i
\(246\) 0 0
\(247\) −7.37860e9 −0.126136
\(248\) 0 0
\(249\) 2.21065e10 0.364438
\(250\) 0 0
\(251\) 9.51298e10i 1.51281i 0.654103 + 0.756406i \(0.273046\pi\)
−0.654103 + 0.756406i \(0.726954\pi\)
\(252\) 0 0
\(253\) − 7.62416e9i − 0.116990i
\(254\) 0 0
\(255\) 4.21655e9 0.0624492
\(256\) 0 0
\(257\) −5.82111e10 −0.832351 −0.416176 0.909284i \(-0.636630\pi\)
−0.416176 + 0.909284i \(0.636630\pi\)
\(258\) 0 0
\(259\) − 2.99520e9i − 0.0413597i
\(260\) 0 0
\(261\) 4.63463e10i 0.618205i
\(262\) 0 0
\(263\) 1.26131e11 1.62563 0.812814 0.582524i \(-0.197935\pi\)
0.812814 + 0.582524i \(0.197935\pi\)
\(264\) 0 0
\(265\) −4.42387e10 −0.551056
\(266\) 0 0
\(267\) 2.19216e10i 0.263980i
\(268\) 0 0
\(269\) − 1.23726e10i − 0.144071i −0.997402 0.0720356i \(-0.977050\pi\)
0.997402 0.0720356i \(-0.0229495\pi\)
\(270\) 0 0
\(271\) 6.40678e10 0.721569 0.360785 0.932649i \(-0.382509\pi\)
0.360785 + 0.932649i \(0.382509\pi\)
\(272\) 0 0
\(273\) 6.90596e8 0.00752475
\(274\) 0 0
\(275\) 1.25820e11i 1.32663i
\(276\) 0 0
\(277\) 1.56675e11i 1.59897i 0.600688 + 0.799484i \(0.294893\pi\)
−0.600688 + 0.799484i \(0.705107\pi\)
\(278\) 0 0
\(279\) 1.51282e10 0.149475
\(280\) 0 0
\(281\) −9.55568e10 −0.914288 −0.457144 0.889393i \(-0.651128\pi\)
−0.457144 + 0.889393i \(0.651128\pi\)
\(282\) 0 0
\(283\) 7.99388e10i 0.740830i 0.928866 + 0.370415i \(0.120785\pi\)
−0.928866 + 0.370415i \(0.879215\pi\)
\(284\) 0 0
\(285\) − 1.90850e10i − 0.171352i
\(286\) 0 0
\(287\) 6.22278e9 0.0541397
\(288\) 0 0
\(289\) −1.06503e11 −0.898093
\(290\) 0 0
\(291\) 3.28970e10i 0.268929i
\(292\) 0 0
\(293\) 9.87325e10i 0.782629i 0.920257 + 0.391315i \(0.127980\pi\)
−0.920257 + 0.391315i \(0.872020\pi\)
\(294\) 0 0
\(295\) −4.28446e10 −0.329380
\(296\) 0 0
\(297\) −3.86754e10 −0.288424
\(298\) 0 0
\(299\) 1.55356e9i 0.0112411i
\(300\) 0 0
\(301\) 1.17334e10i 0.0823900i
\(302\) 0 0
\(303\) 2.90351e10 0.197894
\(304\) 0 0
\(305\) 9.48340e9 0.0627502
\(306\) 0 0
\(307\) 2.01628e11i 1.29547i 0.761864 + 0.647737i \(0.224284\pi\)
−0.761864 + 0.647737i \(0.775716\pi\)
\(308\) 0 0
\(309\) − 2.53309e10i − 0.158065i
\(310\) 0 0
\(311\) 1.25577e11 0.761182 0.380591 0.924744i \(-0.375721\pi\)
0.380591 + 0.924744i \(0.375721\pi\)
\(312\) 0 0
\(313\) 9.72056e10 0.572455 0.286228 0.958162i \(-0.407599\pi\)
0.286228 + 0.958162i \(0.407599\pi\)
\(314\) 0 0
\(315\) 1.78625e9i 0.0102222i
\(316\) 0 0
\(317\) 3.24299e10i 0.180376i 0.995925 + 0.0901880i \(0.0287468\pi\)
−0.995925 + 0.0901880i \(0.971253\pi\)
\(318\) 0 0
\(319\) 5.14073e11 2.77950
\(320\) 0 0
\(321\) 7.52457e10 0.395557
\(322\) 0 0
\(323\) − 5.46989e10i − 0.279620i
\(324\) 0 0
\(325\) − 2.56380e10i − 0.127470i
\(326\) 0 0
\(327\) 1.24984e11 0.604490
\(328\) 0 0
\(329\) −1.99778e10 −0.0940081
\(330\) 0 0
\(331\) 2.13971e11i 0.979781i 0.871784 + 0.489890i \(0.162963\pi\)
−0.871784 + 0.489890i \(0.837037\pi\)
\(332\) 0 0
\(333\) − 3.41802e10i − 0.152326i
\(334\) 0 0
\(335\) 5.16865e10 0.224221
\(336\) 0 0
\(337\) −1.67278e11 −0.706489 −0.353244 0.935531i \(-0.614922\pi\)
−0.353244 + 0.935531i \(0.614922\pi\)
\(338\) 0 0
\(339\) 4.84692e10i 0.199328i
\(340\) 0 0
