Properties

Label 177.12.a.d.1.18
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 1.18
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+31.6204 q^{2} +243.000 q^{3} -1048.15 q^{4} -7920.33 q^{5} +7683.75 q^{6} +55386.7 q^{7} -97901.5 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+31.6204 q^{2} +243.000 q^{3} -1048.15 q^{4} -7920.33 q^{5} +7683.75 q^{6} +55386.7 q^{7} -97901.5 q^{8} +59049.0 q^{9} -250444. q^{10} +749288. q^{11} -254701. q^{12} -855073. q^{13} +1.75135e6 q^{14} -1.92464e6 q^{15} -949065. q^{16} -8.42911e6 q^{17} +1.86715e6 q^{18} -235588. q^{19} +8.30172e6 q^{20} +1.34590e7 q^{21} +2.36928e7 q^{22} +1.10354e7 q^{23} -2.37901e7 q^{24} +1.39036e7 q^{25} -2.70377e7 q^{26} +1.43489e7 q^{27} -5.80537e7 q^{28} -3.07296e7 q^{29} -6.08579e7 q^{30} +9.66575e7 q^{31} +1.70492e8 q^{32} +1.82077e8 q^{33} -2.66531e8 q^{34} -4.38681e8 q^{35} -6.18923e7 q^{36} -7.15175e8 q^{37} -7.44938e6 q^{38} -2.07783e8 q^{39} +7.75412e8 q^{40} +1.96874e8 q^{41} +4.25578e8 q^{42} +1.69435e9 q^{43} -7.85368e8 q^{44} -4.67688e8 q^{45} +3.48942e8 q^{46} -3.75191e8 q^{47} -2.30623e8 q^{48} +1.09036e9 q^{49} +4.39636e8 q^{50} -2.04827e9 q^{51} +8.96246e8 q^{52} -2.44790e9 q^{53} +4.53718e8 q^{54} -5.93461e9 q^{55} -5.42244e9 q^{56} -5.72479e7 q^{57} -9.71681e8 q^{58} +7.14924e8 q^{59} +2.01732e9 q^{60} +1.16084e9 q^{61} +3.05635e9 q^{62} +3.27053e9 q^{63} +7.33472e9 q^{64} +6.77246e9 q^{65} +5.75734e9 q^{66} +3.45178e9 q^{67} +8.83499e9 q^{68} +2.68159e9 q^{69} -1.38713e10 q^{70} +2.89581e10 q^{71} -5.78098e9 q^{72} -3.54355e9 q^{73} -2.26141e10 q^{74} +3.37857e9 q^{75} +2.46932e8 q^{76} +4.15006e10 q^{77} -6.57016e9 q^{78} +3.21817e10 q^{79} +7.51691e9 q^{80} +3.48678e9 q^{81} +6.22523e9 q^{82} +5.97408e10 q^{83} -1.41070e10 q^{84} +6.67613e10 q^{85} +5.35760e10 q^{86} -7.46729e9 q^{87} -7.33564e10 q^{88} +8.65853e10 q^{89} -1.47885e10 q^{90} -4.73596e10 q^{91} -1.15667e10 q^{92} +2.34878e10 q^{93} -1.18637e10 q^{94} +1.86594e9 q^{95} +4.14297e10 q^{96} -1.01633e11 q^{97} +3.44776e10 q^{98} +4.42447e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 31.6204 0.698718 0.349359 0.936989i \(-0.386399\pi\)
0.349359 + 0.936989i \(0.386399\pi\)
\(3\) 243.000 0.577350
\(4\) −1048.15 −0.511793
\(5\) −7920.33 −1.13347 −0.566733 0.823902i \(-0.691793\pi\)
−0.566733 + 0.823902i \(0.691793\pi\)
\(6\) 7683.75 0.403405
\(7\) 55386.7 1.24556 0.622782 0.782395i \(-0.286002\pi\)
0.622782 + 0.782395i \(0.286002\pi\)
\(8\) −97901.5 −1.05632
\(9\) 59049.0 0.333333
\(10\) −250444. −0.791973
\(11\) 749288. 1.40278 0.701389 0.712779i \(-0.252564\pi\)
0.701389 + 0.712779i \(0.252564\pi\)
\(12\) −254701. −0.295484
\(13\) −855073. −0.638726 −0.319363 0.947632i \(-0.603469\pi\)
−0.319363 + 0.947632i \(0.603469\pi\)
\(14\) 1.75135e6 0.870298
\(15\) −1.92464e6 −0.654407
\(16\) −949065. −0.226275
\(17\) −8.42911e6 −1.43983 −0.719917 0.694060i \(-0.755820\pi\)
−0.719917 + 0.694060i \(0.755820\pi\)
\(18\) 1.86715e6 0.232906
\(19\) −235588. −0.0218277 −0.0109139 0.999940i \(-0.503474\pi\)
−0.0109139 + 0.999940i \(0.503474\pi\)
\(20\) 8.30172e6 0.580100
\(21\) 1.34590e7 0.719127
\(22\) 2.36928e7 0.980146
\(23\) 1.10354e7 0.357506 0.178753 0.983894i \(-0.442794\pi\)
0.178753 + 0.983894i \(0.442794\pi\)
\(24\) −2.37901e7 −0.609865
\(25\) 1.39036e7 0.284745
\(26\) −2.70377e7 −0.446289
\(27\) 1.43489e7 0.192450
\(28\) −5.80537e7 −0.637471
\(29\) −3.07296e7 −0.278207 −0.139103 0.990278i \(-0.544422\pi\)
−0.139103 + 0.990278i \(0.544422\pi\)
\(30\) −6.08579e7 −0.457246
\(31\) 9.66575e7 0.606382 0.303191 0.952930i \(-0.401948\pi\)
0.303191 + 0.952930i \(0.401948\pi\)
\(32\) 1.70492e8 0.898215
\(33\) 1.82077e8 0.809894
\(34\) −2.66531e8 −1.00604
\(35\) −4.38681e8 −1.41180
\(36\) −6.18923e7 −0.170598
\(37\) −7.15175e8 −1.69552 −0.847759 0.530381i \(-0.822049\pi\)
−0.847759 + 0.530381i \(0.822049\pi\)
\(38\) −7.44938e6 −0.0152514
\(39\) −2.07783e8 −0.368769
\(40\) 7.75412e8 1.19730
\(41\) 1.96874e8 0.265386 0.132693 0.991157i \(-0.457638\pi\)
0.132693 + 0.991157i \(0.457638\pi\)
\(42\) 4.25578e8 0.502467
\(43\) 1.69435e9 1.75763 0.878813 0.477166i \(-0.158336\pi\)
0.878813 + 0.477166i \(0.158336\pi\)
\(44\) −7.85368e8 −0.717932
\(45\) −4.67688e8 −0.377822
\(46\) 3.48942e8 0.249796
\(47\) −3.75191e8 −0.238624 −0.119312 0.992857i \(-0.538069\pi\)
−0.119312 + 0.992857i \(0.538069\pi\)
\(48\) −2.30623e8 −0.130640
\(49\) 1.09036e9 0.551431
\(50\) 4.39636e8 0.198956
\(51\) −2.04827e9 −0.831289
\(52\) 8.96246e8 0.326896
\(53\) −2.44790e9 −0.804038 −0.402019 0.915631i \(-0.631691\pi\)
−0.402019 + 0.915631i \(0.631691\pi\)
\(54\) 4.53718e8 0.134468
\(55\) −5.93461e9 −1.59000
\(56\) −5.42244e9 −1.31571
\(57\) −5.72479e7 −0.0126022
\(58\) −9.71681e8 −0.194388
\(59\) 7.14924e8 0.130189
\(60\) 2.01732e9 0.334921
\(61\) 1.16084e9 0.175978 0.0879892 0.996121i \(-0.471956\pi\)
0.0879892 + 0.996121i \(0.471956\pi\)
\(62\) 3.05635e9 0.423690
\(63\) 3.27053e9 0.415188
\(64\) 7.33472e9 0.853874
\(65\) 6.77246e9 0.723974
\(66\) 5.75734e9 0.565888
\(67\) 3.45178e9 0.312343 0.156172 0.987730i \(-0.450085\pi\)
0.156172 + 0.987730i \(0.450085\pi\)
\(68\) 8.83499e9 0.736897
\(69\) 2.68159e9 0.206406
\(70\) −1.38713e10 −0.986454
\(71\) 2.89581e10 1.90480 0.952401 0.304848i \(-0.0986057\pi\)
0.952401 + 0.304848i \(0.0986057\pi\)
\(72\) −5.78098e9 −0.352106
\(73\) −3.54355e9 −0.200061 −0.100031 0.994984i \(-0.531894\pi\)
−0.100031 + 0.994984i \(0.531894\pi\)
