Properties

Label 48.22.a.e.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

\(f(q)\) \(=\) \(q+59049.0 q^{3} +3.10995e6 q^{5} -3.63304e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q+59049.0 q^{3} +3.10995e6 q^{5} -3.63304e8 q^{7} +3.48678e9 q^{9} -1.45818e10 q^{11} +1.13351e11 q^{13} +1.83639e11 q^{15} -8.58939e12 q^{17} +2.92029e13 q^{19} -2.14527e13 q^{21} +1.55899e14 q^{23} -4.67165e14 q^{25} +2.05891e14 q^{27} +2.40079e15 q^{29} -2.23982e15 q^{31} -8.61043e14 q^{33} -1.12986e15 q^{35} -3.07851e16 q^{37} +6.69325e15 q^{39} -1.03208e17 q^{41} +1.65557e17 q^{43} +1.08437e16 q^{45} +6.65872e16 q^{47} -4.26556e17 q^{49} -5.07195e17 q^{51} +4.35423e17 q^{53} -4.53488e16 q^{55} +1.72440e18 q^{57} -5.53437e18 q^{59} -7.17621e18 q^{61} -1.26676e18 q^{63} +3.52515e17 q^{65} +1.57554e19 q^{67} +9.20569e18 q^{69} -2.64579e19 q^{71} +1.34712e19 q^{73} -2.75856e19 q^{75} +5.29764e18 q^{77} +1.68861e19 q^{79} +1.21577e19 q^{81} +1.70688e20 q^{83} -2.67126e19 q^{85} +1.41764e20 q^{87} -3.12592e20 q^{89} -4.11808e19 q^{91} -1.32259e20 q^{93} +9.08197e19 q^{95} +9.49015e20 q^{97} -5.08437e19 q^{99} +O(q^{100})\)

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
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) 3.10995e6 0.142419 0.0712096 0.997461i \(-0.477314\pi\)
0.0712096 + 0.997461i \(0.477314\pi\)
\(6\) 0 0
\(7\) −3.63304e8 −0.486117 −0.243058 0.970012i \(-0.578151\pi\)
−0.243058 + 0.970012i \(0.578151\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.45818e10 −0.169508 −0.0847538 0.996402i \(-0.527010\pi\)
−0.0847538 + 0.996402i \(0.527010\pi\)
\(12\) 0 0
\(13\) 1.13351e11 0.228044 0.114022 0.993478i \(-0.463627\pi\)
0.114022 + 0.993478i \(0.463627\pi\)
\(14\) 0 0
\(15\) 1.83639e11 0.0822257
\(16\) 0 0
\(17\) −8.58939e12 −1.03335 −0.516676 0.856181i \(-0.672831\pi\)
−0.516676 + 0.856181i \(0.672831\pi\)
\(18\) 0 0
\(19\) 2.92029e13 1.09273 0.546366 0.837546i \(-0.316011\pi\)
0.546366 + 0.837546i \(0.316011\pi\)
\(20\) 0 0
\(21\) −2.14527e13 −0.280660
\(22\) 0 0
\(23\) 1.55899e14 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(24\) 0 0
\(25\) −4.67165e14 −0.979717
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) 2.40079e15 1.05968 0.529840 0.848097i \(-0.322252\pi\)
0.529840 + 0.848097i \(0.322252\pi\)
\(30\) 0 0
\(31\) −2.23982e15 −0.490812 −0.245406 0.969420i \(-0.578921\pi\)
−0.245406 + 0.969420i \(0.578921\pi\)
\(32\) 0 0
\(33\) −8.61043e14 −0.0978652
\(34\) 0 0
\(35\) −1.12986e15 −0.0692323
\(36\) 0 0
\(37\) −3.07851e16 −1.05250 −0.526250 0.850330i \(-0.676402\pi\)
−0.526250 + 0.850330i \(0.676402\pi\)
\(38\) 0 0
\(39\) 6.69325e15 0.131661
\(40\) 0 0
\(41\) −1.03208e17 −1.20083 −0.600414 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) 0 0
\(43\) 1.65557e17 1.16823 0.584117 0.811670i \(-0.301441\pi\)
0.584117 + 0.811670i \(0.301441\pi\)
\(44\) 0 0
\(45\) 1.08437e16 0.0474730
\(46\) 0 0
\(47\) 6.65872e16 0.184656 0.0923280 0.995729i \(-0.470569\pi\)
0.0923280 + 0.995729i \(0.470569\pi\)
\(48\) 0 0
\(49\) −4.26556e17 −0.763690
\(50\) 0 0
\(51\) −5.07195e17 −0.596607
\(52\) 0 0
\(53\) 4.35423e17 0.341991 0.170995 0.985272i \(-0.445302\pi\)
0.170995 + 0.985272i \(0.445302\pi\)
\(54\) 0 0
\(55\) −4.53488e16 −0.0241411
\(56\) 0 0
\(57\) 1.72440e18 0.630889
\(58\) 0 0
\(59\) −5.53437e18 −1.40968 −0.704842 0.709364i \(-0.748982\pi\)
−0.704842 + 0.709364i \(0.748982\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) 0 0
\(63\) −1.26676e18 −0.162039
\(64\) 0 0
\(65\) 3.52515e17 0.0324779
\(66\) 0 0
\(67\) 1.57554e19 1.05595 0.527977 0.849258i \(-0.322951\pi\)
0.527977 + 0.849258i \(0.322951\pi\)
\(68\) 0 0
\(69\) 9.20569e18 0.453045
\(70\) 0 0
\(71\) −2.64579e19 −0.964588 −0.482294 0.876009i \(-0.660196\pi\)
−0.482294 + 0.876009i \(0.660196\pi\)
\(72\) 0 0
\(73\) 1.34712e19 0.366875 0.183437 0.983031i \(-0.441278\pi\)
0.183437 + 0.983031i \(0.441278\pi\)
\(74\) 0 0
\(75\) −2.75856e19 −0.565640
\(76\) 0 0
\(77\) 5.29764e18 0.0824005
\(78\) 0 0
\(79\) 1.68861e19 0.200653 0.100326 0.994955i \(-0.468011\pi\)
0.100326 + 0.994955i \(0.468011\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) 1.70688e20 1.20749 0.603744 0.797178i \(-0.293675\pi\)
0.603744 + 0.797178i \(0.293675\pi\)
\(84\) 0 0
\(85\) −2.67126e19 −0.147169
\(86\) 0 0
\(87\) 1.41764e20 0.611807
