Properties

Label 15.10.a.d.1.1
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 1.1
Root \(8.26209\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.78626 q^{2} -81.0000 q^{3} -451.374 q^{4} +625.000 q^{5} +630.687 q^{6} +1839.88 q^{7} +7501.08 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-7.78626 q^{2} -81.0000 q^{3} -451.374 q^{4} +625.000 q^{5} +630.687 q^{6} +1839.88 q^{7} +7501.08 q^{8} +6561.00 q^{9} -4866.41 q^{10} +44385.9 q^{11} +36561.3 q^{12} +136584. q^{13} -14325.8 q^{14} -50625.0 q^{15} +172698. q^{16} -253591. q^{17} -51085.7 q^{18} +85435.7 q^{19} -282109. q^{20} -149030. q^{21} -345600. q^{22} -979409. q^{23} -607588. q^{24} +390625. q^{25} -1.06348e6 q^{26} -531441. q^{27} -830473. q^{28} +2.58640e6 q^{29} +394180. q^{30} +8.94787e6 q^{31} -5.18523e6 q^{32} -3.59526e6 q^{33} +1.97452e6 q^{34} +1.14992e6 q^{35} -2.96147e6 q^{36} +1.56064e7 q^{37} -665225. q^{38} -1.10633e7 q^{39} +4.68818e6 q^{40} +2.44893e7 q^{41} +1.16039e6 q^{42} +1.27592e7 q^{43} -2.00346e7 q^{44} +4.10063e6 q^{45} +7.62593e6 q^{46} -6.16764e7 q^{47} -1.39886e7 q^{48} -3.69685e7 q^{49} -3.04151e6 q^{50} +2.05408e7 q^{51} -6.16504e7 q^{52} +5.70418e6 q^{53} +4.13794e6 q^{54} +2.77412e7 q^{55} +1.38011e7 q^{56} -6.92029e6 q^{57} -2.01384e7 q^{58} +8.35095e7 q^{59} +2.28508e7 q^{60} +1.48622e8 q^{61} -6.96704e7 q^{62} +1.20714e7 q^{63} -4.80479e7 q^{64} +8.53649e7 q^{65} +2.79936e7 q^{66} -1.68003e8 q^{67} +1.14464e8 q^{68} +7.93321e7 q^{69} -8.95360e6 q^{70} +2.10986e8 q^{71} +4.92146e7 q^{72} -1.43534e8 q^{73} -1.21515e8 q^{74} -3.16406e7 q^{75} -3.85635e7 q^{76} +8.16646e7 q^{77} +8.61416e7 q^{78} -4.55960e8 q^{79} +1.07936e8 q^{80} +4.30467e7 q^{81} -1.90680e8 q^{82} -3.55106e8 q^{83} +6.72683e7 q^{84} -1.58494e8 q^{85} -9.93465e7 q^{86} -2.09498e8 q^{87} +3.32942e8 q^{88} -4.24540e8 q^{89} -3.19285e7 q^{90} +2.51297e8 q^{91} +4.42080e8 q^{92} -7.24777e8 q^{93} +4.80228e8 q^{94} +5.33973e7 q^{95} +4.20003e8 q^{96} +1.19905e9 q^{97} +2.87846e8 q^{98} +2.91216e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + O(q^{10}) \) \( 2q + 31q^{2} - 162q^{3} + 541q^{4} + 1250q^{5} - 2511q^{6} + 14112q^{7} + 26133q^{8} + 13122q^{9} + 19375q^{10} - 21512q^{11} - 43821q^{12} + 24284q^{13} + 461664q^{14} - 101250q^{15} + 387265q^{16} - 156956q^{17} + 203391q^{18} - 95896q^{19} + 338125q^{20} - 1143072q^{21} - 2901532q^{22} - 735264q^{23} - 2116773q^{24} + 781250q^{25} - 5419166q^{26} - 1062882q^{27} + 11348064q^{28} - 2678212q^{29} - 1569375q^{30} + 10782432q^{31} - 6402523q^{32} + 1742472q^{33} + 5722622q^{34} + 8820000q^{35} + 3549501q^{36} + 21968332q^{37} - 7698404q^{38} - 1967004q^{39} + 16333125q^{40} + 26060372q^{41} - 37394784q^{42} - 7191160q^{43} - 85429972q^{44} + 8201250q^{45} + 17095392q^{46} - 31580240q^{47} - 31368465q^{48} + 73282930q^{49} + 12109375q^{50} + 12713436q^{51} - 173093786q^{52} + 3131116q^{53} - 16474671q^{54} - 13445000q^{55} + 242454240q^{56} + 7767576q^{57} - 224332958q^{58} - 35494664q^{59} - 27388125q^{60} + 341497340q^{61} + 1485504q^{62} + 92588832q^{63} - 205120471q^{64} + 15177500q^{65} + 235024092q^{66} - 288195816q^{67} + 210362042q^{68} + 59556384q^{69} + 288540000q^{70} + 210286064q^{71} + 171458613q^{72} - 232663084q^{73} + 125240234q^{74} - 63281250q^{75} - 218512364q^{76} - 727042176q^{77} + 438952446q^{78} - 24755040q^{79} + 242040625q^{80} + 86093442q^{81} - 129742346q^{82} - 372082152q^{83} - 919193184q^{84} - 98097500q^{85} - 873146372q^{86} + 216935172q^{87} - 894861492q^{88} - 427639116q^{89} + 127119375q^{90} - 1126859328q^{91} + 684362592q^{92} - 873376992q^{93} + 1647543896q^{94} - 59935000q^{95} + 518604363q^{96} + 1771658884q^{97} + 4564085351q^{98} - 141140232q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.78626 −0.344107 −0.172054 0.985088i \(-0.555040\pi\)
−0.172054 + 0.985088i \(0.555040\pi\)
\(3\) −81.0000 −0.577350
\(4\) −451.374 −0.881590
\(5\) 625.000 0.447214
\(6\) 630.687 0.198671
\(7\) 1839.88 0.289633 0.144816 0.989459i \(-0.453741\pi\)
0.144816 + 0.989459i \(0.453741\pi\)
\(8\) 7501.08 0.647469
\(9\) 6561.00 0.333333
\(10\) −4866.41 −0.153890
\(11\) 44385.9 0.914066 0.457033 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(12\) 36561.3 0.508986
\(13\) 136584. 1.32634 0.663169 0.748470i \(-0.269211\pi\)
0.663169 + 0.748470i \(0.269211\pi\)
\(14\) −14325.8 −0.0996648
\(15\) −50625.0 −0.258199
\(16\) 172698. 0.658791
\(17\) −253591. −0.736399 −0.368199 0.929747i \(-0.620026\pi\)
−0.368199 + 0.929747i \(0.620026\pi\)
\(18\) −51085.7 −0.114702
\(19\) 85435.7 0.150400 0.0752001 0.997168i \(-0.476040\pi\)
0.0752001 + 0.997168i \(0.476040\pi\)
\(20\) −282109. −0.394259
\(21\) −149030. −0.167220
\(22\) −345600. −0.314537
\(23\) −979409. −0.729775 −0.364887 0.931052i \(-0.618892\pi\)
−0.364887 + 0.931052i \(0.618892\pi\)
\(24\) −607588. −0.373816
\(25\) 390625. 0.200000
\(26\) −1.06348e6 −0.456403
\(27\) −531441. −0.192450
\(28\) −830473. −0.255337
\(29\) 2.58640e6 0.679054 0.339527 0.940596i \(-0.389733\pi\)
0.339527 + 0.940596i \(0.389733\pi\)
\(30\) 394180. 0.0888482
\(31\) 8.94787e6 1.74017 0.870085 0.492901i \(-0.164064\pi\)
0.870085 + 0.492901i \(0.164064\pi\)
\(32\) −5.18523e6 −0.874164
\(33\) −3.59526e6 −0.527736
\(34\) 1.97452e6 0.253400
\(35\) 1.14992e6 0.129528
\(36\) −2.96147e6 −0.293863
\(37\) 1.56064e7 1.36897 0.684486 0.729026i \(-0.260026\pi\)
0.684486 + 0.729026i \(0.260026\pi\)
\(38\) −665225. −0.0517538
\(39\) −1.10633e7 −0.765761
\(40\) 4.68818e6 0.289557
\(41\) 2.44893e7 1.35347 0.676735 0.736227i \(-0.263394\pi\)
0.676735 + 0.736227i \(0.263394\pi\)
\(42\) 1.16039e6 0.0575415
\(43\) 1.27592e7 0.569135 0.284568 0.958656i \(-0.408150\pi\)
0.284568 + 0.958656i \(0.408150\pi\)
