Properties

Label 177.12.a.b.1.18
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+32.2053 q^{2} +243.000 q^{3} -1010.82 q^{4} -5509.44 q^{5} +7825.90 q^{6} -36737.1 q^{7} -98510.2 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+32.2053 q^{2} +243.000 q^{3} -1010.82 q^{4} -5509.44 q^{5} +7825.90 q^{6} -36737.1 q^{7} -98510.2 q^{8} +59049.0 q^{9} -177433. q^{10} +785481. q^{11} -245628. q^{12} +1.64819e6 q^{13} -1.18313e6 q^{14} -1.33879e6 q^{15} -1.10240e6 q^{16} +14272.3 q^{17} +1.90169e6 q^{18} -8.87395e6 q^{19} +5.56903e6 q^{20} -8.92711e6 q^{21} +2.52967e7 q^{22} +2.97072e7 q^{23} -2.39380e7 q^{24} -1.84742e7 q^{25} +5.30806e7 q^{26} +1.43489e7 q^{27} +3.71344e7 q^{28} +6.76779e7 q^{29} -4.31163e7 q^{30} +6.05970e7 q^{31} +1.66246e8 q^{32} +1.90872e8 q^{33} +459646. q^{34} +2.02401e8 q^{35} -5.96877e7 q^{36} +1.57018e8 q^{37} -2.85789e8 q^{38} +4.00511e8 q^{39} +5.42736e8 q^{40} -1.38293e9 q^{41} -2.87501e8 q^{42} -1.09612e9 q^{43} -7.93977e8 q^{44} -3.25327e8 q^{45} +9.56729e8 q^{46} +1.74265e9 q^{47} -2.67884e8 q^{48} -6.27714e8 q^{49} -5.94967e8 q^{50} +3.46818e6 q^{51} -1.66602e9 q^{52} +3.03335e8 q^{53} +4.62111e8 q^{54} -4.32756e9 q^{55} +3.61898e9 q^{56} -2.15637e9 q^{57} +2.17959e9 q^{58} -7.14924e8 q^{59} +1.35327e9 q^{60} +1.62524e9 q^{61} +1.95155e9 q^{62} -2.16929e9 q^{63} +7.61172e9 q^{64} -9.08062e9 q^{65} +6.14710e9 q^{66} -1.24549e10 q^{67} -1.44267e7 q^{68} +7.21884e9 q^{69} +6.51839e9 q^{70} -8.29797e9 q^{71} -5.81693e9 q^{72} -2.93246e10 q^{73} +5.05681e9 q^{74} -4.48923e9 q^{75} +8.96993e9 q^{76} -2.88563e10 q^{77} +1.28986e10 q^{78} -1.88795e10 q^{79} +6.07363e9 q^{80} +3.48678e9 q^{81} -4.45377e10 q^{82} +2.23593e10 q^{83} +9.02366e9 q^{84} -7.86326e7 q^{85} -3.53009e10 q^{86} +1.64457e10 q^{87} -7.73779e10 q^{88} -9.79467e10 q^{89} -1.04773e10 q^{90} -6.05498e10 q^{91} -3.00285e10 q^{92} +1.47251e10 q^{93} +5.61227e10 q^{94} +4.88905e10 q^{95} +4.03977e10 q^{96} +2.14426e10 q^{97} -2.02157e10 q^{98} +4.63819e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.2053 0.711644 0.355822 0.934554i \(-0.384201\pi\)
0.355822 + 0.934554i \(0.384201\pi\)
\(3\) 243.000 0.577350
\(4\) −1010.82 −0.493562
\(5\) −5509.44 −0.788447 −0.394224 0.919015i \(-0.628986\pi\)
−0.394224 + 0.919015i \(0.628986\pi\)
\(6\) 7825.90 0.410868
\(7\) −36737.1 −0.826162 −0.413081 0.910694i \(-0.635547\pi\)
−0.413081 + 0.910694i \(0.635547\pi\)
\(8\) −98510.2 −1.06289
\(9\) 59049.0 0.333333
\(10\) −177433. −0.561094
\(11\) 785481. 1.47054 0.735269 0.677776i \(-0.237056\pi\)
0.735269 + 0.677776i \(0.237056\pi\)
\(12\) −245628. −0.284958
\(13\) 1.64819e6 1.23117 0.615587 0.788069i \(-0.288919\pi\)
0.615587 + 0.788069i \(0.288919\pi\)
\(14\) −1.18313e6 −0.587934
\(15\) −1.33879e6 −0.455210
\(16\) −1.10240e6 −0.262834
\(17\) 14272.3 0.00243796 0.00121898 0.999999i \(-0.499612\pi\)
0.00121898 + 0.999999i \(0.499612\pi\)
\(18\) 1.90169e6 0.237215
\(19\) −8.87395e6 −0.822190 −0.411095 0.911593i \(-0.634853\pi\)
−0.411095 + 0.911593i \(0.634853\pi\)
\(20\) 5.56903e6 0.389148
\(21\) −8.92711e6 −0.476985
\(22\) 2.52967e7 1.04650
\(23\) 2.97072e7 0.962405 0.481203 0.876609i \(-0.340200\pi\)
0.481203 + 0.876609i \(0.340200\pi\)
\(24\) −2.39380e7 −0.613657
\(25\) −1.84742e7 −0.378351
\(26\) 5.30806e7 0.876158
\(27\) 1.43489e7 0.192450
\(28\) 3.71344e7 0.407763
\(29\) 6.76779e7 0.612714 0.306357 0.951917i \(-0.400890\pi\)
0.306357 + 0.951917i \(0.400890\pi\)
\(30\) −4.31163e7 −0.323948
\(31\) 6.05970e7 0.380156 0.190078 0.981769i \(-0.439126\pi\)
0.190078 + 0.981769i \(0.439126\pi\)
\(32\) 1.66246e8 0.875841
\(33\) 1.90872e8 0.849015
\(34\) 459646. 0.00173496
\(35\) 2.02401e8 0.651385
\(36\) −5.96877e7 −0.164521
\(37\) 1.57018e8 0.372254 0.186127 0.982526i \(-0.440406\pi\)
0.186127 + 0.982526i \(0.440406\pi\)
\(38\) −2.85789e8 −0.585107
\(39\) 4.00511e8 0.710819
\(40\) 5.42736e8 0.838029
\(41\) −1.38293e9 −1.86419 −0.932093 0.362220i \(-0.882019\pi\)
−0.932093 + 0.362220i \(0.882019\pi\)
\(42\) −2.87501e8 −0.339444
\(43\) −1.09612e9 −1.13706 −0.568528 0.822664i \(-0.692487\pi\)
−0.568528 + 0.822664i \(0.692487\pi\)
\(44\) −7.93977e8 −0.725802
\(45\) −3.25327e8 −0.262816
\(46\) 9.56729e8 0.684890
\(47\) 1.74265e9 1.10834 0.554170 0.832404i \(-0.313036\pi\)
0.554170 + 0.832404i \(0.313036\pi\)
\(48\) −2.67884e8 −0.151747
\(49\) −6.27714e8 −0.317456
\(50\) −5.94967e8 −0.269252
\(51\) 3.46818e6 0.00140756
\(52\) −1.66602e9 −0.607661
\(53\) 3.03335e8 0.0996336 0.0498168 0.998758i \(-0.484136\pi\)
0.0498168 + 0.998758i \(0.484136\pi\)
\(54\) 4.62111e8 0.136956
\(55\) −4.32756e9 −1.15944
\(56\) 3.61898e9 0.878116
\(57\) −2.15637e9 −0.474691
\(58\) 2.17959e9 0.436034
\(59\) −7.14924e8 −0.130189
\(60\) 1.35327e9 0.224675
\(61\) 1.62524e9 0.246378 0.123189 0.992383i \(-0.460688\pi\)
0.123189 + 0.992383i \(0.460688\pi\)
\(62\) 1.95155e9 0.270536
\(63\) −2.16929e9 −0.275387
\(64\) 7.61172e9 0.886121
\(65\) −9.08062e9 −0.970716
\(66\) 6.14710e9 0.604197
\(67\) −1.24549e10 −1.12702 −0.563508 0.826111i \(-0.690549\pi\)
−0.563508 + 0.826111i \(0.690549\pi\)
\(68\) −1.44267e7 −0.00120328
\(69\) 7.21884e9 0.555645
\(70\) 6.51839e9 0.463555
\(71\) −8.29797e9 −0.545822 −0.272911 0.962039i \(-0.587986\pi\)
−0.272911 + 0.962039i \(0.587986\pi\)
\(72\) −5.81693e9 −0.354295
\(73\) −2.93246e10 −1.65560 −0.827801 0.561021i \(-0.810409\pi\)
−0.827801 + 0.561021i \(0.810409\pi\)
\(74\) 5.05681e9 0.264912
\(75\) −4.48923e9 −0.218441
\(76\) 8.96993e9 0.405802