\(88\) 0 0
\(89\) −3.12592e20 −1.06263 −0.531317 0.847173i \(-0.678303\pi\)
−0.531317 + 0.847173i \(0.678303\pi\)
\(90\) 0 0
\(91\) −4.11808e19 −0.110856
\(92\) 0 0
\(93\) −1.32259e20 −0.283371
\(94\) 0 0
\(95\) 9.08197e19 0.155626
\(96\) 0 0
\(97\) 9.49015e20 1.30668 0.653341 0.757064i \(-0.273367\pi\)
0.653341 + 0.757064i \(0.273367\pi\)
\(98\) 0 0
\(99\) −5.08437e19 −0.0565025
\(100\) 0 0
\(101\) 1.44798e20 0.130433 0.0652166 0.997871i \(-0.479226\pi\)
0.0652166 + 0.997871i \(0.479226\pi\)
\(102\) 0 0
\(103\) −2.19627e21 −1.61025 −0.805127 0.593102i \(-0.797903\pi\)
−0.805127 + 0.593102i \(0.797903\pi\)
\(104\) 0 0
\(105\) −6.67169e19 −0.0399713
\(106\) 0 0
\(107\) 1.63087e20 0.0801473 0.0400737 0.999197i \(-0.487241\pi\)
0.0400737 + 0.999197i \(0.487241\pi\)
\(108\) 0 0
\(109\) 2.24852e20 0.0909743 0.0454871 0.998965i \(-0.485516\pi\)
0.0454871 + 0.998965i \(0.485516\pi\)
\(110\) 0 0
\(111\) −1.81783e21 −0.607661
\(112\) 0 0
\(113\) −4.24118e21 −1.17534 −0.587670 0.809101i \(-0.699955\pi\)
−0.587670 + 0.809101i \(0.699955\pi\)
\(114\) 0 0
\(115\) 4.84839e20 0.111756
\(116\) 0 0
\(117\) 3.95230e20 0.0760148
\(118\) 0 0
\(119\) 3.12056e21 0.502330
\(120\) 0 0
\(121\) −7.18762e21 −0.971267
\(122\) 0 0
\(123\) −6.09430e21 −0.693299
\(124\) 0 0
\(125\) −2.93580e21 −0.281950
\(126\) 0 0
\(127\) −1.66312e21 −0.135202 −0.0676012 0.997712i \(-0.521535\pi\)
−0.0676012 + 0.997712i \(0.521535\pi\)
\(128\) 0 0
\(129\) 9.77599e21 0.674480
\(130\) 0 0
\(131\) −6.40663e21 −0.376081 −0.188040 0.982161i \(-0.560214\pi\)
−0.188040 + 0.982161i \(0.560214\pi\)
\(132\) 0 0
\(133\) −1.06095e22 −0.531196
\(134\) 0 0
\(135\) 6.40311e20 0.0274086
\(136\) 0 0
\(137\) −1.98314e22 −0.727423 −0.363711 0.931512i \(-0.618491\pi\)
−0.363711 + 0.931512i \(0.618491\pi\)
\(138\) 0 0
\(139\) 5.20143e21 0.163858 0.0819290 0.996638i \(-0.473892\pi\)
0.0819290 + 0.996638i \(0.473892\pi\)
\(140\) 0 0
\(141\) 3.93191e21 0.106611
\(142\) 0 0
\(143\) −1.65286e21 −0.0386552
\(144\) 0 0
\(145\) 7.46633e21 0.150919
\(146\) 0 0
\(147\) −2.51877e22 −0.440917
\(148\) 0 0
\(149\) −7.47631e22 −1.13562 −0.567808 0.823161i \(-0.692208\pi\)
−0.567808 + 0.823161i \(0.692208\pi\)
\(150\) 0 0
\(151\) −1.11044e23 −1.46635 −0.733174 0.680042i \(-0.761962\pi\)
−0.733174 + 0.680042i \(0.761962\pi\)
\(152\) 0 0
\(153\) −2.99493e22 −0.344451
\(154\) 0 0
\(155\) −6.96573e21 −0.0699011
\(156\) 0 0
\(157\) 4.36563e22 0.382913 0.191457 0.981501i \(-0.438679\pi\)
0.191457 + 0.981501i \(0.438679\pi\)
\(158\) 0 0
\(159\) 2.57113e22 0.197449
\(160\) 0 0
\(161\) −5.66388e22 −0.381454
\(162\) 0 0
\(163\) −2.85661e23 −1.68998 −0.844989 0.534783i \(-0.820393\pi\)
−0.844989 + 0.534783i \(0.820393\pi\)
\(164\) 0 0
\(165\) −2.67780e21 −0.0139379
\(166\) 0 0
\(167\) −2.66950e23 −1.22435 −0.612177 0.790720i \(-0.709706\pi\)
−0.612177 + 0.790720i \(0.709706\pi\)
\(168\) 0 0
\(169\) −2.34216e23 −0.947996
\(170\) 0 0
\(171\) 1.01824e23 0.364244
\(172\) 0 0
\(173\) −2.28496e23 −0.723425 −0.361713 0.932290i \(-0.617808\pi\)
−0.361713 + 0.932290i \(0.617808\pi\)
\(174\) 0 0
\(175\) 1.69723e23 0.476257
\(176\) 0 0
\(177\) −3.26799e23 −0.813881
\(178\) 0 0
\(179\) 1.29151e22 0.0285852 0.0142926 0.999898i \(-0.495450\pi\)
0.0142926 + 0.999898i \(0.495450\pi\)
\(180\) 0 0
\(181\) −8.75338e23 −1.72405 −0.862026 0.506863i \(-0.830805\pi\)
−0.862026 + 0.506863i \(0.830805\pi\)
\(182\) 0 0
\(183\) −4.23748e23 −0.743654
\(184\) 0 0
\(185\) −9.57400e22 −0.149896
\(186\) 0 0
\(187\) 1.25249e23 0.175161
\(188\) 0 0
\(189\) −7.48011e22 −0.0935532
\(190\) 0 0
\(191\) −1.33961e24 −1.50013 −0.750066 0.661363i \(-0.769978\pi\)
−0.750066 + 0.661363i \(0.769978\pi\)
\(192\) 0 0
\(193\) −2.13970e23 −0.214783 −0.107392 0.994217i \(-0.534250\pi\)
−0.107392 + 0.994217i \(0.534250\pi\)
\(194\) 0 0
\(195\) 2.08157e22 0.0187511
\(196\) 0 0
\(197\) 1.42895e24 1.15644 0.578220 0.815881i \(-0.303748\pi\)
0.578220 + 0.815881i \(0.303748\pi\)
\(198\) 0 0
\(199\) −7.45856e23 −0.542872 −0.271436 0.962456i \(-0.587499\pi\)
−0.271436 + 0.962456i \(0.587499\pi\)
\(200\) 0 0
\(201\) 9.30344e23 0.609656
\(202\) 0 0
\(203\) −8.72216e23 −0.515129
\(204\) 0 0
\(205\) −3.20970e23 −0.171021
\(206\) 0 0
\(207\) 5.43587e23 0.261565
\(208\) 0 0
\(209\) −4.25832e23 −0.185226
\(210\) 0 0
