Properties

Label 4.34.a.a.1.2
Level 4
Weight 34
Character 4.1
Self dual yes
Analytic conductor 27.593
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5931315524\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 65185566 x - 173679864984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9172.25\) of defining polynomial
Character \(\chi\) \(=\) 4.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.16957e7 q^{3} +6.04966e11 q^{5} +1.12234e14 q^{7} -5.08836e15 q^{9} +O(q^{10})\) \(q+2.16957e7 q^{3} +6.04966e11 q^{5} +1.12234e14 q^{7} -5.08836e15 q^{9} +2.12022e17 q^{11} -2.20110e18 q^{13} +1.31252e19 q^{15} -9.09205e19 q^{17} +5.28541e20 q^{19} +2.43500e21 q^{21} -2.52550e22 q^{23} +2.49568e23 q^{25} -2.31004e23 q^{27} +2.22959e24 q^{29} +5.96021e24 q^{31} +4.59998e24 q^{33} +6.78977e25 q^{35} -1.07477e25 q^{37} -4.77546e25 q^{39} +5.45950e25 q^{41} -7.34669e26 q^{43} -3.07828e27 q^{45} -2.88377e27 q^{47} +4.86547e27 q^{49} -1.97259e27 q^{51} -6.58827e27 q^{53} +1.28266e29 q^{55} +1.14671e28 q^{57} -1.55330e29 q^{59} +5.35869e29 q^{61} -5.71086e29 q^{63} -1.33159e30 q^{65} -5.20034e29 q^{67} -5.47926e29 q^{69} +2.97143e30 q^{71} -5.09019e30 q^{73} +5.41457e30 q^{75} +2.37961e31 q^{77} -1.81467e30 q^{79} +2.32747e31 q^{81} -3.91242e31 q^{83} -5.50038e31 q^{85} +4.83726e31 q^{87} -1.76955e32 q^{89} -2.47039e32 q^{91} +1.29311e32 q^{93} +3.19749e32 q^{95} +8.13793e32 q^{97} -1.07884e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + O(q^{10}) \) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + 227617657302449700q^{11} + 272970442217358762q^{13} - 32142473481241105848q^{15} + 93037188311816716854q^{17} + \)\(13\!\cdots\!28\)\(q^{19} - \)\(22\!\cdots\!52\)\(q^{21} + \)\(11\!\cdots\!96\)\(q^{23} + \)\(25\!\cdots\!21\)\(q^{25} + \)\(11\!\cdots\!24\)\(q^{27} + \)\(28\!\cdots\!78\)\(q^{29} + \)\(15\!\cdots\!28\)\(q^{31} + \)\(51\!\cdots\!80\)\(q^{33} + \)\(10\!\cdots\!48\)\(q^{35} + \)\(34\!\cdots\!42\)\(q^{37} + \)\(13\!\cdots\!68\)\(q^{39} - \)\(67\!\cdots\!62\)\(q^{41} - \)\(15\!\cdots\!20\)\(q^{43} - \)\(82\!\cdots\!86\)\(q^{45} - \)\(72\!\cdots\!76\)\(q^{47} - \)\(47\!\cdots\!09\)\(q^{49} + \)\(24\!\cdots\!36\)\(q^{51} + \)\(54\!\cdots\!34\)\(q^{53} + \)\(72\!\cdots\!80\)\(q^{55} + \)\(27\!\cdots\!48\)\(q^{57} + \)\(20\!\cdots\!76\)\(q^{59} + \)\(59\!\cdots\!66\)\(q^{61} - \)\(12\!\cdots\!68\)\(q^{63} - \)\(22\!\cdots\!12\)\(q^{65} - \)\(29\!\cdots\!28\)\(q^{67} - \)\(15\!\cdots\!16\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} + \)\(13\!\cdots\!02\)\(q^{73} + \)\(20\!\cdots\!28\)\(q^{75} + \)\(20\!\cdots\!20\)\(q^{77} + \)\(43\!\cdots\!04\)\(q^{79} + \)\(60\!\cdots\!27\)\(q^{81} - \)\(71\!\cdots\!64\)\(q^{83} - \)\(13\!\cdots\!84\)\(q^{85} - \)\(25\!\cdots\!92\)\(q^{87} - \)\(19\!\cdots\!98\)\(q^{89} - \)\(38\!\cdots\!08\)\(q^{91} + \)\(55\!\cdots\!88\)\(q^{93} - \)\(20\!\cdots\!76\)\(q^{95} + \)\(56\!\cdots\!22\)\(q^{97} + \)\(22\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

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
\(3\) 2.16957e7 0.290987 0.145494 0.989359i \(-0.453523\pi\)
0.145494 + 0.989359i \(0.453523\pi\)
\(4\) 0 0
\(5\) 6.04966e11 1.77307 0.886535 0.462662i \(-0.153106\pi\)
0.886535 + 0.462662i \(0.153106\pi\)
\(6\) 0 0
\(7\) 1.12234e14 1.27646 0.638229 0.769846i \(-0.279667\pi\)
0.638229 + 0.769846i \(0.279667\pi\)
\(8\) 0 0
\(9\) −5.08836e15 −0.915326
\(10\) 0 0
\(11\) 2.12022e17 1.39124 0.695620 0.718410i \(-0.255130\pi\)
0.695620 + 0.718410i \(0.255130\pi\)
\(12\) 0 0
\(13\) −2.20110e18 −0.917435 −0.458718 0.888582i \(-0.651691\pi\)
−0.458718 + 0.888582i \(0.651691\pi\)
\(14\) 0 0
\(15\) 1.31252e19 0.515940
\(16\) 0 0
\(17\) −9.09205e19 −0.453163 −0.226582 0.973992i \(-0.572755\pi\)
−0.226582 + 0.973992i \(0.572755\pi\)
\(18\) 0 0
\(19\) 5.28541e20 0.420382 0.210191 0.977660i \(-0.432591\pi\)
0.210191 + 0.977660i \(0.432591\pi\)
\(20\) 0 0
\(21\) 2.43500e21 0.371433
\(22\) 0 0
\(23\) −2.52550e22 −0.858693 −0.429346 0.903140i \(-0.641256\pi\)
−0.429346 + 0.903140i \(0.641256\pi\)
\(24\) 0 0
\(25\) 2.49568e23 2.14377
\(26\) 0 0
\(27\) −2.31004e23 −0.557336
\(28\) 0 0
\(29\) 2.22959e24 1.65447 0.827233 0.561859i \(-0.189914\pi\)
0.827233 + 0.561859i \(0.189914\pi\)
\(30\) 0 0
\(31\) 5.96021e24 1.47161 0.735806 0.677192i \(-0.236803\pi\)
0.735806 + 0.677192i \(0.236803\pi\)
\(32\) 0 0
\(33\) 4.59998e24 0.404833
\(34\) 0 0
\(35\) 6.78977e25 2.26325
\(36\) 0 0
\(37\) −1.07477e25 −0.143215 −0.0716074 0.997433i \(-0.522813\pi\)
−0.0716074 + 0.997433i \(0.522813\pi\)
