Properties

Label 15.10.a.b.1.1
Level $15$
Weight $10$
Character 15.1
Self dual yes
Analytic conductor $7.726$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.72553754246\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 15.1

$q$-expansion

\(f(q)\) \(=\) \(q+22.0000 q^{2} -81.0000 q^{3} -28.0000 q^{4} -625.000 q^{5} -1782.00 q^{6} -5988.00 q^{7} -11880.0 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+22.0000 q^{2} -81.0000 q^{3} -28.0000 q^{4} -625.000 q^{5} -1782.00 q^{6} -5988.00 q^{7} -11880.0 q^{8} +6561.00 q^{9} -13750.0 q^{10} -14648.0 q^{11} +2268.00 q^{12} +37906.0 q^{13} -131736. q^{14} +50625.0 q^{15} -247024. q^{16} -441098. q^{17} +144342. q^{18} +441820. q^{19} +17500.0 q^{20} +485028. q^{21} -322256. q^{22} +2.26414e6 q^{23} +962280. q^{24} +390625. q^{25} +833932. q^{26} -531441. q^{27} +167664. q^{28} -1.04935e6 q^{29} +1.11375e6 q^{30} -7.91057e6 q^{31} +648032. q^{32} +1.18649e6 q^{33} -9.70416e6 q^{34} +3.74250e6 q^{35} -183708. q^{36} -2.09926e7 q^{37} +9.72004e6 q^{38} -3.07039e6 q^{39} +7.42500e6 q^{40} +1.32856e7 q^{41} +1.06706e7 q^{42} -2.31308e7 q^{43} +410144. q^{44} -4.10063e6 q^{45} +4.98110e7 q^{46} -1.38737e7 q^{47} +2.00089e7 q^{48} -4.49746e6 q^{49} +8.59375e6 q^{50} +3.57289e7 q^{51} -1.06137e6 q^{52} -5.76352e7 q^{53} -1.16917e7 q^{54} +9.15500e6 q^{55} +7.11374e7 q^{56} -3.57874e7 q^{57} -2.30857e7 q^{58} -3.20421e7 q^{59} -1.41750e6 q^{60} +1.10664e8 q^{61} -1.74032e8 q^{62} -3.92873e7 q^{63} +1.40733e8 q^{64} -2.36913e7 q^{65} +2.61027e7 q^{66} -1.18568e8 q^{67} +1.23507e7 q^{68} -1.83395e8 q^{69} +8.23350e7 q^{70} +2.76680e8 q^{71} -7.79447e7 q^{72} -2.64023e8 q^{73} -4.61836e8 q^{74} -3.16406e7 q^{75} -1.23710e7 q^{76} +8.77122e7 q^{77} -6.75485e7 q^{78} +4.48203e8 q^{79} +1.54390e8 q^{80} +4.30467e7 q^{81} +2.92282e8 q^{82} +8.51016e8 q^{83} -1.35808e7 q^{84} +2.75686e8 q^{85} -5.08877e8 q^{86} +8.49974e7 q^{87} +1.74018e8 q^{88} +1.89895e8 q^{89} -9.02138e7 q^{90} -2.26981e8 q^{91} -6.33958e7 q^{92} +6.40756e8 q^{93} -3.05221e8 q^{94} -2.76138e8 q^{95} -5.24906e7 q^{96} -1.01415e9 q^{97} -9.89442e7 q^{98} -9.61055e7 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.0000 0.972272 0.486136 0.873883i \(-0.338406\pi\)
0.486136 + 0.873883i \(0.338406\pi\)
\(3\) −81.0000 −0.577350
\(4\) −28.0000 −0.0546875
\(5\) −625.000 −0.447214
\(6\) −1782.00 −0.561341
\(7\) −5988.00 −0.942629 −0.471314 0.881965i \(-0.656220\pi\)
−0.471314 + 0.881965i \(0.656220\pi\)
\(8\) −11880.0 −1.02544
\(9\) 6561.00 0.333333
\(10\) −13750.0 −0.434813
\(11\) −14648.0 −0.301655 −0.150828 0.988560i \(-0.548194\pi\)
−0.150828 + 0.988560i \(0.548194\pi\)
\(12\) 2268.00 0.0315738
\(13\) 37906.0 0.368098 0.184049 0.982917i \(-0.441080\pi\)
0.184049 + 0.982917i \(0.441080\pi\)
\(14\) −131736. −0.916491
\(15\) 50625.0 0.258199
\(16\) −247024. −0.942322
\(17\) −441098. −1.28090 −0.640450 0.768000i \(-0.721252\pi\)
−0.640450 + 0.768000i \(0.721252\pi\)
\(18\) 144342. 0.324091
\(19\) 441820. 0.777775 0.388888 0.921285i \(-0.372859\pi\)
0.388888 + 0.921285i \(0.372859\pi\)
\(20\) 17500.0 0.0244570
\(21\) 485028. 0.544227
\(22\) −322256. −0.293291
\(23\) 2.26414e6 1.68705 0.843524 0.537092i \(-0.180477\pi\)
0.843524 + 0.537092i \(0.180477\pi\)
\(24\) 962280. 0.592040
\(25\) 390625. 0.200000
\(26\) 833932. 0.357891
\(27\) −531441. −0.192450
\(28\) 167664. 0.0515500
\(29\) −1.04935e6 −0.275505 −0.137752 0.990467i \(-0.543988\pi\)
−0.137752 + 0.990467i \(0.543988\pi\)
\(30\) 1.11375e6 0.251040
\(31\) −7.91057e6 −1.53844 −0.769219 0.638985i \(-0.779355\pi\)
−0.769219 + 0.638985i \(0.779355\pi\)
\(32\) 648032. 0.109250
\(33\) 1.18649e6 0.174161
\(34\) −9.70416e6 −1.24538
\(35\) 3.74250e6 0.421556
\(36\) −183708. −0.0182292
\(37\) −2.09926e7 −1.84144 −0.920720 0.390224i \(-0.872398\pi\)
−0.920720 + 0.390224i \(0.872398\pi\)
\(38\) 9.72004e6 0.756209
\(39\) −3.07039e6 −0.212521
\(40\) 7.42500e6 0.458592
\(41\) 1.32856e7 0.734265 0.367132 0.930169i \(-0.380340\pi\)
0.367132 + 0.930169i \(0.380340\pi\)
\(42\) 1.06706e7 0.529136
\(43\) −2.31308e7 −1.03177 −0.515884 0.856659i \(-0.672536\pi\)
−0.515884 + 0.856659i \(0.672536\pi\)
\(44\) 410144. 0.0164968
\(45\) −4.10063e6 −0.149071
\(46\) 4.98110e7 1.64027
\(47\) −1.38737e7 −0.414717 −0.207358 0.978265i \(-0.566487\pi\)
−0.207358 + 0.978265i \(0.566487\pi\)
\(48\) 2.00089e7 0.544050
\(49\) −4.49746e6 −0.111451
\(50\) 8.59375e6 0.194454
\(51\) 3.57289e7 0.739527
\(52\) −1.06137e6 −0.0201303
\(53\) −5.76352e7 −1.00334 −0.501668 0.865060i \(-0.667280\pi\)
−0.501668 + 0.865060i \(0.667280\pi\)
\(54\) −1.16917e7 −0.187114
\(55\) 9.15500e6 0.134904
\(56\) 7.11374e7 0.966612
\(57\) −3.57874e7 −0.449049
\(58\) −2.30857e7 −0.267866
\(59\) −3.20421e7 −0.344260 −0.172130 0.985074i \(-0.555065\pi\)
−0.172130 + 0.985074i \(0.555065\pi\)
\(60\) −1.41750e6 −0.0141203
\(61\) 1.10664e8 1.02335 0.511673 0.859180i \(-0.329026\pi\)
0.511673 + 0.859180i \(0.329026\pi\)
\(62\) −1.74032e8 −1.49578
\(63\) −3.92873e7 −0.314210
\(64\) 1.40733e8 1.04854
\(65\) −2.36913e7 −0.164618
\(66\) 2.61027e7 0.169332
\(67\) −1.18568e8 −0.718839 −0.359420 0.933176i \(-0.617025\pi\)
−0.359420 + 0.933176i \(0.617025\pi\)
\(68\) 1.23507e7 0.0700492
\(69\) −1.83395e8 −0.974017
\(70\) 8.23350e7 0.409867
\(71\) 2.76680e8 1.29216 0.646078 0.763272i \(-0.276408\pi\)
0.646078 + 0.763272i \(0.276408\pi\)
\(72\) −7.79447e7 −0.341814
\(73\) −2.64023e8 −1.08815 −0.544076 0.839036i \(-0.683120\pi\)
−0.544076 + 0.839036i \(0.683120\pi\)
\(74\) −4.61836e8 −1.79038