\(211\) 2.65090e24 1.04334 0.521671 0.853147i \(-0.325309\pi\)
0.521671 + 0.853147i \(0.325309\pi\)
\(212\) 0 0
\(213\) −1.56231e24 −0.556905
\(214\) 0 0
\(215\) 5.14875e23 0.166379
\(216\) 0 0
\(217\) 8.13736e23 0.238592
\(218\) 0 0
\(219\) 7.95464e23 0.211815
\(220\) 0 0
\(221\) −9.73614e23 −0.235650
\(222\) 0 0
\(223\) −3.97174e24 −0.874539 −0.437270 0.899330i \(-0.644054\pi\)
−0.437270 + 0.899330i \(0.644054\pi\)
\(224\) 0 0
\(225\) −1.62890e24 −0.326572
\(226\) 0 0
\(227\) 2.14690e24 0.392229 0.196114 0.980581i \(-0.437168\pi\)
0.196114 + 0.980581i \(0.437168\pi\)
\(228\) 0 0
\(229\) 3.66024e24 0.609869 0.304934 0.952373i \(-0.401365\pi\)
0.304934 + 0.952373i \(0.401365\pi\)
\(230\) 0 0
\(231\) 3.12820e23 0.0475739
\(232\) 0 0
\(233\) −2.90921e24 −0.404145 −0.202073 0.979371i \(-0.564768\pi\)
−0.202073 + 0.979371i \(0.564768\pi\)
\(234\) 0 0
\(235\) 2.07083e23 0.0262985
\(236\) 0 0
\(237\) 9.97109e23 0.115847
\(238\) 0 0
\(239\) 4.80224e24 0.510818 0.255409 0.966833i \(-0.417790\pi\)
0.255409 + 0.966833i \(0.417790\pi\)
\(240\) 0 0
\(241\) 7.86263e24 0.766282 0.383141 0.923690i \(-0.374842\pi\)
0.383141 + 0.923690i \(0.374842\pi\)
\(242\) 0 0
\(243\) 7.17898e23 0.0641500
\(244\) 0 0
\(245\) −1.32657e24 −0.108764
\(246\) 0 0
\(247\) 3.31018e24 0.249191
\(248\) 0 0
\(249\) 1.00790e25 0.697144
\(250\) 0 0
\(251\) 1.99525e25 1.26889 0.634443 0.772969i \(-0.281229\pi\)
0.634443 + 0.772969i \(0.281229\pi\)
\(252\) 0 0
\(253\) −2.27330e24 −0.133012
\(254\) 0 0
\(255\) −1.57735e24 −0.0849682
\(256\) 0 0
\(257\) 1.33580e25 0.662894 0.331447 0.943474i \(-0.392463\pi\)
0.331447 + 0.943474i \(0.392463\pi\)
\(258\) 0 0
\(259\) 1.11843e25 0.511638
\(260\) 0 0
\(261\) 8.37103e24 0.353227
\(262\) 0 0
\(263\) 5.83637e24 0.227304 0.113652 0.993521i \(-0.463745\pi\)
0.113652 + 0.993521i \(0.463745\pi\)
\(264\) 0 0
\(265\) 1.35414e24 0.0487060
\(266\) 0 0
\(267\) −1.84583e25 −0.613512
\(268\) 0 0
\(269\) 5.35297e25 1.64511 0.822557 0.568683i \(-0.192547\pi\)
0.822557 + 0.568683i \(0.192547\pi\)
\(270\) 0 0
\(271\) 1.04403e25 0.296849 0.148425 0.988924i \(-0.452580\pi\)
0.148425 + 0.988924i \(0.452580\pi\)
\(272\) 0 0
\(273\) −2.43168e24 −0.0640029
\(274\) 0 0
\(275\) 6.81213e24 0.166069
\(276\) 0 0
\(277\) 3.14884e25 0.711399 0.355699 0.934600i \(-0.384243\pi\)
0.355699 + 0.934600i \(0.384243\pi\)
\(278\) 0 0
\(279\) −7.80977e24 −0.163604
\(280\) 0 0
\(281\) −1.15887e25 −0.225225 −0.112613 0.993639i \(-0.535922\pi\)
−0.112613 + 0.993639i \(0.535922\pi\)
\(282\) 0 0
\(283\) 4.80399e25 0.866652 0.433326 0.901237i \(-0.357340\pi\)
0.433326 + 0.901237i \(0.357340\pi\)
\(284\) 0 0
\(285\) 5.36281e24 0.0898507
\(286\) 0 0
\(287\) 3.74957e25 0.583743
\(288\) 0 0
\(289\) 4.68568e24 0.0678180
\(290\) 0 0
\(291\) 5.60384e25 0.754413
\(292\) 0 0
\(293\) −7.96714e25 −0.998142 −0.499071 0.866561i \(-0.666325\pi\)
−0.499071 + 0.866561i \(0.666325\pi\)
\(294\) 0 0
\(295\) −1.72116e25 −0.200766
\(296\) 0 0
\(297\) −3.00227e24 −0.0326217
\(298\) 0 0
\(299\) 1.76713e25 0.178946
\(300\) 0 0
\(301\) −6.01476e25 −0.567898
\(302\) 0 0
\(303\) 8.55018e24 0.0753056
\(304\) 0 0
\(305\) −2.23176e25 −0.183442
\(306\) 0 0
\(307\) −1.51498e26 −1.16266 −0.581332 0.813666i \(-0.697468\pi\)
−0.581332 + 0.813666i \(0.697468\pi\)
\(308\) 0 0
\(309\) −1.29687e26 −0.929681
\(310\) 0 0
\(311\) −1.41406e26 −0.947292 −0.473646 0.880715i \(-0.657062\pi\)
−0.473646 + 0.880715i \(0.657062\pi\)
\(312\) 0 0
\(313\) 4.99598e25 0.312899 0.156450 0.987686i \(-0.449995\pi\)
0.156450 + 0.987686i \(0.449995\pi\)
\(314\) 0 0
\(315\) −3.93957e24 −0.0230774
\(316\) 0 0
\(317\) 1.81535e26 0.995033 0.497517 0.867454i \(-0.334245\pi\)
0.497517 + 0.867454i \(0.334245\pi\)
\(318\) 0 0
\(319\) −3.50079e25 −0.179624
\(320\) 0 0
\(321\) 9.63011e24 0.0462731
\(322\) 0 0
\(323\) −2.50835e26 −1.12918
\(324\) 0 0
\(325\) −5.29536e25 −0.223419
\(326\) 0 0
\(327\) 1.32773e25 0.0525240
\(328\) 0 0
\(329\) −2.41914e25 −0.0897644
\(330\) 0 0
\(331\) −1.44090e26 −0.501695 −0.250848 0.968027i \(-0.580709\pi\)
−0.250848 + 0.968027i \(0.580709\pi\)
\(332\) 0 0
\(333\) −1.07341e26 −0.350833
\(334\) 0 0
\(335\) 4.89987e25 0.150388
\(336\) 0 0
\(337\) −3.63051e26 −1.04678 −0.523388 0.852095i \(-0.675332\pi\)
−0.523388 + 0.852095i \(0.675332\pi\)
\(338\) 0 0
\(339\) −2.50438e26 −0.678583
\(340\) 0 0
