Properties

Label 43.8.a.b.1.7
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.684341\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.315659 q^{2} +82.9732 q^{3} -127.900 q^{4} +531.642 q^{5} +26.1913 q^{6} -349.841 q^{7} -80.7773 q^{8} +4697.55 q^{9} +O(q^{10})\) \(q+0.315659 q^{2} +82.9732 q^{3} -127.900 q^{4} +531.642 q^{5} +26.1913 q^{6} -349.841 q^{7} -80.7773 q^{8} +4697.55 q^{9} +167.818 q^{10} +3331.87 q^{11} -10612.3 q^{12} -8584.81 q^{13} -110.430 q^{14} +44112.0 q^{15} +16345.7 q^{16} -25786.3 q^{17} +1482.83 q^{18} +16993.6 q^{19} -67997.2 q^{20} -29027.4 q^{21} +1051.73 q^{22} +80723.2 q^{23} -6702.35 q^{24} +204518. q^{25} -2709.87 q^{26} +208308. q^{27} +44744.7 q^{28} -67762.5 q^{29} +13924.4 q^{30} +18887.5 q^{31} +15499.2 q^{32} +276455. q^{33} -8139.67 q^{34} -185990. q^{35} -600818. q^{36} -179478. q^{37} +5364.19 q^{38} -712309. q^{39} -42944.6 q^{40} -611192. q^{41} -9162.77 q^{42} -79507.0 q^{43} -426147. q^{44} +2.49741e6 q^{45} +25481.0 q^{46} -1.18047e6 q^{47} +1.35626e6 q^{48} -701155. q^{49} +64557.9 q^{50} -2.13957e6 q^{51} +1.09800e6 q^{52} +538854. q^{53} +65754.5 q^{54} +1.77136e6 q^{55} +28259.2 q^{56} +1.41001e6 q^{57} -21389.9 q^{58} -1.96283e6 q^{59} -5.64194e6 q^{60} +2.10797e6 q^{61} +5962.00 q^{62} -1.64339e6 q^{63} -2.08736e6 q^{64} -4.56404e6 q^{65} +87265.7 q^{66} -813166. q^{67} +3.29807e6 q^{68} +6.69786e6 q^{69} -58709.4 q^{70} +1.76280e6 q^{71} -379456. q^{72} -3.47624e6 q^{73} -56653.7 q^{74} +1.69695e7 q^{75} -2.17349e6 q^{76} -1.16562e6 q^{77} -224847. q^{78} -3.68544e6 q^{79} +8.69008e6 q^{80} +7.01047e6 q^{81} -192928. q^{82} +3.77929e6 q^{83} +3.71261e6 q^{84} -1.37091e7 q^{85} -25097.1 q^{86} -5.62247e6 q^{87} -269139. q^{88} -2.83826e6 q^{89} +788332. q^{90} +3.00331e6 q^{91} -1.03245e7 q^{92} +1.56715e6 q^{93} -372627. q^{94} +9.03452e6 q^{95} +1.28602e6 q^{96} +4.40344e6 q^{97} -221326. q^{98} +1.56516e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.315659 0.0279006 0.0139503 0.999903i \(-0.495559\pi\)
0.0139503 + 0.999903i \(0.495559\pi\)
\(3\) 82.9732 1.77424 0.887122 0.461535i \(-0.152701\pi\)
0.887122 + 0.461535i \(0.152701\pi\)
\(4\) −127.900 −0.999222
\(5\) 531.642 1.90206 0.951029 0.309100i \(-0.100028\pi\)
0.951029 + 0.309100i \(0.100028\pi\)
\(6\) 26.1913 0.0495025
\(7\) −349.841 −0.385502 −0.192751 0.981248i \(-0.561741\pi\)
−0.192751 + 0.981248i \(0.561741\pi\)
\(8\) −80.7773 −0.0557795
\(9\) 4697.55 2.14794
\(10\) 167.818 0.0530686
\(11\) 3331.87 0.754767 0.377384 0.926057i \(-0.376824\pi\)
0.377384 + 0.926057i \(0.376824\pi\)
\(12\) −10612.3 −1.77286
\(13\) −8584.81 −1.08375 −0.541875 0.840459i \(-0.682285\pi\)
−0.541875 + 0.840459i \(0.682285\pi\)
\(14\) −110.430 −0.0107557
\(15\) 44112.0 3.37472
\(16\) 16345.7 0.997665
\(17\) −25786.3 −1.27297 −0.636484 0.771290i \(-0.719612\pi\)
−0.636484 + 0.771290i \(0.719612\pi\)
\(18\) 1482.83 0.0599289
\(19\) 16993.6 0.568392 0.284196 0.958766i \(-0.408273\pi\)
0.284196 + 0.958766i \(0.408273\pi\)
\(20\) −67997.2 −1.90058
\(21\) −29027.4 −0.683975
\(22\) 1051.73 0.0210585
\(23\) 80723.2 1.38341 0.691705 0.722180i \(-0.256860\pi\)
0.691705 + 0.722180i \(0.256860\pi\)
\(24\) −6702.35 −0.0989664
\(25\) 204518. 2.61783
\(26\) −2709.87 −0.0302373
\(27\) 208308. 2.03673
\(28\) 44744.7 0.385202
\(29\) −67762.5 −0.515937 −0.257968 0.966153i \(-0.583053\pi\)
−0.257968 + 0.966153i \(0.583053\pi\)
\(30\) 13924.4 0.0941566
\(31\) 18887.5 0.113870 0.0569348 0.998378i \(-0.481867\pi\)
0.0569348 + 0.998378i \(0.481867\pi\)
\(32\) 15499.2 0.0836150
\(33\) 276455. 1.33914
\(34\) −8139.67 −0.0355166
\(35\) −185990. −0.733248
\(36\) −600818. −2.14627
\(37\) −179478. −0.582511 −0.291255 0.956645i \(-0.594073\pi\)
−0.291255 + 0.956645i \(0.594073\pi\)
\(38\) 5364.19 0.0158585
\(39\) −712309. −1.92284
\(40\) −42944.6 −0.106096
\(41\) −611192. −1.38495 −0.692475 0.721442i \(-0.743480\pi\)
−0.692475 + 0.721442i \(0.743480\pi\)
\(42\) −9162.77 −0.0190833
\(43\) −79507.0 −0.152499
\(44\) −426147. −0.754180
\(45\) 2.49741e6 4.08551
\(46\) 25481.0 0.0385980
\(47\) −1.18047e6 −1.65849 −0.829245 0.558885i \(-0.811229\pi\)
−0.829245 + 0.558885i \(0.811229\pi\)
\(48\) 1.35626e6 1.77010
\(49\) −701155. −0.851388
\(50\) 64557.9 0.0730390
\(51\) −2.13957e6 −2.25855
\(52\) 1.09800e6 1.08291
\(53\) 538854. 0.497170 0.248585 0.968610i \(-0.420034\pi\)
0.248585 + 0.968610i \(0.420034\pi\)
\(54\) 65754.5 0.0568260
\(55\) 1.77136e6 1.43561
\(56\) 28259.2 0.0215031
\(57\) 1.41001e6 1.00847
\(58\) −21389.9 −0.0143949
\(59\) −1.96283e6 −1.24423 −0.622113 0.782927i \(-0.713726\pi\)
−0.622113 + 0.782927i \(0.713726\pi\)
\(60\) −5.64194e6 −3.37209
\(61\) 2.10797e6 1.18907 0.594537 0.804068i \(-0.297335\pi\)
0.594537 + 0.804068i \(0.297335\pi\)
\(62\) 5962.00 0.00317703
\(63\) −1.64339e6 −0.828037
\(64\) −2.08736e6 −0.995332
\(65\) −4.56404e6 −2.06136
\(66\) 87265.7 0.0373629
\(67\) −813166. −0.330307 −0.165153 0.986268i \(-0.552812\pi\)
−0.165153 + 0.986268i \(0.552812\pi\)
\(68\) 3.29807e6 1.27198
\(69\) 6.69786e6 2.45451
\(70\) −58709.4 −0.0204581
\(71\) 1.76280e6 0.584521 0.292260 0.956339i \(-0.405593\pi\)
0.292260 + 0.956339i \(0.405593\pi\)
\(72\) −379456. −0.119811
\(73\) −3.47624e6 −1.04588 −0.522938 0.852371i \(-0.675164\pi\)
−0.522938 + 0.852371i \(0.675164\pi\)
\(74\) −56653.7 −0.0162524
\(75\) 1.69695e7 4.64467
