Properties

Label 177.12.a.d.1.7
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $28$
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: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-44.5190 q^{2} +243.000 q^{3} -66.0554 q^{4} -2490.42 q^{5} -10818.1 q^{6} -10217.0 q^{7} +94115.7 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-44.5190 q^{2} +243.000 q^{3} -66.0554 q^{4} -2490.42 q^{5} -10818.1 q^{6} -10217.0 q^{7} +94115.7 q^{8} +59049.0 q^{9} +110871. q^{10} -935597. q^{11} -16051.5 q^{12} -1.88159e6 q^{13} +454851. q^{14} -605173. q^{15} -4.05466e6 q^{16} -7.73016e6 q^{17} -2.62880e6 q^{18} +2.90198e6 q^{19} +164506. q^{20} -2.48273e6 q^{21} +4.16519e7 q^{22} +3.68356e7 q^{23} +2.28701e7 q^{24} -4.26259e7 q^{25} +8.37664e7 q^{26} +1.43489e7 q^{27} +674888. q^{28} -1.85891e8 q^{29} +2.69417e7 q^{30} -9.30939e7 q^{31} -1.22395e7 q^{32} -2.27350e8 q^{33} +3.44139e8 q^{34} +2.54446e7 q^{35} -3.90051e6 q^{36} -2.96437e8 q^{37} -1.29193e8 q^{38} -4.57225e8 q^{39} -2.34388e8 q^{40} +3.54502e8 q^{41} +1.10529e8 q^{42} +1.17124e9 q^{43} +6.18013e7 q^{44} -1.47057e8 q^{45} -1.63989e9 q^{46} -3.82655e8 q^{47} -9.85282e8 q^{48} -1.87294e9 q^{49} +1.89766e9 q^{50} -1.87843e9 q^{51} +1.24289e8 q^{52} -1.18844e9 q^{53} -6.38800e8 q^{54} +2.33003e9 q^{55} -9.61580e8 q^{56} +7.05180e8 q^{57} +8.27570e9 q^{58} +7.14924e8 q^{59} +3.99749e7 q^{60} +3.90006e9 q^{61} +4.14445e9 q^{62} -6.03303e8 q^{63} +8.84883e9 q^{64} +4.68594e9 q^{65} +1.01214e10 q^{66} -9.49449e9 q^{67} +5.10619e8 q^{68} +8.95105e9 q^{69} -1.13277e9 q^{70} -2.28335e10 q^{71} +5.55744e9 q^{72} -1.82782e10 q^{73} +1.31971e10 q^{74} -1.03581e10 q^{75} -1.91691e8 q^{76} +9.55900e9 q^{77} +2.03552e10 q^{78} -2.69656e10 q^{79} +1.00978e10 q^{80} +3.48678e9 q^{81} -1.57821e10 q^{82} -5.51453e10 q^{83} +1.63998e8 q^{84} +1.92514e10 q^{85} -5.21423e10 q^{86} -4.51716e10 q^{87} -8.80544e10 q^{88} +2.42061e10 q^{89} +6.54683e9 q^{90} +1.92242e10 q^{91} -2.43319e9 q^{92} -2.26218e10 q^{93} +1.70354e10 q^{94} -7.22715e9 q^{95} -2.97419e9 q^{96} +3.60303e10 q^{97} +8.33815e10 q^{98} -5.52461e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −44.5190 −0.983741 −0.491871 0.870668i \(-0.663687\pi\)
−0.491871 + 0.870668i \(0.663687\pi\)
\(3\) 243.000 0.577350
\(4\) −66.0554 −0.0322536
\(5\) −2490.42 −0.356400 −0.178200 0.983994i \(-0.557027\pi\)
−0.178200 + 0.983994i \(0.557027\pi\)
\(6\) −10818.1 −0.567963
\(7\) −10217.0 −0.229765 −0.114883 0.993379i \(-0.536649\pi\)
−0.114883 + 0.993379i \(0.536649\pi\)
\(8\) 94115.7 1.01547
\(9\) 59049.0 0.333333
\(10\) 110871. 0.350606
\(11\) −935597. −1.75158 −0.875788 0.482695i \(-0.839658\pi\)
−0.875788 + 0.482695i \(0.839658\pi\)
\(12\) −16051.5 −0.0186216
\(13\) −1.88159e6 −1.40552 −0.702758 0.711429i \(-0.748048\pi\)
−0.702758 + 0.711429i \(0.748048\pi\)
\(14\) 454851. 0.226029
\(15\) −605173. −0.205768
\(16\) −4.05466e6 −0.966706
\(17\) −7.73016e6 −1.32044 −0.660221 0.751071i \(-0.729537\pi\)
−0.660221 + 0.751071i \(0.729537\pi\)
\(18\) −2.62880e6 −0.327914
\(19\) 2.90198e6 0.268874 0.134437 0.990922i \(-0.457077\pi\)
0.134437 + 0.990922i \(0.457077\pi\)
\(20\) 164506. 0.0114952
\(21\) −2.48273e6 −0.132655
\(22\) 4.16519e7 1.72310
\(23\) 3.68356e7 1.19334 0.596671 0.802486i \(-0.296490\pi\)
0.596671 + 0.802486i \(0.296490\pi\)
\(24\) 2.28701e7 0.586282
\(25\) −4.26259e7 −0.872979
\(26\) 8.37664e7 1.38266
\(27\) 1.43489e7 0.192450
\(28\) 674888. 0.00741075
\(29\) −1.85891e8 −1.68295 −0.841473 0.540300i \(-0.818311\pi\)
−0.841473 + 0.540300i \(0.818311\pi\)
\(30\) 2.69417e7 0.202422
\(31\) −9.30939e7 −0.584026 −0.292013 0.956414i \(-0.594325\pi\)
−0.292013 + 0.956414i \(0.594325\pi\)
\(32\) −1.22395e7 −0.0644818
\(33\) −2.27350e8 −1.01127
\(34\) 3.44139e8 1.29897
\(35\) 2.54446e7 0.0818883
\(36\) −3.90051e6 −0.0107512
\(37\) −2.96437e8 −0.702786 −0.351393 0.936228i \(-0.614292\pi\)
−0.351393 + 0.936228i \(0.614292\pi\)
\(38\) −1.29193e8 −0.264503
\(39\) −4.57225e8 −0.811475
\(40\) −2.34388e8 −0.361914
\(41\) 3.54502e8 0.477867 0.238933 0.971036i \(-0.423202\pi\)
0.238933 + 0.971036i \(0.423202\pi\)
\(42\) 1.10529e8 0.130498
\(43\) 1.17124e9 1.21498 0.607489 0.794328i \(-0.292177\pi\)
0.607489 + 0.794328i \(0.292177\pi\)
\(44\) 6.18013e7 0.0564947
\(45\) −1.47057e8 −0.118800
\(46\) −1.63989e9 −1.17394
\(47\) −3.82655e8 −0.243371 −0.121685 0.992569i \(-0.538830\pi\)
−0.121685 + 0.992569i \(0.538830\pi\)
\(48\) −9.85282e8 −0.558128
\(49\) −1.87294e9 −0.947208
\(50\) 1.89766e9 0.858785
\(51\) −1.87843e9 −0.762358
\(52\) 1.24289e8 0.0453330
\(53\) −1.18844e9 −0.390356 −0.195178 0.980768i \(-0.562528\pi\)
−0.195178 + 0.980768i \(0.562528\pi\)
\(54\) −6.38800e8 −0.189321
\(55\) 2.33003e9 0.624262
\(56\) −9.61580e8 −0.233320
\(57\) 7.05180e8 0.155235
\(58\) 8.27570e9 1.65558
\(59\) 7.14924e8 0.130189
\(60\) 3.99749e7 0.00663675
\(61\) 3.90006e9 0.591231 0.295615 0.955307i \(-0.404475\pi\)
0.295615 + 0.955307i \(0.404475\pi\)
\(62\) 4.14445e9 0.574530
\(63\) −6.03303e8 −0.0765884
\(64\) 8.84883e9 1.03014
\(65\) 4.68594e9 0.500926
\(66\) 1.01214e10 0.994831
\(67\) −9.49449e9 −0.859133 −0.429566 0.903035i \(-0.641334\pi\)
−0.429566 + 0.903035i \(0.641334\pi\)
\(68\) 5.10619e8 0.0425890
\(69\) 8.95105e9 0.688976
\(70\) −1.13277e9 −0.0805569
\(71\) −2.28335e10 −1.50194 −0.750969 0.660338i \(-0.770413\pi\)
−0.750969 + 0.660338i \(0.770413\pi\)
\(72\) 5.55744e9 0.338490
\(73\) −1.82782e10 −1.03195 −0.515973 0.856605i \(-0.672569\pi\)
