Properties

Label 8.20.a.a.1.2
Level $8$
Weight $20$
Character 8.1
Self dual yes
Analytic conductor $18.305$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.2
Root \(-18.5591\) of defining polynomial
Character \(\chi\) \(=\) 8.1

$q$-expansion

\(f(q)\) \(=\) \(q+22637.5 q^{3} -996804. q^{5} -7.24780e7 q^{7} -6.49805e8 q^{9} +O(q^{10})\) \(q+22637.5 q^{3} -996804. q^{5} -7.24780e7 q^{7} -6.49805e8 q^{9} +4.59555e9 q^{11} -7.68575e9 q^{13} -2.25652e10 q^{15} -6.70499e11 q^{17} -6.03766e11 q^{19} -1.64072e12 q^{21} -1.41235e13 q^{23} -1.80799e13 q^{25} -4.10207e13 q^{27} +1.75566e13 q^{29} +7.97149e13 q^{31} +1.04032e14 q^{33} +7.22464e13 q^{35} +1.12013e15 q^{37} -1.73986e14 q^{39} +3.00183e15 q^{41} -3.41815e15 q^{43} +6.47728e14 q^{45} -1.14830e16 q^{47} -6.14583e15 q^{49} -1.51784e16 q^{51} +2.49535e16 q^{53} -4.58086e15 q^{55} -1.36678e16 q^{57} +7.22986e15 q^{59} +1.29766e17 q^{61} +4.70966e16 q^{63} +7.66119e15 q^{65} +2.74592e17 q^{67} -3.19720e17 q^{69} +1.56270e17 q^{71} -8.21165e17 q^{73} -4.09283e17 q^{75} -3.33076e17 q^{77} +2.09830e17 q^{79} -1.73363e17 q^{81} +3.15127e17 q^{83} +6.68357e17 q^{85} +3.97438e17 q^{87} +2.78627e18 q^{89} +5.57048e17 q^{91} +1.80455e18 q^{93} +6.01837e17 q^{95} +7.58373e18 q^{97} -2.98621e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 27912q^{3} + 1226620q^{5} + 88510512q^{7} + 743186538q^{9} + O(q^{10}) \) \( 2q - 27912q^{3} + 1226620q^{5} + 88510512q^{7} + 743186538q^{9} - 7163787608q^{11} - 10126923604q^{13} - 134958171120q^{15} - 72045078940q^{17} - 3120480472232q^{19} - 9778613729472q^{21} - 14759207090288q^{23} - 32209737998450q^{25} - 52683939132624q^{27} - 30249539245044q^{29} - 123389562777920q^{31} + 698460345860448q^{33} + 430192267170720q^{35} + 2015393170174524q^{37} - 50586287718576q^{39} + 2540784959504244q^{41} - 5633655093389464q^{43} + 3744938810129580q^{45} - 21948339587130336q^{47} + 8372586663064786q^{49} - 45429993200591760q^{51} - 9418125066904676q^{53} - 30726853722003280q^{55} + 113550921657554592q^{57} + 98542449590407624q^{59} + 10292145377839820q^{61} + 271352203396831728q^{63} + 2233432360014760q^{65} + 75753628003984504q^{67} - 287583943397508672q^{69} + 17407052566713776q^{71} - 857508255059832268q^{73} + 304974862137280200q^{75} - 2226194512471875648q^{77} - 226291921444855072q^{79} - 1202809709767720302q^{81} + 767515701460985048q^{83} + 1998974022815558200q^{85} + 2814015710341120464q^{87} + 6092545894435174548q^{89} + 164047654392377376q^{91} + 12071374787086705920q^{93} - 4993887710656339120q^{95} + 1548148249522347076q^{97} - 19366859521479728952q^{99} + O(q^{100}) \)

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
\(3\) 22637.5 0.664013 0.332007 0.943277i \(-0.392274\pi\)
0.332007 + 0.943277i \(0.392274\pi\)
\(4\) 0 0
\(5\) −996804. −0.228242 −0.114121 0.993467i \(-0.536405\pi\)
−0.114121 + 0.993467i \(0.536405\pi\)
\(6\) 0 0
\(7\) −7.24780e7 −0.678852 −0.339426 0.940633i \(-0.610233\pi\)
−0.339426 + 0.940633i \(0.610233\pi\)
\(8\) 0 0
\(9\) −6.49805e8 −0.559087
\(10\) 0 0
\(11\) 4.59555e9 0.587634 0.293817 0.955862i \(-0.405074\pi\)
0.293817 + 0.955862i \(0.405074\pi\)
\(12\) 0 0
\(13\) −7.68575e9 −0.201013 −0.100507 0.994936i \(-0.532046\pi\)
−0.100507 + 0.994936i \(0.532046\pi\)
\(14\) 0 0
\(15\) −2.25652e10 −0.151555
\(16\) 0 0
\(17\) −6.70499e11 −1.37130 −0.685652 0.727930i \(-0.740483\pi\)
−0.685652 + 0.727930i \(0.740483\pi\)
\(18\) 0 0
\(19\) −6.03766e11 −0.429249 −0.214625 0.976697i \(-0.568853\pi\)
−0.214625 + 0.976697i \(0.568853\pi\)
\(20\) 0 0
\(21\) −1.64072e12 −0.450767
\(22\) 0 0
\(23\) −1.41235e13 −1.63503 −0.817517 0.575904i \(-0.804650\pi\)
−0.817517 + 0.575904i \(0.804650\pi\)
\(24\) 0 0
\(25\) −1.80799e13 −0.947906
\(26\) 0 0
\(27\) −4.10207e13 −1.03525
\(28\) 0 0
\(29\) 1.75566e13 0.224729 0.112365 0.993667i \(-0.464158\pi\)
0.112365 + 0.993667i \(0.464158\pi\)
\(30\) 0 0
\(31\) 7.97149e13 0.541506 0.270753 0.962649i \(-0.412727\pi\)
0.270753 + 0.962649i \(0.412727\pi\)
\(32\) 0 0
\(33\) 1.04032e14 0.390197
\(34\) 0 0
\(35\) 7.22464e13 0.154942
\(36\) 0 0
\(37\) 1.12013e15 1.41694 0.708469 0.705742i \(-0.249386\pi\)
0.708469 + 0.705742i \(0.249386\pi\)
\(38\) 0 0
\(39\) −1.73986e14 −0.133475
\(40\) 0 0
\(41\) 3.00183e15 1.43199 0.715995 0.698106i \(-0.245974\pi\)
0.715995 + 0.698106i \(0.245974\pi\)
\(42\) 0 0
\(43\) −3.41815e15 −1.03715 −0.518574 0.855033i \(-0.673537\pi\)
−0.518574 + 0.855033i \(0.673537\pi\)
\(44\) 0 0
\(45\) 6.47728e14 0.127607
\(46\) 0 0