\(38\) 0 0
\(39\) −4.77546e25 −0.266962
\(40\) 0 0
\(41\) 5.45950e25 0.133727 0.0668635 0.997762i \(-0.478701\pi\)
0.0668635 + 0.997762i \(0.478701\pi\)
\(42\) 0 0
\(43\) −7.34669e26 −0.820092 −0.410046 0.912065i \(-0.634487\pi\)
−0.410046 + 0.912065i \(0.634487\pi\)
\(44\) 0 0
\(45\) −3.07828e27 −1.62294
\(46\) 0 0
\(47\) −2.88377e27 −0.741901 −0.370950 0.928653i \(-0.620968\pi\)
−0.370950 + 0.928653i \(0.620968\pi\)
\(48\) 0 0
\(49\) 4.86547e27 0.629346
\(50\) 0 0
\(51\) −1.97259e27 −0.131865
\(52\) 0 0
\(53\) −6.58827e27 −0.233464 −0.116732 0.993163i \(-0.537242\pi\)
−0.116732 + 0.993163i \(0.537242\pi\)
\(54\) 0 0
\(55\) 1.28266e29 2.46676
\(56\) 0 0
\(57\) 1.14671e28 0.122326
\(58\) 0 0
\(59\) −1.55330e29 −0.937990 −0.468995 0.883201i \(-0.655384\pi\)
−0.468995 + 0.883201i \(0.655384\pi\)
\(60\) 0 0
\(61\) 5.35869e29 1.86688 0.933440 0.358733i \(-0.116791\pi\)
0.933440 + 0.358733i \(0.116791\pi\)
\(62\) 0 0
\(63\) −5.71086e29 −1.16838
\(64\) 0 0
\(65\) −1.33159e30 −1.62668
\(66\) 0 0
\(67\) −5.20034e29 −0.385298 −0.192649 0.981268i \(-0.561708\pi\)
−0.192649 + 0.981268i \(0.561708\pi\)
\(68\) 0 0
\(69\) −5.47926e29 −0.249869
\(70\) 0 0
\(71\) 2.97143e30 0.845675 0.422837 0.906206i \(-0.361034\pi\)
0.422837 + 0.906206i \(0.361034\pi\)
\(72\) 0 0
\(73\) −5.09019e30 −0.916024 −0.458012 0.888946i \(-0.651438\pi\)
−0.458012 + 0.888946i \(0.651438\pi\)
\(74\) 0 0
\(75\) 5.41457e30 0.623811
\(76\) 0 0
\(77\) 2.37961e31 1.77586
\(78\) 0 0
\(79\) −1.81467e30 −0.0887054 −0.0443527 0.999016i \(-0.514123\pi\)
−0.0443527 + 0.999016i \(0.514123\pi\)
\(80\) 0 0
\(81\) 2.32747e31 0.753149
\(82\) 0 0
\(83\) −3.91242e31 −0.846559 −0.423279 0.905999i \(-0.639121\pi\)
−0.423279 + 0.905999i \(0.639121\pi\)
\(84\) 0 0
\(85\) −5.50038e31 −0.803490
\(86\) 0 0
\(87\) 4.83726e31 0.481428
\(88\) 0 0
\(89\) −1.76955e32 −1.21040 −0.605199 0.796075i \(-0.706906\pi\)
−0.605199 + 0.796075i \(0.706906\pi\)
\(90\) 0 0
\(91\) −2.47039e32 −1.17107
\(92\) 0 0
\(93\) 1.29311e32 0.428220
\(94\) 0 0
\(95\) 3.19749e32 0.745367
\(96\) 0 0
\(97\) 8.13793e32 1.34518 0.672590 0.740016i \(-0.265182\pi\)
0.672590 + 0.740016i \(0.265182\pi\)
\(98\) 0 0
\(99\) −1.07884e33 −1.27344
\(100\) 0 0
\(101\) 6.19750e32 0.525913 0.262956 0.964808i \(-0.415302\pi\)
0.262956 + 0.964808i \(0.415302\pi\)
\(102\) 0 0
\(103\) 2.96344e33 1.81962 0.909812 0.415020i \(-0.136225\pi\)
0.909812 + 0.415020i \(0.136225\pi\)
\(104\) 0 0
\(105\) 1.47309e33 0.658577
\(106\) 0 0
\(107\) −5.09640e33 −1.66890 −0.834451 0.551082i \(-0.814215\pi\)
−0.834451 + 0.551082i \(0.814215\pi\)
\(108\) 0 0
\(109\) −5.31361e33 −1.28189 −0.640947 0.767585i \(-0.721458\pi\)
−0.640947 + 0.767585i \(0.721458\pi\)
\(110\) 0 0
\(111\) −2.33180e32 −0.0416737
\(112\) 0 0
\(113\) 3.75830e33 0.500262 0.250131 0.968212i \(-0.419526\pi\)
0.250131 + 0.968212i \(0.419526\pi\)
\(114\) 0 0
\(115\) −1.52784e34 −1.52252
\(116\) 0 0
\(117\) 1.12000e34 0.839753
\(118\) 0 0
\(119\) −1.02044e34 −0.578444
\(120\) 0 0
\(121\) 2.17282e34 0.935546
\(122\) 0 0
\(123\) 1.18448e33 0.0389128
\(124\) 0 0
\(125\) 8.05529e34 2.02799
\(126\) 0 0
\(127\) −2.74550e34 −0.531937 −0.265968 0.963982i \(-0.585692\pi\)
−0.265968 + 0.963982i \(0.585692\pi\)
\(128\) 0 0
\(129\) −1.59392e34 −0.238636
\(130\) 0 0
\(131\) −1.61531e34 −0.187620 −0.0938099 0.995590i \(-0.529905\pi\)
−0.0938099 + 0.995590i \(0.529905\pi\)
\(132\) 0 0
\(133\) 5.93203e34 0.536601
\(134\) 0 0
\(135\) −1.39749e35 −0.988194
\(136\) 0 0
\(137\) −1.20499e35 −0.668490 −0.334245 0.942486i \(-0.608481\pi\)
−0.334245 + 0.942486i \(0.608481\pi\)
\(138\) 0 0
\(139\) −1.71862e35 −0.750648 −0.375324 0.926894i \(-0.622469\pi\)
−0.375324 + 0.926894i \(0.622469\pi\)
\(140\) 0 0
\(141\) −6.25655e34 −0.215884
\(142\) 0 0
\(143\) −4.66683e35 −1.27637
\(144\) 0 0
\(145\) 1.34882e36 2.93348
\(146\) 0 0
\(147\) 1.05560e35 0.183132
\(148\) 0 0
\(149\) −3.16022e33 −0.00438676 −0.00219338 0.999998i \(-0.500698\pi\)
−0.00219338 + 0.999998i \(0.500698\pi\)
\(150\) 0 0
\(151\) −8.88992e35 −0.990326 −0.495163 0.868800i \(-0.664892\pi\)
−0.495163 + 0.868800i \(0.664892\pi\)
\(152\) 0 0
\(153\) 4.62636e35 0.414792
\(154\) 0 0
\(155\) 3.60572e36 2.60927
\(156\) 0 0
\(157\) −1.61326e36 −0.944842 −0.472421 0.881373i \(-0.656620\pi\)
−0.472421 + 0.881373i \(0.656620\pi\)
\(158\) 0 0
\(159\) −1.42937e35 −0.0679349
\(160\) 0 0
\(161\) −2.83447e36 −1.09609
\(162\) 0 0
\(163\) −5.86098e36 −1.84874 −0.924368 0.381501i \(-0.875407\pi\)
−0.924368 + 0.381501i \(0.875407\pi\)