\(76\) −2.17349e6 −0.567950
\(77\) −1.16562e6 −0.290965
\(78\) −224847. −0.0536483
\(79\) −3.68544e6 −0.840997 −0.420498 0.907293i \(-0.638145\pi\)
−0.420498 + 0.907293i \(0.638145\pi\)
\(80\) 8.69008e6 1.89762
\(81\) 7.01047e6 1.46571
\(82\) −192928. −0.0386409
\(83\) 3.77929e6 0.725500 0.362750 0.931887i \(-0.381838\pi\)
0.362750 + 0.931887i \(0.381838\pi\)
\(84\) 3.71261e6 0.683443
\(85\) −1.37091e7 −2.42126
\(86\) −25097.1 −0.00425480
\(87\) −5.62247e6 −0.915398
\(88\) −269139. −0.0421005
\(89\) −2.83826e6 −0.426764 −0.213382 0.976969i \(-0.568448\pi\)
−0.213382 + 0.976969i \(0.568448\pi\)
\(90\) 788332. 0.113988
\(91\) 3.00331e6 0.417788
\(92\) −1.03245e7 −1.38233
\(93\) 1.56715e6 0.202032
\(94\) −372627. −0.0462729
\(95\) 9.03452e6 1.08112
\(96\) 1.28602e6 0.148353
\(97\) 4.40344e6 0.489882 0.244941 0.969538i \(-0.421231\pi\)
0.244941 + 0.969538i \(0.421231\pi\)
\(98\) −221326. −0.0237542
\(99\) 1.56516e7 1.62120
\(100\) −2.61579e7 −2.61579
\(101\) −5.88409e6 −0.568269 −0.284135 0.958784i \(-0.591706\pi\)
−0.284135 + 0.958784i \(0.591706\pi\)
\(102\) −675375. −0.0630150
\(103\) 698841. 0.0630156 0.0315078 0.999504i \(-0.489969\pi\)
0.0315078 + 0.999504i \(0.489969\pi\)
\(104\) 693458. 0.0604510
\(105\) −1.54322e7 −1.30096
\(106\) 170094. 0.0138714
\(107\) −2.58664e6 −0.204123 −0.102062 0.994778i \(-0.532544\pi\)
−0.102062 + 0.994778i \(0.532544\pi\)
\(108\) −2.66427e7 −2.03515
\(109\) 1.24939e7 0.924070 0.462035 0.886862i \(-0.347119\pi\)
0.462035 + 0.886862i \(0.347119\pi\)
\(110\) 559146. 0.0400544
\(111\) −1.48918e7 −1.03352
\(112\) −5.71841e6 −0.384602
\(113\) 1.92969e7 1.25810 0.629048 0.777367i \(-0.283445\pi\)
0.629048 + 0.777367i \(0.283445\pi\)
\(114\) 445084. 0.0281368
\(115\) 4.29158e7 2.63133
\(116\) 8.66685e6 0.515535
\(117\) −4.03276e7 −2.32783
\(118\) −619584. −0.0347147
\(119\) 9.02108e6 0.490732
\(120\) −3.56325e6 −0.188240
\(121\) −8.38585e6 −0.430326
\(122\) 665399. 0.0331759
\(123\) −5.07125e7 −2.45724
\(124\) −2.41571e6 −0.113781
\(125\) 6.71957e7 3.07720
\(126\) −518753. −0.0231027
\(127\) −5.00899e6 −0.216989 −0.108494 0.994097i \(-0.534603\pi\)
−0.108494 + 0.994097i \(0.534603\pi\)
\(128\) −2.64279e6 −0.111385
\(129\) −6.59695e6 −0.270570
\(130\) −1.44068e6 −0.0575131
\(131\) −4.18902e7 −1.62803 −0.814016 0.580843i \(-0.802723\pi\)
−0.814016 + 0.580843i \(0.802723\pi\)
\(132\) −3.53588e7 −1.33810
\(133\) −5.94506e6 −0.219117
\(134\) −256683. −0.00921575
\(135\) 1.10745e8 3.87398
\(136\) 2.08295e6 0.0710055
\(137\) 3.89920e7 1.29555 0.647774 0.761833i \(-0.275700\pi\)
0.647774 + 0.761833i \(0.275700\pi\)
\(138\) 2.11424e6 0.0684823
\(139\) 6.03467e7 1.90591 0.952954 0.303115i \(-0.0980267\pi\)
0.952954 + 0.303115i \(0.0980267\pi\)
\(140\) 2.37882e7 0.732677
\(141\) −9.79475e7 −2.94257
\(142\) 556446. 0.0163085
\(143\) −2.86034e7 −0.817978
\(144\) 7.67850e7 2.14293
\(145\) −3.60254e7 −0.981342
\(146\) −1.09731e6 −0.0291806
\(147\) −5.81770e7 −1.51057
\(148\) 2.29552e7 0.582057
\(149\) −1.86557e7 −0.462018 −0.231009 0.972952i \(-0.574203\pi\)
−0.231009 + 0.972952i \(0.574203\pi\)
\(150\) 5.35658e6 0.129589
\(151\) −1.15952e7 −0.274070 −0.137035 0.990566i \(-0.543757\pi\)
−0.137035 + 0.990566i \(0.543757\pi\)
\(152\) −1.37270e6 −0.0317046
\(153\) −1.21132e8 −2.73426
\(154\) −367939. −0.00811809
\(155\) 1.00414e7 0.216587
\(156\) 9.11046e7 1.92134
\(157\) 2.89477e7 0.596988 0.298494 0.954412i \(-0.403516\pi\)
0.298494 + 0.954412i \(0.403516\pi\)
\(158\) −1.16334e6 −0.0234643
\(159\) 4.47104e7 0.882102
\(160\) 8.24001e6 0.159041
\(161\) −2.82403e7 −0.533308
\(162\) 2.21292e6 0.0408943
\(163\) 779647. 0.0141007 0.00705036 0.999975i \(-0.497756\pi\)
0.00705036 + 0.999975i \(0.497756\pi\)
\(164\) 7.81716e7 1.38387
\(165\) 1.46975e8 2.54713
\(166\) 1.19297e6 0.0202419
\(167\) −4.18156e7 −0.694754 −0.347377 0.937726i \(-0.612928\pi\)
−0.347377 + 0.937726i \(0.612928\pi\)
\(168\) 2.34476e6 0.0381518
\(169\) 1.09504e7 0.174513
\(170\) −4.32739e6 −0.0675546
\(171\) 7.98284e7 1.22087
\(172\) 1.01690e7 0.152380
\(173\) 9.94492e7 1.46029 0.730147 0.683291i \(-0.239452\pi\)
0.730147 + 0.683291i \(0.239452\pi\)
\(174\) −1.77478e6 −0.0255401
\(175\) −7.15486e7 −1.00918
\(176\) 5.44618e7 0.753005
\(177\) −1.62862e8 −2.20756
\(178\) −895924. −0.0119070
\(179\) 7.81297e7 1.01819 0.509097 0.860709i \(-0.329979\pi\)
0.509097 + 0.860709i \(0.329979\pi\)
\(180\) −3.19420e8 −4.08233
\(181\) 1.22327e7 0.153337 0.0766684 0.997057i \(-0.475572\pi\)
0.0766684 + 0.997057i \(0.475572\pi\)
\(182\) 948024. 0.0116565
\(183\) 1.74905e8 2.10971
\(184\) −6.52061e6 −0.0771659
\(185\) −9.54177e7 −1.10797
\(186\) 494686. 0.00563682
\(187\) −8.59164e7 −0.960794
\(188\) 1.50983e8 1.65720
\(189\) −7.28747e7 −0.785164
\(190\) 2.85183e6 0.0301638
\(191\) −9.63352e7 −1.00039 −0.500194 0.865914i \(-0.666738\pi\)
−0.500194 + 0.865914i \(0.666738\pi\)
\(192\) −1.73195e8 −1.76596
\(193\) 2.89017e7 0.289383 0.144692 0.989477i \(-0.453781\pi\)
0.144692 + 0.989477i \(0.453781\pi\)
\(194\) 1.38999e6 0.0136680
\(195\) −3.78693e8 −3.65735
\(196\) 8.96779e7 0.850725
\(197\) 4.17794e7 0.389342 0.194671 0.980869i \(-0.437636\pi\)
0.194671 + 0.980869i \(0.437636\pi\)
\(198\) 4.94057e6 0.0452324
\(199\) 5.54068e7 0.498399 0.249199 0.968452i \(-0.419833\pi\)
0.249199 + 0.968452i \(0.419833\pi\)
\(200\) −1.65204e7 −0.146021
