Properties

Label 342.8.a.g.1.2
Level $342$
Weight $8$
Character 342.1
Self dual yes
Analytic conductor $106.836$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(106.835678716\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-12.0797\) of defining polynomial
Character \(\chi\) \(=\) 342.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} +312.472 q^{5} -766.767 q^{7} -512.000 q^{8} +O(q^{10})\) \(q-8.00000 q^{2} +64.0000 q^{4} +312.472 q^{5} -766.767 q^{7} -512.000 q^{8} -2499.78 q^{10} -252.042 q^{11} -1065.19 q^{13} +6134.14 q^{14} +4096.00 q^{16} +18769.7 q^{17} -6859.00 q^{19} +19998.2 q^{20} +2016.33 q^{22} -11559.9 q^{23} +19513.8 q^{25} +8521.55 q^{26} -49073.1 q^{28} +46290.0 q^{29} +46848.2 q^{31} -32768.0 q^{32} -150158. q^{34} -239593. q^{35} -182916. q^{37} +54872.0 q^{38} -159986. q^{40} -819661. q^{41} +477471. q^{43} -16130.7 q^{44} +92478.8 q^{46} -992580. q^{47} -235611. q^{49} -156111. q^{50} -68172.4 q^{52} +852516. q^{53} -78756.0 q^{55} +392585. q^{56} -370320. q^{58} +1.92404e6 q^{59} +209564. q^{61} -374786. q^{62} +262144. q^{64} -332843. q^{65} -2.32447e6 q^{67} +1.20126e6 q^{68} +1.91675e6 q^{70} +5.37237e6 q^{71} -3.71614e6 q^{73} +1.46333e6 q^{74} -438976. q^{76} +193257. q^{77} -1.30851e6 q^{79} +1.27989e6 q^{80} +6.55729e6 q^{82} +6.51683e6 q^{83} +5.86502e6 q^{85} -3.81976e6 q^{86} +129045. q^{88} -3.99586e6 q^{89} +816756. q^{91} -739831. q^{92} +7.94064e6 q^{94} -2.14325e6 q^{95} -6.90391e6 q^{97} +1.88489e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{2} + 128q^{4} - 155q^{5} - 2238q^{7} - 1024q^{8} + O(q^{10}) \) \( 2q - 16q^{2} + 128q^{4} - 155q^{5} - 2238q^{7} - 1024q^{8} + 1240q^{10} + 3295q^{11} - 13427q^{13} + 17904q^{14} + 8192q^{16} + 32256q^{17} - 13718q^{19} - 9920q^{20} - 26360q^{22} + 82525q^{23} + 159919q^{25} + 107416q^{26} - 143232q^{28} + 12749q^{29} + 258944q^{31} - 65536q^{32} - 258048q^{34} + 448167q^{35} - 149260q^{37} + 109744q^{38} + 79360q^{40} - 339130q^{41} - 83869q^{43} + 210880q^{44} - 660200q^{46} - 1471025q^{47} + 1105372q^{49} - 1279352q^{50} - 859328q^{52} + 945643q^{53} - 1736899q^{55} + 1145856q^{56} - 101992q^{58} + 969009q^{59} - 1506755q^{61} - 2071552q^{62} + 524288q^{64} + 5445956q^{65} - 1848219q^{67} + 2064384q^{68} - 3585336q^{70} + 3417184q^{71} - 2499822q^{73} + 1194080q^{74} - 877952q^{76} - 5025267q^{77} + 2636926q^{79} - 634880q^{80} + 2713040q^{82} + 10059354q^{83} - 439425q^{85} + 670952q^{86} - 1687040q^{88} + 3506160q^{89} + 19003851q^{91} + 5281600q^{92} + 11768200q^{94} + 1063145q^{95} + 5893526q^{97} - 8842976q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) 312.472 1.11793 0.558967 0.829190i \(-0.311198\pi\)
0.558967 + 0.829190i \(0.311198\pi\)
\(6\) 0 0
\(7\) −766.767 −0.844929 −0.422465 0.906379i \(-0.638835\pi\)
−0.422465 + 0.906379i \(0.638835\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) −2499.78 −0.790499
\(11\) −252.042 −0.0570950 −0.0285475 0.999592i \(-0.509088\pi\)
−0.0285475 + 0.999592i \(0.509088\pi\)
\(12\) 0 0
\(13\) −1065.19 −0.134471 −0.0672353 0.997737i \(-0.521418\pi\)
−0.0672353 + 0.997737i \(0.521418\pi\)
\(14\) 6134.14 0.597455
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 18769.7 0.926589 0.463295 0.886204i \(-0.346667\pi\)
0.463295 + 0.886204i \(0.346667\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) 19998.2 0.558967
\(21\) 0 0
\(22\) 2016.33 0.0403722
\(23\) −11559.9 −0.198109 −0.0990546 0.995082i \(-0.531582\pi\)
−0.0990546 + 0.995082i \(0.531582\pi\)
\(24\) 0 0
\(25\) 19513.8 0.249777
\(26\) 8521.55 0.0950850
\(27\) 0 0
\(28\) −49073.1 −0.422465
\(29\) 46290.0 0.352448 0.176224 0.984350i \(-0.443612\pi\)
0.176224 + 0.984350i \(0.443612\pi\)
\(30\) 0 0
\(31\) 46848.2 0.282441 0.141220 0.989978i \(-0.454897\pi\)
0.141220 + 0.989978i \(0.454897\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) −150158. −0.655197
\(35\) −239593. −0.944575
\(36\) 0 0
\(37\) −182916. −0.593672 −0.296836 0.954928i \(-0.595932\pi\)
−0.296836 + 0.954928i \(0.595932\pi\)
\(38\) 54872.0 0.162221
\(39\) 0 0
\(40\) −159986. −0.395249
\(41\) −819661. −1.85734 −0.928669 0.370910i \(-0.879046\pi\)
−0.928669 + 0.370910i \(0.879046\pi\)
\(42\) 0 0
\(43\) 477471. 0.915814 0.457907 0.889000i \(-0.348599\pi\)
0.457907 + 0.889000i \(0.348599\pi\)
\(44\) −16130.7 −0.0285475
\(45\) 0 0
\(46\) 92478.8 0.140084
\(47\) −992580. −1.39451 −0.697257 0.716821i \(-0.745596\pi\)
−0.697257 + 0.716821i \(0.745596\pi\)
\(48\) 0 0
\(49\) −235611. −0.286095
\(50\) −156111. −0.176619
\(51\) 0 0
\(52\) −68172.4 −0.0672353
\(53\) 852516. 0.786569 0.393285 0.919417i \(-0.371339\pi\)
0.393285 + 0.919417i \(0.371339\pi\)
\(54\) 0 0
\(55\) −78756.0 −0.0638284
\(56\) 392585. 0.298728
\(57\) 0 0
\(58\) −370320. −0.249218