\(341\) 3.26607e25 0.0831964
\(342\) 0 0
\(343\) 3.57891e26 0.857360
\(344\) 0 0
\(345\) 2.86292e25 0.0645222
\(346\) 0 0
\(347\) 7.09622e25 0.150511 0.0752554 0.997164i \(-0.476023\pi\)
0.0752554 + 0.997164i \(0.476023\pi\)
\(348\) 0 0
\(349\) −7.03939e26 −1.40562 −0.702810 0.711378i \(-0.748072\pi\)
−0.702810 + 0.711378i \(0.748072\pi\)
\(350\) 0 0
\(351\) 2.33379e25 0.0438872
\(352\) 0 0
\(353\) −1.08085e26 −0.191483 −0.0957414 0.995406i \(-0.530522\pi\)
−0.0957414 + 0.995406i \(0.530522\pi\)
\(354\) 0 0
\(355\) −8.22826e25 −0.137376
\(356\) 0 0
\(357\) 1.84266e26 0.290020
\(358\) 0 0
\(359\) 1.67492e26 0.248601 0.124301 0.992245i \(-0.460331\pi\)
0.124301 + 0.992245i \(0.460331\pi\)
\(360\) 0 0
\(361\) 1.38602e26 0.194064
\(362\) 0 0
\(363\) −4.24422e26 −0.560761
\(364\) 0 0
\(365\) 4.18949e25 0.0522500
\(366\) 0 0
\(367\) −9.83667e26 −1.15839 −0.579194 0.815190i \(-0.696633\pi\)
−0.579194 + 0.815190i \(0.696633\pi\)
\(368\) 0 0
\(369\) −3.59863e26 −0.400276
\(370\) 0 0
\(371\) −1.58191e26 −0.166248
\(372\) 0 0
\(373\) 1.00058e26 0.0993824 0.0496912 0.998765i \(-0.484176\pi\)
0.0496912 + 0.998765i \(0.484176\pi\)
\(374\) 0 0
\(375\) −1.73356e26 −0.162784
\(376\) 0 0
\(377\) 2.72131e26 0.241654
\(378\) 0 0
\(379\) −9.23905e25 −0.0776096 −0.0388048 0.999247i \(-0.512355\pi\)
−0.0388048 + 0.999247i \(0.512355\pi\)
\(380\) 0 0
\(381\) −9.82055e25 −0.0780591
\(382\) 0 0
\(383\) 2.13677e27 1.60757 0.803786 0.594919i \(-0.202816\pi\)
0.803786 + 0.594919i \(0.202816\pi\)
\(384\) 0 0
\(385\) 1.64754e25 0.0117354
\(386\) 0 0
\(387\) 5.77263e26 0.389411
\(388\) 0 0
\(389\) −1.25581e27 −0.802516 −0.401258 0.915965i \(-0.631427\pi\)
−0.401258 + 0.915965i \(0.631427\pi\)
\(390\) 0 0
\(391\) −1.33908e27 −0.810868
\(392\) 0 0
\(393\) −3.78305e26 −0.217130
\(394\) 0 0
\(395\) 5.25150e25 0.0285768
\(396\) 0 0
\(397\) 1.78165e27 0.919439 0.459719 0.888064i \(-0.347950\pi\)
0.459719 + 0.888064i \(0.347950\pi\)
\(398\) 0 0
\(399\) −6.26483e26 −0.306686
\(400\) 0 0
\(401\) 1.40181e27 0.651141 0.325570 0.945518i \(-0.394444\pi\)
0.325570 + 0.945518i \(0.394444\pi\)
\(402\) 0 0
\(403\) −2.53885e26 −0.111927
\(404\) 0 0
\(405\) 3.78097e25 0.0158243
\(406\) 0 0
\(407\) 4.48903e26 0.178407
\(408\) 0 0
\(409\) 2.17493e27 0.821015 0.410507 0.911857i \(-0.365352\pi\)
0.410507 + 0.911857i \(0.365352\pi\)
\(410\) 0 0
\(411\) −1.17102e27 −0.419978
\(412\) 0 0
\(413\) 2.01066e27 0.685271
\(414\) 0 0
\(415\) 5.30831e26 0.171969
\(416\) 0 0
\(417\) 3.07139e26 0.0946034
\(418\) 0 0
\(419\) 6.07636e27 1.77990 0.889952 0.456055i \(-0.150738\pi\)
0.889952 + 0.456055i \(0.150738\pi\)
\(420\) 0 0
\(421\) −1.89993e27 −0.529389 −0.264695 0.964332i \(-0.585271\pi\)
−0.264695 + 0.964332i \(0.585271\pi\)
\(422\) 0 0
\(423\) 2.32175e26 0.0615520
\(424\) 0 0
\(425\) 4.01267e27 1.01239
\(426\) 0 0
\(427\) 2.60714e27 0.626141
\(428\) 0 0
\(429\) −9.75999e25 −0.0223176
\(430\) 0 0
\(431\) 8.08572e24 0.00176079 0.000880395 1.00000i \(-0.499720\pi\)
0.000880395 1.00000i \(0.499720\pi\)
\(432\) 0 0
\(433\) −5.60439e27 −1.16253 −0.581267 0.813713i \(-0.697443\pi\)
−0.581267 + 0.813713i \(0.697443\pi\)
\(434\) 0 0
\(435\) 4.40879e26 0.0871330
\(436\) 0 0
\(437\) 4.55272e27 0.857463
\(438\) 0 0
\(439\) 8.51110e27 1.52795 0.763973 0.645248i \(-0.223246\pi\)
0.763973 + 0.645248i \(0.223246\pi\)
\(440\) 0 0
\(441\) −1.48731e27 −0.254563
\(442\) 0 0
\(443\) 6.63134e27 1.08234 0.541168 0.840915i \(-0.317982\pi\)
0.541168 + 0.840915i \(0.317982\pi\)
\(444\) 0 0
\(445\) −9.72147e26 −0.151339
\(446\) 0 0
\(447\) −4.41468e27 −0.655648
\(448\) 0 0
\(449\) 1.30394e28 1.84787 0.923933 0.382555i \(-0.124956\pi\)
0.923933 + 0.382555i \(0.124956\pi\)
\(450\) 0 0
\(451\) 1.50496e27 0.203549
\(452\) 0 0
\(453\) −6.55703e27 −0.846596
\(454\) 0 0
\(455\) −1.28070e26 −0.0157880
\(456\) 0 0
\(457\) 4.72949e27 0.556793 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(458\) 0 0
\(459\) −1.76848e27 −0.198869
\(460\) 0 0
\(461\) 4.92722e27 0.529349 0.264675 0.964338i \(-0.414735\pi\)
0.264675 + 0.964338i \(0.414735\pi\)
\(462\) 0 0
\(463\) −1.20207e28 −1.23404 −0.617021 0.786947i \(-0.711661\pi\)
−0.617021 + 0.786947i \(0.711661\pi\)
\(464\) 0 0
\(465\) −4.11319e26 −0.0403574
\(466\) 0 0
\(467\) −1.09969e28 −1.03144 −0.515719 0.856758i \(-0.672475\pi\)