−0.515973 + 0.856605i \(0.672569\pi\)
\(74\) 1.31971e10 0.691359
\(75\) −1.03581e10 −0.504015
\(76\) −1.91691e8 −0.00867216
\(77\) 9.55900e9 0.402451
\(78\) 2.03552e10 0.798281
\(79\) −2.69656e10 −0.985963 −0.492982 0.870040i \(-0.664093\pi\)
−0.492982 + 0.870040i \(0.664093\pi\)
\(80\) 1.00978e10 0.344534
\(81\) 3.48678e9 0.111111
\(82\) −1.57821e10 −0.470097
\(83\) −5.51453e10 −1.53666 −0.768332 0.640052i \(-0.778913\pi\)
−0.768332 + 0.640052i \(0.778913\pi\)
\(84\) 1.63998e8 0.00427860
\(85\) 1.92514e10 0.470606
\(86\) −5.21423e10 −1.19522
\(87\) −4.51716e10 −0.971649
\(88\) −8.80544e10 −1.77867
\(89\) 2.42061e10 0.459494 0.229747 0.973250i \(-0.426210\pi\)
0.229747 + 0.973250i \(0.426210\pi\)
\(90\) 6.54683e9 0.116869
\(91\) 1.92242e10 0.322938
\(92\) −2.43319e9 −0.0384896
\(93\) −2.26218e10 −0.337187
\(94\) 1.70354e10 0.239414
\(95\) −7.22715e9 −0.0958268
\(96\) −2.97419e9 −0.0372286
\(97\) 3.60303e10 0.426014 0.213007 0.977051i \(-0.431674\pi\)
0.213007 + 0.977051i \(0.431674\pi\)
\(98\) 8.33815e10 0.931807
\(99\) −5.52461e10 −0.583859
\(100\) 2.81567e9 0.0281567
\(101\) 2.80952e9 0.0265990 0.0132995 0.999912i \(-0.495767\pi\)
0.0132995 + 0.999912i \(0.495767\pi\)
\(102\) 8.36258e10 0.749963
\(103\) 6.41454e10 0.545206 0.272603 0.962127i \(-0.412115\pi\)
0.272603 + 0.962127i \(0.412115\pi\)
\(104\) −1.77087e11 −1.42726
\(105\) 6.18305e9 0.0472783
\(106\) 5.29083e10 0.384009
\(107\) −3.34398e10 −0.230490 −0.115245 0.993337i \(-0.536765\pi\)
−0.115245 + 0.993337i \(0.536765\pi\)
\(108\) −9.47823e8 −0.00620721
\(109\) 2.28957e10 0.142530 0.0712652 0.997457i \(-0.477296\pi\)
0.0712652 + 0.997457i \(0.477296\pi\)
\(110\) −1.03731e11 −0.614113
\(111\) −7.20342e10 −0.405754
\(112\) 4.14264e10 0.222115
\(113\) −1.69298e11 −0.864412 −0.432206 0.901775i \(-0.642265\pi\)
−0.432206 + 0.901775i \(0.642265\pi\)
\(114\) −3.13939e10 −0.152711
\(115\) −9.17362e10 −0.425307
\(116\) 1.22791e10 0.0542811
\(117\) −1.11106e11 −0.468505
\(118\) −3.18277e10 −0.128072
\(119\) 7.89790e10 0.303392
\(120\) −5.69563e10 −0.208951
\(121\) 5.90031e11 2.06802
\(122\) −1.73627e11 −0.581618
\(123\) 8.61439e10 0.275897
\(124\) 6.14935e9 0.0188369
\(125\) 2.27759e11 0.667530
\(126\) 2.68585e10 0.0753431
\(127\) −6.07077e11 −1.63051 −0.815255 0.579103i \(-0.803403\pi\)
−0.815255 + 0.579103i \(0.803403\pi\)
\(128\) −3.68875e11 −0.948909
\(129\) 2.84611e11 0.701468
\(130\) −2.08614e11 −0.492781
\(131\) 3.73200e11 0.845180 0.422590 0.906321i \(-0.361121\pi\)
0.422590 + 0.906321i \(0.361121\pi\)
\(132\) 1.50177e10 0.0326172
\(133\) −2.96495e10 −0.0617779
\(134\) 4.22686e11 0.845164
\(135\) −3.57348e10 −0.0685893
\(136\) −7.27529e11 −1.34087
\(137\) 1.05238e11 0.186299 0.0931494 0.995652i \(-0.470307\pi\)
0.0931494 + 0.995652i \(0.470307\pi\)
\(138\) −3.98492e11 −0.677774
\(139\) −9.56452e11 −1.56344 −0.781721 0.623629i \(-0.785658\pi\)
−0.781721 + 0.623629i \(0.785658\pi\)
\(140\) −1.68076e9 −0.00264119
\(141\) −9.29851e10 −0.140510
\(142\) 1.01653e12 1.47752
\(143\) 1.76041e12 2.46187
\(144\) −2.39424e11 −0.322235
\(145\) 4.62948e11 0.599802
\(146\) 8.13726e11 1.01517
\(147\) −4.55124e11 −0.546871
\(148\) 1.95813e10 0.0226674
\(149\) 1.25712e12 1.40234 0.701169 0.712995i \(-0.252662\pi\)
0.701169 + 0.712995i \(0.252662\pi\)
\(150\) 4.61133e11 0.495820
\(151\) −1.06595e12 −1.10500 −0.552501 0.833512i \(-0.686327\pi\)
−0.552501 + 0.833512i \(0.686327\pi\)
\(152\) 2.73122e11 0.273034
\(153\) −4.56458e11 −0.440147
\(154\) −4.25557e11 −0.395908
\(155\) 2.31843e11 0.208147
\(156\) 3.02022e10 0.0261730
\(157\) 1.39049e12 1.16337 0.581686 0.813413i \(-0.302393\pi\)
0.581686 + 0.813413i \(0.302393\pi\)
\(158\) 1.20048e12 0.969933
\(159\) −2.88791e11 −0.225372
\(160\) 3.04814e10 0.0229813
\(161\) −3.76349e11 −0.274188
\(162\) −1.55228e11 −0.109305
\(163\) 4.36425e11 0.297083 0.148542 0.988906i \(-0.452542\pi\)
0.148542 + 0.988906i \(0.452542\pi\)
\(164\) −2.34167e10 −0.0154129
\(165\) 5.66198e11 0.360418
\(166\) 2.45501e12 1.51168
\(167\) −1.88865e12 −1.12515 −0.562575 0.826746i \(-0.690189\pi\)
−0.562575 + 0.826746i \(0.690189\pi\)
\(168\) −2.33664e11 −0.134707
\(169\) 1.74820e12 0.975473
\(170\) −8.57052e11 −0.462954
\(171\) 1.71359e11 0.0896247
\(172\) −7.73665e10 −0.0391874
\(173\) 3.80723e12 1.86791 0.933955 0.357390i \(-0.116333\pi\)
0.933955 + 0.357390i \(0.116333\pi\)
\(174\) 2.01099e12 0.955851
\(175\) 4.35509e11 0.200580
\(176\) 3.79353e12 1.69326
\(177\) 1.73727e11 0.0751646
\(178\) −1.07763e12 −0.452023
\(179\) 1.33826e12 0.544314 0.272157 0.962253i \(-0.412263\pi\)
0.272157 + 0.962253i \(0.412263\pi\)
\(180\) 9.71391e9 0.00383173
\(181\) −4.24390e12 −1.62380 −0.811901 0.583795i \(-0.801567\pi\)
−0.811901 + 0.583795i \(0.801567\pi\)
\(182\) −8.55841e11 −0.317688
\(183\) 9.47714e11 0.341347
\(184\) 3.46681e12 1.21180
\(185\) 7.38254e11 0.250473
\(186\) 1.00710e12 0.331705
\(187\) 7.23232e12 2.31286
\(188\) 2.52764e10 0.00784959
\(189\) −1.46603e11 −0.0442183
\(190\) 3.21746e11 0.0942688
\(191\) 1.11093e12 0.316229 0.158115 0.987421i \(-0.449458\pi\)
0.158115 + 0.987421i \(0.449458\pi\)
\(192\) 2.15027e12 0.594751
\(193\) 4.08585e12 1.09829 0.549145 0.835727i \(-0.314953\pi\)
0.549145 + 0.835727i \(0.314953\pi\)
\(194\) −1.60404e12 −0.419087
\(195\) 1.13868e12 0.289210
\(196\) 1.23718e11 0.0305509
\(197\) 3.53152e12 0.848004 0.424002 0.905661i \(-0.360625\pi\)
0.424002 + 0.905661i \(0.360625\pi\)
\(198\) 2.45950e12 0.574366
\(199\) −5.82785e11 −0.132378 −0.0661891 0.997807i \(-0.521084\pi\)
