Properties

Label 15.10.a.d.1.2
Level 15
Weight 10
Character 15.1
Self dual yes
Analytic conductor 7.726
Analytic rank 0
Dimension 2
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \(x^{2} - x - 60\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.26209\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-expansion

\(f(q)\) \(=\) \(q+38.7863 q^{2} -81.0000 q^{3} +992.374 q^{4} +625.000 q^{5} -3141.69 q^{6} +12272.1 q^{7} +18631.9 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+38.7863 q^{2} -81.0000 q^{3} +992.374 q^{4} +625.000 q^{5} -3141.69 q^{6} +12272.1 q^{7} +18631.9 q^{8} +6561.00 q^{9} +24241.4 q^{10} -65897.9 q^{11} -80382.3 q^{12} -112300. q^{13} +475990. q^{14} -50625.0 q^{15} +214567. q^{16} +96634.7 q^{17} +254477. q^{18} -181332. q^{19} +620234. q^{20} -994042. q^{21} -2.55593e6 q^{22} +244145. q^{23} -1.50919e6 q^{24} +390625. q^{25} -4.35569e6 q^{26} -531441. q^{27} +1.21785e7 q^{28} -5.26461e6 q^{29} -1.96355e6 q^{30} +1.83457e6 q^{31} -1.21730e6 q^{32} +5.33773e6 q^{33} +3.74810e6 q^{34} +7.67008e6 q^{35} +6.51097e6 q^{36} +6.36194e6 q^{37} -7.03318e6 q^{38} +9.09628e6 q^{39} +1.16449e7 q^{40} +1.57111e6 q^{41} -3.85552e7 q^{42} -1.99504e7 q^{43} -6.53953e7 q^{44} +4.10063e6 q^{45} +9.46946e6 q^{46} +3.00961e7 q^{47} -1.73799e7 q^{48} +1.10251e8 q^{49} +1.51509e7 q^{50} -7.82741e6 q^{51} -1.11443e8 q^{52} -2.57306e6 q^{53} -2.06126e7 q^{54} -4.11862e7 q^{55} +2.28653e8 q^{56} +1.46879e7 q^{57} -2.04195e8 q^{58} -1.19004e8 q^{59} -5.02389e7 q^{60} +1.92875e8 q^{61} +7.11559e7 q^{62} +8.05174e7 q^{63} -1.57073e8 q^{64} -7.01874e7 q^{65} +2.07030e8 q^{66} -1.20193e8 q^{67} +9.58978e7 q^{68} -1.97757e7 q^{69} +2.97494e8 q^{70} -699549. q^{71} +1.22244e8 q^{72} -8.91287e7 q^{73} +2.46756e8 q^{74} -3.16406e7 q^{75} -1.79949e8 q^{76} -8.08707e8 q^{77} +3.52811e8 q^{78} +4.31205e8 q^{79} +1.34104e8 q^{80} +4.30467e7 q^{81} +6.09375e7 q^{82} -1.69761e7 q^{83} -9.86461e8 q^{84} +6.03967e7 q^{85} -7.73800e8 q^{86} +4.26433e8 q^{87} -1.22780e9 q^{88} -3.09863e6 q^{89} +1.59048e8 q^{90} -1.37816e9 q^{91} +2.42283e8 q^{92} -1.48600e8 q^{93} +1.16732e9 q^{94} -1.13332e8 q^{95} +9.86009e7 q^{96} +5.72609e8 q^{97} +4.27624e9 q^{98} -4.32356e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + 19375q^{10} - 21512q^{11} - 43821q^{12} + 24284q^{13} + 461664q^{14} - 101250q^{15} + 387265q^{16} - 156956q^{17} + 203391q^{18} - 95896q^{19} + 338125q^{20} - 1143072q^{21} - 2901532q^{22} - 735264q^{23} - 2116773q^{24} + 781250q^{25} - 5419166q^{26} - 1062882q^{27} + 11348064q^{28} - 2678212q^{29} - 1569375q^{30} + 10782432q^{31} - 6402523q^{32} + 1742472q^{33} + 5722622q^{34} + 8820000q^{35} + 3549501q^{36} + 21968332q^{37} - 7698404q^{38} - 1967004q^{39} + 16333125q^{40} + 26060372q^{41} - 37394784q^{42} - 7191160q^{43} - 85429972q^{44} + 8201250q^{45} + 17095392q^{46} - 31580240q^{47} - 31368465q^{48} + 73282930q^{49} + 12109375q^{50} + 12713436q^{51} - 173093786q^{52} + 3131116q^{53} - 16474671q^{54} - 13445000q^{55} + 242454240q^{56} + 7767576q^{57} - 224332958q^{58} - 35494664q^{59} - 27388125q^{60} + 341497340q^{61} + 1485504q^{62} + 92588832q^{63} - 205120471q^{64} + 15177500q^{65} + 235024092q^{66} - 288195816q^{67} + 210362042q^{68} + 59556384q^{69} + 288540000q^{70} + 210286064q^{71} + 171458613q^{72} - 232663084q^{73} + 125240234q^{74} - 63281250q^{75} - 218512364q^{76} - 727042176q^{77} + 438952446q^{78} - 24755040q^{79} + 242040625q^{80} + 86093442q^{81} - 129742346q^{82} - 372082152q^{83} - 919193184q^{84} - 98097500q^{85} - 873146372q^{86} + 216935172q^{87} - 894861492q^{88} - 427639116q^{89} + 127119375q^{90} - 1126859328q^{91} + 684362592q^{92} - 873376992q^{93} + 1647543896q^{94} - 59935000q^{95} + 518604363q^{96} + 1771658884q^{97} + 4564085351q^{98} - 141140232q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 38.7863 1.71413 0.857063 0.515211i \(-0.172286\pi\)
0.857063 + 0.515211i \(0.172286\pi\)
\(3\) −81.0000 −0.577350
\(4\) 992.374 1.93823
\(5\) 625.000 0.447214
\(6\) −3141.69 −0.989652
\(7\) 12272.1 1.93187 0.965936 0.258780i \(-0.0833204\pi\)
0.965936 + 0.258780i \(0.0833204\pi\)
\(8\) 18631.9 1.60825
\(9\) 6561.00 0.333333
\(10\) 24241.4 0.766581
\(11\) −65897.9 −1.35708 −0.678538 0.734565i \(-0.737386\pi\)
−0.678538 + 0.734565i \(0.737386\pi\)
\(12\) −80382.3 −1.11904
\(13\) −112300. −1.09052 −0.545260 0.838267i \(-0.683569\pi\)
−0.545260 + 0.838267i \(0.683569\pi\)
\(14\) 475990. 3.31147
\(15\) −50625.0 −0.258199
\(16\) 214567. 0.818508
\(17\) 96634.7 0.280616 0.140308 0.990108i \(-0.455191\pi\)
0.140308 + 0.990108i \(0.455191\pi\)
\(18\) 254477. 0.571376
\(19\) −181332. −0.319214 −0.159607 0.987181i \(-0.551023\pi\)
−0.159607 + 0.987181i \(0.551023\pi\)
\(20\) 620234. 0.866803
\(21\) −994042. −1.11537
\(22\) −2.55593e6 −2.32620
\(23\) 244145. 0.181916 0.0909582 0.995855i \(-0.471007\pi\)
0.0909582 + 0.995855i \(0.471007\pi\)
\(24\) −1.50919e6 −0.928521
\(25\) 390625. 0.200000
\(26\) −4.35569e6 −1.86929
\(27\) −531441. −0.192450
\(28\) 1.21785e7 3.74442
\(29\) −5.26461e6 −1.38221 −0.691107 0.722752i \(-0.742877\pi\)
−0.691107 + 0.722752i \(0.742877\pi\)
\(30\) −1.96355e6 −0.442586
\(31\) 1.83457e6 0.356784 0.178392 0.983959i \(-0.442910\pi\)
0.178392 + 0.983959i \(0.442910\pi\)
\(32\) −1.21730e6 −0.205221
\(33\) 5.33773e6 0.783508
\(34\) 3.74810e6 0.481012
\(35\) 7.67008e6 0.863960
\(36\) 6.51097e6 0.646077
\(37\) 6.36194e6 0.558061 0.279030 0.960282i \(-0.409987\pi\)
0.279030 + 0.960282i \(0.409987\pi\)
\(38\) −7.03318e6 −0.547174
\(39\) 9.09628e6 0.629612
\(40\) 1.16449e7 0.719230
\(41\) 1.57111e6 0.0868319 0.0434159 0.999057i \(-0.486176\pi\)
0.0434159 + 0.999057i \(0.486176\pi\)