−0.515719 + 0.856758i \(0.672475\pi\)
\(468\) 0 0
\(469\) −5.72402e27 −0.513317
\(470\) 0 0
\(471\) 2.57786e27 0.221075
\(472\) 0 0
\(473\) −2.41413e27 −0.198024
\(474\) 0 0
\(475\) −1.36426e28 −1.07057
\(476\) 0 0
\(477\) 1.51823e27 0.113997
\(478\) 0 0
\(479\) 1.32717e28 0.953684 0.476842 0.878989i \(-0.341781\pi\)
0.476842 + 0.878989i \(0.341781\pi\)
\(480\) 0 0
\(481\) −3.48951e27 −0.240017
\(482\) 0 0
\(483\) −3.34446e27 −0.220233
\(484\) 0 0
\(485\) 2.95139e27 0.186096
\(486\) 0 0
\(487\) −2.62576e28 −1.58563 −0.792813 0.609466i \(-0.791384\pi\)
−0.792813 + 0.609466i \(0.791384\pi\)
\(488\) 0 0
\(489\) −1.68680e28 −0.975710
\(490\) 0 0
\(491\) 2.54066e28 1.40796 0.703982 0.710218i \(-0.251404\pi\)
0.703982 + 0.710218i \(0.251404\pi\)
\(492\) 0 0
\(493\) −2.06213e28 −1.09502
\(494\) 0 0
\(495\) −1.58121e26 −0.00804704
\(496\) 0 0
\(497\) 9.61224e27 0.468903
\(498\) 0 0
\(499\) 1.30048e28 0.608204 0.304102 0.952640i \(-0.401644\pi\)
0.304102 + 0.952640i \(0.401644\pi\)
\(500\) 0 0
\(501\) −1.57631e28 −0.706882
\(502\) 0 0
\(503\) −1.34993e27 −0.0580559 −0.0290280 0.999579i \(-0.509241\pi\)
−0.0290280 + 0.999579i \(0.509241\pi\)
\(504\) 0 0
\(505\) 4.50314e26 0.0185762
\(506\) 0 0
\(507\) −1.38302e28 −0.547326
\(508\) 0 0
\(509\) −4.04902e28 −1.53749 −0.768746 0.639554i \(-0.779119\pi\)
−0.768746 + 0.639554i \(0.779119\pi\)
\(510\) 0 0
\(511\) −4.89416e27 −0.178344
\(512\) 0 0
\(513\) 6.01263e27 0.210296
\(514\) 0 0
\(515\) −6.83028e27 −0.229331
\(516\) 0 0
\(517\) −9.70964e26 −0.0313006
\(518\) 0 0
\(519\) −1.34924e28 −0.417670
\(520\) 0 0
\(521\) 5.40378e28 1.60658 0.803289 0.595590i \(-0.203082\pi\)
0.803289 + 0.595590i \(0.203082\pi\)
\(522\) 0 0
\(523\) −1.54066e28 −0.439988 −0.219994 0.975501i \(-0.570604\pi\)
−0.219994 + 0.975501i \(0.570604\pi\)
\(524\) 0 0
\(525\) 1.00220e28 0.274967
\(526\) 0 0
\(527\) 1.92387e28 0.507182
\(528\) 0 0
\(529\) −1.51670e28 −0.384252
\(530\) 0 0
\(531\) −1.92971e28 −0.469895
\(532\) 0 0
\(533\) −1.16987e28 −0.273842
\(534\) 0 0
\(535\) 5.07192e26 0.0114145
\(536\) 0 0
\(537\) 7.62623e26 0.0165037
\(538\) 0 0
\(539\) 6.21997e27 0.129451
\(540\) 0 0
\(541\) −7.54478e28 −1.51034 −0.755171 0.655528i \(-0.772446\pi\)
−0.755171 + 0.655528i \(0.772446\pi\)
\(542\) 0 0
\(543\) −5.16878e28 −0.995382
\(544\) 0 0
\(545\) 6.99279e26 0.0129565
\(546\) 0 0
\(547\) −7.90524e28 −1.40944 −0.704722 0.709483i \(-0.748928\pi\)
−0.704722 + 0.709483i \(0.748928\pi\)
\(548\) 0 0
\(549\) −2.50219e28 −0.429349
\(550\) 0 0
\(551\) 7.01101e28 1.15795
\(552\) 0 0
\(553\) −6.13480e27 −0.0975408
\(554\) 0 0
\(555\) −5.65335e27 −0.0865426
\(556\) 0 0
\(557\) 1.17729e29 1.73541 0.867707 0.497075i \(-0.165593\pi\)
0.867707 + 0.497075i \(0.165593\pi\)
\(558\) 0 0
\(559\) 1.87660e28 0.266409
\(560\) 0 0
\(561\) 7.39583e27 0.101129
\(562\) 0 0
\(563\) 9.20807e28 1.21291 0.606457 0.795116i \(-0.292590\pi\)
0.606457 + 0.795116i \(0.292590\pi\)
\(564\) 0 0
\(565\) −1.31899e28 −0.167391
\(566\) 0 0
\(567\) −4.41693e27 −0.0540130
\(568\) 0 0
\(569\) −2.71795e27 −0.0320304 −0.0160152 0.999872i \(-0.505098\pi\)
−0.0160152 + 0.999872i \(0.505098\pi\)
\(570\) 0 0
\(571\) −1.28086e28 −0.145487 −0.0727434 0.997351i \(-0.523175\pi\)
−0.0727434 + 0.997351i \(0.523175\pi\)
\(572\) 0 0
\(573\) −7.91029e28 −0.866102
\(574\) 0 0
\(575\) −7.28307e28 −0.768780
\(576\) 0 0
\(577\) 1.49329e29 1.51984 0.759922 0.650015i \(-0.225237\pi\)
0.759922 + 0.650015i \(0.225237\pi\)
\(578\) 0 0
\(579\) −1.26347e28 −0.124005
\(580\) 0 0
\(581\) −6.20116e28 −0.586980
\(582\) 0 0
\(583\) −6.34926e27 −0.0579700
\(584\) 0 0
\(585\) 1.22914e27 0.0108260
\(586\) 0 0
\(587\) −2.62975e28 −0.223468 −0.111734 0.993738i \(-0.535640\pi\)
−0.111734 + 0.993738i \(0.535640\pi\)
\(588\) 0 0
\(589\) −6.54093e28 −0.536326
\(590\) 0 0
\(591\) 8.43783e28 0.667671
\(592\) 0 0
\(593\) 1.92294e29 1.46856 0.734278 0.678849i \(-0.237521\pi\)
0.734278 + 0.678849i \(0.237521\pi\)
\(594\) 0 0
\(595\) 9.70478e27 0.0715414
\(596\) 0 0
\(597\) −4.40421e28 −0.313428
\(598\) 0 0
\(599\) 2.45874e29 1.68940 0.844699 0.535242i \(-0.179780\pi\)
0.844699 + 0.535242i \(0.179780\pi\)
\(600\) 0 0
\(601\) 7.47252e28 0.495776 0.247888 0.968789i \(-0.420264\pi\)
0.247888 + 0.968789i \(0.420264\pi\)
\(602\) 0 0
\(603\) 5.49359e28 0.351985
\(604\) 0 0
\(605\) −2.23531e28 −0.138327