\(47\) −1.14830e16 −1.49667 −0.748336 0.663320i \(-0.769147\pi\)
−0.748336 + 0.663320i \(0.769147\pi\)
\(48\) 0 0
\(49\) −6.14583e15 −0.539160
\(50\) 0 0
\(51\) −1.51784e16 −0.910563
\(52\) 0 0
\(53\) 2.49535e16 1.03875 0.519374 0.854547i \(-0.326165\pi\)
0.519374 + 0.854547i \(0.326165\pi\)
\(54\) 0 0
\(55\) −4.58086e15 −0.134122
\(56\) 0 0
\(57\) −1.36678e16 −0.285027
\(58\) 0 0
\(59\) 7.22986e15 0.108652 0.0543258 0.998523i \(-0.482699\pi\)
0.0543258 + 0.998523i \(0.482699\pi\)
\(60\) 0 0
\(61\) 1.29766e17 1.42078 0.710391 0.703808i \(-0.248518\pi\)
0.710391 + 0.703808i \(0.248518\pi\)
\(62\) 0 0
\(63\) 4.70966e16 0.379537
\(64\) 0 0
\(65\) 7.66119e15 0.0458796
\(66\) 0 0
\(67\) 2.74592e17 1.23304 0.616520 0.787340i \(-0.288542\pi\)
0.616520 + 0.787340i \(0.288542\pi\)
\(68\) 0 0
\(69\) −3.19720e17 −1.08568
\(70\) 0 0
\(71\) 1.56270e17 0.404504 0.202252 0.979334i \(-0.435174\pi\)
0.202252 + 0.979334i \(0.435174\pi\)
\(72\) 0 0
\(73\) −8.21165e17 −1.63254 −0.816269 0.577672i \(-0.803961\pi\)
−0.816269 + 0.577672i \(0.803961\pi\)
\(74\) 0 0
\(75\) −4.09283e17 −0.629422
\(76\) 0 0
\(77\) −3.33076e17 −0.398916
\(78\) 0 0
\(79\) 2.09830e17 0.196974 0.0984871 0.995138i \(-0.468600\pi\)
0.0984871 + 0.995138i \(0.468600\pi\)
\(80\) 0 0
\(81\) −1.73363e17 −0.128336
\(82\) 0 0
\(83\) 3.15127e17 0.185030 0.0925152 0.995711i \(-0.470509\pi\)
0.0925152 + 0.995711i \(0.470509\pi\)
\(84\) 0 0
\(85\) 6.68357e17 0.312989
\(86\) 0 0
\(87\) 3.97438e17 0.149223
\(88\) 0 0
\(89\) 2.78627e18 0.842981 0.421491 0.906833i \(-0.361507\pi\)
0.421491 + 0.906833i \(0.361507\pi\)
\(90\) 0 0
\(91\) 5.57048e17 0.136458
\(92\) 0 0
\(93\) 1.80455e18 0.359567
\(94\) 0 0
\(95\) 6.01837e17 0.0979725
\(96\) 0 0
\(97\) 7.58373e18 1.01286 0.506432 0.862280i \(-0.330964\pi\)
0.506432 + 0.862280i \(0.330964\pi\)
\(98\) 0 0
\(99\) −2.98621e18 −0.328538
\(100\) 0 0
\(101\) −3.62374e18 −0.329689 −0.164845 0.986320i \(-0.552712\pi\)
−0.164845 + 0.986320i \(0.552712\pi\)
\(102\) 0 0
\(103\) −2.07112e19 −1.56405 −0.782026 0.623246i \(-0.785814\pi\)
−0.782026 + 0.623246i \(0.785814\pi\)
\(104\) 0 0
\(105\) 1.63548e18 0.102884
\(106\) 0 0
\(107\) −2.42775e19 −1.27661 −0.638305 0.769784i \(-0.720364\pi\)
−0.638305 + 0.769784i \(0.720364\pi\)
\(108\) 0 0
\(109\) −1.19033e19 −0.524946 −0.262473 0.964939i \(-0.584538\pi\)
−0.262473 + 0.964939i \(0.584538\pi\)
\(110\) 0 0
\(111\) 2.53569e19 0.940866
\(112\) 0 0
\(113\) −2.65952e18 −0.0832833 −0.0416417 0.999133i \(-0.513259\pi\)
−0.0416417 + 0.999133i \(0.513259\pi\)
\(114\) 0 0
\(115\) 1.40783e19 0.373183
\(116\) 0 0
\(117\) 4.99424e18 0.112384
\(118\) 0 0
\(119\) 4.85965e19 0.930912
\(120\) 0 0
\(121\) −4.00400e19 −0.654687
\(122\) 0 0
\(123\) 6.79540e19 0.950860
\(124\) 0 0
\(125\) 3.70346e19 0.444593
\(126\) 0 0
\(127\) 6.57941e19 0.679284 0.339642 0.940555i \(-0.389694\pi\)
0.339642 + 0.940555i \(0.389694\pi\)
\(128\) 0 0
\(129\) −7.73784e19 −0.688680
\(130\) 0 0
\(131\) −3.73472e19 −0.287198 −0.143599 0.989636i \(-0.545868\pi\)
−0.143599 + 0.989636i \(0.545868\pi\)
\(132\) 0 0
\(133\) 4.37598e19 0.291397
\(134\) 0 0
\(135\) 4.08896e19 0.236288
\(136\) 0 0
\(137\) −3.14142e18 −0.0157863 −0.00789315 0.999969i \(-0.502512\pi\)
−0.00789315 + 0.999969i \(0.502512\pi\)
\(138\) 0 0
\(139\) −3.82616e20 −1.67542 −0.837709 0.546117i \(-0.816105\pi\)
−0.837709 + 0.546117i \(0.816105\pi\)
\(140\) 0 0
\(141\) −2.59947e20 −0.993809
\(142\) 0 0
\(143\) −3.53202e19 −0.118122
\(144\) 0 0
\(145\) −1.75005e19 −0.0512926
\(146\) 0 0
\(147\) −1.39126e20 −0.358009
\(148\) 0 0
\(149\) 3.27921e20 0.742163 0.371081 0.928600i \(-0.378987\pi\)
0.371081 + 0.928600i \(0.378987\pi\)
\(150\) 0 0
\(151\) 1.20975e20 0.241220 0.120610 0.992700i \(-0.461515\pi\)
0.120610 + 0.992700i \(0.461515\pi\)
\(152\) 0 0
\(153\) 4.35694e20 0.766677
\(154\) 0 0
\(155\) −7.94601e19 −0.123594
\(156\) 0 0
\(157\) 5.68747e20 0.783199 0.391599 0.920136i \(-0.371922\pi\)
0.391599 + 0.920136i \(0.371922\pi\)
\(158\) 0 0
\(159\) 5.64884e20 0.689742
\(160\) 0 0
\(161\) 1.02364e21 1.10995
\(162\) 0 0
\(163\) −1.77617e21 −1.71279 −0.856393 0.516325i \(-0.827300\pi\)
−0.856393 + 0.516325i \(0.827300\pi\)
\(164\) 0 0
\(165\) −1.03699e20 −0.0890591
\(166\) 0 0
\(167\) 1.55998e21 1.19485 0.597424 0.801926i \(-0.296191\pi\)
0.597424 + 0.801926i \(0.296191\pi\)
\(168\) 0 0
\(169\) −1.40285e21 −0.959594
\(170\) 0 0
\(171\) 3.92330e20 0.239987
\(172\) 0 0
\(173\) −1.55034e21 −0.849159 −0.424580 0.905391i \(-0.639578\pi\)
−0.424580 + 0.905391i \(0.639578\pi\)