−0.0661891 + 0.997807i \(0.521084\pi\)
\(200\) −4.01177e12 −0.886484
\(201\) −2.30716e12 −0.496020
\(202\) −1.25077e11 −0.0261665
\(203\) 1.89925e12 0.386682
\(204\) 1.24080e11 0.0245888
\(205\) −8.82859e11 −0.170312
\(206\) −2.85569e12 −0.536342
\(207\) 2.17511e12 0.397781
\(208\) 7.62919e12 1.35872
\(209\) −2.71508e12 −0.470954
\(210\) −2.75263e11 −0.0465096
\(211\) −5.21749e11 −0.0858832 −0.0429416 0.999078i \(-0.513673\pi\)
−0.0429416 + 0.999078i \(0.513673\pi\)
\(212\) 7.85030e10 0.0125904
\(213\) −5.54854e12 −0.867144
\(214\) 1.48871e12 0.226743
\(215\) −2.91688e12 −0.433018
\(216\) 1.35046e12 0.195427
\(217\) 9.51140e11 0.134189
\(218\) −1.01929e12 −0.140213
\(219\) −4.44159e12 −0.595794
\(220\) −1.53911e11 −0.0201347
\(221\) 1.45450e13 1.85590
\(222\) 3.20689e12 0.399157
\(223\) −1.56293e11 −0.0189785 −0.00948925 0.999955i \(-0.503021\pi\)
−0.00948925 + 0.999955i \(0.503021\pi\)
\(224\) 1.25050e11 0.0148157
\(225\) −2.51702e12 −0.290993
\(226\) 7.53699e12 0.850358
\(227\) 9.18590e12 1.01153 0.505766 0.862671i \(-0.331210\pi\)
0.505766 + 0.862671i \(0.331210\pi\)
\(228\) −4.65810e10 −0.00500688
\(229\) 1.24732e13 1.30883 0.654413 0.756137i \(-0.272916\pi\)
0.654413 + 0.756137i \(0.272916\pi\)
\(230\) 4.08401e12 0.418392
\(231\) 2.32284e12 0.232355
\(232\) −1.74953e13 −1.70898
\(233\) −4.85840e12 −0.463485 −0.231742 0.972777i \(-0.574443\pi\)
−0.231742 + 0.972777i \(0.574443\pi\)
\(234\) 4.94632e12 0.460888
\(235\) 9.52972e11 0.0867375
\(236\) −4.72246e10 −0.00419906
\(237\) −6.55264e12 −0.569246
\(238\) −3.51607e12 −0.298459
\(239\) 2.27996e13 1.89121 0.945604 0.325321i \(-0.105472\pi\)
0.945604 + 0.325321i \(0.105472\pi\)
\(240\) 2.45377e12 0.198917
\(241\) 6.68231e12 0.529460 0.264730 0.964323i \(-0.414717\pi\)
0.264730 + 0.964323i \(0.414717\pi\)
\(242\) −2.62676e13 −2.03440
\(243\) 8.47289e11 0.0641500
\(244\) −2.57620e11 −0.0190693
\(245\) 4.66441e12 0.337585
\(246\) −3.83504e12 −0.271411
\(247\) −5.46032e12 −0.377907
\(248\) −8.76160e12 −0.593061
\(249\) −1.34003e13 −0.887193
\(250\) −1.01396e13 −0.656677
\(251\) −7.74963e12 −0.490993 −0.245497 0.969397i \(-0.578951\pi\)
−0.245497 + 0.969397i \(0.578951\pi\)
\(252\) 3.98515e10 0.00247025
\(253\) −3.44633e13 −2.09023
\(254\) 2.70265e13 1.60400
\(255\) 4.67808e12 0.271704
\(256\) −1.70044e12 −0.0966591
\(257\) 1.22661e13 0.682453 0.341226 0.939981i \(-0.389158\pi\)
0.341226 + 0.939981i \(0.389158\pi\)
\(258\) −1.26706e13 −0.690063
\(259\) 3.02870e12 0.161476
\(260\) −3.09532e11 −0.0161567
\(261\) −1.09767e13 −0.560982
\(262\) −1.66145e13 −0.831438
\(263\) 1.37447e13 0.673561 0.336781 0.941583i \(-0.390662\pi\)
0.336781 + 0.941583i \(0.390662\pi\)
\(264\) −2.13972e13 −1.02692
\(265\) 2.95972e12 0.139123
\(266\) 1.31997e12 0.0607734
\(267\) 5.88208e12 0.265289
\(268\) 6.27162e11 0.0277101
\(269\) 5.26687e12 0.227990 0.113995 0.993481i \(-0.463635\pi\)
0.113995 + 0.993481i \(0.463635\pi\)
\(270\) 1.59088e12 0.0674741
\(271\) −2.16142e13 −0.898273 −0.449136 0.893463i \(-0.648268\pi\)
−0.449136 + 0.893463i \(0.648268\pi\)
\(272\) 3.13432e13 1.27648
\(273\) 4.67147e12 0.186449
\(274\) −4.68510e12 −0.183270
\(275\) 3.98807e13 1.52909
\(276\) −5.91265e11 −0.0222220
\(277\) 1.20207e13 0.442886 0.221443 0.975173i \(-0.428923\pi\)
0.221443 + 0.975173i \(0.428923\pi\)
\(278\) 4.25803e13 1.53802
\(279\) −5.49710e12 −0.194675
\(280\) 2.39474e12 0.0831552
\(281\) −4.23114e12 −0.144070 −0.0720348 0.997402i \(-0.522949\pi\)
−0.0720348 + 0.997402i \(0.522949\pi\)
\(282\) 4.13961e12 0.138226
\(283\) −8.21411e12 −0.268989 −0.134495 0.990914i \(-0.542941\pi\)
−0.134495 + 0.990914i \(0.542941\pi\)
\(284\) 1.50828e12 0.0484429
\(285\) −1.75620e12 −0.0553256
\(286\) −7.83716e13 −2.42184
\(287\) −3.62194e12 −0.109797
\(288\) −7.22727e11 −0.0214939
\(289\) 2.54835e13 0.743568
\(290\) −2.06100e13 −0.590050
\(291\) 8.75537e12 0.245959
\(292\) 1.20737e12 0.0332840
\(293\) 3.03847e12 0.0822020 0.0411010 0.999155i \(-0.486913\pi\)
0.0411010 + 0.999155i \(0.486913\pi\)
\(294\) 2.02617e13 0.537979
\(295\) −1.78046e12 −0.0463994
\(296\) −2.78994e13 −0.713658
\(297\) −1.34248e13 −0.337091
\(298\) −5.59658e13 −1.37954
\(299\) −6.93094e13 −1.67726
\(300\) 6.84208e11 0.0162563
\(301\) −1.19665e13 −0.279159
\(302\) 4.74550e13 1.08704
\(303\) 6.82714e11 0.0153569
\(304\) −1.17665e13 −0.259922
\(305\) −9.71279e12 −0.210715
\(306\) 2.03211e13 0.432991
\(307\) −3.88687e12 −0.0813466 −0.0406733 0.999172i \(-0.512950\pi\)
−0.0406733 + 0.999172i \(0.512950\pi\)
\(308\) −6.31423e11 −0.0129805
\(309\) 1.55873e13 0.314775
\(310\) −1.03214e13 −0.204763
\(311\) 1.35635e13 0.264357 0.132178 0.991226i \(-0.457803\pi\)
0.132178 + 0.991226i \(0.457803\pi\)
\(312\) −4.30321e13 −0.824028
\(313\) −7.66164e13 −1.44154 −0.720772 0.693172i \(-0.756213\pi\)
−0.720772 + 0.693172i \(0.756213\pi\)
\(314\) −6.19031e13 −1.14446
\(315\) 1.50248e12 0.0272961
\(316\) 1.78122e12 0.0318009
\(317\) 3.19725e13 0.560984 0.280492 0.959856i \(-0.409502\pi\)
0.280492 + 0.959856i \(0.409502\pi\)
\(318\) 1.28567e13 0.221708
\(319\) 1.73919e14 2.94781
\(320\) −2.20373e13 −0.367142
\(321\) −8.12586e12 −0.133074
\(322\) 1.67547e13 0.269730
\(323\) −2.24327e13 −0.355033
\(324\) −2.30321e11 −0.00358373
\(325\) 8.02043e13 1.22699
\(326\) −1.94292e13 −0.292253
\(327\) 5.56365e12 0.0822900
\(328\) 3.33642e13 0.485260
\(329\) 3.90958e12 0.0559181
\(330\) −2.52066e13 −0.354558
\(331\) −4.22366e13 −0.584299 −0.292150 0.956373i \(-0.594371\pi\)
