Properties

Label 9.38.a.c.1.3
Level 9
Weight 38
Character 9.1
Self dual yes
Analytic conductor 78.043
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0426343121\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{10}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(31743.2\)
Character \(\chi\) = 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+81067.1 q^{2} -1.30867e11 q^{4} +7.16603e12 q^{5} -5.96869e15 q^{7} -2.17508e16 q^{8} +O(q^{10})\) \(q+81067.1 q^{2} -1.30867e11 q^{4} +7.16603e12 q^{5} -5.96869e15 q^{7} -2.17508e16 q^{8} +5.80929e17 q^{10} -2.06526e19 q^{11} -4.02756e20 q^{13} -4.83865e20 q^{14} +1.62230e22 q^{16} +4.55890e22 q^{17} +1.54990e23 q^{19} -9.37797e23 q^{20} -1.67424e24 q^{22} -2.42540e25 q^{23} -2.14076e25 q^{25} -3.26502e25 q^{26} +7.81105e26 q^{28} -1.52016e27 q^{29} +6.09023e27 q^{31} +4.30456e27 q^{32} +3.69577e27 q^{34} -4.27718e28 q^{35} -6.28281e28 q^{37} +1.25646e28 q^{38} -1.55867e29 q^{40} +8.11303e29 q^{41} +4.69121e29 q^{43} +2.70274e30 q^{44} -1.96620e30 q^{46} -1.02212e31 q^{47} +1.70631e31 q^{49} -1.73545e30 q^{50} +5.27075e31 q^{52} -5.42457e31 q^{53} -1.47997e32 q^{55} +1.29824e32 q^{56} -1.23235e32 q^{58} +5.27739e32 q^{59} -6.19951e31 q^{61} +4.93717e32 q^{62} -1.88071e33 q^{64} -2.88616e33 q^{65} +9.30348e33 q^{67} -5.96611e33 q^{68} -3.46739e33 q^{70} +2.75965e34 q^{71} +2.09192e34 q^{73} -5.09329e33 q^{74} -2.02831e34 q^{76} +1.23269e35 q^{77} +1.20341e34 q^{79} +1.16254e35 q^{80} +6.57700e34 q^{82} +3.25180e34 q^{83} +3.26692e35 q^{85} +3.80303e34 q^{86} +4.49210e35 q^{88} +1.39444e36 q^{89} +2.40392e36 q^{91} +3.17405e36 q^{92} -8.28600e35 q^{94} +1.11066e36 q^{95} -6.18707e36 q^{97} +1.38326e36 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} + O(q^{10}) \) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} - 21511023001649316q^{10} - 20953708852195292976q^{11} + 51830892788989874168q^{13} + \)\(18\!\cdots\!12\)\(q^{14} + \)\(55\!\cdots\!40\)\(q^{16} - \)\(81\!\cdots\!28\)\(q^{17} - \)\(54\!\cdots\!32\)\(q^{19} + \)\(35\!\cdots\!76\)\(q^{20} + \)\(23\!\cdots\!76\)\(q^{22} + \)\(61\!\cdots\!88\)\(q^{23} + \)\(95\!\cdots\!16\)\(q^{25} - \)\(42\!\cdots\!88\)\(q^{26} + \)\(24\!\cdots\!48\)\(q^{28} - \)\(41\!\cdots\!36\)\(q^{29} + \)\(89\!\cdots\!64\)\(q^{31} - \)\(36\!\cdots\!84\)\(q^{32} + \)\(31\!\cdots\!24\)\(q^{34} - \)\(42\!\cdots\!40\)\(q^{35} + \)\(55\!\cdots\!24\)\(q^{37} + \)\(73\!\cdots\!68\)\(q^{38} - \)\(26\!\cdots\!52\)\(q^{40} + \)\(86\!\cdots\!76\)\(q^{41} - \)\(50\!\cdots\!80\)\(q^{43} - \)\(28\!\cdots\!36\)\(q^{44} - \)\(14\!\cdots\!96\)\(q^{46} - \)\(42\!\cdots\!20\)\(q^{47} + \)\(40\!\cdots\!20\)\(q^{49} - \)\(10\!\cdots\!14\)\(q^{50} + \)\(18\!\cdots\!68\)\(q^{52} + \)\(12\!\cdots\!88\)\(q^{53} - \)\(32\!\cdots\!24\)\(q^{55} - \)\(49\!\cdots\!80\)\(q^{56} - \)\(75\!\cdots\!56\)\(q^{58} + \)\(13\!\cdots\!88\)\(q^{59} - \)\(12\!\cdots\!60\)\(q^{61} + \)\(37\!\cdots\!92\)\(q^{62} + \)\(85\!\cdots\!28\)\(q^{64} - \)\(15\!\cdots\!68\)\(q^{65} + \)\(16\!\cdots\!48\)\(q^{67} - \)\(63\!\cdots\!84\)\(q^{68} + \)\(82\!\cdots\!60\)\(q^{70} - \)\(10\!\cdots\!88\)\(q^{71} - \)\(19\!\cdots\!48\)\(q^{73} + \)\(89\!\cdots\!12\)\(q^{74} - \)\(95\!\cdots\!68\)\(q^{76} + \)\(25\!\cdots\!92\)\(q^{77} + \)\(42\!\cdots\!20\)\(q^{79} + \)\(95\!\cdots\!36\)\(q^{80} + \)\(33\!\cdots\!48\)\(q^{82} + \)\(46\!\cdots\!24\)\(q^{83} + \)\(18\!\cdots\!12\)\(q^{85} + \)\(36\!\cdots\!04\)\(q^{86} + \)\(42\!\cdots\!56\)\(q^{88} + \)\(31\!\cdots\!52\)\(q^{89} + \)\(26\!\cdots\!24\)\(q^{91} + \)\(16\!\cdots\!16\)\(q^{92} - \)\(57\!\cdots\!48\)\(q^{94} + \)\(89\!\cdots\!56\)\(q^{95} + \)\(44\!\cdots\!48\)\(q^{97} + \)\(43\!\cdots\!78\)\(q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 81067.1 0.218670 0.109335 0.994005i \(-0.465128\pi\)
0.109335 + 0.994005i \(0.465128\pi\)
\(3\) 0 0
\(4\) −1.30867e11 −0.952183
\(5\) 7.16603e12 0.840105 0.420052 0.907500i \(-0.362012\pi\)
0.420052 + 0.907500i \(0.362012\pi\)
\(6\) 0 0
\(7\) −5.96869e15 −1.38537 −0.692684 0.721241i \(-0.743572\pi\)
−0.692684 + 0.721241i \(0.743572\pi\)
\(8\) −2.17508e16 −0.426885
\(9\) 0 0
\(10\) 5.80929e17 0.183706
\(11\) −2.06526e19 −1.11998 −0.559989 0.828500i \(-0.689194\pi\)
−0.559989 + 0.828500i \(0.689194\pi\)
\(12\) 0 0
\(13\) −4.02756e20 −0.993321 −0.496661 0.867945i \(-0.665441\pi\)
−0.496661 + 0.867945i \(0.665441\pi\)
\(14\) −4.83865e20 −0.302939
\(15\) 0 0
\(16\) 1.62230e22 0.858836
\(17\) 4.55890e22 0.786241 0.393120 0.919487i \(-0.371396\pi\)
0.393120 + 0.919487i \(0.371396\pi\)
\(18\) 0 0
\(19\) 1.54990e23 0.341477 0.170738 0.985316i \(-0.445385\pi\)
0.170738 + 0.985316i \(0.445385\pi\)
\(20\) −9.37797e23 −0.799934
\(21\) 0 0
\(22\) −1.67424e24 −0.244906
\(23\) −2.42540e25 −1.55890 −0.779450 0.626464i \(-0.784501\pi\)
−0.779450 + 0.626464i \(0.784501\pi\)
\(24\) 0 0
\(25\) −2.14076e25 −0.294224
\(26\) −3.26502e25 −0.217210
\(27\) 0 0
\(28\) 7.81105e26 1.31912