\(174\) 0 0
\(175\) 1.31039e21 0.643488
\(176\) 0 0
\(177\) 1.63666e20 0.0721461
\(178\) 0 0
\(179\) −2.89048e21 −1.14516 −0.572579 0.819849i \(-0.694057\pi\)
−0.572579 + 0.819849i \(0.694057\pi\)
\(180\) 0 0
\(181\) −5.15105e21 −1.83633 −0.918163 0.396204i \(-0.870327\pi\)
−0.918163 + 0.396204i \(0.870327\pi\)
\(182\) 0 0
\(183\) 2.93758e21 0.943418
\(184\) 0 0
\(185\) −1.11655e21 −0.323404
\(186\) 0 0
\(187\) −3.08131e21 −0.805824
\(188\) 0 0
\(189\) 2.97310e21 0.702784
\(190\) 0 0
\(191\) 7.17061e21 1.53369 0.766847 0.641830i \(-0.221824\pi\)
0.766847 + 0.641830i \(0.221824\pi\)
\(192\) 0 0
\(193\) −3.25664e21 −0.630921 −0.315461 0.948939i \(-0.602159\pi\)
−0.315461 + 0.948939i \(0.602159\pi\)
\(194\) 0 0
\(195\) 1.73430e20 0.0304646
\(196\) 0 0
\(197\) 1.19004e22 1.89729 0.948643 0.316349i \(-0.102457\pi\)
0.948643 + 0.316349i \(0.102457\pi\)
\(198\) 0 0
\(199\) −2.18989e21 −0.317188 −0.158594 0.987344i \(-0.550696\pi\)
−0.158594 + 0.987344i \(0.550696\pi\)
\(200\) 0 0
\(201\) 6.21607e21 0.818754
\(202\) 0 0
\(203\) −1.27247e21 −0.152558
\(204\) 0 0
\(205\) −2.99224e21 −0.326840
\(206\) 0 0
\(207\) 9.17750e21 0.914126
\(208\) 0 0
\(209\) −2.77464e21 −0.252241
\(210\) 0 0
\(211\) −2.32425e22 −1.93019 −0.965096 0.261898i \(-0.915652\pi\)
−0.965096 + 0.261898i \(0.915652\pi\)
\(212\) 0 0
\(213\) 3.53757e21 0.268596
\(214\) 0 0
\(215\) 3.40723e21 0.236720
\(216\) 0 0
\(217\) −5.77758e21 −0.367602
\(218\) 0 0
\(219\) −1.85891e22 −1.08403
\(220\) 0 0
\(221\) 5.15329e21 0.275650
\(222\) 0 0
\(223\) −3.36614e21 −0.165286 −0.0826429 0.996579i \(-0.526336\pi\)
−0.0826429 + 0.996579i \(0.526336\pi\)
\(224\) 0 0
\(225\) 1.17484e22 0.529961
\(226\) 0 0
\(227\) −3.44209e21 −0.142750 −0.0713751 0.997450i \(-0.522739\pi\)
−0.0713751 + 0.997450i \(0.522739\pi\)
\(228\) 0 0
\(229\) −3.04661e22 −1.16246 −0.581232 0.813738i \(-0.697429\pi\)
−0.581232 + 0.813738i \(0.697429\pi\)
\(230\) 0 0
\(231\) −7.54002e21 −0.264886
\(232\) 0 0
\(233\) 1.01497e22 0.328527 0.164264 0.986416i \(-0.447475\pi\)
0.164264 + 0.986416i \(0.447475\pi\)
\(234\) 0 0
\(235\) 1.14463e22 0.341603
\(236\) 0 0
\(237\) 4.75002e21 0.130793
\(238\) 0 0
\(239\) 5.94716e22 1.51192 0.755961 0.654616i \(-0.227170\pi\)
0.755961 + 0.654616i \(0.227170\pi\)
\(240\) 0 0
\(241\) −6.71172e21 −0.157642 −0.0788210 0.996889i \(-0.525116\pi\)
−0.0788210 + 0.996889i \(0.525116\pi\)
\(242\) 0 0
\(243\) 4.37522e22 0.950037
\(244\) 0 0
\(245\) 6.12619e21 0.123059
\(246\) 0 0
\(247\) 4.64040e21 0.0862847
\(248\) 0 0
\(249\) 7.13368e21 0.122863
\(250\) 0 0
\(251\) 6.83985e22 1.09181 0.545904 0.837848i \(-0.316186\pi\)
0.545904 + 0.837848i \(0.316186\pi\)
\(252\) 0 0
\(253\) −6.49051e22 −0.960802
\(254\) 0 0
\(255\) 1.51299e22 0.207828
\(256\) 0 0
\(257\) −3.76051e22 −0.479603 −0.239801 0.970822i \(-0.577082\pi\)
−0.239801 + 0.970822i \(0.577082\pi\)
\(258\) 0 0
\(259\) −8.11846e22 −0.961892
\(260\) 0 0
\(261\) −1.14084e22 −0.125643
\(262\) 0 0
\(263\) −1.77095e23 −1.81396 −0.906980 0.421173i \(-0.861618\pi\)
−0.906980 + 0.421173i \(0.861618\pi\)
\(264\) 0 0
\(265\) −2.48737e22 −0.237086
\(266\) 0 0
\(267\) 6.30742e22 0.559751
\(268\) 0 0
\(269\) 1.19248e23 0.985837 0.492919 0.870075i \(-0.335930\pi\)
0.492919 + 0.870075i \(0.335930\pi\)
\(270\) 0 0
\(271\) −8.67879e22 −0.668730 −0.334365 0.942444i \(-0.608522\pi\)
−0.334365 + 0.942444i \(0.608522\pi\)
\(272\) 0 0
\(273\) 1.26102e22 0.0906100
\(274\) 0 0
\(275\) −8.30869e22 −0.557021
\(276\) 0 0
\(277\) 7.99312e21 0.0500217 0.0250109 0.999687i \(-0.492038\pi\)
0.0250109 + 0.999687i \(0.492038\pi\)
\(278\) 0 0
\(279\) −5.17991e22 −0.302749
\(280\) 0 0
\(281\) 4.90889e22 0.268085 0.134043 0.990976i \(-0.457204\pi\)
0.134043 + 0.990976i \(0.457204\pi\)
\(282\) 0 0
\(283\) 2.70773e23 1.38240 0.691201 0.722662i \(-0.257082\pi\)
0.691201 + 0.722662i \(0.257082\pi\)
\(284\) 0 0
\(285\) 1.36241e22 0.0650550
\(286\) 0 0
\(287\) −2.17567e23 −0.972109
\(288\) 0 0
\(289\) 2.10497e23 0.880473
\(290\) 0 0
\(291\) 1.71677e23 0.672555
\(292\) 0 0
\(293\) 2.47805e23 0.909635 0.454818 0.890585i \(-0.349704\pi\)
0.454818 + 0.890585i \(0.349704\pi\)
\(294\) 0 0
\(295\) −7.20676e21 −0.0247988
\(296\) 0 0
\(297\) −1.88512e23 −0.608350
\(298\) 0 0
\(299\) 1.08549e23 0.328664
\(300\) 0 0
\(301\) 2.47741e23 0.704070
\(302\) 0 0
\(303\) −8.20325e22 −0.218918
\(304\) 0 0
\(305\) −1.29351e23 −0.324281
\(306\) 0 0
\(307\) 2.71564e23 0.639820 0.319910 0.947448i \(-0.396347\pi\)
0.319910 + 0.947448i \(0.396347\pi\)
