Properties

Label 384.10.d.b.193.5
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 193.5
Root \(43.3581i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.b.193.4

$q$-expansion

\(f(q)\) \(=\) \(q+81.0000i q^{3} -1783.68i q^{5} +4965.03 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q+81.0000i q^{3} -1783.68i q^{5} +4965.03 q^{7} -6561.00 q^{9} +53705.0i q^{11} +54228.9i q^{13} +144478. q^{15} -554662. q^{17} +650852. i q^{19} +402168. i q^{21} -183775. q^{23} -1.22839e6 q^{25} -531441. i q^{27} -2.02445e6i q^{29} +3.43180e6 q^{31} -4.35011e6 q^{33} -8.85603e6i q^{35} +1.30084e6i q^{37} -4.39254e6 q^{39} +157393. q^{41} -2.10193e6i q^{43} +1.17027e7i q^{45} +4.73332e7 q^{47} -1.57021e7 q^{49} -4.49277e7i q^{51} -1.00735e8i q^{53} +9.57926e7 q^{55} -5.27190e7 q^{57} -1.80263e7i q^{59} +5.48498e7i q^{61} -3.25756e7 q^{63} +9.67270e7 q^{65} -2.13387e8i q^{67} -1.48858e7i q^{69} +1.64615e8 q^{71} -7.58505e7 q^{73} -9.94998e7i q^{75} +2.66647e8i q^{77} -4.25335e8 q^{79} +4.30467e7 q^{81} -1.06618e8i q^{83} +9.89341e8i q^{85} +1.63981e8 q^{87} -1.02792e9 q^{89} +2.69248e8i q^{91} +2.77976e8i q^{93} +1.16091e9 q^{95} +6.24259e8 q^{97} -3.52359e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) 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
\(5\) − 1783.68i − 1.27630i −0.769913 0.638149i \(-0.779700\pi\)
0.769913 0.638149i \(-0.220300\pi\)
\(6\) 0 0
\(7\) 4965.03 0.781593 0.390797 0.920477i \(-0.372200\pi\)
0.390797 + 0.920477i \(0.372200\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 53705.0i 1.10598i 0.833187 + 0.552991i \(0.186513\pi\)
−0.833187 + 0.552991i \(0.813487\pi\)
\(12\) 0 0
\(13\) 54228.9i 0.526606i 0.964713 + 0.263303i \(0.0848118\pi\)
−0.964713 + 0.263303i \(0.915188\pi\)
\(14\) 0 0
\(15\) 144478. 0.736871
\(16\) 0 0
\(17\) −554662. −1.61068 −0.805339 0.592815i \(-0.798017\pi\)
−0.805339 + 0.592815i \(0.798017\pi\)
\(18\) 0 0
\(19\) 650852.i 1.14575i 0.819642 + 0.572876i \(0.194172\pi\)
−0.819642 + 0.572876i \(0.805828\pi\)
\(20\) 0 0
\(21\) 402168.i 0.451253i
\(22\) 0 0
\(23\) −183775. −0.136934 −0.0684669 0.997653i \(-0.521811\pi\)
−0.0684669 + 0.997653i \(0.521811\pi\)
\(24\) 0 0
\(25\) −1.22839e6 −0.628937
\(26\) 0 0
\(27\) − 531441.i − 0.192450i
\(28\) 0 0
\(29\) − 2.02445e6i − 0.531516i −0.964040 0.265758i \(-0.914378\pi\)
0.964040 0.265758i \(-0.0856222\pi\)
\(30\) 0 0
\(31\) 3.43180e6 0.667413 0.333706 0.942677i \(-0.391701\pi\)
0.333706 + 0.942677i \(0.391701\pi\)
\(32\) 0 0
\(33\) −4.35011e6 −0.638539
\(34\) 0 0
\(35\) − 8.85603e6i − 0.997546i
\(36\) 0 0
\(37\) 1.30084e6i 0.114108i 0.998371 + 0.0570542i \(0.0181708\pi\)
−0.998371 + 0.0570542i \(0.981829\pi\)
\(38\) 0 0
\(39\) −4.39254e6 −0.304036
\(40\) 0 0
\(41\) 157393. 0.00869877 0.00434938 0.999991i \(-0.498616\pi\)
0.00434938 + 0.999991i \(0.498616\pi\)
\(42\) 0 0
\(43\) − 2.10193e6i − 0.0937583i −0.998901 0.0468792i \(-0.985072\pi\)
0.998901 0.0468792i \(-0.0149276\pi\)
\(44\) 0 0
\(45\) 1.17027e7i 0.425433i
\(46\) 0 0
\(47\) 4.73332e7 1.41490 0.707449 0.706764i \(-0.249846\pi\)
0.707449 + 0.706764i \(0.249846\pi\)
\(48\) 0 0
\(49\) −1.57021e7 −0.389112
\(50\) 0 0
\(51\) − 4.49277e7i − 0.929925i
\(52\) 0 0
\(53\) − 1.00735e8i − 1.75364i −0.480820 0.876819i \(-0.659661\pi\)
0.480820 0.876819i \(-0.340339\pi\)
\(54\) 0 0
\(55\) 9.57926e7 1.41156
\(56\) 0 0
\(57\) −5.27190e7 −0.661501
\(58\) 0 0
\(59\) − 1.80263e7i − 0.193674i −0.995300 0.0968372i \(-0.969127\pi\)
0.995300 0.0968372i \(-0.0308726\pi\)
\(60\) 0 0
\(61\) 5.48498e7i 0.507214i 0.967307 + 0.253607i \(0.0816169\pi\)
−0.967307 + 0.253607i \(0.918383\pi\)
\(62\) 0 0
\(63\) −3.25756e7 −0.260531
\(64\) 0 0
\(65\) 9.67270e7 0.672106
\(66\) 0 0
\(67\) − 2.13387e8i − 1.29369i −0.762621 0.646846i \(-0.776088\pi\)
0.762621 0.646846i \(-0.223912\pi\)
\(68\) 0 0
\(69\) − 1.48858e7i − 0.0790588i
\(70\) 0 0
\(71\) 1.64615e8 0.768786 0.384393 0.923169i \(-0.374411\pi\)
0.384393 + 0.923169i \(0.374411\pi\)
\(72\) 0 0
\(73\) −7.58505e7 −0.312612 −0.156306 0.987709i \(-0.549959\pi\)
−0.156306 + 0.987709i \(0.549959\pi\)
\(74\) 0 0
\(75\) − 9.94998e7i − 0.363117i
\(76\) 0 0
\(77\) 2.66647e8i 0.864428i
\(78\) 0 0
\(79\) −4.25335e8 −1.22860 −0.614298 0.789074i \(-0.710561\pi\)
−0.614298 + 0.789074i \(0.710561\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) − 1.06618e8i − 0.246592i −0.992370 0.123296i \(-0.960654\pi\)