\(59\) 1.92404e6 1.21964 0.609821 0.792539i \(-0.291241\pi\)
0.609821 + 0.792539i \(0.291241\pi\)
\(60\) 0 0
\(61\) 209564. 0.118212 0.0591062 0.998252i \(-0.481175\pi\)
0.0591062 + 0.998252i \(0.481175\pi\)
\(62\) −374786. −0.199716
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −332843. −0.150329
\(66\) 0 0
\(67\) −2.32447e6 −0.944198 −0.472099 0.881546i \(-0.656503\pi\)
−0.472099 + 0.881546i \(0.656503\pi\)
\(68\) 1.20126e6 0.463295
\(69\) 0 0
\(70\) 1.91675e6 0.667916
\(71\) 5.37237e6 1.78140 0.890700 0.454591i \(-0.150215\pi\)
0.890700 + 0.454591i \(0.150215\pi\)
\(72\) 0 0
\(73\) −3.71614e6 −1.11805 −0.559027 0.829150i \(-0.688825\pi\)
−0.559027 + 0.829150i \(0.688825\pi\)
\(74\) 1.46333e6 0.419790
\(75\) 0 0
\(76\) −438976. −0.114708
\(77\) 193257. 0.0482412
\(78\) 0 0
\(79\) −1.30851e6 −0.298596 −0.149298 0.988792i \(-0.547701\pi\)
−0.149298 + 0.988792i \(0.547701\pi\)
\(80\) 1.27989e6 0.279484
\(81\) 0 0
\(82\) 6.55729e6 1.31334
\(83\) 6.51683e6 1.25102 0.625508 0.780218i \(-0.284892\pi\)
0.625508 + 0.780218i \(0.284892\pi\)
\(84\) 0 0
\(85\) 5.86502e6 1.03587
\(86\) −3.81976e6 −0.647578
\(87\) 0 0
\(88\) 129045. 0.0201861
\(89\) −3.99586e6 −0.600822 −0.300411 0.953810i \(-0.597124\pi\)
−0.300411 + 0.953810i \(0.597124\pi\)
\(90\) 0 0
\(91\) 816756. 0.113618
\(92\) −739831. −0.0990546
\(93\) 0 0
\(94\) 7.94064e6 0.986070
\(95\) −2.14325e6 −0.256472
\(96\) 0 0
\(97\) −6.90391e6 −0.768058 −0.384029 0.923321i \(-0.625464\pi\)
−0.384029 + 0.923321i \(0.625464\pi\)
\(98\) 1.88489e6 0.202299
\(99\) 0 0
\(100\) 1.24888e6 0.124888
\(101\) −2.81610e6 −0.271971 −0.135986 0.990711i \(-0.543420\pi\)
−0.135986 + 0.990711i \(0.543420\pi\)
\(102\) 0 0
\(103\) 7.86273e6 0.708995 0.354497 0.935057i \(-0.384652\pi\)
0.354497 + 0.935057i \(0.384652\pi\)
\(104\) 545379. 0.0475425
\(105\) 0 0
\(106\) −6.82013e6 −0.556188
\(107\) −611626. −0.0482662 −0.0241331 0.999709i \(-0.507683\pi\)
−0.0241331 + 0.999709i \(0.507683\pi\)
\(108\) 0 0
\(109\) 5.23210e6 0.386976 0.193488 0.981103i \(-0.438020\pi\)
0.193488 + 0.981103i \(0.438020\pi\)
\(110\) 630048. 0.0451335
\(111\) 0 0
\(112\) −3.14068e6 −0.211232
\(113\) 3.58527e6 0.233747 0.116874 0.993147i \(-0.462713\pi\)
0.116874 + 0.993147i \(0.462713\pi\)
\(114\) 0 0
\(115\) −3.61213e6 −0.221473
\(116\) 2.96256e6 0.176224
\(117\) 0 0
\(118\) −1.53923e7 −0.862418
\(119\) −1.43920e7 −0.782902
\(120\) 0 0
\(121\) −1.94236e7 −0.996740
\(122\) −1.67651e6 −0.0835888
\(123\) 0 0
\(124\) 2.99829e6 0.141220
\(125\) −1.83144e7 −0.838700
\(126\) 0 0
\(127\) −2.99465e7 −1.29728 −0.648639 0.761096i \(-0.724661\pi\)
−0.648639 + 0.761096i \(0.724661\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 0 0
\(130\) 2.66275e6 0.106299
\(131\) 8.11292e6 0.315303 0.157651 0.987495i \(-0.449608\pi\)
0.157651 + 0.987495i \(0.449608\pi\)
\(132\) 0 0
\(133\) 5.25926e6 0.193840
\(134\) 1.85958e7 0.667649
\(135\) 0 0
\(136\) −9.61011e6 −0.327599
\(137\) −1.44157e7 −0.478977 −0.239488 0.970899i \(-0.576980\pi\)
−0.239488 + 0.970899i \(0.576980\pi\)
\(138\) 0 0
\(139\) −3.77446e7 −1.19207 −0.596036 0.802957i \(-0.703259\pi\)
−0.596036 + 0.802957i \(0.703259\pi\)
\(140\) −1.53340e7 −0.472288
\(141\) 0 0
\(142\) −4.29789e7 −1.25964
\(143\) 268473. 0.00767759
\(144\) 0 0
\(145\) 1.44643e7 0.394013
\(146\) 2.97292e7 0.790583
\(147\) 0 0
\(148\) −1.17067e7 −0.296836
\(149\) −7.30679e7 −1.80957 −0.904784 0.425872i \(-0.859967\pi\)
−0.904784 + 0.425872i \(0.859967\pi\)
\(150\) 0 0
\(151\) −4.14056e7 −0.978678 −0.489339 0.872094i \(-0.662762\pi\)
−0.489339 + 0.872094i \(0.662762\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) 0 0
\(154\) −1.54606e6 −0.0341117
\(155\) 1.46388e7 0.315750
\(156\) 0 0
\(157\) 2.93945e7 0.606203 0.303101 0.952958i \(-0.401978\pi\)
0.303101 + 0.952958i \(0.401978\pi\)
\(158\) 1.04681e7 0.211139
\(159\) 0 0
\(160\) −1.02391e7 −0.197625
\(161\) 8.86371e6 0.167388
\(162\) 0 0
\(163\) 9.03924e6 0.163484 0.0817420 0.996654i \(-0.473952\pi\)
0.0817420 + 0.996654i \(0.473952\pi\)
\(164\) −5.24583e7 −0.928669
\(165\) 0 0
\(166\) −5.21346e7 −0.884602
\(167\) 8.16728e7 1.35697 0.678484 0.734615i \(-0.262637\pi\)
0.678484 + 0.734615i \(0.262637\pi\)
\(168\) 0 0
\(169\) −6.16139e7 −0.981918
\(170\) −4.69202e7 −0.732468
\(171\) 0 0
\(172\) 3.05581e7 0.457907
\(173\) −1.33191e8 −1.95575 −0.977874 0.209197i \(-0.932915\pi\)
−0.977874 + 0.209197i \(0.932915\pi\)
\(174\) 0 0
\(175\) −1.49626e7 −0.211044
\(176\) −1.03236e6 −0.0142737
\(177\) 0 0
\(178\) 3.19669e7 0.424845
\(179\) 4.90910e7 0.639759 0.319879 0.947458i \(-0.396358\pi\)
0.319879 + 0.947458i \(0.396358\pi\)
\(180\) 0 0
\(181\) 8.11979e7 1.01782 0.508908 0.860821i \(-0.330049\pi\)
0.508908 + 0.860821i \(0.330049\pi\)