\(77\) −2.88563e10 −1.21490
\(78\) 1.28986e10 0.505850
\(79\) −1.88795e10 −0.690307 −0.345154 0.938546i \(-0.612173\pi\)
−0.345154 + 0.938546i \(0.612173\pi\)
\(80\) 6.07363e9 0.207230
\(81\) 3.48678e9 0.111111
\(82\) −4.45377e10 −1.32664
\(83\) 2.23593e10 0.623057 0.311529 0.950237i \(-0.399159\pi\)
0.311529 + 0.950237i \(0.399159\pi\)
\(84\) 9.02366e9 0.235422
\(85\) −7.86326e7 −0.00192220
\(86\) −3.53009e10 −0.809179
\(87\) 1.64457e10 0.353750
\(88\) −7.73779e10 −1.56301
\(89\) −9.79467e10 −1.85928 −0.929640 0.368470i \(-0.879882\pi\)
−0.929640 + 0.368470i \(0.879882\pi\)
\(90\) −1.04773e10 −0.187031
\(91\) −6.05498e10 −1.01715
\(92\) −3.00285e10 −0.475007
\(93\) 1.47251e10 0.219483
\(94\) 5.61227e10 0.788743
\(95\) 4.88905e10 0.648253
\(96\) 4.03977e10 0.505667
\(97\) 2.14426e10 0.253533 0.126766 0.991933i \(-0.459540\pi\)
0.126766 + 0.991933i \(0.459540\pi\)
\(98\) −2.02157e10 −0.225916
\(99\) 4.63819e10 0.490179
\(100\) 1.86740e10 0.186740
\(101\) −1.43434e11 −1.35795 −0.678977 0.734159i \(-0.737576\pi\)
−0.678977 + 0.734159i \(0.737576\pi\)
\(102\) 1.11694e8 0.00100168
\(103\) −2.08419e10 −0.177147 −0.0885735 0.996070i \(-0.528231\pi\)
−0.0885735 + 0.996070i \(0.528231\pi\)
\(104\) −1.62364e11 −1.30860
\(105\) 4.91834e10 0.376077
\(106\) 9.76902e9 0.0709037
\(107\) 1.99719e11 1.37660 0.688302 0.725424i \(-0.258356\pi\)
0.688302 + 0.725424i \(0.258356\pi\)
\(108\) −1.45041e10 −0.0949861
\(109\) 8.05057e10 0.501165 0.250583 0.968095i \(-0.419378\pi\)
0.250583 + 0.968095i \(0.419378\pi\)
\(110\) −1.39371e11 −0.825109
\(111\) 3.81553e10 0.214921
\(112\) 4.04991e10 0.217143
\(113\) −9.30445e10 −0.475072 −0.237536 0.971379i \(-0.576340\pi\)
−0.237536 + 0.971379i \(0.576340\pi\)
\(114\) −6.94466e10 −0.337811
\(115\) −1.63670e11 −0.758805
\(116\) −6.84099e10 −0.302412
\(117\) 9.73241e10 0.410391
\(118\) −2.30244e10 −0.0926482
\(119\) −5.24324e8 −0.00201415
\(120\) 1.31885e11 0.483836
\(121\) 3.31669e11 1.16248
\(122\) 5.23413e10 0.175334
\(123\) −3.36052e11 −1.07629
\(124\) −6.12524e10 −0.187631
\(125\) 3.70798e11 1.08676
\(126\) −6.98627e10 −0.195978
\(127\) 2.11371e11 0.567709 0.283854 0.958867i \(-0.408387\pi\)
0.283854 + 0.958867i \(0.408387\pi\)
\(128\) −9.53329e10 −0.245238
\(129\) −2.66357e11 −0.656479
\(130\) −2.92445e11 −0.690804
\(131\) −4.56049e8 −0.00103281 −0.000516404 1.00000i \(-0.500164\pi\)
−0.000516404 1.00000i \(0.500164\pi\)
\(132\) −1.92936e11 −0.419042
\(133\) 3.26003e11 0.679262
\(134\) −4.01116e11 −0.802035
\(135\) −7.90545e10 −0.151737
\(136\) −1.40597e9 −0.00259127
\(137\) 7.80465e11 1.38163 0.690813 0.723033i \(-0.257253\pi\)
0.690813 + 0.723033i \(0.257253\pi\)
\(138\) 2.32485e11 0.395422
\(139\) −2.07582e11 −0.339319 −0.169659 0.985503i \(-0.554267\pi\)
−0.169659 + 0.985503i \(0.554267\pi\)
\(140\) −2.04590e11 −0.321499
\(141\) 4.23465e11 0.639900
\(142\) −2.67239e11 −0.388431
\(143\) 1.29462e12 1.81049
\(144\) −6.50959e10 −0.0876113
\(145\) −3.72867e11 −0.483092
\(146\) −9.44408e11 −1.17820
\(147\) −1.52534e11 −0.183283
\(148\) −1.58716e11 −0.183731
\(149\) −1.52018e12 −1.69578 −0.847889 0.530173i \(-0.822127\pi\)
−0.847889 + 0.530173i \(0.822127\pi\)
\(150\) −1.44577e11 −0.155452
\(151\) −8.72188e11 −0.904142 −0.452071 0.891982i \(-0.649315\pi\)
−0.452071 + 0.891982i \(0.649315\pi\)
\(152\) 8.74174e11 0.873893
\(153\) 8.42768e8 0.000812653 0
\(154\) −9.29327e11 −0.864579
\(155\) −3.33856e11 −0.299733
\(156\) −4.04843e11 −0.350833
\(157\) −1.28318e12 −1.07359 −0.536796 0.843712i \(-0.680366\pi\)
−0.536796 + 0.843712i \(0.680366\pi\)
\(158\) −6.08022e11 −0.491253
\(159\) 7.37105e10 0.0575235
\(160\) −9.15920e11 −0.690554
\(161\) −1.09135e12 −0.795103
\(162\) 1.12293e11 0.0790716
\(163\) −1.84584e12 −1.25650 −0.628250 0.778012i \(-0.716228\pi\)
−0.628250 + 0.778012i \(0.716228\pi\)
\(164\) 1.39789e12 0.920092
\(165\) −1.05160e12 −0.669404
\(166\) 7.20088e11 0.443395
\(167\) −2.58949e11 −0.154267 −0.0771335 0.997021i \(-0.524577\pi\)
−0.0771335 + 0.997021i \(0.524577\pi\)
\(168\) 8.79412e11 0.506980
\(169\) 9.24379e11 0.515790
\(170\) −2.53239e9 −0.00136792
\(171\) −5.23998e11 −0.274063
\(172\) 1.10798e12 0.561208
\(173\) 3.88764e12 1.90736 0.953680 0.300822i \(-0.0972611\pi\)
0.953680 + 0.300822i \(0.0972611\pi\)
\(174\) 5.29640e11 0.251745
\(175\) 6.78688e11 0.312580
\(176\) −8.65918e11 −0.386507
\(177\) −1.73727e11 −0.0751646
\(178\) −3.15441e12 −1.32315
\(179\) 1.24427e11 0.0506083 0.0253041 0.999680i \(-0.491945\pi\)
0.0253041 + 0.999680i \(0.491945\pi\)
\(180\) 3.28846e11 0.129716
\(181\) −1.21291e12 −0.464085 −0.232043 0.972706i \(-0.574541\pi\)
−0.232043 + 0.972706i \(0.574541\pi\)
\(182\) −1.95003e12 −0.723849
\(183\) 3.94933e11 0.142247
\(184\) −2.92646e12 −1.02293
\(185\) −8.65080e11 −0.293502
\(186\) 4.74226e11 0.156194
\(187\) 1.12107e10 0.00358511
\(188\) −1.76150e12 −0.547035
\(189\) −5.27137e11 −0.158995
\(190\) 1.57453e12 0.461326
\(191\) −2.18790e12 −0.622793 −0.311397 0.950280i \(-0.600797\pi\)
−0.311397 + 0.950280i \(0.600797\pi\)
\(192\) 1.84965e12 0.511602
\(193\) 2.69207e12 0.723639 0.361819 0.932248i \(-0.382156\pi\)
0.361819 + 0.932248i \(0.382156\pi\)
\(194\) 6.90568e11 0.180425
\(195\) −2.20659e12 −0.560443
\(196\) 6.34503e11 0.156684
\(197\) −4.91992e12 −1.18139 −0.590696 0.806895i \(-0.701147\pi\)
−0.590696 + 0.806895i \(0.701147\pi\)
\(198\) 1.49374e12 0.348833
\(199\) 6.24859e12 1.41935 0.709676 0.704528i \(-0.248841\pi\)
0.709676 + 0.704528i \(0.248841\pi\)