\(341\) − 1.67802e11i − 0.672053i
\(342\) 0 0
\(343\) 4.62117e10 0.180272
\(344\) 0 0
\(345\) −4.01834e9 −0.0152708
\(346\) 0 0
\(347\) 1.93452e11i 0.716293i 0.933665 + 0.358146i \(0.116591\pi\)
−0.933665 + 0.358146i \(0.883409\pi\)
\(348\) 0 0
\(349\) 1.79870e11i 0.649000i 0.945886 + 0.324500i \(0.105196\pi\)
−0.945886 + 0.324500i \(0.894804\pi\)
\(350\) 0 0
\(351\) 7.88083e9 0.0277134
\(352\) 0 0
\(353\) 2.13154e11 0.730646 0.365323 0.930881i \(-0.380959\pi\)
0.365323 + 0.930881i \(0.380959\pi\)
\(354\) 0 0
\(355\) 1.25927e11i 0.420815i
\(356\) 0 0
\(357\) 5.11952e9i 0.0166810i
\(358\) 0 0
\(359\) −1.85269e11 −0.588677 −0.294339 0.955701i \(-0.595099\pi\)
−0.294339 + 0.955701i \(0.595099\pi\)
\(360\) 0 0
\(361\) 7.51089e10 0.232760
\(362\) 0 0
\(363\) 2.37995e11i 0.719428i
\(364\) 0 0
\(365\) 2.39383e10i 0.0705952i
\(366\) 0 0
\(367\) 3.99652e11 1.14997 0.574983 0.818166i \(-0.305009\pi\)
0.574983 + 0.818166i \(0.305009\pi\)
\(368\) 0 0
\(369\) 7.10122e10 0.199395
\(370\) 0 0
\(371\) − 5.37123e10i − 0.147194i
\(372\) 0 0
\(373\) − 3.17540e11i − 0.849394i −0.905335 0.424697i \(-0.860381\pi\)
0.905335 0.424697i \(-0.139619\pi\)
\(374\) 0 0
\(375\) 1.41228e11 0.368791
\(376\) 0 0
\(377\) −1.04752e11 −0.267070
\(378\) 0 0
\(379\) 2.10387e11i 0.523771i 0.965099 + 0.261886i \(0.0843443\pi\)
−0.965099 + 0.261886i \(0.915656\pi\)
\(380\) 0 0
\(381\) 2.04919e11i 0.498219i
\(382\) 0 0
\(383\) 6.26458e11 1.48764 0.743819 0.668381i \(-0.233012\pi\)
0.743819 + 0.668381i \(0.233012\pi\)
\(384\) 0 0
\(385\) 1.98131e10 0.0459599
\(386\) 0 0
\(387\) 1.33897e11i 0.303440i
\(388\) 0 0
\(389\) − 4.71840e11i − 1.04477i −0.852709 0.522386i \(-0.825042\pi\)
0.852709 0.522386i \(-0.174958\pi\)
\(390\) 0 0
\(391\) −1.15169e10 −0.0249194
\(392\) 0 0
\(393\) −1.37864e10 −0.0291532
\(394\) 0 0
\(395\) − 5.88307e9i − 0.0121595i
\(396\) 0 0
\(397\) 4.08614e11i 0.825573i 0.910828 + 0.412786i \(0.135444\pi\)
−0.910828 + 0.412786i \(0.864556\pi\)
\(398\) 0 0
\(399\) 2.31720e10 0.0457705
\(400\) 0 0
\(401\) −4.75227e11 −0.917807 −0.458903 0.888486i \(-0.651758\pi\)
−0.458903 + 0.888486i \(0.651758\pi\)
\(402\) 0 0
\(403\) 3.41928e10i 0.0645747i
\(404\) 0 0
\(405\) 2.03840e10i 0.0376481i
\(406\) 0 0
\(407\) −3.79127e11 −0.684872
\(408\) 0 0
\(409\) 1.47969e11 0.261466 0.130733 0.991418i \(-0.458267\pi\)
0.130733 + 0.991418i \(0.458267\pi\)
\(410\) 0 0
\(411\) − 2.09818e11i − 0.362706i
\(412\) 0 0
\(413\) − 5.20197e10i − 0.0879818i
\(414\) 0 0
\(415\) −1.29237e11 −0.213880
\(416\) 0 0
\(417\) 2.22783e11 0.360802
\(418\) 0 0
\(419\) 8.99050e11i 1.42502i 0.701663 + 0.712509i \(0.252441\pi\)
−0.701663 + 0.712509i \(0.747559\pi\)
\(420\) 0 0
\(421\) 9.67201e10i 0.150054i 0.997182 + 0.0750269i \(0.0239043\pi\)
−0.997182 + 0.0750269i \(0.976096\pi\)
\(422\) 0 0
\(423\) −2.27979e11 −0.346229
\(424\) 0 0
\(425\) 1.90060e11 0.282579
\(426\) 0 0
\(427\) 1.15143e10i 0.0167614i
\(428\) 0 0
\(429\) − 8.74143e10i − 0.124602i
\(430\) 0 0
\(431\) 4.91630e11 0.686263 0.343131 0.939287i \(-0.388512\pi\)
0.343131 + 0.939287i \(0.388512\pi\)
\(432\) 0 0
\(433\) −2.39033e10 −0.0326786 −0.0163393 0.999867i \(-0.505201\pi\)