\(29\) −1.52016e27 −1.34130 −0.670652 0.741772i \(-0.733985\pi\)
−0.670652 + 0.741772i \(0.733985\pi\)
\(30\) 0 0
\(31\) 6.09023e27 1.56474 0.782370 0.622814i \(-0.214011\pi\)
0.782370 + 0.622814i \(0.214011\pi\)
\(32\) 4.30456e27 0.614687
\(33\) 0 0
\(34\) 3.69577e27 0.171928
\(35\) −4.27718e28 −1.16385
\(36\) 0 0
\(37\) −6.28281e28 −0.611536 −0.305768 0.952106i \(-0.598913\pi\)
−0.305768 + 0.952106i \(0.598913\pi\)
\(38\) 1.25646e28 0.0746709
\(39\) 0 0
\(40\) −1.55867e29 −0.358628
\(41\) 8.11303e29 1.18217 0.591087 0.806608i \(-0.298699\pi\)
0.591087 + 0.806608i \(0.298699\pi\)
\(42\) 0 0
\(43\) 4.69121e29 0.283216 0.141608 0.989923i \(-0.454773\pi\)
0.141608 + 0.989923i \(0.454773\pi\)
\(44\) 2.70274e30 1.06642
\(45\) 0 0
\(46\) −1.96620e30 −0.340885
\(47\) −1.02212e31 −1.19039 −0.595196 0.803581i \(-0.702926\pi\)
−0.595196 + 0.803581i \(0.702926\pi\)
\(48\) 0 0
\(49\) 1.70631e31 0.919245
\(50\) −1.73545e30 −0.0643380
\(51\) 0 0
\(52\) 5.27075e31 0.945824
\(53\) −5.42457e31 −0.684323 −0.342162 0.939641i \(-0.611159\pi\)
−0.342162 + 0.939641i \(0.611159\pi\)
\(54\) 0 0
\(55\) −1.47997e32 −0.940898
\(56\) 1.29824e32 0.591392
\(57\) 0 0
\(58\) −1.23235e32 −0.293303
\(59\) 5.27739e32 0.915496 0.457748 0.889082i \(-0.348656\pi\)
0.457748 + 0.889082i \(0.348656\pi\)
\(60\) 0 0
\(61\) −6.19951e31 −0.0580437 −0.0290219 0.999579i \(-0.509239\pi\)
−0.0290219 + 0.999579i \(0.509239\pi\)
\(62\) 4.93717e32 0.342162
\(63\) 0 0
\(64\) −1.88071e33 −0.724423
\(65\) −2.88616e33 −0.834494
\(66\) 0 0
\(67\) 9.30348e33 1.53554 0.767771 0.640725i \(-0.221366\pi\)
0.767771 + 0.640725i \(0.221366\pi\)
\(68\) −5.96611e33 −0.748645
\(69\) 0 0
\(70\) −3.46739e33 −0.254500
\(71\) 2.75965e34 1.55802 0.779012 0.627010i \(-0.215721\pi\)
0.779012 + 0.627010i \(0.215721\pi\)
\(72\) 0 0
\(73\) 2.09192e34 0.706436 0.353218 0.935541i \(-0.385087\pi\)
0.353218 + 0.935541i \(0.385087\pi\)
\(74\) −5.09329e33 −0.133725
\(75\) 0 0
\(76\) −2.02831e34 −0.325149
\(77\) 1.23269e35 1.55158
\(78\) 0 0
\(79\) 1.20341e34 0.0942567 0.0471283 0.998889i \(-0.484993\pi\)
0.0471283 + 0.998889i \(0.484993\pi\)
\(80\) 1.16254e35 0.721512
\(81\) 0 0
\(82\) 6.57700e34 0.258506
\(83\) 3.25180e34 0.102136 0.0510681 0.998695i \(-0.483737\pi\)
0.0510681 + 0.998695i \(0.483737\pi\)
\(84\) 0 0
\(85\) 3.26692e35 0.660525
\(86\) 3.80303e34 0.0619310
\(87\) 0 0
\(88\) 4.49210e35 0.478101
\(89\) 1.39444e36 1.20416 0.602081 0.798435i \(-0.294338\pi\)
0.602081 + 0.798435i \(0.294338\pi\)
\(90\) 0 0
\(91\) 2.40392e36 1.37612
\(92\) 3.17405e36 1.48436
\(93\) 0 0
\(94\) −8.28600e35 −0.260303
\(95\) 1.11066e36 0.286876
\(96\) 0 0
\(97\) −6.18707e36 −1.08695 −0.543473 0.839427i \(-0.682891\pi\)
−0.543473 + 0.839427i \(0.682891\pi\)
\(98\) 1.38326e36 0.201012
\(99\) 0 0
\(100\) 2.80155e36 0.280155
\(101\) 1.95203e37 1.62384 0.811918 0.583772i \(-0.198424\pi\)
0.811918 + 0.583772i \(0.198424\pi\)
\(102\) 0 0
\(103\) 3.11250e37 1.80144 0.900722 0.434397i \(-0.143038\pi\)
0.900722 + 0.434397i \(0.143038\pi\)
\(104\) 8.76026e36 0.424033
\(105\) 0 0
\(106\) −4.39754e36 −0.149641
\(107\) −2.91526e37 −0.833831 −0.416915 0.908945i \(-0.636889\pi\)
−0.416915 + 0.908945i \(0.636889\pi\)
\(108\) 0 0
\(109\) 4.34807e37 0.882890 0.441445 0.897288i \(-0.354466\pi\)
0.441445 + 0.897288i \(0.354466\pi\)
\(110\) −1.19977e37 −0.205747
\(111\) 0 0
\(112\) −9.68298e37 −1.18980
\(113\) −1.30612e38 −1.36155 −0.680774 0.732493i \(-0.738356\pi\)
−0.680774 + 0.732493i \(0.738356\pi\)
\(114\) 0 0
\(115\) −1.73805e38 −1.30964
\(116\) 1.98939e38 1.27717
\(117\) 0 0
\(118\) 4.27822e37 0.200192
\(119\) −2.72107e38 −1.08923
\(120\) 0 0
\(121\) 8.64886e37 0.254349
\(122\) −5.02576e36 −0.0126924
\(123\) 0 0
\(124\) −7.97010e38 −1.48992
\(125\) −6.74805e38 −1.08728
\(126\) 0 0
\(127\) 1.12113e39 1.34675 0.673373 0.739303i \(-0.264845\pi\)
0.673373 + 0.739303i \(0.264845\pi\)
\(128\) −7.44077e38 −0.773096
\(129\) 0 0
\(130\) −2.33973e38 −0.182479
\(131\) −1.51287e39 −1.02396 −0.511980 0.858997i \(-0.671088\pi\)
−0.511980 + 0.858997i \(0.671088\pi\)
\(132\) 0 0
\(133\) −9.25086e38 −0.473071
\(134\) 7.54206e38 0.335777
\(135\) 0 0
\(136\) −9.91598e38 −0.335634
\(137\) 5.50009e38 0.162570 0.0812848 0.996691i \(-0.474098\pi\)
0.0812848 + 0.996691i \(0.474098\pi\)
\(138\) 0 0
\(139\) 2.01159e39 0.454743 0.227372 0.973808i \(-0.426987\pi\)
0.227372 + 0.973808i \(0.426987\pi\)
\(140\) 5.59742e39 1.10820
\(141\) 0 0
\(142\) 2.23717e39 0.340693
\(143\) 8.31793e39 1.11250
\(144\) 0 0
\(145\) −1.08935e40 −1.12684
\(146\) 1.69586e39 0.154477
\(147\) 0 0
\(148\) 8.22213e39 0.582294
\(149\) −1.36577e40 −0.853949 −0.426975 0.904264i \(-0.640421\pi\)
−0.426975 + 0.904264i \(0.640421\pi\)
\(150\) 0 0
\(151\) 3.14967e40 1.53883 0.769415 0.638749i \(-0.220548\pi\)
0.769415 + 0.638749i \(0.220548\pi\)
\(152\) −3.37115e39 −0.145771
\(153\) 0 0
\(154\) 9.99304e39 0.339285
\(155\) 4.36428e40 1.31455
\(156\) 0 0
\(157\) −1.57563e40 −0.374378 −0.187189 0.982324i \(-0.559938\pi\)