\(200\) 1.81990e12 0.402144
\(201\) −3.02655e12 −0.650683
\(202\) −4.61935e12 −0.966380
\(203\) −2.48629e12 −0.506201
\(204\) −3.50569e9 −0.000694716 0
\(205\) 7.61917e12 1.46981
\(206\) −6.71222e11 −0.126066
\(207\) 1.75418e12 0.320802
\(208\) −1.81698e12 −0.323594
\(209\) −6.97032e12 −1.20906
\(210\) 1.58397e12 0.267633
\(211\) −6.22887e12 −1.02531 −0.512656 0.858594i \(-0.671338\pi\)
−0.512656 + 0.858594i \(0.671338\pi\)
\(212\) −3.06616e11 −0.0491754
\(213\) −2.01641e12 −0.315130
\(214\) 6.43202e12 0.979652
\(215\) 6.03901e12 0.896508
\(216\) −1.41351e12 −0.204552
\(217\) −2.22616e12 −0.314071
\(218\) 2.59271e12 0.356651
\(219\) −7.12587e12 −0.955863
\(220\) 4.37437e12 0.572256
\(221\) 2.35236e10 0.00300155
\(222\) 1.22880e12 0.152947
\(223\) −1.60743e13 −1.95188 −0.975942 0.218031i \(-0.930037\pi\)
−0.975942 + 0.218031i \(0.930037\pi\)
\(224\) −6.10738e12 −0.723587
\(225\) −1.09088e12 −0.126117
\(226\) −2.99653e12 −0.338082
\(227\) 1.19496e13 1.31586 0.657931 0.753078i \(-0.271432\pi\)
0.657931 + 0.753078i \(0.271432\pi\)
\(228\) 2.17969e12 0.234290
\(229\) −6.43549e12 −0.675284 −0.337642 0.941275i \(-0.609629\pi\)
−0.337642 + 0.941275i \(0.609629\pi\)
\(230\) −5.27104e12 −0.540000
\(231\) −7.01208e12 −0.701424
\(232\) −6.66696e12 −0.651244
\(233\) 4.91366e12 0.468757 0.234379 0.972145i \(-0.424694\pi\)
0.234379 + 0.972145i \(0.424694\pi\)
\(234\) 3.13436e12 0.292053
\(235\) −9.60104e12 −0.873867
\(236\) 7.22657e11 0.0642564
\(237\) −4.58773e12 −0.398549
\(238\) −1.68860e10 −0.00143336
\(239\) 2.43640e12 0.202097 0.101048 0.994882i \(-0.467780\pi\)
0.101048 + 0.994882i \(0.467780\pi\)
\(240\) 1.47589e12 0.119645
\(241\) −1.61679e13 −1.28103 −0.640517 0.767944i \(-0.721280\pi\)
−0.640517 + 0.767944i \(0.721280\pi\)
\(242\) 1.06815e13 0.827273
\(243\) 8.47289e11 0.0641500
\(244\) −1.64281e12 −0.121603
\(245\) 3.45835e12 0.250297
\(246\) −1.08227e13 −0.765934
\(247\) −1.46260e13 −1.01226
\(248\) −5.96943e12 −0.404062
\(249\) 5.43330e12 0.359722
\(250\) 1.19417e13 0.773384
\(251\) −3.03619e13 −1.92364 −0.961821 0.273680i \(-0.911759\pi\)
−0.961821 + 0.273680i \(0.911759\pi\)
\(252\) 2.19275e12 0.135921
\(253\) 2.33344e13 1.41525
\(254\) 6.80729e12 0.404007
\(255\) −1.91077e10 −0.00110978
\(256\) −1.86590e13 −1.06064
\(257\) 1.34863e13 0.750344 0.375172 0.926955i \(-0.377584\pi\)
0.375172 + 0.926955i \(0.377584\pi\)
\(258\) −8.57812e12 −0.467180
\(259\) −5.76837e12 −0.307542
\(260\) 9.17883e12 0.479109
\(261\) 3.99631e12 0.204238
\(262\) −1.46872e10 −0.000734992 0
\(263\) 3.69926e13 1.81283 0.906417 0.422384i \(-0.138807\pi\)
0.906417 + 0.422384i \(0.138807\pi\)
\(264\) −1.88028e13 −0.902406
\(265\) −1.67121e12 −0.0785558
\(266\) 1.04990e13 0.483393
\(267\) −2.38010e13 −1.07346
\(268\) 1.25896e13 0.556253
\(269\) 1.48939e13 0.644721 0.322361 0.946617i \(-0.395524\pi\)
0.322361 + 0.946617i \(0.395524\pi\)
\(270\) −2.54598e12 −0.107983
\(271\) −3.11925e13 −1.29634 −0.648171 0.761495i \(-0.724466\pi\)
−0.648171 + 0.761495i \(0.724466\pi\)
\(272\) −1.57339e10 −0.000640778 0
\(273\) −1.47136e13 −0.587252
\(274\) 2.51352e13 0.983226
\(275\) −1.45111e13 −0.556380
\(276\) −7.29692e12 −0.274245
\(277\) 2.20689e12 0.0813095 0.0406547 0.999173i \(-0.487056\pi\)
0.0406547 + 0.999173i \(0.487056\pi\)
\(278\) −6.68524e12 −0.241474
\(279\) 3.57819e12 0.126719
\(280\) −1.99385e13 −0.692348
\(281\) −1.29643e13 −0.441433 −0.220717 0.975338i \(-0.570840\pi\)
−0.220717 + 0.975338i \(0.570840\pi\)
\(282\) 1.36378e13 0.455381
\(283\) 8.26712e12 0.270725 0.135363 0.990796i \(-0.456780\pi\)
0.135363 + 0.990796i \(0.456780\pi\)
\(284\) 8.38772e12 0.269397
\(285\) 1.18804e13 0.374269
\(286\) 4.16938e13 1.28842
\(287\) 5.08048e13 1.54012
\(288\) 9.81664e12 0.291947
\(289\) −3.42717e13 −0.999994
\(290\) −1.20083e13 −0.343790
\(291\) 5.21056e12 0.146377
\(292\) 2.96418e13 0.817143
\(293\) −5.87683e13 −1.58990 −0.794952 0.606672i \(-0.792504\pi\)
−0.794952 + 0.606672i \(0.792504\pi\)
\(294\) −4.91242e12 −0.130432
\(295\) 3.93883e12 0.102647
\(296\) −1.54679e13 −0.395663
\(297\) 1.12708e13 0.283005
\(298\) −4.89578e13 −1.20679
\(299\) 4.89631e13 1.18489
\(300\) 4.53778e12 0.107814
\(301\) 4.02682e13 0.939392
\(302\) −2.80891e13 −0.643428
\(303\) −3.48545e13 −0.784015
\(304\) 9.78268e12 0.216099
\(305\) −8.95415e12 −0.194256
\(306\) 2.71416e10 0.000578320 0
\(307\) −2.77624e13 −0.581026 −0.290513 0.956871i \(-0.593826\pi\)
−0.290513 + 0.956871i \(0.593826\pi\)
\(308\) 2.91684e13 0.599630
\(309\) −5.06459e12 −0.102276
\(310\) −1.07519e13 −0.213303
\(311\) −7.79725e13 −1.51971 −0.759853 0.650095i \(-0.774729\pi\)
−0.759853 + 0.650095i \(0.774729\pi\)
\(312\) −3.94544e13 −0.755519
\(313\) 1.06582e13 0.200535 0.100268 0.994961i \(-0.468030\pi\)
0.100268 + 0.994961i \(0.468030\pi\)
\(314\) −4.13252e13 −0.764016
\(315\) 1.19516e13 0.217128
\(316\) 1.90837e13 0.340710
\(317\) 3.75125e13 0.658189 0.329094 0.944297i \(-0.393257\pi\)
0.329094 + 0.944297i \(0.393257\pi\)
\(318\) 2.37387e12 0.0409362
\(319\) 5.31597e13 0.901019
\(320\) −4.19363e13 −0.698659
\(321\) 4.85318e13 0.794783
\(322\) −3.51474e13 −0.565830
\(323\) −1.26652e11 −0.00200446
\(324\) −3.52450e12 −0.0548403
\(325\) −3.04490e13 −0.465816
\(326\) −5.94459e13 −0.894181
\(327\) 1.95629e13 0.289348
\(328\) 1.36233e14 1.98141
\(329\) −6.40200e13 −0.915668
\(330\) −3.38671e13 −0.476377
\(331\) 4.84161e13 0.669785 0.334893 0.942256i \(-0.391300\pi\)
0.334893 + 0.942256i \(0.391300\pi\)