−0.292150 + 0.956373i \(0.594371\pi\)
\(332\) 3.64264e12 0.0495630
\(333\) −1.75043e13 −0.234262
\(334\) 8.40807e13 1.10686
\(335\) 2.36453e13 0.306195
\(336\) 1.00666e13 0.128238
\(337\) −1.09741e14 −1.37533 −0.687663 0.726030i \(-0.741363\pi\)
−0.687663 + 0.726030i \(0.741363\pi\)
\(338\) −7.78284e13 −0.959613
\(339\) −4.11395e13 −0.499069
\(340\) −1.27166e12 −0.0151787
\(341\) 8.70984e13 1.02297
\(342\) −7.62873e12 −0.0881675
\(343\) 3.93382e13 0.447400
\(344\) 1.10232e14 1.23377
\(345\) −2.22919e13 −0.245551
\(346\) −1.69494e14 −1.83754
\(347\) 5.58659e13 0.596121 0.298060 0.954547i \(-0.403660\pi\)
0.298060 + 0.954547i \(0.403660\pi\)
\(348\) 2.98383e12 0.0313392
\(349\) −6.82666e13 −0.705778 −0.352889 0.935665i \(-0.614801\pi\)
−0.352889 + 0.935665i \(0.614801\pi\)
\(350\) −1.93884e13 −0.197319
\(351\) −2.69987e13 −0.270492
\(352\) 1.14512e13 0.112945
\(353\) −1.69275e13 −0.164374 −0.0821868 0.996617i \(-0.526190\pi\)
−0.0821868 + 0.996617i \(0.526190\pi\)
\(354\) −7.73414e12 −0.0739425
\(355\) 5.68651e13 0.535291
\(356\) −1.59894e12 −0.0148203
\(357\) 1.91919e13 0.175163
\(358\) −5.95781e13 −0.535464
\(359\) 2.14398e14 1.89759 0.948793 0.315897i \(-0.102305\pi\)
0.948793 + 0.315897i \(0.102305\pi\)
\(360\) −1.38404e13 −0.120638
\(361\) −1.08069e14 −0.927707
\(362\) 1.88934e14 1.59740
\(363\) 1.43377e14 1.19397
\(364\) −1.26986e12 −0.0104159
\(365\) 4.55204e13 0.367786
\(366\) −4.21913e13 −0.335797
\(367\) −1.01526e14 −0.796001 −0.398001 0.917385i \(-0.630296\pi\)
−0.398001 + 0.917385i \(0.630296\pi\)
\(368\) −1.49356e14 −1.15361
\(369\) 2.09330e13 0.159289
\(370\) −3.28663e13 −0.246401
\(371\) 1.21423e13 0.0896902
\(372\) 1.49429e12 0.0108755
\(373\) −1.91918e13 −0.137631 −0.0688155 0.997629i \(-0.521922\pi\)
−0.0688155 + 0.997629i \(0.521922\pi\)
\(374\) −3.21976e14 −2.27525
\(375\) 5.53455e13 0.385399
\(376\) −3.60138e13 −0.247136
\(377\) 3.49770e14 2.36541
\(378\) 6.52661e12 0.0434994
\(379\) −2.05345e13 −0.134886 −0.0674431 0.997723i \(-0.521484\pi\)
−0.0674431 + 0.997723i \(0.521484\pi\)
\(380\) 4.77392e11 0.00309076
\(381\) −1.47520e14 −0.941375
\(382\) −4.94574e13 −0.311088
\(383\) −1.62004e14 −1.00446 −0.502230 0.864734i \(-0.667487\pi\)
−0.502230 + 0.864734i \(0.667487\pi\)
\(384\) −8.96366e13 −0.547853
\(385\) −2.38059e13 −0.143434
\(386\) −1.81898e14 −1.08043
\(387\) 6.91604e13 0.404993
\(388\) −2.38000e12 −0.0137405
\(389\) −2.51401e13 −0.143102 −0.0715509 0.997437i \(-0.522795\pi\)
−0.0715509 + 0.997437i \(0.522795\pi\)
\(390\) −5.06931e13 −0.284508
\(391\) −2.84745e14 −1.57574
\(392\) −1.76273e14 −0.961862
\(393\) 9.06875e13 0.487965
\(394\) −1.57220e14 −0.834216
\(395\) 6.71557e13 0.351398
\(396\) 3.64930e12 0.0188316
\(397\) −2.69669e14 −1.37241 −0.686205 0.727409i \(-0.740725\pi\)
−0.686205 + 0.727409i \(0.740725\pi\)
\(398\) 2.59450e13 0.130226
\(399\) −7.20482e12 −0.0356675
\(400\) 1.72834e14 0.843914
\(401\) −2.06505e14 −0.994575 −0.497287 0.867586i \(-0.665671\pi\)
−0.497287 + 0.867586i \(0.665671\pi\)
\(402\) 1.02713e14 0.487956
\(403\) 1.75164e14 0.820857
\(404\) −1.85584e11 −0.000857913 0
\(405\) −8.68357e12 −0.0396000
\(406\) −8.45528e13 −0.380395
\(407\) 2.77346e14 1.23098
\(408\) −1.76790e14 −0.774152
\(409\) −7.27752e12 −0.0314416 −0.0157208 0.999876i \(-0.505004\pi\)
−0.0157208 + 0.999876i \(0.505004\pi\)
\(410\) 3.93040e13 0.167543
\(411\) 2.55729e13 0.107560
\(412\) −4.23715e12 −0.0175849
\(413\) −7.30438e12 −0.0299129
\(414\) −9.68336e13 −0.391313
\(415\) 1.37335e14 0.547667
\(416\) 2.30296e13 0.0906301
\(417\) −2.32418e14 −0.902653
\(418\) 1.20873e14 0.463297
\(419\) 8.22109e13 0.310994 0.155497 0.987836i \(-0.450302\pi\)
0.155497 + 0.987836i \(0.450302\pi\)
\(420\) −4.08424e11 −0.00152489
\(421\) 1.37353e14 0.506158 0.253079 0.967446i \(-0.418557\pi\)
0.253079 + 0.967446i \(0.418557\pi\)
\(422\) 2.32278e13 0.0844868
\(423\) −2.25954e13 −0.0811236
\(424\) −1.11851e14 −0.396395
\(425\) 3.29505e14 1.15272
\(426\) 2.47016e14 0.853045
\(427\) −3.98469e13 −0.135844
\(428\) 2.20888e12 0.00743414
\(429\) 4.27779e14 1.42136
\(430\) 1.29856e14 0.425978
\(431\) −5.24678e14 −1.69929 −0.849646 0.527353i \(-0.823184\pi\)
−0.849646 + 0.527353i \(0.823184\pi\)
\(432\) −5.81799e13 −0.186043
\(433\) −2.51830e14 −0.795104 −0.397552 0.917580i \(-0.630140\pi\)
−0.397552 + 0.917580i \(0.630140\pi\)
\(434\) −4.23438e13 −0.132007
\(435\) 1.12496e14 0.346296
\(436\) −1.51238e12 −0.00459712
\(437\) 1.06896e14 0.320859
\(438\) 1.97735e14 0.586107
\(439\) 2.08832e14 0.611281 0.305641 0.952147i \(-0.401129\pi\)
0.305641 + 0.952147i \(0.401129\pi\)
\(440\) 2.19293e14 0.633920
\(441\) −1.10595e14 −0.315736
\(442\) −6.47528e14 −1.82573
\(443\) −2.31456e13 −0.0644538 −0.0322269 0.999481i \(-0.510260\pi\)
−0.0322269 + 0.999481i \(0.510260\pi\)
\(444\) 4.75825e12 0.0130870
\(445\) −6.02834e13 −0.163764
\(446\) 6.95800e12 0.0186699
\(447\) 3.05480e14 0.809640
\(448\) −9.04085e13 −0.236690
\(449\) 4.34904e14 1.12471 0.562353 0.826897i \(-0.309896\pi\)
0.562353 + 0.826897i \(0.309896\pi\)
\(450\) 1.12055e14 0.286262
\(451\) −3.31671e14 −0.837021
\(452\) 1.11831e13 0.0278804
\(453\) −2.59026e14 −0.637973
\(454\) −4.08948e14 −0.995085
\(455\) −4.78763e13 −0.115095
\(456\) 6.63685e13 0.157636
\(457\) −2.73262e14 −0.641269 −0.320634 0.947203i \(-0.603896\pi\)
−0.320634 + 0.947203i \(0.603896\pi\)
\(458\) −5.55294e14 −1.28755
\(459\) −1.10919e14 −0.254119
\(460\) 6.05967e12 0.0137177
\(461\) 1.44656e14 0.323581 0.161790 0.986825i \(-0.448273\pi\)