−0.187189 + 0.982324i \(0.559938\pi\)
\(158\) 9.75569e38 0.0206111
\(159\) 0 0
\(160\) 3.08466e40 0.516401
\(161\) 1.44765e41 2.15965
\(162\) 0 0
\(163\) −9.57266e40 −1.13648 −0.568240 0.822863i \(-0.692376\pi\)
−0.568240 + 0.822863i \(0.692376\pi\)
\(164\) −1.06173e41 −1.12565
\(165\) 0 0
\(166\) 2.63614e39 0.0223342
\(167\) −1.21079e41 −0.917942 −0.458971 0.888451i \(-0.651782\pi\)
−0.458971 + 0.888451i \(0.651782\pi\)
\(168\) 0 0
\(169\) −2.18871e39 −0.0133132
\(170\) 2.64840e40 0.144437
\(171\) 0 0
\(172\) −6.13925e40 −0.269674
\(173\) −2.37365e41 −0.936621 −0.468310 0.883564i \(-0.655137\pi\)
−0.468310 + 0.883564i \(0.655137\pi\)
\(174\) 0 0
\(175\) 1.27775e41 0.407608
\(176\) −3.35046e41 −0.961877
\(177\) 0 0
\(178\) 1.13043e41 0.263315
\(179\) −3.87672e41 −0.814110 −0.407055 0.913404i \(-0.633444\pi\)
−0.407055 + 0.913404i \(0.633444\pi\)
\(180\) 0 0
\(181\) −9.37245e41 −1.60250 −0.801252 0.598326i \(-0.795833\pi\)
−0.801252 + 0.598326i \(0.795833\pi\)
\(182\) 1.94879e41 0.300916
\(183\) 0 0
\(184\) 5.27544e41 0.665470
\(185\) −4.50228e41 −0.513754
\(186\) 0 0
\(187\) −9.41530e41 −0.880572
\(188\) 1.33761e42 1.13347
\(189\) 0 0
\(190\) 9.00381e40 0.0627314
\(191\) 7.76327e41 0.490826 0.245413 0.969419i \(-0.421076\pi\)
0.245413 + 0.969419i \(0.421076\pi\)
\(192\) 0 0
\(193\) 1.83873e42 0.958758 0.479379 0.877608i \(-0.340862\pi\)
0.479379 + 0.877608i \(0.340862\pi\)
\(194\) −5.01568e41 −0.237683
\(195\) 0 0
\(196\) −2.23300e42 −0.875290
\(197\) −2.38170e42 −0.849693 −0.424847 0.905265i \(-0.639672\pi\)
−0.424847 + 0.905265i \(0.639672\pi\)
\(198\) 0 0
\(199\) 3.72548e42 1.10256 0.551278 0.834322i \(-0.314141\pi\)
0.551278 + 0.834322i \(0.314141\pi\)
\(200\) 4.65633e41 0.125600
\(201\) 0 0
\(202\) 1.58246e42 0.355085
\(203\) 9.07338e42 1.85820
\(204\) 0 0
\(205\) 5.81382e42 0.993150
\(206\) 2.52321e42 0.393922
\(207\) 0 0
\(208\) −6.53389e42 −0.853100
\(209\) −3.20093e42 −0.382446
\(210\) 0 0
\(211\) 5.66343e42 0.567354 0.283677 0.958920i \(-0.408446\pi\)
0.283677 + 0.958920i \(0.408446\pi\)
\(212\) 7.09897e42 0.651601
\(213\) 0 0
\(214\) −2.36332e42 −0.182334
\(215\) 3.36173e42 0.237932
\(216\) 0 0
\(217\) −3.63507e43 −2.16774
\(218\) 3.52486e42 0.193062
\(219\) 0 0
\(220\) 1.93679e43 0.895907
\(221\) −1.83612e43 −0.780989
\(222\) 0 0
\(223\) 3.12041e43 1.12350 0.561749 0.827308i \(-0.310129\pi\)
0.561749 + 0.827308i \(0.310129\pi\)
\(224\) −2.56926e43 −0.851567
\(225\) 0 0
\(226\) −1.05884e43 −0.297730
\(227\) 1.69570e43 0.439411 0.219705 0.975566i \(-0.429490\pi\)
0.219705 + 0.975566i \(0.429490\pi\)
\(228\) 0 0
\(229\) −4.70639e42 −0.103689 −0.0518444 0.998655i \(-0.516510\pi\)
−0.0518444 + 0.998655i \(0.516510\pi\)
\(230\) −1.40899e43 −0.286379
\(231\) 0 0
\(232\) 3.30648e43 0.572582
\(233\) −8.91929e43 −1.42641 −0.713207 0.700953i \(-0.752758\pi\)
−0.713207 + 0.700953i \(0.752758\pi\)
\(234\) 0 0
\(235\) −7.32451e43 −1.00005
\(236\) −6.90636e43 −0.871720
\(237\) 0 0
\(238\) −2.20589e43 −0.238183
\(239\) 7.05888e43 0.705301 0.352651 0.935755i \(-0.385280\pi\)
0.352651 + 0.935755i \(0.385280\pi\)
\(240\) 0 0
\(241\) −1.03720e44 −0.888271 −0.444136 0.895960i \(-0.646489\pi\)
−0.444136 + 0.895960i \(0.646489\pi\)
\(242\) 7.01138e42 0.0556185
\(243\) 0 0
\(244\) 8.11311e42 0.0552682
\(245\) 1.22275e44 0.772262
\(246\) 0 0
\(247\) −6.24230e43 −0.339196
\(248\) −1.32467e44 −0.667963
\(249\) 0 0
\(250\) −5.47045e43 −0.237757
\(251\) 1.06389e44 0.429470 0.214735 0.976672i \(-0.431111\pi\)
0.214735 + 0.976672i \(0.431111\pi\)
\(252\) 0 0
\(253\) 5.00907e44 1.74593
\(254\) 9.08865e43 0.294494
\(255\) 0 0
\(256\) 1.98162e44 0.555369
\(257\) 2.17429e44 0.566962 0.283481 0.958978i \(-0.408511\pi\)
0.283481 + 0.958978i \(0.408511\pi\)
\(258\) 0 0
\(259\) 3.75001e44 0.847202
\(260\) 3.77703e44 0.794591
\(261\) 0 0
\(262\) −1.22644e44 −0.223910
\(263\) −5.98155e44 −1.01773 −0.508865 0.860847i \(-0.669935\pi\)
−0.508865 + 0.860847i \(0.669935\pi\)
\(264\) 0 0
\(265\) −3.88726e44 −0.574903
\(266\) −7.49940e43 −0.103447
\(267\) 0 0
\(268\) −1.21752e45 −1.46212
\(269\) 1.15434e45 1.29394 0.646972 0.762514i \(-0.276035\pi\)
0.646972 + 0.762514i \(0.276035\pi\)
\(270\) 0 0
\(271\) 1.52915e45 1.49457 0.747287 0.664501i \(-0.231356\pi\)
0.747287 + 0.664501i \(0.231356\pi\)
\(272\) 7.39589e44 0.675252
\(273\) 0 0
\(274\) 4.45877e43 0.0355492
\(275\) 4.42122e44 0.329524
\(276\) 0 0
\(277\) −5.76088e44 −0.375503 −0.187751 0.982217i \(-0.560120\pi\)
−0.187751 + 0.982217i \(0.560120\pi\)
\(278\) 1.63074e44 0.0994388
\(279\) 0 0
\(280\) 9.30321e44 0.496832
\(281\) −9.77482e44 −0.488699 −0.244350 0.969687i \(-0.578574\pi\)
−0.244350 + 0.969687i \(0.578574\pi\)
\(282\) 0 0
\(283\) −2.00519e45 −0.879237 −0.439619 0.898185i \(-0.644886\pi\)
−0.439619 + 0.898185i \(0.644886\pi\)
\(284\) −3.61147e45 −1.48352
\(285\) 0 0
\(286\) 6.74311e44 0.243270
\(287\) −4.84241e45 −1.63775
\(288\) 0 0
\(289\) −1.28373e45 −0.381826
\(290\) −8.83107e44 −0.246406