\(201\) −6.74710e7 −0.586045
\(202\) −1.85737e6 −0.0158551
\(203\) 2.37061e7 0.198895
\(204\) 2.73652e8 2.25680
\(205\) −3.24935e8 −2.63426
\(206\) 220596. 0.00175817
\(207\) 3.79201e8 2.97149
\(208\) −1.40325e8 −1.08122
\(209\) 5.66205e7 0.429004
\(210\) −4.87131e6 −0.0362976
\(211\) −9.44194e7 −0.691947 −0.345973 0.938244i \(-0.612451\pi\)
−0.345973 + 0.938244i \(0.612451\pi\)
\(212\) −6.89196e7 −0.496783
\(213\) 1.46265e8 1.03708
\(214\) −816497. −0.00569517
\(215\) −4.22692e7 −0.290061
\(216\) −1.68266e7 −0.113608
\(217\) −6.60760e6 −0.0438970
\(218\) 3.94381e6 0.0257821
\(219\) −2.88435e8 −1.85564
\(220\) −2.26557e8 −1.43449
\(221\) 2.21370e8 1.37958
\(222\) −4.70074e6 −0.0288357
\(223\) 1.04942e7 0.0633697 0.0316848 0.999498i \(-0.489913\pi\)
0.0316848 + 0.999498i \(0.489913\pi\)
\(224\) −5.42224e6 −0.0322338
\(225\) 9.60733e8 5.62294
\(226\) 6.09126e6 0.0351016
\(227\) 2.42554e7 0.137632 0.0688158 0.997629i \(-0.478078\pi\)
0.0688158 + 0.997629i \(0.478078\pi\)
\(228\) −1.80341e8 −1.00768
\(229\) 7.20589e7 0.396519 0.198259 0.980150i \(-0.436471\pi\)
0.198259 + 0.980150i \(0.436471\pi\)
\(230\) 1.35468e7 0.0734157
\(231\) −9.67153e7 −0.516242
\(232\) 5.47367e6 0.0287787
\(233\) 6.20516e7 0.321372 0.160686 0.987006i \(-0.448629\pi\)
0.160686 + 0.987006i \(0.448629\pi\)
\(234\) −1.27298e7 −0.0649479
\(235\) −6.27588e8 −3.15455
\(236\) 2.51046e8 1.24326
\(237\) −3.05792e8 −1.49213
\(238\) 2.84759e6 0.0136917
\(239\) 1.05074e8 0.497853 0.248926 0.968522i \(-0.419922\pi\)
0.248926 + 0.968522i \(0.419922\pi\)
\(240\) 7.21044e8 3.36684
\(241\) 3.02273e8 1.39104 0.695520 0.718507i \(-0.255174\pi\)
0.695520 + 0.718507i \(0.255174\pi\)
\(242\) −2.64707e6 −0.0120064
\(243\) 1.26111e8 0.563806
\(244\) −2.69610e8 −1.18815
\(245\) −3.72763e8 −1.61939
\(246\) −1.60079e7 −0.0685584
\(247\) −1.45887e8 −0.615995
\(248\) −1.52568e6 −0.00635158
\(249\) 3.13580e8 1.28721
\(250\) 2.12109e7 0.0858558
\(251\) −1.60365e8 −0.640105 −0.320052 0.947400i \(-0.603701\pi\)
−0.320052 + 0.947400i \(0.603701\pi\)
\(252\) 2.10191e8 0.827392
\(253\) 2.68959e8 1.04415
\(254\) −1.58114e6 −0.00605412
\(255\) −1.13748e9 −4.29590
\(256\) 2.66348e8 0.992225
\(257\) 9.94729e7 0.365543 0.182772 0.983155i \(-0.441493\pi\)
0.182772 + 0.983155i \(0.441493\pi\)
\(258\) −2.08239e6 −0.00754906
\(259\) 6.27885e7 0.224559
\(260\) 5.83743e8 2.05975
\(261\) −3.18318e8 −1.10820
\(262\) −1.32230e7 −0.0454231
\(263\) 1.41037e7 0.0478066 0.0239033 0.999714i \(-0.492391\pi\)
0.0239033 + 0.999714i \(0.492391\pi\)
\(264\) −2.23313e7 −0.0746966
\(265\) 2.86477e8 0.945647
\(266\) −1.87661e6 −0.00611349
\(267\) −2.35500e8 −0.757183
\(268\) 1.04004e8 0.330049
\(269\) 9.31759e7 0.291857 0.145929 0.989295i \(-0.453383\pi\)
0.145929 + 0.989295i \(0.453383\pi\)
\(270\) 3.49578e7 0.108086
\(271\) 4.68939e8 1.43128 0.715639 0.698470i \(-0.246135\pi\)
0.715639 + 0.698470i \(0.246135\pi\)
\(272\) −4.21496e8 −1.27000
\(273\) 2.49195e8 0.741258
\(274\) 1.23082e7 0.0361466
\(275\) 6.81426e8 1.97585
\(276\) −8.56659e8 −2.45260
\(277\) −6.06565e7 −0.171474 −0.0857370 0.996318i \(-0.527324\pi\)
−0.0857370 + 0.996318i \(0.527324\pi\)
\(278\) 1.90490e7 0.0531760
\(279\) 8.87248e7 0.244585
\(280\) 1.50238e7 0.0409002
\(281\) −6.12917e8 −1.64789 −0.823947 0.566667i \(-0.808233\pi\)
−0.823947 + 0.566667i \(0.808233\pi\)
\(282\) −3.09180e7 −0.0820994
\(283\) −6.99501e8 −1.83458 −0.917288 0.398224i \(-0.869627\pi\)
−0.917288 + 0.398224i \(0.869627\pi\)
\(284\) −2.25463e8 −0.584066
\(285\) 7.49623e8 1.91816
\(286\) −9.02894e6 −0.0228221
\(287\) 2.13820e8 0.533901
\(288\) 7.28082e7 0.179600
\(289\) 2.54593e8 0.620446
\(290\) −1.13717e7 −0.0273800
\(291\) 3.65368e8 0.869170
\(292\) 4.44613e8 1.04506
\(293\) 7.00364e8 1.62662 0.813312 0.581828i \(-0.197662\pi\)
0.813312 + 0.581828i \(0.197662\pi\)
\(294\) −1.83641e7 −0.0421458
\(295\) −1.04352e9 −2.36659
\(296\) 1.44977e7 0.0324922
\(297\) 6.94055e8 1.53726
\(298\) −5.88884e6 −0.0128906
\(299\) −6.92994e8 −1.49927
\(300\) −2.17040e9 −4.64105
\(301\) 2.78148e7 0.0587886
\(302\) −3.66015e6 −0.00764671
\(303\) −4.88221e8 −1.00825
\(304\) 2.77773e8 0.567065
\(305\) 1.12068e9 2.26169
\(306\) −3.82365e7 −0.0762875
\(307\) 5.88458e8 1.16073 0.580365 0.814357i \(-0.302910\pi\)
0.580365 + 0.814357i \(0.302910\pi\)
\(308\) 1.49083e8 0.290738
\(309\) 5.79850e7 0.111805
\(310\) 3.16965e6 0.00604290
\(311\) 7.48619e7 0.141123 0.0705617 0.997507i \(-0.477521\pi\)
0.0705617 + 0.997507i \(0.477521\pi\)
\(312\) 5.75384e7 0.107255
\(313\) 2.61454e8 0.481937 0.240969 0.970533i \(-0.422535\pi\)
0.240969 + 0.970533i \(0.422535\pi\)
\(314\) 9.13762e6 0.0166563
\(315\) −8.73697e8 −1.57498
\(316\) 4.71369e8 0.840342
\(317\) −1.08258e9 −1.90877 −0.954384 0.298582i \(-0.903486\pi\)
−0.954384 + 0.298582i \(0.903486\pi\)
\(318\) 1.41133e7 0.0246112
\(319\) −2.25775e8 −0.389412
\(320\) −1.10973e9 −1.89318
\(321\) −2.14622e8 −0.362165
\(322\) −8.91430e6 −0.0148796
\(323\) −4.38202e8 −0.723545
\(324\) −8.96641e8 −1.46457
\(325\) −1.75575e9 −2.83707
\(326\) 246103. 0.000393419 0
\(327\) 1.03666e9 1.63953
\(328\) 4.93704e7 0.0772518
\(329\) 4.12977e8 0.639352
\(330\) 4.63941e7 0.0710663
\(331\) −4.20559e8 −0.637425 −0.318713 0.947851i \(-0.603250\pi\)
−0.318713 + 0.947851i \(0.603250\pi\)
\(332\) −4.83373e8 −0.724935
\(333\) −8.43105e8 −1.25120
\(334\) −1.31995e7 −0.0193841