\(75\) −3.16406e7 −0.115470
\(76\) −1.23710e7 −0.0425346
\(77\) 8.77122e7 0.284349
\(78\) −6.75485e7 −0.206628
\(79\) 4.48203e8 1.29465 0.647325 0.762214i \(-0.275888\pi\)
0.647325 + 0.762214i \(0.275888\pi\)
\(80\) 1.54390e8 0.421419
\(81\) 4.30467e7 0.111111
\(82\) 2.92282e8 0.713905
\(83\) 8.51016e8 1.96828 0.984138 0.177402i \(-0.0567693\pi\)
0.984138 + 0.177402i \(0.0567693\pi\)
\(84\) −1.35808e7 −0.0297624
\(85\) 2.75686e8 0.572836
\(86\) −5.08877e8 −1.00316
\(87\) 8.49974e7 0.159063
\(88\) 1.74018e8 0.309331
\(89\) 1.89895e8 0.320818 0.160409 0.987051i \(-0.448719\pi\)
0.160409 + 0.987051i \(0.448719\pi\)
\(90\) −9.02138e7 −0.144938
\(91\) −2.26981e8 −0.346979
\(92\) −6.33958e7 −0.0922604
\(93\) 6.40756e8 0.888218
\(94\) −3.05221e8 −0.403217
\(95\) −2.76138e8 −0.347832
\(96\) −5.24906e7 −0.0630755
\(97\) −1.01415e9 −1.16313 −0.581566 0.813499i \(-0.697560\pi\)
−0.581566 + 0.813499i \(0.697560\pi\)
\(98\) −9.89442e7 −0.108361
\(99\) −9.61055e7 −0.100552
\(100\) −1.09375e7 −0.0109375
\(101\) −1.31537e9 −1.25777 −0.628885 0.777498i \(-0.716488\pi\)
−0.628885 + 0.777498i \(0.716488\pi\)
\(102\) 7.86037e8 0.719022
\(103\) −1.82797e9 −1.60030 −0.800151 0.599798i \(-0.795248\pi\)
−0.800151 + 0.599798i \(0.795248\pi\)
\(104\) −4.50323e8 −0.377463
\(105\) −3.03143e8 −0.243386
\(106\) −1.26797e9 −0.975515
\(107\) 1.85367e8 0.136712 0.0683559 0.997661i \(-0.478225\pi\)
0.0683559 + 0.997661i \(0.478225\pi\)
\(108\) 1.48803e7 0.0105246
\(109\) 1.97869e9 1.34264 0.671318 0.741169i \(-0.265728\pi\)
0.671318 + 0.741169i \(0.265728\pi\)
\(110\) 2.01410e8 0.131164
\(111\) 1.70040e9 1.06316
\(112\) 1.47918e9 0.888259
\(113\) 1.57875e9 0.910876 0.455438 0.890267i \(-0.349483\pi\)
0.455438 + 0.890267i \(0.349483\pi\)
\(114\) −7.87323e8 −0.436597
\(115\) −1.41508e9 −0.754470
\(116\) 2.93818e7 0.0150667
\(117\) 2.48701e8 0.122699
\(118\) −7.04927e8 −0.334715
\(119\) 2.64129e9 1.20741
\(120\) −6.01425e8 −0.264768
\(121\) −2.14338e9 −0.909004
\(122\) 2.43461e9 0.994970
\(123\) −1.07613e9 −0.423928
\(124\) 2.21496e8 0.0841333
\(125\) −2.44141e8 −0.0894427
\(126\) −8.64320e8 −0.305497
\(127\) 2.40001e9 0.818645 0.409322 0.912390i \(-0.365765\pi\)
0.409322 + 0.912390i \(0.365765\pi\)
\(128\) 2.76433e9 0.910218
\(129\) 1.87359e9 0.595691
\(130\) −5.21207e8 −0.160054
\(131\) −1.96840e9 −0.583971 −0.291986 0.956423i \(-0.594316\pi\)
−0.291986 + 0.956423i \(0.594316\pi\)
\(132\) −3.32217e7 −0.00952442
\(133\) −2.64562e9 −0.733153
\(134\) −2.60850e9 −0.698907
\(135\) 3.32151e8 0.0860663
\(136\) 5.24024e9 1.31349
\(137\) −2.02909e9 −0.492107 −0.246054 0.969256i \(-0.579134\pi\)
−0.246054 + 0.969256i \(0.579134\pi\)
\(138\) −4.03469e9 −0.947009
\(139\) −1.13673e9 −0.258280 −0.129140 0.991626i \(-0.541222\pi\)
−0.129140 + 0.991626i \(0.541222\pi\)
\(140\) −1.04790e8 −0.0230539
\(141\) 1.12377e9 0.239437
\(142\) 6.08695e9 1.25633
\(143\) −5.55247e8 −0.111039
\(144\) −1.62072e9 −0.314107
\(145\) 6.55844e8 0.123210
\(146\) −5.80851e9 −1.05798
\(147\) 3.64295e8 0.0643465
\(148\) 5.87792e8 0.100704
\(149\) −4.73854e9 −0.787601 −0.393800 0.919196i \(-0.628840\pi\)
−0.393800 + 0.919196i \(0.628840\pi\)
\(150\) −6.96094e8 −0.112268
\(151\) 2.26216e9 0.354101 0.177051 0.984202i \(-0.443344\pi\)
0.177051 + 0.984202i \(0.443344\pi\)
\(152\) −5.24882e9 −0.797564
\(153\) −2.89404e9 −0.426966
\(154\) 1.92967e9 0.276465
\(155\) 4.94411e9 0.688010
\(156\) 8.59708e7 0.0116223
\(157\) −1.17889e10 −1.54854 −0.774272 0.632853i \(-0.781884\pi\)
−0.774272 + 0.632853i \(0.781884\pi\)
\(158\) 9.86046e9 1.25875
\(159\) 4.66845e9 0.579276
\(160\) −4.05020e8 −0.0488581
\(161\) −1.35576e10 −1.59026
\(162\) 9.47028e8 0.108030
\(163\) 1.14608e10 1.27166 0.635829 0.771830i \(-0.280658\pi\)
0.635829 + 0.771830i \(0.280658\pi\)
\(164\) −3.71996e8 −0.0401551
\(165\) −7.41555e8 −0.0778871
\(166\) 1.87223e10 1.91370
\(167\) −1.33707e10 −1.33024 −0.665118 0.746738i \(-0.731619\pi\)
−0.665118 + 0.746738i \(0.731619\pi\)
\(168\) −5.76213e9 −0.558074
\(169\) −9.16763e9 −0.864504
\(170\) 6.06510e9 0.556952
\(171\) 2.89878e9 0.259258
\(172\) 6.47661e8 0.0564248
\(173\) 1.06264e10 0.901939 0.450970 0.892539i \(-0.351078\pi\)
0.450970 + 0.892539i \(0.351078\pi\)
\(174\) 1.86994e9 0.154652
\(175\) −2.33906e9 −0.188526
\(176\) 3.61841e9 0.284257
\(177\) 2.59541e9 0.198759
\(178\) 4.17769e9 0.311922
\(179\) 2.61254e10 1.90206 0.951031 0.309094i \(-0.100026\pi\)
0.951031 + 0.309094i \(0.100026\pi\)
\(180\) 1.14817e8 0.00815233
\(181\) 2.34689e9 0.162532 0.0812660 0.996692i \(-0.474104\pi\)
0.0812660 + 0.996692i \(0.474104\pi\)
\(182\) −4.99358e9 −0.337358
\(183\) −8.96379e9 −0.590829
\(184\) −2.68979e10 −1.72997
\(185\) 1.31203e10 0.823517
\(186\) 1.40966e10 0.863589
\(187\) 6.46120e9 0.386390
\(188\) 3.88463e8 0.0226798
\(189\) 3.18227e9 0.181409
\(190\) −6.07503e9 −0.338187
\(191\) 2.24064e10 1.21821 0.609105 0.793089i \(-0.291529\pi\)
0.609105 + 0.793089i \(0.291529\pi\)
\(192\) −1.13994e10 −0.605376
\(193\) −3.65959e10 −1.89856 −0.949282 0.314427i \(-0.898188\pi\)
−0.949282 + 0.314427i \(0.898188\pi\)
\(194\) −2.23113e10 −1.13088
\(195\) 1.91899e9 0.0950424
\(196\) 1.25929e8 0.00609499
\(197\) −5.41546e9 −0.256175 −0.128088 0.991763i \(-0.540884\pi\)
−0.128088 + 0.991763i \(0.540884\pi\)
\(198\) −2.11432e9 −0.0977637
\(199\) 2.62714e8 0.0118753 0.00593764 0.999982i \(-0.498110\pi\)
0.00593764 + 0.999982i \(0.498110\pi\)
\(200\) −4.64063e9 −0.205089
\(201\) 9.60403e9 0.415022
\(202\) −2.89381e10 −1.22289
\(203\) 6.28351e9 0.259699
