Properties

Label 64.20.a.j.1.1
Level $64$
Weight $20$
Character 64.1
Self dual yes
Analytic conductor $146.443$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.442685796\)
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: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.5591\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-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 \(-0.581928\pi\)
−0.254552 + 0.967059i \(0.581928\pi\)
\(6\) 0 0
\(7\) −1.60989e8 −1.50787 −0.753934 0.656950i \(-0.771846\pi\)
−0.753934 + 0.656950i \(0.771846\pi\)
\(8\) 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 \(-0.489837\pi\)
0.0319232 + 0.999490i \(0.489837\pi\)
\(14\) 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 \(-0.488284\pi\)
0.0367988 + 0.999323i \(0.488284\pi\)
\(24\) 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 \(-0.401021\pi\)
0.305966 + 0.952043i \(0.401021\pi\)
\(30\) 0 0
\(31\) 2.03104e14 1.37970 0.689848 0.723954i \(-0.257677\pi\)
0.689848 + 0.723954i \(0.257677\pi\)
\(32\) 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 \(-0.691605\pi\)
−0.566246 + 0.824236i \(0.691605\pi\)
\(38\) 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 \(-0.261105\pi\)
0.682013 + 0.731340i \(0.261105\pi\)
\(48\) 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 \(-0.246244\pi\)
0.715400 + 0.698715i \(0.246244\pi\)
\(54\) 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 \(-0.273070\pi\)
0.654047 + 0.756454i \(0.273070\pi\)
\(62\) 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 \(-0.442480\pi\)
0.179723 + 0.983717i \(0.442480\pi\)
\(72\) 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 \(-0.434378\pi\)
0.204701 + 0.978825i \(0.434378\pi\)
\(80\) 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 \(-0.635700\pi\)
−0.413517 + 0.910496i \(0.635700\pi\)
\(102\) 0 0
\(103\) −1.16083e19 −0.876628 −0.438314 0.898822i \(-0.644424\pi\)
−0.438314 + 0.898822i \(0.644424\pi\)
\(104\) 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 \(-0.469479\pi\)
0.0957385 + 0.995407i \(0.469479\pi\)
\(110\) 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 \(-0.659597\pi\)
−0.480645 + 0.876915i \(0.659597\pi\)
\(128\) 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 \(-0.618608\pi\)
−0.364055 + 0.931377i \(0.618608\pi\)
\(150\) 0 0
\(151\) 1.55272e20 0.309607 0.154804 0.987945i \(-0.450526\pi\)
0.154804 + 0.987945i \(0.450526\pi\)
\(152\) 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 \(-0.398756\pi\)
0.312732 + 0.949841i \(0.398756\pi\)
\(158\) 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 \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(168\) 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 \(-0.560734\pi\)
−0.189646 + 0.981853i \(0.560734\pi\)
\(174\) 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 \(-0.444545\pi\)
0.173338 + 0.984862i \(0.444545\pi\)
\(182\) 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 \(-0.757226\pi\)
−0.722975 + 0.690874i \(0.757226\pi\)
\(192\) 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 \(-0.317911\pi\)
0.541356 + 0.840793i \(0.317911\pi\)
\(198\) 0 0
\(199\) −9.45465e21 −1.36943 −0.684717 0.728809i \(-0.740074\pi\)
−0.684717 + 0.728809i \(0.740074\pi\)
\(200\) 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 \(-0.0932927\pi\)
0.957356 + 0.288910i \(0.0932927\pi\)
\(224\) 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 \(-0.520335\pi\)
−0.0638407 + 0.997960i \(0.520335\pi\)
\(230\) 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 \(-0.462856\pi\)
0.116428 + 0.993199i \(0.462856\pi\)
\(240\) 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 \(-0.614034\pi\)
−0.350635 + 0.936512i \(0.614034\pi\)
\(264\) 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 \(-0.143570\pi\)
0.899995 + 0.435900i \(0.143570\pi\)
\(270\) 0 0
\(271\) 1.88487e23 1.45236 0.726180 0.687505i \(-0.241294\pi\)
0.726180 + 0.687505i \(0.241294\pi\)
\(272\) 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 \(-0.725537\pi\)
−0.650729 + 0.759310i \(0.725537\pi\)
\(278\) 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 \(-0.389465\pi\)
0.340319 + 0.940310i \(0.389465\pi\)
\(294\) 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 \(-0.317612\pi\)
0.542147 + 0.840284i \(0.317612\pi\)
\(312\) 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 \(-0.658646\pi\)
−0.478022 + 0.878348i \(0.658646\pi\)
\(318\) 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 \(-0.404696\pi\)
0.294952 + 0.955512i \(0.404696\pi\)
\(350\) 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 \(-0.373372\pi\)
0.387403 + 0.921911i \(0.373372\pi\)
\(360\) 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 \(-0.415516\pi\)
0.262310 + 0.964984i \(0.415516\pi\)
\(368\) 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 \(-0.716174\pi\)
−0.628118 + 0.778118i \(0.716174\pi\)
\(374\) 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 \(-0.688991\pi\)
−0.559459 + 0.828858i \(0.688991\pi\)
\(384\) 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 \(-0.603944\pi\)
−0.320777 + 0.947155i \(0.603944\pi\)
\(390\) 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 \(-0.361481\pi\)
0.421564 + 0.906799i \(0.361481\pi\)
\(398\) 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 \(-0.449761\pi\)
0.157175 + 0.987571i \(0.449761\pi\)