\(332\) −2.26011e13 −0.307518
\(333\) 9.27174e12 0.124085
\(334\) −8.33953e12 −0.109783
\(335\) 6.86197e13 0.888593
\(336\) 9.84129e12 0.125368
\(337\) 1.23301e13 0.154526 0.0772631 0.997011i \(-0.475382\pi\)
0.0772631 + 0.997011i \(0.475382\pi\)
\(338\) 2.97699e13 0.367059
\(339\) −2.26098e13 −0.274283
\(340\) 7.94831e10 0.000948726 0
\(341\) 4.75978e13 0.559034
\(342\) −1.68755e13 −0.195036
\(343\) 9.57016e13 1.08843
\(344\) 1.07979e14 1.20856
\(345\) −3.97718e13 −0.438097
\(346\) 1.25203e14 1.35736
\(347\) 3.03658e13 0.324021 0.162011 0.986789i \(-0.448202\pi\)
0.162011 + 0.986789i \(0.448202\pi\)
\(348\) −1.66236e13 −0.174598
\(349\) 1.22411e13 0.126555 0.0632775 0.997996i \(-0.479845\pi\)
0.0632775 + 0.997996i \(0.479845\pi\)
\(350\) 2.18574e13 0.222446
\(351\) 2.36498e13 0.236940
\(352\) 1.30583e14 1.28796
\(353\) −1.60954e13 −0.156294 −0.0781468 0.996942i \(-0.524900\pi\)
−0.0781468 + 0.996942i \(0.524900\pi\)
\(354\) −5.59493e12 −0.0534905
\(355\) 4.57172e13 0.430352
\(356\) 9.90060e13 0.917670
\(357\) −1.27411e11 −0.00116287
\(358\) 4.00720e12 0.0360151
\(359\) 3.28329e13 0.290596 0.145298 0.989388i \(-0.453586\pi\)
0.145298 + 0.989388i \(0.453586\pi\)
\(360\) 3.20480e13 0.279343
\(361\) −3.77433e13 −0.324004
\(362\) −3.90623e13 −0.330263
\(363\) 8.05956e13 0.671158
\(364\) 6.12047e13 0.502027
\(365\) 1.61562e14 1.30535
\(366\) 1.27189e13 0.101229
\(367\) 3.18205e13 0.249484 0.124742 0.992189i \(-0.460190\pi\)
0.124742 + 0.992189i \(0.460190\pi\)
\(368\) −3.27493e13 −0.252953
\(369\) −8.16607e13 −0.621395
\(370\) −2.78602e13 −0.208869
\(371\) −1.11437e13 −0.0823135
\(372\) −1.48843e13 −0.108329
\(373\) −3.27361e13 −0.234763 −0.117381 0.993087i \(-0.537450\pi\)
−0.117381 + 0.993087i \(0.537450\pi\)
\(374\) 3.61043e11 0.00255132
\(375\) 9.01039e13 0.627439
\(376\) −1.71669e14 −1.17804
\(377\) 1.11546e14 0.754358
\(378\) −1.69766e13 −0.113148
\(379\) 2.74875e14 1.80559 0.902797 0.430066i \(-0.141510\pi\)
0.902797 + 0.430066i \(0.141510\pi\)
\(380\) −4.94193e13 −0.319953
\(381\) 5.13632e13 0.327767
\(382\) −7.04621e13 −0.443207
\(383\) −3.66325e13 −0.227129 −0.113565 0.993531i \(-0.536227\pi\)
−0.113565 + 0.993531i \(0.536227\pi\)
\(384\) −2.31659e13 −0.141588
\(385\) 1.58982e14 0.957886
\(386\) 8.66992e13 0.514973
\(387\) −6.47248e13 −0.379018
\(388\) −2.16746e13 −0.125134
\(389\) 1.92375e14 1.09503 0.547515 0.836796i \(-0.315574\pi\)
0.547515 + 0.836796i \(0.315574\pi\)
\(390\) −7.10640e13 −0.398836
\(391\) 4.23991e11 0.00234630
\(392\) 6.18362e13 0.337419
\(393\) −1.10820e11 −0.000596292 0
\(394\) −1.58448e14 −0.840730
\(395\) 1.04016e14 0.544271
\(396\) −4.68835e13 −0.241934
\(397\) −3.10067e14 −1.57800 −0.789002 0.614391i \(-0.789402\pi\)
−0.789002 + 0.614391i \(0.789402\pi\)
\(398\) 2.01238e14 1.01007
\(399\) 7.92187e13 0.392172
\(400\) 2.03660e13 0.0994435
\(401\) −4.61347e13 −0.222194 −0.111097 0.993810i \(-0.535437\pi\)
−0.111097 + 0.993810i \(0.535437\pi\)
\(402\) −9.74711e13 −0.463055
\(403\) 9.98756e13 0.468038
\(404\) 1.44986e14 0.670235
\(405\) −1.92102e13 −0.0876052
\(406\) −8.00717e13 −0.360235
\(407\) 1.23334e14 0.547413
\(408\) −3.41651e11 −0.00149607
\(409\) 1.71918e14 0.742751 0.371376 0.928483i \(-0.378886\pi\)
0.371376 + 0.928483i \(0.378886\pi\)
\(410\) 2.45378e14 1.04598
\(411\) 1.89653e14 0.797682
\(412\) 2.10674e13 0.0874331
\(413\) 2.62642e13 0.107557
\(414\) 5.64939e13 0.228297
\(415\) −1.23187e14 −0.491248
\(416\) 2.74005e14 1.07831
\(417\) −5.04424e13 −0.195906
\(418\) −2.24482e14 −0.860421
\(419\) 6.18027e13 0.233792 0.116896 0.993144i \(-0.462706\pi\)
0.116896 + 0.993144i \(0.462706\pi\)
\(420\) −4.97153e13 −0.185618
\(421\) 1.34130e14 0.494282 0.247141 0.968979i \(-0.420509\pi\)
0.247141 + 0.968979i \(0.420509\pi\)
\(422\) −2.00603e14 −0.729657
\(423\) 1.02902e14 0.369446
\(424\) −2.98816e13 −0.105899
\(425\) −2.63670e11 −0.000922404 0
\(426\) −6.49391e13 −0.224261
\(427\) −5.97065e13 −0.203549
\(428\) −2.01879e14 −0.679440
\(429\) 3.14594e14 1.04529
\(430\) 1.94488e14 0.637995
\(431\) −2.17813e14 −0.705439 −0.352719 0.935729i \(-0.614743\pi\)
−0.352719 + 0.935729i \(0.614743\pi\)
\(432\) −1.58183e13 −0.0505824
\(433\) −1.77000e13 −0.0558844 −0.0279422 0.999610i \(-0.508895\pi\)
−0.0279422 + 0.999610i \(0.508895\pi\)
\(434\) −7.16942e13 −0.223507
\(435\) −9.06067e13 −0.278914
\(436\) −8.13765e13 −0.247356
\(437\) −2.63620e14 −0.791280
\(438\) −2.29491e14 −0.680234
\(439\) 3.27938e14 0.959925 0.479963 0.877289i \(-0.340650\pi\)
0.479963 + 0.877289i \(0.340650\pi\)
\(440\) 4.26309e14 1.23235
\(441\) −3.70659e13 −0.105819
\(442\) 7.57585e11 0.00213604
\(443\) 1.89807e13 0.0528557 0.0264278 0.999651i \(-0.491587\pi\)
0.0264278 + 0.999651i \(0.491587\pi\)
\(444\) −3.85680e13 −0.106077
\(445\) 5.39631e14 1.46594
\(446\) −5.17677e14 −1.38905
\(447\) −3.69403e14 −0.979058
\(448\) −2.79632e14 −0.732080
\(449\) −2.96844e14 −0.767668 −0.383834 0.923402i \(-0.625397\pi\)
−0.383834 + 0.923402i \(0.625397\pi\)
\(450\) −3.51322e13 −0.0897505
\(451\) −1.08627e15 −2.74135
\(452\) 9.40509e13 0.234478
\(453\) −2.11942e14 −0.522007
\(454\) 3.84840e14 0.936425
\(455\) 3.33596e14 0.801969
\(456\) 2.12424e14 0.504542
\(457\) 9.36083e13 0.219672 0.109836 0.993950i \(-0.464967\pi\)
0.109836 + 0.993950i \(0.464967\pi\)
\(458\) −2.07257e14 −0.480562
\(459\) 2.04793e11 0.000469185 0
\(460\) 1.65440e14 0.374518
\(461\) −6.28776e14 −1.40650 −0.703252 0.710941i \(-0.748269\pi\)
−0.703252 + 0.710941i \(0.748269\pi\)