0.161790 + 0.986825i \(0.448273\pi\)
\(462\) −1.03410e14 −0.228577
\(463\) 5.13100e14 1.12074 0.560372 0.828241i \(-0.310658\pi\)
0.560372 + 0.828241i \(0.310658\pi\)
\(464\) 7.53726e14 1.62691
\(465\) 5.63379e13 0.120174
\(466\) 2.16291e14 0.455949
\(467\) −6.21431e14 −1.29464 −0.647321 0.762217i \(-0.724111\pi\)
−0.647321 + 0.762217i \(0.724111\pi\)
\(468\) 7.33914e12 0.0151110
\(469\) 9.70052e13 0.197399
\(470\) −4.24254e13 −0.0853272
\(471\) 3.37888e14 0.671673
\(472\) 6.72856e13 0.132203
\(473\) −1.09581e15 −2.12813
\(474\) 2.91717e14 0.559991
\(475\) −1.23699e14 −0.234721
\(476\) −5.21699e12 −0.00978547
\(477\) −7.01763e13 −0.130119
\(478\) −1.01502e15 −1.86046
\(479\) −9.93368e13 −0.179997 −0.0899984 0.995942i \(-0.528686\pi\)
−0.0899984 + 0.995942i \(0.528686\pi\)
\(480\) 7.40698e12 0.0132683
\(481\) 5.57772e14 0.987776
\(482\) −2.97490e14 −0.520851
\(483\) −9.14529e13 −0.158303
\(484\) −3.89747e13 −0.0667012
\(485\) −8.97308e13 −0.151831
\(486\) −3.77205e13 −0.0631070
\(487\) −1.00443e15 −1.66154 −0.830769 0.556617i \(-0.812099\pi\)
−0.830769 + 0.556617i \(0.812099\pi\)
\(488\) 3.67057e14 0.600377
\(489\) 1.06051e14 0.171521
\(490\) −2.07655e14 −0.332096
\(491\) −4.27357e14 −0.675837 −0.337919 0.941175i \(-0.609723\pi\)
−0.337919 + 0.941175i \(0.609723\pi\)
\(492\) −5.69027e12 −0.00889866
\(493\) 1.43697e15 2.22223
\(494\) 2.43088e14 0.371762
\(495\) 1.37586e14 0.208087
\(496\) 3.77464e14 0.564581
\(497\) 2.33290e14 0.345093
\(498\) 5.96569e14 0.872768
\(499\) 1.08268e15 1.56656 0.783281 0.621668i \(-0.213545\pi\)
0.783281 + 0.621668i \(0.213545\pi\)
\(500\) −1.50447e13 −0.0215303
\(501\) −4.58941e14 −0.649605
\(502\) 3.45006e14 0.483010
\(503\) −6.22606e14 −0.862163 −0.431082 0.902313i \(-0.641868\pi\)
−0.431082 + 0.902313i \(0.641868\pi\)
\(504\) −5.67803e13 −0.0777732
\(505\) −6.99690e12 −0.00947988
\(506\) 1.53427e15 2.05624
\(507\) 4.24814e14 0.563190
\(508\) 4.01007e13 0.0525898
\(509\) −7.08898e14 −0.919679 −0.459839 0.888002i \(-0.652093\pi\)
−0.459839 + 0.888002i \(0.652093\pi\)
\(510\) −2.08264e14 −0.267287
\(511\) 1.86748e14 0.237105
\(512\) 8.31158e14 1.04400
\(513\) 4.16402e13 0.0517449
\(514\) −5.46073e14 −0.671357
\(515\) −1.59749e14 −0.194312
\(516\) −1.88001e13 −0.0226249
\(517\) 3.58011e14 0.426283
\(518\) −1.34835e14 −0.158850
\(519\) 9.25158e14 1.07844
\(520\) 4.41021e14 0.508675
\(521\) 1.57927e15 1.80239 0.901196 0.433411i \(-0.142690\pi\)
0.901196 + 0.433411i \(0.142690\pi\)
\(522\) 4.88672e14 0.551861
\(523\) −4.13879e14 −0.462503 −0.231251 0.972894i \(-0.574282\pi\)
−0.231251 + 0.972894i \(0.574282\pi\)
\(524\) −2.46519e13 −0.0272601
\(525\) 1.05829e14 0.115805
\(526\) −6.11899e14 −0.662610
\(527\) 7.19631e14 0.771172
\(528\) 9.21827e14 0.977604
\(529\) 4.04053e14 0.424064
\(530\) −1.31764e14 −0.136861
\(531\) 4.22156e13 0.0433963
\(532\) 1.95851e12 0.00199256
\(533\) −6.67025e14 −0.671649
\(534\) −2.61865e14 −0.260976
\(535\) 8.32791e13 0.0821467
\(536\) −8.93581e14 −0.872424
\(537\) 3.25197e14 0.314260
\(538\) −2.34476e14 −0.224283
\(539\) 1.75232e15 1.65911
\(540\) 2.36048e12 0.00221225
\(541\) 7.77033e14 0.720866 0.360433 0.932785i \(-0.382629\pi\)
0.360433 + 0.932785i \(0.382629\pi\)
\(542\) 9.62244e14 0.883668
\(543\) −1.03127e15 −0.937503
\(544\) 9.46129e13 0.0851445
\(545\) −5.70199e13 −0.0507979
\(546\) −2.07969e14 −0.183417
\(547\) −1.27966e15 −1.11729 −0.558643 0.829408i \(-0.688678\pi\)
−0.558643 + 0.829408i \(0.688678\pi\)
\(548\) −6.95155e12 −0.00600881
\(549\) 2.30294e14 0.197077
\(550\) −1.77545e15 −1.50423
\(551\) −5.39452e14 −0.452500
\(552\) 8.42435e14 0.699635
\(553\) 2.75507e14 0.226540
\(554\) −5.35151e14 −0.435685
\(555\) 1.79396e14 0.144611
\(556\) 6.31788e13 0.0504266
\(557\) −3.96428e14 −0.313300 −0.156650 0.987654i \(-0.550070\pi\)
−0.156650 + 0.987654i \(0.550070\pi\)
\(558\) 2.44726e14 0.191510
\(559\) −2.20378e15 −1.70767
\(560\) −1.03169e14 −0.0791620
\(561\) 1.75745e15 1.33533
\(562\) 1.88366e14 0.141727
\(563\) 1.36730e15 1.01875 0.509374 0.860545i \(-0.329877\pi\)
0.509374 + 0.860545i \(0.329877\pi\)
\(564\) 6.14217e12 0.00453196
\(565\) 4.21624e14 0.308077
\(566\) 3.65684e14 0.264616
\(567\) −3.56245e13 −0.0255295
\(568\) −2.14899e15 −1.52517
\(569\) 2.17587e15 1.52938 0.764690 0.644399i \(-0.222892\pi\)
0.764690 + 0.644399i \(0.222892\pi\)
\(570\) 7.81842e13 0.0544261
\(571\) −1.13698e15 −0.783890 −0.391945 0.919989i \(-0.628198\pi\)
−0.391945 + 0.919989i \(0.628198\pi\)
\(572\) −1.16284e14 −0.0794041
\(573\) 2.69955e14 0.182575
\(574\) 1.61245e14 0.108012
\(575\) −1.57015e15 −1.04176
\(576\) 5.22515e14 0.343380
\(577\) −2.14346e14 −0.139524 −0.0697619 0.997564i \(-0.522224\pi\)
−0.0697619 + 0.997564i \(0.522224\pi\)
\(578\) −1.13450e15 −0.731478
\(579\) 9.92861e14 0.634098
\(580\) −3.05802e13 −0.0193458
\(581\) 5.63419e14 0.353072
\(582\) −3.89781e14 −0.241960
\(583\) 1.11190e15 0.683738
\(584\) −1.72026e15 −1.04791
\(585\) 2.76700e14 0.166975
\(586\) −1.35270e14 −0.0808655
\(587\) 8.32831e14 0.493228 0.246614 0.969114i \(-0.420682\pi\)
0.246614 + 0.969114i \(0.420682\pi\)
\(588\) 3.00634e13 0.0176386
\(589\) −2.70156e14 −0.157029
\(590\) 7.92645e13 0.0456450
\(591\) 8.58160e14 0.489595
\(592\) 1.20195e15 0.679387
\(593\) −1.61872e14 −0.0906507 −0.0453254 0.998972i \(-0.514432\pi\)
−0.0453254 + 0.998972i \(0.514432\pi\)
\(594\) 5.97659e14 0.331610
\(595\) −1.96691e14 −0.108129
\(596\) −8.30396e13 −0.0452304
\(597\) −1.41617e14 −0.0764286