−0.0163393 + 0.999867i \(0.505201\pi\)
\(434\) 0 0
\(435\) − 2.70944e11i − 0.362809i
\(436\) 0 0
\(437\) 5.21277e10i 0.0683757i
\(438\) 0 0
\(439\) −8.29279e11 −1.06564 −0.532820 0.846229i \(-0.678868\pi\)
−0.532820 + 0.846229i \(0.678868\pi\)
\(440\) 0 0
\(441\) 2.62591e11 0.330603
\(442\) 0 0
\(443\) 5.71843e11i 0.705440i 0.935729 + 0.352720i \(0.114743\pi\)
−0.935729 + 0.352720i \(0.885257\pi\)
\(444\) 0 0
\(445\) − 1.28155e11i − 0.154923i
\(446\) 0 0
\(447\) 2.41028e11 0.285551
\(448\) 0 0
\(449\) 1.02914e12 1.19500 0.597500 0.801869i \(-0.296161\pi\)
0.597500 + 0.801869i \(0.296161\pi\)
\(450\) 0 0
\(451\) − 7.87668e11i − 0.896496i
\(452\) 0 0
\(453\) − 8.97468e11i − 1.00133i
\(454\) 0 0
\(455\) −4.03728e9 −0.00441609
\(456\) 0 0
\(457\) −7.44956e11 −0.798928 −0.399464 0.916749i \(-0.630804\pi\)
−0.399464 + 0.916749i \(0.630804\pi\)
\(458\) 0 0
\(459\) 5.84221e10i 0.0614356i
\(460\) 0 0
\(461\) − 1.35785e11i − 0.140023i −0.997546 0.0700115i \(-0.977696\pi\)
0.997546 0.0700115i \(-0.0223036\pi\)
\(462\) 0 0
\(463\) 6.39280e10 0.0646512 0.0323256 0.999477i \(-0.489709\pi\)
0.0323256 + 0.999477i \(0.489709\pi\)
\(464\) 0 0
\(465\) −8.84409e10 −0.0877233
\(466\) 0 0
\(467\) − 8.08293e11i − 0.786399i −0.919453 0.393199i \(-0.871368\pi\)
0.919453 0.393199i \(-0.128632\pi\)
\(468\) 0 0
\(469\) 6.27551e10i 0.0598923i
\(470\) 0 0
\(471\) −9.74591e11 −0.912491
\(472\) 0 0
\(473\) 1.48519e12 1.36429
\(474\) 0 0
\(475\) − 8.60250e11i − 0.775360i
\(476\) 0 0
\(477\) − 6.12946e11i − 0.542112i
\(478\) 0 0
\(479\) 1.38621e12 1.20315 0.601573 0.798818i \(-0.294541\pi\)
0.601573 + 0.798818i \(0.294541\pi\)
\(480\) 0 0
\(481\) 7.72540e10 0.0658064
\(482\) 0 0
\(483\) − 4.87886e9i − 0.00407903i
\(484\) 0 0
\(485\) − 1.92319e11i − 0.157828i
\(486\) 0 0
\(487\) −1.20417e11 −0.0970078 −0.0485039 0.998823i \(-0.515445\pi\)
−0.0485039 + 0.998823i \(0.515445\pi\)
\(488\) 0 0
\(489\) 9.84146e10 0.0778340
\(490\) 0 0
\(491\) − 1.43283e11i − 0.111257i −0.998452 0.0556287i \(-0.982284\pi\)
0.998452 0.0556287i \(-0.0177163\pi\)
\(492\) 0 0
\(493\) − 7.76545e11i − 0.592046i
\(494\) 0 0
\(495\) 2.26100e11 0.169269
\(496\) 0 0
\(497\) −1.52894e11 −0.112405
\(498\) 0 0
\(499\) 1.93465e12i 1.39685i 0.715683 + 0.698426i \(0.246116\pi\)
−0.715683 + 0.698426i \(0.753884\pi\)
\(500\) 0 0
\(501\) 8.20513e11i 0.581856i
\(502\) 0 0
\(503\) −2.14780e12 −1.49602 −0.748010 0.663687i \(-0.768991\pi\)
−0.748010 + 0.663687i \(0.768991\pi\)
\(504\) 0 0
\(505\) −1.69742e11 −0.116139
\(506\) 0 0
\(507\) − 8.41152e11i − 0.565378i
\(508\) 0 0
\(509\) − 4.93770e10i − 0.0326058i −0.999867 0.0163029i \(-0.994810\pi\)
0.999867 0.0163029i \(-0.00518961\pi\)
\(510\) 0 0
\(511\) −2.90646e10 −0.0188569
\(512\) 0 0
\(513\) 2.64431e11 0.168571
\(514\) 0 0
\(515\) 1.48086e11i 0.0927646i
\(516\) 0 0
\(517\) 2.52875e12i 1.55667i
\(518\) 0 0
\(519\) −9.10168e11 −0.550641
\(520\) 0 0
\(521\) −3.06642e12 −1.82331 −0.911657 0.410951i \(-0.865197\pi\)
−0.911657 + 0.410951i \(0.865197\pi\)
\(522\) 0 0
\(523\) − 1.36512e12i − 0.797836i −0.916987 0.398918i \(-0.869386\pi\)
0.916987 0.398918i \(-0.130614\pi\)
\(524\) 0 0
\(525\) 8.05146e10i 0.0462549i