\(204\) −1.00041e9 −0.0404429
\(205\) −8.30348e9 −0.328373
\(206\) −4.02154e10 −1.55593
\(207\) 1.48550e10 0.562349
\(208\) −9.36369e9 −0.346866
\(209\) −6.47178e9 −0.234620
\(210\) −6.66914e9 −0.236637
\(211\) 1.34493e10 0.467121 0.233560 0.972342i \(-0.424962\pi\)
0.233560 + 0.972342i \(0.424962\pi\)
\(212\) 1.61378e9 0.0548699
\(213\) −2.24111e10 −0.746026
\(214\) 4.07808e9 0.132921
\(215\) 1.44567e10 0.461421
\(216\) 6.31352e9 0.197347
\(217\) 4.73685e10 1.45018
\(218\) 4.35312e10 1.30541
\(219\) 2.13859e10 0.628245
\(220\) −2.56340e8 −0.00737759
\(221\) −1.67203e10 −0.471496
\(222\) 3.74087e10 1.03368
\(223\) −2.66463e10 −0.721547 −0.360773 0.932654i \(-0.617487\pi\)
−0.360773 + 0.932654i \(0.617487\pi\)
\(224\) −3.88042e9 −0.102982
\(225\) 2.56289e9 0.0666667
\(226\) 3.47324e10 0.885619
\(227\) 3.36318e10 0.840686 0.420343 0.907365i \(-0.361910\pi\)
0.420343 + 0.907365i \(0.361910\pi\)
\(228\) 1.00205e9 0.0245574
\(229\) 5.00453e10 1.20255 0.601276 0.799042i \(-0.294659\pi\)
0.601276 + 0.799042i \(0.294659\pi\)
\(230\) −3.11319e10 −0.733550
\(231\) −7.10469e9 −0.164169
\(232\) 1.24663e10 0.282515
\(233\) −3.29626e10 −0.732688 −0.366344 0.930479i \(-0.619391\pi\)
−0.366344 + 0.930479i \(0.619391\pi\)
\(234\) 5.47143e9 0.119297
\(235\) 8.67106e9 0.185467
\(236\) 8.97179e8 0.0188267
\(237\) −3.63044e10 −0.747467
\(238\) 5.81085e10 1.17393
\(239\) −7.95422e9 −0.157691 −0.0788455 0.996887i \(-0.525123\pi\)
−0.0788455 + 0.996887i \(0.525123\pi\)
\(240\) −1.25056e10 −0.243306
\(241\) −7.52477e10 −1.43687 −0.718434 0.695595i \(-0.755141\pi\)
−0.718434 + 0.695595i \(0.755141\pi\)
\(242\) −4.71544e10 −0.883799
\(243\) −3.48678e9 −0.0641500
\(244\) −3.09859e9 −0.0559642
\(245\) 2.81091e9 0.0498425
\(246\) −2.36749e10 −0.412173
\(247\) 1.67476e10 0.286297
\(248\) 9.39775e10 1.57758
\(249\) −6.89323e10 −1.13639
\(250\) −5.37109e9 −0.0869626
\(251\) −9.84631e10 −1.56582 −0.782910 0.622135i \(-0.786266\pi\)
−0.782910 + 0.622135i \(0.786266\pi\)
\(252\) 1.10004e9 0.0171833
\(253\) −3.31651e10 −0.508907
\(254\) 5.28001e10 0.795945
\(255\) −2.23306e10 −0.330727
\(256\) −1.12400e10 −0.163563
\(257\) 8.52399e9 0.121883 0.0609416 0.998141i \(-0.480590\pi\)
0.0609416 + 0.998141i \(0.480590\pi\)
\(258\) 4.12190e10 0.579174
\(259\) 1.25703e11 1.73579
\(260\) 6.63355e8 0.00900256
\(261\) −6.88479e9 −0.0918350
\(262\) −4.33047e10 −0.567779
\(263\) 5.90654e9 0.0761259 0.0380630 0.999275i \(-0.487881\pi\)
0.0380630 + 0.999275i \(0.487881\pi\)
\(264\) −1.40955e10 −0.178592
\(265\) 3.60220e10 0.448705
\(266\) −5.82036e10 −0.712824
\(267\) −1.53815e10 −0.185224
\(268\) 3.31991e9 0.0393115
\(269\) −7.15961e10 −0.833689 −0.416845 0.908978i \(-0.636864\pi\)
−0.416845 + 0.908978i \(0.636864\pi\)
\(270\) 7.30731e9 0.0836798
\(271\) −1.24755e11 −1.40506 −0.702530 0.711654i \(-0.747946\pi\)
−0.702530 + 0.711654i \(0.747946\pi\)
\(272\) 1.08962e11 1.20702
\(273\) 1.83855e10 0.200329
\(274\) −4.46401e10 −0.478462
\(275\) −5.72188e9 −0.0603311
\(276\) 5.13506e9 0.0532666
\(277\) 1.12824e11 1.15145 0.575723 0.817645i \(-0.304721\pi\)
0.575723 + 0.817645i \(0.304721\pi\)
\(278\) −2.50080e10 −0.251118
\(279\) −5.19012e10 −0.512813
\(280\) −4.44609e10 −0.432282
\(281\) 8.70208e10 0.832616 0.416308 0.909224i \(-0.363324\pi\)
0.416308 + 0.909224i \(0.363324\pi\)
\(282\) 2.47229e10 0.232798
\(283\) 3.27696e9 0.0303692 0.0151846 0.999885i \(-0.495166\pi\)
0.0151846 + 0.999885i \(0.495166\pi\)
\(284\) −7.74703e9 −0.0706647
\(285\) 2.23671e10 0.200821
\(286\) −1.22154e10 −0.107960
\(287\) −7.95539e10 −0.692139
\(288\) 4.25174e9 0.0364167
\(289\) 7.59796e10 0.640703
\(290\) 1.44286e10 0.119793
\(291\) 8.21461e10 0.671535
\(292\) 7.39265e9 0.0595083
\(293\) −1.49860e11 −1.18791 −0.593953 0.804500i \(-0.702433\pi\)
−0.593953 + 0.804500i \(0.702433\pi\)
\(294\) 8.01448e9 0.0625622
\(295\) 2.00263e10 0.153958
\(296\) 2.49392e11 1.88829
\(297\) 7.78455e9 0.0580536
\(298\) −1.04248e11 −0.765762
\(299\) 8.58243e10 0.620998
\(300\) 8.85937e8 0.00631477
\(301\) 1.38507e11 0.972574
\(302\) 4.97676e10 0.344283
\(303\) 1.06545e11 0.726174
\(304\) −1.09140e11 −0.732915
\(305\) −6.91650e10 −0.457654
\(306\) −6.36690e10 −0.415127
\(307\) −1.84570e11 −1.18587 −0.592937 0.805249i \(-0.702032\pi\)
−0.592937 + 0.805249i \(0.702032\pi\)
\(308\) −2.45594e9 −0.0155503
\(309\) 1.48066e11 0.923935
\(310\) 1.08770e11 0.668933
\(311\) 9.04650e10 0.548351 0.274176 0.961680i \(-0.411595\pi\)
0.274176 + 0.961680i \(0.411595\pi\)
\(312\) 3.64762e10 0.217928
\(313\) 1.07930e10 0.0635615 0.0317808 0.999495i \(-0.489882\pi\)
0.0317808 + 0.999495i \(0.489882\pi\)
\(314\) −2.59355e11 −1.50560
\(315\) 2.45545e10 0.140519
\(316\) −1.25497e10 −0.0708012
\(317\) 4.18319e10 0.232670 0.116335 0.993210i \(-0.462885\pi\)
0.116335 + 0.993210i \(0.462885\pi\)
\(318\) 1.02706e11 0.563214
\(319\) 1.53709e10 0.0831076
\(320\) −8.79581e10 −0.468922
\(321\) −1.50148e10 −0.0789306
\(322\) −2.98268e11 −1.54616
\(323\) −1.94886e11 −0.996252
\(324\) −1.20531e9 −0.00607639
\(325\) 1.48070e10 0.0736195
\(326\) 2.52137e11 1.23640
\(327\) −1.60274e11 −0.775172
\(328\) −1.57832e11 −0.752946
\(329\) 8.30756e10 0.390924
\(330\) −1.63142e10 −0.0757274
\(331\) −1.23310e10 −0.0564640 −0.0282320 0.999601i \(-0.508988\pi\)
−0.0282320 + 0.999601i \(0.508988\pi\)
\(332\) −2.38284e10 −0.107640
\(333\) −1.37732e11 −0.613813
\(334\) −2.94154e11 −1.29335
\(335\) 7.41052e10 0.321475
\(336\) −1.19814e11 −0.512837
\(337\) −2.17976e10 −0.0920606 −0.0460303 0.998940i \(-0.514657\pi\)