\(44\) −2.00346e7 −0.805832
\(45\) 4.10063e6 0.149071
\(46\) 7.62593e6 0.251121
\(47\) −6.16764e7 −1.84365 −0.921825 0.387607i \(-0.873302\pi\)
−0.921825 + 0.387607i \(0.873302\pi\)
\(48\) −1.39886e7 −0.380353
\(49\) −3.69685e7 −0.916113
\(50\) −3.04151e6 −0.0688215
\(51\) 2.05408e7 0.425160
\(52\) −6.16504e7 −1.16929
\(53\) 5.70418e6 0.0993006 0.0496503 0.998767i \(-0.484189\pi\)
0.0496503 + 0.998767i \(0.484189\pi\)
\(54\) 4.13794e6 0.0662235
\(55\) 2.77412e7 0.408783
\(56\) 1.38011e7 0.187528
\(57\) −6.92029e6 −0.0868336
\(58\) −2.01384e7 −0.233668
\(59\) 8.35095e7 0.897226 0.448613 0.893726i \(-0.351918\pi\)
0.448613 + 0.893726i \(0.351918\pi\)
\(60\) 2.28508e7 0.227626
\(61\) 1.48622e8 1.37435 0.687177 0.726490i \(-0.258850\pi\)
0.687177 + 0.726490i \(0.258850\pi\)
\(62\) −6.96704e7 −0.598806
\(63\) 1.20714e7 0.0965443
\(64\) −4.80479e7 −0.357985
\(65\) 8.53649e7 0.593156
\(66\) 2.79936e7 0.181598
\(67\) −1.68003e8 −1.01854 −0.509272 0.860606i \(-0.670085\pi\)
−0.509272 + 0.860606i \(0.670085\pi\)
\(68\) 1.14464e8 0.649202
\(69\) 7.93321e7 0.421336
\(70\) −8.95360e6 −0.0445714
\(71\) 2.10986e8 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(72\) 4.92146e7 0.215823
\(73\) −1.43534e8 −0.591566 −0.295783 0.955255i \(-0.595580\pi\)
−0.295783 + 0.955255i \(0.595580\pi\)
\(74\) −1.21515e8 −0.471074
\(75\) −3.16406e7 −0.115470
\(76\) −3.85635e7 −0.132591
\(77\) 8.16646e7 0.264744
\(78\) 8.61416e7 0.263504
\(79\) −4.55960e8 −1.31706 −0.658529 0.752555i \(-0.728821\pi\)
−0.658529 + 0.752555i \(0.728821\pi\)
\(80\) 1.07936e8 0.294620
\(81\) 4.30467e7 0.111111
\(82\) −1.90680e8 −0.465739
\(83\) −3.55106e8 −0.821309 −0.410655 0.911791i \(-0.634700\pi\)
−0.410655 + 0.911791i \(0.634700\pi\)
\(84\) 6.72683e7 0.147419
\(85\) −1.58494e8 −0.329328
\(86\) −9.93465e7 −0.195844
\(87\) −2.09498e8 −0.392052
\(88\) 3.32942e8 0.591830
\(89\) −4.24540e8 −0.717239 −0.358620 0.933484i \(-0.616752\pi\)
−0.358620 + 0.933484i \(0.616752\pi\)
\(90\) −3.19285e7 −0.0512965
\(91\) 2.51297e8 0.384151
\(92\) 4.42080e8 0.643362
\(93\) −7.24777e8 −1.00469
\(94\) 4.80228e8 0.634413
\(95\) 5.33973e7 0.0672610
\(96\) 4.20003e8 0.504699
\(97\) 1.19905e9 1.37520 0.687598 0.726092i \(-0.258665\pi\)
0.687598 + 0.726092i \(0.258665\pi\)
\(98\) 2.87846e8 0.315241
\(99\) 2.91216e8 0.304689
\(100\) −1.76318e8 −0.176318
\(101\) −1.77085e9 −1.69331 −0.846654 0.532144i \(-0.821387\pi\)
−0.846654 + 0.532144i \(0.821387\pi\)
\(102\) −1.59936e8 −0.146301
\(103\) 3.03322e8 0.265544 0.132772 0.991147i \(-0.457612\pi\)
0.132772 + 0.991147i \(0.457612\pi\)
\(104\) 1.02453e9 0.858763
\(105\) −9.31438e7 −0.0747829
\(106\) −4.44142e7 −0.0341701
\(107\) −1.95414e8 −0.144121 −0.0720607 0.997400i \(-0.522958\pi\)
−0.0720607 + 0.997400i \(0.522958\pi\)
\(108\) 2.39879e8 0.169662
\(109\) 2.31494e9 1.57080 0.785401 0.618988i \(-0.212457\pi\)
0.785401 + 0.618988i \(0.212457\pi\)
\(110\) −2.16000e8 −0.140665
\(111\) −1.26412e9 −0.790377
\(112\) 3.17743e8 0.190808
\(113\) 1.31945e9 0.761270 0.380635 0.924725i \(-0.375706\pi\)
0.380635 + 0.924725i \(0.375706\pi\)
\(114\) 5.38832e7 0.0298801
\(115\) −6.12130e8 −0.326365
\(116\) −1.16743e9 −0.598648
\(117\) 8.96126e8 0.442113
\(118\) −6.50227e8 −0.308742
\(119\) −4.66576e8 −0.213285
\(120\) −3.79742e8 −0.167176
\(121\) −3.87842e8 −0.164483
\(122\) −1.15721e9 −0.472925
\(123\) −1.98363e9 −0.781426
\(124\) −4.03884e9 −1.53412
\(125\) 2.44141e8 0.0894427
\(126\) −9.39914e7 −0.0332216
\(127\) 3.18960e9 1.08798 0.543988 0.839093i \(-0.316914\pi\)
0.543988 + 0.839093i \(0.316914\pi\)
\(128\) 3.02895e9 0.997349
\(129\) −1.03350e9 −0.328590
\(130\) −6.64673e8 −0.204109
\(131\) −5.40115e9 −1.60238 −0.801190 0.598410i \(-0.795799\pi\)
−0.801190 + 0.598410i \(0.795799\pi\)
\(132\) 1.62281e9 0.465247
\(133\) 1.57191e8 0.0435608
\(134\) 1.30811e9 0.350488
\(135\) −3.32151e8 −0.0860663
\(136\) −1.90220e9 −0.476796
\(137\) 1.19903e7 0.00290795 0.00145398 0.999999i \(-0.499537\pi\)
0.00145398 + 0.999999i \(0.499537\pi\)
\(138\) −6.17701e8 −0.144985
\(139\) 9.15482e8 0.208010 0.104005 0.994577i \(-0.466834\pi\)
0.104005 + 0.994577i \(0.466834\pi\)
\(140\) −5.19046e8 −0.114190
\(141\) 4.99578e9 1.06443
\(142\) −1.64279e9 −0.339066
\(143\) 6.06239e9 1.21236
\(144\) 1.13307e9 0.219597
\(145\) 1.61650e9 0.303682
\(146\) 1.11760e9 0.203562
\(147\) 2.99445e9 0.528918
\(148\) −7.04432e9 −1.20687
\(149\) 5.00462e9 0.831827 0.415913 0.909404i \(-0.363462\pi\)
0.415913 + 0.909404i \(0.363462\pi\)
\(150\) 2.46362e8 0.0397341
\(151\) 3.20554e9 0.501770 0.250885 0.968017i \(-0.419278\pi\)
0.250885 + 0.968017i \(0.419278\pi\)
\(152\) 6.40860e8 0.0973794
\(153\) −1.66381e9 −0.245466
\(154\) −6.35862e8 −0.0911002
\(155\) 5.59242e9 0.778228
\(156\) 4.99368e9 0.675088
\(157\) −4.63430e8 −0.0608745 −0.0304373 0.999537i \(-0.509690\pi\)
−0.0304373 + 0.999537i \(0.509690\pi\)
\(158\) 3.55023e9 0.453210
\(159\) −4.62038e8 −0.0573312
\(160\) −3.24077e9 −0.390938
\(161\) −1.80199e9 −0.211367
\(162\) −3.35173e8 −0.0382342
\(163\) −1.27948e10 −1.41968 −0.709840 0.704363i \(-0.751233\pi\)
−0.709840 + 0.704363i \(0.751233\pi\)
\(164\) −1.10538e10 −1.19320
\(165\) −2.24703e9 −0.236011
\(166\) 2.76495e9 0.282619
\(167\) −1.85699e10 −1.84750 −0.923752 0.382992i \(-0.874894\pi\)
−0.923752 + 0.382992i \(0.874894\pi\)
\(168\) −1.11789e9 −0.108269
\(169\) 8.05063e9 0.759171
\(170\) 1.23408e9 0.113324
\(171\) 5.60544e8 0.0501334
\(172\) −5.75917e9 −0.501744
\(173\) 4.90746e9 0.416533 0.208266 0.978072i \(-0.433218\pi\)
0.208266 + 0.978072i \(0.433218\pi\)
\(174\) 1.63121e9 0.134908
\(175\) 7.18702e8 0.0579266
\(176\) 7.66536e9 0.602179
\(177\) −6.76427e9 −0.518014
\(178\) 3.30558e9 0.246807