\(606\) 0 0
\(607\) 1.23466e29 0.738016 0.369008 0.929426i \(-0.379698\pi\)
0.369008 + 0.929426i \(0.379698\pi\)
\(608\) 0 0
\(609\) −5.15035e28 −0.297410
\(610\) 0 0
\(611\) 7.54771e27 0.0421098
\(612\) 0 0
\(613\) −1.59244e29 −0.858476 −0.429238 0.903191i \(-0.641218\pi\)
−0.429238 + 0.903191i \(0.641218\pi\)
\(614\) 0 0
\(615\) −1.89530e28 −0.0987390
\(616\) 0 0
\(617\) 1.23256e29 0.620602 0.310301 0.950638i \(-0.399570\pi\)
0.310301 + 0.950638i \(0.399570\pi\)
\(618\) 0 0
\(619\) 5.18990e28 0.252585 0.126292 0.991993i \(-0.459692\pi\)
0.126292 + 0.991993i \(0.459692\pi\)
\(620\) 0 0
\(621\) 3.20983e28 0.151015
\(622\) 0 0
\(623\) 1.13566e29 0.516564
\(624\) 0 0
\(625\) 2.13632e29 0.939562
\(626\) 0 0
\(627\) −2.51450e28 −0.106940
\(628\) 0 0
\(629\) 2.64425e29 1.08760
\(630\) 0 0
\(631\) 2.05208e28 0.0816366 0.0408183 0.999167i \(-0.487004\pi\)
0.0408183 + 0.999167i \(0.487004\pi\)
\(632\) 0 0
\(633\) 1.56533e29 0.602374
\(634\) 0 0
\(635\) −5.17222e27 −0.0192554
\(636\) 0 0
\(637\) −4.83505e28 −0.174155
\(638\) 0 0
\(639\) −9.22528e28 −0.321529
\(640\) 0 0
\(641\) 5.14342e29 1.73477 0.867386 0.497636i \(-0.165798\pi\)
0.867386 + 0.497636i \(0.165798\pi\)
\(642\) 0 0
\(643\) 8.85766e28 0.289137 0.144569 0.989495i \(-0.453821\pi\)
0.144569 + 0.989495i \(0.453821\pi\)
\(644\) 0 0
\(645\) 3.04028e28 0.0960588
\(646\) 0 0
\(647\) −5.46916e29 −1.67273 −0.836364 0.548174i \(-0.815323\pi\)
−0.836364 + 0.548174i \(0.815323\pi\)
\(648\) 0 0
\(649\) 8.07012e28 0.238952
\(650\) 0 0
\(651\) 4.80503e28 0.137751
\(652\) 0 0
\(653\) −5.66153e29 −1.57161 −0.785806 0.618473i \(-0.787752\pi\)
−0.785806 + 0.618473i \(0.787752\pi\)
\(654\) 0 0
\(655\) −1.99243e28 −0.0535611
\(656\) 0 0
\(657\) 4.69713e28 0.122292
\(658\) 0 0
\(659\) 1.48653e29 0.374867 0.187434 0.982277i \(-0.439983\pi\)
0.187434 + 0.982277i \(0.439983\pi\)
\(660\) 0 0
\(661\) −4.04669e29 −0.988517 −0.494259 0.869315i \(-0.664560\pi\)
−0.494259 + 0.869315i \(0.664560\pi\)
\(662\) 0 0
\(663\) −5.74909e28 −0.136053
\(664\) 0 0
\(665\) −3.29951e28 −0.0756524
\(666\) 0 0
\(667\) 3.74281e29 0.831527
\(668\) 0 0
\(669\) −2.34527e29 −0.504916
\(670\) 0 0
\(671\) 1.04642e29 0.218334
\(672\) 0 0
\(673\) −1.54590e29 −0.312625 −0.156313 0.987708i \(-0.549961\pi\)
−0.156313 + 0.987708i \(0.549961\pi\)
\(674\) 0 0
\(675\) −9.61852e28 −0.188547
\(676\) 0 0
\(677\) −9.88421e29 −1.87828 −0.939142 0.343530i \(-0.888377\pi\)
−0.939142 + 0.343530i \(0.888377\pi\)
\(678\) 0 0
\(679\) −3.44781e29 −0.635200
\(680\) 0 0
\(681\) 1.26772e29 0.226453
\(682\) 0 0
\(683\) 1.41169e29 0.244524 0.122262 0.992498i \(-0.460985\pi\)
0.122262 + 0.992498i \(0.460985\pi\)
\(684\) 0 0
\(685\) −6.16746e28 −0.103599
\(686\) 0 0
\(687\) 2.16133e29 0.352108
\(688\) 0 0
\(689\) 4.93555e28 0.0779891
\(690\) 0 0
\(691\) −7.11585e29 −1.09070 −0.545352 0.838207i \(-0.683604\pi\)
−0.545352 + 0.838207i \(0.683604\pi\)
\(692\) 0 0
\(693\) 1.84717e28 0.0274668
\(694\) 0 0
\(695\) 1.61762e28 0.0233365
\(696\) 0 0
\(697\) 8.86490e29 1.24088
\(698\) 0 0
\(699\) −1.71786e29 −0.233334
\(700\) 0 0
\(701\) −8.80754e29 −1.16096 −0.580478 0.814276i \(-0.697134\pi\)
−0.580478 + 0.814276i \(0.697134\pi\)
\(702\) 0 0
\(703\) −8.99015e29 −1.15010
\(704\) 0 0
\(705\) 1.22280e28 0.0151835
\(706\) 0 0
\(707\) −5.26057e28 −0.0634058
\(708\) 0 0
\(709\) −2.79000e29 −0.326452 −0.163226 0.986589i \(-0.552190\pi\)
−0.163226 + 0.986589i \(0.552190\pi\)
\(710\) 0 0
\(711\) 5.88783e28 0.0668843
\(712\) 0 0
\(713\) −3.49186e29 −0.385139
\(714\) 0 0
\(715\) −5.14032e27 −0.00550524
\(716\) 0 0
\(717\) 2.83568e29 0.294921
\(718\) 0 0
\(719\) −1.21404e30 −1.22625 −0.613124 0.789987i \(-0.710087\pi\)
−0.613124 + 0.789987i \(0.710087\pi\)
\(720\) 0 0
\(721\) 7.97913e29 0.782772
\(722\) 0 0
\(723\) 4.64280e29 0.442413
\(724\) 0 0
\(725\) −1.12157e30 −1.03819
\(726\) 0 0
\(727\) 6.54831e29 0.588868 0.294434 0.955672i \(-0.404869\pi\)
0.294434 + 0.955672i \(0.404869\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −1.42204e30 −1.20720
\(732\) 0 0
\(733\) 2.20665e29 0.182029 0.0910147 0.995850i \(-0.470989\pi\)
0.0910147 + 0.995850i \(0.470989\pi\)
\(734\) 0 0
\(735\) −7.83325e28 −0.0627950
\(736\) 0 0
\(737\) −2.29743e29 −0.178992
\(738\) 0 0
\(739\) 4.07297e29 0.308422 0.154211 0.988038i \(-0.450717\pi\)