−0.0460303 + 0.998940i \(0.514657\pi\)
\(338\) −2.01688e11 −0.840533
\(339\) −1.27878e11 −0.525895
\(340\) −7.71922e9 −0.0313269
\(341\) 1.15874e11 0.464078
\(342\) 6.37732e10 0.252070
\(343\) 2.68568e11 1.04769
\(344\) 2.74793e11 1.05802
\(345\) 1.14622e11 0.435594
\(346\) 2.33780e11 0.876930
\(347\) −8.43613e10 −0.312364 −0.156182 0.987728i \(-0.549919\pi\)
−0.156182 + 0.987728i \(0.549919\pi\)
\(348\) −2.37993e9 −0.00869875
\(349\) −1.23295e9 −0.00444867 −0.00222434 0.999998i \(-0.500708\pi\)
−0.00222434 + 0.999998i \(0.500708\pi\)
\(350\) −5.14594e10 −0.183298
\(351\) −2.01448e10 −0.0708404
\(352\) −9.49237e9 −0.0329559
\(353\) 1.73388e11 0.594336 0.297168 0.954825i \(-0.403958\pi\)
0.297168 + 0.954825i \(0.403958\pi\)
\(354\) 5.70991e10 0.193248
\(355\) −1.72925e11 −0.577869
\(356\) −5.31706e9 −0.0175447
\(357\) −2.13945e11 −0.697100
\(358\) 5.74760e11 1.84932
\(359\) 6.38159e10 0.202770 0.101385 0.994847i \(-0.467673\pi\)
0.101385 + 0.994847i \(0.467673\pi\)
\(360\) 4.87154e10 0.152864
\(361\) −1.27483e11 −0.395066
\(362\) 5.16316e10 0.158025
\(363\) 1.73614e11 0.524814
\(364\) 6.35547e9 0.0189754
\(365\) 1.65015e11 0.486636
\(366\) −1.97203e11 −0.574446
\(367\) 1.12242e11 0.322966 0.161483 0.986876i \(-0.448372\pi\)
0.161483 + 0.986876i \(0.448372\pi\)
\(368\) −5.59296e11 −1.58974
\(369\) 8.71666e10 0.244755
\(370\) 2.88648e11 0.800682
\(371\) 3.45119e11 0.945773
\(372\) −1.79412e10 −0.0485744
\(373\) 7.21004e11 1.92863 0.964313 0.264766i \(-0.0852948\pi\)
0.964313 + 0.264766i \(0.0852948\pi\)
\(374\) 1.42146e11 0.375676
\(375\) 1.97754e10 0.0516398
\(376\) 1.64819e11 0.425268
\(377\) −3.97767e10 −0.101413
\(378\) 7.00099e10 0.176379
\(379\) −2.04331e11 −0.508695 −0.254348 0.967113i \(-0.581861\pi\)
−0.254348 + 0.967113i \(0.581861\pi\)
\(380\) 7.73185e9 0.0190220
\(381\) −1.94400e11 −0.472645
\(382\) 4.92941e11 1.18443
\(383\) −5.47180e11 −1.29938 −0.649689 0.760200i \(-0.725101\pi\)
−0.649689 + 0.760200i \(0.725101\pi\)
\(384\) −2.23911e11 −0.525515
\(385\) −5.48201e10 −0.127165
\(386\) −8.05111e11 −1.84592
\(387\) −1.51761e11 −0.343923
\(388\) 2.83962e10 0.0636088
\(389\) −3.46262e11 −0.766711 −0.383356 0.923601i \(-0.625232\pi\)
−0.383356 + 0.923601i \(0.625232\pi\)
\(390\) 4.22178e10 0.0924070
\(391\) −9.98706e11 −2.16094
\(392\) 5.34299e10 0.114287
\(393\) 1.59440e11 0.337156
\(394\) −1.19140e11 −0.249072
\(395\) −2.80127e11 −0.578985
\(396\) 2.69095e9 0.00549893
\(397\) 2.56758e11 0.518760 0.259380 0.965775i \(-0.416482\pi\)
0.259380 + 0.965775i \(0.416482\pi\)
\(398\) 5.77971e9 0.0115460
\(399\) 2.14295e11 0.423286
\(400\) −9.64938e10 −0.188464
\(401\) 2.01679e10 0.0389504 0.0194752 0.999810i \(-0.493800\pi\)
0.0194752 + 0.999810i \(0.493800\pi\)
\(402\) 2.11289e11 0.403514
\(403\) −2.99858e11 −0.566295
\(404\) 3.68303e10 0.0687843
\(405\) −2.69042e10 −0.0496904
\(406\) 1.38237e11 0.252498
\(407\) 3.07499e11 0.555481
\(408\) −4.24460e11 −0.758343
\(409\) −4.33405e10 −0.0765842 −0.0382921 0.999267i \(-0.512192\pi\)
−0.0382921 + 0.999267i \(0.512192\pi\)
\(410\) −1.82676e11 −0.319268
\(411\) 1.64357e11 0.284118
\(412\) 5.11832e10 0.0875166
\(413\) 1.91868e11 0.324510
\(414\) 3.26810e11 0.546756
\(415\) −5.31885e11 −0.880240
\(416\) 2.45643e10 0.0402147
\(417\) 9.20751e10 0.149118
\(418\) −1.42379e11 −0.228115
\(419\) 5.10680e11 0.809443 0.404721 0.914440i \(-0.367369\pi\)
0.404721 + 0.914440i \(0.367369\pi\)
\(420\) 8.48799e9 0.0133102
\(421\) 3.21228e11 0.498361 0.249181 0.968457i \(-0.419839\pi\)
0.249181 + 0.968457i \(0.419839\pi\)
\(422\) 2.95885e11 0.454168
\(423\) −9.10253e10 −0.138239
\(424\) 6.84706e11 1.02886
\(425\) −1.72304e11 −0.256180
\(426\) −4.93043e11 −0.725340
\(427\) −6.62656e11 −0.964635
\(428\) −5.19029e9 −0.00747643
\(429\) 4.49750e10 0.0641082
\(430\) 3.18048e11 0.448626
\(431\) 4.47617e11 0.624826 0.312413 0.949946i \(-0.398863\pi\)
0.312413 + 0.949946i \(0.398863\pi\)
\(432\) 1.31279e11 0.181350
\(433\) −1.41186e11 −0.193017 −0.0965085 0.995332i \(-0.530767\pi\)
−0.0965085 + 0.995332i \(0.530767\pi\)
\(434\) 1.04211e12 1.40997
\(435\) −5.31233e10 −0.0711351
\(436\) −5.54033e10 −0.0734254
\(437\) 1.00034e12 1.31214
\(438\) 4.70490e11 0.610824
\(439\) −8.50498e11 −1.09291 −0.546453 0.837490i \(-0.684022\pi\)
−0.546453 + 0.837490i \(0.684022\pi\)
\(440\) −1.08761e11 −0.138337
\(441\) −2.95079e10 −0.0371504
\(442\) −3.67846e11 −0.458422
\(443\) 3.03188e11 0.374020 0.187010 0.982358i \(-0.440120\pi\)
0.187010 + 0.982358i \(0.440120\pi\)
\(444\) −4.76111e10 −0.0581413
\(445\) −1.18684e11 −0.143474
\(446\) −5.86218e11 −0.701540
\(447\) 3.83822e11 0.454721
\(448\) −8.42709e11 −0.988386
\(449\) −1.40328e12 −1.62943 −0.814714 0.579863i \(-0.803106\pi\)
−0.814714 + 0.579863i \(0.803106\pi\)
\(450\) 5.63836e10 0.0648181
\(451\) −1.94607e11 −0.221495
\(452\) −4.42049e10 −0.0498136
\(453\) −1.83235e11 −0.204440
\(454\) 7.39899e11 0.817375
\(455\) 1.41863e11 0.155174
\(456\) 4.25155e11 0.460474
\(457\) 3.35354e11 0.359650 0.179825 0.983699i \(-0.442447\pi\)
0.179825 + 0.983699i \(0.442447\pi\)
\(458\) 1.10100e12 1.16921
\(459\) 2.34418e11 0.246509
\(460\) 3.96224e10 0.0412601
\(461\) 1.04275e12 1.07529 0.537645 0.843171i \(-0.319314\pi\)
0.537645 + 0.843171i \(0.319314\pi\)
\(462\) −1.56303e11 −0.159617
\(463\) −9.74084e11 −0.985103 −0.492552 0.870283i \(-0.663936\pi\)
−0.492552 + 0.870283i \(0.663936\pi\)
\(464\) 2.59215e11 0.259614
\(465\) −4.00473e11 −0.397223
\(466\) −7.25176e11 −0.712372
\(467\) −1.97851e11 −0.192492 −0.0962458 0.995358i \(-0.530683\pi\)