0.992370 0.123296i \(-0.0393465\pi\)
\(84\) 0 0
\(85\) 9.89341e8i 2.05570i
\(86\) 0 0
\(87\) 1.63981e8 0.306871
\(88\) 0 0
\(89\) −1.02792e9 −1.73662 −0.868308 0.496025i \(-0.834793\pi\)
−0.868308 + 0.496025i \(0.834793\pi\)
\(90\) 0 0
\(91\) 2.69248e8i 0.411591i
\(92\) 0 0
\(93\) 2.77976e8i 0.385331i
\(94\) 0 0
\(95\) 1.16091e9 1.46232
\(96\) 0 0
\(97\) 6.24259e8 0.715966 0.357983 0.933728i \(-0.383465\pi\)
0.357983 + 0.933728i \(0.383465\pi\)
\(98\) 0 0
\(99\) − 3.52359e8i − 0.368660i
\(100\) 0 0
\(101\) 5.25002e8i 0.502013i 0.967985 + 0.251006i \(0.0807615\pi\)
−0.967985 + 0.251006i \(0.919239\pi\)
\(102\) 0 0
\(103\) −1.09299e9 −0.956859 −0.478429 0.878126i \(-0.658794\pi\)
−0.478429 + 0.878126i \(0.658794\pi\)
\(104\) 0 0
\(105\) 7.17339e8 0.575933
\(106\) 0 0
\(107\) 2.50815e9i 1.84981i 0.380202 + 0.924904i \(0.375855\pi\)
−0.380202 + 0.924904i \(0.624145\pi\)
\(108\) 0 0
\(109\) − 1.99633e9i − 1.35461i −0.735703 0.677304i \(-0.763148\pi\)
0.735703 0.677304i \(-0.236852\pi\)
\(110\) 0 0
\(111\) −1.05368e8 −0.0658805
\(112\) 0 0
\(113\) −1.75634e9 −1.01334 −0.506670 0.862140i \(-0.669124\pi\)
−0.506670 + 0.862140i \(0.669124\pi\)
\(114\) 0 0
\(115\) 3.27796e8i 0.174768i
\(116\) 0 0
\(117\) − 3.55796e8i − 0.175535i
\(118\) 0 0
\(119\) −2.75392e9 −1.25889
\(120\) 0 0
\(121\) −5.26281e8 −0.223195
\(122\) 0 0
\(123\) 1.27488e7i 0.00502224i
\(124\) 0 0
\(125\) − 1.29269e9i − 0.473587i
\(126\) 0 0
\(127\) −4.66916e9 −1.59266 −0.796328 0.604865i \(-0.793227\pi\)
−0.796328 + 0.604865i \(0.793227\pi\)
\(128\) 0 0
\(129\) 1.70256e8 0.0541314
\(130\) 0 0
\(131\) − 5.48941e9i − 1.62857i −0.580468 0.814283i \(-0.697131\pi\)
0.580468 0.814283i \(-0.302869\pi\)
\(132\) 0 0
\(133\) 3.23150e9i 0.895513i
\(134\) 0 0
\(135\) −9.47921e8 −0.245624
\(136\) 0 0
\(137\) −3.05978e9 −0.742075 −0.371037 0.928618i \(-0.620998\pi\)
−0.371037 + 0.928618i \(0.620998\pi\)
\(138\) 0 0
\(139\) 8.06711e9i 1.83295i 0.400089 + 0.916476i \(0.368979\pi\)
−0.400089 + 0.916476i \(0.631021\pi\)
\(140\) 0 0
\(141\) 3.83399e9i 0.816892i
\(142\) 0 0
\(143\) −2.91236e9 −0.582416
\(144\) 0 0
\(145\) −3.61097e9 −0.678373
\(146\) 0 0
\(147\) − 1.27187e9i − 0.224654i
\(148\) 0 0
\(149\) − 6.06220e9i − 1.00761i −0.863817 0.503805i \(-0.831933\pi\)
0.863817 0.503805i \(-0.168067\pi\)
\(150\) 0 0
\(151\) 5.92499e9 0.927451 0.463726 0.885979i \(-0.346512\pi\)
0.463726 + 0.885979i \(0.346512\pi\)
\(152\) 0 0
\(153\) 3.63914e9 0.536892
\(154\) 0 0
\(155\) − 6.12124e9i − 0.851818i
\(156\) 0 0
\(157\) 5.68760e9i 0.747103i 0.927609 + 0.373551i \(0.121860\pi\)
−0.927609 + 0.373551i \(0.878140\pi\)
\(158\) 0 0
\(159\) 8.15956e9 1.01246
\(160\) 0 0
\(161\) −9.12448e8 −0.107027
\(162\) 0 0
\(163\) − 4.56247e9i − 0.506239i −0.967435 0.253120i \(-0.918543\pi\)
0.967435 0.253120i \(-0.0814566\pi\)
\(164\) 0 0
\(165\) 7.75920e9i 0.814966i
\(166\) 0 0
\(167\) −1.22032e9 −0.121409 −0.0607044 0.998156i \(-0.519335\pi\)
−0.0607044 + 0.998156i \(0.519335\pi\)
\(168\) 0 0
\(169\) 7.66373e9 0.722687
\(170\) 0 0
\(171\) − 4.27024e9i − 0.381918i
\(172\) 0 0
\(173\) − 2.24268e10i − 1.90353i −0.306821 0.951767i \(-0.599265\pi\)
0.306821 0.951767i \(-0.400735\pi\)
\(174\) 0 0
\(175\) −6.09901e9 −0.491573
\(176\) 0 0
\(177\) 1.46013e9 0.111818
\(178\) 0 0
\(179\) − 1.72440e10i − 1.25545i −0.778435 0.627725i \(-0.783986\pi\)
0.778435 0.627725i \(-0.216014\pi\)
\(180\) 0 0
\(181\) 2.90161e9i 0.200949i 0.994940 + 0.100474i \(0.0320361\pi\)
−0.994940 + 0.100474i \(0.967964\pi\)
\(182\) 0 0
\(183\) −4.44283e9 −0.292840
\(184\) 0 0
\(185\) 2.32029e9 0.145636
\(186\) 0 0
\(187\) − 2.97882e10i − 1.78138i
\(188\) 0 0
\(189\) − 2.63862e9i − 0.150418i
\(190\) 0 0
\(191\) −1.14739e9 −0.0623824 −0.0311912 0.999513i \(-0.509930\pi\)
−0.0311912 + 0.999513i \(0.509930\pi\)
\(192\) 0 0
\(193\) 3.19936e10 1.65980 0.829900 0.557913i \(-0.188398\pi\)
0.829900 + 0.557913i \(0.188398\pi\)
\(194\) 0 0
\(195\) 7.83489e9i 0.388040i
\(196\) 0 0
\(197\) − 3.38558e9i − 0.160153i −0.996789 0.0800764i \(-0.974484\pi\)
0.996789 0.0800764i \(-0.0255164\pi\)
\(198\) 0 0
\(199\) −7.28755e9 −0.329415 −0.164707 0.986342i \(-0.552668\pi\)
−0.164707 + 0.986342i \(0.552668\pi\)
\(200\) 0 0
\(201\) 1.72843e10 0.746913
\(202\) 0 0
\(203\) − 1.00515e10i − 0.415429i
\(204\) 0 0
\(205\) − 2.80739e8i − 0.0111022i
\(206\) 0 0
\(207\) 1.20575e9 0.0456446
\(208\) 0 0
\(209\) −3.49540e10 −1.26718
\(210\) 0 0