\(462\) −2.25826e14 −0.499165
\(463\) −1.35375e14 −0.295694 −0.147847 0.989010i \(-0.547234\pi\)
−0.147847 + 0.989010i \(0.547234\pi\)
\(464\) −7.46084e13 −0.161042
\(465\) −8.11269e13 −0.173051
\(466\) 1.58246e14 0.333588
\(467\) −6.74862e14 −1.40596 −0.702979 0.711211i \(-0.748147\pi\)
−0.702979 + 0.711211i \(0.748147\pi\)
\(468\) −9.83768e13 −0.202554
\(469\) 4.57558e14 0.931098
\(470\) −3.09205e14 −0.621882
\(471\) −3.11813e14 −0.619839
\(472\) 7.04273e13 0.138376
\(473\) −8.60982e14 −1.67208
\(474\) −1.47749e14 −0.283625
\(475\) 1.63939e14 0.311077
\(476\) 5.29995e11 0.000994108 0
\(477\) 1.79116e13 0.0332112
\(478\) 7.84650e13 0.143821
\(479\) −3.46511e14 −0.627873 −0.313936 0.949444i \(-0.601648\pi\)
−0.313936 + 0.949444i \(0.601648\pi\)
\(480\) −2.22569e14 −0.398692
\(481\) 2.58795e14 0.458309
\(482\) −5.20694e14 −0.911640
\(483\) −2.65199e14 −0.459053
\(484\) −3.35257e14 −0.573757
\(485\) −1.18137e14 −0.199897
\(486\) 2.72872e13 0.0456520
\(487\) −5.80006e14 −0.959453 −0.479726 0.877418i \(-0.659264\pi\)
−0.479726 + 0.877418i \(0.659264\pi\)
\(488\) −1.60102e14 −0.261872
\(489\) −4.48539e14 −0.725440
\(490\) 1.11377e14 0.178122
\(491\) −1.01449e15 −1.60435 −0.802173 0.597092i \(-0.796323\pi\)
−0.802173 + 0.597092i \(0.796323\pi\)
\(492\) 3.39687e14 0.531215
\(493\) 9.65922e11 0.00149377
\(494\) −4.71035e14 −0.720368
\(495\) −2.55538e14 −0.386480
\(496\) −6.68024e13 −0.0999179
\(497\) 3.04843e14 0.450937
\(498\) 1.74981e14 0.255994
\(499\) 1.32024e15 1.91029 0.955147 0.296131i \(-0.0956964\pi\)
0.955147 + 0.296131i \(0.0956964\pi\)
\(500\) −3.74809e14 −0.536382
\(501\) −6.29245e13 −0.0890661
\(502\) −9.77817e14 −1.36895
\(503\) 3.47956e14 0.481838 0.240919 0.970545i \(-0.422551\pi\)
0.240919 + 0.970545i \(0.422551\pi\)
\(504\) 2.13697e14 0.292705
\(505\) 7.90242e14 1.07067
\(506\) 7.51493e14 1.00716
\(507\) 2.24624e14 0.297792
\(508\) −2.13657e14 −0.280200
\(509\) 9.29904e14 1.20640 0.603199 0.797591i \(-0.293893\pi\)
0.603199 + 0.797591i \(0.293893\pi\)
\(510\) −6.15371e11 −0.000789771 0
\(511\) 1.07730e15 1.36780
\(512\) −4.05679e14 −0.509563
\(513\) −1.27331e14 −0.158230
\(514\) 4.34331e14 0.533978
\(515\) 1.14827e14 0.139671
\(516\) 2.69238e14 0.324013
\(517\) 1.36882e15 1.62985
\(518\) −1.85772e14 −0.218861
\(519\) 9.44697e14 1.10122
\(520\) 8.94534e14 1.03176
\(521\) 1.14862e15 1.31090 0.655449 0.755239i \(-0.272479\pi\)
0.655449 + 0.755239i \(0.272479\pi\)
\(522\) 1.28703e14 0.145345
\(523\) −9.44210e14 −1.05514 −0.527569 0.849512i \(-0.676896\pi\)
−0.527569 + 0.849512i \(0.676896\pi\)
\(524\) 4.60981e11 0.000509755 0
\(525\) 1.64921e14 0.180468
\(526\) 1.19136e15 1.29009
\(527\) 8.64861e11 0.000926804 0
\(528\) −2.10418e14 −0.223150
\(529\) −7.02948e13 −0.0737763
\(530\) −5.38218e13 −0.0559038
\(531\) −4.22156e13 −0.0433963
\(532\) −3.29529e14 −0.335258
\(533\) −2.27934e15 −2.29514
\(534\) −7.66521e14 −0.763918
\(535\) −1.10034e15 −1.08538
\(536\) 1.22694e15 1.19789
\(537\) 3.02357e13 0.0292187
\(538\) 4.79664e14 0.458812
\(539\) −4.93057e14 −0.466830
\(540\) 7.99095e13 0.0748915
\(541\) −1.06941e15 −0.992108 −0.496054 0.868292i \(-0.665218\pi\)
−0.496054 + 0.868292i \(0.665218\pi\)
\(542\) −1.00457e15 −0.922534
\(543\) −2.94738e14 −0.267940
\(544\) 2.37271e12 0.00213526
\(545\) −4.43542e14 −0.395142
\(546\) −4.73857e14 −0.417914
\(547\) 2.04326e15 1.78400 0.891999 0.452038i \(-0.149303\pi\)
0.891999 + 0.452038i \(0.149303\pi\)
\(548\) −7.88907e14 −0.681919
\(549\) 9.59686e13 0.0821261
\(550\) −4.67336e14 −0.395944
\(551\) −6.00570e14 −0.503767
\(552\) −7.11129e14 −0.590587
\(553\) 6.93579e14 0.570306
\(554\) 7.10735e13 0.0578634
\(555\) −2.10214e14 −0.169454
\(556\) 2.09827e14 0.167475
\(557\) −2.42812e15 −1.91896 −0.959482 0.281771i \(-0.909078\pi\)
−0.959482 + 0.281771i \(0.909078\pi\)
\(558\) 1.15237e14 0.0901786
\(559\) −1.80662e15 −1.39991
\(560\) −2.23128e14 −0.171206
\(561\) 2.72419e12 0.00206986
\(562\) −4.17520e14 −0.314143
\(563\) −1.28449e15 −0.957049 −0.478524 0.878074i \(-0.658828\pi\)
−0.478524 + 0.878074i \(0.658828\pi\)
\(564\) −4.28045e14 −0.315831
\(565\) 5.12623e14 0.374569
\(566\) 2.66245e14 0.192660
\(567\) −1.28094e14 −0.0917958
\(568\) 8.17435e14 0.580146
\(569\) −1.05248e15 −0.739772 −0.369886 0.929077i \(-0.620603\pi\)
−0.369886 + 0.929077i \(0.620603\pi\)
\(570\) 3.82612e14 0.266346
\(571\) −9.54795e14 −0.658281 −0.329141 0.944281i \(-0.606759\pi\)
−0.329141 + 0.944281i \(0.606759\pi\)
\(572\) −1.30863e15 −0.893589
\(573\) −5.31660e14 −0.359570
\(574\) 1.63619e15 1.09602
\(575\) −5.48815e14 −0.364127
\(576\) 4.49465e14 0.295374
\(577\) 2.48804e15 1.61954 0.809769 0.586749i \(-0.199593\pi\)
0.809769 + 0.586749i \(0.199593\pi\)
\(578\) −1.10373e15 −0.711640
\(579\) 6.54174e14 0.417793
\(580\) 3.76900e14 0.238436
\(581\) −8.21414e14 −0.514746
\(582\) 1.67808e14 0.104168
\(583\) 2.38264e14 0.146515
\(584\) 2.88877e15 1.75972
\(585\) −5.36202e14 −0.323572
\(586\) −1.89265e15 −1.13145
\(587\) 4.96923e14 0.294293 0.147146 0.989115i \(-0.452991\pi\)
0.147146 + 0.989115i \(0.452991\pi\)
\(588\) 1.54184e14 0.0904617
\(589\) −5.37735e14 −0.312560
\(590\) 1.26851e14 0.0730482
\(591\) −1.19554e15 −0.682076
\(592\) −1.73097e14 −0.0978409
\(593\) −1.40763e15 −0.788291 −0.394145 0.919048i \(-0.628959\pi\)
−0.394145 + 0.919048i \(0.628959\pi\)
\(594\) 3.62980e14 0.201399
\(595\) 2.88873e12 0.00158805
\(596\) 1.53662e15 0.836973
\(597\) 1.51841e15 0.819463
\(598\) 1.57687e15 0.843219