−0.0962458 + 0.995358i \(0.530683\pi\)
\(468\) −6.96364e9 −0.00671011
\(469\) 7.09987e11 0.677599
\(470\) 1.90763e11 0.180324
\(471\) 9.54898e11 0.894052
\(472\) 3.80660e11 0.353019
\(473\) 3.38819e11 0.311238
\(474\) −7.98697e11 −0.726741
\(475\) 1.72586e11 0.155555
\(476\) −7.39563e10 −0.0660304
\(477\) −3.78144e11 −0.334445
\(478\) −1.74993e11 −0.153319
\(479\) −9.08731e11 −0.788725 −0.394362 0.918955i \(-0.629035\pi\)
−0.394362 + 0.918955i \(0.629035\pi\)
\(480\) 3.28066e10 0.0282082
\(481\) −7.95744e11 −0.677830
\(482\) −1.65545e12 −1.39703
\(483\) 1.09817e12 0.918136
\(484\) 6.00147e10 0.0497112
\(485\) 6.33843e11 0.520169
\(486\) −7.67093e10 −0.0623713
\(487\) 1.16963e12 0.942254 0.471127 0.882065i \(-0.343847\pi\)
0.471127 + 0.882065i \(0.343847\pi\)
\(488\) −1.31469e12 −1.04938
\(489\) −9.28324e11 −0.734192
\(490\) 6.18401e10 0.0484605
\(491\) −2.49149e12 −1.93460 −0.967301 0.253630i \(-0.918375\pi\)
−0.967301 + 0.253630i \(0.918375\pi\)
\(492\) 3.01317e10 0.0231836
\(493\) 4.62866e11 0.352894
\(494\) 3.68448e11 0.278359
\(495\) 6.00660e10 0.0449681
\(496\) 1.95410e12 1.44970
\(497\) −1.65676e12 −1.21802
\(498\) −1.51651e12 −1.10488
\(499\) −5.57571e11 −0.402576 −0.201288 0.979532i \(-0.564513\pi\)
−0.201288 + 0.979532i \(0.564513\pi\)
\(500\) 6.83594e9 0.00489140
\(501\) 1.08302e12 0.768012
\(502\) −2.16619e12 −1.52240
\(503\) 1.80137e12 1.25472 0.627359 0.778730i \(-0.284136\pi\)
0.627359 + 0.778730i \(0.284136\pi\)
\(504\) 4.66733e11 0.322204
\(505\) 8.22105e11 0.562492
\(506\) −7.29631e11 −0.494796
\(507\) 7.42578e11 0.499122
\(508\) −6.72001e10 −0.0447696
\(509\) −2.40110e12 −1.58555 −0.792774 0.609515i \(-0.791364\pi\)
−0.792774 + 0.609515i \(0.791364\pi\)
\(510\) −4.91273e11 −0.321556
\(511\) 1.58097e12 1.02572
\(512\) −1.66262e12 −1.06925
\(513\) −2.34801e11 −0.149683
\(514\) 1.87528e11 0.118504
\(515\) 1.14248e12 0.715677
\(516\) −5.24606e10 −0.0325769
\(517\) 2.03222e11 0.125102
\(518\) 2.76548e12 1.68766
\(519\) −8.60735e11 −0.520735
\(520\) 2.81452e11 0.168807
\(521\) 1.03008e12 0.612491 0.306246 0.951953i \(-0.400927\pi\)
0.306246 + 0.951953i \(0.400927\pi\)
\(522\) −1.51465e11 −0.0892886
\(523\) 1.06637e12 0.623232 0.311616 0.950208i \(-0.399130\pi\)
0.311616 + 0.950208i \(0.399130\pi\)
\(524\) 5.51151e10 0.0319359
\(525\) 1.89464e11 0.108845
\(526\) 1.29944e11 0.0740151
\(527\) 3.48934e12 1.97058
\(528\) −2.93091e11 −0.164116
\(529\) 3.32516e12 1.84613
\(530\) 7.92484e11 0.436264
\(531\) −2.10228e11 −0.114753
\(532\) 7.40773e10 0.0400943
\(533\) 5.03603e11 0.270281
\(534\) −3.38393e11 −0.180088
\(535\) −1.15855e11 −0.0611394
\(536\) 1.40859e12 0.737129
\(537\) −2.11616e12 −1.09816
\(538\) −1.57511e12 −0.810572
\(539\) 6.58788e10 0.0336199
\(540\) −9.30022e9 −0.00470675
\(541\) 2.51749e12 1.26351 0.631756 0.775167i \(-0.282334\pi\)
0.631756 + 0.775167i \(0.282334\pi\)
\(542\) −2.74460e12 −1.36610
\(543\) −1.90098e11 −0.0938379
\(544\) −2.85846e11 −0.139938
\(545\) −1.23668e12 −0.600445
\(546\) 4.04480e11 0.194774
\(547\) −1.10353e12 −0.527037 −0.263518 0.964654i \(-0.584883\pi\)
−0.263518 + 0.964654i \(0.584883\pi\)
\(548\) 5.68146e10 0.0269121
\(549\) 7.26067e11 0.341115
\(550\) −1.25881e11 −0.0586582
\(551\) −4.63624e11 −0.214281
\(552\) 2.17873e12 0.998799
\(553\) −2.68384e12 −1.22037
\(554\) 2.48213e12 1.11952
\(555\) −1.06275e12 −0.475458
\(556\) 3.18284e10 0.0141247
\(557\) −1.09072e12 −0.480137 −0.240068 0.970756i \(-0.577170\pi\)
−0.240068 + 0.970756i \(0.577170\pi\)
\(558\) −1.14183e12 −0.498593
\(559\) −8.76795e11 −0.379791
\(560\) −9.24487e11 −0.397242
\(561\) −5.23357e11 −0.223083
\(562\) 1.91446e12 0.809529
\(563\) 1.03081e12 0.432403 0.216202 0.976349i \(-0.430633\pi\)
0.216202 + 0.976349i \(0.430633\pi\)
\(564\) −3.14655e10 −0.0130942
\(565\) −9.86717e11 −0.407356
\(566\) 7.20932e10 0.0295271
\(567\) −2.57764e11 −0.104737
\(568\) −3.28695e12 −1.32503
\(569\) −4.06618e12 −1.62623 −0.813113 0.582106i \(-0.802229\pi\)
−0.813113 + 0.582106i \(0.802229\pi\)
\(570\) 4.92077e11 0.195252
\(571\) 9.86144e11 0.388220 0.194110 0.980980i \(-0.437818\pi\)
0.194110 + 0.980980i \(0.437818\pi\)
\(572\) 1.55469e10 0.00607243
\(573\) −1.81492e12 −0.703334
\(574\) −1.75019e12 −0.672947
\(575\) 8.84428e11 0.337409
\(576\) 9.23349e11 0.349514
\(577\) −2.86947e12 −1.07773 −0.538866 0.842391i \(-0.681147\pi\)
−0.538866 + 0.842391i \(0.681147\pi\)
\(578\) 1.67155e12 0.622937
\(579\) 2.96427e12 1.09614
\(580\) −1.83636e10 −0.00673802
\(581\) −5.09588e12 −1.85535
\(582\) 1.80721e12 0.652914
\(583\) 8.44240e11 0.302662
\(584\) 3.13660e12 1.11584
\(585\) −1.55438e11 −0.0548727
\(586\) −3.29692e12 −1.15497
\(587\) 1.14169e12 0.396898 0.198449 0.980111i \(-0.436410\pi\)
0.198449 + 0.980111i \(0.436410\pi\)
\(588\) −1.02002e10 −0.00351895
\(589\) −3.49505e12 −1.19656
\(590\) 4.40579e11 0.149689
\(591\) 4.38652e11 0.147903
\(592\) 5.18567e12 1.73523
\(593\) −2.97176e12 −0.986888 −0.493444 0.869777i \(-0.664262\pi\)
−0.493444 + 0.869777i \(0.664262\pi\)
\(594\) 1.71260e11 0.0564439
\(595\) −1.65081e12 −0.539971
\(596\) 1.32679e11 0.0430719
\(597\) −2.12798e10 −0.00685620
\(598\) 1.88814e12 0.603779
\(599\) 1.77262e12 0.562593 0.281297 0.959621i \(-0.409236\pi\)
0.281297 + 0.959621i \(0.409236\pi\)
\(600\) 3.75891e11 0.118408
\(601\) 1.25838e12 0.393439 0.196719 0.980460i \(-0.436971\pi\)
0.196719 + 0.980460i \(0.436971\pi\)
\(602\) 3.04715e12 0.945606
\(603\) −7.77926e11 −0.239613
\(604\) −6.33405e10 −0.0193649
\(605\) 1.33961e12 0.406519