\(335\) −4.32313e8 −0.628263
\(336\) −4.74474e8 −0.682378
\(337\) 1.18120e9 1.68119 0.840597 0.541662i \(-0.182205\pi\)
0.840597 + 0.541662i \(0.182205\pi\)
\(338\) 3.45660e6 0.00486901
\(339\) 1.60113e9 2.23217
\(340\) 1.75339e9 2.41937
\(341\) 6.29305e7 0.0859450
\(342\) 2.51986e7 0.0340631
\(343\) 5.33401e8 0.713714
\(344\) 6.42236e6 0.00850629
\(345\) 3.56086e9 4.66862
\(346\) 3.13921e7 0.0407431
\(347\) 1.04092e9 1.33741 0.668707 0.743526i \(-0.266848\pi\)
0.668707 + 0.743526i \(0.266848\pi\)
\(348\) 7.19116e8 0.914685
\(349\) −6.58581e8 −0.829316 −0.414658 0.909977i \(-0.636099\pi\)
−0.414658 + 0.909977i \(0.636099\pi\)
\(350\) −2.25850e7 −0.0281567
\(351\) −1.78829e9 −2.20731
\(352\) 5.16412e7 0.0631098
\(353\) 3.03399e8 0.367115 0.183557 0.983009i \(-0.441239\pi\)
0.183557 + 0.983009i \(0.441239\pi\)
\(354\) −5.14089e7 −0.0615923
\(355\) 9.37180e8 1.11179
\(356\) 3.63015e8 0.426431
\(357\) 7.48508e8 0.870678
\(358\) 2.46624e7 0.0284082
\(359\) 1.42234e9 1.62245 0.811226 0.584733i \(-0.198801\pi\)
0.811226 + 0.584733i \(0.198801\pi\)
\(360\) −2.01734e8 −0.227888
\(361\) −6.05089e8 −0.676930
\(362\) 3.86136e6 0.00427819
\(363\) −6.95800e8 −0.763504
\(364\) −3.84125e8 −0.417463
\(365\) −1.84812e9 −1.98932
\(366\) 5.52103e7 0.0588621
\(367\) 5.93187e8 0.626413 0.313206 0.949685i \(-0.398597\pi\)
0.313206 + 0.949685i \(0.398597\pi\)
\(368\) 1.31948e9 1.38018
\(369\) −2.87110e9 −2.97479
\(370\) −3.01195e7 −0.0309130
\(371\) −1.88513e8 −0.191660
\(372\) −2.00439e8 −0.201875
\(373\) −3.41490e8 −0.340719 −0.170360 0.985382i \(-0.554493\pi\)
−0.170360 + 0.985382i \(0.554493\pi\)
\(374\) −2.71203e7 −0.0268067
\(375\) 5.57544e9 5.45971
\(376\) 9.53553e7 0.0925098
\(377\) 5.81728e8 0.559146
\(378\) −2.30036e7 −0.0219066
\(379\) −1.21788e9 −1.14912 −0.574561 0.818462i \(-0.694827\pi\)
−0.574561 + 0.818462i \(0.694827\pi\)
\(380\) −1.15552e9 −1.08027
\(381\) −4.15612e8 −0.384991
\(382\) −3.04091e7 −0.0279114
\(383\) 1.21649e9 1.10640 0.553200 0.833048i \(-0.313406\pi\)
0.553200 + 0.833048i \(0.313406\pi\)
\(384\) −2.19281e8 −0.197625
\(385\) −6.19693e8 −0.553432
\(386\) 9.12310e6 0.00807397
\(387\) −3.73488e8 −0.327558
\(388\) −5.63202e8 −0.489500
\(389\) −2.23009e7 −0.0192087 −0.00960436 0.999954i \(-0.503057\pi\)
−0.00960436 + 0.999954i \(0.503057\pi\)
\(390\) −1.19538e8 −0.102042
\(391\) −2.08155e9 −1.76104
\(392\) 5.66374e7 0.0474900
\(393\) −3.47576e9 −2.88853
\(394\) 1.31881e7 0.0108629
\(395\) −1.95933e9 −1.59962
\(396\) −2.00185e9 −1.61993
\(397\) −8.33419e8 −0.668493 −0.334246 0.942486i \(-0.608482\pi\)
−0.334246 + 0.942486i \(0.608482\pi\)
\(398\) 1.74897e7 0.0139056
\(399\) −4.93280e8 −0.388766
\(400\) 3.34300e9 2.61172
\(401\) −1.28321e9 −0.993786 −0.496893 0.867812i \(-0.665526\pi\)
−0.496893 + 0.867812i \(0.665526\pi\)
\(402\) −2.12978e7 −0.0163510
\(403\) −1.62145e8 −0.123406
\(404\) 7.52577e8 0.567827
\(405\) 3.72706e9 2.78788
\(406\) 7.48304e6 0.00554928
\(407\) −5.97995e8 −0.439660
\(408\) 1.72829e8 0.125981
\(409\) 3.97834e8 0.287522 0.143761 0.989612i \(-0.454080\pi\)
0.143761 + 0.989612i \(0.454080\pi\)
\(410\) −1.02569e8 −0.0734973
\(411\) 3.23529e9 2.29862
\(412\) −8.93820e7 −0.0629665
\(413\) 6.86676e8 0.479652
\(414\) 1.19698e8 0.0829063
\(415\) 2.00923e9 1.37994
\(416\) −1.33058e8 −0.0906177
\(417\) 5.00716e9 3.38155
\(418\) 1.78728e7 0.0119695
\(419\) −1.83847e9 −1.22097 −0.610487 0.792026i \(-0.709026\pi\)
−0.610487 + 0.792026i \(0.709026\pi\)
\(420\) 1.97378e9 1.29995
\(421\) 2.99208e9 1.95428 0.977139 0.212604i \(-0.0681943\pi\)
0.977139 + 0.212604i \(0.0681943\pi\)
\(422\) −2.98044e7 −0.0193057
\(423\) −5.54532e9 −3.56234
\(424\) −4.35272e7 −0.0277319
\(425\) −5.27375e9 −3.33241
\(426\) 4.61701e7 0.0289352
\(427\) −7.37452e8 −0.458391
\(428\) 3.30832e8 0.203965
\(429\) −2.37332e9 −1.45129
\(430\) −1.33427e7 −0.00809288
\(431\) 7.52649e7 0.0452816 0.0226408 0.999744i \(-0.492793\pi\)
0.0226408 + 0.999744i \(0.492793\pi\)
\(432\) 3.40496e9 2.03198
\(433\) 2.36792e9 1.40172 0.700858 0.713301i \(-0.252801\pi\)
0.700858 + 0.713301i \(0.252801\pi\)
\(434\) −2.08575e6 −0.00122475
\(435\) −2.98914e9 −1.74114
\(436\) −1.59797e9 −0.923351
\(437\) 1.37178e9 0.786320
\(438\) −9.10472e7 −0.0517735
\(439\) 1.98503e9 1.11980 0.559901 0.828559i \(-0.310839\pi\)
0.559901 + 0.828559i \(0.310839\pi\)
\(440\) −1.43086e8 −0.0800777
\(441\) −3.29371e9 −1.82873
\(442\) 6.98775e7 0.0384910
\(443\) −2.46053e9 −1.34467 −0.672335 0.740247i \(-0.734709\pi\)
−0.672335 + 0.740247i \(0.734709\pi\)
\(444\) 1.90467e9 1.03271
\(445\) −1.50894e9 −0.811730
\(446\) 3.31259e6 0.00176805
\(447\) −1.54792e9 −0.819734
\(448\) 7.30244e8 0.383703
\(449\) −1.53414e9 −0.799839 −0.399920 0.916550i \(-0.630962\pi\)
−0.399920 + 0.916550i \(0.630962\pi\)
\(450\) 3.03264e8 0.156884
\(451\) −2.03641e9 −1.04531
\(452\) −2.46808e9 −1.25712
\(453\) −9.62095e8 −0.486266
\(454\) 7.65645e6 0.00384000
\(455\) 1.59669e9 0.794657
\(456\) −1.13897e8 −0.0562518
\(457\) 1.63707e9 0.802345 0.401173 0.916002i \(-0.368603\pi\)
0.401173 + 0.916002i \(0.368603\pi\)
\(458\) 2.27461e7 0.0110631
\(459\) −5.37150e9 −2.59269
\(460\) −5.48895e9 −2.62928
\(461\) 3.59530e9 1.70916 0.854579 0.519321i \(-0.173815\pi\)
0.854579 + 0.519321i \(0.173815\pi\)
\(462\) −3.05291e7 −0.0144035
\(463\) −3.31932e9 −1.55423 −0.777115 0.629359i \(-0.783318\pi\)