\(598\) 3.08559e15 1.64999
\(599\) 2.58065e15 1.36735 0.683677 0.729785i \(-0.260380\pi\)
0.683677 + 0.729785i \(0.260380\pi\)
\(600\) −9.74860e14 −0.511812
\(601\) 2.85968e15 1.48767 0.743836 0.668362i \(-0.233004\pi\)
0.743836 + 0.668362i \(0.233004\pi\)
\(602\) 5.32738e14 0.274621
\(603\) −5.60640e14 −0.286378
\(604\) 7.04117e13 0.0356403
\(605\) −1.46943e15 −0.737043
\(606\) −3.03938e13 −0.0151072
\(607\) −1.92232e15 −0.946864 −0.473432 0.880830i \(-0.656985\pi\)
−0.473432 + 0.880830i \(0.656985\pi\)
\(608\) −3.55186e13 −0.0173375
\(609\) 4.61518e14 0.223251
\(610\) 4.32404e14 0.207289
\(611\) 7.19997e14 0.342062
\(612\) 3.01515e13 0.0141963
\(613\) −1.12928e15 −0.526948 −0.263474 0.964667i \(-0.584868\pi\)
−0.263474 + 0.964667i \(0.584868\pi\)
\(614\) 1.73040e14 0.0800240
\(615\) −2.14535e14 −0.0983296
\(616\) 8.99652e14 0.408677
\(617\) 3.52040e15 1.58498 0.792490 0.609884i \(-0.208784\pi\)
0.792490 + 0.609884i \(0.208784\pi\)
\(618\) −6.93933e14 −0.309657
\(619\) 7.60941e14 0.336552 0.168276 0.985740i \(-0.446180\pi\)
0.168276 + 0.985740i \(0.446180\pi\)
\(620\) −1.53145e13 −0.00671349
\(621\) 5.28551e14 0.229659
\(622\) −6.03835e14 −0.260058
\(623\) −2.47314e14 −0.105576
\(624\) 1.85389e15 0.784458
\(625\) 1.51413e15 0.635071
\(626\) 3.41089e15 1.41811
\(627\) −6.59765e14 −0.271905
\(628\) −9.18491e13 −0.0375230
\(629\) 2.29151e15 0.927988
\(630\) −6.68890e13 −0.0268523
\(631\) −1.58045e15 −0.628955 −0.314477 0.949265i \(-0.601829\pi\)
−0.314477 + 0.949265i \(0.601829\pi\)
\(632\) −2.53788e15 −1.00122
\(633\) −1.26785e14 −0.0495847
\(634\) −1.42338e15 −0.551863
\(635\) 1.51188e15 0.581114
\(636\) 1.90762e13 0.00726906
\(637\) 3.52410e15 1.33132
\(638\) −7.74272e15 −2.89988
\(639\) −1.34830e15 −0.500646
\(640\) 9.18655e14 0.338191
\(641\) 1.11628e15 0.407430 0.203715 0.979030i \(-0.434698\pi\)
0.203715 + 0.979030i \(0.434698\pi\)
\(642\) 3.61756e14 0.130910
\(643\) −1.23857e15 −0.444385 −0.222192 0.975003i \(-0.571321\pi\)
−0.222192 + 0.975003i \(0.571321\pi\)
\(644\) 2.48599e13 0.00884356
\(645\) −7.08801e14 −0.250003
\(646\) 9.98684e14 0.349260
\(647\) 4.33083e15 1.50175 0.750874 0.660445i \(-0.229632\pi\)
0.750874 + 0.660445i \(0.229632\pi\)
\(648\) 3.28161e14 0.112830
\(649\) −6.68881e14 −0.228036
\(650\) −3.57062e15 −1.20704
\(651\) 2.31127e14 0.0774739
\(652\) −2.88283e13 −0.00958201
\(653\) −5.76545e15 −1.90025 −0.950125 0.311871i \(-0.899044\pi\)
−0.950125 + 0.311871i \(0.899044\pi\)
\(654\) −2.47688e14 −0.0809520
\(655\) −9.29425e14 −0.301222
\(656\) −1.43738e15 −0.461957
\(657\) −1.07931e15 −0.343982
\(658\) −1.74051e14 −0.0550090
\(659\) 3.43713e15 1.07727 0.538637 0.842538i \(-0.318939\pi\)
0.538637 + 0.842538i \(0.318939\pi\)
\(660\) −3.74004e13 −0.0116248
\(661\) −3.24130e15 −0.999105 −0.499552 0.866284i \(-0.666502\pi\)
−0.499552 + 0.866284i \(0.666502\pi\)
\(662\) 1.88033e15 0.574799
\(663\) 3.53443e15 1.07151
\(664\) −5.19004e15 −1.56044
\(665\) 7.38397e13 0.0220177
\(666\) 7.79275e14 0.230453
\(667\) −6.84742e15 −2.00833
\(668\) 1.24755e14 0.0362901
\(669\) −3.79791e13 −0.0109572
\(670\) −1.05267e15 −0.301217
\(671\) −3.64888e15 −1.03559
\(672\) 3.03873e13 0.00855383
\(673\) 8.79248e14 0.245487 0.122744 0.992438i \(-0.460831\pi\)
0.122744 + 0.992438i \(0.460831\pi\)
\(674\) 4.88558e15 1.35296
\(675\) −6.11635e14 −0.168005
\(676\) −1.15478e14 −0.0314625
\(677\) −2.78076e15 −0.751495 −0.375748 0.926722i \(-0.622614\pi\)
−0.375748 + 0.926722i \(0.622614\pi\)
\(678\) 1.83149e15 0.490954
\(679\) −3.68122e14 −0.0978831
\(680\) 1.81186e15 0.477886
\(681\) 2.23217e15 0.584008
\(682\) −3.87754e15 −1.00633
\(683\) −2.68994e15 −0.692516 −0.346258 0.938139i \(-0.612548\pi\)
−0.346258 + 0.938139i \(0.612548\pi\)
\(684\) −1.13192e13 −0.00289072
\(685\) −2.62087e14 −0.0663969
\(686\) −1.75130e15 −0.440126
\(687\) 3.03098e15 0.755651
\(688\) −4.74897e15 −1.17453
\(689\) 2.23616e15 0.548651
\(690\) 9.92414e14 0.241559
\(691\) 1.62328e15 0.391979 0.195990 0.980606i \(-0.437208\pi\)
0.195990 + 0.980606i \(0.437208\pi\)
\(692\) −2.51488e14 −0.0602469
\(693\) 5.64449e14 0.134150
\(694\) −2.48709e15 −0.586429
\(695\) 2.38197e15 0.557211
\(696\) −4.25135e15 −0.986680
\(697\) −2.74035e15 −0.630996
\(698\) 3.03916e15 0.694303
\(699\) −1.18059e15 −0.267593
\(700\) −2.87677e13 −0.00646943
\(701\) −4.75356e15 −1.06064 −0.530322 0.847796i \(-0.677929\pi\)
−0.530322 + 0.847796i \(0.677929\pi\)
\(702\) 1.20196e15 0.266094
\(703\) −8.60253e14 −0.188961
\(704\) −8.27894e15 −1.80437
\(705\) 2.31572e14 0.0500779
\(706\) 7.53596e14 0.161701
\(707\) −2.87049e13 −0.00611152
\(708\) −1.14756e13 −0.00242433
\(709\) 3.22804e15 0.676681 0.338340 0.941024i \(-0.390134\pi\)
0.338340 + 0.941024i \(0.390134\pi\)
\(710\) −2.53158e15 −0.526588
\(711\) −1.59229e15 −0.328654
\(712\) 2.27817e15 0.466602
\(713\) −3.42917e15 −0.696942
\(714\) −8.54405e14 −0.172315
\(715\) −4.38416e15 −0.877410
\(716\) −8.83994e13 −0.0175561
\(717\) 5.54031e15 1.09189
\(718\) −9.54480e15 −1.86673
\(719\) 3.05161e14 0.0592270 0.0296135 0.999561i \(-0.490572\pi\)
0.0296135 + 0.999561i \(0.490572\pi\)
\(720\) 5.96266e14 0.114845
\(721\) −6.55374e14 −0.125269
\(722\) 4.81112e15 0.912623
\(723\) 1.62380e15 0.305684
\(724\) 2.80333e14 0.0523735
\(725\) 7.92379e15 1.46918
\(726\) −6.38303e15 −1.17456
\(727\) −7.96110e15 −1.45390 −0.726949 0.686691i \(-0.759062\pi\)
−0.726949 + 0.686691i \(0.759062\pi\)
\(728\) 1.80930e15 0.327934
\(729\) 2.05891e14 0.0370370
\(730\) −2.02652e15 −0.361806