\(599\) −1.58628e15 −0.840489 −0.420244 0.907411i \(-0.638056\pi\)
−0.420244 + 0.907411i \(0.638056\pi\)
\(600\) 4.42235e14 0.232178
\(601\) 1.44110e15 0.749695 0.374847 0.927087i \(-0.377695\pi\)
0.374847 + 0.927087i \(0.377695\pi\)
\(602\) 1.29685e15 0.668513
\(603\) −7.35452e14 −0.375672
\(604\) 8.81621e14 0.446251
\(605\) −1.82731e15 −0.916554
\(606\) −1.12250e15 −0.557940
\(607\) 1.02307e15 0.503928 0.251964 0.967737i \(-0.418923\pi\)
0.251964 + 0.967737i \(0.418923\pi\)
\(608\) −1.47525e15 −0.720107
\(609\) −6.04168e14 −0.292255
\(610\) −2.88371e14 −0.138241
\(611\) 2.87223e15 1.36456
\(612\) −8.51883e11 −0.000401095 0
\(613\) 3.57680e15 1.66902 0.834511 0.550991i \(-0.185750\pi\)
0.834511 + 0.550991i \(0.185750\pi\)
\(614\) −8.94097e14 −0.413484
\(615\) 1.85146e15 0.848596
\(616\) 2.84264e15 1.29130
\(617\) −1.98761e15 −0.894876 −0.447438 0.894315i \(-0.647664\pi\)
−0.447438 + 0.894315i \(0.647664\pi\)
\(618\) −1.63107e14 −0.0727840
\(619\) 9.32973e14 0.412639 0.206320 0.978485i \(-0.433851\pi\)
0.206320 + 0.978485i \(0.433851\pi\)
\(620\) 3.37467e14 0.147937
\(621\) 4.26265e14 0.185215
\(622\) −2.51113e15 −1.08149
\(623\) 3.59827e15 1.53607
\(624\) −4.41525e14 −0.186827
\(625\) −1.14083e15 −0.478499
\(626\) 3.43251e14 0.142710
\(627\) −1.69379e15 −0.698052
\(628\) 1.29706e15 0.529885
\(629\) 2.24101e12 0.000907539 0
\(630\) 3.84904e14 0.154518
\(631\) 2.34954e14 0.0935021 0.0467510 0.998907i \(-0.485113\pi\)
0.0467510 + 0.998907i \(0.485113\pi\)
\(632\) 1.85983e15 0.733717
\(633\) −1.51362e15 −0.591964
\(634\) 1.20810e15 0.468396
\(635\) −1.16454e15 −0.447608
\(636\) −7.45077e13 −0.0283914
\(637\) −1.03459e15 −0.390843
\(638\) 1.71203e15 0.641205
\(639\) −4.89987e14 −0.181941
\(640\) 5.25231e14 0.193357
\(641\) −4.01924e15 −1.46698 −0.733490 0.679700i \(-0.762110\pi\)
−0.733490 + 0.679700i \(0.762110\pi\)
\(642\) 1.56298e15 0.565602
\(643\) −3.67581e15 −1.31884 −0.659420 0.751774i \(-0.729198\pi\)
−0.659420 + 0.751774i \(0.729198\pi\)
\(644\) 1.10316e15 0.392433
\(645\) 1.46748e15 0.517599
\(646\) −4.07887e12 −0.00142647
\(647\) −2.25806e15 −0.783001 −0.391500 0.920178i \(-0.628044\pi\)
−0.391500 + 0.920178i \(0.628044\pi\)
\(648\) −3.43484e14 −0.118098
\(649\) −5.61560e14 −0.191448
\(650\) −9.80621e14 −0.331496
\(651\) −5.40956e14 −0.181329
\(652\) 1.86580e15 0.620161
\(653\) −1.04171e15 −0.343339 −0.171669 0.985155i \(-0.554916\pi\)
−0.171669 + 0.985155i \(0.554916\pi\)
\(654\) 6.30030e14 0.205913
\(655\) 2.51257e12 0.000814314 0
\(656\) 1.52455e15 0.489971
\(657\) −1.73159e15 −0.551868
\(658\) −2.06179e15 −0.651630
\(659\) 5.09258e15 1.59613 0.798065 0.602572i \(-0.205857\pi\)
0.798065 + 0.602572i \(0.205857\pi\)
\(660\) 1.06297e15 0.330392
\(661\) 3.62635e15 1.11780 0.558898 0.829237i \(-0.311225\pi\)
0.558898 + 0.829237i \(0.311225\pi\)
\(662\) 1.55926e15 0.476649
\(663\) 5.71623e12 0.00173295
\(664\) −2.20262e15 −0.662238
\(665\) −1.79609e15 −0.535562
\(666\) 2.98600e14 0.0883041
\(667\) 2.01052e15 0.589679
\(668\) 2.61749e14 0.0761404
\(669\) −3.90604e15 −1.12692
\(670\) 2.20992e15 0.632362
\(671\) 1.27659e15 0.362309
\(672\) −1.48409e15 −0.417763
\(673\) −1.04228e15 −0.291005 −0.145502 0.989358i \(-0.546480\pi\)
−0.145502 + 0.989358i \(0.546480\pi\)
\(674\) 3.97095e14 0.109968
\(675\) −2.65084e14 −0.0728137
\(676\) −9.34377e14 −0.254575
\(677\) −2.30069e15 −0.621757 −0.310878 0.950450i \(-0.600623\pi\)
−0.310878 + 0.950450i \(0.600623\pi\)
\(678\) −7.28157e14 −0.195192
\(679\) −7.87740e14 −0.209459
\(680\) 7.74612e12 0.00204308
\(681\) 2.90375e15 0.759713
\(682\) 1.53290e15 0.397833
\(683\) −1.97637e15 −0.508807 −0.254404 0.967098i \(-0.581879\pi\)
−0.254404 + 0.967098i \(0.581879\pi\)
\(684\) 5.29665e14 0.135267
\(685\) −4.29993e15 −1.08934
\(686\) 3.08210e15 0.774577
\(687\) −1.56382e15 −0.389876
\(688\) 1.20837e15 0.298856
\(689\) 4.99955e14 0.122666
\(690\) −1.28086e15 −0.311769
\(691\) 1.81722e15 0.438813 0.219406 0.975634i \(-0.429588\pi\)
0.219406 + 0.975634i \(0.429588\pi\)
\(692\) −3.92969e15 −0.941401
\(693\) −1.70394e15 −0.404968
\(694\) 9.77942e14 0.230588
\(695\) 1.14366e15 0.267535
\(696\) −1.62007e15 −0.375996
\(697\) −1.97377e13 −0.00454480
\(698\) 3.94228e14 0.0900622
\(699\) 1.19402e15 0.270637
\(700\) −6.86028e14 −0.154278
\(701\) 1.47361e14 0.0328801 0.0164401 0.999865i \(-0.494767\pi\)
0.0164401 + 0.999865i \(0.494767\pi\)
\(702\) 7.61649e14 0.168617
\(703\) −1.39337e15 −0.306063
\(704\) 5.97887e15 1.30307
\(705\) −2.33305e15 −0.504527
\(706\) −5.18358e14 −0.111225
\(707\) 5.26935e15 1.12189
\(708\) 1.75606e14 0.0370984
\(709\) 6.07023e14 0.127248 0.0636240 0.997974i \(-0.479734\pi\)
0.0636240 + 0.997974i \(0.479734\pi\)
\(710\) 1.47234e15 0.306257
\(711\) −1.11482e15 −0.230102
\(712\) 9.64875e15 1.97620
\(713\) 1.80016e15 0.365864
\(714\) −4.10331e12 −0.000827549 0
\(715\) −7.13266e15 −1.42747
\(716\) −1.25772e14 −0.0249783
\(717\) 5.92044e14 0.116681
\(718\) 1.05740e15 0.206801
\(719\) −1.92770e15 −0.374137 −0.187069 0.982347i \(-0.559899\pi\)
−0.187069 + 0.982347i \(0.559899\pi\)
\(720\) 3.58642e14 0.0690768
\(721\) 7.65672e14 0.146352
\(722\) −1.21554e15 −0.230576
\(723\) −3.92881e15 −0.739605
\(724\) 1.22603e15 0.229055
\(725\) −1.25029e15 −0.231821
\(726\) 2.59561e15 0.477626
\(727\) 9.80226e15 1.79014 0.895070 0.445926i \(-0.147126\pi\)
0.895070 + 0.445926i \(0.147126\pi\)
\(728\) 5.96477e15 1.08111
\(729\) 2.05891e14 0.0370370
\(730\) 5.20316e15 0.928948
\(731\) −1.56442e13 −0.00277209