\(606\) 2.34399e12 0.706038
\(607\) 1.74535e11 0.0521834 0.0260917 0.999660i \(-0.491694\pi\)
0.0260917 + 0.999660i \(0.491694\pi\)
\(608\) 2.86313e11 0.0849720
\(609\) −5.08964e11 −0.149937
\(610\) −1.52163e12 −0.444964
\(611\) −5.25896e11 −0.152656
\(612\) 8.10332e10 0.0233497
\(613\) 4.63037e12 1.32447 0.662237 0.749294i \(-0.269607\pi\)
0.662237 + 0.749294i \(0.269607\pi\)
\(614\) −4.06054e12 −1.15299
\(615\) 6.72582e11 0.189586
\(616\) −1.04202e12 −0.291584
\(617\) 2.50518e12 0.695914 0.347957 0.937511i \(-0.386876\pi\)
0.347957 + 0.937511i \(0.386876\pi\)
\(618\) 3.25745e12 0.898316
\(619\) 5.44228e11 0.148995 0.0744977 0.997221i \(-0.476265\pi\)
0.0744977 + 0.997221i \(0.476265\pi\)
\(620\) −1.38435e11 −0.0376256
\(621\) −1.20325e12 −0.324672
\(622\) 1.99023e12 0.533146
\(623\) −1.13709e12 −0.302412
\(624\) 7.58459e11 0.200263
\(625\) 1.52588e11 0.0400000
\(626\) 2.37447e11 0.0617991
\(627\) 5.24214e11 0.135458
\(628\) 3.30088e11 0.0846860
\(629\) 9.25978e12 2.35870
\(630\) 5.40200e11 0.136622
\(631\) −6.51975e12 −1.63719 −0.818594 0.574372i \(-0.805246\pi\)
−0.818594 + 0.574372i \(0.805246\pi\)
\(632\) −5.32465e12 −1.32759
\(633\) −1.08939e12 −0.269692
\(634\) 9.20303e11 0.226219
\(635\) −1.50000e12 −0.366109
\(636\) −1.30717e11 −0.0316792
\(637\) −1.70481e11 −0.0410250
\(638\) 3.38159e11 0.0808032
\(639\) 1.81530e12 0.430718
\(640\) −1.72771e12 −0.407062
\(641\) −3.36297e12 −0.786794 −0.393397 0.919369i \(-0.628700\pi\)
−0.393397 + 0.919369i \(0.628700\pi\)
\(642\) −3.30325e11 −0.0767420
\(643\) 1.18082e12 0.272417 0.136208 0.990680i \(-0.456508\pi\)
0.136208 + 0.990680i \(0.456508\pi\)
\(644\) 3.79614e11 0.0869673
\(645\) −1.17099e12 −0.266401
\(646\) −4.28749e12 −0.968628
\(647\) 6.63176e12 1.48785 0.743926 0.668262i \(-0.232962\pi\)
0.743926 + 0.668262i \(0.232962\pi\)
\(648\) −5.11395e11 −0.113938
\(649\) 4.69353e11 0.103848
\(650\) 3.25755e11 0.0715782
\(651\) −3.83685e12 −0.837259
\(652\) −3.20902e11 −0.0695438
\(653\) 8.25189e11 0.177601 0.0888003 0.996049i \(-0.471697\pi\)
0.0888003 + 0.996049i \(0.471697\pi\)
\(654\) −3.52602e12 −0.753677
\(655\) 1.23025e12 0.261160
\(656\) −3.28185e12 −0.691913
\(657\) −1.73226e12 −0.362717
\(658\) 1.82766e12 0.380084
\(659\) −4.40214e11 −0.0909242 −0.0454621 0.998966i \(-0.514476\pi\)
−0.0454621 + 0.998966i \(0.514476\pi\)
\(660\) 2.07635e10 0.00425945
\(661\) 9.77788e12 1.99222 0.996112 0.0880914i \(-0.0280767\pi\)
0.996112 + 0.0880914i \(0.0280767\pi\)
\(662\) −2.71281e11 −0.0548983
\(663\) 1.35434e12 0.272218
\(664\) −1.01101e13 −2.01836
\(665\) 1.65351e12 0.327876
\(666\) −3.03011e12 −0.596793
\(667\) −2.37587e12 −0.464790
\(668\) 3.74378e11 0.0727473
\(669\) 2.15835e12 0.416585
\(670\) 1.63031e12 0.312561
\(671\) −1.62101e12 −0.308698
\(672\) 3.14314e11 0.0594568
\(673\) 4.22591e12 0.794058 0.397029 0.917806i \(-0.370041\pi\)
0.397029 + 0.917806i \(0.370041\pi\)
\(674\) −4.79547e11 −0.0895079
\(675\) −2.07594e11 −0.0384900
\(676\) 2.56694e11 0.0472776
\(677\) −7.56977e12 −1.38495 −0.692474 0.721442i \(-0.743479\pi\)
−0.692474 + 0.721442i \(0.743479\pi\)
\(678\) −2.81333e12 −0.511313
\(679\) 6.07273e12 1.09640
\(680\) −3.27515e12 −0.587410
\(681\) −2.72418e12 −0.485370
\(682\) 2.54923e12 0.451210
\(683\) −1.92029e12 −0.337655 −0.168827 0.985646i \(-0.553998\pi\)
−0.168827 + 0.985646i \(0.553998\pi\)
\(684\) −8.11659e10 −0.0141782
\(685\) 1.26818e12 0.220077
\(686\) 5.90850e12 1.01864
\(687\) −4.05367e12 −0.694293
\(688\) 5.71385e12 0.972257
\(689\) −2.18472e12 −0.369325
\(690\) 2.52168e12 0.423516
\(691\) −3.99839e12 −0.667166 −0.333583 0.942721i \(-0.608258\pi\)
−0.333583 + 0.942721i \(0.608258\pi\)
\(692\) −2.97538e11 −0.0493248
\(693\) 5.75480e11 0.0947830
\(694\) −1.85595e12 −0.303702
\(695\) 7.10456e11 0.115506
\(696\) −1.00977e12 −0.163110
\(697\) −5.86023e12 −0.940519
\(698\) −2.71249e10 −0.00432532
\(699\) 2.66997e12 0.423018
\(700\) 6.54937e10 0.0103100
\(701\) 1.12736e13 1.76332 0.881662 0.471881i \(-0.156425\pi\)
0.881662 + 0.471881i \(0.156425\pi\)
\(702\) −4.43186e11 −0.0688761
\(703\) −9.27493e12 −1.43223
\(704\) −2.06146e12 −0.316299
\(705\) −7.02355e11 −0.107079
\(706\) 3.81453e12 0.577856
\(707\) 7.87643e12 1.18561
\(708\) −7.26715e10 −0.0108696
\(709\) 1.12679e13 1.67469 0.837346 0.546673i \(-0.184106\pi\)
0.837346 + 0.546673i \(0.184106\pi\)
\(710\) −3.80435e12 −0.561846
\(711\) 2.94066e12 0.431550
\(712\) −2.25595e12 −0.328980
\(713\) −1.79106e13 −2.59542
\(714\) −4.70679e12 −0.677770
\(715\) 3.47029e11 0.0496580
\(716\) −7.31512e11 −0.104019
\(717\) 6.44292e11 0.0910429
\(718\) 1.40395e12 0.197148
\(719\) −9.05247e12 −1.26324 −0.631622 0.775277i \(-0.717610\pi\)
−0.631622 + 0.775277i \(0.717610\pi\)
\(720\) 1.01295e12 0.140473
\(721\) 1.09459e13 1.50849
\(722\) −2.80462e12 −0.384111
\(723\) 6.09507e12 0.829576
\(724\) −6.57129e10 −0.00888847
\(725\) −4.09902e11 −0.0551010
\(726\) 3.81951e12 0.510262
\(727\) 1.34408e12 0.178452 0.0892260 0.996011i \(-0.471561\pi\)
0.0892260 + 0.996011i \(0.471561\pi\)
\(728\) 2.69654e12 0.355807
\(729\) 2.82430e11 0.0370370
\(730\) 3.63032e12 0.473143
\(731\) 1.02029e13 1.32159
\(732\) 2.50986e11 0.0323109
\(733\) 6.57401e11 0.0841129 0.0420564 0.999115i \(-0.486609\pi\)
0.0420564 + 0.999115i \(0.486609\pi\)
\(734\) 2.46932e12 0.314011
\(735\) −2.27684e11 −0.0287766
\(736\) 1.46723e12 0.184310
\(737\) 1.73679e12 0.216842
\(738\) 1.91766e12 0.237968
\(739\) 2.15587e12 0.265903 0.132951 0.991123i \(-0.457555\pi\)
0.132951 + 0.991123i \(0.457555\pi\)
