Properties

Label 8.20.a.a.1.1
Level $8$
Weight $20$
Character 8.1
Self dual yes
Analytic conductor $18.305$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3053357245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1453}) \)
Defining polynomial: \(x^{2} - x - 363\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.5591\) of defining polynomial
Character \(\chi\) \(=\) 8.1

$q$-expansion

\(f(q)\) \(=\) \(q-50549.5 q^{3} +2.22342e6 q^{5} +1.60989e8 q^{7} +1.39299e9 q^{9} +O(q^{10})\) \(q-50549.5 q^{3} +2.22342e6 q^{5} +1.60989e8 q^{7} +1.39299e9 q^{9} -1.17593e10 q^{11} -2.44117e9 q^{13} -1.12393e11 q^{15} +5.98454e11 q^{17} -2.51671e12 q^{19} -8.13789e12 q^{21} -6.35737e11 q^{23} -1.41299e13 q^{25} -1.16633e13 q^{27} -4.78062e13 q^{29} -2.03104e14 q^{31} +5.94429e14 q^{33} +3.57946e14 q^{35} +8.95266e14 q^{37} +1.23400e14 q^{39} -4.61047e14 q^{41} -2.21551e15 q^{43} +3.09721e15 q^{45} -1.04653e16 q^{47} +1.45184e16 q^{49} -3.02516e16 q^{51} -3.43716e16 q^{53} -2.61460e16 q^{55} +1.27219e17 q^{57} +9.13126e16 q^{59} -1.19474e17 q^{61} +2.24256e17 q^{63} -5.42776e15 q^{65} -1.98838e17 q^{67} +3.21362e16 q^{69} -1.38863e17 q^{71} -3.63434e16 q^{73} +7.14258e17 q^{75} -1.89312e18 q^{77} -4.36121e17 q^{79} -1.02945e18 q^{81} +4.52389e17 q^{83} +1.33062e18 q^{85} +2.41658e18 q^{87} +3.30628e18 q^{89} -3.93001e17 q^{91} +1.02668e19 q^{93} -5.59572e18 q^{95} -6.03558e18 q^{97} -1.63807e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 27912q^{3} + 1226620q^{5} + 88510512q^{7} + 743186538q^{9} + O(q^{10}) \) \( 2q - 27912q^{3} + 1226620q^{5} + 88510512q^{7} + 743186538q^{9} - 7163787608q^{11} - 10126923604q^{13} - 134958171120q^{15} - 72045078940q^{17} - 3120480472232q^{19} - 9778613729472q^{21} - 14759207090288q^{23} - 32209737998450q^{25} - 52683939132624q^{27} - 30249539245044q^{29} - 123389562777920q^{31} + 698460345860448q^{33} + 430192267170720q^{35} + 2015393170174524q^{37} - 50586287718576q^{39} + 2540784959504244q^{41} - 5633655093389464q^{43} + 3744938810129580q^{45} - 21948339587130336q^{47} + 8372586663064786q^{49} - 45429993200591760q^{51} - 9418125066904676q^{53} - 30726853722003280q^{55} + 113550921657554592q^{57} + 98542449590407624q^{59} + 10292145377839820q^{61} + 271352203396831728q^{63} + 2233432360014760q^{65} + 75753628003984504q^{67} - 287583943397508672q^{69} + 17407052566713776q^{71} - 857508255059832268q^{73} + 304974862137280200q^{75} - 2226194512471875648q^{77} - 226291921444855072q^{79} - 1202809709767720302q^{81} + 767515701460985048q^{83} + 1998974022815558200q^{85} + 2814015710341120464q^{87} + 6092545894435174548q^{89} + 164047654392377376q^{91} + 12071374787086705920q^{93} - 4993887710656339120q^{95} + 1548148249522347076q^{97} - 19366859521479728952q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −50549.5 −1.48274 −0.741370 0.671096i \(-0.765824\pi\)
−0.741370 + 0.671096i \(0.765824\pi\)
\(4\) 0 0
\(5\) 2.22342e6 0.509105 0.254552 0.967059i \(-0.418072\pi\)
0.254552 + 0.967059i \(0.418072\pi\)
\(6\) 0 0
\(7\) 1.60989e8 1.50787 0.753934 0.656950i \(-0.228154\pi\)
0.753934 + 0.656950i \(0.228154\pi\)
\(8\) 0 0
\(9\) 1.39299e9 1.19852
\(10\) 0 0
\(11\) −1.17593e10 −1.50367 −0.751835 0.659352i \(-0.770831\pi\)
−0.751835 + 0.659352i \(0.770831\pi\)
\(12\) 0 0
\(13\) −2.44117e9 −0.0638464 −0.0319232 0.999490i \(-0.510163\pi\)
−0.0319232 + 0.999490i \(0.510163\pi\)
\(14\) 0 0
\(15\) −1.12393e11 −0.754870
\(16\) 0 0
\(17\) 5.98454e11 1.22396 0.611978 0.790874i \(-0.290374\pi\)
0.611978 + 0.790874i \(0.290374\pi\)
\(18\) 0 0
\(19\) −2.51671e12 −1.78927 −0.894633 0.446802i \(-0.852563\pi\)
−0.894633 + 0.446802i \(0.852563\pi\)
\(20\) 0 0
\(21\) −8.13789e12 −2.23578
\(22\) 0 0
\(23\) −6.35737e11 −0.0735975 −0.0367988 0.999323i \(-0.511716\pi\)
−0.0367988 + 0.999323i \(0.511716\pi\)
\(24\) 0 0
\(25\) −1.41299e13 −0.740812
\(26\) 0 0
\(27\) −1.16633e13 −0.294351
\(28\) 0 0
\(29\) −4.78062e13 −0.611932 −0.305966 0.952043i \(-0.598979\pi\)
−0.305966 + 0.952043i \(0.598979\pi\)
\(30\) 0 0
\(31\) −2.03104e14 −1.37970 −0.689848 0.723954i \(-0.742323\pi\)
−0.689848 + 0.723954i \(0.742323\pi\)
\(32\) 0 0
\(33\) 5.94429e14 2.22955
\(34\) 0 0
\(35\) 3.57946e14 0.767664
\(36\) 0 0
\(37\) 8.95266e14 1.13249 0.566246 0.824236i \(-0.308395\pi\)
0.566246 + 0.824236i \(0.308395\pi\)
\(38\) 0 0
\(39\) 1.23400e14 0.0946676
\(40\) 0 0
\(41\) −4.61047e14 −0.219937 −0.109969 0.993935i \(-0.535075\pi\)
−0.109969 + 0.993935i \(0.535075\pi\)
\(42\) 0 0