\(164\) 0 0
\(165\) 2.78283e36 0.717797
\(166\) 0 0
\(167\) 7.82436e35 0.165435 0.0827175 0.996573i \(-0.473640\pi\)
0.0827175 + 0.996573i \(0.473640\pi\)
\(168\) 0 0
\(169\) −9.11269e35 −0.158313
\(170\) 0 0
\(171\) −2.68941e36 −0.384787
\(172\) 0 0
\(173\) −1.05668e37 −1.24791 −0.623957 0.781459i \(-0.714476\pi\)
−0.623957 + 0.781459i \(0.714476\pi\)
\(174\) 0 0
\(175\) 2.80100e37 2.73644
\(176\) 0 0
\(177\) −3.37001e36 −0.272943
\(178\) 0 0
\(179\) 1.19253e37 0.802405 0.401202 0.915989i \(-0.368592\pi\)
0.401202 + 0.915989i \(0.368592\pi\)
\(180\) 0 0
\(181\) 2.20113e36 0.123296 0.0616481 0.998098i \(-0.480364\pi\)
0.0616481 + 0.998098i \(0.480364\pi\)
\(182\) 0 0
\(183\) 1.16261e37 0.543238
\(184\) 0 0
\(185\) −6.50200e36 −0.253930
\(186\) 0 0
\(187\) −1.92772e37 −0.630458
\(188\) 0 0
\(189\) −2.59265e37 −0.711416
\(190\) 0 0
\(191\) 1.41890e37 0.327265 0.163633 0.986521i \(-0.447679\pi\)
0.163633 + 0.986521i \(0.447679\pi\)
\(192\) 0 0
\(193\) −3.71061e36 −0.0720693 −0.0360346 0.999351i \(-0.511473\pi\)
−0.0360346 + 0.999351i \(0.511473\pi\)
\(194\) 0 0
\(195\) −2.88899e37 −0.473342
\(196\) 0 0
\(197\) −4.85371e37 −0.672018 −0.336009 0.941859i \(-0.609077\pi\)
−0.336009 + 0.941859i \(0.609077\pi\)
\(198\) 0 0
\(199\) −1.20691e38 −1.41449 −0.707245 0.706968i \(-0.750062\pi\)
−0.707245 + 0.706968i \(0.750062\pi\)
\(200\) 0 0
\(201\) −1.12825e37 −0.112117
\(202\) 0 0
\(203\) 2.50236e38 2.11186
\(204\) 0 0
\(205\) 3.30281e37 0.237107
\(206\) 0 0
\(207\) 1.28506e38 0.785984
\(208\) 0 0
\(209\) 1.12062e38 0.584852
\(210\) 0 0
\(211\) −8.47609e37 −0.378038 −0.189019 0.981973i \(-0.560531\pi\)
−0.189019 + 0.981973i \(0.560531\pi\)
\(212\) 0 0
\(213\) 6.44675e37 0.246081
\(214\) 0 0
\(215\) −4.44450e38 −1.45408
\(216\) 0 0
\(217\) 6.68938e38 1.87845
\(218\) 0 0
\(219\) −1.10435e38 −0.266551
\(220\) 0 0
\(221\) 2.00126e38 0.415748
\(222\) 0 0
\(223\) −9.55613e38 −1.71101 −0.855504 0.517797i \(-0.826752\pi\)
−0.855504 + 0.517797i \(0.826752\pi\)
\(224\) 0 0
\(225\) −1.26989e39 −1.96225
\(226\) 0 0
\(227\) −1.31422e38 −0.175486 −0.0877430 0.996143i \(-0.527965\pi\)
−0.0877430 + 0.996143i \(0.527965\pi\)
\(228\) 0 0
\(229\) 1.01200e39 1.16922 0.584610 0.811314i \(-0.301247\pi\)
0.584610 + 0.811314i \(0.301247\pi\)
\(230\) 0 0
\(231\) 5.16274e38 0.516752
\(232\) 0 0
\(233\) 4.43939e38 0.385435 0.192717 0.981254i \(-0.438270\pi\)
0.192717 + 0.981254i \(0.438270\pi\)
\(234\) 0 0
\(235\) −1.74458e39 −1.31544
\(236\) 0 0
\(237\) −3.93706e37 −0.0258121
\(238\) 0 0
\(239\) 2.91733e39 1.66502 0.832512 0.554008i \(-0.186902\pi\)
0.832512 + 0.554008i \(0.186902\pi\)
\(240\) 0 0
\(241\) 4.88758e36 0.00243115 0.00121558 0.999999i \(-0.499613\pi\)
0.00121558 + 0.999999i \(0.499613\pi\)
\(242\) 0 0
\(243\) 1.78912e39 0.776492
\(244\) 0 0
\(245\) 2.94344e39 1.11587
\(246\) 0 0
\(247\) −1.16337e39 −0.385674
\(248\) 0 0
\(249\) −8.48829e38 −0.246338
\(250\) 0 0
\(251\) −2.40071e39 −0.610553 −0.305276 0.952264i \(-0.598749\pi\)
−0.305276 + 0.952264i \(0.598749\pi\)
\(252\) 0 0
\(253\) −5.35461e39 −1.19465
\(254\) 0 0
\(255\) −1.19335e39 −0.233805
\(256\) 0 0
\(257\) 1.55258e39 0.267398 0.133699 0.991022i \(-0.457315\pi\)
0.133699 + 0.991022i \(0.457315\pi\)
\(258\) 0 0
\(259\) −1.20626e39 −0.182808
\(260\) 0 0
\(261\) −1.13449e40 −1.51438
\(262\) 0 0
\(263\) 1.41636e40 1.66688 0.833442 0.552607i \(-0.186367\pi\)
0.833442 + 0.552607i \(0.186367\pi\)
\(264\) 0 0
\(265\) −3.98567e39 −0.413947
\(266\) 0 0
\(267\) −3.83917e39 −0.352210
\(268\) 0 0
\(269\) −4.46974e39 −0.362552 −0.181276 0.983432i \(-0.558023\pi\)
−0.181276 + 0.983432i \(0.558023\pi\)
\(270\) 0 0
\(271\) 4.70466e39 0.337703 0.168851 0.985642i \(-0.445994\pi\)
0.168851 + 0.985642i \(0.445994\pi\)
\(272\) 0 0
\(273\) −5.35969e39 −0.340766
\(274\) 0 0
\(275\) 5.29140e40 2.98250
\(276\) 0 0
\(277\) 2.48781e40 1.24423 0.622116 0.782925i \(-0.286273\pi\)
0.622116 + 0.782925i \(0.286273\pi\)
\(278\) 0 0
\(279\) −3.03276e40 −1.34701
\(280\) 0 0
\(281\) 6.82873e38 0.0269579 0.0134789 0.999909i \(-0.495709\pi\)
0.0134789 + 0.999909i \(0.495709\pi\)
\(282\) 0 0
\(283\) −3.13055e40 −1.09937 −0.549686 0.835371i \(-0.685253\pi\)
−0.549686 + 0.835371i \(0.685253\pi\)
\(284\) 0 0
\(285\) 6.93720e39 0.216892
\(286\) 0 0
\(287\) 6.12741e39 0.170697
\(288\) 0 0
\(289\) −3.19880e40 −0.794643
\(290\) 0 0
\(291\) 1.76558e40 0.391430
\(292\) 0 0
\(293\) 4.43484e38 0.00878138 0.00439069 0.999990i \(-0.498602\pi\)
0.00439069 + 0.999990i \(0.498602\pi\)
\(294\) 0 0
\(295\) −9.39696e40 −1.66312