\(732\) −3.99204e14 −0.0702076
\(733\) −1.38846e14 −0.0242360 −0.0121180 0.999927i \(-0.503857\pi\)
−0.0121180 + 0.999927i \(0.503857\pi\)
\(734\) 1.02479e15 0.177544
\(735\) 8.40379e14 0.144509
\(736\) 4.93868e15 0.842914
\(737\) −9.78312e15 −1.65732
\(738\) −2.62991e15 −0.442212
\(739\) −7.72677e15 −1.28960 −0.644798 0.764353i \(-0.723058\pi\)
−0.644798 + 0.764353i \(0.723058\pi\)
\(740\) 8.74436e14 0.144862
\(741\) −3.55411e15 −0.584428
\(742\) −3.58885e14 −0.0585779
\(743\) −5.06696e15 −0.820936 −0.410468 0.911875i \(-0.634635\pi\)
−0.410468 + 0.911875i \(0.634635\pi\)
\(744\) −1.45057e15 −0.233285
\(745\) 8.37532e15 1.33703
\(746\) −1.05428e15 −0.167067
\(747\) 1.32029e15 0.207686
\(748\) −1.13319e13 −0.00176947
\(749\) −7.33710e15 −1.13730
\(750\) 2.90183e15 0.446514
\(751\) 8.33127e15 1.27260 0.636300 0.771442i \(-0.280464\pi\)
0.636300 + 0.771442i \(0.280464\pi\)
\(752\) −1.92111e15 −0.291309
\(753\) −7.37795e15 −1.11061
\(754\) 3.59238e15 0.536834
\(755\) 4.80527e15 0.712868
\(756\) 5.32838e14 0.0784740
\(757\) 1.01977e15 0.149099 0.0745495 0.997217i \(-0.476248\pi\)
0.0745495 + 0.997217i \(0.476248\pi\)
\(758\) 8.85246e15 1.28494
\(759\) 5.67026e15 0.817097
\(760\) −4.81621e15 −0.689018
\(761\) −2.21417e15 −0.314482 −0.157241 0.987560i \(-0.550260\pi\)
−0.157241 + 0.987560i \(0.550260\pi\)
\(762\) 1.65417e15 0.233253
\(763\) −2.95755e15 −0.414044
\(764\) 2.21156e15 0.307387
\(765\) −4.64318e12 −0.000640733 0
\(766\) −1.17976e15 −0.161635
\(767\) −1.17833e15 −0.160285
\(768\) −4.53415e15 −0.612363
\(769\) −1.20942e16 −1.62174 −0.810871 0.585225i \(-0.801006\pi\)
−0.810871 + 0.585225i \(0.801006\pi\)
\(770\) 5.12007e15 0.681674
\(771\) 3.27717e15 0.433212
\(772\) −2.72119e15 −0.357161
\(773\) 2.48533e15 0.323889 0.161945 0.986800i \(-0.448223\pi\)
0.161945 + 0.986800i \(0.448223\pi\)
\(774\) −2.08448e15 −0.269726
\(775\) −1.11948e15 −0.143833
\(776\) −2.11232e15 −0.269476
\(777\) −1.40171e15 −0.177560
\(778\) 6.19551e15 0.779272
\(779\) 1.22720e16 1.53271
\(780\) 2.23046e15 0.276614
\(781\) −6.51790e15 −0.802651
\(782\) 1.36548e13 0.00166973
\(783\) 9.71103e14 0.117917
\(784\) 6.91994e14 0.0834381
\(785\) 7.06960e15 0.846471
\(786\) −3.56899e12 −0.000424348 0
\(787\) −4.29022e15 −0.506545 −0.253273 0.967395i \(-0.581507\pi\)
−0.253273 + 0.967395i \(0.581507\pi\)
\(788\) 4.97313e15 0.583090
\(789\) 8.98920e15 1.04664
\(790\) 3.34986e15 0.387327
\(791\) 3.41818e15 0.392487
\(792\) −4.56909e15 −0.521004
\(793\) 2.67870e15 0.303335
\(794\) −9.98582e15 −1.12298
\(795\) −4.06104e14 −0.0453542
\(796\) −6.31617e15 −0.700539
\(797\) −5.62716e15 −0.619825 −0.309912 0.950765i \(-0.600300\pi\)
−0.309912 + 0.950765i \(0.600300\pi\)
\(798\) 2.55127e15 0.279087
\(799\) 2.48717e13 0.00270208
\(800\) −3.07125e15 −0.331376
\(801\) −5.78365e15 −0.619760
\(802\) −1.48578e15 −0.158123
\(803\) −2.30339e16 −2.43463
\(804\) 3.05928e15 0.321153
\(805\) 6.01275e15 0.626897
\(806\) 3.21653e15 0.333077
\(807\) 3.61923e15 0.372230
\(808\) 1.41297e16 1.44335
\(809\) 1.23571e15 0.125372 0.0626861 0.998033i \(-0.480033\pi\)
0.0626861 + 0.998033i \(0.480033\pi\)
\(810\) −6.18672e14 −0.0623438
\(811\) 1.20094e16 1.20201 0.601004 0.799246i \(-0.294768\pi\)
0.601004 + 0.799246i \(0.294768\pi\)
\(812\) 2.51318e15 0.249842
\(813\) −7.57979e15 −0.748443
\(814\) 3.97203e15 0.389564
\(815\) 1.01695e16 0.990683
\(816\) −3.82334e12 −0.000369953 0
\(817\) 9.72691e15 0.934875
\(818\) 5.53668e15 0.528575
\(819\) −3.57540e15 −0.339050
\(820\) −7.70158e15 −0.725444
\(821\) 1.49862e15 0.140218 0.0701092 0.997539i \(-0.477665\pi\)
0.0701092 + 0.997539i \(0.477665\pi\)
\(822\) 6.10784e15 0.567666
\(823\) −4.57801e15 −0.422647 −0.211323 0.977416i \(-0.567777\pi\)
−0.211323 + 0.977416i \(0.567777\pi\)
\(824\) 2.05314e15 0.188287
\(825\) −3.52620e15 −0.321226
\(826\) 8.45849e14 0.0765425
\(827\) 8.74446e15 0.786055 0.393027 0.919527i \(-0.371428\pi\)
0.393027 + 0.919527i \(0.371428\pi\)
\(828\) −1.77315e15 −0.158336
\(829\) 3.01144e15 0.267131 0.133566 0.991040i \(-0.457357\pi\)
0.133566 + 0.991040i \(0.457357\pi\)
\(830\) −3.96728e15 −0.349593
\(831\) 5.36273e14 0.0469441
\(832\) 1.25456e16 1.09097
\(833\) −8.95894e12 −0.000773944 0
\(834\) −1.62451e15 −0.139415
\(835\) 1.42666e15 0.121631
\(836\) 7.04571e15 0.596747
\(837\) 8.69501e14 0.0731611
\(838\) 1.99038e15 0.166377
\(839\) 2.58723e15 0.214854 0.107427 0.994213i \(-0.465739\pi\)
0.107427 + 0.994213i \(0.465739\pi\)
\(840\) −4.84507e15 −0.399727
\(841\) −7.62022e15 −0.624582
\(842\) 4.31971e15 0.351753
\(843\) −3.15033e15 −0.254862
\(844\) 6.29624e15 0.506055
\(845\) −5.09281e15 −0.406673
\(846\) 3.31399e15 0.262914
\(847\) −1.21846e16 −0.960398
\(848\) −3.34398e14 −0.0261871
\(849\) 2.00891e15 0.156303
\(850\) −8.49158e12 −0.000656424 0
\(851\) 4.66455e15 0.358259
\(852\) 2.03822e15 0.155536
\(853\) −1.46066e16 −1.10746 −0.553732 0.832695i \(-0.686797\pi\)
−0.553732 + 0.832695i \(0.686797\pi\)
\(854\) −1.92287e15 −0.144854
\(855\) 2.88693e15 0.216084
\(856\) −1.96744e16 −1.46317
\(857\) −1.21875e16 −0.900574 −0.450287 0.892884i \(-0.648678\pi\)
−0.450287 + 0.892884i \(0.648678\pi\)
\(858\) 1.01316e16 0.743872
\(859\) 2.33332e16 1.70220 0.851102 0.525001i \(-0.175935\pi\)
0.851102 + 0.525001i \(0.175935\pi\)
\(860\) −6.10432e15 −0.442483
\(861\) 1.23456e16 0.889189
\(862\) −7.01475e15 −0.502022
\(863\) 2.51263e16 1.78677 0.893387 0.449289i \(-0.148323\pi\)
0.893387 + 0.449289i \(0.148323\pi\)