\(182\) −6.53405e6 −0.0803401
\(183\) 0 0
\(184\) 5.91864e6 0.0700422
\(185\) −5.71563e7 −0.663686
\(186\) 0 0
\(187\) −4.73076e6 −0.0529036
\(188\) −6.35251e7 −0.697257
\(189\) 0 0
\(190\) 1.71460e7 0.181353
\(191\) −7.40713e7 −0.769190 −0.384595 0.923086i \(-0.625659\pi\)
−0.384595 + 0.923086i \(0.625659\pi\)
\(192\) 0 0
\(193\) −9.70125e7 −0.971353 −0.485676 0.874139i \(-0.661427\pi\)
−0.485676 + 0.874139i \(0.661427\pi\)
\(194\) 5.52313e7 0.543099
\(195\) 0 0
\(196\) −1.50791e7 −0.143047
\(197\) 4.34770e7 0.405161 0.202581 0.979266i \(-0.435067\pi\)
0.202581 + 0.979266i \(0.435067\pi\)
\(198\) 0 0
\(199\) 3.89651e7 0.350502 0.175251 0.984524i \(-0.443926\pi\)
0.175251 + 0.984524i \(0.443926\pi\)
\(200\) −9.99108e6 −0.0883095
\(201\) 0 0
\(202\) 2.25288e7 0.192313
\(203\) −3.54937e7 −0.297793
\(204\) 0 0
\(205\) −2.56121e8 −2.07638
\(206\) −6.29018e7 −0.501335
\(207\) 0 0
\(208\) −4.36304e6 −0.0336176
\(209\) 1.72875e6 0.0130985
\(210\) 0 0
\(211\) −3.85264e7 −0.282339 −0.141169 0.989985i \(-0.545086\pi\)
−0.141169 + 0.989985i \(0.545086\pi\)
\(212\) 5.45610e7 0.393285
\(213\) 0 0
\(214\) 4.89301e6 0.0341293
\(215\) 1.49196e8 1.02382
\(216\) 0 0
\(217\) −3.59217e7 −0.238642
\(218\) −4.18568e7 −0.273633
\(219\) 0 0
\(220\) −5.04038e6 −0.0319142
\(221\) −1.99934e7 −0.124599
\(222\) 0 0
\(223\) −2.03401e8 −1.22825 −0.614124 0.789209i \(-0.710491\pi\)
−0.614124 + 0.789209i \(0.710491\pi\)
\(224\) 2.51254e7 0.149364
\(225\) 0 0
\(226\) −2.86821e7 −0.165284
\(227\) −2.24944e8 −1.27639 −0.638195 0.769875i \(-0.720319\pi\)
−0.638195 + 0.769875i \(0.720319\pi\)
\(228\) 0 0
\(229\) −2.38882e8 −1.31450 −0.657248 0.753674i \(-0.728280\pi\)
−0.657248 + 0.753674i \(0.728280\pi\)
\(230\) 2.88971e7 0.156605
\(231\) 0 0
\(232\) −2.37005e7 −0.124609
\(233\) −3.23019e8 −1.67295 −0.836475 0.548005i \(-0.815387\pi\)
−0.836475 + 0.548005i \(0.815387\pi\)
\(234\) 0 0
\(235\) −3.10153e8 −1.55897
\(236\) 1.23139e8 0.609821
\(237\) 0 0
\(238\) 1.15136e8 0.553595
\(239\) 5.18160e7 0.245511 0.122756 0.992437i \(-0.460827\pi\)
0.122756 + 0.992437i \(0.460827\pi\)
\(240\) 0 0
\(241\) 1.15926e8 0.533484 0.266742 0.963768i \(-0.414053\pi\)
0.266742 + 0.963768i \(0.414053\pi\)
\(242\) 1.55389e8 0.704802
\(243\) 0 0
\(244\) 1.34121e7 0.0591062
\(245\) −7.36219e7 −0.319835
\(246\) 0 0
\(247\) 7.30617e6 0.0308497
\(248\) −2.39863e7 −0.0998579
\(249\) 0 0
\(250\) 1.46515e8 0.593050
\(251\) 4.03761e7 0.161163 0.0805817 0.996748i \(-0.474322\pi\)
0.0805817 + 0.996748i \(0.474322\pi\)
\(252\) 0 0
\(253\) 2.91356e6 0.0113110
\(254\) 2.39572e8 0.917314
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) −2.67661e8 −0.983602 −0.491801 0.870708i \(-0.663661\pi\)
−0.491801 + 0.870708i \(0.663661\pi\)
\(258\) 0 0
\(259\) 1.40254e8 0.501611
\(260\) −2.13020e7 −0.0751646
\(261\) 0 0
\(262\) −6.49034e7 −0.222953
\(263\) −5.12778e8 −1.73814 −0.869068 0.494692i \(-0.835281\pi\)
−0.869068 + 0.494692i \(0.835281\pi\)
\(264\) 0 0
\(265\) 2.66387e8 0.879333
\(266\) −4.20740e7 −0.137066
\(267\) 0 0
\(268\) −1.48766e8 −0.472099
\(269\) −5.75792e8 −1.80357 −0.901784 0.432187i \(-0.857742\pi\)
−0.901784 + 0.432187i \(0.857742\pi\)
\(270\) 0 0
\(271\) 5.24839e8 1.60189 0.800946 0.598737i \(-0.204330\pi\)
0.800946 + 0.598737i \(0.204330\pi\)
\(272\) 7.68809e7 0.231647
\(273\) 0 0
\(274\) 1.15326e8 0.338688
\(275\) −4.91829e6 −0.0142610
\(276\) 0 0
\(277\) −4.64620e8 −1.31347 −0.656733 0.754123i \(-0.728062\pi\)
−0.656733 + 0.754123i \(0.728062\pi\)
\(278\) 3.01957e8 0.842923
\(279\) 0 0
\(280\) 1.22672e8 0.333958
\(281\) 1.75429e8 0.471660 0.235830 0.971794i \(-0.424219\pi\)
0.235830 + 0.971794i \(0.424219\pi\)
\(282\) 0 0
\(283\) 1.44948e8 0.380154 0.190077 0.981769i \(-0.439126\pi\)
0.190077 + 0.981769i \(0.439126\pi\)
\(284\) 3.43832e8 0.890700
\(285\) 0 0
\(286\) −2.14779e6 −0.00542888
\(287\) 6.28489e8 1.56932
\(288\) 0 0
\(289\) −5.80353e7 −0.141433
\(290\) −1.15715e8 −0.278609
\(291\) 0 0
\(292\) −2.37833e8 −0.559027
\(293\) 4.24522e8 0.985970 0.492985 0.870038i \(-0.335906\pi\)
0.492985 + 0.870038i \(0.335906\pi\)
\(294\) 0 0
\(295\) 6.01210e8 1.36348
\(296\) 9.36532e7 0.209895
\(297\) 0 0
\(298\) 5.84543e8 1.27956
\(299\) 1.23135e7 0.0266399
\(300\) 0 0
\(301\) −3.66109e8 −0.773798
\(302\) 3.31245e8 0.692030
\(303\) 0 0
\(304\) −2.80945e7 −0.0573539
\(305\) 6.54830e7 0.132154
\(306\) 0 0
\(307\) 4.27528e8 0.843297 0.421649 0.906759i \(-0.361452\pi\)
0.421649 + 0.906759i \(0.361452\pi\)
\(308\) 1.23685e7 0.0241206
\(309\) 0 0
\(310\) −1.17110e8 −0.223269
\(311\) 1.70275e8 0.320989 0.160495 0.987037i \(-0.448691\pi\)
0.160495 + 0.987037i \(0.448691\pi\)
\(312\) 0 0
\(313\) −4.02143e8 −0.741269 −0.370634 0.928779i \(-0.620860\pi\)
−0.370634 + 0.928779i \(0.620860\pi\)