\(296\) 0 0
\(297\) −4.89779e40 −0.775387
\(298\) 0 0
\(299\) 5.55889e40 0.787795
\(300\) 0 0
\(301\) −8.24549e40 −1.04681
\(302\) 0 0
\(303\) 1.34459e40 0.153034
\(304\) 0 0
\(305\) 3.24182e41 3.31011
\(306\) 0 0
\(307\) −9.15172e40 −0.838918 −0.419459 0.907774i \(-0.637780\pi\)
−0.419459 + 0.907774i \(0.637780\pi\)
\(308\) 0 0
\(309\) 6.42940e40 0.529487
\(310\) 0 0
\(311\) −4.61111e40 −0.341397 −0.170698 0.985323i \(-0.554602\pi\)
−0.170698 + 0.985323i \(0.554602\pi\)
\(312\) 0 0
\(313\) 8.08893e40 0.538778 0.269389 0.963031i \(-0.413178\pi\)
0.269389 + 0.963031i \(0.413178\pi\)
\(314\) 0 0
\(315\) −3.45488e41 −2.07161
\(316\) 0 0
\(317\) 2.72792e41 1.47351 0.736756 0.676159i \(-0.236357\pi\)
0.736756 + 0.676159i \(0.236357\pi\)
\(318\) 0 0
\(319\) 4.72722e41 2.30176
\(320\) 0 0
\(321\) −1.10570e41 −0.485629
\(322\) 0 0
\(323\) −4.80552e40 −0.190502
\(324\) 0 0
\(325\) −5.49326e41 −1.96677
\(326\) 0 0
\(327\) −1.15283e41 −0.373015
\(328\) 0 0
\(329\) −3.23657e41 −0.947006
\(330\) 0 0
\(331\) 3.98182e40 0.105419 0.0527096 0.998610i \(-0.483214\pi\)
0.0527096 + 0.998610i \(0.483214\pi\)
\(332\) 0 0
\(333\) 5.46882e40 0.131088
\(334\) 0 0
\(335\) −3.14603e41 −0.683161
\(336\) 0 0
\(337\) −1.02227e41 −0.201220 −0.100610 0.994926i \(-0.532079\pi\)
−0.100610 + 0.994926i \(0.532079\pi\)
\(338\) 0 0
\(339\) 8.15391e40 0.145570
\(340\) 0 0
\(341\) 1.26370e42 2.04736
\(342\) 0 0
\(343\) −3.21609e41 −0.473124
\(344\) 0 0
\(345\) −3.31476e41 −0.443034
\(346\) 0 0
\(347\) 1.19238e42 1.44870 0.724348 0.689434i \(-0.242141\pi\)
0.724348 + 0.689434i \(0.242141\pi\)
\(348\) 0 0
\(349\) −9.44292e41 −1.04349 −0.521743 0.853103i \(-0.674718\pi\)
−0.521743 + 0.853103i \(0.674718\pi\)
\(350\) 0 0
\(351\) 5.08463e41 0.511319
\(352\) 0 0
\(353\) 3.78690e41 0.346738 0.173369 0.984857i \(-0.444535\pi\)
0.173369 + 0.984857i \(0.444535\pi\)
\(354\) 0 0
\(355\) 1.79762e42 1.49944
\(356\) 0 0
\(357\) −2.21391e41 −0.168320
\(358\) 0 0
\(359\) −3.89278e41 −0.269899 −0.134950 0.990852i \(-0.543087\pi\)
−0.134950 + 0.990852i \(0.543087\pi\)
\(360\) 0 0
\(361\) −1.30141e42 −0.823279
\(362\) 0 0
\(363\) 4.71410e41 0.272232
\(364\) 0 0
\(365\) −3.07939e42 −1.62417
\(366\) 0 0
\(367\) −1.36076e42 −0.655829 −0.327914 0.944707i \(-0.606346\pi\)
−0.327914 + 0.944707i \(0.606346\pi\)
\(368\) 0 0
\(369\) −2.77799e41 −0.122404
\(370\) 0 0
\(371\) −7.39427e41 −0.298007
\(372\) 0 0
\(373\) 3.42419e42 1.26288 0.631440 0.775425i \(-0.282464\pi\)
0.631440 + 0.775425i \(0.282464\pi\)
\(374\) 0 0
\(375\) 1.74765e42 0.590119
\(376\) 0 0
\(377\) −4.90756e42 −1.51786
\(378\) 0 0
\(379\) −1.97551e42 −0.559928 −0.279964 0.960011i \(-0.590322\pi\)
−0.279964 + 0.960011i \(0.590322\pi\)
\(380\) 0 0
\(381\) −5.95656e41 −0.154787
\(382\) 0 0
\(383\) −4.75888e42 −1.13429 −0.567146 0.823617i \(-0.691952\pi\)
−0.567146 + 0.823617i \(0.691952\pi\)
\(384\) 0 0
\(385\) 1.43958e43 3.14872
\(386\) 0 0
\(387\) 3.73826e42 0.750652
\(388\) 0 0
\(389\) 3.85205e42 0.710434 0.355217 0.934784i \(-0.384407\pi\)
0.355217 + 0.934784i \(0.384407\pi\)
\(390\) 0 0
\(391\) 2.29620e42 0.389128
\(392\) 0 0
\(393\) −3.50453e41 −0.0545949
\(394\) 0 0
\(395\) −1.09781e42 −0.157281
\(396\) 0 0
\(397\) −7.07957e42 −0.933176 −0.466588 0.884475i \(-0.654517\pi\)
−0.466588 + 0.884475i \(0.654517\pi\)
\(398\) 0 0
\(399\) 1.28700e42 0.156144
\(400\) 0 0
\(401\) −3.99621e42 −0.446444 −0.223222 0.974768i \(-0.571657\pi\)
−0.223222 + 0.974768i \(0.571657\pi\)
\(402\) 0 0
\(403\) −1.31190e43 −1.35011
\(404\) 0 0
\(405\) 1.40804e43 1.33539
\(406\) 0 0
\(407\) −2.27875e42 −0.199246
\(408\) 0 0
\(409\) 4.22070e42 0.340369 0.170184 0.985412i \(-0.445564\pi\)
0.170184 + 0.985412i \(0.445564\pi\)
\(410\) 0 0
\(411\) −2.61432e42 −0.194522
\(412\) 0 0
\(413\) −1.74334e43 −1.19731
\(414\) 0 0
\(415\) −2.36688e43 −1.50101
\(416\) 0 0
\(417\) −3.72868e42 −0.218429
\(418\) 0 0
\(419\) −5.25752e42 −0.284610 −0.142305 0.989823i \(-0.545451\pi\)
−0.142305 + 0.989823i \(0.545451\pi\)
\(420\) 0 0
\(421\) 3.56053e43 1.78180 0.890902 0.454195i \(-0.150073\pi\)
0.890902 + 0.454195i \(0.150073\pi\)
\(422\) 0 0
\(423\) 1.46736e43 0.679081
\(424\) 0 0
\(425\) −2.26909e43 −0.971480
\(426\) 0 0
\(427\) 6.01427e43 2.38300
\(428\) 0 0
\(429\) −1.01250e43 −0.371408
\(430\) 0 0
\(431\) −3.50406e43 −1.19041 −0.595205 0.803574i \(-0.702929\pi\)
−0.595205 + 0.803574i \(0.702929\pi\)
\(432\) 0 0
\(433\) 4.65228e43 1.46425 0.732125 0.681170i \(-0.238529\pi\)
0.732125 + 0.681170i \(0.238529\pi\)
\(434\) 0 0
\(435\) 2.92638e43 0.853606