\(42\) −3.85552e7 −1.91188
\(43\) −1.99504e7 −0.889903 −0.444952 0.895555i \(-0.646779\pi\)
−0.444952 + 0.895555i \(0.646779\pi\)
\(44\) −6.53953e7 −2.63033
\(45\) 4.10063e6 0.149071
\(46\) 9.46946e6 0.311828
\(47\) 3.00961e7 0.899643 0.449821 0.893119i \(-0.351488\pi\)
0.449821 + 0.893119i \(0.351488\pi\)
\(48\) −1.73799e7 −0.472566
\(49\) 1.10251e8 2.73213
\(50\) 1.51509e7 0.342825
\(51\) −7.82741e6 −0.162014
\(52\) −1.11443e8 −2.11368
\(53\) −2.57306e6 −0.0447929 −0.0223964 0.999749i \(-0.507130\pi\)
−0.0223964 + 0.999749i \(0.507130\pi\)
\(54\) −2.06126e7 −0.329884
\(55\) −4.11862e7 −0.606903
\(56\) 2.28653e8 3.10693
\(57\) 1.46879e7 0.184299
\(58\) −2.04195e8 −2.36929
\(59\) −1.19004e8 −1.27858 −0.639290 0.768965i \(-0.720772\pi\)
−0.639290 + 0.768965i \(0.720772\pi\)
\(60\) −5.02389e7 −0.500449
\(61\) 1.92875e8 1.78358 0.891790 0.452449i \(-0.149450\pi\)
0.891790 + 0.452449i \(0.149450\pi\)
\(62\) 7.11559e7 0.611573
\(63\) 8.05174e7 0.643958
\(64\) −1.57073e8 −1.17028
\(65\) −7.01874e7 −0.487696
\(66\) 2.07030e8 1.34303
\(67\) −1.20193e8 −0.728691 −0.364345 0.931264i \(-0.618707\pi\)
−0.364345 + 0.931264i \(0.618707\pi\)
\(68\) 9.58978e7 0.543899
\(69\) −1.97757e7 −0.105030
\(70\) 2.97494e8 1.48094
\(71\) −699549. −0.00326705 −0.00163352 0.999999i \(-0.500520\pi\)
−0.00163352 + 0.999999i \(0.500520\pi\)
\(72\) 1.22244e8 0.536082
\(73\) −8.91287e7 −0.367337 −0.183669 0.982988i \(-0.558797\pi\)
−0.183669 + 0.982988i \(0.558797\pi\)
\(74\) 2.46756e8 0.956587
\(75\) −3.16406e7 −0.115470
\(76\) −1.79949e8 −0.618711
\(77\) −8.08707e8 −2.62170
\(78\) 3.52811e8 1.07924
\(79\) 4.31205e8 1.24555 0.622776 0.782400i \(-0.286005\pi\)
0.622776 + 0.782400i \(0.286005\pi\)
\(80\) 1.34104e8 0.366048
\(81\) 4.30467e7 0.111111
\(82\) 6.09375e7 0.148841
\(83\) −1.69761e7 −0.0392633 −0.0196316 0.999807i \(-0.506249\pi\)
−0.0196316 + 0.999807i \(0.506249\pi\)
\(84\) −9.86461e8 −2.16184
\(85\) 6.03967e7 0.125495
\(86\) −7.73800e8 −1.52541
\(87\) 4.26433e8 0.798022
\(88\) −1.22780e9 −2.18251
\(89\) −3.09863e6 −0.00523498 −0.00261749 0.999997i \(-0.500833\pi\)
−0.00261749 + 0.999997i \(0.500833\pi\)
\(90\) 1.59048e8 0.255527
\(91\) −1.37816e9 −2.10675
\(92\) 2.42283e8 0.352596
\(93\) −1.48600e8 −0.205989
\(94\) 1.16732e9 1.54210
\(95\) −1.13332e8 −0.142757
\(96\) 9.86009e7 0.118484
\(97\) 5.72609e8 0.656727 0.328364 0.944551i \(-0.393503\pi\)
0.328364 + 0.944551i \(0.393503\pi\)
\(98\) 4.27624e9 4.68322
\(99\) −4.32356e8 −0.452359
\(100\) 3.87646e8 0.387646
\(101\) 1.78445e9 1.70631 0.853156 0.521655i \(-0.174685\pi\)
0.853156 + 0.521655i \(0.174685\pi\)
\(102\) −3.03596e8 −0.277712
\(103\) 1.16632e9 1.02106 0.510528 0.859861i \(-0.329450\pi\)
0.510528 + 0.859861i \(0.329450\pi\)
\(104\) −2.09236e9 −1.75383
\(105\) −6.21276e8 −0.498807
\(106\) −9.97995e7 −0.0767807
\(107\) −1.72229e9 −1.27022 −0.635109 0.772422i \(-0.719045\pi\)
−0.635109 + 0.772422i \(0.719045\pi\)
\(108\) −5.27388e8 −0.373013
\(109\) 2.08468e9 1.41456 0.707279 0.706934i \(-0.249922\pi\)
0.707279 + 0.706934i \(0.249922\pi\)
\(110\) −1.59746e9 −1.04031
\(111\) −5.15317e8 −0.322197
\(112\) 2.63319e9 1.58125
\(113\) −1.81995e9 −1.05004 −0.525020 0.851090i \(-0.675942\pi\)
−0.525020 + 0.851090i \(0.675942\pi\)
\(114\) 5.69688e8 0.315911
\(115\) 1.52590e8 0.0813555
\(116\) −5.22446e9 −2.67905
\(117\) −7.36799e8 −0.363507
\(118\) −4.61573e9 −2.19165
\(119\) 1.18591e9 0.542115
\(120\) −9.43241e8 −0.415247
\(121\) 1.98458e9 0.841656
\(122\) 7.48092e9 3.05728
\(123\) −1.27260e8 −0.0501324
\(124\) 1.82058e9 0.691530
\(125\) 2.44141e8 0.0894427
\(126\) 3.12297e9 1.10382
\(127\) −3.74913e9 −1.27883 −0.639417 0.768860i \(-0.720824\pi\)
−0.639417 + 0.768860i \(0.720824\pi\)
\(128\) −5.46900e9 −1.80079
\(129\) 1.61598e9 0.513786
\(130\) −2.72231e9 −0.835972
\(131\) −1.96598e9 −0.583256 −0.291628 0.956532i \(-0.594197\pi\)
−0.291628 + 0.956532i \(0.594197\pi\)
\(132\) 5.29702e9 1.51862
\(133\) −2.22532e9 −0.616682
\(134\) −4.66184e9 −1.24907
\(135\) −3.32151e8 −0.0860663
\(136\) 1.80049e9 0.451300
\(137\) −3.63223e8 −0.0880909 −0.0440455 0.999030i \(-0.514025\pi\)
−0.0440455 + 0.999030i \(0.514025\pi\)
\(138\) −7.67026e8 −0.180034
\(139\) 4.55580e9 1.03514 0.517569 0.855642i \(-0.326837\pi\)
0.517569 + 0.855642i \(0.326837\pi\)
\(140\) 7.61159e9 1.67455
\(141\) −2.43779e9 −0.519409
\(142\) −2.71329e7 −0.00560014
\(143\) 7.40032e9 1.47992
\(144\) 1.40777e9 0.272836
\(145\) −3.29038e9 −0.618145
\(146\) −3.45697e9 −0.629662
\(147\) −8.93036e9 −1.57740
\(148\) 6.31342e9 1.08165
\(149\) 8.79830e9 1.46238 0.731191 0.682173i \(-0.238965\pi\)
0.731191 + 0.682173i \(0.238965\pi\)
\(150\) −1.22722e9 −0.197930
\(151\) −2.70702e8 −0.0423736 −0.0211868 0.999776i \(-0.506744\pi\)
−0.0211868 + 0.999776i \(0.506744\pi\)
\(152\) −3.37856e9 −0.513376
\(153\) 6.34020e8 0.0935388
\(154\) −3.13667e10 −4.49392
\(155\) 1.14660e9 0.159559
\(156\) 9.02692e9 1.22033
\(157\) 9.73240e9 1.27841 0.639207 0.769035i \(-0.279263\pi\)
0.639207 + 0.769035i \(0.279263\pi\)
\(158\) 1.67248e10 2.13504
\(159\) 2.08418e8 0.0258612
\(160\) −7.60810e8 −0.0917775
\(161\) 2.99617e9 0.351439
\(162\) 1.66962e9 0.190459
\(163\) 7.76823e9 0.861941 0.430971 0.902366i \(-0.358171\pi\)
0.430971 + 0.902366i \(0.358171\pi\)
\(164\) 1.55913e9 0.168300
\(165\) 3.33608e9 0.350396
\(166\) −6.58440e8 −0.0673023
\(167\) −8.41748e9 −0.837448 −0.418724 0.908114i \(-0.637522\pi\)
−0.418724 + 0.908114i \(0.637522\pi\)
\(168\) −1.85209e10 −1.79379
\(169\) 2.00674e9 0.189235
\(170\) 2.34256e9 0.215115
\(171\) −1.18972e9 −0.106405
\(172\) −1.97982e10 −1.72484
\(173\) 3.59838e8 0.0305422 0.0152711 0.999883i \(-0.495139\pi\)
0.0152711 + 0.999883i \(0.495139\pi\)
\(174\) 1.65398e10 1.36791