0.154211 + 0.988038i \(0.450717\pi\)
\(740\) 0 0
\(741\) 1.95463e29 0.143871
\(742\) 0 0
\(743\) −3.97218e29 −0.284214 −0.142107 0.989851i \(-0.545388\pi\)
−0.142107 + 0.989851i \(0.545388\pi\)
\(744\) 0 0
\(745\) −2.32509e29 −0.161733
\(746\) 0 0
\(747\) 5.95152e29 0.402496
\(748\) 0 0
\(749\) −5.92500e28 −0.0389610
\(750\) 0 0
\(751\) 1.86890e30 1.19500 0.597498 0.801871i \(-0.296162\pi\)
0.597498 + 0.801871i \(0.296162\pi\)
\(752\) 0 0
\(753\) 1.17817e30 0.732592
\(754\) 0 0
\(755\) −3.45341e29 −0.208836
\(756\) 0 0
\(757\) −2.99103e30 −1.75919 −0.879597 0.475720i \(-0.842188\pi\)
−0.879597 + 0.475720i \(0.842188\pi\)
\(758\) 0 0
\(759\) −1.34236e29 −0.0767945
\(760\) 0 0
\(761\) 1.51341e30 0.842206 0.421103 0.907013i \(-0.361643\pi\)
0.421103 + 0.907013i \(0.361643\pi\)
\(762\) 0 0
\(763\) −8.16896e28 −0.0442241
\(764\) 0 0
\(765\) −9.31410e28 −0.0490564
\(766\) 0 0
\(767\) −6.27325e29 −0.321470
\(768\) 0 0
\(769\) 2.53401e30 1.26352 0.631759 0.775165i \(-0.282333\pi\)
0.631759 + 0.775165i \(0.282333\pi\)
\(770\) 0 0
\(771\) 7.88777e29 0.382722
\(772\) 0 0
\(773\) −1.69545e30 −0.800571 −0.400285 0.916390i \(-0.631089\pi\)
−0.400285 + 0.916390i \(0.631089\pi\)
\(774\) 0 0
\(775\) 1.04637e30 0.480857
\(776\) 0 0
\(777\) 6.60424e29 0.295394
\(778\) 0 0
\(779\) −3.01396e30 −1.31218
\(780\) 0 0
\(781\) 3.85804e29 0.163505
\(782\) 0 0
\(783\) 4.94301e29 0.203936
\(784\) 0 0
\(785\) 1.35769e29 0.0545342
\(786\) 0 0
\(787\) 1.87332e30 0.732616 0.366308 0.930494i \(-0.380622\pi\)
0.366308 + 0.930494i \(0.380622\pi\)
\(788\) 0 0
\(789\) 3.44632e29 0.131234
\(790\) 0 0
\(791\) 1.54084e30 0.571353
\(792\) 0 0
\(793\) −8.13429e29 −0.293732
\(794\) 0 0
\(795\) 7.99608e28 0.0281204
\(796\) 0 0
\(797\) −7.79664e29 −0.267051 −0.133526 0.991045i \(-0.542630\pi\)
−0.133526 + 0.991045i \(0.542630\pi\)
\(798\) 0 0
\(799\) −5.71944e29 −0.190815
\(800\) 0 0
\(801\) −1.08994e30 −0.354211
\(802\) 0 0
\(803\) −1.96436e29 −0.0621880
\(804\) 0 0
\(805\) −1.76144e29 −0.0543264
\(806\) 0 0
\(807\) 3.16088e30 0.949807
\(808\) 0 0
\(809\) 8.82262e29 0.258308 0.129154 0.991625i \(-0.458774\pi\)
0.129154 + 0.991625i \(0.458774\pi\)
\(810\) 0 0
\(811\) 2.06044e30 0.587815 0.293907 0.955834i \(-0.405044\pi\)
0.293907 + 0.955834i \(0.405044\pi\)
\(812\) 0 0
\(813\) 6.16490e29 0.171386
\(814\) 0 0
\(815\) −8.88392e29 −0.240685
\(816\) 0 0
\(817\) 4.83476e30 1.27657
\(818\) 0 0
\(819\) −1.43589e29 −0.0369521
\(820\) 0 0
\(821\) −1.83846e30 −0.461160 −0.230580 0.973053i \(-0.574062\pi\)
−0.230580 + 0.973053i \(0.574062\pi\)
\(822\) 0 0
\(823\) −7.73766e29 −0.189196 −0.0945979 0.995516i \(-0.530157\pi\)
−0.0945979 + 0.995516i \(0.530157\pi\)
\(824\) 0 0
\(825\) 4.02249e29 0.0958802
\(826\) 0 0
\(827\) 4.59989e30 1.06891 0.534453 0.845198i \(-0.320518\pi\)
0.534453 + 0.845198i \(0.320518\pi\)
\(828\) 0 0
\(829\) 7.93000e30 1.79660 0.898298 0.439386i \(-0.144804\pi\)
0.898298 + 0.439386i \(0.144804\pi\)
\(830\) 0 0
\(831\) 1.85936e30 0.410726
\(832\) 0 0
\(833\) 3.66386e30 0.789162
\(834\) 0 0
\(835\) −8.30202e29 −0.174372
\(836\) 0 0
\(837\) −4.61159e29 −0.0944569
\(838\) 0 0
\(839\) −4.84033e30 −0.966884 −0.483442 0.875376i \(-0.660614\pi\)
−0.483442 + 0.875376i \(0.660614\pi\)
\(840\) 0 0
\(841\) 6.30944e29 0.122923
\(842\) 0 0
\(843\) −6.84300e29 −0.130034
\(844\) 0 0
\(845\) −7.28400e29 −0.135013
\(846\) 0 0
\(847\) 2.61129e30 0.472149
\(848\) 0 0
\(849\) 2.83671e30 0.500362
\(850\) 0 0
\(851\) −4.79937e30 −0.825893
\(852\) 0 0
\(853\) 2.96903e30 0.498483 0.249242 0.968441i \(-0.419819\pi\)
0.249242 + 0.968441i \(0.419819\pi\)
\(854\) 0 0
\(855\) 3.16669e29 0.0518753
\(856\) 0 0
\(857\) −3.70009e30 −0.591444 −0.295722 0.955274i \(-0.595560\pi\)
−0.295722 + 0.955274i \(0.595560\pi\)
\(858\) 0 0
\(859\) −7.75385e29 −0.120945 −0.0604726 0.998170i \(-0.519261\pi\)
−0.0604726 + 0.998170i \(0.519261\pi\)
\(860\) 0 0
\(861\) 2.21408e30 0.337024
\(862\) 0 0
\(863\) 1.29544e31 1.92443 0.962217 0.272283i \(-0.0877786\pi\)
0.962217 + 0.272283i \(0.0877786\pi\)
\(864\) 0 0
\(865\) −7.10610e29 −0.103030
\(866\) 0 0
\(867\) 2.76685e29 0.0391548
\(868\) 0 0
\(869\) −2.46231e29 −0.0340122
\(870\) 0 0
\(871\) 1.78589e30 0.240805
\(872\) 0 0
\(873\) 3.30901e30 0.435560
\(874\) 0 0
\(875\) 1.06659e30 0.137060
\(876\) 0 0