\(436\) 0 0
\(437\) −1.33483e43 −0.360979
\(438\) 0 0
\(439\) −4.97854e42 −0.124864 −0.0624320 0.998049i \(-0.519886\pi\)
−0.0624320 + 0.998049i \(0.519886\pi\)
\(440\) 0 0
\(441\) −2.47573e43 −0.576057
\(442\) 0 0
\(443\) 6.46629e43 1.39635 0.698174 0.715928i \(-0.253996\pi\)
0.698174 + 0.715928i \(0.253996\pi\)
\(444\) 0 0
\(445\) −1.07052e44 −2.14612
\(446\) 0 0
\(447\) −6.85633e40 −0.00127649
\(448\) 0 0
\(449\) 2.27700e43 0.393820 0.196910 0.980422i \(-0.436909\pi\)
0.196910 + 0.980422i \(0.436909\pi\)
\(450\) 0 0
\(451\) 1.15753e43 0.186046
\(452\) 0 0
\(453\) −1.92874e43 −0.288172
\(454\) 0 0
\(455\) −1.49450e44 −2.07638
\(456\) 0 0
\(457\) −9.89272e43 −1.27849 −0.639247 0.769002i \(-0.720754\pi\)
−0.639247 + 0.769002i \(0.720754\pi\)
\(458\) 0 0
\(459\) 2.10030e43 0.252564
\(460\) 0 0
\(461\) 1.21100e44 1.35544 0.677720 0.735320i \(-0.262968\pi\)
0.677720 + 0.735320i \(0.262968\pi\)
\(462\) 0 0
\(463\) 1.96429e43 0.204701 0.102350 0.994748i \(-0.467364\pi\)
0.102350 + 0.994748i \(0.467364\pi\)
\(464\) 0 0
\(465\) 7.82288e43 0.759264
\(466\) 0 0
\(467\) 1.33961e44 1.21130 0.605650 0.795731i \(-0.292913\pi\)
0.605650 + 0.795731i \(0.292913\pi\)
\(468\) 0 0
\(469\) −5.83655e43 −0.491817
\(470\) 0 0
\(471\) −3.50008e43 −0.274937
\(472\) 0 0
\(473\) −1.55766e44 −1.14094
\(474\) 0 0
\(475\) 1.31907e44 0.901205
\(476\) 0 0
\(477\) 3.35234e43 0.213695
\(478\) 0 0
\(479\) 2.06153e44 1.22646 0.613232 0.789903i \(-0.289869\pi\)
0.613232 + 0.789903i \(0.289869\pi\)
\(480\) 0 0
\(481\) 2.36568e43 0.131390
\(482\) 0 0
\(483\) −6.14959e43 −0.318947
\(484\) 0 0
\(485\) 4.92317e44 2.38510
\(486\) 0 0
\(487\) −1.57095e44 −0.711108 −0.355554 0.934656i \(-0.615708\pi\)
−0.355554 + 0.934656i \(0.615708\pi\)
\(488\) 0 0
\(489\) −1.27158e44 −0.537959
\(490\) 0 0
\(491\) −4.22476e44 −1.67093 −0.835464 0.549545i \(-0.814801\pi\)
−0.835464 + 0.549545i \(0.814801\pi\)
\(492\) 0 0
\(493\) −2.02715e44 −0.749743
\(494\) 0 0
\(495\) −6.52663e44 −2.25789
\(496\) 0 0
\(497\) 3.33496e44 1.07947
\(498\) 0 0
\(499\) 1.23722e44 0.374791 0.187395 0.982285i \(-0.439995\pi\)
0.187395 + 0.982285i \(0.439995\pi\)
\(500\) 0 0
\(501\) 1.69755e43 0.0481395
\(502\) 0 0
\(503\) −1.11832e44 −0.296957 −0.148478 0.988916i \(-0.547438\pi\)
−0.148478 + 0.988916i \(0.547438\pi\)
\(504\) 0 0
\(505\) 3.74927e44 0.932479
\(506\) 0 0
\(507\) −1.97707e43 −0.0460670
\(508\) 0 0
\(509\) −2.64785e44 −0.578162 −0.289081 0.957305i \(-0.593350\pi\)
−0.289081 + 0.957305i \(0.593350\pi\)
\(510\) 0 0
\(511\) −5.71292e44 −1.16927
\(512\) 0 0
\(513\) −1.22095e44 −0.234294
\(514\) 0 0
\(515\) 1.79278e45 3.22632
\(516\) 0 0
\(517\) −6.11423e44 −1.03216
\(518\) 0 0
\(519\) −2.29256e44 −0.363127
\(520\) 0 0
\(521\) 6.23370e44 0.926666 0.463333 0.886184i \(-0.346653\pi\)
0.463333 + 0.886184i \(0.346653\pi\)
\(522\) 0 0
\(523\) 8.27079e44 1.15417 0.577084 0.816685i \(-0.304191\pi\)
0.577084 + 0.816685i \(0.304191\pi\)
\(524\) 0 0
\(525\) 6.07698e44 0.796269
\(526\) 0 0
\(527\) −5.41905e44 −0.666881
\(528\) 0 0
\(529\) −2.27191e44 −0.262647
\(530\) 0 0
\(531\) 7.90376e44 0.858567
\(532\) 0 0
\(533\) −1.20169e44 −0.122686
\(534\) 0 0
\(535\) −3.08314e45 −2.95908
\(536\) 0 0
\(537\) 2.58728e44 0.233489
\(538\) 0 0
\(539\) 1.03159e45 0.875571
\(540\) 0 0
\(541\) 1.03419e45 0.825745 0.412873 0.910789i \(-0.364525\pi\)
0.412873 + 0.910789i \(0.364525\pi\)
\(542\) 0 0
\(543\) 4.77552e43 0.0358776
\(544\) 0 0
\(545\) −3.21455e45 −2.27289
\(546\) 0 0
\(547\) 1.01840e45 0.677844 0.338922 0.940815i \(-0.389938\pi\)
0.338922 + 0.940815i \(0.389938\pi\)
\(548\) 0 0
\(549\) −2.72669e45 −1.70880
\(550\) 0 0
\(551\) 1.17843e45 0.695508
\(552\) 0 0
\(553\) −2.03667e44 −0.113229
\(554\) 0 0
\(555\) −1.41066e44 −0.0738903
\(556\) 0 0
\(557\) −2.74214e44 −0.135357 −0.0676783 0.997707i \(-0.521559\pi\)
−0.0676783 + 0.997707i \(0.521559\pi\)
\(558\) 0 0
\(559\) 1.61708e45 0.752381
\(560\) 0 0
\(561\) −4.18232e44 −0.183455
\(562\) 0 0
\(563\) −1.73021e44 −0.0715666 −0.0357833 0.999360i \(-0.511393\pi\)
−0.0357833 + 0.999360i \(0.511393\pi\)
\(564\) 0 0
\(565\) 2.27364e45 0.886998
\(566\) 0 0
\(567\) 2.61221e45 0.961363
\(568\) 0 0
\(569\) 3.46685e45 1.20388 0.601939 0.798542i \(-0.294395\pi\)
0.601939 + 0.798542i \(0.294395\pi\)
\(570\) 0 0
\(571\) 3.55607e45 1.16540 0.582699 0.812688i \(-0.301997\pi\)
0.582699 + 0.812688i \(0.301997\pi\)
\(572\) 0 0
\(573\) 3.07840e44 0.0952300
\(574\) 0 0
\(575\) −6.30284e45 −1.84084
\(576\) 0 0
\(577\) −1.33881e45 −0.369249 −0.184625 0.982809i \(-0.559107\pi\)
