Properties

Label 38.10.a.d.1.3
Level $38$
Weight $10$
Character 38.1
Self dual yes
Analytic conductor $19.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-26.2676\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000 q^{2} +25.2570 q^{3} +256.000 q^{4} +2126.71 q^{5} -404.112 q^{6} +11469.0 q^{7} -4096.00 q^{8} -19045.1 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} +25.2570 q^{3} +256.000 q^{4} +2126.71 q^{5} -404.112 q^{6} +11469.0 q^{7} -4096.00 q^{8} -19045.1 q^{9} -34027.4 q^{10} -10430.4 q^{11} +6465.79 q^{12} +144659. q^{13} -183503. q^{14} +53714.4 q^{15} +65536.0 q^{16} -654214. q^{17} +304721. q^{18} +130321. q^{19} +544439. q^{20} +289672. q^{21} +166886. q^{22} +1.63517e6 q^{23} -103453. q^{24} +2.56979e6 q^{25} -2.31454e6 q^{26} -978155. q^{27} +2.93605e6 q^{28} +3.60690e6 q^{29} -859431. q^{30} +3.62937e6 q^{31} -1.04858e6 q^{32} -263440. q^{33} +1.04674e7 q^{34} +2.43912e7 q^{35} -4.87554e6 q^{36} -1.06553e7 q^{37} -2.08514e6 q^{38} +3.65364e6 q^{39} -8.71102e6 q^{40} +1.31851e7 q^{41} -4.63474e6 q^{42} -2.73590e6 q^{43} -2.67018e6 q^{44} -4.05035e7 q^{45} -2.61627e7 q^{46} -5.26213e7 q^{47} +1.65524e6 q^{48} +9.11835e7 q^{49} -4.11167e7 q^{50} -1.65235e7 q^{51} +3.70326e7 q^{52} +5.87078e7 q^{53} +1.56505e7 q^{54} -2.21825e7 q^{55} -4.69769e7 q^{56} +3.29152e6 q^{57} -5.77104e7 q^{58} -7.23340e7 q^{59} +1.37509e7 q^{60} -7.46069e7 q^{61} -5.80700e7 q^{62} -2.18427e8 q^{63} +1.67772e7 q^{64} +3.07647e8 q^{65} +4.21504e6 q^{66} +1.12380e8 q^{67} -1.67479e8 q^{68} +4.12994e7 q^{69} -3.90259e8 q^{70} +3.11980e8 q^{71} +7.80087e7 q^{72} +3.47317e6 q^{73} +1.70485e8 q^{74} +6.49052e7 q^{75} +3.33622e7 q^{76} -1.19626e8 q^{77} -5.84582e7 q^{78} +1.96071e8 q^{79} +1.39376e8 q^{80} +3.50159e8 q^{81} -2.10961e8 q^{82} -3.30282e8 q^{83} +7.41559e7 q^{84} -1.39133e9 q^{85} +4.37745e7 q^{86} +9.10995e7 q^{87} +4.27229e7 q^{88} -6.40446e8 q^{89} +6.48055e8 q^{90} +1.65908e9 q^{91} +4.18603e8 q^{92} +9.16671e7 q^{93} +8.41940e8 q^{94} +2.77156e8 q^{95} -2.64839e7 q^{96} +1.56258e9 q^{97} -1.45894e9 q^{98} +1.98648e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + O(q^{10}) \) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + 22320q^{10} - 104249q^{11} + 21504q^{12} + 120486q^{13} - 196912q^{14} - 591090q^{15} + 262144q^{16} - 412139q^{17} - 264608q^{18} + 521284q^{19} - 357120q^{20} + 2437006q^{21} + 1667984q^{22} + 3010300q^{23} - 344064q^{24} + 9760585q^{25} - 1927776q^{26} + 12387978q^{27} + 3150592q^{28} + 6153240q^{29} + 9457440q^{30} + 12774024q^{31} - 4194304q^{32} - 3258022q^{33} + 6594224q^{34} + 9823425q^{35} + 4233728q^{36} + 20506048q^{37} - 8340544q^{38} + 69881444q^{39} + 5713920q^{40} + 11620300q^{41} - 38992096q^{42} + 7698327q^{43} - 26687744q^{44} - 124015815q^{45} - 48164800q^{46} - 31581083q^{47} + 5505024q^{48} + 18970383q^{49} - 156169360q^{50} - 8594812q^{51} + 30844416q^{52} + 72549422q^{53} - 198207648q^{54} + 21332505q^{55} - 50409472q^{56} + 10946964q^{57} - 98451840q^{58} - 149234120q^{59} - 151319040q^{60} + 129004373q^{61} - 204384384q^{62} + 102967551q^{63} + 67108864q^{64} + 124691700q^{65} + 52128352q^{66} + 132595266q^{67} - 105507584q^{68} - 45529972q^{69} - 157174800q^{70} - 47138482q^{71} - 67739648q^{72} - 39332795q^{73} - 328096768q^{74} + 824627010q^{75} + 133448704q^{76} - 165933719q^{77} - 1118103104q^{78} - 307010840q^{79} - 91422720q^{80} + 1305551744q^{81} - 185924800q^{82} - 746568232q^{83} + 623873536q^{84} - 105005985q^{85} - 123173232q^{86} - 82148208q^{87} + 427003904q^{88} + 286943482q^{89} + 1984253040q^{90} + 3155781114q^{91} + 770636800q^{92} + 1151901596q^{93} + 505297328q^{94} - 181797795q^{95} - 88080384q^{96} + 793519958q^{97} - 303526128q^{98} - 1681833809q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) 25.2570 0.180026 0.0900132 0.995941i \(-0.471309\pi\)
0.0900132 + 0.995941i \(0.471309\pi\)
\(4\) 256.000 0.500000
\(5\) 2126.71 1.52175 0.760877 0.648897i \(-0.224769\pi\)
0.760877 + 0.648897i \(0.224769\pi\)
\(6\) −404.112 −0.127298
\(7\) 11469.0 1.80544 0.902720 0.430229i \(-0.141567\pi\)
0.902720 + 0.430229i \(0.141567\pi\)
\(8\) −4096.00 −0.353553
\(9\) −19045.1 −0.967591
\(10\) −34027.4 −1.07604
\(11\) −10430.4 −0.214800 −0.107400 0.994216i \(-0.534252\pi\)
−0.107400 + 0.994216i \(0.534252\pi\)
\(12\) 6465.79 0.0900132
\(13\) 144659. 1.40475 0.702375 0.711807i \(-0.252123\pi\)
0.702375 + 0.711807i \(0.252123\pi\)
\(14\) −183503. −1.27664
\(15\) 53714.4 0.273956
\(16\) 65536.0 0.250000
\(17\) −654214. −1.89976 −0.949882 0.312608i \(-0.898797\pi\)
−0.949882 + 0.312608i \(0.898797\pi\)
\(18\) 304721. 0.684190
\(19\) 130321. 0.229416
\(20\) 544439. 0.760877
\(21\) 289672. 0.325027
\(22\) 166886. 0.151886
\(23\) 1.63517e6 1.21839 0.609196 0.793020i \(-0.291492\pi\)
0.609196 + 0.793020i \(0.291492\pi\)
\(24\) −103453. −0.0636489
\(25\) 2.56979e6 1.31573
\(26\) −2.31454e6 −0.993308
\(27\) −978155. −0.354218
\(28\) 2.93605e6 0.902720
\(29\) 3.60690e6 0.946985 0.473493 0.880798i \(-0.342993\pi\)
0.473493 + 0.880798i \(0.342993\pi\)
\(30\) −859431. −0.193716
\(31\) 3.62937e6 0.705836 0.352918 0.935654i \(-0.385189\pi\)
0.352918 + 0.935654i \(0.385189\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) −263440. −0.0386696
\(34\) 1.04674e7 1.34334
\(35\) 2.43912e7 2.74743
\(36\) −4.87554e6 −0.483795
\(37\) −1.06553e7 −0.934668 −0.467334 0.884081i \(-0.654785\pi\)
−0.467334 + 0.884081i \(0.654785\pi\)
\(38\) −2.08514e6 −0.162221
\(39\) 3.65364e6 0.252892
\(40\) −8.71102e6 −0.538021
\(41\) 1.31851e7 0.728710 0.364355 0.931260i \(-0.381289\pi\)
0.364355 + 0.931260i \(0.381289\pi\)
\(42\) −4.63474e6 −0.229829