\(314\) −2.35156e8 −0.428650
\(315\) 0 0
\(316\) −8.37450e7 −0.149298
\(317\) 1.90630e8 0.336112 0.168056 0.985777i \(-0.446251\pi\)
0.168056 + 0.985777i \(0.446251\pi\)
\(318\) 0 0
\(319\) −1.16670e7 −0.0201230
\(320\) 8.19127e7 0.139742
\(321\) 0 0
\(322\) −7.09097e7 −0.118361
\(323\) −1.28742e8 −0.212574
\(324\) 0 0
\(325\) −2.07860e7 −0.0335876
\(326\) −7.23139e7 −0.115601
\(327\) 0 0
\(328\) 4.19666e8 0.656668
\(329\) 7.61077e8 1.17827
\(330\) 0 0
\(331\) −4.12008e8 −0.624464 −0.312232 0.950006i \(-0.601077\pi\)
−0.312232 + 0.950006i \(0.601077\pi\)
\(332\) 4.17077e8 0.625508
\(333\) 0 0
\(334\) −6.53382e8 −0.959522
\(335\) −7.26333e8 −1.05555
\(336\) 0 0
\(337\) −6.99272e8 −0.995271 −0.497636 0.867386i \(-0.665798\pi\)
−0.497636 + 0.867386i \(0.665798\pi\)
\(338\) 4.92911e8 0.694321
\(339\) 0 0
\(340\) 3.75361e8 0.517933
\(341\) −1.18077e7 −0.0161259
\(342\) 0 0
\(343\) 8.12125e8 1.08666
\(344\) −2.44465e8 −0.323789
\(345\) 0 0
\(346\) 1.06553e9 1.38292
\(347\) 9.29885e8 1.19475 0.597374 0.801963i \(-0.296211\pi\)
0.597374 + 0.801963i \(0.296211\pi\)
\(348\) 0 0
\(349\) −4.43776e8 −0.558823 −0.279412 0.960171i \(-0.590139\pi\)
−0.279412 + 0.960171i \(0.590139\pi\)
\(350\) 1.19700e8 0.149231
\(351\) 0 0
\(352\) 8.25890e6 0.0100931
\(353\) 1.49388e9 1.80761 0.903804 0.427946i \(-0.140763\pi\)
0.903804 + 0.427946i \(0.140763\pi\)
\(354\) 0 0
\(355\) 1.67872e9 1.99149
\(356\) −2.55735e8 −0.300411
\(357\) 0 0
\(358\) −3.92728e8 −0.452378
\(359\) 1.77937e8 0.202972 0.101486 0.994837i \(-0.467640\pi\)
0.101486 + 0.994837i \(0.467640\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) −6.49583e8 −0.719705
\(363\) 0 0
\(364\) 5.22724e7 0.0568090
\(365\) −1.16119e9 −1.24991
\(366\) 0 0
\(367\) −1.37515e9 −1.45218 −0.726088 0.687601i \(-0.758664\pi\)
−0.726088 + 0.687601i \(0.758664\pi\)
\(368\) −4.73492e7 −0.0495273
\(369\) 0 0
\(370\) 4.57250e8 0.469297
\(371\) −6.53681e8 −0.664595
\(372\) 0 0
\(373\) 1.55845e9 1.55493 0.777466 0.628925i \(-0.216505\pi\)
0.777466 + 0.628925i \(0.216505\pi\)
\(374\) 3.78461e7 0.0374085
\(375\) 0 0
\(376\) 5.08201e8 0.493035
\(377\) −4.93079e7 −0.0473938
\(378\) 0 0
\(379\) 8.84549e8 0.834613 0.417306 0.908766i \(-0.362974\pi\)
0.417306 + 0.908766i \(0.362974\pi\)
\(380\) −1.37168e8 −0.128236
\(381\) 0 0
\(382\) 5.92571e8 0.543899
\(383\) −1.44573e9 −1.31490 −0.657449 0.753499i \(-0.728365\pi\)
−0.657449 + 0.753499i \(0.728365\pi\)
\(384\) 0 0
\(385\) 6.03875e7 0.0539305
\(386\) 7.76100e8 0.686850
\(387\) 0 0
\(388\) −4.41850e8 −0.384029
\(389\) 1.02164e9 0.879986 0.439993 0.898001i \(-0.354981\pi\)
0.439993 + 0.898001i \(0.354981\pi\)
\(390\) 0 0
\(391\) −2.16975e8 −0.183566
\(392\) 1.20633e8 0.101150
\(393\) 0 0
\(394\) −3.47816e8 −0.286492
\(395\) −4.08874e8 −0.333811
\(396\) 0 0
\(397\) 1.80050e8 0.144419 0.0722097 0.997389i \(-0.476995\pi\)
0.0722097 + 0.997389i \(0.476995\pi\)
\(398\) −3.11721e8 −0.247842
\(399\) 0 0
\(400\) 7.99286e7 0.0624442
\(401\) −5.06670e8 −0.392392 −0.196196 0.980565i \(-0.562859\pi\)
−0.196196 + 0.980565i \(0.562859\pi\)
\(402\) 0 0
\(403\) −4.99025e7 −0.0379799
\(404\) −1.80230e8 −0.135986
\(405\) 0 0
\(406\) 2.83949e8 0.210572
\(407\) 4.61026e7 0.0338957
\(408\) 0 0
\(409\) 1.14898e9 0.830388 0.415194 0.909733i \(-0.363714\pi\)
0.415194 + 0.909733i \(0.363714\pi\)
\(410\) 2.04897e9 1.46822
\(411\) 0 0
\(412\) 5.03215e8 0.354497
\(413\) −1.47529e9 −1.03051
\(414\) 0 0
\(415\) 2.03633e9 1.39855
\(416\) 3.49043e7 0.0237713
\(417\) 0 0
\(418\) −1.38300e7 −0.00926203
\(419\) −2.02289e9 −1.34346 −0.671729 0.740797i \(-0.734448\pi\)
−0.671729 + 0.740797i \(0.734448\pi\)
\(420\) 0 0
\(421\) 2.55503e9 1.66882 0.834408 0.551147i \(-0.185810\pi\)
0.834408 + 0.551147i \(0.185810\pi\)
\(422\) 3.08211e8 0.199644
\(423\) 0 0
\(424\) −4.36488e8 −0.278094
\(425\) 3.66269e8 0.231441
\(426\) 0 0
\(427\) −1.60687e8 −0.0998811
\(428\) −3.91441e7 −0.0241331
\(429\) 0 0
\(430\) −1.19357e9 −0.723950
\(431\) −1.47316e9 −0.886295 −0.443147 0.896449i \(-0.646138\pi\)
−0.443147 + 0.896449i \(0.646138\pi\)
\(432\) 0 0
\(433\) −9.33243e8 −0.552443 −0.276221 0.961094i \(-0.589082\pi\)
−0.276221 + 0.961094i \(0.589082\pi\)
\(434\) 2.87373e8 0.168746
\(435\) 0 0
\(436\) 3.34855e8 0.193488
\(437\) 7.92890e7 0.0454494
\(438\) 0 0
\(439\) −1.94411e9 −1.09672 −0.548358 0.836244i \(-0.684747\pi\)
−0.548358 + 0.836244i \(0.684747\pi\)
\(440\) 4.03231e7 0.0225668
\(441\) 0 0
\(442\) 1.59947e8 0.0881047
\(443\) 2.35592e9 1.28750 0.643751 0.765235i \(-0.277377\pi\)
0.643751 + 0.765235i \(0.277377\pi\)
\(444\) 0 0
\(445\) −1.24860e9 −0.671679
\(446\) 1.62721e9 0.868503
\(447\) 0 0
\(448\) −2.01003e8 −0.105616
\(449\) 1.83374e9 0.956042 0.478021 0.878349i \(-0.341354\pi\)