\(175\) 4.79380e9 0.386375
\(176\) −1.41395e10 −1.11078
\(177\) 9.63934e9 0.738189
\(178\) −1.20184e8 −0.00897342
\(179\) −8.65338e9 −0.630009 −0.315005 0.949090i \(-0.602006\pi\)
−0.315005 + 0.949090i \(0.602006\pi\)
\(180\) 4.06935e9 0.288934
\(181\) 6.09713e9 0.422252 0.211126 0.977459i \(-0.432287\pi\)
0.211126 + 0.977459i \(0.432287\pi\)
\(182\) −5.34535e10 −3.61123
\(183\) −1.56229e10 −1.02975
\(184\) 4.54888e9 0.292566
\(185\) 3.97621e9 0.249572
\(186\) −5.76363e9 −0.353092
\(187\) −6.36802e9 −0.380818
\(188\) 2.98666e10 1.74371
\(189\) −6.52191e9 −0.371789
\(190\) −4.39574e9 −0.244704
\(191\) −2.96238e10 −1.61061 −0.805306 0.592859i \(-0.797999\pi\)
−0.805306 + 0.592859i \(0.797999\pi\)
\(192\) 1.27229e10 0.675662
\(193\) −4.63438e8 −0.0240427 −0.0120214 0.999928i \(-0.503827\pi\)
−0.0120214 + 0.999928i \(0.503827\pi\)
\(194\) 2.22094e10 1.12571
\(195\) 5.68518e9 0.281571
\(196\) 1.09411e11 5.29550
\(197\) 1.23414e10 0.583803 0.291902 0.956448i \(-0.405712\pi\)
0.291902 + 0.956448i \(0.405712\pi\)
\(198\) −1.67695e10 −0.775400
\(199\) −1.00354e10 −0.453624 −0.226812 0.973939i \(-0.572830\pi\)
−0.226812 + 0.973939i \(0.572830\pi\)
\(200\) 7.27809e9 0.321649
\(201\) 9.73565e9 0.420710
\(202\) 6.92122e10 2.92484
\(203\) −6.46080e10 −2.67026
\(204\) −7.76772e9 −0.314020
\(205\) 9.81943e8 0.0388324
\(206\) 4.52371e10 1.75022
\(207\) 1.60183e9 0.0606388
\(208\) −2.40958e10 −0.892599
\(209\) 1.19494e10 0.433198
\(210\) −2.40970e10 −0.855019
\(211\) −3.56268e10 −1.23739 −0.618693 0.785633i \(-0.712338\pi\)
−0.618693 + 0.785633i \(0.712338\pi\)
\(212\) −2.55344e9 −0.0868189
\(213\) 5.66635e7 0.00188623
\(214\) −6.68011e10 −2.17732
\(215\) −1.24690e10 −0.397977
\(216\) −9.90176e9 −0.309507
\(217\) 2.25140e10 0.689262
\(218\) 8.08571e10 2.42473
\(219\) 7.21943e9 0.212082
\(220\) −4.08721e10 −1.17632
\(221\) −1.08521e10 −0.306018
\(222\) −1.99872e10 −0.552286
\(223\) −3.41646e10 −0.925133 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(224\) −1.49388e10 −0.396460
\(225\) 2.56289e9 0.0666667
\(226\) −7.05889e10 −1.79990
\(227\) 3.64936e10 0.912221 0.456110 0.889923i \(-0.349242\pi\)
0.456110 + 0.889923i \(0.349242\pi\)
\(228\) 1.45759e10 0.357213
\(229\) 3.87446e10 0.931005 0.465502 0.885047i \(-0.345874\pi\)
0.465502 + 0.885047i \(0.345874\pi\)
\(230\) 5.91841e9 0.139454
\(231\) 6.55052e10 1.51364
\(232\) −9.80898e10 −2.22294
\(233\) −2.68717e10 −0.597302 −0.298651 0.954362i \(-0.596537\pi\)
−0.298651 + 0.954362i \(0.596537\pi\)
\(234\) −2.85777e10 −0.623097
\(235\) 1.88101e10 0.402332
\(236\) −1.18097e11 −2.47818
\(237\) −3.49276e10 −0.719120
\(238\) 4.59971e10 0.929254
\(239\) 1.34711e10 0.267063 0.133531 0.991045i \(-0.457368\pi\)
0.133531 + 0.991045i \(0.457368\pi\)
\(240\) −1.08624e10 −0.211338
\(241\) −1.85860e10 −0.354902 −0.177451 0.984130i \(-0.556785\pi\)
−0.177451 + 0.984130i \(0.556785\pi\)
\(242\) 7.69745e10 1.44271
\(243\) −3.48678e9 −0.0641500
\(244\) 1.91405e11 3.45699
\(245\) 6.89071e10 1.22185
\(246\) −4.93593e9 −0.0859333
\(247\) 2.03635e10 0.348110
\(248\) 3.41815e10 0.573797
\(249\) 1.37506e9 0.0226687
\(250\) 9.46930e9 0.153316
\(251\) 4.30327e10 0.684332 0.342166 0.939640i \(-0.388839\pi\)
0.342166 + 0.939640i \(0.388839\pi\)
\(252\) 7.99034e10 1.24814
\(253\) −1.60886e10 −0.246875
\(254\) −1.45415e11 −2.19208
\(255\) −4.89213e9 −0.0724548
\(256\) −1.31701e11 −1.91650
\(257\) −6.26858e8 −0.00896335 −0.00448168 0.999990i \(-0.501427\pi\)
−0.00448168 + 0.999990i \(0.501427\pi\)
\(258\) 6.26778e10 0.880694
\(259\) 7.80745e10 1.07810
\(260\) −6.96521e10 −0.945267
\(261\) −3.45411e10 −0.460738
\(262\) −7.62532e10 −0.999774
\(263\) 6.21446e10 0.800944 0.400472 0.916309i \(-0.368846\pi\)
0.400472 + 0.916309i \(0.368846\pi\)
\(264\) 9.94521e10 1.26007
\(265\) −1.60816e9 −0.0200320
\(266\) −8.63120e10 −1.05707
\(267\) 2.50989e8 0.00302242
\(268\) −1.19277e11 −1.41237
\(269\) −5.44876e10 −0.634472 −0.317236 0.948347i \(-0.602755\pi\)
−0.317236 + 0.948347i \(0.602755\pi\)
\(270\) −1.28829e10 −0.147529
\(271\) −9.74930e10 −1.09802 −0.549012 0.835815i \(-0.684996\pi\)
−0.549012 + 0.835815i \(0.684996\pi\)
\(272\) 2.07346e10 0.229687
\(273\) 1.11631e11 1.21633
\(274\) −1.40881e10 −0.150999
\(275\) −2.57414e10 −0.271415
\(276\) −1.96249e10 −0.203571
\(277\) 1.43195e11 1.46140 0.730702 0.682697i \(-0.239193\pi\)
0.730702 + 0.682697i \(0.239193\pi\)
\(278\) 1.76702e11 1.77436
\(279\) 1.20366e10 0.118928
\(280\) 1.42908e11 1.38946
\(281\) 6.51280e10 0.623146 0.311573 0.950222i \(-0.399144\pi\)
0.311573 + 0.950222i \(0.399144\pi\)
\(282\) −9.45526e10 −0.890333
\(283\) −1.65431e11 −1.53312 −0.766561 0.642171i \(-0.778034\pi\)
−0.766561 + 0.642171i \(0.778034\pi\)
\(284\) −6.94214e8 −0.00633229
\(285\) 9.17992e9 0.0824208
\(286\) 2.87031e11 2.53677
\(287\) 1.92808e10 0.167748
\(288\) −7.98667e9 −0.0684069
\(289\) −1.09250e11 −0.921254
\(290\) −1.27622e11 −1.05958
\(291\) −4.63813e10 −0.379162
\(292\) −8.84490e10 −0.711984
\(293\) 5.64261e10 0.447277 0.223638 0.974672i \(-0.428207\pi\)
0.223638 + 0.974672i \(0.428207\pi\)
\(294\) −3.46375e11 −2.70386
\(295\) −7.43776e10 −0.571799
\(296\) 1.18535e11 0.897499
\(297\) 3.50208e10 0.261169
\(298\) 3.41253e11 2.50671
\(299\) −2.74174e10 −0.198384
\(300\) −3.13993e10 −0.223808
\(301\) −2.44833e11 −1.71918
\(302\) −1.04995e10 −0.0726338
\(303\) −1.44541e11 −0.985140
\(304\) −3.89078e10 −0.261279
\(305\) 1.20547e11 0.797641
\(306\) 2.45913e10 0.160337
\(307\) 1.79360e10 0.115240 0.0576200 0.998339i \(-0.481649\pi\)
0.0576200 + 0.998339i \(0.481649\pi\)
\(308\) −8.02540e11 −5.08146
\(309\) −9.44717e10 −0.589507
\(310\) 4.44725e10 0.273504
\(311\) 3.19953e11 1.93938 0.969692 0.244330i \(-0.0785680\pi\)