\(43\) −2.21551e15 −0.672238 −0.336119 0.941820i \(-0.609114\pi\)
−0.336119 + 0.941820i \(0.609114\pi\)
\(44\) 0 0
\(45\) 3.09721e15 0.610171
\(46\) 0 0
\(47\) −1.04653e16 −1.36403 −0.682013 0.731340i \(-0.738895\pi\)
−0.682013 + 0.731340i \(0.738895\pi\)
\(48\) 0 0
\(49\) 1.45184e16 1.27367
\(50\) 0 0
\(51\) −3.02516e16 −1.81481
\(52\) 0 0
\(53\) −3.43716e16 −1.43080 −0.715400 0.698715i \(-0.753756\pi\)
−0.715400 + 0.698715i \(0.753756\pi\)
\(54\) 0 0
\(55\) −2.61460e16 −0.765525
\(56\) 0 0
\(57\) 1.27219e17 2.65302
\(58\) 0 0
\(59\) 9.13126e16 1.37226 0.686130 0.727479i \(-0.259308\pi\)
0.686130 + 0.727479i \(0.259308\pi\)
\(60\) 0 0
\(61\) −1.19474e17 −1.30809 −0.654047 0.756454i \(-0.726930\pi\)
−0.654047 + 0.756454i \(0.726930\pi\)
\(62\) 0 0
\(63\) 2.24256e17 1.80721
\(64\) 0 0
\(65\) −5.42776e15 −0.0325045
\(66\) 0 0
\(67\) −1.98838e17 −0.892872 −0.446436 0.894816i \(-0.647307\pi\)
−0.446436 + 0.894816i \(0.647307\pi\)
\(68\) 0 0
\(69\) 3.21362e16 0.109126
\(70\) 0 0
\(71\) −1.38863e17 −0.359446 −0.179723 0.983717i \(-0.557520\pi\)
−0.179723 + 0.983717i \(0.557520\pi\)
\(72\) 0 0
\(73\) −3.63434e16 −0.0722533 −0.0361267 0.999347i \(-0.511502\pi\)
−0.0361267 + 0.999347i \(0.511502\pi\)
\(74\) 0 0
\(75\) 7.14258e17 1.09843
\(76\) 0 0
\(77\) −1.89312e18 −2.26734
\(78\) 0 0
\(79\) −4.36121e17 −0.409402 −0.204701 0.978825i \(-0.565622\pi\)
−0.204701 + 0.978825i \(0.565622\pi\)
\(80\) 0 0
\(81\) −1.02945e18 −0.762073
\(82\) 0 0
\(83\) 4.52389e17 0.265626 0.132813 0.991141i \(-0.457599\pi\)
0.132813 + 0.991141i \(0.457599\pi\)
\(84\) 0 0
\(85\) 1.33062e18 0.623123
\(86\) 0 0
\(87\) 2.41658e18 0.907335
\(88\) 0 0
\(89\) 3.30628e18 1.00031 0.500154 0.865936i \(-0.333277\pi\)
0.500154 + 0.865936i \(0.333277\pi\)
\(90\) 0 0
\(91\) −3.93001e17 −0.0962720
\(92\) 0 0
\(93\) 1.02668e19 2.04573
\(94\) 0 0
\(95\) −5.59572e18 −0.910924
\(96\) 0 0
\(97\) −6.03558e18 −0.806097 −0.403049 0.915179i \(-0.632049\pi\)
−0.403049 + 0.915179i \(0.632049\pi\)
\(98\) 0 0
\(99\) −1.63807e19 −1.80217
\(100\) 0 0
\(101\) 9.09027e18 0.827034 0.413517 0.910496i \(-0.364300\pi\)
0.413517 + 0.910496i \(0.364300\pi\)
\(102\) 0 0
\(103\) 1.16083e19 0.876628 0.438314 0.898822i \(-0.355576\pi\)
0.438314 + 0.898822i \(0.355576\pi\)
\(104\) 0 0
\(105\) −1.80940e19 −1.13825
\(106\) 0 0
\(107\) 9.99623e18 0.525642 0.262821 0.964845i \(-0.415347\pi\)
0.262821 + 0.964845i \(0.415347\pi\)
\(108\) 0 0
\(109\) −4.34179e18 −0.191477 −0.0957385 0.995407i \(-0.530521\pi\)
−0.0957385 + 0.995407i \(0.530521\pi\)
\(110\) 0 0
\(111\) −4.52552e19 −1.67919
\(112\) 0 0
\(113\) −9.09237e18 −0.284729 −0.142365 0.989814i \(-0.545471\pi\)
−0.142365 + 0.989814i \(0.545471\pi\)
\(114\) 0 0
\(115\) −1.41351e18 −0.0374689
\(116\) 0 0
\(117\) −3.40053e18 −0.0765210
\(118\) 0 0
\(119\) 9.63443e19 1.84557
\(120\) 0 0
\(121\) 7.71229e19 1.26102
\(122\) 0 0
\(123\) 2.33057e19 0.326110
\(124\) 0 0
\(125\) −7.38252e19 −0.886256
\(126\) 0 0
\(127\) 9.31085e19 0.961289 0.480645 0.876915i \(-0.340403\pi\)
0.480645 + 0.876915i \(0.340403\pi\)
\(128\) 0 0
\(129\) 1.11993e20 0.996753
\(130\) 0 0
\(131\) 1.87606e19 0.144268 0.0721340 0.997395i \(-0.477019\pi\)
0.0721340 + 0.997395i \(0.477019\pi\)
\(132\) 0 0
\(133\) −4.05162e20 −2.69798
\(134\) 0 0
\(135\) −2.59324e19 −0.149855
\(136\) 0 0
\(137\) −1.84318e20 −0.926240 −0.463120 0.886296i \(-0.653270\pi\)
−0.463120 + 0.886296i \(0.653270\pi\)
\(138\) 0 0
\(139\) −1.59539e20 −0.698597 −0.349299 0.937011i \(-0.613580\pi\)
−0.349299 + 0.937011i \(0.613580\pi\)
\(140\) 0 0
\(141\) 5.29017e20 2.02250
\(142\) 0 0
\(143\) 2.87065e19 0.0960038
\(144\) 0 0
\(145\) −1.06293e20 −0.311537
\(146\) 0 0
\(147\) −7.33899e20 −1.88852
\(148\) 0 0
\(149\) 3.21712e20 0.728111 0.364055 0.931377i \(-0.381392\pi\)
0.364055 + 0.931377i \(0.381392\pi\)
\(150\) 0 0
\(151\) −1.55272e20 −0.309607 −0.154804 0.987945i \(-0.549474\pi\)
−0.154804 + 0.987945i \(0.549474\pi\)
\(152\) 0 0
\(153\) 8.33641e20 1.46693
\(154\) 0 0
\(155\) −4.51587e20 −0.702410
\(156\) 0 0
\(157\) −4.54202e20 −0.625464 −0.312732 0.949841i \(-0.601244\pi\)
−0.312732 + 0.949841i \(0.601244\pi\)
\(158\) 0 0
\(159\) 1.73747e21 2.12150
\(160\) 0 0
\(161\) −1.02346e20 −0.110975
\(162\) 0 0
\(163\) 7.72395e20 0.744830 0.372415 0.928066i \(-0.378530\pi\)
0.372415 + 0.928066i \(0.378530\pi\)
\(164\) 0 0
\(165\) 1.32167e21 1.13508
\(166\) 0 0
\(167\) 1.02583e21 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(168\) 0 0
\(169\) −1.45596e21 −0.995924
\(170\) 0 0
\(171\) −3.50576e21 −2.14447