−0.777115 + 0.629359i \(0.783318\pi\)
\(464\) −1.10763e9 −0.514732
\(465\) 8.33163e8 0.384277
\(466\) 1.95872e7 0.00896646
\(467\) 2.20717e9 1.00283 0.501415 0.865207i \(-0.332813\pi\)
0.501415 + 0.865207i \(0.332813\pi\)
\(468\) 5.15791e9 2.32602
\(469\) 2.84478e8 0.127334
\(470\) −1.98104e8 −0.0880138
\(471\) 2.40188e9 1.05920
\(472\) 1.58552e8 0.0694023
\(473\) −2.64907e8 −0.115101
\(474\) −9.65262e7 −0.0416314
\(475\) 3.47550e9 1.48795
\(476\) −1.15380e9 −0.490350
\(477\) 2.53129e9 1.06789
\(478\) 3.31675e7 0.0138904
\(479\) 1.10499e9 0.459394 0.229697 0.973262i \(-0.426226\pi\)
0.229697 + 0.973262i \(0.426226\pi\)
\(480\) 6.83700e8 0.282177
\(481\) 1.54078e9 0.631296
\(482\) 9.54152e7 0.0388108
\(483\) −2.34318e9 −0.946219
\(484\) 1.07255e9 0.429991
\(485\) 2.34105e9 0.931784
\(486\) 3.98080e7 0.0157305
\(487\) 1.39608e9 0.547719 0.273859 0.961770i \(-0.411700\pi\)
0.273859 + 0.961770i \(0.411700\pi\)
\(488\) −1.70276e8 −0.0663260
\(489\) 6.46898e7 0.0250181
\(490\) −1.17666e8 −0.0451820
\(491\) −4.32305e9 −1.64818 −0.824091 0.566458i \(-0.808313\pi\)
−0.824091 + 0.566458i \(0.808313\pi\)
\(492\) 6.48615e9 2.45533
\(493\) 1.74734e9 0.656770
\(494\) −4.60506e7 −0.0171866
\(495\) 8.32104e9 3.08361
\(496\) 3.08730e8 0.113604
\(497\) −6.16700e8 −0.225334
\(498\) 9.89844e7 0.0359140
\(499\) 5.09888e8 0.183706 0.0918528 0.995773i \(-0.470721\pi\)
0.0918528 + 0.995773i \(0.470721\pi\)
\(500\) −8.59435e9 −3.07481
\(501\) −3.46958e9 −1.23266
\(502\) −5.06207e7 −0.0178593
\(503\) −3.07862e9 −1.07862 −0.539310 0.842108i \(-0.681315\pi\)
−0.539310 + 0.842108i \(0.681315\pi\)
\(504\) 1.32749e8 0.0461875
\(505\) −3.12823e9 −1.08088
\(506\) 8.48994e7 0.0291325
\(507\) 9.08591e8 0.309628
\(508\) 6.40652e8 0.216820
\(509\) 3.54159e9 1.19038 0.595191 0.803584i \(-0.297076\pi\)
0.595191 + 0.803584i \(0.297076\pi\)
\(510\) −3.59057e8 −0.119858
\(511\) 1.21613e9 0.403188
\(512\) 4.22353e8 0.139069
\(513\) 3.53991e9 1.15766
\(514\) 3.13996e7 0.0101989
\(515\) 3.71533e8 0.119859
\(516\) 8.43752e8 0.270359
\(517\) −3.93317e9 −1.25177
\(518\) 1.98198e7 0.00626534
\(519\) 8.25162e9 2.59092
\(520\) 3.68671e8 0.114981
\(521\) −2.36861e9 −0.733773 −0.366886 0.930266i \(-0.619576\pi\)
−0.366886 + 0.930266i \(0.619576\pi\)
\(522\) −1.00480e8 −0.0309195
\(523\) 2.62233e9 0.801551 0.400776 0.916176i \(-0.368741\pi\)
0.400776 + 0.916176i \(0.368741\pi\)
\(524\) 5.35777e9 1.62676
\(525\) −5.93662e9 −1.79053
\(526\) 4.45196e6 0.00133383
\(527\) −4.87037e8 −0.144952
\(528\) 4.51887e9 1.33601
\(529\) 3.11142e9 0.913825
\(530\) 9.04291e7 0.0263841
\(531\) −9.22047e9 −2.67253
\(532\) 7.60375e8 0.218946
\(533\) 5.24696e9 1.50094
\(534\) −7.43376e7 −0.0211259
\(535\) −1.37517e9 −0.388255
\(536\) 6.56854e7 0.0184243
\(537\) 6.48267e9 1.80653
\(538\) 2.94118e7 0.00814300
\(539\) −2.33615e9 −0.642600
\(540\) −1.41644e10 −3.87097
\(541\) −3.96326e9 −1.07612 −0.538062 0.842905i \(-0.680843\pi\)
−0.538062 + 0.842905i \(0.680843\pi\)
\(542\) 1.48025e8 0.0399335
\(543\) 1.01498e9 0.272057
\(544\) −3.99666e8 −0.106439
\(545\) 6.64227e9 1.75764
\(546\) 7.86606e7 0.0206815
\(547\) −5.89175e9 −1.53918 −0.769588 0.638540i \(-0.779538\pi\)
−0.769588 + 0.638540i \(0.779538\pi\)
\(548\) −4.98709e9 −1.29454
\(549\) 9.90227e9 2.55406
\(550\) 2.15098e8 0.0551274
\(551\) −1.15153e9 −0.293254
\(552\) −5.41036e8 −0.136911
\(553\) 1.28932e9 0.324206
\(554\) −1.91468e7 −0.00478423
\(555\) −7.91711e9 −1.96581
\(556\) −7.71837e9 −1.90442
\(557\) 9.78407e8 0.239898 0.119949 0.992780i \(-0.461727\pi\)
0.119949 + 0.992780i \(0.461727\pi\)
\(558\) 2.80068e7 0.00682407
\(559\) 6.82552e8 0.165270
\(560\) −3.04014e9 −0.731536
\(561\) −7.12875e9 −1.70468
\(562\) −1.93473e8 −0.0459772
\(563\) 2.19945e9 0.519439 0.259719 0.965684i \(-0.416370\pi\)
0.259719 + 0.965684i \(0.416370\pi\)
\(564\) 1.25275e10 2.94028
\(565\) 1.02591e10 2.39297
\(566\) −2.20804e8 −0.0511858
\(567\) −2.45255e9 −0.565037
\(568\) −1.42395e8 −0.0326043
\(569\) −5.53125e9 −1.25872 −0.629362 0.777112i \(-0.716684\pi\)
−0.629362 + 0.777112i \(0.716684\pi\)
\(570\) 2.36625e8 0.0535179
\(571\) −2.15374e9 −0.484135 −0.242067 0.970259i \(-0.577825\pi\)
−0.242067 + 0.970259i \(0.577825\pi\)
\(572\) 3.65839e9 0.817342
\(573\) −7.99324e9 −1.77493
\(574\) 6.74942e7 0.0148962
\(575\) 1.65093e10 3.62153
\(576\) −9.80549e9 −2.13792
\(577\) −8.51167e9 −1.84459 −0.922294 0.386490i \(-0.873688\pi\)
−0.922294 + 0.386490i \(0.873688\pi\)
\(578\) 8.03646e7 0.0173108
\(579\) 2.39807e9 0.513437
\(580\) 4.60766e9 0.980578
\(581\) −1.32215e9 −0.279682
\(582\) 1.15332e8 0.0242504
\(583\) 1.79539e9 0.375248
\(584\) 2.80802e8 0.0583384
\(585\) −2.14398e10 −4.42767
\(586\) 2.21076e8 0.0453838
\(587\) −6.09853e9 −1.24449 −0.622246 0.782822i \(-0.713779\pi\)
−0.622246 + 0.782822i \(0.713779\pi\)
\(588\) 7.44086e9 1.50939
\(589\) 3.20966e8 0.0647226
\(590\) −3.29397e8 −0.0660294
\(591\) 3.46657e9 0.690787
\(592\) −2.93369e9 −0.581151
\(593\) −6.67019e9 −1.31355 −0.656775 0.754087i \(-0.728080\pi\)
−0.656775 + 0.754087i \(0.728080\pi\)
\(594\) 2.19085e8 0.0428904
\(595\) 4.79598e9 0.933401
\(596\) 2.38607e9 0.461659
\(597\) 4.59728e9 0.884281
\(598\) −2.18750e8 −0.0418306
\(599\) 2.30726e8 0.0438635 0.0219318 0.999759i \(-0.493018\pi\)
0.0219318 + 0.999759i \(0.493018\pi\)
\(600\) −1.37075e9 −0.259077
\(601\) 6.63912e9 1.24753 0.623763 0.781613i \(-0.285603\pi\)