\(43\) −2.73590e6 −0.122037 −0.0610187 0.998137i \(-0.519435\pi\)
−0.0610187 + 0.998137i \(0.519435\pi\)
\(44\) −2.67018e6 −0.107400
\(45\) −4.05035e7 −1.47243
\(46\) −2.61627e7 −0.861533
\(47\) −5.26213e7 −1.57297 −0.786486 0.617608i \(-0.788102\pi\)
−0.786486 + 0.617608i \(0.788102\pi\)
\(48\) 1.65524e6 0.0450066
\(49\) 9.11835e7 2.25961
\(50\) −4.11167e7 −0.930364
\(51\) −1.65235e7 −0.342008
\(52\) 3.70326e7 0.702375
\(53\) 5.87078e7 1.02201 0.511004 0.859578i \(-0.329274\pi\)
0.511004 + 0.859578i \(0.329274\pi\)
\(54\) 1.56505e7 0.250470
\(55\) −2.21825e7 −0.326872
\(56\) −4.69769e7 −0.638319
\(57\) 3.29152e6 0.0413009
\(58\) −5.77104e7 −0.669620
\(59\) −7.23340e7 −0.777157 −0.388578 0.921416i \(-0.627034\pi\)
−0.388578 + 0.921416i \(0.627034\pi\)
\(60\) 1.37509e7 0.136978
\(61\) −7.46069e7 −0.689914 −0.344957 0.938619i \(-0.612106\pi\)
−0.344957 + 0.938619i \(0.612106\pi\)
\(62\) −5.80700e7 −0.499102
\(63\) −2.18427e8 −1.74693
\(64\) 1.67772e7 0.125000
\(65\) 3.07647e8 2.13768
\(66\) 4.21504e6 0.0273435
\(67\) 1.12380e8 0.681322 0.340661 0.940186i \(-0.389349\pi\)
0.340661 + 0.940186i \(0.389349\pi\)
\(68\) −1.67479e8 −0.949882
\(69\) 4.12994e7 0.219343
\(70\) −3.90259e8 −1.94273
\(71\) 3.11980e8 1.45701 0.728507 0.685038i \(-0.240214\pi\)
0.728507 + 0.685038i \(0.240214\pi\)
\(72\) 7.80087e7 0.342095
\(73\) 3.47317e6 0.0143144 0.00715720 0.999974i \(-0.497722\pi\)
0.00715720 + 0.999974i \(0.497722\pi\)
\(74\) 1.70485e8 0.660910
\(75\) 6.49052e7 0.236867
\(76\) 3.33622e7 0.114708
\(77\) −1.19626e8 −0.387808
\(78\) −5.84582e7 −0.178822
\(79\) 1.96071e8 0.566359 0.283179 0.959067i \(-0.408611\pi\)
0.283179 + 0.959067i \(0.408611\pi\)
\(80\) 1.39376e8 0.380438
\(81\) 3.50159e8 0.903822
\(82\) −2.10961e8 −0.515276
\(83\) −3.30282e8 −0.763895 −0.381948 0.924184i \(-0.624747\pi\)
−0.381948 + 0.924184i \(0.624747\pi\)
\(84\) 7.41559e7 0.162513
\(85\) −1.39133e9 −2.89097
\(86\) 4.37745e7 0.0862934
\(87\) 9.10995e7 0.170482
\(88\) 4.27229e7 0.0759431
\(89\) −6.40446e8 −1.08200 −0.541000 0.841023i \(-0.681954\pi\)
−0.541000 + 0.841023i \(0.681954\pi\)
\(90\) 6.48055e8 1.04117
\(91\) 1.65908e9 2.53619
\(92\) 4.18603e8 0.609196
\(93\) 9.16671e7 0.127069
\(94\) 8.41940e8 1.11226
\(95\) 2.77156e8 0.349114
\(96\) −2.64839e7 −0.0318245
\(97\) 1.56258e9 1.79213 0.896064 0.443925i \(-0.146414\pi\)
0.896064 + 0.443925i \(0.146414\pi\)
\(98\) −1.45894e9 −1.59779
\(99\) 1.98648e8 0.207838
\(100\) 6.57866e8 0.657866
\(101\) 2.55498e8 0.244310 0.122155 0.992511i \(-0.461019\pi\)
0.122155 + 0.992511i \(0.461019\pi\)
\(102\) 2.64376e8 0.241836
\(103\) −9.05731e8 −0.792925 −0.396462 0.918051i \(-0.629762\pi\)
−0.396462 + 0.918051i \(0.629762\pi\)
\(104\) −5.92521e8 −0.496654
\(105\) 6.16049e8 0.494610
\(106\) −9.39325e8 −0.722669
\(107\) −3.97489e8 −0.293155 −0.146578 0.989199i \(-0.546826\pi\)
−0.146578 + 0.989199i \(0.546826\pi\)
\(108\) −2.50408e8 −0.177109
\(109\) 5.91761e8 0.401538 0.200769 0.979639i \(-0.435656\pi\)
0.200769 + 0.979639i \(0.435656\pi\)
\(110\) 3.54919e8 0.231133
\(111\) −2.69120e8 −0.168265
\(112\) 7.51630e8 0.451360
\(113\) −2.57125e9 −1.48351 −0.741757 0.670668i \(-0.766007\pi\)
−0.741757 + 0.670668i \(0.766007\pi\)
\(114\) −5.26643e7 −0.0292041
\(115\) 3.47753e9 1.85409
\(116\) 9.23367e8 0.473493
\(117\) −2.75503e9 −1.35922
\(118\) 1.15734e9 0.549533
\(119\) −7.50316e9 −3.42991
\(120\) −2.20014e8 −0.0968579
\(121\) −2.24915e9 −0.953861
\(122\) 1.19371e9 0.487843
\(123\) 3.33015e8 0.131187
\(124\) 9.29120e8 0.352918
\(125\) 1.31147e9 0.480468
\(126\) 3.49484e9 1.23526
\(127\) 1.47739e9 0.503938 0.251969 0.967735i \(-0.418922\pi\)
0.251969 + 0.967735i \(0.418922\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −6.91007e7 −0.0219699
\(130\) −4.92236e9 −1.51157
\(131\) −5.81972e9 −1.72656 −0.863279 0.504726i \(-0.831593\pi\)
−0.863279 + 0.504726i \(0.831593\pi\)
\(132\) −6.74407e7 −0.0193348
\(133\) 1.49465e9 0.414196
\(134\) −1.79808e9 −0.481768
\(135\) −2.08026e9 −0.539033
\(136\) 2.67966e9 0.671668
\(137\) −2.81502e8 −0.0682715 −0.0341358 0.999417i \(-0.510868\pi\)
−0.0341358 + 0.999417i \(0.510868\pi\)
\(138\) −6.60791e8 −0.155099
\(139\) −3.71990e9 −0.845210 −0.422605 0.906314i \(-0.638884\pi\)
−0.422605 + 0.906314i \(0.638884\pi\)
\(140\) 6.24415e9 1.37372
\(141\) −1.32905e9 −0.283176
\(142\) −4.99168e9 −1.03027
\(143\) −1.50884e9 −0.301740
\(144\) −1.24814e9 −0.241898
\(145\) 7.67085e9 1.44108
\(146\) −5.55707e7 −0.0101218
\(147\) 2.30302e9 0.406790
\(148\) −2.72775e9 −0.467334
\(149\) 2.98264e9 0.495750 0.247875 0.968792i \(-0.420268\pi\)
0.247875 + 0.968792i \(0.420268\pi\)
\(150\) −1.03848e9 −0.167490
\(151\) −1.07704e10 −1.68592 −0.842960 0.537977i \(-0.819189\pi\)
−0.842960 + 0.537977i \(0.819189\pi\)
\(152\) −5.33795e8 −0.0811107
\(153\) 1.24596e10 1.83819
\(154\) 1.91401e9 0.274221
\(155\) 7.71864e9 1.07411
\(156\) 9.35332e8 0.126446
\(157\) 4.65738e8 0.0611777 0.0305889 0.999532i \(-0.490262\pi\)
0.0305889 + 0.999532i \(0.490262\pi\)
\(158\) −3.13714e9 −0.400476
\(159\) 1.48278e9 0.183988
\(160\) −2.23002e9 −0.269011
\(161\) 1.87537e10 2.19973
\(162\) −5.60255e9 −0.639099
\(163\) −3.04519e9 −0.337885 −0.168943 0.985626i \(-0.554035\pi\)
−0.168943 + 0.985626i \(0.554035\pi\)
\(164\) 3.37538e9 0.364355
\(165\) −5.60262e8 −0.0588455
\(166\) 5.28452e9 0.540155
\(167\) −5.43187e9 −0.540413 −0.270206 0.962802i \(-0.587092\pi\)
−0.270206 + 0.962802i \(0.587092\pi\)
\(168\) −1.18649e9 −0.114914
\(169\) 1.03216e10 0.973322
\(170\) 2.22612e10 2.04423
\(171\) −2.48197e9 −0.221980
\(172\) −7.00391e8 −0.0610187
\(173\) −5.67736e8 −0.0481880 −0.0240940 0.999710i \(-0.507670\pi\)
−0.0240940 + 0.999710i \(0.507670\pi\)