\(74\) −2.26141e10 −1.18469
\(75\) 3.37857e9 0.164398
\(76\) 2.46932e8 0.0111713
\(77\) 4.15006e10 1.74725
\(78\) −6.57016e9 −0.257665
\(79\) 3.21817e10 1.17668 0.588342 0.808612i \(-0.299781\pi\)
0.588342 + 0.808612i \(0.299781\pi\)
\(80\) 7.51691e9 0.256475
\(81\) 3.48678e9 0.111111
\(82\) 6.22523e9 0.185430
\(83\) 5.97408e10 1.66472 0.832360 0.554235i \(-0.186989\pi\)
0.832360 + 0.554235i \(0.186989\pi\)
\(84\) −1.41070e10 −0.368044
\(85\) 6.67613e10 1.63200
\(86\) 5.35760e10 1.22809
\(87\) −7.46729e9 −0.160623
\(88\) −7.33564e10 −1.48178
\(89\) 8.65853e10 1.64361 0.821806 0.569768i \(-0.192967\pi\)
0.821806 + 0.569768i \(0.192967\pi\)
\(90\) −1.47885e10 −0.263991
\(91\) −4.73596e10 −0.795574
\(92\) −1.15667e10 −0.182969
\(93\) 2.34878e10 0.350095
\(94\) −1.18637e10 −0.166731
\(95\) 1.86594e9 0.0247410
\(96\) 4.14297e10 0.518585
\(97\) −1.01633e11 −1.20168 −0.600841 0.799369i \(-0.705167\pi\)
−0.600841 + 0.799369i \(0.705167\pi\)
\(98\) 3.44776e10 0.385295
\(99\) 4.42447e10 0.467592
\(100\) −1.45730e10 −0.145730
\(101\) 1.49093e11 1.41153 0.705763 0.708448i \(-0.250604\pi\)
0.705763 + 0.708448i \(0.250604\pi\)
\(102\) −6.47671e10 −0.580836
\(103\) 2.23141e10 0.189659 0.0948296 0.995494i \(-0.469769\pi\)
0.0948296 + 0.995494i \(0.469769\pi\)
\(104\) 8.37129e10 0.674697
\(105\) −1.06600e11 −0.815106
\(106\) −7.74035e10 −0.561796
\(107\) 2.06906e11 1.42614 0.713071 0.701091i \(-0.247304\pi\)
0.713071 + 0.701091i \(0.247304\pi\)
\(108\) −1.50398e10 −0.0984946
\(109\) −1.32255e8 −0.000823314 0 −0.000411657 1.00000i \(-0.500131\pi\)
−0.000411657 1.00000i \(0.500131\pi\)
\(110\) −1.87654e11 −1.11096
\(111\) −1.73787e11 −0.978908
\(112\) −5.25656e10 −0.281840
\(113\) −2.69080e11 −1.37388 −0.686942 0.726712i \(-0.741047\pi\)
−0.686942 + 0.726712i \(0.741047\pi\)
\(114\) −1.81020e9 −0.00880541
\(115\) −8.74038e10 −0.405221
\(116\) 3.22093e10 0.142384
\(117\) −5.04912e10 −0.212909
\(118\) 2.26062e10 0.0909653
\(119\) −4.66860e11 −1.79341
\(120\) 1.88425e11 0.691261
\(121\) 2.76120e11 0.967785
\(122\) 3.67063e10 0.122959
\(123\) 4.78404e10 0.153220
\(124\) −1.01312e11 −0.310342
\(125\) 2.76614e11 0.810717
\(126\) 1.03415e11 0.290099
\(127\) 2.62532e11 0.705119 0.352559 0.935789i \(-0.385311\pi\)
0.352559 + 0.935789i \(0.385311\pi\)
\(128\) −1.17242e11 −0.301598
\(129\) 4.11727e11 1.01477
\(130\) 2.14148e11 0.505854
\(131\) 5.48081e11 1.24123 0.620615 0.784115i \(-0.286883\pi\)
0.620615 + 0.784115i \(0.286883\pi\)
\(132\) −1.90844e11 −0.414498
\(133\) −1.30484e10 −0.0271878
\(134\) 1.09147e11 0.218240
\(135\) −1.13648e11 −0.218136
\(136\) 8.25222e11 1.52092
\(137\) −3.83305e11 −0.678550 −0.339275 0.940687i \(-0.610182\pi\)
−0.339275 + 0.940687i \(0.610182\pi\)
\(138\) 8.47930e10 0.144220
\(139\) −8.99134e10 −0.146975 −0.0734874 0.997296i \(-0.523413\pi\)
−0.0734874 + 0.997296i \(0.523413\pi\)
\(140\) 4.59805e11 0.722552
\(141\) −9.11714e10 −0.137770
\(142\) 9.15667e11 1.33092
\(143\) −6.40695e11 −0.895990
\(144\) −5.60413e10 −0.0754249
\(145\) 2.43389e11 0.315338
\(146\) −1.12048e11 −0.139786
\(147\) 2.64957e11 0.318369
\(148\) 7.49612e11 0.867755
\(149\) −9.17759e10 −0.102377 −0.0511887 0.998689i \(-0.516301\pi\)
−0.0511887 + 0.998689i \(0.516301\pi\)
\(150\) 1.06831e11 0.114868
\(151\) −4.78141e11 −0.495658 −0.247829 0.968804i \(-0.579717\pi\)
−0.247829 + 0.968804i \(0.579717\pi\)
\(152\) 2.30644e10 0.0230570
\(153\) −4.97730e11 −0.479945
\(154\) 1.31226e12 1.22084
\(155\) −7.65560e11 −0.687313
\(156\) 2.17788e11 0.188733
\(157\) 2.26002e11 0.189088 0.0945440 0.995521i \(-0.469861\pi\)
0.0945440 + 0.995521i \(0.469861\pi\)
\(158\) 1.01760e12 0.822171
\(159\) −5.94840e11 −0.464211
\(160\) −1.35036e12 −1.01810
\(161\) 6.11213e11 0.445297
\(162\) 1.10253e11 0.0776353
\(163\) 2.24541e12 1.52849 0.764247 0.644924i \(-0.223111\pi\)
0.764247 + 0.644924i \(0.223111\pi\)
\(164\) −2.06354e11 −0.135823
\(165\) −1.44211e12 −0.917987
\(166\) 1.88902e12 1.16317
\(167\) 8.48590e11 0.505542 0.252771 0.967526i \(-0.418658\pi\)
0.252771 + 0.967526i \(0.418658\pi\)
\(168\) −1.31765e12 −0.759626
\(169\) −1.06101e12 −0.592029
\(170\) 2.11102e12 1.14031
\(171\) −1.39112e10 −0.00727590
\(172\) −1.77594e12 −0.899541
\(173\) −2.23867e12 −1.09834 −0.549170 0.835711i \(-0.685056\pi\)
−0.549170 + 0.835711i \(0.685056\pi\)
\(174\) −2.36118e11 −0.112230
\(175\) 7.70072e11 0.354668
\(176\) −7.11123e11 −0.317413
\(177\) 1.73727e11 0.0751646
\(178\) 2.73786e12 1.14842
\(179\) −1.28996e12 −0.524669 −0.262335 0.964977i \(-0.584492\pi\)
−0.262335 + 0.964977i \(0.584492\pi\)
\(180\) 4.90208e11 0.193367
\(181\) 3.31228e12 1.26735 0.633673 0.773601i \(-0.281546\pi\)
0.633673 + 0.773601i \(0.281546\pi\)
\(182\) −1.49753e12 −0.555882
\(183\) 2.82085e11 0.101601
\(184\) −1.08038e12 −0.377640
\(185\) 5.66442e12 1.92181
\(186\) 7.42692e11 0.244618
\(187\) −6.31583e12 −2.01977
\(188\) 3.93257e11 0.122126
\(189\) 7.94739e11 0.239709
\(190\) 5.90016e10 0.0172870
\(191\) 5.17579e12 1.47331 0.736654 0.676270i \(-0.236405\pi\)
0.736654 + 0.676270i \(0.236405\pi\)
\(192\) 1.78234e12 0.492984
\(193\) 4.26973e12 1.14772 0.573859 0.818954i \(-0.305446\pi\)
0.573859 + 0.818954i \(0.305446\pi\)
\(194\) −3.21367e12 −0.839637
\(195\) 1.64571e12 0.417987
\(196\) −1.14286e12 −0.282219
\(197\) −1.76272e12 −0.423273 −0.211636 0.977349i \(-0.567879\pi\)
−0.211636 + 0.977349i \(0.567879\pi\)
\(198\) 1.39903e12 0.326715
\(199\) −1.04809e12 −0.238071 −0.119035 0.992890i \(-0.537980\pi\)