\(422\) 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 \(-0.612472\pi\)
−0.346035 + 0.938222i \(0.612472\pi\)
\(432\) 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 \(-0.699161\pi\)
−0.585651 + 0.810563i \(0.699161\pi\)
\(440\) 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 \(-0.683558\pi\)
−0.545231 + 0.838286i \(0.683558\pi\)
\(462\) 0 0
\(463\) 2.34544e25 1.11482 0.557410 0.830237i \(-0.311795\pi\)
0.557410 + 0.830237i \(0.311795\pi\)
\(464\) 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 \(-0.354851\pi\)
0.440359 + 0.897822i \(0.354851\pi\)
\(480\) 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 \(-0.565635\pi\)
−0.204739 + 0.978817i \(0.565635\pi\)
\(488\) 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 \(-0.433232\pi\)
0.208222 + 0.978082i \(0.433232\pi\)
\(504\) 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 \(-0.670374\pi\)
−0.510053 + 0.860143i \(0.670374\pi\)
\(510\) 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 \(-0.753328\pi\)
−0.714461 + 0.699676i \(0.753328\pi\)
\(542\) 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 \(-0.320462\pi\)
0.534600 + 0.845105i \(0.320462\pi\)
\(558\) 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 \(-0.441673\pi\)
0.182217 + 0.983258i \(0.441673\pi\)
\(600\) 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 \(-0.365551\pi\)
0.409936 + 0.912114i \(0.365551\pi\)
\(608\) 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 \(-0.320849\pi\)
0.533572 + 0.845755i \(0.320849\pi\)
\(614\) 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 \(-0.440188\pi\)
0.186801 + 0.982398i \(0.440188\pi\)
\(632\) 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 \(-0.456955\pi\)
0.134819 + 0.990870i \(0.456955\pi\)
\(648\) 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 \(-0.482298\pi\)
0.0555824 + 0.998454i \(0.482298\pi\)
\(654\) 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 \(-0.581279\pi\)
−0.252578 + 0.967576i \(0.581279\pi\)
\(662\) 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 \(-0.121911\pi\)
0.927550 + 0.373700i \(0.121911\pi\)
\(678\) 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 \(-0.643237\pi\)
−0.434959 + 0.900450i \(0.643237\pi\)
\(702\) 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 \(-0.225505\pi\)
0.759376 + 0.650652i \(0.225505\pi\)
\(710\) 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 \(-0.716996\pi\)
−0.630124 + 0.776495i \(0.716996\pi\)
\(720\) 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 \(-0.555768\pi\)
−0.174305 + 0.984692i \(0.555768\pi\)
\(728\) 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 \(-0.537709\pi\)
−0.118190 + 0.992991i \(0.537709\pi\)
\(734\) 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 \(-0.0645637\pi\)
0.979500 + 0.201445i \(0.0645637\pi\)
\(744\) 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 \(-0.529559\pi\)
−0.0927283 + 0.995691i \(0.529559\pi\)
\(752\) 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 \(-0.839519\pi\)
−0.875578 + 0.483078i \(0.839519\pi\)
\(758\) 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 \(-0.311329\pi\)
0.558625 + 0.829420i \(0.311329\pi\)
\(774\) 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 \(-0.276722\pi\)
0.645326 + 0.763907i \(0.276722\pi\)
\(798\) 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 \(-0.197507\pi\)
0.813595 + 0.581432i \(0.197507\pi\)
\(822\) 0 0
\(823\) −7.55379e27 −1.52008 −0.760040 0.649877i \(-0.774820\pi\)
−0.760040 + 0.649877i \(0.774820\pi\)
\(824\) 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 \(-0.538676\pi\)
−0.121206 + 0.992627i \(0.538676\pi\)
\(830\) 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 \(-0.781277\pi\)
−0.773065 + 0.634327i \(0.781277\pi\)
\(840\) 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 \(-0.342416\pi\)
0.475090 + 0.879937i \(0.342416\pi\)
\(854\) 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 \(-0.278765\pi\)
0.640408 + 0.768035i \(0.278765\pi\)
\(864\) 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 \(-0.435397\pi\)
0.201564 + 0.979475i \(0.435397\pi\)
\(878\) 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 \(-0.248996\pi\)
0.709333 + 0.704874i \(0.248996\pi\)
\(888\) 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 \(-0.737830\pi\)
−0.679561 + 0.733619i \(0.737830\pi\)
\(912\) 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 \(-0.299037\pi\)
0.590230 + 0.807235i \(0.299037\pi\)
\(920\) 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 \(-0.575859\pi\)
−0.236069 + 0.971736i \(0.575859\pi\)
\(942\) 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 \(-0.432848\pi\)
0.209402 + 0.977830i \(0.432848\pi\)
\(968\) 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 \(-0.522203\pi\)
−0.0696966 + 0.997568i \(0.522203\pi\)
\(984\) 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 \(-0.300231\pi\)
0.587198 + 0.809443i \(0.300231\pi\)
\(992\) 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 \(-0.474397\pi\)
0.0803473 + 0.996767i \(0.474397\pi\)
\(998\) 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 64.20.a.j.1.1 2
4.3 odd 2 64.20.a.k.1.2 2
8.3 odd 2 8.20.a.a.1.1 2
8.5 even 2 16.20.a.e.1.2 2
24.11 even 2 72.20.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.20.a.a.1.1 2 8.3 odd 2
16.20.a.e.1.2 2 8.5 even 2
64.20.a.j.1.1 2 1.1 even 1 trivial
64.20.a.k.1.2 2 4.3 odd 2
72.20.a.a.1.1 2 24.11 even 2