\(291\) 0 0
\(292\) −2.73764e45 −0.672656
\(293\) −2.43417e45 −0.561435 −0.280717 0.959790i \(-0.590572\pi\)
−0.280717 + 0.959790i \(0.590572\pi\)
\(294\) 0 0
\(295\) 3.78179e45 0.769113
\(296\) 1.36656e45 0.261055
\(297\) 0 0
\(298\) −1.10719e45 −0.186733
\(299\) 9.76843e45 1.54849
\(300\) 0 0
\(301\) −2.80004e45 −0.392359
\(302\) 2.55334e45 0.336497
\(303\) 0 0
\(304\) 2.51439e45 0.293273
\(305\) −4.44258e44 −0.0487628
\(306\) 0 0
\(307\) −5.23869e45 −0.509522 −0.254761 0.967004i \(-0.581997\pi\)
−0.254761 + 0.967004i \(0.581997\pi\)
\(308\) −1.61318e46 −1.47739
\(309\) 0 0
\(310\) 3.53799e45 0.287452
\(311\) 1.81624e46 1.39029 0.695146 0.718869i \(-0.255340\pi\)
0.695146 + 0.718869i \(0.255340\pi\)
\(312\) 0 0
\(313\) −1.26517e45 −0.0860161 −0.0430081 0.999075i \(-0.513694\pi\)
−0.0430081 + 0.999075i \(0.513694\pi\)
\(314\) −1.27732e45 −0.0818655
\(315\) 0 0
\(316\) −1.57487e45 −0.0897496
\(317\) 1.12945e46 0.607117 0.303558 0.952813i \(-0.401825\pi\)
0.303558 + 0.952813i \(0.401825\pi\)
\(318\) 0 0
\(319\) 3.13952e46 1.50223
\(320\) −1.34772e46 −0.608591
\(321\) 0 0
\(322\) 1.17356e46 0.472252
\(323\) 7.06583e45 0.268483
\(324\) 0 0
\(325\) 8.62204e45 0.292259
\(326\) −7.76028e45 −0.248515
\(327\) 0 0
\(328\) −1.76465e46 −0.504652
\(329\) 6.10069e46 1.64913
\(330\) 0 0
\(331\) −3.60502e46 −0.871145 −0.435572 0.900154i \(-0.643454\pi\)
−0.435572 + 0.900154i \(0.643454\pi\)
\(332\) −4.25554e45 −0.0972524
\(333\) 0 0
\(334\) −9.81555e45 −0.200727
\(335\) 6.66690e46 1.29002
\(336\) 0 0
\(337\) 3.50453e46 0.607403 0.303701 0.952767i \(-0.401777\pi\)
0.303701 + 0.952767i \(0.401777\pi\)
\(338\) −1.77432e44 −0.00291121
\(339\) 0 0
\(340\) −4.27533e46 −0.628940
\(341\) −1.25779e47 −1.75247
\(342\) 0 0
\(343\) 8.94695e45 0.111875
\(344\) −1.02038e46 −0.120901
\(345\) 0 0
\(346\) −1.92425e46 −0.204811
\(347\) −1.32955e47 −1.34156 −0.670781 0.741655i \(-0.734041\pi\)
−0.670781 + 0.741655i \(0.734041\pi\)
\(348\) 0 0
\(349\) 5.30968e46 0.481724 0.240862 0.970559i \(-0.422570\pi\)
0.240862 + 0.970559i \(0.422570\pi\)
\(350\) 1.03584e46 0.0891319
\(351\) 0 0
\(352\) −8.89001e46 −0.688435
\(353\) −9.80590e46 −0.720535 −0.360268 0.932849i \(-0.617315\pi\)
−0.360268 + 0.932849i \(0.617315\pi\)
\(354\) 0 0
\(355\) 1.97757e47 1.30890
\(356\) −1.82487e47 −1.14658
\(357\) 0 0
\(358\) −3.14274e46 −0.178022
\(359\) 3.23469e47 1.74015 0.870073 0.492924i \(-0.164072\pi\)
0.870073 + 0.492924i \(0.164072\pi\)
\(360\) 0 0
\(361\) −1.81986e47 −0.883394
\(362\) −7.59797e46 −0.350420
\(363\) 0 0
\(364\) −3.14594e47 −1.31031
\(365\) 1.49908e47 0.593480
\(366\) 0 0
\(367\) 5.30423e46 0.189802 0.0949009 0.995487i \(-0.469747\pi\)
0.0949009 + 0.995487i \(0.469747\pi\)
\(368\) −3.93472e47 −1.33884
\(369\) 0 0
\(370\) −3.64987e46 −0.112343
\(371\) 3.23775e47 0.948040
\(372\) 0 0
\(373\) 1.73609e47 0.460212 0.230106 0.973166i \(-0.426093\pi\)
0.230106 + 0.973166i \(0.426093\pi\)
\(374\) −7.63272e46 −0.192555
\(375\) 0 0
\(376\) 2.22318e47 0.508160
\(377\) 6.12254e47 1.33235
\(378\) 0 0
\(379\) −7.77135e45 −0.0153345 −0.00766727 0.999971i \(-0.502441\pi\)
−0.00766727 + 0.999971i \(0.502441\pi\)
\(380\) −1.45349e47 −0.273159
\(381\) 0 0
\(382\) 6.29346e46 0.107329
\(383\) 1.37051e47 0.222692 0.111346 0.993782i \(-0.464484\pi\)
0.111346 + 0.993782i \(0.464484\pi\)
\(384\) 0 0
\(385\) 8.83347e47 1.30349
\(386\) 1.49061e47 0.209652
\(387\) 0 0
\(388\) 8.09683e47 1.03497
\(389\) 5.22275e47 0.636549 0.318274 0.947999i \(-0.396897\pi\)
0.318274 + 0.947999i \(0.396897\pi\)
\(390\) 0 0
\(391\) −1.10572e48 −1.22567
\(392\) −3.71137e47 −0.392412
\(393\) 0 0
\(394\) −1.93078e47 −0.185803
\(395\) 8.62367e46 0.0791855
\(396\) 0 0
\(397\) 1.08193e48 0.904844 0.452422 0.891804i \(-0.350560\pi\)
0.452422 + 0.891804i \(0.350560\pi\)
\(398\) 3.02014e47 0.241096
\(399\) 0 0
\(400\) −3.47295e47 −0.252690
\(401\) 1.57080e48 1.09132 0.545659 0.838007i \(-0.316279\pi\)
0.545659 + 0.838007i \(0.316279\pi\)
\(402\) 0 0
\(403\) −2.45287e48 −1.55429
\(404\) −2.55457e48 −1.54619
\(405\) 0 0
\(406\) 7.35553e47 0.406333
\(407\) 1.29756e48 0.684906
\(408\) 0 0
\(409\) 7.85596e47 0.378720 0.189360 0.981908i \(-0.439359\pi\)
0.189360 + 0.981908i \(0.439359\pi\)
\(410\) 4.71309e47 0.217172
\(411\) 0 0
\(412\) −4.07324e48 −1.71530
\(413\) −3.14991e48 −1.26830
\(414\) 0 0
\(415\) 2.33025e47 0.0858051
\(416\) −1.73368e48 −0.610581
\(417\) 0 0
\(418\) −2.59491e47 −0.0836297
\(419\) −4.22926e48 −1.30408 −0.652040 0.758184i \(-0.726087\pi\)
−0.652040 + 0.758184i \(0.726087\pi\)
\(420\) 0 0
\(421\) 4.83549e48 1.36528 0.682640 0.730755i \(-0.260832\pi\)
0.682640 + 0.730755i \(0.260832\pi\)
\(422\) 4.59118e47 0.124063
\(423\) 0 0
\(424\) 1.17989e48 0.292127
\(425\) −9.75953e47 −0.231331
\(426\) 0 0
\(427\) 3.70029e47 0.0804119
\(428\) 3.81512e48 0.793960
\(429\) 0 0
\(430\) 2.72526e47 0.0520286
\(431\) 1.35672e48 0.248119 0.124060 0.992275i \(-0.460409\pi\)
0.124060 + 0.992275i \(0.460409\pi\)
\(432\) 0 0