−0.119035 + 0.992890i \(0.537980\pi\)
\(200\) −1.36118e12 −0.300781
\(201\) 8.38783e11 0.180331
\(202\) 4.71437e12 0.986259
\(203\) −1.70201e12 −0.346524
\(204\) 2.14690e12 0.425448
\(205\) −1.55931e12 −0.300806
\(206\) 7.05579e11 0.132518
\(207\) 6.51627e11 0.119169
\(208\) 8.11520e11 0.144528
\(209\) −1.76523e11 −0.0306194
\(210\) −3.37072e12 −0.569529
\(211\) −9.21253e12 −1.51644 −0.758220 0.651999i \(-0.773931\pi\)
−0.758220 + 0.651999i \(0.773931\pi\)
\(212\) 2.56577e12 0.411501
\(213\) 7.03683e12 1.09974
\(214\) 6.54245e12 0.996472
\(215\) −1.34198e13 −1.99221
\(216\) −1.40478e12 −0.203288
\(217\) 5.35354e12 0.755288
\(218\) −4.18194e9 −0.000575264 0
\(219\) −8.61082e11 −0.115505
\(220\) 6.22037e12 0.813751
\(221\) 7.20750e12 0.919659
\(222\) −5.49522e12 −0.683981
\(223\) 7.77435e12 0.944034 0.472017 0.881590i \(-0.343526\pi\)
0.472017 + 0.881590i \(0.343526\pi\)
\(224\) 9.44301e12 1.11878
\(225\) 8.20991e11 0.0949150
\(226\) −8.50841e12 −0.959958
\(227\) 5.30903e12 0.584620 0.292310 0.956324i \(-0.405576\pi\)
0.292310 + 0.956324i \(0.405576\pi\)
\(228\) 6.00045e10 0.00644974
\(229\) 1.30474e13 1.36908 0.684540 0.728976i \(-0.260003\pi\)
0.684540 + 0.728976i \(0.260003\pi\)
\(230\) −2.76374e12 −0.283135
\(231\) 1.00846e13 1.00878
\(232\) 3.00847e12 0.293875
\(233\) 7.49739e12 0.715242 0.357621 0.933867i \(-0.383588\pi\)
0.357621 + 0.933867i \(0.383588\pi\)
\(234\) −1.59655e12 −0.148763
\(235\) 2.97164e12 0.270472
\(236\) −7.49350e11 −0.0666298
\(237\) 7.82016e12 0.679359
\(238\) −1.47623e13 −1.25309
\(239\) 1.12891e13 0.936424 0.468212 0.883616i \(-0.344898\pi\)
0.468212 + 0.883616i \(0.344898\pi\)
\(240\) 1.82661e12 0.148076
\(241\) −9.51986e12 −0.754287 −0.377144 0.926155i \(-0.623094\pi\)
−0.377144 + 0.926155i \(0.623094\pi\)
\(242\) 8.73102e12 0.676209
\(243\) 8.47289e11 0.0641500
\(244\) −1.21674e12 −0.0900646
\(245\) −8.63601e12 −0.625028
\(246\) 1.51273e12 0.107058
\(247\) 2.01445e11 0.0139419
\(248\) −9.46291e12 −0.640532
\(249\) 1.45170e13 0.961127
\(250\) 8.74664e12 0.566463
\(251\) −2.05017e13 −1.29892 −0.649462 0.760394i \(-0.725006\pi\)
−0.649462 + 0.760394i \(0.725006\pi\)
\(252\) −3.42801e12 −0.212490
\(253\) 8.26866e12 0.501502
\(254\) 8.30137e12 0.492679
\(255\) 1.62230e13 0.942237
\(256\) −1.87287e13 −1.06461
\(257\) −2.52685e13 −1.40588 −0.702940 0.711250i \(-0.748130\pi\)
−0.702940 + 0.711250i \(0.748130\pi\)
\(258\) 1.30190e13 0.709035
\(259\) −3.96112e13 −2.11188
\(260\) −7.09857e12 −0.370525
\(261\) −1.81455e12 −0.0927356
\(262\) 1.73305e13 0.867270
\(263\) 6.25225e12 0.306394 0.153197 0.988196i \(-0.451043\pi\)
0.153197 + 0.988196i \(0.451043\pi\)
\(264\) −1.78256e13 −0.855505
\(265\) 1.93882e13 0.911349
\(266\) −4.12596e11 −0.0189966
\(267\) 2.10402e13 0.948940
\(268\) −3.61799e12 −0.159855
\(269\) −1.23070e13 −0.532740 −0.266370 0.963871i \(-0.585824\pi\)
−0.266370 + 0.963871i \(0.585824\pi\)
\(270\) −3.59360e12 −0.152415
\(271\) 1.97816e13 0.822110 0.411055 0.911611i \(-0.365160\pi\)
0.411055 + 0.911611i \(0.365160\pi\)
\(272\) 7.99977e12 0.325798
\(273\) −1.15084e13 −0.459325
\(274\) −1.21203e13 −0.474115
\(275\) 1.04178e13 0.399434
\(276\) −2.81072e12 −0.105637
\(277\) 1.10910e13 0.408633 0.204316 0.978905i \(-0.434503\pi\)
0.204316 + 0.978905i \(0.434503\pi\)
\(278\) −2.84310e12 −0.102694
\(279\) 5.70753e12 0.202127
\(280\) 4.29475e13 1.49131
\(281\) −3.82184e13 −1.30133 −0.650666 0.759364i \(-0.725510\pi\)
−0.650666 + 0.759364i \(0.725510\pi\)
\(282\) −2.88287e12 −0.0962621
\(283\) −6.07773e12 −0.199029 −0.0995144 0.995036i \(-0.531729\pi\)
−0.0995144 + 0.995036i \(0.531729\pi\)
\(284\) −3.03525e13 −0.974864
\(285\) 4.53422e11 0.0142842
\(286\) −2.02590e13 −0.626045
\(287\) 1.09042e13 0.330555
\(288\) 1.00674e13 0.299405
\(289\) 3.67779e13 1.07312
\(290\) 7.69604e12 0.220332
\(291\) −2.46968e13 −0.693791
\(292\) 3.71418e12 0.102390
\(293\) 5.19893e13 1.40651 0.703254 0.710938i \(-0.251730\pi\)
0.703254 + 0.710938i \(0.251730\pi\)
\(294\) 8.37805e12 0.222450
\(295\) −5.66244e12 −0.147565
\(296\) 7.00167e13 1.79101
\(297\) 1.07515e13 0.269965
\(298\) −2.90199e12 −0.0715329
\(299\) −9.43604e12 −0.228349
\(300\) −3.54125e12 −0.0841375
\(301\) 9.38444e13 2.18924
\(302\) −1.51190e13 −0.346326
\(303\) 3.62295e13 0.814945
\(304\) 2.23588e11 0.00493906
\(305\) −9.19427e12 −0.199466
\(306\) −1.57384e13 −0.335346
\(307\) −1.89550e13 −0.396700 −0.198350 0.980131i \(-0.563558\pi\)
−0.198350 + 0.980131i \(0.563558\pi\)
\(308\) −4.34989e13 −0.894230
\(309\) 5.42232e12 0.109500
\(310\) −2.42073e13 −0.480238
\(311\) −2.99384e13 −0.583507 −0.291754 0.956494i \(-0.594239\pi\)
−0.291754 + 0.956494i \(0.594239\pi\)
\(312\) 2.03422e13 0.389537
\(313\) 4.90106e12 0.0922139 0.0461069 0.998937i \(-0.485319\pi\)
0.0461069 + 0.998937i \(0.485319\pi\)
\(314\) 7.14626e12 0.132119
\(315\) −2.59037e13 −0.470602
\(316\) −3.37313e13 −0.602219
\(317\) 2.55905e13 0.449007 0.224504 0.974473i \(-0.427924\pi\)
0.224504 + 0.974473i \(0.427924\pi\)
\(318\) −1.88091e13 −0.324353
\(319\) −2.30253e13 −0.390262
\(320\) −5.80934e13 −0.967837
\(321\) 5.02782e13 0.823384
\(322\) 1.93268e13 0.311137
\(323\) 1.98580e12 0.0314283
\(324\) −3.65468e12 −0.0568659
\(325\) −1.18886e13 −0.181874
\(326\) 7.10007e13 1.06799
\(327\) −3.21379e10 −0.000475340 0
\(328\) −1.92743e13 −0.280331
\(329\) −2.07806e13 −0.297222
\(330\) −4.56000e13 −0.641414
\(331\) −1.07132e14 −1.48205 −0.741027 0.671475i \(-0.765661\pi\)