−0.184625 + 0.982809i \(0.559107\pi\)
\(578\) 0 0
\(579\) −8.05046e43 −0.0209712
\(580\) 0 0
\(581\) −4.39107e45 −1.08060
\(582\) 0 0
\(583\) −1.39686e45 −0.324804
\(584\) 0 0
\(585\) 6.77562e45 1.48894
\(586\) 0 0
\(587\) 1.27700e45 0.265255 0.132628 0.991166i \(-0.457659\pi\)
0.132628 + 0.991166i \(0.457659\pi\)
\(588\) 0 0
\(589\) 3.15021e45 0.618640
\(590\) 0 0
\(591\) −1.05305e45 −0.195549
\(592\) 0 0
\(593\) −7.49368e45 −1.31611 −0.658056 0.752969i \(-0.728621\pi\)
−0.658056 + 0.752969i \(0.728621\pi\)
\(594\) 0 0
\(595\) −6.17329e45 −1.02562
\(596\) 0 0
\(597\) −2.61849e45 −0.411599
\(598\) 0 0
\(599\) −1.19178e46 −1.77278 −0.886388 0.462943i \(-0.846793\pi\)
−0.886388 + 0.462943i \(0.846793\pi\)
\(600\) 0 0
\(601\) −1.25892e45 −0.177243 −0.0886214 0.996065i \(-0.528246\pi\)
−0.0886214 + 0.996065i \(0.528246\pi\)
\(602\) 0 0
\(603\) 2.64612e45 0.352674
\(604\) 0 0
\(605\) 1.31448e46 1.65879
\(606\) 0 0
\(607\) 5.28539e44 0.0631632 0.0315816 0.999501i \(-0.489946\pi\)
0.0315816 + 0.999501i \(0.489946\pi\)
\(608\) 0 0
\(609\) 5.42905e45 0.614523
\(610\) 0 0
\(611\) 6.34748e45 0.680646
\(612\) 0 0
\(613\) −9.85915e44 −0.100171 −0.0500854 0.998745i \(-0.515949\pi\)
−0.0500854 + 0.998745i \(0.515949\pi\)
\(614\) 0 0
\(615\) 7.16569e44 0.0689952
\(616\) 0 0
\(617\) −2.14287e45 −0.195565 −0.0977826 0.995208i \(-0.531175\pi\)
−0.0977826 + 0.995208i \(0.531175\pi\)
\(618\) 0 0
\(619\) 9.99540e45 0.864778 0.432389 0.901687i \(-0.357671\pi\)
0.432389 + 0.901687i \(0.357671\pi\)
\(620\) 0 0
\(621\) 5.83399e45 0.478580
\(622\) 0 0
\(623\) −1.98604e46 −1.54502
\(624\) 0 0
\(625\) 1.96782e46 1.45199
\(626\) 0 0
\(627\) 2.43128e45 0.170185
\(628\) 0 0
\(629\) 9.77188e44 0.0648996
\(630\) 0 0
\(631\) 1.87168e46 1.17964 0.589818 0.807536i \(-0.299199\pi\)
0.589818 + 0.807536i \(0.299199\pi\)
\(632\) 0 0
\(633\) −1.83895e45 −0.110004
\(634\) 0 0
\(635\) −1.66093e46 −0.943160
\(636\) 0 0
\(637\) −1.07094e46 −0.577385
\(638\) 0 0
\(639\) −1.51197e46 −0.774068
\(640\) 0 0
\(641\) −3.73888e46 −1.81796 −0.908980 0.416841i \(-0.863137\pi\)
−0.908980 + 0.416841i \(0.863137\pi\)
\(642\) 0 0
\(643\) 3.70674e45 0.171203 0.0856015 0.996329i \(-0.472719\pi\)
0.0856015 + 0.996329i \(0.472719\pi\)
\(644\) 0 0
\(645\) −9.64267e45 −0.423118
\(646\) 0 0
\(647\) 3.44855e46 1.43786 0.718928 0.695084i \(-0.244633\pi\)
0.718928 + 0.695084i \(0.244633\pi\)
\(648\) 0 0
\(649\) −3.29335e46 −1.30497
\(650\) 0 0
\(651\) 1.45131e46 0.546606
\(652\) 0 0
\(653\) −8.14942e45 −0.291782 −0.145891 0.989301i \(-0.546605\pi\)
−0.145891 + 0.989301i \(0.546605\pi\)
\(654\) 0 0
\(655\) −9.77205e45 −0.332663
\(656\) 0 0
\(657\) 2.59007e46 0.838461
\(658\) 0 0
\(659\) 2.21169e46 0.680948 0.340474 0.940254i \(-0.389412\pi\)
0.340474 + 0.940254i \(0.389412\pi\)
\(660\) 0 0
\(661\) −1.77870e46 −0.520928 −0.260464 0.965484i \(-0.583876\pi\)
−0.260464 + 0.965484i \(0.583876\pi\)
\(662\) 0 0
\(663\) 4.34187e45 0.120977
\(664\) 0 0
\(665\) 3.58867e46 0.951430
\(666\) 0 0
\(667\) −5.63082e46 −1.42068
\(668\) 0 0
\(669\) −2.07327e46 −0.497881
\(670\) 0 0
\(671\) 1.13616e47 2.59728
\(672\) 0 0
\(673\) 1.15371e46 0.251100 0.125550 0.992087i \(-0.459930\pi\)
0.125550 + 0.992087i \(0.459930\pi\)
\(674\) 0 0
\(675\) −5.76511e46 −1.19480
\(676\) 0 0
\(677\) 8.04683e46 1.58822 0.794112 0.607771i \(-0.207936\pi\)
0.794112 + 0.607771i \(0.207936\pi\)
\(678\) 0 0
\(679\) 9.13352e46 1.71707
\(680\) 0 0
\(681\) −2.85130e45 −0.0510642
\(682\) 0 0
\(683\) −8.01067e46 −1.36687 −0.683435 0.730011i \(-0.739515\pi\)
−0.683435 + 0.730011i \(0.739515\pi\)
\(684\) 0 0
\(685\) −7.28979e46 −1.18528
\(686\) 0 0
\(687\) 2.19562e46 0.340228
\(688\) 0 0
\(689\) 1.45015e46 0.214188
\(690\) 0 0
\(691\) −1.09406e46 −0.154046 −0.0770232 0.997029i \(-0.524542\pi\)
−0.0770232 + 0.997029i \(0.524542\pi\)
\(692\) 0 0
\(693\) −1.21083e47 −1.62549
\(694\) 0 0
\(695\) −1.03971e47 −1.33095
\(696\) 0 0
\(697\) −4.96380e45 −0.0606001
\(698\) 0 0
\(699\) 9.63159e45 0.112157
\(700\) 0 0
\(701\) 6.08980e46 0.676481 0.338240 0.941060i \(-0.390168\pi\)
0.338240 + 0.941060i \(0.390168\pi\)
\(702\) 0 0
\(703\) −5.68061e45 −0.0602049
\(704\) 0 0
\(705\) −3.78500e46 −0.382777
\(706\) 0 0
\(707\) 6.95570e46 0.671306
\(708\) 0 0
\(709\) 1.22329e47 1.12685 0.563425 0.826167i \(-0.309484\pi\)
0.563425 + 0.826167i \(0.309484\pi\)
\(710\) 0 0
\(711\) 9.23368e45 0.0811944
\(712\) 0 0
\(713\) −1.50525e47 −1.26366
\(714\) 0 0
\(715\) −2.82327e47 −2.26310
\(716\) 0 0
\(717\) 6.32937e46 0.484501
\(718\) 0 0