0.969692 + 0.244330i \(0.0785680\pi\)
\(312\) 1.69481e11 1.01257
\(313\) −3.16987e11 −1.86677 −0.933387 0.358872i \(-0.883162\pi\)
−0.933387 + 0.358872i \(0.883162\pi\)
\(314\) 3.77483e11 2.19136
\(315\) 5.03234e10 0.287987
\(316\) 4.27917e11 2.41417
\(317\) −3.08157e11 −1.71398 −0.856990 0.515334i \(-0.827668\pi\)
−0.856990 + 0.515334i \(0.827668\pi\)
\(318\) 8.08376e9 0.0443293
\(319\) 3.46927e11 1.87577
\(320\) −9.81703e10 −0.523366
\(321\) 1.39505e11 0.733361
\(322\) 1.16210e11 0.602412
\(323\) −1.75229e10 −0.0895768
\(324\) 4.27185e10 0.215359
\(325\) −4.38671e10 −0.218104
\(326\) 3.01301e11 1.47748
\(327\) −1.68859e11 −0.816696
\(328\) 2.92728e10 0.139647
\(329\) 3.69343e11 1.73800
\(330\) 1.29394e11 0.600623
\(331\) −1.79184e11 −0.820490 −0.410245 0.911975i \(-0.634557\pi\)
−0.410245 + 0.911975i \(0.634557\pi\)
\(332\) −1.68466e10 −0.0761013
\(333\) 4.17407e10 0.186020
\(334\) −3.26482e11 −1.43549
\(335\) −7.51207e10 −0.325880
\(336\) −2.13288e11 −0.912937
\(337\) −9.73351e10 −0.411088 −0.205544 0.978648i \(-0.565896\pi\)
−0.205544 + 0.978648i \(0.565896\pi\)
\(338\) 7.78340e10 0.324373
\(339\) 1.47416e11 0.606241
\(340\) 5.99361e10 0.243239
\(341\) −1.20894e11 −0.484183
\(342\) −4.61447e10 −0.182391
\(343\) 8.57794e11 3.34626
\(344\) −3.71713e11 −1.43118
\(345\) −1.23598e10 −0.0469706
\(346\) 1.39568e10 0.0523531
\(347\) −3.47750e11 −1.28761 −0.643806 0.765189i \(-0.722646\pi\)
−0.643806 + 0.765189i \(0.722646\pi\)
\(348\) 4.23182e11 1.54675
\(349\) −1.10131e11 −0.397372 −0.198686 0.980063i \(-0.563667\pi\)
−0.198686 + 0.980063i \(0.563667\pi\)
\(350\) 1.85934e11 0.662295
\(351\) 5.96807e10 0.209871
\(352\) 8.02172e10 0.278500
\(353\) −3.40052e11 −1.16563 −0.582814 0.812606i \(-0.698048\pi\)
−0.582814 + 0.812606i \(0.698048\pi\)
\(354\) 3.73874e11 1.26535
\(355\) −4.37218e8 −0.00146107
\(356\) −3.07500e9 −0.0101466
\(357\) −9.60589e10 −0.312990
\(358\) −3.35632e11 −1.07992
\(359\) −2.75063e11 −0.873993 −0.436996 0.899463i \(-0.643958\pi\)
−0.436996 + 0.899463i \(0.643958\pi\)
\(360\) 7.64025e10 0.239743
\(361\) −2.89807e11 −0.898102
\(362\) 2.36485e11 0.723793
\(363\) −1.60751e11 −0.485930
\(364\) −1.36765e12 −4.08336
\(365\) −5.57054e10 −0.164278
\(366\) −6.05954e11 −1.76512
\(367\) −2.12218e11 −0.610639 −0.305320 0.952250i \(-0.598763\pi\)
−0.305320 + 0.952250i \(0.598763\pi\)
\(368\) 5.23854e10 0.148900
\(369\) 1.03081e10 0.0289440
\(370\) 1.54222e11 0.427799
\(371\) −3.15769e10 −0.0865341
\(372\) −1.47467e11 −0.399255
\(373\) 6.08324e11 1.62721 0.813607 0.581415i \(-0.197501\pi\)
0.813607 + 0.581415i \(0.197501\pi\)
\(374\) −2.46992e11 −0.652770
\(375\) −1.97754e10 −0.0516398
\(376\) 5.60748e11 1.44685
\(377\) 5.91215e11 1.50733
\(378\) −2.52960e11 −0.637294
\(379\) 4.64136e11 1.15550 0.577749 0.816214i \(-0.303931\pi\)
0.577749 + 0.816214i \(0.303931\pi\)
\(380\) −1.12468e11 −0.276696
\(381\) 3.03680e11 0.738335
\(382\) −1.14900e12 −2.76079
\(383\) 5.22749e11 1.24136 0.620681 0.784063i \(-0.286856\pi\)
0.620681 + 0.784063i \(0.286856\pi\)
\(384\) 4.42989e11 1.03969
\(385\) −5.05442e11 −1.17246
\(386\) −1.79750e10 −0.0412123
\(387\) −1.30894e11 −0.296634
\(388\) 5.68242e11 1.27289
\(389\) −1.18899e11 −0.263271 −0.131636 0.991298i \(-0.542023\pi\)
−0.131636 + 0.991298i \(0.542023\pi\)
\(390\) 2.20507e11 0.482649
\(391\) 2.35928e10 0.0510487
\(392\) 2.05419e12 4.39394
\(393\) 1.59245e11 0.336743
\(394\) 4.78677e11 1.00071
\(395\) 2.69503e11 0.557028
\(396\) −4.29059e11 −0.876776
\(397\) −2.76954e11 −0.559564 −0.279782 0.960064i \(-0.590262\pi\)
−0.279782 + 0.960064i \(0.590262\pi\)
\(398\) −3.89236e11 −0.777569
\(399\) 1.80251e11 0.356041
\(400\) 8.38152e10 0.163702
\(401\) −1.53482e11 −0.296421 −0.148210 0.988956i \(-0.547351\pi\)
−0.148210 + 0.988956i \(0.547351\pi\)
\(402\) 3.77609e11 0.721150
\(403\) −2.06021e11 −0.389080
\(404\) 1.77084e12 3.30723
\(405\) 2.69042e10 0.0496904
\(406\) −2.50590e12 −4.57717
\(407\) −4.19238e11 −0.757331
\(408\) −1.45840e11 −0.260558
\(409\) −1.52364e11 −0.269232 −0.134616 0.990898i \(-0.542980\pi\)
−0.134616 + 0.990898i \(0.542980\pi\)
\(410\) 3.80859e10 0.0665636
\(411\) 2.94211e10 0.0508593
\(412\) 1.15742e12 1.97904
\(413\) −1.46043e12 −2.47006
\(414\) 6.21291e10 0.103943
\(415\) −1.06101e10 −0.0175591
\(416\) 1.36702e11 0.223797
\(417\) −3.69020e11 −0.597637
\(418\) 4.63472e11 0.742557
\(419\) −9.09824e11 −1.44210 −0.721049 0.692885i \(-0.756340\pi\)
−0.721049 + 0.692885i \(0.756340\pi\)
\(420\) −6.16538e11 −0.966804
\(421\) 9.88529e11 1.53363 0.766814 0.641869i \(-0.221841\pi\)
0.766814 + 0.641869i \(0.221841\pi\)
\(422\) −1.38183e12 −2.12104
\(423\) 1.97461e11 0.299881
\(424\) −4.79411e10 −0.0720380
\(425\) 3.77479e10 0.0561233
\(426\) 2.19776e9 0.00323324
\(427\) 2.36699e12 3.44565
\(428\) −1.70915e12 −2.46198
\(429\) −5.99426e11 −0.854432
\(430\) −4.83625e11 −0.682183
\(431\) 2.03278e11 0.283755 0.141878 0.989884i \(-0.454686\pi\)
0.141878 + 0.989884i \(0.454686\pi\)
\(432\) −1.14030e11 −0.157522
\(433\) −2.10847e11 −0.288251 −0.144126 0.989559i \(-0.546037\pi\)
−0.144126 + 0.989559i \(0.546037\pi\)
\(434\) 8.73234e11 1.18148
\(435\) 2.66521e11 0.356886
\(436\) 2.06879e12 2.74174
\(437\) −4.42712e10 −0.0580704
\(438\) 2.80015e11 0.363536
\(439\) 2.11413e11 0.271670 0.135835 0.990732i \(-0.456628\pi\)
0.135835 + 0.990732i \(0.456628\pi\)
\(440\) −7.67377e11 −0.976050
\(441\) 7.23359e11 0.910711
\(442\) −4.20911e11 −0.524553
\(443\) 1.45517e12 1.79514 0.897569 0.440874i \(-0.145331\pi\)
0.897569 + 0.440874i \(0.145331\pi\)
\(444\) −5.11387e11 −0.624491
\(445\) −1.93665e9 −0.00234115
\(446\) −1.32512e12 −1.58579
\(447\) −7.12663e11 −0.844306
\(448\) −1.92761e12 −2.26084