\(433\) −3.36809e48 −0.565403 −0.282701 0.959208i \(-0.591231\pi\)
−0.282701 + 0.959208i \(0.591231\pi\)
\(434\) −2.94685e48 −0.474021
\(435\) 0 0
\(436\) −5.69019e48 −0.840673
\(437\) −3.75912e48 −0.532328
\(438\) 0 0
\(439\) −3.07628e48 −0.400343 −0.200172 0.979761i \(-0.564150\pi\)
−0.200172 + 0.979761i \(0.564150\pi\)
\(440\) 3.21905e48 0.401655
\(441\) 0 0
\(442\) −1.48849e48 −0.170779
\(443\) 1.21772e49 1.33992 0.669961 0.742396i \(-0.266311\pi\)
0.669961 + 0.742396i \(0.266311\pi\)
\(444\) 0 0
\(445\) 9.99261e48 1.01162
\(446\) 2.52963e48 0.245675
\(447\) 0 0
\(448\) 1.12254e49 1.00359
\(449\) −6.54698e48 −0.561674 −0.280837 0.959756i \(-0.590612\pi\)
−0.280837 + 0.959756i \(0.590612\pi\)
\(450\) 0 0
\(451\) −1.67555e49 −1.32401
\(452\) 1.70929e49 1.29644
\(453\) 0 0
\(454\) 1.37466e48 0.0960861
\(455\) 1.72266e49 1.15608
\(456\) 0 0
\(457\) 1.51332e49 0.936444 0.468222 0.883611i \(-0.344895\pi\)
0.468222 + 0.883611i \(0.344895\pi\)
\(458\) −3.81534e47 −0.0226737
\(459\) 0 0
\(460\) 2.27453e49 1.24702
\(461\) −2.09024e49 −1.10085 −0.550426 0.834884i \(-0.685535\pi\)
−0.550426 + 0.834884i \(0.685535\pi\)
\(462\) 0 0
\(463\) 1.99414e49 0.969412 0.484706 0.874677i \(-0.338927\pi\)
0.484706 + 0.874677i \(0.338927\pi\)
\(464\) −2.46615e49 −1.15196
\(465\) 0 0
\(466\) −7.23061e48 −0.311915
\(467\) 1.49903e49 0.621508 0.310754 0.950490i \(-0.399418\pi\)
0.310754 + 0.950490i \(0.399418\pi\)
\(468\) 0 0
\(469\) −5.55296e49 −2.12729
\(470\) −5.93777e48 −0.218682
\(471\) 0 0
\(472\) −1.14787e49 −0.390811
\(473\) −9.68855e48 −0.317196
\(474\) 0 0
\(475\) −3.31796e48 −0.100471
\(476\) 3.56098e49 1.03715
\(477\) 0 0
\(478\) 5.72243e48 0.154229
\(479\) 1.97535e49 0.512198 0.256099 0.966651i \(-0.417563\pi\)
0.256099 + 0.966651i \(0.417563\pi\)
\(480\) 0 0
\(481\) 2.53044e49 0.607451
\(482\) −8.40825e48 −0.194239
\(483\) 0 0
\(484\) −1.13185e49 −0.242187
\(485\) −4.43367e49 −0.913148
\(486\) 0 0
\(487\) −2.15937e49 −0.412136 −0.206068 0.978538i \(-0.566067\pi\)
−0.206068 + 0.978538i \(0.566067\pi\)
\(488\) 1.34844e48 0.0247780
\(489\) 0 0
\(490\) 9.91248e48 0.168871
\(491\) −4.27399e49 −0.701174 −0.350587 0.936530i \(-0.614018\pi\)
−0.350587 + 0.936530i \(0.614018\pi\)
\(492\) 0 0
\(493\) −6.93028e49 −1.05459
\(494\) −5.06045e48 −0.0741722
\(495\) 0 0
\(496\) 9.88015e49 1.34386
\(497\) −1.64715e50 −2.15844
\(498\) 0 0
\(499\) −6.52314e49 −0.793591 −0.396796 0.917907i \(-0.629878\pi\)
−0.396796 + 0.917907i \(0.629878\pi\)
\(500\) 8.83097e49 1.03529
\(501\) 0 0
\(502\) 8.62466e48 0.0939124
\(503\) −8.97219e49 −0.941653 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(504\) 0 0
\(505\) 1.39883e50 1.36419
\(506\) 4.06071e49 0.381784
\(507\) 0 0
\(508\) −1.46719e50 −1.28235
\(509\) −4.50563e49 −0.379731 −0.189866 0.981810i \(-0.560805\pi\)
−0.189866 + 0.981810i \(0.560805\pi\)
\(510\) 0 0
\(511\) −1.24860e50 −0.978674
\(512\) 1.18330e50 0.894539
\(513\) 0 0
\(514\) 1.76263e49 0.123978
\(515\) 2.23043e50 1.51340
\(516\) 0 0
\(517\) 2.11093e50 1.33321
\(518\) 3.04003e49 0.185258
\(519\) 0 0
\(520\) 6.27763e49 0.356233
\(521\) 5.45339e49 0.298654 0.149327 0.988788i \(-0.452289\pi\)
0.149327 + 0.988788i \(0.452289\pi\)
\(522\) 0 0
\(523\) −1.36223e50 −0.694975 −0.347488 0.937685i \(-0.612965\pi\)
−0.347488 + 0.937685i \(0.612965\pi\)
\(524\) 1.97985e50 0.974998
\(525\) 0 0
\(526\) −4.84907e49 −0.222547
\(527\) 2.77648e50 1.23026
\(528\) 0 0
\(529\) 3.46192e50 1.43017
\(530\) −3.15129e49 −0.125714
\(531\) 0 0
\(532\) 1.21063e50 0.450451
\(533\) −3.26757e50 −1.17428
\(534\) 0 0
\(535\) −2.08909e50 −0.700505
\(536\) −2.02358e50 −0.655499
\(537\) 0 0
\(538\) 9.35787e49 0.282947
\(539\) −3.52397e50 −1.02953
\(540\) 0 0
\(541\) −1.89994e50 −0.518310 −0.259155 0.965836i \(-0.583444\pi\)
−0.259155 + 0.965836i \(0.583444\pi\)
\(542\) 1.23964e50 0.326819
\(543\) 0 0
\(544\) 1.96241e50 0.483292
\(545\) 3.11584e50 0.741720
\(546\) 0 0
\(547\) −1.73538e49 −0.0386037 −0.0193019 0.999814i \(-0.506144\pi\)
−0.0193019 + 0.999814i \(0.506144\pi\)
\(548\) −7.19781e49 −0.154796
\(549\) 0 0
\(550\) 3.58416e49 0.0720571
\(551\) −2.35610e50 −0.458024
\(552\) 0 0
\(553\) −7.18278e49 −0.130580
\(554\) −4.67018e49 −0.0821114
\(555\) 0 0
\(556\) −2.63251e50 −0.432999
\(557\) 9.83753e50 1.56518 0.782591 0.622536i \(-0.213898\pi\)
0.782591 + 0.622536i \(0.213898\pi\)
\(558\) 0 0
\(559\) −1.88941e50 −0.281325
\(560\) −6.93885e50 −0.999560
\(561\) 0 0
\(562\) −7.92416e49 −0.106864
\(563\) −2.99781e50 −0.391200 −0.195600 0.980684i \(-0.562665\pi\)
−0.195600 + 0.980684i \(0.562665\pi\)
\(564\) 0 0
\(565\) −9.35972e50 −1.14384
\(566\) −1.62555e50 −0.192263
\(567\) 0 0
\(568\) −6.00245e50 −0.665096
\(569\) −1.02631e51 −1.10078 −0.550392 0.834906i \(-0.685522\pi\)
−0.550392 + 0.834906i \(0.685522\pi\)
\(570\) 0 0
\(571\) −8.60079e47 −0.000864511 0 −0.000432255 1.00000i \(-0.500138\pi\)
−0.000432255 1.00000i \(0.500138\pi\)
\(572\) −1.08854e51 −1.05930
\(573\) 0 0