\(179\) −1.28930e10 −0.938678 −0.469339 0.883018i \(-0.655508\pi\)
−0.469339 + 0.883018i \(0.655508\pi\)
\(180\) −1.85092e9 −0.131420
\(181\) 2.66233e10 1.84378 0.921889 0.387453i \(-0.126645\pi\)
0.921889 + 0.387453i \(0.126645\pi\)
\(182\) −1.95667e9 −0.132189
\(183\) −1.20384e10 −0.793483
\(184\) −7.34663e9 −0.472506
\(185\) 9.75400e9 0.612223
\(186\) 5.64331e9 0.345721
\(187\) −1.12558e10 −0.673117
\(188\) 2.78391e10 1.62534
\(189\) −9.77786e8 −0.0557399
\(190\) −4.15766e8 −0.0231450
\(191\) −2.72802e10 −1.48319 −0.741595 0.670848i \(-0.765930\pi\)
−0.741595 + 0.670848i \(0.765930\pi\)
\(192\) 3.89188e9 0.206683
\(193\) −9.65442e9 −0.500862 −0.250431 0.968134i \(-0.580572\pi\)
−0.250431 + 0.968134i \(0.580572\pi\)
\(194\) −9.33612e9 −0.473215
\(195\) −6.91455e9 −0.342459
\(196\) 1.66866e10 0.807636
\(197\) 1.21091e10 0.572812 0.286406 0.958108i \(-0.407539\pi\)
0.286406 + 0.958108i \(0.407539\pi\)
\(198\) −2.26748e9 −0.104846
\(199\) 1.89741e10 0.857673 0.428837 0.903382i \(-0.358924\pi\)
0.428837 + 0.903382i \(0.358924\pi\)
\(200\) 2.93011e9 0.129494
\(201\) 1.36082e10 0.588056
\(202\) 1.37883e10 0.582680
\(203\) 4.75866e9 0.196676
\(204\) −9.27161e9 −0.374817
\(205\) 1.53058e10 0.605290
\(206\) −2.36174e9 −0.0913757
\(207\) −6.42590e9 −0.243258
\(208\) 2.35878e10 0.873779
\(209\) 3.79214e9 0.137476
\(210\) 7.25242e8 0.0257333
\(211\) −8.91928e9 −0.309784 −0.154892 0.987931i \(-0.549503\pi\)
−0.154892 + 0.987931i \(0.549503\pi\)
\(212\) −2.57472e9 −0.0875424
\(213\) −1.70898e10 −0.568892
\(214\) 1.52154e9 0.0495933
\(215\) 7.97450e9 0.254525
\(216\) −3.98638e9 −0.124605
\(217\) 1.64630e10 0.504010
\(218\) −1.80248e10 −0.540524
\(219\) 1.16263e10 0.341541
\(220\) −1.25216e10 −0.360379
\(221\) −3.46364e10 −0.976714
\(222\) 9.84275e9 0.271975
\(223\) −4.42755e10 −1.19892 −0.599462 0.800403i \(-0.704619\pi\)
−0.599462 + 0.800403i \(0.704619\pi\)
\(224\) −9.54018e9 −0.253187
\(225\) 2.56289e9 0.0666667
\(226\) −1.02735e10 −0.261959
\(227\) −5.40045e10 −1.34994 −0.674968 0.737847i \(-0.735843\pi\)
−0.674968 + 0.737847i \(0.735843\pi\)
\(228\) 3.12364e9 0.0765516
\(229\) 5.80250e9 0.139430 0.0697149 0.997567i \(-0.477791\pi\)
0.0697149 + 0.997567i \(0.477791\pi\)
\(230\) 4.76621e9 0.112305
\(231\) −6.61483e9 −0.152850
\(232\) 1.94008e10 0.439667
\(233\) 5.41865e10 1.20445 0.602226 0.798326i \(-0.294281\pi\)
0.602226 + 0.798326i \(0.294281\pi\)
\(234\) −6.97747e9 −0.152134
\(235\) −3.85477e10 −0.824505
\(236\) −3.76940e10 −0.790986
\(237\) 3.69328e10 0.760404
\(238\) 3.63288e9 0.0733930
\(239\) 7.91761e10 1.56965 0.784826 0.619716i \(-0.212752\pi\)
0.784826 + 0.619716i \(0.212752\pi\)
\(240\) −8.74284e9 −0.170099
\(241\) −6.14920e10 −1.17420 −0.587100 0.809514i \(-0.699730\pi\)
−0.587100 + 0.809514i \(0.699730\pi\)
\(242\) 3.01984e9 0.0565998
\(243\) −3.48678e9 −0.0641500
\(244\) −6.70841e10 −1.21162
\(245\) −2.31053e10 −0.409698
\(246\) 1.54451e10 0.268894
\(247\) 1.16691e10 0.199481
\(248\) 6.71187e10 1.12671
\(249\) 2.87636e10 0.474183
\(250\) −1.90094e9 −0.0307779
\(251\) 2.89319e10 0.460093 0.230046 0.973180i \(-0.426112\pi\)
0.230046 + 0.973180i \(0.426112\pi\)
\(252\) −5.44873e9 −0.0851125
\(253\) −4.34719e10 −0.667062
\(254\) −2.48350e10 −0.374381
\(255\) 1.28380e10 0.190137
\(256\) 1.01633e9 0.0147896
\(257\) 1.22388e11 1.75001 0.875005 0.484114i \(-0.160858\pi\)
0.875005 + 0.484114i \(0.160858\pi\)
\(258\) 8.04706e9 0.113070
\(259\) 2.87139e10 0.396499
\(260\) −3.85315e10 −0.522921
\(261\) 1.69694e10 0.226351
\(262\) 4.20548e10 0.551391
\(263\) −6.24892e10 −0.805386 −0.402693 0.915335i \(-0.631926\pi\)
−0.402693 + 0.915335i \(0.631926\pi\)
\(264\) −2.69683e10 −0.341693
\(265\) 3.56511e9 0.0444086
\(266\) −1.22393e9 −0.0149896
\(267\) 3.43878e10 0.414098
\(268\) 7.58321e10 0.897938
\(269\) 1.27214e11 1.48132 0.740659 0.671881i \(-0.234513\pi\)
0.740659 + 0.671881i \(0.234513\pi\)
\(270\) 2.58621e9 0.0296161
\(271\) 1.54116e10 0.173574 0.0867871 0.996227i \(-0.472340\pi\)
0.0867871 + 0.996227i \(0.472340\pi\)
\(272\) −4.37946e10 −0.485133
\(273\) −2.03551e10 −0.221790
\(274\) −9.33596e7 −0.00100065
\(275\) 1.73382e10 0.182813
\(276\) −3.58085e10 −0.371445
\(277\) −9.20867e10 −0.939806 −0.469903 0.882718i \(-0.655711\pi\)
−0.469903 + 0.882718i \(0.655711\pi\)
\(278\) −7.12818e9 −0.0715776
\(279\) 5.87070e10 0.580057
\(280\) 8.62567e9 0.0838652
\(281\) −7.78782e10 −0.745139 −0.372570 0.928004i \(-0.621523\pi\)
−0.372570 + 0.928004i \(0.621523\pi\)
\(282\) −3.88985e10 −0.366279
\(283\) 2.92463e10 0.271039 0.135520 0.990775i \(-0.456730\pi\)
0.135520 + 0.990775i \(0.456730\pi\)
\(284\) −9.52334e10 −0.868674
\(285\) −4.32518e9 −0.0388331
\(286\) −4.72034e10 −0.417182
\(287\) 4.50572e10 0.392009
\(288\) −3.40203e10 −0.291388
\(289\) −5.42796e10 −0.457717
\(290\) −1.25865e10 −0.104499
\(291\) −9.71231e10 −0.793970
\(292\) 6.47877e10 0.521519
\(293\) −2.27403e11 −1.80257 −0.901285 0.433227i \(-0.857375\pi\)
−0.901285 + 0.433227i \(0.857375\pi\)
\(294\) −2.33155e10 −0.182005
\(295\) 5.21935e10 0.401252
\(296\) 1.17065e11 0.886368
\(297\) −2.35885e10 −0.175912
\(298\) −3.89673e10 −0.286238
\(299\) −1.33771e11 −0.967927
\(300\) 1.42818e10 0.101797
\(301\) 2.34754e10 0.164840
\(302\) −2.49592e10 −0.172663
\(303\) 1.43439e11 0.977632
\(304\) 1.47546e10 0.0990823
\(305\) 9.28887e10 0.614630
\(306\) 1.29548e10 0.0844668
\(307\) −6.02579e10 −0.387160 −0.193580 0.981084i \(-0.562010\pi\)
−0.193580 + 0.981084i \(0.562010\pi\)
\(308\) −3.68613e10 −0.233395
\(309\) −2.45691e10 −0.153312
\(310\) −4.35440e10 −0.267794
\(311\) 1.36816e11 0.829308 0.414654 0.909979i \(-0.363903\pi\)
0.414654 + 0.909979i \(0.363903\pi\)
\(312\) −8.29866e10 −0.495807