\(174\) −1.45759e9 −0.120549
\(175\) 2.94728e10 2.37548
\(176\) −6.83566e8 −0.0536999
\(177\) −1.82694e9 −0.139909
\(178\) 1.02471e10 0.765089
\(179\) −1.14816e9 −0.0835920 −0.0417960 0.999126i \(-0.513308\pi\)
−0.0417960 + 0.999126i \(0.513308\pi\)
\(180\) −1.03689e10 −0.736217
\(181\) −1.40584e10 −0.973606 −0.486803 0.873512i \(-0.661837\pi\)
−0.486803 + 0.873512i \(0.661837\pi\)
\(182\) −2.65453e10 −1.79336
\(183\) −1.88435e9 −0.124203
\(184\) −6.69765e9 −0.430767
\(185\) −2.26608e10 −1.42233
\(186\) −1.46667e9 −0.0898514
\(187\) 6.82370e9 0.408068
\(188\) −1.34710e10 −0.786486
\(189\) −1.12184e10 −0.639519
\(190\) −4.43449e9 −0.246861
\(191\) −1.97985e10 −1.07642 −0.538209 0.842811i \(-0.680899\pi\)
−0.538209 + 0.842811i \(0.680899\pi\)
\(192\) 4.23742e8 0.0225033
\(193\) 2.34501e10 1.21657 0.608284 0.793719i \(-0.291858\pi\)
0.608284 + 0.793719i \(0.291858\pi\)
\(194\) −2.50013e10 −1.26723
\(195\) 7.77025e9 0.384839
\(196\) 2.33430e10 1.12981
\(197\) 6.29979e8 0.0298008 0.0149004 0.999889i \(-0.495257\pi\)
0.0149004 + 0.999889i \(0.495257\pi\)
\(198\) −3.17836e9 −0.146964
\(199\) −6.30124e9 −0.284831 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(200\) −1.05259e10 −0.465182
\(201\) 2.83838e9 0.122656
\(202\) −4.08797e9 −0.172753
\(203\) 4.13674e10 1.70973
\(204\) −4.23001e9 −0.171004
\(205\) 2.80409e10 1.10892
\(206\) 1.44917e10 0.560682
\(207\) −3.11419e10 −1.17890
\(208\) 9.48034e9 0.351187
\(209\) −1.35930e9 −0.0492784
\(210\) −9.85678e9 −0.349742
\(211\) −4.85098e10 −1.68484 −0.842419 0.538823i \(-0.818869\pi\)
−0.842419 + 0.538823i \(0.818869\pi\)
\(212\) 1.50292e10 0.511004
\(213\) 7.87967e9 0.262301
\(214\) 6.35982e9 0.207292
\(215\) −5.81849e9 −0.185711
\(216\) 4.00652e9 0.125235
\(217\) 4.16252e10 1.27435
\(218\) −9.46818e9 −0.283931
\(219\) 8.77219e7 0.00257697
\(220\) −5.67871e9 −0.163436
\(221\) −9.46377e10 −2.66869
\(222\) 4.30593e9 0.118981
\(223\) 2.29208e10 0.620666 0.310333 0.950628i \(-0.399560\pi\)
0.310333 + 0.950628i \(0.399560\pi\)
\(224\) −1.20261e10 −0.319160
\(225\) −4.89419e10 −1.27309
\(226\) 4.11401e10 1.04900
\(227\) −2.40728e10 −0.601742 −0.300871 0.953665i \(-0.597277\pi\)
−0.300871 + 0.953665i \(0.597277\pi\)
\(228\) 8.42628e8 0.0206504
\(229\) 1.31840e10 0.316801 0.158400 0.987375i \(-0.449366\pi\)
0.158400 + 0.987375i \(0.449366\pi\)
\(230\) −5.56406e10 −1.31104
\(231\) −3.02139e9 −0.0698156
\(232\) −1.47739e10 −0.334810
\(233\) 1.64719e10 0.366135 0.183068 0.983100i \(-0.441397\pi\)
0.183068 + 0.983100i \(0.441397\pi\)
\(234\) 4.40805e10 0.961116
\(235\) −1.11910e11 −2.39367
\(236\) −1.85175e10 −0.388578
\(237\) 4.95217e9 0.101959
\(238\) 1.20051e11 2.42531
\(239\) 6.84337e9 0.135669 0.0678343 0.997697i \(-0.478391\pi\)
0.0678343 + 0.997697i \(0.478391\pi\)
\(240\) 3.52023e9 0.0684889
\(241\) 6.99690e10 1.33607 0.668034 0.744130i \(-0.267136\pi\)
0.668034 + 0.744130i \(0.267136\pi\)
\(242\) 3.59865e10 0.674482
\(243\) 2.80970e10 0.516930
\(244\) −1.90994e10 −0.344957
\(245\) 1.93921e11 3.43857
\(246\) −5.32824e9 −0.0927633
\(247\) 1.88520e10 0.322272
\(248\) −1.48659e10 −0.249551
\(249\) −8.34194e9 −0.137521
\(250\) −2.09836e10 −0.339742
\(251\) −6.10283e10 −0.970509 −0.485254 0.874373i \(-0.661273\pi\)
−0.485254 + 0.874373i \(0.661273\pi\)
\(252\) −5.59174e10 −0.873463
\(253\) −1.70554e10 −0.261710
\(254\) −2.36382e10 −0.356338
\(255\) −3.51407e10 −0.520451
\(256\) 4.29497e9 0.0625000
\(257\) −5.46978e10 −0.782115 −0.391058 0.920366i \(-0.627891\pi\)
−0.391058 + 0.920366i \(0.627891\pi\)
\(258\) 1.10561e9 0.0155351
\(259\) −1.22205e11 −1.68749
\(260\) 7.87578e10 1.06884
\(261\) −6.86937e10 −0.916294
\(262\) 9.31155e10 1.22086
\(263\) 6.53058e9 0.0841688 0.0420844 0.999114i \(-0.486600\pi\)
0.0420844 + 0.999114i \(0.486600\pi\)
\(264\) 1.07905e9 0.0136718
\(265\) 1.24855e11 1.55524
\(266\) −2.39143e10 −0.292881
\(267\) −1.61757e10 −0.194788
\(268\) 2.87693e10 0.340661
\(269\) 1.22877e11 1.43082 0.715409 0.698706i \(-0.246241\pi\)
0.715409 + 0.698706i \(0.246241\pi\)
\(270\) 3.32841e10 0.381154
\(271\) 3.90560e10 0.439872 0.219936 0.975514i \(-0.429415\pi\)
0.219936 + 0.975514i \(0.429415\pi\)
\(272\) −4.28746e10 −0.474941
\(273\) 4.19035e10 0.456581
\(274\) 4.50404e9 0.0482752
\(275\) −2.68039e10 −0.282619
\(276\) 1.05726e10 0.109671
\(277\) −2.55807e10 −0.261068 −0.130534 0.991444i \(-0.541669\pi\)
−0.130534 + 0.991444i \(0.541669\pi\)
\(278\) 5.95184e10 0.597654
\(279\) −6.91217e10 −0.682961
\(280\) −9.99064e10 −0.971365
\(281\) −1.33338e10 −0.127578 −0.0637888 0.997963i \(-0.520318\pi\)
−0.0637888 + 0.997963i \(0.520318\pi\)
\(282\) 2.12649e10 0.200236
\(283\) −1.04221e11 −0.965862 −0.482931 0.875658i \(-0.660428\pi\)
−0.482931 + 0.875658i \(0.660428\pi\)
\(284\) 7.98668e10 0.728507
\(285\) 7.00012e9 0.0628497
\(286\) 2.41415e10 0.213362
\(287\) 1.51219e11 1.31564
\(288\) 1.99702e10 0.171047
\(289\) 3.09408e11 2.60910
\(290\) −1.22734e11 −1.01900
\(291\) 3.94660e10 0.322630
\(292\) 8.89132e8 0.00715720
\(293\) −2.48589e11 −1.97050 −0.985251 0.171114i \(-0.945263\pi\)
−0.985251 + 0.171114i \(0.945263\pi\)
\(294\) −3.68483e10 −0.287644
\(295\) −1.53834e11 −1.18264
\(296\) 4.36440e10 0.330455
\(297\) 1.02025e10 0.0760859
\(298\) −4.77222e10 −0.350548
\(299\) 2.36541e11 1.71154
\(300\) 1.66157e10 0.118433
\(301\) −3.13780e10 −0.220331
\(302\) 1.72327e11 1.19213
\(303\) 6.45312e9 0.0439823
\(304\) 8.54072e9 0.0573539
\(305\) −1.58668e11 −1.04988
\(306\) −1.99353e11 −1.29980
\(307\) 3.35661e10 0.215664 0.107832 0.994169i \(-0.465609\pi\)
0.107832 + 0.994169i \(0.465609\pi\)
\(308\) −3.06242e10 −0.193904
\(309\) −2.28761e10 −0.142747
\(310\) −1.23498e11 −0.759510