\(719\) 4.92469e46 0.360042 0.180021 0.983663i \(-0.442384\pi\)
0.180021 + 0.983663i \(0.442384\pi\)
\(720\) 0 0
\(721\) 3.32599e47 2.32268
\(722\) 0 0
\(723\) 1.06040e44 0.000707435 0
\(724\) 0 0
\(725\) 5.56434e47 3.54680
\(726\) 0 0
\(727\) −1.32921e47 −0.809611 −0.404806 0.914403i \(-0.632661\pi\)
−0.404806 + 0.914403i \(0.632661\pi\)
\(728\) 0 0
\(729\) −9.05689e46 −0.527200
\(730\) 0 0
\(731\) 6.67965e46 0.371635
\(732\) 0 0
\(733\) −3.32699e47 −1.76944 −0.884721 0.466121i \(-0.845651\pi\)
−0.884721 + 0.466121i \(0.845651\pi\)
\(734\) 0 0
\(735\) 6.38602e46 0.324705
\(736\) 0 0
\(737\) −1.10259e47 −0.536042
\(738\) 0 0
\(739\) −1.02617e47 −0.477074 −0.238537 0.971133i \(-0.576668\pi\)
−0.238537 + 0.971133i \(0.576668\pi\)
\(740\) 0 0
\(741\) −2.52403e46 −0.112226
\(742\) 0 0
\(743\) 2.30680e47 0.981059 0.490530 0.871424i \(-0.336803\pi\)
0.490530 + 0.871424i \(0.336803\pi\)
\(744\) 0 0
\(745\) −1.91182e45 −0.00777803
\(746\) 0 0
\(747\) 1.99078e47 0.774878
\(748\) 0 0
\(749\) −5.71989e47 −2.13028
\(750\) 0 0
\(751\) 2.87651e47 1.02519 0.512597 0.858629i \(-0.328683\pi\)
0.512597 + 0.858629i \(0.328683\pi\)
\(752\) 0 0
\(753\) −5.20852e46 −0.177663
\(754\) 0 0
\(755\) −5.37810e47 −1.75592
\(756\) 0 0
\(757\) −1.30633e47 −0.408292 −0.204146 0.978940i \(-0.565442\pi\)
−0.204146 + 0.978940i \(0.565442\pi\)
\(758\) 0 0
\(759\) −1.16172e47 −0.347627
\(760\) 0 0
\(761\) 1.00969e47 0.289295 0.144648 0.989483i \(-0.453795\pi\)
0.144648 + 0.989483i \(0.453795\pi\)
\(762\) 0 0
\(763\) −5.96367e47 −1.63628
\(764\) 0 0
\(765\) 2.79879e47 0.735455
\(766\) 0 0
\(767\) 3.41898e47 0.860545
\(768\) 0 0
\(769\) 4.46600e47 1.07680 0.538399 0.842690i \(-0.319029\pi\)
0.538399 + 0.842690i \(0.319029\pi\)
\(770\) 0 0
\(771\) 3.36844e46 0.0778093
\(772\) 0 0
\(773\) −5.83555e47 −1.29158 −0.645788 0.763516i \(-0.723471\pi\)
−0.645788 + 0.763516i \(0.723471\pi\)
\(774\) 0 0
\(775\) 1.48748e48 3.15480
\(776\) 0 0
\(777\) −2.61707e46 −0.0531947
\(778\) 0 0
\(779\) 2.88557e46 0.0562165
\(780\) 0 0
\(781\) 6.30010e47 1.17654
\(782\) 0 0
\(783\) −5.15043e47 −0.922092
\(784\) 0 0
\(785\) −9.75965e47 −1.67527
\(786\) 0 0
\(787\) 7.26390e47 1.19560 0.597801 0.801644i \(-0.296041\pi\)
0.597801 + 0.801644i \(0.296041\pi\)
\(788\) 0 0
\(789\) 3.07291e47 0.485042
\(790\) 0 0
\(791\) 4.21809e47 0.638563
\(792\) 0 0
\(793\) −1.17950e48 −1.71274
\(794\) 0 0
\(795\) −8.64722e46 −0.120453
\(796\) 0 0
\(797\) 5.44974e47 0.728306 0.364153 0.931339i \(-0.381359\pi\)
0.364153 + 0.931339i \(0.381359\pi\)
\(798\) 0 0
\(799\) 2.62194e47 0.336202
\(800\) 0 0
\(801\) 9.00410e47 1.10791
\(802\) 0 0
\(803\) −1.07923e48 −1.27441
\(804\) 0 0
\(805\) −1.71476e48 −1.94344
\(806\) 0 0
\(807\) −9.69744e46 −0.105498
\(808\) 0 0
\(809\) 7.45844e47 0.778928 0.389464 0.921042i \(-0.372660\pi\)
0.389464 + 0.921042i \(0.372660\pi\)
\(810\) 0 0
\(811\) 2.28203e47 0.228811 0.114406 0.993434i \(-0.463504\pi\)
0.114406 + 0.993434i \(0.463504\pi\)
\(812\) 0 0
\(813\) 1.02071e47 0.0982672
\(814\) 0 0
\(815\) −3.54569e48 −3.27794
\(816\) 0 0
\(817\) −3.88303e47 −0.344752
\(818\) 0 0
\(819\) 1.25702e48 1.07191
\(820\) 0 0
\(821\) 7.49028e47 0.613530 0.306765 0.951785i \(-0.400753\pi\)
0.306765 + 0.951785i \(0.400753\pi\)
\(822\) 0 0
\(823\) −1.29285e48 −1.01730 −0.508652 0.860972i \(-0.669856\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(824\) 0 0
\(825\) 1.14801e48 0.867870
\(826\) 0 0
\(827\) 1.67722e48 1.21828 0.609141 0.793062i \(-0.291514\pi\)
0.609141 + 0.793062i \(0.291514\pi\)
\(828\) 0 0
\(829\) 1.42330e48 0.993451 0.496725 0.867908i \(-0.334536\pi\)
0.496725 + 0.867908i \(0.334536\pi\)
\(830\) 0 0
\(831\) 5.39750e47 0.362056
\(832\) 0 0
\(833\) −4.42371e47 −0.285197
\(834\) 0 0
\(835\) 4.73347e47 0.293328
\(836\) 0 0
\(837\) −1.37683e48 −0.820182
\(838\) 0 0
\(839\) −1.05174e48 −0.602329 −0.301165 0.953572i \(-0.597375\pi\)
−0.301165 + 0.953572i \(0.597375\pi\)
\(840\) 0 0
\(841\) 3.15499e48 1.73726
\(842\) 0 0
\(843\) 1.48154e46 0.00784440
\(844\) 0 0
\(845\) −5.51286e47 −0.280700
\(846\) 0 0
\(847\) 2.43864e48 1.19419
\(848\) 0 0
\(849\) −6.79196e47 −0.319903
\(850\) 0 0
\(851\) 2.71433e47 0.122977
\(852\) 0 0
\(853\) 1.01265e48 0.441368 0.220684 0.975345i \(-0.429171\pi\)
0.220684 + 0.975345i \(0.429171\pi\)
\(854\) 0 0
\(855\) −1.62700e48 −0.682254
\(856\) 0 0
\(857\) 3.70633e48 1.49541 0.747705 0.664031i \(-0.231156\pi\)
0.747705 + 0.664031i \(0.231156\pi\)
\(858\) 0 0
\(859\) −3.78239e48 −1.46852 −0.734259 0.678869i \(-0.762470\pi\)