0.623763 + 0.781613i \(0.285603\pi\)
\(602\) 8.77999e6 0.00164024
\(603\) −3.81989e9 −0.709480
\(604\) 1.48304e9 0.273856
\(605\) −4.45826e9 −0.818506
\(606\) −1.54112e8 −0.0281307
\(607\) −4.67413e9 −0.848282 −0.424141 0.905596i \(-0.639424\pi\)
−0.424141 + 0.905596i \(0.639424\pi\)
\(608\) 2.63387e8 0.0475261
\(609\) 1.96697e9 0.352888
\(610\) 3.53754e8 0.0631025
\(611\) 1.01341e10 1.79739
\(612\) 1.54929e10 2.73213
\(613\) −1.56186e9 −0.273861 −0.136931 0.990581i \(-0.543724\pi\)
−0.136931 + 0.990581i \(0.543724\pi\)
\(614\) 1.85752e8 0.0323850
\(615\) −2.69609e10 −4.67381
\(616\) 9.41558e7 0.0162299
\(617\) 3.07047e9 0.526268 0.263134 0.964759i \(-0.415244\pi\)
0.263134 + 0.964759i \(0.415244\pi\)
\(618\) 1.83035e7 0.00311943
\(619\) −1.00411e9 −0.170162 −0.0850811 0.996374i \(-0.527115\pi\)
−0.0850811 + 0.996374i \(0.527115\pi\)
\(620\) −1.28429e9 −0.216418
\(621\) 1.68153e10 2.81764
\(622\) 2.36308e7 0.00393743
\(623\) 9.92939e8 0.164518
\(624\) −1.16432e10 −1.91835
\(625\) 1.97461e10 3.23519
\(626\) 8.25305e7 0.0134463
\(627\) 4.69798e9 0.761158
\(628\) −3.70242e9 −0.596523
\(629\) 4.62805e9 0.741517
\(630\) −2.75790e8 −0.0439428
\(631\) −1.07127e9 −0.169745 −0.0848725 0.996392i \(-0.527048\pi\)
−0.0848725 + 0.996392i \(0.527048\pi\)
\(632\) 2.97700e8 0.0469104
\(633\) −7.83428e9 −1.22768
\(634\) −3.41727e8 −0.0532558
\(635\) −2.66299e9 −0.412726
\(636\) −5.71848e9 −0.881415
\(637\) 6.01928e9 0.922691
\(638\) −7.12681e7 −0.0108648
\(639\) 8.28086e9 1.25552
\(640\) −1.40502e9 −0.211861
\(641\) 3.42898e9 0.514236 0.257118 0.966380i \(-0.417227\pi\)
0.257118 + 0.966380i \(0.417227\pi\)
\(642\) −6.77474e7 −0.0101046
\(643\) 4.41718e9 0.655249 0.327624 0.944808i \(-0.393752\pi\)
0.327624 + 0.944808i \(0.393752\pi\)
\(644\) 3.61194e9 0.532893
\(645\) −3.50721e9 −0.514640
\(646\) −1.38323e8 −0.0201873
\(647\) −6.14010e9 −0.891272 −0.445636 0.895214i \(-0.647022\pi\)
−0.445636 + 0.895214i \(0.647022\pi\)
\(648\) −5.66287e8 −0.0817568
\(649\) −6.53987e9 −0.939102
\(650\) −5.54218e8 −0.0791559
\(651\) −5.48254e8 −0.0778840
\(652\) −9.97171e7 −0.0140897
\(653\) 7.83011e9 1.10045 0.550227 0.835015i \(-0.314541\pi\)
0.550227 + 0.835015i \(0.314541\pi\)
\(654\) 3.27231e8 0.0457438
\(655\) −2.22706e10 −3.09661
\(656\) −9.99039e9 −1.38172
\(657\) −1.63298e10 −2.24648
\(658\) 1.30360e8 0.0178383
\(659\) 9.87256e9 1.34379 0.671894 0.740647i \(-0.265481\pi\)
0.671894 + 0.740647i \(0.265481\pi\)
\(660\) −1.87982e10 −2.54514
\(661\) 1.73018e9 0.233017 0.116508 0.993190i \(-0.462830\pi\)
0.116508 + 0.993190i \(0.462830\pi\)
\(662\) −1.32753e8 −0.0177845
\(663\) 1.83678e10 2.44771
\(664\) −3.05281e8 −0.0404680
\(665\) −3.16064e9 −0.416773
\(666\) −2.66134e8 −0.0349092
\(667\) −5.47001e9 −0.713752
\(668\) 5.34823e9 0.694213
\(669\) 8.70736e8 0.112433
\(670\) −1.36464e8 −0.0175289
\(671\) 7.02346e9 0.897474
\(672\) −4.49901e8 −0.0571906
\(673\) 1.06458e10 1.34625 0.673127 0.739527i \(-0.264951\pi\)
0.673127 + 0.739527i \(0.264951\pi\)
\(674\) 3.72856e8 0.0469063
\(675\) 4.26028e10 5.33181
\(676\) −1.40056e9 −0.174377
\(677\) 8.45984e9 1.04786 0.523928 0.851763i \(-0.324466\pi\)
0.523928 + 0.851763i \(0.324466\pi\)
\(678\) 5.05411e8 0.0622789
\(679\) −1.54050e9 −0.188851
\(680\) 1.10738e9 0.135057
\(681\) 2.01255e9 0.244192
\(682\) 1.98646e7 0.00239792
\(683\) −7.82807e9 −0.940117 −0.470059 0.882635i \(-0.655767\pi\)
−0.470059 + 0.882635i \(0.655767\pi\)
\(684\) −1.02101e10 −1.21992
\(685\) 2.07298e10 2.46421
\(686\) 1.68373e8 0.0199131
\(687\) 5.97896e9 0.703521
\(688\) −1.29960e9 −0.152143
\(689\) −4.62595e9 −0.538808
\(690\) 1.12402e9 0.130257
\(691\) −8.92999e9 −1.02962 −0.514811 0.857304i \(-0.672138\pi\)
−0.514811 + 0.857304i \(0.672138\pi\)
\(692\) −1.27196e10 −1.45916
\(693\) −5.47557e9 −0.624975
\(694\) 3.28577e8 0.0373147
\(695\) 3.20828e10 3.62515
\(696\) 4.54168e8 0.0510604
\(697\) 1.57604e10 1.76300
\(698\) −2.07887e8 −0.0231384
\(699\) 5.14862e9 0.570192
\(700\) 9.15110e9 1.00839
\(701\) 1.21758e10 1.33501 0.667504 0.744606i \(-0.267363\pi\)
0.667504 + 0.744606i \(0.267363\pi\)
\(702\) −5.64490e8 −0.0615852
\(703\) −3.04997e9 −0.331095
\(704\) −6.95481e9 −0.751244
\(705\) −5.20730e10 −5.59694
\(706\) 9.57706e7 0.0102427
\(707\) 2.05849e9 0.219069
\(708\) 2.08301e10 2.20584
\(709\) −1.53335e10 −1.61577 −0.807885 0.589340i \(-0.799388\pi\)
−0.807885 + 0.589340i \(0.799388\pi\)
\(710\) 2.95830e8 0.0310197
\(711\) −1.73125e10 −1.80641
\(712\) 2.29267e8 0.0238047
\(713\) 1.52466e9 0.157528
\(714\) 2.36274e8 0.0242924
\(715\) −1.52068e10 −1.55584
\(716\) −9.99282e9 −1.01740
\(717\) 8.71829e9 0.883312
\(718\) 4.48974e8 0.0452674
\(719\) 3.82428e8 0.0383706 0.0191853 0.999816i \(-0.493893\pi\)
0.0191853 + 0.999816i \(0.493893\pi\)
\(720\) 4.08221e10 4.07597
\(721\) −2.44483e8 −0.0242926
\(722\) −1.91002e8 −0.0188868
\(723\) 2.50805e10 2.46804
\(724\) −1.56456e9 −0.153217
\(725\) −1.38586e10 −1.35063
\(726\) −2.19636e8 −0.0213022
\(727\) −3.79482e9 −0.366286 −0.183143 0.983086i \(-0.558627\pi\)
−0.183143 + 0.983086i \(0.558627\pi\)
\(728\) −2.42600e8 −0.0233040
\(729\) −4.86810e9 −0.465386
\(730\) −5.83375e8 −0.0555032
\(731\) 2.05019e9 0.194126
\(732\) −2.23704e10 −2.10807
\(733\) −1.11789e10 −1.04842 −0.524211 0.851588i \(-0.675640\pi\)
−0.524211 + 0.851588i \(0.675640\pi\)