\(172\) 0 0
\(173\) 6.92487e20 0.379292 0.189646 0.981853i \(-0.439266\pi\)
0.189646 + 0.981853i \(0.439266\pi\)
\(174\) 0 0
\(175\) −2.27475e21 −1.11705
\(176\) 0 0
\(177\) −4.61581e21 −2.03471
\(178\) 0 0
\(179\) 2.42385e21 0.960290 0.480145 0.877189i \(-0.340584\pi\)
0.480145 + 0.877189i \(0.340584\pi\)
\(180\) 0 0
\(181\) −9.72457e20 −0.346676 −0.173338 0.984862i \(-0.555455\pi\)
−0.173338 + 0.984862i \(0.555455\pi\)
\(182\) 0 0
\(183\) 6.03934e21 1.93956
\(184\) 0 0
\(185\) 1.99056e21 0.576558
\(186\) 0 0
\(187\) −7.03742e21 −1.84043
\(188\) 0 0
\(189\) −1.87765e21 −0.443842
\(190\) 0 0
\(191\) 6.76037e21 1.44595 0.722975 0.690874i \(-0.242774\pi\)
0.722975 + 0.690874i \(0.242774\pi\)
\(192\) 0 0
\(193\) 1.53000e21 0.296413 0.148206 0.988956i \(-0.452650\pi\)
0.148206 + 0.988956i \(0.452650\pi\)
\(194\) 0 0
\(195\) 2.74371e20 0.0481957
\(196\) 0 0
\(197\) −6.79113e21 −1.08271 −0.541356 0.840793i \(-0.682089\pi\)
−0.541356 + 0.840793i \(0.682089\pi\)
\(198\) 0 0
\(199\) 9.45465e21 1.36943 0.684717 0.728809i \(-0.259926\pi\)
0.684717 + 0.728809i \(0.259926\pi\)
\(200\) 0 0
\(201\) 1.00512e22 1.32390
\(202\) 0 0
\(203\) −7.69624e21 −0.922713
\(204\) 0 0
\(205\) −1.02510e21 −0.111971
\(206\) 0 0
\(207\) −8.85577e20 −0.0882080
\(208\) 0 0
\(209\) 2.95949e22 2.69046
\(210\) 0 0
\(211\) −2.14995e22 −1.78544 −0.892721 0.450610i \(-0.851207\pi\)
−0.892721 + 0.450610i \(0.851207\pi\)
\(212\) 0 0
\(213\) 7.01948e21 0.532965
\(214\) 0 0
\(215\) −4.92601e21 −0.342239
\(216\) 0 0
\(217\) −3.26975e22 −2.08040
\(218\) 0 0
\(219\) 1.83714e21 0.107133
\(220\) 0 0
\(221\) −1.46093e21 −0.0781452
\(222\) 0 0
\(223\) −3.89942e22 −1.91471 −0.957356 0.288910i \(-0.906707\pi\)
−0.957356 + 0.288910i \(0.906707\pi\)
\(224\) 0 0
\(225\) −1.96828e22 −0.887877
\(226\) 0 0
\(227\) 1.69117e21 0.0701361 0.0350680 0.999385i \(-0.488835\pi\)
0.0350680 + 0.999385i \(0.488835\pi\)
\(228\) 0 0
\(229\) 3.34630e21 0.127681 0.0638407 0.997960i \(-0.479665\pi\)
0.0638407 + 0.997960i \(0.479665\pi\)
\(230\) 0 0
\(231\) 9.56962e22 3.36187
\(232\) 0 0
\(233\) 1.27609e22 0.413049 0.206524 0.978441i \(-0.433785\pi\)
0.206524 + 0.978441i \(0.433785\pi\)
\(234\) 0 0
\(235\) −2.32688e22 −0.694433
\(236\) 0 0
\(237\) 2.20457e22 0.607037
\(238\) 0 0
\(239\) −9.15938e21 −0.232855 −0.116428 0.993199i \(-0.537144\pi\)
−0.116428 + 0.993199i \(0.537144\pi\)
\(240\) 0 0
\(241\) −4.54596e22 −1.06774 −0.533868 0.845568i \(-0.679262\pi\)
−0.533868 + 0.845568i \(0.679262\pi\)
\(242\) 0 0
\(243\) 6.55938e22 1.42431
\(244\) 0 0
\(245\) 3.22806e22 0.648431
\(246\) 0 0
\(247\) 6.14373e21 0.114238
\(248\) 0 0
\(249\) −2.28680e22 −0.393854
\(250\) 0 0
\(251\) −3.68788e22 −0.588677 −0.294338 0.955701i \(-0.595099\pi\)
−0.294338 + 0.955701i \(0.595099\pi\)
\(252\) 0 0
\(253\) 7.47585e21 0.110666
\(254\) 0 0
\(255\) −6.72621e22 −0.923929
\(256\) 0 0
\(257\) 4.84420e22 0.617813 0.308907 0.951092i \(-0.400037\pi\)
0.308907 + 0.951092i \(0.400037\pi\)
\(258\) 0 0
\(259\) 1.44128e23 1.70765
\(260\) 0 0
\(261\) −6.65936e22 −0.733411
\(262\) 0 0
\(263\) 6.84645e22 0.701271 0.350635 0.936512i \(-0.385966\pi\)
0.350635 + 0.936512i \(0.385966\pi\)
\(264\) 0 0
\(265\) −7.64226e22 −0.728427
\(266\) 0 0
\(267\) −1.67131e23 −1.48320
\(268\) 0 0
\(269\) −2.17729e23 −1.79999 −0.899995 0.435900i \(-0.856430\pi\)
−0.899995 + 0.435900i \(0.856430\pi\)
\(270\) 0 0
\(271\) −1.88487e23 −1.45236 −0.726180 0.687505i \(-0.758706\pi\)
−0.726180 + 0.687505i \(0.758706\pi\)
\(272\) 0 0
\(273\) 1.98660e22 0.142746
\(274\) 0 0
\(275\) 1.66158e23 1.11394
\(276\) 0 0
\(277\) 2.07964e23 1.30146 0.650729 0.759310i \(-0.274463\pi\)
0.650729 + 0.759310i \(0.274463\pi\)
\(278\) 0 0
\(279\) −2.82923e23 −1.65359
\(280\) 0 0
\(281\) 3.33849e23 1.82322 0.911612 0.411052i \(-0.134838\pi\)
0.911612 + 0.411052i \(0.134838\pi\)
\(282\) 0 0
\(283\) −2.26611e22 −0.115694 −0.0578470 0.998325i \(-0.518424\pi\)
−0.0578470 + 0.998325i \(0.518424\pi\)
\(284\) 0 0
\(285\) 2.82861e23 1.35066
\(286\) 0 0
\(287\) −7.42233e22 −0.331636
\(288\) 0 0
\(289\) 1.19075e23 0.498070
\(290\) 0 0
\(291\) 3.05095e23 1.19523
\(292\) 0 0
\(293\) −1.85421e23 −0.680639 −0.340319 0.940310i \(-0.610535\pi\)
−0.340319 + 0.940310i \(0.610535\pi\)
\(294\) 0 0
\(295\) 2.03027e23 0.698625
\(296\) 0 0
\(297\) 1.37152e23 0.442606
\(298\) 0 0
\(299\) 1.55194e21 0.00469894
\(300\) 0 0
\(301\) −3.56671e23 −1.01365
\(302\) 0 0
\(303\) −4.59508e23 −1.22628
\(304\) 0 0
\(305\) −2.65641e23 −0.665958
\(306\) 0 0