\(449\) −2.19470e11 −0.254840 −0.127420 0.991849i \(-0.540670\pi\)
−0.127420 + 0.991849i \(0.540670\pi\)
\(450\) 9.94049e10 0.114275
\(451\) −1.03533e11 −0.117837
\(452\) −1.80607e12 −2.03522
\(453\) 2.19269e10 0.0244644
\(454\) 1.41545e12 1.56366
\(455\) −8.61348e11 −0.942166
\(456\) 2.73663e11 0.296397
\(457\) 7.40816e11 0.794488 0.397244 0.917713i \(-0.369967\pi\)
0.397244 + 0.917713i \(0.369967\pi\)
\(458\) 1.50276e12 1.59586
\(459\) −5.13556e10 −0.0540046
\(460\) 1.51427e11 0.157686
\(461\) 2.80170e11 0.288913 0.144457 0.989511i \(-0.453857\pi\)
0.144457 + 0.989511i \(0.453857\pi\)
\(462\) 2.54070e12 2.59457
\(463\) −5.07092e10 −0.0512828 −0.0256414 0.999671i \(-0.508163\pi\)
−0.0256414 + 0.999671i \(0.508163\pi\)
\(464\) −1.12961e12 −1.13135
\(465\) −9.28749e10 −0.0921213
\(466\) −1.04225e12 −1.02385
\(467\) 1.77228e12 1.72427 0.862135 0.506678i \(-0.169127\pi\)
0.862135 + 0.506678i \(0.169127\pi\)
\(468\) −7.31180e11 −0.704560
\(469\) −1.47503e12 −1.40774
\(470\) 7.29572e11 0.689649
\(471\) −7.88324e11 −0.738092
\(472\) −2.21728e12 −2.05627
\(473\) 1.31469e12 1.20767
\(474\) −1.35471e12 −1.23266
\(475\) −7.08327e10 −0.0638429
\(476\) 1.17687e12 1.05074
\(477\) −1.68819e10 −0.0149310
\(478\) 5.22495e11 0.457779
\(479\) −7.79555e11 −0.676608 −0.338304 0.941037i \(-0.609853\pi\)
−0.338304 + 0.941037i \(0.609853\pi\)
\(480\) 6.16256e10 0.0529877
\(481\) −7.14444e11 −0.608577
\(482\) −7.20880e11 −0.608347
\(483\) −2.42690e11 −0.202904
\(484\) 1.96945e12 1.63132
\(485\) 3.57880e11 0.293697
\(486\) −1.35239e11 −0.109961
\(487\) −1.33958e12 −1.07916 −0.539582 0.841933i \(-0.681418\pi\)
−0.539582 + 0.841933i \(0.681418\pi\)
\(488\) 3.59364e12 2.86844
\(489\) −6.29226e11 −0.497642
\(490\) 2.67265e12 2.09440
\(491\) −9.76349e11 −0.758121 −0.379061 0.925372i \(-0.623753\pi\)
−0.379061 + 0.925372i \(0.623753\pi\)
\(492\) −1.26289e11 −0.0971682
\(493\) −5.08744e11 −0.387872
\(494\) 7.89825e11 0.596705
\(495\) −2.70222e11 −0.202301
\(496\) 3.93637e11 0.292031
\(497\) −8.58495e9 −0.00631152
\(498\) 5.33336e10 0.0388570
\(499\) 1.32057e12 0.953472 0.476736 0.879046i \(-0.341820\pi\)
0.476736 + 0.879046i \(0.341820\pi\)
\(500\) 2.42279e11 0.173361
\(501\) 6.81815e11 0.483501
\(502\) 1.66908e12 1.17303
\(503\) −1.16421e12 −0.810917 −0.405459 0.914113i \(-0.632888\pi\)
−0.405459 + 0.914113i \(0.632888\pi\)
\(504\) 1.50019e12 1.03564
\(505\) 1.11528e12 0.763086
\(506\) −6.24017e11 −0.423174
\(507\) −1.62546e11 −0.109255
\(508\) −3.72054e12 −2.47868
\(509\) 8.32285e11 0.549594 0.274797 0.961502i \(-0.411389\pi\)
0.274797 + 0.961502i \(0.411389\pi\)
\(510\) −1.89747e11 −0.124197
\(511\) −1.09380e12 −0.709649
\(512\) −2.30806e12 −1.48434
\(513\) 9.63671e10 0.0614329
\(514\) −2.43135e10 −0.0153643
\(515\) 7.28949e11 0.456630
\(516\) 1.60366e12 0.995835
\(517\) −1.98327e12 −1.22088
\(518\) 3.02822e12 1.84800
\(519\) −2.91469e10 −0.0176335
\(520\) −1.30773e12 −0.784335
\(521\) 2.43459e12 1.44763 0.723813 0.689997i \(-0.242388\pi\)
0.723813 + 0.689997i \(0.242388\pi\)
\(522\) −1.33972e12 −0.789763
\(523\) −1.07631e12 −0.629043 −0.314522 0.949250i \(-0.601844\pi\)
−0.314522 + 0.949250i \(0.601844\pi\)
\(524\) −1.95099e12 −1.13048
\(525\) −3.88298e11 −0.223073
\(526\) 2.41036e12 1.37292
\(527\) 1.77283e11 0.100119
\(528\) 1.14530e12 0.641308
\(529\) −1.74155e12 −0.966906
\(530\) −6.23747e10 −0.0343374
\(531\) −7.80787e11 −0.426194
\(532\) −2.20835e12 −1.19527
\(533\) −1.76435e11 −0.0946919
\(534\) 9.73494e9 0.00518081
\(535\) −1.07643e12 −0.568059
\(536\) −2.23943e12 −1.17191
\(537\) 7.00923e11 0.363736
\(538\) −2.11337e12 −1.08757
\(539\) −7.26533e12 −3.70771
\(540\) −3.29618e11 −0.166816
\(541\) −3.12532e12 −1.56858 −0.784291 0.620393i \(-0.786973\pi\)
−0.784291 + 0.620393i \(0.786973\pi\)
\(542\) −3.78139e12 −1.88215
\(543\) −4.93867e11 −0.243787
\(544\) −1.17633e11 −0.0575883
\(545\) 1.30293e12 0.632610
\(546\) 4.32974e12 2.08495
\(547\) 1.59061e12 0.759663 0.379832 0.925056i \(-0.375982\pi\)
0.379832 + 0.925056i \(0.375982\pi\)
\(548\) −3.60453e11 −0.170741
\(549\) 1.26546e12 0.594527
\(550\) −9.98411e11 −0.465240
\(551\) 9.54641e11 0.441223
\(552\) −3.68460e11 −0.168913
\(553\) 5.29180e12 2.40625
\(554\) 5.55402e12 2.50503
\(555\) −3.22073e11 −0.144091
\(556\) 4.52106e12 2.00633
\(557\) 3.76617e12 1.65787 0.828937 0.559343i \(-0.188946\pi\)
0.828937 + 0.559343i \(0.188946\pi\)
\(558\) 4.66854e11 0.203858
\(559\) 2.24042e12 0.970458
\(560\) 1.64574e12 0.707158
\(561\) 5.15810e11 0.219865
\(562\) 2.52607e12 1.06815
\(563\) −2.32486e12 −0.975234 −0.487617 0.873058i \(-0.662134\pi\)
−0.487617 + 0.873058i \(0.662134\pi\)
\(564\) −2.41919e12 −1.00673
\(565\) −1.13747e12 −0.469592
\(566\) −6.41643e12 −2.62797
\(567\) 5.28275e11 0.214653
\(568\) −1.30339e10 −0.00525422
\(569\) 3.30244e11 0.132078 0.0660388 0.997817i \(-0.478964\pi\)
0.0660388 + 0.997817i \(0.478964\pi\)
\(570\) 3.56055e11 0.141280
\(571\) 3.83190e12 1.50852 0.754261 0.656575i \(-0.227995\pi\)
0.754261 + 0.656575i \(0.227995\pi\)
\(572\) 7.34388e12 2.86843
\(573\) 2.39953e12 0.929887
\(574\) 7.47832e11 0.287542
\(575\) 9.53690e10 0.0363833
\(576\) −1.03055e12 −0.390094
\(577\) −1.66616e12 −0.625786 −0.312893 0.949788i \(-0.601298\pi\)
−0.312893 + 0.949788i \(0.601298\pi\)
\(578\) −4.23738e12 −1.57915
\(579\) 3.75384e10 0.0138811
\(580\) −3.26529e12 −1.19811
\(581\) −2.08333e11 −0.0758517
\(582\) −1.79896e12 −0.649931
\(583\) 1.69559e11 0.0607874
\(584\) −1.66064e12 −0.590769
\(585\) −4.60499e11 −0.162565
\(586\) 2.18856e12 0.766689
\(587\) 2.00482e12 0.696953 0.348476 0.937318i \(-0.386699\pi\)
0.348476 + 0.937318i \(0.386699\pi\)
\(588\) −8.86226e12 −3.05736
\(589\) −3.32665e11 −0.113891
\(590\) −2.88483e12 −0.980136