\(526\) 0 0
\(527\) −2.53478e11 −0.143150
\(528\) 0 0
\(529\) −1.79018e12 −0.993906
\(530\) 0 0
\(531\) − 5.93630e11i − 0.324034i
\(532\) 0 0
\(533\) 1.60502e11i 0.0861405i
\(534\) 0 0
\(535\) −4.39893e11 −0.232143
\(536\) 0 0
\(537\) −3.02081e11 −0.156761
\(538\) 0 0
\(539\) − 2.91266e12i − 1.48642i
\(540\) 0 0
\(541\) 2.34837e12i 1.17863i 0.807902 + 0.589317i \(0.200603\pi\)
−0.807902 + 0.589317i \(0.799397\pi\)
\(542\) 0 0
\(543\) 1.19112e12 0.587971
\(544\) 0 0
\(545\) −7.30666e11 −0.354760
\(546\) 0 0
\(547\) − 3.90038e12i − 1.86279i −0.364007 0.931396i \(-0.618592\pi\)
0.364007 0.931396i \(-0.381408\pi\)
\(548\) 0 0
\(549\) 1.31397e11i 0.0617317i
\(550\) 0 0
\(551\) −3.51481e12 −1.62450
\(552\) 0 0
\(553\) 7.14292e9 0.00324797
\(554\) 0 0
\(555\) 1.99820e11i 0.0893966i
\(556\) 0 0
\(557\) 9.85631e11i 0.433876i 0.976185 + 0.216938i \(0.0696070\pi\)
−0.976185 + 0.216938i \(0.930393\pi\)
\(558\) 0 0
\(559\) −3.02635e11 −0.131089
\(560\) 0 0
\(561\) 6.48019e11 0.276220
\(562\) 0 0
\(563\) − 3.71992e12i − 1.56044i −0.625508 0.780218i \(-0.715108\pi\)
0.625508 0.780218i \(-0.284892\pi\)
\(564\) 0 0
\(565\) − 2.83355e11i − 0.116980i
\(566\) 0 0
\(567\) −2.47493e10 −0.0100563
\(568\) 0 0
\(569\) 2.34608e12 0.938293 0.469146 0.883120i \(-0.344562\pi\)
0.469146 + 0.883120i \(0.344562\pi\)
\(570\) 0 0
\(571\) 1.58941e12i 0.625710i 0.949801 + 0.312855i \(0.101285\pi\)
−0.949801 + 0.312855i \(0.898715\pi\)
\(572\) 0 0
\(573\) − 1.75598e12i − 0.680493i
\(574\) 0 0
\(575\) −1.81125e11 −0.0690994
\(576\) 0 0
\(577\) −4.20772e12 −1.58036 −0.790179 0.612876i \(-0.790013\pi\)
−0.790179 + 0.612876i \(0.790013\pi\)
\(578\) 0 0
\(579\) 1.98710e12i 0.734794i
\(580\) 0 0
\(581\) − 1.56913e11i − 0.0571302i
\(582\) 0 0
\(583\) −6.79880e12 −2.43738
\(584\) 0 0
\(585\) −4.60720e10 −0.0162643
\(586\) 0 0
\(587\) − 1.63748e12i − 0.569252i −0.958639 0.284626i \(-0.908131\pi\)
0.958639 0.284626i \(-0.0918694\pi\)
\(588\) 0 0
\(589\) 1.14729e12i 0.392786i
\(590\) 0 0
\(591\) 1.17059e12 0.394695
\(592\) 0 0
\(593\) 5.53148e12 1.83694 0.918471 0.395489i \(-0.129425\pi\)
0.918471 + 0.395489i \(0.129425\pi\)
\(594\) 0 0
\(595\) − 2.99292e10i − 0.00978967i
\(596\) 0 0
\(597\) − 2.20040e12i − 0.708954i
\(598\) 0 0
\(599\) −4.16661e12 −1.32240 −0.661199 0.750211i \(-0.729952\pi\)
−0.661199 + 0.750211i \(0.729952\pi\)
\(600\) 0 0
\(601\) −2.90464e12 −0.908149 −0.454075 0.890964i \(-0.650030\pi\)
−0.454075 + 0.890964i \(0.650030\pi\)
\(602\) 0 0
\(603\) 7.16139e11i 0.220582i
\(604\) 0 0
\(605\) − 1.39134e12i − 0.422214i
\(606\) 0 0
\(607\) 4.66728e12 1.39545 0.697726 0.716365i \(-0.254195\pi\)
0.697726 + 0.716365i \(0.254195\pi\)
\(608\) 0 0
\(609\) 3.28967e11 0.0969112
\(610\) 0 0
\(611\) − 5.15278e11i − 0.149574i
\(612\) 0 0
\(613\) − 3.12221e11i − 0.0893078i −0.999003 0.0446539i \(-0.985781\pi\)
0.999003 0.0446539i \(-0.0142185\pi\)
\(614\) 0 0
\(615\) −4.15143e11 −0.117020
\(616\) 0 0
\(617\) 3.59628e12 0.999011 0.499505 0.866311i \(-0.333515\pi\)
0.499505 + 0.866311i \(0.333515\pi\)
\(618\) 0 0
\(619\) − 4.85670e12i − 1.32964i −0.747004 0.664819i \(-0.768509\pi\)
0.747004 0.664819i \(-0.231491\pi\)