\(313\) 2.25535e11 1.32820 0.664101 0.747643i \(-0.268815\pi\)
0.664101 + 0.747643i \(0.268815\pi\)
\(314\) 3.60839e9 0.0209474
\(315\) 7.54465e9 0.0431759
\(316\) 2.05809e11 1.16111
\(317\) −1.28386e11 −0.714089 −0.357045 0.934087i \(-0.616216\pi\)
−0.357045 + 0.934087i \(0.616216\pi\)
\(318\) 3.59755e9 0.0197281
\(319\) 1.14800e11 0.620701
\(320\) −3.00299e10 −0.160096
\(321\) 1.58285e10 0.0832086
\(322\) 1.40308e10 0.0727328
\(323\) −2.16657e10 −0.110755
\(324\) −1.94302e10 −0.0979545
\(325\) 5.33530e10 0.265268
\(326\) 9.96239e10 0.488522
\(327\) −1.87510e11 −0.906902
\(328\) 1.83696e11 0.876329
\(329\) −1.13477e11 −0.533981
\(330\) 1.74960e10 0.0812131
\(331\) 1.15018e10 0.0526673 0.0263337 0.999653i \(-0.491617\pi\)
0.0263337 + 0.999653i \(0.491617\pi\)
\(332\) 1.60286e11 0.724058
\(333\) 1.02394e11 0.456324
\(334\) 1.44590e11 0.635740
\(335\) −1.05002e11 −0.455506
\(336\) −2.57372e10 −0.110163
\(337\) −2.79631e11 −1.18100 −0.590502 0.807036i \(-0.701070\pi\)
−0.590502 + 0.807036i \(0.701070\pi\)
\(338\) −6.26843e10 −0.261236
\(339\) −1.06875e11 −0.439519
\(340\) 7.15402e10 0.290332
\(341\) 3.97159e11 1.59063
\(342\) −4.36454e9 −0.0172513
\(343\) −1.42263e11 −0.554969
\(344\) 9.57078e10 0.368497
\(345\) 4.95826e10 0.188427
\(346\) −3.82107e10 −0.143332
\(347\) 9.07088e10 0.335867 0.167933 0.985798i \(-0.446291\pi\)
0.167933 + 0.985798i \(0.446291\pi\)
\(348\) 9.45621e10 0.345629
\(349\) −3.98825e11 −1.43902 −0.719511 0.694481i \(-0.755634\pi\)
−0.719511 + 0.694481i \(0.755634\pi\)
\(350\) −5.59600e9 −0.0199330
\(351\) −7.25862e10 −0.255254
\(352\) −2.30151e11 −0.799044
\(353\) −4.56192e11 −1.56373 −0.781865 0.623448i \(-0.785731\pi\)
−0.781865 + 0.623448i \(0.785731\pi\)
\(354\) 5.26684e10 0.178252
\(355\) 1.31866e11 0.440662
\(356\) 1.91627e11 0.632311
\(357\) 3.77926e10 0.123140
\(358\) 1.00389e11 0.323006
\(359\) 1.89176e11 0.601094 0.300547 0.953767i \(-0.402831\pi\)
0.300547 + 0.953767i \(0.402831\pi\)
\(360\) 3.07591e10 0.0965190
\(361\) −3.15388e11 −0.977380
\(362\) −2.07296e11 −0.634458
\(363\) 3.14152e10 0.0949643
\(364\) −1.13429e11 −0.338664
\(365\) −8.97090e10 −0.264556
\(366\) 9.37339e10 0.273044
\(367\) −3.17920e10 −0.0914787 −0.0457394 0.998953i \(-0.514564\pi\)
−0.0457394 + 0.998953i \(0.514564\pi\)
\(368\) −1.69142e11 −0.480769
\(369\) 1.60674e11 0.451156
\(370\) −7.59472e10 −0.210671
\(371\) 1.04950e10 0.0287607
\(372\) 3.27146e11 0.885723
\(373\) −4.46388e11 −1.19405 −0.597025 0.802222i \(-0.703651\pi\)
−0.597025 + 0.802222i \(0.703651\pi\)
\(374\) 8.76409e10 0.231625
\(375\) −1.97754e10 −0.0516398
\(376\) −4.62639e11 −1.19371
\(377\) 3.53260e11 0.900655
\(378\) 7.61330e9 0.0191805
\(379\) 3.53467e11 0.879978 0.439989 0.898003i \(-0.354982\pi\)
0.439989 + 0.898003i \(0.354982\pi\)
\(380\) −2.41022e10 −0.0592966
\(381\) −2.58357e11 −0.628143
\(382\) 2.12411e11 0.510377
\(383\) 3.14974e11 0.747963 0.373981 0.927436i \(-0.377992\pi\)
0.373981 + 0.927436i \(0.377992\pi\)
\(384\) −2.45345e11 −0.575820
\(385\) 5.10403e10 0.118397
\(386\) 7.51718e10 0.172350
\(387\) 8.37131e10 0.189712
\(388\) −5.41220e11 −1.21236
\(389\) −4.20729e11 −0.931600 −0.465800 0.884890i \(-0.654233\pi\)
−0.465800 + 0.884890i \(0.654233\pi\)
\(390\) 5.38385e10 0.117843
\(391\) 2.48369e11 0.537405
\(392\) −2.77303e11 −0.593155
\(393\) 4.37493e11 0.925134
\(394\) −9.42843e10 −0.197109
\(395\) −2.84975e11 −0.589007
\(396\) −1.31447e11 −0.268611
\(397\) 5.63093e11 1.13769 0.568844 0.822446i \(-0.307391\pi\)
0.568844 + 0.822446i \(0.307391\pi\)
\(398\) −1.47737e11 −0.295132
\(399\) −1.27325e10 −0.0251498
\(400\) 6.74602e10 0.131758
\(401\) 2.75897e11 0.532841 0.266420 0.963857i \(-0.414159\pi\)
0.266420 + 0.963857i \(0.414159\pi\)
\(402\) −1.05957e11 −0.202355
\(403\) 1.22213e12 2.30805
\(404\) 7.99317e11 1.49280
\(405\) 2.69042e10 0.0496904
\(406\) −3.70521e10 −0.0676778
\(407\) 6.92703e11 1.25133
\(408\) 1.54079e11 0.275278
\(409\) −6.54552e10 −0.115662 −0.0578308 0.998326i \(-0.518418\pi\)
−0.0578308 + 0.998326i \(0.518418\pi\)
\(410\) −1.19175e11 −0.208285
\(411\) −9.71214e8 −0.00167891
\(412\) −1.36912e11 −0.234101
\(413\) 1.53647e11 0.259866
\(414\) 5.00337e10 0.0837069
\(415\) −2.21941e11 −0.367301
\(416\) −7.08218e11 −1.15944
\(417\) −7.41541e10 −0.120094
\(418\) −2.95266e10 −0.0473064
\(419\) 2.70250e10 0.0428354 0.0214177 0.999771i \(-0.493182\pi\)
0.0214177 + 0.999771i \(0.493182\pi\)
\(420\) 4.20427e10 0.0659278
\(421\) −4.29698e11 −0.666644 −0.333322 0.942813i \(-0.608170\pi\)
−0.333322 + 0.942813i \(0.608170\pi\)
\(422\) 6.94478e10 0.106599
\(423\) −4.04659e11 −0.614550
\(424\) 4.27875e10 0.0642940
\(425\) −9.90589e10 −0.147280
\(426\) 1.33066e11 0.195760
\(427\) 2.73446e11 0.398058
\(428\) 8.82048e10 0.127056
\(429\) −4.91054e11 −0.699957
\(430\) −6.20915e10 −0.0875839
\(431\) −9.44700e11 −1.31870 −0.659350 0.751836i \(-0.729169\pi\)
−0.659350 + 0.751836i \(0.729169\pi\)
\(432\) −9.17789e10 −0.126784
\(433\) 2.10762e11 0.288136 0.144068 0.989568i \(-0.453982\pi\)
0.144068 + 0.989568i \(0.453982\pi\)
\(434\) −1.28185e11 −0.173434
\(435\) −1.30936e11 −0.175331
\(436\) −1.04491e12 −1.38480
\(437\) −8.36765e10 −0.109758
\(438\) −9.05253e10 −0.117527
\(439\) 7.69079e11 0.988282 0.494141 0.869382i \(-0.335483\pi\)
0.494141 + 0.869382i \(0.335483\pi\)
\(440\) 2.08089e11 0.264674
\(441\) −2.42550e11 −0.305371
\(442\) 2.69688e11 0.336094
\(443\) −6.49894e11 −0.801726 −0.400863 0.916138i \(-0.631290\pi\)
−0.400863 + 0.916138i \(0.631290\pi\)
\(444\) 5.70590e11 0.696788
\(445\) −2.65338e11 −0.320759
\(446\) 3.44741e11 0.412559
\(447\) −4.05374e11 −0.480255
\(448\) −8.84023e10 −0.103684
\(449\) −4.51858e11 −0.524679 −0.262339 0.964976i \(-0.584494\pi\)
−0.262339 + 0.964976i \(0.584494\pi\)