\(308\) 0 0
\(309\) −4.68850e23 −1.03855
\(310\) 0 0
\(311\) −4.17401e23 −0.869620 −0.434810 0.900522i \(-0.643184\pi\)
−0.434810 + 0.900522i \(0.643184\pi\)
\(312\) 0 0
\(313\) −2.21476e23 −0.434166 −0.217083 0.976153i \(-0.569654\pi\)
−0.217083 + 0.976153i \(0.569654\pi\)
\(314\) 0 0
\(315\) −4.69461e22 −0.0866261
\(316\) 0 0
\(317\) −8.33402e23 −1.44808 −0.724038 0.689760i \(-0.757716\pi\)
−0.724038 + 0.689760i \(0.757716\pi\)
\(318\) 0 0
\(319\) 8.06823e22 0.132059
\(320\) 0 0
\(321\) −5.49582e23 −0.847686
\(322\) 0 0
\(323\) 4.04825e23 0.588631
\(324\) 0 0
\(325\) 1.38957e23 0.190542
\(326\) 0 0
\(327\) −2.69460e23 −0.348571
\(328\) 0 0
\(329\) 8.32267e23 1.01602
\(330\) 0 0
\(331\) 1.25972e24 1.45181 0.725904 0.687796i \(-0.241422\pi\)
0.725904 + 0.687796i \(0.241422\pi\)
\(332\) 0 0
\(333\) −7.27864e23 −0.792191
\(334\) 0 0
\(335\) −2.73714e23 −0.281431
\(336\) 0 0
\(337\) 8.60548e23 0.836163 0.418082 0.908409i \(-0.362703\pi\)
0.418082 + 0.908409i \(0.362703\pi\)
\(338\) 0 0
\(339\) −6.02049e22 −0.0553012
\(340\) 0 0
\(341\) 3.66333e23 0.318207
\(342\) 0 0
\(343\) 1.27161e24 1.04486
\(344\) 0 0
\(345\) 3.18698e23 0.247798
\(346\) 0 0
\(347\) −2.55690e24 −1.88184 −0.940922 0.338623i \(-0.890039\pi\)
−0.940922 + 0.338623i \(0.890039\pi\)
\(348\) 0 0
\(349\) 1.73272e24 1.20750 0.603751 0.797173i \(-0.293672\pi\)
0.603751 + 0.797173i \(0.293672\pi\)
\(350\) 0 0
\(351\) 3.15275e23 0.208100
\(352\) 0 0
\(353\) 7.44110e23 0.465348 0.232674 0.972555i \(-0.425253\pi\)
0.232674 + 0.972555i \(0.425253\pi\)
\(354\) 0 0
\(355\) −1.55771e23 −0.0923246
\(356\) 0 0
\(357\) 1.10010e24 0.618138
\(358\) 0 0
\(359\) −1.67777e24 −0.893995 −0.446997 0.894535i \(-0.647507\pi\)
−0.446997 + 0.894535i \(0.647507\pi\)
\(360\) 0 0
\(361\) −1.61389e24 −0.815745
\(362\) 0 0
\(363\) −9.06407e23 −0.434720
\(364\) 0 0
\(365\) 8.18541e23 0.372613
\(366\) 0 0
\(367\) 1.06458e24 0.460096 0.230048 0.973179i \(-0.426112\pi\)
0.230048 + 0.973179i \(0.426112\pi\)
\(368\) 0 0
\(369\) −1.95060e24 −0.800606
\(370\) 0 0
\(371\) −1.80858e24 −0.705156
\(372\) 0 0
\(373\) −3.55316e24 −1.31638 −0.658189 0.752853i \(-0.728677\pi\)
−0.658189 + 0.752853i \(0.728677\pi\)
\(374\) 0 0
\(375\) 8.38372e23 0.295216
\(376\) 0 0
\(377\) −1.34936e23 −0.0451736
\(378\) 0 0
\(379\) −2.96331e24 −0.943418 −0.471709 0.881754i \(-0.656363\pi\)
−0.471709 + 0.881754i \(0.656363\pi\)
\(380\) 0 0
\(381\) 1.48941e24 0.451054
\(382\) 0 0
\(383\) 4.14872e22 0.0119544 0.00597718 0.999982i \(-0.498097\pi\)
0.00597718 + 0.999982i \(0.498097\pi\)
\(384\) 0 0
\(385\) 3.32012e23 0.0910493
\(386\) 0 0
\(387\) 2.22113e24 0.579855
\(388\) 0 0
\(389\) 1.32009e24 0.328158 0.164079 0.986447i \(-0.447535\pi\)
0.164079 + 0.986447i \(0.447535\pi\)
\(390\) 0 0
\(391\) 9.46978e24 2.24213
\(392\) 0 0
\(393\) −8.45448e23 −0.190703
\(394\) 0 0
\(395\) −2.09159e23 −0.0449577
\(396\) 0 0
\(397\) −2.49987e24 −0.512162 −0.256081 0.966655i \(-0.582431\pi\)
−0.256081 + 0.966655i \(0.582431\pi\)
\(398\) 0 0
\(399\) 9.90612e23 0.193491
\(400\) 0 0
\(401\) 5.22882e24 0.973939 0.486970 0.873419i \(-0.338102\pi\)
0.486970 + 0.873419i \(0.338102\pi\)
\(402\) 0 0
\(403\) −6.12669e23 −0.108850
\(404\) 0 0
\(405\) 1.72809e23 0.0292916
\(406\) 0 0
\(407\) 5.14760e24 0.832641
\(408\) 0 0
\(409\) −3.59253e24 −0.554663 −0.277331 0.960774i \(-0.589450\pi\)
−0.277331 + 0.960774i \(0.589450\pi\)
\(410\) 0 0
\(411\) −7.11139e22 −0.0104823
\(412\) 0 0
\(413\) −5.24006e23 −0.0737583
\(414\) 0 0
\(415\) −3.14120e23 −0.0422317
\(416\) 0 0
\(417\) −8.66148e24 −1.11250
\(418\) 0 0
\(419\) −7.64100e24 −0.937815 −0.468907 0.883247i \(-0.655352\pi\)
−0.468907 + 0.883247i \(0.655352\pi\)
\(420\) 0 0
\(421\) −6.63037e24 −0.777782 −0.388891 0.921284i \(-0.627142\pi\)
−0.388891 + 0.921284i \(0.627142\pi\)
\(422\) 0 0
\(423\) 7.46172e24 0.836769
\(424\) 0 0
\(425\) 1.21225e25 1.29987
\(426\) 0 0
\(427\) −9.40518e24 −0.964500
\(428\) 0 0
\(429\) −7.99562e23 −0.0784347
\(430\) 0 0
\(431\) −8.83021e24 −0.828776 −0.414388 0.910100i \(-0.636004\pi\)
−0.414388 + 0.910100i \(0.636004\pi\)
\(432\) 0 0
\(433\) −1.89602e24 −0.170298 −0.0851488 0.996368i \(-0.527137\pi\)
−0.0851488 + 0.996368i \(0.527137\pi\)
\(434\) 0 0
\(435\) −3.96168e23 −0.0340590
\(436\) 0 0
\(437\) 8.52727e24 0.701837
\(438\) 0 0
\(439\) −2.03577e25 −1.60441 −0.802206 0.597047i \(-0.796340\pi\)
−0.802206 + 0.597047i \(0.796340\pi\)
\(440\) 0 0
\(441\) 3.99359e24 0.301437
\(442\) 0 0
\(443\) −9.63050e24 −0.696327 −0.348164 0.937434i \(-0.613195\pi\)