\(211\) − 1.58433e9i − 0.0550268i −0.999621 0.0275134i \(-0.991241\pi\)
0.999621 0.0275134i \(-0.00875889\pi\)
\(212\) 0 0
\(213\) 1.33338e10i 0.443859i
\(214\) 0 0
\(215\) −3.74917e9 −0.119664
\(216\) 0 0
\(217\) 1.70390e10 0.521645
\(218\) 0 0
\(219\) − 6.14389e9i − 0.180487i
\(220\) 0 0
\(221\) − 3.00787e10i − 0.848192i
\(222\) 0 0
\(223\) −2.82233e10 −0.764250 −0.382125 0.924111i \(-0.624808\pi\)
−0.382125 + 0.924111i \(0.624808\pi\)
\(224\) 0 0
\(225\) 8.05948e9 0.209646
\(226\) 0 0
\(227\) 1.00872e10i 0.252147i 0.992021 + 0.126073i \(0.0402375\pi\)
−0.992021 + 0.126073i \(0.959763\pi\)
\(228\) 0 0
\(229\) − 3.06314e9i − 0.0736051i −0.999323 0.0368025i \(-0.988283\pi\)
0.999323 0.0368025i \(-0.0117173\pi\)
\(230\) 0 0
\(231\) −2.15984e10 −0.499077
\(232\) 0 0
\(233\) −4.21496e10 −0.936896 −0.468448 0.883491i \(-0.655187\pi\)
−0.468448 + 0.883491i \(0.655187\pi\)
\(234\) 0 0
\(235\) − 8.44273e10i − 1.80583i
\(236\) 0 0
\(237\) − 3.44521e10i − 0.709330i
\(238\) 0 0
\(239\) −8.33007e10 −1.65142 −0.825711 0.564093i \(-0.809226\pi\)
−0.825711 + 0.564093i \(0.809226\pi\)
\(240\) 0 0
\(241\) 1.35832e10 0.259373 0.129686 0.991555i \(-0.458603\pi\)
0.129686 + 0.991555i \(0.458603\pi\)
\(242\) 0 0
\(243\) 3.48678e9i 0.0641500i
\(244\) 0 0
\(245\) 2.80075e10i 0.496623i
\(246\) 0 0
\(247\) −3.52950e10 −0.603360
\(248\) 0 0
\(249\) 8.63606e9 0.142370
\(250\) 0 0
\(251\) 9.40336e9i 0.149538i 0.997201 + 0.0747690i \(0.0238219\pi\)
−0.997201 + 0.0747690i \(0.976178\pi\)
\(252\) 0 0
\(253\) − 9.86963e9i − 0.151446i
\(254\) 0 0
\(255\) −8.01366e10 −1.18686
\(256\) 0 0
\(257\) −3.20439e10 −0.458190 −0.229095 0.973404i \(-0.573577\pi\)
−0.229095 + 0.973404i \(0.573577\pi\)
\(258\) 0 0
\(259\) 6.45874e9i 0.0891864i
\(260\) 0 0
\(261\) 1.32824e10i 0.177172i
\(262\) 0 0
\(263\) −8.93217e10 −1.15121 −0.575607 0.817726i \(-0.695234\pi\)
−0.575607 + 0.817726i \(0.695234\pi\)
\(264\) 0 0
\(265\) −1.79680e11 −2.23817
\(266\) 0 0
\(267\) − 8.32615e10i − 1.00264i
\(268\) 0 0
\(269\) − 3.70480e10i − 0.431400i −0.976460 0.215700i \(-0.930797\pi\)
0.976460 0.215700i \(-0.0692033\pi\)
\(270\) 0 0
\(271\) −1.10018e11 −1.23909 −0.619546 0.784960i \(-0.712683\pi\)
−0.619546 + 0.784960i \(0.712683\pi\)
\(272\) 0 0
\(273\) −2.18091e10 −0.237632
\(274\) 0 0
\(275\) − 6.59709e10i − 0.695593i
\(276\) 0 0
\(277\) 8.60128e10i 0.877818i 0.898532 + 0.438909i \(0.144635\pi\)
−0.898532 + 0.438909i \(0.855365\pi\)
\(278\) 0 0
\(279\) −2.25160e10 −0.222471
\(280\) 0 0
\(281\) 1.40155e11 1.34101 0.670503 0.741907i \(-0.266078\pi\)
0.670503 + 0.741907i \(0.266078\pi\)
\(282\) 0 0
\(283\) − 6.86732e10i − 0.636426i −0.948019 0.318213i \(-0.896917\pi\)
0.948019 0.318213i \(-0.103083\pi\)
\(284\) 0 0
\(285\) 9.40339e10i 0.844272i
\(286\) 0 0
\(287\) 7.81461e8 0.00679890
\(288\) 0 0
\(289\) 1.89063e11 1.59428
\(290\) 0 0
\(291\) 5.05650e10i 0.413363i
\(292\) 0 0
\(293\) − 3.90112e10i − 0.309233i −0.987975 0.154616i \(-0.950586\pi\)
0.987975 0.154616i \(-0.0494141\pi\)
\(294\) 0 0
\(295\) −3.21531e10 −0.247186
\(296\) 0 0
\(297\) 2.85410e10 0.212846
\(298\) 0 0
\(299\) − 9.96590e9i − 0.0721101i
\(300\) 0 0
\(301\) − 1.04361e10i − 0.0732809i
\(302\) 0 0
\(303\) −4.25251e10 −0.289837
\(304\) 0 0
\(305\) 9.78346e10 0.647356
\(306\) 0 0
\(307\) − 1.94107e11i − 1.24715i −0.781765 0.623573i \(-0.785680\pi\)
0.781765 0.623573i \(-0.214320\pi\)
\(308\) 0 0
\(309\) − 8.85320e10i − 0.552443i
\(310\) 0 0
\(311\) −1.41999e11 −0.860726 −0.430363 0.902656i \(-0.641614\pi\)
−0.430363 + 0.902656i \(0.641614\pi\)
\(312\) 0 0
\(313\) −1.96211e11 −1.15551 −0.577756 0.816209i \(-0.696072\pi\)
−0.577756 + 0.816209i \(0.696072\pi\)
\(314\) 0 0
\(315\) 5.81044e10i 0.332515i
\(316\) 0 0
\(317\) 3.32729e11i 1.85065i 0.379177 + 0.925324i \(0.376207\pi\)
−0.379177 + 0.925324i \(0.623793\pi\)
\(318\) 0 0
\(319\) 1.08723e11 0.587847
\(320\) 0 0
\(321\) −2.03160e11 −1.06799
\(322\) 0 0
\(323\) − 3.61003e11i − 1.84544i
\(324\) 0 0
\(325\) − 6.66143e10i − 0.331202i
\(326\) 0 0
\(327\) 1.61703e11 0.782084
\(328\) 0 0
\(329\) 2.35011e11 1.10587
\(330\) 0 0
\(331\) − 2.82155e11i − 1.29200i −0.763338 0.646000i \(-0.776441\pi\)
0.763338 0.646000i \(-0.223559\pi\)
\(332\) 0 0
\(333\) − 8.53484e9i − 0.0380361i
\(334\) 0 0
\(335\) −3.80614e11 −1.65114
\(336\) 0 0
\(337\) −2.98725e11 −1.26164 −0.630822 0.775928i \(-0.717282\pi\)
−0.630822 + 0.775928i \(0.717282\pi\)
\(338\) 0 0