\(620\) 0 0
\(621\) − 5.56758e10i − 0.0150229i
\(622\) 0 0
\(623\) 1.55600e11 0.0413821
\(624\) 0 0
\(625\) 2.55111e12 0.668758
\(626\) 0 0
\(627\) − 2.93307e12i − 0.757911i
\(628\) 0 0
\(629\) 5.72699e11i 0.145881i
\(630\) 0 0
\(631\) −6.19329e12 −1.55521 −0.777606 0.628752i \(-0.783566\pi\)
−0.777606 + 0.628752i \(0.783566\pi\)
\(632\) 0 0
\(633\) 2.85484e12 0.706748
\(634\) 0 0
\(635\) − 1.19797e12i − 0.292392i
\(636\) 0 0
\(637\) 5.93509e11i 0.142824i
\(638\) 0 0
\(639\) −1.74477e12 −0.413985
\(640\) 0 0
\(641\) −5.96307e12 −1.39511 −0.697555 0.716531i \(-0.745729\pi\)
−0.697555 + 0.716531i \(0.745729\pi\)
\(642\) 0 0
\(643\) − 7.48485e11i − 0.172677i −0.996266 0.0863383i \(-0.972483\pi\)
0.996266 0.0863383i \(-0.0275166\pi\)
\(644\) 0 0
\(645\) − 7.82775e11i − 0.178081i
\(646\) 0 0
\(647\) −5.21646e12 −1.17032 −0.585162 0.810916i \(-0.698969\pi\)
−0.585162 + 0.810916i \(0.698969\pi\)
\(648\) 0 0
\(649\) −6.58455e12 −1.45688
\(650\) 0 0
\(651\) − 1.07380e11i − 0.0234321i
\(652\) 0 0
\(653\) − 5.05968e12i − 1.08896i −0.838772 0.544482i \(-0.816726\pi\)
0.838772 0.544482i \(-0.183274\pi\)
\(654\) 0 0
\(655\) 8.05967e10 0.0171093
\(656\) 0 0
\(657\) −3.31675e11 −0.0694494
\(658\) 0 0
\(659\) − 1.64097e12i − 0.338936i −0.985536 0.169468i \(-0.945795\pi\)
0.985536 0.169468i \(-0.0542049\pi\)
\(660\) 0 0
\(661\) 9.39337e12i 1.91388i 0.290284 + 0.956941i \(0.406250\pi\)
−0.290284 + 0.956941i \(0.593750\pi\)
\(662\) 0 0
\(663\) −1.32046e11 −0.0265408
\(664\) 0 0
\(665\) −1.35466e11 −0.0268616
\(666\) 0 0
\(667\) 7.40042e11i 0.144774i
\(668\) 0 0
\(669\) − 1.18161e12i − 0.228065i
\(670\) 0 0
\(671\) 1.45745e12 0.277551
\(672\) 0 0
\(673\) 6.84794e11 0.128674 0.0643372 0.997928i \(-0.479507\pi\)
0.0643372 + 0.997928i \(0.479507\pi\)
\(674\) 0 0
\(675\) 9.18804e11i 0.170355i
\(676\) 0 0
\(677\) 2.15344e12i 0.393988i 0.980405 + 0.196994i \(0.0631180\pi\)
−0.980405 + 0.196994i \(0.936882\pi\)
\(678\) 0 0
\(679\) 2.33503e11 0.0421579
\(680\) 0 0
\(681\) 5.10705e12 0.909930
\(682\) 0 0
\(683\) 3.32483e12i 0.584623i 0.956323 + 0.292311i \(0.0944244\pi\)
−0.956323 + 0.292311i \(0.905576\pi\)
\(684\) 0 0
\(685\) 1.22662e12i 0.212863i
\(686\) 0 0
\(687\) −3.98281e12 −0.682157
\(688\) 0 0
\(689\) 1.38538e12 0.234198
\(690\) 0 0
\(691\) − 1.50122e12i − 0.250491i −0.992126 0.125246i \(-0.960028\pi\)
0.992126 0.125246i \(-0.0399719\pi\)
\(692\) 0 0
\(693\) 2.74519e11i 0.0452140i
\(694\) 0 0
\(695\) −1.30241e12 −0.211746
\(696\) 0 0
\(697\) −1.18983e12 −0.190958
\(698\) 0 0
\(699\) − 2.67608e12i − 0.423987i
\(700\) 0 0
\(701\) − 3.02156e12i − 0.472607i −0.971679 0.236304i \(-0.924064\pi\)
0.971679 0.236304i \(-0.0759360\pi\)
\(702\) 0 0
\(703\) 2.59215e12 0.400278
\(704\) 0 0
\(705\) 1.33279e12 0.203193
\(706\) 0 0
\(707\) − 2.06092e11i − 0.0310223i
\(708\) 0 0
\(709\) 3.67440e12i 0.546108i 0.961999 + 0.273054i \(0.0880337\pi\)
−0.961999 + 0.273054i \(0.911966\pi\)
\(710\) 0 0
\(711\) 8.15124e10 0.0119622
\(712\) 0 0
\(713\) 2.41563e11 0.0350047
\(714\) 0 0
\(715\) 5.11031e11i 0.0731257i
\(716\) 0 0
\(717\) − 5.94379e12i − 0.839899i
\(718\) 0 0
\(719\) 9.45558e12 1.31950 0.659748 0.751487i \(-0.270663\pi\)