0.478021 + 0.878349i \(0.341354\pi\)
\(450\) 0 0
\(451\) 2.06589e8 0.106045
\(452\) 2.29457e8 0.116874
\(453\) 0 0
\(454\) 1.79955e9 0.902544
\(455\) 2.55213e8 0.127018
\(456\) 0 0
\(457\) −3.76683e9 −1.84616 −0.923079 0.384611i \(-0.874336\pi\)
−0.923079 + 0.384611i \(0.874336\pi\)
\(458\) 1.91106e9 0.929489
\(459\) 0 0
\(460\) −2.31176e8 −0.110737
\(461\) −3.30560e9 −1.57144 −0.785720 0.618583i \(-0.787707\pi\)
−0.785720 + 0.618583i \(0.787707\pi\)
\(462\) 0 0
\(463\) −3.66326e9 −1.71528 −0.857638 0.514253i \(-0.828069\pi\)
−0.857638 + 0.514253i \(0.828069\pi\)
\(464\) 1.89604e8 0.0881119
\(465\) 0 0
\(466\) 2.58416e9 1.18295
\(467\) 2.22674e9 1.01172 0.505860 0.862616i \(-0.331175\pi\)
0.505860 + 0.862616i \(0.331175\pi\)
\(468\) 0 0
\(469\) 1.78233e9 0.797780
\(470\) 2.48123e9 1.10236
\(471\) 0 0
\(472\) −9.85110e8 −0.431209
\(473\) −1.20342e8 −0.0522883
\(474\) 0 0
\(475\) −1.33845e8 −0.0573028
\(476\) −9.21090e8 −0.391451
\(477\) 0 0
\(478\) −4.14528e8 −0.173603
\(479\) −1.43508e9 −0.596625 −0.298313 0.954468i \(-0.596424\pi\)
−0.298313 + 0.954468i \(0.596424\pi\)
\(480\) 0 0
\(481\) 1.94842e8 0.0798314
\(482\) −9.27409e8 −0.377230
\(483\) 0 0
\(484\) −1.24311e9 −0.498370
\(485\) −2.15728e9 −0.858638
\(486\) 0 0
\(487\) −2.11642e9 −0.830330 −0.415165 0.909746i \(-0.636276\pi\)
−0.415165 + 0.909746i \(0.636276\pi\)
\(488\) −1.07297e8 −0.0417944
\(489\) 0 0
\(490\) 5.88975e8 0.226157
\(491\) −4.07105e9 −1.55211 −0.776053 0.630668i \(-0.782781\pi\)
−0.776053 + 0.630668i \(0.782781\pi\)
\(492\) 0 0
\(493\) 8.68852e8 0.326574
\(494\) −5.84493e7 −0.0218140
\(495\) 0 0
\(496\) 1.91890e8 0.0706102
\(497\) −4.11935e9 −1.50516
\(498\) 0 0
\(499\) 3.93039e9 1.41607 0.708034 0.706178i \(-0.249582\pi\)
0.708034 + 0.706178i \(0.249582\pi\)
\(500\) −1.17212e9 −0.419350
\(501\) 0 0
\(502\) −3.23009e8 −0.113960
\(503\) 1.70030e9 0.595714 0.297857 0.954610i \(-0.403728\pi\)
0.297857 + 0.954610i \(0.403728\pi\)
\(504\) 0 0
\(505\) −8.79952e8 −0.304046
\(506\) −2.33085e7 −0.00799812
\(507\) 0 0
\(508\) −1.91658e9 −0.648639
\(509\) 2.62955e8 0.0883830 0.0441915 0.999023i \(-0.485929\pi\)
0.0441915 + 0.999023i \(0.485929\pi\)
\(510\) 0 0
\(511\) 2.84942e9 0.944676
\(512\) −1.34218e8 −0.0441942
\(513\) 0 0
\(514\) 2.14129e9 0.695512
\(515\) 2.45688e9 0.792609
\(516\) 0 0
\(517\) 2.50171e8 0.0796197
\(518\) −1.12203e9 −0.354693
\(519\) 0 0
\(520\) 1.70416e8 0.0531494
\(521\) 1.86990e9 0.579277 0.289639 0.957136i \(-0.406465\pi\)
0.289639 + 0.957136i \(0.406465\pi\)
\(522\) 0 0
\(523\) −5.49460e9 −1.67950 −0.839750 0.542972i \(-0.817299\pi\)
−0.839750 + 0.542972i \(0.817299\pi\)
\(524\) 5.19227e8 0.157651
\(525\) 0 0
\(526\) 4.10222e9 1.22905
\(527\) 8.79329e8 0.261706
\(528\) 0 0
\(529\) −3.27120e9 −0.960753
\(530\) −2.13110e9 −0.621782
\(531\) 0 0
\(532\) 3.36592e8 0.0969200
\(533\) 8.73098e8 0.249757
\(534\) 0 0
\(535\) −1.91116e8 −0.0539584
\(536\) 1.19013e9 0.333824
\(537\) 0 0
\(538\) 4.60634e9 1.27532
\(539\) 5.93838e7 0.0163346
\(540\) 0 0
\(541\) 6.97892e9 1.89495 0.947475 0.319830i \(-0.103626\pi\)
0.947475 + 0.319830i \(0.103626\pi\)
\(542\) −4.19871e9 −1.13271
\(543\) 0 0
\(544\) −6.15047e8 −0.163799
\(545\) 1.63489e9 0.432613
\(546\) 0 0
\(547\) −5.87859e9 −1.53574 −0.767870 0.640606i \(-0.778683\pi\)
−0.767870 + 0.640606i \(0.778683\pi\)
\(548\) −9.22606e8 −0.239488
\(549\) 0 0
\(550\) 3.93464e7 0.0100841
\(551\) −3.17503e8 −0.0808570
\(552\) 0 0
\(553\) 1.00333e9 0.252292
\(554\) 3.71696e9 0.928761
\(555\) 0 0
\(556\) −2.41565e9 −0.596036
\(557\) 2.03155e9 0.498119 0.249060 0.968488i \(-0.419878\pi\)
0.249060 + 0.968488i \(0.419878\pi\)
\(558\) 0 0
\(559\) −5.08599e8 −0.123150
\(560\) −9.81374e8 −0.236144
\(561\) 0 0
\(562\) −1.40343e9 −0.333514
\(563\) −1.68161e9 −0.397142 −0.198571 0.980087i \(-0.563630\pi\)
−0.198571 + 0.980087i \(0.563630\pi\)
\(564\) 0 0
\(565\) 1.12030e9 0.261314
\(566\) −1.15958e9 −0.268810
\(567\) 0 0
\(568\) −2.75065e9 −0.629820
\(569\) 4.57746e9 1.04167 0.520837 0.853656i \(-0.325620\pi\)
0.520837 + 0.853656i \(0.325620\pi\)
\(570\) 0 0
\(571\) −5.79221e9 −1.30202 −0.651010 0.759069i \(-0.725655\pi\)
−0.651010 + 0.759069i \(0.725655\pi\)
\(572\) 1.71823e7 0.00383879
\(573\) 0 0
\(574\) −5.02791e9 −1.10968
\(575\) −2.25577e8 −0.0494831
\(576\) 0 0
\(577\) −3.74692e9 −0.812005 −0.406003 0.913872i \(-0.633078\pi\)
−0.406003 + 0.913872i \(0.633078\pi\)
\(578\) 4.64282e8 0.100008
\(579\) 0 0
\(580\) 9.25718e8 0.197007
\(581\) −4.99689e9 −1.05702
\(582\) 0 0
\(583\) −2.14869e8 −0.0449091
\(584\) 1.90267e9 0.395292
\(585\) 0 0
\(586\) −3.39618e9 −0.697186
\(587\) 5.69193e9 1.16152 0.580760 0.814075i \(-0.302756\pi\)
0.580760 + 0.814075i \(0.302756\pi\)
\(588\) 0 0
\(589\) −3.21332e8 −0.0647963
\(590\) −4.80968e9 −0.964126