\(574\) −3.92560e50 −0.358127
\(575\) 5.19220e50 0.458666
\(576\) 0 0
\(577\) 1.49637e51 1.23961 0.619806 0.784755i \(-0.287211\pi\)
0.619806 + 0.784755i \(0.287211\pi\)
\(578\) −1.04069e50 −0.0834940
\(579\) 0 0
\(580\) 1.42560e51 1.07295
\(581\) −1.94090e50 −0.141496
\(582\) 0 0
\(583\) 1.12031e51 0.766427
\(584\) −4.55010e50 −0.301567
\(585\) 0 0
\(586\) −1.97331e50 −0.122769
\(587\) 2.21173e51 1.33330 0.666650 0.745371i \(-0.267728\pi\)
0.666650 + 0.745371i \(0.267728\pi\)
\(588\) 0 0
\(589\) 9.43923e50 0.534323
\(590\) 3.06579e50 0.168182
\(591\) 0 0
\(592\) −1.01926e51 −0.525209
\(593\) 9.58294e50 0.478616 0.239308 0.970944i \(-0.423079\pi\)
0.239308 + 0.970944i \(0.423079\pi\)
\(594\) 0 0
\(595\) −1.94993e51 −0.915070
\(596\) 1.78734e51 0.813116
\(597\) 0 0
\(598\) 7.91899e50 0.338608
\(599\) −1.22432e49 −0.00507572 −0.00253786 0.999997i \(-0.500808\pi\)
−0.00253786 + 0.999997i \(0.500808\pi\)
\(600\) 0 0
\(601\) −1.08866e51 −0.424339 −0.212169 0.977233i \(-0.568053\pi\)
−0.212169 + 0.977233i \(0.568053\pi\)
\(602\) −2.26991e50 −0.0857973
\(603\) 0 0
\(604\) −4.12188e51 −1.46525
\(605\) 6.19780e50 0.213680
\(606\) 0 0
\(607\) 4.94921e51 1.60526 0.802629 0.596479i \(-0.203434\pi\)
0.802629 + 0.596479i \(0.203434\pi\)
\(608\) 6.67162e50 0.209901
\(609\) 0 0
\(610\) −3.60148e49 −0.0106630
\(611\) 4.11663e51 1.18244
\(612\) 0 0
\(613\) 3.53515e51 0.955848 0.477924 0.878401i \(-0.341390\pi\)
0.477924 + 0.878401i \(0.341390\pi\)
\(614\) −4.24685e50 −0.111417
\(615\) 0 0
\(616\) −2.68119e51 −0.662346
\(617\) 3.24058e51 0.776868 0.388434 0.921477i \(-0.373016\pi\)
0.388434 + 0.921477i \(0.373016\pi\)
\(618\) 0 0
\(619\) 2.49425e51 0.563201 0.281600 0.959532i \(-0.409135\pi\)
0.281600 + 0.959532i \(0.409135\pi\)
\(620\) −5.71140e51 −1.25169
\(621\) 0 0
\(622\) 1.47237e51 0.304016
\(623\) −8.32299e51 −1.66821
\(624\) 0 0
\(625\) −3.27806e51 −0.619208
\(626\) −1.02564e50 −0.0188092
\(627\) 0 0
\(628\) 2.06198e51 0.356477
\(629\) −2.86427e51 −0.480814
\(630\) 0 0
\(631\) 3.22282e51 0.510145 0.255072 0.966922i \(-0.417901\pi\)
0.255072 + 0.966922i \(0.417901\pi\)
\(632\) −2.61751e50 −0.0402367
\(633\) 0 0
\(634\) 9.15616e50 0.132758
\(635\) 8.03403e51 1.13141
\(636\) 0 0
\(637\) −6.87227e51 −0.913106
\(638\) 2.54512e51 0.328493
\(639\) 0 0
\(640\) −5.33208e51 −0.649482
\(641\) 8.20930e51 0.971477 0.485739 0.874104i \(-0.338551\pi\)
0.485739 + 0.874104i \(0.338551\pi\)
\(642\) 0 0
\(643\) −4.47682e51 −0.500112 −0.250056 0.968231i \(-0.580449\pi\)
−0.250056 + 0.968231i \(0.580449\pi\)
\(644\) −1.89449e52 −2.05638
\(645\) 0 0
\(646\) 5.72807e50 0.0587093
\(647\) −1.90676e52 −1.89919 −0.949594 0.313483i \(-0.898504\pi\)
−0.949594 + 0.313483i \(0.898504\pi\)
\(648\) 0 0
\(649\) −1.08991e52 −1.02533
\(650\) 6.98964e50 0.0639083
\(651\) 0 0
\(652\) 1.25275e52 1.08214
\(653\) 9.48302e51 0.796256 0.398128 0.917330i \(-0.369660\pi\)
0.398128 + 0.917330i \(0.369660\pi\)
\(654\) 0 0
\(655\) −1.08413e52 −0.860234
\(656\) 1.31617e52 1.01529
\(657\) 0 0
\(658\) 4.94566e51 0.360616
\(659\) −1.04759e52 −0.742693 −0.371347 0.928494i \(-0.621104\pi\)
−0.371347 + 0.928494i \(0.621104\pi\)
\(660\) 0 0
\(661\) −8.65976e51 −0.580469 −0.290234 0.956956i \(-0.593733\pi\)
−0.290234 + 0.956956i \(0.593733\pi\)
\(662\) −2.92249e51 −0.190494
\(663\) 0 0
\(664\) −7.07293e50 −0.0436004
\(665\) −6.62919e51 −0.397429
\(666\) 0 0
\(667\) 3.68700e52 2.09096
\(668\) 1.58453e52 0.874049
\(669\) 0 0
\(670\) 5.40467e51 0.282088
\(671\) 1.28036e51 0.0650076
\(672\) 0 0
\(673\) −2.18723e52 −1.05103 −0.525515 0.850784i \(-0.676127\pi\)
−0.525515 + 0.850784i \(0.676127\pi\)
\(674\) 2.84102e51 0.132821
\(675\) 0 0
\(676\) 2.86430e50 0.0126766
\(677\) 1.59031e52 0.684845 0.342422 0.939546i \(-0.388753\pi\)
0.342422 + 0.939546i \(0.388753\pi\)
\(678\) 0 0
\(679\) 3.69287e52 1.50582
\(680\) −7.10582e51 −0.281968
\(681\) 0 0
\(682\) −1.01965e52 −0.383214
\(683\) 3.09230e51 0.113109 0.0565547 0.998400i \(-0.481988\pi\)
0.0565547 + 0.998400i \(0.481988\pi\)
\(684\) 0 0
\(685\) 3.94138e51 0.136576
\(686\) 7.25303e50 0.0244638
\(687\) 0 0
\(688\) 7.61053e51 0.243237
\(689\) 2.18477e52 0.679753
\(690\) 0 0
\(691\) −4.14754e52 −1.22306 −0.611528 0.791222i \(-0.709445\pi\)
−0.611528 + 0.791222i \(0.709445\pi\)
\(692\) 3.10633e52 0.891835
\(693\) 0 0
\(694\) −1.07783e52 −0.293360
\(695\) 1.44151e52 0.382032
\(696\) 0 0
\(697\) 3.69865e52 0.929473
\(698\) 4.30440e51 0.105339
\(699\) 0 0
\(700\) −1.67216e52 −0.388118
\(701\) 2.82631e52 0.638904 0.319452 0.947602i \(-0.396501\pi\)
0.319452 + 0.947602i \(0.396501\pi\)
\(702\) 0 0
\(703\) −9.73771e51 −0.208825
\(704\) 3.88414e52 0.811337
\(705\) 0 0
\(706\) −7.94936e51 −0.157560
\(707\) −1.16511e53 −2.24961
\(708\) 0 0
\(709\) 4.89940e52 0.897817 0.448909 0.893578i \(-0.351813\pi\)
0.448909 + 0.893578i \(0.351813\pi\)
\(710\) 1.60316e52 0.286218
\(711\) 0 0
\(712\) −3.03302e52 −0.514039
\(713\) −1.47712e53 −2.43927
\(714\) 0 0
\(715\) 5.96066e52 0.934614