\(740\) −3.67370e11 −0.0450361
\(741\) −1.35656e12 −0.165294
\(742\) 7.59263e12 0.919548
\(743\) −7.73594e12 −0.931244 −0.465622 0.884984i \(-0.654169\pi\)
−0.465622 + 0.884984i \(0.654169\pi\)
\(744\) −7.61218e12 −0.910817
\(745\) 2.96159e12 0.352226
\(746\) 1.58621e13 1.87515
\(747\) 5.58351e12 0.656092
\(748\) −1.80914e11 −0.0211307
\(749\) −1.10998e12 −0.128869
\(750\) 4.35059e11 0.0502079
\(751\) −1.12448e13 −1.28995 −0.644974 0.764205i \(-0.723132\pi\)
−0.644974 + 0.764205i \(0.723132\pi\)
\(752\) 3.42713e12 0.390797
\(753\) 7.97551e12 0.904027
\(754\) −8.75087e11 −0.0986007
\(755\) −1.41385e12 −0.158359
\(756\) −8.91035e10 −0.00992080
\(757\) −1.18544e13 −1.31205 −0.656024 0.754740i \(-0.727763\pi\)
−0.656024 + 0.754740i \(0.727763\pi\)
\(758\) −4.49528e12 −0.494590
\(759\) 2.68637e12 0.293818
\(760\) 3.28051e12 0.356682
\(761\) −2.24765e12 −0.242939 −0.121470 0.992595i \(-0.538761\pi\)
−0.121470 + 0.992595i \(0.538761\pi\)
\(762\) −4.27681e12 −0.459539
\(763\) −1.18484e13 −1.26561
\(764\) −6.27380e11 −0.0666209
\(765\) 1.80878e12 0.190945
\(766\) −1.20380e13 −1.26335
\(767\) −1.21459e12 −0.126721
\(768\) 9.10436e11 0.0944331
\(769\) −1.44505e13 −1.49010 −0.745049 0.667009i \(-0.767574\pi\)
−0.745049 + 0.667009i \(0.767574\pi\)
\(770\) −1.20604e12 −0.123639
\(771\) −6.90443e11 −0.0703693
\(772\) 1.02469e12 0.103828
\(773\) 5.19022e12 0.522852 0.261426 0.965224i \(-0.415807\pi\)
0.261426 + 0.965224i \(0.415807\pi\)
\(774\) −3.33874e12 −0.334386
\(775\) −3.09007e12 −0.307688
\(776\) 1.20481e13 1.19273
\(777\) −1.01820e13 −1.00216
\(778\) −7.61777e12 −0.745452
\(779\) 5.86983e12 0.571093
\(780\) −5.37318e10 −0.00519763
\(781\) −4.05280e12 −0.389786
\(782\) −2.19715e13 −2.10102
\(783\) 5.57668e11 0.0530209
\(784\) 1.11098e12 0.105023
\(785\) 7.36804e12 0.692530
\(786\) 3.50768e12 0.327807
\(787\) −1.39938e13 −1.30032 −0.650161 0.759797i \(-0.725299\pi\)
−0.650161 + 0.759797i \(0.725299\pi\)
\(788\) 1.51633e11 0.0140096
\(789\) −4.78430e11 −0.0439513
\(790\) −6.16279e12 −0.562931
\(791\) −9.45354e12 −0.858618
\(792\) 1.14173e12 0.103110
\(793\) 4.19483e12 0.376691
\(794\) 5.64867e12 0.504375
\(795\) −2.91778e12 −0.259060
\(796\) −7.35599e9 −0.000649430 0
\(797\) 2.01269e13 1.76691 0.883454 0.468518i \(-0.155212\pi\)
0.883454 + 0.468518i \(0.155212\pi\)
\(798\) 4.71449e12 0.411549
\(799\) 6.11966e12 0.531210
\(800\) 2.53137e11 0.0218500
\(801\) 1.24590e12 0.106939
\(802\) 4.43695e11 0.0378704
\(803\) 3.86741e12 0.328247
\(804\) −2.68913e11 −0.0226965
\(805\) 8.47353e12 0.711185
\(806\) −6.59688e12 −0.550593
\(807\) 5.79928e12 0.481331
\(808\) 1.56266e13 1.28977
\(809\) −8.95775e12 −0.735242 −0.367621 0.929976i \(-0.619828\pi\)
−0.367621 + 0.929976i \(0.619828\pi\)
\(810\) −5.91892e11 −0.0483126
\(811\) −1.76446e12 −0.143225 −0.0716123 0.997433i \(-0.522814\pi\)
−0.0716123 + 0.997433i \(0.522814\pi\)
\(812\) −1.75938e11 −0.0142023
\(813\) 1.01051e13 0.811212
\(814\) 6.76498e12 0.540078
\(815\) −7.16300e12 −0.568703
\(816\) −8.82591e12 −0.696873
\(817\) −1.02196e13 −0.802483
\(818\) −9.53491e11 −0.0744607
\(819\) −1.48922e12 −0.115660
\(820\) 2.32497e11 0.0179579
\(821\) 1.48082e13 1.13752 0.568758 0.822505i \(-0.307424\pi\)
0.568758 + 0.822505i \(0.307424\pi\)
\(822\) 3.61585e12 0.276240
\(823\) −3.39651e12 −0.258067 −0.129034 0.991640i \(-0.541188\pi\)
−0.129034 + 0.991640i \(0.541188\pi\)
\(824\) 2.17163e13 1.64102
\(825\) 4.63472e11 0.0348322
\(826\) 4.22110e12 0.315512
\(827\) 1.88344e13 1.40016 0.700080 0.714065i \(-0.253148\pi\)
0.700080 + 0.714065i \(0.253148\pi\)
\(828\) −4.15940e11 −0.0307535
\(829\) 1.18947e13 0.874695 0.437348 0.899292i \(-0.355918\pi\)
0.437348 + 0.899292i \(0.355918\pi\)
\(830\) −1.17015e13 −0.855833
\(831\) −9.13876e12 −0.664787
\(832\) 5.33462e12 0.385966
\(833\) 1.98382e12 0.142758
\(834\) 2.02565e12 0.144983
\(835\) 8.35666e12 0.594899
\(836\) 1.81210e11 0.0128308
\(837\) 4.20400e12 0.296073
\(838\) 1.12350e13 0.786998
\(839\) −1.22881e13 −0.856165 −0.428083 0.903740i \(-0.640811\pi\)
−0.428083 + 0.903740i \(0.640811\pi\)
\(840\) 3.60133e12 0.249578
\(841\) −1.34060e13 −0.924097
\(842\) 7.06702e12 0.484543
\(843\) −7.04869e12 −0.480711
\(844\) −3.76581e11 −0.0255457
\(845\) 5.72977e12 0.386618
\(846\) −2.00256e12 −0.134406
\(847\) 1.28346e13 0.856853
\(848\) 1.42373e13 0.945465
\(849\) −2.65434e11 −0.0175336
\(850\) −3.79069e12 −0.249076
\(851\) −4.75300e13 −3.10660
\(852\) 6.27510e11 0.0407983
\(853\) −4.61355e12 −0.298376 −0.149188 0.988809i \(-0.547666\pi\)
−0.149188 + 0.988809i \(0.547666\pi\)
\(854\) −1.45784e13 −0.937887
\(855\) −1.81174e12 −0.115944
\(856\) −2.20216e12 −0.140190
\(857\) −4.29363e12 −0.271901 −0.135950 0.990716i \(-0.543409\pi\)
−0.135950 + 0.990716i \(0.543409\pi\)
\(858\) 9.89450e11 0.0623306
\(859\) −6.40428e12 −0.401330 −0.200665 0.979660i \(-0.564310\pi\)
−0.200665 + 0.979660i \(0.564310\pi\)
\(860\) −4.04788e11 −0.0252339
\(861\) 6.44387e12 0.399607
\(862\) 9.84758e12 0.607501
\(863\) −3.24421e12 −0.199095 −0.0995474 0.995033i \(-0.531740\pi\)
−0.0995474 + 0.995033i \(0.531740\pi\)
\(864\) −3.44391e11 −0.0210252
\(865\) −6.64148e12 −0.403359
\(866\) −3.10609e12 −0.187665
\(867\) −6.15435e12 −0.369910
\(868\) −1.32632e12 −0.0793065
\(869\) −6.56527e12 −0.390539
\(870\) −1.16871e12 −0.0691626
\(871\) −4.49445e12 −0.264603
\(872\) −2.35068e13 −1.37680
\(873\) −6.65383e12 −0.387711
\(874\) 2.20075e13 1.27576
\(875\) 1.46191e12 0.0843113
\(876\) −5.98805e11 −0.0343571
\(877\) 2.89711e13 1.65374 0.826871 0.562392i \(-0.190119\pi\)