\(591\) 0 0
\(592\) −7.49226e8 −0.148418
\(593\) −1.45596e9 −0.286720 −0.143360 0.989671i \(-0.545791\pi\)
−0.143360 + 0.989671i \(0.545791\pi\)
\(594\) 0 0
\(595\) −4.49711e9 −0.875233
\(596\) −4.67635e9 −0.904784
\(597\) 0 0
\(598\) −9.85079e7 −0.0188372
\(599\) 6.21735e9 1.18198 0.590992 0.806677i \(-0.298736\pi\)
0.590992 + 0.806677i \(0.298736\pi\)
\(600\) 0 0
\(601\) 9.98776e9 1.87675 0.938377 0.345613i \(-0.112329\pi\)
0.938377 + 0.345613i \(0.112329\pi\)
\(602\) 2.92887e9 0.547158
\(603\) 0 0
\(604\) −2.64996e9 −0.489339
\(605\) −6.06935e9 −1.11429
\(606\) 0 0
\(607\) 3.17827e9 0.576807 0.288404 0.957509i \(-0.406876\pi\)
0.288404 + 0.957509i \(0.406876\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) 0 0
\(610\) −5.23864e8 −0.0934467
\(611\) 1.05729e9 0.187521
\(612\) 0 0
\(613\) −6.34969e8 −0.111337 −0.0556687 0.998449i \(-0.517729\pi\)
−0.0556687 + 0.998449i \(0.517729\pi\)
\(614\) −3.42023e9 −0.596301
\(615\) 0 0
\(616\) −9.89477e7 −0.0170558
\(617\) 4.48835e7 0.00769287 0.00384643 0.999993i \(-0.498776\pi\)
0.00384643 + 0.999993i \(0.498776\pi\)
\(618\) 0 0
\(619\) −9.79132e8 −0.165930 −0.0829648 0.996552i \(-0.526439\pi\)
−0.0829648 + 0.996552i \(0.526439\pi\)
\(620\) 9.36881e8 0.157875
\(621\) 0 0
\(622\) −1.36220e9 −0.226974
\(623\) 3.06390e9 0.507652
\(624\) 0 0
\(625\) −7.24724e9 −1.18739
\(626\) 3.21715e9 0.524156
\(627\) 0 0
\(628\) 1.88125e9 0.303101
\(629\) −3.43330e9 −0.550090
\(630\) 0 0
\(631\) −2.42563e9 −0.384345 −0.192172 0.981361i \(-0.561553\pi\)
−0.192172 + 0.981361i \(0.561553\pi\)
\(632\) 6.69960e8 0.105570
\(633\) 0 0
\(634\) −1.52504e9 −0.237667
\(635\) −9.35744e9 −1.45027
\(636\) 0 0
\(637\) 2.50972e8 0.0384713
\(638\) 9.33361e7 0.0142291
\(639\) 0 0
\(640\) −6.55302e8 −0.0988124
\(641\) −9.23132e8 −0.138440 −0.0692199 0.997601i \(-0.522051\pi\)
−0.0692199 + 0.997601i \(0.522051\pi\)
\(642\) 0 0
\(643\) 6.86334e9 1.01812 0.509058 0.860732i \(-0.329994\pi\)
0.509058 + 0.860732i \(0.329994\pi\)
\(644\) 5.67278e8 0.0836942
\(645\) 0 0
\(646\) 1.02993e9 0.150313
\(647\) −5.02399e8 −0.0729262 −0.0364631 0.999335i \(-0.511609\pi\)
−0.0364631 + 0.999335i \(0.511609\pi\)
\(648\) 0 0
\(649\) −4.84939e8 −0.0696355
\(650\) 1.66288e8 0.0237500
\(651\) 0 0
\(652\) 5.78512e8 0.0817420
\(653\) −4.23523e9 −0.595225 −0.297612 0.954687i \(-0.596190\pi\)
−0.297612 + 0.954687i \(0.596190\pi\)
\(654\) 0 0
\(655\) 2.53506e9 0.352488
\(656\) −3.35733e9 −0.464334
\(657\) 0 0
\(658\) −6.08862e9 −0.833160
\(659\) 1.34554e10 1.83146 0.915731 0.401791i \(-0.131612\pi\)
0.915731 + 0.401791i \(0.131612\pi\)
\(660\) 0 0
\(661\) 1.23518e10 1.66351 0.831753 0.555146i \(-0.187338\pi\)
0.831753 + 0.555146i \(0.187338\pi\)
\(662\) 3.29606e9 0.441563
\(663\) 0 0
\(664\) −3.33662e9 −0.442301
\(665\) 1.64337e9 0.216700
\(666\) 0 0
\(667\) −5.35106e8 −0.0698231
\(668\) 5.22706e9 0.678484
\(669\) 0 0
\(670\) 5.81067e9 0.746387
\(671\) −5.28189e7 −0.00674933
\(672\) 0 0
\(673\) −1.52638e9 −0.193024 −0.0965119 0.995332i \(-0.530769\pi\)
−0.0965119 + 0.995332i \(0.530769\pi\)
\(674\) 5.59417e9 0.703763
\(675\) 0 0
\(676\) −3.94329e9 −0.490959
\(677\) 1.55415e9 0.192500 0.0962502 0.995357i \(-0.469315\pi\)
0.0962502 + 0.995357i \(0.469315\pi\)
\(678\) 0 0
\(679\) 5.29369e9 0.648954
\(680\) −3.00289e9 −0.366234
\(681\) 0 0
\(682\) 9.44616e7 0.0114028
\(683\) 3.21824e9 0.386497 0.193249 0.981150i \(-0.438098\pi\)
0.193249 + 0.981150i \(0.438098\pi\)
\(684\) 0 0
\(685\) −4.50451e9 −0.535465
\(686\) −6.49700e9 −0.768384
\(687\) 0 0
\(688\) 1.95572e9 0.228953
\(689\) −9.08095e8 −0.105770
\(690\) 0 0
\(691\) 9.15999e9 1.05614 0.528071 0.849200i \(-0.322916\pi\)
0.528071 + 0.849200i \(0.322916\pi\)
\(692\) −8.52421e9 −0.977874
\(693\) 0 0
\(694\) −7.43908e9 −0.844814
\(695\) −1.17941e10 −1.33266
\(696\) 0 0
\(697\) −1.53848e10 −1.72099
\(698\) 3.55021e9 0.395148
\(699\) 0 0
\(700\) −9.57604e8 −0.105522
\(701\) 7.97717e9 0.874653 0.437327 0.899303i \(-0.355925\pi\)
0.437327 + 0.899303i \(0.355925\pi\)
\(702\) 0 0
\(703\) 1.25462e9 0.136198
\(704\) −6.60712e7 −0.00713687
\(705\) 0 0
\(706\) −1.19510e10 −1.27817
\(707\) 2.15929e9 0.229796
\(708\) 0 0
\(709\) 7.97010e9 0.839851 0.419925 0.907559i \(-0.362056\pi\)
0.419925 + 0.907559i \(0.362056\pi\)
\(710\) −1.34297e10 −1.40819
\(711\) 0 0
\(712\) 2.04588e9 0.212423
\(713\) −5.41559e8 −0.0559541
\(714\) 0 0
\(715\) 8.38904e7 0.00858304
\(716\) 3.14182e9 0.319879
\(717\) 0 0
\(718\) −1.42350e9 −0.143523
\(719\) −1.00263e10 −1.00598 −0.502991 0.864292i \(-0.667767\pi\)
−0.502991 + 0.864292i \(0.667767\pi\)
\(720\) 0 0
\(721\) −6.02888e9 −0.599050
\(722\) −3.76367e8 −0.0372161
\(723\) 0 0
\(724\) 5.19666e9 0.508908
\(725\) 9.03295e8 0.0880333
\(726\) 0 0