\(311\) 6.13779e10 0.372041 0.186020 0.982546i \(-0.440441\pi\)
0.186020 + 0.982546i \(0.440441\pi\)
\(312\) −1.49653e10 −0.0894108
\(313\) −1.45987e11 −0.859738 −0.429869 0.902891i \(-0.641440\pi\)
−0.429869 + 0.902891i \(0.641440\pi\)
\(314\) −7.45181e9 −0.0432592
\(315\) −4.64533e11 −2.65839
\(316\) 5.01942e10 0.283179
\(317\) 2.34017e10 0.130161 0.0650806 0.997880i \(-0.479270\pi\)
0.0650806 + 0.997880i \(0.479270\pi\)
\(318\) −2.37245e10 −0.130099
\(319\) −3.76214e10 −0.203412
\(320\) 3.56804e10 0.190219
\(321\) −1.00394e10 −0.0527757
\(322\) −3.00059e11 −1.55545
\(323\) −8.52578e10 −0.435836
\(324\) 8.96407e10 0.451911
\(325\) 3.71742e11 1.84828
\(326\) 4.87230e10 0.238921
\(327\) 1.49461e10 0.0722875
\(328\) −5.40060e10 −0.257638
\(329\) −6.03511e11 −2.83991
\(330\) 8.96419e9 0.0416101
\(331\) 1.52176e11 0.696818 0.348409 0.937343i \(-0.386722\pi\)
0.348409 + 0.937343i \(0.386722\pi\)
\(332\) −8.45522e10 −0.381948
\(333\) 2.02931e11 0.904376
\(334\) 8.69100e10 0.382129
\(335\) 2.39000e11 1.03680
\(336\) 1.89839e10 0.0812567
\(337\) 1.05261e11 0.444565 0.222282 0.974982i \(-0.428649\pi\)
0.222282 + 0.974982i \(0.428649\pi\)
\(338\) −1.65146e11 −0.688243
\(339\) −6.49421e10 −0.267072
\(340\) −3.56180e11 −1.44549
\(341\) −3.78558e10 −0.151613
\(342\) 3.97116e10 0.156964
\(343\) 5.82967e11 2.27416
\(344\) 1.12063e10 0.0431467
\(345\) 8.78321e10 0.333785
\(346\) 9.08377e9 0.0340740
\(347\) −2.81412e11 −1.04198 −0.520992 0.853562i \(-0.674438\pi\)
−0.520992 + 0.853562i \(0.674438\pi\)
\(348\) 2.33215e10 0.0852412
\(349\) 3.55958e11 1.28435 0.642176 0.766557i \(-0.278032\pi\)
0.642176 + 0.766557i \(0.278032\pi\)
\(350\) −4.71565e11 −1.67972
\(351\) −1.41498e11 −0.497588
\(352\) 1.09371e10 0.0379715
\(353\) 1.35601e11 0.464813 0.232406 0.972619i \(-0.425340\pi\)
0.232406 + 0.972619i \(0.425340\pi\)
\(354\) 2.92310e10 0.0989303
\(355\) 6.63492e11 2.21722
\(356\) −1.63954e11 −0.541000
\(357\) −1.89507e11 −0.617474
\(358\) 1.83706e10 0.0591085
\(359\) −2.12047e11 −0.673764 −0.336882 0.941547i \(-0.609372\pi\)
−0.336882 + 0.941547i \(0.609372\pi\)
\(360\) 1.65902e11 0.520584
\(361\) 1.69836e10 0.0526316
\(362\) 2.24935e11 0.688443
\(363\) −5.68069e10 −0.171720
\(364\) 4.24725e11 1.26810
\(365\) 7.38644e9 0.0217830
\(366\) 3.01495e10 0.0878245
\(367\) −4.48163e11 −1.28955 −0.644775 0.764372i \(-0.723049\pi\)
−0.644775 + 0.764372i \(0.723049\pi\)
\(368\) 1.07162e11 0.304598
\(369\) −2.51111e11 −0.705093
\(370\) 3.62572e11 1.00574
\(371\) 6.73318e11 1.84517
\(372\) 2.34668e10 0.0635346
\(373\) 1.86916e11 0.499985 0.249993 0.968248i \(-0.419572\pi\)
0.249993 + 0.968248i \(0.419572\pi\)
\(374\) −1.09179e11 −0.288548
\(375\) 3.31239e10 0.0864968
\(376\) 2.15537e11 0.556129
\(377\) 5.21769e11 1.33028
\(378\) 1.79495e11 0.452209
\(379\) −3.54895e11 −0.883535 −0.441768 0.897130i \(-0.645648\pi\)
−0.441768 + 0.897130i \(0.645648\pi\)
\(380\) 7.09518e10 0.174557
\(381\) 3.73143e10 0.0907221
\(382\) 3.16775e11 0.761143
\(383\) 6.87668e10 0.163299 0.0816497 0.996661i \(-0.473981\pi\)
0.0816497 + 0.996661i \(0.473981\pi\)
\(384\) −6.77987e9 −0.0159122
\(385\) −2.54410e11 −0.590147
\(386\) −3.75201e11 −0.860244
\(387\) 5.21055e10 0.118082
\(388\) 4.00020e11 0.896064
\(389\) 7.25930e10 0.160739 0.0803696 0.996765i \(-0.474390\pi\)
0.0803696 + 0.996765i \(0.474390\pi\)
\(390\) −1.24324e11 −0.272122
\(391\) −1.06975e12 −2.31466
\(392\) −3.73488e11 −0.798894
\(393\) −1.46989e11 −0.310826
\(394\) −1.00797e10 −0.0210724
\(395\) 4.16987e11 0.861858
\(396\) 5.08538e10 0.103919
\(397\) 2.87651e11 0.581178 0.290589 0.956848i \(-0.406149\pi\)
0.290589 + 0.956848i \(0.406149\pi\)
\(398\) 1.00820e11 0.201406
\(399\) 3.77503e10 0.0745662
\(400\) 1.68414e11 0.328933
\(401\) −4.88308e11 −0.943071 −0.471535 0.881847i \(-0.656300\pi\)
−0.471535 + 0.881847i \(0.656300\pi\)
\(402\) −4.54141e10 −0.0867308
\(403\) 5.25020e11 0.991524
\(404\) 6.54076e10 0.122155
\(405\) 7.44689e11 1.37539
\(406\) −6.61879e11 −1.20896
\(407\) 1.11139e11 0.200766
\(408\) 6.76802e10 0.120918
\(409\) 5.83827e11 1.03164 0.515821 0.856696i \(-0.327487\pi\)
0.515821 + 0.856696i \(0.327487\pi\)
\(410\) −4.48654e11 −0.784123
\(411\) −7.10990e9 −0.0122907
\(412\) −2.31867e11 −0.396462
\(413\) −8.29596e11 −1.40311
\(414\) 4.98270e11 0.833611
\(415\) −7.02416e11 −1.16246
\(416\) −1.51685e11 −0.248327
\(417\) −9.39535e10 −0.152160
\(418\) 2.17488e10 0.0348451
\(419\) −2.09427e11 −0.331947 −0.165974 0.986130i \(-0.553077\pi\)
−0.165974 + 0.986130i \(0.553077\pi\)
\(420\) 1.57708e11 0.247305
\(421\) −4.21346e11 −0.653687 −0.326844 0.945078i \(-0.605985\pi\)
−0.326844 + 0.945078i \(0.605985\pi\)
\(422\) 7.76157e11 1.19136
\(423\) 1.00218e12 1.52199
\(424\) −2.40467e11 −0.361335
\(425\) −1.68119e12 −2.49958
\(426\) −1.26075e11 −0.185475
\(427\) −8.55664e11 −1.24560
\(428\) −1.01757e11 −0.146578
\(429\) −3.81089e10 −0.0543211
\(430\) 9.30958e10 0.131317
\(431\) −3.62152e11 −0.505525 −0.252763 0.967528i \(-0.581339\pi\)
−0.252763 + 0.967528i \(0.581339\pi\)
\(432\) −6.41044e10 −0.0885545
\(433\) −8.27434e10 −0.113120 −0.0565598 0.998399i \(-0.518013\pi\)
−0.0565598 + 0.998399i \(0.518013\pi\)
\(434\) −6.66002e11 −0.901098
\(435\) 1.93743e11 0.259432
\(436\) 1.51491e11 0.200769
\(437\) 2.13097e11 0.279518
\(438\) −1.40355e9 −0.00182219
\(439\) −6.69663e11 −0.860530 −0.430265 0.902703i \(-0.641580\pi\)
−0.430265 + 0.902703i \(0.641580\pi\)
\(440\) 9.08593e10 0.115567
\(441\) −1.73660e12 −2.18638
\(442\) 1.51420e12 1.88705
\(443\) 5.66791e11 0.699208 0.349604 0.936898i \(-0.386316\pi\)
0.349604 + 0.936898i \(0.386316\pi\)
\(444\) −6.88948e10 −0.0841324
\(445\) −1.36205e12 −1.64654