0.659748 + 0.751487i \(0.270663\pi\)
\(720\) 0 0
\(721\) −1.79799e11 −0.0247787
\(722\) 0 0
\(723\) − 4.48591e12i − 0.610560i
\(724\) 0 0
\(725\) − 1.22127e13i − 1.64169i
\(726\) 0 0
\(727\) −1.44431e13 −1.91760 −0.958798 0.284090i \(-0.908309\pi\)
−0.958798 + 0.284090i \(0.908309\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) − 2.24349e12i − 0.290600i
\(732\) 0 0
\(733\) − 4.09211e12i − 0.523576i −0.965125 0.261788i \(-0.915688\pi\)
0.965125 0.261788i \(-0.0843121\pi\)
\(734\) 0 0
\(735\) −1.53513e12 −0.194023
\(736\) 0 0
\(737\) 7.94342e12 0.991753
\(738\) 0 0
\(739\) − 9.48921e12i − 1.17039i −0.810893 0.585195i \(-0.801018\pi\)
0.810893 0.585195i \(-0.198982\pi\)
\(740\) 0 0
\(741\) 5.97666e11i 0.0728244i
\(742\) 0 0
\(743\) 2.23279e12 0.268781 0.134390 0.990928i \(-0.457092\pi\)
0.134390 + 0.990928i \(0.457092\pi\)
\(744\) 0 0
\(745\) −1.40907e12 −0.167583
\(746\) 0 0
\(747\) − 1.79063e12i − 0.210408i
\(748\) 0 0
\(749\) − 5.34095e11i − 0.0620084i
\(750\) 0 0
\(751\) 1.33252e13 1.52860 0.764298 0.644863i \(-0.223086\pi\)
0.764298 + 0.644863i \(0.223086\pi\)
\(752\) 0 0
\(753\) 7.70552e12 0.873422
\(754\) 0 0
\(755\) 5.24667e12i 0.587656i
\(756\) 0 0
\(757\) 2.46238e12i 0.272536i 0.990672 + 0.136268i \(0.0435108\pi\)
−0.990672 + 0.136268i \(0.956489\pi\)
\(758\) 0 0
\(759\) −6.17557e11 −0.0675443
\(760\) 0 0
\(761\) 7.93783e12 0.857968 0.428984 0.903312i \(-0.358872\pi\)
0.428984 + 0.903312i \(0.358872\pi\)
\(762\) 0 0
\(763\) − 8.87137e11i − 0.0947611i
\(764\) 0 0
\(765\) − 3.41541e11i − 0.0360550i
\(766\) 0 0
\(767\) 1.34172e12 0.139986
\(768\) 0 0
\(769\) −1.58369e13 −1.63305 −0.816527 0.577307i \(-0.804104\pi\)
−0.816527 + 0.577307i \(0.804104\pi\)
\(770\) 0 0
\(771\) 4.71510e12i 0.480558i
\(772\) 0 0
\(773\) − 1.21012e13i − 1.21904i −0.792769 0.609522i \(-0.791361\pi\)
0.792769 0.609522i \(-0.208639\pi\)
\(774\) 0 0
\(775\) −3.98645e12 −0.396943
\(776\) 0 0
\(777\) −2.42611e11 −0.0238790
\(778\) 0 0
\(779\) 5.38542e12i 0.523963i
\(780\) 0 0
\(781\) 1.93530e13i 1.86131i
\(782\) 0 0
\(783\) 3.75405e12 0.356921
\(784\) 0 0
\(785\) 5.69754e12 0.535518
\(786\) 0 0
\(787\) 1.64934e13i 1.53258i 0.642495 + 0.766290i \(0.277899\pi\)
−0.642495 + 0.766290i \(0.722101\pi\)
\(788\) 0 0
\(789\) − 1.02166e13i − 0.938557i
\(790\) 0 0
\(791\) 3.44035e11 0.0312470
\(792\) 0 0
\(793\) −2.96983e11 −0.0266687
\(794\) 0 0
\(795\) 3.58333e12i 0.318152i
\(796\) 0 0
\(797\) 9.14539e12i 0.802860i 0.915890 + 0.401430i \(0.131487\pi\)
−0.915890 + 0.401430i \(0.868513\pi\)
\(798\) 0 0
\(799\) 3.81986e12 0.331579
\(800\) 0 0
\(801\) 1.77565e12 0.152409
\(802\) 0 0
\(803\) 3.67894e12i 0.312250i
\(804\) 0 0
\(805\) 2.85223e10i 0.00239388i
\(806\) 0 0
\(807\) −1.00218e12 −0.0831796
\(808\) 0 0
\(809\) 1.10523e12 0.0907160 0.0453580 0.998971i \(-0.485557\pi\)
0.0453580 + 0.998971i \(0.485557\pi\)
\(810\) 0 0
\(811\) − 5.17595e12i − 0.420142i −0.977686 0.210071i \(-0.932631\pi\)
0.977686 0.210071i \(-0.0673695\pi\)
\(812\) 0 0
\(813\) − 5.18949e12i − 0.416598i
\(814\) 0 0
\(815\) −5.75340e11 −0.0456789
\(816\) 0 0
\(817\) −1.01545e13 −0.797369
\(818\) 0 0