\(339\) − 1.42263e11i − 0.585052i
\(340\) 0 0
\(341\) 1.84305e11i 0.738146i
\(342\) 0 0
\(343\) −2.78318e11 −1.08572
\(344\) 0 0
\(345\) −2.65515e10 −0.100903
\(346\) 0 0
\(347\) − 4.84025e11i − 1.79219i −0.443858 0.896097i \(-0.646391\pi\)
0.443858 0.896097i \(-0.353609\pi\)
\(348\) 0 0
\(349\) 2.44115e10i 0.0880806i 0.999030 + 0.0440403i \(0.0140230\pi\)
−0.999030 + 0.0440403i \(0.985977\pi\)
\(350\) 0 0
\(351\) 2.88194e10 0.101345
\(352\) 0 0
\(353\) −3.28670e11 −1.12661 −0.563306 0.826248i \(-0.690471\pi\)
−0.563306 + 0.826248i \(0.690471\pi\)
\(354\) 0 0
\(355\) − 2.93620e11i − 0.981201i
\(356\) 0 0
\(357\) − 2.23067e11i − 0.726823i
\(358\) 0 0
\(359\) 4.58376e11 1.45646 0.728228 0.685335i \(-0.240344\pi\)
0.728228 + 0.685335i \(0.240344\pi\)
\(360\) 0 0
\(361\) −1.00920e11 −0.312749
\(362\) 0 0
\(363\) − 4.26288e10i − 0.128862i
\(364\) 0 0
\(365\) 1.35293e11i 0.398986i
\(366\) 0 0
\(367\) −5.88051e11 −1.69207 −0.846034 0.533128i \(-0.821016\pi\)
−0.846034 + 0.533128i \(0.821016\pi\)
\(368\) 0 0
\(369\) −1.03265e9 −0.00289959
\(370\) 0 0
\(371\) − 5.00154e11i − 1.37063i
\(372\) 0 0
\(373\) 6.61776e9i 0.0177020i 0.999961 + 0.00885098i \(0.00281739\pi\)
−0.999961 + 0.00885098i \(0.997183\pi\)
\(374\) 0 0
\(375\) 1.04708e11 0.273426
\(376\) 0 0
\(377\) 1.09784e11 0.279899
\(378\) 0 0
\(379\) 5.36022e11i 1.33446i 0.744851 + 0.667231i \(0.232521\pi\)
−0.744851 + 0.667231i \(0.767479\pi\)
\(380\) 0 0
\(381\) − 3.78202e11i − 0.919520i
\(382\) 0 0
\(383\) −5.39762e11 −1.28176 −0.640882 0.767640i \(-0.721431\pi\)
−0.640882 + 0.767640i \(0.721431\pi\)
\(384\) 0 0
\(385\) 4.75613e11 1.10327
\(386\) 0 0
\(387\) 1.37907e10i 0.0312528i
\(388\) 0 0
\(389\) − 1.54973e11i − 0.343150i −0.985171 0.171575i \(-0.945114\pi\)
0.985171 0.171575i \(-0.0548855\pi\)
\(390\) 0 0
\(391\) 1.01933e11 0.220556
\(392\) 0 0
\(393\) 4.44643e11 0.940253
\(394\) 0 0
\(395\) 7.58662e11i 1.56806i
\(396\) 0 0
\(397\) − 4.94662e11i − 0.999427i −0.866191 0.499714i \(-0.833439\pi\)
0.866191 0.499714i \(-0.166561\pi\)
\(398\) 0 0
\(399\) −2.61751e11 −0.517024
\(400\) 0 0
\(401\) 4.84391e11 0.935506 0.467753 0.883859i \(-0.345064\pi\)
0.467753 + 0.883859i \(0.345064\pi\)
\(402\) 0 0
\(403\) 1.86103e11i 0.351463i
\(404\) 0 0
\(405\) − 7.67816e10i − 0.141811i
\(406\) 0 0
\(407\) −6.98619e10 −0.126202
\(408\) 0 0
\(409\) −3.98035e11 −0.703341 −0.351671 0.936124i \(-0.614386\pi\)
−0.351671 + 0.936124i \(0.614386\pi\)
\(410\) 0 0
\(411\) − 2.47842e11i − 0.428437i
\(412\) 0 0
\(413\) − 8.95011e10i − 0.151375i
\(414\) 0 0
\(415\) −1.90172e11 −0.314725
\(416\) 0 0
\(417\) −6.53436e11 −1.05826
\(418\) 0 0
\(419\) 6.97991e10i 0.110633i 0.998469 + 0.0553167i \(0.0176169\pi\)
−0.998469 + 0.0553167i \(0.982383\pi\)
\(420\) 0 0
\(421\) − 3.17674e11i − 0.492847i −0.969162 0.246423i \(-0.920745\pi\)
0.969162 0.246423i \(-0.0792554\pi\)
\(422\) 0 0
\(423\) −3.10553e11 −0.471633
\(424\) 0 0
\(425\) 6.81343e11 1.01301
\(426\) 0 0
\(427\) 2.72331e11i 0.396435i
\(428\) 0 0
\(429\) − 2.35901e11i − 0.336258i
\(430\) 0 0
\(431\) −7.99506e11 −1.11603 −0.558013 0.829832i \(-0.688436\pi\)
−0.558013 + 0.829832i \(0.688436\pi\)
\(432\) 0 0
\(433\) 6.63025e11 0.906430 0.453215 0.891401i \(-0.350277\pi\)
0.453215 + 0.891401i \(0.350277\pi\)
\(434\) 0 0
\(435\) − 2.92489e11i − 0.391659i
\(436\) 0 0
\(437\) − 1.19610e11i − 0.156892i
\(438\) 0 0
\(439\) 4.31042e10 0.0553898 0.0276949 0.999616i \(-0.491183\pi\)
0.0276949 + 0.999616i \(0.491183\pi\)
\(440\) 0 0
\(441\) 1.03021e11 0.129704
\(442\) 0 0
\(443\) − 6.49651e11i − 0.801426i −0.916204 0.400713i \(-0.868762\pi\)
0.916204 0.400713i \(-0.131238\pi\)
\(444\) 0 0
\(445\) 1.83348e12i 2.21644i
\(446\) 0 0
\(447\) 4.91039e11 0.581744
\(448\) 0 0
\(449\) 2.28210e10 0.0264988 0.0132494 0.999912i \(-0.495782\pi\)
0.0132494 + 0.999912i \(0.495782\pi\)
\(450\) 0 0
\(451\) 8.45279e9i 0.00962067i
\(452\) 0 0
\(453\) 4.79924e11i 0.535464i
\(454\) 0 0
\(455\) 4.80253e11 0.525313
\(456\) 0 0
\(457\) −1.44749e12 −1.55237 −0.776183 0.630508i \(-0.782847\pi\)
−0.776183 + 0.630508i \(0.782847\pi\)
\(458\) 0 0
\(459\) 2.94770e11i 0.309975i
\(460\) 0 0
\(461\) − 1.28905e11i − 0.132927i −0.997789 0.0664637i \(-0.978828\pi\)
0.997789 0.0664637i \(-0.0211717\pi\)
\(462\) 0 0
\(463\) −1.71037e12 −1.72972 −0.864860 0.502013i \(-0.832593\pi\)
−0.864860 + 0.502013i \(0.832593\pi\)
\(464\) 0 0
\(465\) 4.95820e11 0.491797
\(466\) 0 0