\(446\) −3.66733e11 −0.438877
\(447\) 7.53325e10 0.0892480
\(448\) 1.92417e11 0.225680
\(449\) 3.81051e11 0.442461 0.221230 0.975222i \(-0.428993\pi\)
0.221230 + 0.975222i \(0.428993\pi\)
\(450\) 7.83070e11 0.900211
\(451\) −1.37525e11 −0.156527
\(452\) −6.58241e11 −0.741757
\(453\) −2.72029e11 −0.303510
\(454\) 3.85165e11 0.425496
\(455\) 3.52840e12 3.85946
\(456\) −1.34821e10 −0.0146021
\(457\) −1.09800e12 −1.17755 −0.588775 0.808297i \(-0.700390\pi\)
−0.588775 + 0.808297i \(0.700390\pi\)
\(458\) −2.10943e11 −0.224012
\(459\) 6.39923e11 0.672931
\(460\) 8.90249e11 0.927046
\(461\) 6.53157e11 0.673540 0.336770 0.941587i \(-0.390665\pi\)
0.336770 + 0.941587i \(0.390665\pi\)
\(462\) 4.83422e10 0.0493671
\(463\) −9.03119e11 −0.913336 −0.456668 0.889637i \(-0.650957\pi\)
−0.456668 + 0.889637i \(0.650957\pi\)
\(464\) 2.36382e11 0.236746
\(465\) 1.94950e11 0.193368
\(466\) −2.63550e11 −0.258897
\(467\) −1.07789e12 −1.04869 −0.524345 0.851506i \(-0.675690\pi\)
−0.524345 + 0.851506i \(0.675690\pi\)
\(468\) −7.05289e11 −0.679611
\(469\) 1.28888e12 1.23009
\(470\) 1.79057e12 1.69258
\(471\) 1.17631e10 0.0110136
\(472\) 2.96280e11 0.274766
\(473\) 2.85365e10 0.0262136
\(474\) −7.92347e10 −0.0720962
\(475\) 3.34898e11 0.301850
\(476\) −1.92081e12 −1.71495
\(477\) −1.11810e12 −0.988886
\(478\) −1.09494e11 −0.0959322
\(479\) −1.13640e12 −0.986329 −0.493165 0.869936i \(-0.664160\pi\)
−0.493165 + 0.869936i \(0.664160\pi\)
\(480\) −5.63237e10 −0.0484290
\(481\) −1.54138e12 −1.31297
\(482\) −1.11950e12 −0.944743
\(483\) 4.73661e11 0.396010
\(484\) −5.75784e11 −0.476931
\(485\) 3.32316e12 2.72718
\(486\) −4.49552e11 −0.365525
\(487\) 2.14926e11 0.173144 0.0865721 0.996246i \(-0.472409\pi\)
0.0865721 + 0.996246i \(0.472409\pi\)
\(488\) 3.05590e11 0.243921
\(489\) −7.69122e10 −0.0608283
\(490\) −3.10274e12 −2.43144
\(491\) 1.98317e12 1.53991 0.769953 0.638100i \(-0.220280\pi\)
0.769953 + 0.638100i \(0.220280\pi\)
\(492\) 8.52519e10 0.0655935
\(493\) −2.35969e12 −1.79905
\(494\) −3.01633e11 −0.227881
\(495\) 4.22467e11 0.316278
\(496\) 2.37855e11 0.176459
\(497\) 3.57809e12 2.63055
\(498\) 1.33471e11 0.0972422
\(499\) 1.66297e12 1.20070 0.600348 0.799739i \(-0.295029\pi\)
0.600348 + 0.799739i \(0.295029\pi\)
\(500\) 3.35737e11 0.240234
\(501\) −1.37193e11 −0.0972885
\(502\) 9.76453e11 0.686253
\(503\) 2.73936e12 1.90806 0.954031 0.299707i \(-0.0968891\pi\)
0.954031 + 0.299707i \(0.0968891\pi\)
\(504\) 8.94679e11 0.617632
\(505\) 5.43372e11 0.371780
\(506\) 2.72887e11 0.185057
\(507\) 2.60693e11 0.175224
\(508\) 3.78211e11 0.251969
\(509\) −1.41595e12 −0.935016 −0.467508 0.883989i \(-0.654848\pi\)
−0.467508 + 0.883989i \(0.654848\pi\)
\(510\) 5.62252e11 0.368015
\(511\) 3.98337e10 0.0258438
\(512\) −6.87195e10 −0.0441942
\(513\) −1.27474e11 −0.0812632
\(514\) 8.75165e11 0.553039
\(515\) −1.92623e12 −1.20664
\(516\) −1.76898e10 −0.0109850
\(517\) 5.48860e11 0.337874
\(518\) 1.95528e12 1.19323
\(519\) −1.43393e10 −0.00867510
\(520\) −1.26012e12 −0.755785
\(521\) −2.76092e12 −1.64166 −0.820832 0.571170i \(-0.806490\pi\)
−0.820832 + 0.571170i \(0.806490\pi\)
\(522\) 1.09910e12 0.647918
\(523\) 5.34558e11 0.312419 0.156209 0.987724i \(-0.450073\pi\)
0.156209 + 0.987724i \(0.450073\pi\)
\(524\) −1.48985e12 −0.863279
\(525\) 7.44395e11 0.427648
\(526\) −1.04489e11 −0.0595163
\(527\) −2.37439e12 −1.34092
\(528\) −1.72648e10 −0.00966739
\(529\) 8.72619e11 0.484478
\(530\) −1.99768e12 −1.09972
\(531\) 1.37761e12 0.751969
\(532\) 3.82630e11 0.207098
\(533\) 1.90733e12 1.02366
\(534\) 2.58812e11 0.137736
\(535\) −8.45346e11 −0.446110
\(536\) −4.60309e11 −0.240884
\(537\) −2.89991e10 −0.0150488
\(538\) −1.96603e12 −1.01174
\(539\) −9.51079e11 −0.485364
\(540\) −5.32546e11 −0.269516
\(541\) −3.40287e12 −1.70788 −0.853941 0.520369i \(-0.825794\pi\)
−0.853941 + 0.520369i \(0.825794\pi\)
\(542\) −6.24896e11 −0.311036
\(543\) −3.55074e11 −0.175275
\(544\) 6.85993e11 0.335834
\(545\) 1.25851e12 0.611042
\(546\) −6.70455e11 −0.322852
\(547\) −1.07691e12 −0.514321 −0.257161 0.966369i \(-0.582787\pi\)
−0.257161 + 0.966369i \(0.582787\pi\)
\(548\) −7.20646e10 −0.0341358
\(549\) 1.42090e12 0.667554
\(550\) 4.28863e11 0.199842
\(551\) 4.70055e11 0.217253
\(552\) −1.69162e11 −0.0775493
\(553\) 2.24873e12 1.02253
\(554\) 4.09292e11 0.184603
\(555\) −5.72342e11 −0.256058
\(556\) −9.52295e11 −0.422605
\(557\) 3.44597e12 1.51692 0.758460 0.651720i \(-0.225952\pi\)
0.758460 + 0.651720i \(0.225952\pi\)
\(558\) 1.10595e12 0.482926
\(559\) −3.95772e11 −0.171432
\(560\) 1.59850e12 0.686858
\(561\) 1.72346e11 0.0734631
\(562\) 2.13340e11 0.0902110
\(563\) 3.12894e12 1.31253 0.656266 0.754530i \(-0.272135\pi\)
0.656266 + 0.754530i \(0.272135\pi\)
\(564\) −3.40238e11 −0.141588
\(565\) −5.46832e12 −2.25754
\(566\) 1.66753e12 0.682968
\(567\) 4.01596e12 1.63180
\(568\) −1.27787e12 −0.515133
\(569\) −3.86048e12 −1.54396 −0.771980 0.635647i \(-0.780734\pi\)
−0.771980 + 0.635647i \(0.780734\pi\)
\(570\) −1.12002e11 −0.0444415
\(571\) 4.04984e12 1.59432 0.797160 0.603768i \(-0.206335\pi\)
0.797160 + 0.603768i \(0.206335\pi\)
\(572\) −3.86264e11 −0.150870
\(573\) −5.00050e11 −0.193784
\(574\) −2.41951e12 −0.930300
\(575\) 4.20204e12 1.60308
\(576\) −3.19523e11 −0.120949
\(577\) 1.81317e11 0.0681001 0.0340500 0.999420i \(-0.489159\pi\)
0.0340500 + 0.999420i \(0.489159\pi\)
\(578\) −4.95053e12 −1.84492
\(579\) 5.92279e11 0.219014
\(580\) 1.96374e12 0.720539
\(581\) −3.78799e12 −1.37917
\(582\) −6.31456e11 −0.228134
\(583\) −6.12345e11 −0.219527
\(584\) −1.42261e10 −0.00506091
\(585\) −5.85917e12 −2.06840
\(586\) 3.97742e12 1.39336