−0.741027 + 0.671475i \(0.765661\pi\)
\(332\) −6.26174e13 −0.851992
\(333\) −4.22303e13 −0.565173
\(334\) 2.68327e13 0.353231
\(335\) −2.73393e13 −0.354030
\(336\) −1.27734e13 −0.162720
\(337\) 8.98851e12 0.112648 0.0563240 0.998413i \(-0.482062\pi\)
0.0563240 + 0.998413i \(0.482062\pi\)
\(338\) −3.35496e13 −0.413661
\(339\) −6.53865e13 −0.793212
\(340\) −6.99760e13 −0.835248
\(341\) 7.24243e13 0.850619
\(342\) −4.39878e11 −0.00508381
\(343\) −4.91262e13 −0.558722
\(344\) −1.65879e14 −1.85661
\(345\) −2.12391e13 −0.233955
\(346\) −7.07876e13 −0.767430
\(347\) 1.31761e14 1.40597 0.702984 0.711205i \(-0.251850\pi\)
0.702984 + 0.711205i \(0.251850\pi\)
\(348\) 7.82686e12 0.0822056
\(349\) −1.70214e14 −1.75977 −0.879886 0.475184i \(-0.842381\pi\)
−0.879886 + 0.475184i \(0.842381\pi\)
\(350\) 2.43500e13 0.247813
\(351\) −1.22694e13 −0.122923
\(352\) 1.27748e14 1.26000
\(353\) 1.79417e14 1.74222 0.871111 0.491086i \(-0.163400\pi\)
0.871111 + 0.491086i \(0.163400\pi\)
\(354\) 5.49330e12 0.0525189
\(355\) −2.29358e14 −2.15903
\(356\) −9.07546e13 −0.841189
\(357\) −1.13447e14 −1.03542
\(358\) −4.07891e13 −0.366596
\(359\) −9.61034e13 −0.850588 −0.425294 0.905055i \(-0.639829\pi\)
−0.425294 + 0.905055i \(0.639829\pi\)
\(360\) 4.57873e13 0.399100
\(361\) −1.16435e14 −0.999524
\(362\) 1.04736e14 0.885518
\(363\) 6.70972e13 0.558751
\(364\) 4.96401e13 0.407170
\(365\) 2.80661e13 0.226762
\(366\) 8.91963e12 0.0709906
\(367\) 7.60650e13 0.596378 0.298189 0.954507i \(-0.403617\pi\)
0.298189 + 0.954507i \(0.403617\pi\)
\(368\) −1.04733e13 −0.0808946
\(369\) 1.16252e13 0.0884619
\(370\) 1.79111e14 1.34281
\(371\) −1.35581e14 −1.00148
\(372\) −2.46188e13 −0.179176
\(373\) −2.23217e14 −1.60077 −0.800385 0.599486i \(-0.795372\pi\)
−0.800385 + 0.599486i \(0.795372\pi\)
\(374\) −1.99709e14 −1.41125
\(375\) 6.72172e13 0.468068
\(376\) 3.67317e13 0.252063
\(377\) 2.62760e13 0.177698
\(378\) 2.51299e13 0.167489
\(379\) −8.76788e13 −0.575942 −0.287971 0.957639i \(-0.592981\pi\)
−0.287971 + 0.957639i \(0.592981\pi\)
\(380\) −1.95578e12 −0.0126623
\(381\) 6.37953e13 0.407100
\(382\) 1.63660e14 1.02943
\(383\) 1.10103e14 0.682665 0.341333 0.939943i \(-0.389122\pi\)
0.341333 + 0.939943i \(0.389122\pi\)
\(384\) −2.84898e13 −0.174128
\(385\) −3.28698e14 −1.98045
\(386\) 1.35011e14 0.801932
\(387\) 1.00050e14 0.585876
\(388\) 1.06527e14 0.615012
\(389\) 1.01968e14 0.580419 0.290209 0.956963i \(-0.406275\pi\)
0.290209 + 0.956963i \(0.406275\pi\)
\(390\) 5.20379e13 0.292055
\(391\) −9.30183e13 −0.514750
\(392\) −1.06748e14 −0.582486
\(393\) 1.33184e14 0.716625
\(394\) −5.57380e13 −0.295748
\(395\) −2.54890e14 −1.33373
\(396\) −4.63752e13 −0.239311
\(397\) 2.02693e14 1.03155 0.515777 0.856723i \(-0.327503\pi\)
0.515777 + 0.856723i \(0.327503\pi\)
\(398\) −3.31410e13 −0.166344
\(399\) −3.17077e12 −0.0156969
\(400\) −1.31954e13 −0.0644306
\(401\) −2.31232e14 −1.11367 −0.556833 0.830625i \(-0.687984\pi\)
−0.556833 + 0.830625i \(0.687984\pi\)
\(402\) 2.65226e13 0.126001
\(403\) −8.26492e13 −0.387312
\(404\) −1.56272e14 −0.722409
\(405\) −2.76165e13 −0.125941
\(406\) −5.38182e13 −0.242123
\(407\) −5.35872e14 −2.37844
\(408\) 2.00529e14 0.878104
\(409\) 8.97745e13 0.387860 0.193930 0.981015i \(-0.437877\pi\)
0.193930 + 0.981015i \(0.437877\pi\)
\(410\) −4.93059e13 −0.210178
\(411\) −9.31432e13 −0.391761
\(412\) −2.33885e13 −0.0970662
\(413\) 3.95973e13 0.162159
\(414\) 2.06047e13 0.0832654
\(415\) −4.73167e14 −1.88690
\(416\) −1.45783e14 −0.573713
\(417\) −2.18490e13 −0.0848560
\(418\) −5.58173e12 −0.0213943
\(419\) 6.50052e13 0.245907 0.122953 0.992412i \(-0.460763\pi\)
0.122953 + 0.992412i \(0.460763\pi\)
\(420\) 1.11733e14 0.417166
\(421\) −2.99663e14 −1.10429 −0.552144 0.833749i \(-0.686190\pi\)
−0.552144 + 0.833749i \(0.686190\pi\)
\(422\) −2.91303e14 −1.05956
\(423\) −2.21546e13 −0.0795413
\(424\) 2.39653e14 0.849319
\(425\) −1.17195e14 −0.409986
\(426\) 2.22507e14 0.768407
\(427\) 6.42953e13 0.219193
\(428\) −2.16869e14 −0.729890
\(429\) −1.55689e14 −0.517300
\(430\) −4.24340e14 −1.39199
\(431\) −3.95410e14 −1.28063 −0.640313 0.768114i \(-0.721195\pi\)
−0.640313 + 0.768114i \(0.721195\pi\)
\(432\) −1.36180e13 −0.0435466
\(433\) 3.39133e14 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(434\) 1.69281e14 0.527733
\(435\) 5.91434e13 0.182060
\(436\) 1.38623e11 0.000421366 0
\(437\) −2.59980e12 −0.00780355
\(438\) −2.72277e13 −0.0807056
\(439\) 4.69140e14 1.37324 0.686622 0.727015i \(-0.259093\pi\)
0.686622 + 0.727015i \(0.259093\pi\)
\(440\) 5.81007e14 1.67954
\(441\) 6.43846e13 0.183810
\(442\) 2.27904e14 0.642583
\(443\) 2.89324e14 0.805682 0.402841 0.915270i \(-0.368023\pi\)
0.402841 + 0.915270i \(0.368023\pi\)
\(444\) 1.82156e14 0.500999
\(445\) −6.85785e14 −1.86298
\(446\) 2.45828e14 0.659613
\(447\) −2.23015e13 −0.0591076
\(448\) 4.06246e14 1.06355
\(449\) 4.27685e14 1.10604 0.553018 0.833169i \(-0.313476\pi\)
0.553018 + 0.833169i \(0.313476\pi\)
\(450\) 2.59601e13 0.0663188
\(451\) 1.47515e14 0.372277
\(452\) 2.82037e14 0.703144
\(453\) −1.16188e14 −0.286169
\(454\) 1.67874e14 0.408484
\(455\) 3.75104e14 0.901756
\(456\) 5.60465e12 0.0133120
\(457\) −5.39035e14 −1.26496 −0.632482 0.774575i \(-0.717964\pi\)
−0.632482 + 0.774575i \(0.717964\pi\)
\(458\) 4.12563e14 0.956600
\(459\) −1.20948e14 −0.277096
\(460\) 9.16125e13 0.207389
\(461\) −2.58635e14 −0.578538 −0.289269 0.957248i \(-0.593412\pi\)
−0.289269 + 0.957248i \(0.593412\pi\)