\(307\) 7.93628e23 1.86983 0.934916 0.354870i \(-0.115475\pi\)
0.934916 + 0.354870i \(0.115475\pi\)
\(308\) 0 0
\(309\) −5.86794e23 −1.29981
\(310\) 0 0
\(311\) −5.20440e23 −1.08429 −0.542147 0.840284i \(-0.682388\pi\)
−0.542147 + 0.840284i \(0.682388\pi\)
\(312\) 0 0
\(313\) −1.65345e23 −0.324130 −0.162065 0.986780i \(-0.551815\pi\)
−0.162065 + 0.986780i \(0.551815\pi\)
\(314\) 0 0
\(315\) 4.98615e23 0.920059
\(316\) 0 0
\(317\) 5.50226e23 0.956044 0.478022 0.878348i \(-0.341354\pi\)
0.478022 + 0.878348i \(0.341354\pi\)
\(318\) 0 0
\(319\) 5.62169e23 0.920143
\(320\) 0 0
\(321\) −5.05304e23 −0.779391
\(322\) 0 0
\(323\) −1.50614e24 −2.18998
\(324\) 0 0
\(325\) 3.44934e22 0.0472982
\(326\) 0 0
\(327\) 2.19475e23 0.283911
\(328\) 0 0
\(329\) −1.68480e24 −2.05677
\(330\) 0 0
\(331\) 5.52973e23 0.637291 0.318646 0.947874i \(-0.396772\pi\)
0.318646 + 0.947874i \(0.396772\pi\)
\(332\) 0 0
\(333\) 1.24710e24 1.35731
\(334\) 0 0
\(335\) −4.42102e23 −0.454565
\(336\) 0 0
\(337\) 5.19323e23 0.504607 0.252303 0.967648i \(-0.418812\pi\)
0.252303 + 0.967648i \(0.418812\pi\)
\(338\) 0 0
\(339\) 4.59615e23 0.422179
\(340\) 0 0
\(341\) 2.38837e24 2.07461
\(342\) 0 0
\(343\) 5.02207e23 0.412657
\(344\) 0 0
\(345\) 7.14524e22 0.0555566
\(346\) 0 0
\(347\) 1.43821e24 1.05850 0.529250 0.848466i \(-0.322473\pi\)
0.529250 + 0.848466i \(0.322473\pi\)
\(348\) 0 0
\(349\) −8.46494e23 −0.589905 −0.294952 0.955512i \(-0.595304\pi\)
−0.294952 + 0.955512i \(0.595304\pi\)
\(350\) 0 0
\(351\) 2.84721e22 0.0187932
\(352\) 0 0
\(353\) 1.34329e24 0.840063 0.420031 0.907510i \(-0.362019\pi\)
0.420031 + 0.907510i \(0.362019\pi\)
\(354\) 0 0
\(355\) −3.08752e23 −0.182996
\(356\) 0 0
\(357\) −4.87015e24 −2.73650
\(358\) 0 0
\(359\) −1.45409e24 −0.774805 −0.387403 0.921911i \(-0.626628\pi\)
−0.387403 + 0.921911i \(0.626628\pi\)
\(360\) 0 0
\(361\) 4.35543e24 2.20147
\(362\) 0 0
\(363\) −3.89852e24 −1.86977
\(364\) 0 0
\(365\) −8.08067e22 −0.0367845
\(366\) 0 0
\(367\) −1.21387e24 −0.524620 −0.262310 0.964984i \(-0.584484\pi\)
−0.262310 + 0.964984i \(0.584484\pi\)
\(368\) 0 0
\(369\) −6.42234e23 −0.263599
\(370\) 0 0
\(371\) −5.53343e24 −2.15746
\(372\) 0 0
\(373\) 3.39082e24 1.25624 0.628118 0.778118i \(-0.283826\pi\)
0.628118 + 0.778118i \(0.283826\pi\)
\(374\) 0 0
\(375\) 3.73183e24 1.31409
\(376\) 0 0
\(377\) 1.16703e23 0.0390696
\(378\) 0 0
\(379\) 4.13450e24 1.31629 0.658144 0.752892i \(-0.271342\pi\)
0.658144 + 0.752892i \(0.271342\pi\)
\(380\) 0 0
\(381\) −4.70659e24 −1.42534
\(382\) 0 0
\(383\) 3.88317e24 1.11892 0.559459 0.828858i \(-0.311009\pi\)
0.559459 + 0.828858i \(0.311009\pi\)
\(384\) 0 0
\(385\) −4.20921e24 −1.15431
\(386\) 0 0
\(387\) −3.08618e24 −0.805689
\(388\) 0 0
\(389\) 2.58080e24 0.641553 0.320777 0.947155i \(-0.396056\pi\)
0.320777 + 0.947155i \(0.396056\pi\)
\(390\) 0 0
\(391\) −3.80460e23 −0.0900802
\(392\) 0 0
\(393\) −9.48340e23 −0.213912
\(394\) 0 0
\(395\) −9.69683e23 −0.208429
\(396\) 0 0
\(397\) −4.11532e24 −0.843129 −0.421564 0.906799i \(-0.638519\pi\)
−0.421564 + 0.906799i \(0.638519\pi\)
\(398\) 0 0
\(399\) 2.04807e25 4.00040
\(400\) 0 0
\(401\) 6.60175e24 1.22967 0.614833 0.788657i \(-0.289223\pi\)
0.614833 + 0.788657i \(0.289223\pi\)
\(402\) 0 0
\(403\) 4.95813e23 0.0880886
\(404\) 0 0
\(405\) −2.28890e24 −0.387975
\(406\) 0 0
\(407\) −1.05277e25 −1.70289
\(408\) 0 0
\(409\) −1.13278e25 −1.74893 −0.874467 0.485084i \(-0.838789\pi\)
−0.874467 + 0.485084i \(0.838789\pi\)
\(410\) 0 0
\(411\) 9.31721e24 1.37337
\(412\) 0 0
\(413\) 1.47003e25 2.06919
\(414\) 0 0
\(415\) 1.00585e24 0.135231
\(416\) 0 0
\(417\) 8.06463e24 1.03584
\(418\) 0 0
\(419\) −1.26882e25 −1.55729 −0.778643 0.627468i \(-0.784092\pi\)
−0.778643 + 0.627468i \(0.784092\pi\)
\(420\) 0 0
\(421\) −2.67974e24 −0.314349 −0.157175 0.987571i \(-0.550239\pi\)
−0.157175 + 0.987571i \(0.550239\pi\)
\(422\) 0 0
\(423\) −1.45781e25 −1.63481
\(424\) 0 0
\(425\) −8.45608e24 −0.906722
\(426\) 0 0
\(427\) −1.92339e25 −1.97244
\(428\) 0 0
\(429\) −1.45110e24 −0.142349
\(430\) 0 0
\(431\) 7.37367e24 0.692069 0.346035 0.938222i \(-0.387528\pi\)
0.346035 + 0.938222i \(0.387528\pi\)
\(432\) 0 0
\(433\) −1.67233e25 −1.50206 −0.751028 0.660271i \(-0.770442\pi\)
−0.751028 + 0.660271i \(0.770442\pi\)
\(434\) 0 0
\(435\) 5.37308e24 0.461929
\(436\) 0 0
\(437\) 1.59997e24 0.131686
\(438\) 0 0
\(439\) 1.48622e25 1.17130 0.585651 0.810563i \(-0.300839\pi\)
0.585651 + 0.810563i \(0.300839\pi\)
\(440\) 0 0
\(441\) 2.02240e25 1.52651
\(442\) 0 0