−0.348164 + 0.937434i \(0.613195\pi\)
\(444\) 0 0
\(445\) −2.77736e24 −0.192403
\(446\) 0 0
\(447\) 7.42330e24 0.492806
\(448\) 0 0
\(449\) 2.24366e25 1.42764 0.713818 0.700331i \(-0.246964\pi\)
0.713818 + 0.700331i \(0.246964\pi\)
\(450\) 0 0
\(451\) 1.37951e25 0.841485
\(452\) 0 0
\(453\) 2.73856e24 0.160173
\(454\) 0 0
\(455\) −5.55268e23 −0.0311454
\(456\) 0 0
\(457\) −1.43601e25 −0.772597 −0.386299 0.922374i \(-0.626247\pi\)
−0.386299 + 0.922374i \(0.626247\pi\)
\(458\) 0 0
\(459\) 2.75043e25 1.41965
\(460\) 0 0
\(461\) 3.20527e25 1.58747 0.793736 0.608262i \(-0.208133\pi\)
0.793736 + 0.608262i \(0.208133\pi\)
\(462\) 0 0
\(463\) 4.99392e24 0.237368 0.118684 0.992932i \(-0.462132\pi\)
0.118684 + 0.992932i \(0.462132\pi\)
\(464\) 0 0
\(465\) −1.79878e24 −0.0820682
\(466\) 0 0
\(467\) −8.24714e24 −0.361238 −0.180619 0.983553i \(-0.557810\pi\)
−0.180619 + 0.983553i \(0.557810\pi\)
\(468\) 0 0
\(469\) −1.99019e25 −0.837051
\(470\) 0 0
\(471\) 1.28750e25 0.520054
\(472\) 0 0
\(473\) −1.57083e25 −0.609463
\(474\) 0 0
\(475\) 1.09160e25 0.406888
\(476\) 0 0
\(477\) −1.62149e25 −0.580750
\(478\) 0 0
\(479\) −2.71323e25 −0.933899 −0.466950 0.884284i \(-0.654647\pi\)
−0.466950 + 0.884284i \(0.654647\pi\)
\(480\) 0 0
\(481\) −8.60902e24 −0.284823
\(482\) 0 0
\(483\) 2.31727e25 0.737019
\(484\) 0 0
\(485\) −7.55949e24 −0.231178
\(486\) 0 0
\(487\) −5.00346e25 −1.47145 −0.735725 0.677281i \(-0.763158\pi\)
−0.735725 + 0.677281i \(0.763158\pi\)
\(488\) 0 0
\(489\) −4.02081e25 −1.13731
\(490\) 0 0
\(491\) 3.80743e25 1.03600 0.517998 0.855382i \(-0.326677\pi\)
0.517998 + 0.855382i \(0.326677\pi\)
\(492\) 0 0
\(493\) −1.17717e25 −0.308172
\(494\) 0 0
\(495\) 2.97667e24 0.0749861
\(496\) 0 0
\(497\) −1.13262e25 −0.274598
\(498\) 0 0
\(499\) 1.94991e25 0.455049 0.227525 0.973772i \(-0.426937\pi\)
0.227525 + 0.973772i \(0.426937\pi\)
\(500\) 0 0
\(501\) 3.53141e25 0.793394
\(502\) 0 0
\(503\) −4.11673e25 −0.890546 −0.445273 0.895395i \(-0.646893\pi\)
−0.445273 + 0.895395i \(0.646893\pi\)
\(504\) 0 0
\(505\) 3.61216e24 0.0752488
\(506\) 0 0
\(507\) −3.17570e25 −0.637183
\(508\) 0 0
\(509\) 6.45581e24 0.124776 0.0623881 0.998052i \(-0.480128\pi\)
0.0623881 + 0.998052i \(0.480128\pi\)
\(510\) 0 0
\(511\) 5.95164e25 1.10825
\(512\) 0 0
\(513\) 2.47669e25 0.444382
\(514\) 0 0
\(515\) 2.06450e25 0.356982
\(516\) 0 0
\(517\) −5.27708e25 −0.879495
\(518\) 0 0
\(519\) −3.50958e25 −0.563853
\(520\) 0 0
\(521\) 7.61781e25 1.17997 0.589986 0.807413i \(-0.299133\pi\)
0.589986 + 0.807413i \(0.299133\pi\)
\(522\) 0 0
\(523\) 7.54461e25 1.12686 0.563431 0.826163i \(-0.309481\pi\)
0.563431 + 0.826163i \(0.309481\pi\)
\(524\) 0 0
\(525\) 2.96640e25 0.427284
\(526\) 0 0
\(527\) −5.34488e25 −0.742569
\(528\) 0 0
\(529\) 1.24857e26 1.67334
\(530\) 0 0
\(531\) −4.69800e24 −0.0607456
\(532\) 0 0
\(533\) −2.30713e25 −0.287849
\(534\) 0 0
\(535\) 2.41999e25 0.291376
\(536\) 0 0
\(537\) −6.54332e25 −0.760400
\(538\) 0 0
\(539\) −2.82435e25 −0.316829
\(540\) 0 0
\(541\) 1.27935e26 1.38553 0.692763 0.721165i \(-0.256393\pi\)
0.692763 + 0.721165i \(0.256393\pi\)
\(542\) 0 0
\(543\) −1.16607e26 −1.21934
\(544\) 0 0
\(545\) 1.18652e25 0.119814
\(546\) 0 0
\(547\) −1.03216e26 −1.00662 −0.503310 0.864106i \(-0.667885\pi\)
−0.503310 + 0.864106i \(0.667885\pi\)
\(548\) 0 0
\(549\) −8.43225e25 −0.794340
\(550\) 0 0
\(551\) −1.06001e25 −0.0964649
\(552\) 0 0
\(553\) −1.52080e25 −0.133716
\(554\) 0 0
\(555\) −2.52759e25 −0.214745
\(556\) 0 0
\(557\) −1.15924e26 −0.951811 −0.475905 0.879497i \(-0.657880\pi\)
−0.475905 + 0.879497i \(0.657880\pi\)
\(558\) 0 0
\(559\) 2.62710e25 0.208480
\(560\) 0 0
\(561\) −6.97532e25 −0.535078
\(562\) 0 0
\(563\) −6.69128e25 −0.496226 −0.248113 0.968731i \(-0.579810\pi\)
−0.248113 + 0.968731i \(0.579810\pi\)
\(564\) 0 0
\(565\) 2.65102e24 0.0190087
\(566\) 0 0
\(567\) 1.25650e25 0.0871210
\(568\) 0 0
\(569\) −1.98116e26 −1.32847 −0.664236 0.747523i \(-0.731243\pi\)
−0.664236 + 0.747523i \(0.731243\pi\)
\(570\) 0 0
\(571\) 2.67644e26 1.73586 0.867930 0.496687i \(-0.165450\pi\)
0.867930 + 0.496687i \(0.165450\pi\)
\(572\) 0 0
\(573\) 1.62325e26 1.01839
\(574\) 0 0
\(575\) 2.55350e26 1.54986
\(576\) 0 0
\(577\) −1.86979e25 −0.109805 −0.0549027 0.998492i \(-0.517485\pi\)
−0.0549027 + 0.998492i \(0.517485\pi\)
\(578\) 0 0
\(579\) −7.37222e25 −0.418940
\(580\) 0 0
\(581\) −2.28398e25 −0.125608
\(582\) 0 0
\(583\) 1.14675e26 0.610403
\(584\) 0 0
\(585\) −4.97828e24 −0.0256507
\(586\) 0 0