\(462\) 3.18880e14 0.704849
\(463\) 8.68735e14 1.89754 0.948772 0.315961i \(-0.102327\pi\)
0.948772 + 0.315961i \(0.102327\pi\)
\(464\) 2.91644e13 0.0629512
\(465\) −1.86031e14 −0.396821
\(466\) 2.37070e14 0.499752
\(467\) 1.04128e14 0.216934 0.108467 0.994100i \(-0.465406\pi\)
0.108467 + 0.994100i \(0.465406\pi\)
\(468\) 5.29224e13 0.108965
\(469\) 1.91183e14 0.389044
\(470\) 9.39642e13 0.188984
\(471\) 5.49185e13 0.109170
\(472\) −6.99921e13 −0.137521
\(473\) 1.26956e15 2.46556
\(474\) 2.47276e14 0.474681
\(475\) −3.27551e12 −0.00621533
\(476\) 4.89341e14 0.917853
\(477\) −1.44546e14 −0.268013
\(478\) 3.56967e14 0.654297
\(479\) 9.78913e13 0.177378 0.0886888 0.996059i \(-0.471732\pi\)
0.0886888 + 0.996059i \(0.471732\pi\)
\(480\) −3.28137e14 −0.587798
\(481\) 6.11526e14 1.08297
\(482\) −3.01021e14 −0.527034
\(483\) 1.48525e14 0.257092
\(484\) −2.89416e14 −0.495306
\(485\) 8.04966e14 1.36207
\(486\) 2.67916e13 0.0448228
\(487\) −1.79576e14 −0.297057 −0.148528 0.988908i \(-0.547454\pi\)
−0.148528 + 0.988908i \(0.547454\pi\)
\(488\) −1.13648e14 −0.185889
\(489\) 5.45635e14 0.882476
\(490\) −2.73074e14 −0.436719
\(491\) −9.44720e14 −1.49401 −0.747007 0.664816i \(-0.768510\pi\)
−0.747007 + 0.664816i \(0.768510\pi\)
\(492\) −5.01440e13 −0.0784172
\(493\) 2.59023e14 0.400572
\(494\) 6.36976e12 0.00974148
\(495\) −3.50433e14 −0.530000
\(496\) −9.17343e13 −0.137209
\(497\) 1.60390e15 2.37255
\(498\) 4.59033e14 0.671556
\(499\) −5.43255e14 −0.786051 −0.393026 0.919528i \(-0.628572\pi\)
−0.393026 + 0.919528i \(0.628572\pi\)
\(500\) −2.89934e14 −0.414919
\(501\) 2.06207e14 0.291875
\(502\) −6.48270e14 −0.907582
\(503\) 5.41766e14 0.750219 0.375109 0.926981i \(-0.377605\pi\)
0.375109 + 0.926981i \(0.377605\pi\)
\(504\) −3.20190e14 −0.438570
\(505\) −1.18086e15 −1.59992
\(506\) 2.61458e14 0.350408
\(507\) −2.57826e14 −0.341808
\(508\) −2.75174e14 −0.360875
\(509\) 8.67796e14 1.12582 0.562911 0.826518i \(-0.309681\pi\)
0.562911 + 0.826518i \(0.309681\pi\)
\(510\) 5.12977e14 0.658358
\(511\) −1.96265e14 −0.249189
\(512\) −3.52098e14 −0.442261
\(513\) −3.38043e12 −0.00420075
\(514\) −7.99001e14 −0.982313
\(515\) −1.76735e14 −0.214972
\(516\) −4.31553e14 −0.519350
\(517\) −2.81126e14 −0.334736
\(518\) −1.25252e15 −1.47561
\(519\) −5.43997e14 −0.634127
\(520\) −6.63034e14 −0.764746
\(521\) 1.74053e15 1.98644 0.993219 0.116259i \(-0.0370904\pi\)
0.993219 + 0.116259i \(0.0370904\pi\)
\(522\) −5.73768e13 −0.0647960
\(523\) 1.55814e15 1.74119 0.870596 0.491998i \(-0.163733\pi\)
0.870596 + 0.491998i \(0.163733\pi\)
\(524\) −5.74472e14 −0.635253
\(525\) 1.87128e14 0.204768
\(526\) 1.97698e14 0.214083
\(527\) −8.14737e14 −0.873090
\(528\) −1.72803e14 −0.183259
\(529\) −8.31030e14 −0.872189
\(530\) 6.13062e14 0.636776
\(531\) 4.22156e13 0.0433963
\(532\) 1.36768e13 0.0139145
\(533\) −1.68342e14 −0.169509
\(534\) 6.65300e14 0.663041
\(535\) −1.63877e15 −1.61648
\(536\) −3.37935e14 −0.329933
\(537\) −3.13461e14 −0.302918
\(538\) −3.89152e14 −0.372235
\(539\) 8.16993e14 0.773535
\(540\) 1.19121e14 0.111640
\(541\) 4.54369e14 0.421526 0.210763 0.977537i \(-0.432405\pi\)
0.210763 + 0.977537i \(0.432405\pi\)
\(542\) 6.25501e14 0.574423
\(543\) 8.04885e14 0.731703
\(544\) −1.43710e15 −1.29328
\(545\) 1.04750e12 0.000933198 0
\(546\) −3.63900e14 −0.320939
\(547\) 2.80242e14 0.244683 0.122341 0.992488i \(-0.460960\pi\)
0.122341 + 0.992488i \(0.460960\pi\)
\(548\) 4.01762e14 0.347277
\(549\) 6.85466e13 0.0586595
\(550\) 3.29414e14 0.279092
\(551\) 7.23952e12 0.00607262
\(552\) −2.62532e14 −0.218031
\(553\) 1.78244e15 1.46564
\(554\) 3.50702e14 0.285519
\(555\) 1.37645e15 1.10956
\(556\) 9.42430e13 0.0752207
\(557\) −9.57487e14 −0.756710 −0.378355 0.925661i \(-0.623510\pi\)
−0.378355 + 0.925661i \(0.623510\pi\)
\(558\) 1.80474e14 0.141230
\(559\) −1.44879e15 −1.12264
\(560\) 4.16337e14 0.319456
\(561\) −1.53475e15 −1.16611
\(562\) −1.20848e15 −0.909264
\(563\) 3.47122e14 0.258634 0.129317 0.991603i \(-0.458721\pi\)
0.129317 + 0.991603i \(0.458721\pi\)
\(564\) 9.55615e13 0.0705095
\(565\) 2.13120e15 1.55725
\(566\) −1.92180e14 −0.139065
\(567\) 1.93121e14 0.138396
\(568\) −2.83505e15 −2.01207
\(569\) 1.78084e15 1.25172 0.625860 0.779936i \(-0.284748\pi\)
0.625860 + 0.779936i \(0.284748\pi\)
\(570\) 1.43374e13 0.00998063
\(571\) −1.77494e15 −1.22373 −0.611866 0.790962i \(-0.709581\pi\)
−0.611866 + 0.790962i \(0.709581\pi\)
\(572\) 6.71546e14 0.458562
\(573\) 1.25772e15 0.850614
\(574\) 3.44795e14 0.230965
\(575\) 1.53431e14 0.101798
\(576\) 4.33108e14 0.284625
\(577\) −1.72513e15 −1.12293 −0.561467 0.827499i \(-0.689763\pi\)
−0.561467 + 0.827499i \(0.689763\pi\)
\(578\) 1.16293e15 0.749810
\(579\) 1.03755e15 0.662636
\(580\) −2.55108e14 −0.161388
\(581\) 3.30884e15 2.07352
\(582\) −7.80921e14 −0.484764
\(583\) −1.83418e15 −1.12789
\(584\) 3.46919e14 0.211328
\(585\) 3.99907e14 0.241325
\(586\) 1.64392e15 0.982753
\(587\) −4.62864e14 −0.274122 −0.137061 0.990563i \(-0.543766\pi\)
−0.137061 + 0.990563i \(0.543766\pi\)
\(588\) −2.77716e14 −0.162939
\(589\) −2.27713e13 −0.0132359
\(590\) −1.79048e14 −0.103106
\(591\) −4.28342e14 −0.244377
\(592\) 6.78747e14 0.383653
\(593\) 1.66652e15 0.933276 0.466638 0.884448i \(-0.345465\pi\)
0.466638 + 0.884448i \(0.345465\pi\)
\(594\) 3.39965e14 0.188629
\(595\) 3.69769e15 2.03277
\(596\) 9.61951e13 0.0523961
\(597\) −2.54686e14 −0.137450
\(598\) −2.98371e14 −0.159551