\(443\) 7.05393e24 0.510030 0.255015 0.966937i \(-0.417920\pi\)
0.255015 + 0.966937i \(0.417920\pi\)
\(444\) 0 0
\(445\) 7.35126e24 0.509262
\(446\) 0 0
\(447\) −1.62624e25 −1.07960
\(448\) 0 0
\(449\) 6.86136e24 0.436586 0.218293 0.975883i \(-0.429951\pi\)
0.218293 + 0.975883i \(0.429951\pi\)
\(450\) 0 0
\(451\) 5.42161e24 0.330713
\(452\) 0 0
\(453\) 7.84891e24 0.459067
\(454\) 0 0
\(455\) −8.73807e23 −0.0490125
\(456\) 0 0
\(457\) −1.33077e25 −0.715975 −0.357988 0.933726i \(-0.616537\pi\)
−0.357988 + 0.933726i \(0.616537\pi\)
\(458\) 0 0
\(459\) −6.97994e24 −0.360272
\(460\) 0 0
\(461\) 2.20176e25 1.09046 0.545231 0.838286i \(-0.316442\pi\)
0.545231 + 0.838286i \(0.316442\pi\)
\(462\) 0 0
\(463\) −2.34544e25 −1.11482 −0.557410 0.830237i \(-0.688205\pi\)
−0.557410 + 0.830237i \(0.688205\pi\)
\(464\) 0 0
\(465\) 2.28275e25 1.04149
\(466\) 0 0
\(467\) 1.76282e25 0.772143 0.386071 0.922469i \(-0.373832\pi\)
0.386071 + 0.922469i \(0.373832\pi\)
\(468\) 0 0
\(469\) −3.20107e25 −1.34633
\(470\) 0 0
\(471\) 2.29597e25 0.927400
\(472\) 0 0
\(473\) 2.60529e25 1.01082
\(474\) 0 0
\(475\) 3.55609e25 1.32551
\(476\) 0 0
\(477\) −4.78793e25 −1.71484
\(478\) 0 0
\(479\) −2.55873e25 −0.880719 −0.440359 0.897822i \(-0.645149\pi\)
−0.440359 + 0.897822i \(0.645149\pi\)
\(480\) 0 0
\(481\) −2.18550e24 −0.0723056
\(482\) 0 0
\(483\) 5.17356e24 0.164548
\(484\) 0 0
\(485\) −1.34197e25 −0.410388
\(486\) 0 0
\(487\) 1.39237e25 0.409478 0.204739 0.978817i \(-0.434365\pi\)
0.204739 + 0.978817i \(0.434365\pi\)
\(488\) 0 0
\(489\) −3.90442e25 −1.10439
\(490\) 0 0
\(491\) −5.82993e24 −0.158631 −0.0793157 0.996850i \(-0.525274\pi\)
−0.0793157 + 0.996850i \(0.525274\pi\)
\(492\) 0 0
\(493\) −2.86098e25 −0.748978
\(494\) 0 0
\(495\) −3.64211e25 −0.917496
\(496\) 0 0
\(497\) −2.23554e25 −0.541997
\(498\) 0 0
\(499\) 1.49503e25 0.348895 0.174447 0.984666i \(-0.444186\pi\)
0.174447 + 0.984666i \(0.444186\pi\)
\(500\) 0 0
\(501\) −5.18554e25 −1.16503
\(502\) 0 0
\(503\) −1.92509e25 −0.416443 −0.208222 0.978082i \(-0.566768\pi\)
−0.208222 + 0.978082i \(0.566768\pi\)
\(504\) 0 0
\(505\) 2.02115e25 0.421047
\(506\) 0 0
\(507\) 7.35981e25 1.47670
\(508\) 0 0
\(509\) 5.27794e25 1.02011 0.510053 0.860143i \(-0.329626\pi\)
0.510053 + 0.860143i \(0.329626\pi\)
\(510\) 0 0
\(511\) −5.85086e24 −0.108949
\(512\) 0 0
\(513\) 2.93531e25 0.526671
\(514\) 0 0
\(515\) 2.58102e25 0.446296
\(516\) 0 0
\(517\) 1.23065e26 2.05104
\(518\) 0 0
\(519\) −3.50049e25 −0.562391
\(520\) 0 0
\(521\) 4.00961e25 0.621074 0.310537 0.950561i \(-0.399491\pi\)
0.310537 + 0.950561i \(0.399491\pi\)
\(522\) 0 0
\(523\) −5.30249e25 −0.791979 −0.395989 0.918255i \(-0.629598\pi\)
−0.395989 + 0.918255i \(0.629598\pi\)
\(524\) 0 0
\(525\) 1.14987e26 1.65629
\(526\) 0 0
\(527\) −1.21549e26 −1.68869
\(528\) 0 0
\(529\) −7.42113e25 −0.994583
\(530\) 0 0
\(531\) 1.27198e26 1.64468
\(532\) 0 0
\(533\) 1.12549e24 0.0140422
\(534\) 0 0
\(535\) 2.22259e25 0.267607
\(536\) 0 0
\(537\) −1.22525e26 −1.42386
\(538\) 0 0
\(539\) −1.70727e26 −1.91518
\(540\) 0 0
\(541\) 1.31942e26 1.42892 0.714461 0.699676i \(-0.246672\pi\)
0.714461 + 0.699676i \(0.246672\pi\)
\(542\) 0 0
\(543\) 4.91572e25 0.514031
\(544\) 0 0
\(545\) −9.65363e24 −0.0974819
\(546\) 0 0
\(547\) 3.18692e25 0.310807 0.155404 0.987851i \(-0.450332\pi\)
0.155404 + 0.987851i \(0.450332\pi\)
\(548\) 0 0
\(549\) −1.66426e26 −1.56778
\(550\) 0 0
\(551\) 1.20314e26 1.09491
\(552\) 0 0
\(553\) −7.02106e25 −0.617325
\(554\) 0 0
\(555\) −1.00622e26 −0.854885
\(556\) 0 0
\(557\) −1.30222e26 −1.06920 −0.534600 0.845105i \(-0.679538\pi\)
−0.534600 + 0.845105i \(0.679538\pi\)
\(558\) 0 0
\(559\) 5.40843e24 0.0429199
\(560\) 0 0
\(561\) 3.55738e26 2.72887
\(562\) 0 0
\(563\) −8.14771e23 −0.00604235 −0.00302118 0.999995i \(-0.500962\pi\)
−0.00302118 + 0.999995i \(0.500962\pi\)
\(564\) 0 0
\(565\) −2.02162e25 −0.144957
\(566\) 0 0
\(567\) −1.65729e26 −1.14911
\(568\) 0 0
\(569\) −1.13811e26 −0.763161 −0.381580 0.924336i \(-0.624620\pi\)
−0.381580 + 0.924336i \(0.624620\pi\)
\(570\) 0 0
\(571\) −6.33817e25 −0.411075 −0.205537 0.978649i \(-0.565894\pi\)
−0.205537 + 0.978649i \(0.565894\pi\)
\(572\) 0 0
\(573\) −3.41733e26 −2.14397
\(574\) 0 0
\(575\) 8.98289e24 0.0545220
\(576\) 0 0
\(577\) 1.26802e26 0.744654 0.372327 0.928102i \(-0.378560\pi\)
0.372327 + 0.928102i \(0.378560\pi\)
\(578\) 0 0
\(579\) −7.73408e25 −0.439503
\(580\) 0 0
\(581\) 7.28295e25 0.400529
\(582\) 0 0
\(583\) 4.04187e26 2.15145
\(584\) 0 0
\(585\) −7.56082e24 −0.0389572