\(731\) −9.05385e15 −1.60431
\(732\) −6.26016e13 −0.0110097
\(733\) −3.48049e15 −0.607531 −0.303765 0.952747i \(-0.598244\pi\)
−0.303765 + 0.952747i \(0.598244\pi\)
\(734\) 4.51984e15 0.783059
\(735\) 1.13345e15 0.194905
\(736\) −4.50848e14 −0.0769488
\(737\) 8.88302e15 1.50484
\(738\) −9.31915e14 −0.156699
\(739\) 4.41230e15 0.736412 0.368206 0.929744i \(-0.379972\pi\)
0.368206 + 0.929744i \(0.379972\pi\)
\(740\) −4.87656e13 −0.00807866
\(741\) −1.32686e15 −0.218185
\(742\) −5.40564e14 −0.0882319
\(743\) 4.00429e14 0.0648765 0.0324382 0.999474i \(-0.489673\pi\)
0.0324382 + 0.999474i \(0.489673\pi\)
\(744\) −2.12907e15 −0.342404
\(745\) −3.13076e15 −0.499793
\(746\) 8.54399e14 0.135393
\(747\) −3.25627e15 −0.512221
\(748\) −4.77734e14 −0.0745980
\(749\) 3.41654e14 0.0529586
\(750\) −2.46393e15 −0.379133
\(751\) −5.50767e15 −0.841294 −0.420647 0.907224i \(-0.638197\pi\)
−0.420647 + 0.907224i \(0.638197\pi\)
\(752\) 1.55153e15 0.235268
\(753\) −1.88316e15 −0.283475
\(754\) −1.55714e16 −2.32695
\(755\) 2.65466e15 0.393823
\(756\) 9.68390e12 0.00142620
\(757\) −6.23781e15 −0.912021 −0.456010 0.889974i \(-0.650722\pi\)
−0.456010 + 0.889974i \(0.650722\pi\)
\(758\) 9.14174e14 0.132693
\(759\) −8.37458e15 −1.20679
\(760\) −6.80188e14 −0.0973093
\(761\) −1.20166e16 −1.70673 −0.853367 0.521311i \(-0.825443\pi\)
−0.853367 + 0.521311i \(0.825443\pi\)
\(762\) 6.56743e15 0.926069
\(763\) −2.33925e14 −0.0327485
\(764\) −7.33828e13 −0.0101995
\(765\) 1.13677e15 0.156869
\(766\) 7.21226e15 0.988128
\(767\) −1.34519e15 −0.182983
\(768\) −4.13208e14 −0.0558062
\(769\) −7.44470e15 −0.998280 −0.499140 0.866521i \(-0.666351\pi\)
−0.499140 + 0.866521i \(0.666351\pi\)
\(770\) 1.05982e15 0.141102
\(771\) 2.98065e15 0.394014
\(772\) −2.69892e14 −0.0354238
\(773\) −1.34774e16 −1.75638 −0.878192 0.478308i \(-0.841250\pi\)
−0.878192 + 0.478308i \(0.841250\pi\)
\(774\) −3.07895e15 −0.398408
\(775\) 3.96821e15 0.509842
\(776\) 3.39102e15 0.432604
\(777\) 7.35973e14 0.0932280
\(778\) 1.11921e15 0.140775
\(779\) 1.02876e15 0.128486
\(780\) −7.52163e13 −0.00932806
\(781\) 2.13630e16 2.63076
\(782\) 1.26766e16 1.55012
\(783\) −2.66734e15 −0.323883
\(784\) 7.59413e15 0.915672
\(785\) −3.46290e15 −0.414626
\(786\) −4.03732e15 −0.480031
\(787\) 6.17125e14 0.0728638 0.0364319 0.999336i \(-0.488401\pi\)
0.0364319 + 0.999336i \(0.488401\pi\)
\(788\) −2.33276e14 −0.0273512
\(789\) 3.33995e15 0.388881
\(790\) −2.98971e15 −0.345684
\(791\) 1.72972e15 0.198612
\(792\) −5.19952e15 −0.592891
\(793\) −7.33829e15 −0.830984
\(794\) 1.20054e16 1.35010
\(795\) 7.19213e14 0.0803227
\(796\) 3.84961e13 0.00426968
\(797\) −8.57078e15 −0.944060 −0.472030 0.881582i \(-0.656479\pi\)
−0.472030 + 0.881582i \(0.656479\pi\)
\(798\) 3.20752e14 0.0350876
\(799\) 2.95798e15 0.321357
\(800\) 5.21718e14 0.0562912
\(801\) 1.42935e15 0.153165
\(802\) 9.19342e15 0.978404
\(803\) 1.71010e16 1.80753
\(804\) 1.52400e14 0.0159985
\(805\) 9.37269e14 0.0977208
\(806\) −7.79814e15 −0.807511
\(807\) 1.27985e15 0.131630
\(808\) 2.64420e14 0.0270105
\(809\) −1.94459e16 −1.97293 −0.986463 0.163981i \(-0.947566\pi\)
−0.986463 + 0.163981i \(0.947566\pi\)
\(810\) 3.86584e14 0.0389562
\(811\) 1.86366e16 1.86531 0.932657 0.360765i \(-0.117484\pi\)
0.932657 + 0.360765i \(0.117484\pi\)
\(812\) −1.25456e14 −0.0124719
\(813\) −5.25225e15 −0.518618
\(814\) −1.23472e16 −1.21097
\(815\) −1.08688e15 −0.105881
\(816\) 7.61639e15 0.736976
\(817\) 3.39890e15 0.326676
\(818\) 3.23988e14 0.0309304
\(819\) 1.13517e15 0.107646
\(820\) 5.83176e13 0.00549317
\(821\) 1.20101e16 1.12373 0.561863 0.827230i \(-0.310085\pi\)
0.561863 + 0.827230i \(0.310085\pi\)
\(822\) −1.13848e15 −0.105811
\(823\) 9.51460e15 0.878398 0.439199 0.898390i \(-0.355262\pi\)
0.439199 + 0.898390i \(0.355262\pi\)
\(824\) 6.03709e15 0.553641
\(825\) 9.69101e15 0.882820
\(826\) 3.25184e14 0.0294265
\(827\) 6.13526e15 0.551509 0.275755 0.961228i \(-0.411072\pi\)
0.275755 + 0.961228i \(0.411072\pi\)
\(828\) −1.43678e14 −0.0128299
\(829\) −2.74031e15 −0.243080 −0.121540 0.992587i \(-0.538783\pi\)
−0.121540 + 0.992587i \(0.538783\pi\)
\(830\) −6.11402e15 −0.538763
\(831\) 2.92103e15 0.255700
\(832\) −1.66498e16 −1.44788
\(833\) 1.44781e16 1.25073
\(834\) 1.03470e16 0.887977
\(835\) 4.70353e15 0.401004
\(836\) 1.79346e14 0.0151900
\(837\) −1.33580e15 −0.112396
\(838\) −3.65995e15 −0.305938
\(839\) −1.56360e16 −1.29848 −0.649240 0.760584i \(-0.724913\pi\)
−0.649240 + 0.760584i \(0.724913\pi\)
\(840\) 5.81922e14 0.0480097
\(841\) 2.23550e16 1.83230
\(842\) −6.11481e15 −0.497928
\(843\) −1.02817e15 −0.0831786
\(844\) 3.44643e13 0.00277004
\(845\) −4.35377e15 −0.347659
\(846\) 1.00592e15 0.0798046
\(847\) −6.02834e15 −0.475159
\(848\) 4.81873e15 0.377359
\(849\) −1.99603e15 −0.155301
\(850\) −1.46693e16 −1.13398
\(851\) −1.09194e16 −0.838664
\(852\) 3.66511e14 0.0279685
\(853\) −1.36871e16 −1.03775 −0.518875 0.854850i \(-0.673649\pi\)
−0.518875 + 0.854850i \(0.673649\pi\)
\(854\) 1.77394e15 0.133635
\(855\) −4.26756e14 −0.0319423
\(856\) −3.14721e15 −0.234056
\(857\) 1.05025e16 0.776065 0.388033 0.921646i \(-0.373155\pi\)
0.388033 + 0.921646i \(0.373155\pi\)
\(858\) −1.90443e16 −1.39825
\(859\) −2.49090e16 −1.81716 −0.908580 0.417710i \(-0.862833\pi\)
−0.908580 + 0.417710i \(0.862833\pi\)
\(860\) 1.92675e14 0.0139664
\(861\) −8.80132e14 −0.0633914
\(862\) 2.33582e16 1.67166
\(863\) 1.65938e16 1.18001 0.590005 0.807400i \(-0.299126\pi\)
0.590005 + 0.807400i \(0.299126\pi\)