\(599\) 1.04977e15 0.556222 0.278111 0.960549i \(-0.410292\pi\)
0.278111 + 0.960549i \(0.410292\pi\)
\(600\) −3.30767e14 −0.173656
\(601\) 9.21075e14 0.479166 0.239583 0.970876i \(-0.422989\pi\)
0.239583 + 0.970876i \(0.422989\pi\)
\(602\) 2.96740e15 1.52966
\(603\) 2.03824e14 0.104114
\(604\) 5.01164e14 0.253675
\(605\) −2.18696e15 −1.09695
\(606\) 1.14559e15 0.569417
\(607\) −3.46584e15 −1.70715 −0.853574 0.520972i \(-0.825569\pi\)
−0.853574 + 0.520972i \(0.825569\pi\)
\(608\) −4.01660e13 −0.0196060
\(609\) −4.13588e14 −0.200066
\(610\) −2.90726e14 −0.139370
\(611\) 3.20815e14 0.152415
\(612\) 5.21697e14 0.245632
\(613\) 6.47040e13 0.0301925 0.0150962 0.999886i \(-0.495195\pi\)
0.0150962 + 0.999886i \(0.495195\pi\)
\(614\) −5.99363e14 −0.277181
\(615\) −3.78912e14 −0.173670
\(616\) −4.06297e15 −1.84565
\(617\) −1.03087e15 −0.464127 −0.232064 0.972701i \(-0.574548\pi\)
−0.232064 + 0.972701i \(0.574548\pi\)
\(618\) 1.71456e14 0.0765095
\(619\) −1.27656e15 −0.564601 −0.282300 0.959326i \(-0.591097\pi\)
−0.282300 + 0.959326i \(0.591097\pi\)
\(620\) 8.02423e14 0.351762
\(621\) 1.58345e14 0.0688021
\(622\) −9.46663e14 −0.407707
\(623\) 4.79567e15 2.04722
\(624\) 1.97199e14 0.0834430
\(625\) −2.86976e15 −1.20367
\(626\) 1.54973e14 0.0644315
\(627\) −4.28951e13 −0.0176781
\(628\) −2.36884e14 −0.0967739
\(629\) 6.02828e15 2.44127
\(630\) −8.19084e14 −0.328818
\(631\) −7.49684e14 −0.298343 −0.149172 0.988811i \(-0.547661\pi\)
−0.149172 + 0.988811i \(0.547661\pi\)
\(632\) −3.15064e15 −1.24295
\(633\) −2.23864e15 −0.875517
\(634\) 8.09182e14 0.313729
\(635\) −2.07934e15 −0.799228
\(636\) 6.23483e14 0.237580
\(637\) −9.32337e14 −0.352213
\(638\) −7.28068e14 −0.272683
\(639\) 1.70995e15 0.634934
\(640\) 9.28596e14 0.341851
\(641\) −2.94694e15 −1.07560 −0.537801 0.843072i \(-0.680745\pi\)
−0.537801 + 0.843072i \(0.680745\pi\)
\(642\) 1.58982e15 0.575313
\(643\) 2.62564e15 0.942053 0.471027 0.882119i \(-0.343884\pi\)
0.471027 + 0.882119i \(0.343884\pi\)
\(644\) −6.40644e14 −0.227900
\(645\) −3.26102e15 −1.15020
\(646\) 6.27916e13 0.0219595
\(647\) −3.82331e14 −0.132576 −0.0662882 0.997801i \(-0.521116\pi\)
−0.0662882 + 0.997801i \(0.521116\pi\)
\(648\) −3.41361e14 −0.117369
\(649\) 5.35684e14 0.182626
\(650\) −3.75921e14 −0.127079
\(651\) 1.30091e15 0.436066
\(652\) −2.35353e15 −0.782273
\(653\) −5.99018e14 −0.197432 −0.0987160 0.995116i \(-0.531474\pi\)
−0.0987160 + 0.995116i \(0.531474\pi\)
\(654\) −1.01621e12 −0.000332129 0
\(655\) −4.34098e15 −1.40689
\(656\) −1.86846e14 −0.0600501
\(657\) −2.09243e14 −0.0666870
\(658\) −6.57090e14 −0.207674
\(659\) −6.11779e15 −1.91745 −0.958727 0.284327i \(-0.908230\pi\)
−0.958727 + 0.284327i \(0.908230\pi\)
\(660\) 1.51155e15 0.469820
\(661\) 2.93249e15 0.903917 0.451959 0.892039i \(-0.350725\pi\)
0.451959 + 0.892039i \(0.350725\pi\)
\(662\) −3.38755e15 −1.03554
\(663\) 1.75142e15 0.530966
\(664\) −5.84871e15 −1.75847
\(665\) 1.03348e14 0.0308165
\(666\) −1.33534e15 −0.394897
\(667\) −3.39112e14 −0.0994607
\(668\) −8.89451e14 −0.258733
\(669\) 1.88917e15 0.545038
\(670\) −8.64478e14 −0.247367
\(671\) 8.69806e14 0.246859
\(672\) 2.29465e15 0.645931
\(673\) 4.96390e15 1.38593 0.692963 0.720973i \(-0.256305\pi\)
0.692963 + 0.720973i \(0.256305\pi\)
\(674\) 2.84220e14 0.0787091
\(675\) 1.99501e14 0.0547992
\(676\) 1.11210e15 0.302996
\(677\) −2.98893e15 −0.807751 −0.403875 0.914814i \(-0.632337\pi\)
−0.403875 + 0.914814i \(0.632337\pi\)
\(678\) −2.06754e15 −0.554232
\(679\) −5.62911e15 −1.49677
\(680\) −6.53603e15 −1.72391
\(681\) 1.29010e15 0.337530
\(682\) 2.29008e15 0.594343
\(683\) 3.57204e15 0.919607 0.459804 0.888021i \(-0.347920\pi\)
0.459804 + 0.888021i \(0.347920\pi\)
\(684\) 1.45811e13 0.00372376
\(685\) 3.03591e15 0.769113
\(686\) −1.55339e15 −0.390389
\(687\) 3.17052e15 0.790438
\(688\) −1.60805e15 −0.397706
\(689\) 2.09313e15 0.513560
\(690\) −6.71589e14 −0.163468
\(691\) 1.69191e15 0.408553 0.204276 0.978913i \(-0.434516\pi\)
0.204276 + 0.978913i \(0.434516\pi\)
\(692\) 2.34647e15 0.562123
\(693\) 2.45057e15 0.582417
\(694\) 4.16634e15 0.982376
\(695\) 7.12144e14 0.166591
\(696\) 7.31059e14 0.169669
\(697\) −1.65947e15 −0.382111
\(698\) −5.38224e15 −1.22958
\(699\) 1.82187e15 0.412945
\(700\) −8.07153e14 −0.181517
\(701\) 1.49899e15 0.334465 0.167232 0.985918i \(-0.446517\pi\)
0.167232 + 0.985918i \(0.446517\pi\)
\(702\) −3.87962e14 −0.0858884
\(703\) 1.68487e14 0.0370093
\(704\) 5.49581e15 1.19779
\(705\) 7.22108e14 0.156157
\(706\) 5.67324e15 1.21732
\(707\) 8.25775e15 1.75815
\(708\) −1.82092e14 −0.0384687
\(709\) 1.36239e14 0.0285592 0.0142796 0.999898i \(-0.495455\pi\)
0.0142796 + 0.999898i \(0.495455\pi\)
\(710\) −7.25239e15 −1.50855
\(711\) 1.90030e15 0.392228
\(712\) −8.47683e15 −1.73618
\(713\) 1.06665e15 0.216785
\(714\) −3.58724e15 −0.723469
\(715\) 5.07452e15 1.01557
\(716\) 1.35208e15 0.268522
\(717\) 2.74326e15 0.540645
\(718\) −3.03882e15 −0.594321
\(719\) −4.57140e15 −0.887239 −0.443619 0.896215i \(-0.646306\pi\)
−0.443619 + 0.896215i \(0.646306\pi\)
\(720\) 4.43866e14 0.0854916
\(721\) 1.23590e15 0.236233
\(722\) −3.68171e15 −0.698385
\(723\) −2.31333e15 −0.435488
\(724\) −3.47178e15 −0.648619
\(725\) −4.27251e14 −0.0792180
\(726\) 2.12164e15 0.390409
\(727\) −9.31384e15 −1.70094 −0.850471 0.526021i \(-0.823683\pi\)
−0.850471 + 0.526021i \(0.823683\pi\)
\(728\) 4.63658e15 0.840379
\(729\) 2.05891e14 0.0370370
\(730\) 8.87460e14 0.158443
\(731\) −1.42819e16 −2.53069