\(586\) 0 0
\(587\) 2.28792e26 1.14125 0.570623 0.821212i \(-0.306702\pi\)
0.570623 + 0.821212i \(0.306702\pi\)
\(588\) 0 0
\(589\) 5.11156e26 2.46864
\(590\) 0 0
\(591\) 3.43288e26 1.60538
\(592\) 0 0
\(593\) −1.87324e26 −0.848348 −0.424174 0.905581i \(-0.639435\pi\)
−0.424174 + 0.905581i \(0.639435\pi\)
\(594\) 0 0
\(595\) 2.14214e26 0.939587
\(596\) 0 0
\(597\) −4.77928e26 −2.03051
\(598\) 0 0
\(599\) −8.85467e25 −0.364433 −0.182217 0.983258i \(-0.558327\pi\)
−0.182217 + 0.983258i \(0.558327\pi\)
\(600\) 0 0
\(601\) −3.61951e25 −0.144325 −0.0721625 0.997393i \(-0.522990\pi\)
−0.0721625 + 0.997393i \(0.522990\pi\)
\(602\) 0 0
\(603\) −2.76980e26 −1.07012
\(604\) 0 0
\(605\) 1.71477e26 0.641992
\(606\) 0 0
\(607\) −2.25964e26 −0.819873 −0.409936 0.912114i \(-0.634449\pi\)
−0.409936 + 0.912114i \(0.634449\pi\)
\(608\) 0 0
\(609\) 3.89041e26 1.36814
\(610\) 0 0
\(611\) 2.55476e25 0.0870882
\(612\) 0 0
\(613\) −3.22922e26 −1.06714 −0.533572 0.845755i \(-0.679151\pi\)
−0.533572 + 0.845755i \(0.679151\pi\)
\(614\) 0 0
\(615\) 5.18184e25 0.166024
\(616\) 0 0
\(617\) −5.57559e26 −1.73213 −0.866067 0.499927i \(-0.833360\pi\)
−0.866067 + 0.499927i \(0.833360\pi\)
\(618\) 0 0
\(619\) −3.69876e26 −1.11428 −0.557141 0.830418i \(-0.688102\pi\)
−0.557141 + 0.830418i \(0.688102\pi\)
\(620\) 0 0
\(621\) 7.41478e24 0.0216635
\(622\) 0 0
\(623\) 5.32273e26 1.50833
\(624\) 0 0
\(625\) 1.05361e26 0.289615
\(626\) 0 0
\(627\) −1.49601e27 −3.98926
\(628\) 0 0
\(629\) 5.35775e26 1.38612
\(630\) 0 0
\(631\) −1.48829e26 −0.373603 −0.186801 0.982398i \(-0.559812\pi\)
−0.186801 + 0.982398i \(0.559812\pi\)
\(632\) 0 0
\(633\) 1.08679e27 2.64735
\(634\) 0 0
\(635\) 2.07020e26 0.489397
\(636\) 0 0
\(637\) −3.54419e25 −0.0813192
\(638\) 0 0
\(639\) −1.93435e26 −0.430802
\(640\) 0 0
\(641\) −2.74129e26 −0.592657 −0.296329 0.955086i \(-0.595762\pi\)
−0.296329 + 0.955086i \(0.595762\pi\)
\(642\) 0 0
\(643\) −2.47762e26 −0.520032 −0.260016 0.965604i \(-0.583728\pi\)
−0.260016 + 0.965604i \(0.583728\pi\)
\(644\) 0 0
\(645\) 2.49007e26 0.507452
\(646\) 0 0
\(647\) −1.36261e26 −0.269638 −0.134819 0.990870i \(-0.543045\pi\)
−0.134819 + 0.990870i \(0.543045\pi\)
\(648\) 0 0
\(649\) −1.07378e27 −2.06343
\(650\) 0 0
\(651\) 1.65284e27 3.08469
\(652\) 0 0
\(653\) −6.13257e25 −0.111165 −0.0555824 0.998454i \(-0.517702\pi\)
−0.0555824 + 0.998454i \(0.517702\pi\)
\(654\) 0 0
\(655\) 4.17128e25 0.0734475
\(656\) 0 0
\(657\) −5.06260e25 −0.0865969
\(658\) 0 0
\(659\) −2.67926e26 −0.445250 −0.222625 0.974904i \(-0.571463\pi\)
−0.222625 + 0.974904i \(0.571463\pi\)
\(660\) 0 0
\(661\) 3.12853e26 0.505157 0.252578 0.967576i \(-0.418721\pi\)
0.252578 + 0.967576i \(0.418721\pi\)
\(662\) 0 0
\(663\) 7.38492e25 0.115869
\(664\) 0 0
\(665\) −9.00848e26 −1.37355
\(666\) 0 0
\(667\) 3.03922e25 0.0450367
\(668\) 0 0
\(669\) 1.97114e27 2.83902
\(670\) 0 0
\(671\) 1.40493e27 1.96694
\(672\) 0 0
\(673\) 7.60450e26 1.03497 0.517485 0.855692i \(-0.326868\pi\)
0.517485 + 0.855692i \(0.326868\pi\)
\(674\) 0 0
\(675\) 1.64801e26 0.218058
\(676\) 0 0
\(677\) −1.44198e27 −1.85510 −0.927550 0.373700i \(-0.878089\pi\)
−0.927550 + 0.373700i \(0.878089\pi\)
\(678\) 0 0
\(679\) −9.71659e26 −1.21549
\(680\) 0 0
\(681\) −8.54878e25 −0.103994
\(682\) 0 0
\(683\) 8.31240e26 0.983399 0.491700 0.870765i \(-0.336376\pi\)
0.491700 + 0.870765i \(0.336376\pi\)
\(684\) 0 0
\(685\) −4.09818e26 −0.471553
\(686\) 0 0
\(687\) −1.69154e26 −0.189318
\(688\) 0 0
\(689\) 8.39069e25 0.0913514
\(690\) 0 0
\(691\) −2.10819e26 −0.223290 −0.111645 0.993748i \(-0.535612\pi\)
−0.111645 + 0.993748i \(0.535612\pi\)
\(692\) 0 0
\(693\) −2.63710e27 −2.71744
\(694\) 0 0
\(695\) −3.54723e26 −0.355659
\(696\) 0 0
\(697\) −2.75915e26 −0.269194
\(698\) 0 0
\(699\) −6.45059e26 −0.612444
\(700\) 0 0
\(701\) 9.41456e26 0.869919 0.434959 0.900450i \(-0.356763\pi\)
0.434959 + 0.900450i \(0.356763\pi\)
\(702\) 0 0
\(703\) −2.25313e27 −2.02633
\(704\) 0 0
\(705\) 1.17623e27 1.02966
\(706\) 0 0
\(707\) 1.46343e27 1.24706
\(708\) 0 0
\(709\) −1.83074e27 −1.51875 −0.759376 0.650652i \(-0.774495\pi\)
−0.759376 + 0.650652i \(0.774495\pi\)
\(710\) 0 0
\(711\) −6.07513e26 −0.490676
\(712\) 0 0
\(713\) 1.29121e26 0.101542
\(714\) 0 0
\(715\) 6.38268e25 0.0488760
\(716\) 0 0
\(717\) 4.63002e26 0.345264
\(718\) 0 0
\(719\) 1.73532e27 1.26025 0.630124 0.776495i \(-0.283004\pi\)
0.630124 + 0.776495i \(0.283004\pi\)
\(720\) 0 0
\(721\) 1.86880e27 1.32184
\(722\) 0 0
\(723\) 2.29796e27 1.58317
\(724\) 0 0