\(450\) −1.99553e10 −0.0229405
\(451\) 1.08698e12 1.23716
\(452\) −5.95564e11 −0.671128
\(453\) −2.59649e11 −0.289697
\(454\) 4.20493e11 0.464523
\(455\) 1.57061e11 0.171797
\(456\) −5.19097e10 −0.0562220
\(457\) −6.18434e11 −0.663240 −0.331620 0.943413i \(-0.607595\pi\)
−0.331620 + 0.943413i \(0.607595\pi\)
\(458\) −4.51798e10 −0.0479788
\(459\) 1.34768e11 0.141720
\(460\) 2.76300e11 0.287720
\(461\) −2.76450e11 −0.285078 −0.142539 0.989789i \(-0.545527\pi\)
−0.142539 + 0.989789i \(0.545527\pi\)
\(462\) 5.15048e10 0.0525967
\(463\) 6.09969e11 0.616870 0.308435 0.951245i \(-0.400195\pi\)
0.308435 + 0.951245i \(0.400195\pi\)
\(464\) 4.46666e11 0.447355
\(465\) −4.52986e11 −0.449310
\(466\) −4.21910e11 −0.414461
\(467\) 2.83260e11 0.275587 0.137794 0.990461i \(-0.455999\pi\)
0.137794 + 0.990461i \(0.455999\pi\)
\(468\) −4.04488e11 −0.389762
\(469\) −3.09104e11 −0.295004
\(470\) 3.00143e11 0.283718
\(471\) 3.75378e10 0.0351459
\(472\) 6.26412e11 0.580926
\(473\) 5.66328e11 0.520227
\(474\) −2.87568e11 −0.261661
\(475\) 3.33733e10 0.0300800
\(476\) 2.10600e11 0.188030
\(477\) 3.74251e10 0.0331002
\(478\) −6.16485e11 −0.540129
\(479\) −1.39149e12 −1.20773 −0.603865 0.797087i \(-0.706373\pi\)
−0.603865 + 0.797087i \(0.706373\pi\)
\(480\) 2.62502e11 0.225708
\(481\) 2.13158e12 1.81572
\(482\) 4.78793e11 0.404051
\(483\) 1.45961e11 0.122033
\(484\) 1.75062e11 0.145007
\(485\) 7.49406e11 0.615006
\(486\) 2.71490e10 0.0220745
\(487\) −1.41532e11 −0.114018 −0.0570092 0.998374i \(-0.518156\pi\)
−0.0570092 + 0.998374i \(0.518156\pi\)
\(488\) 1.11483e12 0.889851
\(489\) 1.03638e12 0.819652
\(490\) 1.79904e11 0.140980
\(491\) −4.66246e11 −0.362034 −0.181017 0.983480i \(-0.557939\pi\)
−0.181017 + 0.983480i \(0.557939\pi\)
\(492\) 8.95359e11 0.688897
\(493\) −6.55887e11 −0.500055
\(494\) −9.08589e10 −0.0686430
\(495\) 1.82010e11 0.136261
\(496\) 1.54528e12 1.14641
\(497\) 3.88188e11 0.285389
\(498\) −2.23961e11 −0.163170
\(499\) −2.21254e12 −1.59749 −0.798746 0.601668i \(-0.794503\pi\)
−0.798746 + 0.601668i \(0.794503\pi\)
\(500\) −1.10199e11 −0.0788518
\(501\) 1.50416e12 1.06666
\(502\) −2.25271e11 −0.158321
\(503\) 3.39574e11 0.236526 0.118263 0.992982i \(-0.462267\pi\)
0.118263 + 0.992982i \(0.462267\pi\)
\(504\) 9.05488e10 0.0625094
\(505\) −1.10678e12 −0.757270
\(506\) 3.38484e11 0.229541
\(507\) −6.52101e11 −0.438308
\(508\) −1.43970e12 −0.959149
\(509\) −5.66691e11 −0.374211 −0.187106 0.982340i \(-0.559911\pi\)
−0.187106 + 0.982340i \(0.559911\pi\)
\(510\) −9.99603e10 −0.0654277
\(511\) −2.64086e11 −0.171337
\(512\) −1.55874e12 −1.00244
\(513\) −4.54040e10 −0.0289445
\(514\) −9.52947e11 −0.602191
\(515\) 1.89576e11 0.118755
\(516\) 4.66493e11 0.289682
\(517\) −2.73756e12 −1.68522
\(518\) −2.23574e11 −0.136438
\(519\) −3.97504e11 −0.240485
\(520\) 6.40329e11 0.384050
\(521\) 4.42970e11 0.263393 0.131697 0.991290i \(-0.457958\pi\)
0.131697 + 0.991290i \(0.457958\pi\)
\(522\) −1.32128e11 −0.0778892
\(523\) −1.44683e11 −0.0845591 −0.0422796 0.999106i \(-0.513462\pi\)
−0.0422796 + 0.999106i \(0.513462\pi\)
\(524\) 2.43794e12 1.41264
\(525\) −5.82149e10 −0.0334439
\(526\) 4.86557e11 0.277139
\(527\) −2.26910e12 −1.28146
\(528\) −6.20894e11 −0.347668
\(529\) −8.41911e11 −0.467429
\(530\) −2.77589e10 −0.0152813
\(531\) 5.47906e11 0.299075
\(532\) −7.09520e10 −0.0384028
\(533\) 3.34484e12 1.79516
\(534\) −2.67752e11 −0.142494
\(535\) −1.22134e11 −0.0644531
\(536\) −1.26020e12 −0.659475
\(537\) 1.04434e12 0.541946
\(538\) −9.90519e11 −0.509733
\(539\) −1.64088e12 −0.837388
\(540\) 1.49924e11 0.0758752
\(541\) −3.10308e11 −0.155742 −0.0778710 0.996963i \(-0.524812\pi\)
−0.0778710 + 0.996963i \(0.524812\pi\)
\(542\) −1.19999e11 −0.0597281
\(543\) −2.15649e12 −1.06451
\(544\) 1.31493e12 0.643733
\(545\) 1.44684e12 0.702484
\(546\) 1.58490e11 0.0763194
\(547\) 1.68502e12 0.804750 0.402375 0.915475i \(-0.368185\pi\)
0.402375 + 0.915475i \(0.368185\pi\)
\(548\) −5.41211e9 −0.00256362
\(549\) 9.75108e11 0.458118
\(550\) −1.35000e11 −0.0629074
\(551\) 2.20971e11 0.102130
\(552\) 5.95077e11 0.272802
\(553\) −8.38911e11 −0.381463
\(554\) 7.17011e11 0.323394
\(555\) −7.90074e11 −0.353467
\(556\) −4.13225e11 −0.183379
\(557\) −2.31328e12 −1.01831 −0.509156 0.860674i \(-0.670042\pi\)
−0.509156 + 0.860674i \(0.670042\pi\)
\(558\) −4.57108e11 −0.199602
\(559\) 1.74270e12 0.754865
\(560\) 1.98590e11 0.0853317
\(561\) 9.11723e11 0.388624
\(562\) 6.06380e11 0.256408
\(563\) 7.03648e11 0.295167 0.147583 0.989050i \(-0.452850\pi\)
0.147583 + 0.989050i \(0.452850\pi\)
\(564\) −2.25497e12 −0.938392
\(565\) 8.24653e11 0.340450
\(566\) −2.27720e11 −0.0932666
\(567\) 7.92007e10 0.0321814
\(568\) 1.58262e12 0.637983
\(569\) 3.21997e12 1.28779 0.643896 0.765113i \(-0.277317\pi\)
0.643896 + 0.765113i \(0.277317\pi\)
\(570\) 3.36770e10 0.0133628
\(571\) −1.21116e12 −0.476801 −0.238401 0.971167i \(-0.576623\pi\)
−0.238401 + 0.971167i \(0.576623\pi\)
\(572\) −2.73641e12 −1.06880
\(573\) 2.20969e12 0.856320
\(574\) −3.50827e11 −0.134893
\(575\) −3.82582e11 −0.145955
\(576\) −3.15242e11 −0.119328
\(577\) 7.30673e11 0.274430 0.137215 0.990541i \(-0.456185\pi\)
0.137215 + 0.990541i \(0.456185\pi\)
\(578\) 4.22635e11 0.157504
\(579\) 7.82008e11 0.289173
\(580\) −7.29646e11 −0.267723
\(581\) −6.53352e11 −0.237878
\(582\) 7.56226e11 0.273211
\(583\) 2.53185e11 0.0907673
\(584\) −1.07666e12 −0.383021
\(585\) 5.60079e11 0.197719
\(586\) 1.77062e12 0.620278
\(587\) 4.55331e12 1.58291 0.791453 0.611230i \(-0.209325\pi\)
0.791453 + 0.611230i \(0.209325\pi\)
\(588\) −1.35162e12 −0.466289
\(589\) 7.64467e11 0.261722
\(590\) −4.06392e11 −0.138074
\(591\) −9.80833e11 −0.330713
\(592\) 2.69520e12 0.901867