\(732\) −2.95668e14 −0.0519988
\(733\) −2.93431e15 −0.512194 −0.256097 0.966651i \(-0.582437\pi\)
−0.256097 + 0.966651i \(0.582437\pi\)
\(734\) 2.40520e15 0.416700
\(735\) −2.09855e15 −0.360860
\(736\) 1.88145e15 0.321118
\(737\) 2.58638e15 0.438148
\(738\) 3.67594e14 0.0618099
\(739\) −5.80834e14 −0.0969410 −0.0484705 0.998825i \(-0.515435\pi\)
−0.0484705 + 0.998825i \(0.515435\pi\)
\(740\) −5.93718e15 −0.983571
\(741\) 4.89511e13 0.00804938
\(742\) −4.28713e15 −0.699753
\(743\) 9.69848e15 1.57132 0.785661 0.618657i \(-0.212323\pi\)
0.785661 + 0.618657i \(0.212323\pi\)
\(744\) −2.29949e15 −0.369811
\(745\) 7.26896e14 0.116041
\(746\) −7.05821e15 −1.11849
\(747\) 3.52763e15 0.554907
\(748\) 6.61995e15 1.03370
\(749\) 1.14599e16 1.77635
\(750\) 2.12543e15 0.327047
\(751\) −2.36608e15 −0.361418 −0.180709 0.983537i \(-0.557839\pi\)
−0.180709 + 0.983537i \(0.557839\pi\)
\(752\) 3.56080e14 0.0539946
\(753\) −4.98191e15 −0.749934
\(754\) 8.30858e14 0.124161
\(755\) 3.78703e15 0.561812
\(756\) −8.33007e14 −0.122681
\(757\) 1.38403e15 0.202357 0.101179 0.994868i \(-0.467739\pi\)
0.101179 + 0.994868i \(0.467739\pi\)
\(758\) −2.77244e15 −0.402421
\(759\) 2.00929e15 0.289542
\(760\) −1.82678e14 −0.0261343
\(761\) −1.12056e16 −1.59155 −0.795776 0.605591i \(-0.792937\pi\)
−0.795776 + 0.605591i \(0.792937\pi\)
\(762\) 2.01723e15 0.284448
\(763\) −7.32515e12 −0.00102549
\(764\) −5.42502e15 −0.754029
\(765\) 3.94219e15 0.544001
\(766\) 3.48151e15 0.476991
\(767\) −6.11312e14 −0.0831550
\(768\) −4.55108e15 −0.614650
\(769\) 8.88072e15 1.19084 0.595420 0.803415i \(-0.296986\pi\)
0.595420 + 0.803415i \(0.296986\pi\)
\(770\) −1.03936e16 −1.38377
\(771\) −6.14026e15 −0.811685
\(772\) −4.47533e15 −0.587395
\(773\) 4.81120e15 0.626997 0.313499 0.949589i \(-0.398499\pi\)
0.313499 + 0.949589i \(0.398499\pi\)
\(774\) 3.16361e15 0.409362
\(775\) 1.34388e15 0.172664
\(776\) 9.95000e15 1.26936
\(777\) −9.62551e15 −1.21929
\(778\) 3.22427e15 0.405549
\(779\) −4.63812e13 −0.00579276
\(780\) −1.72495e15 −0.213923
\(781\) 2.16980e16 2.67201
\(782\) −2.94127e15 −0.359665
\(783\) −4.40936e14 −0.0535409
\(784\) −1.03482e15 −0.124775
\(785\) −1.79001e15 −0.214325
\(786\) 4.21132e15 0.500719
\(787\) 4.41430e15 0.521196 0.260598 0.965447i \(-0.416080\pi\)
0.260598 + 0.965447i \(0.416080\pi\)
\(788\) 1.84760e15 0.216628
\(789\) 1.51930e15 0.176896
\(790\) −8.05971e15 −0.931903
\(791\) −1.49035e16 −1.71126
\(792\) −4.33162e15 −0.493926
\(793\) −9.92606e14 −0.112402
\(794\) 6.40924e15 0.720765
\(795\) 4.71133e15 0.526168
\(796\) 1.09856e15 0.121843
\(797\) 1.28696e16 1.41757 0.708786 0.705424i \(-0.249243\pi\)
0.708786 + 0.705424i \(0.249243\pi\)
\(798\) −1.00261e14 −0.0109677
\(799\) 3.16252e15 0.343579
\(800\) 2.37045e15 0.255762
\(801\) 5.11278e15 0.547871
\(802\) −7.31166e15 −0.778138
\(803\) −2.65514e15 −0.280641
\(804\) −8.79172e14 −0.0922924
\(805\) −4.84101e15 −0.504729
\(806\) −2.61340e15 −0.270622
\(807\) −2.99060e15 −0.307577
\(808\) −1.45964e16 −1.49102
\(809\) −1.48045e15 −0.150203 −0.0751014 0.997176i \(-0.523928\pi\)
−0.0751014 + 0.997176i \(0.523928\pi\)
\(810\) −8.73244e14 −0.0879970
\(811\) 7.61346e14 0.0762021 0.0381011 0.999274i \(-0.487869\pi\)
0.0381011 + 0.999274i \(0.487869\pi\)
\(812\) 1.78397e15 0.177349
\(813\) 4.80692e15 0.474645
\(814\) −1.69445e16 −1.66186
\(815\) −1.77844e16 −1.73250
\(816\) 1.94394e15 0.188100
\(817\) −3.99168e14 −0.0383650
\(818\) 2.83870e15 0.271005
\(819\) −2.79654e15 −0.265191
\(820\) 1.63439e15 0.153950
\(821\) −4.31889e15 −0.404096 −0.202048 0.979376i \(-0.564760\pi\)
−0.202048 + 0.979376i \(0.564760\pi\)
\(822\) −2.94522e15 −0.273730
\(823\) −1.00319e16 −0.926158 −0.463079 0.886317i \(-0.653255\pi\)
−0.463079 + 0.886317i \(0.653255\pi\)
\(824\) −2.18458e15 −0.200340
\(825\) 2.53152e15 0.230613
\(826\) 1.25208e15 0.113303
\(827\) 1.06302e16 0.955563 0.477782 0.878479i \(-0.341441\pi\)
0.477782 + 0.878479i \(0.341441\pi\)
\(828\) −6.83005e14 −0.0609898
\(829\) −2.01838e16 −1.79042 −0.895208 0.445649i \(-0.852973\pi\)
−0.895208 + 0.445649i \(0.852973\pi\)
\(830\) −1.49617e16 −1.31841
\(831\) 2.69512e15 0.235924
\(832\) −6.27172e15 −0.545391
\(833\) −9.19076e15 −0.793969
\(834\) −6.90872e14 −0.0592904
\(835\) −6.72111e15 −0.573014
\(836\) 1.85023e14 0.0156708
\(837\) 1.38693e15 0.116698
\(838\) 2.05549e15 0.171820
\(839\) 1.76431e16 1.46516 0.732578 0.680684i \(-0.238317\pi\)
0.732578 + 0.680684i \(0.238317\pi\)
\(840\) 1.04362e16 0.861010
\(841\) −1.12562e16 −0.922601
\(842\) −9.47546e15 −0.771585
\(843\) −9.28708e15 −0.751325
\(844\) 9.65613e15 0.776104
\(845\) 8.40356e15 0.671045
\(846\) −7.00538e14 −0.0555769
\(847\) 1.52934e16 1.20544
\(848\) 2.32322e15 0.181933
\(849\) −1.47689e15 −0.114909
\(850\) −3.70574e15 −0.286464
\(851\) −7.89222e15 −0.606159
\(852\) −7.37567e15 −0.562838
\(853\) −1.17114e16 −0.887952 −0.443976 0.896039i \(-0.646432\pi\)
−0.443976 + 0.896039i \(0.646432\pi\)
\(854\) 2.03304e15 0.153154
\(855\) 1.10182e14 0.00824699
\(856\) −2.02564e16 −1.50646
\(857\) 1.71522e16 1.26743 0.633717 0.773565i \(-0.281528\pi\)
0.633717 + 0.773565i \(0.281528\pi\)
\(858\) −4.92294e15 −0.361447
\(859\) 1.47994e16 1.07965 0.539824 0.841778i \(-0.318491\pi\)
0.539824 + 0.841778i \(0.318491\pi\)
\(860\) 1.40660e16 1.01960
\(861\) 2.64972e15 0.190846
\(862\) −1.25030e16 −0.894796
\(863\) −1.22051e16 −0.867924 −0.433962 0.900931i \(-0.642885\pi\)
−0.433962 + 0.900931i \(0.642885\pi\)
\(864\) 2.44638e15 0.172862