\(587\) −1.83068e26 −0.913167 −0.456583 0.889681i \(-0.650927\pi\)
−0.456583 + 0.889681i \(0.650927\pi\)
\(588\) 0 0
\(589\) −4.81291e25 −0.232441
\(590\) 0 0
\(591\) 2.69396e26 1.25982
\(592\) 0 0
\(593\) 9.12500e25 0.413250 0.206625 0.978420i \(-0.433752\pi\)
0.206625 + 0.978420i \(0.433752\pi\)
\(594\) 0 0
\(595\) −4.84412e25 −0.212473
\(596\) 0 0
\(597\) −4.95735e25 −0.210617
\(598\) 0 0
\(599\) 2.37311e26 0.976705 0.488353 0.872646i \(-0.337598\pi\)
0.488353 + 0.872646i \(0.337598\pi\)
\(600\) 0 0
\(601\) 2.29560e26 0.915355 0.457677 0.889118i \(-0.348681\pi\)
0.457677 + 0.889118i \(0.348681\pi\)
\(602\) 0 0
\(603\) −1.78431e26 −0.689376
\(604\) 0 0
\(605\) 3.99121e25 0.149427
\(606\) 0 0
\(607\) −1.10962e26 −0.402608 −0.201304 0.979529i \(-0.564518\pi\)
−0.201304 + 0.979529i \(0.564518\pi\)
\(608\) 0 0
\(609\) −2.88055e25 −0.101300
\(610\) 0 0
\(611\) 8.82556e25 0.300851
\(612\) 0 0
\(613\) −1.27027e25 −0.0419779 −0.0209890 0.999780i \(-0.506681\pi\)
−0.0209890 + 0.999780i \(0.506681\pi\)
\(614\) 0 0
\(615\) −6.77368e25 −0.217026
\(616\) 0 0
\(617\) −4.22914e26 −1.31384 −0.656921 0.753960i \(-0.728141\pi\)
−0.656921 + 0.753960i \(0.728141\pi\)
\(618\) 0 0
\(619\) −5.49720e26 −1.65608 −0.828039 0.560671i \(-0.810543\pi\)
−0.828039 + 0.560671i \(0.810543\pi\)
\(620\) 0 0
\(621\) 5.79354e26 1.69268
\(622\) 0 0
\(623\) −2.01943e26 −0.572259
\(624\) 0 0
\(625\) 3.07930e26 0.846431
\(626\) 0 0
\(627\) −6.28108e25 −0.167492
\(628\) 0 0
\(629\) −7.51045e26 −1.94305
\(630\) 0 0
\(631\) 2.83709e25 0.0712187 0.0356093 0.999366i \(-0.488663\pi\)
0.0356093 + 0.999366i \(0.488663\pi\)
\(632\) 0 0
\(633\) −5.26153e26 −1.28167
\(634\) 0 0
\(635\) −6.55838e25 −0.155041
\(636\) 0 0
\(637\) 4.72353e25 0.108378
\(638\) 0 0
\(639\) −1.01545e26 −0.226153
\(640\) 0 0
\(641\) 8.77075e26 1.89620 0.948102 0.317965i \(-0.102999\pi\)
0.948102 + 0.317965i \(0.102999\pi\)
\(642\) 0 0
\(643\) −4.19589e26 −0.880683 −0.440341 0.897830i \(-0.645143\pi\)
−0.440341 + 0.897830i \(0.645143\pi\)
\(644\) 0 0
\(645\) 7.71311e25 0.157185
\(646\) 0 0
\(647\) 3.89932e26 0.771611 0.385805 0.922580i \(-0.373924\pi\)
0.385805 + 0.922580i \(0.373924\pi\)
\(648\) 0 0
\(649\) 3.32252e25 0.0638473
\(650\) 0 0
\(651\) −1.30790e26 −0.244093
\(652\) 0 0
\(653\) 5.24023e26 0.949895 0.474947 0.880014i \(-0.342467\pi\)
0.474947 + 0.880014i \(0.342467\pi\)
\(654\) 0 0
\(655\) 3.72279e25 0.0655505
\(656\) 0 0
\(657\) 5.33597e26 0.912730
\(658\) 0 0
\(659\) 7.67459e26 1.27539 0.637696 0.770288i \(-0.279888\pi\)
0.637696 + 0.770288i \(0.279888\pi\)
\(660\) 0 0
\(661\) −1.32790e26 −0.214413 −0.107206 0.994237i \(-0.534191\pi\)
−0.107206 + 0.994237i \(0.534191\pi\)
\(662\) 0 0
\(663\) 1.16658e26 0.183035
\(664\) 0 0
\(665\) −4.36199e25 −0.0665088
\(666\) 0 0
\(667\) −2.47960e26 −0.367440
\(668\) 0 0
\(669\) −7.62010e25 −0.109752
\(670\) 0 0
\(671\) 5.96346e26 0.834899
\(672\) 0 0
\(673\) −3.37509e26 −0.459349 −0.229674 0.973268i \(-0.573766\pi\)
−0.229674 + 0.973268i \(0.573766\pi\)
\(674\) 0 0
\(675\) 7.41648e26 0.981323
\(676\) 0 0
\(677\) −1.12451e27 −1.44667 −0.723336 0.690496i \(-0.757392\pi\)
−0.723336 + 0.690496i \(0.757392\pi\)
\(678\) 0 0
\(679\) −5.49654e26 −0.687585
\(680\) 0 0
\(681\) −7.79203e25 −0.0947880
\(682\) 0 0
\(683\) 7.05871e26 0.835081 0.417540 0.908658i \(-0.362892\pi\)
0.417540 + 0.908658i \(0.362892\pi\)
\(684\) 0 0
\(685\) 3.13138e24 0.00360309
\(686\) 0 0
\(687\) −6.89676e26 −0.771891
\(688\) 0 0
\(689\) −1.91786e26 −0.208802
\(690\) 0 0
\(691\) −4.70572e25 −0.0498408 −0.0249204 0.999689i \(-0.507933\pi\)
−0.0249204 + 0.999689i \(0.507933\pi\)
\(692\) 0 0
\(693\) 2.16435e26 0.223029
\(694\) 0 0
\(695\) 3.81394e26 0.382400
\(696\) 0 0
\(697\) −2.01273e27 −1.96369
\(698\) 0 0
\(699\) 2.29764e26 0.218147
\(700\) 0 0
\(701\) 5.38593e25 0.0497668 0.0248834 0.999690i \(-0.492079\pi\)
0.0248834 + 0.999690i \(0.492079\pi\)
\(702\) 0 0
\(703\) −6.76295e26 −0.608220
\(704\) 0 0
\(705\) 2.59116e26 0.226829
\(706\) 0 0
\(707\) 2.62642e26 0.223810
\(708\) 0 0
\(709\) −2.80682e26 −0.232849 −0.116425 0.993200i \(-0.537143\pi\)
−0.116425 + 0.993200i \(0.537143\pi\)
\(710\) 0 0
\(711\) −1.36348e26 −0.110126
\(712\) 0 0
\(713\) −1.12585e27 −0.885381
\(714\) 0 0
\(715\) 3.52074e25 0.0269604
\(716\) 0 0
\(717\) 1.34629e27 1.00394
\(718\) 0 0
\(719\) −1.82032e27 −1.32198 −0.660988 0.750397i \(-0.729862\pi\)
−0.660988 + 0.750397i \(0.729862\pi\)
\(720\) 0 0
\(721\) 1.50111e27 1.06176
\(722\) 0 0
\(723\) −1.51937e26 −0.104676
\(724\) 0 0
\(725\) −3.17421e26 −0.213022