\(716\) 5.07334e52 0.775182
\(717\) 0 0
\(718\) 2.62227e52 0.380518
\(719\) −4.55246e52 −0.643816 −0.321908 0.946771i \(-0.604324\pi\)
−0.321908 + 0.946771i \(0.604324\pi\)
\(720\) 0 0
\(721\) −1.85775e53 −2.49566
\(722\) −1.47531e52 −0.193172
\(723\) 0 0
\(724\) 1.22654e53 1.52588
\(725\) 3.25430e52 0.394644
\(726\) 0 0
\(727\) −6.55544e52 −0.755467 −0.377733 0.925914i \(-0.623296\pi\)
−0.377733 + 0.925914i \(0.623296\pi\)
\(728\) −5.22873e52 −0.587442
\(729\) 0 0
\(730\) 1.21526e52 0.129777
\(731\) 2.13868e52 0.222676
\(732\) 0 0
\(733\) 5.46489e52 0.540951 0.270476 0.962727i \(-0.412819\pi\)
0.270476 + 0.962727i \(0.412819\pi\)
\(734\) 4.29999e51 0.0415040
\(735\) 0 0
\(736\) −1.04403e53 −0.958235
\(737\) −1.92141e53 −1.71977
\(738\) 0 0
\(739\) −1.98761e53 −1.69203 −0.846017 0.533156i \(-0.821006\pi\)
−0.846017 + 0.533156i \(0.821006\pi\)
\(740\) 5.89200e52 0.489188
\(741\) 0 0
\(742\) 2.62476e52 0.207308
\(743\) 1.44490e53 1.11313 0.556564 0.830805i \(-0.312120\pi\)
0.556564 + 0.830805i \(0.312120\pi\)
\(744\) 0 0
\(745\) −9.78716e52 −0.717407
\(746\) 1.40740e52 0.100635
\(747\) 0 0
\(748\) 1.23215e53 0.838465
\(749\) 1.74003e53 1.15516
\(750\) 0 0
\(751\) 1.98670e53 1.25544 0.627718 0.778441i \(-0.283989\pi\)
0.627718 + 0.778441i \(0.283989\pi\)
\(752\) −1.65817e53 −1.02235
\(753\) 0 0
\(754\) 4.96337e52 0.291344
\(755\) 2.25706e53 1.29278
\(756\) 0 0
\(757\) 3.23306e53 1.76336 0.881679 0.471850i \(-0.156414\pi\)
0.881679 + 0.471850i \(0.156414\pi\)
\(758\) −6.30001e50 −0.00335321
\(759\) 0 0
\(760\) −2.41578e52 −0.122463
\(761\) 4.26585e52 0.211052 0.105526 0.994417i \(-0.466347\pi\)
0.105526 + 0.994417i \(0.466347\pi\)
\(762\) 0 0
\(763\) −2.59523e53 −1.22313
\(764\) −1.01596e53 −0.467356
\(765\) 0 0
\(766\) 1.11103e52 0.0486962
\(767\) −2.12550e53 −0.909382
\(768\) 0 0
\(769\) 4.26123e53 1.73739 0.868695 0.495347i \(-0.164959\pi\)
0.868695 + 0.495347i \(0.164959\pi\)
\(770\) 7.16104e52 0.285035
\(771\) 0 0
\(772\) −2.40630e53 −0.912913
\(773\) 9.95037e52 0.368569 0.184284 0.982873i \(-0.441003\pi\)
0.184284 + 0.982873i \(0.441003\pi\)
\(774\) 0 0
\(775\) −1.30377e53 −0.460384
\(776\) 1.34574e53 0.464000
\(777\) 0 0
\(778\) 4.23394e52 0.139194
\(779\) 1.25744e53 0.403685
\(780\) 0 0
\(781\) −5.69937e53 −1.74495
\(782\) −8.96373e52 −0.268018
\(783\) 0 0
\(784\) 2.76815e53 0.789481
\(785\) −1.12910e53 −0.314517
\(786\) 0 0
\(787\) 4.51390e53 1.19955 0.599777 0.800167i \(-0.295256\pi\)
0.599777 + 0.800167i \(0.295256\pi\)
\(788\) 3.11686e53 0.809064
\(789\) 0 0
\(790\) 6.99096e51 0.0173155
\(791\) 7.79585e53 1.88625
\(792\) 0 0
\(793\) 2.49689e52 0.0576560
\(794\) 8.77088e52 0.197863
\(795\) 0 0
\(796\) −4.87543e53 −1.04983
\(797\) −3.86364e53 −0.812864 −0.406432 0.913681i \(-0.633227\pi\)
−0.406432 + 0.913681i \(0.633227\pi\)
\(798\) 0 0
\(799\) −4.65973e53 −0.935934
\(800\) −9.21502e52 −0.180856
\(801\) 0 0
\(802\) 1.27341e53 0.238639
\(803\) −4.32036e53 −0.791192
\(804\) 0 0
\(805\) 1.03739e54 1.81433
\(806\) −1.98847e53 −0.339877
\(807\) 0 0
\(808\) −4.24583e53 −0.693190
\(809\) −8.90010e53 −1.42019 −0.710097 0.704104i \(-0.751349\pi\)
−0.710097 + 0.704104i \(0.751349\pi\)
\(810\) 0 0
\(811\) −3.07699e53 −0.469072 −0.234536 0.972107i \(-0.575357\pi\)
−0.234536 + 0.972107i \(0.575357\pi\)
\(812\) −1.18741e54 −1.76935
\(813\) 0 0
\(814\) 1.05189e53 0.149769
\(815\) −6.85980e53 −0.954763
\(816\) 0 0
\(817\) 7.27090e52 0.0967119
\(818\) 6.36860e52 0.0828148
\(819\) 0 0
\(820\) −7.60837e53 −0.945661
\(821\) 1.44038e54 1.75037 0.875185 0.483789i \(-0.160740\pi\)
0.875185 + 0.483789i \(0.160740\pi\)
\(822\) 0 0
\(823\) 7.25265e53 0.842558 0.421279 0.906931i \(-0.361581\pi\)
0.421279 + 0.906931i \(0.361581\pi\)
\(824\) −6.76993e53 −0.769008
\(825\) 0 0
\(826\) −2.55354e53 −0.277339
\(827\) 4.48759e53 0.476608 0.238304 0.971191i \(-0.423408\pi\)
0.238304 + 0.971191i \(0.423408\pi\)
\(828\) 0 0
\(829\) −4.38009e53 −0.444861 −0.222430 0.974949i \(-0.571399\pi\)
−0.222430 + 0.974949i \(0.571399\pi\)
\(830\) 1.88907e52 0.0187630
\(831\) 0 0
\(832\) 7.57466e53 0.719584
\(833\) 7.77892e53 0.722748
\(834\) 0 0
\(835\) −8.67658e53 −0.771167
\(836\) 4.18897e53 0.364159
\(837\) 0 0
\(838\) −3.42854e53 −0.285164
\(839\) 3.60758e53 0.293508 0.146754 0.989173i \(-0.453117\pi\)
0.146754 + 0.989173i \(0.453117\pi\)
\(840\) 0 0
\(841\) 1.02642e54 0.799095
\(842\) 3.91999e53 0.298546
\(843\) 0 0
\(844\) −7.41156e53 −0.540225
\(845\) −1.56843e52 −0.0111845
\(846\) 0 0
\(847\) −5.16224e53 −0.352367
\(848\) −8.80025e53 −0.587722
\(849\) 0 0
\(850\) −7.91177e52 −0.0505852
\(851\) 1.52383e54 0.953323
\(852\) 0 0
\(853\) −3.15083e53 −0.188742 −0.0943708 0.995537i \(-0.530084\pi\)
−0.0943708 + 0.995537i \(0.530084\pi\)
\(854\) 2.99972e52 0.0175837
\(855\) 0 0
\(856\) 6.34093e53 0.355949
\(857\) −9.67011e53 −0.531234 −0.265617 0.964079i \(-0.585576\pi\)
−0.265617 + 0.964079i \(0.585576\pi\)
\(858\) 0 0
\(859\) 3.56564e54 1.87614 0.938068 0.346452i \(-0.112614\pi\)