\(819\) − 5.59383e10i − 0.00434442i
\(820\) 0 0
\(821\) − 1.06367e13i − 0.817076i −0.912741 0.408538i \(-0.866039\pi\)
0.912741 0.408538i \(-0.133961\pi\)
\(822\) 0 0
\(823\) 1.89127e13 1.43699 0.718495 0.695532i \(-0.244831\pi\)
0.718495 + 0.695532i \(0.244831\pi\)
\(824\) 0 0
\(825\) 1.01914e13 0.765932
\(826\) 0 0
\(827\) 6.62866e11i 0.0492777i 0.999696 + 0.0246389i \(0.00784359\pi\)
−0.999696 + 0.0246389i \(0.992156\pi\)
\(828\) 0 0
\(829\) 2.27273e13i 1.67129i 0.549267 + 0.835647i \(0.314907\pi\)
−0.549267 + 0.835647i \(0.685093\pi\)
\(830\) 0 0
\(831\) 1.26906e13 0.923164
\(832\) 0 0
\(833\) −4.39979e12 −0.316614
\(834\) 0 0
\(835\) − 4.79679e12i − 0.341477i
\(836\) 0 0
\(837\) − 1.22539e12i − 0.0862996i
\(838\) 0 0
\(839\) 2.12551e12 0.148093 0.0740463 0.997255i \(-0.476409\pi\)
0.0740463 + 0.997255i \(0.476409\pi\)
\(840\) 0 0
\(841\) −3.53916e13 −2.43960
\(842\) 0 0
\(843\) 7.74010e12i 0.527865i
\(844\) 0 0
\(845\) 4.91745e12i 0.331806i
\(846\) 0 0
\(847\) 1.68929e12 0.112779
\(848\) 0 0
\(849\) 6.47504e12 0.427718
\(850\) 0 0
\(851\) − 5.45778e11i − 0.0356724i
\(852\) 0 0
\(853\) 2.52949e13i 1.63592i 0.575276 + 0.817960i \(0.304895\pi\)
−0.575276 + 0.817960i \(0.695105\pi\)
\(854\) 0 0
\(855\) −1.54588e12 −0.0989303
\(856\) 0 0
\(857\) 2.42903e12 0.153822 0.0769112 0.997038i \(-0.475494\pi\)
0.0769112 + 0.997038i \(0.475494\pi\)
\(858\) 0 0
\(859\) 2.92636e13i 1.83383i 0.399083 + 0.916915i \(0.369329\pi\)
−0.399083 + 0.916915i \(0.630671\pi\)
\(860\) 0 0
\(861\) − 5.04046e11i − 0.0312576i
\(862\) 0 0
\(863\) 1.25262e13 0.768724 0.384362 0.923182i \(-0.374421\pi\)
0.384362 + 0.923182i \(0.374421\pi\)
\(864\) 0 0
\(865\) 5.32092e12 0.323157
\(866\) 0 0
\(867\) 8.62674e12i 0.518514i
\(868\) 0 0
\(869\) − 9.04136e11i − 0.0537830i
\(870\) 0 0
\(871\) −1.61862e12 −0.0952933
\(872\) 0 0
\(873\) 2.66466e12 0.155266
\(874\) 0 0
\(875\) − 1.00244e12i − 0.0578125i
\(876\) 0 0
\(877\) − 6.98911e11i − 0.0398955i −0.999801 0.0199477i \(-0.993650\pi\)
0.999801 0.0199477i \(-0.00634998\pi\)
\(878\) 0 0
\(879\) 7.99733e12 0.451851
\(880\) 0 0
\(881\) −1.80581e13 −1.00991 −0.504953 0.863147i \(-0.668490\pi\)
−0.504953 + 0.863147i \(0.668490\pi\)
\(882\) 0 0
\(883\) 2.54103e13i 1.40665i 0.710867 + 0.703326i \(0.248303\pi\)
−0.710867 + 0.703326i \(0.751697\pi\)
\(884\) 0 0
\(885\) 3.47041e12i 0.190168i
\(886\) 0 0
\(887\) −2.87381e12 −0.155884 −0.0779421 0.996958i \(-0.524835\pi\)
−0.0779421 + 0.996958i \(0.524835\pi\)
\(888\) 0 0
\(889\) 1.45452e12 0.0781019
\(890\) 0 0
\(891\) 3.13271e12i 0.166522i
\(892\) 0 0
\(893\) − 1.72895e13i − 0.909809i
\(894\) 0 0
\(895\) 1.76599e12 0.0919993
\(896\) 0 0
\(897\) 1.25839e11 0.00649004
\(898\) 0 0
\(899\) 1.62878e13i 0.831657i
\(900\) 0 0
\(901\) 1.02701e13i 0.519174i
\(902\) 0 0
\(903\) 9.50405e11 0.0475679
\(904\) 0 0
\(905\) −6.96339e12 −0.345066
\(906\) 0 0
\(907\) 9.81693e10i 0.00481662i 0.999997 + 0.00240831i \(0.000766590\pi\)
−0.999997 + 0.00240831i \(0.999233\pi\)
\(908\) 0 0
\(909\) − 2.35185e12i − 0.114254i
\(910\) 0 0
\(911\) 3.70352e12 0.178148 0.0890741 0.996025i \(-0.471609\pi\)
0.0890741 + 0.996025i \(0.471609\pi\)