\(593\) 3.00074e12 0.996510 0.498255 0.867030i \(-0.333974\pi\)
0.498255 + 0.867030i \(0.333974\pi\)
\(594\) 1.83666e11 0.0605327
\(595\) −2.91610e11 −0.0953841
\(596\) −2.25896e12 −0.733330
\(597\) −1.53690e12 −0.495178
\(598\) 1.04158e12 0.333071
\(599\) −4.03514e12 −1.28067 −0.640336 0.768095i \(-0.721205\pi\)
−0.640336 + 0.768095i \(0.721205\pi\)
\(600\) −2.37339e11 −0.0747633
\(601\) 2.04760e12 0.640192 0.320096 0.947385i \(-0.396285\pi\)
0.320096 + 0.947385i \(0.396285\pi\)
\(602\) −1.82785e11 −0.0567227
\(603\) −1.10227e12 −0.339514
\(604\) −1.44690e12 −0.442355
\(605\) −2.42401e11 −0.0735590
\(606\) −1.11685e12 −0.336410
\(607\) 3.15792e12 0.944175 0.472088 0.881552i \(-0.343501\pi\)
0.472088 + 0.881552i \(0.343501\pi\)
\(608\) −4.43004e11 −0.131474
\(609\) −3.85451e11 −0.113551
\(610\) −7.23256e11 −0.211499
\(611\) −8.42399e12 −2.44530
\(612\) 7.51000e11 0.216401
\(613\) −2.89302e12 −0.827520 −0.413760 0.910386i \(-0.635785\pi\)
−0.413760 + 0.910386i \(0.635785\pi\)
\(614\) 4.69183e11 0.133225
\(615\) −1.23977e12 −0.349464
\(616\) 6.12573e11 0.171413
\(617\) 6.03603e12 1.67675 0.838375 0.545094i \(-0.183506\pi\)
0.838375 + 0.545094i \(0.183506\pi\)
\(618\) 1.91301e11 0.0527558
\(619\) 4.05606e12 1.11044 0.555222 0.831702i \(-0.312633\pi\)
0.555222 + 0.831702i \(0.312633\pi\)
\(620\) −2.52427e12 −0.686078
\(621\) 5.20498e11 0.140445
\(622\) −1.06529e12 −0.285371
\(623\) −7.81102e11 −0.207736
\(624\) −1.91061e12 −0.504477
\(625\) 1.52588e11 0.0400000
\(626\) −1.75607e12 −0.457044
\(627\) −3.07163e11 −0.0793716
\(628\) 2.09180e11 0.0536664
\(629\) −3.95764e12 −1.00811
\(630\) −5.87446e10 −0.0148571
\(631\) −5.34498e12 −1.34219 −0.671095 0.741372i \(-0.734176\pi\)
−0.671095 + 0.741372i \(0.734176\pi\)
\(632\) −3.42020e12 −0.852755
\(633\) 7.22461e11 0.178854
\(634\) 9.99651e11 0.245723
\(635\) 1.99350e12 0.486558
\(636\) 2.08552e11 0.0505426
\(637\) −5.04929e12 −1.21507
\(638\) −8.93859e11 −0.213588
\(639\) 1.38428e12 0.328450
\(640\) 1.89309e12 0.446028
\(641\) 4.81739e12 1.12707 0.563534 0.826093i \(-0.309441\pi\)
0.563534 + 0.826093i \(0.309441\pi\)
\(642\) −1.23245e11 −0.0286327
\(643\) −5.85505e11 −0.135077 −0.0675385 0.997717i \(-0.521515\pi\)
−0.0675385 + 0.997717i \(0.521515\pi\)
\(644\) 8.13372e11 0.186339
\(645\) −6.45935e11 −0.146950
\(646\) 1.68695e11 0.0381114
\(647\) 4.54903e12 1.02059 0.510293 0.860001i \(-0.329537\pi\)
0.510293 + 0.860001i \(0.329537\pi\)
\(648\) 3.22897e11 0.0719410
\(649\) 3.70664e12 0.820124
\(650\) −4.15421e11 −0.0912805
\(651\) −1.33350e12 −0.290991
\(652\) 5.77526e12 1.25158
\(653\) −7.04350e12 −1.51593 −0.757965 0.652295i \(-0.773806\pi\)
−0.757965 + 0.652295i \(0.773806\pi\)
\(654\) 1.46001e12 0.312072
\(655\) −3.37572e12 −0.716606
\(656\) 4.22925e12 0.891653
\(657\) −9.41729e11 −0.197189
\(658\) 8.83561e11 0.183747
\(659\) 5.23559e12 1.08139 0.540694 0.841219i \(-0.318162\pi\)
0.540694 + 0.841219i \(0.318162\pi\)
\(660\) 1.01425e12 0.208065
\(661\) −5.79017e12 −1.17974 −0.589868 0.807500i \(-0.700820\pi\)
−0.589868 + 0.807500i \(0.700820\pi\)
\(662\) −8.95563e10 −0.0181232
\(663\) 2.80555e12 0.563906
\(664\) −2.66368e12 −0.531772
\(665\) 9.82445e10 0.0194810
\(666\) −7.97263e11 −0.157025
\(667\) −2.53314e12 −0.495557
\(668\) 8.38197e12 1.62874
\(669\) 3.58632e12 0.692199
\(670\) 8.17570e11 0.156743
\(671\) 6.59671e12 1.25625
\(672\) 7.72755e11 0.146177
\(673\) −5.92328e10 −0.0111300 −0.00556499 0.999985i \(-0.501771\pi\)
−0.00556499 + 0.999985i \(0.501771\pi\)
\(674\) 2.17728e12 0.406392
\(675\) −2.07594e11 −0.0384900
\(676\) −3.63385e12 −0.669278
\(677\) 4.57177e12 0.836440 0.418220 0.908346i \(-0.362654\pi\)
0.418220 + 0.908346i \(0.362654\pi\)
\(678\) 8.32157e11 0.151242
\(679\) 2.20611e12 0.398302
\(680\) −1.18888e12 −0.213229
\(681\) 4.37436e12 0.779386
\(682\) −3.09238e12 −0.547348
\(683\) −9.71286e12 −1.70787 −0.853934 0.520382i \(-0.825790\pi\)
−0.853934 + 0.520382i \(0.825790\pi\)
\(684\) −2.53015e11 −0.0441971
\(685\) 7.49393e9 0.00130048
\(686\) 1.10770e12 0.190969
\(687\) −4.70003e11 −0.0804998
\(688\) 2.20349e12 0.374941
\(689\) 7.79098e11 0.131706
\(690\) −3.86063e11 −0.0648391
\(691\) 2.23726e12 0.373306 0.186653 0.982426i \(-0.440236\pi\)
0.186653 + 0.982426i \(0.440236\pi\)
\(692\) −2.21510e12 −0.367211
\(693\) 5.35801e11 0.0882478
\(694\) −7.06282e11 −0.115574
\(695\) 5.72176e11 0.0930247
\(696\) −1.57146e12 −0.253842
\(697\) −6.21025e12 −0.996693
\(698\) 3.10535e12 0.495178
\(699\) −4.38911e12 −0.695391
\(700\) −3.24404e11 −0.0510675
\(701\) −2.29477e10 −0.00358929 −0.00179464 0.999998i \(-0.500571\pi\)
−0.00179464 + 0.999998i \(0.500571\pi\)
\(702\) 5.65175e11 0.0878347
\(703\) 1.33334e12 0.205894
\(704\) −2.13265e12 −0.327222
\(705\) 3.12237e12 0.476028
\(706\) 3.55203e12 0.538091
\(707\) −3.25815e12 −0.490438
\(708\) 3.05322e12 0.456676
\(709\) −9.57637e12 −1.42329 −0.711644 0.702540i \(-0.752049\pi\)
−0.711644 + 0.702540i \(0.752049\pi\)
\(710\) −1.02674e12 −0.151635
\(711\) −2.99156e12 −0.439020
\(712\) −3.18451e12 −0.464390
\(713\) −8.76362e12 −1.26993
\(714\) −2.94263e11 −0.0423735
\(715\) 3.78899e12 0.542184
\(716\) 5.81959e12 0.827529
\(717\) −6.41326e12 −0.906239
\(718\) −1.47298e12 −0.206841
\(719\) −5.46390e12 −0.762469 −0.381235 0.924478i \(-0.624501\pi\)
−0.381235 + 0.924478i \(0.624501\pi\)
\(720\) 7.08170e11 0.0982068
\(721\) 5.58075e11 0.0769102
\(722\) 2.45570e12 0.336324
\(723\) 4.98085e12 0.677925
\(724\) −1.20171e13 −1.62546
\(725\) 1.01031e12 0.135811
\(726\) −2.44607e11 −0.0326779
\(727\) 6.59842e12 0.876062 0.438031 0.898960i \(-0.355676\pi\)
0.438031 + 0.898960i \(0.355676\pi\)
\(728\) 1.88500e12 0.248726
\(729\) 2.82430e11 0.0370370
\(730\) 6.98498e11 0.0910358