\(726\) 0 0
\(727\) −1.47227e27 −0.962522 −0.481261 0.876577i \(-0.659821\pi\)
−0.481261 + 0.876577i \(0.659821\pi\)
\(728\) 0 0
\(729\) 1.19193e27 0.759173
\(730\) 0 0
\(731\) 2.29187e27 1.42224
\(732\) 0 0
\(733\) 2.23845e27 1.35350 0.676751 0.736212i \(-0.263387\pi\)
0.676751 + 0.736212i \(0.263387\pi\)
\(734\) 0 0
\(735\) 1.38682e26 0.0817127
\(736\) 0 0
\(737\) 1.26190e27 0.724576
\(738\) 0 0
\(739\) 4.64717e26 0.260056 0.130028 0.991510i \(-0.458493\pi\)
0.130028 + 0.991510i \(0.458493\pi\)
\(740\) 0 0
\(741\) 1.05047e26 0.0572942
\(742\) 0 0
\(743\) 5.63164e26 0.299393 0.149696 0.988732i \(-0.452170\pi\)
0.149696 + 0.988732i \(0.452170\pi\)
\(744\) 0 0
\(745\) −3.26873e26 −0.169392
\(746\) 0 0
\(747\) −2.04771e26 −0.103448
\(748\) 0 0
\(749\) 1.75959e27 0.866629
\(750\) 0 0
\(751\) −6.04104e26 −0.290090 −0.145045 0.989425i \(-0.546333\pi\)
−0.145045 + 0.989425i \(0.546333\pi\)
\(752\) 0 0
\(753\) 1.54837e27 0.724975
\(754\) 0 0
\(755\) −1.20588e26 −0.0550564
\(756\) 0 0
\(757\) −3.91726e27 −1.74410 −0.872051 0.489415i \(-0.837211\pi\)
−0.872051 + 0.489415i \(0.837211\pi\)
\(758\) 0 0
\(759\) −1.46929e27 −0.637985
\(760\) 0 0
\(761\) −3.21509e27 −1.36157 −0.680783 0.732486i \(-0.738360\pi\)
−0.680783 + 0.732486i \(0.738360\pi\)
\(762\) 0 0
\(763\) 8.62725e26 0.356360
\(764\) 0 0
\(765\) −4.34301e26 −0.174988
\(766\) 0 0
\(767\) −5.55669e25 −0.0218404
\(768\) 0 0
\(769\) 3.87198e27 1.48468 0.742339 0.670024i \(-0.233716\pi\)
0.742339 + 0.670024i \(0.233716\pi\)
\(770\) 0 0
\(771\) −8.51285e26 −0.318463
\(772\) 0 0
\(773\) −2.22257e27 −0.811240 −0.405620 0.914042i \(-0.632944\pi\)
−0.405620 + 0.914042i \(0.632944\pi\)
\(774\) 0 0
\(775\) −1.44123e27 −0.513297
\(776\) 0 0
\(777\) −1.83782e27 −0.638709
\(778\) 0 0
\(779\) −1.81240e27 −0.614680
\(780\) 0 0
\(781\) 7.18148e26 0.237700
\(782\) 0 0
\(783\) −7.20184e26 −0.232652
\(784\) 0 0
\(785\) −5.66929e26 −0.178759
\(786\) 0 0
\(787\) 3.32013e27 1.02187 0.510934 0.859620i \(-0.329300\pi\)
0.510934 + 0.859620i \(0.329300\pi\)
\(788\) 0 0
\(789\) −4.00900e27 −1.20449
\(790\) 0 0
\(791\) 1.92757e26 0.0565370
\(792\) 0 0
\(793\) −9.97349e26 −0.285596
\(794\) 0 0
\(795\) −5.63079e26 −0.157428
\(796\) 0 0
\(797\) 3.80480e27 1.03867 0.519335 0.854571i \(-0.326180\pi\)
0.519335 + 0.854571i \(0.326180\pi\)
\(798\) 0 0
\(799\) 7.69936e27 2.05239
\(800\) 0 0
\(801\) −1.81053e27 −0.471300
\(802\) 0 0
\(803\) −3.77370e27 −0.959334
\(804\) 0 0
\(805\) −1.02037e27 −0.253336
\(806\) 0 0
\(807\) 2.69948e27 0.654609
\(808\) 0 0
\(809\) −6.75938e26 −0.160102 −0.0800508 0.996791i \(-0.525508\pi\)
−0.0800508 + 0.996791i \(0.525508\pi\)
\(810\) 0 0
\(811\) 1.86888e27 0.432398 0.216199 0.976349i \(-0.430634\pi\)
0.216199 + 0.976349i \(0.430634\pi\)
\(812\) 0 0
\(813\) −1.96466e27 −0.444045
\(814\) 0 0
\(815\) 1.77050e27 0.390929
\(816\) 0 0
\(817\) 2.06376e27 0.445195
\(818\) 0 0
\(819\) −3.61973e26 −0.0762919
\(820\) 0 0
\(821\) 1.78616e27 0.367841 0.183921 0.982941i \(-0.441121\pi\)
0.183921 + 0.982941i \(0.441121\pi\)
\(822\) 0 0
\(823\) 4.84426e27 0.974831 0.487415 0.873170i \(-0.337940\pi\)
0.487415 + 0.873170i \(0.337940\pi\)
\(824\) 0 0
\(825\) −1.88088e27 −0.369870
\(826\) 0 0
\(827\) 7.46068e26 0.143376 0.0716879 0.997427i \(-0.477161\pi\)
0.0716879 + 0.997427i \(0.477161\pi\)
\(828\) 0 0
\(829\) 9.45101e27 1.77505 0.887524 0.460761i \(-0.152424\pi\)
0.887524 + 0.460761i \(0.152424\pi\)
\(830\) 0 0
\(831\) 1.80944e26 0.0332151
\(832\) 0 0
\(833\) 4.12077e27 0.739352
\(834\) 0 0
\(835\) −1.55499e27 −0.272714
\(836\) 0 0
\(837\) −3.26996e27 −0.560596
\(838\) 0 0
\(839\) −8.87983e27 −1.48822 −0.744108 0.668059i \(-0.767125\pi\)
−0.744108 + 0.668059i \(0.767125\pi\)
\(840\) 0 0
\(841\) −5.79503e27 −0.949497
\(842\) 0 0
\(843\) 1.11125e27 0.178012
\(844\) 0 0
\(845\) 1.39837e27 0.219019
\(846\) 0 0
\(847\) 2.90202e27 0.444435
\(848\) 0 0
\(849\) 6.12963e27 0.917934
\(850\) 0 0
\(851\) −1.58201e28 −2.31674
\(852\) 0 0
\(853\) −8.81716e26 −0.126274 −0.0631369 0.998005i \(-0.520110\pi\)
−0.0631369 + 0.998005i \(0.520110\pi\)
\(854\) 0 0
\(855\) −3.91076e26 −0.0547751
\(856\) 0 0
\(857\) −1.68989e26 −0.0231495 −0.0115747 0.999933i \(-0.503684\pi\)
−0.0115747 + 0.999933i \(0.503684\pi\)
\(858\) 0 0
\(859\) −1.45204e28 −1.94556 −0.972779 0.231735i \(-0.925560\pi\)
−0.972779 + 0.231735i \(0.925560\pi\)
\(860\) 0 0
\(861\) −4.92517e27 −0.645493
\(862\) 0 0
\(863\) 1.45753e27 0.186860 0.0934299 0.995626i \(-0.470217\pi\)
0.0934299 + 0.995626i \(0.470217\pi\)