−0.734259 + 0.678869i \(0.762470\pi\)
\(860\) 0 0
\(861\) 1.32939e47 0.0496706
\(862\) 0 0
\(863\) −4.35247e48 −1.56516 −0.782578 0.622552i \(-0.786096\pi\)
−0.782578 + 0.622552i \(0.786096\pi\)
\(864\) 0 0
\(865\) −6.39258e48 −2.21264
\(866\) 0 0
\(867\) −6.94003e47 −0.231231
\(868\) 0 0
\(869\) −3.84750e47 −0.123410
\(870\) 0 0
\(871\) 1.14465e48 0.353486
\(872\) 0 0
\(873\) −4.14087e48 −1.23128
\(874\) 0 0
\(875\) 9.04077e48 2.58865
\(876\) 0 0
\(877\) 4.09991e48 1.13053 0.565264 0.824910i \(-0.308774\pi\)
0.565264 + 0.824910i \(0.308774\pi\)
\(878\) 0 0
\(879\) 9.62172e45 0.00255527
\(880\) 0 0
\(881\) 2.48110e48 0.634660 0.317330 0.948315i \(-0.397214\pi\)
0.317330 + 0.948315i \(0.397214\pi\)
\(882\) 0 0
\(883\) −6.54154e48 −1.61186 −0.805932 0.592008i \(-0.798336\pi\)
−0.805932 + 0.592008i \(0.798336\pi\)
\(884\) 0 0
\(885\) −2.03874e48 −0.483947
\(886\) 0 0
\(887\) −3.77603e48 −0.863566 −0.431783 0.901977i \(-0.642115\pi\)
−0.431783 + 0.901977i \(0.642115\pi\)
\(888\) 0 0
\(889\) −3.08138e48 −0.678995
\(890\) 0 0
\(891\) 4.93475e48 1.04781
\(892\) 0 0
\(893\) −1.52419e48 −0.311882
\(894\) 0 0
\(895\) 7.21438e48 1.42272
\(896\) 0 0
\(897\) 1.20604e48 0.229238
\(898\) 0 0
\(899\) 1.32888e49 2.43473
\(900\) 0 0
\(901\) 5.99008e47 0.105797
\(902\) 0 0
\(903\) −1.78892e48 −0.304609
\(904\) 0 0
\(905\) 1.33161e48 0.218613
\(906\) 0 0
\(907\) −1.08989e48 −0.172530 −0.0862649 0.996272i \(-0.527493\pi\)
−0.0862649 + 0.996272i \(0.527493\pi\)
\(908\) 0 0
\(909\) −3.15351e48 −0.481382
\(910\) 0 0
\(911\) 5.98086e48 0.880462 0.440231 0.897885i \(-0.354897\pi\)
0.440231 + 0.897885i \(0.354897\pi\)
\(912\) 0 0
\(913\) −8.29520e48 −1.17777
\(914\) 0 0
\(915\) 7.03338e48 0.963199
\(916\) 0 0
\(917\) −1.81292e48 −0.239489
\(918\) 0 0
\(919\) −1.03793e49 −1.32271 −0.661353 0.750075i \(-0.730018\pi\)
−0.661353 + 0.750075i \(0.730018\pi\)
\(920\) 0 0
\(921\) −1.98553e48 −0.244114
\(922\) 0 0
\(923\) −6.54044e48 −0.775852
\(924\) 0 0
\(925\) −2.68229e48 −0.307020
\(926\) 0 0
\(927\) −1.50790e49 −1.66555
\(928\) 0 0
\(929\) 1.49398e49 1.59252 0.796261 0.604953i \(-0.206808\pi\)
0.796261 + 0.604953i \(0.206808\pi\)
\(930\) 0 0
\(931\) 2.57160e48 0.264566
\(932\) 0 0
\(933\) −1.00041e48 −0.0993420
\(934\) 0 0
\(935\) −1.16620e49 −1.11785
\(936\) 0 0
\(937\) 8.00299e48 0.740541 0.370270 0.928924i \(-0.379265\pi\)
0.370270 + 0.928924i \(0.379265\pi\)
\(938\) 0 0
\(939\) 1.75495e48 0.156777
\(940\) 0 0
\(941\) −5.01488e48 −0.432545 −0.216273 0.976333i \(-0.569390\pi\)
−0.216273 + 0.976333i \(0.569390\pi\)
\(942\) 0 0
\(943\) −1.37880e48 −0.114830
\(944\) 0 0
\(945\) −1.56846e49 −1.26139
\(946\) 0 0
\(947\) 7.00367e48 0.543940 0.271970 0.962306i \(-0.412325\pi\)
0.271970 + 0.962306i \(0.412325\pi\)
\(948\) 0 0
\(949\) 1.12040e49 0.840392
\(950\) 0 0
\(951\) 5.91842e48 0.428773
\(952\) 0 0
\(953\) 1.82156e48 0.127471 0.0637353 0.997967i \(-0.479699\pi\)
0.0637353 + 0.997967i \(0.479699\pi\)
\(954\) 0 0
\(955\) 8.58383e48 0.580264
\(956\) 0 0
\(957\) 1.02561e49 0.669782
\(958\) 0 0
\(959\) −1.35241e49 −0.853300
\(960\) 0 0
\(961\) 1.91206e49 1.16564
\(962\) 0 0
\(963\) 2.59323e49 1.52759
\(964\) 0 0
\(965\) −2.24479e48 −0.127784
\(966\) 0 0
\(967\) 4.21827e48 0.232059 0.116029 0.993246i \(-0.462983\pi\)
0.116029 + 0.993246i \(0.462983\pi\)
\(968\) 0 0
\(969\) −1.04259e48 −0.0554336
\(970\) 0 0
\(971\) −1.43768e49 −0.738831 −0.369415 0.929264i \(-0.620442\pi\)
−0.369415 + 0.929264i \(0.620442\pi\)
\(972\) 0 0
\(973\) −1.92888e49 −0.958171
\(974\) 0 0
\(975\) −1.19180e49 −0.572306
\(976\) 0 0
\(977\) 3.59531e49 1.66908 0.834539 0.550949i \(-0.185734\pi\)
0.834539 + 0.550949i \(0.185734\pi\)
\(978\) 0 0
\(979\) −3.75184e49 −1.68395
\(980\) 0 0
\(981\) 2.70375e49 1.17335
\(982\) 0 0
\(983\) −1.32449e49 −0.555797 −0.277898 0.960610i \(-0.589638\pi\)
−0.277898 + 0.960610i \(0.589638\pi\)
\(984\) 0 0
\(985\) −2.93633e49 −1.19153
\(986\) 0 0
\(987\) −7.02198e48 −0.275567
\(988\) 0 0
\(989\) 1.85541e49 0.704207
\(990\) 0 0
\(991\) 4.86373e49 1.78548 0.892738 0.450576i \(-0.148781\pi\)
0.892738 + 0.450576i \(0.148781\pi\)
\(992\) 0 0
\(993\) 8.63886e47 0.0306756
\(994\) 0 0
\(995\) −7.30141e49 −2.50799
\(996\) 0 0
\(997\) −3.00393e49 −0.998206 −0.499103 0.866543i \(-0.666337\pi\)
−0.499103 + 0.866543i \(0.666337\pi\)
\(998\) 0 0
\(999\) 2.48276e48 0.0798187
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 4.34.a.a.1.2 3
4.3 odd 2 16.34.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.34.a.a.1.2 3 1.1 even 1 trivial
16.34.a.d.1.2 3 4.3 odd 2