\(591\) −9.99654e11 −0.337059
\(592\) 1.36506e12 0.456777
\(593\) 1.46848e12 0.487666 0.243833 0.969817i \(-0.421595\pi\)
0.243833 + 0.969817i \(0.421595\pi\)
\(594\) 1.35833e12 0.447678
\(595\) 7.41195e11 0.242441
\(596\) 8.73121e12 2.83443
\(597\) 8.12868e11 0.261900
\(598\) −1.06342e12 −0.340055
\(599\) 5.67302e12 1.80050 0.900251 0.435370i \(-0.143383\pi\)
0.900251 + 0.435370i \(0.143383\pi\)
\(600\) −5.89525e11 −0.185704
\(601\) 2.23747e12 0.699555 0.349778 0.936833i \(-0.386257\pi\)
0.349778 + 0.936833i \(0.386257\pi\)
\(602\) −9.49617e12 −2.94689
\(603\) −7.88587e11 −0.242897
\(604\) −2.68638e11 −0.0821299
\(605\) 1.24036e12 0.376400
\(606\) −5.60619e12 −1.68865
\(607\) 7.87986e11 0.235597 0.117798 0.993038i \(-0.462416\pi\)
0.117798 + 0.993038i \(0.462416\pi\)
\(608\) 2.20734e11 0.0655094
\(609\) 5.23324e12 1.54168
\(610\) 4.67557e12 1.36726
\(611\) −3.37979e12 −0.981079
\(612\) 6.29185e11 0.181300
\(613\) −4.35930e12 −1.24694 −0.623468 0.781848i \(-0.714277\pi\)
−0.623468 + 0.781848i \(0.714277\pi\)
\(614\) 6.95671e11 0.197536
\(615\) −7.95374e10 −0.0224199
\(616\) −1.50678e13 −4.21634
\(617\) −2.04791e12 −0.568889 −0.284444 0.958693i \(-0.591809\pi\)
−0.284444 + 0.958693i \(0.591809\pi\)
\(618\) −3.66421e12 −1.01049
\(619\) −6.31079e12 −1.72773 −0.863865 0.503723i \(-0.831963\pi\)
−0.863865 + 0.503723i \(0.831963\pi\)
\(620\) 1.13786e12 0.309262
\(621\) −1.29748e11 −0.0350098
\(622\) 1.24098e13 3.32435
\(623\) −3.80268e10 −0.0101133
\(624\) 1.95176e12 0.515342
\(625\) 1.52588e11 0.0400000
\(626\) −1.22947e13 −3.19989
\(627\) −9.67899e11 −0.250107
\(628\) 9.65818e12 2.47786
\(629\) 6.14784e11 0.156601
\(630\) 1.95186e12 0.493646
\(631\) 4.94106e12 1.24076 0.620380 0.784301i \(-0.286978\pi\)
0.620380 + 0.784301i \(0.286978\pi\)
\(632\) 8.03418e12 2.00316
\(633\) 2.88577e12 0.714405
\(634\) −1.19523e13 −2.93798
\(635\) −2.34321e12 −0.571912
\(636\) 2.06829e11 0.0501249
\(637\) −1.23812e13 −2.97945
\(638\) 1.34560e13 3.21531
\(639\) −4.58974e9 −0.00108902
\(640\) −3.41813e12 −0.805338
\(641\) −6.55819e11 −0.153434 −0.0767172 0.997053i \(-0.524444\pi\)
−0.0767172 + 0.997053i \(0.524444\pi\)
\(642\) 5.41089e12 1.25707
\(643\) 5.76324e11 0.132959 0.0664794 0.997788i \(-0.478823\pi\)
0.0664794 + 0.997788i \(0.478823\pi\)
\(644\) 2.97332e12 0.681171
\(645\) 1.00999e12 0.229772
\(646\) −6.79649e11 −0.153546
\(647\) −3.52372e12 −0.790556 −0.395278 0.918562i \(-0.629352\pi\)
−0.395278 + 0.918562i \(0.629352\pi\)
\(648\) 8.02043e11 0.178694
\(649\) 7.84212e12 1.73513
\(650\) −1.70144e12 −0.373858
\(651\) −1.82364e12 −0.397945
\(652\) 7.70899e12 1.67064
\(653\) −4.37841e11 −0.0942338 −0.0471169 0.998889i \(-0.515003\pi\)
−0.0471169 + 0.998889i \(0.515003\pi\)
\(654\) −6.54942e12 −1.39992
\(655\) −1.22874e12 −0.260840
\(656\) 3.37108e11 0.0710725
\(657\) −5.84774e11 −0.122446
\(658\) 1.43254e13 2.97914
\(659\) 1.98477e12 0.409945 0.204972 0.978768i \(-0.434290\pi\)
0.204972 + 0.978768i \(0.434290\pi\)
\(660\) 3.31064e12 0.679148
\(661\) 2.80945e12 0.572420 0.286210 0.958167i \(-0.407604\pi\)
0.286210 + 0.958167i \(0.407604\pi\)
\(662\) −6.94988e12 −1.40642
\(663\) 8.79016e11 0.176679
\(664\) −3.16297e11 −0.0631450
\(665\) −1.39083e12 −0.275788
\(666\) 1.61896e12 0.318862
\(667\) −1.28533e12 −0.251447
\(668\) −8.35328e12 −1.62317
\(669\) 2.76733e12 0.534126
\(670\) −2.91365e12 −0.558600
\(671\) −1.27101e13 −2.42045
\(672\) 1.21004e12 0.228896
\(673\) −2.08224e12 −0.391258 −0.195629 0.980678i \(-0.562675\pi\)
−0.195629 + 0.980678i \(0.562675\pi\)
\(674\) −3.77526e12 −0.704657
\(675\) −2.07594e11 −0.0384900
\(676\) 1.99144e12 0.366781
\(677\) −6.44805e12 −1.17972 −0.589861 0.807505i \(-0.700817\pi\)
−0.589861 + 0.807505i \(0.700817\pi\)
\(678\) 5.71770e12 1.03917
\(679\) 7.02712e12 1.26871
\(680\) 1.12531e12 0.201828
\(681\) −2.95598e12 −0.526671
\(682\) −4.68902e12 −0.829952
\(683\) −4.02024e12 −0.706902 −0.353451 0.935453i \(-0.614992\pi\)
−0.353451 + 0.935453i \(0.614992\pi\)
\(684\) −1.18064e12 −0.206237
\(685\) −2.27015e11 −0.0393955
\(686\) 3.32706e13 5.73591
\(687\) −3.13832e12 −0.537516
\(688\) −4.28069e12 −0.728392
\(689\) 2.88954e11 0.0488476
\(690\) −4.79391e11 −0.0805136
\(691\) −8.43146e12 −1.40686 −0.703431 0.710763i \(-0.748350\pi\)
−0.703431 + 0.710763i \(0.748350\pi\)
\(692\) 3.57094e11 0.0591978
\(693\) −5.30592e12 −0.873900
\(694\) −1.34879e13 −2.20713
\(695\) 2.84737e12 0.462927
\(696\) 7.94527e12 1.28342
\(697\) 1.51824e11 0.0243664
\(698\) −4.27159e12 −0.681145
\(699\) 2.17661e12 0.344853
\(700\) 4.75724e12 0.748883
\(701\) 6.55956e12 1.02599 0.512995 0.858391i \(-0.328536\pi\)
0.512995 + 0.858391i \(0.328536\pi\)
\(702\) 2.31479e12 0.359745
\(703\) −1.15362e12 −0.178141
\(704\) 1.03507e13 1.58816
\(705\) −1.52362e12 −0.232287
\(706\) −1.31894e13 −1.99803
\(707\) 2.18990e13 3.29638
\(708\) 9.56583e12 1.43078
\(709\) −5.30959e12 −0.789138 −0.394569 0.918866i \(-0.629106\pi\)
−0.394569 + 0.918866i \(0.629106\pi\)
\(710\) −1.69581e10 −0.00250446
\(711\) 2.82914e12 0.415184
\(712\) −5.77335e10 −0.00841914
\(713\) 4.47899e11 0.0649049
\(714\) −3.72577e12 −0.536505
\(715\) 4.62520e12 0.661840
\(716\) −8.58739e12 −1.22110
\(717\) −1.09116e12 −0.154189
\(718\) −1.06687e13 −1.49813
\(719\) 1.02053e13 1.42411 0.712056 0.702123i \(-0.247764\pi\)
0.712056 + 0.702123i \(0.247764\pi\)
\(720\) 8.79858e11 0.122016
\(721\) 1.43132e13 1.97255
\(722\) −1.12405e13 −1.53946
\(723\) 1.50546e12 0.204903
\(724\) 6.05063e12 0.818422
\(725\) −2.05649e12 −0.276443
\(726\) −6.23493e12 −0.832946
\(727\) −7.45813e12 −0.990206 −0.495103 0.868834i \(-0.664870\pi\)
−0.495103 + 0.868834i \(0.664870\pi\)
\(728\) −2.56777e13 −3.38817
\(729\) 2.82430e11 0.0370370