\(864\) −1.75623e14 −0.0124095
\(865\) −9.48162e15 −0.665724
\(866\) 1.12112e16 0.782176
\(867\) 6.19248e15 0.429299
\(868\) −6.28279e13 −0.00432807
\(869\) 2.52289e16 1.72699
\(870\) −5.00823e15 −0.340665
\(871\) 1.78647e16 1.20752
\(872\) 2.15484e15 0.144735
\(873\) 2.12756e15 0.142005
\(874\) −4.75891e15 −0.315642
\(875\) −2.32702e15 −0.153375
\(876\) 2.93391e14 0.0192165
\(877\) 8.89740e15 0.579116 0.289558 0.957161i \(-0.406492\pi\)
0.289558 + 0.957161i \(0.406492\pi\)
\(878\) −9.29698e15 −0.601343
\(879\) 7.38347e14 0.0474594
\(880\) −9.44749e15 −0.603478
\(881\) 2.46827e16 1.56684 0.783421 0.621491i \(-0.213473\pi\)
0.783421 + 0.621491i \(0.213473\pi\)
\(882\) 4.92359e15 0.310602
\(883\) −5.66978e15 −0.355453 −0.177727 0.984080i \(-0.556874\pi\)
−0.177727 + 0.984080i \(0.556874\pi\)
\(884\) −9.60773e14 −0.0598595
\(885\) −4.32653e14 −0.0267887
\(886\) 1.03042e15 0.0634059
\(887\) 2.27269e16 1.38983 0.694913 0.719094i \(-0.255443\pi\)
0.694913 + 0.719094i \(0.255443\pi\)
\(888\) −6.77955e15 −0.412031
\(889\) 6.20250e15 0.374634
\(890\) 2.68376e15 0.161101
\(891\) −3.26223e15 −0.194620
\(892\) 1.03240e13 0.000612125 0
\(893\) −1.11045e15 −0.0654361
\(894\) −1.35997e16 −0.796476
\(895\) −3.33284e15 −0.193994
\(896\) 3.76879e15 0.218026
\(897\) −1.68422e16 −0.968367
\(898\) −1.93615e16 −1.10642
\(899\) 1.73053e16 0.982883
\(900\) 1.66263e14 0.00938557
\(901\) 9.18685e15 0.515442
\(902\) 1.47657e16 0.823411
\(903\) −2.90787e15 −0.161173
\(904\) −1.59336e16 −0.877785
\(905\) 1.05691e16 0.578723
\(906\) 1.15316e16 0.627601
\(907\) 1.22135e16 0.660692 0.330346 0.943860i \(-0.392835\pi\)
0.330346 + 0.943860i \(0.392835\pi\)
\(908\) −6.06778e14 −0.0326256
\(909\) 1.65899e14 0.00886632
\(910\) 2.13141e15 0.113224
\(911\) 3.02855e16 1.59913 0.799565 0.600579i \(-0.205063\pi\)
0.799565 + 0.600579i \(0.205063\pi\)
\(912\) −2.85927e15 −0.150066
\(913\) 5.15938e16 2.69158
\(914\) 1.21654e16 0.630842
\(915\) −2.36021e15 −0.121656
\(916\) −8.23921e14 −0.0422144
\(917\) −3.81298e15 −0.194193
\(918\) 4.93802e15 0.249988
\(919\) 8.27134e15 0.416237 0.208118 0.978104i \(-0.433266\pi\)
0.208118 + 0.978104i \(0.433266\pi\)
\(920\) −8.63382e15 −0.431887
\(921\) −9.44510e14 −0.0469655
\(922\) −6.43996e15 −0.318320
\(923\) 4.29632e16 2.11100
\(924\) −1.53436e14 −0.00749430
\(925\) 1.26359e16 0.613517
\(926\) −2.28427e16 −1.10252
\(927\) 3.78772e15 0.181735
\(928\) 2.27521e15 0.108519
\(929\) −2.33636e16 −1.10778 −0.553889 0.832590i \(-0.686857\pi\)
−0.553889 + 0.832590i \(0.686857\pi\)
\(930\) −2.50811e15 −0.118220
\(931\) −5.43523e15 −0.254680
\(932\) 3.20923e14 0.0149491
\(933\) 3.29594e15 0.152626
\(934\) 2.76655e16 1.27359
\(935\) −1.80115e16 −0.824303
\(936\) −1.04568e16 −0.475753
\(937\) 2.78969e16 1.26179 0.630897 0.775866i \(-0.282687\pi\)
0.630897 + 0.775866i \(0.282687\pi\)
\(938\) −4.31858e15 −0.194189
\(939\) −1.86178e16 −0.832276
\(940\) −6.29489e13 −0.00279760
\(941\) −4.19435e16 −1.85320 −0.926599 0.376052i \(-0.877281\pi\)
−0.926599 + 0.376052i \(0.877281\pi\)
\(942\) −1.50425e16 −0.660752
\(943\) 1.30583e16 0.570258
\(944\) −2.89877e15 −0.125854
\(945\) 3.65103e14 0.0157594
\(946\) 4.87842e16 2.09353
\(947\) −3.38432e16 −1.44393 −0.721966 0.691928i \(-0.756761\pi\)
−0.721966 + 0.691928i \(0.756761\pi\)
\(948\) 4.32837e14 0.0183602
\(949\) 3.43919e16 1.45042
\(950\) 5.50698e15 0.230905
\(951\) 7.76932e15 0.323884
\(952\) 7.43317e15 0.308085
\(953\) 1.35827e16 0.559724 0.279862 0.960040i \(-0.409711\pi\)
0.279862 + 0.960040i \(0.409711\pi\)
\(954\) 3.12418e15 0.128003
\(955\) −2.76668e15 −0.112704
\(956\) −1.50604e15 −0.0609983
\(957\) 4.22624e16 1.70192
\(958\) 4.42238e15 0.177070
\(959\) −1.07522e15 −0.0428049
\(960\) −5.35507e15 −0.211970
\(961\) −1.67420e16 −0.658914
\(962\) −2.48315e16 −0.971716
\(963\) −1.97458e15 −0.0768300
\(964\) −4.41403e14 −0.0170770
\(965\) −1.01755e16 −0.391431
\(966\) 4.07139e15 0.155729
\(967\) −2.15462e16 −0.819455 −0.409728 0.912208i \(-0.634376\pi\)
−0.409728 + 0.912208i \(0.634376\pi\)
\(968\) 5.55312e16 2.10001
\(969\) −5.45116e15 −0.204978
\(970\) 3.99473e15 0.149363
\(971\) −3.86429e16 −1.43669 −0.718347 0.695685i \(-0.755101\pi\)
−0.718347 + 0.695685i \(0.755101\pi\)
\(972\) −5.59680e13 −0.00206907
\(973\) 9.77206e15 0.359224
\(974\) 4.47162e16 1.63452
\(975\) 1.94897e16 0.708400
\(976\) −1.58134e16 −0.571546
\(977\) −4.19060e16 −1.50611 −0.753055 0.657958i \(-0.771420\pi\)
−0.753055 + 0.657958i \(0.771420\pi\)
\(978\) −4.72131e15 −0.168732
\(979\) −2.26472e16 −0.804839
\(980\) −3.08110e14 −0.0108883
\(981\) 1.35197e15 0.0475101
\(982\) 1.90255e16 0.664849
\(983\) 6.34482e15 0.220483 0.110241 0.993905i \(-0.464838\pi\)
0.110241 + 0.993905i \(0.464838\pi\)
\(984\) 8.10749e15 0.280165
\(985\) −8.79499e15 −0.302229
\(986\) −6.39725e16 −2.18610
\(987\) 9.50028e14 0.0322844
\(988\) 3.60683e14 0.0121889
\(989\) 4.31432e16 1.44988
\(990\) −6.12520e15 −0.204704
\(991\) −5.24907e16 −1.74452 −0.872262 0.489039i \(-0.837348\pi\)
−0.872262 + 0.489039i \(0.837348\pi\)
\(992\) 1.13942e15 0.0376590
\(993\) −1.02635e16 −0.337345
\(994\) −1.03858e16 −0.339482
\(995\) 1.45138e15 0.0471796
\(996\) 8.85162e14 0.0286152
\(997\) −1.01187e16 −0.325313 −0.162656 0.986683i \(-0.552006\pi\)
−0.162656 + 0.986683i \(0.552006\pi\)
\(998\) −4.81999e16 −1.54109
\(999\) −4.25355e15 −0.135251
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.d.1.7 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.7 28 1.1 even 1 trivial