\(587\) 4.63066e11 0.160980 0.0804900 0.996755i \(-0.474351\pi\)
0.0804900 + 0.996755i \(0.474351\pi\)
\(588\) 5.89574e11 0.203395
\(589\) 4.72984e11 0.161930
\(590\) 2.46134e12 0.836253
\(591\) 1.59114e10 0.00536493
\(592\) −6.98305e11 −0.233667
\(593\) 3.90685e12 1.29742 0.648711 0.761035i \(-0.275308\pi\)
0.648711 + 0.761035i \(0.275308\pi\)
\(594\) −1.63241e11 −0.0538008
\(595\) −1.59571e13 −5.21948
\(596\) 7.63555e11 0.247875
\(597\) −1.59150e11 −0.0512771
\(598\) −3.78465e12 −1.21024
\(599\) −4.47991e11 −0.142183 −0.0710916 0.997470i \(-0.522648\pi\)
−0.0710916 + 0.997470i \(0.522648\pi\)
\(600\) −2.65852e11 −0.0837450
\(601\) 2.01818e12 0.630994 0.315497 0.948927i \(-0.397829\pi\)
0.315497 + 0.948927i \(0.397829\pi\)
\(602\) 5.02048e11 0.155798
\(603\) −2.14029e12 −0.659241
\(604\) −2.75723e12 −0.842960
\(605\) −4.78331e12 −1.45154
\(606\) −1.03250e11 −0.0311002
\(607\) 6.23030e12 1.86277 0.931387 0.364032i \(-0.118600\pi\)
0.931387 + 0.364032i \(0.118600\pi\)
\(608\) −1.36651e11 −0.0405554
\(609\) 1.04482e12 0.307796
\(610\) 2.53868e12 0.742376
\(611\) −7.61212e12 −2.20963
\(612\) 3.18965e12 0.919097
\(613\) 3.59461e11 0.102820 0.0514102 0.998678i \(-0.483628\pi\)
0.0514102 + 0.998678i \(0.483628\pi\)
\(614\) −5.37058e11 −0.152498
\(615\) 7.08228e11 0.199634
\(616\) 4.89987e11 0.137111
\(617\) 4.00051e12 1.11130 0.555650 0.831416i \(-0.312469\pi\)
0.555650 + 0.831416i \(0.312469\pi\)
\(618\) 3.66017e11 0.100938
\(619\) −3.72631e12 −1.02017 −0.510084 0.860125i \(-0.670386\pi\)
−0.510084 + 0.860125i \(0.670386\pi\)
\(620\) 1.97597e12 0.537054
\(621\) −1.59945e12 −0.431576
\(622\) −9.82046e11 −0.263072
\(623\) −7.34525e12 −1.95349
\(624\) 2.39445e11 0.0632230
\(625\) −2.22999e12 −0.584580
\(626\) 2.33580e12 0.607926
\(627\) −3.43318e10 −0.00887141
\(628\) 1.19229e11 0.0305889
\(629\) 6.97084e12 1.77565
\(630\) 7.43252e12 1.87977
\(631\) 5.38677e11 0.135268 0.0676342 0.997710i \(-0.478455\pi\)
0.0676342 + 0.997710i \(0.478455\pi\)
\(632\) −8.03107e11 −0.200238
\(633\) −1.22521e12 −0.303315
\(634\) −3.74428e11 −0.0920378
\(635\) 3.14198e12 0.766869
\(636\) 3.79592e11 0.0919942
\(637\) 1.31905e13 3.17419
\(638\) 6.01942e11 0.143834
\(639\) −5.94168e12 −1.40979
\(640\) −5.70886e11 −0.134505
\(641\) 5.87686e12 1.37494 0.687471 0.726212i \(-0.258721\pi\)
0.687471 + 0.726212i \(0.258721\pi\)
\(642\) 1.60630e11 0.0373181
\(643\) 3.03822e12 0.700923 0.350462 0.936577i \(-0.386025\pi\)
0.350462 + 0.936577i \(0.386025\pi\)
\(644\) 4.80094e12 1.09987
\(645\) −1.46957e11 −0.0334328
\(646\) 1.36413e12 0.308182
\(647\) 1.92485e12 0.431844 0.215922 0.976411i \(-0.430724\pi\)
0.215922 + 0.976411i \(0.430724\pi\)
\(648\) −1.43425e12 −0.319549
\(649\) 7.54472e11 0.166933
\(650\) −5.94788e12 −1.30693
\(651\) 1.05133e12 0.229416
\(652\) −7.79567e11 −0.168943
\(653\) −4.80531e12 −1.03422 −0.517109 0.855920i \(-0.672992\pi\)
−0.517109 + 0.855920i \(0.672992\pi\)
\(654\) −2.39138e11 −0.0511150
\(655\) −1.23769e13 −2.62740
\(656\) 8.64097e11 0.182178
\(657\) −6.61468e10 −0.0138505
\(658\) 9.65618e12 2.00812
\(659\) 7.75087e12 1.60091 0.800454 0.599394i \(-0.204592\pi\)
0.800454 + 0.599394i \(0.204592\pi\)
\(660\) −1.43427e11 −0.0294228
\(661\) 4.58458e12 0.934100 0.467050 0.884231i \(-0.345317\pi\)
0.467050 + 0.884231i \(0.345317\pi\)
\(662\) −2.43481e12 −0.492725
\(663\) −2.39026e12 −0.480435
\(664\) 1.35284e12 0.270078
\(665\) 3.17869e12 0.630305
\(666\) −3.24689e12 −0.639490
\(667\) 5.89789e12 1.15380
\(668\) −1.39056e12 −0.270206
\(669\) 5.78910e11 0.111736
\(670\) −3.82400e12 −0.733131
\(671\) 7.78179e11 0.148193
\(672\) −3.03743e11 −0.0574571
\(673\) −7.38830e12 −1.38828 −0.694139 0.719841i \(-0.744215\pi\)
−0.694139 + 0.719841i \(0.744215\pi\)
\(674\) −1.68418e12 −0.314355
\(675\) −2.51365e12 −0.466056
\(676\) 2.64233e12 0.486661
\(677\) −3.48101e12 −0.636878 −0.318439 0.947943i \(-0.603159\pi\)
−0.318439 + 0.947943i \(0.603159\pi\)
\(678\) 1.03907e12 0.188848
\(679\) 1.79212e13 3.23558
\(680\) 5.69887e12 1.02211
\(681\) −6.08006e11 −0.108329
\(682\) 6.05692e11 0.107207
\(683\) 6.60095e12 1.16068 0.580341 0.814373i \(-0.302919\pi\)
0.580341 + 0.814373i \(0.302919\pi\)
\(684\) −6.35385e11 −0.110990
\(685\) −5.98675e11 −0.103892
\(686\) −9.32746e12 −1.60807
\(687\) 3.32987e11 0.0570325
\(688\) −1.79300e11 −0.0305093
\(689\) 8.49259e12 1.43567
\(690\) −1.40531e12 −0.236022
\(691\) −8.94184e12 −1.49202 −0.746012 0.665933i \(-0.768034\pi\)
−0.746012 + 0.665933i \(0.768034\pi\)
\(692\) −1.45340e11 −0.0240940
\(693\) 2.27828e12 0.375239
\(694\) 4.50260e12 0.736793
\(695\) −7.91117e12 −1.28620
\(696\) −3.73144e11 −0.0602746
\(697\) −8.62586e12 −1.38438
\(698\) −5.69532e12 −0.908174
\(699\) 4.16030e11 0.0659140
\(700\) 7.54505e12 1.18774
\(701\) −9.13468e11 −0.142877 −0.0714385 0.997445i \(-0.522759\pi\)
−0.0714385 + 0.997445i \(0.522759\pi\)
\(702\) 2.26398e12 0.351848
\(703\) −1.38861e12 −0.214428
\(704\) −1.74993e11 −0.0268499
\(705\) −2.82652e12 −0.430924
\(706\) −2.16962e12 −0.328672
\(707\) 2.93030e12 0.441088
\(708\) −4.67697e11 −0.0699543
\(709\) 2.22286e12 0.330372 0.165186 0.986262i \(-0.447178\pi\)
0.165186 + 0.986262i \(0.447178\pi\)
\(710\) −1.06159e13 −1.56781
\(711\) −3.73419e12 −0.548003
\(712\) 2.62327e12 0.382545
\(713\) 5.93463e12 0.859985
\(714\) 3.03212e12 0.436620
\(715\) −3.20888e12 −0.459173
\(716\) −2.93930e11 −0.0417960
\(717\) 1.72843e11 0.0244239
\(718\) 3.39276e12 0.476423
\(719\) −1.38107e12 −0.192724 −0.0963622 0.995346i \(-0.530721\pi\)
−0.0963622 + 0.995346i \(0.530721\pi\)
\(720\) −2.65443e12 −0.368109
\(721\) −1.03878e13 −1.43158
\(722\) −2.71737e11 −0.0372161
\(723\) 1.76721e12 0.240528
\(724\) −3.59896e12 −0.486803