0.938068 + 0.346452i \(0.112614\pi\)
\(860\) −4.39940e53 −0.226554
\(861\) 0 0
\(862\) 1.09985e53 0.0542564
\(863\) 2.25029e54 1.08652 0.543262 0.839564i \(-0.317189\pi\)
0.543262 + 0.839564i \(0.317189\pi\)
\(864\) 0 0
\(865\) −1.70096e54 −0.786860
\(866\) −2.73042e53 −0.123637
\(867\) 0 0
\(868\) 4.75711e54 2.06409
\(869\) −2.48535e53 −0.105565
\(870\) 0 0
\(871\) −3.74703e54 −1.52529
\(872\) −9.45740e53 −0.376892
\(873\) 0 0
\(874\) −3.04741e53 −0.116404
\(875\) 4.02770e54 1.50629
\(876\) 0 0
\(877\) 4.14049e54 1.48443 0.742214 0.670163i \(-0.233776\pi\)
0.742214 + 0.670163i \(0.233776\pi\)
\(878\) −2.49385e53 −0.0875432
\(879\) 0 0
\(880\) −2.40095e54 −0.808077
\(881\) −3.03676e54 −1.00082 −0.500410 0.865789i \(-0.666817\pi\)
−0.500410 + 0.865789i \(0.666817\pi\)
\(882\) 0 0
\(883\) 2.77430e54 0.876760 0.438380 0.898790i \(-0.355552\pi\)
0.438380 + 0.898790i \(0.355552\pi\)
\(884\) 2.40288e54 0.743645
\(885\) 0 0
\(886\) 9.87173e53 0.293001
\(887\) 9.39883e53 0.273204 0.136602 0.990626i \(-0.456382\pi\)
0.136602 + 0.990626i \(0.456382\pi\)
\(888\) 0 0
\(889\) −6.69166e54 −1.86574
\(890\) 8.10072e53 0.221212
\(891\) 0 0
\(892\) −4.08359e54 −1.06978
\(893\) −1.58418e54 −0.406491
\(894\) 0 0
\(895\) −2.77807e54 −0.683937
\(896\) 4.44117e54 1.07102
\(897\) 0 0
\(898\) −5.30745e53 −0.122821
\(899\) −9.25814e54 −2.09879
\(900\) 0 0
\(901\) −2.47301e54 −0.538043
\(902\) −1.35832e54 −0.289521
\(903\) 0 0
\(904\) 2.84092e54 0.581224
\(905\) −6.71632e54 −1.34627
\(906\) 0 0
\(907\) 2.17105e54 0.417767 0.208884 0.977941i \(-0.433017\pi\)
0.208884 + 0.977941i \(0.433017\pi\)
\(908\) −2.21911e54 −0.418400
\(909\) 0 0
\(910\) 1.39651e54 0.252801
\(911\) −7.42098e54 −1.31635 −0.658175 0.752865i \(-0.728671\pi\)
−0.658175 + 0.752865i \(0.728671\pi\)
\(912\) 0 0
\(913\) −6.71581e53 −0.114390
\(914\) 1.22681e54 0.204772
\(915\) 0 0
\(916\) 6.15912e53 0.0987307
\(917\) 9.02986e54 1.41856
\(918\) 0 0
\(919\) 6.80622e54 1.02700 0.513498 0.858091i \(-0.328349\pi\)
0.513498 + 0.858091i \(0.328349\pi\)
\(920\) 3.78039e54 0.559065
\(921\) 0 0
\(922\) −1.69450e54 −0.240724
\(923\) −1.11146e55 −1.54762
\(924\) 0 0
\(925\) 1.34500e54 0.179928
\(926\) 1.61660e54 0.211982
\(927\) 0 0
\(928\) −6.54362e54 −0.824481
\(929\) 6.45233e54 0.796940 0.398470 0.917181i \(-0.369541\pi\)
0.398470 + 0.917181i \(0.369541\pi\)
\(930\) 0 0
\(931\) 2.64461e54 0.313901
\(932\) 1.16724e55 1.35821
\(933\) 0 0
\(934\) 1.21522e54 0.135905
\(935\) −6.74703e54 −0.739772
\(936\) 0 0
\(937\) 7.02165e53 0.0740042 0.0370021 0.999315i \(-0.488219\pi\)
0.0370021 + 0.999315i \(0.488219\pi\)
\(938\) −4.50162e54 −0.465175
\(939\) 0 0
\(940\) 9.58538e54 0.952234
\(941\) 8.40171e53 0.0818388 0.0409194 0.999162i \(-0.486971\pi\)
0.0409194 + 0.999162i \(0.486971\pi\)
\(942\) 0 0
\(943\) −1.96773e55 −1.84289
\(944\) 8.56148e54 0.786261
\(945\) 0 0
\(946\) −7.85423e53 −0.0693613
\(947\) 7.39324e54 0.640266 0.320133 0.947373i \(-0.396272\pi\)
0.320133 + 0.947373i \(0.396272\pi\)
\(948\) 0 0
\(949\) −8.42534e54 −0.701718
\(950\) −2.68977e53 −0.0219700
\(951\) 0 0
\(952\) 5.91854e54 0.464977
\(953\) 2.15051e55 1.65700 0.828499 0.559990i \(-0.189195\pi\)
0.828499 + 0.559990i \(0.189195\pi\)
\(954\) 0 0
\(955\) 5.56319e54 0.412345
\(956\) −9.23775e54 −0.671576
\(957\) 0 0
\(958\) 1.60136e54 0.112002
\(959\) −3.28283e54 −0.225219
\(960\) 0 0
\(961\) 2.19419e55 1.44841
\(962\) 2.05135e54 0.132832
\(963\) 0 0
\(964\) 1.35735e55 0.845797
\(965\) 1.31764e55 0.805457
\(966\) 0 0
\(967\) 1.31747e55 0.775085 0.387543 0.921852i \(-0.373324\pi\)
0.387543 + 0.921852i \(0.373324\pi\)
\(968\) −1.88120e54 −0.108578
\(969\) 0 0
\(970\) −3.59425e54 −0.199678
\(971\) −8.47915e54 −0.462164 −0.231082 0.972934i \(-0.574227\pi\)
−0.231082 + 0.972934i \(0.574227\pi\)
\(972\) 0 0
\(973\) −1.20066e55 −0.629987
\(974\) −1.75054e54 −0.0901220
\(975\) 0 0
\(976\) −1.00574e54 −0.0498500
\(977\) −1.06113e55 −0.516080 −0.258040 0.966134i \(-0.583077\pi\)
−0.258040 + 0.966134i \(0.583077\pi\)
\(978\) 0 0
\(979\) −2.87988e55 −1.34863
\(980\) −1.60018e55 −0.735335
\(981\) 0 0
\(982\) −3.46480e54 −0.153326
\(983\) −1.52477e54 −0.0662161 −0.0331081 0.999452i \(-0.510541\pi\)
−0.0331081 + 0.999452i \(0.510541\pi\)
\(984\) 0 0
\(985\) −1.70673e55 −0.713831
\(986\) −5.61818e54 −0.230607
\(987\) 0 0
\(988\) 8.16912e54 0.322977
\(989\) −1.13781e55 −0.441506
\(990\) 0 0
\(991\) 2.88078e55 1.07683 0.538416 0.842679i \(-0.319023\pi\)
0.538416 + 0.842679i \(0.319023\pi\)
\(992\) 2.62157e55 0.961825
\(993\) 0 0
\(994\) −1.33529e55 −0.471986
\(995\) 2.66969e55 0.926262
\(996\) 0 0
\(997\) −2.89753e55 −0.968651 −0.484326 0.874888i \(-0.660935\pi\)
−0.484326 + 0.874888i \(0.660935\pi\)
\(998\) −5.28813e54 −0.173535
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.38.a.c.1.3 4
3.2 odd 2 3.38.a.b.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.38.a.b.1.2 4 3.2 odd 2
9.38.a.c.1.3 4 1.1 even 1 trivial