\(730\) −2.16061e12 −0.281594
\(731\) −1.92790e12 −0.249721
\(732\) −1.55038e13 −1.99589
\(733\) −1.06610e13 −1.36405 −0.682024 0.731329i \(-0.738900\pi\)
−0.682024 + 0.731329i \(0.738900\pi\)
\(734\) −8.23114e12 −1.04671
\(735\) −5.58148e12 −0.705433
\(736\) −2.97196e11 −0.0373330
\(737\) 7.92047e12 0.988889
\(738\) 3.99811e11 0.0496136
\(739\) 2.33179e12 0.287601 0.143800 0.989607i \(-0.454068\pi\)
0.143800 + 0.989607i \(0.454068\pi\)
\(740\) 3.94589e12 0.483729
\(741\) −1.64944e12 −0.200981
\(742\) −1.22475e12 −0.148330
\(743\) −7.80869e12 −0.940001 −0.470001 0.882666i \(-0.655746\pi\)
−0.470001 + 0.882666i \(0.655746\pi\)
\(744\) −2.76870e12 −0.331282
\(745\) 5.49894e12 0.653997
\(746\) 2.35946e13 2.78925
\(747\) −1.11380e11 −0.0130878
\(748\) −6.31946e12 −0.738113
\(749\) −2.11361e13 −2.45390
\(750\) −7.67013e11 −0.0885171
\(751\) 4.20664e12 0.482565 0.241283 0.970455i \(-0.422432\pi\)
0.241283 + 0.970455i \(0.422432\pi\)
\(752\) 6.45763e12 0.736364
\(753\) −3.48565e12 −0.395099
\(754\) 2.29310e13 2.58376
\(755\) −1.69189e11 −0.0189501
\(756\) −6.47217e12 −0.720613
\(757\) −1.54834e13 −1.71370 −0.856848 0.515570i \(-0.827580\pi\)
−0.856848 + 0.515570i \(0.827580\pi\)
\(758\) 1.80021e13 1.98067
\(759\) 1.30318e12 0.142533
\(760\) −2.11160e12 −0.229589
\(761\) 9.77615e12 1.05666 0.528332 0.849038i \(-0.322818\pi\)
0.528332 + 0.849038i \(0.322818\pi\)
\(762\) 1.17786e13 1.26560
\(763\) 2.55835e13 2.73275
\(764\) −2.93979e13 −3.12174
\(765\) 3.96263e11 0.0418318
\(766\) 2.02755e13 2.12785
\(767\) 1.33641e13 1.39432
\(768\) 1.06678e13 1.10649
\(769\) 3.35411e12 0.345867 0.172933 0.984934i \(-0.444675\pi\)
0.172933 + 0.984934i \(0.444675\pi\)
\(770\) −1.96042e13 −2.00974
\(771\) 5.07755e10 0.00517499
\(772\) −4.59904e11 −0.0466003
\(773\) −6.04474e12 −0.608933 −0.304467 0.952523i \(-0.598478\pi\)
−0.304467 + 0.952523i \(0.598478\pi\)
\(774\) −5.07690e12 −0.508469
\(775\) 7.16627e11 0.0713568
\(776\) 1.06688e13 1.05618
\(777\) −6.32403e12 −0.622443
\(778\) −4.61163e12 −0.451280
\(779\) −2.84892e11 −0.0277180
\(780\) 5.64182e12 0.545750
\(781\) 4.60988e10 0.00443364
\(782\) 9.15078e11 0.0875040
\(783\) 2.79783e12 0.266007
\(784\) 2.36563e13 2.23627
\(785\) 6.08275e12 0.571724
\(786\) 6.17651e12 0.577220
\(787\) 4.16503e12 0.387019 0.193509 0.981098i \(-0.438013\pi\)
0.193509 + 0.981098i \(0.438013\pi\)
\(788\) 1.22473e13 1.13155
\(789\) −5.03371e12 −0.462425
\(790\) 1.04530e13 0.954817
\(791\) −2.23346e13 −2.02854
\(792\) −8.05562e12 −0.727504
\(793\) −2.16599e13 −1.94503
\(794\) −1.07420e13 −0.959164
\(795\) 1.30261e11 0.0115655
\(796\) −9.95887e12 −0.879228
\(797\) 2.14334e13 1.88160 0.940802 0.338957i \(-0.110074\pi\)
0.940802 + 0.338957i \(0.110074\pi\)
\(798\) 6.99128e12 0.610300
\(799\) 2.90833e12 0.252454
\(800\) −4.75506e11 −0.0410441
\(801\) −2.03301e10 −0.00174499
\(802\) −5.95301e12 −0.508103
\(803\) 5.87339e12 0.498504
\(804\) 9.66140e12 0.815432
\(805\) 1.87261e12 0.157168
\(806\) −7.99080e12 −0.666933
\(807\) 4.41350e12 0.366313
\(808\) 3.32478e13 2.74417
\(809\) −1.82139e11 −0.0149498 −0.00747490 0.999972i \(-0.502379\pi\)
−0.00747490 + 0.999972i \(0.502379\pi\)
\(810\) 1.04351e12 0.0851756
\(811\) −1.32191e13 −1.07302 −0.536511 0.843894i \(-0.680258\pi\)
−0.536511 + 0.843894i \(0.680258\pi\)
\(812\) −6.41153e13 −5.17558
\(813\) 7.89693e12 0.633944
\(814\) −1.62607e13 −1.29816
\(815\) 4.85514e12 0.385472
\(816\) −1.67950e12 −0.132610
\(817\) 3.61763e12 0.284070
\(818\) −5.90963e12 −0.461498
\(819\) −9.04209e12 −0.702249
\(820\) 9.74455e11 0.0752661
\(821\) −1.15652e13 −0.888398 −0.444199 0.895928i \(-0.646512\pi\)
−0.444199 + 0.895928i \(0.646512\pi\)
\(822\) 1.14113e12 0.0871793
\(823\) 9.35914e12 0.711110 0.355555 0.934655i \(-0.384292\pi\)
0.355555 + 0.934655i \(0.384292\pi\)
\(824\) 2.17307e13 1.64211
\(825\) 2.08505e12 0.156702
\(826\) −5.66448e13 −4.23399
\(827\) −2.25652e13 −1.67751 −0.838753 0.544512i \(-0.816715\pi\)
−0.838753 + 0.544512i \(0.816715\pi\)
\(828\) 1.58962e12 0.117532
\(829\) 1.22484e13 0.900705 0.450353 0.892851i \(-0.351298\pi\)
0.450353 + 0.892851i \(0.351298\pi\)
\(830\) −4.11525e11 −0.0300985
\(831\) −1.15988e13 −0.843742
\(832\) 1.76392e13 1.27622
\(833\) 1.06541e13 0.766681
\(834\) −1.43129e13 −1.02442
\(835\) −5.26092e12 −0.374518
\(836\) 1.18582e13 0.839638
\(837\) −9.74963e11 −0.0686631
\(838\) −3.52887e13 −2.47194
\(839\) 4.67277e12 0.325571 0.162786 0.986661i \(-0.447952\pi\)
0.162786 + 0.986661i \(0.447952\pi\)
\(840\) −1.15756e13 −0.802205
\(841\) 1.32090e13 0.910516
\(842\) 3.83414e13 2.62883
\(843\) −5.27537e12 −0.359773
\(844\) −3.53551e13 −2.39834
\(845\) 1.25421e12 0.0846285
\(846\) 7.65876e12 0.514034
\(847\) 2.43550e13 1.62597
\(848\) −5.52094e11 −0.0366633
\(849\) 1.33999e13 0.885149
\(850\) 1.46410e12 0.0962024
\(851\) 1.55323e12 0.101520
\(852\) 5.62314e10 0.00365595
\(853\) 4.04678e12 0.261721 0.130861 0.991401i \(-0.458226\pi\)
0.130861 + 0.991401i \(0.458226\pi\)
\(854\) 9.18067e13 5.90628
\(855\) −7.43573e11 −0.0475857
\(856\) −3.20895e13 −2.04282
\(857\) −2.35372e13 −1.49053 −0.745267 0.666766i \(-0.767678\pi\)
−0.745267 + 0.666766i \(0.767678\pi\)
\(858\) −2.32495e13 −1.46460
\(859\) 2.19727e13 1.37694 0.688470 0.725265i \(-0.258283\pi\)
0.688470 + 0.725265i \(0.258283\pi\)
\(860\) −1.23739e13 −0.771371
\(861\) −1.56175e12 −0.0968494
\(862\) 7.88441e12 0.486392
\(863\) −2.06939e13 −1.26997 −0.634984 0.772525i \(-0.718993\pi\)
−0.634984 + 0.772525i \(0.718993\pi\)
\(864\) 6.46921e11 0.0394947
\(865\) 2.24899e11 0.0136589
\(866\) −8.17796e12 −0.494099
\(867\) 8.84922e12 0.531887
\(868\) 2.23423e13 1.33595
\(869\) −2.84155e13 −1.69031
\(870\) 1.03374e13 0.611748