\(877\) −1.57355e31 −1.97417 −0.987083 0.160207i \(-0.948784\pi\)
−0.987083 + 0.160207i \(0.948784\pi\)
\(878\) 0 0
\(879\) −4.70452e30 −0.576278
\(880\) 0 0
\(881\) −1.47526e31 −1.76450 −0.882252 0.470778i \(-0.843973\pi\)
−0.882252 + 0.470778i \(0.843973\pi\)
\(882\) 0 0
\(883\) 5.64453e30 0.659235 0.329617 0.944115i \(-0.393080\pi\)
0.329617 + 0.944115i \(0.393080\pi\)
\(884\) 0 0
\(885\) −1.01633e30 −0.115912
\(886\) 0 0
\(887\) −5.89300e30 −0.656354 −0.328177 0.944616i \(-0.606434\pi\)
−0.328177 + 0.944616i \(0.606434\pi\)
\(888\) 0 0
\(889\) 6.04218e29 0.0657242
\(890\) 0 0
\(891\) −1.77281e29 −0.0188342
\(892\) 0 0
\(893\) 1.94454e30 0.201780
\(894\) 0 0
\(895\) 4.01653e28 0.00407107
\(896\) 0 0
\(897\) 1.04347e30 0.103314
\(898\) 0 0
\(899\) −5.37734e30 −0.520104
\(900\) 0 0
\(901\) −3.74002e30 −0.353397
\(902\) 0 0
\(903\) −3.55166e30 −0.327876
\(904\) 0 0
\(905\) −2.72226e30 −0.245538
\(906\) 0 0
\(907\) −8.32782e30 −0.733931 −0.366965 0.930235i \(-0.619603\pi\)
−0.366965 + 0.930235i \(0.619603\pi\)
\(908\) 0 0
\(909\) 5.04879e29 0.0434777
\(910\) 0 0
\(911\) −1.08086e31 −0.909548 −0.454774 0.890607i \(-0.650280\pi\)
−0.454774 + 0.890607i \(0.650280\pi\)
\(912\) 0 0
\(913\) −2.48894e30 −0.204678
\(914\) 0 0
\(915\) −1.31783e30 −0.105911
\(916\) 0 0
\(917\) 2.32755e30 0.182819
\(918\) 0 0
\(919\) −6.55205e30 −0.502996 −0.251498 0.967858i \(-0.580923\pi\)
−0.251498 + 0.967858i \(0.580923\pi\)
\(920\) 0 0
\(921\) −8.94582e30 −0.671265
\(922\) 0 0
\(923\) −2.99902e30 −0.219969
\(924\) 0 0
\(925\) 1.43817e31 1.03115
\(926\) 0 0
\(927\) −7.65791e30 −0.536752
\(928\) 0 0
\(929\) 1.04036e31 0.712883 0.356442 0.934318i \(-0.383990\pi\)
0.356442 + 0.934318i \(0.383990\pi\)
\(930\) 0 0
\(931\) −1.24567e31 −0.834509
\(932\) 0 0
\(933\) −8.34989e30 −0.546919
\(934\) 0 0
\(935\) 3.89518e29 0.0249463
\(936\) 0 0
\(937\) −2.09831e31 −1.31402 −0.657012 0.753880i \(-0.728180\pi\)
−0.657012 + 0.753880i \(0.728180\pi\)
\(938\) 0 0
\(939\) 2.95007e30 0.180653
\(940\) 0 0
\(941\) 2.20967e31 1.32323 0.661617 0.749842i \(-0.269870\pi\)
0.661617 + 0.749842i \(0.269870\pi\)
\(942\) 0 0
\(943\) −1.60900e31 −0.942286
\(944\) 0 0
\(945\) −2.32628e29 −0.0133238
\(946\) 0 0
\(947\) 2.59604e31 1.45424 0.727121 0.686509i \(-0.240858\pi\)
0.727121 + 0.686509i \(0.240858\pi\)
\(948\) 0 0
\(949\) 1.52698e30 0.0836637
\(950\) 0 0
\(951\) 1.07194e31 0.574483
\(952\) 0 0
\(953\) −2.94729e31 −1.54507 −0.772535 0.634973i \(-0.781011\pi\)
−0.772535 + 0.634973i \(0.781011\pi\)
\(954\) 0 0
\(955\) −4.16613e30 −0.213648
\(956\) 0 0
\(957\) −2.06718e30 −0.103706
\(958\) 0 0
\(959\) 7.20482e30 0.353612
\(960\) 0 0
\(961\) −1.58087e31 −0.759103
\(962\) 0 0
\(963\) 5.68648e29 0.0267158
\(964\) 0 0
\(965\) −6.65436e29 −0.0305893
\(966\) 0 0
\(967\) −6.98077e30 −0.313997 −0.156998 0.987599i \(-0.550182\pi\)
−0.156998 + 0.987599i \(0.550182\pi\)
\(968\) 0 0
\(969\) −1.48116e31 −0.651931
\(970\) 0 0
\(971\) 1.30234e30 0.0560946 0.0280473 0.999607i \(-0.491071\pi\)
0.0280473 + 0.999607i \(0.491071\pi\)
\(972\) 0 0
\(973\) −1.88970e30 −0.0796541
\(974\) 0 0
\(975\) −3.12685e30 −0.128991
\(976\) 0 0
\(977\) 3.06599e31 1.23788 0.618939 0.785439i \(-0.287563\pi\)
0.618939 + 0.785439i \(0.287563\pi\)
\(978\) 0 0
\(979\) 4.55817e30 0.180124
\(980\) 0 0
\(981\) 7.84011e29 0.0303248
\(982\) 0 0
\(983\) −8.31325e30 −0.314745 −0.157373 0.987539i \(-0.550302\pi\)
−0.157373 + 0.987539i \(0.550302\pi\)
\(984\) 0 0
\(985\) 4.44398e30 0.164699
\(986\) 0 0
\(987\) −1.42848e30 −0.0518255
\(988\) 0 0
\(989\) 2.58102e31 0.916708
\(990\) 0 0
\(991\) 1.32568e31 0.460962 0.230481 0.973077i \(-0.425970\pi\)
0.230481 + 0.973077i \(0.425970\pi\)
\(992\) 0 0
\(993\) −8.50837e30 −0.289654
\(994\) 0 0
\(995\) −2.31958e30 −0.0773154
\(996\) 0 0
\(997\) 3.42048e31 1.11632 0.558159 0.829734i \(-0.311508\pi\)
0.558159 + 0.829734i \(0.311508\pi\)
\(998\) 0 0
\(999\) −6.33837e30 −0.202554
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 48.22.a.e.1.1 1
4.3 odd 2 3.22.a.a.1.1 1
12.11 even 2 9.22.a.d.1.1 1
20.3 even 4 75.22.b.a.49.2 2
20.7 even 4 75.22.b.a.49.1 2
20.19 odd 2 75.22.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 1 4.3 odd 2
9.22.a.d.1.1 1 12.11 even 2
48.22.a.e.1.1 1 1.1 even 1 trivial
75.22.a.c.1.1 1 20.19 odd 2
75.22.b.a.49.1 2 20.7 even 4
75.22.b.a.49.2 2 20.3 even 4