\(734\) 1.87245e8 0.0174773
\(735\) −3.09293e10 −2.87319
\(736\) 1.25114e9 0.115674
\(737\) −2.70936e9 −0.249305
\(738\) −9.06291e8 −0.0829985
\(739\) −1.77973e10 −1.62218 −0.811091 0.584921i \(-0.801126\pi\)
−0.811091 + 0.584921i \(0.801126\pi\)
\(740\) 1.22040e10 1.10711
\(741\) −1.21047e10 −1.09293
\(742\) −5.95058e7 −0.00534744
\(743\) 2.77646e7 0.00248331 0.00124165 0.999999i \(-0.499605\pi\)
0.00124165 + 0.999999i \(0.499605\pi\)
\(744\) −1.26590e8 −0.0112693
\(745\) −9.91814e9 −0.878786
\(746\) −1.07794e8 −0.00950627
\(747\) 1.77534e10 1.55833
\(748\) 1.09887e10 0.960046
\(749\) 9.04912e8 0.0786901
\(750\) 1.75994e9 0.152329
\(751\) −1.10248e10 −0.949798 −0.474899 0.880040i \(-0.657515\pi\)
−0.474899 + 0.880040i \(0.657515\pi\)
\(752\) −1.92957e10 −1.65462
\(753\) −1.33060e10 −1.13570
\(754\) 1.83628e8 0.0156005
\(755\) −6.16452e9 −0.521296
\(756\) 9.32070e9 0.784553
\(757\) −2.07834e10 −1.74133 −0.870664 0.491879i \(-0.836310\pi\)
−0.870664 + 0.491879i \(0.836310\pi\)
\(758\) −3.84434e8 −0.0320612
\(759\) 2.23164e10 1.85258
\(760\) −7.29784e8 −0.0603041
\(761\) 8.64340e9 0.710949 0.355474 0.934686i \(-0.384319\pi\)
0.355474 + 0.934686i \(0.384319\pi\)
\(762\) −1.31192e8 −0.0107415
\(763\) −4.37087e9 −0.356231
\(764\) 1.23213e10 0.999609
\(765\) −6.43990e10 −5.20072
\(766\) 3.83996e8 0.0308692
\(767\) 1.68505e10 1.34843
\(768\) 2.20998e10 1.76045
\(769\) 2.17566e9 0.172523 0.0862617 0.996273i \(-0.472508\pi\)
0.0862617 + 0.996273i \(0.472508\pi\)
\(770\) −1.95612e8 −0.0154411
\(771\) 8.25359e9 0.648563
\(772\) −3.69654e9 −0.289158
\(773\) 5.58417e9 0.434841 0.217421 0.976078i \(-0.430236\pi\)
0.217421 + 0.976078i \(0.430236\pi\)
\(774\) −1.17895e8 −0.00913907
\(775\) 3.86282e9 0.298091
\(776\) −3.55698e8 −0.0273253
\(777\) 5.20976e9 0.398423
\(778\) −7.03948e6 −0.000535935 0
\(779\) −1.03864e10 −0.787195
\(780\) 4.84350e10 3.65450
\(781\) 5.87343e9 0.441177
\(782\) −6.57061e8 −0.0491340
\(783\) −1.41155e10 −1.05082
\(784\) −1.14609e10 −0.849400
\(785\) 1.53898e10 1.13551
\(786\) −1.09716e9 −0.0805916
\(787\) −7.26404e9 −0.531211 −0.265605 0.964082i \(-0.585572\pi\)
−0.265605 + 0.964082i \(0.585572\pi\)
\(788\) −5.34360e9 −0.389038
\(789\) 1.17023e9 0.0848206
\(790\) −6.18481e8 −0.0446305
\(791\) −6.75085e9 −0.484999
\(792\) −1.26429e9 −0.0904295
\(793\) −1.80965e10 −1.28866
\(794\) −2.63076e8 −0.0186514
\(795\) 2.37699e10 1.67781
\(796\) −7.08654e9 −0.498011
\(797\) 1.81556e10 1.27030 0.635150 0.772389i \(-0.280938\pi\)
0.635150 + 0.772389i \(0.280938\pi\)
\(798\) −1.55709e8 −0.0108468
\(799\) 3.04400e10 2.11120
\(800\) 3.16986e9 0.218890
\(801\) −1.33329e10 −0.916664
\(802\) −4.05058e8 −0.0277272
\(803\) −1.15824e10 −0.789393
\(804\) 8.62956e9 0.585588
\(805\) −1.50137e10 −1.01438
\(806\) −5.11826e7 −0.00344310
\(807\) 7.73110e9 0.517826
\(808\) 4.75301e8 0.0316978
\(809\) −2.32465e10 −1.54361 −0.771806 0.635858i \(-0.780647\pi\)
−0.771806 + 0.635858i \(0.780647\pi\)
\(810\) 1.17648e9 0.0777834
\(811\) −1.13477e10 −0.747024 −0.373512 0.927625i \(-0.621847\pi\)
−0.373512 + 0.927625i \(0.621847\pi\)
\(812\) −3.03201e9 −0.198740
\(813\) 3.89094e10 2.53944
\(814\) −1.88763e8 −0.0122668
\(815\) 4.14493e8 0.0268204
\(816\) −3.49729e10 −2.25328
\(817\) −1.35111e9 −0.0866790
\(818\) 1.25580e8 0.00802202
\(819\) 1.41082e10 0.897385
\(820\) 4.15593e10 2.63221
\(821\) −6.60382e9 −0.416480 −0.208240 0.978078i \(-0.566774\pi\)
−0.208240 + 0.978078i \(0.566774\pi\)
\(822\) 1.02125e9 0.0641328
\(823\) −3.82454e9 −0.239155 −0.119577 0.992825i \(-0.538154\pi\)
−0.119577 + 0.992825i \(0.538154\pi\)
\(824\) −5.64505e7 −0.00351498
\(825\) 5.65401e10 3.50564
\(826\) 2.16756e8 0.0133826
\(827\) 1.37532e10 0.845541 0.422770 0.906237i \(-0.361058\pi\)
0.422770 + 0.906237i \(0.361058\pi\)
\(828\) −4.85000e10 −2.96917
\(829\) 1.83250e10 1.11713 0.558563 0.829462i \(-0.311353\pi\)
0.558563 + 0.829462i \(0.311353\pi\)
\(830\) 6.34232e8 0.0385012
\(831\) −5.03286e9 −0.304237
\(832\) 1.79196e10 1.07869
\(833\) 1.80802e10 1.08379
\(834\) 1.58056e9 0.0943472
\(835\) −2.22309e10 −1.32146
\(836\) −7.24178e9 −0.428670
\(837\) 3.93442e9 0.231922
\(838\) −5.80329e8 −0.0340659
\(839\) 1.70662e10 0.997630 0.498815 0.866708i \(-0.333769\pi\)
0.498815 + 0.866708i \(0.333769\pi\)
\(840\) 1.24657e9 0.0725670
\(841\) −1.26581e10 −0.733809
\(842\) 9.44478e8 0.0545255
\(843\) −5.08556e10 −2.92377
\(844\) 1.20763e10 0.691408
\(845\) 5.82169e9 0.331933
\(846\) −1.75043e9 −0.0993915
\(847\) 2.93371e9 0.165892
\(848\) 8.80796e9 0.496010
\(849\) −5.80398e10 −3.25499
\(850\) −1.66471e9 −0.0929762
\(851\) −1.44880e10 −0.805852
\(852\) −1.87074e10 −1.03628
\(853\) −6.70515e9 −0.369902 −0.184951 0.982748i \(-0.559213\pi\)
−0.184951 + 0.982748i \(0.559213\pi\)
\(854\) −2.32784e8 −0.0127894
\(855\) 4.24401e10 2.32217
\(856\) 2.08942e8 0.0113859
\(857\) 2.45826e9 0.133412 0.0667061 0.997773i \(-0.478751\pi\)
0.0667061 + 0.997773i \(0.478751\pi\)
\(858\) −7.49160e8 −0.0404920
\(859\) −2.04474e10 −1.10068 −0.550340 0.834941i \(-0.685502\pi\)
−0.550340 + 0.834941i \(0.685502\pi\)
\(860\) 5.40625e9 0.289835
\(861\) 1.77413e10 0.947271
\(862\) 2.37581e7 0.00126338
\(863\) −2.86480e10 −1.51725 −0.758624 0.651529i \(-0.774128\pi\)
−0.758624 + 0.651529i \(0.774128\pi\)
\(864\) 3.22861e9 0.170301
\(865\) 5.28713e10 2.77756
\(866\) 7.47457e8 0.0391087
\(867\) 2.11244e10 1.10082