\(727\) −1.65401e10 −1.59650 −0.798248 0.602329i \(-0.794240\pi\)
−0.798248 + 0.602329i \(0.794240\pi\)
\(728\) −4.18179e8 −0.0401701
\(729\) 0 0
\(730\) 9.28953e9 0.883820
\(731\) 8.96200e9 0.848583
\(732\) 0 0
\(733\) 9.81994e9 0.920969 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(734\) 1.10012e10 1.02684
\(735\) 0 0
\(736\) 3.78793e8 0.0350211
\(737\) 5.85864e8 0.0539089
\(738\) 0 0
\(739\) 5.41304e9 0.493385 0.246692 0.969094i \(-0.420656\pi\)
0.246692 + 0.969094i \(0.420656\pi\)
\(740\) −3.65800e9 −0.331843
\(741\) 0 0
\(742\) 5.22945e9 0.469940
\(743\) 8.26105e9 0.738881 0.369440 0.929254i \(-0.379549\pi\)
0.369440 + 0.929254i \(0.379549\pi\)
\(744\) 0 0
\(745\) −2.28317e10 −2.02298
\(746\) −1.24676e10 −1.09950
\(747\) 0 0
\(748\) −3.02768e8 −0.0264518
\(749\) 4.68975e8 0.0407815
\(750\) 0 0
\(751\) −1.44332e10 −1.24343 −0.621715 0.783243i \(-0.713564\pi\)
−0.621715 + 0.783243i \(0.713564\pi\)
\(752\) −4.06561e9 −0.348628
\(753\) 0 0
\(754\) 3.94463e8 0.0335125
\(755\) −1.29381e10 −1.09410
\(756\) 0 0
\(757\) −8.31294e9 −0.696496 −0.348248 0.937402i \(-0.613223\pi\)
−0.348248 + 0.937402i \(0.613223\pi\)
\(758\) −7.07640e9 −0.590160
\(759\) 0 0
\(760\) 1.09734e9 0.0906764
\(761\) 1.96636e10 1.61739 0.808697 0.588226i \(-0.200173\pi\)
0.808697 + 0.588226i \(0.200173\pi\)
\(762\) 0 0
\(763\) −4.01180e9 −0.326967
\(764\) −4.74057e9 −0.384595
\(765\) 0 0
\(766\) 1.15659e10 0.929774
\(767\) −2.04948e9 −0.164006
\(768\) 0 0
\(769\) 1.27102e10 1.00788 0.503942 0.863738i \(-0.331883\pi\)
0.503942 + 0.863738i \(0.331883\pi\)
\(770\) −4.83100e8 −0.0381346
\(771\) 0 0
\(772\) −6.20880e9 −0.485676
\(773\) 2.87614e9 0.223966 0.111983 0.993710i \(-0.464280\pi\)
0.111983 + 0.993710i \(0.464280\pi\)
\(774\) 0 0
\(775\) 9.14188e8 0.0705472
\(776\) 3.53480e9 0.271549
\(777\) 0 0
\(778\) −8.17315e9 −0.622244
\(779\) 5.62206e9 0.426102
\(780\) 0 0
\(781\) −1.35406e9 −0.101709
\(782\) 1.73580e9 0.129801
\(783\) 0 0
\(784\) −9.65063e8 −0.0715236
\(785\) 9.18498e9 0.677695
\(786\) 0 0
\(787\) 5.61813e9 0.410847 0.205424 0.978673i \(-0.434143\pi\)
0.205424 + 0.978673i \(0.434143\pi\)
\(788\) 2.78253e9 0.202581
\(789\) 0 0
\(790\) 3.27100e9 0.236040
\(791\) −2.74906e9 −0.197500
\(792\) 0 0
\(793\) −2.23227e8 −0.0158961
\(794\) −1.44040e9 −0.102120
\(795\) 0 0
\(796\) 2.49377e9 0.175251
\(797\) 4.98926e9 0.349085 0.174543 0.984650i \(-0.444155\pi\)
0.174543 + 0.984650i \(0.444155\pi\)
\(798\) 0 0
\(799\) −1.86305e10 −1.29214
\(800\) −6.39429e8 −0.0441547
\(801\) 0 0
\(802\) 4.05336e9 0.277463
\(803\) 9.36623e8 0.0638352
\(804\) 0 0
\(805\) 2.76966e9 0.187129
\(806\) 3.99220e8 0.0268559
\(807\) 0 0
\(808\) 1.44184e9 0.0961563
\(809\) −2.60176e10 −1.72762 −0.863809 0.503820i \(-0.831927\pi\)
−0.863809 + 0.503820i \(0.831927\pi\)
\(810\) 0 0
\(811\) 1.36033e10 0.895513 0.447756 0.894156i \(-0.352223\pi\)
0.447756 + 0.894156i \(0.352223\pi\)
\(812\) −2.27160e9 −0.148897
\(813\) 0 0
\(814\) −3.68820e8 −0.0239679
\(815\) 2.82451e9 0.182764
\(816\) 0 0
\(817\) −3.27497e9 −0.210102
\(818\) −9.19184e9 −0.587173
\(819\) 0 0
\(820\) −1.63918e10 −1.03819
\(821\) −5.59803e9 −0.353048 −0.176524 0.984296i \(-0.556485\pi\)
−0.176524 + 0.984296i \(0.556485\pi\)
\(822\) 0 0
\(823\) 2.83609e10 1.77346 0.886728 0.462292i \(-0.152973\pi\)
0.886728 + 0.462292i \(0.152973\pi\)
\(824\) −4.02572e9 −0.250667
\(825\) 0 0
\(826\) 1.18023e10 0.728682
\(827\) 1.94087e10 1.19324 0.596619 0.802524i \(-0.296510\pi\)
0.596619 + 0.802524i \(0.296510\pi\)
\(828\) 0 0
\(829\) 1.25701e10 0.766301 0.383150 0.923686i \(-0.374839\pi\)
0.383150 + 0.923686i \(0.374839\pi\)
\(830\) −1.62906e10 −0.988927
\(831\) 0 0
\(832\) −2.79234e8 −0.0168088
\(833\) −4.42236e9 −0.265092
\(834\) 0 0
\(835\) 2.55205e10 1.51700
\(836\) 1.10640e8 0.00654924
\(837\) 0 0
\(838\) 1.61831e10 0.949968
\(839\) −1.95535e10 −1.14303 −0.571515 0.820592i \(-0.693644\pi\)
−0.571515 + 0.820592i \(0.693644\pi\)
\(840\) 0 0
\(841\) −1.51071e10 −0.875781
\(842\) −2.04402e10 −1.18003
\(843\) 0 0
\(844\) −2.46569e9 −0.141169
\(845\) −1.92526e10 −1.09772
\(846\) 0 0
\(847\) 1.48934e10 0.842175
\(848\) 3.49191e9 0.196642
\(849\) 0 0
\(850\) −2.93016e9 −0.163653
\(851\) 2.11449e9 0.117612
\(852\) 0 0
\(853\) 1.14196e10 0.629986 0.314993 0.949094i \(-0.397998\pi\)
0.314993 + 0.949094i \(0.397998\pi\)
\(854\) 1.28550e9 0.0706266
\(855\) 0 0
\(856\) 3.13153e8 0.0170647
\(857\) 1.94401e10 1.05503 0.527516 0.849545i \(-0.323123\pi\)
0.527516 + 0.849545i \(0.323123\pi\)
\(858\) 0 0
\(859\) 1.07841e10 0.580509 0.290254 0.956950i \(-0.406260\pi\)
0.290254 + 0.956950i \(0.406260\pi\)
\(860\) 9.54856e9 0.511910
\(861\) 0 0
\(862\) 1.17852e10 0.626705
\(863\) 8.45236e8 0.0447651 0.0223826 0.999749i \(-0.492875\pi\)