\(467\) 1.91054e12i 1.85879i 0.369092 + 0.929393i \(0.379669\pi\)
−0.369092 + 0.929393i \(0.620331\pi\)
\(468\) 0 0
\(469\) − 1.05947e12i − 1.01114i
\(470\) 0 0
\(471\) −4.60695e11 −0.431340
\(472\) 0 0
\(473\) 1.12884e11 0.103695
\(474\) 0 0
\(475\) − 7.99502e11i − 0.720606i
\(476\) 0 0
\(477\) 6.60924e11i 0.584546i
\(478\) 0 0
\(479\) 1.92498e12 1.67077 0.835385 0.549665i \(-0.185244\pi\)
0.835385 + 0.549665i \(0.185244\pi\)
\(480\) 0 0
\(481\) −7.05433e10 −0.0600901
\(482\) 0 0
\(483\) − 7.39083e10i − 0.0617918i
\(484\) 0 0
\(485\) − 1.11348e12i − 0.913785i
\(486\) 0 0
\(487\) 2.29897e12 1.85205 0.926026 0.377460i \(-0.123202\pi\)
0.926026 + 0.377460i \(0.123202\pi\)
\(488\) 0 0
\(489\) 3.69560e11 0.292277
\(490\) 0 0
\(491\) − 5.35110e11i − 0.415505i −0.978181 0.207753i \(-0.933385\pi\)
0.978181 0.207753i \(-0.0666149\pi\)
\(492\) 0 0
\(493\) 1.12289e12i 0.856101i
\(494\) 0 0
\(495\) −6.28495e11 −0.470521
\(496\) 0 0
\(497\) 8.17317e11 0.600878
\(498\) 0 0
\(499\) 1.06412e12i 0.768312i 0.923268 + 0.384156i \(0.125508\pi\)
−0.923268 + 0.384156i \(0.874492\pi\)
\(500\) 0 0
\(501\) − 9.88461e10i − 0.0700954i
\(502\) 0 0
\(503\) 2.01345e12 1.40244 0.701220 0.712945i \(-0.252639\pi\)
0.701220 + 0.712945i \(0.252639\pi\)
\(504\) 0 0
\(505\) 9.36436e11 0.640718
\(506\) 0 0
\(507\) 6.20762e11i 0.417243i
\(508\) 0 0
\(509\) − 1.34279e12i − 0.886701i −0.896348 0.443350i \(-0.853790\pi\)
0.896348 0.443350i \(-0.146210\pi\)
\(510\) 0 0
\(511\) −3.76600e11 −0.244336
\(512\) 0 0
\(513\) 3.45889e11 0.220500
\(514\) 0 0
\(515\) 1.94954e12i 1.22124i
\(516\) 0 0
\(517\) 2.54203e12i 1.56485i
\(518\) 0 0
\(519\) 1.81657e12 1.09901
\(520\) 0 0
\(521\) −1.74500e12 −1.03759 −0.518796 0.854898i \(-0.673619\pi\)
−0.518796 + 0.854898i \(0.673619\pi\)
\(522\) 0 0
\(523\) − 1.62381e12i − 0.949026i −0.880249 0.474513i \(-0.842624\pi\)
0.880249 0.474513i \(-0.157376\pi\)
\(524\) 0 0
\(525\) − 4.94020e11i − 0.283810i
\(526\) 0 0
\(527\) −1.90349e12 −1.07499
\(528\) 0 0
\(529\) −1.76738e12 −0.981249
\(530\) 0 0
\(531\) 1.18270e11i 0.0645581i
\(532\) 0 0
\(533\) 8.53524e9i 0.00458082i
\(534\) 0 0
\(535\) 4.47374e12 2.36091
\(536\) 0 0
\(537\) 1.39676e12 0.724835
\(538\) 0 0
\(539\) − 8.43280e11i − 0.430351i
\(540\) 0 0
\(541\) − 2.26145e12i − 1.13501i −0.823370 0.567505i \(-0.807909\pi\)
0.823370 0.567505i \(-0.192091\pi\)
\(542\) 0 0
\(543\) −2.35031e11 −0.116018
\(544\) 0 0
\(545\) −3.56082e12 −1.72888
\(546\) 0 0
\(547\) − 2.08228e12i − 0.994480i −0.867613 0.497240i \(-0.834347\pi\)
0.867613 0.497240i \(-0.165653\pi\)
\(548\) 0 0
\(549\) − 3.59870e11i − 0.169071i
\(550\) 0 0
\(551\) 1.31762e12 0.608986
\(552\) 0 0
\(553\) −2.11180e12 −0.960263
\(554\) 0 0
\(555\) 1.87944e11i 0.0840832i
\(556\) 0 0
\(557\) − 1.69657e12i − 0.746831i −0.927664 0.373415i \(-0.878187\pi\)
0.927664 0.373415i \(-0.121813\pi\)
\(558\) 0 0
\(559\) 1.13985e11 0.0493736
\(560\) 0 0
\(561\) 2.41284e12 1.02848
\(562\) 0 0
\(563\) − 1.48379e11i − 0.0622423i −0.999516 0.0311212i \(-0.990092\pi\)
0.999516 0.0311212i \(-0.00990778\pi\)
\(564\) 0 0
\(565\) 3.13275e12i 1.29332i
\(566\) 0 0
\(567\) 2.13728e11 0.0868437
\(568\) 0 0
\(569\) −9.31992e11 −0.372741 −0.186370 0.982480i \(-0.559672\pi\)
−0.186370 + 0.982480i \(0.559672\pi\)
\(570\) 0 0
\(571\) 4.97476e12i 1.95844i 0.202807 + 0.979219i \(0.434993\pi\)
−0.202807 + 0.979219i \(0.565007\pi\)
\(572\) 0 0
\(573\) − 9.29388e10i − 0.0360165i
\(574\) 0 0
\(575\) 2.25748e11 0.0861228
\(576\) 0 0
\(577\) 7.93519e11 0.298034 0.149017 0.988835i \(-0.452389\pi\)
0.149017 + 0.988835i \(0.452389\pi\)
\(578\) 0 0
\(579\) 2.59148e12i 0.958286i
\(580\) 0 0
\(581\) − 5.29362e11i − 0.192735i
\(582\) 0 0
\(583\) 5.40999e12 1.93949
\(584\) 0 0
\(585\) −6.34626e11 −0.224035
\(586\) 0 0
\(587\) 4.52536e11i 0.157319i 0.996902 + 0.0786596i \(0.0250640\pi\)
−0.996902 + 0.0786596i \(0.974936\pi\)
\(588\) 0 0
\(589\) 2.23359e12i 0.764690i
\(590\) 0 0
\(591\) 2.74232e11 0.0924643
\(592\) 0 0
\(593\) −2.09626e12 −0.696144 −0.348072 0.937468i \(-0.613164\pi\)
−0.348072 + 0.937468i \(0.613164\pi\)
\(594\) 0 0
\(595\) 4.91211e12i 1.60672i
\(596\) 0 0
\(597\) − 5.90292e11i − 0.190188i
\(598\) 0 0
\(599\) 2.30531e12 0.731659 0.365829 0.930682i \(-0.380785\pi\)
0.365829 + 0.930682i \(0.380785\pi\)
\(600\) 0 0
\(601\) 2.13269e12 0.666794 0.333397 0.942787i \(-0.391805\pi\)
0.333397 + 0.942787i \(0.391805\pi\)
\(602\) 0 0