\(864\) 0 0
\(865\) 1.54539e27 0.193813
\(866\) 0 0
\(867\) 4.76512e27 0.584645
\(868\) 0 0
\(869\) 9.64282e26 0.115749
\(870\) 0 0
\(871\) −2.11044e27 −0.247857
\(872\) 0 0
\(873\) −4.92794e27 −0.566279
\(874\) 0 0
\(875\) −2.68420e27 −0.301813
\(876\) 0 0
\(877\) 8.44834e27 0.929555 0.464777 0.885428i \(-0.346134\pi\)
0.464777 + 0.885428i \(0.346134\pi\)
\(878\) 0 0
\(879\) 5.60968e27 0.604010
\(880\) 0 0
\(881\) 3.79803e26 0.0400209 0.0200105 0.999800i \(-0.493630\pi\)
0.0200105 + 0.999800i \(0.493630\pi\)
\(882\) 0 0
\(883\) 1.64147e27 0.169280 0.0846400 0.996412i \(-0.473026\pi\)
0.0846400 + 0.996412i \(0.473026\pi\)
\(884\) 0 0
\(885\) −1.63143e26 −0.0164667
\(886\) 0 0
\(887\) 1.53135e28 1.51286 0.756432 0.654072i \(-0.226941\pi\)
0.756432 + 0.654072i \(0.226941\pi\)
\(888\) 0 0
\(889\) −4.76863e27 −0.461133
\(890\) 0 0
\(891\) −7.96696e26 −0.0754144
\(892\) 0 0
\(893\) 6.93306e27 0.642445
\(894\) 0 0
\(895\) 2.88124e27 0.261373
\(896\) 0 0
\(897\) 2.45729e27 0.218237
\(898\) 0 0
\(899\) 1.39952e27 0.121692
\(900\) 0 0
\(901\) −1.67313e28 −1.42444
\(902\) 0 0
\(903\) 5.60823e27 0.467511
\(904\) 0 0
\(905\) 5.13459e27 0.419126
\(906\) 0 0
\(907\) 2.27504e28 1.81853 0.909265 0.416218i \(-0.136645\pi\)
0.909265 + 0.416218i \(0.136645\pi\)
\(908\) 0 0
\(909\) 2.35473e27 0.184325
\(910\) 0 0
\(911\) −1.90023e28 −1.45674 −0.728369 0.685185i \(-0.759721\pi\)
−0.728369 + 0.685185i \(0.759721\pi\)
\(912\) 0 0
\(913\) 1.44818e27 0.108730
\(914\) 0 0
\(915\) −2.92819e27 −0.215327
\(916\) 0 0
\(917\) 2.70685e27 0.194965
\(918\) 0 0
\(919\) 7.01793e27 0.495122 0.247561 0.968872i \(-0.420371\pi\)
0.247561 + 0.968872i \(0.420371\pi\)
\(920\) 0 0
\(921\) 6.14753e27 0.424849
\(922\) 0 0
\(923\) −1.20106e27 −0.0813106
\(924\) 0 0
\(925\) −2.02518e28 −1.34312
\(926\) 0 0
\(927\) 1.34582e28 0.874441
\(928\) 0 0
\(929\) 1.04079e28 0.662545 0.331273 0.943535i \(-0.392522\pi\)
0.331273 + 0.943535i \(0.392522\pi\)
\(930\) 0 0
\(931\) 3.71064e27 0.231434
\(932\) 0 0
\(933\) −9.44891e27 −0.577439
\(934\) 0 0
\(935\) 3.07146e27 0.183923
\(936\) 0 0
\(937\) −2.36290e28 −1.38650 −0.693249 0.720698i \(-0.743821\pi\)
−0.693249 + 0.720698i \(0.743821\pi\)
\(938\) 0 0
\(939\) −5.01366e27 −0.288292
\(940\) 0 0
\(941\) −4.44433e27 −0.250441 −0.125220 0.992129i \(-0.539964\pi\)
−0.125220 + 0.992129i \(0.539964\pi\)
\(942\) 0 0
\(943\) −4.23963e28 −2.34135
\(944\) 0 0
\(945\) −2.96360e27 −0.160405
\(946\) 0 0
\(947\) 1.77081e28 0.939390 0.469695 0.882829i \(-0.344364\pi\)
0.469695 + 0.882829i \(0.344364\pi\)
\(948\) 0 0
\(949\) 6.31127e27 0.328162
\(950\) 0 0
\(951\) −1.88661e28 −0.961542
\(952\) 0 0
\(953\) 4.35882e27 0.217764 0.108882 0.994055i \(-0.465273\pi\)
0.108882 + 0.994055i \(0.465273\pi\)
\(954\) 0 0
\(955\) −7.14769e27 −0.350053
\(956\) 0 0
\(957\) 1.82645e27 0.0876886
\(958\) 0 0
\(959\) 2.27684e26 0.0107166
\(960\) 0 0
\(961\) −1.53162e28 −0.706771
\(962\) 0 0
\(963\) 1.57756e28 0.713735
\(964\) 0 0
\(965\) 3.24623e27 0.144003
\(966\) 0 0
\(967\) −7.37579e27 −0.320817 −0.160408 0.987051i \(-0.551281\pi\)
−0.160408 + 0.987051i \(0.551281\pi\)
\(968\) 0 0
\(969\) 9.16422e27 0.390859
\(970\) 0 0
\(971\) −1.32649e28 −0.554780 −0.277390 0.960757i \(-0.589469\pi\)
−0.277390 + 0.960757i \(0.589469\pi\)
\(972\) 0 0
\(973\) 2.77313e28 1.13736
\(974\) 0 0
\(975\) 3.14565e27 0.126522
\(976\) 0 0
\(977\) 1.57429e28 0.620991 0.310496 0.950575i \(-0.399505\pi\)
0.310496 + 0.950575i \(0.399505\pi\)
\(978\) 0 0
\(979\) 1.28044e28 0.495364
\(980\) 0 0
\(981\) 7.73480e27 0.293490
\(982\) 0 0
\(983\) −2.23838e27 −0.0833059 −0.0416529 0.999132i \(-0.513262\pi\)
−0.0416529 + 0.999132i \(0.513262\pi\)
\(984\) 0 0
\(985\) −1.18624e28 −0.433040
\(986\) 0 0
\(987\) 1.88404e28 0.674649
\(988\) 0 0
\(989\) 4.82761e28 1.69577
\(990\) 0 0
\(991\) −2.72873e27 −0.0940288 −0.0470144 0.998894i \(-0.514971\pi\)
−0.0470144 + 0.998894i \(0.514971\pi\)
\(992\) 0 0
\(993\) 2.85170e28 0.964019
\(994\) 0 0
\(995\) 2.18289e27 0.0723956
\(996\) 0 0
\(997\) 1.56740e28 0.510008 0.255004 0.966940i \(-0.417923\pi\)
0.255004 + 0.966940i \(0.417923\pi\)
\(998\) 0 0
\(999\) −4.59484e28 −1.46689
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.1.2 2
3.2 odd 2 72.20.a.a.1.2 2
4.3 odd 2 16.20.a.e.1.1 2
8.3 odd 2 64.20.a.j.1.2 2
8.5 even 2 64.20.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.20.a.a.1.2 2 1.1 even 1 trivial
16.20.a.e.1.1 2 4.3 odd 2
64.20.a.j.1.2 2 8.3 odd 2
64.20.a.k.1.1 2 8.5 even 2
72.20.a.a.1.2 2 3.2 odd 2