0.0223826 + 0.999749i \(0.492875\pi\)
\(864\) 0 0
\(865\) −4.16184e10 −2.18640
\(866\) 7.46595e9 0.390636
\(867\) 0 0
\(868\) −2.29899e9 −0.119321
\(869\) 3.29800e8 0.0170483
\(870\) 0 0
\(871\) 2.47602e9 0.126967
\(872\) −2.67884e9 −0.136817
\(873\) 0 0
\(874\) −6.34312e8 −0.0321376
\(875\) 1.40428e10 0.708642
\(876\) 0 0
\(877\) 2.17563e10 1.08915 0.544574 0.838713i \(-0.316691\pi\)
0.544574 + 0.838713i \(0.316691\pi\)
\(878\) 1.55529e10 0.775495
\(879\) 0 0
\(880\) −3.22584e8 −0.0159571
\(881\) −2.18056e10 −1.07436 −0.537182 0.843466i \(-0.680511\pi\)
−0.537182 + 0.843466i \(0.680511\pi\)
\(882\) 0 0
\(883\) 2.25662e9 0.110305 0.0551526 0.998478i \(-0.482435\pi\)
0.0551526 + 0.998478i \(0.482435\pi\)
\(884\) −1.27958e9 −0.0622995
\(885\) 0 0
\(886\) −1.88474e10 −0.910401
\(887\) 1.72271e10 0.828857 0.414428 0.910082i \(-0.363982\pi\)
0.414428 + 0.910082i \(0.363982\pi\)
\(888\) 0 0
\(889\) 2.29620e10 1.09611
\(890\) 9.98877e9 0.474949
\(891\) 0 0
\(892\) −1.30177e10 −0.614124
\(893\) 6.80810e9 0.319923
\(894\) 0 0
\(895\) 1.53396e10 0.715208
\(896\) 1.60803e9 0.0746819
\(897\) 0 0
\(898\) −1.46700e10 −0.676023
\(899\) 2.16861e9 0.0995455
\(900\) 0 0
\(901\) 1.60015e10 0.728827
\(902\) −1.65271e9 −0.0749849
\(903\) 0 0
\(904\) −1.83566e9 −0.0826422
\(905\) 2.53721e10 1.13785
\(906\) 0 0
\(907\) 2.32557e10 1.03491 0.517456 0.855710i \(-0.326879\pi\)
0.517456 + 0.855710i \(0.326879\pi\)
\(908\) −1.43964e10 −0.638195
\(909\) 0 0
\(910\) −2.04171e9 −0.0898150
\(911\) 4.22775e10 1.85266 0.926329 0.376715i \(-0.122946\pi\)
0.926329 + 0.376715i \(0.122946\pi\)
\(912\) 0 0
\(913\) −1.64251e9 −0.0714267
\(914\) 3.01346e10 1.30543
\(915\) 0 0
\(916\) −1.52885e10 −0.657248
\(917\) −6.22072e9 −0.266408
\(918\) 0 0
\(919\) 2.74897e10 1.16833 0.584164 0.811635i \(-0.301422\pi\)
0.584164 + 0.811635i \(0.301422\pi\)
\(920\) 1.84941e9 0.0783026
\(921\) 0 0
\(922\) 2.64448e10 1.11118
\(923\) −5.72261e9 −0.239546
\(924\) 0 0
\(925\) −3.56940e9 −0.148286
\(926\) 2.93061e10 1.21288
\(927\) 0 0
\(928\) −1.51683e9 −0.0623045
\(929\) 2.42814e10 0.993614 0.496807 0.867861i \(-0.334506\pi\)
0.496807 + 0.867861i \(0.334506\pi\)
\(930\) 0 0
\(931\) 1.61606e9 0.0656346
\(932\) −2.06732e10 −0.836475
\(933\) 0 0
\(934\) −1.78139e10 −0.715393
\(935\) −1.47823e9 −0.0591427
\(936\) 0 0
\(937\) −3.53708e10 −1.40461 −0.702305 0.711876i \(-0.747846\pi\)
−0.702305 + 0.711876i \(0.747846\pi\)
\(938\) −1.42586e10 −0.564116
\(939\) 0 0
\(940\) −1.98498e10 −0.779487
\(941\) 3.60255e10 1.40944 0.704719 0.709486i \(-0.251073\pi\)
0.704719 + 0.709486i \(0.251073\pi\)
\(942\) 0 0
\(943\) 9.47516e9 0.367956
\(944\) 7.88088e9 0.304911
\(945\) 0 0
\(946\) 9.62740e8 0.0369734
\(947\) 2.20930e10 0.845336 0.422668 0.906285i \(-0.361094\pi\)
0.422668 + 0.906285i \(0.361094\pi\)
\(948\) 0 0
\(949\) 3.95842e9 0.150345
\(950\) 1.07076e9 0.0405192
\(951\) 0 0
\(952\) 7.36872e9 0.276798
\(953\) 2.36409e9 0.0884787 0.0442393 0.999021i \(-0.485914\pi\)
0.0442393 + 0.999021i \(0.485914\pi\)
\(954\) 0 0
\(955\) −2.31452e10 −0.859903
\(956\) 3.31623e9 0.122756
\(957\) 0 0
\(958\) 1.14806e10 0.421878
\(959\) 1.10535e10 0.404702
\(960\) 0 0
\(961\) −2.53179e10 −0.920227
\(962\) −1.55873e9 −0.0564493
\(963\) 0 0
\(964\) 7.41927e9 0.266742
\(965\) −3.03137e10 −1.08591
\(966\) 0 0
\(967\) 2.69098e10 0.957014 0.478507 0.878084i \(-0.341178\pi\)
0.478507 + 0.878084i \(0.341178\pi\)
\(968\) 9.94491e9 0.352401
\(969\) 0 0
\(970\) 1.72582e10 0.607149
\(971\) 4.71594e9 0.165311 0.0826553 0.996578i \(-0.473660\pi\)
0.0826553 + 0.996578i \(0.473660\pi\)
\(972\) 0 0
\(973\) 2.89413e10 1.00722
\(974\) 1.69314e10 0.587132
\(975\) 0 0
\(976\) 8.58375e8 0.0295531
\(977\) −1.79514e10 −0.615839 −0.307919 0.951412i \(-0.599633\pi\)
−0.307919 + 0.951412i \(0.599633\pi\)
\(978\) 0 0
\(979\) 1.00712e9 0.0343039
\(980\) −4.71180e9 −0.159917
\(981\) 0 0
\(982\) 3.25684e10 1.09750
\(983\) −2.39450e10 −0.804039 −0.402019 0.915631i \(-0.631692\pi\)
−0.402019 + 0.915631i \(0.631692\pi\)
\(984\) 0 0
\(985\) 1.35854e10 0.452944
\(986\) −6.95082e9 −0.230923
\(987\) 0 0
\(988\) 4.67595e8 0.0154248
\(989\) −5.51949e9 −0.181431
\(990\) 0 0
\(991\) 1.07956e10 0.352362 0.176181 0.984358i \(-0.443626\pi\)
0.176181 + 0.984358i \(0.443626\pi\)
\(992\) −1.53512e9 −0.0499289
\(993\) 0 0
\(994\) 3.29548e10 1.06431
\(995\) 1.21755e10 0.391838
\(996\) 0 0
\(997\) −3.54933e10 −1.13426 −0.567131 0.823627i \(-0.691947\pi\)
−0.567131 + 0.823627i \(0.691947\pi\)
\(998\) −3.14431e10 −1.00131
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 342.8.a.g.1.2 2
3.2 odd 2 38.8.a.d.1.2 2
12.11 even 2 304.8.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.8.a.d.1.2 2 3.2 odd 2
304.8.a.d.1.1 2 12.11 even 2
342.8.a.g.1.2 2 1.1 even 1 trivial