\(725\) 6.75495e26 0.453326
\(726\) 0 0
\(727\) 5.33232e26 0.348610 0.174305 0.984692i \(-0.444232\pi\)
0.174305 + 0.984692i \(0.444232\pi\)
\(728\) 0 0
\(729\) −2.11925e27 −1.34980
\(730\) 0 0
\(731\) −1.32588e27 −0.822790
\(732\) 0 0
\(733\) 3.90931e26 0.236381 0.118190 0.992991i \(-0.462291\pi\)
0.118190 + 0.992991i \(0.462291\pi\)
\(734\) 0 0
\(735\) −1.63177e27 −0.961455
\(736\) 0 0
\(737\) 2.33820e27 1.34258
\(738\) 0 0
\(739\) 7.87032e26 0.440423 0.220212 0.975452i \(-0.429325\pi\)
0.220212 + 0.975452i \(0.429325\pi\)
\(740\) 0 0
\(741\) −3.10563e26 −0.169385
\(742\) 0 0
\(743\) −3.68492e27 −1.95900 −0.979500 0.201445i \(-0.935436\pi\)
−0.979500 + 0.201445i \(0.935436\pi\)
\(744\) 0 0
\(745\) 7.15302e26 0.370685
\(746\) 0 0
\(747\) 6.30174e26 0.318357
\(748\) 0 0
\(749\) 1.60928e27 0.792600
\(750\) 0 0
\(751\) 3.86208e26 0.185457 0.0927283 0.995691i \(-0.470441\pi\)
0.0927283 + 0.995691i \(0.470441\pi\)
\(752\) 0 0
\(753\) 1.86421e27 0.872855
\(754\) 0 0
\(755\) −3.45235e26 −0.157623
\(756\) 0 0
\(757\) 3.93310e27 1.75116 0.875578 0.483078i \(-0.160481\pi\)
0.875578 + 0.483078i \(0.160481\pi\)
\(758\) 0 0
\(759\) −3.77900e26 −0.164089
\(760\) 0 0
\(761\) −1.44215e27 −0.610742 −0.305371 0.952234i \(-0.598780\pi\)
−0.305371 + 0.952234i \(0.598780\pi\)
\(762\) 0 0
\(763\) −6.98978e26 −0.288722
\(764\) 0 0
\(765\) 1.85354e27 0.746824
\(766\) 0 0
\(767\) −2.22910e26 −0.0876139
\(768\) 0 0
\(769\) −4.77621e27 −1.83140 −0.915701 0.401861i \(-0.868364\pi\)
−0.915701 + 0.401861i \(0.868364\pi\)
\(770\) 0 0
\(771\) −2.44872e27 −0.916056
\(772\) 0 0
\(773\) −3.06095e27 −1.11725 −0.558625 0.829420i \(-0.688671\pi\)
−0.558625 + 0.829420i \(0.688671\pi\)
\(774\) 0 0
\(775\) 2.86984e27 1.02210
\(776\) 0 0
\(777\) −7.28557e27 −2.53200
\(778\) 0 0
\(779\) 1.16032e27 0.393526
\(780\) 0 0
\(781\) 1.63294e27 0.540488
\(782\) 0 0
\(783\) 5.57576e26 0.180122
\(784\) 0 0
\(785\) −1.00988e27 −0.318427
\(786\) 0 0
\(787\) 2.46755e27 0.759461 0.379731 0.925097i \(-0.376017\pi\)
0.379731 + 0.925097i \(0.376017\pi\)
\(788\) 0 0
\(789\) −3.46084e27 −1.03980
\(790\) 0 0
\(791\) −1.46377e27 −0.429334
\(792\) 0 0
\(793\) 2.91656e26 0.0835171
\(794\) 0 0
\(795\) 3.86313e27 1.08007
\(796\) 0 0
\(797\) −4.72784e27 −1.29065 −0.645326 0.763907i \(-0.723278\pi\)
−0.645326 + 0.763907i \(0.723278\pi\)
\(798\) 0 0
\(799\) −6.26301e27 −1.66951
\(800\) 0 0
\(801\) 4.60561e27 1.19889
\(802\) 0 0
\(803\) 4.27374e26 0.108645
\(804\) 0 0
\(805\) −2.27560e26 −0.0564981
\(806\) 0 0
\(807\) 1.10061e28 2.66892
\(808\) 0 0
\(809\) 4.73872e27 1.12241 0.561203 0.827678i \(-0.310339\pi\)
0.561203 + 0.827678i \(0.310339\pi\)
\(810\) 0 0
\(811\) −1.55266e27 −0.359234 −0.179617 0.983737i \(-0.557486\pi\)
−0.179617 + 0.983737i \(0.557486\pi\)
\(812\) 0 0
\(813\) 9.52795e27 2.15347
\(814\) 0 0
\(815\) 1.71736e27 0.379197
\(816\) 0 0
\(817\) 5.57580e27 1.20281
\(818\) 0 0
\(819\) −5.47446e26 −0.115384
\(820\) 0 0
\(821\) −7.90129e27 −1.62719 −0.813595 0.581432i \(-0.802493\pi\)
−0.813595 + 0.581432i \(0.802493\pi\)
\(822\) 0 0
\(823\) 7.55379e27 1.52008 0.760040 0.649877i \(-0.225180\pi\)
0.760040 + 0.649877i \(0.225180\pi\)
\(824\) 0 0
\(825\) −8.39920e27 −1.65168
\(826\) 0 0
\(827\) −9.52722e27 −1.83090 −0.915448 0.402436i \(-0.868164\pi\)
−0.915448 + 0.402436i \(0.868164\pi\)
\(828\) 0 0
\(829\) 1.29069e27 0.242412 0.121206 0.992627i \(-0.461324\pi\)
0.121206 + 0.992627i \(0.461324\pi\)
\(830\) 0 0
\(831\) −1.05125e28 −1.92973
\(832\) 0 0
\(833\) 8.68861e27 1.55892
\(834\) 0 0
\(835\) 2.28086e27 0.400016
\(836\) 0 0
\(837\) 2.36886e27 0.406114
\(838\) 0 0
\(839\) 9.22538e27 1.54613 0.773065 0.634327i \(-0.218723\pi\)
0.773065 + 0.634327i \(0.218723\pi\)
\(840\) 0 0
\(841\) −3.81783e27 −0.625540
\(842\) 0 0
\(843\) −1.68759e28 −2.70337
\(844\) 0 0
\(845\) −3.23722e27 −0.507030
\(846\) 0 0
\(847\) 1.24159e28 1.90145
\(848\) 0 0
\(849\) 1.14551e27 0.171544
\(850\) 0 0
\(851\) −5.69154e26 −0.0833487
\(852\) 0 0
\(853\) −6.63470e27 −0.950180 −0.475090 0.879937i \(-0.657584\pi\)
−0.475090 + 0.879937i \(0.657584\pi\)
\(854\) 0 0
\(855\) −7.79479e27 −1.09176
\(856\) 0 0
\(857\) 2.92147e27 0.400206 0.200103 0.979775i \(-0.435872\pi\)
0.200103 + 0.979775i \(0.435872\pi\)
\(858\) 0 0
\(859\) 8.47684e27 1.13579 0.567896 0.823100i \(-0.307757\pi\)
0.567896 + 0.823100i \(0.307757\pi\)
\(860\) 0 0
\(861\) 3.75195e27 0.491730
\(862\) 0 0
\(863\) −9.99055e27 −1.28082 −0.640408 0.768035i \(-0.721235\pi\)
−0.640408 + 0.768035i \(0.721235\pi\)