\(603\) 1.40003e12i 0.431230i
\(604\) 0 0
\(605\) 9.38718e11i 0.284863i
\(606\) 0 0
\(607\) 3.72283e12 1.11308 0.556538 0.830822i \(-0.312130\pi\)
0.556538 + 0.830822i \(0.312130\pi\)
\(608\) 0 0
\(609\) 8.14168e11 0.239848
\(610\) 0 0
\(611\) 2.56682e12i 0.745093i
\(612\) 0 0
\(613\) 6.70140e12i 1.91687i 0.285305 + 0.958437i \(0.407905\pi\)
−0.285305 + 0.958437i \(0.592095\pi\)
\(614\) 0 0
\(615\) 2.27398e10 0.00640987
\(616\) 0 0
\(617\) 1.02429e12 0.284537 0.142268 0.989828i \(-0.454560\pi\)
0.142268 + 0.989828i \(0.454560\pi\)
\(618\) 0 0
\(619\) 3.15002e12i 0.862393i 0.902258 + 0.431196i \(0.141908\pi\)
−0.902258 + 0.431196i \(0.858092\pi\)
\(620\) 0 0
\(621\) 9.76655e10i 0.0263529i
\(622\) 0 0
\(623\) −5.10365e12 −1.35733
\(624\) 0 0
\(625\) −4.70495e12 −1.23338
\(626\) 0 0
\(627\) − 2.83127e12i − 0.731607i
\(628\) 0 0
\(629\) − 7.21530e11i − 0.183792i
\(630\) 0 0
\(631\) −1.99605e12 −0.501233 −0.250617 0.968086i \(-0.580633\pi\)
−0.250617 + 0.968086i \(0.580633\pi\)
\(632\) 0 0
\(633\) 1.28331e11 0.0317697
\(634\) 0 0
\(635\) 8.32829e12i 2.03270i
\(636\) 0 0
\(637\) − 8.51506e11i − 0.204909i
\(638\) 0 0
\(639\) −1.08004e12 −0.256262
\(640\) 0 0
\(641\) 7.10597e12 1.66250 0.831252 0.555896i \(-0.187625\pi\)
0.831252 + 0.555896i \(0.187625\pi\)
\(642\) 0 0
\(643\) 1.13065e12i 0.260843i 0.991459 + 0.130422i \(0.0416331\pi\)
−0.991459 + 0.130422i \(0.958367\pi\)
\(644\) 0 0
\(645\) − 3.03683e11i − 0.0690878i
\(646\) 0 0
\(647\) 5.14088e12 1.15337 0.576685 0.816967i \(-0.304346\pi\)
0.576685 + 0.816967i \(0.304346\pi\)
\(648\) 0 0
\(649\) 9.68102e11 0.214200
\(650\) 0 0
\(651\) 1.38016e12i 0.301172i
\(652\) 0 0
\(653\) − 5.23628e11i − 0.112697i −0.998411 0.0563486i \(-0.982054\pi\)
0.998411 0.0563486i \(-0.0179458\pi\)
\(654\) 0 0
\(655\) −9.79136e12 −2.07854
\(656\) 0 0
\(657\) 4.97655e11 0.104204
\(658\) 0 0
\(659\) − 2.73828e12i − 0.565579i −0.959182 0.282789i \(-0.908740\pi\)
0.959182 0.282789i \(-0.0912598\pi\)
\(660\) 0 0
\(661\) 1.53582e12i 0.312920i 0.987684 + 0.156460i \(0.0500082\pi\)
−0.987684 + 0.156460i \(0.949992\pi\)
\(662\) 0 0
\(663\) 2.43638e12 0.489704
\(664\) 0 0
\(665\) 5.76396e12 1.14294
\(666\) 0 0
\(667\) 3.72043e11i 0.0727825i
\(668\) 0 0
\(669\) − 2.28609e12i − 0.441240i
\(670\) 0 0
\(671\) −2.94571e12 −0.560969
\(672\) 0 0
\(673\) −9.06140e12 −1.70266 −0.851329 0.524632i \(-0.824203\pi\)
−0.851329 + 0.524632i \(0.824203\pi\)
\(674\) 0 0
\(675\) 6.52818e11i 0.121039i
\(676\) 0 0
\(677\) − 8.23814e12i − 1.50723i −0.657315 0.753616i \(-0.728308\pi\)
0.657315 0.753616i \(-0.271692\pi\)
\(678\) 0 0
\(679\) 3.09947e12 0.559594
\(680\) 0 0
\(681\) −8.17061e11 −0.145577
\(682\) 0 0
\(683\) 1.99109e12i 0.350104i 0.984559 + 0.175052i \(0.0560094\pi\)
−0.984559 + 0.175052i \(0.943991\pi\)
\(684\) 0 0
\(685\) 5.45767e12i 0.947109i
\(686\) 0 0
\(687\) 2.48115e11 0.0424959
\(688\) 0 0
\(689\) 5.46276e12 0.923476
\(690\) 0 0
\(691\) 6.48604e11i 0.108225i 0.998535 + 0.0541126i \(0.0172330\pi\)
−0.998535 + 0.0541126i \(0.982767\pi\)
\(692\) 0 0
\(693\) − 1.74947e12i − 0.288143i
\(694\) 0 0
\(695\) 1.43891e13 2.33939
\(696\) 0 0
\(697\) −8.72999e10 −0.0140109
\(698\) 0 0
\(699\) − 3.41412e12i − 0.540917i
\(700\) 0 0
\(701\) − 9.56476e12i − 1.49604i −0.663677 0.748019i \(-0.731005\pi\)
0.663677 0.748019i \(-0.268995\pi\)
\(702\) 0 0
\(703\) −8.46657e11 −0.130740
\(704\) 0 0
\(705\) 6.83861e12 1.04260
\(706\) 0 0
\(707\) 2.60665e12i 0.392370i
\(708\) 0 0
\(709\) − 5.02818e12i − 0.747314i −0.927567 0.373657i \(-0.878104\pi\)
0.927567 0.373657i \(-0.121896\pi\)
\(710\) 0 0
\(711\) 2.79062e12 0.409532
\(712\) 0 0
\(713\) −6.30679e11 −0.0913914
\(714\) 0 0
\(715\) 5.19473e12i 0.743336i
\(716\) 0 0
\(717\) − 6.74736e12i − 0.953449i
\(718\) 0 0
\(719\) 3.87923e11 0.0541334 0.0270667 0.999634i \(-0.491383\pi\)
0.0270667 + 0.999634i \(0.491383\pi\)
\(720\) 0 0
\(721\) −5.42672e12 −0.747874
\(722\) 0 0
\(723\) 1.10024e12i 0.149749i
\(724\) 0 0
\(725\) 2.48682e12i 0.334290i
\(726\) 0 0
\(727\) −1.21943e12 −0.161902 −0.0809510 0.996718i \(-0.525796\pi\)
−0.0809510 + 0.996718i \(0.525796\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) 1.16586e12i 0.151014i
\(732\) 0 0
\(733\) 1.25774e12i 0.160925i 0.996758 + 0.0804626i \(0.0256397\pi\)
−0.996758 + 0.0804626i \(0.974360\pi\)
\(734\) 0 0
\(735\) −2.26861e12 −0.286725
\(736\) 0 0
\(737\) 1.14599e13 1.43080
\(738\) 0 0