\(725\) 9.26898e12 1.24598
\(726\) 9.08910e11 0.121424
\(727\) −1.01054e13 −1.34168 −0.670838 0.741604i \(-0.734065\pi\)
−0.670838 + 0.741604i \(0.734065\pi\)
\(728\) −6.79561e12 −0.896679
\(729\) −6.18254e12 −0.810761
\(730\) −1.18183e11 −0.0154029
\(731\) 1.78987e12 0.231842
\(732\) −4.82393e11 −0.0621013
\(733\) −2.46166e12 −0.314964 −0.157482 0.987522i \(-0.550338\pi\)
−0.157482 + 0.987522i \(0.550338\pi\)
\(734\) 7.17060e12 0.911850
\(735\) 4.89787e12 0.619034
\(736\) −1.71460e12 −0.215383
\(737\) −1.17217e12 −0.146348
\(738\) 4.01777e12 0.498576
\(739\) 8.80833e12 1.08641 0.543205 0.839600i \(-0.317211\pi\)
0.543205 + 0.839600i \(0.317211\pi\)
\(740\) −5.80115e12 −0.711167
\(741\) 4.76146e11 0.0580174
\(742\) −1.07731e13 −1.30474
\(743\) −2.35539e11 −0.0283539 −0.0141770 0.999900i \(-0.504513\pi\)
−0.0141770 + 0.999900i \(0.504513\pi\)
\(744\) −3.75468e11 −0.0449257
\(745\) 6.34322e12 0.754409
\(746\) −2.99066e12 −0.353543
\(747\) 6.29025e12 0.739138
\(748\) 1.74687e12 0.204034
\(749\) −4.55879e12 −0.529275
\(750\) −5.29982e11 −0.0611625
\(751\) −6.15738e12 −0.706344 −0.353172 0.935558i \(-0.614897\pi\)
−0.353172 + 0.935558i \(0.614897\pi\)
\(752\) −3.44859e12 −0.393243
\(753\) −1.54139e12 −0.174717
\(754\) −8.34831e12 −0.940648
\(755\) −2.29056e13 −2.56555
\(756\) −2.87192e12 −0.319760
\(757\) 1.74954e12 0.193639 0.0968195 0.995302i \(-0.469133\pi\)
0.0968195 + 0.995302i \(0.469133\pi\)
\(758\) 5.67833e12 0.624754
\(759\) −4.30769e11 −0.0471147
\(760\) −1.13523e12 −0.123430
\(761\) −1.52097e13 −1.64395 −0.821976 0.569522i \(-0.807128\pi\)
−0.821976 + 0.569522i \(0.807128\pi\)
\(762\) −5.97029e11 −0.0641502
\(763\) 6.78688e12 0.724953
\(764\) −5.06840e12 −0.538209
\(765\) 2.64979e13 2.79728
\(766\) −1.10027e12 −0.115470
\(767\) −1.04637e13 −1.09171
\(768\) 1.08478e11 0.0112516
\(769\) −9.63035e12 −0.993055 −0.496528 0.868021i \(-0.665392\pi\)
−0.496528 + 0.868021i \(0.665392\pi\)
\(770\) 4.07056e12 0.417297
\(771\) −1.38150e12 −0.140801
\(772\) 6.00322e12 0.608284
\(773\) 1.49129e13 1.50230 0.751148 0.660134i \(-0.229501\pi\)
0.751148 + 0.660134i \(0.229501\pi\)
\(774\) −8.33688e11 −0.0834967
\(775\) 9.32673e12 0.928692
\(776\) −6.40032e12 −0.633613
\(777\) −3.08653e12 −0.303792
\(778\) −1.16149e12 −0.113660
\(779\) 1.71829e12 0.167178
\(780\) 1.98918e12 0.192420
\(781\) −3.25407e12 −0.312966
\(782\) 1.71160e13 1.63671
\(783\) −3.52811e12 −0.335439
\(784\) 5.97580e12 0.564903
\(785\) 9.90492e11 0.0930974
\(786\) 2.35182e12 0.219787
\(787\) 1.51917e13 1.41163 0.705816 0.708396i \(-0.250581\pi\)
0.705816 + 0.708396i \(0.250581\pi\)
\(788\) 1.61275e11 0.0149004
\(789\) 1.64943e11 0.0151526
\(790\) −6.67180e12 −0.609426
\(791\) −2.94896e13 −2.67840
\(792\) −8.13660e11 −0.0734818
\(793\) −1.07925e13 −0.969157
\(794\) −4.60242e12 −0.410955
\(795\) 3.15346e12 0.279985
\(796\) −1.61312e12 −0.142416
\(797\) 1.37336e13 1.20565 0.602827 0.797872i \(-0.294041\pi\)
0.602827 + 0.797872i \(0.294041\pi\)
\(798\) −6.04005e11 −0.0527263
\(799\) 3.44256e13 2.98828
\(800\) −2.69462e12 −0.232591
\(801\) 1.21973e13 1.04693
\(802\) 7.81293e12 0.666852
\(803\) −3.62265e10 −0.00307473
\(804\) 7.26626e11 0.0613280
\(805\) 3.98837e13 3.34745
\(806\) −8.40032e12 −0.701113
\(807\) 3.10349e12 0.257585
\(808\) −1.04652e12 −0.0863767
\(809\) −1.54605e12 −0.126898 −0.0634491 0.997985i \(-0.520210\pi\)
−0.0634491 + 0.997985i \(0.520210\pi\)
\(810\) −1.19150e13 −0.972550
\(811\) 1.50745e13 1.22363 0.611814 0.791002i \(-0.290440\pi\)
0.611814 + 0.791002i \(0.290440\pi\)
\(812\) 1.05901e13 0.854863
\(813\) 9.86438e11 0.0791885
\(814\) −1.77822e12 −0.141963
\(815\) −6.47624e12 −0.514178
\(816\) −1.08288e12 −0.0855019
\(817\) −3.56546e11 −0.0279973
\(818\) −9.34123e12 −0.729482
\(819\) −3.15974e13 −2.45399
\(820\) 7.17846e12 0.554459
\(821\) 7.55550e12 0.580389 0.290194 0.956968i \(-0.406280\pi\)
0.290194 + 0.956968i \(0.406280\pi\)
\(822\) 1.13758e11 0.00869081
\(823\) 3.21334e12 0.244150 0.122075 0.992521i \(-0.461045\pi\)
0.122075 + 0.992521i \(0.461045\pi\)
\(824\) 3.70988e12 0.280341
\(825\) −6.76986e11 −0.0508788
\(826\) 1.32735e13 0.992148
\(827\) 8.79853e12 0.654087 0.327043 0.945009i \(-0.393948\pi\)
0.327043 + 0.945009i \(0.393948\pi\)
\(828\) −7.97233e12 −0.589452
\(829\) 2.39558e13 1.76163 0.880816 0.473459i \(-0.156995\pi\)
0.880816 + 0.473459i \(0.156995\pi\)
\(830\) 1.12387e13 0.821983
\(831\) −6.46092e11 −0.0469991
\(832\) 2.42697e12 0.175594
\(833\) −5.96535e13 −4.29273
\(834\) 1.50326e12 0.107593
\(835\) −1.15520e13 −0.822375
\(836\) −3.47980e11 −0.0246392
\(837\) −3.55009e12 −0.250020
\(838\) 3.35083e12 0.234722
\(839\) −2.38372e13 −1.66083 −0.830417 0.557143i \(-0.811897\pi\)
−0.830417 + 0.557143i \(0.811897\pi\)
\(840\) −2.52334e12 −0.174871
\(841\) −1.49741e12 −0.103219
\(842\) 6.74154e12 0.462227
\(843\) −3.36771e11 −0.0229673
\(844\) −1.24185e13 −0.842419
\(845\) 2.19511e13 1.48116
\(846\) −1.60348e13 −1.07621
\(847\) −2.57955e13 −1.72214
\(848\) 3.84747e12 0.255502
\(849\) −2.63230e12 −0.173881
\(850\) 2.68991e13 1.76747
\(851\) −1.74232e13 −1.13879
\(852\) 2.01720e12 0.131151
\(853\) 1.15908e13 0.749621 0.374811 0.927101i \(-0.377708\pi\)
0.374811 + 0.927101i \(0.377708\pi\)
\(854\) 1.36906e13 0.880771
\(855\) −5.27845e12 −0.337800
\(856\) 1.62811e12 0.103646
\(857\) 2.40055e13 1.52019 0.760093 0.649815i \(-0.225154\pi\)
0.760093 + 0.649815i \(0.225154\pi\)
\(858\) 6.09742e11 0.0384108
\(859\) −1.28254e13 −0.803712 −0.401856 0.915703i \(-0.631635\pi\)
−0.401856 + 0.915703i \(0.631635\pi\)
\(860\) −1.48953e12 −0.0928553
\(861\) 3.81934e12 0.236850
\(862\) 5.79443e12 0.357460
\(863\) 1.10966e13 0.680994 0.340497 0.940246i \(-0.389405\pi\)