\(912\) 0 0
\(913\) −1.98617e13 −0.946014
\(914\) 0 0
\(915\) − 7.68156e11i − 0.0362288i
\(916\) 0 0
\(917\) 9.78564e10i 0.00457012i
\(918\) 0 0
\(919\) 3.79299e13 1.75413 0.877066 0.480370i \(-0.159498\pi\)
0.877066 + 0.480370i \(0.159498\pi\)
\(920\) 0 0
\(921\) 1.63319e13 0.747942
\(922\) 0 0
\(923\) − 3.94354e12i − 0.178846i
\(924\) 0 0
\(925\) 9.00683e12i 0.404515i
\(926\) 0 0
\(927\) −2.05180e12 −0.0912591
\(928\) 0 0
\(929\) 3.49219e13 1.53825 0.769124 0.639099i \(-0.220693\pi\)
0.769124 + 0.639099i \(0.220693\pi\)
\(930\) 0 0
\(931\) 1.99144e13i 0.868747i
\(932\) 0 0
\(933\) − 1.01717e13i − 0.439468i
\(934\) 0 0
\(935\) −3.78837e12 −0.162106
\(936\) 0 0
\(937\) −2.39866e13 −1.01658 −0.508288 0.861187i \(-0.669722\pi\)
−0.508288 + 0.861187i \(0.669722\pi\)
\(938\) 0 0
\(939\) − 7.87365e12i − 0.330507i
\(940\) 0 0
\(941\) 8.10164e12i 0.336837i 0.985716 + 0.168418i \(0.0538660\pi\)
−0.985716 + 0.168418i \(0.946134\pi\)
\(942\) 0 0
\(943\) 1.13390e12 0.0466951
\(944\) 0 0
\(945\) 1.44686e11 0.00590180
\(946\) 0 0
\(947\) 1.66583e12i 0.0673061i 0.999434 + 0.0336531i \(0.0107141\pi\)
−0.999434 + 0.0336531i \(0.989286\pi\)
\(948\) 0 0
\(949\) − 7.49652e11i − 0.0300028i
\(950\) 0 0
\(951\) 2.62682e12 0.104140
\(952\) 0 0
\(953\) 3.98680e13 1.56569 0.782846 0.622215i \(-0.213767\pi\)
0.782846 + 0.622215i \(0.213767\pi\)
\(954\) 0 0
\(955\) 1.02656e13i 0.399365i
\(956\) 0 0
\(957\) − 4.16399e13i − 1.60475i
\(958\) 0 0
\(959\) −1.48929e12 −0.0568587
\(960\) 0 0
\(961\) −2.11230e13 −0.798914
\(962\) 0 0
\(963\) − 6.09490e12i − 0.228375i
\(964\) 0 0
\(965\) − 1.16167e13i − 0.431232i
\(966\) 0 0
\(967\) 3.53153e13 1.29880 0.649402 0.760446i \(-0.275019\pi\)
0.649402 + 0.760446i \(0.275019\pi\)
\(968\) 0 0
\(969\) −4.43061e12 −0.161438
\(970\) 0 0
\(971\) 2.43725e13i 0.879862i 0.898032 + 0.439931i \(0.144997\pi\)
−0.898032 + 0.439931i \(0.855003\pi\)
\(972\) 0 0
\(973\) − 1.58132e12i − 0.0565602i
\(974\) 0 0
\(975\) −2.07668e12 −0.0735951
\(976\) 0 0
\(977\) 3.11200e13 1.09273 0.546366 0.837546i \(-0.316011\pi\)
0.546366 + 0.837546i \(0.316011\pi\)
\(978\) 0 0
\(979\) − 1.96955e13i − 0.685243i
\(980\) 0 0
\(981\) − 1.01237e13i − 0.349002i
\(982\) 0 0
\(983\) 2.69352e13 0.920088 0.460044 0.887896i \(-0.347834\pi\)
0.460044 + 0.887896i \(0.347834\pi\)
\(984\) 0 0
\(985\) −6.84338e12 −0.231637
\(986\) 0 0
\(987\) 1.61820e12i 0.0542756i
\(988\) 0 0
\(989\) 2.13803e12i 0.0710608i
\(990\) 0 0
\(991\) −3.56596e13 −1.17448 −0.587240 0.809413i \(-0.699785\pi\)
−0.587240 + 0.809413i \(0.699785\pi\)
\(992\) 0 0
\(993\) 1.73317e13 0.565677
\(994\) 0 0
\(995\) 1.28638e13i 0.416068i
\(996\) 0 0
\(997\) 3.95731e13i 1.26844i 0.773151 + 0.634222i \(0.218680\pi\)
−0.773151 + 0.634222i \(0.781320\pi\)
\(998\) 0 0
\(999\) −2.76859e12 −0.0879457
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 384.10.d.a.193.3 8
4.3 odd 2 384.10.d.b.193.7 yes 8
8.3 odd 2 384.10.d.b.193.2 yes 8
8.5 even 2 inner 384.10.d.a.193.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.a.193.3 8 1.1 even 1 trivial
384.10.d.a.193.6 yes 8 8.5 even 2 inner
384.10.d.b.193.2 yes 8 8.3 odd 2
384.10.d.b.193.7 yes 8 4.3 odd 2