\(871\) 1.34977e13 0.794652
\(872\) 3.88416e13 2.27496
\(873\) 3.75689e12 0.218909
\(874\) −1.71711e12 −0.0995400
\(875\) 2.99612e12 0.172792
\(876\) 7.16437e12 0.411064
\(877\) −8.18344e12 −0.467130 −0.233565 0.972341i \(-0.575039\pi\)
−0.233565 + 0.972341i \(0.575039\pi\)
\(878\) 8.19991e12 0.465676
\(879\) −4.57052e12 −0.258235
\(880\) −8.83719e12 −0.496755
\(881\) 2.43747e13 1.36316 0.681581 0.731743i \(-0.261293\pi\)
0.681581 + 0.731743i \(0.261293\pi\)
\(882\) 2.80564e13 1.56107
\(883\) −1.17064e13 −0.648039 −0.324019 0.946050i \(-0.605034\pi\)
−0.324019 + 0.946050i \(0.605034\pi\)
\(884\) −1.07693e13 −0.593133
\(885\) 6.02459e12 0.330128
\(886\) 5.64407e13 3.07709
\(887\) 2.83247e13 1.53642 0.768210 0.640198i \(-0.221148\pi\)
0.768210 + 0.640198i \(0.221148\pi\)
\(888\) −9.60134e12 −0.518171
\(889\) −4.60098e13 −2.47055
\(890\) −7.51153e10 −0.00401304
\(891\) −2.83669e12 −0.150786
\(892\) −3.39040e13 −1.79312
\(893\) −5.45738e12 −0.287179
\(894\) −2.76415e13 −1.44725
\(895\) −5.40836e12 −0.281749
\(896\) −6.71163e13 −3.47890
\(897\) 2.22081e12 0.114537
\(898\) −8.51244e12 −0.436828
\(899\) −9.65827e12 −0.493152
\(900\) 2.54335e12 0.129215
\(901\) −2.48647e11 −0.0125696
\(902\) −4.01565e12 −0.201988
\(903\) 1.98315e13 0.992569
\(904\) −3.39091e13 −1.68872
\(905\) 3.81070e12 0.188837
\(906\) 8.50462e11 0.0419351
\(907\) 5.36313e12 0.263139 0.131570 0.991307i \(-0.457998\pi\)
0.131570 + 0.991307i \(0.457998\pi\)
\(908\) 3.62153e13 1.76809
\(909\) 1.17078e13 0.568771
\(910\) −3.34085e13 −1.61499
\(911\) 3.43721e13 1.65338 0.826691 0.562656i \(-0.190220\pi\)
0.826691 + 0.562656i \(0.190220\pi\)
\(912\) 3.15153e12 0.150850
\(913\) 1.11869e12 0.0532833
\(914\) 2.87335e13 1.36185
\(915\) −9.76432e12 −0.460519
\(916\) 3.84492e13 1.80450
\(917\) −2.41268e13 −1.12678
\(918\) −1.99189e12 −0.0925708
\(919\) −2.98037e13 −1.37832 −0.689160 0.724609i \(-0.742020\pi\)
−0.689160 + 0.724609i \(0.742020\pi\)
\(920\) 2.84305e12 0.130840
\(921\) −1.45282e12 −0.0665339
\(922\) 1.08668e13 0.495234
\(923\) 7.85592e10 0.00356278
\(924\) 6.50057e13 2.93378
\(925\) 2.48513e12 0.111612
\(926\) −1.96682e12 −0.0879053
\(927\) 7.65221e12 0.340352
\(928\) 6.40859e12 0.283659
\(929\) −2.03150e13 −0.894843 −0.447421 0.894323i \(-0.647658\pi\)
−0.447421 + 0.894323i \(0.647658\pi\)
\(930\) −3.60227e12 −0.157908
\(931\) −1.99921e13 −0.872136
\(932\) −2.66668e13 −1.15771
\(933\) −2.59162e13 −1.11970
\(934\) 6.87399e13 2.95562
\(935\) −3.98001e12 −0.170307
\(936\) −1.37280e13 −0.584609
\(937\) 2.73553e13 1.15935 0.579674 0.814848i \(-0.303180\pi\)
0.579674 + 0.814848i \(0.303180\pi\)
\(938\) −5.72107e13 −2.41304
\(939\) 2.56759e13 1.07778
\(940\) 1.86666e13 0.779813
\(941\) −4.24123e13 −1.76335 −0.881675 0.471857i \(-0.843584\pi\)
−0.881675 + 0.471857i \(0.843584\pi\)
\(942\) −3.05761e13 −1.26518
\(943\) 3.83578e11 0.0157961
\(944\) −2.55344e13 −1.04653
\(945\) −4.07619e12 −0.166269
\(946\) 5.09918e13 2.07009
\(947\) 9.23741e12 0.373229 0.186614 0.982433i \(-0.440248\pi\)
0.186614 + 0.982433i \(0.440248\pi\)
\(948\) −3.46613e13 −1.39382
\(949\) 1.00091e13 0.400589
\(950\) −2.74734e12 −0.109435
\(951\) 2.49607e13 0.989566
\(952\) 2.20958e13 0.871855
\(953\) 1.11343e13 0.437265 0.218633 0.975807i \(-0.429840\pi\)
0.218633 + 0.975807i \(0.429840\pi\)
\(954\) −6.54784e11 −0.0255936
\(955\) −1.85149e13 −0.720288
\(956\) 1.33684e13 0.517629
\(957\) −2.81011e13 −1.08298
\(958\) −3.02360e13 −1.15979
\(959\) −4.45752e12 −0.170180
\(960\) 7.95180e12 0.302165
\(961\) −2.30740e13 −0.872705
\(962\) −2.77106e13 −1.04318
\(963\) −1.12999e13 −0.423406
\(964\) −1.84442e13 −0.687882
\(965\) −2.89649e11 −0.0107522
\(966\) −9.41304e12 −0.347803
\(967\) −5.48630e12 −0.201772 −0.100886 0.994898i \(-0.532168\pi\)
−0.100886 + 0.994898i \(0.532168\pi\)
\(968\) 3.69766e13 1.35359
\(969\) 1.41936e12 0.0517172
\(970\) 1.38808e13 0.503435
\(971\) −2.07887e13 −0.750484 −0.375242 0.926927i \(-0.622440\pi\)
−0.375242 + 0.926927i \(0.622440\pi\)
\(972\) −3.46019e12 −0.124338
\(973\) 5.59093e13 1.99975
\(974\) −5.19572e13 −1.84982
\(975\) 3.55324e12 0.125922
\(976\) 4.13847e13 1.45987
\(977\) 2.25372e13 0.791362 0.395681 0.918388i \(-0.370509\pi\)
0.395681 + 0.918388i \(0.370509\pi\)
\(978\) −2.44053e13 −0.853022
\(979\) 2.04193e11 0.00710427
\(980\) 6.83816e13 2.36822
\(981\) 1.36776e13 0.471519
\(982\) −3.78689e13 −1.29952
\(983\) −2.20727e13 −0.753989 −0.376994 0.926216i \(-0.623042\pi\)
−0.376994 + 0.926216i \(0.623042\pi\)
\(984\) −2.37110e12 −0.0806253
\(985\) 7.71338e12 0.261085
\(986\) −1.97323e13 −0.664861
\(987\) −2.99168e13 −1.00343
\(988\) 2.02082e13 0.674717
\(989\) −4.87077e12 −0.161888
\(990\) −1.04809e13 −0.346770
\(991\) −5.27788e13 −1.73831 −0.869157 0.494536i \(-0.835338\pi\)
−0.869157 + 0.494536i \(0.835338\pi\)
\(992\) −2.23321e12 −0.0732195
\(993\) 1.45139e13 0.473710
\(994\) −3.32978e11 −0.0108188
\(995\) −6.27213e12 −0.202867
\(996\) 1.36458e12 0.0439371
\(997\) −4.19157e13 −1.34353 −0.671767 0.740763i \(-0.734464\pi\)
−0.671767 + 0.740763i \(0.734464\pi\)
\(998\) 5.12199e13 1.63437
\(999\) −3.38099e12 −0.107399
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.d.1.2 2
3.2 odd 2 45.10.a.d.1.1 2
4.3 odd 2 240.10.a.r.1.1 2
5.2 odd 4 75.10.b.f.49.4 4
5.3 odd 4 75.10.b.f.49.1 4
5.4 even 2 75.10.a.f.1.1 2
15.2 even 4 225.10.b.i.199.1 4
15.8 even 4 225.10.b.i.199.4 4
15.14 odd 2 225.10.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.d.1.2 2 1.1 even 1 trivial
45.10.a.d.1.1 2 3.2 odd 2
75.10.a.f.1.1 2 5.4 even 2
75.10.b.f.49.1 4 5.3 odd 4
75.10.b.f.49.4 4 5.2 odd 4
225.10.a.k.1.2 2 15.14 odd 2
225.10.b.i.199.1 4 15.2 even 4
225.10.b.i.199.4 4 15.8 even 4
240.10.a.r.1.1 2 4.3 odd 2