\(739\) − 1.08922e13i − 1.34343i −0.740812 0.671713i \(-0.765559\pi\)
0.740812 0.671713i \(-0.234441\pi\)
\(740\) 0 0
\(741\) − 2.85889e12i − 0.348350i
\(742\) 0 0
\(743\) 7.60817e12 0.915863 0.457932 0.888987i \(-0.348590\pi\)
0.457932 + 0.888987i \(0.348590\pi\)
\(744\) 0 0
\(745\) −1.08130e13 −1.28601
\(746\) 0 0
\(747\) 6.99521e11i 0.0821974i
\(748\) 0 0
\(749\) 1.24530e13i 1.44580i
\(750\) 0 0
\(751\) 1.34422e13 1.54202 0.771010 0.636823i \(-0.219752\pi\)
0.771010 + 0.636823i \(0.219752\pi\)
\(752\) 0 0
\(753\) −7.61672e11 −0.0863358
\(754\) 0 0
\(755\) − 1.05683e13i − 1.18370i
\(756\) 0 0
\(757\) − 1.22512e12i − 0.135597i −0.997699 0.0677983i \(-0.978403\pi\)
0.997699 0.0677983i \(-0.0215974\pi\)
\(758\) 0 0
\(759\) 7.99440e11 0.0874375
\(760\) 0 0
\(761\) 1.86163e12 0.201216 0.100608 0.994926i \(-0.467921\pi\)
0.100608 + 0.994926i \(0.467921\pi\)
\(762\) 0 0
\(763\) − 9.91186e12i − 1.05875i
\(764\) 0 0
\(765\) − 6.49106e12i − 0.685235i
\(766\) 0 0
\(767\) 9.77545e11 0.101990
\(768\) 0 0
\(769\) −1.61673e13 −1.66713 −0.833564 0.552422i \(-0.813704\pi\)
−0.833564 + 0.552422i \(0.813704\pi\)
\(770\) 0 0
\(771\) − 2.59555e12i − 0.264536i
\(772\) 0 0
\(773\) 1.51715e13i 1.52834i 0.645012 + 0.764172i \(0.276852\pi\)
−0.645012 + 0.764172i \(0.723148\pi\)
\(774\) 0 0
\(775\) −4.21560e12 −0.419761
\(776\) 0 0
\(777\) −5.23158e11 −0.0514918
\(778\) 0 0
\(779\) 1.02439e11i 0.00996664i
\(780\) 0 0
\(781\) 8.84063e12i 0.850263i
\(782\) 0 0
\(783\) −1.07588e12 −0.102290
\(784\) 0 0
\(785\) 1.01449e13 0.953526
\(786\) 0 0
\(787\) 1.28839e13i 1.19718i 0.801055 + 0.598590i \(0.204272\pi\)
−0.801055 + 0.598590i \(0.795728\pi\)
\(788\) 0 0
\(789\) − 7.23506e12i − 0.664654i
\(790\) 0 0
\(791\) −8.72027e12 −0.792019
\(792\) 0 0
\(793\) −2.97444e12 −0.267101
\(794\) 0 0
\(795\) − 1.45540e13i − 1.29221i
\(796\) 0 0
\(797\) 4.41358e12i 0.387462i 0.981055 + 0.193731i \(0.0620589\pi\)
−0.981055 + 0.193731i \(0.937941\pi\)
\(798\) 0 0
\(799\) −2.62539e13 −2.27894
\(800\) 0 0
\(801\) 6.74418e12 0.578872
\(802\) 0 0
\(803\) − 4.07355e12i − 0.345743i
\(804\) 0 0
\(805\) 1.62752e12i 0.136598i
\(806\) 0 0
\(807\) 3.00089e12 0.249069
\(808\) 0 0
\(809\) −1.24867e13 −1.02490 −0.512449 0.858718i \(-0.671262\pi\)
−0.512449 + 0.858718i \(0.671262\pi\)
\(810\) 0 0
\(811\) − 5.75694e12i − 0.467302i −0.972321 0.233651i \(-0.924933\pi\)
0.972321 0.233651i \(-0.0750674\pi\)
\(812\) 0 0
\(813\) − 8.91150e12i − 0.715391i
\(814\) 0 0
\(815\) −8.13799e12 −0.646112
\(816\) 0 0
\(817\) 1.36804e12 0.107424
\(818\) 0 0
\(819\) − 1.76654e12i − 0.137197i
\(820\) 0 0
\(821\) − 1.15601e13i − 0.888009i −0.896025 0.444004i \(-0.853557\pi\)
0.896025 0.444004i \(-0.146443\pi\)
\(822\) 0 0
\(823\) 1.28159e13 0.973758 0.486879 0.873469i \(-0.338135\pi\)
0.486879 + 0.873469i \(0.338135\pi\)
\(824\) 0 0
\(825\) 5.34364e12 0.401601
\(826\) 0 0
\(827\) − 1.88099e13i − 1.39834i −0.714957 0.699169i \(-0.753554\pi\)
0.714957 0.699169i \(-0.246446\pi\)
\(828\) 0 0
\(829\) 1.05174e13i 0.773413i 0.922203 + 0.386707i \(0.126387\pi\)
−0.922203 + 0.386707i \(0.873613\pi\)
\(830\) 0 0
\(831\) −6.96704e12 −0.506808
\(832\) 0 0
\(833\) 8.70935e12 0.626734
\(834\) 0 0
\(835\) 2.17667e12i 0.154954i
\(836\) 0 0
\(837\) − 1.82380e12i − 0.128444i
\(838\) 0 0
\(839\) −2.13441e13 −1.48713 −0.743564 0.668665i \(-0.766866\pi\)
−0.743564 + 0.668665i \(0.766866\pi\)
\(840\) 0 0
\(841\) 1.04087e13 0.717491
\(842\) 0 0
\(843\) 1.13526e13i 0.774230i
\(844\) 0 0
\(845\) − 1.36696e13i − 0.922364i
\(846\) 0 0
\(847\) −2.61300e12 −0.174447
\(848\) 0 0
\(849\) 5.56253e12 0.367441
\(850\) 0 0
\(851\) − 2.39063e11i − 0.0156253i
\(852\) 0 0
\(853\) 1.81634e13i 1.17470i 0.809334 + 0.587348i \(0.199828\pi\)
−0.809334 + 0.587348i \(0.800172\pi\)
\(854\) 0 0
\(855\) −7.61674e12 −0.487441
\(856\) 0 0
\(857\) 2.86744e13 1.81585 0.907927 0.419128i \(-0.137664\pi\)
0.907927 + 0.419128i \(0.137664\pi\)
\(858\) 0 0
\(859\) − 8.97255e12i − 0.562273i −0.959668 0.281136i \(-0.909289\pi\)
0.959668 0.281136i \(-0.0907113\pi\)
\(860\) 0 0
\(861\) 6.32983e10i 0.00392535i
\(862\) 0 0
\(863\) −1.96293e13 −1.20463 −0.602317 0.798257i \(-0.705756\pi\)
−0.602317 + 0.798257i \(0.705756\pi\)
\(864\) 0 0
\(865\) −4.00023e13 −2.42948
\(866\) 0 0
\(867\) 1.53141e13i 0.920459i
\(868\) 0 0
\(869\) − 2.28426e13i − 1.35880i
\(870\) 0 0
\(871\) 1.15717e13 0.681265
\(872\) 0 0
\(873\) −4.09576e12 −0.238655
\(874\) 0 0