\(731\) −3.23561e12 −0.419111
\(732\) 5.43381e12 0.699527
\(733\) 6.91041e11 0.0884170 0.0442085 0.999022i \(-0.485923\pi\)
0.0442085 + 0.999022i \(0.485923\pi\)
\(734\) 2.47541e11 0.0314785
\(735\) 1.87153e12 0.236539
\(736\) 5.07846e12 0.637943
\(737\) −7.45694e12 −0.931016
\(738\) −1.25105e12 −0.155246
\(739\) 7.60189e12 0.937608 0.468804 0.883302i \(-0.344685\pi\)
0.468804 + 0.883302i \(0.344685\pi\)
\(740\) −4.40270e12 −0.539730
\(741\) −9.45200e11 −0.115171
\(742\) −8.17167e10 −0.00989677
\(743\) −2.23002e12 −0.268448 −0.134224 0.990951i \(-0.542854\pi\)
−0.134224 + 0.990951i \(0.542854\pi\)
\(744\) −5.43661e12 −0.650504
\(745\) 3.12789e12 0.372004
\(746\) 3.47569e12 0.410882
\(747\) −2.32985e12 −0.273770
\(748\) 5.08060e12 0.593414
\(749\) −3.59538e11 −0.0417423
\(750\) 1.53976e11 0.0177696
\(751\) 1.65708e12 0.190092 0.0950459 0.995473i \(-0.469700\pi\)
0.0950459 + 0.995473i \(0.469700\pi\)
\(752\) −1.06514e13 −1.21458
\(753\) −2.34348e12 −0.265635
\(754\) −2.75058e12 −0.309922
\(755\) 2.00346e12 0.224398
\(756\) 4.41347e11 0.0491397
\(757\) −1.28420e13 −1.42135 −0.710675 0.703521i \(-0.751610\pi\)
−0.710675 + 0.703521i \(0.751610\pi\)
\(758\) −2.75218e12 −0.302807
\(759\) 3.52122e12 0.385129
\(760\) 4.00538e11 0.0435494
\(761\) 1.61852e13 1.74939 0.874694 0.484676i \(-0.161063\pi\)
0.874694 + 0.484676i \(0.161063\pi\)
\(762\) 2.01164e12 0.216149
\(763\) 4.25921e12 0.454955
\(764\) 1.23136e13 1.30757
\(765\) −1.03988e12 −0.109776
\(766\) −2.45247e12 −0.257380
\(767\) 1.14060e13 1.19002
\(768\) −8.23228e10 −0.00853875
\(769\) 5.30462e12 0.546998 0.273499 0.961872i \(-0.411819\pi\)
0.273499 + 0.961872i \(0.411819\pi\)
\(770\) −3.97414e11 −0.0407413
\(771\) −9.91345e12 −1.01037
\(772\) 4.35775e12 0.441555
\(773\) −1.26188e13 −1.27119 −0.635596 0.772022i \(-0.719245\pi\)
−0.635596 + 0.772022i \(0.719245\pi\)
\(774\) −6.51812e11 −0.0652812
\(775\) 3.49526e12 0.348034
\(776\) 8.99418e12 0.890397
\(777\) −2.32582e12 −0.228919
\(778\) 3.27591e12 0.320571
\(779\) 2.09226e12 0.203562
\(780\) 3.12105e12 0.301908
\(781\) 9.36478e12 0.900675
\(782\) −1.93387e12 −0.184925
\(783\) −1.37452e12 −0.130684
\(784\) −6.38438e12 −0.603527
\(785\) −2.89644e11 −0.0272239
\(786\) −3.40644e12 −0.318346
\(787\) 1.22887e13 1.14188 0.570941 0.820991i \(-0.306579\pi\)
0.570941 + 0.820991i \(0.306579\pi\)
\(788\) −5.46571e12 −0.504985
\(789\) 5.06162e12 0.464990
\(790\) 2.21889e12 0.202682
\(791\) 2.42762e12 0.220489
\(792\) 2.18443e12 0.197277
\(793\) 2.02993e13 1.82286
\(794\) −4.38439e12 −0.391487
\(795\) −2.88774e11 −0.0256393
\(796\) −8.56441e12 −0.756116
\(797\) −3.56495e12 −0.312962 −0.156481 0.987681i \(-0.550015\pi\)
−0.156481 + 0.987681i \(0.550015\pi\)
\(798\) 9.91385e10 0.00865425
\(799\) 1.56405e13 1.35766
\(800\) −2.02548e12 −0.174833
\(801\) −2.78541e12 −0.239080
\(802\) −2.14821e12 −0.183354
\(803\) −6.37090e12 −0.540730
\(804\) −6.14240e12 −0.518425
\(805\) −1.12624e12 −0.0945260
\(806\) −9.51585e12 −0.794218
\(807\) −1.03043e13 −0.855239
\(808\) −1.32833e13 −1.09636
\(809\) −3.11624e12 −0.255778 −0.127889 0.991789i \(-0.540820\pi\)
−0.127889 + 0.991789i \(0.540820\pi\)
\(810\) −2.09483e11 −0.0170988
\(811\) −2.30256e12 −0.186904 −0.0934518 0.995624i \(-0.529790\pi\)
−0.0934518 + 0.995624i \(0.529790\pi\)
\(812\) −2.14793e12 −0.173388
\(813\) −1.24834e12 −0.100213
\(814\) −5.39357e12 −0.430593
\(815\) −7.99677e12 −0.634900
\(816\) 3.54737e12 0.280092
\(817\) 1.09009e12 0.0855980
\(818\) 5.09651e11 0.0398000
\(819\) 1.64876e12 0.128050
\(820\) −6.90864e12 −0.533617
\(821\) −6.28720e12 −0.482962 −0.241481 0.970406i \(-0.577633\pi\)
−0.241481 + 0.970406i \(0.577633\pi\)
\(822\) 7.56212e9 0.000577724 0
\(823\) 3.76056e12 0.285728 0.142864 0.989742i \(-0.454369\pi\)
0.142864 + 0.989742i \(0.454369\pi\)
\(824\) 2.27524e12 0.171932
\(825\) −1.40440e12 −0.105547
\(826\) −1.19634e12 −0.0894219
\(827\) 1.75054e13 1.30136 0.650679 0.759353i \(-0.274484\pi\)
0.650679 + 0.759353i \(0.274484\pi\)
\(828\) 2.90049e12 0.214454
\(829\) −2.42570e12 −0.178378 −0.0891891 0.996015i \(-0.528428\pi\)
−0.0891891 + 0.996015i \(0.528428\pi\)
\(830\) 1.72809e12 0.126391
\(831\) 7.45902e12 0.542597
\(832\) −6.56257e12 −0.474809
\(833\) 9.37486e12 0.674625
\(834\) 5.77383e11 0.0413254
\(835\) −1.16062e13 −0.826229
\(836\) −1.71167e12 −0.121197
\(837\) −4.75526e12 −0.334896
\(838\) −2.10424e11 −0.0147400
\(839\) 9.69596e11 0.0675557 0.0337778 0.999429i \(-0.489246\pi\)
0.0337778 + 0.999429i \(0.489246\pi\)
\(840\) −6.98679e11 −0.0484196
\(841\) −7.81769e12 −0.538885
\(842\) 3.34574e12 0.229397
\(843\) 6.30813e12 0.430206
\(844\) 4.02593e12 0.273102
\(845\) 5.03164e12 0.339512
\(846\) 3.15078e12 0.211471
\(847\) −7.13582e11 −0.0476397
\(848\) 9.85101e11 0.0654183
\(849\) −2.36895e12 −0.156485
\(850\) 7.71298e11 0.0506801
\(851\) −1.52850e13 −0.999042
\(852\) 7.71391e12 0.501529
\(853\) −2.11898e13 −1.37043 −0.685215 0.728340i \(-0.740292\pi\)
−0.685215 + 0.728340i \(0.740292\pi\)
\(854\) −2.12912e12 −0.136975
\(855\) 3.50340e11 0.0224203
\(856\) −1.46582e12 −0.0933142
\(857\) 8.34904e12 0.528717 0.264358 0.964425i \(-0.414840\pi\)
0.264358 + 0.964425i \(0.414840\pi\)
\(858\) 3.82347e12 0.240860
\(859\) −8.24621e12 −0.516755 −0.258378 0.966044i \(-0.583188\pi\)
−0.258378 + 0.966044i \(0.583188\pi\)
\(860\) −3.59948e12 −0.224387
\(861\) −3.64964e12 −0.226326
\(862\) 7.35568e12 0.453775
\(863\) 4.35663e12 0.267364 0.133682 0.991024i \(-0.457320\pi\)
0.133682 + 0.991024i \(0.457320\pi\)
\(864\) 2.75564e12 0.168233
\(865\) 3.06716e12 0.186279
\(866\) −1.64105e12 −0.0991497
\(867\) 4.39665e12 0.264263
\(868\) −7.43096e12 −0.444331
\(869\) −2.02382e13 −1.20388
\(870\) 1.01951e12 0.0603327