0.826871 + 0.562392i \(0.190119\pi\)
\(878\) −1.87109e13 −1.06260
\(879\) 1.21387e13 0.685838
\(880\) −2.26150e12 −0.127123
\(881\) 7.50447e12 0.419690 0.209845 0.977735i \(-0.432704\pi\)
0.209845 + 0.977735i \(0.432704\pi\)
\(882\) −6.49173e11 −0.0361203
\(883\) 2.87141e13 1.58954 0.794772 0.606908i \(-0.207590\pi\)
0.794772 + 0.606908i \(0.207590\pi\)
\(884\) 4.68167e11 0.0257849
\(885\) −1.62213e12 −0.0888877
\(886\) 6.67013e12 0.363649
\(887\) 9.99825e12 0.542335 0.271168 0.962532i \(-0.412590\pi\)
0.271168 + 0.962532i \(0.412590\pi\)
\(888\) −2.02007e13 −1.09021
\(889\) −1.43712e13 −0.771678
\(890\) −2.61106e12 −0.139496
\(891\) −6.30548e11 −0.0335173
\(892\) 7.46095e11 0.0394596
\(893\) −6.12967e12 −0.322556
\(894\) 8.44407e12 0.442113
\(895\) −1.63284e13 −0.850628
\(896\) −1.65528e13 −0.857998
\(897\) −6.95177e12 −0.358533
\(898\) −3.08721e13 −1.58425
\(899\) 8.30095e12 0.423847
\(900\) −7.17609e10 −0.00364583
\(901\) 2.54228e13 1.28517
\(902\) −4.28135e12 −0.215353
\(903\) −1.12191e13 −0.561516
\(904\) −1.87555e13 −0.934052
\(905\) −1.46681e12 −0.0726865
\(906\) −4.03117e12 −0.198772
\(907\) 4.70846e12 0.231018 0.115509 0.993306i \(-0.463150\pi\)
0.115509 + 0.993306i \(0.463150\pi\)
\(908\) −9.41690e11 −0.0459750
\(909\) −8.63013e12 −0.419257
\(910\) 3.12099e12 0.150871
\(911\) 2.69663e13 1.29714 0.648572 0.761154i \(-0.275367\pi\)
0.648572 + 0.761154i \(0.275367\pi\)
\(912\) 8.84035e12 0.423148
\(913\) −1.24657e13 −0.593742
\(914\) 7.37778e12 0.349678
\(915\) 5.60237e12 0.264227
\(916\) −1.40127e12 −0.0657645
\(917\) 1.17868e13 0.550468
\(918\) 5.15719e12 0.239674
\(919\) −3.96618e12 −0.183422 −0.0917112 0.995786i \(-0.529234\pi\)
−0.0917112 + 0.995786i \(0.529234\pi\)
\(920\) 1.68112e13 0.773666
\(921\) 1.49502e13 0.684664
\(922\) 2.29405e13 1.04547
\(923\) 1.04878e13 0.475639
\(924\) 1.98931e11 0.00897799
\(925\) −8.20022e12 −0.368288
\(926\) −2.14298e13 −0.957788
\(927\) −1.19933e13 −0.533434
\(928\) −6.80012e11 −0.0300989
\(929\) 1.96912e13 0.867365 0.433683 0.901066i \(-0.357214\pi\)
0.433683 + 0.901066i \(0.357214\pi\)
\(930\) −8.81040e12 −0.386209
\(931\) −1.98707e12 −0.0866841
\(932\) 9.22952e11 0.0400689
\(933\) −7.32766e12 −0.316591
\(934\) −4.35271e12 −0.187154
\(935\) −4.03825e12 −0.172799
\(936\) −2.95457e12 −0.125821
\(937\) −2.11305e13 −0.895532 −0.447766 0.894151i \(-0.647780\pi\)
−0.447766 + 0.894151i \(0.647780\pi\)
\(938\) 1.56197e13 0.658810
\(939\) −8.74236e11 −0.0366973
\(940\) −2.42790e11 −0.0101427
\(941\) 5.41072e12 0.224958 0.112479 0.993654i \(-0.464121\pi\)
0.112479 + 0.993654i \(0.464121\pi\)
\(942\) 2.10077e13 0.869261
\(943\) 3.00803e13 1.23874
\(944\) 7.91517e12 0.324404
\(945\) −1.98892e12 −0.0811286
\(946\) 7.45403e12 0.302608
\(947\) 3.44259e12 0.139095 0.0695473 0.997579i \(-0.477845\pi\)
0.0695473 + 0.997579i \(0.477845\pi\)
\(948\) 1.01652e12 0.0408771
\(949\) −1.00081e13 −0.400546
\(950\) 3.79689e12 0.151242
\(951\) −3.38839e12 −0.134332
\(952\) −3.13786e13 −1.23813
\(953\) −4.18440e12 −0.164329 −0.0821647 0.996619i \(-0.526183\pi\)
−0.0821647 + 0.996619i \(0.526183\pi\)
\(954\) −8.31918e12 −0.325172
\(955\) −1.40040e13 −0.544800
\(956\) 2.22718e11 0.00862373
\(957\) −1.24504e12 −0.0479822
\(958\) −1.99921e13 −0.766855
\(959\) 1.21502e13 0.463874
\(960\) 7.12461e12 0.270733
\(961\) 3.61375e13 1.36679
\(962\) −1.75064e13 −0.659035
\(963\) 1.21620e12 0.0455706
\(964\) 2.10694e12 0.0785787
\(965\) 2.28725e13 0.849063
\(966\) 2.41597e13 0.892678
\(967\) 1.49800e13 0.550926 0.275463 0.961312i \(-0.411169\pi\)
0.275463 + 0.961312i \(0.411169\pi\)
\(968\) 2.54634e13 0.932132
\(969\) 1.57858e13 0.575186
\(970\) 1.39446e13 0.505745
\(971\) −2.82460e13 −1.01970 −0.509848 0.860264i \(-0.670299\pi\)
−0.509848 + 0.860264i \(0.670299\pi\)
\(972\) 9.76300e10 0.00350820
\(973\) 6.80673e12 0.243462
\(974\) 2.57318e13 0.916127
\(975\) −1.19937e12 −0.0425042
\(976\) −2.73367e13 −0.964321
\(977\) −4.62242e13 −1.62309 −0.811547 0.584287i \(-0.801374\pi\)
−0.811547 + 0.584287i \(0.801374\pi\)
\(978\) −2.04231e13 −0.713835
\(979\) −2.78158e12 −0.0967764
\(980\) −7.87056e10 −0.00272576
\(981\) 1.29822e13 0.447545
\(982\) −5.48127e13 −1.88096
\(983\) 4.80363e13 1.64089 0.820443 0.571728i \(-0.193727\pi\)
0.820443 + 0.571728i \(0.193727\pi\)
\(984\) 1.27844e13 0.434714
\(985\) 3.38466e12 0.114565
\(986\) 1.01831e13 0.343109
\(987\) −6.72913e12 −0.225700
\(988\) −4.68934e11 −0.0156569
\(989\) −5.23712e13 −1.74064
\(990\) 1.32145e12 0.0437213
\(991\) 2.31211e13 0.761512 0.380756 0.924675i \(-0.375664\pi\)
0.380756 + 0.924675i \(0.375664\pi\)
\(992\) −5.12630e12 −0.168074
\(993\) 9.98809e11 0.0325995
\(994\) −3.64487e13 −1.18425
\(995\) −1.64196e11 −0.00531079
\(996\) 1.93010e12 0.0621461
\(997\) −1.35362e13 −0.433878 −0.216939 0.976185i \(-0.569607\pi\)
−0.216939 + 0.976185i \(0.569607\pi\)
\(998\) −1.22666e13 −0.391413
\(999\) 1.11563e13 0.354385
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 15.10.a.b.1.1 1
3.2 odd 2 45.10.a.a.1.1 1
4.3 odd 2 240.10.a.g.1.1 1
5.2 odd 4 75.10.b.b.49.2 2
5.3 odd 4 75.10.b.b.49.1 2
5.4 even 2 75.10.a.a.1.1 1
15.2 even 4 225.10.b.b.199.1 2
15.8 even 4 225.10.b.b.199.2 2
15.14 odd 2 225.10.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.b.1.1 1 1.1 even 1 trivial
45.10.a.a.1.1 1 3.2 odd 2
75.10.a.a.1.1 1 5.4 even 2
75.10.b.b.49.1 2 5.3 odd 4
75.10.b.b.49.2 2 5.2 odd 4
225.10.a.f.1.1 1 15.14 odd 2
225.10.b.b.199.1 2 15.2 even 4
225.10.b.b.199.2 2 15.8 even 4
240.10.a.g.1.1 1 4.3 odd 2