\(875\) − 6.41825e12i − 0.370152i
\(876\) 0 0
\(877\) 1.82741e13i 1.04313i 0.853211 + 0.521565i \(0.174652\pi\)
−0.853211 + 0.521565i \(0.825348\pi\)
\(878\) 0 0
\(879\) 3.15991e12 0.178535
\(880\) 0 0
\(881\) 1.46604e12 0.0819888 0.0409944 0.999159i \(-0.486947\pi\)
0.0409944 + 0.999159i \(0.486947\pi\)
\(882\) 0 0
\(883\) 9.47248e11i 0.0524373i 0.999656 + 0.0262187i \(0.00834661\pi\)
−0.999656 + 0.0262187i \(0.991653\pi\)
\(884\) 0 0
\(885\) − 2.60440e12i − 0.142713i
\(886\) 0 0
\(887\) −5.51475e12 −0.299136 −0.149568 0.988751i \(-0.547788\pi\)
−0.149568 + 0.988751i \(0.547788\pi\)
\(888\) 0 0
\(889\) −2.31825e13 −1.24481
\(890\) 0 0
\(891\) 2.31182e12i 0.122887i
\(892\) 0 0
\(893\) 3.08069e13i 1.62112i
\(894\) 0 0
\(895\) −3.07578e13 −1.60233
\(896\) 0 0
\(897\) 8.07238e11 0.0416328
\(898\) 0 0
\(899\) − 6.94751e12i − 0.354741i
\(900\) 0 0
\(901\) 5.58741e13i 2.82455i
\(902\) 0 0
\(903\) 8.45327e11 0.0423087
\(904\) 0 0
\(905\) 5.17555e12 0.256471
\(906\) 0 0
\(907\) − 3.24027e13i − 1.58982i −0.606725 0.794912i \(-0.707517\pi\)
0.606725 0.794912i \(-0.292483\pi\)
\(908\) 0 0
\(909\) − 3.44454e12i − 0.167338i
\(910\) 0 0
\(911\) −2.85811e12 −0.137482 −0.0687410 0.997635i \(-0.521898\pi\)
−0.0687410 + 0.997635i \(0.521898\pi\)
\(912\) 0 0
\(913\) 5.72592e12 0.272726
\(914\) 0 0
\(915\) 7.92460e12i 0.373751i
\(916\) 0 0
\(917\) − 2.72551e13i − 1.27288i
\(918\) 0 0
\(919\) −1.84172e13 −0.851734 −0.425867 0.904786i \(-0.640031\pi\)
−0.425867 + 0.904786i \(0.640031\pi\)
\(920\) 0 0
\(921\) 1.57226e13 0.720040
\(922\) 0 0
\(923\) 8.92686e12i 0.404847i
\(924\) 0 0
\(925\) − 1.59795e12i − 0.0717670i
\(926\) 0 0
\(927\) 7.17109e12 0.318953
\(928\) 0 0
\(929\) −3.37811e13 −1.48800 −0.744000 0.668179i \(-0.767074\pi\)
−0.744000 + 0.668179i \(0.767074\pi\)
\(930\) 0 0
\(931\) − 1.02197e13i − 0.445826i
\(932\) 0 0
\(933\) − 1.15020e13i − 0.496940i
\(934\) 0 0
\(935\) −5.31326e13 −2.27357
\(936\) 0 0
\(937\) −4.50959e13 −1.91121 −0.955606 0.294649i \(-0.904797\pi\)
−0.955606 + 0.294649i \(0.904797\pi\)
\(938\) 0 0
\(939\) − 1.58931e13i − 0.667136i
\(940\) 0 0
\(941\) − 2.11150e13i − 0.877887i −0.898515 0.438943i \(-0.855353\pi\)
0.898515 0.438943i \(-0.144647\pi\)
\(942\) 0 0
\(943\) −2.89249e10 −0.00119116
\(944\) 0 0
\(945\) −4.70646e12 −0.191978
\(946\) 0 0
\(947\) − 1.89150e12i − 0.0764244i −0.999270 0.0382122i \(-0.987834\pi\)
0.999270 0.0382122i \(-0.0121663\pi\)
\(948\) 0 0
\(949\) − 4.11329e12i − 0.164623i
\(950\) 0 0
\(951\) −2.69510e13 −1.06847
\(952\) 0 0
\(953\) 9.36004e12 0.367587 0.183793 0.982965i \(-0.441162\pi\)
0.183793 + 0.982965i \(0.441162\pi\)
\(954\) 0 0
\(955\) 2.04658e12i 0.0796186i
\(956\) 0 0
\(957\) 8.80658e12i 0.339393i
\(958\) 0 0
\(959\) −1.51919e13 −0.580001
\(960\) 0 0
\(961\) −1.46624e13 −0.554560
\(962\) 0 0
\(963\) − 1.64560e13i − 0.616602i
\(964\) 0 0
\(965\) − 5.70664e13i − 2.11840i
\(966\) 0 0
\(967\) −2.61523e13 −0.961814 −0.480907 0.876772i \(-0.659692\pi\)
−0.480907 + 0.876772i \(0.659692\pi\)
\(968\) 0 0
\(969\) 2.92412e13 1.06546
\(970\) 0 0
\(971\) − 1.18913e13i − 0.429281i −0.976693 0.214640i \(-0.931142\pi\)
0.976693 0.214640i \(-0.0688579\pi\)
\(972\) 0 0
\(973\) 4.00534e13i 1.43262i
\(974\) 0 0
\(975\) 5.39576e12 0.191219
\(976\) 0 0
\(977\) 3.37020e12 0.118340 0.0591699 0.998248i \(-0.481155\pi\)
0.0591699 + 0.998248i \(0.481155\pi\)
\(978\) 0 0
\(979\) − 5.52044e13i − 1.92067i
\(980\) 0 0
\(981\) 1.30979e13i 0.451536i
\(982\) 0 0
\(983\) 4.94392e13 1.68881 0.844404 0.535706i \(-0.179955\pi\)
0.844404 + 0.535706i \(0.179955\pi\)
\(984\) 0 0
\(985\) −6.03879e12 −0.204403
\(986\) 0 0
\(987\) 1.90359e13i 0.638477i
\(988\) 0 0
\(989\) 3.86281e11i 0.0128387i
\(990\) 0 0
\(991\) 3.33144e13 1.09724 0.548619 0.836073i \(-0.315154\pi\)
0.548619 + 0.836073i \(0.315154\pi\)
\(992\) 0 0
\(993\) 2.28546e13 0.745936
\(994\) 0 0
\(995\) 1.29987e13i 0.420431i
\(996\) 0 0
\(997\) − 1.18744e12i − 0.0380612i −0.999819 0.0190306i \(-0.993942\pi\)
0.999819 0.0190306i \(-0.00605799\pi\)
\(998\) 0 0
\(999\) 6.91322e11 0.0219602
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.b.193.5 yes 8
4.3 odd 2 384.10.d.a.193.1 8
8.3 odd 2 384.10.d.a.193.8 yes 8
8.5 even 2 inner 384.10.d.b.193.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.a.193.1 8 4.3 odd 2
384.10.d.a.193.8 yes 8 8.3 odd 2
384.10.d.b.193.4 yes 8 8.5 even 2 inner
384.10.d.b.193.5 yes 8 1.1 even 1 trivial