\(865\) 1.77310e16 1.24493
\(866\) 1.07235e16 0.748149
\(867\) 8.93704e15 0.619568
\(868\) −5.61133e15 −0.386551
\(869\) 2.41134e16 1.65063
\(870\) 1.87014e15 0.127209
\(871\) −2.95152e15 −0.199502
\(872\) 1.29479e13 0.000869680 0
\(873\) −6.00132e15 −0.400561
\(874\) −8.22066e13 −0.00545248
\(875\) 1.53207e16 1.00980
\(876\) 9.02545e14 0.0591148
\(877\) −2.11165e16 −1.37444 −0.687219 0.726450i \(-0.741169\pi\)
−0.687219 + 0.726450i \(0.741169\pi\)
\(878\) 1.48344e16 0.959510
\(879\) 1.26334e16 0.812048
\(880\) 5.63233e15 0.359777
\(881\) 1.99684e16 1.26758 0.633791 0.773505i \(-0.281498\pi\)
0.633791 + 0.773505i \(0.281498\pi\)
\(882\) 2.03587e15 0.128432
\(883\) 1.34077e16 0.840564 0.420282 0.907393i \(-0.361931\pi\)
0.420282 + 0.907393i \(0.361931\pi\)
\(884\) −7.55456e15 −0.470675
\(885\) −1.37597e15 −0.0851965
\(886\) 9.14852e15 0.562944
\(887\) −1.64730e16 −1.00738 −0.503689 0.863885i \(-0.668024\pi\)
−0.503689 + 0.863885i \(0.668024\pi\)
\(888\) 1.70140e16 1.03404
\(889\) 1.45408e16 0.878271
\(890\) −2.16848e16 −1.30170
\(891\) 2.61260e15 0.155864
\(892\) −8.14871e15 −0.483150
\(893\) 8.83904e13 0.00520861
\(894\) −7.05183e14 −0.0412996
\(895\) 1.02169e16 0.594695
\(896\) −6.49365e15 −0.375660
\(897\) −2.29296e15 −0.131837
\(898\) 1.35236e16 0.772807
\(899\) −2.97025e15 −0.168700
\(900\) −8.60524e14 −0.0485768
\(901\) 2.06336e16 1.15768
\(902\) 4.66449e15 0.260117
\(903\) 2.28042e16 1.26396
\(904\) 2.63433e16 1.45126
\(905\) −2.62344e16 −1.43649
\(906\) −3.67391e15 −0.199951
\(907\) 5.02962e15 0.272079 0.136039 0.990703i \(-0.456563\pi\)
0.136039 + 0.990703i \(0.456563\pi\)
\(908\) −5.56468e15 −0.299204
\(909\) 8.80377e15 0.470509
\(910\) 1.18609e16 0.630074
\(911\) 3.83505e15 0.202497 0.101249 0.994861i \(-0.467716\pi\)
0.101249 + 0.994861i \(0.467716\pi\)
\(912\) 5.43320e13 0.00285157
\(913\) 4.47630e16 2.33523
\(914\) −1.70445e16 −0.883853
\(915\) −2.23421e15 −0.115162
\(916\) −1.36757e16 −0.700685
\(917\) 3.03564e16 1.54603
\(918\) −3.82444e15 −0.193612
\(919\) 1.61837e14 0.00814409 0.00407204 0.999992i \(-0.498704\pi\)
0.00407204 + 0.999992i \(0.498704\pi\)
\(920\) 8.55696e15 0.428042
\(921\) −4.60606e15 −0.229035
\(922\) −8.17813e15 −0.404235
\(923\) −2.47613e16 −1.21665
\(924\) −1.05702e16 −0.516284
\(925\) −9.94348e15 −0.482790
\(926\) 2.74697e16 1.32585
\(927\) 1.31762e15 0.0632197
\(928\) −5.23916e15 −0.249889
\(929\) −1.72229e16 −0.816620 −0.408310 0.912843i \(-0.633882\pi\)
−0.408310 + 0.912843i \(0.633882\pi\)
\(930\) −5.88237e15 −0.277266
\(931\) −2.56876e14 −0.0120365
\(932\) −7.85841e15 −0.366056
\(933\) −7.27503e15 −0.336888
\(934\) 3.29258e15 0.151575
\(935\) 5.00234e16 2.28934
\(936\) 4.94316e15 0.224899
\(937\) −2.47773e16 −1.12069 −0.560345 0.828259i \(-0.689331\pi\)
−0.560345 + 0.828259i \(0.689331\pi\)
\(938\) 6.04527e15 0.271832
\(939\) 1.19096e15 0.0532397
\(940\) −3.11473e15 −0.138426
\(941\) −2.69347e16 −1.19006 −0.595031 0.803703i \(-0.702860\pi\)
−0.595031 + 0.803703i \(0.702860\pi\)
\(942\) 1.73654e15 0.0762791
\(943\) 2.17258e15 0.0948771
\(944\) −6.78510e14 −0.0294585
\(945\) −6.29459e15 −0.271702
\(946\) 4.01438e16 1.72273
\(947\) 1.90161e16 0.811328 0.405664 0.914022i \(-0.367040\pi\)
0.405664 + 0.914022i \(0.367040\pi\)
\(948\) −8.19672e15 −0.347691
\(949\) 3.02999e15 0.127784
\(950\) −1.03573e14 −0.00434276
\(951\) 6.21850e15 0.259234
\(952\) 4.57063e16 1.89441
\(953\) −1.78696e15 −0.0736382 −0.0368191 0.999322i \(-0.511723\pi\)
−0.0368191 + 0.999322i \(0.511723\pi\)
\(954\) −4.57060e15 −0.187265
\(955\) −4.09940e16 −1.66994
\(956\) −1.18327e16 −0.479256
\(957\) −5.59515e15 −0.225318
\(958\) 3.09536e15 0.123937
\(959\) −2.12300e16 −0.845178
\(960\) −1.41167e16 −0.558781
\(961\) −1.60658e16 −0.632301
\(962\) 1.93367e16 0.756692
\(963\) 1.22176e16 0.475381
\(964\) 9.97826e15 0.386039
\(965\) −3.38177e16 −1.30090
\(966\) 4.69640e15 0.179635
\(967\) 5.39211e15 0.205075 0.102538 0.994729i \(-0.467304\pi\)
0.102538 + 0.994729i \(0.467304\pi\)
\(968\) −2.70326e16 −1.02229
\(969\) 4.82548e14 0.0181451
\(970\) 2.54533e16 0.951700
\(971\) −9.33620e15 −0.347108 −0.173554 0.984824i \(-0.555525\pi\)
−0.173554 + 0.984824i \(0.555525\pi\)
\(972\) −8.88087e14 −0.0328315
\(973\) −4.98001e15 −0.183067
\(974\) −5.67827e15 −0.207559
\(975\) −2.88892e15 −0.105005
\(976\) −1.10172e15 −0.0398195
\(977\) 4.82539e16 1.73425 0.867127 0.498087i \(-0.165964\pi\)
0.867127 + 0.498087i \(0.165964\pi\)
\(978\) 1.72532e16 0.616602
\(979\) 6.48773e16 2.30562
\(980\) 9.05185e15 0.319885
\(981\) −7.80951e12 −0.000274438 0
\(982\) −2.98724e16 −1.04390
\(983\) 4.12510e16 1.43347 0.716737 0.697343i \(-0.245635\pi\)
0.716737 + 0.697343i \(0.245635\pi\)
\(984\) −4.68365e15 −0.161849
\(985\) 1.39614e16 0.479765
\(986\) 8.19040e15 0.279887
\(987\) −5.04968e15 −0.171601
\(988\) −2.11145e14 −0.00713538
\(989\) 1.86978e16 0.628363
\(990\) −1.10808e16 −0.370321
\(991\) 7.84497e15 0.260727 0.130364 0.991466i \(-0.458386\pi\)
0.130364 + 0.991466i \(0.458386\pi\)
\(992\) 1.64794e16 0.544661
\(993\) −2.60330e16 −0.855665
\(994\) 5.07158e16 1.65775
\(995\) 8.30122e15 0.269845
\(996\) −1.52160e16 −0.491898
\(997\) 3.85622e16 1.23976 0.619881 0.784696i \(-0.287181\pi\)
0.619881 + 0.784696i \(0.287181\pi\)
\(998\) −1.71779e16 −0.549228
\(999\) −1.02620e16 −0.326303
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.1.18 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.18 28 1.1 even 1 trivial