0.340497 + 0.940246i \(0.389405\pi\)
\(864\) 1.02567e12 0.0626175
\(865\) −1.20741e12 −0.0733302
\(866\) 1.32390e12 0.0799877
\(867\) 7.81472e12 0.469707
\(868\) 1.06560e13 0.637173
\(869\) −2.04510e12 −0.121654
\(870\) −3.09988e12 −0.183446
\(871\) 1.62567e13 0.957087
\(872\) −2.42385e12 −0.141965
\(873\) −2.97594e13 −1.73405
\(874\) −3.40955e12 −0.197649
\(875\) 1.50412e13 0.867455
\(876\) 2.24568e10 0.00128848
\(877\) 2.34043e12 0.133597 0.0667986 0.997766i \(-0.478721\pi\)
0.0667986 + 0.997766i \(0.478721\pi\)
\(878\) 1.07146e13 0.608486
\(879\) −6.27860e12 −0.354742
\(880\) −1.45375e12 −0.0817180
\(881\) −1.38455e12 −0.0774315 −0.0387158 0.999250i \(-0.512327\pi\)
−0.0387158 + 0.999250i \(0.512327\pi\)
\(882\) 2.77856e13 1.54600
\(883\) −6.13771e12 −0.339769 −0.169884 0.985464i \(-0.554339\pi\)
−0.169884 + 0.985464i \(0.554339\pi\)
\(884\) −2.42272e13 −1.33435
\(885\) −3.88538e12 −0.212906
\(886\) −9.06866e12 −0.494415
\(887\) 8.64012e12 0.468666 0.234333 0.972156i \(-0.424709\pi\)
0.234333 + 0.972156i \(0.424709\pi\)
\(888\) 1.10232e12 0.0594906
\(889\) 1.69441e13 0.909830
\(890\) 2.17927e13 1.16428
\(891\) −3.65229e12 −0.194141
\(892\) 5.86772e12 0.310333
\(893\) −6.85766e12 −0.360864
\(894\) −1.20532e12 −0.0631079
\(895\) −2.44181e12 −0.127206
\(896\) −3.07868e12 −0.159580
\(897\) 5.97431e12 0.308121
\(898\) −6.09682e12 −0.312867
\(899\) 1.30908e13 0.668417
\(900\) −1.25291e13 −0.636545
\(901\) −3.84075e13 −1.94157
\(902\) 2.20041e12 0.110681
\(903\) −7.92513e11 −0.0396654
\(904\) 1.05319e13 0.524502
\(905\) −2.98983e13 −1.48159
\(906\) 4.35246e12 0.214614
\(907\) −3.36715e13 −1.65207 −0.826037 0.563616i \(-0.809410\pi\)
−0.826037 + 0.563616i \(0.809410\pi\)
\(908\) −6.16263e12 −0.300871
\(909\) −4.86599e12 −0.236392
\(910\) −5.64544e13 −2.72905
\(911\) 2.03568e13 0.979211 0.489606 0.871944i \(-0.337141\pi\)
0.489606 + 0.871944i \(0.337141\pi\)
\(912\) 2.15713e11 0.0103252
\(913\) 3.44497e12 0.164084
\(914\) 1.75680e13 0.832654
\(915\) −4.00747e12 −0.189006
\(916\) 3.37509e12 0.158400
\(917\) −6.67462e13 −3.11720
\(918\) −1.02388e13 −0.475834
\(919\) 2.44953e13 1.13282 0.566412 0.824122i \(-0.308331\pi\)
0.566412 + 0.824122i \(0.308331\pi\)
\(920\) −1.42440e13 −0.655520
\(921\) 8.47779e11 0.0388252
\(922\) −1.04505e13 −0.476265
\(923\) 4.51306e13 2.04674
\(924\) −7.73475e11 −0.0349078
\(925\) −2.73819e13 −1.22977
\(926\) 1.44499e13 0.645826
\(927\) 1.72497e13 0.767226
\(928\) −3.78211e12 −0.167405
\(929\) 2.79588e13 1.23154 0.615768 0.787928i \(-0.288846\pi\)
0.615768 + 0.787928i \(0.288846\pi\)
\(930\) −3.11920e12 −0.136732
\(931\) 1.18831e13 0.518391
\(932\) 4.21680e12 0.183068
\(933\) 1.55022e12 0.0669771
\(934\) 1.72462e13 0.741536
\(935\) 1.45121e13 0.620979
\(936\) 1.12846e13 0.480558
\(937\) −8.07789e12 −0.342349 −0.171175 0.985241i \(-0.554756\pi\)
−0.171175 + 0.985241i \(0.554756\pi\)
\(938\) −2.06221e13 −0.869802
\(939\) −3.68720e12 −0.154775
\(940\) −2.86491e13 −1.19684
\(941\) 1.80335e12 0.0749768 0.0374884 0.999297i \(-0.488064\pi\)
0.0374884 + 0.999297i \(0.488064\pi\)
\(942\) −1.88210e11 −0.00778779
\(943\) 2.15598e13 0.887855
\(944\) −4.74048e12 −0.194289
\(945\) −2.38584e13 −0.973191
\(946\) −4.56584e11 −0.0185358
\(947\) 1.33152e13 0.537989 0.268995 0.963142i \(-0.413309\pi\)
0.268995 + 0.963142i \(0.413309\pi\)
\(948\) 1.26775e12 0.0509797
\(949\) 5.02424e11 0.0201082
\(950\) −5.35836e12 −0.213440
\(951\) 5.91057e11 0.0234324
\(952\) 3.07329e13 1.21266
\(953\) −3.50858e13 −1.37789 −0.688943 0.724815i \(-0.741925\pi\)
−0.688943 + 0.724815i \(0.741925\pi\)
\(954\) 1.78895e13 0.699248
\(955\) −4.21057e13 −1.63804
\(956\) 1.75190e12 0.0678343
\(957\) −9.50203e11 −0.0366195
\(958\) 1.81824e13 0.697440
\(959\) −3.22854e12 −0.123260
\(960\) 9.01179e11 0.0342445
\(961\) −1.32673e13 −0.501795
\(962\) 2.46620e13 0.928413
\(963\) 7.57021e12 0.283654
\(964\) 1.79121e13 0.668034
\(965\) 4.98716e13 1.85132
\(966\) −7.57858e12 −0.280021
\(967\) 3.33371e13 1.22605 0.613026 0.790063i \(-0.289952\pi\)
0.613026 + 0.790063i \(0.289952\pi\)
\(968\) 9.21254e12 0.337241
\(969\) −2.15336e12 −0.0784619
\(970\) −5.31705e13 −1.92840
\(971\) −3.31130e13 −1.19540 −0.597698 0.801722i \(-0.703918\pi\)
−0.597698 + 0.801722i \(0.703918\pi\)
\(972\) 7.19283e12 0.258465
\(973\) −4.26634e13 −1.52598
\(974\) −3.43881e12 −0.122431
\(975\) 9.38909e12 0.332738
\(976\) −4.88944e12 −0.172478
\(977\) −1.28147e12 −0.0449969 −0.0224984 0.999747i \(-0.507162\pi\)
−0.0224984 + 0.999747i \(0.507162\pi\)
\(978\) 1.23060e12 0.0430121
\(979\) 6.68009e12 0.232413
\(980\) 4.96439e13 1.71929
\(981\) −1.12701e13 −0.388525
\(982\) −3.17308e13 −1.08888
\(983\) 3.93077e13 1.34272 0.671362 0.741130i \(-0.265710\pi\)
0.671362 + 0.741130i \(0.265710\pi\)
\(984\) −1.36403e12 −0.0463816
\(985\) 1.33979e12 0.0453495
\(986\) 3.77550e13 1.27212
\(987\) −1.52429e13 −0.511258
\(988\) 4.82612e12 0.161136
\(989\) −4.47366e12 −0.148689
\(990\) −6.75947e12 −0.223642
\(991\) −3.13171e13 −1.03145 −0.515727 0.856753i \(-0.672478\pi\)
−0.515727 + 0.856753i \(0.672478\pi\)
\(992\) −3.80567e12 −0.124775
\(993\) 3.84350e12 0.125446
\(994\) −5.72494e13 −1.86008
\(995\) −1.34010e13 −0.433443
\(996\) −2.13554e12 −0.0687606
\(997\) −8.04007e12 −0.257710 −0.128855 0.991663i \(-0.541130\pi\)
−0.128855 + 0.991663i \(0.541130\pi\)
\(998\) −2.66076e13 −0.849021
\(999\) 1.04225e13 0.331076
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 38.10.a.d.1.3 4
3.2 odd 2 342.10.a.l.1.1 4
4.3 odd 2 304.10.a.e.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.10.a.d.1.3 4 1.1 even 1 trivial
304.10.a.e.1.2 4 4.3 odd 2
342.10.a.l.1.1 4 3.2 odd 2