\(864\) 0 0
\(865\) 1.53969e27 0.193099
\(866\) 0 0
\(867\) −6.01918e27 −0.738509
\(868\) 0 0
\(869\) 5.12850e27 0.615605
\(870\) 0 0
\(871\) 4.85398e26 0.0570066
\(872\) 0 0
\(873\) −8.40751e27 −0.966122
\(874\) 0 0
\(875\) −1.18850e28 −1.33636
\(876\) 0 0
\(877\) −3.66387e27 −0.403129 −0.201564 0.979475i \(-0.564603\pi\)
−0.201564 + 0.979475i \(0.564603\pi\)
\(878\) 0 0
\(879\) 9.37294e27 1.00921
\(880\) 0 0
\(881\) 3.90178e27 0.411141 0.205571 0.978642i \(-0.434095\pi\)
0.205571 + 0.978642i \(0.434095\pi\)
\(882\) 0 0
\(883\) −6.02824e27 −0.621676 −0.310838 0.950463i \(-0.600610\pi\)
−0.310838 + 0.950463i \(0.600610\pi\)
\(884\) 0 0
\(885\) −1.02629e28 −1.03588
\(886\) 0 0
\(887\) −1.43600e28 −1.41867 −0.709333 0.704874i \(-0.751004\pi\)
−0.709333 + 0.704874i \(0.751004\pi\)
\(888\) 0 0
\(889\) 1.49894e28 1.44950
\(890\) 0 0
\(891\) 1.21056e28 1.14591
\(892\) 0 0
\(893\) 2.63382e28 2.44061
\(894\) 0 0
\(895\) 5.38925e27 0.488888
\(896\) 0 0
\(897\) −7.84500e25 −0.00696730
\(898\) 0 0
\(899\) 9.70964e27 0.844279
\(900\) 0 0
\(901\) −2.05698e28 −1.75124
\(902\) 0 0
\(903\) 1.80296e28 1.50297
\(904\) 0 0
\(905\) −2.16218e27 −0.176495
\(906\) 0 0
\(907\) 8.84058e27 0.706662 0.353331 0.935498i \(-0.385049\pi\)
0.353331 + 0.935498i \(0.385049\pi\)
\(908\) 0 0
\(909\) 1.26627e28 0.991215
\(910\) 0 0
\(911\) 1.77289e28 1.35912 0.679561 0.733619i \(-0.262170\pi\)
0.679561 + 0.733619i \(0.262170\pi\)
\(912\) 0 0
\(913\) −5.31979e27 −0.399413
\(914\) 0 0
\(915\) 1.34280e28 0.987442
\(916\) 0 0
\(917\) 3.02024e27 0.217537
\(918\) 0 0
\(919\) −1.67320e28 −1.18046 −0.590230 0.807235i \(-0.700963\pi\)
−0.590230 + 0.807235i \(0.700963\pi\)
\(920\) 0 0
\(921\) −4.01175e28 −2.77247
\(922\) 0 0
\(923\) 3.38989e26 0.0229493
\(924\) 0 0
\(925\) −1.26500e28 −0.838965
\(926\) 0 0
\(927\) 1.61703e28 1.05065
\(928\) 0 0
\(929\) −2.04557e28 −1.30216 −0.651082 0.759008i \(-0.725685\pi\)
−0.651082 + 0.759008i \(0.725685\pi\)
\(930\) 0 0
\(931\) −3.65387e28 −2.27893
\(932\) 0 0
\(933\) 2.63080e28 1.60772
\(934\) 0 0
\(935\) −1.56472e28 −0.936970
\(936\) 0 0
\(937\) −2.17132e27 −0.127408 −0.0637042 0.997969i \(-0.520291\pi\)
−0.0637042 + 0.997969i \(0.520291\pi\)
\(938\) 0 0
\(939\) 8.35809e27 0.480600
\(940\) 0 0
\(941\) 8.37859e27 0.472138 0.236069 0.971736i \(-0.424141\pi\)
0.236069 + 0.971736i \(0.424141\pi\)
\(942\) 0 0
\(943\) 2.93105e26 0.0161868
\(944\) 0 0
\(945\) −4.17482e27 −0.225962
\(946\) 0 0
\(947\) 1.69589e28 0.899648 0.449824 0.893117i \(-0.351487\pi\)
0.449824 + 0.893117i \(0.351487\pi\)
\(948\) 0 0
\(949\) 8.87204e25 0.00461311
\(950\) 0 0
\(951\) −2.78136e28 −1.41756
\(952\) 0 0
\(953\) 3.53298e27 0.176506 0.0882529 0.996098i \(-0.471872\pi\)
0.0882529 + 0.996098i \(0.471872\pi\)
\(954\) 0 0
\(955\) 1.50312e28 0.736140
\(956\) 0 0
\(957\) −2.84173e28 −1.36433
\(958\) 0 0
\(959\) −2.96732e28 −1.39665
\(960\) 0 0
\(961\) 1.95807e28 0.903560
\(962\) 0 0
\(963\) 1.39247e28 0.629992
\(964\) 0 0
\(965\) 3.40184e27 0.150905
\(966\) 0 0
\(967\) −9.62859e27 −0.418805 −0.209402 0.977830i \(-0.567152\pi\)
−0.209402 + 0.977830i \(0.567152\pi\)
\(968\) 0 0
\(969\) 7.61345e28 3.24718
\(970\) 0 0
\(971\) −2.59014e28 −1.08328 −0.541640 0.840611i \(-0.682196\pi\)
−0.541640 + 0.840611i \(0.682196\pi\)
\(972\) 0 0
\(973\) −2.56840e28 −1.05339
\(974\) 0 0
\(975\) −1.74363e27 −0.0701309
\(976\) 0 0
\(977\) 3.99288e28 1.57503 0.787514 0.616297i \(-0.211368\pi\)
0.787514 + 0.616297i \(0.211368\pi\)
\(978\) 0 0
\(979\) −3.88796e28 −1.50413
\(980\) 0 0
\(981\) −6.04807e27 −0.229489
\(982\) 0 0
\(983\) 3.74542e27 0.139393 0.0696966 0.997568i \(-0.477797\pi\)
0.0696966 + 0.997568i \(0.477797\pi\)
\(984\) 0 0
\(985\) −1.50996e28 −0.551214
\(986\) 0 0
\(987\) 8.51656e28 3.04966
\(988\) 0 0
\(989\) 1.40848e27 0.0494750
\(990\) 0 0
\(991\) −3.40811e28 −1.17440 −0.587198 0.809443i \(-0.699769\pi\)
−0.587198 + 0.809443i \(0.699769\pi\)
\(992\) 0 0
\(993\) −2.79525e28 −0.944937
\(994\) 0 0
\(995\) 2.10217e28 0.697186
\(996\) 0 0
\(997\) −4.93862e27 −0.160695 −0.0803473 0.996767i \(-0.525603\pi\)
−0.0803473 + 0.996767i \(0.525603\pi\)
\(998\) 0 0
\(999\) −1.04417e28 −0.333350
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 8.20.a.a.1.1 2
3.2 odd 2 72.20.a.a.1.1 2
4.3 odd 2 16.20.a.e.1.2 2
8.3 odd 2 64.20.a.j.1.1 2
8.5 even 2 64.20.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.20.a.a.1.1 2 1.1 even 1 trivial
16.20.a.e.1.2 2 4.3 odd 2
64.20.a.j.1.1 2 8.3 odd 2
64.20.a.k.1.2 2 8.5 even 2
72.20.a.a.1.1 2 3.2 odd 2