\(868\) 8.45114e8 0.0438628
\(869\) −1.22794e10 −0.634757
\(870\) −9.43549e8 −0.0485789
\(871\) 6.98087e9 0.357970
\(872\) −1.00922e9 −0.0515442
\(873\) 2.06854e10 1.05224
\(874\) 4.33015e8 0.0219388
\(875\) −2.35078e10 −1.18627
\(876\) 3.68910e10 1.85420
\(877\) 2.41157e10 1.20726 0.603631 0.797264i \(-0.293720\pi\)
0.603631 + 0.797264i \(0.293720\pi\)
\(878\) 6.26593e8 0.0312432
\(879\) 5.81114e10 2.88603
\(880\) 2.89542e10 1.43226
\(881\) 8.73417e9 0.430334 0.215167 0.976577i \(-0.430970\pi\)
0.215167 + 0.976577i \(0.430970\pi\)
\(882\) −1.03969e9 −0.0510227
\(883\) −1.22297e10 −0.597795 −0.298898 0.954285i \(-0.596619\pi\)
−0.298898 + 0.954285i \(0.596619\pi\)
\(884\) −2.83133e10 −1.37850
\(885\) −8.65841e10 −4.19891
\(886\) −7.76690e8 −0.0375171
\(887\) −3.43901e9 −0.165463 −0.0827315 0.996572i \(-0.526364\pi\)
−0.0827315 + 0.996572i \(0.526364\pi\)
\(888\) 1.20292e9 0.0576490
\(889\) 1.75235e9 0.0836497
\(890\) −4.76310e8 −0.0226477
\(891\) 2.33579e10 1.10627
\(892\) −1.34221e9 −0.0633204
\(893\) −2.00605e10 −0.942674
\(894\) −4.88616e8 −0.0228711
\(895\) 4.15370e10 1.93667
\(896\) 9.24556e8 0.0429393
\(897\) −5.74999e10 −2.66007
\(898\) −4.84266e8 −0.0223160
\(899\) −1.27986e9 −0.0587495
\(900\) −1.22878e11 −5.61857
\(901\) −1.38950e10 −0.632882
\(902\) −6.42811e8 −0.0291649
\(903\) 2.30788e9 0.104305
\(904\) −1.55875e9 −0.0701759
\(905\) 6.50340e9 0.291655
\(906\) −3.03694e8 −0.0135671
\(907\) 2.11330e10 0.940451 0.470226 0.882546i \(-0.344173\pi\)
0.470226 + 0.882546i \(0.344173\pi\)
\(908\) −3.10228e9 −0.137524
\(909\) −2.76408e10 −1.22061
\(910\) 5.04009e8 0.0221714
\(911\) 3.25882e10 1.42806 0.714029 0.700116i \(-0.246869\pi\)
0.714029 + 0.700116i \(0.246869\pi\)
\(912\) 2.30477e10 1.00611
\(913\) 1.25921e10 0.547583
\(914\) 5.16757e8 0.0223859
\(915\) 9.29866e10 4.01279
\(916\) −9.21636e9 −0.396210
\(917\) 1.46549e10 0.627610
\(918\) −1.69556e9 −0.0723377
\(919\) 2.90516e10 1.23471 0.617356 0.786684i \(-0.288204\pi\)
0.617356 + 0.786684i \(0.288204\pi\)
\(920\) −3.46663e9 −0.146774
\(921\) 4.88262e10 2.05942
\(922\) 1.13489e9 0.0476865
\(923\) −1.51333e10 −0.633474
\(924\) 1.23699e10 0.515840
\(925\) −3.67063e10 −1.52491
\(926\) −1.04777e9 −0.0433639
\(927\) 3.28284e9 0.135354
\(928\) −1.05026e9 −0.0431400
\(929\) 5.17888e9 0.211924 0.105962 0.994370i \(-0.466208\pi\)
0.105962 + 0.994370i \(0.466208\pi\)
\(930\) 2.62996e8 0.0107216
\(931\) −1.19152e10 −0.483922
\(932\) −7.93642e9 −0.321121
\(933\) 6.21153e9 0.250388
\(934\) 6.96715e8 0.0279796
\(935\) −4.56767e10 −1.82749
\(936\) 3.25755e9 0.129845
\(937\) 5.65702e9 0.224646 0.112323 0.993672i \(-0.464171\pi\)
0.112323 + 0.993672i \(0.464171\pi\)
\(938\) 8.97982e7 0.00355269
\(939\) 2.16937e10 0.855075
\(940\) 8.02687e10 3.15209
\(941\) 1.93533e10 0.757168 0.378584 0.925567i \(-0.376411\pi\)
0.378584 + 0.925567i \(0.376411\pi\)
\(942\) 7.58177e8 0.0295524
\(943\) −4.93374e10 −1.91595
\(944\) −3.20838e10 −1.24132
\(945\) −3.87432e10 −1.49343
\(946\) −8.36202e7 −0.00321139
\(947\) 2.30104e10 0.880438 0.440219 0.897890i \(-0.354901\pi\)
0.440219 + 0.897890i \(0.354901\pi\)
\(948\) 3.91110e10 1.49097
\(949\) 2.98429e10 1.13347
\(950\) 1.09707e9 0.0415148
\(951\) −8.98252e10 −3.38662
\(952\) −7.28699e8 −0.0273728
\(953\) −1.01231e10 −0.378867 −0.189434 0.981894i \(-0.560665\pi\)
−0.189434 + 0.981894i \(0.560665\pi\)
\(954\) 7.99026e8 0.0297949
\(955\) −5.12158e10 −1.90280
\(956\) −1.34390e10 −0.497465
\(957\) −1.87333e10 −0.690912
\(958\) 3.48801e8 0.0128174
\(959\) −1.36410e10 −0.499437
\(960\) −9.20778e10 −3.35897
\(961\) −2.71559e10 −0.987034
\(962\) 4.86362e8 0.0176135
\(963\) −1.21509e10 −0.438446
\(964\) −3.86608e10 −1.38996
\(965\) 1.53654e10 0.550424
\(966\) −7.39648e8 −0.0264001
\(967\) 6.28463e8 0.0223505 0.0111752 0.999938i \(-0.496443\pi\)
0.0111752 + 0.999938i \(0.496443\pi\)
\(968\) 6.77386e8 0.0240034
\(969\) −3.63590e10 −1.28375
\(970\) 7.38975e8 0.0259973
\(971\) 2.05584e10 0.720645 0.360322 0.932828i \(-0.382667\pi\)
0.360322 + 0.932828i \(0.382667\pi\)
\(972\) −1.61296e10 −0.563367
\(973\) −2.11117e10 −0.734732
\(974\) 4.40684e8 0.0152817
\(975\) −1.45680e11 −5.03365
\(976\) 3.44563e10 1.18630
\(977\) 4.09891e10 1.40617 0.703084 0.711107i \(-0.251806\pi\)
0.703084 + 0.711107i \(0.251806\pi\)
\(978\) 2.04199e7 0.000698021 0
\(979\) −9.45670e9 −0.322107
\(980\) 4.76765e10 1.61813
\(981\) 5.86907e10 1.98485
\(982\) −1.36461e9 −0.0459853
\(983\) 3.77468e10 1.26748 0.633742 0.773544i \(-0.281518\pi\)
0.633742 + 0.773544i \(0.281518\pi\)
\(984\) 4.09642e9 0.137064
\(985\) 2.22117e10 0.740551
\(986\) 5.51565e8 0.0183243
\(987\) 3.42660e10 1.13437
\(988\) 1.86590e10 0.615515
\(989\) −6.41806e9 −0.210968
\(990\) 2.62662e9 0.0860346
\(991\) −1.40838e10 −0.459688 −0.229844 0.973227i \(-0.573822\pi\)
−0.229844 + 0.973227i \(0.573822\pi\)
\(992\) 2.92740e8 0.00952119
\(993\) −3.48951e10 −1.13095
\(994\) −1.94667e8 −0.00628696
\(995\) 2.94565e10 0.947984
\(996\) −4.01070e10 −1.28621
\(997\) 8.84518e9 0.282666 0.141333 0.989962i \(-0.454861\pi\)
0.141333 + 0.989962i \(0.454861\pi\)
\(998\) 1.60951e8 0.00512550
\(999\) −3.73867e10 −1.18642
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 43.8.a.b.1.7 13
3.2 odd 2 387.8.a.d.1.7 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.7 13 1.1 even 1 trivial
387.8.a.d.1.7 13 3.2 odd 2