\(864\) 2.38544e15 0.168556
\(865\) −2.14187e16 −1.50385
\(866\) −5.70035e14 −0.0397698
\(867\) −8.32802e15 −0.577347
\(868\) 2.25024e15 0.155013
\(869\) −1.48295e16 −1.01512
\(870\) −2.91802e15 −0.198487
\(871\) −2.05281e16 −1.38755
\(872\) −7.93064e15 −0.532681
\(873\) 1.26617e15 0.0845109
\(874\) −8.48996e15 −0.563110
\(875\) −1.36220e16 −0.897838
\(876\) 7.20295e15 0.471778
\(877\) −1.66456e16 −1.08343 −0.541717 0.840561i \(-0.682226\pi\)
−0.541717 + 0.840561i \(0.682226\pi\)
\(878\) 1.05614e16 0.683125
\(879\) −1.42807e16 −0.917932
\(880\) 4.77073e15 0.304740
\(881\) 4.23358e15 0.268745 0.134372 0.990931i \(-0.457098\pi\)
0.134372 + 0.990931i \(0.457098\pi\)
\(882\) −1.19372e15 −0.0753052
\(883\) 1.64624e16 1.03207 0.516035 0.856567i \(-0.327407\pi\)
0.516035 + 0.856567i \(0.327407\pi\)
\(884\) −2.37780e13 −0.00148145
\(885\) 9.57136e14 0.0592633
\(886\) 6.11280e14 0.0376144
\(887\) 7.28714e15 0.445633 0.222817 0.974860i \(-0.428475\pi\)
0.222817 + 0.974860i \(0.428475\pi\)
\(888\) −3.75869e15 −0.228436
\(889\) −7.76516e15 −0.469020
\(890\) 1.73790e16 1.04323
\(891\) 2.73880e15 0.163393
\(892\) 1.62481e16 0.963376
\(893\) −1.54642e16 −0.911265
\(894\) −1.18967e16 −0.696741
\(895\) −6.85521e14 −0.0399020
\(896\) 3.50225e15 0.202606
\(897\) 1.18980e16 0.684096
\(898\) −9.55996e15 −0.546307
\(899\) 4.10108e15 0.232927
\(900\) 1.10268e15 0.0622467
\(901\) 4.32930e12 0.000242902 0
\(902\) −3.49836e16 −1.95087
\(903\) 9.78518e15 0.542358
\(904\) 9.16583e15 0.504947
\(905\) 6.68247e15 0.365906
\(906\) −6.82565e15 −0.371483
\(907\) −3.31414e16 −1.79279 −0.896397 0.443253i \(-0.853824\pi\)
−0.896397 + 0.443253i \(0.853824\pi\)
\(908\) −1.20788e16 −0.649460
\(909\) −8.46964e15 −0.452651
\(910\) 1.07436e16 0.570717
\(911\) 8.72205e15 0.460540 0.230270 0.973127i \(-0.426039\pi\)
0.230270 + 0.973127i \(0.426039\pi\)
\(912\) 2.37719e15 0.124765
\(913\) 1.75628e16 0.916229
\(914\) 3.01469e15 0.156328
\(915\) −2.17586e15 −0.112154
\(916\) 6.50510e15 0.333295
\(917\) 1.67539e13 0.000853267 0
\(918\) 6.59541e12 0.000333893 0
\(919\) 3.28544e16 1.65333 0.826664 0.562696i \(-0.190236\pi\)
0.826664 + 0.562696i \(0.190236\pi\)
\(920\) 1.61231e16 0.806523
\(921\) −6.74626e15 −0.335456
\(922\) −2.02500e16 −1.00093
\(923\) −1.36767e16 −0.672002
\(924\) 7.08792e15 0.346197
\(925\) −2.90077e15 −0.140843
\(926\) −4.35979e15 −0.210429
\(927\) −1.23070e15 −0.0590490
\(928\) 1.12511e16 0.536640
\(929\) 1.96092e16 0.929767 0.464884 0.885372i \(-0.346096\pi\)
0.464884 + 0.885372i \(0.346096\pi\)
\(930\) −2.61272e15 −0.123151
\(931\) 5.57030e15 0.261009
\(932\) −4.96681e15 −0.231361
\(933\) −1.89473e16 −0.877403
\(934\) −2.17342e16 −1.00054
\(935\) −6.17645e13 −0.00282667
\(936\) −9.58742e15 −0.436199
\(937\) −2.05928e16 −0.931426 −0.465713 0.884936i \(-0.654202\pi\)
−0.465713 + 0.884936i \(0.654202\pi\)
\(938\) 1.47358e16 0.662611
\(939\) 2.58994e15 0.115779
\(940\) 9.70489e15 0.431308
\(941\) −3.71778e16 −1.64263 −0.821317 0.570472i \(-0.806760\pi\)
−0.821317 + 0.570472i \(0.806760\pi\)
\(942\) −1.00420e16 −0.441105
\(943\) −4.10829e16 −1.79410
\(944\) 7.88136e14 0.0342180
\(945\) 2.90423e15 0.125359
\(946\) −2.77282e16 −1.18993
\(947\) 1.22223e16 0.521469 0.260734 0.965411i \(-0.416035\pi\)
0.260734 + 0.965411i \(0.416035\pi\)
\(948\) 4.63735e15 0.196709
\(949\) −4.83326e16 −2.03834
\(950\) 5.27971e15 0.221376
\(951\) 9.11554e15 0.380005
\(952\) 5.16513e13 0.00214081
\(953\) 1.36347e16 0.561870 0.280935 0.959727i \(-0.409355\pi\)
0.280935 + 0.959727i \(0.409355\pi\)
\(954\) 5.76851e14 0.0236346
\(955\) 1.20541e16 0.491040
\(956\) −2.46275e15 −0.0997474
\(957\) 1.29178e16 0.520203
\(958\) −1.11595e16 −0.446822
\(959\) −2.86720e16 −1.14145
\(960\) −1.01905e16 −0.403371
\(961\) −2.17365e16 −0.855481
\(962\) 8.33460e15 0.326153
\(963\) 1.17932e16 0.458868
\(964\) 1.63428e16 0.632270
\(965\) −1.48318e16 −0.570551
\(966\) −8.54083e15 −0.326682
\(967\) 3.10975e16 1.18271 0.591356 0.806410i \(-0.298593\pi\)
0.591356 + 0.806410i \(0.298593\pi\)
\(968\) −3.26728e16 −1.23558
\(969\) −3.07764e13 −0.00115728
\(970\) −3.80464e15 −0.142256
\(971\) −3.17570e16 −1.18068 −0.590341 0.807154i \(-0.701007\pi\)
−0.590341 + 0.807154i \(0.701007\pi\)
\(972\) −8.56453e14 −0.0316620
\(973\) 7.62595e15 0.280332
\(974\) −1.86793e16 −0.682789
\(975\) −7.39911e15 −0.268939
\(976\) −1.79167e15 −0.0647565
\(977\) 2.35140e16 0.845098 0.422549 0.906340i \(-0.361135\pi\)
0.422549 + 0.906340i \(0.361135\pi\)
\(978\) −1.44454e16 −0.516255
\(979\) −7.69353e16 −2.73414
\(980\) −3.49576e15 −0.123537
\(981\) 4.75378e15 0.167055
\(982\) −3.26719e16 −1.14172
\(983\) 1.44138e16 0.500881 0.250440 0.968132i \(-0.419425\pi\)
0.250440 + 0.968132i \(0.419425\pi\)
\(984\) 3.31046e16 1.14397
\(985\) 2.71060e16 0.931464
\(986\) 3.11078e13 0.00106303
\(987\) −1.55569e16 −0.528661
\(988\) 1.47842e16 0.499613
\(989\) −3.25626e16 −1.09431
\(990\) −8.22970e15 −0.275036
\(991\) −2.26252e16 −0.751946 −0.375973 0.926631i \(-0.622691\pi\)
−0.375973 + 0.926631i \(0.622691\pi\)
\(992\) 1.00740e16 0.332956
\(993\) 1.17651e16 0.386701
\(994\) 9.81758e15 0.320907
\(995\) −3.44262e16 −1.11908
\(996\) −5.49206e15 −0.177545
\(997\) 3.70510e16 1.19118 0.595589 0.803290i \(-0.296919\pi\)
0.595589 + 0.803290i \(0.296919\pi\)
\(998\) 4.25188e16 1.35945
\(999\) 2.25303e15 0.0716403
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 177.12.a.b.1.18 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.18 27 1.1 even 1 trivial