\(871\) −2.29464e13 −1.35093
\(872\) 1.73646e13 1.01705
\(873\) 7.86697e12 0.458399
\(874\) 6.51527e11 0.0377686
\(875\) 4.49189e11 0.0259055
\(876\) −5.24780e12 −0.301099
\(877\) −1.21873e12 −0.0695680 −0.0347840 0.999395i \(-0.511074\pi\)
−0.0347840 + 0.999395i \(0.511074\pi\)
\(878\) −5.98825e12 −0.340075
\(879\) 1.84197e13 1.04071
\(880\) 4.79085e12 0.269303
\(881\) 3.98577e12 0.222905 0.111453 0.993770i \(-0.464450\pi\)
0.111453 + 0.993770i \(0.464450\pi\)
\(882\) 1.88856e12 0.105080
\(883\) 2.19963e13 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(884\) 1.56340e13 0.861061
\(885\) −4.22767e12 −0.231663
\(886\) 5.06025e12 0.275880
\(887\) 1.39860e13 0.758640 0.379320 0.925266i \(-0.376158\pi\)
0.379320 + 0.925266i \(0.376158\pi\)
\(888\) −9.48226e12 −0.511745
\(889\) 5.86847e12 0.315113
\(890\) 2.06599e12 0.110376
\(891\) 1.91067e12 0.101563
\(892\) 1.99848e13 1.05696
\(893\) −5.26936e12 −0.277285
\(894\) 3.15635e12 0.165259
\(895\) −8.05815e12 −0.419790
\(896\) 5.57290e12 0.288865
\(897\) 1.08355e13 0.558833
\(898\) 3.51829e12 0.180546
\(899\) 2.31428e13 1.18167
\(900\) −1.15682e12 −0.0587727
\(901\) −1.44653e12 −0.0731248
\(902\) −8.46349e12 −0.425716
\(903\) −1.90150e12 −0.0951705
\(904\) 9.89727e12 0.492899
\(905\) 1.66396e13 0.824563
\(906\) 2.02169e12 0.0996869
\(907\) −8.00749e12 −0.392884 −0.196442 0.980515i \(-0.562939\pi\)
−0.196442 + 0.980515i \(0.562939\pi\)
\(908\) 2.43762e13 1.19009
\(909\) −1.16186e13 −0.564436
\(910\) −1.22292e12 −0.0591168
\(911\) −4.09248e13 −1.96858 −0.984291 0.176555i \(-0.943505\pi\)
−0.984291 + 0.176555i \(0.943505\pi\)
\(912\) −1.19512e12 −0.0572052
\(913\) −1.57617e13 −0.750731
\(914\) 4.81529e12 0.228226
\(915\) −7.52398e12 −0.354857
\(916\) −2.61910e12 −0.122920
\(917\) −9.93745e12 −0.464102
\(918\) −1.04934e12 −0.0487669
\(919\) 8.12730e12 0.375860 0.187930 0.982182i \(-0.439822\pi\)
0.187930 + 0.982182i \(0.439822\pi\)
\(920\) −4.59164e12 −0.211311
\(921\) 4.88089e12 0.223527
\(922\) 2.15252e12 0.0980973
\(923\) 2.88172e13 1.30691
\(924\) 2.98576e12 0.134751
\(925\) 6.09625e12 0.273795
\(926\) −4.74938e12 −0.212269
\(927\) 1.99010e12 0.0885147
\(928\) −1.34111e13 −0.593605
\(929\) −4.43436e13 −1.95326 −0.976631 0.214923i \(-0.931050\pi\)
−0.976631 + 0.214923i \(0.931050\pi\)
\(930\) 3.52707e12 0.154611
\(931\) −3.15843e12 −0.137783
\(932\) −2.44584e13 −1.06183
\(933\) −1.10821e13 −0.478801
\(934\) −2.20554e12 −0.0948317
\(935\) −7.03490e12 −0.301027
\(936\) 6.72192e12 0.286254
\(937\) −3.56882e13 −1.51250 −0.756251 0.654281i \(-0.772971\pi\)
−0.756251 + 0.654281i \(0.772971\pi\)
\(938\) 2.40677e12 0.101513
\(939\) −1.82683e13 −0.766838
\(940\) 1.73994e13 0.726875
\(941\) 3.76351e13 1.56473 0.782366 0.622818i \(-0.214012\pi\)
0.782366 + 0.622818i \(0.214012\pi\)
\(942\) −2.92279e11 −0.0120940
\(943\) −2.39850e13 −0.987727
\(944\) 1.44219e13 0.591085
\(945\) −6.11116e11 −0.0249276
\(946\) −4.40958e12 −0.179014
\(947\) −1.01014e13 −0.408139 −0.204070 0.978956i \(-0.565417\pi\)
−0.204070 + 0.978956i \(0.565417\pi\)
\(948\) −1.66705e13 −0.670365
\(949\) −1.96045e13 −0.784616
\(950\) −2.59853e11 −0.0103508
\(951\) 1.03993e13 0.412280
\(952\) −3.49982e12 −0.138096
\(953\) 1.49469e13 0.586993 0.293497 0.955960i \(-0.405181\pi\)
0.293497 + 0.955960i \(0.405181\pi\)
\(954\) −2.91402e11 −0.0113900
\(955\) −1.70501e13 −0.663303
\(956\) −3.57380e13 −1.38379
\(957\) −9.29876e12 −0.358362
\(958\) 1.08345e13 0.415589
\(959\) 2.20607e10 0.000842238 0
\(960\) 2.43243e12 0.0924313
\(961\) 5.36247e13 2.02819
\(962\) −1.65970e13 −0.624803
\(963\) −1.28211e12 −0.0480405
\(964\) 2.77559e13 1.03516
\(965\) −6.03401e12 −0.223992
\(966\) −1.13649e12 −0.0419923
\(967\) 2.42942e13 0.893475 0.446738 0.894665i \(-0.352586\pi\)
0.446738 + 0.894665i \(0.352586\pi\)
\(968\) −2.90924e12 −0.106498
\(969\) 1.75492e12 0.0639441
\(970\) −5.83507e12 −0.211628
\(971\) −1.07032e13 −0.386392 −0.193196 0.981160i \(-0.561885\pi\)
−0.193196 + 0.981160i \(0.561885\pi\)
\(972\) 1.57384e12 0.0565540
\(973\) 1.68437e12 0.0602464
\(974\) 1.10201e12 0.0392346
\(975\) −4.32160e12 −0.153152
\(976\) 2.56667e13 0.905412
\(977\) −1.53038e13 −0.537372 −0.268686 0.963228i \(-0.586589\pi\)
−0.268686 + 0.963228i \(0.586589\pi\)
\(978\) −8.06954e12 −0.282048
\(979\) −1.88436e13 −0.655604
\(980\) 1.04291e13 0.361186
\(981\) 1.51883e13 0.523600
\(982\) 3.63032e12 0.124578
\(983\) 3.27251e13 1.11787 0.558934 0.829212i \(-0.311210\pi\)
0.558934 + 0.829212i \(0.311210\pi\)
\(984\) −1.48794e13 −0.505949
\(985\) 7.56816e12 0.256169
\(986\) 5.10691e12 0.172073
\(987\) 9.19163e12 0.308294
\(988\) −5.26714e12 −0.175861
\(989\) −1.24965e13 −0.415340
\(990\) −1.41718e12 −0.0468884
\(991\) 3.05488e13 1.00615 0.503075 0.864243i \(-0.332202\pi\)
0.503075 + 0.864243i \(0.332202\pi\)
\(992\) −4.63967e13 −1.52119
\(993\) −9.31649e11 −0.0304075
\(994\) −3.02253e12 −0.0982046
\(995\) 1.18588e13 0.383563
\(996\) −1.29831e13 −0.418035
\(997\) 4.87857e13 1.56374 0.781870 0.623441i \(-0.214266\pi\)
0.781870 + 0.623441i \(0.214266\pi\)
\(998\) 1.72274e13 0.549709
\(999\) −8.29388e12 −0.263459
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.1.1 2
3.2 odd 2 45.10.a.d.1.2 2
4.3 odd 2 240.10.a.r.1.2 2
5.2 odd 4 75.10.b.f.49.2 4
5.3 odd 4 75.10.b.f.49.3 4
5.4 even 2 75.10.a.f.1.2 2
15.2 even 4 225.10.b.i.199.3 4
15.8 even 4 225.10.b.i.199.2 4
15.14 odd 2 225.10.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.d.1.1 2 1.1 even 1 trivial
45.10.a.d.1.2 2 3.2 odd 2
75.10.a.f.1.2 2 5.4 even 2
75.10.b.f.49.2 4 5.2 odd 4
75.10.b.f.49.3 4 5.3 odd 4
225.10.a.k.1.1 2 15.14 odd 2
225.10.b.i.199.2 4 15.8 even 4
225.10.b.i.199.3 4 15.2 even 4
240.10.a.r.1.2 2 4.3 odd 2